
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 village plans to build an uncovered cylindrical water storage tank (thickness not considered), with the bottom radius of the tank being $r$ meters, height $h$ meters, and volume $V$ cubic meters. Assuming the construction cost is only related to the surface area, where the cost of constructing the side is $100$ yuan per square meter, and the cost of constructing the bottom is $160$ yuan per square meter, the total construction cost of the water storage tank is $12000\pi$ yuan ($\pi$ is the pi constant).

$(1)$ Express $V$ as a function of $r$, $V(r)$, and determine its domain;

$(2)$ Discuss the monotonicity of the function $V(r)$, and determine the values of $r$ and $h$ when the volume of the water storage tank is maximized.

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