
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passing through the point $(\sqrt{2},\sqrt{3})$, with eccentricity $\frac{\sqrt{2}}{2}$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Given point $A(0,-2)$, point $B$ on the ellipse $(B$ is different from the vertex of the ellipse$)$, $F_{2}$ is the right focus of the ellipse, point $M$ satisfies $3\overrightarrow{OM}=\overrightarrow{OF_{2}}$ (where $O$ is the origin), line $AB$ is tangent to the circle with center at $M$ at point $P$, and $P$ is the midpoint of $AB$. Find the slope of line $AB$.

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