
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 an odd function and an even function defined on \( \mathbf{R} \), respectively. When \( x < 0 \), \( F(x) = f(x) g(x) \) is an increasing function on \( (-\infty, 0) \), and \( g(2) = 0 \). 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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