Linear combinations of integers

Express the greatest common divisor of each of the following pairs of integers as a linear combination of these integers. 34,55

Here's what I have so far

55=34*1+21

34=21*1+13

21=13*1+8

13=8*1+5

8=5*1+3

5=3*1+2

3=2*1+1

2=1*2

Making 1 the greatest common divisor.

I also know that to solve this, you have to solve for the remainders so....

1=3-1*2

1=3-1(5-1*3)

1=3-1((5-1)(8-1*5))

1=3-1((5-1)(8-1)(13-1*8))

1=3-1((5-1)(8-1)(13-1)(21-1*13))

1=3-1((5-1)(8-1)(13-1)(21-1)(34-1*21))

I think I have it right so far, the answer to the question per the back of the book is 1=13*55+(-21)*34. This is where my problem is, I just can't see where the answer is coming from. I think I have it set up, but the end is confusing me.