
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
Determine the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{d} & \frac{1}{c} \\ \frac{1}{b} & \frac{1}{a} \end{pmatrix} \renewcommand{\arraystretch}{1}.
\]

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