

"""simple docstring"""







import heapq
import sys

import numpy as np

__SCREAMING_SNAKE_CASE								=tuple[int, int]







class  UpperCamelCase	:
   def __init__(    self    )			->   List[str]:


        '''simple docstring'''


        lowercase_			:       Dict									=			[]
        lowercase_			:       Union[str, Any]									=			set()
   def     _UpperCAmelCase     (    self    )			->   Any:


        '''simple docstring'''


        if not self.empty():
             return self.elements[0][0]
        else:
             return float('inf'    )
   def     _UpperCAmelCase     (    self    )			->   str:


        '''simple docstring'''


        return len(self.elements    ) == 0
   def     _UpperCAmelCase     (    self  ,__UpperCamelCase  ,__UpperCamelCase    )			->   Optional[Any]:


        '''simple docstring'''


        if item not in self.set:
             heapq.heappush(self.elements  ,(priority, item)    )
             self.set.add(lowerCAmelCase_    )
        else:
             # update
             # print("update", item)
             lowercase_			:       Optional[int]									=			[]
             ((lowercase_)							,						(lowercase_))			:       List[str]									=			heapq.heappop(self.elements    )
             while x != item:
                  temp.append((pri, x)    )
                  ((lowercase_)							,						(lowercase_))			:       Any									=			heapq.heappop(self.elements    )
             temp.append((priority, item)    )
             for pro, xxx in temp:
                  heapq.heappush(self.elements  ,(pro, xxx)    )
   def     _UpperCAmelCase     (    self  ,__UpperCamelCase    )			->   Tuple:


        '''simple docstring'''


        if item in self.set:
             self.set.remove(lowerCAmelCase_    )
             lowercase_			:       str									=			[]
             ((lowercase_)							,						(lowercase_))			:       Any									=			heapq.heappop(self.elements    )
             while x != item:
                  temp.append((pro, x)    )
                  ((lowercase_)							,						(lowercase_))			:       Dict									=			heapq.heappop(self.elements    )
             for prito, yyy in temp:
                  heapq.heappush(self.elements  ,(prito, yyy)    )
   def     _UpperCAmelCase     (    self    )			->   Tuple:


        '''simple docstring'''


        return self.elements[0][1]







   def     _UpperCAmelCase     (    self    )			->   int:


        '''simple docstring'''


        ((lowercase_)							,						(lowercase_))			:       Tuple									=			heapq.heappop(self.elements    )
        self.set.remove(lowerCAmelCase_    )
        return (priority, item)






def     lowercase__( __SCREAMING_SNAKE_CASE							:  Dict  ,    __SCREAMING_SNAKE_CASE							:  List[Any]   ):
     lowercase_			:       Optional[Any]									=			np.array(__lowerCAmelCase   )
     lowercase_			:       Union[str, Any]									=			np.array(__lowerCAmelCase   )
     return np.linalg.norm(a - b   )



def     lowercase__( __SCREAMING_SNAKE_CASE							:  List[Any]  ,    __SCREAMING_SNAKE_CASE							:  Tuple   ):
     return consistent_heuristic(__lowerCAmelCase  ,    __lowerCAmelCase   ) // t



def     lowercase__( __SCREAMING_SNAKE_CASE							:  Any  ,    __SCREAMING_SNAKE_CASE							:  Optional[int]   ):
     return abs(p[0] - goal[0]   ) + abs(p[1] - goal[1]   )



def     lowercase__( __SCREAMING_SNAKE_CASE							:  Optional[Any]  ,    __SCREAMING_SNAKE_CASE							:  Any  ,    __SCREAMING_SNAKE_CASE							:  Optional[int]  ,    __SCREAMING_SNAKE_CASE							:  Union[str, Any]   ):
     lowercase_			:       Optional[int]									=			g_function[start] + Wa * heuristics[i](__lowerCAmelCase  ,    __lowerCAmelCase   )
     return ans



def     lowercase__( __SCREAMING_SNAKE_CASE							:  Union[str, Any]  ,    __SCREAMING_SNAKE_CASE							:  str  ,    __SCREAMING_SNAKE_CASE							:  Dict   ):
     lowercase_			:       Optional[int]									=			np.chararray((n, n)   )
     for i in range(__lowerCAmelCase   ):
          for j in range(__lowerCAmelCase   ):
               lowercase_			:       List[str]									=			'*'

     for i in range(__lowerCAmelCase   ):
          for j in range(__lowerCAmelCase   ):
               if (j, (n - 1) - i) in blocks:
                    lowercase_			:       Dict									=			'#'

     lowercase_			:       Optional[int]									=			'-'
     lowercase_			:       Optional[int]									=			back_pointer[goal]
     while x != start:
          ((lowercase_)							,						(lowercase_))			:       Optional[int]									=			x
          # print(x)
          lowercase_			:       List[Any]									=			'-'
          lowercase_			:       Dict									=			back_pointer[x]
     lowercase_			:       Union[str, Any]									=			'-'

     for i in range(__lowerCAmelCase   ):
          for j in range(__lowerCAmelCase   ):
               if (i, j) == (0, n - 1):
                    print(grid[i][j]  ,    end=' '   )
                    print('<-- End position'  ,    end=' '   )
               else:
                    print(grid[i][j]  ,    end=' '   )
          print()
     print('^'   )
     print('Start position'   )
     print()
     print('# is an obstacle'   )
     print('- is the path taken by algorithm'   )
     print('PATH TAKEN BY THE ALGORITHM IS:-'   )
     lowercase_			:       Union[str, Any]									=			back_pointer[goal]
     while x != start:
          print(__lowerCAmelCase  ,    end=' '   )
          lowercase_			:       Tuple									=			back_pointer[x]
     print(__lowerCAmelCase   )
     sys.exit()



def     lowercase__( __SCREAMING_SNAKE_CASE							:  Optional[int]   ):


     if p[0] < 0 or p[0] > n - 1:
          return False
     if p[1] < 0 or p[1] > n - 1:
          return False
     return True



def     lowercase__( __SCREAMING_SNAKE_CASE							:  List[Any]  ,    __SCREAMING_SNAKE_CASE							:  List[str]  ,    __SCREAMING_SNAKE_CASE							:  Optional[int]  ,    __SCREAMING_SNAKE_CASE							:  str  ,    __SCREAMING_SNAKE_CASE							:  Any  ,    __SCREAMING_SNAKE_CASE							:  Optional[int]  ,    __SCREAMING_SNAKE_CASE							:  Dict  ,    __SCREAMING_SNAKE_CASE							:  str  ,    ):


     for itera in range(__lowerCAmelCase   ):
          open_list[itera].remove_element(__lowerCAmelCase   )
     # print("s", s)
     # print("j", j)
     ((lowercase_)							,						(lowercase_))			:       Any									=			s
     lowercase_			:       Tuple									=			(x - 1, y)
     lowercase_			:       str									=			(x + 1, y)
     lowercase_			:       Optional[Any]									=			(x, y + 1)
     lowercase_			:       int									=			(x, y - 1)

     for neighbours in [left, right, up, down]:
          if neighbours not in blocks:
               if valid(__lowerCAmelCase   ) and neighbours not in visited:
                    # print("neighbour", neighbours)
                    visited.add(__lowerCAmelCase   )
                    lowercase_			:       Optional[Any]									=			-1
                    lowercase_			:       List[str]									=			float('inf'   )

               if valid(__lowerCAmelCase   ) and g_function[neighbours] > g_function[s] + 1:
                    lowercase_			:       Dict									=			g_function[s] + 1
                    lowercase_			:       Union[str, Any]									=			s
                    if neighbours not in close_list_anchor:
                         open_list[0].put(__lowerCAmelCase  ,    key(__lowerCAmelCase  ,    0  ,    __lowerCAmelCase  ,    __lowerCAmelCase   )   )
                         if neighbours not in close_list_inad:
                              for var in range(1  ,    __lowerCAmelCase   ):
                                   if key(__lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase   ) <= Wa * key(
                                       __lowerCAmelCase  ,    0  ,    __lowerCAmelCase  ,    __lowerCAmelCase   ):
                                        open_list[j].put(
                                            __lowerCAmelCase  ,    key(__lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase   )   )



def     lowercase__( ):
     lowercase_			:       Union[str, Any]									=			[]
     for x in range(1  ,    5   ):
          for y in range(1  ,    6   ):
               some_list.append((x, y)   )

     for x in range(15  ,    20   ):
          some_list.append((x, 17)   )

     for x in range(10  ,    19   ):
          for y in range(1  ,    15   ):
               some_list.append((x, y)   )

          # L block
     for x in range(1  ,    4   ):
          for y in range(12  ,    19   ):
               some_list.append((x, y)   )
     for x in range(3  ,    13   ):
          for y in range(16  ,    19   ):
               some_list.append((x, y)   )
     return some_list


