
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 shopping mall is offering a promotional discount to its customers. The rules are as follows:
$(1)$ If the total purchase is not more than $500 yuan, no discount is given.
$(2)$ If the total purchase is more than $500 yuan but not more than $1000 yuan, an $8$ percent discount is given based on the labeled price.
$(3)$ If the total purchase is more than $1000 yuan, a discount of $8$ percent is given for the first $1000 yuan, and a discount of $5$ percent is given for the part exceeding $1000 yuan$. Let $x$ represent the total purchase in yuan, and $y$ represent the actual payment after the discount.
$(1)$ Try to express $y$ as a function of $x$.
$(2)$ A customer actually paid $1600 yuan. How much less did this customer spend in this promotional activity compared to the total purchase?

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