Question 57
Let $a$, $b$, $c$, $d$ be positive real numbers satisfying the system of equations
$$
\begin{cases} a^2 + \frac{1}{b^2} = \frac{1}{2}, \\ b^2 + \frac{4}{c^2} = 8, \\ c^2 + \frac{16}{d^2} = 2, \\ d^2 + \frac{4}{a^2} = 32. \end{cases}
$$
Determine the product $abcd$.
$$
\begin{cases} a^2 + \frac{1}{b^2} = \frac{1}{2}, \\ b^2 + \frac{4}{c^2} = 8, \\ c^2 + \frac{16}{d^2} = 2, \\ d^2 + \frac{4}{a^2} = 32. \end{cases}
$$
Determine the product $abcd$.