
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
Supermarkets A and B sell the same goods at the same price. In order to attract customers, they each offer different discount schemes: at Supermarket A, the excess amount of purchases exceeding $300 is discounted by 20%; at Supermarket B, the excess amount of purchases exceeding $200 is discounted by 10%. Let the customer's expected total purchase amount be $x$ ($x > 300$). The amount paid at Supermarket A is $y_{1}$, and the amount paid at Supermarket B is $y_{2}$.  
$(1)$ Find the functional relationships of $y_{1}$ and $y_{2}$ with respect to $x$.  
$(2)$ Compare which supermarket is more cost-effective for the customer. Explain your reasoning.

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