The value of \( \sum_{k=1}^{13} \frac{1}{\sin \left(\frac{\pi}{4}+\frac{(k-1) \pi}{6}\right) \si...

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The value of \( \sum_{k=1}^{13} \frac{1}{\sin \left(\frac{\pi}{4}+\frac{(k-1) \pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{k \pi}{6}\right)} \) is equal to
(a) \( 3-\sqrt{3} \)
(b) \( 2(3-\sqrt{3}) \)
(c) \( 2(\sqrt{3}-1) \)
(d) \( 2(2+\sqrt{3}) \)
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