
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
The following are statements of four propositions:

① The line $(3+m)x+4y-3+3m=0$ ($m\in\mathbb{R}$) always passes through the fixed point $(-3,3)$;

② For segment AB with endpoint B at $(3,4)$ and A moving on the circle $x^2+y^2=4$, the trajectory equation of the midpoint M of segment AB is $\left(x- \frac{3}{2}\right)^{2}+(y-2)^{2}=1$;

③ Given $M=\{(x,y)|y= \sqrt{1-x^{2}}\}$ and $N=\{(x,y)|y=x+b\}$, if $M\cap N \neq \emptyset$, then $b\in[-\sqrt{2}, \sqrt{2}]$;

④ Given the circle $C: (x-b)^{2}+(y-c)^{2}=a^{2}$ ($a>0, b>0, c>0$) intersects the x-axis and is separate from the y-axis, then the intersection point of the line $ax+by+c=0$ and the line $x+y+1=0$ is in the second quadrant.

Among these statements, the correct ones are __________ (Fill in all the correct conclusion numbers).

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