
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 wants to purchase two types of goods, $A$ and $B$. If they buy 5 items of type $A$ and 3 items of type $B$, the total cost is $450$ yuan. If they buy 10 items of type $A$ and 8 items of type $B$, the total cost is $1000$ yuan.  
$(1)$ How much does each item of type $A$ and type $B$ cost?  
$(2)$ The store decides to purchase enough items of type $A$ and type $B. During sales, it is found that the selling price of type $A$ is $80$ yuan per item, and they can sell 100 items per day. Now, they decide to reduce the price of type $A$ by $1$ yuan per item and sell an additional 20 items. After reducing the price, the store sells more than 200 items of type $A$ per day. Type $B$ sells well, earning $7000$ yuan per day. In order to make the total daily profit from selling both type $A$ and type $B$ equal to $10000$ yuan, how much should the price of type $A$ be reduced per item?

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