
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
The quadratic function \( f(x) = x^2 + mx + n \) has real roots. The inequality \( s \leq (m-1)^2 + (n-1)^2 + (m-n)^2 \) holds for any quadratic function satisfying the above conditions. What is the maximum value of \( s \)?

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