
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 epidemic, to meet the protective needs of the public, a certain pharmacy wants to purchase two types of masks, $A$ and $B$. The cost of each box of type $B$ masks is $10$ yuan less than twice the cost of type $A$ masks. The number of boxes of type $A$ masks purchased with $6000$ yuan is the same as the number of boxes of type $B$ masks purchased with $10000$ yuan.  
$(1)$ What are the costs of each box of type $A$ and type $B$ masks, respectively?  
$(2)$ According to market research, type $B$ masks are more popular. When the selling price of each box of type $B$ masks is $60$ yuan, the daily average sales volume is $100$ boxes. For each increase of $1$ yuan in the selling price of type $B$ masks, the daily average sales volume decreases by $5$ boxes. At what price per box should type $B$ masks be sold to maximize the daily average total profit from selling type $B$ masks? What is the maximum daily average total profit?

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