
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 a circle $C$ with a radius of $\sqrt{2}$, its center lies on the ray $y = -2x (x < 0)$ and is tangent to the line $x + y + 1 = 0$.

(1) Find the equation of circle $C$.
(2) From a point $P(x_0, y_0)$ outside the circle $C$, a tangent line $PM$ is drawn to the circle at point $M$, with $O$ as the origin of the coordinate and $|PM| = |PO|$. Find the minimum area of triangle $PMC$ and the coordinates of point $P$ in this case.

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