
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
In the Cartesian coordinate system, points A(2, 0), B(0, 2), and C(-$\sqrt{3}$, -1) are given.
(1) Find the equation of the circle P passing through points A, B, and C;
(2) If a line l passes through the point (1, 1) and the chord it intercepts on circle P is 2$\sqrt{3}$ in length, find the equation of line l.

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