
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
If \( x, y, z \) are real numbers satisfying
\[
x + \frac{1}{y} = 2y + \frac{2}{z} = 3z + \frac{3}{x} = k \quad \text{and} \quad xyz = 3,
\]
then \( k = \) .

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