
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
A certain commodity has a cost price of $10$ yuan per unit. If it is sold at a price of $20$ yuan per unit, then $360$ units can be sold per month. If it is sold at a price of $30$ yuan per unit, then $60$ units can be sold per month. Assume that the monthly sales quantity $y$ is a linear function of the selling price $x$ (in yuan).<br/>$(1)$ Find the analytical expression of $y$ in terms of $x$;<br/>$(2)$ At what price should the product be sold to maximize the monthly profit? Calculate this maximum profit.

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