More trigonometry I guess?

Geometry Level 4

\[ \large \dfrac1{\cos \alpha} + \dfrac1{\cos\beta} + \dfrac1{\cos \gamma} \geq \dfrac{M \cdot R}{R+r} \]

Let \(r\) and \(R\) denote the incircle and circumcircle, respectively, of an acute-angle triangle with interior angles \(\alpha, \beta \) and \(\gamma\).

Find the maximum value of \(M\) satisfying the inequality above.

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