The Game, is On!: Prequel, Q Learning

A basic Intuition about Q Learning!

In this blog I am trying to give an Intuition about Q Learning . Not a deep article! Just Intuition! For depth , please read Introduction to Reinforcement learning by Richard S. Sutton and Barto

Prerequisite

Neural Network Basics

Introduction

Q learning , can be said one of the most famous -and kind of intuitive- of all Reinforcement learning algorithms. In fact ,the recent all algorithms using Deep learning , are based on the Q learning algorithms. So, to work on recent algorithms, one must have a good idea on Q learning.

Intuition

First , start with the Intuition. Let’s assume , you are in a maze

Okay okay! I admit, it is not a maze. just a house with 5 rooms. And I got it from, this link . Your goal is to get out of this place, no matter where you are. But you don’t know - atleast pretend to - how to get there! After wondering about the map, you stumbled upon a mysterious letter with a lot of numbers in the room.

QMatrix

The matrix has 6 columns and 6 rows. What you will have to do , is to go to the room with highest value. Suppose, you are in room number 2. Then , you will have to move to room number 3 . Then you get out! Look at the picture again! You can try with every state, you are guaranteed to get out of the house, using this matrix! .

In the world of RL, every room is called a state, movement from one state to another is called action. Our game has a very JARGONISH name, Markov Decision Process . Maybe they invented this name to freak everybody out. But in short, this process means, your action from current state never depends on previous state. Practically such processes are impossible, but it helps to simplify problems

Now the question is , how can we get this ?

First , initialize the matrix as Zeros

Then we will apply the Q learning update equation \begin{equation} Q(s_t,a) = Q(s_t,a)+\alpha (Q’(s_{t+1},a)-Q(s_t,a)) \end{equation} Here, $s_t$ is state at time $t$ , $s_{t+1}$ means the next state, $a$ is action. Q(s_t,a_t) means Q matrix value for state $s_t$ and action $a_t$ , $Q’(s_{t+1},a)$ means target Q value with state $s_{t+1}$ and the BEST ACTION for next state. Here $\alpha $ is learning rate}

Before we proceed, let me ask you, does this equation ring a bell ? I mean, haven’t you seen a similar equation ?

Yeah, you got it , it is similar to Gradient descent Equation. If you dont know Gradient descent equation, I am sorry, you wont be able to get the future tutorials. So I suggest you get the basic and working Idea of Neural Networks and Gradient descent algorithms

Now ,How can we get $Q’(s_{t+1},a)$ ?

Using Bellman Equation

\begin{equation} Q’(s_{t+1},a) = r+ \gamma max(Q(s_{t+1},a_t)) \end{equation}

Here, $r$ is reward we get-if we can get - from one state to another state.

It means the target $Q$ value for every state and action is the sum of reward with that state and action, and the maximum $Q$ value of next state multiplied with discount factor $\gamma$. You can get the idea in the later section

**Where did this equation came from ? **

Okay chill! let’s start from the game again ! So suppose , every room has reward, $R_t,R_{t+1},R_{t+2},R_{t+3},R_{t+4},R_{t+5}$.. So obviously , the value of a state will be the expected cumulative reward

\begin{equation} Q(s,a) = R_t + R_{t+1} + R_{t+2}+ R_{t+3}+ R_{t+4}+ R_{t+5} \end{equation}

Suppose, someone comes here, and says, He wants to give more weight to sooner rewards than later rewards. What should we do ? We will introduce, discount factor, $\gamma$ , which is $0<\gamma<1$ ..

\begin{equation} Q(s,a) = R_t + \gamma R_{t+1} + \gamma^2 R_{t+2}+ \gamma^3 R_{t+3}+ \gamma^4 R_{t+4}+ \gamma^5 R_{t+5} \end{equation}

\begin{equation} Q(s,a) = R_t + \gamma [R_{t+1} + \gamma R_{t+2}+ \gamma^2 R_{t+3}+ \gamma^3 R_{t+4}+ \gamma^4 R_{t+5}] \end{equation}

This equation can be rewritten as

\begin{equation} Q(s_t,a) = R_t+\gamma Q(s_{t+1},a_{t+1}) \end{equation}

Suppose, we have some finite discrete actions in our hand, and each resulting $Q$ values of its own, what we will do ? We will try to take the action of maximum $Q$ value!

\begin{equation} Q(s_t,a) = R_t+\gamma max(Q(s_{t+1},a)) \end{equation}

Coding!

Let’s start coding!

I will be using Open Ai gym environment. The Introduction and Installtion of environments are given here

import gym
import numpy as np

Initialization of Environments

I will use the Mountaincar environment by Open AI gym. It is a classic problem invented from 90s. I intend to use this environment for all algorithms .

In this game, your task is to get the car reach that green flag. For every step you will get -1 .So , your job is to reach the goal position with minimum steps. Maximum steps limit is 200.

env = gym.make('MountainCar-v0')
s = env.reset() #Reset the car

env.reset() gives the initial state. State is the position and velocity of the car in a given time

This game’s actions can be 0,1,2 . 0 for left, 1 for doing nothing, 2 for right env.step(action) returns four arguments

next state
reward
terminal , it means if game is over or not
info , for now , it is unnecessary

Hyper Parameters

legal_actions number of actions
actions the actions list
gamma discount factor $\gamma$
lr learning rate $\alpha$
num_episodes number of episodes
epsilon epsilon , to choose random actions
epsilon_decay epsilon decay rate

legal_actions=env.action_space.n
actions = [0,1,2]
gamma =0.99
lr =0.5
num_episodes =30000
epsilon =0.5
epsilon_decay =0.99

Codeblock to discretize the state. Because Q learning doesnt work on continuous state space, we have to convert states into 10 discrete states

N_BINS = [10,10]

MIN_VALUES = [0.6,0.07]
MAX_VALUES = [-1.2,-.07]
BINS = [np.linspace(MIN_VALUES[i], MAX_VALUES[i], N_BINS[i]) for i in range(len(N_BINS))]
rList =[]
def discretize(obs):
       return tuple([int(np.digitize(obs[i], BINS[i])) for i in range(len(N_BINS))])

Q Learning CLass

class QL:
    def __init__(self,Q,policy,
                legal_actions,
                actions,
                gamma,
                lr):
        self.Q = Q #Q matrix
        self.policy =policy
        self.legal_actions=legal_actions
        self.actions = actions
        self.gamma =gamma
        self.lr =lr
    def q_value(self,s,a):
        """Gets the Q value for a certain state and action"""
        if (s,a) in self.Q:
            self.Q[(s,a)]
        else:
            self.Q[s,a]=0
        return self.Q[s,a]
    def action(self,s):
        """Gets the action for cetain state"""
        if s in self.policy:
            return self.policy[s]
        else:
            self.policy[s] = np.random.randint(0,self.legal_actions)
            return self.policy[s]
    def learn(self,s,a,s1,r,done):
        """Updates the Q matrix"""
        if done== False:
            self.Q[(s,a)] =self.q_value(s,a)+ self.lr*(r+self.gamma*max([self.q_value(s1,a1) for a1 in self.actions]) - self.q_value(s,a))
        else:
            self.Q[(s,a)] =self.q_value(s,a)+ self.lr*(r - self.q_value(s,a))
        self.q_values = [self.q_value(s,a1) for a1 in self.actions]
        self.policy[s] = self.actions[self.q_values.index(max(self.q_values))]

