Law Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠤⠔⠒⠒⠚⠉⠉⠁⠀⠀⠈⠉⠉⠉⠒⠒⠐⠦⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠐⢶⣬⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠤⠔⠚⠉⠛⠛⢙⠃
⠀⠀⠀⠀⠀⢀⡟⠯⠉⠿⠿⠶⢶⣴⣄⣀⣠⣀⡀⠀⣀⣠⣤⡴⠶⠒⠉⠁⠀⠀⠀⠀⢀⣀⡴⠖⠋
⠀⠀⠀⠀⠀⢨⣿⣿⣶⣦⣄⣀⠀⠈⠈⠙⠛⣿⡿⠋⠈⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠞⠋⡹⠃
⠀⠀⠀⠀⠀⠀⠙⡟⠛⠻⠿⠿⣯⣶⣶⣄⣀⣻⡇⠀⠀⠀⣀⣠⣤⣶⣿⠟⠋⠀⣀⡔⠊
⠀⠀⠀⠀⠀⠀⠶⢷⣶⣤⣤⣀⠀⠈⠉⠛⠛⠿⠿⣶⣾⣿⣿⣾⠟⠛⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠩⣿⡏⠀⣀⠖⣩⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⣀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣶⣿⣷⣦⣀⣠⣿⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠙⠻⠿⠿⢶⣶⣤⣤⣄⣀⣈⠀⠈⠉⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠾⠟

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⣀⡤⢤⣤⣤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠒⠲⢾⣟⠶⠒⠶⠆
⠀⠀⠀⠀⣟⡛⡏⣿⠠⢶⡾⢟⢓⣲⣾⠷⢦⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⡏⠁⡇⣿⣷⣾⣇⡝⣿⢥⡤⠀⠀⠀⠀⠀⠀⢀⣤⡇⠀⠀⢀⡀⠀⢀⣀⣤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣅⡸⡇⣿⣿⣾⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⡖⠉⠒⠼⣇⣟⠭⢖⣋⡾⠀⠀⠀⠀⣤⣤⠤⠤⠀⢸⡇⠀⢤⣤⡤⢤
⠀⠀⠀⠀⣏⡭⡧⣿⢸⢹⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⢰⡋⠛⠲⢦⣸⡇⡷⢚⣉⡭⡾⠆⠀⠀⠀⠈⠋⠉⠁⠀⠈⣏⠀⠈⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠉⠙⠘⠻⠯⡿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⣿⠛⢉⣡⠾⠛⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠠⣤⡴⠿⣶⠿⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣿⣯⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠠⣠⠒⠒⠀⢀⠀⠀⠀⠀⠀⠰⣟⣿⣟⣶⣛⣿⣳⣿⠿⠿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢻⡟⢿⠛⢿⠛⣿⠛⣿⢻⡇
⠀⠀⠀⠀⠀⡤⠤⣀⡄⠀⢹⡇⢠⡤⢤⣤⠀⠀⠀⠀⢠⣇⣀⣀⣤⣤⣄⣯⣤⣤⣤⣀⠀⢧⠀⠀⠀⠀⢸⡇⣿⠀⣾⠀⣿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⣀⠐⢚⣷⣤⡈⠉⠁⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠆⠀⠀⠀⢸⣁⣿⣀⣿⣀⣿⣀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠯⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡀⣾⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄⣷⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⢁⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣻⡈⣿⢿⣿⣿⣿
⠀⠀⠀⣀⠸⠇⣀⡛⢂⡴⡒⠋⠉⠩⠍⡝⠓⢦⡐⢛⣀⠸⠇⣀⠀⠀⠀⣿⣿⣿⠿⣇⣸⠿⣭⡽⣛⣭⣶⣶⣶⣶⣶⣭⣛⢯⣭⠿⣇⣸⠿⣿⣿
⠀⠀⠀⠙⠃⠠⢉⡔⠁⢂⣬⣶⣿⣿⣿⣶⣮⡂⢌⢣⡉⠤⠘⠋⠀⠀⠀⣿⣿⣿⣧⣼⣟⡿⢫⣾⡿⠛⠉⠁⠀⠈⠉⠛⢿⣷⡝⢿⣻⣧⣼⣿⣿
⠀⠀⠀⣷⠦⢀⡞⡀⣰⣿⣛⡿⠿⠀⠿⢿⣻⣿⣆⢀⢳⡀⠴⣾⠀⠀⠀⣿⣿⣿⡈⣿⡿⢱⢿⠏⠀⠤⢄⣠⣿⣄⡠⠤⠀⠹⡿⡎⣿⣿⢅⣿⣿
⠀⢀⣠⡽⢸⢸⠁⣀⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣀⡈⡇⣿⢧⣄⡀⠀⣿⡿⠿⢃⡇⡇⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⢹⢸⡘⠿⢿
⠀⠀⠉⠢⣼⢸⡄⡀⣿⣯⣭⣽⣿⠀⣿⣯⣭⣽⣿⠀⢠⡇⣷⠔⠋⠀⠀⣿⣿⣶⣽⡃⣷⠿⠿⠀⠚⠶⠓⠀⣿⠀⠚⠷⠃⢀⣿⣿⣾⢼⣿⣾⣿
⠀⠀⠀⠀⠈⠁⢳⡁⠘⢿⣿⣋⣀⣀⣀⣙⣿⡿⠃⢈⡞⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡔⠾⢷⡀⠀⠶⠿⠿⠿⠷⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣝⠻⠿⠿⠿⠟⡋⠐⣠⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣶⣤⣀⣀⣀⣤⣶⣿⢟⣱⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣬⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣛⣛⠿⣛⣛⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣖⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠺⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠢⣔⡁⠈⢢⣄⡈⢻⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠑⢌⡙⣿⠾⣿⣿⡿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣙⣿⣿⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠂⢀⡀⢘⣠⣾⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠙⠳⣦⣟⣾⣿⣿⣿⡿⣫⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠀⠈⠢⣄⣈⣿⣿⣿⣿⠟⠻⢧⣾⡁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡲⣌⣠⣿⣿⡏⠀⠈⠁⠀⠀⠈⠙⢦⡤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣽⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣤⡀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣤⡀⠀⢸⣿⣦
⠀⠀⠀⠲⢲⣶⣶⣶⣶⣶⣦⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟
⠀⠠⣤⣤⣤⣤⣤⣤⣀⣉⣠⣄⣭⣬⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⣽⣿⣿⣿⠮⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⠿⠿⠯⡙⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠂

⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠉⠳⠄
⠀⠀⠀⠀⠀⠔⠀⠀⡠⠘⠄⠀⠀⢈⣄
⠀⠀⠀⠀⠀⢀⡾⢊⡡⠞⠘⡇⠀⠞⠏
⠀⠀⠀⠀⠀⣍⡤⢚⣤⡶⠚⠃⠀⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠈⢉⣷⠞⣷⣷⡙⣀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠘⡉⠹⠃⠀⠉⠃⠀⣤⣤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⢾⢶⣤⣤⣀⡀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⡀⢰⣿⣿⣿⣿⣿⣷⣤⡈⠀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣦⡀⠈⠻⣿⣿⣷⣦⣦⣴⣶⣲⣶⡶⣾⣿⣿⣿⣿⣿⣿⡐⠁⢀⣈⣁⠰⣄
⠀⠀⠀⠰⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣐⠻⣿⣛⡓⠒⠱
⠀⠀⠀⣼⣿⣿⣿⣿⣷⣟⢿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣟⣾⣿⣷⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉⠉⠉⠲⢬⢾⠆
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡨
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⣿⣿⣿⣧⣿⣿⣿⣿⣭⣏⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣍⣛⣿⣿⣿⣿⣿⡿⠿⠷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⢨⡉⠀⠀⠐⢦⣠
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣻⣿⡟⣼⡁⣀⠤⣌⣲⠟
⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣮⡏
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⡟⣿⣿⣿⣿⡀⠀⠀⠐⠐⠒⠶⠀⠀⠀⠘⠃
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣉⣉⣧⣿⣿⣿⣋⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⡤⠀⣤⠵⣥⣦⣤⣼⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣜⠺⣇⣀⣀⡀⠀⣠⡴⠚⢁⡴⡆
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠯⣿⣿⣻⣿⣿⣿⠇⣿⣿⣏⠀⠸⠟⢉⣴⠎⢉⣤⣾⠀⠇⠃
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⣠⡄⣹⣿⠏⢀⣰⠶⠉⣠⣶⣿⡇⢿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⠓⠻⠻⠗⠶⠯⢤⣉⡀⠀⠀⠶⠛⠁⡠⢘⣣⠶⠋⣡⣴⣿⣿⠋⡋⢀⠺⠃
⢰⡿⣿⠿⠿⠋⠛⠛⢿⣷⣬⡀⢨⡄⣷⣰⠀⡄⣿⣔⡡⠃⢛⡾⠋⣡⣴⣿⣿⡿⣟⡃⢤⡆
⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠉⠛⠛⠿⠿⠿⠟⠓⠋⠉⠴⠞⢁⡴⢟⠏⠉⣩⠈⠿⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠠⠂⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣯⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡛⠚⢛⡻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣯⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠳⠶⠶⠶⠂⠠⣿⠀⠐⠲⠶⠶⠶⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣍⣉⣉⣩⣽⣿⠀⣿⣯⣍⣉⣉⣩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣬⣭⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣖⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣫⣶⣦⣠⣴⣮⣝⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⡟⠁⠈⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⠀⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⢤⣤⣤⣤⠄⠀⣿⠀⠠⣤⣤⣤⡤⠄
⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⢀⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠐⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠶⠦⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⠛⠿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣸⣿⣿⣇⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⠿⠛⠛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠀⠀⠀⠀⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣈⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⣀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⣀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⠻⠿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⠿⠿⠋⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣤⣤⣄⣀⣀⣈⣉⣉⣉⣀⣀⣠⣤⣤⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⡿⠟⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⣿⣿⣿⡇⠐⣿⣿⣿⡧⢾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣣⡟⠁⠀⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⢹⣿⣿⣿⢾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠐⣿⣿⣿⣿⣦⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⢿⣧⣿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⣻⣿⣿⣿⣿⣿⣿⡏⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⠟⠁⠀⢸⡀⠀⠀⠘⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣙⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣺⢠⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⢆⠀⠀⠀⠀⠘⢷⣤⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⡿⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⢻⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢰⠀⢹⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⡄⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠰⠀⠀⠁⠀⠀⣇⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡇⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠀⠀⠀⢹⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣤⣄⣀⣶⣶⣿⣿⣿⣿⣷⣶⡧⠀⠀⠀⠘⠀⡀⠀⢸⣿⣿
⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⡄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣁⡀⣀⣀⣀⣆⡃⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣤⣤⣽⣿⣿⣿⣯⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⢦⣀⠂⠀⠀⠀⠀⠀⠀⠰⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣴⣿⣿⣧⡀⢦⢘⣿⣷⣄⠀⠀⠀⠀⠀⢀⠟⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⣤⣾⠿⣿⠟⠻⢳⡎⢹⣋⢹⠭⣦⢀⣠⣀⡠⢄⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠛⠂⠀⠐⠉⠈⠉⠀⠀⠀⠀⠛⠃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⠠⠤⠄⣀⣀⠀⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆
⠀⠀⠀⣴⡶⢷⣦⠀⠀⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣹⣿⣿
⠀⠀⠀⠙⠁⠋⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⡅⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠚⢧⡀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣠⣻⣆⠙⠛⠛⠋⠉⠓
⠀⠀⠀⠀⠀⡀⠀⢈⣷⠄⠘⠷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⠿⡟⠛⡄
⢀⠀⢄⡉⠀⢄⡀⠀⠀⠀⠀⢰⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣅⣂⣿⡀⢃
⢠⠀⠀⠈⠓⠦⣌⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡉⠼⣷⠸
⠈⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣬⣴⣤⠿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠂⡔⣀⢡⠅
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢂⠀⢀⠀⠀⢀⠀⠀⠠⢀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠀⠐⠸⠀⠄⠇⠃⠐⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠇⠀⠆⠁⠀⠀⠀⠸⠈⠀⠀⠱⠈⠘⠀⠂⠀⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡉⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⡠⠋
⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⡀⠀⠄
⠀⠀⠀⠀⠀⠤⣄⠲⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢺⡈⠀⠐⢆⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⢀⡘⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠁⠸⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⢰⠆⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠐⡆⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠀⠀⠳⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⡄⠀⠀⠀⠀⠀⢂⠀⢠
⠀⠀⠀⠀⠀⣾⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠺⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⡀
⠀⠀⠀⠀⡄⢫⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠱⠀⡄⠀⠀⠀⠀⠀⢀⣿⣋⣹⣿⡽⠞⣦
⠀⠀⠀⠰⠀⣸⡤⠀⠤⡄⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠐⠙⠀⠀⠀⠀⠀⠸⢿⣍⣙⣦⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡧⠀⢤⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠲⢶⣷⣾⣿⣷⣶⣶⣶⣦⣤⡀⠀⠀⠀⡤⠤⠤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣶⣶⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣟⣹⣿⣿⠇⠀⠐⠛⠂⠀⠐⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠗⢹⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠤⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣷⣶⣶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⡧⠘⠛⠛⠋⠉⠉⠛⠀⠀⠀⠀⠀⠀⡶⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠗
⠀⠀⠀⠀⠀⠀⢄⣀⣠⣤⣤⣶⢾⡓⠀⠈⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⢾⣟⣉⣉⣥⣼⣿⡝⣧⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠞⠉⢀⠀⠀⢀⣼⠇⣨⡿⡀
⠀⠀⠀⠀⠈⠒⠋⠉⠀⠀⠉⠉⠀⢸⣿⣿⡉⠓⠶⠄⠀⠀⠀⠠⡔⠀⠀⠐⠄⠀⠀⠀⢦⣀⣐⡀⢀⣠⠟⠋⠉⠀⢀⡹⣿⡿⢏⡙⠺⢖⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠌⡀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠰⣾⠛⠛⠒⠛⠿⠀⠀⠙⠻⣷
⠀⠀⡠⢶⡔⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣲⣤⢄
⠀⠀⠒⠒⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⡀⠀⠀⠄⠀⠀⢤⣤⡶⠶⠟⠛⠓⢀⣠⠄⠀⠀⢰⣶⡚⠉⠁⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠄⠴⠀⠀⠈⠳⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⠖⠒⠻⡇⠀⠀⠀⠀⢸⣿⣧⣀⡤⠶⠟⠉⡁
⠀⠀⠀⠀⠀⢠⠀⢰⣋⡩⠖⠒⠋⠀⠀⠀⠀⠛⠿⢯⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠃⠉⠀⠀⠀⠒⠁⠀⠀⠀⠀⢸⢿⣧⠀⣀⠴⠚⠂⠁
⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠂⠀⠀⢀⣠⣤⣤⣴⣾⣷⣤⣤⣤⣀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣟⠖⠚⠛⠉⠁⠀⠉⠻⣿⡿⠋⠁⠀⠉⠉⠛⠒⢞⡏⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⡩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣭⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡥⠆⠀⠄⠀⢠⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⢰⠀⠀⠄⠀⠦⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠴⠀⠀⠶⣤⣾⣤⠶⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⢦⣼⣦⡴⠆⠀⠰⢬⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣆⠀⠀⠀⠀⣀⣤⣤⣤⣶⣾⣿⣿⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⡆⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠃⠀⡆⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡆⠀⠛⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠃⠄⣷⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢰⡇⠄⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠁⣀⡿⠂⠀⠰⢶⠶⠀⠀⠺⣇⡀⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉





⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⣿⣿⣿⣿⣿⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃
⠀⣿⣿⣿⣿⣿⢥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷
⠀⣿⣿⣿⣿⡁⡶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁
⠀⣿⣿⣿⣿⡿⢒⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣿⣷⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⡆
⠀⣿⣿⣿⣿⣇⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆
⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⣿⠃⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠞⠛⣦⣤⡉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⣿⣿⣿⣿⣷⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡁
⠀⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠅
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠅
⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠙⣿⣿⠋⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣿⣿⣴⣾⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⢴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣀⣠⣤⣶⣿⣿⡿⠟⠉⠁⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠉⠛⢿⣿⣿⣶⣤⣄⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠰⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡷⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⢾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠒⢄⡀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠉⠢⣄⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢇⡀⠀⠀⠀⠀⠀⢠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠉⠢⣄⠀⠀⣴⣧⣄⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠑⢾⣿⣿⣿⣷⣦⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠶⠤⠤⠤⠤⠤⠤⠤⠴⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡇⡏⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⢉⣛⣛⣁⣤⣤⣤⣄⠀⣿⠿⠿⠟⠻⠷⣦⠤⢶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢸⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⣼⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⡘⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣿⣿⣿⣿⣿⣿⣿⠇⣇⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⡇⡇⠀⠲⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠇⣇⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠀⣿⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⡟⠛⢻⣿⣿⡟⠛⢻⣿⣿⠟⠛⠛⠿⣿⡟⣤⣤⣤⣤⣤⡄⢠⣤⡄⠀⠀⢀⣠⣤⣤⣄⡀⠀⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⠁⠀⣀⡀⠀⢹⣇⠿⢿⣿⡿⠿⠇⢸⣿⡇⠀⣴⣿⣿⠿⠿⣿⡿⠂⣿⣿⠿⠿⠿⠿⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⡄⠀⠈⠛⠿⣿⣿⠀⢸⣿⡇⠀⠀⢸⣿⡇⢸⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⡄⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⠿⠿⣿⣿⣿⠀⠀⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⣿⣦⣤⡀⠀⠘⣿⠀⢸⣿⡇⠀⠀⢸⣿⡇⢸⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠇⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⡄⠀⠘⠛⠋⠀⢀⣿⣧⠀⠀⠛⠛⠁⠀⣸⣯⠈⠈⠻⠟⠀⠀⣿⠀⢸⣿⡇⠀⠀⢸⣿⡇⠈⢿⣿⣶⣤⣤⣴⣆⠀⣿⣿⣤⣤⣤⣤⠀⠀⣿⡇
⣿⣿⡇⠀⣿⣿⣿⣦⣀⣀⣀⣴⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣦⣀⡀⢀⣠⣾⣿⠀⠸⠿⠇⠀⠀⠸⠿⠇⠀⠀⠙⠻⠿⠿⠿⠋⠀⠿⠿⠿⠿⠿⠿⠀⠀⣿⡇
⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣡⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⢀⠀⠐⡶⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠰⣿⡷
⠀⠀⠀⠀⣺⡟⠓⠀⠿⣷⠖⠐⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢦⣤⣿⣧⣴⠴⠆
⠀⠀⠀⢠⠅⠓⠀⠀⠀⠙⣋⠴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⣾⡄
⠀⠀⣠⡞⣠⡄⠀⠀⡀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡠⠂⠀⣴⣿⠇⠀⠀⠰⢹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹
⠀⠀⠇⠃⠃⢣⠀⠀⠀⠈⠉⡾⡀⠀⠀⢀⡴⠋⠀⣠⣾⠟⠁⠀⠀⠀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⢈⡆⢠⡎⠀⠀⣀⣀⡴⢁⣤⡶⠚⠉⠀⢀⣾⡿⠁
⠀⢰⠟⢷⡸⡧⠀⠀⢘⢟⡴⢡⠊⠀⠀⠀⠀⡼⡋
⠀⠀⠀⠨⠻⡄⡅⢀⠏⡾⠀⡞⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠲⣶⣶⣶⣶⣶⡶
⠀⠀⣰⢸⡧⠙⢄⢸⠃⡇⠰⢳⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⢀⡴⢃⡞⡜⠤⣶⣿⡄⣇⠀⠈⣆⠀⠀⢀⠁
⠏⠀⡞⢰⣇⡩⠝⠋⣇⠸⡄⠀⠘⢆⠀⢸⡆
⠀⠀⠁⠀⠁⠀⠀⡰⠻⡄⠱⠀⠀⠈⠳⣎⠇
⠀⠀⠀⠀⠀⠀⠊⠀⢀⣿⡄⠀⠀⢸⡄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢳⠀⠄⠀⣧⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣆⠠⠄⣰⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠐⠉⠙⢳⡚⠉⠉⠂⠀⠀⠀⣸⣿⣿⠙⣿⡇⠀⠀⠀⠀⣶⣶⣦⠾⢦⣶⣶⡄⠀⠀⠀⠾⠿⢺⡟
⠀⠀⠀⠀⠈⡸⢧⡁⠀⠀⠀⠀⠀⢻⡇⠀⠀⠉⣷⠆⠀⠀⠀⠘⢿⠏⠀⠈⢿⠟⠀⠀⠀⠀⠀⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⢀⡀⢀⣤⣤⡀⢠⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⣴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⢀⡤⢯⣸⣇⡽⢄⡀⠀⠀⠀⠀⢸⡿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠠⢿⣿⣯
⠀⠀⠈⠁⠀⠙⠛⠀⠈⠁⠀⠀⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠒⠀⠀⠀⠀⠀⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣶⣿⠁⠀⠀⠀⠀⣼⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⠎⢿⡄⠀⠀⠀⠀⠀⠀⠀⡤⢾⡟⠹⠃⠀⠀⠀⣆⣿⣿⣧⡄
⠀⠀⠀⣴⣿⣷⣦⠀⠀⠀⠀⠀⢀⡾⠁⠈⡏⠀⢻⣄⠀⠀⠀⠀⠀⠸⠃⣾⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣧⡆
⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠾⠷⠶⠶⠶⠶⠶⠿⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢿⡏⠁⢹⠟
⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⡾⢻⡄⠀⠀⠀⠀⠀⠀⢀⠄⣂⡒⠄⠀⠀⠀⠀⠀⠄⡔⢀⠆
⠀⠀⢀⡀⠐⢼⡧⠂⢀⡀⠀⠀⠀⢀⡾⢁⣆⠻⣆⠀⠀⠀⠀⠀⢸⡞⢻⠟⣶⠀⠀⠀⠀⠀⠀⠁⠀⠤⣊
⠀⠀⠉⠙⠢⣜⣣⠔⠋⠉⠀⠀⢠⡟⠁⠸⡿⠀⠹⣆⠀⠀⠀⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠠⣶⠌⠀⡀⠄
⠀⠀⠀⠘⠛⠁⠈⠹⠃⠀⠀⠀⠛⠛⠛⠛⠒⠛⠓⠛⠃⠀⠀⠀⠀⠀⠉⠁

⠀⢠⢄⠀⠀⢀⣀⠀⣀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⡀⢀⣀⠀⡤⡀⣠⡄⣠⣄⢠⢄⢀⣤⢀⣤⡀⣤⡀⣤⡀⣠⣄⣠⣄
⠀⢈⡉⠀⣀⠈⣉⠀⢈⠀⢈⡀⢀⠀⢈⡁⠀⡀⠀⠀⠀⠉⠀⣱⡁⣹⡁⣘⣃⢈⣎⢈⣋⠈⣛⠀⣹⡁⣹⡁⣙⡁⢈⣋
⠀⢀⡀⠀⡀⠀⣁⠀⠀⠀⢈⡁⢀⡀⢠⡄⢀⡀⢀⡄⠀⠀⠈⣛⡃⠛⡃⢙⡋⢘⣚⢈⣓⠈⣒⠁⣛⡃⣓⡃⢙⡃⢘⡋
⠀⢈⣁⠀⡀⠀⡁⠀⣀⠀⣀⡀⣠⡄⠀⡀⠀⡁⠀⣉⠀⣀⠈⣒⠃⢐⡂⢙⡋⢘⡛⠘⡚⠈⣒⠀⢓⠃⣓⠃⢛⡋⢙⡚
⠀⠈⢩⣀⠁⠀⣁⡀⠀⣀⡈⠁⢙⣡⠈⠀⢀⣅⠀⠉⣀⡉⠈⣳⠃⢻⠃⢻⡟⢘⡟⠘⡟⠈⣛⠃⣻⠃⢳⠃⢳⠃⢹⡟
⠀⠰⣄⡸⠀⢾⣿⠇⠀⢀⣴⠆⠈⢿⡿⠀⠀⣨⠇⠀⢀⡼⠘⠯⠃⠻⠇⠻⠟⠸⠟⠘⠟⠘⠿⠃⠿⠃⠻⠃⠻⠇⠻⠟
⠀⢀⠴⢆⠀⢠⣶⣀⠀⠔⢂⡀⢠⣶⣄⠀⢰⣦⡀⠀⣴⣄⠐⠯⠃⠿⠇⠺⠗⠸⠽⠸⠿⠘⠿⠂⠯⠇⠯⠇⠿⠇⠺⠟
⠀⠀⠒⠞⠁⠘⠛⠋⠀⠐⠞⠁⠈⠛⠛⠀⠐⠚⠃⠀⠛⠛⠐⠯⠆⠮⠅⠾⠗⠺⠽⠸⠽⠰⠽⠂⠏⠆⠯⠇⠿⠇⠺⠷
⠀⠀⠀⠀⠅⠀⠃⠀⠐⠀⠚⠀⠸⠃⠀⠂⠀⠓⠀⠀⠀⠿⠀⣶⣶⠀⠴⠀⢐⡠⠀⠲⢄⠀⠀⠴⠄⣦⠄⢯⠐⢂⠶⠶
⠀⠀⠀⠐⠗⠀⠒⠀⠞⠀⠻⠂⠺⠂⠰⠂⠐⠗⠀⠷⠀⠲⠀⠆⢣⣤⢚⣤⠈⢠⠅⢠⣿⠀⠳⢖⠂⠄⠰⣶⠜⡀⢘⡃
⠀⠰⠆⠐⠵⠀⠶⠀⠄⠄⠀⠀⠰⠆⠰⠆⠀⠂⠀⠲⠀⠶⠀⣃⡀⠀⡈⠃⠘⢐⡲⢆⠉⠀⠑⠤⠂⣐⣦⣀⠀⣿⣿⣿
⠀⠀⣠⡄⠀⠰⡶⠀⠀⢲⠄⠀⠀⡀⠂⠀⢀⡀⠀⠠⣄⡄⠀⠉⠜⠋⡄⢀⣀⡀⠀⠀⣰⡆⠀⠤⢸⣿⣿⣿⡆⣦⡂⠄
⠀⠀⠛⠉⠀⠈⠃⠂⠀⠘⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⠋⠀⠀⣋⠛⣻⡂⠌⣬⡄⢢⣤⡍⠥⣓⣴⢄⣙⠛⣫⠌⢈⡁⣴
⠀⠀⢰⠦⠀⠀⣶⣤⠀⠐⠲⠆⠀⢰⡆⠀⢠⠶⠆⠠⣤⠆⠀⠥⠄⣤⠑⠶⠤⠤⠎⣩⡕⠻⣿⡕⣌⡛⠆⣤⣬⡤⢠⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⢀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠐⣃⠁⠠⠰⢹⣾⣷⢧⡭⢁⣍⠛⠉⣬⠂⠿⣿⠟⣑⡂⣣
⠀⠀⠀⠀⠂⠀⠀⠀⠐⠐⠂⠀⠀⠀⠐⠀⠐⠀⠀⠐⠂⠘⠈⠛⠀⠊⠉⠂⠂⠀⠂⠂⠈⠀⠙⠛⠓⠀⠘⠐⠛⠛⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡴⣶⡆⡠⠊⢀⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠝⠉⢀⡔⠉⠀⠻⢦⡈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⡠⠚⠁⣶⠶⡶⣤⠀⠋⢢⣀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⠔⠉⠀⠀⠀⢹⢺⣿⡟⠀⠀⠀⠌⠳⣄⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⡓⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡀⢯⡉⣹⣉⣇⠀⠀⠀⢇⣠⣇⣹
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠈⠉⠉⠀⡠⠀⠀⠈⡈⠻⣇⣿⣧⣾⠀⠀⠀⢸⣻⢿⣬⠇
⠀⠀⠀⠀⠀⠀⠀⡋⠂⠀⢐⣒⡆⡀⠀⠀⠩⡆⠀⠀⠣⠤⠤⢇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⡃⠀⠀⢢⣤⣭⣭⣅⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⢉⠀⠐⠒⠤⠄⠈⠙⠋⠀⠀⠀⠀⡆⠀⠀⢼⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⣿⢀⠀⢸⡄⠀⢠⠀⡇⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢨⣍⣉⣹⣤⣀⣀⣀⣏⣀⢀⣡⠄⣀⢀⣀⡀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠁⠉⠙⠉⠉⢉⢤⣬⠄⠐⣃⣤⣼⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠇⣤⣔⡠⢀⢀⣠⠤⠴⢚⡯⠥⢂⣶⣦⣿⠿⠟⠛⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠀⠫⠭⠍⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠐⠈⠉⠉⠓⢄⠀⢀⣥⣴⣶⠿⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣸⠀⡀⡄⠄⠀⠁⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣬⣭⡭⠭⡭⣭⡭⢿⣷⣲⡶⣶⣿⣿⣿⠃⠀⠀⢀⠁⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣀⢍⣁⠀⠁⣀⣠⣀⣤⡾⡛
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠠⠂⠠⠀⠌⠀⠀⠀⠒⠀⠀⠈⠙⠻⠾⠿⠿⠿⠿⠛⠡⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠇⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠛⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠉⠁⠀⠀⣸⡏⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⢀⠟⣧⠀⠀⠀⢠⠀⢰⡿⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡆⠀⢀⣿⠀⣿⣧⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⢰⠃⠀⠀⢿⠀⣾⣿⢰⣿⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣤⣤⣤⣶⣿⡿⣣⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⡿⢋⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⣿⣤⠀⠀⣀⣀⣀⣠⣴⣾⣿⣿⣿⠋⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⡇⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠛⣿⣿⣿⣶⡦⠤⢤⣤⣤⣶⣾⣿⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢀⣿⣿⣿⠋⠀⠀⠀⠀⣈⣹⣿⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢀⣴⣿⣿⣿⡏⠀⠀⠀⠘⠛⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⣾⣿⠙⠻⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⡿⢻⣿⣿⡟⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⠃⠀⠀⠀⠀⣾⣿⣿⠀⠃⢸⣿⢻⣷⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⣿⠀⡏⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⣿⠈⣿⡄⢻⣿⣿⡇
⠀⠀⠀⠀⠐⣿⣿⣷⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⢿⠀⣿⡇⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠁⣰⠇⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢉⣽⣷⡀⢀⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠇⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣶⡾⠟⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⣴⣶⣿⡿⠄⠚⠛⢛⣉⣴⡄⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠈⠛⠉⠀⠀⠀⢿⣿⢿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⡀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣸⡇⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⢸⣆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⡄⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⣶⠀⠀⣿⣧⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣿⣿⣿⡏⠀⠀⢸⣿⠀⠀⢰⣿⠀⠀⢻⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⠁⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣿⣿⡏⠀⠀⢀⣿⠃⠀⠀⣿⣿⠀⠀⢸⣿⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣿⠃⠀⠀⣸⡟⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⣿⠀⠀⠀⣿⠇⠀⠀⢸⣿⠀⠀⠀⢠⣿⡇⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⡏⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⢀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣸⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⢀⣾⣿⠁⠀⢠⣿⣿⡇⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣾⣿⣿⣿⡏⢀⣾⣿⡏⠀⣠⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣀⣾⣿⣿⣷
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠀⠀⠀⠀⠀⠠⠀⠀⢀⡇
⠙⠿⣿⣿⡿⠟⠛⠙⠻⢿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠙⠻⢿⡿⠿⠛⠉⠉⠛⠛⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠤⠀⠀⠀⢠⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⢀⡤⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⣠⠞⢦⠀⠀⠄⠀⠀⢸⠀⠀⠔⠈⣇⣠⠚⢦⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢙⣶⠋⠀⠀⠀⠀⠀⢸⠀⢎⠁⠘⣤⠞⢁⡠⠚⠁⠀⡇⠀⠀⢸⠀⠀⢀⡴⠈⢡⡐⠁⠀⠀⠃
⠀⠀⠀⠀⢸⣅⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠈⠳⡀⠀⠀⠀⠀⢸⡀⠀⠑⡄⠀⡴⠋⠀⠀⠀⠀⡇⠀⠀⢸⠀⠰⣁⡠⠉⢆⢈⠦
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⠀⠀⠈⠋⠀⠀⠀⢀⣀⣀⡇⠀⠀⢸⣀⣀⣈⣀⣀⡀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣴⠞⠉⠁⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠖⠀⠀⠀⢠⡏⠀⢶⡀⠀⠀⠀⠀⣀⡤⠘⣧⠀⠀⣾⠁⠀⠰⣄⠀⠀⢀⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠈⣿⡀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⢿⡁⠀⠀⠀⠀⢺⡀⠀⢸⣇⠀⢀⣤⠞⠋⠀⠀⣿⠀⠸⣇⠀⠀⠀⠈⣳⣾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠈⢻⡄⠀⠀⠀⠘⣧⠀⠀⣿⡾⠋⠀⠀⠀⠀⠀⡿⠀⠀⢻⡄⠀⠴⠛⠁⠈⠳⠄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠁⠀⠀⠀⠀⣀⡞⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠦⠤⠤⠴⠚⠉

⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⢀⣠⣶⠇⠀⠀⠀⠀⢶⣦⡀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⣿⣿⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣶⡾⠟⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠘⣿⣦⣀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣆⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡁
⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠙⠿⠗⠀⠀⠀⠀⠀⠀⠘⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⢿⣷⣄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠁⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣄⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⢸⣿⡇⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠘⣿⣷⣄⣠⣾⡿⠃⠀⠀⢰⣶⣄⠀⠀⠀⠀⠀⣠⣴⣶⠟⠃⠀⠀⠀⠀⠻⣿⣷⣄⣴⣿⡿⠋⠁
⠀⠀⠀⠀⣿⣿⣀⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡋⠀⠀⠀⠀⠀⠙⣿⣦⠀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡋
⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠛⢿⣿⣦⡀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠙⠏⠁⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠈⠻⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠠⣦⣘⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⢠⠀⠀⠁⡀⠀⠀⠀⢀⠤⢨
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⢀⡤⡀⠐⠀⡀⣀⡤⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢳⣤⡀⠯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁
⠀⡂⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⡃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣦⠀⠀⠀⣾⣿⣟⣯⣭⡿⡻⣿⣿⣆
⠀⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⠀⠀⢸⢾⣿⣭⢾⣿⣿⣦⣟⠿⣻
⠀⡆⠀⠀⠀⠀⠀⠻⣿⣿⠻⣿⣿⣿⣶⣧⣾⠟⠀⠀⠀⠸⡿⢻⢿⣷⣿⢿⠿⣛⣽⡿⠀⠀⠸⣷⣿⣟⠮⠽⣿⣟⣽⣾⡡
⠀⡅⠀⠀⠀⠀⠀⠀⠛⣷⣾⡿⣽⣿⣿⣿⠟⠀⠀⠀⠀⠀⠁⠢⣜⠛⢿⣐⡿⢿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⣿⠟⠁
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁

⠀⠀⢀⡀⠄⠠⢄⠀⠀⠀⡀⣀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠴⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠤⣞⣦⠄⠀⠀⠀⢠⢆⡀⢀⠀⠀⠀⠀⣤⡓⡧
⠀⠀⠀⢄⡘⠛⠉⠀⠀⠀⠉⡥⢾⠄⠂⠀⠀⢀⡴⠲⢄⠀⠀⠀⢀⣗⣓⠶⡀⠀⠀⠀⣙⠀⣍⠀⠀⠀⠀⠀⢯⡇⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠚⢽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⣠⠜⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠯⢸⡿⠛⠂⠀⠀⠀⡰⡿⣭⠀⠀⠀⠰⢆⠰⢈⡄⠀⠀⠘⡂⠁⢠⡀⠀⠀⠀⣀⡀⡘⣧⠀⠀⠀⠰⡈⢯⡧⣀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠝⠙⠉⠡⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠠⡔⠁⠶⠀⠀⠀⠀⠈⠋⠼⠗⠀⠀⠀⠑⠈⠄⠶⠀⠀⠀⠄⠇⠈⠭⠀⠀⠀⠨⠔⠢⠔⠋

⠀⠀⠀⡀⣤⣀⠠⠀⠀⠀⡀⠀⠆⢀⠀⠀⠀⠀⠀⠠⣐⠀⠀⠀⠰⠦⣤⡤⠀⠀⠀⠀⠠⢤⠀⡀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠠⠄⢀⠄⠀⠀⠀⢀⠀⢰⡀
⠀⠀⠀⠲⠋⠀⠀⠀⠀⠀⢀⡐⠂⡈⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⣟⡀⠀⠀⠀⠀⠄⢨⠉⠄⠀⠀⠀⠀⠈⡖⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠈⠀⢋⣁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠐⠃⠐⠊⠀⠀⠀⠀⠙⠊⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠔⣀⠠⠒⠀⠀⠀⢠⠄⢢⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠰⠧⠜⡄⠀⠀⠀⠀⠠⠤
⠀⠀⠠⠾⣔⡼⠆⠀⠀⠀⠠⢃⠺⠒⠀⠀⠀⠀⠻⠯⠄⠀⠀⠀⠈⠁⠁⠉⠀⠀⠀⠀⠠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠉⡋⢹⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⢂⠀⡒⠀⠀⠀⠈⠁⠿⠀⠂⠀⠀⠀⠁⠖⡀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠏⣸⡄
⠀⠀⠀⠒⠓⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠆⠀⠀⠀⢀⣐⢲⠄
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⣐⠀⢺⠀⠀⠀⠀⠟⠊⠀⠄⠀⠀⠀⠐⢿⣿⠗⠀⠀⠐⠛⠉⠼⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⣠⡤⠤⢄⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠠⢢⣀⠀⠀⠀⠈⠀⠴⠤⡀⠀⠀⠀⡰⠤⠴⠆
⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠈⠑⠒⠁⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠖⡄⢀⠀⠀⠀⡀⢀⣦⢀⡀⠀⠀⠀⠀⢾⢢⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢄⠀⠀⠀⠀⠀⡤⢦⡄
⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⢈⢀⡠⠀⠀⠀⠈⡢⠔⠂⠀⠀⠀⠀⢨⢠⡶⠀⠀⠀⠠⠁⠀⠁⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⢘⠀⢀⣧⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠸⣄⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠉⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⣠⡾⠻⡁⢳⣤⣶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⣀⣴⠟⠉⢀⣠⠿⠛⠁⠀⠹⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠈⢧⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⢀⣤⠾⠻⣄⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢀⡾⣟⠳⣦⣀⡴⢻⡟⠁
⠀⠀⠀⣠⠖⠋⠉⠉⠙⠛⠶⣦⣄⠈⠳⣦⡀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⣾⢦⣀⡿⢮⡆⠀⠉⠛⠋
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠘⣮⠻⡉⠓⠚⠁
⠀⣸⠃⠀⠀⠀⢠⡴⠶⢦⣄⠀⠀⠀⠈⠳⣤⡼⠛⠁⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠞⣻⠇
⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠴⠃
⠀⣿⠀⠀⠀⢸⡿⡄⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢀⣤⣾⠋
⠀⢹⣆⠀⠀⠈⣇⠹⣄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⣧⠘⣧
⠀⠀⢻⡄⠀⠀⠘⣦⠘⢷⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠉⠳⣄⡀⢀⡽⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠈⠙⠓⠚⠛⠋⠁⠀⠀⠀⢀⣿⡾⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⣀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡴⠶⠶⠛⠛⠉⠁⠀⠀⢀⣰⠖⢦⡀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠋⠁⢸⠀⠙⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠶⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⢶⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⡍⠁⠀⠀⡐⠀⠀⠀⠀⢰⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠰⠁⠀⠀⠀⢀⣾⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠁⠄⠘⠀⢀⢺⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠸⡶⠶⠶⠲⠒⠒⠺⠒⠒⣻⠚⠓⠒⠒⠶⠾⠿⠟⢛⡟⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣆⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣎⢃⠂⡔⡰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆⠀⠰⢁⠀⡀⣿⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠘⢧⡀⠀⡴⠀⢻⠀⠀⠀⠀⣠⡞⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠻⣶⣁⠀⢻⠀⠀⢀⡴⠋⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠙⠷⣾⣤⣴⠟⠁⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠉⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣤⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠠⠔⠒⠲⢦⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢀⡞⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿
⠀⠀⠀⢀⡾⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇
⠀⠀⠀⣸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣼⠁
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣸⡿⢦⣀
⠀⠀⠀⣿⠀⠀⠀⠈⠿⣦⣀⣀⣤⡾⠛⠀⠀⢀⣼⡏⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣠⡾⠉⠙⠷⣤⣀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⠎⡽⠀⠀⠀⠀⠀⠙⠳⢦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⠎⡼⠁⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⢃⡞⠀⡰⠒⡄⢠⠋⢱⠆⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢁⠞⠀⡜⢀⡞⡰⠃⣰⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⢀⠞⡰⠁⣰⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣁⡞


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣖⣆⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢠⡶⠛⠛⢶⡄
⠀⠀⠀⢰⣿⣿⣋⣿⡿⠿⠇⠀⠀⠀⠀⠀⣗⠛⠛⠛⠳⡇⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⣿
⠀⠀⢠⣾⡿⠍⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣠⠞⢠⡇⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣬⣍⠁⠀⠀⠀⠀⠀⠈⠳⢦⠴⢾⣅
⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣉⣠⣾⡇⠀⠀⠀⠀⠀⠀⠨⣥⡄⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⣠⣴⣶⣿⢿⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇
⠀⠀⠻⢮⡻⢿⣾⣿⡿⠟⡃⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⣠⣌⣻⣛⣤⣀⡀
⠀⠀⠀⠀⠙⢷⣉⠅⠒⠉⠀⠀⠀⠀⠀⠀⣤⣬⣿⣯⣤⡄⠀⠀⠀⠀⠀⠀⠠⣦⣼⣻⣤⡶⠀⠀⠀⠀⠀⠀⢹⢩⣿⣿⣿⣯⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠏


⠀⠀⠀⢻⡟⣿⣻⡟⣿⠁⠀⠀⠀⠀⠀⢀⣀⣴⣎⣶⣤⣀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⡶⠚⢻⠛⠲⣾
⠀⠀⠀⢸⣷⡏⠋⣷⣿⠀⠀⠀⠀⠀⠀⠈⢹⡏⢹⡏⣿⠉⠀⠀⠀⠀⠀⠀⣿⣿⡍⠈⣽⣿⡇⠀⠀⠀⠀⣠⠁⠐⡀⢸⡀⡰⠁⢱⡀
⠀⠀⠀⢸⣏⣿⣻⣇⣿⠀⠀⠀⠀⠀⠀⠀⣸⣇⣸⣃⣿⡀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣥⣬⣽⠇⠀⠀⠀⠀⠙⢛⣛⣁⣸⣇⣙⣛⠛⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⡄⠀⠀⠀⠀⠀⢰⡖⠒⡒⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡍⢍⢦
⠀⢀⡴⠒⢦⠀⠀⢻⣄⣠⠟⠀⠀⠀⠀⠀⠀⠻⣖⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⣾⢟⡴⠀⠀⠀⠀⠀⠀⠀⠀⢉⡫⢯⢿⢸
⠀⠘⢧⣀⣼⣆⣀⣼⡛⠁⠀⠀⠀⠀⠀⠀⢀⣴⣚⣳⣄⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⠯⢕⡫⢫⠆
⠀⠀⠀⠀⠉⠑⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠹⠛⠶⠂

⠀⠀⠀⠀⠤⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⠁⠀⠀⠔⠀⠄⠀⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠼⠀⠂
⠀⣠⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⢀⠀⠀⠀⠀⠅
⠀⠺⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣾⣿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄⠀⡤⠀⠘⠀⠀⠀⠐⠀⠀⠙⠛⠚⠂⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠈⠀⠄⠀⠀⢻⣿⣿⣿⠏⠀⠀⢀⠔⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⣿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⣄⡀⣀⠀⢀⣠⡀⣀⡀⠀⠘⠀⠘⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠐
⠀⠙⠓⠛⠛⠚⠛⠓⠛⠛⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠠⠒⢄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠛⠋⢀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⠠⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢀⣤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠐⣉⡱⠀⠀⡀⠀⢀⠄
⠀⡄⠀⠀⠀⠀⠀⠃⠜⠀⠄⠈⠛⠁⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢶⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠶⠶⠆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠠⠔⠀⠀⡞⠉⢳⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠔⠀⢀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌
⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣰⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠐⢱⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⡭⣉⣁⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣶⣶⠖⢒⡖⠂⠀⠀⢠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⢄⡤⠤⠤⠦⠤⢬⡦⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣯⣄⡬⣨⣅⢀⣽⢸⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠛⠛⠿⠿⢿⣿⣿⡿⣿⣾⡅⠀⠀⠀⠀⣀⡀⠃⢀⠀⠀⣀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠟⠉⠉⠉⠉⡡⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠒⠤⠤⣤⣤⠴⢋⠡⠀⠀⠀⠀⠀⣀⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠀⠀⠤⠄⠀⣀⣀⡀⢈⠠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⢀⡀⠀⠀⠀⠀⢠⢤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⠀⡜⣟⢓
⠀⠠⠒⠦⡎⠜⢤⠀⠀⠀⠀⠀⠀⠉⣃⠀⢱⣄⠀⠀⠀⠀⠀⣀⠀⣄⢀⡈⠀⠀⠀⠀⠘⠤⠄⢠⠤⠤⠤⠃⠀⠀⠀⠀⠀⡙⠻⠛
⠠⡀⠀⠀⢐⠀⣸⠃⠀⠀⠀⢀⡠⠊⠎⠉⠟⠁⠀⠀⠀⠀⠀⠤⠀⠆⠠⠄⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⡍⠍⡉⠍
⠈⠣⠤⠔⠛⠋⠉⠀⠀⠀⠀⠙⠞⠀⠠⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠧⠤⠧⠤



⣀⣀⣠⣔⢢⣀⣀⡀⠀⠀⠀⢀⣦⠀⣀⠀⢀⠀⠀⠀⠀⠀⢠⠀⢂⢑⠠⢠⠀⠀⠀⠀⠀⠆⡤⣤⣤⣄⠂⠀⠀⠀⠀⠀⠀⠀⣤
⢈⠒⢒⠒⠒⡑⡊⠀⠀⠀⠀⣨⣠⣠⣛⣄⣟⣇⠀⠀⠀⠀⢠⡚⠀⠀⠐⣦⠀⠀⠀⠀⠀⡘⣮⡳⢪⢯⡇⠀⠀⠀⠀⢠⠎⢯⠉⠭⠱
⠨⣇⣸⡀⣀⣇⡁⠀⠀⠀⠀⢩⠤⡄⠤⡤⠤⡍⠀⠀⠀⠀⠈⢌⠢⠤⢊⠌⠀⠀⠀⠀⠀⠁⣿⠠⢸⣸⠁⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠊⠉⠉⠉⠉⠉⠉⠂⠀⠀⠀⠘⠆⠲⠂⠶⠐⠇⠀⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠂⠈⠓⠋⠁⠐⠀⠀⠀⠀⠀⠀⠒⠛⠒



⠀⡏⢀⣤⣀⢸⠀⠀⠀⠀⠀⡫⠮⢺⠴⢕⠱⢽⠀⠀⠀⠀⠀⠀⠀⠿⢀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠆⠀⠀⠀⠀⠀⠀⢀⠀⠴⠁⠴⡤
⠀⡇⠙⠋⠉⢸⠀⠀⠀⠀⠀⡇⠀⠺⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢲⠤⠩⠧⠀⠀⠀⠀⠀⠠⡦⠧⠴⢤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆
⠀⠇⠖⠀⠠⢸⠀⠀⠀⠀⠀⡿⠀⠸⠂⢸⠖⢸⠀⠀⠀⠀⠀⠀⠳⢄⡂⠀⠀⠀⠀⠀⠀⠀⠃⠖⠂⠚⠀⠀⠀⠀⠀⢸⠉⠁⠈⠉⢠
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠈⠈⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈


⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠂⢰⠒⠚⡄⠀⠀⠀⠀⠀⠓⠒⠒⠒⠳⠀⠀⠀⠀⠀⠂⠂⡄⠤⡄⠂⠀⠀⠀⠀⠀⠂⠈⣀⡁⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡠
⠀⢠⠈⠉⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⠯⢳⠐⠀⠀⠀⠀⠀⣀⡘⢄⣀⢇⣀⡀⠀⠀⠀⠀⡄⠀⠧⠤⢤⠧⠀⠀⠀⠀⠀⡠⠒⡄⠉⠔
⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⢻⡀⡘⢈⠀⠀⠀⠀⠀⠀⡔⠚⠘⠒⠄⠀⠀⠀⠀⠀⠈⠂⠄⢂⣀⠀⠀⠀⠀⠀⠰⣑⠊⠐⠭⠄
⣶⣶⡶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⣰⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠚⠒⠄⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⢣⡝⠉⠉⠙⠁
⠀⠀⠀⠐⠻⢿⠇⠉⠙⠲⢦⣄⣀⠀⠀⠀⠀⢸⡇⠨⡀⢀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⢸⡇⠀⢠⣤⣤⠄
⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠦⣤⣩⡤⠾⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡇⠀⠀⠉⠉
⠀⠀⠀⡿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⠄


