Long hair Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⣾⡿⠻⠯⣝⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣴⢿⣿⡗⢌⡛⠶⣝⣾⣯⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣸⣿⢸⣌⢿⣶⢭⣒⢬⡙⢿⣷⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠸⣹⢦⡙⠷⣬⣛⢿⣟⢿⣆⠹⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠙⠳⣍⣓⠶⣬⡓⣿⡆⢻⡄⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡤⠈⢹⣼⣿⣿⠃⣸⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⢄⠀⠀⠳⠦⠞⣻⡿⢃⡴⡣⣾⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣼⠋⣰⢏⢔⣼⣿⣿⡌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⣸⢣⠀⣾⡟⢎⣷⣝⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⢼⢸⣿⣼⣦⠍⡛⠳⠮⣽⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣻⡌⣾⡏⡷⡍⣻⣟⣛⣒⠲⠬⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠸⡇⡘⡧⡹⣿⢝⣿⣿⣷⡶⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠠⠀⢸⢃⣧⡿⣖⠮⢕⡻⠩⣿⣿⣽⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⣀⡤⣫⣾⠻⣌⡲⠭⠥⣒⣵⡿⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠿⢟⣛⠷⢮⡍⠛⠻⣻⡗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⢰⣼⡿⣼⠉⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣝⣷⣶⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢟⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⡒⢦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⠁⠀⠀⠛⠂⠀⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣶⣄⣴⣿⡿⣛⣭⣽⣧⣄⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣀⠴⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠊⠉⠉⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣥⣤⣶⣶⣶⣬⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠈⠻⢿⣿⣿⣿⣮⣭⣭⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⠹⠛⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣶⡘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣄⣀⣀⣀⠠⣤⣤⡀⠀⢢⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣮⣹⣄⡘⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣷⣿⣿⣤⣄⣀⡀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⣐⣲⣤⣶⣶⣿⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠀⣉⣭⣿⣿⣿⣿⡿⠥⢀⡉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣴⢁⡀⢨⣿⢿⠟⠉⠉⠙⠢⣤⡀⠀⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⢧⠹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢿⡄⣸⣰⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⣿⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣿⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡏⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣘⠎⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡛⢝⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣿⣯⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⠹⣿⣍⠙⣄⠁⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣙⢷⣜⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⣟⣻⣿⣧⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣭⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢔⣿⠿⠿⠿⠛⠛⠛⠉⠑⠒
⠀⠀⠀⠀⠀⡠⣰⡟⠏⠀⣠⢖⣫⣤⣄⣤⣀⣙⠛⠷⣮⣽⣲⣤⡀
⠀⠀⠀⠀⢰⠃⣿⢨⠀⠀⣿⣿⠵⣛⣩⣵⣿⣭⣽⣛⠶⣝⢿⣿⡿⡆
⠀⠀⠀⢀⣸⡆⠻⢎⠀⠀⠈⠣⣾⣿⠋⠀⠛⠻⠿⡟⣿⣦⡑⠙⣷⡙⣦⢀
⠀⠀⠀⢸⣯⣈⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠘⠟⣿⣷⡘⣧⢈⢻⣷⣄
⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠇⢹⠈⢷⡙⢿⣷⣄
⠀⢾⡇⢀⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣿⠀⣄⢻⣦⡹⢿⣷⣄
⠀⠈⠁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⠘⣹⣦⡙⢿⣦⣝⢿⣿⣄⡀
⠠⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⡄⣷⣽⣿⣦⣝⢿⣷⣬⡻⢿⣦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡏⣽⣟⢷⣬⣙⠷⣮⡛⢿⣷⣽⡻⣆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡹⣿⣷⣌⣿⣿⣿⣷⣦⣬⡛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡕⠹⣮⡻⢿⣿⣿⠟⡻⠿⢿⣷⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣬⣙⠷⣦⣤⡤⠤⠄⣠⢠⠹⡄⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠿⠿⠶⠖⢲⠶⢿⡵⢟⡄⢧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣂⡠⣴⣶⣶⣖⠶⠖⣊⡙⢋⣵⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣼⡿⠀⣿⣟⣿⣾⣿⢟⣭⠾⢛⡵⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠘⠻⢫⡿⠁⠉⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠁⢦⢦⣀⠀⣙⣀⣀⣀⡤⠖⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢷⣭⣛⠶⠶⠶⣒⡂⢀⣴⡇⠀⠹⣶⣤⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠛⠋⠭⣶⢿⣿⡇⠀⠀⠪⣛⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⢀⢂⠘⣎⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡈⠉⠀⢠⣾⣿⠇⣹⣿⣷⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣿⡾⣛⢉⣥⠞⣰⢟⠞⠉⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣴⡿⢋⣾⣷⣫⢡⡾⠃⢀⡀⠀⠀⡿⣸⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⣿⡟⣵⣷⠏⡴⣴⠫⠂⠀⠸⢃⡿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣧⣿⢿⡼⡇⠃⠀⠀⠀⠀⠞⣡⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢦⡙⠆⢻⣆⠀⠀⠀⠀⠤⠞⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠻⣆⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠜⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠲⢦⡀⠢⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣧⠀⠈⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢰⠀⡟⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⣀⣀⣠⣤⣤⣶⣾⣿⡿⢣⡟⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⣴⣾⣿⣿⣿⣿⣿⡿⠿⠟⣋⣴⡿⠁⠀⠀⢹⡏⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⡿⠟⠋⠉⠁⠀⠀⣠⣴⣾⣿⠟⣤⠴⠖⠒⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠀⠀⠀⣀⣤⣶⣿⣿⡿⠛⠁⠀⠀⣀⣠⣄⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⢀⣴⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⡿⢀⣿⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣿⡇⢸⣿⣧⠀⠀⠀⠀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⣿⣿⣿⣿⣿⡇⠸⣿⣎⠣⡀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣦⡈⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⡀⠙⢿⣿⣦⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣄⠀⠉⠛⢿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣀⠀⠀⠉⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⢠⠂⡆⠀⠉⠛⠻⢿⣿⣿⣷⣦⣀⠀⢤⡀⠈⠙⠿⣿⣦⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⠀⣿⣄⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣤⡙⢢⡀⣄⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢻⣿⣷⣄⠀⠀⠀⠀⣠⣤⣈⣛⣿⣿⣄⠱⠸⣷⡀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⢻⣿⣭⣛⠲⠤⠀⠀⠈⠙⠻⣿⣿⣿⡄⠀⣿⣧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠙⢿⣿⣿⣶⣤⣀⠀⠀⠀⠘⣿⣿⠁⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⡄⢀⠀⠟⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠈⢉⠹⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⠀⠀⠀⠙⢦⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠠⢄⡀⠀⢀⡠⠄⠈⣧⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠴⡦⠌⠀⠈⢤⡦⢰⢹⠈⡎⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⡞⣾⢸⣇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢠⠐⠿⠃⢀⡼⣱⣿⣿⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡎⣿⣶⢣⡞⣱⣿⣿⣿⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡟⣿⣳⢟⣼⣿⣿⣿⡏⢠⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⡿⢃⡾⢃⣾⣿⣿⣿⡟⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠖⢚⣽⡾⠋⢠⡞⢁⣿⣿⣿⣿⡿⢀⢠⡏⠀⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⣰⡟⠀⣾⣿⣿⣿⣿⠇⣾⣼⠀⡇⠸⣧⠹⡄⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⣼⢃⠔⠀⠀⣰⣿⠁⢰⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⡄⠹⣧⡙⢦⣀⡀⠀⣀⠜⣼
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⡀⠹⣿⣦⣈⠻⢦⣬⣭⣭⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⡀⠀⣿⣿⡀⠈⣿⣿⣿⣿⣿⣧⠙⣿⣧⡀⠙⠻⣿⣿⣷⣦⣴⡶⠶⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡘⢿⣷⡀⠘⢿⣮⡛⢿⣿⣷⣌⠙⠻⠦⡀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣽⣿⣦⣈⠻⣿⣦⣄⡉⠉⠛⠂⢀⣀⣤⣶⡾⠟⠋⠁⠀⠀⠀⠉⢻⣄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣄⠉⠛⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣾⡭⢤⠀⠒⠂⠴⠦⣀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠏⢁⣠⡤⠀⠀⠀⠀⢀⠀⠈⢷⣦⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣴⠋⠁⠀⠀⠀⠀⠀⠀⠙⡄⠀⣿⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣯⠀⢸⡀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁⠘⡇⠸⡇⠂⠠⠄⠀⠀⠀⠤⠀⠀⠄⢸⡟⠆⠈⢳⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣮⡵⠟⠋⠐⠀⢙⡄⡳⠾⠛⠦⠀⠀⠀⠐⣛⠶⠒⢘⡧⠈⠢⣄⠀⠉⠑⠒⠒⠲⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⣏⠉⠙⠚⠒⢛⠛⠶⣦⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠜⢴⣴⡇⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⢷⣄⡀⠈⢷⣤⣌⠿⣶⢷⡀⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡄⠈⢻⣇⠀⠀⢧⠠⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠙⠳⣄⣀⠈⠻⣷⣴⣀⢳⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢹⡆⠀⣹⣦⢰⠹⡆⠈⠘⣦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣵⣍⠳⣆⠈⢷⡀⡀⠹⡌⠪⣛⢷⣤⣀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡸⠃⡰⠘⣿⢿⡇⢷⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⡀⢷⠘⣆⢱⣄⡈⠙⠺⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⡠⢊⣴⠞⠁⢠⠇⢸⣷⡜⠀⡠⠓⠁⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡀⠀⠐⡾⠀⢈⢾⣟⣬⠉⠉⠳⣝⡙⢟⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⣠⢚⣱⠋⠁⢀⡴⠋⣠⢺⣿⠇⣼⠁⠀⣠⠂⠀⠀⠀⠀⠀⠀⣠⠎⠀⢼⣧⡤⠞⢡⡇⠀⢿⣇⠨⣧⡀⠀⢸⣷⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡞⢁⣠⠖⣋⡴⢚⣡⡶⣿⢰⡯⢠⠞⠁⠀⠀⠀⠀⢀⣴⠿⡥⠒⠚⠉⠀⠀⠀⣼⠁⠀⢺⣧⠀⢮⠟⠆⠘⣿⡇⠀⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡜⣡⢞⠏⢚⣿⠏⠀⠁⢾⣿⠃⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠣⡀⠀⠀⢠⡇⠀⠀⣸⣿⠀⠀⣼⠀⠀⣿⡇⢠⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⡟⠀⠁⢀⣾⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡼⢀⡃⠀⠀⡞⠀⠀⠀⢣⠁⠀⠀⣿⠀⠀⡎⢷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠤⠊⡾⠁⢀⠔⢠⠄⠁⠽⢧⡀⠀⠀⠀⠀⢀⠉⠓⠒⠊⢳⣼⠃⠀⣼⠁⠀⠀⢂⠌⠀⠀⠠⡏⠘⡈⢧⠈⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠐⡁⠀⡼⠔⣱⠀⠀⠀⠀⠙⠲⠄⠤⠔⠋⠀⠀⠀⠀⣼⡇⠀⢸⠃⠀⠀⠀⡌⠀⠀⣠⢸⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣷⡀⢃⣠⣯⠀⠿⡀⢦⡀⢐⣠⡀⠀⠀⠀⠀⠀⠀⢀⡔⣿⠀⠀⢼⠀⠀⠀⡸⠁⠀⠀⣽⡆⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢹⡍⢮⡨⠻⣦⡘⠿⢤⣬⣓⣿⣿⣷⣄⠀⠀⢀⠔⠉⣼⡇⠀⣆⢸⡀⠀⢠⠁⠀⠈⠀⢸⡄⠀⠀⠀⠀⡆⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠎⠀⠀⢳⣼⣜⡝⠷⠄⡀⠉⠉⠻⣮⡛⣆⡔⠁⡠⠊⡅⢸⢦⡘⢌⢷⡄⠈⢆⠀⠀⠀⠈⠓⠦⣤⡤⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣩⠞⠁⠀⠀⠀⠈⣿⣿⠀⢁⠀⠈⠑⠦⣄⡇⠉⣼⡅⠜⠁⢸⠃⣸⠀⠈⠐⢿⣿⡶⣦⡑⢄⡈⢦⠈⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣰⠋⠀⠀⠀⠀⠀⠀⠱⣟⢠⣸⢧⡀⠀⢀⡼⢿⣆⣾⠈⠓⠤⡘⣿⡗⠀⠀⢰⠀⠁⣹⡀⠉⠲⢝⡒⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠠⠀⠀⠀⣠⡿⠃⠀⢻⡏⠿⠻⠿⠖⢺⡏⣸⠀⠀⠀⠀⠈⠑⠒⠒⠋⠀⠀⣸⠇⠀⠀⠀⠁⣮⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠙⠊⠉⠀⠂⠀⠀⡇⡄⠀⠀⢀⣜⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⢀⣔⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⡷⠧⠶⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠖⠛⠛⠛⠛⢛⣩⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⠇⠀⠀⠋⠁⠀⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠞⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⢋⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⢏⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⠃⠸⡄
⠀⠀⠀⠀⠀⣀⣤⠞⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠙
⠀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠀⠀⢠⠏
⠀⠈⠙⠛⠻⠿⠿⠿⠛⠛⠋⠉⣡⡾⠟⠁⠀⠹⣿⣇⠀⠃⠙⢿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⠇⠸⣿⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠈⠙⠛⠟⠋⠀⠀⢻⣿⢻⡆⠀⠙⠻⠶⠂








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⢀⠀⣀⢀⣠⣀⡀

⠀⠀⠀⠀⢀⡀⠀⠐⠦⣄⣀⠀⠀⠀⣀⠀⢀
⠀⢀⠀⡠⠚⠒⣶⡄⠀⠀⠠⣘⣊⡩⠀⠐⠀⣈⠢⡀⠀⠀⠀⠀⠠⠤⡀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢨⠎⠀⠀⢠⠈⠸⢗⠐⠛⠉⠉⠉⠩⠈⠀⢙⢧⣈⢦⣀⠀⠀⢀⠤⢼⠧⠀⠠⠦⠔⠀⠤⠐⠒⠦⣄⡀⠀⠀⢀⠅
⠀⡆⢀⣉⠀⠈⠲⢄⡄⠢⢀⢀⠀⠀⠀⠳⡄⠑⢦⣍⡛⠫⣉⣀⣉⣠⡮⠴⠂⢀⠉⣛⣉⡛⣒⡾⠶⠤⠤⠬⢍⣀⢤⣀
⠀⠀⠀⠉⠀⠀⠀⢸⠇⡐⠀⢿⠑⠶⠶⢶⣬⣆⡀⠘⠿⣶⡴⠉⠁⠀⠠⡤⠬⡿⠿⠵⢄⠀⠢⠉⠓⠦⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⡏⠠⠁⣤⢈⢗⢋⡀⢆⠻⠿⣿⣗⠢⢌⠹⠓⢶⢛⠟⠲⠶⠾⣄⠒⠢⣕⣄⠀⣑⢀⠀⠀⠈⢉⣤⣀⢀⣀⠀⠀⠀⠀⠀⠔⠐
⠀⠀⠀⠀⠀⠀⠘⡄⡆⠀⢾⠆⠱⣄⠑⠊⡶⠶⣌⠻⠷⣬⡕⣂⠀⠀⠀⠀⠀⠠⠄⠙⠆⡈⠙⠙⠻⣟⣿⣷⣀⡀⢀⡀⠀⠈⠑⠲⢤⣀⡀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠙⠐⠿⣮⣧⡀⠈⠙⠉⠀⠀⠈⢦⠻⢯⡛⠷⠍⠀⠀⠀⡰⠉⠀⠉⠒⠾⡶⢦⣤⠾⣫⣭⣍⠉⠙⠙⠲⠶⠤⠄⠀⠀⠉⠁
⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠟⠛⠿⢟⠛⠛⠁⣀⢈⠢⡈⠛⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⠤⠀⠐⠀⠛⠻⢄⠀⡄⡀⠀⠀⠀⠐
⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠚⢂⣀⡀⠙⠉⠁⠀⠀⠀⣈⠢⡀⠙⠳⣖⠦⣦⡤⠄⣀⣐⡸⠷⣌⠑⠲⣶⡐⠢⢄⢀⣤⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠓⠲⢯⡉⠠⡍⡉⢩⣭⣭⡀⠀⠁⢳⡀⠀⢀⠉⠁⠺⢵⣤⣌⣉⣉⡊⠱⣦⡀⠉⠒⠂⠐⠲⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⢦⠘⠻⠀⠀⠉⠃⡄⣿⣾⣗⢤⣤⠄⠀⠊⠙⠉⠉⠉⠣⠈⠙⢢⠀⠉⠀⠒⢄⣀⣘⣦⣎⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⠀⡘⠀⠈⢣⡀⠀⠀⠙⠤⢾⣿⣷⣤⣁⠀⠀⠀⠠⠒⠀⠀⠇⠀⠈⡗⠀⠡⠌⠶⠤⠤⠤⠄⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠈⠀⠀⠈⠉⢻⠐⠢⠄⠃⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣛⣿⣧⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣡⠀⣴⣾⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣷⣶⣿⡿⠿⢛⣛⣿⣿⣿⣛⣥⣿⣿⣿⣿⣿⡆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣟⣡⣶⢾⣿⣿⣿⠿⢟⣹⣿⣯⠙⠛⠙⠛⣦⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣭⠶⠋⠁⠀⠀⠀⠈⢷⡝⠻⣶⣷⣄⣘⣿⣷⣽⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢡⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠙⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⣟⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⡿⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠛⠛⠻⠷⠶⣦⠀⠀⠀⠀⣴⠶⠿⠛⠛⠉⠉⠙⠀⣿⣿⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⣠⡖⢻⣷⢢⠀⠀⠀⠀⠀⠀⠀⣚⣟⡛⠶⠄⠀⢠⣿⠋⣸⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢠⠻⢻⣧⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣴⣿⡇⠀⢻⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣽⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡄⠀⡿⣸⣇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⡇⣿⠀⢿⡍⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣾⣟⣡⣿⡇⣿⣶⠘⣷⢹⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣿⣿⣿⣿⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣼⣿⣿⠀⣿⣿⡇⣹⢯⡄⠰
⠀⠀⠀⠀⠀⠀⠸⡄⠸⣿⣿⡆⡿⢿⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣴⣿⡿⣧⢹⡌⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣇⠁⠨⣿⣿⢣⡟⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣡⡘⣿⣿⣱⣿⢸⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣧⣰⢹⣇⢻⣦⡀⢹⣷⡏⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢡⣿⣧⠹⣿⣿⣿⣾⡇⢹⣯⢻⡆
⠀⠀⠀⠀⠀⢀⡼⣿⢻⡆⣿⣆⢽⣷⣼⠋⣼⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠘⣿⣷⣝⣿⣿⣿⢁⠸⢧⠀⢿
⠀⠀⠀⠀⠀⢚⣿⢣⣾⣷⣼⣿⣮⣿⣿⣦⣹⣿⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣷⢸⣿⣿⣿⣿⣿⣿⣼⠀⡏⡦⢸⡇
⠀⠀⠀⣠⢾⡏⠸⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣴⢻⡄⣿⡇
⠀⠀⣰⠋⢈⣧⡾⢟⣼⣿⡟⣼⣿⣿⣿⣿⣟⣼⢟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣿⢸⡇⣿⡈⣿⡟⣿⣿⣿⣿⣿⣾⣷
⠀⢰⠁⡠⠋⠀⢀⣾⣿⡿⣸⣯⣿⣿⣿⡿⢋⣵⡿⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠸⣯⢷⣌⣷⡘⣿⣿⣿⣿⣿⣿⡝⢿⡄
⠀⠇⠐⠁⠀⣰⣿⣿⡿⣧⣿⣿⣿⣿⣿⣿⣿⠟⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⢷⣜⢿⣿⡾⣷⢉⡿⠹⣿⣿⣿⣇⠈⢷
⠀⠀⠀⠀⠰⢻⡟⡇⠃⣿⣿⡟⣱⣿⣫⣿⣿⣼⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣌⡙⠻⣿⣿⣷⢸⠇⣿⢹⣿⣿⠀⠈⠣
⠀⠀⠀⠀⠀⣸⠀⠁⠀⠸⢯⣾⣿⣿⣿⣿⣿⣟⣧⢡⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣼⣷⡝⢿⣿⣷⣦⣙⢿⣿⣠⢻⣿⡿⣏⢣
⠀⠀⠀⠀⢠⣿⣦⠀⠀⣰⣿⣿⣸⣿⣿⣿⣿⣿⡎⣷⡿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢸⣿⢸⣿⣮⢻⣷⣜⢿⣿⣿⢻⣼⣿⣿⢡⠀⢣
⠀⠀⠀⠀⠋⢈⠻⣦⡰⣿⢱⡇⣿⡿⣿⣿⣿⣿⣤⠙⣿⠹⡄⠀⠀⠀⠀⠀⠘⣿⡿⠀⢸⡟⣼⣿⣿⣧⣿⣿⣧⢻⡿⡄⢿⣿⣿⠀⣧⠀⢃
⠀⠀⠀⠀⠀⠘⠀⢸⠿⣇⠸⣧⣿⠀⢸⣿⣿⣿⣿⠀⠸⠂⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⡇⣿⣿⡏⣿⡏⢿⣿⣿⢻⣿⣎⢿⣿⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠛⣠⣿⣇⣼⣾⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠃⢸⢠⠟⢀⣿⣿⠁⣿⡇⣸⣿⣿⣼⠉⣿⣶⣿⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⣿⣿⣿⣼⣿⣿⢋⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⡟⠀⣿⢁⡇⢿⣿⡟⣄⣿⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠝⣹⣿⡿⠃⢨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣰⡏⣼⠀⣾⢹⣧⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠠⣿⣷⣿⠀⡏⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣿⣿⢳⡇⠀⠀⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠹⡇⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠾⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢀⣀⣸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣷⡀⠀⠀⢀⣿⣧⣤⣀⡀
⢠⡾⠟⠛⠛⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⣾⡿⢿⣦⣤⣾⡟⠉⠉⠙⢿
⣿⠁⠀⣾⠟⠙⣿⣦⣀⣀⣸⣿⣷⣄⡀⢀⣠⣼⣿⣧⣀⠀⣀⣴⡿⢿⣧⣀⠀⣀⣼⡟⠻⣷⣄⣀⣀⣼⡟⠛⢿⣶⣴⡿⠏⠀⠀⠉⠉⣿⠙⢿⣆
⢻⣷⠾⣿⣦⡀⢸⣿⠛⠟⠋⠁⠈⠛⠻⠟⠛⠉⠀⠙⠻⠿⠟⠋⠁⠀⠙⠻⠿⠛⠋⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣠⣾⡿⠇⣰
⠈⢻⣦⡙⠟⠁⢈⣿⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣾⣇⠙⠟⣁⣴⠟
⠀⠀⠉⠛⠿⠻⣿⠛⣷⡀⠀⠀⠀⢼⣷⣀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⣀⣠⣿⠇⠀⠀⣼⡟⠙⠛⠛⢿⡏
⠀⠀⠀⠀⠀⠀⣿⡇⠘⢿⣄⠀⠀⠀⠘⠛⢿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⣤⡈⣻⡧⠀⠀⠀⠀⠀⠀⠋⠻⠛⠿⠋⠁⠀⢀⣼⠟⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠃⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡶⠿⠃⠀⠀⠀⢀⣠⣤⠴⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⢛⣿⣿⣶⠶⢶⣶⣶⣴⣶⣶⣶⣶⡶⠶⠾⢿⡿⢿⣇⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣾⠃⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⢹⣧⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣠⡿⠻⣧⣤⣤⣼⡿⠿⣶⣤⣤⣤⣤⣶⠾⠋⢱⣶⠷⠾⠟⠙⣷⡀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢰⡿⠁⠀⠀⠉⠁⢹⡇⠀⠀⠀⣸⣿⡄⠀⠀⢀⣼⡟⠀⠀⠀⠀⠘⣷⡀⠈⢿⡆⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣼⡟⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠙⠳⠶⠚⠋⠀⠙⠳⠿⠛⠉⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⢿⣄⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣼⡟⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⢿⣆⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⣼⠏⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⢻⣆⠀⠀⢸⣧
⠀⠀⠀⠀⢸⣿⠀⠀⢀⣾⠋⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠹⣧⠀⠀⢻⣦
⠀⠀⠀⠀⣸⡏⠀⢠⣿⠃⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠹⣷⠀⠀⣿⡆
⠀⠀⠀⢸⡟⠀⠀⣸⣿⣦⣀⣀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⣠⣴⣿⡀⠀⣿⠇
⠀⠀⠀⢸⣧⠀⢰⣿⠁⠈⠉⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠉⠀⢹⣧⠀⣿
⠀⠀⠀⠀⢹⣇⠸⣿⣄⠀⣠⣿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣦⡀⠀⣸⡿⠀⢹⣧
⠀⠀⠀⠀⣸⡏⠀⠈⠛⠛⠛⠁⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠛⠋⠀⠀⠀⢿
⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⢶⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⢶⣶⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣶⡶⠶⠶⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⣿⠉⠉⢹⣿⠉⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⣾⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡇⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⡿⠃⠀⠀⠹⣧⣀⣾⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣠⡤⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠓⠲⠤⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤⠀⠐⠀⠀⠀⠀⣀⣤⡾⠿⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠤⠄⠀⣈⡲⣶⣦⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀⣠⡶⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣲⣿⣿⣿⡿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢶⣾⣿⣷⣾⡆⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠑⠶⣬⣟⣋⠙⠹⠆⠀⠀⠀⠀⠀⠀⡌⢻⣿⣿⣃⣠⡿⠟⡇⠈⣿⡿⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣴⣿⡿⠃⣿⣿⡿⣇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠐⢸⡇⢀⣿⣿⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⣹⠟⠀⠀⣿⣿⡅⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⣼⠙⢠⣿⠷⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⣾⠟⠀⠀⢠⣿⣿⣷⠀⣽⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏⣿⢸⡿⠰⢹⣿⡏⢿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣾⠀⣼⡿⠀⠀⠀⣾⠉⣿⣿⠀⠛⣿⡠⠀⠀⠀⠀⠀⣀⣀⣤⠀⠄⣀⠀⠀⠀⠀⢀⠁⠘⢸⣿⣾⡇⠀⢸⣿⠇⠘⣧⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠀⠀⠀⠀⣿⠀⢹⣿⡇⠀⠋⣿⣦⣄⠀⠀⠀⠉⠙⣟⠛⠈⠉⠀⠀⢀⣠⡾⢠⣾⣿⣿⡟⢀⠀⠸⡇⠀⠀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠀⠀⠀⢸⣿⡄⠘⣿⡗⠀⠀⠉⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣠⠴⠛⣽⡇⢸⣿⣿⡿⠀⢸⠀⡆⠁⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠃⠀⠀⠀⣾⣿⡇⠀⢿⣇⠀⠀⠀⢹⣿⣿⡇⠈⠛⠦⣄⣤⠴⠛⠁⠀⠀⣿⡇⣧⣿⣿⡹⠀⠈⢠⠁⠀⠞⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⢸⢸⣿⣿⠀⢸⡇⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢀⡿⠀⣾⢇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⠇⡀⢸⢸⣿⣿⠀⢸⡇⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣿⠇⣰⠏⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⣴⣆⠀⢸⣿⣿⠀⣞⠀⠀⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣾⣿⣿⣹⡟⢠⢏⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⢿⣿⡤⢿⣿⡿⠀⡧⢠⡆⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠿⣿⣿⠃⣼⣼⣿⣿⠉⠉⠣⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠐⠁⠘⣿⣿⣿⠇⢀⠃⡘⣰⣿⡿⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠁⣿⠛⣿⣿⣿⣿⠄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⠋⠀⣾⠾⣷⡿⠋⠐⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⡉⠀⠀⠀⠀⠈⣿⢸⣿⣿⣿⡟⢠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣼⣯⡷⠋⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣼⣿⣿⣿⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣼⣿⡏⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡇⢠⠁⠀⠀⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠂⠀⠒⠠⣿⡄⢿⣿⣿⣿⠗⠋⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠂⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠒⠐⠀⠀⣾⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⠇⣰⠏⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⠉⣿⡆⣰⡟⠀⠀⠀⠀⠀⡇
⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠏⣴⠏⠀⠀⠀⠀⠸⢤⠄⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢻⣤⣿⣴⣿⡆⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⡇⠀⢀⣠⡴⠋⡁
⠀⠀⠀⠀⠀⡛⠈⠳⠒⠒⠋⠉⠓⠂⠄⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢸⣿⣿⣿⣿⠉⠀⠉⠁⣠⡾⠃
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠸⣿⣿⣿⡇⣀⣠⡤⠾⠋
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⠀⢀⡴⢿⣿⡏⠀⠉⡀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣾⣿⣷⡶⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣭⣍⣉⠻⢿⣿⣧⠀⠐⣤⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣆⠻⣿⣦⣀⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣧⠈⢉⣡⡼⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢋⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢃⣀⠤⣬⡉⠻⠆⠀⠀⠀⣠⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⢸⣿⣿⣿⣟⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⣿⣿⣿⣆⠀⠀⠀⠀⠠⠴⢦⣤⣤⡀⠀⠀⠀⢸⣿⣿⣿⡟⡜⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⠀⢃⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢱⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠀⢸⣿⣿⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸⡏⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢸⣇⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢸⠇⣷⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⣿⢿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⢸⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⡇⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⡆⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⣿⣿⡄⠁⠈⣿⣿⣿⣿⣿⣿⣿
⣄⣰⣿⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⠿⣿⠀⠀⢼⣿⣿⣿⣿⣿⣿
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠀⢳⠀⡄⣸⣿⣿⣿⣿⣿⣿
⡿⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢰⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿
⡟⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿
⠃⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡜⠀⠀⣸⣿⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⡆⠂⠀⠀⢸⡟⠀⠀⠀⠀⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⣰⡎⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡀⢀⢀⡄⢰⡇⣼⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⠏⠀⠀⢰⣾⣷⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⣼⣼⣇⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⠀⡆⢀⣾⣿⣿⠀⠀⠀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠀⠀⠀⠀⢀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⣰⠇⣼⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣤⣤⣤⣤⣤⣼⣧⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣾⣿⣿⣿⣿⣧⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠦⣀⣈⡉⢀⡠⠄⣀⠉⠉⠁⠀⠀⠀⠠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠀⠚⢡⠆⠀⠀⠀⠀⣈⠑⠀⠀⠀⠀⡀⠉⠀⣀⣀⣠⠤⠶⠚⠙⠶⠦⠠⠄
⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⢠⡏⠠⠀⠂⠀⠀⠈⠉⠒⠒⢒⡤⠶⠚⠛⠛⠃⠐⡀⠀⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠃⠘⠇⠀⡤⢀⠀⠀⠀⠢⡐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠤⣶⡛⠡⠀⠀⠀⠀⠂⠀⠈⠀⡀⠈⠓⢂⡀⡐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡘
⠀⠀⠐⠀⠂⢠⠎⠉⠛⠁⠀⠀⠀⠀⠀⢀⠙⠒⢦⣄⣀⡀⠀⠀⠑⠮⡃⠌⡠⠶⠤⠑⠒⠒⠒⠉⠀⠈
⠀⠀⠀⠀⠀⡎⠀⡀⠀⠒⠀⠀⠀⠊⠓⠒⠓⠒⡒⣚⣿⣿⣿⣶⡶⠞⣿⡷⢄⡀
⠀⠰⡀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠀⠀⠑⢛⠲⣤⠀⠋⠉⠉⢀⠌⠀⢩⡿⠰⡇
