
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
In the plane rectangular coordinate system $xOy$, circle $O$ has its center at the origin and passes through point $M(1, \sqrt{3})$.  
$(Ⅰ)$ Find the equation of circle $O$;  
$(Ⅱ)$ If the line $\sqrt{3}x+y-2=0$ intersects circle $O$ at points $A$ and $B$, find the length of the chord $|AB|$.

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