Pendulum clock Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢰⡞⢳⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡋⠉⠀⢸⡀⠀⣨⠯⢿⡀⠀⣸⠛⠋⠉⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣵⠋⢀⠴⠊⠲⢝⡾⣿⢉⣉⣿⣾⠥⠊⠒⢄⠢⡙⢾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠭⠭⠭⠯⠭⠭⠽⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢽⣷⣿⣿⣿⣟⣿⣻⣿⣯⣻⣿⣟⣿⣭⣿⣽⣿⣯⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⡇⡖⢒⣲⠖⣀⠀⠥⠭⣘⡀⠲⢦⡒⢴⢸⣻⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡗⣷⠋⡴⣪⡆⠀⣻⣟⠑⠻⣷⢄⠙⢾⣿⣫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠃⣞⣎⣀⡟⠁⠀⠀⠉⠢⣨⠷⣷⠈⢹⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⢸⣮⣀⣽⢈⠳⢦⡔⠡⠀⢡⣀⣿⡇⢸⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⠘⣿⣠⣽⡘⣜⡁⠘⠀⢀⣏⢉⣿⠇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣿⡻⣉⣽⡦⢥⡤⠴⡉⢻⢿⢿⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⠀⣷⡟⠺⢿⣦⣼⣵⣔⣮⠗⢹⢨⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡇⠀⣿⡇⠀⠀⠀⡉⢯⠀⠀⡀⢸⢸⢠⣼⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⡇⣿⡇⠀⠀⠀⢛⠒⠁⣜⡄⢸⢸⠘⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢣⣿⡇⠀⠀⢀⣸⢡⣘⢺⠃⢸⢸⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠘⣿⡇⣠⣂⣸⣽⣎⣿⠀⠀⢸⢸⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣿⡏⠰⠃⢯⡳⠷⣫⠀⠀⢸⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣿⡇⠀⠀⡾⠃⠐⠿⣄⠀⢸⢸⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡇⠀⣻⢻⠀⡜⡆⠀⠀⠀⢼⡆⣸⢸⣿⢸⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⡀⡿⣎⢧⡘⢼⣦⣀⣠⡾⢡⢏⡏⠸⣸⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⡇⡿⣅⠈⠳⢽⣲⣭⣭⣥⣖⡥⠎⠃⡠⢿⣹⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣻⣇⣃⣈⣓⣒⣀⣋⣏⣋⣉⣁⣀⣴⣾⣿⣻⣗⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣛⣓⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣚⣛⣲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠉⠉⠉⠁⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢀⣀⣀⣀⣀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣶⣿⡿⠿⠿⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠠⢍⡻⢿⣦⣀⣀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⣿⡿⠁⡀⠁⠀⠀⠀⡀⠀⠀⠀⠙⠈⢿⡏⢩⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡿⠀⠌⠀⠀⠀⠀⢠⣇⣀⠀⠀⠈⠄⠘⠇⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡆⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⡀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠣⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⣸⠃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠱⣀⠑⠤⣀⠀⠠⠀⢀⣠⠔⢁⡴⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⠶⢤⣉⡙⢛⣉⡠⠔⠋⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣶⣦⣤⣀⢀⣀⣤⣴⣶⣦⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⡿⠟⠚⢿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣇⠀⢀⣠⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡇⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠰⠤⠤⠤⠤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⣶⣶⠦⠶⠶⠦⠶⠶⠒⠦⠶⠦⠶⠖⢶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣿⠀⠀⢀⠄⠀⠀⡀⠄⠢⠠⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠐⠍⢲⠜⡇⠀⠀⠀⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠋⠔⠀⢀⣸⠄⠀⠀⠀⠈⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢉⠩⢀⡉⠀⠀⠀⠀⠀⠚⣠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⡐⠀⢀⠡⠈⣀⡨⢊⣴⣯⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢯⣝⠷⠂⣠⣬⠄⣠⣟⡽⠛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠈⠀⠚⠇⢀⡼⠋⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⣠⣿⠄⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡔⡇⠀⢀⡼⣯⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⣰⠟⠁⠟⢿⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡞⠁⠀⠈⢂⠀⠓⡄⠀⠀⡴⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢔⠁⢳⡴⠃⠀⠀⣟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢧⠀⠀⠀⠙⠋⠀⠀⠀⠀⠛⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠿⠟⠛⠛⠿⠿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠺⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠟⠛⠒⠒⠀⠀⠀⠻⠗⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⣄⣀⠀⠀⠸⠗⠀⠀⣀⣠⣴⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣠⣿⣿⣄⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣷⣤⡀⠙⠿⣿⣿⠿⠋⢀⣤⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄⠀⠀⣠⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⢻⣿⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠷⠶⠶⢶⣶⡆
⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⢻⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⠏
⠀⠀⠀⢸⣿⠟⠻⠟⠻⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠃
⠀⠀⠀⠘⣿⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣤⣀⣀⣀⣀⣀⣀⣤⣀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⠿⠟⠿⠿⢿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⣠⡶⠞⢋⣡⣤⢴⣶⡶⣤⣤⣉⡛⠷⣦⡀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣇⠀⠀⣠⡾⢋⡴⢾⣯⠉⠀⣠⠛⣧⠀⠈⣽⡟⢷⣌⡙⢦⡀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣟⢀⣼⠋⣴⠋⠀⠈⠁⠀⠀⠉⢰⡏⠀⠀⠀⠀⠀⠙⢷⣌⢻⣆⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣿⣾⢃⣾⠛⠖⠀⠀⠀⠀⠀⡄⠸⡇⠀⠀⠀⠀⠀⠘⠋⢻⣆⢻⣾⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣤⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇⢾⡷⡶⠀⠀⠀⠀⠀⠀⣿⠀⣿⡄⠀⠀⠀⠀⠀⠐⠿⣿⡇⣿⣿⠀⠀⢹⡇
⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣈⢣⡀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⡘⣷⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⡇⠀⠰⠦⣼⡟⣰⡟⣿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⠁⠀⣿⣿⠙⣷⡘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⢀⣴⠟⣱⡟⠀⣿⡃⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇⠈⢿⣦⡙⢷⣄⡛⠀⠀⠀⠀⠀⠀⠀⠘⣣⣴⡿⢫⣾⠏⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡀⠀⣿⣿⡇⠀⠀⠙⠻⣶⣍⡛⠿⠶⣶⣶⣶⣶⠾⠟⣋⣥⡾⠋⠁⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⣇⣀⣀⣀⣀⣀⣉⣛⣻⣷⣶⣶⣶⣶⣿⣟⣋⣁⣀⣀⣀⣀⣀⣀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢻⣿⠿⡿⠻⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⡆
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡄⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⡀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣷⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⣠⡿⠖⠈⠛⠻⢶⣄⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡼⠃⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣟⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣆⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣦⣤⣴⣾⣿⠿⠃⠀⠀⠀⠀⣿⠀⠀⣿⣷
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⣯⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⡿⠛⢿⣄⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⢿⣄⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣷⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⣸⡿⠋⠙⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⡇
⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⣼⡿⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⢻⣦
⡾⠿⣷⣶⣶⣶⣶⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣤⣤⣤⣄⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⡄⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢿⣿⣿⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⡿⠿⣦⣼⣿⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣸⣯⣽⡇⢀⣶⣿⡆
⠀⠀⠀⠀⠀⠰⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣯⣭⣽⣿⣯⣥⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣦⣀⣀⣠⣴⣼⣿⣤⣄⣀⣀⣀⣀⣨⣈⣉⣥⣦⡀⠀⣀⣨⣯⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣏⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⠛⣿⣿⡟⢻⣶⠟⠀⠈⠻⣿⣿⣿⣿⣿⣿⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢸⣿⡾⠻⣿⡟⠛⠻⣤⠀⢠⡐⠻⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣤⣾⣶⣬⠁⡀⢻⣿⠇⠀⠈⣷⣤⣿⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⠛⠻⣯⣉⠀⢀⣾⣿⣦⠀⠀⣿⣭⠛⠛⣿⣿⣿⣿⠃⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⣿⣿⣿⣿⣿⣿⡿⢷⣼⣿⡇⣩⣄⠚⢿⣿⣧⣤⣿⣿⣿⣿⣀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣯⣿⣿⣿⣿⣧⡈⢳⣾⡿⢉⣿⢀⣁⣴⠞⢉⣽⣿⣿⣿⣿⣯⣤⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣄⣀⣽⣯⣉⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⣏⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⡿⢉⣿⣾⣿⣠⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⡿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣉⣥⣾⣿⠟⠉⠉⠙⢻⣿⣿⣿⣿⣧⣽⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⡄⠀⣿⣿⣿⣿⣿⣿⣿⣀⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⡀⠙⣿⡿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣻⣿⣦⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣾⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣴⣿⣶⣿⣿⣦⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣤⣾⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣋⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠙⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠁⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⡿⠿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣴⣶⣶⣶⣶⣶⣶⣶⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⠒⠀⢀⠢⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠘⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⢦⠄⠀⢀⡀⠐⠊⠀⠀⠁⠀⠀⠀⠀⠀⡄⠃⢀⠀⠐⢤⡀⣠⠔⠀⡀⠈⡜
⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡄⠈⠀⠀⠀⠙⠁⠀⠀⠁⠀⢃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠀⠀⠀⣀⠀⠀⠀⢀⠌
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⠀⠀⡀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⠪⡒⠄⠀⠈⠀⠀⢔⠁⢚
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠖⠉⠉⠀⠈⠉⠓⢤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡁⠄⠀⠀⠠⢈⢳⣄⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢠⠀⠀⠀⠀⢳⡀⡀
⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠘⡢⡀⠀⠀⠀⠀⠀⠀⣠⠞⡁⢉⠄⠀⠀⠀⠀⠠⡉⢈⠳⣄⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⣼⠀⠀⠀⠠⠀⡇
⠀⠀⠀⣠⠞⠓⠀⠈⠀⠀⠀⠑⢄⠛⠣⡀⠀⠀⠀⠀⠀⠐⠠⠁⠀⠀⠀⠀⠀⠀⠈⠄⠂⠀⠀⠀⠀⠈⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠂
⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠐⠀⠒⠒⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⡘⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋⠄
⠀⠀⠀⠀⠠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⠓⠒⠶⠒⠚⢉⢴
⠀⠀⠀⠀⠀⠑⢄⠂⠀⠀⠠⠐⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡁⠂⠀⠀⠐⢈⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢀⠌⠁⠀⠀⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⡔⠒⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⢻⠉⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡂⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠔⠉⠀⠀⢰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠐⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⢀⡂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⠉⠉⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣞⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠭⠽⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣴⠣⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢧⠀⠂⠈⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡣⣀⣐⣀⣠⢞⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣫⢽⢲⣶⢿⠛⢯⣭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣩⣿⢯⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⡉⣸⣷⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡂⣶⡿⢽⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⡏⠃⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⢃⢴⣷⣽⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢶⠷⠚⠻⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠈⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⣤⠄⣰⡟⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠙⠻⠋⠾⠟⢁⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢰⣿⣷⣶⣤⣤⡶⣿⢟⢏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣾⣶⣶⣶⣶⣶⣶⣾⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢄⣠⣠⣴⣷⣾⢿⣷⣮⣀⣄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⠛⠛⠛⠛⠛⠛⠙⠛⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠛⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣏⡩⠭⠭⠍⡉⢿⡷⣿⡇⠀⠀⠀⠀⠀⠀⣠⡟⠁⢰⠀⠘⠃⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡁⠂⠀⠁⢀⢀⠠⡇⣿⡇⠀⠀⠀⠀⠀⢠⡿⠀⠐⠠⣀⠀⠀⡰⠊⠈⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠄⠐⢦⡔⠋⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⢸⡇⠐⠂⠀⠈⠑⠞⠀⠀⠐⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠀⠀⠀⠠⢀⡇⣿⡇⠀⠀⠀⠀⠀⠀⢿⡐⠞⢀⠀⠀⠀⠀⡀⠑⠂⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣀⡈⠀⠂⠈⢐⡐⡇⢻⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠁⠀⠂⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢾⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣿⢸⡏⢹⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢸⡇⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣼⣧⣸⣿⡇⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠈⢻⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣼⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⢷⣶⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣛⣻⣿⣟⣿⣿⣻⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠷⠾⠶⠶⠶⠲⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⣶⣿⣶⣤⠀⠀⠀⠀⠀⣤⡾⠋⠙⢷⣤⡀⠀⢲⣾⠿⠿⣷⡖⠀⢀⣠⡶⠿⣦⣀⠀⠠⣶⣿⣿⣿⣶⠄⠀⣠⣴⣶⣶⣄
⠀⠀⠀⢺⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿⠀⠀⢸⡇⠀⠀⠈⡇⠀⢸⢹⣇⣀⣼⢹⠀⠀⡿⠁⠀⠈⢻⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⡟⠁⠀⠈⢻⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠀⠀⢸⡗⢦⡴⢺⡇⠀⢸⣾⠋⣍⢹⣾⠀⠀⣷⣀⠀⣀⣾⠀⠘⢿⣿⣿⣿⣿⠇
⠀⠀⠀⢻⣷⣀⠀⢀⣼⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠘⠳⣼⣮⠞⠃⠀⢸⣼⣦⣭⣾⣼⠀⠀⢹⣿⠟⣿⡏⠀⠀⠀⠉⠛⠉⠁
⠀⠀⣠⣼⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠈⠀⠀⠸⣷⣬⣿⠏⠀⠀⣶⣶⣶⣶⣶⡆
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣶⡶⠀⠀⠀⠀⢠⣴⠾⠿⣦⡤⠀⠀⢀⡴⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⡿⠻⠿⣿⡇⠀⠀⠀⠀⠀⠻⠘⠃⣷⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠘⣿⡶⠷⣾⣟⠀⠀⣴⡾⠛⢷⣦⠀⠀⣿⣽⣿⣿⣿⡇
⠀⠀⠀⢸⡏⠀⠈⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡷⢀⡀⢸⡇⠀⢀⡟⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣰⣦⣀⠀⠀⠀⢸⣧⣀⣂⣼⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⣀⣤⣀⣿⠀⠀⢲⣶⣶⣶⣶⠂
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡽⠟⠛⢿⣥⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⡇⠰⡷⠀⣿⠀⠀⣿⠀⣤⠀⣿⠀⠀⢸⠏⠉⠙⢿
⠀⠀⠀⢸⡇⢠⣶⣄⢸⡇⠀⠀⠀⠀⢾⡇⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⣷⣶⣶⣶⡯⠀⠀⣿⣤⣭⣤⣿⠀⠀⢸⣦⣀⣠⣿
⠀⠀⠀⢸⡇⡎⢿⠁⢸⡇⠀⠀⠀⢠⣼⣿⣦⣴⣾⣯⣄⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢸⡇⠈⢸⣿
⠀⠀⠀⢸⡇⡇⠀⢠⣼⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⠁⠀⢻⠀⠀⢀⡞⠉⠀⠙⢦⠀⠀⣠⠴⠲⠦⣄⠀⠀⢸⣇⢾⢺⣿
⠀⠀⠀⢸⡇⠇⠀⢸⣿⡇⠀⠀⠀⠀⢹⣿⣿⢙⢿⣿⠀⠀⠀⣧⣀⣀⢸⠀⠀⠸⣇⠀⠀⠀⡼⠀⢰⡇⠀⠀⠀⢸⡆⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⢸⡇⠀⠀⠸⢿⡇⠀⠀⠀⠀⢸⣯⠛⠟⢺⣿⠀⠀⠀⡧⠷⠿⢸⠀⠀⠀⢻⣷⣶⣿⡇⠀⠈⢧⣀⣀⣠⡾⠁⠀⠀⠀⣀⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⣿⢀⣄⢸⣿⠀⠀⠀⡇⣠⣄⢸⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⢺⡿⠿⢿⡟⠀⢀⣴⠿⠛⠻⣷⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⠙⠛⣸⣿⠀⠀⠀⣧⣙⣟⣸⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣾⠁⣀⠈⡇⠀⠺⣿⠀⠀⠀⣸⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⡇⠀⠀⢸⣧⣀⣿⡇⠀⠀⢺⡜⠿⢣⡇⠀⠀⣿⣷⣶⣶⣿⡇
⠀⢰⣿⡿⣿⣿⣿⣿⢿⣿⣿⣧⠀⣰⣿⣿⣿⣿⣿⣿⣧⠀⢼⠿⠿⠿⠿⣷⠀⠀⠸⣿⣿⣿⠇⠀⠀⢼⣿⣿⣿⡧⠀⢰⣿⣿⣿⣿⣿⣇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠿⠿⡿⠿⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣿⣏⠉⣉⣙⣋⣀⣰⣟⣩⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡿⠿⠿⠭⠭⠽⠿⠿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣷⢲⢲⢺⢻⢻⣿⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣘⣿⣿⣿⢿⠋⠉⠛⠿⣿⣿⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⠃⠀⠀⠀⠀⠀⠙⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣿⣀⠀⡴⢣⠀⢀⢈⣿⡍⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⣈⣀⣤⣿⣿⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⣿⡟⠛⢿⢻⣿⣿⠿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠈⠇⠀⠀⣻⢠⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠀⠀⠂⠀⠀⢼⠈⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡄⠀⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⡆⠀⠀⣠⢅⡀⠀⣼⠂⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡄⢀⣤⡀⣀⣷⡀⣮⢠⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣼⣿⡇⠘⠀⠀⣹⡿⠃⢿⣘⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣔⣀⡈⠋⠀⠀⢸⢙⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠶⠦⠤⠭⠭⢭⣭⣽⣿⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣦⣄⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢭⣵⣦⣍⣩⣭⣽⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠢⢤⡤⢼⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⠃⠀⠉⠉⠉⠉⠁⠉⠀⠉⠈⠀⠉⠀⠀⠈⠁⠈⠉⠁⠀⠈⠉⠉⠀⠉⠈⠁⠈⠘⣿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠤⠀⠒⠂⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⣀⣀⣀⣀⣀⡀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡰⠋⠀⢀⡤⠚⠋⣀⣀⠀⠀⣀⡀⠙⠲⢄⡀⠈⠳⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠎⠀⢀⡴⠋⡠⠖⣩⡀⠀⠴⠆⠀⠈⡉⠲⣄⠙⢆⠀⠈⢣⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡜⠀⢀⡎⢠⠞⣀⡀⠈⠀⠀⢠⡀⠀⠈⠋⠀⡈⢣⡈⢣⠀⠀⢇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠁⠀⡾⠀⡎⠀⠙⠻⢶⣄⡀⢸⡇⠀⠀⠀⠘⠋⠀⢣⠈⣄⠀⠸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⡅⠀⠁⢶⠄⠀⠀⠈⠻⣟⣣⠀⠀⠀⠀⢰⡆⢸⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⠀⠀⣧⠀⡇⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡸⠀⠟⠀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠘⡄⠰⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡠⠁⡸⠀⠀⡜⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⡄⠀⠘⢆⠈⠢⣀⠛⠁⠀⢤⡄⠀⠺⠁⣠⠞⢁⡔⠁⠀⡰⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢦⡀⠀⠑⠆⣀⠙⠂⠤⠤⠴⠐⠚⠉⣀⠔⠉⠀⢠⠞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠉⠑⠒⠒⠒⠒⠈⠉⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠤⠀⠠⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠶⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠚⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⠀⢀⣀⠀⢀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡀⣀⣀⡀⢸⣿
⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠛⢉⢉⢀⢀⣀⣀⣉⠉⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⢠⣴⢶⡜⠈⠉⠸⠇⠉⠙⠽⠳⢦⣄⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⢠⣦⣙⠀⠀⠀⠀⠀⡜⢢⠀⠀⠀⠀⠈⣩⣶⡄⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣠⡟⠁⠉⠰⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠈⠿⠆⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡍⠃⠀⠀⠀⠈⠻⣦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⢿⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢼⣇⡀⠀⠀⠀⠀⠀⠈⠻⡾⢇⠀⠀⠀⠀⠀⠀⠀⢀⣸⡗⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢭⡍⠁⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠱⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⠏⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠘⡷⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠿⠋⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⡙⠻⢆⣶⣤⣀⣰⡆⣀⡀⡜⣶⠟⢋⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⣌⣉⣙⣉⢁⣋⣁⣡⣤⣾⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠋⠛⠿⠿⠿⣿⣿⠿⠿⠿⠛⠋⠁⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣠⣴⣶⣿⣄⡀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠙⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣨⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣥⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣷⣄⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⣀⠀⣿⠏⢀⣿⡏⠀⣤⣼⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⡿⠟⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⣿⣿⡉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⢻⣾⣿⣶⣿⣦⣤⣴⡿⠿⠿⠶⢶⣾⣿⣿⣿⣿⣄⠙⣷⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣉⣱⣿⣿⣿⠿⠋⠉⣉⣉⣤⣤⣤⣤⣀⣈⠉⠛⢿⣿⣿⣆⡈⠙⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⠹⣿⣿⣿⣿⣿⡉⠻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⣿⠟⠋⠀⢀⣀⣀⡀⠀⠉⠛⢿⣿⣶⡟⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠛⠛⠃⢀⣾⣿⠟⠀⢀⣤⡌⠛⢿⣿⣿⣷⣆⡀⠀⠻⣿⣿⡛⠛⠛⠛⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣀⣠⣿⡟⢳⣤⣼⣿⣿⠃⠘⣿⣄⠀⣿⣿⣷⣶⣶⣶⣶⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢛⣿⡇⢹⣿⣿⠿⠿⠿⠿⠟⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡏⢹⡏⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⣠⣴⠶⠿⠀⠀⠺⡿⠷⠦⠀⠀⠀⠀⠀⣿⡇⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣿⣿⡇⠀⢠⣾⡿⢁⣠⣀⣴⡦⣶⣄⣀⡀⠘⣿⡆⠀⠀⣿⡇⠸⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇⢠⣿⣿⣀⠙⠛⠋⠉⠀⠀⠙⠻⡇⠀⣄⢀⣀⠀⣿⡇⢰⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣾⣿⣼⣿⠄⠀⠀⠀⣠⡄⠀⠀⠘⠟⠈⢻⣿⠀⣿⡇⢠⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇⠈⢹⡿⠁⣤⣶⣶⡿⣿⡿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⢀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⣿⣿⡃⠻⣿⣀⣤⠉⠉⠉⠀⣿⡇⠀⢀⣸⠿⣀⠿⣿⠀⣿⡇⠸⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⡇⠀⢿⣿⣿⡿⢿⠷⢀⣿⡇⠰⣿⠿⣠⣿⡀⠀⠀⣿⡇⢀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⣿⡇⠀⠈⠻⣿⣄⣀⣀⠈⢿⡇⠀⠀⠄⠹⠿⠁⠀⠀⣿⡇⠸⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢿⣿⣇⠀⠀⠀⠈⠙⠛⢿⣦⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⣿⡇⠰⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣹⣍⣙⣿⣿⣿⣿⣿⣿⣿⠃⢠⡀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣼⣿⣿⣿⣍⣉⣿⣿⣿⣿⣟⣉⣉⣉⣋⣉⣀⣽⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⡿⠿⠛⠛⠛⠛⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠐⠃⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣀⣾⣿⡏⠀⠀⣿⡇⠀⢹⣿⠁⠀⢹⣿⣷⡿⠀⣿⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣷⠀⠀⣿⡇⠀⢸⣿⠀⠀⣾⣿⡇⠀⠀⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡿⣿⣿⣿⠀⠀⣿⡇⠀⠘⣿⠀⠀⢻⣿⡇⡄⠀⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⡇⣿⣿⣿⠀⠀⣿⡇⠀⢠⣿⠀⠀⣼⣿⣿⣿⠀⣿⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣇⢹⣿⣿⠀⠀⣿⡇⠀⠀⣿⠀⠀⢸⣿⣿⣇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⢸⣿⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣿⠟⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣾⣿⣿⠀⠀⣿⡿⠀⢻⣧⠀⠀⢸⣿⣤⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣇⢸⣿⣿⣄⣠⣿⡇⠀⣼⣿⣤⣤⣼⡿⢀⡏⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢨⣏⢸⣿⠙⣿⠟⢻⡃⣠⣿⡏⢹⣿⢻⣿⣿⡇⠀⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢸⣿⣤⣿⣷⣾⣿⣿⣿⡇⢼⣿⢸⣿⣿⡇⠀⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣧⢸⣿⣿⣿⡇⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⢙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢼⣿⢀⣿⣿⣿⡇⣿⣿⠘⣿⡇⢸⣿⣸⣿⣿⡿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⠸⣿⣿⣿⣯⣿⣿⣤⣿⡇⢸⣿⢿⣿⣿⣧⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣴⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡿⠟⣉⠉⠉⠻⢿⣿⢹⣿⣿⣿⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢸⣿⣿⡿⣀⠀⠛⠃⠀⠀⠀⠉⢿⣿⣿⡏⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⣿⣏⠀⠉⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢸⣿⢿⣷⡀⠀⠀⠀⢿⡇⢀⣼⢿⣿⣿⣆⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⣼⣿⣿⣿⣿⣦⣤⣤⣤⣼⣿⣿⣴⣿⣿⣿⡆⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⢻⣿⠛⣿⡇⠿⣿⡿⢿⣏⣹⣿⣿⣿⣿⣧⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢸⣿⣾⣿⣿⣶⣿⣷⣼⣿⣿⣿⣿⣿⣿⠿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⠉⢉⢀⣀⣉⣉⡉⢉⣉⣉⣉⣉⣀⣀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠛⡛⠛⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⢹⡿⠿⠛⠛⠿⠿⠿⠿⠟⠛⢿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠶⠶⠶⠶⠿⠿⠿⠿⠖⠺⠟⠿⠿⠗⠒⠿⠿⠿⠛⠿⠿⠿⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣄⣠⣤⠀⢠⣤⠀⢀⣀⡀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠺⡟⠀⢘⠉⠀⠀⢀⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣶⠀⣨⣶⣶⣴⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡍⢻⣿⡏⣹⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡇⠀⠻⠾⢿⣿⣿⠏⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣿⣿⡿⢟⡁⠀⠿⢴⣾⣈⡿⢀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣿⣿⣽⣿⡟⠀⣸⣿⣶⠞⠁⠀⠀⠸⣿⣿⣷⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⣿⡏⠀⠀⠀⠀⠀⠀⣛⣼⣿⡟⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⡷⠀⠀⠀⠀⠀⠀⠻⣿⣿⡗⠛⠿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣥⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣧⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⣾⣿⣿⠿⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠙⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠉⠁⢀⡉⠉⠉⠁⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣻⣿⣛⣻⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠴⠶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⢋⣩⣴⣶⣶⣷⣶⣾⣷⣦⣍⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣤⠶⠛⠉⠀⠀⢈⠙⣁⠛⠒⠸⠽⣛⠿⣶⣄⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⡫⠀⠉⡄⣄⠀⠀⢸⠀⣼⠀⠀⠀⢰⠈⠁⠪⣻⣷⣄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⢯⠊⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⡈⣯⣻⣆⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⣯⠏⠀⠦⣉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠞⠁⠈⢿⣿⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢰⣏⢠⣿⡙⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⢀⡰⡿⠋⠀⠀⠀⡀⠀⢸⣿⡇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⠛⠘⠇⠀⠀⠀⠀⠀⠀⠉⠓⠶⠂⠁⠀⠀⠀⠀⠀⠀⣸⠀⢾⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⠆⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡇⢹⣿⣆⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠀⣰⣿⡿⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡈⢿⣿⣄⠉⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡴⣱⣿⠃⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣌⠻⣎⠣⢄⡀⠘⠀⠀⠀⢰⢖⠀⠀⠀⠀⠋⣀⠠⠊⣴⡿⠃⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠉⠻⢦⣭⡑⠂⢀⣀⣈⣈⣀⣀⣤⣄⣶⣷⣾⠿⣛⡴⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣶⣄⣉⣄⣀⠘⠛⠛⠛⠻⠛⠛⠛⣋⣩⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢻⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⣀⣼⣧⣄⡀⠀⠀⠀⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⢀⡾⠛⠉⠃⠙⣿⡄⠀⠀⣿⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⢸⣿⠃⢸⣿⠓⠀⠐⠛⣼⡇⠀⠀⣿⠂⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠈⢻⣦⡀⠻⢿⣧⣤⣼⠟⠁⢀⣠⡿⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠈⠻⣿⣄⠀⠈⠉⢁⣠⣶⠟⠁⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠛⢷⣦⣴⡿⠟⠁⠀⣀⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⣤⣴⣾⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠌⡤⠈⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠶⣿⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢠⣼⣥⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠲⡀⠀⠀⠈⠛⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⡁⢀⡀⡈⣣⢀⠀⣀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣀⡂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠘⠔⠁⠀⠤⠐⠃⠀⠀⠈⢊⠣⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣄⠀⢀⠴⠚⢉⠙⠒⢤⡀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⢠⠏⠀⠀⢠⢀⠀⠀⢳⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⠉⠉⠉⠽⠿⠿⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⡀⣀⠀⠈⠁⠀⠈⢠⠇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⣄⠐⠀⠀⠀⣠⠏⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡀⠤⡄⣬⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠈⢓⡊⠹⠁⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⢠⡔⠁⠀⢠⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⡇⠀⠀⢰⡄⢀⠀⠀⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠘⠇⠈⡆⠱⠁⠀⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡠⠀⠀⠀⢄⡐⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⢀⠀⡇⡀⡇⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠐⠀⠀⠼⠿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠈⢀⡇⠀⡇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠟⠁⠀⠀⠀⣘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠀⠀⠀⢀⡀⢀⠄⠁⠀⠀⢬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣁⡦⠀⠀⢀⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡁⠀⠐⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠚⢋⣉⡤⠤⠅⢈⠉⠙⠒⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠆⢀⡀⠴⠦⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠐⠴⠒⠒⠀⠐⡄⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⢀⡄⣠⠢⠰⠆⠔⣄⣀⠀⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡮⠀⠀⢀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⡜⢞⠈⢀⡀⣋⡧⡣⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⣀⣀⠢⠴⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣯⡽⠽⠯⠯⠿⠿⠽⠿⠿⢿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠃⠀⠐⣱⠀⠀⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⡀⣰⣿⡿⠟⢰⣷⠘⠿⣿⣷⡀⣸
⠀⠀⠀⠀⠀⠀⠀⣐⡄⠀⠀⡀⠀⠲⡵⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣽⣵⣿⣿⡁⠀⢶⣷⠂⠀⣿⣿⣧⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣰⠆⠀⠺⡠⠢⠀⠰⡠⡄⠠⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣶⣿⣿⣶⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⣀⠭⠤⢄⡀⠀⢀⠈⡁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢠⠎⠀⢀⡀⠀⠈⢧⠀⣙⡅⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⢉⡋⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⡀⠀⠓⠀⠐⢰⠀⣧⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠓⠀⢠⡗⠀⠒⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⢦⡁⠂⠀⠀⣀⠎⠀⡿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⠉⠄⣂⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠠⠀⠨⠙⣒⠫⠁⠀⠀⣳⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢸⠀⢀⠀⡦⢀⡀⠀⠀⣥⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠙⠀⠁⠈⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢽⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠈⠀⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⢼⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠆⠀⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣾⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠘⠀⠀⡀⠀⠀⠃⠀⠀⣒⡂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣥⣤⠸⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠐⠄⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣨⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠄⣀⠠⠦⢀⡀⠠⠀⠈⡁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⢒⠒⠛⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠠⡤⡖⡒⠒⡒⡦⣠⠀⠳⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠛⠯⠥⠤⠥⠝⠛⠀⠀⢠⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⠆⠀⠀⠀⠀⠿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣄⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⢀⣠⡾⣟⠁⠰⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⣀⣤⣾⣿⣗⣨⣿⣦⣌⣿⣿⣿⣿⣿⣿⢸⣿⡟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣷⣤
⠀⠀⠻⠆⢀⣤⣭⢙⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣿⣭⣍⡀⠀⠀⠰⠿⣿⣿⣿⡟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠘⠿⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣤⣄⣀⣤⣿⣿⣿⣿⣴⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⡀⠀⠀⠹⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣿⠋⣸⡿⠁⠀⠀⠀⠀⢀⣀⣀⣠⡤⠀⠀⣽⣿⡿⠀⠀⢄⣨⣶⣿⣿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣟⣿⣿⣿⠷⠶⠿⠿⠛⠛⠛⠛⠙⣫⡿⣅⣮⣾⣫⣤⠦⠂⠈⠹⢿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣴⣦⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⢿⠓⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣆⣠⣦⡜⠛⢂⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡌⣻⣿⣠⣠⣤⠎⠙⠉⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣶⠛⠋⠀⠉⠙⠉⠉⠿⣿⣿⣶⣶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⡀⣷⠃⢀⣠⣦⣤⣤⡦⠤⣤⣿⣿⣿⣿⣿⣿⣥⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⡤⠼⢿⡆⢉⠹⣿⠿⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣀⠀⣿⡄⠀⠀⠀⢀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡿⠋⠁⣠⣦⡿⣹⡔⠻⡄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⠀⢸⣿⣤⠀⣨⠀⢿⠷⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠋⠁⠀⠸⠿⣿⠃⣿⣿⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠈⠙⣇⣠⠏⠘⢺⡟⠋⠂⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠐⢾⣿⣶⢏⣻⣿⣿⣿⣤⣠⣶⠃⠀⢀⣀⠀⠀⢸⡟⠂⠘⠻⣷⣾⡇⠾⠿⠟⠛⠋⠉⣁⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡄⠀⠀⠹⢿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣿⠟⠃⠀⢀⡏⠀⣼⣧⠖⠴⠔⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠠⣖⣿⡖⠀⠀⠘⢿⣿⣄⢛⠛⣿⣿⣿⣿⣿⣿⣿⣏⢽⠿⠛⠁⠀⣤⠞⠀⠀⠙⢿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠚⠛⠛⠻⠦⣿⣄⢳⣾⣆⠀⠘⠛⢛⣿⣿⣧⣤⣉⣹⠏⠉⠁⢀⡘⠻⣷⣤⣴⣾⠋⠀⠀⠀⢐⡈⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡶⠛⣡⠤⠶⠿⢌⠹⣿⡷⠀⠀⠈⠁⠊⠸⠏⠙⠻⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣷⠀⠀⣀⠈⠉⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠥⠶⠋⠁⠀⠀⢀⠀⠡⡟⠁⠀⢿⡏⠀⠀⠀⠀⢻⣿⣿⣿⠟⠄⠀⠀⣼⣿⣷⡵⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⠋⠄⠰⡄⠀⣿⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠐⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡰⠚⠁⠀⠀⠈⠆⣿⠀⣀⡀⠤⠤⣀⠠⠤⢤⢤⣤⣴⣾⣾⣿⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⣿⣞⠁⠀⠀⠀⠈⠳⣦⣦⡀⠀⠉⠛⠷⣶⣷⣿⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⢀⡄⠈⠻⡛⢦⠠⣄⠀⢄⡈⠉⣉⣉⡭⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠈⠀⠀⣄⠙⠾⣧⡀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠛⠟⠳⠤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠤⠄⠠⠤⠤⠤⠄⠰⠢⠖⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⢁⠄⠉⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡷⠋⠀⠘⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠰⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠐⠁⣠⠊⠀⠀⠀⢢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣧⠀⠀⠀⠙⠀⣃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣮⡂⠀⠀⣷⡤⠶⠀⠶⠶⠶⣿⣿⣿⣷⣄⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠞⠁⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡡⠤⠲⢤⣀⢿⠋⠋⠀⠀⠀⠀⠀⣿⡟⢻⣉⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠻⣏⢈⠀⠀⠀⣀⡠⠾⣍⣸⡿⣋⣡⣤⣤⣄⡀⢀⡀⠰⣄⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣄⠀⠀⣰⣿⠐⢃⠀⣆⡈⠀⠀⠙⢁⣼⣿⠷⠚⠙⠺⢿⣦⠁⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠿⠻⠙⠁⠙⣿⠾⠿⠟⠓⠋⠀⠁⠙⢦⡀⠀⢸⣿⣟⠀⠒⠀⡄⣸⣿⣦⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡤⠋⢀⡐⠀⠀⡁⠀⠀⠀⠀⠂⡈⠛⢷⣦⣉⠀⢠⣿⣿⠀⠀⢀⣴⣿⠋⣈⣆⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠈⠀⠀⠙⢸⠁⠠⠀⢈⠸⠶⢸⣦⣽⣦⢸⣿⣿⣗⣿⣿⠟⠀⢀⠘⠻⢧⣀⠾⠿⣦⢻⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⣸⣿⣿⣦⣤⣤⣶⣤⣾⣿⣷⠀⠙⢿⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠡⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡎⢧⢹⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠃⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⣷⠈⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣋⢻⡟⠣⠿⢿⣿⡷⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⢠⣾⣿⣿⠋⣙⣿⣿⣿⡍⣿⡷⠂⡄⠰⣦⣶⣄⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡞⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⠀⢸⣿⣿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⡜⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⠐⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⡀⠂⣴⣤⠐⠃⠀⠀⠀⢠⣤⡶⠂⠀⠀⢠⣜⠏⣀⠀⣾⢋⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣯⡼⣿⣏⠙⠟⣠⣦⡀
⠀⠀⠀⢼⢁⣦⠈⠁⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⡰⠋⢁⡴⠃⠀⠇⢾⣥⣾⡏⢠⣿⣿⡇⠙⣻⢿⣏⡑⠀⡀⠙⢊⣉⣰⣾⡿⠋
⠀⠀⠀⢈⠉⠁⢀⠀⠀⠀⠀⡀⠉⠀⠀⠀⢀⠀⠁⠀⠞⠀⢀⡄⠀⠀⡄⠉⢤⢸⡿⢿⣁⣀⣛⣿⣯⡷⠺⢷⡀⠹⠛⠋⠹⠄
⠀⠀⠘⡵⠚⠋⠀⠀⠀⠀⢀⠤⠒⢀⡰⠟⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣤⣴⣦⣿⡉⠀⢻⡶⠀⠀⡤⠈
⠀⠀⠀⠰⢦⠤⢶⣤⣄⣀⣤⣄⡀⠀⣀⣠⣤⣤⣶⣄⣀⡀⢁⣄⣽⣶⣿⣦⣀⣀⣁⣀⣀⣧⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⢶⣤⣤⣤⣤⣔⣺⠋⠏⠁
⠀⠀⠀⠀⠀⢾⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⠿⡏⣛⣙⠋
⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠛⠛⠛⠀⣾⠛⢫⣤⣿⣦
⠀⠀⠀⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡻⠿⢿⣿⣿⠘⠛⠢⠀⠲⠘⠣⠶⢿⣿⣿⣿⣤⠖⠐⠂
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠙⣧⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡈⢷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣾⠏⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠱⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠁⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠘⠀⠞⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢀⠎⠀⠀⣀⠶⠋⠁⠀⡈⠂⠀⠀⠀⠨⡀⠀⠉⠳⢎⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⠀⠀⠁⠀⣀⢄⣈⠤⠐⠒⠒⠒⠠⠌⣐⢄⡀⠀⠈⢂⠀⢳⠀⡇⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢧⠀⠇⠀⡀⠂⠂⣡⠞⠁⠀⠀⠀⢀⢀⠀⠀⠠⠀⠑⠺⡀⠀⠈⢀⡞⢰⠃⠀⡰⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣄⠟⠁⠀⠀⠠⠶⠀⠘⠚⠀⠀⢰⠀⠀⠀⠘⢦⣀⡈⠀⠀⠀⠈⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⢀⣠⡏⠀⠄⢰⢴⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠈⡆⠉⠉⠉⠉⠉⠉⠀⠋
⠀⠀⠀⠀⠀⡸⠀⠀⠠⡀⠙⣄⠀⠉⠉⢠⠁⠘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⢀⠔⠁⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠉⠑⡴⠁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⠛⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⡆⠈⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢤⠀⡀⠀⡌⠀⠀⠀⠉⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠖⢊⣠⣤⡴⠶
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢒⠀⢀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠁⠀⠀⠓⠀⢀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⡅⢀⠀⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣠⡴⠋⣠⠊⠀⢀⣀⠀⠀⢂⠜⠓⠤⠄⣀⣀⣁⣀⣀⠤⠔⠛⢄⢂⠀⠀⠀⢀⡀⠀⠑⢄⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠁⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠄⢰⡎⠈⢀⠀⢀⠈⠑⣄⠀⣠⠐⢚⣁⣠⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣮⠀⠐⢸⡇⠀⠀⢸⡇⢰⡏⣄⠀⣤⠀⠜⣴⠛⠄⠈⠉⡁⠀⢠⡆⠀⢀⡄⠁⢀⠀⢀⠀⠀⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠸⡇⢠⡀⠈⠇⣼⢁⣛⡀⠘⠀⠾⠁⣾⠀⢘⠛⠁⠀⢸⡇⠀⢸⡇⢠⡟⠀⣶⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠁⠘⣷⡀⢰⣇⡜⠸⣧⣀⡄⣲⡀⠹⠶⠶⠶⠞⠀⢸⡇⠀⢸⡇⢸⠇⢀⣍⣠⣼⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣠⠈⠷⠴⠞⠀⠀⠀⠿⠤⠞⠷⠤⠖⠀⠀⠀⠀⠈⠻⠶⠤⠖⣿⣄⡀⠀⢠⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢩⣽⣿⡆⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢀⣴⣶⣎⡟⢀⣼⣿⣧⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⣴⣿⣿⣿⣿⣦⣬⣭⣿⡏⠀⢀⣀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣾⣿⣏⣽⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣙⣦⣄⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠋⠁⠀⠀⠸⣿⡟⠉⢉⠶⣄⡉⢀⣴⠈⠛⢻⣶⣼⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⡀⠀⠀⠀⠈⠙⣦⣎⡐⢾⣿⣿⣿⣷⡀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡜⠁⣻⣿⣿⣿⣟⡿⢟⡅⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣺⣿⣿⣿⡿⠿⠟⢿⠃⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣰⡟⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏
⠀⠀⠀⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠁⢼⣿⣿⡿⠛⠛⠂⡀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢰⡟⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⡞⣿⣿⣧⠀⠠⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢿⡆⠀⠀⣠⡼⠛⠁
⠀⠀⢠⣿⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣬⣫⣽⣆⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅⠀⠀⠀⠃
⠀⠀⣼⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡿⢏⣴⣿⡿⠿⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⡛⢻⣿⣳⣦⡀⠀⣽⣿⠋⠁⠀⡠⠂⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢺⣭⣿⣿⣷⣜⣿⠋⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠲⡦⢀⠀⠀⠀⠀⠀⠀⢳⠈⢿⢯⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⢤⣀⡀⠀⠀⢀⡀⠀⡇⢠⣿⡿⣛⣿⣿⡿⣿⣿⣿⣿⣷⣦⣀⡸⠃⠀⢀⣠⣴⣤⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢠⠀⠀⠘⣌⣻⡅⠻⣿⣯⣾⣭⣫⣿⣿⣿⣿⣿⣿⣒⣺⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠿⣷⣜⣻⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣯⣽⣿⣿⣿⡛⠛⠋⠉⠡⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢙⡿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⡔⣊⣉⣭⠶⠃⠉⠛⢿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣴⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⣼⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⢀⣠⣀⠀⢑⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡄⠀⣰⢰⢠⡄⣉⠶⣏⠁⢈⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣶⣿⣯⣿⣿⣿⣿⣦⣼⣷⣧⡀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⢠⣿⡈⢨⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣶⠀⠀⠀⡴⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣯⣰⣿⣗⢸⣿⣿⣿⣿⣿⣿⣯⣽⡋⠁⣉⣉⠉⣠⣆⠀⠀⠀⠀⢸⡃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣼⣗⠆⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣟⡁⣷⣋⠖⣹⠭⡗⠉⠁⠀⠀⠀⠀⠀⣗⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠎⢻⡟⠃⣿⣿⢹⣿⣷⣸⣿⣿⣿⢛⡷⢩⡉⠹⣿⠓⠲⠀⠃⠀⡀⠂⡜⠀⠀⢠⡏⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣤⡆⠈⠁⠷⠄⠸⠀⠒⢶⠀⠴⠙⡷⠀⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡗⠒⠚⣿⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠀⢀⣐⣀⡀⢀⠌⠀⠀⠐⡟⠉⣿⣿⣷⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣸⣿⣿⡇⠂⠀⠀⡀⠠⡇⠀⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡇⠀⠰⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠏⠀⣾⣿⣿⣿⣿⡂⠀⠀⠴⢻⡏⢉⣿⣿⣿⣟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡄⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣿⣿⡻⣿⠚⣿⣿⣿⣿⣯⣿⣿⣿⡆⠀⣸⣿⣿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠚⠀⠀⣹⣿⣷⣶⣷⣿⠻⣿⣿⣿⣿⣿⢿⡏⣧⣼⠇⣣⣾⣿⣿⣧⣿⣿⣿⣿⣍⣉⣉⣉⣉⣠⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⢀⢠⣼⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⢻⣸⡇⢿⣿⣧⣿⣾⡏⣿⣳⣿⣿⣿⣿⢻⣛⠟⢿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⢰⣴⣿⣿⣿⣿⣿⣿⣿⡿⣀⠀⣼⣿⣿⣿⢾⣯⢧⣹⣿⣽⣬⡟⣭⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣽⢿⣿⣿⣿⢈⣷⡀⠀⢀⡉⠉⠁⠉⠋⠈⠻⠿⠛⠛⠛⠋⠉⠉⠻⢿⣿⡻⣿⣿⣿⣿⡟⣿
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⢲⣿⣾⣿⣿⣯⣾⣿⢿⣄⠈⠻⣏⣍⢹⠳⣆⠐⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⣿⣝⠂
⠀⢠⢈⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢮⣿⣿⣻⣿⣿⣿⣿⡟⣺⣿⣆⠀⠙⣿⣿⡆⢹⣧⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⡄
⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⠟⢤⣶⣿⣿⣿⣯⡿⠛⣏⣹⣿⣿⣿⣿⣷⡀⠘⢿⢿⡄⣽⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣀⠼⣿⣿⣿⢿⢡⣿⣷⡄⠀⠻⣾⣿⣿⣿⣦⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠛⠛⠛⠿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣿⣿⣿⠟⠃⣿⣿⣆⠀⠀⠀⠀⠈⠉⠁⠀⠱⣄⠀⠀⠀⠀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠙⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢻⡅⢻⡿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠒⠲⠶⠤⢄⣀⠈⠈⠉⠉⠉⠙⠛⠿⠛⠻⠿⣧⡀⣴⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⣻⣿⠿⠟⠿⠿⠿⠿⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⢂⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣉⣉⣩⣭⣤⣄⡀⠀⢰⣶⣦⡄⠀⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⢶⣦⣤⡐⠙⠿⣿⣿⣿⣿⣿⣷⠰⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠶⠒⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠭⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣠⠶⠞⠛⣛⣛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣤⡴⠶⠶⠶⢮⣍⠛⠂⢀⣤⠶⠛⢛⣉⣉⣙⠛⠶⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠳⢶⣤⡀⠀⢈⣁⡴⠚⠋⠉⠉⠙⠛⠷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⢠⡾⣰⠏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠋⠙⠻⣆⣀⣀⣀⠀⠀⠀⠈⢻⡄⠈⣷
⠀⠀⠀⠀⣾⢡⡟⢠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢿⣷⣶⠞⣩⣭⣭⣉⡛⢶⣄⠀⠀⢻⠀⣿
⠀⠀⠀⠀⣿⢸⡃⣿⠀⣤⡤⠴⠶⢚⣩⣤⡀⠀⠀⠴⠾⠿⠶⠿⠿⠿⢿⣿⣦⡙⣧⠀⠘⠀⣿
⠀⠀⠀⠀⠻⠸⣇⠸⣦⠀⠀⠀⠘⣆⢻⣿⠷⡠⢂⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣭⣙⡻⢶⣤⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢃⡸⠿⣦⣄⣀⣈⡤⢖⣫⠞⢋⡭⠶⠛⠉⢉⡉⣉⣉⠉⠛⠻⢿⣿⣷⣮⣙⢷⣄⠙⣦⡀
⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢁⠴⠯⢍⠝⠉⠠⡆⢠⡆⢸⡇⣿⣽⠀⠀⢠⡄⠈⠻⣿⣿⣷⡝⢷⡈⢷⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⡇⠸⢿⠸⣷⠖⠀⢸⡇⠀⢀⡈⠻⣮⢻⣆⢻⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣰⠶⡆⠃⠀⠁⠀⣾⠀⣷⠀⠀⠘⠁⠀⠏⣿⠀⠙⣧⢻⣆⢻⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡐⠀⠀⡇⢿⣀⡟⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⢀⣴⣧⡆⠀⠸⣇⢻⡄⣧
⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡇⠀⠈⠁⠈⠛⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⢿⠘⣧⢸
⠛⢠⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⢰⠛⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⢸⡆⣿⠘
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠸⠶⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆⢀⡼⠁⠀⡀⠀⠀⠀⠀⠀⠘⣧⠅⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⢸⠀⡿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠀⡟⣻⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠏⡇⠀⠀⠆⢀⠇⠘
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡄⠸⣧⠀⢠⡟⢳⠀⠀⣤⣤⣤⠀⠀⣠⠀⠀⢀⠶⠆⠘⠛⡟⠀⠀⠀⠘
⠀⠀⢀⠘⣯⣉⣉⣉⣉⣉⣉⡈⢿⣆⠹⣧⡈⠳⠞⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠸⣆⠀⠀⠀⠃⡀
⠀⠀⠘⣧⠈⢻⣭⣭⣭⣭⣭⣥⣤⠻⣷⣌⢿⣦⡀⠀⠀⢀⡏⠀⠀⡏⢻⠀⢠⣤⠟⠀⢀⡠⠮⡭⠋
⠀⠀⠀⠈⠳⣄⠙⢷⣶⠶⠶⠚⠛⠶⣾⠿⢷⣯⣻⣶⣤⣈⠀⠀⠀⠻⠞⠀⠀⣀⣠⠖⣋⡴⢊⡕
⠀⠀⠀⠀⠀⠉⠳⠦⠉⠻⠃⠀⠀⠀⠘⣧⣀⣉⠛⠿⢿⣿⣿⣿⣷⣶⣾⣿⣿⠿⠞⣋⣥⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠤⠾⣷⣶⣒⣒⡂⠀⠀⠀⠉⣉⣉⣀⣤⠴⠖⠋⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢃⣄⠀⣜⡇⠀⠰⠆⠀⠀⠀⠀⠀⠰⡄⠀⣿⡀⠖⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡽⠞⠋⠉⠭⢝⠢⡀⠀⣖⢲⡤⣄⡤⠒⠒⠛⢧⣞⣳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⡤⢒⡶⢋⣀⣈⣽⣖⣧⡀⠀⠀⠀⠀⠉⠛⢄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢺⢀⡤⣚⣩⠿⠛⠉⠉⠉⠉⠉⠙⠳⢤⣈⢳⣄⡀⠀⠀⢸⡌⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢋⡽⠋⣀⣴⠖⠛⠉⠉⢹⠛⠒⠤⡀⠈⠻⣟⠛⠲⢤⣀⣡⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢣⡀⢠⠙⣅⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢳⡀⢇⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣇⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠂⠀⠀⠀⠀⠸⡊⢂⠀⠀⠀⠀⠀⠀⣀⣧⠀⢹⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢇⡏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⣜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⢍⣒⠤⠀⠰⠀⠀⠀⣀⡴⠊⠁⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⢤⣈⠉⠛⠒⠛⠋⠉⢁⣀⡤⢞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢩⣿⠋⢠⣽⡿⠝⣻⣿⣿⠀⣻⣿⡇⠙⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⣴⡿⠁⢠⣾⡟⠀⢠⣿⣿⠉⠀⣿⣿⡇⠀⢹⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⡟⠀⢠⣾⡿⠀⠀⣾⣿⡇⠀⢀⣿⣿⣿⠀⠀⣿⣿⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣽⡿⠁⢀⡿⣿⡇⠀⢠⣿⣿⠇⠀⢐⣿⣿⣿⠀⠀⢻⣿⣷⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠁⣼⣿⡇⠀⣼⣆⣿⠀⠀⢸⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⢸⣯⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠠⣾⡟⠀⣿⣿⠁⢀⣿⣿⡏⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠨⣟⣿⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⡇⢀⣿⣗⠀⢸⣿⣿⡇⠀⠀⣿⣿⣯⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠐⣿⣿⠀⢸⣿⣿⡇⠀⠀⣿⣿⣯⠀⠀⢸⣿⣿⣿⠁⠀⢐⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⠘⣿⣿⡇⠀⠀⢻⣿⣿⠀⠀⠈⣿⣿⣿⠀⠀⢸⣿⣿⠃⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣄⣹⣿⣤⡀⣿⣿⣿⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⠀⠀⣼⣿⡿⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⢼⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⣴⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣆⣠⣼⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⣿⠇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠛⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠄⠀⢠⣀⡄⣤⣤⣦⣠⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠆⠀⠸⣿⣷⣿⣿⣿⠟⠣⠄⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢱⣀⣤⠮⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠀⠀⠴⣤⣀⡎⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠟⠉⠀⠀⠀⠀⠀⠈⠳⡀⠀⠨⢥⡬⠇⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠛⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⣸⡇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠠⣔⠩⠔⠐⠂⠉⠉⠁⠀⠀⠈⠉⠉⠐⠒⠤⠍⣲⠄⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠄⢊⠗⠊⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠺⣑⠠⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠊⡠⠞⠁⠀⠀⢀⡠⠔⠂⠉⠀⠀⢀⠈⢁⡀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠈⠳⢄⠁⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠁⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠈⠢⣄⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠳⡀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⠄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠃⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⡀⠁⠀⠘⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠘⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⣀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠰⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡹⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣠⠞⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠓⢄⡠⠀⠀⠀⠀⠀⠀⠏⠰⠀⠀⠀⠀⠀⠀⢄⡠⠚⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠠⠄⠀⣀⣀⠠⠄⠊⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠢⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠔⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⠁⠈⠁⠒⠠⠤⠄⣀⣀⣀⡀⠠⠤⠄⠒⠈⠁⠈⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⢷⠾⠷⠾⠷⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢀⡄⣀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣀
⠀⢴⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠄⢳⣍⠉⠒⠺⡅⠀⠀⠀⠀⠀⠄⢃⡄⠀⢀⠀⠀⠀⡀⠀⠀⠀⢢⡀⠀⠈⣌⠘⣶⡂⠤⠠⢠⣤⡤⢤⣠⣀
⡄⣾⣿⡌⣷⢐⡀⠀⢠⠀⣤⠀⣰⡀⣿⡄⠀⡆⣿⠀⠀⠀⠀⠀⢽⡀⢺⡕⣶⡄⣤⣴⣧⣴⠤⠄⢾⣿⣤⣤⣿⡆⣹⡇⠀⣱⢨⢷⢀⠀⢹⣿⠂
⢁⢿⣿⢡⣿⡾⠧⣶⣾⣦⣿⣧⣿⣿⣿⠡⢠⣷⣿⣆⡀⠀⣄⣿⣿⣧⣿⣿⢛⡋⠥⠀⠒⠂⠀⠀⠀⠐⠊⠉⠿⣷⣿⣿⣿⣼⢌⣿⣼⣿⣾⣿⣄
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢷⣷⣿⣿⣿⣿⡷⠚⣋⠭⠵⠿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢿⣿⡺⢟⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠛⢿⣿⡏⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣦⣤⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⡞⠀⢀⣠⣴⠶⠖⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠤⠄⣈⠙⠁⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⡀⠀⠀⠀⡇⢀⡎⠝⠁⡸⠐⠃⡙⠓⣆⠛⣦⠢⠤⠀⠀⣠⣾⠟⠁⢀⣠⡤⢠⣀⠈⠓⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣍⠒⢼⠀⣅⣾⡄⠀⣼⣅⣴⢓⢷⣈⣈⡆⠸⣷⣶⣿⣿⣿⠇⠀⢠⠿⢼⡷⣞⢃⣰⠀⢸⣶⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢿⡈⡸⠤⢩⢿⣇⠀⠡⠠⠹⢿⠿⠳⢠⣷⢠⡏⠉⠉⠉⣿⡀⠀⢤⠄⠘⣷⣯⠷⠤⠀⢸⣍⠉⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠓⠶⠒⠟⣿⣆⠀⠛⣤⣼⡤⣶⠿⢃⡄⣀⣀⣀⣀⣹⣧⡀⠈⠤⣄⣉⣀⠴⠂⢤⠄⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠷⠶⠤⠤⠤⠤⠴⠿⠿⠿⢿⣿⣿⠿⠟⠻⢦⣀⡀⣁⣀⣠⣴⣿⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢣⣰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠞⣱⠃⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠈⢳⠦⡀⠀⠀⠻⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⣀⠀⠠⡧⠴⢤⠇⣠⣒⠛⢁⠎
⠀⠀⠀⠀⠀⠈⠫⣀⣨⠆⠀⠀⠀⠙⠃⠹⣸⠀⠀⠀⣠⣠⢄⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠚⠉⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠔⠊⠙⠲⢍⠀⣴⣒⠃⠁⢈⣍⡷⠀⡠⠚⠉⠉⠉⢢⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⡇⠀⠀⠀⠀⣨⠇⠈⠈⡇⠀⢸⠀⠀⠀⠧⣀⠀⠀⠀⠀⡇⠀⠘⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠧⠤⠴⡂⢫⡤⠴⠒⢲⡃⣠⣼⠋⣱⠖⣲⠏⢨⡓⠒⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⢀⠠⢀⣀⢀⠀⠀⠀⢀⠀⠀⣘⡆⠈⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠊⠁⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⠁⠀⢀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⠀⠀⠓⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠀⠀⠀⢢⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠏⡢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠄⠀⢠⠀⢀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠉⠂⢄⣀⠀⠀⠀⠆⠀⠀⠀⢘⠁⠐⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠤⠶⠤⣄⣉⣉⣉⣛⡤⠬⠥⠦⣄⡄⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢀⢀⣤⣤⣤⣤⣤⣄⣀⡁⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⣪⠒⠉⠉⠀⣄⠀⠉⠉⠙⠺⢕⢦⠀⠋⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣁⠀⣸⠇⠸⠀⠀⠀⠏⠀⠀⢰⠀⠀⠀⣧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣯⣅⠢⡀⠀⡶⢦⢄⣈⡤⣖⣒⣿⣦⣦⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢠⢫⣾⣿⣿⣿⣿⣯⡵⠃⣀⣈⣷⣾⣇⠾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢆⠻⣛⣽⣿⣽⣿⠿⠿⣿⣿⣿⣯⣭⣙⣳⣾⢿⣿⣿⣿⣿⡽⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠈⢛⣽⠟⣫⣵⠞⠛⠋⠉⢹⠛⠛⢿⣟⠿⣷⣽⣿⣯⣟⡻⢃⠃⠀⠀⣀⣠⣤⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠂⢠⡞⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡝⣿⣿⡅⠀⠛⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡟⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣷⡈⣿⣿⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⣼⢡⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡇⢸⡇⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⢸⠀⡇⠁⠀⠀⠀⠀⠸⡋⡀⠀⠀⠀⠀⠀⠀⡀⣿⢸⣿⢸⡇⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉⠀⠀⢻⠸⡆⣇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⢸⡇⡇⠀⠀⠉⠙⠛⠲⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⢃⣾⡟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠘⣦⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢃⣾⠟⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⠄⠈⠳⣿⣷⣭⣒⠤⠀⠰⠀⠀⢀⣠⡴⢟⣥⣾⣷⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣈⣿⢿⣯⣭⣽⣿⣿⣿⣿⣿⣿⢿⣟⠉⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⡉⠉⠉⠉⠉⠉⠀⠀⠹⣻⡄⠀⠙⣆⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⠋⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⣿⠀⠀⠀⣤⣶⠆⠤⠶⡆⠀⢀⡀⠀⢤⣶⠄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠬⠐⡆⠀⠀⠘⠿⠯⠃⠀⠉⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠙⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠐⠐⠀⠁⠠⠇⠀⠠⠀⠁⠐⠆⠉⠀⠀⠀⠀⠀⠀⣠⣶⣾⠏⣡⣴⣿⣿⣦⣀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⢿⠟⢽⡿⠛⠟⠋⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⢄⡄⡆⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⣶⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⢤⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡐⠀⣾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⡀⠂⠀⠀⠀⠀⠀⠀⠸⣿⠛⠛⢺⣧⣀⣿⣇⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠆⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠒⠒⠛⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⢢⠀⠀⠀⠀⠵⠀⢆⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⢸⠂⣸⠀⠀⠈⠁⣴⣾⣹⠁⠀⢸⡏⠀⠀⣀⡀⠒⠒⢠⣤⣤⣈⠃⠀⠀⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠊⠀⠢⠘⠃⣼⣷⡄⠀⠀⠈⠁⠀⠀⠀⠘⠁⠀⠀⠉⠁⠀⣄⠀⠉⠉⢹⡖⠀⠀⣿⡇
⠀⣿⠀⠀⠀⡀⠀⢱⡄⢀⠀⠀⠀⠀⠈⣿⡇⠀⣰⠀⠀⢾⡀⠀⢠⣀⡀⣀⢠⣄⣠⣿⣶⣤⡀⢼⣅⠀⠀⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠛⠃⠀⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠞⠀⠀⣿⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢠⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠐⠂⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⢀⣤⠶⠦⡄⠀⣿
⠀⣿⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢠⠀⣤⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢿⡻⠛⠃⠀⠀⠀⠀⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇

⠀⠀⠀⠀⣀⣠⣤⣴⢶⣤⡶⢦⡤⢦⣀⣤⡀⣀
⠀⠀⣤⣾⣯⣿⣶⣿⣶⡿⣿⣿⣿⣾⣿⣴⣿⣹⣶⢦⣀⣄
⠀⠀⠹⣿⣿⣯⣹⣿⣺⡿⣿⣼⣿⣿⣯⢻⣟⣿⣿⣾⣯⣽⣿⣇⣀
⠀⠀⠐⡇⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⡿⠛⠛⠙⡛⠙⠃⠒⠠⢄⡠⠀⢀⠀⢀⡀⢀⢀⠤⠄⠒⠐⠉⢐⠉⠒⠖⠠⣤
⠀⠀⠀⢠⢐⡀⠄⢩⡈⠉⠈⠉⡈⠋⡀⠁⠀⢀⣂⣀⡤⠌⢧⠝⡄⠸⠁⠶⠈⡡⠀⢠⡄⣴⡆⣀⠀⠶⠀⠿⠀⠛⠤⢯⠴⣷⠰⡆⡄
⠀⠀⠀⠀⠎⣿⠲⠀⠤⡄⠁⡀⠧⢤⠀⠐⠈⠋⠙⢡⠆⠐⢺⠀⡀⠀⠀⠀⢸⡔⣇⣤⠀⣼⡇⣬⣗⠲⣼⠁⠀⠀⠀⠀⠀⢹⠤⠆⣧
⠀⠀⠀⠀⠸⠸⠃⡰⡆⢱⠈⡁⠀⠻⡄⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠄⠀⠀⠸⠁⠀⡽⠀⡏⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠞⣠⢹⣆⢶⣡⡀⠀⢳⠀⠀⠀⠀⠀⡴⠀⠀⡇⠀⠀⠀⠀⡆⢀⢠⡇⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⢸
⠀⠀⠀⠀⠀⠈⠀⢡⡎⢻⡘⣧⣧⠀⠀⢧⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠀⠠⠃⠀⠘⡇⠀⡧⢀⠀⠄⢸⠃⠀⠀⠀⠀⠀⡾⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⢣⠘⣏⢧⠀⠈⢣⠀⠀⠀⢰⡆⠀⠸⡀⠀⠀⢠⠀⠀⢸⠁⢸⣤⣼⠆⠀⢸⠃⠀⠀⠀⠀⡼⢃⢠⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⢃⠘⣌⡄⠐⠘⣦⠀⠀⠀⢁⠀⠀⡇⠀⠀⢸⠀⠀⣿⠀⢸⠀⢰⡇⠀⢸⠀⠀⠀⠀⡼⣁⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠸⡄⢹⡾⡥⠈⠈⢆⠀⠀⠛⠀⠀⢹⡄⠀⢸⠀⠀⢻⠀⢸⠀⠘⠃⠀⣾⠀⠀⢀⡾⣻⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡄⢣⡹⣂⠐⠌⣆⠀⠨⡇⠀⠈⢠⠀⢨⠀⠀⢸⠀⢠⠀⠀⠀⠀⣿⢠⣀⣾⢷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠁⠌⠷⢻⡄⠘⡘⣧⠀⢠⠀⠃⠙⡀⢈⠀⠀⢸⠀⢸⠀⠀⠀⠀⡿⢬⣿⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠐⢃⢿⡀⠐⠘⣧⠘⠀⢠⠀⡇⢸⠀⠀⣸⠀⢿⠀⢀⠀⠀⡇⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢣⠈⡌⢳⠀⢢⠙⡄⡇⠀⠀⢴⢸⠀⠀⣿⠀⢸⠒⠺⠀⠀⣇⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠸⠀⢷⠈⡁⠸⢠⠁⠂⠈⡾⠀⠀⡏⠀⢸⠀⢸⡄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⢃⠼⡗⠀⠀⠻⡠⠀⠀⢷⠀⠀⠀⡄⠈⠀⠰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⠘⡆⢳⡀⠀⠀⡇⠀⠀⠈⡇⠀⢠⡇⠀⡇⠀⣤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠃⡌⣷⠰⠀⢰⠀⠀⠄⠇⠀⠀⢸⠀⢋⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠃⠘⠀⢸⡆⠀⠈⣴⠀⠃⢺⠀⠁⣸⡀⢸⠂⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠇⠀⠀⡄⠈⣿⠐⠇⠻⡄⢠⠸⠒⠀⠈⡧⠀⣟⠀⠠⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣾⠀⠀⢱⠀⠉⡟⠠⠸⠇⠃⡆⡧⠐⠀⢷⠀⢻⡄⠸⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣘⣧⣿⠀⠀⠈⠀⠐⢁⠆⡆⢼⠀⢡⢡⡀⠆⠸⡄⠘⢇⠈⡇⡼⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣧⣱⢊⢡⡇⠀⠀⢀⠀⠀⠸⢆⠁⠀⡆⠈⠸⡇⠁⠀⡇⠀⠸⠀⢸⢀⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢿⣥⣷⡅⠀⢰⡇⠀⠀⣾⠀⠀⠀⣏⠀⠀⢃⠀⠀⢿⠀⠀⢸⡀⠀⡇⠈⣾⡸⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢫⡾⠉⠩⡳⠁⠀⠀⢈⠇⠀⠀⣿⠀⠀⠀⢿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⠀⢹⠠⢿⡄⠾⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡰⢋⣴⡿⠋⠎⡴⠃⠀⠀⠀⣾⠀⠀⢡⠅⠀⠀⠀⢸⡄⠀⠈⡆⠀⠀⣧⠀⠀⢹⠀⠸⡀⠘⣧⠀⠋⡿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⠞⡡⠛⠋⠀⢀⠞⠁⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⢸⡅⠀⠀⡇⠀⠀⢺⠀⠀⠸⡇⠀⡇⠀⢿⠀⠀⢱⠈⠉⣿⣽⣿⡄
⠀⠀⢠⣏⢰⠏⠀⠀⡀⠎⠀⠀⠀⠀⠀⡘⠇⠀⠀⡌⠀⠀⠀⠀⠈⡇⠀⠀⢷⠀⠀⢸⡆⠀⠀⣧⠀⢳⠀⢸⡇⠀⢸⠀⠀⡟⠉⣹⣷
⠀⠀⣿⣵⣶⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠈⡇⠀⠀⢼⠀⢸⠀⠘⣇⠀⠘⡇⠀⠀⡇⢸⣿
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠁⢰⠁⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⡄⠀⣿⠀⠀⡇⠀⠀⡇⡟⢹⠃
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠂⠖⣿⠀⠀⣷⠀⢰⠇⢃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠷⠦⢔⠢⠀⠀⡧⢸⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⢸⠀⠀⣿⠀⠀⣷⠈⠀⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⢃⠀⡁⡾⠀⡀⢀⡇⠀⠀⣸⠀⢸⠀⠀⣟⠠⠀⣏⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⠀⠉⣸⠧⢀⢀⡟⠀⣼⠀⢰⡏⡀⢌⠅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⢀⣀⠨⠅⠉⡇⢀⡪⠅⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠴⠶⠚⠛⠳⠶⠶⣦⣄⡉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⠲⠶⣦⡄⠀⡞⠉⡇⠀⣠⡶⠞⠛⠛⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⠴⠞⠁⠀⠀⠀⢹⡆⣇⠀⡇⣼⠧⣤⣤⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⠒⠳⠿⣤⣀⠀⠉⠛⠶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣤⠾⣋⣤⠴⠶⠚⠛⠛⠛⠛⠻⢦⣄⠀⠈⢳⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠚⠛⠋⣩⡶⢛⣩⠴⠖⠛⠛⠛⠛⠛⠶⣦⣄⠙⢷⣦⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠟⢁⡴⠋⠁⠀⠀⢰⣉⡷⠀⠀⠀⠀⠉⠻⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣾⠋⣰⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠃⡼⠁⠀⢠⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣸⠃⠀⠀⠀⠈⠓⠦⣄⣙⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⢠⡴⡄⠀⠀⠀⠀⠀⠻⣄⡼⣄⠀⠀⠀⠀⠳⠞⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡀⠀⠀⠀⠀⢰⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⣠⠏⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢆⠙⢦⣀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠁⠀⢀⡼⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⠀⠙⠳⠶⣤⣤⣀⣀⣉⣉⣠⣤⣴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢻⡄⠀⠀⠀⠀⠉⠉⠉⠁⣀⣠⣤⡶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠳⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠉⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡖⢦⣀⣣⣤⠱⣶⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠈⠐⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢈⠿⠧⠽⣟⠀⠀⡠⡒⡀
⠀⠀⠀⠀⠀⢀⣤⠤⠒⠛⠛⠚⠀⡠⠁⢀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢵⠀⠀⠀⠀⠀⠁⠨⠾⣿⣷⡀
⠀⠀⠀⠀⠀⡏⢰⠂⠀⠀⠀⠀⠀⢹⣷⠋⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠌⠀⠀⠀⢀⠄⣤⣀⡠⠀⢉⣧⡀
⠀⠀⠀⠀⠀⡇⣟⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠐⣄⠀⠘⠛⠚⠾⠅⣀⠘⠿⠷
⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣟⣋⠠⠀⠀⠀⠒⠂⢄⣠⠙
⠀⠀⠀⠀⠀⣇⢸⠂⠀⠀⠀⢀⠴⢆⠀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⢓⢻⣔⠂⠀⠀⢤⡉⠲⠾⠃⠉⢢⡀
⠀⠀⠀⠠⠀⢻⠘⡄⠀⡀⠠⣸⠀⠀⢢⠀⠀⢠⢠⠆⢰⠒⠀⠄⠀⠒⠂⠀⠈⠂⠀⡛⠀⠸⠘⡯⣄⠀⠀⠰⡄⠀⠀⠠⢊⠁⢳⡀
⠀⠀⠀⠀⢳⢘⡇⢂⠀⠃⣀⣿⡆⠀⢿⢉⣁⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢵⠈⣂⠀⠀⠹⡀⠐⠃⠈⠀⠈⠳
⠀⠀⠀⠀⣸⡄⢳⢸⡀⠎⢉⣁⢇⠀⢰⣼⣿⣧⠀⠲⣶⣦⠈⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢑⠐⠊⢸⡇⣰⣥⡓⠆⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⡘⢡⠄⢹⣦⣈⠀⠀⠀⠊⠀⣀⣀⠉⠋⣦⢀⠈⣻⣆⠀⠀⠀⠀⠀⣠⠉⠃⠊⠁⠀⠀⡀⠀⣿⠛⢠⠙⠚⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠇⠀⠀⠀⣻⣯⣻⢦⣕⡀⠤⠘⡍⠉⠹⠿⠏⢋⡉⠘⣤⣀⣤⠴⡞⠋⠀⠀⠀⠀⠀⠤⠁⠀⣾⠀⠘⠄⠀⠀⠀⠀⠀⠠⠀⠒⢤⣀
⠀⠀⠀⠀⠜⠀⠀⣿⢯⣖⠴⠫⠂⡯⠀⢇⠀⠀⠀⡀⠀⠘⠀⠀⠀⠸⠃⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢢⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀
⠀⠀⠀⠀⠈⠄⠀⣿⠈⠁⢀⠂⠀⡗⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⡝⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢰⠀⡦⢰⠀⣃⠐⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⡀⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠸⠈⡾⠁⠸⠄⠹⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠃⠃⠀⡿
⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠰⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠈⠀⠀⠀⠁
⠀⠠⠦⠄⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⠞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢾⡭⣀⠀⠐⠀⣤⣜⢷⣺⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣼⣍⡁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣾⣳⢷⣜⣻⣦⣤⢠⣤⠀⠀⠀⠅⠠⢠⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠋⠉⠉⠉⠙⠓⠂⠀⠛⠙⠛⠛⠿⣦⣤⣱⡾⠋⠉⠀⠀⡀⠙⠃⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣎⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡷⠖⠚⠛⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠳⠶⢤⡞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⡊⠉⡆⠀⠠⡊⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠔⠊⢁⣀⣠⠠⠬⠯⠀⣀⣀⡈⠑⠢⣀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⢫⠷⠒⠉⠁⣀⠀⣤⣤⢄⣤⢀⣀⠈⠉⠒⠴⣹⠂⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⡠⠚⠁⣀⣔⠶⠙⠋⠉⠉⠁⣄⠉⠉⠘⠻⢷⣠⣀⠈⠓⢄⠈⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⡪⠋⠁⢰⡆⠀⠀⠠⡆⠙⡦⠀⡄⠀⣤⠈⠙⢕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠊⠀⠀⠀⢸⡇⢹⠀⠀⠃⠲⠆⡰⠀⢻⠀⠀⠀⠀⠑⣷⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣯⠃⠰⢆⠀⡠⡈⠁⠈⠀⠀⠀⠀⢠⠁⠀⠉⠀⠀⠰⡄⣤⠘⣵⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣾⠃⠀⠀⢸⠸⣇⠇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠮⠄⠀⠀⠘⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣜⣀⣀⣀⣀⣀⣀⡀⠠⡄⠀⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣇⠀⠋⠈⠞⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠋⠈⠇⠛⠀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⢠⠠⠀⢰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠱⣧⠀⣠⠸⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢀⡀⠀⣼⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠹⣧⡀⠀⠀⠀⠀⣩⠁⠀⠀⢀⠀⠀⠀⢰⡂⠀⠀⠀⠀⢁⣼⠏⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢝⢄⠀⠀⣤⠁⠀⠀⠀⡟⡦⠀⠀⠀⢋⣄⠀⠀⡠⡮⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠯⣦⣄⡀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣡⣴⠟⠊⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⢄⡀⠁⠚⠳⠶⠦⠶⠦⠴⠶⠚⠓⠉⠀⣠⡔⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⠔⠁⠀⠉⠑⠒⠤⠤⢀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠑⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣤⣤⣀⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⡀⢶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⣹⣿⣿⣿⣿⣣⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠻⣿⣿⣯⣤⡀⠀⣠⡾⠉⠀⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣤⣿⠁⠀⠀⣽⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠛⠉⠀⢽⣿⣿⣇⣀⣰⣿⣿⣿⡿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⢀⣀⣤⡇⣿⣿⣽⣟⢛⣿⣿⡛⢵⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⠚⠁⢉⠘⣿⡿⠞⠏⡙⣛⣻⣿⡷⣜⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⡁⡄⢠⠈⢶⡍⠀⠀⠀⡀⠀⠀⢹⣇⠙⣿⣇⢸⣿⣿⣿⣿⣿⣿⠉⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣬⣦⣼⣿⣿⡗⠂⢈⢁⠐⣂⠘⣿⣧⠙⠇⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣾⣿⣦⣾⢿⡆⠀⠚⣿⣷⡝⢦⣼⣿⣿⣿⣿⣿⣿⠋⡀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣥⢶⣿⡇⠀⠀⣿⣿⣇⠀⡙⠻⣿⣿⣿⣿⣿⣴⣿⣾⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡇⢠⣼⣿⡿⠘⢻⡇⠀⠀⣿⡇⢛⡻⠧⣤⣀⣈⡙⠟⠛⠡⠿⢿⣛⣿⡟
⠀⠀⠀⠀⠀⢸⣿⣿⢸⢸⣿⣿⣿⣿⣁⢀⣿⡿⠁⠀⣸⡇⠀⢠⣿⣷⣸⣿⣿⣷⣾⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡿⠸⢸⣿⡿⣿⣿⣿⡿⣿⠃⠀⠀⢸⡇⠀⢸⡏⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡿⠁⠀⣿⣿⣿⣿⡿⢻⠃⠇⠀⠀⠀⠘⡇⠀⠀⣷⠸⠀⠘⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⣿⣿⠿⢃⡞⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⠢⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⠂⡾⠀⠀⠀⠀⠀⠀⠰⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠞⠁⠀⠀⠀⠀⠀⣼⢀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⣼⠁⠀⠀⡀⢳⣻⡄⠀⠀⢰⡎⠀⠀⠀⠀⠀⢀⠀⣬⣙⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣾⣴⣿⣆⡰⠘⣧⣿⣿⣷⡀⠀⣼⣷⣶⣀⣀⣸⣷⡙⣆⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣟⠁⠀⠁⢱⣿⡟⣷⠀⠹⠇⠛⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡈⠿⠟⢛⣻⣿⡤⠀⣤⣼⣿⣿⣾⣤⣴⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣬⣶⣴⣶⣄⣠⣴⣾⣿⣿⡿⠀⣰⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⠛⠛⢻⡇⢠⠿⠿⢿⣿⣿⡿⢯⣿⡻⠿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠈⠀⠋⠂⠐⠘⠀⠈⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣧⣼⡆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠈⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠁⠐⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡁⠀⠂⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢻⡿⠟⠀⠀⠀⠹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠋⠇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠁⣁⣠⡄⠀⡆⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣁⣠⣴⠓⢤⣀⠀⢀⠠⠀⢺⡃
⠀⠀⠀⠀⠄⠀⠐⠀⠀⣉⡙⠋⠛⠉⠀⠀⠀⢹⡟⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡠⠰⣛⣻⡯⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢸⡷⡆⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠐⠛⠛⠛⠁⠀⠀⠀⠀⠀⣤⡞⠀⠀⠀⠀⠀⣾⠗⢁⠀⠀⡰⢢⡄⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⢺⣿⣿⠀⠀⠀⠀⣠⣾⣿⠂⠀⠀⠀⠀⠠⣿⡀⢻⣀⠜⠣⣵⠀⠀⠀⢺⣿⣦
⠀⠀⠀⠀⠀⢀⣸⣿⣿⠀⠀⢀⣴⠟⣿⠋⠀⠀⠀⠀⠀⠰⣽⢿⣾⣧⣤⡾⠃⠀⡀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⣼⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠩⠀⠀⠀⠀⠁⠀⠀⠀⠘⢿⠄
⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠸⣿⢻⡏⠀⠀⠀⠀⠀⠀⠀⢈⡃⡀⠀⠀⠀⠀⠀⠀⠄⠙⠀⠀⠀⠈⡆
⠐⠀⠀⠀⠀⣿⣿⢣⠀⣀⣟⠡⣿⡅⢠⠀⠀⠲⡀⢀⠄⢰⡟⢣⡀⡀⠀⣠⠀⢂⠈⡄⠀⢀⠱⡀⠘⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⣿⡇⠀⣸⡏⠀⣸⠇⠀⢸⠀⠀⠀⠀⠀⠀⠁⠄⠈⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⠀⠹⡀
⠀⠀⠀⠀⢸⣿⠃⢰⣏⡀⢰⠇⡄⠀⡾⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠐⣿⡟⢀⣿⢷⠃⡿⠰⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣿⣧⣼⡏⡄⣸⠷⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀
⠀⠀⠀⠀⣿⣿⣿⣽⢫⡏⠀⠇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠁
⠀⠀⠀⢀⢙⣭⣯⡅⢸⠇⠀⠀⣼⠃⠀⢀⠀⠀⠀⠀⠀⠀⣇⢰⣀⠀⠀⠸⠀⠀⡶⠦⢀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⢀⠐⣿⡿⣿⣤⣿⣦⠀⠀⡇⠀⣠⡞⢲⡄⠀⠀⠀⢠⣿⡄⢟⠳⠦⣤⡄⣧⠑⡀⠈⠀⢺⣤⠂⠓⠀⠀⢠
⠀⠀⢸⣆⠻⣷⣼⣿⣿⣿⠗⣠⣽⡋⣸⠃⢸⠀⢹⡖⠲⠾⣿⡃⢸⠀⡇⢹⠳⣹⣆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⣷⣍⣛⣛⣋⡭⣸⡟⣿⠀⣿⠀⢸⠀⢸⡇⠀⠀⡟⣧⠈⡇⠀⠀⠆⠙⣷⡈⡆⢀⠀⡁
⠀⠀⢸⣿⣿⣿⣿⡟⣹⠇⣿⢧⣿⠀⣿⠁⠸⠀⠀⣇⠀⠀⠗⢻⠀⣷⡄⣆⠈⠀⢻⡇⠹⠀⡀⠳⠀⠃
⠀⠀⣼⣿⠛⠋⠛⢀⡟⢀⡟⢸⡇⢀⣟⡀⢰⡇⠀⠷⠄⢠⠀⡼⠃⢹⠃⢸⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢹⣿⣿⣤⣤⣼⣀⣌⣀⡊⢀⣼⠉⠁⣈⠁⢀⣄⣤⡄⢀⣿⣁⣸⣀⣴⣦⣴⣶⣿⣷⣿⣶⣿⣿⣿⣿⠟⠋
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡛
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠻⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠻⢷⣦⣀⣀⣤⣤⣤⣤⣀⣀⣠⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣷⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠘⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣶⣶⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠻⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠋⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠒⠚⠉⠉⠉⠉⠉⠒⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡝⠶⠛⠻⠿⣷⣶⣄⠀⣤⣤⣀⡠⣴⡬⠾⠟⠚⠳⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⣀⣤⡶⠾⢿⣿⠊⠛⣿⠿⢿⠿⠷⣦⣤⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣏⢻⡀⣨⣟⣁⡀⣴⣿⠄⣈⣳⣄⣼⡿⢛⣿⣶⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠛⡿⣿⣿⣦⣿⠿⠖⢀⣀⣁⣀⣀⡀⠐⠿⣿⣥⣿⣿⠋⠛⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⡹⣫⣿⣋⡴⠞⠋⠉⠉⠀⠄⠉⠉⠙⠲⢦⣝⢻⡙⢯⠒⠺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣩⡾⠋⠀⠀⠀⠀⠀⣰⠺⠀⠀⢠⡞⠂⠈⠳⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⠀⠀⠀⠀⠆⡇⠀⠈⠉⠀⢠⠽⠇⠀⠀⠀⡈⢧⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠃⠀⠀⣾⣷⣀⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢸⡋⠑⠈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡟⠀⡀⣀⠉⠁⠙⠦⠦⣀⣀⢠⡏⠀⠀⠀⠀⠀⢀⡀⣆⢹⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⢰⠂⡿⠀⠀⠀⠀⠀⠈⣷⣾⠀⠀⠀⠀⠀⠀⢈⡇⠛⢸⣷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢿⣿⠞⠀⠀⣀⠀⠀⠀⠀⠠⠋⠈⠣⡀⠀⠀⠀⢀⡄⠀⠂⣼⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⠠⠄⠿⠀⠀⠀⡰⠁⠀⠀⠀⠑⣄⡀⠠⠺⣧⣄⡖⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠃⠐⢆⠀⠈⡅⠀⠀⢠⡆⠀⠀⢨⣦⣀⣶⣿⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⣦⡀⠼⣦⣤⣀⣀⣘⡋⢀⣠⣢⣾⡿⢟⠕⡊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣭⣓⣻⣿⣿⣛⣛⣩⣵⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣌⡻⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣾⠀⣴⣿⣿⣿⡦⠀⠀⠘⣷⡀⠀⢀⣙⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡈⢻⣿⣿⠏⠀⠀⠀⠀⡿⠀⡰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣤⡀⠀⠸⠿⠂⠀⠀⢠⠀⠀⢆⡼⠁⠀⣀⣊⠁⣨⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⣉⠽⠒⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⢤⠉⠀⠀⠀⠀⠀⢠⣄⢁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⢰⣿⣶⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⠏⣀⣉⡀⠀⠈⠁⠀⠀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡻⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣴⣦⠀⠈⠛⠿⣿⠿⢦⡄⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠂⠀⠀⠀⣶⣿⠿⠛⢹⡏⠈⠃⢽⡻⣷⡀⠀⠀⠀⠐⠚⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡤⠜⠛⣡⣶⢟⣿⣿⠀⠀⠀⠀⠀⠙⠀⠀⠖⠀⢠⡤⠶⠦⠜⠃
