"""simple docstring"""






import dataclasses
import json
import sys
import types
from argparse import ArgumentDefaultsHelpFormatter, ArgumentParser, ArgumentTypeError
from copy import copy
from enum import Enum
from inspect import isclass
from pathlib import Path
from typing import Any, Callable, Dict, Iterable, List, Literal, NewType, Optional, Tuple, Union, get_type_hints

import yaml


_A        =     NewType('DataClass', Any)
_A        =     NewType('DataClassType', Any)




def     SCREAMING_SNAKE_CASE       (	__UpperCAmelCase							)				-> Tuple:
				if isinstance(UpperCamelCase__	,       UpperCamelCase__							):
								return v
				if v.lower() in ("yes", "true", "t", "y", "1"):
								return True
				elif v.lower() in ("no", "false", "f", "n", "0"):
								return False
				else:
								raise ArgumentTypeError(
								    F"""Truthy value expected: got {v} but expected one of yes/no, true/false, t/f, y/n, 1/0 (case insensitive)."""							)




def     SCREAMING_SNAKE_CASE       (	__UpperCAmelCase							)				-> Callable[[str], Any]:
				SCREAMING_SNAKE_CASE__			=					{str(UpperCamelCase__							): choice for choice in choices}
				return lambda __UpperCAmelCase							: str_to_choice.get(UpperCamelCase__	,       UpperCamelCase__							)




def     SCREAMING_SNAKE_CASE       (	*,
    __UpperCAmelCase = None	,       __UpperCAmelCase = None	,       __UpperCAmelCase = dataclasses.MISSING	,       __UpperCAmelCase = dataclasses.MISSING	,       __UpperCAmelCase = None	,       **__UpperCAmelCase	,       )				-> dataclasses.Field:
				if metadata is None:
								# Important, don't use as default param in function signature because dict is mutable and shared across function calls
								SCREAMING_SNAKE_CASE__			=					{}
				if aliases is not None:
								SCREAMING_SNAKE_CASE__			=					aliases
				if help is not None:
								SCREAMING_SNAKE_CASE__			=					help

				return dataclasses.field(metadata=UpperCamelCase__	,       default=UpperCamelCase__	,       default_factory=UpperCamelCase__	,       **UpperCamelCase__							)

class lowerCamelCase		(UpperCamelCase__   ):






					'''simple docstring'''


					a 						=		4_2



					def __init__(    self      :					List[Any] ,			_snake_case      :					List[Any] ,			**_snake_case      :					List[Any]		)				->							Optional[int]:
									# To make the default appear when using --help
									if "formatter_class" not in kwargs:
													SCREAMING_SNAKE_CASE__			=					ArgumentDefaultsHelpFormatter
									super().__init__(**__lowerCamelCase		)
									if dataclasses.is_dataclass(__lowerCamelCase		):
													SCREAMING_SNAKE_CASE__			=					[dataclass_types]
									SCREAMING_SNAKE_CASE__			=					list(__lowerCamelCase		)
									for dtype in self.dataclass_types:
													self._add_dataclass_arguments(__lowerCamelCase		)



					@staticmethod
					def       lowerCAmelCase_       (    _snake_case      :					Union[str, Any] ,			_snake_case      :					List[Any]		)				->							List[str]:
									SCREAMING_SNAKE_CASE__			=					F"""--{field.name}"""
									SCREAMING_SNAKE_CASE__			=					field.metadata.copy()
									# field.metadata is not used at all by Data Classes,
									# it is provided as a third-party extension mechanism.
									if isinstance(field.type ,			__lowerCamelCase		):
													raise RuntimeError(
													    "Unresolved type detected, which should have been done with the help of "
													    "`typing.get_type_hints` method by default"		)

									SCREAMING_SNAKE_CASE__			=					kwargs.pop("aliases" ,			[]		)
									if isinstance(__lowerCamelCase ,			__lowerCamelCase		):
													SCREAMING_SNAKE_CASE__			=					[aliases]

