
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \( C \) denote the set of points \( (x, y) \in \mathbb{R}^{2} \) such that \( x^{2} + y^{2} \leq 1 \). A sequence \( A_{i} = (x_{i}, y_{i}) \) (for \( i \geq 0 \)) of points in \( \mathbb{R}^{2} \) is 'centric' if it satisfies the following properties:
- \( A_{0} = (0,0) \) and \( A_{1} = (1,0) \).
- 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 \( A_{2012} = (x, y) \).

Present the answer in LaTex format: \boxed{Your answer}
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