
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
During the "Golden Week" holiday from October 1st to 7th this year, the Drum Tower Shopping Center decided to hold a promotional event for several consecutive days. Originally, a suit is priced at 500 yuan each and a tie at 80 yuan each. During the promotion, two discount plans were offered to customers:
① Buy a suit and get a tie for free;
② Pay 90% of the listed price for both the suit and the tie.
Now a customer wants to buy 20 suits and $x$ ties ($x>20$) from the Drum Tower Shopping Center.
(1) If the customer buys according to plan ①, calculate the payment needed in yuan (express with an algebraic expression including $x$); if the customer buys according to plan ②, calculate the payment needed in yuan (express with an algebraic expression including $x$).
(2) If $x=30$, determine which plan offers a better deal through calculation.
(3) Clever Xiaoming noticed that the sum of the dates of the promotional period is exactly 15. Write down the start and end dates of this promotion directly.

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