
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 the left focus $F(-2,0)$ and eccentricity $\frac{\sqrt{6}}{3}$.<br/>$(Ⅰ)$ Find the standard equation of the ellipse $C$;<br/>$(Ⅱ)$ Let $O$ be the origin, $T$ be a point on the line $x=-3$, and a perpendicular line from $F$ intersecting the ellipse at $P$ and $Q$. When the quadrilateral $OPTQ$ is a parallelogram, find the area of the quadrilateral $OPTQ$.

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