
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
For a function $f(x)$ defined on $\mathbb{R}$, if there exists a function $g(x)=ax+b$ ($a,b$ are constants), such that $f(x) \geqslant g(x)$ holds 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 $f(x)= \begin{cases} \ln x & ,x > 0 \\ 1 & ,x\leqslant 0 \end{cases}$;
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 $f(x)$ whose range is $\mathbb{R}$ 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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