For $k \ge 3$, $P(k, n)$ denotes the number of dots that create a regular $k$-gon with side length $n$. For example, $P(3, 2) = 3$, $P(4, 3) = 9$, $P(5, 4) = 22$. (See the following figure shamelessly stolen from IGMO2022 R2 P1.) Is it possible to find some integers $k \ge 3$, $n_3, n_4, ..., n_k$ all greater than or equal to $2$, such that the following equation hold? $$\frac{1}{P(3, n_3)}-\frac{1}{P(4, n_4)}+\frac{1}{P(5, n_5)}- ... + (-1)^{k+1} \frac{1}{P(k, n_k)}=\frac{1}{2022}$$ (by XDitto#0165)