
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 $f(x),g(x),h(x),m(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by 
$$j(x) = \max\{f(x),g(x),h(x),m(x)\},$$
$$k(x) = \min\{f(x),g(x),h(x),m(x)\}.$$
For each $x$, $j(x)$ is the greatest of these four values; similarly, $k(x)$ is the least.

Shown below is the graph of $y=j(x)$ for $-4 \le x \le 4$.

[asy]
size(150);
rr_cartesian_axes(-5,5,-5,5);
draw((-4,5)--(-1,1)--(1,1)--(4,5),red+1.25);
dot((-1,1),red);
dot((1,1),red);
[/asy]

Let $\ell$ be the length of the graph of $y=k(x)$ for $-4 \le x \le 4$. Compute $\ell^2$.

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