
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 \( P(n) \) be the number of permutations \(\left(a_{1}, \ldots, a_{n}\right)\) of the numbers \((1,2, \ldots, n)\) for which \( k a_{k} \) is a perfect square for all \( 1 \leq k \leq n \). Find with proof the smallest \( n \) such that \( P(n) \) is a multiple of 2010.

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