The three-letter path
Trace DEN through 1 at row 3, column 4; 2 at row 3, column 5; 3 at row 4, column 5. Read in that order, the cells give D → E → N and fill the shortest answer slot with one orthogonally connected trace.
Move right once from the starting cell, then turn downward for the final letter. The path spans two rows and two columns, with one turn and two straight segments. It does not visit an outer edge or corner; after these three cells are assigned, 35 open cells remain for the other five entries.