
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
For each positive integer $n$, find the largest real number $C(n)$ that satisfies the following inequality:
$$
C(n) \leqslant\left|\frac{f(a)-f(b)}{a-b}\right|,
$$

where $f$ is a polynomial of degree $n$, such that $f: \mathbf{Z} \rightarrow \mathbf{Z}$, and $a, b \in \mathbf{Z}$ with $f(a) \neq f(b)$.

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