Q Matrix Parameters

Q - Q table. We will use dictionary data structure.
policy - policy table , it will give us the action for given state

Q = {}
policy ={}
legal_actions =3
QL = QL(Q,policy,legal_actions,actions,gamma,lr)

Training

Psuedocode

get initial state $s_{raw}$
discretize initial state , $s \gets discretize(s_{raw})$
set total reward to zero , $r_{total} \gets 0$
set terminal $d$ to false , $d \gets False$
for each step
- choose action based on epsilon greedy policy
- get next state $s1_{raw} $, reward , $r$, terminal $d$ doing the action
- $s1 \gets discretize(s1_{raw}) $
- $r_{total} \gets r_{total}+r$
- if $d == True $
- - if $r_{total}<-199$
- - - then give $r \gets -100$
- - - Update $Q$ table
- - - break
- else
- - Update $Q$ table
- - break
- $s \gets s1$

rewards=[]

for i in range(num_episodes):
    s_raw= env.reset() #initialize
    s = discretize(s_raw) #discretize the state
    rAll =0 #total reward
    d = False
    j = 0
    for j in range(200):
        
        #epsilon greedy. to choose random actions initially when Q is all zeros
        if np.random.random()< epsilon:
            a = np.random.randint(0,legal_actions)
            epsilon = epsilon*epsilon_decay
        else:
            a =QL.action(s)
        s1_raw,r,d,_ = env.step(a)
        rAll=rAll+r
        s1 = discretize(s1_raw)
        if d:
            if rAll<-199:
                r =-100 #punishment, if the game finishes before reaching the goal , we can give punishment
                QL.learn(s,a,s1,r,d)
                print("Failed! Reward %d"%rAll,end="\r")
            elif rAll>-199:
                print("Better! Reward %d"%rAll)
            rewards.append(rAll)
            break
        QL.learn(s,a,s1,r,d)
        if j==199:
            print("Reward %d after full episode"%(rAll))
            
        s = s1
env.close()  

