The violent earth Emojis & Symbols

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠂
⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠉⠛⠳⣶⠶⡶⠶⢶⣴⠾⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴
⠀⠉⠓⠆⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢹⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣰⡶⠴⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀
⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣼⣷⠀⠀⠀⠀⠀⠰⣷⣤⣶⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⣴⣶⣚⣻⣿⠀⠀⠀⠀⡀⠈⣿⣭⣽⡷⠆⠀⠀⢙⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣄⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣁⣀⣠⣤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠀⠀⠀⢂⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⡾⢶⣿⡟⠻⢶⣾⣶⣬⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢚⠿⠍⠉⠉⠀⡠⢤⣟⠦⣄⡀⠘⠥⡈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠓⠈⠀⠀⢀⡾⠉⣁⣈⣙⠦⠤⠇⠀⠀⠳⣜⠃⢀⣉⣻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠀⠀⠀⠀⠀⣠⠶⠋⠀⠈⣧⠀⠌⠳⠤⠔⠲⠤⣄⠈⠢⠞⠀⠁⣀⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠑⢄⠠⠀⠀⠀⠀⠙⠒⠒⣦⡏⠁⠀⠈⠁⢀⡖⠒⠲⠞⠀⠀⠀⠀⠈⠃⠽⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠸⡀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣮⣿⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢱⢈⣁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣭⠿⣿
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⢏⢳⠀⡤⠔⠚⠛⠀⠐⢦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⡍⠀⣿⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⡏⠧⢤⡄⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠁⠀⡶⣿⠄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠺⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⣋⠥⣿
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠀⢠⡿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⡤⠀⠉⠳⣦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⠀⠤⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠘⠲⠖⢲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⢧⣀⠣⠂⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠈⠙⠦⣤⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠇⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⢦⣿⣤⣤⣤⣾⡶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠠⠂⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠊
⠀⠀⠀⠀⠀⢀⠀⢐⠶⣞⣁⢉⠈⠑⢢⡐⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢶
⠀⠀⠀⡄⠀⠀⠔⠃⣴⠴⡁⠄⠆⢠⣾⠗⠀⢀⡤⠤⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀
⠀⠀⠀⠀⢠⠂⠀⠀⣠⠇⠀⠀⠀⠈⡹⢠⢶⠖⡐⢀⠳⡄⠂⠀⠈
⠀⠈⠀⠀⡄⣀⠠⠊⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠑⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⡷⣼⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣆⡀
⠀⠀⠀⠀⠏⢷⠘⠍⡀⠀⠀⠀⠀⠀⠘⠢⣠⠤⠤⠄⠀⣼
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠑⠤⠄⡀⠀⠀⠀⠀⠀⠀⠆⢰⢯
⠀⠀⠀⠀⠺⠗⢑⢀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡧⠃⠀⠐
⠀⠀⠀⢠⡤⠀⠀⠉⠧⣀⢀⡎⠀⠀⠀⢀⣀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⡿⢿⣯⣝⠛⠛⠋⢉⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢫⣥⡿⢷⣼⣿⣡⣴⣶⣿⣿⣿⡟⢋⣄⠈⠋⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠉⠀⣤⣿⣿⣿⣿⣿⣿⣿⡞⠋⠉⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡯⠴⠿⣿⣦⣆⡀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠡⡀⠐⠦⣤⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣮⣜⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹⠤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⠀⠀⢀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠄⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣰⣿⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠉⢿⡟⠉⠙⠛⠿⠟⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣇⠀⠀⠀⠀⣰⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠸⠇⠘⠋⠀⠀⠀⠰⣿⡷⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣰⣿⣴⡄⠀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠿⠿⣿⣧⡀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡍⠉⠙⠻⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠈⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣴⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⡿⠿⠿⠟⠋⠉⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢢⡇⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡣⠖⠊⣁⣀⡀⢀⣀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⣴⣋⣴⡾⣿⡍⣻⣿⠿⣏⠁⠀⣠⣤⣄⣈⣳⣄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⣾⣿⣿⣥⣾⣿⡄⠉⠀⠀⠡⣶⣬⣯⣾⣿⣿⣿⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣾⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⢀⣾⣻⡟⢿⣿⣿⣿⣿⣿⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣋⠉⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠈⠙⠣⣤⣤⣄⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡀⠀⠸⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⢁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⠀⠘⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⣿⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⡀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣙⢦⡘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠮⣷⣦⣄⣀⠀⠀⠀⢀⣀⡀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡵⠊⠉⠉⠉⠒⠮⡒⢄⠀⠀⠀⠀⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⠤⠶⠾⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣄⡒⠀⠉⡼⠙⢂⠀⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠳⠅⠕⠪⠳⢝⠿⠗⠲⢴⣈⡀⠀⠀⠌⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠰⠀⠆⠀⠚⢄⡻⠀⠁⠁⠒⣄⠀⠀⠀⠀⠀⡀⠀⠁⢳⣄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⡜⠨⠗⡡⠀⠠⢤⣄⡀⠔⠛⣞⣉⢀⠀⢈⢄⢊⢈⠈⡎⠦⠦
⠀⠀⠀⠀⠀⢁⠀⠀⢠⠅⣽⢰⠁⠀⠀⠘⢀⠍⢠⣰⠟⢁⠔⠁⠒⡡⠢⠂⡘⠀⡐⠄⠡
⠀⠀⠀⠀⠀⡀⠀⠀⢸⣴⡀⢗⠀⡄⡤⠄⣤⢼⠫⠵⢂⢅⠖⡁⢈⠔⢀⢄⠄⠐⠀⠘⡆⠁
⠀⠀⠀⠀⠁⡂⠀⠀⠀⠙⠷⡜⠈⣿⣂⣴⡏⠷⣤⠿⢐⣁⡊⠀⠀⠕⠁⡠⢊⠁⠀⠊⢧⢰
⠀⠀⠀⠀⠀⢀⡠⢤⣤⢶⢛⠋⠶⠤⣍⠠⢳⡊⠀⠋⡐⠉⠨⡂⢄⠅⡪⠐⢁⢔⢅⠔⣸⠈
⠀⠀⠀⠀⢻⠹⠷⠞⠚⣚⠈⠀⠒⠉⠀⠉⠓⢭⡕⠋⠱⠘⠓⠪⣄⡤⠌⠂⠦⠁⠕⢞⣼⠇
⠀⠀⠀⠀⠀⠆⢣⡀⣁⢁⠆⢑⠊⠈⠀⠀⠀⡰⠀⡅⠀⠀⠀⠀⠀⠹⢥⣄⠖⡨⣴⢟⡏
⠀⠀⠀⠀⠀⠈⢦⠤⢉⠀⠀⠀⠌⠀⠐⠀⡀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⡃⠴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠂⢨⡀⣠⣆⠙⢶⠄⠁⠀⠀⠀⠀⠀⢰⣥⣬⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢜⡦⠀⢉⡁⠀⢨⠾⡄⠀⠀⢀⡤⢤⣴⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠏⠈⣠⡴⠾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠀⡀⠀⣀⠀⡀⢀⠀⠀⠀⡀⢀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠁⠂⠄⠀⠅⠠⠐⠠⠂⠡⠂⠅⠂⠅⠠⠐⠀⠂⠕⠀⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⢠⡄⠀⠷⡀⠲⣠⠄⠐⣂⡀⠌⣁⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠁⠀⡀⠀⠀⣿⡶⢉⡑⠀⠀⠁⠀⠂⢀⠤⠰
⠀⠀⢀⠀⣄⡀⠀⠀⠀⢀⠈⠙⠅⣂⠀⠀⣜⡿⣓⠀⢋⡋⣢⠹⠀⠈⢀⠀⠑⠬⡀⡀
⠀⠀⡿⣇⠟⠃⢀⡄⠀⠈⡠⢰⣴⢛⣳⣞⢈⣙⣳⡀⠬⠇⡇⡃⠐⠀⠀⠀⠀⠀⠈⣧⡄⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣠⠛⠭⠀⠶⠀⡗⠰⠚⠿⣾⢎⠉⢡⡿⠛⡋⣨⡉⣀⣀⡄⠀⠰⠀⣀⠦⠐⠀⠀⠈⠂⠐⠁⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂
⢀⠠⣼⢷⡆⠰⣾⣴⠦⣴⠤⡼⢋⣉⣘⣶⣾⣿⣛⠙⣧⢅⣀⠃⠄⢠⠀⠐⠀⠀⣀⣀⣀⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢠⠆⠘⠰
⠀⠀⠛⢻⣷⣤⢏⣼⠇⢾⣶⣶⣿⣿⣿⣿⣿⣿⣿⢳⡯⡾⣵⡖⠀⢀⣠⣴⣾⢟⣯⣿⣿⣟⣿⡿⢿⣿⣿⣷⣦⡀⠀⠀⠐⠀⠀⣆
⠀⠠⠲⡾⢹⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⢁⣴⣾⣿⣿⡿⣿⡟⢳⣈⠉⣋⡧⣼⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⡇⠸
⠀⢠⣇⣇⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⢿⡇⣚⠉⠁⠋⠀⠁⢛⠀⠃⠙⠂⣀⣶⠿⠆⣽⣿⢿⣿⣿⣿⣿⣷⣆⠀⡀⢠⢑
⠠⣤⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣯⠘⠋⠑⠈⠀⠀⠀⠂⢰⠀⠀⠀⠀⢫⡙⣿⣿⣿⢿⣓⡿⢿⣿⣿⣿⣿⡄⠹⠀⠀⠄
⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣿⣿⣯⣭⣀⣁⡈⢂⣀⠀⠀⠒⠀⢁⠀⠀⠀⢀⡄⢸⣷⣿⣿⣿⣶⣻⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠻⢋⢹⣿⣿⣿⠿⠛⣛⣩⣴⣾⣿⣿⣿⣿⣿⣿⣳⣌⣿⣷⡑⣲⢆⠰⠂⠀⠀⠄⡀⢀⣿⣿⣻⣯⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡀
⣤⡼⢚⣛⣉⣭⣶⣾⣿⣿⢿⣋⢉⣾⣿⣿⣿⣿⣽⣿⣕⣉⣒⠦⢙⢓⠀⣀⣠⡭⣥⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⣹⣦⣼⣍⣙⡉⣛⣉⣉⣭⣿⣿⣿⣿⣿⣿⣿⣶⣾⣽⣿⣿⣳⣯⣝⣿⣿⣯⣿⣩⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣮⡙⣻⣛⣿⣿⣶⣿⣷⣻⣶
⠸⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⣛⣋⣉⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠁
⢀⡐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣶⣿⣿⠿⠛⠋⠁⠸⠽⡿⠿⠿⠛⠉⠉⠁
⠈⠙⠙⢾⢻⣿⢛⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣛⣥⣿⣿⣿⣿⠿⠟⠋⠀⢀⢔⡐⠆⠀⠘⢒⠀⣀⣀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠘⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠋⠉⠉⠉⠀⢀⡀⠀⢀⣀⣤⠬⣵⠶⠚⠉⠉⠁⠈⠝⢿⣿⣾⢌⠋
⠀⠀⠀⠀⠀⠐⡼⣧⠙⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠐⠙⡋⡵⠚⠙⠃⡐⠀⢀⡀⠀⠀⢀⡠⠚⠉⠀⣸⡽⣿⣝⣺⡧⠁
⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⣑⠂⠀⠚⢥⠖⠀⣀⠀⢂⣊⣉⣥⠥⠴⠛⣿⣛⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣭⣵⣶⣾⣥⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⠪⣥⣶⡶⢿⠟⠛⢛⣿⡛⠿⠷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡁⠀⠀⠀⠋⠀⠻⢾⣶⡉⠉⠉⢹⣶⡿⢿⣿⣿⣲⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠁⠀⠀⠀⣀⣤⡴⢂⣿⠃⠀⠀⣼⣿⣿⣿⡟⠙⠻⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠐⠀⠀⢻⡿⠁⣷⣾⣿⣠⣶⣾⣵⣏⣩⡼⠃⠀⣠⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣠⢻⠏⠀⠀⠀⠐⣷⠀⠈⠀⢀⣀⣻⣿⣁⣉⠛⠏⢉⡟⢿⣶⠠⠿⠀⣀⠀⠀⠀⣄⠘⠣⡀
⠀⠀⠀⠀⠀⠀⢠⢃⡌⠀⠀⠀⠀⠀⠘⠷⢂⣄⣘⣥⣼⣿⠉⠛⠘⣿⣿⣴⡬⢋⡀⠠⠐⠃⢰⣧⡦⠈⢳⠀⠁
⠀⠀⠀⠀⢀⠀⠀⣾⡇⠈⠀⠀⠀⠀⢀⣴⣿⡟⠋⠁⠀⠐⠐⣧⠀⠈⢿⣿⠿⠀⣠⣦⣮⢌⢀⣁⣠⡀⠀⢀⣠⣖⠂
⠀⠀⠀⠀⠈⠀⣸⣿⡷⠀⣾⣷⢲⠰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠐⣠⣾⣿⣄⠨⠿⠟⢻⣿⡿⠿⠟⢃⠀⠀⠉⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣧⡸⠏⣽⣿⣻⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠛⢹⣿⡿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠈⠳⡄⠀⣾⠁
⠀⠀⠀⠀⠀⡄⠛⢻⡿⣿⣆⢹⡟⣷⣦⣽⣷⣾⣆⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⣿⡇
⠀⠀⠀⠀⠀⢰⠀⠀⣁⠻⣿⣮⡝⠉⠛⠿⣿⣿⣿⡃⠀⠀⠀⠀⣸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣿⢀⠆
⠀⠀⠀⠀⠀⢸⣀⠀⠈⠱⣽⡿⢁⣠⠶⠄⣆⠙⠻⣦⠀⠀⠀⢠⡟⢿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠏⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠻⣟⢢⡀⠙⣆⠋⠀⠀⠀⠀⠀⢀⡛⢦⣌⣀⣀⣀⠈⠛⢷⣤⣤⣶⣾⣷⡿⠿⢛⠃⠀⠃⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠌⠁⠈⠠⣿⣄⠃⠀⠀⠀⠀⠨⠀⠀⠀⠙⠻⣿⣡⡀⠀⠀⠙⢛⣿⣿⣷⣠⣏⠀⠀⣦⣾⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡌⢫⠲⠀⠀⠙⣷⡈⣷⢶⡄⠀⠀⠀⠀⢠⣾⣿⠛⠁⢤⠦⡄⠉⢡⣻⡿⠁⠙⠂⣴⡿⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡈⠲⣿⡷⡇⠀⠱⡟⢂⣤⣤⡼⡿⠋⠀⠀⠀⠀⠀⣠⣿⣿⡷⠀⣠⣾⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡿⣷⡇⠀⠀⣠⣼⣿⣿⠁⣀⡄⠀⡴⢒⡛⠉⡉⢩⣿⣷⣎⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠝⠺⣿⣦⣸⣿⣿⣿⣿⣿⡷⠶⡶⠿⠾⢷⣼⣿⣿⠏⠉⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠠⠶⢀⠠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠀⠀⠀⡙⢷⠄⢀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣮⡀⢀⠂⡄⠀⠀⠁⡀⠀⠀⠐⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⢙⢿⣤⢠⣉⡄⠰⠀⠀⢠⡀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠤⢀⣣⣴⡎⡱⡍⡎⠰⣧⠰⠀⠐⠈⢳⡄⠀⠂
⠀⠀⠀⠀⠀⢈⣾⠋⠀⡂⢀⣠⠄⠚⣿⡿⠵⢃⢡⢱⡀⠀⡼⣾⠔⠐⠀⠁⠁⠄⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢠⡿⠋⠁⠀⠉⠺⡍⠙⢆⡀⠐⠒⣏⢰⢨⢣⢀⣼⠁⢀⡀⠄⠙⠆⢀⡉⠁⡀⠀⢈⣤⣄⣀
⠀⠀⠀⠀⠛⠁⡆⠀⠀⠀⠀⠈⠠⠌⣷⣦⡀⢯⣾⠸⣠⡇⡾⠒⡉⠀⠀⠀⠀⣸⣟⠀⢀⡀⠘⠺⠿⠽⠉
⠀⠀⠐⠡⣴⡶⢧⠄⠀⠀⠀⠀⠀⠀⠻⠃⠈⢾⡟⣿⡇⢰⡇⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠄⠐
⢀⠀⠀⡈⠉⠁⣌⠂⡢⣀⣢⣀⠀⣀⠀⠀⠀⢸⣏⢹⡆⣾⠄⠀⠐⣀⠐⠉⣸⡿⠠⠖
