
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
The public security traffic police departments at all levels in Yunfu City remind citizens that they must strictly abide by the regulations of "one helmet per person" when riding a bike. A certain shopping mall in Yunan County simultaneously purchased two types of helmets, $A$ and $B$. It is known that purchasing $3$ type $A$ helmets and $4$ type $B$ helmets costs a total of $288$ yuan; purchasing $6$ type $A$ helmets and $2$ type $B$ helmets costs a total of $306$ yuan.  
$(1)$ What is the cost price of each type of helmet?  
$(2)$ In sales, the shopping mall found that when the selling price of each type $A$ helmet is $50$ yuan, they can sell $100$ helmets per month; when the selling price is increased by $5$ yuan, they sell $10$ fewer helmets per month. Let $x$ represent the selling price of each type $A$ helmet in yuan $(50\leqslant x\leqslant 100)$, and $y$ represent the profit made by the mall from selling type $A$ helmets each month. Find the analytical expression of $y$ in terms of $x$ and determine the maximum profit.

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