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠔⠛⠤⡆⠀⠊⠀⠀⢨⡀⠢⣤⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠬⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠤⢤⣤⣶⣦⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠷⣿⣿⣵⡿⠶⠶⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣭⣾⣟⣛⣿⣿⣿⣿⣤⣶⣿⡶⢮⡻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⣿⣿⣿⣛⠩⠽⠷⣿⡿⠋⠀⠀⠉⠙⠻⣿⣿⣿⣟⢿⣟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣾⣿⠟⣋⣤⣶⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⠹⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣧⣿⣿⣵⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠘⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢹⡎⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣱⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⣿⠈⣧⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣹⢿⣧⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⣿⡇⢹⡄⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢹⣿⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⡿⠁⠀⢷⢿⣮⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠘⣯⣿⠻⣇
⠀⠀⠀⠀⠀⠀⣆⠀⢠⣿⣶⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠀⢹⣿⣷⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⡎⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⢠⠈⣆⢀⣿⣯⢻
⠀⠀⠀⠀⠀⠀⢹⡿⣿⣟⠣⡈⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣾⣽⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠉⠹⣷⣮⣴⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⢠⢀⣸⣿⣧⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⡄⠈⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣦⣿⠿⣿⣿⣆⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢰⣿⢯⣶⣿⣿⣿⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⣿⣿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⣾⣿⣦⡙⣿⣿⡿⣿⣾
⠀⠀⠀⠀⠀⠀⠠⠟⠁⣿⢟⣵⣿⣟⣿⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢸⣿⣿⣧⠙⣿⣿⣷⣿⠁
⠀⠀⠀⠀⣀⠤⣄⣠⣾⣫⣿⣿⡟⢸⢏⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⢸⣿⣿⣟⠄⢹⡏⣿⣯⡳
⠀⠀⠀⠰⢡⠂⡻⣻⣿⣿⣻⠟⠀⣨⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣯⡄⠛⢿⣿⣦⣀⡟⡀⠀⠹⡄
⠀⠀⠰⣿⡿⠳⢁⢿⡿⠣⠃⠀⣴⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⡇⣿⣿⣿⣶⠶⣾⣿⡟⣿⣿⣦⠄⠁
⠀⠀⠀⣻⢁⡴⢉⡜⠁⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡟⣥⣿⣮⡻⣿⣷⢳⢶⣬⣽⣿⠏⠉⢢⡀
⠀⠀⠀⣷⠏⡠⢋⡴⢁⣴⣿⣯⠻⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣞⣾⣿⣿⣿⡿⣮⣿⣀⣧⠙⢿⡏⠀⠀⢸
⠀⠀⢰⣧⡞⣡⡟⣰⣿⣿⣷⣬⢳⣿⡝⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣘⣿⣿⣿⣇⡟⡅⣿⣝⣿⣿⣷⣬⣻⡄
⠀⠀⠘⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⡏⣿⣿⣿⣄⠀⠀⠹⠋⢞⣿⡙⢿⠻⣷⡀
⠀⠀⠀⠙⠀⠸⣿⣿⣷⣄⣿⠌⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠩⣿⡟⢿⡢⠀⠀⠁⠈⢾⣷⣄⠱⣌⡙⣄
⠀⠀⠀⠀⠀⠀⠻⣿⠛⠉⠁⢐⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⣜⢧⡀⢻⠦⠀⠀⠀⠀⠉⠻⢷⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠉⡟⣒⡴⣻⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣿⠏⢣⠀⠈⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠊⠋⠀⠀⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠙⣾⣿⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠀⠀⢁⡽⠀⠾⣹⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢮⣃⠀⠀⣴⢀⡀⠡⢆⠜⠃⠀⢩⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡦⢿⣒⣠⢏⡀⠺⠔⠋⠀⠀⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇⠙⢟⡋⡿⠃⠁⡀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢈⣀⢠⠀⠀⡀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠇⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠐⢼⣴⠀⠀⠀⠰⠀⠀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠀⠀⢸⡆⡼⢣⠀⠀⠀⠇⠀⢰⣷⡛⠋⢁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡄⠀⠀⠄⠀⢺⡇⠀⠇⠀⠠⢠⡦⠈⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣻⠃⠀⠀⠀⠀⣟⡐⠀⠀⠀⠀⠈⢇⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡄⠀⠀⠀⠘⣃⠀⠀⠀⠀⠀⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡄⠀⠘⣿⠀⠀⢀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡄⠙⠏⡀⢀⠈⠀⡴⠀⠰⡅⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠞⢰⡟⡇⠈⡿⠀⠀⠈⠃⣅⡠⣀⣶⣶⣾⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⡟⠀⠀⡀⡻⠆⡀⠀⠘⠆⠳⠚⠋⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢿⣷⠂⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠈⠆⠚⠳⢾⡷⠛⠋⠁⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢠⡆⠀⣸⡗⠀⢀⠀⠀⠀⠦⡘
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⢇⠀⠻⠀⠀⠀⠇⣠⡖⣡⠃⠀⢀⡇⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠇⡀⠀⠄⠀⠶⠟⡄⠧⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠀⣅⠀⠱⡘⢶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⠀⠀⠱⡄⣸⡶⠀⠙⠃⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠲⡄⠙⠟⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠂⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⢥⣉⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⠶⡚⠉⠀⢀⣀⡀⠀⠠⠀⠈⠒⠈⢳⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣐⡽⠛⡩⢐⢎⠔⠉⠀⠀⠀⣄⠁⢀⠃⠀⠀⠇⠀⠀⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠘⠠⠂⣎⢀⡅⠀⠀⠠⠸⡀⡌⠀⠀⠀⠘⠂⢀⡄⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠚⢱⠃⠀⠀⡄⠀⢹⡼⠃⠀⠀⢀⠀⢇⢹⡀⠀⠀⠀⠴⠛⠉⠀⠐⠘⡆⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠞⠁⠀⠀⠱⡸⠀⢦⡀⠑⢦⣀⡙⣦⠀⢖⠊⢧⠈⠢⠙⢦⣀⣀⡤⠖⠛⣑⣤⢰⡇⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⢀⠎⠀⠜⠀⠀⠀⢀⣠⠇⠀⠀⢳⢄⠀⢄⠙⠙⢧⡈⣷⢻⠀⠒⢈⡐⠀⠴⠲⠖⠋⠁⠈⣿⡇⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠘⠀⠀⠐⢄⡴⠊⠉⠀⠀⣠⡇⠘⠈⢡⠀⠃⠀⠀⣗⣿⠘⠀⠂⠄⠈⠑⠲⢆⠄⠀⠀⢘⡏⠇⡘⠢⢀⣄⣄⠇
⠀⠀⠀⠀⠀⠸⡀⣀⡤⡎⠁⠀⠐⠒⡛⠉⠀⠀⠀⣸⢀⠃⠀⢄⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣇⠀⡀⠀⣸⡞⣿
⠀⠀⠀⠀⢀⠎⢿⡀⢿⢝⣋⡉⠉⠤⣤⣀⠀⠀⢀⠣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⡟⢣⡈⠉⢱⡇⠊⠳⢤⢀⡀
⠀⠀⠀⠀⠰⢄⣨⢃⡾⠁⠀⠀⣀⣀⣀⡠⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡇⠀⠀⠘⢤⣀⡀⣀⠉⠱⡀
⠀⠀⠀⣀⠴⠺⠃⡿⡔⠀⡠⠚⠋⣩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠉⡓⢦⠀⠀⠀⠀⠀⠅
⠀⠀⡼⠀⠀⠀⠀⠲⣧⢠⠁⠀⠜⠁⢀⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠣⡐⢦⣀⠀⠸⠄⠀⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠰⠁⢩⠙⠄⡘⠀⢠⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⡄⠉⢷⣼⣀⠀⡼⠀⠈⠄
⠀⠀⠻⡀⠀⠀⠀⢀⣠⠄⢰⠁⣀⠀⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠀⠀⠳⣿⠀⡇⠀⡀
⠀⠀⣠⠶⠶⠚⢋⡟⠀⠀⠀⢄⡘⠷⣄⡘⢦⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⣸⠁⠆⠁⠀⠀⡟⠳⣌⠂⠐⢄
⠀⠀⠀⠄⠀⠀⢼⡀⠀⠀⠤⡀⠀⠀⠀⢻⢈⡿⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢻⠠⠀⠀⠀⠀⢳⣄⠈⢳⣠⠆
⠀⠀⠱⡌⣦⠀⠀⠉⠳⡆⢀⠈⡆⢠⠀⣪⣿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⡜⠧⠀⠄⠐⢀⡠⢻⡷⣾⠁
⠀⠀⠀⢡⣿⠀⠄⠸⠀⡼⢀⡴⠣⢀⠀⣱⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣢⠞⠀⠀⠀⢠⡶⠛⣠⠏⠁⠡
⠀⠉⠙⡋⢁⡴⠂⠀⠀⣱⠋⠀⠀⠀⠀⣆⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⣼⠛⠑⠀⠀⠀⠏⢈⠀⠐⢏⠀⡀
⠀⠀⠀⢰⢌⠄⠀⠀⠀⢻⠄⠀⠀⢠⣇⠈⠉⠒⠶⡄⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠸⡇⡌⡴⠦⠊⠀⣠⣠⡾⠂⠈⠁⠉
⠀⠀⠀⠀⠀⠁⠄⣿⠀⣀⠀⢀⠇⡐⠃⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠀⢧⡆⠀⡀⣚⠁⠀⠀⣤⡑⠀⠀⢠
⠀⠀⠀⠀⠁⣔⠀⠀⠉⢋⡞⢑⠉⠀⠀⠀⡀⠤⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⡷⠈⠀⡄⠀⠀⠈⢢⢈⡆
⠀⠀⠀⠀⠀⠈⠳⡆⢀⠈⢳⣄⡀⠢⣶⠀⡗⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣪⠾⠃⠀⠑⣼⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠐⠉⣧⠯⠑⠙⠶⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡠⢖⣿⠟⠀⢀⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠸⡀⠀⢘⡔⠋⠁⢀⠴⠛⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠃⢰⠁⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠘⡇⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡀⠀⠁⠀⠀⠀⡀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢚⠀⣴⠟⢫⣹⡦⠤⡔⠀⠑⢬⣦⣄⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠄⢾⠁⣠⡞⠀⢹⠁⡀⠀⠀⠀⠀⠀⠀⠚⡀⠀⠙⣣⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⡞⠀⡟⠸⠁⡔⠀⠀⠳⢤⡴⠀⠁⡀⠀⠀⡠⠣⠀⠾⢿⡄⠀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⡠⠊⠀⡂⢀⣤⡃⠈⣇⠀⠁⢢⠀⠣⡀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⣶⡋⠁⣀⠇⢸⡅⠒⠀⠘⠖⠐
⠀⠀⠀⠀⠈⠀⠀⠀⢣⣝⠚⢟⡆⠙⠣⠤⠾⡳⢄⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⡹⠁⠀⠈⠄⡀⠐⣄⡀
⠀⠀⠈⠀⠀⠀⢰⠋⠀⠀⢡⠄⠁⠀⠀⠀⠀⠈⣀⡵⠞⠀⠠⣄⠂⠀⠀⠀⠀⢀⣠⡞⠁⢀⣀⣀⠀⡀⠀⠀⠙⠀⠐⠆⢤⣄
⠀⠀⡀⢀⣀⡴⠋⠈⢢⢠⠞⠀⠠⣦⠀⠀⣀⡜⠁⠀⠀⠘⠀⠀⠀⠀⠀⠘⠋⠉⠚⠲⠖⠀⠀⠉⠂⠄⠀⠀⠀⠀⣆⠀⠀⢹⡝⠄
⠀⠀⠚⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠫⣤⣴⠟⠁⡄⠰⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⡀⠀⢀⠀⡴⠋⠀⢀⠀⠃⢸
⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠄⠜⢇⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠹⡄⠀⠀⢢⠀⠁⠀⠀⠀⢀⡀⠎⢀⠀⠂
⠀⠀⠉⠠⠀⢡⠤⠤⠀⡀⢆⠀⣀⠄⠀⠀⡀⣀⠀⠀⠀⢸⣶⡀⠀⢠⡻⣂⡜⠀⠀⠀⠀⠘⠦⠤⡴⠂⠀⢀⠀⠂⠀⠨⠔⠋⠀⣀⡄
⠀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠋⢁⣀⡀⠀⣼⠃⠀⠀⠘⣾⠯⠀⠈⠀⠀⠀⡈⠀⠀⣃⠀⠀⠀⠀⠠⠀⠀⣴⡤⠊⠁
⠀⡻⡀⢻⡀⠀⠸⡀⠀⠀⣵⠀⠀⠀⠁⠀⢤⠸⣄⣣⠜⠃⠀⠀⠀⢰⠃⠀⠀⠀⠀⠐⠀⠑⠀⢸⣿⠀⠀⠀⠀⢠⡆⠘⠋⠄⠄
⠀⣠⠂⠀⠐⠀⠀⣽⠲⢤⣘⡿⠛⡃⠊⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠐⡆⠀⠀⠀⠀⠀⠻⣦⣀⣠⣴⡏⢠
⠀⢸⠀⠀⠀⣤⠀⣿⠑⠖⠠⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠁⠀⡈
⠀⠀⠐⠀⠀⠻⡄⢸⣆⡀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠀⢤
⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠄



⠀⡀

⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣤⡀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠋⣉⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠙⠻⢦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣯⣴⠶⠛⠋⠉⠴⠶⠶⠟⠛⠛⠛⣛⡻⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠁⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠷⠦⠀⠀⠀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⢀⣤⣴⣾⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⠤⠄⠀⠀⢢⡈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⢀⣀⡤⠂⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠹⣆⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⣠⣴⣾⡿⠋⣀⣤⣾⡿⠿⣛⣯⣿⡿⠛⠉⠀⠀⠀⣀⣀⠀⠀⠀⠘⣷⣄⠀⢻⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣁⠄⠀⢀⣴⣿⣿⡿⢋⣴⣾⡿⠛⣩⣴⣿⠿⠋⢁⣠⣤⣴⡶⠿⠛⠉⠀⠀⢀⠀⠀⠘⣿⣷⣄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⢀⣴⣿⢟⣿⢏⣴⣿⠟⣡⣴⣿⡿⣋⣥⣶⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⣾⣿⢋⣾⢃⣾⠟⣡⣾⣿⣿⣿⣿⠟⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⢷⡀⣿⡼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣾⣿⠃⣾⠇⣾⢋⣼⣿⣿⣿⡿⠋⠡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⢣⢻⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⡿⢸⢣⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⢹⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣾⡇⢠⣿⣿⣿⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠸⣷⣸⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⡇⣾⣿⣿⠋⠉⠉⠙⠛⠿⣶⣶⣤⡄⠀⠀⠀⠀⣤⣤⣶⡾⠟⠛⠉⠉⠉⠂⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣧⣿⣿⡏⠀⡀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⠀⠀⢙⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣖⠁⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠶⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⠖⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣉⣯⣹⣩⣯⣱⣍⣍⣏⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣭⣭⣽⣿⣿⣿⣭⣭⣭⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢈⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠙⠉⣩⣿⣟⠁⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠋⠟⠁⠀⠀⠀⠀⠈⠳⣤⣄⠀⡀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡴⠞⡿⠿⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢾⣦⣢⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢀⣴⠞⠋⣠⠟⠀⠁⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⣄⡀⠀⠱⡀
⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⠀⢀⣾⣶⣶⡟⠁⠀⣼⡏⠀⠀⠀⠀⠁⠈⠛⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢃
⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⠀⢠⣚⣻⣽⡿⠻⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠐
⠀⠈⣸⡇⠀⠀⠀⠀⢴⣾⣿⠟⠋⠟⠉⠁⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌
⠀⠀⠹⣧⠀⠀⠀⠀⠀⠉⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⢹⣆⣄⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢆⡀⠀⠀⠀⠀⠀⢀⢸⣷⠀⠀⠀⡀⢀⠀⠀⠀⣀⣤⡤⠖⠁
⠀⠀⠀⡸⠛⠋⠉⠁⠀⠀⠈⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠹⡄⠀⠀⠀⠀⢸⣿⢃⣴⣴⣿⡷⠷⠶⠚⠛⠉⠁
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢷⣄⠀⠀⠂⠰⠚⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡷⠋⠁
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣬⣳⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠴⠛⠋⠉⠉⠉⠉⠙⠛⠛⢿⣦⣤⣤⣤⡤⠆⠊⠀⠀⠀⠀⡠⠃
⠀⠸⡆⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠷⢶⣦⣴⣦⣤⣦⣾⡇
⠀⠀⢹⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⣿⡟⢹
⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⢸
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⣹⣷⣿⣷⣶⣶⣶⣾⣿⠃⠀⠀⣾
⠀⣼⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⢤⣤⡤⠶⠾⣿⡿⠏⠀⠘⠀⠀⡿⠃
⠀⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠈⠀⠈⠀⢀⣴⠟⠏⠀⠀⠀⣀⠔⠋
⠀⣧⡸⠀⠀⠀⠀⠀⠀⠀⠒⠫⠉⠁⠀⠀⠀⠀⠀⢀⠴⠋⠏⠀⠀⠀⠀⣴⠃
⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠁
⠀⠘⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⣀⡤⡴⠶⠒⠋⠉
⠀⠀⠘⡄⠀⢀⠀⠀⠀⠀⠀⢀⣾⡾⢿⡟⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⡆⣾⠀⠀⠀⠀⢀⠞⠁⠀⠘⣿⠔⡠⢀
⠀⠀⠀⢀⢾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⡟
⠀⠀⡈⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⢸
⠀⢀⡇⡀⠀⣀⣤⠶⠊⢀⡔⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠋⠈⠉⠉⠉⠉⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠀⣀⡀⠤⠀⠐⠂⠒⠒⠒⠂⠤⢄⣀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⡤⠒⠉⠀⠀⢀⡀⠤⠤⠤⠤⠤⠀⣀⠀⠀⠈⠐⠤⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠞⠁⠀⠀⠀⠈⠀⠀⢀⣀⡀⠀⢀⣀⣀⡀⠀⠉⠒⠤⡀⠀⠑⢄⡀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡠⠊⠀⡠⠂⠁⠀⠀⠐⠈⠀⢀⣀⡀⠀⢀⣀⣀⠉⠉⠒⠤⡀⠈⠑⢀⠀⠉⢦⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡔⠁⡠⠊⢀⠤⠊⢀⡠⠔⠊⠉⠁⠀⣀⣀⣀⠀⠀⠉⠑⠂⢄⡀⠑⢄⡀⠑⢄⠀⠱⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡜⠀⡔⠁⡠⠁⣀⠔⠁⢀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⠀⠀⠈⠢⡀⠑⢄⠀⠳⡀⠱⡀⠈⡆
⠀⠀⠀⠀⠀⠀⣀⠼⠤⠴⣀⡜⢀⣎⣀⡔⠁⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠈⠢⠀⠣⠀⠱⡀⢱⠀⢰
⠀⠀⠀⠀⣀⠎⠁⠀⠀⠀⠈⠖⠋⠁⠀⠈⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢱⣀⠷⠤⠷⡀⣦⠚⠁⠒⠒⠒⠐⠢⡀
⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⡠⠒⠠⠔⠚⠁
⠀⠀⠈⠒⠢⠤⠒⠉⠑⠤⠀⣀⣀⣀⡠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⠉⠉⠉⠉⠳⠑⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣠⠴⠒⠛⠉⠀⠀⠀⢸⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡔⣠⡶⢋⡵⢊⣤⠖⠀⠀⠀⠀⢸⠇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣴⠟⣰⣿⢡⣿⠕⡠⠄⠀⠀⠀⠈⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡏⢰⣿⠃⣾⡏⡜⠁⠀⠀⠀⠀⠀⠀⠀⡰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⣷⠀⢻⠀⣿⣿⢰⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⢀⢀⣿⠃⠀⢸⣿⡇⢸⣿⣿⡇⠀⠀⢀⣠⡴⠞⠋⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣻⠵⠂⡀⡼⢻⡇⢸⡿⣿⣿⡄⠀⠙⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡯⠤⠶⠚⠛⠁⠸⣷⣿⠇⢻⣿⣿⣄⠀⠐⠺⢿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡀⠀⠀⠀⠀⣀⡼⠀⣿⡏⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡞⢁⣠⣤⣶⣶⠿⠋⠁⢘⢹⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⣿⣿⣿⠋⣀⣀⣠⣴⡟⢸⠇⠀⡀⠀⣿⡟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣿⣿⡿⠿⠿⠿⠟⠛⠉⠠⠋⠀⠴⠀⢠⣿⠇⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣤⣴⣶⠶⠦⠤⠤⠔⣊⣴⣿⠃⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡎⠀⣴⣿⠛⠁⠀⠀⠀⠀⠉⠉⣩⣴⡟⠀⢶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⡗⠀⠀⠀⢀⣤⣴⣶⡿⠟⠋⡀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣠⣮⣾⣿⠟⠋⠁⢀⣴⢆⣷⡀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠻⣿⣿⠏⠀⣀⣤⣶⣿⢟⣼⣿⡇⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⣾⠿⠛⠋⣡⣾⣿⡿⣱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⢠⠎⣼⣿⣿⣟⣿⡿⠋⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⡿⣸⣿⣿⣿⡿⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣧⢻⣿⣿⢋⡄⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢿⡏⣾⣧⡪⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⣿⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣠⣶⠾⠿⠿⠿⠿⢶⣦⣄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⢀⣀⣠⡏⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠘⣆⠀⠀⢀⣠⣤⣷⣤⣄⡀⠀⢹⣧
⠀⠀⠀⠀⠀⢰⣿⣠⡾⠟⠛⠛⠿⣶⣄⣤⣤⣼⣧⣤⣀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢸⣤⣾⠟⠉⠁⠀⠈⠙⠻⣦⡘⣿
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠈⠛⠉⠁⠰⠶⢭⣙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣠⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⢀⣴⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣤⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠙⠻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣭⣋⣠⣤⣶⣧⡀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠻⠷⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⣥⣶⣾⣿⣶⣦⣌⠹⣦⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⣾⣿⠟⠛⠛⠛⠻⣿⣆⢹⡇⢹⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⣿⡟⠁⠀⢀⡤⠤⠄⠘⣿⣿⠃⠈⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣫⣴⣿⠟⡄⠀⢰⡟⣠⠒⠀⠀⠹⣧⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢋⣡⡜⠁⢠⣿⢱⢃⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠵⠒⠛⠉⣁⣠⣴⡿⣣⠏⠈⠀⠀⢀⡀⠀⠻⢿⠿⣿⡟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣴⣶⣾⡿⠟⠛⣉⠥⢚⡏⠀⠀⠛⠛⠛⠂⠀⠀⠈⣸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢫⣿⣿⡿⠁⠀⢠⣾⣧⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⠃⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣧⠀⠀⢸⣿⣿⡄⢠⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⠃⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠛⢿⣿⣷⣦⣄⡙⠿⣿⡜⠻⠛⠁⠀⠀⣀⣴⣿⠹⣿⣿⡀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡆⠀⠈⢿⣿⣿⣿⣷⠈⣿⣦⣤⠀⠀⠈⠁⠘⣿⣧⠙⢿⣷⡀⢻⣷⡀
⠀⠀⠀⠀⠀⣠⡶⠟⠛⣉⣁⣀⣠⣤⣿⣿⣿⣿⢟⣠⣿⣿⡟⠀⠀⠀⠀⠀⠈⢻⣷⣄⠈⢻⣆⠹⣿⣆
⠀⠀⠀⠀⢸⡏⢠⣾⠿⠟⠛⠛⠛⣛⣋⣭⣵⡾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣿⠀⣿⣿
⠀⠀⠀⠀⠘⢇⢸⣿⡀⠀⡖⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣩⠴⠾⣛⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠈⠈⠻⣷⡘⡇⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠇⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢛⣻⢿⣮⣍⣙⠓⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡫⢔⣤⣶⡟⣽⡲⢕⣨⣍⣛⠳⢮⣝⢮⣙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⡾⢋⣴⠏⠀⣿⡙⢦⣉⡙⠛⠻⠶⣮⣷⣿⣷⡝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⡾⠋⢠⣿⠏⠀⠀⢳⣝⠦⣬⣙⡓⠶⢤⣄⠉⠛⢿⣿⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⠁⠀⣿⣿⠀⠀⠀⠀⠙⠻⢶⣮⣽⣛⠲⢮⣝⣦⡀⠙⣿⡎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⡉⠻⣿⡄⠘⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⣄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠘⣷⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣸⡏⡼⣿⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣰⠟⡰⢱⣿⣿⣧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣴⠏⡴⢡⣿⣿⡝⢿⣧⡘⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠏⡰⢁⣿⣿⢡⡙⢦⣙⠻⠦⣍⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⡏⡄⠁⢸⡿⣟⢸⡉⠓⠻⠿⠷⣶⣬⣭⣛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡇⢸⡄⢸⡇⢿⠸⣯⣑⣒⣒⣂⣒⣀⣀⣈⣉⡁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠳⡌⣷⠘⣇⢧⣙⠫⠭⣭⣤⣤⣤⣤⣌⣉⣙⠲⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢿⡆⠘⣎⢧⠙⣿⣿⣭⢹⣿⣷⣜⣿⣧⢹⡒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⣿⠘⣌⠪⡳⢌⣛⣋⣼⡯⢻⣿⣾⡏⡎⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⠀⡿⣷⡚⠶⠖⣚⡵⢁⡟⣿⣿⢳⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠄⠀⠀⠀⢸⡇⣸⣰⣝⠦⣍⣛⣉⣡⠴⢋⣼⣿⠏⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⡀⠀⠀⣠⠞⣰⣿⣅⠪⢷⣦⣤⣤⣤⣶⢿⣯⡅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣭⣟⣫⣥⡾⢟⡻⢛⡷⡶⣤⡍⠉⠈⠁⢜⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠭⣙⣛⣉⡭⠔⢻⠀⣧⣿⢠⡋⠙⢿⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⣆⣿⣌⠓⠛⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣙⡛⢛⣛⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠉⠐⢄⡀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡘⠀⢀⣀⡀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⣀⣀⠘
⠀⠀⠀⠀⣀⡷⠛⠋⠉⠙⢠⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠀⡠⠊⠉⠙⠛⢷⣀
⠀⡰⠖⠛⠉⠉⠀⠀⢀⣄⡐⠋⠙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⢂⣤⡀⠀⠀⠉⠉⠛⠲⢆
⠀⠁⠀⠀⠀⠀⢠⡞⠉⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠉⣳⠄⠀⠂⠀⠀⠈
⠀⠀⢳⡄⠀⠀⠀⢻⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⢀⣠⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠷⣤⣤⣤⣤⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠋⠳⣤⣤⣤⣤⠾




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠛⠛⠛⠋⠙⠛⠛⠛⠿⢿⣿⣿⣶⣤⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠉⢠⣤⠀⠀⠀⢰⣿⠄⠀⠀⢀⣤⡀⠈⠙⠻⣿⣿⣷⣄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣠⣾⣿⡿⠋⢁⡀⠀⠀⠈⠋⠀⠀⢀⣀⣀⣀⠀⠀⠈⠛⠁⠀⠀⣀⠈⠙⣿⣿⣧⡀⠀⠀⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠻⠟⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠘⠿⠃⠀⠈⠻⣿⣿⣄⠀⠈⠁
⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⣿⡟⢁⣤⡀⠀⠀⣠⠞⠉⠀⣀⣤⣶⣶⣿⣿⣶⣶⣦⣄⡀⠈⠳⣦⡀⠀⠀⣤⡄⠹⣿⣿⣆⠀⠀⣀⡀
⠀⠀⠀⠀⠈⠁⠀⢠⣿⣿⡏⠀⠀⠛⠁⢠⡞⠁⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠻⢿⣦⡀⠈⠳⣄⠀⠛⠁⠀⠸⣿⣿⡄⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⣤⡄⠀⣠⠏⠀⢠⣾⡟⠁⠀⣀⣤⠶⠞⠳⠶⢦⣄⠀⠀⠙⣿⣆⠀⠹⣆⠀⣠⣤⠀⢹⣿⣿⡀⠀⢀⡀
⠀⠀⠀⠻⠀⠀⢸⣿⣿⠁⠀⠙⠁⢠⡏⠀⢠⣿⠏⠀⢀⡼⠋⠀⠀⠀⢀⡀⠀⠙⢷⡄⠀⠘⣿⣇⠀⢹⡄⠘⠋⠀⠈⣿⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⠀⠀⣾⠀⠀⣿⡏⠀⠀⣾⠁⠀⠀⢺⣷⣿⡗⠀⠀⠀⢻⡄⠀⢸⣿⡀⠀⢷⠀⢀⣀⠀⢹⣿⣿
⠀⠀⠰⠖⠀⢀⣿⣿⡇⠘⠛⠃⢠⡇⠀⢸⣿⠁⠀⢸⡇⠀⠀⠀⠈⠿⠟⠁⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⡄⠘⠟⠀⢸⣿⣿⡀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠈⠁⠀⠘⠛⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠃⠀⠈⠁⠀⠀⠀⠀⠛⠛






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣾⣿⣿⣯⠀⠀⠀⠀⣀⣀⡉⣉⣛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣺⣵⠞⠁⢩⡁⢿⠋⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣶⣝⡿⣦⣄
⠀⠀⠀⢀⣤⣶⣿⢿⣿⣿⣿⣿⠃⢠⡅⡄⠳⣌⠐⡀⠀⠀⠀⠀⠈⠻⣿⣍⠻⣧⠙⢿⣿⣿⣧⡀
⠀⠀⠠⠟⡽⢋⣴⠿⡿⠿⣻⣿⡀⠀⠑⣷⡀⡈⠑⠺⢦⣄⠀⠀⠀⠀⠙⣿⣆⠙⣧⠀⠙⢿⣿⣿⣄
⠀⠀⠀⠘⠁⣾⠃⢀⣠⠞⢻⣿⣧⡰⡀⠈⠻⣿⣾⣄⢠⣍⠻⢮⣦⢳⡄⠘⣿⡆⠈⠂⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠁⣿⣷⠟⠁⠀⠀⠻⣿⡻⣿⣦⣀⠀⠙⠻⢦⣝⣷⣄⢻⣧⢻⡄⠸⣿⠀⣄⠀⢲⡘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠈⠁⠈⠙⠻⢷⣶⣤⣄⡈⠙⠻⣷⣿⣾⣷⠀⢿⡇⢸⣆⠘⣧⢹⣿⣿⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⢠⡴⠖⣛⡉⠉⠉⠙⠻⢷⣄⠈⢿⣿⣿⡆⢸⣷⢸⣿⡀⣿⣾⡟⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣦⡀⠀⠀⠀⠰⠟⢿⡿⢻⠿⠋⠀⠀⠹⣧⠈⣿⣿⡇⢸⣿⣼⣿⡇⣿⣿⡇⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⣻⣿⡇⢸⡟⣿⢻⣷⡇⢹⣇⣿⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡇⣿⣿⠀⣼⠇⠟⠀⣿⢀⠈⣸⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⣾⣿⣿⡏⢠⡿⠀⠀⠀⠃⣾⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡏⢰⢻⡟⣿⡿⢀⣾⠃⡄⢀⡆⢰⣧⡞⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣤⣤⠀⠀⠀⡾⠀⡿⠈⣽⣿⠇⣼⡏⣰⠃⣼⣧⣿⢏⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢰⡇⠀⡷⢠⣿⡟⣼⡿⣰⣏⣼⣿⠟⣡⡾⢃⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣧⢸⣿⣱⡿⣱⣿⣾⠟⢁⡴⠏⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⢻⣾⣿⣿⣿⡿⠟⠁⠐⠋⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣞⣿⣿⠟⠋⠀⠀⠀⢀⠄⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡼⣿⡿⢀⡤⠀⢀⡴⢃⡾⣳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⠀⠀⡼⣷⡹⣿⣷⣿⣴⡵⢋⣴⠟⣴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡇⠘⢷⣿⣿⣿⠟⣠⠞⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠎⠻⣿⣿⡞⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣸⣶⡼⢻⣿⡾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣿⣿⡇⣾⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃

⠀⠀⠀⠀⠀⠀⠄⠤⠠⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⢿⣿⠿⣿⣿⡀⠀⠀⠀⠀⢠⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢻⣿⠁⠀⠈⠉⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⢻⣿⣿⠙⠻⢿⣿⡘⢻⣿⣿⠏⠁⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠈⠉⠉⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠹⠃⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡄⠀⠀⠀⠀⣺⡿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⡀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠀⠀⠀⠙⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⡿⠿⡿⠏⠿⢿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠻⠧⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠇⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⣸⣿⠀⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠈⠟⣿⣿⡟⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⢰⣿⣿⡿⠉⠙⠋⠙⠛⢻⣿⣿⣇⠀⠀⠀⠀⠀⠉⢿⡿⡦⠀⠀⠀⠐⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠛⢷⠄⠀⠀⠀⠀⠀⠀⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣶⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠿⠋⠙⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠈⢿⣿⡏⠁⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣼⣏⠃⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠘⠄⠀⠃⠀⠂⠠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢀⠔⠂⠈⠀⠀⠀⠈⠀⠢⡈⠠⠆⠀⡠⠠⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠠⡀⡹⠀⠀⠀⠀⠉⠀⢑
⠀⠀⠀⠀⠠⠤⣀⠁⢠⠀⢀⣄⡀⠀⠀⠀⠀⠀⠉⠈⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠈⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⢣⡀⢄⠀⢀⣀⠜⠉
⠀⠀⠀⠀⠐⠤⠤⠀⠰⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠠⠤⣀⠂⠉⠁
⠀⠀⠀⠀⢀⣀⠤⠭⠤⣣⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⢀⠅⠀⠀⠀⠀⡸⠑⢠⠞⠀⠀⡠⠔⠊⠉⠉⠉⠉⠒⠤⡀⠀⠈⢢
⠀⠀⠀⠘⠤⡄⠀⣀⡠⠒⢡⠋⠀⢠⠊⠀⢀⡠⠤⠒⠦⠤⡀⠀⠈⣦⠀⠀⢱⠊⠉⠢⠤⠤⠠⠄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⠃⠀⠰⠁⠀⡔⢁⣠⠤⠄⠤⣀⠈⠢⡀⠘⣇⠀⠀⡆⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠀⡌⠀⡞⠁⠀⠀⠀⠀⢣⠀⢡⠀⠸⡀⠀⢰⠀⠀⠀⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡨⢄⣀⣇⡰⠀⠀⠀⠀⠀⠀⠀⡀⢨⡀⢀⡀⠀⡼⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⠀⠀⠀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠁⠀⠀⠀⠀⠀⣀⡀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡤⠞⠋⠉⠁⠀⠀⠀⠈⠉⠙⢦⡀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠎⠀⣠⡴⠒⠀⠀⠀⠐⠒⢦⣄⠀⠹⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠹⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⣼⠋⠉⡒⠲⠄⠀⠀⠀⠰⠖⡚⠉⣿⡄⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡏⠀⣿⠋⠹⣿⠂⠀⠀⠀⠀⠀⢸⡿⠉⢻⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢻⣆⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢻⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣰⠃⠀⠀⡆⠐⣄⡀⠀⠀⠈⠀⠀⠀⢀⡜⠁⠀⠀⠹⣆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢰⠃⠀⠀⢰⠃⣤⢹⠈⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡆⠀⢻⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⡞⠀⢿⡆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡇⠀⠈⣇⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠀⠀⠀⡇⠀⠘⣷⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⢻⠀⢹⡆
⠀⠀⠀⠀⠀⢀⡤⠒⠀⠈⣷⡀⠀⠀⠀⠀⠹⡀⠀⠙⣷⡀⠀⠀⠀⠀⢳⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⠀⠑⠦⣄
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠑⠀⠀⠈⢻⣄⠀⠀⠀⠈⢣⠀⠀⠃⠀⠀⠀⡌⠀⢸⡇⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠸⠀⠀⢸⠀⢀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⠂⢀⡠⠅⠰⠒⠀⠐⠆⠤⣈⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠓⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠈⠁⣀⡔⠊⠑⠢⠤⣀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢆⠈⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡁⠀⠀⠏⠙⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⢱⠒⡛⠤⣀⠴⠒⠚
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠒⠒⠒⠾⢄⡉
⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⡀⣈⢆⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⡖⠉⠉⠀⠀⠀⠀⢫⠳⡀⢀⡜⠁⠀⠀⠀⣀⠤⠖⠒⠒⠒⠒⠤⢄⡀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢰⢊⡄⣨⠟⠁⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠈⢣⠀⡀⠄⣀
⠀⠀⠀⠀⠈⠒⠲⣀⢀⡤⠤⠊⠉⣰⡏⠀⠀⡰⠋⠀⠀⣀⠴⠒⠋⠉⠙⠒⠦⣄⠀⠀⠈⢆⠀⠀⠀⢏⠀⠀⠈⠓⠒⠖⠂⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡸⠁⠀⢀⠞⠁⣀⡠⢤⠤⠤⢄⠀⠈⠳⡀⠀⠹⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠁⠀⢠⠃⠀⡶⠋⠀⠀⠀⠀⠀⠑⡄⠀⠱⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠌⠀⠀⡌⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⠠⠀⠀⡀⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠚⠢⠴⠣⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠧⣀⡰⢇⣀⡠⠛⠒⠊⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠴⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠤⠄⠀⠀⠀⠀⢀⣤⡤⠤⠀⠀⢢⣤⣄⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣶⣯⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡏⠬⡙⡻⠮⠿⣇⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⣷⣾⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣾⡟⢹⣿⡿⣇⠀⠀⠛⠋⠀⠈⠀⠀⠀⠀⠀⠂⠸⠿⠯⠴⠟⣹⠀⣿⡟⣿⣿⡄
⠀⠀⠀⠀⡴⠋⣠⠉⠀⢸⣿⡇⠘⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⣿⠇⣿⣿⣷⡀
⠀⠀⠀⠘⠀⡼⠋⠀⠀⡞⣿⣧⠐⣄⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠈⠸⣱⢸⡟⠀⣿⣿⣟⣧
⠀⠀⠀⠀⣸⠋⠀⠀⢸⡇⢻⣿⠀⠁⣦⡀⠀⠀⠀⢤⣤⡀⠤⡀⠀⠀⠀⢀⠀⢃⣿⣼⠃⢀⣿⠁⢿⢸⡄
⠀⠀⠀⠀⢟⠀⠀⠀⣸⡇⠸⣿⠇⠀⠻⣿⣦⡀⠀⠀⠘⠉⠀⠀⠀⣀⢴⡏⣜⣿⣿⠏⡀⠈⠟⠀⢸⣄⡇
⠀⠀⠀⠶⠈⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⣿⣿⡏⠑⢤⡀⢀⣀⠴⠊⠁⣾⢁⢿⣿⡏⢀⠃⡜⠀⡀⢸⡟
⠀⠀⠀⡀⢀⡄⠀⠀⣿⣿⠀⣾⠀⠀⢀⣿⣿⠇⠀⠀⠉⠉⠀⠀⠀⠀⣿⣸⣿⣿⡄⠸⣰⠁⣰⢁⣿⠁
⠀⠀⠀⢀⡈⢀⠀⡁⣷⣿⠀⡟⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡗⢠⡟⢰⢟⣾⡿
⠀⡠⠚⠉⠈⣿⣆⢳⣿⡇⢰⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣿⣇⣿⢡⢯⣾⣿⠃
⠘⠁⠀⠀⠐⠈⢿⣾⣿⠃⡘⠀⢡⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢻⣿⡇⣼⣾⣿⠋⠙⢆
⠀⠀⠀⠈⠀⠀⢸⣿⠇⢠⡗⢣⡿⠋⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⡁⠀⠀⣼⢉⣿⣿⣿⡇⢀⠨⣇
⠀⠀⠀⠀⠀⠀⢸⠃⢠⣿⠶⠋⠀⠀⠀⠈⠑⠒⠠⠤⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⣸⣾⣿⣿⠏⢰⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣾⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣽⡏⣿⣿⣿⡧⠊⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⢠⣤⣐⠻⣿⣿⡇⣿⣧⠀⠙⠻⣿⣿⣷⣶⡄⢀
⠀⠀⠀⠀⠀⢰⣿⠟⢋⣩⣽⠧⠄⠙⢿⣿⣆⠙⢡⣿⣿⠀⢸⣇⢸⣿⣟⢿⣿⠘⣿⣦
⠀⠀⠀⠀⢠⡟⠁⢠⣾⢏⣴⣶⣶⣦⡈⣿⡿⢠⣾⣿⡟⣠⣿⡏⣸⣿⣿⡎⡿⢀⡿⢃⣾⠀⣶⣦⡀
⠀⠀⠀⠀⠸⠀⠀⣾⢃⣾⣿⠿⠿⣿⡇⠿⢡⡿⢛⣿⣴⡿⠋⣰⣿⣿⠿⠃⢀⣠⣴⣿⡿⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⡼⢋⣤⣶⣶⠦⡀⠀⠟⠁⠘⣛⣭⡄⠀⣠⣤⣴⣶⣾⣿⣿⣿⡿⠁⡼⠿⠟⠁
⠀⠀⠀⠀⠀⠀⡸⠀⣰⣿⣿⠿⣯⠃⠀⠀⠀⠀⠀⠉⠉⠉⠁⡀⢨⣭⣭⣍⣩⣭⣥⣴⣶⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣌⡛⢿⣿⣿⣿⢿⣷⡄⠐⢬⣍⡙
⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⣿⣷⠎⠙⠂⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣾⣼⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠦⠀⠀⠀⠀⠙⢿⣿⣷⣌⡻⣷⡆⢼⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣸⢸⡦⣤⡀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⠛⠛⠻⢾⣿⠸⣿⣿
⠀⠀⠀⠀⠀⣤⢸⣿⣿⣷⣿⢸⣷⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡾⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣄⠹⠀⣿⣿⡆
⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⢸⣿⣷⠶⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⢹⣿⣿⣿⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠂⣿⠀⣿⣿⣷
⠀⠀⠀⠀⠀⢹⣿⠘⣿⣿⣿⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⢰⡏⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡟⢰⡀⠸⣿⣿⣆⢲⣟⡁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⡇⣿⡇⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠸⠃⠀⢸⣿⠀⣼⠃⡄⣿⣿⣿⣿⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⢸⣷⠘⠇⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡿⢰⡇⢸⡟⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡇⢸⣿⣧⡀⠀⠈⠻⣯⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⢁⣾⡇⢸⡇⢻⣿⣿⣶⠄⢰⣶⣶⣷⠈⢿⣿⣿⣿⣿⣿⣇⠀⠻⢿⣿⣦⡀⠀⠈⠃⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣼⣿⡇⠈⠇⣾⣿⣿⡟⢀⣾⣿⣿⣿⡇⠸⣿⣿⡟⣿⣿⡿⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣼⣿⣿⡇⠀⠀⣿⢸⡿⢁⣾⣿⣿⣿⣿⣧⠀⣿⣿⡇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢠⡇⢠⠇⠞⣡⣾⣿⣿⣿⣿⣿⡏⠀⣿⡿⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣼⡇⡜⠀⣴⣿⣿⣿⡟⣿⣿⡟⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢰⣿⡇⠀⢀⣿⣿⣿⠏⣰⣿⠏⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣾⣿⠀⠀⣼⣿⡿⠃⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⡏⣰⣿⠟⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⢻⡿⢠⡿⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣴⠾⢿⣿⡿⣦⣄⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡠⠋⠀⣠⣿⠉⣷⡌⣌⢳⣄⠈⠢⣀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⣠⠞⢁⣿⡄⢿⣷⠘⡆⠹⣆⢣⡈⢷⣄⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣀⣤⣴⡶⠞⢋⣡⣴⠟⠹⣧⠈⠁⠑⢿⡄⠹⣧⡣⠀⠈⠛⠶⠤⣍⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠿⠋⣁⣴⣾⠟⠋⠀⠀⠀⠹⣆⠀⠀⠈⢿⡂⠙⣷⣄⠀⠀⠀⢀⣠⠔⠂⢙⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⢠⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⢷⣄⠠⣙⠓⠒⢚⡉⠄⠀⢀⢀⣴⠿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠀⣰⣿⣿⣿⣏⡉⠳⠄⠀⠀⠰⠞⢋⣹⣧⣠⣀⠀⠉⠓⠿⠿⣛⣥⠒⢀⣘⡵⣾⣿⣗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⣿⣿⣿⠿⠿⠷⠀⠀⠀⠀⠞⢉⣿⡿⢷⣝⢿⣷⣶⣿⣿⣿⡛⣟⠻⣿⡿⣎⣿⣮⡁⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⣿⡿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣷⣌⣧⠙⣷⡹⣎⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣿⡿⢡⣿⡀⠀⠀⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢻⣎⣿⠇⣿⡟⢳⡆⢿⣧⢻⡟⢻⡟⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢀⣾⣿⣷⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢘⡼⠋⢠⡟⠘⢸⡇⣾⣿⡜⣿⡌⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⢟⣻⣿⣿⣧⡘⢿⣟⣉⡻⠓⠀⠀⠀⠀⡠⢊⡴⠋⠁⣠⠞⢠⡄⢺⡇⣿⣿⡇⢸⡇⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⠃⢹⡿⢿⣿⣿⣷⡀⠈⠁⠀⠀⠀⢀⣴⣾⡷⠋⠁⣠⠞⢁⡰⠏⢠⣿⠀⣿⣿⣇⡼⢁⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡃⠀⣿⠇⣜⢿⣿⣿⣷⣤⣀⣤⣴⣾⣿⡿⠋⣀⣴⠟⣡⠴⢋⣀⣠⡿⢻⠀⣿⣿⡿⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄⠹⣦⣹⣿⣿⣿⣿⡿⢹⣿⠋⣠⡾⢋⡴⠛⠁⠴⣋⣥⠔⣠⡷⣾⡸⣿⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡿⡄⠘⣧⠐⠘⡿⢾⣿⣿⡿⢃⡾⢁⣼⠟⡵⠋⢀⡴⠛⢉⡴⣻⡟⠁⠀⠈⣷⡹⣿⣿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⠁⢹⡄⠈⢧⠀⠈⣀⣿⠟⢁⣾⡇⣾⢟⣼⠁⣴⠏⠀⣠⡟⠼⣿⠀⠀⢠⠞⠉⢹⣯⢹⡻⣎⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⠇⢠⣿⣿⡄⠈⣧⠀⠟⠁⣰⡿⣿⣧⣿⠘⡇⣸⡏⠀⢰⡿⠀⢰⡇⠀⠀⣍⠀⠀⠀⢻⡀⢳⡈⠳⣤⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠛⠀⣿⠛⣿⡇⠀⢸⡆⣠⣾⣿⠃⣿⣿⣿⣇⢱⣿⡇⠀⣾⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠻⣦⡈⠻⣦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⢹⡇⣸⣴⠿⠋⡌⡟⢸⣿⣿⣿⣿⡌⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⣿⢦⡈⠻⣦⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣤⡼⠟⠋⠁⢠⠞⡼⢁⣿⣿⣿⠸⣿⣧⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⠳⡙⠳⣌⡳⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠉⢀⣀⡤⠖⢁⡞⢡⣾⣿⣿⡟⣠⣿⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⣦⡀⠙⢦⣄⠙⠻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⠛⢛⣉⣩⣤⡾⣫⣴⣿⡿⣱⣿⢇⡇⣿⣿⣿⠀⠀⣿⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣮⡻⣦⣀⠙⠻⢶⣤⣼⠁
⠀⠀⠀⠀⠀⠀⣠⠞⢉⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣿⡟⢸⠇⣿⡏⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⣿⣮⡙⠿⣿⠞⠻⡇
⠀⠀⠀⠀⠀⢠⣿⣾⠿⣛⣭⣥⡴⢾⠶⣶⣶⣛⣉⣴⣿⡿⠁⢏⣼⡿⢠⣿⠀⠀⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⣿⢻⡌⢳⣄⡾⢃⡔
⠀⠀⠀⠀⣠⡞⢹⠏⣸⣿⣿⠏⡔⣵⠰⣿⡏⢻⣿⡿⠛⣀⣤⡿⣿⠃⣾⡏⠀⢀⡇⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣠⣶⡿⠁⠸⢣⡾⢏⡵⠋
⠀⠀⠀⠀⢿⡄⠸⣆⠸⣿⣿⡀⡇⣿⣄⠉⠱⣾⣟⣛⣛⣛⣭⡾⠁⡼⣿⠃⠀⣼⡷⠛⢧⣀⣀⣀⣀⡤⠴⠚⠉⠉⡱⠟⠉⠀⣠⢧⣿⠵⠋
⠀⠀⠀⠀⠀⠙⠳⣬⣑⠿⣿⣷⣵⡈⠳⠮⣥⡛⠛⣛⣻⡿⠋⢀⡞⣵⠃⠀⣰⣿⡇⠘⣷⣤⣈⣁⣤⣤⣤⣤⣤⣤⣤⠖⠒⣉⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⣛⠛⠷⠶⠾⠿⠶⠛⠉⣀⠴⣋⡾⠁⢀⣴⡟⣱⣧⠟⠋⠉⠉⠉⠁⠀⢠⣿⣿⠏⢹⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠳⣬⣙⡲⢤⣄⣀⣀⣀⣠⡤⠞⣋⣤⡶⠻⣿⠀⣿⡀⠀⠠⠒⠋⣉⣉⣩⡉⠽⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣉⠋⢙⣿⣿⡶⠟⠋⠘⢧⡸⠘⠳⠆⣉⣭⠖⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁

⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⢿⠿⣿⣿⣿⣧⡀⠀⠀⠀⢀⢠⣿⣿⣿⣾⣿⣷⣆⡀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⢀⡀⠀⠙⠻⣿⠀⠀⠀⠀⢸⣿⡟⠃⠈⠀⠁⠙⣿⣿⡧⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⡗⠀⠀⠀⣿⣿⣿⣿⡟⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠉⠁⠀⠈⠀⢻⣿⠆⠀⠀⠀⢻⣿⣿⣿⡿⠗⢀⣀⠈⠀⠀⠀⠀⢻⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⣿⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣷⣤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⣤⣤⣶⣴⣤⡄⠀⠀⠀⠀⠀⠀⢄⣠⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣤⡆⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⠂
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣲⣾⡿⠟⠛⠛⢻⣧⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⡏⠁⠐
⠀⠀⢻⣿⣿⡟⡿⡿⠀⠹⡇⠀⠀⠀⠈⠀⣿⠠⠤⠀⠤⠄⠗⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⣿⠃⠀⠀⢀⣤⣶⣾⣿⣿⡃
⠀⠀⠀⠘⠋⠛⠃⠑⠋⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠘⠋⣀⡀⠀⣀⢹⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⢀⠀⢿⣿⡉⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⣿⣿⣿⡿⠫⠛⢿⣿⡄
⠀⠀⠀⠀⠙⣤⣠⣀⡴⠂⠀⠀⠀⠀⠀⠀⢸⣿⠂⠙⠉⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠟⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⡀⢀⠀⠀⠀⠀⠀⣁⣰⣶⣶⣴⣦⡁⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅
⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⢀⣿⠋⠉⠉⠉⠻⣷⡀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣦⡄
⠀⠀⠀⡤⠀⠈⠉⠉⠉⠙⣿⠁⠀⠀⠀⠘⠃⠀⠙⠛⠛⢿⣿⣿⣿⣿⠆⠀⠀⢸⣇⢀⡀⠀⡠⠄⣹⡇⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠈⡿⠻⡿⢻⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⠁
⠀⠀⠀⠂⢀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣰⡇⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠿⠍
⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠖⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣿⣽⣯⣮⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣍⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⡈⢿⣷⡀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢲⣅⠀⠀⠀⠈⢉⣭⣤⣤⣀⡀⠀⢸⣿⣷⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠘⠁⠻⠟⠹⠛⠋⢀⣿⣿⠀⠀⠀⢘⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⣸⡿⢸⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⢀⣴⠟⢡⠏⢠⣿⡟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⣠⠞⢋⠔⢁⣴⣿⠟⠁⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⡀⠀⠀⠀⠀⢰⣿⠏⠀⢠⡞⢁⠔⢁⣴⡿⠋⠁⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠷⠂⠀⠀⣿⡏⠀⣰⠏⡰⢁⣴⠟⠁⢀⣠⣴⡾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⡜⣠⡿⠁⣠⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣷⢠⡿⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠘⣿⣾⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣽⣽⣯⣭⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠐⠒⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⢀⡀⠤⠐⠒⠒⠒⠒⠒⠂⠒⠒⠂⠤⢄⣀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠊⠀⠀⠀⠀⢀⡠⠤⠒⠂⠈⠉⠉⠉⠉⠉⠀⠀⠐⠒⠠⠄⡀⠀⠀⠈⠓⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⠔⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠈⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠠⠃⠀⠀⢀⠎⠀⠀⠀⢀⠔⠊⠀⠀⠀⢀⡀⠤⠄⠀⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠑⠢⣀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡰⠁⠀⠀⡐⠁⠀⠀⠀⠘⠁⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢄⡀⠀⠀⠀⠀⣱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠁⠀⠀⡔⠀⠀⠀⠐⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡤⠖⠁
⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠇⠀⠀⠀⠀⠀⣤⣀⣄⣀⡀⠀⠀⡔⠁
⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠢⣀⠔⠂⠁⠀⠀⠀⠀⠀⠈⢓⣬⠀⠄⣀⡀
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡗⠀⠀⠀⠈⠳⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠡⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⠁
⠀⠀⠈⡾⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠢⢀⡀⠀⢀⣂⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⣀⡠⠤⠚⠀⠙⠢⢤⣀⡀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠀⠀⢀⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣠⠟⣡⣿⣿⣿⣿⣿⣿⡿⠙⢿⣷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢍⣉⣉⣡⣾⣿⣿⣿⢟⣫⣾⠟⠁⠀⠈⢿⣎⢷⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣾⣟⣯⣽⠾⠿⠏⠁⠀⠠⠀⠀⢸⣿⠈⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⡿⣿⣿⠙⠲⠖⠀⠀⠀⠐⠶⠖⣿⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⣾⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⠟⣿⢳⣿⣷⡄⠀⠤⠶⠦⠀⠀⣿⢹⢱⡏⢙⣧⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⡸⣿⣿⣿⣦⣀⠉⢁⠤⠊⢿⣿⣧⣙⠛⠁⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠸⣇⢿⣿⡿⣿⣿⠉⠁⠀⠀⠈⠻⣿⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣄⡈⠻⣿⣿⣮⡛⠷⣤⡀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠻⣿⣶⣬⣙⡿⣿⣦⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠛⠿⣿⡺⣿⣿⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠚⠁⠀⠈⢳⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⢀⣼⣯⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠭⠽⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣶⣴⣶⣶⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣸⣿⡿⣿⡿⢿⣿⣿⣿⣏⣉⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⡴⣽⡿⠃⠹⠁⠀⠉⠙⠋⠈⠉⠋⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⠎⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡎⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⢸⡇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣸⣧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡏⢸⣿⡁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠱⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢈⣿⡿⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠁⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡆⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⣄⡀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧⣿⠁⣦⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⣿⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⠉⠀⠀⢿⣿⠀⣿⣷⡄⠀⠉⠻⣷⣦⣀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠃⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣤⡀⠀⠀⠈⠇⠀⠰⢿⣿⣶⡄⠀⠀⠙⡿⢿⣷⣾⣿⣿⣿⣿⠟⠀⠌⠀⠹⣿⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⢰⡟⢿⣷⢎⠋⠠⠀⠠⠛⢿⣿⣿⣿⠃⢀⠀⢀⡄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⢠⣾⣿⣿⣻⣏⢷⡄⠀⠀⣾⠀⠈⠉⣹⠆⡏⠀⣼⠃⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⢮⣧⠀⠀⠹⠀⠀⠀⢿⡀⣷⠒⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢠⡄⢻⣿⣿⡨⢣⡌⢻⡄⠀⢶⡄⠀⠀⢸⣇⠇⠀⣷⢰⣿⡟⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠿⣟⠂⠙⢿⣇⠘⡇⠀⣿⠀⢸⢧⠀⢀⣿⣿⣄⣾⣸⠘⣿⠁⠋⣾⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣼⢶⡈⠃⠀⠀⠉⠀⠋⠀⢸⠀⣾⠀⠀⣼⣿⣿⣿⣿⡉⠀⠉⣈⣰⠈⣧⠀⠀⣜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⢀⣀⠀⢠⡾⢀⡏⠘⣧⣿⣿⣿⣿⣿⡁⠀⠀⢻⣷⡇⠁⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣿⡇⠘⠃⣼⡇⣾⣿⢿⣿⡟⣿⣟⠁⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠘⠁⠀⠐⣿⣿⣿⡎⢸⣿⣿⣿⡿⠃⠀⢀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠸⣿⣿⡿⠃⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⣛⣛⣛⣛⣳⠶⣤⣀
⠀⠀⠀⠀⠀⢀⣴⢞⣭⢞⣫⡭⠷⠶⠶⠾⣭⣟⡷⣭⡳⣤
⠀⠀⠀⠀⣠⢟⣵⢫⡶⣫⡵⠞⠛⠉⠉⠛⠳⢮⣟⢮⡻⣮⢳⡄
⠀⠀⠀⣰⣏⡾⣱⢏⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡹⣞⢷⢻⡄
⠀⠀⢀⣟⣼⢱⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⡞⣧⣿
⠀⠀⠘⠋⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠀⠙⠃⡀
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⣄
⠀⠂⠄⠠⠤⠀⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠀⠤⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣠⡄⡄⠠⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⣗⢮⣟⢮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⡄⠈⢳⡝⣮⢳⡝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠐⡲⠂⠒⠐⠂⠒⠐⠀⠲⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠛⠛⠁⠀⠀⢿⣸⡏⣿⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣛⣘⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠈⠀⠀⠀⠀⠁⠀⠑⢤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣮⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣄⣀⣤⣀⣠⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣭⣟⠲⢤⣤⣶⡿⠿⡇⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣭⣽⡓⠦⣤⣴⣾⠿⠷⠀⣤⣀
⠀⠀⠀⠀⠀⠀⢸⢹⠸⠋⠛⢿⣷⡄⠀⠀⠀⠉⠙⢿⡌⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠟⠙⠻⣿⣦⠀⠀⠀⠈⠉⠻⣦⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⢻⣿⣄⡀⢀⣀⣠⣤⣭⣖⠒⠶⣦⡄⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠙⣿⣧⣀⠀⣀⣀⣤⣬⣷⡒⠲⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣓⠢⠀⢉⣟⠛⠉⡁⠀⠈⢻⣿⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠈⣹⡛⠋⢉⠀⠀⠙⣿⡇⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢸⡟⣧⡀⢸⣦⡀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠀⠀⣿⢳⣄⠈⣷⣄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⢻⣷⣿⡿⣶⣤⣦⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠹⣷⣾⣿⢷⣦⣶⡴⠞
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢴⣿⡷⠈⣦⡀⣿⡏⡗⠀⢙⠎⣷⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⣾⣿⠆⢱⣄⢸⣿⢹⠀⠈⡓⢹⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⢸⣷⠏⢀⣤⣶⠟⣰⠏⣰⡟⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⣿⡾⠁⣠⣴⡿⢃⡟⢡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣔⣿⣿⡏⠸⢡⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣾⣿⣿⠁⠏⣴⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠙⢿⣮⡻⣄⢻⡅⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠙⢻⣷⣝⢦⡘⣯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣿⡇⠀⠈⣿⠋⣿⡆⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢸⣯⠀⠀⢹⡏⢹⣷⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠉⠛⢦⣔⠁⣠⣾⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡈⠙⠳⣤⡋⢀⣼⣿⡂⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢠⡞⣫⠊⠁⢺⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⡴⢛⡖⠉⠐⣿⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠉⠳⡇⠀⠀⡀⠉⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡏⠛⢾⡀⠀⢀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢧⠀⢻⣆⠀⠓⢄⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⢻⡀⠘⣷⡀⠘⠢⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣠⠞⠃⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡⠴⠛⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢠⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⢂⠠⡈⠦⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠄⠀⠀⠀⠀⠠⡮⠉⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⣾⠂⠈⠢⡀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡾⠂⠀⠀⢠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠼⣇⠈⠀⡏⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠇⠀⠠⡀⠂⢀⡤⠤⡤⢤⣤⣜⣧⣤⣰⣷⡾⠛⠧⠀⠤⠤⣤⡀⠀⠁⠀⣀⡔⠁⠈
⠀⠀⠀⠀⠀⠈⢄⡀⠀⠚⢁⠄⠀⠀⢀⡤⢔⣛⣻⠵⠿⠚⠓⢀⠀⠈⠄⠈⠍⠉⠉⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⡟⠲⠬⡴⠞⠋⠁⡀⠀⠀⠉⠃⠀⠀⡰⠀⠈⠢⣵⣄⠀⣀⢤⣀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢣⡄⠄⠈⠀⠀⠉⠳⠄⠀⠀⣠⠤⠴⠓⠂⠀⠀⠀⠀⠁⠉⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠉⠚⠢⢀⠢⣦⠏⠀⣀⠼⠁⠀⣁⣀⡀⣄⡤⠐⢠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠠⡌⠀⡆⠀⡃⣤⢞⠡⠀⠀⠙⠉⢉⠇⡈⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠁⠀⡼⠁⡼⠀⡰⠀⠀⡠⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⢰⡇⠀⣠⠚⢁⡴⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣼⠀⢑⡼⠀⢂⠅⠀⠃⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡻⠧⣸⠇⠀⠉⣠⠀⠋⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢧⡴⠋⠙⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢠⠊⠀⠀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡠⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡠⡉⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠁⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠄⠀⠇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠉⠉⠉⠁⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠰⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠆⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢠⠀⠀⠀⠀⣆⠀⠀⠀⠸⡄⠀⠀⠈
⠀⣀⠤⢤⠀⠀⠀⢸⠀⠀⠀⠀⠀⡶⠀⠀⠀⠈⡇⠀⠀⠀⢘⠀⠀⠀⠀⢹⠀⠀⠀⠀⡧⡀⠀⢠⡤⠠⢤
⡎⠁⣀⠈⢣⡀⠀⢠⠆⡀⠀⠀⣠⢧⠀⠀⠀⡸⢣⠀⠀⢀⠞⠣⡀⠀⢀⡜⠣⠀⢀⡴⠀⠉⠀⠋⢀⡀
⢇⠀⣷⡀⠈⡁⠒⠂⠀⠉⠚⠂⠁⠀⠐⠒⠋⠀⠀⠑⠒⠋⠀⠀⠉⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡷
⠈⠢⣈⣁⣀⢇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⣰⠄⣉⣠⠔
⠀⠀⠀⠀⢩⠈⢆⠀⠀⠀⠑⠶⠔⠖⠂⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠘⠶⠶⠚⠀⠀⡠⠁⠀⠀⠀⡇
⠀⠀⠀⠀⡘⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠜⠁⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣁⡻⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠐⡶⣤⠤⠄⠀⠀⠀⠀⠠⠤⠤⢰⠖⢏⠀⠱⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⠂⠀⣼⠀⠀⠲⢄⠀⠀⠀⠀⠀⢀⡴⠋⠀⢨⢆⠀⠙⡄⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠃⠀⠜⠀⠑⠂⠒⡂⠁⠒⢒⡒⠈⠁⠀⡌⠉⠁⠈⢃⠀⠘⡄⠀⠀⠀⠀⠀⡄
⠀⠀⢠⡄⠀⠀⠀⠀⢠⠃⠀⡘⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠒⠈⠀⠀⠀⠀⠀⠡⠀⠘⡄⠀⠀⠀⠐
⠀⠀⠈⠁⠀⠀⠀⡠⠃⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠙⡄⠀⠀⠃
⠀⠀⠀⠘⠀⠀⡰⠁⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⢆⠀⠘⡄
⠀⠀⢠⠎⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡀⠀⢰
⠀⠀⢀⠀⠀⡎⠑⠂⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠒⠈⠃⠀⠈
⠀⠀⠈⣡⠀⡆⠀⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠀⠀⡠⠀⠸
⠀⠀⢀⡁⠀⠈⠉⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉
⠀⠀⢨⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠳⣀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠒⠂⠀⢲⠂⠐⠂⠒⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠇⠀⠸⡀⠀⡸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⠈⢉⣁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣼⣿⣿⠁⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⢿⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⠉⠉⢻⣿⡆
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⢻⠏⠘⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⡇⠻⠟⠁⠏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠠⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠷⠀⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠞⢳⣦
⠀⠀⠀⣠⣾⡿⠟⠛⠛⠻⠿⣷⡀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄⡀⠀⠀⢀⡾⠟⠃
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣴⣿⣿⣿⡏⠘⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⢠⣿⠿⠿⠿⠿⠿⠿⢿⡄
⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣸⣿⣿⡛⠇⠀⠈⠻⢿⣿⣿⣿⣿⣏⠉⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠟⣿⣿⣇⠀⠀⠀⠀⠀⠉⢻⣿⣿⡿⠃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⡿⣿⡉⠀⠀⠀⠀⠀⠀⣾⠿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣠⣾⣿⣿⠹⣿⣿⣿⣿⡟⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠉⠉⣿⣿⡄⠀⠀⢀⣾⣿⣿⣿⠀⢹⣿⡟⣿⣿⣿⣿⡆
⠀⠀⢸⣿⣿⡏⠀⠻⣿⡇⠸⠁⢹⣿⡟⠀⠀⠀⣿⣿⡿⡏⢻⣿⠁⠀⠀⢸⣿⣇⠀⠀⠊⣿⣿⣿⠙⠀⠀⡟⠀⠘⣿⣿⣿⠁
⠀⠀⠀⢿⣿⡇⠀⠀⠘⠀⠀⢀⣿⡏⠃⠀⠀⠀⢹⣿⡇⠉⠀⠛⠀⠀⠀⢸⣿⠏⠀⠀⠀⠟⣿⣯⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⣞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠉⢀⣀⡀⠤⠤⠤⠤⠤⢀⣀⡀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣀⠤⠒⠉⠁⣀⣀⣤⠠⠤⠤⣄⣀⣀⠈⠉⠲⢤⡀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠚⠁⣀⠤⠒⠉⠁⣀⣀⠠⡤⠄⣀⣀⠈⠉⠒⠤⣀⠈⠓⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠊⠀⡠⠊⠁⣠⠔⠊⠉⠀⣀⣀⣀⣀⣀⠀⠉⠑⠢⣀⠈⠑⢄⠀⠑⣄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡜⠁⣠⠎⠀⡠⠊⠀⡠⠔⠊⠉⢀⣀⣀⣀⠀⠉⠑⠢⣄⠈⠑⢄⠀⠱⡄⠈⢦⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣜⡀⣰⠃⢀⠞⠀⡠⠊⠀⡠⠖⠋⠁⢀⣀⡀⠈⠙⠲⣄⠀⠑⣄⠈⠣⡀⠘⣄⠀⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠚⠉⠉⠉⠉⠓⢮⡀⡼⠁⢠⠎⠀⡠⠚⠉⠁⠀⠈⠉⠲⢄⠀⠳⡀⠈⢆⠀⡱⠖⠋⠉⠉⠉⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⢀⠇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠱⡀⣈⡞⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢀⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠈⠣⣄⣀⡤⠤⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡴⢤⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠲⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠘⠛⠶⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠾⠷⠦⢤⣤⣄⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⡶⠿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣦⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢹⡆⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣸⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⣀⣀⣠⣴⠞⠁⣠⣿⡄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠶⠛⠉⣉⣀⣀⣉⡉⣉⣉⣉⣁⣀⣤⠾⠋⠀⢻⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡞⠁⠀⠀⠀⠐⠶⠶⠶⢤⡄⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⠖⠀⠀⠀⠙⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣗⠀⠀⠀⠀⠰⠟⢻⣏⡿⠳⣄⠀⠀⠀⠀⠀⢠⡞⢿⣿⡟⠛⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠳⣤⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⢙⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⢻⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠃⠀⠀⠀⠀⠈⠉⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⡾⠻⠶⠛⣰⠶⣬⣿⣏
⠀⠀⠀⠀⠀⠀⣾⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠟⢷⡆⠀⠀⠀⠀⣤⣀⠀⠀⠀⣀⣤⠀⢿⡀⠀⡶⢿⡆⠹⣇⠹⣆
⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⠶⠶⢾⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠈⢳⣬⣷⠀⣿⠀⢿⡀⢿⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢠⣤⣤⣤⡄⠠⣤⡤⠀⣤⣄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡆⢸⡇⠸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣷⠸⣇⠈⣷⠀⣿⠀⢿⡟⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢸⡇⠀⣿⠙⠳⢦⣄⣀⣀⣠⣤⡶⠛⠉⠀⣿⠀⣿⠀⢻⠀⠻⠀⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠘⠛⠛⠛⠒⠛⠛⠛⠛⠛⠋⠀⢸⡇⢠⡟⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠉⠀⠘⠁⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙


⡏⠉⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡏
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⡖⠐⢢⢰⠀⢲⠂⢀⠶⠀⢰⢒⡆⣶⠀⢰⡆⠂⠀⢰⡂⠀⢰⡆⢐⡆⣲⢠⠂⠒⡄⢰⡰⠂⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⣇⣀⡜⢸⡀⢸⠀⡼⠒⢇⢸⣉⡗⣻⣀⣸⣇⣀⠀⢀⣈⠆⢸⡇⢈⡏⢳⠘⣄⣠⠆⣘⠑⣄⣏⣁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⣠⠤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡠⠞⠋⠀⠀⠀⣀⣤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀⠀⠈⠙⠶⣄⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⠴⠋⠀⠀⢀⡤⠖⠋⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠙⠳⢦⡀⠀⠈⠛⢦⡀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢁⡴⠃⠀⠀⣠⠞⠉⠀⢀⣤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠓⠶⣤⣀⠀⠀⠙⠷⣄⠀⠀⠙⢷⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⢠⠞⢀⠀⣰⠋⠀⠀⣠⠞⠁⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠈⢷⣄⠀⠀⠙⣦⡀⠀⠙⣧
⠀⠀⠀⠀⢠⡏⠀⢀⡼⠁⠀⢀⡼⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠙⢦⡀⠀⠈⢻⡄⠀⠘⣧
⠀⠀⠀⠀⡞⠀⢀⡞⠁⠀⢠⠞⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢳⡄⠀⠀⠻⣆⠀⠘⣧
⠀⠀⠀⣸⠁⢀⡾⠀⠀⣰⠋⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠹⣆⠀⠀⢹⡆⠀⠹⡆
⠀⠀⢀⡏⠀⣼⠁⠀⡴⠃⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠘⣦⠀⠀⢻⡄⠀⢿⡀
⠀⠀⢸⠃⣰⣇⣀⣼⠁⢠⠏⢀⣤⠖⠒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣷⡀⠘⣧⢀⡬⠷⠒⠺⢧⣄
⠀⠀⣸⠋⠉⠀⠀⠉⠓⢯⡴⠏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠳⢤⡼⠋⠀⠀⠚⠉⠀⠙⣆
⢠⡴⠟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠙⠲⠶⠞⢻⡄⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢰
⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣰⢦⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣀⣀⣴⣤⣀⡀⣀⣀⡴⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣶
⣿⣿⣇⣤⣍⣤⣮⣬⣧⣥⣾⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡠⠤⣤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣷⠐⡿⢶⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⣼⣿⣯⢸⣟⣽⠀⡿⠻⣿⠿⢌⣳⡀
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣴⣙⠿⢽⠁⢹⣦⣧⡸⣾⠷⠤⠾⠤⣾⣝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹⣦⠼⠓⠋⠉⠀⠀⠀⠀⠀⠰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⢤⠚⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠃⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠌
⢀⡄⢀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠐⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⠠⠲⠴⠈⠒⠂⠥⠴⣤⠄
⣞⠂⠈⠀⢂⠀⠀⠀⠀⠠⠀⠀⡀⢄⣢⡶⠄⣴⡉⠉⠈⠈⠁⠀⠀⠀⠀⠘⠒⠒⠒⣒⡍⠆
⠀⠀⠐⠠⠃⠀⠀⠀⠀⠐⠼⢛⣉⡀⠠⠤⠒⠈⠑⠄⠀⠆⣤⢋⠀⠄⠀⠀⠋⠉⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡄⠀⠀⠒⣲⢆⠀⠀⠀⠈⠤⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠄⠨⠀⠂⡀⠀⠈⠂⢀⠁⣀⠴⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⡱⠄⠈⠁⠑⠒⠒⠛⠋⠁⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠞⠀⣀⠤⠔⠒⠦⡀⠀⠀⠀⠒⠐⠒⠒⠒⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠉⢬⡔⠁⠆⣤⠤⠔⠒⠒⠢⡄⣀⠀⢀⣠⠴⠒⠲⠤⣄⣀⠤⠊
⠀⠀⠀⠀⠠⢂⠀⠿⠜⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣄⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠉⠒⠤⠤⠤⠒⠉⠀⠀⠀⠉⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠒⠛⠉⠉⠀⠀⠈⠑⠦⢄⣀⣀⡠⠔⠉⠀⠀⠉⠙⠒⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣠⠤⠖⠒⠂⠒⠠⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠦⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣀⣒⣒⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⠿⠟⠛⠻⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠿⢻⣿⠇
⠀⠀⠀⠀⠋⠀⣤⠀⠀⠀⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡤⠀⠀⢠⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⠁⠀⠠⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡃⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠐⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣛⣭⣭⣟⡿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠿⠿⠿⠛⣿⣇
⠀⠀⠀⠘⣿⣿⠿⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⢀⠀⢸⡏
⠀⠀⠀⠀⢻⠏⠃⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁⠈⠁⠀⠀⠈⠁⢈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠐⠂⠈⠁
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠀⣠⣾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠉⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠉⠀⠉⢷
⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢃⣀⠀⠀⢀⣘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⠀⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠉⠀⠀⠈⠁⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠈⠢⡀⠀⡀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣦⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⣉⣁⣉⠉⠓⠤⡀⠀⠈⣟⡀⠀⠀⠬⣉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠐⠂⠬⣖⠠⢤⡬⠭⠥⠽⢷⣴⠟⡉⠓⣌⠳⢦⣕⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠚⠒⡑⠫⣑⢤⡙⢶⣄⡉⠀⠒⠀⣸⠀⠘⡀⢸⣄⡀⠙⢷⣌⣢⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠈⠳⡀⠱⡱⡄⠹⡘⡟⠭⢉⣙⢧⠀⣣⢺⠀⢹⠐⡄⠱⡝⡕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢱⡀⠱⢱⠀⢱⠈⣳⢖⠂⠵⣿⠃⡏⣠⡾⡀⢱⠀⠹⡌⣎⢆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡏⡇⠈⡄⠹⣄⠈⠒⠌⢲⡞⠁⠇⢹⠸⠀⠀⢧⠸⡘⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠠⣦⣤⣄⣀⡀⠀⠀⣧⠀⠀⡇⠀⡇⠀⡟⢍⠒⠒⠢⢿⡜⣠⠞⢆⠀⠀⢸⠀⡇⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡀⠀⠀⠀⠀⠉⠙⠳⡄⢾⠀⠁⡇⠀⡇⢀⠘⠢⣍⣀⣢⣄⣷⠁⠀⢸⡶⣄⢸⢀⠃⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠘⠿⠿⠷⠦⠄⠀⠀⠀⠀⢹⠀⠀⡇⠀⠇⠈⡄⢂⢫⡀⠄⠉⡙⢆⢀⡼⡇⢸⡇⡘⢀⠇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠸⡀⠘⡌⠂⡿⡙⠒⠒⠾⢿⠇⠇⠞⠛⢆⠸⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠸⡀⠀⠱⡀⠈⢆⠱⣌⣑⠒⠤⣘⣦⡾⠀⡆⠀⣧⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡘⢆⠀⠘⢤⠈⠀⢸⣷⠌⣉⠉⠙⠳⣄⡇⡜⢸⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢝⡦⢄⠀⢀⡿⠊⢳⣄⣀⣉⣒⣈⣿⡥⠊⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠙⠂⢣⡈⠢⠭⠭⠭⢍⡉⠁⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢌⠢⣙⠒⠶⠶⠶⢤⡈⠑⠒⠋⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠢⠄⣁⠀⠬⢕⡌⢂⡤⢖⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣀⣀⣀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⣐⣒⡂⠀⢦⡀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠃⠚⠻⣯⣙⠛⢷⠦⣽⣸⣿⡷⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠗⠛⠉⠙⢢⡈⢿⡷⠄⢃⢹⣿⣟⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⣷⣸⣿⠀⠸⢸⣿⣿⡿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠀⠀⠠⠒⠠⣿⣿⣿⠧⢀⣿⣿⡿⢁⣴⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠐⠀⠀⠈⠳⠉⠛⠿⣐⢿⣽⠻⣿⣿⣫⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⣷⣅⠿⣿⣿⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⢿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢠⣿⣿⣷⣌⠻⣧⡙⠛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣸⣮⢯⡁⠀⠀⠀⠀⠀⠘⠿⣧⣌⠻⢿⣿⣿⡆⠹⢟⡶⡈⢲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣿⡃⣧⡀⢀⣀⡤⠀⠀⠀⢘⣿⠇⢸⡿⠛⢀⣶⢸⣇⠘⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠟⠟⠛⠉⠁⠀⠀⠀⠀⠀⢠⠞⣩⡤⢿⠁⡴⠛⢃⣾⠏⠀⣰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣣⠘⢦⡑⣿⡟⠁⢠⣾⣿⣿⣿⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣿⣷⠀⢻⢜⣿⣦⣙⠻⣿⣝⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣿⣛⣠⣿⣿⣿⣿⣿⠂⣹⣿⡷⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⣀⣀⣤⡤⠖⠂⠒⠀⠀⠀⠀⣿⣿⣥⣿⢿⣿⣯⣿⠟⢁⡴⢫⡿⠁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⡶⠈⢻⣿⣾⣿⣿⡿⢿⣄⡘⠷⡿⣿⣦⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣤⣤⣤⡴⠶⡾⠛⠋⠉⠀⠀⡸⣿⣧⠘⢿⡄⣠⣽⠿⠀⣽⡟⡫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⢀⣔⣁⣄⡀⠀⣾⡏⠀⣤⡿⠀⣠⡿⠻⠁⠈⠉⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠁⠀⠀⠙⢷⡀⠻⡀⠞⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢤⣤⣦⣤⣤⣤⣤⣤⣤⡴⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⡿⣿⣋⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡏⢿⡇⠻⢷⣿⣟⣻⣿⣶⣯⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⣷⣦⣀⣤⣤⣬⣙⣿⣿⣿⣿⣿⣿⠅⢀⣴⣄
⠀⠀⠀⠀⠀⠀⣿⡿⣿⣇⡉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠿⠟⣻⡇⠀⡀
⠀⠀⠀⠀⢀⡄⠻⣿⣦⣉⣻⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⣿⣿⣿⣶⣶⣾⣿⣶⣿⣧
⠀⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣋⣍⣀⣤⣀⣀⠐⠒⠈⠉⠙⠿⠿⠿⠟⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠈⠉⢁⣽⡿⣿⠟⠙⠉⣀⣽⣿⣭⣤⣄⣀⣄⣩⣭⣽⣿⣿⣿⣿⣾⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢠⣾⡏⢀⣡⣴⣶⣿⣿⡿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣛⡛⠛⠛⠿⣭⣽⣻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣾⣿⢻⡟⠋⣈⣴⣾⣿⣿⣿⡛⠛⠻⠿⡉⠛⣿⣶⣶⣶⣦⣿⣿⣿⠛⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠈⠁⣿⣿⠿⠃⠉⣤⣾⣿⣿⡟⠛⠋⠉⢉⣉⠳⣶⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠇⠀⢸⣿⠀⣤⣾⣿⣿⣿⣍⠋⢉⣵⣶⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡉⠙⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⠿⠿⠿⠛⠛⢶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢠⣿⣿⣿⣿⣿⣿⡶⠦⢀⣠⣾⢋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣯⣭⣥⣤⣾⣿⣿⣷⠃⠀⠀⠀⠀⠠⠶⠶⠶⢶⣶⣤⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠿⠿⠿⣿⣿⡟⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣶⣿⣿⠿⢿⡀⢻⡇⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⢷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⢃⠀⢿⣿⣿⣿⣿⠁⢴⡄⠹⣎⣷⡀⠀⠀⠀⠀⠉⠛⠧⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠡⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⣿⣿⣿⣿⠀⠀⠁⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣆⠀⠀⢿⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⠸⠷⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢸⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣷⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⢻⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣿⣿⡟⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠁⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⠉⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣿⠿⠿⢷⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄


⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣵⠗⠺⣯⣥⣈⣉⣭⣍⣡⣄⣉⣤⣉⣥⣬⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠚⠿⡿⠷⡮⠙⢮⡉⠳⣌⠓⠒⢦⣄⡀⡈⠀⠀⠀⡀⢀⣤⡀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠌⡐⢄⠈⢂⠸⡄⠀⠹⡄⠈⠳⣌⠢⠱⣌⢿⣄⠂⢢⠈⣌⣤⣀⣘⠠⠀⠀⡐⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⠢⡄⡄⢣⢹⡴⠀⠙⠀⠀⠈⡆⢠⡋⠘⠿⡁⢤⢀⠿⣿⣿⠊⣵⠀⠀⢡⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⡘⢿⡄⠎⣇⢣⠀⠐⠀⠀⢸⠀⠗⠈⠢⡘⠒⠒⠠⠘⣘⡞⠃⢀⠚⠀⠀⢀⠠⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠁⠠⠀⢸⣦⡑⢤⣼⡄⠆⠀⠀⠀⠀⣀⡠⠿⠂⠙⠀⡀⢐⠉⠉⠁⠾⢤⡔⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⠦⠀⠀⢸⡿⡎⢗⢨⡓⢌⡄⠀⠀⠀⠋⠠⠌⢀⠀⢀⣠⠋⠀⠀⠀⠉⢠⠻⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡷⡈⣎⠻⡂⠙⡀⠀⠀⠄⠸⣛⠁⣰⣤⣾⣿⡗⠀⠒⠀⠔⠃⢠⠁⠲⠆
⠀⠀⠀⠀⠀⠀⢓⠀⠁⠀⠀⠀⠀⢀⠔⠱⡙⣜⡿⢠⢇⠀⠀⣦⠀⠌⠀⠉⡉⠁⠿⠟⢟⠃⠀⢀⡐⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠤⢦⠀⠀⠀⠀⠀⠁⠀⠀⢟⠮⡻⣿⣧⠢⣡⠹⡄⠓⠰⠀⣁⡠⣀⡀⠀⡇⡔⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠⠄⠀⠀⠘⣄⠀⢹⣿⣷⣌⠣⡙⢦⣃⢀⣀⠀⠉⠀⠠⠤⠈⢢⣄⣠
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣨⣷⡤⠤⠛⡛⢅⡉⠛⠈⠝⠳⠭⣦⡭⢭⠫⡿⡤⢿⠄⡀⠁⠈⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⣹⢩⡈⠓⢐⠁⢌⡀⢨⠓⠞⠃⢀⠀⠱⠀⠀⠀⡅⣠⠈⢀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⢉⡇⠀⠀⢂⠀⠈⠻⡖⠀⢠⠜⠀⠙⢀⡞⢹⠏⣉⠐⡆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠙⡄⠀⠺⠀⠀⠀⠹⡀⠀⠀⢰⣷⣾⠤⠐⠁⠰⠀⣘⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⠉⢽⣿⣿⢆⠀⠀⠀⠤⠠⠀⠂⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⢉⠃⠀⠂⠀⡀⣠⠖⣅⠁⠀⠀⠙⢃⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠿⠃⠀⠀⠀⠀⠀⠀⢀⡐⠇⠀⠀⠀⢹⠀⠀⠀⠀⡌⡐⣀⣣⣄⡓⠃⠤
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠘⠁⠀⠰⠠⡂⣠⣓⠠⠶⠈⢑⡒⢾⣿⡿⠁⣍⠈⠂
⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⢂⡱⢄⣤⣤⠹⣐⠒⠄⠂⠻⠐⣮⣄⡀⠃
⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠡⠰⢌⡠⢙⣿⣿⠯⠆⠁⣼⡁⠒⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠦⢤⠀⠊⣁⡀⠀⠙⠀⠀⢅⡐⣧⣗⠦⢋⠑⡎⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢠⡥⠄⠀⠀⠀⡠⠀⠀⠉⢀⡄⠤⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠁⠱⠤⢤⣯⣼⣯⣄⠀⡄⠀⠓⢸⣇⠉⠁⠀⠀⠢⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠿⠿⡿⠋⠈⠐⠀⣈⡅⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢢⣀⠀⠰⠀⡆⢀⠰⠀⠀⠠⠆⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⡀⠀⢻⠁⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⢈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣾⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣟⣛⣛⣛⣛⠾⠿⣽⣂⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣯⣭⣭⣭⣭⣿⣛⡷⣶⣍⡛⢦⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣭⣍⣙⡻⠿⣿⣶⣍⠻⢦⡙⢦⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠦⠄⠀⠉⠛⠿⣿⣷⣬⢻⣿⣆⢦⠙⠂⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⣦⡙⢿⣿⣿⣿⡞⣷⡱⡄⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠹⣿⣿⡇⢹⣧⢹⡼⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⡄⣿⣿⡇⣾⣿⡄⣿⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⡿⢡⣿⣿⢡⣿⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣡⣿⣿⢏⣾⡿⢡⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣳⣿⠋⣠⠟⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢏⣼⢟⡅⢈⡵⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣾⣏⣾⠁⣾⢣⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⡘⣿⣾⣿⡈⠀⢈⢅⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢷⣅⠙⣿⣿⣧⡈⢂⣾⣞⣣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣜⢿⡌⠻⣷⣦⡙⢿⣿⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⣹⣷⣌⡀⠢⢝⡻⣦⠹⣆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢩⣨⡾⢟⣿⣷⡜⠿⣻⢨⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⠀⠈⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⡍⢛⣀⢀⣀⠀⡀⡀⣠⢄⡀⡀⠀⢀⣀⠀⣀⡀⣀⡰⠂⠀⢠⠆⣀⡀⢀⢀⣴⡃⡠⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠚⠹⠗⠋⠻⠚⠛⠓⠇⢛⡿⠀⠀⠋⠻⠚⠻⠛⠛⠇⠀⢰⡷⠛⠻⠒⠞⠚⠻⠙⠗⠻⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠈⣁⡠⠤⠤⠤⠤⠤⠄⢀⡉⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⠔⠊⣁⠠⠄⠒⠒⠒⠒⠂⠀⠤⢀⡉⠒⠤⡈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢁⠔⠉⡠⠒⢉⠠⠐⠒⣀⣉⠉⣉⣁⡐⠒⠤⢈⠑⠢⡀⠐⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡔⢁⡴⢋⡠⠊⡡⠔⣊⠥⠤⠐⠒⠒⠠⠤⢉⡒⠤⡉⠢⡈⠐⠀⠑⡄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠎⢠⠊⡠⠊⡴⢋⠔⣉⠄⠒⠉⠉⠁⠉⠉⠑⠂⢌⡐⢌⠐⠈⠢⡈⠢⠈⢀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠎⢰⠁⡜⢠⠊⡔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠌⢂⠐⡀⠁⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠼⠤⢇⣘⡠⠧⠮⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡌⠄⠤⢀⡀⠤⠨⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠑⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢮⣄⣀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⢐⠀⠸⢫⡴⠊⠶⢄⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠀⠀⠈⠉⠀⠀⠐⠿⡷⠀⢀⣀⡤⠀⠀⢸⣷⡂⢿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⢅⠀⠀⠀⠀⠀⠀⠀⠈⢛⣋⢠⡧⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠸⠛⠆⡀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠟⣡⡀
⠀⠀⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⡆⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⢎⢡⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⡻⠃
⠀⠀⠀⠀⠀⢀⣽⣦⠂⠀⠀⠀⠀⣀⠀⠀⠘⠪⡮⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢻⣿⣆⠊⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠁⠤⠶
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣈⣯⣔⡀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠘⠺⠒⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠼⠟⠤⠂⠀⠀⢤⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⡀⠤⣀⠂⢠⣠⣤⡴⢫⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠹⣧⡀⢠⡖⠀⠐⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠌⠀⠒⠉⠇⠉⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⠖⠂⠀⠁⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠟⠛⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢴⠛⡇⠀⠀⡆⠀⠀⠀⠈⠠⠿⣷⣾⣿⠆
⠀⠀⠀⠀⠀⠈⡳⠮⢤⡠⣾⢩⣾⣷⣄⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡙⠃⠀⠀⠀⠀⠀⠽⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⡔⣿⢾⣁⣿⣯⣽⠄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠠⢴⣤⠔⠃⠀⠀⠀⠀⠀⣿⣇⣤⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠱⡶⠻⡞⣷⡫⢙⣅⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣀⢰⡎⠰⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠓⡸⢲⣹⣶⠄⠀⠀⠘⢿⣦⠀⠀⣀⣤⡀⠀⠀⠄⢀⣰⢉⡏⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⣶⡈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣉⣉⡁⠀⠀⠀⡀⠄⠊⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠋⠃⠠⠀⣄⡴⠀⠀⠉⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠉⠉⢉⣀⣀⣀⣀⣀⡉⠉⠉⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⢁⣠⣤⠶⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⣤⣄⡀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⣀⡴⠞⠋⠁⣀⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣤⣀⡈⠙⠳⢦⣄⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⠞⠉⢀⣤⠶⠛⠉⣀⣠⣤⠴⠶⠶⠶⠛⠶⠶⠶⠶⣤⣄⣀⠉⠙⠳⣦⣀⠈⠻⢦⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣴⠟⠁⣠⡾⠋⢀⣠⡶⠛⠉⢁⣀⣤⣤⠴⠶⠶⠶⠦⣤⣤⣀⡈⠉⠛⠶⣤⡀⠙⠳⣄⡀⠙⢦⡀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⣠⡟⠁⣠⠟⠁⣠⡾⠃⢀⡴⠛⠁⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠈⠙⢶⣄⠈⠻⣄⠀⠻⣆⠀⠹⣆
⠀⠀⠀⠀⠀⣰⠏⠀⣴⠏⠀⣴⠋⢀⣴⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢷⡀⠘⢷⡀⠘⣧⡀⠹⣦
⠀⠀⠀⠀⢠⡟⠀⣸⠇⢀⡾⠁⢠⡾⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢻⣄⠈⢷⡀⠘⣷⠀⠹⣆
⠀⠀⠀⠀⣿⠀⢰⡏⠀⣼⠃⢀⡿⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢻⡄⠈⣷⠀⠸⣇⠀⢻⡄
⠀⠀⠀⢸⡇⠀⣾⠀⢰⡏⠀⣼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⠀⠸⣇⠀⢿⡀⠘⣇
⠀⠀⠀⣼⠁⢠⣯⣤⣼⣁⣀⣟⣀⣼⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣿⣤⣼⡇⠀⣿
⠀⣠⣤⠿⣶⠋⠁⠀⠀⠉⠛⠟⠉⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾⠛⠛⠛⠋⠁⠀⠀⠀⠈⠙⢷⣿
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣄
⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠈⠙⠛⢷⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡷⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠞
⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⣤⠾⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⡶⠶⠋⠛⠲⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧
⠀⢠⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣰⣿⣿⣿⣿⡟⠋⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣶⣿⣟⣿⣿⣿⣿⡇
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢼⣿⣿⠏⠁⠉⠙⣿⣿⢿⣿⡆⠀⢸⣿⣿⣿⣿⡿⠡⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⣾⢷⡾⠿⣿⠿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⠛⠻⠃⠉⠉⠂⠁⠁⠀⠘⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⢸⣿⣿⣿⣏⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⠋⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⢡⠟⠀⠀⠘⢻⣿⣿⣿⣆⠀⠀⠰⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⣿⣿⣿⣿⡆⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠈⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣿⣿⣿⣦⡀⠀⠀⣰⣶⣶⣿⣿⣶⣶⡄⢸⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⠿⣿⣿⡿⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠿⠿⠿⠀⠀⠀⠀⠀⣹⣿⠟⠉⠉⠛⢿⢿⣿⡇⢀⣿⣿⣿⣿⣿⢿⡿⣿⣿⢸⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⡗⠐⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⡼⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⢳⡿⠁⣿⣿⡷⣌⠹⠧⠘⠂⠈⠉⢸⣿⣿⡆
⠀⠀⠀⠙⢿⡟⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣞⠀⠀⠀⠀⠀⠀⠄⠘⠋⠧⠻
⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⢾⣿⣿⡟
⠀⠀⠀⢀⣠⣴⣶⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢰⡿⠋⠀⠀⢀⣀⣀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢿⣿⣿⡿⠁⠀⢻⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⢿⣿⣿⣆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣸⣿⣿⡿⡿⠉⠉⠙⣿⣿⣿⣦
⠀⠀⢻⣿⣿⠁⠈⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣻⡿⡟⠏⠀⠀⠉⣿⡿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢾⣿⣿⡿⠿⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠈⣿⣯⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⢱⡏⠀⠀⢻⣿⣿⡿⠙⠋⠉⠩⠉⣿⣿⣿⠁⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⣿⠇
⠀⠀⠀⠈⠿⠄⠀⠀⠀⠀⠘⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⣼⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⠂⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣋⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣯⣭⣿⣿⣽⣿⣯⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⣀⠔⠁⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⡴⢄⡀⠀⢀⠜⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⠙⠋⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠚⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⣚⣓⣒⣒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⢀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⠉⠉⠉⠉⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠎
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⠲⢄⢀⠔⠉⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⣀⡀⠤⠒⠊⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⣀⣀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠒⠒⠒⠀⠐⠚⠃⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣁⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⢀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠉⠉⠉⠀⠒⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠁⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⠀⠀⠀⠀⠀⣴⠚⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⢀⣾⠗⠀⠀⣀⠀⠀⠀⠀⠓⠀⢤⡾⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣯⣀⢠⡉⠀⢀⠁⢸⣇⠀⠙⠀⠀⠐⠀⠀⣴⣶⣶⣶⣶⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠁⣄⠡⣀⠘⠉⠃⣙⠀⠘⠀⠀⠀⠀⠀⠈⢻⡉⣹⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⢸⣿⣶⢌⣿⣦⣄⠈⠀⢠⠀⠀⠀⠀⠐⣦⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⢸⣿⡟⡈⠻⣿⡯⣌⠁⡚⢀⠄⠰⠀⠇⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠃⣀⠌⠁⡌⡀⠉⢸⣶⣆⣼⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠉⠄⠀⠀⠀⠁⠘⠆⠀⠹⣿⠃⣤⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⢠⡖⣗⠛⠻⠛⠒⣶⣄⢹⠀⣾⣿⣮⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢔⣥⣿⣟⡛⠃⠀⠀⠛⢛⣿⣀⣶⣿⣯⣿⣦⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠉⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢐⡂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⣿⣿⢀⡴⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣤⡇⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⣿⣿⠿⢿⣿⣿⣿⡿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠛⠁