Example 1: During a promotional period, a mall stipulates that all goods in the mall are sold at 80% of the marked price; at the same time, when customers spend a certain amount in the mall, they receive a corresponding amount of coupons according to the following scheme:
\begin{tabular}{|c|c|c|c|c|c|}
\hline Range of spending amount (CNY) & {$[200,400)$} & {$[400,500)$} & {$[500,700)$} & {$[700,900)$} & $\cdots$ \\
\hline Amount of coupons received (CNY) & 30 & 60 & 100 & 130 & $\cdots$ \\
\hline
\end{tabular}

According to the above promotional method, customers can receive double discounts when shopping in the mall. For example, if a customer buys a product marked at 400 CNY, the spending amount is 320 CNY, and the discount received is: $400 \times 0.2 + 30 = 110$ CNY. Suppose the discount obtained by purchasing a product is:
(1) If a customer buys a product marked at 1000 CNY, what is the discount rate the customer receives?
(2) For products with marked prices in the range $[500,800]$ CNY, what marked price of the product should a customer buy to receive a discount rate of no less than $\frac{1}{3}$?
(2002 Shanghai College Entrance Examination Question)
Solution (1) $\frac{1000 \times 0.2+130}{1000}=33 \%$.
(2) Let the marked price of the item be $x$ yuan, then $500 \leqslant x \leqslant 800$, consumption amount: $400 \leqslant 0.8 x \leqslant 640$, when $0.8 x \in[400,500)$, then $\frac{0.2 x+60}{x} \geqslant \frac{1}{3}$, no solution; when $0.8 x \in[500,640]$, then $\frac{0.2 x+100}{x} \geqslant \frac{1}{3}$. Solving this, we get $x \in[625,750]$. That is, when the customer purchases an item with a marked price in the range of $[625,750]$ yuan, they can receive a discount rate of no less than $\frac{1}{3}$. !Comment This problem tests the knowledge points of piecewise functions and inequalities.