
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 a sequence $\{a\_n\}$ with 4 terms, satisfying $a\_1 > a\_2 > a\_3 > a\_4 \geqslant 0$, for any $i, j$ ($1 \leqslant i \leqslant j \leqslant 4$ and $i,j \in \mathbb{N}^*$), $a\_i - a\_j$ is still a term in the sequence $\{a\_n\}$. Determine the correct proposition(s) in the following options:
1. The sequence $\{a\_n\}$ is definitely an arithmetic sequence.
2. There exist $1 \leqslant i < j \leqslant 4$ such that $i a\_i = j a\_j$.
3. The sequence $\{a\_n\}$ definitely contains a term equal to $0$.

Please write down the number(s) of the correct proposition(s).

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