
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 finite sequence of three-digit integers has the property that for each term, its units digit is the hundreds digit of the next term, and this pattern loops back from the last term to the first term. For instance, a sequence might start with 312, move to 256, then to 639, and finally loops back with 392. Let $S$ be the sum of all terms in the sequence. What is the largest prime factor that always divides $S$?
A) 3
B) 11
C) 13
D) 37
E) 43

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