
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
Find the largest integer  $N \in \{1, 2, \ldots , 2019 \}$  such that there exists a polynomial  $P(x)$  with integer coefficients satisfying the following property: for each positive integer  $k$ ,  $P^k(0)$  is divisible by  $2020$  if and only if  $k$  is divisible by  $N$ .  Here  $P^k$  means  $P$  applied  $k$  times, so  $P^1(0)=P(0), P^2(0)=P(P(0)),$  etc.

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