
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 odd function $f(x)$ defined on $\mathbb{R}$ is monotonically decreasing on $(-\infty, 0)$, and the even function $g(x)$ defined on $\mathbb{R}$ is monotonically increasing on $(-\infty, 0]$, with $f(1) = g(1) = 0$, determine the range of values of $x$ for which $f(x)g(x) > 0$.

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