The border clue
The border clue leads to FRONTIER. Begin at row 5, column 5 for F, then trace upward through row 4, column 5 and row 3, column 5 for R and O. Turn left to row 3, column 4 for N, then go up to row 2, column 4 for T and row 1, column 4 for I. Finish by moving right to row 1, column 5 for E and down to row 2, column 5 for R.
This route has four upward moves, then one move each down, left, and right. It spans all five rows but only two columns, which is a useful way to distinguish it from the other routes when you are checking your own trace. Six of its eight cells visit an edge, including two corner visits, so the path keeps close to the board boundary without remaining straight.
For a position-by-position check, use: 1 at row 5, column 5; 2 at row 4, column 5; 3 at row 3, column 5; 4 at row 3, column 4; 5 at row 2, column 4; 6 at row 1, column 4; 7 at row 1, column 5; 8 at row 2, column 5. After this route is assigned, 16 open cells remain for the other two answers.