TxO Math Bowl 2024 - Individuals A - Problem 10

In \triangle ABC, AB = 3, BC = 5, and AC = 7. Let the incircle of ABC be \omega_0. Circle \omega_1 is drawn inside \triangle ABC and outside \omega_0, such that \omega_1 is tangent to \omega_0, BC, and AC. If the radius of \omega_1 can be expressed as \frac{a\sqrt{b} - c\sqrt{d}}{e}, where a, b, c, d, e \in \mathbb{Z}^+, b and d are not divisible by the square of any prime, and \gcd(a, c, e) = 1, find a + b + c + d + e.