
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 $A\left(-\sqrt{2},0\right)$ and $B\left(\sqrt{2},0\right)$ on a plane is a constant value $\left(-\frac{1}{2}\right)$.

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

(2) Suppose line $l:y=kx+1$ intersects 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}
user