
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 shopping mall purchased two types of products, A and B, totaling 600 boxes. After selling all the products, type A made a profit of $40,000, and type B made a profit of $160,000. Type A made $200 less profit per box compared to type B.<br/>$(1)$ Find the profit per box for type A and type B.<br/>$(2)$ According to the market situation, the shopping mall wants to reduce the price of type A products for sale while increasing the price of type B products. According to the survey, for every $5 decrease in the price of type A products, two more boxes are sold, and for every $5 increase in the price of type B products, two fewer boxes are sold. With the condition that the total sales volume remains the same, at what price reduction for type A products will the shopping mall maximize its profit? What is the maximum profit?

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