
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 is preparing to purchase two types of clothing, A and B, for sale. The cost price of type A clothing is $160$ yuan per piece, and the selling price is $220$ yuan; the cost price of type B clothing is $120$ yuan per piece, and the selling price is $160$ yuan. It is planned to purchase a total of $100$ pieces of both types of clothing, with at least $60$ pieces of type A. Let $x$ represent the number of type A clothing purchased. When all the clothing is sold out, the mall will make a profit of $y$ yuan. <br/>$(1)$ Find the functional relationship between $y$ and $x$; <br/>$(2)$ If the total cost of purchasing 100 pieces of clothing does not exceed $15000$ yuan, what is the maximum profit? <br/>$(3)$ Under the condition in $(2)$, the store offers a discount of $a$ yuan per piece for type A clothing ($0 \lt a \lt 20$), and the cost price of type B clothing is reduced by $b$ yuan per piece, with the selling price remaining the same. It is known that $a-b=4$. If the maximum profit is $4950$ yuan, determine the value of $a$ directly.

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