Vectors - Areas of Triangles and Parallelograms Using Vectors

Relative to the origin \(O\), the position vectors of \(A\), \(B\) and \(C\)  are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively where \(\mathbf{c}=5\mathbf{a}-\mathbf{b}\). Given that \(\mathbf{a}\) is a unit vector, \(\left| \mathbf{b} \right|=\sqrt{3}\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{1}{3}\pi \), find the exact area of triangle \(ABC\).