python3 <<'PY'
import math
a=0.25; b=1.0/12
def F(M,N=40):
    m11,m12=M[0]; m21,m22=M[1]
    alpha=m11+m12+m21+m22
    if alpha<1e-15: return 0.0
    mu1=m11+m12; nu1=m11+m21
    best=-1.0
    for iq in range(1,N):
        q=alpha*iq/N; q2=alpha-q
        lo=max(0.0,q-(alpha-mu1)); hi=min(q,mu1)
        for ir in range(1,N):
            r=lo+(hi-lo)*ir/N
            den=r*(alpha-q)/q-(mu1-r)
            if abs(den)<1e-14: continue
            c=(m11*(alpha-q)-(mu1-r)*nu1)/den
            if c<-1e-8 or c>nu1+1e-8 or c>q+1e-8 or nu1-c>q2+1e-8: continue
            c=min(max(c,0.0),nu1)
            r1=r/q; s1=c/q; r2=(mu1-r)/q2; s2=(nu1-c)/q2
            val=q*(math.sqrt(max(0,m11*r1*s1))+math.sqrt(max(0,m12*r1*(1-s1)))+math.sqrt(max(0,m21*(1-r1)*s1))+math.sqrt(max(0,m22*(1-r1)*(1-s1))))+q2*(math.sqrt(max(0,m11*r2*s2))+math.sqrt(max(0,m12*r2*(1-s2)))+math.sqrt(max(0,m21*(1-r2)*s2))+math.sqrt(max(0,m22*(1-r2)*(1-s2))))
            if val>best: best=val
    return best
def G(y1,y2,y3):
    M=[[a,b-y1],[b,a-y2]]
    gam=y1+y2+y3
    ct=math.sqrt(gam)*(math.sqrt(max(0,b*y1))+math.sqrt(max(0,a*y2))+math.sqrt(max(0,a*y3)))
    bt=(a+b-y3)**1.5
    return F(M)+ct+bt
besty=None; best=-1
N1=5; N2=5; N3=5
for i1 in range(N1+1):
  y1=b*i1/N1
  for i2 in range(N2+1):
    y2=a*i2/N2
    for i3 in range(N3+1):
      y3=a*i3/N3
      g=G(y1,y2,y3)
      if g>best: best=g; besty=(y1,y2,y3)
print("best",best,besty,"1-best",1-best)
PY