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 numTeams(self, rating: List[int]) -> int:\n#         target=3\n#         self.count=0\n#         self.lst=[]\n#         def isSolution(candidate_count,candid):\n#             if candidate_count==target and ((candid[0]<candid[1]<candid[2]) or (candid[0]>candid[1]>candid[2])):\n#                 return True\n#             # if candidate_count==target:return False\n            \n#         def backtrack(index,rating,candidates,candidate_count):\n#             if isSolution(candidate_count,candidates):\n#                 self.count+=1\n#                 return \n#             if candidate_count<3:\n#                 for i in range(index,len(rating)):\n#                     candidates.append(rating[i])\n#                     backtrack(i+1,rating,candidates,candidate_count+1)\n#                     candidates.pop()\n        \n#         directions=[1,-1]\n#         #-1 means decreasing order\n#         # for direction in directions:\n#         backtrack(0,rating,[],0)\n#         # return len(self.lst)\n#         return self.count\n        \n        \n        \n        \n        target=3\n        self.count=0\n        self.lst=[]\n        def isSolution(candidate_count):\n            if candidate_count==target:\n                return True\n            \n        def isViable(rating,i,candidates,candidate_count,direction):\n            if i==0 or candidate_count==0:return True\n            \n            if direction==-1 and rating[i]<candidates[-1]:\n                return True\n            if direction==1 and rating[i]>candidates[-1]:\n                return True\n            return False\n            \n            \n        def backtrack(index,rating,candidates,candidate_count,direction):\n            if isSolution(candidate_count):\n                self.count+=1\n                return \n             \n            for i in range(index,len(rating)):\n                if isViable(rating,i,candidates,candidate_count,direction):\n                    candidates.append(rating[i])\n                    backtrack(i+1,rating,candidates,candidate_count+1,direction)\n                    candidates.pop()\n        \n        directions=[1,-1]\n        #-1 means decreasing order\n        for direction in directions:\n            backtrack(0,rating,[],0,direction)\n        # return len(self.lst)\n        return self.count\n                \n                \n                \n            \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: 7776