
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 moving point P $(x, y)$ and two fixed points A $(-3, 0)$ and B $(3, 0)$. It is known that $\frac{|PA|}{|PB|} = 2$, where $|PA|$ and $|PB|$ represent the distances from the point P to points A and B, respectively.
(1) Find the equation of the trajectory $\Gamma$ of point P.
(2) A moving point Q lies on the line $y - x - 1 = 0$, and QM, QN are two tangents to the trajectory $\Gamma$, where M and N are the tangent points, and C is the center of trajectory $\Gamma$. Find the minimum area of quadrilateral OMCN and the equation of line MN at that moment.

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