
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 $C:\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ ($b\gt 0$) with its right focus at $F$, two lines passing through $F$ and tangent to the circle $O:{{x}^{2}}+{{y}^{2}}={{r}^{2}}\left( r\gt 0 \right)$ 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 $\sqrt{3}+1$.

1. Find the equations of ellipse $C$ and circle $O.
2. If a line $l:y=kx+m$ (where $k\lt {}0$ and $m\gt 0$) does not pass through point $F$ and 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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