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def bisect_left(
    a: List[T], x: S, lo: int = 0, hi: Optional[int] = None, T: type, S: type
) -> int:
    """
    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.

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

def bisect_right(
    a: List[T], x: S, lo: int = 0, hi: Optional[int] = None, T: type, S: type
) -> int:
    """
    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.

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

def insort_left(
    a: List[T], x: S, lo: int = 0, hi: Optional[int] = None, T: type, S: type
):
    """
    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.

    Default values: lo=0, hi=None
    For now seq will use len(a) instead of None
    """
    lo = bisect_left(a, x, lo, hi)
    a.insert(lo, x)

def insort_right(
    a: List[T], x: S, lo: int = 0, hi: Optional[int] = None, T: type, S: type
):
    """
    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.

    Default values: lo=0, hi=None
    For now seq will use len(a) instead of None
    """
    lo = bisect_right(a, x, lo, hi)

    if lo == len(a):
        a.append(x)
    else:
        a.insert(lo, x)

bisect = bisect_right
insort = insort_right
