Cute name Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡶⠶⣿⣿⣿⣿⣿⣿⡿⠷⣀⡀
⠀⠀⠀⠀⠀⣠⣿⢰⠗⠸⢿⡿⢿⣿⡿⠐⢷⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠔⠒⠺⠿⠟⠀⠂⠊⡊
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⢰⣿⣿⠀⣀⡛⠁⠀⡄⠀⠘⣃⡀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣘⠋⠀⢠⠀⠀⢛⣀⠸⡇
⠀⠀⠀⠀⠀⠛⡿⠂⠀⠁⠀⠀⠀⠀⠀⡈⠀⠺⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠈⠉⠀⠈⠀⠀⠀⢈⠉⠐⢀
⠀⠀⠀⠐⠈⠁⠡⠀⠀⠈⠈⠀⠀⠈⠈⠂⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠂⠉⠉⠄⠀⡠⠁⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠷⠘⠈⠏⡏⠏⠘⠲⢶⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⠂⠁⠀⠈⠀⠓⢶⡄
⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠸⠂⣀⡛⠁⠀⡄⠀⢙⣃⡀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⠃⠀⣠⠀⠘⣛⣀
⠀⠀⠀⠀⠀⠀⡀⠂⠉⠁⠀⠀⠀⠀⢀⡉⠁⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠈⢁⠀⠀⠀⠀⠀⡨⠁⠘
⠀⠀⠀⡐⠉⠀⠃⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠠⠉⠈⠐⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠜⠁⠀⠀⠀⠀⠡⠀⢀⠌⠀⠁

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠷⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠋⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠒⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠚⣉⣻⣀⣠⠤⢶⠖⠚⠛⠋⠉⣻⠓⠒⠲⢤⣀⡀⣸⣉⡙⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢋⡡⠞⠉⠀⠀⠈⠳⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠉⠳⢄⠙⢦⡇
⠀⠀⠀⠀⠀⠀⢠⠤⠶⢒⣒⡲⢤⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⡤⢖⣒⡒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡘⢿⣿⣿⡿⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⣿⣿⣿⡿⢃⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢭⣽⡇⠀⠀⠀⠀⢿⣧⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣼⣿⠀⠀⠀⠀⢸⣫⠭⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣠⣌⣉⠁⢀⡔⢒⠢⠤⠤⠖⡒⢦⡀⠈⣉⣥⣄⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣈⠛⠛⠛⠛⠁⣸⠂⠈⠀⠀⠀⠀⠉⠘⡇⠈⠛⠛⠛⠛⢀⡤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠬⢦⣀⠀⠀⠀⢿⠀⠀⢠⣤⣤⠄⠀⢀⡟⠀⠀⠀⣀⡴⠯⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠑⡄⠀⠀⠑⠦⢤⣀⣠⡤⠴⠋⠀⠀⢠⠞⠁⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⡇⠀⠀⠀⠀⠀⠀⣹⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⢿⠀⠀⠀⠀⠀⠀⢈⡗⠒⠒⠲⢤⡀
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⢟⠛⠛⠛⠛⠛⠛⡻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠟⠉⠀⠺⠭⠟⠀⢀⡀⠀⠻⠭⠗⠀⠉⠻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⣿⣷⣦⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⣴⣾⣿
⠀⠀⠀⠀⠘⣿⣿⣷⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⠄⠀⠀⠠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠉

⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣶⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣛⡻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣱⡟⠛⢿⡿⠋⠉⣷⡻⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⢟⣅⣶⣆⠀⠀⠁⠀⢠⣿⣷⡹⣿⣿⣿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⢱⣿⢹⣿⣿⣷⣄⠀⣠⣶⣝⢿⣷⢻⣿⢫⣾⣿⠿⣛⣭⣭⣭⣭⣭⣛⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣧⣡⡿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⢠⡹⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⢩⣝⣱⡎⣿⣷⡝⢟⣼⣿⣿⣿⣿⣿⣿⣿⡻⣿⣯⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣇⣿⣿⣿⡹⣿⣿⣿⣮⣛⠿⣹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⡏⠳⣄⢿⣿⡟⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡸⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢴⣷⣿⢦⣻⢇⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣷⢹⣿⣿⣧⢻⣿⣿⣿⣏⠉⣽⣿⣿⣿⣿⢻⣿⣿⣿⠠⣿⣮⣿⣿⣷⡎⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡹⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡝⣿⢢⢰⣴⣿⣿⣿⢏⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣜⢿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⣮⡳⣼⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣧⡸⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣿⣧⡻⣿⣿⣿⣿⣿⢟⣷⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣷⡝⣿⣿⡟⣵⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣎⣫⣾⣿⢟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⠿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣦⠀⠀⠀⠀⢀⣠⣴⠞⠛⠉⣭⣭⠉⠉⢛⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢀⣴⣋⡉⠉⢀⡄⠀⢉⣀⣀⠀⠛⠓⠀⣀⠙⠳⣤⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣵⣮⢹⡄⠈⠁⢠⡏⣵⣬⢳⡄⠀⠀⠙⠃⢀⡈⠻⣤⠾⠋⢉⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡟⠷⣽⣫⠾⠁⠀⠀⠘⢧⣙⣫⡼⠁⠀⢀⣀⣀⠀⠁⠀⠀⠰⠟⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠑⣾⠁⠀⠀⢰⡴⢚⣓⢦⣴⠂⠈⠀⠀⠀⠀⢭⠥⠙⣧⠀⠀⠀⠀⡛⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⣿⠀⠀⠀⢸⣧⣉⣉⣴⣧⠀⠀⠀⠀⠀⠀⣙⡓⣠⡏⠀⠀⣠⡀⠙⠳⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣰⡏⠙⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⠃⠀⠀⠀⠀⢀⡀⠀⠀⠠⣄⠀⠀⠀⠶⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠛⠀⠀⠀⠉⠀⠀⡀⠀⠀⠠⠦⠀⢀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⢦⣄⡀⠙⠃⠀⠀⠙⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠓⠒⠿⠟⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣶⣀⣀⣀⣦⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣻⡇⠀⣼⡟⠉⢹⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠉⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠟⠋⠀⠀⠀⠠⠾⠛⠁

⠾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠛⠻⠶⢒⠉⢩⠉⠓⠓⠚⢻⣄⣤⡴⠶⠞⠛⠛⠓⠒⠲⠦⢄⡀
⠀⠀⠀⣀⣠⠶⠶⠓⠉⠁⠀⠀⠀⠀⠀⠀⠘⠿⣧⡀⢀⣖⠻⠧⠀⢹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡼⣋⣉⣉⡙⣻⠖⠋⠉⠉⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠟⠋⠛⠧⣴⣶⣿⣿⣶⡄⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⣿⡟⠁⠾⠆⠀⣀⠤⠒⠊⠉⠉⢁⢈⢙⠛⢿⣿⣷⣄⠀⠀⠀⠉⠛⢿⣿⣿⠀⠘⠄⠀⢀⠀⠐⡐⢦⠀⠀⠀⣇
⢸⣿⡇⢸⣦⡴⠊⠁⠀⠀⠀⢀⣴⠞⣛⢳⣔⡀⠈⠻⢟⠀⠀⠀⠀⠀⠀⠈⠓⢤⡌⠀⠠⠱⠂⠀⠻⡆⠃⠀⠀⢳
⠘⣿⢷⣈⡟⠀⠀⠀⠀⠀⠀⢸⡇⢾⣿⡷⢻⡄⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠘⢄⠠⠤⠴⢁⠆⠀⠀⢸
⠀⠘⣿⣿⠁⠀⡠⠤⢄⠀⠀⢊⡻⣦⣭⣤⠿⠁⢀⣠⡴⠞⠛⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠓⠀⠀⠀⠀⠀⠀⢸
⢀⣾⣿⡇⢀⠊⢠⣤⡄⠑⠀⣠⣶⠊⢀⡐⢀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠸⣿⣿⡇⢸⠀⢻⡿⠇⢠⡇⢈⠉⣀⣴⠿⣾⣿⡇⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠰⠤⡀⠀⠀⢸
⠀⠙⠻⣷⠀⠳⠤⠤⠒⢹⣇⠉⠛⢋⡁⣀⣿⡿⠀⢸⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠀⠀⠰⠶⠸⣄⠀⢸
⠀⢀⡄⣼⣆⠀⠀⠀⠀⠘⣿⡦⠂⠀⠁⠀⠙⢧⠀⠿⠟⠿⡷⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⠃⠀⠀⢠⡶⠶⣤⠳⣸
⠀⢸⣧⠀⠉⠳⣤⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⣄⢠⡄⣤⢀⡄⠀⠀⠀⠀⠀⠀⢈⢅⠒⣼⢃⣿
⠀⠸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠳⡄⠀⠠⠤⠤⠀⠀⢿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃⣴⠚⢹
⣴⢊⠟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠠⣼⠀⠀⢠⠌⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠠⠙⣼
⢻⣎⠄⣠⠶⠒⠓⠲⣄⠀⠀⠀⠳⢤⣄⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢤⠀⣽
⠈⢻⣼⡏⠀⠀⠀⠠⢸⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⠙⡆⢻⡇
⢀⡴⣯⡷⣤⣤⣤⡶⠋⠀⠀⢖⠞⠁⠀⠀⢀⡠⢤⡄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠙⣇
⢋⠀⣨⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣾⣁⣀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⡇⠀⢿⣧
⠀⠉⣸⢩⡆⠀⠀⠀⠀⣠⠴⠺⣧⣀⣠⣿⠿⠛⠛⠛⠛⢻⠏⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⠀⠀⢾⣿
⠀⣴⡿⠈⠿⡄⢠⡖⠋⣷⣤⣼⠿⠛⠛⠁⠀⢀⡰⠔⣊⣁⣀⣴⠦⢷⠴⢶⣿⣧⣀⣀⢼⣷⣄⣰⡇⠘⠋⠀⠀⢾⣿
⠀⣿⡇⢸⣆⢺⣿⣿⣿⡿⠿⢅⡀⢀⠠⢒⣪⠥⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣀⢩⣿⣿⡇⣠⠀⠀⠀⢰⣿
⠀⠹⣿⣄⣿⡶⣿⡌⠛⠦⠄⠚⠉⠀⣚⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾⣯⢠⣿⡄⠀⠀⢰⣿
⠀⢰⡏⠀⠀⠀⠈⢳⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⣶⡇⠀⠀⣿⡏
⠀⣸⣧⣤⣤⣤⣄⠀⢳⠀⠀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠠⣿⠀⡇
⠀⣿⣿⣿⣿⣿⣿⠀⠀⢧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⣇⣰⡽⢠⠃
⠀⠙⠿⣿⠟⠉⢀⣀⢀⣿⠿⠂⠀⠀⡠⢂⣥⣶⣶⣶⣤⡀⡄⠀⠀⠀⠀⢠⣵⣿⣿⣿⣿⣮⡱⡄⠀⠀⠙⣏⡐⣿⠄
⠀⠀⠀⠹⡦⠤⠋⢨⣿⠁⠀⠀⠀⡸⣱⣿⣿⣿⣿⣿⣿⣿⣶⠂⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆⢦⠀⡄⣿⣇⣿⠂

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣴⡾⠷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣦⣤⣤⣤
⠀⠀⠀⠀⢀⣠⣽⡿⢟⣛⡁⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠿⠿⠛⠛⠙⠋⢙⠉⠙⢟⠻⡿⠋
