
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
Given a function $f(x)$, if for any $x_1, x_2$ within its domain, when $f(x_1) = f(x_2)$ it always implies $x_1 = x_2$, then $f(x)$ is called an injective function. For example, the function $f(x) = x$ is injective, but the function $f(x) = x^2$ is not injective. Consider the following propositions:  
① The function $$f(x) = \begin{cases} 
\log_{2}x & \text{for } x \geq 2, \\
x - 1 & \text{for } x < 2 
\end{cases}$$ is injective;  
② For $a > -2$, the function $$f(x) = \frac {x^2 + ax + 1}{x}$$ is injective on the interval $(0, +\infty)$;  
③ If $f(x)$ is an injective function within its domain and $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$;  
④ If $f(x)$ is an injective function and is differentiable within its domain, then there definitely exists an $x_0$ within its domain such that $f'(x_0) = 0$.  
Indicate the correct propositions by filling in their numbers.

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