
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 \( X \) be the set of residues modulo 17. We regard two members of \( X \) as adjacent if they differ by 1, so 0 and 16 are adjacent. We say that a permutation of \( X \) is dispersive if it never maps two adjacent values to two adjacent values, and connective if it always maps two adjacent values to two adjacent values. What is the largest \( N \) for which we can find a permutation \( p \) on \( X \) such that \( p \), \( p^2 \), ..., \( p^{N-1} \) are all dispersive and \( p^N \) is connective?

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