
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 an odd function $f(x)$ defined on $\mathbb{R}$, with its derivative $f′(x)$, it always holds that $xf′(x) < f(-x)$ when $x \in (0, +\infty)$. If $g(x) = xf(x)$, determine the range of real numbers $x$ that satisfy $g(1) > g(1-2x)$.

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