
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
Observe the following operation:  
If \(\overline{\mathrm{abc}}\) is a three-digit number, then since \(\overline{\mathrm{abc}} = 100a + 10b + c = 99a + 9b + (a + b + c)\),  
Therefore, if \(a + b + c\) is divisible by 9, \(\overline{\mathrm{abc}}\) is divisible by 9.  
This conclusion can be generalized to any number of digits.

Using this conclusion, solve the following questions:
1. \(N\) is a 2011-digit number, and each digit is 2. Find the remainder when \(N\) is divided by 9.
2. \(N\) is an \(n\)-digit number, and each digit is 7. \(n\) leaves a remainder of 3 when divided by 9. Find the remainder when \(N\) is divided by 9.

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