
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
S is the set of all ( $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$ ) where  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  are integers such that  $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$ . Find the largest  $k$  which divides abcdef for all members of  $S$ .

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