
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 $f(x)$ defined on $\mathbb{R}$ is said to have a supporting function $g(x) = ax + b$ (where $a$ and $b$ are constants) if $f(x) \geq g(x)$ holds for all real numbers $x$. Consider the following propositions:

1. The function $g(x) = -2$ is a supporting function of $f(x) = \begin{cases} \ln x, & x > 0 \\ 1, & x \leq 0 \end{cases}$;
2. The function $g(x) = x - 1$ is a supporting function of $f(x) = x + \sin x$;
3. If $g(x) = ax$ is a supporting function of $f(x) = e^x$, then the range of $a$ is $[0, e]$;
4. A function $f(x)$ with a range of $\mathbb{R}$ does not have a supporting function.

The sequence numbers of all correct propositions are $\_\_\_\_\_\_\_\_$.

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