TxO Math Bowl 2024 - Guts Contest - Set 5 Problem 3

Two circles centered at O_1 and O_2 intersect at A and B, and a common external tangent closer to A intersects the two circles at C, D, where C is on the circle with center O_1. The extension of segment BA intersects CD at E. If O_1O_2 = 10, O_1C = 6, and O_2D = 14, then AE can be written as a - b\sqrt{c}, where a, b, and c are positive integers with c not divisible by the square of any prime. Find a + b + c.