A squirrel of mass $m$ climbs slowly on a thin straight vertical rod of length $L$. Mass of the rod is negligible as compared to that of the squirrel and size of the squirrel is negligible as compared to the length $l$ that it climbs on the rod. Due to weight of the squirrel, the rod bends at its lower end through an angle $\theta$ and due the elasticity of the material of the rod a restoring torque $C\theta$ is developed. If $C=2mgL$, find the maximum length the squirrel can climb on the rod.
I tried to solve this problem by balancing the torque of the weight of the squirrel and the restoring torqe
$$mgl\sin\theta=2mgL\theta$$ but this equation gives me $l\approx2L$. But as the length of the rod is $L$, so it means that squirrel would have climbed the whole rod.
But when the squirrel just started climbing the rod, restoring torque must have been $2mgL\theta$ and torque of squirrel's weight is $mgl\theta$ ($\sin\theta\approx\theta $ for small $\theta$). Which is clearly less than the restoring torque. So according to me the squirrel must not be able to climb the rod at all, which is not the case. So aren't the two things contradicting each other? What is the reason for this contradiction?