
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 clothing store is preparing to purchase two types of clothing, type A and type B. Type A costs $80$ yuan per piece and is sold for $120$ yuan; type B costs $60$ yuan per piece and is sold for $90$ yuan. The plan is to purchase a total of $100$ pieces of both types of clothing and sell them all, with at least $65$ pieces being type A.<br/>$(1)$ If the cost of purchasing these $100$ pieces of clothing cannot exceed $7500$ yuan, how many pieces of type A clothing can be purchased at most?<br/>$(2)$ Under the condition in $(1)$, if the store offers a discount of $a$ yuan per piece for type A clothing (where $0 < a < 10$) and the price of type B clothing remains the same, how should the store purchase the clothing to maximize profit?

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