
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
How many triples \((x, y, z) \in \mathbb{R}^{3}\) satisfy the following system of equations:
\[ 
\begin{array}{l} 
x = 2018 - 2019 \cdot \operatorname{sign}(y + z) \\
y = 2018 - 2019 \cdot \operatorname{sign}(z + x) \\
z = 2018 - 2019 \cdot \operatorname{sign}(x + y)
\end{array} 
\]
where for any real number \(\alpha\), \(\operatorname{sign}(\alpha)\) is defined as:
\[ 
\operatorname{sign}(\alpha) = 
\begin{cases} 
+1 & \text{if } \alpha > 0 \\
0 & \text{if } \alpha = 0 \\
-1 & \text{if } \alpha < 0 
\end{cases} 
\]

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