python3 <<'PY'
import math,random
a=0.25; b=1.0/12
def bestK(y1,y2,K=4,N=150000):
    alpha=2*a+2*b-y1-y2; mu1=a+b-y1; nu1=a+b
    best=-1
    for _ in range(N):
      w=[random.random() for i in range(K)]; sw=sum(w); q=[alpha*v/sw for v in w]
      u=[random.random() for i in range(K)]; su=sum(u); aa=[mu1*v/su for v in u]
      if any(aa[i]>q[i]+1e-12 for i in range(K)): continue
      r=[aa[i]/q[i] for i in range(K)]
      c1=random.random()*min(q[0],nu1); c2=random.random()*min(q[1],nu1-c1)
      R=nu1-c1-c2; T=a-r[0]*c1-r[1]*c2
      if abs(r[2]-r[3])<1e-14: continue
      c3=(T-r[3]*R)/(r[2]-r[3]); c4=R-c3
      cc=[c1,c2,c3,c4]
      if any(cc[i]<-1e-10 or cc[i]>q[i]+1e-10 for i in range(K)): continue
      val=0.0
      for i in range(K):
        qi=q[i]; ai=aa[i]; ci=cc[i]
        bi=qi-ai; di=qi-ci
        val+=math.sqrt(max(0,a*ai*ci))+math.sqrt(max(0,b*ai*di))+math.sqrt(max(0,b*bi*ci))+math.sqrt(max(0,a*bi*di))
      if val>best: best=val
    return best
for y1,y2 in [(0.035,0.095),(0.02,0.22),(0.0,0.0),(0.0,0.2)]:
  print(y1,y2,bestK(y1,y2,4,100000))
PY