Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:4/4
K:Amix
['AeeA e2 Ae eAef gfeg', 'Aeeg aged efge a2 eg', 'geBg fded geBg f2 de', 'AB e2']  
X:1
L:1/8
M:4/4
K:Amix
|: e2 | AeeA e2 Ae | eAef gfeg | GddG B2 Gd | dGBd e2 de | AeeA e2 Ae | eAef gfeg | Ggge aged | 
 e2 A2 A2 :: e2 | Aeeg aged | efge a2 eg | GddG d2 Gd | dGBd e2 de | Aeeg aged | efge a2 eg | 
 Ggge aged | e2 A2 A2 :: ef | geBg fded | geBg f2 de | dBGd BG B2 | GdBG dBAB | geBg fded | 
 geBg f2 de | Ggge aged | e2 A2 A2 :: e2 | AB e2 dBGB | AB e2 d2 Bd | gfed BAGd | dGBd e2 de | 
 AB e2 dBGB | AB e2 d2 Bd | Ggge aged | e2 A2 A2 :|  