You are given a positive integer k. You are also given:

a 2D integer array rowConditions of size n where rowConditions[i] = [above_i, below_i], and
a 2D integer array colConditions of size m where colConditions[i] = [left_i, right_i].

The two arrays contain integers from 1 to k.

You have to build a k x k matrix that contains each of the numbers from 1 to k exactly once. The remaining cells should have the value 0.

The matrix should also satisfy the following conditions:

The number above_i should appear in a row that is strictly above the row at which the number below_i appears for all i from 0 to n - 1.
The number left_i should appear in a column that is strictly left of the column at which the number right_i appears for all i from 0 to m - 1.

Return any matrix that satisfies the conditions. If no answer exists, return an empty matrix.

Example 1:

Input: k = 3, rowConditions = [[1,2],[3,2]], colConditions = [[2,1],[3,2]]
Output: [[3,0,0],[0,0,1],[0,2,0]]
Explanation: The diagram above shows a valid example of a matrix that satisfies all the conditions.
The row conditions are the following:
- Number 1 is in row 1, and number 2 is in row 2, so 1 is above 2 in the matrix.
- Number 3 is in row 0, and number 2 is in row 2, so 3 is above 2 in the matrix.
The column conditions are the following:
- Number 2 is in column 1, and number 1 is in column 2, so 2 is left of 1 in the matrix.
- Number 3 is in column 0, and number 2 is in column 1, so 3 is left of 2 in the matrix.
Note that there may be multiple correct answers.

Example 2:

Input: k = 3, rowConditions = [[1,2],[2,3],[3,1],[2,3]], colConditions = [[2,1]]
Output: []
Explanation: From the first two conditions, 3 has to be below 1 but the third conditions needs 3 to be above 1 to be satisfied.
No matrix can satisfy all the conditions, so we return the empty matrix.

Constraints:

2 <= k <= 400
1 <= rowConditions.length, colConditions.length <= 10⁴
rowConditions[i].length == colConditions[i].length == 2
1 <= above_i, below_i, left_i, right_i <= k
above_i != below_i
left_i != right_i

Build a Matrix With Conditions

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