
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 \( k \) such that: if \( F(x) \) is an integer-coefficient polynomial and satisfies the condition: for any \( C \in \{0,1, \cdots, k+1\} \), \( 0 \leqslant F(c) \leqslant k \), then
\[
F(0)=F(1)=\cdots=F(k+1).
\]

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