
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
Qingxi School decided to purchase a batch of sports equipment to adapt to the new requirements of the high school entrance examination. After checking on Tmall, the school found that a certain brand of soccer ball is priced at $100$ yuan each, and a jump rope is priced at $20$ yuan each. There are two online stores, $A$ and $B$, both offering free shipping and their own discount schemes.<br/>$A$ store: Buy one soccer ball and get one jump rope for free;<br/>$B$ store: Both soccer balls and jump ropes are priced at $90%$ of the original price.<br/>It is known that the school wants to buy $30$ soccer balls and $x$ jump ropes $\left(x \gt 30\right)$.<br/>$(1)$ If purchased from store $A$, the payment will be ______ yuan, and if purchased from store $B$, the payment will be ______ yuan; (express using algebraic expressions with $x$)<br/>$(2)$ When $x=100$, calculate and determine which store is more cost-effective to purchase from?<br/>$(3)$ When $x=100$, design a more cost-effective purchasing plan and calculate the total payment?

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