
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ with left and right foci at ${F}_{1}(-\sqrt{3},0)$, ${F}_{2}(\sqrt{3},0)$, point $P$ lies on $C$, and $cos∠{F}_{1}P{F}_{2}=-\frac{1}{3}$, the area of $\triangle F_{1}PF_{2}$ is $\sqrt{2}$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ Let $O$ be the origin, if the line $l$ is tangent to $C$ at point $Q$, $OT$ is perpendicular to $l$ with foot at point $T$, find the maximum value of $|QT|$.

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