
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 \( F(x) = \dfrac{f(x)}{g(x)} \) be an even function on \( (-\infty, 0) \cup (0, \infty) \), where when \( x < 0 \), \( f''(x)g(x) - f(x)g''(x) > 0 \), and \( f(2) = 0 \). Determine the solution set of the inequality \( F(x) < 0 \).

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