
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 $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively. When $x < 0$, $f'(x)g(x)+f(x)g'(x) > 0$, and given that $g(-3)=0$, determine the solution set for the inequality $f(x)g(x) < 0$.

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