
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
During this year's summer vacation, there is a self-driving tour convoy consisting of 31 vehicles of the same model, each approximately 5m in length (calculated as 5m), travelling to Qinghai. During the trip, the convoy passes through a tunnel that is 2725m long (the speed through the tunnel cannot exceed 20m/s), moving at a constant speed. Let the convoy's speed be $x$ m/s. For safety and traffic flow considerations, when $0 < x \leq 12$, the distance between adjacent vehicles must be maintained at 20m, and when $12 < x \leq 20$, it must be $ (\frac{1}{6}x^{2}+ \frac{1}{3}x)$ m. The time $y$ (in seconds) that it takes from the front of the first car entering the tunnel to the rear of the 31st car leaving the tunnel is to be calculated.
(Ⅰ) Express $y$ as a function of $x$;
(Ⅱ) Determine the minimum value of the time $y$ for the convoy to pass through the tunnel and the corresponding speed of the convoy.

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