A $10$-bus power system consists of four generator buses indexed as $G1, G2, G3, G4$ and six load buses indexed as $L1, L2, L3, L4, L5, L6$. The generator-bus $G1$ is considered as slack bus, and the load buses $L3$ and $L4$ are voltage controlled buses. The generator at bus $G2$ cannot supply the required reactive power demand, and hence it is operating at its maximum reactive power limit. The number of non-linear equations required for solving the load flow problem using Newton - Raphson method in polar form is ___________.