Grapevine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠈⢢⠀⠄⣀⠀⠀⠀⠀⠀⠀⢥⡀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡿⢠⢄⣰⡶⣛⢀⣠⣾⣠⠆⣀⠤⢰⠀⠈⢻⠆⣠⢀⢀⠀⠀⢀⠚⠓⠐⠀⠠⣆⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣄⠀⢀⢀⠀⢀⡮⠁⠤⠓⠶⣄⢠⣯⡀⠉⣿⣿⣿⣿⡏⠁⢠⠜⣶⡃⠰⠴⢿⣤⣄⣀⠥⡴⠤⣀⠻⠀⠀⣈⠙⠀⠀⢀⣤⣀⣶⢟⣀⡀
⠀⠀⠀⠀⠈⠁⠀⠁⠈⠙⢿⣟⠑⢀⣤⣮⠉⠈⢰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠛⠩⠴⠂⣄⣾⠛⠡⠾⢧⡤⢀⡹⠛⢁⠠⠾⠒⡀⡟⠖⡉⠯⠀⣤⡀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠐⢃⠈⢿⠏⢠⣀⣤⣿⣿⠏⢸⣿⣿⠻⠗⠀⡃⠀⠀⠴⠉⠀⣀⣤⣦⣼⡟⠄⣴⣾⣿⣿⣿⣦⠉⠉⠙⠢⠀⢒⠐⠃⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⠈⠀⠲⠼⢿⣧⠍⢀⠂⠀⠛⠁⠑⠂⠁⢀⣤⡀⡔⠠⠀⠁⠚⢻⡿⠓⠀⠸⣿⣿⣿⣿⡧⠔⠒⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠈⣵⣾⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠛⠀⡂⡈⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠎⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠠⡀⠀⡀⠄⠒⠒⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣁⡀⠖⡰⣾⠋⠁⣑⠲⠄⠀⠀⠑⢢⣠⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢈⣔⢛⡿⠧⠶⠖⠷⢹⡄⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⢹⠠⠾⢺⣟⣴⣦⣾⣖⣧⢸⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡖⠔⣂⣰⠴⡉⢀⠄⠳⠀⠀⠀⢠⠀⢀⡾⡜⠿⣻⠉⠛⠋⢀⢰⠟⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⣀⠃⣇⠈⠀⢨⣤⣄⡀⠙⠸⣿⡞⠋⠀⠀⣀⣰⠿⡃⠐⠈⠔⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠠⠙⡖⠷⢶⠚⠀⠀⠈⠉⠳⠶⣤⡤⠴⠾⠟⠋⠀⢤⡇⢠⠖⠋⠙⠃⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠃⠄⠀⡄⠀⠀⠀⠀⠀⢰⠋⠀⠀⡀⠀⠠⠒⠒⠙⠳⡀⠁⠤⠜⠁
⠀⠀⠀⠀⠄⠀⠐⠠⠀⣠⠟⠉⠂⠐⠉⠀⠀⠀⠀⠀⠠⠸⣄⠀⡀⣄⠀⠀⡀⢀⠄⠀⣸⣀⠀⡆
⠀⠀⠀⡂⠀⠀⠀⠀⡿⠸⢀⠀⠀⠀⠀⠀⡀⢤⠈⢸⠷⣶⣇⢸⣅⠘⣒⠐⠉⠈⣆⠋⢡⠙⡏⠁
⠀⠀⠀⠈⡪⠀⠋⠟⠆⠀⠘⢂⠀⠀⠠⠊⡁⢸⡶⠛⣐⣅⣸⠃⣽⢀⣇⢈⠆⣢⠀⠂⠤⠴⠁
⠀⠀⠀⠀⠀⡠⠚⡇⡠⣀⠀⡏⠀⢠⠳⣴⠏⢈⣇⣶⡃⢹⢋⣶⠙⡿⠚⠛⡲⢿
⠀⠀⠀⠀⠑⠀⣂⠁⠀⢃⣁⠤⣀⠈⠚⢃⣼⠟⡿⠋⡿⠛⠟⢳⠾⠶⠸⡆⢥⣸⠇⠀⠀⠄⢀
⠀⠀⠀⠀⠄⠀⣀⠬⢁⣟⡁⠀⠀⠻⠀⠈⠙⢒⣿⡬⠳⡾⢶⠿⡜⢰⣠⣅⣸⠀⡁⠠⠆⠀⠀⠀⡆
⠀⠀⠀⠀⠠⠀⠀⢀⢨⢄⡠⡄⠠⠚⠀⠀⠀⢮⣸⣇⣐⣇⣼⣄⣿⣼⢇⣸⢏⠿⢷⢄⠀⡰⢀⠄
⠀⠀⠀⠀⠀⠁⠀⠩⡀⢈⣧⣉⡄⠀⠀⠀⠀⠈⠉⢥⣿⣀⣏⠸⠥⢈⠀⠁⠀⠐⠀⠀⡟⡤⠄⡀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠄⠠⠆⠀⠀⠀⠀⠀⠀⠈⠁⣻⠉⡟⠑⠁⠀⠀⢂⠐⠀⠊⠔⠁⠀⢸⢂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⡇⠞⠀⠀⠀⠀⠑⠒⠰⠃⠁⠀⠡⣈⡧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠰⣿⣇⣸⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡄⣿⣿⠇⠀⠀⣴⠛⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣏⠉⠀⢀⣰⠏⠀⠀⢰⣿⡏⠁
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⢿⣿⣿⣿⣶⣬⡟⣀
⠀⠀⠀⠀⠀⠀⠈⠿⠟⢱⣾⣿⣿⣿⣿⡟⠀⠁⣀⡴⠞⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠁⠈⣾⠃⠀⢰⡏⠀⠠⠀⠀⠀⣿⣤⣤⣄⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣸⠇⠀⠐⢤⠀⡤⢿⣿⣿⣿⣿⠿⣫⣴⣶⣶⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⠀⠀⣿⠀⠀⡰⠋⠰⢄⠀⣤⣶⣿⣎⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠳⣀⣿⡠⠞⠁⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠐⠚⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠿⢛⣿⣿⣿⣿⣿⡟⠻⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢠⠶⠚⠓⠦⡀⠀⢰⣾⣿⣿⣿⣿⡻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣾⠁⠑⠀⠀⠀⠀⣰⠇⠘⢆⠀⢀⠃⢠⠊⠁⠀⡇⠀⢺⣿⣿⣿⠻⡟⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠈⠑⠂⣤⠤⠴⠒⠉⠀⠀⠈⢂⠀⠀⠈⠳⠀⠞⠁⠀⢸⣿⣿⣯⡀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠐⣿⣿⠆⣠⣤⣄⣀⣰⣶⣶⣄⣀⣀⣉⣿⣿⣋⣡⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⣀⣈⣹⣷⣟⢻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣽⣿⠟⢛⣩⣿⣿⣿⡿⣿⣿⣿⣿⣿⣭⡛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣟⢋⣀⣿⣿⣿⡟⢿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⢻⣿⣷⠿⣿⡟⠁⠘⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠉⢙⣿⣏⢸⣿⣿⣿⠀⣸⠁⠀⠀⠈⠁⣨⣭⣝⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿⣿⣬⣿⠟⠋⠀⣿⠀⠇⠀⠀⠞⠁⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣿⡿⣿⠿⠀⠀⢸⡇⠀⣄⠀⠀⠳⠄⠠⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠛⠋⠀⠀⠀⠀⠀⣾⠁⠀⠇⠐⠄⣀⣀⣀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⢰⡏⠀⣘⣀⠀⠀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠉⠛⢦⣾⣧⢰⣿⣿⣷⠀⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⡿⢣⣤⡝⢻⣿⣶⣶⣿⡿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠒⠖⠊⠁⣸⡇⠘⠿⢧⣼⣿⣿⢻⣿⣷⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣿⣿⣷⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⢸⣿⠀⠀⢰⣿⣯⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⠃⢸⣯⠀⠀⠀⢉⢷⣾⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠶⢾⣿⡤⠴⠞⠋⠀⢙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⢿⣿⣙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣼⣿⡿⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣷⠘⣧
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠀⠀⠹⡄⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢰⠏⠰⠏⠇⣠⠀⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠚⠁⠀⠉⠀⠁⠈⠀⠉

⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡀⠀⡀⠀⠘⢦⡀⠀⠀⠠⢂⡩⠤⠤⠄⢀⠀⣄⡀⠰⠀⠀⣀⣀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠦⣈⡅⢀⡀⠀⢀⠀⡁⠁⠀⠀⠀⠀⠀⠀⠀⢈⠟⠋⠁⣷⠢⠤⠔⢩⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠁⠀⣘⣆⠀⡠⠐⠂⠀⠀⢀⠔⠁⠀⠀⠀⠁⠂⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠔⢠⠊⠐⠛⣀⡀⠠⠤⠷⢆⣀⠡⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠂⠄⠄
⠀⠀⠀⠀⠀⠀⢀⠲⡎⢑⡈⠙⣉⡡⡔⠲⠲⠴⢍⣑⠀⠀⠀⠀⠀⠀⠀⡀⣀⣣⠀⠀⢀⡐⠓⣬⠁
⠀⠀⠀⠀⠀⠠⣀⠝⢙⣉⣉⠉⠛⠓⠋⠑⠚⡶⠲⣀⠀⠀⠐⠀⠀⠀⠀⠀⢨⣈⠷⠾⢤⡜⢻⡉⠹⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣋⣀⣍⣉⠙⠊⠀⠀⡋⢑⠖⠉⠒⠀⠀⠀⠀⠀⠀⠀⠔⢢⡤⠶⢎⡱⡎⢉⠷⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢋⣀⣭⣨⡛⠀⠀⠀⠚⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠂⠠⠦⠼⠟⢦⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⠀⠐⠶⠶⠒⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠈⡹⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⡠⠏⣉⢀⡕⠢⠄⢠⡔⠐⠥⠄⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⣀⡠⠤⠤⠀⠠⠤⠐⠠⠂
⠀⠀⠀⠁⠂⠨⠤⣄⣻⣈⠑⠚⠶⡼⠓⠾⠂⢔⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠉⢀⠄⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡀⢪⣀⣈⠀⠱⠓⢤⠶⢶⢛⡶⠒⢤⢤⡤⠔⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠩⠤⠽⣳⣤⡾⠭⠑⠶⢦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡌⠀⠘⠰⢤⠀⢀⣄⡀⠀⠀⡀⡀⠘⠁⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠁⠀⠂⢀⣀⣀⣶⣜⡿⣾⣇⠤⠚⠍⠁⠀⠂⠀⠀⠈⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⢀⠴⠶⢋⠀⠐⢿⢞⣿⣞⡿⠀⠂⠀⠀⠀⠄⠀⠀⠀⠁⠀⠠⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠴⠢⠤⠴⠲⣏⠀⠀⠀⠀⠐⠀⠀⠻⣇⠿⠄⠈⠄⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡔⡁⢠⠀⠀⠀⠀⠀⠓⡀⢰⣷⣴⡄⠄⠀⣩⡿⠂⠀⠀⠈
⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢺⣿⣟⡟⣷⠏⠀⠀⠀⢨⡕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⢻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣳⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣽⣿⣿⣭⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⣀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡎⠣⠡⠘⡶⠀⢠⡑⡀⠀⠀⠠⡴⢋⠉⣱⠖⢱⣷⣼⢓⣶⠀⢀⠀⢀⠀⠀⠠⠄⠀⢀⡀
⠀⠀⠀⠀⠀⣆⠄⠀⠐⠾⡁⠐⠃⠀⠆⣤⠹⠇⡤⢼⣀⠀⢤⡗⠂⢀⣠⢖⣿⡇⠚⣴⣎⣱⠮⢬⢗⠂⠞⠁⢀⣲⣭⡀⢰⡻⠊⢣⡆
⠀⠀⠀⠀⠀⡼⠢⣤⣤⡴⠶⢶⣤⣤⠴⠳⢦⣄⣉⣶⣇⡶⠶⣵⣶⣿⡀⢸⣿⣧⣬⢾⣿⡟⢳⠉⠜⢁⣦⡖⠛⢳⠀⢽⠛⠒⠲⠚⠓⠒⠁
⠀⠀⠀⠀⠈⠀⠾⡀⠃⠀⠠⡎⢴⣿⡆⢖⡩⠎⠀⣴⣷⡀⢀⠒⡉⠜⠗⣺⠟⠛⢦⣀⢻⣿⡟⠤⠚⠉⣿⣿⠈⠉⠀⠘⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⠿⠓⠁⢀⠢⢫⡌⢿⡇⠀⠀⠀⠐⣾⠃⠂⢯⠈⣧⢻⠛⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠃⠀⠀⠀⠀⠀⠀⢠⠇⠈⢃⠀⠉⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣷⣿⣷⠦⣤⣤⣀⣠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠀⠀⠀⠛⠛⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠔⠒⢦⡀⣠⢒⠀⠠⢄⡰⣀⡂⠀⡍⠒⢄⡀
⠀⠀⠀⢀⣠⠈⠀⠀⢨⠋⠀⠇⠀⠀⠀⠀⠈⠙⢻⠷⠶⠶⠤⠔⠀⡔⠆⢴⣁⠘⠓⡢
⠀⠀⠀⠋⠀⠀⠀⡰⠁⠀⠃⠀⠀⠀⠀⢀⡀⢀⣼⣀⣀⣀⠀⢀⡠⠔⠢⣐⢏⡃⠠⡹⡀
⠀⠀⠀⢰⠀⠀⠀⢳⠀⣀⡤⠴⠴⡊⠙⣁⠉⠹⢀⠀⠈⠉⠙⡿⠓⠖⣶⣿⠈⠱⢆⠀⡅
⠀⠀⠀⢸⢀⣀⣠⣾⠋⠁⠀⠈⠀⠈⢳⡮⠤⠀⠈⡕⠀⡴⠋⢱⢪⠁⠀⠺⣟⠒⣌⣰⠂⣠⠀⠠⢄⠀⠐⠉⠃⢢
⠀⠀⠀⡼⠉⠻⢅⡇⠀⠀⠈⢂⠀⠀⠂⠘⡂⠀⠠⣊⠖⣔⡶⢼⠾⣽⡇⡉⠘⢄⡑⡈⢻⡄⠀⠈⠘⠻⢧⠀⠀⢀⡱
⠀⠀⠀⢠⠐⢀⠜⠀⠀⢀⡐⢨⡀⠁⠀⠀⢁⡠⠊⡋⢀⠮⡗⡚⢷⠲⣧⢟⡻⠃⠜⣁⠔⠧⠀⠈⠀⠡⠀⠁⡀⢸
⠀⠀⠀⡏⠈⠘⢦⠀⠀⢨⣯⣵⣍⣧⣄⣸⡹⣵⣦⠁⠈⢻⡛⢿⣟⣯⡿⡝⢄⠀⠀⠀⠀⠈⠂⣀⠀⢀⠀⠀⠀⡾
⠀⠀⠀⡇⡇⡐⠁⠀⠀⠀⢧⡴⢿⣭⢞⡏⠀⠢⠀⠀⠀⠨⣭⠧⣭⡕⢀⠌⠠⠂⠀⠀⠀⠀⠈⠇⠲⡄⡰⠊⠑
⠀⠀⠀⡇⠇⡇⠀⠀⠀⠘⢷⡓⣎⡻⣾⠅⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⢈⠛⠀⠐⠈⠑⠂⠀⠠⢼⣄⣈⣀⡄⡑⠐⡄
⢤⣦⡀⠃⣶⠁⠀⠀⠀⠀⠈⣟⣻⣓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠄⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠙⠓⠤⠀⠐
⠈⣧⠝⠀⣟⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢢⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈
⠀⠈⣧⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⡀⠀⠀⠀⠀⠀⠀⠳⣆⣠⠀⠀⢀⡍
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠤⠾⠜⡼⢟⣧⣶⡼⢂
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠁⣉⠻
⠀⠀⠘⡆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠦⢄⠀⠐⠄
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡔⢁⠀⠀⠸⣟⢑⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢠⠋⠙⠒⠁⠀⠀⣤⢹⠢⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢐⡈⢾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠁⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠢⠲⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠒⠋⠓⢀⠘⠀⠀⠀⠀⠀⠀⠈⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠋⠀⠀⠀⠈⠀⠜⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠐⢄⡀⠀⠐⢄⠀⠀⠀⠀⠈⠀⠀⠀⢠⠎⠈⠉⠙⠄⡶⠊⠙⠲⣤
⠀⠀⠀⠀⣐⠉⠀⠀⠀⣀⠤⠤⠤⠵⠦⠤⠤⠦⣀⣀⢠⡂⣀⡠⠔⠁⠀⠀⠀⠀⠀⠈⢿⠉⣳⠈⣧
⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢛⡉⠉⠁⠈⠉⠀⠀⠀⠀⠀⢸⡟⢁⣰⠇⢰⡆
⠀⠀⠀⠀⠀⠈⠙⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠤⣴⡏⠀⠀⠱⣄⣀⡀⠀⡔⢲⡤⠖⠲⣎⠉⠉⢁⣠⣞⣁
⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⢡⠀⠀⠁⠘⠋⠓⣆⣀⣼⠛⠻⣿⠙⠋⠉⠑⠚⠓⠤⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠿⣀⠀⠀⠀⠀⠠⠄⣴⡎⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⠀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⡜⠳⠄⠀⢰⠀⠀⠀⠀⠰⡞⡏⠑⢄⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠀⣠⠎⠃⠀⠀⠀⠀⠀⢃⠘⠀⠀⠸⠆⠀⠀⠀⢰⡇⠈⠑⠋⠀⠀⠀⠐⣀⡴⠁⠀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠀⠀⡴⠋⢀⣠⣄⣤⣦⣀⡄⢠⠋⠀⣀⣀⣤⡒⠐⣚⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡀⢀⣄⡴⣦⡴⠳⢦⡾⢛⣿⣿⣦⢀⠀⢀⣽⠷⣾⣿⣿⣿⣿⣿⣿⣿⡇⠐⣿⣿⣿⣿⣿⣿⣷⣦⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠰⠚⠗⠒⠋⠉⠁⠩⠀⠀⢀⣀⣤⡆⢸⣿⣿⡏⠉⢹⣿⣿⣶⠈⠉⢿⢿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⠟⣽⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢀⣾⣿⣿⣧⠀⠙⠋⠙⢁⣾⠻⣿⣿⢦⠤⢸⣿⣅⣈⠛⠙⢻⣿⣷⣾⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⡀
⠀⠀⠀⡔⠃⠀⠀⢀⡠⠤⠬⠶⠄⠤⠽⡆⢸⣿⣿⣿⣿⣶⢲⣿⣍⢹⣷⣍⠛⠀⣠⣾⢿⣿⣿⣦⣀⡤⠫⠈⡉⠛⣿⣿⣿⣿⣯⣍⣉⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠟⠉⢉⡉⢹⣿⣿⡇⠀⢹⣿⣿⠿⠻⣾⠻⣿⣟⠋⠀⡀⠛⠤⠆⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠈⠉⠳⠂⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣶⣶⣿⡧⠘⠿⣿⠛⠀⣸⣿⣿⣄⡀⢸⣧⣀⡀⠠⣴⣁⠄⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡄⢽⣿⣿⣿⡏⣾⣿⣿⠃⠤⣀⣠⣴⡋⠻⠿⠟⠉⢠⣿⣿⣇⠀⠀⢻⣾⣿⣶⡆⠀⠀⠀⠀⠠⡈⠛⠿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⠀⠀⠘⡀⢸⣿⣿⣿⡇⠹⣿⣿⠷⠀⢸⣿⣿⠷⣦⣤⣤⣶⣿⡏⠉⢻⡟⠋⢈⣿⣿⣿⣿⣷⠦⠀⠀⠀⠰⠀⠀⠹⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠰⠞⠑⣆⡈⢀⣀⣾⣄⠈⠁⢀⣠⡟⣿⣿⣷⠄⢸⠏⢿⣿⡷⠖⢸⣧⠔⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣦⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣇⠀⠠⣴⣿⣶⣾⣿⡿⠟⣹⣿⣿⡿⠛⢿⣿⣦⣀⠙⢁⣠⣾⢷⣄⢉⣁⣠⣼⣿⠋⠹⣿⠿⠟⠻⠗⡴⠓⠔⠓⠉⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠐⣿⣿⣿⠟⣉⣠⡄⠘⣿⣿⣧⡀⢠⣿⣿⣿⡛⠛⣿⣿⣷⡉⠉⢻⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠛⠉⣡⠞⠉⣀⣀⢄⡀⠀⢀⣠⡟⠙⠟⠋⠉⣠⠟⣿⣿⡿⠒⢠⣿⣿⣿⣿⣿⠂⠀⠀⠠⡖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⣨⠇⢈⡿⢻⣿⣿⡗⠒⢲⣾⣿⣦⣀⣉⣁⣠⣾⣿⣿⣿⠿⣧⠀⠀⠠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢷⡘⠧⡤⠞⢑⡀⠛⠟⠛⠂⣼⣿⣷⡄⢹⣿⣿⣿⣿⣿⣿⠇⠠⠉⠁⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠖⠋⠉⢰⣦⣴⣾⣀⡉⢀⣠⣮⡘⠿⣿⠟⠟⠻⠶⢦⣀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠉⠻⣿⣿⣿⣄⡀⣿⠀⠀⠀⠀⠀⠈⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠆⢰⣷⡙⠋⠉⡡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣤⡴⣿⣿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⣿⣿⣿⢸⣿⣿⣇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠙⠟⠙⢂⠙⠛⠉⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣶⣏⢰⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⠋⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠋⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⡀⠀⢄⡀
⠀⠀⠀⠀⠀⢦⠀⠀⠊⠀⠈⢣⡐⠮⡝⠀⠈⠓⠮⠒⣧⣀⣰⡀⢈⡒⠄⢀⠀⠀⠀⠀⡠⠄⣄
⠀⠀⠀⠀⠀⣸⠴⠊⠀⠀⢀⠎⠀⡀⠁⠀⠀⠀⠀⠀⠀⠉⠟⠙⠛⠏⠙⠀⠀⠎⠑⠚⣤⣈⠁⠨⡃
⠀⠀⠀⠀⠄⠉⠀⠀⠀⢠⠃⠀⠀⠁⠀⠀⠀⣀⡠⢄⠀⡰⡂⠠⣤⣀⡀⣀⠀⠈⠑⢤⠟⣭⠒⠠⡣
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⡠⠔⠲⠲⣉⠈⣤⡄⠀⠧⡀⠀⠀⠀⢙⡟⢉⠓⡯⣾⢇⠀⠅⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣄⡀⢀⢰⠟⠀⠀⠀⠋⠉⠀⢰⡶⠤⡀⠄⢨⡂⢀⡞⠁⣃⢻⠘⠁⠒⠏⠳⢲⢄⡔⠀⡤⠀⠠⢄⠀⠀⠈⢙⠀⡆
⠀⠀⠀⠀⠀⣼⠋⠸⠧⣸⠀⠀⠀⠙⠂⠀⠠⠂⠀⠐⠀⠀⢐⡡⢪⣖⠦⣽⠶⣿⣇⡈⠀⠆⡈⠒⠜⠧⡄⠀⠉⠚⢻⠯⡀⠀⠀⢉⠆
⠀⠀⠀⠀⠀⢠⠀⠂⡠⠁⠀⠀⢀⠀⣰⠀⠁⠀⠀⠙⣀⠀⢘⠃⢠⣝⠷⠞⢷⠾⣽⡴⢓⡶⠄⣒⣀⠼⡅⠀⡈⠀⢀⠀⠁⢒⠀⡅
⠀⠀⠀⠀⠀⠘⢀⠐⢥⡀⠀⠀⣸⡗⣩⠑⣄⣀⢠⠊⣋⣤⣸⠀⠳⠶⠶⣷⠞⢞⣻⡗⠁⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⢸⢀⠞⠁⠀⠀⢙⣏⣩⣟⢓⣍⣟⠃⡨⠍⠁⠀⠀⢹⢓⣋⣛⣩⠅⣨⠀⠀⠀⠀⠀⠀⠀⠐⠋⢤⠈⠀⡄⠒⡌
⠀⠀⠀⠀⠀⡄⡀⢸⠀⠀⠀⠀⣦⣭⣧⡽⢯⣵⡂⠀⠀⠀⠀⠀⠀⠈⠻⡭⠉⠁⠐⢤⠄⢀⡠⠀⠄⣀⢀⠀⠀⠀⠱⠈⠀⠠⠄⡀
⠀⠀⢀⠀⠀⡇⡇⠇⠀⠀⠀⠀⠀⠳⡦⢳⣮⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡏⢳⣧⠚⠀⠘⡄
⠀⠀⢻⠏⢤⠁⡇⠆⠀⠀⠀⠀⠀⠐⠷⠾⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⠲⢠⠄⡏⠠⠈⠀⠄⢨
⠀⠀⠈⢿⡂⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡆⠀⠀⠀⠀⠀⠀⠑⠀⠈⠇⠀⠀⠀⢠⣬⠁
⠀⠀⠀⠀⢗⢺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡇⠀⠀⠀⠀⠀⠀⢀⠘⢷⣴⡀⠀⠀⡬
⠀⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⠀⠖⠛⠘⠒⢻⢷⡿⡒⣾⡄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠀⠨⡉⠳
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢬
⠀⠀⠀⠀⠀⣇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡀⠄⠠⣿⠓⣢⣶
⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⡰⠘⠒⠄⠊⠀⠀⣀⡎⡧⠑⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠌⣠⠇⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠁⠒⢡⠣⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⡴⠒⢀⡤⠀⠀⠒⠒⠐⠂⠀⠀⠀⠤⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠒⠤⢬⡇⣤⣸⣀⠀⠰⠶⠖⣠⠀⠉⠛⠋⠁⠒⠒⠀⠀⠉⠀⠀⣼⠗⠒⠒⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠶⠿⠄⢀⣀⢨⣭⣥⣐⣦⡀⢀⣅⣤⣤⡤⠄⠴⠶⠶⠒⢰⢟⠀⡀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠋⠉⠉⣎⡩⢿⡉⢉⠒⠂⠶⣤⣨⣬⣧⠤⠄⢸⠰⣾⡿⣟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣲⠋⡀⣨⠀⢄⣻⣿⡙⡧⣏⢾⡼⡽⠃⠀⠙⠲⣄⡀⠀⢘⡇⢰⣪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠘⠘⢶⡃⠀⠀⠐⠀⠚⢺⢿⡗⣿⡻⢯⡞⠁⠀⠀⠀⠀⠈⠻⣷⣼⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄⠁⠁⠀⠀⢿⠐⠃⠄⠀⠁⣀⠀⠈⢽⡟⢼⡛⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠤⠀⠠⠀⠐⠈⢁⠊⠀⠁⠐⠀⢔⠀⢫⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡄⠈⠀⠩⡌⠑⠠⠀⠂⠁⠀⠀⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣰⠀⠀⠀⠐⠀⢀⠀⠃⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡐⠂⠱⡀⠀⡴⠻⡄⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⢥⢄⠀⡀⢋⡠⠔⠋⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢁⠀⠄⠀⠽⠉⠧⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡠⠄⠌⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠄⠀⠀⠁⠀⠀⠀⠀⠜⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠃⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢔⠀⠤⠀⠐⢡⠀⡀⡀⠀⠀⠀⢀⠄⢀⢌⣦⠔⠠⣠⠈⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠌⠐⢦⣄⣅⠈⠀⣀⢀⡀⠀⢀⢄⠹⣄⠈⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⢀⠀⢱⡄⠈⠌⠀⠀⠟⠂⠀⠉⠉⠿⡉⠀⠢⠀⡀⠀⠘⢂⠀⠀⡠
