
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 junior high school soccer team needs to purchase two brands of soccer balls, $A$ and $B$. It is known that the unit price of brand $A$ soccer balls is $10$ yuan higher than the unit price of brand $B$ soccer balls. If $2$ brand $A$ soccer balls and $3$ brand $B$ soccer balls together cost $220$ yuan, find:<br/>
$(1)$ The unit prices of brand $A$ and brand $B$ soccer balls;<br/>
$(2)$ The total cost $W$ of purchasing $60$ brand $A$ and brand $B$ soccer balls, with $m$ brand $A$ soccer balls purchased. Write down the relationship between the total cost $W$ and the number of brand $A$ soccer balls $m$;<br/>
$(3)$ Under the condition in $(2)$, if the total cost of purchasing both brands of soccer balls does not exceed $2850$ yuan, and the quantity of brand $A$ soccer balls purchased is not less than $43, how many different purchasing plans does the soccer team have? Which purchasing plan has the lowest total cost, and what is the minimum cost?

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