
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
The matrix
\[\begin{pmatrix} a & 4 \\ -9 & d \end{pmatrix}\]
is its own inverse, for some real numbers $a$ and $d.$ Find the number of possible pairs $(a, d)$.

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