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.
I want to find the minimum possible value of this expression, so I need to think about how to make it as negative as possible. The expression has three negative signs, which means I can make three pairs of terms negative. For example, $1 + 2 + 3 - (4 + 5) + 6 - 7$ is one way to pair up the negative signs with two terms each. I notice that the largest pair of terms is $5+6=11$, so if I want to minimize the expression, I should definitely make that pair negative. That means I need to put parentheses around 5 and 6, like this: $1 + 2 + 3 - 4 + (5 + 6)$. Now I have two negative signs left, and four terms to choose from. I want to make the largest pairs of terms negative, so I look at the remaining options: $1+2$, $2+3$, $3-4$, and $-4+1$. The largest pair is $2+3=5$, so I should make that pair negative as well. That means I need to put parentheses around 2 and 3, like this: $1 + (2 + 3) - 4 + (5 + 6)$. Now I have one negative sign left, and two pairs of terms to choose from: $1+(2+3)$ and $-4+(5+6)$. I want to make the larger pair negative, so I compare the values of these pairs: $1+(2+3)=6$ and $-4+(5+6)=7$. The larger pair is $-4+(5+6)=7$, so I should make that pair negative as well. That means I need to put parentheses around -4 and (5+6), like this: $1 + (2 + 3) - (4 + (5 + 6))$. This is the final placement of parentheses that minimizes the expression. I can now simplify it by evaluating the terms inside the parentheses first, and then applying the order of operations: $1 + (2 + 3) - (4 + (5 + 6)) = 1 + 5 - (4 + 11) = 1 + 5 - 15 = 6 - 15 = -9$. 