
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 store purchased 60 boxes of type A fruits and 40 boxes of type B fruits before the "May Day" holiday. After selling all the fruits, they made a total profit of $1300. Type A fruits made $5 more profit per box than type B fruits.  
$(1)$ Find the profit per box for type A and type B fruits.  
$(2)$ After selling all the type A and type B fruits, in preparation for the "May Day" holiday, the store purchased another batch of type A fruits. With the original profit per box unchanged, they can sell 100 boxes per day on average. If they adjust the price and reduce it by $1, they can sell an additional 20 boxes per day. What is the optimal price reduction that maximizes the store's profit, and what is the maximum profit?

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