
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 triples $(x, y, z)$ of real numbers that satisfy
\begin{align*}
x &= 2023 - 2024 \operatorname{sign}(y^2 - z^2), \\
y &= 2023 - 2024 \operatorname{sign}(x^2 - z^2), \\
z &= 2023 - 2024 \operatorname{sign}(x^2 - y^2).
\end{align*}
Note: For a real number $a$:
\[
\operatorname{sign}(a) = \left\{
\begin{array}{cl}
1 & \text{if } a > 0, \\
0 & \text{if } a = 0, \\
-1 & \text{if } a < 0.
\end{array}
\right.
\]

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