
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 of four-digit integers has the property such that each term's thousands, hundreds, and tens digits respectively become the hundreds, tens, and units digits of the next term in the sequence, and this pattern holds for all terms. The sequence wraps around so that the relevant digits of the last term match back to the start. If $S$ denotes the sum of all terms in the sequence, find the largest prime factor that always divides $S$.
A) 11
B) 31
C) 101
D) 37
E) 97

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