Let $a_n$ be a geometric sequence with first term $a_1 = 1$ and complex common ratio $r = e^{\frac{i\pi}{3}} = \cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right)$. Let $S_n = \sum_{k=1}^n a_k$ be the sum of the first $n$ terms. Which of the following statements is/are correct?