There are n soldiers standing in a line. Each soldier is assigned a unique rating value.
You have to form a team of 3 soldiers amongst them under the following rules:

Choose 3 soldiers with index (i, j, k) with rating (rating[i], rating[j], rating[k]).
A team is valid if:  (rating[i] < rating[j] < rating[k]) or (rating[i] > rating[j] > rating[k]) where (0 <= i < j < k < n).

Return the number of teams you can form given the conditions. (soldiers can be part of multiple teams).
 
Example 1:
Input: rating = [2,5,3,4,1]
Output: 3
Explanation: We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1). 

Example 2:
Input: rating = [2,1,3]
Output: 0
Explanation: We can't form any team given the conditions.

Example 3:
Input: rating = [1,2,3,4]
Output: 4

 
Constraints:

n == rating.length
1 <= n <= 200
1 <= rating[i] <= 10^5

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n    def increment_index(self, nums, index):\n        index += 1\n        while index < len(nums):\n            nums[index] += 1\n            index += (index & -index)\n\n    def prefix_sum(self, nums, index):\n        index += 1\n        current_sum = 0\n        while index > 0:\n            current_sum += nums[index]\n            index -= (index & -index)\n        return current_sum\n\n    def numTeams(self, rating):\n        if len(rating) < 3:\n            return 0\n\n        n = len(rating)\n        sorted_nums = rating.copy()\n        sorted_nums.sort()\n\n        index = {}\n        for i in range(n):\n            index[sorted_nums[i]] = i\n\n        fenwick_tree = [0] * (len(sorted_nums) + 1)\n\n        lesser_before = [0] * n\n        for i in range(n):\n            rate_i = rating[i]\n            index_i = index[rate_i]\n            lesser_before[i] = self.prefix_sum(fenwick_tree, index_i)\n            self.increment_index(fenwick_tree, index[rating[i]])\n\n        for i in range(len(fenwick_tree)):\n            fenwick_tree[i] = 0\n\n        lesser_after = [0] * n\n        for i in range(n - 1, -1, -1):\n            rate_i = rating[i]\n            index_i = index[rate_i]\n            lesser_after[i] = self.prefix_sum(fenwick_tree, index_i)\n            self.increment_index(fenwick_tree, index[rating[i]])\n\n        num_teams = 0\n        for i in range(n - 1):\n            num_teams += lesser_before[i] * (n - 1 - i - lesser_after[i])\n            num_teams += (i - lesser_before[i]) * lesser_after[i]\n\n        return num_teams\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numTeams'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6750