
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 $C$: $\frac{{x}^{2}}{2}+y^{2}=1$ with left and right foci $F_{1}$ and $F_{2}$, a line $l$ passing through point $A(0,2)$ intersects the ellipse $C$ at two distinct points $P$ and $Q$.  
$(1)$ If the line $l$ passes through $F_{2}$, find the perimeter of $\triangle F_{1}PQ$;  
$(2)$ If a circle with diameter $PQ$ passes through point $F_{2}$, find the equation of line $l$;  
$(3)$ If $\overrightarrow{AQ}=\lambda \overrightarrow{AP}$, find the range of real number $\lambda$.

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