
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). Let the radius of the base of the water tank be $r$ meters, the height be $h$ meters, and the volume be $V m^{2}$. Assuming the construction cost is only related to the surface area, the construction cost of the side surface is $100$ yuan/$m^{3}$, and the construction cost of the base is $160$ yuan/$m^{2}$, the total construction cost of this water tank is $12,000\pi$ yuan ($\pi$ is the pi constant).

$(1)$ Express $V$ as a function of $r$, $V(r)$, and find the domain of this function;

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

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