
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 $C: \frac{x^2}{a^2} + \frac{y^2}{4} = 1 (a > 2)$, the line $l: y=kx+1 (k\neq 0)$ intersects ellipse $C$ at points $A$ and $B$, and $D$ is the midpoint of $AB$.
$(1)$ If the product of the slopes of line $l$ and line $OD$ (where $O$ is the origin) is $-\frac{1}{2}$, find the equation of the ellipse $C$.
$(2)$ Under the condition of $(1)$, determine whether there exists a fixed point $M$ on the y-axis such that $\angle AMO=\angle BMO$ (where $O$ is the origin) when $k$ varies. If it does exist, find the coordinates of point $M$; if not, explain why.

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