Vectors \(\mathbf{i}\) and \(\mathbf{j}\) are vectors parallel to the \(x\)-axis and \(y\)-axis respectively.

Given that

\(\mathbf{a}=2\mathbf{i}+3\mathbf{j},\qquad \mathbf{b}=\mathbf{i}-5\mathbf{j},\qquad \mathbf{c}=3\mathbf{i}+11\mathbf{j},\)

find

(i) the exact value of \(|\mathbf{a}+\mathbf{c}|\),

(ii) the value of \(m\) such that \(\mathbf{a}+m\mathbf{b}\) is parallel to \(\mathbf{j}\),

(iii) the value of \(n\) such that \(n\mathbf{a}-\mathbf{b}=\mathbf{c}\).