
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 \( S \) be the set of all ordered tuples \( (a, b, c, d, e, f) \) where \( a, b, c, d, e, f \) are integers and \( a^{2} + b^{2} + c^{2} + d^{2} + e^{2} = f^{2} \). Find the greatest \( k \) such that \( k \) divides \( a b c d e f \) for every element in \( S \).

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