Boom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢠⡦⠀⠀⣠⡼⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⢻⡀⠀⣀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠡⠀⢹⣿⣦⡀⠀⠀⠀⠀⠁⠰⠃⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⡀⠈⠀⠀⠈⣿⡎⠛⢦⡀⠀⣠⡀⢁⣾⠏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠲⢦⣤⣿⡇⠀⠀⠀⠐⣿⢳⣼⠏⠀⠀⠹⣷⡦⠤⣀⠀⠔⣾⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠬⣿⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠹⠿⠛⠉⠀⢠⡟⠘⠃⠀⢀⡴⠛⠃
⠀⠀⠀⠠⢤⣤⣤⣄⣀⣀⠀⠈⡻⣿⣿⡛⠛⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠥⠤⢶⣶⠖⠀⢠⣤
⠀⠀⠀⠤⢤⣄⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠈⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⡻⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠋⠉⠉⠩⠛⠉
⠀⠀⠀⠀⠘⠃⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠆
⠀⠀⠀⣀⣤⡤⠚⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣶⣤⣤⣀⡀⠀⠀⠄
⠀⠀⠘⠋⠉⠉⠉⠉⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠑⠚⢛⡿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⡤⠶⠤⠂⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠂⠀⠠⠞⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣽⣿⣶⣶⣖⡺⠛⠉⠉⠙⠃
⠀⠀⠀⠠⠐⠙⠉⠁⠀⠀⠀⠀⠀⠠⣿⠀⠀⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⡶⠾⠟⠻⠻⠟⣫⣍⡓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡤⠒⠉⠭⠈⢿⡀⠀⢀⣶⡿⢤⣶⣦⣀⠀⠀⢀⡟⠿⣶⣦⡀⠙⢿⡆⠐⠒⠂⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣴⠟⠁⡀⠈⢧⠀⣾⡿⠀⠈⠃⠀⠙⠳⣄⢸⣧⠀⠀⠉⠙⠒⠚⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⡞⢰⡗⠀⠘⢷⡿⠃⠀⠀⠀⠀⣠⠶⢌⠻⢿⡇⢰⡌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡇⠀⣤⡀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⢿⡌⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠠⠠⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠠⠠⠁⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠌⠂⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣀⡾⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⢀⣴⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠐⢦⣄⣾⠏⠀⠀⠀⠴⠶⠶⠶⠷⣦⣴⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⠁
⠐⠄⠀⠹⣧⠀⢀⣼⠃⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠉⣿
⠐⠀⠀⠀⠈⢷⣿⡁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣁⣴⠟⠀⠀⠀⠀⣀
⡀⠀⠀⠀⠀⠀⠈⠛⠳⢶⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠉⠛⠳⢶⣄⢀⣴⠞⠛⠻⣧⠈⠂⠀⠀⠻⣦⢑⣬⡶⠛⠁⠀⠀⠀⠀⢘⣿
⠻⣦⠀⠲⣆⠀⠀⠀⠀⠀⢍⢷⡈⡀⠀⠀⣿⠀⠀⣼⠋⣵⣤⡙⢷⡀⠀⠀⠀⠠⢀⣽⠟⣠⣾⣷⠄⣿⠅⡀⠀⠀⠀⠹⣿⠁⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠘⣧⣀⣹⣧⠀⢀⠐⢌⠂⢸⣷⢰⠰⠀⣿⠀⠀⣿⢠⣿⣿⣷⠈⣷⠀⠀⠀⣴⠟⢡⡾⣻⣿⣿⡆⣿⠂⠌⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⡈⢺⡇⠀⠀⠀⠂
⠀⠀⠈⠉⠉⠙⢷⣄⡑⢄⣢⣴⢿⡜⠀⢸⡇⢀⣰⠏⣼⣿⣿⣿⡆⣿⢀⠀⢀⣿⢀⣿⡅⠙⣿⣿⡇⣿⡃⠀⠈⠘⠀⠀⠀⣿⠀⠀⠀⠀⢀⢀⣠⡿⠷⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣾⣿⣿⣮⣿⡄⢸⣇⢸⣿⣼⣿⣿⣿⣿⣷⣌⠛⠛⠛⣡⣾⠃⡀⢸⣿⣿⣇⠹⣧⣤⣀⣁⣠⣴⠾⠻⣦⣀⢀⣤⠞⠋⠉⢀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⣿⣼⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⢠⠁⣾⣿⠿⣿⣶⣤⣭⣭⣯⣭⣴⣾⣿⣦⣝⢿⡁⠀⠀⠀⣿⠆⠀⠀⣿
⠙⠛⠳⡶⣤⣄⠀⡐⢀⣿⣿⣿⣿⣿⣿⣛⣻⣯⣾⠿⠛⠛⠛⠒⠿⣿⣿⣿⣿⠃⠠⠀⡼⢋⣁⣠⣬⣤⡀⣴⠶⠿⢷⣶⡌⢹⣿⣿⡞⣷⠀⣀⣴⡟⠁⠀⠀⠹⣦
⠈⠀⠌⢀⠔⠙⢷⣄⣾⠃⣿⣿⣿⣿⣿⡿⢿⠟⣡⡶⠟⠛⠛⢷⣦⡈⣿⣿⠇⣘⠀⣴⡆⢻⡉⠁⠀⠈⠿⠏⠀⠀⠀⣿⣷⢸⣿⣿⣷⢸⡿⠋
⠀⠀⠀⠁⠐⡡⠀⢻⣿⠀⣿⣿⡿⢋⣡⣤⡄⣴⠏⢀⣤⣤⡀⠀⣿⣧⢸⠟⡴⠃⣼⠟⣡⣬⣥⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⢘⣿⣿⡧⣸⠇⡀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⢀⣀⠀⣽⡇⣻⣿⡀⣿⡉⠀⣿⡏⠀⣾⣟⣽⠃⢀⣿⣿⣠⣼⣿⣄⡀⠸⣿⣿⣿⣿⣿⣦⠀⠀⣶⡄⠀⢹⣿⠀⣿⠟⣼⠏⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⡀⢠⡾⠋⢰⣿⣿⣇⠸⣇⠀⠈⠀⠘⠿⠟⠁⢀⣼⣿⢯⣿⣿⣿⣿⣽⡄⠸⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⢸⣿⡇⠋⣼⠃⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠈⠛⠾⣦⡶⠟⠛⠶⢶⡄⢿⣿⡆⢿⡄⠀⠀⠀⠀⠠⣶⣿⣿⣯⡻⣿⣿⣿⣷⣿⣿⠀⣿⣿⣿⣿⣽⡇⣿⣿⣷⠀⠸⣿⡇⢸⡇⣀⣀⣀⣀⣀⣠⣴⢟⠃
⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⣿⣿⣇⠘⣧⠀⠀⠀⠀⠀⣀⣀⠉⠻⣿⡝⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢱⡟⠛⣿⡀⠀⣿⣇⢸⡟⠉⠭⠉⠉⠉⠉⠹⣦
⠀⠀⠀⠀⠀⢀⣾⠃⣰⣾⣿⣿⣿⣿⡆⢻⡄⠀⠀⠀⠘⢿⡛⣿⡆⠘⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⠏⣿⣿⠀⢸⡇⠀⢻⣿⠈⢷⣄⡀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠐⠠⠬⣿⢸⣿⣿⣿⣿⣿⣿⣧⠈⣷⠀⠀⠀⢀⡈⠙⠛⠁⢠⡟⣿⣿⣿⠟⠁⠾⣿⡿⢛⠁⠀⣿⣿⡀⡀⢿⣤⣼⣿⡇⢰⣌⠻⣶⡂⠀⣁⠀⠸⣇
⠀⠀⠄⠀⢀⣀⣿⣄⣀⣋⣉⣙⠻⣿⣿⡀⢹⡇⠀⠀⠘⣿⠶⣤⠶⠟⠀⠉⢉⣠⡀⠀⣀⡿⠃⢿⣤⣴⣿⣿⢀⣷⠌⠙⣛⡉⠁⠿⣿⣷⣌⢿⡄⢀⡀⠀⠻⣦⣀
⠀⠀⠀⠀⠀⣰⡏⠉⠉⠉⣉⣽⡇⢸⣿⣀⡀⠻⠶⠛⠛⠋⠀⣀⣠⣶⣞⣶⣿⣯⣖⢰⣯⣞⡆⠈⠙⠛⢋⣁⠜⡁⠐⢀⡤⢤⣔⡂⠈⣿⣿⡎⣿⠀⠀⠀⢠⡟⠙
⠻⣦⡀⠀⠀⣿⠀⠀⠀⣴⢏⣿⠁⣿⣿⣿⣿⣄⣠⣶⣐⣾⣿⣿⣟⣿⣿⣿⣿⣿⣯⣄⠉⠛⠖⠚⠉⠦⠀⡅⠈⢁⡼⣫⣶⡶⣦⣉⠛⠿⠿⢃⣿⠀⠀⠀⢼⡇
⠀⠈⣿⡄⠀⣿⠀⠀⣸⠏⢨⣏⢠⣿⣿⣿⣿⣿⣿⡿⠛⠿⠋⠛⠿⣿⣿⣿⣿⣿⣿⡿⠛⣛⣉⣛⠛⢶⣶⣴⣾⣿⣿⡏⠀⢻⣌⢉⠻⠶⢶⠛⢃⠈⠀⠀⠈⠻⣦
⠀⢊⠜⠻⣤⣿⠀⠀⢹⡆⠀⢻⡄⠺⣿⣿⣿⡿⢫⡶⠟⢻⠛⣷⣦⡈⠻⣿⣿⣿⠟⢁⣾⠏⠉⢙⡿⢶⡘⢿⣿⣿⣿⠃⠀⠀⠙⣦⡀⠀⠀⠑⠀⠁⠀⠀⠀⠠⠉
⠀⠀⢢⣀⣼⠇⠀⢀⡾⣷⠀⡌⠻⣦⣙⣙⣫⣴⠟⠁⠀⠈⣸⠇⠈⢿⡄⠙⠛⢁⣠⣾⡏⠀⠀⣾⠃⠈⢿⣄⠙⠋⣿⠀⠀⠀⠀⠈⠻⣦⣤⣤⣀⡀
⠀⠰⠟⠉⢀⣀⣤⡾⠁⢸⡇⠀⠄⠈⢙⠉⠉⠀⠀⠀⠰⠀⣿⠀⠀⠈⠛⠛⠛⣿⢋⠚⠀⠀⣼⠇⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⢶⣄⡀
⠀⠀⣠⡾⠋⠉⣁⣠⣴⠟⠷⣤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣽⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⣦⣄⡀⠀⠉⠻⣦
⠀⣰⠏⠀⠀⠞⠋⢹⣇⠀⠀⠈⠙⠛⠶⡄⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣸⠏⠻⣦⠀⠈⢻⣶⣤⣤⣶⢶⣤⠀⠀⠀⠀⣸⠏⠀⠀⣾⠁⠀⠉⠻⣦
⠾⠃⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⠟⠁⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⣼⠃⠁⠀⠀⠀⠈⠀⠀⠀⢰⠟⠀⠀⠀⠈⠀⠀⠀⠀⠘⢷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠶⣶⡿⠁⠙⠛⢩⡻⠂⣤⡀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⡀⠀⣀⢳⡀⢀⠉⡀⢻⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⣡⠬⡍⠁⢨⠄⣁⣸⠘⠣⣤⠥⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡤⣤⣶⣿⣶⣭⣠⣿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠉⠈⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡼⠷⠼⠇
⠀⠀⠀⠀⠀⠀⠀⢦⣼⡁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣽⣿⣿⣿⡄⠀⠦⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠒⠛⡶⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣦⣾⣿⣿⣿⣿⣿⣧⣀⣴⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣦⠀⠀⠀⢀⠀⠀⠠⣤⣸⡿⢿⣿⡏⢙⣿⡿⢿⣿⣿⢿⠿⣯⣻⡇
⠀⠀⠀⠀⠠⢀⠀⠀⠀⣸⢛⣿⣿⣿⣿⡆⢠⣴⠃⠀⠀⠀⠀⠹⡷⠿⠛⠿⠿⠟⢿⣿⠿⠿⢀⣠⣿⣯⠅
⠀⠀⠀⠀⠀⠀⠙⢶⣤⣽⣾⣿⡟⡋⠈⠇⢄⢧⣀⠄⠀⠀⠀⢠⠷⠛⠉⢣⡴⠶⣄⡴⠛⠳⢾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢩⢹⣰⡚⡟⡗⡈⣦⣸⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠐⠒⢆⠶⢱⠅⠟⠐⣫⣭⢿⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠿⣿⣷⡾⠛⢧⠀⠈
⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠔⠋⢹⠈⡏⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠈⣿⣤⣶⣿⣿⠀⣿⠷⠶⠦⣄⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠢⣄⣠⣤⣤⡤⣿⣿⡿⠛⠛⠛⠁⠀⠀⠀⠈⢧⣀⢐⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣌⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢰⣀⣀⠀⠀⠀⠀⠀⠉⠩⠟⠲⠶⢤⡀⠈⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠈⠉⠙⢿⡿⠷⠖⣒⡯⠝⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠟⣿⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⡟⠀⠀⠘⠆⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⠙⣇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢰⢀⠀⢸⠇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠂⠀⠀⠙⠌⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣼⠏⣠⠟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣦⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⡾⠁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡿⢋⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⠟⣩⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡫⣖⠙⠻⠿⠟⠛⢶⣤⣤⣶⣧⣀⠀⠀⠀⢀⣡⣴⡛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠁⠀⠀⠀⠀⠀⠉⣿⡟⠛⠛⠋⠙⠛⠉⠁⠀⠱⡜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠸⠁⠙⠢⡄⠃⠄
⠀⠀⠀⠀⢀⠔⠠⡀⡀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⡇⠀⠐⡁⠀⠀⠠⡀⣀⠀⠀⠀⠁
⠀⠀⢠⠒⠚⠀⠀⠀⠈⠀⠀⠀⠀⠠⠤⡀⠀⠀⡤⠖⠒⠢⡀⠀⠀⢰⠤⠜⠂⠊⠀⠱⠥⠘⠃⡀⡀⢀⠂⢈⠅
⠀⠀⠸⢄⠀⠀⡠⠒⠀⠀⡠⠖⠒⠒⠢⣄⢀⡸⣀⠀⠀⠀⠈⠊⠉⠙⢄⣠⠄⠀⢄⣠⢤⠮⡀⠉⢀⢂⠔⠁
⠀⠀⠀⠈⠓⠋⠀⠑⠀⡞⠀⠀⠀⢀⢀⠈⠁⠀⠀⠑⠒⠚⠁⠀⠀⣀⣈⣁⠀⠀⠀⠑⡫⢔⣒⣚⡳⣅⢖⡀⠀⠐⢒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠼⠀⡀⡸⠢⠴⣄⡼⡃⠀⢀⡤⢒⢒⠋⠅⠒⠢⡑⢄⡠⢊⠔⠁⠀⠀⣿⠌⡘⠀⠀⢀⡾⢡⠃
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠁⠘⠂⠉⠁⠀⡠⠔⠐⠷⡶⢣⠊⠁⠀⠘⠂⠀⠀⠙⢮⠔⠁⠀⠀⠀⢰⣿⢀⠃⠀⢀⣾⢣⣏⠀⢀
⠀⠀⠀⠀⠦⠀⢾⠀⠀⡠⠖⣒⠢⢤⠎⠔⠁⠀⠑⢦⠁⠀⣴⣦⠀⠄⠀⢠⡀⠀⠀⣠⡄⠀⠀⢸⡏⠘⠀⠀⣼⢃⠇⠀⡄⠑⡄
⠀⠀⠀⠀⠂⠀⢈⠗⡩⠂⣁⠀⠈⣧⠈⠀⣴⡶⠀⠈⠀⠸⣏⠜⠀⡇⠀⢸⡟⠶⣾⠿⠀⠀⠀⣿⠇⠃⠀⣼⢃⠎⠀⢀⠱⢴⠃
⠀⠀⠉⠢⠀⠀⠸⡸⡀⠘⠿⠃⠼⠯⣆⠀⠻⠔⠀⢰⠀⠀⠀⠀⠀⡇⠀⠈⡧⢴⣶⢸⠀⠀⢀⣿⢸⣤⣼⢏⣾⠴⠖⠧⠀⡠
⠀⠈⠙⠄⠀⠀⠀⢇⢡⠀⢀⣤⢆⠀⠸⠀⠀⠀⠀⣸⣧⠀⠀⢀⣴⣇⠀⠀⣷⠈⠹⡎⠀⠀⢸⡿⠁⠈⡄⣎⠀⠀⠀⡏⠉
⠀⠀⠔⡥⢄⠀⡀⠘⡌⡄⠀⠛⠀⠀⢀⣷⠤⣤⣴⠟⣡⣝⣛⣛⣩⢬⢛⣛⣋⠌⠰⣙⠻⠷⠿⠟⢦⡾⢃⠗⠀⠈⠋
⠀⠀⠀⡇⠐⠁⢸⡎⢳⢰⠀⠀⢀⣤⠾⣡⠖⠲⠖⠊⠀⠀⠉⣅⡎⣀⠀⠀⠀⠀⠀⠀⠉⣍⡉⠉⠑⠚⠉⠀⠀⠀⡀
⠀⠀⠀⠈⠲⡤⠀⠁⠨⢆⡲⢞⣋⡥⠞⠁⠀⠀⠀⠄⠀⠀⠀⠁⠠⠞⢲⢀⢤⠀⡖⠲⠀⠀⢫⡀⠀⠀⣠⠤⠴⠊
⠀⠀⠀⠀⢰⠀⠀⢀⡤⠀⠉⠉⠁⠀⠀⠀⠌⠀⠀⣀⠆⠀⠀⠀⠀⠀⢸⠒⠀⢻⠀⠀⢀⡠⠈⠀⡂⠀⢘⡀
⠀⠀⠀⠀⠀⠣⣅⢀⣀⢀⠀⠀⠀⠀⢠⡆⠀⠀⢀⢄⡀⢀⠤⠤⠂⠀⠀⠀⠠⠀⠑⠒⠋⠢⣄⣀⣀⡴⠋⠱⡄
⠀⠀⠀⠀⠔⠒⠁⠈⠀⠀⠙⠲⠤⠆⠈⠈⠙⠒⢏⠀⠈⠁⠀⠀⠠⠤⡠⠴⠀⠀⣠⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠙⠒⠒⠒⠁⠀⠀⠀⠀⠎⠈⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢄⠀⢀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⢀⡤⠉⠈⢙⠀⠀⠀⢠⣀⡀⠀⠀⠀⠄⠀⡀⠀⣸⠟⠙⣆⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢦⡀
⠀⠀⠀⠀⠸⠀⠢⠀⠘⢦⠀⠀⠀⢿⡍⠉⠓⠒⠤⠤⠤⠞⠁⠀⠀⠈⢆⠀⡀⠀⠀⠀⠀⣀⠴⠚⠡⠂⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠳⡀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⡥⠀⠈⠀⢀⣀⡤⠦⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⠐⢌⢨⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠤⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡑⠄⠢⣀
⠀⠀⠀⢀⡔⠂⠴⠒⠋⠁⠀⡈⢠⠆⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣼⣷⠘⢆⠀⠈⠳⣄
⠀⠀⠀⠘⢀⠀⢠⢄⣠⠔⠊⠀⣾⣠⡶⠟⠋⠁⠊⠻⣦⠀⠀⠀⣴⠟⠑⠀⠀⠀⠀⠉⠁⠈⢳⡀⠐⢪⠀⡆
⠀⠀⠀⠀⠈⠢⠤⠈⠁⠀⠀⠠⠟⠉⠀⠀⠀⠐⠀⠀⠈⠳⣄⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠚⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠃⠀⡞⠉⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⢠⡞⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡰⢆⠀⠀⠀⢀⡀⠀⠀⢀⡾⢹⣇⣠⠴⢾⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠁⠀⠀⡄⠢⡀⠀⠀⣿⢷⡀⡰⠁⠈⢣⣀⢠⠎⠱⡄⣰⠿⡤⠯⣿⠀⠀⢸⠀⠀⠀⠀⣠⠔⣉⠔⠀⠀⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢆⣱⣄⣴⣏⡍⢻⡀⠀⢀⣼⢛⣧⡀⢀⡼⠋⣛⡫⠏⣿⠀⠀⣸⣤⣶⣄⣊⣴⣊⠀⠀⠀⠈⠢⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⡷⡈⣿⣷⣒⣒⣳⣄⣾⣳⡓⣒⣳⣟⣶⠞⠁⠙⢳⣺⣖⣛⣽⣾⡿⢋⡡⢾⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠀⢸⠀⠑⠢⣟⢧⣜⣿⣻⣖⣒⣒⣛⣒⣾⣦⣒⠒⣓⣛⣶⣤⣤⢟⢒⣞⣛⡞⣻⣧⣤⠶⣟⣛⣛⣹⣏⣀⣀⠠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣙⣾⣤⡀⢠⣷⣶⣽⡿⣳⡿⢶⣶⠞⠉⠀⠀⠉⠳⣶⠋⠀⠀⠙⠷⠒⣦⡶⠛⠛⠻⣶⣶⣿⣿⣷⠏⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠫⠬⠮⠽⢻⡿⠟⠛⠻⠋⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⢿⢏⡀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⢻⡭⠭⠭⠭⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⣠⣴⣶⣤⣄⢀⣶⣶⡄⣾⣿⣆⠀⠀⢀⣿⠤⠤⠤⠭⠔⠳⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠼⠷⢾⠯⠭⣭⠟⠉⣠⣶⣿⣿⣷⡆⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣯⠭⠭⠭⢭⡽⠫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣮⣼⣿⣿⣾⠃⠀⠀⢹⣿⣧⣿⣿⣇⣸⣿⣿⠟⢻⢿⠿⡿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣇⣯⣥⢛⣶⣮⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢶⡋⠁⣘⣷⣛⣛⣻⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣾⣶⣧⣽⣶⣿⣿⣿⣿⡟⠙⠏⢻⣿⣿⣇⢀⢀⣾⣳⣳⡟⠉⠙⣼⢒⣈⣭⠿⠓
⠀⠀⠀⠀⢀⣀⣤⣴⣟⣛⣛⡶⠲⠶⣖⣻⣶⠂⠀⣿⣿⣤⣼⣿⣿⡿⣿⣿⣿⣿⠟⠛⠿⣿⡿⠟⠻⠿⠃⠀⠀⠀⠛⠛⠉⠀⣿⣐⣒⣒⣇⣰⡾⠟⢿⡉
⠀⠀⠀⠈⠛⠷⣷⣒⡚⢛⢧⣀⠀⣰⠓⣺⠇⠀⠀⠻⠿⠟⠛⠛⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠀⠀⠀⠀⠀⣿⢚⣚⡚⠒⣚⢳⡄⠀⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣷⣞⣉⡕⡦⣿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⣷⣄⣀⣤⢿⡶⠾⢶⢞⢦⣤⠾⣷⣶⠷⡶⡤⣷⠶⠿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠉⢻⡾⣩⡽⠷⣤⡤⠾⣆⠀⠀⡾⢧⣄⣀⣀⡴⠯⠳⠶⠶⠯⠭⠯⠭⣍⣯⡭⠭⣿⠛⠿⠛⡯⠿⣽⢷⣽⡿⠿⣭⣏⡹⢷⡀
⠀⠀⠀⠀⠀⠠⠾⠅⠒⠒⢒⣶⣿⢟⡿⠿⢭⣡⠬⠭⠬⠷⠟⠭⠭⠭⠽⠭⢭⡩⢭⠭⠭⠭⠭⠭⠥⣿⣿⡟⠽⠏⠀⡀⢠⣟⠭⢬⣿⣽⢿⡀⠀⢏⠉⠛⠻⠆
⠀⣠⠤⣤⠊⢱⠀⠀⠀⢂⡽⣫⣿⡯⡿⠛⢉⡿⣭⣭⠭⠭⠭⠭⠭⠭⡭⠽⠋⠉⠉⣿⠩⠭⠭⠭⢭⡭⠭⠭⠥⢤⣦⣤⢴⡏⠻⣯⡙⠿⠿⣿⢶⠚
⡞⠁⠀⠀⠀⠀⠀⡠⢒⠿⠋⠉⠀⢰⡡⠖⠹⠿⠉⣿⣺⣆⣗⣤⣶⣊⣧⣀⠀⢀⣞⣇⡿⢷⣜⣰⡟⣷⣑⣊⢢⡟⠀⠹⠿⠉⠚⠀⠙⠳⣤⣸⡷⡑⢄⠀⠀⢀
⠙⡲⠀⢰⡀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠀⠀⠀⠿⠚⣿⢒⣶⠟⠁⠘⣷⣒⣛⣶⣛⣓⣾⠁⡼⠙⠋⢆⢽⣖⣒⡟⢣⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠿⠈⠃⠀⠀⣸
⠠⡇⠀⢠⡇⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢱⠀⢠⠎⢹⣖⣒⣒⣲⡟⠀⠱⠃⠀⠀⠀⠘⣷⡾⠁⠈⢧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⡄⠀⣷
⠀⠙⠦⢤⠑⠒⠒⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⠀⠀⢻⡖⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢀⣠⠴⠁
⠀⠀⠀⠈⠣⣄⣠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡤⠏
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⡐⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢦⣀⠈⠀⠃⠀⠀⣼⣄⣀⡴⠏⠀⢹⣄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣤⡿⠿⠿⠿⠷⠾⣧⣙⠋⠀⠀⠀⠀⠻⣦⣀⡴⠞⣷⠀⠀⢀⣀⡤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢀⣤⡶⠶⠿⠿⠷⢶⣿⡛⠛⠉⠁⢠⡟
⠀⠀⠀⠀⠀⠈⢳⣦⡤⣤⣀⣀⣴⡾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⣤⣤⣼⡇⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⢿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣷⠖⠛⠛⣿⡎⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣮⣕⢶⡽⠿⠶⣶⡍⣿⣇⠀⠀⣿⡇⣯⣠⡼⠋⠁
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⢤⣀⠀⠀⢀⣤⣶⣒⣒⠦⣄⡏⣿⠀⠀⠀⠹⣾⠁⠀⠀⢹⡏⣿⢻⡆⠀⣿⡇⣿⠋
⣀⣤⡤⢴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡽⠚⠛⠛⠳⣮⣹⢆⡟⠋⠉⠀⠉⠻⣦⡹⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⣻⠶⠿⣧⣻⣤⠤
⠀⠉⠓⢿⡇⠀⢀⢤⣔⣶⣶⣬⣭⣗⡲⣮⡟⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠘⣷⢹⡇⠀⠀⠀⠀⠀⡀⠀⢸⣿⢸⣏⢿⣄⣠⣿⢇⣷⠐⠒⠈
⠀⠀⠀⠈⢷⠀⣷⣏⠉⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⢰⣶⡄⠀⢶⣿⠀⠀⢸⣿⡇⠀⠀⣿⣟⣧⠀⠘⣷⣄⣰⣿⣀⣸⣿⢸⡏⠪⣽⣯⣵⡟⢷⣤⡀
⠀⠀⠀⢀⣀⣿⢇⣿⡀⠀⠀⢠⡄⠀⢸⣇⢿⡄⠀⠘⠿⠇⠀⣸⡼⣧⠀⠀⠛⠃⠀⣰⣿⢻⣿⠿⠿⢟⣛⣿⣛⣛⣿⣯⠾⠁⠀⠀⠀⣸⡇⢀⣈⣹⣷
⠀⠀⠀⠀⠈⣿⠘⣼⣇⠀⠀⠈⣁⠀⠻⣿⢦⡻⣦⣄⣀⣠⣴⡿⣳⣌⠻⣦⣤⣴⣾⢿⣷⠟⠉⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠉⠉
⠀⠀⠀⣀⡼⢿⣆⢫⢿⡀⠀⠀⠋⢀⣰⣿⢸⡟⠢⢭⣟⣯⡭⠾⠋⠀⠙⠲⠾⠷⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣾⣿⣄⡀
⣀⣤⣾⠭⠶⠾⣿⣾⡼⣷⣶⣶⠿⢿⣻⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉⢹⣿⠀⠀⠈
⠁⠄⠀⠂⠀⠀⠀⠈⢛⣶⠶⣾⠿⠟⠋⠈⠻⣷⣄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠤⠴⠛⣰⠇⠀⠀⠠⠴⢲⠟⠋⠁⠀⠀⠀⠀⢻⡆
⠀⠀⢀⣠⠖⠀⠀⡠⠾⠓⠺⣿⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠴⠶⢷⣄⠀⠀⠢⠶⠶⠛⠁⠀⠀⠀⠀⣠⠏⣀⣠⣴⣶⣶⣴⣤⣤⣻⣦⡀
⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⡀⠀⣀⣀⣠⣴⠟⣡⣼⣿⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡴⠞⠛⠛⢻⣇⣠⡴⣦⡤⡀⠀⠀⠀⠀⢀⠀⣹⣿⣿⣿⣟⠋⠉⠀⢠⡿⠁⠙⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠁⠀⠀⠀⠀⣸⠛⠁⠀⠈⠻⣷⠀⠀⣠⡼⠿⣖⠏⠉⠉⡏⠛⠿⣶⡀⢸⡇⠀⠀⠀⠀⠀⠐⣴⣍⡲⢦⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡼⠋⠀⠀⠙⠀⠀⠀⢰⠀⠀⠈⠹⣾⡇⠀⠀⠀⠀⠀⠀⢹⠎⡡⠄⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⢀⡬
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀
⣿⡟⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⠀⠤⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠂⠀⠀⠀⠀⠘⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠐⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠃⠀⠐⡄⠀⢹⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠉⠀⠀⠀⠉⠉⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠔⠙⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠡⣀⠴⠢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⠂⠁⣀⣠⠄⠀⠀⢀⣀⠄
⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠓⢄⣀⣀⡠⠋⠦⡀⠀⠀⡠⠔⠉⠀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠆⠀⣀⠜⠁
⠀⠀⠀⠁⠀⠀⠢⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⢀⡠⠄⠀⠈⠁
⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡴⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⢀⡇⠀⡞⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⣾⡇⠀⠈⠑⠒⣲⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡉⠉⠀⠠⡇⠀⠀⠀⠀⣼⠀⢀⠀⠀⢠⡄⠀⡟⢧⣼⠃⢿⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⡇⠀⠀⢀⣴⠏⢠⠛⣷⠀⡏⢻⡀⠇⠘⠃⠀⠸⠀⠀⠈⠦⡀
⠀⠀⠀⠠⠶⠶⠿⣭⣉⡀⠀⠀⠀⠀⣇⣠⣶⠟⠁⠀⢸⠀⣸⠀⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠠⠂⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢹⣁⣼⠁⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠒⠫⠤⡀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⢀⣀⣀⣀⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠏⠀⠀⠀⠀⠉⠙⢧⡀
⠀⠀⢀⠄⠂⠂⠀⡴⠞⠛⠉⠀⢀⡤⠤⣄⠀⠀⣀⣀⡀⠀⠀⣀⠤⢤⣳⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⣒⠉⠉⠘⣄⡀
⠀⠐⠈⠀⠀⠰⠒⠈⠀⠀⠀⠀⠀⠀⠀⢸⡠⠊⠁⠀⢹⡠⠊⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⢲⠀⠀⠀⠈⡄
⠀⠀⡀⠀⠀⠀⠢⠊⠑⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠄⠑⠀⠀⠀⠀⢀⡞
⠀⠀⠣⠤⣀⠀⠀⡄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠧⠀⠄

⠀⠀⠀⠲⣄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠈⢳⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⡠⠚⠁
⠀⠀⠀⠀⢤⡀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠜⢠⠎⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠀⠰⣄⠀⢳⣄⠀⠀⠹⡄⢠⡀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢀⣴⠃⠀⢀⠔⣁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠈⠳⢤⣀⡙⡀⣇⣀⡤⠞⠓⠚⠉⠢⢤⣤⡤⠖⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠦⡤⠚⠀⠀⠀⠀⠀⠉⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢴⠖
⠀⠀⠀⠀⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣇⢸⡇⢂⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠓⢶⣖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⠇⠀⠉⠉⢀⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠨⣳⣤⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⣸⡇⠈⡇⡿⢸⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠀⠀⠀⡏⣸⡀⢀⡾⢷⡄⢠⣾⣆⣧⢻⡟⡇⠀⡇⡆⢸⠀⠀⠀⠀⢈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠶⠞⣁⠀⠀⠀⣿⠟⣷⢸⡁⠀⢿⣾⠀⣿⢻⠈⠀⣿⠀⠇⠃⠘⠀⠀⣀⣴⣁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⠀⠀⠀⣧⣤⡟⠘⣧⣀⡿⠘⠿⠋⠈⠀⠀⠀⠀⢀⡄⢸⠀⠀⠒⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⢿⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⢦
⠀⠀⠀⠀⠀⣠⡴⠟⠓⠊⠒⡄⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⠀⠀⠀⣀⣀⣀⣸⠁⠀⠉⠙⢷⣄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣰⠗⢚⢏⡾⢄⠀⢠⠞⠉⠀⠀⠈⢿⠁⠈⠳⡄⡞⠀⠀⠀⠙⠂⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⢧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢄⠢⣈⠲⢤⡀
⠀⠀⢀⡴⢦⠀⠀⡠⠂⡰⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢦⠈⠳⡄⠁
⠀⠀⣯⠀⠈⢽⡋⠀⣜⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⢦
⠀⢀⡗⠀⢀⠀⠑⠀⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳
⠀⠀⠉⠻⣅⣠⣄⣀⣄⡠⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⢀⠀⠀⠀⠀⢀⡰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡉⢄⠀⠀⠀⣼⠁⠀⠀⡤⡄⡄⠀⠀⢀⡼⠆
⠋⢀⡀⠀⠀⣀⡐⡄⠀⠀⢐⠀⠀⠀⠀⢹⢠⠀⠀⠸⡇⠀⠀⠀⠀⡗⡄⠀⠀⣸⠍⠀⠀⠀⠀⠀⠀⠀⣤⠊
⠘⠂⡀⠀⠀⠀⠝⠁⠠⠃⠰⠀⠀⠀⠀⠨⡶⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠀⠥⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠀⣀⠁⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠐⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠀⡌
⠀⠀⠀⠠⠃⠀⠀⠁⠹⠀⠀⠀⠘⢄⠀⠀⠀⣰⠁⠘⠄⠀⢀⡀⠀⠀⠒⠛⠀⠀⠀⠚⠒⠀⠀⢀⣾⠃⠀⠀⡘
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⡠⡔⠀⠀⠈⠉⠈⠁⠀⠀⠈⠒⠚⠛⠛⢻⡋⠙⠓⠲⢻⢛⠁⠀⢠⠛⠋⠀⠀⠈⠁
⠀⠀⠀⠐⡐⠄⠀⢂⠀⠀⠪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠲⣄⠁⢂⠖⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠒⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠢⣌⢦⣠⣈⣦⡶⠚⠛⣿⣿⣤⣶⡞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠷⠖⣀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⢄⣠⡿⠛⢛⣋⣩⣄⡀⠀⠈⠉⠘⠿⢷⣾⣯⣶⣋⠀⠀⠀⠀⠐⢾⣭⣀⠀⢠⠛⣦⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣦⣤⡴⠛⠋⠉⠉⠀⠀⠀⠙⠢⣄⣠⠴⠚⠉⠉⠁⠉⠙⠒⠦⣀⠀⢰⡆⠀⠙⣦⡏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣄⠀⠀⠈⠀⠀⠛
⠀⠀⠀⢀⡀⠀⠀⣀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣀
⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠁
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣀⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⡀
⠀⠀⠈⠛⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠃⠘⠋
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠻⣷⣦⣀⡀
⠀⠀⠀⢰⣶⡆⠈⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣤⣿⣯⣍⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠰⣿⠦⣄⠀⠐
⠀⠀⠀⠀⣀⡀⢠⠟⢶⡀⠘⠳⢶⣤⣀⣀⣤⣤⣶⡾⢿⣿⣧⣀⣀⣠⣀⣀⠀⣀⣴⡿⠟⢋⣁⣤⣿⣗⠂
⠀⠀⢸⠉⠁⠙⢋⣠⠤⠟⠂⠀⠀⠀⠉⡟⠋⠉⠀⠐⠩⠋⠟⡿⣻⣽⣽⣿⢿⠿⣿⡿⠦⣿⡏⠙⠳⡌⠑⠂
⠀⠀⢸⣀⡀⠀⢈⣿⡆⡀⠘⡏⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠟⠉⠃⠀⠀⠈⠁⠀⠙⢷⠀⢀
⠀⠀⠀⠀⠙⠓⠛⠁⠀⢠⡞⠛⠚⢳⡆⠀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⠀⡀⠀⢀⣀⡀⠀⠀⠰⠠⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠉⠀⠉⠈⠉⠉⠲⢏⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡲⠆⠀⠀⡄⣤⠀⣀⣠⠰⠀⠆⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠆⠀⠈⡯⣧⢰⠷⠰⠿⠏⠇⠟⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠦⡄⢀⡠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠞⠳⣤⣀⣠⣤⣤⣤⣀⣀⡠⠔⠉⠧⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⣀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⠄⢠⡄⠈⡷⣠⢼⡔⠄
⠀⠀⠀⠈⣉⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠢⢤⣀⣴⣿⣿⣿⣿⣷⣦⠔⠁⢸⡇⠀⠀⠀⢀⠀⠘⣹⢸⡗⢉⢂⢠⢃⠧⡘⣩⢛⠊
⠀⠀⠀⠸⠀⠒⠀⢀⡀⠃⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠓⠀⠀⡉⠙⠋⣉⡉⢿⠋⢹⠃⠀⠾⠇⠠⢉⡴⠁⠀⢸⢹⣘⠇⣌⡆⢬⠇⢀⣻⠷⡟
⠀⠀⠀⠀⠈⠀⠁⠄⠀⠀⠀⠀⠀⠈⠒⠦⣄⡀⠀⠀⠈⠀⠀⠀⣧⠀⠸⠿⠃⣼⢀⠀⢀⠀⠀⢀⠘⠻⠤⢄⡀⠀⠁⠀⠀⠈⠑⠒⠒⠋⠹⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠀⠀⠀⠀⠀⠀⠲⣵⣶⣾⣿⡾⠇⠈⠀⠊⢀⠈
⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⢘⠀⡀⠀⠀⠀⠴⠁⠀⠐⠒⠒⣶⠀⠀⢀⣠⠉⠿⠁⠀⣰⠟⠲⠦⠼⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠆⠁⢰⡌⠀⠁⣴⠀⠐⠁⠀⠀⠀⠀⠀⢹⣀⠴⠋⠈⢧⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢨⠤⠞⠓⠋⠙⠒⢧⡀
⠀⠀⠀⠒⠄⠈⢷⠄⣀⣼⠉⠒⠊⠵⢀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠣⠊⠀⠀⠀⠀⠠⠀⡄⠀⠀⡀⠀⠀⣀⡴⠃⠀⠀⠀⠈⢠⠐⢘⡽⠂
⠀⢀⡀⠂⠐⠻⡏⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠦⠀⠀⠀⢑⠏⢣⡁⣄⠂⡀⠀⠀⢑⠏
⠀⠀⠀⠀⢤⠺⢧⠁⠀⠀⠀⠀⠀⣀⣾⡁⠀⠀⠠⠒⠀⠒⠐⠆⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠲⠀⠀⢐⠄⠀⠀⠙⠦⢤⡀⠤⡀⠀⠄⡇⠰
⠀⠀⠐⡠⠀⠠⠾⠗⣦⣠⣄⡤⠨⣅⠀⠤⣀⡔⠀⠀⠀⣀⠀⠈⢦⢀⠠⠄⠀⠀⣀⠴⠊⠙⠒⠁⠀⡠⠎
⠀⠀⠀⠀⠀⡀⢠⠀⠛⠁⠀⠓⠀⣀⠀⠀⠀⡇⠀⣠⠠⠘⣤⠀⠀⠀⢰⠒⠀⠁⠀⠀⠓⠲⠤⠂⠊⠀⡠⠐⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⠀⠈⣓⠤⠐⠫⠅⣀⡀⢐⡘⠂⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠀⠀⠀⠀⠀⣀⠀⠀⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠂⠄⣒⢡⠉⠙⠋⠀⠀⣠⠼⠃⠀⠀⠀⠀⠀⠀⠠⠀⢁⠠⡎⣷⢰⠖⢹⡟⢨⣳⡇⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⣄⡤⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡟⠁⠮⠵⠋⠈⠉⠈⠀⣩⠓⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠉⠂⠀⠀⢀⣀⣤⣁⣀⡀⣡⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁⠀⠀⠀⠂⠈⠉⠉⠀⠂

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢛⣛⣋⣉⣉⣉⡛⠓
⠀⣀⠴⠒⣋⣭⠭⠤⠤⠤⠤⢬⣉⠳⣄⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⣄⠀⠀⣠⠖⠛⢉⣉⡉⠓⢦⣀⣤⡤⠤⠤⠤⣄⠀⣀⣤⣤⣀⡀⢸⡇⢸⠁⠀⠀⠀⠀⡇⢸
⡞⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡌⢳⡴⢋⠔⠚⠉⠉⠉⠓⠢⣌⠳⠞⢁⠔⠋⠉⠀⠈⠙⠢⡉⢠⠒⠒⠒⠐⠞⠾⣡⠒⠲⢦⢳⠈⡇⠀⠀⠀⠀⠀⠘⠀⣾
⢹⠈⠀⠀⠀⠀⡤⠀⠤⡀⠀⠀⠀⠀⢹⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢂⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡼⠀⠀⠀⠀⢣⡔⠁⠀⠀⠘⡌⣇⣧⢰⠀⠀⠀⠀⠁⠀⡇
⠘⡇⡁⠀⠀⠀⢣⣀⣠⠇⠀⠀⠀⠀⢸⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣿⢸⠀⠀⠀⢠⠀⢸⠃
⠀⢳⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⢰⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠁⠀⠀⠀⢰⠛⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢸⠀⠀⠀⡄⠀⡟
⠀⢸⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⡀⠀⠀⠀⠘⠤⠇⠀⠀⠀⠀⠘⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠑⠒⠒⢁⡴⠃
⠀⠀⣇⠇⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡆⠀⣾⠉⣷⢘⡇
⠀⠀⢹⠀⠀⠀⠀⠀⢃⢀⣨⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠢⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢰⠶⠦⠾⠻⠀⠈⠁⠀⠀⢿⠠⣬⣿⣥⠞⠁
⠀⠀⢸⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠋⠢⢀⠀⠀⠀⣀⣀⠤⢀⡤⠛⠲⣤⣁⣒⣒⣒⠈⡟⠀⠀⠀⢀⢋⡞⠛⢶⠈⠇⠀⠀⠀⠀⠘⠀⢻
⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⢤⣄⣀⣀⣉⡭⠴⠚⠉⠀⠀⠀⠀⠀⠉⠉⢹⠀⠧⠤⠤⠤⠜⣸⠁⠀⠘⣇⠸⣀⣀⣀⡀⢠⠅⢸
⠀⠀⠀⢧⡓⠦⠤⠤⠤⠤⠴⠒⢊⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠶⠖⠒⠃⠀⠀⠀⠸⠦⢤⣤⡤⠤⠤⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠦⠏⠀⠀⠀⠀⠉⠉⠉⠀⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠀⠀⡖⢦⠀⣠⢦⡀⢀⣤⣄⠀⣀⠀⢀⠀⢀⠀⠨⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢰⣇⡼⢀⡏⠀⡇⣼⠁⢸⡄⣿⣆⣾⡇⢸⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⡏⢹⣼⡇⠀⡇⣿⠀⢸⢻⡇⠿⢹⡇⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠀⠀⠘⠧⠞⠁⠷⠜⠁⠘⢦⠞⠸⠁⠀⠸⡇⢠⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠙⠦⠴⡀⠀⠀⠀⢀⡔⠢⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⠙⠛⠛⡁⠀⡀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸⠀⣧⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢸⡄⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢒⣒⠒⠈⠀⠀⠃⠀⠀⠀⠈⠒⠶⠶⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⣏⣠⠤⡀⠀⣀⢄⡀⠀⠀⠐⠲⠤⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠖⠦⣤⠤⢬⡉⠀⣠⡈⠉⢀⡔⠦⣄⡤⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠙⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠉⠀⠀⠀⠀⠀⢘⣇⠧⡄⠀⠀⣴⣶⡄⠀⣠⣾⣶
⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⡙⢉⣽⣿⣿⣿⣿⣿⣷⣤⡀⣠⣶⠿⣿⣿⣷⣦⠀⢀⣤⡶⣾⣿⣶⣄⠀⣰⣿⣿⣧⢰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣤⠄⢻⣿⣷⣿⣿⠟⠉⠀⠀⠈⠙⣿⣿⡿⠁⠀⢻⣌⠹⣿⣷⣿⠋⠀⠨⣿⡛⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⠀⣺⣿⣿⢻⡏⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢈⠀⠀⣿⣿⡇⠀⠀⢠⠄⠀⢸⣿⣿⣿⢿⣿⣿⡟⢻⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⡏⠘⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⠀⠀⠀⠀⣰⣿⣿⣇⠀⠀⠈⠀⠀⣾⣿⣿⡏⠈⢻⡟⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⢻⠿⣿⣿⣿⣅⣀⣀⣀⣀⣀⣤⣾⣿⡿⢻⣿⣦⣄⣠⣴⣿⠟⠹⣿⣦⣀⣀⣠⣾⣿⣿⡿⠀⠀⠘⠃⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠘⠀⢹⡟⢿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠙⠛⢿⣿⠟⠉⠀⠀⠈⢻⣿⡿⠿⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢹⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⢀⡠⠄⠀⡀⢀⡊⠀⠀⠠⠀⠂⢀⠀⠀⢀⣀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡎⠀⠀⡄⢈⢀⡀⠂⠈⢀⣠⠄⠅⡀⠀⡩⠐⠁⣑⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠄⠒⠁⠈⠨⢤⡔⠋⠀⠠⠀⠈⢎⡂⠀⠀⠀⣿⠀⠀⢀⣟⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢦⡀⠐⠂⢀⠄⠀⠀⠐⠆⢠⡶⠀⠀⢠⡀⢀⡤⠀⢀⡏⠀⢀⡞⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠆⢸⣇⠀⡿⠀⠂⠈⠃⠀⠀⢸⡏⠍⠁⠀⢸⢇⣀⡾⣤⡤⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⠆⠀⡀⠀⢀⡶⢀⣀⣴⣄⣀⡇⠈⣀⣀⣸⠃⢩⢰⠁⢀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠆⢠⢠⠀⠈⠀⣠⢏⢴⣻⠕⠓⠛⡖⠀⠀⠀⠀⠊⠍⠉⠙⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠀⠆⣒⡫⠕⠁⠀⠀⠀⠀⠈⠡⠇⡠⢄⡐⠂⠈⢃⠀⠀⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠒⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⢡⠠⢄⠀⠀⡠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠈⠒⠤⠈⠁⠒⢊⡀⠀⠀⠀⠤⠄⢀⡀⠀⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⢸⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⢦⣀⣠⠀⠀⠀⢀⣠⣶⡿⠿⠾⠿⢷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⠦⡔⢋⣿⣶⣶⣶⣾⣿⣿⢋⣠⠀⠀⠀⠀⠙⢿⠿⢿⣧⡀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠑⣌⠛⢿⡿⠛⢩⢉⡽⠋⠉⠉⠉⠂⠀⠀⠀⠐⢄⣀⣹⣿⣷⠶⣩⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣷⡏⠀⠀⠇⠊⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠠⢽⣿⢿⣷⣾⣿⡆⠁⠀⠀⠀⠀⣀⣠⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣠⡆⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⠦⠀⠀⠀⠈⠂⠀⠙⢻⣿⣤⣴⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⠟⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢄⢸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠿⢷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡏⠐⠁⠉⡏⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣬⣇⠀⠀⠀⠀⠈⠉⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣟⣒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⣯⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡿⠻⢷⣬⡈⠓⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⣴⣿⣿⣷⣄⡀⣀⠈⠦⣒⡶⣄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⢉⣾⡟⠁⠀⠀⠈⠙⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠹⣌⠻⠿⣿⣿⣿⠿⢷⡂⠈⠉⠉⠝⢲⣶⣖⡛⠂⠀⠀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈⠙⠒⠂⠀⠙⠻⠶⠿⠟⢲⠲⠋⠛⢿⣿⣿⣿⣿⣿⠿⠛⢮⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠀⠐⢤⣀⣉⣉⣁⡤⠂⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⡴⠁⢱⠀⠀⠊⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⢠⠶⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⣠⠀⠀⠀⠉⠀⡜⠀⠀⠀⢣⠀⠀⢀⠀⠀⢀⠔⠁⠸⠀⠀⡞⠁⠀⠀⠀⠀⠀⣾⠼⠁
⠀⠀⠀⠀⢰⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢦⠀⢀⡜⠀⠀⠀⠀⠈⢧⠔⠉⡆⡠⠋⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢵⣃⡀⠀⠀⠀⡀⠀⢀⠀⠀⡇⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠓⠒⢒⠖⠀⠠⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠉⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠉⠉⠉⠀⠀⢀⣀⣀⠀⠀⢠⣴⣤⡀⢠⣾⢿⣆⠀⣶⣆⠀⢠⣶⠀⠀⠀⠀⠀⠀⢀⣠⠬⠛⠁
⠀⠀⠀⠀⢰⣿⡷⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⣿⡟⢻⣷⠀⣿⠃⢹⣧⣾⡇⠀⣿⢠⣿⢿⣦⣿⣿⠀⠀⣀⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠉⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⢻⣷⣿⣧⡘⣿⠀⢸⣿⢿⡇⢀⣿⢻⡟⠈⠻⢫⣿⠀⠀⠙⢆⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠮⢅⣀⡀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣹⣿⢻⣧⣼⠏⠈⠿⠿⠋⢿⠇⠀⠀⠘⠿⠀⠀⠐⣍
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠉⠉⡱⠆⠀⠀⠀⠘⠿⠿⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠴⠯⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢧
⠀⠀⡀⢠⢤⡆⠴⠒⠂⠀⠀⠀⠀⠀⠸⠓⠒⠉⡇⠀⠀⠀⢀⡀⠀⠀⡠⡀⠀⢠⠛⢦⠃⠈⠑⢤⡀⠀⠈⡆⠀⠈⡦⣀
⠀⠀⠈⠁⠀⠀⣀⠤⣒⠆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠔⠉⢃⢀⠞⠀⢣⠀⡜⠀⠀⠀⠀⠀⠀⠉⠢⣄⠘⡄⠀⠀⠉⠷⡄
⠀⠀⠀⠀⠀⠸⠝⠊⠁⠀⠀⢀⠄⠀⠀⠀⢀⠇⢀⠔⠁⠀⠀⠘⠃⠀⠀⠘⣆⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣜⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣢⠄⠀⠀⠀⢸⣛⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠔⠂⠀⠀⠀⠀⠀⠀⠈⠯⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠐⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣳⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣷⡀⠴⠚⠛⢺⡄⠀⢀⠀⠘⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣫⠔⣋
⠀⠀⠀⠀⠀⠀⠈⠑⠕⠤⡀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠙⠟⠀⡄⠀⣸⡗⠉⠉⠉⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡭⢒⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⢀⠀⢀⣤⣁⠀⠀⠀⢀⠀⡦⢷⢚⡏⢹⣤⡄⠘⠛⣰⢻⡄⠀⠀⠀⠀⠀⢀⡤⣺⠕⡊⠑⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡏⠁⠈⠻⣟⠲⢶⣋⡀⡸⠈⣿⠃⠸⣇⣨⠝⢰⠃⠸⠁⡀⠀⣀⣴⡺⡓⠍⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡵⠀⠀⠘⢿⠣⠤⠬⣿⠁⠀⠈⠳⡶⠋⠀⠀⡟⣲⣴⡇⡷⡮⡥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡷⠔⠒⠒⠚⠺⣇⠀⢰⠉⢧⠀⠀⡸⠹⡄⠀⠀⢻⣭⠌⠧⡎⠐⢲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⣶⡇⠀⢰⠂⢲⠀⢸⣤⠇⢠⡈⢧⢠⠃⠀⢱⡀⢀⡜⠁⡷⢤⠃⠀⡯⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⣽⠀⠈⠁⠁⠠⠞⢏⢀⠎⢡⠈⢯⠀⣄⠀⢳⠋⠀⠀⡇⡸⠀⢸⠓⠦⢤⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠛⠛⡆⠀⢠⠤⢤⡀⠀⠙⡂⠈⠁⠈⢶⠉⢢⠀⡔⠃⠀⢇⡇⠀⡇⠀⡔⠉⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢤⠀⣧⠀⠸⠦⠔⠃⠀⣠⠋⢉⠛⡄⠈⢆⠀⢛⠒⠃⠀⢸⣧⣼⣀⣀⠙⢦⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡒⢻⣄⣤⣤⡤⣄⡞⠁⠀⢈⠵⠿⡖⠚⠀⢸⠶⢦⣀⡁⠀⡁⠀⠈⠉⠐⠊⡵⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠉⠘⠹⠻⣆⣤⡇⠀⣴⠁⢀⣴⠟⣦⡀⢸⠟⠐⠋⠉⠙⠳⠀⡤⠤⠤⠶⠁
⠀⠀⠀⠀⠀⠴⠚⠍⠀⠀⠀⠀⠀⣆⠀⠀⣠⣇⠀⢀⠞⠉⣧⣾⡿⠓⠏⠱⣜⠀⡀⠘⢄⠀⣀⠼⠗⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⣦⡊⣀⣴⢻⡿⠔⠂⠙⠒⠚⣿⢀⠀⠀⠉⠀⠀⠀⠀⠐⠕⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡹⡈⠘⠁⠀⠀⠀⠀⠀⠈⣏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢍⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠐⣄⢀⡀⢠⠀⣄⣿⣼⣧⣼⠀⣠⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣄⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠉⠀⠈⢷⣼⣷⣿⠮⠿⠉⠀⠀⠻⠿⠛⣿⣿⣀⣤⡔⣀⡤⠄⠀⠀⠀⠀⣀⡤⠖⠚⠉⢀⣦⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠲⣾⣟⣠⡬⠵⠖⠒⠲⣄⠀⠀⠀⠀⢉⣩⠿⠿⠿⢯⣖⡂⠀⠀⠀⠀⠘⠻⠷⣆⠀⡞⠘⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠘⠛⠀⠀⠘⣾⠁⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⣄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣤⠀⠀⠀⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠻⠯⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠺⠇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⣿⣤⡄
⠀⠀⠀⠀⣤⣤⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⣸⣿⣟⠓⠒
⠀⠀⠀⠀⠻⠟⠀⠀⠈⠻⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣻⣿⡛⠛⠛⣓
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⢳⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠀⣹⣯⣍⠓
⠀⠀⢀⣠⠤⠴⣤⡎⠀⠻⣦⡀⠈⠙⠻⠶⣴⣶⡿⠿⠟⢛⡿⣻⢿⣶⣶⣶⡶⣶⣤⣤⣾⣿⣟⡉⢠⣶⣾⣿⠛⢦⣀
⠀⠀⢰⠃⠀⠀⠉⢶⣖⠛⠉⢁⣠⣤⣤⡀⠺⠀⠀⠀⠀⠁⠊⠑⠋⠞⢹⢿⣿⣿⠿⠿⠉⢿⡏⠙⢾⣯⠀⠈⠓⠄⠈⠑
⠀⠀⠸⠦⣄⣀⢀⣠⡿⠠⠄⢈⣧⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠏⠞⠀⠙⠀⠀⠀⠀⠁⠀⠀⠻⡆⠀⣄
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢰⠏⠉⠙⠙⡿⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡞⠢⡀⠀⠀⠀⠀⠀⠀⢠⢢⡀⠀⠀⠀⠀⠀⠀⠀⡐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⡀⠀⠀⠀⠀⢠⠀⠀⠈⠢⡀⠀⢀⡀⢠⠃⠀⠱⡀⠀⠀⠀⠀⠀⡔⠀⠈⡀⠀⠀⠀⡠⠛
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡄⠀⠀⠀⠀⠈⠢⠂⠙⠃⠀⠀⠀⠑⡄⠀⢀⣀⠜⠀⠀⠀⢡⠀⡠⠊
⠀⠀⠀⠀⠈⠐⠦⣀⠤⣰⠀⠀⠀⠀⠀⠈⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠃⠉⠀⠀⠀⠀⠈⠞⠀⠀⠀⢀
⢒⠢⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢠
⠀⠈⡄⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⣰⠐⠂
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠋⠁⠀⡇⠀⠀⠇
⠀⠀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡄⠠⠂⠀⠀⠀⢰
⠀⠀⠰⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⠀⢈⡤⠒⠢⠄⢀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⡜
⠀⠀⠀⢃⠀⠀⠀⠘⠢⠋⠀⠀⠀⠘⢸⠀⠀⡚⠉⠉⢲⠀⠀⠀⠃⠀⠈⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡀⢼⠁
⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⠎⠠⢀⠃⠀⠀⢄⠀⠈⣦⠀⠀⡇⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢂⠀⡀⠈⠀⠀⠀⠀⡇⡎⢁⠆
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠢⡀⠀⠉⠈⠀⠀⢸⠁⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠚
⠀⠀⠀⠀⠀⢃⠀⠀⠀⢣⡠⠠⢉⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠴⡎⠒⠶⡞⠳⣶⡶⠶⠚⢷
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⠣⠤⠞⠀⠀⡀⠉⠀⠀⠤⠄⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣵⠀⠠⠆⢀⠦⠃⠀⠀⠃⠀⠛⠇⣠⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠄⠊⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡀⠀⠖⠀⠀⠐⠀⠀⢧⡆⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠈⠦⢀⣋⣀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠄⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠐⠀⠀⠊⠀⠀⠀⠐⠀⠀⠀⠈⠂⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⣖⡐⠀⠀⡰⢄⠀⢠⠓⡄⠀⡰⠉⢄⠀⡰⠉⡆⠀⠜⠑⡄⣠⢦⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡆⠙⢦⡀⠇⠈⠣⠆⠀⠘⠴⠁⠀⠈⠶⠁⠀⠘⠊⠀⠀⠉⠁⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡒⠒⠒⠈⠐⢺⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣧⡯⡡⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠗⣰⡏⠀⠀⠀⠀⠀⠀⡻⠀⠘⠪⣶⠦⢤⣤⣤⣶⣭⣜⠀⠀⣾⠈⠉⡓⡆
⠀⠀⠴⠞⠉⠀⠀⠀⠉⢷⡀⢨⣿⠁⠀⠀⢾⠀⠀⠀⢹⡀⠀⢠⡏⠀⠀⢸⠿⠇⠀⠉⣏⣖⡍⠀⠠⢱
⠀⠀⠰⣱⠀⠸⠀⡅⠀⡘⣧⣻⡎⠀⠀⠀⡁⡀⠀⠀⢘⡇⠀⢼⠃⢀⠄⢸⡌⠀⠀⢀⣡⣟⠀⠀⣌⡆⣠⠄
⠀⠀⠀⠅⡂⠀⠋⠀⠐⢵⣏⣰⠇⠀⠀⢰⠀⡇⠀⠀⠈⣷⠀⢿⠀⢸⠂⠘⠀⡼⠀⢸⡘⣭⠀⢐⠸⠊
⠀⠀⠀⠸⣡⠀⠰⠒⢄⠀⢙⢻⠀⠀⠀⠈⠀⠁⠀⠀⠀⢙⠀⡃⠀⡸⡁⠀⡰⠃⠀⡢⣙⠀⠀⣌⠃
⠀⠀⠀⠀⡕⠀⠀⠒⠃⢀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⡅⠀⡅⠇⡠⢁⠀⠀⣵⢯⠚⠻⢜⠳⣤⢀⡀
⠀⠀⢀⣠⢶⠣⠠⠤⢐⠭⣶⠁⠀⠀⢸⡭⠀⠌⠀⠀⠀⠈⣰⠨⠭⠜⠐⢲⡚⠤⠜⣼⢗⠀⢠⢹⠅⠐⠒⠊⠙
⠀⠀⠀⠀⠈⠁⠉⠫⡙⣤⡗⠀⠀⠀⣤⠆⠀⢨⢃⡀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠉⠙⠉⠒⢬⢩⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡿⠃⠤⠠⠤⢧⠀⠀⠀⠂⠀⠡⠍⠭⠇⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⣆⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠐⡛⣉⡉⠉⢉⣙⠒⢄⠀⠀⠀⣀⡀⠤⠄⣀⠀⠀⢀⠔⡡⠒⠐⠀⠒⠐⠢⠼⠤
⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠑⡌⣂⠤⡪⠔⠒⠀⠒⠢⢝⠢⡬⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠊⠀⠀⠀⠀⠀⠀⠀⠑⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⢀⢀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢷⣂⣀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀⢀⣿⠿⣚⣷⣟⡷⠈⠉⠀⠀⠒⡿⠒⢠⣤⣤⣶⣤⣄⣀⠈⠁⣰⣿⠟⠻⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠼⡟⠃⢀⡆⠀⠀⢀⣀⣤⣤⣯⣟⣈⣿⣿⣿⢿⣿⣿⣶⣴⣿⣿⣷⣷⣤⣿⣿⠉⠙⠿⣿⣿⣷⣦⣿⡏⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣷⣶⣴⣉⢀⣶⣿⠟⠉⠙⠻⣿⣿⡿⠋⠀⠀⠈⠙⢿⣿⡿⠉⠻⣿⣿⣿⡏⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠛⠻⣿⣿⡏⠀⢀⣄⠀⢹⣿⠇⠀⢸⣿⡄⠀⠘⣿⡇⠀⠀⠈⢿⣿⡇⠀⡀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣾⣿⣿⣦⠀⠙⣿⡇⠀⣿⣿⡆⠀⢿⡄⠀⣿⣿⣿⡀⠀⣿⡇⠀⢠⡀⠀⠹⠃⠀⣷⠀⠀⠀⠹⣿⣿⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢹⣿⣿⠏⠀⢀⣿⡇⠀⢹⣿⣷⠀⢸⡇⠀⢸⣿⣿⡇⠀⢹⡇⠀⢸⣿⠀⠀⠀⢰⣿⣧⠀⠀⠀⢹⣿⡀⠀⢀⣾⣿⣏⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠻⢿⣧⠀⠘⣿⣿⠀⠈⣧⠀⠈⣿⣿⡇⠀⢸⡇⠀⠸⣿⡇⠀⠀⢸⣿⣿⣇⠀⠀⢈⣿⠿⠿⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠰⣶⣶⣦⣄⠀⠀⢻⡄⠀⢿⣿⠆⠀⣿⡆⠀⠸⠿⠁⠀⢸⡇⠀⠀⣿⣿⡄⠀⢸⣿⣿⣿⣧⣀⣸⣷⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⣿⣿⣿⣿⠇⠀⠀⣧⡀⠈⠋⠀⢠⣿⣿⡀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⣤⣼⣿⣿⠋⢿⣿⣿⣿⣷⣶⣾⣿⣿⣯⠃
⠀⠀⠀⠀⠀⠀⣻⣿⣇⠀⠀⠙⠛⠋⠁⠀⢀⣾⣿⣷⠀⠀⣀⣾⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣶⣿⠋⠛⢿⣿⣿⠃⠀⠀⠉⠙⠁⠙⢿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣤⣤⣤⣤⣾⣿⡿⢿⣿⣿⣿⣿⣿⠇⠘⠿⣿⣿⣿⣿⠿⠳⢭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠿⠿⠿⠟⠉⠈⠳⠿⠿⠿⠟⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠻⣦⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢠⣀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⢠⠋⢠⠞⠁⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠉⠑⠀⢠⡀⠀⠀⢦⡀⠀⠀⠀⠸⣆⠀⢀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠁⡰⠃⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣨⠛⢦⡀⠀⠀⠹⡆⠘⣧⠀⠀⠀⢀⣿⠀⣠⢿⡀⠀⠀⠀⠀⠀⣀⡴⢻⡇⠀⠀⠀⠊⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠳⢤⣄⣘⣄⣹⣄⡤⠖⠋⠀⠉⠁⠀⠉⠓⠢⠤⠴⠚⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⢴⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⡆⢰⣿⠀⡎
⠀⠀⠀⠀⠀⠶⣤⣀⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⢸⣿⠀⠘⠢⠤⣀⣀⣠⣤⣤⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⣿⢸⡟⠀⠀⠀⠀⠀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⣷⡤⠀⠀⠀⠠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⣾⡇⠀⣿⠀⡿⢸⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠛⣿⠁⠀⠀⠀⡏⢉⣿⠀⠀⣴⣿⣦⡀⠀⣴⣶⡀⣿⢹⣶⡿⡇⠀⣿⠀⡇⢸⡇⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⣧⣾⣿⡄⢰⡏⠀⠙⣧⢸⠏⠙⣷⢹⠀⠿⠀⣿⠀⣿⠀⡇⠸⠁⠀⠀⠀⠀⡠⠚⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢦⠀⠀⠀⠀⣿⠉⢈⣿⢸⡇⠀⠀⣿⢹⣄⣠⡿⢸⡇⠀⠀⠿⠀⠻⠀⠀⠀⡀⠀⠀⠒⢶⠧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠊⠀⠀⠀⠀⣿⣤⣾⠇⠀⢿⣤⣤⡟⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⡀⢠⡄⢸⡇⠀⠀⠘⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠀⠀⣿⡟⠋⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠘⠃⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣹⣦
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠁⠉⢳⠀⠀⢀⡄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⢀⡤⠤⢤⣀⣾⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⢠⣿⠖⢛⡟⣳⢧⡀⠀⠀⡴⠋⠉⠀⠀⠀⠈⢳⠋⠀⠈⠳⡄⢰⠏⠀⠀⠀⠈⠙⠆⢀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡞⠀⠁⠀⠙⣆⣸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⣾⠀⠀⠀⠀⠀⠀⠀⡄⠱⣄⠐⠤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠳⣄⠈⠳⠦
⠀⠀⠀⣠⠞⠛⣆⡀⢀⣤⠊⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠳⡄
⠀⠀⢸⣅⠀⠀⢉⣼⠋⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠘⣦
⠀⠀⢸⣅⠀⠀⠀⠘⠣⠀⠘⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⢷⡀
⠀⠀⠸⣥⣤⡴⠂⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠳⠤⠶⠦⠤⠶⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢀⣠⠶⢃⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢸⠀⠂⠀⢀⣠⠶⠋⠁⠀⠸⠀⠀⠞⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠙⠳⠦⣤⣀⢀⣠⠖⠋⣀⡤⠚⠙⠀⠀⠇⠀⠐⠋⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠛⠒⠦⢤⣄⣉⢉⡠⠖⠋⠁⠀⠀⠀⠀⠀⢃⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠂⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⣀⣈⣉⣁⣤⠀⡇⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠎⠁⡇⠈⠀⠀⢀⣀⣀⡀⠀⢀⣀⣠⠤⢤⡀⠀⣀⣀⣉⡉⠀⣀⣀⠀⣇⣀⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⣀⠂⡖⠋⠉⠀⠀⢹⣦⡏⠀⠀⠀⢸⣿⡞⠉⠁⢀⡏⠉⠉⢹⣷⣈⣉⣉⡉⠉⠉⠙⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢰⡚⠉⠁⠀⡇⠀⢠⡆⠀⣼⣿⠀⢀⠀⠀⣸⣿⠁⠀⠀⠘⠀⠀⠀⣼⣿⠀⠉⠉⠉⢉⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠘⢦⠱⡄⠀⠀⡇⠀⠀⠀⠈⢹⡇⠀⢸⠀⠀⢿⠇⠀⢰⠁⠀⡀⠀⠀⣿⡏⠀⠀⠀⢠⢃⡞
⠀⠀⠀⠀⠀⠐⠋⠉⠁⡤⠄⠀⢀⡴⠋⣠⠞⠀⠀⡇⠀⠰⠟⠀⡼⠀⠀⠀⠀⢀⠏⠀⢀⡟⠀⢰⠇⠀⢰⣿⡇⠀⠀⠐⢇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠾⠥⢤⣤⣤⡇⠀⠀⣠⣾⣃⣀⣤⣯⣤⣾⣤⣤⣼⡇⢀⣿⣤⣤⣼⣿⠀⠀⠴⠖⠚⠃⣀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠶⠶⠦⠤⢤⠀⢳⣾⣿⡿⠟⠹⠿⠛⠛⠛⠛⠙⠉⠉⠹⠿⠿⠛⠛⠛⠋⠀⠀⢀⠰⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⠀⠀⠀⣀⣀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⣀⠉⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢩⠔⠀⠀⡇⠀⠘⠉⠁⣀⣤⡈⠣⡀⠀⠀⡰⠋⠙⠲⢤⣀⠀⠀⣠⢚⣛⠒⠒⠒⠀⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣠⡴⠖⠋⠉⠀⠹⣄⠘⠦⠊⠀⣠⠛⠳⢦⣌⡙⠚⣡⠋⠉⠉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⢀⠄⡠⠀⡀⠘⢦⠀⠀⡴⠁⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⠃⠀⠁⠀⠀⠳⠞⠀⠀⠀⠹⡌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷

⠀⠀⢠⣤⣤⡀⢀⣠⣤⡄⠀⢀⣀⣀⡀⣀⢀⡀⣀⢀⣀⣀⣀⠀⣀⣀⣀⣀⣀⠀⠀⡀
⠀⠀⢸⣿⣿⣷⣭⣽⣿⡷⣴⣿⣷⣾⠿⠋⢈⣡⣭⣬⡉⡟⠿⣷⡽⣿⣿⣿⠿⣾⣛⣋⣭⣵⣿⣿⣿⣿⡶⢶⣶⣿⣿
⠀⠀⢸⣿⣧⢾⣿⡏⢹⣷⣿⣿⣿⠁⣴⣿⡿⠿⡛⠙⣳⣷⣷⣿⣿⣿⣧⣼⣿⣿⣋⣡⡿⠁⠀⢈⣿⡏⠀⠀⠈⠛⠻
⠀⠀⢸⡟⣷⣿⣿⣿⣾⡿⣷⣿⣿⣿⣏⡽⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣷⣿⠋⣿⣿⣿⣿⣶⣿⣿⣿⢻⠀⢠⣄⣀⣠
⠀⠀⢸⣷⣿⣿⣿⣿⡟⠓⣿⣏⣿⣿⠋⠁⠀⢀⣤⣿⣟⢻⣿⣿⣟⣿⣇⣀⣉⣉⣹⢻⣍⢰⣻⣿⣶⢾⣃⠢⢿⣿⣿
⠀⠀⢸⣿⣯⣾⣿⣿⣧⡀⢸⣿⣿⣿⡀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⢸⣯⢠⡌⢰⡏⡎⣿⢽⡎⢉⠁
⠀⠀⢸⣽⣿⣿⠁⢤⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⣼⣻⡿⣿⣿⣿⡿⠿⣿⣿⡿⡟⠸⡟⣿⡇⢸⡇⡇⠏⠜⠇⢘⣷
⠀⠀⢸⣿⡏⠸⢟⣞⢹⣿⣿⣿⡿⣿⣿⣷⡿⠇⠿⣿⣿⣷⣿⣿⣿⣿⣀⣿⠟⢻⣷⠀⡇⣿⡇⢸⡅⣇⣌⠧⢘⡈⣿
⠀⠀⢸⢸⣧⣴⣤⢤⣼⣿⣿⠂⢰⣿⢙⣡⣠⣴⠠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣤⠸⠟⣁⢣⣿⡇⢸⡇⣿⠌⣉⢰⠂⠘
⠀⠀⢸⣻⣿⣯⣽⡯⢿⣿⡟⠀⠚⣿⣬⣿⣿⣿⡰⠿⢿⣿⣿⢿⣘⣽⣿⣿⡿⣿⡛⣿⣾⣿⡇⢸⠁⣿⠂⠘⠀⡇⢠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠏⠽⠤⠿⢭⣭⣿⣛⣐⣠⡼⠿⠿⢿⣿⣾⣿⣿⣿⣿⣇⣿⣿⣿⣇⠸⣀⣿⡁⢠⡆⠁⢀
⠀⠀⢸⡇⣿⣆⡜⠻⠦⣶⡾⣿⠿⠶⠦⠼⠛⠿⠿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇⣟⢻⣇
⠀⠀⢸⣿⣿⣿⣶⣤⣬⣶⡀⣠⣄⣠⠄⠄⢰⣶⡶⠆⢹⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣼⣿⡿⠢
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣾⣿⣿⠿⠿⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⠀⢰⢦⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⣀⡀⢯⡇⠘⠚⠃⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠄⠄⠥⢨⣎⠔⠀⠀⠀⠀⠀⠀⠐⠃⠈⠡⠞⣻⢃⣤⣄
⠀⠀⢀⣼⠓⠁⠀⠀⠀⢀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣾⣦
⠀⠀⠈⠁⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠑⠉⠁
⠀⠀⠀⠀⠀⡀⠀⢀⣀⡉⠀⠠⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣦⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⣀⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢾⣿⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⣴⡎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡆
⠀⡀⠀⠀⠀⠐⢠⠀⠄⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⢠⣦⡴⠁⠀⠀⠀⠀⣀⡼⠙⢞⠙⠣⠄⠀⠀⠀⢀⠀⠀⠀⠄
⠀⢷⠄⣀⣹⣬⣠⣤⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡏⢂⣼⠖⠂⠀⠀⠀⠀⠉⠀⢀⠈⢈⣼⠰⠂⠀⠀⠈⣧⣀⣜⣒⢓⡒⠢⠂⣀⣴⡾
⠀⠀⠳⣬⣥⣤⣤⡀⢈⣁⣤⠼⠁⠀⠀⠀⠀⠀⠖⠛⠀⠀⠀⠀⠀⠀⠀⠚⢯⣤⡤⢴⡏⠁⠀⠀⠀⠀⠈⠉⠙⠉⠉⠛⠒⠛⠋
⠀⠀⠀⠀⠀⠛⡟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⡔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⠒⠀⠒⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⣤⣠⣤⡄⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⠧⠘⠧⠋⠟⠛⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠘⠈⠁⢠⡠⢂⡤⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣶⣿⣶⣀⣰⣿⣷⣤⣶⣦⣦⣦⣤⣾⣿⣿⣷⣀⣷⣾⣷⣀⣀⣀⣀⣀⣀⣀⣶⣷⣆⣠⣴⣿⣶⣄⣠⣤⣿⣟⣠⣤⣤⣄⣀⣀⣀⣀
⣿⣿⠉⡉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡟
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢀⠤⠶⢤⡄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣫⠔
⠀⠀⠀⠀⠀⠀⠩⢓⢄⠀⠀⠀⠀⠀⠀⠀⠆⠉⠻⠆⢀⠀⢸⣗⠒⠉⢲⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡫⣒⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡠⠀⠀⢠⣌⡀⠀⠀⡀⢰⠾⡖⣿⢻⣤⡀⠚⢳⠞⡄⠀⠀⠀⠀⢀⣠⣖⠿⠪⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⢙⣿⡳⢶⣅⣀⠃⢻⡇⠸⣇⠄⠁⡎⠐⠃⠄⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠤⢂⡀⢀⣹⡙⠒⢺⢇⠀⠀⠉⣟⠁⠀⢸⣩⡶⡇⢟⣛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠉⠉⣁⣀⠉⢣⢀⡎⠈⢆⠀⡜⠈⡆⠀⠀⡷⢧⣙⡇⠀⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡇⠀⠧⠜⠁⣸⡞⢀⠶⠈⢦⠁⠀⠸⣀⠞⠀⠈⢹⠁⢰⣍⣀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠶⢷⠀⠀⣀⣀⠀⠑⢾⠀⠇⠈⢦⠛⡄⠉⡰⠀⢸⡸⠀⡎⠀⡬⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣀⠘⡄⠀⣇⠠⠃⠀⡸⠒⠲⠄⠘⡄⠘⢮⢴⠀⢸⣇⣰⡀⠀⠣⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⣧⣀⣠⣤⣤⡖⠁⠀⣩⠾⣆⠚⠀⡾⠶⣀⣈⠉⠫⠈⠉⠑⠊⢹⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠂⠈⠉⠁⠉⠛⣦⢤⠅⢠⣞⠀⣴⡏⣢⡀⣿⠃⠊⠉⠙⠳⢀⡆⠤⠴⠋
⠀⠀⠀⠀⠰⡊⠉⠀⠀⠀⠀⠐⢄⣀⠤⢧⠀⡰⠋⢹⣾⠿⢉⣃⠑⣼⣠⠀⠣⢀⠠⠞⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠥⡴⡏⣿⠉⠁⠀⠉⠙⢯⡆⠀⠀⠀⠀⠀⠀⠁⠩⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠁⠀⠀⠀⠀⠀⠀⠘⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⡵⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡓⠘⣡⠶⠛⠛⠛⠻⢶⣤⡶⠟⠋⠉⠙⠻⢶⣄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⣴⢟⡉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⡆⠀⣴⣊⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣉⣠⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣠⠞⠉⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⣀⠀⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠄⠀⡏⠀⠀⠀⠀⠀⣿⡏⢻⣷⡀⠀⣸⣿⣇⠀⠀⣿⣿⡆⢰⣿⣇⠀⢰⣿⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⣠⠁⠀⠀⠀⠀⠀⣿⣧⣾⡿⠁⠀⣿⠿⣿⠀⠀⣿⣿⣿⣿⢿⣿⠀⢸⣿⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣄⠀⠀⠀⠀⢠⣿⠿⢿⣦⡀⣸⣿⣶⣿⡆⢸⣿⠃⠻⠏⢸⣿⠀⢸⡿⠀⠀⠀⠀⠀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣌⡀⠀⠀⠀⢸⣿⣄⣼⣿⢃⣿⠇⠀⢻⣧⢸⣿⠀⠀⠀⠘⣿⡀⢠⡄⠀⠀⠀⠀⢠⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⠉⢿⠀⠀⠈⠙⠉⠉⠁⠀⠉⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠚⢁⠄⠸⣦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣦⣄⣀⣀⣠⡴⠲⣄⡀⢀⣠⡀⠦⡀⠀⠀⣠⠎⠙⠒⠋⠀⠈⠲⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⡴⡀⡈⠉⠉⠁⠈⠳⠶⠶⠛⠁⠀⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠸⠀⠉⠢⠌⠐
