<Question>
The expression $$1 + 2 + 3 - 4 + 5 + 6$$is equivalent to 13. If we add parentheses in various locations, we can change this value. For example, $1+2+3-(4+5)+6=1+2+3-9+6=6-9+6=3$. If we only add parentheses (and not rearrange the order of the terms), what is the minimum possible value this expression can obtain? (For the purposes of this problem, we cannot change addition or subtraction into multiplication. For example, $ (1+2+3)(-4)+5+6 $ is not a valid placement of parentheses.)

Enter your answer as a single value.
</Question>

<think>
<Plan_1>
Okay, so I need to figure out the minimum possible value for the expression 1 + 2 + 3 - 4 + 5 + 6 by adding parentheses in different places. The catch is that I can't rearrange the order of the terms, just insert parentheses to change the grouping of operations. The original expression without any parentheses is 1 + 2 + 3 - 4 + 5 + 6, which equals 13. The problem gives an example where adding parentheses can change the value: 1+2+3-(4+5)+6 equals 3. Now, my task is to find the minimum possible value by strategically placing parentheses.

First, I should recall that parentheses change the order of operations, so wherever I place them, the operations inside the parentheses will be calculated first. Since the operations here are all addition and subtraction, the key is to group numbers in such a way that larger negative values are created or minimized positive contributions.

Let me start by analyzing the original expression:

<prompt_1.1> Analyze the impact of parentheses on the expression by considering different groupings of additions and subtractions. </prompt_1.1>

Let me note that the negative sign before the 4 affects the entire sum inside the parentheses. So, if I can group subsequent numbers after a subtraction sign in a way that their total is as large as possible, subtracting a larger number would decrease the overall total.
Alternatively, if I can group numbers after a subtraction to create a negative sum, that would effectively add to the total. Wait, but since subtraction is involved, maybe grouping in a way that creates larger negative numbers when subtracted?

Wait, perhaps another approach: To minimize the overall expression, we want to subtract as much as possible and add as little as possible. Since the original expression has some subtractions, but mainly additions, the strategy might be to create larger negative terms by grouping numbers that follow a subtraction sign.

For example, in the given example, after the 3, there's a subtraction, then (4+5). So 4+5 is 9, then 3 - 9 + 6.
So 3 -9 is -6, then +6 gives 0. Wait, but the example says 3. Wait, let me check that again.

<prompt_1.2> Verify the example calculation and identify the pattern of grouping. </prompt_1.2>

Wait, but how do we get a lower value? Let's see. Let me think of other groupings.

First, let's list all possible ways to insert parentheses into the expression 1 + 2 + 3 - 4 + 5 + 6. Since parentheses can be added around any consecutive subexpression, but we can't change the order of operations beyond that. So, the expression is a sequence of terms: 1, +2, +3, -4, +5, +6. So, the operations are between 1 and 2: +, between 2 and 3: +, between 3 and 4: -, between 4 and 5: +, between 5 and 6: +.

Therefore, the possible places to insert parentheses are between the terms, grouping consecutive terms into a subexpression. Each subexpression can be grouped starting at any point and ending at any point after that, as long as it's a consecutive sequence.

So, for example, we can group 1+2, or 1+2+3, or 1+2+3-4, etc. Similarly, after each term, we can start a new grouping.

But since we can only add parentheses, not rearrange the terms, the key is to figure out where to place the parentheses to minimize the total.

One strategy is to maximize the negative impact by creating large negative numbers that are subtracted. Since subtraction can be used to turn a group of numbers into a large negative, which when subtracted again, would add positively.
Wait, maybe not. Let me think.

Alternatively, perhaps we can create a large negative number in the middle of the expression, which when subtracted, reduces the total.

Let me try to break down the expression into parts. Let's denote the expression as:

1 + 2 + 3 - 4 + 5 + 6

We can consider different groupings. Let's list possible groupings:

1. ((1+2)+3)-4+5+6

<prompt_1.3> Evaluate different groupings to find the minimum value. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
1 + 2 + 3 - 4 + 5 + 6

The operations here are all additions and a single subtraction. The subtraction is only between 3 and 4. If I add parentheses, I can group different parts of the expression. However, since the subtraction is only once, unless I can create more subtractions by grouping, but the original expression only has one subtraction. Wait, but when we add parentheses, the subtraction sign applies to the entire grouped expression that follows. So, for example, if I have 1 + 2 + (3 - 4) + 5 + 6, that would be 1 + 2 + (-1) + 5 + 6 = 13. But in the example given, they have 1 + 2 + 3 - (4 + 5) + 6, which is 1 + 2 + 3 - 9 + 6 = 6 - 9 + 6 = 3. So here, the subtraction is applied to the group (4 + 5), turning it into -9, which is then added to the previous sum.
</execution_1.1>

<execution_1.2>
The example given is 1+2+3-(4+5)+6. Let's compute that:

1+2+3 = 6

4+5 = 9

Then 6 - 9 + 6 = (6 -9) +6 = (-3) +6 = 3.
</execution_1.2>

<execution_1.3>
2. (1+2)+(3-4)+5+6

3. 1+(2+3)-4+5+6

4. 1+2+(3-4)+5+6

5. 1+2+3-(4+5)+6

6. 1+2+3-4+(5+6)

7. (1+2)+(3-(4+5))+6

8. 1+((2+3)-4)+5+6

9. 1+(2+(3-4))+5+6

10. 1+2+((3-4)+5)+6

11. 1+2+(3-(4+5+6))

Hmm, so these are some possible groupings. Let me compute each of them:

1. ((1+2)+3)-4+5+6

First, (1+2)=3, then +3=6. Then 6-4=2, +5=7, +6=13. So back to original.

2. (1+2)+(3-4)+5+6

(1+2)=3, (3-4)=-1. Then 3 + (-1) =2, +5=7, +6=13. Same as original.

3. 1+(2+3)-4+5+6

1 + (5) =6, 6-4=2, +5=7, +6=13.

4. 1+2+(3-4)+5+6

Same as above: 1+2=3, (3-4)=-1, 3-1=2, +5=7, +6=13.

5. 1+2+3-(4+5)+6

Given example: 1+2+3=6, 4+5=9, 6-9= -3, -3+6=3.

6. 1+2+3-4+(5+6)

1+2+3=6, 6-4=2, 5+6=11, 2+11=13.

7. (1+2)+(3-(4+5))+6

(1+2)=3, (4+5)=9, 3-9=-6, -6+6=0.

8. 1+((2+3)-4)+5+6

(2+3)=5, 5-4=1, 1+1=2, 2+5=7, 7+6=13.

9. 1+(2+(3-4))+5+6

(3-4)=-1, 2+(-1)=1, 1+1=2, 2+5=7, 7+6=13.

10. 1+2+((3-4)+5)+6

(3-4)=-1, -1+5=4, 1+2=3, 3+4=7, 7+6=13.

11. 1+2+(3-(4+5+6))

3 - (4+5+6)=3-15=-12, 1+2=3, 3 + (-12)= -9.

So here, grouping 3 - (4+5+6) gives 3 -15= -12, then 1+2=3, 3 + (-12)= -9. So that's a value of -9. That's lower than the previous example's 3.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.3:
Let me check if there are other groupings that can lead to even lower values.

Let's consider grouping more terms. For example, grouping from the start: (1+2+3-4+5)+6.

<prompt_2.1> Explore more complex groupings and calculate their values. </prompt_2.1>

Another possibility: 1+2+(3-4+5+6).

<prompt_2.2> Evaluate additional groupings to find the minimum value. </prompt_2.2>

Another grouping: (1+2+3)-(4+5+6).

<prompt_2.3> Confirm if negative values can be achieved with different groupings. </prompt_2.3>

Another grouping: 1+(2+3-4+5+6).

<prompt_2.4> Verify the possibility of achieving negative values with different groupings. </prompt_2.4>

Another possibility: (1+2+3-4)+(5+6).

<prompt_2.5> Confirm if even more terms can be grouped to achieve lower values. </prompt_2.5>

Wait, let me verify if grouping more terms can lead to even lower values. For example, grouping 1+2+3-4+5+6 as (1+2+3-4+5)+6=7+6=13.

<prompt_2.6> Verify the impact of grouping on the expression and confirm the minimum value. </prompt_2.6>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's compute that:

1+2=3, 3+3=6, 6-4=2, 2+5=7, 7+6=13. So that's 13 again.

Alternatively, grouping (1+2+3)-(4+5+6): 6 -15= -9. Wait, that's the same as grouping 3 - (4+5+6). Wait, if we group (1+2+3)-(4+5+6), that's 6 -15= -9. So that's another way to get -9. So that's the same as before.

Wait, but if we compute (1+2+3) first, which is 6, then subtract (4+5+6)=15, so 6-15=-9. So that's another grouping that gives -9.
</execution_2.1>

<execution_2.2>
Let's compute:

(1+2+3-4+5) +6. Compute inside first: 1+2=3, +3=6, -4=2, +5=7. Then 7+6=13.

Alternatively, 1+2+(3-4+5+6). Compute inside: 3-4=-1, +5=4, +6=10. Then 1+2=3, +10=13.

Another grouping: (1+2)+(3-4)+5+6. Wait, we already did that earlier, which gave 0.
</execution_2.2>

<execution_2.3>
Let's compute that:

(1+2+3-4+5) +6 = (7 -4 +5) +6 = (3 +5) +6 = 8 +6 =14. Then 14 +6=20? Wait, no, wait. Wait, the original expression is 1+2+3-4+5+6. If we group (1+2+3-4+5)+6, that would be (1+2+3-4+5)= (6 +3 -4 +5)= (9 -4 +5)=10, then +6=16. Wait, but that's different. Wait, let me compute step by step:

1+2=3; 3+3=6; 6-4=2; 2+5=7; 7+6=13. Wait, but if we group as (1+2+3-4+5)+6, let's compute inside the parentheses first:

1+2=3; 3+3=6; 6-4=2; 2+5=7. Then 7 +6=13. So same result. So grouping all those terms doesn't change the result. Hmm.

Alternatively, grouping 1+2+3-4 as (1+2+3-4)=2, then +5+6=13. So same as before.

Wait, but if we group 3-(4+5+6), as in example 11, we get 3-15=-12, then 1+2=3, 3 + (-12)= -9.
</execution_2.3>

<execution_2.4>
Let's compute:

(1+2+3)=6, (6-4)=2, (2+5)=7, (7+6)=13. So same as original.

Compute (1+2+3)-(4+5+6): (6) - (15) = -9.

Compute 1+(2+3-4+5+6): 2+3=5, 5-4=1, 1+5=6, 6+6=12. So 1+12=13.

Another grouping: ((1+2)+3-4+5)+6. Let's compute:

(1+2)=3, 3+3=6, 6-4=2, 2+5=7, 7+6=13.

Another idea: 1+2+(3-4)+5+6. As before, that's 3 + (-1) +5 +6=13. Not helpful.

Wait, maybe grouping 3 -4 +5 together? Like 1+2+(3-4+5)+6. Let's compute:

3-4= -1, -1+5=4. So 1+2+4+6=13.
</execution_2.4>

<execution_2.5>
Let's compute:

1+2=3, 3+3=6, 6-4=2, 2+5=7, 7+6=13.

Alternatively, 1+(2+3-4+5+6)=1 + (6) =7.

Alternatively, (1+2+3)-(4+5+6)=6 -15= -9.

Alternatively, 1+2+(3-4+5+6)=3 + (4)=7.

Alternatively, (1+2+3-4)+(5+6)= (6-4)=2 +11=13.

So the only way to get -9 is by grouping either 3-(4+5+6) or (1+2+3)-(4+5+6). Let's check those:

First, 3-(4+5+6)=3-15=-12. Then, the entire expression becomes 1+2+(-12)+6. Wait, wait, hold on. Wait, original expression is 1 + 2 + 3 - 4 + 5 + 6. If we group 3 - (4 +5 +6), then it's 1 + 2 + (3 - (4 +5 +6)) +6. Wait, no. Let me parse that correctly.

If we group 3 - (4+5+6), then the expression becomes 1 + 2 + (3 - (4+5+6)) +6. Wait, but the original expression is 1 +2 +3 -4 +5 +6. So inserting parentheses around 3 - (4+5+6) would require that after 3, we have a subtraction, then the group. So the expression would be 1 + 2 + (3 - (4 +5 +6)) +6. Let's compute that:

3 - (4+5+6) =3 -15= -12

Then 1 + 2 + (-12) +6 = 3 -12 +6= -9 +6= -3.

Wait, but that's different from the previous calculation. Wait, so in this case, the parentheses are around 3 - (4+5+6), which is part of the expression. So the entire expression would be 1 +2 + [3 - (4+5+6)] +6. So that's 1+2=3, then 3 + [3 -15]=3 + (-12)= -9, then -9 +6= -3. So that gives -3, which is higher than -9.

Alternatively, if we group (1+2+3) - (4+5+6). Let's compute that:

(1+2+3)=6, (4+5+6)=15, so 6 -15= -9.

So that's another way to get -9. So depending on how the parentheses are placed, we can get -9. So that seems to be a lower value.
</execution_2.5>

<execution_2.6>
Alternatively, (1+2+3)-(4+5+6)=6-15=-9.

Wait, same as before. So regardless of how we group, if the total of the first three terms is 6, and the last three terms grouped as 4+5+6=15, then 6-15=-9. Alternatively, if we group differently, but the total would be same. Wait, but if we group differently, maybe we can get more negative. Wait, for example, if we do 1+2+(3-4)+5+6=1+2-1+5+6=13. Not helpful.

Alternatively, 1+2+3-4+5+6=13. If we group 1+2+3-(4+5+6)=6-15=-9. So that's the same as before. So regardless of how we group the middle terms, if the total of 4+5 is 9 or 4+5+6 is 15, then subtracting that from the sum before gives the minimal value. Wait, but what if we group even more terms? For example, 1+2+3-4+5+6. If we group 1+2+3-4+5+6 as (1+2+3-4+5+6)=13. Alternatively, if we group 1+2+3-4+5+6 as 1+2+3-(4+5+6)=6-15=-9. So that's the key. So the maximum we can subtract is 15, leading to -9.
</execution_2.6>
</Execution_2>

<Plan_3>
Based on execution_2.6:
But let me check if there is a way to get a more negative value.

Suppose we group 1+2+3-4 as (1+2+3-4)=2, then +5+6=13. Not helpful.

<prompt_3.1> Verify if even more terms can be grouped to achieve lower values. </prompt_3.1>

Alternatively, grouping 1+2+3-4+5 as (1+2+3-4+5)=7, then +6=13. Still 13.

<prompt_3.2> Confirm if even more terms can be grouped to achieve lower values. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Alternatively, if we group 1+2+3-4 as 2, then 2+5=7, +6=13. Same.

Alternatively, group 1+2 as 3, then 3+3=6, 6-4=2, +5=7, +6=13. Same.

Alternatively, group 1+2+3-4+5 as 7, then +6=13.
</execution_3.1>

<execution_3.2>
Alternatively, if we group 1+2+3-4 as 2, then + (5+6)=13. So 2 +11=13.

Alternatively, 1+ (2+3-4+5+6)=1 + (8)=9. Then 9 + something? Wait, no. Wait, the original expression is 1 +2 +3 -4 +5 +6. If we group 2+3-4+5+6 as (2+3-4+5+6)=8, then 1 +8=9. But the original expression is 1 + (2+3-4+5+6) + nothing else. Wait, but that would be 1 + (2+3-4+5+6) =1 +8=9. Which is higher than 13. Wait, that's not helpful.

Wait, perhaps another approach: To minimize the expression, we need to maximize the negative impact. Since the original expression has a total sum of 13 (1+2+3+5+6=17, minus 4 gives 13). To minimize, we need to subtract as much as possible. The maximum negative we can get is by grouping all the positive numbers after a subtraction sign. For example, if we have a subtraction followed by all the remaining terms grouped. So, 1 +2 +3 - (4+5+6). Let's compute that:

4+5+6=15, so 1+2+3=6, 6 -15= -9.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.2:
Therefore, the minimal possible value is -9.

But let me check once more if there's another grouping that can lead to a lower value. For example, what if we group 1+2+3-4+5 as (1+2+3-4+5)=7, then +6=13. Not helpful.

<prompt_4.1> Verify if any other grouping can lead to a lower value than -9. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Alternatively, if we group 1+2+3-4 as 2, then +5+6=13. Alternatively, 1+2 as 3, then 3+3=6, 6-4=2, +5=7, +6=13. All these groupings result in 13 or higher. The only way to get a negative value is by grouping a subtraction with a positive sum that is large enough to make the result negative. The maximum sum we can subtract is 15 (4+5+6), leading to 6-15=-9. Alternatively, grouping 3-(4+5+6)=3-15=-12, but then the expression becomes 1+2+(-12)+6= -3. Wait, that's not as low as -9. So grouping 3 - (4+5+6) gives -3, which is higher than -9. Therefore, the best grouping is (1+2+3) - (4+5+6)=6-15=-9. Therefore, the minimal possible value is -9.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Wait, but let me confirm once more. Let's parse the expression with the grouping (1+2+3)-(4+5+6). The original expression is 1 +2 +3 -4 +5 +6. By inserting parentheses as (1+2+3)-(4+5+6), we are effectively computing (1+2+3) - (4+5+6). Let's compute that:

<prompt_5.1> Finalize and confirm the minimal possible value by considering all possible groupings. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
1+2+3=6, 4+5+6=15, so 6-15=-9. Correct.

Alternatively, if we group 3-(4+5+6), as in 1+2+3-(4+5+6). Then 3 -15=-12, then 1+2=3, 3 + (-12)= -9, then +6? Wait, no. Wait, the original expression is 1+2+3-4+5+6. If we group 3 - (4+5+6), then it's 1+2+ [3 - (4+5+6)]. So 1+2=3, then 3 + (-12)= -9. So that's -9. So same as before. So regardless of where the parentheses are placed, as long as the total sum of 4+5+6=15 is subtracted from the sum of 1+2+3=6, we get -9. Alternatively, if we group 1+2+3 as a group and 4+5+6 as another, then subtract the second group from the first. So (1+2+3)-(4+5+6)=6-15=-9.

Is there a way to get a lower value? For example, if we can subtract a larger number than 15. But 4+5+6=15 is the maximum consecutive grouping starting from 4. Since the expression is 1 +2 +3 -4 +5 +6, after the subtraction at position 3 (after 3), we have +5+6. So if we group -4+5+6 as -(4+5+6)= -15. Wait, but that would be equivalent to 1+2+3-15. Which is 6-15=-9. So same result.

Alternatively, if we can group more terms, but since the original expression only has up to 6, the maximum we can group after the subtraction is 4+5+6=15. Therefore, subtracting 15 from the sum of 1+2+3=6 gives -9. So that seems to be the minimal possible value.

Therefore, after considering all possible groupings and verifying their results, the minimum possible value of the expression by adding parentheses is -9.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the minimum possible value this expression can obtain is $\boxed{-9}$.
</Final_answer>
</think>