
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 supermarket offers discounts to customers during the Spring Festival, as follows:

| One-time Purchase | Discount Method |
|-------------------|-----------------|
| Less than $200 | No discount |
| Less than $500 but not less than $200 | 10% discount |
| $500 or more | 10% discount on the first $500 and 20% discount on the amount exceeding $500 |

$(1)$ If Mr. Wang makes a one-time purchase of $400, he will actually pay ______ yuan. If he makes a one-time purchase of $600, he will actually pay ______ yuan.  
$(2)$ If a customer makes a one-time purchase of $x$ yuan at this supermarket, when $x$ is less than $500$ but not less than $200$, the actual payment will be ______ yuan. When $x$ is greater than or equal to $500$, the actual payment will be ______ yuan. (Express using algebraic expressions with $x$).  
$(3)$ If Mr. Wang's total purchase amount for two transactions is $820$, and the amount of the first purchase is $a$ yuan $(200 < a < 300)$, express in terms of $a$ how much Mr. Wang actually paid for the two transactions.

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