
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 clothing factory produces jackets and $T$-shirts, with each jacket priced at $100$ yuan and each $T$-shirt priced at $60$ yuan. During a promotional period, the factory offers two discount options to customers:<br/>① Buy one jacket and get one $T$-shirt for free;<br/>② Both the jacket and $T$-shirt are paid at $80\%$ of the original price.<br/>Now, a customer wants to buy 30 jackets and $x$ $T$-shirts from the factory $\left(x \gt 30\right)$.<br/>$(1)$ If the customer chooses option ① to purchase, the customer needs to pay ______ yuan for the jackets and ______ yuan for the $T$-shirts (expressed in terms of $x$); if the customer chooses option ② to purchase, the customer needs to pay ______ yuan for the jackets and ______ yuan for the $T$-shirts (expressed in terms of $x$);<br/>$(2)$ If $x=40$, by calculation, determine which option, ① or ②, is more cost-effective?<br/>$(3)$ If both discount options can be used simultaneously, when $x=40$, can you provide a more cost-effective purchasing plan? Please write down your purchasing plan and explain the reason.

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