
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 $C$: $\frac{{x}^{2}}{3}+\frac{{y}^{2}}{{b}^{2}}=1\left(b \gt 0\right)$ with the right focus $F$, two lines passing through $F$ are tangent to the circle $O$: $x^{2}+y^{2}=r^{2}(r \gt 0)$ at points $A$ and $B$, forming a right triangle $\triangle ABF$. It is also known that the maximum distance between a point on ellipse $C$ and a point on circle $O$ is $\sqrt{3}+1$. <br/>$(1)$ Find the equations of ellipse $C$ and circle $O$; <br/>$(2)$ If a line $l: y=kx+m$ ($k \lt 0, m \gt 0$) that does not pass through point $F$ is tangent to circle $O$ and intersects ellipse $C$ at points $P$ and $Q$, find the perimeter of $\triangle FPQ$.

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