
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 shopping mall purchased a total of 100 pieces of two types of clothing, $A$ and $B$. It is known that the cost of purchasing these 100 pieces of clothing must not exceed 7500 yuan, and that the number of type $A$ clothing purchased is not less than 65. The cost and selling price of the clothing are shown in the table below:

| Clothing | Cost (yuan/piece) | Selling Price (yuan/piece) |
|----------|--------------------|---------------------------|
| $A$      | $80$               | $120$                     |
| $B$      | $60$               | $90$                      |

Let $x$ represent the number of type $A$ clothing purchased. If all the purchased type $A$ and type $B$ clothing are sold out, answer the following questions based on the information in the table:
1. Find the relationship between the total profit $y$ yuan and the number of type $A$ clothing purchased $x$, and write down the range of values for $x$.
2. If the shopping mall offers a discount of $a$ yuan (where $0 < a < 20$) on each piece of type $A$ clothing for a promotional activity while keeping the selling price of type $B$ clothing unchanged, how should the shopping mall adjust the quantities of type $A$ and type $B$ clothing purchased to maximize the total profit $y$?

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