Consider a simple symmetric random walk on $\mathbb{Z}^2$ starting at the origin $(0, 0)$. Let $A_n$ be the event that the random walk returns to the origin at time $2n$. Find a closed-form expression for the probability $P(A_n)$ that the walk returns to the origin for the *first time* at time $2n$. All variables should be enclosed in LaTeX format, e.g. '$\mathbb{Z}^2$', 'origin $(0, 0)$', '$A_n$', '$n$', '$2n$', '$P(A_n)$', and 'for the *first time* at time $2n$' should be formatted