'''simple docstring'''




from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload


class       UpperCAmelCase__  :






   """simple docstring"""
   def __init__(self       ,				_a = None				)    ->     None:
        if components is None:
             lowercase_					:       Dict     				=					[]
        lowercase_					:       Optional[int]     				=					list(_a				)
   def __len__(self				)    ->     int:
        return len(self.__components				)
   def __str__(self				)    ->     str:
        return "(" + ",".join(map(_a       ,				self.__components				)				) + ")"
   def __add__(self       ,				_a				)    ->     Vector:
        lowercase_					:       List[Any]     				=					len(self				)
        if size == len(_a				):
             lowercase_					:       List[str]     				=					[self.__components[i] + other.component(_a				) for i in range(_a				)]
             return Vector(_a				)
        else:
             raise Exception('must have the same size'				)
   def __sub__(self       ,				_a				)    ->     Vector:
        lowercase_					:       Optional[int]     				=					len(self				)
        if size == len(_a				):
             lowercase_					:       Any     				=					[self.__components[i] - other.component(_a				) for i in range(_a				)]
             return Vector(_a				)
        else:  # error case
             raise Exception('must have the same size'				)
   @overload
   def __mul__(self       ,				_a				)    ->     Vector:
        ...
   @overload
   def __mul__(self       ,				_a				)    ->     float:
        ...
   def __mul__(self       ,				_a				)    ->     float | Vector:
        if isinstance(_a       ,				(float, int)				):
             lowercase_					:       Optional[int]     				=					[c * other for c in self.__components]
             return Vector(_a				)
        elif isinstance(_a       ,				_a				) and len(self				) == len(_a				):
             lowercase_					:       Union[str, Any]     				=					len(self				)
             lowercase_					:       str     				=					[self.__components[i] * other.component(_a				) for i in range(_a				)]
             return sum(_a				)
        else:  # error case
             raise Exception('invalid operand!'				)
   def 		_lowerCamelCase (self				)    ->     Vector:
        return Vector(self.__components				)
   def 		_lowerCamelCase (self       ,				_a				)    ->     float:
        if isinstance(_a       ,				_a				) and -len(self.__components				) <= i < len(self.__components				):
             return self.__components[i]
        else:
             raise Exception('index out of range'				)
   def 		_lowerCamelCase (self       ,				_a       ,				_a				)    ->     None:
        assert -len(self.__components				) <= pos < len(self.__components				)
        lowercase_					:       Dict     				=					value
   def 		_lowerCamelCase (self				)    ->     float:
        if len(self.__components				) == 0:
             raise Exception('Vector is empty'				)
        lowercase_					:       str     				=					[c**2 for c in self.__components]
        return math.sqrt(sum(_a				)				)





   def 		_lowerCamelCase (self       ,				_a       ,				_a = False				)    ->     float:
        lowercase_					:       Any     				=					self * other
        lowercase_					:       Union[str, Any]     				=					self.euclidean_length() * other.euclidean_length()
        if deg:
             return math.degrees(math.acos(num / den				)				)
        else:
             return math.acos(num / den				)



def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_							):
     assert isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)
     return Vector([0] * dimension							)

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							):
     assert isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							) and (isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							))
     lowercase_					:       Dict     				=					[0] * dimension
     lowercase_					:       Optional[Any]     				=					1
     return Vector(SCREAMING_SNAKE_CASE_							)

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							):
     assert (
         isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)
         and isinstance(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)
         and (isinstance(SCREAMING_SNAKE_CASE_  ,       (int, float)							))
     )
     return x * scalar + y

def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							):
     random.seed(SCREAMING_SNAKE_CASE_							)
     lowercase_					:       Optional[int]     				=					[random.randint(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							) for _ in range(SCREAMING_SNAKE_CASE_							)]
     return Vector(SCREAMING_SNAKE_CASE_							)


class       UpperCAmelCase__  :






   """simple docstring"""
   def __init__(self       ,				_a       ,				_a       ,				_a				)    ->     None:
        lowercase_					:       List[Any]     				=					matrix
        lowercase_					:       Optional[Any]     				=					w
        lowercase_					:       List[Any]     				=					h
   def __str__(self				)    ->     str:
        lowercase_					:       Optional[Any]     				=					''
        for i in range(self.__height				):
             ans += "|"
             for j in range(self.__width				):
                  if j < self.__width - 1:
                       ans += str(self.__matrix[i][j]				) + ","
                  else:
                       ans += str(self.__matrix[i][j]				) + "|\n"
        return ans
   def __add__(self       ,				_a				)    ->     Matrix:
        if self.__width == other.width() and self.__height == other.height():
             lowercase_					:       List[str]     				=					[]
             for i in range(self.__height				):
                  lowercase_					:       Tuple     				=					[
                      self.__matrix[i][j] + other.component(_a       ,				_a				)
                      for j in range(self.__width				)
                  ]
                  matrix.append(_a				)
             return Matrix(_a       ,				self.__width       ,				self.__height				)
        else:
             raise Exception('matrix must have the same dimension!'				)
   def __sub__(self       ,				_a				)    ->     Matrix:
        if self.__width == other.width() and self.__height == other.height():
             lowercase_					:       Dict     				=					[]
             for i in range(self.__height				):
                  lowercase_					:       Tuple     				=					[
                      self.__matrix[i][j] - other.component(_a       ,				_a				)
                      for j in range(self.__width				)
                  ]
                  matrix.append(_a				)
             return Matrix(_a       ,				self.__width       ,				self.__height				)
        else:
             raise Exception('matrices must have the same dimension!'				)
   @overload
   def __mul__(self       ,				_a				)    ->     Matrix:
        ...
   @overload
   def __mul__(self       ,				_a				)    ->     Vector:
        ...
   def __mul__(self       ,				_a				)    ->     Vector | Matrix:
        if isinstance(_a       ,				_a				):  # matrix-vector
             if len(_a				) == self.__width:
                  lowercase_					:       Optional[Any]     				=					zero_vector(self.__height				)
                  for i in range(self.__height				):
                       lowercase_					:       str     				=					[
                           self.__matrix[i][j] * other.component(_a				)
                           for j in range(self.__width				)
                       ]
                       ans.change_component(_a       ,				sum(_a				)				)
                  return ans
             else:
                  raise Exception(
                      'vector must have the same size as the '
                      'number of columns of the matrix!'				)
        elif isinstance(_a       ,				(int, float)				):  # matrix-scalar
             lowercase_					:       Union[str, Any]     				=					[
                 [self.__matrix[i][j] * other for j in range(self.__width				)]
                 for i in range(self.__height				)
             ]
             return Matrix(_a       ,				self.__width       ,				self.__height				)
        return None
   def 		_lowerCamelCase (self				)    ->     int:
        return self.__height
   def 		_lowerCamelCase (self				)    ->     int:
        return self.__width
   def 		_lowerCamelCase (self       ,				_a       ,				_a				)    ->     float:
        if 0 <= x < self.__height and 0 <= y < self.__width:
             return self.__matrix[x][y]
        else:
             raise Exception('change_component: indices out of bounds'				)
   def 		_lowerCamelCase (self       ,				_a       ,				_a       ,				_a				)    ->     None:
        if 0 <= x < self.__height and 0 <= y < self.__width:
             lowercase_					:       Optional[Any]     				=					value
        else:
             raise Exception('change_component: indices out of bounds'				)
   def 		_lowerCamelCase (self       ,				_a       ,				_a				)    ->     float:
        if self.__height != self.__width:
             raise Exception('Matrix is not square'				)
        lowercase_					:       Dict     				=					self.__matrix[:x] + self.__matrix[x + 1 :]
        for i in range(len(_a				)				):
             lowercase_					:       Tuple     				=					minor[i][:y] + minor[i][y + 1 :]
        return Matrix(_a       ,				self.__width - 1       ,				self.__height - 1				).determinant()
   def 		_lowerCamelCase (self       ,				_a       ,				_a				)    ->     float:
        if self.__height != self.__width:
             raise Exception('Matrix is not square'				)
        if 0 <= x < self.__height and 0 <= y < self.__width:
             return (-1) ** (x + y) * self.minor(_a       ,				_a				)
        else:
             raise Exception('Indices out of bounds'				)





   def 		_lowerCamelCase (self				)    ->     float:
        if self.__height != self.__width:
             raise Exception('Matrix is not square'				)
        if self.__height < 1:
             raise Exception('Matrix has no element'				)
        elif self.__height == 1:
             return self.__matrix[0][0]
        elif self.__height == 2:
             return (
                 self.__matrix[0][0] * self.__matrix[1][1]
                 - self.__matrix[0][1] * self.__matrix[1][0]
             )
        else:
             lowercase_					:       int     				=					[
                 self.__matrix[0][y] * self.cofactor(0       ,				_a				) for y in range(self.__width				)
             ]
             return sum(_a				)



def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_							):
     lowercase_					:       list[list[float]]     				=					[[0] * n for _ in range(SCREAMING_SNAKE_CASE_							)]
     return Matrix(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)






def _UpperCamelCase   (       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							):
     random.seed(SCREAMING_SNAKE_CASE_							)
     lowercase_					:       list[list[float]]     				=					[
         [random.randint(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							) for _ in range(SCREAMING_SNAKE_CASE_							)] for _ in range(SCREAMING_SNAKE_CASE_							)
     ]
     return Matrix(SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_  ,       SCREAMING_SNAKE_CASE_							)





'''simple docstring'''




class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   pass



class       UpperCAmelCase__  (       _snake_case			):






   """simple docstring"""
   pass



class       UpperCAmelCase__  :






   """simple docstring"""
   def __init__(self				)    ->     List[str]:
        lowercase_					:       Optional[int]     				=					[
            [],
            [],
            [],
        ]
   def 		_lowerCamelCase (self       ,				_a       ,				_a				)    ->     None:
        try:
             if len(self.queues[priority]				) >= 100:
                  raise OverflowError('Maximum queue size is 100'				)
             self.queues[priority].append(_a				)
        except IndexError:
             raise ValueError('Valid priorities are 0, 1, and 2'				)
   def 		_lowerCamelCase (self				)    ->     int:
        for queue in self.queues:
             if queue:
                  return queue.pop(0				)
        raise UnderFlowError('All queues are empty'				)





   def __str__(self				)    ->     str:
        return "\n".join(f'''Priority {i}: {q}''' for i, q in enumerate(self.queues				)				)



class       UpperCAmelCase__  :






   """simple docstring"""
   def __init__(self				)    ->     str:
        lowercase_					:       List[Any]     				=					[]
   def 		_lowerCamelCase (self       ,				_a				)    ->     None:
        if len(self.queue				) == 100:
             raise OverFlowError('Maximum queue size is 100'				)
        self.queue.append(_a				)
   def 		_lowerCamelCase (self				)    ->     int:
        if not self.queue:
             raise UnderFlowError('The queue is empty'				)
        else:
             lowercase_					:       Tuple     				=					min(self.queue				)
             self.queue.remove(_a				)
             return data





   def __str__(self				)    ->     str:
        return str(self.queue				)



def _UpperCamelCase   (       ):
     lowercase_					:       Any     				=					FixedPriorityQueue()
     fpq.enqueue(0  ,       10							)
     fpq.enqueue(1  ,       70							)
     fpq.enqueue(0  ,       100							)
     fpq.enqueue(2  ,       1							)
     fpq.enqueue(2  ,       5							)
     fpq.enqueue(1  ,       7							)
     fpq.enqueue(2  ,       4							)
     fpq.enqueue(1  ,       64							)
     fpq.enqueue(0  ,       128							)
     print(SCREAMING_SNAKE_CASE_							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(SCREAMING_SNAKE_CASE_							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)
     print(fpq.dequeue()							)

def _UpperCamelCase   (       ):
     lowercase_					:       Optional[int]     				=					ElementPriorityQueue()
     epq.enqueue(10							)
     epq.enqueue(70							)
     epq.enqueue(100							)
     epq.enqueue(1							)
     epq.enqueue(5							)
     epq.enqueue(7							)
     epq.enqueue(4							)
     epq.enqueue(64							)
     epq.enqueue(128							)
     print(SCREAMING_SNAKE_CASE_							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(SCREAMING_SNAKE_CASE_							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)
     print(epq.dequeue()							)


if __name__ == "__main__":
 fixed_priority_queue()
 element_priority_queue()




