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:Bb
['', '(Bd)(FB) .D.F.B.d (fb)(df) .B.d.F.B', '', 'd>fB>f d>Bd>f b2 =e>gc>g e>gc>g =e>ce>g b>ge>c']  
X:1
L:1/8
M:2/2
K:Bb
 vF2"^Segno" |: (Bd)(FB) .D.F.B.d | (fb)(df) .B.d.F.B | (ce)(cB) .A.F.C.F | .A.F.c.A (ag)(ec) | 
 (Bd)(FB) .D.F.B.d | (fb)(df) .B.d.F.B | (3(vc=Bc e>)c .A.F.G.A |1 B2 [dF]2 [BD]2 F2 :|2 
 B2 [dF]2 [BD]2"^Fermat:a" |] (cB) |]: vA>fc>f A>fc>f | A>FA>c f2 (f=e) | d>fG>f d>fB>f | 
 d>Bd>f b2 (3(b"^3"gf) | =e>gc>g e>gc>g | =e>ce>g b>ge>c |({g} f=e) (3(fga) b>ge>c |1 
 f2 [fA]2 [fA]2 (cB) :|2 (f=efg) (f_ecA)"^Segno" |]  