
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 domain of $f(x)$ is $R$. If there exists a constant $M \gt 0$ such that $|f(x)|\leqslant M|x|$ holds for all real numbers $x$, then $f(x)$ is called an $F$ function. The following functions are given:<br/>①$f(x)=2x$;<br/>②$f(x)=x^{2}+1$;<br/>③$f(x)=\sqrt{2}(sinx+cosx)$;<br/>④$f(x)=\frac{x}{{{x^2}-x+1}}$;<br/>⑤$f(x)$ is an odd function defined on the set of real numbers $R$, and for all $x_{1}$, $x_{2}$, $|f(x_{1})-f(x_{2})|\leqslant 2|x_{1}-x_{2}|$. The functions that are $F$ functions are ____.

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