
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 \((a, b, c)\) of real numbers such that the equation \(|a x+b y+c z|+|b x+c y+a z|+|c x+a y+b z|=|x|+|y|+|z|\) holds for all real numbers \(x, y, z\).

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