
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
After the highly anticipated 2022 Qatar World Cup, to meet market demand, a certain merchant plans to purchase two types of jerseys, $A$ and $B$. According to a survey, the number of $A$ jerseys that can be purchased for $30,000$ yuan is three times the number of $B$ jerseys that can be purchased for $9,000$ yuan. The cost price of one $A$ jersey is $20$ yuan more than the cost price of one $B$ jersey.  
$(1)$ Find the cost price of one $A$ jersey and one $B$ jersey.  
$(2)$ If the merchant purchases a total of $210$ jerseys of types $A$ and $B$ for a trial sale, where the number of $A$ jerseys is no more than twice the number of $B$ jerseys and is not less than $100$, and all jerseys are sold out. Given that the selling price of one $A$ jersey is $320$ yuan and one $B$ jersey is $280$ yuan, and defining the purchase of $A$ jerseys as $m$, find the analytical expression of the function between the profit $W$ of selling these goods and $m$, and write down the range of values for $m$.  
$(3)$ Under the conditions of $(2)$, the merchant decides to support children in impoverished mountainous areas by deducting $a$ yuan from the profit of each $A$ jersey sold during the trial sale. Find the maximum profit the merchant can obtain after selling all the jerseys and supporting the children in impoverished mountainous areas.

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