Let \triangle ABC be an equilateral triangle with side length 1 and let \Gamma be the circle tangent to AB and AC at B and C, respectively. Let P be on side AB and Q be on side AC so that PQ \parallel BC, and the circle through A, P, and Q is tangent to \Gamma. If the area of \triangle APQ can be written in the form \frac{\sqrt{a}}{b} for positive integers a and b, where a is not divisible by the square of any prime, find a + b.