
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
A sequence ‘seq’ of three-digit integers has the property that each digit’s position shifts according to a cyclic permutation in the next term of the sequence. The hundreds, tens, and units digits get permuted cyclically in each step through a specific permutation (e.g., $(1 2 3)$ or $(1 3 2)$). Let $S$ be the sum of all the terms in ‘seq’. Determine the largest prime factor that always divides $S$ given any permutation of digits.
A) 3
B) 7
C) 13
D) 37
E) 43

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