
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
To build a rectangular pool without a cover with a volume of $8$ cubic meters and a depth of $2$ meters, where the cost of the pool bottom is $120$ yuan per square meter and the cost of the pool walls is $80$ yuan per square meter.<br/>$(1)$ Find the functional relationship between the total cost $y$ yuan of the pool and the width $x$ meters of the pool bottom;<br/>$(2)$ For what width of the pool bottom in meters can the total cost of the pool be minimized? What is the minimum cost?

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