Physical therapy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⣴⣞⠃⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⣤⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⡾⠃
⠀⠀⠀⠀⠈⢿⣦⣘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣹⠿⣋⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣹⡿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣰⡟⢿⡆
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠛⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠈⠛



⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠇
⠀⠀⠀⠀⠀⣘⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⣴⣾⡆⢈⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⢧⣄⡀
⠀⠀⠀⠀⣾⣿⣿⡄⠔⠑⢢⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⣿⣿⣿⠿⢿⠗⠀⠀⠀⠀⠀⠈⠋⠻⣷⣬⣭⣵⡿⠃⠀⠀⠀⠀⠀⠀⠀⠤⠤⠯⠬⣩⠤⠭⠭⠄
⠀⠀⠀⠰⠿⠷⠻⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠿⠿⠿⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛


⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⣙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣏⣀
⠀⠀⠀⠠⠶⠟⣿⣿⡟⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠍⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⡟⠛⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡟⢿⡇
⠀⠀⠀⠀⠀⢼⠏⠈⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⢸⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⣀
⠀⠀⠀⣀⣀⡀⣀⣀⣠⣄⡾⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⡁⢾⣿⢇⣴⡟⠛
⠀⠀⠘⠛⠛⠋⣿⡟⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⠻⢷⠄⠀⠀⠀⠀⠀⠀⢠⣤⣄⣿⣇⣀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣿⠋
⠀⠀⠀⠀⠾⠿⠿⠃⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠷⠖⠀⠀⠀⠀⠀⠀⠈⠛⠉⠛⠛⠛⠛⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠴⠶⠬⠿⠽⠴⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⡀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⣾⣷⠄⣴⣦⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣼⣿⣿⡶⣾⣿⣷⡄⠀⠀⠀⠀⣴⣮⠁⣰⣿⣿⣦⡀⠀⠀⠀⠀⢠⣶⡍⠁⣴⣮⠉
⠀⠀⣾⢻⣿⣿⢻⣿⣿⣿⠃⠀⠀⠀⢰⣿⣧⡴⠟⣿⣿⡏⠙⠳⠖⠀⠀⣾⣿⠻⠶⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣴⣦⡄
⠀⠀⠛⢸⣿⣿⠙⣿⣿⣿⠀⠐⢲⠒⠒⣿⣿⠓⠒⣿⣿⡗⠒⠒⢲⠒⠀⠙⣿⡄⠀⢻⣷⡘⠓⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⠀⠀⢸⠀⠀⣼⣿⡆⠀⣿⣿⡇⠀⠀⢸⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠠⠀⠀⠀⢀⡴⣭⣥⢠⡀⠀⣶⣶⠄
⠀⠀⠄⠈⠁⣿⠀⠛⢻⡿⠀⠀⢸⠀⢸⡿⠹⣷⠀⣿⢿⡇⠀⠀⢸⠀⠀⠀⣿⠃⠀⣾⡇⠀⠀⠀⡀⠀⢠⡏⢡⣈⣁⠞⢿⣶⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⢠⡄⠘⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣀⣀⡀⢿⣿⣿⡿⠷⣷⣿⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⡇
⠀⠀⠀⣤⡄⠀⠀⠀⣰⣿⡀⠀⠀⠀⣠⣴⡏⠀⠀⠀⠀⠈⢹⡏⠡⠠⣿⣿⡇⠸⣿⣿⣿⣿⡀⠀⠀⠀⠴⠿⠶⠷⠾⠶⠿⠎⠋⠿⠇
⠀⠀⠀⣿⣗⠀⠀⠀⣿⣿⠛⢦⣄⣀⠴⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣷
⠀⠀⠀⣿⡟⠉⠉⠉⢹⣿⣏⠉⠉⠀⠀⣿⡇⠀⠀⠀⠀⢀⡼⢧⡀⠀⣿⢿⡇⠀⠸⣿⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣿⡇⠀⠀⢀⣼⡟⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠿⠿
⠀⠀⠀⣿⣇⣀⣀⡻⣋⣀⣻⣃⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⢘⡡⠛⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣧
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢿⡿⠃⠀⠀⠀⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣾⡗⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠘⣿⣿⣦⣄⠀⠀⠀⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠙⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣿⢹⡟
⠀⢠⣞⣳⡄⠀⠠⣽⣿⣿⠻⣷⣄⣀⣤⣤⣰⣿⣇⠀⠀⠿⠿⠿⠻⠿⠟⠻⠃⠀⣶⣷⡄
⠀⠸⢿⡿⠇⣀⣀⣀⣿⣀⣀⣈⣩⣭⣟⣋⣻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⡐⠿⢷⣶⣦⣤⣤⣤⣤⣤⣤⣠⣴⣄
⠀⠀⠀⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⢹⣿⠛⠛⢿⡈⠛⠋
⠀⠀⠀⠘⢦⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣯⢸⣿⠀⠀⠀⠀⢶⣶⣾⡿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠠⡆
⠀⠀⠀⠀⣒⡶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠑⢸⡀
⠀⠀⠀⠀⢸⢛⣱⡆⣤⡀⠀⠀⠀⠀⠀⢼⣲⡀⣀⠠⢊⣹⡆⠀⠀⠀⠀⠀⠀⢲⣿⡖⠀⠀⠀⠀⠀⠀⠀⠁⠈⣴⣶⠀⢹⡆
⠀⡀⣾⣿⡿⣻⣻⣛⡀⠀⠀⠀⠀⠀⠀⠈⠉⢣⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⢿⠀⢹⢱
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣀⠀⠀⠀⠀⠀⢀⠀⣘⢇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⢀⠞⢿⡿⠀⠀⠀⠀⠀⠀⠋⢿⣿⠙⠃⠀⠀⠀⠀⠀⢀⡀⢞⣻⠇⠵⠒⠂⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠄⠀⢀⠀⡸⠁⢠⠞⠂⣵⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⠧⠿⡿⡿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⣷⠈⣀⣠
⢠⣀⣙⣁⣛⣛⣋⣀⣀⣟⣗⡀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡄⠀⠀⣈⣋⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡶⠀⠀⢀⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠠⡖⠒⠊⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠙⠃⠐⠒⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⡀
⠀⠀⣯⣤⠀⢸⢿⠉⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣾⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣂⠀⠀⠀⠀⠀⢀⣷⣿⢦⡆
⠀⢿⣿⡟⣇⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⠾⢀⡀⠀⠏⡇⠀⠀⠀⠀⣰⣷⣼⣿⣷⣧⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⠋⡓⠂
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⣼⣿⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⢶⣶⣿
⢿⣿⠏⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⢿⠗⠀⠀⠀⠀⠀⠀⣿⣧⠀⣿⣥⡋
⠈⠻⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⡶⠟⠃


⠀⣀⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣤⡀⠀⣠⡶⠶⡾⣧⠀⠀⠀⠀⠀⠀⣸⡧⠄⢀⣠⣾⣿⡆⠀⠀⠀⠀⠀⠀⢀⣴⡿⣦⡀⣀
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡆⢠⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠁⢸⣿⠃⠀⠀⠀⠾⠿⠿⠿⠿⠷⠿⠷⣿
⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠛⠀⠀⢸⣿


⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣄
⣾⡏⠷⡤⠤⠤⠤⢠⡄⠀⠀⠀⠀⠀⠀⠘⡟⢳⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢛⠓⣄
⣿⡇⠀⡃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⠃⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⠀⢩
⠉⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣠⣤⣄⠰⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢰⣖⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠈⠛⠛⢿⣻⣻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠶
⠀⠀⠀⠑⣼⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠙⠛⠻⣷
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠿⢿⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠞⠈

⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⣀⣤⠴⠀⣀⡀⠀⠀⠀⠀⠀⠠⣶⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢤⣄⡀
⠀⠀⠰⠖⢻⣿⠳⠆⠀⠀⠀⠀⠈⠘⢷⣜⣻⠃⠀⠀⠀⠀⢀⢠⣾⣏⣡⣤⣀⠄⠀⠀⠀⠀⢠⣶⡿⣶⠀⠀⠀⠀⠀⠞⢳⣌⣛⣧⡾⠀⠁
⠀⠀⠀⢀⡿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⠈⠈⠃⠛⠋⢦⡈⠀⠀⠀⠀⠀⠘⣻⣿⣽⠇⠀⠀⠀⠀⠀⡠⢌⡿⢿⡠⠄
⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⢀⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⣘⣟⢙⣿⣋⣹⡏⠀⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠈⠽⣦⣻⣟⣤⡾⠅⠀⠀⠀⠀⠀⣘⣿⣃⠀⠀⠀⠀⠀⠀⣠⣴⣟⢣⣿⢷
⠀⢰⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⢰⣟⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠛⠈⢿⣿⡿⠙⠛⠀⠀⠀⠘⠻⢿⣷⣿⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⢠⣾⣧⠀⢠⣤⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠄⠀⠀⠀⠀⠀⣀⢀⣸⣿⣃⡀⣀⠀⠀⠀⠀⠀⡼⣿⠷⣤⡀
⠀⠐⠒⠶⢺⡿⠻⡏⠁⠀⠀⠀⠀⠒⠻⣶⠉⠒⠂⠀⠀⠀⠀⠀⣼⠛⣷⠀⠀⠀⠀⠀⠈⢫⠉⣿⣿⡏⢩⠉⠀⠀⠀⠐⠉⢳⡷⣶⠊⠑
⠀⠀⠀⠛⠛⠃⠀⠃⠀⠀⠀⠀⠀⠀⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣥⣤⠄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠋⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⣴⣶⠻⠃⢻⡂⠀⠀⠀⣠⡔⢠⣶⡄⣹⠓⠆⠀⠀⠀⠀⢠⡄⠀⣴⠀⠀⠀⠀⠀⢀⣶⡶⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⢻⣟⢾⠛⣠⣼⣷⠀⠀⠀⠁⠙⢷⣿⣾⠋⠀⠀⠀⠀⣴⣦⣾⣥⣼⣏⣀⣀⠀⠀⠀⢹⣿⣿⢾⣿⣿⠉⠀⠀⠀⠀⡔⠙⢀⣿⣿⡆
⠀⠈⠉⠉⠈⠈⠉⠉⠋⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠐⠛⠛⠋⠛⠃

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢘⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⠄⠀⠀⠀⠀⠰⠾⠷⠆⠀⠀⠀⠀⠀⠀⠔⢉⣬⡙⢦⠀⠀⠀⠀⠀⢀⣴⣾⠛⠁
⠀⠀⡀⠰⠯⠨⠷⠀⡀⠀⠀⠀⠛⠛⢻⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⡆⣨⣯⣄⢘⠃⠀⠀⠀⠀⢸⡏⠙⠓⢠
⠀⠀⠁⠋⠆⠀⠎⠃⠁⠀⠀⠀⠀⠙⠛⠋⠀⠙⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠑⠢⠤⠔⠁⠀⠀⠀⠀⠀⠸⠗⠀⠃⠹

⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⣽⠂⠀⠀⠀⠀⠀⣋⠷⣀⣠⣤⣤⡀⠀⠀⠀⠀⢬⡷⣽⢮⡵⣭⠆
⠀⠀⠀⠀⠀⢀⢴⡦⠴⠀⠀⠀⢰⣦⡏⠀⢥⣾⡆⠀⠀⠀⠻⠖⣉⣉⣁⣹⣇⣀⡀⠀⠀⢼⡂⢿⣴⣇⣾⡄⠀⠀⢀⠈⠲⠊⠲⠂⠲⠂⢢
⡄⠀⠀⠀⠐⠎⠺⡷⣄⠀⠀⠀⠨⡇⣀⣀⡠⡏⡇⠀⠀⢸⢩⠂⠐⢦⠗⠾⢾⠆⡇⠀⠀⠀⡠⢤⠡⠀⡀⡤⠀⠀⠀⡿⣏⡹⠍⠹⢉⡹⠉
⡃⢰⠆⠀⠀⢰⣶⠟⠁⠀⠀⠀⠀⣿⣮⣿⣦⡗⡇⠀⠀⠸⠈⠃⠒⠃⠑⢺⣟⣆⡇⠀⠀⠀⠗⠒⢶⠤⡏⣿⠀⠀⠀⢑⣾⣶⣶⡄⡏⠓⠒
⠻⠋⠉⠉⠳⠋⠀⠀⠀⠀⠀⠀⠺⠇⠀⠋⠀⠻⠃⠀⠀⠘⠀⠠⠄⠤⠀⠄⠀⠄⠃⠀⠀⠀⠓⠒⠛⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠙⠣
⡔⠢⠰⠲⡤⠒⢆⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠤⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀
⠻⡟⠘⣟⠉⢿⠃⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⢠⠻⠞⣤⠑⡕⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⢸⢽⠳⠶⠶⠶⢞⠏
⠀⡴⠉⡭⠭⠭⡉⢢⠼⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢫⠢⣌⡾⣳⣩⣜⡝⠀⠀⠀⠫⢧⢥⣮⠉⣤⡼⠓⡄⠀⠈⠀⠥⣁⢉⡉⠭⠚
⠀⠃⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⢞⠡⠠⠄⢋⠹⣆⠀⠀⢸⡜⣓⠒⡄⠉⠀⠈⡧⡇⠀⣏⣦⡔⡚⡛⠆⣽⣵⣀⠇⠀⠀⠀⠀⢸⠀⡇
⠰⣆⢀⡑⣒⢚⡀⣠⠇⠀⢰⡏⠍⠉⠉⠉⠉⠉⠋⣷⠀⠸⡼⠒⠚⠀⠀⠀⠀⢥⠇⠀⢧⡥⠘⠒⠓⠁⢭⡼⠁⠀⠀⠀⠀⠀⢸⢠⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠛⠂⣿⣀⣼⡇⢸⣟⣛⡃⢸⣟⣻⡷⠀⣸⢿⣆⠀⣿⣛⣿⠄⠻⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⣿⠉⢹⡇⢸⣯⣭⡅⢸⡏⢻⣦⣰⡿⠾⣿⡄⣿⠋⠉⠀⠀⣿⡇


