Let $A_n$ be a sequence of subsets of a universal set $U = [0, 1]$ defined for each $n \in \mathbb{N}$ (starting at 1) as follows: $A_n = [0, \frac{1}{2}]$ if $n$ is odd, and $A_n = [\frac{1}{2}, 1]$ if $n$ is even. Determine the symmetric difference of the limit superior and limit inferior of this sequence, defined as $(\limsup_{n \to \infty} A_n) \Delta (\liminf_{n \to \infty} A_n)$, where $\limsup_{n \to \infty} A_n = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} A_n$ and $\liminf_{n \to \infty} A_n = \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} A_n$.