
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]


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Given a function $f(x)$ defined on $\mathbb{R}$, if for any two distinct real numbers $x\_1$, $x\_2$, the inequality $x\_1f(x\_1)+x\_2f(x\_2) > x\_1f(x\_2)+x\_2f(x\_1)$ holds, then the function $f(x)$ is called an "H function". Determine the serial numbers of the following functions that are "H functions":

1. $y=-x^{3}+x+1$;
2. $y=3x-2(\sin x-\cos x)$;
3. $y=e^{x}+1$;
4. $f(x)= \begin{cases} \ln |x|,x\neq 0 \\ 0,x=0 \end{cases}$

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