The sides of a triangle inscribed in a given circle subtend angles \(\alpha\), \(\beta\) and \(\gamma\) at the centre. Then find the minimum value of the arithmetic mean of \(\cos\left(\alpha+\frac{\pi}{2}\right)\), \(\cos\left(\beta+\frac{\pi}{2}\right)\) and \(\cos\left(\gamma+\frac{\pi}{2}\right)\).

If the minimum value can be expressed as \( - \sqrt{ \frac{a}{b} } \), where \(a\) and \(b\) are coprime positive integers. What is the value of \( a + b \)?

This problem is part of the set Trigonometry.

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