
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 an ellipse $G:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and passing through the point $(2,\sqrt{2})$. <br/>$(Ⅰ)$ Find the equation of the ellipse $G$; <br/>$(Ⅱ)$ A line $l$ with slope $k (k\neq 0)$ passes through point $M(0,1)$ intersects the ellipse $G$ at points $A$ and $B$. Does there exist a point $N$ on the $y$-axis such that $\angle ANM=\angle BNM$ (point $N$ is not coincident with point $M$)? If yes, find the coordinates of point $N$; if not, explain why.

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