Recurring Decimals

The decimal representation of an integer can be turned into another integer by rearranging its digits. We use this to build a sequence.

You are given a non-negative integer $a_0$ and a digit count $L$. From $a_i$ we obtain $a_{i+1}$ by the following rules.

1. Write $a_i$ in decimal using exactly $L$ digits, adding leading zeros if necessary. For example, written with six digits, $2012$ becomes $002012$.
2. Rearrange the digits to form the largest possible integer and the smallest possible integer. In the example above, the largest is $221000$ and the smallest is $000122 = 122$.
3. Obtain $a_{i+1}$ by subtracting the smallest from the largest. In the example above, $221000 - 122 = 220878$.

Repeating this computation gives a sequence $a_0, a_1, a_2, \dots$.

For example, starting from $83268$ with $L = 6$ you get:

• $a_0 = 083268$
• $a_1 = 886320 - 023688 = 862632$
• $a_2 = 866322 - 223668 = 642654$
• $a_3 = 665442 - 244566 = 420876$
• $a_4 = 876420 - 024678 = 851742$
• $a_5 = 875421 - 124578 = 750843$
• $a_6 = 875430 - 034578 = 840852$
• $a_7 = 885420 - 024588 = 860832$
• $a_8 = 886320 - 023688 = 862632$
• $\dots$

Because the number of digits is fixed, some value must eventually repeat, so there is always a pair $i > j$ with $a_i = a_j$. In the example, $(i = 8, j = 1)$ works because $a_8 = a_1 = 862632$.

Write a program that, given $a_0$ and $L$, finds the smallest $i$ for which $a_i = a_j$ holds for some $j < i$.

The input consists of several datasets. Each dataset is a single line containing two integers $a_0$ and $L$ separated by a space, where $1 \le L \le 6$ and $0 \le a_0 < 10^L$.

A line containing two zeros marks the end of the input and is not a dataset.

For each dataset, find the smallest $i$ satisfying $a_i = a_j$ with $i > j$, and print one line with the three integers $j$, $a_i$, and $i - j$, separated by single spaces. Suppress leading zeros and do not print any extra characters.

You may assume that this $i$ is never greater than $20$.