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