Better! Reward -165
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Better! Reward -165
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Better! Reward -161
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Better! Reward -164
Better! Reward -146
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Better! Reward -149
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Better! Reward -161
Better! Reward -161
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Better! Reward -177
Better! Reward -179
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Better! Reward -175
Better! Reward -178
Better! Reward -172
Better! Reward -177
Better! Reward -164
Better! Reward -161
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Better! Reward -165
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Better! Reward -174
Better! Reward -165
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Better! Reward -179
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Better! Reward -162
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Better! Reward -171
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Better! Reward -168
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Better! Reward -168
Better! Reward -164
Better! Reward -169
Better! Reward -172
Better! Reward -166
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Better! Reward -166
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Better! Reward -163
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Better! Reward -163
Better! Reward -165
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Better! Reward -164
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Better! Reward -159
Better! Reward -177
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Better! Reward -161
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Better! Reward -177
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Better! Reward -165
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Better! Reward -177
Better! Reward -163
Better! Reward -166
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Better! Reward -162
Better! Reward -172
Better! Reward -164
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Better! Reward -165
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Better! Reward -160
Better! Reward -169
Better! Reward -161
Better! Reward -164
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Better! Reward -164
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Better! Reward -160
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Better! Reward -164
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Better! Reward -138
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Better! Reward -147
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Better! Reward -147
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Better! Reward -147
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Better! Reward -144
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Better! Reward -149
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Better! Reward -146
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Better! Reward -147
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Better! Reward -154
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Better! Reward -144
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Better! Reward -145
Better! Reward -149
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Better! Reward -150
Better! Reward -141
Better! Reward -153
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Better! Reward -147
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Better! Reward -145
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Better! Reward -144
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Better! Reward -143
Better! Reward -147
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Better! Reward -145
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Better! Reward -149
Better! Reward -149
Better! Reward -145
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Better! Reward -146
Better! Reward -147
Better! Reward -149
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Better! Reward -151
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Better! Reward -169
Better! Reward -153
Better! Reward -152
Better! Reward -150
Better! Reward -144
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Better! Reward -149
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Better! Reward -139
Better! Reward -153
Better! Reward -149
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Better! Reward -153
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Better! Reward -149
Better! Reward -141
Better! Reward -144
Better! Reward -145
Better! Reward -149
Better! Reward -144
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Better! Reward -145
Better! Reward -151
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Better! Reward -151
Better! Reward -146
Better! Reward -143
Better! Reward -147
Better! Reward -147
Better! Reward -169
Better! Reward -153
Better! Reward -149
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Better! Reward -155
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Better! Reward -145
Better! Reward -147
Better! Reward -153
Better! Reward -147
Better! Reward -151
Better! Reward -154
Better! Reward -145
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Better! Reward -146
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Better! Reward -139
Better! Reward -149
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Better! Reward -143
Better! Reward -146
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Better! Reward -149
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Better! Reward -155
Better! Reward -149
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Better! Reward -153
Better! Reward -151
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Better! Reward -149
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Better! Reward -152
Better! Reward -149
Better! Reward -146
Better! Reward -149
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Better! Reward -147
Better! Reward -150
Better! Reward -149
Better! Reward -141
Better! Reward -148
Better! Reward -154
Better! Reward -150
Better! Reward -151
Better! Reward -152
Better! Reward -148
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Better! Reward -145
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Better! Reward -154
Better! Reward -146
Better! Reward -146
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Better! Reward -145
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Better! Reward -170
Better! Reward -146
Better! Reward -164
Better! Reward -176
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Better! Reward -144
Better! Reward -153
Better! Reward -141
Better! Reward -149
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Better! Reward -146
Better! Reward -146
Better! Reward -147
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Better! Reward -153
Better! Reward -147
Better! Reward -151
Better! Reward -152
Better! Reward -169
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Better! Reward -151
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Better! Reward -170
Better! Reward -154
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Better! Reward -154
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Better! Reward -160
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Better! Reward -161
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Better! Reward -158
Better! Reward -160
Better! Reward -161
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Better! Reward -160
Better! Reward -159
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Better! Reward -167
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Better! Reward -167
Better! Reward -166
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Better! Reward -166
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Better! Reward -166
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Better! Reward -169
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Better! Reward -167
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Better! Reward -167
Better! Reward -168
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Better! Reward -166
Better! Reward -178
Better! Reward -166
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Better! Reward -166
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Better! Reward -168
Better! Reward -171
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Better! Reward -169
Better! Reward -171
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Better! Reward -165
Better! Reward -169
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Better! Reward -169
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Better! Reward -170
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Better! Reward -170
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Better! Reward -165
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Better! Reward -162
Better! Reward -165
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Better! Reward -157
Better! Reward -159
Better! Reward -159
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Better! Reward -157
Better! Reward -158
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Better! Reward -159
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Better! Reward -158
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Better! Reward -158
Better! Reward -157
Better! Reward -158
Better! Reward -159
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Better! Reward -158
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Better! Reward -159
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Better! Reward -158
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Better! Reward -158
Better! Reward -158
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Better! Reward -158
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Better! Reward -157
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Better! Reward -157
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Better! Reward -158
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Better! Reward -160
Better! Reward -158
Better! Reward -160
Better! Reward -158
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Better! Reward -164
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Better! Reward -168
Better! Reward -165
Better! Reward -168
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Better! Reward -171
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Better! Reward -169
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Better! Reward -163
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Better! Reward -167
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Better! Reward -168
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Better! Reward -170
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Better! Reward -173
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Better! Reward -171
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Better! Reward -173
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Better! Reward -177
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Better! Reward -166
Better! Reward -166
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Better! Reward -171
Better! Reward -171
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Better! Reward -171
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Better! Reward -173
Better! Reward -165
Better! Reward -165
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Better! Reward -166
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Better! Reward -165
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Better! Reward -166
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Better! Reward -165
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Better! Reward -165
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Better! Reward -165
Better! Reward -169
Better! Reward -165
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Better! Reward -145
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Better! Reward -147
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Better! Reward -146
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Better! Reward -135
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Better! Reward -144
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Better! Reward -163
Better! Reward -159
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Better! Reward -157
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Better! Reward -160
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Better! Reward -160
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Better! Reward -149
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Better! Reward -149
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Better! Reward -155
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Better! Reward -144
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Better! Reward -146
Better! Reward -148
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Better! Reward -155
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Better! Reward -154
Better! Reward -153
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Better! Reward -152
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Better! Reward -149
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Better! Reward -146
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Better! Reward -157
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Better! Reward -149
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Better! Reward -149
