Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. The water in the tank is $25\%$ of the tank's capacity.  The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by a perfect cube greater than 1.  What is $a+b$?

[asy]
size(150);
defaultpen(linewidth(.8pt)+fontsize(8pt));

draw(shift(0,96)*yscale(0.5)*Circle((0,0),16));
draw((-16,96)--(0,0)--(16,96)--(0,96));

draw(scale(0.75)*shift(0,96)*yscale(0.5)*Circle((0,0),16));

draw((-18,72)--(-20,72)--(-20,0)--(-18,0));
label("water's height",(-20,36),W);

draw((20,96)--(22,96)--(22,0)--(20,0));
label("96'",(22,48),E);

label("16'",(8,96),S);
[/asy]
user