Overlapping Annuli

Just from the diagram, can you work out the radius of the smaller circles?
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The diagram shows eight circles of two different sizes.
Image
Overlapping Annuli


The circles are arranged in concentric pairs so that the centres form a square.

Each larger circle touches one other larger circle and two smaller circles.

The larger circles have radius 1.


What is the radius of each smaller circle?


This problem is taken from the UKMT Mathematical Challenges.