Given an integer array A, and an integer target, return the number of tuples i, j, k  such that i < j < k and A[i] + A[j] + A[k] == target.
As the answer can be very large, return it modulo 109 + 7.
 
Example 1:
Input: A = [1,1,2,2,3,3,4,4,5,5], target = 8
Output: 20
Explanation: 
Enumerating by the values (A[i], A[j], A[k]):
(1, 2, 5) occurs 8 times;
(1, 3, 4) occurs 8 times;
(2, 2, 4) occurs 2 times;
(2, 3, 3) occurs 2 times.

Example 2:
Input: A = [1,1,2,2,2,2], target = 5
Output: 12
Explanation: 
A[i] = 1, A[j] = A[k] = 2 occurs 12 times:
We choose one 1 from [1,1] in 2 ways,
and two 2s from [2,2,2,2] in 6 ways.

 
Constraints:

3 <= A.length <= 3000
0 <= A[i] <= 100
0 <= target <= 300

```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 threeSumMulti(self, A: List[int], target: int) -> int:\n#         # 1 enumerating\n#         # 2. backtracking\n#         if len(A)<3:\n#             return 0\n        \n#         A.sort()\n#         self.target = target\n#         self.backtrace(A, [])\n#         #print(A,self.target, self.allPaths)\n#         return len(self.allPaths)\n    \n#     def __init__(self):\n#         self.allPaths = []\n    \n#     def backtrace(self, nums, path):\n#         if len(path) == 3:\n#             if sum(path)==self.target:\n#                 self.allPaths.append(path)\n#             return\n# #         if len(path) > 3:\n# #             return\n#         #print(len(nums), path)\n#         for i in range(len(nums)):\n#             if sum(path) <= self.target:\n#                 uPath = path + [nums[i]]\n#                 #print(uPath)\n#                 # allocate remaining\n#                 uNums = nums[i+1:]\n#                 self.backtrace(uNums, uPath)\nimport math\nimport collections\nclass Solution:\n\n    def threeSumMulti(self, A: List[int], target: int) -> int:\n        c1 = collections.Counter(A)\n        arr = []\n        triples = set()\n        for i in sorted(c1.keys()):\n            arr += [i] * min(3, c1[i])\n            \n        # find candidates from arr\n        for i in range(len(arr)-2):\n            j = i + 1\n            k = len(arr) - 1\n            twoSum = target - arr[i]\n            \n            while j<k:\n                tmp = arr[j] + arr[k]\n                if tmp == twoSum:\n                    triples.add((arr[i], arr[j], arr[k]))\n                    k -= 1\n                elif tmp >= twoSum:\n                    while k-1 > 0 and arr[k-1] == arr[k]:\n                        k -= 1\n                    k -= 1\n                else:\n                    while j+1 < len(arr)-1 and arr[j+1] == arr[j]:\n                        j += 1\n                    j += 1\n        # count the combinations\n        tot = 0\n        mod = 10**9 + 7\n        #print(triples)\n        for t in triples:\n            c2 = collections.Counter(t)\n            combs = 1\n            for k in list(c2.keys()):\n                combs *= math.comb(c1[k], c2[k])\n                combs %= mod\n            tot += combs\n            tot %= mod\n        return tot\n\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'threeSumMulti'
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: 13201