Vectors in Space · Superchallenge problem

Vectors $\mathbf a, \mathbf b, \mathbf c$ in space satisfy $\lVert\mathbf a\rVert + 2\lVert\mathbf b\rVert + 3\lVert\mathbf c\rVert = 18$. Find the largest possible value of the scalar triple product $\mathbf a \cdot (\mathbf b \times \mathbf c)$.