
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
Given real numbers \( x, y, z \) satisfy the following system of equations:
\[ 
\begin{cases} 
\log_2\left( xyz - 6 + \log_5 x \right) = 4, \\ 
\log_3\left( xyz - 6 + \log_5 y \right) = 3, \\ 
\log_4\left( xyz - 6 + \log_5 z \right) = 2. 
\end{cases} 
\]

Then, \(\left| \log_5 x \right| + \left| \log_5 y \right| + \left| \log_5 z \right| = \) ______

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