"""Bisection algorithms."""

def insort_right(a, x, lo=0, hi=None):
 """Insert item x in list a, and keep it sorted assuming a is sorted.

 If x is already in a, insert it to the right of the rightmost x.

 Optional args lo (default 0) and hi (default len(a)) bound the
 slice of a to be searched.
 """

 lo = bisect_right(a, x, lo, hi)
 a.insert(lo, x)

def bisect_right(a, x, lo=0, hi=None):
 """Return the index where to insert item x in list a, assuming a is sorted.

 The return value i is such that all e in a[:i] have e <= x, and all e in
 a[i:] have e > x. So if x already appears in the list, a.insert(x) will
 insert just after the rightmost x already there.

 Optional args lo (default 0) and hi (default len(a)) bound the
 slice of a to be searched.
 """

 if lo < 0:
 raise ValueError('lo must be non-negative')
 if hi is None:
 hi = len(a)
 while lo < hi:
 mid = (lo+hi)//2
 if x < a[mid]: hi = mid
 else: lo = mid+1
 return lo

def insort_left(a, x, lo=0, hi=None):
 """Insert item x in list a, and keep it sorted assuming a is sorted.

 If x is already in a, insert it to the left of the leftmost x.

 Optional args lo (default 0) and hi (default len(a)) bound the
 slice of a to be searched.
 """

 lo = bisect_left(a, x, lo, hi)
 a.insert(lo, x)


def bisect_left(a, x, lo=0, hi=None):
 """Return the index where to insert item x in list a, assuming a is sorted.

 The return value i is such that all e in a[:i] have e < x, and all e in
 a[i:] have e >= x. So if x already appears in the list, a.insert(x) will
 insert just before the leftmost x already there.

 Optional args lo (default 0) and hi (default len(a)) bound the
 slice of a to be searched.
 """

 if lo < 0:
 raise ValueError('lo must be non-negative')
 if hi is None:
 hi = len(a)
 while lo < hi:
 mid = (lo+hi)//2
 if a[mid] < x: lo = mid+1
 else: hi = mid
 return lo

# Overwrite above definitions with a fast C implementation
try:
 from _bisect import *
except ImportError:
 pass

# Create aliases
bisect = bisect_right
insort = insort_right