Problem 3. Let $n \ge 3$ be an integer. There are $n$ colored lamps arranged in a circle. Pressing the button of a lamp once changes its color as follows:
$$ \text{green} \rightarrow \text{red}, \text{red} \rightarrow \text{blue}, \text{blue} \rightarrow \text{green}. $$
Initially, all lamps are colored red. Aladdin makes moves on these lamps. Each move consists of the following three steps:
$\bullet$ he chooses a lamp $L$, without pressing its button;
$\bullet$ he presses the button of the clockwise neighbor of $L$ once;
$\bullet$ he presses the button of the counterclockwise neighbor of $L$ twice.
For each $n$, determine the maximum possible number of lamps that are colored green simultaneously, after finitely many moves.
