I discovered an interesting fact the other day, that you can start with 0, and reach ANY positive rational number by selectively using two different actions:

The two actions are as follows:

a) Adding 1, such that x becomes (x+1)

b) Taking the reciprocal, such that x becomes (1/x)

For example, trying to reach 2/5:

starting with 0,

Action a makes it 1

Action a makes it 2

Action b makes it 1/2

Action a makes it 1 1/2

Action a makes it 2 1/2

Finally action b makes it 2/5

Pi is famously not a rational number, however 355/113 is a great approximation to it. Starting with 0, how many steps will it take to reach 355/113?