Consider a uniform rod of mass \(M\) and length \(L,\) free to rotate around a frictionless axis passing through its center and going into the page. Initially, the rod is stationary in the horizontal position, as shown in the diagram below.

Now, a small bullet of mass \(m\) moving with velocity \(v\) hits the rod at its extreme end and sticks to it. The system rotates vertically through some angle \(\theta\) before it momentarily comes to rest. If this angle can be expressed (in degrees) as \[\theta = \alpha + \arcsin \left(\frac {\beta mv^2}{( M+\gamma m)gL}\right), \] where \(g\) denotes the gravitational acceleration and \(\alpha\), \(\beta\), and \(\gamma\) are positive integer constants with \(\alpha\) in degrees, then find the value of \(\alpha + \beta + \gamma \).

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