
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 $3n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $2n$-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 $c$. What is the greatest possible value of $a + b + c$ for which there is at least one value of $n$ such that $C_n - B_n = A_n^2$?

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