
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
Suppose $p(x), q(x), r(x)$ are linear functions, and $m(x)$ and $n(x)$ are defined by:
$$m(x) = \max\{p(x), q(x), r(x)\},$$
$$n(x) = \min\{p(x), q(x), r(x)\}.$$
Given that for $-4 \le x \le 4$, the graph of $y=m(x)$ is a piecewise linear function with vertices at $(-4, 6)$, $(-1, 1)$, and $(4, 6)$. 

Calculate the square of the total length $\ell$ of the graph of $y=n(x)$ over the interval $-4 \le x \le 4$.

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