
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
To welcome the shopping peak during the "May Day" holiday, a certain brand specialty store is preparing to purchase two types of shirts, A and B, with the cost and selling price of the two types of shirts as shown in the table below:

|         | A     | B     |
|---------|-------|-------|
| Cost ($) | $m$   | $m-10$|
| Price ($)| $260$ | $180$ |

If the cost of three A shirts and two B shirts is $480.

$(1)$ Find the cost of each A and B shirt.

$(2)$ If the specialty store needs to purchase a total of 300 A and B shirts, and the number of A shirts does not exceed 110, how many purchasing plans are there for the store to make a total profit of at least $34,000?

$(3)$ Under the conditions of $(2)$, the specialty store plans to conduct a promotional sale on A shirts, deciding to offer a discount of $a$ dollars per A shirt $(60 < a < 80)$ while keeping the price of B shirts unchanged. How should the store purchase to maximize profit?

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