Let $$\overrightarrow{a} = 2\widehat{i} + \widehat{j} - \widehat{k}$$ and $$\overrightarrow{b} = \widehat{i} + 2\widehat{j} + \widehat{k}$$ be two vectors. Consider a vector $$\overrightarrow{c} = \alpha \overrightarrow{a} + \beta \overrightarrow{b}, \alpha, \beta \in R.$$ If the projection of $$\overrightarrow{c}$$ on the vector $$\left(\overrightarrow{a} + \overrightarrow{b}\right)$$ is $$3\sqrt{2},$$ then the minimum value of $$\left(\overrightarrow{c} - \left(\overrightarrow{a} \times \overrightarrow{b}\right)\right) . \overrightarrow{c}$$ equals _________ .