
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 is preparing to purchase two types of water purifiers, $A$ and $B$. The cost of each $A$ water purifier is $600$ yuan less than the cost of each $B$ water purifier. The mall spends $36,000$ yuan to purchase the $A$ water purifiers, which is twice the number of $B$ water purifiers that can be purchased with $27,000$ yuan. The selling prices of each $A$ and $B$ water purifier are $1,350$ yuan and $2,100$ yuan, respectively. Please answer the following questions:<br/>$(1)$ What is the cost of each $A$ and $B$ water purifier?<br/>$(2)$ If the shopping mall uses $60,000$ yuan to purchase both $A$ and $B$ water purifiers, with the condition that the number of $B$ water purifiers purchased does not exceed $8$, and let $a$ represent the number of $A$ water purifiers purchased, how many purchasing plans does the shopping mall have?<br/>$(3)$ Under the condition in $(2)$, to promote sales, the mall introduces a promotion where customers can draw a prize for each water purifier purchased, and the winning customers will receive a filter for the same water purifier. The cost of each filter for $A$ and $B$ water purifiers is $400$ yuan and $500$ yuan, respectively. If all the water purifiers are sold out and the mall still makes a profit of $5,250$ yuan after deducting the cost of the prizes, how many filters in total were given away for both types of water purifiers? Please provide the answer directly.

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