
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
On October 28, 2016, after nearly half a century of service, the Nanjing Yangtze River Bridge finally took a break, having played an indispensable role in improving the traffic conditions of our city. According to relevant data, under normal circumstances, the traffic speed $v$ (unit: km/h) on the bridge is a function of the traffic density $x$ (unit: vehicles/km). When the traffic density on the bridge reaches 280 vehicles/km, it causes congestion, and at this time, the traffic speed is 0; when the traffic density does not exceed 30 vehicles/km, the traffic speed is 50 km/h. Studies have shown that when $30 \leq x \leq 280$, the traffic speed $v$ is a linear function of the traffic density $x$.
(1) When $0 \leq x \leq 280$, find the expression for the function $v(x)$;
(2) When the traffic density $x$ is at what value, the traffic flow (the number of vehicles passing a certain observation point on the bridge per unit time, unit: vehicles/hour) $f(x) = x \cdot v(x)$ can reach its maximum, and find the maximum value.

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