The five-letter path
Trace QUOTE from row 5, column 5. Move up to row 4, column 5, then continue left through row 4, column 4, row 4, column 3, and row 4, column 2. The letters appear as Q → U → O → T → E, completing the spelling after one upward move and three consecutive leftward moves. The starting cell has one open orthogonal neighbor, while the finishing cell has three.
Follow these cells for a numbered check: 1 at row 5, column 5; 2 at row 4, column 5; 3 at row 4, column 4; 4 at row 4, column 3; 5 at row 4, column 2. This route makes one turn and consists of two straight segments. It spans two rows and four columns, visits two edge cells including one corner, accounts for five of the 18 open cells, and leaves 13 cells for the other two paths.