Form a cycle with a permutation of the first \(n\) positive integers. The cycle is called *Prime Cycle* if all neighboring pairs sum up to be a prime. The two **distinct** prime cycles for \(n=6\) are:

- \(1,4,3,2,5,6\)
- \(1,6,5,2,3,4\)

The permutation \(3,2,5,6,1,4\) is considered the same as the first sequence.

How many distinct prime cycles are there for \(n=16\)?

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