
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
The cross-sectional area of a plane passing through the diagonal \( BD_{1} \) of a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) is \( S \). Let \( S_{\max} \) and \( S_{\min} \) denote the maximum and minimum values of \( S \), respectively. Find the value of \( \frac{S_{\max}}{S_{\min}} \).

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