
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
For a polynomial \( P(x) \) with integer coefficients, let \( P^{1}(x) = P(x) \) and \( P^{k+1}(x) = P(P^{k}(x)) \) for \( k \geq 1 \). Find all positive integers \( n \) for which there exists a polynomial \( P(x) \) with integer coefficients such that for every integer \( m \geq 1 \), the numbers \( P^{m}(1), \ldots, P^{m}(n) \) leave exactly \( \left\lceil n / 2^{m} \right\rceil \) distinct remainders when divided by \( n \).

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