
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
Let \( p \) be a prime number and \( J \) be a \( 2 \times 2 \) matrix \(\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\) with elements belonging to the set \(\{0, 1, 2, \ldots, p-1\}\), satisfying the following congruences:

\[
\begin{array}{ll}
a+d \equiv 1 & (\bmod p), \\
ad - bc \equiv 0 & (\bmod p).
\end{array}
\]

Determine the number of such matrices \( J \).

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