
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
For a positive integer \( n \) and nonzero digits \( a \), \( b \), and \( c \), let \( A_n \) be the \( n \)-digit integer each of whose digits is equal to \( a \); let \( B_n \) be the \( n \)-digit integer each of whose digits is equal to \( b \), and let \( C_n \) be the \( 2n \)-digit (not \( n \)-digit) integer each of whose digits is equal to \( c \). What is the greatest possible value of \( a + b + c \) for which there are at least two values of \( n \) such that \( C_n - B_n = 2A_n^2 \)?

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