
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
Let  $S$  be the set of positive integers not divisible by  $p^4$  for all primes  $p$ . Anastasia and Bananastasia play a game.

At the beginning, Anastasia writes down the positive integer  $N$  on the board. Then the players take moves in turn; Bananastasia moves first. On any move of his, Bananastasia replaces the number  $n$  on the blackboard with a number of the form  $n-a$ , where  $a\in S$  is a positive integer. On any move of hers, Anastasia replaces the number  $n$  on the blackboard with a number of the form  $n^k$ , where  $k$  is a positive integer. Bananastasia wins if the number on the board becomes zero.

Compute the second-smallest possible value of  $N$  for which Anastasia can prevent Bananastasia from winning.

*Proposed by Brandon Wang and Vincent Huang*

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