Problem I
Mixing Drinks

Pia is a famous bartender at the hip Stockholm night club Supernova. One of her most impressive feats is the mixing a series of drinks using each of the $N$ distinct drink ingredients in the bar exactly once. She does this in the following way.

First, Pia chooses a number of drinks to make. Each of the drink ingredients are then lined up in front of her in order $1, 2, \dots , N$. For the first drink, she uses some positive number $K$ of ingredients starting from the left, i.e. $1, 2, ..., K$. For the next drink, she uses some positive number $L$ of ingredients starting from the first unused ingredient, i.e. $K + 1, K + 2, \dots , K + L$. She continues this process until the final drink, which uses some set of ingredients $N - M, N - M + 1, \dots , N$.

However, not every pair of ingredients work well in a drink. For example, milk and water would not go very well together. She may not include a bad pair of ingredients in any drink.

So far, she has managed to make a different set of drinks every night. For how many nights can she mix a new set of drinks? We call two sets of drinks different if they do not consist of the exact same drinks (though they are allowed to have drinks in common).


The first line of the input contains two integers $1 \le N \le 100\, 000$ and $0 \le P \le 100\, 000$, the number of ingredients and bad pairs of ingredients.

Each of the next $P$ lines contains two integers $1 \le a \not= b \le N$, two ingredients that do not work well together in a drink. The same pair of ingredients may appear multiple times in this list.


Output a single integer, the number of nights Pia can construct a different set of drinks. Since this number may be large, output the remainder when divided by $10^9 + 7$.

Sample Input 1 Sample Output 1
5 3
1 3
4 5
2 4
Sample Input 2 Sample Output 2
5 0
CPU Time limit 1 second
Memory limit 1024 MB
Statistics Show
License Creative Commons License (cc by-sa)

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