									SCREAMING_SNAKE_CASE__			=					getattr(field.type ,			"__origin__" ,			field.type		)
									if origin_type is Union or (hasattr(__lowerCamelCase ,			"UnionType"		) and isinstance(__lowerCamelCase ,			types.UnionType		)):
													if str not in field.type.__args__ and (
													    len(field.type.__args__		) != 2 or type(__lowerCamelCase		) not in field.type.__args__
													):
																	raise ValueError(
																	    "Only `Union[X, NoneType]` (i.e., `Optional[X]`) is allowed for `Union` because"
																	    " the argument parser only supports one type per argument."
																	    F""" Problem encountered in field '{field.name}'."""		)
													if type(__lowerCamelCase		) not in field.type.__args__:
																	# filter `str` in Union
																	SCREAMING_SNAKE_CASE__			=					field.type.__args__[0] if field.type.__args__[1] == str else field.type.__args__[1]
																	SCREAMING_SNAKE_CASE__			=					getattr(field.type ,			"__origin__" ,			field.type		)
													elif bool not in field.type.__args__:
																	# filter `NoneType` in Union (except for `Union[bool, NoneType]`)
																	SCREAMING_SNAKE_CASE__			=					(
																	    field.type.__args__[0] if isinstance(__lowerCamelCase ,			field.type.__args__[1]		) else field.type.__args__[1]
																	)
																	SCREAMING_SNAKE_CASE__			=					getattr(field.type ,			"__origin__" ,			field.type		)

        # A variable to store kwargs for a boolean field, if needed
        # so that we can init a `no_*` complement argument (see below)
									SCREAMING_SNAKE_CASE__			=					{}
									if origin_type is Literal or (isinstance(field.type ,			__lowerCamelCase		) and issubclass(field.type ,			__lowerCamelCase		)):
													if origin_type is Literal:
																	SCREAMING_SNAKE_CASE__			=					field.type.__args__
													else:
																	SCREAMING_SNAKE_CASE__			=					[x.value for x in field.type]

													SCREAMING_SNAKE_CASE__			=					make_choice_type_function(kwargs["choices"]		)

													if field.default is not dataclasses.MISSING:
																	SCREAMING_SNAKE_CASE__			=					field.default
													else:
																	SCREAMING_SNAKE_CASE__			=					True
									elif field.type is bool or field.type == Optional[bool]:
													# Copy the currect kwargs to use to instantiate a `no_*` complement argument below.
													# We do not initialize it here because the `no_*` alternative must be instantiated after the real argument
													SCREAMING_SNAKE_CASE__			=					copy(__lowerCamelCase		)

													# Hack because type=bool in argparse does not behave as we want.
													SCREAMING_SNAKE_CASE__			=					string_to_bool
													if field.type is bool or (field.default is not None and field.default is not dataclasses.MISSING):
																	# Default value is False if we have no default when of type bool.
																	SCREAMING_SNAKE_CASE__			=					False if field.default is dataclasses.MISSING else field.default
																	# This is the value that will get picked if we don't include --field_name in any way
																	SCREAMING_SNAKE_CASE__			=					default
																	# This tells argparse we accept 0 or 1 value after --field_name
																	SCREAMING_SNAKE_CASE__			=					"?"
																	# This is the value that will get picked if we do --field_name (without value)
																	SCREAMING_SNAKE_CASE__			=					True
									elif isclass(__lowerCamelCase		) and issubclass(__lowerCamelCase ,			__lowerCamelCase		):
													SCREAMING_SNAKE_CASE__			=					field.type.__args__[0]
													SCREAMING_SNAKE_CASE__			=					"+"
													if field.default_factory is not dataclasses.MISSING:
																	SCREAMING_SNAKE_CASE__			=					field.default_factory()
													elif field.default is dataclasses.MISSING:
																	SCREAMING_SNAKE_CASE__			=					True
									else:
													SCREAMING_SNAKE_CASE__			=					field.type
													if field.default is not dataclasses.MISSING:
																	SCREAMING_SNAKE_CASE__			=					field.default
													elif field.default_factory is not dataclasses.MISSING:
																	SCREAMING_SNAKE_CASE__			=					field.default_factory()
													else:
																	SCREAMING_SNAKE_CASE__			=					True
									parser.add_argument(__lowerCamelCase ,			*__lowerCamelCase ,			**__lowerCamelCase		)

