
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
Given that \(m\) and \(n\) are constants, the parabola \(C: y = (t^2 + t + 1)x^2 - 2(m+t)^2 x + (t^2 + 3mt + n)\) passes through the point \(P(1,0)\) for any \(t \in \mathbb{R}\).

(I) Find the values of \(m\) and \(n\).

(II) For which value of \(t\) does the chord obtained by the parabola \(C\) when it intersects the \(x\)-axis have the maximum length? What is the maximum length?

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