HMMT 2012 G10

Let $C$ denote the set of points $(x, y) \in \mathbb{R}^{2}$ such that $x^{2}+y^{2} \leq 1$. A sequence $A_{i}=\left(x_{i}, y_{i}\right) \mid i \geq 0$ of points in $\mathbb{R}^{2}$ is 'centric' if it satisfies the following properties:

• $A_{0}=\left(x_{0}, y_{0}\right)=(0,0), A_{1}=\left(x_{1}, y_{1}\right)=(1,0)$.
• For all $n \geq 0$, the circumcenter of triangle $A_{n} A_{n+1} A_{n+2}$ lies in $C$.

Let $K$ be the maximum value of $x_{2012}^{2}+y_{2012}^{2}$ over all centric sequences. Find all points $(x, y)$ such that $x^{2}+y^{2}=K$ and there exists a centric sequence such that $A_{2012}=(x, y)$.