Question 54
A lattice point in the plane is a point of the form $(n, m)$, where $n$ and $m$ are integers. Consider a set $S$ of lattice points. We construct the transform of $S$, denoted by $S'$, by the following rule: the pair $(n, m)$ is in $S'$ if and only if any of $(n, m-1)$, $(n, m+1)$, $(n-1, m)$, $(n+1, m)$, and $(n, m)$ is in $S$. How many elements are in the set obtained by successively transforming $\{(0,0)\}$ 14 times?