Vectors in Space · Superchallenge problem

Unit vectors $\mathbf a, \mathbf b, \mathbf c$ are such that the angle between $\mathbf a$ and $\mathbf b$ is $\dfrac{\pi}{3}$ and the angle between $\mathbf b$ and $\mathbf c$ is $\dfrac{\pi}{4}$ (the angle between $\mathbf a$ and $\mathbf c$ is free). Find the largest possible volume of the parallelepiped spanned by $\mathbf a, \mathbf b, \mathbf c$.