Loop the loops - Hard

In the world of pencil puzzles, there are many puzzle types where you have to draw a loop on a lattice grid, including Country Road, Masyu, Pure Loop, Slalom, and Yajilin. In most of these puzzles (and all of the linked above), the loop visits some of the cells, passing through the cells' centers, and may not use a cell more than once (which also means no intersections, no touching itself, etc).

Formally, on a polyomino PP, a loop is a sequence of n4n \ge 4 squares (a1,a2,a3,,an)(a_1, a_2, a_3, \ldots, a_n) such that all squares aia_i are in PP, aia_i and ai+1a_{i+1} share a side for all valid ii, ana_n and a1a_1 also share a side, and all squares in the loop are distinct. Loops are cyclic (it can start from any square in the loop) and don't have any orientation (reversing the loop doesn't matter), thus (a1,a2,a3,a4),(a2,a3,a4,a1),(a4,a3,a2,a1)(a_1, a_2, a_3, a_4), (a_2, a_3, a_4, a_1), (a_4, a_3, a_2, a_1) all describe the same loop.

Determine the number of loops on a 3×303 \times 30 rectangle.

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