Let $z$ be a nonzero complex number, and let $\alpha = |z|$ and $\beta$ be the argument of $z$, where $0 \leq \beta < 2\pi$ (where $\pi$ is the circumference ratio). For any positive integer $n$, let the real numbers $x_{n}$ and $y_{n}$ be the real and imaginary parts of $z^{n}$, respectively. Select the correct options. (1) If $\alpha = 1$ and $\beta = \frac{3\pi}{7}$, then $x_{10} = x_{3}$ (2) If $y_{3} = 0$, then $y_{6} = 0$ (3) If $x_{3} = 1$, then $x_{6} = 1$ (4) If the sequence $\langle y_{n} \rangle$ converges, then $\alpha \leq 1$ (5) If the sequence $\langle x_{n} \rangle$ converges, then the sequence $\langle y_{n} \rangle$ also converges
Let $z$ be a nonzero complex number, and let $\alpha = |z|$ and $\beta$ be the argument of $z$, where $0 \leq \beta < 2\pi$ (where $\pi$ is the circumference ratio). For any positive integer $n$, let the real numbers $x_{n}$ and $y_{n}$ be the real and imaginary parts of $z^{n}$, respectively. Select the correct options.\\
(1) If $\alpha = 1$ and $\beta = \frac{3\pi}{7}$, then $x_{10} = x_{3}$\\
(2) If $y_{3} = 0$, then $y_{6} = 0$\\
(3) If $x_{3} = 1$, then $x_{6} = 1$\\
(4) If the sequence $\langle y_{n} \rangle$ converges, then $\alpha \leq 1$\\
(5) If the sequence $\langle x_{n} \rangle$ converges, then the sequence $\langle y_{n} \rangle$ also converges