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡊⠀⠈⣶⣶⣦⣤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠿⡆⠀⠀⠀⠀⠀⠀⠀⢸⠉⣉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⢰⡏⣿⣿⢹⣿⡏⣿⣿⢹⡇
⠀⠀⠀⣠⡿⠿⣧⡀⣿⠋⠀⠈⢻⡄⠀⠀⠀⢸⠀⠒⠒⠒⠒⠀⣷⣶⡄⠀⠀⠀⠀⠀⢸⠀⣛⠈⠉⠁⠀⡇⠀⠀⠀⠀⠀⣸⢡⡇⣿⢸⣿⡇⣿⣿⢸⡇
⠀⠀⢰⡏⠀⠀⠈⣷⠹⣦⣄⣤⡾⠁⠀⠀⠀⢸⠀⣭⣭⡭⢉⣴⡟⠛⠁⠀⠀⠀⠀⠀⢸⠀⣋⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠾⠃⠿⠾⡿⠷⠿⡿⠾⠇
⠀⠀⠀⠻⢦⣤⠾⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⣀⣀⣀⣰⣛⣀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠋⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⠯⠿⠀⠿⣽⠿⢯⡿⠿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⣀⣀⣀⡀⢸⠒⠲⡄⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⢻⢿⣿⡇⠀⠀⠀⠀⠀⠠⠴⠒⠋⢽⠍⠓⠲⠤
⠀⠀⠀⢸⠀⠖⠒⠐⠆⢸⠀⢰⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠂⢸⡇⠀⠀⠀⠀⠀⠀⣿⣽⢿⠈⠀⠈⣹⡇⠀⠀⠀⠀⠀⢸⡏⣿⢩⡏⣽⢩⡏⣿
⠀⠀⠀⢸⠀⣩⠍⠁⢀⣸⠀⡏⠀⠀⠀⠀⠀⠀⣿⠠⣦⣼⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣻⣆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡧⣿⣾⣷⣿⣾⣷⣿
⠀⠀⠀⣨⠾⣥⣤⣄⡿⠋⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠈⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⠰⠶⡆⠀⠀⠀⠀⠀⠼⣷⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⢰⣶⣶⣶⡶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣭⢭⡻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣉⡿⠀⠀⠀⠀⠀⠀⠀⠰⣿⡸⣿⣼⢇⣿⣷⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⢨⣯⣿⣯⣯⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠠⠐⢰⠋⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣷⣿⣿⣿⠂⠀⠀⠀⠀⠀⢠⠉⣿⣿⣿⣿⣿⣯⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠉⠉⠉⠢⠀⡴⢇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⠁⠛⠿⣿⣿⠿⠋
⠀⠀⠈⠛⠛⠛⠋⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡇⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠁⢱⡁⠁⠀⠀⠀⠛⠧⣑⢋⠜⠋⠀⠀⠀⠀⢠⠓⢤⠤
⣛⡛⠛⠛⠁⠀⢀⡔⠓⠊⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⢠⣀⣱⠄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠙⠛
⢛⡁⠀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠠⠀⠤⠀⠀⠀⠀⠢⠂⠶⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⢤⡄⠠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠠⣄⠀⠀⠀⠀⣤⢞⢭⢕⢦⡀⠀⠀⠀⠲⢦⣀⢀⣀⣀⢳⡄⠀⠀⠀⣤⠧⠆⣄⠀⠀⠀⠀⠀⠀⠰⠤⠤⢤⡴⠿⠀⠀⠀⣠⡀
⠉⠉⠟⠃⠀⠀⠀⠈⠳⣕⠵⠋⠀⠀⠀⠀⠊⠋⠀⠀⠀⠀⠊⠀⠀⠀⠀⠋⢦⠎⠙⠀⠃⠛⠀⠀⠀⢀⣤⠤⢤⣇⠀⠀⠀⠀⢥⠟
⠀⣴⠶⠂⠀⠀⣀⠀⡀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢧⠀⠀⠀⠁⠀⠀⠀⠁⢈⢀⠈⠈⠀⠀⠀⢠⣴⣀⠤
⡀⠈⠋⠀⠀⠀⠈⠀⠀⠈⠁⢀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠈⣈⢀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⢶⠀⠐⡶⠀⣀⡀⢀⡀⠀⠀⠀⣰⢤⠞⠙⣤⢤⡀
⠀⠀⠀⣄⣄⡄⣄⣤⡄⣄⣄⡄⠀⠀⠀⢀⡜⢧⢠⠏⣆⣰⠻⡀⠉⢁⣁⠁⠀⠀⣼⣥⠤⠤⠤⠤⠤⣷⡀⠀⠀⢀⣄⣀⢀⣀⣠⣀⠤
⠀⠀⠀⠁⢁⢁⠁⠁⣁⣁⣁⡁⠀⠀⠀⠺⠶⠚⠛⠒⠚⠓⠀⠓⠀⠀⠁⠀⠀⠀⢠⡄⠀⢤⢠⣄⠤⢠⠀⠀⠀⢈⢉⢈⠈⢀⠉⠁⠉
⠀⠀⠀⠀⠈⠈⢀⡀⠉⡉⠉⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠠⠤⠀⠄⠀⠀⠀⠀⠈⠀⠈⠀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠶⠶⠶⠖⠀⠀⠀⢰⡄⣴⢠⡆
⢢⣠⠀⠀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠀⠀⠁⠂⠀⠀⠰⠁⠀⠀⠀⠀⣧⢹⣸⠃⠀⠀⠀⠀⠀⠸⣸⣇⠏⠀⠀⠀⠀⠀⠀⠀⣠⣤⠔⠁
⠓⠛⠀⠀⠀⠀⠣⠅⡠⠤⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠀⠁⠀⠀⠀⠀⠈⠲⠃
⠖⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠐⠀⠀⠀⠰⠛⠓⠆⠀⠁⠀⠀⠀⠀⠐⠂⠰⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣡⡝⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢤⣄⠀⠀⠀⠀⡀⠐⠀⠂⠀⠄⢀⠀⠀⢠⡶⡘⣣⣆⠀⠀⠀⠠⠔⠒⢢⡀⠀⠀⣠⣄⣤⣤⣤⣤⣤⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⣤⡤
⠋⣿⠀⠀⠐⠀⢀⢤⠀⠤⢄⡄⠀⠄⠀⠀⠀⠙⠁⠁⠀⠀⠀⣗⠰⠷⠀⡇⠀⠀⠀⠀⠈⠶⠁⠁⠀⠀⠀⢀⣞⡶⣷⡄⠀⠀⠀⡏⢙
⠽⠿⠀⠀⠐⢀⠀⠐⠒⠒⠉⠁⠀⠄⠀⠀⠀⠆⠆⠀⠀⠀⠀⠈⠒⠒⠊⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠙⡒⠒⣒⠃⠀⠀⠀⠿⠯
⠈⠀⠀⠀⠀⠀⠀⠃⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠙⠁⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⢀⣀⠁⡀⠀⡀⣀
⣀⠀⠀⠀⠀⠀⠁⠚⠁⠀⠀⠐⠀⠀⠀⠀⢀⣄
⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⡨⠳⣄⠀⠀⠀⢀⣆⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠰⣏⣆⣀⠀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠈⢿⣿⣦⡀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢰⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠋⠀⠀⣠⣾⣷⣄⠀⠀⠙⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⡿⠟⠋⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠙⠻⢿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿
⢠⣿⡏⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣀⣤⣾⡿⠿⠿⠛⠛⠛⠛⠿⠿⣿⣷⣦⣀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢹⣿
⢸⣿⠇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿
⣸⣿⠀⠀⣼⠋⠉⠉⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⠉⠉⠙⣧⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿
⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸
⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⢹⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⢸⣿⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣿
⠘⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⢿⣷⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡿
⠀⠸⣿⡆⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣿⠇
⠀⠀⢿⣷⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣾⡿
⠀⠀⠘⣿⣇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣸⣿⠃
⠀⠀⠀⢹⣿⡄⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⣿⡏
⠀⠀⠀⠀⢻⣿⡄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⣿⡟
⠀⠀⠀⠀⠀⢻⣿⡄⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀⠀⠉⠛⢶⣤⣀⠀⠀⣀⣤⡶⠛⠉⠀⢀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣀⠀⠈⠙⠛⠛⠋⠁⠀⣀⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⣠⣤⣶⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⣭⠽⠶⠶⠿⢭⣟⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣄⣀⠀⠀⡏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣠⣤⣄⣈⠉⠙⡃⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⣤⣌⣉⠙⠒⠇⠈⣧⣀⡉⠙⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣦⣼⣌⣙⠛⣶⠀⢻⡀⠀⠀⢀⣠⠤⠖⢒⡟⢠⠟⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣀⡴⠶⠛⠛⠳⢶⣤⡀⠈⠉⠙⠛⠋⠀⣸⠇⢠⡴⠋⢠⣴⠿⢋⣴⣻⠄⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⢰⣿⣴⠋⠀⠀⠘⠻⠶⠿⠛⠁⡀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡿⣻⠃⠀⠀⠀⠀⠀⣴⡶⠟⠛⠛⠳⣄⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⢀⡾⢃⡟⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠙⠲⠶⠶⠾⠟⠋⠀⠀⢀⣴⡿⠁⢸⡇⠀⠀⠀⠀⠀⠈⠻⠶⠞⠛⠉⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠻⠶⣦⣤⣄⣀⠀⠀⢀⣠⣴⠟⠉⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠁⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣦⠀⠀⠈⢉⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠄⣀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣀⣤⣤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⡾⠉⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⣠⡤⠶⠒⠋⠙⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⡞⣼⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⢰⡟⠁⠀⠀⠀⠀⠀⢹⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⡼⣱⢣⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢀⣤⣴⠶⠛⠋⠁⠀⠘⡆⠘⢷⣤⣤⣄⣀⣀⠀⢨⡗⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⡸⢡⢧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣠⡾⠁⣀⣤⣤⣬⣉⡉⠛⠛⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡏⣰⢃⠏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠹⣷⡀⠀⠀⣠⡾⠋⢠⡾⠋⠀⠀⠀⠈⠉⠑⢦⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⢠⡟⢠⢇⠎⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠘⠿⢶⠾⠋⠀⣠⡿⠁⠀⠀⠀⠀⣀⣤⡴⠞⠀⠀⢀⣾⠃
⠀⠀⠀⠀⢀⡾⢁⠏⡜⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣁⣠⣴⡶⠟⠛⠉⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⣼⠃⡌⡘⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣀⣤⣴⡶⠟⠋
⠀⠀⠀⣸⠇⡘⠠⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡀⣀⣀⣠⣤⣶⠶⠟⠛⠋⠉
⠀⠀⠀⠹⡄⢀⣡⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠘⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣶⣿⠶⣶⠷⠾⠿⢷⣶⡶⠶⣾⠾⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠿⠛⠉⠀⣶⣭⣛⠿⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣾⣭⣭⣭⣭⣭⣿⣿⣿⡛⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⢻⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟
⠀⠀⠀⠀⠀⠀⢸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⡻⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣭⣻⠿⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢉⠛⠉⠂⡀⠀⠀⠀⠀⠀⡔⢏⣸⢀⠝⢆
⠀⠀⠀⠀⠀⢀⠞⠀⠑⡀⠀⠀⠀⠀⠀⠀⠉⣠⠁⠀⠈⡀⠉⠀⠀⠀⠀⠀⡖⢒⢿⢗⠒⢺
⠀⠀⠀⠀⢠⠎⢤⣤⡤⠘⡄⠀⠀⠀⠀⠀⢦⠁⠄⠀⠴⠀⡴⠀⠀⠀⠀⠀⠘⠥⠸⡄⠝⠃
⠀⠀⠀⢠⠃⠀⠀⠀⠔⠂⠈⢂⠀⠀⠀⠀⠀⠑⠂⠦⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄
⠀⠀⠠⠣⠤⠤⠄⠀⠄⠠⠠⠤⠇⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢠⣸⠞⠒⠓⠪⣤⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠋⠉⢻⡇
⠀⠀⠀⠀⢀⢴⠞⠉⠁⠀⠀⠀⠀⠀⠀⣈⣻⠁⠒⠀⠈⠓⢸⣇⡀⠄⠀⠀⠁⠀⠀⠀⠀⠘⠃⡸⠃
⠀⠀⠀⢠⠇⡮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢠⠄⠀⣸⠗⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢸⡀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢳⠦⠴⣏⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠳⣌⠳⠤⣀⣀⣀⡴⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢿⣾⡁⢲⡆
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠊⠁⠀⠀⠀⡈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢀⢨⡇⠄⠔⡬⠁
⠀⠀⠀⠀⢀⣠⠤⠤⢤⣄⠀⠀⢀⢀⡱⢌⡀⡄⠀⠀⢀⡴⠉⠉⠀⡄⠀⠀⠈⢌⠧⣞⡙⣂⠾⡭⠁
⠀⠀⣴⠿⢭⣖⠊⣟⡍⣪⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠻⣧⣄⣀⣴⡟⠀⠀⠀⠀⠘⠸⠙⠙⠆⠀⠂
⠀⠀⠀⢔⣀⣀⡽⠿⡏⠀⠀⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⢈⣉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⡍⠱⠀⠀⠀⠀⠀⠀⡠⠤⠀⡀
⠀⠀⠀⠀⡰⠊⢹⠉⠢⡀⠀⠀⠀⠀⢦⣆⡰⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⠀⠀⠀⠼⠇⠀⠀⠀⠿⠃⠀⠀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠐⠎⣸⢘⠶⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⢂
⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠛⠛⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢀⣤⣄⡀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⢿⣯⣿⣿⣷⡶⠾⠛⠿⣷⣶⡶⠾⠿⠷⢶⣦⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣽⡿⠋⠀⢀⣀⣠⣾⠟⠁⠀⠀⣀⠀⢀⣨⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⠀⠀⢀⣾⡟⠀⠀⢀⣿⢿⣿⠏⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⢸⣿⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣻⠏⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣵⠋⠀⣼⣿⣆⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⢀⣼⠏⠈⠻⣷⣦⡤⠤⠤⠬⠿⣷⣦⣄⣀⡤⠖⢊⣡⣴⡾⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠃⢀⣾⠃⠀⠀⠀⠀⢹⣧⣤⣤⣤⣤⣾⣿⠟⢻⣷⣿⡟⠉⠹⣧⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣿⢣⡿⠁⠀⠀⠀⠀⠀⠈⣿⡇⠀⣹⠀⣿⠆⠀⠀⠉⠛⣿⡷⠀⠹⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢻⠘⣿⠄⠀⠀⠀⠈⢿⣧⣄⠀⢻⡆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠈⢻⡆⠀⠀⠀⠀⠈⠙⠿⣦⡀⢻⡄⠘⣷⡀
⠀⠀⠀⠀⠀⢠⣿⡟⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣻⣷⠀⢿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⢸⣆⣿⠇⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠈⣧⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⠶⠒⠒⠒⠁⠀⠀⠀⠀⠐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠊⠽⣒⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡨⢕⡂⠭⡀⠈⠙⠲⣤⡀⠠⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠒⠥⣒⠭⢀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠐⠚⠛⠲⢦⣄⣀⡀⢀⣠⡟⣀⣀⣤⡤⠶⠚⠛⠛⠛⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⢯⠻⣍⣉⣩⣭⣭⣭⣭⠭⠽⠿⠛⠛⠋⠉⠉⠉⢉⣩⡝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢧⣟⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢀⣠⡤⠴⠒⢻⣧⡴⠖⠒⠚⠛⠛⡻⠿⠿⠶⠦⠤⠤⠤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣘⣧
⢠⡾⠋⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠟⠍⠁⠉⠵⠴⠶⠶⠀⠀⠀⠀⢁⠟⢤⣀
⡟⠀⠀⠀⠉⠒⠀⠸⣇⠀⠀⠀⠀⠀⠤⠤⠤⠭⢁⣐⣒⣒⣒⣤⡬⠭⠭⠭⠄⠀⠀⢸⠀⠀⠈⠙⠶⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠭⣭⣽⣿⣶⣶⡀⣂⡈⣉⣅⣡⣤⣬⣅⠐⠒⢲⣗⣂⠘⢷⡶⠞⠛⡿⠋
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣀⠉⠉⢉⣉⡭⠉⠙⠛⠛⠋⠙⠛⠛⠛⠛⠛⠃⠀⢸⡇
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⠁
⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡘⢷⡤⠀⠀⠀⠀⠤⠤⠴⢖⣒⣛⣫⡭⠭⠶⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠚⠉⢀⣠⠖⠋⠙⠲⣄⡀⠙⠓⠦⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⣠⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣀⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠙⠦⠴⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣠⠖⠋



