
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
Let a finite sequence of four-digit integers, where the thousands and hundreds digits of each term are, respectively, the hundreds and tens digits of the next term. Additionally, the hundred and tens digits of the last term are, respectively, the thousands and hundreds digits of the first term, but each term is reduced by 1 from the successor. For instance, the sequence might begin with 3128, 1287, and 2874, and end with 8741. Let $T$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $T$?
A) 11
B) 37
C) 101
D) 403
E) 1111

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