The position vectors of the points \(A, B\) and \(C\) with respect to an origin \(O\) are the unit vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(OB\) is perpendicular to \(OC\), \(m = \cos \widehat{AOC}\), \(n = \cos \widehat{AOB}\), and \(m+n \neq 1\). The vector \(\mathbf{r}\), where \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\), is perpendicular to the plane \(ABC\). Show that

\[\lambda = \dfrac{(1-m)(1+m-n)}{1-m-n}\]

and find a similar expression for \(\mu\).

It is further given that \(\widehat{AOB}=\widehat{AOC}=120^\circ\). The point \(N\) is the foot of the perpendicular from \(O\) on the plane \(ABC\). Find the position vector of \(N\), and deduce that the position vector of the reflection of \(O\) in the plane \(ABC\) is

\[\dfrac{4}{5}\mathbf{a}+ \dfrac{3}{5}\mathbf{b} + \dfrac{3}{5}\mathbf{c}.\]