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Better! Reward -153
Better! Reward -155
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Better! Reward -155
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Better! Reward -155
Better! Reward -154
Better! Reward -155
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Better! Reward -153
Better! Reward -155
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Better! Reward -149
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Better! Reward -149
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Better! Reward -154
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Better! Reward -154
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Better! Reward -154
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Better! Reward -154
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Better! Reward -154
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Better! Reward -155
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Better! Reward -156
Better! Reward -153
Better! Reward -159
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Better! Reward -158
Better! Reward -159
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Better! Reward -156
Better! Reward -158
Better! Reward -156
Better! Reward -158
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Better! Reward -153
Better! Reward -154
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Better! Reward -156
Better! Reward -155
Better! Reward -156
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Better! Reward -154
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Better! Reward -153
Better! Reward -151
Better! Reward -155
Better! Reward -153
Better! Reward -159
Better! Reward -151
Better! Reward -159
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Better! Reward -154
Better! Reward -155
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Better! Reward -155
Better! Reward -163
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Better! Reward -154
Better! Reward -157
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Better! Reward -181
Better! Reward -179
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Better! Reward -179
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Better! Reward -179
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Better! Reward -177
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Better! Reward -176
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Better! Reward -185
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Better! Reward -187
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Better! Reward -185
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Better! Reward -189
Better! Reward -185
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Better! Reward -164
Better! Reward -164
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Better! Reward -171
Better! Reward -181
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Better! Reward -181
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Better! Reward -179
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Better! Reward -180
Better! Reward -180
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Better! Reward -179
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Better! Reward -188
Better! Reward -174
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Better! Reward -171
Better! Reward -175
Better! Reward -187
Better! Reward -176
Better! Reward -171
Better! Reward -165
Better! Reward -173
Better! Reward -171
Better! Reward -189
Better! Reward -167
Better! Reward -189
Better! Reward -170
Better! Reward -178
Better! Reward -170
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Better! Reward -189
Better! Reward -171
Better! Reward -174
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Better! Reward -186
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Better! Reward -180
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Better! Reward -177
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Better! Reward -197
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Better! Reward -175
Better! Reward -176
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Better! Reward -177
Better! Reward -175
Better! Reward -175
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Better! Reward -180
Better! Reward -196
Better! Reward -175
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Better! Reward -177
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Better! Reward -176
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Better! Reward -175
Better! Reward -177
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Better! Reward -176
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Better! Reward -175
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Better! Reward -177
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Better! Reward -176
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Better! Reward -172
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Better! Reward -181
Better! Reward -183
Better! Reward -176
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Better! Reward -176
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Better! Reward -180
Better! Reward -176
Better! Reward -175
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Better! Reward -183
Better! Reward -172
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Better! Reward -182
Better! Reward -183
Better! Reward -171
Better! Reward -185
Better! Reward -196
Better! Reward -182
Better! Reward -194
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Better! Reward -177
Better! Reward -194
Better! Reward -177
Better! Reward -172
Better! Reward -195
Better! Reward -178
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Better! Reward -177
Better! Reward -177
Better! Reward -177
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Better! Reward -176
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Better! Reward -176
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Better! Reward -175
Better! Reward -181
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Better! Reward -175
Better! Reward -183
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Better! Reward -181
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Better! Reward -195
Better! Reward -175
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Better! Reward -183
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Better! Reward -183
Better! Reward -175
Better! Reward -195
Better! Reward -176
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Better! Reward -181
Better! Reward -194
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Better! Reward -177
Better! Reward -175
Better! Reward -177
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Better! Reward -176
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Better! Reward -177
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Better! Reward -183
Better! Reward -182
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Better! Reward -177
Better! Reward -177
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Better! Reward -177
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Better! Reward -176
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Better! Reward -173
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Better! Reward -170
Better! Reward -170
Better! Reward -176
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Better! Reward -180
Better! Reward -166
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Better! Reward -173
Better! Reward -176
Better! Reward -178
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Better! Reward -191
Better! Reward -187
Better! Reward -172
Better! Reward -174
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Better! Reward -173
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Better! Reward -170
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Better! Reward -179
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Better! Reward -179
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Better! Reward -176
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Better! Reward -176
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Better! Reward -169
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Better! Reward -169
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Better! Reward -168
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Better! Reward -180
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Better! Reward -180
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Better! Reward -167
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Better! Reward -179
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Better! Reward -179
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Better! Reward -158
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Better! Reward -176
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Better! Reward -159
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Better! Reward -176
Better! Reward -101
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Better! Reward -167
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Better! Reward -167
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Better! Reward -178
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Better! Reward -169
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Better! Reward -168
Better! Reward -167
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Better! Reward -179
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Better! Reward -170
Better! Reward -174
Better! Reward -180
Better! Reward -169
Better! Reward -169
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Better! Reward -163
Better! Reward -169
Better! Reward -167
Better! Reward -167
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Better! Reward -163
Better! Reward -159
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Better! Reward -158
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Better! Reward -169
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Better! Reward -159
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Better! Reward -174
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Better! Reward -101
Better! Reward -174
Better! Reward -174
Better! Reward -159
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Better! Reward -158
Better! Reward -176
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Better! Reward -167
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Better! Reward -168
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Better! Reward -170
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Better! Reward -170
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Better! Reward -179
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Better! Reward -169
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Better! Reward -159
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Better! Reward -159
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Better! Reward -167
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Better! Reward -169
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Better! Reward -159
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Better! Reward -180
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Better! Reward -101
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Better! Reward -159
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Better! Reward -170
Better! Reward -100
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Better! Reward -168
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Better! Reward -179
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Better! Reward -101
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Better! Reward -164
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Better! Reward -177
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Better! Reward -104
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Better! Reward -167
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Better! Reward -165
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Better! Reward -167
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Better! Reward -159
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Better! Reward -167
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Better! Reward -166
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Better! Reward -165
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Better! Reward -168
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Better! Reward -159
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Better! Reward -165
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Better! Reward -159
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Better! Reward -159
Better! Reward -179
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Better! Reward -160
Better! Reward -172
Better! Reward -159
Better! Reward -170
Better! Reward -179
Better! Reward -185
Better! Reward -163
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Better! Reward -160
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Better! Reward -159
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Better! Reward -169
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Better! Reward -180
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Better! Reward -163
Better! Reward -175
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Better! Reward -166
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Better! Reward -163
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Better! Reward -169
Better! Reward -177
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Better! Reward -172
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Better! Reward -165
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Better! Reward -160
Better! Reward -168
Better! Reward -162
Better! Reward -180
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Better! Reward -179
Better! Reward -161
Better! Reward -169
Better! Reward -179
Better! Reward -168
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Better! Reward -168
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Better! Reward -168
Better! Reward -180
Better! Reward -159
Better! Reward -180
Better! Reward -175
Better! Reward -164
Better! Reward -168
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Better! Reward -164
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Better! Reward -172
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Better! Reward -159
Better! Reward -179
Better! Reward -164
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Better! Reward -161
Better! Reward -177
Better! Reward -180
Better! Reward -174
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Better! Reward -179
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Better! Reward -164
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Better! Reward -177
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Better! Reward -164
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Better! Reward -179
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Better! Reward -160
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Better! Reward -179
Better! Reward -168
Better! Reward -160
Better! Reward -180
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Better! Reward -162
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Better! Reward -167
Better! Reward -178
Better! Reward -170
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Better! Reward -178
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Better! Reward -171
Better! Reward -160
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Better! Reward -164
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Better! Reward -166
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Better! Reward -160
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Better! Reward -178
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Better! Reward -159
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Better! Reward -163
Better! Reward -171
Better! Reward -179
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Better! Reward -164
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Better! Reward -166
Better! Reward -163
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Better! Reward -170
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Better! Reward -174
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Better! Reward -161
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Better! Reward -161
Better! Reward -171
Better! Reward -163
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Better! Reward -165
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Better! Reward -173
Better! Reward -165
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Better! Reward -164
Better! Reward -160
Better! Reward -165
Better! Reward -162
Better! Reward -163
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Better! Reward -160
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Better! Reward -164
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Better! Reward -165
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Better! Reward -168
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Better! Reward -170
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Better! Reward -166
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Better! Reward -169
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Better! Reward -159
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Better! Reward -164
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Better! Reward -163
Better! Reward -171
Better! Reward -163
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Better! Reward -162
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Better! Reward -170
Better! Reward -173
Better! Reward -165
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Better! Reward -159
Better! Reward -173
Better! Reward -159
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Better! Reward -164
Better! Reward -180
Better! Reward -161
Better! Reward -177
Better! Reward -164
Better! Reward -167
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Better! Reward -160
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Better! Reward -180
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Better! Reward -163
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Better! Reward -160
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Better! Reward -180
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Better! Reward -164
Better! Reward -167
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Better! Reward -164
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Better! Reward -179
Better! Reward -160
Better! Reward -161
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Better! Reward -179
Better! Reward -163
Better! Reward -164
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Better! Reward -167
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Better! Reward -181
Better! Reward -159
Better! Reward -160
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Better! Reward -160
Better! Reward -163
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Better! Reward -164
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Better! Reward -165
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Better! Reward -161
Better! Reward -171
Better! Reward -163
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Better! Reward -159
Better! Reward -166
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Better! Reward -163
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Better! Reward -170
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Better! Reward -159
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Better! Reward -159
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Better! Reward -171
Better! Reward -159
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Better! Reward -160
Better! Reward -159
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Better! Reward -165
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Better! Reward -169
Better! Reward -159
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Better! Reward -167
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Better! Reward -170
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Better! Reward -164
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Better! Reward -166
Better! Reward -179
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Better! Reward -186
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Better! Reward -187
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Better! Reward -173
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Better! Reward -188
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Better! Reward -183
Better! Reward -188
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Better! Reward -188
Better! Reward -172
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Better! Reward -176
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Better! Reward -184
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Better! Reward -189
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Better! Reward -180
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Better! Reward -179
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Better! Reward -173
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Better! Reward -188
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Better! Reward -177
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Better! Reward -180
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Better! Reward -173
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Better! Reward -176
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Better! Reward -172
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Better! Reward -178
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Better! Reward -183
Better! Reward -181
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Better! Reward -167
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Better! Reward -155
Better! Reward -171
Better! Reward -188
Better! Reward -180
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Better! Reward -180
Better! Reward -172
Better! Reward -188
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Better! Reward -181
Better! Reward -178
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Better! Reward -180
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Better! Reward -188
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Better! Reward -185
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Better! Reward -180
Better! Reward -178
Better! Reward -179
Better! Reward -172
Better! Reward -182
Better! Reward -185
Better! Reward -166
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Better! Reward -186
Better! Reward -175
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Better! Reward -175
Better! Reward -187
Better! Reward -189
Better! Reward -182
Better! Reward -181
Better! Reward -186
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Better! Reward -175
Better! Reward -171
Better! Reward -182
Better! Reward -184
Better! Reward -177
Better! Reward -169
Better! Reward -182
Better! Reward -182
Better! Reward -175
Better! Reward -166
Better! Reward -188
Better! Reward -184
Better! Reward -178
Better! Reward -188
Better! Reward -181
Better! Reward -185
Better! Reward -173
Better! Reward -183
Better! Reward -178
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Better! Reward -185
Better! Reward -174
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Better! Reward -185
Better! Reward -188
Better! Reward -173
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Better! Reward -175
Better! Reward -177
Better! Reward -172
Better! Reward -182
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Better! Reward -185
Better! Reward -184
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Better! Reward -169
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Better! Reward -187
Better! Reward -177
Better! Reward -185
Better! Reward -187
Better! Reward -182
Better! Reward -173
Better! Reward -186
Better! Reward -172
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Better! Reward -166
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Better! Reward -180
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Better! Reward -188
Better! Reward -189
Better! Reward -189
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Better! Reward -189
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Better! Reward -179
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Better! Reward -180
Better! Reward -183
Better! Reward -174
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Better! Reward -187
Better! Reward -187
Better! Reward -189
Better! Reward -174
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Better! Reward -185
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Better! Reward -176
Better! Reward -188
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Better! Reward -188
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Better! Reward -185
Better! Reward -181
Better! Reward -177
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Better! Reward -175
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Better! Reward -172
Better! Reward -175
Better! Reward -176
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Better! Reward -175
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Better! Reward -190
Better! Reward -191
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Better! Reward -182
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Better! Reward -190
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Better! Reward -189
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Better! Reward -185
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Better! Reward -190
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Better! Reward -186
Better! Reward -181
Better! Reward -168
Better! Reward -171
Better! Reward -190
Better! Reward -171
Better! Reward -165
Better! Reward -159
Better! Reward -171
Better! Reward -168
Better! Reward -190
Better! Reward -183
Better! Reward -189
Better! Reward -189
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Better! Reward -181
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Better! Reward -185
Better! Reward -185
Better! Reward -189
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Better! Reward -189
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Better! Reward -189
Better! Reward -182
Better! Reward -191
Better! Reward -191
Better! Reward -189
Better! Reward -189
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Better! Reward -195
Better! Reward -187
Better! Reward -189
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Better! Reward -190
Better! Reward -192
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Better! Reward -187
Better! Reward -190
Better! Reward -164
Better! Reward -169
Better! Reward -169
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Better! Reward -165
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Better! Reward -171
Better! Reward -168
Better! Reward -169
Better! Reward -169
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Better! Reward -171
Better! Reward -169
Better! Reward -171
Better! Reward -167
