
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
Abby and Brian play the following game: They first choose a positive integer \( N \). Then they write numbers on a blackboard in turn. Abby starts by writing a 1. Thereafter, when one of them has written the number \( n \), the other writes down either \( n+1 \) or \( 2n \), provided that the number is not greater than \( N \). The player who writes \( N \) on the blackboard wins.

(a) Determine which player has a winning strategy if \( N=2011 \).

(b) Find the number of positive integers \( N \leq 2011 \) for which Brian has a winning strategy.

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