
from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__lowerCAmelCase      :					str   	=       20

# Time between time steps in seconds
__lowerCAmelCase      :					List[str]   	=       INTERVAL / 1000




class 	UpperCAmelCase_      :


  '''simple docstring'''







  def __init__(			self      :			Any      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float = 1.0      ,							UpperCamelCase__      :			float = 1.0      ,							UpperCamelCase__      :			str = "blue"      ,							)      ->     None:







    """simple docstring"""






    __magic_name__    					=	position_x
    __magic_name__    					=	position_y
    __magic_name__    					=	velocity_x
    __magic_name__    					=	velocity_y
    __magic_name__    					=	mass
    __magic_name__    					=	size
    __magic_name__    					=	color
  @property
  def 							_lowercase							(			self      :			List[str]   )      ->     tuple[float, float]:







    """simple docstring"""






    return self.position_x, self.position_y
  @property
  def 							_lowercase							(			self      :			List[Any]   )      ->     tuple[float, float]:







    """simple docstring"""






    return self.velocity_x, self.velocity_y
  def 							_lowercase							(			self      :			Optional[Any]      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float      ,							UpperCamelCase__      :			float   )      ->     None:







    """simple docstring"""






    self.velocity_x += force_x * delta_time
    self.velocity_y += force_y * delta_time





  def 							_lowercase							(			self      :			int      ,							UpperCamelCase__      :			float   )      ->     None:







    """simple docstring"""






    self.position_x += self.velocity_x * delta_time
    self.position_y += self.velocity_y * delta_time







class 	UpperCAmelCase_      :


  '''simple docstring'''







  def __init__(			self      :			str      ,							UpperCamelCase__      :			list[Body]      ,							UpperCamelCase__      :			float = 1.0      ,							UpperCamelCase__      :			float = 1.0      ,							UpperCamelCase__      :			float = 0.0      ,							)      ->     None:







    """simple docstring"""






    __magic_name__    					=	bodies
    __magic_name__    					=	gravitation_constant
    __magic_name__    					=	time_factor
    __magic_name__    					=	softening_factor
  def __len__(			self      :			Union[str, Any]   )      ->     int:







    """simple docstring"""






    return len(self.bodies   )





  def 							_lowercase							(			self      :			List[str]      ,							UpperCamelCase__      :			float   )      ->     None:







    """simple docstring"""






    for bodya in self.bodies:
      __magic_name__    					=	0.0
      __magic_name__    					=	0.0
      for bodya in self.bodies:
        if bodya != bodya:
          __magic_name__    					=	bodya.position_x - bodya.position_x
          __magic_name__    					=	bodya.position_y - bodya.position_y

          # Calculation of the distance using Pythagoras's theorem
          # Extra factor due to the softening technique
          __magic_name__    					=	(dif_x**2 + dif_y**2 + self.softening_factor) ** (
              1 / 2
          )

          # Newton's law of universal gravitation.
          force_x += (
              self.gravitation_constant * bodya.mass * dif_x / distance**3
          )
          force_y += (
              self.gravitation_constant * bodya.mass * dif_y / distance**3
          )

            # Update the body's velocity once all the force components have been added
      bodya.update_velocity(UpperCamelCase__      ,							UpperCamelCase__      ,							delta_time * self.time_factor   )

    # Update the positions only after all the velocities have been updated
    for body in self.bodies:
      body.update_position(delta_time * self.time_factor   )







def a__   (	A_,					A_,					A_ ):






  '''simple docstring'''





  body_system.update_system(A_ )

  # Update the positions of the patches
  for patch, body in zip(A_,					body_system.bodies ):
    __magic_name__    					=	(body.position_x, body.position_y)

def a__   (	A_,					A_,					A_ = -1,					A_ = 1,					A_ = -1,					A_ = 1,					):






  '''simple docstring'''





  __magic_name__    					=	plt.figure()
  fig.canvas.set_window_title(A_ )
  __magic_name__    					=	plt.axes(
      xlim=(x_start, x_end),					ylim=(y_start, y_end) )  # Set section to be plotted
  plt.gca().set_aspect("""equal""" )  # Fix aspect ratio

  # Each body is drawn as a patch by the plt-function
  __magic_name__    					=	[
      plt.Circle((body.position_x, body.position_y),					body.size,					fc=body.color )
      for body in body_system.bodies
  ]

  for patch in patches:
    ax.add_patch(A_ )

  # Function called at each step of the animation
  def update(A_ ) -> list[plt.Circle]:
    update_step(A_,					A_,					A_ )
    return patches

  __magic_name__    					=	animation.FuncAnimation(  # noqa: F841
      A_,					A_,					interval=A_,					blit=A_ )

  plt.show()

def a__   (	):






