
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 set $A \subseteq \mathbb{R}$, for any $a, b, c \in A$, the operation "⊕" has the following properties:  
(1) $a \oplus b \in A$;  
(2) $a \oplus a = 0$;  
(3) $(a \oplus b) \oplus c = a \oplus c + b \oplus c + c$  
Given the following propositions:  
① $0 \in A$  
② If $1 \in A$, then $(1 \oplus 1) \oplus 1 = 0$;  
③ If $a \in A$, and $a \oplus 0 = a$, then $a = 0$;  
④ If $a, b, c \in A$, and $a \oplus 0 = a$, $a \oplus b = c \oplus b$, then $a = c$.  
Among these, the correct propositions are (Fill in the correct proposition numbers).

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