
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
Let $M(x_1, y_1)$ and $N(x_2, y_2)$ be two distinct points, and let line $l: ax+by+c=0$. Define $\delta= \frac{ax_1+by_1+c}{ax_2+by_2+c}$. Consider the following propositions:  
① Regardless of the value of $\delta$, point $N$ is not on line $l$;  
② If line $l$ perpendicularly bisects segment $MN$, then $\delta=1$;  
③ If $\delta=-1$, then line $l$ passes through the midpoint of segment $MN$;  
④ If $\delta>1$, then points $M$ and $N$ are on the same side of line $l$, and $l$ intersects the extension of segment $MN$.  
The correct propositions are \_\_\_\_\_\_ (list all correct proposition numbers).

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