
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 a function $f(x)$ defined on $\mathbb{R}$ satisfies $x_1f(x_1) + x_2f(x_2) > x_1f(x_2) + x_2f(x_1)$ for any $x_1 \neq x_2$, then the function is called an "H function". Given the following functions:  
① $f(x) = 3x + 1$  
② $f(x) = \left(\frac{1}{2}\right)^{x+1}$  
③ $f(x) = x^2 + 1$  
④ $f(x) = \begin{cases} - \frac{1}{x}, & x < -1 \\ x^2 + 4x + 5, & x \geq -1 \end{cases}$  
Identify which of the above functions are "H functions". (Fill in the sequence number)

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