
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]


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Given the following propositions:  
① There does not exist real numbers $a$, $b$ such that the domain and range of $f(x) = \lg(x^2 + bx + c)$ are all real numbers;  
② The graph of the function $y = f(x + 2)$ is symmetric to the graph of the function $y = f(2 - x)$ about the line $x = 2$;  
③ The equation $\ln x + x = 4$ has exactly one real root;  
④ $a = -1$ is a sufficient but not necessary condition for the equation $a^2x^2 + (a + 2)y^2 + 2ax + a = 0$ to represent a circle. Among these, the true propositions are numbered as follows.

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