#find_steady: a recursive solution
#   with an envelope function
def find_steady(L):
    ''' find L[i]==i or return None, recursive '''
    return find_steady_rec2(L,0,len(L)-1)

def find_steady_rec2(L,l,r):
    if l>r:
        return None
    mid = (l+r)//2
    if L[mid]==mid:
        return mid
    elif L[mid]<mid:#go right
        return find_steady_rec2(L,mid+1,r)
    else:           #go left
        return find_steady_rec2(L,l,mid-1)



#find_steady: a recursive solution
def find_steady_iter(L):
    ''' find L[i]==i or return None, iterative '''
    l = 0
    r = len(L) - 1
    while l <= r:
        mid = (l+r)//2
        if L[mid] == mid:
            return mid
        elif L[mid] < mid:
            l = mid+1
        else:
            r = mid-1
    return None