
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 any positive integer $k$, let $S(k)$ denote the sum of the digits of $k$ in its decimal representation. Find all integer-coefficient polynomials $P(x)$ such that for any positive integer $n \geqslant 2016$, $P(n)$ is a positive integer and satisfies
$$
S(P(n)) = P(S(n)).

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