⠀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⠀⠀⣀⠩⡙⢋⡙⢻⠿⣶⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡷⠶⠿⠿⠿⢷⣤⣄⡐⠣⠜⡸⢿⢿⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠻⡿⠆⠲⠶⣶⣤⣈⠛⢿⣦⡃⢦⠉⣞⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠟⠿⢷⣦⣀⠀⠀⠙⢻⣷⡀⠹⣿⡤⣃⠌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠉⢿⣦⣡⣐⣈⠻⣿⣄⣻⣷⡡⠎⡌⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⠟⠙⠋⠛⠻⣿⣿⣿⠛⡰⠑⡌⢆⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣼⣿⣄⡀⠀⠀⢀⣼⠏⠀⠀⢀⡀⠀⠈⢿⣏⠴⣁⢋⠴⣈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠀⢀⣾⣏⠒⡌⠒⡌⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣯⣿⠿⣟⣷⣤⣄⣀⣀⣠⣾⣿⠏⡜⠠⢍⠰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣽⣿⣿⣿⣿⣏⡻⣿⡶⣶⣮⣟⠿⣿⣿⣿⢛⡙⠫⠜⡰⢉⢆⣩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⢻⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⣽⣷⠂⡜⢡⠓⠤⢃⣺⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡏⢿⡇⠀⠈⠻⣿⣏⣹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠆⣉⠦⡑⢌⡡⣹⣿⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠛⣍⠫⡝⠁⡀⠂⠀⠌⠙⠿⢷⣧⣿⢛⡿⣿⣿⣿⣿⣿⠻⣈⢅⠒⡡⢆⣱⣽⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣉⠜⣠⠃⡍⠀⠀⠄⠁⡀⠂⠀⠀⠛⠛⣛⣟⣿⣿⣿⣿⢏⡱⡐⢊⠬⣁⣻⣿⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠳⢠⠚⢤⡙⠀⢀⠁⠂⠐⠀⠠⠁⠂⠠⠀⣸⣿⠏⢓⠨⢰⡈⡔⢡⠎⡰⢰⣻⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢌⠣⢃⡚⡔⠀⠐⠀⠄⠈⡀⠌⠀⡐⠀⠁⣸⡿⢁⠎⡄⢣⠢⠔⡌⡅⠎⡴⣿⣿⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠡⢌⢊⠱⡼⠀⠐⠈⠀⠄⠂⠀⠄⠂⠀⠄⣽⠿⡁⢎⠒⡍⢆⠱⢊⢔⢨⡑⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⡠⢃⠎⢢⡽⠋⠀⠐⡈⠀⠄⠂⠁⡀⢐⣬⣾⠏⡰⢁⠎⡱⢈⠆⡱⢈⠌⡲⣾⣿⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⢨⠑⡌⣊⠷⠀⢀⠁⠄⠐⠀⡀⢂⣰⣼⡿⠋⡔⣈⠆⣍⠢⡑⢌⠢⡑⠬⡘⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢢⠑⢢⠑⡼⠋⠀⡀⠂⠀⠄⠂⠐⣤⣿⠟⡡⢢⠑⡌⢦⡘⢄⠣⠜⡠⠓⡌⣅⢳⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡿⣁⠫⣔⠋⠄⠐⠀⠠⠀⢁⠠⢀⡿⢏⠰⠌⡔⡡⠚⣄⠣⡘⢌⠦⣡⠑⢣⠜⣤⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣋⠔⣢⢁⡞⠀⠠⠀⢈⠠⠀⠁⡀⢀⠾⡑⢌⢢⢉⡔⢡⠣⢄⠣⠜⣬⣞⡇⠜⡡⢚⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⢛⡡⠚⡄⡞⠀⠐⠀⡈⠀⡀⠄⠁⢀⣾⠡⠜⣠⠒⡌⡰⢁⢎⠰⣉⢒⠰⣈⠔⡣⣘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠣⢌⠡⢒⠱⣸⠁⠀⠌⠀⡀⠁⠀⠄⠈⣰⠎⠱⡘⢄⠣⠔⡡⢎⠢⡑⢤⢊⠱⢄⡊⢥⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢉⠬⡑⢌⡘⢂⡟⠋⠀⡈⠀⡐⠀⢈⠀⠄⣲⠋⡔⢣⠘⠤⠓⡌⡑⢢⠑⡌⡒⡌⡱⢂⣹⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⡡⢋⠔⣡⢂⣼⠟⠀⢀⠐⠠⠐⠀⡀⢂⣠⠞⡍⢒⠌⢢⡉⠖⡩⢐⡡⢃⡜⠤⡑⢢⡑⣮⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡁⢎⠰⢡⢊⣶⡿⠃⠀⡀⠂⠀⠄⠐⣀⡴⠞⣉⠒⠬⢌⡊⠥⡘⡰⢁⠣⠔⢣⠘⡄⢣⢡⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⣿⣿⡿⠐⣌⡑⢢⠉⠉⢀⠀⠄⡀⠄⠁⡀⢂⡾⢃⠜⡠⢍⠒⡆⣉⠦⡑⡰⢉⠦⡙⢄⠣⡘⢤⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⣾⠿⡟⠛⡅⢣⡐⡼⠃⠀⡁⠠⠀⠄⢀⠠⠐⣴⡟⢠⠃⡜⢰⡈⢣⠔⡂⠖⣡⢑⡊⡔⡁⢎⠰⣡⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⣿⡿⢠⠉⢧⠘⣴⠚⠀⡀⠁⠀⠄⠂⠀⢄⣴⡿⢃⠜⢢⡉⠜⢢⠘⠤⠚⡌⣒⢡⠒⡌⠴⡉⣤⣋⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⡿⢁⢣⠚⡤⣋⠆⠀⡐⠀⢈⠠⠐⠀⣱⡾⢋⠄⡃⢎⠱⡨⢜⠡⢎⡑⠣⡔⢡⢊⠱⣈⠦⣱⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢠⣿⠏⡔⡉⠦⠱⣜⠁⠀⠄⠀⠌⠀⠀⠤⣜⠣⣐⠡⣊⠱⢌⡒⢡⢊⡜⢢⢘⠡⢌⠡⢊⡑⢢⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣼⣿⢃⡘⢄⠣⡙⡄⠠⠈⠀⠂⢀⢂⡙⠎⡄⢣⠄⠳⣀⠳⠌⠦⡑⣮⣞⠆⣡⠊⢆⢡⢣⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡯⠐⡌⢢⠱⢱⢨⡐⣤⣡⠶⡝⢊⠅⢣⢘⡐⢊⡱⢀⠣⠜⢢⠙⢣⠛⡰⡀⠧⣘⣲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡇⢣⠘⠤⣉⠆⢦⡑⠢⡔⡡⠜⣂⠎⡅⢎⡰⢡⠢⢍⢢⡙⢆⡉⠦⣁⠣⣌⣳⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣿⡠⢋⠔⡂⢎⠳⣌⠳⡄⢓⡸⠰⢌⡸⠐⡔⠣⡜⢢⠡⠜⡂⢅⠣⢆⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢿⣷⣏⣰⠁⠎⡔⢂⠓⡸⢐⡂⠳⡌⡰⡉⡔⢣⠑⡌⡒⢡⡜⣬⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⢿⣮⣝⣢⣉⠦⣉⠔⠣⣌⠱⡐⢡⢚⡴⣥⣏⣴⣭⣷⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣾⣷⣦⢷⣽⣶⣯⣿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
