
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 circle $C$: $(x-2)^{2}+y^{2}=3$.  
(I) If a line $l$ passes through the fixed point $(-1,0)$ and has an inclination angle $\alpha=30^{\circ}$, and intersects circle $C$ at points $A$ and $B$, find the coordinates of the midpoint $P$ of segment $AB$;  
(II) From a point $P$ outside circle $C$, draw a tangent to circle $C$ at point $M$, with $O$ being the origin, and $|PM|=|PO|$, find the coordinates of point $P$ that make $|PM|$ the smallest.

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