
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 "flowing water fish cage" aquaculture technology is characterized by high breeding density and good economic benefits. Studies have shown that for a certain type of fish under specific conditions, the average growth rate $v$ (unit: kg/year) is a function of the breeding density $x$ (unit: tails/cubic meter). When $x$ does not exceed 4 tails/cubic meter, the value of $v$ is 2 kg/year; when $4 < x \leqslant 20$, $v$ is a linear function of $x$, and when $x$ reaches 20 tails/cubic meter, due to reasons such as lack of oxygen, the value of $v$ is 0 kg/year.
1. When $0 < x \leqslant 20$, find the function expression of $v$ with respect to $x$;
2. When the breeding density $x$ is how much, can the annual growth amount of fish (unit: kg/cubic meter) reach the maximum? And find the maximum value.

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