#John Kim
#Use whatever you like

P = Primes()
var('x1,x2,x3')
var('x')

#Definition of the function 
def multivariate_leading_term(f):
    Lst = []
    L = list(f.iterator())
    for i in L:
        Lst.append((i/i.substitute(x1=1,x2=1,x3=1)).substitute(x1=P.unrank(0),x2 =P.unrank(1),x3=P.unrank(2)))
    ind=Lst.index(max(Lst))
    return L[ind]


def Di(L):
    Diff=[]
    for i in range(len(L)-1):
        Diff.append(L[i+1]-L[i])
    return Diff

def IsPol(L):
    Diff=L
    for i in range(len(L)-4):
        Diff=Di(Diff)
        if set(Diff)==set([0]):
            return i
        return 'FAIL'

def GuessPol(L):
    deg=IsPol(L)
    if deg=='FAIL':
        return 'FAIL'
    a=list(var('a_%d' % i) for i in range(deg+1))
    P=sum(a[k]*x^k for k in range(deg+1))
    eq=list(P.substitute(x=i)==L[i] for i in range(len(L)))
    sol=solve(eq,a,solution_dict=True)
    return P.substitute(sol[0])