⠂⠀⠀⠠⠄⠀⠱⠖⠒⠊⠉⠉⠀⡟⠉⠀⠀⠁⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠐⠉⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄
⠀⠀⠀⠰⠀⠀⠀⠀⢑⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠘

⠀⠀⠀⠀⠀⠾⡯⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠶⡀⠀⠀⠀⠠⠀⠐⠂⠆⠀⢺⠧⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠶⣆⠸
⠀⠐⠒⠒⠒⠶⣖⠉⣠⡤⡶⢤⣤⢤⡤⣄⣀⠀⣀⣀⣤⡤⠬⣽⡶⠗⠒⠒⠶⠦⠶⡦⢀⣠⣤⣶⣤⣟⠃⠀⠀⠀⠸⠇⠀⠳⣷⠀⡤⠂⣊⠉
⠀⢀⠰⠞⠓⢶⠟⠚⡿⠂⡀⢀⠈⠆⠴⠞⠛⣓⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⡼⠆⡀⠀⠑⠀⠀⢰⢀⠘⠈⠀⠰⣿⣿⡀
⢐⡉⠚⡇⠄⠈⣷⠘⠋⡘⠣⣸⣷⡞⠀⠰⣾⣿⣿⣿⣿⣧⠘⠒⠒⠲⣶⡀⠉⠉⠉⣹⣿⣿⣿⣿⠀⠀⣀⡤⢶⡄⠀⣠⠀⠈⠀⠀⠀⠀⠃⠀⠁
⠋⢁⣀⣀⣿⠷⡟⠛⣰⣤⣶⣿⣿⠑⠲⠶⢿⣿⣿⣿⣿⡿⢐⡀⡀⢀⣛⣃⣀⣤⣴⣿⣿⡿⠛⠒⠛⠁⠀⠀⠀⠀⠲⣤⠀⣰⡀⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢶⠞⠁⠈⢻⣿⣿⡿⠇⠀⠀⠀⠻⡿⣿⣿⣿⠁⠈⠉⣉⣿⡍⠀⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢾⣠⢾⣇⡏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡃⠀⠀⠀⠀⠐⢟⣿⣿⡦⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠉⢠⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡣⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⠉⣀⣴⠍⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣿⠀⠀⠙⡾⠐⠋⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⠀⠀⢸⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⠈⣿⠀⠀⣵⡀⢀⣼⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡟⠚⠁⢹⣛⠘⣌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠐⢿⠀⢠⡞⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⡾⢀⣼⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⣇⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠂⡁⠋⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠄⣿⡶⠋⢹⢹⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢂⣤⣿⡷⢄⡞⢻⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⡎⠀⠸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠁⠀⢠⣧⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣠⡇⠈⠻⣿⠀⠀⠈⢻⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⠾⠿⠻⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢈⣇⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠹⠗⢚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢧⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡖⠁⢸⣿⣅⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣾⢈⣿⣿⣿⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢈⣿⣿⣿⣿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠁⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⡴⡏⠀⠀⣿⣿⣟⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡃⠀⠀⠉⢠⣬⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⣛⡆⠀⢠⡋⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠧⠥⠁⠀⡼⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠀⠀⠀⡿⣦⠜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠋⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⠆⠀⠀⠀⠀⠀⠠⠶⠃⢲⡤⠤⠤⢀⠀⠀⠀⣀⢤⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢈⣶⠶⢛⠀⠀⠀⢮⣾⠇
⠀⠀⢀⡼⢋⡀⣰⣶⣶⡆⠦⢤⣔⡬⡽⣭⣿⣷⠦⠬⢬⡭⡵⣾⡿⣍⢳⡑⣤⣦⡴⠖⢾⣿⣷⢹⢷⣤⠭⠬⡶⣾⣾⣥⣛⣏⣡⣤⣤⣖⠒⠂⠈⠁
⠀⠀⠈⠁⣿⡿⢿⠟⣿⠗⠀⠀⣿⠖⢾⣿⢹⡞⠣⠴⣾⣷⠆⡛⢷⠻⠀⡗⣟⡿⠤⣤⠌⠻⠋⠇⣼⣿⡆⢀⣸⣿⢏⡽⠀⢾⣷⢚⣿⡝⡇⠀⠀⡟
⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⢹⣹⣿⣟⡆⢻⠇⠀⠁⠑⠀⣸⢁⡞⠀⣞⣵⣷⣴⠀⠀⠘⡿⠀⠀⠈⠟⠉⠁⠀⠈⠃⠀⠀⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣧⠙⠁⠀⠀⠀⠀⠀⢠⢻⡟⠀⠀⠻⣿⡭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⢿⣵⣿⠿⣿⣦⣤⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠛⡿⠋⠁

⠀⠀⠀⠀⠀⠀⢀⠔⠒⢦⢀⣀⡔⡆⠀⢠⣄⡴⣂⡒⠀⢈⠑⠢⣀
⠀⠀⠀⠸⠀⣀⠌⠀⠀⢈⡗⠊⠹⠀⠀⠀⠈⠁⠈⠛⠾⣾⢦⣴⠦⣁⠰⠀⡠⢤⢀⡞⠁⠜⠠⢄
⠀⠀⠀⢰⠋⠁⠀⠀⢠⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠉⠀⠀⠀⣱⢄⠀⢽⠳⢀⠀⢣
⠀⠀⠀⠀⡀⠀⠀⠀⣇⠀⠀⠀⢀⡀⣠⡤⠶⠊⠒⠲⠏⠑⠒⠒⠲⢦⣴⠏⣀⣀⣱⡏⠽⣄⠈⠊⠂
⠀⠀⠀⠀⡇⠀⠀⠀⣹⣠⠔⠚⠉⡍⠑⢦⣈⠺⠀⠀⠑⠄⡀⠀⢀⠴⡇⣠⠅⠜⢛⣧⣄⢈⠳⢈⡃⠀⠀⠀⠀⠀⠀⢀⠤⡄⡀
⠀⠀⠀⠀⣇⡶⣄⠖⡟⠀⠀⠀⡀⠀⠀⠀⡙⢃⠓⠒⠂⢘⡅⠰⠃⡀⢼⠸⣤⡀⢠⡙⡎⠩⡖⢾⣀⠜⠃⠈⣙⣆⣀⡀⠀⠀⠈⣆
⠀⠀⠀⠰⠃⢀⠙⢲⠃⠀⠀⠀⠈⡀⢀⠈⠀⠈⡄⠀⠀⣧⠘⠱⣮⡛⢺⡛⠳⢇⢠⣀⡐⡤⢉⠌⠹⡷⠀⠀⡀⠁⡟⠢⡀⠀⢀⡬⠂
⠀⠀⠀⢠⡆⡄⡠⠃⠀⠀⠀⣄⡑⢥⡈⠀⠀⠀⣁⣤⠊⣨⠀⡸⠪⢛⣋⢻⡛⡿⡞⣩⠝⠓⠬⠤⠔⠳⠀⠈⠀⠀⠆⠀⠈⠄⢰⠂
⠀⠀⠀⢸⠁⠃⠓⡦⠀⠀⢠⣿⣩⣦⣙⣆⣄⣼⡸⣤⣦⡰⠀⠙⣛⠛⢿⣏⣫⣽⢏⠡⡀⠀⠀⠀⠀⠀⢳⢀⡀⠀⢁⠀⠀⠀⠸
⠀⠀⠀⢸⢰⠀⡜⠁⠀⠀⠀⢽⣀⣾⣏⣱⡬⡍⠢⠒⠄⠀⠀⠀⢸⣩⣧⣩⣵⠂⡠⠃⠸⠀⠀⠀⠀⠀⠀⠫⠑⢆⠈⡰⠊⠱⠁
⠀⠀⠀⢸⢸⢸⠃⠀⠀⠀⠰⢶⠽⢴⠛⣧⣞⡁⠀⠀⠀⠀⠀⠀⠀⠹⠦⠂⠀⠈⠳⠂⠀⠤⠒⠒⠤⠀⠀⣐⣆⠀⠃⠀⡀⠲⠠⡄
⠀⣀⠀⠸⢸⡸⠀⠀⠀⠀⠀⠀⢳⠾⣿⣬⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉⢿⠆⡁⠀⠸⡀
⠹⣏⠳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠙⠖⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠈⠀⠆⠈⠢⠀⣨
⠀⠹⡗⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢡⣀⠀⠀⠀⠄⠀⠠⠽
⠀⠀⢹⡲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢤⠤⠂⠀⠀⠀⠀⠀⡀⡙⢦⣦⡀⠀⠀⡔
⠀⠀⠸⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠀⠀⠀⠰⠚⠓⣛⠞⡯⣶⡟⡶⣣⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠄⠁⢌⠙⢧
⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⡄⠀⠀⠄⢨
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢁⠀⠀⢸⣟⠒⠤⡞
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢋⠜⠲⠮⠔⠁⠀⠀⢐⢽⣎⡃⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠊⠀⠀⠀⠀⠀⠀⢠⠋⣸⠀⠠⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠐⠖⠘⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠤⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠂⠐⠓⡄⠀⠈⠁⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⣄⡀⠀⢀⣤
⠀⠼⠀⠨⠁⠈⢃⠀⠀⠀⠀⠀⢀⠴⠀⠓⠂⠐⠀⠂⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠇⠀⣀⠀⠀⠆⢉⠒⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠠⠚⠀⠀⠀⠁⠀⠀⠠⠀⠢⠠⠀⠄⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣇
⠃⢄⡀⣥⠀⠀⣚⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠠⠼⠃⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠀⠀⣠⠖⠀⠀⠀⠢⠀⠀⠀⠈⠀⡀
⠀⠀⠉⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⡀⠄⠐⠃⠀⠀⠀⠀⠀⢀⠀⠀⠁⢀⠀⠀⠠⠀⠀⡀⠉⠞⠅⠀⠀⠀⠀⠀⠐⣀⡀⠀⡤⣷⠑⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣈⡆⢑⢠⡞⡄⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⣀⠐⠀⢹⠌⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡄⠶⠀⠓⢏⠑⠀⠍⠋⠃⠈⡁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡠⠃⢀⢠⠖⠀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠔⠒⠀⠀⠀⠈⠉⣠⠀⠈⠀⠀⠀⢀⡄⠀⠀⠀⠀⡄⠀⢦⣄⣞⢳⡂⠂⠐⠄⣙⡇⣀⢠⠊⠘⣄⡠⢀⣀⠤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠄⠀⠀⠄⠛⢩⠀⠀⠀⠀⠀⠀⠇⠈⡀⠔⠁⠀⡀⢄⡜⠉⠈⠁⠃⠰⠈⠛⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠈⠄⠀⠀⠀⠂⠂⢀⠀⠡⠀⣡⡻⢱⡤⡎⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠀⢀⡉⠀⡄⠀⠀⠃⠀⠙⠀⠀⠀⠨⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⡀⠠⠉⠀⠁⠀⠀⡀⢀⠀⠆⠀⢺⡄⠀⣠⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠈⠉⠃⠀⠀⠁⠀⢀⡬⡁⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢎⠁⠀⠀⠀⠀⡀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢸⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠸⢆⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠤⡀⠠⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠐⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡁⠈⠀⠀⠀⠀⠀⠁⢠⣀⠤⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠚⠂⠀⢀⡄⢶⡃⠀⠀⠀⠀⠂⠠⡥⢙⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠤⡸⣷⠏⢀⡁⢀⠈⠀⠀⡁⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢈⢰⡈⠀⡀⢤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠙⣄⠡⠦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠁⠀⠀⠀⠘⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠍⠁⠀⠀⠰⢋⣠⡶⠀⠈⠉⠛⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠗⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠿⠀⠀⠀⠀⣹⠀⢠⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠘⣧⣘⠃⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣴⣆⣤⣴⠄⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⣻⡿⢿⡕⠀⢀⣴⠂⠐⢿⡿⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡿⢿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠉⣠⡾⣫⣾⡿⠃⢰⣿⡁⠀⠀⠈⣠⣾⣿⡇⠀⠀⠀⠙⢶⣟⣩⣴⣾⢟⣿⠏⠉⠛⢿⣿⠆⠒
⠀⠀⠀⣠⠖⠋⠀⢀⣾⣿⡿⠋⠁⠀⠀⠉⠀⠀⠀⢀⣼⣿⣿⣿⡄⠀⠀⢀⣴⣾⣿⢣⣾⣻⠟⠁⣀⣴⠞⠑
⠀⡠⠾⣤⣀⣀⣠⣿⣟⠉⠀⠀⠀⠀⠀⢁⠀⠀⠀⣾⠿⠿⠟⢻⠃⢀⣴⣿⣿⣿⣿⢋⠞⠁⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⢀
⡀⢀⣠⢶⣼⣿⣿⠿⠟⠀⡠⠀⠀⠀⢠⠂⠀⠀⠀⣨⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠁⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣟⣀⣲⠶⠶⠶⠶⢶⠄
⠻⣿⡛⠻⣿⣿⣿⣿⣿⣿⣷⠀⣠⢀⡏⠀⢀⢾⣿⢋⠀⠀⠀⠀⠀⠛⣉⠭⢟⡚⠋⠉⣥⣥⠤⠴⠶⠶⠶⠶⠶⣤⣤⣤⣬⣉⣉⡉⠙⠓⠒⠀⠀⠀⣀
⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣧⢀⡇⠈⡇⠀⠉⠚⠃⠈⠀⠀⠀⠀⠀⣡⣴⣿⣛⡠⠤⠤⠐⠒⠒⠒⠒⠚⠁⣉⣉⣁⣈⣛⣦⣶⣿⣿⣿⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⡎⠁⢠⡧⠀⠀⡄⢲⡄⣰⢀⠀⠀⡐⠛⢉⡁⠀⠀⢤⣴⠶⠶⠷⠛⠛⠻⠷⠒⠶⠼⠿⢿⢿⣿⡿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⡾⠁⠀⠀⢣⣼⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⢀⣀⡤⡖⠋⠀⢀⡞⠁⠠⠀⠀⢸⣿⣿⡟⢰⠂⠀⣀⠤⠶⣶⣾⠶⠾⠽⠛⠛⠛⠋⠛⣛⡛⢛⣙⢿⣿⣿⣿⣿⠏⣐⣒⡠⠤
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣠⠎⣠⡄⠠⠂⠀⣸⣿⣿⢃⣯⡖⢋⠠⢂⣉⣀⣤⠀⠒⠒⠤⣤⡀⠠⣬⣙⠻⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠖⠁⠀⠀⠠⢰⠛⠿⠃⢸⣿⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣷⣿⣿⣿⣿⣿⠷⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⢤⡏⠀⠀⠲⠈⣿⣷⡦⠴⠄⠻⣷⣶⣤⣀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⡻⢦⠀⠀⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⣄⣙⡓⢊⠻⠣⠩⠥⠄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣮⠙⡆⠀⠀⠀⢰⡟⢶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡄⠙⠛⠿⢿⡷⠶⣶⣤⣤⣶⠂⣬⣿⣿⣿⣿⣿⣿⠎⠀⠀⢀⣀⠈⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠷⣄⡀⠤⠠⢶⣶⣄⠉⠛⢿⡿⢶⣬⣛⢿⡿⢿⠟⠁⠀⠀⠀⠉⠉⠛⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠨⠿⣷⣤⣤⣈⡙⠳⣶⣤⣬⣶⣎⡙⠿⣶⣦⣤⣤⡄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⡀⠀⠀⠀⣀⠤⠴⠆⢶⣮⣍⣛⠛⠿⣿⣿⣿⣯⣹⣿⢷⣄⡙⠿⠋⠀⠀⠀⠀⢀⠠⠐⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠃⠀⠀⠠⠀⠀⠐⠟⠛⠋⠁⠀⠀⣀⣀⡉⠉⠉⠉⠆⠈⠉⠀⠀⠀⠀⠀⠀⠚⣍⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣤⡶⠿⠛⡉⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣈⣭⣍⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠟⠀⠀⠀⣠⡤⡀⠀⠀⠀⢠⣴⠟⠋⠁⣀⠄⣈⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣉⣠⣤⣠⣼⡿⠟⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⢀⡀⣤⣴⣿⡟⠉⠀⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⢛⣛⣯⣭⣤⣴⣶⣶⣶⡷⢀⣨⣽⣿⣥⡴⠶⠖⠛⠃⠀⠀⢆⣉⣠⣶⠶⠄⠀⠀⣤⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⣀⠀⡴⡏⠙⢛⣿⣿⣟⣻⣿⣏⡑⠈⠁⠘⢋⡱⠶⠚⠀⠀⠀⠀⠀⠀⣈⣩⡿⠗⠀⠂⠀⠈⠉⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠉⠒⠋⠩⣈⡉⠉⠉⠉⠉⠉⠁⠄⠠⠀⠘⠋⠁⠘⠀⠀⠀⣠⠴⠚⠉⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣈⠉⠳⢄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢉⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦⡀⠀⠀⠈⢀⣤⡶⠾⠿⠛⠁⠀⠀⣀⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉

⠀⢐⡀⠀⣂⠀⠀⠀⠄⠀⠀⠀⢜⠀⠀⠆⠀⠀⠀⢤⢀⠠⡀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠠⠄⠄⣠⢀⠄⡀
⠠⠈⠂⠐⣈⠂⠀⡄⠂⠄⠀⠁⠉⠀⠂⡥⠆⠑⣈⢰⣇⡠⠋⠢⡀⣿⣰⣧⣝⣀⣇⣷⣅⣿⢀⠔⠀⠀⡤⠀⠠⡄⠀⢠⡔
⠀⡀⢀⠀⠄⠄⡀⢔⢊⠌⠀⡨⣀⠀⠀⠰⠀⠰⠢⢓⠀⢳⡄⠐⠀⠙⠛⠛⢻⡿⢛⡛⠛⠁⢨⠀⡳⠆⠀⣼⡄⢡⣶⡄⠁⢳⡇
⠀⠁⠁⠂⠀⠸⠅⠃⠀⡂⢀⠦⢀⠀⠁⢉⠀⠀⠾⠂⢠⠂⠀⡠⠊⠖⠀⡀⠈⠀⢠⡄⠀⠈⠀⠁⠠⠘⠠⣀⠔⠢⣀⠄⠁⠤
⠀⡠⢂⠅⠰⢈⢐⠀⡨⠇⣀⠪⠂⠁⢀⠰⢁⠀⠀⠠⠂⢰⡁⠄⠀⠀⠀⢧⢀⠀⡇⢩⠀⡞⢰
⠠⠠⠠⢐⠀⠂⠂⠂⠰⠃⠀⠄⠡⠂⠂⠀⠠⠃⠠⠄⠠⠊⠧⠤⠤⠀⠂⠈⠄⠀⠀⢸⠀⠇⢸⠀⡀⡤⠀⠠⡢⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡀⠐⢖⠂⠡⠀⠁⠀⠀⠔⠀⠈⠀⠀⠈⣹⡄⠀⠀⠐⠤⠘⠀⡄⢼⠀⠀⠈⢀⠀⠀⠀⢀⠅⠘⢂
⢀⠄⠀⢠⢀⠀⠀⣉⣂⠐⠻⠲⡶⠲⡶⠆⠀⠰⢆⠂⢒⠶⡀⠁⡁⠈⠁⡀⠀⡐⠈⠀⠁⢀⠀⠀⠀⣄⠀⢠⠀⠀⠌⠈⡀
⢀⣤⣤⣄⡏⢄⢾⣿⣿⠀⠀⠀⠹⠜⡀⠈⠤⠂⠈⠳⠎⠀⠈⠒⠀⠀⠂⠂⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠘⠈⠑⠠⠆⠀⠀⠠⠘
⠀⢿⣿⡟⠀⠀⠈⠋⠙⢆⠀⠀⠀⢀⢆⠀⠀⠠⠀⢀⠀⠲⠲⠒⠒⠒⠖⡤⢄⠀⠀⠀⠀⠞⠉⠀⠐⠄⠀⠀⠆⠀⠀⠀⣀⡀
⠀⠀⢖⠀⠀⠀⡠⠒⡀⠘⠣⢀⠠⠼⠸⠄⢀⠀⠀⡞⠀⢀⠀⢀⠐⠀⡆⠀⢘⠆⠘⠋⠁⠀⠀⠀⡀⠈⢆⠈⠀⠀⠃⡀⠉
⠀⠂⠛⠑⠀⠀⠅⠀⠀⢂⠀⠔⡀⠀⠀⠀⢰⡀⠀⡁⢈⠁⠄⠁⠈⠀⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⡀⡀⠀⠁⠀⠈⠂⠁⠀⠐
⠀⠀⠀⢰⡄⠸⡀⠀⠀⡘⢸⠠⠄⣁⠀⡀⠡⢸⠀⢃⢄⠀⠐⠐⠢⠀⠠⠄⢀⠇⠀⠀⠀⠀⠀⠤⠀⠸⠄⠀⠀⡀⠈⠁⠈⠂
⠈⡐⢠⠃⠱⡀⢡⠀⢀⠀⡸⣀⠀⠀⠀⠀⢐⠸⠀⠀⠘⠀⠤⠤⠤⠤⠄⠐⠁⠀⠰⠆⢰⣶⠞⠉⠁⠀⠙⢶⣶⠀⠀⠐⠈⠄⠁
⠀⠀⠎⠀⠀⠰⠀⠉⠈⠀⣈⠂⢩⠄⠀⠬⠀⠊⠀⣄⠀⠀⠀⢀⠀⠢⠉⠁⢐⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⢸⠁⢀⠠⠀⠒⢐⠢
⠀⢸⢀⡀⣀⠀⢇⠀⣸⠀⠉⠀⠙⠁⠀⠀⣠⠊⡀⠊⡀⣀⡂⠘⡬⢄⣲⠅⠈⡀⠓⠁⠀⠈⢀⡀⠀⠀⣠⠊⠀⢀⠘⠂⢒⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⢀⢐⠀⠀⡰⠁⠀⠀⢠⡅⠃⠀⠀⠀⠀⠀⡀⢀⠅⠀⠴⠢⠀⣤⣬⣷⣯⣤⣇⠀⠀⠑⠀⠐⢂⠁
⠀⢀⡀⠙⢋⢄⠀⠈⠢⠈⠈⠈⠀⠀⠐⠧⠐⠂⠉⢀⡀⡀⠀⠀⠀⠉⠐⠒⠁⠐⡿⠠⡤⠤⡄⡉⠉⣉⣉⣀⣀⢀⠀⠠⠄⠺⠃
⢀⢮⣈⠻⠧⣔⠢⡄⠐⠠⠀⠀⠀⢀⠆⢀⠀⢠⠞⠁⠀⠰⠞⠉⠁⠀⠀⠀⠂⡀⠀⢠⣀⣀⡃⠀⠀⠀⠀⣰⠀⠀⠀⡀
⠘⠉⠀⠤⠀⠐⠀⠅⠐⠀⠊⠀⠈⢂⣀⡈⠀⠐⠀⠁⠂⢐⠆⠼⠀⠷⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠈⠁⠈⠁⠉⠀⠁
⠈⠢⡀⠀⠀⣀⠤⠀⠚⠒⠀⠀⠂⠀⠙⠀⠀⠃⠒⠀⠀⠌⢸⠀⣇⠀⠍⠀⠀⠀⠘⠄⢱⠀⢉⠀⠀⠎⠂⠰⡰⢢⠚⠁
⠀⡠⠬⠉⠀⠀⢄⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠌⠀⠀⠲⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠂⠐⠃⠀⠊⠀⠐⠂⠀⠈
⠀⠀⠀⡖⠲⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣰⡌⢄⡸⠀⠀⡂⢀⠀⢀⠂⡂⠀⠚⠈⠇⢹⠈⠐⠄⠀⠡⠀⢡⠋
⠀⠒⠀⠀⠀⠀⠚⠀⠁⠂⠤⠁⠁⠀⠈⠉⠈⠒⠒⠒⠒⠒⠀⠂⠀⠁⠀⠑⠀⠈⠂⠘⠑⠃⠐⠂⠐⠁⠊⠐⠒⠀⠀⠀⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⡀⠀⣤⣤⣠⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡎⠁⠀⢠⠫⣿⢿⠀⠀⢹⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⣼⣷⠀⠀⠃⠀⢀⡏⠀⣀⣸⣋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣍⣿⣯⣕⣧⢸⠀⣄⡘⠀⠈⢛⡟⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣯⣷⢹⡾⢨⠿⠏⠓⢀⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⢸⡾⠁⠀⠀⣰⣫⣾⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠃⠀⢀⡎⢛⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢀⠀⣸⣥⠎⠉⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣾⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⣤⣼⣦⣻⢷⡄⠀⠀⠀⠀⣀⣀⠠⠤⢤⣞⣿⢿⣿⣟⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣼⣿⣿⠃⢈⡻⣿⣿⠀⣀⢸⣿⢇⠀⠀⠀⠀⢿⣟⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣟⣃⠀⢈⢲⡦⣿⡆⢸⡙⣏⣹⣇⣄⣠⡴⠟⢉⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⡝⠘⠻⡧⠀⠀⠀⢸⡇⠀⣴⠙⢿⠿⠛⠁⠀⠀⠈⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠰⢿⣿⣿⡜⣿⣷⡆⠀⠃⣮⣀⠀⣸⠘⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣾⣿⣿⣷⣮⠛⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣄
⠀⠀⠀⠀⠀⠀⢛⣿⣧⣿⣿⣿⣿⣿⣿⡛⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⣇⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⣙⣻⡇⠀⠀⠙⠷⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣷⣾⣿⣿⣿⣿
⠀⠀⢠⠀⠀⣴⣾⡿⢏⣿⠃⠀⣰⣿⡇⠀⠐⠋⣀⡼⠛⠙⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⢀⣾⠟⣷⣿⣿⣧⣀⣼⣿⣷⣾⣾⣶⣿⣟⡋⢀⡀⠀⢻⣷⡀⢀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠟⠛⠻⠿⠛⠏⠰⠿⠿⠻⠿⠍⠥⠯⠭⠼⠧⠾⠿⠦⠽⠿⢷⡶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⡴⠘⡄⢀⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⢸⠁⢧⢰⠁⠹⡀⠀⣷⠃⢰⣣⠖⠁⡤⠖⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡀⠈⠺⡄⠈⢿⠀⠀⠹⡄⢹⠀⡼⠁⠀⢰⠀⢀⣾⠤⠖⣲⠶
⠀⠀⠀⠀⠀⢠⡀⠀⢃⠀⠉⠒⢤⣙⢦⣸⡄⠠⠀⢻⣼⡾⠁⠀⠀⡼⢀⡞⠁⣠⣞⣁⡀
⠀⠀⠀⠀⠀⠈⢏⠓⠮⣷⡀⠑⢄⠈⠻⣿⣧⣠⣧⣼⣿⣧⣴⣇⣰⣷⣯⠴⠚⠉⠀⢀⣆⣀
⠀⠀⠀⠀⠀⡤⢬⣷⣄⣀⡙⢦⡀⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⠔⠈⣀⠔⠁⠀⣀⠵
⠀⠀⠀⠀⠀⠱⣄⡠⠤⠬⠭⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣞⣁⠤⠖⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠊⢄⡀⠀⠀⠤⠤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⠓⠒⠶⢶⣒⠒⠋
⠀⠀⠀⠀⣤⠤⠤⠭⠟⠒⠶⢦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁⠀⠀⣀⡈⠕
⠀⠀⠀⠀⠈⠓⢦⣤⠴⢚⡽⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣭⣉⡉⠉⠀⠙⢆
