


from __future__ import annotations


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE       ):
  A_      :				List[Any]		     =					str(SCREAMING_SNAKE_CASE       )
  return len(SCREAMING_SNAKE_CASE       ) == 9 and set(SCREAMING_SNAKE_CASE       ) == set('''123456789'''       )


def   _SCREAMING_SNAKE_CASE	( ):
  for base_num in range(9_999 ,			4_999 ,			-1       ):
    A_      :				Any		     =					100_002 * base_num
    if is_9_pandigital(SCREAMING_SNAKE_CASE       ):
      return candidate

  for base_num in range(333 ,			99 ,			-1       ):
    A_      :				Tuple		     =					1_002_003 * base_num
    if is_9_pandigital(SCREAMING_SNAKE_CASE       ):
      return candidate

  return None


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








from math import isclose, sqrt


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ):
  A_      :				List[str]		     =					point_y / 4 / point_x
  A_      :				Union[str, Any]		     =					2 * normal_gradient / (1 + normal_gradient * normal_gradient)
  A_      :				int		     =					(1 - normal_gradient * normal_gradient) / (
      1 + normal_gradient * normal_gradient
  )
  A_      :				List[str]		     =					(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

  # to find the next point, solve the simultaeneous equations:
  # y^2 + 4x^2 = 100
  # y - b = m * (x - a)
  # ==> A x^2 + B x + C = 0
  A_      :				List[str]		     =					outgoing_gradient**2 + 4
  A_      :				List[str]		     =					2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
  A_      :				Any		     =					(point_y - outgoing_gradient * point_x) ** 2 - 100

  A_      :				str		     =					(
      -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)
  A_      :				Any		     =					(
      -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)

  # two solutions, one of which is our input point
  A_      :				int		     =					x_minus if isclose(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ) else x_plus
  A_      :				Any		     =					point_y + outgoing_gradient * (next_x - point_x)

  return next_x, next_y, outgoing_gradient


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE = 1.4 ,			SCREAMING_SNAKE_CASE = -9.6       ):
  A_      :				int		     =					0
  A_      :				float		     =					first_x_coord
  A_      :				float		     =					first_y_coord
  A_      :				float		     =					(1_0.1 - point_y) / (0.0 - point_x)

  while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
    A_     ,       A_     ,       A_      :				List[str]		     =					next_point(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )
    num_reflections += 1

  return num_reflections


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




