Stacks of Flapjacks

Time Limit: 1000 ms Memory Limit: 10000 KiB

Problem Description

Background

Stacks and Queues are  often considered the bread and butter of  data
structures and find use in architecture, parsing,  operating systems,
and discrete  event simulation.   Stacks  are also  important in  the
theory of formal languages.
This problem  involves both butter and  sustenance in the form  of
pancakes rather than bread in addition to a finicky server  who flips

pancakes according to a unique, but complete set of rules.

The Problem

Given a stack of pancakes, you are to write a program  that indicates
how the  stack can be sorted  so that the largest  pancake is on  the
bottom  and the  smallest pancake  is on  the top.    The size  of  a
pancake is given by the pancake's diameter.  All pancakes in  a stack
have different diameters.
Sorting a  stack is done  by a sequence of  pancake flips''.   A
flip consists of inserting a spatula between two pancakes in  a stack

and flipping (reversing)  all the pancakes on the spatula  (reversing

---
the sub-stack).   A flip is specified  by giving the position of  the
pancake on  the bottom of  the sub-stack to  be flipped (relative  to
the whole stack).   The pancake on the bottom of the whole stack  has
position 1 and  the pancake on the top of  a stack of n pancakes  has

position n.
A  stack is specified by  giving the diameter  of each pancake  in
the stack in the order in which the pancakes appear.
For  example,  consider the  three stacks  of pancakes  below  (in
which pancake 8 is the top-most pancake of the left stack):

8           7           2
4           6           5
6           4           8
7           8           4
5           5           6
2           2           7

The stack on the left  can be transformed to the stack in the  middle
via flip(3).    The middle stack  can be transformed  into the  right
stack via the command flip(1).

Input

The input consists of a  sequence of stacks of pancakes.  Each  stack

will consist of between 1 and 30 pancakes and each pancake  will have
an integer diameter  between 1 and 100.   The input is terminated  by
end-of-file.  Each stack is given as a single line of input  with the
top pancake on a stack appearing first on a line, the  bottom pancake
appearing last, and all pancakes separated by a space.

Output

For  each stack  of pancakes,  the output  should echo  the  original
stack on one  line, followed by some  sequence of flips that  results
in the stack  of pancakes being sorted  so that the largest  diameter

pancake is  on the bottom and  the smallest on top.   For each  stack
the sequence  of flips  should be terminated  by a  0 (indicating  no
more flips necessary).  Once a stack is sorted, no more  flips should

Sample Input

1 2 3 4 5
5 4 3 2 1
5 1 2 3 4

Sample Output

1 2 3 4 5
0
5 4 3 2 1
1 0
5 1 2 3 4
1 2 0