Arcs and Chords and Segments, Oh My!  
 
The Story  
There is an arc of a certain circle called arc AXC. Its length is known and its length is less than the circumference of the circle. X is a point on the arc that is at its midpoint. B is a point on chord AC such that it is the midpoint of the chord. We also know that segment BX is perpendicular to chord AC and that segment BX has a length less than the radius.
Find the length of the radius of the circle.
 
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