Grandfather Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⣤⣤⣤⣄⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⣄⠐⣭⣾⡿⣿⣿⣟⡲⢥⡪⢿⣟⣿⣽⣧⣰⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠿⠿⣷⡘⢦⡀⢭⣿⣿⣿⠿⣿⡷⣯⣿⢷⣿⢾⣿⣿⠏⢠⣄⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠋⠁⠀⠀⠀⠀⠉⠱⠿⢷⣬⡻⣿⣿⣦⡝⢷⣭⣝⠛⠫⠙⠂⠀⠁⠛⠂⢬⣄
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧
⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢸⣿⡀
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢹⣞⣿⡇
⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠀⠀⠀⡄⠈⣯⣿⣿
⠀⠀⠀⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠠⠸⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢷
⠀⠀⠀⣄⣈⢿⣿⡏⠀⠀⠀⠀⠀⠀⣤⣶⢦⣤⣄⣀⣠⣶⣄⠀⡄⠀⣴⣦⣤⣤⣤⣦⣤⣀⣠⠀⠀⠀⣾⣿⠏⡘
⠀⠀⠀⢸⡿⠄⠻⠁⠀⠀⠀⠀⠠⠈⣁⣤⣤⣶⣿⣿⣿⣯⡈⠀⠀⢠⣼⣼⣿⣿⣽⣤⣤⣉⠓⡀⠀⠀⣿⡿⠐
⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⢀⠐⣶⡾⠛⠛⠛⠳⣿⣿⣿⡿⠁⠀⠀⣿⣿⣿⣿⣿⠟⠛⠊⠛⠆⠙⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠐⣪⣤⣴⣿⠿⠋⠀⠀⠀⠀⠻⡻⠿⢿⣿⣤⣤⡶⠎⠉⠙⠛⠀⡟⢀
⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣰⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠐⠠⣧⣀⠀⠀⠀⠀⠀⠀⠀⡄⠡⣿⠇
⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠓⠠⡀⠀⠀⢀⣒⣾⡿⠁⠙⣿⣿⣷⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⢀⠙⠶⠿⠿⠟⠁⡀⣸⣆⣌⠻⣿⠄⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣿⣿⠿⠿⠿⣿⣧⣤⣴⣴⣷⡾⠿⠛⠛⢿⣇⢹⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⡀⠈⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣸⠏⣴⠃⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣀⣤⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⣼⠇⠸⠃⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣟⠿⣇⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⠀⡰⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡻⠟⠒⢤⣄⠘⠂⠂⢀⣄⣀⣤⣴⠿⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠐⣤⣴⣶⠾⢛⣿⣟⠉⢹⣿⢋⣴⣿⣵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢼⣏⠙⢷⣌⢿⣿⡄⠈⣿⣿⢯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⢾⣿⡀⢹⣿⣿⣿⣧⢸⡿⣿⣿⡿⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⣿⣿⣿⣿⣼⣿⣿⠋⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠉⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡶⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⢻⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠙⣷⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠶⣤⡿⠁⠀⠀⢀⣴⠶⠶⠶⣤⠀⠀⣤⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣸⣷⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣀⢀⡾⠃⠀⠀⣴⠋⣀⣤⡶⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣤⣀⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣹⣧⠶⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⢻⡇⠀⠀⠠⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⢸⡇⠀⠀⠀⠀⠀⢙⣿⠤⠄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠶⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⣀⠀⣿⠀⠀⠰⠞⠓⠀⢿⣇⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⣿⡟⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⡈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡽⢷⡟⠀⠀⠀⠰⠋⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠐⢦⡉⠙⠃⠀⠀⠀⠀⠀⠀⣿⣏⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⠷⢶⣤⣶⣤⠶⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⣷⢀⣾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠁⠀⠀⠀⠀⠠⣤⣤⡶⠞⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠘⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⡏⠀⢠⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠷⠤⠿⠷⠶⠿⢦⣤⣤⣷⣀⣴⣤⡿⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠻⠷⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⣠⣴⠾⠛⠁⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢨⡿⠷⢦⣄⣀⣀⣀⣀⣀⣠⣤⡶⠟⢻⡿⠛⠛⠉⠉⢠⣶⣀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠙⣯⣭⡿⠛⠉⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣄⣤⢴⡟⣁⣭⡿⣦⡀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⢀⣤⠶⢛⣻⡿⠒⠒⠢⣤⣾⣧⣤⡤⢤⣀⣿⠀⠀⠀⠀⠀⠀⠉⠁⣼⠋⠙⣋⠀⣿⠻⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⢀⠀⠀⠀⠀⢻⡆
⢠⣴⡾⠁⣴⠋⢁⡤⠞⠂⠀⠀⠉⢣⡈⠛⣦⡉⢿⡄⠀⠀⠀⠀⠀⠀⠀⢹⡷⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣛⡛⡿⠆⠀⠀⣸⡇
⣯⢰⡇⢸⡇⢀⡿⠁⢀⣤⣤⣄⠀⠈⣿⠀⠘⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠉⠙⠋⠀⠀⠀⢠⣿⠃
⢻⡌⢳⡈⣷⢘⡇⠀⣼⠃⠀⢘⣧⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣠⡿⠁
⠀⠙⠻⣿⠿⠿⣿⡶⠿⢶⡟⢋⣡⠾⠃⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣰⡟⠁
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠈⠛⠛⠳⢦⣤⣤⠴⠟⠁⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠸⣿
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢿⣇
⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⡿⢻⡇
⠀⠀⠀⢻⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠁⠀⠀⠘⣡⡿
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠻⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠾⠛⠋⠉⠀⢹⡷⠤⢤⣤⠾⠋
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠞⠛⠛⠛⠖⠀⠀⠀⠈⡇⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠳⢶⣄⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢉⣿⡄
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣿⠀⠀⠀⠀⠀⢀⡾⠛⠉⠉⠉⠉⠛⢶⣤⣤⡴⠛⠉⢷⡀
⠀⠀⠀⣾⢷⣦⡾⢷⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠁
⠀⠀⠀⣟⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣶⣤⣤⣤⠞⠁⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠛⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⢀⣀⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠈⠛⢷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡤⡠⠒⠉⠉⠢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⡶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⣄⡠⢶⡀⣀⣀⡟⣣⠄⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠚⠁⠀⠊⠁⠀⠀⠑⠋⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣟⣭⠿⠇⠀⡠⠞⢂⡞⠒⣦⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠖⠚⠉⠀⢀⡞⢀⠜⡏⠀⠀⠤⣌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⣀⡠⠔⣯⠱⠧⡜⠀⢠⠀⠀⠈⠲⣀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⡀⡀⠀⠉⠳⣄⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢇⠱⡀⠀⠀⠘⡆⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡼⠀⢱⠀⠀⠀⢹⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⠞⠀⡇⠈⡄⠀⠀⣾⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⡔⠁⠀⠀⠁⠀⠇⠀⢀⠏⠁⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣀⠠⡀⠀⠊⠀⠀⠀⠀⢠⡀⡀⠀⢺⡀⠀⠀⠀⢸⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡀⠈⢷⠍⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⡶⠛⠒⠒⠒⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⢰⠁⠈⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠙⡀⢰⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠦⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠀⠀⣇⠼⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡁⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢳⡄⠀⠐⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣱⢠⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠇⠀⠀⠀⠀⠀⠀⢠⡞⠉⢀⡈⠩⠻⠾⣄⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⠀⠀⠀⠀⠀⠉⣉⣡⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠉⠉⠁
⣿⣿⡿⠻⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠋⡭⠽⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣿⢧⣴⠿⠟⠧⣤⣼⣾⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⢿⢐⡿⠀⠀⠀⠀⠀⢹⣶⠋⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣸⡮⣤⣾⣷⣦⣰⣶⣶⣿⡀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢿⠀⡜⠧⠽⠇⢸⣍⣿⣿⡇⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⡉⢰⡎⣙⢋⣁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣦⡀⠐⠊⣠⡿⣠⣤⣤⢤⣄⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢧⡀⠻⣷⣾⡿⣯⢍⣹⣧⣼⣿⣟⡇⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⣀⣤⠶⣿⣽⣆⠀⠙⢦⡽⣿⣿⣏⢉⡀⠀⠙⢻⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⣞⣯⣿⣿⣿⠀⣿⣿⣿⣆⠀⣰⢷⣿⣿⠁⠘⠃⢠⠛⠋⣹⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟⣥⢸⣎⣿⣿⣿⠀⣿⣿⣿⣿⣷⣿⣼⣿⣷⣦⡀⠠⡟⢀⣾⣿⣧⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡿⢸⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣟⢦⣴⣾⣿⣿⣿⡆⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⡟⠋⣟⣠⣼⣆⢹⠧⣄⣿⠙⢳⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⣼⣿⣿⣶⣿⣿⠏⠀⢠⢸⡅⠀⠉⠁⠀⠀⣿⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢠⠟⡛⢿⣿⣿⡿⢀⣀⣼⠿⠷⠶⣤⡤⠶⠴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⣿⣿⣿⡿⣤⣾⣿⣧⣴⣿⣶⣾⣿⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣯⣤⣿⣿⣇⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠘⠒⠚⠒⠒⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡔⠁⢸⠀⠀⢀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠰⠁⠙⣄⠀⡇⠀⢸⠀⡰⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢣⠀⠘⢄⠀⠀⢄⡠⠁⢀⡜⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠱⡀⠀⠀⠀⠈⠀⠀⠘⠔⠋⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠰⠶⠶⠤⡀⠀⠀⠀⠀⠀⠃⡔⠀⠈⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⣄⣉⠒⠒⠒⠠⣄⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠰⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠀⠀⡧⠤⠀⡠⠀⠑⠴⠤⠲⡤⠖⠁
⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⠀⡠⠊⣠⡄⠀⠀⠀⠀⠀⠀⣶⡄⡑⠀⠇⠀⠀⠠⠔⠀⠀⢠⠃⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⡌⠁⠀⠑⠤⠀⠀⠀⢈⠤⠄⠛⠓⠀⢀⠀⡀⠀⠒⠋⠁⢁⡄⠀⠀⣀⢤⠀⠀⠀⠆⠀⠀⠀⠀⢀⠃
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠉⢀⡄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⡠⠀⢸⠄⠄⡀⠀⠀⠀⠀⢀⠆
⠀⡜⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠢⡀⠉⠒⠂⠐⠀⠀⠀⠀⠈⠁⠀⣁⠠⠐⠈⠀⢀⠶⠀⠀⠈⠓⠤⡀⠴⠋
⠀⡀⠀⠀⠀⠀⠀⡰⠁⠀⠈⢧⠀⠈⠒⠂⠴⡖⠂⠤⠐⠆⠀⠉⠀⠁⠀⠀⣀⠔⠉⠀⠃
⠀⢇⠀⠀⠀⠀⠠⣇⠀⠀⢀⠃⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⢸
⠀⠈⡄⠀⠀⠀⠀⠈⠆⠀⠘⡀⠀⠀⠀⠁⠂⠶⠀⡀⠀⠀⠀⡠⠷⠉⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠘⡄⠀⠀⠀⠀⠘⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠢⣰⢤⠁⠀⠀⠀⠀⡀⠀⠀⠀⣇
⠀⠀⠀⣉⠀⠀⠀⠀⢰⡇⢀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⡊⢀⠇⠀⠀⠀⠀⠇⠀⠀⡀⠈⠳⣄
⠀⠀⠀⢯⣀⣀⠀⠤⠐⠃⠁⡅⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠢⡀⠐⠀⠈⠣⡀
⠀⠀⠠⠊⠁⠀⠀⠀⠠⠒⠋⠤⡃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⢃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐
⠀⠀⢇⡀⠀⠀⠀⠀⢠⡀⢀⡀⠈⠄⠈⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⢨⠉⡲⠂⠂⠀⠁⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⠊⠘⠀⠁⠀⠐⠂⠀⣻
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠒⡇⠀⠓⠒⠒⠂⠐⠈⠉⡠⠋⠀⠀⠑⢤⠀⠢⠀⠀⠉⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠑⠄⠀⢀⠀⠈⡄⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⡄⠀⡅⠀⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⢇⣀⠤⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢸⠀⢨⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⢈⠀⠀⠂⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠀⠘⠀⠀⣀⠒⠀⠀⠀⡤⠄⠀⠀⡠⠊⠁⠐⠶⠀⠀⢴⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠦⠀⠁⠄⠀⢠⠼⠃⠀⠈⢆⠣⡀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃⠑⣄⡠⠖⠀⠀⠀⠀⢀⠌⠀⠉⠊⠁⠀⠀⠀⠀⠈⠁⠀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠯⠀⠈⠁⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⠀⣠⠤⠤⠤⠄⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠇⠹⣿⠀⠀⠀⠀⡴⠓⠀⠀⠤⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠀⠀⠀⠉⡝⠀⠀⠀⠀⠠⠀⠀⢀⡐⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡄⠀⣀⣘⠷⠂⠀⠷⣟⣠⠀⢰⠆⢹⡂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣾⠋⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⣠⡔⠀⠀⠀⠀⣀⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠙⢦⡀⠀⠀⠁⣄⣀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣦⡀⠉⠓⠤⣄⠀⢹⡟⠳⢦⣄⣀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⢻⡀⠀⠀⢸⣿⣿⣶⣤⣠⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⢀⠀⠀⠀⡀⠈⢳⡀⠀⠈⣯⠙⠛⣿⠏⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠂⠀⡀⠀⠌⠀⠀⠀⡛⢦⣄⠨⠙⠋⠁⢀⡜⢣⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠃⠀⠻⡆⠀⠀⠀⠀⠁⠀⠈⢷⡀⠀⠀⠀⠙⡄⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢈⠀⢡⠀⠀⠀⠀⠇⠀⠀⡸⠛⠲⣤⣤⠶⠁⠀⠻⠲⡀
⠀⠀⠀⠀⠀⡞⠀⠠⠀⠀⠀⠀⠀⠀⠑⢤⡆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠙⣆
⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣜⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⢧
⠀⠀⠀⠀⡌⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢣
⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⢧
⠀⠀⠀⠀⠇⠀⣺⠀⠀⠀⠀⠀⠂⠀⠀⠈⡏⠉⠉⠁⠘⠓⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢷⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡗⢤⣄⣠⣤⢴⠦⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠱⠀⠈⣧
