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:Ador
['', 'c2 Ac BAGB AGEF', 'agef g2']  
X:1
L:1/8
M:4/4
K:Ador
 ed |: c2 Ac BAGB | AGEF GE D2 | EA A/A/A ABcd | eaaf gfed | c2 Ac BAGB | AGEF GEDF | 
 EA A/A/A ABce |1 dBGB A2 ed :|2 dBGB A2 eg |: a3 b ageg | agbg agef | g2 gf gd d/d/d | gabg aged | 
 eaab ageg | agbg agef | g2 ge a3 ag | b2 ba gfed :|  