
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 Zhengzhou Metro has experienced a transformation from non-existence to existence, from single line to multiple lines, from point to surface, from surface to network, forming a grid operation, sharing the passenger flow of buses, alleviating urban traffic pressure, and stimulating new vitality in the city. It is known that after a certain line is opened, the train departure time interval (in minutes) satisfies $2\leqslant t\leqslant 20$. According to market research and calculations, the passenger capacity of the train is related to the departure time interval $t$. When $10\leqslant t\leqslant 20$, the train is at full capacity with 1200 passengers. When $2\leqslant t \lt 10$, the passenger capacity decreases, and the decrease in the number of passengers is proportional to $\left(10-t\right)^{2}$. Additionally, when the departure time interval is 2 minutes, the passenger capacity is 560 people. Let the train's passenger capacity be denoted as $p\left(t\right)$.
  
$(1)$ Find the expression for $p\left(t\right)$.
  
$(2)$ If the net profit per minute for this line is $Q(t)=\frac{6p(t)-3360}{t}-360$ (unit: yuan), determine at what departure time interval the net profit per minute for this line is maximized, and find the maximum value.

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