Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \cdots$).

Now we rotate $P$ clockwise above $A_1$ with angle $\frac{\pi}{n}$ we have points $P_1$, rotate $P_1$ clockwise above $A_2$ with angle $\frac{\pi}{n}$ we have points $P_2$ $,\cdots,$ rotate $P_k$ clockwise above $A_{k+1}$ with angle $\frac{\pi}{n}$ we have points $P_{k+1}$....

By my computation, I proposed a conjecture that:

Exist $k$ such that $P_{k+1}\equiv P$

My question: I am looking for a proof of the conjecture above and is the conjecture true for Euclidean space (and non-Euclidean geometries)?