python3 - <<'PY'
import math,random
ln2=math.log(2)
def f(x): return 0.0 if x<=0 else -x*math.log(x)/ln2
def H(x): return f(x)+f(1-x)
target=(3,0.8957,0.5896,0.1762)
def obj(v,K):
    ps=list(v[:K]); ps=[max(x,0) for x in ps]; ss=sum(ps)
    if ss<1e-9: return -1e9
    ps=[x/ss for x in ps]
    b=[0,0,0,0]; ent=0
    for i in range(K):
        al=min(max(v[K+2*i],0.0001),0.9999); be=min(max(v[K+2*i+1],0.0001),0.9999)
        ent+=ps[i]*(H(al)+H(be))
        b[0]+=ps[i]*al*be; b[1]+=ps[i]*al*(1-be); b[2]+=ps[i]*(1-al)*be; b[3]+=ps[i]*(1-al)*(1-be)
    err=sum((b[j]-target[j])**2 for j in range(4))
    return ent-1000*err
def golden(fun,a,b,tol=1e-8):
    g=(math.sqrt(5)-1)/2
    c=b-g*(b-a); d=a+g*(b-a); fc=fun(c); fd=fun(d)
    while b-a>tol:
        if fc>fd: b,d,fd=d,c,fc; c=b-g*(b-a); fc=fun(c)
        else: a,c,fc=c,d,fd; d=a+g*(b-a); fd=fun(d)
    return (a+b)/2
for K in (2,3):
    best=None
    random.seed(2)
    for st in range(30):
        v=[random.random() for _ in range(K)]+[random.uniform(.05,.95) for _ in range(2*K)]
        for it in range(30):
            for j in range(len(v)):
                def g(x,j=j): vv=v.copy(); vv[j]=x; return obj(vv,K)
                v[j]=golden(g,max(0.0001,v[j]-0.5),min(0.9999,v[j]+0.5))
        val=obj(v,K)
        if best is None or val>best[0]: best=(val,v)
    print(K,best)
PY