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠄⠀⠠⠿⠖⢾⣧⣿⣯⠁⠀⠀⠀⢘⠀⠀⠀⡀
⠀⠀⠀⠀⠸⣿⡝⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⢻⡽⠇⣀⣤⡶⠛⠹⠇⠀⢠⠀⠀⢻⡦⠄
⠀⠀⠀⠀⠀⠀⠙⠓⠿⢄⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⡠⣾⢁⢸⡾⠻⢟⣶⣟⠀⠀⡀⠈⠀⠀⠛⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⢰⠟⠀⠀⠈⠁⠘⠀⣿⠀⠀⠈⠀⣿⣞⡆⠰⣄⣘⣿⡙⠛⠛⠛⠿⡦
⠀⠀⠀⠀⠀⠀⠠⠦⠾⠋⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⣿⣶⠈⡅⠀⠀⠀⠀⠙⢿⡇⠀⠈⠙⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⢤⣜⣗⡛⠉⣸⠇⠀⣠⣄⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢻⣿⣿⣷⢾⣿⣟⢫⡿⢀⣿⣦⠀⠜⠉⢿⣿⡷⠀⢠⣴⢿⠶⠀⢀⣤⣀⠤⣀⡀
⠀⠀⠀⠀⣠⡶⠿⢛⣛⠟⠢⣤⣴⣬⣼⣿⣿⡄⠙⠻⠇⠙⢸⠁⢸⣧⠀⠀⠸⠟⠁⠀⠸⠡⠀⠐⠻⢿⣯⣾⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣩⣷⣶⠚⠀⠓⠄⠸⡀⠈⢿⡎⠀⠴⠚⠿⡷⣾⣿⣷⣦⠀⠀⠈⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠁⠀⠀⠀⠀⠣⠀⠀⡵⢀⣤⣤⣶⣟⣓⣿⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠄⡘⠀⠀⠈⠙⠛⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣯⣩⣭⣯⣥⣍⣽⣿⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣭⣭⣭⣿⣿⣿⣭⣭⣭⣭⣭⣯⣭⣭⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⡾⠵⠒⢒⢒⣒⣒⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠈⠁⣒⣛⣛⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠇⠀⠀⣠⠉⠙⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡶⠋⠀⠀⢹⠄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣠⠆⡀⠘⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⣯⢾⣁⣀⣱⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣀⡈⠂⡠⠚⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣿⠀⢸⠋⠁⠀⠀⢙⡇⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠠⠲⠆⠛⠀⠈⠒⠤⠶⠞⠉⠛⠒⠛⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠖⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢠⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢓⠦⡀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠼⡀⠈⡶⠁⢀⠔⠉⢉⡇⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠱⡞⠀⠀⠚⠰⠴⠎⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠒⠤⠄⡨⢇⠀⠘⢆⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⣠⣀⣀⠤⠄⠒⣒⠠⠭⠿⢿⢿⠇⠀⠀⠀⠀⠘⣆⠀⠈⠣⡀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠤⠤⠤⠴⠃⠀⠀⠰⣎⠁⠀⠀⠀⢀⣸⣾⡠⠤⢤⠀⠀⠀⠸⡄⢀⡤⠾⠚⢻⢳⡶⠆⠀⠀⠀⠀⢀⠔⢋⣨⠆
⠙⠒⠒⠛⣄⡀⠀⢀⣴⡠⠷⠀⠀⢰⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⠾⡴⢡⠀⢃⠀⠀⠀⠀⠀⠈⢆⢹⢜⡫⠤⣪⡧⣀
⠀⠀⠀⠀⠀⣇⡴⠋⠀⠀⠀⠀⢠⠋⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢰⠃⡄⢀⣤⡷⠀⠀⠀⠀⠀⡏⣳⠁⢠⠆⢸⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⡸⠀⣼⠀⣷⠻⣿⡀⠀⣀⠤⢒⠩⠐⠛⠷⢀⡰⣇⣀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠒⠤⡴⠃⠀⠙⣾⡁⠀⠹⡙⣨⠔⠊⠁⠀⠀⠀⠀⠀⠘⣆⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠘⠤⣀⠀⢀⡜⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠘⣷⠾⡄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠀⠀⣿⣇⠹⡄⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠀⠀⢠⡋⣨⣆⣷⣰⣯⡤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠷⠾⠿⠷⢿⣿⣟⣋⣙⣛⡿⠂⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤⡄
⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡾⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡄
⠀⠀⠀⢀⣻⣿⣟⡀⠀⠀⠀⠀⠀⠈⠉⢉⣿⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡾⠿⢶⣿⠄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⢿⡿⠿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⣯⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠘⠃⠀⠀⠈⠙⠻⣿⣷⣄⠀⠀⠀⠀⠀⣿⠛⣿⠛⣿⡇
⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠀⠀⣼⣿⡿⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂⠈⢿⣿⣧⠀⠀⠀⢀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠻⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠛⣟⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿
⠀⠀⠀⣤⣀⣤⣀⣤⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⢀⣴⣶⣶⣶⣶⡄
⠀⠀⠀⡟⠁⡀⠈⣿⠀⠀⠀⠀⢻⣿⣷⡀⠠⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠄⢀⣾⣿⡟⠀⠀⠀⢀⣿⠀⠉⢻⣿⡇
⠀⠀⠀⣳⣄⣀⡠⣏⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⠀⠀⢠⡄⠀⠀⢀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠸⣿⣤⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣾⣷⣶⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⢉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀
⠀⠀⠀⢤⣾⣿⣦⡄⠀⠀⠀⠀⠀⠀⢤⣶⣿⣦⡤⠀⠀⠀⠀⠀⢰⡟⢿⡿⣿⢻⡇⠀⠀⠀⠀⢀⣿⡟⢿⢻⣿⡇
⠀⠀⠈⢻⢾⡿⢦⣿⠀⠀⠀⠀⠀⠈⣻⣾⣿⣷⣿⠁⠀⠀⠀⠀⢸⡇⣠⡄⣤⢸⡇⠀⠀⠀⠀⠐⣿⡇⣠⢸⣿⡇
⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢂⣉⣉⣉⣉⡉⠐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⣠⠖⠋⣉⣁⣤⣤⣌⣉⠓⠦⣌⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⠁⠀⠀⣿⠅⠀⣿⣿⣿⠀⠀⠈⢦⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⣿⡄⠘⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⣿⣄⠀⠀⣿⡷⣶⣿⣿⡿⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣼⠂⣠⣿⣿⣤⣴⣶⣿⡿⠋⠀⠀⠀⠀⣴⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣛⣻⣿⣿⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⣸⠇⠀⠉⣱⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠉⠉⠁⠁⠀⠀⠀⠈⠉⠉⠙⠓⠲⢤⣀⡀⠐⡿⢶⣦⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⣀⣤⣤⣶⣶⡶⠚⠛⠛⣛⣓⣲⣶⠦⣤⣄⡀⠀⠉⠛⠦⣄⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⣠⡴⠞⠛⠛⠉⠀⠀⠀⠀⠀⢀⣤⠀⡀⠀⡉⠁⠛⠙⠿⢿⡶⣄⡀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⣴⣟⠕⠀⢀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃⠈⠐⠀⠄⠀⠈⠓⠽⡳⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⣾⡿⠋⠀⣠⠺⣄⠀⠀⠠⠆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠴⠄⠀⣠⠟⠀⡀⠈⠻⣝⢦⡀⠈⠳⡄
⠀⠀⠀⠀⣰⠏⠀⢀⣼⡿⠋⢀⠰⠁⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠐⢀⠈⢫⡻⣄⠀⠙⣦
⠀⠀⠀⣼⠃⠀⣠⣿⡟⠁⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣀⠙⣞⢦⠀⠈⢧⡀
⠀⠀⣸⠃⠀⣰⣿⡟⠀⣬⣁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠸⣭⠂⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⣀⣤⠘⢯⢷⡀⠈⢷
⠀⢠⠇⠀⢰⣿⡿⠀⡔⠀⠉⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠠⢆⠈⢏⣷⠀⠸⣇
⢀⡟⠀⠀⣿⣿⠃⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡌⠀⠀⢸⡇⠀⣿
⢸⠇⠀⢸⣿⡏⢀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⢀⣠⣤⠶⠞⠛⠉⠁⠀⠠⠀⠀⣷⠀⣿
⣿⠀⠀⣾⣿⠁⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣦⣠⠤⠀⣀⣠⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢹⠀⣿
⣿⠀⠀⣿⣿⠀⢸⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡞⠉⢻⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⡅⠀⢸⠀⣿
⣿⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠚⠛⠻⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡅⢠⣿⢰⣿
⢻⡄⠀⢹⣿⡆⠈⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⣾⣽⣾⣿
⠘⣇⠀⠘⣿⣧⠀⢫⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢀⣏⣿⣿⣿
⠀⢻⡀⠀⢻⣿⣆⠈⢦⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠴⠀⡼⣼⣿⣿⣿
⠀⠀⢷⠀⠈⢿⣿⣆⠈⣧⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠁⡰⣱⣿⣿⣿⠇
⠀⠀⠈⢧⠀⠈⢿⣿⣦⠈⢶⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠀⡔⣵⣿⣿⣿⠏
⠀⠀⠀⠈⢷⡀⠀⠻⣿⣷⡀⠙⢶⡠⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡀⢠⡼⠃⣠⢊⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠈⢻⣄⠀⠈⢿⣿⣶⣄⠙⠮⣤⢂⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠜⠋⣠⢞⣵⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠿⣿⣷⣦⡈⠙⠳⢥⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⣠⡰⠖⠋⢀⡤⣪⡵⣻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠛⠿⣿⣷⣦⣄⡀⠉⠑⠓⠒⠢⠤⠿⠤⠜⠒⠒⠋⢉⣀⡤⢖⣫⡵⢞⣩⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠈⠙⠛⠿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣔⣚⣯⣭⠷⠞⣋⣥⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣤⣀⡀⠀⠈⠉⠉⠉⢛⣋⣉⣉⣭⣭⣴⣶⣾⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⠛⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⣠⠤⣴⣲⣽⠇⠀⠀⢀⡀⠀⠀⠀⠠⣤⠖⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣠⠤⣒⣺⣭⣷⣚⣻⣭⣥⣤⣤⣤⣤⣟⣙⢻⡆⠀⠀⢠⣴⡶⠶⢦⣄⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⣯⠵⠞⢋⣩⡼⠟⠋⠉⠀⠀⣀⣀⣀⣀⠀⠈⠉⠙⠳⢦⣄⠈⢷⡱⡄⠀⠈⠛⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣠⠞⠋⠁⢀⡠⠴⣒⣯⣭⠭⠶⠦⣬⣭⣙⡒⠦⣀⠀⠈⠳⣦⡻⣜⣆⠀⠀⠈⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠁⢀⡠⣚⡥⠖⠋⠉⠀⠀⢠⡆⠀⠀⡤⢨⣝⠳⣬⡳⣄⠀⠈⢳⣿⣮⢦⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⡴⣫⠞⠉⠀⠀⠻⠂⠀⠀⠀⠀⠀⢈⡁⠶⡟⠀⠈⠻⣎⢣⡀⠀⠹⣏⢷⡳⡀⠀⣸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢏⡼⠃⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⣀⡘⢧⠹⡄⠀⢹⡎⢳⣝⣤⠏⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢣⡞⠁⠲⠞⠋⠀⠀⠀⠀⠀⠀⠀⢰⣏⡾⠁⠀⠀⠀⠀⠉⠁⠈⣧⢹⡀⠀⣿⠀⠙⠁
⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠠⡇⡇⢀⣠⠀⠀⠀⠀⢠⡾⠋⠛⣲⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢈⡇⡹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⣧⠀⠁⠀⠀⠀⠀⠀⠛⠶⠛⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⢇⡇⠀⣼⠁⡶
⠀⠀⠀⠀⠀⡀⠸⣇⠀⠸⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⠀⢠⡟⡼⠀⢠⡟⠀⠃
⠀⠀⠀⠀⠀⠹⠀⢻⡄⠀⢳⣻⡄⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣰⡿⢣⡟⡼⠁⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠳⣽⣦⡀⠀⠀⣤⣶⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣵⢫⠞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡻⣄⠀⠈⠻⣿⢦⣀⠻⠽⠂⠀⠰⠇⠀⠀⠘⠃⣀⣠⠶⣋⠕⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠐⣯⣀⣽⠟⠻⣦⡀⠈⠙⠯⣟⣓⠶⠶⠶⠦⠤⠶⠖⣛⡭⠔⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠷⣤⣀⡀⠈⠉⠉⠉⠛⠛⠋⠉⠀⣀⣠⡴⠞⠉⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠶⠶⠶⢺⣿⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⡤⢾⣃⢠⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣦⣴⣶⣾⣾⡇⠀⠀⡀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣯⣿⣿⣿⣿⣿⣿⣦⣬⣁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣽⣷⣶⣿⣿⣿⣿⣿⣿⣇⠹
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡛⡟⠙⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠻⢟⣿⡟⠉⠻⣷⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⢋⡭⠛⣿⠟⠛⠶⣒⠀⠗⢺⡿⠋
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣻⣔⡹⠁⠀⠀⠀⠀⠑⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⢿⡇⠠⠶⠀⠈⠋⠀⣾⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⢿⣾⢿⡄⠀⠀⠀⠀⠀⡺⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⢟⠯⣿⠆⠀⠀⠀⠸⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠂⠈⣻⣛⡄⠀⠀⠀⠀⢩⠄⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡏⠀⣀⠂⠉⡉⠠⠀⠄⠀⠀⠈⢧⠄⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠐⢤⣌⠹⣡⠀⠀⠀⣆⠀⠀⠈⣆⠰⠀⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣡⣄⣀⡈⠲⠄⠛⠀⠀⢻⣆⡀⢀⣿⣞⣦⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⡀⢑⣆⠀⠀⣾⠛⣎⣻⣧⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠁⠀⢀⡄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣹⣦⣀⣨⣤⣿⡟⢿⡿⢟⣿⡷⠄
⠀⠀⠀⠀⠐⠀⠀⣈⠀⠈⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣿⣿⡾⠛⠻⣿
⠀⠀⠀⠀⠀⠀⣀⣤⠞⢿⣿⣥⣎⣿⡿⣻⣿⣶⣿⣿⡿⠋⠀⠀⠀⠀⠸⣏⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⢡⣾⣿⡿⣿⣿⣿⣿⣿⠏⠁⠙⢿⣿⣾⣄⣀⣀⡀⣠⠤⣵⠦⠄
⠀⠀⠀⡄⠀⠀⣀⣿⣿⡗⣼⣿⣿⡿⡿⣿⣆⠀⠰⣄⠈⠛⠿⣉⣉⣥⣤⣶⡲⢿⡇⠀⠈⠂
⠀⠀⠠⠀⠀⢠⣽⣿⣿⣿⣿⣟⣿⣇⢰⣿⣿⣿⣦⣤⡛⠂⠀⠈⠳⢤⣊⠍⠁⡼⡁⢀⠀⡄
⠀⠀⠘⠀⢀⣿⢿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⡿⣿⣿⣿⣶⠀⠀⠀⠀⠈⠀⠛⠳⠅⠤⢾⠁
⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠠⣤⣤⡀⣶⣤⣤⡀
⠀⣴⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣿⣿⣿⣿⣿⢿⣿⣿⣿⣦⡄⢀⠀⣿⣿⠿⣿⠁⠈⢣⡠⢤⡄
⠀⣶⣴⣾⣿⣿⣿⡟⣾⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣷⡄⠀⣸⡿⠘⣿⠀⢀⣀⠀⣀⣀
⠀⠛⢿⣿⣿⠛⢻⣇⣽⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⠀⠈⣿⣿⣿⣯⠁
⠀⠀⣾⣿⣷⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⢷⠀⢺⡇⠀⣿⣿⣿⣿⣿
⠀⢺⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠇⠘⣀⠘⡇⠀⣿⣿⣿⣿⣿⡀
⠀⠈⢻⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⠀⣿⡀⠇⠀⢻⣿⣿⣿⣿⡆
⠀⠀⡼⣿⡇⠀⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣾⣧⠀⠀⢸⣷⣿⣿⡿⣀
⠀⠀⣄⣼⡧⠀⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢹⣾⠇⠹⣿⣿⡋⢲⡟
⠀⠠⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⡶⠄⠄⢹⣿⣿⣿⡟
⠀⠀⠸⠛⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠂⣀⣾⣿⣿⣿⣯⡀
⠀⠀⠀⢀⠅⠀⢿⣽⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠀⠀⢸⡱⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣯⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠶⠇⠠⠏⢿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠵⠞⠛⠛⠛⠋⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⣀⠔⢊⠩⠔⠒⠃⠈⠘⠒⢄⡈⣧⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠈⠀⠰⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⡆⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⣠⣡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣷⡀⠀⠤⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠁⠀⠀⠀⠈⠢⡀⠀⠀⢀⠤⢤⠞⠉⠁⠀⠀⠉⠲⢤⡌⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⡇⠀⣀⣠⣤⢶⣶⣾⣿⡷⠀⠀⠀⠑⢲⡗⢤⣀⠀⠀⠀⠀⠀⠈⢑⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣆⢹⡘⣿⡽⠟⢶⢲⠲⠶⢶⣿⣶⣤⣬⣱⣦⣄⡀⠀⠀⣔⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⡿⢿⣛⡭⠖⠛⠋⠉⢉⡍⠛⠒⠯⣽⢿⣷⣿⣿⠛⠻⢿⡒⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢻⣅⠀⠀⠀⠙⢦⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡴⠣⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠃⠈⢣⡙⣆⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡧⠄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢑⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⢠⡄⠀⠀⠀⢀⣠⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡇⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢠⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢷⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⣠⡎⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠴⢿⣷⣋⡄⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⣌⣒⣤⠼⣀⣀⣀⠠⠤⢤⠐⣐⣩⣶⠞⣋⣉⠁⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⣮⠟⠉⠛⠶⢿⣶⣾⣾⣿⣿⡻⠟⠛⠒⣰⠁⠿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⡄⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣾⣿⡿⣁⢸⠇
⠀⠀⠀⠀⢀⢀⣿⣿⡇⠁⠀⢴⡆⠐
⠀⠀⢀⡀⢀⠌⡻⠟⠁⠀⠄
⠀⠀⠀⠽⠀⠈⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠈⠙⢿⣦⡄⠀⠀⠀⠀⠀⢾⡿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⠚⠏⠐⠉⠇⠀⠀⠀⠀⠀⠘⠇⠐⠀⢹⣿
