
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 \( f(x) = mx^2 + (2n + 1)x - m - 2 \) (where \( m, n \in \mathbb{R} \) and \( m \neq 0 \)) have at least one root in the interval \([3, 4]\). Find the minimum value of \( m^2 + n^2 \).

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