Confused - Valuing bond using forward rates

Can someone where I going wrong? Using Kaplan notes.
Given the following spot and forward rates what is the closest value of a 4 year 10% annual pay $1000 par value bond: Current 1 year spot rate is 5.5% 1y1y is 7.63% 2y1y is 12.18% 3y1y is 15.5%
A. $996
B. $1009
I calculated the spot rate as (1+r)^4 = 1.0551.07631.1218*1.1550 Which gave me spot rate as 10.13% and the PV as 996
But the answer is 100/(1.055) + 100/(1.055) (1.0763) + 100/(1.055) (1.0763) (1.1218) + 1100/(1.055) (1.0763) (1.1218) (1.155) = 1009.
(My intuition is S4 be the spot rate of for 4 years which gives a particular return when we convert into forward rates of each, effectively we will be calculating using spot rate (s1) and forward rates for each year from now [1y1y,2y1y,3y1y] which is equivalent to S4).
Can someone please help me in understanding this?

You need the spot rates at each of time 1, 2, 3, and 4. The spot rate for 4 years is indeed 10.13, but that only applies to any cash flows at time 4: you need the specific spot rates for times 1, 2, and 3. You know the 1 year spot rate is 5.5%, so you can derive the remaining rates using the same process as for the 4 year rate.

Thank you mate
Can you correct If have understood wrong, so spot rate for 4 years is equivalent to (1+s1)(1+1y1y)(1+2y1y)(1+3y1y) only when there are no payments made.

Yes, an n year spot rate is only for cash flows occuring at time n. Using implied forward rates is basically a reinvestment process: investing at n year spot or using some combo of shorter spot rates and reinvesting to get to time n should produce the same end value.