Given an array of integers nums and an integer target.
Return the number of non-empty subsequences of nums such that the sum of the minimum and maximum element on it is less or equal than target.
Since the answer may be too large, return it modulo 10^9 + 7.
 
Example 1:
Input: nums = [3,5,6,7], target = 9
Output: 4
Explanation: There are 4 subsequences that satisfy the condition.
[3] -> Min value + max value <= target (3 + 3 <= 9)
[3,5] -> (3 + 5 <= 9)
[3,5,6] -> (3 + 6 <= 9)
[3,6] -> (3 + 6 <= 9)

Example 2:
Input: nums = [3,3,6,8], target = 10
Output: 6
Explanation: There are 6 subsequences that satisfy the condition. (nums can have repeated numbers).
[3] , [3] , [3,3], [3,6] , [3,6] , [3,3,6]
Example 3:
Input: nums = [2,3,3,4,6,7], target = 12
Output: 61
Explanation: There are 63 non-empty subsequences, two of them don't satisfy the condition ([6,7], [7]).
Number of valid subsequences (63 - 2 = 61).

Example 4:
Input: nums = [5,2,4,1,7,6,8], target = 16
Output: 127
Explanation: All non-empty subset satisfy the condition (2^7 - 1) = 127
 
Constraints:

1 <= nums.length <= 10^5
1 <= nums[i] <= 10^6
1 <= target <= 10^6

```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    #Version 1: Sliding window\n    #Count the mismatch count\n    #TC: O(nlogn), SC: O(1)\n    '''\n    def numSubseq(self, nums: List[int], target: int) -> int:\n        from bisect import bisect_left\n        M = 10**9 + 7\n        mismatch = 0\n        nums.sort()\n        base = 0\n        for k in range(len(nums)):\n            if nums[k]*2 > target:\n                # while base > 0 and nums[base] + nums[k] > target:\n                #     base -= 1\n                # while base < k and nums[base] + nums[k] <= target:\n                #     base += 1\n                base = bisect_left(nums, target - nums[k]+1)\n                mismatch += pow(2, k-base, M)\n        return (pow(2, len(nums), M) - 1 - mismatch) % M\n    '''\n        \n    #Version 2: Sliding window\n    #Consider the valid count\n    #TC: O(nlogn), SC: O(1)\n    '''\n    def numSubseq(self, nums: List[int], target: int) -> int:\n        from bisect import bisect_left\n        M = 10**9 + 7\n        pows = [1]*(len(nums)+1)\n        for k in range(1, len(pows)):\n            pows[k] = (pows[k-1]<<1) % M\n        ans = 0\n        nums.sort()\n        base = 0\n        for k in range(len(nums)):\n            if nums[k]*2 <= target:\n                ans += pow(2, k, M)\n            else:\n                # while base > 0 and nums[base] + nums[k] > target:\n                #     base -= 1\n                # while base < k and nums[base] + nums[k] <= target:\n                #     base += 1\n                base = bisect_left(nums, target - nums[k]+1)\n                ans += (pow(2, base, M) - 1) * pow(2, k-base, M)\n            #print(k, base, ans)\n        return ans % M\n    '''\n    \n    #Version 2: Two sum\n    #Once the window is valid, we can add the number of subsequences with left or right number\n    #TC: O(nlogn), SC: O(1)\n    def numSubseq(self, nums: List[int], target: int) -> int:\n        M = 10**9 + 7\n        pows = [1]*(len(nums)+1)\n        for k in range(1, len(pows)):\n            pows[k] = (pows[k-1]<<1) % M\n        ans = 0\n        nums.sort()\n        left = 0\n        right = len(nums) - 1\n        while left <= right:\n            total = nums[left] + nums[right]\n            if total > target:\n                right -= 1\n            else:\n                ans = (ans + pows[right-left]) % M\n                left += 1\n        return ans\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numSubseq'
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: 5684