Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ in $C$. Also, let $\{C_{11},C_{12}\}$ and $\{C_{21},C_{22}\}$ be the sets of two arbitrary tangent circles with radius $n$ in $C_1$ and $C_2$, respectively. Is there a finite number of points with size $k$ in the circle $C$ such that each $C_1$ and $C_2$ contains the odd number of points and each $C_{11}$, $C_{12}$, $C_{21}$ and $C_{22}$ contains the even number of points?

In the following I drew a time-fixed picture of the problem, since actually the circles can revolve in the original circles by the condition of being tangent in each time:

Motivation: Actually, one of my friends is working on the effects of critical points in the bounded area in the plane. He is engineer and do not like abstract mathematics. Based on his problem, I determine some special points in the bounded area and I give some properties to these points. After that, we define these points with their properties in the computer and we compute some special parameters of the phenomenon on that bounded area by some simple line integral and other mathematical tools. Actually, I helped him for all of this procedure. But in my private time, I abstracted this problem to the version which you can see.

In my opinion, for arbitrary $k$ the answer is no, and if these points exist, I think they have symmetry in the plane.

I think in general, the bellow claim is true:

Let $S$ be a set of points which is distributed in the circle $C$ such a way that any two tangent circles $C_1$ and $C_2$ in $C$ contain the even number of points of $S$. Then $S$ contains even number of points.

I appreciate any answer or helpful comment.