Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
Q:1/2=100
M:2/2
K:A
['e2 cA f2 dB', 'A2 B2 GE c2 AF dcBA GBeg']  
X:1
L:1/8
Q:1/2=100
M:2/2
K:A
|: e2 cA f2 dB | ceac BGFE | e2 cA f2 dB | gefg a2 A2 :: B2 GE c2 AF | dcBA GBeg | a2 ec f2 dB | 
 ceEG A2 A,2 :|  