Vectors - Perpendicular Vectors Using the Scalar Product

Relative to the origin \(O\), the position vectors of \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively where \(\mathbf{c}=3\mathbf{a}-2\mathbf{b}\).
Given that \(\mathbf{a}\) is unit vector, \(\left| \mathbf{b} \right|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{2}{3}\pi \), find the exact value of \(\lambda \) such that \(\mathbf{a}\) is perpendicular to \(3\mathbf{b}-4\mathbf{a}+\lambda \mathbf{c}\).[3]