Two hunters with one bullet each walking to a target

Two hunters, A and B, start together at $x=0$ and walk side by side at the same speed towards a target at $x=1$. Each has a single bullet and may fire at any moment. From position $x$, A hits with probability $x$ and B hits with probability $x^2$; both know these functions and each sees when the other fires. Whoever is the only one to hit wins the prize. If they fire at the same instant and both hit or both miss, the contest is replayed from scratch. Both play optimally. A's probability of winning has the form $\dfrac{\sqrt a-b}{c}$ with positive, pairwise coprime integers $a,b,c$. Find $a+b+c$.

$8$. Whoever fires first and misses lets the other walk up to $x=1$ and hit for sure. So if A fires first at $x$ he wins with probability $x$, and if B fires first A wins with probability $1-x^2$. Each waits until these balance, $x=1-x^2$, i.e. $x=\tfrac{\sqrt5-1}{2}$, which is A's winning chance; hence $5+1+2=8$.