
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
When riding an electric bicycle, wearing a safety helmet can protect the head and reduce injuries. A certain store sells two types of safety helmets, A and B, at wholesale prices of $30$ per unit and $20$ per unit, respectively. When sold, the unit price of helmet A is $15 more expensive than that of helmet B. One day, the sales revenue of helmet A is $450, and the sales revenue of helmet B is $600. At this time, the sales volume of helmet B is exactly twice that of helmet A.<br/>$(1)$ Find the unit prices of helmets A and B;<br/>$(2)$ If the store is prepared to purchase a total of $100$ helmets of these two types with no more than $2350$ dollars, how many helmets of type A can be purchased at most?

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