Thirsty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⡆⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢃⣦⣍⣛⡛⠟⠻⢛⣛⣭⣶⣿⣿⡇⣿⣿⠸⠿⠿⠿⣿⣿⣿⣿⡇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢛⢋⣶⡘⣿⣿⣿⣿⣿⣿⣮⣙⡛⠿⠿⠟⣴⣿⣿⠰⣿⣿⣷⣮⣙⡛⢛⡃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⣿⡘⠿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⡜⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠇⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡿⠿⣡⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣧⣘⣛⣛⣛⣛⣛⣛⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠻⠿⠟⣛⠉⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣦⣝⢿⣿⣿⣿⣿⣿⡿⣋⣴⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡛⠛⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡭⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣴⣶⣲⣿⡇⢸⣷⣒⣒⣦⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⣽⡾⠟⠛⠉⠉⣿⡆⢸⡏⠉⢉⠽⢛⡿⣿⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⠏⠁⢀⣠⡴⠶⠶⣿⡇⢸⡷⢲⠶⠾⣥⣍⣀⠈⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡻⣄⣘⣩⣀⠴⡂⠈⣿⠓⢚⡏⠀⠀⠀⠀⠈⢉⣠⢟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⢽⣻⠶⣯⣄⣀⣿⣃⣸⣇⣀⣤⣤⣴⣶⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠓⠮⣍⣛⡒⢊⣛⣉⢙⢏⣐⠛⢋⣩⡴⠚⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠉⠉⣉⠉⢉⡍⠈⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠿⠀⠘⠇⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣴⣦⡀⣾⣷⠀⢿⠂⠻⣿⢇⣠⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⠿⠿⢃⣬⣁⠸⣿⠇⣴⣦⠸⣿⡿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⡆⢻⣿⡿⣴⣶⡤⣾⠉⣰⣶⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣷⡤⣾⣶⣈⡋⢰⣿⡦⣈⡛⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢙⣛⠁⣉⡹⣿⡿⣴⣿⡄⠻⠟⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡻⠿⠸⣿⡿⢠⣶⡝⣻⡑⠿⠃⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣬⣭⣤⣭⣴⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⠿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣆
⠀⣰⣿⣿⠿⠻⠿⠟⠀⢀⣀⡈⠉⠀⢸⣿⣿⠿⠿⠿⢿⢿⣿
⢠⣿⠿⠃⠀⠀⠰⠶⠾⠋⠛⡙⣷⣤⣿⣿⠏⠀⠀⠀⠀⠀⡟
⣼⠁⠀⣠⣤⣶⣤⣿⣶⣿⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠸⡀
⣿⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠤⠀⠀⠚⣧⠀⢀⡀
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣠⣤⠄⠀⠀⢸⡿⠋⠁⣙⠢⡀
⠀⠹⣿⣿⣿⠛⣛⣛⠻⣿⣇⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⡠⠊⠀⡠⠈⡰⠃⠈⠳⣄
⠀⠀⠹⣿⣇⠘⠁⣉⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡴⠂⠀⡀⠼⠂⡝⢯⡒⠢⣄
⠀⠀⠀⠈⢻⣄⠀⠈⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢰⠀⠊⠀⠰⠉⢀⡤⠚⠹⠀⠉⠢⡀⠙⣄
⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠈⠓⣜⢆
⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⣴⡾⠋⠻⣦
⠀⠀⠀⠀⠀⣠⠾⠛⠶⢤⣀⡀⠈⠉⠛⠲⢦⣤⣄⣀⣤⠎⠀⠀⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⡀⠱⡄
⠀⠀⠀⢠⣞⣡⡤⢤⡀⠀⠈⠙⠳⣤⡀⣰⣿⡿⠟⠉⠀⠀⠀⠀⠀⣀⣴⣿⡿⣿⣷⣄⡀⠸⠁⠀⠀⢀⡴⠁⠀⠘⣆
⠀⠀⢀⡿⠉⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢻⡟⠀⢻⣿⣿⣿⣶⣤⣤⣴⡏⠀⠀⠀⠀⠸
⠀⠀⣼⡵⠞⠁⣀⡤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⣧
⠀⢠⠟⠁⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⡋⠀⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠹⣷⡀
⣰⠏⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡛⠿⣿⣿⣷⣶⣶⣿⣷⡶⠾
⣿⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠛⠉⠉
⠘⢧⣸⣿⣦⣄⡀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠻⣿⣿⣿⣿⣿⣶⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢹⡿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡼⠁⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢐⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠓⡌⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠙⡄⠀⡄⠀⡹⠣⡈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⣇⢀⣃⠔⢁⠎⠁⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠉⠑⣶⣶⢮⠄⣀⡏⠀⡼⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠉⠁⠒⠒⠒⠒⡉⠉⠉⠹⣤⡔⠊⠀⡘⢰⣡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠄⡠⠀⠀⣀⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠛⣡⠖⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡒⠒⠀⠚⣠⠄⠈⢄⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡈⢰⣀⣀⠀⠀⠈⠚⠣⡀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⢸⣉⠉⠀⠀⢀⡐⢂⠀⡗⠚⠒⠒⠊⠉⡇⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⢸⠁⠀⡄⠀⠀⠹⡀⢨⠃⡓⢪⡀⠩⢻⠇⠀⠀⠀⠀⢘⡒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠘⡏⠉⢛⠉⠁⠀⠳⢸⢒⠓⢲⠗⠒⢺⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⡄⠤⠒⠀⠀⢹
⠀⠀⠀⠀⣧⣀⣈⣀⡠⠔⣁⡸⢸⠀⣞⠀⡠⡟⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠇⠀⡘⠀⠀⣘⠇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠇⡈⠈⠀⣿⠀⡇⠀⠀⠀⠀⠀⡘⠉⠑⠆⠀⠀⡀⠸⠀⠯⠨⣁⠃
⠀⠀⠀⠀⠸⡀⠘⠃⠀⠀⠀⢀⠇⢠⡄⠀⢸⠁⠀⠀⠀⠀⣜⠀⠀⡟⠓⠒⠀⠉⠁⠄⢀⡤⠃
⠀⠀⠀⠀⠀⡇⠠⣦⠐⠀⠰⢸⢠⡤⠀⠀⢸⠀⠀⠀⣀⡀⣀⠟⠛⠗⢲⠶⠶⡖⠒⠒⠈
⠀⠀⠀⠀⠀⢹⡀⠐⠷⠀⣄⡜⠀⠀⠂⢀⡏⠀⠀⢞⢝⣙⡻⣖⢲⡞⢘⠀⠀⠈⢢⡀
⣀⣀⣀⣀⣀⣀⣈⣙⣒⣲⣶⣶⣖⣒⣊⣉⣀⣀⣀⣀⣀⣁⣈⣯⣧⣯⣎⣁⣀⣤⣖⣁⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠛⠉⠉⠉⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡽⠚⠁⢀⣠⢚⣡⣶⠴⣶⣶⠶⠛⠛⠛⠛⣶⣄⡀⠀⠀⠚⢛⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣩⢞⣡⡾⢋⡷⠟⠉⠁⠘⠋⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣥⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢼⣷⢻⠀⡴⠛⠁⠀⠀⠀⠀⠁⠀⠉⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢲⣤⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠈⣿⠦⠀⣀⣀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠸⠟⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢠⣏⡴⠞⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⢰⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢩⠟⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠤⢀⣀⠤⠈⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⣀⣠⡴⠞⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡯⡍⠉⠀⠀⠀⣿⠏⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢼⣇⣀⣀⣤⠶⠛⠁⢠⠎⠈⣙⣦⣀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⢧⠀⠀⠈⠉⠉⠀⢀⡠⠖⣡⠖⠋⠉⠀⠉⠳⣦
⠀⠀⢀⣴⣟⢛⡛⠉⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣴⣾⣽⠋⠀⠀⠁⠒⠒⠒⠒⠈⠁⢀⡜⠁⠀⠀⢀⣠⠤⠀⠸⢧⡄
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⡶⢋⠿⠤⠤⠤⣤⣤⣤⡴⢋⣿⠖⠫⠘⢹⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⠟⠀⠀⠀⠀⠊⠻⢦⡀⣀
⢾⡁⢰⡶⠶⣄⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢀⡤⠶⡶⠶⠶⣦⡅⠀⠀⠀⡿⠀⠀⠀⢀⠔⢈⠀⠀⠉⠉⢳⣄
⠈⠉⠉⠀⠀⢈⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⣏⣤⠶⠛⢷⣤⣤⣯⡤⠞⠋⠁⠀⠈⠻⣶⠖⠦⣤⣤⡤⠴⠟⠋⠉⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⢠⠖⠋⢀⣀⣀⣀⣸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠻⣞⠉⠀⠀⠀⢀⡄⠀⠀⠈⢻⡄⠈⠑