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

⢼⠦⣀⡇⠀⠀⠀⣀⣀⣀⣀⡀⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠈⣁⠛⠶⠖⠀⠀⠀⠀⠀⠀⢾
⠼⠤⠞⠁⠀⠀⠀⠁⠉⠉⠁⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣤⡇⢀⣀⡄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⠀⣠⡾⡿⣶⣾⣿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠹⡇⠁⠈⣡⣋⡤⠟⠁
⢋⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠋⠉⠀⠀⠀⠀⣠
⠠⢈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠙⡃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠿⠥
⠀⢁⠀⠀⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣶⣶⣤⣄
⠢⠴⠀⠀⢠⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠻⡿⡍⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⡀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠧⠀⡀⠤⣀⠀⢰⡅⠰⠊⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠁⠄⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⠰⠶⠇⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠈⠊⠐
⣤⣤⠀⣠⣶⠀⣠⠶⡇⣀⡴⠶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠤⡦⢀⠀⠀⠀⠀⣯⣭⣭⣭⣥⠉⣶⣷
⠀⠘⠾⠋⠻⠟⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠀⡄⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⣀⠠⠃⠀⠀⠀⠀⣰⠀⢠⡆⢀⣠⡄
⢺⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢻⠀⠏⠠⣭⣅⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⢀⣤⣄⠈⠙⠿⣶⣤⣄⡀
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠛⠉⢀⣀⡤⠞⠋⢹⠉⠛⠦⣄⣀⠈⠙⠛⠻⠿⠶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⣿⠇⠀⢠⣤⣤⣤⣤⣤⠤⠶⠖⠚⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠉⠛⠒⠶⠦⢤⣤⣤⣤⣤⣤⠀⠀⢿⡇
⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⡇
⠀⠀⠀⣿⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⡇
⠀⠀⠀⢿⡇⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿⠀⠀⣿⠃
⠀⠀⠀⢸⣷⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣿
⠀⠀⠀⠈⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⡇
⠀⠀⠀⠀⠸⣷⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠹⣷⡄⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠉⠛⠶⢤⣄⣀⠀⠀⢸⠀⠀⢀⣀⣤⠴⠞⠋⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢶⣶⣤⣤⣤⣴⣶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠛⠛⠛⠙⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠋⠉⠉⠛⠒⠒⠓⠀⠠⢤⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠖⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠷⠒⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠖⢿⣿⣿⠛⠉⢩⠏⠁⠀⠀⠀⡶⠚⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣴⣤⣦⣤⣤⣤⣤⣤⣤⠤⣤⡤⣠⠤⠤⠤⠶⠶⠦⢤⣦⣤⡤⠤⠤⠴⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣛⣛⡟⣁⣈⣃⣹⡀⠀⢙⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡛⣻⣿⣿⣿⣿⣟⣺⣿⣿⣾⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠈⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⢯⣿⣿⣿⣦⣤⣤⣤⣤⣴⣶⠶⣶⣶⣶⣶⠶⠶⠶⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀⡠⠤⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠠⠀⠤⣀⣀⣀⣀⡀⠀⠀⣠⠴⠒⣉⠁⠀⠀⠀⠀⠀⠠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠙⠁⢸⠃⠀⠀⠀⠀⠀⢀⣴⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠒⠠⠤⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠂⠤⠀⣀⣀⡀⢘⠄⠦⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠄⠀⠠⠀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⢻⣿⠀⠀⢠⣾⣿⣿⣿⡿⢻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⡌⠄⠀⠀⠀⠀⠀⢀⣴⡦
⠀⠀⠀⠀⣿⣏⠙⠟⢁⣴⣿⣿⠀⠀⢺⣯⡙⠿⢋⣴⣿⣿⠀⠀⠀⠀⢷⣄⣴⠟⠁⠀⠀⠀⠐⠀⠺⣦⣴⠟⠁⠀⠀⠀⠰⣷⣄⣴⡿⠋
⠀⠀⠀⠀⣿⣿⣷⣶⣿⣿⣿⣿⠀⠀⠈⢿⣿⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠡⡀⠈⠁⠀⠀⡠⠃⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠔⠀⠀⠀⠂⠄
⠀⠀⠀⠀⣿⣿⣍⠻⠋⣹⣿⣿⠀⠀⢰⣿⣏⠙⠟⣩⣿⣷⠀⠀⠀⠀⠰⣶⣤⡶⠂⠀⡁⠀⢀⠁⠐⢷⣄⣴⠆⠀⠂⠀⠀⠸⣷⣤⣾⠟
⠀⠀⠀⠀⣿⣿⠋⣠⣌⢻⣿⣿⠀⠀⢸⣿⡟⣡⣄⠙⣿⡿⠀⠀⠀⠀⢠⡾⠻⣷⡄⠀⡁⠀⠘⡀⢀⣴⠟⢷⡄⠀⡀⠀⠀⣠⣾⠿⣷⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⡿⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⣀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⠞⠃⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢻⡆⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⢀⣀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⡀⠈⣭⠟⢙⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⠟⢧⣸⣠⠟⢷
⠀⠀⠀⢷⣄⣸⢋⠉⣨⠇⠀⠀⠀⠀⠀⠂⠀⢀⡴⠾⠿⣛⣿⣿⣷⣄⠀⠀⠀⠀⠸⡏⡹⢿⢟⠛⣿
⠀⠀⠀⠀⠉⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣶⣿⣿⡟⢿⡆⠀⠀⠀⠀⠙⠦⣾⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⠀⠀⢘⣿⣿⣿⠁
⠀⠀⠀⣀⣀⣦⣴⣀⡀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠈⢿⣿⠋⠚⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣀⣀
⠀⠀⢀⣬⣟⠁⠈⣻⣅⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠉⠙⣿⠏
⠀⠀⠀⢀⠿⢶⡖⠿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢷⡤⣴⢯⣧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡔⢺⡷⡀⠀⠀⠀⠀⠀⢀⠜⠈⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢷⣿⡤⢻⠆⠀⠀⠀⠀⠈⡀⢀⣠⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡭⣾⢵⡾⠋⠀⠀⠀⠀⠀⠀⠰⠸⣧⣼⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠲⠦⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠠⠤⣍⣁⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⢥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠶⠦⣤⡴⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⣹⣀⡀
⠀⠀⠀⠀⢤⣀⠚⠛⠛⠓⠲⠦⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡏⢀⣠⣀⡀⠀⠀⠀⢀⣠⣤⡶⠟⣋⣁⠀⠀⠐⠂⠄
⠀⠀⠀⠀⠘⡇⠉⠁⠒⠦⣄⣀⠀⠀⠀⠉⠉⠛⠒⠶⠦⣤⣤⣀⡀⠀⠀⠀⠀⠘⢸⡇⠀⠀⠙⣿⣦⡶⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠉⠙⠳⠦⣤⣀⡀⠀⠀⠀⣀⣀⣀⣭⠽⠿⠒⠶⠦⣤⣧⣴⠾⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⢀⣤⡬⢛⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠀⢒⣚⣫⣿⡟⠖
⢀⣿⠋⠙⠛⠻⠷⣦⣍⣛⠷⢦⣄⡀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠴⠒⠒⣊⣉⣉⡤⠤⠴⠒⠚⠛⠋⠉⠁⣿
⢸⣏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣷⣦⣤⣀⠞⢰⣀⣠⣤⣤⡶⠶⠿⢛⣛⣫⣭⣤⣤⠶⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻
⠈⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣭⣶⡶⠾⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣸⣤⠄
⠀⠀⠉⠛⠻⠷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠞⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⠦⣤⣄⡲⠄⢀⣀⣀⣤⣤⠴⠾⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠂⠘⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⣿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣏⣽⣿⣷⣭⣿⣿⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⣀⠄⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⢀⡇⢹⡇⢸⡇⢸⡇⠀⣯⠀⣿⠀⠀⠀⠀⠀⡤⣀⣀⡀⠀⠀⣀⡤⠖⠉⠀⠀⣀⡀⠀⢁⡤
⠀⠀⠀⢠⠧⢀⣠⣤⣶⣶⣤⣿⣤⣤⣤⣤⣀⣀⣸⡄⠀⠀⠀⢸⡇⢸⡇⢸⡇⢸⡇⠀⣿⠀⣿⠀⠀⠀⠠⠦⣄⣀⣉⣹⢶⠾⠷⠦⣄⣀⡠⠊⢁⡤⠞⠉
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢀⣸⣧⣼⣧⣼⣧⣼⣧⣤⣿⣤⣿⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠤⣤⣯⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣾⣷⣤⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡯⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣢⠰⢶⣦⣤⡉⢻⢳⠞⢻⠁
⠀⠀⠀⠀⠀⢘⢳⣤⣄⡀⠀⣀⠀⠀⢈⡿⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣉⣷⢇⣘⣧⣦⣿⢸⠀⢹⠁
⠀⠀⠀⠀⠀⠰⢧⣑⡢⢌⣧⣼⠒⠉⠁⣦⡸⠇⠀⠀⠀⠀⠀⢠⢀⣀⣀⠀⠀⡇⠀⢠⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣯⣟⣿⢸⡧⢘⠁
⠀⠀⠀⠀⠀⠻⡶⣭⣉⡛⣇⣹⡤⠶⠛⢉⣹⠄⠀⠀⠀⠀⠀⠈⠒⠛⠋⠀⢸⡇⠀⠈⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣼⣿⡟⡇⣿⣸⢺⢙⠁
⠀⠀⠀⠀⠀⠛⠻⠦⣌⣉⢻⣇⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⡿⠇⠇⠛⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠴⣯⣤⡶⠿⠉⢱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠽⢶⣤⣽⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠼⣧⣄⡠⠾⠻⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⠟⠷⢄⣠⣽⠦⠄⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⡤⠂⢀⠟⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠻⡀⠸⢤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⢀⡾⡟⠉⠀⠀⣤⡀⠀⣀⠀⠀⡀⠀⢀⣤⠀⠀⠉⠻⢷⡀⠀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣾⡇⠈⣧⣥⠀⠀⠀⢸⡇⠀⢾⡁⣀⡷⢀⣸⠇⠀⠀⠀⣈⣼⠁⢸⣷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢆⣸⣅⠷⣄⣀⣌⢉⣅⣈⣉⣯⣥⣼⣯⣱⣀⣠⡾⣨⣇⡰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠀⣿⣧⡤⠞⠁⣽⣿⠈⠣⣤⣬⣿⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⡗⢀⡸⢃⡈⠤⢘⡇⠁⢩⣛⡏⣿⣻⡯⢿⣏⡇⠤⢁⡘⡇⡀⢚⣿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⡈⠠⣼⣌⣰⠤⠛⣥⣶⠿⠀⣀⣄⠀⠀⣿⢈⣐⣸⡇⠿⣶⣼⠛⠴⣆⣡⣧⠄⢁⣱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢻⠞⠋⣩⠿⢠⡀⢉⣿⡛⠀⠐⠁⠀⠀⣿⠀⠈⢺⡇⢚⣿⡈⢀⡄⠿⣍⠙⠷⡟⠀⣀
⠀⠀⠀⠀⠀⠀⣀⠄⣉⠛⠉⣠⣾⠗⣦⣼⠟⣋⠛⠣⢌⣁⣀⠄⠀⢻⠀⣂⣨⡭⠜⠛⣙⠻⣧⣴⠺⣷⣄⠉⠛⣉⠠⣀
⠀⠀⠀⠀⠀⡾⢿⠀⠻⣀⣠⣮⡍⢽⡟⠡⠞⠁⠉⠉⠈⠉⠉⠙⠳⠞⠋⠉⠉⠁⠉⠈⠈⠳⠌⢻⡯⠉⣵⣄⣀⠟⠀⡿⢷
⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⠇⠀⣠⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣄⠀⠸⣿⣿⣷⡀⠀⠈
⠀⠀⠀⠀⠀⠐⣊⣯⣭⡁⠀⠀⣰⡿⢛⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⢿⣆⠀⠀⢀⣭⣽⣛⠂
⠀⠀⠀⠀⢀⡜⠛⢉⣽⡿⢃⡾⠁⣵⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣮⠈⢷⡘⢿⣯⡉⠛⢣⡀
⠀⠀⠀⠚⢩⣧⣴⡉⠛⠉⡏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢹⠈⠛⢉⡦⣼⡍⠓
⠀⠀⢲⡴⡟⠁⢘⠏⢠⡶⠶⢻⡗⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢼⠛⠶⢶⡄⠹⡓⠈⢻⢦⡞
⠀⠀⠘⣆⣀⣶⠇⠐⠊⠠⠰⣤⡗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣤⠆⠄⢑⠂⠸⣶⣀⣼⠃
⠀⠀⠀⠈⠻⠿⡶⢴⣶⣤⢠⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⡄⣤⣶⡦⢶⠿⠟⠁
⠀⠀⠀⠀⠀⠠⣦⢀⣿⣿⣮⡁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢨⣵⣿⣿⡀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠍⠹⢁⡻⣿⡆⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢰⣿⢟⡉⠏⠩
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⢿⣷⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⡿⠾⡇⠀⠀⢀
⠀⠀⢀⡄⠈⢀⣿⣄⢰⣤⡆⠀⢻⡿⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣾⠿⡟⠀⢰⣤⡆⣠⣿⡀⠁⠠⡀
⠀⠀⠘⠟⠛⠙⢿⣿⢮⠀⢤⣀⢀⣀⣸⣦⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⣶⣇⣀⣀⣀⡦⠀⠴⣿⡿⠋⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠸⠙⠇⠀⣏⠈⢿⠟⠛⣿⣦⡈⠢⣄⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⣠⠔⢁⣴⣿⠛⠻⡿⠉⣹⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⣨⣿⡿⠴⡿⠋⠻⡿⣄⠈⢻⣦⣤⣀⡀⢀⣈⣤⣴⡟⠁⣠⢿⠟⠙⣿⠦⢿⣿⣅⠀⠙⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠸⠆⢰⠗⠀⣀⠀⣿⣶⠿⣈⣿⣀⣥⣤⣤⣿⣁⠻⣶⣿⡀⣀⠀⠺⡆⠰⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠽⠾⠿⠟⠃⠙⠛⠉⠉⠉⠙⠛⠋⠘⠻⠿⠥⠯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⠶⠖⠒⠂⠀⠀⠀⠀⠀⠐⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠩⢑⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠌⣑⠢⠍⠛⠶⢦⣄⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠋⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠬⢐⠪⢕⡂⠉⠒⠠⠀⠀⠈⠻⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠉⠓⢯⣑⠢⠅⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⢶⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⠉⠉⠙⠳⠦⣤⣀⣀⣀⣠⡾⣁⣀⣤⡤⠶⠞⠛⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⡿⣟⢿⣛⣛⣛⣛⣋⣉⣉⣉⣭⣭⣭⠿⠿⠟⠛⠛⠉⠉⠉⠉⣉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠁⠹⣆⣿⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠈⢳⡀
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⢁⣀⣤⣤⣼⣿⣶⣶⣶⣶⣛⣛⣛⡋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⢀⣤⠶⠛⠋⠉⠀⠠⣿⠟⠉⠀⠀⠀⠀⠀⠲⠬⠭⠭⠭⠭⠭⢉⣉⣛⣛⣛⣛⣓⠒⠒⠒⠲⠶⠶⣶⠿⠂
⣰⠟⠁⠀⢄⣀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⠉⠉⠉⠁⠀⠐⢒⣒⣒⣂⡀⠀⠀⠀⢰⠉⠓⠦⣄⡀
⡟⠀⠀⠀⠀⠀⠁⠀⠀⢷⡀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠠⠬⠭⠭⠭⡭⠁⠒⠒⠒⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣳⣦
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠯⣭⣭⣭⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⠤⠀⠀⠀⠀⠀⠀⣀⠀⠈⢲⡶⠚⠛⢹⠋⠁
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⡀⠀⠀⠈⠉⣉⣩⡭⠭⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⡾
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣂⣀⣀⡤⠤⠀⢀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡘⢿⡂⠄⠀⠀⠀⠠⠔⠒⢒⣒⣛⣫⣭⠭⠶⠶⠒⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠛⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠛⠉⠀⠈⠙⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢰⣾⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⡻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠈⣿⣷⣝⡻⣾⣿⣾⣝⡿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⢀⣠⣴⣿⣻⣿⣿⠆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣟⣿⣯⣿⣮⣛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⣀⣤⣶⣟⣯⣽⣿⡟⠋⠁