⠀⠀⣤⡍⣶⢽⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⡕⠀⠀⠀⠀⠀⠀⠺⣾⠧⠀⣿⠆
⠀⠀⣼⢿⠻⣽⢧⡀⠀⠀⠀⡀⣀⢀⠀⠄⠠⠤⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠉⠉⠶⠀⠀⠀⠀⠀⣨⡛⡧⠈⡧⡦⠄
⠀⠀⠀⢼⡄⠛⠙⠁⠀⠀⠘⠄⡸⠻⣤⠛⡤⠤⣠⡬⠀⢀⣬⠷⣯⠷⣮⠷⠠⠦⡀⠀⠀⠀⠘⣇⣀⣀⡏⠀⠀⠀⠀⠀⠈⢹⡃⢁⢡⠶⠟
⠀⠀⢠⣸⣗⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⢸⠓⡞⡍⡦⠾⡭⠭⠆⡇⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠐⠟⢛⣦⡄⡷
⠀⠀⠛⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠃⠀⠀⠘⠁⠃⠓⠋⠉⠋⠁⠛⠃⠀⠀⠀⠀⠓⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠲⣤⠀⣀⠄⢄
⢠⠦⡄⠀⠀⠀⢀⡴⢄⠀⢀⡄⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡍⣲⣊⡝⣶⣪⣱⠀⠀⠀⠑⠚⠋⣇⢯⠻⣻
⣾⣆⣟⣂⠴⢒⠚⠓⠾⠀⠈⡌⠱⡄⠠⣌⠈⡗⠢⣄⠀⢀⣀⠀⠤⢬⡡⡇⡠⡁⡄⠀⠈⠉⠀⠉⠈⠀⠉⣞⡦⡀⠀⠀⠀⠾⢅⣤⠀⡇
⣿⡋⠒⣙⠀⢸⠀⠀⢸⠀⢰⠷⠶⠷⠦⢬⡑⡗⠒⠀⠀⠈⠛⠒⠒⢲⠣⣏⠗⣇⠇⠀⣯⠋⣭⢫⣽⣽⠀⢻⡗⠁⠀⠀⠀⠀⢀⣹⣄⣇⡀
⣿⡷⠒⠊⠂⠁⠒⠊⣛⠀⢸⣀⣀⣁⣉⣉⣉⣙⣲⣦⠀⠀⠀⠀⠀⠈⢩⣟⣁⣑⡆⠀⢳⡿⠛⠛⢾⣿⠁⠈⠀⠀⠀⠀⠀⠀⣾⠛⣷⠓⢻
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⣮⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⣭⠄⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡼⣄⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⡠⢷⡄⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢛⣻⠀⢸⡃⠀⠄
⠀⠀⠀⠀⠀⢠⣤⣶⣾⠿⡿⡜⠁⠀⠀⠀⠀⠀⠙⠻⡞⠊⢰⠛⠃⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⠀⢸⢣⠀⡆
⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠘⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⠀⠀⠀⠀⠀⠀⢀⡻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡘⢻⡿⠃⠀⠀⠀⠀⠘⠉⣿⠁⠀⠀⠀⠀⠀⠀⣀⡐⣙⠃⢴⠒⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠠⠀⣀⡀⣎⣀⡞⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠐⠾⠚⣿⡾⠓⠛⠻⠆⠀⠀⠀⠀⠀⣹⡄⠥⠤
⠀⠀⠀⠀⠐⠤⠌⠠⠭⠭⠤⠄⠸⠸⠄⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠐⣇⣀⠔⣶⣶⠀⠀⠀⠀⠀⠀⢸⠂⠀⢘⡥⠤⢰⣂⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⣇⡀⠀⣽⡿⠀⠀⠀⠀⠀⠀⠈⠆⠀⢈⡁⠀⠀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⢠⣷⣆⡀
⠀⠀⠀⠀⠀⣴⣽⡿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡁⣄⣬⠇⠀⠀⡿⡇⠀⠀⠀⠀⣤⣄⣤⣤⣟⣾⣷⠀⠀⠀⠀⠠⣿⣿⡿⠯⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠙⠻⢿⣯⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢤⡀⠨⠄⠘⢹⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢧⣂⡄⠀⠘⣿⣾⣾⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⢥⠊⠀⠈⠻⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠠⠀⠀⢠⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢸⡅⢀⠜⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡂⠈⠉⢀⢩⢹⡇⠀⠀⠀⢀⢠⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⡟⠀⠀⠸⣇⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣚⣟⣿⢧⠀⣐⠲⢃⡼⠀⠀⠀⢠⡘⢧⡿⣾⠆⠦⡄⢱⡄⠀⠀⠸⡁⣴⠀⠀⠀⠀⠛⣳⠀⠀⠀⢀⣿
⡄⠀⡤⠐⠒⢺⡉⠙⢁⡞⡀⠀⣷⠻⡤⣄⡀⠀⠀⢳⠈⢠⡏⠀⠀⠀⣸⠃⠀⠀⡄⠀⢸⡇⠀⢠⣷⣵⡿⠤⢤⣤⣼⡟
⣇⠈⣇⠀⣠⠞⣿⠀⢸⡄⣙⣦⢈⡆⠵⠀⠙⢦⡀⢸⠀⢸⡇⠀⠄⢠⡟⠀⠀⢰⠀⠀⣾⠙⢲⣤⣉⠉⣱⠀⠀⢠⡿⣧
⠘⣆⠘⣾⠃⠀⣿⠀⠀⡇⠀⠈⠿⠁⠀⢀⣲⠀⡻⡏⠀⢸⡇⠘⢠⡿⠀⠀⠀⡆⠀⣰⠇⠀⠈⣇⠈⠉⡟⠀⠀⣾⠁⢸⣆
⠀⠘⣆⠘⣦⡤⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⡎⠀⡇⠀⢠⣿⢀⡿⠁⠀⠀⡸⠀⢀⡟⠀⠀⠀⣙⠦⠊⠀⠀⣼⠃⠤⢚⣿
⠀⠀⠸⣆⠈⢧⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢸⣏⡾⠁⠀⠀⠠⠁⠀⡾⠁⣠⠔⠊⠀⠀⢀⣤⠞⠁⠶⠚⠋⢻⡀
⠀⠀⠀⡏⢧⠈⢻⣄⢀⣿⠄⠠⠤⢒⣢⠴⡟⠁⠐⠦⠤⡟⢹⠇⠀⠀⠀⣤⠄⠚⠓⠉⠀⢀⣠⡴⢞⡫⠤⣔⠂⠀⠀⠀⣸⠃
⠀⠀⠀⣷⠈⢷⡀⢻⣾⠃⠀⠈⢉⣁⣤⣴⣇⠀⠀⠀⠀⣃⠽⠦⠒⢲⣖⣥⡀⢂⣤⠖⠋⠉⢀⡴⠋⠀⠀⠀⠀⠀⢀⡴⠋⠐⡒⠒⠒⢶⡄
⠀⠀⠀⢹⠀⢸⣷⡄⢻⡆⠀⠊⢁⣠⠴⠖⠛⠛⠋⠉⢻⡄⠀⠀⠀⠀⢘⠛⠚⣏⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⣴⠟⠓⠒⢲⡈⠳⠶⣉⣉⣑⣢
⠀⠀⠀⢸⡇⢼⠀⢘⣿⣇⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢸⡇⡄⢻⡆⢀⠞⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⢸⣗⣠⡤⠴⠶⠛⠉
⢰⣶⢤⡘⡇⢸⡖⠉⢻⡟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡇⢿⣇⠈⣾⡄⢸⡇⢁⠘⣿⡀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⣀⣤⠶⠚⠁
⠀⠁⠀⠉⠻⣌⣷⡀⢸⡟⠹⢦⡀⠀⠀⠒⣾⠀⢰⠀⣇⠸⣿⡞⠹⣇⠈⣧⢸⡀⠹⣇⠀⠀⠀⠀⠁⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠁⠢⡈⠻⡇⢸⡇⠢⡈⠳⣄⡴⠞⣿⠀⢸⠀⢻⡀⢻⡇⠀⢿⡀⢻⠈⡆⠀⠙⣶⠒⠀⣦⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠈⠢⣄⠼⢷⣒⠳⣄⠙⣧⡀⢿⠀⢸⠄⠈⠃⢀⡷⣶⠚⠃⣶⠀⣠⠊⡐⣽⠀⢠⡟⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠉⠳⣌⢦⠈⠻⣄⠀⠀⠆⢀⣠⠋⡘⣹⠀⠀⣿⠠⠃⠐⣰⡏⠄⢚⡃⠄⢺⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠛⢷⡄⠙⢷⡀⠔⠩⠃⢰⢠⡟⠐⣼⡣⠔⢒⠩⠅⣐⣬⡴⠶⠛⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠙⢦⣈⣽⠐⣄⢠⠧⠿⢓⠩⠁⣐⣨⣴⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡘⠟⠀⣈⠀⢐⣨⣥⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢼⣿⡿⠋⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣽⣧
⠀⠀⠀⠀⢀⡶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡀⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠘⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⠀⢀⡤⠖⠛⣇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠻⣿⣥⡞⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⣿⣿⣷⣄⣠⣤⣾⠟
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠘⠟⢻⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡟⠁
⠀⠀⠀⠠⣾⣀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠻⣦⣄⣀⠀⠀⠀⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣤⣤⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠿⢻⣿⣿⣿⣿⠘⠿⣦⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣾⠳⣄⠀⠀⠀⠀⠀⢹⣿⣷⣶⣦⣾⣿⣿⣿⣿⣷⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣿⣠⡏⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣷⣾⡿⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣰⣦⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣿⣿⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⡿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⢾⣿⣿⡶⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣼⣿⣿⣷⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⠤⠠⠤⠠⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠂⠉⠀⠀⣀⠀⣀⢀⣀⠀⠀⠀⠙⠀⠤⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⣀⠠⠆⠉⠁⠁⠀⠀⠀⠀⠈⠉⠐⠆⠀⠀⣠⠞⢁⡾⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠀⠠⠆⠁⢀⣠⠴⠒⠛⠉⠉⠉⠃⠀⠀⢀⡤⢤⡞⠁⡀⢸⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠚⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠚⣏⠙⢲⡉⠀⡴⠃⠀⠐⠦⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠞⠀⢠⠉⠀⣰⠋⠀⠀⠀⠀⠀⣠⢾⡙⢦⡈⠳⣄⠉⢺⡁⠀⢀⠀⠀⠇⠀⠘⣂
⠀⠀⠀⠀⠀⠘⠀⢀⡌⠀⣸⠃⠀⠀⠀⠀⣠⢾⠙⢆⠙⢦⡙⠂⠈⠓⢦⠇⠀⠘⣆⠀⠈⣃⠀⢙⡀
⠀⠀⠀⠀⠀⡋⠀⢰⠀⠀⠇⠀⠀⠀⡠⢎⠹⣄⠳⢄⠙⠂⠀⠀⢀⡴⠋⠀⠀⠀⢹⠀⠀⢙⠀⠈⡅
⠀⠀⠀⠀⠀⡃⠀⢰⠀⠀⠀⣄⡠⢾⠀⠈⠳⣌⠓⠀⠀⠀⢀⠄⠀⠀⢀⡀⠀⠀⢸⠀⠀⢘⠀⠀⡅
⠀⠀⠀⠀⠀⣃⠀⠰⠀⠀⠀⠘⢧⠀⠳⣌⠳⣄⠀⠀⢀⡴⢋⡴⠛⣉⣩⣉⡙⠲⣄⠀⠀⣘⠀⢀⡅
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⠚⣳⣄⠈⠳⢬⣧⠴⠋⢠⠏⢠⠋⠁⠀⠀⠙⢦⠘⣆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⠲⣤⠊⣠⠞⢡⠜⠓⠦⣴⠋⠀⠀⢸⠀⢿⣠⢤⣄⡤⣄⣸⠀⢹⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠳⣄⠜⠁⢠⣀⠀⠈⠁⠀⠀⠸⣆⠸⣭⡛⠵⢙⣫⡗⢀⡏⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣬⡗⠀⠀⠈⠙⠲⠦⠤⠤⠤⠘⠦⣄⣙⠛⢛⣀⡴⠋⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠠⠄⢀⡀⢀⠀⢀⡀⣀⠀⠉⠉⠉⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢠⣄⢀⠀⠀⠈⠁⠉⠁⠀⠀⢀⡄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠛⠀⠛⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠸⠦⠀⠀⠀⠀⠀⣀⠴⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⠀⢤⠀⢀⡀⢀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠢⡀⠀⠄⠀⠀⠀⠀⠀⠃⠀⠆⠀⠤⠀⢠⠀⢈⠁⠘⠁⠘⠃⢐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⢠⠃⠜⠀⠀⠀⠀⠀⠀⠋⠐⠂⠀⠦⠀⠤⠀⢨⡁⢈⡀⠘⠁⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠐⠊⠹⡖⠀⠀⠀⠀⠀⠠⣂⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠈⠁⠠⠀⢀⠶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠐⠤⠂⣓⣀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠐⠒⢖⠠⠤⠀⣀⠀⠀⠀⣁⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⢸⠃⠸⣧⠀⠠⠀⠀⠀⠀⠀⠈⠀⣀⡀⠈⠁⠀⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠚⠀⠄⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠴⠒⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡲⠞⢺⠉⠁⠤⡀
⠀⠀⠀⠀⠀⠀⣔⣮⠇⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠂⠀⠓⠦⡀⠈
⠀⠀⠀⠀⠀⡐⠉⠃⠀⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠀⠀⠂⠀⠀⠀⠈⠓⠂⠉⠉⢢⠄⠍⠉
⠀⠀⠀⠀⠀⠃⢀⣴⣤⢀⠀⠀⠀⠀⠀⢠⡤⢤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠄⠖⠓⠒⠉
⠀⠀⠀⠀⠀⠃⠘⠩⠞⠂⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⣶⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠄⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠉⠉⠉⠉⠉⠁⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡖⠐⠒⠢⠤⠄⠀⣀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠖⠲⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⢄⣀⣀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠀⠀⠀⠀⠀⠀⠻⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⡟⠉⠋⠀⠈⠑⠢⣄⠀⠀⠀⣠⠜⢻⠰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⠁⢸⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠇⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀⠠⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⡀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡌⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⠀⡀⢾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢐⣇⢀⡤⠀⠒⠂⢉⡧⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⡶⠶⠁⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢀⡤⠚⠉⠀⠀⠀⣀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⡸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⣄⠀⢀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠁⢀⠾⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡆⣀⡜⠀⠀⠀⠀⠀⠀⠸⡆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⠋⠀⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⠂⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠸⣄⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣠⠜⢧⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠠⠔⠒⠈⠣⣄⡠⠷⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠫⡄⠀⠀⠀⠠⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺
⠀⠀⠈⢦⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⠠⠟
⠀⠀⠀⠀⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⡀⠤⠤⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢀⡀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣾⣄⣈⡀⠀⠀⠴⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣷⣴⣆⡠⠶⢶⣦⣄⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠀⠀⠀⢀⣶⣿⣿⡿⠿⠟⢹⠁⠀⠀⠉⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠘⣧⠀⠀⠀⠀⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⢀⢜⣿⣿⡇⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠟⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠄

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⣾⣟⣛⣿⣦⢠⣿⣿⣿⣿⡆⠀⠀⠀⣀⣤⠀⢀⡀⠀⠀⠀⠀⠀⢠⣶⠒⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣶⠒⠀⠀⠀⢀⡀
⠀⠀⢿⣯⣾⣿⡟⠸⣿⣷⣿⣿⠇⠀⠀⠐⠋⠛⠋⠛⠛⠛⠛⠿⠛⠀⠀⠃⠙⠛⠛⠛⠛⢿⣻⠟⠃⠀⠛⠛⠿⠛⠃⠐⠓⠚⠛⠿⠛⠛⠛⠛⠛⠻⠟⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣤⣄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⣴⣶⡄⣠⠞⠁⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣠⣄⢰⠋⠀⠄
⠀⠀⠀⢀⣬⣿⣿⣁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣯⡾⠋⠀⠀⠀⠀⢠⣶⢿⣶⠀⣰⣾⣷⣆⢀⣶⣿⣶⡄⠀⠀⠀⠀⣿⡿⠞⠃⢤⠄⠂
⠀⠀⠀⢸⣿⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⣸⣿⢘⣿⡇⣿⡇⢹⣿⢸⣿⠁⣿⡇⠀⠀⢠⣾⡿⣤⣀⣠⣴⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣤⣤⣤⠀⣿⣿⡟⠀⠀⠀⠀⣀⣻⣿⣿⣿⣡⢿⣿⣿⡟⠘⢿⣿⣿⠃⠀⠀⢺⣿⣁⠈⠉⠁⢰
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠌⠀⣄⠈⡟⠻⠀⣿⣿⣇⠀⠀⠀⠸⣿⣿⣿⣿⡟⠁⠈⣿⣿⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⡿⣷⠀⠀⢸
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠡⢀⢁⡠⠁⠠⠀⣿⢿⡷⠀⠀⠀⢰⣿⣿⡿⠿⠇⠀⠀⠿⠿⠀⠀⠸⡿⠇⠀⠀⠀⢠⣿⠃⢻⣇⠀⢘⠀⠀⠀⠠⠀⠘⠀⠈
⠀⠀⠀⠉⠉⣉⡀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢀⣀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⢀⡀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠈⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢿⡿⢂⡾⠋⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⢀⣻⡿⠃⠀⣰⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⠁⠀⣠⡄⠀⠀⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⠛⠾⣏⣈⣋⣉⣹⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠳⠾⠋⣴⣶⣴⠿⣿⣿⡿⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⣿⡿⠛⠿⣿⠃⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣶⣶⣶⣿⣿⣶⣶⣿⣿⣿⣶⡆
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⢠⣋⡆⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠿⠻⠇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣷⣤⣶⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠘⠛⠛⠛⣋
⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⡀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠀⠀⠰⣿⡷⠀⠀⢺⣿⠇
⠀⠀⠀⠀⢠⣿⡞⠿⠗⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢷⣼⣷⣦⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣾⣿⡶⠶⣤⣤⣴⣿⣆
⠀⠀⢀⣴⣿⢿⣿⣶⠞⠀⠀⢀⣴⣾⡿⠏⠻⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠋⠀⠙⠲⠟⢻⣿⣿⡀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢸⣿⣿⠷⢤⡄⠀⢀⣿⣿
⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⠹⠧⣤⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⡿⠆⠀⢸⣿⣿⠀⠀⠀⠰⢿⣿⠁
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠈⣿⡄⢰⡟⣙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢻⡇⠘⢷⣶⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡿⢿⠇⠀⠀⠀⠀⠀⠺⠇⠀⠀⠸⠟⠿⠀⠀⠀⠀⠘⠿
⠀⠀⠀⢈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣧⣄⣤⣤⣤⣴⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠉⠻⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠙⠿⠛⠀⢸⣿⣿⠋⠀⠀⠀⠈⠙⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⡾⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⢀⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣤⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⣰⣿⣽⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠟⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⣿⡏⢻⣿⣿⣿⣿⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣀⣠⣾⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣇⠀⡿⣿⣿⡿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⢸⣿⠁⠀⢀⣿⠃⠀⠀⠀⠀⠤⠶⠶⠴⠶⠶⠶⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⣼⣿⣄⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⣠⡿⠋⣀⣻⣿⣿⣿⣇⡈⠻⣷⡀⠀⠠⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢼⡿⠿⠟⠋⠉⣿⣿⡿⠉⠛⠻⠿⣿⠆⠀⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠟⠛⠛⠻⠶⢦⠀⠀⠨⡿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⡟⣿⡇
⠀⣴⣿⣿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢿⣿⣿⣿⠀⣿⠇
⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣯⠀⣿
⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⠀⠀⢀⣠⡿⢠⣿⣿⣿⣿⣄⢿⣄⣀
⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢌⣷⣤⣿⣻⣭⣶⣿⣿⣿⣿⣿⣿⣷⣭⣝⣿⣦⣦⣤
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷

⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⠿⣿⣿⠿⠿⠿⣿⣿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣿⣾⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠚⡆⠀⠀⠄⠓⠁⠑⠀⠐⠀⠠
⠀⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⡇⠀⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠆⠀⠀⠑⠒⠊⠀⠀⠀⢀⣦⠀⢢⢀⠢⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠎⠀⠺⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⣄⡀⢀
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣁⠰⠄
⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠆⠀⠀⠀⠰⠇
⠀⠀⠈⠉⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠊⠔⠁⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⡄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠀⠀⠈⠈⡀⠀⠀⠀⠠⠼
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠘⠀⡃⠈⡀⠄⠀⡀⠁⠐⠀⠀⠠⠀⠀⢃⠀⠀⠀⡄⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠠⠄⠄⠈⠠⠃⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠊⠑⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠄⡰⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠢⣀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⠃⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢐⠆⠀⠀⠀⡀⢂⠀⠀⠀⣮⣁⣠⣤⠆⠀⠀⠀⡆
⠀⠀⠀⠀⡄⢘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠀⠀⠀⠈⢀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⠀⠶⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⡉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠆⠐⠚⠛⠓⠀⠀⠀⢈⡉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠄⠠⠂⠈⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢀⠴⡁
⠀⠀⠀⢦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠐⠀⠀⢠⠀⠀⠀⠤⠴⠂⠀⠀⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣶⡀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡧⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣇⢀⣠⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣠⣴⣿⡿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠻⠿⠛⠉⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠤⠤⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⣀⣹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠃⠀⢀⠀⠈⡇
⠀⠀⠀⠀⣀⡀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠚⠀⠀⡇
⠀⠀⢰⠋⠁⠱⣄⠀⡇⠀⢀⡀⠀⠸⡟⠲⣄⠀⢠⡴⠒⠋⠹⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇
⠀⠀⠈⣇⠀⠀⢈⡟⠁⠀⡞⠱⣄⣠⠷⠀⢸⡄⠀⢧⠀⠀⠀⡗⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠙⠲⣄⠈⢧⠀⠀⢻⡀⠈⠀⠀⠀⣼⠁⠀⠈⠓⡶⠚⢧⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⣰⠛⣆⠈⣧⡤⡄⠉⠒⠶⡄⠀⠘⣆⠀⠀⠀⣧⠀⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢰⠃⢠⡞⠋⠁⠀⢹⠀⠀⠀⠹⡄⠀⠘⣆⠀⠀⢻⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⡏⠀⠀⠳⢤⡀⠀⠙⠳⣄⠀⠀⠹⡄⠀⠘⣆⠀⠘⡆⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⢼⣿⣿⣤⡀⠀⠀⠀⢀⡾⣿⣿⠶⠂⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢠⢿⣿⡇
⠀⠀⠀⠀⡸⣿⣿⡆⠉⠃⠀⢀⠞⠀⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣏⠙⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⢠⠃⢼⣿⡛⣄
⠀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⣾⠿⣧
⠀⠀⠀⢠⡞⠁⠀⣿⠀⠀⠀⠀⢀⡾⠁⠀⣷⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⠀⢠⡟⠁⢸⣇⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⢠⡾⠃⠀⢻⡄
⠀⠀⢠⣟⠀⠀⠀⠸⣧⣀⠀⠠⣾⠁⠀⠀⠸⣦⡀⠠⣿⡁⠀⠀⢹⣆⡀⠀⣾⠀⠀⠀⢹⣄⠀⠘⡟⠁⠀⣯⡀⠀⢠⣞⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢸⣿⣿⣿⣿⣿⣿⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⢸⣿⣿⣿⣿⣿⣯⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣿⣿⣥⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣽⣿⣷⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⡟⣿⣿⣿⣿⣿⣿⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⣠⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⢻⣿⣿⣿⡇⠘⣿⣧⡀
⠀⠀⠀⢀⣼⣿⠿⠁⠀⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣧⡀⠀⠀⠀⢀⣴⣿⠟⠁⢠⣿⣿⣿⣿⣧⡀⠈⢿⣿⣄
⠀⠀⢰⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡄⠀⣰⣿⣿⣥⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣬⣿⣷⡄
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠀⠀⠀⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠛⢳⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⣄
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⣎⢧⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣀⣼⣾⣇⠀⠀⡾⠁⠀⣠⠞⢳⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣿⣿⣿⣿⣿⡆⠐⣇⢀⡜⠁⠀⣸⠃
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣿⣿⣿⣿⠇⠀⠈⠻⠤⠤⠖⠃⠀⠀⣀⣀
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠉⢳⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠞⠉⠉⣹⢦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣦⠀⣠⠴⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡞⠀⠀⣏⠀⢠⠞⠁⠈⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣸⠁⠀⠀⠙⢶⣋⣀⣠⠼⠁⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣇⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⠒⠶⠒⠋⠀⠀⢰⡇⠀⠀⠙⠒⠶⠒⠋⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠲⣤⣀⣠⡴⠂⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⣄⣀⣀⣀⣀⣀⣀⣤⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣾⣧⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠘⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈
⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠷⠆⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂
⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡾⠀⠀⠀⢠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣾⣗⡄⠀⠠⠀⢀⠀⠘⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⠟⣽⡻⡿⠿⣿⣿⣯⡼⠀⠂⡸⠗⠀⠀⠁⠀⠀⠠⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢧⡿⢨⠏⣼⡇⢰⠏⢸⠓⡇⢠⢒⡐⠶⢤⡀⠂⢀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⡎⣌⣞⢀⡏⠀⡎⢀⡇⠸⣪⣗⢡⠀⠀⠀⠛⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⡿⢳⣾⠧⣼⣁⣸⠁⣾⠇⡇⣿⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⡿⢰⠇⢠⡏⢸⠏⢳⡏⣿⣿⣾⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⡟⠈⡜⢀⡟⠀⡾⢠⡟⢠⢇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣼⢁⣸⡇⣼⠀⣰⠃⣾⡁⡞⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣱⡇⣼⡟⢸⢟⢣⡏⣼⡟⣸⢱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⣜⣙⣿⠳⡾⣤⣼⢠⡟⢩⢇⣿⡿
⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣯⣽⣿⣿⣷⣶⣇⣽⠉⡟⣾⣿⠃
⠀⠀⠀⠀⠀⢰⡹⣧⣉⢾⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠙⢮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠻⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⡀⣀⢀⠀⢀⠀⣠⡀⣠⠀⣀⠀⣀⢀⢀⡀⡠⢀⡀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠃⠁⠋⠛⠀⠘⠀⠋⠃⠋⠀⠛⠃⠃⠘⠛⠃⠋⠐⠙⠃⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡠⠎⠣⣄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠁⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⢦⡔⢩⠔⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣀⣶⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠘⠬⠵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠻⠶⠟
⠀⠀⠀⠀⠀⠀⡉⠉⢉⡉⠉⢉⡉⠉⢉⡉⠉⢉⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⣉⡉⠉⣉⡉⠉⣉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⡈⠉⣁⠈⢉⠁⢈⡉⠁⡈⠉⢁⠀⠀⠀⠀⠀⠀⠀⢀⡉⢉⡀⠉⣉⠀⣉⠉⢀⡉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠠⠄⠠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠤⠀⠠⠄⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠀⠀⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⠀⠀⣴⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣀⣀⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠃⢀⣾⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠶⠦⣞⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⢠⣦⠀⠀⣀⡤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠱⣿⣶⣶⣶⣿⣿⣿⣿⡿⢫⣿⣿⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⡿⠋⢠⣿⣿⣿⣿⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠻⠿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠳⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⡆⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⡾⠛⣿⡀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⣴⡟⠻⣧
⠀⠀⠀⠘⢷⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣴⠟



⣿⣿⣿⣻⣛⣟⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣿

⠀⢠⡤⠀⠀⢀⠀⠀⠀⢠⠀⠀⢰⠆
⠀⠀⠀⢀⠀⠈⠀⠀⠀⠈⠀⠀⢀⠀⠀⠒⠀⠠⣤⠖
⠠⣤⠀⠉⠀⠀⠀⠘⠂⢰⡖⠀⠸⠇⠰⢤⠄⠀⢄⡀
⠀⠀⠤⣤⠆⢠⡄⠀⠀⠠⣤⠖⢀⠀⠃⠀⠀⢠⠈⠁⠀⣶⠆
⠰⢤⣤⠄⠀⠐⠃⢀⡀⠀⠀⢠⡄⠃⢀⡈⠁⢸⡧⠀⠐⢮⠆
⠀⡌⠀⠀⠀⣦⠀⠀⠀⢰⠀⠘⠇⠀⠿⠀⠀⠀⠁⠀⢄⡀⠀⠸⠃
⠘⠻⠀⠀⠀⠚⠀⠀⢀⡀⠀⠀⢰⡴⠂⠘⠋⠀⠄⠀⠀⠀⢤⠠⣄
⠀⠀⠀⡀⠀⠀⠀⣀⠈⠓⠀⢀⠈⢁⡀⠀⡀⠀⣀⠐⣶⠀⠙⠀⠀⠀⢂⣰⣴⣆⣤
⠀⠀⠀⠙⠁⠀⠀⠈⠀⠀⡄⠀⣀⣘⣋⠀⡙⠀⠋⢠⡁⠀⡶⠀⠀⠚⣨⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠶⠒⠀⠀⠈⠋⠁⠈⠓⠀⣦⠄⣷⣤⠀⢠⡄⠙⢛⢚⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⢸⡄⠀⢠⠷⢰⡅⠀⢀⡀⠸⢃⡀⠸⠂⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡴⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣈⡀⢐⣨⣭⠤⣶⣾⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠰⠃⣀⡉⠀⣀⠀⠉⣴⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⡛⠀⠀⠘⠏⢀⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡄⠀⠤⢠⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⢹⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣰⣷⡾⣿⠛⣁⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣙⢿⣿⣾⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠟⣿⣿⣃⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣾⣦⣤⣤⣾⣤⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⡟⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣵⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠾⠿⠯⡿⠟⠛⠻⠿⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠧⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣯⣯⣿⣭⣿⣿⠉⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⡀⣠⣾⣿⣿⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡿⣿⣿⢿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⣿⣿⣿⣾⣽⣯⣿⣾⣿⣿⣿⣾⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠦⠬⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢴⡴⠂⠀⠇⠀⠀⢨⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠒⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠹⢣⡀⠓⠃⡀⠀⠀⠙⠇⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡁⢸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡀⠋⠖⠁⠀⠀⠀⠁⠀⠀⢀⠀⠳⠀⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠘⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠛⠋⠀⣤⠑⠀⠀⡀⠈⢂⠘⠛⡆⠐⡉⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢈⠢⡄
⠀⠀⠀⠀⠀⢄⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠾⠁⠄⢤⢳⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠦⣤⡅⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠁⠂⡀⢸⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀
⠀⠀⠀⠀⡹⠃⣋⠀⠀⠀⠀⠀⠀⠻⡘⠀⣠⠖⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠩
⠀⠀⠀⠀⠀⣒⡋⠀⠀⠀⠀⠀⠀⡠⠋⠈⠁⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂
⠀⡀⠀⠠⡋⠉⢵⠒⠒⠐⠂⠀⠚⢧⡀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣉⣀
⠀⣿⠢⣀⠀⠀⡬⠂⠀⠀⠀⠀⠀⠀⢹⡢⠖⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉
⠀⣿⠀⠈⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠀⠀⠀⢀⠀⣂⡀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠚
⠀⡇⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠘⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠄⡀
⠀⠁⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⠀⠀⠀⠈⠒⠀⢀⠀⣀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠐⠄⠲⢀⠀⠀⠀⠀⠀⠀⢈⣀⡶⠀⢠⢰⡄⠁⠀⠊⡀⠀⠈⠉⠀⠀⡈⠁⠀⣀⠀⠀⠀⠀⠀⠴⠤⢄⡀⠢⠀⠀⠀⠀⢸
⠀⣠⢠⢶⣶⣤⣄⡀⠀⠀⠀⠀⢀⠌⠁⠠⠠⡀⠀⠈⠉⠈⣳⡀⠀⡠⠁⠀⠀⠀⠙⠲⣌⠉⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⠄⠀⠀⢧⠚⠀⡀⠀⠀⠈⠀⠀⠘⠒
⠀⠛⡹⢢⣉⣿⠀⠉⠛⠒⠒⠠⠈⡄⠀⠠⠄⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣤⡸⣤⠂⠀⢀⠀⡆
⠀⢾⠴⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡁⣀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣌⠁⠀⠀⠀⠀⢣⣤⠚⠀⠞⠁⠈⠀⠀⠴⠀⠃
⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⣼⠀⠀⠀⠁⠀⠙⠘⠟⠆⡀⠀⠠⠄⠠⠄⠀⠀⠀⢀⡐⠀⠀⠒⠢⠘⡀⠀⠀⠘⠂⠠⢸⡀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠤⢻⢄⠀⢠⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣢⠐⠤⠀⠈⠀⠀⠀⢀⠀⠡⠠⠐⢦⣀⠰⠆⠀⠉⠁⠀⣨⡄⠀⠀⠐⠂
⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠐⢿⡈⢦⡘⠂⠀⠀⠠⠀⡆⠃⠀⠀⢀⣀⣀⣀⣀⢀⠐⢄⡀⠀⠀⠀⠀⠁⢡⠀⠈⠃⠀⣠⡀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠒⠂⢤⠀⠓⠀⠱⣄⠀⢤⣄⡄⡇⡖⢠⡆⠹⢥⣔⡤⠭⠙⠃⠀⠉⠂⢄⠀⠀⠀⢠⢀⡀⠀⠰⠯⢶⠛⠅⠰⠁⠀⠈⠁⠠
⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⢦⡀⠈⢉⡃⢍⢀⡀⣠⣤⣄⠀⢀⠀⢀⠀⠀⠂⣠⠀⠂⠀⠀⠀⠀⠢⣄⠀⠈⠉⠉⠀⠀⡀⠀⢀⣀⠀⠑⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠻⡇⡾⠦⠤⠌⠖⠊⠁⠈⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⣄⠀⠈⠓⠒⠒⠒⠀⠈⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⢰⠀⠀⠀⠀⠑⠂⡀⠀⢀⡀⠀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡔⠒⠀⠀⠈⠉⠓
⠀⠀⠀⠀⢀⠶⠆⢀⣤⡀⣠⠀⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠓⢄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠓⠒⠐
⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⡀⠀⣀⠀⢀⠀⡀⠀⣀⣀⢀⢀⠀⡀⠀⣀⠀⢀⠀⡄⡀⠀⡀⠀⠀⢀⣀⡀⢀⡀⠀⣀⠀⠈⣀⣀⢀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⡶⠶⠶⠶⢶⡴⠶⠶⠦⣤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣀⣀⣀⣛⣀⣀⣀⣘⣃⣀⣀⣀⣋⣀⣀⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⡀⠀⠀⢸⡇⠀⠀⢀⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢙⡴⠋⠉⠉⠙⢶⡋⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣀⡀⡟⠀⠀⠀⠀⠀⠀⢻⣀⣀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠀⢧⠀⠀⠀⠀⠀⠀⣼⠀⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡬⠳⢤⣄⣠⡤⠞⢥⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠀⠀⠀⢰⡆⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠶⠶⠶⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⠀⠄⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⠀⠀⠀⢀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⢀⡀⠀⠀⡼⠀⠂⠀⠀⢀⠍⢻⠆
⠀⠀⢸⣿⣶⣤⣤⠀⠀⠟⠀⠀⢀⠔⠊⢶⡴⠋⠝⠀⠀⠀⣷⣦⣤⣤⣴⣾⠀⠀⠀⠪⠙⠆⠃⣰⠃⠀⠀⠀⠀⠀⠁⡞
⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠈⡁⢀⡤⠞⠁⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠈⠁⠀⠀⢀⣣⡀⣠⠟⠉⠀⠀⠀⠘⠀⢃⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣮⣀⡀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢉⡉⠉⠀⠀⠀⠀⢹⣦⡀⠀⠈⠉⠛⠲⠦⣶⡿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⠀⣄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⠐
⠀⠀⠀⠠⠀⠀⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠄⠀⠀⠜⠀⠀⠀⠩⠵⠀⠀⠀⠻⠀⠰⠀⢢⠀⠀⠀⣴⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⣾⣿⢿⡆⠀⢠⠀⡈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡂⠀⢰⢦⠴⣶⠙⢰⠳⡄⡏⡔⠱⢰⡆⠐⠀⠀⠀⢠⡄⠀⡠⠀⢐⡇⠀⠀⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⡞⠂⢸⠀⡇⡇⡇⢠⣸⢼⠀⠀⠀⠀⠰⠀⠁⠽⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠚⠋⠐⠓⠘⠊⠋⠈⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣖⣲⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠁⠀⠐⠈⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢄⡴⠊⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⡜⡖⠤⠤⢤
⠀⠀⡀⠀⢀⡋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠁⠀⠀⠀⠀⠀⠈⠑⢒⣂⠛
⠀⠀⠈⠒⢭⡰⢂⡌⠰⣀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⡤⢈⡥⠊⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠐⠅⠈⠋⠁⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠰⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠈⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡖⣋⠐⠄
⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⣰⣄⠙⢦⠀⠀⠀⠀⠀⠱⣀⠠⠌⣿⠻⠗
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⡀⠀⠀⠶⠆⠀⠀⠀⠈⠙⠓⠀⡐⠀⠀⠀⠀⠀⠘⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣴⣶⣶⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢸⣿⣿⠟⣋⣛⢿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⣿⣿⣿⠸⣿⡿⢨⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⣤⣤⡾⢛⣵⣿⣿⣿⣿⣷⡆⣶⣿⣿⢟⣥⣶⣶⣦⡝⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣭⠍⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟⣵⣿⣿⣿⣿⣿⣿⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣵⣶⣝⢿⣿⠿⣛⡻⠏⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⡇⣴⣿⣿⠿⣳⡌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡅⣿⡿⣩⣾⣿⡗⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣤⣬⣭⡛⢿⣿⣿⣿⣷⠌⠻⠿⠿⢛⣵⣌⠻⠿⠿⢛⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⠛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣛⠻⠿⣿⣿⠇⣘⣛⣛⣛⣛⣛⣛⣋⣥⣴⣶⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣶⣿⣿⣿⣷⣦⣴⣾⣶⣴⣶⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⣀⠤⢄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢳⠀⠀⠀⢰⠇⠀⠈⢣⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣌⠁⠀⡾⠀⠀⠀⠸⡎⠁⢀⡜⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠑⠈⠀⠀⠈⠉⢦⡈⠒⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣾⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⠉⢈⡻⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠈⠛⠿⢶⣶⣤⣤⣤⣠⣤⣤⣤⡶⠞⢿⣇⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠉⢹⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣄⠀⣹⠆
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡴⠚⠋⠛⠓⣶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⠉⢳⡀
⠀⠀⠀⠀⠀⣰⠟⠀⣴⠏⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢳⡀⠙⠳⢤⣀
⠀⠀⠀⣠⠴⠋⢠⡞⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠳⠦⣄⠉⠛⠶⣄⣀⣀
⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠉⠳⠦⡀⠀⠈⠙⢦⣄
⣶⡋⢀⣴⣄⢰⡏⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣤⣄⣈
⠈⠛⠋⠁⠉⠋⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠶⣦⣤⣄⠀⠀⠀⠀⠀⠹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠠⠀⡈⢿⣆⠀⠀⡀⠄⢀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠻⠷⣦⣄⣸⣿⣶⣀⣠⡶⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠀⠀⣠⡾⠉⠛⠁⠻⠟⠻⣦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠉⠁⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣀⣀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⠿⠟⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⢀⣀⣀⠠⣶⡿⠉⠉⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠿⠂⢀⣀⣀⣀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⠋⠈⠙⠻⣦⣤⠀⠀⠀⣤⣤⡴⠛⠋⠀⠀⠀⠀⠈⢻⣿⠿⢿⣶⣦⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠉⠉⠉⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⣿⠇⠀⣿⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢴⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⣿⡇⠀⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠈⡇⠀⢠⣿
⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠛⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⣿
⠀⠀⠠⣾⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⣷⡆
⠀⠀⠀⠉⠁⠙⡿⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣤⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⣤⣤⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡖⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣥⡀⠀⠀⠀⢀⣶⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠀⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠋⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠿⠋⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠁
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠛⠋
⣀⣀⣉⣉⣉⣈⣹⣿⣿⣿⣿⣋⣁⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⢶⣦⡀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⣿⡄⠀⢸⣿⢸⡇⠀⠀⢀⣴⠟⠉⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡇⢸⡇⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⡇⢸⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡇⢸⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣼⠏⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⣧⣾⠁⠘⣿⣸⡟⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠸⣷⡀
⠀⠀⠀⠀⣸⡏⢸⣿⣉⡇⠀⠀⠀⠀⠀⣏⣹⣿⠀⠘⣿⢁⣾⣷⣜⢿⡇⢸⣿⣉⡇⠀⠀⠀⠀⠰⣍⣹⡿⠀⠹⣷
⠀⠀⠀⢠⡿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣿⠟⠁⠈⠻⢿⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢻⣧
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⡄⠀⠀⠠⣶⣄⣀⣀⣀⣠⣄⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⣇⠀⠀⠀⢀⣀⠀⠀⣀⣤⡾⠃⠀⠀⠀⢠⣿
⠀⠀⠀⢹⣧⡀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣤⣄⣀⣀⣀⡀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣮⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠋⠁⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣄⠀⠀⠀⢈⣍⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠻⠟⡀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⠓⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⣶⠒⢚⡿⢿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⣹⢟⡭⠖⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⡇⠀⡏⠀⠀⠀⠀⢤⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢸⡅⠀⡇⠀⠀⠀⠀⢸⠙⢦⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠁⢀⡤⢾⡇⠠⡇⠀⠀⠀⠀⢸⡆⠈⢧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢈⡽⠶⠚⠉⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⢧⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣸⡇⠀⠀⢠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠉⠉⢹⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡶⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡞⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠙⢦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⣰⠃⠀⠀⠳⢴⡎⠙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠉⠀⠀⠀⠀⠀⣹⢶⠞⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣺⣁⣼⠁⠀⠀⠀⠀⠀⠀⡿⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⣀⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⢧⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠾⠿⠆⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢶⣼⡴⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⣯⣧⣤
⠠⣾⣷⡄⠀⠀⠀⠀⠐⠟⠻⠟⠟⠀⠀⠀⠀⠈⡽⢳⢫⡍⠀⠀⠀⠀⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⢠⣾⣆⠀⠀⠀⠀⠀⠀⠲⣶⡶⠂⠀⠀⠀⠀⠀⢶⣄⣽⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⢿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠸⣦⢸⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎
⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⠀⠀⠀⠀⠠⣦⣿⣧⡦⠀⠀⠀⠀⠀⢀⣄⠘⠃⠀⠀⠀⠀⢀⣾⣿⣆⡀⠀⠀⠀⠀⠀⣸⣃⡀
⠀⠀⠀⠀⠐⠾⣿⡿⠗⠀⠀⠀⠀⠀⡛⠛⡃⠀⠀⠀⠀⠀⢨⡉⠉⢩⠀⠀⠀⠀⠀⠸⡀⢓⠆
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⡟⣛⣛⣛⡻⢶⣤⣾⣿⠿⠚⠉⢉⣳⣦⣀
⠀⠀⠀⠀⠀⠀⣷⢿⣙⣿⣿⣦⡙⢿⣿⣦⠀⠀⠘⢌⣻⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢻⣸⣿⣿⣿⡛⣿⡄⢿⡚⠁⠀⢤⣀⠀⢀⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⣿⡇⢸⡗⠚⠀⠀⠈⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣾⣿⣿⣷⢸⣇⣀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠴⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣽⡟⣸⠛⠭⣉⣩⠿⢇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⠻⣿⣿⡇⣿⡀⠀⠀⠀⢸⡜⢦⠀⠀⠀⠀⠀⡴⣻⠃⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⣾⣿⣿⣷⡏⡇⠀⠀⠀⠀⢷⡈⢳⣤⣤⣴⣾⠴⢿⡤⠶⠶⠖⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣤⣾⣿⡇⢷⠀⠀⠀⠀⠸⣷⣤⣸⡏⢉⠀⡀⠄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⡿⣿⡏⣧⠸⠓⢺⡋⡟⠠⢾⣅⠀⣻⠈⣀⣠⡤⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠒⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⣤⣼⣷⣙⣛⣟⢛⣿⣳⠖⠋⠙⢯⣍⣿⠽⠛⣷⣶⣶⣶⣾⣿⣶⣤⣤⣤⣤⣤⣤⡾⡉⠉⠓⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣟⠛⣿⣿⡟⣟⠁⢀⠠⠘⣏⢡⡀⠀⠄⢻⠈⠏⠀⡉⠀⡏⠀⣧⠀⢠⡇⠀⡇⠘⢆⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡛⠟⣿⣿⠿⠟⣙⣮⡽⠶⣄⡴⢿⣌⣤⣤⠶⣞⣀⣼⣀⣧⣠⣇⣀⣿⣀⣨⣇⣀⣇⣀⣀⣷⣟⣀
⠀⠀⠀⠀⡀⠤⣀⡤⠶⠛⠉⢻⠸⣿⣇⢾⠋⣉⠁⡀⢠⠈⢷⣿⠿⡟⠛⣻⡏⠀⠀⣿⣽⣿⣿⠀⢠⠏⣩⣥⣾⣿⠿⢿⣷⣾⣭⣝⢳⡄
⠀⠀⠀⠀⢺⡋⢁⠀⠀⠀⠂⠘⣇⢻⡛⡞⣇⢉⣀⣠⡤⠶⡞⠢⠐⢙⡿⣿⡇⠀⠀⣿⣿⣷⡇⠀⣿⢸⣧⣽⣿⣿⣷⣿⡿⢿⣷⣼⣇⣷
⠀⠀⠀⠀⢦⢳⡀⣰⠖⠒⠒⠛⠛⠉⢹⣗⣈⣭⣥⣠⣷⣀⡤⣖⡻⠽⣿⣿⡇⠀⠀⣿⣿⣿⠇⠀⣿⢸⠟⢿⣿⣭⣿⣿⣷⣾⣿⡛⣿⣿
⠀⠀⠀⠀⠁⠉⣋⢹⠀⠚⠀⠀⠈⠀⢸⢋⠀⠘⣯⣭⣥⣭⣿⣭⣭⣤⡭⠽⣇⠀⠀⡟⡿⢽⣀⣀⣿⠬⢥⡽⣿⣿⠿⢿⣿⣿⣷⣶⡖⣿
⠀⠀⠀⠀⠀⠀⠉⠻⠶⣶⢻⣿⡟⣟⠋⣁⣀⣀⣿⠾⣟⢻⣿⠿⢷⣤⣿⡷⣻⡄⠐⠧⠖⠲⠤⠬⣿⢸⣛⣉⡻⢿⠶⣛⣛⣻⣥⠽⣓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠽⠞⠲⠾⣭⣭⠴⠞⠛⠓⠾⠿⠶⠛⠙⠛⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠛⠛⠉⠉⠉⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⢠⣄⠀⠀⢸⣿⡇⢾⣿⣦⣰⣿⡿⠀⠀⡟⢻
⠀⠀⠀⢀⠀⡘⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⣤⢄⢿⣿⠇⠀⢸⣿⡇⠀⢹⣿⣿⡟⠁⠀⠀⡇⢸
⠀⠀⢀⣾⡞⠀⠀⠀⢀⠔⢲⠀⢀⠠⠶⠂⠀⠀⠸⢿⣿⣶⡄⠀⠀⢸⣿⡇⣰⣿⡿⢿⣿⣦⠀⠀⣧⣼
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠀⠀⠸⢿⠇⠘⠛⠀⠀⠙⠋⠀⠀⢿⡿
⠀⠀⠁⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⣔⣂⠁⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⢠⡀⢸⡇
⠀⠀⢠⠤⠤⡄⠀⠁⠀⠀⠀⠀⠀⣴⡾⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠔⠀⢥⡴⢸⠁
⠀⠀⠈⠈⠉⠁⠀⢈⠓⠶⠚⠇⢸⠋⣠⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠐⢃⠠⣴⠌⠁
⠀⠀⠐⠢⠤⠀⡆⠀⠀⢱⣶⠂⠀⠀⠫⠊⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣤⠲⠆⢽⠆⠁
⠀⢀⡅⠀⠀⠀⡇⢈⠀⢘⠦⠀⠀⠀⢰⣶⣤⣤⣤⣴⣾⠀⠠⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⢀⡀⠒⡀
⠀⠈⠢⠶⠒⠚⠁⠀⠒⠺⠕⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳
⠀⡀⠐⢀⠀⡄⠀⠀⠀⠰⢄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠃⠀⠠⠒
⠀⠻⠶⢎⣠⡼⠀⠀⠙⠒⢫⡄⠐⠊⠀⠀⠀⠀⠀⠐⢄⠀⠶⣦⣶⣄⡊⠀⠀⠀⠂⠀⠉⠁⢀⠀⠁⠘⢸
⠀⠰⣤⣤⠁⡀⠀⢀⠀⠘⢒⠀⢤⠄⠀⢀⡘⠛⢀⣿⠈⠀⠐⣫⣿⣿⡿⠀⠀⠀⣀⣤⣴⣦⠀⠀⠀⠅⡇
⠀⠀⠀⠀⠀⠙⠛⠫⣄⠠⠿⡗⠀⠀⠀⠘⠻⠖⠈⠁⢀⡀⠀⠀⠚⠁⠀⠀⠻⣿⣏⣹⣿⡟⠀⠀⢨⣽⠇
⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡄⠈⠲⣄⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⢰⣶⠆⠲⠿⠟⠙⠙⣷⣶⡄⣼⣿
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⢢⡖⠘⢷⣄⠀⡾⠟⠉⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠈⠀⠇⢠⢐⡴⠮⠀⠋⢸⡏⠉⢹⠀⠀⡠⣿⣄⠀⠀⠸⣼⠃
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣏⣿⣹⡇⠀⠀⠰⠬⣕⠻⠦⣵⡚⠣⢾⡇⠀⢸⠀⢰⣵⣿⣿⠀⠀⢠⡏
⠀⠑⠒⠒⠙⠿⠶⠿⠃⠈⠁⠉⠈⠁⠀⠀⠀⠐⠊⠠⠄⠒⠀⠤⠘⠿⠿⠟⠀⠀⠙⠛⠋⠀⠀⠸⠃

⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⢸⣣⣷⣹⠆⠀⠀⠘⢿⣷⣤⠶⠛⠉⣿⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣼⡟⠁
⠀⠀⠈⣷⣿⣿⠀⠀⠀⠀⢸⣿⣇⣠⣄⣠⣽⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⢿⡃
⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣷⣶⠆⠀⢀⣤⣀⣤⣿⡇⠀⠀⠘⣧
⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠸⣿⠉⠙⠛⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⢀⣠⣈⣿⣧⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠒⠶⣾⣷⠶⠖⠒⠒⠀⠀⢸⡏⠙⠛⠻⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠃⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣄
⠀⠀⢀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣟
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣴⣶⠂⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⡀⠠⠴⠶⠾⠿⠿⠿⠿⠿⠟⠋⠙⢦⣀
⠀⠀⠀⠀⠹⢷⣶⣴⠶⣶⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⣤⣤⡀⠀⠈⠛⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠤⠤⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣏⡀
⠀⠀⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⣾⡿⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡼⢿⣿⣿⢿⡄
⠀⠀⠀⢹⡿⢿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣼⣿⢻⣷⠀⠀⠀⠘⠛⣿⣿⣿⡞⠛
⠀⠀⠀⣿⡇⠘⠿⠇⠰⠖⠶⠶⠿⠃⠀⠀⠹⠦⠀⠀⠀⢀⣴⡿⠟⠃⠀⠙⣷⡀⠠⣶⣿⣿⡿⣿⣿⣷⠆
⠀⠀⠀⠹⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⢸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⢀⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣿⡇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣹⣿⣠⣾⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣤⣄⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⡀⠀⢀⣼⠟⣿⣿⣿⣿⢻⣧⡀⠀⣀⠀⠀⠀⠀⢀⣿⠸⣿⡿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢀⡀
⠀⠀⠀⠻⢿⣿⣥⣴⣿⣿⣿⣿⣤⣬⣿⡿⠟⠀⠀⠀⠀⠐⣿⣦⣿⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣿⣿⣷⣾⣿⡟
⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⢿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣿⣶⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣋⣽⣿⣿⡇⠀⠀⠀⠀⢸⣇⣀⠀⠀⢿⣿⣿⣶⣀⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣿⣿⣿⡿⠛⠛⠋⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣶⣦⣶⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢶⣶⣶⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠿⠈⠙⢿⣿⣿⣿⣦⡈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡙⠻⣿⣿⣿⣷⣄⠰⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⢠⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠆⠈⠻⢿⣿⡿⠀⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷
⣾⣿⣿⠃⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠘⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠈⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣭⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⢀⣠⣿⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣻⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⠍⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣆⠀⠀⠀⠀⡾⣾⣿⣿⣯
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠺⢸⡿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⢻⣿⣿⠛
⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⠙⠲⠄⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠷
⠀⠀⠀⠀⣼⠟⠈⢿⡄⠀⠀⠀⠀⢠⡿⠁⠙⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣿⠟⣿⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⣸⠏⠀⠀⠈⠻⣤⣤⠀⢠⡿⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢠⡟⠁⠀⢻⡀⠀⠀⠀⠀⣰⡿⠀⠙⣷⡀
⠀⠀⠀⠛⠓⠀⠀⠀⠀⠉⠀⠀⠘⠳⠂⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠛⠓⠀⠀⠈⠉⠁⠀⠀⠐⠛⠁⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣸⣿⢰⣟⣛⢸⡟⠻⣆⣿⢠⡞⠛⠂⢰⡿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⠏⠿⠸⠷⠶⠸⠷⠾⠃⠻⠈⠳⠶⠂⠿⠛⠻⠸⠷⠆
⠀⠀⠀⠀⠀⢠⣤⣤⢠⣤⣤⡀⣤⣤⠀⢠⣤⠀⣤⣤⡄⣤⡀⣠⡄⢠⣤⡄⣠⡄⢠⡀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡷⣾⠁⣿⣒⢀⣿⣽⣇⢸⡇⠀⣿⢷⠟⡇⢸⣗⡂⣿⠻⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠁⠉⠉⠉⠁⠀⠉⠈⠁⠀⠉⠈⠀⠁⠈⠉⠉⠈⠀⠈⠁⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⣧⡀⠀⣀⣀⣤⣤⣴⣄⣠⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠤⢘⡇⠶⠛⠉⠁⠀⠀⣈⣉⣛⠂⣾⠁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢸⠀⠒⢸⡇⠀⠀⣾⢻⣶⠟⠉⢉⣽⠿⣿⠀⠀⠀⣿⠀⣀
⠀⠀⠀⢀⣀⣶⠶⠟⠛⠉⢸⣀⣛⣸⡇⠀⢠⡟⠘⣧⢀⣴⠟⣡⡞⠹⡟⠛⠛⣿⠈⠛⠳⢶⣆⣀⡀
⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣼⣏⣿⣹⣇⠀⢸⡇⠀⣿⠞⣠⡾⠋⠀⣸⠇⠀⠀⣿⠀⠀⠀⠀⠈⠙⣧⡀
⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⢩⣭⣿⣭⡅⠀⠸⣧⣤⡿⠿⠯⢠⡄⠚⠻⢦⣤⡴⠋⠀⠀⠀⠀⠀⠀⠘⣯
⠀⢀⣿⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠈⣹⣆⢀⡾⠋⢀⣿⡀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠲⠶⣤⣀⣤⠶⠖⠶⢦⡀⠀⠀⣰⠷
⠀⠀⠉⠙⠸⡷⠶⠛⠉⠛⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢀⡈⠛⠁⠀⠀⠀⠀⢹⡆⠼⠋
⠀⠀⠀⠀⠀⣷⠶⠶⠶⠶⠶⢶⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡾⣧⠀⠀⣠⣄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⣷⠂⠀⠀⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⠛⠛⠁⢿⣴⠟⠋⠙⢛⣿⠟⠛
⠀⠀⠀⠀⠀⣀⣽⣟⣛⣛⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠸⠏⠀⢀⣴⠟⠁
⠀⣠⡶⠚⠛⠹⢧⣍⡿⣯⣡⠿⠙⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣴⠟⠁
⢰⡏⠀⠀⠀⠀⠀⠹⣧⡿⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣠⠶⠶⠶⠦⠤⣾⡟⠛⠛
⣼⠁⠀⣄⠀⠀⠀⠀⢹⠁⢸⡏⠉⢹⡇⠀⢨⡀⠀⠀⣾⠛⢷⣄⣤⠦⠦⠴⠞⠁⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⢸⠀⠈⠛⠛⠛⡁⠀⢠⡄⠀⠀⠈⠳⣄⠙⠷⠶⠶⠶⠶⠀⠀⠀⠀⢀⣠⣼⡇
⠋⠀⠀⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠃⠀⠈⠃⣤⣀⣀⣤⡙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋⠁⠹⠧⠶⠶
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣥⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠻⠷⠶⠦⣤⡄⠀⠀⠀⠀⠀⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢿⣿⠶⠶⠦⠀⠀⠀⠀⠙⣷⠒⠶⢠⣤⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⠀⠈⢳⡀⠀⠀⠈⠙⢷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡄⠀⠀⡠⠀⠐⠒⠀⠀⢠⡄⠆⠀⠀⢠⡄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⢶⣶⠄⠈⠀⠀⠀⠀⠀⠀⠀⠒⣾⡶⠀⠀⠀⠀⠀⠈⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠁⠀⠀⠀⠂⠀⠀⠈⠁⠔⠻⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢶⡿⣧⠉⠘⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡿⠁⠉⠉⠁⠈⣰⡆⠋⠀⠀⠉⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠈⢃⣾⠀⠀⠀⠚⠁⠙⠄⠀⠀⠀⠊⣿⣹⠇⠀⠀⠀⠀⠀⣰⠏⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⡟⠀⠀⠸⠃⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⡇⠀⠀⠀⠻⡈⢧⣀⡀⠁⠀⣀⣠⡞⣡⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⣸⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠙⠷⠶⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠋⢻⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣞⣁⡶⢸⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠙⠿⣧⣼⣯⣸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢦⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠶⣶⣄⡈⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣘⣻⣷⣦⣴⣟⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠛⠇⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠒⢊⣉⣉⡉⠉⢉⠙⣿⣿⣇⣶⣼⠟⠁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢠⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠛⠛⠛⣛⣛⣉⣙⣿⣻⣿⣴⣶⢾⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠘⡏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢸⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠤⠤⠤⠤⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠐⠒⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣄⣠⣤⣤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣋⣀⠴⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠊⠉⠉⠀⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢴⣶⣶⣶⡤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣴⣾⣿⣏⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⠿⠿⠿⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢲⣶⣴⣿⣯⣿⣿⣿⣿⡏⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⢃⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⢀⠀⠙⠛⣿⡿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠉⢹⣇⣨⣽⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠴⠵⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⠛⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠟⠁⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠔⠂⠈⢿⣦⣀⠀⠁⠀⣀⡀⠀⠀⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⠶⠶⣦⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⠀⠀⠙⢿⣦⡀⠀⠸⣷⡀⠀⠤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣄⣈⣻⣦⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠘⠉⠀⢀⡀⣀⣠⣾⣿⣿⣥⣴⡾⠿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠂⠀⠘⣿⣿⣿⣿⠛⠉⠉⠀⠀⠂⠊⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠋⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠄⠄⣠⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠉⠲⢂
⠀⠀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⣰⠁⠀⠈⢦⣀⠀⠀⠀⠀⠀⡠⠚⢅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠀⢉⡆⠀⡤⠘⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡤⠼⣄⣀⣀⠀⠀⠀⠀⠀⢀⢠⠄⠀⢤⠔⠏⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡀⢙⠦⣀⠀⠀⠀⠀⡇
⠀⢠⣞⠁⠀⡧⠀⠈⠁⠀⠉⡉⠉⠁⠜⠀⠀⠀⠀⡤⡀⠀⠀⠙⠛⠋⠒⠂⠉⠈⣍⠀⠀⠚⠳⣄⠀⢠⠃
⠀⢀⠃⣀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠇⠀⢀⠀⡠⠁⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡨⣧⠋⠢⡀
⠀⠀⠀⠸⠊⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡫⢛⠁⠀⠀⠀⣠⠀⠡⡀⠐⠆⠂⠀⠀⠀⠀⠀⣩⡕⠙⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⣓⢤⣤⣀⣠⡤⣤⠒⠙⠢⣼⠻⠀⡰⡴⠀⠀⠀⡠⠰⣶⣤⣤⡤⠐⡶⠮⣖⡓⡄⠘⠢⠔⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡷⠀⣠⠒⠦⡙⢍⠈⠚⣶⠏⠀⠀⠀⠀⠀⠁⠀⡿⠀⠀⠀⠀⡇⠀⠈⡂⣹⠆⢠⡄
⠀⠀⠀⠀⠀⠀⡇⡤⠚⠀⠀⡇⠀⠀⠠⡀⠙⡄⠈⢆⠀⠀⠀⠀⠀⡆⣞⠁⠀⠀⠀⠀⡘⠶⡈⠡⠸⠈⣤⠃
⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠑⠤⡀⠀⠀⣠⡶⠐⠈⠄⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⡐⠀⠘⠈⠈⢹
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠳⡤⠤⢤⣤⣤⣀⠀⢀⠃⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠂⠤⣤⣀⣀⣀⣀⡠⣶⡻⣑⠒⡁⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠀⠀⠸⡵⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡟⢀⠇⢐⠉⠙⢮⡀⠀⠀⡀⠀⠀⠀⠀⡠⠉⠀⢹⡄⠊⡠⠉⠑⠠⡀
⠀⠀⠀⠀⠀⡠⡴⠣⣌⡢⠤⡼⠳⣮⠔⠉⡄⠀⠀⢱⠀⠁⠀⡰⠖⠆⠺⠱⢄⣀⡆⠘⡯⠀⠀⠀⠀⢸
⠀⠀⠀⠠⡎⠀⠑⠄⠈⠁⠀⡄⠀⠘⡓⠊⠀⠀⢀⡜⠀⠀⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⠁⠀⢠⠋⠀⠲⠄⠂
⠀⠀⠀⠀⢣⡀⢀⢀⠀⢀⡰⠀⠀⠀⠙⠢⠤⠒⠉⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⠎⠀⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣟⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣾⣿⡿⠁⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⣼⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣷⠀⠈⠙⠛⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠋⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢿⣿⣶⣴⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣾⣿⣿⣿
⠀⠀⠀⢸⣿⡿⣿⡇⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢰⡀⠸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣷⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠸⠀⢠⣤⣭⣿⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣠⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⠿⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠛⠓⠒⠀⠀⠘⠓⠒⠒⠀⠀⠀⠒⠒⠚⠀⠀⠘⠓⠒⠛⠀⠀⠘⠛⠓⠛




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⢀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⠀⢀⣠⣀⠀⢠⣤⣤⣌⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠀⣤⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠀⠈⠉⠸⢻⣿⠄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣉⠀⠀⠀⠜⠀⠀⠀⠀⠀⠂⠀⠊⠀⠁⠀⠐⠶⠀⠀⢠⡄⠀⠀⠀⡀⠈⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠈⠇⠄⠀⢯⠀⠀⠀⠁⠀⠀⣆⣀⡠⠀⠀⠀⠀⠀⠀⠙⢉⠀⠀⠀⢁⠸⠀⠀⢀⡀⠵
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠠⠄⠀⣗⠘⠀⠀⠣⠀⠀⠹⣾⡇⢀⠀⠀⠀⣰⠇⢀⢀⠀⠀⣼⠉⠀⠈⡀⢀⡄⢡⠶
⠀⠀⠀⠀⠀⠀⠀⠘⠙⡆⢠⠄⣗⢠⠙⠀⠀⠐⠀⡀⢿⣰⣻⠀⠈⠅⠁⠀⣼⢋⠦⠙⣏⠁⠀⠂⠈⣦⠘⠋⢰
⠀⠀⠀⠀⠀⠀⠀⢀⡶⡌⢨⠄⡺⢿⡆⡀⢠⠀⠀⠸⠘⠡⠴⠀⠀⡖⢠⣼⡗⡯⢠⣾⡟⣷⡄⠀⠀⠀⠹⢂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣠⣾⡇⣼⣿⡋⠀⠀⣀⡇⣦⠀⢠⠀⢈⠳⠺⣍⡻⢷⡘⠜⣷⡈⢷⣴⠂⢀⣐⠜⠲
⠀⠀⠀⠀⠀⠀⠀⡄⠘⢿⢿⠥⡿⠜⢻⣷⡄⡈⢸⠃⠹⢦⣦⠀⠸⢁⠀⠈⢙⣿⣿⣀⡄⢧⣾⢽⣦⠮⢬⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣬⣿⠿⣷⠞⢺⡏⣿⢱⣿⠄⠴⣿⣿⠀⢰⡟⠛⣠⠆⢿⣯⣈⠁⣘⣁⣼⡎⠓⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠈⣥⣘⣿⣿⣿⠐⢪⣁⡼⢿⡎⢸⠗⡇⡨⠀⢰⣇⣤⠟⣿⣾⢻⢗⡆⡹⠋⠟⠑⢮⣷⣰⣬
⠀⠀⠀⠀⠀⠀⠀⢏⣿⣿⣯⣝⣿⣁⢌⢧⣿⣿⣿⣿⡶⠡⠴⠀⢸⣻⣿⣿⣿⣼⣌⠩⢫⣷⡆⢰⣄⡾⢁⡟⢻⠂
⠀⠀⠀⠀⠀⠀⠀⡛⣿⣿⢿⣿⣁⣀⢾⣼⣿⣿⣛⣿⠈⢳⡂⠀⢸⣿⣻⢸⣿⡿⢋⣼⢉⣹⣿⣾⣿⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠀⠉⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣋⣉⣩⣤⣤⣾⣿⠿⠟⠛⣻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡶⢾⣿⠛⠛⣉⣉⣥⡤⠼⠿⠒⠛⢋⣉⣡⣽⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢉⣩⣤⠤⠶⠾⠟⢛⣉⣉⣤⣤⣴⡶⠶⠿⠛⠛⠋⠉⠁⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡴⠋⢀⣤⣶⠶⠾⠟⠛⠛⠉⠉⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡇⢠⣿⠋⠀⠀⠀⠀⠀⢀⡴⠛⣩⣤⣤⠀⠈⠳⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⢠⡞⠀⠀⣿⠀⣿⣀⣤⡄⢻⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⣾⢠⡖⠛⠋⠀⢈⣀⣸⡇⣸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⣿⠘⠗⠚⣿⠀⣿⠁⠀⣠⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⣿⠀⠀⠀⠀⠀⠘⣦⡀⠀⠻⠶⠛⣀⡴⠋⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠛⠉⠀⢀⣀⣀⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠶⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠾⠟⠛⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠛⠋⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⢻⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠠⢬⣉⠾⠇⠺⡿⣿⣿⣧⠕⠀⣀⠀⠀⢴⣾⣷⣮⡙⠲⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣄⡶⠂⠀⠀⠀⠀⠀⣀⣼⣿⣶⣶⣶⣤⣤⣄⡲⠢⣿⡶⢷⡆⠐⠚⠓⠈⠉⠘⠙⠛⠐⠂⢻⣿⠟⠛⠿⣿⣿⣿⣿⣿⡛⢃⣀⣀⠈⡉⢻⠛⠛⢻⣿⣿
⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⣿⣿⣿⡿⠿⣿⣿⣦⣴⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠉⠙⠃⠠⠤⠄⠀⠁⠀⠀⠠⠿⣿⣿
⠹⠟⠛⠿⠿⠷⠦⠤⠶⠦⠬⠁⠁⠐⠀⠉⠛⠏⠀⠀⠈⠙⠛⠓⠀⠀⢠⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣴⢦⣶⠾⠶⠿⠿⠿⠿⠞⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠤⠤⠤⠤⠤⠶⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣶⣤⣤⣤⣼⣿⣇⣻⣿⣿⣿⣿⣿⣿⢸⣿⣷⣴⣶⣶⡿⢟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠄⣀⡀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⣄⡀
⠀⠀⠀⡴⢶⣿⡿⢦⠀⠀⠀⠀⠀⠉⠘⣷⣄⠿⠟⠀⠀⠀⠀⠀⠀⢠⣿⡞⢁⣀⡀⠄⠀⠀⠀⠀⠀⢠⣾⡿⣤⠀⠀⠀⠀⠀⠠⠻⣅⡺⢿⣢⡾⠉⠁
⠀⠀⠀⠀⣸⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣆⠀⠀⠀⠀⠘⠙⠟⠿⠟⣭⠙⠈⠀⠀⠀⠀⠀⠾⣾⡷⣻⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡿
⠀⠀⠀⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠁⠁


⠀⠀⢠⣶⡦⣶⣦⣤⣤⡤⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠰⣆⠀⣶⣶⠀⣰⡦⠀⠀⠀⠀⠀⠀⢴⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠓⣤⡀
⠀⠀⣴⣾⢤⣼⣷⣶⡿⠀⠀⠀⠀⠀⠀⣠⣽⣭⣀⠀⠀⠀⠀⠀⠀⠙⢷⣮⣽⡾⠋⠁⠀⠀⠀⢀⣀⣤⣴⣿⣦⢄⣄⠀⠀⠀⠀⣴⣾⣿⢆⣼⢿⠕
⠀⠀⠿⠿⠿⠿⠿⠈⠀⠀⠀⠀⠀⠀⢸⣯⣽⣿⣿⠄⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠈⠁⠈⠿⠿⠏⠀⠉⠀⠀⠀⠀⠈⠙⠿⠿⠓⠉


⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶
⠀⠀⠀⠀⣴⣷⡆⠀⣴⡄⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠶⠄⠀⠀⠀⠀⠀⢀⡀⣀⣻⣟⣀⠀⣀⠀⠀⠀⠀⠀⢰⣿⣿⠷⣤
⠀⠐⠶⠶⢰⣿⠿⢿⠈⠁⠀⠀⠀⠐⠒⠺⣶⡉⠒⠒⠀⠀⠀⠀⠀⠀⣽⠛⣷⠀⠀⠀⠀⠀⠀⠙⢉⠛⣿⣿⠋⢩⠛⠀⠀⠀⠀⠂⠙⣾⢷⡖⠊⠃
⠀⠀⠀⠲⠾⠟⠁⠸⠂⠀⠀⠀⠀⠀⠀⠸⢿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣯⣤⡄⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠘⠃⠼⠃


⠀⠀⠀⣀⣠⣰⡦⢠⣦⠀⠀⠀⠀⠀⣠⠀⣠⣄⠰⣤⣀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣄⣤⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⢸⣿⣿⡷⡄⡀⣿⡆⠀⠀⠐⠞⢳⣤⣛⣫⣾⠃⠀⠀⠀⠀⣀⡀⣾⠁⢠⣿⠀⠀⠀⠀⠀⠀⣬⣽⣧⣤⣭⣭⡁⠀⠀⠀⠀⣀⡤⣌⢹⣿⣦⡀
⠀⠠⠾⠿⠸⠇⠾⠿⢿⠇⠀⠀⠀⠀⠀⠉⣿⣿⡇⠀⠀⠀⠀⠀⠻⠛⠻⠿⠿⠓⠶⠖⠀⠀⠀⠀⠙⠻⠛⠙⠻⠟⠀⠀⠀⠀⠀⠘⣠⣤⣾⢯⣿⡇


⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⣠⡿⣅⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡾⠿⠀⠀⠀⠀⢸⠙⠛⠛⠋⡇⠀⠀⠀⠀⠀⠐⠃⢰⣶⠈⢣⡀⠀⠀⠀⠀⠀⣴⣿⣏⡉
⠀⠀⡄⢀⠿⠀⠾⢇⢠⡀⠀⠀⠀⠉⢉⣙⣿⠉⢹⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠠⠶⠷⠄⢨⠁⠀⠀⠀⠀⠀⣿⠀⢩⠉⣠
⠀⠀⠀⠉⠐⠀⠘⠉⠈⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠑⠲⠖⠒⠁⠀⠀⠀⠀⠀⠀⠟⠂⠘⠀⠹

⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣢⢲⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠸⣿⠁⢺
⠀⠀⠀⠀⠀⣀⣀⣼⣵⣷⠴⠀⠀⠀⠀⠀⢯⢦⠤⠐⡭⠗⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⡀⠘⣿⠀⣼⡀⠈
⠀⠀⠀⠀⠠⠿⠿⠯⠿⠎⠦⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠃⠀⠛⠀⠄⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣃⡀⠀⠀⠀⠀⡀⣘⣃⢀⠀⠀⠀⠀⠀⠀⣀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡊⣿⡃⠀⠀⠀⠀⠈⢸⡏⠈⠀⠀⠀⠀⠠⡄⣮⣟⡘⣅⣁⠄⠀⠀⠀⠀⢰⢧
⠀⠀⠀⠀⡀⠰⠆⣧⡴⠃⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⢹⣽⠉⠉⠉⠁⠀⠀⠀⠀⠼⠆⠒⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⡠⣬⣆⠀⠀⠀⠀⠀⢸⠀⠀⣛⣀⠀⣬⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡿⠀⠀⠀⠀⠀⠘⠀⠀⣉⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⢠⣷⣀⡀
⠀⠀⠀⠀⢰⣿⡿⠀⡿⡇⠀⠀⠀⠀⠀⢀⡠⠤⠼⢀⠀⠿⡇⠀⠀⠀⢠⣦⣴⣦⣟⣾⣇⠀⠀⠀⠀⣿⣿⠟⡅
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡌⢻⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣿⣶⣿⣴⣤⣤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠉⢻⡿⠟⠿⠇⠀⠈⢻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠁⠀⠀⠀⠀⢀⣠⣿⣿⣿⣷⣦⣀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡦⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣿⣷⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠷⠲⠖⠂⠀⢀⣈⡋⠓⠀⠀⠀⠉⠛⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⢦⡤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣄⣀⠀⣰⣆⠀⡀⠀⣼⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣯⣴⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠋⠘⠋⠉⢛⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣄⣤⣝⠈⠁⠀⠀⠀⠆⣴⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣠⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⣿⣿⣷⣺⣠⡏⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⢀⣄⣀⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⡀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠀⠙⢿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⡇⢀⠀⢀⠸⠿⠿⠷⠦⠤⠄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣰⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣶⣦⡀⢠⡾⠋⠁⠀⠀⠀⣿⠀⠀⠀⠈⠛⢷⡄⣸⡟⢃⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⣤⣶⡿⠷⠿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠗⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣶⣶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣴⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⢀⣤⣤⣤⣤⣤⣤⡀⢰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃⠈⠉⠉⠉⠉⠉⠉⠁⠘⢿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣴⣾⣿⣿⣿⣶⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡀⠀⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⢛⡛⣷⡀
⠀⠀⠀⢀⣼⠿⠿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠈⠁⣸⡇
⠀⠀⢀⣿⠁⠶⠶⠆⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠻⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠃⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⠟⠋⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⡾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠘⠻⣦⣀⠀⠀⣀⣀⣀⣤⣶⠶⢶
⠀⣀⣀⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⣠⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⢈
⣾⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢿
⢻⣦⣀⣀⣀⣀⣀⡀⠀⠀⠰⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⣿⠆⠀⠀⣾⣿⣯⣥⣤⡿
⠀⠸⣿⡉⢉⣩⡿⣿⡄⠀⠀⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⠀⢀⣿⠁⠀⠉⠁
⠀⠀⠈⠛⠛⠉⠀⠈⠻⣶⡶⠟⠁⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠻⢶⣄⡈⠻⣦⡀⠀⠀⠀⣠⣴⠿⢷⡄⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠉⠛⢷⣾⠿⣦⣴⠞⠋⠁⠀⠈⢿⣆⢀⣴⡞⣫⣝⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⢻⣦⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣿⣟⡵⣫⡿⠛⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢳⠀⠀⢹⠟⣹⠟⣺⣿⣉⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣾⡿⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢣⡀⠈⠛⠛⢶⡋⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⣡⢮⣾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣵⡟⠁⠀⢀⣤⡾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠉⣉⠉⡉⢉⡉⢉⣉⣉⣉⠀⠀⡀⣀⠀⣀⣀⣀⠈⣻⣷⣶⡿⠛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⡓⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠆
⠀⠀⠀⠀⠠⠐⠐⢃⠀⠀⠀⠀⠀⠀⠀⢤⡟⠐⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠑⠆⡀⠀⠀⠀⠀⠀⠐⢈⣤⡆⠀⢀⡆
⠀⠀⠀⠀⠀⢠⢨⡟⠆⠀⠀⠀⠀⠒⣲⠁⣹⠁⠀⠀⠀⠀⠀⠀⠙⡉⣇⣀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠈⠁⢌⠂⠀⠀⠀⠀⢈⠋⡆⢨⣐⠕
⠀⠀⠀⠀⠐⢲⠁⠘⠁⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⢦⣀⣋⠑⠑⠀⠀⠀⠠⠦⠕⢺⠀⣠⠀⠀⠚⠃⠀⠀⠀⠘⢾⠀⢠
⠀⠀⠀⠀⠀⢘⠐⠚⠀⠀⠀⠀⠀⠀⢀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠘⢯⠝⠲⠄⠀⠀⠀⠀⠀⠀⠈⠇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣐⠂⠷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⡌⠑⡈⢄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⠀⠈⠫⡂⠀⠀⠀⡰⢁⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠰⠤⠀⠀⠀⠀⠀⠀⠐⠞⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠅⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠀⠀⠀⠀⠀⠀⢸⡁⡍⠀⠀⠀⠀⠀⠀⡎⢻⠟⠀⠀⠀⠀⠀⠀⠀⠞⢻
⠀⠀⢀⡀⠀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣧⠀⠀⠀⠀⠀⠚⢻⠓⠄⠀⠀⠀⠀⠀⢉⠿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠙⠄
⠀⠀⠀⠻⡂⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠂⠟⠃⠀⠀⠀⠀⠐⡆⣛⢤⠂⠀⠀⠀⠀⠐⡿⣠⢧⠀⠀⠀⠀⢉⣅⠘⠇⠐⠤
⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠉⢂⠀⠀⠀⠀⠀⢠⡃⠘⢑⠂⠀⠀⠀⠀⠰⡅⡀⡀⠆⠀⠀⠀⠀⠁⢚⠀⡰
⠀⠀⠀⠀⠀⣹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠐⠇⢰⠈⠀⠀⠀⠀⠀⠀⡗⢀⠂⠀⠀⠀⠀⠀⠀⠈⠘⢳⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡌⢂⠀⠀⠀⠀⠀⠀⠸⡀⠘
⠀⠀⠀⠀⠘⠀⢆⠐⡀⠀⠀⠯⡤⠮⠤⡴⠀⠀⠀⠉⠈⠫⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠙⡌⢆⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⡀⠐⠀⠣⣐⠄⠀⠠⣶⠔⢆⠧⠄⠀⠀⠀⣆⠀⢡⡄⠀⠀⠀⠀⡄⢸⠀⡀⠀⠀⠀⠀⠈⡠⠀⠈⡼⠂⠀⠀⠀⣠⢀⠂⠠⠀⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀
⠀⠀⠀⢀⡀⠉⠉⢀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠆⠀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠹⠆⠰⠏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡘⠿⢃⡀
⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠛⠞⠛⠀⠀⠀⠀⠀⠈⠛⠀⠛⠁

⠀⠀⠀⢠⣤⡀⠂⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠰⠶⣶⠖⠒⠀⠀⠀⠀⠀⣶⠐⠄⡀⠀⠀⠀⠀⠀⣶⣶⣶⡦
⠀⠀⠀⢸⣿⣇⣀⢀⡀⠀⠀⠀⠀⣟⠛⠛⠛⣧⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⣲⠀⠀⠀⠀⠀⠙⠛⠉⢻⡀
⠀⠀⠀⠘⠛⠛⠒⠒⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠚⠀⠀⠀⠀⠀⠚⠚⠒⠛⠓

⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠤⣶⣶⡦⠄⠀⠀⠀⠀⢀⣴⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⢸⠰⠌⡇⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠸⠿⢿⣿⣷⠀⠀⠀⠀⠀⠥⠤⠧⠬⠄⠀⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⣴⣆⣀⣿⣇⠀⠀⠀⠀⢀⡿⠇⣾⣷⠀⠀⠀⠀⠀⢀⡿⠇⢐⠿⠇
⠀⢀⣾⣿⣿⣿⣿⣿⠇⠀⠀⢠⣿⣇⣼⣿⣿⡗⢦⣄⠀⢀⣿⣧⣴⣿⣇⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠘⢻⣿⣿⢻⣿⣿⠀⠰⡶⠶⣿⡿⠶⣿⣿⡶⠶⢶⠆⠈⢿⡆⠘⢿⡝⠒⠀⠀⠀⠀⢩⣿⡍
⠀⠀⠸⡿⣿⢸⣿⣿⠀⠀⡇⠀⣿⣿⠀⣿⣿⡇⠀⢸⠀⠀⢸⡇⠀⣼⠇⠀⠀⠀⠀⣠⡴⣿⢦⣄⠀⣶⡆
⠀⠀⠀⠀⠛⠈⠉⠛⠀⠀⠃⠘⠃⠙⠃⠛⠛⠁⠀⠘⠀⠀⠺⠃⠀⠟⠀⠀⠀⠀⢀⡏⠰⠐⠊⢹⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣗⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣶⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⢿⣿⣿⢷⣶⣿⣇⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿
⠀⠀⢰⣦⠀⠀⢰⣿⣄⠀⠀⠠⣶⠁⠀⠀⠀⠈⣿⠁⢸⣿⣿⠸⣿⣿⣿⣄⠀⠀⠰⠿⠾⠿⠿⠿⠎⠙⠟
⠀⠀⢸⣿⠠⠄⠼⣿⡬⠷⠔⠚⣿⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⠀⢸⣿⣿⣿
⠀⠀⢸⣿⠀⠀⢠⣿⣧⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠀⠈⠋⠋⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢠⡄⢴⣶
⠀⠀⢸⣿⢀⠰⠟⠁⠿⠀⠀⢀⣿⠀⠀⠀⠀⠀⣒⠖⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣾⣷⡄
⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⡁⠀⠀⣤⡀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡟⠃⠀⠀⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣷⠀⣿⣧⣀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⡿⠿
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣿⡟⣿
⠀⢰⣳⡆⠀⠠⢿⠿⠻⣷⣀⣤⣤⣾⣿⡀⠘⠛⠛⠛⠛⠛⠋⢀⣾⡷⠀⠀⣀
⠀⠘⢟⠃⠶⠶⡾⠶⠶⠶⠶⠿⠷⢿⠿⠷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣤⡤⠙⠛⠛⠿⢶⣶⣶⣶⢴⣿⠄
⠀⠀⠘⣄⣀⣠⣇⣀⣀⣀⣀⣀⣀⣼⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⣿⠀⠀⠀⢠⣤⣼⡇⠀⠸⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣦⠀⠀⠀⣀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠃⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡝⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⢀⣏⠀⠀⠻⠄⠀⠀⠀⠖⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠁⠀⠀⠀⠀⢤⣤⠀⠀⠀⠸⠛⠀⠀⠀⢠⠄⠀⢀⡀⠀⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⢀⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠙⠟⠃⠀⠀⠉⠀⠀⢠⣇⠀⠀⠀⠀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⡈⠙⠋⠀⠀⠀⠛⠟⠁⠀⠀⠐⠀⠀⢠⣿⠀⠀⢀⣤⣆⠀⠠⠤⠀⠀⠀⠀⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⢠⡶⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⢀⡄⠀⠀⢠⠀⠀⠙⠂⠀⠀⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⡆⠈⣀⡤⢀⡄⠀⠈⠀⠀⠆⠈⠛⠉⠀⣀⡴⠀⠀⠀⠈⠛⠂⠀⣿⠄⠀⠀⠴⠶⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠗⠀⠈⠁⠸⠃⠀⠀⣄⡤⠀⠀⠀⠀⠈⠙⠁⠀⠀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣤⠀⠀⢠⡄⠀⠀⠘⠃⠀⠤⠞⠀⠀⠀⠀⠒⠀⠈⠀⠀⠘⠁⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣯⣿⡄⠀⠘⠛⠂⠰⡾⠀⠀⠀⢀⠀⠲⢶⡶⠂⠀⠘⠀⢀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣭⡇⠀⡀⠀⠀⠈⣀⡤⠂⢠⣾⠃⠀⠀⢀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣟⣿⠀⠛⠁⠂⠀⡀⠈⠁⠀⠀⠈⠀⠀⠀⣿⠀⠀⠁⢀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣯⣸⠇⠀⣠⡴⠃⠘⠃⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠚⠷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠶⣿⣿⠀⠀⣌⠁⠀⠈⠙⠟⡁⠈⠁⠀⠘⠻⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⡇⠀⣀⠈⠀⠀⠀⢲⡞⠁⠀⣤⠀⠀⠀⠀⡆⠀⠘⠃⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⠀⠀⠰⠦⠀⣤⡀⠀⠀⠀⠀⢠⣤⡀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣄⠀⢀⣠⠀⠀⠀⠀⠓⠀⠀⠉⠁⢀⡖⠀⠀⠱⡦⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⠀⠙⠀⠀⠀⠀⡀⠀⠀⢀⠴⠿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿⣿⣿⣟⣚⡁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⡀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣅⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣥⣿⣿⣆⣶⣶⣦⣀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣤⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⢷⡄⠀⣾⣿⣿⣟⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣆⣩⣿⣅⢼⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡯⣻⣿⠃⠀⠙⣿⣿⣿⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣿⣿⣿⣿⣾⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣬⡄⠀⢈⣵⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣾⣿⣿⣿⣽⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡿⠀⢘⣻⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⣭⣍⠁⠀⠸⣿⣟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣿⣿⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠆⣴⡦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⢿⠿⠋⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⢾⣽⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣦⡀⣠⢶⢹⣿⣿⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣿⡿⣿⠀⠀⠀⠀⢿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠉⢁⣘⢸⣿⣿⣿⣿⣿⣿⣿⣷⡴⠀⠀⠀⠀⠀⠀⠀⠀⣺⢷⢿⣇⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠀⢸⣀⡏⢛⣿⣿⣻⣯⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡏⠀⠀⠀⠶⣾⣿⣹⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡇⠀⠉⠻⠿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠾⠇⠀⠀⠀⠻⠛⠉⠙⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣄
⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠻⣦⡀
⠀⠀⠀⠀⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⢸⣿⣇
⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⡀⠀⠈⠛⠻⣿⣿⣿⣦⣈⣿⣿
⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣽⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⣰⣿⣿⣿⣿⣿⡟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠈⣹⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠋⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣆
⠀⢸⣿⣿⣿⣿⣿⠇⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣯⣤⣤⣤⣤⣤⣤⣉⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠈⠉⢹⣿⣿⣧⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠃
⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠃⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠟⠁
⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠁
⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⢨⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣄
⠈⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷
⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠆
⣸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⡀
⢻⣿⣿⣿⣷⣶⣶⣷⣶⣶⣶⣶⣶⣶⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⢹
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣶⣾⣿⣿⣿⣷⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠻⠶⠒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣦⣀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⢠⣼⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠉⠛⠁⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠻⠿⠿⠟⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠿⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾
⢳⡀⠉⠛⠶⣤⣤⣀⣀⣀⣀⣤⣤⣤⡶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢦⣤⣤⣄⣀⣀⣀⣠⣤⡴⠞⠋⠁⣰
⠀⣹⢦⣀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⣠⢾⡁
⠀⠻⣄⠈⠉⠛⠒⠒⠶⠶⠶⠒⠒⠉⠁⠀⠀⠘⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠚⠋⠉⢀⡼⠃
⠀⠀⠈⢻⣶⣤⣤⣤⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⣌⠳⣀⠀⢀⣤⣄⠀⢀⣴⢋⡜⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⣤⣤⣴⣾⠋
⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡟⠁⠀⠀⠙⣯⠴⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠈⠛⢷⡶⠤⠤⠤⠒⠊⠉⠀⢀⡠⠀⠀⠀⠈⣇⠀⠀⠀⢀⡏⠀⠀⠀⠀⡀⠀⠈⠁⠒⠒⠤⠤⠴⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣄⣀⣤⡴⠞⠋⠀⣀⡤⠶⠶⠾⢷⡶⣶⠿⠶⠶⠦⣤⣀⠀⠑⠦⣤⣀⣀⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⣤⡤⠖⢋⣤⣴⣦⣤⡀⠀⡇⡇⠀⣠⣤⣶⣦⣍⡓⠦⣤⣤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⢿⣾⣿⣆⡇⢇⣾⣿⣾⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣤⡀⠈⠙⠉⡇⡏⠙⠉⠀⣠⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣤⡇⣧⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠉⡏⡏⠙⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⡇⡇⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⡇⡇⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⡏⡛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⠀⡇⡇⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⡇⡇⠀⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣇⣥⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⡆⡏⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⡇⡇⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣇⣧⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⡇⡇⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⡇⡇⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡇⣣⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⡏⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⡇⡇⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⡇⡇⠀⠠⠞
⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣿⣭⣭⣭⣭⣬⣭⣽⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣥⣽⣿⣧⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠖⠒⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆⠈⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣰⡀⠹⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢼⠀⢆⠀⢰⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢣⠘⠀⡇
⠀⠀⠀⠀⠀⢀⠁⡀⢀⠔⢀⠠⡄⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠈⠒⠂⠛⢠⠉⢆⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⡦⠀⠈⠀⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠰⠆⠀⠀⠁⠀⠠⡀⠀⢀⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡤⠀⠀⢠⡄⠀⠸⠆⠐⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⡀⠚⣧⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠊⠀⢰⡀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⡤⠄⠲⢧⡀⠀⠀⠀⢀⡙⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⡠⠤⠺⠛⠭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣃⣝⠀⡀⠀⠀⠠⡀⠀⠀⠓⠊⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠁⠀⠰⠀⠀⠐⣄⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⠜⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢄⢰⡉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠰⠠⣀⠀⠀⠀⠀⠀⠀⡀⡔⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⠈⠀⠀⠐⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡄⠰⠦⠄⠀⠀⠀⢠⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢠⡀⢄⡒
⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⢤⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡐⠲⠰⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠈⠂⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⢀⠎⠀⠀⠀⠟⢠⠂⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢒⣒⡲⢤⠄⠀⢀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠏⠀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢇⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢐⠄⣸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣸⠀⠀⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⢀⠃⠡⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠄⠀⠀⠡⡀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣟⠳⢄⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠌⠀⠀⠀⠀⠱⡀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⠀⠀⠙⠢⣄⢸⠀⠀⢀
⠀⠀⣀⠠⠤⢄⡀⠀⢀⡴⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⣎⠉⠀⠀⠈
⠀⠘⢃⠺⢦⣄⠙⠐⠧⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠈⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢸⠀⠀⠀⠀⡿⠀⠀⢀⠆
⠀⠀⠀⠑⠀⠑⠳⣄⠀⠀⠀⠑⢆⠈⠀⠀⠀⠀⠀⡠⠔⠚⠀⠀⠀⢠⣢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠀⠃⠉⡆⠀⠀⡼⠁⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠐⠀⡉⣂⡀⣠⠋⠀⠀⠀⠀⠀⠸⣠⣤⣂⣀⣡⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠨⠤⠀⠀⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠃⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠀⢀⠠⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸
⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠐⠠⠦⠂⠒⠂⠀⠤⠂⠦⠤⡤⠴⠤⢤⡴
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠳⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢼⠀⢡
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠂⠀⠀⢈⠀⡁⠠⠀⠀⠀⠀⠈⢿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠁⠀⠀⠠⠀⠀⠀⠀⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⢧⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠐⢄⠄⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠈⠓⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⣉⠁⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⡀⠀⠂⡇⠰⠀⢲⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁⠀⠁⢈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠻⣿⣿⡿⠿⠉⠁⠀⣉⢀
⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⢀⣠⣤⣦⣤⣀⡀⠀⠀⠀⠀⣠⣤⣴⣤⣄⡀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠃⢸⠏⠐⠀⡉⠩⠉⠉⠉⠋⣽⡆
⠀⠀⢠⠋⠙⠛⠛⠋⠙⢷⠀⠀⡠⣿⠙⠛⠙⠋⠉⣶⡀⠀⠀⣿⠋⠛⠙⠛⠙⣳⣄⠀⠀⠀⠀⠀⠾⢐⠀⠀⠀⠀⠀⠀⠀⣿⠛⠚⠃⠐⢻⠇⣶⠀⠀⠇⠀⣿
⠀⠈⢹⠂⢀⠀⠀⠀⠀⢻⡆⠀⢹⣿⠀⡄⢀⠀⠀⣿⡇⠀⢸⣿⠀⡀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⢚⣮⠀⠀⠀⠀⠀⠀⠀⣿⢀⡄⣤⢀⡸⣆⢿⡄⣠⣤⡆⣹⠄
⠀⠀⠘⢦⣜⣤⣤⣤⣤⠞⠀⠀⠈⢿⣄⣤⣼⣤⣤⡟⠀⠀⠀⢿⣤⣧⣤⣤⣤⡿⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠿⣼⣧⣿⣼⣧⣿⣼⣤⣿⣼⣧⠼⠂
⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⡄
⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⣿⠀⠀⣾⡿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⢿⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠘⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠰⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⡀⠀⠀⠀⠀⣼⠏⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢀⣴⠿⠛⠉⠙⣿⡄⠀⠀⣼⡟⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢿⣆⠀⠀⢠⣿⠋⠀⠀⠀⠀⠈⢻⣄⢰⡟⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠈⢻⣦⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⠀⢠⣾⢷⣦⣀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⠀⣴⡿⠁⠀⠈⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⡇⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⣿⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⢸⣿
⣿⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠁⠀⢸⣿
⠻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠏







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⡀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⣿⠿⠛⢿⡅⢻⡇⣿⣹⠆⢂⡌⢸⠀⢈⠁⠀⠀⠁⠰⠁⠀⠷⠠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣾⣿⣶⣾⣷⣾⣿⣶⣾⣶⣾⣿⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠓⣿⡁⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⡀⢰⣶⡆⢂⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣴⣏⡀
⠀⠀⠀⠀⠈⢿⣶⣭⣤⡀⠀⠀⠀⠀⠀⠀⠉⠿⣶⣽⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢹⣿⣆
⠀⠀⠀⠀⠀⠸⣿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⡿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠁⠹⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡟⠁
⠀⠀⠀⢀⣴⣭⠉⠠⢀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣿⣥⣤⣤⡀⠀⠀⠀⠀⠠⢶⣅⡸⠿⢇⣼⠋⠁⠀⠀⠀⠀⠀⠀⢸⠻⡿⠛⠶⠄
⠀⠀⠀⣸⣿⢻⣧⣡⣄⠃⠀⠀⠀⠀⠀⠀⢹⠈⢹⣿⡏⠁⡏⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⢉⣭⢻⡏⠉⠉
⠀⠀⠀⠈⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠺⠁


⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⢀⣠⣬⣯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠷⢤
⠀⠀⠀⠈⠉⣸⣿⡩⠉⠁⠀⠀⠀⠀⠀⠀⠀⢘⡿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⣿⣽⠓⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⣤
⠀⠀⠀⠀⢠⡿⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⢸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣤⡀⢦⣤⡄
⠀⠀⢤⣤⡤⣴⣶⣶⣶⠙⠁⠀⠀⠀⠀⠀⠀⣠⣾⠿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣸⡏⠀⠀⠀⠀⠀⠀⠐⠛⢦⣜⣛⣵⡟
⠀⠀⠀⢠⣤⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣬⣭⡄⠀⠀⠀⠀⠀⠺⠟⠾⠿⠾⠿⠱⠶⠆⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣾⣿⣿⣿⣶⣶⣄⣤⡀
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢻⣿⣶⣾⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⠛⠛⠛⠃⠀⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠘⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣄⠃⢴⣤⡤⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⢻⣿⡿⢟⡛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠋⠀⠀⠀⠀⡆⠀⠀⣀⠈⠙⠛⠉⠀⠀⠀⠁⢰⡏⣹⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⣡⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠙⠛⠒⠲⠒⠒⠚⠃⠀⠀⠀⢀⣠⡾⠉⠉⠉⠛⠛⠛⠻⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⣀⣀⣀⣀⣠⣤⣴⠟⣋⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⢻⠉⠉⢀⡠⠛⣧⠍⣸⡛⠽⣟⢲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣯⠀⠙⠒⠂⠉⣴⠊⣠⡖⠉⠛⣦⠈⣿⡚⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣰⣇⣴⠿⣤⣷⣶⠁⣠⣟⡀⠀⠀⠸⡇⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣤⣀⣰⡟⢰⠇⠀⠀⠀⢏⣉⠉⠒⠋⠈⠹⠀⠀⠀⢱⡞⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠇⠙⠺⣿⣿⣄⠇⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣤⡀⠀⢸⣿⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠠⣦⣼⠗⢤⣤⣀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣀⠀⢈⣿⠀⢸⡿⢠⠏⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⠀⠀⠀⠀⠈⡃⠹⣸⠿⠋⢠⣆⠘⣅⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢻⣷⠀⠀⢸⡗⢻⡃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣿⠀⠀⠀⠁⣼⣡⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣸⣇⣀⣀⠴⣦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡸⣄⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⢫⣤⠃⣽⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡟⠁⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⠛⠶⢦⣤⣤⣤⣤⣤⣤⣤⣴⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠶⣾⣍⠀⢀⣈⣉⣁⣤⠛⢶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⣀⠁⠈⠉⠉⠷⠤⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡅⠈⠉⠁⠀⠀⣀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠⡏⠀⠀⠀⠀⠀⠀⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣸⠇⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⡰⠋⠈⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣦⠀⢀⡇⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⢀⠻⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢈⡇⠸⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠘⡀⣠⠟⠁⠀⠙⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⡸⠀⠀⠀⠁⠟⠀⠀⠀⠀⢧⡀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⣦⣄⢸⠇⠀⠀⠀⠀⠘⢧⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄
⠀⠀⢀⠂⠉⢦⠀⠀⡀⠀⠀⠀⠊⢀⣠⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠠⠀⠄⠠⡀⠀⠀⠀⠂⠂
⠀⠀⢀⠀⠀⠘⣆⠀⡇⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠀⠒⢩⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠄
⠀⠸⠈⠤⠒⠀⠘⡦⠆⠘⠣⠀⠀⢤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠃⠄⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⢶⣶⠤⠴⠂⠃
⠀⠈⡤⠯⢨⠄⠀⠠⡏⠙⠄⠀⠐⡆⠙⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⢳⢀⠀⠸⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠮⠿⠂
⠀⠀⡁⠠⠫⠂⠀⠀⠁⠀⠀⠘⠇⠙⠀⣆⠀⠀⠀⠀⠦⡆⠀⡄⠉⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠒
⠀⢠⠁⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⢰⠀⠁⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠁⠀⠀⠀⠀⠀⡴⢁⠀⠀⠀⡀⡆
⠀⡆⠈⠀⠀⢀⠂⡀⠂⠀⠀⡇⠀⡄⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡰⠂⠘⣆⠀⠀⠀⠀⣼⠀⠀⡷⠤⡎⠀⠱
⠀⠁⠀⠀⠀⡬⠘⠁⠀⠀⠀⠃⠀⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠸⠀⠀⠀⡌⠀⠀⠀⠘⢆⠀⠀⠀⠈⠓⠋⠀⠀⠙⠛
⠀⠄⠀⠁⡆⠀⠀⡀⠀⠀⠀⠀⠠⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠠⠐⠈⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨
⠀⠀⡐⠋⠈⠙⢿⣄⡴⠶⠶⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠔⢈⠍⢭⡲⠄
⠀⠐⠀⣀⢠⠆⠀⡟⠀⠀⢠⡆⡇⠀⣰⠟⠁⠀⠀⢶⡆⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡶⠖⠚⠚⠒⠒⠒⠚⠛⠒⠒⠆⠀⠀⠀⠀⠀⠀⠖
⠀⠘⠒⠹⠊⡇⢰⣇⣤⡤⢿⡰⠃⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣲⡆⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠤⠤⠖
⠀⠀⠈⢠⡀⠈⢸⠏⠀⢀⠔⠁⠀⠰⡇⠀⠀⠀⠀⣸⠃⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠟⠃⠀⠀⠀⣠
⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⢿⡀⠀⠀⢰⡏⠀⠀⠀⡀⣒⠍⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠈⠳⣶⠶⠋⠀⠀⠀⠀⢲⡟⠀⠀⢀⠃⠀⠀⠉⠙⠀⠈⠈⠉⠉⠉⠉⠉⠀⠁⠀⠀⠓⠒⠒⠒⠒⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣼⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠄⠄⠀⠀⠂⠃
⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡴⠃⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠀⠀⠀⠀⠀⠀⠠⣴⣿⣶⠀⠀⠠
⠀⠀⠀⠀⡠⠇⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠴⠚⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣄⠀⠀⠆⡨⠀⢀⠀⠀⠀⠰⢤⢄⣀⣛⢀⣀⡠⠄
⠀⠀⠀⠰⠇⠀⠀⠀⠀⠘⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⣀⠀⠀⠀⡠⠜⠀⠀⠀⠰⣤⣀⣀⣀⣤⣤⣤⠄
⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⢸⡟⠁⠀⠀⠁⠀⢀
⠀⣠⢔⣊⡀⠠⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠀⠀⢀⠜⠁⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢀⠄
⠘⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠂
⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⣀⣄⣠⣄⣀⣄⣤⣀⣀⣠⣠⣄⣉⣉⣀⣀⣄⣀⣀⣀⣀⣀⣀⣄⣠⣤⣄⣠⣀⣀⣀⣀⣀⣀⣀⣠⣀⣄⣠⣀⣀⣀⣀⣀⣀⣠⣀⣀⣄⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⠑⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠆⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⣤⣦⣰⣷⠁⣵⠃⠀⠀⠰⠀⢳⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣿⣿⣿⣿⡿⠀⢸⡄⠀⠀⠀⠀⢸⣿⣶⣦⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠉⠀⠀⠀⠻⣦⡀⠀⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣋⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡔⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣝⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠶⠦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠀⠉⠻⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣾⣅⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⡴⠃
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠙⠋⠉⠀⠀⠈⠉⠉⠀⠀⠈⠀⠁⠀⠉⠀⠉⠉⠈
⠀⠀⠀⠁


