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
['A/A/A ag']  
X:1
L:1/8
M:4/4
K:Ador
|:"Am" AAAB eAAB |"Am" AAAe"G" dBGB |"Am" AAAB ABdA |"Am" eAdA"G" BGGB :: A/A/A ag aAAB | 
"Am" A/A/A af"G" gGGB |"Am" A/A/A ag aged |"C" e/f/g"D" fd"G" gGGB :|  