⠀⠀⠀⠀⠀⢖⡉⢀⠔⠉⠠⠔⠊⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠧⡀⠙⣎⡉⠙⢿⡓⠚
⠀⠀⠀⠀⠀⠀⠉⠁⠀⡠⢤⡶⠋⣡⣾⠃⢸⠛⢻⣿⠟⢿⠋⢻⢿⠦⣄⠈⠂⠈⢻⡲⢤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡼⢁⡇⠀⠂⢠⠏⡟⡆⠈⡀⢸⡌⠳⣌⠑⠢⢄⣀⢳⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠶⠯⠖⠚⡿⠁⢸⠀⠀⡰⡟⠀⣇⠹⡀⠀⢸⠷⡀⠘⡶⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠴⠚⠠⠊⢠⠇⢠⢿⠀⠑⡀⡸⠀⢳⢀⡇⠈⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠁⠸⡆⠞⠀⠁⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⡶⠆⠀⠀⠰⠂⠀⠃⠈⠀⠀⠚⠦⠈⡴⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠐⡆⢀⠀⠁⣀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠀⠀⠀⠐⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⠈⠌⠦⠐⠌⠁⠀⠀⣀⠀⠂⠀⠀⠀⠓⠒⠀⠀⠠⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣭⠴⠆⠘⡆⠣⠀⠐⠀⠀⠀⢀⡀⠀⠐⠤⠀⢈⠒⠤⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢀⡀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠂⢌⠢⠀⠚⡈⠓⠀⠀⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢠⠐⠀⣔⠀⠙⠇⠐⢤⣀⡧⠀⠀⢡⠘⠀⠀⠱⠂⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⠀⠈⠓⠈⢲⡀⢼⠀⡗⢀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠦⠀⠼⠡⠂⠘⠐⠁⢠⠎⡄⠀⠐⠁⣀⠀⠀⠀⠀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃⠀⠰⠀⠀⠛⢆⠀⢊⠴⠃⠀⢀⠸⠀⠀⠓⠄⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⢘⡀⠁⠠⢠⡗⠄⠀⠀⠀⠀⠣⠤⡤⣀⣀⠤⠃⠠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠪⠀⠘⣇⠀⠀⠀⠀⠂⠤⠀⠀⠠⠤⠄⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⠀⢓⠄⢎⡼
⠀⠀⠀⠀⠀⢀⠄⢠⡤⢠⡄⣤⢤⡄⠀⣤⢠⣄⠠⠠⠀⢀⠄⠠⣠⢠⠀⠀⠀⠁⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠁⠁⠀⠁⠀⠉⠀⠈⠀⠈⠀⠉⠀⠁⠈⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣄⡀⢠⣤⣤⣤⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⣀⣀⣀⣁⣀⣀⣀⣀⣠⡤⠖
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣶⠖
⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣶
⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠃⠀⠀⠀⣀⣀⣀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠿⠟⠋⠀⠀⠀⠀⢠⣖
⠀⠀⠀⣈⣿⣁⠀⠀⠰⣤⠉⠀⠰⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠿⠿⠿⣿⣇⠀⠀⠘⠀⣆⣀⠠⠀⣤⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠰⠟⠉⢹⡇⠀⠀⠀⠀⠀⣼⠁⢸⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠉⠃⡈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⢀
⠀⠀⠀⠀⢀⠀⠀⣴⣤⣤⡀⠀⣿⣠⣤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠂⠀⠀⠀⠀⢙⠇⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⣠⣷⣦⣄⡘⢿⠿⢷⠀⣿⣿⣿⠏⡀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⣿⡇⠀⠀⠀⠀⠀⠀⣬⣁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠸⣿⠄⣿
⠀⢠⣴⢛⡛⠛⠛⠁⠀⠀⢀⡀⠉⣶⣤⣶⣷⣤⡀⠙⢿⣧⡀⠀⠀⠿⠀⢸⣦⠀⣴⣦⣼⣃⠀⠀⠀⠀⠀⠀⠛⣙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠛
⠘⣿⣿⣾⡏⢀⠀⠀⠀⠀⠀⠃⠀⠉⠘⠛⠉⠁⠀⠀⠀⠈⠻⠞⢿⡗⠀⠀⠋⣀⣿⣿⠟⠛⠻⣦⠀⠀⠀⠀⠘⣿⡦⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰
⠠⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⡄⢷⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠛⢿⡗⠀⠀⠀⠿⣿⣆⣰⣜
⠀⠀⠉⠻⠀⠀⠀⠀⠀⠰⣾⣿⡇⠀⠀⠀⠀⠀⠀⡀⣧⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⠬⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠃⠀⠀⠶⠂⠀⣠⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⣶⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣾⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘
⢸⡁⠀⠀⠁⠀⣀⣠⣤⣤⣀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⠃⠀⠀⣤⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠛⠃⠀⢶⡄⠀⠀⠀⠀⠀⣰⣆
⢸⣷⡀⠀⠀⠀⣽⣿⣿⣗⠀⠀⠀⠀⠸⠛⣿⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠆⠈⠁⠀⠀⠀⠀⠸⣿⣿⡆
⠈⢹⣷⡆⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⢀⣾⡁⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒
⢸⣟⢁⣌⣅⠀⠀⠀⠈⠻⢷⣦⢴⣿⣸⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠋⠁
⢈⣉⠉⠻⢿⡇⠀⠀⠀⠀⠀⠀⠘⠋⣩⡄⠀⣀⡠⠘⠉⠀⣹⣿⠿⠿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠐⣷⠀⠘⢠⠄⠀⢈⣿⣂⠀⠀⠀⠀⠀⠀⠐
⠸⠿⣿⣦⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠉⠆⢩⡄⠀⠀⠀⠀⠙⠉⠛⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣿⢻⣧⠀⠀⠀⠀⠠⣤⣄⠀⢾⠁⠀⠀⠀⠀⠈⣥⡀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡇⢰⣶⣶⣷⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⢹⣷⣾⠇⠀⠀⠀⠀⠀⣿⣷⡈⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣍⣛⡛⣧⠟⠉⠛⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠹
⠠⠀⠀⢀⡄⠀⢀⡀⢀⣀⣀⣀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⠏⠉⠈⠉⠻⣿⡿⠛⠿⠃⠀⠀⠀⠀⢹⣏⣽⡆
⠀⠀⠀⣼⣣⣶⣾⣿⣿⣿⣿⣿⣷⡶⠈⡅⠀⠀⠀⠀⠀⠀⠀⠈⢁⣄⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣿⣠⣤⠀⠀⠀⠀⠀⠀⣁⣀⡴⣶
⠀⠠⢾⡿⠿⠿⠿⠿⡿⠿⠟⠋⠁⠙⠰⠀⠀⠀⠀⠀⠀⠀⢾⢿⠿⠿⠿⢶⡦⡤⠏⠀⠀⠀⠀⠀⠸⠹⡿⣿⠀⠀⠀⠀⠀⠈⠉⠻⣧⢷⠀⠀⠀⠀⣶

⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣿⢹⡃⢠⡶⠀⠀⢀⠀⠀⠀⠀⢠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡆
⠀⠀⠀⢰⣿⠉⡇⣼⠇⠀⠀⠎⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⣿⠀⠀⣿⠀⢀⣾⣀⠉
⠀⠀⠀⢸⣿⠀⢠⣿⣄⣼⣿⡿⠁⠀⣀
⠀⠀⠀⠘⣿⠀⣾⢿⣿⣿⣿⠃⠀⠀⠙
⠀⠀⠀⠀⣿⠀⣿⠀⣿⣿⡿
⠀⠀⠀⢀⣿⣤⣿⠀⣿⣿⠇
⠀⠀⠀⢨⡟⢿⣿⢀⣿⣿
⠀⠀⠀⢸⣇⣿⠟⢸⣿⡇
⠀⠀⠀⠈⠛⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣤⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠋⣿⡇⠀⠚⠛⣩⣁⣤⣬⣙⡿⠾⠂⠀⠀⠘⢻⡟⣉⣩⣉⡉⠉⠳⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢼⡇⣾⠏⠀⠀⠸⠿⡿⠍⠩⠄⠙⠟⠀⠀⠀⢀⡾⠋⢙⣟⠻⠿⠓⠖⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣄⣘⣶⣚⣀⣈⡦⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠐⠏⠉⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠃⠈⡀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎⠀⠀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠆⠀⠌⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠄⢹⣿
⠀⠀⠀⠀⣀⠔⠋⠤⠀⠈⠠⠆⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⡀⣠⠆⢠⠊⠀⠀⡇⠀⠀⠀⠀⡤⠊⠀⢄⠠⠀⠌⠀⠘⣿
⠀⠀⠀⠌⠀⣀⡀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⢂⠈⢄⢀⠏⠀⠲⠄⠃⠀⠀⠀⢘⢘⠀⠀⠈⣄⣤⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⢁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⢣⡀⠀⠜⢄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠁⢁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⢋⠂⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡔⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⢸⡴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⡠⡾⠆⠀⠀⠀⠀⠀⢀⠀⠄⠀⡠⠊⢠⠂⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠔⠜⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡆⠀⢸⠇
⠀⠀⠀⠀⠐⠊⠉⠀⢠⣿⠇⠀⠀⣿⠀⠀⠀⠀⠀⠐⠊⠉⠃⠤⠤⠌⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠂⠠⠀⠤⠟⠋⠀⠀⠀⠛⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡶⣆⣠⡴⣦⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⣤⡷⢦⣀⡾⠁⣼⠋⠀⢸⠀⣿⠖⡷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣦⣄⣿⠀⡈⣧⠀⠙⢧⣾⠃⠀⠀⣼⡾⠁⢠⡿⠋⣿⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣦⡐⢽⡌⠛⢦⣱⣿⠀⢀⠈⣿⠀⡀⣸⡟⡀⣴⠋⠀⠀⡟⠉⢉⣟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⠉⠛⠻⣄⢀⠙⢿⣧⠸⣆⣿⢠⣇⣿⣵⣾⠁⠀⠀⢀⡷⠞⠋⠉⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡉⠉⠓⠦⣤⡀⡈⢿⡳⣤⣻⣦⣿⣿⣿⣿⣿⣿⣧⢀⡜⢀⣞⣀⣀⣠⣾⣋⡉⢉⡟
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠦⣄⡀⠀⣿⣿⣴⣿⣿⣯⢽⢽⣺⣷⣟⣟⢿⣿⣫⠶⢛⡉⠉⠉⠉⣩⣟⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠧⣄⡀⠀⠈⠉⠛⣛⣿⣿⢯⣿⣬⣽⣮⡯⣷⣗⡯⣯⣿⣗⣊⣁⣀⣤⠴⠛⠁⠀⢈⡭⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⢛⣻⣿⣿⣿⣳⡿⣽⣲⣻⣟⣿⣯⣟⣿⡻⣿⣿⡿⠿⢿⣶⡶⠶⠶⢯⣄
⠀⠀⠀⠀⠀⠀⠈⣛⣶⠶⠖⠚⢋⣩⣿⣿⣿⣷⣞⣟⢛⡜⣿⢯⣷⣿⣻⡾⣹⣿⣿⣒⡒⠂⣈⣻⣷⢦⣀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⣩⠟⠛⢛⡿⠋⠀⣀⣀⣤⣿⣿⣽⣟⡾⡯⣟⣟⣯⣿⣿⡏⠀⠘⣷⣾⡋⠉⠀⣀⡤⣿⡯⠿⠆
⠀⠀⠀⠀⠀⠀⠚⠓⠒⣶⠟⠒⢛⣉⣭⠟⠉⢁⣸⣏⠙⠻⣷⣿⣿⣟⠉⠉⢳⣾⠿⠿⣤⣉⠛⢿⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡞⠋⢩⡟⠁⠀⠉⠁⠀⢙⣷⡶⢿⠋⢻⢻⣿⡶⠦⠿⣗⠒⠦⡈⠛⣦⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⠋⠉⣙⣛⣛⣿⡏⠸⠁⣿⠀⢸⢸⡇⠳⣄⠀⢿⣤⣤⣤⣤⣬⣟⠋⠁
⢠⣤⣤⣤⠤⠤⠶⠶⠶⠤⠤⢭⣿⣍⣉⠉⠉⠀⠈⢧⡀⠀⣿⠀⠎⢸⠗⢦⣌⠀⠸⡆⠀⠀⠀⠀⠉
⠀⠙⢦⡀⠀⠙⠛⠒⠶⢦⣤⣄⣀⠈⡍⠛⠶⣄⡀⠈⠛⠶⣿⠀⠀⣾⠀⠀⠈⠙⠓⠿
⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠉⠉⠙⠛⢷⣦⣀⠈⢳⡄⠀⠀⠘⣦⣀⣿
⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠈⠛⠉⠉⠉⠈⠙⢷⣄⢻⣄⠀⠀⣿⠻⣿⠀⠀⠀⠀⢀⣠⠶⠛⠛⠛⢛⡛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⣀⣨⣿⠟⢧⡀⢸⠀⢻⠀⠀⠀⣰⠟⠁⠀⠀⣠⣞⣩⣤⠤⠴⠤⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠁⠙⢷⣄⠻⣾⠃⢸⡆⠀⢰⠏⠀⣠⣴⡞⠋⠉⠛⠓⠂⠀⠀⠀⠀⢀⣠⡤⠶⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠙⠀⢸⣇⣤⣿⡴⠟⠁⠈⠉⠙⠒⠂⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⠋⣡⡿⠷⢦⣄⣀⡀⠀⠀⣀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⡴⠋⠀⠀⠀⠈⠉⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠛⠿⠛⡿⠻⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⣿⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠸⠃⡘⢀⠘⠷⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⡲⠋⠁⠀⠀⠀⢿⡠⡀⡄⠀⡰⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢴⣾⣷⣎⡀⠀⠀⠐⢹⠱⡀⣧⠀⠟⣹⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⣌⢿⣿⡷⡀⠀⢠⠸⠀⢿⣻⣴⡢⠇⢁⢀⡛⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠈⠳⢻⣿⣯⣶⡋⠐⠱⢲⣏⣸⣷⡀⠐⣾⡷⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⢹⡄⠀⠈⠑⠐⠚⠿⣿⡀⡘⠌⣿⡿⠁⢸⣧⣴⣟⡑⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠈⢷⡈⠐⢒⠀⠠⠘⢲⣾⣿⡷⠰⣿⡇⠀⡌⣿⣿⠎⠓⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡎⢄⠀⠀⠸⡗⠐⢁⠀⡀⠀⠀⠙⢿⣿⣤⣽⣥⣄⣿⡿⠇⠐⠈⠀⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣦⡀⢠⣟⠀⠉⠁⠈⠠⢶⣿⣿⣿⣿⣿⣿⣿⢿⡛⢏⠐⠎⠁⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣿⠀⠁⠐⠀⠀⠈⢙⣚⣿⣿⣿⣿⣿⣧⠀⠀⠐⢋⣽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⢿⣿⣶⣬⣤⣄⣤⠔⣉⡽⣿⠏⢹⡏⠈⠙⠛⠶⠛⠋⠁⠀⣀⣤⣤⣴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣼⡟⠛⢻⡿⠻⠿⢯⣶⣷⣴⣏⣀⣸⡇⠀⠀⠀⠀⠀⣠⡖⢫⣵⡾⣟⡖⣊⣩⣴⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣿⣧⡰⣠⠚⢀⠀⡀⠈⣿⠿⣭⡉⠁⣙⢛⣛⣿⣶⣾⣷⣤⣎⣥⣵⠛⣿⣏⡽⠋⣠⢧⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⢶⣿⣶⣶⣿⣿⡟⢋⣽⣏⠙⣿⣶⣿⡟⢹⣿⠳⣭⡉⠉⠁⣿⣼⣿⣿⣟⢁⣽⡞⢻⣧⣾⣾⢁⠠⠄⠚⠒⢶⣦
⠀⠀⠀⠀⠀⠀⠀⣟⣽⣟⠠⣻⣟⠀⣸⣯⡿⢿⣯⣍⣿⣿⡋⣿⣟⣧⣸⣿⡀⠈⢷⡄⢸⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⠉⣷⣖⠦⠈⠉⠀⣣⡀
⠀⠀⠀⠀⠀⢀⣶⡟⣿⣩⠿⠋⠈⣅⢻⣿⣿⣿⡭⢻⣿⣿⣷⣿⣿⣿⣿⠿⢟⢻⣦⢹⣿⣿⡟⠻⣿⠿⠻⣿⣿⣧⡼⣿⣿⣿⢋⡖⠀⠀⠀⠀⠉⠶⣄
⠀⠀⠀⠀⢀⡼⠛⠿⠃⠐⢦⣤⣙⣿⣿⣟⠉⠉⣿⣹⣫⣿⣯⣿⣾⣟⣹⣆⣬⣾⣿⣽⠋⣟⣿⣾⣷⣦⣿⣭⣼⣿⡷⣾⣿⣿⣿⠆⣁⠀⠀⠀⠀⠀⢋
⠀⠀⢠⡔⠊⠁⠠⢀⠈⠙⣼⣿⣿⣿⣿⣿⡷⠿⢿⡗⢻⠷⣧⣭⣿⡿⠑⣽⣿⡇⢾⣇⢀⣥⣿⣿⣿⢧⢰⣿⣿⣯⣧⡿⣿⣿⣿⣷⣴⣀⣈⠀⠀⠀⠀⢥⡀
⠀⡰⠁⢀⠱⣔⣶⣼⣿⣾⣿⣿⣿⣏⠀⣸⣯⢘⣾⣷⢟⣼⢻⠓⣻⣷⣾⡿⠿⣶⣾⡿⢻⠳⢞⣿⣿⠶⣿⣿⢋⢙⣿⣿⣿⣿⡌⠻⣿⢛⠿⣧⣦⣦⡀⠘⠆
⠰⣅⣤⠶⣾⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⣿⣯⣼⣿⢾⢼⣿⣿⡃⢠⣸⣿⣿⣆⣾⣄⣈⣿⡿⢲⣿⡿⠿⣿⣿⠿⠏⠀⠀⠀⠸⡄⠁⠈⠳⠙⠟⠋⠀⣉⣤
⠀⠀⢰⠂⠁⢀⠁⠽⡋⣤⣿⣿⣿⣿⣿⡧⣄⣼⣿⣝⣻⡻⣴⢿⡿⠿⠿⣾⢟⢁⣿⡟⣿⣿⣋⣽⡞⣻⣯⣀⣴⣿⣿⣿⠯⠉⠉⠄⠀⠶⡀⠀⢀⠰⠛⠉⠉⠁
⠀⢀⡘⡠⠄⠀⠉⢀⡰⠾⢿⣿⣿⡿⣻⢶⣿⣿⣿⣿⣿⣿⡞⠛⠄⠐⣂⣾⣿⣿⣿⢀⣿⣿⡟⢋⣽⣿⡋⣼⣿⠉⣽⣿⣿⣦⠀⠂⠀⠀⠙⢦⡈⠇
⠀⠈⡃⠀⠀⠀⣀⣠⣄⣐⣩⣽⡯⢟⡵⢟⢙⣿⣿⣿⣿⡿⣌⣭⣶⣬⢿⡟⠁⢀⣿⣧⣠⣿⣿⣿⣿⣿⣿⣯⣿⣷⠿⠿⢿⣧⠠⣄⡠⠄⠀⣠⠤
⠠⡎⠀⠀⠀⠐⢈⣹⣾⣿⠏⠁⠗⠋⢀⢥⡿⣿⣿⣿⣿⣿⣾⢿⣽⡷⣏⣈⠶⣾⣏⡉⢻⣿⣀⣼⣿⡿⣿⣿⠿⢮⣶⣰⣾⣿⣄⠀⠐⠉⠲⢦
⢠⠋⠀⢀⡤⠪⢉⣤⣿⣥⠀⠀⠀⠀⠠⣫⠘⣿⣸⠀⠁⣾⡛⠻⢷⣿⣻⣇⢀⡈⣾⣷⣿⣽⣿⣿⣿⣁⡈⢹⠂⠀⣈⢃⠉⠻⣻⣶⣖⡂⠀⠄
⢈⠀⠐⠈⢀⡴⣿⠿⣂⠃⠀⠀⣨⠀⠊⣡⣾⣿⠊⠀⠄⣿⣧⣵⠀⣹⣟⣿⣶⣾⢿⣿⣿⣟⣹⣿⣟⣸⠷⣤⣉⡴⠺⣮⠤⣀⢳⠡⠹⣯⣴⠂
⢨⡃⢀⣴⠟⠀⣋⡴⠛⣷⠀⠀⡯⠰⣹⣿⣿⣽⡇⠁⢱⡿⡛⠛⠱⣆⠎⠄⠐⣮⣦⣫⡿⣿⣋⠹⣇⡅⠀⠀⠈⠁⠀⠀⠀⠈⠉⠹⠶⠾⠇
⢈⣇⠟⢋⣤⠴⠋⠀⠀⠀⠉⣀⠀⢚⣿⣿⣿⡏⢥⣰⡿⠁⢷⣝⠚⣻⠷⣠⣿⣯⠛⣿⣾⠉⠉⠛⢿⡆
⣼⡿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠛⢤⠈⠋⠻⠇⣧⡾⠉⠁⠀⢰⣌⢹⢣⠀⢸⣿⣧⣤⣿⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⠶⠛⠁⠀⠀⠀⠀⠈⠛⠚⠣⢽⡿⠋⢻⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣻⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡬⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⣁⡃⠀⠀⠀⠀⠀⢱⠚⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠺⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⡀⠀⠀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢀⠠⠀⠀⠐⠀⠑⢄⠈⠂
⠀⠀⠀⠀⠀⢀⠀⢤⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠔⢁⠄⠀⠈⠀⠀⠄⠀⠀⠠⠀⠔⠀⠀⢸⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡠⢂⡄⠀⢈⣨⡄⠉⠛⢄⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣐⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠈⠀⣀⡣⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐⠞⠉
⠀⠀⢀⠂⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠈⡆⠀⡠
⠀⠀⡀⠁⠈⠀⠐⠀⠀⢳⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⡯⡊⢀⠴⠁⠀⡀
⢸⡂⠧⠀⠀⠀⠀⠀⠀⢄⠁⠠⡄⠀⠀⠀⠗⢄⠀⢀⡀⠀⠀⠊⢀⡌⡀⠀⠀⠀⠀⢀⠇⠀⠁⠀⠈⠀⠀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠔⠑⠐⣦⣤⠀⠘⡀⠐⠁⠀⠀⠀⡠⠊⠉⠀⠐⠀⠀⠒⡤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⢑⠶⠀⡀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢀⣤⢀⣤⠀⠀⢀⡀⠀⠀⠀⣠⡄⠀⣴⠀⢀⣄
⠀⠀⠀⠀⠀⣠⠀⠀⡀⠀⠠⠶⢤⣄⡠⠀⡲⠒⡦⠈⠒⣶⠟⢥⣘⡎⠉⠐⢀⠧⣤⣄⣬⢕⣒⣭⡌⠛⠒⡊⠡⡞⠻⡐⠀⣀⣠⠰⠦⠀⠀⣀⠀⢠⡀
⠀⠀⠐⢶⡀⠙⠀⢤⡟⠆⣀⠚⢃⡬⢡⠚⣧⡘⠃⠐⢓⡖⣥⡀⢴⠎⢉⡛⠓⠛⠚⢉⡰⣉⠀⠀⢠⡝⣞⠋⠀⠛⢠⠍⢠⠌⢄⠛⢂⣀⠞⢥⠄⡘⠁⣰⠖
⠀⠀⠘⡫⣝⠒⠊⣓⡒⠚⠁⠂⠁⠀⠟⠙⠇⠀⠀⠀⠸⠃⠉⠁⠀⠀⠙⠃⠀⠀⠀⠉⠀⠉⠀⠀⠉⠁⠻⠀⠀⠀⠀⠾⠙⠇⠀⠁⠂⠙⠒⣂⡊⠒⢚⡭⡛⠂
⠀⠀⠜⣇⠙⣠⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢤⡘⢁⡝⠄
⠀⠀⠀⠙⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠁
⠀⠰⢤⡤⠀⣬⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⡄⠠⣤⠴
⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠁
⠀⠀⠀⢠⡆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣦
⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⢀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧
⠀⠀⠀⠸⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣤⣍⣤⣮⣬⣧⣥⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢦⡀⢀⣠⠴⡄⠀⢠⡄⡠⠆⠠⠒⠐⣘⠑⠢⡀
⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠀⣿⠬⠛⡾⠀⠀⠀⠉⠑⠉⠻⠶⣴⣯⣄⣠⣬⣉⠀⡄⠀⣀⡀⠀⡴⠃⢰⠦⢄
⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⡴⠃⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠈⠙⠒⠊⠁⠀⠘⢇⠈⠈⣳⢆⡀⠀⠰⡅
⠀⠀⠠⠈⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⣀⣠⣾⣄⣤⣤⣄⡀⢀⣤⠖⠉⠑⢤⡼⢢⡄⠁⠢⠢⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⣀⡤⠴⠤⣟⠁⠙⣀⡄⠀⠹⣀⠀⠀⠀⠀⠉⣻⠛⠈⠗⢒⣺⡿⠈⢙⢆⠀⢠⠅
⠀⠀⠀⠀⢸⠀⢀⠀⢀⣼⠖⠋⠁⠀⠰⠀⠀⠙⢶⣬⣁⡀⠀⠈⣣⠄⠀⣠⠞⠹⡤⢞⠠⢁⠤⢿⡷⡤⣄⠃⡰⠀⢀⡀⠀⣀⡀⠀⢀⠔⠛⠆⠠⡀
⠀⠀⠀⠀⢸⠸⢻⣮⠋⡟⠀⠀⠀⠰⡄⠀⠀⢀⠜⠙⡄⠈⠈⠁⡸⢆⡰⢁⣤⡐⣇⣹⠶⡄⠰⠅⢧⠀⢺⡉⢛⣤⠊⠁⠐⠢⢽⣤⣦⠂⠀⠐⠀⠳⡀
⠀⠀⠀⠀⠋⠀⡠⠙⢱⠃⠀⠀⠀⠀⢸⡀⢠⠂⠀⠀⢡⠀⠀⢀⣶⠊⠐⢷⣎⢉⣿⡉⠿⢇⢴⣀⣀⡱⠂⣰⠂⠹⡟⠀⠀⡀⠀⢸⠉⠳⣀⠀⠠⡔⠁
⠀⠀⠀⠀⣰⢀⠁⡴⠃⠀⠀⠀⢰⣉⡣⢇⡀⠀⠀⠀⢀⣤⡔⠁⣆⠀⡸⠓⡟⣉⡘⣟⣛⣿⡞⢡⠽⠉⠲⠥⠤⠊⠳⠀⠀⠄⠀⠰⠀⠀⠈⠄⠰⡇
⠀⠀⠀⠀⡏⠘⠀⠳⣤⠀⠀⢀⣸⣏⣱⣌⣻⣄⣀⢰⠃⣧⣐⣄⡃⠀⠙⢻⠛⢿⣿⡙⣏⣽⢏⠣⡀⠀⠀⠀⠀⠀⠀⢳⠀⡀⠀⠘⡀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⡇⢰⢀⡔⠁⠀⠀⠀⢉⣏⠉⢿⡏⢫⣌⣛⠫⡠⢎⠀⠁⠀⠀⢸⡓⣻⣉⣋⣣⠅⢈⠇⠈⡆⠀⠀⠀⠀⠀⠈⢻⠹⢤⠀⠁⡠⠘⢢⠃
⠀⠀⠀⠀⡇⡘⢸⠀⠀⠀⠀⠀⣤⣙⣯⣈⣿⣏⣱⢎⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣩⠋⠁⠰⣥⡄⠈⢀⡠⠤⣄⡀⠀⠀⠀⡆⠀⠩⠊⠁⢀⢤⣀⡀
⠀⠀⠀⠀⡇⡇⡸⠀⠀⠀⠀⠀⠉⢫⣌⣽⣴⠛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠉⠀⠀⠀⠈⠉⠉⠑⠻⣶⠶⢶⣴⠠⠀⠀⢹⡀
⢤⣶⢄⢀⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⣏⣉⣟⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠈⠀⠙⠁⠶⡀⠀⠳
⠀⣣⣌⠛⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠁⠈⠀⡅⠀⠈⠁⢰
⠀⠈⢷⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡄⠀⠀⠘⠂⠀⢐⠒
⠀⠀⠘⣏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠂⠀⠀⠀⠀⠀⠀⡀⢨⠹⣾⣧⢀⠀⠀⡞
⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠄⠀⠀⠀⠰⠋⠛⢊⡷⢛⣯⣶⡏⡷⣋⡄
⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠄⠁⢌⠉⢺
⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣀⠁⠀⠀⢄
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⠀⣳⡶⡤⡀⢠
⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣄⡠⠁⡰⠂⠉⠈⢘⢷⡐⢮⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠊⡸⠁⠀⠉⠉⠀⠀⠀⠀⡼⢸⠃⠖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⢴⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠙⠁⣌⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠉⠂⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣷⣤⣄⣀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⠉⢻⣿⣿⣿⣿⣿⣿⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡇⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⣠⣥⣶⣾⣿⣿⣿⣷⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣿⣯⣿⡆⠀⠀⠸⣿⠿⠿⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣄⣀⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣶⣶⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠄⠴⠖⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣶⣤⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣿⣶
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢤⣵⣼⣿⣿⣿⣷⣿⣿⣿⡿⠟
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢃⡀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣙⣷⣼⡇⠀⠙⡋⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢹⣿⣿⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⡏⠸⠋⣼⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⡟⠁⠀⡈⢻⣿⡿⢛⣿⠃⠁⠘⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣧⣤⣤⣶⣶⣾⣿⣤⣼⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡥⠤⣟⣙⠛⣛⣛⣛⡛⣋⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠋⢠⠀⢀⠆⠀⢹⠁⠀⡠⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠓⠦⣴⠁⠙⢢⡆⠀⠀⠘⣦⠎⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠓⡄⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⣰⠔⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢃⠀⠀⠀⠈⡄⢰⠀⠀⠀⠋⠀⠀⠀⡰⠁⠀⢠⠃⠀⢀⡠⠤⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠠⡒⠉⠉⠉⠳⡡⡀⠀⠀⢸⠀⢣⠀⠀⢰⠀⢀⡞⠀⠀⡰⠁⢀⠴⠋⠈⠁⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣈⡢⣀⠀⠀⠘⢌⡢⡀⣀⡳⠰⠷⠖⠈⡗⠛⢇⡴⠊⡠⠊⠁
⠀⠀⠀⠀⠀⣠⠔⠉⠁⠀⠀⠑⠢⢄⡀⢹⠛⠉⠀⠀⠈⠀⠀⠀⠀⠀⠰⣴⢁⣤⠶⠒⠒⠚⠉⡨⠁
⠀⠀⠀⢀⣚⠢⢤⡤⢀⣀⡀⠀⠒⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⡏⣀⣀⡀⠀⢀⡔⠉⠂
⠀⠀⠀⠘⡀⠉⠁⠀⠀⠀⠈⠙⠒⠚⡧⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⠀⣹⠉⠀⠀⠈⡱⠛⢍⠑⠢⠥
⠀⠀⠀⠀⠈⠑⠢⣀⣀⠠⠤⠀⠀⢤⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣯⣠⠤⠖⠉⠲⡄
⠀⠀⠀⠀⠀⠀⣴⡾⠒⠂⠈⠉⠉⣉⡭⡷⣠⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠑⢄⡀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣼⠟⠀⠀⢀⡠⠖⣉⡤⠚⠋⢉⡽⠖⠶⣤⠴⣴⠖⢇⠀⠀⠀⠈⡻⠙⡁⠀⠐⠒
⠀⠀⠀⠀⠀⡋⢀⡤⡖⠉⡠⢊⠍⠀⠀⣠⠋⠀⢀⡾⠁⠀⠁⠀⢸⠉⠉⠉⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠛⠁⢰⣧⠋⢀⠎⠀⢠⠊⡇⠀⢀⠏⠆⠀⠀⠸⠀⠈⡧⣀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠸⠀⡰⠁⠀⢹⠀⡈⠀⠘⡀⠀⢠⠀⠀⢃⠀⠉⠑⠀⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠰⠁⠄⢠⠋⢦⡇⠀⠀⡜⢢⠘⡧⡀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠁⠀⠀⠱⡀⡸⠀⠀⠉⠁⠈⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠓⠃⢁⠀⣤⠀⠀⠀⡘⢁⠈⠆⣐⣀⡠⡄⠀⠀⠀⠰⡶⣄⠠⣀⠀⢀⠀⠀⠀⢀⠰⡞⠋⠀⠀⠀⠐⠚⠀⡀
⠀⠀⠠⠋⢀⣤⣿⠿⢣⠐⠒⣦⡒⠂⢶⠞⢻⠑⠒⣶⡚⢻⡏⢱⠘⡆⣦⡶⠊⠉⢧⣹⠄⠓⣶⡒⠒⠛⣽⡟⠲⠈⣤⡖⢶⡟⡄⠉⠀⢀
⠀⠀⠀⠨⢜⡁⠏⠐⠊⠀⠀⠿⠁⠁⢡⣴⣉⣀⢹⣿⠋⠀⢘⠀⢠⠇⡸⠋⡩⢂⡀⡁⠈⠘⣿⡇⠀⠠⣋⠠⠃⠀⢻⠏⠀⢑⠐⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠋⠨⠀⠋⠀⠀⠀⠀⡸⣴⠃⠀⢓⠟⢅⠄⠀⠀⠙⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⢸⡇⠀⠀⠈⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⣠⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠘⠻⡿⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡇⠀⢠⡴⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣿⣿⣤⣼⣿⣯⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢰⣿⣿⡗⠈⠛⠛⠛⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡲⣶⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⢉⣿⡿⣯⢹⢿⣧⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣩⠿⡟⢃⣽⣯⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣯⣼⣾⡟⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠿⢿⣿⣤⣭⡍⢉⣵⣿⣾⡟⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⣛⣿⣶⣯⣝⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⣿⣿⣿⣧⣿⣿⣿⣿⣤⣾⣯⣿⣹⡍⡙⢿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣽⣯⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣾⣿⣾⣿⣥⣾⣷⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠛⣛⣿⣾⣏⣡⣼⣇⠿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣻⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠟⠹⠊⠁⣿⣿⣿⢶⣜⣁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⡍⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣶⣷⣾⣿⠙⣿⣧⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣹⣿⣧⡍⢿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣧⣈⠙⢷⣿⣿⣿⢿⢾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⡿⠋⠐⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠛⠋⠀⠀⢀⣤⣿⣿⣿⣿⡟⢻⣿⣧⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⠁⠀⠈⠻⣿⠟⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⢀⣿⣿⠉⠉⠙⠛⢻⡟⣷⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⢀⣄⡀⢹⣿⣿⣿⣦⣿⡿⠁⣀⣀⣸⣾⣿⣟⠃⠀⣙⣷⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣛⣿⣿⡀⢿⣿⣿⣿⡿⠛⢠⣿⠟⣣⢂⢙⣿⠆⠀⡜⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠐⠀⠻⠇⠀⣾⢯⣄⣿⣝⣉⡾⣟⠀⣴⣇⣼⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠏⠁⠀⠀⠀⠀⠀⠺⣤⣾⢻⣿⡿⢛⣡⣿⣾⡟⢹⡷⣿⢹⡇⢿⠟⣿⣿⣿⡏⠛⠿⠂
⠀⠀⠀⠀⠀⢘⠋⠀⠀⢠⣦⣰⡆⠰⣮⣡⣯⣼⣙⣇⣸⣿⣿⣿⣿⣵⡿⣿⣿⡇⠀⠀⠙⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣛⢫⣴⣿⣞⣓⣿⣿⣿⣿⣿⢿⡿⣟⣿⣿⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⠟⠃⠀⠈⠙⢿⣿⠉⠙⡿⢻⡿⡟⠛⠟⠁⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠼⢳⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⠷⣧⡀⠀⠀⢸⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢘⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⢥⠆⣤⣿⠁⠀⠀⢸⡧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠒⢦⡀⣀⣠⢶⠀⠀⣤⣀⢴⣠⠒⠐⣈⠑⠢⣀
⠀⠀⠀⠀⠆⡀⡰⠀⠀⠀⣹⠚⠉⡏⠀⠀⠀⠉⠂⠙⠳⢶⣿⣤⣴⣤⣁⡰⠄⣀⢤⠀⡔⠉⠸⠦⢤
⠀⠀⠀⢠⡞⡋⠀⠀⠀⡔⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠉⠀⠁⠀⠀⢑⣄⠈⢹⣷⠄⠀⠚⡄
⠀⠀⠐⠀⡄⠃⠀⠀⢸⠀⠀⠀⠀⠀⡀⣀⣠⡤⠲⠤⢴⠏⠒⠲⠶⠦⣤⣴⣞⢁⣀⢱⣾⢱⣆⡁⠓⠡
⠀⠀⠀⠀⡇⠃⠀⠀⢸⣆⣠⠴⠒⢙⠛⢥⣀⠠⡴⠀⠐⢢⡀⠀⠀⠀⡠⣟⢀⡍⠹⠿⢿⣄⠸⢳⡄⢺⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣧⣧⣄⣄⢼⠋⠀⠀⠠⠈⠀⠀⠹⡿⡷⠦⠠⠄⢸⡁⢠⠞⠀⣹⢹⣌⡰⢊⠝⡏⠒⣣⢤⡏⢀⠶⠒⡒⠦⡀⣠⡀⠀⣉⢑⡀
⠀⠀⠀⢠⠇⠁⠛⠆⡜⠀⠀⠀⠈⠃⠀⢀⠚⠀⢙⠀⠀⠀⣎⠵⢣⣘⠷⢼⠶⢯⡿⢘⠁⠙⢄⡜⢂⠟⣧⡄⠀⠈⠙⢹⠿⣅⠀⠀⠀⣙⠄
⠀⠀⠀⠀⡄⢰⠁⡴⠁⠀⠀⢀⠠⢜⡄⠚⠀⠀⠀⢋⣀⠔⠛⠀⢰⣝⠷⠞⢷⠶⣼⣨⢗⣻⠮⣈⠇⢀⠼⡅⠀⠚⠀⢨⡀⠈⠒⠀⢸⠅
⠀⠀⠀⢰⡇⡄⠾⣄⠀⠀⠀⢸⡗⢊⡳⢦⠀⠀⡔⢻⢣⡄⠋⠀⢷⣶⢾⣷⡾⢷⢻⣿⢮⠁⠀⠀⠈⠁⠀⠣⡀⠀⠀⠰⠀⠀⠀⠁⣸
⠀⠀⠀⢸⠀⡇⢀⠞⠀⠀⠀⢻⡟⢻⣿⠺⣟⣳⠷⡈⡙⡉⠃⠀⠀⢻⠶⡟⠻⢺⡻⠈⡆⠑⡄⠀⠀⠀⠀⠀⠳⡞⣄⠀⠃⢀⠴⢤⠇
⠀⠀⠀⢸⠀⠃⡏⠀⠀⠀⠀⣌⣛⣟⣻⣿⡛⡽⠃⠈⠀⠁⠀⠀⠀⠘⢿⣛⡻⠛⠀⣎⣀⠈⠁⣀⣀⡀⠀⠀⠀⢣⠀⢳⠔⠂⢀⣈
⠀⠀⠀⢸⢸⢠⡇⠀⠀⠀⠀⠉⢿⣍⣭⡼⢻⠗⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠃⠂⠊⠀⠀⠈⠉⠉⠐⠺⣶⡴⢦⣤⠡⠄⠊⢣
⢬⡶⣄⠈⢸⣸⠀⠀⠀⠀⠀⠀⠠⣯⣹⣋⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠑⠰⠛⠑⠤⡀⠈⢦
⠘⣧⣌⠃⢸⡟⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⢠⡐⠁⠈⠀⠇⠀⠈⠀⢾
⠀⠈⣗⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⠀⡀⠈⠀⠀⠊⠋
⠀⠀⢸⡍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡒⠁⠀⠀⠀⠀⢀⣤⢲⣉⣿⣷⢠⣀⠐⡇
⠀⠀⠀⡇⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠉⠈⠀⡎⠩⠓⠿⡽⢽⣼⡄
⠀⠀⠀⣇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⢲⠈⢷
⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠒⣄⣀⠈⠢⠸
⠀⠀⠀⢸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡜⠁⡀⠠⠀⢺⣟⢙⡢⡏
⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠌⡔⠚⠛⠓⠊⠀⠀⢀⡰⢹⢦⣃⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⡔⠓⣹⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠲⠊⡜⠄⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⣠⠼⠛⢅⢸⠉⡦⣖⠂⠀⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⡰⠁⠀⠀⠈⣿⠊⠀⠈⠱⣄⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠀⢀⠃⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⢆⠀⠀⠠⡆⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡵⠋⡠⠋⠀⡜⠀⢀⣄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠐⠁⠀⢸⠀⡱⡜⠙⢆⡀⢠⠤⣄⠀⠀⠀⢀⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⡁⠀⣰⠋⠙⠳⡇⠘⣄⣀⢽⠚⠃⠘⣆⡤⠀⠘⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠁⢱⠤⠤⣿⠉⠀⣇⠈⠳⡤⠞⢂⠀⠈⠢⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠻⣀⣸⣄⠀⢈⡆⡔⠋⠁⢹⠀⠀⡸⢦⡀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡜⠩⡅⠈⠹⡍⠀⠘⡶⢶⠋⠁⢸⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣩⠖⢶⠉⠒⠲⡁⠈⣳⠒⠉⢲⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣄⠘⣦⣀⠠⡏⠉⠙⢤⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⢼⠉⠙⣇⠀⠈⡦⠠⡞⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣱⡖⠊⢳⠊⠁⢀⣇⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠁⡷⠤⠼⡗⠂⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠛⢣⣀⣀⡼⠉⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡴⠋⠙⢤⣤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠙⡋⠉⠉⢙⠉⢻⣏⠉⢻⢉⡏⣻
⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⣀⠃⣿⠂⠈⢸⣷⠏
⣿⠀⠀⠀⠠⡆⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⠰⡏⠀⠀⣼⠇
⣿⠀⠀⠀⠀⠠⢄⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢀⠀⣠⡄⢀⡿
⣿⠒⠃⠒⢢⣤⡶⢴⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢀⡇⠀⠀⢀⡋⣠⡟⠁⣾⠃
⣿⠀⠀⠀⠀⠈⠙⠿⢶⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢉⣀⠀⠀⠀⠀⡎⠁⠀⠀⠀⢱⠀⠀⠒⠁⠀⢀⣼⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠴⠀⣶⠀⢠⢀⢀⠀⠀⠀⣘⠉⢀⣇⡸⠇⠀⠤⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣿⣿⣷⣀⡀
⣿⠀⠀⠁⢄⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠁⠀⠀⠈⠘⠦⢀⣼⠟⠛⣿⡷⠟⡈⠁⡤⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⡈
⣿⠆⠀⡀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⡆⠀⠀⢀⠀⠈⠁⣆⣼⢏⠀⠸⣿⢀⣱⢀⡀⠀⠀⠀⢇⣀⠀⠀⠀⠀⠀⣽⣿⣿⠈⢽⡆
⣿⠀⣀⣦⢤⡠⠀⢸⣤⠀⠀⠰⠀⠀⠀⠀⠀⠄⠐⠈⠀⠀⠀⢿⣁⣼⡖⠐⠛⢯⣬⣷⣽⣀⠴⡖⠚⠿⠀⠀⠀⡀⠀⣾⣿⣿⡄⠠⠄
⣿⠐⢯⣅⢸⡄⣾⣾⠇⣼⠗⡾⢮⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣾⡟⠁⠈⠀⢘⡛⣻⡀⠀⢴⢡⠀⠀⠀⣀⡄⣠⡾⠿⠼⢷⠴⠉
⣿⠈⠉⠁⠛⠚⠛⠻⢿⣾⣤⣼⣬⣀⣙⠒⡧⢤⢀⢠⣠⣄⣠⡀⠀⢻⣿⢭⣷⠙⠟⡟⣃⡲⠤⠥⢬⡀⠤⠴⣉⠁⠬⠂⢀⠀⠑⢂⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣯⠟⠛⠓⠢⣬⢹⣅⢉⢈⣧⠶⠾⠛⣶⣔⡶⣤⡔⠉⠀⠀⠀⠀⠱⡀⡔⢡⣶⣄⡀⠀⠉⠲⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣶⡶⠗⠚⣿⣿⣿⣏⡟⠉⠉⠰⢄⠀⠀⣿⠋⠀⠀⡈⣿⣷⠀⠀⠈⠉
⣿⠀⠀⠀⠀⣠⠄⠠⠄⢤⡔⠒⠲⢶⣦⠄⣀⠀⠀⠀⠰⠀⠀⠈⠃⠀⠀⠀⠲⢿⣈⣡⣄⠀⠀⣠⣤⣼⣿⠛⡉⠟⣷⢈⣿⡶⢂⣾⡀
⣿⠀⠀⠀⠀⠈⠀⠀⢀⠈⠙⠳⠷⣞⣧⣹⣗⣄⡀⠀⠀⠛⠃⠀⠀⠣⢄⣠⡦⠾⣉⠁⢨⣦⢴⣎⣿⣿⣿⠐⠂⢀⣿⠘⡿⣧⢚⣉⣿
⣿⣶⣤⣄⣀⡀⣀⣀⠸⣭⣡⡶⠠⣤⠄⣀⡀⠉⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣼⣿⢿⣿⠿⠛⣻⢶⡶⣿⣿⣘⣧⣿⣯⣟⢻
⣿⣿⣿⣟⡿⠛⠡⡌⢀⠨⢿⡋⠀⣨⢦⡙⠋⠀⣄⣐⣤⡀⣤⢤⢤⣴⣧⠀⠻⢶⣶⣾⣏⣯⣾⡇⣶⢊⠁⠸⣷⠾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⠾⠁⠀⠀⠑⣤⡰⡗⣷⣶⣼⣦⣍⡡⣦⡜⢏⣌⣰⣲⣆⢀⣿⣿⣦⣄⣿⣾⣿⣿⣽⢯⡿⠁⡏⠀⣰⡏⢰⣿⣇⠉⠛⠿⣿⣿
⣿⣨⠟⠸⡇⠀⠀⠀⠈⢷⠋⢼⣯⣾⣉⣩⡤⢻⣰⣏⡾⣽⣟⡁⠈⠛⣿⣿⣿⣧⡌⠙⢻⢯⣶⠀⠰⡇⢰⡿⢶⢿⣿⣿⣤⠒⢂⣨⡿
⣿⠃⡄⠀⢁⠀⠀⠀⠘⣾⣇⣾⣿⣿⢉⠁⠀⣸⣾⠛⢝⠳⡏⠁⠀⠐⠙⢻⣿⣧⠻⣶⣌⣺⠃⠀⣴⠾⢇⣰⣾⣸⡆⠉⠻⣇⢼⣿
⣿⢳⡀⣄⠀⠀⠀⠀⠄⣹⣿⣿⣿⡟⣿⣷⣾⣹⣿⣾⡷⣗⡉⣂⣀⠀⠀⠘⢛⣿⡿⠛⠀⢷⡀⠸⢿⠀⠀⡝⢿⡏⣿⡦⣤⣽⡾⡏
⣿⡐⠙⢻⣦⡀⢀⠀⠀⠤⠛⣿⣯⣟⣻⣿⣸⡏⣇⣼⢰⠉⠻⡁⠈⠀⠀⣤⡘⢳⢃⣀⣀⣃⣁⣺⡿⠀⢀⡁⠈⢧⠸⣿⡉⠀⣁⣀⣰
⣿⡇⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠛⠽⢿⣛⣬⢉⣠⡟⢿⣧⡀⠱⠄⣠⣬⣵⣼⡌⠻⡟⢿⣿⣿⡇⠀⠀⠁⠀⢈⡶⣯⣿⣿⣿⣿⢿
⣿⠀⠀⠀⠈⠉⢠⣜⣿⣿⡶⢶⣤⣐⠀⣀⣽⣿⢸⠷⢤⠶⢿⣳⠄⠀⠀⠉⠁⠈⣃⢠⣴⣾⣿⣿⠇⠀⢀⠀⣠⣿⣇⢸⣏⠙⣿⣿⢸
⣿⡆⠀⠀⠀⠈⠱⠿⣿⣧⣷⢾⣧⡽⣞⣻⣅⣨⡼⣵⣿⠴⣾⠉⣄⡀⠀⣀⣠⣶⡏⣹⣹⣷⣿⣿⠀⠸⡀⢠⣽⢟⡇⢸⡏⠀⢿⣿⢻
⣿⠇⠀⠀⠀⠀⠀⠘⢻⣷⣾⡇⣼⠄⡈⢉⣿⡟⣿⠟⢿⣤⣐⣤⢬⡿⣟⣻⣾⣯⡇⣼⢹⡿⣿⣿⠀⠀⢇⣰⡷⢻⢃⣿⠅⠀⠘⣿⢸
⣿⣨⠀⠀⠀⠀⣠⠶⠶⠿⣿⣧⡳⣤⠞⣿⢷⡀⣿⣴⣾⣁⠈⠅⡉⢻⣿⣿⣷⣺⡇⢻⢈⣿⣿⣿⠀⠀⠈⢡⡀⢈⣼⡟⠀⠀⢸⣿⢀
⣿⣧⣤⣤⣤⣾⣿⡄⠂⠀⣾⣿⣿⣿⣾⠻⢿⡇⣼⣗⣻⣿⣶⣴⡿⠛⠋⠉⠉⣹⡇⢸⢸⣿⣿⡇⠀⠀⠀⢸⠑⡌⣿⠃⠀⠀⠈⣿⣽
⣿⢀⣦⣽⣯⣿⣿⡅⠀⢰⡿⣝⢛⣿⣿⣿⣾⡇⣿⠿⠿⣿⣿⣯⡍⠀⠀⠀⢀⣰⡇⢸⢸⣿⣿⠀⠀⠀⠀⡜⠀⣼⣇⣤⠄⣤⣼⣿⢿
⣿⣼⣿⡿⠿⠿⠿⣇⠀⠘⣿⡻⠾⠟⠿⣿⣾⣁⣹⣛⠟⠿⠜⠛⠷⣀⠂⢀⣈⣾⡇⢸⢸⣿⡿⠀⠀⠀⢰⡿⠠⠟⠉⠠⢊⣾⣯⣿⢾
⣿⠟⡿⠛⠛⠛⠓⣿⣆⠀⠻⣷⠀⠀⠀⠈⢿⣿⠿⣯⠍⠓⢚⠀⠀⠙⢿⣿⣿⣿⣧⠛⡘⢿⡇⠀⠀⢠⣼⡇⣔⢀⢌⣵⡾⣿⣿⣿⣻
⣿⣶⠋⠀⠀⠀⠀⢠⣾⣷⣄⣻⣧⡀⠀⠀⠀⢽⡶⠗⠁⠀⠁⠀⠈⠂⣀⣙⣻⣿⣿⣿⣇⣿⠃⠀⠀⣩⠿⢣⡇⣴⣿⣿⡿⢙⣿⣿⣿
⡿⠋⠀⠀⠀⠀⣐⣯⣿⣿⠟⠁⢛⣿⣿⣿⣿⣿⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠈⠉⡙⠛⠛⠛⠒⠾⠷⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠁⠉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣤⡄
⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣞⢷⣤⣾⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⣿⣄⢸⡏⣀
⠀⠀⠀⠀⢠⡏⣶⣶⣿⣿⣶⣷⢸⡆⠀⠀⠀⠀⠀⡟⣸⣿⣿⣄⠈⡇⠀⠀⠀⠀⠀⠀⣼⠁⢀⣀⣈⣻⣷⣅⠘⣧
⠀⠀⠀⠀⢸⢱⣿⣿⣿⣿⣿⣿⡆⡇⠀⠀⠀⠀⢠⣧⣿⣿⣿⣿⣦⢷⠀⠀⠀⠀⠀⢸⢇⣄⣘⣛⣻⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⡸⣿⣿⣿⣿⣿⣿⢇⡇⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⡏⣇⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⡇⣶⣤⡌⡇
⠀⠀⠀⠀⠀⢧⡹⣿⣿⣿⣿⢟⡼⠁⠀⠀⠀⠸⡏⣿⣿⣿⣿⣿⣿⠇⡿⠀⠀⠀⠀⠀⢷⡻⣿⣿⣿⣿⣿⣿⢏⡾⠀⠀⠀⠀⢸⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣭⣭⡶⠛⠀⠀⠀⠀⠀⠀⠙⢮⣛⡻⠿⣟⣫⠞⠁⠀⠀⠀⠀⠀⠈⠻⢮⣝⣛⣛⣫⡵⠏⠀⠀⠀⠀⠀⠸⡼⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣻⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡿⣣⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣶⠄⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣧⡇
⠀⠀⣀⠀⢀⣠⣀⡀⠀⢴⣦⡌⣧⡼⢋⡉⠉⠉⠁⠀⠀⢠⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣬⣜⣿⣥⣾⢃⡀⠀⠀⠀⠀⡆⣿⣿⣿⣿⢠
⠀⠀⠈⠛⢎⢻⣿⣿⣷⣶⣤⣿⢇⡴⠋⠁⠀⠀⠀⠀⠀⣼⢀⣤⣄⠀⠘⡇⠀⠀⠀⠀⡏⣴⣷⣶⣤⣥⣿⣿⣿⣿⣧⢱⠀⠀⠀⠀⡅⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠳⡙⣿⣿⣿⣿⣯⠎⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⢧⡻⢿⣿⣿⣿⣿⣿⣿⡿⢏⡼⠀⠀⠀⠀⣇⣿⣿⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠙⣎⠿⠟⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠙⠲⣍⡻⢿⣿⢟⣩⠖⠋⠀⠀⠀⠀⠀⢸⡸⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣿⣿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⣮⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣛⣛⣋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠓⠒

⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡠⠀⠀⠀⠀⢀⡦⢄
⠀⠀⠀⢸⠀⡸⠁⠙⢉⠇⠠⠀⣠⢐⠚⣓⠢⢀⠴⠂⠀⣴⢿⡭⠭⡻⣧⠀⡀⠀⠀⠇⢰⠙⠒⡉⡂
⠀⠀⠀⢸⣰⠁⠀⠀⣌⠀⠐⠄⡁⠬⣶⡟⠁⣇⡄⠀⠸⢟⡸⡀⠀⡀⣸⠀⠀⠒⠊⢊⠓⠘⠉⢏⡿
⠀⠀⠀⢀⠏⠀⠀⠀⠘⡄⠀⠀⠉⢄⡀⢀⡜⠁⡂⠁⡄⠑⢯⢮⣯⠾⠋⡀⡀⠀⠀⠈⠦⠀⠠⠼⠃
⠀⠀⠀⠈⠉⠉⠉⠉⡉⢁⠀⡀⢠⣤⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⡌⣳⡁⢸⠂⡒⠄⠀⠀⠀⠒⡱
⠀⠀⠀⠀⠀⡲⢠⡀⠂⠠⠃⢱⡟⣷⡎⡆⠐⠂⢰⠀⢰⠅⠀⠀⠀⠈⠁⠉⠁⢸⠀⡃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢵⠐⠃⠀⣀⢀⢀⢉⡒⠊⠀⠀⠀⠈⠀⠀⡀⠀⡀⢀⠀⡀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠱
⠀⠀⠀⠀⠀⠀⠤⣸⣆⠹⢯⣟⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠇⠑⠈⠠⠥⠀⠀⠀⠀⠀⢀⠰⠄⡀
⠀⠀⠀⠀⠀⢀⣀⡈⣁⡀⠀⠀⠀⡀⠀⠀⠙⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡍⣔⠁⠀⠀⠈⡴⡆
⠀⠀⠀⠀⡊⠁⠀⠀⠀⣈⠄⢀⠙⢋⡇⠀⠳⠀⣉⣀⣀⠈⢸⢓⣲⢀⢨⣿⣔⡁⠀⢿⠀⣴⣶⡀⠃⣤
⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠸⠤⠼⠆⠀⠀⠂⠡⣀⡠⠒⠉⠉⡀⠘⠿⠟⠉⠀⢁⠀⠈⠛⠞⠁⡀⠽
⠀⠀⠀⢸⡑⢒⠒⠠⢂⡄⠀⠰⢪⢍⢢⠀⠀⢶⠀⡀⠀⠀⠀⡼⣁⣥⠀⠀⠐⠉⠂⠉⢢⣀⣀⡜⠁
⠀⠀⠀⠘⠁⠀⠒⠒⠂⠝⠀⠰⢌⡈⠜⠐⠟⠀⠐⠘⠀⠀⠀⠀⠃⠔⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡆⠰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣶⣄⡀⠀⢀⠀⠀⢠⣴⣾⡀⠈⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠾⢿⣿⣷
⣤⣤⣤⡀⢠⣤⡀⢶⣿⣿⣿⣿⣿⠀⠈⢀⣤⡜⠙⠟⠋⠀⢀⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠁⠀⣈⣉⠁
⠹⣿⣿⣿⣾⣿⡧⢈⣻⣿⣿⣯⡬⠴⣿⣾⣿⣿⣷⢄⣴⣶⣜⠋⠿⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⢉⢀⡈⣿⣿⠗
⢠⣬⣿⣿⠍⠉⠀⢀⣉⠛⠛⣿⣤⣀⡻⠟⠿⡿⠀⠈⠿⠿⠃⠀⠸⣿⣿⡇⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡸⠿⠇⠉⠉⢠⣤⣿⡦
⠈⠙⠻⠟⢀⣿⣦⣼⣿⣿⣧⣹⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣀⡀⢀⣌⠛⠛⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠻⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⢀⣿⣆⣴⣄⣀
⠀⠀⠀⠀⠈⢿⠟⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠟⠃⠀⢼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⣶⣷⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⣿⣿⣯⠄⠀⢸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⠟⠁⠀⠀⠀⢿⢿⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⣀⣤⣾⣧
⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠻⠿⠹⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⣀⡀⠀⠀⣠⣠⣴
⠀⠀⠀⠁⠀⠀⠀⢠⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣤⠀⠀⣦⣤⠰⣿⣯⣿⣶⣿⠇⠀⠀⢺⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠠⣦⣴⣟⠉⠀⠘⠿⠟⠛⠋⠀⢀⣴⣶⡤⠓⠀⣰⣶⡶⠀⠀⠘⠂
⠀⠀⠀⣠⣦⡀⠀⣀⣀⢅⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣀⣶⣶⣆⠀⢸⣿⢿⣷⡄⠀⠻⠟⠻⠆
⠀⠀⠀⠘⠛⠁⢼⣿⣷⣶⠄⠻⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡗⠀⠈⠉⠉⠿⠃⠠⣿⣿⣾⣿⣿⣿⡦⠙⠁⠀⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⢀⣤⣦⣤⡄⢰⣿⡿⣹⣿⣾⡅⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠠⣶⣿⣿⣿⠗⠀⠁⠀⠘⠿⣿⠃⠀⠀⠀⠀⠀⢱⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡇⢰⠀⣾⠀⢰⡇⡇⣴⢢⣞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⢧⠠⢸⡾⣘⣿⣸⣇⣿⣷⣿⢠⣳⡟⣼⣧⣿⡾⣄⣴⣶⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢧⡠⠿⣷⣿⣷⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣿⣿⣾⣫⣿⣷⣟⣴⣣⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠢⣬⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣥⡾⣡⠾⣃⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣤⣾⣿⣿⡿⢿⣿⠟⠉⠉⢄⣠⣴⣦⣧⡜⡠⠙⣻⣿⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡭⢖⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠟⠋⠁⠀⢸⣿⡤⠀⠀⣸⣟⠉⠁⠀⣿⡁⠀⢸⣿⡇⠀⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⠾⠟⢋⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣿⡆⠀⡠⡺⣿⣧⣠⣾⡟⠓⠀⢀⡿⠀⠀⠀⣀⡠⣀⣼⣿⣿⡿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠤⠖⠛⢯⣀⡀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣪⢖⠁⠈⠉⠈⢦⠐⢠⠞⠁⠠⣚⣭⣿⣿⡿⠿⣋⢍⠉⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠹⡻⠷⣶⣤⣴⣾⣛⣀⣙⣿⣮⣤⣄⣀⣤⣼⣿⣷⣶⣿⣿⣿⠛⠏⠉⠙⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠁⠉⠙⠉⠛⠛⠛⡛⠻⠛⠻⡟⣿⠻⠟⠻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈








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

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠿⠛⠻⠆⠀⠀⣁⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⠀⣴⣷⣄
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢾⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠺⢢⣄⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⠴⠒⠒⠒⢧⡀⠀⠀⠀⠈⡝⢿⡿⠛⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠸⣌⠂⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⡀⠹⡄⠀⠲⢄⣧⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⢀⠀⠀⢀⡀⠀⠣⡄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠀⠀⡨⢀⡸⢦⡀⠀⠙⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠱⡀⠀⢻⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠙⠦⣀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢷⠀⠈⣋⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⠰⢤⣀⡘⠷⣄⣠⣤⠬⠷⢦⣌⣂⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡆⢀⣩⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠈⠉⣹⠟⠛⠛⠛⠛⠛⠛⠛⠛⢦⣄⣀⡀⠀⠀⠀⠐⠀⢁⢸⡇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡿⠛⠦⠤⠤⣼⠧⠄⠀⠀⣀⣤⣉⠉⠀⠀⠀⠑⡍⢆⠀⠀⠀⠀⠠⠃⢸⡇⠀⠀⠀⢀⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⣼⠋⠀⠀⠉⠀⠀⠀⠀⠸⡈⠀⠀⣀⣀⡆⠀⢸⡇⠀⢠⡾⠋⠁⠄⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠐⠈⢣⠀⠀⠀⢀⣇⠀⢸⡇⢠⡏⠀⡌⠀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⣴⣶⣶⣷⢺⣿⣿⣎⡇⠈⠀⠀⠁⠄⡀⠐⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠤⠐⠊⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣟⣯⣾⠛⢿⡄⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⡃⠀⠀⣙⢶⣄⠀⠘⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣬⣷⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠤⠶⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣽⣽⣷⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣿⣿⣷⣾⣿⣶⣶⣾⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠖⠊⠙⠒⢄⡘⠢⡥⠀⣀⣤⠤⢤⣀⠀⠠⠂⢲⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⢀⠞⠱⠇⠀⠀⠈⠳⣇⠀⠥⠇⠀⠀⠴⠂⠀⠀⠠⠖⠒⠂⠀⠀⠀⠀⠀⠀⡤
⡀⠀⠀⠀⠀⢠⡤⠤⠊⠀⢀⠎⢀⠞⡀⠁⠀⠀⠀⠀⣑⠦⠤⠤⣼⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⣠⠤⠤⣘⠀⠀⠀⠠
⠀⠂⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⡎⢱⠁⣆⠰⠀⠀⠀⠁⣀⠤⢄⠘⢤⣀⣀⠔⢫⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢅⠸⡉⠀⠀⠀⠀⠘⠁⠀⠀⡃⠀⠀⢁⢄⠀⠀⢄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠀⠀⠀⢀⠀⠀⠀⠴⠁⠀⠲⠭⠐⠀⠀⠀⢀⣀⢀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠒⠤⢧⠜⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⡀⠈⠂⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠐⠀⠊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠑⠤⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠈⡄⠀⠀⠀⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢁⡁⣱⠀⠑⠢⣌⠣⢅⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠸⣅⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣼⣿⣥⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠶⠖⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⢀⡤⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣽⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⠀⡀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠠⠏⣼⠏⠀⠀⣼⣁⡄⠀⠃⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣿⡇⠀⠀⠀⣽⠆⠀⠐⡁⠄⠈⠀⠀⠈⣼⡇⠀⠀⠀⠁⠀⠀⠀⡉⣰⠼⠚⠀⠀⢀⣈⣿⣄⣀
⠀⠀⢀⣾⡗⠀⢀⡴⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢲⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠨⠉⠁⠉⢹⡿
⠀⠀⠀⠉⠀⠀⢙⢠⣛⣇⠀⠀⠀⠀⠀⠀⢰⡄⠀⠸⠀⠀⠀⢼⣿⡀⢈⠀⣸⣿⡇⠀⠀⠀⠠⠀⠀⢠⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⣟⣿⣿⠀⠐⣦⢀⢰⣶⣶⡆⠀⠀⠄⠀⠀⠈⣿⡇⠈⠀⠉⢹⡇⠀⠴⠀⢰⠀⣠⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣴⠄⠈⣏⡉⢛⣿⠁⠀⠀⠘⠸⢻⠿⠃⠰⠈⠁⠀⠀⠀⢹⡏⠀⠀⠀⢸⡇⠀⠶⣤⠞⠀⣿⢺⣿⣿⣷⠇⠀⠀⠀⠂⠀⠀⢄
⠀⠰⢦⣖⠂⠀⢀⠀⢸⣿⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢀⣄⣼⡇⢀⢀⣀⣸⡇⠀⠀⠁⠀⠀⢸⡄⣿⣿⣿⠀⣀⣠⣦⣄⣀⣀⣈⣳⣄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠀⠀⢀⡀⣨⣿⠇⣠⠀⠀⠀⢀⠀⠀⠠⣀⠀⢨⠸⣿⣿⣧⠸⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⠇⢿⣿⣿⠀⠻⣯⣽⣭⣽⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢀⠤⠀⠀⢀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⠀⠀⠈⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠣⡄⢱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠹⡀⢆⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠊⠀⠀⠀⠁⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢧⢀⣀⣀⡀⠀⠀⠀⠀⠀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠁⣸⠋⠀⠀⠉⠓⢦⡴⠞⠛⠓⢦⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⠋⠀⠀⠉⠛⢶⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣼⣠⠄⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤
⠀⢠⠏⠀⠉⠢⣀⣀⡤⠒⠒⢢⡝⠋⠉⠓⠦⣤⢞⡠⠤⠴⣏⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢠⠤⠂⠠⠒⠂⠀⠀⠁
⠀⢸⣤⠤⠤⠤⠤⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡎⠀⠀⠀⠀⠙⡄⠀⠀⣠⠟⠙⠲⡄⠀⠀⠁⠈
⢰⠋⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⢧⣀⠀⠀⠀⣨⡅⠀⠀⠀⠀⠀⣽⠒⠚⠁⠀⠀⠀⣿⠐⢃
⢹⡀⠀⠀⠀⣀⣤⠶⠿⠦⠀⠀⣠⠌⠛⠛⠋⠁⢙⡶⠖⠒⠦⣴⠛⠉⠉⠛⠲⢤⡜⠃⠀⠀⠀⡀
⢀⡷⢦⣤⡼⠉⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠙⣆⠁
⡾⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⣸
⣧⠀⠀⠀⠈⢓⡦⠤⠤⠤⠄⠀⠳⠤⣀⠤⠀⠀⠈⠳⢤⣀⣠⠖⠉⠙⠲⠤⢤⠤⠞⠁
⠘⠧⣄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⢸⣤⣤⡤⠞
⠀⠀⠀⠈⢳⣄⣀⣀⣠⡴⠃⠘⢦⡀⠀⣀⡼⢉⡷⢤⣀⣀⣀⣠⡜
⠀⠀⠀⣰⠋⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⡞⠀⠀⠈⠉⠉⠁⠙⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠘⢧⣀⠀⠀⠀⣧⡀⠀⠀⠀⢀⠴⡀⠞⠉⠉⠒⢤⣤⠴⠞
⠀⠀⠀⠀⠀⠈⣹⠟⠉⠁⢉⡟⠒⠉⠳⣄⡏⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⠏⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⢀⡇⠑⢦⣄⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⢧⡀⠀⠀⢀⡼⢄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢩⠖⠚⠁⠀⠀⢳⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⡰⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠛⢻⣥⡄⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⡟⠀⠚⢛⣿⠶⠆⠈⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⡈⢣⡄⠚⠛⢧⣄⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⡵⣋⠼⠅⢜⣛⡉⠉⠀⠀⠼⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡶⠂⠀⠀⠀⠀⠈⠉⢤⡀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣿⣿⡷⢶⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⢠⣧⣀⡤⠀⠀⢈⣵⣔⢂⠤⠀⠀⠈⢿⣷⠀⠛⢿⣇⡼⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣦⠀⠀⠈⠹⠙⠒⠊⠀⠀⠪⠀⠻⠇⠉⣸⣿⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠃⠈⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⣶⣯⠁⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢠⡶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⡋⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⡆⠀⠀⠐⠞⠁⠀⠀⠀⠀⢀⣤⣤⣀⠤⣶⣿⠟⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠣⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠀⣾⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⠁⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⡇⣹⣾⣧⣴⡆⠀⠀⠀⢠⣀⠀⠀⣀⡴⠛⢀⠏⠀⠀⠀⢀⣱⠄⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢸⣿⣿⡿⣭⣿⣿⠀⡠⠀⠟⠛⡻⠁⠀⢀⣴⠋⠀⠀⡠⠚⠁⠀⠀⠐⢀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠿⢻⣿⠟⠁⠀⠀⠀⣰⠁⠤⠒⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡁⠀⡾⣼⡟⠀⠀⠀⠀⢀⡄⠠⠂⠁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⢰⡇⣶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾⠃⣠⣴⣄⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠈⠀⠀⢸⣿⡿⠃⡞⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡼⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⠀⠰⣄⠐⠀⠀⠓⠶⣾⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠑⠀⠈⠀⠀⠔⠌⠛⠿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⣿⡈⠀⠀⠀⠀⠀⠈⠀⠀⣠⠖⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⣤⣤⣶⡾⢃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠻⠐⠆⠁⠀⠀⣀⠀⠙⢦⣄⠻⠤⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠆⠀⠈⣠⣀⠻⣦⣿⡟⠶⣯⠻⣆⡿⠁
⠀⠀⠀⠀⠀⠒⠒⠛⠚⠿⠟⠛⠛⠛⠛⠛⠓⠀⠀⠀⢀⠰⡇⠀⠀⠀⠀⠀⠀⠘⢷⡌⠻⢷⠀⠈⣿⡶⢿⣆⠁⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡄⠀⠀⠀⠀⠀⠓⠈⡋⠀⡘⠷⠀⠈⠁⠘⡆⣆⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠰⣶⣛⣿⣿⣷⣦⣤⠀⠛⣸⠇⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠙⣿⣯⣭⣿⣿⣿⣷⠜⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢈⣹⣿⣷⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢀⣷⠀⠐⡆⠀⡆⠀⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡟⠀⠀⠀⠀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗

⠀⠀⠀⠀⣠⡄⠐⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⠀⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠂⢀⡄⠀⠀⠀⢠⠀⠀⠀⠿⠀⠀⠀⠀⠈⠓⠀⠀⠘⣿⣦⣾⡏⠀⠀⣠⣡⣄
⠀⠀⠀⠀⣠⣄⠀⠀⢡⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠙⠿⠁⠀⠀⡒⠄
⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠉⠀⣴⣶⣶⠀⠀⢸⣿⡇⠀⠀⠰⢿⣿⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠈⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢹⡇
⠀⠀⠀⠀⠀⢄⠀⠀⠤⠀⠀⢀⠀⢻⠀⢹⡇⠀⠻⠛⣇⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠟⠀⠸⠇

⣶⣶⡶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠋⠂⠀⠀⢠⠝⢷⣄⠀⠀⠀⠠⢈⡐⡀⠀⠀⠄⠐⢠⠀⠀⠈⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⣛⠀⠀⠀⠒⠁
⠀⠀⠀⠈⣉⠀⠀⠈⠄⢠⡑⠀⠀⠀⠄⠈⠀⡀⠀⠈⠂⠀⢀⠀⠠⠀⡀⡀⠀⠀⠀⣀⢁⠀⠀⢐⡆⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢡⣥⠀⠀⠀⡾⢹⠳⠄⠀⠀⠇⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⡁⠀⠀⢀⣀⠀⠀⠀⠈⡍⠀⠀⠐⠈⠂
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠃⡴⠀⠀⠀⠀⢦⠀⠀⠀⠀⠰⠀⠀⠂⠀⠈⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢺⡂⠀⠀⢉⠁
⠀⠀⠀⠀⢤⠀⠄⠀⠀⢫⠟⠀⡄⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⠠⠀⠁⠌⠀⠀⠀⠠⢠⠀⠀⠀⠀⠰⢇⡀⠀⠈⡉
⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡄⠀⠀⠀⠐⠀⠈⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠐⠘⠂⠀⠀⠀⠀⠸⠖⠀⠀⢡⠁
⠀⠀⢀⣙⠓⠀⠀⠠⠀⠈⠃⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠄⠀⠀⠀⢉⢈⠀⠀⠀⠀⠚⡂⠀⠀⠆
⠀⠀⠀⣁⠀⠀⠀⡄⢐⡇⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⢈⡇⠀⠀⠰
⠀⠀⣠⣀⠀⠀⠀⠀⠸⢷⠀⠀⠀⠀⠀⠤⠐⠋⡀⠀⠀⠀⠘⢄⠀⠀⠀⠂⢁⠈⠀⠀⠤⠀⠀⠀⠀⠀⠰⠷⠀⠀⠴⠄
⠀⠀⠠⡆⠄⠀⡀⠀⣷⠀⠀⠀⠀⠀⠀⠄⠞⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠄⠀⠀⠀⢀⡃⠀⠀⢠
⠀⠀⠠⠴⠀⠀⠀⠄⠹⠦⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠠⠅⠀⠀⡄
⠀⠀⠀⢀⡀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠐⠈⣄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠐⠠⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠐⠀⠀⠀⠀⠀⡎⠙⢷⢄⠀⠀⠀⠂⠸⢄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠂⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠒⡈⠀⠀⠀⠘⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠈⠀⡁⠀⠀⠀⠄⠀⠸⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⢠⣶⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⡿⢿⣿⣧
⠀⠀⢠⣿⣿⡏⠉⠙⠛⠻⠿⢿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⡿⠿⠿⠛⠉⠉⠀⠀⣿⣿⡄
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⡿⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠈⣿
⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿
⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿
⣿⣿⡇⠀⣴⣌⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣁⣾⠆⢀⣿
⣿⣿⡇⠀⢹⣿⣿⣶⣤⣌⡉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣉⣤⣶⣾⣿⣿⠀⢨⣿
⢻⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠘⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿
⠀⣿⣿⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿
⠀⢹⣿⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⡏
⠀⠈⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⡿⠁
⠀⠀⠸⣿⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⠏
⠀⠀⠀⠀⠹⣿⣧⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠘⣿⣷⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⡀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢀⣤⣾⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⣤⡀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣀⣤⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⣿⠇⠀⠈⣿⣿⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣤⣤⠊⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠂⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠄⠀⠤
⠀⠀⠀⠀⠀⢠⠒⠀⢠⣤⣤⣧⣤⡈⠉⢤⡄⠀⠀⠀⠀⠀⠀⢻⡁⠉⠑⠀⠀⠹⣄⠀⠀⠀⢰⠀⣄⠀⠲⠛⠀⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠁⣴⣏⠤⡑⠦⠉⠠⠀⠰⡑⠀⠀⢀⡴⠛⣦⡈⠭⢓⠀⠐⠇⠀⠈⠛⣄⡀⠁⠀⠹⣄⢀⠐⠀⢀⣨⠍
⠀⠀⠀⠀⣻⠁⠚⢿⣿⡶⢦⢴⡄⣹⢶⠳⠁⠀⠀⠀⢁⡴⠆⠂⠀⣲⠁⡁⠐⣸⠇⢀⣀⠉⠗⠲⠄⠀⠀⢦⡈⢋
⠀⠀⠀⠀⠈⢶⠌⠹⣯⣵⡊⢻⣦⡎⠹⠋⠀⠀⠀⠀⡉⡠⠊⠀⠄⠃⢠⣷⣾⠩⠒⢢⣼⡃⠂⠀⠊⠙⢶⣄⡉
⠀⠀⠀⠀⠀⠀⢂⢀⡈⢷⣟⣹⣧⣨⡟⠀⠀⠀⠀⠀⠠⠐⠒⠊⢣⢴⣿⣯⡱⢴⠁⣄⡿⡀⠀⠙⠖⠀⣀⠊⡀⡀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⡭⣟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠒⡼⣇⡶⡋⠀⢰⣀⠘⠀⠈⣾⡇⢁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣏⣿⠁⠙⠒⠁⠀⠀⢀⣴⠋⠀⠚⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣹⠁⠀⠀⠀⣦⣶⠶⠁⠈
⠀⠀⠀⠀⠀⠀⠄⢂⠈⠀⡗⢀⡔⡶⡩⠢⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠨⠄⠒⠊⣉⣵⡿⣿⣏⣼⠀⠉⢦
⠀⠀⠀⠐⠊⡡⠒⡎⣕⠘⣟⠟⢓⠲⡁⡁⠀⠀⢀⠀⠀⠀⠀⠀⢺⡞⢻⠁⠠⢠⣴⡽⡀⠥⠀⠌⣠⡙⡆⡐⠘
⠀⠀⠀⡁⠠⠴⡈⣃⣾⡟⠛⡓⡈⢊⢻⣼⡍⠉⠀⠀⠀⠀⠀⢐⣾⠀⠇⢘⣿⠀⣿⣷⣞⣦⣬⡆⣦⠰⣾⡞
⠀⠀⠀⠀⠀⠀⢻⣿⣷⠛⣲⠉⢙⠉⡟⠋⠀⠀⠀⠀⠀⠀⠀⠞⣿⠀⠐⡄⢹⠀⢸⣿⡿⠀⠉⡁⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠿⠏⢹⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠧⢾⣷⡲⠄⠀⠻⣷⢷⡔⣗⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣤⢿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢢⠉⠻⠚⠋⠀⠀⢿⣾⣋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡋⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠁⢄⠓⠀⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⢻⡅⠈⢇⡴⠛⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠈⣛⣓⣚⣓⡰⢦⣴⣚⣵⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠻⢿⠿⠛⠿⠷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⣀⢀⣀⡄
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⣠⡝⠒⠶⠶⠶⠶⠶⠖⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣌⠙⠟⠁
⠀⠀⠀⠘⠿⢿⣷⣶⣦⣤⣤⣴⣶⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠙⠗⠲⠶⠶⠶⠶⠲⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠘⠀⠇⡇
⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓⠂⠀⢠⡿⡀
⠀⠀⠀⠰⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠠⢦⣉⣉⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠻⢦⣬⣭⣭⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⣤⠾⠉⠉

⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢂⠀⢀
⠀⠀⠀⠀⢰⢁⢂⠐⢀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣘⣀⡀⠐
⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠉⢠⣶⣿⠛⢃⠅
⠀⠀⠀⠀⠀⠀⠐⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣐⠐⠂⡀⡀⠀⢀⠁⡈⢿⠯⡁⢘⡙⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠤⠀⠀⠀⠀⠄⠤⣀⣨⣤⣤⣄⡁⠘⢄⠀⠁⠱⠠⠌⠋⠿⣇
⠀⠀⠀⠀⠀⡀⠀⢻⡀⠈⢀⣀⡁⠀⢈⡄⠢⡾⣽⣿⣷⣿⣷⠬⠨⡂⠀⠀⠀⠀⠀⠀⢻⠀⢀⡀⠀⠀⡀
⠀⢀⠐⠀⠀⡒⠀⣸⠁⠀⠀⠀⠀⠀⠀⠂⡸⢿⣿⣿⣿⣿⣿⠀⡀⠄⠀⠀⠀⠀⠄⠀⠘⡶⠁⠐⠠⠀⠂
⠀⠀⠈⠉⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠂⡤⠔⠟⠛⠛⠉⠀⠱⠀⠀⠠⢀⣀⡀⠌⠀⣰⡇
⠀⠀⠀⠀⠀⠠⠀⢸⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⢠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠂⠀⠀⠀⡀⠀⠀⠀⠠⡂⠀⠇⠀⡢
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠀⠀⡿⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠂⠀⠀⡆⣀⠀⠀⠀⠀⠀⠆⠘⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢠⠤⠁⡁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠆⠀⠀⡀⠀⢀⡀
⠀⠀⢸⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠙⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣀
⠀⠀⠀⠀⠈⠛⠲⠦⠤⠴⠶⣿⠏⠉⠉⠙⣦⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠑⡄⣠⣾
⠀⠀⠀⠀⠀⠀⣠⠖⠒⠂⠀⠀⠀⠀⠀⠀⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⢀⡴⠃⠉⠁
⠀⠀⠀⠀⠾⠃⠘⠢⣄⣀⡀⠀⠀⢀⣱⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⡉⠉⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⣄⠀⣿⠀⢠⣿⠇⠀⠀⢶⣦⠀⠀⢸⣦
⣀⡓⠀⠤⠰⠂⠀⠀⠶⠆⠀⠀⣀⣴⢟⣀⠀⠀⠀⠀⠀⠘⠟⠂⣿⡆⠀⠀⠀⢀⠀⠈⠉⠀⠀⠈⠉
⣿⣿⣿⣷⣶⣶⣶⣶⣶⡶⠾⠛⠉⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⢾⡷⠀⠀⠀⠀⠀⠤⢀⣀
⠘⣋⢉⡋⡉⠉⣭⠀⠀⠠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⠻⠿⢶⣶⣶⣶⣶⡶⠾⠛⠋
⡀⠙⠛⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣽⡇⠉⠛⢿⣶⣄⠀⠀⢀⣶⡟⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⢸⣿⡇⠀⠀⠙⣦
⠀⠀⠀⠀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠸⠿⣤⡀⠀⠀⠈⣇⠀⠀⢀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⢈⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠉⢢⡀⠀⢸⣤⠞⠋⣉⣀⣀⣈⠙⠷⣄
⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⣄⠀⢀⡇⠀⠀⠀⠀⡷⠀⡞⣡⠖⠋⣹⡿⠋⠉⠓⢆⠘⣆
⠀⢀⣾⠃⠀⠀⠀⠀⠀⣠⣴⠶⠶⠶⣶⣦⣴⠏⢠⠤⣤⠹⣆⡼⠁⠀⣀⡴⠚⢁⣴⡟⠁⠀⢰⡿⢰⣿⠛⠛⢸⡇⢹
⠀⣼⠃⠀⠀⠀⠀⠀⣼⠋⠀⠈⠛⢦⠈⢻⣿⠀⠀⠀⠈⠁⣿⣿⠖⢋⣁⡤⠶⣿⣏⠀⠀⠀⢸⣇⠈⠻⣿⣿⠟⣡⣾
⢰⣿⡶⠿⠿⠿⣿⣶⡇⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢀⣿⡿⢿⣿⠁⠀⠀⠀⠙⡇⠀⠀⠀⠙⢷⣶⣶⣶⠿⠛⠁
⠉⠀⠀⠀⠀⠀⢘⣿⣿⣦⣀⣀⣀⣠⡖⠉⠸⣷⣤⣠⣤⡞⠁⣠⣤⡈⢧⡀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠀⠉⢻⣿⠀⠀⠀⠀⠈⢹⣿⠀⠈⠁⠈⢹⠈⣿⣿⠟⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢿⠀⠀⣀⣀⣀⣾⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠐⠛⢮⢻⡆
⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⣼⣧⡾⠋⠉⠙⢿⣿⣄⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⣀⣀⣴⣿⡿⠀⠚⠛⠻⣆⢹⡿⠳⢶⣾⡟⠉⠛⢧⡀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⠀⠀⠀⠀⠹⢸⡇⠀⠈⢿⡆⠀⠀⠀⠹⣶⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⢀⣾⠁⠀⣠⠟⠻⢦⣀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣴⣿⠿⠿⣿⠁⠴⠻⢦⡙⡍⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⠁⠀⢰⣇⠀⠀⠀⠈⠇⣷⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⡄⠀⠀⠀⢀⡿⠀⠀⢠⡿⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⢻⣦⣤⠤⣾⣿⣷⣶⣎⡰⠖⠒⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣾⠟⠁⠀⠈⣿⠁⠒⠶⣝⢦⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢧⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⢀⣼⡿⢷⡀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠙⢶⣄⣠⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡂⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠰⣶
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠠
⠀⠀⠅⠀⠀⠀⠀⣶⡶⠸⡆⠀⣸⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⢦⡀⠀⢻⠟⠀⠙⢦⣽⣥⠾⠁⠀⣦⠀⢀⣇⠀⢰⠆⠀⣦⠀⢀⣆⠀⠀⠀⢠⡄⠀⣠⣇⠀⣾⠀⠀⢰⣀⠀⠀⣀⠀⡀⠀⠀⢀⣄⠀⢀⡆
⠀⠈⠀⣀⣻⣋⣠⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢦⣙⣟⣤⠟⠀⠀⠘⢦⣙⣟⣠⣄⡀⠀⠹⣦⣙⣯⡴⠃⠀⠀⣙⣿⣤⠴⠋⠀⠙⠶⣤⣘⣟⣠⠞⠁
⠀⠀⢸⡍⣿⣿⠀⠀⠀⠀⢀⣼⠋⠻⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠙⠂⠀⠀⢹⣿⡇⠀⠀⣞⠹⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡇⢀⡀
⠀⠀⠀⠙⣿⣿⠀⠀⠀⢠⡾⠁⠀⠀⠙⣧⠀⠀⠀⣿⢻⡅⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠙⠀⣿⠻⣆⠀⠀⠀⠀⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡿⡏
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⠀⠀⠐⠠⠌
⠀⠀⠀⠀⢠⡾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣵⣄⣀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⣿⢋⡿⠋⠙⠁⠀⠀⠀⠀⠀⢐⣯⣽⡛⣿⡏⠛⠁⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠳⠀⠘⠀⢰⡃
⠀⠀⠀⠀⠀⠀⣿⡌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⠛⢷⣄⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡄⠄⠀⣠⣤⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⡀⠀⢈⣻⡆⠘⢿⣿⣿⣿⣷⡿⠁⠀⠂⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠟⠁⠀⠀⠀⠀⣾⣿⡄⠀⠀⣾⣿⣇⠀⠀⡌⠻⣷⣶⣸⣿⣿⣏⣀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⣦⣤⡻⢿⣋⣠⡄⢟⢆⢸⡿⣿⣻⡟⣿⣿⣿⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠐⢿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣯⡿⣧⣬⣭⣥⣴⡶⠟⠋
⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣶⣿⣿⣟⣯⣴⣮⣥⣼⣿⢿⣿⠀⣿⣿⣿⣿⡀
⠀⠀⠀⢙⣛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢛⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢸⣿⣻⣷⣸⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⡷⠶⠶⠾⠀⠀⠸⠶⠶⢶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢸⣿⠙⣿⣹⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠿⣿⣿⣿⢸⣿⢸⣿⢹⣿⢻⣿⠟⠘⠿⠘⢛⣿⠟⣿⡏⠁
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⣿⠁⠉⢸⣿⢸⣿⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⢸⣿⠀⠀⠈⠉⠀⠉⢸⣿⢸⣿⠀⠀⠀⠀⠐⠿⠃⠿⠃
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⠿⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠈⠉⠉⠁⠠⠤⠤⠤⣉⣉⡁⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣤⣤⣤⠤⠶⠋⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⢻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠖⠛⠉⢀⡐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡿⠿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠶⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀⠀⢰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣦⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡁⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⡴⢻⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⢷⣄⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⠛⠂⠀⠀⣿⠇⠀⣀⡴⠋⠀⢸⣿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠉⠛⠷⢶⣥⡀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠒⢛⣁⠴⠞⠉⠀⠀⢀⣾⢋⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣⡈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⡴⠟⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠋⢉⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠻⢯⡉⠉⢉⡽⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡤⢠⡄⠀⠁⢀⡀⢀⢄⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢸⣄⠈⢀⣰⠏⠀⠳⢦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⠲⠖⣶⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠴⠦⣄⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⢀⡠⠞⠙⠶⣤⣤⣤⣤⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⢀⠖⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⡤⠊⠀⡀⠈⠢⠀⡔⠒⠙⠑⠊⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣇⠀⠀⠀⠀⢂⠀⠄⠑⣄⠀⠱⠀⠀⡄⠋⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣿⠠⢢⡤⠒⢒⣤⣂⢀⡠⣄⠙⠀⡀⢔⡀⠀⢤⠞⠓⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣭⣇⠀⠀⠀⠀⢰⠉⠄⠒⠒⡠⢤⠀⠐⠀⠠⣀⡀⠐⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⢳⡄⠀⠀⠀⠈⢀⠀⢂⡜⢡⠀⠀⠀⡒⠀⠀⠈⠑⡒⠶⣀⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⢻⡶⢄⡀⠀⠀⠙⣀⡠⠚⠒⡄⠀⠋⠐⠘⡑⠀⠀⢀⣈⢬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇⢀⠼⣧⠊⠉⠙⡍⠻⣷⠴⠒⠳⡜⠓⢆⠀⠤⠄⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢸⣛⡓⣇⠀⣸⣦⣤⣼⣇⠀⡇⠈⢹⣖⡿⢦⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⡈⣧⠀⠀⠀⠀⠀⠀⣰⡘⣼⠴⡏⠀⢹⡏⠀⠀⠹⠑⠶⡊⡜⠉⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠌⢧⠋⠱⡀⠀⠀⠀⢠⣿⠟⠹⣄⣓⡶⠞⠙⢷⢴⣧⡀⢠⡿⠛⢦⡾⠥⡀
⠀⠀⠀⠀⡤⠔⠀⠒⠲⠀⠐⠘⢌⠀⠄⠱⣀⡠⠞⣿⡟⠀⠀⢌⠉⢱⣀⣀⠌⡼⠋⠙⢿⣇⠀⣠⣧⣀⡄
⠀⠀⡴⢎⠁⠀⠑⠄⠑⠢⠤⡀⢄⠁⠀⣀⡹⠀⣠⢿⣇⠀⠀⠈⠓⠋⠀⠘⡗⢳⢄⣠⢿⡟⢿⡉⠀⢸
⠐⠮⠀⠴⠋⠀⠛⢈⠓⠂⡀⠐⠀⡒⢍⠩⠉⢢⢸⡜⠙⢧⡀⠀⠀⠳⠤⠔⠁⢃⠀⠀⢈⣯⣿⢏⣸⡥⣀
⠀⠀⠀⠀⠙⠦⣔⠁⠀⠊⢠⡶⢐⠇⠄⠱⡤⠈⣮⠁⠀⠨⡅⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠐⢹⢾⡁⠀⠈⣱
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠊⡙⠈⡜⠃⠀⠀⠂⠼⠈⠁⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡒⢋⣦⡟⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠑⠀⣀⠆⠀⠀⢀⢠⢸⣀⡗⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠸⢿⠳⠄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡐⠚⠁⠀⠀⠀⢸⢀⡀⠙⡟⠁⣹⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⣄⡇⣼⠁⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠃⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⠷⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠈⣷⣿⣿⣷⣿⡴⠂⠀⠀⢀⠼⡽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠷⢦⣄⡀⢿⣷⣾⣿⣿⣿⣿⣷⣴⣶⡤⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣂⡀⠑⠲⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⠿⠟⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣭⣤⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡨⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠿⠉⠀⢀⠔⠫⠟⣿⣿⠿⣿⡟⣿⣿⡏⠛⣗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠲⣿⠃⠀⠀⣴⡟⠀⢸⣧⠀⠈⠻⢇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠁⠀⠀⠻⢃⠀⣾⡀⠱⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠘⠀⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⣡⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣸⡇⠀⠀⣀⠀⠙⣿⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣿⣧⡀⠊⠀⠀⠠⣝⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⣉⡆⠙⠟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⠃⠀⠐⢠⣬⣿⣦⢲⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⠟⠲⢄⡀⠙⠛⠋⢸⡟⡆⠀⠀⢀⣴⣾⢿⢿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠲⢄⠀⢸⣧⡇⠀⠀⣬⣿⣷⣦⣄⢀⠙⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠁⢸⡏⠀⠀⠀⠛⠻⢿⣉⡙⠫⣦⣄⡰⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡝⠛⠛⠛⠓⠀⠀⢺⡗⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣦⣿⣤⠿⠷⠺⠷⠿⣶⣼⣿⠀⢀⣀⣀⠀⠀⠀⠀⠀⠙⠻⢶⣟⣿⣿⣷⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠋⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⠐⢬⡍⠛⣷⣆⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠶⠀⠰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣫⠀⠀⠀⢣⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡶⡀⠀⠀⠀⢨⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣼⣿⣷⡄⠈⢷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠆⠳⣝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⣀⡀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⡴⠀⠀⠀⣿⣿⣿⣦⣄⣤⢾⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣠⠔⠁⠀⠀⠀⠈⠪⣟⣻⣛⡯⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀⠀⠀⠀⠀⠀⢀⣀⣼⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⣷⣤⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⣠⠎⠀⠀⠀⠀⠀⠻⣻⠿⢿⡷⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇
⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢩⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⢪⣛⣫⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠒⠂⠀⠐⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠒⠂⠀⠀⠀⠀⠀⠐⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢉⣉⣉⣩⣍⣻⢷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠀⢲⣶⣶⣿⣧⣬⠻⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⢿⣿⠿⠿⠟⠛⠛⠻⣿⡟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣴⣠⣼⡏⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⣤⣉⣻⣿⠀⠀⠀⠀⢀⣠⣤⣀⣻⣿⣿⣏⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠑⣿⣟⣿⣿⣿⣦⠀⢸⠿⢿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⢿⣿⡟⠙⠀⠀⠀⠀⠀⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⣤⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⣄⡀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠛⠛⠋⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠟⠁⠺⠏⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⡟⠀⠀⠀⣠⠀⠀⠀⢰⣿⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠸⣿⣿⣿⣿⠇⠀⠀⠀⣿⠀⠀⠀⠀⢻⡄⣤⣶⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢠⣿⡟⠃⢡⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣤⣤⣍⣉⣉⣁⣀⣀⣀⣠⣤⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢠⣿⠻⢈⡉⠀⠀⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣿⣻⣿⣿⣵⡆⠐⠟⢉⣭⣽⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⣾⣿⣿⡟⢀⣀⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⣀⣶⢿⣹⡆⣿⣿⣟⣴⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡇⣸⡄⠀⢀⠀⠀⡀⠴⠋⢠⣿⣿⠃⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⢠⣿⣿⣠⡏⣠⡶⠋⠀⠀⢨⡿⠋⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⣁⣠⣞⣿⣿⣿⣿⣿⣿⠋⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⠿⠿⠛⠛⠛⠉⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⡿⢟⣋⣿⠉⠁⠀⠀⢀⣤⣶⣦⣀⣀⠀⠀⠠⠤⠤⠤⠤⣤⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠘⠛⠛⠛⠛⠛⠒⠒⣺⣿⣧⣤⣶⣶⣿⣿⠿⠿⠿⠟⠛⠻⠿⠶⠶⠶⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠰⢶⣿⣿⣿⣿⣿⣿⣉⣭⣷⠄⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢉⠉⠁⠲⣄⠀⢀⡤⠒⠉⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠀⠐⠂⠀⢰⠪⠀⣶⡿⠛⠋⠐⠲⠦⣤⡀⠀⠀⠈⠑⢦⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⡛⠡⠬⣑⣇⣿⣧⣔⠀⠰⣐⣂⡤⢇⠀⠀⠀⠀⢀⢯⡽⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠀⢠⡶⢆⢁⡼⠛⠚⢛⢞⠇⡳⠊⢎⡇⠀⠀⠠⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡦⢀⡼⡾⢾⡿⣡⠴⡦⠴⢷⡲⠿⡄⣻⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⢀⣀⢀⣠⡤⠤⡀⢠⠤⠏⠉⡉⠳⡄⠀⠀⠀⠀⢸⠀⠔⠀⢀⡎⢹⡵⢿⢶⠿⢖⣾⣱⠧⠘⢀⣼⣾⠃
