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Group Lockers

/problems/rupc19.grouplockers/file/statement/en/img-0001.jpg
A set of lockers.
Sets of lockers can be found at different locations around the university. Each set is arranged into columns of equal width, with between one and four lockers in each column. At the start of each semester, students get the opportunity to rent one of these lockers for the entire semester; this can be very handy for students that would otherwise have to carry stacks of books between their classes.

Upon hearing this, a group of $n$ friends decides that they each want their own locker. None of them have any specific requirements about the size or location of their locker, except that — being such good friends — they want their lockers to be close to each other. In fact, they insist that all of their lockers belong to the same set of lockers. Within that set, they want to minimize the distance required to walk between the leftmost and rightmost lockers that belong to the friends.

Given a candidate set of lockers, and the number of lockers that are available for rent in each column, can you help the friends determine this distance if they choose their $n$ lockers optimally?

Input

The input consists of:

  • One line with two integers $n$ and $m$ ($1 \le n \le 10^9$, $1 \le m \le 5\cdot 10^5$), the number of friends and the number of columns in the candidate set of lockers.

  • One line with a string of $m$ digits $c_1c_2\ldots c_ m$ ($0 \le c_ i \le 4$), where $c_ i$ represents the number of available lockers in column $i$.

Output

Output the distance, measured in number of columns, required to walk between the leftmost and rightmost lockers that belong to the friends, assuming they rent the $n$ lockers that minimize this distance. If it is not possible to rent $n$ lockers, output “impossible”.

Sample Input 1 Sample Output 1
3 10
0200110102
2
Sample Input 2 Sample Output 2
5 7
0401330
1
Sample Input 3 Sample Output 3
4 3
224
0
Sample Input 4 Sample Output 4
10 10
0301010400
impossible
CPU Time limit 1 second
Memory limit 1024 MB
Authors
Arnar Bjarni Arnarson, Bjarki Ágúst Guðmundsson, and Unnar Freyr Erlendsson
Source Reykjavík University Spring Contest 2019
License Creative Commons License (cc by-sa)

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