
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
 $A$  and  $B$  play a game, given an integer  $N$ ,  $A$  writes down  $1$  first, then every player sees the last number written and if it is  $n$  then in his turn he writes  $n+1$  or  $2n$ , but his number cannot be bigger than  $N$ . The player who writes  $N$  wins. For which values of  $N$  does  $B$  win?

*Proposed by A. Slinko & S. Marshall, New Zealand*

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