
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
A tunnel has a length of $2150m$. The speed limit through the tunnel is $20m/s$. A convoy of $55$ vehicles, each with a length of $10m$, passes through the tunnel at a constant speed. The maximum speed this model of vehicle can travel is $40m/s$. Let $xm/s$ be the speed of the convoy. For safety and traffic flow, when $0 < x \leq 10$, a distance of $20m$ is maintained between adjacent vehicles; when $10 < x \leq 20$, a distance of $(\frac{1}{6}x^{2}+ \frac{1}{3}x)m$ is maintained between adjacent vehicles. Let $y(s)$ be the time from when the first vehicle enters the tunnel until the last vehicle leaves the tunnel.
(1) Express $y$ as a function of $x$;
(2) Find the minimum time $y$ for the convoy to pass through the tunnel and the speed of the convoy at this time. $(\sqrt{3} \approx 1.73)$

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