Let \(\left\{A_{n}ight\}_{n=1}^{\infty}\) be a sequence of events from \(\mathcal{F}\), a \(\sigma\)-field on the sample space \(\Omega=(0,1)\), defined by

\[A_{n}= \begin{cases}\left(\frac{1}{3}, \frac{2}{3}ight) & \text { if } n \text { is even } \\ \left(\frac{1}{4}, \frac{3}{4}ight) & \text { if } n \text { is odd }\end{cases}\]

for all \(n \in \mathbb{N}\). Compute

\[\begin{aligned}& \liminf _{n ightarrow \infty} A_{n}, \\& \limsup _{n ightarrow \infty} A_{n},\end{aligned}\]

and determine if the limit of the sequence \(\left\{A_{n}ight\}_{n=1}^{\infty}\) exists.