
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 players play a game, starting with a pile of \( N \) tokens. On each player's turn, they must remove \( 2^{n} \) tokens from the pile for some nonnegative integer \( n \). If a player cannot make a move, they lose. For how many \( N \) between 1 and 2019 (inclusive) does the first player have a winning strategy?

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