
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
On the board is written an integer  $N \geq 2$ . Two players  $A$  and  $B$  play in turn, starting with  $A$ . Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. 
Determine the smallest even number  $N$  requires you to play at least  $2015$  times to win ( $B$  shifts are not counted).

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