
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
Definition: If the function $y=f(x)$ has a point $x_{0}(a < x_{0} < b)$ in the given interval $[a,b]$ within its domain, satisfying $f(x_{0})= \dfrac {f(b)-f(a)}{b-a}$, then the function $y=f(x)$ is called an "average value function" on $[a,b]$, and $x_{0}$ is its mean value point.  
For example, $y=|x|$ is an "average value function" on $[-2,2]$, and $0$ is its mean value point. Given the following propositions:  
$(1)$ The function $f(x)=\sin x-1$ is an "average value function" on $[-\pi,\pi]$;  
$(2)$ If $y=f(x)$ is an "average value function" on $[a,b]$, then its mean value point $x_{0}\leqslant \dfrac {a+b}{2}$;  
$(3)$ If the function $f(x)=x^{2}+mx-1$ is an "average value function" on $[-1,1]$, then the real number $m\in(-2,0)$;  
$(4)$ If $f(x)=\ln x$ is an "average value function" on the interval $[a,b](b > a\geqslant 1)$, and $x_{0}$ is one of its mean value points, then $\ln x_{0} < \dfrac {1}{ \sqrt {ab}}$.  
Among these propositions, the true ones are __________ (Write down the numbers of all true propositions).

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