If the area of the bounded region $R=\left\{\left(x,y\right):\max \left\{0,{\log }_{e}x\right\}\le y\le 2^x,\frac{1}{2}\le x\le 2\right\}$ is, $\alpha {\left({\log }_{e}2\right)}^{-1}+\beta \left({\log }_{\mathrm{e}}2\right)+\gamma$ then the value of ${\left(\alpha +\beta -2\gamma \right)}^2$ is equal to: