
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 $\{a_{n}\}$ be an arithmetic sequence with all terms being integers, the first term $a_{1}=3$, and for any positive integer $n$, there always exists a positive integer $m$ such that $a_{1}+a_{2}+\cdots+a_{n}=a_{m}$. Then, which of the following statements about the common difference $d$ of this sequence are correct? Select the correct options:  
① The common difference $d$ can be $1$;  
② The common difference $d$ cannot be $1$;  
③ There are a finite number of common differences $d$ that satisfy the conditions;  
④ There are infinitely many common differences $d$ that satisfy the conditions.

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