
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 prepare for the shopping peak of International Women's Day, a certain cosmetics brand store is planning to purchase two types of cosmetics, A and B. The purchase price and selling price of the two types of cosmetics A and B are as follows:

| Type | A | B |
|------|---|---|
| Purchase Price (in $/item) | $m$ | $n$ |
| Selling Price (in $/item) | $250$ | $200$ |

If 3 items of type A and 4 items of type B are purchased, the total cost is $620$ yuan; if 5 items of type A and 3 items of type B are purchased, the total cost is $740$ yuan.

$(1)$ Find the values of $m$ and $n$.

$(2)$ In order for the total cost of purchasing 200 items of type A and B not to exceed $18100$ yuan, and the total profit after selling all items is not less than $27000$ yuan, how should the store purchase the items to maximize profit? Also, find the maximum profit.

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