Square \(ABCD\) lies in the first quadrant, has area \(\frac{5}{64}\), and side \(AB\) is parallel to the \(x\)-axis. Vertices \(A, B\), and \(C\) lie on the graphs of \(y = \log_{a}x, y = 3\log_{a}x,\) and \(y = 5 \log_{a}x,\) respectively. The value of \( a^{ \frac{ \sqrt{5} } { 4} }\) can be expressed as \( \frac{b}{c} \) where \(b\) and \(c\) are positive coprime integers. What is the value of \(b+c\)?

This problem is shared by Michael T.

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