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

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