
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 point $A(-1,0)$ and circle $C: x^{2}+y^{2}-2x-2\sqrt{3}y+3=0$.
1. Find the length of the tangent line from point $A$ to circle $C$.
2. Line $l_{1}$ passes through point $A$ and intersects circle $C$ at points $P$ and $Q$, with $\overrightarrow {AP}=\overrightarrow {PQ}$. Find the slope $k$ of line $l_{1}$.
3. Points $M$ and $N$ lie on the line $l_{2}: x=1$. For any point $R$ on circle $C$, it satisfies $RN=\sqrt{3}RM$. Find the coordinates of points $M$ and $N$.

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