
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
The subway project in Jinan is progressing rapidly, and after the subway is opened, it will bring convenience to the citizens. It is known that after a certain route is opened, the train departure time interval $t$ (unit: minutes) satisfies $2 \leq t \leq 20$. According to market research, the passenger capacity of the train is related to the departure time interval $t$. When $10 \leq t \leq 20$, the train is at full capacity with 500 passengers. When $2 \leq t < 10$, the passenger capacity decreases, and the decrease in the number of passengers is directly proportional to $(10-t)^2$. Additionally, when the departure time interval is 2 minutes, the passenger capacity is 372 passengers. Let the passenger capacity of the train be denoted as $p(t)$.

$(1)$ Find the expression for $p(t)$ and determine the passenger capacity of the train when the departure time interval is 5 minutes.

$(2)$ If the net profit per minute for this route is $Q(t) = \frac{{8p(t) - 2656}}{t} - 60$ (in dollars), determine at what time interval the net profit per minute for this route is maximized, and find the maximum value.

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