
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
During the Spring Festival, a supermarket plans to purchase two types of pre-made vegetable gift boxes, $A$ and $B$. It is known that the number of boxes of type $A$ that can be purchased with $2000$ yuan is the same as the number of boxes of type $B$ that can be purchased with $1600$ yuan. The unit price of type $B$ pre-made vegetable gift boxes is $20$ yuan less than the unit price of type $A$ pre-made vegetable gift boxes.  
$(1)$ Find the unit prices of the two types of pre-made vegetable gift boxes.  
$(2)$ The supermarket plans to purchase a total of $50$ boxes of type $A$ and type $B$, and the total cost does not exceed $4600$ yuan. Find the maximum number of boxes of type $A$ pre-made vegetable gift boxes that can be purchased.

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