⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡀⠀⣀⢀⣄⡀⠆⠴⠀⠂⣀⠐⡀⠒⡀⠀⢂⡄⢂⣐⡀⣀⢃⡐⡂⢂⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡈⠃⠅⣯⠪⡱⣥⡮⢿⣯⣤⣎⡣⡏⢘⡥⣍⡷⣍⠷⣭⣯⣟⠿⣿⣟⡷⣷⢮⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡑⡸⡔⢯⣛⠳⢎⡵⣺⢼⡳⢯⡗⢧⡊⡵⣈⠶⣈⠷⡼⣹⢭⣟⢷⣿⣽⢯⡿⣜⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠑⡌⠳⡌⡝⣎⠶⣭⢞⡽⢯⡞⣣⠽⡰⢥⡚⣍⢞⡱⣏⡷⣽⣻⡾⣯⣿⢿⣽⣺⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢠⠡⡓⣌⡳⣜⡱⣏⡾⡵⢯⣞⡿⣜⢧⡛⡵⢣⠽⣘⢯⡳⣽⡹⣗⣿⣽⣿⢿⣿⢷⣻⣞⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢢⠘⡤⢣⠝⣤⡑⣌⡕⣋⠷⢝⠻⣼⡽⣞⡧⣟⣼⢳⠯⣝⠮⣝⣺⣭⢽⣮⣿⣾⣿⡿⠿⠛⠛⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⠘⡄⢣⠜⣣⠞⣤⣲⠀⢒⡒⠎⠀⠠⡪⠞⣻⡽⠺⠁⡨⠑⡈⣉⡲⠝⣬⠽⠝⠓⠉⠀⠀⠀⢀⠀⠀⠀⡀⢀⣠⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠋⠲⠥⠽⣣⢁⡿⣹⡅⣌⠀⢕⣋⠀⣡⡘⠷⠭⠊⠈⠉⠀⠀⠀⠀⡀⠐⠀⠈⠀⢀⣠⣶⣿⢿⣟⣿⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄⠁⠈⣠⣴⣿⣿⣿⢿⣟⣿⡷⣽⣶⣾⢿
⠀⠀⠀ ⠀⠠⢄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀ ⣴⣾⢿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣽⡿⣾⣽⣄
⠀ ⠀⡈⠔⡀⢃⠎⣰⢉⡓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⣿⣿⣿⡻⠾⠿⠿⠾⠟⠛⠚⠛⠛⠙⠉⠉⠉⠁⠀⠀
⠀ ⢀⠂⡈⠔⡈⠆⡜⡌⢇⢣⢎⡝⡱⢦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⡾⣟⡿⢿⡿⣟⣷⢀
⠀⠀⠠⢀⠂⠄⡑⡈⠴⠉⠔⠸⠌⠃⠚⠘⠉⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠻⣝⢿⣻⣽⢿⣽⢯⣗⣻⣿⣟⡿⣟⣿⣽⡿⣽⣻⢿⣿⡿
⠀⠀⠈ ⠀⠀⠀⠀⠀⠀⡀⢀⡀⢄⠠⢠⡀⢆⠶⣐⠖⣲⢭⢯⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⡱⣻⠽⣞⡷⣯⣟⡾⣿⢿⡷⣯⣽⣻⣟⣯⢿⣝⡷⣏⡿⣾⣟
⠀⠀ ⡀⠄⡐⠠⢈⠐⢡⠐⡡⠘⠤⢉⠒⡘⣌⠣⢥⢫⠥⣏⠾⣹⠾⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⣌⡳⣭⢻⡼⣽⡞⣵⡿⣯⣿⣻⣯⢷⡿⣞⣽⣻⢮⡽⡽⣽⣻
⠀⠀⡀⢀⠐⠠⠁⠂⠌⡀⠎⠠⢉⠔⡡⢡⠑⢢⠍⣆⢣⢛⡬⢳⢭⡻⢮⣟⣷⡄⠀⠀⠀⠀⠀⠀⢀⡀⠓⢦⠱⣎⠷⣹⠶⣻⣽⡻⣽⣞⡿⣮⣿⣳⡟⣮⣽⢳⢯⡟⣭
⠀ ⡀⠀⠂⡀⠡⢈⠐⠠⠐⡈⢐⠁⢢⠁⢆⡉⠦⢩⡐⢣⠓⡜⡣⣏⡽⣳⡞⣾⣻⣆⠀⠀⠀⠀⠀⠠⠐⡉⢆⠳⣌⠻⣥⢻⡱⣧⢟⣧⢿⡝⣯⣖⣯⡟⣧⢯⢏⡷⣝
⠀⡀⠄⠁⢀⠂⠄⡈⠄⠡⠐⡈⢐⠂⢌⠰⠈⢆⡡⡘⣅⠫⢜⡱⢬⠳⣵⣚⡧⡷⢿⡄⠀⠀⠀⠀⠀⠀⡁⠘⡌⠱⣌⠳⣌⢧⢻⡜⡹⢾⢭⣟⣳⡽⣲⢿⡜⣯⢞⣪⠀ ⠀⠠⠈⢀⠀⢂⠐⠀⡁⢂⠐⡈⠐⡈⠄⡉⠄⢢⠑⡤⠓⡬⣘⠥⣛⠶⡱⢯⣻⢭⣟⢦⠀⠀⠀⠀⠐⡐⢈⠱⡠⡙⢤⠣⣏⢞⡹⣏⠾⡼⢶⣫⡽⣞⡽⣜⠯⡶⣍⠷⠀
⠐⠀⢀⠂⠠⠀⡐⠠⠐⢀⠂⠄⠡⠐⡐⠠⢉⢂⠡⢂⡑⠲⣌⠲⡍⢞⡭⣳⢭⡳⣞⡃⠀⠀⠀⠀⠁⠐⡈⠔⠡⡜⣨⠓⡜⡎⣵⢋⣞⡳⣝⠶⡽⢎⡷⣎⠟⡴⣋