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

(Ⅰ) Try to find the trajectory equation $C$ of the moving point $P$.

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

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