Question 12
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6,$ $BC = 5 = DA,$ and $CD = 4.$ Draw circles of radius $3$ centered at $A$ and $B,$ and circles of radius $2$ centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k + m\sqrt{n}}{p},$ where $k,$ $m,$ $n,$ and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k + m + n + p.$