Gwen loves most numbers. In fact, she loves every number that is not a multiple of $n$ (she really hates the number $n$). For her friends’ birthdays this year, Gwen has decided to draw each of them a sequence of $n-1$ flowers. Each of the flowers will contain between $1$ and $n-1$ flower petals (inclusive). Because of her hatred of multiples of $n$, the total number of petals in any non-empty contiguous subsequence of flowers cannot be a multiple of $n$. For example, if $n = 5$, then the top two paintings are valid, while the bottom painting is not valid since the second, third and fourth flowers have a total of $10$ petals. (The top two images are Sample Input $3$ and $4$.)

Gwen wants her paintings to be unique, so no two paintings will have the same sequence of flowers. To keep track of this, Gwen recorded each painting as a sequence of $n-1$ numbers specifying the number of petals in each flower from left to right. She has written down all valid sequences of length $n-1$ in lexicographical order. A sequence $a_1,a_2,\dots , a_{n-1}$ is lexicographically smaller than $b_1, b_2, \dots , b_{n-1}$ if there exists an index $k$ such that $a_ i = b_ i$ for $i < k$ and $a_ k < b_ k$.

What is the $k$th sequence on Gwen’s list?

-----Input-----
The input consists of a single line containing two integers $n$ ($2 \leq n \leq 1000$), which is Gwen’s hated number, and $k$ ($1 \leq k \leq 10^{18}$), which is the index of the valid sequence in question if all valid sequences were ordered lexicographically. It is guaranteed that there exist at least $k$ valid sequences for this value of $n$.

-----Output-----
Display the $k$th sequence on Gwen’s list.

-----Examples-----
Sample Input 1:
4 3
Sample Output 1:
2 1 2

Sample Input 2:
2 1
Sample Output 2:
1

Sample Input 3:
5 22
Sample Output 3:
4 3 4 2

Source: kattis | Difficulty: HARD


Solutions: