
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
Consider a cyclic sequence of four-digit integers where each integer has the property that the hundreds, tens, and units digits respectively become the thousands, hundreds, and tens digits of the next term, and the tens and units digits of the last term are respectively the thousands and hundreds digits of the first term. If the sequence contains 4 terms, and each term is increased by 11 before summing, determine the largest prime factor that always divides the sum of these modified terms.
A) 11
B) 44
C) 101
D) 1001

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