python3 - <<'PY'
import math,random
ln2=math.log(2);c=2/ln2
def HB(pA,pC,m,A,B):
    if m<3 or pA<9 or pC<3 or abs(pA+pC+m-20)>1e-6: return -1e9
    if A<3/m or B<3/m or A>6/m or B>4/m or A+B<1+2/m or A+B>1+3/m or A>1 or B>1: return -1e9
    def He(p,q):
        if p<=q: return 0
        r=(p-q)/2
        return -(q/p*math.log(q/p)+2*r/p*math.log(r/p))/ln2
    HA=He(pA,9);HC=He(pC,3)
    bb=2-c*(2*abs(3/m-A*B)+(A-.5)**2+(B-.5)**2)
    return (pA*HA+pC*HC+m*bb)/20
def golden(fun,a,b,tol=1e-9):
    g=(math.sqrt(5)-1)/2;cc=b-g*(b-a);dd=a+g*(b-a);fc=fun(cc);fd=fun(dd)
    while b-a>tol:
        if fc>fd: b,dd,fd=dd,cc,fc;cc=b-g*(b-a);fc=fun(cc)
        else: a,cc,fc=cc,dd,fd;dd=a+g*(b-a);fd=fun(dd)
    return (a+b)/2
best=(-1,None)
random.seed(5)
for st in range(30):
    m=random.uniform(3,8);pA=random.uniform(9,20-m-3);pC=20-m-pA
    if pC<3: continue
    A=random.uniform(3/m,min(1,6/m));B=random.uniform(3/m,min(1,4/m))
    v=[pA,pC,m,A,B]
    for it in range(20):
        for j in range(5):
            lo=max(0.0001,v[j]-1);hi=min(1,v[j]+1) if j>=3 else v[j]+1
            def g(x,j=j): vv=v.copy();vv[j]=x;return HB(vv[0],vv[1],vv[2],vv[3],vv[4])
            v[j]=golden(g,lo,hi)
        val=HB(*v)
        if val>best[0]: best=(val,v)
print(best)
print(2.1822287-best[0])
PY