__SCREAMING_SNAKE_CASE								={0: consistent_heuristic, 1: heuristic_a, 2: heuristic_a}

__SCREAMING_SNAKE_CASE								=[
    (0, 1),
    (1, 1),
    (2, 1),
    (3, 1),
    (4, 1),
    (5, 1),
    (6, 1),
    (7, 1),
    (8, 1),
    (9, 1),
    (10, 1),
    (11, 1),
    (12, 1),
    (13, 1),
    (14, 1),
    (15, 1),
    (16, 1),
    (17, 1),
    (18, 1),
    (19, 1),
]
__SCREAMING_SNAKE_CASE								=make_common_ground()


__SCREAMING_SNAKE_CASE								=blocks_blk
# hyper parameters
__SCREAMING_SNAKE_CASE								=1
__SCREAMING_SNAKE_CASE								=1
__SCREAMING_SNAKE_CASE								=20
__SCREAMING_SNAKE_CASE								=3  # one consistent and two other inconsistent

# start and end destination
__SCREAMING_SNAKE_CASE								=(0, 0)
__SCREAMING_SNAKE_CASE								=(n - 1, n - 1)

__SCREAMING_SNAKE_CASE								=1



def     lowercase__( __SCREAMING_SNAKE_CASE							:  str  ,    __SCREAMING_SNAKE_CASE							:  int  ,    __SCREAMING_SNAKE_CASE							:  Optional[int]   ):
     lowercase_			:       Any									=			{start: 0, goal: float('inf'   )}
     lowercase_			:       List[str]									=			{start: -1, goal: -1}
     lowercase_			:       Optional[Any]									=			[]
     lowercase_			:       Union[str, Any]									=			set()

     for i in range(__lowerCAmelCase   ):
          open_list.append(PriorityQueue()   )
          open_list[i].put(__lowerCAmelCase  ,    key(__lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase   )   )

     lowercase_			:       str									=			[]
     lowercase_			:       List[Any]									=			[]
     while open_list[0].minkey() < float('inf'   ):
          for i in range(1  ,    __lowerCAmelCase   ):
               # print(open_list[0].minkey(), open_list[i].minkey())
               if open_list[i].minkey() <= Wa * open_list[0].minkey():
                    global t
                    t += 1
                    if g_function[goal] <= open_list[i].minkey():
                         if g_function[goal] < float('inf'   ):
                              do_something(__lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase   )
                    else:
                         lowercase_							,						lowercase_			:       List[str]									=			open_list[i].top_show()
                         visited.add(__lowerCAmelCase   )
                         expand_state(
                             __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    )
                         close_list_inad.append(__lowerCAmelCase   )
               else:
                    if g_function[goal] <= open_list[0].minkey():
                         if g_function[goal] < float('inf'   ):
                              do_something(__lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase   )
                    else:
                         lowercase_			:       int									=			open_list[0].top_show()
                         visited.add(__lowerCAmelCase   )
                         expand_state(
                             __lowerCAmelCase  ,    0  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    __lowerCAmelCase  ,    )
                         close_list_anchor.append(__lowerCAmelCase   )
     print('No path found to goal'   )
     print()
     for i in range(n - 1  ,    -1  ,    -1   ):
          for j in range(__lowerCAmelCase   ):
               if (j, i) in blocks:
                    print('#'  ,    end=' '   )
               elif (j, i) in back_pointer:
                    if (j, i) == (n - 1, n - 1):
                         print('*'  ,    end=' '   )
                    else:
                         print('-'  ,    end=' '   )
               else:
                    print('*'  ,    end=' '   )
               if (j, i) == (n - 1, n - 1):
                    print('<-- End position'  ,    end=' '   )
          print()
     print('^'   )
     print('Start position'   )
     print()
     print('# is an obstacle'   )
     print('- is the path taken by algorithm'   )


if __name__ == "__main__":
       multi_a_star(start, goal, n_heuristic)


"""simple docstring"""







from math import sqrt



def     lowercase__( __SCREAMING_SNAKE_CASE							:  int   ):

     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or number % 2 == 0 or number % 3 == 0:
          # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
          return False

     # All primes number are in format of 6k +/- 1
     for i in range(5  ,    int(sqrt(__SCREAMING_SNAKE_CASE   ) + 1   )  ,    6   ):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True



def     lowercase__( __SCREAMING_SNAKE_CASE							:  int = 1_00_01   ):
     lowercase_			:       str									=			0
     lowercase_			:       Optional[Any]									=			1
     while count != nth and number < 3:
          number += 1
          if is_prime(__SCREAMING_SNAKE_CASE   ):
               count += 1
     while count != nth:
          number += 2
          if is_prime(__SCREAMING_SNAKE_CASE   ):
               count += 1
     return number


if __name__ == "__main__":
       print(F"{solution() = }")