⠀⠀⠠⣇⢀⡴⠉⠀⠉⠉⠀⠻⠟⠛⠶⢤⣀⣀⠀⠀⠀⣀⣠⡾⠋⡀⢀⣀⣀⣀⣀⡀⠀⠙⢷⣤⠤⠚⠉⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠛⣋⣾⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠙⢄⠀⠂⠹⣦⡤⠶⠚⢁⡀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣼⡟⠀⠈⠉⣿⢧⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠻⣶⠒⢉⣀⣴⠟⠛⣠⣞⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠺⠍⠁⣀⣘⣡⣈⡉⠝⣷⠀⢰⡟⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠁⠈⠛⢛⢡⡟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠙⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣱⠀⢰⡇⢠⡀⡴⠀⠀⡟⠀⣿⠇⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡲⢾⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⠁⠁⠀⠀⠉⠉⠀⢸⣇⣠⣿⣀⣀⣀⣤⣤⣬⣧⡀⠀⠀⠀⠀⢀⠴⠋⠀⢀⣴⣯⣤⣤⣤⣤⣼⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⡀⠀⠀⠀⠆⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⠔⠁⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⠌⡀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣬⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣁⣉⣉⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠉⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠧⣀
⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⣞⠛⡆⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢹⢹⡅⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⡜⠙⠛⠙⡄⠀⠀⠀⠀⠀⠀⠙⡄⢀⡤⠤⡄
⠀⠸⡇⠀⠀⠀⠀⠀⠀⣸⣶⣶⣶⣶⣿⠀⡀⠀⢀⠀⠀⠀⣻⢸⣀⣠⢇⢤⡀
⠀⠀⠙⠲⡄⠀⠀⠀⠀⡿⠿⠿⠿⠿⢿⠀⡧⠤⢼⠀⠀⢠⠏⠀⠀⠀⣘⣚⣡⣄⣀⣀⣀
⠀⠀⠀⠸⡅⠀⠀⠀⠀⠙⠒⠒⠒⠚⠊⠀⠙⠛⠀⠀⣰⠁⠀⡠⠖⠋⠀⠀⠀⠀⠈⠳⣍⠙⢦⣄
⠀⠀⠀⠀⠙⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠨⣃⠀⢹⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⣤⣤⠤⠴⠤⠤⠤⠖⠁⠀⡞⠀⠀⠁⣀⣠⡤⠀⠀⠀⣤⣀⡀⠑⠄⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢈⣙⣋⠡⠀⠀⢈⣛⣛⣓⣂⠀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⠤⠤⠂⠀⠀⠀⣀⣠⣤⣀⣈⡍⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣠⠴⠟⠋⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⣀⣤⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠉⠁⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠔⠊⠉⠉⠒⠢⡀⠀⠀⠀⠀⠀⠀⢠⠃⡤⠚⠓⢀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⢶
⠀⢀⡎⠀⠀⠀⠀⢀⠀⠹⠤⣤⣤⣤⣤⣴⢃⡎⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢚⡘⠉⠉⠘⢤⡀
⠀⠸⠇⠀⠀⠀⠑⠀⢤⣴⣿⣿⣿⣿⣿⣣⣿⡶⢗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠁⢆⢠⠀⠀⠀⠀⠀⠘⣄⡀
⠀⠀⠡⡀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠐⢀⡤⠀⠀⢀⡯
⠀⠀⠀⠈⠙⠒⣡⠗⢤⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢀⣀⣤⣄⣀⣴⣿⡇⢀⣎⣉
⠀⠀⠀⠀⠀⡔⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡆⠀⠀⠀⠀⠀⠀⡔⠊⠈⠀⠚⠿⠟⠛⠛⠛⠛⠧⣠⡀⣨⠃⠀⠀⠄
⠀⠀⠀⠀⠘⢄⠀⡲⠓⠄⠀⠀⠀⠀⠀⠀⠈⡧⢀⠆⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠳
⠀⠀⠀⠀⠀⠀⡜⠁⢀⡄⡀⠀⠀⠀⠀⠀⠀⢰⠳⢴⡏⠒⢆⣀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⡴⠤⣀⣀⣠⠔⠀⢀
⠀⠀⠀⠀⠀⠘⠀⢐⡛⠁⢨⠂⠀⠀⠀⢀⣀⣼⣀⡀⡀⠀⠀⠈⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠨⡁⠀⣉⠭⠄⣀⣁⡠⠞⠀⠀⠹⡇⠀⠜⠉⠉⠲⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⠖⢄⠀⢀⡖⠒⣄⢀⡀⢰⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⡄⠀⠀⣠⡗⡜⡀⠀⠈⠁⠙⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢢⣠⣎⠈⠳⠁⢡⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⢦⠀⠀⠈⠢⠤⠜⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⡦⡀⠀⠀⠀⢻⣀⡀⡠⠞⠀⠀⠀⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠇⠈⠢⡀⠀⠘⢇⡀⠀⢀⡴⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢰⠀⠀⠈⢦⡀⠀⢏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢇⠀⠀⠀⠙⣄⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠆⠀⠈⢦⠀⠀⠀⢘⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣔⣋⠩⠤⠤⠤⠄⠐⠒⠃⠀⠀⠀⠉⠒⠤⣀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡋⡴⣶⣀⣨⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡮⠒⠴⣦⣀⠙⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠓⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢄⡀⠀⠀⠀⢀⠔⠈⠡⣀⡠⠊⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣤⠀⠀⠀⠀⠈⠑⠲⢤⠋⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠄⣾⠟⡀⢱⠤⠤⠄⠀⠀⣀⠆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠐⠓⢅⠀⠀⠀⠈⠑⢆⢄⡀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⡈⠓⢄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠰⣯⡁⠀⠀⠀⠀⠀⢀⠜⠢⡀⠀⠀⠀⠀⢸⠂⠈⠑⣤⢶⠇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢳⠀⠀⢘
⠀⠀⠀⠀⠀⠀⢸⡟⢦⣄⠀⠀⠀⠀⠐⡄⠑⠬⠦⡒⠕⢊⣠⣶⣋⠡⢎⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢀⡂⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⢦⠀⠈⠢⠊⣀⡤⠖⠋⠁⠀⢀⡤⠈⠑⠦⣄⡀⠀⠀⢀⠀⠀⣠⢃⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠙⣦⡀⣠⣴⠚⠉⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢁⡠⠛⢺⡱⠀⡘⠊⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠋⠁⢸⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⣀⣤⣐⣊⣉⣀⣠⡞⠦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠲⡞⠉⣿⠀⠀⠉⢹⡇⠸⣧
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠀⠀⡘⢇⠀⠀⠀⠘⡀⣿⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠈⢦⠀⠀⠀⢣⣿⠀⠀⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢸⡄⢀⣰⠧⠤⣤⠃⠀⠈⢆⠀⠀⠈⠇⠀⠀⠀⠘⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢸⡏⢉⣀⣀⣤⡼⠀⠀⠀⠈⢆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠤⠖⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣁⣤⠞⠁

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

⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⢀⣤⣴⣶⣶⣤⡀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⣴⡿⠛⣿⣿⣿⠛⣿⣆⠀⣰⣿⠛⣿⣿⣿⠛⢻⣦⠀⢠⣿⠛⢻⣿⣿⠛⢻⣧⠀⢠⣾⠛⢻⣿⣿⡟⢻⣷⡀⢀⣾⡟⢻⣿⣿⡟⠻⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆⣿⣿⣷⣿⣿⣿⣷⣿⣿⡇⣾⣿⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣿⡇
⠀⠀⢿⣿⣾⣿⣿⣿⣷⣿⡟⠀⢻⣷⡻⣿⣿⣿⣿⣽⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠸⣿⣟⣿⣿⣿⣝⣿⣿⠃⠘⣿⣿⣫⣿⣿⣽⣿⣿⠇
⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠙⠿⢾⣿⣷⠿⠋⠀⠀⠀⠙⠻⢿⣿⣿⠿⠛⠁⠀⠀⠈⠻⢿⣿⣿⠿⠟⠁⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣆⠀⣼⣿⡄⣠⣾⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣷⣼⡿⠿⠿⢿⣿⠿⠿⢿⣷⣴⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⢟⣫⣵⠶⡟⣻⣏⣿⡛⣟⡷⢶⣭⡻⢷⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⢟⣵⣿⢯⣯⠶⠛⠛⠋⠉⠛⠛⠷⣯⡾⣿⣷⣝⢿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢣⣿⣿⣿⠋⠁⠀⣀⠀⠀⣀⠀⠀⡀⠀⠙⢿⣿⣿⣦⢻⣿⣁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢣⣿⣿⡿⠁⣾⡇⢰⠟⣧⢸⠟⡇⣼⢻⢀⡏⠀⠹⣿⣿⣧⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡏⣾⣿⣿⠁⠀⢸⡇⢸⠀⣿⣿⠀⣷⢹⣼⣸⢱⢶⠀⢿⣿⣿⡸⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⣿⣤⣿⠀⠀⣸⣇⢸⣦⡟⢹⣤⡏⠀⢀⡏⢹⣸⠀⢸⣦⣼⡇⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⢿⣿⣿⡄⢀⡉⡉⣀⣉⡁⡈⡉⢁⣀⡈⣁⣈⠉⠀⣾⣿⣿⢹⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⡜⣿⣿⣷⡈⠷⠛⠀⠫⠷⠃⠷⠢⠀⠀⠣⠏⠃⣼⣿⣿⡏⣼⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡜⢿⣿⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⢟⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣮⡻⢮⣿⢻⡷⣦⣤⣤⣤⣤⣤⡶⢛⣽⣯⠟⣡⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⣹⣿⣶⣝⣛⠶⣧⣧⣽⣎⣿⣤⡷⢞⣫⣵⣿⣿⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⣿⣿⡿⣶⣶⣶⣶⣶⣶⢿⣿⣿⠏⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠸⣿⠋⠈⢿⡟⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠉⠈⠁⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⣅⣠⠄⠀⠀⠀⠠⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢴⠗⠛⠀⠀⠀⣤⡀⣼⠿⠶⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠠⠠⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠹⠏⡄⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣏⣀⣧⠧⠄⠒⠀⠀⠀⢆⡀⠀⠀⠀⢱⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠜⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠥⣄⠀⢸
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡀⢀⡀⠀⠀⠀⠀⢤⣿⣄⣠⣿⡄⠀⢸⡇⠀⠀⢀⡀⠀⠀⢀⡤⣞⡛⣉⠀⢲⡀⠀⠀⠀⣀⣤⣾⣧⣾
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠈⣀⣀⠀⠀⣷⣦⣰⣿⣿⣿⣿⣿⣿⣆⠀⢻⡄⠀⠀⠀⠀⠀⣿⣷⣶⡞⠉⠛⠑⡶⠊⠀⠚⠫⠽⠿⣿⠏
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠘⠁⠀⠀⠀⠙⢿⢃⣀⢹⣿⣿⣿⣿⣿⡆⠀⠻⣄⠀⣀⠄⠀⣿⣿⣿⠁⠀⢀⡜⠁⠀⠴⠒⠉⠁⣴⠋
⢰⣿⣿⣿⣿⣿⢻⣿⠟⣻⣿⣿⣿⣿⣀⡄⠀⠀⠉⠁⠀⠀⠀⠀⢃⣾⣿⣿⣿⣿⣿⣿⡀⠀⠙⢷⣄⠀⢠⣿⣿⠉⠀⠀⠈⠀⠀⠀⠀⢀⣤⠞⠁
⣾⣿⣿⣿⣿⠧⣼⡁⢼⠛⠛⢿⣿⣿⠀⠤⠐⠀⠀⠄⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⠾⣿⣇⣀⣀⣀⣀⣀⣠⡤⠞⠋⠁
⣿⣿⣿⣿⣿⡈⠤⠇⠈⠀⣿⣖⢻⣿⡄⠀⠀⡀⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣷⠀⢀⠀⢀⠈⠋⢸⣿⣿⣄⣀⡀⣀⠆⣩⣾⣿⠿⠛⠉⠨⠿⣿⣿⣿⣿⡇
⢹⣿⣿⣿⣿⣿⡄⠀⡱⠁⠀⣠⣿⣿⢿⣿⣿⣦⣤⣾⡚⢋⣰⠆⠀⠀⠴⠒⠚⠛⢿⣿⠇
⠘⣿⣿⣿⣿⡿⠟⠂⠀⠀⠚⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠘⣿⣿⡿⠁⠈⠉⠀⠀⠘⠿⠋⠻⠋⠁⠀⠌⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣴⣶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⣠⣤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⣠⣯⣯⣿⣫⣽⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣉⡀⠀⠈⠀⠀⠉⢿⣿⠀⠀⠀⠐⢦⡄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡥⣬⣭⣧⡀⠀⣀⣿⣷⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⡟⢻⣿⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠉⠉⣉⣁⡀⠀⠛⡛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢲⡊⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠇⠀⠀⠡⣄⣀⠀⣛⣦⣶⣿⣿⣿⣿⣶⣿⣿⣿⣶⣞⠀⠀⢀⡿⠀⠀⠀⠀⢀⠴⡆⠀⠀⠀⠉⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣏⠙⢿⣍⣉⣉⣉⣽⡟⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⢂⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠛⠿⢿⣿⢟⣰⣿⣿⣆⠀⠀⠀⠀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣦⣾⣿⣿⣿⣿⣿⣿⣷⣀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠛⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠿⣿⣿⣿⣿⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣬⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠻⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⣠⣾⣿⣿⣿⣶⣄⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⢀⣴⣾⣿⣿⣿⣦⡀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⢀⣿⣇⣨⣿⣿⣅⣸⣿⡀⣼⣿⣀⣿⣿⣯⣨⣿⡆⢰⣿⣅⣸⣿⣿⣀⣽⣷⠀⣾⣯⣨⣿⣿⣇⣸⣿⡄⣸⣿⣄⣽⣿⣯⣀⣿⣧
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⢿⣿⡿⣯⣽⡷
⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠈⢿⣮⣿⣿⣿⣿⠟⠁⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠙⢿⣾⣿⣿⣿⣿⠟⠀⠈⠻⣷⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⢠⣴⣶⣿⣷⣶⣾⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣶⣶⣆⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣴⣾⣿⣿⣟⣛⡉⠙⣋⣻⡇⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣽⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⣠⣀⣀⣀⡀
⢀⣴⡿⠛⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣴⣴⣤⣤
⢾⣿⠀⠀⠙⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣯⣁⣉⣀⣽⣿⣿⣦⠀⠀⠀⠀⠻⣿⣶⣤⣭⣥⣶⣾⡿
