Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \\ \vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k} \) and \( \vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, \\ x \in \mathbb{R}. \) If \( \vec{d} \) is the unit vector in the direction of \( \vec{b} + \vec{c} \) such that \( \vec{a} \cdot \vec{d} = 1 \), then \( (\vec{a} \times \vec{b}) \cdot \vec{c} \) is equal to:
