jmc

Let $a=-1-3i$ and $b=7+8i.$ Then $z$ lies on the circle centered at $a$ with radius 1, and $w$ lies on the circle centered at $b$ with radius 3.



By the Triangle Inequality, $$|a-z|+|z-w|+|w-b|\ge |a-b|,$$so $$|z-w|\ge |a-b|-|a-z|-|w-b|.$$We have that $|a-b|=|(-1-3i)-(7+8i)=|-8-11i|=\sqrt{185}.$ Also, $|a-z|=1$ and $|w-b|=3,$ so $$|z-w|\ge \sqrt{185}-4.$$Equality occurs when $z$ and $w$ are the intersections of the circles with the line segments connecting $a$ and $b.$



Hence, the smallest possible value of $|z-w|$ is $\boxed{\sqrt{185}-4}.$