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