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Better! Reward -171
Better! Reward -170
Better! Reward -168
Better! Reward -170
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Better! Reward -167
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Better! Reward -167
Better! Reward -165
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Better! Reward -165
Better! Reward -169
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Better! Reward -165
Better! Reward -165
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Better! Reward -157
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Better! Reward -157
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Better! Reward -157
Better! Reward -157
Better! Reward -155
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Better! Reward -157
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Better! Reward -161
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Better! Reward -154
Better! Reward -164
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Better! Reward -165
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Better! Reward -157
Better! Reward -158
Better! Reward -157
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Better! Reward -153
Better! Reward -155
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Better! Reward -157
Better! Reward -158
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Better! Reward -158
Better! Reward -158
Better! Reward -158
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Better! Reward -157
Better! Reward -164
Better! Reward -154
Better! Reward -156
Better! Reward -158
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Better! Reward -161
Better! Reward -159
Better! Reward -153
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Better! Reward -153
Better! Reward -158
Better! Reward -158
Better! Reward -157
Better! Reward -154
Better! Reward -157
Better! Reward -158
Better! Reward -154
Better! Reward -159
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Better! Reward -160
Better! Reward -164
Better! Reward -153
Better! Reward -157
Better! Reward -159
Better! Reward -154
Better! Reward -160
Better! Reward -159
Better! Reward -159
Better! Reward -157
Better! Reward -158
Better! Reward -158
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Better! Reward -158
Better! Reward -158
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Better! Reward -157
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Better! Reward -158
Better! Reward -157
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Better! Reward -152
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Better! Reward -152
Better! Reward -157
Better! Reward -160
Better! Reward -164
Better! Reward -154
Better! Reward -157
Better! Reward -159
Better! Reward -155
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Better! Reward -154
Better! Reward -159
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Better! Reward -158
Better! Reward -157
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Better! Reward -158
Better! Reward -160
Better! Reward -159
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Better! Reward -157
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Better! Reward -160
Better! Reward -157
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Better! Reward -154
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Better! Reward -165
Better! Reward -165
Better! Reward -160
Better! Reward -163
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Better! Reward -157
Better! Reward -164
Better! Reward -161
Better! Reward -165
Better! Reward -157
Better! Reward -161
Better! Reward -165
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Better! Reward -158
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Better! Reward -161
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Better! Reward -155
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Better! Reward -159
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Better! Reward -160
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Better! Reward -154
Better! Reward -157
Better! Reward -154
Better! Reward -165
Better! Reward -155
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Better! Reward -158
Better! Reward -159
Better! Reward -152
Better! Reward -152
Better! Reward -154
Better! Reward -164
Better! Reward -162
Better! Reward -157
Better! Reward -153
Better! Reward -163
Better! Reward -159
Better! Reward -161
Better! Reward -159
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Better! Reward -152
Better! Reward -157
Better! Reward -151
Better! Reward -161
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Better! Reward -165
Better! Reward -163
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Better! Reward -155
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Better! Reward -153
Better! Reward -157
Better! Reward -154
Better! Reward -159
Better! Reward -151
Better! Reward -157
Better! Reward -154
Better! Reward -158
Better! Reward -165
Better! Reward -154
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Better! Reward -159
Better! Reward -160
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Better! Reward -159
Better! Reward -153
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Better! Reward -151
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Better! Reward -157
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Better! Reward -155
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Better! Reward -154
Better! Reward -157
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Better! Reward -158
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Better! Reward -158
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Better! Reward -158
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Better! Reward -158
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Better! Reward -159
Better! Reward -158
Better! Reward -159
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Better! Reward -158
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Better! Reward -164
Better! Reward -157
Better! Reward -158
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Better! Reward -158
Better! Reward -157
Better! Reward -157
Better! Reward -159
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Better! Reward -154
Better! Reward -154
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Better! Reward -154
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Better! Reward -158
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Better! Reward -158
Better! Reward -159
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Better! Reward -158
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Better! Reward -157
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Better! Reward -158
Better! Reward -157
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Better! Reward -163
Better! Reward -155
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Better! Reward -165
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Better! Reward -160
Better! Reward -157
Better! Reward -155
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Better! Reward -152
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Better! Reward -152
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Better! Reward -154
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Better! Reward -152
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Better! Reward -154
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Better! Reward -165
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Better! Reward -158
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Better! Reward -158
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Better! Reward -154
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Better! Reward -165
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Better! Reward -165
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Better! Reward -157
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Better! Reward -159
Better! Reward -165
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Better! Reward -181
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Better! Reward -154
Better! Reward -157
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Better! Reward -162
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Better! Reward -154
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Better! Reward -151
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Better! Reward -159
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Better! Reward -161
Better! Reward -161
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Better! Reward -161
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Better! Reward -154
Better! Reward -153
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Better! Reward -159
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Better! Reward -161
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Better! Reward -170
Better! Reward -164
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Better! Reward -153
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Better! Reward -155
Better! Reward -154
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Better! Reward -154
Better! Reward -155
Better! Reward -155
Better! Reward -154
Better! Reward -156
Better! Reward -156
Better! Reward -148
Better! Reward -155
Better! Reward -154
Better! Reward -155
Better! Reward -154
Better! Reward -155
Better! Reward -154
Better! Reward -154
Better! Reward -147
Better! Reward -154
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Better! Reward -156
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Better! Reward -155
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Better! Reward -156
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Better! Reward -160
Better! Reward -149
Better! Reward -157
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Better! Reward -168
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Better! Reward -161
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Better! Reward -162
Better! Reward -168
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Better! Reward -168
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Better! Reward -167
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Better! Reward -166
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Better! Reward -168
Better! Reward -165
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Better! Reward -166
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Better! Reward -160
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Better! Reward -161
Better! Reward -164
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Better! Reward -165
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Better! Reward -159
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Better! Reward -156
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Better! Reward -158
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Better! Reward -164
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Better! Reward -161
Better! Reward -166
Better! Reward -167
Better! Reward -165
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Better! Reward -166
Better! Reward -159
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Better! Reward -165
Better! Reward -160
Better! Reward -167
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Better! Reward -161
Better! Reward -162
Better! Reward -165
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Better! Reward -158
Better! Reward -163
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Better! Reward -161
Better! Reward -162
Better! Reward -157
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Better! Reward -162
Better! Reward -169
Better! Reward -164
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Better! Reward -166
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Better! Reward -161
Better! Reward -164
Better! Reward -165
Better! Reward -161
Better! Reward -157
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Better! Reward -165
Better! Reward -156
Better! Reward -162
Better! Reward -165
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Better! Reward -165
Better! Reward -161
Better! Reward -163
Better! Reward -160
Better! Reward -167
Better! Reward -168
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Better! Reward -165
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Better! Reward -161
Better! Reward -166
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Better! Reward -188
Better! Reward -167
Better! Reward -159
Better! Reward -161
Better! Reward -161
Better! Reward -166
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Better! Reward -159
Better! Reward -164
Better! Reward -165
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Better! Reward -166
Better! Reward -165
Better! Reward -164
Better! Reward -162
Better! Reward -188
Better! Reward -166
Better! Reward -164
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Better! Reward -160
Better! Reward -163
Better! Reward -161
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Better! Reward -163
Better! Reward -160
Better! Reward -165
Better! Reward -161
Better! Reward -156
Better! Reward -157
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Better! Reward -164
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Better! Reward -164
Better! Reward -166
Better! Reward -164
Better! Reward -165
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Better! Reward -167
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Better! Reward -164
Better! Reward -160
Better! Reward -161
Better! Reward -159
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Better! Reward -161
Better! Reward -162
