
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
Let \( n \) be the number
\[ 
\left(\frac{999999999 \ldots 999}{20069 \text{'s}}\right)^{2} - \left(\frac{666666666 \ldots 666}{20066 \text{'s}}\right).
\]
Find the remainder when \( n \) is divided by 11.

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