A circle and a pentagon

Geometry Level 3

Pictured here is a \({\color{Blue} \text{regular pentagon}}\) which has one vertex on a \({\color{Purple} \text{unit circle}}\), and the \({\color{Green} \text{green line segments}}\) connect vertices of the pentagon:

What is the length of the \({\color{Orange} \text{orange line segment}}\)?

Provide your answer to 3 decimal places. If you think not enough information is given, submit -1 as your answer.

Note: The circle isn't necessarily tangent to any of the line segments drawn.


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