
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
Let $f(x)$ be a function defined on $\mathbb{R}$. If there exists a function $g(x) = ax + b$ (where $a$ and $b$ are constants) such that $f(x) \geq g(x)$ for all real numbers $x$, then $g(x)$ is called a supporting function of $f(x)$. Consider the following propositions:
1. The function $g(x) = -2$ is a supporting function of $f(x) = \begin{cases} \ln x & \text{for } x > 0 \\ 1 & \text{for } x \leq 0 \end{cases}$.
2. The function $g(x) = x - 1$ is a supporting function of $f(x) = x + \sin x$.
3. If the function $g(x) = ax$ is a supporting function of $f(x) = e^x$, then the range of $a$ is $[0, e]$.
4. A function $f(x)$ with the range $\mathbb{R}$ does not have a supporting function.

The number of correct propositions is ________.

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