
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 Double 11 period, as follows:

| One-time Purchase | Discount Method |
|-------------------|-----------------|
| Less than $200 | No discount |
| Less than $500 but not less than $200 | 20% off |
| $500 or more | 20% off for the first $500 and 30% off for the amount exceeding $500 |

$(1)$ If Mr. Wang makes a one-time purchase of $600, he actually pays ____ yuan. If Mr. Wang actually pays $160, then Mr. Wang's one-time purchase could be ____ yuan;  
$(2)$ If a customer makes a one-time purchase of $x$ yuan at the supermarket, when $x$ is less than $500$ but not less than $200$, the actual payment is ____ yuan, and when $x$ is greater than or equal to $500$, the actual payment is ____ yuan (expressed in algebraic terms containing $x$ and simplified);  
$(3)$ If Mr. Wang goes shopping at the supermarket for a total of $900 over two days, with the original price of shopping on the first day being $a$ yuan $(200 < a < 300)$, express in algebraic terms how much Mr. Wang actually paid for shopping over these two days. When $a = 250$ yuan, how much discount did Mr. Wang get over the two days?

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