python3 - <<'PY'
import math
ln2=math.log(2)
c=2/ln2
def f(x): return 0.0 if x<=0 else -x*math.log(x)/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
    HA=-(9/pA*math.log(9/pA)+2*((pA-9)/2/pA)*math.log((pA-9)/2/pA))/ln2 if pA>9 else 0
    HC=-(3/pC*math.log(3/pC)+2*((pC-3)/2/pC)*math.log((pC-3)/2/pC))/ln2 if pC>3 else 0
    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-10):
    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)
for pA0 in [10,12,14,16]:
  for m0 in [3.5,4.5,5.5,6.5]:
    pC0=20-pA0-m0
    if pC0<3: continue
    v=[pA0,pC0,m0,min(.9,6/m0),min(.9,4/m0)]
    for it in range(30):
        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('bound',best)
print('I_lower',2.1822287-best[0])
PY