[THOUGHT] The point A(a, b) satisfies b = log_{16}(8a + 2), which rearranges to 4a + 1 = 2^{4b - 1}.
The reflection (b, a) lies on OB, introducing parameter s = 1/t where c = s b, d = s a.
Midpoint gives a + s b = 77/4 and s a + b = 133/4.
Solving yields a(s) = (77 - 133 s)/(4 (1 - s^2)), b(s) = (133 - 77 s)/(4 (1 - s^2)).
Define diffA(s) = 4 a(s) + 1 - 2^{4 b(s) - 1} = 0 to find s.
Use fsolve with guess 1.8 (since s > 133/77 ≈ 1.727 for positive a,b).
Then compute a, b, c = s b, d = s a, verify B curve: s a + 1/2 == 4^{s b - 1}, and a * b. [/THOUGHT] [PYTHON] import numpy as np from scipy.optimize import fsolve import math

def a_func(s): den = 1 - s**2 num = 77 - 133 * s return (1/4) * num / den

def b_func(s): den = 1 - s**2 num = 133 - 77 * s return (1/4) * num / den

def diffA(s): a = a_func(s) b = b_func(s) left = 4 * a + 1 right = 2 ** (4 * b - 1) return left - right

Solve for s

s_sol = fsolve(diffA, 1.8)[0] print('s:', s_sol)

a = a_func(s_sol) b = b_func(s_sol) print('a:', a) print('b:', b)

c = s_sol * b d = s_sol * a print('c:', c) print('d:', d)

Verify A curve

print('Verify A: 4a+1 =', 4a + 1) print('2^(4b-1) =', 2**(4b - 1))

Verify B curve

print('Verify B: 4^(c-1) - 0.5 =', 4**(c - 1) - 0.5) print('d =', d)

Midpoint verify

print('Mid x:', (a + c)/2) print('Mid y:', (b + d)/2)

a * b as fraction

product = a * b print('a * b =', product) print('As fraction:', f"{product.numerator}/{product.denominator}" if hasattr(product, 'numerator') else product) [/PYTHON]

[THOUGHT] The Python output confirms s=2, a=63/4, b=7/4, a×b=441/16 with p=441, q=16 coprime, so p+q=457. [/THOUGHT] \boxed{457}

Explanation

The reasoning derives from the problem constraints and verifies solutions algebraically. The midpoint condition gives (a + c)/2 = 77/8 and (b + d)/2 = 133/8. Since (b, a) lies on OB, c = s b and d = s a for some s > 1. Substituting yields a + s b = 77/4 and s a + b = 133/4.

Solving the system:
a = (77 - 133 s) / [4 (1 - s²)]
b = (133 - 77 s) / [4 (1 - s²)]

Point A satisfies b = log₁₆(8a + 2), or equivalently 4a + 1 = 2^(4b - 1). Substituting a(s) and b(s) into this equation defines f(s) = 4a(s) + 1 - 2^(4b(s) - 1) = 0.

Numerical solution via fsolve yields s ≈ 2. Substituting s = 2 exactly:
a = (77 - 133×2) / [4(1 - 4)] = (-189) / (-12) = 63/4
b = (133 - 77×2) / [4(1 - 4)] = (-21) / (-12) = 7/4

Verification:

• For A: 4(63/4) + 1 = 64, and 2^(4×7/4 - 1) = 2^6 = 64.
• c = 2×7/4 = 7/2, d = 2×63/4 = 63/2. For B: 4^(7/2 - 1) - 1/2 = 4^{1.5} - 0.5 = (2^2)^{1.5} - 0.5 = 2^3 - 0.5 = 32 - 0.5 = 31.5 = 63/2.
• Midpoint: (63/4 + 7/2)/2 = 77/8, (7/4 + 63/2)/2 = 133/8.

Thus, a × b = (63/4) × (7/4) = 441/16. Here, 441 = 3²×7² and 16 = 2⁴ are coprime, so p = 441, q = 16, and p + q = 457. The Python output aligns with these exact values, confirming the result.