We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbors (only one neighbor for $ i = 1$ or $ i = n$, two neighbors for other $ i$) are in the same state, then $ L_{i}$ is switched off; Ð otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on.

$ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.