⠂⠀⠀⠀⠀⠈⠁⣀⣡⡀⠀⣿⠢⡈⠁⠠⠀⠘⣿⢸⣇⣿⠖⠋⠁⠀⠁⠀⣹⠃⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠈⠈⠁⠉⠀⠀⠀⠉⢹⡖⠈⠳⠦⢬⣌⣿⢹⡏⢹⠀⠀⡀⢀⠠⠎⠀⠀⠐⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣼⠇⣸⡟⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠻⠲⠿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣋⣴⣿⣷⣶⣸⣿⣿⣷⣶⡀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠻⠞⠁⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠽⠈⠴⣷⡆⠀⠀⣼⣿⣿⣿⣿⣿⣻⠿⠋⠁⠙⠟⠯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣰⢈⣷⠀⣴⣿⣿⡿⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠙⠀⣹⣿⣾⣿⡟⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢀⣀⣾⣿⣿⣿⡿⠚⠛⠗⠏⠀⢀⠀⠀⠀⠀⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡷⣾⣽⠛⠻⣿⣿⣧⣀⡀⠀⠀⣰⣿⣧⣾⡦⡿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄⠀⢀⣿⣿⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⠟⠓⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣶⣬⣿⣿⣿⣿⣿⣿⣥⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⣠⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣦⣤⡀
⠀⠀⢠⣿⣿⣿⠟⠓⠲⠄⠀⠀⠀⠀⠀⣰⡟⠉⠉⠉⠁⡀⠠⡀⠀⠀⠀⠀⣴⣇⢠⣴⣾⣿⣿⣿⣦
⠀⠀⠛⢿⣿⣿⣤⣀⠀⠀⢠⣷⠀⠀⠀⣿⣄⣤⠀⠀⠀⠈⣹⣿⡆⠀⠀⢰⣿⣿⣿⣿⠉⠛⢿⣿⣿⡇
⠀⠀⢄⠀⣿⣿⣿⣿⠀⢀⣿⡟⠀⠀⠀⣿⣿⣿⡇⠀⠀⣸⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⣼⣿⠇
⠀⠀⠈⠺⣿⣿⣿⣿⣶⡿⠏⠀⠀⠀⠀⠈⠻⣿⣿⣤⣴⣿⡿⠋⠀⠀⠀⠀⠘⠿⣿⣿⣇⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠄⠀⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⡠⢄⠦⠾⣷⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠂⠒⢄
⠀⠀⡐⠀⠀⠀⣴⣷⣮⣽⣿⣆⠀⠀⠀⠌⣴⣿⣿⣿⣿⣿⡿⢳⡄⠀⠀⠀⠊⠹⣏⡉⠀⠀⠀⠀⠑⡀
⠀⠀⣷⣄⡀⠀⠙⠛⣿⣿⡿⢸⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⠃⠀⠁⠀⠀⢸⠀⠀⠀⢸⣿⣶⣤⡀⠀⠁
⠀⠀⢻⣿⠁⠀⠀⠀⣿⡟⠀⡈⠀⠀⠀⠄⠀⠀⢸⣿⣿⢇⠀⠀⠄⠀⠀⠈⡀⠀⠀⠀⢻⣿⡿⠃⠀⠄
⠀⠀⠀⠻⢄⠀⢀⣀⣁⡠⠊⠀⠀⠀⠀⠈⠢⢀⠀⠛⠁⢀⠠⠊⠀⠀⠀⠀⠈⠢⢀⠀⠘⠉⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⣠⠤⠤⠤⣀⠼⠇⠀⠀⢀⡠⠋⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣎⣀⠖⠚⠉⡢⠈⢇⢀⠔⣥⠒⠆⠀⠀⠀⠀⠈⡽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⡳⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⣀⣇⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠚⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢥⠙⡌⢲⡄⠀⠙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠃⠀⣀⠤⠤⢄⠀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⡠⠃⠀⠀⠀⠙⣿⣿⣿⣿
⣿⣿⣿⡿⠄⠀⡎⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠉⠀⡤⢄⡀⠀⠀⠈⢿⣿⣿
⣿⣿⡟⠈⢄⠀⠓⢲⠀⠀⠆⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣢⣴⣃⣀⣘⠚⠤⣄⡈⢿⣿
⣿⡿⠀⠀⠀⠹⡀⢧⠀⠀⠀⠀⠐⠊⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠉⠢⣄⣉⣾⣿
⣿⠃⠀⠀⠀⠀⠸⣻⡀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⡇⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⡇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⣧⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠛⡆
⣿⡀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⣾⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⣾⣿⣿
⣿⣿⣿⣿⣷⡀⠀⠀⠀⡇⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⢡⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣵⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠩⣥⡌⠉⠁⣹⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠰⠶⠋⠐⣿⣧⣶⣿⢿⣿⣿⣿⣿⡟⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⣤⣷⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣌⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡶⠀⠀⠈⠻⢿⣿⣿⣿⣆⢷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⣸⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⢀⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣜⣿⣿⣿⣿⣿⠼⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠻⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣦⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠘⠛⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠸⠇⣯⠙⠁⠀⠀⠀⠀⠀⠀⠙⠉⣽⠉⡏⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⢿⠋⠋⠹⣿⡟⠛⠋⢹⠁⠀⠀⠀⠘⠁⠛⢻⠛⠁
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠀⢰⡶⣶⢶⡶⢶⡶⣶⡶⠶⠄⠀⠠⠶⠶⣶⠶⢶⠶⣶⢶⣶⠄⠀⠀⠀⠶⠶⡶⡖⠶⣶⠶⡶
⠀⠀⠈⠉⡏⢹⠀⠀⠘⠿⠃⠈⠀⠀⠀⠀⣾⢰⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠲⠀⢸⠀⠦⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣧⠛⠚
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠛

⠀⠘⠒⠲⠶⠒⡖⢲⡖⠶⠖⢲⠶⢶⣶⣶⣶⡶⠶⡶⣶⣶⣶⣶⡶⠃⠀⠲⡶⢶⠒⡖⢲⠒⠖⡶⠀⠀⠀⠘⠲⠒⢶⠶⢶⠒⠒⠙
⠀⠀⠀⠀⠀⠀⣇⠸⠇⠀⠀⢸⡇⠀⠀⠘⠻⠃⠀⠀⢸⠉⡇⠈⠀⠀⠀⠀⠀⢼⠆⣧⠘⠰⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⢀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠈⠉⠹⠏⠙⠋⢹⠉⠏⠙⠁⠀⠀⠁⠙⠋⡏⢸⠉⠋⠉⡏⢻⠉⠛⠉⠟⠉⠉⠀⠉⠈⠛⢹⠩⡏⠁⠀⠉⠈⠉⠻⠛⡟⠉⠉⠁
⠀⠀⠀⠘⠀⠀⠀⠛⠀⡇⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠰⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⠖⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⢨⣿⠀⠉⢻⣿⠁⠀⠀⠀⠀⠀⠘⢿⡿⠁⣿⠁⠀⠀⠀⠀⠀⠀⠸⣿⡟⠁⠹⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⣿⠛⠿⠟⢻⣿⠿⣿⡿⠟⠁⠀⠘⠋⠉⠿⠛⢻⠟⠁⠀⠈⢻⡟⣿⠛⢿⣿⠉⠀⠈⠹⠻⣿⠻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⠀⠀⠀⣿



⠀⠀⠀⠐⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⡶⠂⠀⠀⠰⠶⠶⣶⣶⣶⣶⣶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠉⠉⠈⠻⠟⠉⠙⠋⠁⢹⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⠀⠀⠀⠈⠻⣿⠟⠻⣿⠋⠻⠟⠙⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⠿⣿⠿⣿⠟⢿⡿⠛⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠸⠏⠀⠻⠏⠀⠀⠀⡿⠃⠀⣿⠀⠃⠀⠈⠁⠀⠻
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⠤⣤⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠙⠉⠃⠁⠀⠱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠄⠀⠀⠙⠋⠙⠊⠣⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣿⣿⣷⣸⣿⣿⣿⢾⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢿⣿⣿⣿⣿⡻⢿⣿⠷⠜⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠷⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⢲
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠶⢆
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣯⡽⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⣔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣴⣶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣶⣾⣶⣶⣴⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠚⠩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⢠⠄⡄⠐⠄⠀⠀⠀⠀⠲⢾⣶⣤⣀⣀⡀⠑⠒⣤⣄⣰⠷⠀⠀⢀⣤⠀⡄⠀⠀⠀⠂⡆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠚⠀⣁⡀⠀⠀⢀⡀⢤⣘⡛⠿⠯⠈⢷⠇⣀⠾⠋⣉⢀⣶⣭⣀⡏⠂⡁⢀⠀⠀⠂⠢⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⢠⠠⠄⠀⠀⢉⡆⠉⠓⣶⠀⠀⠡⠞⠁⠀⠀⠀⠀⠴⡶⢫⣧⠀⠅⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠐⠈⡆⡀⠙⠉⣩⠶⠛⠋⡉⢉⣿⣶⡟⣴⢿⠀⠎⡐⠊⣤⢠⡴⢤⣄⠀⢀⠉⠛⠐⢀⠀⠀⠀⡀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⢀⣴⡯⢂⡠⣴⣚⣿⣿⣭⣥⡾⢃⣿⢠⡤⣀⣀⣀⠈⢡⣤⣦⣤⣄⡄⢀⠀⠀⠀⠀⠀⡅⢄⠀⠀⠀⡇⡇
⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣴⡟⡡⠂⣷⣾⣋⣍⣀⣀⣀⠈⠀⠚⠿⢋⣤⣬⣭⣥⢔⠲⠮⡙⢿⠂⠀⠀⠘⢀⠀⠀⠀⠀⢸⠈⠀⠀⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸⡏⡌⢰⣤⠞⠛⣛⠛⠛⠻⠿⢿⣶⣶⡖⠙⠯⣽⣿⣷⣯⡓⢦⡈⡌⠆⠀⢰⠀⠈⠀⠀⠀⠀⠐⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣷⡿⠁⠠⣤⣭⣽⣿⣶⣦⣤⣤⣶⠀⠀⡠⣦⢙⢟⠆⣻⡆⡇⢠⠀⠀⠂⠰⠀⠙⠀⠈⠀⠀⠀⠈⠀⠈⡀
⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⣿⣇⣠⡞⣽⠿⠷⠮⠉⡻⣮⢹⣿⠀⠀⢹⡿⣾⣡⣾⡿⢋⡀⠀⠀⡈⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⡦
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢇⣳⠙⠿⣿⠾⣣⠆⠰⠦⠀⡀⣽⡆⢧⠀⠀⡎⢣⡬⠋⡀⠐⠉⡡⠂⡴⠁⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠀⠱⢇
⠀⠃⠀⠠⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠚⢯⣀⠀⠀⢀⡴⠋⠀⠀⠐⠈⠚⢿⡲⢦⡀⠹⣠⡤⣎⡄⢁⣤⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⢸
⠀⢦⠀⠀⠃⠄⠀⠀⢸⠀⠀⢃⠀⠀⠀⠀⠀⠉⣉⣀⣀⣀⣒⣤⣶⠎⡠⠄⡉⠀⠘⢦⠙⣿⣣⣄⠃⠀⠀⠸⣯⡀⠀⠀⠀⠘⠂⢀⠀⢀⠁⠃
⠀⠘⠀⠀⠸⠑⠀⠀⠀⡂⢁⠘⡀⠀⢁⡇⢨⣇⡀⠈⣽⢋⡔⢿⠉⢰⢴⡃⠀⠄⠂⡈⢳⡘⠇⠹⣄⠐⠀⠀⠈⠉⠒⠆⠀⠀⣤⢼⠀⠀⠀⠈
⠀⠀⠂⠀⠀⠂⠀⠀⠀⠰⠀⠀⢇⠀⢢⠁⢄⡹⢋⠔⣡⢛⣀⣉⣢⣄⠷⢌⢢⡀⡀⠙⠈⣿⡞⣦⣍⢉⣙⠲⢦⣄⠀⠀⠡⠀⢉⠈⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢧⠸⡄⢸⡇⠎⣴⣁⣤⠴⣯⢌⠭⣭⡉⢷⣿⠀⣿⣺⣮⠀⡥⡿⠄⢀⢷⠚⣿⢦⣿⣆⠀⠐⢆⠈⠀⠀⠀⠀⠀⠂
⢠⢠⠠⡀⠀⠰⠀⠀⠀⠀⠰⠘⡀⢻⠁⡄⠢⣭⠁⠈⠞⠑⣦⠚⠉⢸⣎⣿⠀⣛⣧⠠⢥⡇⠙⠂⠠⣳⣾⣿⣾⠻⣿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⢀
⠸⠆⡆⢇⠁⠀⠃⠀⠀⠀⠄⢄⠁⠈⣦⣄⠃⠀⠀⢀⢼⣆⠀⡟⠳⡌⢷⡸⡄⠻⢮⡓⢻⢟⣁⡤⠚⠁⠀⠈⢻⣷⣽⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠃⢠⢰⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠈⠙⠓⢶⠖⣡⡄⠁⣼⠇⠀⠈⠀⠉⠂⠰⠶⠾⡛⠛⠉⣀⣤⣤⡀⠀⠀⡇⣸⠇⠀⢠⠀⠀⠀⢦⠀⠀⠁⠀⠲
⠠⠰⠀⠀⡇⠀⢠⡓⢤⡀⠠⠀⠀⠀⠀⠈⠱⠌⢫⣰⢧⣄⣿⣧⣠⠀⢀⡀⢀⡴⢂⢤⣭⣭⣉⠉⠙⣻⣿⣆⣀⣧⣿⠀⠀⢸⠀⠀⠆⠠
⠀⠀⠇⡀⢸⡈⠢⡁⠦⠉⣦⢁⠀⠀⠀⡇⢰⣾⠋⠉⠛⠠⡟⣸⣿⢧⣮⣗⣋⣐⣶⠶⠿⣿⡏⠻⢿⣯⣠⣭⣞⡿⣵⠀⠀⠸⣄⠀⠀⠐⡃
⠀⢠⠰⠸⠀⢳⠀⠃⠀⠀⠏⠀⠀⠀⠀⠸⣿⣌⣩⣭⣵⠀⠘⢻⣟⢠⣄⣻⣿⡷⢭⣶⡺⠛⣧⢠⣄⡀⠙⠷⠿⠟⡁⠀⠀⠀⠉⣦⠈⠁⠰
⠀⠈⡈⠃⠃⢸⠅⠀⠀⠀⠀⠀⢀⡀⣀⡀⠨⢉⡛⢳⣿⠃⡻⡾⣧⣄⠘⢿⡋⢭⡛⠻⣿⠈⡄⡀⠀⠸⡀⠀⠀⠂⠠⠀⠀⠀⠀⢸⠀⢠⠀⡀⠀⠈
⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⢀⡞⡉⠛⡵⣿⣷⣮⠶⠿⢑⣘⠈⠀⠋⣻⣻⣿⡇⠈⠃⣠⡟⠳⠿⠧⢦⡖⣇⡀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠠⠄⠁
⠀⠀⠐⠀⡀⠀⠆⠀⠀⠀⢸⠁⣱⠇⣘⠛⢓⡲⢂⡞⠋⣉⣀⣀⠽⢟⣽⣿⣄⠀⠀⠉⠁⠀⣀⣴⠞⠀⣨⡿⠀⠈⠀⠀⠀⠀⠀⠀⢳⠀⠲⠘
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡄⠘⣼⣟⠻⣿⣄⣘⣛⣋⣁⣤⣶⣛⣩⣿⣯⣿⡆⠈⢀⣄⣠⣤⣴⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠆
⠀⠀⠀⢡⠀⠐⠀⠀⠀⠀⠀⠀⠳⣿⡪⠙⡋⠛⠛⠛⢛⣛⣭⣭⠭⠭⠭⠤⡹⣷⣤⣀⠉⠛⠛⠿⡶⠴⠖⠚⠻⣦⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢄
⠀⠀⠀⠈⠀⡀⠀⢀⠀⠀⠀⠀⠀⠙⢯⡦⡈⠉⠳⣦⣄⠉⠉⢿⣿⣿⣷⠀⠐⣌⡙⠛⠻⠿⣿⣿⣿⣿⣿⡀⠀⣛⠀⠀⠀⠀⠀⠠⠀⠸⠄⠀⠘⡆⠳
⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠡⠷⡄⠠⠆⢻⠿⠇⠀⠺⣾⡿⠏⠀⠀⠘⠻⠷⢦⠀⠀⠹⠏⠻⠸⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢹⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⠿⠿⠛⠿⣿⣿⣿⠋⠉⢹⣿⣿⡿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣿⡿⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠀⠀⢹⣿⣿⣿⣿
⠿⠛⠛⠛⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⡇⠀⠀⠈⠛⠟⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠐⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⣿⠏⠁⠀⠀⠈⠙⣿⡿⠁
⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠁⠀⠈⣿⣿⣿⣿⠛⠉⠙⠻⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⣹⣿⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿
⠀⠀⢠⣿⣿⡇⠀⢰⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠹⠿⠟
⠀⠀⣸⣿⣿⡇⠀⠘⠻⠟⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠘⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠠⠀⠀⠀⠘⡶⣄⠍⠁⠀⠀⠒⣄⠨⡉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠠⡀⡀⠰⠋⣠⡄⢠⠀⠀⠀⠀⠋⠈⠁⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡗⢡⠈⣈⠏⠁⠀⠁⡂⠀⠎⠀⠀⢠⠄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡆⠀⠀⠉⠉⢈⡅⢀⠄⡠⠀⠀⠃⠀⢀⠔⡀⢀⡾⢦⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣆⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⡀⢤⡐⠆⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠙⠠⣀⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⣠⠀⠀⠈⠙⠒⠒⠲⡀⠀⠀⠀⠀⠀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⡓⡠⡠⠒⠀⢠⠪⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠐⢹⠀⠀⠀⠀⢀⡎⠔⠀⡤⢤⢅⠤⣤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠠⠐⠀⠀⡀⠀⠀⠠⠁⠀⠀⠀⣃⣒⠐⢂⠃⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠐⠀⠀⠂⡀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣘⢻⡯⢽⣋⠁⠉⠉⠐⣶⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⣩⠟⡻⣼⠯⣋⢍⣇⠀⠀⣀⡼⣠⢔⡫⢍⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠸⡔⡹⠋⠱⡮⢗⡩⣿⣠⡾⢕⡫⢖⡩⢒⡵⢚⡠⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠉⠁⠠⣖⣛⢝⡪⢕⡭⢒⡡⢖⡭⠚⣁⡴⠉⢙⣏⡔⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⡩⢖⡭⢚⡥⢒⡥⠚⣡⣴⡾⡝⢇⣠⣄⠉⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢷⠀⠀⠀⠀⠀⠀⠀⣞⡩⢚⠥⣊⠵⢚⡡⢖⡩⢚⣾⡷⢏⡨⠵⠋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣻⡆⠀⠀⣤⡠⡞⢧⡽⢊⡥⢚⡡⢖⡩⢒⣩⡶⢛⡡⡞⠽⢀⣀⢀⠤⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⣛⠶⡄⠙⢻⢖⡡⠒⡡⠖⡩⣔⢥⡪⢙⡡⢖⠕⢺⡁⢰⡡⢊⠧⣋⣿⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢊⠵⣊⠼⢦⡈⠛⠐⠋⢴⡭⡺⠗⡩⢖⡩⢚⡡⡺⠟⣩⠟⠔⠥⠞⠉⠁⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⢚⡡⠖⣹⠁⠀⠀⠀⠀⠐⠲⢯⣖⡡⢒⡥⢊⠔⣩⡎⠀⠀⠀⠀⠀⠐⣦⣴⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣩⠔⣋⠽⢣⣄⠀⠀⠀⠀⠀⢰⣣⠔⡡⢚⡵⣪⢞⢧⣀⣀⠀⠀⠀⠀⢸⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣊⡡⢊⡡⠚⡩⢷⡀⠀⢀⣏⠔⣪⢔⡵⣪⠖⡡⠊⣁⠜⣧⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣉⠔⣩⠔⣉⣿⢰⢋⡥⢊⠕⡩⢚⡵⠋⡠⠊⡡⢊⡹⣤⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠥⣊⡡⢚⣼⣯⠞⡡⢊⠔⢁⡴⢊⡴⢊⡰⢫⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠏⢼⣁⣴⣋⣴⡋⠴⠟⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⣀⢀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠐⠉⠐⠉⠒⠑⠋⠘⠁⠊⠐⠁⠚⠁⠚⠁⠚⠁⠚⠁⠈⠀⠉⠀⠁

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⢀⣾⠟⠿⠿⠀⠀⠘⠟⣁⡀⢠⠀⠘⠛⣩⣿⣷⡀⠰⠋⠉⢩⣿⡄⠀⠀⡿⡿⠟⠂⠀⠀⠟⠉⣃
⠀⠀⠈⠿⠀⠀⠀⡤⠀⠀⠀⢻⡿⠀⠀⠄⠰⢷⣿⠋⠀⠀⣿⠆⠈⢹⠇⠀⠀⢁⣄⠀⠀⠸⣿⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠈⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣠⣄⣄⣶⢠⣾⣴⡾⣿⣆⠀⣤⣶⠖⠀⠀⠀⣠⣴⣷⡖⠀⠀⢠⣿⢁⢀⣠⣄⢰⣿⣤⣶⣽⣆
⠀⠀⠀⣐⣿⣿⣿⡟⠘⣿⣿⡄⣨⡿⠀⢻⣏⡈⠁⣶⠃⢻⡈⣹⣷⣆⠀⠸⣿⡿⠻⣿⡟⠀⠁⠉⠁⢸⡟
⠀⠀⠀⠈⢙⣛⠋⠁⠀⠈⢛⣛⠋⠀⠀⠈⠙⣓⠊⠁⠀⠀⠉⣛⡛⠁⠀⠀⠈⢃⣀⠈⠀⠀⠀⢀⣐⠉
⠀⠀⠀⣴⠟⠿⢿⠀⠀⠐⠟⣋⠀⢠⠀⠐⠛⣫⣿⣷⡀⠰⠟⠉⢩⣷⡄⠀⠀⣿⣿⠟⠆⠀⠀⠿⠛⡃
⠀⠀⠈⡿⠀⠀⠀⡠⠀⠀⠀⢻⡷⠀⠀⠄⢰⣿⣿⠋⠀⠀⣷⠆⠈⢻⠇⠀⠀⢁⣄⠀⠀⠸⣿⣶⣶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡚⠛⢋⣭⣉⠙⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⡏⠀⠘⡆⠈⢦⡀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢴⡯⠷⣄⠀⣜⣡⠴⠚⠁⠀⠀⠀⠀⡤⠴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠲⠤⣄⣀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠱⣌⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣤⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠙⢿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠙⠦⠤⣄⡀⠀⠀⠀⠀⢸⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⠲⣄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⡜⠁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠙⣄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣀⣀⣠⣤⡤⠶⠚⠉




⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣻⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠢⡤⠤⠤⠒⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⢀⣠⠏⠠⣄⣀⣀⡴⠀⠀⠀⠈⠳⠖⠦⣤⡀
⠀⠀⠀⠀⠀⠀⢻⡀⢀⠀⢀⡼⢀⣤⡤⠞⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠃⠀⢉⡽⠁⠁⠀⠀⢀⠀⠀⠑⢤⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⠀⢸⡗⠶⠶⠋⠈⠒⠒⠓⠢⠖⠃⠀⠑⠲⢦⣨⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠓⠲⠂⠾⠯⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠴⠖⠊⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠒⠒⠒⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⢀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠢⠤⠤⠤⣤⡀⠈⠑⠊⠉⠁⠀⠱⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⠳⢤⠀⠀⠀⠀⠀⠀⠈⠓⠶⡶⠄⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠸⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⡣
⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠉⠉⠓⣦⠀⠀⠀⠀⢀⣀⣤⡴⠖⠒⠒⠲⠶⠶⠒⠋⠁
⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⠤⠤⠤⠤⠔⠒⠒⠤⣄⡀
⠀⢀⣤⠞⠋⠘⢦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⢀⣀⣀⣀⣀⣠⡤⠴⠶⠤⠤⢤⣄⣀⡀⠀⠀⢙⡆
⣰⣯⣤⣤⣤⣤⣤⣥⣶⣶⣶⣦⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣤⣄⣀⣀⣀⡀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠈⠛⣿⠛⠻⣿⡿⠋⠀⠈⢻⣿⡟⣻⠟⠛⠁⠀⠀⠙⢻⡟⠛⢿⠏⠙⠟⠉⢿⡿⠻⣿⠿⢿⣿⡿⣿⣿⣿⠋⠻⣿⣿⠿⣿⠛⡿⠛⢿⡿⠙⣿⠿⠟⠋
⠀⠀⠀⠀⠛⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠟⠀⠙⠋⠀⠀⠀⡿⠁⠀⣿⠀⠁⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠙⣿⠟⠻⣿⡿⠿⣿⣿⡿⠃⠀⠀⠘⢿⣿⡿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠻⠏⠀⠙⠋⠀⠘⣿⠏⠙⢿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⠀⢸⡏⠀⠙⢿⡇⠀⠀⠀⠀⠀⠉⠀⢹⡏⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠉⠁⠈⢻⡿⠋⠛⠿⠟⠛⣿⣿⠟⠻⠿⠟⠛⢻⣿⠟⠻⠿⠋⠁⠉⠉⠀⠀⠀⠀⠉⠉⠈⠹⡿⠛⣿⠟⢻⡏⠁⠉⠉⠀⠀⢹⣿⠋⠙⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠀⣾⠟⠉⠑⠂⠀⠠⣴⣿⣷⣦⣤⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣽⣿⡇⠀⠀⠀⠀⠀⠐⢻⣶⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠈⠁⠀⠀⠀⠀⠀⢰⣾⣟⣛⡿⢿⣯⣽⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣴⣯⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⠟⠋⣽⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⡀
⠀⠀⠀⠀⠸⠺⣇⣈⣭⣩⡇⠀⠀⠀⠀⠀⠈⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠹⡟⠃
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣷⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠾⠿⠿⠿⠿⠿⠟⠛⠋⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠂⠀⠀⠂⠀⠂⠊⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⡀
⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠁⠒⠀⠀⠘⠂⠤⢠⡀⠀⠀⠈⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⡀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠈⠈⠉⠉⠉⠈⠀⠀⠀⠈⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⣀⣀⣠⣤⣤⣤⣄⣈⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢶⡾⠿⠟⢛⡋⠉⠉⣉⣷⣍⠛⢛⣛⣿⣿⣶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠘⠀⠀⠡⠷⣾⣏⠁⠀⠀⠘⣿⣿⠿⢷⣿⣿⡟⣶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⣀⣰⠆⢹⡿⠀⠀⠀⢰⣿⠿⣾⣿⡿⠛⠿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⠋⠀⠀⠀⠀⢰⣾⡿⠋⣧⣀⣿⡇⢀⣠⣴⣟⣡⠛⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⠋⠀⠀⠀⢀⡄⠀⠀⠟⠁⠀⠛⣿⣿⣿⠛⠿⠿⡟⠹⠧⣿⣧⢀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠸⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢤⡤⠾⣿⣧⣤⡄⢀⣀⣀⡴⢉⢻⣿⠇⠛⠁⠀⡴⠀⢀⠀⠀⢧⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠏⣠⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣠⣄⣫⣴⡶⠿⠇⠈⠁⠈⢻⣿⣷⣾⡔⠁⡀⠀⠀⠈⠁⠐⣯⡾⠆⠀⠹⠄
⠀⠀⠀⠀⠀⢀⠀⡄⣰⣿⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⠉⠀⠀⠀⠀⠐⣀⠀⠀⠙⣿⣿⠿⠀⢀⣴⣴⡮⣄⠀⣀⢀⣠⠀⠀⢀⣠⣴⡔
⠀⠀⠀⠀⠀⠘⠀⢀⣿⣿⡆⠀⣴⣶⠶⣄⢰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣇⡀⢀⣿⡷⠿⢷⣾⣷⣿⣿⠿⠂⠀⠀⠙⠙⣷
⠀⠀⠀⠀⠀⠀⠀⢼⡧⣿⡇⢰⡿⢻⡷⠿⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⣩⣿⣿⠋⠁⠀⠀⠀⠘⠛⠿⠀⠀⠀⠙⢦⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣦⡁⠼⣿⣿⡎⠉⠻⣿⢓⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠆⠈⠀⢻⠛⣿⣿⠀⣿⣨⣿⣿⣶⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢸⡯
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣀⠹⢿⣷⣤⡉⠀⠉⠹⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⣸⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠉⠁⣌⣷⠏⠀⣠⠶⠄⠰⠈⠙⠻⣄⠀⠀⠀⠀⢠⡟⠻⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣶⣾⠏⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣎⠦⡀⠀⠘⣇⠰⠉⠀⠀⠀⠀⠀⠀⠀⠻⢄⡀⠀⠀⠀⠀⠀⠻⠿⣠⣀⣀⣤⣤⣶⣦⡶⢼⠿⠟⠁⠀⠂⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠈⠳⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠙⢯⣿⡿⠀⠀⠀⠉⠉⠙⠿⣿⣿⣿⣯⡀⣴⠂⠀⠀⡀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠙⠻⣦⡀⠠⣄⣀⠀⠀⠈⠀⠀⠀⠀⠀⣈⣿⣶⡦⠀⠀⠀⠀⠀⠾⠛⣿⣿⡇⠷⣆⠀⢀⣽⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⣘⠀⠀⠀⠈⣿⠀⣿⠛⢿⠂⡀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠚⠁⠀⠀⠀⣆⣹⡏⠀⠀⠉⣠⣞⡟⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠲⣌⡕⡇⠀⠀⠻⡋⠀⣤⣤⡶⠿⠿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠀⢀⡴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡭⣷⡔⠀⠀⠀⢠⣴⣿⡿⠏⠀⠀⣀⠀⢠⠤⠒⡒⠉⠉⠁⣩⣿⣧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⠳⣼⣿⣦⡀⢾⣿⣿⣿⣷⣶⣼⣭⣤⣤⣤⣤⣬⣤⣀⣶⣶⣿⡉⠁⠋⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠾⣿⣿⡿⠟⠛⠋⠉⠀⠉⠀⢈⣀⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⠿⠿⠻⠿⠿⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠻⣅⣀⣀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⢳⡀⠈⠉⠉⠁⠀⠀⠀⠙⢧⡀⠀⢀⣤⠶⠶⢤⣀⠉⠻⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣾⠁⠀⠀⠀⠉⠙⠛⠉⠙⠻⣦⡀
⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠚⠻⣷⡀
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠹⣷⡀
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠹⣧
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⢻⣇
⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⣿
⣿⠃⠙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠈⠳⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣇⡴⠶⢦⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠳⣦⣠⡾⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣇⠀⢠⡴⠟⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠈⣿⡀⠹⡄⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠈⠓⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠹⣧⠴⠃⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠻⣧⡀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⡉⠛⣧⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠹⣷⡀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⢷⡀⣸⠇⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⡼⠃⢿⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠘⢷⣤⣀⡀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⡉⠉⠛⠛⢦⡌⠳⠶⠶⠶⣦⣄⣠⡤⢤⣤⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠶⠶⣿⣶⣤⣤⣤⣴⣶⠶⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⢸⠇⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⢸⡄⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠒⠀⢀⠀⠀⠆⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠀⢀⡟⠳⡄⠀⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⢻⠆⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠒⠦⣄⣀⢰⢷⣃⡤⠖⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡞⠛⠒⠭⣽⢸⡥⠖⠛⠛⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠦⣀⡀⠉⠉⢁⣀⡤⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢸⠀⡦⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢠⡜⠶⠃⠀⠀⠀⠀⠀⢳⡀⣸⠐⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡤⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⠉⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⢋⠆⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⠞⠉⣉⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡁⠀⠀⠀⡰⠃⢀⠞⠁⠘⣎⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠷⣌⣓⠒⠦⠤⢤⣄⣀⣀⣀⣀⡤⠤⠴⠒⣋⡴⠳⠤⠚⠉⠳⢠⠏⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⠄⠀⠀⣿⢭⠙⠛⠒⠂⠀⠆⠀⠀⠀⠀⠚⠛⢉⣹⡇⠀⠀⠸⣶⡀⣾⠀⣀⣀⠼⣡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⢮⣀⠀⠰⣍⡛⣒⠒⠆⠰⠀⠠⠤⠤⠄⠀⠀⠴⠖⢚⡽⠀⠀⣚⡵⢃⡼⠀⠓⢦⣸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣉⡓⠒⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⡤⠬⠥⠶⠒⢚⣉⡥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡗⠒⠒⠒⠶⠤⠤⠤⠤⠤⠶⠒⠒⢲⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠭⢭⣀⣀⣀⣀⣀⣀⣀⣘⣀⣛⣉⡭⠽⠃

⣿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢤⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠳⡖⠒⠳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣮⣀⣠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⠦⡐⠚⢆⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣽⡻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠈⢦⠤⠳⡄⠐⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⣟⣿⣿⣿⣿⣧⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠥⠼⠦⠤⢶⡀⢓⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⠖⠒⠒⠒⠒⠤⣈⠛⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠿⣿⣿⣿⣟⣯⣻⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠦⠀⠀⠀⠀⠀⠀⠈⠳⡌⡳
⠀⠀⠀⠀⠀⣠⣶⠾⢿⣟⣿⣿⠿⣿⣿⣿⣿⡿⠿⣿⡿⣟⣺⣿⠿⣟⣻⠿⠶⠀⠀⠀⠀⣀⡤⢤⠶⠊⠀⠀⠀⠀⠒⡀⡄⠀⣤⠆⠸⣷⡂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⢈⣽⣿⣟⡸⠿⠷⠶⠾⢽⣿⣷⣾⡿⠿⠛⣿⣿⣿⠿⠿⠆⠀⠀⠀⠀⠺⡁⠀⠙⠦⠄⣀⣀⣀⠀⠀⠙⣁⣀⣀⣀⣤⠴⣟⣠⣤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣾⣿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⢳⡀⠀⠀⠀⢀⣀⣀⣀⣠⠵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⠛⠉⣴⡿⠋⠀⣿⡇⠈⠻⣷⡌⠙⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣄⣀⣾⡟⠁⠀⠀⣿⡇⠀⠀⠘⣿⣆⣀⣬⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠉⠛⣿⡿⢷⣶⣶⣶⣿⣷⣶⣶⣶⠿⣿⡟⠋⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⡿⠿⣿⣿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⢿⣿⡿⠿⢿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⣀⣤⣿⣷⡾⠿⠿⠿⣿⡿⠿⠿⠿⣷⣿⣧⣄⡀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠋⠉⢿⣧⡀⠀⠀⣿⡇⠀⠀⢠⣿⠏⠉⢛⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠻⣷⣄⠀⣿⡇⢀⣴⡿⢃⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁






⡿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠚⢁⣤⣾⠗⠚⠛⠛⠻⢿⣿⣿⣯⣙⢟⣻⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣶⠿⠠⠾⠟⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⠶⣬⠰⡄⠀⠻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⣡⡾⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⡾⠏⢷⣄⠀⠉⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠡⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠟⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡎⠳⡀
⠀⠀⠀⠀⠀⢠⡿⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠠⣤⡼⠟⠋⠀⢠⡙⡄
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠔⢺⠐⠖⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣦⢠⠀⠈⠈⢻⡄
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⣷
⠀⠀⠀⣸⣿⢡⡀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⣠⠆⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⢠⣿⢻⣇
⠀⠀⢀⣿⡏⠘⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢀⡀⠈⢠⣶⡀⠀⠀⠀⣿⣿⣷⣤⣀⣼⣿⣇⣿
⠀⠀⢸⡿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠋⢀⡤⣬⠓⠲⣆⠀⣶⡿⢷⡄⠠⠙⣷⡤⠀⠀⠈⠿⣟⠙⠛⠛⣿⣿⢿
⠀⠀⢸⣇⠠⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣥⢤⣙⢻⣆⢤⡄⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣿⢀⠀⢰⣿⣿⣾⡆
⠀⠀⢸⣿⡰⣷⡀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠾⣯⣽⣿⣾⣿⣶⣿⡇⠀⠀⠀⠰⣦⣤⡦⣴⣿⣿⣮⠀⢸⣿⣿⣿
⠀⠀⠸⣿⣧⣹⣷⣄⣹⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠉⠉⠉⠁⠠⣤⡀⠀⠀⠀⠉⠀⢸⣿⣿⣿⣧⡻⣻⣿⣿
⠀⠀⠀⣿⣿⣝⠿⠿⢿⣦⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠘
⠀⠀⠀⢸⣿⣿⡧⡖⠊⠙⠳⠨⣿⣄⡀⠀⠀⠀⢀⣀⣠⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠷⣴⠟⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⢱⠆⡀⠀⣠⠈⠻⣿⣿⣶⣤⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡟⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⡿⣷⣿⣻⠛⠆⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣧⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡻⠤⠓⠀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⠛⠺⠗⠒⠲⠈⠻⢿⣿⣿⣿⣿⣿⣷⡱⢶⣦⣀⡢⠤⣤⣀⣠⠠⠀⢀⣠⠦⠶⣿⣿⡁⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⡏⠋⢰⡟⠁⠀⠀⢻⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠿⠟⠀⠀⠀⠀⡜⢷⡴⣿⠟⣱⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡘⣿⢷⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠁⣼⣫⣿⣯⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡻⡿⠂⠈⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⣠⢞⣷⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣀⣼⣷⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⠿⠟⠛⠛⠻⠿⡿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣶⣷⡄⣀⡀⣠⡄⢠⣆⣿⣿⣿⣿⣿⣿⣧⣤⢀⡄⠀⠀⠀⠀⣠⣀⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⣀⠀⣶⠀⣠⡀⢰⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⡆⢴⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢸⣿⣤⣿⢀⣿⣿⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⠀⣿⣿⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢀⣴⣻⣶⡀⠀⠀⠀⠀⠀⣮⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠐⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⠀⠀⠀⠀⠀⣼⣿⣿⣫⣭⠀⠀⠀⠀⠀⣠⣤⣾⣿⡯
⠀⠀⠭⣚⡫⠭⣙⣓⠅⠀⠀⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⣀⡀⠀⠀⠘⣧⣿⡿⠋
⠀⠀⠉⠒⠂⠉⠒⠒⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⢄
⠀⠀⠀⠈⢙⣚⡚⠃⠀⠀⠀⠀⠀⢢⣶⣯⣿⣇⠀⠀⠀⠀⠀⡷⡇⡔⡇⡀⡀⡀⠀⠀⠀⡴⠚⠉⠁⡴⠀⢳
⠀⠀⠀⠰⠿⠿⠟⠀⠀⠀⠀⠀⠀⢹⣛⣻⣛⣻⠁⠀⠀⠀⠀⣉⠏⣇⣯⡯⡏⡁⠀⠀⠀⠳⠦⠤⠾⡿⠳⠚
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠌⠉⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠈⠁⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣌⡄⠀⠀⠀⠀⠀⢠⣿⣿⣀⣄⣀⣠⣤
⠀⠀⠀⠀⠀⢴⣿⠓⠀⠀⠀⠀⠀⠠⣿⣿⡿⢃⠀⠀⠀⠀⠀⣿⣿⣿⣛⡎⠀⠀⠀⠀⠈⠙⣹⣿⣯⡝⠛⠋
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠘⠛⢿⠾⠧⠂⠀⠀⠀⠀⠛⠛⠟⠛⠋⠻⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⢀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣰⠟⢿⠿⠛⠃⠀⠀⠀⠀⣀⡤⠤⣽⡟⠀⠀⠀⠀⠀⢸⡥⠀⠀⢰⡷⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣄
⠀⠀⢸⣿⣇⣿⡆⠀⠀⠀⠀⠀⢰⡟⢀⡀⢼⣷⠀⠀⠀⠀⠀⣼⣧⣠⣴⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⡿
⠀⠀⣾⣿⣿⠿⠇⠀⠀⠀⠀⠀⠈⠆⡘⢻⡧⠃⠀⠀⠀⠀⠈⠻⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠐⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠟⠀⣀⣀⣀⣀⣀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⣠⡴⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠐⠷⣤⡈⠙⠳⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⣠⠖⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣍⠁⠀⠀⣠⠀⠀⠀⣠⡟⢀⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣇⠀⠰⠟⢠⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣠⡶⣠⡾⠃⠀⠀⢸⡆⠀⠀⣼⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⢀⡼⠋⠘⠋⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠳⢦
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⢹⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠻⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁
⠀⠀⠀⠀⢀⣠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⠀⣀⣀⣤⠶⠞⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⢿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠉⠉
⠈⠙⠛⠒⠚⠛⠛⠋⠉⠉⠉⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄⣀⣠⣤⠴⠶⠛⠋⠉⠉⠛⠛⠛⠛⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠥⠁⠀⡀⠀⣀⠀⢀⣠⣴⡀⠤⠀⠀⠀⠉⣒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠄⠚⠐⠂⠐⣻⠼⣂⡄⠀⠀⠀⠈⠉⡆⡠⢬⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢄⢱⠈⠠⡌⠀⠁⢘⡾⠁⣥⡄⠱⡀⠀⠀⠀⠀⠎⠑⠙⠁⠀⢂⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠁⠀⢸⡰⠁⠒⠒⢞⠽⠓⠀⡓⢤⠀⠀⡰⠀⠀⠁⠀⢀⠢⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢆⠀⠀⠀⠹⢤⡼⠠⡲⡈⠈⠀⠀⠀⠐⢄⡔⠅⠀⠐⣠⠙⠁⢀⡖⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡸⠁⠀⠀⠀⠀⠐⠀⢗⠀⡰⠊⠤⠂⠀⠀⠈⠠⢀⠔⠁⠌⣠⠟⠙⠷⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⣇⠀⠀⠀⠀⠐⠁⠞⠀⠔⠐⠀⠁⠀⠀⠔⠀⠀⢊⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⢸⡀⢠⡐⠰⡤⠚⠅⠠⡄⠢⠂⠢⡢⠀⠊⠀⠑⠁⢀⣰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⢬⠙⢆⢐⠁⠊⠐⠀⢂⡀⢀⠔⠠⠀⠂⠀⡁⠀⠀⠁⠘⠢⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠄⢂⠀⠈⠓⠢⠦⡄⣈⣐⡁⠰⠀⠄⠀⠀⠀⡠⠀⠨⠊⡐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⢐⠀⣢⢣⠀⠀⠁⠀⠈⠀⠀⠉⢫⡀⠀⠀⠀⡊⠀⡀⢐⡎⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⢨⠂⡔⡥⢚⠀⠀⢠⢋⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠀⠐⢹⠀⠂⠀⠀⠀⠀⡔⣣⠬⠄⠈⠀⡁⠁⠁⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⡀⢁⠠⡀⠈⠀⠆⠀⠀⠀⢠⣚⠉⠀⠠⡐⠄⡈⡔⡡⠁⡆⡂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁⠀⠃⠈⠁⠁⢐⢉⠂⡄⠠⢀⡈⠀⠀⠀⠀⠂⠈⠪⠊⠀⠠⠀⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⡅⠀⠀⠄⠋⠀⢀⠇⠀⠁⣄⠆⠀⠂⣶⠉⠀⡆⠀⠀⠀⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⠀⠨⠐⠀⠀⠀⠀⠰⠀⠀⢠⠀⠅⠀⠀⠀⠀⡀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠀⠀⠁⠄⡆⠀⠀⢀⡂⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡫⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠊⠑⠀⠀⠉⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠐⠁⠈⠀⠈⠀⠀⠀⠈⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣷⣉⠛⠻⠿⠿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣟⣿⣟⣻⣿⣿⣿⠿⠿⠿⠛⣿⣶⠙⣿⡗⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⣋⡿⢿⣿⣿⣿⣿⣄⡀⢀⣤⣜⠿⠟⠁⠙⠻⣿⣿⠓⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣭⣴⣾⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣷⣶⣀⠀⠀⠀⠁⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⣿⣿⣿⣿⣿⣿⠟⠙⠋⢹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡋⢀⡀⠀⠀⠱⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⣆⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠙⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠯⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢀⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢶⣶⣦⣤⣤⣿⣹⣤⣤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠚⠉⠉⠉⠉⠉⠉⢑⣲⠋⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⠠⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠁⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠃⠤⣄⠀⠀⣠⣞⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢏⠉⠏⠙⠛⠲⣀⠀⠀⠀⠀⠀⠀⣷⢦⡘⡄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣴⣖⠲⠖⠒⡶⢺⣷⡄⢧⡂⢀⡠⣴⣭⡱⣦⠀⠀⠀⣸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⡅⢀⠼⠁⠚⠒⢄⠈⢔⣒⠟⠁⠈⠉⠢⡝⠉⣼⢓⣚⣦⠀⣄⣿⣴⣤⣤⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡤⣄⣳⠀⢨⣗⡁⡀⢄⣤⣮⣴⡞⠀⠀⢿⣾⠻⣍⠓⠛⠛⠋⣼⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡅⣼⣿⠀⠄⢿⣿⠤⣀⡘⣿⠟⠁⠰⠀⢸⠛⢋⡽⡇⠀⠀⣠⠿⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠄⢍⢣⡘⣿⣤⣤⣦⣿⣿⣯⠁⡙⢧⣢⢔⠀⡎⢰⡿⣄⣧⢀⣴⠋⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡄⠙⠙⢛⠁⡄⣷⣹⣧⠈⠈⠉⠙⠾⠲⠋⠀⠈⢟⣛⠁⠀⡆⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢰⣷⠛⠃⠘⠿⠹⠟⠀⠰⠿⠷⢶⣦⣵⣆⣜⡢⠞⢰⣆⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⡄⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣹⢻⣿⣷⣶⣤⡟⢹⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡃⠸⣧⣁⣤⣨⣰⣅⡆⢡⢢⢠⣰⣇⣾⣿⣿⠏⡙⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⣿⢻⡿⠿⡿⠛⢻⣿⣼⣿⢹⢿⠇⣧⠾⠁⢇⡇⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡇⠀⡇⠀⢸⠹⢿⡟⠘⢘⠘⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡇⠀⢿⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣀⣀⠀⣤⡀⢤⣤⡖⢦⠤⠠⣤⣤⣄⢀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠈⢏⠦⡸⠒⠾⠓⠓⠈⠧⡋⡝⢦⠉⡌⣤⠉⠳⡄⢩⢌⠏⠑⠣⠁⢒⠊⠡⢚⡁⠠⠀⡌⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⠒⠁⠀⠀⠉⠀⠀⠁⠀⠈⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠂⠃⠒⠒⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣠⣴⣶⣶⣷⣿⣶⠷⠶⠄⠤⢄⡀⠀⠀⠀⠀⠊⡀⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠄⣰⡦⣄⣀⢀⡠⢶⠿⠿⠛⠛⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⢀⡐⣀⣄⢸⠂⠀⠃
⠀⠀⠀⠀⠀⠀⣔⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣠⣄⠀⢠⣄⠠⠶⢄⠠⠶⡄⠀⠀⠀⠀⠀⠁⠨⢡⢁⢁⡆⠀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⡋⠀⠈⠁⠀⠀⢠⡤⠤⡄⠀⠰⠆⠀⠀⠀⠀⠀⠐⠄⢀⡰⣆⠀⠉⠀⠀⠀⠀⡠⠄⢀
⠀⠀⠀⠀⠀⣿⣿⡟⠋⣿⡿⠿⠿⡋⠉⠀⠁⠀⢀⡂⠔⠀⠀⠀⠀⠒⢤⡀⠑⣤⠛⠑⠀⢀⡀⠀⠀⠀⠀⠁⠀⠱⠀⠀⠀⢀⠜⠁⣐⠞
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠁⠀⠀⡐⠀⢀⠀⠀⠀⠁⠀⠀⢠⣔⠒⢤⠁⠠⠸⠲⣾⣆⢸⠀⢸⡇⢀⣾⠳⢴⡿⢧⣔⠀⠀⢀⠊⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⡇⠀⢀⡈⠀⣠⢈⡉⠱⡄⠈⠸⣾⡇⠀⢸⠀⠙⠁⠈⠂⠋⠀⠉⠀⠀⠀⣷⡞⣿⠀⡠⠉⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠃⠀⠀⡁⡐⠀⠀⣀⣈⠉⠻⡔⠻⠀⢀⣟⠀⢀⣔⣀⠀⠈⠃⠀⠀⠔⡉⠉⠐⡀⠘⣿⣿⢠⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠐⡀⠛⠀⠀⠂⠐⢠⡤⡹⠼⢏⣷⢶⠁⠈⠰⠊⠁⠈⠁⢀⡘⠃⣤⡀⡆⠘⠀⠧⣀⠄⡀⣠⣿⡟⠉⠀⠀⠨⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⠢⣂⣩⣶⠹⢫⠤⠞⠀⠀⠀⡐⢩⣤⣄⡀⡗⠀⢨⡛⣧⡄⢀⠀⠐⢪⡄⣹⠏⠀⠀⠀⠀⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⡽⣟⣴⣶⡆⣧⠀⡀⠄⠀⠀⢈⠀⠀⠙⢻⣤⣚⢋⣹⣥⣾⣤⣯⣾⠗⠉⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢕⠀⠀⠀⠀⣟⣉⣥⡶⣛⣾⣥⢦⣦⣄⣇⠀⠀⠀⠀⠉⠙⠛⠛⠻⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢻⣿⣯⣿⠿⢿⣮⠽⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠡⢴⡾⣿⣾⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠟⠿⡿⣿⣽⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠙⠛⠉⠛⠆⣀⠀⠀⠀⠀⠠⣤⢠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⢤⣠⣤⣄⣀⣄⡀⣤⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⢛⣟⣻⣟⡻⢿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⣴⠜⠀⢄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠶⢼⣿⣄⠙⠻⠓⣾⣿⣷⣄⡀
⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣿⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⡿⠶⣄⠀⠱⠟⠈⠕⣤
⠀⠀⠀⠐⣉⣙⣿⡿⣿⣿⣯⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣿⡿⠛⠃⠀⠀⠀⠀⠈⢐⣈⣷⡀
⠀⠀⠀⠀⠙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⣭⢿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠸⣿⣿⣿⣿⣿⠿⠻⠉⢻⡁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣌⠈⣷
⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠹⢻⣯⣿⣿⠀⢠⡄⠐⠲⢤⢀⣀⠀⠀⠀⠀⠀⢠⣄⢠⣗⣿⡇
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢉⡛⠛⠻⣿⣤⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⢻⣿⣷
⠀⠀⢰⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣴⣿⣿⣿⣿⣦⣄⠀⠀⠀⠠⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⠴⣿⣿⡿⠀⠀⠀⣀⡀
⠀⠀⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣶⣾⣯⣀
⠀⠀⠀⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠖⡠⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⢟⠋⠉⠃
⠀⠀⠀⠀⠀⡿⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠦⣀⣦⣴⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠈⠿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣮⣯⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⠽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣯⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠿⢿⣿⠿⠯⠀⠀⠉⠓⠀⠃⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠉⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡀⠀⣠⣴⣶⣶⣄⠀⠀⠀⠀⣰⣿⣿⣶⣄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣰⢡⣾⣿⣿⣿⣿⣿⡆⠀⠀⢄⣿⣿⣿⣿⣿⣷⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⠇⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡏⠀⢿⣿⣿⠿⠿⠛⠋⢀⣿⣿⣿⣿⡌⠙⠻⢿⣿⡿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⡊⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢰⣄⠀⠀⠀⠻⠿⠋⠿⠃⠈⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠠⢤⣶⣤⡀⠛⠀⠀⠰⠆⢤⢦⡐⢠⣄⣷⢰⣶⣶⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠞⠻⠇⠀⣀⠖⡆⣄⣸⣈⣧⣈⣜⡉⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣾⡇⣤⣾⣿⡆⠀⠋⡀⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⡇⣿⣿⣿⣇⢰⡆⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣄⣿⣿⣿⣿⡇⢹⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⡇⠛⢿⣿⣿⠟⡇⠸⠛⠇⠈⠐⠀⠋⠹⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢈⣇⢠⣡⠀⠀⠀⠃⢀⠄⠀⠀⠠⠀⠀⠀⠈⠉⠙⣏⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠋⠀⠚⠛⠊⠃⠀⠀⠀⢤⣈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠺⠿⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣀⣠⣤⣴⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⣄⣀⣀⣀⠀⠀⠀⠀⠀⢸⣻⡇
⠀⢰⣾⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣌⣙⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣷⡆⠀⢸⣿⡇
⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠛⠋⠉⠁⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠶⢢⣤⢤⠴⠒⠦⣠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢖⠿⡿⠉⠁⠀⠀⣠⣀⣶⣤⠆⠁⠀⠀⠈⠈⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⡋⠋⠀⠀⣇⠀⠀⣠⡀⠉⠉⠁⠀⠠⣦⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⢌⣴⠆⠀⣀⠘⡳⣾⢿⣴⣤⢄⣀⣀⠀⣤⣥⠈⢿⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣾⣾⣿⣦⣾⣿⣿⣟⣼⠦⣄⡀⠀⠙⠛⠛⠛⠿⣦⡆⠉⢻⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣣⡶⢈⡁⠀⠁⠀⢀⣀⠀⢈⣧⣀⣠⡴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠻⠿⡿⠙⢩⣿⢻⣿⣿⣴⠏⣁⣤⠶⠟⠛⠙⠋⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡆⠰⣿⠀⣾⣧⡿⣿⣿⡏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢷⠀⠿⠀⠉⣿⠃⣿⢿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡘⠇⠀⠀⠀⠋⠀⠁⣽⠏⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⠠⠒⠉⠙⠒⠛⠁⠀⠹⣦⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠉⠛⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠦
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠘⢯⣄⣀⣙⠲⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⡴⠀⠈⠁
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣦⡀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣮⡇⠀⠀⠀⠢⠠⠒⠐⢄
⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⡴⠓⣆⠀⠀⠀⣀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣀⣀⡈
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠈⠉⠀⠀⠹⡇
⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⡜⢃
⠀⢸⣟⠑⢦⡤⠤⣄⣀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣰⠋⠀⣸
⠀⠀⢿⠀⠀⠀⠀⠈⠉⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢹⡆⠀⡟
⠀⠀⠘⣧⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡄⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢀⡾⠀⡸⠁
⠀⠀⠀⠘⢷⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡾⠁⠀⠀⠀⠀⡰⠋⢀⣴⠃
⠀⠀⠀⠀⠈⢷⡄⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠁⠀⠀⠀⠀⠸⣇⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠛⢦⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡖⠦⣀⠀⡀⠄⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠙⢦⡀⠀⠀⡇⠀⠀⣤⠶⠶⠟⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠮⣻⣷⣤⡇⢀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣷⣻⣿⣿⣷⣟⣟⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠴⠀⠀⠀⠀⡤⢀⠄⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⢄⡀
⠀⢀⠀⠀⠠⠆⠀⠀⠀⠚⠁⠀⠀⡇⠀⠀⠀⠐⠈⠁⠂⠢⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠰⢜⡀⠀⠐⡄⠀⠀⠀⠀⠈⢲⡖⠚
⠀⠈⡁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜⠊⠋⠉⠀⠀⠀⡁⠀⠀⠀⠀⠀⠂⢂⣡⡀⠀⢀⣀⣈⣀
⠀⢶⠱⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠄⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠁⠀⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠀⠀⠐⠉⠀⠉⠁⠀⠀⠘⠀⠄
⠀⠀⠑⠓⠉⠉⠉⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣶⡆⣶⡆⠀⠰⠶⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠖⠉⠉⠁⠀⠀⠀⠐⡰⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⢰⠈⠀⠁⡆
⠀⠀⠀⠙⠲⣦⠄⠁⠀⠀⣀⠀⠈⠉⠀⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠈⠄⠴⠀⠀⠰⠈⠟⠃⠀⢀⠀⠀⠀⢠⣶⣶⣿⠿⠶⠂⠠⡀⠀⠄⠄⠢
⠀⠀⣠⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠀⣂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠂⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠘⠁⠀⠀⠳⠶⠞
⠀⠘⠋⢀⠄⠦⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠁⠀⠀⠀⠠⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠁⡤⠠⠠⠀⠁
⠀⠂⠀⠀⠀⡀⠀⠄⠐⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠠⠐⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡀⡀⠀⢀⢀⡀⠀⠀⠀⠠⠁⠂
⠀⠈⠐⠒⠈⠀⠀⢀⠄⢸⡇⠀⠀⠀⣠⡼⠀⠀⠀⢰⠾⠀⠁⠐⠀⠂⠐⠁⠀⠀⠀⠐⠀⢘⠇⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⠀⠀⠀⠢⢐
⠀⣀⣀⣤⣤⣤⠄⠈⠀⢸⡇⠀⠀⡀⠄⠀⠀⠀⠀⠀⠋⠐⡅⠤⠀⠀⢐⠀⠰⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡄
⠀⢼⣿⣿⣿⣿⡗⠀⠀⣸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣾⣷⠀⢠⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠿⠛⠛⠛⠃⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠈⠻⠟⠀⠀⠙⠃⠀⠀⠀⠨⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠐⠒⠒⠒⠂⠐⠒⠐⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠋⡡⠔⠋⠁⠉⣹⠀⠀⠀⠀⠐⠒⠀⢠⡞⠁⠈⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢟⠆⢀⠔⠉⠐⠀⣀⠠⠊⠀⠀⠀⠀⣀⡀⠀⠀⠀⠧⢄⠢⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⡤⠚⠰⡧⠒⠒⢫⡄⢢⠀⠀⠀⢠⠖⠉⠈⠉⠹⣆⠞⠉⠚⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢠⠤⠼⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠰⣃⠀⣾⠧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠅⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣘⡛⣋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠱⢼⠇⢀⡀⢀⣠⡀⠀⠀⠀⢀⣄⡀⠀⠀⠀⢰⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢀⠴⠊⠉⠁⠀⠑⢆⠀⢠⢪⠒⠚⠀⠀⠀⠘⠢⡀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⣜⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠍⠥⠴⠒⠠⡄⠀⠀⠨⠗⠊⠈⠣⠤⠂⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡀
⠀⠀⢠⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠘⠢⠄⡀⡴⠶⠤⢀⣄⡀⢇⠀⠀⠀⠀⢰⠦⡀⠀⣤⠤⣄⠀⠀⠀⠸⠘⡇
⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣄⢇⠀⠀⠀⠀⠳⠴⠚⢓⠀⠈⠙⣄⠀⡼⠀⡇
⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢳⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠆⠰⡄
⠀⠀⢠⠀⠷⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡑⢆⠀⢀⣠⠞⠁⠀⠀⠀⠀⠑⢂⡇⡆
⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⢄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡛⠥⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠇⠀⠈⢂⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠃⠊⠀⠀⠀⠀⠀⢠⠃⡘
⠀⠀⠀⠸⡄⠀⠀⡧⠀⠀⠀⠀⠀⠐⠤⡙⠒⠂⠀⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠹⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎
⠀⠀⠀⠀⠀⠱⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠤⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡜⠁⡌⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣎⡤⠈⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠈⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠋⠉⠀⠀⠀⣀⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⡞⠙⠛⠛⠻⣟⠓⣆⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡏⠀⢀⡀⠀⠈⠙⡄⢀⡴⠾⠁⠘⢷⠾⠋⠙⢶⣄⠘⢧⣹⠆
⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠁⢠⡁⠤⠚⠛⠤⣀⢀⠀⠈⠓⣶⡄⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⡀
⠀⠀⠀⠀⢀⣴⠋⣀⠉⠉⢳⠏⠀⠰⠤⣄⡀⠀⠀⡄⣰⠖⠋⠀⠀⠀⠀⠰⠀⠀⠀⣈⠉⠉⠙⠳⣄
⠀⠀⢀⡴⠾⢧⣀⠀⠙⣀⠸⢄⡤⠤⣄⠀⠙⡆⠀⣴⢙⡶⠀⠀⠀⠀⠈⠃⢀⡔⢀⠟⠀⣀⣀⠀⠹⣄
⠀⠀⣏⠀⠀⣼⣤⡀⢀⣁⠀⠀⠀⠀⣀⡀⢰⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢈⡉⠥⠚⠁⠀⢸⡆⠀⠘⣧
⠀⢀⣿⠂⠀⠀⠀⡇⣼⠋⠳⡾⠛⠟⠙⢗⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡀⠀⢀⣀⡴⠋⠀⠀⠀⠹⣧
⢰⡏⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠈⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⢻⡄
⣼⠃⣰⠋⠙⠀⢠⡏⢸⠇⠀⢀⡀⠀⠀⠀⠓⠋⠁⠀⠀⠀⠀⡆⡝⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠐⠀⠀⠀⠸⣧⡀
⢻⣴⠋⠀⠀⠀⠈⠓⠋⡀⡆⠀⠳⢤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠷⠁⠀⠀⠀⠀⠀⢨⣷⣾⣆⣠⣤⣄⠀⢱⠀⣏⢻⡄
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠀⢻
⢰⡟⠀⠀⠀⠀⣆⠀⢸⣷⡀⢰⣧⣄⠰⣆⠀⠀⡀⠀⢠⣇⠀⣠⠃⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠘
⠈⣷⠀⢀⡀⢀⣹⢿⣿⣿⡇⢿⣿⣿⣟⣾⣷⣶⣿⣴⡟⠓⠚⠁⠀⣷⣄⣿⠉⠣⠴⠁⠈⠙⣿⣿⣿⣿⠇⠀⠀⠠⣼
⠀⣽⠀⠀⠈⠉⣿⢸⣟⣿⣇⣸⡟⠋⠛⣿⣇⡿⠋⠉⡠⠄⠘⢦⣴⣿⣿⠃⠀⣠⣤⣀⠀⠀⢸⣿⣿⣇⠀⢠⣀⠀⣿
⣸⠃⠀⠀⠠⠾⠛⣾⠷⣬⣻⣿⡷⣾⠿⣏⠉⠀⢀⠞⣠⣶⢰⣌⣿⣿⣿⡄⠘⢵⣦⣹⠆⢀⣼⣿⣧⣿⠆⠘⠙⠆⢸
⢻⣆⣀⠀⠀⠀⠀⠘⢷⠴⠿⠛⠻⠇⠀⠀⠀⠀⠂⣰⣿⡇⣾⣿⠸⣯⠻⢿⣷⣤⣬⣤⣶⣿⡿⠻⣄
⠀⠈⠙⣧⡇⠀⠀⠀⢘⠇⠀⠀⠀⠀⣠⠤⠄⢀⣼⣿⣿⢁⣿⣿⠀⠻⠛⠛⠿⠿⣿⠋⠀⠻⠀⠀⠀⠀⠀⢰⡟⣆⣠
⠀⠀⣰⡟⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡟⠟⠋⢼⣿⣿⡇⠀⠀⢀⣀⠀⠁⠀⢤⣤⠀⠀⠀⣰⣦⠈⢧⠈⠁
⠀⠀⢿⡀⠀⡿⠛⣷⠀⠀⠀⠀⢀⠀⠀⠰⠇⠀⢀⣄⠀⠲⣾⠿⣡⠀⣼⠿⠃⠀⠀⠀⠀⠀⣤⣀⣼⠋⢸⣀⣼
⠀⠀⠈⣷⠀⣿⠀⢿⡀⢠⡗⣦⢸⡇⠀⠀⢠⡀⠈⠀⠀⠀⢀⡶⣀⠀⠉⠀⣠⠀⡾⠀⠀⠀⣿⠋⠁⠀⠈⠉⠁
⠀⠀⠀⣿⠀⣇⠀⠀⣿⠀⡉⠋⠘⠀⢰⡟⣼⣇⠀⣞⣷⢘⠙⣿⠘⣷⠀⠘⡇⣸⠗⣴⠻⡆⣿⠀⠀⣼⢧⡄
⠀⠀⠀⣿⡀⢸⡆⢠⡏⢰⣇⣰⡶⡄⢸⣷⣿⣿⠀⣿⣿⣾⣿⣿⣶⣿⡗⢧⢻⣿⣤⠟⢸⠃⢹⡆⠀⣧⠀⡇
⠀⠀⠀⠙⠻⠟⠁⢸⠇⢸⡟⢻⣇⠀⣼⡿⢻⣿⠀⣿⣯⠀⢰⡏⢹⣿⣿⡘⠃⠹⡇⠀⠘⠷⠾⠁⠀⢷⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢿⠘⣾⠇⢀⡟⠀⠛⠃⠞⣿⠀⡟⠸⣷⠿⣇⡼⢧⡼⠃⢀⡄⢿⡀⠀⡀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⡞⢹⠀⠘⣟⢸⡇⠈⢻⠸⡶⣄⠀⢸⡄⢻⠀⠉⠀⠋⠀⠀⠀⠀⣸⣿⣰⠇⢸⠉⣆
⠀⠀⠀⠀⡇⢈⡇⢀⡏⢸⠀⠀⣿⠀⣿⣿⢄⢸⡇⡆⢀⣄⠀⢠⡿⢶⡄⣾⠉⠀⠁⠀⢸⣀⡿
⠀⠀⠀⠀⠻⠟⠁⠀⣷⢸⡄⠀⠛⠿⢿⢟⣆⣸⡇⢠⡼⢻⡇⢸⡇⣸⠃⣿⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⢾⠘⣏⣹⠇⢸⠀⣸⠇⣸⡇⠸⣄⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠃⠀⠀⠀⠘⠷⠏⣿⠀⢸⠇⠘⡿⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⢠⡿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⢄⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠢⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣦⡀⢀⣤⡀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⢄⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⢱⣶⣷⡆⠀⠀⠂⠒⣸
⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⣿⣿⣿⣿⣿⡄⣀⣀⠀⠀⣼⣿⣿⣷⠀⣤⣤⣄⠀⡀⡀⠀⠀⣾⣿⣿⣿⡇⣿⣿⣿⣇⠀⢸⣶⣤⣿
⣿⣿⡁⠘⣿⣿⡇⠀⠀⠘⠠⠔⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⠀⠈⠁⠰⣿⣿⣿⣿⠀⣿⣿⣿⣧⢀⣴⣶⡄⣿⣿⣿⣿⠀⢸⣿⣿⣟⣀⣼⣿⣿⣿
⣿⣿⣇⣠⣿⣿⡇⠀⠀⠀⢠⣤⠀⣿⣿⡀⠸⣿⣿⣿⣿⣿⠀⣾⣇⠀⣿⣿⣿⡏⠀⢿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⢠⣿⡀⠀⣿⠀⢻⣿⣇⣰⣿⣿⣿⣿⡟⠀⢹⣿⣀⣿⣿⣿⡇⠀⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢸⣿⣦⣴⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠀⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣇⢄⢀⣿⣿⣿⣿⣿⣿⣿⡝⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠹⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠙⢷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢿⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⡀⠀⠛⢦⣀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢶⣤⣄⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⣠⡀⠀⢸⣆
⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣶⡏⠁⠀⠀⠈⠙⠿⠶⠿⠿⠋⠉⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠀⠹⢿⣄⠀⠀⣼⡏⠿⣇⣸⡿
⠀⠀⢻⣧⣴⠾⢷⣦⠀⠀⠀⣼⡟⢳⣆⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠈⢻⣧
⠀⠀⣼⡟⠁⠀⢸⡿⠀⠀⣾⡏⠀⢸⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⢀⣿⡇⠀⠀⢸⡟
⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⡟⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠘⢷⡀
⠀⠀⣿⡇⠀⢠⣿⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣧
⠀⢠⡿⠁⠀⢸⣿⠀⠀⠀⣿⡇⠀⠈⠻⢷⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢨⡅⠀⠀⢸⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠈⢻⣆⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡿
⠀⣿⡇⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣠⠁⠀⠀⠀⠀⠀⢠⣿⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⠿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⠀⠻⢇⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⡀
⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⣴⠿⣿⣶⠀⢀⣼⠟⠉⠉⠹⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⣇⣀⡀⠀⠀⠀⣷⠀⢸⣿⠂⢸⡇⠀⠀⠀⠀⠀⢿⣿⣀⣠⣶⠶⠀⠶⣶⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣶⣿⠛⣿⣶⣤⣄⠀⠀⢀⣄⣀⣨
⣿⠛⠛⣿⣄⠀⠀⣿⠀⠸⣿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⣄⠀⣼⡟⠉⠀⠉⠉⠙⢿⣧⡀⠙⠋⠉⠙
⠟⠀⠀⢸⣿⠀⠀⣿⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⡀⠀⠸⣿⣄⣰⣿
⠀⢻⠀⠀⠉⠛⠉⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿
⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠈⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⢰⡇⠀⠀⠀⠀⣶⠾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣷⡄⠀⠀⠀⢀⡟
⠀⠀⠀⢱⡄⠀⠀⢀⣿⠀⣸⡇⠀⠀⢠⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⠀⠀⠀⠀⡄⠀⣿⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠿⣤⣤⡾⠁⠀⣿⡇⠀⠀⡀⠀⠀⠀⠙⠛⠛⠛⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⢿⡿⠛⠛⠛⠋⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⣿⡄⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠻⣇⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠀⢀⣠⣷⣶⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠧⠌⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠲⠀⠑⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣤⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠳⡀⠙⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣽⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢓⡀⢙⡀⢡⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣽⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⠭⠥⠤⠭⣙⠲⣆⡺
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣻⣻⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⡄⠀⠀⠀⠀⠈⠑⢌⢣
⠀⠀⠀⠀⠠⣶⣺⣿⣻⣿⡿⢿⣿⣿⡿⢿⣿⢟⣛⣿⡿⣯⡿⠷⠂⠀⠀⢀⡠⢤⠖⠊⠀⠀⠀⠐⣀⡀⠠⠔⠈⠷⣒⠀⠀⡀
⠀⠀⠀⠀⠀⠰⢾⣿⣥⣿⣿⣟⣛⣛⣛⣛⣛⣋⣹⣿⣟⠛⠛⠀⠀⠀⠀⠃⠄⠈⢑⣲⣤⣤⡤⠤⠬⠤⠤⠤⠖⠒⡯⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠤⠤⠤⠖⠒⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣽⣭⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣽⣽⣽⣭⣯⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢹⢳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⢠⣷⡘⠀⣿⡇⠀⠀⣀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢻⣿⣤⣾⣿⣿⣿⡀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⢢⣄⠀⠀⢐⣄⣀⣠⣼⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡇⠀⢀⣴⡆⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠙⣿⣶⣶⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠹⠗
⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⢀⣀⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡆⢨⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣼⣥⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠁
⠀⠐⠿⠟⠛⠛⠛⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣧⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣁⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠙⠃⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠿⠛⠻⠟⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⣿⡁
⠀⠀⠀⠀⠀⡾⠛⠉⠀⠀⠀⠀⠀⠀⣰⡿⠋⠻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⢯⡈⢳⣾⣷⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⢄⠀⣰⡿⠉⠙⣿⣿⣿⣶⣿⣿⣿⣿⣿⣏⠻⣮⣿⣶⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠏⠀⠀⠐⢿⣿⡿⠻⣿⠿⠟⣿⠋⠙⠀⠈⠻⣿⣿⣿⠟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠻⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⣿⣿⣿⣿⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠉⠀⠀⠀⠀⠀⢀⠀⢄⠒⠀⠂⠰⣿⣷⣿⣿⣿⣿⣿⡇⠀⠠⢀⣠⠄
⠀⠐⠠⠀⠀⠀⠘⢿⡿⢀⠃⠀⠀⣰⠎⣿⣿⢿⣿⣿⣿⣷⣄⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣧⡔⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⣈⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣦⣿⡅⠀⠂⣴⣤⣦⣶⣤⢄⣾⠗⠁
⠀⠀⠀⠀⠀⠠⢎⣽⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⣿⣿⣧⣴⡇⠸⣿⣿⣿⣿⣿⣡⡄
⠀⠀⠀⠀⠀⠀⠘⢻⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⡟⠁⢠⣤⣭⣟⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠
⠀⠀⠀⠀⠀⠠⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣯⡉⣩⣽⣿⣿⣿⣿⠭⣾⠃
⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⢿⣿⡟⣶⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡧⣶⣿⣿⣧⠀⠉⢘⣿⢻⣿⣿⣿⢸⣿⡇⠷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠿⣿⢸⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆⠀⢸⣿⢰⣿⣿⠉⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠹⣋⣿⣷⣿⠀⣿⣿⠿⢿⣿⣿⡟⢿⣿⣿⡇⠀⣾⡏⣿⣿⣿⠨⣦⠟
⠀⠀⠀⠀⠀⢠⠠⠾⣧⣶⣾⣿⣿⣁⣷⡤⠹⠀⠀⢈⣉⣙⣃⡀⢹⠏⠀⢀⡿⠃⢸⡇⡿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣍⣿⣗⠀⠀⢠⣿⣿⣿⣿⣿⣿⣄⠀⠀⣇⠀⢸⡷⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⡟⣯⠉⠀⠰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⣿⡘⡟⠀⠱⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣹⠻⣿⣷⣾⡄⠀⣠⡿⣯⣾⣿⣿⣿⣿⣿⣿⣇⣿⠃⡇⠀⡄⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⣼⣿⡿⠛⠙⣿⢋⣿⣿⠟⢿⢷⣿⡏⣿⣿⠙⢈⣧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡇⠀⠀⡏⢸⣿⣯⣀⣸⣸⣿⣤⣿⣟⣶⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡃⢸⣿⡏⠈⠉⢹⡿⢸⣿⡟⣏⢠⡆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢸⣿⡇⠀⠀⢸⣷⠈⣿⡇⡿⠘⡇⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠁⢰⣿⠇⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠆⠀⢀⠀⠁⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠀⠀⠀⠀⠀⠸⠀⠀⠀⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⢴⡄⠀⠀⠀⣀⢔⣒⣳⠀⠀⠀⠀⡠⢚⣩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠋⠀⠀⠀⣠⠾⠋⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠷⠉⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⡠⠚⢹⠇⠀⠀⠀⢀⡴⢺⠆⠀⠀⠀⡰⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⣒⣥⠔⠁⠀⠀⠀⢀⡴⢫⠇⠀⢀⠤⠴⢊⡼⠀⠀⠀⢀⠞⢀⡴⠋⢀⣔⣊⣑⣉⡠⠎⠀⠀⠀⢰⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠰⠏⠀⠀⠀⠀⠀⠠⠾⠦⠒⠉⠀⠰⠗⠒⠒⠉⡀⣀⠀⢠⠏⡔⠁⠀⠀⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⠀⠀⠀⢀⡠⠖⢛⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠒⠁⡠⢃⠞⣠⡀⠀⠀⠀⠀⡠⢦⠀⠀⢀⡠⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣌⠙⡇⠀⠀⠀⡠⢋⠄⠈⣡⠤⠀⠀⣀⠤⠒⢩⡍⠉⠒⠦⠤⠤⠤⠊⢀⠎⢠⡴⠀⢶⠀⢀⠎⡀⠁⣠⠊⢭⠄⠀⠀⢀⠔⡆⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⣱⠃⠀⢸⠀⡇⣀⠀⠄⣰⠃⠰⠊⡀⢤⡴⠋⣡⠴⠒⣺⠆⠀⢀⣤⡤⢤⠤⠒⠁⠀⠀⡠⢢⠋⠀⡎⢰⠁⠀⣷⠁⠀⠀⠀⠀⢬⡘⠀⡰⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠀⡸⢸⠁⠇⡌⢰⡇⠀⡞⠀⡰⠋⡠⢺⢁⠤⠀⠁⢀⠴⡟⠁⠀⠴⠊⠀⡠⠔⠊⠀⡁⠀⠀⡇⢸⠀⠀⠛⠀⢀⡔⡄⠀⠀⠀⡌⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⡰⠀⣾⠃⢰⠁⡼⠁⡠⢇⠀⡱⢠⠇⡴⠋⢀⡴⠒⠉⡐⠹⣪⠂⠀⠔⡜⠀⠀⢠⣇⢸⠀⢋⡆⠀⠘⡇⠋⠀⠀⢰⠁⡎⠀⡠⢲
⠀⠀⠀⠀⠀⠐⠂⠀⣴⢷⠇⢠⠇⣠⣿⡶⣧⢞⠁⠾⠒⡻⠟⣥⢀⣾⣤⣀⣮⣤⣀⣰⠃⢰⡟⠁⠀⠀⠁⠀⠀⠀⡇⢸⠀⠟⠀⠀⠀⢧⡄⠀⠀⡏⢸⠃⢸⡵⠋
⠀⠀⠀⠀⠀⢀⡄⣼⠃⢸⣴⡧⠾⣻⠋⠈⠁⣉⣠⠀⣠⠴⠚⡡⠛⠯⠉⠁⠀⠀⠈⢋⡀⢸⡇⠀⠀⣰⠦⠺⠁⠸⠁⡎⠀⢀⡴⠂⠀⣸⡇⠀⠀⣧⢸⡀⠾
⠀⠀⠀⠀⠀⢘⡁⢻⣴⠋⠀⠀⠀⠁⠀⢀⣀⣤⠿⠾⠧⠀⠀⠀⠂⠀⠀⣠⠦⠄⠀⠸⠃⡼⡟⠻⢯⡀⠀⡄⠐⠁⢰⠁⢀⠏⠀⠀⢠⢣⠇⠀⠀⢸⠈⡇⠀⠠⡄
⠀⠀⠀⠀⢸⢸⣇⣼⡇⠀⠀⠀⠀⢀⡞⠉⠀⢀⡤⠤⠠⠀⠀⠀⠀⠙⢿⠁⠰⢄⠀⠀⠜⠀⠃⠀⠀⠛⣋⣥⣄⣠⡏⢀⡎⠀⠀⡠⢃⡞⠀⠀⠠⢫⠀⠁⠀⡔
⠀⠀⠀⠀⡸⣼⠿⠻⠀⠀⣠⠴⣦⣼⡀⠀⢀⣀⠈⢁⡄⠲⡋⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡶⢀⡼⠛⠛⢿⡠⢺⠁⡠⠞⣠⠎⠀⣀⠔⢡⠂⠀⡄
⠀⠀⠀⣰⢱⠇⠀⠀⠀⠈⣠⠖⠋⠉⠉⠉⠻⠁⣌⠭⠐⢀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠀⠀⠀⠀⢠⡟⠋⠁⠀⠀⢸⡧⠞⠉⠀⣴⠥⠒⠞⢁⡴⠃⠀⢠⠃⠀⣰⢛
⠀⠀⢰⣧⣀⣀⡀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⣀⡀⠀⠀⠀⠀⠀⠈⠹⣆⣀⣀⠀⠀⠆⠀⢠⠀⠀⠀⣶⣦⠀⡟⠁⠀⣬⠴⢊⡴⢻⡇⡾⢠⠔⣣⠞
⢀⡴⠋⡉⠉⠀⠀⠀⣼⠧⠦⠀⠀⠀⣀⡄⠀⠀⠀⡠⠴⠋⠁⠈⠀⠀⠀⡶⠆⠀⠀⠈⢯⠉⠀⠀⠀⠀⢸⣄⠀⠀⠉⢀⣀⡙⠚⠋⠁⣠⠊⠀⡿⣸⢡⡷⠊⠁
⡼⠁⠂⠀⠔⠇⢠⡞⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠒⠲⢿⡄⠀⠀⠀⠚⠉⠉⠃⠀⠀⠀⣿⡿⠃⠀⣰⠟⠀⢰⠟⠁⠈
⣧⠀⠁⠆⠀⢸⣼⡁⠀⠀⣆⣀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠈⠙⢲⣄⡤⠄⠀⠀⠀⣠⠠⢄⠉⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣾⠃⠀⠂⠀⠀⢀⡠⡀
⣹⠀⠃⠂⠀⠀⢻⡿⠚⠋⠉⠀⠀⠀⠀⠘⢦⣴⠟⣡⠒⠖⢤⠀⡏⠀⠀⠀⡤⠩⠤⠔⠆⣧⠠⠖⠀⠀⠀⢸⡄⠀⣶⠖⠛⠩⠖⣁⠀⠀⠘⠻⠭⠭⠔⠁
⡿⠀⠀⠲⣞⠀⢸⡇⠀⠀⢄⠀⢀⡀⠀⠀⢸⠇⠀⣧⠒⠓⠒⠀⠁⠀⠀⠤⣄⠀⠀⠠⡀⠀⠀⣠⠀⠀⠀⣼⠷⠛⠛⢂⡡⢚⠥⠖⠁
⢷⠀⠀⠀⠈⢦⣘⣇⡀⠀⢈⣷⢿⡇⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⢠⠟⠛⠉⢁⢀⡤⠾⠥⠴⠁⡴⢋⠔⠁⠀⢀⣀
⠈⢷⣤⣦⠀⠀⠨⣿⡇⠀⠘⠁⠀⠙⢄⠀⠀⠀⠀⣠⡬⠷⠦⣤⣤⡴⠂⠀⡰⠀⢠⠏⠀⠀⠚⢋⣙⣒⣒⡒⢂⣡⠔⠁⠀⠞⠀⠒⠚
⠀⠀⠈⠛⢧⣀⢀⣘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⣠⠾⠷⠶⠞⠋⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷⢦⣤⣤⣼⣦⣄⣀⣀⣈⣽⣦⣀⣀⡴⠷⠤⠤⠶⠚⠁
⣶⣶⣶⢶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠁⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠰⢮⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠩⠈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡗⠛⠒⠐⠀⣄⠀⠀⠁⠀⠀⠀⢀⠊⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣠⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⢇⡘⣸⠇⠀⣤⠑⠀⠑⠄⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣀⣀⠀⢪⢣⡀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠒⠘⠡⠊⠘⣔⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠐⢄⢀⠈⠘⠠⠧⠊⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠡⡀⠀⠸⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢣⡖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⡀⢠⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⠀⠀⣀⡀⢀⣀⠀⠀⠀⠠⢀⡄⠀⠀⡀⠂
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠜⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠐⠍⢑⢏⢦⠀⣄⠑⠀⠀⠂⠈⣇⠈⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠚⠉⠈⠁⠀⢏⡀⠀⡑⠮⠐⢦⠄⠀⠀⠘⠀⢁
⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⣀⠀⠘
⠀⠀⠀⠃⠘⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⠂⣶
⠀⠀⠀⠀⠀⠻⠀⢰⠁⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢀⠔⢡
⠀⠀⠀⡇⠀⠀⠑⢄⠒⠹⡈⠉⠊⠖⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢳⠀⠈
