
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 study interest group of the second year of a certain school decided to conduct a market sales survey on the "famous brand" $A$ series during a social practice activity. They randomly selected a shopping mall for the survey. Through a study of a phase of the $A$ series of this brand, it was found that the daily sales volume $f(x)$ (unit: kilograms) of the $A$ series is approximately related to the selling price $x$ (in yuan per kilogram) by the equation $f(x)=\frac{a}{x-4}+10(x-7)^2$, where $4 \lt x \lt 7$, and $a$ is a constant. It is known that when the selling price is $6$ yuan per kilogram, $15$ kilograms of the $A$ series can be sold daily.<br/>$(1)$ Find the analytical expression of the function $f(x)$.<br/>$(2)$ If the cost of the $A$ series is $4$ yuan per kilogram, determine the value of the selling price $x$ that maximizes the profit obtained by the shopping mall from the daily sales of the $A$ series.

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