Starting with three numbers $a, b, c$, at each step do one of the three
operations:
- change $a$ to $2(b + c) - a$;
- change $b$ to $2(c + a) - b$;
- change $c$ to $2(a + b) - c$;
Define $f(a, b, c)$ to be the minimum number of steps required for one number
to become zero. If this is not possible then $f(a, b, c) = 0$.
For example, $f(6, 10, 35) = 3$:
$$(6, 10, 35) \to (6, 10, -3) \to (8, 10, -3) \to (8, 0, -3).$$
However, $f(6, 10, 36) = 0$ as no series of operations leads to a zero number.
Also define $\displaystyle F(a, b) = \sum_{c = 1}^\infty f(a, b, c)$.
You are given $F(6, 10) = 17$ and $F(36, 100) = 179$.
Find $\displaystyle \sum_{k = 1}^{18} F(6^k, 10^k)$.