
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 $100 | No discount |
| Between $100 and $300 | 10% discount |
| $300 or more | 10% discount on the first $300 and 20% discount on the amount exceeding $300 |

$(1)$ A customer makes a one-time purchase of $500$, the actual payment is ______ yuan;  
$(2)$ If a customer makes a one-time purchase of $x$ yuan at the supermarket, when $x$ is less than $300$ but not less than $100$, the actual payment is $0.9x$ yuan, and when $x$ is greater than or equal to $300$, the actual payment is ______ yuan (expressed in terms of $x$);  
$(3)$ If a customer makes two purchases totaling $620$, with the first purchase amounting to $a$ yuan $(100 < a < 300)$, how much does the customer actually pay for the two purchases (expressed in terms of $a$)?

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