Let $S$ be a circle with centre $O$. A chord $AB$, not a diameter, divides $S$ into two regions $R_1$ and $R_2$ such that $O$ belongs to $R_2$. Let $S_1$ be a circle with centre in $R_1$, touching $AB$ at $X$ and $S$ internally. Let $S_2$ be a circle with centre in $R_2$, touching $AB$ at $Y$, the circle $S$ internally and passing through the centre of $S$. The point $X$ lies on the diameter passing through the centre of $S_2$ and $\angle YXO=30^\circ$. If the radius of $S_2$ is 100 then what is the radius of $S_1$?
I have tried this for over an hour now but I can't get the right answer, which is 60.
After some construction and taking the sine of given angle I got $XY=100\sqrt3$ but radius of circle is still out of reach.