
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 $\{a_n\}$: $a_1, a_2, \ldots, a_n$ ($0 \leq a_1 < a_2 \ldots < a_n$), when $n \geq 3$, it has property P: for any $i, j$ ($1 \leq i < j \leq n$), at least one of the two numbers $a_j + a_i$ and $a_j - a_i$ is an element of the sequence. Now, consider the following four propositions:  
① The sequence 0, 1, 3 has property P;  
② The sequence 0, 2, 4, 6 has property P;  
③ If the sequence $\{a_n\}$ has property P, then $a_1 = 0$;  
④ If the sequence $a_1, a_2, a_3$ ($0 \leq a_1 < a_2 < a_3$) has property P, then $a_1 + a_3 = 2a_2$.  
Among these, the correct propositions are numbered as follows. (List all correct proposition numbers)

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