
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
Determine all the triples \((a, b, c)\) of positive real numbers such that the system
$$
\begin{aligned}
a x + b y - c z & = 0, \\
a \sqrt{1 - x^{2}} + b \sqrt{1 - y^{2}} - c \sqrt{1 - z^{2}} & = 0,
\end{aligned}
$$
is compatible in the set of real numbers, and then find all its real solutions.

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