
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
A function $y=f(x)$ is defined on $(0,+\infty)$ with its inverse function $y=f^{-1}(x)$. If $g(x) = \begin{cases} f(x), & x > 0 \\ 3^x-1, & x \leqslant 0 \end{cases}$ is an odd function, find the value(s) of $x$ that satisfy $f^{-1}(x)=2$.

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