
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 line passing through the fixed point $T\left(t,0\right)$ intersects the ellipse at points $P$ and $Q$, where $t\in \left(0,a\right)$.<br/>$(1)$ If the length of the minor axis of the ellipse is $2\sqrt{3}$ and it passes through the point $(-1$,$\frac{3}{2})$, find the equation of the ellipse;<br/>$(2)$ For the ellipse in $(1)$, if $t=\sqrt{3}$, find the maximum area of $\triangle OPQ$;<br/>$(3)$ Does there exist a point $S\left(s,0\right)$ on the $x$-axis such that $\angle PST=\angle QST$? If yes, find the relationship between $s$ and $t$; if not, explain the reason.

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