⠀⠀⠀⠀⠠⠒⠢⠤⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡃⠀⢚⡀⠀⠀⠢⠤⢄⡀⠀⠈
⠀⠀⡐⠉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠠⠤⠀⠀⡠⠒⠉⠐⢠⠀⠀⢸⠂⠘⠁⠊⠀⠑⠒⢈⢡⣀⠀⠠⠀⡸⠤⠄⠐
⠀⠀⠑⢄⠀⢀⡖⠃⠀⣠⠖⠋⠉⠙⠢⣀⡰⠧⡀⠀⠀⡀⠉⠀⠈⢢⡠⠒⠒⠢⡠⣶⠶⡄⢄⠠⠡⠊
⠀⠀⠀⠈⠉⠁⠀⠀⢠⠇⢀⠀⠀⣄⠄⠈⠀⣀⢈⠉⠉⠀⠀⢀⣤⠠⠤⡀⠀⠀⢈⠕⡂⠉⠭⡑⢴⢩⠉⠉⠉⣍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡃⠠⣠⠏⠒⠻⠴⠋⠀⢀⠔⡋⠭⢉⡒⠀⠑⢎⢢⡔⡡⠊⠀⠀⢠⣿⠀⠇⠀⠀⣰⠏⡜
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠸⠁⠀⠀⡠⢒⠩⢙⠻⢣⠊⠀⠀⠀⠃⠀⠀⠈⠳⠊⠀⠀⠀⠀⢸⡟⢸⠀⠀⣰⠏⡼⠅⢀⠄
⠀⠀⠀⠀⠓⠀⢧⢀⡠⢚⠩⠭⣐⠚⡐⠁⠀⠀⠙⠇⠀⣴⠟⠆⠀⠀⢰⣄⠀⢀⣴⠂⠀⠀⣼⠇⠆⠀⢠⡟⡜⠀⠰⡁⢱
⠀⠠⢤⠀⠁⠀⠰⠡⠊⢡⣤⠀⣸⣧⠀⢰⡟⡃⢰⠀⠀⠻⠊⠀⠀⠀⠈⣏⢙⡛⢻⠀⠀⠀⣿⢠⠀⢠⡟⡘⠀⣀⠄⠙⠏
⠀⠀⣀⠁⠀⠀⠰⡰⠀⠘⠁⠀⠛⠲⠀⠈⠛⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⣿⢨⠸⠈⠀⠀⢸⣏⠸⠶⠟⣰⠟⠙⢈⣃⡤⠂
⠀⠀⢀⠁⠀⠀⠀⢇⢃⠀⢰⣟⠆⠀⢡⡀⠀⠀⣰⡿⠠⣀⣠⣾⢧⣤⣤⣿⢸⠇⢥⣄⣀⣸⣏⠀⣸⡎⡇⠀⣀⡼
⠀⠀⢉⡏⢰⠄⢄⡸⡌⡄⠀⠁⠀⢀⣼⢋⣒⠚⣋⡔⠓⠮⠥⢶⠒⠒⠒⠒⠊⠈⠢⠭⡍⠙⠬⣛⣋⠜⠁
⠀⠀⠀⢧⡀⠀⠸⠀⢣⢡⣀⣤⡶⢛⡵⠋⠈⠉⠁⠀⠀⠀⠴⠛⣰⣀⠀⠀⠀⢀⣀⠀⠲⡤⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⢨⠋⠀⠀⠈⠳⠭⠴⠚⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢸⣴⠋⣦⠇⠈⠀⠀⠹⠀⡀⠀⢺⠉⠁
⠀⠀⠀⠀⠸⡄⠄⠒⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠘⠁⠀⠘⢄⣀⠶⡊⠀⠀⠀⢀⠴⣄
⠀⠀⠀⠀⣀⣨⠇⠔⠂⠱⢄⣀⣀⠄⠟⠦⢤⣤⠎⠒⠖⠊⠈⠀⠀⠀⢀⠈⠐⠈⠀⠀⠈⠓⠒⠚⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⡄⠀⠈⠲⢄⣀⣀⠤⠀⠈⠉⠉⡠⠰⠚⢀⣈⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠐

⢦⡈⠳⣄⠈⢷⣌⢻⣷⠀⠙⢆⠈⢧⠀⢳⡀⠀⠸⡆⠀⠀⢿⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠸⠀⠀⢰⠃⠀⠀⣸⠏⠀⠠⠁⡴⠃⣠⡞⣡⣿⡿
⡀⠙⢦⡈⠳⣄⠙⢧⡙⢷⡀⠈⠀⠀⠃⠀⢣⠀⠀⠻⠀⠀⢸⡄⠀⠀⠀⢈⣿⠁⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣴⠃⠀⠀⢀⠞⢀⡼⢋⣾⠟⠋⢀
⣿⣦⣀⠉⠢⡈⠳⡄⠙⢆⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠁⣠⢋⣴⠛⠁⢀⠔⠋
⠈⠛⢿⣷⡀⠈⠂⠈⠢⠀⠳⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠜⣡⠞⠀⠀⠐⠁⣠⠖
⡁⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠠⠊⢁⣤
⡛⠶⣄⡀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⡀⠁⠀⠙⠲⢄⡀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡤
⠛⠲⢤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠊⠁
⠑⠂⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⠴
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣈⣁⣀⣀
⠿⠿⠭⣩⣉⠉⠉⠉⠉⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⢉⠉⡙⢛
⠛⠒⠒⠒⠂⠀⠐⠀⠂⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⡿⢹⣿⠩⠉⠀⠆⠐⣐⣂
⠌⠡⠡⠈⠄⠠⠠⠀⠄⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⢸⣿⢁⠁⠈⡈⢁⢁⠈
⣀⣀⣂⣐⣀⣀⢂⠀⡀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢸⣿⠂⠂⠐⠐⠀⠂⠐
⠠⠀⠄⠠⠠⠀⠄⠀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠄⢠⠁⠈⠌⠠
⢦⡖⡐⢂⢂⠀⡀⠀⢀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⠀⠂⠂⠐⣐⣀
⠀⠠⢀⠀⠄⠀⠠⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠡⡌⠄⠠⠡⠈
⠿⠿⠿⠿⠾⠿⠷⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠿⠿⠷⠶⠶
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠈⠀⣿⣧⠤⠒⡁⢀⡆⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣀
⡴⠞⣿⣿⣷⠁⠀⣾⡇⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⡑⠢
⣶⠋⣿⣿⣿⣧⣼⣿⣷⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠈⠢⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠑⣄⠀⠀⠱⣄⡈⠑⢄⡈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢧⡀⠀⠈⠻⣦⡀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠄⠀⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢣⠠⡀⠀⠀⠀⠈⢳⡀⠀⠀⠹⣦⡀⡀⠘⢿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⡟⠉⢩⠏⡜⠀⠀⠀⠀⢰⡇⠀⠀⣿⡇⠀⡄⠀⠀⠀⠀⠘⡆⢣⠀⢀⠀⠀⠀⢻⣆⠀⠀⠈⢿⣎⠢⡀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⣧⠀⠀⠀⠀⠀⠰⠷⠀⠀⣴⡞⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠘⣇⠀⢠⡀⠀⠀⠀⠀⠀⠘⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡈⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⡘⢁⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⠦⢀⠀⠀⠀⢿⡇⠉⠓⠄⠀⣴⡄⢰⣿⠁⠹⣷⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠉⠛⠶⢾⡇⠀⠀⠀⠈⠿⠹⣿⠃⠀⠀⠹⣿⣐⣦⠄⠘⢹⡇⣴⠂⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣬⣿⣗⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢀⡿⠁⠁⠀⣠⠾⠋
⠀⠀⠀⠘⠓⠶⠶⠶⠖⠀⠀⠈⠽⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣶⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠟⠋⠀⠰⣶⠄
⠀⠀⠀⠚⠛⠳⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣀⣀⣠⠤⢤⠀⣤⡤⠄
⠀⠀⠀⠀⢠⠄⠀⠀⠀⣀⠤⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠊
⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⡛⠛⠁
⠀⠀⠠⠾⠿⠿⠶⠖⠒⠒⠒⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠷⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⣥⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⣠⣾⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠿⣭⣉⠉⠀⠒⠚⠓⠛⠆
⠀⠀⠀⠀⢀⠶⠞⠋⠁⠀⠀⠀⠉⢰⡏⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⣿⣭⠍⠵⠻⠿⠛⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡀⠴⢛⡋⢿⡀⠀⠀⣠⣶⡀⣤⣄⠀⠀⠀⢀⡤⣤⣄⡀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⢀⣴⠆⠀⠈⢧⠀⢰⣿⠃⠘⠏⠉⠳⢦⡀⢸⣇⠀⠉⠛⠶⠠⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⣡⠆⣾⠀⠈⢷⣾⠏⠀⠀⠀⠀⣠⢤⡙⢾⣿⠄⢦⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠋⠀⡿⠀⢀⡀⠁⠀⠀⠀⠀⠀⠳⠤⠏⠀⠈⠀⠘⣧⠙⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢸⡇⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⢦⣠⠏⢻⣀⡤⠤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢾⠀⠀⠙⠀⠈⠁⠀⠀⠀⠀⠈⠙⠶⠚⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢴⠀⠀⠀⢘⣦⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣯⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⡆⠀⠀⠀⡏⠀⠀⢸⠀⠀⣼⠉⠉⢸⣧⣤⠶⢺⠟⠁
⠀⠀⠀⠀⠀⠙⢿⡓⠶⢿⡀⠀⢠⣄⠀⠙⢦⡀⠀⠀⠀⠀⡿⠀⠀⠀⣿⠀⠀⢀⡇⠀⠀⢸⡇⢀⡏⠀⠀⢸⡇⠀⠰⣏⡀
⠀⠀⠀⠀⠀⠀⠀⣱⡄⠸⡇⠀⠸⡟⢦⡀⠀⠙⣆⠀⠀⢠⡇⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⢸⡇⢸⠇⠀⠀⠘⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⣷⠀⠀⣧⠀⠻⣄⠀⠈⢷⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⢸⡇⣾⠀⠀⠀⠀⡇⠀⠀⠀⠈⣉⡭⠟⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡀⠀⢻⠀⠀⣿⠀⠀⣼⠀⢸⠀⠀⠀⠀⢸⡆⠀⣸⠀⠀⠀⢸⣇⡏⠀⠀⠀⠀⡇⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠸⡇⠀⠸⣧⠞⠁⠀⠀⣿⠀⣿⠀⠀⣶⠀⠘⡇⠀⣿⠀⠀⠀⢸⣿⠃⠀⣴⠀⠀⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣷⠀⠀⠀⠀⠀⣠⡴⠋⠀⡏⠀⠀⣿⡄⠀⡇⠀⡿⠀⣀⠀⢸⡿⠀⢀⣿⠀⠀⡇⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⢸⡀⠀⠀⢴⡛⠁⠀⠀⢸⡇⠀⢠⡿⡇⠀⣷⠀⡇⠀⣿⠀⢸⡇⠀⢸⡏⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠘⡇⠀⠀⠀⠙⠦⣄⠀⢸⠁⠀⢸⡇⣇⠀⣿⢸⡇⠀⣿⡄⠈⠁⠀⣿⡇⠀⠀⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠀⠀⣿⠀⢀⣄⠀⠀⠈⠳⣾⠀⠀⣸⠀⣿⠀⢸⢸⠃⢸⡏⡇⢀⣀⣸⢿⡇⠀⠀⡇⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣠⣾⠥⠖⠛⢦⣼⡀⠀⣏⠳⣄⡀⠀⠈⣷⠀⣿⣠⡽⠀⢸⣿⠀⢸⠇⠉⠉⠁⠀⢸⠀⠀⣀⣇⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⠀⢈⡿⠀⠀⡇⠀⠉⢀⣀⡀⢸⣿⢀⣸⠀⠀⠀⠀⠀⠈⠉⢩⠏⠉⠙⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⡷⠋⠀⠀⣠⡇⠀⣾⠉⠘⡇⠈⡟⠉⠉⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⣠⢾⡏⠀⢠⡷⠤⠴⢧⣤⡗⠀⢠⣄⢠⡏⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠁⠸⠷⠶⠚⠀⠀⠀⠀⠀⢻⡄⣸⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠈⠁



⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⡶⠶⠶⠶
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠶⠶⣤⣀⣀⣀⣠⣤⡶⠞⠋⠀⠘⣿⠀⠠⡠
⠶⠶⠟⠋⢻⣆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠳⣤⣄⣀⣠⡟⠉⠀⠀⠀⠀⢀⡼⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣉⠟⠛⢳⡞⠛⢛⣿⣿⣿⣧⡀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡴⠋⠁⠀⠀⠘⡇⠀⡙⣿⣿⣿⡏⢳⡆⠀⠀⠀⠀⠀⣠⠶⠟⠛⠛⠛⠛⠷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁
⠀⠀⡼⠁⠀⠀⠀⠀⠢⢹⡄⠀⠈⠉⠉⠁⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⢤⡈⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣇⠆⠈
⢀⡼⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⡼⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠞⠁⠀⠀⠀⠀⠀⢀⣿⠁⢰⡆⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠈⡇⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢦⣾⠟⠀⣧⠀⡇⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠈⣯⠀⣸⠇⠀⡃⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡉⠒⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠤⠄⣀⡀⣿⣷⢋⡄⠀⠁⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠲⠾⠛⣋
⠀⠀⠀⠀⠀⠈⠙⠿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠿⢟⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠡⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣀⣴⣾⣿⣿⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠿⣣⣴⠆
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠹⣦⡀⠀⠀⢀⣤⣿⣿⣿⡿⠋⣡⠟⠉⠁
⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣷⠀⠀⠉⠙⠛⠛⢿⡁⠘⠃
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠶⠿⣿⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⠇⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠴⠛⠋⠉⠴⠛⠁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⠀⠀⢀⡰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡁⠂⠀⠀⠀⡼⠁⠀⠀⡤⡄⠅⠀⠀⢀⡜⠄
⠋⢀⡀⠀⠀⢀⡐⡀⠀⠀⠐⠀⠀⠀⠀⢱⢠⠀⠀⠘⡇⠀⠀⠀⠀⡇⠄⠀⠀⣸⠍⠀⠀⠀⠀⠀⠀⠀⣤⠂
⠐⠂⡀⠀⠀⠀⠝⠁⠠⠃⠐⠀⠀⠀⠀⠸⡴⠀⠀⠀⠁⠀⠀⠀⢀⡅⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠐⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⣶⠀⠀⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠁⠀⠀⠁⠹⠀⠀⠀⡘⢄⠀⠀⠀⣰⠁⠐⡄⠀⠀⠀⢀⡠⣞⣑⠀⠀⢀⣚⡉⠀⠀⢀⣸⠃⠀⠀⡐
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⡀⡔⠀⠀⠈⠁⠈⠀⠀⠀⠈⠒⠒⠐⠒⢡⡁⠀⠀⠢⢨⢀⠀⠀⢠⢂
⠀⠀⠀⠀⡐⠀⠀⢀⠀⠄⠪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠲⣄⠀⢂⠏⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠈⠉⠁⠈⠉⠃⠾⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠄⠀⣠⡜⢂⡀⢠⣰⡀⣤⣄⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠘⠑⠋⠉⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⢈⠶⣀⠀⢀⡀⢀⠀⠀⢀⠄⢾⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⢙⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⣿⣧⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄⣠⡀⠀⠄⠈⠆⣄⡄⠄
⠀⠀⠀⢄⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⣆⠑⠢⢴⣿⠿⢿⣿⣷⡆⠀⣾⠀⣀⠀⠀⠀⢸⠰⠏⢠⠃⠠⠁⣈⣎⡅
⠀⠀⠈⠡⠀⠄⠐⠀⠀⠀⠂⢄⡀⠀⠀⠀⠀⠄⠀⢰⡀⠁⣶⠀⡏⠈⠁⠘⠉⢀⣴⡊⠀⠀⠘⠣⠁⠲⠀⢃⡤⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡄⠀⠀⠀⠈⠡⣢⣭⣴⣧⡦⠐⠀⠆⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⡄⢀⠀⢰⠀⠀⠀⠀⠰⠃⠀⠄⠀⣦⠀⠀⣈⠙⠿⠉⢀⠖⠦⠤⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢀⠈⠀⠀⣆⠈⢠⠆⠀⠁⠀⠀⠀⠀⢸⡤⠊⠀⢣⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⠁⠂⡈⠊⣦
⠀⠀⠈⠁⣀⡷⠢⠼⠁⠉⠀⢃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡤⠞⠅⠀⠀⠠⠀⢀⠼⠉
⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⢨⡓⠀⠀⢀⠄⠀⢄⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡄⠈⠀⠀⠀⠈⡈⠀⠂⠀⢀⠀⠈⣀
⠀⠀⢀⠈⢠⠴⢤⣀⡀⡀⢸⡈⢁⡀⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⡠⠔⠺⠅⠀⢀⠄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⡠⠘⠋⠀⠃⠀⡀⠀⠀⡁⢀⠄⠐⢦⠀⠀⠐⠒⠀⠁⠀⠒⠢⠐⠀⠁⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣩⠄⠀⠁⠠⣄⣠⠌⠀⠀⡰⠀⠀⠀⠀⠀⠀⢀⠄⠈⡀⢀⣀⣀⣠⣀⡀⡖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⡀⠀⡀⡰⠚⠁⠀⠀⠀⠀⠀⠄⠀⠂⣫⠿⡾⣡⠟⠵⠼⠁⢧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠛⠀⠉⠁⠀⣀⠀⡐⠃⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠂⠉⠉⠉⠉⠉⠓
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡘⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠀⠉⡆⠀⠀⠀⣀⡤⠴⠒⠢⢤⡈⠂⠁
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠴⠢⠄⢰⠒⠋⠊⡅⠤⢺⠀⡀⠀⡠⠀⠀⠀⢸⠀⢠⡖⣿⠓⡄⠀⡇⠀⠀⣴⠃⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀
⠀⠀⠀⠈⠠⣄⢐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⢀⠋⠀⠀⠀⠀⢸⠀⠈⠛⠿⠟⠁⠀⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⢀⣀⠸⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠲⠀⠀⠀⠀⢸⠀⠀⣀⣤⣀⠀⠀⡇⠀⠀⠀⠉⢳⡆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⡈⢄⠀⠀⠀⠀⢸⠀⠠⢿⡿⠇⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⢠⠗⠀⠢⠄⠠⢫⠔⠃⣸⠙⠀⠠⣄⠀⠀⠀⠁⠈⠁⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠒⠁⠀⠀⠀⢰⠀⠀⠀⠀⠛⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢲⣿⡦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠴⠒⠚⠲⢼⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠈⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠹⡄⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⡔⡇
⠀⠀⠀⡏⠀⠀⠀⠀⠶⠠⠆⠠⠄⠀⠀⠀⠀⢱⠀⣀⣀⡀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢓⠦⣄⡴⠉⠓⠚⠉⠀⢧⡀⠀⢀⡀
⠀⠄⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠒⢤⡈⠱⠹⠂⠀⠀⠀⠀⠀⠦⠒⠋⠀⠀⠀⠀⣀⠀⢀⡀⡀⠉⠁⢼
⠀⠀⠀⠈⠀⣽⢠⠶⠶⠤⠤⠤⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣴⣶⡶⠀⣰⡟⠀⠨⠟⠇⠀⠀⠘⢆⠁
⠀⠀⠀⠀⣰⠷⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣀⡴⠃⠀⠀⠀⠀⠠⠴⠪⣄⡀⠀⠈⠛⠁⠀⠘⠀⠀⠈⠉⠁⠀⣀⡀⠀⠁⢦⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⢶⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠔⢦⢀⠤⢤⠀⣀⠀⢠⠄⢄⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⡾⠋⢣⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠍⠊⠀⠢⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠀⡼⢤⣀⣄⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢠⠉⠀⠀⠂⠀⠀⠀⡀⣤⠊⠀⠀⠀⠀⠀⠀⠘⠓⢦⠐⠀⠀⠀⠀⠀⠀⣀⢄⣠⡋⠁⠈⠙⢇⠰⠦⡀
⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⣀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠠⠊⠄⠒⠒⠢⣀⠔⠠⢅⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢠⠆⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠉⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠳⠴⣀⣀⣠⣤⠄⠐⠛⠈⠁⠀⠀⠀⠀⠀⠈⢺⠄⣑⢰⣀⢀⡀⠰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠁⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡟⠀⠈⠑⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠰⠇⠀⠀⣠⠤⢄⣠⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⠒⠂⠀⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡤⠂⠀⠀⢸⠁⢀⠔⢁⢦⠀⢸⠁⠀⠈⢧⠤⠤⣄⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠘⣗⠒⠚⠉⠀⠀⠀⠀⣸⠴⠋⠒⠊⣸⠀⡾⠀⠀⠀⡏⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠔
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡼⠁⠀⠀⣀⡴⠃⠀⡇⠀⠀⢸⠁⠀⢀⠇⠸⡄⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠀⣿⣇⣀⠘⣇⢀⠀⣸⣀⠀⣿⠋⠀⡇⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⠇⠀⠀⠐⠿⣿⣆⣱⣿⣀⣣⣿⣤⣶⣷⡀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸⡀⠀⠀⠀⠇⠈⠉⠙⢿⠋⠉⠉⠉⠣⠀⠀⢰⠇⠀⠀⠀⠈⣆
⠀⠀⠀⠤⢾⣁⣀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⡠⠍⠙⠲⣄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⠔⠒⠒⠒⠒⠷
⠀⠀⠀⢀⠔⠀⠀⠀⠑⠘⣇⠀⠀⠀⠈⢧⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⢠⠏⠀⢰⠁
⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣽⠤⠄⢀⣠⣬⡗⠚⠉⠁⠀⠀⠀⠀⠠⢴⣇⠀⠀⢸
⠀⠀⠀⠀⠈⠂⠈⠒⠁⢰⣧⠶⠛⠉⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠀⠀⠤⠞⠀⢸⠀⠀⠀⢠⢠⠀⠀⠀⢸⠀⠀⠈⢻⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠈⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠴⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢙⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠸⣛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⠟⣻⣿⣿⣿⣿
