
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]


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Given that the domain of the function $f(x)$ is $\mathbb{R}$, and $f(x)$ is not a constant function, consider the following propositions:

1. The function $g(x) = f(x) + f(-x)$ is always an even function;
2. If $f(x) + f(2 - x) = 0$ for any $x \in \mathbb{R}$, then $f(x)$ is a periodic function with a period of $2$;
3. If $f(x)$ is an odd function, and $f(x) + f(2 + x) = 0$ for any $x \in \mathbb{R}$, then the equation of the axis of symmetry of the graph of $f(x)$ is $x = 2n + 1 (n \in \mathbb{Z})$;
4. For any $x_1, x_2 \in \mathbb{R}$, and $x_1 \neq x_2$, if $\frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0$ always holds, then $f(x)$ is an increasing function on $\mathbb{R}$.

Find the serial numbers of all correct propositions.

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