
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]


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A certain store plans to purchase a certain number of items of two types, $A$ and $B$, for this year's Christmas. The quantity of type $A$ items that can be purchased for $480$ yuan is $\frac{3}{4}$ of the quantity of type $B$ items that can be purchased for the same amount. It is known that each item of type $A$ costs $4$ yuan more than each item of type $B$.  
$(1)$ Find the cost of purchasing one item of type $A$ and one item of type $B.  
$(2)$ If the store purchases a total of $200$ items of types $A$ and $B$ at once, and the total cost does not exceed $3000$ yuan, what is the minimum number of items of type $B$ that must be purchased?

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