
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with one focus at $F_{1}(-1,0)$ and the vertex at $B_{1}$, the distance from the origin $O$ to the line $B_{1}F_{1}$ is $\frac{\sqrt{3}}{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ If point $T$ lies on the circle $x^{2}+y^{2}=2$, point $A$ is the right vertex of the ellipse, does there exist a line $l$ passing through point $A$ that intersects the ellipse $C$ at point $B$ (different from point $A$), such that $\overrightarrow{OT}=\frac{\sqrt{14}}{7}(\overrightarrow{OA}+\overrightarrow{OB})$ holds? If yes, find the equation of the line $l$; if not, explain the reason.

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