
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
Find the number of ordered nonuples \( (a, b, c, d, e, f, g, h, i) \) of real numbers such that:
\[
\begin{pmatrix} 
a & b & c \\ 
d & e & f \\ 
g & h & i 
\end{pmatrix}^{-1} = 
\begin{pmatrix} 
\frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ 
\frac{1}{d} & \frac{1}{e} & \frac{1}{f} \\ 
\frac{1}{g} & \frac{1}{h} & \frac{1}{i} 
\end{pmatrix}.
\]

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