⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠛⠉⠉⠉⠉⠉⠛⠻⠶⣤⣀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⠉⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠙⠷⣦⣤⣄⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢠⡾⠂⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀
⠀⠀⠀⢀⣴⠟⠁⠀⣀⣤⠶⠞⠛⠻⠶⣤⡀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢤⣶⢻⡟⣶⢠⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢠⡟⠁⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣌⣛⡋⠀⠀⠀⠸⢧⡴⠛⠓⠞⠁⠀⠀⠀⠈⠛⠛⢈⣥⠟⠁⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠀
⠀⠀⣾⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡽⠟⠀⠀⠀⠀⠀⢀⣤⣤⣴⣤⣄⣀⣤⣤⠶⠞⠛⠁⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣤⣤⣤⡶⠾⠋⠁⠀⠀
⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣀⣀⣀⣀⣤⣤⣴⡟⠁⠘⣧⠀⠙⠻⠋⠛⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢋⣩⣽⠿⣯⣽⡇⠸⠋⠀⠀⣸⡟⠷⣤⡀⠀⠀⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠚⠛⠋⠉⠀⣰⡟⠉⢀⡀⠀⣿⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠘⠷⣤⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢈⣻⢦⣄⣀⣀⠀⢠⣴⣶⠞⠛⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠈⠛⢷⣤⣤⡾⠃

                     ⣠⢤⣀
   ⢰⠋⠉⠒⢄  ⣠⠤⠔⢒⠎    ⠈⠳⣤⢤⡀
 ⢀⡏      ⠔⠉            ⢄    ⠘⢦⠈⠑⢤⡀  ⢀⣤⣤⡀
 ⢸⢠⢆                    ⠈⡆    ⢈⠆    ⠙⢔⠋⢀⡼⠃
 ⠸⢨⠊                    ⠈⠃              ⠈⢯⠁
 ⡰⠁                                          ⠈⡆
⣰⠁⢀⣀                                          ⣷
⡇⣂⢸⣿⠂⠠⣀⡠  ⢰⣿⣷  ⣀                      ⡏
⢳⡀               ⠉⠁⠐⠄                  ⢀⡜⠁
 ⠙⢤⡀                                  ⣀⡴⠋
     ⠉⠉⠒⠒⠦⠤⠤⣤⣤⣤⣤⣤⠤⠤⠶⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠔⠁⣠⡾⠃⢐⡩⠐⡊⣵⡻⣏⢿⡹⢯⡝⣯⣛⢿⡽⣫⢻⢯⣿⡹⢯⢿⣿⣽⠀⠙⢦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣠⡾⡿⢁⣴⢟⡴⣋⣼⣳⡝⣮⢮⡝⣧⠏⣶⢹⣎⡿⣵⢫⢷⡹⣿⣭⣛⣦⣳⣷⣄⠀⢵⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣾⠇⢀⣼⢯⡽⣵⡻⣝⢮⡳⣝⣾⣳⠽⣎⢷⡹⣎⢷⣎⠷⣮⡝⣷⢫⡞⣵⢫⣷⣯⣶⣙⡞⣿⣦⠈⠹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⠟⠁⢀⣾⣿⣹⢾⣝⡳⣝⣮⢷⣽⣷⣫⠿⡽⣎⢷⣏⣾⣿⡝⣶⡹⢾⣏⢾⡱⣏⢿⣞⡽⣻⢾⣵⣯⣷⡀⢳⠀⠀⠀
