
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 four-digit integers has the property that the hundreds and tens digits of each term are, respectively, the thousands and hundreds digits of the next term, but the units digit repeats for all terms. The sequence cycles completely, such that the hundreds and tens digits of the last term are, respectively, the thousands and hundreds digits of the first term. Find the largest prime factor that always divides the sum of all the terms in the sequence.

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