
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
Improving the vehicle passage capacity of a mountain tunnel can effectively alleviate traffic conditions. Generally, the vehicle flow speed $v$ (unit: kilometers per hour) in the tunnel is a function of the vehicle flow density $x$ (unit: vehicles per kilometer, representing the number of vehicles per kilometer of road). When the vehicle flow density in the tunnel reaches $210$ vehicles per kilometer, congestion occurs, and the vehicle flow speed at this time is $0$. When the vehicle flow density does not exceed $30$ vehicles per kilometer, the vehicle flow speed is $60$ kilometers per hour. Research shows that when $30 \leqslant x \leqslant 210$, the vehicle flow speed $v$ is a linear function of the vehicle flow density $x$.

(I) Find the expression of the function $v(x)$ when $0 \leqslant x \leqslant 210$;

(II) Find the vehicle flow density $x$ that can achieve the maximum vehicle flow $f(x) = x \cdot v(x)$ (unit: number of vehicles per hour), and find the maximum value.

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