USACO #$3$ Bessie has woken up on a strange planet. In this planet, there are N $(1<=$ N $<=104)$ months, with a$1,...,$ aN days, respectively $(1<=$ ai $<=4*109,$ all ai are integers$).$ In addition, on the planet, there are also weeks, where each week is L days, with L being a positive integer. Interestingly, Bessie knows the following:

For the correct L$,$ each month is at least $4$ weeks long.

For the correct L$,$ there are at most $3$ distinct values of ai mod L$.$

Unfortunately, Bessie has forgotten what L is$!$ Help her by printing the sum of all possible values of L$.$

Note that the large size of integers involved in this problem may require the use of $64-$bit integer data types $($e$.$g$.,$ a $$long long$$ in C$/$C$++).$

Input Format $($input arrives from the terminal $/$ stdin$)$:

The first line contains a single integer N$.$ The second line contains N space$-$separated integers, a$1,...,$ aN$.$

Output Format $($print output to the terminal $/$ stout$)$:

A single integer, the sum of all possible values of L

Sample Input:

$12$

$312831303130313130313031$

Sample Output:

$28$

The possible values of L are $1,2,3,4,5,6,$ and $7.$ For example, L $=7$ is valid because each month is at least length $4*7=28$ days long, and each month is either $0,2,$ or $3$ mod $7.$

Sample Input:

$4$

$31352829$

Sample Output:

$23$

The possible values of L are $1,2,3,4,6,$ and $7.$ For example, L $=6$ is valid because each month is at least $4*6=24$ days long, and each month is either $1,4,$ or $5$ mod $6.$

Scoring:

Inputs $3$~$4$: $1<=$ ai $<=106$

Inputs $5-14$: No additional constraints

Please show me that your code shows the correct output for each of the following sample input