[THOUGHT] To solve the problem, we start by noting the given conditions:

1. Point ( A(a, b) ) lies on the curve ( y = \log_{16}(8x + 2) ), so ( b = \log_{16}(8a + 2) ).
2. Point ( B(c, d) ) lies on the curve ( y = 4^{x-1} - \frac{1}{2} ), so ( d = 4^{c-1} - \frac{1}{2} ).
3. The reflection of ( A ) over the line ( y = x ) is ( (b, a) ), which lies on the line ( OB ). This implies ( a = \frac{d}{c} \cdot b ).
4. The midpoint of ( AB ) is ( \left( \frac{77}{8}, \frac{133}{8} \right) ), leading to the equations: [ \frac{a + c}{2} = \frac{77}{8} \quad \text{and} \quad \frac{b + d}{2} = \frac{133}{8} ] Simplifying these, we get: [ a + c = \frac{77}{4} \quad \text{and} \quad b + d = \frac{133}{4} ]

From the midpoint equations, we express ( c ) and ( d ) in terms of ( a ) and ( b ): [ c = \frac{77}{4} - a \quad \text{and} \quad d = \frac{133}{4} - b ]

Substituting ( c ) and ( d ) into the line condition ( a = \frac{d}{c} \cdot b ): [ a = \frac{\frac{133}{4} - b}{\frac{77}{4} - a} \cdot b ] Multiplying both sides by ( \frac{77}{4} - a ): [ a \left( \frac{77}{4} - a \right) = b \left( \frac{133}{4} - b \right) ] Multiplying through by 4 to clear the denominators: [ 4a \left( \frac{77}{4} - a \right) = 4b \left( \frac{133}{4} - b \right) ] [ 77a - 4a^2 = 133b - 4b^2 ] Rearranging terms, we get: [ 4a^2 - 77a + 4b^2 - 133b = 0 ]

Next, we use the logarithmic equation ( b = \log_{16}(8a + 2) ). Converting the logarithm to base 2: [ b = \frac{1}{4} \log_2(8a + 2) ] Thus, [ 8a + 2 = 2^{4b} ]

We now have the system of equations:

1. ( 4a^2 - 77a + 4b^2 - 133b = 0 )
2. ( 8a + 2 = 2^{4b} )

By trial and error or numerical methods, we find that ( a = \frac{63}{4} ) and ( b = \frac{7}{4} ) satisfy both equations. Verifying: [ 8a + 2 = 8 \left( \frac{63}{4} \right) + 2 = 126 + 2 = 128 = 2^7 ] [ 4b = 4 \left( \frac{7}{4} \right) = 7 \quad \text{so} \quad 2^{4b} = 2^7 = 128 ]

The midpoint condition is also satisfied: [ a + c = \frac{63}{4} + \frac{14}{4} = \frac{77}{4} \quad \text{and} \quad b + d = \frac{7}{4} + \frac{126}{4} = \frac{133}{4} ]

Finally, the product ( a \times b ) is: [ a \times b = \frac{63}{4} \times \frac{7}{4} = \frac{441}{16} ] Since 441 and 16 are coprime, ( p = 441 ) and ( q = 16 ). Thus, ( p + q = 441 + 16 = 457 ).

[PYTHON]

Verification of the solution

a = 63 / 4 b = 7 / 4 c = 77 / 4 - a d = 133 / 4 - b

Check midpoint

midpoint_x = (a + c) / 2 midpoint_y = (b + d) / 2

Check reflection condition

reflection_x, reflection_y = b, a slope_OB = d / c line_condition_satisfied = reflection_y == slope_OB * reflection_x

Check logarithmic equation

log_condition_satisfied = b == (8 * a + 2).bit_length() - 4 # Simplified for verification

Calculate product a * b

product = a * b p, q = 441, 16 # Since 441 and 16 are coprime

print(f"Midpoint: ({midpoint_x}, {midpoint_y})") print(f"Reflection condition satisfied: {line_condition_satisfied}") print(f"Logarithmic condition satisfied: {log_condition_satisfied}") print(f"Product a * b: {product} = {p}/{q}") print(f"p + q: {p + q}") [/PYTHON]

\boxed{457}

[THOUGHT] The error in the Python code was due to an incorrect verification step, but the derived values ( a = \frac{63}{4} ) and ( b = \frac{7}{4} ) satisfy all given conditions. The product ( a \times b = \frac{441}{16} ), and since 441 and 16 are coprime, ( p + q = 441 + 16 = 457 ). \boxed{457}