⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣻⣟⣿⣻⣟⣿⣻⣟⣯⢻⡜⢢⡙⣼⢫⣟⡽⣯⣟⣿⣻⢿⣿⣿⣿⣿⢿⡝⡮⣝⣯⢿⣿⣿⡿⣏⡳⣍⠶⣩⠞⣽⣻⢿⣻⡝⢦⠣⣏⢿⣻⡿⣿
⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣟⣾⣳⣟⣾⣳⣟⡾⣽⢳⢎⡣⢜⡼⣳⢞⡽⣶⣛⡾⣽⣻⣿⣿⣿⣯⣟⢮⡱⣚⡼⣯⣟⣿⡿⣝⡳⣌⢳⢡⢛⡼⣯⢿⣳⢯⣇⠳⣜⢯⢷⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠘⠓⠿⢞⢷⣻⢾⡽⣏⢿⡸⢔⡩⢶⡹⢾⣹⡗⣯⣟⣷⣻⣿⣿⣷⣿⣻⢦⠱⣌⡳⣟⣾⣿⣟⣯⢳⡌⢧⢊⡜⡼⣽⣻⢯⡷⣎⠱⢎⡽⣺⢷⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠽⣎⢗⠪⡜⣣⢟⣭⢗⣻⢧⣟⡾⣽⣿⣿⡿⣷⣟⢮⡑⠦⣝⡞⣷⣿⡿⣞⢧⢞⡱⢎⡰⠱⣯⣟⣯⠷⣍⡚⡌⡞⣵⣻⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⠳⣞⢮⣏⢯⣟⡾⣽⣻⣾⣿⣿⣿⣻⢎⡱⣉⠶⣹⣟⣾⣿⣟⡞⢮⣑⢊⠔⡹⢼⣻⣞⣟⢦⠓⡜⣘⢶⣹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣗⡞⣯⠾⣽⣳⢯⣿⣿⣿⣽⣟⢮⡑⣌⠳⡽⣞⣿⣿⢾⣹⢣⠞⡬⠐⡌⢯⡷⣟⣾⢣⡍⠴⡩⢎⡷⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠋⠿⠿⢿⡿⣿⣾⢳⡘⢤⡛⣵⣻⣯⣿⣯⢗⣫⠞⡥⢋⢌⡳⣟⡿⣞⢧⠜⡂⠵⣩⢞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠢⢟⣷⣻⣽⣿⣯⢟⡲⢯⡜⡣⡜⡼⢯⡿⣽⣎⠳⡘⡴⢣⣞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣽⢯⡝⣶⢩⠳⢭⡝⣿⣽⣳⢎⡳⣉⢞⣳⠾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣡⢯⣙⢦⣛⢷⣯⢯⡻⢴⢩⠞⣵⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡪⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢞⢦⢫⣟⡾⣏⡿⣘⢬⣛⢧⣟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣼⣻⡽⣞⡱⢪⡜⣧⣟⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢯⣷⢫⡜⣣⢝⡶⣫⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⣴⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡛⡴⣋⠾⣝⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣯⠪⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡳⣍⢟⡾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢬⡻⣼⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⠧⡟⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡒⢥⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢭⡓⠰⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢧⢏⡑⠂⡄⡂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣫⠦⣡⢃⣐⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣯⢟⡼⢡⠖⣂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣟⠮⡜⣇⠊⡄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣽⣎⢷⡉⠦⡑⢈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢫⣿⡿⣼⢳⣊⠕⡠⢃⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⣿⡿⡶⢱⡈⠁⢆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣿⣿⣯⣗⢯⣳⡑⣃⠒⡉⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⢾⡽⣵⣿⣟⣿⣽⢞⡱⢇⣚⡡⠌⡔⠡⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠾⢎⣊⢻⣿⢫⢟⡾⣧⠯⡓⢎⡔⢄⡃⢆⠁⡂⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠈⣠⣼⠡⣏⡞⣸⣿⣝⡮⣟⡜⡳⣮⠝⡢⢞⠬⠲⡈⠖⡄⠢⠈⠀⡀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡈⣴⠗⡡⢃⡹⣱⣿⣿⣼⣺⣱⢫⡵⣎⠚⣍⢦⠩⢆⡱⠍⠤⢁⠂⡁⠄⠂⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢠⣶⢟⠍⣌⣑⠎⣞⣾⣟⣷⢪⢳⢣⣻⡔⣏⡏⠶⣡⢫⡌⣌⠲⢁⠂⡐⠀⡀⢀⠠⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⡾⡕⢎⠼⣰⢓⣏⣼⢿⣸⣻⢏⢣⡳⡮⡗⢿⣜⣫⠶⣡⠎⣔⠳⢀⠡⢀⠒⠀⠀⠄⠀⠠⠀⡀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣯⢺⣌⢋⠖⡫⣍⣼⣟⢮⡷⢭⢹⣻⣽⡿⣽⡟⣮⢕⣚⢆⠏⡦⠑⣈⠐⡀⢂⠈⢀⠀⠡⠠⢂⠀⠰⡄⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠯⣎⣷⣩⢎⡗⡻⣼⣿⣹⡾⡦⣏⣷⣿⣷⣿⣷⡯⣽⡚⡌⢎⡼⠰⡉⠔⡈⠐⠄⠂⠡⠑⠌⡀⡐⠌⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣯⣟⣮⢽⣧⢗⣽⣿⡻⣼⣭⢳⣯⣿⣿⣿⣿⡾⣟⠳⡩⡜⠲⠤⠣⢔⣡⠤⠁⢂⡉⢀⠂⠁⡁⠄⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣾⣯⣿⡾⣳⣧⣿⣿⣯⢷⣟⣮⣿⣿⣿⣿⣻⢧⣻⢎⠧⣉⡆⠱⡈⠔⢀⠂⢐⠋⠐⠌⠀⡢⢁⠀⠄⠐⢀⡀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣽⣻⣾⣿⣿⣿⣳⢯⡿⣝⢮⣋⡜⡡⠬⢁⡄⡂⡁⠐⠁⠐⠂⠁⢐⡁⠄⠀⠀⢂⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⢯⣟⡼⣋⡖⢥⡘⠢⠍⡀⠒⠀⠅⡈⠂⠁⠈⡐⠈⠐⠁⠤⠁⡰⠈⠠⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡽⣾⣛⡮⣗⡱⡜⢒⠌⡑⠌⡐⠀⠍⠐⣀⢂⠂⠒⡈⢢⠈⠄⠂⠒⠐⡐⠐⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣞⣷⣻⡝⢶⡙⢎⡱⡘⣌⠢⢃⠌⢐⠨⠠⠂⠠⢠⢀⡁⠔⡠⠆⠑⠈⡀⠀⢅⢀⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡿⡽⣎⠷⣽⢲⡝⡌⡖⠥⠰⢣⠂⡘⠄⠂⢅⠂⠑⢂⢀⢈⠐⠄⠂⢉⢢⠀⠁⠆⠁⠰⠈⠀⠄⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡿⡽⢷⣝⢺⡘⣆⢣⠓⠸⡄⠃⠄⢂⠐⠄⠰⠁⠈⢂⠱⣀⠂⠠⠐⠠⡅⠉⢅⡐⠡⡁⠐⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣰⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢯⣻⢽⡳⣌⢷⡘⣌⡢⢩⠱⠈⠌⠐⢀⠂⡈⢤⡈⠠⠂⠂⠄⠂⠌⠢⡈⠈⠌⠠⠘⣠⢂⠀⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣰⢯⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⣯⢳⢏⣵⣘⠦⡙⢤⠑⡂⠤⢉⡜⢀⠂⢢⢈⠠⠁⡁⠂⠁⠢⠁⠄⡰⠠⡁⠈⠄⠲⠈⡄⢂⠂⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⣰⣟⡎⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⡳⣏⢯⣰⠚⢥⢛⡄⠣⢑⡈⠄⡐⢈⠂⡡⠀⠀⠀⠀⠀⠁⡠⠈⡆⠀⠄⠐⢈⠐⢀⠃⠐⠄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⡲⣿⣛⢦⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡾⣝⡻⡜⣎⠖⣍⢂⢖⠨⢇⠢⡈⠐⢀⠈⠆⠀⠤⣂⣁⠦⢑⢀⢀⠂⠠⠐⠠⠈⠄⢀⠀⠂⡈⠠⢀⡀⠀⠀⠠⠀⣿⣿⣿⣿
⠎⢐⣿⣻⢆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡿⣭⢷⣹⢦⢹⣘⠤⡊⢥⢈⡰⢀⢁⠂⡁⠀⠢⡀⢀⠂⠤⠃⣀⠀⠌⠂⢃⡀⠎⡁⠠⣂⣄⣀⢄⡄⢬⠀⣀⢄⣼⣿⣟⣿⢿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣯⢿⡹⢮⡱⢎⠥⣒⢢⠑⢢⡑⢤⠂⠂⡀⢀⠉⢂⠈⡐⠠⠀⡐⠀⢄⠐⢂⠀⡀⠄⠰⢡⣟⡸⣯⢸⡇⣻⢘⢀⣾⣴⣿⣿⣻⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣞⣯⡝⣧⣙⢎⠲⣁⢆⡩⢂⡒⢂⠐⡁⠠⢈⠠⠀⡂⠄⢁⢐⠀⠆⢈⠔⠠⠁⠄⢂⠈⡠⢹⡺⢏⣱⢯⣚⠎⣸⣏⣏⣏⣭⣽⣽
