
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 the following propositions:
1. If for any $x_1, x_2 \in \mathbb{R}$ with $x_1 \neq x_2$, the function $f(x)$ satisfies $(x_1 - x_2)[f(x_1) - f(x_2)] < 0$, then $f(x)$ is a decreasing function on $\mathbb{R}$.
2. If for any $x \in \mathbb{R}$, the function $f(x)$ satisfies $f(x) = -f(2+x)$, then $f(x)$ is a periodic function.
3. The graph of the function $y = f(x)$ cannot coincide with the graph of the function $y = f(x+1) - 2$.
4. For any real number $x$, if $f(-x) = -f(x)$, $g(-x) = g(x)$, and when $x > 0$, $f'(x) > 0$ and $g'(x) > 0$, then for $x < 0$, $f'(x) > g'(x)$.

Indicate the correct propositions by filling in the correct proposition numbers.

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