
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 sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.  
$(1)$ Find the cost of each ball of brand A and brand B.  
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?  
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?

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