
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
Given a function $f(x)$ defined on $\mathbb{R}$, if for any $x\_1 \neq x\_2$, it satisfies $x\_1 f(x\_1) + x\_2 f(x\_2) > x\_1 f(x\_2) + x\_2 f(x\_1)$, then the function is called an "H function". Determine which of the following functions are "H functions".
\[
\begin{aligned}
&\text{(1)} \quad f(x) = 3x + 1 \\
&\text{(2)} \quad f(x) = \left(\frac{1}{2}\right)^{x+1} \\
&\text{(3)} \quad f(x) = x^2 + 1 \\
&\text{(4)} \quad f(x) =
\begin{cases}
-\frac{1}{x}, & x < -1 \\
x^2 + 4x + 5, & x \geq -1
\end{cases}
\end{aligned}
\]

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