
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 \(g(3) = 0\). Determine the solution set of the inequality \(f(x)g(x) < 0\).

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