2019 AIME I Problem 13

Triangle ABC has side lengths AB=4, BC=5, and CA=6. Points D and E are on ray AB with AB<AD<AE. The point F \neq C is a point of intersection of the circumcircles of \triangle ACD and \triangle EBC satisfying DF=2 and EF=7. Then BE can be expressed as \tfrac{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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.