Super mario 64 Emojis & Symbols

⠀⠀⠀⠀⠀⢀⠄⠈⣠⣄⣨⡞⠀⢸⡆⠠⣾⠀⣀⠀⢢⠀⠀⢀⣶⠀⠀⣀⠐⢠⡀
⠀⠀⠀⠀⠀⢹⡿⠆⠀⠉⣺⡇⠀⠾⠃⠀⣿⠀⠛⠋⣸⠀⢘⣳⣿⡆⠀⠿⠋⢈⡇⠀⠀⡴⢩⣤⣾⠀⠀⠀⡰⢋⠹
⠀⠀⠀⠀⠘⣿⣶⣤⣴⣾⠟⣷⣤⣤⣤⢾⣇⠀⣶⣾⣿⣄⣨⣥⣤⣇⡀⢸⣦⣀⣽⣦⢰⣇⢠⣶⣶⡝⣆⡞⠰⠻⠆⠣⡀
⠀⠀⠀⠀⡰⠁⠀⢸⠏⠈⠉⠹⣧⡹⠋⠙⠛⢿⣿⡏⠉⠈⠉⠛⠻⢿⣿⠛⠛⣿⡿⠇⢸⣿⣶⣬⣭⣴⣿⣿⣿⣿⣷⣼⠃
⠀⠀⠀⢰⠁⠀⠀⠀⢀⠀⠀⠀⢏⠀⣠⢾⠀⠘⣿⡇⠀⠀⣷⡦⠀⠘⣭⠀⠀⢿⡠⠐⠚⠋⠛⠻⣿⠃⠉⠉⠉⠿⠿⠃
⠀⠀⢠⠃⠀⠸⣆⢀⣾⣷⡀⠀⠸⡌⠛⠺⠀⠀⢹⡇⠀⠀⠋⠀⠠⣶⣿⠀⠀⢸⡇⠀⢠⣶⣤⡀⠀⢹⡆
⠀⢀⠇⠀⠀⢸⣿⣿⣿⣷⡇⣀⣠⣷⣦⢤⡀⠀⠀⢣⠀⠀⣦⡀⠀⠈⣹⠀⠀⠀⣧⡀⠀⠛⠉⠀⠀⢸⠁
⠀⣾⣐⣶⣶⣾⣿⡟⡿⢻⣿⡸⣿⣯⣿⢿⣧⣶⡄⣼⣷⣾⣿⢹⣶⡟⣿⣦⣠⣤⣿⣿⣦⢤⣤⣶⡦⣺
⠀⠈⠻⠿⠿⠞⠿⠃⠀⠈⠻⠓⠉⠁⠀⠀⠷⠻⠃⠻⠀⠛⠋⠙⠧⠂⠿⠏⠸⠳⠟⠺⠯⠞⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠁⢀⡤⠤⢤⡉⠻⣦⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⡴⠁⠀⠀⠀⠈⣢⣽⠿⠛⠉⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⡇⠀⠀⠀⣠⡾⠋⠁⢀⡠⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠙⢄⣠⡾⠋⠀⣀⡴⠋⠀⠀⠀⠀⣰⠟⠻⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠻⣦
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⡾⠋⠀⢀⣼⣋⣀⣴⣶⣦⣀⣼⠏⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⠟⠁⠀⢠⡿⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣰⠟⣤⣶⣶⣟⠉⠉⠉⠟⠁⠉⢻⡇⠀⠀⠀⠀⠀⢠⡾⠛⠻⢿⣷⣤⣾⠋⠀⠀⢠⣿⠿⠋⠉⠀⣹⣿⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣴⠏⢸⡿⠉⣀⣄⡀⠀⠀⢠⡋⠙⢦⣿⡀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠹⣿⣁⠀⠀⠈⠻⣅⠀⣠⠶⠛⠉⠉⣿
⠀⠀⠀⣴⠛⠃⠀⠀⠀⣸⠟⠛⠛⠃⢸⠁⠀⠈⢦⠀⠘⣗⣦⣼⠿⠷⠶⢶⣄⠀⠀⠘⢷⡄⠀⠀⠀⠈⠳⡀⠀⠀⠈⠻⡅⢀⣤⠶⠞⠻⣦
⠀⠀⢸⡇⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠸⣵⣲⡄⠈⡇⢠⡿⠋⠀⠀⠀⠀⠀⠹⣧⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⣴⠟
⠀⠀⠈⣷⡀⠀⠀⠀⣿⠈⠳⣄⣀⠀⠀⠹⣿⣇⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⢸⡿⣦⡴⠾⠿⣆⠀⠀⢹⣧⠀⣠⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢸⣯⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⣿⠁⣿⠁⢠⣤⣌⠳⢤⣼⠇⢠⣿⣿⣷⣶⣤⣼⣷⣦⣤⣤⣤⣴⣾⣿⡆⢀⡾⠛⢷⡄⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠁
⠀⠀⠀⣿⠀⢿⡀⠀⠀⠼⠷⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣠⠟⠁⠀⠀⠙⢷⣤⣀⠀⢀⣽⣿
⠀⠀⠀⣿⡄⠈⠳⣄⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣋⣀⣙⠿⢿⣟⡼⠋⠀⠀⠀⠀⠀⠀⢈⣿⠛⠛⢹⡟
⠀⠀⠀⢸⣇⠀⠀⠈⠛⠲⠶⢶⡟⠳⣄⠀⠀⠀⢹⡷⠤⠬⢭⣽⠇⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣸⡇
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣸⠇⠀⠹⣆⠀⠀⠀⢳⣄⠀⣀⠏⠀⣸⠏⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠸⣧⣀⣴⣦⣿⡆⠀⠀⠀⠉⠉⠁⣀⡼⣯⡀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠈⠉⣀⣴⠾⢿⣄⣀⣀⣀⣘⣉⣽⣇⠈⠻⣄⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣤⠤⣾⠛⢧⡀⠀⠻⣌⠉⠉⠉⠉⠀⠸⡄⠀⠙⢷⣠⡟⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠈⢷⠾⣷⣀⣀⠘⣦⡀⠀⠀⠀⠀⠹⡄⠀⠈⢿⣇⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣼⠃⠘⣯⠉⢿⡋⠙⣦⣀⣀⣠⣤⣿⡄⡾⠛⣿⣦⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣇⣀⣠⡟⠀⢸⡇⠀⢸⣯⣭⣭⣤⣴⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠶⠿⣦⣤⡾⠋⠉⠁

⠀⠀⠀⡃⠰⠄⠀⠀⠷⠀⠀⠉⠒⠠⠒⣩⠞⠁⠀⣤⡀⠀⢀⣤⠀⠀⠀⠀⠈⠙⠲⣏
⠀⠀⠀⠑⠀⠒⠒⠒⠒⠒⠐⢀⡀⢀⡜⠁⠀⠀⡼⠁⠑⠴⠋⡈⣇⠀⠀⠀⠀⠀⠀⠈⠑⢄⣄
⠙⠒⠒⠒⠦⣄⣀⣀⣀⣠⠴⠋⢉⡏⠀⠀⠀⢰⢃⡞⠳⣤⠎⢳⣘⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⡳⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⠀⠀⠀⣀⣈⠭⠴⠖⠒⠒⠒⠒⠶⠤⢤⣀⡀⠀⠀⠀⠀⠀⠱⡀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠴⠊⢁⡠⠤⠖⠒⠒⠒⠒⠒⠶⠤⢤⣀⡉⠓⢤⡀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⣠⠚⢁⣀⣀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠉⢳⠦⣝⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣀⣠⠤⢤⡀⠀⠀⢹⠎⡽⣼⡿⠻⢿⣷⠀⠀⠀⣾⡿⠛⠉⠙⢿⡄⠘⢦⡀⠙⠲⣄⠀⡇⣀
⠀⣠⠖⠒⠏⠀⠈⢂⠀⣳⠀⢀⡞⢻⠇⠋⠀⢀⣶⡁⠀⠀⠀⠀⠠⠀⣴⣆⠀⠁⠀⠈⢧⠀⡴⠋⢹⠋⠉⠙⠋⠳⣤⣄⡀
⡼⠑⠀⠀⠀⠀⠀⢼⠉⠉⠉⠙⣆⢸⠀⠀⡆⣰⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⢸⣾⡷⠚⠛⠒⢲⡀⠀⠀⠀⠀⠹
⣇⠀⠀⠀⠀⠀⣀⠜⢦⣀⠀⢀⡾⡞⠀⡀⢁⡿⠛⠉⠉⠉⠑⢦⡀⢸⣿⡿⢿⡄⠀⠀⠈⢹⡀⢀⠀⢀⣼⡁
⠈⣗⠒⠒⠂⠁⠀⠀⠀⢀⡠⠎⢸⡁⢸⠿⡾⠀⠀⠀⠀⠀⠀⠀⢳⠚⠋⠀⢾⡇⢀⠀⠀⠀⢳⣀⠉⠉⠀⠈⠑⠢⠤⠤⣴
⠀⠘⢦⠀⠀⠀⠀⠀⠀⢯⠀⠀⠸⡝⠺⡇⢷⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣰⡿⠚⠁⠀⠀⢀⣞⣩⡷⠢⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢻⣀⣠⡤⠴⠒⠚⠛⠦⢤⣀⣳⡀⠹⢮⡓⢤⣤⣤⣤⣴⠋⣀⣠⣼⠋⠀⠀⠀⠀⢀⣼⣩⠴⠓⠦⢤⣀⣀⠀⢈⡇
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⢻⡍⣉⣁⣈⡉⠭⢄⠏⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣄⡀⠀⠀⣀⣀⡠⠀⠀⢀⠀⣀⣴⠊⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⡦⢀⠀⢀⡄⠈⠓⠦⢤⣀⣀⡀⠀⠀⡟⢸⡇⠙⣦⣀⠀⠀⠀⣀⡤⠾⡏⠁⡏⠀⠀⠀⡶⠴⠖⠒⠋⠁
⠀⢘⡄⠉⠉⣼⣼⡭⠁⠁⢰⠉⠉⠉⠙⡇⠈⡇⠀⣇⣈⡭⠭⠭⠥⠤⢬⣇⠀⡇⣠⠔⠚⠉⠉⠙⠲⣄
⠔⠁⣠⢄⠀⠿⣉⣤⠀⣠⣄⣑⣤⠀⠀⡇⣀⣧⢖⡉⠀⠀⠀⠀⠀⠀⠀⡠⠛⢻⠃⡠⠆⠉⠐⢤⠀⠈⣆
⠀⣰⣡⡴⠞⠉⡷⠞⡾⢧⡈⢳⠀⠀⠀⣏⠁⢇⢀⡅⠀⠀⠀⠀⠀⠀⠀⠱⣄⡜⡰⠃⠄⠀⠁⠂⢥⠀⢸
⣠⠏⢀⠟⠒⠊⠠⣀⣀⠗⠙⢪⡀⠀⠀⢹⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠂⠠⢀⠈⠀⡎⢠⡏⢀⡀⣉⢀⡀⣉
⡋⠉⡵⠒⠢⢀⠄⠠⡀⠀⠀⣆⠙⡆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣎⠈⠀⠀⠄⠈⢸⠗⠁⠀⢀⠀⢀⠀⡁⢀
⣼⣀⢇⠀⣠⢣⡀⢀⠋⡁⢀⣈⡾⠃⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢣⡀⠀⠀⠄⣠⡏⣀⠀⡀⢀⠀⠀⠀⡀⢀
⠀⠈⠉⣉⡏⠉⠩⣹⠉⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡝⠒⠛⠓⠒⠚⠁⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠒⠢⡄⠀⠀⠀⢀⣄⠀⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣿⠿⠛⠿⢿⣿⠗⣡⡠⡄⢀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣯⣀⣀⣀⣀⡷⠛⠓⢲⣤⠌⠣⢀⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⡿⠿⠿⢿⣿⣷⠀⠀⠀⠹⣏⠣⡤⠁⠅⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⠁⡇⠀⠀⠼⣿⣉⣧⠀⠀⠀⠹⡄⢁⠀⠰⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢂⢁⡏⠀⣇⠀⠀⠀⠈⢻⣿⣧⡀⠒⠉⠉⢳⠀⠃⣹⣇⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠢⡻⠀⠀⠀⠀⡘⠰⣿⡄⠀⠀⠀⠓⣅⠘⢉⠿⢿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⣿⠃⠀⠀⠀⠀⢸⠀⠌⠀⠀⡿⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⠀⢀⠀⠛⢮⣴⣌⣌⡇⠀⠘⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⣰⣦⠶⣶⠈⢻⣿⣿⣿⣿⣿⠟⠁⣿⣿⡿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣶⠐⢿⣿⣿⣶⡝⢸⡆⢻⣿⣿⡿⠃⣸⢸⡿⠟⠁⣿⣿⣿⣿⣭⣥⣄
⠀⠀⠘⢻⣿⣿⣦⣀⣀⠘⢿⣿⣿⣿⠀⠀⠙⠿⣷⣼⠟⠟⠘⠛⠛⠋⠸⠏⠈⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠖
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣃⣀
⢀⣀⣀⣉⣽⣿⣿⣿⣿⠟⠛⢿⢿⣿⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣟⣀⣀⣤⣤⣤
⣾⣿⣿⣿⣿⡿⠟⠀⠉⠀⠀⠀⠀⠹⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣤⣤⣶⣶⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡆⠀⢀⣤⠤⠄⠀⠀⠀⠐⠛⠛⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⢩⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣽⣿⡿⠁⠀⢀⠀⠀⠀⣵⣆⠂⠀⣀⣷⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣿⣿⠟⠁⠀⠀⢰⠇⠀⣘⣿⣿⣷⣾⣿⣿⣥⠀⠀⢲⡄⠀⠂⠀⢬⣙⠻⠿⣿⡿⣃⣼⣿⣿
⣿⣿⣿⠋⣿⣿⣯⣿⣿⣿⡿⠛⠁⠀⠀⠀⢾⠏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠑⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣯⡛
⣿⣿⣿⠄⢫⣭⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⢹⣿⣿⣿⣟⡅⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣬⠍⣠⣤⠖⠁⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠈⠀⡀⠀⠀⢀⡀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠈
⡟⠿⢛⣿⢟⣵⣾⠟⠁⠀⠀⢀⣴⠿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁
⠀⣴⠟⢁⣾⠋⠁⣀⡴⠂⠁⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠉⠓⠲
⡞⠡⣠⡟⠁⢀⣾⡟⠁⠀⢸⡿⠃⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠈⠻⣿⣦⠈⢳⡀
⢠⡇⠟⠀⠀⣼⡟⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⢹⡾⢟⣛⣛⠁⠀⠀⠈⠛⠛⠿⣇⠀⠀⠀⠀⠀⠀⠈⠙⢷⠀⠁⠰⣄
⠸⠿⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢛⠩⠭⠄⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡿⣠
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡷⠂⠀⠀⠀⠉⠉⢀⠀⠀⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢷⡀
⠀⢀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠀⠀⠀⣠⣴⣿⣷
⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣄⠀⠀⠺⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣦⣠⣤⣿⣿⡿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣿⣤⣀⡘⢿⡛⠉
⣿⣿⣡⣭⣛⢿⣿⣿⣯⠿⠛⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡾⣷⠻⣿⣿⣛⣢
⠉⠉⠉⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠐⡀⠀⠆⠂⠄⠠⠀⠀⠀⢂⠢⠀⢲⠀⠀⠀⠐⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠈⠀⠀⠀⠀⣀⣀⡴⠋⠁⠠⡈⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⢉⡉⠉⠒⠦⣄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠤⠀⠉⠻⣆⠀⠀⠀⠀⠀⢸⡇⠂⠁⠀⠀⢈⠁⢢⡀⠑⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⣀⡥⠤⠤⣤⣀⡘⡆⠀⠀⠀⠀⠀⠙⠦⢄⡞⠀⠀⠁⠀⠎⢢⡈⠆⢸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⢀⡤⠚⣉⣠⠴⠖⠀⠠⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣸⡇⣤⠀⠀⣠⠀⠀⠛⡬⠤⣀⠇
⠀⠀⠀⠀⠀⠀⢸⠋⠀⠁⠋⢐⠋⠁⠉⢄⠉⠀⠰⢖⠈⠂⠘⣧⠀⠀⠀⠀⠸⡴⠋⠉⠘⠚⠋⠀⠀⠀⢠⠀⡏
⠀⠀⠀⢀⡴⢦⢈⡖⡉⠓⠀⠈⠀⣴⣄⣄⠀⢀⡐⠊⣧⣠⡴⠋⠀⠀⠀⠀⠀⠷⢄⣀⠀⠐⠂⢲⠂⣠⣠⠴⠁
⣠⠖⠦⣼⠀⢘⡿⡅⠳⠤⠴⢊⠀⠙⠛⠉⠀⠻⡴⠃⡸⢧⠀⠀⠀⠀⠀⠀⠀⠑⠤⡞⡓⠤⠒⢁⣞⣈⡇
⢻⡀⠀⠹⡄⢈⡧⢍⡶⠂⠀⠠⠀⠲⡀⠀⠀⢀⠇⢰⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢦⠄⣠⡾⢻⠟⢦⣄⣀⣀
⠀⠙⡶⠒⢾⠃⠀⡈⣯⣻⣤⡀⠑⢖⢀⣉⣉⠅⢈⡽⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⡃⢀⠄⢠⡇⣀⠀⡈⣀⡼
⠀⠠⡏⠁⠢⡶⠒⠁⢋⡇⠉⠉⠑⡄⠑⡊⠡⡻⡧⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢰⡎⠸⠃⢀⠜⢢⠀⠀⠀⣿
⠀⠀⠙⠦⢝⣡⣄⣤⠞⠠⢤⠀⠀⢰⠀⣐⡆⠓⡘⠪⠱⡄⣠⡰⠖⠶⣤⡞⢁⠟⠑⠁⠀⠒⠋⢄⣯⡛⠀⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⡰⠛⡾⡀⡁⠀⠘⣰⡤⠋⢆⡈⠂⠀⠁⠑⣼⠀⠀⠀⠀⠀⠀⠀⢹⡝⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠉⠀⠀⠑⠚⠁⠀⠀⢸⡇⠀⠀⠉⠓⣦⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢧⣀⡇⠘⢧⡀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⡔⠊⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢳⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢹⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣛⣁⠀⠀⠀⣼⡇⠀⠀⢀⣀⠧⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠅⣀⠀⢸⠀⠀⠀⢀⣘⢷⣄⡀
⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠉⠒⢼⣧⠤⠊⠉⠀⠀⠀⢹⡀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠉⢳⠠⠔⠊⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⢀⢀⡠⣼⢧⣀⣀⣀⣀⣀⣠⡼⠁⠀⠀⠀⠸⣇⡤⠀⣀⣠⣤⣼⡂⢄⣀⣀⠀⣀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠋⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠈⠙⠛⠒⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⠐⡠⢄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠴⠾⣋⣹⣌⠂⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠐⠢⣄⣩⣷⣤⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣰⡶⠏⠉⢘⣋⡛⢧⣙⣦⠀⠘⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⣦⣵⠀⠀⢰⣷⠀⣠⠞⢣⣀⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⢀⣺⣧⡄⢣⡠⢣⡄⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣶⣿⡿⢿⡋⠀⠀⠀⠈⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣌⡁⠙⢒⡉⠀⣀⣠⣾⣉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠙⠈⠈⣓⣦⣴⡺⣑⡬⠂⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢲⡀⡀⢠⠋⠀⠙⢦⠵⠋⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⣧⣸⣤⣀⠀⠀⠀⠀⣠⢖⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠟⠋⠀⠀⢹⣷⢶⡶⠚⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠑⢶⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠸⡄⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠓⠲⢤⣴⠃⣀⣤⠴⠒⠛⠛⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢸⠋⠈⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣒⠒⠒⣒⣒⣚⣧⢤⣀⣀⣀⣀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠓⠲⠶⠴⠶⠚⠁

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

⣾⣿⣿⣻⣻⣻⣟⣝⣟⣟⣿⣻⣟⢀⠀⠀⠀⠀⢀⡠⠐⠐⠋⣁⣈⡐⠋⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠂⠀⠀⠀⠐⠁⠀⣠⣾⣿⣿⣿⣿⣿⣶⡄⠀⠁⢢⡠⠄⠤⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠹⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣠⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠉⣛⣛⣿⣿⣯⣥⣴⣶⣿⣿⣿⣿⣿⣿⡇⠠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣤⣧⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⣷⣾⣿⣿⣿⣿⣿⣿⠿⢛⡋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⡿⢹⡆⠀⠙⢿⣿⣷⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠛⠛⠛⠛⠻⢿⣧⠄⠘⠟⠀⠀⠀⠿⢸⣿⣿⣿⣿⣿⣇⢸⣧⣤⡄⣼⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠈⣰⣧⣧⣄⠀⠀⣠⣤⣶⣿⣷⣆⠈⢡⣤⣦⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣦⣴⣮⠻⣿⣿⣿⣏⠉⠙⣱⣿⠉⢱⣿⡏⠻⠉⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣹⣿⣿⣿⡟⢹⡎⠙⠿⣿⣿⢿⣿⣿⣿⣿⣿⣴⡿⠈⠁⢻⣧⣿⣿⣿⣿⣧⡀⠛⠉⠀⠀⠙⠃⠀⠀⣿⣻⣻⣿⡇
⠀⠀⠀⠀⠙⢛⣿⣿⠁⠻⠃⠀⠀⢿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣠⣼⡿⡎⣻⣿⡧
⠀⠀⠀⠀⠀⠀⣻⣿⡄⠀⠀⠶⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⣀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⢀⠻⣿⣧⡀⢻⣿⠂⢀⣰⣿⣿⡟⠉⠛⢋⣿⣯⠉⠉⣿⣿⡝⢿⠿⢻⣿⣿⢿⣿⠿⣿⣿⡿⢟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣮⣿⣿⣿⡏⠄⣸⣷⠀⠀⠸⡿⠟⠀⠀⠻⢿⡇⠘⠀⣾⣿⣳⡄⠀⠙⠋⣉⣿⣿⣿⣿⢛⠛⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠉⠙⠿⣿⣿⣿⣇⠀⠻⠇⣶⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⡜⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣀⠀⠻⣿⣿⣦⣀⢀⣀⣀⣴⣿⣿⠿⠃⣨⣿⣿⣿⠿⣿⡿⢿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⣿⣧⣀⠈⠙⠛⠿⠿⣿⣿⢿⠛⠉⣈⣼⣿⣿⣿⠛⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣷⣦⣄⣀⣀⣈⣥⣮⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣀⢀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣾⠟⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠈⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣶⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⣄⠰⢆⣠⡀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠹⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠖⠛⠃⠀⠄⠀⠀⠀⠀⢿⡿⢇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⠀⡠⠊⢸⡇
⡄⠀⠀⠀⠀⠀⠺⠟⠛⠛⠟⠃⠀⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠰⠖⠠⠄⠀⠔⠀⠄⠀⠄⠀⡄⠚⡄⠀⠘⠇⠀⠀⠀⠄⠀⠀⠄⠀⠀⠠
⡇⢰⠂⠒⠐⠒⠐⠃⢲⠂⠓⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢰⠂⠒⠐⠒⠐⠃⢰⠀⠓⠐⠂⠐⠂⢲
⡇⠈⠀⠐⠀⠁⠀⡂⠈⠀⠂⠀⠁⠘⠂⠐⠂⠐⠂⠐⠂⠐⠂⠐⠂⠐⠂⠐⠂⠐⠂⠐⠂⠀⠂⠀⡇⠈⠀⠂⠀⠁⠀⠂⠈⠀⡂⠀⠁⠐⠀⠈
⣏⣹⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⣁⣉⣉⣉⣈⣉⣉⣉⣉⣉⣉⣉⣉⣹⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢡⠤⣤⡀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣱⡏⠟⣷⣄⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⠚⠻⠟⠟⢿⣿⡿⢿⣶⣤⣄⡀⠈⠻⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠡⠊⣵⡷⣦⠄⠊⠹⣿⡰⠒⢏⠉⠑⠫⣕⠦⣀⠀⠀⠀⣱⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠻⣄⠀⢿⣰⠋⢢⣀⠀⠋⠇⠀⣾⣷⠀⠀⠀⠉⣾⣝⣦⡾⠁⢀⡀⢻
⠀⠀⠀⠀⠀⠀⣴⠞⠛⢤⡀⠀⠈⠳⣄⢻⣤⡴⠿⠂⠀⢂⠈⠜⠿⡄⢤⡀⣼⣿⣿⣿⠇⠀⢸⠁⢸⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢀⡾⠫⣤⣤⣀⢹⠷⠚⠓⣾⠟⠁⠀⠀⠀⠀⠈⠳⠤⢞⣡⣾⣿⠛⠿⢿⠟⠀⠀⠈⣠⠿⠛⠛⠉⠉⠉⠉⠙⠳⣤
⠀⠀⠀⠀⢸⡁⠀⠀⠀⣹⠋⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠓⠦⣄
⠀⠀⠀⠀⢸⡇⠐⠛⠉⡟⠀⠀⠐⣾⣷⡀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡟⢯⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡀⠀⠹⣧⡀
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠻⣄⢀⣠⠟⢻⣿⣷⣶⣶⣿⣿⡿⠟⠋⡹⠁⡜⢳⡀⡀⢀⣴⠃⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠐⠚⠉⠀⠀⡀⣦⠙⢦
⠀⠀⠀⠀⠀⠈⣷⠠⡀⠀⣨⠟⢁⡴⠃⣹⠟⢧⠈⠉⠁⢀⠀⠾⣥⣎⠤⢺⣧⡷⠟⢻⠀⠀⠀⠀⢠⡶⣏⠈⣷⠄⠀⠀⠀⠀⢸⠀⠗⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠙⠶⣬⣿⡁⠀⠋⢀⡴⠋⠀⠀⠑⢦⣤⣬⣷⣄⡈⠙⠚⣫⠾⡇⠀⠘⣇⠀⠀⠀⠙⠁⢻⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⢯⡀⠀⠀⠀⠀⣾⠀⠀⢿⠈⠙⠛⠛⠉⠀⡇⠀⠀⠘⡄⠀⠀⢠⠀⢸⡇⠹⣾⠇⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣿⠀⠀⠘⣆⠀⡀⠄⠒⢺⡇⣰⠒⢢⢹⡆⠤⠊⠀⢸⡇⢸⡁⠀⢐⣧⠀⠀⣰⠗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢿⡄⡔⠑⢮⢫⡀⠀⠀⢸⡇⣇⠀⢾⡏⡇⠀⠀⠀⢸⡷⠾⠷⠶⠾⣭⣛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣵⣀⣃⣇⣳⠀⠀⠈⣇⠘⠶⠛⣠⡟⠀⠀⠀⣸⠷⠂⠐⠠⢄⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣶⣛⠛⠻⢯⣀⠈⠉⠓⠦⡀⠈⠛⠶⠞⠋⠀⠀⠀⢠⡿⠶⢦⣄⠀⠀⠈⠳⣄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠤⠄⠤⠤⣀⠉⠙⠲⢤⡈⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢹⡆⠀⠀⠀⠸⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠔⠂⠀⠐⠢⠌⡑⠢⣄⠙⢦⡀⢳⡀⢀⡏⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⡼⠁⠀⢀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⠀⠀⠀⠀⠀⠑⠀⡑⢄⠙⣼⣃⣾⣠⣤⣴⣾⣿⣋⣀⣀⣀⣤⠴⠛⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⡰⠃⠈⢎⢣⢸⡟⠛⠉⠀⠒⠒⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢒⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠉⠉⠛⠲⣤⣀⣆⣿⠃⠀⠀⠠⠄⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠝⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢀⠀⠀⠀⠀⠀⠀⠀⡦⠙⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢉⢑⡢⠀⠀⠱⣀⣀⣀⣀⣖⣄⢷⠀⠀⠐⠠⠤⣤⠞⢯⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠉⠈⠹⠄⢈⠙⢳⠖⠉⡁⠠⠀⠉⠓⢮⣄⡀⠀⠀⡴⠛⣀⠈⡆
⠀⠀⠀⠀⠀⢰⡁⠀⠈⠹⣒⡒⠒⠊⠀⢱⠃⠀⠀⡀⠀⠀⡄⠀⠈⣧⡈⠀⠄⠘⠚⠀⠠⠂⣲⠤⠔⠒⠢⣄
⠀⠀⠀⠀⠀⠸⠔⠲⢶⡔⡚⠁⠉⠀⠀⠘⣤⣀⣠⣤⠤⢤⠤⠬⢶⣏⣀⠤⠀⠀⢸⠂⠗⠊⠀⢀⡀⠀⠤⢼
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠁⠸⣀⣀⡀⠀⠘⣼⡆⠸⠇⠀⠸⠃⠀⢺⠃⠘⡇⢲⡀⠸⡎⠁⠈⠉⢈⠁⠀⡄⠀⡇
⠀⡎⠙⣀⠂⠀⡾⠀⠀⠀⠈⠫⠚⠁⠂⠀⢀⡧⣀⣆⠀⠀⠀⣠⠟⣢⠜⠉⣀⣐⠀⠱⠂⠀⠀⠈⠐⠳⠋⠉
⠀⠱⡀⠀⠀⢀⡁⢦⡀⠀⠀⠀⠀⢀⡤⠾⢅⠀⠀⠝⠢⠀⠚⡈⣦⠇⢀⠎⠐⠁⠀⠀⠂⡀⠦⠦⠄⠴⠁
⠀⠀⠈⢄⡠⠋⠙⢄⠉⠓⠤⢄⡴⠋⠀⣀⠀⢻⠔⢆⡀⠀⠀⠙⢫⠀⡈⠀⠀⠆⠄⢀⢈⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠤⢞⠠⡀⢄⡨⠀⡨⠃⠀⢀⠤⢆⠀⠀⢳⡅⠀⢸⡒⠀⢈⢉⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⢀⡸⠪⠮⠋⠑⠉⠀⣦⠠⢾⠀⠈⠆⠀⠀⠱⣄⢠⡌⠒⠁⢀⡀⠀⠀⠼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⢠⣀⠀⣀⠘⣄⡄⠀⠀⠀⠈⠢⢤⡐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠠⠂⠋⠓⠻⠀⠒⠖⠗⠷⠀⠀⠒⠀⠵⠢⠀⠀⠀⠀⠀⠱⣤⣤⣴⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣉⡤⠤⣄⡉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠛⠻⠿⢾⣯⡰⠋⠀⠀⠀⠀⠙⢆⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠀⠀⠀⠀⠀⡀⠈⠛⢶⣄⠀⠀⠀⠀⢸⡆⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⢻⣧⠀⠀⢀⣴⡾⠛⠁⢿⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢷⣄⠀⠀⢸⠃⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣠⣼⡟⠀⠘⣷⡀⠀⠀⠀⠹⣷⣴⠟⠁⠀⠀⠀⠸⣧⠀⠀⣠⣶⣤⣀⣀⣙⣷⡀⠀⠙⢷⣴⠃⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⣾⠋⠈⣿⣴⠶⠛⠛⠛⠒⠆⠀⠹⣷⡀⠀⠀⠀⠀⠀⠹⣧⣰⡟⠛⠛⢯⠉⠉⠉⣻⣶⣶⣶⣝⢧⡀⠀⠀⠀⠀⠀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⡀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⢨⡟⡴⠊⢳⡄⠀⠀⢈⣭⣭⣿⡙⠿⠈⢷⡄⠀⠀⠀⠀⢸⣿⠘⢷⡄
⠀⠀⠀⠀⢸⡏⠙⢷⣄⡙⣷⣤⣤⣤⠀⠀⠀⠀⠀⢈⣿⣄⣠⣶⠟⠛⠛⠛⠻⣧⣴⣿⠁⠀⣴⠋⠀⠀⢿⠻⢦⣀⣀⢻⡄⠀⠀⠀⠀⣿⠀⠈⢿⡄
⠀⠀⠀⠀⠘⣷⣄⠀⠉⢙⣿⠁⠀⠻⠀⠀⠀⠀⠀⠘⠁⢹⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⠀⢠⡇⠀⣾⣹⡾⠀⠀⠉⠉⠙⢷⡀⠀⠀⠀⣿⣷⣄⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣿⠿⠁⠀⠀⣀⣠⣴⠿⣧⠀⠀⠀⠸⠀⠙⣷⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⡤⠄⠀⠀⠀⢀⣴⠟⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣤⣤⣶⣾⡆⣾⠋⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣠⣤⡾⠟⠁⠀⠀⢿⣿⣷⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⢳⡏⠀⢀⣠⠶⠞⠛⠳⣦⡀⠀⢀⡾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡋⠻⣷⣄⠀⠀⠀⠀⠈⣿⡘⠿⠿⠻⣿⣿⣿⠟⠻⣿⣿⠋⠀⠀⠀⠘⠷⠶⠋⢁⣤⠤⠀⠀⠈⣷⠾⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠻⣦⡀⠀⠀⠀⠘⢷⣄⠀⠉⢻⣛⣛⣛⣉⣭⡿⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⠀⠀⠀⢠⡿⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣙⣷⡀⠀⠻⣅⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣴⠟⠁⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⣴⠟⠋⣉⣷⣄⠀⠈⠉⠉⠁⣀⣤⣤⣤⣤⣤⣶⡟⠛⠾⠿⣿⡋⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢀⣀⣴⡿⠋⠀⢀⣾⠋⣀⣉⣛⣷⣶⣶⠟⠉⢁⡶⠋⠁⠈⠉⠛⢷⣶⡾⠛⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡟⠋⠉⣿⢀⣾⠛⠉⠉⠉⠉⢉⡉⠿⠛⠶⣤⡿⠶⣦⣀⣀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣖⣱⢋⡾⠋⠉⠀⠀⠀⠀⣈⡉⠁⠀⠀⠈⠃⠀⢸⣿⠛⠛⠓⠀⠀⠀⠙⣷⡄⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⡃⠀⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣀⡀⠀⠀⠀⣀⡴⠀⢀⣾⣁⣀⣀⠀⠀⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⢷⣶⠿⠋⠉⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⠟⣩⣿⣿⣦⣤⣿⣿⢋⣴⣀⡌⢻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⣸⣷⣶⣶⣿⣿⣿⣧⣾⣙⠛⣷⢸⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣤⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠈⠙⠛⢿⣿⣭⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡒⢚⣠⣶⠶⠾⣿⣿⠁⠀⠀⣼⣟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⠉⠂⠀⢠⣿⠇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣄⠀⢰⣾⣿⣿⣿⠟⠛⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣀⣀⣀⣴⣶⣾⣿⣿⣿⠀⠈⢿⣿⠟⠁⠀⢀⣠⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠠⡟⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣶⣤⣤⣽⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠻⣿⣿⣮⣻⣿⠿⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠰⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡜⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠠⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣷⠀⠐⣦⠀⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣽⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⡼⠀⣇⢈⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⣒⣲⣿⠞⡁⠀⠀⢳⢀⣀⣇⣀⠧⠼⢤⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢇⠻⡜⢁⠎⣠⣤⣀⡏⠣⣀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⣹⣡⣿⠊⠉⠐⣿⠶⠒⠺⡇⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠻⡇⠀⠰⣦⠀⠋⠀⠀⠀⢸⠒⠦⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡼⠞⠀⠀⠀⠨⠁⣠⣀⣀⣤⣿⡂⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡟⠒⠊⣢⠻⠿⣿⢿⣿⢟⠋⢠⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢠⠋⠩⣽⠲⠜⠀⠀⠘⡀⠈⡟⢰⣗⠲⢧⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⣀⡠⢤⡤⠤⣤⣈⣨⠴⣣⠖⠉⠺⣶⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠈⠳⣰⠊⠻⣧⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠺⣉⠉⠑⠚⠋⠉⠢⡀⡠⠖⣏⣠⢛⡿⠾⠼⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠚⠲⠬⠝⠁⠀⠀⠀⠐⢯⡀⠀⡏⠵⠋⠀⣀⡀⠈⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢉⠗⢀⣀⣀⣠⠤⠖⠠⠴⠃⠀⢱⣄⡔⢉⣉⣉⠑⢿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣔⡭⠒⠒⢍⢪⡖⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡏⠀⠀⠀⠀⣿⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠧⢄⣀⠀⡰⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⠝⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡎⠈⠓⠤⠤⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⡁⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣄⣀⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡚⠭⠅⡈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣰⣼⠞⢳⡘⡆⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠀⠍⣉⠙⠓⢥⡀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢠⣤⡴⠒⢻⠿⢿⡏⠳⣄⠀⠈⣧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠉⣅⠀⡐⢘⣦⠉⠓⢛⢧⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠛⠿⠀⠁⣸⡿⠀⢰⠉⢠⠗⠲⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣤⣤⣿⣿⠇⠘⠶⠡⡎⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⣾⠿⣿⢟⣻⠄⠀⠀⠀⣀⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢦⣀⡈⠩⠄⢀⣠⠤⡾⢥⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠁⠀⢈⣙⣲⠾⡙⣎⡶⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⣆⢀⠀⡴⠁⠀⠙⠢⠜⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⢻⣄⣳⣤⡀⠀⠀⠀⠀⢀⡜⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠟⠉⠁⠀⢿⣷⠶⡶⠚⠉⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠦⣄⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⣄⣀⠀⢈⡇⠀⠀⢀⣀⣀⣼⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠈⠛⢻⡤⢶⠞⠋⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣭⣭⣭⠭⠥⠤⢿⡒⡢⠤⠤⠤⠤⠴⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠉⠉⠉⠉⠉⠉⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠢⢄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠉⠙⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠉⠁⠸⠀⠘⢁⡛⢻⣏⠁⠀⠀⠀⠈⣠⣿⣿⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⡠⠁
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡜⠁⠀⠀⢾⣷⣾⣿⠀⠀⠀⠀⠰⠛⢋⣴⠉⢉⠡⠈⠙⠁⠀⠀⠀⠀⠘⠛⠤⢄⣠⠾⢀⡀
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡑⢿⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠆
⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣣⣄⠀⠀⠀⡀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠈⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠉⠀⠀⠰
⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣸⣯⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡜⠁⠈⠀⠀⢰⠇
⣼⠇⠀⠀⠀⠀⠀⠈⠉⠶⠶⠃⡘⠁⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣃⣀⣇⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠀⠀⠀⣠⠜⠁
⢣⡄⠲⣄⠀⠀⠀⠀⠀⢀⣴⡏⠉⠉⠙⢄⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠟⠒⠒⠊⠁
⠈⡟⠢⢀⣉⠉⠉⣉⡙⠋⠉⡇⠀⠀⠀⠈⠣⡌⠙⠁⣤⣄⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⠎
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠦⡀⠈⠛⠛⠛⠻⠿⠿⠿⠟⠁⠈⠙⠋⠁⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠑⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠜⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⡾⠿⠽⣆⡀
⠀⠀⠀⠀⠉⠁⠒⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢀⣤⡀⣀⣠⣀⠔⠂⠉⠀⠀⠀⠀⠀⠈⠁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠐⠈⢣⡀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠏⠀⠀⠀⠈⠓⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠈⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⢠⡿⠁⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠀⠒⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠉⠀⠴⠒⠶⠶⠆⠘⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠶⠤⢀⣀⣀⣀⣀⣀⠠⠤⠞⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠁⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢲⣆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠉⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡁⠀⠂⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠄⠀⣀⡀⠀⠴⠊⣹⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⣀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠋⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠘⠠⠤⠤⠤⠤⠤⠀⠛⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢀⣤⣤⣀⣀⣠⡀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣠⣶⡿⢿⣦⡀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠻⡿⠿⠟⠛⠛⠛⠒⠚⠛⢻⠟⠉⠉⠛⠻⣷⡄
⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀
⠀⠀⠀⠀⢸⣿⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢀⡁⠀⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢿⣧⠀⠈⢂⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠸⡀⠁⠀⠀⠀⢀⡊⠀⠁⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠉⠢⡀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣭⣀⣀⣤⡐⠃⠠⠤⣠⠀⠀⠀⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢷⣤⡀⠀⠈⠐⠠⡀⠀⠀⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠀⠀⠈⢻⢿⣿⣿⣿⣿⠟⣡⡄⠀⠀⠀⣉⠓⠆⠀⠐⠘⠧⡀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠐⠁⠀⣻⣿⣿⠿⠐⠛⠣⡀⠀⢰⣿⡇⠀⠀⠀⠄⠀⠀⣀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⢄⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠤⠀⡐⠁⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢄⣀⠀⣀⠌⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⣠⣀⠀⠀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠃⠀⠆⠀⡀⠜⠙⢿⣿⣿⣷⣶⣾⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⣀⣠⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣰⣶⣦⡀⠀⠀⠀⠻⣿⣏⠙⠟⠿⠋⡙⠋⠁⠀⠀⠀⠀⡠⠂⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢀⣿⡏⠙⢿⣶⣄⠀⠀⠹⣿⠷⠨⣐⠀⠀⠀⠀⠀⣀⣤⣾⣶⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⣼⡿⠀⠀⠀⠈⠻⣿⡦⢄⣰⠀⠀⠀⠈⠉⠉⠉⠁⠈⠻⣿⣶⣍⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠃⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⡬⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢦⠹⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⣦⣠⣄⣀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠛⠷⠤⠤⢀⡀⠀⠀⠀⠀⢀⠤⠊⠠⠋⠙⠻⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⡈⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣌⡐⠂⠤⠀⠀⠀⠠⢔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣂⣀⣀⣀⣀⣀⣤⣴⣾⠿⠃⠈⠙⠻⠿⢷⣶⣶⣶⠾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢁⢔⣲⣄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⣿⣘⣽⣦⣄⠀⠀⠈⠑⢦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣊⢍⠛⠉⠉⣹⣿⡟⠻⣿⢶⡦⢄⡀⠙⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠊⢰⡟⢻⠖⠁⠀⣿⡔⠉⢛⣄⠈⠑⠪⢕⡦⣀⠀⢀⡿⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠙⢦⠈⢧⡇⠈⣦⣄⠀⠁⠀⢸⣿⡄⠀⠀⢠⣿⣮⣿⡏⠀⣤⠄⢳
⠀⠀⠀⠀⠀⣠⡟⠁⠈⠲⣤⣀⣈⡓⣤⡷⠚⠋⠉⠀⠐⣄⠀⣉⡇⣰⣦⣿⣿⣿⣿⠀⠀⠫⢀⣇⣠⣤⠤⠶⠶⢤⣄⡀
⠀⠀⠀⠀⣰⠋⠘⠛⠲⢦⠞⠁⠀⢹⠏⠀⠀⠀⠀⠀⠀⢈⣭⣭⣶⣿⡿⠀⠉⠛⠁⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢀⣙⣦⣀
⠀⠀⠀⠀⢹⡄⢀⡤⠤⡎⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠈⢳⡄
⠀⠀⠀⠀⢺⡀⠀⠀⠀⡇⠀⠀⢸⠿⣷⣤⣀⣀⣠⣴⣿⣿⡿⠛⠏⢹⢦⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣠⡤⠖⠀⠀⢙⠳⣄
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠙⢲⡖⠋⡴⠛⣿⣿⠛⠻⠟⠋⠁⣠⠊⡰⠃⣸⡆⣧⡴⣏⠀⠀⠀⠀⢀⣤⠞⢻⣇⠀⠀⠀⠀⢀⡄⢦⠘⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢧⣈⣲⡶⠋⠠⠚⢀⡴⠃⠈⠳⣄⡀⢀⣦⡈⠙⠷⢤⠞⣹⠉⠀⢸⡀⠀⠀⠀⢿⠘⡇⢸⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠦⡤⢴⡋⠀⠀⠀⠀⢸⠋⠉⢻⠙⠳⠶⠶⠞⠁⡇⠀⠀⢳⠀⠀⠀⢸⠀⣿⠸⣦⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠀⠄⢰⡇⣠⠄⣌⢧⡄⠠⠆⠀⢹⠀⣾⠁⠀⣤⠀⠀⠀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⡇⡠⠀⢌⢾⡁⠀⠀⢸⡇⡏⠀⣼⡼⡆⠀⠀⠀⣸⣤⣽⣤⣤⣟⡳⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣱⠀⡎⣇⢳⠀⠀⠘⣇⠳⠤⠟⢰⡇⠀⠀⠀⣿⠤⠤⠀⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣖⠚⠻⢯⡉⠉⠙⠳⢄⡀⠘⠶⠶⠶⠋⠀⠀⠀⣸⡷⠤⣄⠀⠀⠈⠓⢤⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠤⠤⠤⣈⠉⠙⠲⢤⡈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⣷⠀⠀⠀⠈⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠂⠀⠐⠂⠬⡁⠢⢄⠙⢦⡀⢲⡀⢀⡇⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⢠⠏⠀⠀⣀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠁⠠⡙⢦⠙⣾⣃⣾⣤⣤⣴⣾⣿⣁⣀⣀⣠⡤⠞⠋⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⢤⣄⣠⠃⠈⠢⠱⣸⡟⠉⠁⠐⠒⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠍⠉⠓⢦⣤⣥⣿⣃⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠒⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢛⠳⣦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣼⡫⡡⡉⠻⣦
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠞⣦⣯⣺⣷⣾⣿⣽⠛⢶⣄⠀⢠⡴⡟⢹⠏⢹⡶⠒⠺⠧⢼⡅⡐⠀⠘⣧
⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⠧⠓⠋⢉⣠⣤⣾⡿⣦⡀⣿⢴⡞⢆⠑⠈⠠⠞⢻⣿⣷⡲⢤⡈⠢⡀⠀⢹⡆
⠀⠀⠀⣀⣼⠁⠀⡠⠊⢁⣠⣔⠊⠡⣿⣏⣉⡠⢠⣼⠏⠈⠛⢦⡀⠀⢀⡞⠉⠉⠻⠁⣼⠈⢒⢴⡀⠈⠙⣆
⠀⠀⣼⠋⠀⣀⡜⠹⡉⠁⠀⢿⠀⠐⠉⠀⠈⢻⡾⠋⠀⠀⠀⠀⠹⢯⡤⢷⣄⣀⣀⣀⣀⡀⢇⡡⠗⠢⣴⠏
⠀⠀⠹⣶⠉⢬⡙⢄⡸⠃⡖⠢⠤⣄⣀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠻⡌⠲⣶⡤⣀⡴⠃⢀⡪⠁⣠⠇
⠀⠀⠀⢿⣄⠀⠃⡀⠀⠀⠈⣦⣀⣀⣉⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⣗⠁⢀⣼⣭⡿⠋⠁
⠀⠀⠀⢀⣩⡿⠿⠿⢶⣤⣀⣱⣤⣾⣿⠟⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠋⡝⣹⠛⠶⣤⣀⣀⣀⣀⡤⡦⣄
⠀⣠⡾⠛⠓⢤⣠⠖⠉⠈⠀⣙⣧⠟⢹⢷⣾⠋⠉⠛⣷⡄⠀⠀⠀⠀⠀⣼⣹⢠⠟⢲⡇⠑⣄⡀⢸⠉⠉⣛⠀⣂⠩
⣼⢋⡄⣀⠀⠐⣯⠄⠠⠞⠰⡁⢈⠇⠈⢧⡇⠀⠀⢠⣻⡏⠀⠀⠀⠀⠀⢸⡁⠈⠓⠊⠈⠒⢻⡟⠷⣦⡀⠀⣤⣤⠴
⢻⣄⠀⠁⢠⣦⠿⢦⣄⠀⠀⠈⠁⠀⠀⣼⠁⠀⢠⣷⡟⠁⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢀⣼⣅⠀⠀⠙⢦⣸⠇
⠀⠉⠛⣶⡞⠁⠀⠈⢸⣇⣀⣀⣠⣴⠞⠛⠷⠶⢷⠟⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣳⠶⣖⡋⠁⠈⣧
⠀⠀⢠⡟⡇⠀⠀⢀⣼⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡜⠁⠀⠀⣰⡟
⠀⠀⠈⠿⣧⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡾⣦⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡤⠶⠒⠲⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠋⠀⠀⠄⡈⠀⢻⡄⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠸⣄⣄⣀⡀⣀⣠⡼⠁⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⠤⠖⠊⠉⠉⠉⠁⠀⠈⠉⠉⠐⠢⢤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣞⠁⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠳⠄⢠⠇
⠀⢀⣀⠀⠀⢀⡠⡤⢤⣀⡀⢀⡔⠋⣿⡏⠉⠋⢉⣹⣧⠀⠀⠀⠀⣴⢿⣛⠻⠉⠛⡛⣳⠯⣄
⣞⡉⠉⠉⠓⠀⠢⠉⠀⣩⠟⠹⡇⠈⣇⢀⠀⠀⡏⢠⣘⡇⠀⠀⢠⣇⡀⠘⡆⠀⡰⠀⣶⠀⢸
⠈⠉⠙⢢⡀⠀⠀⢸⠀⡇⠀⠀⠹⡄⣅⠇⠀⠀⢧⠸⠿⡇⠀⠀⠘⣿⠃⢠⠁⠀⠓⠄⡿⢀⡟
⠀⠀⠀⠀⢳⠀⠀⠘⡄⠣⡀⠀⠀⠈⣯⠀⣿⣦⣄⣱⠛⠉⠀⠈⠙⠻⣟⣁⣤⣶⠀⢺⠖⠋
⠀⠀⠀⣠⠞⠀⢀⣀⣨⠶⠞⠋⠉⠑⢿⡄⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⣽⣿⡿⠋⢀⡞
⠀⠀⠀⠣⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣹⢦⡀⠙⠛⠢⣄⣀⣀⣤⠞⡿⠿⢦⣠⣾⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡏⠈⠓⣤⡀⠀⠀⠀⢀⣼⡀⠀⣀⡭⠀⠋⠉⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢗⡀⡀⢸⡇⢉⣛⣒⣋⡥⠞⠉⣹⠋⠀⢀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⡏⠀⢸⣸⢠⡏⠉⠉⠁⠀⠀⠀⡇⠀⢀⠎⠀⢀⡔⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠈⠙⢶⡛⠁⠈⠳⠤⠤⠖⠒⢧⣄⣙⠒⠚⣄⣠⡞⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⡼⢃⢔⣊⠫⠭⠑⠒⢤⡀⢿⠳⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⣟⠙⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠁⠁⠀⠀⠀⠀⠀⠰⣵⣸⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⡄
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢸⡀⡟⠛⠓⠚⠛⠙⠲⣤⣀⠀⠀⠀⠀⢀⣠⠟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣐⣂⣀⣀⢀⣷⠘⡷⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠫⠥⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠔⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⠛⠛⠋⠛⠻⢿⣷⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⢀⡴⣏⠉⡹⡓⢦⣻⣿⡄⢀⣠⣴⣿⠟⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⡞⢠⡅⡘⢡⡘⠄⠻⢿⣿⣿⣿⣛⡛⠻⣶⡌⠛⠛⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢣⠘⢇⣉⣥⣶⡾⠿⠛⠛⠛⠉⠛⠛⠻⢿⣧⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢀⣤⡷⠟⠛⠉⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣄⣀⣤⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⢀⣠⡶⠟⢋⢻⣿⣿⡿⠿⠛⠁⠀⠉⠉⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⣀⣤⡾⠛⠁⠀⢰⣿⣧⢻⣧⣀⣀⡀⠀⣀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠻⠀⠀⠀⣠⣿⣿⣤⣤⠶⠞⠛⠉⣿⣿⡆⠀⠀⠈⣿⡿⠛⠛⠛⠛⢿⣷⣦⣠⣾⣿⠏
⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⣴⡿⠋⢻⣆⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⢿⣿⡇⢀⣴⡶⠿⠛⢿⣧⡀⠈⠻⢷⣄⠀⠀⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⠀⠠⣄⠀⠻⣷⣤⣀⣾⠟⠀⢸⣟⠻⣷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⢨⣿⣏⠀⠀⠀⠸⢿⡄⠈⠛⠿⠛⠀⠀⠸⣿⣄⡀⠈⠉⠉⠉⠻⣶⣶⣦⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣀⣀⣠⡀⠀⢀⣀⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣤⣶⣦⡀⠀⠀⠀⠀⠀⢸⡿⠛⣿⢻⣿⣾⣿⠟⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠈⢿⣄⣼⣿⣿⡿⣫⣾⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⡿⠟⠛⠉⠉⠁⠀⢀⣽⡿⢿⣶⣶⣤⣤⣿⠿⢿⣿⣿⣿⡏⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⣰⣿⡿⠿⠿⠿⢶⣤⡀⠀⣀⣴⠾⠟⠋⠀⢸⣿⠀⠀⠙⣷⣤⣤⣿⡧⠙⣿⣿⣶⣾⣿⠟⠉⠉⠉⠻⣿⣷⣤⡀
⠀⠀⣠⣶⣿⠟⠓⠀⠀⠀⠀⠈⠻⣿⣿⠁⠀⠀⠀⠀⣸⡏⢀⣶⠿⠻⢿⣅⠀⠀⢸⣏⠙⣿⣿⠇⠀⠀⠀⠀⠀⢸⡿⣿⣷
⠀⣰⣿⡟⢀⣴⠀⠀⡀⠀⠀⠈⢻⣿⣧⣀⠀⠀⢀⣰⡟⠀⣿⡏⠀⠀⠀⢻⡆⠀⠘⣿⣆⣸⣿⡆⠀⠀⠀⠀⠀⣸⠃⣿⣿
⠀⣿⣿⠀⢾⠟⢀⣾⡇⠀⠀⠀⠀⢿⡟⠛⠿⠟⠛⠉⠀⠀⠹⣷⣄⣀⣤⡿⠁⠀⠀⠈⠻⣿⠟⠀⠀⠀⠀⠀⢠⠏⣸⣿⡏
⠀⣿⣿⡀⠀⠀⠈⠋⠀⠀⣠⡖⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢀⢟⣾⣿⡟
⠀⠘⣿⣷⣄⠀⠀⠀⠀⢰⡿⣿⡿⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⠀⠀⠀⠀⣠⢏⣾⣿⠏
⠀⠀⠈⠻⢿⣷⣶⣤⣤⣶⡾⠟⠀⠀⠀⣾⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⣿⣷⣦⣴⣾⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣹⣿⢏⡇⠀⠀⠀⠀⢸⡇⠀⣿⣿⣶⣤⣤⣤⣤⣤⣤⣶⣾⡿⠟⠋⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⢠⣿⡟⢸⠁⠀⠀⠀⠀⠸⠁⠀⣿⣿⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⠇⣿⠀⠀⠀⠀⠀⠀⢠⣼⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣇⣿⡀⠀⠀⠀⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠉⢉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⣷⣄⣄⡀⠈⠳⣄⠀⠀⠀⠀⠀⡴⢲⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣠⣯⡴⠒⠚⠛⠷⣄⠘⡆⠀⢮⠉⢾⡁⢾⣁⠜⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⡀⢸⣀⠀⡸⣷⢧⠀⠀⡇⠀⠀⠀⠉⠒⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⡧⡀⢸⠋⠉⠉⢃⣵⡇⢸⠃⣸⢌⠀⠀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠑⢿⣾⣦⣤⣴⣿⡿⡟⢁⡴⠁⠀⢉⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⢿⠛⠷⠂⡛⢠⡧⠊⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣑⣦⡄⠠⡏⢧⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢞⣭⠋⠁⠀⡈⢻⡆⠀⡇⡈⣧⣸⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⡸⢻⣟⠉⢻⣄⢈⡏⠁⠙⢦⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⡬⠞⠓⢯⡿⠀⠀⠉⠉⠀⠀⠀⠀⣯⡀⠉⠒⠗⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡀⠀⠀⠀⠀⡟⠉⠳⡄⠀⠈⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣰⠃⠀⠀⠀⢠⠇⠀⢀⣼⣀⣀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣞⡀⣀⠞⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣸⠀⠀⠀⠀⢸⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠉⠓⢄⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀⡀⠀⠀⠀⠙⠢⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢌⣑⠂⠤⣀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⣿⣷⣦⡭⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⡴⠚⠉⠉⠙⠲⣄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⡞⢀⣀⣀⠀⠀⠀⠸⡆⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⡴⠞⠛⠉⠉⠉⠉⠛⠛⠶⣧⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡞⢡⣾⠿⣿⡟⠛⠛⠛⣲⣶⣿⣶⡝⢷⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣴⠞⣡⠖⠲⡅⠀⠀⢀⡯⠶⢌⠙⣷⣄⢹⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣿⠀⡇⠀⣴⣿⣀⠀⡞⠀⣠⣼⡇⠀⠙⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣽⡞⢀⡷⠋⠉⠉⠉⠙⠳⡄⣿⣿⡇⠀⢀⣠⠞⣇⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣿⣷⣾⡁⠀⠀⠀⠀⠀⠀⢹⠞⠋⣀⣴⣜⡇⢰⠏⣉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠋⠓⢣⡾⠁
⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠉⢿⣿⣿⢿⣿⣿⣿⡿⣿⠿⠁⠀⠀⠀⠸⠀⢀⡼
⠀⠀⠀⠀⣼⠃⠈⠻⣦⣀⣀⣤⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢿⡭⠭⠭⢽⠁⠀⠀⠀⢀⡴⢤⡴⠶⠛⠁
⠀⠀⠀⠀⢹⣆⠀⠀⠈⢹⠏⢡⣽⡷⠶⠶⠒⠚⢛⡿⠟⣷⣄⠈⠣⣤⠴⠃⠀⣀⣤⢶⣿⣶⡚⠁
⠀⢀⣤⠞⠉⠹⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡴⠋⠀⣼⠃⠈⠛⠶⠶⢶⡶⠛⠉⠀⣼⠁⠈⠙⠳⣤⡀
⢠⣿⡵⠞⣧⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣠⡿⢤⡀⢰⣇⣀⣀⣀⣀⣀⡞⢁⣀⠀⠀⡟⠀⠀⠀⠀⠈⠙⢦
⣼⢋⡴⠚⢹⡄⠀⠀⠀⠀⣦⣀⣉⡿⢩⡏⠀⢇⡇⡿⠁⠀⠀⠉⠉⣿⡴⠋⠉⠳⣄⡇⢤⡀⠀⠀⠀⠀⠈
⠙⢿⣠⡞⠁⠀⢀⣀⣴⠞⠉⠉⠉⠀⢸⣷⡤⢞⡼⠃⠀⠀⠀⠀⠀⣿⢧⠀⠘⡠⣻⢷⣄⣿⠶⠀⠀⠀⣰
⠀⠀⠉⠛⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢦⣍⣭⠴⠋⠀⠻⢻⡆⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡟⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣾⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠛⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠐⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢡⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠊⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣥⣀⡀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣀⣀⣄⣤⣬⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠈⠙⢷⣦⣤⣴⣯⣤⢶⠟⠉⠉⠀⠀⠉⠉⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠶⠦⠴⠶⠶⠶⠶⠶⠚⠛⣷⠤⣄⣀⣀⣀⠀⣀⣀⣀⣠⣼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠖⠛⠛⠙⠛⠛⠷⠦⣤⣉⣉⣉⣉⣉⣉⣠⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⣀⠤⠤⣀⡀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢠⡞⠉⣶⠂⠈⠙⢦⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⡎⠀⠀⣿⢤⣤⣄⢸⡇⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⢸⣦⡤⠿⠶⠶⣥⣼⡃⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⡤⢊⣡⣤⣴⣶⣶⣦⣤⣤⣉⠲⡄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢈⣴⣿⣭⣤⣄⠀⠀⢀⣤⣬⣽⡳⣮⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣴⠋⠾⠟⠋⠉⠙⠁⠀⠛⠋⡉⠛⠻⡎⢻⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡼⠋⠀⠀⠠⠈⡉⠳⠀⠀⡜⠂⢀⠁⠀⠀⠈⢻⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠲⢧⠆⠀⠁⠈⣴⢾⡇⠀⣿⣶⡀⠀⠀⠀⢰⡄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠛⢀⠀⠀⠀⢻⣿⣇⣸⣿⣿⠡⠀⠀⠐⠀⠆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠻⣦⡀⠈⣀⠛⠉⠉⠈⠙⠻⣠⠁⠀⣠⡤⠆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣤⡇⠀⠀⠀⠀⠀⠀⠘⣀⣼⣿⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣖⣒⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⠉⠤⠁⠀⠉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⣀⠀⣀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠋⢻⡏⠙⠛⠛⠛⠉⢹⡟⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢈⠀⠀⣷⠀⠀⠀⠀⠀⣼⠀⠀⣼⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣾⠀⠀⢸⡄⠀⠀⠀⢀⡇⠀⠀⣿⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢠⡀⠀⡏⠀⠀⢸⡇⠀⠀⠀⣸⠃⠀⠀⢿⠀⢠⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣸⠃⣸⠏⠁⠀⠀⣿⠷⠶⢶⣿⢴⠉⠰⢸⠀⢸⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣿⠶⠛⠲⠤⡤⠞⠃⠀⠀⠈⠳⢮⣀⡠⠜⠷⣾⣆⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡃⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡟⣦⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠉⢈⡉⠓⢦⡀⣼⠋⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⠀⠹⣆⠀⠀⢀⣀⣼⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢧⣦⣀⡀⠀⠳⡄⣻⠁⠀⠀⣿⡄⠈⠙⠓⠶⠦⢤⣤⣤⡤⠴⠶⠚⠋⠁⢰⡟⠀⠀⠸⣦⡶⠋⡁⠐⠀⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⢡⡄⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⡔⠁⠀⢀⡀⣴⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠀⡁⢰⡇⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡿⠀⠠⡀⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠟⠛⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣧⠀⠇⠀⠀⢀⢀⣼⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⠛⠛⠛⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠲⣤⡀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠶⠶⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠞⣿⠀⠀⣿⠒⠂⣀⡴⠊⠁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠀⠀⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢀⣀⡠⠄⠀⣰⠏⠀⠀⢿⣀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠶⠶⠛⠛⠿⠒⠛⠋⠀⠀⠀⠈⠛⠶⠶⠟⠛⠶⣤⣭⣥⣥⣤⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⣹⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢁⣾⣿⣶⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢤⣄⠀⠤⣀⠀⢀⡤⠛⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠈⢹⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⠓⠦⣀⠙⠒⢬⠙⠉⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢧⡈⠳⣤⣉⠳⠄⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⡸⢒⣫⣽⣷⠀⠀⠀⠀⠙⣿⣿⡿⠟⠻⣿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠉⢻⣦⣄⣀⠀⣠⢊⣴⣿⡿⠛⢿⣦⡀⠀⢠⣾⣿⢁⡖⢁⣼⡏⠀⠀⢀⣾⣯⣍⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣣⣾⣿⡏⠀⠀⠀⠙⣷⣄⣻⣿⢇⣾⣴⣿⣿⣇⠀⢠⡿⣡⣴⣿⠇⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡄⠀⠀⠀⠈⣿⣿⣷⡾⠋⠁⠀⠈⢻⣿⣿⢡⣿⣿⡿⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⣸⣿⣿⣧⣀⠀⠀⠀⣰⣿⣿⣿⣿⢛⣁⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⢻⡟⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⡿⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠸⣷⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⢀⣾⣿⣹⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠙⣷⡄⠸⣿⣿⣿⣿⡿⠟⠉⠁⠀⣠⣴⣿⣿⣿⣾⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣿⣿⣿⣧⠈⣍⣉⣉⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⡿⠿⣧⢹⣿⣿⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠟⠋⠀⠀⢹⡎⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠹⣦⡹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡄⠀⠀⠀⠀⠀⣽⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠀⠀⠀⣺⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⠀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡿⠟⠿⢿⠛⠛⢁⡀⠻⣿⣿⣿⣿⣷⣄⡀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠟⠛⠛⠉⠁⠐⣤⣾⠿⣟⠿⣟⠥⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⠔⣲⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠋⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⡼⠁⠀⢻⣯⠼⠓⡛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢠⡄⠀⢠⡄⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⣸⠁⠀⣴⡟⣳⡄⠼⣛⢦⠂⠀⢠⡾⡙⡄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠁⠀⠑⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣛
⠀⠀⠀⡇⠀⢸⠃⠉⠰⡷⣠⠋⠀⠉⡄⢻⡗⠀⠈⠒⢊⠗⠀⠀⠈⣇⢀⡄⣀⣀⣠⡴⠋⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢸⡁⢀⣀⠀⠀⠟⠛⠛⠲⣤⡾⠳⣸⠋⠁⢑⡴⠃⠀⢀⡠⠤⣀⠀⠀⠀⠀⠁⠀⠀⠸⡀⠀⡎⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠑
⠀⠀⠀⠙⡏⠘⢱⡀⠈⠉⠙⢭⡽⢠⠛⠓⠶⢶⡾⠋⠉⠒⣇⢀⣴⢾⢷⡀⠀⠀⠀⠀⢏⡙⢧⠠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆
⠀⠀⠀⠀⠈⠒⠺⣍⣖⣶⠂⠒⠺⣙⠄⠀⠀⢸⠁⢀⠶⡛⢻⡏⢦⣸⣆⣧⠤⠀⠀⠀⡀⣳⠀⠀⠀⢹⣀⠄⢄⠀⠀⠀⠀⠀⠀⠀⡔⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠉⢲⡄⢸⠀⠀⠀⠑⠾⢁⠴⠋⠁⠀⠀⠀⠀⢀⠀⠉⠢⡀⠀⠀⣁⠤⠜⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⣄⣴⠏⠀⢀⢧⡄⠀⠀⡴⠃⠀⠀⠀⠀⠀⠐⠷⠘⠇⠀⠀⠈⢦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⡄⠐⠀⣰⠛⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⠁⠀⠀⠈⣲⣵⢃⠤⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠰⡉⣇⠀⢣⠀⠀⢸⡆⠀⠀⠀⡟⠃⠈⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⢜⠳⡈⢆⠘⡦⣀⡤⠧⡀⠀⠀⢳⠆⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠆
⠀⠀⠀⠀⠈⠓⢌⠂⠀⠁⢸⠀⡠⠧⠖⠲⠊⠉⠢⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⡀⠀⠛⠀⣀⣀⣀⠠⠠⡌⠐⠢⠄⠀⠹⡲⠦⠤⣄⣀⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⠆⠙⢝⢦⡀⠀⢄⠀⠀⢀⠈⠀⠀⠄⡑⠢⠐⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢁⡽⠉⠉⠙⠪⢀⢀⣀⣀⠤⠜⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⡀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣀⣤⡶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢠⡾⠋⠁⣴⣶⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠁⠀⣼⣿⣟⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠿⠿⠿⠿⠿⠿⣿⣷⣿⣤⣀⡻⠿⣿⣿⡷⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣍⣁⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠉⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⣨⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⣼⠟⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⢘⣿⢫⣾⣷⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⣠⣶⠟⠛⠉⠉⠉⠉⠛⠻⢷⣾⣷⣿⣿⠃⠀⠀⠀⠀⠀⣀⠀⠈⠳⣦⡀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢸⡟
⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠙⠶⣄⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⠿⣷⣄
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⣿⣇⠀⠀⣸⠏⠀⠈⢻⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠋⠀⠀⠀⠀⣠⣶⠶⣶⣄⠀⠀⢀⣼⣿⡄⠀⠃⠀⠀⠀⠀⢻
⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠙⠿⠿⠟⠉⢻⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠹⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⠀⠀⠀⠀⣼
⠀⢰⣿⠉⠛⠿⢶⣶⣶⣤⣴⣶⣶⣆⣠⣤⣤⣤⣴⡶⠶⠶⢶⡄⠀⠀⠀⠻⢦⣄⣀⣀⣀⣠⣴⠾⠛⠉⠉⠙⣿⡄⢀⣼⡿
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⣀⡤⠄⠀⠀⢸⣿⡿⠋
⠀⠀⠘⢿⣦⣄⣀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠈⠉⠉⠉⣿⡄⠙⢷⣦⣄⣀⠀⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⣯⠙⠟⠻⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠘⢷⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠈⠻⢷⣶⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠙⠞⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⢠⣀⢀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠙⠷⢶⠶⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣤⣤⣤⣶⡶⠶⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠰⢆⠀⡠⠦⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢠⢹⠾⣝⠿⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢦⡌⢀⣈⣀⣹⡄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡠⠤⠘⠂⠀⠀⠉⠉⠁⠀⠈⠓⠠⠤⣀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣠⠋⠀⠀⢠⣤⣶⣶⡊⠉⠉⠉⠉⠉⢉⣿⣿⣶⣦⣄⡀⠀⠉⡄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠃⢀⡠⢾⡿⠛⠉⠛⠷⡀⠀⠀⠀⢠⠟⠋⠉⠙⠻⡌⠙⢷⡄⢰⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠀⠶⡟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⢻⠀⡆⠀⠀⠀⠀⢸⣿⣿⣇⡀⠀⠀⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠃⠀⠀⠀⠀⠈⣿⠛⠉⠁⠀⠈⠉⠻⣿⠃⠀⠀⠀⠀⢈⣀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢻⡴⠊⢀⣶⣄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⣠⣶⡄⠀⠉⣼⠃⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠸⡇⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠺⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⡇⠀⠀⠙⢻⣿⣿⣷⣄⡀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠋⠀⠀⠀⡿⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠛⣻⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠘⠉⠙⠲⣦⡀⠀⠀⠀⠀⠊⠁⢀⡴⠚⠋⢹⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡇⠀⠀⠀⣉⠙⠶⠦⠤⠤⠶⠞⢻⠀⠀⠀⠀⡆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⣤⢀⣀⣀⣀⣀⣀⣀⣨⡄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⡌⠁⠈⠂⡌⠀⠈⠉⠉⠉⠉⠀⠈⠀⠎⠀⠉⢿⣄⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⢎⣃⣀⣀⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⣎⠿⣆⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣴⠏⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⣀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢀⣨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠙⠛⠛⠛⠋⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠓⠛⠛⠉⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⣾⢉⠛⠒⠶⠶⠶⠶⠚⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠗⠶⠶⠶⠶⠶⠖⠛⣷⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠠⠅⠀⢰⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⢸⠀⠈⡆⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡴⠟⣆⣀⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠈⠁⠀⠀⠀⠀⣼⠀⣇⣀⣀⠾⢦⣀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⠶⠋⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢤⣄⣀⣀⣀⠀⠀⠀⠀⣠⡷⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠙⠳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠛⢯⣔⣂⣀⡤⣀⣠⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⣤⣤⣤⣽⠿⠋

⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠹⣏
⠀⠀⠀⠈⠸⠄⠀⠀⠀⠂⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⣀⣤⣀⡀⢀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠈
⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣾⣿⣿⣿⣿⣶⣤⣾⣿⣿⣷⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠁⠈⠀⠀⠀⠀⢿⠀⠀⠀⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢤⢁⢀⣀⠀⠈⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐
⠀⠀⠠⣄⠀⠠⡄⠀⣼⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥
⠀⠀⠀⢫⣆⡸⣌⠀⠈⠁⠀⠢⠀⠀⢘⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⠙⢿⡟
⠀⣠⠀⣨⣝⠃⢹⣧⡀⠀⠀⡀⠀⠀⢬⣙⣻⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⡡⠀⠀⠀⠀⠀⡒⠀⠙⠀⠀⠈⠉⣅⣈⠃
⠀⢻⣿⣿⣿⣷⣄⢁⢻⡈⠉⣿⡆⠉⠀⠝⠻⠿⠿⠿⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣬⣿⣧⠄
⠀⠀⠉⢻⣿⣿⣿⣷⣿⣀⠀⠈⢣⣄⠀⠀⠀⠈⣠⢦⡄⠀⠒⣺⣿⣿⣿⣿⣿⣿⣿⣿⡙⠀⠀⠃⠀⠀⠀⠀⢨⣥⠈⣴⣿⠿
⠀⠀⠀⢬⣿⣿⣿⣬⣿⢯⣷⡀⠠⢿⣷⣲⠀⣦⣥⠞⣠⣶⣾⣟⣥⣿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⠞⢀⣀⠁⢠⣤⣿⠀⠴
⣶⣦⠀⠺⣿⣿⡿⢿⣿⣾⣿⣿⣧⣀⢀⣑⣶⣾⣧⣿⣿⣿⣽⡟⠿⡆⣤⣶⣿⣿⣿⣿⣿⣷⣤⠠⣶⠋⣠⡾⢛⣥⠀⠉⠉⡀
⠉⢁⠀⠘⣻⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⢻⣿⣟⣙⣿⣿⣿⣿⣾⣷⣿⡾⣿⣿⣿⣿⣿⣿⣯⡶⣴⣿⣏⣴⣿⣵⣶⣿⠈⣋
⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⡿⡿⠛⠟⠫⣿⢿⠿⢿⡿⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠷
⠀⠀⣘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣶⣤⣾⣟⣾⣷⣯⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣯⣯⣁⣀
⠀⠀⠈⠓⠢⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠘
⠀⠀⣀⡀⠀⢭⣡⣾⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣶
⠒⠶⠶⢶⣄⣨⠍⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣼⡿
⠀⠀⠀⠻⢿⣌⣀⡠⣀⣠⣶⣿⣿⣭⣟⣩⠗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣴⡆⠛⠁
⠀⠀⠘⢃⣀⠈⠹⣶⣉⠉⠹⠋⠁⠈⠛⠁⢀⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢟⠻⣿⢿⣤⡰⢒⣾⣿⣿⠂
⠀⠀⠀⠈⠙⠋⠉⠉⠉⠖⠋⠉⠁⠀⠀⠀⣬⣽⣿⣿⣿⠛⣟⠛⠸⣿⣿⣿⣿⣷⣮⣟⠡⣬⠉⠉⠁⡈⢳⣀⡀⠈⠋⠙⠛⠀⣧
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠿⠀⠙⠙⢯⣍⡙⠎⠉⢀⣼⣙⣳⠀⠈⠙⠉⢻⣤⠀⢀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠈⠃⡀⠂⠀⠀⠙⠛⣿⠃⢠⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⣶⡄⠀⠊⢻⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠈⡵⠾⠟⠋
⠀⠀⠓⠶⠶⠦⠄⠀⠙⠛⠶⠾⠿⣶⣦⣽⣿⡿⡷⣶⣤⠸⠶⠄⠀⢀⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⢛⡙⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣠⣤⣤⣤⣴⣿⣿⠇⢠⡿⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⢉⠉⠛⢷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⠄⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡀⠀⠙⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡀⠀⠀⢻⣶⠛⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⢧⡀⣾⠃⠀⠀⠀⠀⢻⣿⣿⣿⠋⠉⠻⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣇⣸⡷⠶⠖⠀⠀⢀⣿⣿⡗⢊⣱⡄⠀⢀⡡⢤⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⠦⣴⠾⠉⠉⠀⠀⠀⣠⡶⠿⢷⣿⣤⣿⡿⠀⢠⠋⣠⣤⣿⠀⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠈⠙⢿⡇⠀⡇⢠⣿⣻⡿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡶⠆⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⣿⡿⠁⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣤⣴⣿⣤⡀⠀⠀⠀⠀⣠⣴⣀⣀⣉⣭⡄⠀⢸⣿⣿⣿⣿⢋⣡⡄⠀⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⠛⠋⢱⡿⠃⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣾⣿⡄⢻⣟⣋⣩⣤⣿⠁⠀⠀⠀⠀⢀⣼⡿⠶⠶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣿⡏⠀⠀⠀⠀⣰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⠉⠛⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣸⣏⣉⠉⠁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠟⣭⣿⣿⣿⣿⣿⡿⠛⠁⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠟⠋⠉⢠⡴⠄⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⣾⣻⣿⣿⣿⡿⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⣧⢠⡟⠛⠛⠻⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣠⣀⠉⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣶⢫⡾⠀⠀⠀⢰⣏⡶⠛⠳⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⢋⣾⠃⠀⠀⠀⣿⣿⠀⠀⣸⣏⣿⠻⣿⡿⣿⣿⠉⠉⠉⠉⠉⠉⠉⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠋⠁⠀⠀⠀⠀⢹⣏⠳⠶⠾⣹⡟⠀⠹⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠠⣶⣤⡀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣤⣤⣤⣤⣭⣛⠲⠟⠋⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣿⣥⣀⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣴⡾⠛⠉⠁⠀⠀⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⣀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⣴⣿⣴⠾⠿⠿⠷⣶⣤⡙⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⣀⣤⣾⣏
⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣷⡜⢿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣾⣿⣦⣶⣶⠶⠿⠛⠉⠀⠹⣧⡀
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⢻⣿⣿⣿⣿⡆⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠐⠀⢻⣆⢻⣿⣿⣿⣿⣄⣼⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠙⠿⣦⣄⠀⣀⣈⣠⣤⣤⣿⣆⢻⣿⣿⣿⣿⠋⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣝⠛⠛⠉⠉⠈⠹⣧⣿⣿⣿⠏⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⣀⣠⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⠿⠷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢋⣤⣤⣤⣄⠀⠀⠉⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⣴⠛⠁⠀⠈⠙⠻⣆⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⠶⠶⠶⠿⣷⣾⣄⣀⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⣤⣄⣀⠉⠉⠻⠷⣦⣄⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣶⣄⣈⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⣴⣿⣿⣶⡶⣶⣿⣿⣷⣦⣤⣽⠿⣦⡀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⣰⡿⠋⠈⠛⠀⠀⠀⠼⠟⠛⠿⣿⣷⡈⠻⣦⡀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠿⣠⠖⢶⡄⠀⠀⠀⣠⡴⠶⣤⠈⢻⣷⣤⣈⣿⣄⠀⠀⠀⠀⠀⠿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⣆⣠⣾⠀⠀⣴⠏⡀⢤⡨⣷⠀⠀⠉⠛⠛⢻⣧⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠿⠿⣧⣼⣯⡽⠀⠰⡇⠀⢱⡾⠿⣿⠀⠀⠀⠀⠀⣼⠿⣷⡀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠙⠻⣧⠀⠈⢃⡜⢸⣿⣾⡟⠀⠀⠀⢀⣼⠏⠀⣹⣷⣶⣶⣄⡀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠱⣶⠹⠋⠀⢀⣴⠾⠛⠁⣠⡾⠋⠁⠀⠈⠹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢿⣤⣠⣾⠏⢀⣶⠖⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⠀⠀⠀⠀⢀⣠⣴⣿⣦⣤⣤⣶⣿⣿⡆⠀⠀⠉⠉⠀⠀⣿⠟⠒⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠆⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣄⣀⣀⣤⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣏⠀⢰⣆⣘⣛⣉⣿⡿⠁⠀⠀⠀⠀⠀⣰⡟⢛⣿⠛⠉
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢿⡏⠉⢉⣼⠃⠀⠀⠀⠀⠀⢰⡟⠀⠈⢻⣆
⣼⡟⠛⢷⡄⢀⣴⣶⣄⠀⢀⣀⠀⠀⠀⠈⢿⡄⠀⠙⠶⠞⠃⠀⠀⠀⠀⠀⢰⣿⠁⠠⣀⣀⣽⡇
⣿⠁⠀⠀⢿⡿⠁⠈⢻⣶⠟⠛⢷⣤⣶⣦⡈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⠾⠟⠋⠉
⣿⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⣯⠀⠈⣿⠀⠈⠛⢻⡿⠁⠀⠀⠀⢀⣠⣿⣀
⠸⣷⠀⠀⠘⢿⣄⣀⣾⣿⡀⠀⣸⡏⠀⠀⣸⣷⡶⣾⠟⠻⠿⠿⠛⣿⡿⠛⠉⢹⣿⣦⡀
⠀⣿⠇⠀⠀⠀⠙⠛⠁⠘⠷⡶⠿⣷⣀⣠⡿⢡⣶⡇⠀⠀⠀⣠⠞⠋⠀⠀⣰⡟⠁⠈⢻⣄
⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⣴⣿⣿⡏⠀⣠⡾⠋⠀⠀⠀⣴⠟⠀⠀⠀⠀⠹⣧⡀
⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣾⣟⣛⣁⢀⣼⠏⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⢀⣿⠷⣄⠀⠀⠀⠀⢠⣴⢿⣤⠀⣸⡟⠛⠛⠛⣿⠋⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠘⣿⣄⣀⡀⠀⠀⢰⡟⠀⢠⡟⢠⣿⠃⠀⠀⣴⡏⡴⠛⠛⢳⡄⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⣼⡏⠙⠛⠿⠿⣿⠀⠀⢀⣿⣸⡇⠀⠀⠀⣸⣿⠀⠀⠀⢾⡁⢸⣿⣆⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⣾⡇⠀⠀⠀⠀⢸⣷⠶⢛⣼⡟⠀⠀⠀⠀⣿⡟⣦⣀⣀⣠⠃⣼⡏⠻⣦⢹⣧⡀⠀⠀⠀⠀⠀⣀⡴⠶⠈⠛⠛⠳⣦
⠀⠀⠸⣷⡀⠀⠀⠀⣸⡿⠶⠿⠋⠀⠀⠀⠀⠀⠘⢷⣤⣉⣉⣡⣾⠏⠀⠀⠙⠿⢿⣷⣄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠻⠷⣦⣤⣀⡀
⠀⠀⠀⠙⢷⣦⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⠻⣦⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠟⠛⠻⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠈⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠉⠉⠀⠀⠀⠀⠊⡉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⠄⢁⠤⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠈⠻⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⢁⠐⠊⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠈⢻⡄⠉⠉⠉⠛⠷⠦⣤⣤⠾⣧⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠙⠷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣀⣴⠟⠛⠋⠉⠉⠻⣦⣀⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⡀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⣇⠀⠀⠀⠀⠀⣀⣠⡶⠶⠶⠚⠛⠛⠻⠷⣦⣀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠘⣿⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⠠⠤⠴⠆⣿⡟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠻⣯⣴⡶⠶⣶⡶⢶⣦⣬⣽⡿⠿⠿⠷⠶⠶⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣁⡴⠤⠀⣀⣀⣀⣀⡀⠀⠤⢤⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣯⣭⣭⣭⣿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⢸⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⡀⠈⠀⣇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀⡼⣇⠀⠀⠀⠀⠀⠁⣸⡾⢿⢠⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠸⢱⡟⣤⡀⠀⠀⠀⠐⢿⡇⠛⣾⡷⢦⣤⠠⢿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄⡡⢾⡷⢹⣇⠀⠀⠀⠀⢸⠁⠀⠉⣗⣲⠀⢸⠶⣧⡈⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⡋⠀⠀⠀⠀⢾⣤⠀⢰⣿⡿⠓⢨⠠⣿⠁⡀⢠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⣶⣶⣶⣶⣶⡆⠀⠀⣀⠁⣿⠀⠀⠀⠀⣼⠁⠀⠀⢿⠛⠀⠀⡴⠋⠀⢹⣾⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⢻⣿⣿⡇⠀⠀⣿⠀⣸⣧⠀⠀⠀⢸⠠⠀⠀⣿⣔⠂⠀⠀⠀⠀⢸⠿⠛⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⣿⣷⣜⣿⣿⡇⣴⢰⡾⣻⣿⣿⡃⢀⡇⢸⡀⠀⢸⠛⡭⠭⢁⣀⠀⠀⠦⠀⠢⡈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⣾⣷⣿⣿⡿⠋⢀⣾⣻⢷⠟⠉⢀⡙⣧⣾⣇⠀⠈⠀⠀⡄⢠⢀⠀⠈⠙⡇⠀⢱⡘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⣄⣿⡿⠿⠿⠲⠶⠁⠠⠀⢺⣀⣷⣬⣷⡾⠻⠦⡠⢾⡀⡀⠃⠘⡈⠀⠀⢀⡇⢀⢞⡁⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⢀⠀⠀⠀⠀⠚⢃⣴⣿⠩⣿⣧⣷⣖⣾⢁⢘⣞⡷⠚⢀⣳⣿⣦⡸⣷⠃⢸⡗⠊⢻⡀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠀⠀⠀⠀⠠⠾⣿⡟⠛⣿⣿⡿⣿⡁⠀⣿⠾⣿⠟⢈⣤⣽⣿⣿⠁⡟⢠⣿⣥⣤⡉⠆⠀⠟⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠂⠀⠀⠀⠈⠀⢰⣿⣿⠐⠿⢛⣴⡏⠀⠘⢷⣿⣿⣿⣿⣿⣉⣵⠟⣧⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⡸⠂⠀⠀⣿⣿⡦⠤⣜⠿⢧⠀⡀⠈⣿⣿⠁⢹⣿⢉⣁⣾⠋⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣶⣀⠀⠀⠀⠀⠈⠁⢀⣼⣿⡇⠀⣄⢀⣠⡍⢀⣿⣄⢠⠛⢄⣈⡉⠛⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁⣄⠀⠀⠀⢀⠾⠿⠛⢷⣷⣃⡚⣋⡴⠋⢹⡇⠀⠀⠀⣽⠗⡼⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠀⠀⣠⡾⠀⠀⠀⠐⠿⠉⠉⢿⠁⠀⣸⠁⠀⡀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠉⠁⠀⠀⠀⠀⠀⢀⢀⠟⠀⠀⠃⠀⣰⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠘⡇⠰⢆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⢇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐
⠀⠠
⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣤⣤⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢋⣤⣤⣤⣤⣤⣾⣿⡿⠻⣿⣟⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣶⡟⠿⠿⠟⢛⣿⣿⣿⠁⠀⢻⣿⣿⡟⠛⠛⠛⠛⠃
⠀⠀⠀⠰⠟⢻⣿⣁⣈⣻⣿⣿⣀⣀⣀⣸⣿⡟⢁⣀⣀⣈⣿⣿⣿⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⢀⣀⡴⠖⠿⡄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠁⠀⠀⣽⣿⡇⠀⠀⠈⣿⠋⠛⠋⠋⠙⠋⠉⠀⠀⠀⠈⠉⠙⣷⣴⠛⠁⢀⣤⡴⠇
⠀⠀⠀⠀⣸⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣿⡇⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⢴⣏⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠙⠻⣿⠛⠉⠉⠛⠻⠿⠁⠀⠀⠀⣾⠀⠀⠀⢸⣿⠀⡿⠀⠀⠀⣿⣾⣷⣶⡿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠷⣄⠀⠘⠛⣶⠏⢉⡿
⠀⠀⠀⠀⠀⠀⠿⠷⣄⠀⠀⠀⠀⠀⣀⣴⣾⣿⡆⠶⣶⣾⡿⠞⠁⠀⠀⢀⣿⣧⠾⠋⠀⠀⠀⢀⡀⠀⠀⠀⣿⡇⠀⠀⠈⠻⠦⣴⡟⠀⠘⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⢉⣽⡿⣇⡀⠀⠀⠀⠀⠀⠀⣠⣤⡉⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡜⠁⢠⡀⠀⢷⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣾⠿⠟⠁⠈⠻⠷⣶⣶⣶⣶⡿⠿⠋⠙⠷⢶⣶⣾⠿⠛⠿⣷⣶⣶⣶⣿⡇⠀⠀⠀⠀⣤⣁⢀⡿⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢉⠀⢈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⠶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣤⠜⣛⠛⠛⠀⠀⠈⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⢸⡗⠺⠭⣛⡦⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠚⠛⠛⠻⣦⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⠙⠷⣄⡀
⠀⠀⠀⠀⢸⡏⢰⣟⡫⠔⠹⢛⣤⡀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡾⠁⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣠⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠘⣿⡿⣽⢃⡾⣃⣻⢈⣛⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⣧⡠⠔⠓⠄⠀⠈⠉⠀⠀⠀⠤⠀⠉⠚⠚⠳⠆⠀⡀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠘⣷⡿⣛⣷⣍⣹⣤⠸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⣘⠛⣷⠄⢰⡶⠃⠐⠂⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠙⠠⣀
⠀⠀⠀⠀⠀⣸⡿⠿⢿⣛⣿⣿⣿⠻⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢈⣿⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⢠⠄⠀⠓⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⣾⣷⡿⠻⣿⡿⠁⠈⢻⡏⢹⣆⠀⠈⠄⠀⠀⠀⣀⡀⠀⢀⠀⠀⣼⡟⠀⠀⣀⠈⠀⠀⠀⠀⠀⣼⠟⡿⢷⡆⠀⠀⠀⠀⠐⡂⣤⡀⠀⢂⢀⣼
⠀⠀⠀⠀⢿⣿⠀⢠⣼⣇⠁⠀⣼⢿⡇⠘⢦⡄⠴⠂⠀⠚⠉⠀⠀⠉⠀⢺⣿⠃⢀⣾⣿⣦⣀⠠⠀⠒⠚⠛⠺⠧⣾⠁⠀⠀⠀⠀⠀⠇⠸⠟⣤⠴⢿⡁
⠀⠀⠀⠀⠀⢿⣄⡀⠙⠻⠓⠾⠋⠀⠀⢀⣾⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣤⡘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⢻
⠀⠀⠀⠀⣇⠀⠉⠁⢠⡀⠀⢀⣴⣾⢿⣿⠟⢸⣧⣀⣀⣀⣀⣠⣤⣴⡞⠁⣾⣿⣷⣌⠛⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣾⣿⡇⠀⣾⠏⢰⠀⣸
⠀⠀⠀⠀⠀⠐⠒⠋⠀⣀⠐⠀⠀⢠⣄⣩⡖⢻⢿⣙⣿⣻⣿⣿⣿⣿⣷⠄⠛⠛⠛⢿⣿⣶⣤⣙⠛⠿⣶⣦⣤⣤⣤⣤⣴⣿⣿⣿⡿⠁⢸⡟⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣾⠙⣿⢖⡋⠅⠀⠃⢠⡟⠉⠁⠈⠙⢿⣿⠃⠀⠀⠠⠄⠀⠿⠛⠛⠛⠻⢷⣾⣥⣬⣝⣛⣉⣉⠉⠉⠉⣀⡄⠈⣿⣶⠋⠁
⠀⠀⠀⢀⠀⠀⢄⠀⠀⠸⠄⠁⢀⠞⠳⠛⣷⠂⣾⠁⠀⠀⠀⣠⠘⠟⣠⣤⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠇⢀⡿⠃
⠀⠀⠀⢼⠀⠀⠀⠋⠀⠀⠀⠀⠀⠐⠃⢼⣇⣼⣿⠀⠀⠀⠨⠉⠿⢀⡇⠘⠀⠀⠀⠀⠀⠈⢦⣄⣀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠛⠁⠒⠿⣇
⠀⠀⠀⠈⢣⡄⣨⡇⢀⡷⠀⠦⠀⠀⢀⠀⠉⣿⠏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡀⠃⠀⠀⠈⠹⡏⠁⠐⠒⠒⠒⠀⣤⣤⠆⠀⠁⡀⠀⢸⡄
⠀⠀⠀⠀⣆⠙⠋⢀⣾⠁⠀⣼⡆⠈⣍⣱⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠀⠀⠀⠀⢸⣦⣤⠀⠒⠒⠀⣿⣿⣧⠀⠀⠀⠀⢸
⠀⠀⣴⠀⠻⠋⠀⠀⠉⠀⠀⠉⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⡧⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⣰⠛
⠀⠀⠛⠤⠊⠀⠠⡀⢷⡞⠃⠀⢠⡤⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡘⠀⡂⢀⣠⣼⢿⣿⣧⡀⠀⠀⣠⠟⠱⣦
⠀⠀⠀⠀⠈⠐⡖⠛⣡⠍⠙⠉⣡⣤⣉⣳⡀⠀⠀⠐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡎⠀⣠⡞⠃⠀⠙⣄⠀⠟⠛⢻⣿⠿⣶⣾⠻⣦⠀⠈⠓⡂
⠀⠀⠀⠀⠀⢩⣽⣿⣿⣶⢠⣾⠿⢿⣿⣿⠛⠲⢦⡀⠸⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠁⣞⠉⢀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⢙⣿⠛⢧⠀⠘⡇
⠀⠀⠀⢸⠀⢸⡿⠁⢀⣈⣿⣡⠀⠀⠞⠻⢦⡤⠼⣷⣄⠙⢿⠟⠉⠀⠀⠀⠀⠀⠈⠿⠃⣰⠉⠉⡿⠀⠀⢸⠀⠁⠠⡎⠁⠀⠀⢠⡿⠋⠀⠈⣶⠚⡇
⠀⠀⠀⠈⢤⣚⠃⠀⠈⠻⢿⠋⠀⠀⠆⠀⣏⠀⠄⣸⣿⠓⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠟⠀⠀⠀⠀⠞⠀⠀⣤⣏⠀⢀⠷⠊⠀⠀⠀⠐⠄⢘⡇
⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢻⡆⠀⠙⠷⠒⠏⠈⠉⠀⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⣠⠐⠂⠀⠀⣠⠾⡋⠉⢁⣀⠐⢀⠀⡀⠀⠀⠘⡇
⠀⠀⠀⣠⡆⠀⣶⢀⣄⠀⠄⠀⢠⡀⣶⣾⣄⠙⣄⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠺⠦⣤⡴⣾⣇⡴⢿⠞⠋⠀⢩⣤⡀⠙⢷⣤⣼⠃
⠀⠀⣮⠀⢀⣤⡹⢿⣋⠁⠀⢀⣤⣤⡀⢙⣿⣿⣉⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣤⣄⣵⡟⠈⢱⡏⣴⣦⠐⠛⠛⠃⣀⡄⣸⠇⡀
⠀⠀⠁⠲⣿⠛⠁⣀⡉⠁⠙⠿⠛⠻⣷⡿⠛⣿⠋⠓⠲⣦⣠⡄⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠉⠉⠉⠛⠿⣿⣿⠀⠀⡳⣯⣯⠶⢶⡶⣶⣟⣃⡸⠗⠉⢲
⠀⠀⠀⠀⠈⠁⠐⢧⠀⢀⠐⠁⠀⠶⣏⠶⠀⢸⠱⠎⠁⢴⢀⡟⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠀⠈⠉⠁⠀⠈⢿⡄⣞⠁⠘⡷⣤⣤⣤⣌⡿⢿⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢥⠀⠠⠀⠐⠀⠀⠙⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠁⠂⣀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡏⢹⣦⠙⠀⠀⠠⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆⣀⠂⠀⠀⠀⠀⡠⠐⠂⡴⠂⢹⣄⠀⠀⠀⠀⢀⣽⣇⠀⠀⠀⠐⠂⠀⠀⠀⠀⣸⡇⠺⠀⠁⠔⣶⣦⠄⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠷⠀⠀⠀⢾⠁⢀⡀⠁⠀⠀⠙⠲⠶⠶⣾⣿⣿⣿⣦⣄⣀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠈⠁⠈⠀⠁⠀⠀⠴
⠀⠀⠀⠀⠀⣀⠀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠋⠉⠀⠤⠀⠀⠤⠄⠀⠀⠤⠄
⠀⠀⠀⠀⠐⣿⣦⡿⢸⡏⣿⣼⠆⠈⠉⣻⠀⣰⣿⣏⡀⠘⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠐⠒⠒⠒
⠀⠀⠀⠀⠐⠟⠈⠳⠘⠷⠷⠾⠂⠒⠚⠛⠃⠀⠘⠃⠀⠘⠁⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣖⠦⠜⠁⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠓⠠⢄⠀⠀⠈⢢⡀⠀⠀⠀⠀⢀⢦⠀⢱⠀⠀⠀⠰⠄⢀⣀⣀⣤⢎⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠒⢶⡃⠘⠒⢥⢢⣄⣀⠵⠒⠛⠛⠛⠛⠷⣏⡐⡅⠀⠀⠀⠀⠀⢀⣼⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠲⠂⣼⡀⠀⠀⠀⠉⡄⠴⢀⢂⡞⠁⠀⠠⠂⠀⠀⢀⠀⠀⠙⣾⠦⢤⡄⠀⠀⢏⢁⡠⠒⠒⠜⢀
⠀⠀⠀⠀⠀⠀⠀⢮⡀⠀⠀⠀⠐⢏⣋⡉⡛⠋⠀⠀⢀⡎⠀⠀⠀⢁⡀⠀⠀⣠⠀⠀⠀⢸⠄⡀⠀⠀⡀⠈⠋⠀⠀⠰⠋⢩⣳⠤⠒⠚⠛⠒⢦
⠀⠀⠀⠀⠀⠀⠀⢮⣠⠤⢤⣤⣠⠤⠓⠀⠀⠁⢀⠀⠠⢇⡀⠀⠁⢀⣀⣀⣀⣀⣀⣉⣢⣯⠊⠀⣀⡀⠀⠀⢀⠷⠒⡣⠔⠋⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠗⠈⢢⠀⠀⠀⠀⠀⠁⡎⢝⠉⢙⣯⠁⠀⠀⣦⠀⠀⢩⢻⠏⢻⡄⢄⡀⠀⢘⣆⡸⠀⠄⠶⠶⠒⠆⠀⠀⠀⠀⢱⡀
⠀⠀⣀⣀⠀⠀⠀⠀⠚⡂⠀⠀⢀⡀⣸⣠⢦⡤⠂⠉⠀⠙⢼⠃⠄⠁⠀⠀⠀⠁⠀⠀⢸⡏⠀⢀⣼⣀⠽⡄⠀⢫⡀⠑⠀⠀⠀⢀⡄⢀⠀⣁⣀⠼⠁
⠀⢸⠀⠘⣄⡈⠀⠨⡯⠁⠀⢀⠥⠒⠘⡳⠊⠁⠉⠑⢄⠀⣈⠧⣀⠘⡀⠀⠀⠀⠀⣠⠞⠛⣲⠞⠁⠀⣀⣘⡀⠈⢗⠀⠠⠀⢠⠈⠀⠛⢞⠁
⠀⠘⡄⠀⠀⠁⠀⠀⠑⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣟⠁⠀⠈⢙⠦⢤⡀⠴⠋⠙⣦⢸⠀⠀⡔⠉⠐⠁⠀⠀⠠⠆⠀⠢⡀⣄⡀⠀⡸
⠀⠀⠈⢢⠀⠀⢱⠔⠶⡀⠑⠲⢄⠀⠀⠀⠀⣰⠟⠁⠀⠀⢹⣤⠂⡘⠀⠐⠀⠀⠈⣆⣿⠋⠀⡘⠀⠀⠀⡀⠀⠀⠀⠀⠋⡆⠈⠈⠉⠉
⠀⠀⠀⠀⠑⠤⠊⠀⠀⠙⢦⡀⠀⠉⠒⢲⠏⠁⠀⠀⠄⡄⠀⢹⡇⠘⠆⠀⠀⠀⠀⠀⠈⢧⠀⡇⠀⠀⡐⠁⠂⠀⡴⣠⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠦⠤⡏⠀⣄⠣⣄⣀⠇⢀⡄⡝⠀⠀⢠⡤⠶⡁⠀⠀⠈⢧⡇⠀⠀⡾⡉⠀⠀⢸⢀⠆⠀⢻⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢸⢌⢪⡱⠖⠓⠖⠋⠀⣇⠀⢀⣾⠀⠀⠹⡀⠀⠀⠈⢳⡄⠀⢠⠙⠢⠞⠁⠁⠀⠀⠀⠉⣨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣁⡜⠀⠀⠀⠇⠀⠀⠀⠀⠹⡆⠈⢃⠀⠀⠂⠊⠓⢄⠀⠀⢄⡈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠒⠢⠤⣀⣹⠤⠂⠀⠀⠀⠀⠀⠀⠉⡚⣆⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋
⠀⠀⠀⢆⠄⠄⠐⡆⠢⠆⠠⠠⡰⢂⠔⣂⠐⠀⠄⠦⠀⣀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⠀⠠⠀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣉⡁⠈⠉⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣼⡟⡱⢋⣧⠍⠑⣆⠀⠀⠹⣿⣆
⠀⠀⠀⣀⣠⣿⠋⠉⢻⡟⠉⠙⣿⣤⣤⣶⣶⣶⣿⣱⢃⣾⣏⡀⠀⢸⠀⠀⠀⠘⣿⡆
⠀⠀⣾⠏⠉⢿⠀⠀⢸⠇⠀⠠⣿⣯⣥⣤⣀⡀⠀⠈⠉⠛⠿⣏⡠⠋⠀⠀⠀⠀⠹⣿⡀
⢠⣶⠿⢆⠀⠈⠃⠀⠀⠀⣀⡼⠙⠿⣿⣿⣿⣿⠟⠲⢤⣀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢿⣇
⢸⣧⡀⠀⠑⠀⠀⠀⠀⠈⠀⠀⣀⣼⣿⣬⣿⣿⣿⠀⠀⠈⡙⠦⣄⠀⠻⣄⠀⠀⠀⢸⣿⣦⡀
⠀⠙⠻⢷⣦⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠉⠙⠏⠀⠀⣾⡇⠀⠈⠛⠦⢽⣆⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠙⢷⣦⡤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢠⠖⠦⠔⢻⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⠼⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⡼⠓⠒⠶⣄⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠘⣯⡙⠛⠒⠒⠚⠛⠛⠛⠋⢹⡆⠀⠈⠉⣡⠤⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⠙⢻⣟⠻⣦⣄⣀⣀⣴⠟⠀⠀⠀⡀⠣⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢻⣆⠳⣀⡏⠁⠀⠀⢀⣴⠟⠻⣶⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⣧⡀⠀⠀⠀⢠⣿⣥⣴⣦⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡴⣿⠟⠓⡶⢖⣿⣯⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢏⡞⠁⢀⡞⠀⣼⢯⠀⠉⠛⠿⣶⣤⣀⣀⣠⣄⣀⣀⣠⣴⠾⢻⠷⢶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢳⠈⠉⣛⠯⣀⢰⡇⠈⠳⣄⠀⠀⠀⠉⢙⡿⠋⠙⠛⢉⣭⠀⠀⡁⠤⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠏⠀⣼⠁⠀⠈⣿⣧⠀⠀⢹⣷⣤⡀⠀⡼⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠠⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠙⢦⣤⠴⠃⠘⠷⣤⣠⣿⣿⠻⢿⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠻⢿⣦⣀⠀⠀⠲⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠈⠛⣷⡄⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡿⣶⡄⠀⠀⠀⠀⠀⠈⠻⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠻⠷⢶⣶⣶⣶⠾⠟⠋⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠒⠛⠀⣸⡷⠤⠶⠋⠀⠀⢀⡏⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡴⢣⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⣛⣛⣭⣿⣓⠲⠖⣒⣫⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣤⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣶⣶⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠈⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡬⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠈⢹⣇⠀⠀⠀⠀⠀⠈⠙⠻⣶⡶⠞⠻⣿⣦⡀⣠⠄⠀⠀⠀⠀⠐⠻⣿⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡷⠀⡽⠋⠉⠙⠉⠉⠉⠉⠉⠙⠛⠒⠶⡖⠚⠻⡟⢛⠛⠒⠂⠠⢄⡀⠀⠈⠿⠛⠉
⠀⠀⠀⢰⠀⣀⣀⢠⠀⠀⠀⢰⣶⣟⠛⠻⢿⣿⠿⣿⣯⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁
⠀⠀⢰⢎⢰⣿⣿⣼⣷⡶⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒
⠀⠀⢸⣼⣿⣿⣿⣿⡿⣁⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑
⠀⠀⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡄⠀⠠⢴⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡀⠀⠀⣧⣀⡴⠸⠶⣶⣶⣤⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⠀⠀⠉⠉⠐⠂⢀⣩⢿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣶⣶⣤⣾⣬⣦⣽⣿⣿⣿⣿⣶⣆
⠀⠀⠘⠋⢹⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠐⠂⠀⠈⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠙⠛⠟⠻⣿⣿⣿⣿⣿⣿⠻⣿⣿⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠁⠀⠀⠉⠓⣶⣥⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣀⣀
⠀⠀⢰⣮⣼⠏⢠⣄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⠛⡃⠀⢻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠘⠓⠐

⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⡀
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠤⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠄⠀⠀⠀⠀⠐⡾⠹⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⣶⠀⠀⣼⠰⠄⠀⠀⠀⠀⠀⠀⢤⠞⠙⣢⠄⠀⠀⠀⠀⠀⠀⢀⡴⣳⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸⡀⠀⢉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠧⠦⠇⠀⠀⠀⡰⢳⠀⢔⡋⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣧⠀⠀⢃⠀⠀⠀⡼⠁⠉⠢⣀⡤⠢⢄⡀⠀⢘⣆⣰⠇⣯⣤⣄⡒⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠈⢣⡀⠈⣆⣀⡠⠧⠀⠤⣤⡙⠀⠀⠀⠱⡄⠸⡯⣽⢸⠧⠄⣮⣱⠋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡿⡀⢈⡷⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⡚⠒⠛⠚⠧⠾⣬⡀⣸⡿⢉⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⣆⠀⠘⡼⠋⠀⠀⡴⠊⡩⡍⠉⠢⣄⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⠉⠳⢮⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠷⠀⠉⠻⣄⢰⠀⠀⠀⠰⢡⢪⠶⣩⡍⠣⠘⡆⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣀⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡿⠶⠀⠀⠚⠾⠇⣄⡀⡇⣼⢀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡤⠖⠊⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⡴⠂⣉⡀⠳⣤⣴⡒⢤⢀⠀⠀⠈⠙⢤⡞⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣀⠔⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⢀⠎⠁⠀⡰⣡⠔⣭⣾⠀⠉⠢⣄⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠘⡆⠉⠀⢀⡠⠤⠄⣀⣀⡀⠀⠀⣀⣀⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⠀⠰⠱⠡⢪⣾⠃⠀⢠⣾⣤⡻⢦⣄⠀⠈⢦⠀⠀⠀⠀⠀⢃⠀⣠⠎⡀⠀⠀⠀⠀⣠⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡇⡇⢠⠘⡟⠀⢀⠞⡡⢋⣽⡆⠘⠟⠓⠤⡆⠀⠀⠀⠀⢸⠺⢅⣀⣭⠖⠂⠐⢻⣥⠤⠄⠚⢷⠶⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠴⠒⠚⠂⠿⣄⠀⡘⠀⡇⠻⠿⡇⠀⠀⠀⣠⠃⣠⠤⠤⢄⡼⠴⠶⣠⡈⠉⠉⠉⠁⠀⠀⠀⠀⠨⡆⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠱⡶⠄⠈⠶⠞⠁⠀⠀⡜⠁⢰⠁⢀⠀⠀⠸⣄⠀⠘⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⢷⠦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⡤⠶⠚⠦⠀⠀⡟⢠⠃⠰⠋⠀⠀⡰⠈⢣⠀⢠⣇⣠⣤⣤⣄⠀⠀⠀⠀⢸⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⣄⡀⠀⠀⢀⡠⠞⠀⠀⠀⠀⢀⠞⠀⠈⠶⠃⠀⠀⠀⣀⡞⠁⣀⡠⠶⠚⠛⠒⠤⡈⠹⣧⡀⠀⠀⢸⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠔⠁⠈⢫⢹⣿⣅⣠⣖⡤⠤⠜⠉⠀⠀⠀⢀⡴⠶⡖⠒⠊⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠫⡢⢀⡾⠊⠁
⠀⠀⠀⠀⠀⠀⡡⠀⠀⠑⢲⡄⠀⠀⠀⠣⣙⠛⠙⠛⠀⠀⠀⠀⣀⣠⠔⠉⠀⠀⠘⢦⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢞⡇
⠀⠀⠀⠀⠀⣀⢠⠰⠋⠀⠉⠀⠀⠀⠀⠀⢹⡇⠘⠓⠀⡖⠚⠉⢸⠃⠀⠀⠀⠀⠀⠀⠈⠓⠶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠉⠀⠲⠄⠈⠀⢀⣤⠄⠀⠀⠀⠈⢿⣦⠀⢀⠃⠀⠀⢰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⣀⢄⡀⢀⡰⠃⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⡠⢇⠀⠀⠀⠀⠈⠳⡝⠚⠀⠀⠀⠘⢇⠀⠀⢑⣢⢄⡀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠿⠤⢉⠉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠘⠃⠤⠓⠀⠀⠠⡀⠀⠈⠓⢤⣀⠀⠀⠀⠉⠉⣀⣠⠴⠛⠲⢄⡀⠀⡰⠁⢠⠞⠀⠀⠀⠀⠈⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠑⠂⠀⠀⠀⠀⠘⠦⡀⠀⠀⠈⠁⠘⠛⠛⠈⠉⠀⠀⠀⠀⠀⠉⠲⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣇⠀⠀⠀⠀⠀⠲⣖⡠⠄⣀⣬⡰
⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⣀⣀⣤⠤⠤⠴⠒⠚⠉⠁⠈⠑⠒⠀⢤⡀⠀⡸⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠉⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⣀⣀⣀⠀⠀⠀⠉⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠿⢿⣿⣿⣦⡀⠀⠀⣼⡿⢃⡴⠛⣉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠉⠻⣿⣶⣄
⠀⠀⠀⢀⣴⣿⣿⠿⠛⠛⢁⣴⡶⠿⠿⢿⣿⣶⣼⣿⠃⡞⠀⣴⣿⣆⣴⣿⠀⢹⡀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⢠⣿⣿⠋⠀⠀⠀⢠⣿⣧⣴⣶⠶⠶⠿⠿⠿⣿⣼⣅⣾⡟⢹⠿⢻⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⣿⣿⡇⠀⠀⠀⠀⣼⣟⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣜⣋⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⣿⣿⡇⠀⠤⣤⣤⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠋⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢰⣿⡇⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠈⢿⣿⡋⠙⠉⠁⣠⣴⣶⡶⠶⣿⣷⣿⣿⡇⠀⠀⠀⠀⢠⣶⡏⠙⠳⢶⣤⣄⣀⣀⣈⣻⣆⠀⠀⠀⠀⠀⢸⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⣾⡿⠋⠀⠀⠀⠀⠀⠉⠛⢆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠉⣿⠙⣷⡀⠀⠀⠀⠸⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⣠⡶⠟⠁⠀⠀⢀⣾⠟⠛⠛⠶⣦⡀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠞⢻⣷⠀⠀⠻⣦⠀⠀⠀⣠⠟⠀⠀⢀⡀⠀⠈⠹⣦⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⣀⡀⠀⠀⣀⣠⡴⠛⠛⠛⠉⠉⠀⠀⢠⡿⠀⠀⠀⠻⢶⠴⠞⠁⠀⣠⣾⠉⠁⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣀⣤⡀⠀⠀⢀⡀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣍⠙⣿⡷⢾⠟⡻⠛⠛⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣦⣤⣤⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡈⢷⣌⠻⣆⡀⠀⢠⡟⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣶⣾⣿⣿⣟⡋⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⢻⣿⣦⣽⢷⣬⣉⣛⣋⣀⣀⣀⣤⣴⣾⣿⣿⠉⠁⠀⠀⠉⠉⠙⠻⠿⣿⣦⣄⡀
⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⢿⣿⣷⣄⠀⣀⣠⣾⣿⠟⣡⡾⠋⠉⠛⣻⠟⠉⠉⠁⣼⠁⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣶⣶⣤⡀
⠀⣠⣴⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⠃⠐⠻⠶⠶⠤⣾⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⢷⣦⡄⢀⣤⡶⠟⠋⠉⠉⠈⠙⢿⣿⡄
⢰⣿⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⣼⠀⠀⠀⠀⣴⠞⠛⠛⠳⣆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣶⡄
⢸⣿⣿⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⡟⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡆⠀⠘⣧⡀⠀⠀⠀⠀⠀⢸⣿⣧⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢿⣿
⠘⣿⣿⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⠛⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣸⠇⠀⠀⠈⠛⠶⠦⢤⠤⠶⢛⡿⠁⠀⠀⠀⠀⠀⣶⡄⠀⢻⣷⡄⠈⣿
⠀⠹⣿⣿⡄⠸⣆⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠸⣿⠆⠀⠉⠁⠀⣿
⠀⠀⠙⢿⣿⣦⠹⣆⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⣷⣀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⢻⣿⣇⠘⣆⠀⠀⠀⠀⠀⢀⣽⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠁⠀⠀⠈⠛⣿⡟⠻⠃⠀⠀⠀⠀⠀⣠⣼⣿⠟
⠀⠀⠀⠀⠈⣿⣿⣦⣈⣷⣦⣤⣤⣶⣿⡿⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠈⣇⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠛⠛⠛⠋⠉⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠸⡄⢿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣇⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⢀⣀⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⢀⣤⣤⣤⣬⡙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⣆⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⠆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣠⣤⣾⠿⠿⠿⠟⠛⠿⠿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣴⠾⠛⠋⢁⣀⣠⠤⠤⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⢁⣤⣴⠟⠋⢉⣁⡀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⣠⡾⢛⣷⣶⣾⣿⡿⠿⠿⢷⣿⢿⣿⣆⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡏⠀⠀⠀⠀⠀⣠⡾⠋⢠⣿⠟⠋⠉⠙⠃⠀⠀⠀⠁⣀⡈⢻⣤⣤⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠙⠀⠀⠀⠀⢀⣴⣿⣤⡾⠿⠋⢠⡶⠋⠳⢆⠀⠀⠀⡞⠉⠑⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⣿⣤⡖⡄⠘⣇⠀⢠⡿⣶⠋⣠⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢀⡀⣀⣿⠉⢱⡄⠀⠀⠀⠀⣿⣉⣿⡄⠀⢸⠀⠈⣳⣿⠿⠛⠉⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣾⠟⠛⠻⢷⣄⠈⠻⣦⣄⡀⠀⠘⢿⡿⣀⢹⠆⠀⠰⠏⠁⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣀⣀⠙⢷⣄⠀⠉⣿⡆⠀⠀⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⣹⣷⡀⠙⠷⠿⠋⠀⢀⣶⣦⣀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠈⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡿⢷⡀⠀⠀⠀⠀⠙⠻⣿⡿⠿⠿⠁⣠⠘⠻⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⢿⡄⠀⠀⠀⠀⠀⢹⡇⠛⠚⢿⡟⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠻⢦⡤⠊⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⣴⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣶⡶⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣤⣄⣠⡾⠿⣦⣼⠋⠙⢿⣾⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠰⣶⠾⠟⠁⢰⡟⠁⢹⣿⠀⠀⢸⡇⠀⠀⢸⡏⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠛⠻⣷⡶⠶⠶⠶⠿⢷⣶⣤⣾⠁⠀⠸⣿⠀⠀⢸⣿⡀⢀⣼⠇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠙⢿⣄⠀⠈⠙⢦⡀⠀⠀⠀⣱⣌⠹⣧⣀⣠⡿⢷⣴⠟⠉⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢻⣄⠀⠀⠀⠙⣦⡀⠀⢿⣿⣷⡌⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⢻⣄⠀⡙⢛⣿⡄⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡿⠛⠛⠛⣿⡀⢠⡼⢷⣄⠀⠀⠀⠀⢠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠛⠛⠳⠄⣻⡆⠀⠀⢹⣧⢈⡁⠀⠸⣇⠀⠀⢀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣸⣿⡆⢸⠇⠀⠀⠀⣿⡁⠀⠀⠀⣿⣼⣇⠀⠀⣺⠟⠛⠛⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠋⠁⠰⠶⣄⡀⠀⠀⠀⠀⢀⣼⣃⣼⠋⢸⣇⠑⢤⣀⣠⠞⣻⡇⠀⠀⠀⠘⣿⣝⠳⢶⡿⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠞⠛⠃⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⣠⣿⡿⠛⠃⠀⠀⠹⣷⣄⣀⣡⣾⠟⠀⠀⠀⠀⠀⠈⠛⠛⠻⣧⡀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⣠⣤⡶⠟⠋⠙⠋⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠋⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣴⠿⠁
⠀⢀⣠⠶⠛⠁⠠⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣰⠟⢁⡀⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⣱⣾⠓⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠻⣇⠀⢿⡀⠀⠀⢀⣀⣠⣤⣤⠤⢴⡆⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠙⠶⠞⠛⠶⠚⠋⠉⠁⠀⠀⢀⡾⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣰⠟⠛⠛⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣀⣠⣤⡾⠋⠁⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠧⠶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣸⡿⠶⠶⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠾⢿⡆⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠶⢶⠾⠿⢿⣿⣉⣁⣴⢶⡦⣤⣶⣆⣹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠚⠛⠛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣠⠤⠀⠀⠀⠀⣀⠀⠀⠤⢤⣘⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢾⣿⣯⣭⣭⣿⠿⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⠈⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢐⢂⡀⠂⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡈⠑⠈⢂⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠉⠐⠂⠠⢀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠐⠁⢀⣀⣀⣄⠀⠀⠀⣀⣀⣀⠀⠀⠠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⣀⣴⠿⠻⢧⡀⠀⠀⢠⠾⠛⠿⡟⠦⣀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⡐⠚⠁⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠂⣠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡁⠀⠀⠀⠀⢺⣿⠇⠀⠀⠾⣿⡇⠀⠀⠀⠀⠀⡀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠔⢠⣄⠀⠸⠁⠀⠀⠀⠀⠘⠂⢀⣠⡄⠘⣠⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣻⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⣀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠻⠿⢷⣦⣤⣤⣴⣿⠿⠟⠀⠀⢀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣄⢀⠀⠈⠉⠁⠉⠉⠁⠀⠀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠉⠑⡆⣀⠀⠀⢀⣠⠖⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⠀⠀⠀⡀⠀⠉⠉⠁⢀⡀⠀⠀⠁⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣠⠊⠁⠂⠁⠉⠉⠉⠉⠀⢠⠀⠀⢢⡀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡴⠃⠤⠴⠂⠀⠀⠀⠀⠀⠀⠁⠢⠤⠁⠹⡄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⣀⡉⠉⢁⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠙⠉⠉⢁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠉⠉⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡝⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢣⠀⠀⠄⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠁⡠⢆⡀⡧⢨⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⡇⣎⠀⡰⢄⠈⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⣀⠴⠊⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠒⠂⠠⠤⠔⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠛⠳⣄⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠑⠢⠤⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⡀⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠁⠠⠂⠀⠁⠀⠀⠘⠐⢈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣴⣾⣿⣶⣄⠀⠀⠠⠀⠶⠖⠁⣦⡀⠠⠂⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⣤⡀⡀⠀⠈⠃⢀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣉⣻⣾⡛⣄⡀⢀⠀⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⣠⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣟⢻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡟⠠⡾⠉⠷⢻⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣀⠀⠸⠇⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢛⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⣾⣿⣿⡏⠉⢉⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡟⠀⣰⣿⡅⠀⠀⠞⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⢻⣿⠃⠀⠀⠀⣴⣟⣷⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣧⣤⣶⣶⣤⣤⣤⣴⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠶⠒⠤⣤⣄⠀⠀⠘⣿⣿⡟⠀⠀⠀⣠⣿⡿⠋⠉⠙⢻⣿⣿⣿⣷⣍⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠈⠹⣧⠀⠀⠈⠁⣀⠀⠀⢸⣿⣿⠃⣠⠶⠒⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⡟⠛⠀⠀⠀⠒⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣤⣤⣶⣿⣿⡇⠀⢰⣿⠇⠈⠉⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣦⣴⣾⣿⣿⣿⣿⠿⠟⢁⣐⣮⣫⣄⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⣷⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠈⢿⣿⠻⣿⣿⣿⣽⣿⣿⣋⣐⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⣀⣠⢤⡄⢻⣭⣿⣇⠀⠻⣷⣦⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⢾⣿⡿⠾⠓⠺⣿⣿⣏⣲⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠛⢃⠀⣴⣦⢾⣏⣻⣷⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠙⠛⢻⠋
