
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 any function defined on a symmetric interval about the origin, it can be expressed as the sum (or difference) of an odd function and an even function. Let $f(x)$ be any function defined on $\mathbb{R}$, and let $F(x) = \frac{f(x) + f(-x)}{2}$ and $G(x) = \frac{f(x) - f(-x)}{2}$. Determine the parity (even or odd) of $F(x)$ and $G(x)$. Now, express the function $f(x) = \ln(e^x + 1)$ as the sum of an odd function $g(x)$ and an even function $h(x)$. Find $g(x) = \_\_\_\_\_\_\_\_$.

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