
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
Define the functions $f(x)$ and $g(x)$ on $\mathbb{R}$ such that for any $x$, $f(x) + f(-x) = 0$ and $g(x) = g(x+4)$. Given that $f(-2) = g(-2) = 6$ and $f(f(2) + g(2)) + g(f(-2) + g(-2)) = -2 + 2g(4)$, find the value of $g(0)$.

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