### 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

be made.

#### 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