There is a real number sequence $\left\langle a_{n} \right\rangle$, where $a_{n} = \cos\left(n\pi - \frac{\pi}{6}\right)$, and $n$ is a positive integer. Select the correct options.
(1) $a_{1} = -\frac{1}{2}$
(2) $a_{2} = a_{3}$
(3) $a_{4} = a_{24}$
(4) $\left\langle a_{n} \right\rangle$ is a convergent sequence, and $\lim_{n \rightarrow \infty} a_{n} < 1$
(5) $\sum_{n=1}^{\infty} \left(a_{n}\right)^{n} = 3 - 2\sqrt{3}$