
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
Given point P(-4, 0) lies on circle O: $x^2 + y^2 = r^2$ ($r > 0$), and line $l$ intersects circle O at points A and B. Additionally, line $l$ intersects circle C: $(x + 1)^2 + (y + 1)^2 = 2$ at points M and N.
(1) Find the equation of circle O.
(2) If point M is the midpoint of segment AB, and $|PM| = |PN|$, find the equation of line $l$.

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