# Problem A

Same Digits (Hard)

Two positive integers, $x$ and $y$, are called
*digit-preserving* if their product, $xy$, contains exactly the same digits
as $x$ and $y$ contain together, including
repetitions. For example, if $x =
807$ and $y = 984$,
then their product, $794088$, contains $\textrm{one}~ 7$, $\textrm{one}~ 9$, $\textrm{one}~ 4$, $\textrm{one}~ 0$, and $\textrm{two}~ 8\textrm{'s}$, which is
exactly the same set of digits and corresponding frequencies as
in $807$ and $984$ combined.

Given an interval, $[A,B]$, find all digit-preserving
pairs, $x, y$, in the
interval, **with the additional requirement that
their product must also be in the same interval**, i.e., all
three of $x, y, xy$ are in
$[A,B]$. To avoid
double-counting, you can assume $x \leq y$ (this avoids, for example,
treating $(807, 984)$ and
$(984, 807)$ as different
digit-preserving pairs).

## Input

The input consists of a single line containing two space-separated integers, $A$ and $B$, with $1 \leq A \leq B \leq 200\, 000$.

## Output

First output a single line containing “$n$ `digit-preserving pair(s)`”, where $n$ is the number of digit-preserving
pairs in $[A,B]$ (as
described above). Then output $n$ lines, each of which contains one
of the digit-preserving pairs and the corresponding product.
Carefully format your output as in the sample output (note the
single space separating adjacent tokens). These lines should be
sorted by increasing value $\textrm{of}~ x$, breaking ties by
increasing value $\textrm{of}~
y$.

Sample Input 1 | Sample Output 1 |
---|---|

1 1000 |
3 digit-preserving pair(s) x = 3, y = 51, xy = 153 x = 6, y = 21, xy = 126 x = 8, y = 86, xy = 688 |

Sample Input 2 | Sample Output 2 |
---|---|

1000 2000 |
0 digit-preserving pair(s) |