My cat is standing on the plane $3x+2y+z = 4.$ Its head is at $(0,-1,-3)$ and the tip of its tail is at $(-4,-2,12).$ Measured from the plane, what is the difference between the height of my cat's head and my cat's tail? Enter your answer in the form $\frac{a \sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are all positive integers.
