
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 deals with two types of products, $A$ and $B$. The cost price of each unit of product $A$ is $40$ yuan, and the selling price is $60$ yuan. The selling price of each unit of product $B$ is $80$ yuan, with a profit margin of $60\%$.<br/>$(1)$ The profit margin per unit of product $A$ is ______, and the cost price per unit of product $B$ is ______ yuan;<br/>$(2)$ If the shopping mall purchases a total of $50$ units of products $A$ and $B$ together, with a total cost of $2200$ yuan, how many units of each product did the shopping mall purchase?

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