
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's consider a function $f(x)$ defined on $\mathbb{R}$. If for any two unequal real numbers $x_1$ and $x_2$ where $x_1 < x_2$, the inequality $x_1f(x_1) + x_2f(x_2) > x_1f(x_2) + x_2f(x_1)$ holds, then function $f(x)$ is referred to as an "H function". Identify all indexes of the functions below that are "H functions":  
① $y = e^x + x$; ② $y = x^2$; ③ $y = 3x - \sin x$; ④ $$y = \begin{cases} \ln|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$$

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