General survivor formula for the sword circle

People numbered $1$ to $N$ stand in a circle, and $X$ denotes the largest power of $2$ not exceeding $N$. Person $1$ holds a sword, kills the next living person to their left, and hands the sword to the next living person after that; this repeats until only one person remains. The survivor's number can be written as $aN+bX+c$ for constants $a,b,c$. Find $a+2b+3c$.

$1$. When the circle size is a power of $2$, the current sword-holder survives, since each pass halves the group cleanly. So after $N-X$ eliminations (persons $2,4,\dots,2(N-X)$), the holder, person $2(N-X)+1$, survives. Hence $a=2$, $b=-2$, $c=1$ and $a+2b+3c=1$.