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠛⠀⠁⠀⢸⡇⠀⠠⠀⡄⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠇⠀⠸⡆
⠀⠀⠀⢸⡀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⣰⠎⠰⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠀⠀⢹
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠚⠛⠛⠛⠓⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣤⠴⠶⠃⠀⢹⠀⢸⠀⠀⣸⣄⠀⢀⣀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⣼⢰⠉⣉⠉⠠⠀⠀⠀⠈⣱⡀⣠⣀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢦⢉⠀⠐⡇⢸⡀⡶⠏⣀⡁⢈
⠀⠀⠀⠀⢸⡀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠈⠙⣁⠘⡖⠊⠉⠁⠀⠈⠁⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢺⣅⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠶⠶⠶⢶⠦⠶⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠦⣄⣀⣀⣀⡄⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡄⠀⢷⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠒⠶⠚⠛⠉⠀⠀⠘⡇⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⣀⠀⠀⢸⠀⡇
⠀⠀⢀⣀⠤⠄⠒⠛⠁⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠺⣇⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠾⣗⢀⡋⠁⢙⡆
⠲⢯⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠖⠚⠧⠔⠚⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠤⠄⣀⣀⣀⣀⣀⣀⣠⣤⡼⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠛⠉⠛⠛⠛⠛⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⣾⠛⢿⣿⣿⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⢿⠀⠟⠀⠀⠀⠙⠛⠫⣤⣤⠈⠉⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠛⠲⣤⣤⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠤⣶⣤⣄⢀⡀⢹⣦⠀⠀⠀⠀⠀⠹⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⣶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠠⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⠀⣄⠀⠀⠀⠀⣀⠀⠀⢲⠆⠙⠀⠀⠀⠀⠖⠀⠀⠀⣀⠀⠀⠐⢢⠀⠀⠀⣀⢀⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣿⠁⠀⠀⠉⠀⠀⠘⢿⣿⣿⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠴⡿⠟⠁⠀⠀⠀⢘⠀⠀⠀⠀⠁⠀⠠⢿⡿⠛⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠻⠷⠶⠖⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠒⠲⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⢿⠇⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡶⠶⣦⣤⠄⢀⠀⠀⢀⣀⠀⣠⠤⣶⡶⠖⠛⣿⡏⢻⠆⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢰⣿⠁⠀⠀⠀⠀⠀⢀⣿⡁⠉⠀⠀⠀⠀⠀⣀⣾⡏⣷⡏⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠏⣸⡆⠀⠀⠀⠴⠋⠀⠙⣀⠀⠀⠀⠠⠎⢹⣏⡹⠇⠀⠀⠀⠀⠀⠀⢠⡾⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⣿⡇⠀⠀⢰⡟⢡⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡴⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣿⠀⠀⠀⠈⢣⡜⢣⡄⠀⠀⠻⠀⠀⠀⠀⢀⣴⠟⣱⠎⠁⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣿⠀⠀⠀⠀⠈⢳⡌⠙⢦⣤⣀⠀⢠⣤⡴⠛⣥⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠙⠛⠋⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⣀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⣾⠋⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠒⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢡⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢷⣶⣤⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣟⠙⣦⡤⣤⣄⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⣠⡿⠟⠁⠀⠀⢠⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⢰⣿⣀⣴⡇⣸⠄⣼⠋⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠉⠉⠉⠻⣇⣀⣿⡀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⠶⢦⣄⣀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⠁⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣠⣤⣄⠀⠈⠙⢷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠾⠃⠀⠀⠀⠉⠻⢦⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⡄⠀⠀⠀⠀⠀⠀⢤⣤⡿⢿⠆⢼⣇⠀⠀⠀⠀⠀⢀⡼⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣆⣀⣠⣴⠟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⡄⣀⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠀⠈⠙⢛⣛⡁⠠⣴⡦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⢿⡿⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠟⠈⡿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣸⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠛⠛⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣤⣤⣲⣶⣶⣶⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣤⡴⠶⠈⣉⣉⣀⣴⣦⡀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⣉⡉⠁⠉⢉⣉⣻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡤⠶⠈⠉⠉⠉⠁⠶⠿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⡀⣀⣀⣀⣀⡀⢀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠉⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⠁⠀⠀⠀⣴⣿⣭⣿⣦⠀⠀⠀⠀⠀⣶⣿⣿⣷⣆⠀⠀⠈⢳⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⣀⣼⣃⣀⣀⣀⣼⠟⠋⠉⢶⣄⢀⣀⣀⠀⣠⠾⠋⠲⣦⣉⡀⢀⠀⢘⣧⡀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡯⢽⣿⣛⡿⠛⠛⣿⣿⣏⣴⠛⠛⢶⣿⠋⠉⠉⢻⣧⠾⠛⢷⢈⣿⣿⡿⠟⠛⠿⣿⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⡼⣿⡟⠁⠀⠀⠙⠟⠻⢦⣤⣤⣼⡟⠀⠀⠀⠈⢻⣤⣀⣠⡾⠻⠏⠁⠀⠀⠀⠈⣿⣿⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣈⣻⡗⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠉⠉⠀⠀⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⣤⣄⣀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⣀⣀⣤⣴⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠋⠀⣽⡇⠙⢷⣤⣀⣀⣀⣀⣀⣠⣴⡞⠋⢙⣿⠀⠈⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠁⠀⠀⠀⠀⢻⡇⠀⠀⠻⣧⡉⠉⠉⢉⣼⠟⠀⠀⢸⡏⠀⠀⠀⠀⠈⠙⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⣦⣴⡟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣧⣀⡴⠶⠛⠋⠉⠛⠶⢦⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⢀⣞⠁⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢻⡦⠶⠶⢶⣿⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢷⣤⣤⠤⠴⢾⡏
⠀⠀⠀⠀⣸⠇⠀⠀⣸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢹⡇⠀⠀⠘⣧
⠀⠀⠀⢠⣿⠀⠀⢰⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣷⡄⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣇⠀⠀⢻⡀
⠀⠀⠀⣾⠁⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢿⡀⠀⠘⣧
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠘⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠀⠀⠀⠀⠸⣧⠀⠀⣿
⠀⠀⣰⠏⠀⠰⣏⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠶⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠞⠛⠛⢹⡏⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⡄
⠀⣰⡟⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣠⡴⠋⠀⠀⠈⢳
⢰⣯⣀⢀⡄⢠⣤⣻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣯⣀
⣿⡏⣿⢸⡇⢸⡿⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡟⢰⣿⡀⠀⣀⣀⣴
⠙⢗⣿⣸⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡇⢸⠀⠉⠀⠉⠉⠁
⠀⠘⠋⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⣧⡀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣿⣤⣤⣤⣤⣤⣄⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⢻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣯⣄⣀⣀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢤⣤⣤⣀⣀⣀⣀⣀⠀⢀⣀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠉⢻⡷⣏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣏⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠂⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢀⣀⣀⣀⠀⢀⣀⣤⠤⠶⠚⠙⢿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⠏⠁⢠⡽⠛⠉⠁⠀⠀⠠⣤⠀⠀⢳⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣜⣀⡔⢩⣽⡝⢦⡠⣤⠾⣽⣭⠑⣄⣈⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠏⠈⠧⣀⣈⣠⣾⠂⢸⣆⣈⣀⣠⠋⠈⢩⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣼⣋⣭⣻⣯⣌⣹⣄⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⢦⣄⡀⠀⠀⢦⣀⢀⡴⠀⠀⢀⣠⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡀⢸⣏⣉⣩⠶⣶⠶⠷⣶⠶⣛⡉⣹⠀⢸⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠸⡇⠀⠉⠙⢯⣄⠈⢳⡼⠃⢀⡽⠋⠋⠀⡿⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣧⠀⠀⠀⠀⠙⢦⢰⠀⣴⣋⣀⣀⣀⣠⡇⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⢿⡾⢹⣅⣀⣀⣀⣹⢻⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠉⠈⠉⠉⠀⠸⣇⣠⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢻⡖⠒⠒⠲⠶⠶⠶⠿⠶⠶⠶⠶⠒⠒⠚⣿⠋⠙⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣼⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠙⠲⢶⡶⠒⠋⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠⡇⢸⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠈⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠳⠤⣤⣤⠴⢿⠀⠀⣿⠦⢤⣤⠤⠞⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣠⣤⣶⡿⠃⠀⠿⣷⣦⣤⣀⣀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⡜⠀⠀⠀⠀⣀⠠⠴⠲⢄⠀⠀⠀⠀⠀⡔⠂⠙⠃⠲⠦⣀⠀⠘⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⡀⠠⠀⠀⢀⡤⠚⠁⠀⠀⣀⠜⠀⠀⠀⠀⠀⠳⢤⣄⣀⡀⠀⠉⣆⠀⡇⠂⠒⢄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡧⠁⣀⡈⢀⠀⢠⠟⠀⠀⣠⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠳⠌⡉⠐⠾⠀⡆⣡⠂⢨⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡌⠸⣦⠘⠒⣂⠝⢋⣁⣉⡁⠀⠀⠀⠀⠀⠀⡠⠤⣶⢤⣄⡈⠂⠀⣼⠃⢹⠆⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡆⠀⠈⢳⣄⢁⣎⣘⣷⣯⡤⢥⡀⠀⣀⣀⣠⠒⠲⠿⠿⠶⠿⡥⢼⠏⠰⡘⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣳⠇⠀⠀⠙⠙⣄⠀⠀⠀⡀⠀⣸⠋⠉⠉⠻⡀⠀⠀⢀⡠⠞⠀⣸⠀⠀⢱⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⡄⠀⠀⠀⠂⠀⠙⠓⠛⠒⠈⠁⠀⠀⠀⠀⠈⠀⢻⡉⠉⠉⠉⠁⠀⢠⡏⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠉⠆⠀⠀⣀⠴⠒⠉⠉⠀⠀⠀⣀⠀⢀⡠⠒⠒⠊⠉⠑⠢⣄⠀⠀⠆⢰⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣈⣧⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡨⠤⠤⢄⡀⠀⠀⠀⠀⠀⣀⣾⣧⣀⠀⢀⣠⠔⠊⠁⠀⠈⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⢁⣈⣁⣀⡩⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⣀⠴⠊⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡏⢣⡀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⠁⠀⡠⠊⢸⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣇⠔⠹⣄⠑⣄⠀⠀⠀⠀⠑⠄⠀⠠⠤⠤⠤⠄⡠⠔⠃⠀⠀⢀⠔⠁⠀⡜⠰⡎⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠹⠀⠀⠹⣦⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⡔⠀⡜⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢻⠀⠀⠱⡑⡄⠀⡜⠛⠒⢄⠀⠀⠀⠀⠀⠀⢀⡤⠎⠤⡀⠀⢀⠞⠀⡴⠁⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢱⡄⠀⠸⢌⠺⠀⠀⠀⠀⠱⡀⡀⠀⢀⡴⠋⠀⠀⠀⠈⡤⠋⠀⡔⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠁⠈⢢⠀⠀⠀⠀⠀⠘⠪⡴⠋⠀⠀⠀⠀⢀⠜⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠑⣄⠀⠀⠀⠀⡜⠐⠁⠀⠀⢀⡴⠃⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠈⠢⡀⠀⢸⠀⡀⠀⢀⡴⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢘⠢⣅⠀⢁⠔⠋⠀⣸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣸⠀⠈⠒⠁⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠖⠚⠉⠉⠉⠉⠉⠉⠐⠶⠦⣀⣀⡀⢀⣀⣀⡴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠲⣄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣶⠉⠱⠀⠀⠀⠀⠀⠰⠛⢶⡄⠀⠀⠀⡁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠀⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢃⠀⢀⣼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣴⠀⠀⠀⠀⣶⠄⠉⠉⠀⠀⠀⠀⢠⡆⠁⠀⠀⢴⡖⠀⠈⢠⣼⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠀⠀⢸⠇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠘⠇⠈⡄⠀⠀⠀⣤⠔⠒⠒⠒⠒⠒⠒⠛⠒⠒⠲⢤⡀⠀⢠⠜⠇⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢄⠀⠏⠀⠀⠀⠀⠩⣉⠉⠛⠿⠛⠒⢦⡀⠹⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠻⠇⠀⠀⠀⠀⠀⢈⣀⣤⠄⠀⠀⣈⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠤⠤⠄⠁⠤⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠢⠔⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⡄
⠀⠀⢸⠁⠀⠀⠀⠀⡐⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠃
⠀⠀⡌⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⡀⠀⠀⠀⠸⡄
⠀⠀⡇⠀⠀⠀⢀⡟⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⠀⠀⡅
⠀⠀⠃⠀⠀⠀⡾⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡆
⠀⢸⠀⣀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⡀⢀⣀⣼
⠀⢸⡀⠈⠁⣾⠀⠀⠀⠙⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠀⠀⠀⠸⠋⠁⠀⣹
⠀⠈⠙⠳⠶⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠉⠉⠀⠀⣼⠀⠀⠀⠀⠀⣶⠐⠊⠉
⡴⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠸⢁
⠱⣤⣀⡀⠀⣿⡌⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠶⠒⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠛⠀⠁⠐⠂
⠀⠀⠉⠉⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠁⣼⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⢀⣨⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⣈⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠈⠇⠀⠤⣀⣀⣤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⡤⠤⠤⠶⠞⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣰⣁⣠⢟⣟⣀⣀⣜⢻⣂⣈⣆⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢨⢏⠈⠛⠥⠼⠀⠘⠧⠼⠛⠁⠙⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠈⠒⢴⠖⢿⡛⣻⠖⠖⠢⠴⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠓⠤⠛⠣⠔⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡟⠢⡤⠄⣀⣀⠠⢤⠖⢹⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢠⡆⠀⠱⠐⠊⠑⠂⠎⠀⢰⡄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣠⠞⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠆⠀⡆⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⡇⠈⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡸⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠇⠀⠀⠫⢲⣷⡆⠤⠤⠤⠤⠤⠤⢠⣾⡒⢽⠀⠀⢀⠇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢄⣾⠀⠀⢰⠀⠀⠀⠁⢉⠭⠭⡍⠁⠀⠀⠀⢸⠀⢠⣧⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠀⠀⠀⢘⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⡀⠀⢸⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡇⠀⠀⠈⡀⠀⠀⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠘⠀⠀⠀⠀⢸⠀⠀⡌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⠈⢉⣁⠇⠀⠀⠀⠀⠘⢆⣉⠉⠈⠁⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⠂

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⠴⠦⣄⣀⣀⣀⣀⣀⣀⡤⠤⠦⠤⣀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢠⡶⠿⢶⠀⠀⠀⠀⡴⠾⢶⣄⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠤⢧⠀⢠⠇⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢇⠀⡾⠤⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣷⠋⠀⢒⡶⠈⠁⠀⢱⠀⠀⠀⠐⠁⠀⠑⢶⡒⠀⠀⢸⢾⠁⠀⢹
⠀⠀⠀⠀⠀⠀⢳⡀⠀⡟⡆⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡀⣠⠖⠉⢉⣉⣩⣭⣭⣭⣉⣉⠉⠑⢦⠀⠀⡼⠁⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⠇⢀⠞⠉⠀⠙⠢⠤⠖⠁⠈⠑⣄⠈⣧⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⢤⣀⡀⠛⠟⠃⠀⢀⣠⠼⠀⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠉⠉⠉⠉⠉⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⢂⣈⣉⣉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠤⠂⠋⠀⠀⠀⠀⠈⠢⡄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠤⠛⠋⠉⠈⠉⢉⡐⠢⠀⠀⠈⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⠀⠀⠀⣑⠄⠀⠎⠉⠙⢳⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠴⢴⠇⠀⠢⠘⣛⣁⠼⠂⠘⢌⠗⠀⠤⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠤⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⢃⠀⡴⠁⠀⠀⠀⠀⠈⢰⠀⠀⢀⠃⠀⠀⠸⢖⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢨⠴⠦⠀⢀⡄⠀⠀⠒⠛⣤⠐⠁⠀⠀⠀⢰⡏⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⢀⡔⠦⡀⠀⠀⠀⠀⠢⠄⣄⠀⠀⣜⢀
⠀⠀⠀⣠⡤⠒⠒⠋⠓⠢⠤⣤⢴⡊⠁⠐⠠⠘⠓⠂⠤⠤⠔⠊⠁⡠⢊⠝⠉⠉⠉⠙⢲⣄
⠀⢀⡾⠉⡇⠀⠀⠀⠀⠀⠀⠈⢢⠹⡁⠳⣄⣀⣀⡄⠀⢀⡤⠲⠏⣠⠋⠀⠀⠀⠀⠀⣸⠈⡆
⠀⡾⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠾⠁⠉⡅⠀⠀⢈⠶⢁⠞⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⠄
⢸⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢦⠀⡇⢀⡴⢂⠴⠉⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡀
⡎⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⠗⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠁
⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠶⠞⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰
⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⡴⠒⢉⡤⢤⡔⢲⠖⢢⣙⢶⡤⠄⣀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠇⢀⡀⢀⡠⠊⠀⢑⠴⣉⡠⣾⠀⢿⠀⢸⠉⣦⢈⣑⣏⠀⠀⠀⢠⡄⠀⠀⠀⡇
⡇⠀⠀⣠⡴⠛⠀⠈⠉⠱⣄⠀⠳⣴⠁⢀⠿⠤⢿⣀⠾⣤⣿⡀⠉⡄⠈⠑⢤⣤⣠⠀⠀⢰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠊⢃⠀⢸⡄⠀⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⠀⠋⠙⠀⢀⡎
⢳⣀⣀⣀⡤⠤⠖⠚⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⠢⡀⠀⠀⠀⢀⡾
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠙⠶⣤⡴⠛⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠉⠛⢳⡄⠀⠀⠀⠀⠀⣿⠀⢸⣀⣤⣤⣄⣀⣠⠁
⠀⠀⠀⣿⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠄⠀⠤⠤⢄⣿⠀⢸⡇⠀⠀⠀⠀⠉⣳
⠀⠀⠀⢹⡖⢸⠃⠀⠀⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢸⡇⠀⢠⠇⠀⠀⠙⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠂⣿⠀⣸⠀⠀⠀⠀⠀⠉⢸
⠀⠀⠀⠸⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢰⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢻⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠶⠼⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⠉
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⣿⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣾⠀⣿⠀⣿⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡟⠀⣿⠀⣿⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⣀⣿⢰⡇⠀⣿⣀⣽⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⡷⠺⠁⠀⠈⡟⠛⠛⠛⠛⠉⠛⠛⠓⠲⠶⣤⣀
⠀⠀⠀⠀⠀⢠⡎⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⢸⣧⠀⠀⠀⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨

⠀⠀⠀⠀⢠⡟
⠀⠀⠀⡰⠁⡇
⠀⠀⡰⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠒⠒⠒⠤⢄⣀
⠀⢠⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⡘⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠙⢄
⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂⠁⠐⡀⠀⠀⠀⠀⠀⠑⢄
⠀⢾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠘⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⠀⡀⢀⣉⣠⡜⠁⠙⣆⡀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⢱⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠖⠚⠋⠉⠉⠀⠀⠀⣸⠀⠀⠀⣀⠴⠒⠺⡄⠀⠀⢨⠉⠉⠉⠉⠛⠓⠒⠒⠂⠵⠄
⠀⠀⠀⢇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠒⠒⢲⣤⡤⠦⢾⡶⠛⢩⠃⠀⠀⢠⣇⠀⢀⡜
⠀⠀⠀⠈⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠐⠚⢤⠤⠠⠤⢚⠝⠛⡇⣘⣲⡁⠀⠈⢗⣀⡴⠚⢯⡩⠽⣁⡀
⠀⠀⠀⠀⠸⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠴⠁⣠⠠⠐⣁⣠⣬⣿⣿⣿⣿⣷⣶⣤⣀⡆⠀⠀⠀⠀⠉⠉⠉⠉⠒⠢⣄
⠀⠀⠀⠀⡠⠷⠒⡷⠤⡀⢀⣀⡀⠀⢀⣀⡀⠀⣠⣾⣿⣿⣿⣿⢿⠀⠀⢹⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⡎⠀⠀⢸⠀⠀⢩⡁⠀⠈⢏⣉⣁⣀⢹⣿⣿⣿⣿⣿⡏⠈⢣⣤⣬⣤⣤⣥⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⢀⡤⠶⠦⣄⣀⣀
⠀⠀⢸⠀⠀⠀⠈⢦⡼⠆⢃⠀⠀⢸⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢀⠏⠀⢰⠀⢀⣇⣀⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡞⠀⠀⠀⢠⠏⠙⡄
⠀⠀⠀⢆⠀⠀⠀⠈⠓⢦⣿⣀⣀⣸⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣼⣀⣤⣾⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣤⣶⣿⠀⠠⠁⠀⠀⢀⠇⠄⢰⢿⣦⠀⠀⠈⢓⢆
⠀⠀⠀⠀⠉⠐⠒⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢇⡀⠀⡘⢶⠖⠁⠀⠳⠑⢤⣠⣖⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⠀⠀⠀⠁⠀⠙⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠐⢩⠛⠙⠋⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣓⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⢸⢨⠁⠀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠈⠉⠒⠒⠆⠀⢸⠀⠉⠉⠉⠑⠛⠉⡇⠘⠀⠀⢀⡽⠋⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠉⠑⠒⠤⡜⠀⠀⠈⠉⠓⠒⠒⠒⠒⢚⠃⠀⠀⢾⠁⢀⠔⠁⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠰⠤⢤⣀⡀⠀⢰⠃⠀⠁⣄⠀⣀⡀⠀⠀⠀⢸⢰⠀⠀⠀⠳⠋⠀⡠⢊⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠳⢶⣤⣍⡓⠊⠀⠀⠀⡏⠀⠀⠈⠉⠉⠑⡇⡄⠀⠀⠀⠀⠘⡫⢶⠋⠉⠉⠒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⡼⢿⠀⠈⠙⠒⠶⢄⡇⠀⠀⠀⠀⠀⢰⠁⠃⠀⣀⠤⠒⠋⠀⡸⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠓⠉⠁⠀⠀⠀⣼⡁⠀⢀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣹⠿⡏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⢿⣃⣔⠫⠊⣿⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠀⣉⡛⠉⠀⠀⠈⠉⠙⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠃⠈⠑⠒⠒⠴⠦⠤⠤⠤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡉⠲⢤⣤⠤⠤⣤⠖⠈⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⢙⣳⣿⠀⠀⠈⣷⣛⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢈⣡⡴⠟⠒⠚⠚⠷⣄⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠁⠀⢧⣀⣠⠶⠊⠉⠀⠀⣀⠀⠀⠀⠀⠀⠉⠢⣤⠟⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢠⣤⣬⠁⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢼⣿⣿⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⡀⠀⠀⠀⠉⠙⠻⢷⣶⣤⣀⠈⠻⠿⠋⠀⠀⠀⣀⣀⣤⣤⡶⠖⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⠀⠀⠀⣶⠿⠟⠋⠀⠀⠀⠀⠀⠀⣴⣧⣄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⣤⡄⠀⠀⠁⢩⡄⠀⠈⠛⠓⠀⡶⢶⠀⢲⡖⠀⠠⠤⠤⠤⠤⠤⣤⠄⠀⠀⠹⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⣿⣇⡠⠀⠀⠘⣇⠀⠀⠀⠀⠀⠑⢚⠀⡾⠻⠀⣉⡉⠉⠃⢀⠾⠳⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠿⠏⠁⠀⠀⠀⠈⠁⠲⠶⠶⠖⣊⡵⠎⠀⠀⠀⠢⠉⠉⠉⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠉⣠⣊⣙⣶⣙⣆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡗⠀⠀⠀⠀⠀⠀⢀⠤⣆⢀⣠⠴⠞⠻⢭⣁⣀⣠⠏⠙⠳⢦⣄⠀⠀⠙⠻⠿⠛⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡞⠉⠉⠉⠉⠉⠳⠀⠀⠀⣧⠴⠛⠉⠀⠀⠀⠀⠀⠈⢉⣤⣤⣀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠀⠀⠠⠂⠀⠂⠁⠀⢴⣿⣷⡄⠀⠀⠀⠀⠀⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢺⡉⢁⠀⠀⠀⠀⠐⠢⠀⢀⣀⣈⠛⠛⠁⠀⠀⠀⠀⢀⣀⠀⠠⠄⠀⠀⠀⠀⢠⣤⢀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢏⠀⢸⡒⠁⠐⠦⠄⠠⣄⣉⠀⣉⡉⢉⣉⣁⠀⣀⡤⠀⠤⠤⠴⠂⠀⠀⠻⣿⡦⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⢆⠀⠀⠀⢠⣤⣶⣴⡄⠀⠀⠀⠀⠀⠀⣶⣦⣦⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⠈⠛⠓⢲⡄⠘⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠛⠟⠋⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⠙⠒⠒⠲⢶⡶⠶⠶⠶⠦⠤⣤⠶⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠀⣨⠂⠀⠀⠀⠀⢀⡑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣻⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⠀⣴⣩⡤⠿⠂⠀⠀⠀⠀⠀⠐⠻⠤⣍⣳⠀⡟⠉⠉⢹⡀
⠀⠀⠀⠀⠀⢸⡷⢶⡀⢸⠀⠋⠁⢀⣤⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠙⠀⣟⠀⡴⢶⡇
⠀⠀⠀⠀⠀⣿⠀⢺⣇⠸⡆⠀⠑⠢⠉⠀⠰⠔⠒⠢⠄⠀⠈⠕⠚⠀⢀⡇⢸⣷⠀⣿
⠀⠀⠀⠀⠀⠘⠷⣬⣏⠀⠙⠶⠶⠒⢶⣃⡀⠀⠀⠀⢀⣘⡶⠖⠶⠶⠋⠀⢸⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡀⠉⠉⠓⠒⠒⠉⠉⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠲⠶⠤⠤⠤⠶⠖⠋⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⣠⠶⠋⢡⡟⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢻⠻⣦⡀
⠀⠀⣴⠃⠀⠀⣼⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠀⠀⠀⡆⠀⠙⠷⣤⡀
⠀⣸⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠳⣄
⢀⡿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢹⡶⣦⣀⡀⠀⠈⢷
⣸⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⠀⢸⠇⣼⢀⡟⢳⠀⢸
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡟⢦⢿⣠⣏⣼⣡⣿⡴⠏
⣿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎⠙⠲⠶⣶⠏⣹⢹⠁
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸
⣿⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⢸
⠈⣿⠉⠉⠉⢻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠺⠤⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⢿⢸
⠀⣿⠀⢠⣶⠸⣧⠉⠙⠲⠶⠶⣤⣤⣤⣤⣄⣀⣿⣤⣄⣤⣤⣤⣤⡶⠶⠞⠛⠉⣹⠏⠀⠀⣿⢸
⠀⠸⣧⠸⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⢸
⠀⠀⠈⠳⡿⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⠶⠶⠦⣤⣤⣤⣿⣦⣤⣤⠴⠶⠶⠶⢶⣾⡁⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⢿⣼
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠠⠆⠀⠤⠠⠄⠤⠀⠄⠤⠠⠠⠤⠠⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⣶⠛⣦⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣴⣿⣷⣿⣿⣍⣿⣾⡛⣻⣿⣉⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⠀⠐⢿⡿⠛⠛⠛⠋⠉⠉⠻⣿⣥⣿⣉⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠻⠇⠀⠗⠋⠀⠀⠀⠋⠀⠀⠀⣀⣤⡤⠀⠙⣿⣾⣏⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⣆⠘⢋⠕⣶⡤⠀⢿⠿⠻⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡹⠀⠀⠀⠀⠀⠀⠁⠖⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⠠⠦⠀⠀⠀⠀⠀⢰⣏⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣒⠒⠂⠀⠀⠀⣠⣾⣿⠟⣿⣴⡀
⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⣤⣴⠞⠋⠸⡏⣴⣩⡷⣿
⠀⠀⠀⢠⠁⠀⠀⠀⠀⣾⣀⣀⣀⣠⣤⣤⣄⣀⠀⢀⣀⢀⣠⣤⣤⣀⠀⠀⣽⡿⠛⠁⠀⠀⠀⢳⣾⣭⣷⠏
⠀⠀⠀⢸⡆⠀⠀⠀⠁⠈⠉⠉⠉⠀⠀⠀⠀⠉⠙⠻⢍⣻⡁⠀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠙⡿⣷
⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣷⠀⠀⠘⠿⣻⡋⠛⠻⣄⠀⠀⠀⢠⡾⠁⣸⠷⣄
⠀⠀⠀⠀⠉⠛⠿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⢠⡟⠀⠀⠀⠀⣀⣀⣴⣶⣸⣆⡴⠞⠋⠀⢰⠏⠀⠈⠓⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢶⣤⣤⣤⣹⣻⣿⡃⣘⠀⣶⠀⠀⠀⠉⠉⠀⢸⢧⣿⠞⠉⢻⡄⢾⠀⠀⡠⠀⠀⠀⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣟⣴⡟⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠑⠿⣠⠞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⢹⡟⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⠄⠀⠀⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠄⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠙⠀⡀⢸⣿⣀⣠⡤⠶⠶⣄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠹⡿⠓⠚⣛⣓⠢⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠈⢢⠘⣧⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣷⣤⠶⠿⠿⣿⣯⡒⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢲⡀⠘⡇⣿⠀⠀⠀⠀⠀⠈⠀⣻⡧⠞⠋⠀⠀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⢄⠘⢦⠘⢾⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠛⠄⠀⠀⣃⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠋⠉⠀⠀⠀⠀⢀⣀⣀⣤⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⢀⡠⠤⠖⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢂⠈⡀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣼⠁⣶⣶⣆⡄⠀⠂⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣶⣿⡿⢿⣿⣖⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠠⠙⠄
⠀⠀⢀⣠⣴⠶⠆⠀⠉⠑⠲⢤⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠀⢠⡇⠀⠀⠀⠀⣂⠈⡄
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⠳⠖⠚⠉⠀⠀⠀⠀⠀⠐⢦⡀⠀⢸⡇⠀⠀⠀⠀⣿⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣶⡾⣷⡀⠀⠀⣴⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⢻⣦⡀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⢿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⡿⣿⡿⠃⡄
⠀⠀⠀⠀⢀⣠⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠚⠛⠹⠛⠉⢁⣄⠸⣿⣿⣿⣿⣷⡼⠁
⠀⠀⠀⢠⣿⣿⣿⠟⠁⠐⢻⣿⠋⠀⠀⠈⠙⠻⢿⣿⣷⣤⣀⣀⣠⣶⣿⣿⡿⠿⠾⠿⠟⠋
⠀⠀⠀⠉⠁⠛⠋⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⡽⠋⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣈⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠒⠚⠛⠉⠀⠛⣆⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠉⠒⠄⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠤⢄⣀⡀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠑⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣠⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⡀⠀⠀⠀⢠⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠧⠀⠀⠀⡴⠒⠿⠀⠀⠘⠃⠃⠀⠀⡀⠚⠘⠀⠀⠠⠗⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡯⠴⠂⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⢀⡝⢂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠑⠦⠤⠴⠊⠀⠀⠂⣠⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⡀⠀⠀⡤⠴⠒⠋⠁⠀⠀⠀⠀⣼⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⡀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡜⠀⢀⡝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠘⠿⠾⢿⣦⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣴⣮⣛⠿⠷⣤⣤⣤⣴⠿⢛⣩⣶⣄⠀⢸⡇⠀⠘⠁⠀⠀⠀⢈⣿⠉⠉⠉⠁⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣵⣾⣿⣿⣿⣿⣻⣿⠿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠰⠂⠀⠈⠃⠀⠀⠀⠀⢸⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣻⣿⣿⣾⣿⡟⠁⠀⢠⡍⠁⠹⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢳⡀⠀⢣⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⣯⠉⠀⠀⠀⠈⠋⠀⠀⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢦⣸⠤⠊⠀⠀⠀⠀⠀⠀⢀⡠⠒⠖⠋
⠀⠀⠀⠀⢹⣿⣿⣾⠋⠉⠀⠀⠸⠇⠀⠰⠆⠀⠀⠀⢠⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠑⠲⡄⢠⡤⠤⠴⠚⠉
⠀⠀⠀⠀⠘⣟⣿⠉⠀⠀⠀⠀⠀⠀⠀⢶⡖⡏⠀⣠⠾⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠊⠋⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⢡⡀⢀⣀
⠀⠀⠀⠀⠀⠈⠻⢷⣟⣷⣄⣀⠀⠀⠀⠀⣩⠵⠚⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠘⢿⠂⠀⠀⠀⠀⠀⠀⢳⢀⠁⠌⠀⣵⡽⠮⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⢉⣉⡉⠉⠀⠉⠀⢄⠀⠀⠀⠀⢸⡟⠟⢻⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠲⢄⠘⣿⡭⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠾⣧⣄⠀⠀⠀⠀⠙⢦⣤⠴⠚⠁⢀⣠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⠈⠟⠐⢕⡰⣙⠿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣽⡿⠀⠀⠐⢤⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⢁⡄⠒⣂⠖⡰⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣈⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠉⠉⠐⠒⣤⢀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠤⣤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡦⡇⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⣄⠰⠆⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠩⡒⠝⠜⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜⠉⠑⠂⠬⠭⠥⡍⠛⠛⡀⠀⢸⠀⠀⠀⠃⠀⠀⡔⠒⠂⠠⠤⠤⠤⠐⠒⠂⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⡄⢸⠀⠀⠚⠁⠀⠀⠈⠙⠀⠀⡇⢠⠼⡀⠀⠀⣣⣀⢠⠁⠀⠰⠛⠀⠀⠀⠛⠂⠀⢰⣀⡸⡀
⢀⡴⣞⡶⠀⠀⠀⠀⢸⠀⢦⠆⠀⠀⠀⠒⢀⣀⠒⠂⠀⠀⠰⣔⠀⡇⠀⠀⡁⢨⡝⠁⠒⢲⠉⢱⠒⡎⠉⡖⠂⠉⢅⠃⠰⠀⠀⠀⠀⠀⣴⣲⢦
⢿⡾⢁⠔⠣⢄⣀⡤⢄⠑⡆⠀⠀⠠⠢⣀⣁⣀⣀⠔⠂⠀⠀⠰⠊⠀⠀⠀⠑⢼⠀⠀⠀⠄⢉⡉⠉⢉⡑⠤⠀⠀⢸⠔⠛⣄⡀⠀⣀⠼⠀⠙⢯
⠀⠈⠉⠲⢄⢀⠏⠀⠜⠑⠰⡄⠀⠀⠀⠈⠒⠂⠁⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⣜⡀⠐⠛⠈⠙⢄⣀⠔⠊⠉
⠀⠀⠀⠀⠀⢏⢀⠎⠀⠀⡠⠊⡳⠤⢤⢄⣈⣀⣠⣤⠤⡚⡍⠲⡒⠒⠒⣖⠴⠒⡞⠒⢄⣀⣀⠀⠀⢀⣀⣠⠤⠚⣅⠀⠀⠀⠀⡄⡸
⠀⠀⠀⠀⠀⠀⠉⠒⠄⣠⠁⠀⠇⡆⡆⠀⣿⣻⠆⢈⠀⠇⡁⠀⢱⡔⡟⠁⠀⡌⠀⠀⢰⠀⠣⡉⠉⢁⠜⠁⡇⠀⠈⡆⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠁⠈⠉⠡⠆⠉⠉⠀⠰⢠⠀⠀⡀⡀⠀⠰⠀⠀⠀⠀⠓⠀⠚⣿⠓⠤⠐⠁⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⢸⠀⠀⠀⠠⠆⠀⠀⠀⢀⢸⠀⠀⡟⠾⠃⢹⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⠀⠀⠠⠆⠀⠀⠀⠈⢸⠀⢰⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⠈⡀⠀⠀⠠⠆⠀⠀⠀⠰⠘⠀⡆⠀⠀⠀⠀⢇⠈⠉⠉⠀⠉⠉⠀⠈⠉⠀⠈⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡇⣇⠀⠀⠀⠀⠀⠀⢀⡆⣥⠴⠀⠀⠀⠀⠀⠆⠈⠉⠈⠁⠐⠒⠒⠒⠂⠈⠁⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠈⠉⠉⠉⠉⠉⠈⠁⠙⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠤⢠⢧⠤⠤⢎⡀⠀⠀⠀⠀⠀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⣤⠤⠾⠰⠦⠤⣤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠈⢲⣳⣞⣶⣾⠴⣷⣶⣁⢶⡏⠉
⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣾⠂⠀⠃⣼⣧
⠀⠀⠀⠀⢀⠞⢻⡇⠀⢹⣿⣇
⠀⠀⠀⠀⣿⣷⣾⡇⠀⠸⣿⣿
⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣁⡌⠙⣦⠀⠀⠀⠀⢀⣴⠿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠘⣧⠀⠀⠀⡼⢋⣀⠀⠀⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠸⠲⢄⠀⠂⠀⠉⠀⠚⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⣶⣬⠀⠀⠈⢧⣀⠘⠟⠻⢆⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣈⣿⠀⠀⠀⠈⢻⣧⣤⣴⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠉⡿⠿⠟⠒⠙⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⣿⣇⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⢿⣿⣦⣀⣀⡴⠁⠀⠀⠀⠀⠀⢂⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠈⣽⡯⠰⣿⣦⣄⣀⣀⣭⣿⣦⡐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⠀⢀⣀⣀⡈⠙⠿⣿⣿⣿⣿⠃⠻⣦⣀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠀⠉⠻⢶⣦⣶⡿⣿⠃⠀⠀⠈⠙⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣩⣿⣿⡾⠃⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⢶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠲⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡎⢦⣤⣤⣤⣤⡰⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠐⣿⣿⣿⣟⢥⡹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣘⣿⣿⣿⣿⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠉⣻⣿⣿⠃⠀⠀⠀⠀⠀⠻⢿⠈⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣃⣻⣤⣭⣭⣭⣬⣥⣭⣥⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡄⠤⠠⠄⢀
⠀⠀⢠⣄⣶⠶⢤⣤⡤⠾⡄⠷⣲⣤⣿⣼⣼⠿⠶⠞⣇⠠⠄⠰⡢⣀⡤⡶⣼⣦⢠⡀⠤⠴⠖⢶⣀⣠⠄⣀⠆⡀⢀
⠀⠘⢛⣯⠍⠋⣀⡞⣊⣀⣬⢩⣿⠋⠀⠈⠀⢀⢠⡧⣿⠀⠀⣳⢏⣺⠉⣯⣻⠻⠻⢧⡀⠀⠀⠀⡘⢀⠉⣈⠿⠽⣉⣖⠳⠶⡠
⠀⡾⠋⠈⠿⠚⠻⡟⢻⣄⠀⠈⣇⠀⣠⠖⠆⠛⢺⠃⢷⠈⠋⠁⢘⠈⠁⢿⢩⣇⣀⡀⠁⢉⠛⣋⠙⡶⠟⠋⠳⠶⣎⠉⠙⠠⡿⠄
⣰⢅⡐⡄⢀⣄⢰⣵⣶⠿⣆⠚⠙⢤⠀⠀⠴⡱⠈⠳⣜⡅⠀⢈⠉⢠⠄⢠⡔⠇⠈⠁⢠⣀⠀⠀⢀⢁⡀⠠⢤⡀⠱⢢⡄⠠⠶⢄
⡉⢉⢿⠁⠸⠾⢿⣼⣟⡣⢝⣶⣤⣤⣦⣄⣴⣤⣁⠀⣸⣧⣀⠁⣨⣿⣞⣫⡭⠽⣦⠠⠤⣿⣶⣀⣆⢛⣠⡨⡌⠁⠀⠈⠠⠤⠀⠁
⡇⢀⡎⠀⡀⢰⠋⡔⠈⢻⣿⠚⠉⠉⠛⡍⠹⢿⣍⢚⠿⡶⢶⣿⣿⡃⠈⠁⢀⣼⣍⣓⣢⠜⣫⡽⠿⣿⡉⣗⢆⣀⠚⠀⠀⠐⢠
⠀⢨⡔⠛⢁⠜⣴⠆⢀⣿⡟⢶⣶⡶⠀⠹⠿⣾⠉⠛⠶⢼⡦⢹⣻⣯⡄⠺⠗⢻⣫⣿⡇⣸⡿⠁⠀⠈⠏⢿⣰⣅⠀⠠⠀⠀⢠⠅⣆
⠰⠶⢦⠔⠤⡄⠀⠀⠀⢉⣽⠆⠈⠀⣶⡄⠀⢑⠶⠉⠶⣶⠶⣾⣿⠃⠛⢁⠀⡈⡎⢉⣿⣚⠁⠀⣀⢀⠀⠘⣿⣧⢦⡆⣶⡐⢃⣃⠞
⠀⠡⡀⠀⠀⢦⠀⠀⠀⠈⠙⣤⣀⢀⣀⣄⠠⣊⡑⠳⠀⠀⠀⢚⡅⡘⠀⣀⣉⠉⣀⢸⡆⠻⠗⠂⠈⠉⠉⢬⡿⢺⣿⠀⠀⠙⡀⠏
⠀⠄⠈⡖⠒⢚⠤⠤⢤⡤⠖⠀⠿⣷⡶⢦⣎⣭⣀⣴⣧⡀⢠⢾⣧⣀⠈⠈⠃⣺⡯⢼⣧⢄⡀⠀⠦⢆⣸⠆⠒⠉⠁⠀⠀⠄⡉⠁
⠀⣀⡘⢆⢄⣀⣤⣄⣀⠿⣄⢶⠈⡏⣿⢻⢀⣩⡤⠿⣟⣿⡶⠫⡿⡿⡿⢦⣾⠉⠀⠀⠀⠀⠹⣆⡤⢤⢄⢰⣾⡿⠿⢦⣌⠓⠃⢠⢣
⠀⠘⢧⣶⠼⠈⠈⡍⢧⠚⢁⠀⠈⠁⠹⣿⣉⣀⠀⠀⣠⣭⡓⡬⣀⣲⠤⢤⣟⠀⠀⢀⣀⠀⡄⡈⠙⡿⢟⣿⡯⢝⠷⣞⣙⣖⡚⠈⢰
⠀⠠⢨⣧⠀⠀⠀⠀⠉⠄⠀⠠⡀⠀⠠⢸⡿⠿⠋⠀⠛⠿⠂⣿⠇⠠⣤⣼⡟⠋⡄⠈⠛⠉⠀⠅⡸⣥⢿⡉⠁⢠⣿⡍⠉⣙⠃⢄⣰
⠀⠀⠀⡄⠀⠀⠀⠀⣀⣦⡤⠰⠛⠻⡿⠋⢯⡀⠠⠤⠄⠀⢠⢅⠀⠀⠉⢉⠳⡔⠚⠂⠀⠀⠀⠈⠁⠀⠀⢃⢀⣷⣦⠆⢀⠋⠀⠂⠁
⠀⠀⠀⣾⠢⠶⢊⠩⢑⠃⠄⠀⠀⠀⠀⠀⢱⡄⡀⢄⡲⠀⡄⠀⠀⠀⠀⢄⣲⣮⠀⠂⢀⠀⠀⡀⠀⠀⠀⣸⢣⣄⡤⠄⠀⠺⠄⠂⡄
⠀⠀⡘⡧⢼⣷⡰⣴⠆⡿⠃⠀⠏⠁⠀⠀⢘⣭⠵⠶⠷⢿⣾⡓⢤⣄⡄⣄⣿⣟⣶⣶⡇⠰⠻⠿⠵⢀⣶⠋⠶⡳⣦⣀⠀⠶⠿⠿⡃
⠀⠀⠁⢰⠾⡏⣀⣈⠀⡴⠶⢦⠄⠀⠀⡰⠿⠋⢉⠱⠶⠸⠊⣷⢺⣯⢐⠀⣹⣿⣾⠀⠁⠀⠀⠀⠀⠀⣻⠑⠃⢈⣼⠗⠷⠖⢶⣿⣧
⠀⠀⠀⠈⠠⠳⡙⠸⢤⡜⠀⠈⢢⡄⢞⣭⡘⠿⠿⠶⠄⢲⠺⢿⠁⠀⢸⣤⣲⠁⢹⣖⠂⣠⣄⡐⡀⣸⠿⣶⠄⣈⢀⠀⠀⢠⣟⡼⠝
⠀⠀⠀⠀⠦⠈⣀⢤⡈⣋⡁⡴⢊⣙⢛⣽⡀⠀⠘⠰⠆⠈⠷⠰⠁⠀⢠⣤⣤⠀⣼⠱⣶⣭⣡⣠⡰⡿⠀⠀⠀⠋⢩⠄⠀⡈⡁
⠀⠀⠀⠐⢄⠳⣁⠀⠀⠙⠁⠔⠙⠋⣤⠄⠀⠀⠀⢶⠀⠠⠔⡰⠀⠀⠀⠀⠁⠐⣣⢭⠙⣯⣭⣻⡇⠊⠚⠂⣤⠀⠠⠄⢊⠠⠁⠉
⠀⠀⠀⠀⠀⠁⠚⠛⢀⠘⠀⡰⠀⢀⠐⠶⢂⠔⠛⠒⠠⠅⠊⠀⠀⠀⠒⠒⠐⠈⠀⠀⠘⠠⠨⠄⠁⠀⠀⠊⠀⠑⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⣤⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣰⠋⢸⡀⠀⠷⠞⢛⣿⣤⠤⠴⠶⠶⠦⠤⠤⢤⣄⠀⣠⠏⠀⣠⣶⢀⣴⡛⠁
⠀⠀⠀⠀⠀⠀⢰⣾⡇⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠀⠺⣅⡸⠀⠈⡿
⠀⠀⠀⠀⠀⢠⡴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⣀⣀⣀⣀⣐⠚⠿⠴⠋⠀⠀⠀⠀⠀⠀⠸⠭⠭⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⠀⠀⠸⡉⠉⠉⠉⡏⠉⡏⠉⠉⢙⡷⠂⠀⠀⠀⠰⣖⠛⠛⠛⢻⡓⠒⡖⠒⠒⢒⡆⠀⠞⠶⠦⣄
⠀⠀⢠⡖⠚⠛⠀⠀⠀⠈⠓⠶⠤⣽⣼⡥⠴⠒⠉⠀⢠⡤⠴⢲⠀⠈⠓⠦⣄⣀⣧⣠⣇⣀⡤⠊⠀⠀⠀⣀⣠⠟
⠀⠀⣤⠟⠓⠂⠀⠀⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣾⠀⠀⠀⠀⠀⠈⠉⢉⡉⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠙⠶⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠞⠲⠃⠀⠀⠀⠀⠀⠉⠷⠀⠀⢀⣠⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠾⠴⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠒⠒⠛⠋⠉


⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠀⠒⠒⠎⠉⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⠒⠂⠐⠺⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠶⣄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡴⢋⢉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡠⠐⠠⡑⠀⠀⠀⠀⠔⢀⠀⠀⣸⠁⠀⠈⠃⢣⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣼⠁⠋⣿⣻⠄⠆⠀⠈⢹⠀⣀⠔⠁⠀⠀⠒⣁⠎⠀⠀⠀⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢦⣰⡬⠟⠁⠀⠀⠀⠚⠈⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠢⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⣆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣄⡀⣀⣠⠄⠈⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠇⠀⠀⠀⠀⠀⢹⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠇⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡞⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣀⣀⣀⣀⡠⠤⠖⠉⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠏⠁⠀⢀⠈⠁⠀⠀⠀⠈⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠤⠄⠐⠂⠉⠀⢠⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⢀⣠⠔⠉⠀⠈⠉⡗⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢠⠁⡄⠀⢰⡠⠂⠀⢻⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⡯⣄⣣⠀⠫⠤⣤⠶⡼⣦⢀⣤⣤⡴⠁⠀⠀⠀⠀⠀⡀⠐⢁⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠘⡏⠁⠈⠑⠒⠳⠀⠸⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡙⢁⠄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠋
⠀⠁⢸⠀⠀⠀⠀⠢⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣣⠂⡠⠊⠀⢀⠀⠙⠦⢄⣀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⡴⠁⠀⡠⡛⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠒⠁⠣⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠉⠀⠀⠀⠀⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠁⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠐⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⣇⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡤⠤⠤⠤⠤⠤⠤⠤⣤⡤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠚
⠀⠀⠀⠸⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⣀⡀⠤⠔⠒⠉⠀⠀⢨⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠔⢒⡒⠚⠉
⠀⢤⣤⠴⠃⠀⠀⠀⠘⠃⠒⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⡀⣤⠰⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣶⣦⣬⣭⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠉⠀⢰⠀⡠⢀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⡀⣀⠀⡆⠀⠉⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⢄⣉⣡⣍⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠷⠂⠐⠾⠿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣡⣌⣉⡠⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⡀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣰⠢⠧⠴⠜⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠛⢦⣼⣧⡶⠛⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿
⣿⣿⠃⠀⠀⠀⠀⠀⣿⠿⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠿⣿⠀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣷⣶⣶⣆⣀⠀⠀⠀⠀⠀⢀⣠⠀⣀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⣄⡀⠀⠀⠀⠀⠀⣀⣰⣶⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡇⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⠤⣀⡀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠔⠒⠒⠒⠉⠉⠑⠂⠒⠄⣀
⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠀⠀⠘⠀⠉⠈⠉⠐⠢⢖⠁⠈⠑⢦⡀
⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⣱⢄⣀⣠⠚⠀⢀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢁
⠀⠀⠀⡸⠀⠀⠀⠈⡖⠀⠉⠀⠀⠀⠀⠑⠲⡁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠆
⠀⠀⠀⢇⠀⠀⠠⠊⠀⢠⠤⠀⠀⠀⠰⢦⠀⠙⢄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠔⠁⠘⠢⢀⡀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⡎⠲⠆⡦⢄⡀⢀⠄⢤⡀⣀⠄⢄⠀⣀⠼⡆⠺⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⠀⡴⠒⠈⠋⠁⠀⠀⠀⠛⠂⠉⠐⡄⠀⣰⡀
⠀⠀⠀⢇⠀⠶⠁⠀⠀⠙⠄⢠⠥⢽⠄⠔⠁⠀⠀⠱⡇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⣠⠁⠀⠀⢠⣠⠀⠀⢤⡄⠀⠀⠀⢡⣘⠀⠱
⠀⠀⠀⠀⠙⡂⠀⠀⠀⠚⠤⢀⣀⣀⠠⠜⠂⠀⠀⠀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢸⠁⠀⠀⢀⠄⠀⠊⠑⠀⠀⢀⠀⠀⠀⢡⡤⠃
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠉⠒⠒⠊⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢀⠤⡶⠋⠠⠄⣀⣀⠀⠀⠀⢀⣀⠠⠄⠚⠳⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⢠⠃⠌⠀⠀⠸⡀⠀⢀⠝⣣⡉⠣⡄⠀⢀⠃⠀⠱⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⠎⠀⢈⠓⢖⠒⠤⠤⠔⡲⠒⡉⠀⠈⢫⠐⢄
⠀⠀⠇⢸⠀⠀⠀⠀⠑⠔⠁⠀⢈⡀⠀⠈⠓⠃⠀⠀⠀⢡⠀⠈⢢⠀⠀⠀⠀⠀⣰⠁⠀⠀⡌⠀⠀⠸⡀⠈⠢⣄⣠⠔⠁⢀⠇⠀⠀⠀⠃⠀⠑
⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⢢⡀⠀⠡⡀⠀⠀⡄⢠⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠰⠆⠈⠉⠁⠀⠀⠀⠀⢀⠀⠀⢰⠤
⠀⡆⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠑⣄⣀⠧⣠⠞⠙⣆⣑⣲⠞⡀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠜⢠
⢀⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⣄⠞⠁⡴⠋⠀⠀⠁⠀⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⢬⠣⢾⠁
⢸⣀⣀⡎⢡⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⢿⣿⣧⠃⠀⠀⠀⠀⠀⢣⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⠆⠰⠀⠈
⡤⠤⠤⡇⠈⢆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢸⠉⠁⠒⠐⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⣤⠒⠀⢨
⢇⣀⡀⠂⠀⠀⠇⠁⠉⠉⠀⠒⠒⠒⠀⠈⠉⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠄⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣸⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠤⠤⠿⡤⠤⠴⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⣞⣤⣠⣀⡴⠇⢧⣄⣠⣤⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣖⣡⣑⣪⡷⠺⣕⣊⣑⣍⡿⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠰⠛⢀⠠⠔⠒⠒⢒⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠀⠀⣆⡡⠔⢈⣡⠄⠤⠄⠐⠚⠂⠀⣀⠀⠴⠦⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⣁⡴⡈⠻⠳⠶⠶⠶⠾⠦⢉⠳⠞⠛⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠋⠠⣽⡋⠴⡙⠁⠰⠀⠘⠋⠉⢀⣍⠉⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢤⠁⠐⠠⠌⠒⠉⠀⠀⠂⠁⠀⠀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⣠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⡤⣆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⠀⠠⠐⠀⠀⠀⡰⢄⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠉⠰⠦⠄⠀⠚⠳⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠒⠀⠀⠠⠀⠀⠀⠀⠁⠐⠂⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡤⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⡀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⡄⠉⠈⠉⣂⠰⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠠⠀⠒⡠⠈⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠁⠀⡤⣄⡠⣀⠀⠀⠀⠠
⠠⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠍⠆⠁⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠣⡑⡁⢴⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⣈⡄⠠⠉⠁⠀⢀⠈⠒⠐⠁⠀⠄⠀⠀⠀⠀⢀⠠⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠘⠀⠤⠀⢀⡠⠋⠃⠀⠀⢨⣄⡈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠐⢐⠊⠀⠀⠀⢇⠀⠀⠀⢀⣤⡒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠂⠀⠁⠤⣀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠒⢀⠀⠀⠀⠀⠀⠀⣈⠠⠒⠁⠀⠈⡇⢄⡠⠒⠒⡴⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠴⡁⠘⠀⣤⠤⠬⡐⠀⠈⠀⠘⢿⣷⣄⠀⠀⠀⠀⠉⠀⠀⠋⠀⠀⠀⠀⠀⢸⠗⠀⣤⡀⠀⠀⠃⠜⠀⡐⠉⠙⠀⠐⢀⡄⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠀⠘⠃⠄⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⢀⡀⠀⠆⠀⠀⠀⠀⠀⠀⢐⡟⢁⠀⠀⡞⠀⠰⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠋⠐⠁⠌⢈⠃⢉⠒⠠⠤⠘⠓⠂⠤⡈⡉⠿⠅⡀⠀⠰⠂⠀⠀⠀⠠⠤⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠸⠇⠀⠐⠀⠀⠀⠁⠀⠄⠀⠀⠀⠁⠐⠀⠴⢁⠌⠁⡃⠀⠀⠀⢠⠌⠁⠀⠀⠁⠀⠀⠐⠤⠄⠂⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡬⡌⢈⠂⡀⠀⠀⠀⠐⠀⡠⠀⢀⠀⡠⠔⣀⠴⢡⠀⡈⣁⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣳⡄⠢⠀⠀⡀⠀⠀⢀⡐⣛⠛⠻⠓⡻⠂⠤⠀⠀⠀⠈⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠒⠐⠒⠉⠀⡄⢋⠌⣀⠈⢀⠈⢀⡴⠠⢖⡺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠉⠀⠙⠓⠈⠛⠂⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣀⣤⣀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠙⡆⢀⠁⠠⠀⠀⢠⣼⣿⣿⣷⣌⠋⣿⣧⣰⣷⢶⣿⣄⢰⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠹⣾⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⢷⠀⠇⠓
⠀⠀⠀⠀⠀⠀⠈⠣⠘⠿⠿⠓⠀⠠⣤⣐⣿⣿⡿⣋⠙⢟⣿⢿⡟⠿⠛⠻⢧⣁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠄⣘⡿⠉⠁⠻⠛⠁⣿⣿⠊⠁⠀⠀⠀⠀⠘⣛⡶⢦⡀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠀⢠⣞⣥⡆⡀⠷⠀⠀⢶⣿⠿⠃⠀⠀⠀⠀⠀⠀⢻⡟⢺⠳⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠉⠠⣿⠁⠀⠀⣸⠃⠀⠀⠀⠀⣄⠀⠀⠀⠀⢷⣸⣇⠁⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣟⣾⡿⢷⣿⢶⣿⣤⡄⠀⠅⠀⠀⠀⠀⠀⣋⣀⣀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣹⣿⠑⢬⡙⠛⣿⢻⣠⣴⠀⠀⢀⡀⠀⣼⣟⣻⡿⣳⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢈⣴⣯⡼⠋⠀⠀⢀⠞⠁⣾⠋⠤⠶⠚⠃⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿⣧⢀⣿⣿⣽⣇⢴⣤⠶⣤⣤⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⣴⣿⣿⣿⣿⣿⣿⣿⣷⣼⣥⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣋⡉⠛⠉⢉⠛⠛⣳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡛⠻⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⢛⣏⠙⠒⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣆⡀⣤⣀⣤⣾⣤⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠘⠛⠋⠉⠀⠐⠲⠒⠛⠿⢻⣦
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣄⣤⡀⠀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠏⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⡟⢻⠅⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⣿⡟⠁⠨⠁⠐⣿⣦⠀⠄⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢰⣾⣿⣿⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡶⠶⠶⢶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⣀⣀⣀⣰⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⡀⠀⠀⠀⠀⡀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⠞⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⢀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠉⠙⠻⠿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⣿⣧⠀⢀⣾⠋⠁⠀⠀⠀⣠⡈⢻⣄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄⣀⣤⠖⠀⠀⢸⣿⣠⡿⠁⠀⠀⢘⣶⣿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡖⠒⠶⣦⣿⣿⣿⡟⠉⠀⠀⠀⢀⣼⣿⣿⣧⡀⢦⣤⣿⣿⣿⠟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣌⡻⣿⣿⣷⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣴⣿⣿⠟⣵⣾⣷⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣍⡉⠙⠛⠛⠉⠁⠈⠛⢿⡿⠃⣸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢹⣿⡸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡛⠻⣶⣄⠀⠀⠀⠀⠀⠀⢳⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠘⣿⣷⡹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⣿⣿⣿⣿⣿⢿⣶⢿⣿⣷⣤⣀⣠⣾⣦⣼⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣜⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢠⣿⠟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣶⣶⣶⠗⠀⠀⠀⣼⠃⠀⣾⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠘⣿⣷⣤⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⣼⠃⠀⢰⣿⣿⣄⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠠⢶⣦⡀⢸⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⣿⣿⣿⣷⣝⣛⠿⠿⢿⣿⣿⣧⣤⣤⣄⠀⠻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⢛⣠⡈⢻⣿⣟⣓⣺⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣷⣄⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⡿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣿⡇⠱⠀⠀⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⡀⢠⠀⣿⣿⣿⡜⡿⢡⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⠀⣿⣿⣷⣿⡆⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣿⠻⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠹⡇⠘⢿⣿⡇⠀⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⢦⡙⠇⠀⢹⣿⣿⠏⢠⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣄⢸⣿⡟⠀⠘⣹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⢰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⢏⡛⢿⠉⡝⠙⣏⠏⣹⠟⡯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡒⠐⠒⠲⠤⢤⣔⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⢤⣶⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢻⠁⠋⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⢸⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡇⡀⢤⡐⠂⡒⠂⠀⢀⣙⣿⣿⣿⣧⡆⠀⠀⠀⠙⠦⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣠⣤⣬⠿⠿⠿⠗⠃⠓⠐⠺⠿⠿⢿⣿⣿⣿⣿⣀⣀⣀⣀⣠⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣬⣭⣭⣭⣿⡿⠿⠛⠛⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠻⢿⣿⣿⣿⣿⣿⣿⡭⠟⢡⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠉⡃⣠⢲⠶⠶⣶⣿⣿⣿⣿⣧⡭⣍⣾⣿⣿⣿⡩⠂⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢣⢢⢸⣧⡀⠈⠙⠧⣴⣿⠋⠀⢹⣿⣿⣛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢼⣿⠻⠄⠀⠀⠀⠉⡲⢢⣴⣿⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⢠⣾⣥⣶⣎⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢿⣧⢀⣷⣿⡏⠁⢲⣶⣶⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⣠⣿⣿⡏⣾⡏⠻⣷⡼⠷⠶⠋⠰⠿⠟⢛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣠⡞⠁⣹⡟⢸⣿⣷⠀⠘⣿⣦⣦⣦⣾⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠚⠋⡿⠀⠀⢻⠃⠸⣿⣿⣷⡀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⢃⣀⣠⣨⠀⠀⠻⣿⣿⣿⣄⠀⠘⡿⣠⣿⣿⣿⣿⣿⣽⣿⣿⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⡿⢿⣤⣤⣀⠈⠻⣿⣿⠦⣀⠰⢯⣿⠟⠉⠉⢉⣈⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣬⣿⣿⣿⣦⡈⠻⣷⣄⠀⡾⢋⣠⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠿⠦⣽⣿⠿⠵⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠅⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣜⣛⠁⠀⣀⣀⣀⡀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣷⠀⢻⠿⣿⠅⠀⠀⣼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⣿⣧⣶⣶⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠦⠤⠛⠟⠀⢀⣾⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠉⠁⠀⣰⣷⣿⠏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣤⣾⡿⣯⡝⠀⠀⠈⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿⣿⣿⣿⣿⡟⣹⣿⠁⠀⠀⢀⣠⣶⠊⣡⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠇⣿⣋⣩⣽⣿⣿⣇⣿⣿⡶⠤⠴⠟⢻⠃⢰⠇⠀⠀⠀⠙⠓⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠀⠉⠩⠀⢸⠀⠀⠈⠉⠻⣿⣿⠁⠀⠀⠀⢀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠑⡯⠀⠀⠀⠀⠀⡸⠀⢰⠀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢠⡏⠀⠀⡆⠸⣿⡇⠀⠀⠀⠘⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠃⠀⢼⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣾⡷⠀⠀⠁⠘⢿⣷⣀⡅⠀⠀⢣⡀⠀⠀⢠⣷⡀⠀⠀⠀⡜⠀⠀⣿⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠩⣿⣿⣿⡄⠀⠀⠀⠱⢤⣀⣼⣿⡃⠀⣠⠞⠀⠀⢀⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⣀⣀⣤⣿⠟⠉⠛⠋⠁⣀⣀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠛⠉⠙⠛⢧⣄⡀⠀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣿⡏⠀⠀⣠⣾⣿⣿⣿⣿⣵⣦⣤⣤⣤⣄⡤⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣍⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⢀⠠⠃⡸⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡅⡼⠁
⠀⠀⠀⢿⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡾⠋⡾
⠀⠀⠀⢸⣇⢰⣶⣶⣄⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣉⣱⣒⣛⣳⣾⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠚⠉⠉⠉⠀⣿⣿⡿⠟⠛⠉⠁⢠⠉⠻⠟⠻⢿⣿⣿⣿⣿⣿⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠶⠓⠒⠛⠚⠚⠓⠒⠒⠒⠒⠢⠤⠅⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠲⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣖⣋⠉⠉⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠚⠧⣀⠙⣦⣤⣤⣤⣤⣴⠏⠁⠈⠉⠉⠉⠉⠲⣄⠙⣦⣀
⠀⠀⠀⠀⢰⢻⡹⠝⣦⢾⣩⢤⡏⠀⣰⠀⠀⠀⠀⢰⡄⠀⠈⡗⠛⣶⠶⠶⠶⣟⢀⣤⠀⠀⣤⡤⠀⠀⠈⣧⣸⠈⣧
⠀⠀⠀⠀⢫⡃⠀⠀⠀⠘⡇⢸⡇⠀⠉⠀⠰⡄⠀⠈⠀⠀⠀⠀⣴⣿⠟⡦⣄⣽⠘⠉⠀⠀⠉⠈⠃⠀⠀⠁⣸⡇⣿
⠀⠀⠀⠀⠀⠑⣄⡀⣀⠴⠳⢤⣧⠀⡠⠊⠀⠀⠀⡀⠙⢆⠀⠀⣯⡀⠀⠀⢸⣿⠀⡔⠁⢀⡀⠀⠑⢆⠀⢠⠏⡇⣿
⠀⠀⠀⠀⠀⡼⢡⠿⣅⠀⠀⠀⠈⠳⢤⣀⠈⠛⠋⠁⠀⣀⣤⡞⠁⣽⢦⠞⠁⠈⠳⣄⣀⡙⠋⣀⣀⣠⠴⣏⠀⡇⢻
⠀⠀⠀⢀⡞⢡⠏⠀⠈⠲⣄⠀⠀⠀⠀⡼⠉⠉⠉⠉⠉⠁⠀⠙⢦⣺⠈⢳⡀⠀⢰⠃⠈⠉⠉⠉⢹⠁⠀⠈⡇⡇⢸
⠀⠀⢀⡞⠀⠻⠶⠶⠶⠶⠶⠿⠶⠶⠶⠷⠶⠖⠴⠖⠶⠶⠶⠶⠸⠻⠤⠴⠝⠷⠿⠦⠴⠦⠤⠤⠾⠦⠴⠴⠟⠁⢸
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾
⢠⢏⡴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻
⡏⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀
⣇⣘⣦⣤⣇⣀⣠⣤⠤⠤⢤⣤⡤⠤⢤⣤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠖⠒⠒⠒⠒⢒⡶⠶⠒⠲⢶⣒⠒⠚⠂⢸⠀⠀⠀⠀⠲⠒⠒⠒⠂⠀⠀⣨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⣄⣀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡤⣤⢤⡀⠈⢦⠀⠀⢸⠄⠀⠀⠀⠒⠒⠚⠛⠋⠠⣄⡸
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡞⠎⠉⢪⣇⠀⢹⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡇⠀⢾⡎⠀⢙⢹⠀⢸⣆⣀⡼⠁
⠙⠒⠒⠒⠒⠒⠒⠛⠛⣇⠀⠳⣕⣐⡯⠃⠀⡾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡀⠈⠪⠴⠵⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠻⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢣⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣶⣶⣦⣤⣤⣄⡠⢄⡬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⡀⠀⠀⢇
⠀⢠⣦⢶⠒⣀⣤⡀⢐⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡯⣶⡵⣂⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣬⣓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠁⠀⠀⠀⠀⢀⣤⣀⡀⠄
⠀⠀⠀⠀⠀⡐⡰⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⠿⣻⠂
⠀⠀⠀⢀⢠⣷⣻⣿⡿⢿⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣽⢷⣶⡔
⠀⠀⠀⠶⠿⠿⠛⣧⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⠻⣿⣿⣿⣿⣯⣟⣿⡿⠻⠛⠛⢉⣥⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⡹⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠈⠙⢻⣿⣿⡅⠀⠀⠀⠀⢢⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣿⣿⣿⣿⣿⣿⡿⣿⠏⠀⠀⠀⠀⣽⣿⡷⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⡯⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠭⣽⣿⣿⣿⣿⣟⠭⠊⠁⠀⠀⠀⠀⠀⣽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠉⠚⠩⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⣞⡽⣻⠽⡫⠅⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠩⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣴⣶⣶⣿⣿⣶⠄⠀⠀⠀⠀⠀⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣻⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡝⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⢎⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡷⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠵⠫⠊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠁⠀⠀⠀⢰⣠⣄⠖⣀⣀⣴⠀⠒⠀⢠⢂⡠⠒⣶⡂⢆⡀⠀⠒⢀⠐⠀⠀⣲⠀⣆⠐⠂⣲⡐⢀⡆⢰⠀⡆⠀⡴⠀⡰
⠀⠠⣄⠘⣃⡄⠀⠀⠀⠁⠈⠀⠀⠈⠉⠁⠁⠀⠈⠂⠈⠂⠁⠂⠀⠁⠀⠀⠈⠐⠁⠀⠊⠁⠉⠈⠂⠉⠈⠀⠀⠉⠀⠑⠀⠉⠀⠀⠀⠀⠀⠀⠋⠀⠶
⠤⠤⠤⠿⠴⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠤⠶⠤⠦⠤⠤⠴⠤⠤⠤⠤
⠀⠀⠀⠀⠀⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⣠⠤⣀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⣄
⠀⠀⢀⡄⠉⠀⠀⠀⠀⠀⠁⢆⠀⠀⢀⠤⠀⢄⠀⢠⣏⠀⠀⠀⠱⠀⠀⠸⠿⠀⠈⠑⠂⠒⠁⠀⠀⠁⠀⠀⠀⠀⢠⠂⠁⠀⠂⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠀⠀⠀⠈⠉⡠⠀⠒⠁⠀⢬⡷⠀⠀⢼⣦⠄⠀⠀⢀⡠⠂⠠⠤⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⢂⡀⡰⠃⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠁⠀⠸⠀⠀⠇⠰⢀⡀⠀⠀⣀⠐⠀⢀⠂
⠀⠀⠑⡄⠀⠘⠃⠰⠇⠀⠀⢂⡀⠀⠀⣀⠜⢣⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠆⠑⠀⡼⠒⠋⠀⠀⠉⠀⠰⠀⠉
⠀⠀⠀⠘⡀⠀⠀⠀⠀⣶⡦⢄⠈⠀⠈⠀⢀⡰⠃⠔⠀⠀⠀⠀⠀⠄⠀⣀⣠⣄⣐⣀⠀⠄⠀⠀⢀⠔⠀⡰⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠜⠀⡤⠊⢉⠠⠀⢸⠀⠀⠀⡄⠑⡄⠀⠀⠀⠀⡀⠠⠄⣀⠴⠂⠉⡤⠒⠉⠰⠁⠀⠈⠒⠻⠟⣀⠐⠀⠐⢦⡀⢀⠄
⠀⠠⠚⠂⣆⠀⠀⢀⠀⠀⡈⠀⢘⠀⠀⠃⠀⠈⠁⠢⣀⢀⡀⠀⠱⢄⣀⡠⠊⠀⠀⢠⠉⠲⠄⢸⠀⠀⠀⠐⣄⠀⠀⡀⠔⠘⠋⠒⡄⠀⠀⠁⠁
⠀⠁⠤⠔⢸⠀⠀⠀⠀⠀⠡⣀⠘⢄⠀⢣⣀⡔⠁⠀⠀⢀⠀⢀⡴⠖⠒⠂⡄⠀⠀⢀⠀⠀⠸⠀⠉⠐⠀⠀⢢⠉⠁⠀⠀⠀⠢⠤⠂⣠⡄⡏⠉⠓⡄
⠀⠀⠀⠀⢨⠀⢠⠖⠀⠑⠀⠈⢣⠀⠀⣀⡉⠀⠀⡀⢀⠉⠀⡎⠀⠀⢱⠀⠈⢦⡀⠈⠓⠚⣁⠔⠂⠒⠒⢄⠀⠄⠀⠀⡔⠀⠂⠀⢏⣙⣁⠈⢢⠀⠘
⠀⡠⠂⠒⡄⢉⠖⠉⠑⠄⠘⠀⠎⠀⠊⠀⡀⠀⡰⠀⠀⡷⠀⠐⣀⡰⠂⠠⠒⠒⢏⠐⠒⠊⠁⡠⠈⠉⠢⡀⠱⠈⠢⠤⠃⠀⡈⠀⠀⠀⠀⠉⠁⠀⠀⡆
⠀⠑⢄⣀⡟⠁⢀⠀⢀⠈⠢⡀⠀⠈⣥⠔⠁⡠⠃⠀⢀⠓⡐⠀⠉⠀⠁⠀⠀⠀⠀⠁⢤⡀⣠⠁⠀⠀⢀⠇⠀⠗⠀⢀⡀⠚⣁⠀⠀⠀⠀⠁⠀⠀⢠⠁
⠀⠀⠀⡰⠁⠀⠀⠀⣌⠉⠂⠀⢈⡡⠤⣌⠉⠀⠀⡠⠎⡀⠃⠀⠀⠀⠐⠂⠐⠂⠀⠄⡈⠁⢹⠀⠒⠈⠁⠀⡴⠂⡔⢁⠄⠉⠻⠁⠀⠄⠀⠀⠀⠔⠁
⠀⠀⡜⠁⠀⠀⠀⠀⠿⠁⠙⠀⡏⠠⣶⠄⠀⠒⠈⢡⠊⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠠⠀⠚⡀⠀⢀⠊⠁⠀⠀⠀⡄⠀⠀⠀⠀⠘⠦⠀⣀⣀⠈⠂
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠃⡀⠈⠂⡀⠀⠀⠀⢱⠀⠀⠆⠀⠀⠙⠋⠘⠛⠀⠀⠠⠀⠀⠈⠑⠢⡕⠢⡀⠀⢣⠈⠒⠀⡄⠀⠀⠁⠐⠄⠀⠽⠆
⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠆⠀⢰⠀⢀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠤⠄⠠⠌
⠀⠀⠀⢱⠀⠠⠤⠒⠂⠀⠀⢠⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⢀⠀⠀⠀⠛⠀⠀⠀⠰⠿⠗⠈⢠⠃⠀⠀⢰⠀⠨⡁⢀⠄⠴⠓⠂⢄
⠀⠀⠀⠈⡄⠀⠀⠀⠀⣀⢤⡠⠤⠄⣀⠠⠃⢀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣀⣀⠔⢋⠁⠐⠂⠸⠀⣀⠤⠤⠤⢀⢣⠀⠀⠡
⠀⠀⠀⠀⠈⠁⠒⠀⠉⠀⢈⠀⠀⡠⠃⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢰⠀⡃⡁⠘⠢⠀⠀⠀⠀⠈⠛⠁⠀⠈⠒⠒⠰⠁⠰⠀⠀⠀⠀⠀⢰⠉⠈⠁
⠀⠀⢠⠔⠂⠀⢈⡈⣀⢀⡠⢖⠋⠀⠎⠀⠲⣀⠜⢀⠀⠀⠀⠀⠀⢸⠀⡥⡄⢘⡁⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⡖⠢⡄⠓⠀⡁⠀⠁⠀⢄⣀⣀
⠀⠀⠈⢄⠀⠀⠸⠀⠠⠀⠥⠈⠤⡀⠈⠉⠉⡃⠠⠿⠃⠀⠠⠀⠀⢨⠀⡆⠀⢨⣁⠀⠀⠄⠀⡰⠃⢀⠆⠉⠁⠀⣡⣤⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠰⠀⠀⠀⠀⣰⣦⠀⠀⢀⡀⢰⣦⠀⡠⠄⠶⢄⡀⡇⠀⠀⠀⠀⢉⠉⠀⠀⠀⣀⣤⡀
⠀⠀⢀⠚⠀⠈⠀⠀⠀⠀⠀⠀⢰⣶⠔⠀⡸⠀⠀⡇⠀⡤⠂⠉⠀⠀⠀⠹⠏⠀⡹⠸⠀⢠⡄⠀⠐⢵⢄⡀⠀⠀⢸⠀⠀⠀⠘⠈⠡⠠⠄⠀⠀⠀⠸
⠀⠠⠁⠀⠀⠀⠀⠙⠃⠚⠁⠀⠀⠀⡠⠘⢀⡠⠖⠀⠘⠀⠀⠀⡠⠄⠊⠁⢀⠜⠁⠀⠇⠈⡇⠀⠀⠀⡆⠙⠄⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⠁
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡜⠀⠀⡤⠀⠀⢯⡀⠀⡁⠀⠀⠀⠘⣄⠀⠀⡀⠀⠱⡀⠀⢠⡅⠀⡀⠀⠀⠣⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⣎⠀⠀⢀⡴⠉⡆⠱⣄⠀⠀⠀⠀⠑⠀⣀⠀⢀⣉⠐⠁⠀⡐⠀⠀⢀⣀⣈⠒⠠⠤⠄⠒⣀⠄⠒⠠⡀
⠀⠀⠑⠒⠒⠒⠂⠠⣀⠀⠀⠀⠀⡆⠀⠐⢄⡈⠁⠈⠉⠠⢀⢃⠀⠈⠑⠒⠒⠒⠈⣡⠐⢠⡀⠀⢣⠀⢰⠀⠀⡐⠁⠀⠈⡆⠀⠔⠉⠙⠋⠰⡀⠀⢰
⠀⠀⢀⠒⠉⠈⠁⠢⡈⠑⠠⠄⠰⢁⣀⡀⠀⠈⠢⠀⠇⠀⠈⢋⠃⣰⣆⠐⠆⠀⢰⠁⠀⢀⡅⠀⡘⠀⢠⠀⢠⠁⠀⠀⢀⠇⠠⡀⠀⠠⡀⠀⡠⠀⡘
⠀⠀⣄⡄⠠⡀⠀⠀⠈⠑⠒⠂⡉⢀⡀⠈⠢⡀⠀⡇⠈⠑⠢⡈⢆⠉⣹⠀⠐⠄⠀⣦⡄⠋⠀⢠⠁⠀⠈⠦⢘⣀⢀⠔⠃⠀⠀⠿⠃⠀⠈⠉⠀⡐⠁
⠀⠀⠛⣄⠀⣨⠀⠀⠀⠀⣶⠎⠀⠀⠈⠰⡀⠂⠀⡇⢀⠤⠀⢾⢄⠁⠀⠀⠀⠀⠀⠀⠀⢢⡀⢸⠀⢀⠤⠤⡀⠋⡀⠀⢀⣲⡖⣖⣲⣒⣖⠀⡐
⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠰⠠⠄⠠⠌⠀⡗⠀⠀⣤⠀⣤⠀⠀⠀⠈⠆⠱⣇⠀⢀⡐⠀⢠⠀⠀⠉⠁⠉⠉⠉⠉⠀⠇
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠶⣦⠀⠀⠀⠸⠀⡇⠀⠑⣄⠀⡤⠊⠀⠀⡀⢉⠁⡉⠀⠀⡀⠀⠘⠀⠩⡈⠁⠀⢀⡘⠀⠀⠀⠓⠛⠒⠺⠀⠀⠰
⠀⠀⡠⠃⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠈⢂⠑⠀⠒⢂⠀⠈⠁⠁⢠⣤⠀⠠⠚⠀⠀⠈⠉⣉⡁⠀⠀⠀⠀⠘⠗⠺⠂⠀⠀⠀⠦
⠀⠘⡁⠀⠒⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠠⠀⠀⠀⠸⠀⠀⡴⠃⠀⠀⠀⠀⠈⠹⠀⠣⠀⠀⠤⠐⠉⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠈⠒⠈⠉⠉⠒⠤⠄⠀⠀⠀⠴⠊⠀⠀⠀⠀⠁⠒⠊⠀⠀⠀⠑⠂⠤⠤⠤⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⣤⠴⠞⣛⣉⡙⣦⡀
⠀⠀⠀⠀⠀⣾⡟⠛⠉⣉⣉⣻⣾⠹⡆
⠀⠀⠀⠀⢀⡟⠴⠂⠀⣀⣀⣈⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⠄⠀⠀⣀
⠀⠀⠀⠀⢸⡇⠐⠋⠡⣤⣀⡈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⠀⠠⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⢸⠁⠀⠀⠈⠙⣯⣴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢸⠀⣰⠀⠀⠀⣸⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠸⣇⠘⠓⢶⣚⣫⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⠞⣳⣒⡫⠍⠂
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠙⠽⣄⠻⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠺⠿⢶⣤⣾⣤⣄
⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠀⠘⢦⠙⣦⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣌⢳⡄⠙⢶⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠹⣆⠀⠀⠀⠀⠈⢷⣄⠀⣸⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢻⣄⠀⠻⣦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⠻⣿⢠⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠳⢦⣈⠻⢶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠀⢀⠈⢷⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⠰⣄⠈⣿⡷⣄⢀⣀⠀⠀⠀⠀⠸⣇⠹⡆⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠈⡇⣸⠀⠘
⠀⠀⠀⠀⠀⠀⠸⣇⠚⠀⣿⠈⠚⠛⠒⠂⠀⠀⠀⠀⣿⡿⡅⠦⣤⣤⣤⣴⠾⠋⢀⣴⡣⠀⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⣦⡀⠀⠀⠀⠀⠤⠤⠞⠁⣧⠀⠀⠀⠀⠀⣀⢄⣀⢯⠏⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⡛⠛⠶⠶⠶⠖⠂⠀⠀⠏⢀⡆⠀⠀⠀⠈⠳⠥⠬⡇⠀⠀⣾⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠙⢦⡀⠀⠀⠀⠀⢠⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃⠀⠈⠻⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣙⠓⠒⠒⠀⢀⣠⣤⣤⣤⣤⣤⣤⣶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡉⠉⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠈⠷⣄⠀⠀⠀⠀⠀⠁⠒⠤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠤⠛⡱⠶⠖⠀⠀⠀⠀⠀⠈⠁⠒⠀⠄⠠⡤⠔⠒⠒⠤⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⠤⢴⠖⠒⠺⠿⠶⠦⢄⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡐⠀⠀⣌⠀⠀⠃⠀⠀⠁⠉⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠉⠙⠛⠲⠤⣤⣤⣤⣤⠴⠖⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢁⡨⡧⡀⠀⠀⠀⠀⠀⡀⠀⣇⠀⠈⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡤⠄⠀⠀⣠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠉⠅⠀⠃⡍⠀⠀⠀⠀⠀⠈⠳⠄⣉⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣬⣭⣅⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⢦⡀⠀⠀⢄⡀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠔⠀⡠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢦⠀⠑⢄⠀⠀⠈⠛⠒⠚⠋⠁⢀⡴⠋⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡙⠲⢤⣤⣤⡤⠤⠖⠉⠈⠁⢹⣤⣄⠘⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⡀⠀⠀⠈⠛⢦⡞⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇⠈⠆⣸⠧⠬⠂
⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠙⣆⠙⢦⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠁⠀⠀⠀⢢⣡⠈⣆⠀⠀⠀⣠⡾⠉
⠀⠀⠀⢀⠃⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠛⠀⠀⠀⠈⠉⠛⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⠀⠀⠢⢄⡀⠀⠀⠀⠉⣲⣷⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢦
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣬⣭⠶⠶⠟⠛⠛⠓⠒⠄⠀⠀⠀⠀⠐⢒⡷⠶⠶⠧⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠚⢿⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠶⠒⠛⠛⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠈⠳⣤⣤⡖⢡⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠒⠒⠒⠒⠂⠈⢶⣤⡀⠐⢤⡀⠀⠀⠀⠀⠀⠰⣌⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⢲⣄⡀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡄⠀⠀⠈⣶⡶⠆⠈⢙⣾⠿⠷⢶⣾⣿⣶⣄⢀⣙⣶⠆⠀⠲⣄⡀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⣠⡼⢻⠏⠉⠛⣇⠁⠁⣽⠟⠢⣄⣀⠈⠉⠉⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠘⠛⢫⠉⠁⠀⢠⣿⣤⢴⡏⠀⠼⡇⠀⠀⠀⠈⠃⡀⠀⠈⠉⠀⠉⠙⠣⠤⢤⡀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦⣄⡀⠀⠀⠀⠘⣇⠀⠀⠛⡏⢁⣨⣷⠀⣰⣷⡏⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠠⣴⠟⢦⣀⣼⣿⣿⣿⠃⠀⣿⣿⣧⡀⠀⠀⠀⠀⢀⡈⠱⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠁⠐⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠛⠛⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢁⣀⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡄⠀⠀⠈⠉⠉⠈⠉⠄⡰⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⢀⣷⡀⠀⠀⡀⣴⢶⣴⣿⣶⣄⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸⣿⡿⠂⣤⣄⢀⣼⣿⣿⣿⣿⣷⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⢈⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⣿⣿⣿⣿⣿⣯⠀⠈⠉⠛⠉⣿⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡼⣾⣿⣿⣏⣿⣿⣿⠟⠀⠀⠀⢠⣇⠘⠇⠑⠁⠂⠠
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣽⣿⢣⣾⣿⣿⣿⠏⠉⠁⠀⠀⠀⢀⠜⣿⣦⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠸⢃⣿⣿⣿⢹⡧⢀⣠⡤⣶⣶⡾⠃⠀⠈⠻⣿⡀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠠⠂⠀⠀⠈⢘⡟⣻⣿⡿⠟⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⡀⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠙⠀⠀⠀⠀⠠⠞⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⢠⡖⠀⣀⣴⡟⢠⡾⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠿⠛⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣾⣷⣶⣶⣴⣴⣶⣾⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⣿⣿⣛⣿⣿⣿⣿⣿⣋⣽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢋⣙⠉⢿⠿⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢏⣀⢾⣿⣧⣇⢰⣷⣿⣿⣿⣯⢿⡗⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣗⢿⣿⣿⣿⣿⣾⣿⣿⣿⡏⠛⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⢷⣝⣿⣿⣿⡿⠻⠿⠎⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣅⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢰⠃⠀⣀⠄⠀⠀⠀⠀⠀⠀⣷⡄⠀⠘⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⣀⠀⣿⣶⣿
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣿⣿⡿⠲⠶⢶⣤⣀
⠀⠀⠀⠀⠀⣴⠟⣽⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣍⠛⠿⣶⣄⠀⠀⢈⣿⠀⣠⣦⣦⠀⠀⠀⠀⠀⣀⡴⠟⣿⣿⠟⠙⣿⣿⣿⣿⢳⣿⣿⠿⡎⠹⣷
⠀⠀⣠⣴⣾⠏⢰⣿⣷⠉⠀⠀⢹⣿⣿⡟⢿⣿⣿⣿⣶⣤⣽⣷⣄⣀⣽⣿⣯⣼⣽⣷⣤⣀⣠⣞⣫⣴⣿⡿⠋⠀⣰⣿⣿⣿⣿⠀⠈⠻⣿⣷⠀⠹⣦⣀⡀⠀⡀
⠀⣰⡿⣿⠏⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣷⣀⡉⠻⠿⠿⣿⣿⣿⣿⣿⡟⠛⠃⠼⠟⠻⡿⢿⣿⣿⣿⣿⣅⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠈⣿⡄⠀⠹⢟⠛⠛⠻
⣰⠟⣸⠃⠀⠀⢸⣿⠁⠀⠀⠀⣿⠿⣿⣿⣿⡉⠛⠢⠤⣿⣿⣿⣿⢾⠟⠀⠀⠀⠀⠰⢀⣼⣿⣿⡿⠿⠋⢁⣼⣿⣿⠟⢹⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⠈
⠏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⡇⠹⣿⣿⣷⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⢀⣾⣿⠿⠁⠀⣿⣿⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠠⣿⣿⡄⠘⣿⣿⡆⠀⡄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⠀⢀⣼⣿⣿⣿⠇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⢰⣿⣿⣿⣼⣿⣿⣿⠀⣿⠀⠀⠀⠀⠛⠀⢀⠀⠀⢀⡀⠰⠀⠀⠀⢠⡿⠁⣿⡏⢠⣾⣿⣿⢸⣿⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠻
⡄⠀⠀⠀⠀⠀⠀⣧⠀⠀⢠⡜⣿⣿⡟⢿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠈⢳⡶⠞⠁⠀⠀⠀⠀⠀⠀⢀⣿⣁⣿⣿⣿⠇⣾⣏⠀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠸⣇
⣷⡀⠀⠀⠀⠀⢀⢹⡀⢰⣿⣧⢻⣿⣿⡌⢿⣿⣿⣿⣿⠀⠀⠀⠄⠠⠤⠤⠴⠁⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣷⡟
⢹⣧⠘⣆⠀⠀⢸⣿⡇⣽⣿⣿⣸⣿⣿⣿⡌⣿⣿⣧⣹⣇⠀⣀⣤⣶⣾⣿⣿⣿⣿⡿⣿⣿⣷⣶⡆⠀⣸⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡞⣧
⣾⣿⣯⣿⣄⣀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢀⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⠁⢀⡴⠀⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢀⣾⡿⣻⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⡏⣠⣿⡗⢀⣿⣿⡏⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠿⠋⢠⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⡟⣴⣿⡟⢀⣾⣿⡿⣼⣿
⢹⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡟⠁⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣾⣿⣟⣿⣟⣿
⣈⣿⡷⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣯⠙⣿⣿⣿⣿⣿⣧⣾⣿⣿⠁⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣷⣿⣿⠿⠟
⣿⣿⣿⣿⣿⣧⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⡏⢠⡀⠦⠀⣿⣿⣿⣽⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⡿⣿⣿⣿⡆⠙⣿⣿⣳⣿⣿⡟⢠⠯⡘⠀⢸⣿⣿⠉⣿⠃⢸⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣷⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣷⣿⣷⣤⣄⣾⣿⣿⣾⣿⣶⣿⣿⣿⣿⠿⡀⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣋⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣸
⣴⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠐⣊⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠒⠉⠉⠁⢇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢍⡑⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣭⠭⠭⠙⠀⠀⠀⠀⠀⠀⣀⢤⣒⠬⣛⡏⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢀⣀⣣⠃⠀⠀⠀⠀⣀⡠⣔⡪⠝⠊⠁⣠⣾⠗⢹⣃⣀⣀⡀
⠀⠀⠀⠀⡠⢒⠭⢒⠈⠉⠉⢑⣶⠭⡲⢄⢤⡲⠍⠒⠉⣿⠀⠀⡠⡪⢚⠥⣒⠭⡵⡖⠒⠒⢭⣕⠦⡀
⠀⠀⢀⢎⠜⠁⡀⠀⠀⠀⢀⣾⠏⠀⠈⠢⡣⡀⠀⣀⠤⢿⡠⡪⠊⡴⡡⠊⠀⠀⢧⢣⠀⠀⠂⠀⠙⢮⢢
⠀⠀⡎⠞⠀⠀⡈⠢⠀⢀⣾⡏⠀⠀⠀⠄⢳⢡⠎⠀⣀⡀⠈⠆⡘⡼⠀⠀⠁⢄⠘⡾⡀⠀⠀⠈⠀⡀⢧⢡
⠀⢸⢸⠄⠀⠀⠀⠁⢰⡞⢻⣴⣾⠶⠦⠬⢽⣼⠓⢴⡷⠃⠀⡼⡃⡁⠀⠀⠁⠀⠀⡶⢷⡀⠐⠀⠁⠀⢀⡎⡆
⠀⠈⡜⣄⠀⠀⠀⠀⠀⠉⠋⠠⡀⠀⠀⠀⢰⢰⠉⠢⡸⠖⠊⠀⡇⡅⠀⠀⠀⠀⡈⠛⠚⢃⠂⠀⠀⠀⢀⢇⠃
⠀⠀⠱⡱⡀⠀⠀⠀⠈⠀⠀⠂⠈⠢⠀⣠⢣⠃⠀⠀⠀⠀⠀⠀⠱⡹⡀⠀⡀⠈⠐⠀⠀⠀⠁⠀⠀⢀⠞⡜
⠀⠀⠀⠘⢮⡣⢄⡈⠀⠀⠀⠀⣀⡠⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠦⡀⠠⠀⠀⠀⠀⠀⢀⡱⣪⠎
⠀⠀⠀⠀⠀⠈⠐⠪⠭⠬⠥⠭⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠕⣒⣒⣒⣒⡪⠕⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣒⠀⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⢟⢷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⠳⠞⢿⡟⠫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣦⢺⣿⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣏⡡⠄⢺⣿⣿⣿⣿⣷⠹⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⡀⣀⡀⠀⢀⡾⠿⣿⣷⠀⡆
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈⠉⠐⢴⣦⣿⣿⣿⢟⣋⠑⠢⠒⠋⠉⢐⣢⣤⡤⠶⠒⠒⠶⣖⣉⡀⠀⠀⣤⣶⣾⣷⣿⠿⢶⣶⠮⣵⣿⣯⡼⠁
⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⢀⣠⠀⠀⠀⠀⠹⣿⢇⣼⣿⣶⡦⠀⢰⣶⣎⠉⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢸⣿⡃⠀⠀⣠⣾⣧⡀⠀⠀⠀⣠⠟⡿⢿⣿⡿⣰⠓⣸⣿⣿⡷⠦⢴⣶⡿⠟⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠈⣿⡇⠀⠰⠟⠀⠘⣿⣶⣄⣼⣵⣷⣿⡿⠿⢷⣡⣾⠟⠉
⠀⠀⠀⠀⢸⣿⢧⠀⠀⠀⠀⠀⣿⠿⠋⠀⣨⡝⠉⠀⠀⢠⣿⠏
⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠰⠋⠀⠀⠀⣸⡿⢳⣶⣦⠴⠋⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⡔⠀⠀⣼⡿⠁⠈⣿⠃⡜
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣤⣤⣀⣴⠿⠚⠂⢀⣿⡜
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠋⢉⡴⠶⠒⠀⣾⠁
⠀⠀⠀⠀⠀⢈⣿⣿⣿⣯⠀⠀⡰⠶⠦⢤⣏⡟
⠀⠀⠀⠀⢀⣾⣭⣽⣛⣛⣿⠿⢗⠒⣒⣻⡋
⠀⠀⠀⠀⠋⠀⠀⠈⠈⠃⠛⠙⠛⠙⠙⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠛⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠛⠛⠛⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠉⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠂
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣃⣀⣀⣀⣀⣀⣀⣀⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠉⠉⠉⠙⠁⠒⠀⠄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠁⠘⠆⠀⡠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⢀⠀⠱⡀⣿⠀⢀⠀⡔⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢢⠀⠑⠏⠀⠸⠜⠀⢠⠊⡠⠖⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣀⣀⢣⡀⠀⠀⠀⠀⠀⠈⠉⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠶⠛⠋⠀⠀⠀⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⢅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠘⠒⠢⡀⠀⠀⡙⠢⣀⡀⣠⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠌⢁⡀⠀⠀⠀⠀⠀⣠⡀⢢⠀⠀⠀⠀⠀⡰⠁⠈⡲⠁⠀⠀⠀⠠⠁
⠀⠀⠀⠀⢀⠔⠂⠁⢀⠀⠀⠀⠀⡂⡀⠿⠇⠀⠀⠀⠀⠠⠿⠃⠂⡀⠀⠀⠀⡅⠀⠀⢰⠀⠀⠀⠀⠀⡂
⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠄⠁⠀⠀⡁⢐⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠹⠀⢣⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡠⠁⢨⢁⡀⠀⠀⠀⠀⠀⡜
⠀⡜⠀⠀⠀⠀⠀⢀⣀⣀⡆⠠⡀⠁⠐⠀⠀⠀⠀⠠⠀⠀⠀⣀⡠⠔⠈⠀⢀⢦⠀⠀⠑⠢⢀⠤⠊
⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠷⡀⠈⠐⠀⠶⠒⠀⠄⠒⠂⠉⠁⠀⠀⠀⡠⠔⠁⠀⡆
⠀⢇⠀⠀⠀⠀⠸⡀⠀⠀⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⡇
⠀⠘⡄⠀⠀⠀⠀⢹⠀⠀⢡⠀⠀⠀⠀⠐⠓⠠⡀⠀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⡆⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⢲⡗⢆⠀⠀⠀⠀⠤⠀⠀⠀⢅
⠀⠀⠀⡇⠀⠀⠀⠀⣇⡠⠄⠈⠀⠀⠀⢀⠂⠈⠀⠘⠤⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠑⢄
⠀⠀⠀⡳⠐⠂⠀⠀⢁⡠⠊⡰⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠁⠠⠼⠀⠀⠱
⠀⠀⠊⠀⠀⠀⠀⠐⠁⠀⠉⠣⡀⠠⠀⠀⠁⠀⠀⠀⠀⠑⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⡑⠂⣀⠀⣀⠤⠋⢀⠂⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⠤⠤⠤⠤⠤⠤⡄⠂⠁
⠀⠀⠑⠒⠁⠀⠀⠀⠀⠘⠦⣅⠀⠧⢤⠤⠀⠄⠐⠀⡰⠂⠈⠢⢀⠀⡀⠀⠀⠂⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠠⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡄⠀⠁⠀⠁⠀⡔⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⢀⣠⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠾⠀⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⢄⠀⠀⠂⠀⠈⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⢈⠀⢀⡀⠃⠀⠀⢸⠤⣀⠀⡠⠊⠉⠐⠒⠂⠀⡖⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠳⡀⠀⠀⠀⠖⠃⠀⠀⢣⠐⣄⢀⡀⠀⠀⠁⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠚⠁⠈⠀⠀⡠⠊⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢱⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠚⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠷⠒⣶⣶⣶⡶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⡿⠋⠁⡄⡀⠀⠀⣴⣿⣦⣶⣿⣷⡦⣭⣛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⣴⣿⣁⣠⣾⣾⣴⣿⣿⣿⣿⣿⡟⠛⠋⠉⠙⠻⣿⣦⣙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⣾⡿⡫⠚⠿⣿⢿⠿⠛⠉⠉⠀⠀⠈⣿⡆⠀⢴⣶⣶⣼⣿⢿⣷⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣼⡿⡫⠊⠀⠀⢸⠁⠀⠀⢀⡀⠀⠀⠀⢀⣿⠃⠰⣿⣿⣿⣿⣿⡗⢝⢿⣧⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⡟⡝⠁⠀⠀⢀⡎⠠⠀⣀⣤⣶⣤⣄⡀⠈⢹⣿⣿⣿⣿⣿⣿⣿⠁⠈⢫⢻⣷⡜⣧
⠀⠀⠀⠀⠀⠀⠀⣸⢇⣾⡟⡜⠀⠀⠀⢸⣿⣿⣷⣼⣿⡿⣿⣿⣿⠿⠛⠛⣿⣿⠿⣻⣭⢹⡏⠀⠀⠀⢣⢻⣷⡸⡇
⠀⠀⠀⠀⠀⠀⠀⡟⣸⣿⢹⠀⠀⠀⠀⠘⢿⣾⠟⠀⠻⣿⣿⡿⠋⠉⡀⠀⣿⡟⠀⢳⡄⣸⠃⠀⠀⠀⠈⡏⣿⣇⢻
⠀⠀⠀⠀⠀⠀⢸⡇⣿⡟⡎⠀⠀⠀⠀⠀⠀⢯⣀⣤⢦⠆⠉⠐⢞⠉⣀⣴⣿⣷⠞⣸⣾⠋⠀⠀⠀⠀⠀⢱⢿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢶⣦⣄⠈⣸⣿⣿⣿⣿⠳⣿⡇⠀⠀⠀⠀⠀⠀⢸⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⣇⡇⠀⠀⠀⠐⢿⣿⣿⣯⣭⣛⣷⣤⣌⣙⣛⣻⣿⣿⣿⡇⢸⣅⠀⠀⠀⠀⠀⠀⣸⣾⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⢹⣿⣿⠀⠀⠀⠀⣰⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡼⢫⣇⠀⠀⠀⠀⢀⣧⣿⡏⣼
⠀⠀⠀⠀⠀⠀⠀⢹⡎⣿⣿⣧⡀⠀⢠⣿⣿⣰⢄⣈⣇⡀⢸⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⠟⣳⡀⠀⢀⣾⣿⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⡿⠃⡰⡷⠙⣿⣿⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣴⣿⣿⣥⣞⣼⣯⣿⣿⣿⡟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⠿⣿⣿⣿⣿⣿⣿⣿⣅⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣝⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣩⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠬⢭⣭⣭⠭⠴⠶⠖⠛⠉⠁

⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⣦⣦⠀⠀⠀⠀⣀⢠⣤⢀⡀⠀⠀⠀⠀⣤⣠⠆⠀⠀⠀⢤⣠⣄⡤
⠀⠀⣼⣿⣷⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠐⣾⣿⡷⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠉⢻⣿⠉⠀⠀⠀⢀⡴⣾⡏⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⠀⢠⠏⢧⠀⠀⠀⠀⠀⡾⠹⡄⠀⠀⠀⠀⢰⠟⢧⠀⠀⠀⠀⢀⡟⢻⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠀⠈⠁
⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⢠⡄⠀⠀⠀⢀⣀⣰⠆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣀⣶⢀⡀
⠀⠀⢛⣿⠑⠀⠀⠀⠀⠐⣿⡛⠆⠀⠀⠀⠀⣼⣿⡶⠇⠀⠀⠀⢨⣿⠿⠖⠀⠀⠀⠘⢿⡟⠋⠀⠀⠀⠐⢻⣿⠳⠂⠀⠀⢀⠚⣹⣟⠋
⠀⠀⠞⠙⠧⠀⠀⠀⠀⠠⠟⠹⠄⠀⠀⠀⠴⠻⠏⠙⠂⠀⠀⠠⠞⠽⠁⠀⠀⠀⠀⠴⠋⠹⠆⠀⠀⠀⠠⠞⠸⠃⠀⠀⠀⠈⠙⠋⠉⠧
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣠⣿⣄⠀⠀⠀⠀⢀⣼⣧⡴⠀⠀⠀⠀⠀⣰⣦⡖⠀⠀⢠⣄⣠⣿⣀⡄⠀⠀⠈⢷⣷⠞⠀⠀⠀⠀⣠⣿⠞⠁⠀⠀⠀⠀⣼⣥⠆
⠀⠐⢫⠿⡟⠂⠀⠀⠀⢋⡟⢷⠀⠀⠀⠀⠀⢋⣿⣧⡀⠀⢀⣠⠴⠿⣇⠀⠀⠀⠀⠀⣸⢿⣄⠀⠀⠀⠈⢹⣷⡄⠀⠀⠀⠀⠸⡿⣦
⠀⠀⠛⠀⠙⠀⠀⠀⠀⠘⠁⠋⠀⠀⠀⠀⠚⠋⠀⠚⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠐⠃⠘⠁⠀⠀⠀⠀⠚⠘⠁⠀⠀⠀⠀⠘⠃⠁
⠀⠀⠀⣤⠀⠀⠀⠀⠀⢰⣠⣄⡆⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢦⣴⣤⠆⠀⠀⠀⢰⣠⣦⡰⠀⠀⠀⠀⠠⣦⡴⠀⠀⠀⠀⢢⣰⣆⡖
⠀⠠⣴⣿⡦⠄⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣹⣾⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢾⣿⡁⠀⠀⠀⠀⣀⣻⡏
⠀⠀⢰⠟⡇⠀⠀⠀⠀⠸⠏⡽⠃⠀⠀⠀⢀⡼⠻⣿⠇⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⢠⠇⠟⠀⠀⠀⠀⠉⠛⠃
⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⣀⣰⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡦⠀⠀⠀⠀⢀⣀⣴⠀⠀⠀⠀⠀⣀⡶⠀⠀⠀⠀⠀⣀⣀⡶⠀⠀⠀⠀⢀⣀⢀⣄
⠀⠀⢡⣿⡙⠁⠀⠀⠀⢈⣿⢷⣺⠂⠀⠀⠘⢹⣟⠷⠂⠀⠀⠀⢫⣿⡛⠃⠀⠀⠀⠟⣻⣛⠛⠀⠀⠀⠘⢫⣿⡷⠆⠀⠀⠀⢻⣿⡿⠿⠃
⠀⠰⠋⠸⠇⠀⠀⠀⠠⠞⠽⠉⠉⠀⠀⠀⠤⠞⠽⠀⠀⠀⠀⠀⠾⠁⠻⠀⠀⠀⠀⠀⡝⡍⠀⠀⠀⠀⢠⠟⠽⠃⠀⠀⠀⠰⠾⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⣤⢀⡄⠀⠀⠀⠀⢠⡄
⠀⡤⢤⣼⡧⠤⠀⠀⠀⣤⣿⣶⡦⠀⠀⠀⠀⣤⣿⠃⠀⠀⠀⠀⠤⣾⡷⠆⠀⠀⠀⠀⣨⣿⠃⠀⠀⠀⢠⢾⣿⠉⠀⠀⠀⠀⠠⣾⣷⡄
⠀⠙⢻⡏⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⢹⣟⡦⠀⠀⠀⠀⠐⠛⡿⠀⠀⠀⠀⠘⣹⢛⡦⠀⠀⠀⠀⢈⣿⠆⠀⠀⠀⠀⠘⠛⠿⠃
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⡄⣠⠀⠀⠀⣠⣀⢠⡄⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢶⠀⡀
⠀⠠⢴⣾⠧⠆⠀⠀⠀⡶⣾⠏⠁⠀⠀⠀⢷⣾⠟⠛⠋⠀⠀⠀⣴⡟⠻⠓⠀⠀⠀⢀⣤⣯⣤⠆⠀⠀⠀⡴⣿⣆⡀⠀⠀⠀⠺⣿⣏⠁
⠀⠘⠻⠛⡆⠀⠀⠀⠀⢠⣟⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡿⢻⡄⠀⠀⠀⠀⣨⡼⠻⢦⡀⠀⠀⠀⣀⡿⣄⠀⠀⠀⠀⠀⣿⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢀⡀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢹⡧⡷⢿⢿⡇⠀⡇⣿⠄⠀⠀⠀⠐⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣷⡀⢰⡇⠀⢰⣴⢿⣧⢰⡶⣦⠀⢶⡆⣴⣶⡄⢠⡾⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⢻⡿⠛⢿⡿⢸⡿⣿⡄⢹⡇⢸⣿⡏⢸⡇⣿⠃⢻⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠁⠀⢸⡇⠀⣸⡇⣼⠇⢹⡇⢸⡇⢸⣿⠁⢸⡇⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⡆⠀⢸⡇⠀⣿⡿⠋⠀⢿⡿⣾⡇⢸⡟⠀⣾⡇⣿⡐⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣾⡟⠀⠘⣧⡀⢹⡟⠲⠀⠘⢧⡿⠉⢸⡇⠀⠿⠇⠸⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠳⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⠉⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢹⠄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠿⠴⣟⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢰⡿⠿⢿⣿⣦⣀⣀⣠⣤⠶⠶⠾⠂⠀⠀⠀⠀⢸⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠕⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠞⠁⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣉⣉⡋⠀⠀⠀⠀⠀⣼⠆⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⢏⡴⠋⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠛⠉⠀⠀⠀⠀⠀⠀⠘⢡⠞⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢠⡾⢁⡞⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠀⠀⠙⠛⠋
⠀⠀⠀⠀⢠⡟⢀⡿⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠙⠆
⠀⠀⠀⢠⡟⠀⠸⠁⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠙⣀⡀⠀⠀⠀⣠⣴⡾⠃
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⣠⠞⠋⠉⠉⠉⠉⠙⢦⣤⠖⠋⠉⠀⠀⠀⠀⠈⠉⠛⠲⣄⡀⢹⣤⠤⠶⠚⠋⠁
⠀⠀⢸⡇⠀⠀⠀⢰⡆⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⡲⠞⠋⠉
⠀⠀⢸⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠸⣇⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠙⠳⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠉⠙⠻⣧⣄⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠟⠀⠀⠀⠙⠛⠷⢶⣶⣶⣶⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣦⣤⣄⣀⣀⣀⡀⣀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⠟⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⢿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⢉⣀⣤⡴⠖⠚⠋⠉⠉⠉⠙⡓⠦⣤⣀⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⣀⣴⣾⡿⢋⣡⣴⡾⠟⢛⣉⠉⠓⠲⣌⡳⠌⢻⠿⣷⣦⣀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣤⣾⢟⢟⣵⣶⣿⣿⣿⣿⣿⣯⡷⣚⣉⣉⣈⣀⣀⠀⣷⠀⠉⠻⣷⣤⡈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⣰⣿⠋⠀⠸⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣾⠀⠀⠀⠀⠙⣿⣄⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢀⣾⡟⠁⠀⠀⠀⠈⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⢻⣿⣿⣷⣾⣿⣇⠀⠀⠀⠀⠀⠈⢻⣧⡀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⣾⠏⠀⠀⠀⠀⢀⡴⠋⢁⣀⣀⣀⣤⣤⣴⣶⣿⣿⠋⠉⠹⣯⣭⣯⡀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣾⡟⠀⠀⠀⠀⠀⠉⠙⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠶⠛⠉⣱⠞⢿⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⣇⣘⣿⣿⡏⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⢾⣡⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣼⣿⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣤⣤⣭⣿⣿⡿⠁⢐⣖⠀⢰⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢰⣦⢟⣡⣿⣧⣤⣄⣈⣉⣩⣵⠏⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣟⡉⠉⠉⢻⠀⢀⡼⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢸⣿⣿⡿⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠹⡝⠻⠆⢀⣼⡷⣿⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠤⣿⣿⣿⡇⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢿⣿⣿⣥⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⠃⣁⣤⣶⣿⠟⣁⢵⣻⣿⣿⣿⣿⠹⣷⣦⣤⣤⣤⣿⣿⡿⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⢿⣿⣷⡄⠀⠀⣀⣤⣶⢿⣿⣿⣿⡿⣿⣿⣿⠷⣪⣵⣿⣿⣿⣿⡟⠈⠂⢻⣿⣿⣿⣿⣿⡿⠁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡈⠻⣿⠴⠟⣫⡿⠋⠕⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⠿⠄⠀⠀⠀⢹⣿⣿⣿⠟⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣧⣴⣾⣿⣿⡶⢆⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿⣟⣡⣾⣵⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢛⣛⣥⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄⢿⣿⣿⣿⢟⣫⣭⣶⣶⣿⣿⢿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠐⣾⠊⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⢿⢆⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⠀⠀⢠⠃⢸⠘⣆⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⢸⠀⠘⡆⢠⣊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢠⠃⠀⠀⢸⠀⠀⠘⡄⠀⢉⡝⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⡠⠤⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣀⣠⠃⠀⠀⠀⢸⠇⠀⠀⠸⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⠢⠁⠒⢲⠠⡆⠐⠒⡡⠔⠁⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⢅⠀⠘⠀⡇⠀⡎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠰⣦⠤⠤⠤⠼⠆
⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢨⡇⢀⠎⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠑⠒⢄⠀⠀⠀⣀⢠⠊⠀⠸⠈⠇⠈⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠳⡱⡀⠀⠀⢀⠜⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠃⠰⢏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠠⢤⠠⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⣀⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⣰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠙⠢⢄⡀
⠀⠀⠀⣀⡀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢈⣑⠢⠤⣀
⠀⠀⠸⣀⡨⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣀⠤⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⠇⠀⠀⣉⡱⠂
⠀⠀⠀⣸⣊⣁⣀⣀⣀⡀⠀⠀⠀⠀⣀⣠⣴⡮⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⣧⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠈⠉⠉⠉⠉⠁⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠰⢦⡀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠒⠉⠀⠀⠀⠀⠸⠋⠁⠀⠈⠉⠁⣤⣤⠀⠀⠹⡄⢈⡷⢄⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⠇⠀⢴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡅⢈⡇⠀⠀⡖⢶⠃⠈⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢣⡼⠃⠀⠀⣰⠄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣦⠋⠀⠀⠀⠀⠋⠐⠋⠃⠀⣠⠀⠀⠀⠀⣸⡀⠀⣠⡞⠁⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣠⡤⡀⠀⠘⢆⣸⠀⠀⠀⠀⠀⠢⣤⣤⣶⡾⢁⣠⣤⡶⠂⠉⠙⢏⠉⠉⠏⠙⠢⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠉⠀⣇⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⡡⠜⠓⢉⡀⠀⠀⢱⡄⠀⠨⠀⠀⢸⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠤⢄⣀⣀⠉⠁⠒⢢⠖⠂⠷⣦⡄⠠⠔⠊⠁⢀⡀⠈⠑⢆⡀⠀⠁⠀⢀⠀⠀⡌⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⢤⠏⠀⠀⢠⠋⠀⠀⠀⠀⢀⡉⠑⢢⡀⠈⢡⠀⠰⠀⢸⠀⢀⠃⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⢸⠀⠀⠀⠀⠀⢀⠐⣦⣀⢱⡀⢸⠀⠄⢠⠂⢠⣉⣜⣁⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⢰⠇⠀⠀⠉⠈⠙⠛⠛⠓⠚⠓⠛⠉⠉⠀⢱⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⠀⠀⣀⣤⠄⠀⠀⠀⠀⠤⣤⡀⠀⠀⠸⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⡄⠀⡸⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢲⠀⢀⠼⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⢸⠋⠁⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠑⣿⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠈⢇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠄⡈⠀⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠎⠁⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠈⠳⣤⠜⠣⢾⠀⠀⠀⠀⠀⣠⠄⠀⠀⠈⠀⠀⠁⠀⠀⠢⡀⠀⠀⠀⠀⣰⠤⠛⢤⡴⠋⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠈⠢⣤⠟⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠘⣄⡴⠋⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣴⣁⣀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣀⣀⣱⡄⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠉⠉⢉⡿⠦⣀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⠴⠛⠻⡉⠉⠀⠀⠀⠀⠙⡄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠉⠙⡶⠐⠦⢀⣀⣀⠠⠖⠀⢻⠉⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠆⣀⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣤⣦⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠈⢧⢀⣀⡀⠶⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢺⡷⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠀⢬⠭⠁⠀⠐⠒⢒⡒⠒⠒⠀⠀⠩⠭⠄⠐⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠷⠆⠤⠠⡾⢻⠦⠀⠀⠶⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⣀⣀⣀⣀⠀⡆⢸⡄⣀⣀⣀⣀⣈⣳⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣹⣶⣶⣶⣦⣭⣍⣡⣤⣙⣭⣥⣶⣶⣶⣽⣅⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣃⣼⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣿⣿⡛⠿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⢟⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⣾⣿⣿⣿⣷⠀⠀⣀⣀⣀⡠⣶⣾⣿⣷⣶⣤⣴⣤⣼⣿⣿⣿⣿⣧⣤⣦⣤⣶⣾⣿⣷⣶⢄⣀⣀⣀⠀⠀⣾⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠉⠉⠙⠿⢿⣿⣿⣿⡿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⢿⣿⣿⣿⡿⠿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢰⣾⠟⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢾⣿⣄⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣀⣀⣠⣼⣷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣤⣀⠀⠀⣀⠙⠛⠛⠛⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢀⠈⠻⠛⠛⠻⣿⡀⠀⠀⠀⣠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠁⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠈⠉⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡴⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡄⠀⠀⠀⢰⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠀⠀⣀⣀⣀⣀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠛⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⣀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣿⣷⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠒⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠄⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⠀⠹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠏⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣥⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣠⣼⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⣿⣿⣿⠿⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣀⡀⠀⢀⣀⠀⢀⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿⣶⣶⣶⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽
⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣸⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⡿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣶⣾⣿⣿⠿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⠁⡼⡄⠀⠀⠀⠶⣶⣶⣶⣿⣟⠉⢿⣿⣏⠁⡀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⡀
⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⡁⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⡿⠿⠟⠉⠙⢹⣷⣴⣄⣤⣄⡀
⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠟⠁⠀⠀⠉⢻⣿⣿⣶⣤⣶⣾⣦⣶⣶⣿⣿⣿⠛⠉
⣤⡿⠜⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣽⠟⣿⣶⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⠋⠘⣿⡏⠙⢻⣿⣿⣿⣄⣠⢾⣿⣦⣴⡿⣿⣿⣿⣿⡿⠛⠿⣿⡳⣿⣿⡇⠀⠀⠀⠀⠰⣼⠆
⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠛⢳⣿⣷⣦⣬⣿⢤⣞⢻⡿⣹⡿⡷⣴⣿⣶⠿⣸⣽⣿⣛⣿⣤⡰⣴⣾⣹⠾⣟⠅⠀⠀⠀⠀⠘⣿⣦
⣊⠀⠐⡇⠀⠀⠀⠀⠀⠀⠈⣀⣄⣤⣿⡿⣿⡯⢿⠛⠿⢾⣿⣾⣄⠠⣽⣻⣯⣿⡿⠟⢫⣮⣻⡿⠟⣽⠿⠶⣷⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⡟⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡿⠼⣿⠳⢦⣤⣀⣘⣻⣿⣿⣿⣿⡿⣿⣵⣰⣾⣿⠿⠩⣦⡴⢿⡾⡘⢻⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄⢗⡉⢱⣂⣯⣵⣶⣽⡿⠿⠟⣻⣿⣿⣭⣍⣛⠛⠿⣷⣿⣖⣛⣿⣉⣹⣓⡋⢹⡷⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣾⣿⡇⠀⣤⠀⠀⠀⠐⢿⣷⣿⠷⠟⢫⡭⡟⠁⢤⣶⠖⠀⠻⠿⣿⣿⣿⣯⣤⡀⠀⠉⠉⣿⣿⣻⣿⣿⣷⣾⣷⡦⠀⠀⠀⠀⠀⠀⠀⠈
⢰⣿⣿⢂⣾⣿⠀⠀⠀⠀⠘⠿⢿⣷⢶⣿⣿⣥⣤⣜⠋⣴⣶⣾⣿⣿⣿⣿⣯⣿⡿⠈⢻⠲⠛⠚⠊⢻⣼⣿⠿⠟⠂
⣼⣿⡿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣿⣯⡻⣧⣶⣿⣿⣿⣿⣿⠏⠉⠉⢀⣀⣀⣈⣁⣶⣶⣷⠟⢟⣡⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⣿⢿⡟⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣾⠯⢽⡿⢫⣷⣿⠟⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⣄⣿⡄
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠉⠀⠶⠀⠀⣿⠀⠛⣿⡿⣻⣿⠋⡘⣿⢿⠉⣿⣧⣶⣞⠋⣿⣿⣿⠀⠚⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠿⠆⠀⠠⠄⣿⠀⣠⣼⣿⣿⣿⣧⣇⢸⠈⠀⣿⣿⣿⣿⡟⠋⢸⣯⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣙⠴⠿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠓⠀⢸⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⢠⠁⠀⣾⣿⡇⠀⢸⣿⣿⣿⠹⣿⣿⡀⣻⣿⠟⠁⠈⠀⠀⢸⡇⠈⠉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣯⣿
⣿⣿⣿⣿⣀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⠀⢸⣿⣿⣿⠀⢽⣿⣿⡿⣧⣀⣀⡤⠀⠀⣸⠇⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⣿⣿⠀⢺⡏⠋⠀⣿⣿⣿⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠈⠛⠛⠻⠀⠘⠇⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣭⣿⣿⣿⣽⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠿⠛⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢤⣀⠀⠀⠀⣀⣤⠤⠀⢹⣟⣽⠇
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⠙⠟⠀⠀⠀⠀⠛⠉⠀⠘⡿⢸⡄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡸⡇⢻⣿⣦⡀
⠀⠀⠀⣾⡿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣀⣀⠀⠀⠀⠁⠀⠀⢻⣿⣿⣦⡀
⠀⠀⢰⣿⣷⠀⠔⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⡶⠋⠀⢀⡴⠀⠀⠀⠈⣿⣿⣿⣿⣦
⠀⠀⣾⣿⣿⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣁⠀⢀⠒⠄⢹⣿⣿⣿⣿⣧
⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⠓⠀⠀⡀⠀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠦⡄⠀⠈⠻⣿⣿⣿⣿⣧
⠀⢸⣿⡟⣿⣷⠀⠀⠀⣠⢇⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢹⣿⣿⣿⣿
⠀⠘⣿⡆⠙⢿⠇⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⢹⢿⣦⣸⣷⣶⢚⠀⠀⠀⡀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠚⠁⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠘⠷⣮⣭⣭⣅⣠⣀⣠⣤⣤⣴⣶⣌⡛⠷⠶⠶⠖⠊⢀⣰⣿⣿⣿⣿⣿⣿⣭⣍⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠛⠛⣻⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⢠⡄⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠁⠀⠋⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣦⣴⣶⠏⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠘⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⠘⠁⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠹⣇⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠈⠣⠀⠀⠀⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⣿⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⡴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠦⡀⣰⠋⠀⠀⠈⢦⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠳⡌⢧⠤⠤⢤⣤⣸⠎⠡⡀⢀⡠⠒⣉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠠⡞⠚⠒⠒⠘⠻⠿⢋⡒⡼⠁⡠⠊⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠜⠒⠙⠢⣀⠀⠀⠀⠀⠀⠀⠉⣘⠦⠀⢨⢸⠁
⢠⣴⢲⠒⠤⢄⣀⡠⡤⠊⠁⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠂⠈⠁⠀⠀⠀⠁⠐⢄⡀
⠸⣏⠎⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠤⠈⠉⠁⠀⠂⠰⣒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⢇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⡞⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣀⠀⠀⠀⠀⠘⡀⠀⢀⠞⠀⢀⡴⠋
⠀⠀⠀⠀⢀⠜⠀⠀⢠⠊⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⠀⠀⠀⠀⣇⠴⠃⢀⠴⠋
⠀⠀⠀⠀⠘⠦⢽⠉⡉⡁⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡃⠀⠀⠀⣧⠤⠊⠁
⠀⠀⠀⠀⠀⠑⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠡⠤⣎⠁⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⠟⠑⠒⠒⠠⠴⠦⣌⠲⡒⠈⢹⣁⣠⣀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠼⠀⠿⡀⠀⠀⠀⠀⠀⠀⣸⠀⠧⠠⠄⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡡⢢⠄⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⢀⡞⠁⠀⠀⠀⠠⡀⠠⡈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠿⠻⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣁⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠉⠀⠀⠐⣷⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠛⠀⠀⢀⣶⣯⢞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠈⠛⠒⠢⢤⣤⣤⣤⣽⣟⣫⠭⣶⣾⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣴⡋⠀⢬⣿⣷⠶⢶⣶⣦⣷⣶⠾⢿⠵⠖⣋⣩⠴⠖⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⠁⢹⠦⠄⣉⣁⣀⣀⡤⢦⠀⣧⠀⢸⣏⡭⠥⠖⢒⣋⡥⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠙⠶⠋⠀⠀⠀⠀⠀⠉⠀⣸⠀⡟⠀⢸⡗⠒⢚⣋⣉⡥⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢦⣤⡀⠀⠀⠀⠀⠐⠦⠏⣰⣿⣿⣿⣿⣿⣯⣥⣶⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣷⡛⠋⠀⠀⠀⠀⠀⢀⣠⡾⢻⠏⣹⣿⡉⠀⢠⠇⠀⢸⠈⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⠤⣤⣤⣶⠟⢋⡴⠋⣠⠏⢹⣧⡴⠋⠀⢀⡟⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢁⡴⠋⢀⡴⠋⠴⣿⠟⠃⠀⣠⠟⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠙⠠⣔⡋⢀⡴⠚⠁⢀⣠⠞⠁⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢻⣿⠃⠀⠀⠀⠉⠉⠐⠒⠛⠛⠓⠒⢻⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣟⣿⣶⣤⣤⣴⣶⣖⣶⣶⣖⣒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠈⠙⠛⠋⠉⠀⠈




⣀⣀⡀⢀⡀⣀⣀⣀⣀⣀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢉⡓⠦⣴⠶⠶⣶⠛⢉⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣩⡿⣿⠀⠀⢸⡿⠅⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢻⠀⠀⠀⢀⣠⡴⠖⠋⠉⠛⠉⠓⢄⡀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⢦⠴⠚⠉⠁⠀⢠⡤⠀⠀⠀⠀⠀⠈⠓⠊⠈⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣴⣶⣄⡀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠐⠶⢶⣤⣄⡀⠀⠘⢿⣿⠿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡄⠀⠀⢀⣤⣶⡾⠿⠛⠋⠉⠀⠀⣠⣶⣤⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡉⠀⠐⠒⢶⠶⠶⠦⠤⠤⠤⢤⠡⢠⣤⢠⣉⣀⣀⣀⣀⣀⣀⡀⠀⢻⣿⣿⠂⢃
⠀⠀⠀⠀⠀⢀⠂⠀⠀⣾⣿⣿⡆⠀⠈⢿⠀⠀⠀⠀⠀⠀⠜⢁⣼⢷⢀⠒⠒⠒⠦⣤⠿⠉⠁⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠙⠿⠟⠁⠀⠀⠈⠑⠤⠤⠤⠤⠔⣪⠟⠁⠀⠀⠁⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠈⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢡⢞⣓⣄⣐⡀⠀⠀⠀⠀⠀⢰⣿⣿⡇⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⡌⠀⡆⠀⠀⠀⠀⠀⠀⡠⢤⠀⣀⡤⠖⠻⢍⣉⣁⡽⠋⠓⠦⣄⡀⠀⠈⠻⠿⠟⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠹⡞⠉⠉⠉⠉⠉⡗⠀⠀⣘⠤⠚⠉⠁⠀⠀⠀⠀⠈⢡⣤⣄⡀⠀⠀⠉⠓⢄⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⢸⣅⣀⡀⠀⠀⣸⠁⠐⠈⠀⢀⣿⣿⣦⠀⠀⠀⠀⠀⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠸⡅⣀⠀⢀⠀⣉⠀⠒⠄⠠⣈⣙⣉⣁⣀⣀⣀⡀⣠⡄⠠⠄⠂⠀⠀⠀⢀⣴⣶⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⡈⠈⠁⠒⠒⠠⠤⠠⠤⠤⠤⠤⠤⠤⠤⠄⡐⠒⠒⠂⠀⠀⠘⠿⠿⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢏⣢⡀⠀⠀⣼⣿⣿⣶⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢰⠚⠁⠀⠀⠀⢉⣽⣀⣘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠉⢉⣀⡤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢸⠀⡄⢀⠀⢸⡏⠀⠀⠈⠉⠙⠫⣍⠉⠉⠉⠉⠻⣍⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⢦⡼⠁⠀⠀⠀⠀⢀⣴⣁⣀⠀⠀⠀⣴⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