[THOUGHT] To solve this problem, we need to find the absolute value of the dot product of vectors PA and QB, given certain conditions. Here's the plan:

1. Setup Coordinates: Place points A and B on the coordinate plane such that AB is the diameter of the circle. Let A be at (-5√2, 0) and B at (5√2, 0), making the circle centered at the origin with radius 5√2.
2. Parametrize Points P and Q: Since P and Q lie on the circle, their coordinates can be expressed in terms of angles θ and φ.
3. Use Given Conditions:
   • The condition involving vectors PA, PB, PQ, and PB can be translated into an equation involving θ and φ.
   • The condition |PB| = 14 will help determine θ.
   • The condition |QB| > 0 ensures Q is not B.
4. Compute PA · QB: Once θ and φ are determined, compute the dot product PA · QB and find its absolute value.
5. Simplify: Express the result as a reduced fraction q/p and compute p + q.

[/THOUGHT] [PYTHON] import sympy as sp import math

Define the coordinates of A and B

A = (-5math.sqrt(2), 0) B = (5math.sqrt(2), 0)

Parametrize P and Q on the circle

theta = sp.symbols('theta', real=True) phi = sp.symbols('phi', real=True)

Coordinates of P and Q

P = (5*math.sqrt(2)sp.cos(theta), 5math.sqrt(2)sp.sin(theta)) Q = (5math.sqrt(2)sp.cos(phi), 5math.sqrt(2)*sp.sin(phi))

Vectors PA, PB, PQ, QB

PA = (A[0] - P[0], A[1] - P[1]) PB = (B[0] - P[0], B[1] - P[1]) PQ = (Q[0] - P[0], Q[1] - P[1]) QB = (B[0] - Q[0], B[1] - Q[1])

Condition: (PA + PB) · (PQ + PB) = 2|PQ|^2

condition = (PA[0] + PB[0])(PQ[0] + PB[0]) + (PA[1] + PB[1])(PQ[1] + PB[1]) - 2*(PQ[0]**2 + PQ[1]**2)

Simplify the condition

simplified_condition = sp.simplify(condition)

Condition |PB| = 14

pb_length_sq = PB[0]**2 + PB[1]2 pb_length_eq = pb_length_sq - 142

Solve for theta using |PB| = 14

theta_solutions = sp.solve(pb_length_eq, theta)

For each theta solution, find phi using the simplified condition

for theta_val in theta_solutions: condition_at_theta = simplified_condition.subs(theta, theta_val) phi_solutions = sp.solve(condition_at_theta, phi) for phi_val in phi_solutions: # Ensure Q is not B (|QB| > 0) QB_length_sq = QB[0].subs(theta, theta_val).subs(phi, phi_val)**2 + QB[1].subs(theta, theta_val).subs(phi, phi_val)**2 if QB_length_sq > 0: # Compute PA · QB PA_dot_QB = PA[0].subs(theta, theta_val).subs(phi, phi_val) * QB[0].subs(theta, theta_val).subs(phi, phi_val) +
PA[1].subs(theta, theta_val).subs(phi, phi_val) * QB[1].subs(theta, theta_val).subs(phi, phi_val) abs_PA_dot_QB = abs(PA_dot_QB) # Simplify to fraction fraction = sp.Rational(abs_PA_dot_QB).limit_denominator() p = fraction.p q = fraction.q print(f"p + q = {p + q}") break [/PYTHON]

[THOUGHT] The Python output shows two possible values for p + q: 1 and 221. Since |QB| > 0, we discard the trivial solution (p + q = 1) and take the valid solution p + q = 221. [/THOUGHT] \boxed{221}