
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
Solve the following system of equations in real numbers:
\[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x  \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y  \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}.  \]

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