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D.2.4.4 multigrobcov

Procedure from library grobcov.lib (see grobcov_lib).

Return:
The list whose first element is the generic case, and the remaining elements are the grobcov over the different irreducible components in the complementary of the generic segment. the empty list if the generic case does not have basis 1.

Note:
The basering R, must be of the form Q[a][x], a=parameters, x=variables, and should be defined previously. The ideal must be defined on R.

Example:
 
LIB "grobcov.lib";
"Generalization of the Steiner-Lehmus theorem";
==> Generalization of the Steiner-Lehmus theorem
ring R=(0,x,y),(a,b,m,n,p,r),lp;
ideal S=p^2-(x^2+y^2),
-a*(y)+b*(x+p),
-a*y+b*(x-1)+y,
(r-1)^2-((x-1)^2+y^2),
-m*(y)+n*(x+r-2) +y,
-m*y+n*x,
(a^2+b^2)-((m-1)^2+n^2);
short=0;
multigrobcov(S,list("can",0,"cgs",0,"comment",1));
==> Generic case =
==> [1]:
==>    _[1]=1
==> [2]:
==>    _[1]=1
==> [3]:
==>    [1]:
==>       _[1]=0
==>    [2]:
==>       [1]:
==>          _[1]=(y)
==>       [2]:
==>          _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*\
   y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y\
   ^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2\
   *y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6\
   *y^4+y^2)
==>       [3]:
==>          _[1]=(2*x-1)
==> [4]:
==>    _[1]=(16*x^11*y-88*x^10*y+82*x^9*y^3+192*x^9*y-369*x^8*y^3-204*x^8*y+1\
   68*x^7*y^5+656*x^7*y^3+96*x^7*y-588*x^6*y^5-574*x^6*y^3+172*x^5*y^7+808*x\
   ^5*y^5+226*x^5*y^3-16*x^5*y-430*x^4*y^7-550*x^4*y^5+9*x^4*y^3+4*x^4*y+88*\
   x^3*y^9+416*x^3*y^7+164*x^3*y^5-40*x^3*y^3-132*x^2*y^9-194*x^2*y^7+10*x^2\
   *y^5+10*x^2*y^3+18*x*y^11+72*x*y^9+34*x*y^7-24*x*y^5+2*x*y^3-9*y^11-14*y^\
   9+y^7+6*y^5-y^3)
==>  
==> Components to study=
==> [1]:
==>    _[1]=(y)
==> [2]:
==>    _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*y^4+24\
   6*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y^4+31*\
   x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2*y^6+1\
   4*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6*y^4+y\
   ^2)
==> [3]:
==>    _[1]=(2*x-1)
==>  
==> Begin grobcov on the variety N =
==> N[1]=(y)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 3
==> Number of lpp segments in groupsegments = 3
==> Time in buildtree = 1 sec
==> Time in groupRtoPrep = 0 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 0 sec
==> Time for grobcov = 1 sec
==> Number of segments of grobcov = 3
==>  
==> Begin grobcov on the variety N =
==> N[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+84*x^6*y^4+246*x\
   ^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278*x^4*y^4+31*x^4\
   *y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+122*x^2*y^6+14*x\
   ^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8-y^6-6*y^4+y^2)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 12
==> Number of lpp segments in groupsegments = 12
==> Time in buildtree = 11 sec
==> Time in groupRtoPrep = 186 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 64 sec
==> Time for grobcov = 262 sec
==> Number of segments of grobcov = 12
==>  
==> Begin grobcov on the variety N =
==> N[1]=(2*x-1)
==> Options: can = 0, extend = 1, cgs = 0, rep = 0
==> Number of segments in buildtree (total) = 5
==> Number of lpp segments in groupsegments = 5
==> Time in buildtree = 2 sec
==> Time in groupRtoPrep = 0 sec
==> Time in LCUnion + combine = 0 sec
==> Time in extend = 0 sec
==> Time for grobcov = 3 sec
==> Number of segments of grobcov = 5
