Let $x$ be a real number in the interval $(0, \frac{\pi}{2})$ such that $\frac{1}{\sin x \cos x} + 2\cot 2x = \frac{1}{2}$. Evaluate $\frac{1}{\sin x \cos x} - 2\cot 2x$.

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