
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
\[
\begin{align*}
x &= 1000 - 1001 \operatorname{sign}(y + z - 1), \\
y &= 1000 - 1001 \operatorname{sign}(x + z + 2), \\
z &= 1000 - 1001 \operatorname{sign}(x + y - 3).
\end{align*}
\]

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