
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
In the Cartesian coordinate plane $(xOy)$, point $B$ is symmetric to point $A(-1, 1)$ about the origin $O$. $P$ is a moving point such that the product of the slopes of lines $AP$ and $BP$ is equal to $-\frac{1}{3}$.

(I) Find the trajectory equation of the moving point $P$;

(II) Assuming that lines $AP$ and $BP$ intersect lines $x = 3$ at points $M$ and $N$ respectively, does there exist a point $P(x_0, y_0)$ such that the areas of triangles $PAB$ and $PMN$ are equal? If so, find the x-coordinate of point $P$ ($x_0$); if not, explain the reason.

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