Number Square

In Number Square you place the numbers $1, 2, \ldots, N$ into an $N \times N$ grid so that every number appears exactly once in each row and exactly once in each column (the grid is a Latin square).

In addition, for some pairs of neighbouring cells you are told which of the two must hold the larger number. Given the pre-filled cells and these ordering constraints, fill in the whole grid so that all of them are satisfied.

The first line contains a single integer $N$ ($1 \le N < 10$).

The next $2N - 1$ lines each contain exactly $2N - 1$ characters and describe the starting position. Using 0-based indices, cell $(r, c)$ of the board is located at line $2r$, column $2c$.

• On the even-indexed lines ($0, 2, 4, \ldots$): the even-indexed columns hold the board cells — a digit 1–9 for a pre-filled cell, or . if it is empty. The odd-indexed columns hold the horizontal ordering signs between two side-by-side cells: <, >, or . when there is no constraint.
• On the odd-indexed lines ($1, 3, \ldots$): the even-indexed columns hold the vertical ordering signs between two stacked cells: ^, V, or . when there is no constraint. The odd-indexed columns are always ..

Each sign is an inequality that opens toward the larger number: < means the left cell is smaller than the right one, > means the left cell is larger, ^ means the upper cell is smaller than the lower one, and V means the upper cell is larger.

It is guaranteed that at least one valid filling exists. Print the completed board as $N$ lines, each containing $N$ integers separated by single spaces.

If more than one valid filling exists, print the lexicographically smallest one, where a completed board is compared as the sequence obtained by reading its cells row by row from top to bottom and, within each row, from left to right.