XOR Infinitude

Let $X_1,X_2,\dots \sim \text{Bernoulli}(p)$ IID, where $0 < p < 1$. Define $Y_n = \displaystyle \bigoplus_{i=1}^n X_i = X_1 \oplus \dots \oplus X_n$, where $a \oplus b = 0$ if $a = b$ and $a \oplus b = 1$ otherwise. When $p = 1/3$, find $\displaystyle \lim_{n \rightarrow \infty} \mathbb{P}\left( Y_n = 1 \right)$.