
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 $A$ denote the set of functions whose range is $R$, and $B$ denote the set of functions $\phi(x)$ that have the following property: for the function $\phi(x)$, there exists a positive number $M$ such that the range of $\phi(x)$ is contained in the interval $[-M,M]$. For example, when $\phi_{1}(x)=x^{3}$, $\phi_{2}(x)=\sin x$, then $\phi_{1}(x) \in A$, $\phi_{2}(x) \in B$. Consider the following propositions:  
$(1)$ For a function $f(x)$ with domain $D$, the necessary and sufficient condition for "$f(x) \in A$" is "for every $b \in R$, there exists $a \in D$ such that $f(a)=b$";  
$(2)$ The necessary and sufficient condition for a function $f(x) \in B$ is that $f(x)$ has both a maximum and a minimum value;  
$(3)$ If the functions $f(x)$ and $g(x)$ have the same domain, and $f(x) \in A$, $g(x) \in B$, then $f(x)+g(x) \notin B$;  
$(4)$ If the function $f(x)=a\ln (x+2)+ \frac{x}{x^{2}+1} (x > -2,a \in R)$ has a maximum value, then $f(x) \in B$.  
Among these propositions, the true ones are __________. (Write down the numbers of all true propositions)

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