
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 the function \( y = 4x^2 - 4kx + (k - 1) \) within the range \(-1 \leq k \leq 1\), find the value of \( m + n \) for the maximum and minimum length of the segment intercepted by its graph on the \( x \)-axis.

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