⠘⢿⣷⣤⣤⣤⡤⣾⣿⠃⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟
⠀⠀⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣠⣄
⢠⣶⠿⠛⠻⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠰⣿⣿⣛⣛⣛⣛⣛⣿⣿⡆⠀⠀⠀⠀⣴⣿⣟⡙⠛⠛⣛⣿⣷
⠀⠉⠻⣾⣶⣶⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣟⠛⠛⠙⢛⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣭⣉⣩⣽⣿⣿
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠠⣾⣿⣯⣿⣿⣽⣿⣿⡦⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠠⣶⣿⣿⣛⣿⣟⣿⣿⣷
⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢴⠦⡀⠀⠀⠀⠀⠀⠀⣀⢴
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠦⠤⠤⠤⠤⡾⠃⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⢀⡀⠀⠀⠀⠀⠀⡼⢄⣀⣀⡠⠤⠔⠒⠋⠉⠉⠉⠉⠑⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠽⣧⢀⠈⡄⢉⠁⢤⡼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⢦⡉⠉⣁⡴⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢣
⠀⠀⠐⢏⠉⠉⠀⠀⠉⢻⠀⠈⠉⠁⠀⠁⠀⢠⠊⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⡆⠀⠀⠀⣠⠬⢧⡀⠀⠀⠀⠀⠀⢎⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⢡⠀⠀⠀⣏⡉⠉⠉⠉⠑⠒⠒⠂⠘⢄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠳⡄
⠀⠀⠀⠀⠀⢣⡀⠀⠉⠓⠉⠉⠀⠀⠈⠉⠉⠉⠙⠓⠤⣀⠀⠉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⠄⠒⠒⠒⠂⠤⠤⠄⠀⠠⠤⠤⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠉⠉⠁⠀⠉⠉⠉⢙⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣷⠀⣿⠛⠛⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⢿⡄⠀⢸⡇⠀⢰⡟⠀⠀⢀⣴⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠿⠟⠛⠛⠻⠿⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠠⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠀⣿⠀⠀⠀⠀⣿⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠈⠉⠋⠀⠀⠀⣰⡟⠀⠀⣀⣠⣿⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢻⣷⢶⣤⣤⡶⣾⡛⠉⠻⣦⡀⠀⠀⠀⣼⠏⠀⢠⡾⠋⠉⠉⠉⠛⢷⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠾⠃⠀⢸⡇⠀⠘⠷⠀⠀⠈⠻⠆⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⠻⣦⣤⣀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠉⠉⠉⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢁⠄⠈⠀⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣶⠀⠀⣶⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠀⠀⢳⡀⠀⠀⠀⢤⣀⣀⣀⠄⠊⠀⢀⠌⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠉⠒⠒⣒⢒⠈⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠌⢰⣿⣦⠱⡄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠌⠀⠈⠻⠃⠀⠘⢆⢀⡀⢄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡠⠲⡆⠠⠤⠤⠸⣆⡠⠂⠈⠆⠦⡀⠙⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢃⠀⠐⠤⢸⠁⠀⠠⠐⢸⣷⡈⠉⠁⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠬⢄⡒⠥⡂⢄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣁⣀⣀⠀⡠⠀⠀⠀⠀⠀⣀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⠈⠑⢬⡢⡙⢦
⠀⠀⠀⣇⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⠆⠱⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢒⡀⣡
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠻⣿⣿⣿⠃⣀⡀⠀⠀⠀⠀⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠤⠤⠐⠒⠀⠀⠉⠁⠈⠑⠒⠤⣀⡀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣠⠃⠀⢀⠠⡉⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠱⡕⡄⠀⠀⠈⠉⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⢀⣀⠀⠤⠔⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⡀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠉⠀⠐⠉⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⣀⡤⠤⢀⡀⠄⢠⠄⣀⡀⠀⠈⢤⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠈⠻⣔⠒⠒⠚⠋⠤⠄⡀⠀⠀⠀⢀⣀⠄⠃⠚⠓⠂⠹⠋⠀⠀⠀⠻⠟⠒⠒⠒⠐⠂⠈⠛⠃⠀⠀⠬⠓⠉⠉⠀⠒⠀⢒⠓⠃
