
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>b>0)$ with two foci $F_{1}(-1,0)$ and $F_{2}(1,0)$, passing through point $A(\sqrt{2},-\frac{\sqrt{6}}{2})$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ If the left vertex of the ellipse is $B$, is there a point $P$ on the ellipse such that the area of $\triangle PBF_{2}$ is $\frac{3\sqrt{3}}{2}$? If yes, find the coordinates of point $P$; if not, explain the reason.

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