
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 $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$ with its right focus $F\left(c,0\right)$ and $a \gt b \gt c \gt 0$. Let $D$ be one endpoint of the minor axis, and the distance from the origin $O$ to the line $DF$ is $\frac{\sqrt{3}}{2}$. A line passing through the origin and not coinciding with the $x$-axis intersects the ellipse $E$ at points $C$ and $G$, such that $|\overrightarrow{GF}|+|\overrightarrow{CF}|=4$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ Determine whether there exists a line $l$ passing through the point $P\left(2,1\right)$ intersecting the ellipse $E$ at two distinct points $A$ and $B$ such that $\overrightarrow{OP}^{2}=4\overrightarrow{PA}\cdot \overrightarrow{PB}$ holds. If it exists, 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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