
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
Split the numbers \(1, 2, \ldots, 10\) into two groups and let \(P_{1}\) be the product of the first group and \(P_{2}\) the product of the second group. If \(P_{1}\) is a multiple of \(P_{2}\), find the minimum value of \(\frac{P_{1}}{\P_{2}}\).

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