Five, let $A=\{1,2,3, \cdots, 17\}$. For any function $f: A \rightarrow A$, denote
$$
f^{[1]}(x)=f(x), f^{[k+1]}(x)=f\left(f^{[k]}(x)\right)
$$
$(k \in \mathbb{N})$. Find the natural number $M$ such that:
$$
\begin{array}{l}
\text { (1) When } m<M, 1 \leqslant i \leqslant 16, \text { we have } \\
f^{[m]}(i+1)-f^{[m]}(i) \neq \pm 1(\bmod 17), \\
f^{[m]}(1)-f^{[m]}(17) \neq \pm 1(\bmod 17) ;
\end{array}
$$
(2) When $1 \leqslant i \leqslant 16$, we have
$$
\begin{array}{l}
f^{[M]}(i+1)-f^{[M]}(i) \\
\quad \equiv 1 \text { or }-1(\bmod 17), \\
\quad f^{[M]}(1)-f^{[M]}(17) \equiv 1 \text { or }-1(\bmod 17) .
\end{array}
$$

Determine the maximum possible value of $M$ for all functions $f$ that satisfy the above conditions, and prove your conclusion.