Consider a point P moving on the coordinate plane every second according to the following rules (i), (ii).

(i) Initially, P is at the point $(2, 1)$.

(ii) When P is at the point $(a, b)$ at some moment, one second later P is at:
\begin{itemize}
    \item with probability $\dfrac{1}{3}$, the point symmetric to $(a, b)$ with respect to the $x$-axis
    \item with probability $\dfrac{1}{3}$, the point symmetric to $(a, b)$ with respect to the $y$-axis
    \item with probability $\dfrac{1}{6}$, the point symmetric to $(a, b)$ with respect to the line $y = x$
    \item with probability $\dfrac{1}{6}$, the point symmetric to $(a, b)$ with respect to the line $y = -x$
\end{itemize}

Answer the following questions. For question (1), you only need to state the conclusion.

(1) Find all possible coordinates of points that P can occupy.

(2) Let $n$ be a positive integer. Show that the probability that P is at the point $(2, 1)$ after $n$ seconds from the start equals the probability that P is at the point $(-2, -1)$ after $n$ seconds from the start.

(3) Let $n$ be a positive integer. Find the probability that P is at the point $(2, 1)$ after $n$ seconds from the start.