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠰⠶⣶⠶⠀⣶⠶⠶⣦⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⣿⠀⠀⣿⣀⣀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⣿⠀⠀⣿⠉⠉⠁⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠟⠀⠀⠛⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤⣀⡤⡄⡤⠄⣤⢠⡄⠀⡄⡤⣄⡤⡄⡄⣤⣠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣰⣷⡗⡇⣟⡀⡿⡾⡇⠠⡇⡗⠋⣙⡦⣇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠛⠛⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠉⠛⢛⣋⣉⡉⠉
⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣿⢾⡆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣄⠀⠀⢾⡛⣿⠛⠡⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠉⠙⣿⢇⣸⡻⠟⠁⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⣿⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⢀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠈⠅⠈⣻⡿⠂⠉⠀⠀⢰⡾⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⢿⡀⠀⣾⠇⠠⣶⣦⣤⠀⢠⣶⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⢴⣿⡆⠀⠀⠋⢻⣷⡀⠀⠢⠉⠀⠀⠄⠋⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⢧⠈⢿⣄⠙⢃⣠⣿⣿⣟⣀⠈⠁⣾⠁
⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠁⢴⣿⣿⡿⠂⠀⠉⣼⡇⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⠋⠘⣦⣀⠹⠧⠘⠋⠉⠙⢿⡁⠀⣀⡉
⣿⣵⠀⠀⠀⠀⠀⠛⠛⡟⣳⠀⠁⣾⣬⡙⢻⡄⠐⠚⣯⠐⠂⢠⣿⣿⣾⠛⠛⠛⠂⣀⠀⠀⠀⠀⠈⠙⠛⣼⠟⢀⣿⣿⠣⠶⠂⢻⡧⢘⡉⣇
⣿⡿⢟⠀⠀⠀⠀⠀⢚⣿⠋⠀⣄⢠⢹⣿⣿⣷⣶⢶⣀⣤⡄⢸⣿⠛⠛⠃⠀⠈⠳⠉⢹⣿⡆⠀⣠⡶⠐⢉⣀⠈⢉⣿⣧⡀⠀⠛⠠⣾⡧⡯
⣿⡿⠛⢧⡄⠀⠀⠀⢸⡇⠀⠀⢠⣤⣾⡅⠘⠙⡟⠛⢱⣶⣶⣿⠁⢀⣀⣀⣀⠀⠀⠛⠺⣿⣷⠀⣠⣶⠆⠈⠻⣠⣇⠘⠋⠀⣠⣾⣦⡄⠀⣠⡄
⣿⣥⣤⠀⠉⠀⠀⠀⢸⣿⡆⢀⣸⣟⣿⡇⠀⢀⣁⠤⠀⠀⠀⣀⣠⡈⠝⠛⠛⠛⠓⢤⡀⠈⠃⠀⣿⠇⠀⣀⣴⡟⠛⠷⠶⠿⣿⠟⠛⠀⣠⣿⣇
⣿⣿⣿⣿⣷⣆⡀⠘⣾⣿⡀⣼⣿⣿⣽⣷⡾⢋⣤⣄⣶⠖⠛⣉⣯⣿⡶⠠⠠⠄⠀⠀⣧⡀⠀⠰⠇⣠⠞⠋⠈⠑⢶⣶⣶⣶⠇⠀⣀⣸⣿⣿⣿
⣿⡟⣿⣿⣿⣿⡷⢶⣿⣿⠷⣿⣿⣿⣿⣯⣡⣤⣀⠈⠉⠀⢄⡙⢷⣿⣿⣷⠀⠀⢀⣶⣿⡇⠀⠀⠈⣁⡠⣦⡀⠠⠴⣻⣿⣿⠀⠰⣿⣿⣿⣿⡿
⣿⣶⣟⣛⣁⡀⢁⣀⠿⡻⠶⠟⠛⠛⠿⠋⠉⠉⠛⠃⠄⠀⠀⠈⠃⠼⠛⣾⣷⣠⣿⣿⣿⡇⠀⠀⠀⠏⠀⠉⢀⠄⢾⣿⣿⡁⠰⢷⣏⢸⣿⣿⣿
⣿⢲⣿⣿⣿⣿⣾⣭⣦⣤⣶⣶⣶⣶⠶⣀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⡇⠀⠀⠀⢠⡌⠁⣀⠀⣾⡟⢛⣁⣰⣾⣿⣿⣿⡿⠏
⣿⣈⣿⣿⣿⣟⣿⣿⣿⢿⣿⣿⣷⣿⣾⣿⠿⣿⣟⡄⠀⣿⡀⠐⠀⠀⠶⠶⠶⣤⣍⢿⣿⣄⣤⣾⠂⣿⣴⣾⣯⡙⠲⠾⣿⡿⠿⣿⣿⣤⣿⣷⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡇⢸⣿⣷⣀⠀⢀⡀⢦⣤⠮⠁⢸⣿⣿⡿⢃⡼⠋⣤⣤⣴⣿⡿⠿⠿⠷⠶⣶⣿⡇⠸⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣣⣤⣶⡾⣿⣿⣿⣷⣮⠉⠉⠉⣠⣬⣝⣂⣤⣶⣶⠦⠜⠉⣠⣮⣴⣾⣿⣿⠟⠉⣠⡶⠶⠄⠀⡀⠘⠇⠀⢻⣿
⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣹⣿⣦⣼⣿⣿⣿⣿⣿⣶⣞⣹⣻⡿⢿⠿⠟⠻⣶⣶⣾⣿⣿⣿⣿⣼⠛⢀⠈⢠⣴⣶⣦⣄⠀⠀⠀⠛⢨⣿
⣿⣶⣴⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⣿⡦⢤⣤⡄⣼⣿⣿⣾⡿⠻⣿⣿⠀⠀⠀⠈⠉⠉⢿⣿⣧⡀⠠⣶⠘⣿
⣿⣿⣿⣿⣿⣿⣿⣇⡴⠃⢸⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣇⠰⣿⣿⣿⠂⢤⣴⣾⣿⡷⣟⣁⣤⣌⠻⣇⠀⠀⣴⢠⡴⠿⣿⣿⣿⣰⣿⡀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣼⣿⣿⣿⣿⠰⣿⣿⣿⣿⠟⠋⠉⠉⠀⠈⠛⣧⡀⠀⣷⣾⣿⡷⣈⣧⣿⣿⣷⣤⡘⢛⣶⣧⣄⢀⣿⣿⣿⡿⠿⣇⠈
⣷⢨⣽⣿⠿⣿⣿⠋⠳⣄⣽⣿⣿⣿⣿⣧⣸⣿⣿⣷⣀⡀⠀⠀⠀⠈⠁⠈⠻⠶⠟⢿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⠿⠛⠛⢋⡙⠏⠉⠃⣠⡋
⣿⣿⠿⣿⠆⠉⡀⠀⠀⢙⣿⡿⠟⢻⣿⣿⣇⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⡀⣴⣦⠀⠰⢿⣿⣿⣾⢻⣿⣿⣿⡟⢡⡀⠀⠀⠉⠙⠀⠀⢠⡿⣴⣗
⣿⣿⡶⠋⠀⠘⢷⣾⡷⣈⣥⠀⣦⣼⡟⠀⠛⣿⣿⣿⣿⣿⣟⣿⣿⠟⢩⡅⢨⡿⠟⣳⣶⣼⣽⡟⠘⣿⣿⡿⠀⠙⡷⠀⠀⠀⠀⠀⠀⢸⣄⣿⡟
⣿⡿⠃⠀⠀⠀⣠⣿⣷⣼⣿⣾⣿⣧⡀⠰⢿⣿⣿⣿⣿⡟⢿⡾⠃⠀⢀⣴⣾⣧⡾⣿⣿⣿⣿⡇⢸⣿⣿⣅⠀⢸⣿⡘⠷⡀⠀⠀⠀⠬⣿⣿⡄
⣿⡟⠀⠀⠁⢸⠟⠁⢈⣧⣈⣿⣿⣿⠁⢸⢸⣿⣿⣿⣿⣷⣾⣁⣤⠀⡸⣿⣿⣿⣿⣿⣿⡷⢿⣿⣄⠛⠿⣾⣷⣼⣿⣷⣄⣹⣤⣀⢻⣤⣿⣿⡿
⣿⡇⠀⠀⠰⢏⠀⠈⣿⣿⣿⣿⣿⣿⠀⠘⠈⣿⣿⣿⠟⣽⣿⣋⣩⡿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⡔⢢⣼⣾⣿⣿⠣⡜⠛⠈⠛⠻⣿⣿⣿⣇
⣿⣿⡆⠀⠳⠊⠀⠀⣿⡿⣿⣿⣿⣿⠀⣦⠀⣿⡿⣯⡇⢀⠉⣿⠛⢿⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⡿⠋⣽⠁⠘⣴⢷⡀⢀⣈⠉⠙⣿
⣿⣿⣿⡄⠘⠀⠀⣴⣿⡇⢹⣿⣿⣿⣀⠋⠀⣿⡇⣿⠀⠘⠀⣿⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⢀⣤⣤⣤⣶⣾⠧⣸⣿⠀⠀⢸
⣿⣿⡏⠀⠙⠷⣶⣿⠿⠃⠈⠏⢹⣿⣿⣛⣾⣿⣷⣿⠀⣀⣰⣏⠸⣦⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣤⡉⠉⠙⠻⣿⣷⣾⠟⠁⠀⠀⠸
⣿⡟⣃⢲⡄⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⢀⣤⣴⣿⣿⣿⣿⠋⠀⠈⠀⠀⠀⠀⢠⣄⠩⠝⠀⠄⠶⠛⢻⡇⠀⠀⠀⠀⢸
⣿⣿⣿⣶⡇⠀⠀⠄⠀⠀⠐⣷⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢾⠟⠛⠙⠛⠿⠿⣿⣾⣥⣿⠟⠛⠳⡟⠉⠀⢀⣴⠞⠀⠀⠰⡇⠀⠀⠀⢶⣸
⡿⢻⡟⢹⣿⣿⣶⣶⣶⣶⣶⣿⣾⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⡀⠀⠉⠉⠻⢦⡀⠀⠇⣠⣿⠿⠇⠀⠀⣠⢼⣷⡄⠀⠀⢸⣿
⣿⢈⢡⣿⣧⠟⠛⣿⠙⠛⠋⠙⠛⠃⠀⠀⠀⠀⠈⢠⡄⠤⠀⠐⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠇⢀⡉⠉⠶⢶⣶⠞⣵⠟⠉⣤⣀⣠⣾⣿
⣿⣿⣿⣿⣽⠀⢹⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡆⠻⠇⠀⠘⣤⠀⣠⣿⣯⣤⣐⠂⠀⣠⡞⠁⠀⢰⣽⣿⣽⣿⣿
⣿⣿⣿⣿⣿⠀⢾⣟⠃⠠⠀⠀⠀⣤⣀⣱⣟⡁⠰⢄⠀⠀⠀⠀⣀⢠⡼⢧⡾⠃⣶⠆⠀⠀⠉⠀⢸⣿⠿⠟⠛⠛⠋⠉⠉⠀⠴⣿⡁⣿⣿⠛⣿
⣿⠘⣿⠛⣿⠀⣼⣟⠀⠀⠐⠃⠀⣿⣿⡟⣿⣷⢠⠀⠀⠘⠻⠦⠿⠟⠛⣬⡟⢸⡟⠀⠀⠀⠀⠀⠈⣿⠀⡀⠀⣠⡀⠀⠹⠗⠀⠛⠛⠛⠛⠛⠋
⣿⣰⣿⠀⣿⠀⣿⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣷⡰⢿⡖⠀⣴⣶⣿⡟⠁⣸⠇⠀⢠⡀⠀⠀⠀⢿⣿⣿⣷⣿⣧⣤⣤⣤⣀⣠⣤⣶⣶⣶⣶
⡁⣿⣿⣶⡟⠀⣿⣿⠈⠀⢀⡀⠀⣿⣿⣿⣿⣿⣧⠘⠇⢸⡿⣦⣤⣉⣉⠉⣰⡟⠀⠀⠘⣷⠀⢰⠀⢸⣿⢿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿
⣿⠈⠁⠈⣿⠀⣿⡟⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⡇⠀⢻⡆⠉⠉⠁⠀⣉⣿⡇⠀⠀⠀⣿⠀⠸⡀⠀⣏⣰⣿⣿⣷⣾⣷⠿⣋⣹⣿⣿⣿⠈⣿
⣿⠀⠀⠀⠁⠀⣿⣿⣀⠀⠀⠀⠠⣿⣿⣿⣿⣟⠹⣿⣷⣄⡀⠀⣠⣴⠾⣻⣿⠃⠀⠀⠀⣿⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿
⣿⣰⡀⠀⠀⢠⡿⠛⠛⠶⢄⠀⠀⣿⣿⠿⡟⢿⠶⣏⣹⣿⣿⡿⢛⣭⣶⣿⣿⠀⢀⡀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠙⠣⣄⠀⣠⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣯⡀⢀⡔⢋⣀⠀⠲⠿⠿⠟⣼⣿
⠿⠄⠀⠀⠀⠒⠀⠀⠀⠲⠊⠀⠚⠛⠉⠉⠉⠩⡄⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⣤⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⣟⠉⠀⠀⠀⠀⠀⠀⣨⣉
⠒⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣀⣀⣀⣀⣉⣉
⢁⣸⣿⣶⣆⣀⣀⣀⣀⡈⣳⡀⡀⣀⡀⠀⠀⣀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣥⣤⣠⣀⣀⣠⣤⣤⣤⣬⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿
⣿⣿⣿⣿⠟⠉⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