
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 assertion holds:
If  $F(x)$  is polynomial with integer coefficients ehich satisfies  $F(c) \leq k$  for all  $c \in \{0,1, \cdots,k+1 \}$ , then \[F(0)= F(1) = \cdots =F(k+1).\]

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