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Geometry Level 5

Let $$P = \tan \dfrac \theta2 + 2\tan \theta + 4\tan2\theta + 8\cot4\theta$$ and given that $$\dfrac1{1- \cos\theta + i \sin\theta}$$ can be expressed in the form of $$a + bi$$, where $$a$$ and $$b$$ are real numbers with $$i = \sqrt{-1}$$. Compute $$\dfrac{a\cdot b}{ P }$$.

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