
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 that the product of the slopes of the lines connecting a moving point $P$ to two fixed points $A(-\sqrt{2},0)$ and $B(\sqrt{2},0)$ on a plane is a constant value $(-\frac{1}{2})$.

(1) Find the trajectory equation of the moving point $P$.

(2) Let line $l: y = kx + 1$ intersect with curve $C$ at points $M$ and $N$. When $|MN| = \frac{4\sqrt{2}}{3}$, find the equation of line $l$.

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