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 $F(a,b)=\sum _{c=1}^{\mathrm{\infty }}f(a,b,c)$. You are given $F(6,10)=17$ and $F(36,100)=179$.

Find ${\displaystyle \sum _{k=1}^{18}F(6^k,{10}^k)}$.