Vector Algebra

Let \(\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k\), \(\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k\) and \(\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k\) be three vectors such that, \(\left| {\overrightarrow b \times \overrightarrow c } \right| = 5\sqrt 3\) and \({\overrightarrow a }\) is perpendicular to \({\overrightarrow b }\). Then the greatest amongst the values of \({\left| {\overrightarrow a } \right|^2}\) is _____________.