Solve the programming task below in a Python markdown code block.
For a binary string S, define its beauty to be the maximum number of [substrings] S can be partitioned into, such that each substring has strictly more ones than zeroes. For example, the beauty of 11011 is 3, as it can be partitioned as [110][1][1]. If such a partition doesn't exist, the beauty is defined to be 0. 

Also, define the happiness of a binary string to be the maximum beauty over all its substrings. For example, the happiness of the string S = 010011011 is 3, which is the beauty of its substring 11011.

You are given N binary strings S_{1}, S_{2}, \ldots, S_{N}. You have to process Q queries:
- Each query consists of two integers L, R, where 1 ≤ L ≤ R ≤ N.
- The result of the query is the maximum possible happiness of S_{p_{1}} + S_{p_{2}} + \ldots S_{p_{(R - L + 1)}}, where p is some permutation of \{L, L + 1, \ldots, R - 1, R\} and + denotes the concatenation operation.

------ Input Format ------ 

The first line of input contains two space-separated integers N and Q, the number of strings and the number of queries. Then, 2N + Q lines follow:
- For each 1 ≤ i ≤ N, S_{i} is represented by two lines. The first line contains the length of S_{i} and the second line contains S_{i} itself.
- Each of the next Q lines contains two integers, L and R, denoting the query parameters

------ Output Format ------ 

For each query, print the maximum possible happiness of the concatenation of some permutation of the given interval on a new line.

------ Constraints ------ 

$1 ≤ N, Q ≤ 10^{5}$
$|S_{i}| ≥ 1$ for each valid $i$.
- Every string consists of characters which are either '0' or '1'.
- The sum of the lengths of all the input strings doesn't exceed $10^{6}$.

------ subtasks ------ 

Subtask 1 (30 points): The sum of $R L + 1$ over all queries doesn't exceed $10^{6}$.
Subtask 3 (70 points): No further constraints.

----- Sample Input 1 ------ 
4 1
1
0
2
11
3
010
4
0110
2 4
----- Sample Output 1 ------ 
3
----- explanation 1 ------ 
It is optimal to concatenate as $S_{3} + S_{4} + S_{2}$ to get $010011011$ with a happiness value of $3$.