
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 a domain $A$. It is called an injective function if for any $x_1, x_2 \in A$, whenever $f(x_1) = f(x_2)$, it implies that $x_1 = x_2$. For example, the function $f(x) = 2x + 1$ is injective over its entire domain $\mathbb{R}$. Consider the following statements:

① The function $f(x) = x^2$ for all $x \in \mathbb{R}$ is injective.  
② If $f(x)$ is injective and for any $x_1, x_2 \in A$ with $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$.  
③ If $f: A \rightarrow B$ is an injective function, then for any $b \in B$, there is at most one pre-image.  
④ If the function $f(x)$ is monotonic on some interval, then $f(x)$ is necessarily injective.

Identify all true statements among these. (List the numbers of all true statements)

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