Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
M:2/4
K:Amix
['ag eg', 'eg e/d/c AA/B/ e/d/c A2 A2 e2', 'AA/B/ dA/B/ ce d/c/A e2 e2 d/A/c']  
X:1
L:1/8
M:2/4
K:Amix
|:"A" ag eg | e/d/c Aa- | ag eg | a>b aa- ||"A" ag eg | e/d/c AA/B/ | ce"E" e/d/c |"A" A2 A2 :: 
"A" e2 e2 | d/A/c AA/B/ | ce"D" dA/B/ |"A" ce d/c/A ||"A" e2 e2 | d/A/c AA/B/ | ce"E" e/d/c | 
"A" A2 A2 :|  