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Let \( A = \left\{ \frac{1967 + 1686 i \sin\theta}{7 - 3 i \cos\theta} : \theta \in \mathbb{R} \right\} \). If \( A \) contains exactly one positive integer \( n \), then the value of \( n \) is _____.

Let \( z \) be a complex number satisfying \( |z|^3+2z^2+4\overline{z}-8=0 \), where \( \overline{z} \) denotes the complex conjugate of \( z \). Let the imaginary part of \( z \) be nonzero.
Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let \( A_{1}, A_{2}, A_{3}, \ldots, A_{8} \) be the vertices of a regular octagon that lie on a circle of radius \( 2 \). Let \( P \) be a point on the circle and let \( PA_{i} \) denote the distance between the points \( P \) and \( A_{i} \) for \( i=1,2,\ldots,8 \). If \( P \) varies over the circle, then the maximum value of the product \( PA_{1}\cdot PA_{2}\cdots PA_{8} \), is: