
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 shopping mall has a promotional rule: All goods in the mall are sold at $80\%$ of their original price. Additionally, when customers spend a certain amount in the mall, they receive vouchers according to the following scheme:
\begin{tabular}{|c|c|c|c|c|c|}
\hline Spending Amount (Yuan) Range & {$[200,400)$} & {$[400,500)$} & {$[500,700)$} & {$[700,900)$} & $\cdots$ \\
\hline Voucher Amount (Yuan) & 30 & 60 & 100 & 130 & $\cdots$ \\
\hline
\end{tabular}

Based on the above promotional method, customers can receive double discounts when shopping in the mall. For example, purchasing an item priced at 400 Yuan will cost 320 Yuan, and the total discount received is: $400 \times 0.2+30=110$ Yuan.

1. What is the discount rate for a customer purchasing an item priced at 1000 Yuan?
2. For items priced within the range of $[500,800]$ Yuan, what price should a customer choose to achieve a discount rate of at least $\frac{1}{3}$?

(Shanghai Gaokao 2002)

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