python3 <<'PY'
import math
a=0.25; b=1.0/12
def Aval(y1,y2,q,r):
    alpha=2*a+2*b-y1-y2; mu1=a+b-y1; nu1=a+b; q2=alpha-q
    if q<=0 or q2<=0: return -1
    lo=max(0.0,q-(alpha-mu1)); hi=min(q,mu1)
    if r<lo-1e-12 or r>hi+1e-12: return -1
    den=r*q2/q-(mu1-r)
    if abs(den)<1e-14: return -1
    c=(a*q2-(mu1-r)*nu1)/den
    if c<-1e-10 or c>q+1e-10 or c>nu1+1e-10 or nu1-c>q2+1e-10: return -1
    r2=mu1-r; c2=nu1-c
    return (math.sqrt(max(0,a*r*c))+math.sqrt(max(0,b*r*(q-c)))+math.sqrt(max(0,b*(q-r)*c))+math.sqrt(max(0,a*(q-r)*(q-c)))+math.sqrt(max(0,a*r2*c2))+math.sqrt(max(0,b*r2*(q2-c2)))+math.sqrt(max(0,b*(q2-r2)*c2))+math.sqrt(max(0,a*(q2-r2)*(q2-c2))))
def best2(y1,y2,N=300):
    alpha=2*a+2*b-y1-y2; mu1=a+b-y1; mu2=alpha-mu1
    best=-1; arg=None
    for iq in range(1,N):
      q=alpha*iq/N
      lo=max(0.0,q-mu2); hi=min(q,mu1)
      for ir in range(N+1):
        r=lo+(hi-lo)*ir/N
        v=Aval(y1,y2,q,r)
        if v>best: best=v; arg=(q,r)
    return best,arg
y1=0.024697867630391723; y2=0.23045484476568612
for N in (100,200,400):
  print(N,best2(y1,y2,N))
PY