from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
lowerCAmelCase      :				Dict     			=20

# Time between time steps in seconds
lowerCAmelCase      :				List[Any]     			=INTERVAL / 1000




class 						__snake_case						:





      '''simple docstring'''






      def __init__(       self  :					Optional[Any]     ,	_UpperCamelCase  :					float     ,	_UpperCamelCase  :					float     ,	_UpperCamelCase  :					float     ,	_UpperCamelCase  :					float     ,	_UpperCamelCase  :					float = 1.0     ,	_UpperCamelCase  :					float = 1.0     ,	_UpperCamelCase  :					str = "blue"     ,	)					->None:


          """simple docstring"""


          _lowerCamelCase  :		str			    =    position_x
          _lowerCamelCase  :		str			    =    position_y
          _lowerCamelCase  :		List[str]			    =    velocity_x
          _lowerCamelCase  :		Optional[int]			    =    velocity_y
          _lowerCamelCase  :		str			    =    mass
          _lowerCamelCase  :		int			    =    size
          _lowerCamelCase  :		Union[str, Any]			    =    color



      @property
      def 						_SCREAMING_SNAKE_CASE     (       self  :					Union[str, Any])					->tuple[float, float]:


          """simple docstring"""


          return self.position_x, self.position_y



      @property
      def 						_SCREAMING_SNAKE_CASE     (       self  :					List[Any])					->tuple[float, float]:


          """simple docstring"""


          return self.velocity_x, self.velocity_y



      def 						_SCREAMING_SNAKE_CASE     (       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 						_SCREAMING_SNAKE_CASE     (       self  :					List[Any]     ,	_UpperCamelCase  :					float)					->None:


          """simple docstring"""


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





class 						__snake_case						:





      '''simple docstring'''






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


          """simple docstring"""


          _lowerCamelCase  :		int			    =    bodies
          _lowerCamelCase  :		int			    =    gravitation_constant
          _lowerCamelCase  :		List[Any]			    =    time_factor
          _lowerCamelCase  :		int			    =    softening_factor



      def __len__(       self  :					Optional[int])					->int:


          """simple docstring"""


          return len(self.bodies)




      def 						_SCREAMING_SNAKE_CASE     (       self  :					int     ,	_UpperCamelCase  :					float)					->None:


          """simple docstring"""


          for bodya in self.bodies:
              _lowerCamelCase  :		Tuple			    =    0.0
              _lowerCamelCase  :		Any			    =    0.0
              for bodya in self.bodies:
                  if bodya != bodya:
                      _lowerCamelCase  :		List[str]			    =    bodya.position_x - bodya.position_x
                      _lowerCamelCase  :		str			    =    bodya.position_y - bodya.position_y

                      # Calculation of the distance using Pythagoras's theorem
                      # Extra factor due to the softening technique
                      _lowerCamelCase  :		Any			    =    (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						):
        _lowerCamelCase  :		List[str]			    =    (body.position_x, body.position_y)







def     A__	(       __A ,			__A ,			__A = -1 ,			__A = 1 ,			__A = -1 ,			__A = 1 ,			):







    '''simple docstring'''
    _lowerCamelCase  :		Any			    =    plt.figure()
    fig.canvas.set_window_title(__A						)
    _lowerCamelCase  :		Dict			    =    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
    _lowerCamelCase  :		List[Any]			    =    [
        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

    _lowerCamelCase  :		Union[str, Any]			    =    animation.FuncAnimation(  # noqa: F841
        __A ,			__A ,			interval=__A ,			blit=__A						)

    plt.show()







def     A__	(       ):







    '''simple docstring'''
    _lowerCamelCase  :		Dict			    =    0.9_700_436
    _lowerCamelCase  :		Union[str, Any]			    =    -0.24_308_753
    _lowerCamelCase  :		Tuple			    =    0.466_203_685
    _lowerCamelCase  :		Any			    =    0.43_236_573

    _lowerCamelCase  :		Optional[Any]			    =    [
        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'''
    _lowerCamelCase  :		Optional[int]			    =    7.34_76E22
    _lowerCamelCase  :		int			    =    5.9_72E24
    _lowerCamelCase  :		int			    =    1_022
    _lowerCamelCase  :		Optional[Any]			    =    384_399_000
    _lowerCamelCase  :		Any			    =    6.6_74E-11

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

    _lowerCamelCase  :		Optional[Any]			    =    Body(-earth_moon_distance ,			0 ,			0 ,			__A ,			__A ,			10_000_000 ,			"""grey"""						)
    _lowerCamelCase  :		Dict			    =    Body(0 ,			0 ,			0 ,			__A ,			__A ,			50_000_000 ,			"""blue"""						)
    return BodySystem([earth, moon] ,			__A ,			time_factor=1_000_000						)







def     A__	(       ):







    '''simple docstring'''
    _lowerCamelCase  :		str			    =    []
    for _ in range(10						):
        _lowerCamelCase  :		Optional[Any]			    =    random.uniform(-0.5 ,			0.5						)
        _lowerCamelCase  :		List[Any]			    =    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)





from __future__ import annotations

import sys







def     A__	(       __A						):







    '''simple docstring'''
    if n < 0:
        raise ValueError("""Negative arguments are not supported"""						)
    return _fib(__A						)[0]







def     A__	(       __A						):







    '''simple docstring'''
    if n == 0:  # (F(0), F(1))
        return (0, 1)

    # F(2n) = F(n)[2F(n+1) − F(n)]
    # F(2n+1) = F(n+1)^2+F(n)^2
    _lowerCamelCase    ,				_lowerCamelCase  :		List[Any]			    =    _fib(n // 2						)
    _lowerCamelCase  :		List[Any]			    =    a * (b * 2 - a)
    _lowerCamelCase  :		Optional[int]			    =    a * a + b * b
    return (d, c + d) if n % 2 else (c, d)


if __name__ == "__main__":
    lowerCAmelCase      :				Union[str, Any]     			=int(sys.argv[1])
    print(F"""fibonacci({n}) is {fibonacci(n)}""")




