
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
1. Given that the center of circle C is the intersection point of the line $x - y + 1 = 0$ and the x-axis, and it is tangent to the line $x + y + 3 = 0$, find the standard equation of circle C.
2. Given circle C: $x^2 + (y - 3)^2 = 4$, and line l passes through point A($-1$, $0$) and intersects circle C at points P and Q. If $|PQ| = 2\sqrt{3}$, find the equation of line l.

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