									# Add a complement `no_*` argument for a boolean field AFTER the initial field has already been added.
									# Order is important for arguments with the same destination!
									# We use a copy of earlier kwargs because the original kwargs have changed a lot before reaching down
									# here and we do not need those changes/additional keys.
									if field.default is True and (field.type is bool or field.type == Optional[bool]):
													SCREAMING_SNAKE_CASE__			=					False
													parser.add_argument(F"""--no_{field.name}""" ,			action="store_false" ,			dest=field.name ,			**__lowerCamelCase		)



					def       lowerCAmelCase_       (    self      :					Optional[Any] ,			_snake_case      :					Any		)				->							Any:
									if hasattr(__lowerCamelCase ,			"_argument_group_name"		):
													SCREAMING_SNAKE_CASE__			=					self.add_argument_group(dtype._argument_group_name		)
									else:
													SCREAMING_SNAKE_CASE__			=					self

									try:
													SCREAMING_SNAKE_CASE__			=					get_type_hints(__lowerCamelCase		)
									except NameError:
													raise RuntimeError(
													    F"""Type resolution failed for {dtype}. Try declaring the class in global scope or """
													    "removing line of `from __future__ import annotations` which opts in Postponed "
													    "Evaluation of Annotations (PEP 563)"		)
									except TypeError as ex:
													# Remove this block when we drop Python 3.9 support
													if sys.version_info[:2] < (3, 10) and "unsupported operand type(s) for |" in str(__lowerCamelCase		):
																	SCREAMING_SNAKE_CASE__			=					".".join(map(__lowerCamelCase ,			sys.version_info[:3]		)		)
																	raise RuntimeError(
																	    F"""Type resolution failed for {dtype} on Python {python_version}. Try removing """
																	    "line of `from __future__ import annotations` which opts in union types as "
																	    "`X | Y` (PEP 604) via Postponed Evaluation of Annotations (PEP 563). To "
																	    "support Python versions that lower than 3.10, you need to use "
																	    "`typing.Union[X, Y]` instead of `X | Y` and `typing.Optional[X]` instead of "
																	    "`X | None`."		) from ex
													raise

									for field in dataclasses.fields(__lowerCamelCase		):
													if not field.init:
																	continue
													SCREAMING_SNAKE_CASE__			=					type_hints[field.name]
													self._parse_dataclass_field(__lowerCamelCase ,			__lowerCamelCase		)



					def       lowerCAmelCase_       (    self      :					Tuple ,			_snake_case      :					str=None ,			_snake_case      :					List[Any]=False ,			_snake_case      :					Any=True ,			_snake_case      :					List[str]=None ,			_snake_case      :					Union[str, Any]=None ,			)				->							Any:

									if args_file_flag or args_filename or (look_for_args_file and len(sys.argv		)):
													SCREAMING_SNAKE_CASE__			=					[]

													if args_filename:
																	args_files.append(Path(__lowerCamelCase		)		)
													elif look_for_args_file and len(sys.argv		):
																	args_files.append(Path(sys.argv[0]		).with_suffix(".args"		)		)

													# args files specified via command line flag should overwrite default args files so we add them last
													if args_file_flag:
																	# Create special parser just to extract the args_file_flag values
																	SCREAMING_SNAKE_CASE__			=					ArgumentParser()
																	args_file_parser.add_argument(__lowerCamelCase ,			type=__lowerCamelCase ,			action="append"		)

																	# Use only remaining args for further parsing (remove the args_file_flag)
																	SCREAMING_SNAKE_CASE__	,							SCREAMING_SNAKE_CASE__			=					args_file_parser.parse_known_args(args=__lowerCamelCase		)
																	SCREAMING_SNAKE_CASE__			=					vars(__lowerCamelCase		).get(args_file_flag.lstrip("-"		) ,			__lowerCamelCase		)

																	if cmd_args_file_paths:
																					args_files.extend([Path(__lowerCamelCase		) for p in cmd_args_file_paths]		)

													SCREAMING_SNAKE_CASE__			=					[]
													for args_file in args_files:
																	if args_file.exists():
																					file_args += args_file.read_text().split()

            # in case of duplicate arguments the last one has precedence
            # args specified via the command line should overwrite args from files, so we add them last
													SCREAMING_SNAKE_CASE__			=					file_args + args if args is not None else file_args + sys.argv[1:]
									SCREAMING_SNAKE_CASE__	,							SCREAMING_SNAKE_CASE__			=					self.parse_known_args(args=__lowerCamelCase		)
									SCREAMING_SNAKE_CASE__			=					[]
									for dtype in self.dataclass_types:
													SCREAMING_SNAKE_CASE__			=					{f.name for f in dataclasses.fields(__lowerCamelCase		) if f.init}
													SCREAMING_SNAKE_CASE__			=					{k: v for k, v in vars(__lowerCamelCase		).items() if k in keys}
													for k in keys:
																	delattr(__lowerCamelCase ,			__lowerCamelCase		)
													SCREAMING_SNAKE_CASE__			=					dtype(**__lowerCamelCase		)
													outputs.append(__lowerCamelCase		)
									if len(namespace.__dict__		) > 0:
													# additional namespace.
													outputs.append(__lowerCamelCase		)
									if return_remaining_strings:
													return (*outputs, remaining_args)
									else:
													if remaining_args:
																	raise ValueError(F"""Some specified arguments are not used by the HfArgumentParser: {remaining_args}"""		)

													return (*outputs,)



					def       lowerCAmelCase_       (    self      :					Dict ,			_snake_case      :					Dict ,			_snake_case      :					Tuple = False		)				->							List[Any]:
									SCREAMING_SNAKE_CASE__			=					set(args.keys()		)
									SCREAMING_SNAKE_CASE__			=					[]
									for dtype in self.dataclass_types:
													SCREAMING_SNAKE_CASE__			=					{f.name for f in dataclasses.fields(__lowerCamelCase		) if f.init}
													SCREAMING_SNAKE_CASE__			=					{k: v for k, v in args.items() if k in keys}
													unused_keys.difference_update(inputs.keys()		)
													SCREAMING_SNAKE_CASE__			=					dtype(**__lowerCamelCase		)
													outputs.append(__lowerCamelCase		)
									if not allow_extra_keys and unused_keys:
													raise ValueError(F"""Some keys are not used by the HfArgumentParser: {sorted(__lowerCamelCase		)}"""		)
									return tuple(__lowerCamelCase		)



					def       lowerCAmelCase_       (    self      :					int ,			_snake_case      :					Tuple ,			_snake_case      :					Dict = False		)				->							Union[str, Any]:
									with open(Path(__lowerCamelCase		) ,			encoding="utf-8"		) as open_json_file:
													SCREAMING_SNAKE_CASE__			=					json.loads(open_json_file.read()		)
									SCREAMING_SNAKE_CASE__			=					self.parse_dict(__lowerCamelCase ,			allow_extra_keys=__lowerCamelCase		)
									return tuple(__lowerCamelCase		)



					def       lowerCAmelCase_       (    self      :					Union[str, Any] ,			_snake_case      :					Optional[Any] ,			_snake_case      :					Optional[Any] = False		)				->							str:
									SCREAMING_SNAKE_CASE__			=					self.parse_dict(yaml.safe_load(Path(__lowerCamelCase		).read_text()		) ,			allow_extra_keys=__lowerCamelCase		)
									return tuple(__lowerCamelCase		)











from __future__ import annotations

from sys import maxsize
from typing import Generic, TypeVar

a__: Tuple							   =						TypeVar('T')


def       UpperCamelCase__(			UpperCamelCase__     :			int    )->int:
       return (position - 1) // 2


def       UpperCamelCase__(			UpperCamelCase__     :			int    )->int:
       return (2 * position) + 1


def       UpperCamelCase__(			UpperCamelCase__     :			int    )->int:
       return (2 * position) + 2




class        SCREAMING_SNAKE_CASE__					(		Generic[T] ):



 def __init__(				self					):
        A__		   =		[]
        A__		   =		{}
        A__		   =		0



 def __len__(				self					):
        return self.elements



 def __repr__(				self					):
        return str(self.heap					)



 def 							UpperCamelCase     (				self					):
        # Check if the priority queue is empty
        return self.elements == 0



 def 							UpperCamelCase     (				self,__lowerCamelCase,__lowerCamelCase					):
        # Add an element with given priority to the queue
        self.heap.append((elem, weight)					)
        A__		   =		self.elements
        self.elements += 1
        self._bubble_up(__lowerCamelCase					)



 def 							UpperCamelCase     (				self					):
        # Remove and return the element with lowest weight (highest priority)
        if self.elements > 1:
               self._swap_nodes(0,self.elements - 1					)
        A__   ,      A__		   =		self.heap.pop()
        del self.position_map[elem]
        self.elements -= 1
        if self.elements > 0:
               A__   ,      A__		   =		self.heap[0]
               self._bubble_down(__lowerCamelCase					)
        return elem



 def 							UpperCamelCase     (				self,__lowerCamelCase,__lowerCamelCase					):
        # Update the weight of the given key
        A__		   =		self.position_map[elem]
        A__		   =		(elem, weight)
        if position > 0:
               A__		   =		get_parent_position(__lowerCamelCase					)
               A__   ,      A__		   =		self.heap[parent_position]
               if parent_weight > weight:
                      self._bubble_up(__lowerCamelCase					)
               else:
                      self._bubble_down(__lowerCamelCase					)
        else:
               self._bubble_down(__lowerCamelCase					)



