
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
In general, the speed of traffic flow $v$ (unit: kilometers per hour) on a bridge is a function of traffic density $x$ (unit: vehicles per kilometer). When the traffic density on the bridge reaches $220$ vehicles per kilometer, congestion will occur, and the traffic speed will be $0$; when the traffic density does not exceed $20$ vehicles per kilometer, the traffic speed is $100$ kilometers per hour. Studies have shown that when $20\leqslant x\leqslant 220$, the traffic speed $v$ is a linear function of traffic density $x$. <br/>$(Ⅰ)$ When $0\leqslant x\leqslant 220$, find the analytical expression of the function $v(x)$; <br/>$(Ⅱ)$ Let the traffic flow $f(x)=x\cdot v(x)$ $(0\leqslant x\leqslant 220)$. Write down the analytical expression of $f(x)$, determine the value of traffic density $x$ at which $f(x)$ attains its maximum value, and find the maximum value of $f(x)$. <br/>("Traffic flow" refers to the number of vehicles passing through a certain observation point on the bridge per unit time, unit: vehicles per hour).

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