Description

There are $n$ students in Little O's grade, numbered $1 \sim n$. The school surveyed $m$ activities. Each student is interested in some activities and not in others. Let $a_{i,j}$ indicate whether student $i$ is interested in activity $j$, where $a_{i,j} = 1$ means interested and $a_{i,j} = 0$ means not interested.

For any two students $i$ and $j$, their number of common interests is the count of activities $k$ where $a_{i,k} = a_{j,k} = 1$, i.e., the number of activities both students are interested in.

Each student $x$ will find all students $y$ who have the maximum number of common interests with $x$ and send them friendship invitations. If multiple students $y$ satisfy this condition, $x$ will send invitations to all of them.

Little O (student $1$) wants to receive as many invitations as possible. He can choose one activity $i$ where he is not initially interested (i.e., $a_{1,i} = 0$) and change it to $a_{1,i} = 1$. He may also choose not to modify any activity. Note: At most one modification is allowed.

What is the maximum number of students who will send invitations to Little O after making at most one modification?

Input Format

The input consists of $n+1$ lines:

• Line $1$: Two integers $n$ and $m$, representing the number of students and activities.
• Lines $2$ to $n+1$: Each line contains $m$ integers. The $j$-th integer in line $i+1$ represents $a_{i,j}$.

Output Format

Output one integer: the maximum number of students who will invite Little O.

3 3
0 0 0 
1 0 1
0 1 1

2

4 3
0 0 0 
1 0 1
0 1 1
1 1 1

0

Hint

Sample #1 Explanation

Initially, students $2$ and $3$ each have $1$ common interest with each other (activity $3$). Their common interests with Little O are $0$. Modifications:

• No modification: $0$ invitations.
• Modify $a_{1,1}$ to $1$: Student $2$ invites Little O ($1$ invitation).
• Modify $a_{1,2}$ to $1$: Student $3$ invites Little O ($1$ invitation).
• Modify $a_{1,3}$ to $1$: Both students $2$ and $3$ invite Little O ($2$ invitations).

Thus, the maximum is $2$.

Sample #2 Explanation

Adding student $4$ (who has $2$ common interests with students $2$ and $3$). Little O's maximum common interests with any student after modification cannot exceed $1$, so the answer is $0$.

For $100\%$ data:

• $2 \le n \le 500$
• $1 \le m \le 500$
• $a_{i,j} \in \{0,1\}$

Special Properties:

• A: For all $1 \le i \le m$, $a_{1,i} = 0$.
• B: For all $1 \le i \le n$, $a_{i,1} = 0$.