minimum value of vector expression If $\vec{a}$ and $\vec{b}$ are unit vectors and $\vec{c}$ satisfies  $$\vec{c}=-\vec{a}-(\vec{a} \cdot \vec{b})\vec{b} -\vec{a} \times\vec{b}$$ 
Then the minimum absolute value of  $(\vec{a} \times\vec{c} ) \cdot \vec{b}$ 
my attempt--
$(\vec{a} \times\vec{c} ) \cdot \vec{b}$ can be written down as  $\vec{a} \cdot(\vec{c}  \times \vec{b})$ and thus i calculated $\vec{c}  \times \vec{b}$ and then took the dot product with $\vec{a}$ but could not proceed to find out the minimum value.Please help me with this one.Thanks.

Hint:
$$(\vec{a} \times\vec{c} ) \cdot \vec{b} = -(\vec{c} \times\vec{a} ) \cdot \vec{b}
   = \vec{a}\cdot(\vec{a}\times\vec{b})+(\vec{a}\cdot\vec{b})\vec{b}\times(\vec{a}\times\vec{b}) + (\vec{a}\times\vec{b})\cdot(\vec{a}\times\vec{b}).$$
Simplify the first two terms.
