Triangle \triangle A B C has sidelengths A B=10, A C=14, and, B C=16. Circle \omega_{1} is tangent to rays \overrightarrow{A B}, \overrightarrow{A C} and passes through B. Circle \omega_{2} is tangent to rays \overrightarrow{A B}, \overrightarrow{A C} and passes through C. Let \omega_{1}, \omega_{2} intersect at points X, Y. The square of the perimeter of triangle \triangle A X Y is equal to \frac{a+b \sqrt{c}}{d}, where a, b, c, and, d are positive integers such that a and d are relatively prime, and c is not divisible by the square of any prime. Find a+b+c+d.