⠀⠀⠠⢾⣿⠿⠋⢰⠁⢶⠿⠟⢐⡀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡰⠀⢾⡿⠀⠰⣿⠆⠀⠁⣷
⠀⠀⠀⠀⠹⡀⠀⠈⠓⡐⠀⣴⣯⣿⠭⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⡙⠁⠁⡀⣀⣀⡉⣀⣖⢈⠄⡟
⠀⠀⠀⠀⠀⢧⡆⢑⡪⡵⠋⠉⠉⠙⠒⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠛⠛⣿⠛⢻⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⢠⣧⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣼⣧⣛⣢
⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⣿⣿⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠈⠁
⠀⠀⠀⣠⢋⠀⠀⠀⣶⣦⡀⣬⣛⠻⢿⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢹⡁⠎⠀⠀⣈⡉⠀⠙⠛⠃⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠙⠒⣭⢗⣀⠤⠤⠄⡀⢀⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣯⣸⡇⠉⣿⡙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠓⠞⠉⠙⠦⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⣀⣠⣴⠾⠭⢕⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣊⠕⠒⠈⠐⢲⣌⠣⡀⢀⡼⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠁⠀⠀⠀⠀⣿⡇⢱⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⢿⣧⡏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⢈⡟⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⣰⣽⣛⣛⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⡄⠐⢃⠐⠶⣢⢠⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⡇⠀⠘⠀⢺⠅⢺⢿⡏⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠤⠼⠴⠤⢤⢰⠀⠐⠧⢸⢺⡷⠤⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢽⣿⣿⣟⣞⣖⣷⣲⣶⣶⣶⣿⠈⠛⠻⠻⢟⢼⣻⣳⣲⣲⣆⣖⣶⣻⣃⣖⣶⡦⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡯⣯⡭⡡⣼⡗⣀⡀⣀⣀⢀⣸⠀⠀⠀⠈⢟⢸⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢕⣟⡇⢗⢍⢬⡎⠀⡇⢥⡧⢸⢸⠾⡻⢛⡖⡶⠳⢾⣿⡇⠀⡞⣉⡓⢦⠀⠀⠀⠀⠘⡿⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠵⡯⡧⡿⡕⡕⠁⠀⣇⡖⣆⣸⢼⣺⣓⡆⣇⠓⠲⣼⡐⠓⣦⢧⡓⠞⡸⠀⠀⠀⠀⠀⣯⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡪⣟⣯⢏⡮⠊⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⢉⠉⣉⠀⣉⠉⠁⡀⠉⠉⠀⠀⠀⠀⠀⠀⣗⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⡗⣗⣿⠕⠀⠀⠀⠀⠀⠈⠈⠙⠀⠈⠉⠛⠋⠉⠙⠋⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⡠⢒⣊⣍⣒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣯⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⡜⡴⠋⠁⠀⠉⢳⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢘
⡇⢣⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠠⡆⢻⣇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡽⣿⠸⡆
⠙⣌⢦⠀⠀⠀⠀⠀⠈⠺⣷⢦⣀⡀⠀⠀⣠⡇⢸⣻⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣸⣽⢠⠃
⢀⡈⠓⣽⡂⠶⠤⣀⣤⡀⠠⢽⣿⣛⣯⣭⣾⣧⠘⠿⢽⣶⣶⣦⣤⣶⣶⣶⣲⣖⣖⣶⣲⣖⣖⣶⣲⣲⣲⡖⣶⣶⣶⣶⣖⣶⣶⣾⡿⠟⢃⡼⡀
⠀⠀⠁⠋⠙⠙⠋⠛⠻⠛⠓⠋⠑⠩⠒⠩⠝⠭⡻⠖⡦⢤⠤⡤⢤⡤⢤⠤⣤⠤⡤⢤⡤⢤⡤⣤⢤⡤⣤⣤⢥⡤⣤⢤⣤⡤⢤⣤⠴⡶⠿⠾⠪⠊⠕⠵⠲⠢⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣾⣿⣿⣿⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⢀⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⠈⠻⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠃⢰⣿⡿⠃⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠻⣿⣷⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠘⠉⠀⠀⠀⠈⠉⠉⠁⠀⠉⠛⠛⠋⠀⠀⠀⠈⠛⠀⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠁⠀⠀⠀⣴⣲⣦⠀⠀⠀⠀⠀⠀⠀⢠⣖⣲⡄⠀⠀⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⣀⣘⡛⠋⠀⠀⠀⣤⡄⠀⠀⠈⠛⣛⣀⡀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠘⠻⠟⠛⠀⠀⠒⠐⠋⠓⠐⠂⠀⠘⠛⠿⠛⠀⣠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠛⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠉⠉⠻⣿⠟⠉
⠀⠀⠀⠀⢀⠤⠐⠖⠲⡇⠀⠀⠀⠀⠀⡷⠲⠲⠒⠒⠒⠒⠲⠖⠒⠲⡇⠀⠀⠀⠀⠀⡿⠲⠖⠠⢄
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠱⣄⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡰⠃⠀⠀⠀⠀⢱
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⠣⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠀⢤⣄⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⢿⢧⠖⠢⣤⠿⠾⣷⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣴⣶⣶⡞⣠⣾⣅⣀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣓⣻⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡿⣿⣿⠡⣽⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⠨⠀⠍⠀⣽⠉⡇⠈⡉⠨⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣂⡇⣄⠙⠀⠀⠀⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⡖⣋⢹⠛⢀⣥⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣿⡛⠛⠛⠻⠟⢃⣻⠞⠛⠻⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣉⣉⣙⣷⢶⣾⣒⣛⠿⠛⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⡀⠈⠙⣒⣦⣤⡶⠒⠒⠲⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣝⡛⠛⡟⠛⠃⢐⣒⡂⢀⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣦⣄⢨⣦⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⣲⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⣦⠀⠀⠀⠀⠀⠀⠀⡆⠤⡤⢻⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻
⠀⠀⠀⢿⣤⠿⡇⠀⠀⠀⠀⠀⠀⠀⢩⣤⡤⠜⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⡞⠀⠀⠀⠀⠀⠀⠀⣿⣉⣁⣜⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠟
⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠒⣦⡀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⣠⣾⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⣼⠋⠙⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣄⡀⠀⠝⠀⢡⠀⠀⠀⠳⠒⠾⠟⠛⠲⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄
⠀⡟⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠠⠬⠭⢤⠴⠖⠛⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠀⠐⠀⢀
⢀⣇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠻⠟⠿⠛⠃⠀⢘⠓⠒⠶⠶⠶⠖⠒⠒⡃
⠈⡏⠉⠛⠉⠛⠋⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⣀⣀⣀⠠⠤⢴⡇⠀⠀⠁⠀⠀⣀⣀⠀⠀⠺
⠀⢻⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣦⣤⣤⣄⣤⣤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠙⡄⠀⣾⡇⠀⠀⠀⠀⡂⠀⠈⠆
⠀⠘⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠄⠀⣿⡇⠀⠀⠀⠀⠁⠐⠂⠀⠀⣰
⠀⢀⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠖⠀⠐⠒⠛⠃⠀⠀⠀⣸⠈⠓⠆⠀⠀⠀⠀⢻⡇⠀⠀⡀⠀⠀⠀⠀⢀⣄⣿
⢠⣾⠉⠉⠘⠁⠉⠀⣹⠀⠀⠐⠓⠀⠠⠦⠤⠄⠀⠛⠀⠀⠀⠈⢀⠀⠀⠀⠀⢀⣡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠉⠉⠉⠉⢠
⠀⠻⢟⣳⣒⡲⠆⠞⠃⠀⠀⠀⠠⢄⣀⣀⣀⣠⠤⠒⠀⠀⠀⠀⢀⣈⣙⣿⣿⠿⠃⠀⠀⠙⠶⣤⣀⣀⣀⣀⡤⠴⠃⠀⠀⢆⠘⢦⣀⢀⠼⠀⡨






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

⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢀
⠀⠀⠀⠀⠀⠴⡯⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣁⢌⠀⠀⢀⠆⠀⠀⠀⠣⠁⢠⠀⠀⠀⠒⠦⠨⠒⠁
⠀⠀⠠⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢀⣨⢲⣿⠾⢢⣶⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⣤⣠⡇⠀⠀⠀⠀⠀⠘⠀⠍⠀⠂⠉⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠉⠉
⠀⠀⠀⠊⠒⠀⢀⣦⠀⠼⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠠⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠘⠆⠠⠁⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠰⡘⡤⠒⠈⠁⠀⠀⡖⢈⡆⠀⠀⠀⠀⠀⢠⢆⣄⡀
⠀⠀⠀⠙⠀⠍⠀⠀⡀⡀⠃⣤⢀⡄⠀⠀⠚⠁⠀⠀⠀⣑⠄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⢠⣽⠀⠀⠀⠀⢀
⠀⢤⡠⡀⠀⠃⠀⠀⣀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠄
⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠂⠂⠀⢠⠆⠀⢈⠁⢀⠤⣴⠃⠀⠘⠀⢴⣄⠀⠀⠀⠈⠀⠘⠉⠉⠁⡀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠘⠆⠀⠈⠀⠘⠛⠋⠀⠁⠀⠀⠈⠉⠀⠀⠀⠈⢀⡁⠀⠀⠀⠈⠀⠀⠀⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠓⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠰⠀⠀⠐⡀⠀⠀⠀⢈⠀⠀⠀⡀⠀⠠⠀⠢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠘⠆⣠⣞⣧⢀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡆⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠰⠂⡾⠋⣉⠞⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠫⠛⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠦
⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠁⠘⠒⠛⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⢰⡖⠂⠀⠀⠀⠐
⠀⠀⠀⠎⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣿⠇⠀⠀⠀⠠
⠀⠀⠀⠒⠀⠠⠄⠀⠈⡀⠀⢢⠶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠓⠀⠇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠆⠠⠦⠿⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⢀⡁⠀⠀⣨⡄⠀⣼⣿⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣦⣿⠟⠉⣹⢻⠟⢰⠿⡯⠊⠀⠀⢀⣠⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠋⢠⣷⠃⠀⠀⠀⢀⠴⠛⠛⠝⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠸⠋⡀⠀⢠⡶⢋⡀⠀⠀⢸⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢠⣼⣧⡀⣾⠁⠀⢀⣀⠜⢻⡟⠀⠝⠓⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⠀⠉⠉⢉⡳⠦⢶⣿⡀⠄⣈⠓⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢺⡋⠈⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⢦⡄⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣟⠁⠀⠀⢧⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣸⠁⠈⠀⠀⠈⠿⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢐⠃⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠠⠀⠐⠈⠠⡄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣿⠀⠀⠀⠀⠐⢘⠉⠢⢄⢳⠀⠀⢀⢀⣀⠀⢠⠄⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⠁⢀⡇⠀⠀⠀⢀⡀⠀⠒⠀⠢⠘⢀⣠⣼⡏⠀⠁⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡋⠈⣿⡇⠀⠀⠀⠈⠀⠁⠀⢸⣶⣯⡍⢿⢺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⣾⠇⠀⠀⠀⣀⠀⠀⠀⢸⣿⡞⢁⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⣸⡿⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⣧⢨⡄⠀⣀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡯⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⡛⣁⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⡍⣛⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⢿⣟⣿⣿⣗⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⣦⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⣏⣰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣅⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠉⠙⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⡿⠁⡀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠆⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣏⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣴⣶⣾⣷⣶⣶⣤⣶⡆⠉⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣤⣄
⠀⠀⠀⠀⡾⣥⠀⠀⢘⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⡿⣌⠳⢦⣤⠤⠶⠶⠚⠛⢛⣛⣿⣿⣿⣿⣿⠂⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢰⣷⡌⠙⠒⠦⠀⠀⠠⠖⠿⠛⣉⠝⠛⠛⠿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⡈⢻⣦⡄⠀⠀⠀⠀⠐⠃⢸⠃⠀⠀⠀⠀⠀⠁⠀⣸⠉⠉⢉⠛⢿⣿⣿⣿⣯⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣳⣄⡉⠛⠊⠀⠀⠀⠀⠀⠀⣦⣷⣦⣄⣀⢀⣠⣴⡃⠀⠀⠠⠀⠀⠓⠄⠀⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⡇⠈⠻⣟⣃⡀⢠⠤⢤⣤⡀⠀⠉⠛⠿⢿⣿⣿⣿⠃⠀⠈⠓⠦⠂⠀⠀⡠⠀⠁⡀⣿⠟⠻⢿⣿⣿⡇
⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠰⠟⠁⠠⠀⠀⠿⣋⠀⠀⢠⠟⠛⠃
⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⣈⣽⠖⠀⠀⢠⡿⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢿⣍⡉⠙⠛⠻⠿⡷⣶⡤⣤⠾⡿⠟⠠⠂⠀⠈⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⡀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢟⡔⢡⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠀⢛⢞⠞⢿⠿⠃⠞⢻⣿⠻⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠟⠘⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣷⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠹⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠀⠀⢀⠄⢸⠆⠠⠀⠀⡀⢒⡀⢠⠆⠃⢀⠔⢠⠟⠀⠀⠋⠀⠌⠀⡴⠀⢐⠂⡐
⠀⠀⢀⠔⠀⢀⣅⢀⠏⢠⠉⣅⢠⢆⡀⠶⠙⠄⠊⠀⠓⠈⠒⠋⠀⠐⠉⠀⠉⠒⠁⠀⠀⠐⠀⠘⠅⠤⠋⡴⠃⠇⢀⠆⢀⠛⠁⡀
⠀⠀⠀⠀⢀⣀⡠⠈⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠐⠧⢴⠃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⢠⡘⠃⣄⠛⠀⠀⠁⠀⠉⠀⠘⠁⠀⠁⠀⠛⢠⡘⠃⣄⠀⠀⠀⢤⠀⠅⢤⡄⠈⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⢠⡄⢨⠀⡄
⠀⠀⠀⢈⡑⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⡁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠶⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃
⠀⠀⠀⢠⡌⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠃⣤⠀⠀⠀⢠⠈⡉⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢉⠁⡄
⠀⠀⠀⠀⠐⠓⠀⠛⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠆⠀⠀⠰⠂⠀⠀⠄⠀⠁⠠

⠀⠀⠀⣒⣒⣒⣒⡒⠲⠒⠶⠶⠖⠒⠒⠲⠶⠒⠲⠒⣒⣒⣒⣒⠂⠀⠀⠠⠶⠔⠴⠴⠂⠂⠀⠴⠴⠴⠦⠦⠆⠀⠐⠐⠴⠦⠢⠢⠄
⠀⠀⠀⠤⠬⠬⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠭⠤⠬⠄⠀⠀⠨⠭⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⠭⠅
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠁⠀⠀⢐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⢭⣭⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣭⣭⠀⠀⠀⠨⠭⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⠅
⠀⠀⠀⣒⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⣒⡂⠀⠀⢐⣏⣪⣪⣁⣄⠀⢀⣀⡤⣠⣄⣤⣄⠀⠀⣠⣸⣟⣕⣝⡃

⠀⠀⠀⡀⣀⢠⡀⣄⢠⡀⡀⣀⢀⡀⣄⢠⣀⡀⣠⢀⡄⡄⢠⢀⡀⠀⠀⣀⡀⢀⣀⠀⣠⠀⢀⡀⢀⣄⠀⣠⡀⢀⡀⠀⣄⠀⣠⡀
⠀⠀⠀⡦⣶⢴⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡶⢶⠴⡆⠀⠀⢈⡹⢎⣉⠞⠁⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠉⠶⣏⡵⠆
⠀⠀⠀⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⡆⠀⠀⢙⣳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡆
⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⡇⠀⠀⠲⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⡁
⠀⠀⠀⠶⢾⢶⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⢾⠶⡆⠀⠀⢴⣎⣱⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⣷⣎⠁
⠀⠀⠀⠃⠛⠘⠃⠓⠘⠐⠃⠛⠘⠃⠓⠘⠙⠃⠛⠘⠂⠓⠘⠐⠃⠀⠀⠀⠘⠃⠀⠛⠀⠘⠀⠘⠛⠀⠛⠀⠙⠃⠐⠛⠀⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠈⠄⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣯⣽⡆⠀⠀⠘⠃⠀⠀⠀⣾⣭⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⣭⣍⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣥⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⠉⠉⠉⠀⠀⠘⢿⡿⠃⠀⠀⠈⠉⠉⣁⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠙⢦
⠀⠀⠀⠀⣠⠖⠒⠒⠒⣇⠀⠀⠀⠀⠀⣸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠀⠀⠀⠀⠀⢸⠓⠒⠒⠲⣄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠘⠦⣀⣀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⡤⠋⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠑⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠚⠉⠓⢦⡀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⡀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⡠⡰⠄⠀⠀⠘⣶⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠌⠀⠈⠻⠃⢀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⣠⡶⠀⣀⢠⠀⠰⢦⣄⠘⠃⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠋⠉⠑⢦⡀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⢠⣤⣦⡠⠀⠠⠀⣘⣀⣀⠯
⠀⠀⠀⠀⠀⠐⣌⢉⢹⡆⢠⠄⠉⠀⢀⣤⣂⠀⠀⠀⠀⣧⣹⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⣦⣠⠀⠀⠸⠃⠀⠀⠀⠀⣾⠏⠀⠈⠃⣠⡶⡷⣮⡊⠁
⠀⠀⠀⣰⢶⣤⠿⠋⠉⠉⠛⠀⠀⢰⡤⢻⣧⠀⠀⠀⠀⠾⠛⠘⠀⠀⠐⠑⠀⠀⣰⣰⣢⣼⣿⡻⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠈⠙⠀⠠⠟⠁
⠀⢀⣬⠿⣾⠋⠀⠀⠀⣀⠀⠀⠀⠼⠟⠾⠿⣇⠀⠀⠀⠀⠀⢠⡤⣴⣖⠀⠀⢠⠉⠉⠙⠛⠛⣇⠀⠀⠀⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣠
⠀⣴⣿⣶⡏⠀⠀⠀⣘⣿⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠛⠉⡅⠀⠀⣺⠀⠀⡀⠀⠀⠻⠀⠀⠀⢿⡖⠁⠀⠀⠀⢠⣒⡆⠀⠀⠀⠀⡿
⠀⠉⢛⣽⡇⠀⠀⠀⢽⡟⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠀⠋⠀⠀⢐⠀⠀⠀⣠⡁⠀⠀⠀⢰⡶⢯⠉⠀⠀⠀⢸⡿⣇⣤
⠀⠀⠉⡘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠈⠟⠀⠀⠀⠀⠀⠐⠒⠒⡀⠀⠀⠚⠀⠀⠀⢀⣴⡇⠀⠀⠀⡙⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣤⣍⠉
⠀⠀⠀⠠⠌⠴⠀⡤⠠⣆⣡⡄⢤⢼⢇⡀⠀⠀⡀⠀⠀⡤⠤⣄⣀⣀⢇⣀⠀⠀⠀⢸⡄⠀⢿⢧⢄⡄⣀⢪⡼⣦⣄⢀⠀⠀⢀⡰⠀⢀⣀⣸⣧⡀⣄
⠀⠀⠘⡠⠊⠐⠃⠀⠀⠨⡄⡘⠛⠛⠰⣾⣷⣞⣿⣿⢋⡿⡠⢼⣬⣛⡦⠀⠀⠀⠀⢠⡆⠀⠀⠮⢿⢿⡻⠮⠹⡢⢭⣝⡠⠛⢄⠔⠽⠸⢿⡟⣿⡇
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢹⣂⡀⠀⠀⣩⡌⢫⡙⢁⡆⢜⢻⣿⣈⣭⠐⡿⠏⢶⠄⡨⣇⣶⣀⡴⣯⠈⣽⢦⡁⢉⣷⡶⢾⡋⠉⠀⠀⠀⠀⠰⣾⠃
⠀⠀⠈⠦⣀⣀⡈⠀⠈⣀⣤⠭⡼⠏⠗⣍⣷⣿⣿⢼⣧⣾⡿⠛⠉⠉⠀⢾⠇⡀⢀⡉⠈⠉⠍⠀⠘⠴⠃⠀⠛⠞⢿⣷⣮⣶⣴⢂⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠈⠒⠚⠀⠀⠀⠀⠀⠈⠈⠀⠈⠁⠂⢸⠏⠙⠃⠀⠀⠀⠴⠄⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠑⠂⠈⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣰⣿⡆⢰⣿⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣶⣶⣿⠯⠿⠓⠚⠿⠿⣻⢶⡆⣀⣄⣠
⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⢈⢿⣤⢿⠗⠉⠀⢀⣀⠀⠀⠀⠀⠀⠑⢮⣫⣥⠟⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠹⠓⠛⠛⢻⡷⠁⠀⠀⠀⠀⠀⡁⠄⣠⠄⠀⠀⠀⠹⣽⠟⠳⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢯⡓⠒⢻⠁⠀⠀⠀⠀⠀⣼⠃⣼⠃⠀⣤⡤⠀⠀⢸⡑⠖⠔⢢
⠀⠀⠀⠀⠀⠀⠀⣀⣰⡛⢆⢸⠀⠀⠀⠀⠀⠴⢃⡰⠃⠀⠀⠀⠀⠀⠀⢠⡇⡠⢾⡓
⠀⠀⠀⠀⠀⠀⠸⢧⡁⣞⠿⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢸⣹⡷⡍⢛⣲
⠀⠀⠀⠀⠀⠀⠀⣸⡿⣾⣤⣶⡳⡀⠀⠀⠠⡘⠿⢈⣠⠊⠀⠀⠀⠀⣠⣫⣅⣘⣵⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⢠⡿⢷⣝⢦⡀⠀⠈⠉⠉⠀⠀⠀⠀⣠⢞⡵⣿⡍⠉⠉⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠁⠷⢮⣕⣒⣦⠤⢤⣔⣒⣭⣶⠟⠻⠞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢿⡿⠀⣿⡿⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠴⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡏⠀⠀⣅⣠⠀⠀⢹⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⠄⠒⠒⠒⠛⠛⠓⠒⠒⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠴⠶⠶⠶⠶⠶⠤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄
⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠀⠀⠀⠈⠛⠷⢶⣶⠶⠾⠋⠁⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆
⢀⣾⡿⠃⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⣠⣴⡾⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⣠⣤⠀⠀⠀⠀⠀⠘⢿⣷
⣼⣿⠃⠀⠀⠀⣠⣶⠿⠋⢁⣤⣶⠿⠋⠁⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⡿⠛⠁⠀⢀⣤⣶⠟⠋⠁⠀⢀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠉⣀⣴⡾⠟⠉⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⣤⣶⠟⠋⠁⠀⣠⣴⡾⠛⠉⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠰⠾⠛⠉⠀⠀⣀⣴⡾⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⢀⣠⣴⠿⠛⠁⠀⠀⣠⣴⡾⠛⠉⠀⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⠿⠛⠁⠀⣀⣤⡾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⠿⠋⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠠⣾⠟⠋⠀⠀⣀⣤⡾⠟⠋⠀⠀⣠⣴⡿⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣾⠟⠋⠀⢀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠀⢀⣤⣶⠟⠋⠁⠀⢀⣠⣴⠿⠛⠁⠀⢀⣠⣴⡾⠛⠉⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣶⠟⠋⠁⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣶⠟⠋⠁⠀⢀⣠⣶⠿⠋⠁⢀⣠⣴⠿⠋⠁⢀⣠⡄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠁⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⡿⠛⠁⠀⣠⣴⡿⠛⠉⠀⣀⣴⡾⠟⠋⠀⣀⣴⡆⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⢀⣤⣶⠟⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣴⡿⠛⠉⠀⠀⣠⣴⡾⠛⠉⠀⣀⣤⡾⠟⠉⠀⣀⣤⣶⠟⠋⠁⣀⣤⣶⠟⠋⠁⣀⣤⣶⠇⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠁⠀⢀⣠⣴⠿⠛⠁⢀⣠⣴⠾⠛⠉⢀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠟⠉⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣤⡾⠟⠋⠁⢀⣤⣾⠟⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠃⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠉⠀⣀⣴⡾⠟⠋⠀⣤⣴⣾⠿⠋⠀⣠⣤⣾⠟⠋⠀⣠⣴⣾⠿⠋⢀⣠⣴⣾⠿⠋⠀⣠⣤⠀⠀⠀⠠⣿
⢹⣿⠀⠀⠀⢀⣶⠿⠋⠁⢀⣤⣰⠘⠉⠉⢀⣀⡀⠿⠟⠉⢀⣠⣶⡿⠟⠉⢀⣤⣶⡿⠟⠋⢀⣠⣴⡿⠟⠁⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⣀⣤⣶⠟⠛⠉⢀⣀⡿⠛⠛⠁⠀⣀⠾⠟⠋⠁⠀⣠⡾⠟⠋⠉⣀⣴⣾⠿⠛⠁⣀⣤⡶⠀⠀⠀⢀⣿
⢸⣿⠀⠀⠀⢠⡾⠟⠋⠀⣠⣴⡾⠟⠋⠀⣠⣴⡿⠟⠋⠀⣠⣴⡿⠟⠋⠁⣠⣴⡾⠟⠋⠁⣠⣴⡾⠟⠋⠁⠀⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠇⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠰⠟⠋⠀⣀⣤⣾⠟⠋⠀⣀⣤⣶⠟⠋⠁⢀⣤⣶⠟⠋⠀⢀⣤⣾⠟⠋⠀⣀⣴⡾⠟⠋⢀⡀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⢀⣠⣴⡾⠛⠁⢀⣤⣶⠟⠃⠀⠀⠀⠀⣿
⢸⣿⡄⠀⠀⠘⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⣀⣤⣶⠟⠋⠁⣀⣴⡾⠛⠉⣀⣤⡆⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⣀⣴⡾⠟⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠉⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠁⠀⣠⣴⠿⠛⠁⠀⣠⣴⡿⠛⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠛⠁⣀⣴⡾⠃⠀⠀⠀⢸⣿
⠸⣿⣇⠀⠀⠀⠐⠿⠋⠁⢀⣤⣶⠟⠋⠁⣀⣤⡾⠟⠋⠀⣠⣴⡾⠛⠉⣀⣤⣶⠟⠋⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡄⠀⠀⠀⠀⠀⠾⠛⠉⢀⣠⣴⡾⠛⠉⢀⣤⣶⠟⠋⢁⣠⣴⡾⠛⠉⣀⣤⡶⠟⠋⢁⣤⠖⠃⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠐⠿⠋⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠋⠁⣠⣴⡾⠛⠉⠀⠤⠞⠋⠁⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠈⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠉⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⣀⣀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣀⣀⠀⠙⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠇⠀⣴⣿⣿⡿⠆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⣿⣿⣿⣦⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⣿⠟⠉⠀⠀⠀⠉⠻⠿⠿⠿⠛⠁⠙⢿⣿⣿⣿⠿⠋⠀⠀⠀⠉⠻⣿⠀⢀⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣀⣈⣉⠁⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠈⣉⣁⣀⣀⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⠛⠿⠿⠿⠿⠃⠀⠘⠲⠦⠴⠞⠳⠦⠴⠖⠃⠀⠘⠿⠿⠿⠿⠛⠀⠀⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠷⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠾⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠹⣿⠉
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⢺⡁⠀⠀⠀⠀⠀⠀⠀⣹⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠒⠒⠶⢦⣀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠤⠤⠀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡤⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠘⡆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠋⠉⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠈⢯⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⢦⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡇⢀
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢧⣠⠞⠀⢠⠀⡤⠂⠔
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⡄⠀⠀⠀⠀⠸⠀⠀⣀⣈⠀⠘⠀⠀⣡⡂
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡞⠉⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠒⢉⡽⠀⠀⠀⢠⢃⡠⠞⠀⢃⠀⠀⢀⡠⠒
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣇⠠⠁⢍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠉⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠈⠦⣀⠀⢀⣠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠉⠉⠀⠈⠑⠦⣄⣀⣀⣀⣀⣀⣀⣀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣧⣀⠉⢄⠔⣰⣀⠄⠂⠉⠀⠀⢄⡀⠀⠀⠀⠀⠚⠉⡩⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⠀⠀⡰⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠀⢱⠀⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⣱⠤⠒⠉⠀⠀⠀⣄⠀⠀⠀⠀⠀⡎⠀⠀⢸⠂⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠒⠘⠣⣄⠀⠀⠸⠀⠀⠀⠸⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣀⣠⠴⠋⠓⢤⣀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠠⣦⠀⠀⠀⠾⣿⠆⠀⠀⢀⣀⣀⠀⣦⠀⠀⣶⡀⠀⠀⠀⠀⠲⡄⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡼⠇⠀⢸⡟⠀⠀⣿⠃⠀⣀⣩⠴⠖⠚⠉⣩⡉⠉⠀⣿⠃⠀⢸⡗⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⠀⠀⠀⠈⣇⠀⠀⢻⡀⠈⠁⠀⠀⠀⠾⢦⠋⠀⠀⠀⢸⡄⠀⢸⡇⢀⡌⠀⠠⡀⣄⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⣺⡧⠀⠀⠀⣿⠀⠀⢸⡇⠀⣴⣶⣶⣦⣤⣈⣀⡀⠀⠀⢸⡇⠀⠀⣷⠘⠟⠀⠀⠛⠉⠁⢀⡀⠈⣧⢰⣅
⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⡠⠀⢨⠀⠀⣰⣿⣧⣹⡀⠀⠈⣇⠀⠀⠀⣀⣉⣭⡿⠿⠿⠿⠟⠀⣧⠀⠀⢿⠀⠀⣴⠀⠀⣾⠀⠘⣷⡄⣿⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠟⢰⣧⠀⠘⡆⢰⠋⠀⠛⢿⣷⣶⣶⣿⡶⠿⠟⠛⠉⠀⢠⢰⣰⠀⠀⠀⢿⠀⢰⢸⡆⡼⣹⠀⠀⢸⡆⠀⢸⡆⢸⡄⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠈⠉⠀⢀⣠⣼⣷⣄⡀⠀⣧⠀⠀⢹⡄⢦⠀⠀⣀⣀⡘⠋⠉⠀⠀⠀⢸⡆⢸⡸⣧⡇⣿⠀⠀⣸⣷⡀⠀⡷⢸⡇⠀⢻⡅
⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⣴⡾⠛⠉⠉⠉⠛⠃⠀⣿⠀⠀⢸⡇⠈⠀⣾⡛⠛⠛⠿⠿⣶⣶⡦⠸⡇⠀⠻⣿⠀⠹⣿⡿⠃⠘⠿⠿⠃⠀⠇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠈⢷⣶⣾⠟⡋⠀⢠⣦⣤⣦⠀⠀⠀⣸⠀⠀⠈⣧⠀⠀⠙⣿⣦⣴⠶⠟⠛⠉⠁⠀⡷⠀⠀⢹⠀⠀⠀⣠⣶⡄⠀⠀⣤⠀⣠⣦⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⠉⢀⣥⣶⡶⠿⠃⠀⠀⠀⣿⣠⣴⡾⠟⣿⣷⠀⠀⣀⣤⣶⠟⠁⠀⠀⢸⡇⠀⢰⠋⠙⢿⣶⠾⠻⠿⠋⠙⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣴⡿⠋⠅⠀⠀⠻⠗⢀⣴⡿⠟⢡⠀⠀⠈⠁⢠⣾⠟⠉⡜⠀⠀⠀⠀⠘⠇⢀⣸⣦⡀⠀⠀⠀⠀⢀⣤⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣴⡿⠋⠀⢸⠀⠀⠀⢀⣴⡿⠋⠠⠀⢸⠀⠀⢀⣴⡟⢁⠀⢰⠀⠀⠀⠀⣠⣴⡿⠟⠛⠛⠿⠇⠀⠀⡠⠋⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠸⣦⣤⣾⡿⠋⠀⠀⣧⠀⠘⣦⣴⡿⠋⠀⣾⡄⠘⣧⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⡿⠿⠿⠿⠿⣿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⢫⣍⠁⠀⠀⠀⡴⢶⣶⠆⠀⢠⣤⡈⠉⠛⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠀⠀⣀⢸⡿⣷⣄⠀⠀⠐⠚⠁⢀⠀⣿⡿⢿⣦⡀⠀⠀⠀⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⣀⡤⠖⠋⠁⠘⣷⠈⠻⣧⡀⠀⠳⣆⠈⠁⢹⡇⠀⠙⣿⣆⠀⠀⣶⡀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⡰⠚⠁⠀⠀⣀⢄⠀⣿⡀⠀⠹⣷⠀⠀⢘⣦⠀⢸⣇⠀⣀⣬⣿⣧⣤⣭⣄⣀⣀⣀⣙⢿⣷⣄
⠀⠀⠀⢀⣼⡿⠋⣠⠊⠀⠙⠀⠀⠀⠳⠛⠊⠀⢹⡇⠀⠀⢻⣇⠀⢯⠀⠀⠘⣿⢸⣟⠛⠛⢿⡟⠛⠛⣻⣿⣿⠿⠁⠙⣿⣦
⠀⠀⢀⣾⡟⠁⣰⡇⣠⠀⠀⣀⡀⠀⠀⣠⣤⡀⠘⣿⠈⢧⠈⣿⠀⠈⠆⠀⠀⣿⡀⠛⠿⣦⣼⣿⣶⠟⠋⠁⠠⣄⠀⠀⠈⢿⣷⡀
⠀⢀⣾⡟⠀⠀⡻⠿⠃⠀⠀⠻⠃⠀⡼⠋⠙⣧⠀⣿⡄⠈⠀⣿⣆⡠⠖⠊⠁⣿⡇⠀⠀⣼⠟⣿⣄⣀⣀⣀⠀⠈⠇⠀⣤⠈⢿⣷
⠀⣼⡿⠀⢀⠜⠁⠀⣀⠀⠀⠀⠀⣀⣧⣤⣤⡹⣷⣼⣇⠀⠀⢻⡇⠀⠀⣀⣀⣼⣷⣴⣾⣿⠿⢿⡿⠿⠿⠿⠿⣷⣦⠀⣿⣇⠈⣿⣇
⢰⣿⠃⢠⠃⠀⠻⣄⠉⠀⣠⣶⠿⠛⢻⣿⣙⣿⠈⠙⣿⠿⣿⣿⡿⠿⠿⠿⠛⢻⣿⢹⠁⣀⠀⢸⡇⠀⠀⣤⡀⠀⠙⡇⣿⠉⠀⢸⣿
⣼⡿⠀⡏⠀⠀⠀⢉⣤⡾⠋⠁⠀⠀⠀⠙⢛⠁⠀⠀⣿⠀⠀⢸⠀⠀⣀⣖⡄⠈⣿⣾⠀⣌⣠⢸⣧⠀⢘⠶⠟⠀⢀⠇⣿⠀⠀⠀⣿
⣿⡇⠀⠳⣤⣤⣴⡿⠋⠀⣴⣤⣠⣷⣦⠀⠸⣷⣶⡿⠃⠀⢀⣯⣤⣄⣉⠀⠀⠀⣿⡸⣧⡀⠀⢸⣿⠊⠁⠀⣀⣴⠏⠀⣿⡆⢬⡥⢻
⣿⡇⠀⠀⡈⠉⠁⢰⠁⠀⠉⠋⠀⢈⣁⡤⠀⢹⡇⠠⡄⠀⠘⠛⠛⠛⠿⢿⣶⣤⣿⡇⠙⢿⣶⣼⣿⣤⣶⡾⠛⠁⡆⡆⢸⡇⠀⠀⢸
⣿⡇⠀⠨⠿⠂⠀⡏⠀⢀⣠⠴⠚⠉⠀⢀⣠⡾⠀⠀⠙⠀⣠⣴⣶⣶⡶⠾⠿⠿⢿⡿⠿⣷⣿⢛⣿⠉⠀⠀⠀⠓⠉⠃⢸⡇⠀⠀⣸
⢻⣷⠀⢳⠀⢠⠀⣷⣄⠉⠀⠀⣀⣤⣶⠿⠋⢴⣄⠀⣀⣴⢿⣟⠻⣷⠀⠀⠀⠀⢸⡇⠀⢀⡀⢸⡏⢰⡆⠀⢠⣦⠀⠀⢸⡇⠀⠀⣿
⠸⣿⡄⠀⢇⠃⠀⠈⠻⠿⠿⠿⠟⠋⠁⠀⠀⠈⣿⠿⠟⠁⠈⠻⣷⣿⠀⠘⠂⠀⢸⠇⠀⢾⣧⢸⠇⢸⣷⠀⠀⣿⡁⠀⢸⡇⠀⢰⣿
⠀⢻⣧⠀⢸⠀⢠⣄⠀⠶⠀⢠⣶⡄⠀⠀⡴⠁⣸⠀⢠⡀⠀⠀⠀⢀⣀⣤⣴⣶⠟⣠⠖⢻⠇⠘⠀⠀⣿⠀⠀⢻⣧⠀⢸⡇⢀⣾⠏
⠀⠀⢻⣧⢸⣆⠀⠘⡆⠀⠀⠈⠙⠁⠀⣼⣴⡾⠃⠀⠈⠁⣠⣶⡿⠟⠛⠉⠉⠀⣾⣥⣴⢿⡄⠀⠀⢀⢿⣇⠀⠀⣿⠀⠸⢁⣾⡟
⠀⠀⠀⢻⣷⣿⣷⣦⣤⣤⣤⣤⣴⣶⢿⠛⠁⢠⠀⠀⣠⡾⠟⠁⠀⠀⠀⠰⡄⠀⡉⠉⠀⠸⣷⣦⣴⡞⠈⠻⢿⡿⠏⠀⣠⣿⠏
⠀⠀⠀⠀⠙⢿⣮⡉⠛⠛⠛⢩⡉⠀⠀⢧⣄⣠⣴⡾⠋⠀⠀⠀⠘⠀⢀⠉⠁⠀⣿⡆⠘⠳⡌⠉⠁⢠⡀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠙⠄⢠⠊⠉⠛⠉⠀⠀⣠⣴⣶⡿⢿⣿⣷⣤⣀⢹⣇⠀⠀⠉⠀⠀⣈⣡⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣴⣅⡀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⢻⣿⡟⠻⠿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡏⠛⠛⠻⣿⡿⠿⠷⢶⣆⠀⠀⣠⣆⠉⢀⣶⣀⠘⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠷⣶⠿⠋⠀⠀⠀⠘⠿⠿⠟⠉⠿⠿⠋⠙⠿⠿⠃⠀⣈⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠒⠉⠁⢀⣠⣄⣀⣠⠀⣾⣦⠄⡤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠓⠦⡀⠀⠀⠀⠀⠀⢀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠓⣆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣖⠲⣄⠀⣰⡿⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⣰⠃⠀⢣⡼⠀⠀⢱⣿⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣼⡏⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣰⠃⠀⠀⠀⢀⠀⠀⠀⢠⢻⡆⠀⢀⡀⠀⣿⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡴⠃⣀⣴⠀⠀⣿⠀⠀⢠⠇⢸⡇⠀⢈⡁⠀⣿⠀⠀⠀⢰⡆⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠀⢀⣼⠇⣿⠀⠀⣿⡴⣿⡞⠀⣾⡇⠀⣸⡇⢀⡏⠀⠀⠀⢸⡇⣾⣀⡤
⠀⠀⠀⠀⡀⠀⠀⢸⣧⠴⠒⢉⡟⢰⡇⠀⢰⣿⠁⣿⠃⠀⣿⠁⢀⣿⣠⠞⣇⠀⠀⢠⠋⢋⣼⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠐⠚⠿⠆⣀⠈⠷⠋⣧⢠⠏⠉⠀⠛⠀⠀⣿⢀⡞⠈⠁⠀⢿⠀⣠⠃⣴⣃⠼⠄⠐⠂⠉⠁
⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢹⡏⠐⡦⠀⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⢠⣴⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⠇⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠁⠈⠉⠀⡏⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡾⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠈⡇⡠⠧⣰⠀⠀⠒⡤⣠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠀⠤⠄⣀⢠⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠐⠄⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣄⢀⠀⠀⠀⠀⠀⠉⠢⡀⠑⢤⠔⢄⡈⠀⢀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠰⡀⠒⠤⡤⠁⠀⠁⠒⠒⠒⠀⠀⠀⠁⠀⣨⣀⠸⢀⡏⠀⠷⣒⠊⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠑⢄⠀⠣⠀⠂⠤⣀⡀⠀⠀⢀⠠⠓⠀⠘⠒⠀⠠⠘⠚⠃⠁⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠂⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⣠⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡁⣛⣀⣈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣄⣀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⡟⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣿⡟⠛⠻⠿⣿⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⠃⢰⣿⡿⠀⠀⠀⠀⠀⠻⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠈⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⣿⣟⠀⠀⠀⠀⠈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⡇⠀⢸⣿⣇⡀⠀⠀⠀⠀⣈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⢸⣿⣿⣷⠯⠭⠀⠾⠛⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢸⣿⣿⡆⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⢸⣿⣿⡀⠀⠀⠀⠀⠐⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠁⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⢸⣿⣧⠀⠀⠀⠈⣽⠋⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡦⠀⠸⣿⡏⠀⠀⠀⢠⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠀⠀⣿⣿⡿⠒⠐⠋⣻⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⣿⣿⣷⡀⠀⠤⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⣿⡆⠀⣿⣿⣿⡇⢤⣖⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠲⠦⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⣿
⠀⠀⠀⠀⢰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠛⠉⠉⠙⠳⠶⢤⡀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠠⣰⠇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣠⠞⠁⠀⠀⠀⡠⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠃⢡⠀⡴⠊⣈⠑⡄⠀⢠⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣵⠇
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⡇⠀⠀⢃⠀⢅⠉⠀⠀⡄⠰⢣⢸⠀⠀⠖⠢⣀⢄⠀⠀⠀⠀⢹⠃⠘⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⡇⢀⠀⠀⠑⢄⠳⡀⠰⠀⢈⡈⠈⡇⠀⢆⠀⠁⡰⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠘⣧⢄⠀⠀⠀⠀⠀⢠⠈⠒⠊⠀⡸⢀⣀⠀⢨⠀⡇⡄⢠⠃⢱⠀⢡⠀⠀⠑⠞⠁⠀⠀⠀⠀⢠⣲⠏
⠀⠀⠀⠀⠀⠈⠳⣗⠄⠀⠀⠀⠀⠢⠤⠀⠔⠁⠸⡀⠉⢁⠔⠁⠥⠜⠀⠈⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠧⣄
⠀⠀⠀⠀⠀⠀⢠⡶⠛⠻⡆⠀⠀⠀⠉⠉⠉⠛⢦⣴⠶⢤⣤⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠉⠛⠷⠵⢶⣦⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢾⠁⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⡄⠀⢠⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣡⠇
⠀⠀⠀⠀⠀⢠⣞⣀⣀⡘⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠼⣧
⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⡶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⡟⢿⣿⣽⡿⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣴⣿⣷⣾⣇⣸⣟⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⢿⣧⣭⣾⣿⣿⣾⣿⣿⣄⢹⣆
⠀⠀⠀⠀⠀⠀⣰⣉⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⣩⢻⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣯⣾⣿⢿⣛⠟⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⢿⣿⣠⣿⠻⣿⣿⣿⣿⣿⣿⣶⡟⢻⢧⡀⠀⠀⠀⠀⢀⡯⣭⣻⣿⣿⣽⣿⣿⣿⣿⣿⣜⣿⣏⠹⡍
⠀⠀⠀⠀⠀⠀⣶⣾⣷⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣦⣥⡾⣷⡀⠀⠀⢠⣿⣿⣿⣿⣎⣿⣿⣯⣿⣿⣿⣿⣿⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⢠⣼⡏⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣘⣪⣷⡀⢀⣿⣱⣹⣿⣿⣬⣿⣿⣿⣾⣿⣿⣿⣿⣼⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠘⡟⣿⣿⢿⣿⣿⣿⣾⣿⣿⣸⣿⣿⣿⣿⠉⡍⣷⡞⢹⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⢻⣯⣹⠃
⠀⠀⠀⠀⠀⠀⠀⢻⠿⣿⢛⣹⣾⣿⣿⢺⡟⣻⣿⣿⢿⣻⣷⡶⣿⣷⣿⠻⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣯⣏⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣽⣿⣿⣿⡻⣷⣿⣿⣿⣿⣿⢹⣦⣾⡿⣿⡇⣿⣿⢧⣿⣿⣿⣯⣾⣿⡳⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣽⣿⣿⡻⣝⡷⣭⡙⣿⣻⣿⡟⣼⣇⣸⣇⢿⣇⣝⣳⣿⣿⣿⣿⠟⣿⠟⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⣻⣿⣿⣿⣿⣿⣟⣿⣿⣾⣿⣿⣿⣿⣮⡿⣿⡿⣿⢟⠟⣳⣟⣹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⢟⣿⣟⢿⣿⣭⣯⣽⣿⣿⣿⠻⣿⣿⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣻⣿⣟⣿⣿⣿⣿⣼⣿⣿⣿⣿⣧⣿⣿⡷⣻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠻⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠯⢽⣿⣿⣿⣿⣾⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢻⢿⣿⠉⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣵⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁







⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⢋⡙⢻⡉⣹⠛⣻⡻⢉⡿⢻⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣿⣿⣿⣯⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢸⣿⣿⣿⣿⣷⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠿⠿⠿⠿⠟⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⢠⡤⠤⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣰⠁⠀⠀⠈⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢸⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⢠⠏⠹⡄⠀⠀⠀⠀⡴⠊⠉⠉⠉⠲⡄
⠀⢹⡀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⠃⠀⠀⠙⣄⠀⠀⠀⠣⣀⠀⠀⢀⡠⠃
⠀⠀⠙⠒⠒⠊⠁⠀⠀⠀⠘⠒⠒⠒⠒⠒⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠐⠓⠒⠒⠒⠒⠚⠂⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠰⠂⠀⠀⠀⠰⠂⠀⠠⠰⠀⠰⠀⠀⠀⠀⠆⠄⠆⠀⠰⠀⠀⠀⠀⠀⠀⠐⠰
⠀⠀⢀⡠⢤⡀⠀⠀⠀⠀⠀⣠⠤⠤⠤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⣴⠋⠀⠀⠈⢳⠄⠀⠀⡰⠁⠀⠀⠀⠘⣆⠀⠀⠀⡞⠀⠀⠀⠘⡆⠀⠀⠀⢠⠋⠉⠉⠉⡽⠀⠀⠀⠀⢠⠏⠈⢧
⠀⠸⡄⠀⠀⠀⡌⠀⠀⠀⠘⣄⠀⠀⠀⡰⠃⠀⠀⢸⣁⣀⣀⣀⣀⣱⠀⠀⠠⠧⠤⠤⠤⠴⠁⠀⠀⠀⠀⠈⢦⣠⠏
⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠒⠀⠐⠂⠘⠀⠀⠀⠀⠒⠐⠒⠒⠘⠀⠀⠀⠀⠃⠂⠀⠀⠀⠘⠀⠐⠒⠒⠐⠐⠀⠐⠒⠒⠚⠀⠐⠂⠒⠒⠘⠐
⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⡖⠒⢲⠀⠀⠀⠀⢠⠖⠒⠦⠖⠒⢦⠀⠀⠀⠀⢀⠔⣲⠖⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠲⣞⠁⠈⢙⡶⠀⠀⠀⢸⠉⠁⠀⠈⠉⡇⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢰⠃⠀⠀⠀⠀⠀⢠⠞⠉⠉⠙⢦
⠀⠀⣸⡠⢤⣸⠀⠀⠀⠀⠘⠛⣇⣀⣸⠛⠁⠀⠀⠈⠳⣄⣀⣀⠴⠃⠀⠀⠀⠀⠱⣄⣣⣀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠚⠁
⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⠉⠉⣉⣈⡉⠀⠁
⠀⠀⢠⠊⠙⢦⠀⠀⠀⠀⢀⡔⣋⣭⡑⢦⠀⠀⠀⠀⡰⠊⢹⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⣆⠀⠀⠀⠀⠀⠀⡏⠈⡇
⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⢸⠰⡇⠀⢸⢈⡇⠀⠀⠰⡇⠀⠘⠶⠒⡆⠀⠀⠀⡴⠛⠀⠀⠛⢦⠀⠀⠀⠀⢲⠃⠀⢳⠂
⠀⠀⠳⣄⣀⠴⠁⠀⠀⠀⠀⠳⢌⣉⡥⠞⠀⠀⠀⠀⠑⢤⣀⣠⠜⠁⠀⠀⠀⠣⣀⡴⢤⣀⠜⠀⠀⠀⠀⠀⢳⡰⠃
⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⢀⣀⢀⢀⠀⠀⠀⠀⠀⣀⣀⣁⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠂⠀⠴⠒⠚⠛⠛⠛⠛⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠦⢤⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠘⠲⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣰⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⢦⡠⠏⠴⠂⣀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⣨⠽⠛⢶⡀⠀⠀⠹⡄⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡞⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣰⠋
⡀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠣⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡃⠈⠉⠳⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⠁
⢳⡀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠛⢧⠀⠀⠀⠀⢀⠤⣄⡀⠀⢠⡇⠀⠀⠀⢀⣠⡴⠖⠂
⠀⠳⣄⠀⠈⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠐⠦⠄⠤⠒⠁⠀⠀⠹⡆⢸⠀⠀⠀⣰⠟⠉
⠀⠀⠈⠑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠠⢤⡀⠀⠀⠀⠀⢀⣰⢒⡤⢦⠀⠀⠀⢹⢘⡀⠀⣰⠃
⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⡴⠁⠁⠈⣠⣃⡤⡀⠀⠀⠄⠀⠀⡏
⠀⠀⣰⠃⠈⠀⠉⢧⠀⠀⠀⠀⠲⢤⡀⠀⢀⠀⠀⠀⠀⠈⡹⠃⠀⠀⠀⢉⣥⢾⡁⠀⠀⠀⠀⠸⡇
⠀⡞⠙⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⣀⡤⠊⠀⠀⠀⣀⣠⠔⠒⠋⠀⠀⠀⠀⠀⠀⠃
⠀⡇⠀⠀⠀⠀⠀⠈⡟⢦⡀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⣠⠞⠁⠘⠆
⠀⡇⠀⠀⠀⠀⠁⠀⠸⣄⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⡤⠖⠋⠁
⠀⠀⠳⣄
⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠿⠿⠿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠒⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠈⠉⠙⠉⠙⠉⡛⠉⠁⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣠⣀⣀⠀⣀⣀⣇⣸⣤⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠮⢭⣭⣭⣭⣭⡭⠥⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠡⠤⠔⠒⠒⠒⠒⠒⠒⠠⠼⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠑⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠫⢶⣢⣤⣤⣍⣉⣁⣀⣀⣀⣀⣀⣩⠭⠕⣂⡠⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⣈⡉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⣾⣶⠃⣼⣷⠚⣷⠖⢰⣿⠃⢸⣭⣟⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢀⡀⠀⠀⠀⠀⠉⠉⠀⠛⠀⠘⠛⠂⠈⠀⠀⠀⢀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠘⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠤⢄⣉⣉⡐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠤⢀⣀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠶⠶⠶⠒⠒⠋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠓⠊⠉⠛⠓⠒⠒⠒⣒⣶⣖⣒⣚⣯⠝⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡂⢤⣀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⣀⣤⠖⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢮⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣋⣥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠏⠛⠟⠉⡛⠛⣟⠛⣛⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⢸⠿⢿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣤⣴⣾⣷⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠠⣀⢸⣿⣿⣿⡿⠟⠀⠀⣠⣶⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠉⠁⠀⠀⠀⢰⣿⣯⣤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⡀⠀⣶⣶⣄⣸⣿⣿⣿⣿⣿⡆⠀⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⠻⠿⢿⣿⡿⠿⠿⣿⣿⡿⡿⠗
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠤⠄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠟⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⠀⠀⠐⡆⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠄
⠀⠀⠀⠀⢀⡻⠿⠛⠋⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣶⣿⣶⣾⣿⣿⣿⣿⣷⡿⠿⠿⠶⡁⠰⠶⠖
⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⣄⣀⢂⡀⣉⣁⣽⡿⣟⣽⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣃⣀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣿⣿⣿⣿⠟⠿⠿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣟⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠋⠉⡠⠛⢦⠀⠀⢰⡛⠻⠥⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠴⠛⢀⣏⣠⠤⠶⠶⢤⡀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠞⠋⠀⠀⠾⣒⠄⢹⢻⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⠎⠀⠀⠀⢀⣀⠀⠐⣝⡞⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⣾⡉⣢⡀⠉⠁⣢⡰⠃⠁⢀⡼⠁
⠀⠀⠀⠀⢀⡴⠦⣀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢸⣿⣿⣏⣀⡦⣀⠝⢁⡤⠞⠋
⠀⠀⠀⢰⠋⠀⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⠧⠞⠛⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⡼⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣤⡄⣀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠙⠢⠄⢀⡀⠀⠀⠀⠀⠈⢻⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠙⠒⠦⣤⣤⣤⣤⣤⠤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣿⡁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠊⠁⠉⠣⡄
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠡⠾⠀⢠⡄⠀⢻⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠚⠀⡀⠀⡀⠈⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⠁⠀⢻⣿⣿⡿⠓⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢯⣄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠚⠀⠲⠒⠴⠤⠦⠶⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠶⠶⠶⠶⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⢿⣶⣦⣌⣒⢾⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠈⠉⠁⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠑⠢⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠖⢛⡇
⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠉⠒⠒⠶⠶⠦⠖⠒⠒⠉⠉⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠐⠠⠤⠀⠀⠠⠤⠐⠒⠂⠒⠂⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠑⠀⠀⢀⣀⡀⠠⠄⠒⠒⠒⠂⠀⠤⠤⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⡇⡀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢀⢀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⠤⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠑⠂⠰⠶⠆⠐⠂⠉⠉⠁⠐⠀⢀⡝⠀⢰⣯⣀⠀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⣀⡀⢨⣧⡉⠉⠉⠉⠉⢉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠓⠂⠚⠁⠉⢫⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣬⣯⣠⣍⣽⣍⣽⣯⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⡔⣄⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣳⡦⢠⡟⠉⣿⡏⠻⣦⠰⣾⣣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢬⣽⣿⣦⡘⣧⡴⣿⠷⣴⡏⣠⣾⣿⣥⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣤⢍⣿⣟⣛⣻⡶⠟⠋⣉⡥⠀⠑⣼⠘⠁⠢⣍⠉⠛⠷⣿⣛⣛⣿⣍⢥⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠠⢔⠋⠜⠋⠭⣥⡤⢤⡆⠋⠀⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⢠⠍⠉⠓⠌⠓⠤⠈
⠀⠀⠀⠀⢀⡔⣠⡤⢩⡾⣽⠛⠂⣿⠀⢸⡟⠻⣦⣾⣻⣦⣾⠛⢱⣞⣳⡆⠀⢸⡶⣿⣛⠀⠀⠀⢤⣄⢢
⠀⠀⠀⠀⠒⢰⠏⠀⠀⠃⠀⠀⠀⠻⠀⠘⠃⠀⠋⠛⠒⠃⠛⠀⠈⠓⠚⠁⠀⠘⠛⠓⠛⠁⠀⠀⠐⢿⡆⠒
⠀⠀⠀⠀⠛⠸⠀⠀⠀⣶⡖⠒⢦⢀⡴⡶⠢⡀⢲⡶⡖⢂⡀⢲⠖⣶⣶⠒⠒⡖⣶⣶⠒⠲⡄⠀⡠⢌⠇⠛
⠀⠀⠀⠀⡽⡧⠀⠂⠀⣟⣧⣠⠜⣿⣸⠀⠀⢹⠘⡷⣇⡸⢧⣸⠀⣿⣿⠴⡆⠀⣿⣿⠤⡴⠃⠀⠑⠀⠼⢧
⠀⠀⠀⣠⣶⣔⢢⠀⠀⣯⣇⠀⠀⠙⢿⣧⣠⠏⠀⣿⣿⠇⠘⡿⠀⣿⣿⣀⣀⣆⣿⣿⡀⠘⢆⠀⠀⡔⣢⣴⣄
⠀⠀⠘⠁⣀⠘⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠃⣀⠈⠂
⠀⠀⠀⠐⠋⠀⠀⠋⢹⡆⠀⢿⡿⠿⠀⣴⢠⡖⣾⠀⢀⡿⢱⠟⡷⣠⣞⡆⠀⠸⠷⣿⠇⠀⢸⡏⠙⠀⠀⠙⠂
⠀⠀⠀⠀⢀⡀⡀⠐⠋⠀⠀⠈⠛⠁⠀⠛⠛⠀⠛⠁⠘⠓⠋⠘⠓⠙⠓⠊⠀⠀⠘⠋⢀⡤⠀⠙⠂⢀
⠀⠀⠀⣠⠋⡘⠃⠀⠀⠀⢠⡞⠻⡆⢴⣒⣶⣰⡞⢳⡖⢲⣦⣶⣲⣦⠀⠀⣴⠞⢶⡄⢿⡗⠀⠀⠀⠘⢃⠙⡄
⠀⠀⠀⣿⠸⠺⠇⡀⢀⣀⠸⠇⠀⠿⠻⠥⠿⠿⠂⠸⠇⠸⠟⠿⠭⠟⠀⠀⠻⢦⠾⠃⠸⠇⣀⡀⢀⠸⠇⠇⣿
⠀⠀⠀⠻⣤⣀⠴⠏⠈⢛⡓⠀⡀⠀⣀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡛⠁⠹⠦⣀⣤⠟
⠀⠀⠀⠀⠈⠁⢄⡀⢀⠨⡟⠀⠀⢾⡉⣼⠏⣴⣟⣿⢠⢾⡇⢠⡿⣰⡟⢠⢾⡇⣠⢄⠈⠻⢆⠀⢀⣠⡉⠁
⠀⠀⠀⠀⠀⠘⠶⢤⣨⠔⠀⠀⣠⠴⣾⢟⣋⣻⣶⣖⣳⣾⠗⠻⠗⠻⠗⠳⠾⠛⢁⣰⠃⠀⣮⣠⠤⡞
⠀⠀⠀⠀⠀⠀⠈⠒⢦⡄⠒⢈⣻⣾⣿⣿⣽⡿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⠙⠒⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠉⠉⠉⠛⠉⠀⠀⢈⣷⢲⢷⡶⡆⢔⠇⠺⢟⠆⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠃⣀⡴⢦⣄⠈⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣀⣶⣛⠀⠀⣙⣷⣄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⠀⠀⡀⠻⠛⢹⡏⠙⣷⣿⡋⢹⣿⠙⠿⢀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⠆⠀⣠⡾⢿⡄⠀⠸⣧⣼⡇⠸⣷⣼⠿⠀⠀⣿⠷⣤⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⠀⣿⠀⢽⠷⣾⣷⣟⣙⣷⣾⣋⣻⣷⣿⡆⣿⠀⣿⠀⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣿⠀⣿⠓⢮⢰⡿⠻⣧⣼⠟⢿⣦⣩⠚⣿⠃⣿⡸⣯⡟⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣻⡄⣿⠀⣿⠀⢠⢻⡇⢸⡏⢹⣿⠀⣿⢧⠀⣿⠀⢸⠶⢿⡇⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠁⣿⣠⣿⠖⠁⢸⡇⢸⡇⢸⣿⠀⣿⠈⠓⢿⣤⣼⡇⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢸⣿⠟⣿⠀⣿⡄⢀⣾⡇⢸⡇⢸⣿⠀⣿⠀⠀⢿⡇⢸⡟⢿⡇⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⢸⡿⢰⣿⠀⠋⣾⠋⢸⡇⢸⡇⢸⣿⠀⣿⠀⡀⠀⠁⢸⡇⠼⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣬⣤⣸⣿⠀⣼⡟⠀⣼⡇⢸⡇⢸⣿⠀⣿⠀⣿⡦⠀⢸⣇⣤⣶⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⠿⠿⠾⠿⠿⠿⠿⠾⠿⠀⣿⠀⣿⡇⠀⢸⣇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡾⠃⢀⣶⣤⠀⣠⡆⢰⡆⢠⣤⠀⠃⠀⣿⡇⠀⠈⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⡷⠿⠷⠾⠷⠾⠿⠶⠶⠶⠿⠇⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠉⠉⠈⠉⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⡶⠂⠀⠀⠀⠙⠳⣄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⣶⠀⠀⣸⡥⠒⠚⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠋⠀⠀⠙⡦⠺⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠊⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⢀⠀⠀⡟⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⢦⡀⠀⠀⠀⠘⠓⢦⣤⣤⣤⣤⡴⠛⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠲⣄⣀⡀⠀⠀⠉⠓⠚⠚⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⢀⡠⠈⠉⠉⢛⡖⠲⣖⠒⠋⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠙⠦⣄⡀⢸⠁⢀⣨⠗⠚⠙⣦⠀⢠⡏⢺⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠠⠤⡖⠛⡶⢧⡘⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠘⠓⠢⢕⡠⣄⣀⣀⣀⣀⣀⣀⣀⣤⡀⣀⠀⢲⠞⠁⠀⠉⠖⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠂⠀⠀⠀⠈⠓⠦⣬⣉⣉⡉⠁⠀⠘⣌⣙⡶⠻⡦⠤⠤⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠳⢄⠀⡠⣀⡀⠀⠀⠉⠀⠀⣘⣶⠒⠐⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠛⠐⠁⢋⡗⠀⢀⡴⠋⠀⠘⠆⠈⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠦⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⣸⠤⠖⠛⠦⠤⠤⣀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡶⠒⠻⠤⢾⣀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢻⠉⠀⠀⠀⠀⠀⠉⠳⣄⢠⠇⠀⠀⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⠏⠀⠀⠀⠀⢠⠞⠉⠙⢾⣿⣢⢀⡞⠉⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣀⣀⡀⠀⣔⠛⡇⢠⠴⠶⢤⡀⢸⡀⣠⣦⡀⢧⠻⡟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣠⠟⠉⠉⠉⠙⠿⣷⠁⡇⠀⢠⣤⡹⡄⢳⣽⠿⠷⠞⠀⠙⢦⡀⠀⠀⠀⣰⠋
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡆⠹⣄⠙⣿⠟⠋⠀⠀⠀⢀⣀⣤⣄⣀⠑⣄⡴⠛⠁
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⢁⣀⣠⣤⠶⠟⢿⡇⠀⣸⣿⣿⣮⡳⣶⠋⠙⡆
⠀⠀⠀⠀⠘⠳⣤⣀⣀⣠⣤⡇⠀⠀⣼⠋⠉⢹⣇⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⡜⣦⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣼⠀⠀⣿⣷⣶⣿⣿⣿⣿⣿⠿⠛⠛⠉⠀⠉⠛⣇⢸⡦⠃
⠀⠀⠀⠀⠀⢀⡖⠒⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠢⣄⠀⠀⠀⠀⢸⠋⠉⠉⠉⠛⠛⢷
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠖⠋⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣎⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠙⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⡼⠁⠀⠀⠱⠇⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣠⠁⠀⠀⠀⠇⠀⠀⠀⠈⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢃⠀⠀⠀⢸⠢⠤⠤⠞⠉⠂⠄⢴⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡜⠢⠤⠴⡇⠀⠀⠀⣄⠀⠀⠀⡸⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠒⠊⠈⡆⠀⠀⡷⠄⠀⢴⠁⠈⠱⡞⠁⠀⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⣅⣀⠼⡇⠀⠀⣸⠀⠀⠀⡘⠐⠲⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠞⢏⡤⠆⠺⡓⠀⠚⢹⡄⣀⡴⠁⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⢸⣀⠠⣦⡀⠀⣸⠦⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠼⢄⣀⣠⡏⠀⠀⡃⠈⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠃⠀⠉⢲⠖⠙⠦⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⢄⡴⢆⠀⢀⣼⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠉⢉⠇⠑⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⠲⡔⠫⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⠴⡀⠀⢀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣨⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⢀⣰⣷⡶⠟⠛⠉⠀⠀⡀⢠⠈⠳⣤⡘⢷⡄⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢶⡀⣿⠀⢸⣿⣿⣿⡿⠿⢶⣒⣠⠴⣫⢞⡟⢠⠐⡌⢻⣾⣿⣦⣄⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢀⣷⠀⠀⠙⢶⣿⣿⣶⣶⣶⣾⣿⡿⠿⣛⣫⣵⠞⣡⡞⣰⢸⡇⠹⢄⠹⣿⣿⣿⣷⣿⣧⠀⠘⡆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣾⣦⣉⣛⣛⣋⣭⣭⣭⣴⣶⣾⡿⠟⢋⣴⡾⠋⢠⠇⣾⣿⠀⠘⣆⠹⣿⣿⣜⠿⣿⣦⠀⢿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣾⡿⠋⠀⣴⠏⣼⣿⣿⡄⠀⣿⡀⢹⣿⣝⢆⠹⣿⣇⢸⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⡿⢛⣵⣿⡿⠛⣁⣴⣾⠿⠋⢁⠄⣠⡾⢋⣾⣿⣏⣿⡇⠀⢸⡇⠈⣧⢹⡄⠀⢸⡿⠘⢸⠇
⠀⠀⠀⠀⣀⠀⢀⣀⣤⣾⣿⣿⠿⢋⣴⣿⠟⢉⣴⡾⠛⠉⢀⡠⠚⣡⣾⢏⣴⣿⣿⡟⢸⣿⠀⠀⣿⡇⠀⣿⠀⣿⠀⠘⠃⢺⠟⠀⠀⠀⠀⢴⢤⣀⡄
⠀⠀⠀⠀⠈⠉⠉⢛⣿⢫⡾⢋⡴⠫⠛⣡⠖⠋⠁⠀⠐⠊⣡⣴⡿⢋⣵⣾⣿⣿⡟⢀⣿⡏⠀⢁⣿⣱⢀⡟⠀⣿⡇⠀⣆⢠⡈⠁⠀⢷⡄⣼⣿⣿⣃
⠀⠀⠀⠀⠀⠀⢀⣾⢣⡟⠀⠊⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠛⣡⣴⣿⢟⣽⣿⠏⢀⣾⡟⠀⠊⣼⣿⠇⣸⠃⠀⣿⡏⠀⠨⣳⣝⡻⣶⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡾⠁⡞⠀⠀⠀⠀⠀⠀⠒⠋⠛⢋⣩⣤⣶⣿⠿⢋⣴⣿⠟⠁⣠⣿⡟⠀⠀⣸⣿⠏⡰⠃⠀⢰⢟⡇⠂⢀⠈⠛⢿⣿⣿⣿⣛⣋⡤
⠀⠀⠀⠀⠀⢸⠇⡀⠀⠀⠀⠀⢽⣖⣒⣒⣿⣿⡿⠿⠛⣉⣤⣾⡿⠟⠁⣠⣾⡿⠋⠀⢀⣴⣿⠃⠀⠀⠀⣠⠏⠈⡼⠀⠸⡄⠀⡀⣿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⣱⠀⡌⡀⠀⢸⠁⠈⣁⣀⣤⣴⣶⠿⠟⠋⠁⣠⣴⣾⡿⠋⠀⠀⣠⣾⠟⠁⠀⣀⣤⠞⠁⢀⡼⠁⠀⠀⣇⠀⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠃⠀⠚⠦⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⠛⠁⠀⣀⣴⣿⠟⠁⠀⠀⠈⠁⠀⣀⡴⠟⠀⠀⠀⠀⣿⠀⣿⢸⡷
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠒⡲⠿⠿⠿⠟⠛⠋⠙⠋⠴⠶⠾⠛⠛⠉⠀⠀⠀⠀⠠⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⣀⣘⢸⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⡟⣚⣀
⠀⠀⠀⠀⢠⣿⣿⠿⡷⢿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣄⠀⠀⠀⠀⠀⡟⢠⡟⠾⢿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⠀⢀⡈⠣⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠋⢀⠀⢦⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣤⡿⠿⣦⡀⠀⠀⢀⢠⣶⣤⣾⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢿⣿⣷⣤⣦⢀⠀⠀⠀⢠⡾⠿⣷⣼⣿⡏
⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⢹⡇⠀⠀⠨⠿⢿⡿⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⠿⠯⠀⠀⠀⣿⠁⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣈⣡⡄⠀⠀⠰⠾⠧⠀⢽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠟⠿⠶⠂⠀⠀⢠⣉⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡗⠀⠀⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⣒⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠦⠤⠀⠀⠀⠀⠀⠀⠒⠾⠿⠛⠋

⠀⠀⠀⠀⠀⣎⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠰⣃⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢺⠆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠈⡆
⠀⠀⠀⠀⠀⣇⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⣾
⠀⠀⠀⠀⢀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⢖⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⢀⣴⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⠖⣿⠆⠘⣿⣷⣦⠀⠀⠀⠀⠀⠀⣼⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⢼⠂
⠀⠀⢸⡏⣿⣿⡏⣿⣿⡿⣷⠀⠀⠀⣿⠀⣿⠄⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⢰⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣻⡀⢸⣧⡄
⠀⠀⢸⡇⣿⣿⣷⣿⣿⡷⣿⠀⠀⠀⠋⠉⠉⠁⠉⠉⠉⢛⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢠⠟⣿⣏⢹⣿⣿⡀
⠀⠀⢸⠇⠿⠿⠿⣿⢿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⢳⠂⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢀⡟⢠⣿⣿⢾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠀⣶⣀⣤⣿⣿⣇⡀⠀⠀⣼⠁⢸⣿⣿⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣿⠀⢨⣿⡟⢸⣿⣿⣿⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣦⣤⣤⣤⣤⣤⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠛⠂⠘⠛⠃⠒⠛⠛⠛⠓
⠀⠀⢸⡇⣿⣿⣶⣿⣿⣿⣿⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⢸⣿⣿⢾⣿⣿⣿⡇⠀⢰⣦⣤⠤⣀⣄⣤⣤⣤⣤⣤⡖
⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣾⣿⣷⣼⣿⣿⣿⡇⠀⠈⣿⡇⠀⢈⣯⡅⠀⣿⣿⣿⠇
⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣧⣀⣤⣿⢶⣤⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠻⢿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠙⠻⠷⣶⣶⣶⡶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠠⣤⣤⡤⢤⣤⣤⢠⡄⢤⠤⢤⢤⢤⡄⣤⢤⠀⣤⣤⣤⣤⡄⢠⢤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁⠉⠙⠛⠚⠂⠙⠉⠈⠋⠋⠓⠉⠘⠀⠙⠋⠋⠙⠃⠈⠋⠉⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡆⠀⠀⠀⣤⣤⣤⣥⣤⡄⣿⡿⡿⣿⢀⡀⠁⡀⠀⢀⠀⠀⠀⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⣰⡇⠙⢿⣿⣿⠟⠁⠈⠛⠛⠁⠘⣿⣿⣿⣿⡿⠱⠆⠀⠘⢿⣿⣿⡿⠏
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠂⠐⠀⡂⢀⣀⣀⣀⣀⠀⠀⠈⠉⠛⠉
⠛⠋⠁⠀⠀⠀⠀⠀⠀⢠⣤⣠⣤⣤⣤⣤⡄⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣆⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⢿⣿⣿⣿⣿⠗⠤⠄⠌⠤⠄⢿⣿⣿⢿⣿⡗⠀⢈⠛⠛⠛⠀⠀⠀⠀⠀⠙⠿⣿
⠀⢠⣤⣤⣤⣤⣤⡆⠀⠄⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢀⡂⠈⠛⠿⠟⠋⠀⠘⠛⠷⢬⣦⠀⠀⡀⢄⠒⡠⡀
⠀⠈⠻⣿⣿⣿⠟⠁⠀⢠⢤⣶⢶⣶⣶⡆⣟⣿⣿⣛⣿⡇⠀⠀⠀⠀⠀⠐⣿⣶⣿⣿⡟⠀⢰⣤⣦⣤⣴⣤
⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠀⠙⠋⠋⠋⠛⠁⢶⣶⠶⡶⣶⡆⠈⠛⠛⠛⠁⠀⠈⠻⣿⣿⡿⠃
⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠨
⣶⣿⣿⡟⠀⠀⠀⢀⡀⡄⡀⠀⠀⢶⣶⠀⢶⡶⢶⢶⣶⣶⠀⡀⡀⣀⢀⠀⠶⠊⠀⠀⠀⠀⠀⠀⠀⠐
⠛⠛⠋⠀⠀⠀⠀⢏⣂⣱⣬⡆⠀⠀⠉⠀⠈⠳⢿⣾⠿⠋⠀⠹⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣻⣿⡿⣧⡟⠀⠙⢦⡀⠀⠀⠀⢀⣠⠴⢶⣶⡞⠀⠻⣄
⠀⠀⠀⠀⠀⢀⡼⣿⣿⡿⢿⡟⢀⣀⣄⣀⣹⣶⣤⣶⣿⣷⣶⢾⡿⣰⣶⣦⠙⢦⡀
⠀⠀⣀⠀⢠⡿⣶⣿⣿⡇⠈⣷⢟⣩⣿⣿⢷⣜⠛⠛⠛⣛⣽⣿⣿⣿⣿⣿⢠⣾⣿⣷⡄⢀⣠⡆
⠀⠀⠻⢓⣿⣤⣻⣿⣿⢃⣾⣿⣿⣿⣤⣿⣀⣿⣿⢰⡿⣿⣀⣿⣀⣽⡿⣿⣿⠃⠀⢹⣿⡛⠉
⠀⠀⠀⠘⣿⣄⣝⣯⣏⣾⣽⡿⢠⣾⣿⣿⣻⣿⡟⠈⢿⣯⣿⣿⣿⣾⣿⣽⣣⣶⣿⡆⢿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⣿⣾⡺⣎⢿⡜⠻⣿⣯⣿⣿⣤⣤⣬⣿⣿⣟⡛⣿⣯⣿⣿⣿⣭⢿⡞⣷⠶⠿
⠀⠀⠀⡿⠶⢾⣿⣿⣿⣷⣎⠳⣿⢷⣤⣿⣿⣿⣽⢿⣉⣈⣿⣽⣿⣿⣿⣯⣿⣤⣾⣿⣧⣽⣦⡀
⠀⠀⠀⠀⢠⡟⠁⢻⣿⢿⣿⣷⠈⢙⣿⣿⡿⣯⡛⠛⣉⡽⢻⣿⣥⣬⣿⡿⢿⣿⠯⠭⢭⣭⠿⠛⠛⠃⣠
⠀⠀⠀⠀⠈⢷⡀⠈⣿⣀⣿⣿⡷⣻⠛⣷⠀⠀⠙⠛⠋⢠⡏⣿⠏⠈⣯⢹⡎⠛⠚⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠻⣦⣿⡟⣿⣽⢱⡇⠈⠁⠀⠀⠀⠀⠀⠘⠷⡿⠷⠶⠟⠾⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⢠⣤⡴⢻⣿⣼⠏⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢹⡇⢠⣿⠿⣿⠟⣿⢿⣿⡿⠻⣿⣿⣿⡿⢀⣾⠿⣿⡿⢻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⡇⣸⣧⣤⣯⣠⣏⣼⣼⣇⣶⣿⣿⣿⡇⣼⣧⣤⣿⡠⢻⣿⡜⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⢻⣿⡟⢻⡿⢛⣹⡟⠟⣹⣿⣿⠛⣿⠛⢿⢻⠋⡙⢿⣿⢻⠛⡟⣻⢻⠟⡟⣿⡉⣻⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣂⣽⣘⣛⣠⣄⣳⣜⣛⣇⣦⣹⣿⣇⣤⣘⣀⣦⣸⣆⣫⣾⣿⣆⣴⣠⣿⣰⣆⣇⣿⣇⣿⣐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⠿⠿⣿⠿⠿

⣄⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⢸
⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⢰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⠀⠠⠤⠤⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠙⠲⠤⠤⠀⡀
⠀⠀⣾⡃⣻⡟⢻⣿⣿⠛⣿⣿⡟⢛⣿⣿⠛⣿⣟⠛⣿⠟⢻⣿⠿⣿⣉⣳⠀⠉⠉⠁⢈⠀⠀⣀⣉⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠀⠀⠀⠈⠋⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣿⡶⠆⠀⠀⠀⠀⠠⢀⠀⠀⠄⠤⢠⠠⠀⠐⠀⡤⠀⠠⠄⠀⢰⠶⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢀⡀⡀⢀⠀⠀⠘
⠀⠀⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⠀⠆
⠀⠀⣿⣋⣳⣶⠒⣶⣆⣀⣶⣶⣆⣰⣶⣶⣀⣶⣶⣆⣰⣦⣤⣆⣰⣎⣩⣿⢠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠠⡤⠀⠐
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠤⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢀⠀⠀⣀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠍⠉⠹⢷⣄
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠐⠄⠠⠀⠀⠂⠖⠠⠐⠀⠐⠂⠂⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠠⠀⢀⡈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⠀⠀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠋⢛⠉⠋⠋⣏⢉⣩⢠⠀⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠇⠇⠅⠺⢸⠁⠀⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⣷⠶⡞⠓⠛⠋⢻⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⡏⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⢀⡼⠁⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠀⠀⢰⡟⢉⣉⣉⣉⣩⣭⣭⣥⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠈⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢠⡇⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡇⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠻⣦⣉⣛⣛⣛⣛⡛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⣁⣤⣶⣾⣿⣿⣷⣶⣤⣍⠛⢿⣿⣿⣿⣿⣿⣿⠿⠛⢋⣉⣥⣤⣤⣤⣍⣉⠻⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣼⣿⣿⡟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⠟⢋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⣷⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣆
⠀⢀⣾⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡄
⠀⣸⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠻⠿⠟⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣷
⢀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡹⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠶⣄⠉⢃⣽⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⡝⣿⣙⣿⣿⣆⢠⣼⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⡘⣿⣆⠙⠿⢸⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠁⢻⣿⣆⢶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣧⠈⠻⠿⢞⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⠙⢿⢘⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢠⠘⣿⡇⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢰⡈⠚⣧⡘⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⢿⣧⠀⢳⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣼⡏⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣿⠿⠛⣉⣤⣶⣌⣛⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢂⣉⡉⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠛⠛⠛⣛⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡛⠛⠛⠛⠛⢻⣿⣿
⣿⣿⡏⠀⣀⣤⣶⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣄⠀⠹⣿⣄⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣾⠿⠋⠁⢀⣠⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢿⣧⡀⠈⢿⣧⡀⠀⢸⣿⣿
⣿⣿⡏⢀⣠⣴⠾⠛⢉⣹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡻⣿⣆⠀⠻⣿⣄⢸⣿⣿
⣿⣿⣿⠿⠋⣁⣤⣶⡿⠟⠉⠀⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠙⢿⣮⢿⣷⣄⠙⣿⣿⣿⣿
⣿⣿⣧⣴⣿⠿⠛⣁⣠⣤⣶⣿⡿⠋⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⢿⣷⡀⠈⢿⣷⡙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣫⣴⣶⣿⣿⣻⣿⡿⢋⣠⡶⠟⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠙⢿⣦⣌⣿⣷⡈⠻⢿⣿⣿⣿
⣿⣿⡿⠛⠉⠀⣨⣿⡿⣋⣴⣿⣿⣿⣶⣿⠟⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⠄⠈⠻⣷⣄⠀⢀⡙⠻⣿⣿⡇⠀⠀⣸⣿⣿
⣿⣿⡇⠀⠀⠚⠿⢫⡾⠛⣡⣿⠃⠀⣠⣴⡿⠋⢁⣀⣄⡀⠀⠀⠀⠀⢠⣈⠻⣿⠷⣤⣄⠈⠻⣿⠿⠿⠄⠄⠉⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠿⠿⠆⣾⣿⡽⣟⡴⣆⠂⠀⣠⣌⢿⠷⡌⠳⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⡁⠀⠘⢟⣉⣉⡀⠐⡀⢀⣀⠀⢀⣀⠀⠀⢀⣀⠁⠀⣬⡌⢸⣿⣿
⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⢀⠀⣘⠻⠏⡿⣼⡇⣸⣿⠇⣼⡏⠀⣰⡿⢻⡇⣼⣿⠁⣼⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⡇⠙⠁⠀⠘⣫⢟⣀⠀⠁⠁⠛⠷⢠⣿⡀⢰⣿⣁⣿⢳⣿⠃⣼⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘⠷⣾⣷⡝⠿⠀⠀⠀⠀⢼⣭⠟⢡⣄⠀⠀⠀⠀⠉⠛⠈⠿⠿⠃⣿⣿⣾⣿⢻⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠟⢿⣷⡀⣯⣽⠿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⣥⠀⠀⠀⠀⠀⠀⠀⠂⠛⠛⢿⠃⢸⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⣸⣿⣠⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿
⣿⣿⣏⣀⣰⣿⣶⡄⢸⣿⠀⠙⢿⣦⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣶⢶⣶⣿⡿⢫⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿
⣿⣿⡿⣿⣿⣿⠿⣿⣎⢿⣧⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⠿⢋⣴⠿⣿⣦⣴⡶⠆⡀⠀⠀⠀⠂⠀⢸⣿⣿
⣿⣿⡇⠈⢿⣿⡄⠙⣿⣮⠻⣷⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⣴⣿⣯⣾⠿⠛⣉⣴⡿⠟⠀⣀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⢻⣿⣄⠈⢿⣷⡌⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠋⠉⣠⣴⣾⠟⠋⢀⣴⣾⠟⠐⢠⠀⢸⣿⣿
⣿⣿⡇⢠⡄⠀⢻⣿⣆⠀⢻⣿⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠁⣠⣶⡿⠋⠁⢀⣴⡾⠖⢼⣿⣿
⣿⣿⣷⣤⣤⣤⣤⣿⣿⣦⣤⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣶⣿⣿⣿⣶⣤⣶⣿⣿⣤⣤⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