
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
In order to boost domestic demand and promote economic development, a certain store is holding a promotion. There is no discount for purchases under $200. For purchases over $200 but not exceeding $500, a 10% discount off the original price is given. For purchases over $500, there is a 20% discount on the first $500 and a 50% discount on the amount exceeding $500. Mr. Wang made two purchases, the first one for $100 and the second one for $432.  
$(1)$ The discount for the first purchase is ______. (Fill in "yes" or "no")  
$(2)$ Find the original price of the items purchased in the second transaction.  
$(3)$ If Mr. Wang combines these two purchases into one, how much money can he save?

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