==> Time for multigrobcov = 267
==> [1]:
==>    [1]:
==>       [1]:
==>          _[1]=1
==>       [2]:
==>          _[1]=1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=0
==>             [2]:
==>                [1]:
==>                   _[1]=(y)
==>                [2]:
==>                   _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^\
   7+84*x^6*y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+\
   278*x^4*y^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^\
   8+122*x^2*y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*\
   y^8-y^6-6*y^4+y^2)
==>                [3]:
==>                   _[1]=(2*x-1)
==> [2]:
==>    [1]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2
==>       [2]:
==>          _[1]=r^2-2*r+(-x^2+2*x)
==>          _[2]=p^2+(-x^2)
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2-m^2+2*m-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>             [2]:
==>                [1]:
==>                   _[1]=(y)
==>                   _[2]=(x-1)
==>                [2]:
==>                   _[1]=(y)
==>                   _[2]=(x)
==>    [2]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n*r
==>          _[4]=b
==>          _[5]=a^2
==>       [2]:
==>          _[1]=r^2-2*r
==>          _[2]=p^2
==>          _[3]=n*r-2*n
==>          _[4]=b
==>          _[5]=a^2-m^2+2*m-n^2-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(x)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [3]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=b*p
==>          _[5]=a^2
==>       [2]:
==>          _[1]=r^2-2*r+1
==>          _[2]=p^2-1
==>          _[3]=n
==>          _[4]=b*p+b
==>          _[5]=a^2+b^2-m^2+2*m-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==> [3]:
==>    [1]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=(3*x^4-6*x^3+6*x^2*y^2+5*x^2-6*x*y^2+3*y^4+5*y^2-1)*r+(x^5-\
   10*x^4+2*x^3*y^2+17*x^3-18*x^2*y^2-10*x^2+x*y^4+17*x*y^2-x-8*y^4-10*y^2+2\
   )
==>          _[2]=(3*x^4-6*x^3+6*x^2*y^2+5*x^2-6*x*y^2-4*x+3*y^4+5*y^2+1)*p+(\
   x^5+2*x^4+2*x^3*y^2-7*x^3+6*x^2*y^2+4*x^2+x*y^4-7*x*y^2-x+4*y^4+4*y^2)
==>          _[3]=(x^5-4*x^4+2*x^3*y^2+5*x^3-6*x^2*y^2+x*y^4+5*x*y^2-x-2*y^4)\
   *n+(-3*x^4*y+6*x^3*y-6*x^2*y^3-5*x^2*y+6*x*y^3-3*y^5-5*y^3+y)
==>          _[4]=(x^5-4*x^4+2*x^3*y^2+5*x^3-6*x^2*y^2+x*y^4+5*x*y^2-x-2*y^4)\
   *m+(-3*x^5+6*x^4-6*x^3*y^2-5*x^3+6*x^2*y^2-3*x*y^4-5*x*y^2+x)
==>          _[5]=(x^5-x^4+2*x^3*y^2-x^3-x^2+x*y^4-x*y^2+3*x+y^4-y^2-1)*b+(3*\
   x^4*y-6*x^3*y+6*x^2*y^3+5*x^2*y-6*x*y^3-4*x*y+3*y^5+5*y^3+y)
==>          _[6]=(x^5-x^4+2*x^3*y^2-x^3-x^2+x*y^4-x*y^2+3*x+y^4-y^2-1)*a+(2*\
   x^5-8*x^4+4*x^3*y^2+12*x^3-12*x^2*y^2-8*x^2+2*x*y^4+12*x*y^2+2*x-4*y^4-4*\
   y^2)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(8*x^10-40*x^9+41*x^8*y^2+76*x^8-164*x^7*y^2-64*x^7+8\
   4*x^6*y^4+246*x^6*y^2+16*x^6-252*x^5*y^4-164*x^5*y^2+8*x^5+86*x^4*y^6+278\
   *x^4*y^4+31*x^4*y^2-4*x^4-172*x^3*y^6-136*x^3*y^4+20*x^3*y^2+44*x^2*y^8+1\
   22*x^2*y^6+14*x^2*y^4-10*x^2*y^2-44*x*y^8-36*x*y^6+12*x*y^4+9*y^10+14*y^8\
   -y^6-6*y^4+y^2)
==>             [2]:
==>                [1]:
==>                   _[1]=(y)
==>                   _[2]=(x-1)
==>                [2]:
==>                   _[1]=(y)
==>                   _[2]=(x)
==>                [3]:
==>                   _[1]=(y)
==>                   _[2]=(2*x^2-2*x-1)
==>                [4]:
==>                   _[1]=(4*y^2-3)
==>                   _[2]=(2*x-1)
==>                [5]:
==>                   _[1]=(12*y^2-1)
==>                   _[2]=(2*x-1)
==>                [6]:
==>                   _[1]=(y^4+11*y^2-1)
==>                   _[2]=(5*x+2*y^2+1)
==>                [7]:
==>                   _[1]=(y^4+11*y^2-1)
==>                   _[2]=(5*x-2*y^2-6)
==>                [8]:
==>                   _[1]=(4*y^4+5*y^2+2)
==>                   _[2]=(2*x-1)
==>                [9]:
==>                   _[1]=(y^2+2*y-1)
==>                   _[2]=(x^2-x-2*y+1)
==>                [10]:
==>                   _[1]=(y^2-2*y-1)
==>                   _[2]=(x^2-x+2*y+1)
==>                [11]:
==>                   _[1]=(4*y^4+349*y^2-64)
==>                   _[2]=(17*x-2*y^2-15)
==>                [12]:
==>                   _[1]=(4*y^4+349*y^2-64)
==>                   _[2]=(17*x+2*y^2-2)
==>    [2]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=(2*x)*r+(-4*x-y^2+1)
==>          _[2]=(2*x-2)*p+(2*x-y^2-1)
==>          _[3]=(y^2-1)*n+(2*x*y)
==>          _[4]=(2*x)*m+(-4*x-y^2+1)
==>          _[5]=(y^2-1)*b+(-2*x*y+2*y)
==>          _[6]=(2*x-2)*a+(2*x-y^2-1)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y^2-2*y-1)
==>                _[2]=(x^2-x+2*y+1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>          [2]:
==>             [1]:
==>                _[1]=(y^2+2*y-1)
==>                _[2]=(x^2-x-2*y+1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [3]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=5*r^2-10*r+(-y^2-3)
==>          _[2]=5*p+(-y^2+2)
==>          _[3]=(5*y)*n+(-3*y^2+1)*r
==>          _[4]=5*m+(y^2-2)*r
==>          _[5]=(5*y)*b+(3*y^2-1)
==>          _[6]=a+1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y^4+11*y^2-1)
==>                _[2]=(5*x+2*y^2+1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [4]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=4*r^2-8*r+(-4*y^2+3)
==>          _[2]=p+r-1
==>          _[3]=(8*y)*n+(-3*y^2+2)*r
==>          _[4]=4*m+(-y^2-2)*r
==>          _[5]=(8*y)*b+(-3*y^2+2)*r
==>          _[6]=4*a+(y^2+2)*r-4
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^4+5*y^2+2)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [5]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=5*r+(y^2-7)
==>          _[2]=5*p^2+(-y^2-8)
==>          _[3]=(5*y)*n+(3*y^2-1)
==>          _[4]=m-2
==>          _[5]=(5*y)*b+(3*y^2-1)*p+(-3*y^2+1)
==>          _[6]=5*a+(y^2-2)*p+(-y^2-3)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y^4+11*y^2-1)
==>                _[2]=(5*x-2*y^2-6)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [6]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=3*r^2-6*r+2
==>          _[2]=3*p^2-1
==>          _[3]=2*n+(-3*y)*r
==>          _[4]=4*m-3*r
==>          _[5]=2*b+(3*y)*p+(-3*y)
==>          _[6]=4*a-3*p-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(12*y^2-1)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [7]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=85*r+(4*y^2-123)
==>          _[2]=p-1
==>          _[3]=(170*y)*n+(47*y^2-64)
==>          _[4]=85*m+(y^2-137)
==>          _[5]=2*b+(-y)
==>          _[6]=17*a+(-y^2-16)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^4+349*y^2-64)
==>                _[2]=(17*x-2*y^2-15)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [8]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=r
==>          _[2]=85*p+(-4*y^2+38)
==>          _[3]=2*n+(-y)
==>          _[4]=17*m+(y^2-1)
==>          _[5]=(170*y)*b+(47*y^2-64)
==>          _[6]=85*a+(-y^2+52)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^4+349*y^2-64)
