The French gourmet Frank is a very well-respected gourmet; his job involves going around to various restaurants, eating their food and then writing a review about the restaurant. But he carries a dark secret: he is actually only interested in eating as much as possible and in any order!
However, Frank understands that a true gourmet can’t eat on just any manner, like starting with his dessert and finishing with a salad. Therefore, he asks for your help in constructing a list with all the different ways he could arrange his dishes during a serving, so he can pick the best order.
During the serving, Frank has $M$ minutes to eat. The restaurant offers $N$ different courses, and he can eat any number of portions of each course. For course $i$, it takes $T_ i$ minutes to eat a portion of it. Frank wants to eat something during every of the $M$ minutes he has, and he wants to finish all the portions that he is served. He never wants to start eating a new portion until he finished the last one.
Your task is to write a program to compute the number of ways he could arrange his meal, given these restrictions.
The first line contains an integer $M$, the number of minutes, where $1 \le M \le 1\, 000$.
The second line contains an integer $N$, the number of courses, where $1 \le N \le 30$.
Afterwards, $N$ lines follow with an integer $T_ i$ on each line, the time it takes to each every dish, where $1 \le T_ i \le 200$.
The program should output the number of ways in which Frank can eat during exactly $M$ minutes.
The answer will always be at most 2 billion ($2\, 000\, 000\, 000$).
Your program will be tested on 5 test cases. For each test cases you pass, you will get 20 points.
For test cases worth 60 points, the answer will be less than 2 million ($2\, 000\, 000$).
Explanation of Sample 1
Frank is going to eat during 10 minutes and have two courses to choose from, for example a sallad (takes 3 minutes to eat) and pie (7 minutes). There are only two ways he can eat them: first pie and then sallad or vice versa. If he only eats sallad he will wither finish too quickly (3 portions = 9 minutes) or too late (4 portions = 12 minutes).
|Sample Input 1||Sample Output 1|
10 2 7 3
|Sample Input 2||Sample Output 2|
12 6 1 3 3 5 6 12