from fractions import Fraction as Q
from itertools import permutations

def run(load,check):
 # Each coordinate is a*m+b in K units; i cancels from differences.
 pts={'S':(1,-12),'F':(1,-10),'R':(0,5),'N':(0,7)};solutions=[];failures=[]
 orders=[p for p in permutations(pts) if p.index('S')<p.index('F') and p.index('R')<p.index('N')]
 for p in orders:
  gaps=[(pts[b][0]-pts[a][0],pts[b][1]-pts[a][1]) for a,b in zip(p,p[1:])]
  eq=[(gaps[k][0]-gaps[k+1][0],gaps[k][1]-gaps[k+1][1]) for k in [0,1]]
  cand={-Q(b,a) for a,b in eq if a};ok=len(cand)==1 and all(a or b==0 for a,b in eq)
  if ok:
   m=next(iter(cand));vals=[a*m+b for a,b in gaps];ok=len(set(vals))==1 and vals[0]>0
  if ok:solutions.append({'order':p,'W_K':str(m),'spacing_K':str(vals[0]),'coefficients':[str(pts[k][0]*m+pts[k][1]) for k in p]})
  else:failures.append({'order':p,'equations':eq,'status':'inconsistent or nonpositive spacing'})
 check('six order domains',len(orders)==6)
 check('four exact real solutions',sorted(Q(r['W_K']) for r in solutions)==[13,16,18,21])
 check('interlace solutions',sorted(Q(r['W_K']) for r in solutions if Q(r['spacing_K'])==1)==[16,18])
 check('separated solutions',sorted(Q(r['W_K']) for r in solutions if Q(r['spacing_K'])==2)==[13,21])
 check('common annual offset cancels',all(r['S']+12*529==r['i'] for r in load('C287_DATA.json')['rows']))
 return {'solutions':sorted(solutions,key=lambda r:Q(r['W_K'])),'failed_orders':failures,'method':'All order types solved exactly overQ; no chosen parameter scan','retained_bridge_matches':[13,16],'diagnostic_only_bridges':[18,21],'new_bridge_adopted':False}