
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 functions $f(x)$ and $g(x)$ have domains $D_{J}$ and $D_{E}$, respectively. Given that $D_{J} \subsetneq D_{E}$, if for any $x \in D_{J}$, $g(x) = f(x)$ holds, then $g(x)$ is called an extension of $f(x)$ on $D_{E}$. Assume $f(x) = x\ln{x}$ for $x > 0$, and $g(x)$ is an extension of $f(x)$ on $(-\infty, 0) \cup (0, +\infty)$, and $g(x)$ is an odd function, then $g(x) =$ ______. Assume $f(x) = 2^x - 1$ for $x \leq 0$, and $g(x)$ is an extension of $f(x)$ on $\mathbb{R}$, and $g(x)$ is an even function, then $g(x) =$ ______.

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