
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
The number \( A \) in decimal notation has the form \( A = \overline{7a631b} \), where \( a \) and \( b \) are non-zero digits. The number \( B \) is obtained by summing all the six-digit, distinct numbers derived from \( A \) by cyclic permutations of its digits (the first digit moves to the second place, the second to the third, and so on, the last digit moves to the first place). How many such numbers \( A \) exist, for which \( B \) is divisible by 121? Find the largest such \( A \).

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