$\begin{array}{l} {a^{2} (b+c)=b^{2} (a+c)} \\ {a^{2} b-b^{2} a+a^{2} c-b^{2} c=0} \\ {\left(a-b\right)\left(ab+ac+bc\right)=0} \\ {ab+ac+bc=0} \\ {-b(a+c)=ac{\rm \; \; \; (*)}} \end{array}$ $\begin{array}{l} {b^{2} \left(a^{2} +c^{2} \right)+ac\left(a+c\right)} \\ {b^{2} \left(a^{2} +c^{2} \right)-b\left(a+c\right).\left(a+c\right)} \\ {-2abc} \end{array}$ $\begin{array}{l} {b^{2} (a+c)=2013} \\ {b.\left(-ac\right)=2013} \end{array}$ \[b^{2} \left(a^{2} +c^{2} \right)+ac\left(a+c\right)=4026\]