
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
Let \( n \) be a strictly positive integer. Let \( m_{n} \) be the largest real number satisfying the following property: "Let \( x_{0}, x_{1}, \ldots, x_{n} \) be mutually distinct integers. For any monic polynomial \( F(x) \) of degree \( n \) with integer coefficients, at least one of the numbers \( \left|F\left(x_{0}\right)\right|, \left|F\left(x_{1}\right)\right|, \ldots,\left|F\left(x_{n}\right)\right| \) is greater than or equal to \( m_{n} \)."

Find the problem corresponding to the integer part of \( m_{9} \).

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