
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 of subway departure time $t$ (unit: minutes) satisfies $2 \leq t \leq 20$. According to measurements, the passenger capacity of the subway is related to the departure time 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 in the number of passengers is directly proportional to $(10-t)^2$, and the passenger capacity when the departure time interval is 2 minutes 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 is the departure time interval at which the net profit per minute of this line is maximized, and what is the maximum net profit per minute?

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