Rule 30 is a fascinating one-dimensional and simple cellular automaton giving raise to some quite chaotic behaviour. This is the starting point of this question.

Let $\{0,1\}^\mathbb{Z}$ denote the set of all functions $x:\mathbb{Z}\to \{0,1\}$. Let $+$ denote addition modulo $2$ on $\{0,1\}$ (corresponding to ${\sf XOR}$ in computer science).

Define a function $f:\{0,1\}^\mathbb{Z} \to \{0,1\}^\mathbb{Z}$ by $x \mapsto f(x)$ where $f(x):\mathbb{Z}\to\{0,1\}$ is defined by $$f(x)(i) = x(i) + x(i-1) + x(i+1) \text{ for all }i\in \mathbb{Z}.$$ Inductively define $f^{(0)}(x) = x$, and $f^{(n+1)}(x) = f(f^{(n)}(x))$ for all integers $n\geq 0$.

Question. For what integers $n>1$ is there $x\in \{0,1\}^\mathbb{Z}$ such that $f^{(n)}(x) = x$ but $f^{(k)}(x) \neq x$ for all integers $k\geq 1$ with $k<n$?