Problem of the Day #22 (Challenge)

POTD Challenge December 11, 2023
Author: munch

Given a 16 \times 12 grid, color exactly 96 squares such that two squares touching at a single vertex may not both be colored. (It is possible for two squares sharing an edge to both be colored).

Let K be the number of possible colorings in this grid, where two colorings are considered to be distinct if they differ by a rotation or reflection of the square. Find the value of \sqrt{K}.