
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
Mathematical operation refers to the ability to solve mathematical problems based on clear operation objects and operation rules. Because of operations, the power of numbers is infinite; without operations, numbers are just symbols. Logarithmic operation and exponential operation are two important types of operations.  
$(1)$ Try to calculate the value of $\frac{{\log 3}}{{\log 4}}\left(\frac{{\log 8}}{{\log 9}+\frac{{\log 16}}{{\log 27}}\right)$ using the properties of logarithmic operations.  
$(2)$ Given that $x$, $y$, and $z$ are positive numbers, if $3^{x}=4^{y}=6^{z}$, find the value of $\frac{y}{z}-\frac{y}{x}$.

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