<Question>
Determine the modulo 4 remainder of the following sum: $$ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12. $$
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the remainder when the sum of numbers from 1 to 12 is divided by 4. Hmm, modulo 4 remainder. Alright, first step, maybe I should calculate the total sum of those numbers. Then once I have the total, I can divide it by 4 and find the remainder.

<prompt_1.1> Calculate the sum of numbers from 1 to 12 as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
How do I sum numbers from 1 to 12? Oh, right, there's a formula for the sum of the first n natural numbers. It's n(n + 1)/2. Let me check if that works here. So n is 12 in this case. Plugging into the formula: 12 times 13 divided by 2. Let me compute that. 12 times 13 is 156, and dividing by 2 gives 78. So the sum is 78.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to find 78 modulo 4. That means I need to divide 78 by 4 and find the remainder.

<prompt_2.1> Find the remainder when 78 is divided by 4 as per the above plan. </prompt_2.1>

Wait, but maybe I should verify this another way in case I made a mistake with the formula. Let me add the numbers manually to confirm the sum is indeed 78.

<prompt_2.2> Manually add the numbers from 1 to 12 to verify the sum as per the above plan. </prompt_2.2>

Alternatively, maybe there's another way to compute the modulo without calculating the entire sum. Since modulo 4 is concerned, perhaps I can find the remainder of each number from 1 to 12 when divided by 4, sum those remainders, and then take that total modulo 4. Let me try that method to cross-verify.

<prompt_2.3> Find the remainder of each number from 1 to 12 modulo 4 and sum them as per the above plan. </prompt_2.3>
-----

Just to make sure there's no mistake in the initial sum calculation, let's add the numbers step by step:

<prompt_2.4> Add the numbers step by step to verify the sum as per the above plan. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Let me do that division. 4 times 19 is 76, because 4 times 20 is 80, which is too big. So 4 times 19 is 76. Then 78 minus 76 is 2. So the remainder should be 2.
</execution_2.1>

<execution_2.2>
Let's see: 1+2 is 3, plus 3 is 6, plus 4 is 10, plus 5 is 15, plus 6 is 21, plus 7 is 28, plus 8 is 36, plus 9 is 45, plus 10 is 55, plus 11 is 66, plus 12 is 78. Yep, that matches. So the sum is definitely 78.
</execution_2.2>

<execution_2.3>
Let's list the numbers 1 to 12 and their remainders when divided by 4:

1 divided by 4 is 0 with remainder 1.

2 divided by 4 is 0 with remainder 2.

3 divided by 4 is 0 with remainder 3.

4 divided by 4 is 1 with remainder 0.

5 divided by 4 is 1 with remainder 1.

6 divided by 4 is 1 with remainder 2.

7 divided by 4 is 1 with remainder 3.

8 divided by 4 is 2 with remainder 0.

9 divided by 4 is 2 with remainder 1.

10 divided by 4 is 2 with remainder 2.

11 divided by 4 is 2 with remainder 3.

12 divided by 4 is 3 with remainder 0.

So listing the remainders: 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0.
</execution_2.3>

<execution_2.4>
1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 9 = 45

45 + 10 = 55

55 + 11 = 66

66 + 12 = 78.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.3:
Now, summing these remainders: Let's add them up step by step.

<prompt_3.1> Sum the remainders obtained in the previous step as per the above plan. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with 1 + 2 = 3

3 + 3 = 6

6 + 0 = 6

6 + 1 = 7

7 + 2 = 9

9 + 3 = 12

12 + 0 = 12

12 + 1 = 13

13 + 2 = 15

15 + 3 = 18

18 + 0 = 18.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, let me check that again. The remainders are: 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0. Let me group them:

<prompt_4.1> Recalculate the sum of remainders by grouping them as per the above plan. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
There are three full groups of 1,2,3. Each group sums to 1+2+3=6. There are four such groups? Wait, no. Wait, from the list: first three numbers 1,2,3; then next three 0,1,2; then next three 3,0,1; then last three 2,3,0. Wait, actually, the remainders are:

First four remainders: 1,2,3,0

Then next four:1,2,3,0

Then last four:1,2,3,0. Wait, no, original list is 12 numbers. Let me count:

1. 1

2. 2

3. 3

4. 0

5. 1

6. 2

7. 3

8. 0

9. 1

10. 2

11. 3

12. 0

So that's four groups of three: each group of four numbers. Each group of four numbers: first group 1,2,3,0; second group 1,2,3,0; third group 1,2,3,0. Wait, but 12 numbers make three groups of four. Wait, perhaps I should think of it as three cycles of 1,2,3,0. So each cycle sums to 1+2+3+0=6. There are three cycles, so 6*3=18. That's the total sum of remainders, which is 18.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Then, 18 modulo 4. Let's divide 18 by 4.

<prompt_5.1> Find the remainder when 18 is divided by 4 as per the above plan. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
4*4=16, remainder 2. So again, the remainder is 2.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
So both methods give the same result. Therefore, the remainder is 2. Hmm, so that's consistent.

<prompt_6.1> Compare the results from both methods and conclude as per the above plan. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
Wait, but let me think again if there's any other way this could go wrong. For example, did I correctly apply the modulo to each term? Let's see. For each number from 1 to 12, when divided by 4, the remainders are as I listed. Then summing those gives 18, which modulo 4 is 2. The other method, summing all numbers to 78, which modulo 4 is also 2. So both methods agree.
</execution_6.1>
</Execution_6>

<Final_answer>
Therefore, the modulo 4 remainder of the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12$ is $\boxed{2}$.
</Final_answer>
</think>