
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 a certain store, the cost price of a product is $30$ yuan per item, and the selling price is $40$ yuan per item. The store can sell $48$ items per day. In order to reduce inventory quickly, the store decides to reduce the price for promotion.<br/>$(1)$ If the selling price decreases to $32.4$ yuan per item after two consecutive reductions of the same percentage, find the percentage of the two reductions;<br/>$(2)$ After investigation, if for every $1$ yuan reduction in the price of the product, the store can sell an additional $8$ items per day, then in order to make a profit of $504$ yuan per day, how much should the price be reduced per item?

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