Show that the number \( 4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right) \) is an integer and determine it.

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