
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 such that
\begin{align*}
x &= 3000 - 3001 \operatorname{sign}(y + z + 3), \\
y &= 3000 - 3001 \operatorname{sign}(x + z + 3), \\
z &= 3000 - 3001 \operatorname{sign}(x + y + 3).
\end{align*}
Note:
\[\operatorname{sign} (a) = \left\{
\begin{array}{cl}
1 & \text{if $a > 0$}, \\
0 & \text{if $a = 0$}, \\
-1 & \text{if $a < 0$}.
\end{array}
\right.\]

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