
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 online store implemented a discount shopping promotion for customers during the "National Day" event in 2023, with the following rules:<br/>① If the total purchase is within $400$ yuan, a discount of 10% off the original price will be given;<br/>② If the total purchase exceeds $400$ yuan, customers can first enjoy the discount policy of "Tmall" of $50$ yuan off for every $400$ yuan spent ($100$ yuan off for every $800$ yuan spent, and so on, with no limit), and then receive a 20% discount.<br/>$(1)$ Xiao Hua's mother purchased a vacuum cleaner priced at $720$ yuan from the online store. How much should she pay?<br/>$(2)$ Teacher Xu first bought a desk lamp for her daughter from the online store for $180. Later, she realized that the juicer at home was broken, so she spent $824 to buy a juicer from the same online store. If Teacher Xu makes a one-time purchase, how much does she need to pay in total?<br/>$(3)$ A customer made two purchases at the online store with a total price of $900 for the items. If the price of the items purchased in the first transaction is $a$ yuan and is less than the price of the items purchased in the second transaction, find the total amount the customer paid for the two transactions. (Express the total amount using an algebraic expression involving $a$)

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