Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:2/2
K:D
['EG G2', 'c AcGc AcGc', 'abag e2 dB cded eaag']  
X:1
L:1/8
M:2/2
K:D
 vE2 E2 EG G2 | AcGc AcGc | E2 E2 EG G2 | (3(ABc).G.A ED D2 | E2 C2 EG G2 | AcGc AcGc | 
 E2 C2 EG G2 | (3(ABc).G.A ED D2 |]"^Segno" e2- e2 d3 c | AcGc AcGc | e2 (3.e.e.e d2 c2 | 
 (3(ABc)Gc ED D2 | e2- e2 d3 c | AcGc AcGc | eddB cBAG | (3(ABc).G.A ED D2 |] ef{a}gf gagf | 
 eaag abag | e2 dB cded | eaag ed d2 | ef{a}gf gagf | ea{b}ag abag | e2 dB cded | 
 eaag ed d2"^Segno" |]  