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⢃⣼⣿⡟⣹⣿⣿
⣿⡇⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇⣼⣿⣿⢘⣿⣿⣿
⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣷⣟⣻⣿⣼⣿⣿⣿
⣿⡇⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡟⠾⠿⠿⠿⠿⠿⣿⣿⣿⣿
⣿⡇⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡀⠀⠀⠀⠀⠈⠃⠘⣿
⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⡿⠿⠿⠟⠋⠁⠠⣶⣴⣾⣷⣤⣀⡀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠛⠛⠛⠉⠁⢠⣿
⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣷⡘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡿⠿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⡇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠬⢻⣿⠦⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿
⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⣇⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿
⣿⣿⣦⠀⣠⣿⣿⣿⣿⠉⢻⠘⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⡏⣰⣿⣿⣿⣿⡿⢀⠆⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⠁⣿⣿⣿⡿⠟⢋⡎⠀⠀⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣿⣿⣷⣿⣋⣡⣴⣾⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⢿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢰⣿⣿⣿⣿⣿⡿⠇⢊⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⠤⠐⠈⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⢻⣿⡿⠟⠋⠁⣨⠞⢹⣿⣿⣿⣿⣿⣿⣿⠌⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⡾⠚⠉⠀⣀⣴⣾⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣾⡏⣿⡿⣿
⣿⡏⠀⠀⠈⠹⡟⢠⠞⠁⠀⢀⣴⣾⣿⣿⡏⠀⠀⢸⠟⠻⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⣼⡏⣿
⣿⡇⠀⠀⠀⣸⣷⠋⠀⠀⢠⠏⠀⠹⡿⣏⡅⠀⠀⠘⠀⠀⠈⠁⠀⠈⠉⢠⣾⣷⣤⣀⡀⠀⣀⣾⣿⣿⣷⣾⣿⣿⣿
⣿⡇⠀⠀⠀⡿⠃⠀⠀⠠⢃⣀⣤⠤⣬⣉⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⢸⠁⠀⠠⠔⠛⠉⠁⠀⠀⠀⠈⠙⠳⢦⣍⠓⢄⠀⠀⠀⠀⠀⠀⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣾⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣿⣶⣷⣶⣶⣶⣶⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠴⠒⠚⠉⠉⠉⠙⠢⡄⠀⠀⢀⡴⠒⠚⠓⠲⢤⡀⠀⢀⡠⠴⠖⠦⣄⢀⡤⠤⣄⣀⠀⠀⠀⠀⠀⢠⡞⠓⠒⠦⠤⣄
⢸⡅⠀⠀⠠⠤⡄⠀⠀⠀⢹⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⣴⠋⠀⠀⠀⠀⠈⢻⠀⠀⠀⠘⣧⠋⠙⠒⣶⠃⡇⠀⠀⠀⠀⡼
⡎⠁⠀⠀⠀⠉⠀⠀⠀⣠⠏⢰⠃⠀⠀⢀⠤⡀⠀⠀⠀⡟⠀⠀⣀⡀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⢨⠀⡇⠀⠀⠀⢰⠃
⡇⠸⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢇⠀⠀⠀⠉⠁⠀⠀⠀⡇⠀⠀⠑⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⡏
⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢀⡠⣺⠳⢄⡀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⣸
⠀⣇⢸⠀⠀⠀⠰⣉⠆⠀⠀⠀⠀⢸⣗⠒⠒⠈⣁⡴⠳⣄⣀⣀⣠⡼⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⣠⠇
⠀⢸⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⡼⡸⠀⠀⠀⡰⠁⢣⡔⡺⠇⠀⠀⠀⠸⠒⠚⢣
⠀⠀⣇⢠⠀⠀⠀⠀⠀⠀⠀⡀⠎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢇⣀⡀⡰⠳⣀⣎⣴⠁⡄⠀⠀⠀⢰⣀⣀⡜
⠀⠀⢸⠈⢆⣀⣀⡠⠤⠔⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣸⠃⠀⠈⠁⢸⡀⠱⠤⠤⠤⣼⠴⠋
⠀⠀⠈⠣⣄⣀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⢱⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠲⠋⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠖⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠙⢣
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠂⠀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⢀⣀⣤⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⣀⠄⠙⠦⣠⣀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⣄⠈⢶⣤⣴⣶⣶⣶⣶⣦⣴⣾⠿⠛⠛⠛⠛⠻⣿⣾⣅⡤⠊⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⠟⠉⠁⠀⠀⠈⢙⣛⣁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣧⣤⣀⠀⣠⣴⣾⠿⠛⣻⣿⡄⠀⢀⣴⣶⣶⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠋⠉⠉⠉⠉⠙⣿⣿⣿⠏⠁⠀⠀⠉⠻⣿⣆⣿⡟⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⡿⢋⣡⠄⠀⣾⣿⣿⡿⠃⣠⣾⣿⣿⠏⠀⣰⣿⣷⠀⠀⣿⣿⣿⠇⠀⠀⢸⣿⡿⢀⣀⣤⣶⡿⠿⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠀⠀⠿⠛⣡⣴⣿⢛⣩⣿⡟⠀⣴⠿⠿⠿⠀⠀⣿⣿⡿⠀⢠⣄⠀⠻⠿⠿⠛⢋⣡⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠐⠛⠛⠛⠻⢿⣿⠉⠀⣀⣤⣴⣶⣶⠂⢠⣿⣿⠇⠀⣾⣿⣷⣦⣶⣶⣿⡿⠁⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡆⠀⢾⣿⣿⠿⠃⢸⣿⠇⢰⣿⠋⠁⢸⣿⠀⣼⣿⣿⠀⣸⣿⠁⠀⠉⠁⣾⡿⠁⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠘⢋⣡⣴⣾⢿⣿⣠⣿⡏⠀⠀⠘⢿⣾⠿⠛⠿⠿⠟⠃⠀⠀⠀⢼⣿⣥⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣼⡿⠛⢿⣷⣶⡝⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡼⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠠⠚⢀⡵⠛⢿⣷⣤⣄⣀⣀⣤⣶⣿⣶⣤⣠⣤⣾⡿⢯⡈⠳⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠉⠙⢦⠐⠁⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠉⠙⠛⠋⠉⠀⠀⠙⠂⣠⠒⠛⠁⠈⢱⣀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠄⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠙⡶⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣠⠷⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣤⠴⣀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠁⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⢀⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣀⡀⠀⠀⠀⠀⢀⣾⡿⣷⣄⠀⠀⢀⣴⣿⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣷⣦⣤⣀⣼⡿⠀⠈⢻⣷⣴⣿⡿⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠙⠛⠿⣿⣿⠁⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⣷⣴⣶⣶⣾⣿⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢶⡛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠂⣁⠠⠄⣛⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣟⡋
⠀⠀⠀⠀⠀⡀⢨⡅⠲⢘⣃⣽⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠼⠟⠛
⠀⠀⠀⢀⠀⡤⠰⢆⠙⣤⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣏⢉⠃
⠀⠀⠀⠈⠰⠀⠃⡈⠰⣰⣿⣥⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣐⠂⡈⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⣼⣿⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠲⠶⣦⠤⣤⣽⣅⠰⢀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣀⣤⠞⠋⣷⠀⠀⠀⠀⣼⠳⣄⡀⠀⠀⢸⡆⠆⢉⠰⢈⠠⠌⡁⠬⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⡴⠛⠉⠀⠀⠀⢸⡄⠀⠀⣸⠃⠀⠈⠙⢦⣀⠀⣷⠐⢀⠐⢈⠀⠂⡁⠰⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⡀⠀⠀⠀⠘⣇⠀⣰⠃⠀⠀⠀⠀⠀⠉⠳⣼⡇⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⢀⠀⠀⠀⢿⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢁⣐⣦⣐⢄⢀⢌⢡⡐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠤⣍⣡⣐⡒⢤⣙⣿⣿⣿⣏⠄⢹⣎⡌⢂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⢒⣥⣴⣿⣛⠻⢯⡻⢶⡘⣿⡟⣿⣿⣷⢸⢻⣿⣸⢜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣾⣿⣭⣥⣬⣽⣷⣿⣦⣻⣜⣷⣿⣿⣯⣯⣾⡟⠱⣷⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢻⣿⣿⣿⡿⠋⠀⢸⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣭⡬⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⢹⣿⡟⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣮⡒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢹⣿⠁⣀⣴⣏⣀⡀⠀⠀⠀⠀⣀⣠⣤⣄⣹⣿⣿⣏⢳
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣿⠘⣾⣿⠀⢁⣀⣀⡉⠙⠀⠀⠀⠈⢁⣤⣄⡈⠉⣟⢿⣟⠾⡀
⠀⠀⠀⠀⠀⡞⢁⣴⣶⣿⢟⢏⣼⡿⣿⠀⠈⠅⠛⠁⠀⠀⠀⠈⠀⠀⠛⠠⠉⠀⢸⣦⣽⣷⣥
⠀⠀⠀⠀⠀⠁⣿⢛⡩⠤⠞⠻⢿⣿⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠿⣝⢟⠷⣄
⠀⠀⠀⠀⠀⢸⣧⠏⠀⠀⡀⢀⣼⣿⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⠋⠠⢹⢿⣷⣜⠆
⠀⠀⠀⠀⠀⢸⡟⢀⢔⣽⠿⢛⢋⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡹⣧⢧⠢⡳⣍⠛⠵⣄
⠀⠀⠀⠀⠀⢸⡇⣴⣫⠓⣩⠔⠋⡼⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣿⠸⣿⡦⣌⢣⡀⠘
⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠧⣴⠾⠋⣼⢻⣿⡗⢄⠀⠀⠈⠉⠀⠀⠀⣠⣾⣿⠟⣿⡇⡞⠞⣿⠀⠌⡇
⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡠⢢⢆⣴⣫⣿⣿⡇⠀⠑⠤⢀⣀⣀⡤⠞⠁⢸⡏⢸⣿⣼⢱⠋⠈⠆⡶⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⢾⣿⣿⣿⣵⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⢯⠿⠀⠀⡓⢘⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠱⠀⠀⠐⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⡔⠐⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢉⠀⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡔⠁⠀⠀⠀⠀⠀⣤⡀⠀⠠⢢⠀⠀⠀⡀⠀⠀⣠⣟⣷⡠⠖⡇⠀⠀⠀⠀⠀⡠⠀⠀⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠄⠀⠀⢄⢢⠀⢀⣟⣷⡠⠁⠀⢳⣤⠞⠈⢢⣼⣣⣗⣿⠀⠀⡇⠀⠀⢀⠔⡨⠐⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠳⣾⡟⣿⣷⡀⣠⢿⣿⠣⣠⢟⡿⠿⣽⣻⣇⡤⣿⣾⡿⠖⣻⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠓⠤⣰⣮⡑⣾⣿⣾⣦⣿⢯⣶⣷⢶⣷⣿⣤⣀⣰⣷⣵⣾⡿⣿⣀⣡⣼⡴⣶⣿⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⠾⣄⡀⢸⡟⢟⠟⣻⡿⣿⡿⠋⠉⠉⠻⣿⠟⠁⠈⢿⠿⢿⡿⠟⠛⢿⣿⣿⣿⣿⠋⠉⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⣿⣿⣿⣿⠟⠛⠛⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⢿⣁⣀⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣿⣛⣛⣻⡧⠀⣀⣀⣀⠀⢀⣤⣤⣤⣄⣠⣶⣶⣶⣄⣼⣿⣦⣾⣿⡄⠀⢠⣿⣚⣛⣋⣛⢛⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⠁⠐⣿⣟⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢿⣿⣿⣿⣋⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⢻⣿⣿⣿⡁⠀⠀⣿⣿⣿⣿⣷⣿⣿⣇⣸⣗⣻⣻⡄⣿⣿⣿⣿⠻⡟⣿⣿⣧⠀⠀⣼⣿⣿⠖⢻⣿⣿⣶⣦⠤
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣶⣻⡽⠯⣽⣷⣤⠂⢸⣿⣉⣽⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⠿⠿⠏⠀⠀⠸⠟⠛⠀⢾⢻⣏⣿⣠⡶⠿⣟⠉
⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⢾⣧⣄⣠⣿⣾⠃⠀⠘⠟⠛⠛⠋⠁⠀⠉⠁⡀⠀⠀⠀⡀⠀⠀⢠⣤⣴⠀⠀⠀⢀⣾⣶⣾⣶⣾⣷⣄⣈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠿⣯⣟⣟⢿⡀⠀⠀⡀⠀⠀⣄⠀⠀⠀⢀⣧⡀⠀⣰⣷⣦⣴⣞⣀⣛⣞⣛⣶⣛⣛⣿⣛⣛⡻⣛⣻⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣊⣀⣀⣠⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⢿⣿⣿⣿⣿⡙⠻⡿⢮⣷⡀
⠀⠀⠀⠀⣀⣀⠤⠌⠀⠀⠠⣲⣟⣿⣯⠵⢿⣟⣟⣛⣟⣺⣿⣟⣛⣯⠟⠛⢳⣟⣟⣿⣝⣿⣻⣿⣯⣻⣟⠺⡿⢮⡛⢿⣿⣷⣤⠧⠀⠀⠁
⠀⠀⢠⠎⠀⠈⠀⠀⢀⠄⡾⠛⠋⢁⣇⠔⠺⡟⢻⣿⣿⣿⣿⣿⣿⡀⠀⢠⣾⣿⢿⡿⣿⢿⣿⣿⢿⠏⠙⠷⠇⠰⠉⠲⣼⣻⡦⡢⠀⠀⢀
⠀⠀⠈⢳⠆⠠⡀⠀⠐⠈⠀⠀⠀⠈⠁⠀⠀⠓⢻⣗⣴⠟⠁⢻⣶⡿⡶⣿⣶⠇⡸⠛⠛⠼⣷⢦⡟⢆⢸⠀⠀⠀⠀⠀⠀⠙⠷⠈⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠰⡅⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⡀⠔⠉⢿⣻⣟⡿⠁⠈⠃⠀⠀⠀⢻⡟⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡗
⠀⠀⠀⠀⠉⠳⡈⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣷⡾⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠖⠊
⠀⠀⠀⠀⠀⠀⠈⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢀⡀⠀⠀⠀⠀⣄⠀⠈⢳⡀⠀⢀⡠⠖⠒⠒⠒⠷⣜⡠⠗⠒⠒⢦⡀⣠⠴⠒⠒⠲⢤⡀⠀⠀⠀⢀⡴⠚⠁⠀⣀
⠀⠀⠀⠙⠳⢤⣀⠀⠈⠳⣄⠀⠣⣠⠏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠋⠀⣀⣀⠀⠀⠀⠹⣔⣠⠔⠉⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠈⠓⠄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣀⠀⠀⣫⡵⠖⢋⣁⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠒⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠉⠙⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠹⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⣠⢞⡡⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠉⠁⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⣖⡾⢧⣂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⣀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠳⣅⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠙⣩⠏⣽⠿⣢⣤⣤⡴⣦⡀⠀⠀⠀⠀⣠⡷⡤⣤⠶⢞⠥⡡⠈⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡴⠃⣠⠋⠀⡼⠁⠛⠎⠓⠙⢳⢶⢖⡻⣻⣏⢿⠙⠏⠊⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⠏⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠊⠐⠈⠇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⢀⣀⣠⣤⣦⣤⣤⣀⣥⣤⣅⣈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄⠠⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⢿⣿⣦⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⡚⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⣿⢫⣾⣿⡿⠋⠁⠀⠀⠈⠉⠉⠙⠻⣷⡝⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠔⠒⢄⢠⡟⣯⡿⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣿⡇⠀⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⣀⣠⣀⣀⣀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀⠈⠢⠄⠊⢸⣇⣿⢱⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠈⠈⠐⠈⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⢸⡿⡏⣾⣿⣃⣉⣳⡦⠀⠀⢞⣫⣤⣤⡀⠀⠿⠋⣿⠁
⠀⠀⠠⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣧⡏⡏⠉⠛⠋⠈⠀⠀⠀⠘⠛⠈⠀⠀⠘⠀⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡟⣸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⡆
⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⢀⣷⣴⣿⣿
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠺⣿⣽⡿⠃⠀⠀⣠⣾⣿⣿⣿⠹⠀⠀⠐
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧
⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⣻⡻⣿⡿⠿⡇⢲⠀⠒⠊⠁⣿⣿⣹⣿⡏⣿⣿⣿⣀⡈
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠉⠙⠻⠿⠛⠀⢠⠃⠀⠁⠀⠀⠀⠈⠛⢿⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠒⠒⠒⠒⠀⠁⠉⠀⠘⠟⠀⠀⠀⠀⠀⢀⣹⡏⣿⣇⡞⣿⣷
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠀⠀⠠⠤⠤⠀⠒⠂⠈⡤⠀⠀⠀⠀⠀⠀⠤⠎⠠⠿⣏⠀⠙⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠀⠀⠀⠀⠤⠄⢐⠀⠐⠆⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⡀⠄⠂⢺⣿⡄⠀⠀⠈⠂⠢⠌⠉⠀⠀⠀⠀⢀⣴⣿⡉
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢂⣹⣿⣅⢁⣤⡀⢀⠀⠐⢀⣊⣹⣿⣇⢀⣿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⡲⢤⣤⣀⣤⡤⠔⠊⢐⠈⠒⠈⠋⠉⢣⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣿⣿⡿⢿⣦⣣⣧⣿⣿⣿⠿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⣸⠇⢆⠀⠀⠀⡼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠑⢄⡀⠀⠀⢠⠏⠀⠈⠢⠤⠚⠁⠈⢆⠀⠀⠀⠀⢀⣀⠤⠚⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⣬⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠓⠚⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠶⣦⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡠⠖⠉⠁⠀⠙⣷⣠⠖⠋⠙⠻⣶⠁⠀⠀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣉⣀⠐⠒⠒⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠈⠉⢹⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⠀⠀⠀⠀⠀⠈⣠⣣⡘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠋⠽⠁⠙⠦⡑⢤⣀⣀⠀⠀⠀⣀⣠⠐⣈⠹⠀⠀⠀⡌⡎⠙⠳⢶⡤⡀
⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠳⢦⣄⠀⠀⢰⢠⡏⡆⠀⠀⠀⢠⡀⠀⠀⠀⠸⢰⠀⠀⠀⠈⢃⠋⠉⠛⠻⡟⡴⠩⠛⠻⠷⡤⢄⢰⠹⠀⠀⠀⡌⡐
⠀⠀⠀⡎⢏⠀⠀⠀⡀⣀⠀⠀⠙⡵⠀⢠⢸⢡⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡌⡄⠀⠀⠈⠉⠀⠀⠀⠀⠁⢯⠁⠀⠀⠀⠀⠀⠆⠆⠀⠀⢰⠠⠃
⠀⠀⠀⠰⠘⡀⠀⠈⡄⠀⡇⠀⠀⡇⣇⠜⡬⡆⠀⠀⠀⠀⠇⢁⠀⠀⠀⠀⢁⠃⠀⠀⠘⠀⠀⢠⠀⠀⠁⠆⠀⠀⠀⠀⡆⢀⢰⠀⠀⠀⠆⡂⢀⣀
⠀⠀⠀⠀⠇⢃⠀⠀⠣⠚⠀⠀⡔⢠⠟⢊⢡⠁⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠸⢸⠀⠀⠀⠃⠀⢸⠀⠀⠟⠀⢠⡇⠀⢠⠀⡋⡈⠀⠀⠰⣨⠥⠊⠁
⠀⠀⠀⠀⠘⠘⡀⠀⠀⠀⠀⠀⠈⠹⠶⣜⡈⠀⠀⠀⠀⣐⣀⣀⡄⠀⠀⠀⠀⡆⡄⢠⢤⠀⠀⣿⠀⠀⠀⢀⡾⠀⠀⠸⠀⠁⠇⠀⠀⠃⡏
⠀⠀⠀⠀⠀⠇⢃⠀⠀⡏⠈⢲⠀⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢁⠈⡀⠀⢀⠘⠀⠀⠀⡆⠇⠀⠀⡇⡄⡀⡀⠀⠸⢰
⠀⠀⠀⠀⠀⠸⠘⡀⠀⠰⠤⠊⠀⠀⡌⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠇⠁⠀⢸⠀⡀⠀⡸⢰⠀⠀⢰⢀⡏⠤⠕⢲⠃⡿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⠀⢀⡠⢊⡀⠃⠀⠀⠀⠀⠰⠒⠒⠂⠰⠀⠀⠀⠀⠀⡇⢈⠒⠂⠄⡠⠬⢈⡁⡆⠀⠀⡌⢸⣠⠀⠀⠀⡄⡷⠒⢙⣉⡉⠠⠤⡤
