
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 helmet for safety and a seatbelt for security. A certain store sells a batch of helmets for $80$ yuan each, selling $200$ helmets per month. During the "Creating a Civilized City" period, the plan is to reduce the price of the helmets for sale, but without incurring losses and not reducing the price by less than $10. After investigation, it was found that for every $1$ reduction in price, an additional $10$ helmets could be sold per month. It is known that the cost of each helmet is $50$ yuan.<br/>$(1)$ When the monthly profit is $5250$ yuan, find the selling price of each helmet at that time.<br/>$(2)$ At what price per helmet will the monthly sales profit be maximized? What is the maximum profit?

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