
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
To encourage college graduates to start their own businesses, a certain city government has implemented relevant policies. Local companies provide products for college graduates to sell independently, and the government also provides a certain subsidy to college graduates. Wang Hua invested in selling a certain brand of clothing according to the relevant policies. It is known that the cost price of this brand of clothing is $100$ yuan per piece, the government subsidizes $20$ yuan per piece, and the relationship between the monthly sales volume $y$ (pieces) and the selling price $x$ (yuan) satisfies a linear function: $y=-3x+900$.

$(1)$ If Wang Hua sets the selling price at $160$ yuan, how much does the government subsidize each month?

$(2)$ Let Wang Hua's total monthly revenue be $w$ (yuan). At what selling price will the total revenue be maximized, and what is the maximum total revenue per month? (Monthly total revenue $=$ monthly sales profit $+$ monthly government subsidy)

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