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system
Let \( A = \{0, 1, \cdots, 2016\} \). If there exists a surjective function \( f: \mathbf{N} \rightarrow A \) such that for any \( i \in \mathbf{N} \),
\[ 
f(i+2017) = f(i), 
\]
we call \( f \) a "harmonic function."

Define:
\[ 
\begin{array}{l}
f^{(1)}(x) = f(x), \\
f^{(k+1)}(x) = f(f^{(k)}(x)) \quad (k \in \mathbf{N}_{+}).
\end{array}
\]

Let \( f \) be a "harmonic mapping" satisfying the conditions: there exists a positive integer \( M \) such that
1. For \( m < M \), if \( i, j \in \mathbf{N} \) and \( i \equiv j + 1 \pmod{2017} \), then \( f^{(m)}(i) - f^{(m)}(j) \not\equiv \pm 1 \pmod{2017} \);
2. If \( i, j \in \mathbf{N} \) and \( i \equiv j + 1 \pmod{2017} \), then \( f^{(M)}(i) - f^{(M)}(j) \equiv \pm 1 \pmod{2017} \).

Find the maximum possible value of \( M \).

Present the answer in LaTex format: \boxed{Your answer}
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