<Question>
If $0.\overline{1331}$ is written as a fraction $\frac{a}{b}$ with $a$ and $b$ being positive integers with a greatest common divisor of 1, what is $a+b$?
</Question>

<think>
<Plan_1>
Okay, so I need to convert the repeating decimal 0. \overline{1331} into a fraction and then find the sum of the numerator and denominator after simplifying it. Hmm, repeating decimals can sometimes be tricky, but I remember there's a method to do this. Let me try to recall the steps.

First, I know that for a repeating decimal like 0. \overline{xyz}, where "xyz" repeats indefinitely, the way to convert it into a fraction is to set it equal to a variable, multiply both sides by 10^n where n is the number of repeating digits, and then subtract the original equation to eliminate the repeating part. Let me try applying that here.

The repeating part here is "1331", which has four digits. So, let me denote x = 0. \overline{1331}. Then, to move the decimal point past the repeating part, I need to multiply x by 10,000 (since there are four digits).

<prompt_1.1> Set up the equation for the repeating decimal 0. \overline{1331} and plan the steps to convert it into a fraction. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, 10,000x = 1331. \overline{1331}. Then, if I subtract x from 10,000x, the repeating parts should cancel out. Let's write that out:

10,000x = 1331.13311331... 
- x = 0.13311331... 
-----------------------------
9,999x = 1331

So, 9,999x = 1331. Therefore, x = 1331 / 9999.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but now I need to check if this fraction can be simplified further. To do that, I have to find the greatest common divisor (GCD) of 1331 and 9999.

Let me start by factoring both numbers.

<prompt_2.1> Find the GCD of 1331 and 9999 by factoring both numbers. </prompt_2.1>
-----

Wait, but let me double-check my calculations to make sure I didn't make any mistakes. Let's verify the GCD again.

<prompt_2.2> Re-verify the GCD calculation of 1331 and 9999. </prompt_2.2>

Alternatively, maybe there's another way to factor these numbers. Let me try prime factorization.

<prompt_2.3> Perform prime factorization of 1331 and 9999 to confirm the GCD. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Starting with 1331. Hmm, I remember that 11^3 is 1331 because 11*11=121, 121*11=1331. Yes, so 1331 is 11 cubed. So, 1331 = 11^3.

Now, 9999. Let's see. 9999 is 10,000 - 1, which is 10^4 - 1. There's a formula for factoring numbers like that. 10^4 - 1 can be factored as (10^2)^2 - 1^2 = (10^2 - 1)(10^2 + 1) = 99 * 101. So, 9999 = 99 * 101. Then, 99 is 9*11, which is 3^2 * 11. So, putting it all together, 9999 = 3^2 * 11 * 101.

So, the prime factors of 1331 are 11^3, and the prime factors of 9999 are 3^2 * 11 * 101. The only common factor is 11. Therefore, the GCD of 1331 and 9999 is 11.
</execution_2.1>

<execution_2.2>
Using the Euclidean algorithm:

9999 divided by 1331 is 7 with a remainder. 1331*7 = 9317. 9999 - 9317 = 682.

Then, 1331 divided by 682. 682*1 = 682. 1331 - 682 = 649.

682 divided by 649 is 1 with a remainder of 33.

649 divided by 33. 33*19 = 627. 649 - 627 = 22.

33 divided by 22 is 1 with a remainder of 11.

22 divided by 11 is 2 with no remainder. So, the GCD is 11.
</execution_2.2>

<execution_2.3>
Starting with 1331. I know that 11^3 is 1331 because 11*11=121, 121*11=1331. So 1331 is 11^3.

Now, 9999. Let's factor 9999. Hmm, 9999. Let's divide by 9 first: 9999 ÷ 9 = 1111. So 9999 = 9 * 1111. Now, 1111. Let's check if 1111 is prime. Wait, 1111 ÷ 11 = 101, because 11*101=1111. So 9999 = 9 * 11 * 101. But 9 is 3^2, so overall, 9999 = 3^2 * 11 * 101.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Therefore, we can divide both the numerator and the denominator by 11 to simplify the fraction.

1331 ÷ 11 = 121 (since 11*121 = 1331).

9999 ÷ 11 = 909 (since 11*909 = 9999). Wait, let me check that multiplication: 11*900 = 9900, and 11*9 = 99, so 9900 + 99 = 9999. Yep, that's correct.

So, the simplified fraction is 121/909. Now, we need to check if 121 and 909 have any common factors.

<prompt_3.1> Check if the fraction 121/909 can be simplified further by finding the GCD. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Let's check. 121 is 11^2. Let's factor 909. 909 ÷ 9 = 101, so 909 = 9 * 101 = 3^2 * 101. So, 121 is 11^2 and 909 is 3^2 * 101. There are no common prime factors, so the GCD of 121 and 909 is 1. Therefore, the fraction 121/909 is in its simplest form.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, but hold on. Let me verify this again. Because 121 is 11 squared, and 909 divided by 11 would be 82.636...
So, 11 doesn't divide 909. Let me confirm: 11*82 = 902, 909 - 902 = 7. So, no, 11 isn't a factor of 909.

Therefore, a = 121 and b = 909.
Hence, a + b = 121 + 909 = 1030.

But wait, let me double-check the original steps to ensure I didn't make any mistakes.

<prompt_4.1> Re-evaluate the steps to ensure the fraction is correct. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Starting with x = 0. \overline{1331}, then 10,000x = 1331. \overline{1331}, subtracting gives 9,999x = 1331, so x = 1331/9999. Then, simplifying by dividing numerator and denominator by GCD 11, resulting in 121/909. Yes, that seems right. Then 121 + 909 is 1030.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But just to be absolutely thorough, let me verify the decimal conversion. Let's compute 121 divided by 909 and see if it equals 0. \overline{1331}.

Alternatively, maybe there's a mistake in the GCD calculation. Let me use the Euclidean algorithm again on 121 and 909.

<prompt_5.1> Re-verify the GCD of 121 and 909 using the Euclidean algorithm. </prompt_5.1>

Alternatively, maybe I made a mistake in the initial step. Let me think. If the repeating decimal is 0. \overline{1331}, then the number of repeating digits is 4, so we multiply by 10,000.

<prompt_5.2> Re-evaluate the initial steps and confirm the solution. </prompt_5.2>
</Plan_5>

<Execution_5>
<execution_5.1>
Applying the Euclidean algorithm:

909 divided by 121. 121*7 = 847. 909 - 847 = 62. So, remainder 62.

Then, 121 divided by 62. 62*1 = 62. 121 - 62 = 59. Remainder 59.

62 divided by 59. 59*1 = 59. 62 - 59 = 3. Remainder 3.

59 divided by 3. 3*19 = 57. 59 - 57 = 2. Remainder 2.

3 divided by 2. 2*1 = 2. 3 - 2 = 1. Remainder 1.

2 divided by 1. 1*2 = 2. Remainder 0. So, GCD is 1. Correct. Therefore, 121 and 909 are coprime.
</execution_5.1>

<execution_5.2>
Wait, but 1331 is a four-digit number? Wait, 1331 is four digits? Wait, 1-3-3-1, that's four digits. So, yes, the repeating block is four digits, so multiplying by 10,000 is correct. Then subtracting gives 9,999x = 1331, so x = 1331/9999. That seems right. Then simplifying by dividing numerator and denominator by 11 gives 121/909, which can't be simplified further. So, a + b = 121 + 909 = 1030.
</execution_5.2>
</Execution_5>

<Plan_6>
Based on execution_5.2:
I think that's the answer.

**Final Answer**
The value of \(a + b\) is \boxed{1030}\).
</Plan_6>

<Execution_6>
No prompt to execute.
</Execution_6>

<Final_answer>
Therefore, the final answer is $a+b = 121 + 909 = \boxed{1030}$.
</Final_answer>
</think>