
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 certain fruit market deals with a high-end fruit. The original price is $50$ yuan per kilogram. After two consecutive price reductions, the price becomes $32$ yuan per kilogram. If the percentage of reduction is the same each time, find:<br/>$(1)$ the percentage of reduction each time.<br/>$(2)$ If there is a profit of $10$ yuan per kilogram, and the market can sell $500$ kilograms per day, the market decides to increase the price by $1$ yuan per kilogram. The daily sales volume will decrease by $20$ kilograms. The market needs to ensure a daily profit of $6000$ yuan and reduce inventory quickly. How much should the price increase per kilogram?<br/>$(3)$ Under the conditions of $(2)$, to maximize the daily profit of the market, how much should the price increase per kilogram? What is the maximum daily profit?

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