Minimum Subject Weight Requirement

There are \(n\) students (numbered \(1\) to \(n\)) taking an exam in \(m\) subjects (numbered \(1\) to \(m\)). Student \(i\) scores \(a_{i,j}\) in subject \(j\). You want to rank the students based on a weighted total score. To do so, you assign non‐negative real weights \(p_1, p_2, \dots, p_m\) to the subjects such that \(\sum_{j=1}^{m} p_j = 1\). The weighted total score for student \(i\) is defined as

[ S_i = \sum_{j=1}^{m} a_{i,j},p_j. ]

The students are then ranked in descending order of \(S_i\) (ties are considered equal). However, the teacher of each subject \(k\) requires that in the final ranking there is no pair of distinct students \((x, y)\) such that:

• Student \(x\) is strictly ranked higher than student \(y\), yet \(a_{x,k} < a_{y,k}\);

You are free to choose the weights \(p_j\), but you want to maximize the flexibility in assigning the other weights. Therefore, for each subject \(k\) \( (1 \le k \le m)\), determine the minimum value of \(p_k\) (modulo \(998244353\)) that guarantees no matter how the other weights are chosen (subject to being nonnegative and summing to \(1-p_k\)), the teacher \(k\)'s requirement is satisfied.

More formally, for each subject \(k\), you need to compute the minimum \(p_k\) such that for any choice of \(p_j \ge 0\) (for \(j \neq k\)) satisfying \(\sum_{j\neq k} p_j = 1 - p_k\), the following holds: For every pair of distinct students \(x,y\) with \(a_{x,k} < a_{y,k}\), it is guaranteed that

[ p_k(a_{y,k}-a_{x,k}) \ge (1-p_k),D_{x,y,k}, ]

where

[ D_{x,y,k} = \max_{\substack{1\le j \le m\ j\neq k}} \max(0,,a_{x,j}-a_{y,j}). ]

This inequality can be rearranged into

[ p_k \ge \frac{D_{x,y,k}}{(a_{y,k}-a_{x,k})+D_{x,y,k}}. ]

You must choose \(p_k\) so that the inequality holds for every pair \((x, y)\) with \(a_{x,k} < a_{y,k}\). If no such pair exists for a subject \(k\), the answer for that subject is \(0\).

All answers should be given modulo \(998244353\).

inputFormat

The first line contains two integers \(n\) and \(m\) \((1 \leq n, m \leq \text{constraints})\), representing the number of students and the number of subjects respectively.

Then \(n\) lines follow, each containing \(m\) integers. The \(j\)-th integer in the \(i\)-th line represents \(a_{i,j}\), the score of student \(i\) in subject \(j\).

outputFormat

Output \(m\) integers separated by spaces. The \(k\)-th integer is the minimum required value of \(p_k\) modulo \(998244353\).

sample

3 2
1 2
2 1
1 1

499122177 499122177