
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 circle C passes through the fixed point (1, 0) and is tangent to the line $x = -1$.

(1) Find the equation of the trajectory of the center C of the circle;  
(2) Suppose a line $\ell$ passing through the fixed point M (-4, 0) intersects the trajectory of the center C at two points A and B, with the origin as O. Let $\angle xOA = \alpha$ and $\angle xOB = \beta$. Investigate whether $\alpha + \beta$ is a constant value. If it is, find the constant value; if not, explain why.

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