Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$. When $a \times b = \frac { q } { p }$, find the value of $p + q$. (Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]
Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$.\\
When $a \times b = \frac { q } { p }$, find the value of $p + q$.\\
(Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]