
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 the following four propositions:  
① The necessary and sufficient condition for the function $f(x) = x|x| + bx + c$ to be an odd function is $c = 0$;  
② The inverse function of $y = 2^{-x}$ is $y = -\log_{2}x$;  
③ If the range of the function $f(x) = \lg(x^2 + ax - a)$ is $\mathbb{R}$, then $a \leq -4$ or $a \geq 0$;  
④ If the function $y = f(x - 1)$ is an even function, then the graph of the function $y = f(x)$ is symmetric about the line $x = 1$.  
Among these, the correct propositions are numbered as follows:

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