
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 peak of summer vacation travel, a souvenir shop decides to purchase two types of souvenirs, $A$ and $B$. If they buy 7 pieces of type $A$ and 4 pieces of type $B$, it will cost $760$ yuan; if they buy 5 pieces of type $A$ and 8 pieces of type $B$, it will cost $800$ yuan. <br/>$(1)$ Find the cost of each piece of type $A$ and type $B$. <br/>$(2)$ If the shop decides to purchase a total of $100$ pieces of these two types of souvenirs, considering market demand and capital turnover, the cost of these $100$ pieces of souvenirs should be no less than $7000$ yuan, but not exceed $7200$ yuan. How many purchasing plans does the shop have? <br/>$(3)$ If the profit per piece of type $A$ souvenir is $30$ yuan, and the profit per piece of type $B$ souvenir is $20$ yuan, which purchasing plan from part $(2)$ will yield the maximum profit? What is the maximum profit?

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