
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
The function $f(x)=\frac{x}{1+|x|}$, where $x\in \mathbb{R}$, is given. Four conclusions are provided as follows:  
① The range of $f(x)$ is $(-1,1)$.  
② For any $x_{1}$, $x_{2}\in \mathbb{R}$ with $x_{1}\neq x_{2}$, $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} \gt 0$.  
③ For any $x_{1}$, $x_{2}\in (0,+\infty)$ with $x_{1}\neq x_{2}$, $\frac{f(x_{1})+f(x_{2})}{2} \gt f(\frac{x_{1}+x_{2}}{2})$.  
④ Define $f_{1}(x)=f(x)$, $f_{n+1}(x)=f(f_{n}(x))$, where $n\in \mathbb{N}^{*}$. Then $f_{10}(\frac{1}{2})=\frac{1}{12}$.  
The correct conclusions are those with the following numbers: ______.

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