
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 factory plans to build an open-top rectangular prism water storage tank with a volume of $4800$ cubic meters and a depth of $3$ meters. The cost per square meter of the tank bottom is $150$ yuan, and the cost per square meter of the tank wall is $120$ yuan. Let the length of the rectangular bottom be $x$ meters. $(1)$ Calculate the area of the bottom and express the area of the tank walls in terms of $x$; 

$(2)$ How should the tank be designed to minimize the total construction cost? What is the minimum cost?

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