
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 the fishing operation of a fishing boat, the main expenses incurred are fuel costs and labor costs. It is known that the fuel cost per hour of the fishing boat is directly proportional to the cube of the boat's speed. When the boat's speed is $10$ knots per hour, the fuel cost is $600$ yuan per hour, and the labor cost is $4050$ yuan per hour. Let the sailing speed of the fishing boat be $v$ (knots per hour), satisfying $0\leqslant v\leqslant 30$. Let $q$ yuan denote the main cost of the fishing boat sailing for one hour (main cost = fuel cost + labor cost), and let $p$ yuan denote the main cost generated by sailing $1$ nautical mile.<br/>$(1)$ Express the main cost $q$ yuan of sailing for one hour in terms of the sailing speed $v$ (knots per hour);<br/>$(2)$ Express the main cost $p$ yuan generated by sailing $1$ nautical mile in terms of the sailing speed $v$ (knots per hour);<br/>$(3)$ Find the minimum value of the main cost $p$ (yuan) generated by sailing $1$ nautical mile, and determine the corresponding sailing speed $v$ (knots per hour).

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