Octo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠎⠤⠤⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡓⠆⠀⠀⠀⡅⣠⣴⡞⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠱⠏⠀⢀⣤⣾⣾⠟⠁⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣱⠧⣴⣶⣿⣿⠟⠁⠈⡐⣋⠁⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⢟⣻⣿⣿⣋⣥⡴⡞⠟⠉⢠⠀⠀⢂⣠⡾⠃
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⠄⢠⣤⣠⣤⠶⠾⠛⣛⣉⣭⣶⣾⣿⠽⠛⢛⠩⠐⠁⠀⢀⠀⠈⣛⣿⣭⣁
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣼⣿⠶⠶⠶⠿⠟⠛⠛⠛⠛⠛⠛⠓⢊⣀⣀⣠⣬⣴⣶⠿⠛⠋⠉⠛⢉⠃
⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠟⠿⣿⣿⣿⣿⣷⠛⢉⣉⠤⠴⠶⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠿⣯⣥⣤⣼⣥⣀⣠⣤⡛
⠀⠀⠀⠈⠻⢿⣿⣇⠀⠀⠀⠀⠀⠛⣿⣿⡿⣀⠘⠫⠭⣐⡂⠤⣤⣀⣒⣿⣿⣿⣿⣷⣶⠜⣶⢿⡿⠿⠛⠛⠛⠶⠾⠿⠿⠿⠷⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠠⠶⠛⠋⠀⠀⠈⠦⠀⠀⠈⠉⠐⠚⠛⠿⠿⠿⢿⣟⣻⡿⢶⣾⣭⣶⣦⣤⡀⠀⠀⠀⢀⠀⠠⡀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣍⠙⡛⠿⣿⣷⣄⠀⠈⠒⠤⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢳⣤⣄⠀⠈⢫⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠸⣿⣿⡃⣠⣾⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⣸⡼⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠴⠒⠚⢀⠀⠀⢀⡀⣰⣷⣿⣆⡀⣀⡄⠀⠈⠲⠃⠀⠀⠀⠀⠀⠚⠉⢻⣿⣆
⠀⣀⠀⠀⠠⠈⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠺⠁⠀⠀⠒⠉⣿⣿⠉⢡⡿⠂⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⣰⣿⡿
⠀⠈⠀⠒⠠⢈⡁⠠⠐⠀⠀⠀⣀⣀⢤⣴⠺⣶⡀⠀⢀⡀⠀⣿⣿⣷⣾⣧⠀⠀⠀⠂⠐⢁⠀⠀⠀⠻⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠒⠰⠆⠀⠀⠀⠈⣇⠻⢚⣈⣁⣥⣶⣺⠈⠃⠀⠿⠿⠿⠿⠏⠀⠀⣚⡠⠰⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⢀⠁⠀⠈⠁⣀⠀⠀⣠⡄⠙⠜⠃⠙⠉⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠫⠀⠀⠀⠀⠀⠀⠀⠀⣀⢒⢐
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠀⣀⣴⡄⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⡆⢀⣦⠀⠀⠀⠀⠀⠘⠛⢧⠈⠢⠈⠀⠀⠀⠀⠀⠀⢄⡄⠀⠠⠄⠀⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⢨⣀⣂⣠⠄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠀⠀⠉⠀⠀⠀⠀⠄
⠀⢀⡀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠈⠙⠉⠉⠀⠈⠘⠂⠈⠀⠀⠐
⠀⠘⠃⠀⠀⣶⣆⣼⡿⠇⠀⠀⠀⢠⠀⣤⢠⡆⠄⠀⡄⡤⣠⠀⠀⢠⠤⢠⠤⢀⢰⠀⠀⠀⠀⠀⣠⢀⡀
⠀⠀⠀⠀⡀⠈⠉⠁⠀⠀⠀⠀⠀⢸⠦⡏⡾⣧⢠⢰⠀⠁⡇⡾⣿⢸⠆⢸⠆⢸⣿⠀⠀⠀⠀⠀⣤⡈⠁⠀⠀⢴⡀⢼
⠀⠀⠀⢀⢠⠀⠀⠀⢀⣠⡀⠀⠀⠸⠀⠙⠃⠸⠈⠸⠤⠛⠀⠁⠋⠈⠁⠘⠒⠈⠙⠀⡠⠂⠂⠀⠀⠀⠀⠊⠀⢰⡆⠘
⠀⠀⠀⠀⠡⠀⠀⠀⠈⠁⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠳⠺⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣿⣷⡆⠀⠀⣂⠀⠀⡄⠰⣠⠀⠀⡼⠀⠀⡴⠋⠀⠀⡀⠄⣴⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡌⠹⣿⠀⠀⠸⠛⠵⠀⠘⠲⠅⠀⡼⠁⡠⠊⠀⠀⠀⢠⣷⣄⠉⠀⠀⠀⠀⠙
⠀⠀⠀⢩⠄⠀⠀⠀⠒⢲⠞⣉⠀⠠⠂⠓⠄⠀⠛⠁⢠⣖⡅⠆⠀⠀⠀⠇⠀⠁⠈⣁⣤⡔⠀⢀⣾⣿⣿⠷
⠀⠀⠤⠀⠇⠀⢀⣀⠈⠂⠈⠀⠀⠀⣂⣐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⠀⠀⠀⢻⣿⣦⡿⢦⡄
⠀⢣⠀⠂⠀⠈⢙⡁⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣀⣼⣤⣿⣇⣸⣇
⠀⠀⠁⢀⠀⠈⠀⠉⠁⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⢠⠀⠀⢀⡔⠂⠁⠀⣰⡿⣟⣻⣿⣿⣿⣿⣿⣆
⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠂⠐⠃⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⣴⡄⠀⢀⠚⠉⠂⠀⠀⠀⢻⣽⣶⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠹⠇⠀⠀⡤⢀⠦⠀⠀⠀⣼⣿⣿⡿⠟⠻⣿⡿⠖
⠀⠀⠐⠁⠀⠁⠀⠀⠀⠑⡂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠞⠛⠁⠀⠈⠉⠁⠀⠀⠀⢾⣿⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⢀⠀⡄⡀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠇⡼⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠃⠂⠃⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⡿⢠⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⢠⣴⣶⣶⣿⣄⠀⠀⢀⣑⠦⣴⣿⠖⢋⡁⠀⡷⠃⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⢋⠏⡸⠈⠐⡌⢳⣠⣿⠉⡷⢸⣿⢠⣿⠙⣞⡏⢱⣤⡿⠋⢱⢓⡦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠁⠘⡀⡇⢀⡟⣿⠀⢀⣾⡇⢸⢿⣀⡿⣇⢸⠋⠳⠴⠃⢺⣇⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠫⢂⣼⠤⠥⠧⠊⠀⠈⡓⣍⠘⣧⣸⠀⢉⣀⡙⠋⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠈⠃⠀⠋⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠉⠀⠐⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢻⡄
⠀⠀⠀⠀⠀⣰⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⢀⠀⠀⣷
⠀⠀⠀⠀⠀⣿⣶⣄⠀⣤⣭⣉⠉⢰⣾⣏⠵⢖⣯⡀⠀⣾
⠀⠀⠀⠀⠀⡾⢻⣥⣮⣿⣿⣿⣷⢿⣯⠀⠀⣈⡀⠉⢄⡏⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢻⡾⣿⡧⠉⠉⢉⣡⣬⣟⣛⣋⣭⣷⣾⡟⠀⠀⠀⠀⠀⣠⡴⠛⠉⣀⣀⣀⡀⠉⠳⣦⠀⠀⠀⠀⠈⢯⢳⡄
⠀⠀⠀⠀⣠⢾⡃⣠⡄⢠⣶⣿⣿⡿⣿⣿⣿⠿⣟⠋⠀⠀⢀⣠⡴⠛⠁⠀⢀⣨⡴⠖⠶⠾⣵⡀⠈⠳⡄⠀⠀⠀⢸⡇⡇
⠀⠀⣴⠟⢁⡼⢡⠇⠁⣰⢯⠏⠻⡇⢆⠳⠽⣆⠻⠿⠿⠛⠋⢁⣀⣠⣤⣴⠿⠋⠀⠀⣤⢤⡀⠙⢦⡀⠙⢦⣄⣤⠞⣡⠇
⠀⣼⠁⢠⡞⠀⠎⠀⣰⣿⠋⣰⠀⠀⣾⠀⠀⠈⠛⠛⠿⢋⡉⠉⠉⠙⠻⡅⠀⠀⠀⠀⠈⠳⣍⠳⣄⠙⠶⠤⠤⠴⠞⠋
⠀⢿⠀⣼⠀⠀⠀⢸⡿⠁⢠⣿⠀⠀⢹⣦⣀⠀⡀⢀⣀⣀⣤⣤⡀⠀⠀⢹⠛⠓⢦⡀⠀⠀⠈⢧⠘⡆
⠀⠘⢧⣽⢀⠀⠀⠈⡇⠀⢸⢻⢸⡀⢀⡹⠿⠿⠶⢶⣒⠋⠉⠉⢙⣮⢧⠀⢧⢀⠈⢷⡀⠀⠀⣼⠀⡗⠀⠀⢀⣤⠴⠶⠶⢤⣄
⠀⠀⠀⠙⣧⢷⣄⠀⠹⣄⠈⠘⣮⠗⠋⠀⣠⣶⣦⣤⣈⠳⣶⢛⣉⣹⣯⡂⠀⠓⠿⣄⣹⡶⠞⠁⣸⣧⡴⠞⢉⣤⣶⣶⣶⣦⡈⢳⡀
⠀⠀⠀⠀⠘⢧⣉⠂⠀⠈⠉⠉⠀⢀⣴⡟⠛⠛⢛⣿⡾⣧⠸⡟⠛⠉⠙⢯⣶⣦⣤⣄⣤⣤⣤⡞⠉⢁⣀⣴⡿⠗⠋⠁⠀⠈⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠉⠳⠶⢤⡤⢴⡞⠋⠉⠉⠛⠛⠉⣠⣿⣿⡆⣷⠀⠀⠀⠀⠈⠉⠙⢿⡏⠹⣏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⠶⠞⠉⢸⣿⢃⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⣆⠀⠀⠀⢠⢞⡇⠀⠀⠀⠀⠻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠾⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡈⠳⠶⠞⢫⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣽⣷⣶⡲⢤⣀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⣿⣿⣿⣿⣷⣝⡻⢶⠍⠂⠀⡀⠀⠀⠀⠀⠀⠀⣴⠎⠉⠈⠙⣆⠀⠀⠀⠀⠀⠰⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣦⣭⣴⣄⡀⠀⠀⢿⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢇⡀⠀⢸⣿⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⡄⠠⢿⣿⢀⣾⣿⣿⠿⠷⠶⣦⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣘⣵⣿⣿⠋⠈⠓⠲⠤
⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢀⣤⣦⣤⣄⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⠊⠀⠀⠀⣴⣾⣿⣷⣮⣙⠫⢄⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⠀⠀⠀⣾⠋⠈
⠀⡼⠁⠀⠀⠀⢸⠋⠉⠀⠀⢸⡟⠻⣶⣮⣍⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⢻⣿⣿⡇⠀⠀⠛⠀⠀⢠
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠙⠛⠛⠛⢿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢣⡀⠀⣿⣿⣇⠀⠀⠀⠐⠀⠈⠓⠲⠶⠤⡀
⠈⢿⣄⠀⠀⢀⠀⢢⣴⣶⣤⡀⠈⠑⢄⠀⠀⠀⠀⠀⠸⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠿⣿⣿⣧⠱⠀⠘⢿⠿⠦⠄⠀⠘⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠛⠿⠶⠿⢷⡄⠀⠙⠻⣿⣷⣦⣜⣀⠀⢀⣀⠔⣁⣾⣿⡟⢼⣿⣿⡄⢿⡀⠀⠀⠀⠀⠀⢻⣿⡇⠁⠀⠀⠁⠂⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⡌⠻⣿⣿⣷⣦⣥⣶⣿⣿⣿⡿⠁⠈⠻⣿⣷⣦⡝⢄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣠⣦⡶⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠢⠀⠈⠙⠻⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠈⠻⢿⣿⣶⣭⡢⡀⠀⠀⢹⣿⣿⣇⠀⠀⠺⣿⡄⠀⠀⠀⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣽⡆⠀⠀⢿⣿⣿⡄⠀⠀⡙⠻⠀⠀⠀⢹⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠸⡆⠀⠀⠀⠀⠀⠀⠈⠉⡆⠀⠀⢀⣠⡀⠠⣄⠀⠀⠈⢻⣿⡟⠀⠀⠈⠻⢿⣿⣿⣿⣷⡄⠀⠀⠀⠈⠙⠒⠋
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠠⠇⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠟⠋⠀⠀⠙⣧⠀⠀⣸⣿⣿⠁⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠚⠃⢀⠀⢸⡟⠀⠰⣟⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢽⣷⣹⣟⣷⠿⠟⠊




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣎⡷⠀⠀⠀⠀⡠⢄⠀⠀⣀⣀⣀⠀⠉⠡⠑⣄⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠐⠄⠀⢀⣤⢻⢺⡖⠋⠉⠀⠀⠈⢀⡀⣈⣸⢋⠴⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆⠀⠘⠧⠑⢊⡍⡽⠀⠀⠀⠀⠀⠀⠀⡟⠨⢹⡜⠶⠞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣂⢄⢀⠀⠁⢀⠄⡤⢄⠉⢙⠇⠀⠀⠀⠀⠈⠀⣀⣧⠀⠈⠟⡶⢦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡏⠻⠉⡠⠀⠀⠁⢰⡀⠀⡰⠚⠀⠀⠀⠉⠀⣀⣴⡿⢀⠛⠁⢀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠀⠼⠷⠀⠸⣂⢀⣉⣨⡍⠀⠀⠀⣠⢄⢈⣿⡿⠁⠀⠀⢤⠊⠀⢐⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⣶⢠⠄⠐⢙⢿⣿⡟⠁⠀⠀⠀⣠⣾⡿⢛⠀⠀⠀⠀⠀⠱⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡹⠀⠐⢁⣾⡿⠟⠀⡀⡀⠀⠀⠙⣩⡷⠎⠁⠖⠤⠔⠀⠀⡈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⡀⠀⠉⠀⠀⣀⣀⢀⣼⠞⠀⠀⠀⠀⠙⢏⠐⠦⣠⣄⢀⠄⠀⠑⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠬⡉⣿⠛⣿⣉⠝⠋⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⠈⠫⡒⠐⠈⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⡨⠋⣹⠖⠋⠀⠀⠀⣀⣀⣀⠀⠃⠀⠀⠀⠐⣀⠠⡀⠈⢆⠒⠋⠁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢃⠜⠁⢀⣤⠴⢚⠫⠉⠱⠀⣱⣀⠀⠀⠀⠀⠙⢃⡄⠀⠀⠐⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡅⠀⠀⠶⣏⡀⠀⢀⠀⠹⣁⡈⠁⢸⠂⠀⠀⠀⠀⢈⣇⢰⣄⣀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠀⡀⠀⡈⠉⠉⠉⠉⠑⢦⡀⠀⠠⠀⡀⠚⢦⠀⠈⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢰⣀⠨⣈⡵⠐⣆⣀⠀⠀⠀⠈⠉⠀⠠⣀⣂⠾⢠⡴⢁⡯⠔⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠒⣃⠀⠤⠰⠀⠀⣠⠀⠂⣸⡇⠐⢠⡇⠈⢃⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠧⣀⡠⠀⢀⠈⠉⢄⣠⡌⢉⠙⡟⠀⠄⡽⠛⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⣡⡀⠔⠀⠀⠀⠀⠑⠦⠔⠙⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠀⠀⠀⣀⡠⠄⢀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠸⠁⠀⠀⠀⢹⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⢠⡀⠀⡄⣿⣀⡐⣲⣿⣿⣦⣶⣾⣿⣿⣽⣿⣭⣤⣶⣤⣤⣿⣴⣶⣶⣄⠀⠸⠓⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢀⣀⣰⡇⣴⣧⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⡟⣿⣷⣄⡀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹⣦⡂⢻⣿⢳⣿⣿⣿⣏⣹⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣿⣿⡇⣿⣿⣿⡿⢋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣻⣿⢸⣿⣾⣿⣿⣿⣿⡏⣿⣿⣸⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣸⣿⣿⡿⣿⣿⣿⣤⣾⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣹⣷⣿⡟⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣾⣿⡿⠉⢙⣿⡄⣿⣻⣿⣿⣿⣿⣿⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⣭⣿⣿⣧⢿⣿⢹⣿⣿⣿⣿⣌⢿⡿⠋⣿⣿⣿⣿⣿⣿⣧⡻⠿⠾⠟⢡⣾⣿⣿⣇⣿⣇⢿⣿⣿⣿⣏⢹⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢾⣿⣿⣿⢸⣿⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠘⡇⠈⠁⠀⠀⠀⠀⠚⣻⣿⣿⡿⣿⣿⠼⣿⣿⣿⣿⡄⠈⢻
⠀⠀⠀⠀⠀⠀⠀⣄⣾⣤⣾⣿⣿⣿⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣧⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠻⠛⡁⠀⣩⣴⣶⢻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⢠⠀⠀⣿⣿⡷⣴⣿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⠇
⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠈⠛⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠘⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠸⡋⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠞⠻⠛⠁⠛⠉⠀⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⢙⣷⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣄⡙⠀⢁⡀⠀⣴⡀⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠟⠛⠉⠐⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉

⠀⠀⠀⠠
⠀⢀⠘⣅⡴⢀⠀⠀⠀⢂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠦⣼⡡⠊⠀⠀⠀⢹⠎⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠐⠿⢶⣿⣶⡦⠐⡄⠀⢀⠀⠆
⠀⠀⢰⡟⠀⣴⣁⣄⣠⣾⣀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢸⠀⡟⢡⡆⠀⠑⠲
⠛⠻⣾⡇⣼⠗⠋⣩⡿⠉⠉⠁⠀⠛⠛⠛⠀⠶⢶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⣄⠸⡀⢹⡷⠚⠀⠀⢀⢀
⠀⠀⠸⣿⡟⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⣾⣆⠀⠀⢃⡴⠯
⠀⠀⠠⢹⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⣰⣿⣿⣆⢠⣿⣿⣆⠀⠀⠀⠀⠙⠛⢿⣷⣶⣦⠄⠈⢿⣄⣸⡟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏
⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡔⣿⡿⠿⢿⣿⣮⣿⡏⠉⣿⣿⠉⠉⣿⣇⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⠖⠀⠀⢠⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣷⣶⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⣿⣿⡇⠚⠋⠁⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⡟⠛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⢻⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣇⣀⣼⣿⡦⢴⠄⢴⣠⡆⠀⡄⢰⣿⣿
⠀⠀⠀⢸⣿⣿⣧⣴⣄⣀⠀⣤⣀⢸⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇⢸⣴⣾⣿⣿⣿⣇⣾⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡎⠙⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠲⣇⡀
⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣶⣧⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⢀⡠⢐⠊⠃⠀⠀⠀⠀⠀⣷⣾⡄⢀⠴⠊⠀⠀⠀⠀⡐⠒⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⡀⠄⠀⠀⠀⣿⡏⠓⠁⠀⠀⡀⡀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣿⠀⠗⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠠⠔⠈⠋⠒⠐⣿⠄⠐⠈⠀⠠⠄⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠀⠀⠀⠤⠂⠈⠓⠀⢼⡯⠠⠐⠉⠀⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢺⡗⠐⠁⠈⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠤⠠⠠⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⡌⠀⢀⡤⠀⠀⠀⠀⠀⠀⡀⠄⠠⡤⠀⠒⠂⠈⠁⠅⢠⠊⠀⠀⠈⢀⠰⠂⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠈⠀⡜⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢰⠀⢠⠃⠀⠀⠀⠀⠐⣀⠎⠀⠀⠀⡇⢀⡀⠀⡄⠀⢰⠃⢀⠔⠐⡀⠀⡆⣀⡘⠀⡰⠀⢸⠀⠀⡁⠈⠁⠀⢠⠁⠰⠁⢂⠀⠀⠀⠀⡆
⠀⠀⠀⡀⠀⡆⠀⢀⠊⠁⠄⠀⠘⠀⠀⡴⠚⠉⠀⢠⠀⢁⠀⠆⠀⠆⠀⠀⡇⠀⡃⠉⠀⠊⠀⠀⢸⠀⠀⠀⠀⠑⠀⡈⠀⡀⠀⡘⠀⠀⡠⠀⢀
⠀⠀⠀⡇⠀⡇⠀⠘⠀⠀⠘⠀⠀⠃⠰⠂⠀⠀⠀⠘⡀⠈⡀⠃⠀⠀⠀⢀⠃⠀⡇⡤⠀⠀⠀⠀⡀⠀⢰⠂⠀⠈⠀⡇⠀⠈⠉⢀⠠⠚⠁⠀⢸
⠀⠀⠀⡇⠀⡇⠀⠰⠀⠀⢠⠀⠀⢰⠀⡀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠁⠀⠊⠀⢠⠁⠁⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⠰⠀⠀⠇⠀⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠃⠀⢡⠀⠀⠁⠠⠎⠀⠀⢘⠀⠁⠀⠊⢢⠀⢰⠀⠀⠣⡀⠀⠀⠀⢀⠼⡀⠘⠚⠀⠀⡸⠇⠀⠈⠀⠀⢄⠆⠀⡌⢰⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠸⠀⠀⢢⡀⠀⠀⠀⠀⠀⡎⠐⠄⠀⠀⠀⠃⠸⠄⠀⠂⠑⠀⠀⡤⢋⠀⠧⠠⠀⠀⠊⠸⢀⠀⠀⠀⢀⡞⠀⠰⠀⠈⠀⠀⢠⠀⠀⠀⠇
⠀⠀⠀⠀⢃⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⢠⡄⠀⣐⠀⠀⢀⣀⠀⣀⢡⣋⡠⠄⢄⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠸⠀⠀⡘
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠉⠀⡰⢛⠁⢠⢂⠅⠉⠉⠀⠀⢄⠀⢦⠀⣠⣒⣁⡀⢒⠁⠠⠀⡀⠀⠀⠁⠀⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢄⡘⢉⢀⡀⣉⠀⢠⠄⠾⠁⠀⢀⠃⠉⠀⠀⠀⠈⡀⠀⠂⠀⡆⠀⠀⢀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠰⠋⢀⠉⠈⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⢠⠀⠀⠀⠀⠸⠀⢸⠀⡀⡀⠀⠉⠀⠁⠉⠁⠈⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠠⢇⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⡘⡜⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠘⢠⠇⠈⠄⠂⠈⡄⠀⠀⠀⠀⢠⠃⢀⠃⢠⠁⠀⠺⠘⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠈⠀⠀⠈⠂⡀⠘⡄⠀⠘⠠⠃⡠⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓⠊⠀⠊⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠂⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣦⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣠⣤⣄⣠⣿⣦⣅⣤⣤⣄
⠀⠀⠀⠀⢸⣿⣿⣍⣿⣿⣯⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣷⣿⣇⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣻⣿⣿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡉⠙⠋⠉⠛⠉⣹⣿⠀⠀⠀⠀⠀⠀⣴⣿⣤⣥⣬⣤⣿⣧⡀⠀⠀⠀⠀⠸⣿⣍⡛⠛⠿⠿⠛⢋⣹⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣷⣾⣿⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠻⠿⢿⣷⣾⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶
⠀⠀⠀⠀⣴⣶⣶⣾⣿⣿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣿⣷⣾⣿⣷⣦
⠀⠀⠀⠀⠻⣿⣿⢿⣿⣿⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⡟⢿⣿⢛⣏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⢀⣿⣿⡿⠋⠙⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠟⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⡿⢿⣿⣿⣇
⠀⠀⠀⢰⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⣆⣠⣤⣠⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠙⠉⠉⠛⠛⠛⠉⠁⠉⠉⠀⠀⠀⠀⠀⠘⢿⣿⡿⣿⣿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣂⣽⣀⡀
⠀⠀⠀⠀⣰⣾⣿⣾⣿⣿⣿⣾⣿⣦⠀⠀⠀⠀⠀⠀⣴⣶⣶⣴⣾⣦⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣿⣿⣧⣈⣿⣿⣏⣠⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣅⣨⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣽⡇
⠀⠀⠀⠀⢹⣿⠉⠉⠛⠋⠋⠉⢹⣿⠀⠀⠀⠀⠀⢻⣿⠏⢻⡿⠿⠿⠛⢻⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⡯⢸⣿⣿⡀
⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣾⣿⠃⠀⠀⠀⠀⠀⠘⣿⣧⣤⣤⣤⣾⣤⣾⡿⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠉⠉⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⠟⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⢀⣀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠘⣿⣆⠠⡆⠤⠖⠉⢀⣼⡟⠛⣦⡀
⠀⠀⠀⠀⠀⣠⡤⠶⣤⣀⠀⢹⡿⣤⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣼⡿⠈⠀⢀⣠⣾⠿⠋⠀⠀⢸⡓
⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⣆⠀⠱⡽⣧⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⣰⣿⣿⡇⠀⣰⣿⠟⠁⠀⠀⠀⠀⡰⡯
⠀⠀⠀⠀⠀⠳⣤⡀⠀⢘⢻⠀⠀⢳⣿⠆⠀⠀⠀⢿⣽⣿⣿⣶⣤⣴⣿⡿⣿⠃⣤⢹⡏⠀⠀⠀⠀⣠⢞⡵⠁
⠀⠀⠀⠀⠀⠀⢀⡥⠀⡼⡾⠀⢀⢿⣿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⠃⠀⢻⡄⢳⣀⣠⡴⢋⠵⠊
⠀⠀⠀⠀⠀⢔⠉⠀⣰⣱⠃⢀⣾⣾⠇⠀⠀⠀⠀⠀⢠⣿⣿⡟⠛⢿⣿⣧⠀⠀⠀⠹⣦⣈⠻⣾⠁
⠀⠀⠀⠀⠀⠈⠀⣰⣿⠃⢠⣶⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⣠⣾⣿⣾⠁⠀⠀⣰⡏⢹⠷⣤⡙⠶⣄
⠀⠀⠀⠀⠀⢀⡼⣿⠋⢠⣿⢿⠏⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⡄⢽⠀⠈⠹⣆⠙⢧⡀
⠀⠀⠀⠀⢀⡟⣵⠃⠀⣼⢻⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⣷⡄⠘⢿⣄⡓⠤⣄⣸⣧⠀⢷
⠀⠀⠀⠀⠶⣼⠃⠀⢶⡏⣾⠀⠀⠀⠀⠀⠀⢠⣿⣹⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⢷⠀⠀⠙⠻⠷⢦⣮⣿⡀⢘⣷⣄
⠀⠀⢀⣛⢱⡏⠀⠀⣾⡇⢿⡀⠀⠀⠀⠀⣠⡟⢠⣿⣷⣾⡿⠿⠿⠿⢿⣿⣿⣿⣺⣧⠀⠀⠀⠀⠀⢀⡟⠀⣸⣿⡈⢳⡄
⠀⠀⢠⣅⣿⠁⠀⠀⢹⣿⠙⣷⣄⣀⣠⣼⣿⣿⣿⣿⡭⢶⣞⣿⣿⣿⣶⣦⣌⡙⠻⢿⣦⡀⠀⠀⣠⠞⠁⢠⠆⠈⣿⡀⢶
⠀⠀⢰⠃⣿⠀⠀⠀⠀⠰⠞⣆⣩⡉⢫⣭⣖⢾⣫⣽⣷⠏⣽⣿⣿⣽⣟⣿⣿⡿⢇⣤⢪⡽⣿⣟⠁⢠⣴⠛⠀⢀⣿⠃⣿⠁
⠀⠀⠺⡗⣿⡄⠀⠀⠀⣠⡾⠟⠛⠿⠗⢸⣯⣿⣿⠟⢁⣼⣿⣿⣿⣷⠹⣭⡽⣷⣮⡙⠻⢾⣿⣿⣿⣏⣀⣤⣶⡿⠃⣼⠏
⠀⠀⠀⢹⣛⣷⣄⢀⡾⠋⣀⠾⠏⠊⣀⣴⣿⡿⠋⣀⣺⢫⣿⣿⣿⣿⣷⠘⣷⣦⠙⢿⣧⣦⣬⣍⣉⣙⢛⣋⣥⣴⠛⠃
⠀⠀⠀⠀⠸⠥⣽⡿⠁⣴⣶⣶⣶⣿⡿⠟⣉⣤⣾⡟⢁⢬⣿⣿⣛⣿⣷⣗⡈⠻⣿⢆⡉⠛⠿⣿⣾⣿⣉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⣴⣿⢯⡭⢥⡄⠶⣪⣿⠿⠋⣰⡧⠛⠉⠀⠀⠀⠈⠛⠿⣄⠙⢿⣿⣶⣦⣠⡍⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⡀⣾⠀⠀⣀⣤⣴⡾⠛⣁⡴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠻⣿⣆⠈⠋⠹⠦⢮⣉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠘⢷⣎⡷⢿⣟⣫⡥⠴⠎⠉⠀⢀⡴⠓⠺⢷⣦⡀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣆⠀⠀⠀⠈⠹⣆⠈⣷⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣻⣷⣿⣿⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠹⣝⢄⠀⠀⠀⠀⠀⠀⠘⢳⠘⣿⡀⠀⠀⠀⠀⠈⡧⢸⡇
⠀⠀⠀⠀⣰⡿⢣⡞⠋⠀⠈⣿⡁⠀⠀⠀⣀⣀⡀⢝⠳⢄⠀⠀⣸⡞⣿⠿⢷⣶⣄⡀⠀⠘⡇⣿⡇⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⣿⢠⡏⠀⠀⣠⡾⠏⠀⡠⣞⠽⠛⠙⣿⠀⣇⣼⠀⣼⠟⢷⢻⡀⠀⠉⠻⣿⣦⣘⠗⣽⡇⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⢻⣌⣧⣀⢠⣿⣤⣔⡾⠞⠁⠀⢀⣤⠋⠀⠀⠀⢰⡟⠀⠘⣆⠳⣄⡀⠀⠀⣹⡟⣡⣿⣤⢠⡤⠶⣛⡶⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⡄⠀⠀⣰⠏⠁⢀⣀⠀⠀⠸⣧⠀⠀⠘⠳⣬⣉⠚⢛⣡⡴⠋⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣀⢀⡿⠀⠀⠘⠢⠒⠉⢬⠀⠀⠀⠑⠷⣀⣀⣀⣈⣭⣍⠉⣡⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠹⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢓⡄⠀⠀⠀⠀⣜⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡼⢁⣀⣀⡀
⠀⠀⠀⣠⠖⢋⣉⣉⣄⣉⣉⡧⠀⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡇⠀⠀⠀⠀⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⣀⠤⡇⠠⣏⣉⣁⣌⣉⣉⠓⢦⡀
⠀⢀⠞⣡⠞⠋⠁⠀⠀⠉⠉⡇⠀⣷⡀⠉⠢⣀⠀⠀⠀⠀⠀⢹⣌⠀⠀⠀⠀⠀⢈⣾⠁⠀⠀⠀⠀⢀⡠⠊⠁⣰⡇⠀⡏⠉⠀⠀⠀⠉⠛⢦⡙⢆
⠀⢸⢰⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⢩⠹⢆⡀⠈⠑⠤⣀⣀⡠⢎⠁⠀⠀⠀⠀⠀⠀⢙⡦⣀⣀⡠⠔⠉⠀⣠⠾⢩⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⠈⠄⢇⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠳⢌⣙⠲⢤⣄⣤⡤⠞⣱⠀⠀⠀⠀⠀⠀⠀⢰⡙⠦⣄⣄⣤⠴⢚⣁⠴⠃⢀⡾⠁⠀⠀⠀⠀⠀⠀⢀⠏⡌
⠀⠀⠈⠘⡄⠀⠀⠀⢀⣠⠤⠤⠼⠷⣤⣀⢈⠉⡉⢁⣀⣤⣶⠃⢸⠀⠀⢀⠀⠀⢸⠀⢳⣦⣄⣀⠉⡉⢉⢀⣠⡴⠿⠤⠤⢤⣀⠀⠀⠀⠀⡜⠈
⠀⠀⠀⠀⡇⠀⢀⡴⢋⡤⠖⠒⠲⢤⣀⠉⠻⢯⣙⣉⡩⠔⠁⠀⣽⠀⠀⢸⠀⠀⢸⡅⠀⠑⢬⣉⣙⣫⠿⠋⢁⣠⠴⠒⠒⠦⣌⠳⣄⠀⠀⡇
⠀⠀⠀⠐⠀⠀⡞⢰⠋⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⢀⣀⣠⡤⠛⢹⠀⠀⢨⠀⠀⢨⠙⠣⣤⣀⣀⢀⣀⣠⠖⠋⠀⠀⠀⠀⠀⠈⢳⠘⡆⠀⠐
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⡰⠃⠀⢠⠏⢧⠀⠀⠳⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⢀⡴⠋⠀⠈⠳⣄⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜
⠀⠀⠀⠀⠀⠀⠀⠈⠪⢣⠀⠀⠀⠀⠀⠀⣠⠤⠚⠉⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⢠⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡴⠋⢁⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⠉⠳⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣶⣤⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⣶⣦⡀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢿⣶⣶⣾⣯⣀⡀⠀⣰⣾⣿⣿⠟⢀⣴⣶⣿⣿⣿⣶⡄⠀⠆⢀⠙⢿⣆⠀⠆⠀⠀⠤⠀⣀
⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⢿⠿⣿⣧⠈⠛⠛⠋⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠊⣀⠿⣷⡀
⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⣿⣿⣶⣤⣤⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣄⣠⠴⠒⠉⠹⠦⠒⠂⠉⠉⠽⠦⠐⠒⠒⠠⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠀⢀⡀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠧⠒⣲⣤⠽⠟⠻⢷⣄⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡄⠀⠀⠈⠚⠠⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠐
⠀⠀⠀⠀⠀⢀⣤⣀⣤⣄⠀⠀⠀⢰⣦⣶⡤⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⢰⣾⣿⣿⣿⡿⠂⠀⠀⠀⠀⣀⡤⠞⠁⠀⢠⠊
⠀⠀⠀⠀⠀⣿⣿⡿⠀⠈⠻⣶⠛⠙⠿⠛⠡⠀⠀⠀⠀⠠⠉⢿⡿⣶⣤⣄⣈⣉⣉⣉⣩⣤⣤⣶⣶⡿⠟⠉⡀⠀⠲⡀⣿⣀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢾⠀⢠⡾⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠠⠠⠄⢤⣤⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣦⡿⠁⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠘⠒⠂⠛⠀⠀⠈⠂⣀⠀⠁⣤⠀⠒⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠹⣷⡔⣿⠉⠉⠉⠁⠀⢀⠀⡀⠀⣤⠀⠀⠳⠀⠉⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⢻⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠼⣿⡇⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⡤⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⡨⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢹⣿⡁⠀⠀⠀⠀⠀⢀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠛⠺⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡘⢀⠄⢰⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣷⣿⣿⣿⣿⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣷⣮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠏⠀⠀⣤⣿⢈⣴⣲⡜⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣟⣯⣭⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡄⠀⢀⣾⣿⢡⡴⢉⣨⣿⢸⣷⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣟⡜⣿⣯⠖⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣾⠃⠀⠰⠀⠀⠀⠈⠙⣄⣸⢸⣿⣇⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⢛⡭⣺⠷⣌⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⣤⣼⡇⠀⠀⠀⠀⠀⠈⠱⣶⣾⠿⢮⣽⣾⠟⠓⠲⠬⣭⣶⣖⢹⠟⠸⣏⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣶⣶⢺⣿⣿⡇⠀⠀⠀⣀⠀⠀⠀⠉⠻⠦⠀⠨⣉⣓⠶⣖⣫⣵⡿⡟⢀⠄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣤⠶⣿⣿⣧⡄⠀⣴⠸⣧⡀⠀⣦⡀⣦⣦⣄⣈⠙⠛⢛⡻⠭⠊⠠⠏⢹⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⣠⠏⣿⣧⣿⣳⣾⣷⣙⣿⣿⣿⣿⣿⣷⣝⠇⢀⠞⣁⢀⠄⢠⢹⣶⠃⢀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠋⣠⣾⠟⢹⢸⣿⣯⢿⡵⣿⣿⣿⣮⣿⢿⣿⠅⠀⠉⢣⢸⣀⡉⢀⣀⠿⢿⣦⡶⠋⢀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢀⣾⡿⠃⠀⡿⣸⡿⠈⣧⢻⣌⢿⣿⣿⣮⢷⣌⠳⣄⠀⠀⠁⢳⠶⢿⠿⢿⡄⠙⢿⣶⢈⡄⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡟⡾⠀⠀⣰⢣⡿⠁⢀⣿⣎⣿⣯⣿⣿⣯⡊⠙⣧⡹⡷⢦⡀⠀⡙⣯⣹⣵⣽⠀⢸⡷⢘⠇⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢸⣇⠳⢴⣞⡵⠋⠀⠀⠸⣿⢻⣿⣿⣏⠀⠐⠃⠈⠋⠂⡘⢾⣷⠀⠛⢧⡁⣰⢏⡴⣫⠦⢉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠉⠙⠋⠁⡄⠀⠀⢠⡼⢋⣯⣿⡁⣠⠔⢊⣿⣿⣷⡀⠉⢠⠃⠀⢠⣿⣯⠶⠥⠤⠤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣈⣘⠐⣀⠚⢈⣵⣶⣿⣇⣿⣿⠋⠀⣡⣆⢾⠟⣿⣿⡇⠘⡠⡣⢀⣼⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡽⣞⡿⠿⠿⠿⠿⢿⣿⣿⣯⡃⢰⣿⣯⢪⡤⠊⠁⢐⡠⣾⢃⣼⣭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠉⠀⠀⠀⠀⠀⠒⠛⡻⠿⡷⢡⢈⠯⢚⣥⠶⡢⣴⡿⠟⣱⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡤⠔⠒⠒⠢⣴⢳⡤⠚⠒⠛⢲⣿⡟⢡⢞⡭⠐⠉⢀⣉⣰⡾⠿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⡆⣴⡿⠻⣷⠈⢿⢠⡆⠀⠀⣸⣿⣇⢻⠋⣤⡑⢶⣿⣿⣿⡋⠀⢨⣿⣌⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡿⣸⣟⠛⠧⠇⡄⢀⣼⣿⡇⠀⣸⠉⢿⣿⣄⣉⡝⠋⡀⠻⢻⡽⠒⠓⠲⣌⣿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢷⣕⠏⠫⠤⢊⣤⣾⢣⡿⠀⣼⢁⡴⠾⠟⠃⢐⣸⣅⡀⠖⣀⠙⢦⣄⣠⣿⡿⠏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⣶⠾⠿⠋⣿⢿⣇⢸⣇⣼⣀⠊⣀⣈⣹⡇⢸⣿⣧⣁⡘⠄⡄⠉⢁⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣾⣷⠸⣅⣐⣠⢯⡴⠁⣸⡇⠘⡏⢽⠉⠓⠶⠳⠤⠟⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣿⣿⢿⡄⠙⠯⠥⠞⢁⣴⡿⠀⠀⠹⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡼⠏⠻⣝⣶⣿⣶⣾⡿⠟⠁⠀⠀⠀⠈⠉
⡀⣀⣀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄⠀⠀⢀⣀⠀⠴⠀⠀⠀⠀⠀⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⡺⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡞⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠁⠒⠀⠀⠀⠀⠀⣠⣶⣶⠀⠀⠀⢉⡠⠊⠁⡑⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠇⠀⠀⠀⠘⡀⠀⢠⡄⠀⠈⠛⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⡃⠀⠀⠀⠀⠉⠛⠛⠿⠛⠛⠿⣿⣿
⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠇⠀⠀⣠⠀⠀⢀⡀⠄⠀⠀⠠⠺⠀⢠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠄⠀⢸⣷
⣿⣿⣿⡿⠁⢀⠀⠀⠀⠀⠀⠆⠀⠀⡤⠈⠀⠀⣸⠀⠀⠈⠀⠠⠀⠀⠸⡆⠀⠀⢃⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⣸⣿
⣿⣿⣿⠁⣰⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠀⠀⠀⢘⠀⢉⠀⠘⢃⢣⠀⠀⢢⡠⢰⠀⠀⠀⠀⠀⢀⣰⣿⣿
⣿⣿⡇⢠⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⠀⡘⠀⠴⠀⠈⠸⡇⢀⠀⠀⠳⡌⠀⠐⣦⣶⣾⣿⣿⣿⣿
⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣲⣷⡄⠀⠀⠀⠈⠲⣾⣿⣿⣯⣭⠉⣿⣿
⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠠⡆⠀⠀⠀⠄⠁⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁⠚⠛
⣿⣿⣧⢘⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⢻⣿⣶⠙⣿⣿⣿⣦⡀
⣿⣿⣿⡌⢿⡿⠛⢉⣀⣀⣀⣤⢀⣾⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⢠⣿⣿⣿⣿⣿⣶⣄⡈⠀⠀⠄
⣿⣿⣿⡷⠀⠰⣾⣿⡟⣿⣿⣿⠸⡟⠇⠀⠀⣿⣇⠀⠀⠀⢸⣷⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣶
⣿⣿⣿⡇⣼⣷⡶⠽⣷⣿⣿⣿⣷⣤⠄⠀⢰⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⣾⡏⢠⣶⣮⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣷⡀⠀⠀⠸⣿⣿⣿⣿⣾⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣛⣷⣿⡇⢸⣿⣿⣿⣿⣿⡟⠁⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠈⠛⠿⠿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡠⢛⢟⡛⡟⠉⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢄⡀⡀⠠⠤⠤⠶⠶⠒⠲⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⡞⠋⠙⡄⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠂⠀⢥⠀⠀⠀⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⡶⠀⠀⠀⠀⠘⣄⡀⡄⠀⠀⢀⢴⠁⠀⠀⠀⠈⠙⠲⣤⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠀⠀⠀⠀⠤⠂⢹⡀⡇⠀⣰⡟⠈⡇⠀⢠⠄⠀⠀⡷⠛⠁⢠⣇⣤⠶⠒⢚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣀⣀⡀⣀⠀⠸⣷⣧⠞⣿⡇⠀⢰⢀⠎⢣⠀⠀⣷⠆⠀⣾⣩⠤⠶⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠔⠒⠉⠉⠁⠀⣀⠼⠀⠀⠿⠉⠀⠙⠁⠀⠰⠟⠃⠈⢇⠀⡇⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣆⣤⡾⣶⡏⠉⣽⠿⣦
⠀⠀⠀⠀⠀⣀⣠⠤⣤⡤⠴⠒⢦⣰⠦⣄⡀⠀⢀⠤⠰⡀⠈⠙⠒⠦⠤⢤⣤⣀⣠⠼⠛⠛⠉⠉⢁⠀⠈⠁⠀⢿⡀
⠀⢤⠄⠀⠉⠀⠀⠀⠀⡠⡄⠀⠘⣏⠀⠀⠈⠉⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⠤⢾⠋⠔⢊⣻⣤⢀⡞⠀⠀⠀⠀⣀⡽⠂
⠀⢿⡄⠀⠀⡤⠀⣸⠁⠰⠇⠀⢶⣿⠀⠀⣷⠰⠷⣤⠠⠁⠀⠀⠀⠀⣠⣿⣆⢺⣣⠔⢋⡿⠀⠘⠷⣶⠶⢶⡿⠉
⠀⠀⢿⠖⣺⠃⠀⢉⠄⢢⠀⠀⠈⢧⠀⠀⢀⣤⡀⠟⠀⠀⠀⠀⢀⡾⢋⡴⢫⣿⠀⣠⠎⠀⠀⠀⢠⠇⠀⢸⡇⠳⠠
⠀⠀⡠⢴⠏⠀⠀⣸⣦⠊⠀⠀⢀⣴⠁⠀⢻⡀⠀⠀⠀⠀⠀⢠⣾⣵⡏⣀⣀⣉⣫⣅⣀⠀⠀⢀⣸⡤⢄⣸⡇⠐⢚⠂
⠀⢸⠃⠘⠀⠀⠀⠀⠀⠀⠀⣴⡿⣟⣀⣀⣈⡇⠀⠀⠀⠀⢰⡏⠀⠀⠉⠉⠁⠀⠀⠀⠈⠛⠻⠟⠉⠀⠀⠈⠳⣄
⠀⡾⠀⢀⣸⠛⠛⠛⠛⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⡧⣤⡀⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣀⣀⣀⣸⣿
⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⣿⠏⠉⣹⣶⡄⠀⠀⢀⣴⠞⠛⠛⠉⠉⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠶⣄⠀⠈⠛⠒⠘⠿⢿⣴⣠⠶⠿⣷⡀⠀⠀⠀⣀⣿⡃⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠉⠀⣀⣸⡟⠁⠀⠀⠀⠀⠀⠘⣿⡷⢶⣄⠈⠁⢀⠔⠋⠁⠈⠉⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠷⠤⠖⠶⠾⠃⠀⠀⠀⠀⢀⣀⣀⣶⢿⣇⠀⡟⢧⠀⠻⣆⠀⠀⢀⣀⣼⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡶⠛⠉⣉⣩⣤⣴⠟⠀⠹⡎⣧⠀⠺⣶⡾⠝⠋⣉⢙⠯⣻⣽⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢿⣄⠀⠀⠀⡀⠈⢀⣀⣤⠾⠋⣠⠛⠉⠁⠀⠀⠀⠀⠀⢰⡏⣿⢠⢬⣞⠃⠀⠀⣩⡈⠈⣾⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⠉⠉⠛⠛⠛⠋⠉⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠙⠷⣿⣞⢧⣄⣰⣔⣠⣴⡿⢋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠉⠙⠚⠛⣻⣛⣡⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣼⣧⠀⠀⢸⣿⢸⣇⠀⠀⢠⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⣿⣿⢸⡇⠿⢿⢸⣿⣿⣿⣿⣿⣿⡃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣛⣷⢰⣿⣿⣾⣿⢸⣟⣛⣛⣸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣿⣸⣷⣶⣶⠶⠚⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡶⢲⣴⣿⣿⣿⢱⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣽⣽⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⠾⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢐⣒⣲⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠸⠿⠶⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣽⣼⡙⣿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠲⣾⣧⣿⠸⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣭⡝⡿⠿⠿⠶⢤⣄⣀⣀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣭⡽⠇⣿⣿⢿⣿⢸⣯⣭⣭⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢈⣿⣿⣿⢸⡇⣶⣾⢸⣿⣿⣿⣿⣿⣿⡁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢻⠏⠀⠀⢸⣿⢸⡏⠀⠀⠘⣿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠐⠂⠤⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⡠⢶⡖⠁⠀⠸⠵⠀⠀⠀⠀⠰⣹⠀⠀⠀⠉⠑⢦⠀⣰⠩⢈⠒⢢
⠀⠀⠀⠀⢠⣾⠋⢀⡴⠚⡏⠁⠢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠟⢳⡽⡿⣉⠐⡏
⠀⠀⠀⠀⠸⣿⢠⡿⠁⠠⠃⠀⡰⢳⡄⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⡇⠀⠀⠀⠻⠂⠑⠒⠚⠁
⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠊⠁⠀⣿⠀⠀⠂⠀⣿⡇⠐⠂⠀⠀⠀⠀⢠⣿⠀⡔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⣿⣠⡤⢠⠀⣿⡇⠀⠀⣠⡤⠠⡀⢸⣿⡡⠠⣤⡀⠀⣠⡤⢤⠛⡖⢶⣤
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⣸⠟⣿⡅⠈⠀⣿⡇⠀⢸⣿⠀⠀⡇⢸⣿⠀⠀⣿⡇⢰⣿⠀⠀⢀⠇⢸⣿⠀⡄
⠀⠀⠀⢰⠢⢄⠀⠙⠦⠄⠀⠠⠞⠁⠀⠙⠓⠀⠈⠈⠛⠒⠁⠙⠓⠚⠐⠉⠛⠈⠛⠋⠒⠁⠙⠓⠐⠁⠀⠈⠛⠊⠀⡀⣄
⠀⠀⠀⠀⠐⠞⠓⠢⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠂⠀⠀⠀⠀⢠⠦⠤⠒⡄⠀⢀⣀⣀⡔⣟⡀⢀⡤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⢐⣹⠀⣀⣈⣳⡀⠀⠀⠀⠀⡰⢊⠏⢹⠉⢱⢢⠀⠀⠀⢈⡆⠑⣼⡜⡜⢡⠋⢠⠃⠈⡟⣞⢇⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠁⠴⢾⠓⣲⡏⠩⠂⡇⢸⠀⡇⠀⢀⢋⠀⠧⢴⡆⠉⠈⣧⠽⣇⢈⠀⠞⠀⢀⠇⣯⠿⠦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⠀⠀⠀⠈⠚⠂⠓⠀⠚⠉⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠀⠙⠂⠚⠉⠀⠑⠈⠁⠁

⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢄⣀
⠀⠀⠀⡴⠉⠠⠀⠀⠀⠁⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠈⠀⠑⣄
⠀⠀⠀⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⣀⣀⣈⠑⠦⢄⣀⣀⠀⡀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠢⣌⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⣙⡿⣌⠑⠢⢄⣀⣨⢏⣩⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢟⠛⠀⢱⡀⠀⡀⠀⠀⠀⣠⣶⣥⡀
⠀⠀⠀⡠⣟⡓⠒⠛⠛⠛⣁⠔⢤⡉⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⠉⠛⠉⠁⠀⣠⡀⠀⠺⣀⣙⡷⡹
⠀⠀⠘⠧⠽⠋⠓⠒⠚⢉⢔⣶⠄⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠳⢤⡤⠴⠣⠩⠞⠢⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⡒⣒⣡⠎⠀⣠⠶⢦⡀⠀⣠⠤⡄⠠⢤⡤⠄⢀⡤⠶⢤⡀
⠀⠀⠀⢀⡖⢲⠶⣖⢲⠀⠀⠈⠁⠀⠀⢰⡇⠀⢀⡇⢸⡃⠀⠀⠀⢸⡇⠀⢸⠁⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣵⣁⠀⠉⣾⠀⠀⠀⠀⠀⠀⠀⠓⠶⠚⠁⠈⠓⠖⠃⠀⠘⠃⠀⠈⠳⠶⠋⠀⠀⠀⠀⢀⠊
⠀⠀⠀⢸⣿⣿⡧⠀⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢀
⠀⠀⠀⣸⠈⠉⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠨⠁
⠀⠀⠀⣇⠀⠀⣀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀
⠀⠀⠀⠉⠻⣿⠈⠶⠛⣧⠀⠀⠀⠀⠀⠀⢠⠂⠒⠇⠀⠘⠛⠁⠈⠀⠀⢴⣁⡀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠉
⠀⠀⠀⠀⠰⣏⡄⠀⢠⢻⡆⠀⠀⠀⠀⠀⠈⠢⠔⡚⠝⢀⡔⢢⠀⠐⣗⠢⠤⠃⠀⠀⠀⠀⠀⠀⠀⢘⡃⡀
⠀⠀⠀⠀⠀⠀⣧⠀⠏⢹⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⣰⠁⢼⡀⠀⣨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⠦⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⡿⠟⠟⡟⠿⠿⢿⣿⣿⣿⣿⡟⣝⠟⣽⡏⡟⣝⠟⢻⢻⡏⢻⣿⠟⠹⣿⣿⣿⠝⠉⠛⣿⣿⠟⠋
⣿⠟⠛⠛⠛⠟⠛⠿⠿⠛⠛⠛⠛⠻⠷⠷⠓⠛⠟⠓⠓⠓⠺⠾⠗⠺⠃⠀⠀⠿⠿⠻⠀⠀⠀⠸⢿⠀⢀
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿
⣿⡆⢀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⡀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⢐⡀⠈⠀⢐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠒⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠤
⣿⡏⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠
⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠂⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⢸
⣿⡄⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠐⠀⠀⠁⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠂⠀⠀⠀⠀⢰⠀⠐⠂⠑⠃⠀⠀⠀⠀⠀⢠
⣿⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⢸
⣿⣇⢀⢄⠀⡂⠀⠀⣠⣀⠀⠀⢰⠀⠠⡀⣄⠀⠀⣀⣄⠀⠁⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⢀⢀⠀⣀⢸
⣿⡇⠀⠇⠀⠀⠀⢀⢄⣠⡀⠀⠀⠀⠲⠀⠀⠀⠀⢠⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣿⣇⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠖⠆⠀⠀⠀⠀⠀⢀⠀⠠⡤⡄⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⡤⠄⣠⣿
⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣶⣶⣒⣟⣛⣛⣒⣶⣒⣿⣟⣳⣒⣖⣶⣲⣖⣚⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢨⢚⣧⣼⠇⣸⡃⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠛⡿⠆⣼⣷⣿⣏⣣⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢛⡄⠈⠁⣘⣿⣿⣿⠓⣿⣽⣿⣿
⠀⠀⠀⢠⡀⠀⠀⠘⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⣴⣾⡿⢿⣿⣿⣜⣿⣟⣿⣿
⠀⠀⠀⠸⡟⠦⣀⣀⡀⠀⠈⢳⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠊⠀⠠⠁⠈⣁⡤⠁⢙⣿⣿⣾⣿⣼⢿⣿
⠀⠀⠀⠘⣿⠢⣬⣭⣭⣯⣽⣾⣿⠶⣶⣤⡴⢄⣀⣀⣀⣒⠙⠀⢀⣀⡀⢈⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠹⠷⣶⣿⣿⣿⣿⣭⡿⠿⠿⠿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠟⠛⣿⣿⣤⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⠛⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣏⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠰⠂⢤⡤⢀⣿⣿⣍⣉⢉⠿⢩⣥⣤⣾⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣀⠰⠲⠠⢆⠀⡈⢿⣿⣿⣷⡨⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠀⠀⠀⠊⠈⠁⠿⠟⠿⢿⡄⢻⡿⣿⡏⣿⣻⣿⣿⢿⡿⣟⢿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣷⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠚⡛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣙⡇⠀⠀⢀⠺⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠙⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠹⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⠀⢀⣘⣾⡷⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢴⣂⠀⡀
⠀⠀⠀⠀⠙⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠦⢶⡥⠀⠀⠸⣥⣎⣁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣇⡴⣻⣄⠀⣺⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠉⠁⠀⢠⠄
⠀⠀⠀⠀⢰⡄⣿⡏⢙⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐⠻⢶⡄⠀⠰⣅⠠⠦
⠀⣈⣙⣿⣾⢃⠈⠀⠛⠀⢻⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠜⠀⠀⣧⡈⠛⠀⣀⡀⠀⡀
⠀⠙⢛⣿⡟⠋⠀⢰⠀⣤⡄⠑⣆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣆⣸⠀⠀⠀⡌⠀⠠⠀⢧⠤
⠀⠲⣶⠟⠁⠀⠀⠈⢷⡾⣷⣶⡏⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠂⠄⠰⠂⠒⠁⣀⣈⠐⠂
⠀⠈⠉⠀⠈⢃⠀⠉⣠⡟⠋⠀⠹⠀⢶⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⣶⣋⢑⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠏⣤⣄⠦⣀⠃⣄⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠘⢿⡷⢖⠀⣰⢒⣄⢿⣾⡿⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠊⢠⠏⠱⠒⡉⠀⠀⠦⣠⣾⣿⡟⠀⣛⡲⢵⣦⠀⢠⣴⣾⡯⠅⠀⢸⣷⠈⢀⠂⢿⡧⠚⠂⡘⠉⠐⣌⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⡌⣻⣧⡅⢸⣿⣷⣶⣯⣽⣿⣿⠶⣿⣿⢿⣶⣶⣷⣾⢿⡗⠀⠀⢺⠉⠓⠀⠙⢎⠃⡈
⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠐⠛⠻⠿⢷⠈⠛⠛⠛⠛⣿⠟⣼⠀⢻⡛⠇⠈⠛⠛⠛⠻⡶⠓⠀⠀⠉⠀⠀⠀⠠⠏
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠛⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠙⠁⠀⠀⠀⢀⡾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⢸⠀⢩⣄⢁⡤⡄⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢀⣀⣀⣀⡀⠀⠀⡴⠚⣹⣤⣄⡀⠀⠀⢀⣤⣤⡤⠴⢦⡄⠀⢸⡀
⠀⠀⠀⢸⠀⢻⠘⠋⢰⣧⢤⠤⠖⠋⠉⣙⣯⣭⡑⠲⠤⡼⠁⢘⣛⣶⠋⠉⢳⣤⢭⣶⠶⢷⣄⣠⠴⠛⢦⣤⢴⠋⠰⣧⣆⠸⡿⢅⠀⢸⡇
⠀⠀⠀⢸⡆⢸⠗⠒⣻⢥⡀⠀⠀⠀⢸⣥⡄⢺⠋⠀⠀⠹⢤⣬⠏⠁⠀⠰⡏⠀⠀⢼⠃⠀⠉⠉⠀⠀⢀⣴⠋⠳⡄⠀⠙⠳⠇⠀⡇⠸⠇
⠀⠀⠀⢸⡇⣿⠉⡟⠁⠸⠧⣄⠀⠀⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠳⠾⠇⠀⠀⠀⠀⠀⠈⣏⣠⠶⠃⠀⠀⣠⠶⠚⠷⡀
⠀⠀⠀⢸⡇⠹⣆⠻⠾⠛⠶⠛⠀⠀⢠⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢷⡆⠀⡶⡟
⠀⠀⠀⢸⡇⢠⣾⢿⡶⣄⡠⣄⠰⣟⡿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⢡⠃⡀
⠀⠀⠀⢸⠁⣿⠛⠓⢃⡟⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⡇
⠀⠀⠀⢸⠀⠈⣴⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡌⠑⣟⡆⠇
⠀⠀⠀⢸⠀⢠⡷⠘⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡠⠳⡼⢸
⠀⠀⠀⢸⡄⠘⠷⠾⠻⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣦⣠⠞
⠀⠀⠀⢸⡇⠀⠀⣀⡴⠟⠉⡇⢰⠿⡆⣯⣇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡄⠀⡇
⠀⠀⠀⠸⡇⠀⢰⡏⠀⠀⠀⠛⠋⠀⠛⠋⠉⢀⣠⣶⣤⣤⣴⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡦⠁
⠀⠀⠀⠀⡇⣴⠾⠟⠛⠛⠛⠿⠶⣦⡀⠀⣴⠟⢡⡞⢡⠞⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣀⣈⣓⣘⡁⠀⠉⠀⠁⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠞⠛⠉⠉⠉⣉⣻⣮⣿⣟⠁⣀⡐⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡒⢦⡀⠀⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⣿⢀⡴⠋⠁⠀⠀⠀⣠⡴⠟⠋⠉⠁⠈⠙⢻⣎⠙⢷⣄⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡜⠀⢀⡴⢶⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠎⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢿⡀⣠⡿⠀⣾⡀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⢸⣷⠀⠀⠀⠀⠠⣖⠋⠙⠋⢈⣇⠈⢻⢷⣤⣄⣀⡽⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠻⣇⣸⠋⠙⢦⡟⠀⣈⣭⣄⠀⣸⠁
⠀⠀⠀⠀⢸⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀⣿⠀⢀⣿⠧⠦⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣇⡞⢀⣀⠚⠁⣿
⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣸⡟⣠⣿⠋⠀⠀⠸⡇⠀⠀⠀⣠⠶⡆⠀⠀⠀⢰⢯⡴⠋⢽⣀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⣰⡿⠁⠀⠹⣆⢠⣄⣀⣹⣄⠀⢀⠃⠀⠟⠛⢶⣖⠁⢼⢁⣄⠀⠘⣆⣿
⠀⠀⠀⠀⣾⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⠀⠀⠀⠈⠛⠋⠉⠉⠈⠉⠉⠳⣄⡀⠻⡏⠁⢀⣘⣋⣈⠙⠓⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣤⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣼⠁
⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢬⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣽⡇⠀⠀⠀⠀⢸⣮⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⡥⠤⠤⠤⠤⢤⣀
⠀⠀⢀⡴⠊⣁⡤⠶⠶⠴⢦⣴⡟⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠐⣇⠿⠏⠀⠀⠀⠀⠹⠿⣸⠂⠀⠀⠀⠀⠀⠀⢀⡤⠖⡋⠀⢹⣦⠤⠦⠶⠶⢤⣈⠙⢦⡀
⠀⢀⠎⣠⠟⠁⠀⠀⠀⠀⠀⠀⡇⠀⢹⣤⠀⠉⠢⡀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢀⠔⠉⠀⣤⡇⠀⢸⠀⠀⠀⠀⠀⠀⠈⠻⣆⠱⡀
⠀⠸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡈⠻⣄⡀⠈⠓⠦⠤⠤⠔⡣⡀⠀⠀⠀⠀⠀⠀⠀⢜⠢⠤⠤⠴⠚⠁⢀⣠⠟⢁⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠇
⠀⠀⢣⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⠦⣌⡛⠷⣤⣤⣤⠶⠋⡸⠀⠀⠀⠀⠀⠀⠀⠀⢇⠙⠶⣤⣤⣤⠾⢛⣡⠴⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡏⡘
⠀⠀⠀⠄⢇⠀⠀⠀⠀⢀⡤⠴⠶⠾⠷⣦⣄⡀⡈⢉⠀⣀⣠⣤⡾⠃⢸⠀⠀⠀⠀⠀⠀⡇⠘⢷⣦⣄⣀⠀⡉⢁⢀⣠⣤⠾⠷⠶⠦⢤⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢸⠀⠀⢀⠞⢁⡤⠖⠖⠶⢤⣀⠉⠛⢯⣛⡛⣉⡩⠞⠀⠀⣽⠄⠀⠠⡆⠀⠠⣯⠀⠀⠳⢍⣉⢛⣛⡽⠛⠉⢀⡤⠶⠒⠲⢤⡈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠂⠀⢠⠏⡴⠋⠀⠀⠀⠀⠀⠈⠳⢦⣄⠀⠀⢀⠀⣄⣬⠞⢹⠀⠀⢠⡁⠀⠀⡍⠳⣥⣠⠀⠀⠈⠀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠙⢧⠹⡄⠀⠐
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⣠⠃⠀⢀⡾⢷⡀⠀⠘⣄⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⣠⠞⠁⠈⠳⣄⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⣀⡴⠞⠁⠀⠀⠀⠀⠈⠳⢦⣀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⢀⡤⠞⠉⢀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⡀⠉⠳⢤⡀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣰⠋⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠙⣆⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⢿⠧⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢹⠑⠲⢅⠀⢸⠀⢠⠵⠂⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⢸⠀⢸⠀⣷⠀⠀⠐⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢸⠀⡧⠀⠀⢀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠢⣀⢸⠀⢸⡀⡿⢀⡠⠿⠧⠈⠙⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⡀⠝⠀⣼⡄⠻⢤⡀⠀⠀⠀⠀⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡄⠀⢀⠠⠐⠁⡀⠐⠊⢉⠉⠓⠦⣈⠑⠦⣄⠀⣠⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠤⠒⠁⡀⠄⠊⢹⠙⠢⢄⡀⠉⠲⢄⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠊⠀⢀⠔⠈⠀⠀⠀⣼⣆⠀⠀⠈⠓⢄⠀⠈⢲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⡀⠔⠀⠉⠉⠑⠢⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⡀⠀⠈





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠉⠉⠉⡽⠉⠋⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠁⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠱⠀⠀⠀⠀⢀⡴⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢀⡴⠋⠁⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢀⡴⠋⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣰⠏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⡴⠋⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣴⠃⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡠⠋⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣰⠇⠀⣀⣤⠞⠛⠙⠉⠓⠲⣄⡀⠀⠀⠀⠀⣀⠀⠀⢰⠋⢻⣤⠞⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⡏⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⣄⡠⢴⡏⣩⠗⢠⠏⢀⡔⠻⣧⡀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢻⡄⢺⣍⣁⡴⠋⢠⠎⠀⠀⠘⣷⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢠⡇⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⢸⡇⠀⠉⠀⠀⢠⡏⠀⠀⠀⠀⢸⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢈⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣸⠇⠀⠀⢿⡄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⣀⡴⠋⠀⠀⠀⠈⠻⠤⡀⠴⠚⠁
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠈
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠛⢮⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠂⠀⠀⠀⢀⣿⠃⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⢰⡏⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⣿⠀⠀⠀⠀⢠⠔⠁⡟⠀⠀⠀⠀⠹⡄⠀⠀⠰⠆⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡴⠛⠁⠀⢀⣴⠖⢲⡄⠀⠀⢠⡾⠉⠻⣦⣀⡤⠖⠂⠉⠉⠉⠓
⠻⣄⠠⠴⠚⠁⠀⢠⡇⠀⠀⠀⠀⠀⢱⠀⠀⣤⡀⠀⠀⠀⠈⠻⣦⣀⣠⠤⠆⠀⢺⣿⢁⡀⠀⠀⠀⣾⣧⠰⠚⢀⣠⠶⠋⠀⢀⡴⠊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⡇⠀⣿⡇⠀⠀⠀⢀⡤⠚⠙⢷⣄⠀⠀⢸⣿⠋⠙⠦⡠⠞⠉⠛⠒⠚⠋⠁⠀⠀⡰⠋⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⠀⠛⠁⠀⠀⡴⠋⠀⠀⠀⠀⠙⣷⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢀⡠⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣦⣀⢀⠀⠀⣸⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡿⣯⡷⠿⢯⡿⡇⢤⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⢿⡷⣏⠰⡷⠀⣿⣿⠆⢻⡃
⠀⠀⠀⠀⠀⠀⠀⣷⣝⡷⣶⣞⣷⡇⣠⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠟⠉⢨⣴⣿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡼⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡘⣃⣦⡀⠀⠀⠱⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠀⠀⠀⠀⠙⣄⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢸⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠇⠀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠑⢆⡠⠅⠐⠊⠁⠠⢠⣤⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠀⠻⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢦⣄⠀⣠⡤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⡶⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣹⣷⠞⠛⠻⣯⡉⣉⣉⣉⣉⣉⣹⣧⣴⠾⠟⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⣿⠃⢀⡀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣄⡴⠚⠛⠻⣏⢹⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⡏⠀⠀⣠⣾⡁⠀⠀⠀⣽⠋⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯
⠀⠀⠀⠀⠀⢻⡀⠀⠿⡼⢷⣤⣤⠶⠋⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣤⣤⣄
⠀⠀⠘⠛⠛⢻⡟⠛⠂⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣼⣷⡤⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⢲⣿⠷⠦⠄⢀⣴⠶⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣾⡁⠀⠀⣴⠏⠀⠀⠀⣀⢸⡇
⠀⠀⠀⠀⢠⡴⠞⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠈⠙⠳⠤⣟⠀⢀⣀⣴⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣋⠁⠀⠀⠀⠀⣠⡶⣫⣤⣍⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣤⡾⢋⡟⠉⠉⣿⠉⠉⠉⠉⠉⠁⣿⠀⠀⢹⣆⢙⣷⠶⠦⣤⡾⣫⡾⠋
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠈⢛⡿⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢹⣿⠁⠀⠀⠘⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⡴⢦⣤⠶⢦⡄⠀⠀⠀⣾⠁⠀⠀⠀⣰⠇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣄⣠⡴⠋⠀⠀⢀⣸⡛⣻⣇⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠻⣄⡀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡶⠛⠉⠙⠉⠉⣙⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣧⡴⠞⠛⠳⠾⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠞⠛⣻⡟⠁⠀⠘⢻⡾⠻⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⣀⣽⣇⣀⣀⣀⣀⣠⣤⡾⠏⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠩⠭⠩⣉⣉⣉⣉⠀⢀⣀⣀⣀⡄⠄⠤⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⣖⡶⢷⣲⣏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣋⠿⠯⠭⠽⢑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠘⠛⠛⠟⠛⠃⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠈⠀⢠⢿⣷⣶⣴⣦⣴⡿⠿⠀⠀⠁⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣿⣿⣾⣿⣿⣿⣿⣷⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⣴⣿⣿⡿⡿⣛⡟⢻⣻⢿⢿⣿⣿⣦⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠀⠈⠙⠛⠋⠉⠑⠂⠀⡐⡛⣚⢂⠀⠐⠈⠉⠙⠛⠋⠁⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⠤⡄⠄⠁⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠈⡠⢠⢤⣀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣆⣬⠋⠔⠀⠀⠀⠀⢁⠂⠀⠀⠐⡈⠀⠀⠀⠀⠪⠙⣥⣰⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠈⠉⠙⡄
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠳⠒⠀⢲⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠖⠐⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⣀⡰⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠈⠉⠁⠉⠙⠲⡄⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠉⠙⠳⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣿⠁⠀⠀⠀⢀⡴⠋⠁⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⣸⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠑⣾⠁⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢀⡴⠲⣦⠀⠀⣠⠞⠙⣆⢠⣿⠀⠀⠀⢰⣿⠈⠁⠀⢩⠞⠉⠙⢦⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⡾⠀⢠⣿⠀⢰⣿⠀⣠⠏⠀⣿⡀⠀⠀⢸⣏⠀⠀⠀⢸⣿⣀⡀⣸⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⠿⡤⠞⠿⣎⣁⡤⠊⠹⣧⡀⢀⠜⢿⣄⢀⡠⠻⣧⣀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⠤⠤⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣤⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⣠⠖⠲⣦⠀⠀⣠⡴⠶⣤⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣼⡇⠀⠀⣼⡇⠀⠀⡼⠉⣿⡰⠁⠀⢠⣿⠀⣼⠋⠀⢠⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢠⣿⡁⠀⣰⠁⢸⣿⠇⠀⠀⣸⡇⠀⣿⡂⠂⠁⠀⠀⡴⠃⠀⠀⢀⡠⠀⠀⠀⠒⢤⡀
⠀⠀⠀⣠⠄⠒⠐⠒⢤⡀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⠷⠤⠊⠙⠧⠞⠁⠀⠘⠛⠀⠀⠀⢻⡇⠀⠈⠳⠦⡤⠴⠋⠀⠀⣀⡔⠁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⢸⣇⠀⠀⣀⠄⠰⠶⠦⢤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡏⠀⠀⠀⢀⠠⠀⠀⠸⡧⠤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢣⠀⠀⢳⡦⠂⠀⠀⠀⢀⡴⠂⠈⢳⡀
⠀⠀⣇⠀⠀⡔⠁⠀⠀⠀⠀⡇⠀⠀⠈⠒⢾⣇⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡜⠀⣠⠊⠻⣄⠀⠀⠒⢅⠀⠀⠀⢀⡇
⠀⠀⠸⡀⠸⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⣿⠉⠑⠦⢄⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⢠⠃⠀⠀⠈⠳⢤⣀⠀⠱⡀⣀⠞
⠀⠀⠀⠱⣄⠙⠦⠤⠤⠚⠁⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻
⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠐⢚⣶⡤⠖⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⡤⠴⡏⠈⠀⠙⢳⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢾⠀⣾⡎⣭⡍⢣⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡚⠣⣉⡡⢎⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢐⠒⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⢧⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡎⠸⡀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢭⡑⡄⠀⢀⡜⢸⠀⠀⠑⠦⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⣣⠇⠀⣸⠉⡾⢄⠀⠀⡔⣫⣭⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡉⠲⣄⡟⠀⠀⠺⣇⠀⠹⡹⣾⡃
⠀⠀⠀⠀⠀⢤⣄⡠⠤⢴⠁⢀⡼⣋⡁⢇⠀⠀⠀⢈⣉⡛⢦⡯⢇⣀⡀⠀⢠⠒⡖
⠀⠀⠀⠀⠀⠸⣬⡵⠶⣦⡶⢯⡀⠀⢈⢹⡦⣴⣊⣁⣀⣀⣀⣷⣒⣋⣉⣏⣙⣁⣽⠀⣰⣧⣶⠤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠁⠉⠉⠉⠁⠀⠁⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⢙⡆⠀⠀⣠⠔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣼⠃⠀⣼⠃⠀⡿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣾⠇⠀⢀⡼⠁⠀⢰⠇⠀⣼⠃⢠⡾⠋⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣿⢀⠴⠋⠀⠀⠀⣿⠀⣴⠃⢰⣿⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠤⣾⡟⠁⢀⣤⣄⠀⢠⣏⡼⠁⠀⣿⣧⠞⠁⢀⣶⠏⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⣿⠗⠋⠀⣸⡟⠀⠀⠀⣿⠁⠀⠀⣸⣿⣀⣾⠀⠀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⠛⠦⠴⠚⠉⣷⠀⠀⠀⣿⠀⠀⣰⠇⠈⠉⠁⠀⣼⠋⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠈⠉⠳⣦⡀⠙⠃⠀⢸⣧⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⠳⠚⠁⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠈⠻⠄⠀⠀⠀⢀⣶⣦⠀⠉⢑⣁⣠⠤⢤⡾⢿⠦⢤⣀⡀⠉⠉⠉⢙⣿⠉⠉⣿⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣠⡄⠀⠀⢸⣿⠀⠘⠢⠄⣀⣀⣀⣀⣼⣿⠗⠚⠉⠉⠀⠀⢀⣿⣣⠏⠀⠀⠈⢹⡆⠀⠀⣸⡏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣴⣿⠏⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣸⣿⠁⠀⠀⠀⢀⣀⠀⠀⢠⡿⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⣸⣿⡏⠀⠀⠀⡀⣼⡿⠒⢹⣿⠏⠻⠃⠀⣸⣿⡟⠀⣠⡶⣶⠒⠉⣿⡇⢀⣀⡀⣰⣿⣻⡟⢠⣿⠃⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⠇⠀⢸⣿⡀⠀⣠⡾⢿⣿⡇⠀⣿⡇⣿⠀⠀⣿⣿⠟⢀⡟⣿⣟⠁⣠⡿⠃⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⢀⣴⣿⠋⠀⠀⠀⠛⠷⠞⠋⠀⠸⣿⣧⠀⠙⠛⠁⠀⠀⠻⠟⠒⠉⠀⠙⠛⠚⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡍⠈⠁⣷⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⢄⡀⠀⠑⠂⠈⠀⣀⠀⣀⣠⣀⣠⡤
⠀⠀⠀⠀⡀⠀⣒⣫⠥⠞⢿⣀⢈⡙⠻⣿⡿⢟⠻⣟⠛⡳⠾⣄⠀⠀⣂⡠⠄⠒⠐⢓⠞⠁⢀⣀⡠⣡⡿⠟⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⣦⠶⠞⠉⠀⠀⠀⠀⠀⠘⢇⠀⠀⣀⣼⠀⢰⣦⣷⠀⠈⠋⠀⠀⠀⠀⠀⡔⡉⣴⠶⢉⡿⠝⠋
⠀⠀⠀⠁⠀⠀⠠⠂⢀⡴⠋⠀⠀⠀⠁⠚⠑⠠⣴⠂⢸⡽⠄⠀⠀⠀⠀⠀⠀⠀⠈⣇⢬⠀⠉⠂
⠀⠀⠀⣄⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠶⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣛⣛⡿⣻⣿⡛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠁⠀⠀⢈⣠⡶⠿⠿⣿⣿⠛⠛⢿⣿⣿⠟⠳⠦⣀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⠾⠋⠀⠀⠀⢰⡟⠁⠀⠀⠈⢻⣟⠂⠀⠀⠈⠳⣄⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⡔⠟⣠⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢻⣦⠄⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡾⢫⣾⣟⣻⣄⠀⢠⣯⡀⠀⠀⠀⠀⠀⠈⣿⡐⣶⣿⣦⡀⠀⢷⡀⠀⠀⠀⠀⠈⢻⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⣷⡛⠀⢺⣾⣿⣷⣼⣿⠁⠀⠀⠀⠀⠀⠀⣸⣹⡿⠟⢛⣿⣦⠈⣷⠄⠀⠀⠀⠀⠀⢻⣿
⣿⣿⠇⠀⠀⠀⠀⠀⣠⡿⢟⣃⣄⡻⣿⠁⣨⣿⣿⡃⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⢴⣿⡿⠑⢸⡶⠀⠀⠀⠀⠀⠀⣿
⣿⡿⠀⠀⠀⠀⠀⠀⢸⠉⡁⠉⠉⠉⠉⠀⢨⣿⡏⡇⠀⠀⠀⠀⠀⠛⠛⣿⡊⠋⠉⠘⠛⠓⠛⣿⣧⠀⠀⠀⠀⠈⢿
⣿⢃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣼⣿⡛⠀⢀⣾⣿⣶⡆⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸
⣿⠸⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣿⣿⡟⢠⣿⣿⡟⣿⣷⠀⠀⢿⡯⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸
⣿⡰⠀⠀⠀⠀⠘⢠⣼⣿⣶⠀⠀⠀⠀⢀⣿⣿⣵⣿⣿⣿⣄⣼⣿⣧⠀⣼⣿⡇⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⢸
⣿⡄⣆⠀⠀⠀⠐⢾⣿⣟⢻⣶⣄⣄⠀⠸⢿⣟⠷⣦⠀⠉⠀⠀⠀⠀⠐⣿⣿⠷⠁⢠⣄⣸⣾⡏⣽⡇⠀⠀⠀⠀⢸
⣿⣧⢙⡄⠀⠀⠀⠻⠇⣀⣏⣩⢼⢿⣿⣶⢿⣿⠿⣦⣴⣠⣤⣀⣄⣀⣜⣿⣿⣶⣼⣿⡿⢻⣿⣿⣿⡇⠀⠀⠀⠀⣾
⣿⣿⣧⣧⠀⠀⠀⠀⠓⢛⣿⣷⣞⠀⢸⣿⡯⠛⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⠏⠈⣿⣿⣤⣼⣿⣿⡏⡴⠀⠀⠀⣸⣿
⣿⣿⣿⣯⡀⠀⠀⠀⠀⠘⣯⡽⢿⣶⣿⣿⣧⠀⣿⣿⡄⢠⣿⣿⡀⣰⣿⡿⣧⣴⡿⢿⡟⢹⣿⣿⡤⠉⠀⠀⣴⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣆⠀⠋⠉⠳⠻⣷⠶⣧⡿⣾⡽⣿⡼⣴⣞⣿⡜⠙⠃⠀⢠⣿⡟⡾⢀⠃⢀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠻⣷⣄⠀⠀⠀⢸⡇⠁⠀⠈⠀⠉⠀⣈⣿⠀⠀⠀⠀⣰⣿⣿⣿⣾⣞⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡠⣤⣈⡻⠧⣀⣀⡛⣿⣤⣀⠀⠀⢀⣸⣿⡿⣧⣤⣢⣾⡿⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⠃⠀⠄⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢄⡠⢄⢀⠀⠀⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣄⠦⠀⠃⠀⢁⠁⠀⠀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⢠⠂⡸⢘⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠞⣳⠀⠐⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠛⢰⠚⠀⠀⠀⠀⠀⡠⠐⣐⣝⡰⣀⣀⢠⠏⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠌⢸⠀⠀⠀⠀⠌⢀⠌⠀⣰⡿⠏⢀⠄⡐⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠄⢂⠀⠀⠌⢰⣾⡀⡰⣿⣷⠀⠔⣔⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣⠌⠀⠀⢻⡏⠀⠈⢿⣷⠔⡉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠆⠀⢠⠈⢷⠀⠀⠈⢿⡆⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠇⠀⠘⣧⠀⠀⠈⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠔⠁⡄⠀⠇⠘⠧⣀⠀⢀⣠⠞⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢰⡏⠁⠀⠩⠉⠉⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣼⠁⠀⠀⢰⣀⣠⣤⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢸⣿⠀⠀⠀⠀⠀⠄⠀⠑⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠒⠂⠂⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠐⠒⠒⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⢂⣀⠀⠀⠀⢣
⠀⠀⠀⠠⠊⠀⠀⠀⢀⡀⣀⠀⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠰⠏⣦⠀⠁⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡏⠀⠀⠀⠀⢃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣽⠇⠀⠀⠀⠀⠈
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡃⠀⠀⣀⣀⠀⠀⡇⠀⣿⠠⠀⢸⠀⢀⠀⠤⢀⠀⠀⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⢸⠁⠀⢀⣠⡀⠈⣄⡇⠀⣿⠀⠀⣼⠀⠀⢀⣤⡄⠑⠄⠀⠀⡩⠩⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⢸⡆⠀⣨⣭⣤⣶⡿⡇⠀⣿⠀⠀⣿⡇⠀⣿⠋⠃⠀⣰⠀⡈⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢸⢇⠀⠙⠓⠉⡗⡆⠇⠀⣿⠀⢀⣿⢣⡢⠈⠁⢀⠀⡟⠀⢧⣂⣀⣤⠇⠀⠀⠀⡐
⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀⣸⠃⢄⣀⣾⠈⢷⡤⠤⢴⣶⠟⢰⣶⡿⠰⣾⡿⠈⠻⠿⠿⠽⠋⠀⠀⠈⠛⠛⠁⠀⠀⠀⠈
⠀⠀⠀⢂⠀⠀⠀⠀⠈⠛⠉⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠄⠠⠀⠠⠀⠠⠐⠠⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠑⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠄
⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣠⡆
⠀⠀⠀⠀⠉⠻⠟⠀⠀⠀⠀⠀⠘⠓⢀⠐⠓⢀⠀⠀⠀⢤⣦⣤⡄⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠳⠶⠖⠞⠋⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⡀⠀⠀⠀⢰⣬⣤⠀⠀⠀⠀⠠⠶⠶⠶⠄⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⡆⢠⠇⣀⠗⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠄
⠀⠀⠀⢀⡧⣾⡽⢁⣶⡄⠀⠀⠀⢠⣿⣿⣿⣿⡅⠛⢸⣿⠿⠋⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢿⣿⣿⡿
⠀⠀⠀⣼⣔⣟⣡⣷⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⢀⣴⡂⢸⡟⠀⠀⠀⠀⣠⡀⢸⡃⠀⠀⠀⠈⠛⠻⠿⠿⠇
⠀⠀⣴⣧⠾⠉⠻⣨⠁⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣠⡴⠯⢶⣬⡇⠀⠀⠀⠈⠉⠃⠸⡄
⠀⠀⢿⡇⠀⣠⣞⡷⠣⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡆⣿⠀⣀⢸⣇⣶⣿⣟⡁⢸⠎⠃⠀⠠⠄⠠⠤⠴⠶⣿⣤⠤⠤
⠀⠀⢸⣷⠞⠛⣃⣠⠯⠀⠀⠀⠀⣾⢰⡿⢀⣠⠀⣧⣿⣀⣏⣠⠟⠁⠀⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⣀⣄
⠀⠀⠠⣿⠀⣼⠋⠀⠀⠀⠀⠀⠀⣿⠛⣿⣼⢿⠟⣷⠌⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⣿⡿⠁⠀⠀⠠⣶⡄⠀⠀⣿⠀⠻⠋⠀⠀⠁⠀⠀⠀⠀⠂⠈⠴⠈⠀⠀⠀⠀⠀⢄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⢸⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⡌⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋⣈⠛⡇⠀⠀⠀⢰⠀⠀⣴⠀⠀⠶⠀⠀⢾⣿⡀
⠀⢀⣿⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢧⡶⢶⠾⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⣤⣄⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⡈⠀⠂⠀⠀⠀⠀⢿⡿
⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡀⠀⠀⢉⠹⠶⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠡⡁⢀⣈⣾⢥⣄⠀⠈⠁
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢈⠣⡄⢉⡉⡩⠙⠶⡟⠲⣆
⠀⠀⠀⠹⣿⣻⠏⣠⠾⠋⠉⠙⣷⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠂⠀⠀⠁⠼⣄⣁⠀⠐⢢⡷⠴⣇
⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣘⣷⡀⠈⣳⠄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⣄⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠿⠷⠿⠿⠗⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⣿⣿⣿⣿⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠉⠋⠹⠟⠛⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⢠⠃⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡿⠒⠒⠶⠀⠀⣿⠀⠀⠀⠀⣦⠀⠀⠀⠀⣴⠞⠲⣤⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⣻⡄
⠀⠀⢸⠀⠀⠀⠀⠀⣿⠶⠤⠤⣾⠀⠀⠀⡿⠛⠛⠛⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⡀⠀⠀⠀⠂⢸⢇
⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠀⠀⠆⣼⣿
⠀⠀⢻⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣧⣀⣀⣀⠀⠀⢿⣀⣀⣀⠀⣿⣀⣀⣀⣀⠹⣦⣀⣠⠟⠀⠀⠀⠀⠀⡏⡇
⠀⠀⠸⡀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⣸⠃
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⠏
⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠊
⠀⠀⠀⠀⠀⠈⠻⢶⡤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠭⢷⡀⠀⠀⠀⡰⢶⠲⡖⠦⠲⠖⠶⠲⠖⠖⠒⠒⣂⣋⣉⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⠞⣡⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠒⡓⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢬⡶⠛⠳⢆⡀⢙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣖⠖⣶⣤⡀⠀⠀⠘⡀⢻⣟⢦⡀⠀⠙⣖⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣨⣞⣶⢶⠦⠄⠉⢹⡄⠀⠀⠁⢤⢉⠛⣻⠀⠀⠘⣆⢀⡴⠓⠉⠀⠀⠀⠃⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢛⠁⣾⣹⡆⠇⢸⠀⣼⣧⣄⡀⠀⠀⠉⠉⠁⠀⠀⠀⠘⡏⢀⣦⣶⠖⠛⠳⢤⡈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣄⠈⣴⢿⣌⣧⣀⠟⢀⡾⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠌⢸⣇⣾⠁⠈⡀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⡀⢈⡷⠋⡝⣛⣿⣾⠃⠀⠻⢤⡭⠶⠊⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⠆⠴⣿⡏⣇⠀⢡⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠘⢃⡞⠀⣀⠁⠈⠛⣿⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢀⠠⠸⠇⢻⠀⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⢳⡾⠀⠀⣍⠀⢀⣾⡇⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠄⠰⣠⢾⠄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⡇⠀⢂⡻⢶⡕⠙⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠑⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⢺⠀⠀⠈⢹⠀⠀⠀⢷⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢠⠏⢀⠀⠲⠓⢀⠦⢤⠀⣸⠃
⠀⠀⡄⠀⣹⡆⠀⠀⢸⠀⠀⠀⠈⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡟⠀⣴⠶⢛⠛⠛⠾⣄⡈⠉⢄⡀
⠀⠀⢨⠀⠤⣇⠀⠀⠀⢇⠀⠀⠀⢸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⣮⡐⣾⢸⢠⡋⣴⣿⢷⡄⠀⠀⢹⠿⣄⠀⠂⣄
⠀⠀⠐⠂⠈⢹⡄⠀⠀⠈⢆⠀⠀⠐⣧⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡜⡿⠃⠀⣤⠻⢿⣧⢾⠃⠀⠀⣸⠀⠘⢧⡠⠀⠆
⠀⠀⠀⠇⠀⠁⢷⠀⠀⠀⠀⠢⡀⠀⠈⢷⣀⣿⢣⣾⣧⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⡟⠀⠀⠀⢳⡞⠧⠀⠀⠀⣰⠃⠀⠀⠀⢿⣆⡐
⠀⢀⡠⠒⠀⡐⠨⣧⡀⠀⠀⠀⠘⢦⠀⠀⠉⢯⠈⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⣼⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢸⡷⢿
⠀⠁⣀⣿⠯⣍⠲⠘⣷⣄⠀⠀⠀⠸⡇⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣹⠦⠖⠊⠁⠀⠀⠀⠀⠀⠀⣸⠇⠄
⠅⠀⣼⣇⣴⢼⡇⠏⠿⠺⠳⣄⣀⢰⡇⠈⢂⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠰⠏⠀⠠
⡄⠀⢹⡙⠏⠹⢫⣾⠂⠒⠀⠈⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⣀⣤⣤⡄⠀⠀⠂⡄
⠁⠀⢘⢷⡀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣾⡵⠛⠁⠙⣧⠀⠈⡁
⠀⠂⠺⣿⡷⣄⡀⠀⠀⠀⠉⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⡿⠀⠀⠁
⠀⠀⠀⢷⡀⠀⠉⠓⠒⠶⡶⡞⠠⠄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⡤⢺⣧⣴⣷⠎⣹⠃⠁⢀
⠀⠀⠀⠀⠁⠀⠀⠠⠀⡀⣀⡀⣆⠀⠄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠙⢯⣀⡉⠉⢉⣠⣼⢏⠁⠀⡉
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣴⠿⠐⠳⡜⠆⠲⠶⠒⠒⠲⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⠏⢩⠭⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢀⣈⢀⠀⠀⠀⠀⠀⣸⠄⡀⠄⠰⠂⢀⣀⣀⣤⣤⣈⣶⣄⠀⡈⠀⢁⠐⡠⠴⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠥⠀⠀⢣⠘⠷⠿⠧⠉⠀⠀⠀⠀⣠⡏⠀⠁⢀⡾⠚⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⠉⠀⠀⠀⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠳⣄⠀⠀⠀⠀⢀⣠⠞⠓⣀⠴⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠡⠀⠈⠀⠀⡉⠒⠚⢛⡛⡐⠃⠋⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢀⠀⠉⠀⠀⠀⠠⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠀⠀⠢⡀⠀⠀⠀⣠⠞⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢸⠀⢀⡾⠁⠀⠀⠀⢠⠧⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣾⠁⠀⠀⢀⡠⠄⠤⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⣄⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⠏⠀⠀⠀⣹⠀⠀⠀⢤⡀
⠀⢠⠔⠈⠀⠀⠀⠉⠲⣄⠀⠈⢧⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⣿⠀⠀⠀⣀⡃⠀⠀⠀⠀⡇
⢰⠃⠀⢀⡠⠄⠀⠀⠀⠈⢦⠀⠘⡇⠸⣆⢀⡔⢹⡆⠀⣴⠉⢹⠄⣿⠀⠀⢸⡇⠀⠀⢰⡁⠈⢳⠀⢀⡴⠁⢀⠀⠀⠀⠠⢄
⠸⣀⡞⠉⠀⠀⠀⠀⠀⠀⠸⡆⢠⠇⠀⠹⡾⠀⢸⣇⣠⣿⠀⠊⡠⢿⡀⠀⣸⣇⠀⢀⣾⡗⢂⡞⠀⠁⠀⢠⠃⠀⠀⠀⠀⠀⡹⣆
⢠⡟⢤⡀⠀⠀⠀⠀⠀⠀⢀⡧⢊⡤⠤⢤⣄⡀⠀⠈⡀⠀⠁⠀⡀⠀⢉⡨⠀⢉⠉⠁⠈⠙⠉⣾⠙⡆⠀⠘⢄⠀⠀⠀⠄⠊⠀⠘⣆
⣾⠀⠀⠈⠑⠒⠂⠀⠀⠀⢸⠃⠈⠀⠀⠀⢀⡏⠉⣡⠤⢤⡀⠸⡄⢰⠋⠀⠀⣰⠀⣀⣀⠀⠀⢿⣄⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣠⠞⠉⠳⠀⣸⠃⠐⣧⠒⠀⢹⠀⢸⡟⠒⠀⠀⢠⡟⠁⠀⡁⠀⠀⡘⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣆⠀⠀⠀⠀⠀⠀⢠⠞⠀⣿⠀⠀⠀⢀⣿⠀⢰⡏⠑⠀⣼⠂⢹⣇⡴⠒⢦⢸⡇⠀⠀⠀⡠⠊⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠙⢦⣀⡀⢀⣠⠔⠉⠀⠀⠈⠓⠒⠂⠁⢹⡇⠀⠙⠒⠚⠁⠀⠸⡿⠀⢀⠟⠀⠉⡃⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢾⡓⣄⡴⠋⠀⠀⢀⠇⠀⠀⠀⠀⠘⣧⡀⠀⢀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⣵⠀⠤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠐⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢢⠈⠀⠀⢿⠃⡖⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠉⢀⠀⠘⠀⠱⡀⠈⣄⠆⠀⠀⠀⠈⠀⠀⣀⡰⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠤⣈⠁⠀⠀⠘⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠐⠀⠁⠀⢆⠑⣆
⠀⠀⠀⠀⠘⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣴⣶⣥⣀⡀⠀⠀⠀⠀⠢⣄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⡀⠀⠘⡿
⠀⠀⠀⠀⠀⢀⡉⠑⠈⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠘⠡⢋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⠀⠀⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⣀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⢠⠊⠀⡀⠈⡆
⠀⠀⠀⠀⠀⣀⣤⡀⢨⡃⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⡏⠀⠀⡀⠀⠀⠀⠣⠀⠑⠁⢠⠃
⠀⠀⠀⠁⠀⠉⠻⠷⡀⠁⠀⠀⠀⣿⡟⡻⣿⡏⣻⢿⠸⡻⠱⣧⡽⠀⠠⠚⠀⠀⡀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠉⠀⠀⢹⢸⢇⡃⣶⣻⣲⣤⣷⣿⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⡀⠴⠶
⠀⠀⠀⠀⡌⠀⠠⠄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠙⠇
⠀⠀⠀⠀⢇⠀⠂⢠⠀⠒⠂⠐⠂⠀⠀⠈⠛⠻⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⠐⠀⠀⡀⢠⠀⡄⠀⠀⠁⠀⠀⠀⠀⢄⠀⡑⡀⠀⠀⢀⠀⠐⡄⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠁⠀⠀⠀⠠⠀⠀⠄⢁⠰⡀⠠⣄⠁⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠾⠈⠁⠀⠀⠀⠀⠀⠀⠂⢰⣄⠁⠀⠀⠀⢀⣈⡐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣠⠖⠒⠒⠦⣴⠋⢳⣀⣀⣤
⠀⠀⠀⠀⠀⡏⠉⠉⣿⡄⠀⠘⢦⡀⣴⠛⠓⢦⢸⠁⠀⣤⠉⢻⠀⠘⠋⠀⡽⢧⠀⠉⣀⡬⠇
⠀⠀⠀⠀⠀⢻⠀⠀⠈⠁⠀⠀⠈⡇⡏⠀⠆⠈⢿⡄⠀⢀⣠⠾⡇⠀⢰⠊⠁⣼⠀⡞⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠀⣤⡄⠀⠀⢷⡇⢠⠶⡄⢘⣧⣤⣼⣤⣀⡙⢚⡭⠤⢤⣨⠟⢧⡀⣀⣤⣄⡤⠤⢤⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⡀⠀⢹⠻⣤⣤⣼⣉⠉⢠⠼⠟⠚⢻⡇⠀⣄⠙⣾⠀⢀⠀⢿⢦⡀⠙⠁⣠⣿⡀⠀⠤⣞⠀⠀⠀⢠⡟⠉⠙⣇
⠀⠀⠀⢀⡤⢦⡀⠓⢚⡯⠤⣄⣿⠉⠙⡇⠘⢲⡄⢰⢿⡇⠀⠿⠃⣿⠀⣈⡀⠈⣇⣷⠀⡏⠁⣏⠉⠃⠀⢈⡇⣤⠶⣼⡇⠀⠀⣿⣠⣄⡀
⣾⠛⠉⣿⡀⠀⠛⣶⠏⠀⣤⡄⢻⡀⠀⠷⠶⡞⠃⠘⢻⣧⣤⡤⠞⠳⠤⠏⠹⠴⠞⠛⠚⠃⠀⠙⠲⠤⠴⠚⢸⡇⠀⠈⠃⠀⠀⠟⠁⠀⣿
⢹⡄⠀⠈⠁⠀⠀⢸⣆⠀⠹⠃⢸⣇⣀⣤⠴⠟⠛⠋⠉⠀⠀⠀⣴⠛⠶⣄⠀⣠⠖⠛⣦⠀⠀⠀⢠⡴⠶⠶⠶⠛⠒⡂⠈⠀⠈⠉⠒⢛⠛⠛⠛
⠀⢿⡀⠀⢶⣆⠀⠈⣿⠲⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣏⣀⣀⡭⠉⢅⣀⣀⣻⣤⣤⡀⢸⡇⠀⠂⠀⠀⠊⢆⠀⠀⠀⠀⠄⠁⠀⠀⡀
⠀⠈⣧⣀⣸⠟⠖⠒⢋⣤⣀⡴⣆⢀⣄⡀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣧⠈⡗⠀⠀⠀⠀⠠⠚
⠀⠀⠈⠉⠁⠀⡴⠦⣾⠀⣿⠁⣿⠋⣸⣃⠀⠀⠀⠀⢺⠥⠭⠬⠤⠤⢬⠀⣄⠀⠇⠒⠒⠒⠒⣿⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⢠⣷⣄⠘⣇⢸⢠⠏⡰⢋⣽⣃⣀⣠⣤⠸⡇⠀⠀⠀⠀⢲⠀⠉⢀⢸⣄⠀⠀⠀⣿⠀⣿⠀⠄⠁⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⢀⣤⣬⡷⠾⠷⠾⠛⠈⠉⠋⠉⠉⠉⠁⠀⢸⠀⣧⠀⠀⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠀⣿⠀⢻⠂⠀⡀⠈⠀⠀⠄⠀⠀⠰⠂⠀⢀
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣿⠀⢸⡄⠂⠀⡠⠀⠀⠀⠀⠀⠸⢀⠐⠁
⠀⠀⠀⠸⡗⠂⠈⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⠀⠀⡀⢒⠀⠀⠘⠁⠀⠠
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠈⠀⢸⠀⠀⢀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⡀⠀⠀⣿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⡇⠀⠈⣇⣀⣊⣀⣀⠀⢀⡀⢀⣈⢀⣀⡐⠁⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⡀⣸⣿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠈⡇⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⠆⠀⠀⠀⠀⢀⣠⣄⡀⢸⣿⡇⢰⠇⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⢀⠞⡆⠀⢠⣿⡟⠈⡷⢸⣿⡇⡼⠀⣿⣿⠀⠁⠀⢠⣸⣏⣻⣦⣴⠗
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⣼⠀⣿⠀⢿⣿⣁⠴⠁⠸⣿⣿⠁⠀⢻⣿⡆⠀⠀⣿⡇⠉⠙⣿⠁
⠀⠀⠀⠀⠀⣠⠔⢻⣿⣷⡇⠀⣿⠀⢸⣿⣧⡀⠀⣠⢿⣿⡄⠀⢘⣿⣧⡀⣠⢿⣧⣀⣰⡟
⠀⠀⠀⠀⠋⠁⠀⠀⢿⣿⡧⠀⢿⣦⠎⠈⠙⠛⠋⠁⠈⠻⠿⠖⠋⠈⠛⠛⠁⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣴⣶⣄⠀⣤⣤⢤⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⢼⣿⣿⣿⡿⠀⠉⠉⠉⠉⠀⠺⣿⠆⠀⠀⢀⢐⡳⠥⣥⡀⢀⣤⡴⠟⠁⠀⠀⠦⠀⠀⠀⠈⣣⣼⡁
⠀⠀⠀⢉⡉⠹⠀⠀⠀⣴⠉⣳⠀⠀⠀⠀⠀⠀⠁⣾⡄⠀⠻⣾⢦⠁⣀⢀⡀⠀⡀⠀⠀⠀⣀⠀⢉⣈⢡⡄⣾⡄
⠀⠀⡰⣿⣿⣿⡶⠖⠋⢹⠟⠋⠄⣨⣦⣼⢤⡆⠀⢿⠇⢀⣀⠛⡸⠸⠡⠚⠉⠼⠙⢀⣀⢸⣭⡧⣿⣙⣯⠻⠟⣷
⠀⢀⡇⠘⢿⣁⣀⠀⣠⣏⠀⣇⠘⢥⣌⣫⡤⠃⠀⠀⠙⠛⠛⠋⣁⢀⡀⣤⢠⡾⢷⡌⣿⣀⣭⡁⠹⣿⡿⠀⢀⠈⣀
⠀⠈⣇⠠⠁⠙⠻⢿⣿⣿⡇⠘⠶⣶⡀⠀⠀⢤⡴⢚⣤⣤⣾⣇⣿⣾⣿⣿⡾⣷⣼⡟⠹⠿⠛⠁⠀⢰⡶⡿⢿⠀⠕
⠀⠀⠙⣶⣄⣀⣀⣀⡼⠛⠁⠀⡀⢁⣠⣴⣶⢄⣠⡼⣯⠙⣉⣿⡙⡷⠻⠙⠋⣨⢁⠀⠀⠈⣇⠀⢀⣸⡷⢦⣼⣤⣤
⢠⣾⣿⣷⣬⣽⣿⣅⠠⠔⠊⠀⠂⠀⠉⢻⣇⢻⣷⣗⠙⠿⠟⠉⠁⢠⠀⠀⢄⠁⠈⣃⠀⠘⠉⠀⠈⠻⣿⣿⣿⣿⠃
⠨⣿⡇⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⢀⠀⠈⠿⠈⠛⠋⠁⢀⣰⢤⣀⠀⠃⠀⠈⠀⠋⢁⣤⣶⣿⣿⣷⠂⠀⠙⠿⠃
⠐⣿⣰⠀⠀⠀⡸⠀⠀⠱⠄⡤⠔⠁⢀⣀⠤⠐⠛⠀⣀⣸⣿⣿⡶⣦⣤⡄⠀⠀⠀⠈⠻⣿⣿⣿⠏⠀⡞⠳⣤
⠀⠀⠉⠉⠉⠉⠀⠀⠀⢀⡀⠠⠖⠚⣉⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠘⣈⠀⡀⠀⢀⠈⠻⣯⠀⢰⣿⣤⡄⢾⣯
⠀⠀⠀⠀⠀⠀⠐⣠⡂⠀⣧⣴⣶⣿⡿⠿⠿⠟⠛⠛⡉⠉⢹⣿⠏⠀⠀⠀⠈⢀⣀⠠⣾⣷⠀⢿⠀⠸⠟⠋⠀⣻⠇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣩⡭⢁⡀⠀⠀⠀⣤⣀⣼⠃⠀⣀⠀⠘⢸⡀⠀⡤⢜⠿⠃⣠⠔⢒⡙⠒⢄⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡀⠀⡔⢛⣤⣼⣒⣦⣽⣮⠠⢀⠈⠛⠃⠀⠀⡀⠈⠑⣎⠱⢤⣇⢸⠀⡜⢡⣾⣿⣿⣿⣶⡆⠀⢵⣚⡄
⠀⠀⢠⣶⡈⠃⢸⡿⠟⠛⢿⠉⠀⢀⣈⠳⣇⠁⢰⢀⣴⠿⣿⢿⣿⣜⠳⢦⣿⣯⠀⣇⠸⣿⣿⣿⣿⡿⠃⠀⢸⣿⡇
⠀⠐⢻⡿⠷⠀⣼⣦⡦⢀⣼⣷⣦⣈⢉⡸⠻⣧⠸⠠⣿⠿⣿⢸⢹⣿⠀⠸⣟⠝⢷⣼⣦⣉⣛⣛⣉⠤⠚⣖⣦⡀
⠀⠀⢸⡇⠀⢠⡿⠙⠲⣼⣿⣿⣿⡿⢫⠀⠀⢹⡇⠅⣿⢙⢻⣿⣿⣿⡇⠀⠹⣮⣮⣿⠷⣢⣉⣭⣭⣭⣵⢹⣿⣻⡄
⠀⠀⢸⡇⠀⣸⣇⠀⠀⠳⢭⣝⣋⠤⠎⠀⢀⡜⠗⣆⢹⡟⠟⡿⡟⣿⣇⢸⠀⠈⠛⢻⡜⢕⡂⣙⣏⣈⠜⢀⣿⣿⡇
⠀⠀⣸⣇⢸⠁⡧⣷⡀⠀⢰⠋⠁⢈⠃⠀⢸⣿⡗⣼⢾⣿⡚⣳⣛⣿⣿⣌⠂⠀⠀⠈⣷⣶⣦⣤⣶⣶⣾⣿⣿⡁⡇
⠀⠀⡿⣿⠈⠧⠇⣛⣅⢀⠘⢲⠀⠸⢤⠀⠀⣤⠀⠀⢸⣿⣧⣿⢭⡽⣿⡎⡆⠀⠀⠀⢸⣭⠽⢿⠛⣿⣷⣺⣅⡓⠁
⠀⣸⡟⣿⣆⠀⢘⡛⢩⡆⢀⠼⠴⣠⠃⠈⢻⣿⠟⠀⢸⣿⢿⢸⢠⠃⣿⡇⡣⠀⠀⣴⢿⣿⣿⣿⡆⣿⣿⣿⣿⡿⡆
⠀⠉⠉⠉⠉⠃⠘⠁⠀⠀⠀⠀⠘⠃⠀⠀⠈⠈⠀⠀⠈⠻⠶⠾⠿⠿⠟⠀⠁⠀⠀⠛⠛⠛⠛⠛⠁⠉⠋⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⣶⣤⣄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡴⡿⢹⣿⣕⡄⢀⣴⣟⣿⢿⣿⡟⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⠿⣝⣧⣿⣷⣿⡿⣿⡿⠭⠿⣿⣽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠪⣿⢿⠟⠁⠀⠀⠀⠀⠉⠺⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠆⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣠⠴⡄⠤⡀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠂⠀⠀⠉⡏⠉⠉⠉⡟⠁⠀⡴⠋⢉⡇⠀⡇⠀⠀⢸⠀⠀⡰⢁⡔⠁⠀⠸⡄⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢰⠗⠚⠉⠀⠀⡇⠀⠀⢸⠀⠀⡗⠉⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠘⣆⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢧⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠙⠒⠤⠴⠊⠀⠀⣎⣱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠣⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⣠⢀⣾⡛⣣⣷⢦⡔⠉⠁⣨⣽⣶⡀⠀⢀⣴⢻⣴⠶⢤⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣤⣽⣷⣿⣟⣩⢤⣰⡿⣾⡇⠰⠿⡍⣹⣯⡩⢶⡯⣭⣉⣁⡂⢄⢇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⡆⡇⢠⠃⠀⠈⢃⠈⠸⠷⡬⢥⣻⠟⢐⠱⡐⡃⢕⣤⣷⣶⣧⡊⢽⡳⢜⣵⡿⣄⠀⠀⢠
⠀⠀⠀⣀⠤⠒⠒⠒⠟⣯⡤⡷⢺⠀⠀⠀⠈⠻⢶⡶⣿⣿⡇⢠⠮⣤⣠⣜⡀⣉⣿⡝⣿⣿⣿⡿⢸⠁⢘⡾⣷⣤⡾
⠀⢠⠚⡡⠒⠈⠉⠙⡿⣟⣿⣁⣺⠛⠒⠢⠤⢴⣺⣁⣿⣿⣏⣀⢷⡄⠹⢹⣷⣤⡀⢻⣶⢶⣾⣿⣿⣶⣿⡇
⠀⣬⢸⠁⢀⠤⢄⠀⠈⠚⠛⠋⠉⠑⢲⣦⡤⣤⣬⣿⣿⣿⢞⣿⡿⢋⣊⡉⢻⡿⠋⢡⣶⠦⠟⣿⣧⣍⠹⣿⠆
⠀⠘⣞⠦⣧⣧⢈⠇⠀⠀⠀⠀⠀⠶⢫⡋⠀⠀⢈⠟⣡⣶⣿⣿⢺⣯⠟⢿⣘⡆⠀⠈⠙⠛⠋⠛⠏⠛
⠀⠀⠈⠑⠒⠒⠁⠀⠀⠀⠀⠀⠀⠃⡻⠃⠀⠀⠃⣀⣁⡿⠋⢳⢎⡻⠇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢸⣆⡤⣤⠀⢀⣤⡀⠀⢀⡀⠀⠀⠀⢀⡀⡤⣄⠀⣴⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠛⠀⡿⢀⣏⣀⡷⠀⢸⢇⠞⢹⡦⣾⠏⢀⣾⠻⡇⠀⣸⡇⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠓⠊⠈⠉⠧⠄⢹⡟⣴⡽⠃⣿⠀⠷⠏⠀⣉⡁⢼⠧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡽⠀⠀⠀⠀⢀⡴⡄⠀⢸⠇⠀⠀⠀⠛⠀⠀⡴⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠉⠀⣀⣀⠀⢀⣞⠜⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠳⠄⠤⠞⠁
⠀⠀⠀⠀⠀⡔⢹⡇⣠⣄⠘⣧⣄⣷⣾⠁⠀⠶⠀⣠⢤⣾⠁⢀⠴⣄⠀⢠⡄⠀⡀⢰⡋⠿⠀⣸⠂
⠀⠀⠀⠀⠀⠛⠀⡷⢀⡟⠀⣏⢀⠏⣿⠀⣰⠇⢰⣇⣰⡇⠀⣿⣠⡏⠀⣿⠀⣰⠏⡘⢷⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠜⠉⠉⠀⠈⠉⠘⠂⠊⠉⠘⢧⡴⠋⠀⠓⠊⠈⣹⠏⢾⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⣤⣤⣀⡀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠉⠙⠛⢶⣴⣿⣒⣉⣉⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠖⠒⠉⢠⠋⣬⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃



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

⠀⠀⠀⣀⡤⠤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⣰⢛⣁⣠⣤⣶⣁⠝⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡵⣫⠑⣿⡆
⠀⣇⣼⠋⠀⠀⠈⢻⠼⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢾⠁⠙⠾⠟⠁
⠀⢻⣿⣄⣀⣠⢀⡟⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡄
⠀⠀⠙⠒⠋⠁⡼⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⢷⡀
⠀⠀⠀⠀⢀⡞⠁⠄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠹⣄⠰⠳⡀
⠀⠀⠀⠀⡾⠁⠀⢰⠏⠀⠀⠀⠀⢀⠴⠚⠋⠉⠉⠉⠩⠛⠲⣄⠀⠀⠹⣆⠀⡻⣄
⠀⠀⠀⣼⠁⠆⢀⡏⠀⠀⠀⠀⣰⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⣇⠈⠸⣆⠀⡠⠒⣈⣁⠢⡀
⠀⠀⢠⡏⠰⠂⣼⠀⠀⠀⠀⢀⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⡴⢤⣿⠀⠀⠀⢸⡄⠀⠺⡎⢰⠟⠁⠀⣷⡇
⠀⠀⢸⡇⠄⠀⣯⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣿⠀⠀⠀⠈⣇⠀⢸⠃⡿⠀⠘⠞⠋
⠀⠀⢸⡏⠀⠀⠹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⢸⠀⣷
⠀⠀⠀⣧⠠⠀⠀⢹⣦⣄⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠇⠀⢸⠀⢹⡀
⠀⠀⠀⠘⢷⡌⠁⠩⠄⠁⠅⢐⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠸⣄⠀⣀⡴⠋⠀⠀⣼⠀⠨⡇
⠀⠀⠀⠀⠀⠙⠿⢶⣤⣥⣶⡿⠃⠀⢀⠐⠐⡆⠀⢰⡄⠀⠙⢦⡉⠉⠉⠀⠀⠀⣰⠟⠀⠀⡗
⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⢁⡀⣀⣤⡎⠀⢰⡇⠀⠀⢻⣆⣀⠈⠙⠓⠶⠤⠴⠞⠁⠀⢀⡽⠃
⠀⠀⠀⠀⠀⡾⠁⠀⣰⠾⣋⣙⠛⢿⡇⠀⢸⡇⠀⠀⢸⡟⠻⣤⣠⠄⠀⢀⡀⣀⣠⡶⠋⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠸⡇⠀⡾⢷⠋⠀⠈⢻⡄⢻⡄⢸⡇⠀⠀⢸⣇⠀⠀⠙⢿⣟⠛⠉⠉⠀⠀⠀⢰⣿⠻⡛⢿⣆
⠀⠀⠀⠀⠀⢻⡀⠹⢌⡀⠀⠀⣼⢇⢸⠇⣸⠇⠀⠀⣸⣿⣷⣶⣤⣤⣉⠛⠶⣄⡀⠀⠀⠀⠛⠶⠃⣸⣿
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠓⠫⣭⣳⠟⠀⡿⠀⠀⣰⠟⠈⠛⠿⣶⣍⡟⠻⢦⣄⠙⠷⣶⠶⠶⢶⠟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡾⠟⠋⠁⢀⡾⠃⠀⣴⠏⠀⢀⣤⠖⠎⠛⢿⡻⠿⠿⣷⡄⠙⢮⡴⠶⠋
⠀⠀⠀⠀⢀⡤⠶⠚⠋⠁⢀⣀⡤⢴⡿⠁⠠⡾⠁⠀⢰⢋⡴⠚⠉⠛⣆⢳⠀⠀⠀⠙⣆⠈⣆
⠀⠀⠀⣰⠋⣠⡴⣶⢾⡿⢭⡁⠀⢸⡇⠀⢸⡇⠀⠀⠸⣸⡄⠀⠀⠀⣸⢸⢠⢆⣤⠀⢸⡀⢸⡀
⠀⠀⠀⡇⠀⣏⠐⣧⠋⠙⢷⢹⡄⠈⣇⠀⠈⢧⡀⠀⠀⠉⠉⠁⢀⡴⣣⠟⣿⡇⠸⠃⢸⠇⢸
⠀⠀⠀⠻⣄⠻⢤⣀⢀⣠⡜⢸⠃⠀⠘⢧⡀⠁⠉⠓⢶⠦⠶⠞⣋⣴⠟⠀⢻⡳⠤⠤⠎⢠⠇
⠀⠀⠀⠀⠉⠓⠦⢬⣥⡤⠔⠃⠀⠀⠀⠀⠙⠓⠶⢤⣤⣤⣤⠞⠋⠁⠀⠀⠀⠈⠑⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣤⣄⣤⣄⣿⠀⠀⠀⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠈⠻⣿⡿⢻⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠁⠀⠀⣈⠄⢸⡇⠈⠀⢸⡇⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢤⠀⢰⡇⠀⢸⠇⠀⠀⢸⠃⠀⠈⣤⠀⢹⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⣼⠀⠘⠳⠄⢺⡆⠀⠠⢻⡀⠀⢘⡇⢀⡾⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠃⠀⠀⠀⠑⠀⠀⠈⣇⣀⠂⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠠⠤⠤⠤⠔⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠉⠁⠀⠉⠐⠀⠀⠋⠆⠀⢰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢠⠂⠀⠐⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠈⠀⠂⠀⠀⠀⠀⡀⠀⠀⢀⠞⠀⡠⠈⠀⠀⠀⠀⠀⠀⢉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⢠⠀⢘⢠⠀⠳⠁⠰⠀⠠⠀⠈⠀⠀⠂⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⠀⠆⠀⠄⠀⠀⠀⣂⠡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠎⠀⠀⠀⠐⠀⠀⠀⡐⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢀⠀⠀⠀⣀⡀⠀⠀⢁⡀⠁⠀⠁⠀⠑⢌⠁⠀⠉⠀⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠆⠐⣄⠀⠀⠀⢀⠀⠁⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⢀⠜⠁⠀⠀⠀⠑⠄⠀⠀⠀⠀⠂⠀⠀⢃⠀⠐
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠠⠀⠀⠀⢀⠀⠀⠢⠀⠀⠀⠀⠀⡠⠄⠠⠙⠤⡐⠠⡐⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠄⠀⠂⠀⠡⢒⠀⠂⠀⠀⢂⠄⠀⠀⠀⠠⠂⠐⣶⡤⢤⡀⡀⠀⠀⠀⠀⠨⠆⠀⠀⠤⣄⣠⠀⠄⠀⠀⠀⠠⠀⡈⣤
⠀⠀⠆⠈⠄⠤⡤⠔⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⢤⡗⠁⣛⠘⡄⠀⠀⠀⢀⠀⠀⠀⠀⠶⠤⠤⠀⠄⠀⠉⠀⢀⡄⡗⢨⠷⠆
⠀⢀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠀⠂⠀⡈⠀⡀⠀⠀⠀⡠⠋⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠲⠐⠀⠒⠒⠂⠉⠘⠁⠪⢂
⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠀⣌⠉⢉⠁⠀⠀⠀⡠⠀⠀⠀⠠⠔⠀⠀⠀⠀⢀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠈⠦⡀⠀⠀⠀⠈⠑⠤⣀⡀⢀⣀⣀⢔⡈⠤⠒⢉⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠄
⠀⠀⠀⠀⠀⢄⠈⠂⠀⢈⠀⠁⠐⠒⠒⠒⠒⠒⠈⠉⠉⠀⠠⠀⠀⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠈⠀⠈⠁⠀⢈⣁⡀⠤⠄⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣔⣐⡙
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⡈⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⡐⣲⠻⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠠⡒⠃⠀⠀⠀⠸⡀⠀⣀⣀⡀⠤⢠⣇⠐⠂⠒⠀⠀⢢⣇⠶⠠⠠⠤⠤⣄⠀⠋⠗⠀⠀⠀⠀⠀⠀⠀⢀⣜⠔⡰⠂⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢀⣀⠤⢲⣇⢈⣀⢀⠤⠴⠀⠛⠂⠂⠐⠐⠂⠐⠓⠒⠀⠂⠤⠤⠟⠀⠑⡚⠳⠀⣀⣀⣀⣀⠐⠃⠛⢙⣿⣦⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⣀⢤⡲⠁⣄⡄⠦⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠪⠩⠙⠋⠁⠀⠚⠉⠉⠙⠍⠉⠉
⠀⠀⠀⠀⠀⣐⡬⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⣿⣿⣿⠏⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠂⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⣠⣤⠤⠀⣴⣄⠀⠀⠙⢿⣿⣿
⣿⣿⡏⠀⠀⣿⣿⣠⣾⠀⢀⣤⡄⢀⣤⡤⢀⣀⡀⠸⣿⣿⡀⣿⡍⡟⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢠⣄⠀⠀⠀⢿⣿⣸⢇⣿⣿⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⠀⣹⣿⢷⣧⡈⠻⣧⣾⡻⣷⣟⣿⠋⠀⠀⢹⡿⣷⡾⣦⢾⣤⢶⣼⣿⠤⣿⠀⠈⣿⣿⣵⣦⠀⠀⢰⢿⡛⡋⣿⠋⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⣨⣭⣿⣰⣼⡏⢺⣯⣯⣹⣾⡏⣍⣿⠀⠘⣸⡏⣰⣄⢹⡋⠛⡛⣦⢸⡟⢻⣄⣂⣿⢠⣅⣿⠁⠀⣦⣿⣾⣿⠿⣧⣤⠀⠀⠀⢿⣿
⣿⣿⠀⠀⢠⣷⣞⢻⣿⡿⢹⣾⢿⣻⣿⣼⢿⣯⣷⣴⣶⠻⢿⣇⣿⠞⣷⣶⢳⡿⣷⡜⣷⣧⣾⠿⣿⣹⠾⣳⣌⣿⣿⣬⣯⣴⣿⢿⡀⠀⠀⢸⣿
⣿⣿⠀⠀⣼⠛⣿⣶⣿⣵⣾⠻⣧⣿⡟⢻⣦⣼⣵⡟⣿⣧⣈⣻⠋⣰⡟⠹⣏⣴⣿⠻⣿⠛⣿⣤⣙⣏⣠⣿⢿⡟⣿⣿⣿⣿⣿⣾⡇⠀⠀⣿⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣟⡛⣿⣿⣟⣻⣿⣿⣿⠷⣿⣿⣿⣿⣿⣿⠖⠺⣿⣿⣿⡚⣿⣳⣿⣿⣿⣿⣿⣿⣼⣧⣽⠻⣿⣿⡿⣿⠃⠀⠀⣿⣿
⣿⣿⠀⠀⠿⡿⠿⢿⣿⣿⡿⠷⢿⣿⡏⠿⢿⣿⡿⢠⣴⣿⣿⣿⣿⣿⣸⣿⣽⣿⠟⢳⣿⣿⢿⣿⣿⣿⣿⣷⡼⡇⠯⠀⣿⡷⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠈⢻⣷⡄⢠⣿⣿⡿⠙⠛⣿⣿⣿⡟⠙⣿⣿⡿⢠⣿⡟⠀⠀⢹⣿⣿⡟⠙⠛⠃⠀⢰⣿⡇⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠾⠟⠃⢸⣿⣿⠃⠀⠀⣾⣿⣿⡏⢰⣿⡿⠁⠘⠛⠛⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⡧⠀⠀⠀⠀⣀⣿⣿
⣿⣿⣿⣦⣤⣤⣤⣿⣿⣷⠄⠀⠀⠀⠀⣀⣼⣿⣿⡄⠀⠀⣿⣿⣿⢠⣿⣿⣿⡄⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢠⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⣤⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣤⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣷⣶⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠒⠀⡅⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢
⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⠀⠢⡈⣦
⠀⠀⠀⠀⠀⡠⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⡤⣉⠁⠈⠈⠻⡌⢣⣀⢄⣀
⠀⠀⠀⠀⢸⠀⠈⡆⠀⢀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⢰⢆⢸⣾⣿⣦⣤⡄⠀⠸⠈⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠀⢀⠐⠋⠀⠤⠒⠉⠉⠙⢆⠀⠀⠙⢆⡥⣼⢋⠁⢙⣿⡿⠃⠀⣸⠁⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠐⢦⠺⠅⠑⠈⠀⠀⣾⠆⠱⠘⣆⠀⠀⠾⠟⢩⡷⢯⡂⣤⣨⠀⠐⠀⠀⢀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⡦⠀⠆⢱⠀⠀⣟⠘⢧⠀⣿⡶⡖⠺⠠⠒⢀⣺⠁⠉⠃⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠂⠌⡁⠀⠈⠳⢌⣡⠟⠟⣦⣀⣠⡖⢉⣀⠄⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⢢⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⢇⠁⠚⢻⠙⠓⡦⢤⠀⠀⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⡀⠾⠛⠀⠀⠀⡖⠈⠨⠏⠆⠡⣾⠇
⠀⠀⠀⠀⠀⡇⢐⠂⠔⠂⢀⣤⣶⠆⣤⢶⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠂⢉⠀⢴⢧⠎⠠⢴⠞⠋⠀⠀⠀⠀⢀⣤⣶⣾⡶
⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⣿⡷⠊⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠁⠀⠀⠀⣀⣤⣄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠁⠀⠀⣀⣴⣿⠉⡙⡿⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢀⣴⣻⠿⠿⣿⠟⠁⠀⠀⣠⢾⣿⠟⡝⠂⠀⡃⠀⢠⡅⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢠⣿⠋⠀⣠⠟⠁⠀⢀⣴⣯⣟⢹⣿⠧⡟⠦⠄⡇⠀⢸⡉⣳⡆⠈⣧
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⣿⡇⣠⠞⠁⠀⢀⣔⡏⠋⠘⡯⢼⣿⡆⠹⡦⠀⠱⣄⠀⠉⠉⠀⡴⠣⠂
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠈⢩⠞⠁⠀⢀⡴⠛⢸⡤⡀⠤⡃⢈⣸⣄⢈⣷⣎⠀⠀⠀⢀⣄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⣠⠋⠀⠀⡠⠊⢀⠔⣽⡶⡦⠸⠗⢦⠘⢁⠡⠰⡿⡋⠲⠄⣠⠠⠢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠰⢲⡰⠟⠀⠠⢄⠀⠙⠀⠘⢻⢒⡂⠤⠶⡑⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢠⠏⠀⢀⡤⣶⣿⠆⠖⠀⠰⢳⡂⠻⣇⠀⢔⠌⡢⠄⡀⠀⠑⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⢡⡐⢛⣩⡦⠟⢡⣞⣐⡄⡐⠓⠚⡖⠈⠘⠆⢀⠉⠶⢤⣄⣂⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⠦⠿⠓⠉⣠⠾⠽⠉⣨⡙⢣⣀⠀⠹⡀⠀⠀⠀⠀⠓⠘⠛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⢄⠀⠀⠐⠫⣽⠀⠙⠀⠻⠈⠑⢽⡇⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠑⠐⠂⠛⠺⠄⠂⠁⠂⠀⠛⠉⠁⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠮⢀⠍⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⢸⢹⠀⠀⣿⢹⠀⢰⠋⠷⠘⡋⡇⢸⢹⠀⠈⢹⠀⡎⢹⠀⡏⡇⣻⢹⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢺⢴⡓⢻⡜⠋⠉⢩⢼⠉⠉⡤⡜⠉⣷⠏⠙⠼⠉⢡⢼⠉⢥⢈⠉⡧⡍⠸⠰⠄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠀⡀⠀⠀⠌⠀⡀⠄⠄⡀⠀⢀⡀⠐⠁⠈⠐⢀⠂⡀⠀⠀⢀⡄⠈⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⢆⠀⠁⠂⠀⠁⠀⠃⠀⢁⠈⠁⢀⠀⠀⠄⠂⠀⠨⠁⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠐⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⠀⠀⣀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢃⠈⠘⣁⣈⠈⣀⣈⠨⠀⠀⢸⡀⡀⠠⠀⣰⢀⣀⣠⠀⠀⢀⠀⡁⠸⢨⡀⢈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⡀⣀⣀⠀⣀⢀⠀⡀⣀⠀⣀⡀⠀⣀⡀⠀⣀⡀⠀⣀⠁⠀⣀⡀⠀⣀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⠀⢅⣀⡀⢀⣀⠀⢀⣀⠀⠀⠨⠘⣀⡸⢨⣀⣠⠀⣁⡀⠀⢅⡈⠀⣀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡀⢀⣀⣀⢀⠀⠀⢀⠤⡀⠠⢤⠀⡠⠤⠀⣀⢀⠀⡀⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⡄⠠⡀⠀⠈⠄⠀⠈⠄⠄⠠⢠⠀⠧⠀⠇⠧⠤⠀⠦⠤⠀⠤⠴⠀⠇⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣄⠀⠀⢀⣀⡈⠉⠉⠙⠛⠉⠉⠩⠭⠉⠉⠉⠉⠉⠀⠀⠀⠤⠤⠀⠀⠀⠈⠉⠉⠄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣄⣀⣀⣐⣒⣒⣒⣒⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣀⣠⡴⠖⠚⠋⠉⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄⡀
⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⣼⡡⠂⣠⢤⠀⠀⠀⠀⠀⠀⣠⢶⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠙⠆⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⡠⠔⢺⡏⠀⣼⡧⠞⠀⠀⠀⠀⠀⣼⠃⡘⠀⣼⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⡧⣴⡟⠀⢀⣴⠶⡄⠀⣸⣯⠀⠀⣼⡇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠇⠀⣾⡇⠔⠃⢠⣿⠃⠀⢰⡿⠉⠀⢠⣶⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣼⠇⢸⣿⠀⠠⢿⡁⢀⠔⠀⣿⠀⠀⢾⡇⠀⢠⣿⡇⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣁⡴⠋⠀⠈⠛⠚⠁⠀⠁⠀⠀⠀⠙⠂⠀⠈⠛⠒⠋⠻⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣴⠏⡰⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣿⣤⠄⣸⣯⠞⠁⠀⠀⢀⣤⢦⠀⢙⡾⢥⡄⠀⢀⣴⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢠⣿⢁⠔⣿⠀⢀⣾⡣⠊⣠⠎⢠⣿⠇⠀⣿⠇⠊⢀⠠⠂⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡴⢹⣿⠊⢰⣿⣠⠞⠿⠤⠚⠁⠀⠈⠿⠴⠚⠻⠶⠚⠁⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⣀⠀⠀⠀⠈⠉⠀⠈⢁⣀⣤⣭⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠒⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠈⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⡦⠸⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠐⠒⠒⠛⠛⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⡾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡟
⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠐⣦⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠹⢦⣤⣤⣴⣶⠆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠂⠀⠀⠀⠀⠀⠀⢸⣿⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡾⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⡾⢡⣶⡿⣦⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⢠⣿⡇⢠⡾⢡⣿⠏⢰⡇⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⣿⠀⠀⠀⣀⡀⠀⢠⣾⢏⣽⠇⢸⣿⣵⠟⠁⣼⡟⣠⡟⢠⣶⠶⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿⠀⢸⣿⡀⣠⡾⣿⡇⠀⣼⣿⠟⠁⠀⣾⣿⠁⠀⠀⣿⣿⠋⠀⢻⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⣿⣷⠟⢰⣿⠁⠀⣿⣿⠀⢀⣾⢿⣿⠀⠀⢀⣿⡇⠀⠀⢸⣿⣰⡏⠀⠀⣀
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠙⠋⠀⢸⣿⣀⣾⠛⠿⠿⠟⠁⠸⣿⣄⣤⡾⢿⣧⣀⣴⠟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⣠⠀⠀⠈⠛⠛⠁⠀⡀⠀⠀⠀⠀⠙⠛⢉⣤⠀⠉⠋⠁⢀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠠⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠐⠆⠀⠀⢻⡆⠀⢀⣶⡄⠀⠀⠁⣠⣴⠶⠶⠷⠶⣾⣿⣦⣤⣶⠾⠛⣡⣤⡀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⣍⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⣾⡟⢹⡇⢠⣿⣿⣠⣶
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⣰⣆⠀⠀⢸⣷⠀⣸⣿⠀⠀⣿⡇⠀⠀⣀⣀⠀⠀⣿⡇⠀⠀⠀⣼⣿⣵⡟⠁⢀⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣿⡇⠀⠀⢸⣿⠀⣿⣿⠀⠀⣿⡇⢀⣼⣿⣿⠀⢸⣿⠃⠀⠀⠀⣿⡿⠋⠀⣠⡿⠃⣼⣿⠁⣀⠀⠀⠶
⠀⠀⠀⠃⠀⠀⢸⣿⡇⠀⠀⣠⡿⣿⡇⠀⠀⢸⡿⠀⣿⣿⠀⣸⣿⣇⣾⢣⣿⡏⠀⢸⣿⠀⠀⠀⢀⣿⣷⣠⣾⠟⠁⢠⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣾⠟⠀⣿⣧⠀⠀⣼⠇⠀⢻⣿⣴⡿⣿⣿⠇⣼⣿⠀⠀⣼⣿⠀⠀⢀⣾⠋⠙⠉⠁⠀⠀⠸⣿⣇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠘⠿⢷⠾⠋⠀⠀⠀⠈⠉⠀⢿⠟⠀⣿⡏⠀⣸⢿⣿⡆⣠⡾⠃⠀⠀⠀⣤⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡾⠋⠀⠛⠿⠛⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠴⠾⢶⣦⣤⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠻⣦⡀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠉⠀⠀⠄⠀⢉⡙⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⠳⠛⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢭⠀⠀⢀⠲⠀⠠⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢠⡀⠈⠄⠄⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣇⠀⠆⠀⠁⠉⠰⣤⠀⡱⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⡎⠀⠀⠀⠸⣤⠀⠀⢠⠀⠀⢀⠄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⣋⡤⠴⠋⠈⠎⠉⠀⠀⢠⣼⣿⠁⠈⠰
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠜⠣⠁⠀⠀⠀⠀⢁⠩⠂⣴⠀⢶⠙⣦⡀⢠⠉⠀⢀⣔⠀⠀⠀⣠⢤⣦⠚⠋⠠⠀⠈⠁⠀⠀⢻⣑⠚⠂⢀⣱⢒⣩⡾⠉
⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠀⠀⡠⠒⠁⠈⢀⣴⠟⠛⠲⢶⠂⠀⠸⢯⠀⠀⠈⠟⠑⠿⠊⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢣⡀⠃⠤⢰⠋⠉⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⣰⠯⠃⠀⠀⠀⢸⠂⠀⠀⠀⠑⠄⠀⠈⠀⠤⡀⠀⠀⠀⠁⣤⠃⣰⠄⠰⠀⠀⠀⢂⢤⠉⢠⡌⠕⢀⣰⡄⡠⠚⠢
⠀⠀⠀⠀⠀⡀⠰⢷⡆⠀⠀⡀⡢⠄⠚⠻⠂⠈⠫⣿⠷⣴⠞⠀⠀⠀⠀⠂⠐⢄⠀⠀⠢⠀⢠⠀⠆⣂⠉⢁⡠⡄⠀⠄⠠⠔⠁⠀⠀⠁⡤⠤⠠
⠀⠀⣔⣨⣤⣉⣆⠀⠀⠀⢐⣁⣀⣠⠞⠘⠛⣀⠀⠀⣙⡚⢄⠀⠀⠀⠀⠀⠀⠀⢒⢠⡖⡣⠈⠌⡥⡉⠄⠈⠀⠀⠈⡄⠄⠠⠀⣀⠒⠛⠁⠀⠀⢡
⠀⠒⣈⠁⠀⠉⠈⡁⣀⠰⡈⠁⠋⠁⠀⠀⠾⠟⠀⠺⢛⣇⠚⠀⠀⠀⠀⠀⠀⢀⡀⠂⠁⠀⠀⠀⠑⢄⡀⠢⠀⠀⠀⢠⡁⠀⠠⠀⠀⠀⠀⠀⠀⢠
⢐⣀⠂⠄⠀⠤⠄⡛⠑⠂⠈⠉⠨⠕⠄⡀⠀⠀⢂⠠⠂⠀⠀⠀⠠⠽⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠲⠀⠰⠋⠀⣔⠒⡡⠤⣀⠠⠠⢆
⢃⠀⣈⠋⠒⢒⡨⠄⠀⠀⠀⠔⠀⠀⠀⣤⣟⣀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠅⠀⢨⡈⡿⠓⠦⠒⠂⡄
⠐⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠀⠝⠉⣷⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠢⠄⠀⠀⠀⠀⠀⠀⠉⢵⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠦⢦⡶⡇⠀⠀⠀⠉⠁⠀⠉⠙⣮⣈⠀⠒⣄⠀⠀⡠⠊⢱⣆⠀⠰⠒⠋⠙⠁⠀⠠⡀⠀⠀⠀⠀⠀⢈⠉⠀⠈
⠀⠀⠈⠢⠀⠀⠀⠀⠠⣎⡃⠁⠀⠆⠓⠂⠐⠀⠀⠀⠀⠀⠀⠀⡀⠻⠿⠯⠯⠦⢀⡢⠄⠀⠈⠛⠁⠀⠈⠓⢫⡄⠀⠙⠀⡀⢠⡀⠂
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡐⠁⠈⠧⣜⠠⠔⠀⠀⠀⠛⠉⡶⠉⠁⠘⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠒⠄⠠⠄⢀⡉⢤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⢀⡉⠘⠃⡀⢒⡈⢡⡚⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⢹⡇⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⣴⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣴⠋⡏⠀⢀⣾⠃⠀⣤⠴⠶⢛⣿⠋⠉⠁⠀⠀⠀⢀⡼⣣⠏
⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⡿⠁⣠⠶⠶⢀⡴⣿⠁⢀⣤⡶⢶⡄⠀⣼⡷⠃⢀⣠⢶⡶⠀⣠⠴⣷
⠀⠀⠀⠀⠀⣿⢀⣠⡾⠋⠀⢸⣁⣠⠴⠋⠀⢿⡀⠈⣿⢀⡼⠃⠀⣿⣺⠟⠻⣟⣩⡴⠚⠁⣼⣃⡴⠋
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠢⠄⠔⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑
⠀⠀⠀⠀⠀⠀⣄⡀⠑⠓⢜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠁
⠀⠀⠀⠖⠀⠠⡆⠁⠀⢠⠀⢧⢠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠄⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠒⢤⣌⢂⠌⠓⠾⢿⢷⠾⠋⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠀⣄⣀⡆
⠀⠀⠀⢊⠁⠖⢂⡉⣻⠀⠰⠁⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⡅⠀⠰⠁⠤⠖⠀⠠⣀
⠀⠀⠂⠤⠂⠀⠉⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⣴⠒⡚⢀⠀⠈⠁⠀⠘⠉
⠀⢠⠚⠀⢁⠶⠺⠑⠃⢠⠠⠀⢀⠀⠄⠀⠒⠤⣤⡜⠂⠀⠠⠐⠂⠀⠈⣀⠘⠢⠀⠐⠀⠀⠂⠐⠋⠈⡁⠲⠆⣉⠠
⠀⠀⠀⡀⠚⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⡞⠛⠸⠀⠀⠀⠒⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠐⠓⠀⠘⠀⡉⠰⣦⡐
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠄⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣎⡷⠀⠀⠀⠀⡠⢄⠀⠀⣀⣀⣀⠀⠉⠡⠑⣄⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠐⠄⠀⢀⣤⢻⢺⡖⠋⠉⠀⠀⠈⢀⡀⣈⣸⢋⠴⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆⠀⠘⠧⠑⢊⡍⡽⠀⠀⠀⠀⠀⠀⠀⡟⠨⢹⡜⠶⠞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣂⢄⢀⠀⠁⢀⠄⡤⢄⠉⢙⠇⠀⠀⠀⠀⠈⠀⣀⣧⠀⠈⠟⡶⢦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡏⠻⠉⡠⠀⠀⠁⢰⡀⠀⡰⠚⠀⠀⠀⠉⠀⣀⣴⡿⢀⠛⠁⢀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠀⠼⠷⠀⠸⣂⢀⣉⣨⡍⠀⠀⠀⣠⢄⢈⣿⡿⠁⠀⠀⢤⠊⠀⢐⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⣶⢠⠄⠐⢙⢿⣿⡟⠁⠀⠀⠀⣠⣾⡿⢛⠀⠀⠀⠀⠀⠱⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡹⠀⠐⢁⣾⡿⠟⠀⡀⡀⠀⠀⠙⣩⡷⠎⠁⠖⠤⠔⠀⠀⡈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⡀⠀⠉⠀⠀⣀⣀⢀⣼⠞⠀⠀⠀⠀⠙⢏⠐⠦⣠⣄⢀⠄⠀⠑⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠬⡉⣿⠛⣿⣉⠝⠋⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⠈⠫⡒⠐⠈⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⡨⠋⣹⠖⠋⠀⠀⠀⣀⣀⣀⠀⠃⠀⠀⠀⠐⣀⠠⡀⠈⢆⠒⠋⠁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢃⠜⠁⢀⣤⠴⢚⠫⠉⠱⠀⣱⣀⠀⠀⠀⠀⠙⢃⡄⠀⠀⠐⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡅⠀⠀⠶⣏⡀⠀⢀⠀⠹⣁⡈⠁⢸⠂⠀⠀⠀⠀⢈⣇⢰⣄⣀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠀⡀⠀⡈⠉⠉⠉⠉⠑⢦⡀⠀⠠⠀⡀⠚⢦⠀⠈⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢰⣀⠨⣈⡵⠐⣆⣀⠀⠀⠀⠈⠉⠀⠠⣀⣂⠾⢠⡴⢁⡯⠔⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠒⣃⠀⠤⠰⠀⠀⣠⠀⠂⣸⡇⠐⢠⡇⠈⢃⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠧⣀⡠⠀⢀⠈⠉⢄⣠⡌⢉⠙⡟⠀⠄⡽⠛⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⣡⡀⠔⠀⠀⠀⠀⠑⠦⠔⠙⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⠃⠀⣸⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⡜⠀⢠⣿⠁⡜⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡼⠁⠀⣼⡏⣸⠃⣼⡿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣠⠊⠀⠀⠀⠀⢀⣴⠶⠄⠀⢸⣿⠜⠀⠀⠀⣿⡟⠁⢠⣿⡇⠀⠙⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠞⠁⣠⣄⠀⠀⢀⣿⠏⠀⠀⠀⣸⣏⠀⠀⠀⢠⣿⡁⠀⠀⢻⣇⢀⡎⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣿⠀⢀⡞⢹⣿⠀⠀⢸⣿⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⡇⡄⠀⠀⠙⠛⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣿⠀⠎⠀⣼⣿⠀⠀⠸⣿⠁⠀⢠⠇⠈⣿⡄⠀⡀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⡇⠀⠀⣿⡏⠀⠀⠈⢻⣇⢀⡞⠀⠀⠹⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢹⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠠⠁⠀⠀⢀⣴⡟⠂⢰⣷⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠈⠻⠇⣴⣿⢂⣠⠔⠂⠀⠀⣾⣯⠔⠀⠀⠀⠀⣾⣟⠠⠀⣾⣟⢀⣼⣿⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣰⣿⢻⠀⢀⣀⣀⢀⣤⣼⣿⠷⠋⣴⡶⠄⠀⣸⣿⢁⡴⢶⡄⠀⠰⣿⠃⠀⠀⠀⠉⢡⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣿⡇⡘⢰⣿⠟⠛⠋⣹⣿⠃⠀⣼⡿⠀⢀⠀⢹⣿⠟⠀⢸⣿⠀⠀⢿⣄⡜⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠹⡧⠁⣿⠇⠀⠀⢰⣿⡏⠀⢠⣿⠛⢲⠋⠀⢸⣿⠀⡠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⣰⠋⠉⢿⡀⣠⠎⢸⣿⠀⠀⠘⢿⣤⠏⠀⠀⠀⠻⠀⣡⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣴⠃⠀⠀⠀⠉⠁⠀⠘⣿⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠃







⣷⣶⣶⣾⣷⡿⣾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⡶⡾⣿⠿⠿⢶⣶⡾⠿⠶⠾⠷⠿⠿⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡀⠂⠀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣤⡶⠶⠶⠶⠶⠶⢶⣾⠟⠉⠉⠙⣿⡶⠟⠛⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣠⣤⣤⣹⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⢸⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢠⣾⠋⣻⡿⠿⠿⢶⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠠⠼⣿⣷⡟⢀⡀⠀⠀⠙⣿⣶⣶⡶⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣤⣾⣿⠷⠈⠁⢾⡿⢀⣿⠿⣦⡀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣄⠳⢢⣶⣶⣿⣧⣀⣾⠇⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⡃⣿⣷⠉⠋⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢾⡷⠂⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠹⣿⣆⣀⣀⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣀⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠛⠿⣿⣿⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠾⠁⠹⣇⣀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⡤⠒⠂⣄⠀⠀⠀⠀⠀⠘⢿⠁⠀⠀⠀⠠⣴⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⣾⠏⠀⠀⠉⠛⠒⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠙⠳⢦⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣾⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣶⡶⠿⠟⠉⣿⠇⠀⠀⢹⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⡟⠉⠁⠸⣯⡝⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠉⢿⣆⡀⠀⠈⣾⡏⠀⠀⠀⠀⠹⣄⣀⡀⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⢻⣷⠆⠀⠀⢽⣿⠟⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠰⣿⣥⣷⣴⡟⠀⠀⢀⣀⣀⠀⠈⠛⠷⠶⠴⠖⠋⠁⠀⠀⠀⠀⠀⠈⣿⡀⣴⣷⣾⠛⠂⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⣾⢿⣿⡟⠀⠀⢰⡏⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠛⣦⠀⠸⣿⣿⣿⠙⠁⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⡠⠒⣺⡟⠀⠀⠀⣈⣷⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠟⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢀⡤⠀⣰⠟⠀⣠⡾⠛⠋⠉⠉⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢿⣆⠀⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⡀⠈⢿⣶⣤⣿⣀⣰⠏⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⢀⢀⢤⡄⠚⠛⢛⣿⢿⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠸⣷⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣠⠞⢨⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⠀⢠⣶⡆⠀⠀⢀⡄⢠⣾⣿⢀⣤⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠁⠀⠸⣿⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⢿⠟⣿⣤⡿⠿⡿⠛⠛⠛⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⢸⣿⡀⡀⠂⠀⠁⠀⠀⠀⠀⠀⢀⣿⠿⠇⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⣤⠻⣿⣷⢤⡼⠃⠀⠀⠀⠀⣠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⠿⣷⣧⣬⣦⣤⣥⣾⠟⠁⠈⠻⢷⣦⣤⣀⣀⣀⣤⣤⣤⣶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠀⡄⠂⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠠⣰⠀⠀⠀⠔⠉⣀⣀⠅⠁⠀⠀⠀⠀⠀⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⠀⢀⣧⡠⠟⠛⠉⠀⠀⠀⠀⠀⢠⠀⡀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙⢯⣦⣾⠞⠀⠀⠀⠀⢀⡤⠆⠀⡀⠘⠁⠉⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣿⣷⣄⡀⠀⠙⢯⢦⣀⠀⢀⡴⡥⠀⢨⣡⣴⡿⢿⣇⣤⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠠⣿⣿⣷⡀⣀⠙⠎⢻⣟⡈⠄⣠⣾⣟⣆⢸⠏⠂⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⠂⠓⣀⢹⣿⣼⠦⢆⠺⠿⣿⣾⣿⠿⠷⠉⣼⣧⢤⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⢫⡰⣽⠁⠀⡠⣻⢈⢻⠿⣏⣥⠤⢞⠫⢌⢿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣵⠟⣦⢠⠀⡿⣮⣿⣿⡟⠀⠬⡄⢠⣖⣦⢰⠏⠙⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣽⡟⡏⠉⠀⠈⠙⣀⡈⠁⠘⣴⢀⠀⠒⠳⢄⠁⡁⠀⠈⡀⢄⢍⣡⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠅⠉⠏⢳⣿⣵⡞⠛⢲⣦⡉⢳⣀⢀⣿⠪⢧⣴⡀⠚⣅⠚⠶⢮⡷⢊⣄⡀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡻⠧⣾⠞⠉⢠⠼⢻⣿⣿⡔⣿⣿⣿⣶⣦⢤⣒⡡⠥⠐⠉⠙⠟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣯⣲⠞⢫⣾⣾⣿⣿⣿⣿⣯⡿⣿⡿⢺⣀⣀⡀⠤⠐⠂⠤⢡⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣶⡗⢹⣿⣿⣿⣿⢧⡟⡼⣿⠁⢠⠒⣂⣐⡀⢐⠁⠦⡄⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢹⣷⣿⡆⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⢱⡖⠁⠐⠊⢴⣞⣁⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⢳⢻⣯⣿⣶⣟⣿⣿⠥⠚⣩⠂⢠⠏⢈⡷⠐⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠈⣿⣿⡿⣿⡿⣿⣏⢐⡄⣇⠀⠈⠂⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠻⠮⠽⠛⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡉

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠖⣖⠑⠠
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⣠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠏⠛⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣄⡀⢀⠀⣀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠘⢶⣾⣿⠋⠸⢛⣙⢻⣤⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋
⢀⣦⡼⠟⠉⠁⢰⠈⠁⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡆
⠀⠁⠀⠄⠀⠐⣧⣲⡄⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣡⠞⠜⢻⠌⠉⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠈⠟
⠀⠀⠀⠤⠊⠉⢘⣶⡎⢠⣄⠀⣀⣼⣷⠀⠄⠀⠀⠀⣴⠀⠀⠀⠀⠀⠰⠂⠀⠢⠶⣀⠿⠇⠀⠀⠒
⠀⠀⠀⠐⠚⠻⡆⠸⢻⡿⠛⠛⠁⠠⠿⣅⣤⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠯⠉
⠀⠀⠀⠈⠁⠀⢘⠿⣯⠷⡠⠀⢀⠀⠐⠋⢹⣶⠊⡤⣀⠀⠀⠶⣠⢀⡤⠖⠂⣂⠠⡄⠀⠀⠘⢯⡍
⠀⠀⠀⠀⠀⠐⠛⢾⣿⣦⠀⠀⠀⠀⠀⠀⡸⢷⠶⡟⠛⠿⢦⣶⡾⣏⠀⠀⡰⠧⠸⠇⠀⣴⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠃⠀⠉⡍⠀⠀⢠⡿⠀⠁⡱⢾⣧⠀⠛⡹⠄⠎⠥⠔⠀⡔⠆⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠅⣠⣤⢷⣤⣄⠘⠳⠁⢄⠘⢼⣯⠋⠀⠙⠇⢤⣤⣀⠐⡉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠻⠪⢿⡏⢡⡔⠿⠆⣴⠋⠅⠙⠀⠀⢲⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠀⠀⠘⠁⠀⡎⠛⠣⢄⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⣠⣴⡶⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠉⠛⣿⠟⠁⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⣴⡾⠿⠶⣶⣶⣤⣴⣶⡾⠿⠟⣿⠇⠀⢠⡶⣶⡾⠷⠶⣦⡴⠾⠛⠻⣦
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⠀⠀⠸⣧⣿⠀⠀⠀⢘⡷⢶⡀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣠⣴⠿⣦⣄⣤⣾⠷⠟⠁⢰⡟
⠀⠀⠀⠀⠀⠀⢿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠷⠶⠶⠛⣿⡄
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠞⠛⠃
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⣸⣿⣀⣀⡀
⠀⠀⠀⠀⠀⢈⣿⣀⣀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠈⢹⡿⠉⠉⠁
⠀⠀⠀⠲⠛⠛⢻⣏⠉⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⣶⠛⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠷⣦⣤
⠀⠀⠀⠀⠀⢀⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣥⣀
⠀⠀⠀⠀⠀⠈⠀⠀⣹⣿⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣟⠀⠀⠉⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⢉⣻⣿⣿⣶⣶⣶⣶⣶⣶⡶⠾⠿⣿⠛⢻⣯⠀⠈⢿⣦⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣏⢡⣿⠃⠀⢻⣇⡀⠀⠀⢀⣼⡟⠀⠀⠹⣧⡀⠀⣻⣷⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢈⣿⡿⠟⠛⠿⣶⣍⠛⠛⠛⠛⢉⣴⡾⠟⠛⠿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡄⢠⣾⠋⠀⠀⠀⠀⠈⢻⣧⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⣠⣿⣧⣤⣤⣤⣿⣧⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⡶⠿⠛⠉⠉⠉⠉⠉⠉⠙⠛⠿⠶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⣧⠀⣰⠏⠉⣧⠀⢄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠇⢰⡏⠀⢠⠇⠀⣬⢩⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⡞⠀⣿⠀⢠⠎⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡠⠊⠀⢸⣇⡴⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡀⠀⢀⢴⡄⠀⣰⠏⣸⠃⢀⣿⠁⠀⠀⣾⠇⠀⠀⢠⡏⣴⠛⣆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢸⣿⡇⢠⠃⢸⡇⢠⣿⠋⠀⠀⣸⣿⠀⠀⢠⣿⠀⠀⠀⣿⠃⠈⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⣿⣧⠇⠀⣾⠃⢸⣇⠀⠀⡰⠁⢿⡀⢠⠎⢻⡄⢀⡞⣿⡄⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢻⣿⠀⠀⣿⣠⠃⠙⠂⠊⠀⠀⠀⠙⠁⠀⠀⠉⠉⠀⠈⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢼⣄⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⡀⠀⠀⠀⢠⠎⠀⣸⠀⠘⠽⠷⠽⡇⠀⢠⡞⠁⢸⠃⢀⡴⠋⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢉⡂⣀⠀⢠⡏⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⠎⢠⡿⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⢈⣄⠘⠃⠉⠀⣿⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇⠔⠁⢀⣾⠃⠀⡠⠋⠀⠀⡀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡎⢀⠁⠀⠀⠀⢰⡿⠁⠀⠀⠀⣴⠃⢠⡄⠀⠀⣿⡇⠀⠀⢸⣿⠤⠊⠀⠀⠠⠆⠄⢐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⣾⡇⠀⠀⣀⢴⡿⠀⣼⡇⠀⢠⣿⡇⠀⠀⣼⡇⠀⠀⠀⠀⡔⠀⠀⢉⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⡷⡆⠈⠁⢸⡇⢰⢹⡇⢀⡏⢸⡇⠀⠀⣿⣇⠀⠀⠀⡼⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠸⣧⢱⠀⠀⠈⠳⠋⠘⣧⡞⠀⠈⢿⣀⡼⠘⣿⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⢷⠀⠀⠀⠀⡀⠀⠀⠀⡰⠚⠂⢀⡀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣼⠆⠀⠀⣼⠁⠀⠁⡔⠁⠀⡤⢇⠁⢀⠤⠀⢀⡀⠀⠠⠀⠊⠁
⠀⠀⠀⠀⠀⠀⢀⣻⣇⣀⡴⢃⣠⣤⠂⠀⡤⠀⠸⠀⠀⠘⠔⠃⢀⠂⠀⠐⢇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠠⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠠⠄⠀⠬⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠐⠀⠄⠀⠠⠄⠀⠈⠀⠐⠀⠀⢀⠀⠀⢀⠀⠰⠀⠠⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⠂⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠄⠀⢀⠀⠀⠀⢠⠐⠀⠄⠂⠀⠀⠀⠠⠀⠀⠐
⠀⠀⠈⠀⠀⠤⠄⠀⡄⠀⠘⠁⠀⠁⠀⠂⠀⠁⠀⠀⠀⠐⠀⠆⠀⠘⠠⠄⠀⢀⠀⠀⠀⠀⡄⢀⠀⠂
⠀⢀⠀⠰⠀⠀⡉⠀⠀⠀⠀⠈⠠⠀⡐⠂⠁⢀⠁⠁⠠⠀⠁⠀⠀⠀⠃⠀⡀⠈⠁⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⣀⡀⡀⢀⠁⠀⠠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠐⠀⠀⠠⠀⠀⠀⠀⠈⠀⢂⠀⠀⠂
⠀⢐⠀⠈⠀⠈⠀⠀⠐⠂⠀⠀⠀⠆⣀⠀⠄⠀⢀⠀⠁⠀⠀⠀⠐⠀⠠⠀⠀⢐⡂⠐⠀⢀
⠀⠀⠀⠂⠐⠨⠀⠀⠀⠐⠀⠈⠑⠀⠐⠀⠀⠀⠀⠄⠀⢀⠈⢈⠀⠀⡀⠠⠀⠂⠀⠀⠀⠄⠄⠀⠄⡌⠁
⠀⠀⠀⠀⠁⠀⠀⢀⠐⠂⠀⠠⠀⠈⠀⠀⠄⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠁⠐⢂⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⢀⠀⠠⢄⠠⠀⠀⠀⠀⠀⠀⢐⠂⠴⠀⣀⠀⠃⠰⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠐⠈⠄⠠⠀⠈⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡀⠁⠀⠀⢀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⢠⠀⠰⠀⠀⠁⠰⠀⠀⠀⠀⡀⠊⠉⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡐⠂⡩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠐⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⠠⢤⢾⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢎⠙⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠑
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠁⠉⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠠
⠀⠂⠈⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⢀⣇⢠⠠⠄⠀⡀⢠⣸⢀⡄⡀⢀⣀⣀⠀⢰⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠞⡰⠃⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⣀⣬⠦⠂⠪⠉⡁⣀⠄⠿⠄⠄⠄⠀⣀⠀⢽⣅⣀⣀⢀⢀⢹⡧⠀⢤⠤⢤⡀⠀⠀⠀⠀⢀⠴⢫⣿⠾⢥⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠄⢂⢋⣡⠍⠗⠒⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠲⢤⣧⣶⣴⡴⠖⢀⢠⠤⠴⠦⣶⠿⠦⠞⠁
⠀⠀⠀⠀⠀⠀⡚⢏⣳⠁⠘⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢄
⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⠦⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠘⣦⡀⢀⢦⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠠⠃⢠⠁⠾⠠⠤⢈⡉⠔⠁
⠀⠀⠀⠀⠀⢰⠹⠚⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠃⠀⠀⠀⣀⠔⠋⠉⠳⡄
⠀⠀⠀⠀⠀⠨⣇⣤⠄⠤⠱⠦⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⢠⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠀⠀⠁⠰⠁⠀⠀⠀⠀⠅⠒⠀⠄⠐⠉⠈⠉⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠈⢑⣶⠶⢁⠀⠀⢀⣦⡀⠀⢁⢀⠁⠰⠄⠤⠤⠤⡆⠲⠆⢌⣆⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⢾⡄⠰⠒⡰⠋⠁⠀⢀⡄⠀⠀⠳⠉⠑⣈⠑⠂⢋⠀⠇⠀⣠⠄⠀⠀⠙⠨⣋⢂⠀⠁⠦
⠀⠠⠖⠒⠠⡀⠀⣨⠁⡈⠁⢣⡀⣰⣛⠆⣇⠀⠀⠀⡄⠀⠻⡆⠀⠀⢱⡄⠀⠗⡆⠀⠀⠀⠀⠉⠉⠧⠶⠖
⠀⠀⠀⠀⠐⠲⠖⠤⠔⠁⠀⠀⠈⠉⠀⠀⣹⠀⢠⠀⠜⡒⠊⠀⠀⠀⠀⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠉⡠⠂⠠⠶⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠩⠛⠉⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡁⠀⠤⣀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠙

⠀⠀⠀⠀⠀⠀⡴⠛⠉⠹⠿⠛⠿⠿⠷⠤⠄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢄⣴⣦⣀⣠⣀⣤⣖⣒⣒⣶⣇⣄⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⡏⢉⣉⣉⣉⣉⣉⣉⣉⣉⣓⣒⠔⠴⠢⠶⠶⠒⠄⠠⠂⠀⠂⠉⠩⠭⠍⠭⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⣿⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⣷⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡰⣻⠗
⠀⠀⠀⠀⠀⠀⠀⣉⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣟⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣿
⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡟
⠀⠀⠀⠀⠀⠀⠁⣙⣸⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡏⢠⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⡀⢐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡸⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢱⠈⣦⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣺
⠀⠀⠀⠀⠀⠀⢸⠀⠿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⣆
⠀⠀⠀⠀⠀⠀⢸⠀⠂⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠸⡇⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠙
⠀⠀⠀⠀⠀⠀⡤⣴⡀⠀⠹⠤⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡇⢸⡂⠀⣀⣀⣈⣱⣌⠑⠦⠠⠄⠶⠷⠄⠤⠤⢠⣴⣀⣂⠥⠶⠶⠶⠶⠶⠤⠶⠶⠶⠶⠶⡄⠤⢀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠡⣈⣉⣋⣋⡿⠊⠁⠉⠳⠤⠤⠤⠤⠤⠤⠔⠋⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠲⣤⣒⣺⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤⣤⡶⠔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⠤⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡉⢩⠂⣀⣀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠓⢉⠤⠈⠩⡥⠁⡞⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣜⠀⠀⠃⠀⡆⠉⠀⡱⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠫⡀⠀⠀⣀⣀⠠⠀⡔⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢳⡀⠀⠀⠁⢀⡰⢱⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣀⠀⢿⠛⠚⠚⠉⡀⢀⡿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣽⠀⠀⠀⠀⠿⠨⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢠⠃⠏⢌⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠉⠀⠳⡁⠀⠀⠀⠀⢈⡲⢈⠁⠀⠉⠑⢲⡀⠀⠀⠀⠀⠀⠀⢀⠇⢱⠃⠘⠐⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠠⠀⠈⠑⠒⠒⢶⣯⡀⠈⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⡎⠀⠸⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠀⣾⡷⠀⠀⠘⠛⠁⠇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⠀⠀⠘⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⡰⠁⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⢠⡇⠀⠀⡄⠀⠀⣀⣀⠀⠀⠸⠀⠀⠀⠀⢀⡠⠖⠋⢧⠀⠀⡰⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠀⡇⠀⠀⠼⠿⠁⠀⡇⠀⠀⠀⢿⠋⠀⠀⠀⠈⣆⡐⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠁⠀⠀⢀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⣠⡀⠀⠀⢸⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡌⠀⠀⠀⠈⠀⠙⠁⠀⠀⣾⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡇⠀⠀⡄⠀⡇⠀⠀⠀⠀⣿⠀⠀⠐⢄⠀⠀⢣⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡇⠀⡔⠀⠀⣷⠒⠂⠈⠛⢻⠀⠀⠀⠀⠱⣄⡀⢣⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣀⡥⠊⠀⠀⠀⢿⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⠈⠙⠻⢽⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⡇⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡟⠀⠀⠀⠀⠀⠘⡇⠈⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢈⣀⣀⣀⣤⣤⣄⣠⡇⠀⠈⠀⠀⢸⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⡏⠀⠀⠀⠀⠉⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠙⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠘⠒⠂⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠀⠀⢀⠠⢤⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠓⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠉⢠⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠱⠀⣀⠀⠀⠤⢸⡄⠂⠀⠀⠐⢶⠔⠀⠠⠤⠄⢀⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠤⠐⡺⡅⠤⠀⠒⠊⠈⠉⠀⠀⠀⠁⠈⠉⠀⠀⠒⠂⠀⠀⠆⡁⠂⠤⢠⠤⠂⢀⡓⣌⠒⢄⡀
⠀⠀⠀⠀⠀⠒⣸⠇⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⢀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠑⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠉⠙⢷⣄⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣰⠿⡄⠀⣰⡏⢰
⠀⠀⠀⠀⠀⠀⢸⣀⣀⡄⠀⠀⠈⢿⣆⠀⠀⠀⠀⣾⡃⠊⠀⠀⠀⠀⠀⢰⡿⠀⡇⠀⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⣿⡆⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⣾⡇⠀⠄⢀⣿⠀⡈⠀⠀⠀⢀⠤⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠔⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⠀⢸⣿⠀⡇⠀⠀⠀⡎⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢹⣿⠀⢠⣿⠁⠀⠀⣴⡟⠢⠀⢸⣿⠠⠁⠀⠈⣿⡼⠀⠀⠀⠀⢇⣼⡇⠀⠀⠀⢻⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⡿⠀⢸⣿⠀⠀⢠⣿⠄⠂⠀⢸⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣄⠀⠀⣀⣼⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⣠⡿⠃⠀⢸⣿⠀⠀⠘⣿⡆⠀⢀⢾⣿⠀⠀⠀⠀⣿⣇⠀⠀⠀⣰⣿⡟⠀⠉⠉⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⣿⠀⠀⠀⡟⠿⠶⠋⢸⣿⠀⠀⠀⢀⠘⢿⣄⠀⣴⠃⣿⡇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢰⠃⠀⠀⠀⠈⣿⠀⠀⠀⡜⠀⠈⠙⠛⠁⠀⢻⣿⣄⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣾⠀⠀⠀⠀⠀⣿⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣴⠇⠀⠀⠀⠀⠀⠘⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢠⣶⣿⠛⠙⢳⡄⠀⠀⠀⢠⣶⣿⠏⠙⠦⠀⠀⣸⣷⠀⠀⠀⠀⠀⣶⠀⠀⣰⣿⠏⠀⠈⠳⣄⠀⠀⣠⣾⡟⠁⠈⠙⣆⠀⠀⣿⣷
⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⣿⠀⣼⣿⠏⠀⠀⠀⠀⢹⡄⢰⣿⡟⠀⠀⠀⠀⠸⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠲⣤⣀⠀⠀⣾⣿⡿⠀⠀⠀⠀⢠⢰⣿⡏⠀⠀⠀⠀⢸⣿⢰⣿⡿⠀⠀⠀⠀⠀⢸⣇⣿⣿⠁⠀⠀⠀⠀⢠⡇⣸⣿⡿
⠀⠀⠀⣶⡀⠀⠀⠀⠀⢿⣷⡀⣿⣿⠇⠀⠀⠀⢀⡟⢸⣿⡟⠛⠛⠛⠛⢻⣿⢸⣿⡇⠀⠀⠀⠀⢀⡾⢸⣿⡟⠀⠀⠀⠀⠀⣼⠃⣿⣿⡇
⠀⠀⠀⠹⣧⡀⠀⠀⣠⣿⣿⠃⢻⣿⡆⠀⠀⣠⠞⠁⣼⣿⡇⠀⠀⠀⠀⣸⣿⠘⣿⣧⠀⠀⠀⣠⡾⠁⠀⢿⣷⠀⠀⠀⢀⡼⠃⠀⣿⣿⠁
⠀⠀⠀⠀⠉⠛⠒⠟⠛⠋⠁⠀⠀⠉⠛⠒⠚⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠙⠛⠀⠈⠛⠳⠖⠚⠉⠀⠀⠀⠈⠙⠳⠶⠚⠉⠀⠀⠀⠙⠛⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⠑⠒⠂⠉⠙⠻⠶⣦⣄⠀⠀⠀⠀⠀⠀⠒⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢀⠶⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠈⢻⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡀⠀⢀⢀⣀⣀⣴⣿⠏
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⣻⣞⣭⣙⡿⣦⣄⠀⠹⣽⣗⢆⠀⠀⠀⠀⠀⠠⣄⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠁⠀⠀⠀⡰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠋⠉⠉⠈⠉⠛⠲⣽⣷⣄⢱⢟⣿⡇⠀⢀⡴⣯⣟⡿⠋⠉⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠸⡗⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠄⠶⠒⠲⠀⠈⢷⣿⡚⣟⣿⣿⣠⣬⣿⡟⠋⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣄⣸⣿⠁⠀⣠⡸⠉⢠⢤⣶⡶⠗⠉⠉⠛⢧⣿⣿⣿⣿⣿⣷⢃⣤⣤⡶⠶⣏⣹⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⣾⠟⠉⢝⣄⡈⠙⡻⣾⣿⣟⣳⣮⡶⠊⠟⣷⠿⠀⠈⠙⣷⣦⠀⢻⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠻⣿⣷⣆⠀⠀⠀⠀⠀⠒⠠⡀⠀⠀⡿
⠀⢀⡟⠁⠀⠀⠀⠉⠙⠛⠉⠀⠈⠈⠁⠀⢀⣼⣸⡧⠁⠀⠠⣶⣿⡿⠚⣿⣿⡟⠿⢋⣓⣓⡶⣦⣄⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣀⡼⠃
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡽⠋⠀⠀⣺⡿⠹⠋⢠⣾⣿⣷⣾⡿⠛⠛⢛⣿⣿⣿⣧⣄⠀⠀⠻⣿⣷⣤⢄⣀⣤⣶⡶⠊⡅
⠀⢨⠀⠀⠀⡤⣾⠿⠿⢷⣦⣀⣀⣂⡁⠄⠚⠁⠀⣠⡀⠉⠁⠀⠀⠋⣼⡿⣻⣿⠀⠀⠀⢸⣿⠟⠌⢿⣿⣇⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⣸⠁
⠀⠘⡄⠀⢸⣿⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠄⠋⠀⠀⠀⠀⠀⢰⣿⠿⣉⠏⠀⠀⠀⠈⣿⡻⡞⠈⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠁⠀⠘⣿⡀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠤⢴⣿⣿⡉⠁⠀⠀⠀⠀⠀⢻⣾⣇⢀⣟⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇
⠀⠀⠀⠀⠐⠌⢻⣦⡀⠤⠔⠁⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡜⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣈⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⠴⣺⠏
⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢷⣧⠄⠀⠀⠀⢀⣴⣘⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⡟⣗⠳⣶⢤⣀⣀⣠⣴⣫⡿⠃
⠀⠈⢂⡀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠈⢛⠌⠠⠀⣌⢯⣿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡃⠀⠉⠙⠓⠛⠛⠋⠉
⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠷⢙⢭⡿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠂⠀⠀⠐⠀⠀⠂⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣔⣐⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄
⡈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢠⡐⣲⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠇⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡶⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⡀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⡒⠃⠀⠀⠀⠸⡀⠀⣀⣀⡀⡤⢠⣧⠐⠂⠂⠀⠀⢢⣧⠲⠀⠠⠤⢄⣀⣀⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⢜⠜⡱⠃⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⢀⣀⠤⢲⣇⢈⡀⣀⠤⠴⠂⠛⠂⠀⠐⠒⠂⠐⠓⠒⠐⠂⠤⠤⠂⠀⠓⠚⠓⠀⣀⠀⠀⠀⠐⠃⠛⠛⠛⣦⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⣀⢤⠶⠁⣄⡦⠦⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠻⠛⠋⠁⠐⠞⠉⠉⠛⠟⠛⠉
⠀⠀⠀⠀⠀⣀⡮⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡈⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⢻⡍⠁⠀⠀⣀⠀⠀⠀⢦⣾⡼⢦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠈⠁⣴⠀⠈⣿⠂⠈⠡⠘⡆⠐⢀⠀⣼⣏⡗⣹⣀⠰⣠⠄
⠀⠀⠀⠀⠀⠀⠀⣤⠈⠇⠀⠀⠰⢠⣆⠃⡟⢀⠀⡨⢵⣿⣿⣾⣻⣟⠦⠚⣿⠍⡐⠱⠀⠁
⠀⠀⠰⢷⣤⡄⢢⡏⣼⠀⣀⠀⢄⠈⡿⣆⣻⠀⡀⢦⢸⣿⣹⢏⡹⢋⣱⣾⡧⠚
⠀⢀⠘⠻⣅⣻⡜⡀⣯⠀⠈⡶⠀⠀⣀⠌⣿⡾⡁⣾⠀⢨⣿⢸⣿⢛⠏⠁
⠀⢉⠒⠶⠞⠛⡟⢱⣊⢷⡜⣱⠆⠼⢷⡋⢹⣆⣹⣯⣴⣾⣷⣿⣿
⠀⢤⣀⣀⡀⣻⢹⣦⠻⣿⢰⠋⠎⢱⣾⣿⣆⣽⣏⣿⣍⣿⡏⠃⡘⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠺⠉⢨⣿⣼⡟⢙⣷⣿⢏⣹⣾⣧⠦⣿⣿⣻⡟⢡⣼⣟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁
⠀⠀⠀⠛⢿⣿⣌⢫⠆⣿⡟⢹⣽⠷⣞⢁⠾⣿⣿⣿⣿⠟⡁⠀⠀⡔⢀⢀⡠⠀⠞
⠠⠄⠀⣙⣻⣴⠿⠿⠶⢾⣧⣼⢇⡶⠼⢀⣷⠹⣿⣿⣿⠛⠀⠀⣸⡷⠉⠨⠀⠀⠀⠀⠀⠐
⠀⠀⠚⠉⠐⠛⠀⣀⠀⠀⠙⣿⣄⡼⠂⢻⠹⣇⢿⣿⣻⣦⡶⠾⠁
⡀⠀⠀⠠⣀⡀⣀⠈⠰⢇⠄⠀⡙⢿⣦⣄⣿⣿⡟⣻⣷⣿⠇
⡀⠀⠀⠀⢤⠉⢃⡸⣤⡃⠲⣈⡤⢸⣿⣿⢷⣏⣷⣿⣿⡿⢀
⣇⠀⠀⡄⡨⠋⡹⢇⣌⣷⡄⠘⣧⡼⣿⣿⣴⣿⣿⡿⣿⣷⡿⢶⠀⠂
⠚⣦⡀⡿⢀⣀⣤⣠⣄⣦⣽⣾⣯⠵⢿⣿⣿⣿⣿⣷⣿⡋⡕⣊⣠⣀⠠⠄⠀⠀⠀⠀⢡⠰⠀⢀
⠀⠀⠙⣇⠠⢿⡿⠚⢠⣹⡦⠘⠷⠶⣿⣿⣽⣿⣷⣿⡏⢟⣿⡿⣫⢒⣤⡆⠀⠀⠀⡴⠀⠀⠀⠁
⠀⠐⠰⡼⣧⣀⠷⢦⡀⠙⣆⠛⢤⡰⢸⣿⣏⣿⢿⣿⣿⣿⠕⠋⢉⣿⣇⣭⣷⣤⣤⣄⠐⠀⠀⢂
⢂⣖⣠⡴⠛⠻⢶⣾⣷⣶⣿⣦⣿⣄⣾⣿⣿⣿⣿⣏⣼⣷⠾⠿⠿⣯⡙⢄⠴⠤⠆⠀⠀⠀⠀⠀⠑
⠀⠸⠉⠀⠘⢁⠟⡱⢾⡇⠀⠉⠛⢻⣯⡿⠿⣹⣿⣿⠟⠀⠀⠀⠀⠈⠻⢿⣴⣶⡷⠆
⠀⠀⠀⠀⠉⠉⠃⠀⠀⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣧⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣻⣿⣿⣿⣿⢿⣶⡶⣆⣄⣀⣀⣀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⢀⢤⡤⣤⠤⠤⠄
⣤⣤⣾⠛⠿⡉⠹⡿⣟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠿⣿⣿⠿⡿⢿⣿⣿⣿⡿⣟⠶⠜⠉⠑⠛⠋⠙⠟⠛⠿⢿⣟⠻⠛⠢⠉⠛
⠉⠻⠆⠻⠃⢡⣿⣿⣿⣿⣏⣿⠏⣻⠉⢿⠃⠒⠀⠗⠨⠹⠄⠐⠀⠀⢻⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠
⠀⠀⠀⠐⠀⠊⠉⣉⠀⢹⡦⢙⠚⠛⠂⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠹⢰⣆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠁⠉⠖⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠂⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⢏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠶⣽⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣠⣶⣌⠻⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⢿⣿⠿⣿⣿⣶⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡴⠋⢁⣶⠿⣿⣿⣮⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣯⡁⢻⣤⣿⣿⣦⠞⢛⠲⣻⣾⡇⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⣾⣾⠛⡋⣖⠚⠳⠶⣉⣉⡠⠜⣻⠄⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡠⠞⠋⠀⣾⣿⡟⠳⠶⢞⣛⠛⢟⣾⣧⣴⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠆⠚⡛⠋⠈⣦⢠⣴⣾⣿⣿⣿⡷⠒⢾⡿⣿⣼⣿⣛⠛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡞⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣗⡐⠛⠳⠤⢥⣛⠻⠛⣻⣍⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠫⢟⣿⠟⠋⣱⡿⠿⣶⠟⠛⠻⣷⡶⢆⣠⣛⣁⡀⢠⣶⢦⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠧⠀⠘⣆⠉⢀⣤⣴⠿⠓⠾⣿⣝⠓⠒⠊⢿⣥⡀⠙⠛⠿⠿⠥⠿⠄⣀⣠⡄
⠀⠀⠀⠀⡆⣶⣦⣠⠀⠻⢦⣦⣷⡦⢰⣼⡶⡟⠁⣠⣶⣻⣦⠀⠙⣿⣶⠀⠀⡨⠻⣿⣷⣦⣴⣤⡟⠚⠉⠛
⠀⠀⠀⠸⣷⣿⢫⠏⠀⠀⠀⠀⠻⣷⣿⠋⠁⢐⣿⣯⡿⠺⣿⡇⠀⠈⢿⣷⣧⣑⡰⠃⠈⠻⢯⣉⠀⠀⠀⠀⣠⡠⣶
⠀⠀⠀⢠⠏⢱⣏⡄⣾⣿⣷⣤⠾⠛⢁⣄⢿⣿⣿⠛⠛⠚⣿⣿⣇⣠⡀⢉⡛⣿⢽⡛⠲⣦⣤⣨⣕⠲⠶⣿⣿⣷⣿⠂
⠀⠀⠀⢨⠀⡾⠘⣷⡹⣿⣿⡇⣦⣤⠿⠿⣯⣿⣿⣿⣿⡾⠿⠷⠿⣿⡇⣿⣿⡷⢀⡵⠈⠁⠀⠉⣿⣟⣿⣿⠛⢀⣯⣭⠁
⠀⠀⠀⢸⡆⢳⡴⢻⠇⠉⣉⣁⣼⢯⣶⣿⠿⠿⣿⣿⠏⠀⣀⣀⡀⠿⠱⠿⣿⣧⣾⣴⡿⠿⣦⠀⣇⣼⣠⡟⡠⡸⠁
⠀⢀⡴⣾⣧⠀⢻⣿⣶⣿⡟⠉⢹⣿⣿⡧⠀⠀⢿⢻⣠⡞⠋⠉⡿⠓⠒⠲⣾⣿⣿⡇⠀⠀⡿⣠⠗⣛⣭⣾⡴⠧⢄
⠀⠀⠘⠻⠛⢢⣼⢹⠿⠋⢀⠀⣺⣿⣿⣌⡀⡀⣸⡧⣾⣇⠠⠀⣧⢀⠀⠸⢾⣿⡙⢃⠈⢁⠛⣿⣯⣵⠟⠓⢻⣶⢦⠄⠂
⠀⠀⠀⠀⡴⣞⠻⡟⠷⢿⣖⣭⠋⢛⣏⠁⠁⠒⠛⠁⣸⢳⣥⣤⡾⢏⢠⡤⢾⣿⣿⣦⣭⣁⠔⣉⠹⠆⣸⠥⠈⠁
⠀⠀⠀⢸⣃⢉⣄⣸⣦⣀⡈⣭⡚⠛⠿⣦⣤⣀⣤⣴⣿⣿⣿⣿⣄⣴⠉⣦⣴⣿⣿⠿⠟⠛⠈⠀⠀⢀⣷⣄⠀⢁
⠀⠀⠀⣼⢵⠋⣼⣿⣿⠟⠁⣯⡝⣷⠀⠀⠀⠉⠙⠛⠛⠛⢋⡙⡉⠉⠀⠉⠁⠀⠀⠀⠀⠀⣀⣀⠀⣸⠿⠋⣀⠜
⠀⠀⠐⣧⠿⡄⢈⡎⡉⠀⠀⠻⠼⣿⣼⢳⡆⢰⠠⡄⡄⢸⠈⠁⡧⡄⡤⢠⠴⣂⡇⣾⣿⣷⠿⠘⢠⠏⠉⡏
⠀⠀⢰⡏⠀⣿⠾⠀⠉⠓⠲⠤⠤⣄⣀⡈⠁⠘⠀⠓⠇⠘⠀⠀⠓⠛⠓⠈⠁⠉⠁⡉⣀⣀⠤⠒⠙⣆⡜⠁
⠀⠀⠀⢻⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⠢⠤⠤⠤⠤⠤⠤⠄⠒⠚⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⣀⣀⠀⢸⡟⠟⠝⣅⠀⠴⣶⠰⢌⢠⠀⠀⠐⠖⠀⢀⢠⣤⡄⠀⣀⣀⣀⣂
⠀⠀⠀⡠⠔⠍⡃⢰⣶⣶⣾⣟⣿⣿⡇⠀⢿⣷⣾⣿⣿⠂⠀⠋⠎⠊⢞⠰⠩⢛⣳⡇⡀⠀⠀⠀⢸⣧⠀⠘⣿⡟⠀⠛⠛⠛⠛⠁
⠀⠀⠀⡥⢒⡡⡄⠸⣿⣿⣿⣿⣿⣿⡇⠀⠈⠻⣿⠟⠁⠀⣤⣦⣥⣶⣷⡄⠁⠀⢰⡿⢳⡀⢠⠀⠘⣁⠀⡂⢉⣴⣾⠿⠿⣷⣦⡀
⠀⠀⢰⣀⠅⢊⠅⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⣴⡟⠉⠉⠉⠙⢻⣦⡀⠞⢰⠀⠁⠀⠀⢸⣿⠇⠁⣾⡿⣱⣾⣷⣦⡝⣿⡆
⠀⠀⢸⠠⢒⠥⠒⢡⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⣾⣿⠀⠀⠐⠖⠐⠂⢸⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢻⣿⣿⣿⢳⣿⡇
⠀⠀⢸⠒⡡⢊⠸⢸⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣱⠀⠈⢿⣷⣄⣀⡃⣀⣴⠟⠀⠀⠸⠀⠀⢀⠀⠀⠀⠃⠀⠘⠿⣷⣮⣭⣾⡿⠋
⠀⠀⠸⠁⣊⡔⣀⡄⠀⠀⠀⠀⠀⡀⡇⠀⡄⢄⠃⡀⠀⠀⠈⠻⡿⢿⠟⡟⠀⠀⠀⢰⠀⠀⠈⢀⡀⠀⣀⣀⢀⣀⢀⠌⡍⡅⠀⡤⠆
⠀⠀⠀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠂⠐⠁⠀⠈⠢⣿⠔⠂⠀⠀⠈⠓⠀⠒⠇⢠⢦⡄⠘⠀⠠⡶⡦⠙⠛⠓⠛⠒⠒⠂⠒⠂⠊⠊⠁
⠀⠀⠈⠁⠀⠀⣀⠄⣢⣤⣖⡤⡀⠀⠀⠤⡀⠀⣄⠔⢀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠓⡀
⠀⠀⠀⠠⢀⠞⠰⠊⠀⠉⠻⣿⣮⣄⠀⠀⠀⠸⡧⠞⠁⠀⠐⢀⣀⣤⣬⣭⣬⣭⣤⣤⣤⣤⣴⣶⣶⣶⣶⡦⠈⡆⠀⠁⠀⠒⠀⢸⡄⠄
⠀⠀⠀⠀⡅⠲⠁⠀⠀⠀⠀⠹⣿⣿⣀⠀⢰⣶⣧⣴⠀⠀⠀⢸⣿⠉⠉⠁⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⣿⣧⢠⡇⠀⠀⠄⠀⢠⠟
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠸⣿⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣭⣻⣧⠀⢿⣿⢀⡇⠀⠀⠀⣶⡇⢠⣤
⠀⠀⠀⠰⠐⠆⠀⠀⠀⠀⠀⠘⠘⣿⣿⡇⠀⠀⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⣽⣻⣿⢿⡟⠀⢸⣿⠀⡇⠀⠀⠀⣿⣧⠀⠁
⠀⠀⠀⢐⠈⢓⠒⠤⣄⡀⢀⠤⠐⠛⢿⣿⡀⠀⠸⣿⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⢸⣿⢀⡅⢠⡄⢠⣿⣿
⠀⠀⠀⠠⠐⠈⢿⣦⡈⢿⣆⠀⣰⣿⣆⢿⣧⠀⠀⠉⠁⠀⢀⠸⠿⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⢸⡇⠀⠀⠀⠉⠉
⠀⠀⠀⠀⣴⣾⣦⠻⣿⡌⣿⣰⣿⣿⡿⣸⣿⠀⠃⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠒⠶⣶⣶⣶⣶⢖⣂⣀⣀⣈⣉⣐⣄⠀⠀⠈⠀⠄
⠀⠀⠀⢸⣿⣿⣿⣇⢹⡇⣿⣿⣿⠟⣡⣿⡟⠀⠀⠠⠀⠸⢶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠄⠀⠁
⠀⠀⠀⠈⢿⣿⣿⡟⢸⣧⠿⣦⣶⣾⠿⡿⠀⠀⠀⠋⠀⠀⠀⠀⠀⣤⡶⠄⣤⠶⠀⣄⠶⠀⣤⣤⡀⢀⠀⠀⠀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠐⠀⠉⠉⠚⠉⠁⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢸⣿⣷⠀⠀⠘⢻⣿⠛⣿⡿⠛⣿⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⠀⠀⠐⠁⠁⠀⠚⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠹⣧⠀⣿⡇⠀⣿⠛⣷
⠀⠀⠀⠀⠀⢀⢀⣀⣈⣤⣀⡤⠤⠠⢌⣀⣤⢤⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡄⠀⣿⡆⢸⣿⠀⣿⡾⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⢰⣰⣿⣿⣿⣧⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣠⣿⠀⣾⡏⢸⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⢿⠛⣈⠙⠿⣿⡿⠃⢠⠀⠂⠀⠀⢀⣀⣀⠀⡀⡀⡁⠀⠀⠀⠀⢰⣿⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠺⠴⠤⠰⠤⠤⠴⠴⠵⠴⠶⠾⠟⠛⠀⠀⠈⠉⠉⠙⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠚⠤⠒⠒⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠬⠯⠤⠒⠒⠂⠉⠳⡄⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣦⠀⡀⢠⠁⠀⡤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠒⠒⠶⢤⣤⡀⠀⠀⠀⠀⠀⠀⣿⡙⠙⣠⡼⢣⠞⢀⣼⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⣷⢦⠤⣤⣴⣿⣿⣷⣿⣋⣵⣿⣶⣿⡉⠉⠉⠒⢀
⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⣀⣀⣠⣀⡀⢀⣹⠿⣿⣿⠿⣿⣳⣦⡀⠀⠂
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣿⣿⠏⠹⣃⣿⣿⣦⡙⠦⠁⠉⠀⠀⠀⠀⠁⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡄⠀⠀⢀⣀⡀⠀⢠⣤⠞⠡⠀⡡⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡐⠧⠊⠀⠀⠐⣿⡟⠋⠈⠁⡀⠴⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢷⠖⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⡋⠀⠀⠀⠉⠢⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⡠⠁⢀⣾⠏⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢦⠀⠀⣾⣿⢀⠜⠁⢀⣾⡏⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⣿⣿⣿⢀⠔⠐⣦⡀⠀⣰⣿⡟⢀⠎⠀⢰⣿⡿⠁⠀⠀⣼⣿⠁⡔⠁⠀⣠⣾⡿⠋⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⢸⣿⣿⠏⠀⢸⣿⣇⠀⣿⣿⠇⠁⠀⠀⣼⣿⡇⠀⠀⢠⣿⡿⠊⠀⠀⣼⣿⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⢸⣿⣿⠀⢻⣿⡆⠀⠀⡠⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⢠⣿⣿⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢸⣿⣿⠀⢰⠋⠛⠒⠊⠁⣿⣿⠀⠀⢀⣿⣿⠃⠀⠀⠘⣿⣿⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⡌⠀⠀⠀⠀⠀⢻⣿⡀⢀⠎⣿⣿⠀⠀⠀⡰⠙⠿⠷⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢻⣿⡄⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠋