
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 sells two types of products, $A$ and $B$, both with a cost price of $20$ yuan each. An investigation found that if 20 units of type $A$ and 10 units of type $B$ are sold, the total sales amount is $840$ yuan; if 10 units of type $A$ and 15 units of type $B$ are sold, the total sales amount is $660$ yuan. <br/>$(1)$ Find the selling prices of products $A$ and $B$; <br/>$(2)$ After market research, it was found that if product $A$ is sold at the original price, 40 units can be sold. For every $1$ yuan reduction in the selling price of product $A$, the sales volume can increase by 10 units. The selling price of product $B$ remains unchanged, and the selling price of product $A$ is not lower than that of product $B$. Let the price of product $A$ be reduced by $m$ yuan. If the sales volumes of products $A$ and $B$ are equal, find the value of $m$ that maximizes the total profit obtained from selling products $A$ and $B. What is the maximum profit?

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