





import copy
import re




class 			__UpperCamelCase    :
     __A	:  Any							  =   """hp"""
     __A	:  Optional[int]							  =   {}
     __A	:  Tuple							  =   None





     @classmethod
     def 		UpperCamelCase(     cls				,							_UpperCamelCase				,							_UpperCamelCase					):
        _UpperCAmelCase           =   prefix
        _UpperCAmelCase           =   defaults
        cls.build_naming_info()





     @staticmethod
     def 		UpperCamelCase(     _UpperCamelCase				,							_UpperCamelCase					):
        if len(_UpperCamelCase					) == 0:
           return ""
        _UpperCAmelCase           =   None
        if any(char.isdigit() for char in word					):
           raise Exception(f'''Parameters should not contain numbers: \'{word}\' contains a number'''					)
        if word in info["short_word"]:
           return info["short_word"][word]
        for prefix_len in range(1				,							len(_UpperCamelCase					) + 1					):
           _UpperCAmelCase           =   word[:prefix_len]
           if prefix in info["reverse_short_word"]:
              continue
           else:
              _UpperCAmelCase           =   prefix
              break

        if short_word is None:
           # Paranoid fallback
           def int_to_alphabetic(_UpperCamelCase					):
              _UpperCAmelCase           =   ''''''
              while integer != 0:
                 _UpperCAmelCase           =   chr(ord('''A'''					) + integer % 10					) + s
                 integer //= 10
              return s

           _UpperCAmelCase           =   0
           while True:
              _UpperCAmelCase           =   word + '''#''' + int_to_alphabetic(_UpperCamelCase					)
              if sword in info["reverse_short_word"]:
                 continue
              else:
                 _UpperCAmelCase           =   sword
                 break

        _UpperCAmelCase           =   short_word
        _UpperCAmelCase           =   word
        return short_word





     @staticmethod
     def 		UpperCamelCase(     _UpperCamelCase				,							_UpperCamelCase					):
        _UpperCAmelCase           =   param_name.split('''_'''					)

        _UpperCAmelCase           =   [TrialShortNamer.shortname_for_word(_UpperCamelCase				,							_UpperCamelCase					) for word in words]

        # We try to create a separatorless short name, but if there is a collision we have to fallback
        # to a separated short name
        _UpperCAmelCase           =   ['''''', '''_''']

        for separator in separators:
           _UpperCAmelCase           =   separator.join(_UpperCamelCase					)
           if shortname not in info["reverse_short_param"]:
              _UpperCAmelCase           =   shortname
              _UpperCAmelCase           =   param_name
              return shortname

        return param_name





     @staticmethod
     def 		UpperCamelCase(     _UpperCamelCase				,							_UpperCamelCase					):
        _UpperCAmelCase           =   TrialShortNamer.shortname_for_key(_UpperCamelCase				,							_UpperCamelCase					)
        _UpperCAmelCase           =   short_name
        _UpperCAmelCase           =   param_name





     @classmethod
     def 		UpperCamelCase(     cls					):
        if cls.NAMING_INFO is not None:
           return

        _UpperCAmelCase           =   {
            '''short_word''': {},
            '''reverse_short_word''': {},
            '''short_param''': {},
            '''reverse_short_param''': {},
        }

        _UpperCAmelCase           =   list(cls.DEFAULTS.keys()					)

        for k in field_keys:
           cls.add_new_param_name(_UpperCamelCase				,							_UpperCamelCase					)

        _UpperCAmelCase           =   info





     @classmethod
     def 		UpperCamelCase(     cls				,							_UpperCamelCase					):
        cls.build_naming_info()
        assert cls.PREFIX is not None
        _UpperCAmelCase           =   [copy.copy(cls.PREFIX					)]

        for k, v in params.items():
           if k not in cls.DEFAULTS:
              raise Exception(f'''You should provide a default value for the param name {k} with value {v}'''					)
           if v == cls.DEFAULTS[k]:
              # The default value is not added to the name
              continue

           _UpperCAmelCase           =   cls.NAMING_INFO['''short_param'''][k]

           if isinstance(_UpperCamelCase				,							_UpperCamelCase					):
              _UpperCAmelCase           =   1 if v else 0

           _UpperCAmelCase           =   '''''' if isinstance(_UpperCamelCase				,							(int, float)					) else '''-'''
           _UpperCAmelCase           =   f'''{key}{sep}{v}'''
           name.append(_UpperCamelCase					)

        return "_".join(_UpperCamelCase					)





     @classmethod
     def 		UpperCamelCase(     cls				,							_UpperCamelCase					):
        _UpperCAmelCase           =   repr[len(cls.PREFIX					) + 1 :]
        if repr == "":
           _UpperCAmelCase           =   []
        else:
           _UpperCAmelCase           =   repr.split('''_'''					)

        _UpperCAmelCase           =   {}

        for value in values:
           if "-" in value:
              _UpperCAmelCase       ,      _UpperCAmelCase           =   value.split('''-'''					)
           else:
              _UpperCAmelCase           =   re.sub('''[0-9.]'''				,							''''''				,							_UpperCamelCase					)
              _UpperCAmelCase           =   float(re.sub('''[^0-9.]'''				,							''''''				,							_UpperCamelCase					)					)

           _UpperCAmelCase           =   cls.NAMING_INFO['''reverse_short_param'''][p_k]

           _UpperCAmelCase           =   p_v

        for k in cls.DEFAULTS:
           if k not in parameters:
              _UpperCAmelCase           =   cls.DEFAULTS[k]

        return parameters


"""simple docstring"""





from collections import deque
from math import floor
from random import random
from time import time







class 							SCREAMING_SNAKE_CASE   :






					"""simple docstring"""



					def __init__(			self     :							Union[str, Any]						):
						lowerCAmelCase__      :		List[str]    							=      {}







					def 			__lowerCAmelCase					(			self     :							Optional[Any]  ,lowercase_     :							Optional[Any]  ,lowercase_     :							str  ,lowercase_     :							List[str]=1						):
						if self.graph.get(lowercase_						):
							if self.graph[u].count([w, v]						) == 0:
								self.graph[u].append([w, v]						)
						else:
							lowerCAmelCase__      :		Optional[int]    							=      [[w, v]]
						if not self.graph.get(lowercase_						):
							lowerCAmelCase__      :		Union[str, Any]    							=      []







					def 			__lowerCAmelCase					(			self     :							List[Any]						):
						return list(self.graph						)







					def 			__lowerCAmelCase					(			self     :							Dict  ,lowercase_     :							Tuple  ,lowercase_     :							Tuple						):
						if self.graph.get(lowercase_						):
							for _ in self.graph[u]:
								if _[1] == v:
									self.graph[u].remove(lowercase_						)







					def 			__lowerCAmelCase					(			self     :							int  ,lowercase_     :							Any=-2  ,lowercase_     :							List[Any]=-1						):
						if s == d:
							return []
						lowerCAmelCase__      :		int    							=      []
						lowerCAmelCase__      :		List[Any]    							=      []
						if s == -2:
							lowerCAmelCase__      :		int    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		int    							=      s

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		List[Any]    							=      s
								for node in self.graph[s]:
									if visited.count(node[1]						) < 1:
										if node[1] == d:
											visited.append(lowercase_						)
											return visited
										else:
											stack.append(node[1]						)
											visited.append(node[1]						)
											lowerCAmelCase__      :		Optional[int]    							=      node[1]
											break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		str    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		Union[str, Any]    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return visited







					def 			__lowerCAmelCase					(			self     :							Dict  ,lowercase_     :							Optional[Any]=-1						):
						if c == -1:
							lowerCAmelCase__      :		Any    							=      floor(random() * 1_0_0_0_0						) + 1_0
						for i in range(lowercase_						):
							# every vertex has max 100 edges
							for _ in range(floor(random() * 1_0_2						) + 1						):
								lowerCAmelCase__      :		Any    							=      floor(random() * c						) + 1
								if n != i:
									self.add_pair(lowercase_  ,lowercase_  ,1						)







					def 			__lowerCAmelCase					(			self     :							Optional[Any]  ,lowercase_     :							Union[str, Any]=-2						):
						lowerCAmelCase__      :		Tuple    							=      deque()
						lowerCAmelCase__      :		Tuple    							=      []
						if s == -2:
							lowerCAmelCase__      :		Any    							=      list(self.graph						)[0]
						d.append(lowercase_						)
						visited.append(lowercase_						)
						while d:
							lowerCAmelCase__      :		Dict    							=      d.popleft()
							if len(self.graph[s]						) != 0:
								for node in self.graph[s]:
									if visited.count(node[1]						) < 1:
										d.append(node[1]						)
										visited.append(node[1]						)
						return visited







					def 			__lowerCAmelCase					(			self     :							Union[str, Any]  ,lowercase_     :							Optional[Any]						):
						lowerCAmelCase__      :		Optional[Any]    							=      0
						for x in self.graph:
							for y in self.graph[x]:
								if y[1] == u:
									count += 1
						return count







					def 			__lowerCAmelCase					(			self     :							str  ,lowercase_     :							Optional[int]						):
						return len(self.graph[u]						)







					def 			__lowerCAmelCase					(			self     :							List[str]  ,lowercase_     :							List[str]=-2						):
						lowerCAmelCase__      :		Union[str, Any]    							=      []
						lowerCAmelCase__      :		Any    							=      []
						if s == -2:
							lowerCAmelCase__      :		Any    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		Tuple    							=      s
						lowerCAmelCase__      :		Optional[int]    							=      []

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		Optional[Any]    							=      s
								for node in self.graph[s]:
									if visited.count(node[1]						) < 1:
										stack.append(node[1]						)
										visited.append(node[1]						)
										lowerCAmelCase__      :		Dict    							=      node[1]
										break

            # check if all the children are visited
							if s == ss:
								sorted_nodes.append(stack.pop()						)
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		Union[str, Any]    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		int    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return sorted_nodes







					def 			__lowerCAmelCase					(			self     :							List[Any]						):
						lowerCAmelCase__      :		str    							=      []
						lowerCAmelCase__      :		List[str]    							=      []
						lowerCAmelCase__      :		int    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		List[Any]    							=      -2
						lowerCAmelCase__      :		str    							=      []
						lowerCAmelCase__      :		Optional[Any]    							=      s
						lowerCAmelCase__      :		Union[str, Any]    							=      False
						lowerCAmelCase__      :		Optional[int]    							=      set()

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		List[str]    							=      s
								for node in self.graph[s]:
									if (
									    visited.count(node[1]						) > 0
									    and node[1] != parent
									    and indirect_parents.count(node[1]						) > 0
									    and not on_the_way_back
									):
										lowerCAmelCase__      :		List[Any]    							=      len(lowercase_						) - 1
										while len_stack >= 0:
											if stack[len_stack] == node[1]:
												anticipating_nodes.add(node[1]						)
												break
											else:
												anticipating_nodes.add(stack[len_stack]						)
												len_stack -= 1
									if visited.count(node[1]						) < 1:
										stack.append(node[1]						)
										visited.append(node[1]						)
										lowerCAmelCase__      :		Optional[int]    							=      node[1]
										break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								lowerCAmelCase__      :		str    							=      True
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		Tuple    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		int    							=      False
								indirect_parents.append(lowercase_						)
								lowerCAmelCase__      :		str    							=      s
								lowerCAmelCase__      :		Tuple    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return list(lowercase_						)







					def 			__lowerCAmelCase					(			self     :							int						):
						lowerCAmelCase__      :		Tuple    							=      []
						lowerCAmelCase__      :		Tuple    							=      []
						lowerCAmelCase__      :		Any    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		List[str]    							=      -2
						lowerCAmelCase__      :		int    							=      []
						lowerCAmelCase__      :		int    							=      s
						lowerCAmelCase__      :		List[Any]    							=      False
						lowerCAmelCase__      :		List[str]    							=      set()

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		Optional[int]    							=      s
								for node in self.graph[s]:
									if (
									    visited.count(node[1]						) > 0
									    and node[1] != parent
									    and indirect_parents.count(node[1]						) > 0
									    and not on_the_way_back
									):
										lowerCAmelCase__      :		List[Any]    							=      len(lowercase_						) - 1
										while len_stack_minus_one >= 0:
											if stack[len_stack_minus_one] == node[1]:
												anticipating_nodes.add(node[1]						)
												break
											else:
												return True
									if visited.count(node[1]						) < 1:
										stack.append(node[1]						)
										visited.append(node[1]						)
										lowerCAmelCase__      :		Optional[Any]    							=      node[1]
										break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								lowerCAmelCase__      :		List[str]    							=      True
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		Dict    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		Union[str, Any]    							=      False
								indirect_parents.append(lowercase_						)
								lowerCAmelCase__      :		Tuple    							=      s
								lowerCAmelCase__      :		Union[str, Any]    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return False







					def 			__lowerCAmelCase					(			self     :							Any  ,lowercase_     :							List[Any]=-2  ,lowercase_     :							Union[str, Any]=-1						):
						lowerCAmelCase__      :		Optional[int]    							=      time()
						self.dfs(lowercase_  ,lowercase_						)
						lowerCAmelCase__      :		Any    							=      time()
						return end - begin







					def 			__lowerCAmelCase					(			self     :							Any  ,lowercase_     :							Dict=-2						):
						lowerCAmelCase__      :		List[str]    							=      time()
						self.bfs(lowercase_						)
						lowerCAmelCase__      :		List[Any]    							=      time()
						return end - begin







class 							SCREAMING_SNAKE_CASE   :






					"""simple docstring"""



					def __init__(			self     :							Union[str, Any]						):
						lowerCAmelCase__      :		Tuple    							=      {}







					def 			__lowerCAmelCase					(			self     :							List[Any]  ,lowercase_     :							Optional[int]  ,lowercase_     :							Dict  ,lowercase_     :							Any=1						):
						# check if the u exists
						if self.graph.get(lowercase_						):
							# if there already is a edge
							if self.graph[u].count([w, v]						) == 0:
								self.graph[u].append([w, v]						)
						else:
							# if u does not exist
							lowerCAmelCase__      :		List[Any]    							=      [[w, v]]
						# add the other way
						if self.graph.get(lowercase_						):
							# if there already is a edge
							if self.graph[v].count([w, u]						) == 0:
								self.graph[v].append([w, u]						)
						else:
							# if u does not exist
							lowerCAmelCase__      :		int    							=      [[w, u]]







					def 			__lowerCAmelCase					(			self     :							Dict  ,lowercase_     :							Union[str, Any]  ,lowercase_     :							Dict						):
						if self.graph.get(lowercase_						):
							for _ in self.graph[u]:
								if _[1] == v:
									self.graph[u].remove(lowercase_						)
        # the other way round
						if self.graph.get(lowercase_						):
							for _ in self.graph[v]:
								if _[1] == u:
									self.graph[v].remove(lowercase_						)







					def 			__lowerCAmelCase					(			self     :							int  ,lowercase_     :							Optional[int]=-2  ,lowercase_     :							str=-1						):
						if s == d:
							return []
						lowerCAmelCase__      :		Optional[Any]    							=      []
						lowerCAmelCase__      :		Any    							=      []
						if s == -2:
							lowerCAmelCase__      :		Tuple    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		int    							=      s

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		Optional[int]    							=      s
								for node in self.graph[s]:
									if visited.count(node[1]						) < 1:
										if node[1] == d:
											visited.append(lowercase_						)
											return visited
										else:
											stack.append(node[1]						)
											visited.append(node[1]						)
											lowerCAmelCase__      :		List[str]    							=      node[1]
											break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		str    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		Dict    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return visited







					def 			__lowerCAmelCase					(			self     :							Union[str, Any]  ,lowercase_     :							List[Any]=-1						):
						if c == -1:
							lowerCAmelCase__      :		str    							=      floor(random() * 1_0_0_0_0						) + 1_0
						for i in range(lowercase_						):
							# every vertex has max 100 edges
							for _ in range(floor(random() * 1_0_2						) + 1						):
								lowerCAmelCase__      :		List[str]    							=      floor(random() * c						) + 1
								if n != i:
									self.add_pair(lowercase_  ,lowercase_  ,1						)







					def 			__lowerCAmelCase					(			self     :							Optional[Any]  ,lowercase_     :							List[Any]=-2						):
						lowerCAmelCase__      :		Tuple    							=      deque()
						lowerCAmelCase__      :		List[Any]    							=      []
						if s == -2:
							lowerCAmelCase__      :		Tuple    							=      list(self.graph						)[0]
						d.append(lowercase_						)
						visited.append(lowercase_						)
						while d:
							lowerCAmelCase__      :		Any    							=      d.popleft()
							if len(self.graph[s]						) != 0:
								for node in self.graph[s]:
									if visited.count(node[1]						) < 1:
										d.append(node[1]						)
										visited.append(node[1]						)
						return visited







					def 			__lowerCAmelCase					(			self     :							Any  ,lowercase_     :							Any						):
						return len(self.graph[u]						)







					def 			__lowerCAmelCase					(			self     :							Any						):
						lowerCAmelCase__      :		List[Any]    							=      []
						lowerCAmelCase__      :		int    							=      []
						lowerCAmelCase__      :		List[Any]    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		Optional[int]    							=      -2
						lowerCAmelCase__      :		List[str]    							=      []
						lowerCAmelCase__      :		Tuple    							=      s
						lowerCAmelCase__      :		Tuple    							=      False
						lowerCAmelCase__      :		Union[str, Any]    							=      set()

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		Union[str, Any]    							=      s
								for node in self.graph[s]:
									if (
									    visited.count(node[1]						) > 0
									    and node[1] != parent
									    and indirect_parents.count(node[1]						) > 0
									    and not on_the_way_back
									):
										lowerCAmelCase__      :		Tuple    							=      len(lowercase_						) - 1
										while len_stack >= 0:
											if stack[len_stack] == node[1]:
												anticipating_nodes.add(node[1]						)
												break
											else:
												anticipating_nodes.add(stack[len_stack]						)
												len_stack -= 1
									if visited.count(node[1]						) < 1:
										stack.append(node[1]						)
										visited.append(node[1]						)
										lowerCAmelCase__      :		Optional[int]    							=      node[1]
										break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								lowerCAmelCase__      :		Optional[int]    							=      True
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		str    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		Tuple    							=      False
								indirect_parents.append(lowercase_						)
								lowerCAmelCase__      :		Optional[int]    							=      s
								lowerCAmelCase__      :		Optional[int]    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return list(lowercase_						)







					def 			__lowerCAmelCase					(			self     :							Optional[Any]						):
						lowerCAmelCase__      :		Optional[Any]    							=      []
						lowerCAmelCase__      :		Union[str, Any]    							=      []
						lowerCAmelCase__      :		Tuple    							=      list(self.graph						)[0]
						stack.append(lowercase_						)
						visited.append(lowercase_						)
						lowerCAmelCase__      :		Dict    							=      -2
						lowerCAmelCase__      :		Union[str, Any]    							=      []
						lowerCAmelCase__      :		Optional[Any]    							=      s
						lowerCAmelCase__      :		Any    							=      False
						lowerCAmelCase__      :		Optional[int]    							=      set()

						while True:
							# check if there is any non isolated nodes
							if len(self.graph[s]						) != 0:
								lowerCAmelCase__      :		str    							=      s
								for node in self.graph[s]:
									if (
									    visited.count(node[1]						) > 0
									    and node[1] != parent
									    and indirect_parents.count(node[1]						) > 0
									    and not on_the_way_back
									):
										lowerCAmelCase__      :		str    							=      len(lowercase_						) - 1
										while len_stack_minus_one >= 0:
											if stack[len_stack_minus_one] == node[1]:
												anticipating_nodes.add(node[1]						)
												break
											else:
												return True
									if visited.count(node[1]						) < 1:
										stack.append(node[1]						)
										visited.append(node[1]						)
										lowerCAmelCase__      :		List[str]    							=      node[1]
										break

            # check if all the children are visited
							if s == ss:
								stack.pop()
								lowerCAmelCase__      :		List[Any]    							=      True
								if len(lowercase_						) != 0:
									lowerCAmelCase__      :		List[str]    							=      stack[len(lowercase_						) - 1]
							else:
								lowerCAmelCase__      :		List[str]    							=      False
								indirect_parents.append(lowercase_						)
								lowerCAmelCase__      :		Optional[int]    							=      s
								lowerCAmelCase__      :		List[Any]    							=      ss

							# check if se have reached the starting point
							if len(lowercase_						) == 0:
								return False







					def 			__lowerCAmelCase					(			self     :							str						):
						return list(self.graph						)







					def 			__lowerCAmelCase					(			self     :							Any  ,lowercase_     :							Optional[int]=-2  ,lowercase_     :							Any=-1						):
						lowerCAmelCase__      :		Dict    							=      time()
						self.dfs(lowercase_  ,lowercase_						)
						lowerCAmelCase__      :		List[str]    							=      time()
						return end - begin







					def 			__lowerCAmelCase					(			self     :							Union[str, Any]  ,lowercase_     :							int=-2						):
						lowerCAmelCase__      :		Dict    							=      time()
						self.bfs(lowercase_						)
						lowerCAmelCase__      :		Any    							=      time()
						return end - begin




