Puzzle of the Week #544 - Exact Expression

a, b and c are each real numbers between 0 and 1.

 

I have a number ‘a’ and I want to find an exact expression for it.

I take the reciprocal and get that 1/a = 1+b.

I subtract the 1 and take the reciprocal. I find that 1/b = 3+c.

I subtract the 3 and take the reciprocal. I discover that 1/c = 4+b.

 

What is the exact value of a?