
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 the line $l: kx-y-k+1=0$ (where $k$ is a constant) and the circle $C: \left(x-1\right)^{2}+y^{2}=4$, the following four conclusions are provided:<br/>① When $k$ changes, the line $l$ always passes through the fixed point $\left(-1,1\right)$;<br/>② The line $l$ and the circle $C$ may have no common points;<br/>③ If the line $l$ intersects the circle $C$ at two distinct points $M$ and $N$, then the minimum length of segment $MN$ is $2\sqrt{3}$;<br/>④ For any real number $k$, there are no two points symmetric about the line $l$ on the circle $C.<br/>The correct conclusions are ____ (Write down the numbers of all correct conclusions).

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