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠈⠂⠀⢀⡠⠄⠠⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠈⠁⠊⠁⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡀⠐⠁⣀⡄⠐⡐⠂⠒⡴⠀⢩⠔⠈⢰⠑⢄⠀⠄⠀⠀⠀⠀⠀⢀⠄⠊⢀⣀⡄⠰⠒⠒⢰⠆⠒⠰⡆⠄⡀⠑⢤⡀
⠀⠀⠀⠐⠚⠁⠲⢭⣜⢆⣀⡶⡄⠀⠇⠀⣸⠷⠂⠈⠀⠈⠉⡩⠒⠀⠀⠀⠻⣋⠚⠈⠁⠀⣁⣉⣀⠒⠊⡁⣀⣈⣁⣀⠉⠐⠒⠚⡻
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⣖⠀⠁⠀⠀⠉⠐⠛⠒⠈⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠤⣀⡠⠔⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠘⠈⠀⡄⠀⠒⠢⠆⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⢠⠂⠁⠀⠀⠀⠀⣓⣦⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣠⠤⠈⡇⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠶⡄⠀⠈⠁⠀⠀⠀⠑⠂⠂⠐⠁⠀⠀⠀⠈⠁⠀⢒⡳⠆⠀⠀⠈⠀⠀⠀⣀⠈⠀⠉⠀⠂⠐⢣⠀⢀⠁⠒⡈⠀⡜⠂
⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠠⠼⠀⢀⡥⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠀⠀⠠⠤⠤⢀⡀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂


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

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⢸⣿⡆⠀⠀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⣦⠀⠻⠛⠇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣤⣀⠈⢉⣤⣶⣿⣿⣿⣶⣤⡉⠀⣠⣴⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠟⠀⡉⠻⢿⣿⣿⣿⡿⠟⢋⡀⠛⠻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠀⢠⣤⣤⠀⠺⣿⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡀⠠⠄⠀⠀⣠⡿⠋⠀⠀⠘⣿⡇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⣀⠄⠊⠀⠀⠀⠀⠹⣤⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠟⠀⠀⠀⠀⠈
⠀⠀⠠⡀⠀⠀⠀⠱⣄⡎⠀⠀⠀⠀⠀⠸⡀⠀⣿⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⢳⣠⠁⠀⠀⠈⠙⠶⣤⣠⠀⠀⠀⠙⣦⡏⠀⠀⠀⢠⠎⠀⢀⡠⠂
⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠹⣦⠀⠀⣠⡟⠀⠀⢀⣠⠞⠉⠉
⠀⡀⠀⠀⠀⣀⡤⠞⠋⠉⠉⠉⠙⢿⣧⣼⣯⣴⠶⠿⠻⠷⢤⣀⠀⠀⠀⠀⣀⠠⠄
⠀⠈⠉⠈⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⣏⠛⠛⠉⠉⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠋⣿⠉⠛⠶⣤⣤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠸⠀⠀⠀⠀⠙⢆⠐⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣴⠆⠰⠟⣶⠒⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠠⢢⣠⣸⣧⣾⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣟⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡠⠤⠤⠌⣉⠐⠀⣀⢺⡿⣻⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠊⠁⠁⣴⡶⠶⡶⠶⠲⠤⠄⠸⠿⢡⡿⠋⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⢀⠀⠀⢷⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⣾⠀⢸⣏⣑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣇⣸⠃⠸⠿⢼⠀⠀⠀⠰⠶⠒⠒⠂⠂⠀⠀⠀⠀⠀⡆⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢸⣿⢿⣿⣷⠀⠀⠀⠘⠀⠀⠐⠚⠓⠀⣶⡶⣶⡄⢀⠀⠀⠀⣵⡀⠀⢹⠐⡀⠀⠀⠀⠀⠀⢠⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⡛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⡾⣾⣷⡄⠸⣷⣀⠀⠀⢢⣤⣦⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣇⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣳⣤⡈⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⣿⠀⠀⠀⣈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡀⠸⠸⡿⠀⢰⡇⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣆⠀⠀⠀⠉⠉⠓⠒⠶⠂⠄⠀⠀⠀⠀⠀⢀⣾⡿⣿⠃⠀⠀⠃⠀