
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 $A(-\sqrt{3}, 0)$ and $B(\sqrt{3}, 0)$, a moving point $P$ satisfies the condition that the product of the slopes of lines $PA$ and $PB$ is a constant $-\frac{2}{3}$. Determine the trajectory curve $C$ of the moving point $P$.
(1) Find the equation of curve $C$;
(2) Draw two lines perpendicular to each other through point $A$, intersecting curve $C$ at two distinct points $M$ and $N$. Determine whether line $MN$ passes through a fixed point. If it does, find the coordinates of the fixed point. If not, explain the reason.

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