
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
Among the following propositions, the correct ones are \_\_\_\_\_\_ (fill in all the correct proposition numbers on the line):  
① If the sequence $\{a_n\}$ is an arithmetic sequence, and $a_m+a_n=a_s+a_t$ ($m, n, s, t \in \mathbb{N}^*$), then $m+n=s+t$;  
② If $S_n$ is the sum of the first $n$ terms of the arithmetic sequence $\{a_n\}$, then $S_n$, $S_{2n}-S_n$, $S_{3n}-S_{2n}$ form an arithmetic sequence;  
③ If $S_n$ is the sum of the first $n$ terms of the geometric sequence $\{a_n\}$, then $S_n$, $S_{2n}-S_n$, $S_{3n}-S_{2n}$ form a geometric sequence;  
④ If $S_n$ is the sum of the first $n$ terms of the geometric sequence $\{a_n\}$, and $S_n=Aq^n+B$; (where $A, B$ are non-zero constants, $n \in \mathbb{N}^*$), then $A+B=0$.

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