Dinghy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠐⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠤⢸⡞⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢂⡇⢳⡐⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡅⢬⢧⠀⠀⠀⠀⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢸⠀⠀⠀⠡⠀⢰⠀⠁⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⠘⠀⠀⠀⠀⠐⠈⡇⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠤⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠭⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⠄⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣠⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠙⢧⡀⡉⢉⢉⣉⢉⡁⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠼⠘⡇⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠈⢳⡌⠿⢻⠃⣫⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠵⠃⠖⣽⠀⠀⠀⠀⠀⠀⠘⢻⢳⡀⠀⠀⠀⠀⠑⡄⠁⠊⢁⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠈⢎⢿⣄⣀⠀⠀⠀⠘⢦⠀⠈⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⢀⠾⡇⢸⣀⣀⣀⣀⣀⣀⣠⣸⣮⣙⣛⠿⠷⣦⣠⣀⣳⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣚⣒⣒⣛⣒⣓⣘⣋⣿⣿⣿⣿⣿⣙⣉⠙⠛⠻⢿⣭⣽⣿⣷⣶⠾⠛⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠿⠿⠿⠿⣯⣿⣿⣿⣿⣿⣿⣟⣛⣻⣶⠶⢶⣶⣶⣶⣾⡿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣿⠏⠉⠡⠤⠿⠿⠿⣿⣿⣿⣶⠤⡬⠉⠉⠉⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠒⠀⠀⠒⠚⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢠⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⡟⠀⠀⠀⡌⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣴⡿⠀⠀⠀⢀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢰⣿⠇⠀⠀⠀⡘⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣿⡟⠀⠀⠀⠀⡇⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿⡇⠀⠀⠀⢸⠁⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠄⢰⣿⡇⠀⠀⠀⡘⠀⠀⡅⢸⡟⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠘⣿⣧⠀⠀⢀⠇⠀⢀⠀⢸⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣿⣿⡀⠀⢸⠀⠀⠈⠙⢸⡿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠈⢾⣿⣧⠀⡇⠀⢀⠃⠀⢺⣿⡄⢿⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⣼⣿⣿⣶⠇⠀⢸⠀⠀⣼⣿⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⠀⠈⠾⣿⣿⡄⠀⡆⠀⠀⠈⣿⡇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡓⠢⡉⠁⠀⠀⠙⣿⣿⣀⡅⠀⠀⠀⣿⣷⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⡀⠀⡀⠘⣿⣿⡀⠀⠀⣐⣿⣯⠀⠀⢈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠉⠲⣿⠆⡙⣿⣷⠀⠀⢽⣿⣿⠇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠙⠻⢟⣿⣧⠀⠂⢸⣿⣷⣆⠀⢀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⠀⢀⠰⣿⣿⣿⣿⡷⡄⠘⣿⣿⣿⣶⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⢀⣠⣾⣿⣿⡿⠟⠃⠀⠘⡿⠛⠛⠛⠛⣛⣛⡛⠛⠛⢣⣶⣶⠂⠠⠄⠄⠀⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣯⣥⣤⣴⣦⣤⣤⣧⡄⠀⠰⠾⠿⣿⣿⣦⣴⣿⣿⣷⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣉⡉⠛⠿⠻⢿⡻⠿⠭⣭⣿⣟⣾⣟⣸⣯⣿⣭⣭⣭⣽⣯⣽⠿⠿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠷⣶⣿⣿⣷⣾⣷⣷⣶⣦⣼⣧⣤⣤⣈⣋⣤⣤⣬⣡⣤⣤⣥⣤⣈⣀⣆⣙⣿⣾⣷⡶⣶⣶⣆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢛⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠋⠉⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣿⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡞⠀⣿⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⣠⣿⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⡀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠿⠿⠛⢿⠀⢀⠀⡀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠙⠛⠛⠛⠛⠛⠉⠉⢸⡇⠀⠀⠀⠀⢀⠀⠡⡀⠀⠈⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣁⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡤⠴⠒⢛⣿⠏
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⣰⣟⣥⠶⠒⠒⠶⢤⣀
⣀⣀⣀⣀⣠⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠷⠶⠒⠚⠛⠉⠉⠉⠀⢀⣀⡠⠴⠒⠂⠀⠀⠀⠀⢀⡼⠟⠛⠛⢷⡄⡠⠤⢄⠙⣦
⠙⢿⣿⣿⣿⡛⠛⠛⠉⢉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣾⣡⡾⢻⠀⠤⠼⠃⣿
⠀⠀⠙⢿⣿⣿⣦⠠⠤⠼⢦⠤⠤⠤⠄⠒⠒⠒⠒⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⣀⣠⠤⠖⠂⠀⠀⠀⣀⡤⠾⠷⠖⠲⠶⠶⣤⣤⠤⢼⣦⣤⣤⠾⠗⢦
⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⣤⣤⠤⠤⠤⠴⠒⠒⠋⠉⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⣠⣤⣄⡀⠀⠀⠀⠙⣷⡄⣀⡉⢳⡄⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠊⠁⠀⠀⠀⢀⡾⠁⢀⣬⣷⠀⠀⠀⣰⡿⠱⣧⣠⠾⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⢤⣈⣉⠛⠓⠲⠶⠤⠤⠤⠤⠴⠶⠖⠒⣛⣩⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠉⢁⣠⣴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢄⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⣫⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢡⠀⠀⠀⠀⢀⣴⢟⣵⢿⣫⡾⣻⡿⠟⡡⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠀⢀⣤⠾⢋⡴⢛⡵⠟⣩⠞⣋⡴⣋⠴⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⣿⢋⣡⠞⣩⠾⣋⡴⣋⠵⢊⡥⢚⣡⠾⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢹⣋⡴⢞⡡⣚⠕⢊⣥⠞⣉⡴⢟⡡⢞⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠀⠀⠀⠀⠀⢫⣴⡭⢊⣥⠞⢋⣴⠞⣋⠴⣫⡴⢋⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢋⠔⢉⣠⡾⢋⡥⢊⡵⢞⣩⠶⣫⣾
⣛⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣟⠯⢓⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⣡⠖⣩⠖⣩⡶⣋⣵⠞⠛⠛
⣭⣙⠳⢦⣝⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⢶⣿⠿⠟⢋⣩⠴⠚⣋⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⡵⣛⡵⢟⡵⠞⣋⣤⣴⡶⠟
⣤⣍⡛⠶⣌⡛⠦⣍⠛⢷⣤⡖⠒⠒⢚⣛⡻⠭⠟⢚⣉⡡⠤⠒⣉⣡⣴⣶⡿⠿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡿⢟⣭⠾⠛⠋⣉⣤⣴⣾
⡳⢮⣝⠓⢮⣝⠷⣬⡙⢦⣉⠻⣷⣤⠤⠖⠒⣚⣋⣭⣤⣶⣾⠿⠟⢛⣉⠥⠖⣊⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣤⡴⠾⠛⣋⣭⣿⡿
⣭⡑⠪⣙⠲⢍⡛⢦⣙⠷⣌⡳⣄⠙⢿⣟⠋⠩⠝⢒⣋⠥⠤⠒⣊⡩⠴⠒⠋⢁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⣤⡶⠿⣻⣿⢏⡾
⣍⠙⠷⣦⣌⠒⠉⡓⠍⠳⣼⣽⡢⣕⠤⡙⢷⡈⢡⡤⠴⠒⢊⡉⣀⣀⡤⠶⢚⣩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⢁⣠⣾⡟⣡⠟⣠
⣤⣅⣂⠌⡙⠷⣦⣈⠐⠤⡂⠙⢿⣮⣓⣦⡢⡙⢆⠀⠤⠔⢒⡉⠩⣄⣲⡾⠛⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⣻⡿⣫⠞⢁⡾⢃
⣶⢬⣙⡻⠶⣤⡈⠙⠻⢦⣍⠢⣕⢽⡿⣷⣝⢦⢊⣃⣬⣥⠶⠖⠛⠯⣥⣈⡙⠛⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⣫⠞⢁⡴⢋⣴⠟
⠻⠿⠿⣯⣷⣮⣝⠳⢦⣀⣙⣻⣦⣵⣼⠶⠞⠛⣯⣍⣁⡈⠛⠿⣶⣤⣀⠉⠙⠳⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢁⡴⢋⣴⠟⢁⣴
⣍⡙⠒⢺⣛⠛⠛⠛⠛⠋⠉⢉⣠⣤⣬⣉⣉⣒⠢⠬⢭⣙⣛⣶⣤⣈⠉⠛⠶⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⡶⠖⠒⣶⠟⡋⢴⡟⣡⡶⢟⣥
⡓⢿⣟⡶⢤⣉⡛⠲⠶⢤⣤⣤⣀⣀⣈⣉⣉⣉⣙⠛⠶⢦⣭⣉⣛⠻⠿⣷⣶⣤⣽⣶⣶⣦⣉⡭⠭⠀⠀⠘⠷⠾⠫⠤⠶⠯⢤⣾⣇⣸⢘⡡⠖⣋⣤
⠻⢦⣈⠛⠳⣮⣍⣛⠓⠶⠶⣤⣭⣭⣉⣉⣉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⡿⣇⢸⠄⠀⠀⠀⠀⠁⠈⣉⡉⠥⠤⠐⠚⠒⠒⠒⠚⢻⠒⢋⣉⣠
⣙⠦⣭⣓⠦⣤⣈⡙⠛⠶⠯⢤⣒⣒⣚⣻⠿⠿⠿⣿⣿⣭⣩⣿⣭⣿⣿⣽⣾⡏⠀⣾⡀⡇⠀⠀⢀⠀⠀⠁⠰⠂⠀⣈⣉⣤⣶⡶⠶⠿⠛⢛⣋⣉⣥
⣬⣙⠲⢬⣭⣛⡿⠿⣿⣷⣶⣦⣤⣤⣭⣭⣭⣭⣵⠶⠶⠾⣻⣿⠿⣛⣭⣬⠿⠿⣾⣾⣧⢳⣶⣞⣤⣀⣠⣴⣷⡾⠿⠛⠋⣉⣠⣤⣶⣶⠿⠟⣛⣉⣉
⠺⢿⣿⣿⣷⣿⡿⠿⠿⠟⣛⣛⣻⣿⣶⣶⣲⣶⣶⠾⢟⣋⠩⠤⠴⠶⠶⠒⢚⣋⣉⣿⠟⣾⣽⣿⣿⠿⠛⠉⣀⣠⣴⣶⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⢉
⣽⣷⣦⣭⣭⣭⣟⣛⣛⣛⣻⣶⣾⠿⠟⠋⣉⣤⣤⣒⣒⣒⣂⣀⣉⣉⣉⣭⡭⢍⣲⣽⠿⠟⠋⢉⣠⣤⣶⣿⠿⠟⠛⢋⣉⣡⣤⠤⠶⠖⠒⠛⠛⠛⠋
⣲⣬⣭⣽⣷⢶⣾⣟⣛⣩⠭⠒⠒⢉⣉⣉⠭⠭⠤⠭⠭⠬⠤⢤⣴⣶⣶⣶⠾⠛⢉⣀⣤⣶⠿⠛⢋⣉⣥⠴⠖⠚⠋⠉⠁⠀⣀⣀⣀⣤⣤⣤⣤⡶⠶
⠒⠒⢒⣚⣛⡩⠽⢒⣩⡥⠶⢚⣋⣥⠤⠶⠖⠒⠒⠒⠒⠒⣲⡶⠞⠛⣉⣤⡶⠾⠛⣉⣠⠴⠖⣛⣉⣡⣤⣤⡶⠶⠾⠛⠛⣛⣉⣉⣭⣥⣤⣤⣤⣤⣤
⣙⣉⣭⣽⠦⠖⢛⣉⠤⢒⣚⣭⣤⣶⣶⠿⠛⠛⣻⣿⠷⣛⣩⣤⠶⠟⣋⣥⣤⣒⣫⣭⠶⠾⠛⢛⣉⣩⣤⣤⡶⠶⠟⠛⢛⣋⣉⣭⡥⠤⠴⠶⠶⠶⠶
⣵⣛⣡⡴⢒⣩⣵⣶⣿⣿⣯⣍⣉⡉⣉⣿⢿⣛⣭⠶⢟⣋⣭⣶⣺⣿⠿⠟⢛⣉⣥⣴⠶⠾⠟⢛⣋⣉⡥⠤⠖⠒⣛⣉⣉⣡⣤⡤⠶⠶⠶⠶⠶⠶⠶
⣉⣉⣉⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠛⠻⠿⠯⢵⣶⣚⣭⠷⠛⠉⠥⣴⣶⣟⣉⣉⣥⡶⠶⠛⣋⣉⣥⡴⠶⠞⢛⣋⣉⣭⠤⡤⠶⠶⠶⠶⠟⠛⠳⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠒⠒⠶⠶⢸⣿⠂⠤⢤⣄⢸⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⢸⣇⣀⣀⣠⣤⣶⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⣿⣷⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢠⠉⠀⣀⣀⣀⡀⠠⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⣩⣭⣝⣻⣿⠀⠀⠀⣀⠀⠠⠀⠈⠩⠦⠄⠐⠒⠒
⠀⠀⠐⠁⣀⣀⣀⣠⣤⣽⣿⣿⣿⣿⣷⣭⣽⣷⣶⣾⣿⠟⣻⣿⠿⡿⣟⣟⣭⣉⣥⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣛⣷⣂⡀⡀⠒⠒⠒⠒⠒⠒
⠀⠀⢀⣬⣭⣁⡈⠉⠙⢿⣿⣴⣶⣶⣾⣿⣻⣿⣿⠿⠿⠴⠿⣿⣏⣁⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⠶⠿⣿⣿⣿⣿⣭⣤⣤⣤⡤⠀⠒⠒⠂
⠐⠛⠚⠛⠛⠿⠿⠿⠿⠿⢿⡉⠉⠉⠉⠒⠛⠛⢛⣿⣿⣿⠛⠋⣭⣿⣿⣿⣿⠾⠶⠶⠉⠉⠉⣙⣻⣟⣛⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠈⠛⠲⠶⠶⠒⠀⣀⣄⣉⡙⠛⠛⠛⠃⠀⠀⠘⠿⠿⠖⠂⣓⢻⣿⣿⠁⠀⠠⠤⠤⠤⠤⠤⠴⠒⠒⠒⠚⠛⠛⠃⢀⣤⠤⢤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠒⠀⢀⣤⣤⣄⣈⡉⠉⠉⣿⣯⠁⠀⠀⢠⣤⣀⣀⠀⠶⠶⠚⠋⠉⢡⣤⣄⣀⣀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣶⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠲⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢷⡦⠤⠤⠤⠀⠀⢀⡀⠀⠘⢆⠀⠈⢆⠰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠉⠓⠢⠄⣀⠀⠀⢨⠁⠒⠮⢧⡀⠀⠓⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠓⠒⠒⠛⠛⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠑⣄⡤⠞⠁
