
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 factory's cost for processing a product is $30$ yuan per kilogram. According to market research, when the wholesale price is set at $48$ yuan per kilogram, the factory can sell $500$ kilograms per day. In order to increase market share, while ensuring profitability, the factory adopts a price reduction measure. For every $1$ yuan reduction in price per kilogram, the daily sales volume can increase by $50$ kilograms. <br/>$(1)$ Find the functional relationship between the factory's daily profit $y$ yuan and the price reduction per kilogram $x$ yuan. <br/>$(2)$ When the price is reduced by how many yuan per kilogram, the factory's daily profit is maximized, and what is the maximum profit?

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