
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 sells a pair of sports shoes and a pair of sports socks. The sports shoes are priced at $200$ yuan per pair, and the sports socks are priced at $40$ yuan per pair. The mall decides to launch a promotion during which two discount options are offered to customers:<br/>Option 1: Buy one pair of sports shoes and get one pair of sports socks for free;<br/>Option 2: Pay $90\%$ of the original price for both the sports shoes and sports socks.<br/>Now, a customer wants to buy $20$ pairs of sports shoes and $x$ pairs of sports socks at the mall $\left(x \gt 20\right)$.<br/>$(1)$ If the customer chooses Option 1, how much money does the customer need to pay? (Express in simplified algebraic expression with $x$)<br/>If the customer chooses Option 2, how much money does the customer need to pay? (Express in simplified algebraic expression with $x$)<br/>$(2)$ When $x=30$, calculate and determine which of the two purchasing options saves money.

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