Vectors in Space · Superchallenge problem

Vectors $\mathbf a$ and $\mathbf b$ have $\lVert\mathbf a\rVert = 2$, $\lVert\mathbf b\rVert = 3$ and $\mathbf a\cdot\mathbf b = 3$. Let $\mathbf n = \mathbf a \times \mathbf b$. Find the volume of the parallelepiped spanned by $\mathbf a$, $\mathbf n$ and $\mathbf b \times \mathbf n$.