Let P be a plane passing through the points (1, 0, 1), (1, $-$2, 1) and (0, 1, $-$2). Let a vector $\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k$ be such that $\overrightarrow a$ is parallel to the plane P, perpendicular to $(\widehat i + 2\widehat j + 3\widehat k)$ and $\overrightarrow a \,.\,(\widehat i + \widehat j + 2\widehat k) = 2$, then ${(\alpha - \beta + \gamma )^2}$ equals ____________.