
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 sells two types of computers, type $A$ and type $B$. The profit per unit for type $A$ computer is $400$ yuan, and for type $B$ computer is $500$ yuan. The store plans to purchase a total of $100$ units of both types of computers at once, with the condition that the quantity of type $B$ computers purchased does not exceed three times the quantity of type $A$ computers. Let $x$ be the number of type $A$ computers purchased. The total profit from selling these $100$ computers is denoted as $y$ yuan.  
$(1)$ Find the expression of $y$ in terms of $x$.  
$(2)$ For how many units of type $A$ computers purchased will the total profit be maximized? What is the maximum profit?

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