
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two people  $A$  and  $B$  play the following game: They take from  $\{0, 1, 2, 3,..., 1024\}$  alternately   $512$ ,  $256$ ,  $128$ ,  $64$ ,  $32$ ,  $16$ ,  $8$ ,  $4$ ,  $2$ ,  $1$ , numbers away where  $A$  first removes  $512$  numbers,  $B$  removes  $256$  numbers etc. Two numbers  $a, b$  remain ( $a < b$ ).  $B$  pays  $A$  the amount  $b - a$ .  $A$  would like to win as much as possible,  $B$  would like to lose as little as possible. What profit does  $A$  make if does every player play optimally according to their goals? The result must be justified.

Present the answer in LaTex format: \boxed{Your answer}
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