⠀⠀⠀⢡⠀⠀⠀⠀⠁⠒⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠢⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡐
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠰⠀⡀⠐
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢼⢈⡟
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠃⢐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⢀⠔⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⢁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⡀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠁⠹⣧⢻⣿⠒⣦⠀⠀⠀⠀⠀⠷⡉⢦⣄
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⢀⣠⠆⢻⣿⣷⡀⢸⣿⣯⢻⣧⡹⣷⡄⣀⠀⠀⠀⠙⢆⠈⢳⡄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠙⠓⠈⠋⠛⠉⡈⠉⠀⠈⠛⠃⢛⠃⠙⠂⠀⠀⠀⢸⠀⢹⠅
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢽⠇
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠃⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡇⠀⠀⠀⠀⣷⢰⡏
⠀⠀⠀⠀⠀⢀⣴⣿⡿⣧⡷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣿⡇⠀⢀⣠⣲⠾⣞⣧⠶⠶⣤⠖⡄
⠀⠀⠀⠀⠀⣀⡼⠏⠀⢻⠆⠙⠉⡛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢑⣾⣿⡇⠀⠘⠿⠶⠖⠛⠷⠲⠶⠒⠂⠁
⠀⠀⠠⠊⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠰⣶⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢊⣐⣿⣿⣿⣿⡇
⠀⠀⢠⣴⣶⣦⣼⣿⣿⣿⣿⣷⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠐⠀⠀⠐
⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠉⠉⠙⠛⠋⠛⠋⠁⠀⠀⠀⠀⠀⠀⡙⢔⡀⠈⠉⠉⠉⠉⠉⠉⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠿⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠭⠉⠛⣛⡛⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠿⠿
⠀⠀⠐⣶⣄⣾⣿⡟⢤⣴⡿⢻⣧⣤⣤⣴⠿⣆⢀⠀⠀⠀⢲⠒⣈⡈⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠠⠄⠀⠀⠄
⠀⠀⣰⣎⣿⣟⡆⢳⡆⣻⣧⣶⣸⣿⣻⡏⣴⡄⣾⣇⠀⠀⠰⡔⡒⠂⠀⠀⠀⠰⠄⠀⠀⠀⢀⡠⠞⠀⠀⠀⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠤⠤⠤⠤⠴⠖⠒⠚⠉⠀⠀⠀⠠⠤⠅
⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡔⠒⢒⣤⣌⠁⠀⠒⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣩⣫⠿⣫⣶⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⣄⠞⠁⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣀⢫⢀⣴⣦⡀⠀⠀⣿⣿⣿⣿⡿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣜⡡⠟⢋⣴⡀⠁⣿⣇⠀⢀⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⡀⠀⠀⠠⣤⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠛⣀⠙⠟⠀⠀⠹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡁⠀⢸⣿⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⡙⠿⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣴⣿⡿⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡿⠿⣟⠻⣿⣿⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣏⣀⣀⠈⠁⠙⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡟⠉⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣴⣶⣶⡾⣿⣿⣿⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣷⠗⠀⠀⠹⡇
⠀⠀⠀⠀⠸⡀⠀⠈⠻⢄⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣩⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠟⢡⡾⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠋⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⠀⠀⠀⠠⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⡠⠀⠀⠀⠀⢀⠄⠀⠤⣾⠔⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠
⠀⣠⣶⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⢛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠴⠂⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠈⠉⠃⠀⠀⠀⢠⠀⠐⠀⠀⠀⢾⠀⠈⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⢀⡖
⠂⠐⠐⠀⠐⠀⠀⠺⠬⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠅
⠄⣀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠁⠀⠉
⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⢀⠠⢀⠀⠀⠀⠐⠀⠃
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣊⣘⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠑⠦⠀⠀⠀⡠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀
⠀⠀⢀⣰⡧⠀⢜⡰⢤⣆⠀⠀⢀⣤⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⢾⣴⣶⣾⣷⣿⣿⣿⣄⢼⣿⣷⣶⣦⠀⠀⢀⣶⠀⠀⠀⢠⣶⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⢀⡼⠋⣀⢀⠔⢄
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠇⠀⠀⠀⢞⣤⣀⣴⣿⣿⣿⣿⣿⣦⣤⣀⣀⠀⠨⢷⣀⣙⣁⣀⣿⣀⣘⠃⣠⣆⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠋⠃⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⣿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⣰⣶⣾⣿⣿⣾⣿⣿⣷⢸⣿⡄⠀⠀⠀⠀⣴⣿⣿⠋⣉⣀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⡔⠀⣠⠀⠼
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡯⠛⣿⢛⣻⣿⣇⠀⠀⠀⠐⣿⣿⣇⠀⡽⠋⠈⡠⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⢾⣷⠘⠛
⠀⠼⣧⠀⠻⢿⢿⡙⢯⣄⡟⢷⣮⣛⡿⣫⡄⠀⠀⠀⢹⣿⣿⣷⣦⣥⣵⣤⣿⣨⣤⣴⡆⠀⠀⠀⠀⣤⡆⠀⠀⠀⣾⣤⡏
⠀⠀⠟⠦⠤⠼⠾⣿⣦⣾⣿⣿⣿⣿⣿⣿⠷⠤⠀⠀⠸⠿⠟⠿⣿⣿⣿⠿⠛⠛⠿⠿⠿⠿⠓⠀⠀⢩⢷⡒⢶⣶⣿⣿⣶⡶⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣠⢀⡠
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠈⠳⡀⠀⠀⠀⠀⠀⢠⣦⡰⢢⣶⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣯⣍⢻⠄
⠀⠀⠀⠀⠀⠐⠋⣤⠀⠀⣤⠘⠂⠀⣀⠀⠀⠘⣿⣿⣧⣐⡒⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⡸⠿⠟⣨⡶⠄
⠀⠀⠀⠀⠀⢤⠦⣯⢤⣤⣭⣤⡤⢠⡫⠀⠀⠀⣿⣿⣿⣿⣿⡤⠎⢨⡑⠒⣀⣀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠛⠉⠀⠠⡄
⠀⠀⢀⣀⡀⣀⡀⣿⣸⣟⣺⣆⣨⣍⣍⣭⡄⠀⣿⣿⣿⣿⣿⣇⡀⣼⣭⣛⣛⣛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣟⡀
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿⠿⠷⠉⠉⠉⠒⠿⠀⠀⠀⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⡢⠄⠉⡁⠀⠀⠀⢀⠉⢀⠰⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⠁⡠⠔⠅⠀⡀⠔⠁⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⡄⠊⠶⠔⠐⣡⢠⠀⠀⢀⠔⢈⠉⠈⡦⠊⠐⠃⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⡔⢁⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⡏⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠒⠒⠲⠒⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠂⠈⢄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢦⡇⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠈⢨⠑⡀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣀⠔⠊⠉⠀⠉⢆⢀⠾⡖⠋⠀⠀⠈⠢⣨⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⡸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠐⠊⠉⠀⢀⠀⠐⢉⡀⠑⠊⢘⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢐⢰
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠉⠈⠁⢊⣆⠀⠀⠈⢯⡡⢨⠉⠑⢆⠀⡠⠈
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⢠⠆⠀⠀⠇⠰
⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⣄⠤⠖⠁⠀⠀⠀⠈⠱⢣
⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⢡⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠆⠇⠀⠀⠀⠀⠄
⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠅⠂⠄⠠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠈⢠⠃
⠀⠀⠀⠱⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⣌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠒⡁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠐⠄⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠊⠁⡣⠀⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⣈⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⣀⠤⠒⠁⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠤⠀⣀⣀⡀⠀⠀⣀⣀⠠⠤⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠲⣀⡱⣔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢨⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀
⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠰⣄
⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟
⠀⠂⠿⢻⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⡀⠀⢀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣦⣾⣻⣟⣿⠃
⠀⢠⣄⣨⣿⣛⡿⠛⠉⠉⠉⠙⠛⠛⠛⠿⠟⣿⣟⢿⣿⣿⣿⣿⢿⡿⣻⡿⣿⠿⣻⡟⠋⠉⠉⠉⠀⠙⢫⡍⣿⠈⣿
⢰⣾⣿⡿⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢯⣽⢉⣩⣿⣷⣶⠸⠟⠁⠀⢨⡇⠀⠀⠀⠀⠀⢀⠏⠀⠻⣿⣿⡄
⠀⣻⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⠋⢽⣿⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠊⠀⢀⣘⡿⠿⡾
⠀⠻⢿⠙⢇⣄⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣾⡇⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠐⠁⠀⣔⠛⣿⡷⠊
⠀⠀⠈⠀⠈⠛⠀⠋⡙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢫⣿⡃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠐⠁⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣬⢫⠜⠲⠤⣤⡀⡀⠀⠀⠀⠀⠀⠛⢠⣿⠁⠀⠀⢠⠃⠀⠀⠀⡀⠰⣄⣀⣴⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠞⣋⣉⣦⣄⠰⣦⡀⣠⠉⣄⠀⣀⠔⠁⣠⣤⣠⡀⡄⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡤⢍⣧⠞⡷⣻⡄⣿⣤⣿⡆⣤⣮⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡈⢻⣿⡛⣩⡇⢙⡁⢣⢾⣿⣿⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣵⢇⠉⣇⣙⣅⣸⣿⣟⠘⣭⢩⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠛⣛⣋⢝⣋⣹⠯⠿⠟⠿⠟⠁⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠉⠉⣏⠀⠀⣦⣀⠀⢀⣄⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠛⠀⢶⠆⣶⠀⠀⢀⠻⣄⠀⠙⣶⣶⣾⡟⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢳⣴⢣⣶⠴⣧⣁⣀⣿⢡⣟⣀⣠⣬⣿⣶⣶⠺⠟⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢿⠸⡟⠀⠀⣿⣹⡿⢦⣿⣿⡟⢿⡟⠻⠿⠃⠀⠃⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⣶⣜⡿⢿⠀⣀⣀⣼⣿⠉⣝⠿⣿⣇⠴⠾⠳⢦⡀⠀⠀⠶⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⡤⠿⣿⣹⡇⢸⠟⠋⠉⠁⣽⣦⢛⡄⠰⣿⣶⠀⢠⡄⠙⢂⡌⠰⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⣼⣧⣼⡿⠊⢻⣷⡿⠀⣤⡄⠀⠿⠁⠈⣧⠀⠸⡏⠀⠩⠀⢀⠀⠀⠘⣣⡄⠀⣀⡀⠠
⠀⢀⠀⣿⣷⣶⣿⣿⣿⣿⠟⠉⠉⠉⣥⠂⠈⠃⢀⣀⠀⠀⢙⡀⢸⣧⣤⠴⠟⠚⠂⣤⣴⠇⠹⠟⠃⠄⠞⠛⠁⣤⠗⡀
⠴⣾⠗⢙⡿⣻⠿⠿⠛⠓⠂⠀⠀⠀⢈⠁⠀⠙⠉⠉⠛⢀⠈⠡⠘⠉⠀⣠⣴⣾⣿⡷⠤⠀⠒⠒⠒⠒⠒⠶⢻⣋⢠⣤
⠀⠈⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣄⣠⣄⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢿⡿⢍⣩⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣴⡷⢬⣶⣟⣿⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣵⣴⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠈⠀⠀⠀⠐⢠⠍⠅⠂⠈⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠀⠀⠃⠀⠀⡁⠀⠀⢇⣦⣿⠀⠀⠀⠀⠀⠀⡀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠐⠃⢀⡠⠒⣅⢰⠒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠓⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡴⢀⣯⠖⢤⠕⠋⠀⠄⢠⠄⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⡀⠳⡁⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⡽⠔⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⢀⠀⠀⠀⠀⠃⢂
⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣤⡜⣀⣦⣀⡀⠀⠀⠠⡶⣔⡀⠠⠢⣇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡁⢆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢾⡕⠉⠀⠈⢻⢬⡦⡴⠖⠢⠀⠃⠀⢑⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠘⡀
⠀⠀⠀⠀⠀⡄⠀⠀⣰⠋⠉⠁⠽⠀⠀⠀⠃⠘⠚⠛⡃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠑⠄⠇
⠀⠀⠀⠀⠀⡁⢠⡈⠉⠀⠀⠀⠀⠈⠒⠋⠒⠠⠔⣐⠀⠀⠀⠀⡒⠴⡤⣄⠄⠶⠤⣂⣀⠀⠀⠀⠁⠀⠛⡄⠆⢓
⠀⠀⠀⠀⠀⡄⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡵⣄⠀⠀⠐⠒⠂⠁⡤⠀⠀⠉⠙⢤⡄⠆⠀⠀⠀⠐⡤⣆
⠀⠀⠀⠀⠀⠃⢸⡣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡘⢶⣄⠀⠀⢀⠄⠁⠀⠀⠀⠀⠈⠣⠀⡄⠀⠀⠤⢦⡜
⠀⠀⠀⠀⠀⠈⠀⢻⣥⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠱⠀⣿⡶⠔⡋⠀⠀⠀⠀⢂⠀⠀⠀⠀⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠡⠀⠳⠿⢶⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢩⠁⠀⠀⠐⢈⡠⠀⠀⠄⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠆⠀⠈⣗⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠆⠀⠀⠂⠀⢁⠤⠀⡀⠄⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⡄⠈⢻⣤⡄⠀⠀⠀⡀⠾⠀⢀⠆⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡂⠀⢹⣷⡃⠀⠀⠙⠂⢄⠘⢅⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⡻⣤⢀⠀⠀⠀⠈⡠⠞⠀⣆⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣤⣴⣴⣇⠐⠀⣉⣀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣿⠎⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⣠⣶⡞⠀⠀⠀⠀⠹⣎⢄
