
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 school plans to purchase some table tennis rackets and table tennis balls to enrich students' extracurricular activities. It is known that the price of a pair of table tennis rackets is $50, and the price of a box of table tennis balls is $20. It is now understood that two stationery stores are both running promotional activities: Store A: buy one pair of table tennis rackets and get one box of table tennis balls for free; Store B: all items are discounted by 20%. If the school plans to purchase 10 pairs of table tennis rackets and $x$ boxes of table tennis balls $(x > 10)$, find:

$(1)$ Express the total cost of purchasing rackets and balls at Store A and Store B with algebraic expressions involving $x$.

$(2)$ If the school plans to purchase 40 boxes of table tennis balls and choose to buy from either Store A or Store B, determine which store is more cost-effective.

$(3)$ Under the condition in $(2)$, if the school can also choose to purchase from both Store A and Store B simultaneously, design the most cost-effective purchasing plan.

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