
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 that two groups, A and B, are arranged in ascending order: Group A: $27$, $28$, $37$, $m$, $40$, $50$; Group B: $24$, $n$, $34$, $43$, $48$, $52$, if the 30th percentile and the 50th percentile of these two groups correspond to the same values, determine the ratio $\frac{m}{n}$.

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