
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 all triples \((x, y, z)\) of real numbers that are solutions to the system of equations

$$
\left.\begin{array}{l}
3 \cdot 2^{y}-1=2^{x}+2^{-x} \\
3 \cdot 2^{z}-1=2^{y}+2^{-y} \\
3 \cdot 2^{x}-1=2^{z}+2^{-z}
\end{array}\right\}
$$

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