
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
After the opening of Line 4 of the Shenyang Metro, it will bring convenience to the residents of Heping and Changbai Island. It is known that after the line is opened, the interval $t$ (in minutes) between subway trains satisfies $2 \leq t \leq 20$. According to measurements, the passenger capacity of the subway is related to the interval $t$. When $10 \leq t \leq 20$, the subway is at full capacity with 1300 passengers. When $2 \leq t < 10$, the passenger capacity decreases, and the decrease is proportional to the square of $(10-t)$. Additionally, when the interval is 2 minutes, the passenger capacity is 660 people. <br/> $(Ⅰ)$ Write the function expression of $p$ in terms of $t$; <br/> $(Ⅱ)$ If the net profit per minute of this line is $Q = \frac{{6p-3960}}{t}-350$ (in yuan), what interval of time will maximize the net profit per minute, and what is the maximum net profit per minute?

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