Let $f,g:(1,\infty ) \to $ $\mathbb{R}$ be defined as $f(x) = \frac{{2x + 3}}{{5x + 2}}$ and $g(x) = \frac{{2 - 3x}}{{1 - x}}$. If the range of the function fog : [2,4] $\to$ $\mathbb{R}$ is $[\alpha ,\beta ]$ , then $\frac{1}{{\beta - \alpha }}$ is equal to
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