⠀⠀⠀⠀⡐⠁⠀⢀⣾⡿⢶⣻⡾⣭⣟⡼⣮⠟⢹⣷⢫⡿⣽⢎⡿⣞⡾⢹⣟⢶⡹⣏⣟⢮⡳⣝⠾⣿⡜⣧⢻⢮⢷⣻⢷⣸⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⢻⡽⣯⢿⡵⣻⢞⡵⢀⣾⢞⡯⣽⡞⣭⢿⣽⠇⢹⣯⢻⡷⣭⣟⡮⢷⣭⢻⣽⣿⢾⡽⣾⢷⣻⢿⣏⠀⠀⠀
⠀⠀⠀⠀⠀⡜⣼⡿⣭⢿⣹⢯⣷⡿⠕⠉⠀⠀⣿⢞⡽⢧⠿⣜⣯⡿⠃⢫⣯⢷⡟⣞⣿⢼⡳⣎⠷⣾⣏⢾⡹⣞⣳⢯⣿⣿⠀⠀⠀
⠀⠀⠀⠀⣰⣿⡿⣽⣏⢾⣽⣿⢿⣿⣿⣶⣄⠀⠘⣯⡻⣭⣛⢷⠟⠀⠀⠀⢻⡾⣝⣾⣻⢮⢷⣭⢻⡼⣛⣮⣷⢿⣟⢿⣳⡿⢤⡀⠀
⠀⠀⢀⠜⢹⣿⣽⣳⢯⣿⡿⠉⣲⣿⣿⡟⣿⡆⠀⠙⣿⣶⣭⢿⠀⢀⡤⠀⠀⣿⣣⣿⡽⣏⡾⣎⢷⣹⢯⡷⣏⡿⣞⢯⡷⡇⠘⣟⣧
⠀⢠⠎⠀⣼⣟⡾⣽⣿⣿⡇⠀⣿⡿⣟⣿⡿⠀⠀⠀⠈⠉⠙⠛⠂⣾⣿⣿⣿⣾⣿⢧⡿⣽⢺⡽⣧⢻⣯⢷⡿⣹⣞⣻⡼⡇⠀⠘⣾
⢠⠃⠀⣰⣟⣾⣽⣳⣟⡏⠻⠀⠙⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⣿⢻⣿⣻⣽⣳⢏⣷⣻⣞⣯⣽⣳⣭⢷⣻⡇⠀⠀⠱
⡎⠀⢰⣿⣹⡞⣷⢯⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡹⣻⣿⣿⠀⣿⣿⣾⡽⣟⡾⣵⣻⡼⡶⣏⡾⢧⣳⢿⠀⠀⠀
⣇⢠⣿⡷⠉⠟⠛⠻⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠯⣘⠋⠀⠛⠋⠀⠐⠾⣿⣳⢯⡷⣿⡽⣹⣏⢷⣻⠇⡀⠀
⠻⣿⡟⠀⠀⡔⠛⠒⠒⠺⠷⣤⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠈⠈⣻⡟⠉⣿⣳⣟⣳⡞⣿⡄⠙⠓⠄
⠀⠈⠳⣀⡜⠀⠀⣠⣤⣤⡴⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠚⠿⠒⣺⡏⠀⣼⢿⡽⣯⡷⣽⢯⡇⠀⠀⠀
⠀⠀⠀⠈⢆⠀⣰⣿⢿⡽⡇⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠻⠄⢄⣠⡈⠉⢹⣦⣤⣴⣾⠏⠀⡔⠁⣼⡿⣹⣿⡞⣿⡇⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠹⣌⠛⣿⠷⠀⣿⠟⡷⢄⣀⠀⢀⢀⣀⣠⡜⠀⣰⣾⣟⣿⣻⠀⠐⣿⡽⠛⠁⢀⠞⠀⣸⣟⡷⣏⣷⣻⢷⣻⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠈⠳⡀⠀⣰⣷⣤⣧⣤⣬⠍⠉⠉⠀⢺⠀⠠⣿⣳⣟⠾⠃⠀⠴⠋⠀⡠⠔⠁⠀⣠⣿⢾⣹⡽⢾⡽⣯⢿⡇⠀⠀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠙⣶⠃⠈⣿⠿⠍⠀⠀⠈⢀⣠⣼⠀⠀⠛⠋⠁⠀⢠⡀⠐⠒⠉⠀⢀⣤⣾⣿⣽⣞⡷⠿⠯⣿⡽⣯⣷⠀⠀
⠀⠀⠀⠀⡸⠀⠀⠀⠀⢠⠃⠀⠀⡇⠑⣒⡲⠖⠛⠉⠀⢸⠀⠀⢀⣀⣀⣀⣔⠳⣔⣤⠤⠖⠛⠉⠉⠀⣀⣀⣤⣤⣴⣿⢻⣷⣻⡆⠀
⠀⠀⠀⠀⠃⠀⠀⠀⣠⠃⠀⠀⡔⠛⠋⢣⠀⠀⠀⠀⠀⣸⢀⠖⠉⠀⠀⠀⠉⠉⠉⠙⠀⣀⣤⣶⢾⣟⡿⣽⡯⣷⣻⣼⣻⢞⡷⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⡎⠀⢀⡀⠈⡆⠀⠀⠀⢀⢿⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⣻⢿⢮⡷⣞⡽⣞⣿⣵⣻⢶⢯⣻⣽⣳⣧
⠀⠀⠀⠀⠀⠀⡹⠋⠀⠔⢹⠉⠈⠉⠚⣊⣅⠀⢀⣰⠃⠘⡄⠀⠀⠀⢀⡔⢿⣿⢿⣞⣯⣽⢻⣿⣞⡽⣞⡽⣻⡶⣏⡿⣏⣷⣳⢯⣟
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠠⠁⠘⣤⡺⠃⠀⣠⢳⡀⠀⠀⠀⢇⠈⠙⢆⠹⣷⢯⣟⢾⣻⡾⣭⣻⢽⣻⣳⡽⣛⣮⢷⣻⣞
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡜⠁⠀⠀⢫⠉⠳⢤⣄⠤⠘⣦⠀⠘⢷⣿⣻⣞⣯⢷⣻⢷⣯⣳⢯⣷⣹⣻⡼⣛⣦⡙