
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 $k$ that satisfy the following proposition: If $F(x)$ is a polynomial with integer coefficients and meets the condition that 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