# Balanced Diet

Every day, Danny buys one sweet from the candy store and
eats it. The store has $m$
types of sweets, numbered from $1$ to $m$. Danny knows that a balanced diet
is important and is applying this concept to his sweet
purchasing. To each sweet type $i$, he has assigned a *target
fraction*, which is a real number $f_ i$ ($0 < f_ i \le 1$). He wants the
fraction of sweets of type $i$ among all sweets he has eaten to
be roughly equal to $f_
i$. To be more precise, let $s_ i$ denote the number of sweets of
type $i$ that Danny has
eaten, and let $n = \sum _{i=1}^
m s_ i$. We say the set of sweets is *balanced*
if for every $i$ we
have

Danny has been buying and eating sweets for a while and during this entire time the set of sweets has been balanced. He is now wondering how many more sweets he can buy while still fulfilling this condition. Given the target fractions $f_ i$ and the sequence of sweets he has eaten so far, determine how many more sweets he can buy and eat so that at any time the set of sweets is balanced.

## Input

The input consists of three lines. The first line contains two integers $m$ ($1 \le m \le 10^5$), which is the number of types of sweets, and $k$ ($0 \le k \le 10^5$), which is the number of sweets Danny has already eaten.

The second line contains $m$ positive integers $a_1, \ldots , a_ m$. These numbers are proportional to $f_1, \ldots , f_ m$, that is, $\displaystyle f_ i = \frac{a_ i}{\sum _{j = 1}^ m a_ j}$. It is guaranteed that the sum of all $a_ j$ is no larger than $10^5$.

The third line contains $k$ integers $b_1, \ldots , b_ k$ ($1 \le b_ i \le m$), where each $b_ i$ denotes the type of sweet Danny bought and ate on the $i^\text {th}$ day. It is guaranteed that every prefix of this sequence (including the whole sequence) is balanced.

## Output

Display the maximum number of additional sweets that Danny
can buy and eat while keeping his diet continuously balanced.
If there is no upper limit on the number of sweets, display the
word `forever`.

Sample Input 1 | Sample Output 1 |
---|---|

6 5 2 1 6 3 5 3 1 2 5 3 5 |
1 |

Sample Input 2 | Sample Output 2 |
---|---|

6 4 2 1 6 3 5 3 1 2 5 3 |
forever |