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2017-04-15 19:00 CEST

NAIPC 2017

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2017-04-16 00:00 CEST
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Problem F
Incremental Double Free Strings

A string is called double free if no two adjacent letters are the same.

A string is called $k$-incremental if for all values of $j$ in the range $[1,k]$, there exists exactly one character with $j$ occurrences, and the string’s length is $1+2+3+\ldots +(k-1)+k$. For example, if $k=3$, then a $3$-incremental string should have one character appear once, another twice, another three times, in any order, for a total string length of $6$.

A string is both $k$-incremental and double free if it meets both these criteria. Now consider examining all such strings of lowercase letters for a given $k$ in alphabetical order. Consider the following examples.

$k=2$: aba, aca, ada, …, aya, aza, bab, bcb, bdb, …, zxz, zyz

$k=3$: ababac, ababad, …, ababay, ababaz, ababca, …, zyzyzx

What is the $n^\mathrm {th}$ string in an alphabetized list of all $k$-incremental, double free strings?

Input

Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. There will be exactly one line of input. It will contain two integers, $k$ and $n$ ($1 \le k \le 26, 1 \le n \le 10^{18}$), which is asking for the $n^\mathrm {th}$ string in the alphabetically sorted list of all $k$-incremental, double free strings.

Output

Output the $n^\mathrm {th}$ $k$-incremental, double free string in the alphabetized list. If no such string exists, output $-1$.

Sample Input 1 Sample Output 1
2 650
zyz
Sample Input 2 Sample Output 2
2 651
-1
Sample Input 3 Sample Output 3
5 12345678901234
yuzczuyuyuzuyci