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Better! Reward -161
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Better! Reward -161
Better! Reward -165
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Better! Reward -166
Better! Reward -162
Better! Reward -157
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Better! Reward -166
Better! Reward -160
Better! Reward -160
Better! Reward -158
Better! Reward -167
Better! Reward -158
Better! Reward -157
Better! Reward -161
Better! Reward -160
Better! Reward -156
Better! Reward -156
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Better! Reward -159
Better! Reward -165
Better! Reward -160
Better! Reward -164
Better! Reward -162
Better! Reward -158
Better! Reward -163
Better! Reward -165
Better! Reward -160
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Better! Reward -164
Better! Reward -167
Better! Reward -160
Better! Reward -167
Better! Reward -166
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Better! Reward -159
Better! Reward -160
Better! Reward -159
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Better! Reward -163
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Better! Reward -164
Better! Reward -166
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Better! Reward -157
Better! Reward -161
Better! Reward -157
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Better! Reward -164
Better! Reward -162
Better! Reward -164
Better! Reward -165
Better! Reward -161
Better! Reward -168
Better! Reward -163
Better! Reward -161
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Better! Reward -164
Better! Reward -159
Better! Reward -161
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Better! Reward -160
Better! Reward -160
Better! Reward -161
Better! Reward -162
Better! Reward -169
Better! Reward -159
Better! Reward -165
Better! Reward -157
Better! Reward -160
Better! Reward -159
Better! Reward -159
Better! Reward -165
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Better! Reward -160
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Better! Reward -162
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Better! Reward -159
Better! Reward -160
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Better! Reward -159
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Better! Reward -161
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Better! Reward -158
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Better! Reward -164
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Better! Reward -161
Better! Reward -159
Better! Reward -159
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Better! Reward -165
Better! Reward -158
Better! Reward -158
Better! Reward -166
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Better! Reward -157
Better! Reward -161
Better! Reward -161
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Better! Reward -161
Better! Reward -164
Better! Reward -157
Better! Reward -161
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Better! Reward -159
Better! Reward -165
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Better! Reward -164
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Better! Reward -160
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Better! Reward -162
Better! Reward -163
Better! Reward -157
Better! Reward -161
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Better! Reward -159
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Better! Reward -157
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Better! Reward -160
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Better! Reward -165
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Better! Reward -165
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Better! Reward -160
Better! Reward -160
Better! Reward -157
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Better! Reward -159
Better! Reward -159
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Better! Reward -159
Better! Reward -161
Better! Reward -164
Better! Reward -160
Better! Reward -166
Better! Reward -165
Better! Reward -161
Better! Reward -160
Better! Reward -161
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Better! Reward -157
Better! Reward -162
Better! Reward -162
Better! Reward -161
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Better! Reward -167
Better! Reward -161
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Better! Reward -164
Better! Reward -161
Better! Reward -167
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Better! Reward -163
Better! Reward -164
Better! Reward -164
Better! Reward -164
Better! Reward -165
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Better! Reward -157
Better! Reward -164
Better! Reward -159
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Better! Reward -159
Better! Reward -160
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Better! Reward -159
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Better! Reward -166
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Better! Reward -163
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Better! Reward -162
Better! Reward -157
Better! Reward -162
Better! Reward -161
Better! Reward -166
Better! Reward -162
Better! Reward -162
Better! Reward -189
Better! Reward -163
Better! Reward -160
Better! Reward -161
Better! Reward -160
Better! Reward -157
Better! Reward -160
Better! Reward -164
Better! Reward -165
Better! Reward -161
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Better! Reward -162
Better! Reward -165
Better! Reward -163
Better! Reward -161
Better! Reward -161
Better! Reward -165
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Better! Reward -165
Better! Reward -156
Better! Reward -160
Better! Reward -157
Better! Reward -160
Better! Reward -161
Better! Reward -159
Better! Reward -166
Better! Reward -159
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Better! Reward -160
Better! Reward -157
Better! Reward -164
Better! Reward -187
Better! Reward -160
Better! Reward -164
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Better! Reward -164
Better! Reward -164
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Better! Reward -165
Better! Reward -163
Better! Reward -159
Better! Reward -156
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Better! Reward -163
Better! Reward -165
Better! Reward -161
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Better! Reward -158
Better! Reward -162
Better! Reward -160
Better! Reward -160
Better! Reward -162
Better! Reward -165
Better! Reward -163
Better! Reward -163
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Better! Reward -157
Better! Reward -161
Better! Reward -163
Better! Reward -162
Better! Reward -167
Better! Reward -161
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Better! Reward -164
Better! Reward -167
Better! Reward -160
Better! Reward -160
Better! Reward -158
Better! Reward -161
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Better! Reward -163
Better! Reward -161
Better! Reward -161
Better! Reward -161
Better! Reward -161
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Better! Reward -161
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Better! Reward -166
Better! Reward -159
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Better! Reward -163
Better! Reward -165
Better! Reward -156
Better! Reward -159
Better! Reward -163
Better! Reward -167
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Better! Reward -163
Better! Reward -167
Better! Reward -165
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Better! Reward -165
Better! Reward -165
Better! Reward -156
Better! Reward -161
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Better! Reward -157
Better! Reward -163
Better! Reward -163
Better! Reward -158
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Better! Reward -161
Better! Reward -163
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Better! Reward -161
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Better! Reward -164
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Better! Reward -165
Better! Reward -164
Better! Reward -165
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Better! Reward -161
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Better! Reward -167
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Better! Reward -161
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Better! Reward -161
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Better! Reward -164
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Better! Reward -157
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Better! Reward -167
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Better! Reward -161
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Better! Reward -159
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Better! Reward -159
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Better! Reward -164
Better! Reward -162
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Better! Reward -164
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Better! Reward -165
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Better! Reward -164
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Better! Reward -165
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Better! Reward -164
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Better! Reward -161
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Better! Reward -165
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Better! Reward -170
Better! Reward -163
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Better! Reward -165
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Better! Reward -165
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Better! Reward -170
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Better! Reward -171
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Better! Reward -174
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Better! Reward -165
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Better! Reward -166
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Better! Reward -170
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Better! Reward -167
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Better! Reward -167
Better! Reward -168
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Better! Reward -170
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Better! Reward -168
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Better! Reward -166
Better! Reward -168
Better! Reward -167
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Better! Reward -176
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Better! Reward -170
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Better! Reward -183
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Better! Reward -167
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Better! Reward -168
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Better! Reward -165
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Better! Reward -179
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Better! Reward -157
Better! Reward -165
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Better! Reward -159
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Better! Reward -157
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Better! Reward -167
Better! Reward -167
Better! Reward -168
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Better! Reward -176
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Better! Reward -159
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Better! Reward -167
Better! Reward -165
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Better! Reward -166
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Better! Reward -159
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Better! Reward -167
Better! Reward -159
Better! Reward -160
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Better! Reward -159
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Better! Reward -174
Better! Reward -165
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Better! Reward -176
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Better! Reward -167
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Better! Reward -170
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Better! Reward -171
Better! Reward -171
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Better! Reward -159
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Better! Reward -176
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Better! Reward -176
Better! Reward -176
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Better! Reward -177
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Better! Reward -176
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Better! Reward -177
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Better! Reward -176
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Better! Reward -175
Better! Reward -178
Better! Reward -176
Better! Reward -177
Better! Reward -176
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Better! Reward -179
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Better! Reward -175
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Better! Reward -177
Better! Reward -177
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Better! Reward -175
Better! Reward -178
Better! Reward -178
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Better! Reward -176
Better! Reward -176
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Better! Reward -177
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Better! Reward -174
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Better! Reward -178
Better! Reward -178