⠀⠀⠀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣮⣛⢿⣽⣳⣮⡛⢿⣷⣤⣀⣠⣴⣿⠿⠛⠉⠀⠀⣠⣴⡾⣟⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣯⡻⢷⣝⡳⣬⡻⣿⠟⠋⠀⠀⢀⣠⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣞⢷⡝⣿⣿⣿⣿⣿⣿⣷⣽⡿⣾⣝⣿⠀⣀⣴⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣈⣿⣿⣿⣝⢷⡿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⣿⣿⠋⠉
⠀⢀⣴⣾⡿⠟⠛⢿⣿⣿⣮⡻⣝⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢰⣿⣿⣦⣄
⠀⣿⣿⢿⣷⣄⠀⠀⠻⣿⣿⣷⡝⣾⣷⣽⣿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢙⣿⣷⡄
⠀⣿⣿⠀⠹⢿⣦⡀⠀⠘⢿⣿⣿⣾⢳⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⠋
⠀⠘⣿⣦⡀⠀⠹⣿⣦⡀⠈⠻⣿⣿⣷⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣽⣿⣷⣄
⠀⠀⢠⣿⣿⣦⡀⠈⠻⣿⣦⡀⠹⣿⣿⣿⣧⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣁⣿⣿⡿⠃
⠀⠀⢸⣿⣿⣿⣿⣦⡀⠈⠻⣿⣦⡈⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⢀⣠⣶⣿⣿⡏⠉
⠀⠀⢸⣿⡙⣿⣿⣿⣿⣦⡀⠈⠻⣿⣦⡹⣿⡿⠿⠿⠛⠛⠉⠉⠉⣀⣤⣶⣿⠿⠋⠁⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠸⣿⣷⡜⢿⣿⣿⣿⣿⣆⠀⠈⠻⣷⣦⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢿⣿⣻⣎⢿⣿⣿⣿⣿⣷⣄⠀⠘⢻⣷⣤⣴⣾⡿⠟⠋⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⢿⣷⡻⣎⢻⣿⣿⣿⣿⣿⣷⣄⠀⢹⣿⠋⠁⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣠⣴⣿⣷⡹⣇⠻⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢿⣿⣯⡁
⢀⣴⣿⣿⣿⣿⣿⣷⡘⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣀
⣼⣿⡿⠻⣿⣿⣿⣿⣷⡜⢷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⡿⣦⡈⠻⣿⣿⣿⣿⣌⢷⡘⣿⠿⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣧⡈⢳⣄⠘⢿⣿⣿⣿⣦⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡏
⣿⣿⡟⠻⣦⠙⢷⣄⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣸⣿
⢹⣿⣿⣦⡈⢷⣄⠙⢷⣄⠻⣿⣿⣿⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⣿⣿
⠘⣿⣿⣏⠻⣄⠙⢷⣄⠙⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠹⣿⣿⣦⡙⢷⣄⠙⢧⡈⠻⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣀
⠀⠀⠙⣿⣿⣿⣄⠙⢷⡀⠹⣦⡈⠻⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷
⠀⠀⠀⠈⠻⣿⣿⣷⣄⠙⣦⡈⠻⣦⠈⢳⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣌⠻⣦⡈⠳⣄⠙⢷⣄⠻⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡈⠻⣄⠙⢷⣄⠙⢦⡈⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡙⢷⣄⠙⢷⡀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⠙⢷⣄⠙⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠙⢦⣸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⠻⣷⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣿⠀⣠⣴⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠒⠠⣀
⠀⠀⠀⠀⠀⣰⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⢴⣤⣄⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣧⡔⠒⠒⢤⡇⠱⡀
⠀⠀⠀⢀⣘⠻⠃⡀⠛⣣⣄⠀⠀⠀⠀⠀⠈⢿⡿⠍⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⣿⣶⣾⣟⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⡷⢻⡛⢿⣷⣿⠟⢛⠳⠇
⠀⠀⠀⣾⣿⠃⠸⠿⢀⠙⣿⡆⠀⠀⠀⠀⠀⢘⣿⣒⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠋⠀⣹⣿⡀⠀⠀⠀⠀⠀⢣⠀⠣⡀⣿⡇⡠⠃⢠⠃
⠀⠀⠀⠘⠿⠿⠿⠆⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠿⣷⡆⠈⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠓⠤⣞⣋⣛⡦⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢂
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡀⢰⣦
⠀⠀⠘⠻⣿⣿⡿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⣠⣴⡄⠀⠀⠀⠀⠀⠀⢸⣿⡟⣉⠉⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡿⢃⡄
⠀⠀⠀⣀⠀⡀⢀⡈⢉⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣶⣼⠿⠿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣇⠛⠓⢂⣾⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⣾⠃
⠀⠀⠀⠉⣄⠁⡌⢁⡈⢡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⢀
⠀⠀⠀⠀⠀⠶⢿⣿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣶⣶⣦⠐⠀⠀⠀⠀⠀⠀⡀⣀⣠⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣾⣿⡇⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⣿⠟⠀⡁⠀⠀⠀⠀⠈⠿⢿⠿⠿⠿⠉⢵
⠀⠀⠀⠀⢤⣤⣞⣛⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣉⠉⠉⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⢉
⠀⠀⠀⠀⠀⢿⣻⣼⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣷⠀⡇⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣠⣴⣧⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠁⠀⠇⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢼⠀⠺⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⣄⠀⢻⠏⢀⣤⠄⠀⠀⠀⠀⠀⠀⠰⠉⠃⠑⢺⠊⢉⠰⢀⠀⠀⠀⠀⠀⠠⢖⣯⣷⣿⣿⣷⣯⣗⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡟⠃⠀⢀⣴⠂
⠀⠀⠀⠀⠘⢷⣜⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢺⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⡿⠋⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⢀⣤⣤⡀⣠⣤⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣿⣿⣿⣿⣿⠟⠛⢿⡿⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠛⣿⠛⠛⠁⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣤⠆⢦⣤⣴⣿⣿⣿⣿⣷⣄⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⣰⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣹⣿⣿⣿⣿⣿⡿⠁⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣶⣷⣮⣿⣯⣶⣶⡄⠀⠀⣼⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠁⠀⠀⠉⠉⢻⣿⣿⡟⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⠿⠛⣽⠙⠿⠿⠃⠀⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⣤⡆⣶⣄⣀⣴⣷⣿⣿⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠦⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣥⣬⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡆⢀⢠⣦⡀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⣿⣇⠀⠀⠀⠀⠀⠠⠃⠈⣿⣿⡿⠟⢛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣫⡷⢟⣛⣛⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⣿⡇⢰⣿⡿⠀⠀⣀⡀⠖⣃⣀⣴⣿⣯⣶⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣀⣿⣿⣿⣿⣾⣿⣿⠟⢃⠄⣷⣄⢀⢸⣿⣶⠀⣶⣿⠇⠀⠀⢉⠛⠛⠉⠉⣸⣿⣛⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣷⣿⣿⡯⢀⣤⣶⣿⣼⡿⢙⣃⣿⡏⠀⣿⣿⢀⡂⣲⣿⣿⣶⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣋⠏⣰⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣟⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⣿⣇⢸⣿⣿⣿⡟⡿⢿⣿⣦⣾⣿⣿⢶⡿⣷⣶⣾⣷⠛⡟⣿⣿⣿⣿⣆⢳⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⣴⣿⡿⣿⡇⢀⣾⣿⣿⣿⣿⣯⢸⡇⡟⠿⣿⣿⡆⠀⣿⣿⣿⣷⡏⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢩⣿⢿⣷⣿⡿⢿⠿⣇⣘⣿⣯⣿⣿⣷⣿⣹⣷⣿⣴⣬⣿⣥⣄⣿⡿⣿⣿⠷⣏⡈⠻⣦⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣏⣿⡟⠻⠿⠿⠟⠛⠛⠉⠈⠙⠛⠿⠷⠿⠿⠿⠟⠋⠁⠉⠙⠛⠻⠿⠿⠟⢻⣿⠆⠈⡀
⢀⡀⠀⠀⠀⢰⡇⠀⠀⠀⠂⢿⣿⣞⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣳⣆⠀⠁⠀⠀⢸⡄⠀⠀⠀⢀
⠀⠙⢶⡄⠀⣿⣿⡄⠀⠀⠀⠘⢿⣿⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⡆⠀⠀⢠⣿⣿⠀⣠⡾⠋
⠀⠀⠀⠈⠳⣿⣿⣿⣄⡠⠊⠡⡘⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠙⢄⣠⣿⣿⣿⠞⠁
⠀⣀⡀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠈⠻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠈⣿⣿⡟⠁⠀⠀⢀⣀
⠀⠀⠙⠛⠶⠤⣈⣿⠏⠲⣄⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣠⠖⠹⣿⣁⠤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠢⢷⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠔⠚⠉⢻⡀
⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⢾⣿⣿⣦⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣤⣾⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣛⡀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⢀⣛⣩⣅
⠀⠀⠀⠀⠐⢤⣤⣤⡞⠉⠻⢿⣷⠋⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠹⣾⣿⠟⠋⠰⣦⣀⡤⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣶⣤⣀⡀⠈⠐⠂⠄⠀⠀⠀⠀⠀⠀⠁⣀⣠⣤⣾⡿⠟⠓⣴⠂⠉⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠀⠀⠒⠀⠒⠠⠄⠘⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣲⣦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡼⠛⠉⠁⠀⠀⠘⢿⡯⠉⠐⠒⠒⠈⠓⠀⠢⠤⠴⠖⠒⠒⠒⠊⢹⡿⠃⠀⠀⠈⠉⠛⢆⠆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠶⢶⣒⣒⣒⣛⣛⣛⣉⡋⢙⣓⣒⣒⣒⣒⣒⣒⣒⠒⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡿⠋⣉⣉⣉⣉⣉⣙⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠏⣿⢩⣿⡏⣽⠃⣼⠁⣿⡟⣸⣿⢸⡇⠀⠀⠀⠀⠤⠤⠤⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⠀⣧⠘⢫⠇⠏⡇⢃⡆⠛⠰⣿⡏⣿⠀⠀⠉⠀⠀⠀⠒⠒⠂⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⣷⣿⣶⣾⣶⣾⣶⣾⣷⣾⣷⣾⣷⢻⡇⠀⢐⣀⣀⣂⣀⣀⣀⣘⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠶⠶⣿⣿⣭⣽⣿⣿⣿⣛⣛⣛⣛⣿⠿⠿⣿⣾⣷⣀⣬⣭⣭⣥⣴⣶⣶⣶⡶⠿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⡾⢋⣉⣉⣉⣀⡀⠀⠠⠄⠄⢠⣭⣥⡀⠙⣋⡉⠙⠛⠛⠋⠙⠿⢛⣛⣋⣉⣩⣭⠭⠭⠀⣐⣆⣀⣺⣼⣿⣦⡿⠟
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠉⠉⢉⣉⡙⠛⠛⠛⠒⠶⠶⠶⠦⣤⣤⣭⣍⣀⣀⣈⣉⣭⣭⡴⢶⠶⠒⠛⠛⠉⠉⠉⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⢿⣒⣀⡀⠀⠀⠠⠤⠌⢁⢀⠀⠀⠀⠀⠒⠒⠀⠠⣼⡉⣁⡀⠀⠀⠈⢉⡉⠀⠠⠤⠀⠀⢀⣀⣀⠰⠦⠼⠢⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⠶⣤⣤⣭⣉⣉⣐⡒⠒⠲⢶⡴⠞⠛⠛⠛⡻⠷⣬⣏⠀⠀⢀⣀⣀⠠⠤⠄⠐⢒⣂⣈⣭⣥⣤⣶⠶⠾⢿⣛⣻⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⢁⠄⠨⣛⠿⣭⣑⠢⢍⣛⠾⣤⣤⣴⣶⣾⣿⣿⣟⣻⣯⣭⣷⠶⠶⠟⠛⠛⠉⠉⠅⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⣿⣽⠋⠿⢹⣿⣿⠃⡌⠀⠀⠀⠉⠐⠨⢝⡳⠦⣍⡒⠭⣛⠿⣿⣿⣿⠋⠉⠉⠀⠀⢀⠀⠨⠅⢀⡀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣹⣿⣠⣿⢰⣟⣿⠇⡘⠀⣄⣰⣠⢀⢀⡀⠀⠀⠉⠒⠭⣓⢦⣝⡲⣭⡻⢦⣄⠀⣀⠤⠔⠀⠀⠀⢀⣀⣭⠴⢞⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⣿⣿⡏⠰⠀⠹⠏⠋⠛⠘⠋⠷⣟⡀⠀⠀⠀⠀⡙⢨⠏⠉⠟⣸⠏⢠⣐⣤⡤⢶⣚⣩⣭⣴⠾⠟⢿⣉
⠀⠀⢀⣠⣤⣶⣾⣿⣿⣇⣀⡀⠉⠉⣹⡟⢀⠁⠀⠀⢠⣠⢄⡀⠀⠀⠉⣴⢤⡄⠀⡀⢀⡏⣐⡜⣰⣿⣻⣭⣷⡿⠿⠛⠋⣉⣠⣤⣶⣾⣿⣿⣿⣿⣶
⢠⣾⡛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠆⡀⢀⠄⡈⣉⠛⠷⢏⣴⣠⠐⠻⠂⠠⢀⡞⠀⡹⢰⡿⠛⠋⣉⣠⣤⣶⠾⣟⣿⣿⣾⠿⢟⣫⣭⠶⡿⠋
⣿⣿⣿⣿⣿⣶⣶⣤⣭⣍⣛⣛⢿⣿⢁⠼⣶⡀⠀⠀⠁⠈⠋⠂⢤⡈⠁⠀⠀⢀⠀⡾⠀⢰⢡⣿⣷⣾⣿⣿⣿⣿⣿⠿⣟⣫⣥⡶⠞⠛⠉⠐⣘⡧⢤
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡜⠀⠹⢿⣦⣾⣦⠀⠀⡀⠀⠡⣀⠀⠀⠀⣼⠁⢠⢇⣾⣿⣿⣿⡿⢛⣫⣥⡶⠟⠛⠉⠀⣀⣤⠶⣚⣫⣽⠖⠋
⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⠏⡼⠁⠀⢀⢀⡉⢛⣿⢿⣿⣿⣀⠀⠈⡄⡘⣰⠃⢀⠎⣾⣿⣛⣳⣿⣿⠋⠉⢀⣀⡤⠴⢚⣩⣵⡶⠛⠋⠉
⠀⠀⠀⣠⣤⣤⣴⡏⣿⠦⣼⡟⠰⠁⠀⢀⡀⠈⠻⢾⣿⣁⠀⠙⠛⠛⠛⠠⢰⠇⠀⡞⣼⣿⣿⣿⣿⣿⣯⠴⣚⣫⣥⣶⠿⠟⣻⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⢡⠃⠀⠀⠈⣙⠂⢤⡀⠈⠉⠙⠂⠀⠀⢀⢣⡟⠀⡸⢰⣿⠿⠿⠿⣿⣿⣿⡿⠿⠛⢉⠠⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣄⠀⠀⠀⠀⢀⣙⠒⠌⢑⠢⢄⠀⠀⠀⡌⡾⠁⢠⢃⣿⠟⠓⠒⣶⠛⣉⠡⠀⢀⡠⠔⢂⣩⠤⢚⣻⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡘⣼⠃⠀⡎⣾⣏⣁⠂⠈⢫⠅⠀⢀⣠⠥⠒⣈⣥⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣤⣤⣤⠴⢦⣄⡀⠀⢰⠏⠀⡸⣸⣿⣤⣄⣉⠉⠙⠓⢋⣡⣴⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⢈⠙⠶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡾⠿⠛⠉⠁
⠀⠀⢀⣤⠶⠚⠛⠉⠉⠀⠀⠀⠀⢀⡀⢻⠤⠦⡴⢜⡶⠛⠚⠁⠀⠀⠉⠻⢿⣿⣿⣿⠿⠟⠋⠁
⠀⢠⡟⠡⡀⠢⣀⠀⠀⢤⡐⡦⠽⠒⠋⠁⠀⠀⠀⠀⣄⠀⣄⠲⣤⣷⠤⠀⠀⠈⠙⠷⣦⣄⡀
⠀⣾⠃⡄⠈⠳⢌⠀⠠⡀⠁⠀⠀⢀⡀⢰⣌⣳⢬⠷⠚⠓⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⣿⡀⠿⣇⣠⢀⡉⠢⣀⠀⠀⠀⠈⠟⠒⠁⠀⠀⠀⠀⡠⠤⢔⣒⢢⡭⠶⠖⠚⠋⠁⠀⠀⠀⠀⢈⡛⢷⡀
⠀⠈⠻⣦⡘⠹⢾⣱⠆⠀⠑⢤⡙⠢⠀⠀⠀⠀⠀⠀⠦⠒⠓⠉⡥⠶⠒⠛⠀⣀⣀⡤⠴⠖⢚⣋⣭⣥⡾⠃
⠀⠀⠀⠈⠻⣶⣄⠙⠀⣖⣠⢀⠈⠲⢄⠐⢦⡀⠀⠀⠀⠀⣀⣀⠠⠤⢐⣛⣨⣥⡴⠶⠚⠛⠉⠉⣴⣟
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡉⠙⢿⣶⣠⠀⠑⠦⡀⣠⠀⣀⣭⣥⠶⠶⠛⠛⠉⠁⠀⠀⠈⠁⠰⠆⠂⢻⡿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠈⢻⠀⣠⡀⣸⠃⣾⣏⡀⠠⠤⠐⠂⠀⠄⠂⠀⠀⣀⡠⠤⠖⢒⣋⣡⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠘⢁⡟⢸⣿⠤⠐⠒⠀⣀⡐⠤⠔⣒⣉⣭⣴⡶⠾⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣼⡇⠘⠿⠐⣒⣉⣥⣴⡶⠾⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣩⡴⠶⠒⠶⠮⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡞⠁⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠤⠒⠒⢤⣀⣤⣤⣤⣤⣯⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⣹⣿⣞⠛⠓⣦⠈⠳⣤⢤⡤⣤⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠸⣅⣈⣻⣗⡶⣿⠃⠀⢸⣡⣞⡡⠞⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡿⠟⠉⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⠋⢠⠏⠆⠀⢠⣴⠋⠀⠀⠀⢰⠀⠸⡄⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡞⠀⡞⠀⣠⠴⠋⠈⠢⣤⠴⠢⢄⢣⠀⠹⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠁⢰⠃⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢣⢣⠀⠹⡄⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡏⢀⡏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢳⠀⠹⡄⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⠁⣼⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⡀⠹⡜⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠇⢠⠇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀⠹⡼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⠀⡞⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⡀⠹⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣸⢳⠇⢸⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣻⡄⢹⡞
⠀⠀⠀⠀⠀⠀⠀⠹⣟⣠⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠸⢧⣤⡄⠀⠀⠀⠀⠀⠊⠉⡂⠀⠀⠀⠀⠀⢰⣿⡧
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢺⣿⡷⠀⠀⠀⠀⠀⠀⢤⡏⠀⠀⠀⠀⠀⠀⠺⠟⠂⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁

⠀⠀⠀⠀⣔⢐⠄⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠘⠛⠘⠃⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠙⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠰⠲⠂⠀⠀⠀⠀⠀⠈⠙⠿⠃⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠻⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⢛⣡⣍⠛⠿⣿⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠛⢛⣋⣭⣴⣾⣿⣿⣿⣿⣶⣦⣍⣙⡛⠻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣙⠻⠿⣿⣿⠿⠛⣩⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣦⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠒⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠖⠐⠂⠄⠤⣤⣀
⠀⠀⠀⠀⠀⢠⣢⠿⠒⠒⠃⠏⠥⠤⢤⣄⣒⣂⣀⡤⣀⡄⢀⣠⣤⣤⣤⣶⢦⢴⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠐⢀⣤⣄⣀⠈⠉⠉⢹⢏⠉⠩⠋⢈⠄⢁⣁⣁⣻⡀
⠀⠀⠀⠀⠀⠰⣟⠛⠖⠶⠶⠤⣤⣤⣄⣀⡀⠀⢀⠀⠀⠸⣄⣠⣠⣦⣶⣾⣾⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⢘⠷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⢙⣙⣛⣿⣻⣿⣽⣥⣴⢶⣴⣽⣦
⠀⠀⠀⠀⠀⠀⠅⡀⢸⢷⣶⣶⠶⠶⠾⠻⠟⡛⠛⢝⢍⢜⡽⡩⠡⡠⠀⠐⠑⢔⠨⡻⣧
⠀⠀⠀⠀⠀⠀⢈⣎⠌⠃⢸⡄⠐⠀⠀⠀⠈⠊⡠⡠⠐⠄⠔⢄⢤⢈⢈⣊⣢⣂⣰⢾⡗⣀⣦
⠀⠀⠀⣠⢔⠊⠁⠈⠻⣤⣼⣴⣵⣵⣴⣷⣶⣶⣾⠿⢾⠿⠿⠿⠿⠿⠿⠛⠻⠛⣛⣿⣾⡟⠁
⠀⢀⣾⠷⠒⠻⠿⠦⢶⣀⣴⣉⡭⣅⣀⣀⣀⠒⠂⠉⠓⠂⠀⠀⠀⠀⠀⠁⣠⣾⢟⠃⠈⢇
⠀⢸⣿⠀⠀⣎⡴⠀⠀⠀⠀⠠⠉⡉⠉⠛⠒⠺⠥⠭⣴⣂⣤⣤⢄⣠⣴⠞⡋⠉⢀⣨⣾⠿⠙⠂⠠⢄
⠀⠀⠉⠓⠒⢺⣧⣤⣬⣤⣀⡀⠒⠀⠀⠤⠒⠀⠀⠀⠀⠀⠈⢉⠟⠏⠸⠀⢰⣴⡿⠛⠑⠀⢀⣠⣶⡟
⠀⠀⠀⢠⣶⢿⡿⣻⡤⠉⡉⠛⠛⠟⠲⠦⢤⣤⣤⣀⣀⠀⠀⠸⡇⠀⣠⠾⠟⠉⢀⣠⣶⡾⡛⠵⣺
⠀⠀⠀⢰⡇⠀⠀⠀⠉⢑⠲⠤⣅⣀⠠⢀⡀⠀⠀⠉⠉⠙⠻⠖⠿⠿⣯⢤⣖⡿⢟⠋⡛⢀⠠⠂⣹⣅⡀
⠀⠀⠀⠘⣷⢄⠀⠀⢀⠀⠀⠰⠤⡈⠉⠒⠦⢅⣒⠠⠀⢠⣀⣠⣴⡾⢻⠉⠍⠀⠢⠘⣄⣤⣶⠟⠛⠀⢐⠒
⠀⠀⠀⠀⠀⠙⠣⢤⣀⡀⠃⠢⢄⡀⠀⠀⠀⠀⠈⠉⠳⠚⠋⠫⠀⠀⢁⢈⢠⡴⣾⡯⠟⠉⢁⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⢤⢄⡍⠉⠓⠠⠀⠀⠀⡷⠡⠀⠑⣐⣬⣶⡯⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⢔⡂⠤⢳⣤⣾⣾⠿⠋⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠟⠉⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠒⠛⠛⠛⠛⠛⠓⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⢀⣠⣤⣤⡴⠶⠶⠶⠶⣶⣤⣤⡈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣈⡙⠳⣄⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣚⣭⣤⠬⢭⣷⣦⠹⢦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠙⡇⠀⠳⡄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢹⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢰⡇⠀⠀⡴⣛⣭⣭⣗⢦⡀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠶⠋⠀⠀⠀⠈⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⢸⢡⡏⠀⠀⠈⠳⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠸⡞⡇⠀⠀⠀⠀⠈⢳⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⢧⠀⠀⠹⡝⢄⠀⠀⠀⠀⠀⠙⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⢦⡳⡄⠀⠀⠀⠀⠈⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⡀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠙⢬⡦⣄⣀⠀⢀⡼⣸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⢇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡿⠀⠀⠀⢀⡤⢶⡄⠈⠳⢤⡀⠀⠀⠉⠓⠿⠭⠭⠞⠁⠀⠀⠀⠀⠀⣠⡴⠋⠀⢀⡴⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⠟⠀⠀⣠⠖⠋⣠⠏⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠁⠀⢀⡴⠋⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠟⠀⠀⠀⢀⡴⠊⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠴⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢀⣠⠞⠉⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣀⡴⡶⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠖⠋⣡⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠶⠛⠁⠀⣠⠴⠋⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⣤⠿⠶⣤⣤⣈⣉⣉⣉⣉⣁⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣴⠟⠁⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠁
⠀⠀⣀⣤⣄⣠⡴⠏⠀⢀⡴⠋⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣡⠾⠋
⣠⠾⠋⠀⠀⠀⠀⣠⠶⠋⢀⡰⠊⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⡶⠟⠁
⣿⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣤⠞⠋
⣿⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠴⣋⡴⠋⠁
⣿⡤⠊⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡠⠚⣡⡾⠋
⣿⣧⣤⣎⠁⠀⠀⠀⠀⠀⠀⣠⠴⢋⣴⠟⠋
⠈⠛⠶⣬⣝⣓⠲⠦⠤⡴⠛⣡⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢠⠎⠀⠀⠀⠉⠁⠂⠠⣄
⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⣰⠃⠀⠀⠀⠀⣀⢔⡪⠒⡋
⠀⠀⠀⠀⠀⠀⠘⣔⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠁⠀⢀⡠⣔⠭⠂⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢈⠛⠛⠛⠚⠿⣒⡠⣉⡉⠉⠉⠀⠉⠉⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠼⠟⢀⠤⡠⠕⠊⠀⠀⠀⣀⣔⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁⠋⠓⠲⢴⣂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡪⠓⠉⠀⠀⢀⣠⡴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠥⣴⣀⠠⠄⣀⠀⠀⠀⠀⠀⠉⠐⠘⠍⠔⢂⠠⠄⣀⢀⡠⣐⠍⠂⠁⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣖⣁⠀⠀⠀⠀⠈⠁⠒⠪⠵⣶⣀⡠⢀⡀⠀⠀⠀⠀⠈⠁⠒⠿⠊⠀⠀⠀⣀⡴⠞⡟⠁
⠀⠀⠀⠀⠀⠀⡟⠀⠒⠒⠒⠛⠉⠩⠭⠶⠒⠒⠶⠤⠬⠷⠶⢯⣖⣂⣤⡀⢀⡀⠀⢀⣠⡴⠛⠉⡠⠊
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠒⠿⠏⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠷⠦⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠒⠒⠚⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⠿⠛⠛⠉⠉⠀⠀⠈⠉⠙⠛⠻⠿⠷⣶⣶⣤⣄⣀⡀
⠀⣀⣀⣤⣤⣶⣶⡿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣈⡭⠉⠛⠛⣿⣷⣶⣶⣶⣶⣦⣤⡀
⢿⣿⣯⣙⡛⠻⠿⣵⣶⣠⣤⣄⡀⢀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠂⠉⣁⣤⣴⡾⠿⠟⠛⠋⣉⣉⡉⠉⠉⢻⣦⡀
⠀⠈⠙⠻⣿⣷⣶⣤⣌⣉⠙⠛⠿⢿⣿⣶⣯⣶⣾⣋⣄⣠⠴⠂⠀⢀⣀⣤⠴⠖⠛⠉⣁⣠⣤⣴⠶⠟⠛⠉⠙⠻⣿⣆⠀⢿⣇
⠀⠀⠀⠀⢹⣿⣿⣿⣛⠿⢿⣶⣦⣤⣀⡉⠙⠛⠿⢿⣿⣷⠾⠟⠛⠉⢁⣀⣤⣴⣾⡿⠟⠋⣉⣀⣤⡤⠀⠀⠀⠀⣿⡿⠀⣸⣿
⢀⣠⣤⣶⣿⣿⣿⣿⠿⡳⠦⣄⡉⠉⠛⠻⠿⣶⣦⣤⣿⣧⣤⣴⣶⡿⠿⠟⢋⣉⣤⣴⣶⢿⠟⣋⡛⠶⣄⡀⣡⣴⡿⠃⢀⣿⡇
⣾⣅⠀⠘⠻⢿⣿⣯⣙⠲⠥⣀⡉⠙⠲⠦⣄⣀⡈⣙⢻⣿⣿⡉⠀⠰⠚⠛⠋⠻⣯⡄⠲⣮⠳⡝⢯⡑⠌⢻⣟⠉⠀⢀⣾⡟
⠙⠛⠿⣷⣤⣄⡈⠙⠻⢿⣷⣦⣍⡓⠲⠤⣄⣈⠙⠛⠿⣿⣿⣧⠀⠀⠐⠋⠀⢀⣈⣿⣦⡀⠀⠈⠈⢧⠀⠀⢻⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠉⢻⣿⢷⣦⣄⡈⠙⠛⠿⣿⣶⣤⣉⠙⠳⢶⣾⣿⡇⣀⣤⣴⡶⠟⠛⠉⢀⣹⣷⡄⠀⠀⠈⠀⠀⢸⣷⡄⠀⠉⠉⠙⠓⠲⣤⡀
⠀⠀⢶⣾⠛⢻⣿⠘⡜⢻⣿⣷⣦⣄⡀⠈⠙⠻⢿⣶⣾⣿⠿⠟⠛⢉⣀⣤⣴⣶⠿⠟⠛⠉⢿⡄⠀⠀⠀⠀⢸⣿⣟⣩⣥⣶⣶⣾⡿⠿⠃
⠀⠀⠘⣿⣦⠘⣿⡀⠀⠛⢱⠏⣿⢿⣿⣷⣦⣄⡀⠀⣡⣤⣶⣶⣿⠿⠛⠉⠁⠐⠉⠀⢀⣠⣿⡇⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠘⣿⣧⡈⠻⣶⣤⡀⠀⠁⠾⢹⡟⣿⣿⣿⣿⣿⣿⡟⡋⢀⠀⠀⣀⣠⣴⣶⡿⠿⣛⣿⣷⣶⡿⠟⠛⣛⣡⣽⣿⣿⣏⡛⠻⠶⣦⣄
⠀⣀⣠⠤⢾⣿⣷⡄⠈⠙⢿⣷⣤⡀⠺⢠⠇⣿⠁⣿⠟⢻⣿⣶⢋⣾⡿⠟⢫⡭⢖⣨⠭⠞⠋⢉⠀⠐⠲⠟⠛⢋⣉⣤⣾⠿⠛⠀⣀⣬⣿
⠈⢿⣧⠐⣿⡍⢿⣿⣄⠀⠀⠉⠻⢿⣷⣄⣸⡇⢸⣿⠀⠛⣿⡇⢸⡏⠀⠀⠀⢈⣡⣤⡶⠾⠛⣁⣠⣤⡴⠶⠞⢛⣉⣥⣤⣶⣾⠿⠛⠋⠁
⠀⠈⢿⣆⠹⣿⣌⠻⣿⣦⠀⠀⠀⠀⠉⠛⠿⢷⣾⣿⠀⠀⣿⡇⠸⣿⣶⡶⣟⣛⣭⣤⠶⠞⢛⣉⣭⣤⣶⣶⠿⠿⢛⠻⣿⣍
⠀⠀⠈⢿⣆⠹⣿⣦⠈⢿⣷⡀⠳⡄⠀⠠⠀⠒⠘⠿⣷⣦⣼⣷⣀⠈⠛⠛⠛⢉⣡⣤⣶⣿⠿⢿⣛⡩⠥⠄⠀⠀⠈⠷⠌⠻⣷⣆
⠀⠀⠀⠈⢿⡆⠹⣿⣇⠀⢻⣿⣄⠀⢀⡀⠀⠀⠀⠰⠦⠌⠙⠿⠿⠿⠿⠿⠟⠛⣛⣉⣁⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡄⠹⣿⡄
⠀⠀⠀⠀⠘⣿⡄⠹⣿⡆⠄⠹⣿⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣿⠿⠿⠿⠿⢟⣛⣛⣛⣛⣩⣭⡭⠭⠭⠑⣀⣀⠉⢹⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⡄⠀⠹⣿⣿⣿⡛⠛⠉⠉⠉⠉⠐⠲⠶⠶⠾⠿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠁⠩⠦⢀⣼⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⣿⡄⢠⣿⣿⣿⣟⡒⠀⠠⠤⠄⠀⢒⣒⣒⣒⣒⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⢁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢘⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⣉⣁⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠋⠉⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠉⠻⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢀⣤⣄⠀⠈⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⣾⣏⣿⣿⣶⣦⣴⣶⢿⣿⣷⣤⣤⣴⣶⣶⣶⣤⣤⣀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠈⢻⣿⡿⠟⠉⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠉⣻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⢀⣴⣾⣾⣿⠋⠀⠀⢠⣤⣶⣤⣤⣾⣿⠟⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣄⠀⠀⠀⢰⣿⠏⠀⠀⠀⢸⣟⣿⣿⠇⠀⠀⢰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡄⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠈⠿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⡟⠁⢿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢣⡞⠀⢀⣼⣿⣆⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⠀⢀⣾⠏⠙⢿⣷⣤⣀⣀⣀⣀⣈⣻⣿⣦⣀⠀⠀⣀⡤⠴⠚⣉⣤⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠟⠉⢠⣿⠃⠀⠀⠀⠉⢻⣿⣁⣀⣀⣀⣀⣉⣻⣿⣿⣿⣧⣤⣶⠿⠛⢿⡆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡄⣰⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⡟⠛⢻⠛⢻⣟⠋⠀⠸⠿⣿⣷⣤⡀⠈⣿⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⣼⡟⠀⠀⠀⠀⠈⣽⣿⠃⠀⠘⣷⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⣿⡷⠀⠀⠀⠀⠀⠸⣿⣶⡄⠀⠸⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⢏⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡆⠙⣿⠀⠀⠀⠀⠀⠀⠀⠛⢻⣶⡀⠹⣧⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢸⡇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⠄⢹⡆⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢸⡇⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⠀⠀⢻⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢸⡇⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠈⢿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣈⣇⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀
⠀⠀⠀⠀⢀⣠⡤⣤⡀⠀⠀⠀⠀⣤⣼⣧⣤⠀⠀⠀⠀⠙⢶⣶⣶⢶⡶⠋⠀⠀⠀⢠⢸⣿⣿⣿⢀⡄⠀⠀⠀⠀⠀⣁⠉⢷⣆
⠀⠀⠀⠶⡁⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡏⠉⢹⡏⠀⠀⠀⠀⠘⣿⡿⠛⢿⣿⠃⠀⠀⠀⠀⠀⣠⣤⣿⣿⡆
⠀⠀⠀⠀⠈⠓⠒⠚⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠻⣧⣤⣼⠏⠀⠀⠀⠀⠀⠘⣿⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠘⠉⠘⠉⠁⠃⠀⠀⠀⠁⠃⠁⠋⠈⠋⠀⠀⠀⠙⠋⠊⠙⠛⠀⠀⠀⠀⠐⠘⠘⠁⠛⠋⠁⠀⠀⠀⠀⠘⠚⠃⠓⠙

