You are installing a billboard and want it to have the largest height.  The billboard will have two steel supports, one on each side.  Each steel support must be an equal height.
You have a collection of rods which can be welded together.  For example, if you have rods of lengths 1, 2, and 3, you can weld them together to make a support of length 6.
Return the largest possible height of your billboard installation.  If you cannot support the billboard, return 0.
 
Example 1:
Input: [1,2,3,6]
Output: 6
Explanation: We have two disjoint subsets {1,2,3} and {6}, which have the same sum = 6.


Example 2:
Input: [1,2,3,4,5,6]
Output: 10
Explanation: We have two disjoint subsets {2,3,5} and {4,6}, which have the same sum = 10.



Example 3:
Input: [1,2]
Output: 0
Explanation: The billboard cannot be supported, so we return 0.


 
Note:

0 <= rods.length <= 20
1 <= rods[i] <= 1000
The sum of rods is at most 5000.

```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 exclu(self,nums):\n        # whether we can find two elements x and y in nums so that x & y == 0\n        n=len(nums)\n        if n<=1: return False\n        if n<=4:\n            for x in nums:\n                for y in nums:\n                    if x & y ==0: return True\n            return False\n\n        set0,set1=set(),set()\n        for x in nums:\n            if x & 1 : set1.add(x>>1)\n            else: set0.add(x>>1)\n        if set0:\n            if self.exclu(set0): return True\n            if set1:  return self.exclu2(set0,set1)\n        return False\n        \n    def exclu2(self,set0,set1):\n        # whether we can find x in set0 and y in set1 so that x & y == 0\n        # set0 and set1 are not empty\n        if len(set0)==0 or len(set1)==0: return False\n        if len(set1)==1:\n            x=next(iter(set1))\n            for y in set0: \n                if x & y ==0: return True\n            return False\n\n        if len(set0)==1:\n            x=next(iter(set0))\n            for y in set1: \n                if x & y ==0: return True\n            return False\n\n        if len(set0)<=8 and len(set1)<=8:\n            for x in set0:\n                for y in set1:\n                    if x & y ==0: return True\n            return False\n\n        set00,setx0,set01,set11=set(), set(), set(), set() \n        for x in set0:\n            setx0.add(x>>1)\n            if x & 1 ==0: set00.add(x>>1)\n        for x in set1:\n            if x & 1 : set11.add(x>>1)\n            else: set01.add(x>>1)\n        \n        if set00 and set11 and self.exclu2(set00,set11): return True\n        if set01 and setx0 and self.exclu2(set01,setx0): return True\n        return False\n\n    def tallestBillboard(self, rods: List[int]) -> int:\n        n=len(rods)\n        if n<=1: return 0\n        \n        t=sum(rods)\n        m=t//2\n        infos=[[] for _ in range(m+1)]\n        infos[0]=[0]\n        \n        b=1\n        for x in rods:\n            # print(f'x={x}')\n            for y in range(m-x,-1,-1):\n                comb=infos[y]\n                if comb:\n                    infos[x+y]+=[z+b for z in comb]\n            b*=2\n            # print(infos)\n        \n        for y in range(m,0,-1):\n            comb,n=infos[y],len(infos[y])\n            # print( (y, n, comb)  )\n            if n>=2:\n                # print( (y, n)  )\n                # print(n)\n                if self.exclu(comb): return y\n        return 0\n            \n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'tallestBillboard'
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: 6692