
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
Let the mean of the samples $x_{1}$, $x_{2}$, …, $x_{m}$ be $\overline{x}$, and the mean of the samples $y_{1}$, $y_{2}$, …, $y_{n}$ be $\overline{y}$ ($\overline{x} \neq \overline{y}$). If the mean of the combined samples $x_{1}$, $x_{2}$, …, $x_{m}$, $y_{1}$, $y_{2}$, …, $y_{n}$ is $\overline{z} = \frac{1}{4}\overline{x} + \frac{3}{4}\overline{y}$, then calculate the value of $\frac{m}{n}$.

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