
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
The domain of the function $f(x)$ is $A$. 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 statements:

① The function $f(x) = x^2$ ($x \in \mathbb{R}$) is injective;

② If $f(x)$ is injective, and $x_1, x_2 \in A$ with $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$;

③ If $f: A \rightarrow B$ is injective, then for any $b \in B$, it has at most one pre-image;

④ If the function $f(x)$ is monotonic in some interval, then $f(x)$ is necessarily injective.

The true statements are __________. (Write down the numbers of all true statements)

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