Consider an obtuse angled triangle \(ABC\) in which the difference between the largest and the smallest angle is \(\frac…

Consider an obtuse angled triangle \(ABC\) in which the difference between the largest and the smallest angle is \(\frac{\pi }{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1 .

Then the inradius of the triangle \(ABC\) is ____