python3 <<'PY'
import math,random
a=0.25; b=1.0/12
y1=0.025697868008998203; y2=0.22745484489188828
def Ared(y1,y2,A):
    return A
def bestK(K,N=200000):
    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)]
      c=[0.0]*K
      rem=nu1
      for i in range(K-2):
        ci=random.random()*min(q[i],rem)
        c[i]=ci; rem-=ci
      # solve last two for sum rem and m11 a
      r1=r[K-2]; r2=r[K-1]
      if abs(r1-r2)<1e-14: continue
      T=a-sum(r[i]*c[i] for i in range(K-2))
      c1=(T-r2*rem)/(r1-r2); c2=rem-c1
      c[K-2]=c1; c[K-1]=c2
      if any(c[i]<-1e-10 or c[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=c[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
print("K2 approx",0.304319)
for K in (3,4,5):
  print(K,bestK(K,100000))
PY