
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 $f(x)$ is an increasing function defined on the set of real numbers $\mathbb{R}$ and $f(1)=0$. The function $g(x)$ is increasing on $(-\infty, 1]$ and decreasing on $[1, +\infty)$, and $g(4) = g(0) = 0$. Find the set $\{x \mid f(x)g(x) \geq 0\}$.

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