Construct a musical piece that follows the designated lettered structure and integrates the provided motif throughout.
X:1
L:1/8
Q:1/2=112
M:2/2
K:F
['c d2']  
X:1
L:1/8
Q:1/2=112
M:2/2
K:F
|:"^A" c2 |"F" f2 e2 f2 c2 |"^/" f4 c3 c |"Bb" d2 (BA)"C7" G2 c2 |"F" A2 F4 ::"^B" A2 | 
"Bb" B3 c d2 (cB) |"F" A3 G F2 A2 |"Bb" B3 c d2 (cB) |"F" A3 B c2 (BA) |"Gm" GFGA BABc | 
"Bb" d3 e"F/A" f2 A2 |"Gm" B2 A2"C7" G2 c2 |"F" A2 F4 :|  