code: 
  
  
  
  
  
  import numpy as np
  
  def     a						(      _lowercase   :     Tuple       )     ->Union[str, Any]:
  		UpperCamelCase__         =	a.shape
  		UpperCamelCase__         =	min(lowercase__ ,						lowercase__       )
  		UpperCamelCase__         =	np.eye(lowercase__       )
  		UpperCamelCase__         =	a.copy()
  
  		for k in range(t - 1       ):
  				# select a column of modified matrix A':
  				UpperCamelCase__         =	r[k:, [k]]
  				# construct first basis vector
  				UpperCamelCase__         =	np.zeros_like(lowercase__       )
  				UpperCamelCase__         =	1.0
  				# determine scaling factor
  				UpperCamelCase__         =	np.linalg.norm(lowercase__       )
  				# construct vector v for Householder reflection
  				UpperCamelCase__         =	x + np.sign(x[0]       ) * alpha * ea
  				v /= np.linalg.norm(lowercase__       )
  
  				# construct the Householder matrix
  				UpperCamelCase__         =	np.eye(m - k       ) - 2.0 * v @ v.T
  				# pad with ones and zeros as necessary
  				UpperCamelCase__         =	np.block([[np.eye(lowercase__       ), np.zeros((k, m - k)       )], [np.zeros((m - k, k)       ), q_k]]       )
  
  				UpperCamelCase__         =	q @ q_k.T
  				UpperCamelCase__         =	q_k @ r
  
  		return q, r
  
  
  if __name__ == "__main__":
  				import doctest
  
  				doctest.testmod()
  
  
  
  
  
  
style_context: 
  
  
  
  
  
  def     a						(      _lowercase   :     int ,						_lowercase   :     int       )     ->int:
  		UpperCamelCase__         =	1.0
  		for i in range(1 ,						r + 1       ):
  				ret *= (n + 1 - i) / i
  		return round(_lowercase       )
  
  def     a						(      _lowercase   :     int       )     ->int:
  		return choose(8 + n ,						_lowercase       ) + choose(9 + n ,						_lowercase       ) - 1_0
  
  def     a						(      _lowercase   :     int       )     ->int:
  		return sum(non_bouncy_exact(_lowercase       ) for i in range(1 ,						n + 1       )       )
  
  def     a						(      _lowercase   :     int = 1_0_0       )     ->int:
  		return non_bouncy_upto(_lowercase       )
  
  
  if __name__ == "__main__":
  				print(f"""{solution() = }""")
  
  
  
  
  
  