
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
Let function $f(x)$ be defined on $\mathbb{R}$, satisfying $f(1 + x) = f(1 - x)$. Also, let $g_1(x) = f(x + 3)$, and $g_2(x) = f(3 - x)$. Consider the following four statements:

1. The graph of $f(x)$ is symmetrical about the line $x = 1$, and the graphs of $g_1(x)$ and $g_2(x)$ are symmetrical about the line $x = 3$.
2. The graph of $f(x)$ is symmetrical about the line $x = 1$, and the graphs of $g_1(x)$ and $g_2(x)$ are symmetrical about the line $x = 0$.
3. The function $f(x)$ has a period of 4, and both $g_1(x)$ and $g_2(x)$ have periods of 2.
4. The graph of $f(x)$ is symmetrical about the line $x = 2$, and the graphs of $g_1(x)$ and $g_2(x)$ are symmetrical about the line $x = 3$.

Provide the correct proposition numbers.

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