In modern gadgets, it's crucial to trim down the fat and get rid of unnecessary features, like a headphone jack. The same trend applies to the calculator industry.
In their pursuit for minimalism, Cosio calculator company started to produce calculators that have a single display and can only compute trigonometric functions and their inverses .
Initially, the calculator's display shows the number . After that, for each of the functions listed above, you can press a button that applies that function to the displayed number. If the operation is inapplicable or produces infinity, then the calculator breaks and stops responding.
You took it as a challenge to figure out what you can achieve using this calculator. Find a way to compute using at most operations.
输入格式
The only line contains two integers and ().
输出格式
In the first line, print a single integer --- the number of button presses in your solution ().
In the second line, print the applied operations in order, separated by spaces.
The solution will be checked with a program in C++ using the standard 64-bit floating-point type: . Your answer will be considered correct if the sequence of actions does not cause an error, and in the end the calculator displays with an absolute error of at most .
You do not have to find the shortest solution. Any solution satisfying the constraints will be accepted.
样例
样例输入 1
1 1
样例输出 1
4
atan cos sin asin
样例输入 2
2 1
样例输出 2
11
cos atan sin atan sin atan sin atan sin acos tan
