Let $a_n$ be a sequence defined by $a_1 = 1, a_2 = 1,$ and $a_n = a_{n-1} + a_{n-2}$ for $n > 2$. Consider the sequence $S_n = \text{gcd}(a_n, a_{n+k})$ for a fixed positive integer $k$. Which of the following statements is true for all $n \ge 1$ regarding the properties of this GCD?