⠀⠀⠀⠀⠈⢡⠀⡀⠀⠠⣔⢢⣤⣾⠂⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠇⠘⠤⠏⠁⡌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⢣⠀⠀⠀⢀⣑⡼⢂⡆⠀⠀⠀⠀⢀⠄⣴⠄⠀⠈⣀⢰⣧⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠟⠓⠒⡦⢿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣷
⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⡇⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠂⠀⠈⢁⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⣤⠶⠛⣿⢿⣿⣶⠀⠀⠘⠆⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⣴⠀⣱⠟⢸⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠐⠀⠀⠀⠀⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠰⠇⢀⣿⠀⢸⣿⠃⠀⠀⠀⠆⠀⠄⠀⠀⡄⢳⣀⠀⠀⠀⢞⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣸⣿⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⣿⣿⣧⡎⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⢋⠇⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠘⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡶⢿⣿⣿⣟⣿⡿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠹⣃⠀⠀⠰⣚⠄
⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠰⣯⣹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠗⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⣿⣿⣿⠇⠀⠀⠼⣷⠆⠀⣼⠇⠀⠀⢀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⠿⣧⡟⠀⠀⣠⠐⠃⠤⠄⠌⢠⣦⠠⠘⠦⠈⠳⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠸⢿⣶⣤⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⠹⠋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⣇⣸⣷⡞⢀⣤⣄⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠁⠀⣨⣿⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⡄⠀⠀⢫⣿⣿⣯⣤⣾⣷⡟⠛⠐⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⣿⣷⣀⡀⠛⠙⣿⣧⣀⣤⣤⣄⡀⠋⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⣦⣄⣿⣿⣶⣾⣾⣹⣿⣿⡟⢁⣿⣶⣸⣿⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣤⣤⣤⣴⣿⣿⠻⣿⣧⣬⣹⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣷⠀⢸⣿⣿⣿⣧⣼⣿⣿⡏⠈⠙⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣼⣿⣿⢿⡿⠿⠛⠁⢠⠀⠀⣤⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⢳⣾⣿⠆⠀⠰⠟⠃⠀⠛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢸⣿⣦⠀⠀⠀⠀⠒⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣮⣻⣿⣍⣅⠀⠀⠀⠘⠛⠁⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡁⠀⣀⣀⣀⣠⣾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣛⠉⠙⠻⣿⣋
⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣠⣄⣠⠖⠛⢧⠛⠛⠛⠛⢻⡇⢿⡻⠷⠄⢀⣀⣀⣤⣤
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣄⣻⠛⠛⣄⡒⢶⡿⠀⠘⠀⠀⢸⣇⠀⡀⠰⣤⣼⣿⡉⢻⣿⠆
⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠟⣩⠄⣩⠬⠟⢹⡅⠀⠀⠀⠀⢸⡟⠙⠶⠀⠀⠀⣉⡓⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠻⣾⡧⢀⣴⠺⠁⠀⠀⠀⠀⢨⣿⠆⠀⢟⢃⡿⠋⢁⣬⣅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⣆⠈⡇⠻⢿⡇⠀⠀⠀⠀⠀⠀⢿⡀⠀⠠⠋⣡⣾⠟⠉⠉⣼⡆
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⣴⣿⠻⣆⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⢄⡀⠀⣰⣿⣯⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠉⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡄⠙⡀⠐⣿⠁⠀⠀⠀⠀⠀⠀⢸⡄⢃⣼⣿⣿⠟⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠘⢠⡟⠉⠁⠠⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⢻⠐⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠟⠙⠛⢀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠋⡀⠁⠀⠦⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡿⠀⠚⠀⣹⣧⡀⠀⠀⠀⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠆⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣾⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠠⠄⢹⠃⢀⡈⢠⣄⣀⣰⣿⣧⣭⣋⢉⣿⣿⣿⣿⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⠈⠁⠲⣬⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⢠⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣦⣀⣴⣦⣴⣿⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠇⠀⢰⣦⣤⣀⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠙⠻⢿⣶⢿⣿⣿⢉⣿⣿⠛⠛⢫⡽⢿⠛⠒⠈⢿⡏⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣻⣿⣴⣿⣄⣠⣿⣿⣦⣼⣿⡿⠀⠀⠁⠘⠁⠀⠉⠉⠈⠁⠈⢉⠉⠀⠀⠀⠀⠀⠀⠈⠃⠀⣠⣤⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠲⠰⢿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠋⠀⠳⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠷⠶⠂⠘⢡⣿⣿⣿⣿⣷⣾⣻⡿⠟⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣶⣦⣽⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢷⠿⣿⣿⣿⣿⣿⣿⣻⣿⣤⣾⣿⣦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⣦⣤⡛⠻⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠀⢰⣶⣶⣝⣿⣿⣿⣿⣏⠉⣿⣿⠃⠀⠀⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⡾⠉⠛⠃⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣈⡶⠛⣿⡉⣿⣿⣿⣿⣷⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠸⣧⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠿⢻⣷⣾⣿⣧⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠛⣿⡷⢿⣿⣿⣿⡟⢿⣿⠃⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⣀⠚⢃⡘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣾⣿⣿⣿⢁⣾⡯⠛⡄⢠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣠⣴⡖⢛⣧⠴⠋⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠟⣿⣿⣿⣿⣟⠂⠀⠀⠸⠏⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢿⣏⠘⢿⣷⠈⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡾⢛⣿⣿⣿⣿⣦⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⣀⣠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣾⣷⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⣠⣾⣦⣾⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⡀⠂⣿⡉⢻⠏⢻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⡿⠂⠈⠀⠀⣴⠀⢿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠆⠀⠀⢼⣧⠛⡄⢻⡞⣿⣿⣿⣿⣿⣿⣿⣿⣸⡧⣿⣿⢹⣿⣿⣿⡃⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣻⣦⡉⠻⣤⣿⣿⣿⣿⣿⣿⣿⠀⠿⠈⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠈⢻⣿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⣹⣿⡿⠁⠀⠀⠀⠀⠀⠘⢯⡙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣻⣿⣿⣿⣿⢿⣿⣿⣿⡟⢛⠻⣿⠟⠚⠿⠿⢿⠚⠛⠻⡾⠛⠿⣾⠟⠿⣿⣿⡿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣬⣽⣿⣿⣟⣀⣻⣻⣱⣀⣀⣤⣆⡿⣆⣀⣀⣘⡆⢀⣉⡳⡄⠀⡘⢿⣄⠀⠈⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢌⠙⢍⠙⠻⣍⠉⠉⠛⢻⡟⠉⠉⠛⢿⡻⠛⠏⢻⠛⠛⡟⠛⡟⡟⠛⠛⠛⣿⠟⠙⠛⡿⡿⠛⣻⠿⣿⠿⠛⠛⡿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠑⢄⠈⢧⠂⢄⣠⣙⣄⠀⠀⠈⢇⠀⠀⢈⠀⠀⡇⠸⠁⡏⠀⠊⡰⠁⠀⣠⡜⣫⠔⣋⠴⠊⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠐⢄⠉⠚⠻⣦⡀⠀⢨⡃⠀⠘⡀⠀⣇⠀⠠⠀⠀⢪⣞⠅⣪⣝⡩⠔⠋⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠠⡀⠀⠈⠛⠲⣄⢱⠀⠐⡁⠀⣿⡀⣸⢀⣴⣿⢑⢯⠞⠁⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢀⣷⣀⣿⣴⣿⡿⠛⠋⠀⠊⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⢙⡻⣿⡟⠁⠀⠀⢀⠎⠀⠀⡠⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡁⠀⠀⡀⢀⣼⣠⣧⣿⡄⠀⢠⠃⠄⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠱⣸⣇⣸⣬⡻⣷⣤⣣⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣽⣿⣿⣾⣷⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⡿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⠶⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠛⠶⣄⠀⠀⠀⣴⡶⢦⡀⠀⣠⡴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢠⣾⡟⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠂⢀⣠⡴⣞⣩⡤⠶⠒⠚⠛⠛⠓⠒⠶⢤⣄⡈⠳⣤⡀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠚⣩⡵⠛⣩⡤⠶⠚⠛⠉⠉⠉⠉⠙⠓⠲⢤⣍⠓⢦⣝⠲⣄⡈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⠳⣄⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠹⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⡿⠟⣿⠟⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣷⣤⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡀⠹⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢦⣙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⣌⡙⠲⠦⢤⣤⣤⣤⣤⠤⠴⠖⢋⣡⢴⣞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠛⠉⠉⠛⠒⠶⠶⠦⠴⠶⠶⠒⠛⠉⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⣡⣠⡤⣤⣍⣉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢰⠞⠁⣀⣀⣀⣀⠈⠑⢄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⡟⠀⠛⠉⠀⢀⢠⢳⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣇⠸⢠⠄⢀⣠⣸⢸⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢆⣳⠙⠭⡝⢫⠟⠔⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠤⡀⠀⠀⠀⠀⠀⠙⠳⢯⠀⠀⢂⣺⡦⠦⠞⠁⠀⠀⠀⠀⠀⠀⡔⠕⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡸⠯⠽⠭⠿⣁⣀⣀⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠳⠠⢀⠄⡤⠦⣇⠠⠔⠂⢉⢀⢀⢤⣠⠀⣄⣀⣀⡀⠀⠉⠁⠒⠢⢤⣗⠒⡶⣄⠰⢈⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡈⣾⣁⠔⣿⣊⢵⣸⡽⠼⠒⠒⠉⠉⠉⠙⠛⠛⠻⠶⣦⣤⣀⠀⠈⠑⠺⣟⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠋⣨⢎⡶⠋⠁⠀⠀⠁⠀⡀⣡⠠⡀⠀⠀⠀⠀⠀⠨⠉⠛⠶⣤⡀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡴⣽⠋⠀⢈⡂⡤⠀⠀⠀⠙⠉⠈⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⣠⡉⠢⡀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣜⡼⠃⠀⠀⠀⠁⠉⠀⠀⠀⠀⠰⡆⠠⠄⠀⠀⠀⠀⠀⠀⣠⣾⢟⢁⡀⠜⢆⠀⡘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣘⢾⠁⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⢼⣷⠄⢀⠘⠀⠀⠀⣰⣾⠟⠁⠀⠉⠁⠀⠘⣆⠕⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣿⠃⠈⠀⠻⠺⠃⠀⠀⠀⠀⡐⠆⠀⠥⠀⠈⠁⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡤⠈⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠞⡅⠐⢣⠔⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆⠀⠀⢀⡀⠀⠀⠀⠔⠀⠀⠀⠠⣿⠿⢟⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠂⢸⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠂⠘⠃⠃⠀⠀⢂⠔⠀⠀⢠⣴⡿⠻⠟⠃⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⣹⣴⡅⣤⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⡆⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠁⠀⠀⠀⠀⡀⠀⠀⡟⢉⠋⣓⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢳⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠂⠀⠀⠀⠀⠛⠈⠉⠓⢘⠋⡋⠘⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠳⡈⠠⠈⠛⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡲⢋⢞⢹⠟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠢⣔⠀⠄⠀⠀⠀⠠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡒⠀⠀⢀⣴⣕⠟⢡⡛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⣀⠈⠑⠢⢀⠀⢀⡀⠁⠀⠀⠀⠀⠲⢔⡄⠀⠀⠀⠉⠀⠐⣀⣴⢟⢑⠕⣥⠣⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⣀⠉⠐⠠⠈⡀⢀⡀⠀⠉⢠⠀⠀⣀⣀⠤⣔⡩⢊⠔⢓⡡⢎⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠺⢭⣗⡒⡤⣤⣁⣀⠐⢏⠉⠠⠠⠠⡠⠲⣋⣰⠵⡚⣫⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠺⠯⠯⠭⠿⠽⠭⠯⠵⠘⠒⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠡⠊⠔⠨⠀⡡⢊⠄⡠⠂⠔⡠⢂⠔⡠⢀⠔⡠⠀⠔⡠⢂⠔⡠⢀⠄⡠⢀⠀⠠⡀⢂⠄⡠⢂⠄⡀⡠⢀⠄⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠐⠁⠊⠐⠁⠊⠈⠀⠁⠈⠀⠁⠈⠈⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢢⠶⢶⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠤⢀⠀⠀⢠⣰⣴⣠⠀⠀⠀⣥⣤⣀⠀⠀⠀⠀⠀⢠⢼⢟⡌
⠀⠀⠀⠀⠺⠞⢿⣿⠉⠛⠀⠀⠀⢀⣤⣴⡤⣀⠀⠀⠀⠱⠤⠴⠆⠉⠀⠀⠀⣿⣿⠏⠀⠀⠴⡿⠟⠀⠁⠀⠀⠀⠀⠀⠀⢿⡅⠆
⠀⠀⠀⠀⠀⠀⠈⡿⢀⠀⠀⠀⠀⣿⣟⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡤⠄⡀⠀⠀⠀⠀⠀⠀⠩⣿⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠘⠿⡿⢿⣽⠁⠀⠀⠀⢹⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠚⢛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡄⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢄⡆⠀⡽⠚⠃⠀⠀⠂⣳⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣷⡿⠄⠀⠠⡀⡇⣠⣂⠢⡀⠀⠀⠃⠀⠀⠀⠀⠐⠉⠁⡈⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠁⠀⠀⠀⠀⠀⢀⣠⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠶⠤⠀⠀⠀⣀⣀⠀⠀⠀⡄⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠘⠃⢶⠄⠀⠀⠛⠛⠀⠀⣄⣷⠀⠏⠀⠀⠀⠀⠀⢀⠀⠐⣠⣦⡄⠀⠀⣨⣄
⠀⡆⠀⠀⠀⠀⠒⢨⣡⠤⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠯⠾⣄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠈⣹⡉⢀
⠀⡁⠀⠀⠀⠀⠂⢸⣅⠀⠀⠀⠸⠁⠀⠀⠀⠀⠰⡗⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⠀⠀⠀⠐⠛⠋
⠀⡁
⠀⠄
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣤⣄⡀⠀⣠⣤⢶⣶⣤⣤⣠⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠽⣶⣶⣶⣽⣿⣿⡿⢓⣽⣾⣿⣿⡏⢻⣿⣿⡏⢠⡟⣫⠹⣿⣿⠿⣶⣤⡆
⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣬⣿⣿⠿⣋⣴⣿⣿⣿⣿⣿⡇⣾⣿⠿⡗⠘⣧⣛⡢⠋⠀⢀⡞⠋⠁
⠀⠀⠀⠀⠀⠀⣤⣤⡶⣾⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⠏⢜⠞⠋⠀⠁⠢⣀⠀⠀⠀⠀⠁⠛⠓⡶
⠀⠀⠀⠀⠀⣀⣠⣤⣾⡿⢃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⠂⠀⠀⠀⠈⡇⠀⠀⢘⠀⢀⠎
⠀⠀⠀⠀⠀⠀⣊⣽⡿⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠂⠀⣀⣤⡘⠢⢤⣠⠞⠁
⠀⠀⠀⠀⠔⠋⢻⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣷⢄⡀⠀⠂⠀⠩⠟⠋
⠀⠀⠀⠀⢀⣴⣾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣇⡀⠀⠀⠋
⠀⠀⠀⠀⢐⣨⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⣻⡅⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⢀⣀⣶⣷⡞⠁
⠀⠀⠀⠀⠔⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⡟⡿⠃⣸⣷⣾⣿⠿⠋
⠀⠀⠀⠀⠀⡼⡇⣿⣿⣿⣿⣿⣿⣿⣿⠏⣥⠹⣟⣬⣽⢶⡿⠋⠁
⠀⠀⠀⠀⠀⢀⣷⢹⣿⣿⣿⣿⣿⣿⣿⡁⠨⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢰⠀⢹⡌⣿⣿⣿⣿⣿⣿⣏⢌⡐⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠿⠿⣿⣯⡻⣟⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢢⡆⢰⡰⣿⣿⣿⣿⣿⣿⡶⡡⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢮⡻⡻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠈⢿⡄⠽⠹⣿⣿⣿⣿⣿⡵⣽⣃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⢘⢿⣳⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⣷⡟⢿⣿⣿⣿⣿⣧⣷⠄⠠⣴⡿⡿⢋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢸⣿⡇⠀⠀⠀⢀⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡃⢶⡝⣿⣿⣿⣿⣿⣿⢀⠈⠛⠉⠁⠛⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣻⣧⣦⣾⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠐⠸⡟⠦⣝⡿⣿⣿⣿⣿⣖⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠐⠳⣿⡋⠀⠙⠻⠿⣿⣿⣾⣷⣤⣄⣀⣀⣀⣀⣠⣖⣁⡀⣠⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⠶⠶⣤⣝⣫⣍⣿⣻⣿⣛⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠟⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠏⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢨⡅⠀⢠⡉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠉⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⣤⣀⣀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣤⣤⣾⣟⣛⡒⠶⣤⡀⣤⣀⣐⡄⣰⠞⠉⣁⣤⣶⣶⣿⣿⣿⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣷⠞⠃⠉⢻⡏⠁⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣟⢦⠈⢳⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠃⣰⠃⣿⣿⣿⣿⣿⡿⢋⣠⡴⢶⣚⣻⣿⣿⣿⣿⣶⣦⣭⣿⡿⣿⣿⣿⣿⢸⡇⠘⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⠀⢿⣿⡿⢛⣿⠟⣫⣴⣾⠿⠟⠛⣛⠛⠛⠛⠿⣿⣿⣿⣷⣤⡉⠛⠿⣼⠇⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣤⠞⠋⣠⠋⣡⣾⠿⢋⡀⠀⠀⠀⣿⠀⠀⠀⠠⠆⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⡿⠁⠀⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡟⢠⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠿⠇⢸⢿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣼⣿⡄⠉⠁⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡆⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⢃⣿⠇⢀⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⡄⠘⣿⣿⣷⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣾⠏⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠻⣄⠈⢿⣿⣿⣦⣀⠀⣤⠀⠀⠀⣠⡀⠀⠀⠐⢂⣤⠞⣱⡿⠋⠀⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⠿⣿⣿⣷⣦⣤⣀⣀⣘⣁⣠⡤⠶⣋⣵⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⣿⠿⢿⣿⣿⣿⣿⣿⣶⣾⠿⠟⠉⠹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠈⠉⠉⠉⠀⠠⠤⠖


