
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
The "Yanzhao Flavors Hebei Agricultural Products" promotion campaign is launching. The cost price of a certain product is $30$ yuan per item, and the selling price is $40$ yuan per item, with a daily sales volume of $48$ items. In order to reduce inventory quickly, the mall decides to reduce prices for promotion.
$(1)$ If the selling price of the product drops to $32.4$ yuan per item after two consecutive reductions of the same percentage, find the percentage reduction each time;
$(2)$ After investigation, if the product sells $4$ more items per day for every $0.5$ yuan reduction in price. If you want to make a profit of $504$ yuan per day and reduce inventory quickly, how much should the price be reduced per item?

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