
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$ , let  $ c(n)$  be the largest real number such that
\[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\]
for all triples  $ (f, a, b)$  such that

-- $ f$  is a polynomial of degree  $ n$  taking integers to integers, and
-- $ a, b$  are integers with  $ f(a) \neq f(b)$ .

Find  $ c(n)$ .

*Shaunak Kishore.*

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