
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 all positive integers \( k \) for which the following statement is true:
If \( F(x) \) is a polynomial with integer coefficients satisfying the condition
\[ 0 \leq F(c) \leq k \quad \text { for each } c \in \{0,1, \ldots, k+1\}, \]
then \( F(0) = F(1) = \cdots = F(k+1) \).

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