
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 school plans to purchase two types of teaching equipment, $A$ and $B$. It is known that the price of type $A$ equipment is $40$ yuan each. For type $B$ equipment, if the quantity purchased does not exceed $25$ units, it will be purchased at the original price. If the quantity purchased exceeds $25$ units, the excess units will be discounted by $b$ yuan each on top of the original price. The school has calculated that purchasing $30$ units of type $B$ equipment will cost $1425$ yuan, and purchasing $50$ units will cost $2125 yuan. <br/> $(1)$ Find the selling price of type $B$ equipment and the value of $b$. <br/> $(2)$ The school wants to purchase a total of $90$ units of type $A and type $B$ equipment. They require the quantity of type $B$ equipment to be no less than $15$ units and no more than twice the quantity of type $A$ equipment. Help the school make a decision on how to allocate the purchase quantities to minimize the total purchase cost. What is the minimum total purchase cost in yuan?

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