⠀⠀⢀⡀⢰⡆⣀⠀⠀⠀⠀⠀⠀⡴⢒⡒⢄⠀⠀⠀⠀⢀⡀⠀⢀⣠⣄⡄⠀⠀⠀⠀⣠⠤⡤⠤⡀
⠀⢀⣤⣽⣾⣾⣯⣤⠀⠀⠀⢀⠘⢆⢸⣁⡼⢀⠀⠀⠀⢸⡇⠶⢸⠰⠂⠀⠀⠀⠀⡸⢗⡚⠚⢖⡺⡆⠀⠀⠀⣀⠀⣼⣷⠀⢀⡀
⠀⠀⢀⡶⢻⡟⢶⠀⠀⠀⠀⠈⠳⠦⣽⡥⠶⠋⠀⠀⠀⠀⠉⠭⢹⢩⠍⡇⠀⠀⠀⢡⣠⣳⣤⡚⣀⠇⠀⠀⢨⣿⣴⡿⢿⣷⣿⣧
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠀⠀⠒⠒⠚⠀⠀⠃⠀⠀⠀⠀⠙⠓⠒⠚⠁⠀⠀⠀⠘⠀⠛⠃⠀⠛⠃⠘
⠀⢠⣄⣀⣤⣤⢀⣀⠄⠀⠀⠀⣄⢠⡠⡀⣀⣤⠀⠀⠀⠀⣄⣀⢄⣠⡠⠄⠀⠀⠀⠀⢠⡄⡀⡀⢠⠀⠀⠀⠀⠀⢠⣠⣀⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠤⠠⠢⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⣠⠖⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⠦⠆⠀⠀⠀⠀⠀⠀⠀⣬⣿⣤
⠀⠀⠒⠄⠀⠀⠈⡐⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠰⣿⠀⠀⢼⣖⠀⠀⠀⠀⠀⡌⠽⡧⡞⢹⡄⠀⠀⠀⠀⣐⣿⣿⣿⣀
⠀⠀⠈⠀⡄⠢⠆⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⢦⡀⠈⠀⠀⠀⠀⠀⠀⠹⣤⠷⠓⠋⠀⠀⠀⠀⠀⠻⠿⡿⠿⠿
⠀⠀⠀⡶⠤⠦⠤⠀⠀⠀⠀⠀⠶⠦⠤⠄⠤⠤⠦⠀⠀⠀⠀⠤⠦⠤⠤⠀⠀⠀⠀⠶⠠⠤⠦⠤⠠⠤⠄⠀⠰⠦⠤⠤⠰⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠤⠤⠤⢤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⣶⣶⣶⣾⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠒⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡆⠐⠀⠂⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠁⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡛⠉⠉⠉⠉⠉⢉⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣤⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⢲⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠿⠿⠄⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⠿⢿⠿⡿⡿⡿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⡠⠄⠀⠀⣀⠀⢀⡀⠀⣀⠀⠂⠀⢿⣛⣻
⠀⠀⠀⠀⠀⢀⡀⡑⠀⠀⠊⣑⠀⠀⢨⠀⢹⠁⠀⠁⠀⠸⠀⠈⠇⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⢀⡀⠀⠀⢈⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠃⠒⢲⠃⠒⠑
⠀⠀⠀⣴⠀⠀⠈⠀⠀⠔⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠲⠔⠀⠘⠀⠀⠢⠆⠀⠀⣀
⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠤⠄⠄⠤⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠚
⠀⢰⡂⢸⡇⠀⠀⠀⠀⣾⣿⡄⠀⢹⡄⠀⣸⣿⡄⢀⣿⠏⢻⣧⠀⣼⡟⠀⣿⠛⠛⠃⠀⣿⡟⠻⣷⡀⡆
⠀⣇⠀⢸⡇⠀⠀⠀⣸⡟⠘⣷⠀⠸⣷⢠⡿⠹⣇⢸⡟⠀⠀⠻⣿⡟⠀⠀⣿⣶⢶⠄⠀⣿⣧⣴⠟⠁⣸
⠀⠙⠤⢸⣇⣀⣀⢰⡿⠛⠛⢻⣇⠀⢻⣿⠃⠀⢿⣿⠃⠀⠀⠀⣿⠀⠀⠀⣿⣀⣀⣀⠀⣿⠀⠹⣧⡤⠃
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⡀⠉⠉⠉⠉⠀⠉⢀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣷⡀⠀⠀⢈⠐⠰⠄⠄⠀⠀⠀⣸⠷⠀⠄
⠀⠀⠀⠀⠀⠀⠛⡂⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠘⢾⣽⡽⠃⠀⠀⠈⠉⠀⠀⠀⠀⠙⠖⠉
⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠂⠀⠀⠶⠶⠶⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠐⠂⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠁⠀⠘⠀⠀⠀⠁⠀⠀⠂⠉⠉⠲
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠀⠐⠒⠀⠉⠉⠁⠐⠒⠒⠢⠤⢤⣄⣀⣀⡀
⠀⠀⠀⢠⠴⠶⠀⣉⡉⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠶⢤⣤⡄
⠀⠀⠀⡆⠄⠀⠀⠀⠀⠉⠉⠑⠒⠲⠤⠤⣄⣒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠊⠉⡟
⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠦⠤⣄⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠠⢤⣾⡛⠛⠒⠒⠦⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⣀⡀⠀⠀⠀⣠⣤⣶⣛⣋⡉⠭⠵⠒⢆
⠀⠀⠀⢸⡅⠉⠙⠒⠦⣄⡀⠀⠀⠈⠉⠙⠓⠲⠦⢤⣤⣀⡀⠀⠀⠀⣷⠀⢀⣙⣿⡶⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀⣠⣤⠤⠴⠖⠚⠛⠉⠛⠓⠾⠾⠛⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢰⣾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣧⣄
⣰⡿⠿⠶⣦⣍⣛⢷⣦⣄⡀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠒⣒⣛⣭⣭⠽⠶⠚⠛⣿⠁
⣿⠁⠀⠀⠀⠈⠉⠛⠛⠿⢿⣷⣦⣄⡀⡼⢡⢀⣀⣀⣠⣤⠴⠶⢒⣚⣋⣩⣭⠥⠶⠖⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠇
⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣯⣭⣴⠶⠶⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡤
⠀⠉⠙⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⡀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠾⣿⣤⡶⠶⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄

⠀⡀⠀⢠⠀⠀⠠⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄
⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠄⠠
⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠆⠂⠀⠱⠤⠤⠤⠌⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠉⠒⠀⠀⠉⠀⠀⢀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠑⡄⠀⠀⠀⠀⠀⢀⠠⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡆⠁⡉⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈
⡄⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⠒⠒⠲⠀⠀⠐⠊⠑⠂⠀⠀⠑⠀⠀⡀
⠓⠐⠂⠀⢀⠀⠀⠀⢰⠖⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠘⠀⠀⠃⠂⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⠀⠀⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠃⠀⠁⠀⠀⠘⠀⠁⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠁
⠀⠠⠤⠀⠀⠀⠰⠒⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠄⠀⠀⠀⠀⠀⠀⠀⣅⢀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠄⠀⠀⢈⠄⠔⠁⠀⠀⢲
⠀⠠⠄⠀⠀⠀⠰⠐⠆⠉⠀⠁⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⠠⠀⠂⠀⠠⠀⠐⠤⠀⠀⠠⠁⠀⠀⠀⠀⠀⠾
⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠄⠀⠠⠤⠤⠄⣀⠀⠀⠀⠀⠤
⢁⢀⢀⠹⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠋⡃⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂
⠘⠘⠘⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡁⠀⠀⠀⠐⠠⡄⠀⠀⠀⠀⠀⠘⠀⠃⠄⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣀⡤⠶⠚⡟⠲⠦⣄⣀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢀⡿⡏⠙⠋⠉⠁⠀⠀⠀⡇⠀⠀⠀⠉⠉⠛⢉⡿⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣸⠁⢹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠃⢹⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢀⡏⠀⠈⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡏⠀⠈⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠁⠀⠀⢹⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢰⡟⠒⠒⠒⠚⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡟⠒⠒⠒⠚⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠳⢤⣄⣤⠴⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⢤⣀⣠⠴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠴⠖⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⣦⣄⡀
⠀⠀⠀⠀⣴⠟⣁⣤⠶⠶⢦⣄⡉⠻⣦
⠀⠀⢠⡾⠁⣼⠋⠀⠀⠀⠀⠈⢻⡄⢹⡇
⠀⢠⡿⠁⠰⣇⠀⠀⠀⠀⠀⠀⠀⣷⢀⣿
⠀⣾⠁⢀⣤⠿⠆⠀⠀⠀⠀⠀⣰⡟⢸⡇
⢸⡇⢰⡟⠀⠀⠀⠀⠀⠐⣶⠛⠉⢀⣾⣦⣄⡀
⢸⡇⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⡏⠀⠈⠛⠳⢶⣤⣀
⠘⣧⠘⣧⡀⠀⠀⠀⠀⣠⠏⢀⡾⠋⠛⠶⣤⣀⡀⠀⠈⠉⠛⠷⣦⣄⣀
⠀⠙⢦⣈⠙⠳⠶⠶⠛⢁⣴⠟⠁⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⠙⠻⢶⣤⣀⡀
⠀⠀⠀⠙⠛⠶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠉⠛⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⣀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠉⠛⣶⠋⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣴⠃⢠⠟⠛⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠻⢦⣄⣀⠀⠈⠙⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣍⠻⠶⣤⣀⡀⠀⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠦⠼⠃⠀⠀⢉⣙⠳⣦⣄⡀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣀⡀⠀⠀⠺⣏⡙⠋⠈⠙⠳⠶⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⣼⠋⢙⣷⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⣾⠃⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟

⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣸⣶⣄⡀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⡠⠒⠉⠀⣶
⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈⠙⠳⠮⣝⣷⣤⡀⣀⡴⢈⣀⡀⣀⡀⣠⠔⠈⠀⠀⠀⣰⠟⠃
⠀⠀⠀⣸⣧⣤⣄⣀⣀⠀⠀⠀⠀⠃⠀⠀⣀⣠⣤⠔⠂⢻⡇⠀⠀⠀⠀⠀⠀⠉⡻⠿⠿⣟⡛⠿⠭⣉⡉⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⣿⣷⣲⣖⣒⣩⣭⣵⣦⣄⣤⣶⣿⣿⣿⣶⣾⣿⣾⣿⠀⠀⠀⠀⠀⠀⠊⣀⠀⠀⠀⠉⠓⠦⣄⣹⡄⢀⡤⠊
⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠉⠈⠉⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡀⠀⠀⠀⠀⠀⠲⠂⢄⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⣑⡭⠆
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠢⠄⠈⠙⠷⣄⡙⠢⣄⡠⣀⠤⠒⢋⡡⠔⠂⡠⠤⠄⠠⢀⣀⡀
⠀⠀⠀⠰⣏⠙⠢⡀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⢀⣔⡵⣛⡥⠂⠀⠀⠈⠛⢷⣼⣛⣥⠔⠊⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠈⢀⡴
⠀⠀⠀⣤⡈⠢⡀⠈⠢⡀⠀⠀⠀⠙⠶⣤⡀⠀⢠⣿⣿⣚⣭⣿⣷⡀⠀⠀⠀⠀⠈⠁⠀⠓⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⢋⠟
⠀⠀⠀⠙⢧⡀⠈⢤⠀⠀⠔⠒⣒⢂⣀⣈⣻⣤⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⡁⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠙⠤⠵⠶⠿⣛⣿⠿⠛⠛⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⣈⣉⠙⠒⠒⠤⠤⠎⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡶⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠉⠙⠒⠢⠴⠊

⠀⠀⠀⠀⢀⣤⡶⠚⠛⠛⠛⠻⠶⣤⣄⣀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣴⠟⠁⠀⣀⠤⠤⠤⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⣸⡏⠀⠀⣾⡿⠛⠻⣦⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⣿⡁⡀⠸⣿⠀⠀⠀⢹⣷⡏⠀⠀⠀⣶⠶⠛⠛⠛⠓⠶⣄⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣿⡇⢳⠀⠙⢷⣤⣤⠾⠋⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢻⣆
⠀⠀⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⢹⣿⠁⠀⠀⢀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠆⣿
⠀⢀⣿⠃⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠌⣼⡟
⠀⣼⠃⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣼⡿
⣸⡏⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⣿⣿⠁
⣿⡇⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢩⣕⢦⠀⠀⠀⠀⠀⢿⣿⡀
⣿⣧⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠁⠀⠈⡄⢿⡓⢅⢦⡀⠀⠀⠈⠻⢷⣄⣤⣄⡀
⠸⣿⢣⡀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠳⡌⠢⡑⢄⠀⠀⠀⠀⠀⠁⢹⣷⡀
⠀⠙⣷⣕⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠀⢀⡴⠓⡜⢦⡑⢌⠣⡀⠀⠀⠀⡠⢸⣿⡇
⠀⠀⠈⠙⠻⠿⢾⣿⣍⣡⣠⣤⣤⣴⣶⣶⣤⣍⣳⣄⠀⠻⡀⠀⠈⢦⡱⣌⠢⡀⢄⠀⠀⢣⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆⠀⠘⢆⠀⠀⠳⡌⢳⡔⢄⠢⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠈⠣⡀⠀⠈⢆⠙⣦⠱⡐⢄⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣤⡙⢄⠀⠀⠱⡈⢳⡌⢂⠳⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⢻⣆⠓⢀⣀⣈⠢⡙⢦⡑⠜⢆⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣷⣶⣬⡇⠙⢮⠳⣄⠢⡳⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣷⠀⠀⠱⣌⢦⡀⢈⠂⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣭⡑⠢⡈⠣⡙⢦⠱⡕⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣧⠹⠀⠈⢪⠳⣌⢮⡢⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠑⢜⢧⡑⣄⢆⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣴⣀⣀⣀⡀⠢⡙⣎⠢⡑⣄⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣬⣥⠀⠘⢎⠳⣔⢌⠢⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣅⡉⡺⠳⡜⢧⡁⡀⢄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⠀⠈⢦⡙⣮⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣳⣼⣿⠏