⠀⠀⠀⠀⠀⠘⡖⠐⠲⢤⡸⠋⠉⡶⠀⡤⠒⠀⠙⢢⠀⠀⠀⠈⣆⣀⢠⣼⡑⢦⣚⣃⠸⠀⠀⠉⠁⢀⡔⢘⡟⠁⢀⡠⠴⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢯⡀⠠⠀⠙⢄⣀⠗⢘⢃⠉⠀⠀⢸⣤⣤⣄⡀⠀⠑⠬⢻⣽⡟⠋⠀⠀⠀⠀⣀⣐⣽⢿⢏⠴⠊⠁⠠⠒⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⢠⡈⢻⣤⣼⣊⣢⣈⠷⠈⠈⠉⠙⠛⠷⣦⣀⡤⠭⠄⠀⣤⣐⣪⡽⠟⠉⣀⠘⡟⠀⠀⣀⡤⠒⠲⠦⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠁⠀⣧⠈⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡻⠛⠉⢩⠍⠁⠀⢀⠀⠈⠻⣇⣠⠞⠁⣀⠀⠀⠀⣴⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠁⠰⠄⠆⢱⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢰⠀⠀⠀⠴⠚⠚⠋⠀⠙⠿⡉⠀⠰⠦⡔⠋
⠀⢀⡀⠰⠂⠤⡀⠀⠀⢀⣴⠖⠒⠤⢀⠠⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠘⠀⠠⠂⠀⠀⠀⢁⡀⠀⠀⢸⠈⠁⠀⢲
⢠⠈⠀⠀⡀⠀⠈⢢⡠⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⠁⣀⣼⠆⢈⡷⠊⢹⡄⠀⠀⡰⠤⠴⠁⢀⡴⡟⠳⢦⡥⡿
⡅⠀⠀⠀⠀⠀⠀⣸⠓⠚⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠶⠀⢠⡿⠛⣷⠾⣄⠘⣷⡀⠈⣛⡉⡉⠀⠀⠘⢧⠉⠀⠰⡀⠘⡅
⠈⠳⣀⠐⠚⠛⣿⢧⡀⠀⠀⠳⠄⠀⠀⠀⢀⠖⠛⡃⢀⢻⣶⢾⢁⡼⡃⢀⣼⠋⢨⣻⠀⣽⠀⢨⠶⠒⢄⠀⠢⢀⣀⣄⡞⠁
⠀⠰⠁⠤⢄⣠⡄⠀⠀⠀⠢⢠⠼⠀⠀⢀⣎⠀⣠⡷⠀⢿⠁⢠⣱⠞⠻⣶⠿⡀⠼⣿⣴⣇⣶⣭⡆⠘⣼⠀⠀⠀⠀⠉
⠀⢀⢀⠴⠊⢩⠇⡤⠒⣤⣀⢸⡇⠀⠀⡎⠈⣿⠟⡇⢀⣸⣧⣽⣿⠀⢠⡇⣠⣿⠀⣹⡗⠈⣹⠀⣩⠉⢧⡀
⠀⠈⠳⠀⠒⠒⠃⠀⠀⡇⠀⢀⡠⣀⠀⠙⠛⢛⠀⣼⡿⠻⣿⠋⢉⡗⠑⢿⠚⠳⣼⣦⣶⠤⢽⡤⢭⡀⢸⡇⠀⢀⣀⠀⠤⢀
⠀⠀⢀⠄⠉⠉⠉⠒⠒⡷⠚⣁⠀⠈⠁⣆⠀⠈⠻⢻⣄⣀⣿⡘⠾⣥⣀⢦⣀⣶⡀⣰⢠⠀⠞⠇⠀⡻⠉⠖⢋⠀⡀⠀⠀⠈⠑⠠⡄
⠀⢐⠁⠀⠀⠔⠉⠓⡶⠷⢖⡀⠠⠄⠀⢉⠀⠀⠀⠀⠠⡽⢻⡿⠃⠙⡟⠉⢿⠋⠳⡅⠘⣤⠿⢳⣴⣃⠀⡆⡑⠁⠉⠀⢀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⡀⡰⣄⠀⢤⡍⣠⠔⠋⠀⠀⠀⠀⢸⡄⢸⣿⣐⠇⣿⡠⣿⣀⣸⣧⣜⣿⣀⢔⡇⣨⠿⢿⡔⡀⠁⠀⠜⣀⣴⠁
⠀⠀⠐⠒⠁⠐⢚⠀⠁⠘⣎⠐⠓⢢⠀⠀⠀⠀⠀⠀⠀⠙⠟⠇⢨⣿⠉⢹⡏⠉⡛⠁⠙⠁⠈⠋⡸⠁⠀⠂⠀⠈⣶⡉⣀⡠⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠈⢣⠁⠴⢻⠛⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠾⣖⣺⣧⣀⣕⠁⡸⠀⠀⠀⡇⠀⠀⠀⠀⡠⠃⡼⠋⠉⠲⣔⠃⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠜⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⢸⡇⠀⡟⠀⢹⠀⠀⠀⠀⠑⡀⠘⠀⠈⠁⡞⠁⠀⠀⢐⡿⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠋⠱⠷⡇⢴⠎⠀⠀⠀⠀⠀⠙⢦⠤⣄⣠⠗⠈⠂⠀⠸⠀⠈⠶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣾⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡄⠀⡀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠀⣠⠓⠖⠴⠚⢥⣤⣷⣶⠤⠒⣆⡀⠻⣠⡤⠄⢀⠀⠉⢡⠄⠐⢤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠚⠂⠰⢶⡴⠅⠈⣡⠙⠊⣿⣦⣾⣿⣿⠋⠀⠘⠽⣣⢌⢹⡿⢗⣮⣉⠉⡮⠄⢀⣦⠀⡤⡜⣶⠩⣒⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠉⢀⠹⡟⢀⣠⣿⣿⠋⣿⣟⠶⠀⠃⠀⠠⠈⢉⣠⣤⣾⠧⣤⣷⣾⣷⡄⠋⠓⠄⢰⠘⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣄⣐⠚⠿⠊⠄⠀⠈⠁⠂⠀⠰⠆⢒⠀⠀⠈⠻⠋⠀⠸⣿⡿⣿⠖⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠿⠃⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠃⠀⡆⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣇⢄⢀⣡⣤⣤⣤⣷⣀⡅⣴⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠢⠀⠀⠀⠈⠲⣬⣿⣿⣯⣿⣿⡯⣿⣿⣿⣿⡀⡀⠄
⠀⠀⠀⠈⡤⠀⠤⠀⠀⠀⠀⣀⠀⠀⠢⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⡹⣿⣷⡄⡀
⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣯⢠⠔⠚⠁⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢯⣽⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⡟⠒⠀⠀⠀⠒⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠿⣙⠂⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠊⢀⠿⣿⣾⣿⣿⣾⣿⡿⢿⠛⠀⠀⠉⠑⠀⠀⠀⠀⠠⠄⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⡀⠈⠈⢸⠏⠉⠛⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⡰⢄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠃⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠃⠀⠀⠀⠅⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠿⣿⡿⠿⠿⠿⠿⣾⣿⣿⣿⣿⣿⡍⣽⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢫⣿⣿⠘⣿⣟⠻⠿⠇
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣿⣿⣿⣹⣿⣤⢻⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣏⠘⣻⣟⣈⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢧⠀⠼⣏⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣤⣿⣿⠏⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠀⢸⡤⠘⠃⡀⠀⠀⠀⠀⠀⠀⢼⣿⡷⢻⠿⢣⣿⣿⣽⣷⣦
⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠁⠀⠀⠀⣹⡏⠀⠈⣧⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣟⣬⣟⢛⣿⢿⠟
⠀⠀⠀⠀⠀⠂⡴⠂⠀⠀⠀⠀⢀⢲⠀⢰⡆⠈⠳⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣟⢸⣯⣿⠗
⠀⠀⠀⠀⠀⠈⢁⢀⠀⠀⠀⠀⠐⠎⠉⣻⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⢀⢄⠜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣂⠻
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠰⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣖⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣟⣻

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⡼⠁
⣿⣿⣿⣿⣿⣿⣮⣍⡛⠛⢋⣩⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠑⠶⣦⣤⡶⠖⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⣀
⣿⣿⣿⣿⣿⣭⣖⣒⣂⣀⣀⣒⣻⣯⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠒⠯⠿⠿⠿⠿⠿⠄⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠰⡆⠀⠀⢦⡀⢦⡀⠐⣦⠰⠆⠀⠀⠀⢶⡤⡶⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠋⠙⠛⢶⣦
⠀⠀⢻⣄⠀⠈⢳⣌⠟⢀⡠⠶⠶⠿⠶⣆⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠚⢋⡿⠀⠀⠀⠀⠀⠀⠀⢠⣴⠂⠛⠻⣦⣼⡉⠑⢄
⠀⠀⠈⢿⢄⢀⠌⢁⡴⠃⠀⠀⠀⠀⠀⠙⢯⣰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠃⢘⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠬⡿⠅⠀⠘⢃⠀⢰⡃⠉⠉⡀⠂⢄⣀
⠀⠀⠀⠈⢾⠃⣰⡿⠃⠀⢀⣤⣤⢐⢀⠔⢨⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢊⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠾⠀⠀⠀⠀⠈⡄⠈⡇⠀⠀⡀⠀⠀⣥
⠀⣀⣀⠀⠈⣰⡿⠀⢗⡀⠀⠘⠂⢹⣻⣄⣾⣧⣘⣧⠀⢀⠠⢤⣄⠀⡀⣰⠛⠇⠊⣽⣿⡿⠃⣴⠆⣁⠀⠀⠈⢀⠀⠀⢠⡆⠀⠀⠐⠀⠰⢀⡙⠀⢠⠃⠀⢀⡅
⢈⡇⢿⠀⣤⣿⠇⠀⠈⠁⠀⠀⠰⠆⠹⣿⣿⠙⢗⠆⣀⣰⠀⠀⠙⠂⣟⣠⣄⠀⠔⠁⠟⠁⠠⠋⠀⠀⠀⢠⣤⣤⣤⡄⠀⠓⢦⣤⣈⣀⠂⠀⢹⣇⣸⡄⡄⢿⡧
⠀⡇⢿⡶⠋⠁⠀⠀⠰⠀⠀⠦⠠⢾⣆⠙⣏⠛⠎⠢⠿⡏⠀⠀⡰⢾⣿⣿⠁⢀⣠⠁⠀⠀⠀⣄⠀⠀⣴⣿⣿⣿⠋⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠻⠶⣟⠸⡇
⠀⠳⢸⡇⢀⡤⠤⠟⢠⡀⠀⠀⠐⡄⢹⣄⡈⢧⡘⠀⡘⢀⡀⠀⢀⠀⣉⡉⠋⠼⠏⠀⠀⢸⡿⠿⠀⢰⣿⣿⣿⣿⠁⠀⡠⠾⠛⠷⣿⣿⣿⣿⣿⣦⣄⣠⣿⣼⡇
⠀⡷⢠⣤⣎⠁⠤⠈⠑⠃⠠⠀⠀⠘⠈⠹⣿⡆⠻⣶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠀⠀⢀⣾⣿⣿⣿⣿⣀⣾⣷⢯⣿⣶⣿⣿⣿⣿⣿⣿⠲⠿⣽⣮⣽
⠐⡿⢿⡀⠡⣀⢀⡀⠀⠀⡀⢀⣀⣤⣧⣠⣜⡿⢠⡄⠀⢠⡀⢀⣤⢤⡤⠄⠐⠐⠛⠛⠓⢶⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆⣨⣸⣿⣿
⠀⣿⢸⡇⠐⣾⣿⣿⡀⠀⣿⠞⠙⠂⠾⡠⣲⣿⡿⠈⠒⠞⠻⣮⣿⣜⠷⠀⠀⠀⢀⡴⠂⢨⣿⡟⡿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠉
⠀⠈⣿⠀⠃⠙⡁⢻⠃⠀⢸⣷⣶⡾⠇⣴⣽⣿⣇⡨⣏⡇⠀⠈⢻⣽⣇⠀⠀⣴⠋⡀⠀⣶⠹⣿⣸⣿⣏⣴⡟⢿⣿⣷⣾⣾⣿⣿⡿⠁⢿⣿⣿⣿⣿⡄
⠀⢀⡿⠀⠀⢠⢹⡟⢀⠀⢸⣿⠬⣻⣶⡿⣿⠻⣧⠀⢸⣧⠀⠀⣸⡇⢩⠀⡰⠃⠀⠀⣸⠃⡇⠈⣻⡇⣹⣾⣿⣷⣶⣯⣿⣿⣿⠟⡄⠀⣼⢻⣿⣿⣿⣧⠀⠐⢂
⠀⠈⣷⠀⠀⠀⠨⢁⣰⠆⢸⡧⠀⠃⢚⠛⠻⣄⢹⡆⢸⣏⠠⣾⣏⣀⣼⡆⠁⣀⢠⠷⣏⣴⣁⣠⡼⢯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⡇⠀⣿⣿⣿⣻⣷⠺⠿
⠀⠀⣿⠀⠀⠀⠀⠈⠙⣦⣾⠁⣠⣴⣿⣧⣠⢘⡇⢻⡎⢣⡀⠈⡷⠻⣬⡇⠛⠉⣠⠉⠋⣼⠉⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠁⠀⠸⣿⡿⢿⠏⢤⡤
⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟⢻⢿⣿⣧⣄⡊⣻⣼⣷⣬⡿⣦⠀⢠⡏⠀⠈⡤⢦⣀⣿⠀⣤⡿⠀⢀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⣠⡿⠀⠀⠀⠀⠀⢺⠁⣾⣶⣿⠽⢿⡌⢻⠆⢻⣿⣟⣿⣆⣸⡀⠀⠀⡇⠀⠉⣾⡴⣡⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⡟⠲⢤⣄⡀⠀⠀⠀⢻
⢰⣏⠀⠀⠀⠀⠀⠀⣟⠀⢾⠙⠣⠶⠛⢣⠈⡷⠸⣿⣿⡿⠛⢻⡇⠀⠀⡇⠀⠀⢻⡀⠈⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠃⠀⠀⠀⠉⠙⢲⣾⣿⣆⣀
⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⢨⣀⢤⡾⠋⠉⢾⣥⠀⡏⢹⡇⠀⠀⣇⠀⠘⠓⠀⠀⠘⣇⠀⡿⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⠀⠘⡟⠀⠉⠈
⠀⢸⡷⠀⠀⠀⠀⠀⡿⢢⣾⡇⠐⣷⠀⠀⡀⢻⣠⣧⠸⡇⠀⠀⢻⡄⢠⡜⠀⠀⠀⠹⣦⡗⣸⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡏⠓⠶⠤⣄⣀⡀⢰⣀
⠀⢸⡟⠀⠀⠀⠀⠀⣰⡯⣸⡇⠀⢿⠀⠈⠁⠠⠻⣿⢸⡇⠀⠀⠈⢷⠀⡇⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⣐⡀⠀⠀⠀⢀⣮⡿⠘⠛
⠀⢸⡏⠀⠀⠀⠀⠀⢿⣗⢺⡓⢆⣾⠆⢠⠄⢠⠀⢹⣿⡇⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠀⢸⡇⢈⠀⠀⠀⠀⠈⠉⡵⠊
⠀⢸⡇⠀⠀⠀⠀⢠⢸⡇⢸⠅⠉⢹⡇⢀⠀⢀⠀⠀⠹⡇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣾⡏⠀⣿⡇⠀⠀⠀⠀⡄⠀⠀⠁⠀⠄
⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⣼⣀⠁⠀⣷⠐⠀⠀⠀⠀⡀⢱⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡇⡆⠀⠀⠀⠸⢀⣀
⠀⢸⣇⠀⠀⠀⠀⠀⡐⢿⣼⠃⠀⠀⣻⡀⠀⡶⠀⡴⠀⠀⠃⠀⠀⠀⢀⢿⡇⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⢶⡿⣇⣦⣤⣀⣀⣋⡀⠀⡙⠃
⠀⠈⡉⠀⠀⣀⡤⠀⠤⢿⣿⠀⠀⠛⢹⡇⣼⠃⢠⠀⡘⣀⣯⠀⠀⠀⠈⢶⠕⠚⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⣿⣞⣻⢛⣁⣀⣀⠉⠉⢹⣋⣿⠁⢈
⢶⡄⠱⢶⡞⠋⠙⠞⠷⢿⣿⣂⢄⠀⢉⣷⣟⠀⣀⡀⣀⣉⣴⣦⣤⡤⠈⠘⢣⣴⠤⣿⣿⣿⣿⣿⣿⠿⣫⣵⣿⣿⣿⣿⣿⣿⠿⡟⢸⠿⠷⠟⠳⣶⣤⣴⣶⠿⠛
⠾⠛⠋⠉⠉⣙⣶⣶⣤⡌⠀⠁⠛⠁⠉⠀⣠⡏⠉⠈⣿⡋⢰⣇⣈⠶⣐⣲⣧⣤⠀⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠐⠶⠀⢀⣀⡀⣀⣨⢹⣼⣄⣀
⠀⢠⣶⣿⠛⠛⠛⣛⣿⠁⠀⠀⠀⢀⣴⣮⠉⠙⢛⣶⠏⣔⢾⡷⠟⠹⢿⣁⣄⠀⢸⣿⡏⣿⣿⣿⣷⡸⣿⣿⣿⣿⢻⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⣀⠤
⠀⣸⣿⠛⢛⡙⣽⣯⣭⠆⠀⠀⣶⠛⣋⣣⠴⠞⠛⠉⢷⣍⣁⠉⠚⢢⣤⣤⣀⣀⣸⣿⡇⢸⣿⣿⣿⣷⡘⠿⠿⢣⣿⣿⡿⣿⣀⠀⠀⠀⠀⠀⠀⢀⡌⠰⡁⢀⠄
⠐⢿⣿⣿⣯⣽⣿⠇⠀⠀⡀⠀⣾⡿⢁⣁⣀⣀⣠⣤⠤⠹⠟⠛⠛⠁⠉⠉⠉⠙⣿⣿⣿⢸⣿⣿⢿⣿⣷⣤⣴⣿⣿⣿⡇⣿⡏⠉⠛⠛⠒⠒⠚⣋⢼⣆
⠀⠈⠙⠲⠟⢻⣿⠀⢀⡌⠀⡘⠃⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣇⢿⣿⢦⡀⠀⢈⡔⣪⠕⠂⠈⠳⣤⡀
⠀⠀⠀⠀⠀⣿⣿⡔⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢸⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⢸⣿⣎⡡⢾⡥⠊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠈⢿⣧⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣾⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⡎⣿⣷⠘⠀⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⠆⠀⠈⠂⡈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣷⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⣿⣿⣿⣿⡿⠿⡿⠿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠘⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡞⢻⡏⠀⠾⠿⢷⡴⠏⠀⠇⠐⠀⠐⠐⠒⠀⠒⠂⠰⠟⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⡼⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⣩⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⠜⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⡤⠀⠐⠪⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣈⠤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⣀⠤⠔⠀⠀⠀⠀⠰⢀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘
⠀⠀⠀⠀⣠⡴⡇
⠀⠀⠀⡸⠅⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⠴⠒⢈⣩⠽⠓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠐⠄
⠀⠀⠀⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠤⣌⠀⡞⠁⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰
⠀⠀⠀⠇⡞⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⢮⡶⠸⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⠔
⠀⠀⠀⡃⠀⡀⠂⠀⠀⠀⠀⠀⠀⠐⠦⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠟⣣⢙⡴⠶⣖⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣖⠀⠀⠀⠀⣰
⠀⠀⠀⡇⠀⡾⠁⠀⠉⠀⠀⠀⠀⠀⠺⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠂⠉⠀⠀⠀⠐⠃⡀
⠀⠀⢀⠄⠄⢀⠀⠀⠀⡀⢀⠄⠀⠀⠑⠖⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠘⠂⢄⠀⠱⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠷⢠⠀⢀⠀⠀⠘⠂⢦⡀⠑⣄
⠀⠀⠀⠄⠀⠉⠀⣿⠄⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠒⠋⢰⠋⠁⠀⠀⠀⠀⠉⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠄⠀⠀⠂⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣈⠳⢄⡀⢀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠚⠐⡂⠀⣄⠖⠀⠛⠣⠄⠰⠀⡽⠁⠀⠋⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠍⠀⠀⠀⠐⠂⣥⠠⠤⠠⠤⠖⠀⠀⠃⠀⢀⡰⢆⠀⡛⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢥⡈⠁⠶⣀⠀⢀⡀⠖⠀⠰⠦⢠⠄⠀⠀⠀⠰⠂⢀⡀⠀⠀⠀⠂⠀⠰⠂
⠀⠀⠀⠀⣀⠀⣤⠀⠘⡼⠂⠌⠦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⢨⡄⠈⠁⠰⠶⠀⢀⡚
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠆
⡀⠀⢠⠂⡇⠀⠀⢀
⡝⢀⢇⣇⢣⠀⢀⠎⢦
⠀⢺⢸⣿⣸⠀⡼⠸⠂⠳⡀⠀⠀⡴⡇⠀⡠⡎
⠀⠈⢺⣿⠃⠀⢡⠀⡀⠀⠹⣀⠞⠀⣧⠞⠀⠁⢀
⡀⠀⠈⡏⠀⠀⠀⡄⢻⣤⣶⠋⠀⠀⣯⣼⠀⡇⡸⠄⠀⠀⡔
⠀⢀⣸⠁⠀⠀⠆⢣⢸⡿⠁⠀⠀⠀⣿⣯⢀⡽⠁⡱⠊⢰⠁⢀⡤
⢀⢸⠏⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣼⡷⢋⡴⠋⠀⠀⣿⡊⢠⢀⠰⠂⠀⣠⠂
⢸⡼⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⣿⡴⠋⠀⠀⠀⢠⡿⠗⠋⠀⠀⡥⠎⡅
⠸⣷⢸⠀⠀⠀⠀⣄⣷⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⢠⡷⡹⠁⠀⢀
⣟⣿⣼⣦⣸⢂⣰⣿⣧⠀⢀⡃⠀⣼⡄⡠⠀⠀⠀⣼⣤⠄⠄⣀⡠⠞⠛⠁⣠⢖⠥⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣟⡕⡡⡠⢡⣾⡿⠟⢛⡉⠀⠀⠀⣠⢞⣑⡁
⢿⣿⣫⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣟⣕⡴⣋⡥⢔⣠⠔⠀⠀⢀⡞⡉⢉⡌⠀⠰⠊
⣟⣎⣳⡗⣿⣿⣟⣏⣥⣿⢿⣿⣿⣿⣿⣿⣿⡿⢚⠉⠀⠀⣀⡴⠛⠛⠂⠉⢉⠬⠋⠁
⣾⣿⣿⣿⣿⣿⣶⡷⣯⣞⣿⢺⢿⣿⣿⣿⠿⠟⣋⣡⣴⣾⣧⣀⣀⣀⣤⣾⣁⣀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣯⣭⣾⣿⣿⣿⡟⣛⣩⠍⠉⠉⠉⠉⢁⣀⡈⠉⠉⠉⣉⣩⣭⠖⢊⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⡷⣿⣿⣿⣿⣿⣛⣃⡩⠍⢀⡈⠄⠀⣀⣴⡾⠖⠂⠔⠂⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣫⣿⣻⢿⣿⣿⠿⠿⠧⠒⢒⣈⣡⣤⣶⣉⣉⣉⠒⠚⠤⠆⠒⠒⡨⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣯⡼⣹⣿⣿⣷⣶⣾⣿⣿⡢⠬⠍⣙⠓⠶⣔⣒⣂⣀⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠀⠀⠀⠀⠐⡍⠀⠀⠋⠃⠀⠀⠀⡤⠠⠀⠀⠀⠀⠨⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢤⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠦⢤⡄⠀⠀⠀⡀⠀⠀⠄⠉⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⣄⠀⠀⠀⣠⠐⡀⠀⠀⢀⠔⠁⠐⠀⠀⠀⠀⢀⠄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢄⠁⢂⡤⠐⠀⡆⠀⠘⠉⡐⠂⠤⡰⢶⣀⠈⠒⠃⠀⠂⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠚⡌⠀⠀⠀⠐⢡⠀⠀⠀⠁⠀⠐⠚⠀⠐⠐⠃⠀⠊⣀⠀⠀⠘⠁⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⣀⠠⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢁⣀⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⡀⠀⢠⠀⠀⠀⠑⣀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠉⠀⡀⠈⠈⠀⠀⠀⢀⡤⠤⠀⠉⠀⡀⠙⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠃⣰⡟⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢠⠱⣄⠐⣿⠀⣀⠀⠖⣧⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠴⢦⡀⠘⠓⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠉⢿⣆⠈⣵⣿⣿⢣⠀⢘⣤⣤⠃⠘⠀⠐⢂
⠀⠀⠀⠀⠀⢀⡤⠶⠄⠁⠀⠀⠀⠀⠀⠉⢻⡄⠀⢀⡈⣷⣼⡛⢀⣤⠐⡠⠰⠀⠀⠀⠀⠀⠀⠀⠃⠟⠛⠛⢻⡿⠁⠀⠴⢋⣤⠂⠉⠈⠀⠀⠙⠐⢦
⠀⠀⠀⠀⠄⠈⠀⣀⣀⣀⣤⡀⠀⠀⠀⠐⠀⠙⢠⠀⠑⣿⣿⣥⠈⣀⡀⠅⣆⣴⠂⠘⠛⢫⣄⣈⣁⣠⣵⣶⣾⣤⣤⣴⣀⣠⣀⣤⣤⣴⣶⣦⡀⠀⠀⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠻⠿⢷⣤⣄⡀⠀⠀⠀⠀⠀⣽⠟⠋⣠⣴⠴⠦⠤⠌⠛⠻⠷⢿⣯⣁⣀⠤⠻⣿⣟⠿⣿⣍⡙⢛⡙⠟⢫⢉⣧⣀⡀⠀⢀⠔
⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣶⣶⣄⣀⣸⣄⣠⣽⠏⠀⠀⠛⠲⠆⢀⣲⣿⣿⠟⠉⠀⣠⣿⠀⠀⠈⠿⣿⣷⣍⡀⠶⢭⣙⣦⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠁⣀⡀⣀⠀⠤⠈⠁⣀⣿⣿⣯⠿⣿⠧⠒⠒⣠⣤⠶⣾⡿⠛⠋⢀⠀⠀⠀⢉⣿⠛⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠉⠉⠉⠁⠀⣠⣴⣿⣿⠋⠿⠁⢀⠃⢀⡴⠟⠋⠀⠐⠋⠀⠁⣠⡞⢀⠄⠀⣾⠇⠀⠀⢀⠄⠀⠸⢿⣿⣿⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⢀⣿⣿⣿⡿⠋⠀⣰⡀⠀⡇⠀⠈⠀⠀⣀⣀⢀⡀⢂⡴⢋⠴⠋⠀⣸⡿⠀⠀⠀⢸⡄⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠢
⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⢀⠐⢡⣎⣠⣧⣴⣦⠰⠦⠸⠛⢿⠾⡿⠃⠀⠀⠀⢰⣿⠁⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⠀⠀⠀⠀⢀⣶⠞⠙⠁⠀⠀⠀⣠⠆⠀⣰⠛⠛⠿⠿⣧⣄⣀⣀⣰⠋⠐⢧⠀⠁⠄⢀⣿⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⡀⣾⢇⠰⠀⠀⠀⠀⠀⠂⠙⠿⣿⣿⣿⠀⠀⢨⠀⠀⠀⢻⡟⠀⠀⠀⠀⣸⣷⣀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢈⡄
⠀⠀⠀⠐⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠸⣇⣀⣤⢃⠀⠀⠀⢀⣤⠿⠴⣦⣜⣷⡂⠀⠨⠀⠀⠀⠀⣾⡿⣿⣿⠒⡠⢂⠀⠀⠀⠘⣿⠀⠀⠀⢀
⠀⠀⠀⠀⡰⠛⢁⡤⢤⠀⠖⠂⠤⠐⠊⠀⡇⠀⠀⠹⢿⣿⣷⣵⣦⣴⠋⠀⠀⠀⠀⣹⡟⢧⡄⠀⠀⢀⣴⣾⣿⡇⣼⣿⠀⠀⠈⠁⠀⠀⠁⢸⡆⠠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⡀⠀⠀⠀⠈⢻⣿⠃⠀⠀⠀⠀⢠⣿⡆⠀⠙⢦⣴⡛⢿⣿⣿⣠⣿⡟⠀⠀⠀⠀⠀⢀⡀⠈⣿⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣧⣂⣀⣀⠜⠁⢀⡠⢀⠀⠀⣰⣾⡿⠁⠀⠀⣼⣿⡇⣽⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡆⡻⢿⣿⡟⠀⠀⠈⢿⣿⣿⣿⣿⠿⣯⣦⣴⣶⣿⡟⠀⣾⣿⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠂⠀⠀⠀⡀⠉⠉⠉⠀⠀⣻⡿⢿⣟⡉⣀⣤⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠙⣮⣤⣿⡇⠀⠀⠀⠀⣰⣿⡇⠈⣿⣷⣿⣿⣿⣷⣆⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣠⡀⡈⣿⣿⠷⣼⣤⣶⣶⣿⣿⠁⠀⢹⣿⡆⠀⠰⣿⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠟⠁⠀⠙⢻⠿⠛⠛⢃⣀⣀⣸⣿⣇⠠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣵⣦⣤⠃⠀⠀⠀⢨⣿⣿⣿⣿⠓⠟⠻⠋⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠻⢿⠿⠀⠀⠀⠀⢸⣿⠏⠸⣿⣷⣀⣠⣄⣥⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⣠⡄⠁⠀⢧⣔⣠⣶⣿⡿⠀⠀⣿⣿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇⡀⠀⠀⣹⠟⠛⠋⠀⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷⣾⣾⡇⠀⠀⠀⢠⣿⡟⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠋⣡⢀⡀⣀⣰⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣶⣶⠈⠾⢿⣿⣿⣿⡟⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠇⠀⢀⣀⣰⣿⡃⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠿⠛⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠙⠀⠀⠀⠀⠀⡔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⡄
⠀⠀⠀⠈⠁⠀⠀⠀⢠⠁⠀⠐⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠤⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⢏⡋⠀⠀⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠂⠀⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠉⠓⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠠⠀⠈⡇⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡇⠀⡠⠔⠀⠂⠀⠳⢤⣀⣀⠄⢹⠀⠀⠀⠀⠀⡌⣀⡤⢂⠀⡀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢰⡇⡠⠸⠏⠀⢠⠀⠀⠀⠠⢄⣉⡷⢦
⠀⠀⢻⣸⡇⠀⠺⠀⠀⠀⠈⢳⣦⠀⡀⠀⠀⠀⠀⠀⠀⢯⠀⢧⣘⠀⠐⢀⠄⢀⡜⣇⠀⠀⠀⠀⠀⣧⠈⠀⠒⠒⠒⠒⠒⠒⠚⠉⢻⡆⡾
⠀⠀⠀⠻⣯⣒⠀⠀⠠⠤⠒⢛⣡⠞⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠒⠒⠒⠛⠉⡸⡟⠀⠀⠀⠀⠀⠹⣆⠀⠴⡎⠀⢀⢄⠀⠀⢰⠋⣰⠇
⠀⠀⠀⠀⠀⠉⡽⢿⠒⡚⢯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣀⣀⣀⣀⡀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢐⣫⠞⠃
⠀⠀⠀⠀⠀⠀⠃⢀⠌⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⡝⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣁
⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⣉⡀
⠀⠀⠀⡄⠈⠀⠀⠀⢸⠱⢒⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠋⠀⠀⠀⠀⠉⠉⡒⢶⣤
⠀⠀⠀⠘⠶⣤⣀⣀⣀⣉⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠂⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣄⣀⠀⠀⠀⠀⠓⢚⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⠦⠀⢚⠀⢨⠀⢾⢃⢀⠈⠀⠀⢲⡆⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⢇⠈⠃⠀⠀⠀⠀⣀⢀⠀⢈⣿⣷⠀⢀⣼⠿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⠴⠂⠁⠀⣀⡀⣠⣴⠼⠛⠁⢠⣼⡟⠋⢠⡄⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠶⡶⠿⠿⠛⠋⣉⣀⣠⣴⣾⣿⡏⣿⡄⠸⢧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⣿⣿⣵⣶⣴⣶⣶⣦⣿⠿⠿⠿⣿⣭⡉⠀⢈⣁⣄⡹⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⣷⠟⢀⣈⣽⣿⡛⠉⢹⣯⣵⡄⣼⡟⠉⠉⠣⡀⠈⠉⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⣬⡶⠃⠘⡋⠉⢻⣤⠞⠋⢿⠁⠈⢻⠷⢤⡀⠹⣿⡅⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⠁⠤⠼⣇⡀⢻⠿⠆⠀⢩⡰⣄⣼⣿⣜⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠀⠠⡆⠰⡇⠋⠾⠻⡏⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⢶⣶⡀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠷⠶⣦⣤⡘⣳⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣋⣼⣷⣶⣶⣦⡤⠭⠭⢥⣀⠀⢨⣿⠀⠀⠀⠀⢣⡞⠉⣀⠄⢊⣠⣶⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣋⡉⠝⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⣿⡇⠀⢠⣾⣿⣿⣷⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂⣀⠀⡀⢀⣼⠟⠁⢼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢒⣠⡿⠟⠛⠛⢿⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠞⠝⠁⠀⠀⠀⠀⠘⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⣠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⢿⡿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢥⠀⠂⠈⡑
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠶⢆⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⡅⣹⣴
⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠂⣰⠞⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⡀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⢀⡀⢀⣤⣀⠀⣀⠀⠀⠀⠂⠀⠀⢀⡔⢁⠀⢀⢀⢀⠀⢀⠀⠀⠀⠀⠀⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠈⢃⠀⠔⠒⠁⠀⠰⠦⠄⡀⠀⠀⣀⠰⠦⠄⠐⠀⠀⠀⠀⠀⠀⠂
⠀⢀⠀⠁⠈⣀⠀⠞⢀⢀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠇⠀⠀⣀⢀⣐⡤⠐⠒⠞⠀⠉⠉⠀⠈
⠀⠀⠁⠈⠀⠈⠐⡞⠁⠄⠈⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⢩⠐⠫⢸⡇⡄⠀⠀⠈⠉⠈
⠀⠀⣀⠄⠂⠀⠈⡁⠛⡀⠀⠉⢄⡄⣼⡄⡀⠐⠦⣀⣪⢠⠋⣷⠖⠛
⡤⢆⠀⠀⠀⠀⠸⠁⠠⠀⣠⠨⠉⡰⢶⡠⡀⠀⠀⣰⢅⣥⡞⠋

⠀⠀⡤⠠⣤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠓⠄⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠋⠁⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠃⠃
⠀⠀⠉⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢳⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠄⡀⢀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢰⠺⣄⠀⠀⠀⠈⠀⠐⠃⣹⢾⠀⠀⠀⠀⠀⠧⣀⠀⠈⠀⠉⡆⠀⣹⡼
⠀⣿⡖⡁⠘⠀⠀⡢⢆⣾⠀⠀⠀⠀⠀⠀⠈⠆⠈⠙⠒⠒⠒⠒⢚⠉⠁⡾⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⣰⠿⠁
⠀⠘⢯⡉⠘⠀⠐⠀⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⢑⠦⢭⣙⣛⣉⣥⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠝⠛⠛⠻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢀⡀⠀⠀⠈⠁⠁⠐⠣⣦⡄⠀⠀⠀⠀⠀⠀⠀⡀⢄⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠂
⠈⠐⣆⡀⠀⠀⠀⠀⣀⣑⡯⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠨⠀⡈⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⣿⣿⣿⣿⣿⠷⠶⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢢⡀⣀⣿⣿⣿⣿⣧⡀⣀⠎⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣰⣷⣴⣿⣿⠿⡿⠿⢿⠿⢿⣿⣦⣿⣄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⡕⠉⠁⠀⠉⠉⠉⠁⠉⠉⢫⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⣿⣿⣿⡇⠀⠀⠀⢀⣾⣷⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⠀⠀⠀⠀⢹⣿⣿⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⣠⣴⣾⣿⣿⣿⣯⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠃⠀⠀⠀⠸⡟⠉⠉⢙⡏⠉⠉⠻⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡔⡀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢣⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣻⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⡇⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣿⡻⠀⠀⠀⠀⠀⠀⠀⢘⣧⣀⡏⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠌⢿⣧⡀⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⣵⠀⠀⠀⠀⠀⠀⠀⣮⣽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡘⠀⠈⠙⣿⣯⡙⠛⠷⣤⣀⠀⠀⠀⠀⠻⣿⣿⣷⣧⣶⣴⣦⣶⣶⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⣩⣙⢷⣤⣌⠻⡷⠄⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣷⣦⣄⣀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢲⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠛⠉⠉⠉⠉⠀⠉⣩⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⢻⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣼⣿⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⢐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⣷⡀⠀⢀⢄⠒⠤⠜⠖⡠
⠀⠀⠀⠀⣤⡀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⠁⠀⠀⠀⠀⠀⠀⡼⠉⠉⣉⠉⠳⡀⢆⠁⠀⠀⠀⠀⠈⠊⠂
⠀⠀⠀⠀⠿⣷⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⠟⠀⠀⢓⡈⠀⠀⠀⠀⠀⠀⠸⢀⢀⡀
⠀⠀⠀⠀⠀⠸⠦⠶⠄⠈⣿⣧⣄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢃⢃⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠢⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⣴⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡶⠀⠀⢰⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢠⣦⡈⢻⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⠿⢂⣤⣾⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣮⠉⠙⠻⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠿⠿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⣀⣀⣶⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡿⣿⣿⢹⣧⣿⣀⠉⠻⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⠿⢟⣿⣿⣿⣿⣿⣛⣋⣉⣠⣀⣀⣀⣸⣿⣿⣿⣶⣿⣿⣿⡟⠻⠁
⠀⠀⠠⣤⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡷⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡟⠻⢿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠪⢝⡻⣶⣤⣨⣿⣿⣿⣿⣾⣿⣿⢛⣴⢿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣾⣷⣼⣤⣨⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠉⠑⠮⣿⣿⣿⣽⣯⣿⣿⣿⣿⣏⠀⣿⠀⠈⢹⣿⡍⣻⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣟⣿⣿⣿⣿⣿⣿⣤⣿⣶⣤⣼⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡛⠛⠙⢿⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠸⢶⡟⢃⠀⠐⠘⣿⣿⣿⣿⣿⡟⢉⣹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣾⣇⠠⣶⡖⠉⣶⣧⢸⣿⡟⠛⣿⣝⢛⣹⣿⣦⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣠⣽⢀⣸⣿⠿⠋⢹⣾⣿⣷⣾⣟⠁⠘⣿⡟⠉⢹⣿⣧⣉⣈⠙⠟⠙⠻⠿⠛⠿⡿⢿⡿⢿⡿
⠀⠀⠀⠀⠈⠛⣿⣿⠿⢿⣶⣤⢾⣿⠛⡇⣈⣿⣿⣿⡿⠁⠰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠿⣿⡿⠰⣾⣿⡇⣠⡿⠿⢿⣿⠛⠉⣹⣷⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣻⣧⢹⣿⣇⣤⣾⣿⠷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣀⣻⠛⢿⣿⣇⢠⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠿⣿⣾⣿⣇⣤⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣷⣶⣿⡿⣾⡟⠛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢽⣉⣿⣧⡼⠿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠛⣷⣶⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢶⣾⣿⡟⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⣶⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣶⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⣎⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⡗⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⣖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣒⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠬⠭⠍⣵⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠀⠀⣀⣠⣋⡛⠒⠒⠖⠴⡇⠀⠀⠀⠀⠈⠁⠛⠃⠀⠀⠀⠀⠀⠉⠲⠤⠼⠾⣀⣀⡄⠀⠀⠀⠀⢠⠞⠉⠀⠉⠙⠢⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⡉⠉⡉⠐⠒⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⣠⠒⠒⠹⡄
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠞⠁⠡⢀⡤⠤⢷
⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢾⡏⠉⠙⠲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠤⠞⠛⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣸⠀⠀⢠⣶⡄⠀⠀⢸⣧⠀⠀⣀⡗⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣰⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⣰⠃
⠀⠀⠀⠀⡤⠠⢤⣠⠔⠒⢢⠀⠀⠀⠀⣾⡷⠤⠤⠤⡏⠀⠀⣾⠇⢷⠀⠀⠘⣿⠒⠋⠁⢠⡶⠒⠢⣄⠀⠀⢿⡇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢙⠾⡅
⠀⠀⠀⠀⢇⠀⠀⠁⠀⠀⣸⠀⠀⠀⢐⣿⠇⠀⠀⢰⡇⠀⣼⣯⣤⣼⡇⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠈⡇⠀⠈⣿⡞⠁⠀⠀⠀⣠⢄⣠⢤⠀⠀⠀⠳⣏
⠀⠀⠀⠀⠈⠳⣀⠀⠀⡰⠃⠀⠀⠀⠸⣿⠀⠀⠀⢸⠀⢰⣿⠀⠀⠀⢹⠀⠀⢿⡆⠀⠀⢸⣇⣀⣠⡴⠃⠀⠀⣾⡇⠀⠀⠀⠀⠙⢄⣡⠞⠀⠀⠀⠀⢸⡄
⠀⠀⢠⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⣼⡏⠉⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠻⠃⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣴⡞⠉⠓⢦⠀⠀⠀⢸⣧⠀⠀⣰⡞⠛⠲⡄⠀⠉⠉⣿⡏⠉⠀⣀⠀⠀⠀⠀⢸⡇⠀⠙⢦⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⣠⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣸⠀⠀⠀⣼⡇⠀⠀⣿⡇⢀⣠⠇⠀⠀⠀⣿⠇⠀⢸⣿⠀⠀⢸⠇⢸⣷⠀⠀⠘⡇⠀⠀⠀⠀⠈⣿⡄⠀⢠⠇⠀⠀⢸
⠀⠀⢀⡀⠀⠀⠀⢸⣷⠶⠚⠧⣄⠀⠀⣿⠇⠀⠀⣿⡿⣍⠁⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣼⠀⠈⣿⠀⠀⢠⡇⠀⣰⣿⡀⠀⠘⣿⣀⡞⠀⠀⠀⢸
⠀⠀⠁⠑⠙⠆⠀⠈⣿⠀⠀⠀⢸⠀⠠⠿⠄⠀⠀⣿⡇⠈⠳⠀⠀⠀⠀⠿⠀⠀⣿⡇⠀⠀⡿⠀⠀⣿⣀⡠⠏⠀⣰⡟⠀⡇⠀⠀⠸⣿⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⠞⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⡇⠀⠀⠙⠉⠀⠀⣰⡟⠉⠙⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠃⢀⡶⡄⠀⠀⠰⠟⠀⠀⠀⠾⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣓⡆
⠀⠀⠀⠀⠀⢀⣠⡀⠀⣀⡉⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⠄⠀⠀⠀⠀⠀⠀⣞⠑⠞⢱
⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⠉⢳⠦⣄⡀⠀⠞⠀⠀⠠⠖⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⣠⠤⠴⢆⠀⠀⠘⢦⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⡏⠑⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⣋⣉⣩⣤⡤⠖⠒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠
⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠀⠸⡆⠁⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣁⡨⠤⠔⠒⠋⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠙⠁⠀⠀⠀⠀⢸⡀⢀⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠉⠀⠀⠀⣀⣠⢤⡤⢴⣛⡦
⠀⢀⡤⢤⡀⠀⠀⠀⠀⢠⠗⠋⠀⠀⠀⠀⠀⠖⢦⣀⣀⡀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢸⣓⣮⣽⠯⠽⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤
⠚⠉⠀⠀⠙⠢⠤⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠀⠚⢢⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⢠⣀⠀⢀⣠
⠀⠚⠀⠈⠀⠙⡀⠀⠀⠀⠀⠀⠖⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠠⢀⠀⠀⠀⠈⠁⡀⠐
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠁⡀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠢
⠉⠠⡤⣆⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠒⠁⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠁⠀⡄⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠁⠀⢀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⢠⠄⠀⠚⠋⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡨⠀⠀⢣⠦⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣠⠈⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢆⠈⠀⣀⠀⢤⠃⠁⡠⢠⡁⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠀⠀⠀⠠⠜⠂⠀⠀⠀⠀⠸⠁⠀⢀⡄⠉⠀⡈⡻⠗⠘⠂⠄⠀⠲⠎⠀⠀⠀⠀⠂⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠃⠐⠀⠠⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⡜⠁⠀⠀⠀⠂⠀⠀⠀⠉⢐⠰⢢⣀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠒⠀⡀⠀⠑⠀⠐⠊⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠐⠀⠀⠀⠈⠁⠁⠀⠀⠄⠀⡴⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠒⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣤⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠀⠀⢠⠤⡞⠀⠀⠀⠀⠀⠠⠍⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⠈⡞⢁⡀⠀⠀⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢄⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⡀⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⡶⢾⣿⡷⣶⣶⣶⣶⣶