⠀⠀⠀⠠⠶⠍⣥⠘⢂⣈⣉⣡⠔⠊⡕⠃⠀⠀⠀⠀⡆⡋⠈⠉⡆⡆⠀⠀⠀⠀⢠⢉⠈⠁⠒⠃⠀⠀⠧⢌⣉⠒⢁⣧⠳⠄⣀⢠⢁⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠢⠀⠡⣰⠰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⢁⠀⠀⠀⠀⠘⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠂⢤⣡⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇⣆⣀⣀⠀⠀⡘⠰⠀⠀⠀⠸⠬⠌⠁⣈⣙⣙⣁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡜⢢⣤⡤⠤⠄⢉⣁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢉⣉⣉⣉⣉⠙⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⣉⣠⠠⠤⠤⠤⠤⠤⣄⠉⠢⡀⠀⠀⠀⢀⡠⠤⠄⠒⠤⢤⣀⠀⠀⠀⡰⢁⠔⠉⠁⠀⠀⠀⠀⠀⠉⠁⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢱⢀⡠⢊⡡⠒⠊⠉⠉⠁⠒⢌⡑⠦⣠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃




⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⣀⠄⠀⢀⡠⠤⠐⠒⠊⢀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠚⠉⣀⡤⠖⠉⠠⠀⠒⠒⠂⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⣤⡶⠛⠡⠤⠤⠒⠒⠂⠘⠻⣷⡀⠴⠥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡠⢔⣥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠔⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠂⡀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠥⣐⡵⠟⠓⠊⠉⠽⣄⣀⣠⠌⠙⠒⠚⠳⣤⡡⡰⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⡠⠐⠀⠈⠇⠀⠀⠀⠀⠀⡟⠉⣇⠀⠀⠀⠀⠀⡟⠁⢳⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡠⠞⠠⠀⠸⠄⠀⠀⠀⠜⠀⠀⠈⠢⠀⢀⠀⠴⠇⠀⢰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠃⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠊⠈⠑⠀⡃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣷⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠤⠀⠀⠀⢀⠄⠀⢠⢠⠛⢅
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡸⡈⠀⠀⠀⢸⡟⢠⠈⠲⡀⠀⠀⠀⢀⠔⠁⣇⠘⠀⠂⢃⢡⢁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠠⠐⠁⠃⢸⠀⠀⠀⠁⠀⠈⠀⠀⠀⡟⡌⡄⠀⠘⣐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⡡⠂⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠠⡁
⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄⠀⠂⡩⠒⠀⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡂⠐⠂⣀⠁⣃⠰⣒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠐⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠘⠀⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠼⣿⡷⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣴⠈⠀⢀⡾⠁⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠳⣄⠀⠀⢸⣦⡀⠀⠀⢀⡾⠁⠀⠀⣿⠀⠀⠀⣰⣆⠀⡸⠃⠀⡰⠋⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠙⢷⡄⢸⡇⠙⢦⣀⡾⠁⠀⠀⠀⢸⡇⣠⠞⠁⣿⡰⠁⠀⠈⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⢾⡇⠀⠀⠙⠃⠀⠀⠀⠀⠸⡟⠁⠀⠀⢻⡃⠀⣀⣚⣁⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⣀⣀⢹⣆⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠉⢉⣡⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠲⠶⠶⠤⣽⣦⡉⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⢠⣴⣆⠀⠀⠶⠶⠶⠖⠒⠒⠒⠊⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⢛⢐
⠀⠀⠀⠈⠙⠁⠀⠀⢠⣀⡄⢀⣠⡤⠴⢖⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠷⠈⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⢶⠾⠻⠋⠍⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣡⣤⣤⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢿⣿⢽⢨⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⠀⢻⣞⢐⢐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⣿⣿⣿⡆⠀⣄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⣽⠭⢽⣿⣼⣿⣆⠀⠀⢠⣿⣿⣿⣿⡯⡯⠿⠽⠬⠹⠣⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣡⡇⠉⢸⣿⣿⣿⣿⣆⠀⣸⣿⣻⣿⣿⡏⡅⡁⠉⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡟⠀⢀⣘⣘⢚⡂⡓⣿⣆⣿⣛⢚⣘⡒⡂⡂⡂⣒⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠸⠃⠀⢼⣿⣿⠀⠀⠂⠆⢿⡿⠰⠰⠐⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠍⠍⠨⠨⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠙⠶⣄⡀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠘⢧⡀⠀⠀⠀⠀⣀⣠⠴⠖⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠛⠷⠶⠞⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢅⠒⠲⠤⢤⣄⣠⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠢⠤⠤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠛⠛⠉⠉⠉⠁⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⢤⣄⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⢀⣤⣤⣦⣤⣄⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⠑⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠾⠛⠉⠁⠀⠉⠛⠿⣦⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⢰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠐⢄⠀⣀⠀⠀⠀⠀⣠⣶⣿⠼⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡢⣀⠚⣻⣥⣤⣴⣾⣿⠟⣁⠄⠀⠀⠈⠻⡿⢿⣦⡀⠀⣀⣀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠙⢮⡛⢿⡿⠋⡉⣩⠟⠉⠉⠙⠀⠀⠀⠀⠢⣀⣹⣿⣷⠖⡙⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡏⠀⠀⠓⠁⠀⠀⢀⣀⣄⡀⠀⠀⠀⠬⣿⠿⢷⣾⣿⡄⠀⠀⠀⣀⣀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⡷⠄⠀⠀⠀⡠⠚⠉⠁⠀⠈⠑⠂⠀⠀⠀⠁⠀⠈⢻⣷⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⢢⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⠋⠜⠣⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⢿⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣆⠀⠀⠀⠀⠘⠢⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡄⣠⡇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠿⣬⡙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣚⢵⣾⣿⣷⣄⣠⡀⠑⢄⣀⣦⡀⠀⠀⠀⣀⠆⠀⠀⠐⢉⣿⡏⠀⠀⠀⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢧⣙⠻⠿⢿⣿⠛⢷⣀⠀⠁⢛⣻⣶⣏⣉⡀⣀⣠⣴⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠛⠛⠛⠉⢛⠋⠙⠻⠿⠿⠿⠿⠛⠉⠳⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠓⠦⠤⠴⠖⠁⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑



⡶⡶⢶⠶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⡶⣶⡶⠶⣶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢀⣴⣾⣿⢿⣿⣿⡿⣠⣿⣿⡟⠀⠀⠀⣰⣿⣿⡿⠿⣿⣷⣤⣾⣿⡿⢿⣿⣿⣦⠀⢀⣴⣶⣦⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠈⠛⠋⠀⣼⣿⣿⣿⣿⣿⣯⣤⣄⡀⣸⣿⡿⠋⠀⠀⢹⣿⣿⣿⠏⠐⠂⢻⣿⣿⡆⢸⣿⣿⣿⡇⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⡿⠿⠿⠛⠿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠈⣿⣿⣧⣾⣿⣿⣿⣇⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣏⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠸⣿⣿⡄⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⣴⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢻⣿⣿⡄⠀⠀⠀⠸⣿⠏
⠀⠀⠀⢀⣾⡿⠀⠀⠻⣿⠇⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀⣼⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠃⠹⣿⣿⡟⠀⠈⢿⣿⣿⡀⠀⢀⣤⣤⣀
⠀⠀⠀⠈⣿⣿⣤⣀⣀⣟⣠⣤⣾⣿⣿⡿⠋⠘⠻⣿⣿⣿⣿⠟⠁⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠁⠀⠀⠀⠘⢿⣿⣷⡀⢿⣿⣿⣿⠂
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠈⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢳⠀⢀⠀⠀⠀⠀⣀⡤⠶⠚⠉⢉⡏⢀⣀⡤⢶⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⡏⠙⠲⠴⠏⠀⠀⠘⠗⠋⠳⠴⠚⠉⠁⠀⠀⠀⠀⠈⠛⠉⠀⠀⣈⠁⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⢀⣴⣾⣶⡀⠀⣴⣿⣦⡀⠀⢀⣾⢿⣷⠘⠓⠦⠤⢤⣀⣀⡀
⠀⠀⠀⣀⣀⣀⣼⠀⣰⠟⣿⣿⣿⣿⣷⠀⠀⠀⣾⣹⣿⣿⣧⠀⣟⣸⣿⣿⣄⣼⣧⣿⣿⡇⠀⠀⠀⠀⠀⣠⠿⠃
⠀⠀⠀⠹⣄⠀⠀⠀⣿⣼⡟⢻⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠈⠱⡄⠀⣿⣿⡇⢸⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠹⣄
⠀⠀⠀⠀⠀⡼⠁⠀⣿⣿⣿⣿⣿⣟⠁⠀⢸⣿⣿⡏⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⡇⠀⣴⠾⠃
⠀⠀⠀⢀⡾⠁⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⡇⠀⢻
⠀⠀⠀⠚⠷⢶⡄⠀⢿⣿⡏⢹⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⡇⠀⠀⠀⠀⢻⣿⡇⠀⢸⡆
⠀⠀⠀⠀⠀⢼⣁⠀⢸⣿⣿⣿⣿⣿⡿⠸⣿⣿⡟⠉⠻⣿⡿⠀⠈⠻⠟⠀⠀⢀⡀⠀⠀⠈⠀⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠈⡇⠘⠻⠿⠟⠛⠉⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣤⠋⠉⠙⠦⣄⣸⠃
⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⢀⣴⡆⠀⠀⣠⢦⡀⣰⢦⡀⠀⠀⣷⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣼⣠⠞⠙⠟⠹⣧⡴⠋⠁⠀⠙⠁⠀⠙⢦⣀⡇⠈
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⣕⣽⣯⣮⢔⠁⠀⠀⠀⠀⢑⣭⡮⡿⣗⡁⠀⠀⠀⠀⠀⠈⠨⣺⣿⣷⡅⠈⠀⠀⠀⠀⠀⠈⢕
⣻⣷⠖⠄⠀⠀⠀⠀⠀⠀⡺⣿⣿⠷⠄⠤⠀⢀⠐⠀⠁⠀⠀⠀⠀⠉⠢⡄⠀⢀⠠⠀⠄⢽⣿⡁⠈⢼⡄⠀⠀⠀⠀⠄⠸
⣿⣿⠕⠅⠀⠀⠀⠀⠀⠀⣪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢔⢵⣿⣿⠿⠛⠃⠀⠀⠀⠀⠐⢪
⣿⣿⣸⠑⠚⣄⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣟⠁⡀⠀⠀⠀⠀⡀⡈⢘
⣿⣿⡃⢦⣶⡿⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠀⠀⠀⠀⠀⠈⢱
⣿⣿⠕⠄⠈⠀⠀⠀⠀⣠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠄⠐⠺
⣿⣿⡗⠅⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣷⠀⠀⠀⠐⢺
⣿⣿⡣⡀⢀⠀⠀⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⡀⡀⢝
⣿⣿⠎⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠃⠀⠀⠀⠈⢰
⣿⣿⠕⠄⠀⠀⠀⠡⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⡿⠟⠁⢠⡀⠀⠀⠔⠸
⣿⣟⡁⠀⠀⠀⠀⠐⠉⢪⣄⣀⣀⣀⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣯⡖⠀⠳⠀⢨⣆⠀⠐⢺
⣿⣿⡣⡀⢀⠀⠀⠀⠀⣀⢝⢿⡿⡿⢟⠟⠉⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⡣⣺⣿⣗⡅⡠⡀⠙⠛⠃⡀⡈⢑
⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⡵⣿⣯⣮⡀⠀⠀⠀⠀⠙⠟⢠⣶⣶⣶⢶⣿⣿⠿⠛⠁⠀⢀⣾⣿⣷⡄⢈⠀⠀⠀⠀⠀⢀⢱
⣿⣿⠕⠄⠀⡤⡀⠀⠀⠀⡪⣿⣟⢗⠨⠀⠀⠀⠀⣈⢀⠘⠿⠋⣡⠀⣷⡆⠀⢀⠀⠀⠔⢽⣿⡯⠂⠐⠀⠀⠀⠀⠀⠐⠺
⣿⣿⡓⣃⡠⠀⠈⠀⣽⡄⢺⣿⣿⡇⠘⠀⠀⠀⠀⢁⠂⣤⠀⣴⣿⡄⡿⠗⠀⢀⣢⡔⠓⣻⣿⣿⠒⠀⠀⠀⠀⠀⠀⠐⢺
⣿⣿⡢⡋⠲⡄⢀⣀⣻⣀⢕⢿⡯⡪⢀⠀⠀⠀⡀⢃⢜⣾⡿⣿⣟⢥⣠⣴⡇⣾⣿⠇⡨⣪⣟⣕⢄⡨⠀⠀⠀⠀⡀⡠⢝
⣿⣿⢄⠄⠀⠑⠻⠋⠉⠉⡱⣿⣿⢮⡤⡔⠀⠀⣢⣢⣿⣟⣽⣿⡧⡁⠀⢿⠣⢰⡿⠀⢄⢾⣿⡷⣰⠂⠀⣀⠀⠀⠀⢀⡰
⣿⣿⠕⠅⠀⠀⠀⠀⠀⠀⡪⣻⣗⣵⠎⠒⣿⡿⠟⠹⠫⢐⢟⣿⡯⠢⠀⠐⡤⣾⠃⡐⠔⢽⣿⡿⡓⣄⣤⣿⠀⠀⠀⠐⠺
⣿⣿⡓⡃⠀⠀⠀⠀⠀⠀⢜⣿⣿⣗⣼⠞⠋⠀⠀⠀⢈⢐⣿⣿⣿⠂⠂⠃⠀⣿⣾⠀⠃⣻⣿⣟⠓⠁⠋⠀⠀⠀⠀⠀⢸
⣿⣿⡪⡀⢀⠀⠀⠀⠀⢀⢝⣿⡯⡫⢕⠀⠀⠀⡠⢀⡔⢭⡻⣿⣗⢅⢠⠀⣼⣿⡏⠨⡨⣺⣿⣕⠅⡨⠀⠀⠀⠀⡀⡨⢝
⣿⣯⢄⠄⠀⠀⠀⠀⠀⠀⡱⣿⣿⢦⡠⠀⠀⠘⢤⣬⣷⣨⢼⣿⡵⡁⠀⢠⠟⣻⠁⠀⢄⢼⣿⡧⡄⢄⠀⠀⠀⠀⠀⢀⡰
⣿⣿⠕⠅⠀⠀⠀⠀⠀⠀⡪⣿⣗⢗⠨⠀⠀⠀⠀⠉⠨⠒⢵⣿⡯⠂⠀⠐⠀⠏⠀⠐⠔⢽⣿⡯⠂⠐⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣃⣃⣀⣀⣀⣀⣀⣀⣜⣿⣿⣧⣘⣀⣀⣀⣀⣀⣈⣠⣻⣿⣟⣀⣀⣀⣀⣀⣀⣀⣃⣻⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣸
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣤⣤⣤⣄⠀⠀⠀⣠⣀⡀⠀⢀⣤⣤⠀⢀⣀⣤⣤⡀⠀⣀⣀⡀⡀⠀⠀⢠⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⣀⠀⡀
⠀⠿⣿⣿⣿⠿⠃⠈⢻⣿⣿⣿⠃⠘⣿⠿⠃⢸⣿⣿⣿⣿⠰⣿⣿⣿⣧⠀⢰⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡾⣿⢸⣿⡇⣿⣿⣿⢸⡆
⠀⣠⣤⣤⡈⠀⢀⣤⣤⣬⣤⣤⠀⠀⣶⣦⠀⠀⠛⠛⠛⠁⠀⠈⠛⢋⠁⠀⠀⠈⠈⠋⠉⠀⠀⠀⠘⠿⠛⠛⠛⠋⠉⠉⠉⠀⢸⡿⣿⣼⣿⣧⣿⣿⣿⢸⡇
⠈⠻⢿⣿⣃⡀⠀⠿⠿⠿⢿⠟⢰⣿⣿⣿⡄⢀⣤⣤⣄⡀⢰⣿⣿⣿⣿⣿⣆⠀⣶⣾⣿⣿⣾⣧⠀⠀⠀⢄⢀⡀⠀⠀⠀⠀⢸⣿⠟⠛⠘⠋⠁⠋⠛⢼
⠀⠰⣿⣿⣿⣿⡧⢠⣶⣶⣶⣶⡆⠉⠟⠋⠐⣿⣿⣿⡿⠗⠀⠙⠻⠿⠟⠛⠁⠀⠋⠛⠻⠟⠙⠁⠀⠀⢰⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⣀⢀⡀⣠⣀⣤⣤
⢀⣀⣈⠉⠉⠉⠀⠘⢛⡛⠛⠟⠁⣴⣶⣶⣶⣤⠈⠉⣁⣀⣀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣤⡀⠀⠀⠀⠙⠛⠟⠛⠋⠀⠀⣠⣴⣦⣿⣻⣿⣧⣿⣿⡿⣿
⠘⠛⣻⣧⣴⣤⣤⣼⢿⡿⠗⠀⠀⠻⠿⢿⡿⠋⠀⢸⣿⣿⣿⣿⣿⣿⡿⠐⢿⣿⣿⣿⣿⣿⣿⡷⠀⣤⣴⣶⣶⣶⣿⣿⡇⠀⣿⡿⢛⣿⣿⡇⣿⣿⣿⡇⣿
⠀⠀⠙⠛⠿⠿⣟⠋⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠘⠛⠛⠛⠛⠛⣟⠃⠀⠀⠀⠈⠉⠙⠏⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠿⠃⠀⠛⠋⠀⠙⠁⠛⠃⠿
⠀⢴⣿⣿⣷⠆⠀⠀⠸⡿⣿⢿⣿⣿⢿⣿⠿⠀⣶⣶⣿⣶⣿⠃⠀⠀⠀⢠⣾⣶⣶⣦⣶⣶⣤⣤⠀⣿⣿⣿⣿⡿⠿⠿⠁⣠⡄⣴⣴⢤⣤⣄⣀⡀
⠀⠀⠉⠉⣩⣦⣤⣤⣄⠀⠀⢀⣀⣀⣀⡄⠀⠀⠈⢉⠏⣉⣩⣥⣤⣄⡀⠸⢿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⣧⣿⢿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠛⠿⢿⡿⠛⢰⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠺⣿⣿⣿⣿⣿⡿⠂⠀⠈⠁⠀⠀⠈⠉⠘⠀⠀⢀⣀⣀⣀⣀⣀⡀⠿⠛⠛⠉⠁⠁⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠻⢿⣿⣿⣿⠿⢋⣀⣀⣀⣀⣀⣈⠹⠉⠁⠀⠀⣤⣶⣶⣿⣶⣦⣀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⡀⣶⡆⣶⣶⣤⣤⡀
⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⡿⠄⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣽⣿⣧⣿⣿⣿⣧⡅⣷
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠃⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠻⡅⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁⠁⠀⠀⠀⠀⠉⠉⠉⠉⠋⠀⠀⡿⠞⠛⠋⠋⠉⠘⠉⠛⠇⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢀⣀⣀⣠⣄⣤⣀⣀⡀
⠀⠀⠀⡠⠃⠁⠁⠑⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣆⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢶⣿⣿⣿⣿⣿⣿⣿⣏⣻⣽⣿⣶
⠀⠀⠀⠉⠦⢄⠤⢀⠋⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⡠⠀⠀⣸⣿⣿⣿⣿⣿⣿⠁⠀⠛⠋⠉⠉⠉⠉⠀⠸⠀⠘⡿⠛⠈⠉⠈⠁⠈⠉⠛⠛⠿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠁⠈⠁⠉⠀⠈
⣿⣿⠿⢿⠿⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠐⠲⠤⠀⠀⠀⢀⣠⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⢀⣀⣤⡤⣄⠀⠀⠠⣟⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢉⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣈⣋⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠾⣛⡛⢛⣻⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣟⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡌⢿⣷⣶⣤⣤⣀⣀⣀⣀⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣠⡇⠀⠈⠉⠛⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣤⣀⣀⣀⣈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⢄⣀⣀⣤⣤⣤⣤⣶⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣴⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡐⢠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⠁⠀⠀⠀⠀⠀⠀⣼⣧⢀⠀⣀⠂⣜⡣⢌⡴⢠⠂⠄
⠀⠀⠀⣀⠄⠂⢄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡌⡐⢢⣟⣾⣙⢾⡽⡣⢘⡰⢁⠂
⠀⣠⠚⠁⠀⠀⣀⠃⠀⠀⠘⣷⣄⡀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⢰⠇⠀⠈⣗⣼⣳⣾⣷⠋⠛⠘⢛⢾⣑⢢⡜⠠
⠀⠑⡀⠀⠀⡂⠌⠳⡀⠀⠀⠘⣆⠙⠦⣄⠀⠀⠀⡟⠙⢦⡀⢠⠏⠀⠀⠀⢹⡶⣟⣾⣵⣃⣤⡰⣯⡿⣥⠓⠌⠁
⠀⠀⠘⠶⠆⠙⠳⠦⣈⠀⠀⠀⠘⣆⠀⠈⠑⢦⣸⠇⠀⠀⠱⠏⠀⠀⠀⠀⠸⡷⢯⡿⣿⣟⣾⣻⢯⡝⠣⢍⠐⢠⠞⠉⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠘⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠉⢹⡿⢯⠛⠬⠁⢆⣠⠶⠃⠄⠀⠀⢹
⠀⣀⠀⠀⠀⡀⠀⠀⠀⠲⣶⡶⠤⢤⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡘⠣⠌⠠⠘⢉⡀⠤⠤⠶⠀⢀⠈
⠀⣀⣀⣆⣀⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⣤⣁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⣹⣿⡟⠉⠀⠀⠀⠀⠀⠀⢀⣠⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⣀
⠀⠁⠀⠁⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠶⠶⠖⠒⠚⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠶⠿⠿⠷⠶⠶⠦⠤⠤⢤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣴⠞⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⠶⣌⠑⠢⢤⣀⡠⠤⣄⠀⠀⠀⣤⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⢀⡶⣿⠶⠟⣩⣿⠁⠀⠀⡰⡆⠀⢀⣴⡀⠀⠀⢸⡇⠀⠀⠀⠑⢦⡐⠈⠁⠀⠈⢢⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⣐⡦⠞⠁⣠⣾⣽⡇⠀⣠⢾⡕⢻⣠⠞⠀⠙⣦⠀⠘⡇⠀⠀⠀⠀⠀⠙⠂⠀⠀⢠⠼⠁
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠚⠉⠔⢋⣾⢿⣿⡿⢀⡾⣯⠓⡌⡹⠋⠀⠀⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠓⠂⠀⠒⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠐⠟⣽⣿⣿⣷⢿⡿⢣⣟⠤⢁⠂⠀⠀⠀⠀⠀⠹⣿⡂
⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⣀⣼⢳⣟⡿⣯⠟⡽⣳⠎⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢉⠙⠠⢁⠎⡑⢣⡜⠰⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠌⠁⠀⠀⠀⠀⠀⠀⠀⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠜⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠙⢢⠄⠀⠀⡠⢤⠞⠻⠀⠀⢠⠇⠀⠀⣶
⠀⠀⢤⡀⠀⠀⣀⠀⠈⢢⠀⢸⣿⠁⠀⠀⢰⠀⠀⠀⢸⡆⠀⠀⡾⠀⠀⠀⣷⡏⠀⠀⢸⡇⡾⠀⠀⣸⠇
⠀⠀⠘⣇⠀⠘⣿⠇⠀⢸⣇⠼⡟⠀⠀⠀⣾⡆⠀⠀⠀⣇⠀⢠⡇⠀⠀⠀⡟⠀⠀⠀⢸⢃⡇⠀⢠⣟⣀
⠀⠀⠀⢻⡀⠀⠁⠀⠀⠻⡀⢰⠃⠀⠀⢀⣿⡇⠀⠀⠀⢻⠀⣸⠀⠀⡇⠀⠀⢠⠀⠀⣿⢸⠃⠀⣼⠋⠁
⠀⠀⠀⠘⣇⠀⠰⣶⡆⠀⢸⡾⠀⠀⠀⠀⠁⠁⠀⠀⠀⢸⡆⡟⠀⢰⡇⠀⢀⡏⠀⢠⡇⣼⠀⢠⡟
⠀⠀⠀⠀⢿⠀⠀⠋⠁⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⢸⣇⣀⣾⠃⠀⣸⢳⡏⠛⢺⡃⠀⣀
⠀⠀⠀⣀⣸⣇⣤⣤⣴⠿⣻⠀⠀⠀⢀⡿⠛⢻⡇⠀⠀⠀⢻⠛⠿⠿⠉⠛⣿⣤⣄⡿⣸⡀⠀⣸⠿⠟⠛⠛⠃
⠀⠀⠀⠀⠈⠉⠉⠻⣷⡄⡏⠀⠀⠀⢸⠇⠀⠈⣇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠉⠉⠃⠙⠻⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣄⣀⣾⠀⠀⠀⠛⠛⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣼⣿⣿⣿⣿⣯⣦⣄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣠⡾⠟⠛⠛⠛⠛⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠙⢿⡆⣠⣶⠿⠛⠛⠛⠻⢷⣦⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣯⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⣰⡿⢿⣿⣿⣿⣿⣷⡄⠀⠀⣠⠾⣟⠉⢹⣿⠀⢀⣴⣿⣿⣿⣷⣄⣤⣾⣿⣿⣦⣤⣴⣿⣿⣶⡟⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣴⣿⣿⣶⣾⡟⠉⠉⣿⣿⣿⠿⠟⠛⢿⣿⣿⣿⡏⣿⣿⡆⣿⣇
