
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 group of high school seniors at a school is conducting research-based learning, with the main task being to conduct market research on a certain product. Through a period of investigation, it was found that the daily sales volume $f(x)$ (unit: kilograms) of the product is approximately related to the selling price $x$ (unit: yuan per kilogram) by the equation $f(x)=\frac{a}{x-3}+b(x-6)^{2}$, where $3 \lt x \lt 6$, and $a$ and $b$ are constants. It is known that when the selling price is $4.5$ yuan per kilogram, 22 kilograms can be sold daily, and when the selling price is $5$ yuan per kilogram, 11 kilograms can be sold daily.<br/>$(1)$ Find the analytical expression of $f(x)$;<br/>$(2)$ If the cost of the product is $3$ yuan per kilogram, determine the value of the selling price $x$ that maximizes the daily profit for the business.

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