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:G
['G2 g2 d2 (cB) cBAG F2 (ED)', 'c2 (BA)']  
X:1
L:1/8
M:2/2
K:G
 G2 g2 d2 (cB) | cBAG F2 (ED) | GDGB AFED | GDGB AFED | G2 g2 d2 (cB) | cBAG F2 (ED) | 
 c2 (BA) B2 (AG) | D2 F2 G4 :: d2 g2 f2 (ed) | e2 (fd) c2 (BA) | dAdf ecBA | dAdf ecBA | 
 d2 g2 f2 ed | e2 fd c2 (BA) | c2 (BA) B2 (AG) | D2 F2 G4 :|  