Mancala

*Mancala* is a family of board games played around
the world, sometimes called *sowing* games, or
*count-and-capture* games, which describes the game
play. One simple variant is a solitaire game called
*Tchoukaillon* which was described by Véronique
Gautheron. *Tchoukaillon* is played on a board with an
arbitrary number of bins numbered $1, 2, \ldots $, containing
$b[1], b[2], \ldots $
counters respectively and an extra empty bin called the Roumba
on the left.

A single play consists on choosing a bin, $n$, for which $b[n] = n$ (indicated by the darker
circles in the diagram) and distributing the counters one per
bin to the bins to the left including the *Roumba*
(getting the next diagram below in the figure above). If there
is no bin where $b[n] =
n$, then the board is a losing board.

If there is a sequence of plays which takes the initial
board distribution to one in which every counter is in the
*Roumba*, the initial distribution is called a winnable
board. In the example above, $0,1,3,\ldots $ is a winnable board
(the “$\ldots $” indicates
all the bins to the right of bin $3$ contain $0$). For each total number of
counters, there is a unique distribution of the counters to
bins to make a winnable board for that total count (so
$0,1,3,\ldots $ is the
only winnable board with $4$ counters).

Write a program which finds the winnable board for a total count input.

The first line of input contains a single integer $P$, ($1 \le P \le 200$), which is the number of data sets that follow. Each data set should be processed identically and independently.

Each data set consists of a single line of input. It contains the data set number, $K$, followed by a single space, followed by the total count $N$ ($1 \le N \le 2000$) of the winnable board to be found.

For each data set there will be multiple lines of output. The first line of output contains the data set number, $K$, followed by a single space, followed by the index of the last bin, $B$, with a non-zero count. Input will be chosen so that $B$ will be no more than $80$. The first line of output for each dataset is followed by the bin counts $b[1], b[2], \ldots , b[B]$, 10 per line separated by single spaces.

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

3 1 4 2 57 3 500 |
1 3 0 1 3 2 12 1 2 2 2 2 6 2 4 6 8 10 12 3 39 0 2 2 1 3 2 2 2 6 7 5 0 6 12 2 6 10 14 18 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 |