
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
A teacher intends to give children a task of the following kind. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will provide them with \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \) and separately will communicate the value of the expression \( P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right) \). Based on this data, the children must identify the polynomial that the teacher might have thought of. For what minimum \( k \) will the teacher be able to compose a problem of this type such that the polynomial found by the children necessarily coincides with the one the teacher had in mind?

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