
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
A non-empty set $G$ with operation $\oplus$ satisfies: ① For any $a, b \in G$, $a \oplus b \in G$; ② There exists $e \in G$ such that for all $a \in G$, $a \oplus e = e \oplus a = a$, then $G$ with operation $\oplus$ is called a harmonic set. Now consider the following propositions:  
① $G=\{a+bi | a, b \text{ are even numbers}\}$, $\oplus$ is the multiplication of complex numbers, then $G$ is a harmonic set;  
② $G=\{\text{Quadratic trinomials}\}$, $\oplus$ is the addition of polynomials, then $G$ is not a harmonic set;  
③ If $\oplus$ is the addition of real numbers, $G \subseteq \mathbb{R}$ and $G$ is a harmonic set, then $G$ is either $\{0\}$ or an infinite set;  
④ If $\oplus$ is the multiplication of real numbers, $G \subseteq \mathbb{R}$ and $G$ is a harmonic set, then $G$ is either $\{0\}$ or an infinite set. The correct propositions are    .

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