
"""simple docstring"""







from __future__ import annotations




class 				A__ :

		def __init__(   self ,	_SCREAMING_SNAKE_CASE				):
			__lowerCAmelCase    :		Union[str, Any]		    =   TypeError(
			    'Matrices must be formed from a list of zero or more lists containing at '
			    'least one and the same number of values, each of which must be of type '
			    'int or float.'				)
			if len(_SCREAMING_SNAKE_CASE				) != 0:
				__lowerCAmelCase    :		List[str]		    =   len(rows[0]				)
				if cols == 0:
					raise error
				for row in rows:
					if len(_SCREAMING_SNAKE_CASE				) != cols:
						raise error
					for value in row:
						if not isinstance(_SCREAMING_SNAKE_CASE ,	(int, float)				):
							raise error
				__lowerCAmelCase    :		List[str]		    =   rows
			else:
				__lowerCAmelCase    :		List[str]		    =   []

		def 							__lowerCamelCase							(   self				):
			return [[row[i] for row in self.rows] for i in range(len(self.rows[0]				)				)]

		@property
		def 							__lowerCamelCase							(   self				):
			return len(self.rows				)

		@property
		def 							__lowerCamelCase							(   self				):
			return len(self.rows[0]				)

		@property
		def 							__lowerCamelCase							(   self				):
			return (self.num_rows, self.num_columns)

		@property
		def 							__lowerCamelCase							(   self				):
			return self.order[0] == self.order[1]

		def 							__lowerCamelCase							(   self				):
			__lowerCAmelCase    :		int		    =   [
			    [0 if column_num != row_num else 1 for column_num in range(self.num_rows				)]
			    for row_num in range(self.num_rows				)
			]
			return Matrix(_SCREAMING_SNAKE_CASE				)

		def 							__lowerCamelCase							(   self				):
			if not self.is_square:
				return 0
			if self.order == (0, 0):
				return 1
			if self.order == (1, 1):
				return int(self.rows[0][0]				)
			if self.order == (2, 2):
				return int(
				    (self.rows[0][0] * self.rows[1][1])
				    - (self.rows[0][1] * self.rows[1][0])				)
			else:
				return sum(
				    self.rows[0][column] * self.cofactors().rows[0][column]
				    for column in range(self.num_columns				)				)

		def 							__lowerCamelCase							(   self				):
			return bool(self.determinant()				)

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
			__lowerCAmelCase    :		List[str]		    =   [
			    [
			        self.rows[other_row][other_column]
			        for other_column in range(self.num_columns				)
			        if other_column != column
			    ]
			    for other_row in range(self.num_rows				)
			    if other_row != row
			]
			return Matrix(_SCREAMING_SNAKE_CASE				).determinant()

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
			if (row + column) % 2 == 0:
				return self.get_minor(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				)
			return -1 * self.get_minor(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				)

		def 							__lowerCamelCase							(   self				):
			return Matrix(
			    [
			        [self.get_minor(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				) for column in range(self.num_columns				)]
			        for row in range(self.num_rows				)
			    ]				)

		def 							__lowerCamelCase							(   self				):
			return Matrix(
			    [
			        [
			            self.minors().rows[row][column]
			            if (row + column) % 2 == 0
			            else self.minors().rows[row][column] * -1
			            for column in range(self.minors().num_columns				)
			        ]
			        for row in range(self.minors().num_rows				)
			    ]				)

		def 							__lowerCamelCase							(   self				):
			__lowerCAmelCase    :		Optional[int]		    =   [
			    [self.cofactors().rows[column][row] for column in range(self.num_columns				)]
			    for row in range(self.num_rows				)
			]
			return Matrix(_SCREAMING_SNAKE_CASE				)

		def 							__lowerCamelCase							(   self				):
			__lowerCAmelCase    :		List[str]		    =   self.determinant()
			if not determinant:
				raise TypeError('Only matrices with a non-zero determinant have an inverse'				)
			return self.adjugate() * (1 / determinant)

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

		def __str__(   self				):
			if self.num_rows == 0:
				return "[]"
			if self.num_rows == 1:
				return "[[" + ". ".join(str(self.rows[0]				)				) + "]]"
			return (
			    "["
			    + "\n ".join(
			        [
			            '[' + '. '.join([str(_SCREAMING_SNAKE_CASE				) for value in row]				) + '.]'
			            for row in self.rows
			        ]				)
			    + "]"
			)

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE = None				):
			__lowerCAmelCase    :		Optional[int]		    =   TypeError('Row must be a list containing all ints and/or floats'				)
			if not isinstance(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
				raise type_error
			for value in row:
				if not isinstance(_SCREAMING_SNAKE_CASE ,	(int, float)				):
					raise type_error
			if len(_SCREAMING_SNAKE_CASE				) != self.num_columns:
				raise ValueError(
				    'Row must be equal in length to the other rows in the matrix'				)
			if position is None:
				self.rows.append(_SCREAMING_SNAKE_CASE				)
			else:
				__lowerCAmelCase    :		List[Any]		    =   self.rows[0:position] + [row] + self.rows[position:]

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE = None				):
			__lowerCAmelCase    :		str		    =   TypeError(
			    'Column must be a list containing all ints and/or floats'				)
			if not isinstance(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
				raise type_error
			for value in column:
				if not isinstance(_SCREAMING_SNAKE_CASE ,	(int, float)				):
					raise type_error
			if len(_SCREAMING_SNAKE_CASE				) != self.num_rows:
				raise ValueError(
				    'Column must be equal in length to the other columns in the matrix'				)
			if position is None:
				__lowerCAmelCase    :		Optional[int]		    =   [self.rows[i] + [column[i]] for i in range(self.num_rows				)]
			else:
				__lowerCAmelCase    :		Optional[int]		    =   [
				    self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
				    for i in range(self.num_rows				)
				]

		def __eq__(   self ,	_SCREAMING_SNAKE_CASE				):
			if not isinstance(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
				return NotImplemented
			return self.rows == other.rows

		def __ne__(   self ,	_SCREAMING_SNAKE_CASE				):
			return not self == other

		def __neg__(   self				):
			return self * -1

		def __add__(   self ,	_SCREAMING_SNAKE_CASE				):
			if self.order != other.order:
				raise ValueError('Addition requires matrices of the same order'				)
			return Matrix(
			    [
			        [self.rows[i][j] + other.rows[i][j] for j in range(self.num_columns				)]
			        for i in range(self.num_rows				)
			    ]				)

		def __sub__(   self ,	_SCREAMING_SNAKE_CASE				):
			if self.order != other.order:
				raise ValueError('Subtraction requires matrices of the same order'				)
			return Matrix(
			    [
			        [self.rows[i][j] - other.rows[i][j] for j in range(self.num_columns				)]
			        for i in range(self.num_rows				)
			    ]				)

		def __mul__(   self ,	_SCREAMING_SNAKE_CASE				):
			if isinstance(_SCREAMING_SNAKE_CASE ,	(int, float)				):
				return Matrix(
				    [[int(element * other				) for element in row] for row in self.rows]				)
			elif isinstance(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
				if self.num_columns != other.num_rows:
					raise ValueError(
					    'The number of columns in the first matrix must '
					    'be equal to the number of rows in the second'				)
				return Matrix(
				    [
				        [Matrix.dot_product(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				) for column in other.columns()]
				        for row in self.rows
				    ]				)
			else:
				raise TypeError(
				    'A Matrix can only be multiplied by an int, float, or another matrix'				)

		def __pow__(   self ,	_SCREAMING_SNAKE_CASE				):
			if not isinstance(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
				raise TypeError('A Matrix can only be raised to the power of an int'				)
			if not self.is_square:
				raise ValueError('Only square matrices can be raised to a power'				)
			if other == 0:
				return self.identity()
			if other < 0:
				if self.is_invertable():
					return self.inverse() ** (-other)
				raise ValueError(
				    'Only invertable matrices can be raised to a negative power'				)
			__lowerCAmelCase    :		Tuple		    =   self
			for _ in range(other - 1				):
				result *= self
			return result







		@classmethod
		def 							__lowerCamelCase							(   cls ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
			return sum(row[i] * column[i] for i in range(len(_SCREAMING_SNAKE_CASE				)				)				)


if __name__ == "__main__":
							import doctest

							doctest.testmod()

"""simple docstring"""







from __future__ import annotations

from collections import Counter
from random import random




class 				A__ :

		def __init__(   self				):
			__lowerCAmelCase    :		Any		    =   {}

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE				):
			__lowerCAmelCase    :		Dict		    =   {}

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
			if nodea not in self.connections:
				self.add_node(_SCREAMING_SNAKE_CASE				)
			if nodea not in self.connections:
				self.add_node(_SCREAMING_SNAKE_CASE				)
			__lowerCAmelCase    :		Optional[Any]		    =   probability

		def 							__lowerCamelCase							(   self				):
			return list(self.connections				)







		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE				):
			__lowerCAmelCase    :		List[str]		    =   0
			__lowerCAmelCase    :		List[Any]		    =   random()

			for dest in self.connections[node]:
				current_probability += self.connections[node][dest]
				if current_probability > random_value:
					return dest
			return ""



def 					__lowerCAmelCase       (_UpperCamelCase ,						_UpperCamelCase ,						_UpperCamelCase	):
	__lowerCAmelCase    :		Optional[Any]		    =   MarkovChainGraphUndirectedUnweighted()

	for nodea, nodea, probability in transitions:
		graph.add_transition_probability(_UpperCamelCase ,						_UpperCamelCase ,						_UpperCamelCase	)

	__lowerCAmelCase    :		str		    =   Counter(graph.get_nodes()	)
	__lowerCAmelCase    :		Tuple		    =   start

	for _ in range(_UpperCamelCase	):
		__lowerCAmelCase    :		int		    =   graph.transition(_UpperCamelCase	)
		visited[node] += 1

	return visited


if __name__ == "__main__":
							import doctest

							doctest.testmod()