⠀⠀⠀⠀⠀⠀⢼⢸⣿⣗⣀⡟⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⢹⡿⠭⠍⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣮⣿⣿⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠸⠇⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠀⠀⠀⠓⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢦⣤⣤⣄⣤⡆⣦⣤⣤⣤⣤⣶⣴⣶⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⢾⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⣿⣿⣀⢀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣥⣤⣀⡉⢉⣭⣉⡉⢡⡤⠿⠿⣍⣉⣭⣥⡍⠁⠸⣏⣻⣿⣿⣯⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇⢾⣷⠆⢽⣿⣿⣷⣴⣶⣿⣿⣿⣉⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢹⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⡇⢼⣅⠺⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠄⠈⠻⢛⠉⠉⠉⠿⢿⠿⠟⢻⠏⢳⡾⢻⣿⣿⣿⠿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠛⠛⠉⠉⠉⠛⠛⠿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣴⠾⠛⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡿⠁⢠⣾⣧⠀⢀⣴⡄⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣾⡇⢠⣿⠁⢿⣶⠟⣿⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⣧⣤⣤⣿⣧⣿⡃⠀⠀⠁⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣠⣴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠶⣴⣿⣄⣀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⢠⣾⣏⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠻⣿⣍⡉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠉⠛⠷⣶⣤⣀⣀⠀⠀⠀⢀⣾⠋⠀⠉⠙⠛⠿⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢷⣿⢃⣾⣿⣆⠀⠀⠀⠀⠀⠉⠙⢻⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠈⣿⡛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠈⠙⠻⠷⣦⣤⣤⣤⣤⣶⠾⣿⠿⣦⡀⠀⠀⠀⠀⠘⠷⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠛⠉⠛⠛⠻⠿⣧⣄⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣸⡟⠈⠛⠛⠛⠿⠶⣶⣄⡀⢸⡟
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢸⣿⡁⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣤⡶⠀⠀⠉⢻⣾⠇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣀⣠⡶⠟⠋⢿⡆⠀⠀⠀⠙⠻⣷⣀⣴⡿⠃⠀⠀⢠⣾⣿⣥⣤⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣤⣶⠾⠛⠉⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⣿⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣶⢶⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠷⣦⣄⣀⣀⣀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣠⣄⣀⢀⣠⣾⣟⡻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣋⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠙⠛⣛⣿⠿⠛⠛⠿⠶⢶⣶⣶⣶⣶⡶⠶⣿⣿⣿⠋⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠹⣿⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣴⠟⠻⣦⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⣿⠀⠀⠀⠹⣷⡀⠀⣀⣤⣤⣤⣤⣤⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⠁⠀⠀⣿⠂⠀⠀⠀⣴⡿⠋⠉⠉⠙⠻⣦⡀⢹⣧⠀⠀⠀⣸⣿⠟⠋⠉⠀⠀⠀⠉⠉⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢸⣿⡇⠀⠀⣰⡟⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢹⣷⠀⢻⣆⢀⣾⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⡿⣧⣤⣾⠟⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⣇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣀⣴⠟⠁⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⢠⣿⣧⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⣤⣘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⣆⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⠉⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣤⣄⣿⡆⠀⠀⠀⠙⠿⠁⠀⠈⠻⠏⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣤⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⠛⠛⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠾⠛⠛⠻⠷⢶⣶⣶⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠟⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣉⣛⠛⠛⠛⠛⠛⢿⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠟⢿⣤⡙⠻⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠟⢋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣶⣦⣀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⣿⠿⠟⠻⣿⣿⣧⠀⢀⣶⣿⣿⣿⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠄⠁⣤⡀⠀⣿⣿⠋⠀⠸⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣷⠀⠙⠃⠀⠀⠀⠉⢻⣿⣿⣿⣟⣁⣼⡿⠂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀⠳⣤⣶⣿⡿⠁⠀⣠⣦⣠⡾⣿⣭⣛⢿⡿⠿⠿⢿⣿⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⡟⡛⢋⣤⣶⣯⣭⣥⣶⣿⣿⡿⣣⣾⣿⣿⣿⣷⣿⣆⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣖⠼⣿⣿⣜⡣⣶⣼⡿⣻⣿⣿⣯⣿⣿⠛⠺⢿⣿⣿⣿⣷⣆⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠓⡿⢿⣿⣿⣇⣸⣦⣼⣿⣽⣿⣿⣿⡻⠗⣴⠟⠻⠏⢹⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣙⡛⢿⣿⡿⠻⢿⣿⣿⣿⣿⣷⣄⡻⣤⣀⣤⣭⡄⣼⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣾⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠟⠁⣉⣸⣁⠟⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣻⣟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣀⣴⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⡋⣛⠉⠙⡃⢀⢀⠀⠀⣀⣀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⠿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⠿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠄⢠⢛⣄⣠⡛⡄⠠⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢀⢂⠋⠱⠏⢱⣘⣤⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡦⠽⠤⠤⠤⠤⠭⠴⢄⡀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡔⠃⠀⠀⣀⣠⡤⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⠀⠀⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡎⠀⢀⣴⣾⠿⠻⢿⣄⠀⠀⠀⠀⣴⠟⠛⠻⣯⠙⢦⣄⠈⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡴⠾⠉⠈⠀⠀⠀⡀⠌⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠙⠷⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢠⠞⠉⢻⠀⣆⠀⠀⢀⠀⣴⣶⣾⡀⠀⠀⢸⣶⣶⡆⠀⠀⠀⠀⠀⠂⠘⠉⠙⣆
⠀⠀⠀⠀⠀⠀⢺⠀⠀⢺⠀⠉⠀⠀⠀⠀⢹⡿⠛⠉⠀⠀⠉⠻⣿⠃⠀⠀⠀⠀⢈⡀⢰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠹⡴⠋⢀⣶⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡃⢀⣠⣶⡀⠈⢡⡞⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⡇⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⢰⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⠀⠉⢻⣿⣿⣷⣄⡀⠀⢀⣀⣴⣿⣿⣿⠟⠁⠀⠀⣸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠉⠉⠉⠿⠿⠿⠿⠿⠿⠏⠉⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⢶⣤⣀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣀⣤⡶⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⡀⠀⠈⠙⣦⣀⠀⠀⠀⠁⣀⣤⡞⠉⠀⢁⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇⠀⠀⠀⡇⠈⠉⠉⠉⠉⠁⢸⠃⠀⠀⠘⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢠⡠⠴⢤⠀⡗⠒⠲⠶⠶⠒⠒⠻⠃⠠⠠⢀⡇⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⢀⢹⣦⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢠⡿⠉⠒⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠁⠘⣧⠀⠀⠀⠀⠀⠈⣧
⠀⢀⣰⡏⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢘⣧⡀
⠀⢯⠀⠉⠑⠲⠶⠶⠖⢛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠒⠒⠶⠶⠚⠉⠀⡿
⠀⣾⠛⠲⠶⠶⠶⠶⠖⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⡷⠶⠤⠤⠤⠶⠶⠛⣧
⡸⠁⠀⠂⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢧⡀⠀⠀⢀⡀⢀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡄⠀⠀⣀⣀⠀⠀⠀⢀
⠈⢳⡀⠀⠒⠀⢨⡆⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⢠⠇⠀⢰⡅⠀⠇⠀⢀⡿
⠀⠀⠛⠦⠤⠴⠛⠳⠦⠼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠈⠁⠀⠀⠀⠀⢠⡇⠘⢦⡤⠞⠳⢦⣄⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⠤⠞⠋⠁⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠉⠉⠑⠒⠒⠒⠶⠶⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠻⢕⣂⠤⠀⣀⡀⢠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢖⣀⣀⣐⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣷⣼⢟⣩⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣀⣼⢿⣿⣿⣿⡿⡿⢗⣻⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡈⠉⠁⣉⣽⡿⠿⠟⠃⢬⣛⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢙⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⢀⣠⡤⠤⣌⡓⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢘⣿⣿⠁⢀⣀⣀⣀⡀⢀⣹⣧⠀⠁⠀⠀⢹⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣶⣦⣿⣿⣿⣿⡻⡌⠉⠀⣷⠀⠀⢠⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣩⣴⣤⣤⣿⠀⢀⡎⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣷⡧⣤⣄⠀⠹⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣠⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣤⡴⠞⠛⠻⣾⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠻⡆⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣋⡉⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣌⣵⡴⣎⢳⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⢛⠓⠲⠿⣤⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢀⣉⣱⣦⣴⣿⡷⢄⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⡿⠿⠀⠀⢫⢭⡙⠧⣄⣵⣄⠀⠘⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣧⣼⡆⠠⠁⢴⣷⠀⢀⡴⠛⣆⣀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠃⠀⠲⢼⣧⡄⢸⡀⡰⣉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⢀⣀⣴⣶⣿⣿⠟⠁⠈⠉⠀⠟⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡻⠟⠻⣟⣩⠿⠂⠀⠀⢀⡴⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠻⠦⣄⡀⠒⠂⣠⣴⠲⣟⠲⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⠐⠀⠀⠀⡠⠉⡿⠂⡉⢀⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢣⠀⡀⢠⠏⠀⠀⠙⠂⠚⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠙⢃⣘⣦⣀⠀⠀⠀⠀⠀⢀⡔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⠿⠃⠈⠀⠘⣿⣶⣦⡴⠖⠛⠶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠓⠲⢤⣄⣼⠃⢀⣠⡤⠖⠒⠚⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣻⠋⠙⠋⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⣀⣀⣀⣠⡤⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠤⠖⠒⠒⠚⢿⣉⣗⣒⡒⠒⢒⣚⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⢀⠀⣀⢀⣀⠀⢀⠀⣀⢀⣀⣀⣀⡀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⠀⢠⡀⢀⠄⡀⢀⠀⡀⠀⠀⠀⠀⠀⢠⠀⣄⠀⢀⣀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠱⠀⣀⠘⡄⠀⠉⠉⠑⠀⠀⠀⡀⢤⡈⢢⠀⢨⠃⠀⠀⡀⠀⡄⠉⢈⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣰⠟⠁⠀⠒⠠⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣶⠿⠛⠋⠉⠚⠓⠶⣤⣀⠇⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⡠⠄⠐⠒⠒⠠⠤⣀⠉⠓⢦⡀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠁⠀⠀⢀⠄⠒⠈⠁⠂⢈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⡀⣀⣀⣀⡌⠳⣦⡀⠙⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣈⣤⣤⣤⣤⣄⣈⡂⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡾⠀⠀⠀⠀⠀⡀⠷⠸⣆⠈⠆⠀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣠⠶⠛⠉⢉⣀⣀⠀⠀⠀⠉⠙⠳⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡂⠀⠀⠀⢈⠀⠀⠸⢧⡘⣀⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠟⠀⡴⠋⣁⡤⠖⣤⠏⠉⡥⠀⠀⠀⠉⠂⢄⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⣠⠚⣇⣼⡇⣀⡀⣤⡟⠀⠄⠀⠀⠛⠉⠑⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⠀⡜⡠⢊⠟⠀⠀⠀⠐⡄⠀⠀⠀⠘⢢⡇⠀⠣⠀⢹⡇⠀⠀⠀⠀⠀⠀⢹⡞⠁⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⢰⠁⢀⡇
⠀⠀⠀⠀⠀⣠⡤⣄⠀⢻⠞⠉⠛⠆⠀⢸⠀⠀⣶⡄⢠⢸⠀⠀⣀⠐⠄⢱⣄⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⢈⢷⣄⡀⠀⠀⡀⠖⠰⢶⡆⠀⣀⣀⣠⠞⠁
⠀⣀⣄⡀⢰⡏⠀⠘⣧⣿⠀⢳⡀⠀⣀⣸⠀⠀⢿⣿⠿⠓⠠⢤⣿⢀⡖⠌⣿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣈⣉⣽⣅⣀⣀⣠⠞⢡⡞⠹⣏⠁
⣾⠋⠈⠙⢾⡇⠀⠀⡏⠹⣄⠈⠉⡓⠉⠰⠦⡀⡔⠀⠀⠀⠀⠀⢹⠟⠐⣤⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠦⠝⠉⠀⢠⣯⡤⡤⠟
⠹⣆⠀⠀⠈⢧⠀⢀⡷⠰⣌⣿⠉⠀⠀⠀⣤⠀⠙⢄⠀⠀⠀⢀⡼⠄⢲⡏⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣲⠖⢀⣴⢿⢋⡟⠛⣳⢦⣀⣀⣀⡀
⠀⠈⠳⣦⡤⠤⣷⠏⠀⠀⠸⣏⠻⣍⣄⠀⠈⠢⡤⢄⠉⠉⠉⡉⠀⠜⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣰⠃⠀⣩⠁⢋⣬⠀⡎⠀⠀⠋⠀⢈⡗
⠀⠀⢘⡧⠤⣀⠸⣀⢀⣴⠃⡟⣿⠛⠒⠷⠦⣄⠐⠆⣛⣲⠞⣠⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢄⡟⢒⣴⠁⠀⢰⣏⠒⠓⠀⠊⢠⣞⠋
⠀⠀⠸⣆⠀⡀⢹⡏⠉⠀⠀⢁⡿⠀⠀⠀⠀⠘⣇⠒⠠⡔⠈⢣⠙⢟⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣇⣸⣿⠃⠜⠁⠀⣰⠋⡘⣇⡀⠀⢀⣠⣿
⠀⠀⠀⠘⠳⢬⣉⣠⡄⣀⣴⡟⠠⠤⣤⠀⠀⠀⢹⡀⠀⣸⡂⠉⢣⠘⠣⡈⢳⡀⣀⣔⠲⡛⠟⢦⡴⡟⠁⢸⠛⠔⠋⠀⠐⠊⠛⠉⣆⣹⣎⠋⠉⠉⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣿⠀⠀⣠⠿⣴⡏⠈⠇⠀⠀⢛⠄⡅⢠⠛⣇⠈⠓⠄⠀⠀⠇⠳⡀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣏⠳⣦⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠦⠞⠁⠀⢿⡓⢲⡇⠀⠀⠀⠓⣿⠃⠀⠘⠳⢤⣀⡀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢹⠆⠀⣀⠼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⣀⣀⡷⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠙⣧⠘⠲⢦⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⡼⠉⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠥⠄⠠⠤⣄⡀⠀⢿⡇⠀⠀⢀⣠⠤⠖⠚⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠋⠲⢤⣀⣸⠃⠀⠀⠀⣠⡴⠖⠚⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠉⠓⢾⣧⠤⠖⠋⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠈⢹⡗⠒⠒⠋⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇⠀⠀⠀⠀⠀⢻⡇⣀⡀⠀⠀⠀⢀⣀⣸⣧⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢭⣁⣀⣒⣒⣒⣂⣩⡭⠞⠈⠓⠲⠿⠷⠶⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠳⠶⢶⣶⡶⠶⠿⠛⠛⠛⠶⣬⣽⣗⣒⣒⣪⣭⡵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠾⠗⠻⠾⠋⠙⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⢠⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠃⠀⢀⠄⠀⠠⠀⠀⠀⢠⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠘⠃⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣶⣶⠶⠶⢦⣄⣀⠈⠛⢷⡄⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣌⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⢰⣿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⣿⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⣀⣀⣀⡀⠀⠠⠄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⡀⢈⠶⠀⠁⠀⠀⠠⣶⡿⠿⠛⠿⠇⠀⠀⠀⠀⠈⢿⣿⣷⣶⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⣼⠛⠀⠀⠀⢠⣄⠀⠀⠛⠁⠀⢰⣶⣄⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢠⡿⠁⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⢿⣯⣿⡆⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⣇⣀⠀⠀⠀⣀⠀⠀⢠⣿⡄⠀⠀⡘⢻⣿⡿⠀⠀⢀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠛⠓⠖⠖⠛⣷⡖⠈⠉⠁⠀⠀⠀⠀⠉⠁⢀⣴⣿⣦⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣶⣶⣾⣿⠏⢉⣡⡄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⣧⣀⠙⠻⠇⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠁⠈⠙⢷⣬⣽⡷⠶⣤⣤⣤⣄⡀⠀⠈⠁⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠤⠀⠀⠀⠙⣧⡀⠀⣠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⢏⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠐⠄⠀⠀⠀⠚⠋⡀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡀⠶⠄⢀⡀⠀⠀⠀⠈⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠊⠀⡀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠠⠈⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣄⠂⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣴⠶⠟⠉⠉⠉⠛⠳⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡏⠀⠉⠛⠓⠚⠛⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡿⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⠈⣧⡴⠟⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠷⠶⠶⠾⠟⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⡀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠐⢤⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠻⠛⠟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⡀⠘⢦⣈⣷⣼⣇⠀⠀⠀⢀⣠⣤⠶⠖⢛⣛⣛⡛⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⣦⣶⠐⠒⠰⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠹⣦⡾⠉⠀⠀⢹⣧⡴⠞⠋⠁⠀⢠⠎⣥⠶⢦⣠⡕⢄⠻⣆⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣧⡀⢀⡿⢧⣤⠴⠀⣸⠏⠀⠀⣀⡤⣴⡚⠒⠒⠒⠦⣄⡻⢸⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠙⠋⠁⠀⠀⣀⡴⠋⠀⣠⠾⠧⠞⠛⠿⡶⠶⠦⣄⡀⠙⢯⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠙⠂⠀⠀⠀⠀⠈⢉⡏⢹⡟⣷⠀⢀⠖⢉⢳⡀⠀⠀⠰⠟⠿⣷⢳⡀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⡤⠶⢿⠀⡟⢠⠞⠀⡞⠀⣱⢶⡇⠀⣰⡋⠍⢳⠹⣦⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠘⣧⠗⢣⣾⣄⢹⡼⠛⠉⠙⠳⣯⡟⠄⡸⠀⠀⣸⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣧⠀⠀⠀⠀⠀⡟⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠈⣯⠔⠁⢠⡞⢹⡦⣞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⠀⠀⠀⠀⢷⠀⠀⠙⢿⢿⣷⣤⣀⣀⣀⣴⣷⣶⣾⣧⡇⣼⠆⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣆⠀⠀⠀⠈⢧⡀⠀⢸⣦⣍⣉⠻⠿⢿⣿⣿⠟⠉⢀⡟⠁⠀⡼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⡦⠀⢀⡴⠋⢹⡄⠘⣏⠀⠉⢻⣿⠞⠀⠀⠀⣀⡾⠷⠶⠛⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣼⠃⣰⠋⠀⠀⠀⡿⢦⣀⡉⠉⢉⣀⣤⠴⠶⣞⠉⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡏⡿⣡⠖⠒⣄⢰⣧⣀⠈⠉⠉⠉⣹⣇⣀⠀⡏⠳⣄⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡃⢀⢄⡟⣼⢀⡼⠛⠛⠚⠋⠉⠉⠉⠉⠓⢦⣘⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣯⣼⠃⣼⠁⠀⡴⠀⢰⠀⢠⠀⠀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣴⡇⠀⣿⠀⢸⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣸⣦⣠⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⡀⠀⠀⠀⠀⢀⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠻⢿⣿⣦⡈⠻⣶⣷⢸⣿⠀⣿⣿⠀⢀⣠⡾⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡐⠙⢺⠀⣿⣿⣿⡄⠈⣿⣾⠿⠤⠙⠃⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠻⢿⣿⣿⣶⣴⣿⣿⣿⣿⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢠⣆⠀⠘⢛⠻⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠰⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠁⠀⠀⡿⠃⠀⣠⡟⠛⠻⣶⣦⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡆⢀⡿⠀⠀⢠⣿⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣾⣿⡿⠋⠁⣀⣶⣤⣶⣿⣿⣿⣿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⣷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠋⠁⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠛⠉⠀⢀⣀⡤⠤⢤⣈⠛⣿⣿⣄⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⡴⠋⣦⡀⣠⠳⡉⠳⣌⢿⣿⡆⢀⣤⣶⣿⣿⣟⠛⠻⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⣼⠃⢰⠀⠘⠃⣀⠙⡄⠸⠟⣿⣿⣿⣿⡿⠛⠛⠛⢿⣶⡈⠋⠛⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡀⠛⠒⣉⣧⣴⣶⣶⣿⠿⠿⠿⠿⠿⠿⣷⣶⣿⡗⠀⠀⠀⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣈⣽⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣶⣬⣿⣿⠆⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣴⣤⠶⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠉⢿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⣾⣿⡆⢿⣿⡁⠀⠀⠀⠀⢀⠀⠀⠀⣀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠁⠀⠀⠀⠀⣠⣾⡿⠃⠀⢀⣀⣤⣴⡾⠟⣿⣷⣄⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⣷⣶⣄⡉⠂⢋⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠙⠀⠀⠀⠀⣴⣿⣿⣿⠛⠛⠛⠛⠉⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⢘⡿⠋⠉⠀⠀⠀⠈⠙⢿⣿⣷⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠋⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⣤⣶⣾⠿⠿⣿⣷⡀⠀⠈⠻⣷⣦⡀⠀⠀⢀⡀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣼⣿⠋⠁⠀⠀⠀⠈⢿⣿⣄⠀⠀⣤⣿⣿⠀⠀⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠈⠛⢶⣄⠀⠙⢿⣷⣶⣾⡿⠀⠀⠸⣿⡆⠉⠛⠿⠿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡅⠀⠀⠀⠀⠛⠻⣇⠀⠀⠉⠛⠋⠀⠀⠀⠀⢻⣿⣄⣀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⠾⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⡀⢀⣠⡀⠀⠀⠀⡀⠀⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣶⣶⣾⣿⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠈⢻⣴⣿⣿⣿⡿⢃⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠿⠿⣿⣷⣤⣀⠀⠀⠀⠀⠻⣷⣴⣾⣿⣿⣿⢋⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣾⣿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣼⣿⡟⢿⣿⣷⣶⣶⣶⣾⣿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⣴⣶⣶⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠀⢸⣿⡆⠀⠀⠹⣷⣄⢀⣀⣿⣿⡌⠻⣿⣾⣷⣦⣼⣿⣿⠟⠋⠉⠛⠛⢿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣭⡀⠀⠀⠈⠉⠛⢿⣦⣤⣾⡿⠉⠁⠀⠀⠀⠀⢸⣿⠃⠀⣀⣤⣼⣿⣿⠛⠛⠉⠁⢠⣿⠛⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦
⠀⠀⢠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡙⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⡟⠀⣴⣿⠋⠉⠉⠻⣧⡀⠀⠀⢸⣿⡄⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿
⠀⢀⣿⣿⡏⠀⣠⣾⠀⠀⢀⠀⠀⠀⠀⠈⢿⣿⣿⣄⡀⠀⠀⢀⣠⣾⠟⠁⢸⣿⡇⠀⠀⠀⠀⢹⣷⠀⠀⠈⣿⣷⣄⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⣿⣿
⠀⣸⣿⣿⠀⢰⣿⠏⠀⣴⣿⠀⠀⠀⠀⠀⠈⣿⣿⠻⠿⡿⠿⠿⠛⠁⠀⠀⠘⣿⣷⡀⠀⢀⣠⣿⠇⠀⠀⠀⠈⠻⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⡟⢀⣿⣿⡿
⠀⣿⣿⣿⠀⠀⠁⠀⠘⠿⠋⠀⠀⠀⣠⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣾⣿⡿⠁
⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣠⣴⣿⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⢀⡾⣱⣿⣿⡟⠁
⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠸⠟⣻⣿⠟⠉⠛⠿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣇⠀⠀⠀⠀⠀⣠⡞⣰⣿⣿⡏
⠀⠀⠀⠈⠻⣿⣿⣷⣶⣤⣤⣤⣶⣾⠟⠁⠀⠀⠀⣰⡿⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣿⣿⣷⣶⣤⣴⣾⣿⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⢹⡏⠀⠀⠀⠀⠀⠀⣿⠁⠀⣻⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⡿⠟⠉⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⣾⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢀⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿⣿⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣤⣤⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⢛⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠶⠾⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢸⣙⣉⣉⠉⣟⣉⣉⣉⢉⡿⠀⠀⣀⠙⡍⢻⣷⠀⠀⠀⠆⢸⣿⡇⠰⠀⢰⡆⠠⠄⢀⡶⠀⠀⠀⡐⠒⠀⠀⠐⠀⠀⡛⢻⢻⣀⣛⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠟⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⣍⡁⠨⢄⠀⡄⣉⣀⣨⢉⣭⣭⣶⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⣿
⠀⠀⠀⠀⢀⣤⣤⣤⡆⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠳⣶⣦⠀⠀⠀⠀⠰⢺⣿⣿
⠀⠀⠀⠀⠸⡤⠾⢿⡿⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠿⢿⠟⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⢀⠓⣣⡮⣷⣤⠄⠀⠀⠀⠀⠈⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡿⢿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡊⣡⣴⣹⣴⡶⢶⠀⠀⠈
⠀⠀⠀⠀⠈⠈⣿⣾⣻⣯⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣄⠀⠀⠀⣘⠋⣡⡬⣷⣤⡄⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣻⣦
⠀⠀⠀⣠⣴⣾⣿⣿⣿⠟⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢹⣷⣿⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠿⠟
⠀⠀⠘⠿⣿⡿⢿⠿⠁⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⢠⣶⣿⣿⡟⠀⢀⡤⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠸⠿⠿⠷⠶⠿⠿⠿⠦⠀⠉⠀⠀⠀⠀⠈⠿⠿⠶⠶⠾⠿⠷⠄

⠀⠀⠀⢀⠴⠚⠛⠓⠦⣄⢀⣀⣀
⠀⠀⣰⠃⠀⡔⢧⡈⣱⠞⡉⠁⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀
⠀⢠⡇⠀⠀⠣⣀⡿⢡⢎⣠⣄⠀⢸⠀⠀⠀⠀⢀⣀⡀⣠⠋⣠⠎⣡⢿⡀⢀⡤⢄⡤⡀
⠀⢸⠀⠀⠀⢀⣾⣴⠿⠁⢈⠈⣳⡏⣀⣀⠀⠀⢺⡀⠙⠃⠀⠁⠘⠁⣰⠃⠉⡇⢸⠀⢹⠀⠀⠀⣠⠤⣄⡀⢀⣠⡤⣤⣀
⠀⢸⡀⠀⠀⣸⣹⠇⠊⢑⣄⢢⡽⠋⠉⠉⢳⡀⠀⢳⠀⠄⡀⠀⠀⠀⠛⠙⡦⠷⠜⠂⢸⡤⣄⢸⠁⠀⠀⡙⢻⡷⣌⡐⡌⠳⡄
⠀⡼⠇⠀⠀⣏⠁⠀⠑⢌⡻⠀⠁⠀⠀⠀⣸⠃⠀⡼⠄⠀⠀⠁⠀⡴⠒⢻⠀⠀⠀⠀⠀⢀⠼⠛⣆⣠⣤⡈⢦⠙⣼⢃⡆⠀⠸⡄
⠀⢧⠀⠀⣰⣇⡙⠂⢲⡀⣤⣤⣴⣦⣴⣾⡿⠀⢠⠷⢄⡀⠀⢒⡎⠀⠀⠘⢆⠀⠀⠀⢐⡷⠒⠲⣾⡗⠈⠉⢻⣧⣘⡏⠀⠀⠀⣧
⠀⠈⠙⢺⠁⠠⢌⠓⠋⠀⠙⢿⣿⣟⣱⠘⡆⡰⠋⠀⠀⢉⡟⠉⠀⠀⠀⠀⠘⢦⠤⠶⣿⠀⠀⠀⠈⠏⢀⣶⠐⠄⣿⢻⡀⠀⠀⣿
⠀⠀⠀⠈⠳⣄⣀⡴⢗⢦⡀⠀⠣⣀⠝⣸⡏⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣿⣦⣀⣀⣀⠀⠫⠡⠔⠁⠘⠮⡇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⣉⢦⣤⠖⡾⠁⡇⠀⠀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣇⠙⣛⣿⣿⣿⣶⠾⠃⡖⠒⠒⡇⠀⠐⢣
⢠⠖⠒⠒⠦⠼⠁⢱⠈⢉⡤⡞⢄⢸⠃⠀⠘⢦⡀⠀⢹⢆⠀⠀⠀⠀⠀⠀⠀⣀⡸⠦⠼⣆⢹⠖⠓⡺⠁⠀⠸⠤⢔⣊⠉⢳⠀⣸⠃
⠈⠓⣶⡂⠀⡄⠀⣸⠟⠁⠀⠹⡄⠛⠀⠀⠀⠀⠉⠓⠚⠈⢧⠀⠀⠀⠀⣰⠆⣠⡦⠚⠯⣙⣦⣁⠈⣁⣀⣀⣀⣀⠀⠀⣠⠞⠚⠁
⠀⢰⡧⠀⠐⠀⠀⣇⠀⢀⡤⠚⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡴⠉⠉⠉⠀⠀⣠⡾⠁⡌⢩⠟⠉⠀⠉⠳⣟⠉⠁
⠠⣏⣀⠤⣀⠀⠀⣈⣡⠿⠓⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣸⠁⠀⠀⠀⠀⠀⣧⣣⣄⡺⣇⠐⡦⣄⠀⢀⡼⠓⠶⠤⠒⠒⢓⡦⣀
⠀⠈⠀⠀⠈⠉⠉⢉⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿⠀⠀⠀⠀⢠⠞⠉⠀⠈⠑⢾⢺⡇⠈⢳⣏⠄⠀⠀⠀⠠⣖⣋⡠⢮
⠀⠀⠀⠀⠀⠀⣰⠋⢣⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡄⠀⠀⠀⢸⡴⠒⠒⠲⢤⡠⣰⠇⠀⢏⣀⣀⣄⡀⠀⠀⠈⠀⢤⠴
⠀⠀⠀⠀⠀⠀⣿⠀⠘⣇⠀⠈⠙⠓⢦⣤⣄⡀⠀⠀⠀⠀⡇⠀⠀⠈⣇⠀⠠⣀⣏⠔⣂⣀⣒⢤⢳⣻⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠢⠬⠇
⠀⠀⠀⠀⠀⠀⠳⡱⢄⠈⠓⠲⠶⠒⠋⡇⣼⠀⠀⠀⠀⢰⡇⠀⠀⠀⢹⡀⠀⠘⣯⠊⠀⠀⠀⠱⡱⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠯⠶⡖⠒⢒⡻⢁⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⣧⠀⠀⠹⡀⠀⠀⠀⠀⣳⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠼⠇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⡆⠀⠀⠙⣤⠒⠚⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠚⠑⠲⢤⡀⢸⡇⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣈⡷⢲⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠒⢕⢦⠀⠙⣦⡇⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠁⠀⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢣⠱⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠀⠉⠀⠳⣱⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣿⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡁⢀⠀⠉⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡐⠁⡀⢠⠈⠄⠀⠀⠀⢀
⠀⠀⠀⢓⠀⠉⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠧⢴⣀⢂⠀⠘⠀⠀⠀⠀⠡
⠀⠀⠀⠘⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠂⠀⢉⠙⠂⢆⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⡾⢷⡄⠀⣱⣶⣦⡕⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⡁⢄⠀⠀⢉⠀⡉⢿⠄⠑⡀⠀⠀⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣬⣾⠀⠐⢀⣲⣾⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠙⠀⠠⢀⢿⡟⠀⠀⡠⠃⡨⠂⠠⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡀⠀⠎⠠⠊⡀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠉⡀⠉⠂⣄⠀⠀⠐⠀⠀⠀⠀⢢⣄⣀⣀⣀⣤⣶⣶⣶⣿⣧⠀⠀⠀⠘⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⢅⢀⡀⠁⡀⠘⣧⠀⠀⠡⡀⢀⡀⢼⣿⣿⣿⠟⠻⢿⣿⣿⠿⠃⠀⠀⢀⡀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⢃⠀⠈⠢⠔⠂⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⢃⠈⠟⠀⢂⠃⠀⠀⠀⡐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢄⠈⠒⠁⠀⠀⠀⢠⡀⢠⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠑⣠⠀⠀⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠁⠀⣠⡴⠊⡰⠉⠀⡡⠚⠁⢀⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⠀⠄⠀⠉⠀⠑⠺⠃⠀⠎⠀⢀⠄⠀⠀⢀⠂⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠴⢀⠀⡘⠀⢰⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⢀⠁⠀⡄⠎⠀⠑⠀⡅⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣲⠀⠀⠆⠀⠀⠃⠃⠀⠐⢠⠈⠀⢠⠀⠀⠀⢀⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠀⠁⠀⠀⠀⠈⠂⠀⠐⠁⠀⠀⠈⡳⣀⠈⠀⠀⠀⠀⠀⠈⣙⠶⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠂⠈⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠔⠠⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢄⡀⠀⠀⠐⠒⠀⠤⠀⣀⣀⣄⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡔⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠈⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠒⠀⠤⠄⠂⠉⠐⠒⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠅⠒⠒⠒⠂⠈⠍⠁⠀⠀⠰⠶⠶⠶⠸⠗⠶⠆⠒⠖⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠟⢢⡀⢠⠞⢳⡀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡎⢠⣄⠙⢁⣤⠀⢣⡀⠀⠀⡇
⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⢻⠀⢀⡞⠀⡞⠀⠳⠊⠀⢧⠀⢣⠀⠀⡇
⠀⢀⠞⠀⠰⣄⣨⢗⠋⠙⢦⠀⠈⢆⠈⠑⠾⢁⣀⢀⡀⠀⠘⠶⠋⠀⡼⠁
⠀⡼⠀⠀⠢⡽⣱⠡⡶⡄⡸⠀⠀⠈⡳⢄⡰⠋⠜⡡⠷⠀⡤⢢⠤⡞
⠀⡇⠀⠀⡼⡟⠩⢀⠀⢺⡧⠐⢄⠀⠱⡀⠁⠀⠀⠀⡼⢤⣠⢸⠀⠇⠀⢠⠚⠓⠦⣴⡖⠓⠢⣄
⠀⡴⠀⢀⡟⠁⠀⢸⡇⠋⠀⠀⢀⠇⢀⠇⠀⠑⠀⣠⠤⡞⠁⠀⠈⠉⣨⠾⡀⣠⡐⣌⢯⡗⡀⠈⡆
⠘⣄⢀⡼⢍⡀⢲⠠⣤⣴⣶⢶⡿⠀⡸⠢⣄⣀⡲⠁⠀⠹⡄⠀⠀⡸⠒⠲⣿⠂⠉⠻⣦⣏⠀⠀⢸
⠀⠈⠹⡄⠒⠈⣁⠀⠙⠿⠩⡆⣥⠞⠁⠀⢠⠏⠀⠀⠀⠀⠙⡞⢛⣇⠀⠀⠈⢠⠞⠉⠸⡙⡄⠀⢸
⠀⠀⠀⠈⠑⠉⢱⠙⣦⣀⣡⡜⡇⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢧⢸⠙⠷⣶⣶⣤⣭⢄⠤⠨⠇⠀⢬
⣠⠒⠲⠤⠎⢹⡈⣪⢵⢧⢰⠀⠘⢦⡀⠘⢦⠀⠀⠀⠀⠀⠀⡸⠄⢷⠸⠏⢉⠋⠀⠪⠤⣖⠙⠆⢨⠇
⠈⢹⠇⢠⠂⡸⠋⠀⢈⣦⡉⠀⠀⠀⠈⠉⠀⢧⠀⠀⢀⡾⠤⠖⠉⠢⡷⠤⢤⡤⠤⢤⣄⡠⠴⠋⠁
⢠⠊⣀⠀⠀⠘⢤⣴⣍⡴⠁⠀⠀⠀⠀⠀⠀⠘⡆⢀⠏⠀⠀⠀⠀⡎⣀⢡⠊⣄⡀⠀⢙⣦⣀⣀⡠⢤⡀
⠈⠉⠁⠉⠒⠛⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⢀⠔⠉⠉⠣⡕⢎⠉⢲⡋⠀⠀⠀⢄⢚⡡
⠀⠀⠀⠀⠀⠎⢱⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⡞⠀⠈⡆⠀⠀⢸⠖⠊⠑⢢⣠⡎⠀⠧⠤⠤⢄⡀⢀⠁⢦
⠀⠀⠀⠀⠘⣧⡈⠦⣀⣀⣩⢿⢲⠆⠀⠀⢀⡇⠀⠀⢧⠀⠐⣷⡠⠒⠊⠐⣽⡆⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠙⠿⢦⣤⠤⢤⡞⣼⠀⠀⠀⢸⠃⠀⠀⠘⡄⠀⠻⡀⠀⠀⠀⣼⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠇⠀⠀⠀⡼⠀⠀⠀⠀⢹⠀⠀⠱⡄⠒⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⡉⡉⠢⡀⡇⠀⠀⠀⠀⠘⢦⣤⡤⠜⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠈⠞⢄⠘⣇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⢪⡆⢸⠀⠀⠀⠀⠀⠀⠈⠂⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⣿⡼