Suppose $\alpha, \beta,$ and $\gamma$ are the roots of polynomial $x^{3} + px^{2} + qx + 2 = 0$ and that $\alpha \beta + 1 = 0$. What is the value of $2p + q + 100$?

This problem is shared by Prakkash M, who came across it in his Mathematical Olympiad Training.

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