There is a positive integer $s$ such that there are $s$ solutions to the equation
\[ 64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46 \]
in the interval \( (0, \frac{\pi}{2}) \), all of the form \( \frac{m_k}{n_k}\pi \), where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, \ldots, s$. Find \((m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + \cdots + (m_s + n_s)\).

Let's think step by step and output the final answer within \boxed{}.
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