Mathematics Question from JEE Main 2023 · 24 January, Shift 2

Let $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + \lambda \mathbf{k}, \, \mathbf{b} = 3\mathbf{i} - 5\mathbf{j} - \lambda \mathbf{k}, \, \mathbf{a} \cdot \mathbf{c} = 7, \, 2\mathbf{b} \cdot \mathbf{c} + 43 = 0, \, \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c}$. Then $|\mathbf{a} \cdot \mathbf{b}|$ is equal to: