2020 Fermat Problem 24

In the diagram, the circle with centre X is tangent to the largest circle and passes through the centre of the largest circle. The circles with centres Y and Z are each tangent to the other three circles, as shown. The circle with centre X has radius 1. The circles with centres Y and Z each have radius r. The value of r is closest to

\textbf{(A)}\ 0.93\quad \textbf{(B)}\ 0.91\quad \textbf{(C)}\ 0.95\quad \textbf{(D)}\ 0.87\quad \textbf{(E)}\ 0.89

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.