Let ACDB be a cyclic quadrilateral with circumcircle \omega. Let AC = 5, CD = 6, and DB = 7. Suppose that there is a unique point P on \omega such that \overline{PC} intersects \overline{AB} at a point P_1 and \overline{PD} intersects \overline{AB} at a point P_2, such that AP_1 = 3 and P_2B = 4. Let Q be the unique point on \omega such that \overline{QC} intersects \overline{AB} at a point Q_1, \overline{QD} intersects \overline{AB} at a point Q_2, Q_1 is closer to B than P_1 is to B, and P_2Q_2 = 2. The length of P_1Q_1 can be written as \frac{p}{q} where p and q are relatively prime positive integers. Find p + q.