
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 circle P: $(x-1)^2 + y^2 = 8$, with center C, a moving circle passes through point M(-1,0) and is tangent to circle P.
(1) Find the trajectory equation of the moving circle's center;
(2) If line $y = kx + m$ intersects with the trajectory at points A and B, and $k_{OA} \cdot k_{OB} = -\frac{1}{2}$, determine whether the area of $\triangle AOB$ is a fixed value. If it is, find the fixed value; if not, explain why. (Note: O represents the origin)

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