
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 company purchased a batch of $A$ and $B$ type domestic chips, where the unit price of type $A$ chips is $9$ yuan less than the unit price of type $B$ chips. It is known that the company purchased an equal quantity of type $A$ chips for $3120$ yuan and type $B$ chips for $4200$ yuan.<br/>$(1)$ Find the unit price of type $A$ and type $B$ chips purchased by the company.<br/>$(2)$ If the company purchased a total of $200$ chips of both types, and the quantity of type $A$ chips purchased does not exceed $\frac{1}{3}$ of the quantity of type $B$ chips purchased, but is not less than $\frac{1}{4}$ of the quantity of type $B$ chips purchased, find how to make the purchase to minimize the total cost. What is the minimum cost in yuan?

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