If $x_r = \cos(\pi/3^r) - i\sin(\pi/3^r)$ (where $i = \sqrt{-1}$),then the value of $x_1 \cdot x_2 \cdot x_3 \cdots \infty$ is:

If $z = \frac{7 - i}{3 - 4i}$,then $z^{14} = $

If $\tan (u + iv) = i$,then the value of $v$ is

The number of solutions for $z^3+\bar{z}=0$ is

If $e^{i x}$ is a solution of the equation $z^n+p_1 z^{n-1}+p_2 z^{n-2}+\ldots+p_n=0$,where $p_i$ are real $(i=1, 2, \ldots, n)$,then $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0) = $ (Note: The constant term in the equation is $p_n$ and the coefficient of $z^0$ is $1$ if we normalize,but here the equation is given as $z^n + p_1 z^{n-1} + \ldots + p_n = 0$. Let us assume the constant term is $p_n$. The expression to evaluate is $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0)$). Given the standard form,find the value of $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x$.

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