
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 $f(x), g(x), h(x), p(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by:
$$j(x) = \max\{f(x), g(x), h(x), p(x)\},$$
$$k(x) = \min\{f(x), g(x), h(x), p(x)\}.$$
This means that, for each $x$, $j(x)$ will be the maximum and $k(x)$ will be the minimum of these four values.

Suppose further that the points where $k(x)$ changes its definition over the interval $-4\le x \le 4$ are $x = -3, x = 0, x = 3$. Calculate $\ell^2$, where $\ell$ is the length of the graph of $y = k(x)$ over the interval $-4 \le x \le 4$.

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