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

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