
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$  for each  $ c\in \{0,1,\ldots,k \plus{} 1\}$ , then   $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$ .

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