
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
In the rectangular coordinate system $(xOy)$, an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a > b > 0)$ has one focus at $F_{1}(-\sqrt{3},0)$, and $M(1,y)\ (y > 0)$ is a point on the ellipse. The area of $\triangle MOF_{1}$ is $\frac{3}{4}$.
(1) Find the standard equation of the ellipse $(C)$;
(2) Does there exist a line $(l)$ passing through point $A(2,0)$ that intersects the ellipse $(C)$ at point $B$, such that $\overrightarrow{OT}=\frac{\sqrt{5}}{5}(\overrightarrow{OA}+\overrightarrow{OB})$? If yes, find the equation of line $(l)$; if not, explain the reason. Here, point $T$ lies on the circle $x^2+y^2=1$.

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