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡟⣉⣽⣿⣿⣇⣀⣴⣿⣿⣿⣄⠀⠀⣸⣿⣿⣿⣧⢸⣿⡇⣽⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⠸⣿⣿⣿⣿⠿⠙⠿⣿⣿⠛⠁⠻⣿⣿⣿⣿⣿⢿⣿⣿⠸⣿⡇⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠸⢿⢸⠁⠀⠀⠀⠀⠉⡉⡏⠁⠀⠀⠀⠉⠀⠀⠀⠀⢿⣏⠉⠉⠀⠈⣿⡏⠀⠙⠁⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠐⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⠻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢎⡵⠀⢀⠎⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡇⠀⠀⣠⡜⠙⠀⠀⠀⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡤⣄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠘⡄⠀⠀⣠⠀⠀⢀⠞⠁⢰⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡵⠊
⠀⠀⠀⠀⠀⡆⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄⠀⢠⠃⠀⠀⠀⠀⠀⠱⠴⠋⠸⡀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠫⣾⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠒⠒⢲⠆⠀⢠⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⢩⠑⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⣴⣶⣦⡀⠀⣴⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠴⠖
⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣶⡿⠿⣶⡄⢠⣿⠋⢿⣇⢸⡿⠀⢻⣇⢸⣿⣿⡄⣼⣿⡇⠀⠀⠀⢀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠻⠷⠁⠀⠀⠀⢀⡠⠜⠃⠀⠀⠀⠀⢻⣧⣴⣿⠃⢸⣿⠀⢸⣿⣿⡇⠀⢸⣿⣼⡇⠹⣿⡟⣿⡇⠀⠐⢎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠿⣶⡸⣿⡀⢸⣿⠸⣿⣀⣾⠇⣿⠇⠀⠈⠀⣿⡇⠀⠀⣈⠵⠀⠀⠀⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⠘⣿⣀⣠⣿⠇⠻⢷⠿⠃⠀⠉⠛⠉⠈⠛⠀⠀⠀⠀⠙⠁⠀⣀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠰⠯⠄⠐⠒⢒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⢄⠀⠀⠀⠀⠘⡆
⠀⠀⣀⢀⡤⣰⠆⠖⠒⠂⠀⠀⠀⠀⠀⠀⠞⠒⠒⠉⡹⠀⠀⠀⠀⢀⠀⠀⠀⡠⡄⠀⠀⡸⠑⢤⠁⠀⠙⠦⡀⠀⠀⠸⡄⠀⠈⡧⡄⡀
⠀⠀⠈⠙⠀⠀⠀⢀⡠⢔⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠔⠉⡇⢀⡜⠁⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢱⠀⠀⠀⠘⢱⣤
⠀⠀⠀⠀⠀⠀⢞⡧⠖⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⡏⠀⢀⠔⠁⠀⠀⠰⠊⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠢⣧⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠸⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠃⠀⠀⠀⠀⡱⠶⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁

⠀⠀⠀⠀⠀⣀⡀⠀⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠃⠀⠀⢦⠀⠀⠂⠀⢀⡀⠀⠀⠀⠈⠄⠄⢀⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⢹⡿⣶⣄⠀⡠⢐⣈⡾⠁⢻⡀⠀⠀⠀⠀⣀⣤⡄⠀⢀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠛⢿⣶⣿⡿⠁⠀⠘⣿⣦⣤⣶⡿⠛⡽⠁⠀⠊⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣼⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠉⠉⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⣻⠷⠂
⠀⠀⠀⠀⠀⣀⣲⣾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⣀⣈⣛⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣃⣀⣀⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠙⣦⠀⠀⢀⣠⣄⡀⠀⠀⣠⠶⠶⢤⣀⠀⣷⡀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠄⠀⠀⠀⠀⣿⢀⣴⠫⢭⢩⠹⢦⣼⠁⠀⠀⠀⠈⠙⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠹⠋⠀⠈⡐⡐⠀⠀⠁⠀⠀⡄⠠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⠓⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠇⠀⠓⠀⠱⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠢⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠻⠿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⢿⣿⣿⣿⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⡿⠿⠛⠛⠛⠛⠿⠆⠀⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⢛⣛⣛⣿⣿⣤⣀⢀⣠⡾⣋⣭⣴⣶⣶⣾⣿⣿⣷⣄⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣶⣶⣬⣙⠷⣤⡀⢀⣠⣴⣶⣞⣛⣛⠻⢿⣿⣿
⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻
⣿⣳⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣼⣿⡿⠛⢻⣿⣿⣿⠀⠀⣿⣿⣿⡿⠻⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣿⣿⣿⣧
⣿⣿⣿⣿⣇⠀⠀⢻⣿⣿⡿⢿⣿⣿⡿⠀⠀⢰⣿⠟⠀⢀⣾⣿⣿⣿⠀⠀⣿⣿⣿⣇⠀⠘⢿⣿⡀⠀⠸⣿⣿⣿⠿⣿⣿⣿⠃⠀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⢻⣿⠃⠈⢿⣿⠃⠀⢀⣿⡏⠀⢀⣾⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣆⠀⠈⣿⣧⠀⠀⢹⣿⡟⠀⢻⣿⠏⠀⢀⣾⣿⣿⣿⣿
⠉⠹⣿⣿⣿⣿⣧⠀⠀⠋⠀⣤⠘⠟⠀⠀⣼⣿⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡿⣿⠀⠀⢸⣿⡆⠀⠈⡟⢀⡆⠀⠋⠀⢀⣾⣿⣿⣿⣿⡟
⠳⣄⣸⢿⣿⣿⣿⣧⠀⠀⣰⣿⣆⠀⠀⢰⣿⣿⠀⠀⢸⣿⣾⡿⠁⠀⠀⠀⠀⠀⢻⣿⡽⣆⠀⢸⣿⣷⠀⠀⢀⣾⣿⡀⠀⢠⣿⣿⣿⣿⣿⣏⡴
⡀⠀⠀⠀⢿⣿⣿⣿⣧⣠⣿⣿⣿⡆⢀⣿⣿⣿⡆⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⣾⣿⣿⣇⠀⣼⣿⣿⣧⣠⣿⣿⣿⣿⡟⠁
⣷⣦⣀⣀⣼⣏⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠏⢀⣀⠀⠀⠀⠀⠀⠄⣀⡀⢹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠟⣿⣧⣀⣀⣴
⣿⣿⣿⣿⣿⡟⢧⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⡄⠛⠿⠿⠆⠀⠀⠰⠟⠿⠛⡼⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⡾⣹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠈⠳⡄⠀⠀⠈⠛⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⣠⠟⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡆⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣟⠛⢻⣛⣛⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣛⠛⠛⠛⢛⣻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣫⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⢿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⠿⣫⣾⡿⣫⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⠫⢕⡫⢛⡭⠚⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢀⣤⠞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠛⣄⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡈⠉⠢⡴⠁⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡜⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣠⠞⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣴⣏⣠⠤⠶⠶⠶⠶⠶⣄⡀⢀⠃⠀⠀⢀⣠⡤⠦⢤⣤⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠟⣋⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣠⡶⠤⣤⣯⣀
⠀⠀⠀⢠⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡎⠀⢠⠖⠉⠀⠀⠀⠀⠀⠉⠢⡀⢠⠖⠋⠁⠀⠈⠙⢦⡾⠉⠉⠙⠓⣦⢸⡧⠄⣀⡀⣰⠟⡇⠀⠀⠀⠉⠉⣶
⠀⠀⣰⣿⡿⡀⠀⠀⠀⢪⡱⠀⠀⠀⠀⢠⠇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠹⠏⠀⠀⠀⢹⠃⠘⡇⠀⡀⠀⠀⣸⡇
⠀⣰⡿⢠⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢸⠀⠀⠀⠀⡔⢠⠀⠀⠀⠀⢸⠁⠀⢀⡤⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⢠⣿⣧
⢠⡿⠁⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠻⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⣸⠀⠀⠀⠉⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⣼⠀⢿
⣾⡇⠀⠀⢇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠴⠿⣄⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠃⠀⢸
⣿⡇⠀⠀⠸⡄⢡⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⢹⡶⠤⠤⠴⠖⠊⢁⣾⡀⠀⠀⠒⠂⣀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡄⠀⢀⡞
⢿⡇⠀⠀⠀⢧⠘⡀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⢼⡷⠦⠤⠤⠴⠚⠉⠀⠉⠓⠒⠒⣻⠏⠇⠀⠀⠀⠀⠟⢆⠀⣀⣴⠀⠀⠀⠀⠀⣋⡈⢩⡟
⢸⣷⠀⠀⠀⢸⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡘⠀⠀⠀⢠⡎⠀⢈⠟⠡⠉⠀⠀⠀⠀⠀⠉⠀⠀⢹⠀⠀⠀⢸
⠀⢿⡆⠀⠀⠀⣇⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠧⠤⣀⣠⡟⠹⣤⣮⠴⡇⢀⡄⠀⠀⠀⠀⣶⣀⣠⠞⠀⠀⢀⣾
⠀⠈⢿⡄⠀⠀⢹⠀⠷⣀⣀⡀⠤⠤⠒⠊⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣀⡟⠀⠀⠀⠀⠀⣯⠀⠓⠒⠒⠀⣴⡦⠞⠁⠀⠀⠀⣼⣿
⠀⠀⠈⢿⣄⠀⠈⢧⣄⣀⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠖⠃⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠹⣷⡀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⣤⣶⡿⠿⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣠⣴⡿⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠻⠷⣦⣠⡾⠋⠁⠀⠀⠀⠀⠀⠙⣷⣄⣠⣤⣤⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀⡾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⢀⣾⠛⠉⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠈⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⡀
⠀⠀⠀⠀⣿⡀⠀⣀⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⣹⣧⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⣴⠿⠋⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢰⡿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡘⣦⣾⣿⣷⣄⡀⢀⡆⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⣿⠃⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀
⣿⡆⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠘⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠉⠙⣻⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⢹⠟⠛⣿⣿⡿⠿⢿⣇⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠙⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣰⠟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢠⣿⣇⣠⣴⠾⠋
⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣴⠿⠛⣿⠉⠁
⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⣶⡶⠿⠋⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣴⣶⠿⠟⠁⠀⠈⠿⣦⣄⣀⣠⣴⣾⠟⠻⢶⣤⣄⣀⣀⣀⣀⣠⣴⡿⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣀⣴⣿⣿⣿⣿⣷⡀⠀⢀⣀⣤⣤⣶⣾⡇⠀⠀⠀⠐⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⠆
⠀⠀⠀⠀⠀⢠⡖⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⣿⠟⠛⠛⢫⡀
⠀⠀⠀⠀⢤⣼⡇⠀⠀⠀⣤⡀⠀⠀⢸⣿⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠙⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⣿⣧⣀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣠⣼⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢻⣀⣀⣀⣸⣄⣠⣇⣀⣀⣀⣿⡏⠉⠉⢻⠉⠁
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⣠⣿⣿⣀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠳⣶⣦⡄
⠀⠀⠀⣠⣤⣤⣤⠄⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠙⠛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠁⠀⠈⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⣿⣿⡿⢘⣿⠿⠿⢿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠘⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠿⠧⠁⠀⠈⣉⡑⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠐⣇⠀⣐⠦⠀⢠⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠟⠀⠠⠤⠔⠀⢸⢿⡭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣷⡀⠀⠀⢀⣤⣤⠀⠈⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀⢠⣾⣷⣾⣷⣄⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⢹⡟⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠘⣧⠀⠀⠀⠘⣿⣿⣝⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⣿⣿⣧⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠤⠄⠀⠀⠞⠛⠋⠉⠉⠙⠂⢻⣿⣿⢧⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣀⡀⠀⠀⠀⠀⢠⣶⠒⠢⡤⠀⠘⡿⣿⣆⠈⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢹⣿⡄⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⡀⠀⢸⣋⣉⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣻⣿⡇
⠀⠀⠀⠀⠀⢀⣿⡿⢸⣿⣧⠀⢸⡿⠝⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣷⢸⣿⣿⣆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⢹⣿⣆⣿⣿⣿⣦⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣌⠻⣮⡙⠻⠿⠿⠟⢋⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠝⠻⠿⠿⠷⠂⠀⠀⠀⠀⠐⢛⣽⣿⣿⣿⣿⢟⣵⣿⠟⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠛⠋⠁⠈⠉⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠦⢤⡀⠀⣸⡟⠀⢠⡄⠀⢳⠀⠀⡴⠒⠒⣦⠀⢠⠤⠤⢤⠀⡞⠀⠀⣰⡇
⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⢀⡀⠀⠀⠹⣤⣿⠁⠀⢼⠇⠀⠈⢧⡼⠁⠀⠀⢸⣠⠏⠀⠀⢸⣾⠁⠀⢠⠏
⠀⠀⠀⣀⣷⡀⠀⠀⠀⠐⣿⣿⡿⠀⠀⢠⣿⡇⠀⠀⣀⣀⡀⠀⠘⠃⠀⢠⠀⠈⠏⠀⢀⠀⠀⣿⠀⠀⣼
⠀⠀⠀⢻⣿⣷⣶⡄⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣠⠀⠀⠀⣾⡆⠀⠀⠀⣿⠀⠀⠸⠟⠛⢯⡀
⠀⠀⠀⠀⠻⠿⣿⣷⡀⠀⠀⠀⣤⣶⣿⣷⠀⠀⠀⣿⠁⢻⣿⣿⠀⠀⢀⣿⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠹⣍⡹⠋⠀⠀⣸⠋⠀⠀⢹⣿⣤⣤⡼⢿⣿⣷⣶⡟⣿⣿⣄⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠉⠀⠀⣠⣼⠇⠀⠀⠀⠸⡿⠿⠿⠇⠀⠸⠿⠮⠀⠸⢿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⣦⡀⠲⠀⢀⡀⠈⢸⡇⣴⢰⣇⠀⢠⠄⠘⠁⢀⠄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣄⠘⣿⣦⡀⠀⢷⣄⡘⢡⣿⣸⣿⣰⣿⠀⢀⣴⡿⠠⠟⠁⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣈⡀⠈⢿⣿⣦⡈⣿⣿⡾⠋⠛⠉⠉⢻⣴⢿⣿⠃⣀⣤⡖⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠷⠤⠀⢹⣿⣷⣾⣿⠆⠹⠟⠉⠁⠀⠀⠀⠀⠀⠁⠈⠻⠞⣿⣯⣴⣞⣥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣤⣤⣈⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠉⠀⠚⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠌⠻⣅⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡀⠛⠛⠉⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⣠⣤⡶⠞⠿⠒⠋⠉⠙⠂⠀⠠⠶⠳⢦⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⢛⣿⣿⣿⡶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠞⠁⠈⠲⢄⡀⢤⣬⣿⠟⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣹⡻⠄⠀⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠦⣀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣌⡀
⠀⠀⠀⠀⠀⠻⣆⠱⣄⠀⣄⠀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠈⠑⠢⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣄⣹⣷⡾⠖⢋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢄
⠀⠀⠀⢤⡄⠐⢰⣟⠋⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠧⠄
⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡶⠶⠶⠲⣄⠀⠶
⠀⠀⠀⠚⠛⢋⠉⢻⣿⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠿⣌⣹⣛⣋⣡⣴⣖
⠀⠀⠀⠉⠀⠌⠀⠋⠙⣽⢿⣦⣀⣠⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠿⢉⣋⡉⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟⡸⠙⣿⠹⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣶⣤⣾⣿⣿⣿⣿⢤⣄⣀⣒⣒⣖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠀⠀⠂⠋⠰⠀⢷⠙⠠⠀⠀⠀⠀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡴⠶⠀⠈⣥⡴⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣄⣼⣿⣿⣿⣿⡿⣿⣒⣒⠶⠶⠶⣤⠂⡤⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠛⣿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠛⠃⠒⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠽⠟⠛⠛⠋⠉⠁⠁⠰⠾⠿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⠄⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠇⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠙⠛⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣠⠏
⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⡴⠶⠶⢶⣄⠀⠀⠀⠘⠁⢀⠞⠁
⠀⠀⠘⠻⠋⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠉⠀⠀⠀⠀⠘⢧⢀⣤⠶⠛⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠙⠢⣀⣤⣤⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⣾⠁⠀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠞⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⣰⡧⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⣦⣀⣀⣀⣀⣤⡞⣿⢿⣦⣀⠀⠀⣀⣴⡟⠒⠙⠛⠒⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⡉⣙⢙⣋⣩⢹⡏⢉⢹⣏⣿⣿⡟⣻⡉⡃⢈⠀⠁⠉⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠤⠀⠄⠩⣿⠇⢸⡄⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠐⣿⠂⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡁⠈⣿⡁⢘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠠⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⠃⠙⢷⣎⠱⢌⠰⢈⠰⢈⠤⡁⠄⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠆⡁⣾⠀⠀⠀⠹⣷⣎⠰⢈⠴⣉⠶⣉⠦⡁⠆⡁⠄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠐⣀⠒⣀⠖⣁⡏⠀⠀⣀⠀⠈⠻⣷⣉⠶⣉⠶⣉⠶⣁⣆⣡⡶⠷⠛⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠒⣀⠒⣉⠒⣹⡇⠀⢰⣯⡣⡀⠀⠘⢿⣶⣩⣶⠿⠞⠋⠉⠀⠀⠀⢠⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⢠⠒⣠⠒⣤⢓⣽⠁⠀⢸⣙⣿⣿⢦⡀⠀⠉⠁⠀⣀⣀⣠⣴⠂⣀⡠⠟⠐⠂⣴⣤⣤⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣴⡦⠾⠶⠞⠛⠛⠉⠉⠉⠀⠀⠾⣏⣿⣿⡕⣿⣾⡄⠀⠀⠀⠳⣧⠃⣼⡟⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠻⣄⠀⠀⠀⣀⣠⣤⢴⣿⡇⠀⠀⠈⠻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢼⡿⠀⠀⣠⠀⠀⠘⣿⣿⡀⠀⠰
⠀⠀⠀⠀⡠⠐⠒⠈⠉⠀⠀⠈⠻⢮⣷⣹⣿⠁⠀⢠⡄⠀⠘⢿⣿⣿⣿⠀⠀⢸⣆⠀⠀⠁⠀⣰⣿⠀⠀⠘⢿⣿⣇⣀⠄
⠀⠀⠀⢿⣷⡀⠀⠰⣶⣿⣦⠀⠀⠀⠑⢽⣿⠀⠀⠼⠿⠂⠀⠀⠹⣿⣿⡇⠀⠘⣿⣷⡀⠀⣰⣿⣿⠀⠀⢘⣾⣿⠁
⠀⠀⠀⠈⢿⣷⡀⠀⠙⠋⠁⠀⠈⠁⠢⢸⣿⠀⠀⣀⣠⣤⣴⣄⠀⠈⢿⣧⣀⣀⡿⣿⣿⠿⠉⣿⣿⣶⣶⠛⠿⠿⠿⠋
⠀⠀⠀⠀⠀⢿⣷⡄⠀⠰⣾⣿⣿⠀⠀⢹⣿⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⡿⠿⠿⣻⣴⣶⣶⣄⠀⠈⠻⣦⡙⠤
⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠙⠋⠁⠀⣠⣾⣷⣶⣾⠁⣐⣷⣿⣿⣿⣧⣮⠏⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠻⣦⡈
⠀⠀⠀⠀⠀⠀⠠⢻⣿⣄⣠⣤⣶⣿⣿⠻⠟⠉⠋⠁⠀⠹⣼⣿⣿⣏⡟⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢈⢻⡿⠟⠋⠉⠁⠀⠀⢀⣀⣤⣤⡀⠀⠹⣏⣿⣽⠃⠀⢰⡿⡹⢎⡱⢎⡱⢌⠱⢈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣀⣠⣴⠶⡛⢏⡹⢎⡹⢿⡄⠀⠘⠗⡟⠀⠀⣾⠶⡁⢎⡱⢎⡱⢎⠱⢈
⠀⠀⠀⠀⠀⠀⠀⡾⠶⠚⠛⠉⠒⡁⠒⡁⢆⡑⢎⡱⢎⡹⣆⠀⠈⠁⠀⢸⣏⠶⡁⠎⡑⢎⡱⢈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠂⡁⠂⡑⢂⡐⢊⡐⢻⣧⠀⠀⠀⡾⣁⠆⡁⠆⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠂⡐⢂⠐⢀⠘⢷⡀⢸⠗⡀⠂⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠈⠳⡟


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

⠀⠀⠀⠀⠀⠀⠀⡰⢼⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣶⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠏⠀⠀⠀⣰⢺⡆⠀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⣠⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠙⠀⢿⡄⣌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡿⠃⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠸⡌⠀⠰⠆⠀⠀⠀⠀⠀⠈⠃⡴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠂⢠⠄
⠀⠀⠀⠃⠀⠀⠈⠈⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣀⠀⠀⠈⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠗
⠀⠀⠀⠁⢷⣤⡀⠈⢦⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠀⠀⠙⠳⣶⡶⠖⠲⣶⣄⣀⣀
⠀⠀⠀⢀⠀⠉⠛⢦⡈⢿⣧⢠⡤⠤⠠⣤⠶⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⢀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠈⠛⠶⢦⣝⣦⡄⠻⡒⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣤⣿⣟⣛⣦⠀⠀⠘⠃
⠀⠀⣀⣄⠀⠒⠀⠈⠛⠁⠤⡿⣻⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣀⠉⠭⣥⣀⣀
⠀⢨⢁⡿⠀⠀⠀⠀⠀⠁⣠⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡆⠀⠀⠉⠙⠻⢂
⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠰⣁⣱
⠀⠀⠀⠈⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠛⠋
⠀⠀⡿⠶⠶⠶⠖⠂⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⢠⡤⠴⠾⠀⡀
⠀⠀⠘⠂⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣠⣠⣤⣶⣾⣿⠏⠤⣤⣤⣄⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⢿⣿⣶⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣶⣴⣾⣿⡿⠿⠿⠿⠛⠁⠈⣶⢄⠈⠉⠛⠛⠷⡇
⠀⢼⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠟⠛⠋⠤⠠⠈⠉⠁⠀⠀⡀⠈⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⣎⣱⠂
⠀⠈⠂⠁⠀⠀⠀⠰⠶⢟⡛⡶⠶⠶⡎⠉⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⠦⠀⠀⠀⢀⡀⠉⠃
⠀⠀⠀⠀⠀⢴⠄⠈⣉⡴⢢⠖⢠⠇⠃⠀⠀⠀⠀⠀⠀⣠⡟⠁⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢠⣇⢀⡅
⠀⠀⠀⠀⠀⠀⣴⠿⠋⠀⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⢣⠀⠀⠀⠀⠀⠀⣴⠐⢃⠹⣦⣀⡤⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢂⡆⠀⠉⠋⠀⠀⠀⠀⠀⠀⠙⠓⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢹⡛⠃⠠⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣄⢲⠂
⠀⠀⠀⠀⠀⠀⠉⠅⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⢀⣀⠀⣀⢀⡀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠢⢝⣾⣿⣿⠋⠁⠀⠀⠉⠛⠛⠦⢍⠳⣭⠈⠁⡇⣇⢎⡴⣢⣦⡀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣞⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠳⣎⡗⣼⠮⠾⠏⠀⠀⠈⠀⠀⠃⣀⣀⠀⠀⠀⠀⢈⠁⠀⢀⣀⠄⠁
⠀⠀⠀⠀⠤⠂⠐⠀⡀⢁⣢⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠟⣲⣿⣯⡀⠀⣀⣀⣠⣶⣶⣾⣿⣿⡿⠀⠀⠀⠀⠸⣄⡰⠂⠀⢠⠄
⠀⠀⠀⠀⣤⣶⣶⣦⣤⣿⢡⣛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡫⡓⣸⣿⣿⢿⣟⣫⡿⠶⠞⠛⠛⣻⠟⠀⠀⠀⠀⡾⠿⠒⠲⠴⢿⡇
⠀⠀⠀⢲⣾⣿⣿⣷⣤⣿⣗⠢⢀⠛⠛⣦⣤⣤⡶⣦⡀⢠⡤⢾⠞⣵⠇⠀⢹⣶⠟⣅⡀⠀⠀⢠⡴⠢⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠃
⠀⠀⠈⢧⡙⣿⣿⣿⣿⣿⣽⠁⠀⠈⠂⠀⠈⠀⠀⠀⠈⠀⠠⠌⠜⠃⠒⢸⡆⣿⠙⣧⠑⢾⡶⠊
⠀⠀⠀⠀⢙⠷⣍⣹⡿⠟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠁⠤⠤⡤⠀⠀⠀⠀⠀⠀⢀⠤⢀
⠀⢀⠀⢀⣀⠀⢩⣥⠄⠀⠀⠀⠀⠀⠒⠶⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠋⠃⠀⢠⡠⠄⠊⠀⠀⠀⠐⠚⠒⢤⡄
⠀⠈⢀⡞⠁⣴⠋⢁⡴⠾⠛⠙⠙⠛⠳⠤⡌⠻⣄⠈⠷⢈⠀⠠⢄⣨⣦⣶⣾⣷⣾⣷⠖⠀⠔⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⡌⠀⠂
⠀⠀⡾⠀⣼⠃⣠⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠸⠆⠀⣠⣴⣷⣾⣿⣿⣿⣿⣿⣿⠁⠉⠁⠘⣬⡖⣷⣾⠷⣤⣾⢳⡾⢷⣇⣾⠇
⠀⢸⡇⢠⡇⣠⣴⣶⣿⣷⢸⣿⣿⢿⣿⣿⣧⣶⣦⣶⠀⠙⡿⠟⣿⣿⣿⣿⡿⠖⠐⠀⠀⢱⡄⠻⠛⠋⠋⠀⠉⡁⠀⠁⠈⢉⣻⠇⠀⠀⡆
⠀⠈⠁⠸⡇⢿⠷⢛⣿⣭⣭⠿⢯⣬⢿⣽⣿⣻⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡅⠁⠀⢹⠿⣦⡐⠀⠂⢠⣤⠐⣠⡴⠮⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠛⠐⠻⣏⡑⢠⠀⣀⣀⠀⡀⢀⣨⠟⢻⠇⡀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢁⠀⠀⣀⠀⠀⠛⠒⠻⠉⠋⢋⠉⠀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⠐⢀⣈⠻⢾⣦⣥⣬⣴⡶⠟⢉⣴⠏⠠⠀⠀⠀⠀⠁⠀⠀⠠⠤⢄⠁⠀⠀⠀⣿⣻⣷⠂⣄⠀⠀⠀⠀⠀⠀⠀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠈⠈⠠⠀⣤⠖⠋⣡⠀⠁⡤⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⢻⣿⣿⣿⡗⢄
⠀⢠⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠯⠽⠇⠆⢺⢸⠋⡇⡔⢠⠄⣴⠀⠄⠈⠧⢿⣶⣿⣿⡄⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣰⡄⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠞⢠⡅⠀⠀⠀⠈⢀⠙⢷⣾⣧⠀⢡⡔⠀⡀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣷⢀⣴⣷⣾⣿⣿⡖⣄⣀⠀⣀⣤⢀⡄⣠⠤⠚⠀⠀⠀⠾⣦⠦⡀⠉⢹⠲⣜⣿⠻⢿⣆⠀⠈⣡⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣧⠀⠀⠰⢸⣿⣿⠀⠀⠀⠈⣿⣿⡇⠀⠀⣱⡆⠚⠁⣸⡇⠀⠀⠀⠀⠀⠶⣤⠀⠞⣶⣼⣷⣽⣟⡓⠄⠙⠂⣈⣉⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠘⢿⣿⠀⠀⠀⠀⣿⣿⡇⠀⢀⣿⡇⠀⢀⣿⡇⠀⠀⡀⠀⠾⠶⠦⠀⢣⣿⠀⢏⠹⠿⠋⡀⢁⣬⣉⣉⡁
⠀⢀⠀⣤⣄⣿⣿⣿⠃⠐⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇⢠⣾⣿⣇⣀⢀⡇⠀⠀⠐⠛⣠⡄⠛⢧⠀⠀⣰⣮⣵⡷⠦⣬⣍⡁
⠀⠘⠉⠛⣿⣿⣿⢿⣷⠂⢄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⡿⠿⠋⠉⠀⠀⠀⠠⠛⣡⠞⣩⢒⡎⡆⡝⣏⣙⣡⠖⠉⠉⠃
⠀⢠⣴⣿⣋⣩⠿⣦⡼⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠻⠃⠀⠀⠀⠀⠀⢠⣶⣟⣁⡼⠇⠾⠴⠧⠻⡜⠉
⠀⠀⡞⡿⠁⢉⠀⠀⢙⣆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⢀⡀
⠀⢸⣠⣇⠀⠀⠷⠀⠀⣿⠀⠀⠀⢲⠂⠐⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠈⢦⣀⣴⣤⣤⣿⠀⠀⠀⠐
⠀⠘⢿⣷⣀⠀⠀⣀⣰⠏⠀⠀⢠⠟⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⢌⢠⣰⣶⣼⢰⣦⣼⡇⢹⡿⠸⢿⠄
⠀⠀⠀⠙⠙⠛⠺⠏⠀⢀⠄⠀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢳⣶⣶⣶⣶⣮⣿⣿⠘⠟⠀⠉⠉⠀⠀⠀⡠⣐⣶⣦⣶⣦⣾⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠇⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣤⣭⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠁⠀⠀⠀⠀⡀⠂⠱⢀⣿⣾⠀⠀⣿⠀⣶⣠⡆⣠⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣮⠥⠤⠤⠭⣭⡉⠉⢩⣿⠉⢉⡉⢉⡁
⠀⠀⠀⠀⡀⢠⡀⣀⢿⣦⣶⣿⣿⣿⡇⢺⡇⠀⣏⠉⠋⠙⠛⠻⠾⠧⠾⣧⣌⣄⢀⡄⠀⢀⠀⠉⣠⡤⠤⢤⣤⡈⠙⢦⡀⠹⣇⠈⣷⠀⠃
⠀⠀⠀⠀⠻⣦⣾⡛⠿⠿⠟⠉⠀⢰⣀⣴⡇⠀⠈⠉⠁⠒⠒⠒⠤⢤⣤⣀⣀⣈⡉⠉⠉⠋⢠⡾⠉⣠⣤⣤⠀⠛⠉⠉⠓⠀⣹⡄⠸⡇
⠀⠀⠀⠀⠀⠈⢻⠟⠠⠶⣄⣠⣤⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢨⣿⠉⢿⡇⣀⠀⢸⡁⢸⡇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠙⠁⠈⠁⠀⠻⣇⣘⣢⣴⣦⣴⣦⣧⣼⣖⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⠂⡀⠀⠀⠀⡰⠁⠀⠀⣀⡀⢃⠀⠀⠀⢰⡫
⠜⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠌⠀⠀⠀⠘⢆⢀⠀⠀⢰⠃⠀⠀⠀⠁⢷⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠠⠈⠁⠀⠀⠐⢲⡌⠀⡠⠀⡅⠀⠀⠀⠀⢸⠘⠀⠀⢀⠂⠀⠀⠀⠀⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢼⠇⠀⠀⠐⠀⠀⠀⢀⣰⠋⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢀⡁
⠀⠀⠀⠡⠀⠀⠀⠒⠢⡀⠀⠀⠀⣅⠁⠀⠀⠀⣠⠊⢃⠀⠀⠀⠀⠀⠀⢠⣬⡄⠀⠀⢀⠭⠇⠀⠀⠀⣴⠂⠀⠀⢰⠁
⠀⠀⠀⠀⣌⠀⠀⠀⠈⠁⠀⠀⡀⠀⠁⢤⠈⠠⠊⠀⠀⠢⣀⢀⠀⣀⡦⡛⠰⢀⠀⠀⠖⠒⠀⠀⢀⠚⠋⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠠⢔⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠢⢀⠤⠀⢡⠃⠢⣀⡠⡎
⠀⠀⠀⠀⢃⠁⠈⠁⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠨⢘⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣚⠽⠓⠚⠛⠿⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣖⣶⣶⠤⣄⡀⠀⠀⠀⠀⠊⠜⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠉⠀⠀⠈⠉⠒⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⣿⣧⡀
⠀⠀⠀⢀⢈⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣿⣿⡄
⠀⠀⠀⣌⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⢰⡀⠀⢲⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⣷⡀⢀⡟⢷⠀⣿⡄⠀⣸⣧⠀⢸⡀⠀⠀⠀⠀⠀⠘⣿⣿⠆
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣷⠿⠀⠀⡿⠈⣧⢸⡇⠘⣇⢿⢿⡀⢻⣿⡆⠘⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⣿⣸⣦⡀⣿⠀⢸⡜⡇⠀⣿⢸⡇⢷⣸⡇⣿⡀⠃⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⣧⢹⡀⠸⡇⢷⠀⢸⠈⣧⠈⢿⡇⠘⣇⢤⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⠈⣧⣰⡇⠘⣧⣾⠀⠻⠀⠈⠁⠀⠈⠘⠀⢀⣠⡾⣫⠞⠁
⠀⠀⠀⠈⢿⣿⢦⡀⠀⠀⠀⠀⠀⢿⣴⡟⠀⠘⠿⠇⠀⠈⠋⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠩⠊⠁
⠀⠀⠀⠀⠀⠻⣷⣝⢦⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠝⠊⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣮⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣆⡠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣭⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠳⣤⡀⢀⢀⠀⡀⣠⠞⠹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⢰⠇⠀⠀⠙⢦⣀⡠⠚⠁⠀⠀⢿⠀⠀⡀⠀⢈⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠶⠦⣤⣀⣁⡞⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠵⠖⠚⠉⠉⢠⢏⣀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣇⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠾⣾⣧⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⠀⢠⡾⠉⠉⠀⠀⠹⣄⠀⠀⠀⣀⡤⠴⠲⣄⠀⢰⡟⠀⠀⠀⢿⣿⡆⠀⠈⣧⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⠦⠴⠒⠛⠉⠁⠀⣼⡇⠀⠀⠀⠀⠀⠈⢦⡀⢰⣿⡄⠀⠀⠈⠳⣾⡇⠀⡀⠀⠘⣿⣿⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠳⢄⡀⢀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣦⠀⠀⠀⠱⡄⢿⣧⠀⠀⢦⡀⠈⠀⢠⣧⠀⠀⢻⣿⣷⡶⠾⢧
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠋⠉⠉⠉⢳⡀⢀⣿⠁⠀⠀⣼⣿⡷⠀⠀⠀⠘⢿⣿⡄⠀⠸⣷⣄⣀⣼⣿⡄⠀⠘⣿⣷⣀⣠⣼⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠠⣶⣶⡆⠀⣸⡇⢸⣿⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠻⣷⠀⠀⢻⣿⠿⠟⢿⣷⣶⣶⣿⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠉⠁⣀⡀⠈⠉⢾⡟⠀⠀⠀⣴⣶⣾⣿⣿⣧⡀⣀⣠⣼⣶⣶⣾⡆⠀⢀⣈⡿⠿⠿⠓⠙⠉⠈
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠘⣿⣿⡿⠀⠀⣸⡇⠀⠀⢀⡟⠉⠁⠀⠘⢿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠈⣯⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠉⠀⣀⣠⣴⣿⣷⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣿⣿⣿⣋⢭⠿⠻⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡼⠩⠛⠛⠖⠶⠠⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⡉⠋⠍⠀⠀⠀⠀⢷⠀⠀⢀⣤⠞⠉⡳⣄⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠴⠋⡀⠀⠀⠀⠀⡱⢦⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⢀⡴⠾⠿⣷⡎⠒⠦⢤⣄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠤⣿⣷⠄⣴⢧⢉⢆⠀⣄⠀⠀⢌⢳⣀⠱⡐⢦⡀
⠀⠀⣉⣴⢦⣤⣍⣽⡔⣄⢀⡈⠁⣈⠂⠀⠢⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡯⢐⣏⣿⢃⣦⣿⡎⢸⠀⡜⣶⡀⠈⠳⣿⢇⠹⣯⡳
⠀⠀⢇⢠⠆⢹⣿⣬⡛⡌⢮⡛⠗⠦⣑⠍⣓⠀⠑⠄⠀⠀⢠⣾⣿⡟⠉⠙⠻⣟⡋⠸⠽⡟⡇⡇⠁⢰⣮⠘⢆⢂⠊⢫⢣⢨⢻
⠀⠀⢀⠼⠓⢈⣁⣀⠬⠻⣖⠒⠶⣶⣶⣶⣶⣶⣶⣶⠶⢶⢯⣾⠏⠀⠀⠘⠃⢘⣯⠓⣦⣄⠈⠈⠢⠀⠙⢳⡀⡇⠱⡈⠇⡏⡇
⢠⣲⡾⡖⡉⢍⠑⠙⣆⠀⢠⡝⣦⡤⠤⠽⣿⣿⣿⣧⣴⢳⡿⡡⠄⣀⣄⡀⠀⠙⣻⣶⣿⣿⠃⠀⠀⣸⣾⠀⠃⣷⠀⢧⢀⡇⢳
⠹⡀⡰⢹⡀⠈⠃⠀⠘⡆⠀⠀⡜⣠⣶⠿⠟⢛⣋⡛⠿⣿⣮⠭⢙⣧⡉⡇⢀⣴⣿⣿⣸⡏⠀⢀⢰⠉⢠⠀⡾⢿⡀⠨⡄⢧⢄
⠀⠹⡄⢸⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⡥⠤⡮⠿⠿⠤⠾⣽⠃⠀⠀⠙⠿⠟⠿⠿⣽⣿⡿⠀⠘⢸⠸⢠⣗⠀⠁⠘⢸⡄⡜⣆⢸
⠀⠀⠘⡎⢀⠬⢠⠋⢉⡿⣿⣷⣶⣶⣶⣦⣤⣤⣴⣶⡶⠶⠶⢶⠦⠤⣤⣴⠶⠆⠀⠈⣷⠀⡀⡘⡀⠈⠾⡃⠀⠀⠸⡄⣿⣹⡸
⠀⠀⠀⢞⠝⣲⢻⣐⡱⡃⠀⠀⠆⡽⠛⠻⣿⣿⣿⠟⣗⠂⡰⠡⣠⣀⡈⢹⡧⠀⠀⣠⣿⡀⠃⣁⣷⢠⠀⣷⣾⠀⠀⡇⢿⣼⠁
⠀⠀⠀⠈⢧⠙⢶⣻⣿⣥⣄⡴⡀⢡⢄⣀⢀⣴⣿⣶⡿⡆⠀⡉⢄⠎⣿⠐⠀⢸⠛⢻⣇⡇⡐⠛⣽⣬⡆⠀⠺⢦⡀⢠⢸⢧
⠀⠀⠀⠀⠈⢷⡘⢯⡻⣽⡩⠑⠀⢀⢉⡻⣿⣯⣤⢼⡇⢸⠚⠻⣶⣺⣭⠎⢈⡎⠈⣏⣽⡄⣄⡀⡏⡇⠈⠀⠀⠸⣧⠘⠈⡎
⠀⠀⠀⠀⢀⡜⢻⡌⢷⣆⠙⠦⣄⡻⣸⣅⠻⠉⢉⢍⢠⣼⣷⡂⠀⢠⠏⠀⠜⠄⠀⢉⡟⠈⠋⠰⡅⠃⢀⠆⠀⠀⣿⠀⠀⡇
⠀⠀⡜⠛⠉⠹⡄⠇⣰⣿⣹⡆⠘⢿⣿⠹⠒⠒⢧⡘⠷⣽⠿⢃⣼⠃⢲⡜⢲⡶⣶⡟⣳⡖⠀⠀⢑⡷⢨⣄⡀⢀⡟⡀⠀⠁⡄
⠀⠀⠘⢁⠀⠀⢹⠀⠘⠿⠿⠷⠤⣀⡼⢷⡀⠀⠐⠧⣄⡀⢠⡞⢁⣴⠟⠁⣿⡙⡶⢑⣳⡬⡦⣺⠀⣶⡾⣏⡇⡸⢱⠇⠀⠊
⠀⠀⠀⠀⠀⠀⠚⠦⣀⠀⢉⣷⣴⣏⢾⣿⡷⠀⠀⠀⠚⣡⡋⠉⠀⠁⠀⢸⣿⣿⠁⡩⢖⣪⡶⢗⠈⠁⠰⣺⢿⡃⣾⡆⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠾⢿⣿⠀⢦⣰⡇⠀⠀⢀⡼⠁⠘⠠⡤⡄⠀⠀⠉⠛⠰⣤⠞⠁⣴⣿⠗⢛⡿⡍⢀⠛⣹⣷⡃⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢋⡁⠶⠿⠿⠲⡂⢝⣧⣛⣸⣏⢬⡷⠶⠶⡬⡢⡁⠠⣄⠘⠋⢁⣤⣾⡟⣡⣾⠟⠉⡤⢂⠞⡿⣿⠿⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠋⠙⠦⡄⠉⢰⡋⢉⡉⢰⣾⠀⣀⠸⠧⢇⠔⡀⣾⡄⢀⠻⣿⡟⣰⡟⢁⡞⡽⡵⠙⢉⡴⠃⡔⠙⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠸⣿⢟⣷⢞⡏⢸⠀⠰⢊⠵⣪⢴⣭⠳⣾⣦⢸⣿⠋⠀⠈⢨⡾⢁⠀⠀⠁⡨⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢧⠀⠀⠠⠩⠠⢀⠀⢸⢠⠐⡌⡄⡇⠹⣫⡾⢻⡿⠟⠁⠀⠒⠉⢁⣠⠄⣢⡢⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠨⠅⡀⠀⣀⡹⠄⠘⠉⣸⣟⣞⠀⡼⠁⡀⢫⠔⠀⠀⠀⠀⠀⠉⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠵⠋⡪⠖⢁⠔⠁⠀⠀⠠⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⠐⠠⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⣤⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣴⠟⠈⣧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢙⣶⣤⣤⣦⣤⣠⣏⠻⡶⠛⠁⠀⠀⠘⣧⡀⠀⢀⡤⢾⠁⠀⠀⠀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⣿⣷⡾⠛⠉⠉⠁⠀⠉⠉⠛⠳⣦⡀⠀⠀⠀⣀⣨⣿⣿⣏⣀⣿⣤⣤⡴⠶⠛⢹⡟
⠀⠀⠀⠀⠀⠢⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠶⠛⠉⠉⠉⠉⠉⠙⠻⢦⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠈⠻⣟⡓⠒⢶⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠿⢶⣿⣤⡤⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢿⣛⣛⣒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⢤⣷⣟⠉⠉⠁⣿⡇⣿⠤⣤⣤⣤
⠀⠀⠀⠀⠀⢀⣴⡶⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣾⣭⠿⠿⣭⢛⡶⠛⠛⢻⣷⢸⣿⣆⠀⠀⣿⡇⣿⣠⡶⠋
⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣄⡀⠀⣀⢤⣒⣶⣒⣒⠦⣤⡼⣧⠀⠀⠀⠹⣿⠁⠀⠀⢨⣿⢸⣧⢻⡄⠀⣿⡇⣿⠋
⣀⣤⣤⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣮⠞⠛⠋⠛⠷⣦⣹⢮⡿⠋⠉⠀⠈⠙⢷⣌⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡎⣷⠶⠿⣧⡻⣶⡦⠀⠄
⠈⠉⠛⢾⣿⠀⠀⣀⡤⣔⣲⣶⣬⣵⣒⡲⣴⣿⠁⠀⠀⠀⠀⠀⠈⡟⣾⠀⠀⠀⠀⠀⠀⠀⢻⣇⢿⡄⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⣿⡿⣧⣀⣠⣿⡇⣿⠤⠔⠒
⠀⠀⠀⠀⢻⡀⢸⣿⠛⠉⠁⠀⠀⠀⠉⢻⡇⣿⠀⠀⠀⣶⣦⠀⠀⡗⣿⠀⠀⠠⣿⣷⠀⠀⢸⣿⢸⡇⠀⠸⣷⡄⢠⣿⡀⣀⣿⡇⣿⠑⢬⣛⣛⣯⡾⢷⣄⡀
⠀⠀⠀⠀⠀⢱⣾⣸⣇⠀⠀⠀⢠⣄⠀⢀⣷⢻⡄⠀⠀⠻⣿⠀⠀⢹⣹⣆⠀⠀⠹⠟⠀⠀⣼⡿⣸⣿⣶⣾⡿⠿⠿⢟⡿⣿⣛⣣⡿⠀⠀⠀⠁⢸⡇⠀⠈⣛⣳
⠀⠀⠀⠈⠋⣿⡇⢣⣿⡀⠀⠀⠘⠋⠀⢼⣿⢦⠻⣦⡀⠀⠀⢀⣠⣿⢧⡙⢷⣤⣀⣀⣤⣾⡿⣱⡟⠓⠶⠖⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⢠⣿⣷⠶⠟⠛⠋
⠀⠀⠀⠀⢀⣿⣇⠘⡼⣇⠀⠀⠀⠿⠀⢀⣿⡇⣷⢬⣙⠻⠿⠿⣛⣥⠞⠉⠲⢭⣛⣛⣫⣵⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡋
⠀⠀⣠⣶⣋⣈⣿⣦⣣⢿⣄⣠⣤⣤⣶⣿⢟⣵⠏⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳⡶⠿⠛⢻⣿⠛⠛⠒
⠖⠛⠛⡉⠉⠉⠙⠛⠿⣬⣛⣛⣯⣭⣽⡶⠟⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢡⡇⠀⠀⠀⢀⣀⣠⣴⠾⠋⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣋⣹⣿⠋⠁⠀⠀⠀⠙⠿⣶⣄⣀⡀⠀⠀⠘⣄⠀⠀⠀⠈⠉⢉⣁⣴⠏⠀⠀⠀⠈⠉⣹⠏⠀⠀⠀⣀⢀⠀⠀⠹⣧
⠀⠀⢠⡶⠋⠁⠀⠰⠋⠉⠉⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠙⢷⣄⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠃⣴⣿⠿⠟⠛⠛⠛⠻⠷⢮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣤⣤⣤⣤⣴⡶⠟⠋⣰⡶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡵⠞⠛⠉⠉⠉⡿⣤⡶⠻⣿⣆⡀⠀⠀⠀⠀⣠⡄⢠⣿⣽⣯⣍⣩⡄⠀⠀⢰⡿⠀⠀⠙⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠈⢻⣷⠀⠀⣠⡾⠛⣷⡟⠀⠀⠈⡄⠉⠻⢿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠸⡗⣬⡛⠶⠖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡾⠃⠀⠀⠘⠀⠀⠀⠀⢇⠀⠀⠀⠙⢾⡧⠀⠀⠀⠀⠀⠀⠀⠿⢈⡥⠠⠞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠁⠁
⠀⠐⠒⠃

⠀⠀⠀⠀⠀⢀⣄⣠⣄⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⡸⣷⠏⠋⠙⡇⠀⠀⢀⡺⠛⡓⠓⠋⢷⠀⠀⠀⣠⠟⠟⠆⣄⠀⠀⠀⢸⡟⠁⠻⢿⡆
⠀⠀⠀⢰⣏⠁⠀⠀⢰⣿⠀⠀⢻⠑⠁⠁⢀⣼⣾⠃⠀⢰⠇⠀⠀⢰⢿⡄⠀⠀⢸⢧⠀⠁⠀⣇⡆
⠀⠀⠀⡼⣻⣧⣀⢰⣏⠟⠀⠀⢸⣦⣈⣄⣾⣭⡏⠀⠀⠸⠆⠀⣀⠀⣿⠁⠀⠀⢸⣞⣦⣆⣠⣿⠋
⠀⠀⠀⠀⠀⠙⠻⣿⢸⡆⠀⠀⢸⠳⣸⡟⡀⠛⠁⠀⠀⠀⢿⡀⠀⣺⡦⠀⠀⠀⠀⣿⡟⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁⠙