<Question>
The sum of the digits of a two-digit number is $13.$ The difference between the number and the number with its digits reversed is $27.$ What is the sum of the original number and the number with its digits reversed?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find a two-digit number where the sum of its digits is 13. Then, when I subtract the reversed number from the original, the difference is 27. Finally, I have to find the sum of the original number and the reversed number. Hmm, let's break this down step by step.

First, let me represent the original two-digit number. Let's say the tens digit is 'a' and the units digit is 'b'. So, the original number can be written as 10a + b. When we reverse the digits, the tens digit becomes 'b' and the units digit becomes 'a', so the reversed number is 10b + a.

<prompt_1.1> Set up the equations for the original number, reversed number, and their difference using the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The problem states that the sum of the digits is 13, so that gives me the equation a + b = 13. Got that. Then, the difference between the original number and the reversed number is 27. So, subtracting the reversed number from the original number: (10a + b) - (10b + a) = 27. Let me simplify that equation. 

Expanding the subtraction: 10a + b - 10b - a = 27. Combine like terms: (10a - a) + (b - 10b) = 27, which simplifies to 9a - 9b = 27. I can factor out the 9: 9(a - b) = 27. Dividing both sides by 9 gives a - b = 3. So now I have two equations:

1. a + b = 13
2. a - b = 3

I need to solve these two equations simultaneously. Let's add them together. Adding the left sides: (a + b) + (a - b) = 13 + 3. That simplifies to 2a = 16, so a = 8. Then, substituting a = 8 into the first equation, 8 + b = 13, so b = 5. Therefore, the original number is 10a + b = 10*8 + 5 = 85. The reversed number is 10b + a = 10*5 + 8 = 58.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, the problem asks for the sum of the original number and the reversed number. So, 85 + 58. Let me calculate that.
85 + 58 is 143.

Wait, let me double-check everything to make sure I didn't make a mistake.

<prompt_2.1> Verify the solution by checking the sum of the digits, the difference between the original and reversed numbers, and their sum. </prompt_2.1>

But let me think again—could there be another possible two-digit number that fits the criteria? Let's see. Since a and b are digits, they must be between 0 and 9. Given that a + b = 13, the possible pairs (a, b) are (4,9), (5,8), (6,7), (7,6), (8,5), (9,4).

<prompt_2.2> Check if there are other possible two-digit numbers that satisfy the given conditions. </prompt_2.2>

So, the answer should be 143.
I think that's correct. Let me just verify once more with the numbers. Original:85, reversed:58. 85 -58=27.

<prompt_2.3> Perform a final verification of the solution using the original and reversed numbers. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
First, the sum of the digits: 8 + 5 = 13. That's correct. Then, the difference: 85 - 58 = 27. Yes, that's right. And the sum of the original and reversed numbers: 85 + 58 = 143.
</execution_2.1>

<execution_2.2>
Let's check which of these pairs satisfy a - b = 3. 

For (8,5): a - b = 8 -5 =3. That works. The other pairs: (9,4) would give 9-4=5, which is not 3. (7,6) gives 1, (6,7) gives -1, (5,8) gives -3, (4,9) gives -5. So only (8,5) works. Therefore, 85 is the original number, 58 reversed. Sum is 85+58=143.
</execution_2.2>

<execution_2.3>
Yes. 85+58=143. Correct.
</execution_2.3>
</Execution_2>

<Final_answer>
Therefore, the sum of the original number and the number with its digits reversed is $\boxed{143}$.
</Final_answer>
</think>