
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
Given \( n \in \mathbf{N}_{+}, k \in \mathbf{N}, P \) denotes the set of all \( n \)-th degree polynomials with complex coefficients and leading coefficient 1. For \( F \in P \), find the minimum value of
$$
|F(0)|^{2} + |F(1)|^{2} + \cdots + |F(n+k)|^{2}.
$$

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