
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 $f(x)$ and an even function $g(x)$ (where $g(x) \neq 0$) defined on $\mathbb{R}$, when $x < 0$, it satisfies $f'(x)g(x) > g'(x)f(x)$, and $f(-3)=0$, solve the inequality $f(x)g(x) \leqslant 0$.

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