
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 system $x0y$, given points $A(-\sqrt{2},0)$, $B(\sqrt{2},0)$, and $E$ as a moving point, the product of the slopes of line $EA$ and line $EB$ is $-\frac{1}{2}$.
- (Ⅰ) Find the equation of the trajectory $C$ of the moving point $E$.
- (Ⅱ) Suppose a line $l$ passing through point $F(1,0)$ intersects the curve $C$ at two distinct points $M$ and $N$. If point $P$ is on the $y$-axis, and $|PM|=|PN|$, find the range of the $y$-coordinate of point $P$.

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