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If $$\int_{\pi/6}^{\pi/4} \left(\cot\left(x - \frac{\pi}{3}\right)\cot\left(x + \frac{\pi}{3}\right) + 1\right) dx = \alpha \log_e(\sqrt{3} - 1)$$, then $$9\alpha^2$$ is equal to __________.
Correct Answer: 12
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