⠀⠀⠀⠀⠀⠠⠒⠉⠀⢀⡠⠴⠒⠛⠛⣉⣉⣙⣛⣓⠲⠮⢭⣟⣳⠶⢦⣤⣄⣀⣀⣀⣀⣠⣾⡷⠖⠛⣉⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠀⠀⠀⠉⠉⠙⠓⠲⠭⣍⣓⠲⠦⠭⢭⣭⣭⠤⠤⠶⠚⠋⣁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠒⠒⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⡟⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⢸⠇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡏⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⠃⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⢃⣿⠃⢰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⢻⣟⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠶⠾⠛⠛⠛⢛⣛⣛⣛⣛⣛⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣛⣉⣉⣥⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡟⠉⢻⡟⠿⠿⠿⠻⢿⠻⠿⠿⢿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠰⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣶⣦⣾⣿⣿⣿⣷⣤⣦⣤⣤⣠⠂⠀⠀⠀⠀⢀⡄⠀⠀⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⡀⣀⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠤⠶⠾⠿⠿⢿⠟⠀⠀⠀⣟⢀⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⢶⣾⣿⣿⣿⣿⡇
⠀⣦⠄⢀⣀⣀⣀⠀⠠⣶⡦⣤⣤⣀⣀⣀⣀⣀⡐⠒⠛⠛⣏⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⢻⣿⣿⡿⠄⠀⠀⠀⠀⢤⢰⡆⣸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠞⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣿⡿⠓⠠⠾⠐⣿⡇
⠀⣿⠉⢻⣿⣿⣿⣿⣿⣿⠓⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⠀⠀⠤⠶⢾⡇
⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣟⣛⣻⣿⣿⡿⠿⣿⡿⠿⢿⡿⠿⢿⡿⢿⡿⠿⠿⣿⠿⣿⣿⣿⣿⠆⠀⠀⠀⠰⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠉⠛⠃⠀⠠⠭⠅⠈⢩⡟⠉⠉⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠄⡄⢀⣄⣠⣏⣸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⢺⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣦⣤⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⡀⢨⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⢀⣄⣀⣿⣟⠛⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠳⠖⠀⠀⣈⣯⣌⡙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣧⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣯⣤⣌⣉⡉⠛⠛⠛⠛⠛⠻⠿⠷⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣭⣽⣿⣛⣿⣿⣿⡟⠶⠶⠶⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⢠⣀⡄⢷⣾⠾⡅⠀⠀⠀⠀⠀⠀⠀⢚⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⡇⠀⠀⠀⠄⠀⠀⢀⠎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢴⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⠤
⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣦⣤⣇⣀⠄⠀⠀⢀⣼⣿⡿⠀⠀⢀⣀⠄⠄⠐⢻⣿⣿⢑⣾⠄⠀⠒
⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠭⣖⣊⡭⠽⢟⣿⣹⣿⣿⣷⣶⣾⣿⣛⣁⠀⡁⠤⠠⢖⠒⢀⣈⣻⠃⠘⡁⠀⠀⠒⠀⠤⠄⠁
⠀⠀⠀⠀⡀⠬⠢⠴⠒⠤⠤⢽⣿⣿⣿⣷⣲⣶⣦⡤⠤⡤⠤⢤⣄⣀⣀⣈⣨⣑⣮⣧⣿⣶⡿⠤⠀⢱⠀⠌⠉⠡⠤⠤
⠀⠉⠁⠭⠿⠶⠶⠶⣤⣤⣬⡍⠉⠙⠛⠻⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠑⠢⠄⢸⣄⣀⣀⠀⠤⠀⣀
⠀⠀⠀⠐⠚⠉⠓⠀⠀⠘⠛⠩⠟⠓⠂⠐⠓⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠛⠢⠴⠐⠒⠒⠉⠳⠀⢠⣀⣀⣤⡄⠤⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣿⣿⠿⠿⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣾⣷⣿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣤⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣧⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⣿⠀⠘⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣻⣆⡄⠀⠀⣿⡆⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠅⠀⠀⠿⠃⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣤⣠⡅⢀⣤⠀⠰⣀⣀⣽⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠟⣿⣿⢿⣯⣷⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠂⢈⣀⣠⣶⣿⣧⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⡼⠛⢰⣛⡩⠠⣶⣫⣽⣿⣿⣽⡿⠟⠉⠁⠀⠀⣼⣻⣿⣶⣯⡛⠓
⠀⠀⠀⠀⠀⠀⣀⣼⣧⣠⣼⣷⣾⣿⣿⠿⠟⠋⠉⠀⠀⠀⢄⠀⠃⠋⢨⡾⠛⠛⠃⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡙⣟⣛⠛⠋⠉⠉⠁⠀⣠⣠⣶⣶⣶⣶⡿⠿⠀⠀⣶⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⡟⢁⣴⣶⣿⢟⡕⠙⣽⡿⠿⠿⠧⠀⢺⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠿⠮⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⠀⢸⣏⣟⣽⣿⣽⣿⣽⣿⣯⣿⣫⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⣿⢸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢀⣿⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⢶⡇⣿⢸⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⠇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣋⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⠉⠉⠉⢹⡆⣿⢸⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡿⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⢸⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠈⢧⡀
⠀⠀⠀⠀⢠⡏⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⢸⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⣼⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⣀⡤⠶⠛⠒⠛⠓⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠶⠦⠤⣤⣀⣀⣀⠀⠀⣀⣤⣚⣁⣀⣀
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢀⡼
⢯⣤⣤⣤⠤⠤⠤⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠁⢸⡇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⣿⠀⠸⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡀⠀⠀⠀⠀⣿⠀⠀⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢴⡿⢴⠀⠀⠀⣿⡇⠀⠀⠀⠀⠣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠀⠀⠚⠁⠀⠀⠐⠀⠀⠑⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣴⣠⣶⢆⣠⠅⢀⣴⡀⠀⢂⣀⣠⣽⣾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⢿⣯⣿⢾⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠠⠀⠉⠀⢀⣥⣶⣿⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⣶⡆⣤⠶⠚⢉⣩⣻⢟⣿⣿⡟⢿⣿⣿⠿⠛⠋⠉⠉⣹⢿⣿⣫⣔⡲⢶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⣷⣒⣩⣤⣾⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⢠⠀⡴⠻⣺⣿⣿⣟⢏
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣶⣶⣿⣿⡿⠿⠟⠛⠉⠀⢀⠀⠀⠀⣀⣥⠄⠀⠈⣀⢰⣟⠀⠀⠈⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠤⣄⣶⣤⣀⣀⠀⠀⠀⠀⢀⣠⣦⢾⣿⣿⣿⣿⣟⡛⠁⠀⣰⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⡿⠉⣠⣬⣿⣿⠟⡡⠋⢡⠾⠿⠛⠻⠷⠢⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣝⠿⡷⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣶⠒⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣠⣿⣶⣷⣾⣿⣶⣿⣶⣿⣾⣷⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢺⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⢻⠀⣀⡼⠀⠀⠀⠠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⡀⣿⣤⣬⣯⣦⣥⣧⣄⣢⣽⣶⡶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣟⣠⣼⣿⣿⡏⣠⣤⣶⠿⣟⣿⣿⣿⣿⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣟⣿⠟⠉⠀⢰⣿⣿⣿⣿⡏⢻⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤⡀⠀⠀⢉⢿⣿⣿⣿⣹⠇⢀⣤⡾⠟⠉⣼⣿⣿⢀⣼⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⡿⠀⢀⣰⡿⢛⠁⠈⣩⡶⠋⠁⣀⣸⣿⣿⣿⠇⣾⣿⡯⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣼⣿⣷⡾⢛⡅⢸⣿⣿⠛⠁⣴⣾⣿⣿⣿⠟⠛⣿⣷⣿⣿⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠍⣽⣿⣎⣾⣿⣿⣾⣿⣭⣅⣠⣼⡟⠛⠁⣀⣅⣴⣿⣿⣿⣿⣿⣦⣀⠀⠖
⠀⠀⠀⠈⢿⣦⣵⣦⣀⣤⣀⣶⣶⣦⣌⠙⠿⣿⣿⣤⣿⠏⠙⠻⢿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠃⠀⠙⠛⠋⠻⣿⣯⣅⣀
⠀⠀⠀⠀⠀⠉⠋⠉⠿⣿⠿⣿⢿⣿⣿⣿⡿⠿⢿⣿⡿⠟⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣋⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⠁⢹⡿⠛⡟⢻⣛⡟⣿⡿⡻⣿⢻⡟⣿⣿⢛⢿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠂⠒⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠰⡧⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢢⡄⠀⠈⠀⢀⡇⠀⣀⠠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢍⣭⡝⢛⣒⢦⣈⣀⡀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠛⠀⠄⢠⡄⢈⣑⣠⣴⠎
⠀⠀⠀⠀⠀⠠⠤⠶⣤⣤⣤⡀⣀⣀⣤⣀⠘⣿⣿⣾⣿⣶⣬⣭⣽⣏⣉⣾⣀⣐⣃⣀⣶⣤⣤⣽⡶⠶⠛⢻⡟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⢐⣶⣶⣶⣮⣿⣟⣻⣿⣿⣿⣿⠿⣿⣿⠫⣿⣿⣧⠖⢦⣭⣭⣭⠨⢭⣥⣶⣢⣤⣿⣟⠛⢻⣶⣶⠏⢀⣒⠂⠀⠀⠁
⠀⠀⠉⠀⠀⠚⠤⠀⠙⠛⢛⡛⠡⠐⠺⠿⠶⠦⣀⣬⠛⠛⠿⠿⣒⡾⠛⠉⠠⣾⣿⣥⢂⠀⢩⡍⠀⣉⠉⠒⠂⠉⠉⠀⠀⠘⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠲⠂⠘⠛⠛⠻⠤⠀⠀⠐⠒⠀⢲⡀⠉⠉⠩⠤⠄⠀⠀⠉⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣭⣭⣽⣤⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠷⡶⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡇
⠀⠄⢤⠶⢤⡀⢠⣷⡤⢴⡆⣸⣇⣠⣤⡄⣐⣀⣠⣀⣠⣀⡀⡏⡀⠀⠀⣠⣰⡞⢁⠀⠀⠀⡄⠀⢀⠀⢸⣖⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠠⠹⣀⣼⣤⣬⣿⣦⣌⣤⣹⣯⣬⣭⠉⠿⣿⠙⣩⣹⣿⡗⢭⣶⣶⣖⣾⠻⣿⣿⣤⡼⠯⠷⣶⢸⠂⠛⠛⠠⠀⠤⠀⠀⡀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⣭⡇⠀⠀⠉⢙⠋⠉⠋⠁⠿⡟⠛⠛⣿⠒⠾⢿⣿⠟⠛⠛⢛⣛⣿⣿⣿⣿⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠂⠠⠴⠄⠠⠄⠀⠈⡈⠈⠀⠀⠀⠀⠀⠀⣄⡀⡤⠌⠙⡟⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇⠀⡇
⠀⠀⣼⢻⠂⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡤⠀⠐⠇
⠀⠤⠾⢿⣯⣭⡁⠈⠁⠀⠀⠀⠂⠀⠀⠀⣶⡶⣄⢰⣶⠦⠀⠀⠀⢠⣤⣧⣀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⣻⡎⠀⣀⣀⣀⣀⣀⣀⣤⡤⣤⣤⣶⣲
⠀⠀⠀⠀⢼⣿⡟⡉⣠⣀⣄⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣦⣀⡀⠀⠀⡈⠉⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠿⠿⢭⣭⣶⣶⠿⠿⠿⠿⠿⠛⡟
⠀⠀⠀⠀⠈⠛⠻⣤⣾⠿⢿⡾⠚⠛⠉⠉⠀⢴⠟⠛⠛⠉⠉⠁⠈⠉⠃⡤⠀⠀⠀⢀⣠⣿⣿⠿⠟⠓⠀⠀⠀⣀⡀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣶⣶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣴⣾⠯⠉⠛⠛⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠫⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⡟⠛⠿⠟⠋⠀⠀⠀⠀⠀⢻⡏⠉⠁⠀⠈⠉⠀⠐⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠛⠩⢤⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠩⠽⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣯⠐⢤⡀⠀⠀⠀⠀⠀⢀⣈⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⣿⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⡏⡃⠀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⡇⡇⠀⣠⡾⠋⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢸⡇⢡⣼⠏⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡠⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⢀⡔⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⡠⢊⠔⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢀⠜⡡⠊⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡴⢁⠞⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⡰⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡎⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠄⠀⠠⠤⠤⠤⠤⣼⡿⠤⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠛⠛⠛⠛⣉⣡⣤⠤⠤⢭⣭⣭⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠈⠉⡉⠈⠉⠉⠉⠋⡉⠉⠑⠉⠐⠉⠒⢉⠒⢈⠐⢊⠑⠊⠐⠉⠑⠉⠉⡉⠈⠉⠉⠉⠈⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⢨⣁⣄⢀⣈⣉⠈⠁⢀⣀⣀⣀⣉⡁⠉⠀⠉⠁⡉⣁⣀⣀⣀⠀⠉⢈⣉⣀⢀⣄⣩⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠤⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠆⡀⠁⠀⠀⠀⢄⠀⠰⣄⠀⠀⠀⢀⣀⡀⠔⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠃⠀⠁⠀⠀⠀⠐⢄⡑⠂⢈⣂⣠⠍⠊⠁⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠉⠉⠂⢐⡛⣀⠐⢄⠀⢀⡠⠴⣶⡾⠋⠉⠛⠉⠀⠀⠀⣀⠠⠔⢊⠕⠁
⠀⠀⠀⠀⠀⠸⣅⡀⠀⡠⠤⠂⠁⢂⣈⣤⢔⠊⠡⠖⠂⠉⠁⠀⠀⠀⣀⡠⢴⡲⠭⣒⠤⠚⠁
⠀⠀⠀⠴⠊⠁⠀⣀⠄⠁⣠⠖⠉⠉⠑⠒⠉⠀⠀⢀⣀⠠⠄⢒⠈⠍⣂⠬⠐⠊⠁
⠀⠀⠀⠀⠀⠠⠚⠁⢠⠞⠁⠀⠀⠀⠂⠀⠤⠒⢈⠁⠀⣠⠔⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⢀⡠⠄⠀⡠⠊⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠊⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠐⠢⢄⡈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠑⠒⠤⢀⠉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⣀⠀⠀⠀⠈⠒⠠⢀⠀⠑⠠⣀
⠀⠀⠀⠀⠸⠆⠀⣧⠀⢀⠀⠉⠒⠤⣀⠀⠀⠀⣈⠒⠄⢀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⢢⡀⢛⠒⠄⠁⠉⠁⢀⣈⡀⠀⠀⢅⠀⠠⢬⠌⠒⠤⡀⠐
⠀⠀⠀⠀⠀⠘⣇⠀⢄⠀⠁⠀⠑⢄⠀⠙⠦⣀⠀⠀⠂⢄⠀⠀⠀⠀⠑⠢⣀⡀⠤⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⡉⡐⠌⢦⡀⠀⠀⠑⢄⠀⠈⠳⢤⡀⠀⠀⠀⠀⣀⠀⠀⠌⠁⠤⡀⠁⠢⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠑⡌⠛⣀⡤⢀⡐⠄⠀⠒⠀⠀⠀⠉⠁⠀⠀⠀⠊⠀⠀⠀⣀⣠⡀⠑⠢⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠙⠿⣭⡀⡀⢄⠀⠲⠄⠀⠀⠀⠢⡑⠀⡀⠀⠀⠉⠉⢄⡀⠀⠀⠁⠢⠄⠈⠨⢦⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠙⠛⠦⣦⣄⠀⠀⢄⠀⠂⠈⠑⠄⠐⠠⣀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⡌⠀⡌⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠉⠙⠒⠤⣶⣄⡀⠀⠀⠀⠀⠈⠀⠀⠸⠀⢀⠁⠀⠀⠀⢰⠁⠰⠀⢀⣀⡀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢦⣄⣀⠀⠀⠀⠀⠈⠀⠤⢀⣤⣶⣶⡯⠶⠾⠭⠶⠶⠶⠒⠰⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢤⣴⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⣈⠑⠢⣄⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⡑⢄⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋



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

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣄⡀⢈⣛⡛⠿⣿⡿⠛⠛⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣽⣿⢿⣿⣿⣏⢻⣷⣶⣿⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠘⠿⠲⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣠⣿⢷⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣧⢀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠿⠃⣸⡿⠿⠟⠹⠗
⢸⣿⣿⣿⣿⣾⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣬⣿⣿⣟⠿⡿⠛⢻⣿⣿⠀⠀⠁
⢸⣿⣿⣿⣯⣽⣀⡈⠏⠉⠀⣤⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⡦⠤⢴⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠤⢤⣿⠇⢀⠀⠀⠀⠄⣀⣀⣀⣴⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⡿⠽⡿⠿⠛⠛⠉⠉⠉⠉⢸⡿⠀⡈⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣾⣇⣶⡇⠠⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣉⣀⣀⣸⣃⣼⣿⣿⣿⣹⣽⣯⣽⣿⣿⣿⣿⣯⣽⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠉⠉⠉⢩⣉⣉⣭⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡀⡀⢠⣐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣧⡀⠀⠀⠲⣟⣻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣂⣿⣿⣽⣟⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⣧⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣽⡄⠀⢿⡿⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠔⣿⠀⠈⢇⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣿⡇⠀⠈⠂⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣠⣀⠀⠀⠀⠀⣀⣀⠀⠴⠇⠀⠀⠀⢸⣧⠀⣾⣿⣷⠀⠀⠐⡄⠀⠀⠈⣿⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠟⠋⠹⣦⠀⠀⠹⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⣿⣷⡀⠀⠀⣠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢿⣇⣷⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿⣿⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇⠀⡄⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠀⢰⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣬⣟⣿⣛⣛⣻⠿⠿⠷⣤⣤⣠⠀⠀⠀⠀⠀⣀⠜⠀⠀⢀⠄⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠛⠛⠶⠶⠦⢦⣤⣾⡏⠀⢀⣴⣿⣶⣶⣿⣄⡀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠉⠉⠛⠿⢿⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⢠⣿⣯⢴⣤⢀⣘⣅⣀⠀⢀⠀⡀⠀⠀⠈⠉⠀⠀⣀⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⢠⠀⠦⣤⣤⣾⣿⡿⠿⠿⠿⠍⠉⠋⠀⠉⠉⢉⣭⣬⠩⣭⡭⣩⣿⣟⣡⣀⣶⠶⠶⠟⢻⢽⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠒⡐⠒⠒⠚⠛⠋⠿⠟⠉⠛⠛⠁⠀⢈⣁⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡛⠛⣿⣿⠛⠿⠷⠶⠖⠒⠒⠠⠤⠤⠤⠀⠄⠠⠀⠀⠀⠀⠼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠻⣟⠿⣿⡿⠦⠔⠛⠛⠓⠆⣠⣤⣶⠶⠖⠒⠒⠒⠶⠶⣶⣶⠾⠛⠉⠉⠁⢲⡴⠦⠰⠦
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢋⣉⠡⠤⠀⣀⣀⣀⡀⣉⣉⣉⣀⣀⠠⠄⠂⠀⠀⢈⣉⣤⡴⢶⣚⣋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠛⠛⠛⠓⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒



⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠁⠁⠃⠀⠀⠀⠀⠒⠀⠃⠀⠀⠀⠘⠀⠃⠀⠀⠀⠚⠁⠀⠀⠀⠐⠂⠀⠀⠈⠚⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠑⠒⠒⠒⠒⢸⣿⠀⠤⠤⢹⠀⠀⠀⠀⠀⢀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣳⣄⣀⣸⣤⣤⣶⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠒⠀⠀⠀⠀⡀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢟⡻⢿⣇⠀⠀⠀⠀⠀⠠⠤⣀⠀⠠
⠀⠀⠀⢀⣀⣉⣩⣵⣶⣶⣾⣿⣭⣿⣭⣿⣿⢿⣿⢿⠿⣿⣛⣛⣯⣥⣤⣴⣾⣿⣿⣿⣿⣲⣄⠀⠀⠶⠶⠶⠶
⠀⠀⣤⣭⣅⣉⣉⣻⣷⣶⣶⣾⣟⣿⣿⣿⡷⢾⣿⣯⣥⣭⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣯⣥⠤⠤⠄⠑⠒
⠈⠉⢉⣉⣙⣋⠉⠛⠻⢤⣀⣀⣩⠉⠉⢻⣿⣏⣉⢸⣿⣿⡿⠙⠛⠋⠀⠀⠚⠿⢿⣛⣿⣿⡆⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠉⠁⠀⠈⠛⠲⠦⠤⣤⠄⠀⠀⠈⠁⠐⠒⠺⣿⡇⠀⠈⠉⠉⠉⠉⣩⣥⠴⢀⣄⡀⠀⠉⢉⣉⡉⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠛⠁⠀⠀⠈⠉⠉⠑⠂⠀⠀⠀⠀⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠟⠻⠛⡻⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠚⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⢀⢟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠡⠀⠐⠒⠀⢄⡀⠀⠀⠀⢀⢰⠀⠆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢛⢸⠀⠘⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢃⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⠀⠀⠘⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠁⢨⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢠⠃⠈⢸⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢸⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠸⢘⠀⡌⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⢀⡇⠀⠀⠀⠀⣀⡀⠤⠤⠄⠀⠒⠒⠒⠀⠠⠤⠄⣀⡱⣈⠈⣞⡠⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠂⠤⣀⠀⠀⠀⠀⢳
⠀⢰⢀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠈⡄
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸
⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢀⣀⠠⠶⢴
⠰⠊⠤⠀⢠⣄⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣄⠀⠤⠤⠄⠐⠒⠂⠉⠉⠀⠀⠀⡰⠁
⠀⢡⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠉⠒⠲⠦⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠰⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣷⢄⡀⠀⣿⠀⠀⠸⣿⡆⢀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢷⠘⡄⠙⠲⣿⡀⠀⢀⡟⠍⠀⠑⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⠀⢒⣿⠁⠀⢨⠀⠀⠀⠀⠉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⡇⠀⡸⠀⠀⠀⠀⠐⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀⡿⡎⣦⠁⠀⠀⠀⠀⠀⠀⠽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢸⣿⣇⠇⠀⠀⠀⠀⠀⠀⠀⢻⡿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡷⠋⠀⠀⠀⠀⡁⣸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡀⠀⠀⠀⠀⠀⠀⠰⡒⠮⠢⠂⠘
⠀⠀⠀⠀⠀⠀⠀⣨⡅⠀⠀⠀⠀⢀⣈⣹⣿⣧⡤⢄⣀⠀⠀⠀⠀⠀⢺⣾⡀⡇⠀⠀⠀⠀⠀⣫⡤
⠀⠀⠀⠀⠀⠀⣴⡿⠀⣀⠄⠁⠀⠀⠉⢹⣯⣥⣮⣟⢛⣑⣨⡁⢀⠀⢸⣿⣦⣠⠀⠈⣡⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠸⠝⢻⢟⣀⡒⠀⠒⠒⠛⠻⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⣩⢝⣉⣵⣶⣿⣿⣿⡿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⡏⠉⣿⣶⠂⢀⣠⣤⣿⣾⣿⣿⡿⠟⠙⠉⠻⠃⡠⠃
⣀⣄⣀⡀⠠⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣾⠟⡂⣰⡆⢘⣿⡇⢀⣄
⣿⣿⣿⠿⠿⠀⠑⠓⠄⡠⣦⣄⡘⠻⣿⣿⣿⡿⣿⣿⡿⠟⣿⣿⣿⣄⠉⣈⣱⣶⣿⣿⣿⡃⢸⡿⠛⢸⣿⣷⣤⣄⣀⡤⠐⡂
⠉⠉⠉⠑⠂⠀⠀⠘⠷⣦⣤⣈⠙⠻⢿⣾⣶⣿⡿⢿⣷⣾⣿⠿⠛⠻⠿⣿⡟⠿⠿⠿⠿⠃⠘⠃⡀⢸⣿⣿⣎⠛⠿⢿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠦⠄⠈⠁⠈⠙⠳⠶⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡋⠀⠘⢷⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⡁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠒⠒⠒⠶⠦⠤⠤⠤⢤⡄⣴⡆⢀⣀⡀⣿⠉⠉⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⢹⠀⣀⣀⣠⣴⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣀⠄⠀⢠⠤⢤⣄⣂⣀⣹⡿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢉⣡⣵⣶⣭⣿⣃⡀⠀⠀⠀⢀⣈⣙⡒⠀⠉⠉
⠀⠀⠀⠤⠴⠶⠾⢷⣿⠟⣛⣛⢿⣿⣿⣿⣿⣿⢋⣿⣯⡭⠽⠿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠤⠀⠀⠈⣉⣉⠁
⠤⢤⢶⣿⣶⣶⣴⣾⣽⡿⠿⠿⠟⣫⡿⣿⣿⣭⣭⡽⠿⣶⣾⣿⣿⣿⣟⡻⠿⠿⠿⢯⠽⢿⣿⣿⣿⠛⠛⠛⠀⠀⠀⠁
⠀⠀⠰⠤⣬⣥⠄⠀⣉⡙⠲⠶⠴⠶⠀⠀⠻⢿⣷⠤⠌⠿⣿⣿⠋⠈⢉⣀⣀⣀⣉⣉⠭⠭⠭⠿⠿⠄⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠶⠆⠀⣀⣀⣀⣀⠉⠛⢻⣿⠁⠀⠀⣀⣀⣀⠠⠤⠴⠖⠒⣰⣂⣀⣈⣀⣀⣤⡈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣣⣿⣧⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢼⡇⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⢸⡇⣷⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠛⠋⠉⠉⠉⠉⢹⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡿⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⣿⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⣿⠀⠀⢀⣴⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⢇⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣿⠀⣰⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣷⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇⣿⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⣀⣬⣿⠦⠴⠶⠤⠤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣤⣀⣀⣀⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⠶⠶⠶⠿⠓⠒⠛⠛⢳⡆
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠻⠶⠴⠶⠶⠒⠒⠛⠓⠒⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⢸⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⢀⡴⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⣴⠋⢀⣴⠒⢺⣟⣛⣻⣉⣯⣉⣏⣩⣏⣹⣿⣓⡖⢦⣄⣀⣠⣤⡶⠶⠿⢻⠛⠛⡟⢉⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣇⠀⢻⣴⣫⢿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⡼⡿⠛⡟⠉⠁⢰⠀⣀⣀⣼⣤⣴⠿⠿⠿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡰⣿⠟⠁⢸⡆⠀⠀⠀⠀⢀⣠⣤⠶⠟⢛⣉⠀⠀⣇⣠⡧⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣾⠟⠁⠀⠀⠀⣇⣀⣤⡶⠟⢋⢹⠠⣀⣠⡿⠶⣛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⢀⣠⣶⣿⠋⠉⣆⣠⣤⠾⠛⠋⢉⠡⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⣠⣴⢾⠋⠁⢐⣿⡴⠞⠋⠉⠠⣐⠀⠀⠀⠁⠉⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢘⣧⡶⠛⡏⠀⣸⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠋⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠰⣟⣿⢰⣠⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣁⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡼⠄⢀⡾⢠⡟
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠥⠩⠑⢁⡞
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⣠⣐⣭⣿⣤⡽⣶⣶⣮⣍⣉⣉⣙⣦⣀⣖⣍⣉⡉⣳⡤⠒⠂⠉⠉⠁⠐⠂⣄⣀⣤⡷⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠈⠀⠘⠓⠉⠉⠉⠉⠉⠉⠙⠋⠙⠋⠛⢛⣻⡿⠟⠛⠟⡻⢭⣵⣤⣦⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣽⠃⠀⢀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⣾⠃⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣼⡟⠀⠀⠀⠆⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣿⠃⠀⠀⢸⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⣸⡃⢀⢠⣿⠀⠀⠀⡌⠀⢸⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠸⣿⡄⠀⠀⠃⠀⡀⢸⣇⢿⡄
⠀⠀⠀⠀⠀⢰⣇⡀⠀⢴⣿⣇⠀⢸⠀⢀⠈⠙⣿⠸⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠘⣿⣿⡄⡆⠀⠸⠀⢙⣿⠈⢻⡇
⠀⠀⠀⠀⠀⢸⣃⠀⠀⠀⢛⣿⣿⡁⠀⠇⠀⠘⣿⡄⠸⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠤⠐⠂⠀⠙⣿⣷⣠⠀⠀⠀⣹⣇⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⣿⠆⠉⠀⡀⢀⠘⣿⣿⠀⠀⣀⣿⡏⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠸⣧⠄⠀⠉⠻⣗⣈⣿⣧⠀⠻⢿⣿⡁⠀⠸⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣶⡀⠀⠀⢀⣩⣽⣿⣧⠀⢸⣿⣿⣆⣀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⠀⠀⣈⣿⣿⣿⠿⠓⢠⡿⠿⠿⠿⠿⠿⠿⠦⢄⣠⣤⠀⡀⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣩⣅⣤⣠⣤⣧⡀⠀⠴⢿⣿⣦⣤⣾⣿⣇⣀⣧⣤⣤⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠭⣉⠛⠟⠛⣿⠿⠯⣽⡿⢷⣶⣼⣿⣯⣭⣭⣽⡭⠿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣤⣽⣶⣶⣥⣶⣾⣶⣦⣴⣦⣼⣿⣿⣿⣿⣿⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠸⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡟⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⠃⠀⠀⠀⠀⠀⢀⡾⢃⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡿⠀⠀⠀⠀⠀⣠⣿⣴⠿⠋⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⢿⡇⢀⡎⠀⣠⣾⠿⠋⠁⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⡿⢠⡾⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⣿⠋⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⣿⠇⢰⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣠⢿⠏⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠐⢡⡏⠀⠞⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢠⡟⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢠⡟⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢠⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⢁⣠⡾⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⡀⢀⡾⠃⠀⠀⠀⠀⠀⢀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣀⠄⠀⠀⠀⠀⠀⢀⣷⡾⠁⠀⠀⠀⠀⠀⠀⠀⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣛⣁⣀⣤⣤⠴⢶⣾⣟⣉⣀⣀⡀⠀⠀⠀⠀⠀⣀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⣠⣶⣿⠿⠿⠛⠛⠛⠉⠉⠀⠀⣠⠟⠉⠛⠛⠛⠛⠓⠚⠛⢛⡛⢿⣯⣥⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠉⠛⠓⠶⢶⣶⣾⣷⣶⣿⡿⠿⢿⡉⠉⢿⡉⢉⣟⠘⠷⣤⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣛⠛⠻⢿⡿⠿⠿⠿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠉⠢⠉⣛⣻⣷⣶⣦⣤⣤⣀⣈⡛⠓⠶⠶⠶⠤⠤⣤⣤⣤⣤⣤⣍⠈⠛⢀⣀⣤⠇⠀⢀⡄⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠉⠉⠉⠙⠛⢿⣿⣟⡛⠛⠻⠷⠶⢶⣶⣶⣶⣶⣶⣶⣶⣞⣛⣋⣉⣀⣠⣴⠞⢀⠠⠖⢛⣩⣥⣴⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⠁⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠋⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠟⠛⠋⠁⣀⣠⣶⠾⠟⠋⠁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⣀⣀⠀⠀⠀⠈⠻⢿⣟⡳⠶⠞⠃⠀⢀⣸⡟⠈⠛⠻⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣷⢄⠀⠀⣿⠀⠀⠘⣿⡦⢀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⡀⠙⠢⣿⠀⠀⠀⡟⠀⠀⠑⢄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⣿⠁⠀⢠⠀⠀⠀⠀⠈⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠘⠀⠀⠀⠀⠀⠉⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⢰⠀⡃⠀⠀⠀⠀⠀⠀⣻⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⣦⠁⠀⠀⠀⠀⠀⠀⠭⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠇⠀⠀⠀⠀⠀⠀⠀⢨⡵⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡦⠂⠀⠀⠀⠀⠀⢸⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⢲⣳⠈⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠀⢈⣸⣟⣇⡠⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠇⠀⠀⠀⠀⠀⢨⣤
⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⠄⠀⠀⠀⠉⢹⣯⣥⣊⣟⢛⣐⣀⠀⠀⠀⢸⣿⣦⡀⠀⠈⣁⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠸⠝⣰⢊⡀⠒⠐⠒⠒⠙⠺⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⣨⠍⣉⣥⣶⣿⣿⣿⡿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⡍⠉⢽⣶⠂⢀⣀⣤⣿⣾⣿⣿⡿⠟⠙⠉⠻⠃⡠⠁
⣀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣶⠗⠂⣰⡄⢐⣿⡆⢀⣄
⣿⣿⣿⠿⠿⠀⠁⠒⠀⠠⣦⣄⡘⠻⣿⣿⣿⣿⣿⣿⡯⠟⣿⣽⣿⣄⠁⣈⣡⣶⣾⣿⣿⡇⢸⡿⠛⢸⣿⣦⣤⣄⣀⡠⠀⡀
⠉⠉⠉⠑⠀⠀⠀⠘⠷⣦⣄⡈⠙⠻⢷⣶⣶⣿⡿⣿⣶⣾⣿⠿⠛⠿⠿⣿⡛⠿⠿⠿⠿⠃⠘⠃⠀⢸⣿⣿⡎⠛⠿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠆⠀⠈⠁⠈⠙⠳⠶⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡂⠀⠘⢷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠛⠒⠦⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠶⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⢹⡜⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣿⠀⠈⣇⠸⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣿⠀⠀⢹⠀⢹⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣿⠀⠀⢸⡄⠀⢧⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡄⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠘⡇⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣿⠀⠀⢸⠁⠀⠀⡇⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⠀⡇⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⣸⠁⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣇⣿⢰⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣀⠀⠀⠀⣿⣷⣯⡤⠤⣤⣀⡀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⣾⠭⣵⣶⣶⡿⠭⠷⠿⠒⠒⠚⠛⠋⠉⣉⣳⣿⣿⣿⡶⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠉⣉⣀⣀⣀⣤⣤⣤⣴⣶⣶⣾⣿⡿⠿⠟⠛⠉⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⣁⣀⣀⣠⣤⠤⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣤⣤⣤⡤⠤⠤⠤⠴⠶⠶⠖⠒⠚⠛⠋⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣿⣿⡆⠀⠀⣟⠓⠂⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⣼⣿⣿⣿⡿⣥⡀⠀⠈⠫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⣿⡿
⠀⠀⣼⣿⣿⣿⣿⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢀⣿⣿⣿⣿⡟⣽⣿⣿⣇⠀⠀⠀⠀⢠⡴⠟⢷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡄
⢰⣀⣀⣤⣼⣤⣼⣭⣭⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⡟⢁⠀⠀⠀⠀⠀⠘⢷⣶⣶⣶⣶⣷⣿⣷⣤⣴⣶
⠈⢙⣋⣉⣿⣉⣛⣉⣝⣋⣙⠉⠀⠀⠀⣓⣫⣝⣛⣍⣛⣫⣑⣊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤
⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠛⠛⠋⢩⠟
⢠⣄⡀⣿⡟⠁⠀⠀⣽⠀⠀⠀⠀⣀⣤⡄⠀⠀⣸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣄
⠀⠙⠛⢿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠀⠀⠀⡟⣿⣾⠀⠀⠲⣶⣶⣤⣴⣶⣶⣴⣿⣿⣶⣶⣶⣶⣶⡶
⠀⠀⠰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⡁⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣇⣀
⠀⠀⣾⣿⣷⡀⠀⠀⣤⣭⣥⠀⠀⠀⠀⠀⠬⠴⠮⠭⠴⠮⠤⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠙⠛⠟⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡦
⠀⠀⠀⠀⢘⡃⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣐⣚⣍⣓⣚⣉⣚⣀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⣠⡆
⠀⠀⢀⣴⣿⣿⣦⡀⣿⣿⣿⠀⠀⠀⠀⠀⠈⢁⣀⣉⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣾⣿⡤⠴
⠀⠀⠀⣿⡏⢹⣿⠀⣿⣿⣿⠀⠀⠀⠀⣀⣀⣰⣿⣭⣿⣿⣿⣿⡆⠀⠀⢲⣶⣶⣶⣶⣯⣥⣤⣤⣭⣥⣤
⣀⣤⣤⣿⣷⣾⣿⣀⣿⣿⣿⣧⣄⠀⠀⠈⠿⠿⣿⣿⣽⣤⣧⣤⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠛⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠄⠠⠤⠤⠤⠼⢦⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠎⠂⣀⣠⠤⠤⡀⠀⠀⠰⡆⡀⢐⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⣸⠀⠀⠠⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢻⠂⠀⠀⡄⣤⡥⠤⠄⠰⡄⡤⠤⢶⠄⠸⢦⠤⠰⠌⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⣾⠀⠃⢀⠇⠔⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠠⣀⣤⡅⢀⠠⠀⠀⠀⠠⠼⠿⠧⠅⠈⠄⠅⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡻⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣿⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠰⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠠⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣿⡜⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⣴⢀⣀⣀⡀⠂⠀⠀⠀⣠⡤⠷⠞⠉
⠀⠀⠀⠀⠀⠠⠤⠴⣴⣀⣀⣀⣀⣀⠀⠀⠀⠈⠩⠏⠁⢽⡇⣥⡌⡀⠄⠀⠀⢀⢀⣹⡇⠈⣿⢿⣿⠃⠀⡀⣀⣀⣌⣩⣷⡒⠛⠈⢳⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⠛⠛⢹⠿⢿⡿⡷⠿⠧⠿⠶⠶⠿⠿⢿⡶⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡏⣩⠞⣁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠷⠀⣳⣶⠬⠽⣿⣯⣽⣭⡯⢿⣯⡭⠿⠟⠬⡭⠭⠭⠭⠭⠭⠨⠅⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣏⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣮⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⣷⣾⣟⠛⣒⣒⠂⠊⠀⣇⠀⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⠍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⣛⣛⣛⣛⣟⣛⣛⣛⣵⣿⣿⣛⣛⡉⢹⡏⠛⠛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⢶⣶⣶⣶⣾⣷⣶⡾⢯⣥⣤⣤⣤⣤⣤⡜⠧⠤⠤⠤⢬⢯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡇⠀⠉⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠘⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⠀⠀⠀⠀⢀⣨⠍⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠇⠀⠀⣠⢞⣯⢞⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣉⣹⣿⣿⣿⣿⡇⣿⣿⠏⢀⡴⣫⣿⣿⢋⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⠃⠀⣸⣿⣿⣿⣿⣿⣿⢣⣿⢋⡴⣫⣾⣿⣿⢃⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢀⣻⣿⣿⣿⣿⣿⢏⣾⣵⣋⣬⣥⣖⡒⠀⠈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⠃⣰⣿⠿⢿⡿⠛⢻⡫⢞⣫⣽⡿⣟⣫⢵⠃⣴⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠋⠉⠀⠀⢸⢷⣺⣵⣾⣿⣯⣷⣿⣿⣯⡏⢰⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣠⡿⠽⠧⠤⠤⠤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⢩⣶⣶⣿⣉⣭⢉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣰⣿⡟
⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⡽⠁⣰⣿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠟⢀⣴⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁⠐⠋⢉⣩⣭⣭⣉⣁
⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⢿⣿⣿⣭⣭⣤⣤⣤⡶⠶⠶⠖⠒⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡃⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣧⠠⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⠉⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠬⠖⠒⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠖⠚⠋⠉⠁⡏⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡇⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⣠⣀⠤⠔⢲⠈⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡥⠴⠒⠚⠙⡏⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢧⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠼⠀⠀⠐⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢢⡾⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣦⡤⡖⠉⢰⠟⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣇⠤⠴⠆⣻⠅⠉⣁⣤⡧⣎⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠐⠋⠉⢸⡀⠄⢈⣤⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣒⣩⣴⣾⣿⣿⡿⠿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢝⡻⠿⠟⢛⣻⣋⠻⢚⣷⡖⣿⣿⣭⠛⣿⣿⣿⠷⠶⣤⣶⣶⣖⡒
⠀⠀⠀⠀⠋⠁⠀⢀⠘⢋⣡⣽⣭⣽⣿⣧⠚⢻⣯⣴⣾⣟⣛⠻⢛⣫⣶⠈⠠⣙⣛⣛⠛⢿⣿⡿⠶⣿⣯⣟⡛⠡⡤
⠀⠀⠀⠀⠐⠛⠃⠈⠉⠭⣥⠄⠨⣿⣋⡉⠉⠤⠟⢉⣉⡉⠑⠂⢰⡿⣋⠀⠐⢶⣭⣿⣠⠤⠶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠘⠛⠻⣿⠷⠆⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠐⠒⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠶⢢⣴⠄⠀⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢸⡀⠀⢀⡀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⠢⣀⢸⡇⠀⢸⣿⡆⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠈⠆⠈⣻⡇⠀⢰⠟⠀⠀⠱⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⢸⠐⠀⠀⠀⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡈⠀⠀⠀⠀⠀⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠥⢧⠁⠀⠀⠀⠀⠀⢿⢏
⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠀⠀⣸⣯⡝⠀⠀⠀⠀⠀⠀⣸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢅⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣧⢰⠀⠀⠀⠀⢀⡄⠋⠁⠀⠁⠁
⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠐⠶⣿⠿⡚⣥⡤⠄⠀⠀⠀⢸⣿⡈⠀⠠⠄⢀⣤⡿⠁
⠀⠀⠀⠀⢠⡿⢧⠖⠡⠤⠤⠤⠦⣿⣿⣿⣿⠀⠐⠃⠀⠐⢚⡻⢛⣤⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⢻⠛⢻⣦⠀⢀⣠⣼⣿⣿⣿⠿⠛⠛⠿⠁⡸
⣤⣤⣄⡀⢀⠀⠀⢀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡵⠾⣂⣰⡆⣸⣿⠀⣤
⣿⣿⣿⠛⠁⠈⢑⡀⠿⠦⣬⣛⣿⣿⣿⣿⣿⣟⣻⣿⣿⣧⣀⣤⣾⣿⣿⣿⡅⣿⠋⢹⣿⣷⣦⣤⣤⣖⠅
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣄⡙⠻⠿⠿⢿⣟⣿⠿⠛⠉⠉⠙⠻⠍⠙⠛⠋⠁⠈⠀⠘⣿⣿⡇⠉⣿⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠄⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣲⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⣟⣵⣋⣁⣀⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢬⣙⠲⢤⣈⡩⠤⠶⠒⠒⠒⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⣹⢂⡇⣀⣀⣠⠤⢤⡄⡖⠒⢮⣳⡄⠀⠀⠀⠀⠀⣿⠤⠶⣈⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠾⣍⣓⠦⣤⠇⣼⢰⠁⢸⣿⠀⠀⢷⠹⣄⣤⠷⠙⢦⠀⠀⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢲⣟⣉⣀⣀⣀⣤⡬⠽⠶⣤⣇⡘⠒⠚⠉⣉⣉⣉⣠⣤⠤⠤⠤⠘⠳⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⣹⣋⠔⠊⠁⣸⠉⠉⡏⠀⠀⢱⡀⠀⠘⢦⢻⡉⠉⣉⣉⠉⣀⣀⠀⢠⠤⡀⢰⠒⡌⠳⣤⢼⣧⣤⡀
⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠻⣄⡀⠀⠀⠇⠀⠀⢃⠀⠀⢀⣇⣀⡠⠜⠊⢧⣀⣓⣚⠀⠓⠚⠀⠘⠒⠁⢈⣉⣀⣀⣬⠧⠔⠋⢻⡄
⠀⠀⢠⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣻⡿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⡉⣉⣭⣍⢩⠵⢦⠀⠀⠀⠀⢀⡹⣤⣀
⠀⠀⠈⠙⢿⣟⡛⠛⠛⠛⠛⠛⠋⠉⣩⣭⠤⠴⠖⠒⣦⠀⣀⣀⣀⢀⣤⣤⡄⣴⠶⠖⠒⠛⠛⡇⣏⣉⡇⢧⣤⠼⠘⠒⡚⡁⠀⣠⣤⣼⣷⠟⣸
⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⣀⣀⠀⠀⠀⠙⠲⣄⠀⠀⢘⡆⢳⣀⣸⡏⣧⡤⠇⠸⠶⠒⠒⠛⠋⠁⠈⠁⠀⢀⣰⣲⣿⣾⡶⠛⣟⢉⡤⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠉⠙⠛⠿⢶⣤⡀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣠⣶⡶⠶⢶⢿⣿⣿⠿⠧⠨⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡃⠀⠀⠀⠀⠀⠉⠓⠶⣦⣴⣿⠶⠶⠶⠶⠶⣿⡟⢿⣭⣽⣿⣾⡷⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⠞⠛⠿⠟⠂⠸⠦⠔⠶⠖⠲⠿⠝⠛⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⢶⣶⣿⠿⠿⠷⠶⠤⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢸⣿⣟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠻⣷⠶⠿⠶⠤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣸⡀⠉⠉⢉⣙⡛⠛⠛⠛⠻⠟⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣤⣦⣴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣄⣠⣍⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣄⣠⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠶⠲⠒⠒⠚⠓⠒⢲⣯⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⡟⠛⠓⠖⠶⠶⠶⠶⠒⠒⢶⣶⠞⠛⠒⠒⠒⠲⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⣭⣭⣭⣭⣭⣉⣉⡹⠿⣿⣿⣿⣿⣿⣯⣉⣩⣭⣭⣤⣬⣭⣭⣭⣭⣭⣭⡍⢉⢩⠉⠉⠉⠉
⣀⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣴⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠂⠈⠂⢇⡀⢀⣸⣷⣠⠀⡀⣠⣇⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣇⠘⠒⠇⡀⣀⢡⣤⡤⠀⠶⠌⠓⠛⠚⠛⠛⠘⠈⠉⠉
⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠈⢿⣿⣛⠛⠓⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⣆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠤⠤⢤⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⡄
⠄⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⡀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣰⣤⠀⠀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒
⠀⠀⠉⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠀⠈⠀⠀⠀⠠⠀⠁
⠀⠀⠀⡄⠀⢀⠈⠉⡁⠀⠀⡌⢁⠀⠀⠀⠀⠀⠀⠚⠓⠶⠶⠦⠄⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣐⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⠿⢛⠿⢿⢿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⢂⢀⣉⣀⣀⠀⠀⢐⡤⢀⣀⣀⣉⣑⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣥⠞⣻⣿⣿⣿⠟⢀⣾⣿⣿⣿⣯⠤⠤⠤⠈⠀⠀⣈⣒⡒⠀⠒⢒⡒⠂⠫⠉⢩⠉⠉⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠊⢐⣛⣛⣛⣉⠀⠉⠉⢡⠀⣠⣄⣶⣲⣆⣴⣶⡧⠤⠤⠤⠤⠤⠤⠤⠄⣙⣛⣓⣒⣦⠷⠖⠂
⠀⠀⣀⣄⠀⣀⡤⣖⠯⠤⠤⠛⠓⠒⠒⠛⠛⠉⠉⠉⠉⠉⢉⣀⣒⣒⣒⣒⡂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠋⠁⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠈⠉⠉⠙⠻⣦⣶⣤⣤⣠⣤⣤⣤⣴⠷⢶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠤⠀⠠⠤⠀⠀⠤⣄⣀⠠⣄⣀⡀⣀⣀⣤⣄⣀⣠⡀⢀⣀⣄⣀⣉⡷⠘⠁⠀⠀⠀⠈⠋⠉⠉⠻⣦⠤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⠀⠀⢀⡠⢔⠝⠉⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢸⣠⡖⣭⠊⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠕⠀⠀⠀⠀⢸⢹⣷⣤⣑⢤⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⣼⠘⣿⣿⣿⣿⣾⣶⣄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⡁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⢼⡄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠷⠾⣄⣀⣀⡀
⠀⠀⠀⢠⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⡿⢷⡄⡇⠀⠀⠀⠀⠀⠀⣶⣶⡒⢖⠒⡖⠂⠀⡤⠦⢤⠀⠤
⠀⠒⠛⠛⠒⠒⠒⢲⣦⣴⣶⣶⣤⣤⣴⣧⣼⣧⣯⣭⣉⣉⣹⣇⣀⣿⣿⣿⣿⣷⣿⣶⣶⣷⣶⣼⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠣⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⢀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⣀⠀⣿⠽⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡄⠀⠀⠀⡜⠀⠀⠠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⣬⣿⠉⣕⣦⣤⣀⣀⣀⣀⠀⢈⡇⠀⠀⠈⢀⠀⠠⣀⣱⣤⡶⠃
⠀⠀⠀⠀⣀⣒⡒⢶⣶⣦⣤⣤⣦⣄⣸⣿⣿⣿⣿⠶⠶⠿⠤⠿⠤⠼⠤⠾⠷⣶⣿⣟⣉⣉⣋⠝⣁
⠀⠢⠄⠀⣻⡟⠛⣿⣯⣽⠟⢛⣿⣏⣙⠛⠃⣿⣿⣿⣍⣻⣿⠿⠈⣶⣿⢗⠛⠿⠯⠀⠿⢿⡋⠠⠟⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠐⠶⠂⠨⠥⠀⣬⠉⠷⠿⢤⣀⠀⠐⣢⠀⢤⠈⠛⠛⠃⡀⠈⠁⠒⠂⠀⠠⠀⠀⠀⠀⠉⠈
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⢠⣀⡀⢱⣶⡦⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠠⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢈⣼⣄⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⣴⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢴⣿⣷⣾⣶⣤⣄⣀⡇⠀⠀⠀⠀⠀⢸⣶⣿⡇⠀⠀⠀⣀⢀⠀⠭⣶⣾⡟⣬⡇
⠀⠀⠀⠈⠉⡤⠀⠠⠄⠈⢻⡯⠿⣛⣿⣿⣿⣿⣿⣶⣦⣤⣼⣿⡿⠟⠐⢐⢀⣉⣡⢤⠔⠀⢻⡿⠁⠟⠁⠀⠀⠄⠀⣀
⠀⠀⠀⠀⠀⣐⢀⣊⡭⣀⣁⣹⣿⣷⣾⣿⢭⣍⣉⣙⣉⣛⢉⠉⣍⣉⢡⠉⠤⡒⣨⣵⣚⣟⣼⣇⠂⠸⡀⠀⠴⠶⢀
⠀⠒⠂⢒⣖⣒⣒⣤⣀⣀⣘⠚⠛⠻⠿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⠿⠿⠿⠜⡂⢠⡧⠌⠄⠀⢉⡉⠁
⠀⠀⠀⠥⠤⠶⠦⠀⠉⠹⠿⠿⠶⠤⠤⠀⠤⠈⠉⠩⠉⠙⠛⠋⠙⠻⠟⠛⠻⠏⣉⡄⣤⡤⠴⠢⡌⠀⡉⠉⢉⡀⢀⣉⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠲⠖⠋⠁⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠈⠈⠙⣧⠴⠶⡄⠀⠀⠸⠷⠞⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠱⠀⠀⢠⣤⣤⣤⣿⣭⣭⣿⣭⣭⣭⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣭⣭⣭⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⣿⣿⣿⡉⣻⣿⣿⡏⢙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠿⣶⣶⣶⣾⣿⣭⣿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣶⣾⣯⣭⣭⣭⣿⣿⣿⡭⠽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠽⡜⠙⢛⠒⠤⣄⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠠⠔⠐⠚⠛⠋⠉⠉⠉⠋⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⣠⡴⠉⠀⠄⠀⠒⠂⢀⣀⣀⣤⣤⣤⣤⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⠠
⠀⠀⠀⠀⠈⠙⠛⠛⣷⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠰⠞⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠓⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠐⠂⠈⢥⢈⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣶⣶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠁⠉⠉⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⠟⠋
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣠⣠⣤⣽⣭⣭⣿⣿⣿⠟⠁
⠀⠙⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠍⠀⢈⡻⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⣸⣿⠖⠋⠉⢛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡏⠉⠙⠛⠛⠉⠙⠛⠛⠋⠉⢉⢍⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⡇⢀⣼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡎⠀⠀⠀⠀⢀⣀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠙⠚⠚⠛⠉⠉⠁⠀⢀⢢⠃⠀⠀⠀⠀⠀⠀⣈⡟⠉⠉⠉⠉⠉⠉⢉⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⡀⢤⠄⠀⢀⢣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⠤⠤⠤⠴⢶⣦⠀⠒⠂⣊⠜⢀⡐⠒⠒⠒⠂⠉⠉⠁⠀⠉⢸⠋⠀⠀⣌⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢀⠀⠠⢄⠀⠈⡀⠀⡸⠀⠀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠌⢫⠉⠉⠙⠛⠋⠀⠀⠀⠐⠒⠈⠂⠀⡨⢻⠊⠀⠉⠀⢠⣷⣿⣿⣟⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⠀⠀⠀⡸⠀⣀⡠⠔⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣀⣀⠤⠒⠋⠉⠀⢀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⢶⡒⠛⠒⠦⡉⠓⢤⣀⡴⠃⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠒⠒⠉⠉⠁⠀⠀⣀⠠⠔⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠘⢏⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠒⠈⠵⢦⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠈⢷⠀⠀⠊⠀⠀⠀⠀⢀⣄⠈⠁⠉⠀⢠⣞⡆⠀⠀⠀⣀⠑⠉⠂⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠣⣀⠀⠀⠀⠀⠀⠑⠘⢀⣀⡀⠤⠤⠀⠐⠒⠈⠉⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡒⠀⠀⠀⠒⠒⠒⠒⠂⠈⠉⠉⠉⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠠⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢖⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢸⣿⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠲⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢸⣿⡇⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢸⣿⢸⠀⠀⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢸⣿⢸⠀⠀⣠⡾⠋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣿⠈⣤⣾⠋⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⣈⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢸⣿⠀⡟⠁⠀⠀⠀⠀⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⢀⡴⢁⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⣠⠊⡴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⢀⠞⢡⠞⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡷⢃⡴⠁⠀⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡧⠊⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣯⡭⠥⠴⠶⠒⠒⢒⣒⣒⣚⣛⣒⣒⣒⣒⣒⣒⣒⣀⣀⡀⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣤⠤⢵⡤⠤⡤⢄⡠⢤⣀⢤⣀⠤⣀⡠⢄⣀⠤⣀⡤⣀⡤⢄⡠⠤⠤⢤⡠⠴⣧⠤⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⠤⠄⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠴⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠤⠴⠦⠤⠀⠀⠠⠤⠶⠤⠴⠤⠄⠀⠀⠀⠀⠠⠤⠦⠤⠶⠤⠀⠀⠀⠤⠴⠦⠤⠢⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣴⡾⠇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠀⠒⠚⠓⠀⠶⠤⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⣾⡧⠀⢀⠀⢀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣾⠿⠿⣓⣀⣠⣤⣄⣰⣾⣶⡄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣷⣿⡠⢏⠠⣾⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠻⠆⣠⠀⠀⡄⢸⠁⠀⠀⣦⣼⣿⢿⠁⢸⣷⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡀⠀⠀⠀⡀⠀⠀⠄⠀⠀⢀⠀⠀⠀⣾⣿⠀⢸⡇⢘⠀⢀⠠⣻⣿⠋⢸⡇⣾⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣥⡵⣀⣴⣡⣀⣀⡼⣆⣀⣀⣠⣄⡀⠀⢸⣿⡀⢸⡇⢸⠁⢼⣾⠟⠁⠀⢠⣇⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠶⠄⠤⠦⠤⢄⣠⣟⣿⣿⢿⣿⣿⣿⣿⣶⣦⣿⠇⣿⠁⣸⣼⠟⠁⠀⠐⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡤⠊⢀⡞⣽⡿⠀⠉⠉⠛⠛⢻⣿⣿⢷⣿⣷⠿⠁⠀⢀⠀⠀⡄⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⠞⢁⠀⡰⠏⡠⣿⡇⠄⠀⠀⠀⠀⠿⣿⢿⣿⣿⠁⠔⢀⠀⠈⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⡴⠋⠀⣠⢢⡞⠁⢠⣰⣿⡇⢀⠁⠦⠤⠀⢸⡃⢿⣟⣿⠀⡀⠀⠐⢄⠀⠀⠠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡤⠚⠁⠀⠀⢠⠇⣸⠇⠀⠤⢿⣿⣷⢠⠀⠃⠀⠀⢸⡇⣽⡏⣿⠀⡃⠀⠀⠀⠧⡀⢰⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣡⠖⠁⠀⠀⠀⠀⡠⠂⢰⡏⠀⠀⠆⣼⢻⣿⠀⠂⣄⡀⠀⢈⠁⣿⣇⣿⠀⡇⠀⠀⠀⠀⢸⣾⡝⢿⣯
⠀⠀⠀⠀⠀⠀⠀⡠⣊⠴⠋⠀⠀⠀⠀⠀⣀⡾⠀⢰⣟⠃⠀⠰⢡⣿⢨⣿⠀⡇⠀⠈⠀⠙⠻⢹⡏⣿⠀⠀⠂⠀⠀⠀⢈⡹⢷⣾⡷
⠀⠀⠀⠀⠀⣠⠼⠊⠁⠀⠀⠀⢀⢀⡀⡼⠁⠀⠐⠈⡋⠀⠀⡀⢺⡟⢸⣿⠄⣇⠀⠀⠀⠼⢶⢻⠀⣿⠀⠀⠀⠆⠀⢁⣿⣿⣼⣿⡿
⠀⠀⠤⣤⣾⣥⣤⣆⣨⣶⣶⣶⣷⣿⣷⣷⣤⣐⣋⠰⠀⢀⢀⣔⣿⠇⠀⣿⡇⣯⣤⣤⢤⣮⣷⣻⠄⣿⡆⠀⠆⠀⢱⡾⣿⣿⣿⣼⣷
⠀⠀⠀⠀⠈⠋⠉⠋⠛⠲⠴⣯⣉⣼⣿⣿⣿⢿⣿⠃⣠⢤⣿⣿⠟⠀⣈⣿⡇⢿⡀⠛⢺⢿⢿⣿⠀⣿⣏⡄⡄⠘⠂⣰⣿⣷⣿⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⢶⣿⣿⡦⡴⢶⣾⣤⣿⣦⣤⣤⣿⣧⣸⣧⣠⣮⡈⠾⠧⠤⣿⡿⠅⣤⣴⢾⣟⡻⢛⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣭⣟⠳⣀⡻⠥⠉⣩⣉⣛⣻⣻⣛⣛⣿⣿⠻⠿⠿⠿⣿⣿⠿⡿⣿⣿⣿⡟⠻⠟
⠀⠀⠀⠀⣀⣠⠀⠰⣦⣄⠀⠀⠀⣀⣄⡀⠀⢳⡤⠈⠉⠙⠓⠛⠉⠁⣛⣿⣽⡿⠿⣿⠿⢛⣿⣿⠿⠿⡿⠟⢙⢻⣋⣉⣀⣀⣤⣤⣤⣤⣤⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣄⠀⠤⠍⠙⠢⠄⠁⠀⠀⠀⠠⠀⠀⠀⢛⡛⣒⣐⣚⡚⠿⠿⠋⣿⣿⣿⣖⣪⠟⠻⠿⣿⣿⣿⠿⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠈⠀⠬⢭⡷⢶⡤⣀⣈⣭⠀⠤⠐⠒⠒⠒⠒⠒

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣟⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⣿⡇⠠⠤⠤⣿⠀⠀⠀⠀⠀⠀⢀⣠⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣦⡀⢀⣸⣀⣤⣤⣴⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀
⠀⠀⢀⠒⠂⠰⠶⠦⢤⣅⣤⣌⣻⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣁⣤⣶⣿⣶⣿⣯⣤⡀⠒⠂⣐⣂⣀⣘⣃⡀
⠀⠀⠀⠴⠶⠒⠶⠿⣿⣟⣻⣛⣛⣿⣿⣿⣿⣿⣟⣠⣿⣯⠕⠹⢷⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⡾⠗⠂⠀⠀⢤⣤⣤
⠠⠤⠽⠿⢿⣶⣶⣶⣯⣿⠛⠛⠛⣛⠿⠿⣿⣾⣭⣭⠿⢟⣿⣿⣿⣟⣿⣛⡛⠛⠻⠻⢧⠶⡿⣿⣿⣿⡛⠋⠉⠉⠀⠀⠠⠤⠄
⠀⠀⠒⠦⢤⣤⠤⠀⢀⣉⠓⠶⠶⠶⠖⠀⠀⠻⠿⡧⠤⠸⠿⣿⣿⠁⠀⢀⣀⣀⣀⣈⡭⠭⠭⠭⠽⠿⠇⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠶⠀⠀⣀⣀⣀⣈⠉⠙⣻⣿⠁⠀⠀⣀⣀⣀⠠⠤⠴⠖⠛⢁⣒⣀⣀⣀⣀⣠⣄⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⠀⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⣹⡄⠀⢿⣿⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⣿⡀⠈⠟⠁⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣾⡇⠀⠈⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⢸⣿⣧⠀⠀⠀⡀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠈⠀⠀⢀⡀⠤⠖⠂⠉⠉⠁⠀⠀⠀⢸⡿⣠⣼⡿⢿⣄⠀⠀⠰⡀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⣿⣆⠀⠀⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⣿⡿⢣⠀⠀⢠⣿⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠙⣧⣼⠃⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⣰⠄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠦⠭⠿⠭⠽⠟⣟⣛⣛⣳⠶⠦⠄⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⠄⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠙⠛⠛⠳⠶⠶⠶⣾⡇⠀⢀⣴⣿⣿⣾⣾⣧⣄⣀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠁⠈⠙⠻⠿⠿⣽⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢰⣶⡇⣶⣤⢀⣈⣅⣀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⡄⠤⠤⣤⣴⣿⣿⠿⠿⠿⠏⠉⠉⠁⠈⠉⠁⢉⣭⣄⠉⣭⠭⢩⣿⢯⣉⣄⣴⡶⠶⠟⠛⡫⢭⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣛⣛⣻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠲⠆⠴⠶⠶⠒⠛⠚⠹⠟⠛⠛⠉⠉⠀⠀⣉⣀⣤⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠛⠻⣿⡞⠿⠶⠶⠶⠖⠂⠀⠠⡀⢀⡀⢀⠀⢀⡀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠽⠿⠿⠾⣿⣷⠠⠔⠒⠒⠶⠄⣀⣀⣰⣶⠶⠀⠒⠠⠤⣤⣤⣴⡴⠞⠛⠉⠁⠒⣤⣤⠤⠤⠤
⠀⠀⠀⠀⠀⠈⠛⠛⠓⠚⠛⠛⠛⠋⠉⠀⣀⠀⠀⠀⠒⠛⠛⠋⠁⠀⢀⡀⠀⠀⠀⠀⠚⣉⣀⣀⣐⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠿⠿⠶⠶⠬⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠚⠉⠁⠈⠉⠉⠙⠓⠶⠦⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⢀⣀⠤⠤⢒⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⡻⠘⠋⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠒⠋⣈⣁⠥⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⡄⠤⠤⠄⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠤⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣀⡀⠀⣀⡀⠤⠤⠐⠂⠉⠉⠀⠈⠁⢉⢴⠁⠀⠆⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠤⠶⠖⠎⠀⡟⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠡⡆⠀⢀⠀⢸
⠀⠀⠀⢀⣦⣶⣶⣟⡉⠭⠥⠤⠤⠤⠴⠒⠒⠒⠚⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⡌⠀⠆
⠀⠀⠀⠌⠉⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡎⠀⢰⠀⡜
⠀⠀⢰⠒⠀⠀⠤⠤⠀⠀⣁⣈⣉⠉⠉⠉⠉⠉⠀⠀⢉⡉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⢠⠂⡰
⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠆⢀⠃⡰⠁
⠀⠀⠀⢷⠂⠠⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠌⡰⠁
⠀⠀⠀⡈⠆⠀⠀⠀⢀⠀⢀⣈⡁⠀⠐⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠋⠐⢶⣔⢁
⠀⠀⠀⠀⠉⠉⠀⠈⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⢀⠄⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠈⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢼⣤⣤⣮⣤⠾⠤⠴⠒⠐⠒⣀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠿⠇⠀⠀⠉⠉⠉⠈⠁⠀⠀⣤⣴⣶⠶⠟⠛⠉⠑⠦⠄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣷⣶⣒⣂⣀⣤⣀⣀⣤⣄⠄⠠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣄⡀⢐⠲⠶⢶⣶⡶⣶⣶⣶⡶⢶⠖⠁
⠀⠀⠀⠀⠀⣠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⢛⣓⡛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⠞⣛⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠞⠋
⠀⠀⠀⠀⠻⡏⢹⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⢀⡼⢃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢀⠀⠀⣠⣾⡿⠋⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡞⢀⡼⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡟⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡼⡿⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡼⠁⠘⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⡼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⡼⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⣡⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠛⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⣴⠟⣁⣠⠀⠀⠀⠀⠀⣸⣷⠏⠀⠀⠀⠀⠀⠀⠸⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⢀⣴⣿⣿⣿⣯⣥⡤⠶⠶⢻⣿⣯⣤⣤⣄⣀⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃
⠀⠀⠴⠿⠟⢋⣉⣉⣀⣀⣀⣀⣴⣋⣀⣀⣭⣭⡭⣭⠭⠤⡌⣿⠗⠒⠠⠤⠤⠀⠀⢀⣀⣀⣠⣀⣠⣤⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡻⡟⠻⠿⣯⣭⣀⣀⠘⠂⠸⢧⣌⣙⣀⣀⣀⠀⠀⠀⠀⠤⢼⠆⠈⠉⣯⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢰⡶⠛⠛⠛⠿⣿⣿⡷⠶⠶⢤⣤⣬⣭⣭⣭⣭⣭⣥⡴⠶⠖⠋⢁⣠⡴⠁⡀⠴⠒⣒⣩⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠁⠀⠀⠀⠙⢿⣷⣄⡀⠀⠚⠉⠉⠙⢻⣿⣿⣿⣿⠿⠿⠛⠉⣀⣠⣴⠾⠛⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠀⠀⠀⠀⠙⠻⣿⣛⡒⠚⠀⢀⣠⡿⠈⠛⠻⠿⠿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠈⠇⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢀⡾⢁⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⣰⣿⣶⠟⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⣰⠀⢠⣾⠟⠋⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣏⣼⣃⡴⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣿⠃⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⣿⠃⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠰⣫⠇⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣰⠏⠀⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⡏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣰⠏⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⣰⠏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⢉⣠⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠋⠉⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⡀⢀⡾⠃⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢀⣤⡞⣉⣠⠄⠀⠀⠀⠀⣀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢳⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣯⣥⣤⠶⠶⢚⣿⣿⣥⣤⣤⣤⣀⣀⣀⣀⣤⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠴⠿⠿⣯⣭⣍⣁⣀⣀⣀⣠⣏⣀⣈⣭⣭⣭⣭⡭⠭⢥⣙⣿⠶⠶⠦⠀⠀⠀⣤⣤⣤⣄⣤⣤⣤⣤⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⡻⠿⣿⣯⣉⣀⠀⠳⠀⠾⣤⣉⣓⣀⡀⠀⠀⠀⠠⠤⠤⢼⡦⠈⠉⣻⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⢈⣶⠾⠛⠛⠿⢿⣿⣷⠶⢶⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⠴⠶⠛⠁⣀⣴⠃⢀⡤⠴⢒⣚⣡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣤⡖⠀⠀⠀⠉⠻⣷⣦⣀⠀⠀⠟⠉⠉⠛⢿⣿⣿⣿⣶⡶⠿⠿⠛⠁⣈⣤⣶⠿⠛⠉⠁⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠈⠻⢿⣷⡶⠶⠞⠀⠀⣨⡿⠙⠻⠿⣷⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢉⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣷⠀⠀⠀⠀⠻⡍⠀⠀⠀⠀⠹⣇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸⠂⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⣽⡀⠀⠀⢄
⠀⠀⠀⢴⣤⠀⠀⠀⣬⣄⠀⠀⠘⠓⠃⢠⣤⣸⡈⡍⠛⠓⠐⠦⢌⣿⣇⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠁⠀⠂⡄⠀⠉⠃⠲⡄⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠀⠿⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠃⠀⢲⠀⠀⡧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⣀⣀⣀⠀⡀⣀⠀⠀⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⡀⠀⠳⡀⡀⠘⠢⡀⠀⢀⠄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢳⣷⣷⣠⣰⣖⣛⡶⠿⠛⠄
⠀⠀⠠⠤⠤⠀⠀⣀⠀⠀⢠⡄⢼⡇⢀⣤⣤⠴⠤⠄⠤⣶⣒⣒⣣⠼⣻⣿⣿⣶⣶⣾⣿⠛⠃
⠀⠀⠀⠨⣧⣴⣾⣒⣖⣛⣙⣁⣸⣇⣠⣤⣭⣖⣾⣶⣿⣭⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⠿⢿⣿⠿⡿⠿⡟⠟⠻⠟⣝⡛⠛⠛⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠃⠀⡀⠈⠌⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠰⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠤⠔⠠⠂⠐⠂⠖⠲⠀⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⠀⠀⣠⠆⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠂⡀⠀⠒⠀⠐⠛⠀⠀⠠⠀⢤⠀⠀⠂⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠐⠦⠐⠲⠄⠰⠆⠀⠶⣤⠤⠈⠆⠀⢀⣀⣀⠀⠀⠀⢠⡌⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀⠀⠀⡌⠀⠈⡇⠀⠀⢸⠃⠀⠐⠀⠀⣀⣤⠀⢀⡀⠀⣀⠤⢄⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠖⠖⠂⣦⠂⠀⠾⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣤⣿⣷⣒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣿⣦⣽⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⢛⣿⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠋⠑⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⡟⢻⣿⣿⠛⠻⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⣛⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⠏⠀⠀⠀⠀⢰⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠁⠐⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⠀⢀⡀⡀⣠⣀⠀⠀⠀⠀⠃⠠⠀⠀⠀⣸⡧⢀⠂⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠙⠙⠛⠛⠛⠦⢀⠀⠀⠀⡇⠀⠀⣠⣿⡿⠁⣦⣋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⢸⠀⡁⢀⣐⣿⠋⢸⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣓⠚⡃⠚⠒⢢⣼⣷⣤⣤⣦⣤⢀⢿⡆⣼⠀⣧⣼⠟⠁⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠝⢛⢿⣿⠛⠛⠻⢿⣿⣿⣧⣿⣴⡟⠁⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⡠⠚⠁⡰⠋⣼⡇⠀⠀⠀⠀⢎⣿⣿⡟⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⠔⠃⢀⠎⡞⠀⢄⣿⡇⠀⠀⠀⠀⢩⠸⣿⡇⢠⠀⠈⢠⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢊⠔⠊⠀⠀⠀⠊⣰⠁⠀⣈⡿⣿⠰⠀⠀⠀⢸⢸⡇⣧⢠⠀⠀⠀⠑⣠⠯⣽
⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠠⠀⣠⡃⠀⠀⢸⡇⣿⠠⠈⠐⠀⠐⣾⡟⣿⠈⠀⠀⠀⠀⠻⣇⣿
⠀⠀⠀⢀⡠⠖⠉⠀⠀⢀⢀⠀⠈⠀⡐⠈⠁⠀⠀⣿⠁⣿⠀⡆⠀⠀⡒⢺⡇⣿⠀⠀⠀⠀⣰⣳⣼⣿⠂
⠀⠐⠒⠿⡶⣶⣶⡿⢿⠿⠷⣶⣷⣦⣥⡃⠀⣰⣼⡟⠀⣿⠂⡇⢦⢰⣾⣿⠃⣿⡀⠀⢀⠹⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠻⣭⣿⣏⣠⣭⣛⣿⡁⣀⣿⡇⢟⠀⣈⢉⣿⢀⣿⣗⢀⣀⣰⠿⠿⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⢣⣹⠋⢙⣛⣛⣻⣿⣻⣿⡿⠷⠾⠷⣿⡷⢾⣿⣾⣿⠺⠗
⠀⠀⠀⠠⠤⠀⠲⣤⠀⠀⠠⣤⡀⠙⡤⠈⠉⠙⠋⠉⢨⣽⠿⠟⢻⣟⣿⣿⣟⣻⠟⣉⣻⣭⣤⣤⣴⠦⢤⣴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠬⠛⠒⠀⠤⢟⡉⠘⠛⠛⠱⠋⢉⠙⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉
⣀⣤⣀⣀⣀⣄⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠒⠭⣑⡲⢧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠉⠑⠲⠭⣒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⠽⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠖⠲⠦⠬⢭⣭⣑⣒⡒⠶⠤⠤⣄⣀⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠦⠭⠭⣍⣛⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠈⡏⠉⢹⠉⠑⠒⠒⠢⠤⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢋⣹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⣤⠶⠚⣋⡥⠖⠉⠀⢻
⠀⠀⠀⠀⠀⠀⣰⠧⣍⣉⡛⠛⠛⠒⠲⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠾⠴⠶⠷⠶⠖⠚⢛⣋⣩⠤⠶⠚⠉⠁⣀⠄⠂⡠⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠙⠓⠒⠶⠶⠤⠤⠤⢤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠒⠚⠛⠉⠉⠁⠀⣀⠠⠔⠒⠉⠀⠀⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠄⠀⠂⡀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⠀⣤⣀⠴⡀
⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠃⡀⠀⠈
⠊⠋⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠑⠠⠄
⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢖⣼⠷⣄⠀⠖⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠂⠰⠶⠀⠀⠋⠁⠙⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡤⠖⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠄⠉⠀⢠⡿⠃
⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠟⠁⠀⢰⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⣿⣿⣷⣶⣤⣶⣿⡟⠀⠀⠀⠀⠀⢈⣰⣶⣀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⣶⣶⣦⣤⣿⣀⡀
⢠⣤⣶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣶⡾⠿⠿⣿⠛⠛⢻⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠘⢿⣦⡑⣄⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠷⠶⢶⣿⣧⣄⣀⣿⡀⠀⢸⣿⡇⠀⠀⣿⠀⠀⢸⡏⠉⢹⣿⣿⡇
⠀⠀⠙⣿⣌⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⣿⣷⣿⣶⣶⣾⣧⣤⣼⣿⣿⡇
⠀⠀⠀⠈⢿⣷⡙⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⣤⣤⡌⠉⢉⣙⡛⠛⠛⠻⠿⠷⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠈⣿⣷⠸⣟⣛⠿⠿⠿⢿⣶⣶⣶⣦⣤⣤⣄⣀⣀⣈⠙⠛⠁⠀⠿⠿⠇⠀⢾⣿⡇⠀⣿⣿⡆⠀⣿⣷⠀⢀⡇
⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣬⣭⣭⣭⣛⣛⣛⣛⠿⠿⠷⢶⣶⣶⣦⣤⣤⣤⣤⣬⣭⣤⣀⣉⣉⣀⡼
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣯⣭⣭⣭⣭⡭⠁
⠀⠀⠀⠀⠀⢈⣭⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⣛⣛⣛⡛⠿⢿⣿⣿⣿⣿⡿⠿⠟⣛⣋⣭⣥⣬⣙⣛⣛⣛⣛⣛⣛⠿⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠠⢴⣶⣾⣿⣿⣿⣿⣿⣿⣷⢶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⠆⢦⣤⣦⠐⠿⠿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⡸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡠⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢉⣹⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣄⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣭⣶⣤⣤⣆⣀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠂⢹⣏⣽⣶⠀⠀⠀⠠⠄⠀⠀⠠⠤⢆⣀⣴⣿⠱⢟⠋
⠀⠀⠀⠀⣀⣀⡀⠀⠠⠄⠀⠐⢟⣻⢿⣿⣿⣿⣿⣶⣶⣞⣀⡀⠂⡀⠀⠀⢠⣼⣿⣿⡟⠀⠀⣀⣀⡠⠦⠐⠒⠉⢹⣿⣿⡇⢸⡿⠂⠀⠐⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠶⠆⠀⢚⣒⡀⣈⢯⣟⠶⠭⢍⣟⣟⣿⣯⣿⣿⣿⣷⣶⣴⣿⣿⣟⠛⢉⣁⣁⢠⡤⠴⡖⠲⠋⡀⠈⢿⡟⠀⠸⡁⠀⠀⠀⠲⠀⢠⣤⠉
⠀⠀⠀⠀⠀⠀⣀⣂⣀⣂⣤⣀⣀⣈⡻⣿⣿⣶⡶⢶⣤⣥⣉⣉⣙⣉⣉⡉⠉⠉⣍⣉⣉⣠⠠⠦⠞⠃⣬⣑⣒⣫⣩⣿⡅⠂⠀⢇⠀⠁⠶⠶⠆
⠀⠦⠤⠀⠤⠦⠤⠴⠆⠁⠀⠀⠉⠩⠽⠿⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣼⣧⣿⣿⣶⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣧⡘⠅⠀⣼⣀⣉⡀⠀⠈⠉⠉⠁
⠀⠀⠀⠐⠚⠛⣛⣛⠋⠙⠛⠛⣿⣶⣆⣀⣀⠀⠉⠉⠙⠛⠻⠻⠟⠛⠿⢿⣿⠿⢿⣿⣿⣿⣟⣻⣿⠿⠛⠉⠁⠠⠄⣀⠍⠒⠀⠻⠒⠶⠤⠄⠀⠉⠒⠠⠤
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠩⠭⠉⠉⠁⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠈⠓⠲⠶⠅⠚⠋⠙⠃⠙⠂⠀⢀⡀⣀⣲⠦⠄⠰⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠙⠒⠒⠤⠤⣄⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡇⡇⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢸⡇⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢸⡏⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣿⠆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢁⠀⣿⠄⠀⠀⠀⠀⢘⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⢀⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⡄⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢻⢆
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠣⡣⡀
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡕⢄
⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠢⡀
⡀⡠⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠣⡀
⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⣀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⠶⠚
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⠇⢀⣀⣀⣤⣤⣤⣤⡤⠴⠒⠒⠚⠋⠉⠉⠉
⠀⠀⠀⠀⢽⣦⡶⠤⣄⣀⣀⣸⢻⠃⠀⠀⣸⣿⡛⢯⣛⣟⣿⣿⠭⠗⠛⠲⢀⣤⣤⣤⣀⡄
⠀⠀⠀⠀⠀⢿⣛⠷⠶⣬⣭⣛⣛⣛⣋⣈⣛⢛⣳⣼⣉⣁⣼⣷⣤⣤⣷⣤⣶⡾⢿⣛⣯⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠙⠯⠷⣶⣦⣭⣭⣭⣭⣭⣭⣿⣿⣽⣿⣿⣿⣭⣽⠿⠷⠒⢺⠟⠛⠉⠉⠉⠛⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⣿⣿⠃⠀⣿⠀⢀⠀⠀⠀⠀⠐⠀⢹⢹⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣼⣏⣀⠀⣿⠀⠈⣿⣷⣶⣦⣀⡀⢸⡀⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣞⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠘⣿⣿⣿⣞⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⠀⣾⣿⣿⣿⣿⣿⣿⠏⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠳
⣀⣀⡠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠣⢼⠜⠛⠉⠉⠉⠉⠁⠀⠀⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠘⡇⠀⠀⣄⠀⠀⠀⢀⠀⢻⣀⣀⣀⣶⣶⣶⣶⣶⡟⠋⠉⠉⠉⠉
⠀⠀⠀⠀⢿⣿⣾⣶⣶⣶⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