
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 that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(x+2) = 1$, $f(x-4) - g(x) = 3$. If the graph of $y = f(x)$ is symmetric about the line $x=1$ and $f(-1) = 0$, there are four conclusions:<br/>① $g(1) = 1$;<br/>② $4$ is the period of $g(x)$;<br/>③ The graph of $g(x)$ is symmetric about $(4,-1)$;<br/>④ $g(2) = -1$. The correct conclusions are ______ (fill in the corresponding numbers).

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