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠹⣟⢃⡀
⢀⣀⠤⣤⣾⣯⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠲⠀⠀⠀⢠
⠉⠀⠀⠀⠀⠉⠙⠲⣶⣦⣄⠀⠨⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣄⡀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⣠⣸⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡿⠟⢠⡾⠟⠁⠛⠦⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠉⠉⠹⣿⣿⣄⡀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⣾⣿⣿⢧⡤⠤⠤⢤⣤⣄⣤
⠀⠀⠀⠈⢻⣿⣧⣤⠀⠴⠤⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠸⣿⢧⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢿⣟⡄⠀⣀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⢞⣿⣿⡿⠛⠛⠿⣷⣦⣶⣄⣠⡀
⠀⠀⠀⠀⠀⠸⣿⣷⣐⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⢹⡀⠉⠙⠻⣧⡢⡀
⠀⠀⠀⠀⣠⡶⠝⠉⠛⠛⢳⠶⠷⠷⣶⣦⠤⡤⢤⣴⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⣼⣡⠀⠀⠀⠈⠉⠉⠛
⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠇⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢠⣟⡇
⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣿⡟
⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣧⡇
⢣⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⣠⡴⠋⠙⢧⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡑⡄⠀⠉⠁⠀⠀⠉⠚⢻⡌⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠻⢄⣦⡶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⣀⣀⠀⠒⢡⣄⣀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠉⢄⠠⢠⠤⠐⠤⠎⠀⠀⠝⠛⠃⠤⣉⠒⠄⡀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠐⠀⠀⠀⠀⣈⣤⠆⠪⡵⡀⢀⠃⠃⠈⡆⠀⠀⠀⠉⠦⢌⠒⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⠀⠀⠀⠸⣀⠀⠀⠀⡠⠭⢋⡘⢤⡵⡇⠀⠀⠀⡰⠒⠀⢱⢄⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠢⠜⠀⠀⠚⠋⢿⠀⠳⣄⣢⠞⠀⠀⡀⢄⠌⠓⣄⠢⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠋⠀⠀⠀⠀⠀⠀⢀⠅⠎⠌⢀⡟⢂⠐⡀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⠖⠲⠶⢄⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠂⠈⠀⠸⢤⡾⡄⠰⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢣⠀⠀⢰⠋⠀⠀⠀⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⠕⠁⡀⠐⡠⡥⠻⠿⡀⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢹⠆⠀⠀⢮⣤⠀⠀⠀⠀⠀⠀⢀⡠⠊⢔⠕⡁⡠⠀⢊⢘⠅⠀⠀⠀⠅⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣄⠂⢸⡆⠀⠈⠃⠀⢳⡀⠀⠀⡠⢔⠡⣊⠔⡡⡈⢈⢄⡑⠋⠁⠂⠂⣀⣀⡀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠱⠦⣀⣀⠀⢀⠠⢒⠥⢂⠥⢊⠔⡀⢊⢔⠅⠈⢇⠀⠀⠀⣸⠉⠌⠆⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⢮⡔⠨⢀⠨⢐⠁⣈⠔⡡⠂⠠⠈⣓⣴⣾⢡⣤⠶⡆⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⡂⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠐⡨⠔⣈⠄⣈⢔⡡⠢⡾⠋⠀⠀⠀⠀⠠⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⠀⠤⠄⢂⡨⠅⡞⠀⠀⠀⠀⠀⠀⠀⢰⢒⡊⠅⡐⠅⢒⠩⡠⢑⠠⣺⠁⠀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠌⠩⠀⠢⢕⡃⠀⠀⠀⠀⠀⠀⣠⠭⢀⠀⠁⡀⠈⡐⠅⢊⠤⣪⠌⠀⠀⢀⡀⢀⠀⠠⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠂⠀⠐⡄⠀⠀⠀⠀⡀⠔⠓⠈⠄⠒⠥⠐⠉⠄⡪⢔⠨⢈⢗⢶⡀⡗⠠⠃⡠⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣒⣨⠆⠀⣠⠊⠉⠀⠒⠨⠄⡒⠬⢐⡈⠄⡂⠥⡂⢔⠡⣾⣁⠖⠀⠊⠀⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⣔⣀⠠⠄⠂⠈⠅⢒⠈⠅⡂⠅⠂⠅⣪⡅⠜⢋⠡⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⢉⠘⠃⠬⢍⣉⠒⠒⠀⠶⠒⠀⠒⠛⢂⣉⠠⠔⠂⠁⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣙⣒⣶⠶⡆⠐⠐⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠹⠾⠮⣕⡉⠂⢀⠩⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣔⡃⠤⠄⠀⢤⣀⣀⣅⣈⣄⣪⢕⣬⣢⣽⢦⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠀⠛⠩⠒⠲⠒⠲⠖⠶⠆⡶⠼⠼⠶⠞⠺⠛⠁⠊⠀⠂⠀⠊⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠉⠙⠛⠛⠛⠉⠁⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢳⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠛⣄
⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣡⡆
⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠄
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⢸⡇⢀⡀⠀⠀⠀⣀⠀⠀⠀⡸⠋⠀⠉⠛⠦⠔⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⡾⠗⣼⡇
⠀⢸⡷⠊⢻⠄⠀⣾⠙⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⡑⠀⠘⠇
⠀⢸⡇⠀⣌⠀⠀⡇⠠⡇⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠃⠀⠀⢰
⠀⢸⠃⠀⡿⠀⠀⡀⠀⢡⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⣧⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠠⡇⠀⠀⠛⣤⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡄⠀⠀⠀⢠⡟
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠁⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⠶⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡀⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⡧
⡷
⣧
⣿⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⡀
⣿⡀⠀⠀⢰⡏⢹⡆⢸⠏⠀⠈⠹⣆⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⢲⣤⠀⠀⠀⣀
⡏⠙⣷⡀⢸⡇⢸⣁⡆⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⣀⡜⠋⠉⠉⠙⢧⡀⠋⠉
⠀⠀⢹⡇⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⠙⠉
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠐⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠋
⠀⠀⠆⠀⠀⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠆⠀⠀⢠
⠀⠀⢠⡀⢠⡄⠀⡏⠀⠀⠶⠶⠦⠄⢀⣄⡀⠀⠠⠀⠀⠀⠀⠀⣠⡄⠀⠴⠶⠖⠈⠀⠀⡁⢸⡆⠀⠀⠸
⠀⠀⠀⠀⠉⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠳⣤⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣀⣠⡄⠶⠞⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠉⠙⠛⠋⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠳⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢟⣋⣁⣠⣤⣀⠉⠓⣄⠀⠀⠀⠀⡠⠴⠚⢯⡭⠗⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢧⣀⣼⣄⠀⠀⠀⢸⡇⠀⠘⢦⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠭⡳⣄
⠀⠀⠀⠀⠀⣀⡴⠋⠠⠂⠀⢀⣠⠜⢈⡿⣶⡶⣯⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡽⠈⠳⣄
⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠻⣅⣀⡼⠁⠀⠀⠀⠈⠉⠒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢀⣀⠈⢳⡀
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠋⢹⣿⠂⠿⡄
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢶⠯⣠⠖⢧⡀⣀⣻⡀
⢀⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠙⣷
⣸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠃⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠈⢷⣄⡄⠀⠀⠀⠀⣀⠢⢤⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠈⠹⣦⠀⠀⢸⠁⠀⠀⡉⢋⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡴⠛⠳⣄⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠹⣄⡀⠈⠓⠒⠊⢷⡘⠳⠦⣤⡭⠭⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢆⠀⠀⠙⡆⣸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠢⣄⡀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠆⠀⠀⢑⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢮⡁⠀⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠉⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠈⢿⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠘⢿⣄⠀⠀⠀⠰⡙⡄⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠉⣷⠀⠀⢀⡴⠶⠟⣀⣤⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⢤⣤⣾⣤⣤⣾⣶⡾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡾⡿⠉⡀⠁⠈⠉⠉⠉⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡙⠙⠒⠶⣿⣷⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⣈⣹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⠗⠀⣻⡟⠃⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡏⢼⣿⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠶⣾⣿⣇⡀⣰⣿⣿⣿⣌⣩⣽⣿⣿⣿⡿⠋⠀⣀⠀⣼⠟⠚⠃⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠘⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣠⡀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣔⣈⡿⠃⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣰⠿⢿⣧⣱⡀⡄⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⢿⡲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠛⠿⠗⠿⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⠺⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣍⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣻⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⣿⣿⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡸⠛⣹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣥⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠐⢅⡀⠉⠢⢄
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡿⣧⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠐⡶⠆⠀⠀⠈⠳⡀
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⢣⠀⠀⡟⠀⠀⠀⠀⠀⠘⢄
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⢰⣿⠗⠉⠀⠀⠀⠀⠀⠀⢄
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⢰⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡼⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠴⠃⠿⠃
⡇⠀⡔⠈⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⢀⡴⠉⠀⠈⢣⠀⠀⠀⠀⠀⠀⢠⠖⠠⠒⠂⠀⠀⡠⠂⡀
⡇⠀⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⡜⠁⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⣇⣼⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣠⡃⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⢦⣀⠀⠐⠁⠀⠀⠀⢠⡾⢹⡽⣆
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⡆⠀⠀⠀⠀⠀⠀⠀⢹⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠻⠤⠁
⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⢆⣣⡑⠢⡀⠀⢀⡀⠤⠂⢠⠇⣀⣀⣤⣴⡂⢀⠀⣀⡀⢀⣰⣤⢰
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠙⣏⣁⠠⡏⠁⠀⠀⣼⡟⢻⠋⢀⡽⠾⠦⣭⠁⣈⡏⠉⢁⡎
⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⣟⣀⣒⣒⡋⠈⠉⠶⣛⣁⠀⠀⠀⠀⠛⣒⡁⠀⠀⠞
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠓⠒⠛⠛⠻⠟⠛⠛⠃⢡⠖⢢⣠⠀⠁⠀⠞
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⠁⠀⠈⠁⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣀⣀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠄⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠤⠤⠤⢠⠀⡀⣤⠠⠄⠤⠀⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠄⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠞⢻⠁⢀⠀⠚⠒⢡⠀⡀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢕⡵⢀⡈⡲⡞⠄⠢⢀⣰⡥⠐⢄⣆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣾⣟⣾⣟⡴⡩⠒⠒⢤⣄⠈⣣⣀⡩⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡍⡟⢰⢿⣾⣷⡵⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢇⠁⠈⡟⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠀⣀⡠⠐⠯⠖⠒⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⣍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠐⠯⢑⡂⠠⠤⠤⠤⣤⠤⠒⠊⠉⠁⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⣈⠄⠀⠰⡡⠤⡀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢲⣣⣀⠠⠔⠁⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡠⠤⠄⠒⠀⠉⠀⠀⣀⠀⢀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠐⠂⢠⠓⠒⠒⠒⠒⢄⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⡔⠀⠀⠸⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⢄⠀⠀⡎⠀⠀⢀⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⢄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠦⠜⠀⠀⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⠢⡀⠀⠀⠀⠆⠀⠀⢀⠤⡀⢀⠤⠂⠐⢒⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢌⡢⣀⢸⠀⠰⠒⠋⠀⠈⠁⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠚⠂⠀⠒⠒⠂⠁