
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 function $f(x)$ has domain $A$. If $x_1, x_2 \in A$ and $f(x_1) = f(x_2)$ always implies $x_1 = x_2$, then $f(x)$ is called an injective function. For example, the function $f(x) = 2x + 1$ ($x \in \mathbb{R}$) is injective. Consider the following propositions:
① The function $f(x) = x^2$ ($x \in \mathbb{R}$) is injective;
② If $f(x)$ is an injective function, and $x_1, x_2 \in A$ with $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$;
③ If $f: A \to B$ is an injective function, then for any $b \in B$, there is at most one element in $A$ that corresponds to $b$;
④ If the function $f(x)$ is monotonic on some interval, then $f(x)$ is necessarily an injective function.
Which of the propositions are correct? Write down all the correct proposition numbers.

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