
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 $a(x), b(x), c(x)$ are linear functions, and $m(x)$ and $n(x)$ are defined by:
$$m(x) = \max\{a(x), b(x), c(x)\},$$
$$n(x) = \min\{a(x), b(x), c(x)\}.$$
Given the graph of $y=m(x)$ for the interval $-4 \leq x \leq 4$, which shows three line segments meeting at points $(-2, 1)$ and $(2, 1)$, compute the value of $\ell^2$, where $\ell$ is the length of the graph of $y=n(x)$ over the same interval.

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