
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 a promotion in a certain shopping mall, all items are sold at 80% of the marked price. Customers receive vouchers based on the amount they spend in the following scheme:

| Range of Amount Spent (in yuan) | $[200,400)$ | $[400,500)$ | $[500,700)$ | $[700,900)$ | $\ldots$ |
|---------------------------------|-------------|-------------|-------------|-------------|---------|
| Voucher Amount (in yuan)        | $30$        | $60$        | $100$       | $130$       | $\ldots$ |

According to the above promotion method, customers can enjoy double discounts when shopping in the mall. For example, if a customer buys an item with a marked price of 400 yuan, the amount spent is 320 yuan, and the discount received is: $400 \times 0.2 + 30 = 110$ yuan. Let the discount rate obtained by purchasing goods be equal to (discount amount received when purchasing goods) / (marked price of the goods). The questions are as follows:

$(1)$ If a customer buys items with marked prices of 500 yuan and 1000 yuan respectively, what are the discount rates they will receive?

$(2)$ For items priced between 500 and 1000 yuan, what is the minimum marked price of an item that a customer can purchase to receive a discount rate of at least $\frac{1}{3}$? What is the maximum discount rate that can be obtained by purchasing an item with a certain marked price?

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