
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 any  $n\in\mathbb N$ , denote by  $a_n$  the sum  $2+22+222+\cdots+22\ldots2$ , where the last summand consists of  $n$  digits of  $2$ . Determine the greatest  $n$  for which  $a_n$  contains exactly  $222$  digits of  $2$ .

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