
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 factory is preparing to produce two types of epidemic prevention supplies, denoted as $A$ and $B$. It is known that the cost per box of type $A$ is $500$ yuan more than the cost per box of type $B$. After calculation, it is found that the number of boxes of type $A$ that can be produced with 6000 yuan is equal to the number of boxes of type $B$ that can be produced with 4500 yuan. Please answer the following questions:<br/>$(1)$ Find the cost per box of type $A$ and type $B$ epidemic prevention supplies;<br/>$(2)$ The factory plans to produce a total of 50 boxes of type $A$ and type $B$ epidemic prevention supplies simultaneously with no more than 90000 yuan, with no more than 25 boxes of type $B. How many production plans are there?<br/>$(3)$ To expand production, the manufacturer plans to use the same cost as the lowest cost in $(2)$ to purchase two types of equipment, denoted as type A and type B (buying both types). If the cost of one unit of type A equipment is 2500 yuan and the cost of one unit of type B equipment is 3500 yuan, how many purchasing plans are there? How many units of type A and type B equipment can be purchased at most? (Please provide the answers directly)

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