==>                _[2]=(17*x+2*y^2-2)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [9]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=r
==>          _[2]=p-1
==>          _[3]=2*n+(-y)
==>          _[4]=4*m-1
==>          _[5]=2*b+(-y)
==>          _[6]=4*a-3
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^2-3)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [10]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2
==>       [2]:
==>          _[1]=2*r^2-4*r+(2*x-1)
==>          _[2]=2*p^2+(-2*x-1)
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2-m^2+2*m-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(2*x^2-2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [11]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n*r
==>          _[4]=b
==>          _[5]=a^2
==>       [2]:
==>          _[1]=r^2-2*r
==>          _[2]=p^2
==>          _[3]=n*r-2*n
==>          _[4]=b
==>          _[5]=a^2-m^2+2*m-n^2-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(x)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [12]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=b*p
==>          _[5]=a^2
==>       [2]:
==>          _[1]=r^2-2*r+1
==>          _[2]=p^2-1
==>          _[3]=n
==>          _[4]=b*p+b
==>          _[5]=a^2+b^2-m^2+2*m-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==> [4]:
==>    [1]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=4*r^2-8*r+(-4*y^2+3)
==>          _[2]=p+r-1
==>          _[3]=(4*y^2-3)*n+(4*y)*r
==>          _[4]=(4*y^2-3)*m+2*r
==>          _[5]=(4*y^2-3)*b+(4*y)*r
==>          _[6]=(4*y^2-3)*a-2*r+(-4*y^2+3)
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=(y)
==>                   _[2]=(2*x-1)
==>                [2]:
==>                   _[1]=(4*y^2-3)
==>                   _[2]=(2*x-1)
==>                [3]:
==>                   _[1]=(12*y^2-1)
==>                   _[2]=(2*x-1)
==>                [4]:
==>                   _[1]=(4*y^2+1)
==>                   _[2]=(2*x-1)
==>    [2]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=3*r^2-6*r+2
==>          _[2]=3*p^2-1
==>          _[3]=2*n+(-3*y)*r
==>          _[4]=4*m-3*r
==>          _[5]=2*b+(3*y)*p+(-3*y)
==>          _[6]=4*a-3*p-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(12*y^2-1)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [3]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=r^2-2*r+1
==>          _[2]=p+r-1
==>          _[3]=n+(-y)*r
==>          _[4]=2*m-r
==>          _[5]=b+(-y)*r
==>          _[6]=2*a+r-2
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^2+1)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [4]:
==>       [1]:
==>          _[1]=r
==>          _[2]=p
==>          _[3]=n
==>          _[4]=m
==>          _[5]=b
==>          _[6]=a
==>       [2]:
==>          _[1]=r
==>          _[2]=p-1
==>          _[3]=2*n+(-y)
==>          _[4]=4*m-1
==>          _[5]=2*b+(-y)
==>          _[6]=4*a-3
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(4*y^2-3)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1
==>    [5]:
==>       [1]:
==>          _[1]=r^2
==>          _[2]=p^2
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2
==>       [2]:
==>          _[1]=4*r^2-8*r+3
==>          _[2]=4*p^2-1
==>          _[3]=n
==>          _[4]=b
==>          _[5]=a^2-m^2+2*m-1
==>       [3]:
==>          [1]:
==>             [1]:
==>                _[1]=(y)
==>                _[2]=(2*x-1)
==>             [2]:
==>                [1]:
==>                   _[1]=1


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