
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
If for a function $f(x)$ defined on the interval $D$, for any two distinct real numbers $x_1, x_2$ within the interval $D$, it holds that $x_1f(x_1)+x_2f(x_2) > x_1f(x_2)+x_2f(x_1)$, then the function $f(x)$ is called an "H function" on the interval $D$. Given the following functions and their corresponding intervals:

1. $f(x)= \frac{1}{3}x^3- \frac{1}{2}x^2+ \frac{1}{2}x \quad (x \in \mathbb{R})$;
2. $f(x)=3x+\cos x-\sin x, \quad x \in \left(0, \frac{\pi}{2}\right)$;
3. $f(x)=(x+1)e^{-x}, \quad x \in (-\infty,1)$;
4. $f(x)=x\ln x, \quad x \in \left(0, \frac{1}{e}\right)$;

The sequence numbers of the functions that are "H functions" on their interval $D$ are __________. (Write down all correct sequence numbers)

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