 def 							UpperCamelCase     (				self,__lowerCamelCase					):
        # Place a node at the proper position (upward movement) [to be used internally
        # only]
        A__		   =		self.position_map[elem]
        if curr_pos == 0:
               return None
        A__		   =		get_parent_position(__lowerCamelCase					)
        A__   ,      A__		   =		self.heap[curr_pos]
        A__   ,      A__		   =		self.heap[parent_position]
        if parent_weight > weight:
               self._swap_nodes(__lowerCamelCase,__lowerCamelCase					)
               return self._bubble_up(__lowerCamelCase					)
        return None



 def 							UpperCamelCase     (				self,__lowerCamelCase					):
        # Place a node at the proper position (downward movement) [to be used
        # internally only]
        A__		   =		self.position_map[elem]
        A__   ,      A__		   =		self.heap[curr_pos]
        A__		   =		get_child_left_position(__lowerCamelCase					)
        A__		   =		get_child_right_position(__lowerCamelCase					)
        if child_left_position < self.elements and child_right_position < self.elements:
               A__   ,      A__		   =		self.heap[child_left_position]
               A__   ,      A__		   =		self.heap[child_right_position]
               if child_right_weight < child_left_weight and child_right_weight < weight:
                      self._swap_nodes(__lowerCamelCase,__lowerCamelCase					)
                      return self._bubble_down(__lowerCamelCase					)
        if child_left_position < self.elements:
               A__   ,      A__		   =		self.heap[child_left_position]
               if child_left_weight < weight:
                      self._swap_nodes(__lowerCamelCase,__lowerCamelCase					)
                      return self._bubble_down(__lowerCamelCase					)
        else:
               return None
        if child_right_position < self.elements:
               A__   ,      A__		   =		self.heap[child_right_position]
               if child_right_weight < weight:
                      self._swap_nodes(__lowerCamelCase,__lowerCamelCase					)
                      return self._bubble_down(__lowerCamelCase					)
        return None




 def 							UpperCamelCase     (				self,__lowerCamelCase,__lowerCamelCase					):
        # Swap the nodes at the given positions
        A__		   =		self.heap[nodea_pos][0]
        A__		   =		self.heap[nodea_pos][0]
        A__   ,      A__		   =		(
            self.heap[nodea_pos],
            self.heap[nodea_pos],
        )
        A__		   =		nodea_pos
        A__		   =		nodea_pos




class        SCREAMING_SNAKE_CASE__					(		Generic[T] ):



 def __init__(				self					):
        A__		   =		{}
        A__		   =		0



 def __repr__(				self					):
        return str(self.connections					)



 def __len__(				self					):
        return self.nodes



 def 							UpperCamelCase     (				self,__lowerCamelCase					):
        # Add a node in the graph if it is not in the graph
        if node not in self.connections:
               A__		   =		{}
               self.nodes += 1




 def 							UpperCamelCase     (				self,__lowerCamelCase,__lowerCamelCase,__lowerCamelCase					):
        # Add an edge between 2 nodes in the graph
        self.add_node(__lowerCamelCase					)
        self.add_node(__lowerCamelCase					)
        A__		   =		weight
        A__		   =		weight


def       UpperCamelCase__(			UpperCamelCase__     :			GraphUndirectedWeighted[T]					,	)->tuple[dict[T, int], dict[T, T | None]]:
       A__		   =		{node: maxsize for node in graph.connections}
       A__		   =		{node: None for node in graph.connections}

       A__		   =		MinPriorityQueue()
       for node, weight in dist.items():
              priority_queue.push(UpperCamelCase__					,	UpperCamelCase__    )

       if priority_queue.is_empty():
              return dist, parent

       # initialization
       A__		   =		priority_queue.extract_min()
       A__		   =		0
       for neighbour in graph.connections[node]:
              if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
                     A__		   =		dist[node] + graph.connections[node][neighbour]
                     priority_queue.update_key(UpperCamelCase__					,	dist[neighbour]    )
                     A__		   =		node

    # running prim's algorithm
       while not priority_queue.is_empty():
              A__		   =		priority_queue.extract_min()
              for neighbour in graph.connections[node]:
                     if dist[neighbour] > dist[node] + graph.connections[node][neighbour]:
                            A__		   =		dist[node] + graph.connections[node][neighbour]
                            priority_queue.update_key(UpperCamelCase__					,	dist[neighbour]    )
                            A__		   =		node
       return dist, parent





