
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 integer each of whose digits is equal to  $c$ . Find the greatest possible value of  $a + b + c$  for which there are at least two values of  $n$  such that  $C_n - B_n = A_n^2$.

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