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Better! Reward -178
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Better! Reward -175
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Better! Reward -174
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Better! Reward -178
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Better! Reward -177
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Better! Reward -176
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Better! Reward -178
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Better! Reward -177
Better! Reward -178
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Better! Reward -178
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Better! Reward -173
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Better! Reward -175
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Better! Reward -175
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Better! Reward -176
Better! Reward -175
Better! Reward -178
Better! Reward -175
Better! Reward -175
Better! Reward -177
Better! Reward -169
Better! Reward -174
Better! Reward -171
Better! Reward -174
Better! Reward -192
Better! Reward -175
Better! Reward -188
Better! Reward -178
Better! Reward -173
Better! Reward -176
Better! Reward -186
Better! Reward -177
Better! Reward -176
Better! Reward -177
Better! Reward -174
Better! Reward -177
Better! Reward -177
Better! Reward -176
Better! Reward -173
Better! Reward -178
Better! Reward -178
Better! Reward -176
Better! Reward -174
Better! Reward -173
Better! Reward -176
Better! Reward -173
Better! Reward -175
Better! Reward -178
Better! Reward -176
Better! Reward -177
Better! Reward -170
Better! Reward -176
Better! Reward -175
Better! Reward -177
Better! Reward -175
Better! Reward -174
Better! Reward -178
Better! Reward -174
Better! Reward -178
Better! Reward -175
Better! Reward -175
Better! Reward -174
Better! Reward -177
Better! Reward -176
Better! Reward -177
Better! Reward -177
Better! Reward -178
Better! Reward -177
Better! Reward -174
Better! Reward -176
Better! Reward -179
Better! Reward -178
Better! Reward -175
Better! Reward -185
Better! Reward -175
Better! Reward -177
Better! Reward -175
Better! Reward -178
Better! Reward -175
Better! Reward -177
Better! Reward -180
Better! Reward -177
Better! Reward -187
Better! Reward -176
Better! Reward -175
Better! Reward -177
Better! Reward -172
Better! Reward -179
Better! Reward -177
Better! Reward -177
Better! Reward -175
Better! Reward -169
Better! Reward -175
Better! Reward -178
Better! Reward -177
Better! Reward -178
Better! Reward -175
Better! Reward -178
Better! Reward -175
Better! Reward -173
Better! Reward -178
Better! Reward -174
Better! Reward -178
Better! Reward -175
Better! Reward -177
Better! Reward -177
Better! Reward -174
Better! Reward -176
Better! Reward -178
Better! Reward -176
Better! Reward -177
Better! Reward -175
Better! Reward -180
Better! Reward -177
Better! Reward -175
Better! Reward -175
Better! Reward -177
Better! Reward -178
Better! Reward -177
Better! Reward -176
Better! Reward -174
Better! Reward -174
Better! Reward -183
Better! Reward -176
Better! Reward -177
Better! Reward -182
Better! Reward -174
Better! Reward -178
Better! Reward -177
Better! Reward -173
Better! Reward -172
Better! Reward -175
Better! Reward -178
Better! Reward -177
Better! Reward -178
Better! Reward -174
Better! Reward -174
Better! Reward -173
Better! Reward -177
Better! Reward -175
Better! Reward -175
Better! Reward -171
Better! Reward -176
Better! Reward -176
Better! Reward -176
Better! Reward -173
Better! Reward -175
Better! Reward -172
Better! Reward -173
Better! Reward -174
Better! Reward -175
Better! Reward -177
Better! Reward -176
Better! Reward -177
Better! Reward -175
Better! Reward -168
Better! Reward -175
Better! Reward -178
Better! Reward -174
Better! Reward -177
Better! Reward -176
Better! Reward -171
Better! Reward -176
Better! Reward -176
Better! Reward -177
Better! Reward -177
Better! Reward -173
Better! Reward -172
Better! Reward -175
Better! Reward -180
Better! Reward -174
Better! Reward -176
Better! Reward -179
Better! Reward -169
Better! Reward -180
Better! Reward -183
Better! Reward -181
Better! Reward -168
Better! Reward -181
Better! Reward -180
Better! Reward -178
Better! Reward -182
Better! Reward -174
Better! Reward -181
Better! Reward -177
Better! Reward -177
Better! Reward -179
Better! Reward -179
Better! Reward -179
Better! Reward -179
Better! Reward -179
Better! Reward -179
Better! Reward -182
Better! Reward -174
Better! Reward -180
Better! Reward -183
Better! Reward -174
Better! Reward -175
Better! Reward -175
Better! Reward -174
Better! Reward -183
Better! Reward -181
Better! Reward -178
Better! Reward -178
Better! Reward -179
Better! Reward -175
Better! Reward -179
Better! Reward -174
Better! Reward -180
Better! Reward -181
Better! Reward -179
Better! Reward -181
Better! Reward -180
Better! Reward -182
Better! Reward -178
Better! Reward -179
Better! Reward -179
Better! Reward -183
Better! Reward -171
Better! Reward -175
Better! Reward -180
Better! Reward -180
Better! Reward -178
Better! Reward -180
Better! Reward -180
Better! Reward -180
Better! Reward -180
Better! Reward -181
Better! Reward -177
Better! Reward -176
Better! Reward -173
Better! Reward -178
Better! Reward -172
Better! Reward -175
Better! Reward -174
Better! Reward -177
Better! Reward -176
Better! Reward -175
Better! Reward -177
Better! Reward -178
Better! Reward -175
Better! Reward -180
Better! Reward -176
Better! Reward -183
Better! Reward -180
Better! Reward -181
Better! Reward -177
Better! Reward -182
Better! Reward -180
Better! Reward -178
Better! Reward -179
Better! Reward -173
Better! Reward -177
Better! Reward -177
Better! Reward -174
Better! Reward -173
Better! Reward -174
Better! Reward -175
Better! Reward -172
Better! Reward -176
Better! Reward -176
Better! Reward -176
Better! Reward -176
Better! Reward -177
Better! Reward -175
Better! Reward -176
Better! Reward -172
Better! Reward -177
Better! Reward -182
Better! Reward -181
Better! Reward -179
Better! Reward -173
Better! Reward -181
Better! Reward -177
Better! Reward -179
Better! Reward -177
Better! Reward -181
Better! Reward -179
Better! Reward -188
Better! Reward -179
Better! Reward -179
Better! Reward -187
Better! Reward -178
Better! Reward -179
Better! Reward -173
Better! Reward -177
Better! Reward -176
Better! Reward -176
Better! Reward -174
Better! Reward -177
Better! Reward -176
Better! Reward -174
Better! Reward -177
Better! Reward -171
Better! Reward -176
Better! Reward -177
Better! Reward -174
Better! Reward -174
Better! Reward -178
Better! Reward -174
Better! Reward -178
Better! Reward -177
Better! Reward -176
Better! Reward -175
Better! Reward -172
Better! Reward -177
Better! Reward -177
Better! Reward -177
Better! Reward -174
Better! Reward -176
Better! Reward -181
Better! Reward -181
Better! Reward -179
Better! Reward -178
Better! Reward -178
Better! Reward -176
Better! Reward -179
Better! Reward -182
Better! Reward -176
Better! Reward -175
Better! Reward -174
Better! Reward -175
Better! Reward -179
Better! Reward -178
Better! Reward -180
Better! Reward -180
Better! Reward -184
Better! Reward -176
Better! Reward -181
Better! Reward -183
Better! Reward -171
Better! Reward -169
Better! Reward -181
Better! Reward -175
Better! Reward -185
Better! Reward -177
Better! Reward -173
Better! Reward -176
Better! Reward -176
Better! Reward -177
Better! Reward -175
Better! Reward -176
Better! Reward -175
Better! Reward -177
Better! Reward -174
Better! Reward -178
Better! Reward -174
Better! Reward -176
Better! Reward -177
Better! Reward -179
Better! Reward -176
Better! Reward -171
Better! Reward -177
Better! Reward -178
Better! Reward -175
Better! Reward -174
Better! Reward -174
Better! Reward -174
Better! Reward -171
Better! Reward -164
Better! Reward -170
Better! Reward -173
Better! Reward -170
Better! Reward -172
Better! Reward -171
Better! Reward -174
Better! Reward -172
Better! Reward -171
Better! Reward -172
Better! Reward -169
Better! Reward -171
Better! Reward -166
Better! Reward -168
Better! Reward -172
Better! Reward -168
Better! Reward -168
Better! Reward -170
Better! Reward -173
Better! Reward -169
Better! Reward -179
Better! Reward -168
Better! Reward -174
Better! Reward -172
Better! Reward -197
Better! Reward -177
Better! Reward -177
Better! Reward -172
Better! Reward -172
Better! Reward -170
Better! Reward -171
Better! Reward -170
Better! Reward -196
Better! Reward -175
Better! Reward -170
Better! Reward -171
Better! Reward -171
Better! Reward -179
Better! Reward -170
Better! Reward -172
Better! Reward -168
Better! Reward -173
Better! Reward -168
Better! Reward -171
Better! Reward -181
Better! Reward -190
Better! Reward -171
Better! Reward -172
Better! Reward -175
Better! Reward -173
Better! Reward -184
Better! Reward -172
Better! Reward -172
Better! Reward -168
Better! Reward -173
Better! Reward -170
Better! Reward -171
Better! Reward -191
Better! Reward -169
Better! Reward -170
Better! Reward -167
Better! Reward -174
Better! Reward -167
Better! Reward -172
Better! Reward -171
Better! Reward -170
Better! Reward -172
Better! Reward -167
Better! Reward -173
Better! Reward -173
Better! Reward -170
Better! Reward -170
Better! Reward -171
Better! Reward -170
Better! Reward -170
Better! Reward -174
Better! Reward -169
Better! Reward -170
Better! Reward -173
Better! Reward -168
Better! Reward -172
Better! Reward -172
Better! Reward -170
Better! Reward -170
Better! Reward -174
Better! Reward -169
Better! Reward -169
Better! Reward -167
Better! Reward -171
Better! Reward -170
Better! Reward -168
Better! Reward -167
Better! Reward -167
Better! Reward -176
Better! Reward -171
Better! Reward -173
Better! Reward -172
Better! Reward -173
Better! Reward -171
Better! Reward -173
Better! Reward -173
Better! Reward -171
Better! Reward -175
Better! Reward -167
Better! Reward -171
Better! Reward -172
Better! Reward -175
Better! Reward -177
Better! Reward -183
Better! Reward -179
Better! Reward -172
Better! Reward -170
Better! Reward -181
Better! Reward -181
Better! Reward -168
Better! Reward -179
Better! Reward -180
Better! Reward -181
Better! Reward -190
Better! Reward -196
Better! Reward -198
Better! Reward -192
Better! Reward -194
Better! Reward -196
Better! Reward -196
Better! Reward -194
Better! Reward -187
Better! Reward -194
Better! Reward -197
Better! Reward -194
Better! Reward -196
Better! Reward -196
Better! Reward -190
Better! Reward -191
Better! Reward -146
Better! Reward -157
Better! Reward -146
Better! Reward -156
Better! Reward -167
Better! Reward -167
Better! Reward -157
Better! Reward -146
Better! Reward -159
Better! Reward -155
Better! Reward -141
Better! Reward -132
Better! Reward -153
Better! Reward -149
Better! Reward -170
Better! Reward -144
Better! Reward -167
Better! Reward -150
Better! Reward -170
Better! Reward -145
Better! Reward -158
Better! Reward -150
Better! Reward -153
Better! Reward -153
Better! Reward -145
Better! Reward -150
Better! Reward -151
Better! Reward -157
Better! Reward -163
Better! Reward -170
Better! Reward -155
Better! Reward -146
Better! Reward -150
Better! Reward -165
Better! Reward -166
Better! Reward -163
Better! Reward -154
Better! Reward -162
Better! Reward -145
Better! Reward -163
Better! Reward -165
Better! Reward -152
Better! Reward -147
Better! Reward -146
Better! Reward -164
Better! Reward -150
Better! Reward -163
Better! Reward -163
Better! Reward -166
Better! Reward -164
Better! Reward -163
Better! Reward -155
Better! Reward -163
Better! Reward -155
Better! Reward -163
Better! Reward -163
Better! Reward -152
Better! Reward -160
Better! Reward -158
Better! Reward -146
Better! Reward -131
Better! Reward -136
Better! Reward -138
Better! Reward -141
Better! Reward -148
Better! Reward -149
Better! Reward -156
Better! Reward -139
Better! Reward -151
Better! Reward -150
Better! Reward -154
Better! Reward -159
Better! Reward -154
Better! Reward -155
Better! Reward -159
Better! Reward -147
Better! Reward -142
Better! Reward -149
Better! Reward -169
Better! Reward -135
Better! Reward -137
Better! Reward -146
Better! Reward -142
Better! Reward -142
Better! Reward -174
Better! Reward -196
Better! Reward -170
Better! Reward -171
Better! Reward -142
Better! Reward -150
Better! Reward -185
Better! Reward -149
Better! Reward -148
Better! Reward -143
Better! Reward -146
Better! Reward -147
Better! Reward -154
Better! Reward -150
Better! Reward -147
Better! Reward -146
Better! Reward -143
Better! Reward -149
Better! Reward -150
Better! Reward -144
Better! Reward -150
Better! Reward -143
Better! Reward -151
Better! Reward -147
Better! Reward -149
Better! Reward -150
Better! Reward -150
Better! Reward -143
Better! Reward -184
Better! Reward -143
Better! Reward -150
Better! Reward -150
Better! Reward -150
Better! Reward -149
Better! Reward -149
Better! Reward -150
Better! Reward -143
Better! Reward -150
Better! Reward -143
Better! Reward -149
Better! Reward -149
Better! Reward -146
Better! Reward -145
Better! Reward -149
Better! Reward -147
Better! Reward -141
Better! Reward -146
Better! Reward -151
Better! Reward -184
Better! Reward -141
Better! Reward -145
Better! Reward -146
Better! Reward -149
Better! Reward -147
Better! Reward -192
Better! Reward -191

Lets plot the reward vs episodes!

moving average function

def moving_average(a, n=100) :
    ret = np.cumsum(a, dtype=float)
    ret[n:] = ret[n:] - ret[:-n]
    return ret[n - 1:] / n

scores_ma_QL=moving_average(rewards,n=100)

from matplotlib import pyplot as plt

# plot the scores
plt.plot(np.arange(len(scores_ma_QL)), scores_ma_QL)
plt.ylabel('Score')
plt.xlabel('Episode')

Text(0.5, 0, 'Episode')

png

Next blog is the real issue, Deep Q Network