We are given a 2-dimensional grid. "." is an empty cell, "#" is a wall, "@" is the starting point, ("a", "b", ...) are keys, and ("A", "B", ...) are locks.
We start at the starting point, and one move consists of walking one space in one of the 4 cardinal directions.  We cannot walk outside the grid, or walk into a wall.  If we walk over a key, we pick it up.  We can't walk over a lock unless we have the corresponding key.
For some 1 <= K <= 6, there is exactly one lowercase and one uppercase letter of the first K letters of the English alphabet in the grid.  This means that there is exactly one key for each lock, and one lock for each key; and also that the letters used to represent the keys and locks were chosen in the same order as the English alphabet.
Return the lowest number of moves to acquire all keys.  If it's impossible, return -1.
 

Example 1:
Input: ["@.a.#","###.#","b.A.B"]
Output: 8


Example 2:
Input: ["@..aA","..B#.","....b"]
Output: 6


 
Note:

1 <= grid.length <= 30
1 <= grid[0].length <= 30
grid[i][j] contains only '.', '#', '@', 'a'-'f' and 'A'-'F'
The number of keys is in [1, 6].  Each key has a different letter and opens exactly one lock.

```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 = 'from collections import defaultdict, deque\nimport heapq, itertools\n\nclass Solution:\n    def shortestPathAllKeys(self, grid: List[str]) -> int:\n        # Brute Force + Permutations\n        # Time  complexity: O(R x C x A x A!), where R, C are the dimensions of the grid,\n        # and A is the maximum number of keys.\n        # Space compleixty: O(R x C + A!)\n        # R, C = map(len, (grid, grid[0]))\n        # location = {v: (r, c)\n        #             for r, row in enumerate(grid)\n        #             for c, v in enumerate(row)\n        #             if v not in \'.#\'}\n\n        # def neighbors(r, c):\n        #     for cr, cc in (r - 1, c), (r, c - 1), (r + 1, c), (r, c + 1):\n        #         if 0 <= cr < R and 0 <= cc < C:\n        #             yield cr, cc\n\n        # def bfs(source, target, keys=()):\n        #     sr, sc = location[source]\n        #     tr, tc = location[target]\n        #     seen = [[False] * C for _ in range(R)]\n        #     seen[sr][sc] = True\n        #     queue = deque([(sr, sc, 0)])\n        #     while queue:\n        #         r, c, d = queue.popleft()\n        #         if r == tr and c == tc: return d\n        #         for cr, cc in neighbors(r, c):\n        #             if not seen[cr][cc] and grid[cr][cc] != \'#\':\n        #                 if grid[cr][cc].isupper() and grid[cr][cc].lower() not in keys:\n        #                     continue\n        #                 queue.append((cr, cc, d + 1))\n        #                 seen[cr][cc] = True\n        #     return float(\\"inf\\")\n\n        # ans = float(\\"inf\\")\n        # keys = \\"\\".join(chr(ord(\'a\') + i) for i in range(len(location) // 2))\n        # for cand in itertools.permutations(keys):\n        #     bns = 0\n        #     for i, target in enumerate(cand):\n        #         source = cand[i - 1] if i > 0 else \'@\'\n        #         d = bfs(source, target, cand[:i])\n        #         bns += d\n        #         if bns >= ans: break\n        #     else:\n        #         ans = bns\n\n        # return ans if ans < float(\\"inf\\") else -1\n\n\n\n        # Points of Interest + Dijkstra\n        # Time  complexity: O(RC(2A + 1) + ElogN), where R, C are the dimensions of the grid, \n        # and A is the maximum number of keys, N = (2A + 1) x 2^A is the number of nodes when we\n        # perform Dijkstra\'s, and E = N x (2A + 1) is the maximum number of edges.\n        # Space complexity: ON(N)\n        R, C = len(grid), len(grid[0])\n\n        # The points of interest\n        location = {v: (r, c)\n                    for r, row in enumerate(grid)\n                    for c, v in enumerate(row)\n                    if v not in \'.#\'}\n\n        def neighbors(r, c):\n            for cr, cc in (r - 1, c), (r + 1, c), (r, c - 1), (r, c + 1):\n                if 0 <= cr < R and 0 <= cc < C:\n                    yield cr, cc\n\n        # The distance from source to each point of interest\n        def bfs_from(source):\n            r, c = location[source]\n            seen = [[False] * C for _ in range(R)]\n            seen[r][c] = True\n            queue = deque([(r, c, 0)])\n            dist = {}\n            while queue:\n                r, c, d = queue.popleft()\n                if source != grid[r][c] != \'.\':\n                    dist[grid[r][c]] = d\n                    continue # Stop walking from here if we reach a point of interest\n                for cr, cc in neighbors(r, c):\n                    if grid[cr][cc] != \'#\' and not seen[cr][cc]:\n                        seen[cr][cc] = True\n                        queue.append((cr, cc, d+1))\n            return dist        \n\n        dists = {place: bfs_from(place) for place in location}\n        target_state = 2 ** sum(p.islower() for p in location) - 1\n\n        #Dijkstra\n        pq = [(0, \'@\', 0)]\n        final_dist = defaultdict(lambda: float(\'inf\'))\n        final_dist[\'@\', 0] = 0\n        while pq:\n            d, place, state = heapq.heappop(pq)\n            if final_dist[place, state] < d: continue\n            if state == target_state: return d\n            for destination, d2 in list(dists[place].items()):\n                state2 = state\n                if destination.islower(): #key\n                    state2 |= 1 << ord(destination) - ord(\'a\')\n                elif destination.isupper(): #lock\n                    if not(state & (1 << ord(destination) - ord(\'A\'))): #no key\n                        continue\n\n                if d + d2 < final_dist[destination, state2]:\n                    final_dist[destination, state2] = d + d2\n                    heapq.heappush(pq, (d+d2, destination, state2))\n\n        return -1\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'shortestPathAllKeys'
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: 12114