Let \(\vec {a} = 2\hat{i} - 5\hat{j} + 5\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k} \). If \( \vec{c} \) is a vector such that \( 2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0} \) and \( (\vec{a} - \vec{b}) \cdot \vec{c} = -97 \), then \( \left|\vec{c} \times \hat{k}\right|^{2} \) is equal to:
