Lawn Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠋⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣻⣿⣆⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣿⡿⠣⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣄⠀⠘⢤⡄⠀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠼⢿⣟⣼⣿⣦⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢋⣾⣿⣿⣿⣿⣿⡗⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢀⣇⣀⣠⣸⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠋⠉⠁⠈⠈⣷⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠯⠤⠤⠀⣀⣀⣠⠟⣿⡇⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⡟⠀⢸⠁⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣴⡞⣷⣶⣶⣶⡶⠒⢲⣾⠀⠀⠨⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣿⣿⣿⣿⣿⣿⣿⡿⠃⣰⣿⣿⡆⠀⣤⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⣿⣿⣾⣽⣏⣹⣿⣿⣟⣁⣴⣿⣿⡿⠃⠀⡇⣿⣿⡏⠙⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡏⢹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠐⢹⣿⡇⠀⠀⠉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⠖⠛⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣭⣿⣿⣿⣿⣿⡏⢳⠀⠀⠘⣻⣿⣿⠀⠀⠰⠿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⢰⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⢏⣭⣶⣿⣿⣿⣿⣿⡇⢹⣿⡇⣸⠀⢠⣾⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠷⠶⠿⣿⣿⣿⣿⡏⣾⣿⠇⠈⣿⣿⣿⡿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⡄⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠞⠁

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡶⠿⠿⠟⣿⣿⣶⣶⣶⡾⣿⣿⡿⠿⠿⣿⣿⣿⠟⠛⠿⠿⣿⣶⢶⡶⠶⢶⣶⣶⡷⢾⡿⣿⣿⡿⠟⠛⠛⠿⠿⠿⠿⠿⣶

⣶⣶⣶⣾⣷⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣿⣿⣾⣦⣤⣶⣶⣶⣷⣶⣶⣦⣶⣶⣶⣶⣶⣷⣶⣶

⣤⡀⡄⣠⠀⢀⢀⠀⢀⠀⣀⠀⠀⢠⣠⡄⢀⣀⡄⠀⠀⠀⣀⢤⣠⣀⣀⠀⢀⢀⠀⠀⠠⡀⠀⠀⣀⣠⡄⢀⣀⡀⠀⡀⢀
⣷⣿⣿⣿⣿⣾⣎⣿⣾⣦⣿⣾⣿⣧⣿⣧⣼⣿⣿⣷⣿⣿⣿⣾⣷⣿⣿⣿⣿⣼⣷⣿⣼⣿⣾⣿⣧⣿⣇⣾⣿⣷⣾⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣠⣆⣷⣄⣠⣀⣰⣄⣾⣀⣄⣀⣦⣸⣇⣀⣀⣠⣆⣲⣆⣠⣀⣸⣄⣶⣀⣄⣀⣴⣸⣆⣠⣀⣠⣆⣰⣇⣀⣀⣸⣄⣶
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠐⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠴⠒⡀⠠⠀⡤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠋
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣰⢰⣄⣖⡀⣠⡆⣰⢦⣤
⣴⣄⣧⣆⣶⣰⣠⣾⣶⣿⣿⣾⣿⣿⣾⣿⣿⣤⣾⣷⣷⣦⣿⣿⣶⣾⣿⣷⣾⣿⣿⣿⣿⣎⣿⣷⣿⣾⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠄⠀⠆⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢤⡉⠂⠀⠀⠀⠀⢨⢀⠀⠀⢡⠀⠀⠸⠀⠀⠀⠀⠜⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⡀⠀⠤⠤⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠁⠸⡄⡄⠀⠘⡄⠀⠀⠀⠀⠀⡔⢀⡌⡀
⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡆⠘⣀⠀⠀⠀⠀⠀⠀⡄⠇⢀⠀⠀⠀⠀⣁⡎⠀⠈
⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠄⣧⢐⠁⡂⢰⠅⠀⠀⠀⠙⢆⠘⠃⠀⠀⢘⡸⠀⠀⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢠⢀⠀⠀⠀⠄⠄⠀⠀⠀⢹⡿⢄⣿⣸⣄⠀⠀⡼⠀⠸⢧⡄⠈⠀⣸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⡄⠀⠀⠀⣎⡸⣃⠂⢧⣀⡀⠀⠀⣾⣿⡞⣿⣿⣧⣴⣾⢷⠄⢀⡟⣷⠀⣶⡞⠒⠒⢤⣦
⠀⠀⡄⠀⢠⠀⠀⠀⠀⡀⠀⠀⣧⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⢢⠀⢸⣞⣦⢰⢻⡇⡼⣀⠔⡽⣿⣷⣤⣿⣿⣿⣸⣿⣷⣿⣿⣿⣿⣿⡗⣻⣲⣿⣓⣦⠖⢀⣼
⠀⠀⣇⢰⣾⡄⠀⠀⠤⣧⢰⠦⣹⣧⣠⣤⢤⣘⣀⡏⠀⠀⠰⠀⢈⣇⣷⣿⣿⣼⣾⣿⣿⣿⡗⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣉⠞⠚⣶
⣽⣦⣼⣶⣿⣿⣶⣤⣶⣿⣍⣷⣿⣿⣿⣿⣾⣽⣿⣿⣲⣦⣵⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢶⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣿⡃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠩⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⢀⠀⠀⠀⠀⠘⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠙⠪⣗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠙⠯⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠒⠀⠊⠑⢽⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣀⠀⠀⠀⠀⢠⣀⣀⣀⠊⠒⠿⣶⡌⢁⣐⡷
⠀⠀⠀⢀⣀⣀⣀⣀⣽⢿⣼⣷⣠⣀⣀⣠⣿⣷⣾⣷⣤⣄⠀⠀⠀⠀⠀⣀⢐⡀⠤⠀⣀⣿⡏⠉⣷⣀⣀⣀⣀⣄⣾⠛⢢
⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠙⠛⠓⠓⠛⠚⠓⠋
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠚⣉⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠑⡤⠞⡆⠀⠀⠀⢺⡄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣄⡁⢠⢃⡀⡀⠀⣤⠋⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣴⣿⣯⡽⣧⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡿⠹⣿⣧⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣀⣻⣿⣦⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡞⠉⢹⣿⣿⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⢆⣸⣿⣿⢀⣤⣿⣿⣿⠟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣜⢿⣿⣿⣿⠿⡿⢿⣿⣿⠦⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣏⡸⣿⣿⣛⣻⣦⣗⣸⣿⣿⣦⡤⣼⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⣏⢱⣿⢿⣿⣿⡿⠻⣻⣿⣿⣷⡖⠒⠛⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣤⣜⣋⠁⠀⠙⠿⡇⠆⣿⣿⣿⣿⣏⣀⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠳⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢀⡇⢰⣤⣴⠰⠀⢤⡇⡄⠀⣤⠀⡆⡄⢀⢠⣇⠀⢰⣦⢸⣧⣴⣷⣦⣴⣾⣼⣿⣤⢸⣧⠦⠀⠀⣧⡀⢀⡆⢀⡇⡄⣷⢠⡄⢠⡐⣆⡇
⢸⣷⣿⣿⣿⢰⣧⣾⣇⣧⣸⣿⣿⣷⣿⣾⣼⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣿⣿⣾⣷⣿⣿⢿⣿⣿⣿⣾⣧⣧⣿⣰
⣾⣿⣿⣿⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⠀⠀⠀⢀⠀⣀⡄⢀⠀⠀⡠⡀⢀⣀⡄⢀⢀⣄⡀⣆⠀⠀⢀⠀⢀⣀⡄⡀⠀⠀⡀⠀⣀⠀⣄⡀⣠⢀⣠⠀⢠⠀⢀⠀⠀⣄⣠⣀⠀⢀⡄
⣠⣿⣀⣾⣷⣟⣷⣿⣿⣾⣷⣶⣿⣯⣿⣾⣿⣿⣿⣿⣼⣷⣾⣷⣼⣾⣿⣿⣿⣦⣿⣶⣿⣷⣷⣿⣿⣿⣷⣧⣿⣦⣼⢲⣾⣿⣿⣿⣦⣾⣷
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉


⠀⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢰⣧⣧⢀⡄⠀⣿⣇⢀⡆⠀⣷⢸⢰⣤⡀⠀⡄⢠⣾⡀⣀⣸⢠⡀⣤⡆⡀⠀⠀⡇⠀⣧⠀⣰⢨⠀⡇⣠⡇⢠⢰⡆⠀⢠⡆⠀⢠⣼⡇⢠
⢸⣿⣿⣿⡇⡄⣿⣿⣸⣿⣄⣿⣿⣿⣿⣷⢹⡏⣾⣿⣧⣿⣿⣾⣇⣿⣷⣿⣧⡇⣷⣰⣿⡀⣿⣸⣷⣷⣿⡟⣿⣾⣿⡄⣸⣧⡇⣿⣿⣿⣿
⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣧⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀⠀⠀⠀⣠⠀⠐⡄⡀⠀⠀⡄⢠⡆⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⢀⡀⣠⠂
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠰⡜⣷⡏⠁⡰⣾⡇⡌⣆⢻⡗⣄⠈⣿⡾⣵⡀⠀⠀⢀⢀⠀⣆⣿⡖⡀⠘⣷⡟⡴⡀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⣶⡞⠀⢧⣧⠀⢰⢣⢱⣿⣷⣴⢫⣟⣷⢻⢟⡞⣿⡙⣤⣿⣷⠹⢰⡆⢀⣾⡎⢰⢿⣿⣯⠱⣔⣿⣷⢱⢱⠀⢠⣧⠇⠈⣶⡆
⠀⠀⢰⢸⡇⡆⣸⣧⠃⣼⣼⣾⣿⣿⣷⣿⡿⣿⣾⣿⣿⣿⣷⢿⣿⣿⡇⣿⡇⡘⣼⡆⣿⣾⣿⣿⣷⣿⣿⣿⣞⣼⡄⢆⣿⡄⡆⣿⣐
⠀⠀⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠆⡀⠀⢀⠀⠘⠁
⠀⠀⠀⠀⠘⢁⡤
⠀⠀⠀⠀⢧⠠⠒⢸
⠀⠀⠀⠀⣀⡇⠀⣰
⠀⠀⣀⡀⠙⢷⣰⣿⢀⡾⠁
⠀⠈⠉⠻⣶⣄⢿⢸⠛⠁
⠀⠀⠀⠀⢙⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⣀⠀⠈⣿⣷⢈⡇⠀⣀⡄⠀⠀⡄⢀⢦⠀⣠⠆⢀⡀⠀⠸⣿⣦⠀⠀⠀⡀⢠⠀⠀⢠⣀⡄⢀⣰
⠈⢪⣷⢆⢹⣿⣿⣷⣷⣿⡇⣧⣸⣷⡞⣿⣶⣿⣾⣿⣿⣶⣄⣿⣿⡆⣴⣷⢳⣾⣇⣼⢸⣿⣿⣼⣿⣶⣼⢠⡄
⠀⠀⢿⡏⣿⣿⣿⣿⡿⣿⣧⣿⣿⢹⣷⢻⣿⣿⣿⣿⣿⣿⡸⣿⣿⣷⣿⡟⣾⡟⣿⣿⣼⣿⢻⣿⣿⣾⣿⣼⡆

⠀⠀⠀⠀⠀⣄⠀⢄⠀⠀⡀⠀⠀⡄⠀⣄⠀⠠⡀⠀⠀⡄⠀⠀⢠⡀⢠⠀⠀⣄⠀⠀⢠⠀⢠⠀⠀⣄⠀⠀⣀
⠀⠀⠀⠀⣰⠿⡆⢸⡆⢀⣷⠀⣼⢻⡄⢼⡆⠀⣷⠀⣾⢻⡀⢀⡟⣷⠈⣧⢠⣸⡆⢰⡿⣆⢸⣧⠀⢸⡆⢰⠿⣆
⠀⠀⠀⢀⣿⣦⣿⣼⣿⢸⣿⣷⣯⣼⣧⢸⣿⢠⣾⣷⣯⣼⣇⣼⣷⣽⣦⣿⡆⣿⣿⣿⣴⣿⡀⣿⡆⣼⣷⣿⣤⣿⡀
⠀⠀⠀⢸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣷⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣻⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠤⣄⡀⠑⣄⠀⠀⠀⢀⠀⠀⡰⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢻⡄⠘⣆⠀⠀⢸⣇⢠⡇⠀⣠⠂⢀⡞⠁⠀⢀⣀⡴⠁⣠⡴⠚⠉⢀⢀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⡀⣹⡾⢁⡼⣿⣸⡇⢠⣿⡀⣼⣿⡄⠀⣰⣿⢁⣾⣯⡾⠋⠀⣸⡏⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡐⣦⣸⣷⣿⣇⡞⢀⣿⣿⡇⣾⣿⣿⡏⢹⣿⣰⣿⣿⣼⣿⢋⠀⣶⢰⣿⣿⠏⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡸⣷⣼⣿⣿⣿⣿⣿⡇⣿⣿⣷⣿⡇⣿⣷⣸⣿⣿⡇⣿⣿⣷⣿⢀⣿⣿⣿⣿⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠐⣦⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⢸⣿⣿⣿⣿⠧⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⠀⢰⠇⡜⠀⢠⠄⠀⠀⣆⠀⢸⡈⢧⠀⣰⠀⠀⠀⢸⠀⢀⡀⢸⡄⢠⠏⢰⠀⡀⡆⠀⠀⠀⣠⠴⠂
⠀⠀⠀⢀⡀⢤⠈⡇⢰⡄⢸⢀⣿⣰⡇⡘⣾⣠⣼⢠⣹⣸⡀⣧⣸⡇⣿⠀⡆⢀⡏⣷⠛⡆⢸⡇⣾⣠⡾⣸⢸⣣⠀⣢⡾⠃
⠀⠀⠀⠀⠹⣦⣷⣹⡀⣧⣸⣼⣿⣿⣷⣧⣿⣻⣿⡇⣿⣧⣷⣿⣿⣧⣿⣸⡇⣼⣿⣿⡞⣿⣼⣧⣿⣿⣷⣏⣿⣏⣼⣿⡵⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣆⡀⠙⢶⣄⠀⠀⣴⠆⣷⣄⠈⣇⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣿⣷⣌⢿⣦⣼⣿⢀⣿⣿⣀⣿⣿⣿⣷⠀⣠⡔⣠⡴⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣿⣾⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢠⡀⠀⡄⠀⠀⢀⠀⠀⠀⡀⢀⡠⢠⠀⢀⠀⠀⡀⠀⡄⢄⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢳⣦⣄⢼⣿⣌⣧⣿⣼⡇⣌⣲⣯⣀⢰⡀⣇⣾⣵⣇⣶⣧⢴⣦⣽⣶⣸⣬⣧⣸⢀⡆⡼⣡⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠬⢶⡆⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠆⢳⣄⣦⠂⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡀⠐⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡦⠀⢀⢲⣖⡲⡈⢀⣴⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⡀⡈⣠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣱⡰⠃⢼⣿⢹⣅⣼⡯⡀⠀⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠠⢤⡠⡈⠡⠀⠎⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡬⠀⢫⡛⠼⡟⣿⣡⣼⡯⡐⣯⣸⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢀⣀⡀⠀⢷⠀⠀⠀⠀⠀⢤⠀⠀⠄⢙⡆⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡟⠀⠈⠀⣈⠗⣦⣼⡍⣱⣿⡟⣯⠿⢛⡿⣌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣰⣟⣳⡀⢸⡀⠀⠀⠀⠒⠊⠠⣽⣿⡬⠁⠀⠀⠀⠀⠐⣻⠆⠀⠀⠲⢫⣀⠌⣈⠿⣀⠀⢸⡶⢾⣿⡿⣡⠆⣭⡥⡀
⠀⠀⠀⠀⠀⠀⠀⣌⠉⠢⡌⢻⠀⠑⢄⡗⢤⡀⠀⠐⠀⠀⠚⠻⠿⠂⠆⠀⠀⠀⠀⣿⡃⣻⡿⠀⠘⢣⣛⠋⡖⢯⡶⡾⡿⠶⢿⡏⡡⣮⠛⢂⡀⡆
⠀⠀⠀⠀⠀⠀⢀⣈⡙⢦⠙⠉⠀⠀⣹⣥⡯⠹⣄⡀⠀⠔⣒⣒⣶⠶⣒⣍⠲⡄⠀⣾⣡⡛⣿⡀⣾⠁⠉⠳⢀⠈⠟⢢⣼⢷⣀⣳⢟⠋⠀⢀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡙⡌⠳⡴⠶⢞⠛⡀⢨⢷⣄⢳⣴⠴⣼⣿⣾⣶⣿⣿⠀⢹⡆⣸⠋⡁⣿⡟⠁⡄⡄⣦⢸⢢⡤⠀⠋⣷⢉⡎⣏⢀⢠⡀
⠀⠀⠀⢀⢀⠖⡔⣿⣗⡁⠠⠇⠀⠁⡆⠱⣽⡾⠛⢨⠯⠇⠀⠀⠀⠹⣭⠬⠄⠃⠘⠛⠳⠘⠚⠿⡇⠁⡇⣿⢸⢰⣬⣷⣦⣿⣽⣀⣿⢷⡿⣡
⠀⠀⠴⠛⠀⠀⠘⠿⠧⠄⠒⠂⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠘⠙⠛⠿⠚⠩⣧⠟⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣎⠉⢡⣠⠀⠀⠀⠀⠠⡂⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⡆⠀⣠⣤⣴⣿
⠀⠀⠀⠀⢐⠀⡐⡿⢿⣦⡟⠀⢐⡔⠀⠀⣾⠁⢠⢀⡀⡀⠀⠙⠀⠀⢠⠠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢷⠀⠀⠀⠀⢠⣀⣀⣤⢠⣾⣧⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠐⢣⣯⡴⢻⣷⣄⠸⣧⡆⣸⣿⡷⣻⡟⣾⡟⣴⡆⣠⠀⣘⠃⣿⣗⣤⡄⣶⠀⢁⡄⢀⢞⠀⣀⢩⡆⢠⠀⣴⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⣧⣆⣠⣼⣥⣾⣷⣴⣾⣿⣿⣾⣿⢧⣿⣿⣧⣿⣷⣿⣿⣸⣽⢻⡟⢹⣷⣿⣿⣿⣷⣟⣦⣾⣴⢾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠈
⠀⠀⠀⠈⠉⠙⠲⢄⠀⡇⣸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠊⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠔⠁⣀⣠⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣧⠇⠀⡇⠀⠀⠀⠀⡔⠀⢠⠣⠚⠁⠀⠀⠀⠀⠀⣠⠞⠀⢠⣰⠋⢠⡾⠋⠀⣠⠋
⠀⠀⠀⠀⢠⠠⡀⡀⠀⣿⠀⠀⡇⡔⠀⠀⡼⠀⢀⡟⡰⡀⠀⡴⢀⣴⡮⢾⠃⠀⠀⡼⡇⣰⡟⠁⠀⣰⠃
⠀⠀⠀⢦⢸⠀⢣⢱⠀⣿⣰⠀⣿⠁⡼⣸⠁⠀⣼⢡⢡⢇⡞⣰⢟⡜⢧⡏⠀⣀⣾⢓⣿⢧⣧⡀⣰⠇
⠀⠀⠀⠘⣿⠱⡜⡞⡇⣿⡏⢀⣿⢰⢧⡿⣀⣼⡏⡎⣼⡞⣼⣯⣾⢁⣼⡄⣼⣹⢁⣿⣿⠀⣿⣿⣯⡀⡄
⠀⠀⠀⠀⣿⣀⣇⣇⣿⡇⣷⣿⣿⣾⣼⣇⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇⣾⣿⣿⣠⣿⡿⡈⣷⡇⡀⢠
⠀⠀⠀⢰⡏⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣸⣿⣿⣏⣿⣿⢿⣿⣇⣧⣧⣸
⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢹⣇⢿⣿⣿⣿⣿⣿⢿⡟⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡾⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠈⠉⠉⠁⠀⠈⠈⠁⠀⠈⠁⠁⠁⠁⠁

⠀⠀⡶⣤⢄⠀⠀⣀⣤⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⡘⠉⠳⣴⣦⠞⠉⢓⡲⠆⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠙⣿⣟⣷⣶⣻⣿⠛⠀⠈⢷⣿⣶⣤⣴⡾⣷⠏⠀⠀⠉⢶⣀⣼⢣⣀⣶⠉⠀⠀⠙⢃⢼⡿⢾⡾⣿⢆⠑
⠀⠀⠀⢿⣿⠕⠃⢟⣿⠂⠀⠀⠀⢌⣙⡛⣛⣁⠄⠀⠀⠀⠀⠨⠛⠁⠈⠛⠇⠀⠀⠀⠀⠈⠻⠯⠁⠩⠿⠋
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠠⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣠⣴⣄⠀⠀⢀⣤⣤⣄⣤⣤⣄⣤⣤⢤⣤⣤⠀⠸⣿⣿⠓⢠⡄⠚⣿⣿⠇
⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⣀⣴⣺⣿⣿⣿⣟⣢⣄⠉⠛⠻⣿⢿⠿⡿⣿⠛⠋⠉⠀⠀⠈⢻⣆⡘⢃⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⣯⠿⢯⠿⠛⠋⠁⠀⠀⠑⠚⠀⠓⠈⠀⠀⠀⠀⠀⠀⢨⠛⠋⠙⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀
⠀⠀⢄⠒⢠⣤⠐⠀⡔⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢿⣿⠿⡿⠃⡀
⠀⠀⠀⠠⠤⠃⠤⠀⠀⢠⡄⠀⣠⠀⠀⠀⠄⠡⠄⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠙⠺⠻⠚⠁⠀⠀⠀⠠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⣠⣤⣤⣤⠤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⢀⣶⣴⡦⣤⣀
⠀⠀⣿⣏⢨⡿⠷⠠⠉⠀⠀⢠⣭⣝⣲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡞⠩⣥⠀⠀⣀⣀⠀⠘⢻⡦⢸⣿
⠀⠀⢻⣧⣙⠧⠈⣀⠀⠀⠘⠿⢋⠀⠀⣉⣠⣷⡦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣡⡀⠀⡩⠛⠀⠀⠠⢤⣘⢮⣅⣾⠟
⠀⠀⠀⠙⢟⣷⣤⣟⠢⡤⠀⠀⢨⣿⡞⣻⣏⣛⠀⠪⣦⡀⣠⣀⢀⣾⢟⢕⡋⠩⠟⣻⣿⠄⠀⠒⠶⠌⣳⣦⢿⡽⠋
⠀⠀⠀⠀⠀⠑⢏⠧⣒⡛⠥⢔⠂⣽⣿⠦⠤⠬⢝⣳⣦⣽⣆⣷⡿⠷⢛⣉⠀⠒⠲⣿⣷⡄⣒⠠⢉⣶⠼⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢛⣧⣶⣉⠄⢚⠉⠩⠤⠒⠀⠘⢊⢘⡹⣯⠃⡁⠉⠀⠀⣀⠭⠉⠐⣢⣭⡷⣾⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢝⣿⠜⠀⡄⠐⠀⠀⡀⠀⢁⠜⢩⠀⡀⠈⠀⢅⠀⢀⠁⠢⠈⠰⢿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⠄⢚⠄⡠⠄⠊⢀⠀⠌⠀⠘⠀⢠⠀⠑⡀⠑⠀⢬⠠⡇⠂⢥⣿⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣏⠔⡠⠋⢠⠃⢠⠀⡆⡜⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠈⡆⢈⢦⡲⣽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣿⣠⢃⡎⡖⡸⣸⠠⠡⠀⠀⠀⠀⠀⠀⠀⠀⢧⢡⢸⠘⣜⣇⡽⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢏⡿⣲⢧⣷⡿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⢾⡖⣷⢻⡍⠗⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⣴⠲⡉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡂⠔⠡⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣕⡒⡄⠈⠳⡄⠀⡀⢻⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠪⠤⠏⠌⠀⠁⢐⣨⡎⡇⠀⠘⠇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠉⠓⠢⠀⠀⠀⠀⠈⠄⠥⢢⢹⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠏⣮⠍⢳⠱⠀⠀⠀⠀⠀⠐⠈⢁⡸⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢺⠀⠑⠒⠫⠈⠀⠀⠀⠀⠠⡀⢀⠁⢃⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠳⣄⠀⠀⣟⣻⢰⡍⢣⠀⠀⠀⠠⠄⢀⣴⣶⣾⡿⠟⠛⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡅⣴⣾⣾⡟⢦⠀⢻⣬⡆⠐⠁⠀⣀⠤⠒⣽⡿⠟⠁⠀⠀⠀⠀⢠⢾⡍⠋⣹⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣳⠗⢌⣧⡱⣹⣾⣇⠀⠙⠛⠒⠚⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⣟⢳⣾⡊⣀⡼⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣏⢰⣶⣾⣿⠙⣏⠘⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣫⣧⡼⢯⡀⠘⣟⣧
⠀⠀⠀⠀⠀⠀⣼⣟⡾⢈⣻⣦⠙⣧⠸⣶⣷⢹⠀⢰⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣽⣿⡔⣩⣧⠐⣩⢾⢧
⠀⠀⠀⠀⠀⠀⠸⣿⡃⣤⣿⣿⡂⠀⢳⣿⡿⣏⡇⠀⠁⠀⠀⠏⠀⠀⠀⣀⠀⠀⠀⡶⣶⣛⢢⢹⢎⡁⠼⣏⣠⠎⣹⡆
⠀⠀⠀⠀⣀⣀⣀⣈⡓⢿⣿⠁⠠⠀⠀⠻⣇⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢡⡥⢲⡹⣼⣋⡤⠒⠛⣥⠀⢰⠧⣄
⠀⠀⡴⢿⡿⣿⣌⢙⠯⠙⣿⢷⣤⣶⣤⡀⢠⡜⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⡸⣿⣯⡴⠖⠚⣁⣔⣀⠀⠚⢲⡄
⠀⢸⣓⡟⢠⣬⣍⣴⠛⠫⠉⣛⠧⣿⣷⢝⣮⣻⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⠐⠁⠋⣹⢦⠖⠋⠐⢛⣧⣤⠤⠾⠧⡄
⠀⠸⡷⢻⡀⠶⢂⣜⢛⠶⢶⣖⣻⣿⡿⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⡀⠀⠙⣮⡳⢯⠁⠀⠘⣷⠀⠀⠒⠛⢀⣀⣀
⠀⠀⠘⠳⢽⣤⡿⠭⠟⠛⠉⠁⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡦⢤⣬⣻⣭⢧⣀⣀⣽⣏⠁⢴⢲⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠉⠙⠳⢦⣼⡉⢿⣚⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁

⣿⠿⢿⠿⣿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢻⣿⣿⡿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠶⢦⣤⣄⣀⠖⡛⠛⠛⠛⠛⠛⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠷⠦⣶⣄⣀⠉⠢⣄⠀⠀⠀⠀⠀⠈⠙⠻⢷⣷⣤⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣀⠀⠀⠀⠉⠙⠿⣦⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠠⠄⡀⠀⠀⠀⠠⠽⣟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⡠⠔⠒⠂⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠛⠛⠛⠛⠛⠶⢦⣤⣳⡀⠀⠀⠀⠐⠊⠉⠀⠤⠄⠀⠞⠐⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢥⣤⣀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣷⠀⠀⣠⡶⠿⠶⠾⠶⣶⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠄⣙⣧⠀⠀⠀⠀⣼⢫⣀⣨⣥⣤⣤⣭⣗⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣀⠀⠀⢈⡃⠀⠀⢸⣷⠟⠋⠉⠀⠀⠀⠀⣉⠽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠻⢿⣦⣀⣿⡄⠀⠘⢿⠿⢶⡄⠀⠀⠀⡠⠤⠒⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣽⠿⢟⣛⣉⣉⣩⣭⣙⣶⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣀⣉⣉⣉⠲⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⠭⠖⠊⠉⠁⠀⠀⠀⠀⡬⣻⣿⣿⠷⠾⠟⠻⠶⢷⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡿⠗⠢⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠺⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠀⠀⢸⡆⠀⠀⠀⠀⣰⣿⣄⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢠⠏⠀⠈⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⢹⡄⠀⠀⠀⠀⠈⢀⡠⠘⠯⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠖⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣦⣄⠀⠀⠀⢤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠
⠈⠳⣝⠿⣦⣄⠀⣠⣾⢿⡿⠃⠘⣿⡖⣤⡀⠀⢀⢤⣐⡿⠃⠘⡟⠀⣵⣤⡾⡀⢻⠁⠐⣖⣂⡀⠀⠀⢀⣀⣴⠀⠈⠻⣧⢼⠆⡻⠏
⠀⠀⠙⠷⣨⠻⣿⠿⣵⠟⠁⠀⠀⠈⠿⣽⣿⣿⢿⢷⠟⠁⠀⠀⢀⠀⠁⠋⠀⠀⡄⠀⠀⠘⠌⠉⢳⠞⠉⠱⠁⠀⠀⠀⠀⠈
⢰⠦⣤⡀⠙⠗⠉⠘⢃⣤⣖⡗⠀⠀⠐⡇⠀⠀⠈⠸⠃⠀⠀⠀⠈⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⢰⠞⠳⠀⠀⠀⠀⢀⢠⢼⢤⠄⡀⠉
⠀⠐⢿⡛⠷⣤⣤⡾⠛⣩⠎⠀⠀⠀⠠⠑⠁⠀⠙⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⢠⣴⡖⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠙⣾⡏⠟⣽⡿⠃⠀⠀⢀⣤⡴⠂⢠⡀⠤⣠⣀⠀⣴⡶⣤⣄⣠⣶⠖⣶⠆⠀⠀⠘⣷⡽⣡⡞⠀⠀⠻⢾⣤⣄⣀⣠⣴⡶⠖
⠀⢶⣓⣦⡈⠀⠀⢈⣴⣒⠆⠀⠀⠉⠻⢦⠼⣤⡶⠛⠉⠀⠈⠳⡝⢿⡿⢫⡞⠁⠀⠀⠀⠀⠻⠃⠟⠃⠀⠀⠀⠀⠋⠁⣯⠉⠋
⠀⠀⢻⡙⢿⣄⣤⡿⢛⡟⠀⠀⠀⡤⠄⠀⠀⠀⠠⠤⠀⠀⠀⠀⡹⠓⠚⢏⠀⠀⠀⠠⣖⡤⢀⠀⠀⠀⠀⣄⢤
⠀⠀⠀⢠⣌⠹⡟⣁⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢶⡶⠂⡀⠀⠀⡠⠐⢲⡶⠀⠹⢵⠋⠠⠶⠈⢧⢹⠃⠀⢤⡀⠀⠷⠀⣀⠖
⠀⠀⠀⠘⠿⠇⠙⠻⠃⠀⠀⠀⠀⠀⣄⠀⠀⢀⣰⠀⠀⠈⡇⠀⠈⢢⡄⠀⠀⣸⠁⠀⠀⠀⣦⣀⠦⣠⣤⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠰⡦⠘⠀⠀⠀⠀⠡⢶⠀⠀⠀⡀⠉⠁⠀⠈⠁⠀⢀⠀⢩⠀⠀⢸⣇⠀⠀⡅⠀⠀⠀⠀⠐⠁⠀⠑⠁⠀⠀⠀⠀⢃⢶⣄⢦⠌
⠀⠀⠃⠈⠁⠀⠀⠈⠀⠁⠀⠀⠀⠳⣤⣄⣤⣤⣤⣤⠞⠁⠈⠳⠠⠞⠙⠤⠔⠁⠀⠀⢰⣶⣤⣄⠀⠀⠀⣠⣤⣶⡆⠐⡧⠛⣾
⠀⠀⠀⠒⠀⣶⡆⠀⠶⠀⠀⠀⠀⠀⠈⠉⠘⡇⠈⠁⠀⠀⠀⣾⠗⠀⠀⠀⠀⠂⠸⣿⡄⢻⣿⡉⢣⣤⠞⠁⣼⡟
⠀⠀⠀⠀⠘⠀⠘⠄⠀⠀⠻⣿⣻⣶⣦⡀⠀⠀⣠⣤⡖⣺⡿⠊⣄⠀⠀⡦⠀⠀⢠⠃⠀⠀⢛⡗⠈⠟⠐⢶⠛⢁⣤⡴⢶⡶⣦⣄
⠀⠀⢀⣤⠀⢀⡀⠀⣤⠀⠀⠀⠙⠻⠋⠿⣦⣤⠷⠽⠟⠉⠀⠀⠀⠁⢀⠀⡀⢐⠀⠀⠀⠀⠈⠻⡶⠐⣶⠞⠀⠛⠛⣿⡏⢹⡿⠿⠷
⠀⠀⠀⠈⢀⡀⠁⠔⠁⠀⠀⠀⠀⠀⣎⣂⡘⢧⣠⡗⠀⠀⠀⠀⠀⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣯⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⣼⣷⠦⡀⠀⠀⠀⠀⠀⣀⢶⣾⡧⠀⢸⣿⠷⢆⠀⠀⠀⠀⠀⠀⣀⠴⣶⣶⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⠀⠀⠀⢀⣴⣾⣿
⢸⡾⢿⣛⣃⡀⠀⢀⣘⣛⡻⢿⡇⠀⠈⣿⠿⠿⢷⣄⠀⠀⢀⡶⠿⠿⢾⡏⠀⠸⣯⠘⢣⣼⣶⣤⣀⣄⣠⣴⣶⡜⠃⡽⠀⠀⠸⣿⣟⠀⠱⣀⡀⠁⠈⣿⡿
⠘⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠁⠀⠀⠫⠀⣀⠀⠙⢦⣤⠏⠀⠀⠀⡸⠁⠀⠀⠈⢀⠿⢿⠿⢛⣿⣿⡟⠻⣿⢿⡌⠁⠀⠀⠀⢉⣿⠖⠚⣿⠓⠒⢿⣏
⠀⠘⣿⣿⡿⠃⣿⠘⢿⣿⣿⠃⠀⠀⠀⢠⠚⠀⠀⠀⠸⡇⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠁⠀⠀⠸⣿⣹⡿⠀⡀⠋⠀⠀⠀⠀⠀⠸⣇⡀⢀⠈⢠⠀⣀⡿
⠀⠀⠈⠻⠃⠀⠀⠀⠸⡟⠁⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠏⠀⠈⠻⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃
⣴⣿⠶⠀⠀⠀⠀⠀⠀⠀⠚⠿⣷⠀⠀⢺⣶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠆⠀⠀⠀⠀⢄⠀⠀⠀⠀⡠⠂⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶
⠏⠀⣀⣤⠴⠀⠀⠀⠢⣤⣄⠀⠘⠀⠀⠀⢇⣇⢹⣷⣄⠀⠀⠀⣠⡾⢑⣄⡇⠀⠀⠀⠀⠀⠀⢣⡀⢀⠒⠀⠀⠀⠀⠀⠸⠋⠀⢀⣀⠄⠀⠀⠤⣀⣀⠀⠙
⠀⣸⣿⣿⠄⠀⣧⠀⠠⣿⣿⠇⠀⠀⠀⠀⠸⠿⣤⠭⢽⡦⣤⢾⡻⠥⣘⠳⠀⠀⠀⠀⠀⢠⣼⣿⣶⣾⣿⣤⠀⠀⠀⠀⠀⠠⣾⣿⣅⠀⢰⡀⢀⣼⣿⣗
⢸⠙⢿⣿⣤⡾⣿⢶⣤⣿⡿⠃⠇⠀⠀⠀⠀⠞⢈⣤⡘⠛⠟⠛⢁⣄⠉⣣⠀⠀⠀⠀⠠⣏⠈⠟⠘⠘⠟⢘⡇⠀⠀⠀⠀⡞⢿⣿⣇⣠⣾⢦⣄⣹⣿⠷⢱
⠀⠀⠀⠉⠛⠁⠀⠈⠻⠉⠀⠈⠀⠀⠀⠀⠀⠘⢀⣤⡿⠂⠀⠲⣿⣆⡰⠁⠀⠀⠀⠀⣠⠛⠻⠀⠀⠀⠹⠛⢅⠀⠀⠀⠀⠁⠀⠙⠿⠏⠘⠈⢿⠟⠃
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡟⣩⡶⠗⠂⠀⠀⠀⠀⠀⠐⠿⢶⣍⠇⠀⠀⠈⢶⡀⠀⠀⠀⠀⠀⢠⡔⠉⠀⠀⢾⣷⣢⣄⠀⠀⠀⠀⠀⣠⣴⣼⡷⠀⣿⠉⣂⣀⠀⠀⠀⠀⠀⠀⡀⠈⢛
⢿⠉⢀⣠⣤⣄⣄⣤⣄⣤⣦⣄⡀⢙⡇⠀⢀⣀⣸⣿⣦⠀⠀⢀⣴⣿⣋⣀⡀⠀⠘⣿⡍⠫⣷⡀⠀⢠⣮⠝⣹⣿⠃⠀⠘⣯⠻⣿⣦⡀⠀⠀⣠⣾⣿⢠⡟
⠈⢛⣛⡿⣏⢯⣹⣿⣹⣏⡞⣿⢉⡋⠀⠀⠘⢿⣇⣈⣿⣧⣦⣾⣿⣁⣹⡿⠀⠀⠀⠈⣻⡄⠛⣿⣿⡟⠛⢠⡛⠁⠀⠀⠀⠹⢷⠈⣻⣷⣴⣴⣿⠋⠨⡿⠁
⠀⢾⣇⣿⣿⣾⡇⡏⢻⣼⣻⡿⡼⠇⠀⠀⠀⣰⣾⣿⣿⢿⣿⢿⣿⣿⣶⡄⠀⠀⠀⠀⣻⣇⢸⠃⡇⢹⢀⣼⡗⠀⠀⠀⠀⠀⣴⣿⠛⠛⢻⠋⢻⠻⣤
⠀⠈⠻⠶⠯⠋⠀⠁⠀⠹⠶⠞⠋⠀⠀⠀⠀⢻⣦⡼⠃⠈⡋⠀⡹⢧⣾⠇⠀⠀⠀⢀⠍⠿⠀⠀⠀⠀⠹⠏⢆⠀⠀⠀⠀⠀⠈⠻⠟⠀⠘⠀⢻⠏⠃
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠂⠀⠃⡶⢄⠀⠀⠀⡀⢴⡜⠀⠄⠀⠀⠘⣿⣷⣦⡀⠀⠀⠀⣠⣶⣿⡿⠀⠀⢰⡓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆⠀⠀⣿⣿⡿⢂⣀⠀⠀⠀⠀⣀⠠⣶⣶
⠀⠀⠀⠀⠉⢳⣶⣖⠉⠀⠀⠀⠀⠀⠀⠀⡿⣠⠀⢈⠀⣄⠀⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠇⠀⠀⠘⣿⠴⡟⠁⠀⢰⡆⠀⠙⢿⢈⡿
⠀⠀⣀⠹⠀⠚⣿⠻⠰⢏⢀⠄⠀⠀⠀⠀⠉⣹⣶⠾⣿⣿⣿⠷⢶⣊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣷⠞⠉⢹⠇⠙⢶⣿⡌⠁
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢿⣇⣠⡶⠛⠴⡆⢀⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⢿⠀⠀⠀⠸⠁⠀⠀⢙⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⡴⡶⡛⣻⠉⠉⣿⢍⢟⢻⢶⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⡌⡜⡰⢸⡄⠀⣿⡌⣆⣣⠷⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠀⠀⠀⠙⠲⠭⣝⣛⣻⠁⠀⣸⠭⠽⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡟⠀⠀⠀⠀⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⡇⠀⢠⠇⢀⡤⠖⠊⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢶⣟⣹⡟⠉⣟⢒⣶⡄⢸⠄⠀⠀⠀⡞⠀⠀⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⡇⠀⣯⣭⠭⠴⠋⠀⠀⠀⡼⠁⠀⢰⠃⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⡏⠀⠀⠀⠀⠀⡝⠙⡶⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⢀⡏⠀⠀⡾⠀⠀⠀⠀⢀⡞⠀⣸⠁⠀⣀⡠⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢧⠀⠀⠀⠀⣸⠀⠀⢠⡇⠀⠀⠀⢠⠎⠀⣰⠃⢰⢋⣁⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣄⠀⠳⡀⠈⢳⡀⠀⠀⡏⠀⠀⢸⠀⠀⠀⢠⠏⠀⣰⠃⠀⠈⠮⣓⣮⠋⣱⠦⣄⠸⡄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠙⢦⠹⡄⠀⠹⣄⢸⡇⠀⠀⢸⠀⠀⠀⡟⠀⢠⠇⠀⠀⠀⠀⡸⠃⣰⠯⠭⠭⠞⠁
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⢀⣀⡠⠞⠀⠹⡄⠀⠘⢾⡇⠀⠀⢸⡀⠀⣸⠁⠀⢸⠀⠀⢀⡤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⢇⠀⠀⠀⠀⢳⡀⠀⠘⡇⠀⠀⠈⣇⠀⡟⠀⠀⣿⣠⠖⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠳⢤⣀⣀⡀⢧⠀⠀⢳⠀⠀⠀⢹⡄⡇⠀⠀⢸⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣹⠺⡆⠀⠈⢧⡀⠀⣰⠃⢧⠀⠀⢸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⢧⣀⣀⡞⠉⠉⠁⠀⠘⠦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢈⣡⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⡿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢤⡀⠒⠶⠾⢦⣔⢾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⣤⣤⣤⣈⠻⢶⣶⣦⣶⣶⣤⣄⣤⣤⣴⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣿⣿⣿⣿⣧⡄⠀⠉⠛⢻⣿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠶⢂⣤⣴⣶⢶⣾⣶⣶⣦⣾⠀⢠⡿⠛⣻⣟⣠⣿⠷⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡌⢿⡿⠟⠀⠃⠀⠈⣠⣾⣩⣹⡿⠑⡆⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⣶⠄⠀⠀⠀⠀⠀⠿⠿⠻⠛⠏⠁⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢦⠀⠸⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⡰⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣠⣶⣧⣄⠃⠀⠀⣸⣶⡀⢸⣄⠀⢀⣠⠎⠀⠊⠁
⠀⠀⠀⠀⠂⠀⢣⡀⣰⡈⠛⢹⣿⣯⠤⢦⣿⣿⣷⣼⣧⣖⣼⡃⠄
⠀⠀⠀⠀⠀⠀⠻⠋⠉⣿⣿⣻⣿⣿⣷⣄⣿⣿⢉⠙⡀⠟⠋
⠀⠀⠀⠀⠀⠀⠺⣟⠁⠈⢻⢛⣾⣿⠋⡙⣛⠋
⠀⠀⠀⠀⠀⠀⠈⠟⠓⠀⢠⠨⠅⠈⠁⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣸⣄⠀⢆⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠭⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠧⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡞⠛⠛⠣⣼⣿⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢿⣿⡤⠛⠛⠛⣾⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣯⠉⠉⠛⣿⠃⠀⣠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀⠀⢻⡟⠉⠉⢩⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣉⣉⣙⣇⣸⣿⡿⢉⣿⣷⡀⠀⠃⠀⠀⠃⠀⣰⣿⣏⠹⣿⣿⣀⣟⣉⣉⣉⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠉⣀⣀⣍⣿⣿⡀⢸⣿⣿⢿⡄⢡⡀⡠⠁⣼⢿⣿⣿⠀⣸⣿⣯⣅⣀⡀⢹⣟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣀⣀⣀⣤⠿⠿⢿⣿⣧⣈⣿⣘⣿⣟⣸⣏⣠⢿⣿⠿⠿⢷⣤⣀⣀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣇⣠⠤⠤⠶⠒⢾⣿⠦⠬⣭⣿⠁⢹⣯⡭⠴⢾⣿⠗⠒⠦⠤⠤⣀⣿⣽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣦⡤⠴⠶⠒⠚⢻⣶⣶⢾⣿⠁⢸⣿⢶⣶⣾⠛⠒⠒⠶⠦⣤⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⠴⢒⠞⠉⣿⠃⢀⡾⣿⣗⣿⡿⣆⠀⢻⡏⠙⢖⠲⠤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣼⣃⣤⡏⠀⠀⣿⢠⡞⢰⠏⣗⡏⢷⠘⣦⣸⡇⠀⠈⣧⣄⣻⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠻⡅⣸⣇⣀⣠⣿⣻⢀⡞⠀⠻⠃⠘⣆⢸⣽⣧⣀⣀⣿⡀⡽⢻⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢾⣾⣷⠁⡜⠉⡏⠈⣿⣽⠃⠀⠀⠀⠀⢻⣭⡏⠈⡏⠙⡄⢹⣿⣿⠿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣿⣼⣿⠷⣴⡷⢾⣷⣿⠀⠀⠀⠀⠀⢸⣷⣿⠶⣧⡴⢿⣿⣼⣿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⢹⣿⣿⣲⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣯⣿⣷⣿⣿⣿⢡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣶⣽⡿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⣿⣵⡾⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡤⡀⠀⠀⠀⠀⠀⠀⠸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣪⢓⠄⠀⠀⠀⠀⠀⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠫⠈⢀⠀⠀⠀⣰⣿⡿⣧⣀⠤⠴⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣞⣿⣸⠃⠸⠀⠀⠀⢸⣿⣡⠞⣿⠖⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢫⣼⣏⠀⣼⡇⢀⠄⠚⢻⣅⣀⡀⣀⠀⠐⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠶⢽⣷⣼⣿⣿⣷⣤⠂⢀⢿⡏⢀⡀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣿⣿⣯⣿⣿⣿⣵⣏⠀⠙⠲⠶⣶⣥⡶⢿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣤⣤⣬⣿⣿⣿⣿⣿⠏⢹⣿⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡀⠉⠃⠐⠀⢰⣾⠿⠛⠉⠉⠙⣻⣧⡾⠟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣁⠤⠀⠀⣠⠔⠋⠀⠀⠀⠀⢤⣸⣿⣿⣿⣿⠃⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣿⡿⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡝⣋⡏⢨⡀⠠⣠⣾⣠⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠲⣼⣯⣙⣿⣟⣿⣿⣿⣿⡿⠁⠀⢰⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠀⠀⠀⠀⠀⠈⠿⣿⣹⣿⣏⣠⣿⠿⠀⠀⠀⡄⠛⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⡴⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠻⡁⠀⠀⠀⠀⠁⣴⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⣀⠢⠀⠀⠀⠀⠠⣾⣿⣿⣧⠀⠀⢀⠀⠴⢦⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⣿⣖⣲⣴⣾⣵⣟⣈⡿⠧⣀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢬⣩⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢉⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⢀⣿⣿⡃⠀⠀⠀⠀⠈⢹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣥⣶⠿⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣏⣽⣦⣼⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⠭⠙⠉⠉⠙⠓⢄
⠀⠀⠀⠀⣀⣤⣤⡤⣤⣴⣷⣿⣽⣿⣿⣿⣿⣻⡿⣿⣇⠀⠀⠀⠀⣀⠤⠒⣒⢀⣴⡂⠁⡡⠋⠁⠀⠀⠐⠲⠢⢄⠀⠂
⠀⠀⣰⣿⣿⣷⣾⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣼⡏⠀⠀⠀⡼⠁⠀⠀⠀⠈⠁⠠⢊⠀⠘⠀⠀⠀⠀⠀⠄⠈⡀
⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⡀⠶⠄⠀⠀⠐⠁
⠀⠀⠘⢿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⠀⠀⠀⠈⠙⡄⠀⠉⠀⠂
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⢐⣠⡀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⣶⣢⣴⣦⣄⣀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡁⣀⢀⠈⠉⠳⠛⠀⠂⢤⢴⣶⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣭⣼⣿⣿⡆⠀⠀⠀⠀⠀⠈⠃⠀⠀⠨⠉⠖⢄⡀⠀⠀⠀⠛⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣿⣟⣿⣿⢿⣿⣿⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠆⠤⢤⠤⢤⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠛⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠑⠀⠀⠀⠂⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠄⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⣿⣿⣿⣿⣷⣦⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡇⢠⡄⢸⠀⠀⠂⡛⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠛⠸⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠨⣛⣦⣷⣾⣴⡀⢰⠟⠋⠁⡈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠄⡄⠘⡔⡠⡄⢊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⡶⠂⡀⢰⠀⠀⡀
⠀⠀⠀⠀⣰⠐⡋⡈⠳⡠⠒⠃⠰⢈⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠃⠺⣀⠴⢂⠋⠀⡐⢠
⠀⠀⠀⠀⠸⣈⡇⠑⠒⠶⠀⡅⢘⠘⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠞⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢒⠈⠈⠄⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠠⢄⢠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠊⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⣀⡰⠎⣡⠀⠀⠀⠁⠀⠀⠀⠀⢢⣎⠀⡄⢀⠀⠀⠀⠀⠁⢠⡄⢀⠀⢈
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡆⠈⢅⡀⣖⠙⠇⢀⣄⣠⢨⡇⠈⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠏⠲⠄⠀⠁⠊⠀⠙⠒⠂⣠⠗⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⡶⣶⡦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⠁⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢺⣏⢹⡇⣩⣿⣫⡾⠃
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣋⠈⠁⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⣀⣠⣴⣿⣿⡔⠋⢀⣼⣿⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣗⠙⢷⠆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡟⣿⢿⠇⠀⢠⣾⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⣌⣀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣿⣿⣟⢿⣿⣌⠂⠀⢠⢸⣿⣘⠇
⠀⠀⠀⠀⢀⣠⢺⣿⣿⣿⡄⣷⣿⣿⣧⡽⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣆⡹⠟⠛⠻⠈⠃⢿⣿⡧⠀⣠⣿⣿⠧⠃
⠀⠀⠀⢠⣜⣿⣿⣿⡟⠁⠘⠋⠹⣿⣿⢇⢰⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⠿⠦⢉⣁⣀⣠⣀⠠⠄⢀⣉⣟⠞⢻⣿⡟⣺⡀
⠀⠀⠀⢘⡉⢿⣿⣿⣷⡶⢀⣠⣤⣬⣽⣿⣿⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⡶⡿⢿⢿⣿⣟⢿⣿⣿⣿⣿⣦⡀⠀⠗⠉⠀⠓⠄
⠀⠀⠀⠀⠁⠈⠛⠉⠟⠃⠻⠿⠿⠟⠫⠌⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠷⠛⠛⠉⠉⠁⠈⠋⠖⠋⠽⣉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢿⣝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠳⠾⠁⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠉⠁⢑⣶⠄⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣶⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⡀⢁⡂⠀⠀⢀⠀⠀⣠⡀⠀⠒⣀⠘⢛⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⠥⢠⣶⣿⡃⠀⠀⢥⣄⢺⠭⠞⠉⠀⠀⠀⠀⠀⠀⣠⣶⣿⣛⡷
⠀⠀⣾⣿⡟⢶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⢀⠀⠱⠀⠵⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠟⣡⣿⠐⡟
⠀⠀⠙⣿⣿⣴⠿⠿⠟⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⠇⣾⣿⣹⠇
⠀⠀⠀⠘⣿⣷⠒⠒⠂⠀⠄⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣭⡛⢲⡿⣧⣼
⠀⠀⠀⠀⠘⣿⡏⠉⣁⣠⣤⣤⣤⣀⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣷⣶⣷⣤⣿⡈⣿⢤⣄
⠀⠀⠀⠀⠀⣿⡛⠻⠟⠉⠍⠉⣡⣾⣿⣿⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣈⠿⢹⢿⣿⣿⣷⣤⡟⣿⠟⠛⠋
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠞⠀⣾⣿⡿⠁⢿⡆⠀⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣾⠀⢶⣿⣿⣿⣗⣿
⠀⠀⠀⠀⠀⠀⠹⠿⣷⡶⣾⡟⠁⣀⠂⢸⡇⠀⠁⢀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⢹⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠈⠁⠐⢿⡇⢰⠀⢾⠘⣽⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⢿⣿⡿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣦⣼⣷⣾⣦⣾⣾⣯⡍⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠠⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⠻⢿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢻⡛⠦⣀⣀⡴⠛⡿⠀⠀⡤⢀⠄⠀⠀⠀⠠⣄⢠⡄⠀⢸⣷⣄⠀⠀⢀⣴⣿⠀⠀⠀⣶⣶⣦⡀⠀⢀⣴⣶⣶⠀⠀⢻⣶⣄⠀⠀⠀⠀⡠⣶⡟
⠀⠀⠀⠈⠃⠀⠸⢇⠀⢐⠃⠀⠀⠱⠁⠀⠀⣴⡀⠀⠀⠛⠁⠀⠀⣿⢻⣧⣠⣿⣳⡇⠀⠀⠀⢾⣟⣹⣿⣴⣿⣏⣻⡗⠀⠀⠈⣷⣿⣷⣄⣠⣾⣿⣾⠁
⠀⠀⠀⠀⢹⣤⠂⠀⣤⡞⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢈⣯⣿⢿⣯⣏⠀⠀⠀⠀⠠⣿⣬⡿⠿⣿⣭⣿⠀⠀⠀⠀⠈⣽⣿⠟⠻⣿⣯⠁
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠹⡟⠀⠀⠀⠀⠀⠙⠛⠇⠀⠸⠛⠋⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⢠⣴⡆⠀⣤⠤⢄⠀⠀⠀⠀⢀⠤⢤⡄⠀⣶⣤⡀⠀⠀⠀⢀⣴⣶⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⣿⣿⢗⣤⣴⣦⣤⡲⢿⣿⠀⠀⢫⣴⣾⣤⣾⣶⣌⠇⠀⠀⣀⠀⠹⣄⣀⡶⠁⠀⡀⠀⠀⠘⣿⢿⣆⣀⣰⡿⣿⠇⠀⠀⢸⡿⢷⢀⠀⠀⣀⡾⢿⡇
⠀⠀⠈⢷⣸⣿⣿⣿⣿⣟⡼⠃⠀⠀⠈⢉⣿⢿⢿⡏⠉⠀⠀⠀⢀⠐⠀⢈⣏⠀⠐⢀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⢉⢊⣉⣿⣿⣃⡙⠈
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠘⠃⠀⠈⠃⠀⠀⠀⠀⠀⢦⠀⡞⠹⡄⢠⠆⠀⠀⠀⠀⣿⣿⠉⠀⠉⣿⣿⠀⠀⠀⠀⢸⡸⠿⠁⠀⠿⠇⡆
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠉⠉⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉
⠀⠘⣇⠉⠀⠄⠀⠀⢀⠔⠉⢠⠇⣶⣦⣄⠀⠀⠀⢠⣴⣶⠄⠸⡟⡷⢀⠀⠀⢀⢸⡟⡿⠀⠰⡾⠂⠀⠀⠀⢀⠠⠐⣿⠄⠀⣤⠄⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠈⢦⣴⣾⣦⣰⣿⣶⡴⠋⠀⢸⡟⢹⣷⡤⣼⣿⠙⣿⠀⠀⠃⠀⡜⢱⣶⠙⡄⠀⠁⠀⠀⠅⠀⠒⠃⣴⠘⠓⠀⠨⠀⠀⠁⢠⠀⠀⢰⡆⠀⠀⡄⠘
⠀⠀⠀⢀⣼⣿⡿⢿⣿⣧⡄⠀⠀⠀⢴⠈⠿⠻⠻⠇⢰⠀⠀⠀⠠⡀⠀⠈⠏⠀⠀⡰⠀⠀⠀⠀⠈⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠈⡿⠞⠁⠀⠁⢺⠅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣤⡀⠀⠀⠀⠀⣠⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⢤⢤⣄⠀⠀⠀⠀⣠⡤⣤
⠀⠀⠘⣿⣋⣴⣄⢠⣴⣝⣿⡇⠀⠘⣿⢚⣦⣀⣠⡟⣽⡇⠀⠀⡇⣧⠀⠀⠀⠀⢠⡇⡇⠈⢻⡿⣷⣄⠀⠀⠀⣠⣾⢿⡟⠀⠘⡍⠈⢳⡄⢠⣾⠁⢉⠇
⠀⠀⠀⢻⡟⠛⠻⡿⠛⠻⣿⠀⠀⠀⣡⣾⣿⢿⣿⣿⣬⠀⠀⠀⠁⠸⡿⢷⣷⡿⡿⠀⠁⠀⠀⠻⠌⣻⣷⢶⣾⣟⠡⠟⠀⠀⠀⠘⣡⣿⣿⣿⣷⣌⠃
⠀⠀⠀⠘⠷⣤⠂⠐⣤⡾⠃⠀⠀⠀⢹⢉⡛⠀⠘⣃⣻⠀⠀⠀⠐⣄⠃⠁⠂⠀⢃⡔⠀⠀⠀⠀⢾⣅⡉⠚⢉⣩⡷⠀⠀⠀⠀⠈⠢⠝⠀⠀⠫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣾⠀⢀⡏⡹⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡿⢖⠂⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⠀⠃⠲⡛⠿⣦⣤⠄⠀⠀⠀⠀⠀⠀⢀⠴⠻⣶⠾⠳⢽⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣲⠀⠀⠀⠙⣆⠀⠙⢷⡀⠀⠀⠀⢀⠔⠁⢀⠼⠓⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣃⠈⢀⣀⣉⣕⣦⣄⡙⢦⣶⡷⠃⣀⣒⣉⣀⣀⣀⢀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢛⢻⠿⠟⢉⡙⠫⡭⣿⣿⣷⣯⣋⡉⠉⠛⠿⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠀⠀⡐⠊⡠⠂⠐⠸⣿⠘⠂⠠⡈⠲⠀⠀⠘⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠁⠒⠀⡐⠀⢀⣿⠀⠠⠠⠈⠒⢤⠀⠀⢐⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢢⡄⡀⠀⠀⠠⠀⠄⠈⣿⠀⢀⠂⠡⠀⢂⠁⣈⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢇⣞⣦⣷⡾⠀⠀⠁⠀⠀⠺⢦⣷⣾⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠈⡰⠒⠀⢀⣠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⠇⢨⣿⡷⠀⢸⣿⠀⠀⢀⠀⠀⠀⠀⣴⣦⠠⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠁
⠀⠀⠀⠈⠋⠹⣶⣶⠸⣿⡀⠀⢼⣿⣭⠀⠀⠀⠀⣴⣤⣹⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀
⠀⠀⠀⠀⠀⢰⣿⠏⠀⠈⠃⠀⠀⠈⠉⠀⠀⠀⣴⡎⣵⡿⠟⠀⠲⠀⠈⠛⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊
⠄⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠈⠁
⡀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢀⠀⠀⠢⠄⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠆⠀⠀⠀⠀⠰⠀⠈⠉⠠⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠴⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠦⠤⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀
⠀⠀⠀⠀⡠⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⡀
⠀⠀⠀⠀⣀⢀⡀⠀⢀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣴⡾⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⣀⠀⠀⣼
⠃⠀⠀⡾⡥⠾⠃⠀⣿⣿⠇⠀⠀⠀⠠⣀⠄⠂⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠈⢉⣿⠟⣙⣿⠲⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⠿⠂
⠰⡄⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢃⡃⠀⠀⠀⠠⠀⠀⠸
⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠌
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠚⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⠀⠀⠀⠂⠀⠘⢢⠀⡀⠀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢀⠀⠂⠀⢂⠀⠄⠀⠀⢈⡠⠊⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠄⢀⡀⠠⠒⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠄⠲⠀⠘⠀⠀⠀⡤⠀⠒⠒⠈⠉⠁⠀⠐⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠂⢂⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠠⠞⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠉⠀⠨⠁⠒⠂⠐⠢⢄⣹⡒⢤⡀⠀⢀⡀⢤⠤⠖⠒⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠈⢉⠁⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠀⡀⢀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠱⣄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠂⠀⠀⠀⠀⠀⡎⠀⠔⠀⠀⢠⡀⠀⠀⠱⡄⠀⠀⠀⠀⠉⠀⠀⢀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠄⠀⠀⠀⠀⢃⠘⠁⠀⣀⠁⡁⠠⢄⣜⡅⠀⠀⠀⠀⠀⢃⡄⠀⠀⢈⣦⠔⠉⠠⢀⣠⠂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⢶⡡⣤⢁⠿⢄⡜⠈⠀⠀⠀⢀⠠⠀⠈⢌⠁⠢⠼⣘⡄⠔⡁⠋⢰⠖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⢁⠌⠓⠨⣳⡂⠀⢵⠂⠀⠀⢐⠃⠆⠠⠂⢈⠕⠒⠒⢬⠽⠏⢬⠀⠺⢅⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⢣⡀⠀⢀⠈⠈⠉⡹⠢⠥⠶⢄⠁⢠⢓⠆⠺⠄⣀⣠⡆⠀⠀⡰⢄⠀⢀⣧⠃⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⢀⢲⣀⢠⠉⠉⢩⠉⠒⢶⢅⡀⢠⡃⢀⡀⠘⠬⠄⢸⡀⠀⣠⠋⠙⡁⠀⢀⠿⡁⠀⢱⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡄⠀⡱⡖⠈⢦⣠⠼⠀⠈⣡⡃⢠⠁⠠⡀⢸⠉⡗⠉⠣⡀⣀⠝⠲⠁⠀⠈⡒⢣⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣀⡈⢩⡀⠘⢲⡂⠑⢄⣉⡝⢁⣜⠃⠀⠀⢱⠚⢆⡸⠦⣤⠌⠑⢄⣠⠖⠐⡎⠀⣰⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⠙⡈⢋⠧⡈⣈⢇⣀⣌⣡⠏⠀⠀⠀⠈⢦⣘⢤⡠⠶⣀⣀⡴⠓⣄⠴⢛⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⣙⡀⠀⣠⠄⢢⡀⢾⠔⠁⠀⠀⠀⠀⠀⠀⠡⡀⠱⣄⣈⡆⠀⢐⡀⡠⢛⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠄⠓⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⠬⠯⢁⣉⠽⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠚⠉⠉⢉⡭⠭⠭⢭⡛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⡡⠤⢤⡀⠀⠀⠀⠀⠈⠆⡀⣀⡠⠟⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⠤⠤⣀⠀⠑⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡔⠉⠀⠀⠀⠈⡧⠀⠀⢠⠖⠉⠉⠉⠉⠉⠙⡆⠀⠀⢠⠞⠉⠉⠉⠉⠲⡀⠈⢦
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠣⠤⠤⠤⠖⠊⠁⠀⠀⡇⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠱⣄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠑⠒⠒⠉⠉⠉⠀⠀⢀⣠⣀⡀⠓⠤⠤⠖⠚⠁⠀⠀⢀⠤⣌⢧
⠀⠀⠀⢰⠃⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢘⡞⡆
⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⣀⡤⠔⠋⣀⣀⣠⡤⢤⡤⢤⢤⠤⢤⠤⣤⠤⢤⣤⣀⡀⠀⢰⠂⠒⢦⠀⠈⠉⠀⡇
⠀⠀⠀⢸⡄⠀⠀⠀⠁⠀⠈⠉⣠⡴⠚⠛⢅⡀⠈⢦⠀⡇⢸⢸⢀⡆⡴⠁⡠⢊⠴⢊⡩⠓⢦⡑⠦⠞⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡀⠞⠉⢱⠀⣠⠚⠡⢤⣉⡒⠦⢌⡓⢄⠣⣡⢈⣘⣸⡜⣡⡮⢖⡥⣚⡥⠔⠊⣁⡙⣦⠀⢀⠖⣳⡼⠁
⠀⠀⠀⠀⠀⠘⠢⡀⠁⠀⣇⣉⣉⣐⣒⣺⣷⣶⣦⡭⠟⠋⠉⠉⠉⠉⠙⠛⢯⣭⡶⠶⠟⠛⡒⠒⢸⡇⠐⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⣦⡐⠒⢒⣒⡦⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣭⣉⣉⣩⣭⡴⠾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠋⠉

⠀⠀⠀⢰⣦⣄⠀⢀⣴⡶⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⡀⠀⢠⢤⡀⠀⠀⠀⢀⡤⡄⠀⢠⣤⡀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⢿⠹⣶⡾⢹⠃⠀⠀⠀⠿⣈⣑⣴⣞⣁⠹⠃⠀⠈⠾⣬⣢⣤⣴⣧⠿⠃⠀⠘⣿⣿⣦⣀⣼⣿⡿
⠀⠀⠀⠰⣏⡖⠙⠑⣊⡇⠀⠀⠀⢸⣟⣻⠛⢿⣛⡷⠀⠀⠀⢰⣿⡮⠿⢵⣿⡆⠀⠀⠀⢾⣿⡜⠛⣣⣿⡇
⠀⠀⠀⠐⠙⠁⠀⠀⠙⠑⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉

⠀⠀⠀⣶⣶⢄⠀⢀⢴⣶⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠠⠀⠀⠀⠆⢻⡇⠀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⠈⣿⣟⣿⣟⣿⡏⠀⠀⠀⠀⣀⠀⢿⠀⢀⠀⠀⠀⠀⣁⠤⢸⣿⡇⠠⣌⠀⠀⠰⣄⣀⣭⣿⣍⣀⣠⠆
⠀⠀⠀⠈⢿⠟⠈⠘⢿⠟⠀⠀⠀⠀⠁⠀⠘⠀⠀⠁⠀⠀⠀⠹⣦⠟⠉⠻⣴⠏⠀⠀⠀⠀⠋⠝⠙⠭⠉


⠀⠀⠀⢾⠒⢦⣤⣤⠆⢺⠆⠀⠰⢖⠋⠲⣴⠶⠊⢒⠶⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣴⣦⡈
⠀⠀⠀⠈⣷⡤⠿⢧⢴⠏⠀⠀⠀⠀⠹⣶⠟⢦⡞⠀⠀⠀⠀⠀⠀⣤⠈⣠⡀⠀⠀⠀⠀⠸⣿⣛⣿⣋⣿⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢶⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠉⣿⠛⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣾⣧⣾⣿⣾⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣼⠋⣸⡏⣠⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠛⢿⣟⣿⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠹⠁⡜⠁⠀⣠⠞⠻⣯⣽⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⡧⡞⢀⡴⠋⠁⢀⡴⠟⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⢀⡟⠁⠀⣠⠖⠋⠀⠀⣨⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⣿⣿⠀⠀⢸⣁⠴⠊⠁⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⣾⣿⣻⣿⠶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⠇⠀⢀⡟⠁⠀⠀⢀⣴⣿⡿⠋⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠘⢿⡿⣟⠛⣦⡿⠳⢭⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡼⠁⠀⢀⣴⣿⣿⣿⡿⠛⠉⠙⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣳⣾⡟⠲⢶⣿⠿⠭⡛⢿⣿⣿⣶⣤⣘⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡇⠀⢰⠃⢀⣴⣿⣿⡿⠟⠁⣀⠴⠒⠉⢀⣀⡬⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⡿⣿⠋⠙⠒⠲⣏⠀⠉⠛⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣯⣾⣿⠟⣩⣾⣤⣒⣉⣠⡤⠒⠋⠁⣀⣀⣀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣳⡦⠤⠴⠖⠛⠲⢤⣄⣀⠀⠉⠙⠻⢿⣦⣄⣰⣿⣷⣾⣶⡿⠟⢋⡴⠚⠁⠀⠀⠉⠉⢉⣙⣶⣞⡉⠉⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡠⠤⠔⠒⠋⠉⠉⠙⠛⠒⠒⠺⢿⣿⣿⣿⣿⣷⣴⡾⠭⠤⠤⠤⠴⠶⠶⢞⣉⠀⠀⠀⠉⠙⠲⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⠟⢋⡽⣿⣮⣿⣿⡲⢦⣄⣀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣰⣧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣟⣋⣩⠤⠞⠉⡸⠹⢿⣿⣿⣍⡲⢤⡉⠉⠒⠶⢤⣄⣀⠀⠀⢠⡿⣷⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢈⡽⢿⡁⠀⠀⢰⠃⠀⡇⢳⡙⣿⣍⠓⠩⢷⣄⡀⠀⠀⠉⢙⣾⢿⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠋⠀⣼⢇⠀⢀⡏⠀⠀⢷⠀⢧⠸⣿⣦⠀⠀⠈⠉⠓⠶⠶⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⠃⠘⡷⡏⠱⡀⠀⠸⡄⠈⣧⣟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣇⠘⢆⠙⢄⠀⢳⠀⠘⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⣬⣷⣨⣦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣻⣻⣿⣯⣝⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣭⣽⣭⣿⣯⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⠈⠲⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠂⡀⠀⠀⠀⠀⠀⡠⠤⡄⠁⢈⡠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠲⠀⠀⢀⠔⠉⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡒⠃⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠋⠁⠐⠐⠀⠀⢰⡠⠉⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⢠⠐⠐⠏⠈⠀⠟⠀⠐⠄⡄⠀⠀⢀⡄⠀⠀⠀⢀⣖⡖⠀⠀⢀⣌⠀⠀⠀⠀⠀⠀⠹⠂
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢐⣤⠀⠀⠠⠐⠁⠈⠠⠀⠠⠀⠈⡨⡇⠀⣤⠁⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⣠⡀⠀⠑⠰⠀⠀⠀⠀⢠⠤⠒⠈⠀⠘⠓⠤⣶⠀⠂⣴⣳⣠⢠⠁⠀⠀⠀⠀⠐⡆⠀⠀⠀⢤⠀⠀⠀⣀
⠀⠀⠀⠀⠁⠄⡀⠀⠀⠀⠀⠒⠞⡁⠀⠀⠀⠀⠀⠀⠀⠌⠳⣆⠉⢀⠎⡬⠀⠀⣀⢀⣻⠌⠵⠊⠀⠀⠈⠀⠀⠀⠛⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⢠⡂⠀⠀⠀⠐⠀⠀⠠⠀⠀⡤⠀⠈⠳⣞⠒⣻⣦⠞⠁⠂⠀⠀⠀⣤⠂⠀⠀⠀⠀⢀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠘⣟⠻⢣⠀⠀⠀⠀⠀⠈⠀⠀⠩⠀⠐⠚⠠⢄⠀⠀⣀⠀⠀⠚⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠟⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀⠀⢗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠐⠀⠀⠐⠀⠀⠠⠀⢀⡄⢀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣄⠠⠘
⠀⠀⠀⠀⠀⠀⢦⡀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⢀⠡⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠃⠀⠄⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠐⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣋⣽⣿⣧⠴⣶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣰⣿⣆⣀⣠⣤⣁⣤⣠⣶⣄⣀⣄⣉⣈⠁⢀⣀⠐⠂⠈⠛⣦⣄
⠀⠀⠀⠀⣰⣏⡴⡛⠛⠉⠀⠈⠪⣻⠁⠙⡭⡹⢹⣹⢹⡙⡏⣉⢋⢋⠟⡯⢴⣦⣀⡉⡤⠈⢣⡀
⠀⠀⠀⠀⡟⠁⠀⠀⠄⢀⠂⠄⡐⠈⢶⣄⠌⣱⡾⣏⣧⣕⣼⣏⣡⣪⢊⡴⠓⠉⡹⠶⣗⣄⠀⢱⡀
⠀⠀⠀⠀⢷⡀⡀⠀⠀⠠⠬⣒⣬⣑⣦⣜⡷⠿⠷⢿⣿⣿⣿⣿⣥⣿⣥⢤⣴⣮⣴⣵⣮⠿⣌⣋⠡
⠀⠀⠀⠀⠀⠻⣤⠒⣒⠺⠷⢶⣬⣽⣻⣿⡇⠀⠀⢰⡟⢿⣿⣶⣦⣤⣄⣀⡤⠍⠉⢉⡬⠟⢻⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⣥⠶⢒⣛⣻⣿⠀⠀⠀⢸⠀⢸⣧⣙⠡⡉⠙⡻⠽⢍⠛⡻⠶⠄⣼⡛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⠀⠀⠀⠀⡆⢻⣿⣿⡀⠀⠀⠈⠂⡀⣑⣌⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⡇⣼⠻⢽⣧⠴⠖⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠠⣶⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠂⣤⡀⡻⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⣵⣿⣾⣿⣿⣷⣮⣬⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⡀⣤⣀⣼⠟⠋⠁⠸⠸⡟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣗⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⣀⣻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⢿⢿⠻⠏⠿⠿⢿⢿⣷⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⠛⣉⣡⣌⣤⣤⣤⣴⣶⣶⣄⣄⠁⠛⠙⠿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢋⣈⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣬⣍⡻⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣟⠋⣟⠈⣋⠡⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⢟⡀⣽⡧⠈⠘⠁⠴⢧⣬⣿⡿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣛⣽⣧⣶⣙⣿⣿⡆⢰⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠏⣇⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣂⣉⣙⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠀⠁⢸⡟⠻⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿⠿⣿⣿⣿⠿⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣡⣤⣶⣶⣤⣄⣀⠢⡀
⠀⠀⣠⢔⣾⣿⡿⠋⢀⣉⣤⣶⣤⣴⣦⣴⣶⣤⣮⣙⡻⠆⠀⠀⠀⠀⠀⣧⠀⢠⣿⠀⠀⠀⠀⢀⣠⠀⠀⡰⠶⠿⠿⣿⠿⠿⠿⠿⠿⠿⢿⣷⣝⢢⡀
⠀⢰⣵⢿⣫⣥⣤⣾⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡇⠐⢸⣿⡀⠀⠀⠀⣿⣷⣶⣷⣶⣶⣶⣷⣶⣷⣶⣛⣿⣻⣗⣾⠶⠤⣬⡙⣆
⠀⡟⢋⣼⣿⣿⣿⣻⣻⣤⣬⣥⣿⣿⣾⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⡁⠀⠀⠐⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⣖⠠
⠀⠰⣽⣿⣿⣿⠿⠿⠛⠉⠛⠻⡯⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⢹⡇⠀⠀⠀⢋⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⠿⠟⢻⣿⣿⣿⣿⣷⣞⠃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢡⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⠀⠀⡷⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠀⠉⢻⡾⡇⢰⣾⣧⣶⣿⣿⣿⣿⣿⣿⣆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⢸⣷⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠇⠸⠟⠉⠙⠻⠿⢿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣴⣾⣹⣟⡻⢿⣛⣿⣻⣦⣄⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣣⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣾⡷⢿⡿⠿⣿⡿⢛⣿⣿⣛⣋⣋⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣿⠠⠈⠳⡀⠀⠀⠀⠀⠠⢽⣠⠀⣷⣾⣿⣿⣿⠿⣿⠎⡿⣿⣿⡿⠿⠇⠀⠀⣿⡇⣀⠀⠀⠀⠀⢠⡀⣟⠉
⠀⠀⢸⠇⠃⣠⠀⠀⠀⠀⢒⠀⢁⠀⠘⣿⣧⠸⣗⠀⠀⠀⠀⢈⢻⣠⡄⣿⡙⣿⣿⣿⠀⣹⢷⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⠄⠀⠀⠀⠘⡿⠟
⠀⠀⠠⡴⠆⠉⢀⣄⠀⠀⠘⠁⠀⠁⠀⠈⢿⣦⡙⡄⠀⠀⠀⢉⢹⣀⣀⣿⠃⠁⠋⢻⠀⣿⢸⠀⢠⣿⡀⠀⠀⢀⣾⡯⠀⡰⠀⠀⠐⣿⣾⣷
⠀⠀⠀⠀⣤⣤⣼⡍⠀⠀⠰⠡⠤⠁⢁⠀⠀⢻⡗⠈⢮⣂⠀⢉⣹⡅⠉⣿⠂⠀⠀⢸⠀⡿⢸⠀⠲⣿⡅⠀⢀⣾⡿⠁⢰⠃⢀⣤⣄⣸⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠛⠉⠀⣤⡖⠀⠀⣲⠀⠘⣀⠀⠀⢻⡀⠘⠿⢣⠈⠙⡧⠄⣿⠀⢸⠁⣼⢀⣷⡟⠀⣛⡇⠀⢀⣾⡏⢠⠰⡞⠀⠀⠀⣿⠛⢋⠀⠒⠡⡀
⠀⠀⠀⠀⠀⠀⠒⢃⡱⠁⠤⡄⠰⣂⢼⠿⠀⠀⣀⢷⠾⢤⡈⣷⣴⣧⣤⣿⢠⢺⣧⣿⣾⠃⡇⠀⠉⡟⢳⣿⠟⠀⠀⡘⠀⢸⣇⣠⡶⢀⡈⠛⡀⠒⠑
⠀⠀⠀⠐⠶⠀⠰⢣⠕⠀⣺⣏⠈⢛⠂⢀⣤⢀⣽⣷⠶⢼⡗⣼⢿⣿⣿⣯⡼⢸⣿⡹⡟⢠⡍⠙⢀⣧⣾⢏⡜⠀⡾⢠⡄⠾⣾⡇⠀⢺⠔⡠⢁⢀⡀
⠀⠀⠐⠷⠈⢠⡄⠉⢙⡚⣥⢾⡧⠰⢂⣌⢙⠉⣰⣟⠀⣺⣿⣸⣷⣿⣿⣿⡇⣿⣿⣷⢂⡼⢳⣾⣿⡟⢇⡾⠑⢸⠋⠀⡗⠀⠘⣇⣟⡀⢛⢀⡘⠚⠃
⠀⠀⠀⠀⠀⠛⢼⠧⠘⡻⠏⡄⢱⠀⠀⣻⢦⣤⣽⣯⠀⣿⠏⠁⠉⣿⣿⣿⣿⣿⠋⣡⣾⣧⣿⣿⡿⢀⢾⠃⢠⣾⣷⣴⣳⡾⢄⠈⠀⡟⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣁⢐⠀⢛⠀⡽⠃⣩⣤⣿⣿⣧⣠⣿⣦⣠⡀⣿⠋⠈⣿⡧⢾⣿⣿⣿⣿⣿⠃⠀⠠⢀⣼⣿⣿⣿⣿⣿⣾⣧⣼⣇⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣶⣧⣶⣿⣿⣿⣻⣟⣹⣿⡉⠈⢷⣿⣆⣼⠏⠀⠀⢠⣾⣿⣿⣟⠀⠄⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡷⢦⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⣮⢹⣿⣠⣴⣶⣿⣿⣿⣿⣿⣤⣶⣿⡿⣿⣿⢿⣿⣿⡿⠿⠿⠋⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠛⠿⣿⣉⣻⡿⢿⣿⣿⡿⠧⣀⣀⣾⣎⠹⣯⣀⡀⢹⠈⠛⠛⠀⠈⠉⠋⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⣄⣁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣵⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠁⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣱⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⢀⡴⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠘⠋⠀⠀⠴⠚⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀⡴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠄⢀⣰⣆⣼⣿⣿⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⡇
⠀⠀⠀⠀⠀⢀⣠⣤⠤⢴⣶⠾⠟⠛⠋⠉⠉⠁⠈⠉⠙⠛⠛⠛⣛⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣏
⠀⠀⠀⣴⡾⢛⠩⠀⠲⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡿
⠀⠀⠰⣿⣖⠃⣴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠹⢿⣿⣧⣤⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡿⠛⢻⣿⣿⣿⣿⡟⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⠻⣷⡀⠈⠙⠢⠀⠀⠀⠀⠀⠀⢈⡿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣤⣤⣴⠇⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠹⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢥⣄⣠⣾⡴⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⡇⠀⣼⠀⠀⢠⠀⠀⠀⠀⢠⡀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣦⣤⣾⣇⣀⣦⣤⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⢿⡟⠇⠀⣤⡀⠤⡴⣴⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠉⠁⠀⠀⠁⠀⡄⠀⢁⠀⠀⠀⠒⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠈⠀⠀⠸⠄⣶⣇⣄
⠀⠀⠀⠀⠀⠀⢤⠀⢻⠘⡀⠘⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠙⠏⠁
⠀⠀⠀⠀⠀⠀⢀⡆⠘⢁⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠰⡄⡄
⠀⠀⠀⠀⢸⡴⠈⢠⡆⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢤⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢀⣾⡟⢀⠈⡁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠆⢀⣀⣴⣶⣶⠶⠒⠀⡤⢸⠀⢸⣿⡇
⠀⠀⠀⢸⣟⠃⠆⠀⠁⠀⠀⠀⡸⣶⠂⢀⣤⣤⡤⠶⣶⣺⣿⡿⢹⠵⠛⠁⠀⠀⠔⠀⠀⠃⢸⣿⣧
⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⣀⣼⣧⡴⢞⣩⣿⣿⣾⣾⣿⡿⠟⢭⣶⣾⣶⣤⡤⠂⠀⠀⠀⠱⢼⣿⡇
⠀⠀⠀⠘⣿⡇⣀⠀⠀⢸⣿⣿⣟⡉⠉⠉⠓⣹⣛⣥⣤⣤⣴⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⢿⣿⡀⠀⠀⠈⠛⠻⠛⠿⠦⠠⠞⠛⠛⠛⢿⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠈⣿⣷⡄⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⣀⠕⠀⠂⠀⠀⠀⢀⣸⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣷⣨⠃⢄⡑⠀⠢⡄⠀⠀⠀⠀⠆⢰⡅⠀⣾⣿⡀⠀⠀⠀⠀⠞⢹⡏⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣜⠁⢦⢰⣿⣇⣤⣤⡀⠁⠀⢻⠆⠘⠿⢷⣶⣤⣄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⣻⣿⣯⡿⣿⡭⠛⢺⣇⠀⢚⡡⠀⢀⡟⢩⣧⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢀⠁⣷⣿⣿⣿⣿⣄⣴⠾⠷⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠐⢶⣾⣾⣶⣷⣾⠀⣿⠿⢾⣿⠟⠛⠁⠀⠀⠰⣯⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣴⣶⣤
⠀⠀⠈⠑⠶⠖⠃⠀⠀⠈⠀⠐⠂⣤⣶⣦⡀⣠⣄⣀⠀⠀⠐⣹⡇⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣮⠣⠀⣠⡄⡞⣿⠂⠀⠰⣿⣿⣿⣽
⠀⠀⠀⢀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠙⠿⠿⢻⣥⣬⣏⣙⣳⠀⠀⣠⡤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠂⠻⣷⢰⣿⠸⡇⢿⡅⠀⠀⠻⢿⣿⡿
⠤⠐⠒⠛⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠻⢿⣿⣧⠀⠀⠘⠛⠁⠘⢯⠟⠀⠴⠀⠀⠲⠶⣤⣤⣤⣾⡅⢀⣙⡛⣿⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⠈⣿⣿⣦⠈⣀⠂⠠⢽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠂⢨⣼⣿⣾⣿⡝⣿⢻⣏⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠈⠛⠃⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠂⠀⣻⣿⠟⡹⠿⣷⢿⣿⡛⣭⣿⣵⣠
⠀⢀⣴⣶⣶⣖⠀⠀⠀⠀⠀⠀⢀⡾⣶⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠊⠀⠀⠈⠉⠈⠻⠓⠋⣬⣿
⠀⠘⠻⠿⠛⠁⠀⣀⣤⣶⣦⡀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣒⡒⠒⠂⠄⠀⠐⣻⣿
⠰⣔⣄⠀⡀⠀⢸⣿⡯⢿⣿⡇⠀⠀⠀⠀⢠⡶⣦⢠⡶⠆⢠⡶⢤⢠⡶⢦⢠⡄⣴⠶⠀⣴⠦⠀⠀⠀⠈⠻⣙⠛⠿⠿⠿⣷⣦⣁⠀⠺⡗
⣀⠀⠻⠮⢠⠀⠀⠙⠳⢼⣿⡇⠀⠀⠀⠀⠸⢯⠽⠘⢯⡭⠸⠟⠯⠸⠟⠯⠸⠇⢿⣭⠀⢭⡽⠂⠀⢰⠶⠀⠈⠛⠶⠶⠶⠒⠛⠛⠃⠀⠻
⠷⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠈
⠤⠀⠀⠀⠀⠀⠀⠘⢲⢀⠞⡃⠶⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⢁⠁⣀⣀⢀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀
⠀⠘⠀⢀⠸⠟⠹⢯⠆⠉⣤⣌⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡟⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⢠⠀⠀⢐⣿⣷⡁
⡀⠀⠐⠁⣠⣀⡀⠀⠀⠀⠈⠉⠀⢀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠸⣿⣿⡇
⠙⠆⠀⠀⠈⠋⠁⠀⠁⠀⠀⠀⢐⠛⠻⣟⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠠⣄⠀⠀⠀⠘⣿⡿⠁⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠷⠆⣤⣄⢠⣶⡆⠀⠀⠀⠀⠀⣾⠙⣿⣆⠙⡆⠀⠀⡔⠁⠀⠀⠀⠠⢼⣏⣷⣥⣀⠀⢀⣀⣠⣤⡤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠀⠉⢀⠴⠂⣭⣥⣴⣿⡆⣿⣷⢀⠇⠀⠘⠀⠀⠀⠀⠀⢠⣿⡟⠝⠻⣿⡄⢸⣾⡿⢻⡟⡃⡘
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣄⡁⠽⡟⠀⠀⠀⣡⣶⣿⣿⣿⣿⣿⠛⢻⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⣿⣿⣦⣽⠀⠘⢻⣻⡿⣻⣶⡽
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠁⠐⠿⠇⠀⠚⢝⣁⣉⣈⣠⣴⣾⣿⡟⠁⠂⠀⠀⠠⠊⠀⠀⣺⣷⣶⡀⠈⠙⠛⠃⢠⣶⣦⡀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⠀⠀⠀⣀⡤⣤⠀⠀⠀⠀⠀⠀⣤⡄
⠰⣄⠀⠀⢰⣦⣄⠀⠀⣠⣴⡶⠀⠀⢀⡷⠘⠇⠀⠀⠀⢀⡄⠀⠀⠀⠸⠃⠀⠀⠹⣦⠀⢳⣄⠀⣴⠇⢀⣼⠃⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠈⠓⢤⡀⢈⣩⣽⣿⣍⡁⠀⡠⠖⠉⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠙⣧⡈⢻⣿⠋⢠⣾⠁⠀⠀⠀⠀⠀⠀⠰⣟⢻⣄
⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⠁⣀⡴⠃⠀⠀⠀⠀⠀⠀⢿⣷⠆⠇⠴⣿⡟⠀⠀⠀⠀⠀⠈⠷⠞⣃⣮⡿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢀⣄⡁⠀⠀⠀⠀⠀⣑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃
⠀⠀⢠⣤⡄⠀⠀⠀⡀⠀⠀⠠⣤⡄⠀⠀⠀⠸⣟⣷⣄⠀⠀⣠⣾⣻⠏⠀⠀⠀⠀⢻⣯⣷⣄⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠻⡤⣠⡼⣧⣤⡠⡞⠋⠀⠀⠀⠀⠀⠻⡿⠿⢷⡶⠿⠿⠿⠀⠀⠀⠀⠀⠀⢹⣿⡏⡟⢽⣿⠃⠀⠀⠀⠀⠀⢠⠀⡀⢈⡃
⠀⠀⠀⠀⠀⠷⡶⠆⠑⢶⡾⠇⠀⠀⠀⠀⠀⠀⠰⠀⠠⢚⡟⠄⠀⠆⠀⠀⠀⠀⠀⠀⣨⠿⠃⠃⠛⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⠾⠛⠉
⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠻⣿⢷⣦⣀⠀⠀⢀⣴⡾⣿⠟⠀⣶⡲⣦⣄⡀⠀⠀⠀⢀⣤⡶⣲⡦⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠘⣧⠀⢠⡄⠀⣰⠏⠀⠀⠀⠀⠀⠙⢷⡝⠿⢶⡶⠿⢫⡾⠋⠀⠀⠘⢿⠊⠻⣷⡄⠀⢰⡿⠋⢘⡟⠁⠀⠀⠀⠄⠂⠐
⠀⠀⠀⠀⠰⣇⠀⠙⡟⠁⣠⡇⠀⠀⠀⠀⠀⠀⠸⣿⣅⠸⠇⣠⣿⠇⠀⠀⠀⠀⠈⢳⣤⣘⢶⣷⢞⡡⣤⡏⠀⠀⠀⠀⠀⢰⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠘⣟⡿⠁⠀⠸⣗⡿⠁⠀⠀⠀⠀⠀⠀⠳⠾⠅⠐
⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠈⠹⣿⣿⣷⡄⢀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣦⣶⣴⣶⣶⠆⠀⠀⠀⠀⣀⠂⡀⠁⠄
⠀⠀⠀⠀⠙⡟⢻⣿⣿⣟⢻⠛⠁⠀⠀⠀⠀⠀⡆⠀⠀⠘⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⣤⠈⠀⠏⠈⢠⡄⠀⠀⠀⠀⠀⠹⠷⣆⣶⣗⢇
⠀⠀⠀⠀⠀⠹⠤⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠀⠠⠖⠃⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠘

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⡤⣘⣁⡀⠀⢄⡤⡞⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⡈⢀⠋⠐⠋⢣⡭⣤⡾⠁⢀⣀⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠀⢶⡰⣤⠅⠃⣦⡗⢀⢂⡤⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⢉⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠙⠄⠃⠸⣧⠘⠢⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⢻⡶⠓⠆⠀⠋⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠺⠃⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⡄⠀⡀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠈⠂⠀⠀⠀⠀⣼⣅⠂⠀⠼⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠁⠰⠀⠀⠀⠀⣀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠚⡤⠀⠄⠀⠀⡀⣠⠄⢸⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡇⢀⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠⠞⠠⢶⣦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⣈⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠒⠀⢠⠀⠀⠀⡀⠊⠀⠀⠐⠂⠈⠀⠀⠀⠀⠑⠀⠆

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

⣰⠋⢻⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⢻⠋⢳
⣿⣷⣯⣶⣿⣿⣷⡘⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⣱⣿⣿⣷⣮⣷⣿
⢻⣿⡓⢼⣿⣷⣮⣿⠿⠟⠛⠍⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠝⠛⠿⢿⣯⣶⣿⣿⠔⣻⣿
⠀⢻⣿⣿⣿⣿⣟⠛⠦⢄⣀⣀⣶⣌⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣏⡴⣆⣀⣀⠤⠞⢛⣻⣿⣿⣿⣿⠃
⠀⠀⢻⣿⣟⠛⢧⣀⣉⣿⣽⣯⣄⣹⣾⣦⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣾⣾⣁⣬⣿⣽⣏⣁⣠⠝⢛⣿⣿⠃
⠀⠀⠈⢿⣿⡖⠻⣿⣿⣿⣿⠛⠛⠛⡏⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠈⡟⠛⠛⠻⣿⣿⣿⡿⠓⣾⣿⠏
⠀⠀⠀⠈⣿⣏⣹⣿⠋⠀⠀⠀⠀⠀⠙⠢⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠠⠚⠁⠀⠀⠀⠀⠈⢻⣿⣉⣿⡏
⠀⠀⠀⠀⢻⡟⢻⣙⠦⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⢿⣷⣄⣶⣆⣴⣿⠟⠁⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠠⢞⣹⠛⣿⠃
⠀⠀⠀⠀⠀⣷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⡶⠶⠛⢻⣿⣿⠛⠳⠶⣦⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⠟⡻⢯⠉⠉⠛⢉⠽⠋⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠻⢍⠙⠋⠉⢭⠻⡛⢿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⢛⣿⠊⠁⠀⠀⠀⠴⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⡷⠄⠀⠀⠀⠉⢺⣟⢣
⠀⠀⠀⠀⠀⠀⠀⢸⡋⢫⡤⠐⠀⠀⠀⠞⠛⠛⠈⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠘⠛⠛⠆⠀⠀⠁⠠⣬⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠻⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡷⢤⡄⠀⠀⢀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⠀⠀⠀⣤⠴⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⡶⣶⡦⠶⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⠦⣴⡶⣾⣿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⠶⠦⢤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠉⠒⠄⠀⠀⠉⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠰⢶⣄⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⡀⠄⠂⠊⠀⠀⠀⠀⠀⠐⠛⠻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⢀⣠⡤⠄⠀⠢⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣤⠴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⠀⣠⣺⣵⠾⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣀⣀⣤⣤⣤⣴⣶⠎⠉⢷⣤⣾⣿⢟⡁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⢀⣰⡾⣿⡿⡆⠀⢸⣿⣟⣩⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠊⠄⠇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠈⠈⢿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⠇⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠀⠛⠿⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣠⣄⣀⣈⡑⠠⠀⠀⠀⠀⠀⠐⢀⣴⠏⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢽⠀⠹⣦⡿⠓⠼⠖⠀⠀⣮⣿⠔⠈⠀⠀⠀⠂⠺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠄⡲⣶⡞⠉⢱⡶⠚⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢹⡀⠀⠀⢈⣧⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠳⠆⠴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⡾⠃⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⡇⢀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡀⠒⢤⠾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠈⠀⠀⠀⠀⠀⠉⠒⠤⠶⠖⠹⡎⠀⠀⠀⠀⠀⢀⢠⡶⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠠⣀⠀⠀⠀⠠⠀⠀⠀⠐⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡦⠄⣀⣀⣀⣵⣆⢀⠤
⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⣀⠈⢸⡇⠀⠀⠀⠁⠒⠀⢶⡤⠄⡉⣬⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠠⠝
⠀⠀⠐⠒⠂⠀⠀⠀⣠⡤⠤⠝⠶⢶⠞⠁⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠶⠐⠀⠤⠄⠀⠀⠀⠀⠀⡀
⠀⣤⠄⠠⣤⠴⠚⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⢋⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⣧⠀⠀⠀⠀⠀⠀⠀⠲⢆⣀⠀⠠⡄
⠈⢁⢠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡄⠈⣽⣾⠃⠈⠙⣆⠀⠀⠀⠀⠀⡴⠋⠀⢈⡆⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⡄
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢨⠍⠉⠀⠀⠀⢀⡾⠂⠀⠀⠀⠈⠁⠀⠀⢸⠇⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⠆⠂⠄⢀⣰⡞⠀⠀⠀⠀⠀⠀⠀⠙⢴⠏⠀⠠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⢘⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⢠⣨⠟⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠛⠉⠀⠀⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠞⣀⡘⠒⠢⠄⠀⣠⣤⣄⡀⠉⠁⠀⠈⠁⠐⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⡌⠀⠰⠋⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⢦⠘⠛⠛⠆⠄⠀⣰⠇⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠢⡄⠈⣆⠀⠀⠀⢤⠀⠀⠀⠀⣴⠿⠅⠐⠀⠀⣤⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡶⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠻⠦⣄⠀⠀⠀⠀⠞⢁⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠉⢀⠠⣀⣀⣀⣀⣀⣀⡴⠛⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠸⠓⢄⢀⠀⣀⣁⣀⣠⡀⠀⠉⠁⣠⡀
⠀⠀⠠⠀⠒⠀⠀⠀⠀⠄⠉⢉⡽⠉⠀⠀⠑⠤⠀⣤⢠⣤⠷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠟⠉⠁⠀⠀⠉⠋⠁⠠⠤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⡀⢀⡞⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠈⠁⢸⣀⣤⢄⡀⠀⢦⣄⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠈⠙⠻⠶⠻⣟⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⠀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣻⣿⠆⢼⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠆⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠈⠑⠒⡟⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⣠⠝⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⡶⢀⠀⠀⢀⠐⡆⠀⠀⠛⣴⠂⠀⠀⠀⠀⠀⣲⠘⠀⢻⠉⠀⢄⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠘⡢⠄⢸⡇⠠⠔⠃⠀⠀⠸⡧⡇⡖⡔⡆⡣⢽⠀⠀⠀⠘⠀⠀⠠⡆⠀⠀⠛⠀⠀⠈⣇⠀⠀⢲⡇⠀⢀⡟
⠀⠀⠀⠐⣇⠀⠘⠃⠀⣸⠂⠀⠀⠀⢾⣇⠙⡿⢁⣾⡆⠀⠀⠀⠠⣦⡀⠀⠃⠀⣰⡆⠀⠀⠀⢨⠋⣾⠹⠹⡮⢹
⠀⠀⠀⠀⣸⠛⠉⠉⠛⣇⠀⠀⠀⠀⠈⠻⠄⠁⠸⠋⠀⠀⠀⠀⠀⠈⡿⠓⠘⠻⡋⠀⠀⠀⠀⠀⠳⡼⠁⠸⢶⠋
⠀⠀⣀⡀⠁⠀⠀⠀⠀⠈⢀⣀⠀⢀⢀⡀⠀⠀⠀⠀⢀⡀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣆⠀⠀⠀⠀⠀⠀⡶
⠀⠀⢻⡀⠀⠀⢀⡀⠀⠀⢀⠏⠀⠸⡄⡼⠷⣄⡤⠻⠤⣴⠃⠀⢿⠀⠀⠀⣀⠀⠀⠀⣼⠀⠀⠸⣶⡄⢠⠀⣴⣾⠁
⠀⠀⠀⢳⡄⠀⢸⠃⠀⢰⡟⠀⠀⠀⢳⢤⠐⢼⠵⢀⡄⡏⠀⠀⠸⣄⠀⠀⢿⠀⠀⣠⠇⠀⠀⢠⠁⠀⠸⠀⠀⢁
⠀⠀⠀⠈⠿⣦⡄⢠⡴⠟⠀⠀⠀⠀⠈⠻⣙⠀⢐⣫⠎⠀⠀⠀⠀⠘⢤⡀⠀⢀⡤⠃⠀⠀⠀⠈⢧⣤⠀⢠⣤⠞
⠀⠀⢠⠀⡀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⢦⠑⠁⢠⡀⠘⢊⡼⠁⠀⢀⡀⠀⠀⣾⡗⠀⠀⢀⡀⠀⠀⠀⡖⠐⢀⡀⠂⡆⠀⠀⠀⠀⠀⡀⣾⡇⡀
⠀⠀⠀⠨⡇⠀⢸⠇⠀⢨⠀⠀⠀⠘⠂⠀⠀⠸⠁⠀⠀⠚⠁⠀⠀⠀⡍⠀⠸⠃⠀⡡⠀⠀⠀⠀⣤⣙⣊⣛⣡⡄
⠀⠀⠀⠀⠙⠒⠀⠐⠖⠋⠀⠶⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠤⠆⠀⠘⣆⠀⠀⡴⠁⠀⠀⠀⣼⠓⣮⣵⣯⡔⢻⡆
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠘⡿⢡⡶⠀⠀⠀⠐⣷⠈⣿⠀⠸⡟⠳⢢⣄⣀⣤⠲⠛⡗⠀⢸⡟⠓⢀⠀⠠⠚⢻⡇⠀⠛⣿⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠈⣧⠀⢸⡇⢀⡾⠀⠀⠀⠀⢻⡀⣾⠈⡇⢸⡆⡸⠃⠀⠀⣇⠀⠀⣶⠀⠀⣸⠃⠀⠀⠹⣇⠀⢸⠇⢀⡼⠃
⠀⠀⠀⠰⣌⠓⠚⠓⢛⣠⠀⠀⠀⠀⠀⣦⣀⠀⠀⢀⣠⡆⠀⠀⠀⢸⠁⡀⠙⢀⡀⣽⠀⠀⠀⠀⢿⡀⡼⣄⣨⡇
⠀⠀⠀⠀⠉⠛⠀⠈⠛⠉⠀⠀⠀⠀⠀⠘⠉⠀⠀⠈⠙⠀⠀⠀⠀⠈⠛⠁⠀⠀⠛⠁⠀⠀⠀⠀⠈⠉⠁⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠛⠛⠉⠉⠉⠉⢹⣏⠉⠛⠛⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠆⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠙⣷⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣴⠿⠁⠉⣻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢿⣆⠀⠀⠀⣠⡿⠀⠀⢀⣤⣶⣶⣦⣄⠉⠉⠀⠀⠀⣾⠏⠁⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠙⠻⠿⠟⠛⠁⠀⢀⣾⠋⠀⠀⠈⢹⣧⠀⠀⠀⠘⣿⡀⠀⠀⠀⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⣄⣀⣀⣀⣸⣿⡀⠀⠀⠀⠈⠻⠶⣶⠾⠟⠁⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣀⣿⣧⣶⣶⠿⠿⣿⠛⠛⢻⡟⠛⠛⣿⠟⠛⠛⢻⡟⠻⠿⢷⣶⣶⣤⣄⣀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡿⠿⣿⡉⠁⢻⡄⠀⢻⡀⠀⠘⠇⠀⠀⠹⠀⠀⠀⢸⠃⠀⠀⡾⠀⠀⢨⡟⠛⠻⣷⣦⣄⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠈⠳⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡼⠋⠈⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⡉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣾⣇⣀⡀⣠⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠰⣄⢀⣘⣧⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣷⣿⡟⠀⠀⠀⠀⢀⠈⣿⣷⣾⣿⡿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠸⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠈⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢰⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⡄⡸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡗⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣸⢱⠁⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⢰⠃⣼⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⢀⡏⠀⠃⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠈⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣄⣀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠐⠀⢉⠑⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠫⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠈⠄⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⢀⣄⣸⣃⠀⢀⡠⣀⡠⣀⣀⠌⣀⣀⡠⣀⣠⣀⣌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢛⡉⠉⠉⠈⠉⢻⠀⠈⠉⠀⠀⢀⣋⣀⣠⠠⠭⠝⢛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⣏⠤⠤⠢⡤⢄⣈⡇⠀⠀⠀⠐⢈⠀⠀⢀⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⣠⠤⣶⠤⢤⣤⣄⣀⣈⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢣⠀⠱⠊⠉⠙⠢⢬⠀⠀⡏⡉⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢀⠏⢠⡢⠀⢀⣤⠴⠆⣧⠀⣷⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⢀⠎⠀⢆⡤⢾⡇⠀⢷⣳⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠐⢁⡎⠀⠀⠈⠒⠤⡇⠀⢸⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⠀⠀⠀⠀⡘⣇⠀⠀⠀⠀⢠⠇⠀⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡃⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠸⣤⣄⣠⢇⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠌⠃⢷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠦⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠁⣄⠤⠀⠉⠀⠀⠀⠀⠿⠁⢀⣀⠠⠀⠀⠄⡀⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⠒⠂⢈⢀⠁⠀⠄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠐⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⠄⠒⠛⠛⠋
⠀⠀⠀⣠⠴⠂⠀⠀⠉⠘⠀⠘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢶⠿⡿⣁⣤⡄⠈⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢠⠃⠘⠏⠀⠀⠝⠁⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠈⠤⠀⡀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠈⠀⠀⠀⠀⢀⢀⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠃⠠⠬⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣦⠤⢤⣯⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠱⣀⣀⣀⢄⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢞⠄⠀⠚⠃⠀⠈⢃⠁⠀⠈⠐⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣀⡤⢤⢆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠌⠫
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠟⠿⠿⢿⣿⣿⣿⣷⣷⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢀⠀⠀⠉⠀⠐⠀⠀⠉⠈⠉⠙⠿⠿⠿⠛⠟⠛⠻⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⢀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡎⠀⠀⢿⣿⣿⣿⡇
⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣻⣿⣿⣿⡀⠠⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣇⠀⠀⠻⠿⠁⣤⣴⡆⠀⠀⠀⠀⠀⣀⣐⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠙⠋⣀⣤⣰⣶⣶⡿⠟⠃⠀⠀⢀⣤⣜⠉⢽⡷⣿⣋⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠍⠁⢸⣿⣿⠀⠀⠀⠀⠴⣷⣽⡋⠉⠀⠀⠈⠉⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢚⣿⡆⠀⠀⠀⢶⡦⢀⡉⠀⠰⢦⡀⣠⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⢿⣷⠀⠀⠀⠀⠀⠉⠁⠠⠄⠈⣃⠙⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠀⠀⠀⠀⠀⠒⢲⣦⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡄⠀⠀⡀⠀⠀⠾⠃⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢺⣿⣧⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⣿⣿⡄⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⣿⣇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⠍⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢷⢫⢺⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⢶⡟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣽⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⣾⣿⠀⠀⠀⡀⠀⢛⡀⠀⣀⠿⣦⣀⡠⠷⡀⠀⠀⠀⠀⡀⠈⢉⠩⢛⡗⠀⠀⠀⠈⠲⠽⠾⠿⠃⠀⠀⢿⣣
⠋⠀⠀⣀⠀⠸⣖⠶⠛⣘⠃⠀⠀⡍⠀⠀⣿⡆⠀⠀⠀⠀⠀⡦⡆⠉⣳⡔⠐⣆⣤⣄⠀⠀⠀⠀⠀⠀⠈⠻⠼⠿⠿
⣴⢦⡀⠁⠤⠞⠋⢧⡀⢀⣼⠶⠾⠟⠲⠶⠞⠁⠀⠀⢀⠀⠚⢿⣷⣦⣴⠋⠙⡿⣯⣽⣷⠀⠀⠀⠠⠄⢠⣤⣀
⠡⠞⠁⠀⠀⠀⠀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣖⣿⣿⡏⠉⣷⣿⣟⠛⠛⠃⠀⠀⠀⠀⢠⡀⠙⢮⢹
⠀⣀⡀⠀⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⣿⣴⣿⣿⣋⣿⡇⠀⠀⠂⢀⠤⠒⠈⣳⣄⠸⠏
⠳⡈⠉⢦⠀⠙⠛⠉⡀⠀⠀⠀⡀⢀⡱⠴⠒⠒⢒⣦⡄⠀⠀⡁⠲⣾⢤⣿⠟⠉⠀⠀⡠⠀⠀⠀⢀⣀⣿⠏
⠀⢡⠀⢸⠇⠀⠀⣸⣆⠀⠀⠀⠳⠀⠀⠀⠀⠈⠉⠀⠉⠦⠀⠳⠿⠟⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠋⡁
⠀⠸⠀⡝⠀⠀⠀⠙⢿⣶⣄⣤⡶⠀⠀⠀⢴⣶⣄⠀⠀⠀⠀⠀⠠⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡠⣄⠄⠀⠀⠢
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⣿⣄⣘⣿⠀⠀⠀⠀⠁⠂⠀⠉⠳⣻⠄⠀⠀⠀⠀⠀⢤⡉⡉⣋⣶⣤⣀⠄
⠀⠀⠃⠀⠀⢠⣄⣤⣴⠄⠛⠉⠀⠀⠀⠈⠙⠋⠁⠀⠀⢀⡇⠀⠀⠀⠀⣐⣿⡇⠀⠀⣦⠀⠸⡀⣀⡰⣼⣾⡏⠐⠚
⢀⠀⠀⠀⢠⡅⠩⠋⠉⠙⢶⣂⣤⣀⡤⠀⠀⠀⣀⡀⠀⠨⣷⣷⣶⣶⣾⡿⠋⠀⡠⠀⠁⠀⠀⠙⠿⠿⠟⢿⣧⣢⣼
⠀⠢⣄⣠⣴⠇⠀⠀⡱⠤⠈⠙⠓⠂⠀⠀⣴⡄⢅⣻⣆⠀⠈⠉⠉⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠉
⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡛⣡⡅⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⣷⣀
⣿⠂⠀⠀⠀⠀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⡀⠀⠙⠐⢂⠀⢸⣤⠀⣤⠀⠀⠀⠈⠓⠚⢻
⠁⠀⠀⠀⠀⣐⡀⢤⡂⠸⢧⠆⠀⠀⠀⠀⠠⠐⠤⠄⠀⠀⠀⠀⠘⣟⡒⠤⣄⡾⣀⠼⠷⣶⢶⡄⠃⠀⠀⠀⠀⣠⣽
⠀⣀⠀⠀⠀⠘⠻⣽⣯⣷⢶⠠⠀⠀⢀⠄⢄⠈⠀⢀⡐⠀⠣⠀⠀⠈⠉⠉⠉⠀⣧⣀⣠⣿⠇⡿⠃⠀⠀⠈⢺⢲⣛
⠈⣿⡷⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡙⣦⢤⣄⠀⠀⠀⢀⠆⠀⠘⠻⠿⠿⠞⠁⠀⠀⠀⠀⡀⠈⢻
⠻⡉⢷⣬⡿⠃⠀⠀⠀⠀⠀⢀⣉⢠⠀⢶⣬⣄⣀⡿⠿⣏⡀⣻⣧⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⢤⠔⠒⢦⠀⠈⠐⠚
⣾⠝⣶⡄⠀⠀⠀⠀⠀⣀⣶⣴⡋⣿⠟⠻⣧⣨⣿⣧⡀⣙⣿⠛⣿⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠸⣷⣆⣼⡃
⠂⠴⠟⠉⠀⢀⠀⠐⠶⢿⣈⡏⠙⠛⠿⣤⠟⠉⢀⠙⡛⠛⠋⠋⠁⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠈⠉⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⣠⣤⠠⣶⡀⠀⢀⠀⡀⡤⠠
⠂⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⡀⠀⣦⠀⠀⠀⠀⠀⠀⠤⡀⠀⠂⢀⣧⣆⠘⠀⠀⠢⠤⠠⡔⢚⡇⠀⠀⣽⡿⠆⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⢀⣐⣁⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣀⢨⣿⡿⠇⠒⠀⠘⡿⠦⡀⠀⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢘⡉⠁⠀⢰⡀⠀⠘⠁⠀⠀⠀⠀⠀⠰⣃⠻⠁⡀⣫⢕⣪⠁⠀⠀⠈⢛⡖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠐⢀⡤⢉⠱⣄⠀⠀⠀⠀⠀⠀⠀⢠⢠⠂⠄⢋⡐⠃⠁⠀⡀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⢀⠀⠈⠀⡀⠐⢹⠆⠀⠀⠀⠀⠀⠀⠰⢴⠆⠔⠉⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠠⠃⠀⠁⠈⠠⠂⠀⠀⡎⢸⠑⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠤⠮⣤⣔⣁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⡀⠀⠐⢀⠀⠐⠁⢀⡀⠐⡀⢢⠓⠀⠀⠀⠰⠤⣆⡤⢌⡫⠚⠉⠉⠀⠰⠀⠁⠀⠏⠠⡄
⠀⠀⠀⠀⠀⡈⠀⢀⡀⠁⠠⠂⠀⡈⠀⠀⠁⠀⠀⠀⠐⣃⠄⡴⢈⢙⠋⠀⠰⠀⠀⠇⠀⡀⠈⠀⠘⠐⠄⠂
⠀⠀⠀⠀⡈⣠⠀⢉⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⢡⠠⢿⠘⠀⢠⠀⢀⠀⠀⠂⠀⠀⠀⠀⠁⢼
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠂⠞⠂⠀⡈⠀⠠⠀⠰⠀⢰⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠀⡃⠀⠀⠀⠁⠀⡄⠀⡄⢠⠄⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⢀⠄⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠈⠰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣘⢃⡀⠀⠀⠀⣐⠋⠨⠍⠉⠐⠌⠆⠀⠀⠀⠄⢄⣀
⠀⠀⠀⠈⠀⠀⠀⠉⠀⠉⠁⠉⠁⠉⠀⠀⠀⡀⠀⠀⠀⠀⢐⠲⣷⣦⢄⡀⠀⠀⠀⠉⠛⢿⣵⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡠⣠⠅⠀⠀⠰⠆⠁⢠⠊⠡⠀⠀⠬⠁⠀⠀⠀⠀⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠿⡿⡃⢰⡄⠰⠖⡄⠶⡀⠀⠊⢐⢰⠄⠀⠀⠀⠀⠘⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⠁⠀⣼⠁⢸⠄⠘⠉⠗⢤⠰⠀⢘⠀⡘⡖⠀⣤⢴⡾⣵⣾⡄⠀⢀⣵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠐⢥⠘⢀⠀⠆⠩⡅⠾⠀⠐⠝⠈⣀⠁⠀⣀⠉⢠⠀⢉⣻⢸⣁⠋⠹⢹⡝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠘⠀⢠⠀⢠⠀⡀⠰⠆⠀⠀⠀⠈⠁⠀⢠⡄⠄⠠⠐⡠⣑⡾⢣⡘⠇⡌⠴⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⡄⠂⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠋⣤⠁⠉⠘⡈⢁⠃⠆⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠁⠈⡔⠊⠁

⣶⣾⣿⣿⠲⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢒⣿⣿⣿⡿⢶
⢿⣸⣿⣿⣿⠳⠆⢹⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣷⠀⢘⣿⣿⣿⣧⡿
⠈⢳⣼⣿⣿⣷⣄⢾⣿⡇⠈⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠀⠀⣿⡿⢴⣺⣿⣿⣧⡟⠁
⠀⠀⢳⣹⣿⣿⣤⣀⠺⠧⠀⢸⣿⡟⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢹⣿⣿⣄⡤⠘⠛⢻⣿⣿⣯⡾⠁
⠀⠀⠀⢻⣙⣿⣿⡙⠣⠀⣀⠘⣿⡏⠀⣸⣿⣿⣦⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⢿⣿⡁⢊⣤⣼⣿⣿⣿⣼⠇
⠀⠀⠀⠀⢿⣸⣿⣿⡟⠆⠀⠀⢹⣧⠀⣿⣿⡟⠙⢿⣦⣄⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⡾⠋⠉⢿⣿⡄⢠⠞⠉⠀⠤⢌⣽⣿⣿⣥⡟
⠀⠀⠀⠀⠘⣏⢹⣿⣿⣦⠶⣬⠉⠉⠑⢿⣿⠁⠀⢰⣿⣿⣦⡘⠢⢌⠢⡀⠀⠀⡠⠞⣁⠔⣢⣾⣿⣶⠀⠀⠈⣿⡏⠀⠀⠀⡐⠶⣿⣿⣿⣇⣾⠁
⠀⠀⠀⠀⠀⠹⣞⠻⣿⣿⣇⣤⣒⣒⠈⣾⠗⠐⠀⣿⣿⣿⠙⣿⣦⡈⡆⢰⣿⣿⣄⡰⣡⣾⠟⢛⣿⣿⡂⠄⡀⡝⠁⠀⠀⢀⣠⣴⣿⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠻⣦⠿⣿⣿⣍⣉⠀⠀⠊⠁⠀⠀⣻⣿⠇⢐⣨⣿⣿⣮⣪⣽⣯⣿⣿⣿⣗⠀⠀⢹⣿⣇⠀⠈⠁⠀⣀⡐⢈⣹⣿⣿⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⣿⣿⣿⠟⣃⠀⠤⠄⣰⣿⠟⠀⢀⡘⠿⢿⣿⣿⣿⣿⣽⣿⡛⠢⣤⣦⡾⠻⢿⣦⣤⣀⡀⢘⣻⣿⠿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠓⣒⣂⠈⠀⢀⢹⣧⠐⠂⢻⠿⣿⣿⣿⣿⣿⠃⠀⢸⣿⠇⢀⡀⢸⣷⣤⡉⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠠⠒⠃⠀⢻⡄⠀⡀⢠⢙⣿⣿⡿⠏⠂⠀⢸⡇⠀⠈⡝⠶⣿⣅⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⢛⣿⣏⡀⠀⢠⠀⠀⠈⣓⢀⠀⡆⣿⢿⣿⠐⣷⠈⠄⢸⠃⠀⠀⣴⠤⣾⣿⢿⡃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⡔⠉⢧⠀⠀⢾⠃⠀⠁⡇⣼⣿⠏⡙⡆⠀⣞⠀⠀⣠⡟⠐⢿⡇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⣿⣧⣶⡀⠠⠠⠀⢸⡇⢰⠀⡇⠹⡿⠀⡷⡄⠀⡇⠀⠺⠋⢤⣄⣼⡷⣾⠇⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣴⣿⣿⣿⣿⢀⣤⡀⡀⢸⡇⠸⠂⡆⠀⠀⢠⠟⡇⣸⡁⠀⣇⡐⠈⢹⡟⣴⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⢹⣿⣿⣿⡿⠁⡀⢈⣇⠰⢓⠃⠀⠀⠘⢀⠁⣿⣧⡀⠈⢹⣿⣿⣧⠏⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡿⣿⢻⣷⡿⢿⡿⢿⡿⣛⡃⠀⠀⢘⣛⣰⣿⣿⣿⣶⣟⣵⢾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⢳⣟⣁⣾⣇⣰⣿⠉⠁⠀⠀⠈⢹⠋⡹⣿⠇⠸⣿⣠⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⣿⡿⠛⠻⡟⣋⣿⠀⠀⠀⠀⠀⠸⣿⡻⣿⠟⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢽⣿⣶⣦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⠉⠉⠉⠉⠈⠉⢉⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠫⢈⣈⣧⣶⣿⣿⣷⣶⣶⣕⠂⡁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡝⣶⣿⡿⠛⢹⣿⣿⣿⣶⣾⢷⠎⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⠤⠜⣾⣟⡇⠀⣸⣿⣿⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠤⡸⣧⣀⣿⣯⡻⡟⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⣀⣀⣴⣾⡻⣏⣋⣝⣟⡧⠠⠄⠠⠎⢼⣿⣿⣿⣿⣿⣖⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡼⠟⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠁⢠⠀⠐⠂⣶⣿⣭⣯⡷⢷⣿⣿⣡⣤⡀
⠀⠀⠀⠀⠀⠈⠀⠈⠩⣿⣿⣿⣿⣿⣯⠁⠀⠀⠐⠀⠀⢠⡟⣿⣿⣵⣿⣿⣦⣄⡈⠩⢻⣿⣷⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⠈⠟⠃⠀⠀⠀⢀⣴⣄⠃⠀⣫⣿⣿⡟⣿⣿⠏⠀⠀⣼⣿⣿⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⣤⣶⣿⣿⣿⣦⡉⠀⠀⠸⠡⡟⠀⢁⣠⣾⣿⣯⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠨⡐⠿⢿⣿⡽⠟⠁⣰⡟⠐⠁⠀⠺⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠏⠁⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡁⠀⠀⠀⠂⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⢸⢿⠃⠀⠀⠎⠀⠀⠀⠛⠿⣿⣿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡄⢠⡀⣾⣿⠊⣀⡟⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⣧⠘⣷⣿⢄⣿⣿⣰⣿⠃⡀⠀⠀⠀⠐⢤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣿⣏⢸⣼⣿⢧⣿⣿⡏⠀⠀⣀⣠⣤⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣤⢿⡿⠛⢻⢿⡋⢹⡟⠳⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠙⢉⡱⠟⠻⣇⡹⠀⢈⡏⠙⢿⠛⣿⡟⡟⢁⣾⠀⠀⣆⣼⠂⠘⠣⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠉⠑⠀⣀⠍⠛⢀⠀⣰⠂⢈⡆⡸⢧⠀⠀⠁⠀⡜⠉⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⣸⠀⢣⠀⢆⠈⠤⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡋⠈⡉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⡛⠻⢿⡿⣦⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣁⠈⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠿⠿⣦⢌⣹⡯⣇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠯⣶⣦⡌⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡉⠀⠀⠀⠀⢘⡷⢶⣙⠯⡍
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠤⠻⠈⠂⢌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠘⣷⣲⣟⡡⢾⣉⠵⠾⣯⣽⠃
⠀⠀⠀⠀⢀⣤⣞⣿⣿⠛⢻⣙⣷⣶⡀⢪⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠉⠀⠊⣨⠁⠛⣥⠀⠿⠒⠾⢋⣲⠄
⠀⠀⣴⣶⣯⣿⣿⣿⠿⣓⣬⣿⣿⣿⣿⣟⠛⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡿⣥⢀⠀⠠⣤⣤⠀⠐⠛⠀⢶⣤⠬⡇
⠀⢶⣏⣿⣿⣿⣿⡇⠀⠀⠉⠀⠙⣿⣿⢟⡄⣼⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣼⡁⠉⠈⠝⡀⠀⡹⢁⣀⠶⠂⣨⡗⠲⡏
⠀⢛⡑⢿⣿⣿⣿⣿⡿⢂⣠⣤⣤⣤⣽⣿⣿⣿⡿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⠀⠉⠀⡀⠈⠴⣥⣀⣈⠱⡦⠀⢻⣛⣻⠃
⠀⠈⠃⠈⠻⠏⠹⠿⠃⠿⠿⣿⠿⠿⠋⡽⠋⠉⠁⡺⠃⠀⠀⠀⠀⠀⠀⢠⣤⣼⣃⣻⡔⠒⣉⣄⡐⠛⠢⠤⠽⣗⡀⣤⠿⣬⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⠌⣿⣤⣴⡯⠉⠉⠉⠭⣅⠓⠒⠶⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠙⣿⡿⡿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣋⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡆⠀⠀⠀⠀⢹⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠟⣉⣿⠧⢾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡻⠄⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣾⡯⣼⣿⠓⣿
⠀⠀⠀⢀⣠⡤⠤⢿⡟⠟⠂⠀⠀⠀⠀⠀⠀⣿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣝⣻⡿⢔⣿⣿⣩⡇
⠀⢠⠞⠻⠿⠋⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⡿⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣍⣹⣆⠉⣷⡝⢿⡇
⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣸⢇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⡀⠈⢿⣿⢿⣿⣿⣧⣾⡋⢻⣦⣀⣤⣤⣤
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢠⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣔⠓⢺⠹⠟⣿⣿⣿⣿⣉⣿⠉
⠀⠈⣃⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣯⣯
⠀⠀⠈⠳⢆⣠⡄⠀⠀⠐⠂⣠⣤⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠤⠸⢿⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡆⠉⠉⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠢⡄⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡟⠢⠄⡠⠔⠒⠲⠓⠰⠅⣵⡴⠋⠀⠀⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢀⡁⣝⠦⠄⠉⡳⣄⣈⠓⡄⠢⡀⠱⣔⣠⣜⣉⣑⣒⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢄⣀⠀⠀⠈⡱⠒⠒⠒⠂⠉⠈⠉⠙⠒⠓⠊⠁⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠒⠀⠁⠀⠀⠀⣏⣸⠄⠠⢀⣀⣉⡌⢌⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠑⠄⠀⡏⠤⣇⠀⠀⠈⠀⠀⠃⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⣠⠀⢡⣀⠄⠀⠀⠄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠁⠈⠘⠀⠀⢠⠀⠄⠁⣸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⢡⣀⠠⠤⠘⡋⠀⢀⠁⢨⠉⠦⠀⠈⠀⠸⠀⠀⠃⠀⠀⠀⠠⠀⠐⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠦⠤⣀⠔⠋⡩⠘⠒⠃⠀⠀⡄⠀⠀⠇⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⢀⠡⡀⠀⡀⠀⠈⠀⠀⠀⠀⢀⠜⠀⠀⠈⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⡀⠁⠀⠀⠀⠂⡀⠐⠀⠐⠢⡀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⣐⠁⠰⠀⠀⠀⢠⠀⠀⠀⢀⠊⠘⠦⣄⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠒⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⡀⠔⠁

⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⢠⣿⣿⣿⣿⣿⣿⣿⠛⠙⠓⠲⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀
⢸⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣀⡤⣖⠾⠋⠉⠀⠀⡉⠿⢿⣿⣿⣿⣿⣿⣿
⠸⠿⠿⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠳⠄⣓⠙⠢⢄⡀⠀⠀⠈⢢⠀⠀⠀⠀⠠⠊⠀⠀⠀⣀⠴⢚⠝⣫⡆⠥⢀⡀⠐⠀⠁⠀⠀⠉⠉⠛⠻⢿⣿⣿
⠀⢖⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣉⠲⣄⠀⠀⠁⣤⣤⠐⠀⠀⢀⠔⣊⣴⠴⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠛⠻⠿⡟
⠀⠘⠄⠀⠀⠀⢺⣿⡟⠀⠀⠈⠀⠀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠙⡢⣑⠄⢰⣿⣿⡆⣠⣾⣶⠟⠉⠀⠀⠀⠀⠀⢀⣀⡠⠔⠂⠀⠀⢀⣶⣶⣶⠀⠀⠠⢤
⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠤⢄⣀⣀⠻⣛⣿⣿⣿⣿⢛⣁⣤⣄⣤⡤⠶⠶⠞⠏⢉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠏⠀⠀⢠⠇
⠀⠀⠀⢃⠀⠀⠀⠀⠀⣰⣿⣷⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⢤⡬⣽⣽⣿⡿⢿⡿⣍⢍⠭⣩⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠉⠉⠛⠢⣶⣷⣆⢀⣀⡀⠠⠤⠀⠂⠔⣲⣿⣿⣿⡗⢈⡟⠣⢽⡒⠒⠶⠬⢷⣔⣀⠀⠀⠀⡀⣀⣀⣿⣿⣿⡦⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⢴⣶⣿⠫⠅⠈⠀⣀⠀⠀⠀⠀⠈⠀⣈⣿⣿⣶⢾⣿⣆⠀⠉⠀⠄⠠⡀⢀⣀⡉⣀⠀⢹⣽⣯⡙⠉⠉⢀⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⢻⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⡞⠉⡸⣿⣟⣻⢻⢌⣣⢀⠀⠀⠀⠈⠉⠐⢤⠀⠈⠃⠻⣿⣿⡿⢅⡉⠉
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠙⠀⢐⠃⣿⣯⣭⠈⠏⢄⠘⠃⠄⠀⠀⠀⠀⠀⠁⠀⠐⠀⢹⡿⠁⠠⠹⡄
⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠁⠥⠋⠀⣿⣯⣽⠴⠐⡈⠂⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻
⠀⠀⠀⠀⠀⢸⠂⢀⠀⡄⠀⠁⠀⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⠀⣰⢸⣿⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠈⠳⡁⠀⠁⠀⠀⠀⠀⠀⠾⠿⠀⠀⠀⠀⠀⠀⠀⠆⢠⠇⠸⣿⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠈⠸⣆⡀⠀⢀⠀⠈⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠉⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠖⠀⡀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣅⡀⠀⠰⠀⠀⠆⠀⠀⠀⠈⢀⡠⠋⠀⠀⣀⠐⠃⠀⠈⢇⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣤⣀⣀⡀⣀⣠⡶⠋⠀⠀⠀⠀⠳⠚⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣀⡵⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣷⣿⣿⣿⣶⣿⣾⣓⡞⠀⠤⠐⣤⡄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣬⣶⡿⠟⠶⣛⠯⢴⣖⢠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢺⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠂⠀⠈⣝⣺⣝⠛⠷⣥⣽⡖⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠁⠀⠼⠛⠅⠘⣿⠟⠉⠈⠙⠹⠿⠿⣿⠟⠋⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠙⠛⠻⢯⡬⣑⡢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⢷⣤⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣖⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣑⣦⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣶⣤⣤⣤⠤⣄⣀⣒⣒⣃⣀⣀⣀⣈⢿⠿⣿⣿⣿⡄⠀⠀⠀⠀⣀⣀⣀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣷⣾⣾⣿⣿⣿⣿⣿⡿⠛⠟⠛⠀⠀⠹⣿⣿⢷⣾⡿⢜⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⡶⠒⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣆⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡛⠆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣙⣟⣛⡻⣋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡵⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡂⣠⠶⠆⠲⠀⠺⢿⣿⣿⣶⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠉⣻⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠻⣿⡎⠸⠀⠀⠀⠀⡅
⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⣼⣿⡙⠁⠀⠀⡞⠉⠁⠿⣷⣦⣦⣀⣠⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢸⡄
⠀⣰⣿⡏⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⡇⠀⠀⠀⣼⣿⡿⠻⢿⣤⣤⣼⣿⣧⢤⣴⣄⡀⠀⠀⠉⢆
⠀⣿⣿⠀⠀⠀⣠⡴⣾⣿⣿⡇⠀⠀⠀⠀⠸⡄⠀⣼⣿⡏⠀⠀⠀⠸⣿⣿⣿⡇⠈⠈⠹⠇⠀⠀⠀⠀⠳⠀⠀⠀⠀⡚⢰⡏
⠀⢻⣿⣀⣹⣿⣿⠀⠀⠙⢿⣷⣦⣦⣤⣀⣸⣯⣼⣿⣁⠀⠠⠀⢀⣀⣿⣿⣧⡀⠞⠛⠻⣦⠀⠀⡀⠀⠀⢁⠀⠀⣈⠃⣾⠀⢀⡤⠀⠀⠁
⠀⠀⠉⠉⠀⢻⣿⡀⢀⣠⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⢧⣼⣿⣿⣿⣿⣿⣦⣀⡀⠃⣠⣄⣤⣧⣾⣾⣤⣀⣁⣠⣿⠀⣥⣴⣠⣠⡄⢀⣀⠄
⠀⠀⠀⠀⠀⠠⢿⢿⣿⡟⣟⠋⣋⡀⢘⡿⢿⠿⠿⠟⠛⢻⢿⠻⢿⠡⠾⠛⠛⢿⢿⣶⣷⣶⣿⣿⣾⣿⣿⣿⣿⣯⣭⣿⠹⡾⠋⢹⠿⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⠈⠁⠁⠀⠋⠈⠛⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣻⣷⡌⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠄⢋⡽⢿⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢿⣶⡡⠔⠻⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣰⡋⠀⠀⠀⠙⠺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣴⣿⠗⠀⡀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⣼⣿⣿⣿⠟⠋⠀⠊⢨⠷⣼⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⣀⣀⡤⠤⠤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⢀⣯⣿⡋⢻⣗
⠀⠀⠀⠀⢀⣴⣾⠁⣾⣿⡿⠿⣿⣆⣀⣠⣴⣿⢾⡿⢛⣿⣿⣿⣿⡍⠉⠛⢿⡛⠲⣷⠦⢼⡏⢘⡿⠂
⠀⠀⠀⠀⠀⢿⣿⣄⠙⣿⣶⣞⡿⠋⡡⠛⠁⡤⠊⣴⠟⣿⡻⠈⠻⣿⣆⠀⠀⠈⠁⡈⢑⠿⢁⡼⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣬⠛⢿⣴⠈⠀⠀⠀⡠⣾⠏⢨⡟⢧⠀⠀⠙⢿⣆⠀⠀⠀⢀⣡⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣿⡽⠂⠀⡀⣶⣴⣅⠀⢰⡇⠈⠆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣷⡼⢿⣿⣾⢹⣿⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⣈⡿⠺⠟⢛⣁⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⡀⠀⠀⡴⣣⠴⠣⠀⢠⡏⠂⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡴⠉⠀⢸⠀⠀⠈⠒⢼⣿⣅⠀⡼⠀⢸⢀⢃⠔⠉⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡞⠀⠀⣠⢼⣲⠂⠀⠀⢸⣍⠻⠛⣦⡀⡎⢸⣷⠖⠲⣠⠤⢄⠘⣄⠀⠀⠀⢰⢆
⠀⠀⠀⠀⠠⡇⠀⠀⠀⢸⣷⡡⠀⠀⡘⢇⠀⢀⠉⠙⡿⠋⠁⠀⢀⠏⠀⠀⠻⣄⠱⡀⠀⠈⠀⡗⢄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⢿⣿⠖⢩⣷⡮⠭⠉⢷⣦⣇⠀⠀⠀⡏⠛⠁⠀⠀⠈⡶⠃⢐⠀⢸⣷⡬⠖⠋⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠃⠀⠊⠉⢻⣷⣶⠳⠤⢍⠳⡄⢾⠀⡄⢀⣠⣶⠟⠓⠤⢄⣆⢸⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⢀⠄⠚⡝⢻⡠⠀⠀⠓⠛⢻⣿⣿⡿⠋⢻⠀⠀⠀⠀⢿⣬⡝⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⡀⠀⠉⢸⠇⠀⢀⡄⢠⠞⠍⡞⠀⠀⢸⣖⣄⣀⣴⠋⠉⠓⢦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠔⠉⠉⣽⠁⠀⠈⠛⠚⡇⠀⠀⢘⣟⠛⣿⢸⣤⠢⠈⣢⡇⠀⠀⠀⢤⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠬⠷⣄⡀⣀⣧⣸⣄⠀⣾⣷⢰⣿⢸⣤⣟⠻⡁⠀⣀⡄⠀⠀⣧⠴⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠲⢿⡟⠳⡌⢙⠟⠀⠘⣿⠿⣷⢦⣿⣦⡃⣸⣿⣄⣠⡼⠯⠥⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠱⣄⣼⠸⠀⠀⡼⡇⢠⠃⢀⡿⠀⢻⢷⡄⣀⣀⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠷⣹⠙⠻⣿⡁⠀⠾⠞⢿⣟⣉⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⢧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠃⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⠀⠁⠀⠀⠀⠀⢠⠎⠁⠀⠀⣀⣹⠗⡄
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠁⢀⣖⠈
⠀⠀⠐⣌⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢢⣅⣀⠂
⠀⠀⢦⡈⠹⣆⠀⠀⠀⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠀⠀⠀⡀⠔⠉⠹
⠀⠀⢠⡉⠓⢀⣀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⠞⠀⠛⠿⢦⣭⣤⠤⠂
⠀⠀⠀⠙⠠⣈⣻⡄⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⣀⠩⠿⠦⣶⠀⠀⠀⠀⠀⠀⠀⢀⢀⠐⠂
⠀⠀⢀⠔⠻⠟⠻⣂⡆⠀⠑⣴⣣⠔⠒⠀⠀⠉⠉⠁⠩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠎⣸⣇⠺⠿⠀⠁⠀⠀⠈⡧⢀⣀⠀⡀⠀⠠⠄
⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠛
⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⡤⠄⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠉⠀⠤⠀⠤⠤⠚⠁⠀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣟⣡⠶⠖⠓⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡎⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠙⠏⠀⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠎⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠠⢇⠀⠉⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⣪⡾⠟⠻⢤⣀⣀⣀⠠⠛⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠒⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣠⠔⠣⠤⣀⣠⣤⠶⡿⠁⠀⠀⠀⠀⠈⠉⠛⠃⠤⣤⣄⡀⠀⠀⠄
⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠗⠤⠤⠤⠤⢔⣒⣈⡽⠋⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠠⡄
⠀⠠⠼⣣⠄⠀⠀⠀⡀⠀⠀⠀⠈⢿⣆⠀⣄⣀⠀⢀⠤⠀⠀⠀⢠⡴⠊⢁⠈⠉⠛⠦⣠⠇
⠀⣦⣴⠀⠀⠀⠀⠈⠙⡤⠤⠤⠤⠄⣁⠀⠀⠀⠛⠳⢄⣀⣀⣠⢋⢀⡰⠃⠀⠀⠀⠀⢛
⠀⢿⣡⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣞⡁⠀⠀⠀⠀⠀⠀⢻⠉⠹⠾⠚⣷⡀⠀⠀⠀⠀⢸⣆⡀
⠀⠈⠃⠑⠂⠀⠀⣠⣇⣠⡤⠞⠋⠈⢲⠄⠀⠀⠀⠀⠀⣸⠖⣺⡶⢠⣼⡏⠀⠀⣀⣤⡾⠉⢿⣿⡛⠉⠉⠙⠀⠂
⠀⠀⠀⠞⠒⠂⠀⠀⠈⠈⠁⠀⠘⠃⣸⡇⠀⠀⠀⠀⠐⣻⠃⠀⠸⢦⣌⣈⡙⣉⠽⠚⠁⠀⠸⡎⠙⠢⣀⠀⠀⠀⠀⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⢤⣅⡀⠀⠀⠀⣨⠋⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢰⣇⠀⠀⠙⢷⡄
⠀⠀⠈⡷⡀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢿⠀⠐⠀⠀⠉⠠⡄
⠀⠀⠀⠀⠁⠐⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⠙⠦⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⢿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠷⣦⡀⠀⢀⣼⣿⣿⠗⣀⣀⣀⣀⠀⠀⠀⠰⠷⡞⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡅⠀⠀⠀⠈⢻⡄⣾⢟⣩⠖⠛⠉⠉⠉⠉⣹⡷⠂⠀⠐⡿⢀⣠⠞⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣿⡇⣦⠀⠀⠀⠀⡧⢫⠞⠁⠀⣀⣤⠄⠀⣼⠏⣤⠀⠀⢀⡗⣉⠀⠀⣼⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣇⣿⣧⢰⠀⠀⠃⠀⢀⢦⣾⣫⣤⡄⣴⠋⢠⣿⡆⢀⢈⡾⠃⠀⡰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠻⣿⡆⣇⠀⠀⣄⣾⣾⡿⠟⠁⠊⠀⠀⢸⣿⣿⣼⣨⣾⠁⠈⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠂⠀⠈⢃⡿⠴⠚⣿⡏⠉⠀⠀⠀⠈⠙⠓⢮⣻⠇⣻⣿⣷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⡴⠦⣬⣶⣏⣀⣤⣾⡿⢁⡶⠒⠈⠡⢤⣀⣀⢰⣽⣾⡿⠀⠀⣴⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣤⣀⠤⣬⣽⡿⢿⠟⢃⣤⣭⣍⠉⠉⢠⣛⡳⠦⣤⣤⣶⣾⣿⣿⣷⠟⣠⣤⣤⠸⣷⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⣻⣿⣱⢟⣵⢸⠁⠀⠀⢸⣿⣿⡷⠀⠀⢈⣩⣤⢶⣿⡽⣿⡻⣿⣧⡿⠀⠀⢀⣿⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⡿⣿⡈⠀⠀⣸⠘⠿⣟⣥⠀⠀⢉⠛⠧⣽⣦⡀⠀⠈⠊⣿⡇⠀⡄⣾⢏⣴⡿⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡏⢏⣿⣬⣁⣀⣠⠏⠀⣀⣭⠷⠾⢿⡜⣷⡀⠀⠙⠿⣧⠱⡄⡏⢠⣾⠱⠏⢚⣵⡾⠿⠟⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣉⣉⠁⠈⠳⢿⣿⣵⢋⡴⠞⢩⣶⠾⠿⢸⡇⣿⣷⠀⠀⢀⣈⠃⠚⣵⠛⣇⣀⣚⡛⠛⠛⠉⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣍⠉⠉⠉⠙⠓⢦⡀⠈⢀⡞⣴⣧⡞⠁⠀⠀⢸⡇⣿⣿⠀⣴⣿⣿⠇⡀⠻⣤⣿⡟⢷⣤⣴⣆⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣇⣿⡀⣤⡠⣀⠀⢀⠁⠀⠀⣸⣿⢋⠔⠀⠀⠀⣼⢇⣿⢃⣠⣿⠟⢡⣾⡇⣷⢬⡙⠛⠶⡀⠉⠉⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡻⡸⣷⣿⣷⣌⢳⡌⣆⠀⠀⠿⣵⡿⠀⠀⢀⣼⣫⡾⣣⣿⡿⠉⡏⡿⠛⠈⠁⢀⡇⢸⣷⡍⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣙⢮⣟⣻⣿⣳⡿⢋⡴⠖⠶⣦⡙⣦⠖⠋⠚⢩⣌⣛⠋⠀⠀⣧⠀⠀⠀⣠⡞⣷⢈⢷⣍⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣶⢯⡀⢌⣉⣛⡛⠃⣿⡀⠀⠀⢸⣧⢐⠚⠛⠻⠿⢓⣮⣝⢶⣄⠹⠷⠾⠟⠉⢸⣿⣯⡳⣮⣳⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣿⣿⢟⡿⣫⣶⣿⣿⡟⣡⣆⠹⣿⣶⣶⣿⠇⣼⣿⠶⠦⠬⠭⠍⠙⠓⠹⣆⠀⠀⠀⠀⠀⠙⢿⣿⡎⠙⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢫⡿⣁⣠⣤⠞⠩⠞⠁⣸⡷⢠⣭⣭⢡⡈⠻⢦⣄⡀⠀⠀⠀⠀⣀⡴⣛⣤⣶⡟⠀⠀⠀⢀⠙⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡿⠋⠁⠀⠀⠀⣰⡿⢁⡿⠉⣿⢸⣿⣆⢀⢯⣝⣛⣛⣿⠍⠀⠾⠿⣟⣻⣃⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠻⠇⠀⠀⢀⣠⡾⣟⣵⢺⡇⠀⠸⠸⡿⣿⡌⡇⠿⠿⠛⠁⠀⠂⠉⠉⠉⢛⣿⡟⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⢸⣿⣿⡸⣇⠀⠀⠀⠃⢹⢧⡇⠐⢶⣤⣤⣤⣤⣴⣶⣾⠟⠋⠀⣠⡟⣴⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠙⢆⠀⠀⠀⠞⣾⠁⠀⠀⠛⠿⣦⣉⠉⠉⠀⠀⣀⡴⠋⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠳⣦⣠⣾⠃⠀⠀⠀⠀⠁⠒⠤⠤⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠉⠉⠉
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⡄⠀⠘⡄⠀⠀⠀⠀⠀⠢⠈⠳⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠐⢀⠀⢀⠢⡀⠠⡀⢳⡄⠀⢀⡠⠂⠈⠑⢢⡄⠚⠂⠘⠳⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠙⢆⡀⠀⠀⠀⢘⠀⠑⠢⣈⠦⣸⣤⣤⠿⠛⣿⠀⡤⢤⠀⠀⡇⣀⣩⠷⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠌⠓⠤⠤⢴⣶⢺⠟⠋⡟⡈⡋⠀⠀⠀⠘⣄⡉⠊⣀⠼⢫⠇⠀⠀⠀⠀⠉⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠠⠖⢃⣿⠃⠀⡴⠉⠁⠉⠱⡄⠀⢟⠏⢠⠉⢀⣦⠏⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⢠⢲⠇⠀⠸⠁⠀⢀⡀⢀⡜⠉⠁⠀⣀⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠘⠀⠠⠀⠘⠢⠵⠾⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠁⠀⠴⠊⠉⠀⡀
⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⡀⠀⠀⡄⠀⠀⣀⢄⠀⠄⡀
⠀⠀⠀⠀⠀⠢⣠⠔⠉⠉⠐⢢⡔⡡⢒⠆⠈⠳⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠈⢄⠂⠈⢀⡴⠃⠀⢰⠑⠄⠃⠀⠘
⠀⠀⠀⠀⠈⢠⠃⠀⡴⠠⠀⡸⠒⠛⠓⠤⡀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⣄⠀⣀⢀⡼
⠀⠀⠀⠀⠀⠀⢣⡀⠑⠂⡙⠀⣀⡀⠀⠀⢰⡾⠋⠒⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠉⠢⡀⠀⡠⠒⠀⠒⢌⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢳⡀⠌⠙⠀⢀⡼⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⢘⠀⠠⠂⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⠼⠃⠈⢤⣈⣉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣄⣾⣿⣿⣾⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢿⣿⣿⣿⣟⢿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⣻⣷⣄⠀⠀⠀⠘⠿⢿⣷⣿⣿⣿⠿⣿⠿⠟⠛⠋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⣠⣶⣿⠿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⣿⣿⣿⣯⣾⣿⡗⠀⣠⣾⡟⣿⣷⣇⠀⠀⣿⣼⣿⣿⣷⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡏⠹⡏⠉⠉⢠⣾⣿⣷⣦⣾⣿⣿⣧⢰⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣦⡀⠀⡇⠀⡇⠀⣰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣇⢿⣿⣾⣿⣿⣿⣾⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣷⣄⢛⡀⠿⠀⣿⣿⣿⣿⣿⣜⣿⣿⣿⣿⣿⣸⠉⠉⡹⠉⠉⢹⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣼⣿⠷⡿⠿⠟⡈⠃⠀⣀⠈⠉⠉⠉⡟⠉⠛⡟⠙⠋⠁⡿⠀⢀⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⡖⠢⢯⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⢀⡇⠀⣸⠈⠀⢠⣇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣤⡀⠸⡄⠀⡆⠀⠀⢀⡆⠀⢸⢀⣤⣶⣌⠀⢠⠇⢀⣴⣿⣿⢛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⣿⣿⣷⣇⠀⢸⠀⠀⢸⠁⢀⣴⣿⣶⣿⣿⡆⠎⠀⠸⣿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡿⡿⠛⡄⠀⣦⣤⡜⠀⢸⡟⡿⠿⣾⢿⡿⠀⠀⢠⠏⠄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⡄⣱⡀⢇⢰⣿⣽⣿⡆⢸⢃⠃⠀⣧⠞⠁⠀⢠⠎⡸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣆⢳⠈⢷⠸⠀⢘⠁⡇⠀⣾⠎⠀⡾⠁⠀⢀⢢⠺⣄⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⢿⡁⢣⠈⢳⡇⢸⠀⡇⢀⡟⠀⣸⠃⠀⢔⡵⠃⠀⠀⡳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠳⡈⢧⠀⠙⢾⠀⣇⠜⠀⣰⣏⠀⠄⠁⠒⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣳⡆⠀⠀⠈⠂⠙⠀⠓⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠔⠊⠀⠀⠀⣴⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡿⠇⢀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⢀⠀⠻⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⢀⣶⣿⣿⣷⣆⠀⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠁⢸⣿⣿⣿⣿⣿⠀⠁⠀⠙⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁

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