
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
The "living water net cage" fish farming technique has the characteristics of high farming density and good economic benefits. Research shows that when a certain type of fish is farmed using the "living water net cage" method, the average growth rate $v$ (in kilograms per year) of each fish is a function of the farming density $x$ (in tails per cubic meter). When $x$ does not exceed $4$ (tails per cubic meter), the value of $v$ is $2$ (kilograms per year); when $4 \leqslant x \leqslant 20$, $v$ is a first-degree function of $x$; when $x$ reaches $20$ (tails per cubic meter), due to lack of oxygen and other reasons, the value of $v$ is $0$ (kilograms per year).
(1) Find the expression of the function $v(x)$ when $0 < x \leqslant 20$;
(2) Find the farming density $x$ when the annual growth $f(x) = x \cdot v(x)$ (in kilograms per cubic meter) of the fish can reach the maximum, and find the maximum value.

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