  '''simple docstring'''





  __magic_name__    					=	0.9700436
  __magic_name__    					=	-0.24308753
  __magic_name__    					=	0.466203685
  __magic_name__    					=	0.43236573

  __magic_name__    					=	[
      Body(A_,					A_,					A_,					A_,					size=0.2,					color="""red""" ),
      Body(-position_x,					-position_y,					A_,					A_,					size=0.2,					color="""green""" ),
      Body(0,					0,					-2 * velocity_x,					-2 * velocity_y,					size=0.2,					color="""blue""" ),
  ]
  return BodySystem(A_,					time_factor=3 )

def a__   (	):






  '''simple docstring'''





  __magic_name__    					=	7.3476e22
  __magic_name__    					=	5.972e24
  __magic_name__    					=	1022
  __magic_name__    					=	384399000
  __magic_name__    					=	6.674e-11

  # Calculation of the respective velocities so that total impulse is zero,
  # i.e. the two bodies together don't move
  __magic_name__    					=	earth_mass * velocity_dif / (earth_mass + moon_mass)
  __magic_name__    					=	moon_velocity - velocity_dif

  __magic_name__    					=	Body(-earth_moon_distance,					0,					0,					A_,					A_,					10000000,					"""grey""" )
  __magic_name__    					=	Body(0,					0,					0,					A_,					A_,					50000000,					"""blue""" )
  return BodySystem([earth, moon],					A_,					time_factor=1000000 )

def a__   (	):






  '''simple docstring'''





  __magic_name__    					=	[]
  for _ in range(10 ):
    __magic_name__    					=	random.uniform(-0.5,					0.5 )
    __magic_name__    					=	random.uniform(-0.5,					0.5 )

    # Bodies are created pairwise with opposite velocities so that the
    # total impulse remains zero
    bodies.append(
        Body(
            random.uniform(-0.5,					0.5 ),					random.uniform(-0.5,					0.5 ),					A_,					A_,					size=0.05,					) )
    bodies.append(
        Body(
            random.uniform(-0.5,					0.5 ),					random.uniform(-0.5,					0.5 ),					-velocity_x,					-velocity_y,					size=0.05,					) )
  return BodySystem(A_,					0.01,					10,					0.1 )


if __name__ == "__main__":
    plot('Figure-8 solution to the 3-body-problem', example_a(), -2, 2, -2, 2)
    plot(
        'Moon\'s orbit around the earth',
        example_a(),
        -430000000,
        430000000,
        -430000000,
        430000000,
    )
    plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)



import sys

sys.set_int_max_str_digits(0)  # type: ignore

def a__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	[0] * 11
  __magic_name__    					=	[0] * 11

  # mark last 9 numbers
  for _ in range(9 ):
    __magic_name__    					=	1
    __magic_name__    					=	number // 10
  # flag
  __magic_name__    					=	True

  # check last 9 numbers for pandigitality

  for x in range(9 ):
    if not check_last[x + 1]:
      __magic_name__    					=	False
  if not f:
    return f

  # mark first 9 numbers
  __magic_name__    					=	int(str(A_ )[:9] )

  for _ in range(9 ):
    __magic_name__    					=	1
    __magic_name__    					=	number // 10

  # check first 9 numbers for pandigitality

  for x in range(9 ):
    if not check_front[x + 1]:
      __magic_name__    					=	False
  return f

def a__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	[0] * 11

  # mark last 9 numbers
  for _ in range(9 ):
    __magic_name__    					=	1
    __magic_name__    					=	number // 10
  # flag
  __magic_name__    					=	True

  # check last 9 numbers for pandigitality

  for x in range(9 ):
    if not check_last[x + 1]:
      __magic_name__    					=	False
  return f

def a__   (	):






  '''simple docstring'''





  __magic_name__    					=	1
  __magic_name__    					=	1
  __magic_name__    					=	2
  # temporary Fibonacci numbers

  __magic_name__    					=	1
  __magic_name__    					=	1
  __magic_name__    					=	2
  # temporary Fibonacci numbers mod 1e9

  # mod m=1e9, done for fast optimisation
  __magic_name__    					=	[0] * 1000000
  __magic_name__    					=	1000000000

  for x in range(1000000 ):
    __magic_name__    					=	(aa + ba) % m
    __magic_name__    					=	ba % m
    __magic_name__    					=	ca % m
    if checka(A_ ):
      __magic_name__    					=	1

  for x in range(1000000 ):
    __magic_name__    					=	a + b
    __magic_name__    					=	b
    __magic_name__    					=	c
    # perform check only if in tocheck
    if tocheck[x + 3] and check(A_ ):
      return x + 3  # first 2 already done
  return -1


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

