
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 points A (-1, 0), B (1, 0), and lines AM, BM intersect at point M, with $k_{MA} \times k_{MB} = -2$.  
(1) Find the equation of the trajectory C of point M;  
(2) A line PQ passes through a fixed point F (0, 1) and intersects the curve C at points P and Q. Does the area of $\triangle OPQ$ have a maximum value? If it exists, find the maximum area of $\triangle OPQ$. If it does not exist, explain why.

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