The reflection of point A(a, b) across the line y = x is A'(b, a). Given that A' lies on the line OB (where O is the origin) and the midpoint of AB is (77/8, 133/8), the conditions lead to the system of equations:

• ( a + \frac{b}{t} = \frac{77}{4} )
• ( b + \frac{a}{t} = \frac{133}{4} )

Solving for a and b in terms of t, and using the curve equation ( b = \log_{16}(8a + 2) ), the value ( t = \frac{1}{2} ) satisfies all conditions. Substituting ( t = \frac{1}{2} ):

• ( a = \frac{63}{4} )
• ( b = \frac{7}{4} )

The product ( a \times b = \frac{63}{4} \times \frac{7}{4} = \frac{441}{16} ). The fraction ( \frac{441}{16} ) is in simplest terms since 441 and 16 are coprime (441 = ( 21^2 ), 16 = ( 2^4 ), no common factors). Thus, ( p = 441 ), ( q = 16 ), and ( p + q = 457 ).

\boxed{457}

[THOUGHT] We need to verify the solution using Python.

[PYTHON]

Verify the solution

a = 63/4 b = 7/4 product = a * b print(product) [/PYTHON]

\boxed{457}