
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 wants to purchase two types of household appliances, $A$ and $B$. It is known that the cost price of type $B$ appliances is $100$ yuan more per unit than the cost price of type $A$ appliances. After calculation, the number of units of type $A$ appliances that can be purchased with $10,000$ yuan is the same as the number of units of type $B$ appliances that can be purchased with $12,000$ yuan. Please answer the following questions:  
$(1)$ What are the cost prices per unit of these two types of household appliances?  
$(2)$ If the shopping mall wants to purchase a total of $100$ units of the two types of household appliances, with the total amount not exceeding $53,500$ yuan, and the number of type $A$ appliances not exceeding $67$ units, what are the possible purchasing schemes for the shopping mall?  
$(3)$ Under the conditions in $(2)$, if the selling prices of type $A$ and type $B$ appliances are $600$ yuan and $750$ yuan per unit, respectively, and the shopping mall takes out a total of $10$ units of the two types of appliances to reward outstanding employees, and after selling the remaining appliances, the mall still makes a profit of $5,050$ yuan, please directly write down the number of units of type $B$ appliances among these $10$ units.

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