⣿⣿⠿⢿⡿⢿⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣴⡿⠋⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣎⢻⢉⠶⢵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢮⡵⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣙⢡⣪⣏⣦⣤⡤⣤⣴⣴⣶⠷⠖⠒⠞⠛⠣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡁⣦⣵⣾⠿⠛⠋⠉⢸⡟⡱⣻⣳⢳⠁⣠⠀⠀⢀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢩⣶⣿⠟⠋⠀⠀⢴⣤⣀⡈⢷⣷⣿⣿⣼⣾⡇⢀⣴⠟⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡽⠋⠁⠀⠈⠂⠚⢥⣽⣛⠟⣾⣿⣿⣿⣿⠏⠱⣿⣿⣀⣤⠤⠗⠒⠒⠚⠺⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠖⠁⠈⠀⠀⠀⠀⠀⠀⠀⢉⣫⣴⣿⣿⣿⡛⡁⢰⠷⠤⠤⠤⠴⠤⠤⠤⠤⠴⢿⣷⣤⡀
⠀⠀⢴⣶⣾⣵⠽⣿⣶⠒⢒⣖⣏⡄⢀⣰⣀⣤⣤⣤⣤⠶⢶⡿⢿⠿⠛⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠉⠛⠲⣶⢿⣷⣁⡩⠛⢋⠉⠉⠛⢟⣃⣤⡤⠐⠂⠐⠊⠉⠀⠀⠀⠠⡤⠄⢀⠒⠄
⠀⠀⠀⠀⠀⣼⢷⣾⡿⠋⠀⣀⡋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠠⠀⠀⠉⠀⠘⠀⢀⠀⠀⠀⠀⠉⣠⠴⠒⠂
⠀⠀⠀⠀⢸⣿⠟⠋⠀⠀⠚⠀⠁⠂⠋⠒⠐⠖⠚⣠⣉⡉⠁⠈⠀⠀⠀⠤⠀⠠⠀⠀⠁⠠⠴⠒⠚
⠀⠀⠀⠀⠈⠀⠀⢀⠈⢀⡀⠂⠈⠀⠀⠀⠀⠀⠀⠤⠥⠭⠽⠶⠶⠶⠶⠖⠶⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢰⠛⡁⠨⢦⡉⢻⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠻⣄⠈⣆⡅⣻⣬⠉⠹⠞⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠃⠀⠀⠀⠹⣿⡿⣷⣿⢿⠃⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠠⠀⣰⣿⣶⠇⠺⣧⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⢁⡜⢠⠀⠉⢿⣻⣶⡀⢀⠻⡜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣿⣞⣿⡿⠀⠀⠈⠉⠉⠀⠈⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⠦⠔⠂⠀⠀⠀⢠⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠰⢧⢀⢶⣌⡹⣇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⣧⠀⣤⣦⣥⠀⠀⠀⢠⡼⣿⠀⡿⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⡟⠛⡩⠀⠀⠀⢀⣴⡴⢹⡁
⠀⠀⠀⢰⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠁⠀⠀⡀⢈⡉⢛⠙⠃
⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠀⠀⠀⡀⠀⠣⠀⠿⠛⠻
⠀⠀⠀⡘⠃⠀⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠀⠀⠀⠀⠀⠂⢧⡀⣡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢸⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢉⡬⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⣇
⠀⠀⠀⠤⠁⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠞⠉⠃
⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠣
⠀⠀⠀⠸⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⡄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⢀⡠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⡀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠋⠠⠔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⣠⡄⣤⠀⠀⠄⠀⠤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠡⠱
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠦⠀⠀⠀⠀⠠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠢⢤⣤⡆
⠀⠀⠀⠀⣠⡄⢾⣤⣠⠄⠀⠠⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠐⣤⡀⠀⠁
⠀⠀⠀⠀⠛⠣⠀⠈⠉⠑⠒⠒⠛⠋⢀⣴⣤⣶⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢆⣀⠀⠀⠀⠀⠸⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⡶⠚⠋⠉⠉⣿⣿⣷⣤⡴⠶⠀⠀⠀⢰⡄⠀⢦⡀⡀⢀⡈⠉⠓⠶⢤⡶⠟⠊⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠾⠒⠀⠀⠀⠐⠿⠽⢿⡀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⢀⡾⠿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢳⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⠉⠀⠀⠀⠉⠑⠦⡀⢔⠉⠉⣉⣩⠆⠀⣠⠖⠋⠁⠀⠀⠈⠙⠲⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠉⢹⠁⠀⠀⠸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣠⠴⠊⣁⣠⠤⠤⠼⠦⠤⠤⢤⣀⡉⠓⢤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⡤⠖⣻⠿⠓⠉⠀⣀⣀⡤⠤⠤⠤⣀⣀⡀⠈⠑⠺⢿⡓⠢⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⡤⠚⠁⣀⠤⠒⠉⠁⠀⠀⠀⡀⣀⠀⠀⠉⠑⠲⢄⡀⠉⠲⣄⠉⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⠞⠁⠀⠀⢀⣤⠀⠀⠀⣏⣹⠃⠀⠀⣀⠀⠀⠙⢦⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠁⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠙⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠎⠀⠀⣰⡔⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡞⠀⠀⠀⠙⠳⠾⣀⢰⣦⣄⠀⠀⢰⠇⠀⠀⣀⣤⣶⠟⠻⠷⠀⠀⠘⡆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⢀⡤⡄⠀⠀⠈⠉⠛⠿⣿⣦⣼⣤⣴⡾⠟⠉⠀⠀⠀⠀⢠⡀⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠈⠓⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣘⠇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢣⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣆⠀⠀⢠⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢧⠀⠀⠘⠛⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠳⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣴⡀⠀⠀⠀⢙⣳⠀⠀⠀⡰⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠷⠏⠀⠀⠀⠈⠁⢀⡠⠊⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠈⠙⠲⠤⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣹⠶⢤⣀⡀⠈⠉⠉⠉⠉⠉⠀⢀⣠⠤⠾⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡜⠁⠀⠀⠀⠈⠉⠉⠓⠒⠊⠉⠉⠁⠀⠀⠀⠙⣌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣴⣶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡾⢿⣿⣯⣭⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⣿⠟⠿⠿⣿⣿⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⢸⣏⣀⠀⣴⡄⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⢀⡉⠉⠀⠈⠀⠀⣿⣿⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠻⣦⠀⠁⠀⢠⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠼⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣷⣄⡂⠀⠀⣤⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠂⣠⠌⠠⠄⠼⣿⡟⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⣰⠂⠀⠠⣄⡀⠀⣼⣇⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⠀⠹⣧⠀⠀⠘⠃⠀⢹⣿⠀⠸⠟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⢀⠸⣷⡀⠀⠀⠆⠈⣿⡇⢀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢀⣤⠄⣤⡙⣿⣷⣦⣶⣼⣿⣿⣾⡶⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠋⠀⠀⠛⠛⠉⠙⠿⠿⣿⣿⣿⠟⢣⠶⡆⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠁⠀⠀⠀⠀⠀⢀⡴⠀⣰⠋⠀⠀⠠⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠠⠎⠀⠠⠇⠘⢣⡀⠀⠀⠀⢤⣤⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣷⡿⠆⢁⣴⠟⠀⡤⠀⠀⣀⡀⢀⣾⣿⣆⣠⡄⢼⣿⡆⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⣿⠏⠁⠐⠋⠀⢀⣠⡀⠀⢸⡏⠉⠿⣿⣿⠸⣿⡅⠀⠛⠁⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡆⠀⣰⡞⠁⠀⠈⣿⠁⠀⣾⠇⠀⡀⢻⣿⠀⣿⣧⠀⣄⠀⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣽⣿⡿⠋⢀⣴⠏⠀⠀⣀⣼⣿⠀⢰⣿⡄⠀⠅⢸⣿⡆⠾⣿⣆⠙⠇⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣋⣁⣤⣾⡿⠃⠀⢀⣠⣄⠡⠄⢀⣿⣿⡇⠀⠘⠘⣿⡇⠀⠿⣿⠀⣾⣤⠀⠸⣿⣄
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣼⣿⣿⡛⢛⣿⣿⣿⣋⠀⠰⠦⡊⡉⢻⡁⠄⣾⡟⣿⣇⢸⡀⣶⣿⣷⠀⢿⣿⣇⠉⠀⠆⢸⣿⣿⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢡⣾⣿⣿⠿⠛⠛⠻⣿⠟⠉⠀⢘⠘⢷⡟⠋⠀⣼⣿⣳⣿⠋⠘⠁⠀⢿⣿⠀⠳⠟⣿⣆⠀⣰⣏⢹⣿⣧⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢂⣿⣧⣾⠟⣛⡛⢻⣿⠏⠀⠀⠀⠊⢁⠀⠁⠀⢠⡾⠿⢿⠿⠀⠀⠀⠀⢸⣿⡀⠀⠂⠻⣿⡆⠙⢿⡃⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣾⣿⣿⣻⣿⣻⣿⠟⠁⠀⠀⠠⡀⣶⢸⠗⠀⢀⡛⠃⠀⠉⠀⠀⠀⠈⠉⠀⣿⡇⠀⠙⣤⢻⣿⡄⠀⠻⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⢀⣴⠶⢂⣀⣀⣾⠿⠏⠀⢀⣿⣿⡖⢰⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠀⠻⣿⡄⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠟⣥⣾⠟⠋⣠⠟⠛⣩⡅⠀⠀⣀⣾⣿⣿⠁⢸⠀⠀⠀⠀⠦⠀⠀⣿⣷⠀⡌⠃⠀⠀⣻⣿⣆⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣱⡾⠋⠁⠀⠀⠈⢷⣶⡿⢥⣤⣾⣿⠟⣿⣿⡄⠀⠀⠀⠀⠀⣠⣧⠀⣿⣿⠀⣿⡀⠀⠀⢻⣿⣿⣦⡸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⠋⣀⠀⠂⠀⠀⣰⡾⠛⠃⣀⣤⣿⡿⢸⡟⠘⣷⡆⠀⠀⠀⠀⢈⡁⠀⣿⣿⠀⢿⠿⣧⡀⠀⣿⣿⡿⣷⡝⣿⣿
⠀⠀⠀⠀⠀⢻⣾⣿⣛⡟⢋⣶⣿⠿⡄⠀⢾⠛⠉⢁⡴⣶⣿⡿⢻⡇⢸⡇⠀⢻⣇⠀⠀⠀⠀⠘⠁⠀⣿⣿⡂⣾⠓⢿⣷⠆⠛⠛⣿⠙⢿⣾⣿⣧
⠀⠀⠀⠀⠀⣨⣿⣿⠻⣥⣿⠦⢙⠧⠰⡄⢀⠀⠀⢀⣴⡿⢻⣷⡌⠀⠀⠀⢀⠈⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢹⠀⠀⠀⠀⠈⣛⣿⡇⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⡯⣿⣿⣿⠶⣿⡃⠰⠀⢀⣤⣾⠟⠁⣴⣿⠹⣧⡀⢲⣤⠀⠒⡁⠀⠀⠂⠀⠀⠀⠀⣿⣿⡇⢸⡇⠀⠀⠄⠸⣿⣿⣟⢠⡍⢻⣿⡇
⠀⠀⠀⠀⣴⡛⣿⣷⣤⡛⢛⣶⣏⣡⡶⣠⣿⣿⡇⢺⣄⠘⣿⠀⢻⣷⡞⢻⡟⠛⠻⣦⠀⠀⢀⡀⠀⠀⣿⢸⡇⠼⣿⡇⠀⠀⣲⡉⢛⣣⣼⣧⠀⠿⣁⡆
⠀⠀⠀⢠⣿⣿⣌⠙⠟⠁⠠⠿⠟⢋⣴⣿⣿⣿⠃⡀⢻⣦⠉⠀⣿⣿⣷⡄⠈⠀⠣⣀⣴⣤⡮⣤⠀⠀⣿⠘⡇⠀⣘⢿⣶⠀⠙⠐⢠⠿⠿⠟⠀⢰⣿⠁
⠀⠀⠀⣾⣿⣿⣿⣶⣶⣶⣶⣿⣆⠘⢿⣺⣿⣿⠆⢹⣶⣉⣷⣴⣿⣿⣿⡻⣦⡀⠈⠉⠁⣠⣏⠉⡅⢠⣿⠀⣧⠀⣩⢛⡿⠋⢀⣴⠆⠀⢰⣿⠀⠐⣿⠻
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣌⣿⡆⠀⣿⣿⣇⠀⠈⠻⢿⣿⣿⣿⣿⡟⢰⣿⣿⣶⣄⠀⠀⠀⠸⠇⢈⣿⠀⣿⠀⠉⠉⢁⣠⠞⠁⠀⠀⣸⣿⠆⠀⣿⣧⣄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢿⣇⠙⠃⢰⣿⣿⣿⣆⠀⠀⠀⢻⣿⣿⣿⠃⢸⣿⡇⠈⣻⣿⣶⣤⣄⡀⠐⣿⠀⣿⣀⣠⡶⠿⣭⠀⢀⡤⠞⢹⣿⣿⠆⢸⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡿⠟⠁⠀⠙⠂⠀⢻⣿⣿⣿⣿⡄⠠⠀⠀⢿⣿⣿⡀⣿⠏⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠀⠸⠛⠁⠀⣼⣿⠀⢉⣤⠀⢸⣿⡇⠀⠘⣿⡿
⠀⠀⠈⠛⠛⠛⠻⣧⣶⣦⣄⣈⣁⣤⣿⣿⣿⣿⣿⣷⡄⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠐⠂⣠⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⡀⠤⠄⠀⢸⣿⠇⠀⠀⣿⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⢿⢛⣿⣿⣿⣿⣽⣿⣝⣿⣟⣀⣤⣴⣿⣿⣿⣶⣶⣦⣤⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣄⣀⣀⣠⣼⣿⣦⠴⠞⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣭⣹⣿⡟⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⠿⠿⣷⣶⣿⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠒⠛⠉⠈⠉⠉⣈⡉⠉⠙⠛⠉⠛⠉⠿⣿⠿⠿⢿⡿⢼⣿⣤⣴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣩⠭⢭⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣀⣀⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣉⣹⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠛⠉⠉⢉⣉⢉⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⢁⡠⠴⠒⠊⠉⠉⠸⡇⠈⠉⠑⠒⠤⣀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⡠⠞⠙⠆⠀⠀⣀⣠⣤⠤⠥⢤⣄⣀⡀⠀⠋⠙⠲⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣠⠎⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀⠀⠈⠳⣄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⡃⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢦⡀⢀⡼⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢰⠃⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⣿⠇⠀⠀⢳⡀⠀⠘⡆⠘⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠁⠀⢰⡏⠀⠀⠀⠀⢇⠀⠀⢹⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠏⢳⡤⠠⡄⠀⢠⡟⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠛⠉⢸⠀⠀⠀⠀⠀⢰⡏⠀⠘⠧⠤⢯⣠⡟⠀⠀⠀⠀⠀⠀⢸⠈⠉⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⠀⠀⠸⡄⠀⠀⠀⢠⡟⠀⠀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣇⠀⠀⢳⡀⠀⠀⣾⣥⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡟⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠹⡤⠂⠀⠳⣄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢉⡟⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠙⣆⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⣠⠞⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡈⠳⣄⠀⠀⡀⠈⠙⠓⠒⠶⡶⠖⠒⠛⠉⠀⠠⣀⣠⠞⠁⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣌⡙⠺⢤⣀⣀⠀⠀⠀⣇⠀⣀⣀⣤⠴⠚⠉⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣉⣉⡉⠉⠉⠉⣉⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡴⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⢦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⣰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⣄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢈⡷⠟⠚⠋⠉⠓⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡖⠟⠋⠙⠒⡿⣾⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢫⠂⠁⠀⠀⠀⢀⣠⠤⠾⠂⣀⣀⣀⣀⣀⣀⠘⠧⢤⣄⡀⠀⠀⠀⠈⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⣀⣴⠚⣉⡤⠖⠚⣉⠩⠤⠄⠤⠤⠉⣉⠓⠲⢤⣉⠓⣦⡀⠀⠀⠀⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⢑⡴⠋⡡⠴⡈⠁⠀⠀⢰⠀⡀⠀⠀⠀⠀⢑⠢⢈⠛⢤⡋⠳⣄⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠁⡴⠋⡠⠊⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢆⠈⢣⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡞⢡⠚⠠⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⡈⢧⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠃⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⡠⢄⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⠆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⠀⠀⡀⢀⡔⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡁⠢⣀⡐⠀⠀⠀⠀⠀⠙⠤⠁⠀⠀⠀⡀⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢉⡷⢤⣉⡐⠀⠀⠠⠤⠤⠄⠀⠐⢂⣁⠤⢾⣁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣡⠎⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⠻⣦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠛⠒⣶⠀⢰⡖⠚⠃⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⢹⡆⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⣾⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⢶⣸⠶⢖⣛⣩⣭⣭⣭⣭⣙⣓⠶⢾⣰⠶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡵⣾⠋⠁⠀⠀⢸⠀⠀⠀⠉⢻⡶⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡾⠋⠀⠘⠇⠀⠀⠀⠈⠀⠀⠀⠀⠟⠀⠈⠻⣮⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣏⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠇⠀⠙⠃⠀⠀⠀⢿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⣧⣹⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠶⠶⠂⠀⠀⠀⠀⠀⠘⣏⠀⢈⣟⣛⣛⣿⡷⠶⠶⠲⠶⢾⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣆⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⡾⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠀⢠⠇⠀⠀⠀⢀⠀⠀⠀⠀⢧⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡓⢯⣄⣀⠀⠀⢸⠀⠀⢀⣀⡬⠗⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣭⣍⣙⣛⣛⣉⣭⡥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