
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 that $f(x)$ and $g(x)$ are odd functions, the solution set for $f(x)>0$ is $\left(a^2, b\right)$, and the solution set for $g(x)>0$ is $\left(\frac{a^2}{2}, \frac{b}{2}\right)$ (where $a$ and $b$ are constants, and $b > 2a^2$), find the solution set for $f(x) \cdot g(x)>0$.

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