Santa and an invisible elf play a hide and seek game in the Euclidean plane. Firstly, the elf chooses $3$ points, $A_1$, $A_2$ and $A_3$. These points are known to Santa. Also, we define $A_s = A_{s-3}$ for all $s \ge 4$. Then the elf chooses a point $P_0$ such that the distant between $P_0$ and $A_1$ is $100$. $P_0$ is his original position, and it is not known to Santa. In the beginning of round $n$, Santa chooses a number $\theta_n$ between $45$ to $90$, and then the elf will move to point $P_n$, which is defined as the point where $P_{n-1}$ is rotated $\theta^o_n$ anti-clockwise about $A_n$. The point is not known to Santa since the elf is invisible. Santa will then choose an area to scan using an elf detector. The detector can scan a circular area of radius of $1$. If the invisible elf is within the area of scanning (inside the circle or on the edge of the circle) of the detector, then Santa wins. Does there exist a winning strategy to ensure that Santa can win within $2022$ rounds? Prove your claim. Note : Assume the invisible elf is a point, i.e. he has no area.