
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 the function $f(x)$ satisfies that for any two distinct real numbers $x_1, x_2 \in (a, b)$, it always 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 a "G" function on the interval $(a, b)$.

Consider the following propositions:

1. The function $f(x) = \sin x$ is a "G" function on $(0, \frac{\pi}{2})$.

2. The function $f(x) = \tan x$ is a "G" function on $(0, \frac{\pi}{2})$.

3. The function $f(x) = \begin{cases} 2^x, & x \geqslant 1 \\ 2x + 1, & x < 1 \end{cases}$ is a "G" function on $(-3, 6)$.

4. If the function $f(x) = \cos x$ is a "G" function on $(0, \frac{\pi}{2})$.

The correct propositions are          .

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