
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  $N\leqslant2011$  for which Brian has a winning strategy.

(This is based on ISL 2004, Problem C5.)

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