
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( A = \{1, 2, \cdots, 17\} \). For the mapping \( f: A \rightarrow A \), define
\[ f^{(1)}(x) = f(x), \]
\[ f^{(k+1)}(x) = f(f^{(k)}(x)), \quad k \in \mathbb{N}. \]

Consider a bijective mapping \( f \) from \( A \) to \( A \) that satisfies the following conditions: there exists a natural number \( M \) such that:

1. For \( m < M \) and \( 1 \leq i \leq 16 \),
\[ 
\begin{cases}
f^{(m)}(i+1) - f^{(m)}(i) \not\equiv \pm 1 \pmod{17}, \\
f^{(m)}(1) - f^{(m)}(17) \not\equiv \pm 1 \pmod{17};
\end{cases}
\]

2. For \( 1 \leq i \leq 16 \),
\[ 
\begin{cases}
f^{(M)}(i+1) - f^{(M)}(i) \equiv \pm 1 \pmod{17}, \\
f^{(M)}(1) - f^{(M)}(17) \equiv \pm 1 \pmod{17}.
\end{cases}
\]

Determine the maximum possible value of \( M \) for all functions \( f \) that satisfy the above conditions, and provide a proof for your conclusion.

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