
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 supermarket purchases two types of goods, $A$ and $B$. Buying 4 items of type $A$ costs $10$ yuan less than buying 5 items of type $B$. Buying 20 items of type $A$ and 10 items of type $B$ costs a total of $160$ yuan.  
$(1)$ Find the cost price per item of goods $A$ and $B$ respectively.  
$(2)$ If the store purchases a total of 200 items of goods $A$ and $B$, both priced at $10$ yuan for sale. After selling some of the items, the store reduces the price by 20% to sell all remaining items. The number of items sold at $10$ yuan is 30 less than the number of items purchased for type $A$. If the store makes a profit of at least $640$ yuan from selling goods $A$ and $B, find the minimum number of items of type $A$ purchased.

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