
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 purchases two types of products, $A$ and $B$, with their purchase prices and selling prices as shown in the table below (profit = selling price - purchase price):

|          | $A$ type | $B$ type |
|----------|----------|----------|
| Purchase Price (in $/item) | $40$ | $28$ |
| Selling Price (in $/item) | $55$ | $40$ |

$(1)$ The store purchases a total of $80$ items of types $A$ and $B$, with $x$ items of type $A$ purchased. Find the total profit the store can make after selling all items of types $A$ and $B$ in terms of $x$.  
$(2)$ The store plans to adjust the selling price of type $B$ for sale. If sold at the original price, an average of $4$ items can be sold per day. After investigation, it was found that for every $1$ yuan reduction in selling price, an additional $2$ items can be sold per day. What should be the selling price per item of type $B$ in order to achieve an average daily sales profit of $96$ yuan for type $B$?

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