
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 store decides to purchase two types of souvenirs, $A$ and $B$, for sale. It is known that the cost price of each type $A$ souvenir is $30$ yuan higher than that of each type $B$ souvenir. It is found that the quantity of type $A$ souvenirs purchased with $1000$ yuan is the same as the quantity of type $B$ souvenirs purchased with $400$ yuan.  
$(1)$ Find the cost price of each type $A$ and $B$ souvenir.  
$(2)$ Through market research, the store has compiled a table showing the relationship between the selling price $x$ (in yuan per item) and the quantity sold of type $A$ souvenirs as follows:

| Selling Price $x$ (yuan per item) | $50\leqslant x\leqslant 60$ | $60 \lt x\leqslant 80$ |
|-----------------------------------|-----------------------------|-----------------------|
| Quantity Sold (items)              | $100$                       | $400-5x$              |

① For what value of $x$ is the profit maximized when selling type $A$ souvenirs, and what is the maximum profit?  
② The store purchases a total of $200$ items of type $A$ and $B$ souvenirs, with the quantity of type $A$ souvenirs being less than the quantity of type $B$ souvenirs but not less than $60$ items. If the selling price of type $B$ souvenirs is $30$ yuan per item, find the maximum profit the store can obtain after selling all type $A$ and $B$ souvenirs.

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