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⣿⣦⠀⠐⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣥⣶⠏⠈⠀⠀⠀⡀⠀⡸⡀⡀⠀⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠴⠿⠟⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣴⡿⢋⡥⠊⠀⠀⠀⣰⡇⠀⢠⡇⣿⡶⠈⠢⢺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢨⣶⣶⡖⠒⠖⠒⠛⠉⠀⠀⠡⠖⠊⠁⢀⡤⠊⣀⢴⠟⠀⠰⠟⢰⡯⡧⠐⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠅⠀⠂⠀⠀⠀⠀⠀⠶⣦⣶⠛⢃⣀⣈⣁⢀⠁⠄⠀⠐⠋⠸⢀⡼⢃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⠛⠉⠀⠀⠀⠠⠀⠀⠈⠙⢾⣍⡁⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠥⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢴⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⢈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⢀⣤⣄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠟⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⢸⣇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣤⣄⡀⠀⠀⠀⠀⣀⣈⣉⣀⣀⡴⠋⠉⣷⠀⠀⠀⠀⡿⢩⠭⣍⠓⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠤⠤⠤⡀⢻⠋⠉⠉⣉⣉⠉⠉⠀⠀⠘⠂⠒⠉⠉⠉⠉⠙⠃⠸⣀⣠⠀⣏
⠀⠀⠀⠀⠀⢠⡞⠩⢬⠀⣁⠤⠔⠇⢸⠉⠉⠋⠀⠀⠱⠊⠉⠙⠲⠒⠛⠲⠞⠛⠒⠉⠉⠳⢢⣀⣠⠌⢉⡧
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠉⠙⠒⠚⠘⠦⠤⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠒⠒⠤⢌⣉⣑⡒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⢒⣒⣚⣉⠭⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠙⠒⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⣦⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⣶⡀⢸⡇⠀⠈⠁⠀⠰⠿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣾⣿⣶⣤⡀⠀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡇⠀⠾⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠻⠿⠛⠋⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣠⡄⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⡇⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢀⣿⣿⡇⠸⣧⡀⠀⢀⡀⢀⣀⣤⣤⣬⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣾⣿⣿⣷⣆⠙⣿⡀⠀⠙⠛⠛⠛⠁⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠲⢶⣶⣶⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⠿⣿⣿⠿⠋⣰⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡞⡄⠀⡰⡄⠀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⢸⠈⢦⡜⠀⠸⡔⠁⠳⠜⠰⣆⠌⣽⠀⣠⢶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠢⣄⣸⠀⠈⠃⠀⢀⡀⣀⣤⢤⠤⡄⣤⣀⣉⠰⠸⢞⢵⢙⣀⢔⣹⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⢄⣀⢘⡀⠀⠉⣀⢤⣲⠬⠗⠓⠛⠊⠉⠉⠉⠉⠛⠛⠿⢵⣳⢥⣁⠕⡍⣺⠤⡪⣭⠀⢀⡀
⠀⠀⠀⠀⠀⣀⡀⠈⣆⠀⠉⠁⣠⣺⠝⠋⢀⠀⠀⠴⠒⠀⠰⡀⢰⠙⠂⠒⡦⠄⠈⠙⠺⢽⣲⢬⣎⠆⢷⢊⢵⠃⠀⡀
⠀⠀⠀⠀⠀⠩⡈⠉⠉⠀⣠⣺⠞⠁⢠⡀⠛⠤⢄⠘⡎⠀⠏⠙⠘⣤⠄⠀⡁⠀⢀⠁⡠⣄⠈⠑⢮⣗⢭⡑⡿⡒⡹⠃
⠀⠀⠐⢖⢖⠒⠚⠀⢀⣞⠞⠁⡠⡂⠀⠑⢄⠀⠚⠀⣐⣠⡤⡤⡴⡶⡤⣤⣀⠀⠁⠸⡀⠎⣠⡆⠀⠈⠻⣟⣮⢚⢗⠒⣨⠃
⠀⣀⣀⣠⣱⠄⠀⢠⣻⠊⠠⣄⣀⡈⠳⠄⢠⢠⢵⣻⠝⣚⢛⢉⢉⢍⢉⢉⠛⠿⣶⣄⡀⠐⢀⡧⠔⠁⠀⢊⢾⢧⡣⠺⠧⠤⢤⠄
⠀⠉⠢⡂⡀⠀⢠⣷⠃⡠⠀⠘⢮⡉⠁⢀⡶⡧⠋⠀⠀⠀⠈⠀⠈⠈⠀⠈⠈⠑⢌⡙⢽⣦⠀⠀⠀⠀⠀⠀⠈⢯⢧⢕⢵⣟⣁⡀
⢀⡤⠄⠞⠂⠀⣞⠃⠀⠁⠈⠀⢹⠀⠨⣸⡿⠁⢀⣀⠀⠀⣠⠤⣀⠀⠀⡴⡖⣄⠀⠀⠈⢻⣷⠀⠀⠀⡶⡀⠀⠼⡝⡆⢺⢀⡶⠞⠁
⠀⠑⠪⣀⠄⢰⢾⠀⠀⠀⠀⠀⠀⠀⢀⢯⠃⡴⢩⢤⡇⢸⢳⣶⢎⢧⠸⢉⡾⡮⣇⠀⣀⠀⢻⡇⠻⡉⢠⣿⠝⠁⣯⢵⢐⢝⣑⠢⣤
⠰⢖⣊⡉⠀⠸⠿⠀⠀⠀⢠⠀⠀⠐⢸⢿⠀⠓⢳⡇⡇⢸⢺⡇⢳⣻⡀⡞⡇⢸⣽⠘⢾⠃⢸⢿⠀⣯⠝⢾⡄⠀⢸⢽⠀⡻⣟⡋⠁
⠀⢀⣉⡦⠀⢸⣷⠀⣀⡠⠃⠣⢤⡄⠺⣿⠀⠀⢀⡣⣛⠸⡮⣧⣸⡺⡇⢡⢽⣆⢼⡇⡜⠯⢺⣿⠀⠉⠀⠀⠀⠀⢸⢽⠎⣴⢧⣮⠵
⠛⠫⢔⣒⠀⢸⣻⠀⠉⡆⢰⡿⡏⠀⠁⢯⣆⠀⠈⠚⣽⡀⠻⣕⣫⡾⠁⠈⠣⠥⠟⠁⠀⡀⣾⠇⠀⣠⠤⣄⠀⠂⣎⡿⢺⢪⠫⣀
⠀⢀⡤⠊⠁⠀⠿⡆⠀⠷⠁⠉⠓⠀⠈⢌⣿⣆⠀⠓⠋⠁⠀⠀⠀⠀⠀⠂⢀⠠⢀⠀⢂⣼⡟⠀⡀⠑⠢⡬⢀⢵⣶⡵⣡⡥⠷⠿⠓
⠀⠉⠀⠒⡲⠁⠘⡽⣄⠀⠀⠀⠀⠀⠀⠀⠩⠱⣷⣤⣀⡀⠀⠀⠄⠌⠄⠌⠆⠂⣀⢴⡿⠋⠀⠐⢧⡆⣠⠀⢊⢯⡇⣴⣑⣍⢦⡀
⠀⠀⢠⣎⣀⣴⡀⠘⢯⢆⠀⠀⠀⠀⠔⠉⠃⡀⠀⠉⠛⠻⠽⣿⣖⣖⣶⣲⣾⠯⠟⠉⠀⠀⢮⣠⠀⡈⢁⣀⣯⣏⣬⣌⣯⡉⠉⠁
⠀⠀⠀⠀⢀⠞⡀⠀⠈⠻⡵⢄⠀⠸⠔⠃⡜⢀⠎⢀⡆⠀⡤⢄⠀⢀⡀⠀⡀⠀⠀⠐⠤⠀⠀⠐⠊⣰⣺⢯⢟⣾⢷⠻⠿⠟
⠀⠀⠀⠀⠚⠒⠓⣹⠃⢀⠈⠫⢗⡦⣀⠀⠓⠃⡴⠛⡆⢀⠓⡎⠀⣨⢷⠀⠋⠲⣧⠀⠈⠃⠀⠠⣫⢾⢿⢇⢾⣝⣏⣇
⠀⠀⠀⠀⠀⠀⠰⠧⠐⠊⡇⠀⠀⠉⠫⠽⣲⣤⣀⡀⠀⠀⠀⠁⠀⠃⠈⠂⠘⠀⡈⠀⢐⣀⢴⡻⢾⢡⣎⡻⠌⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠔⢻⠀⣠⣀⠀⢝⣞⠻⠯⠧⣖⣶⣤⢤⢤⡤⣶⣲⡖⠿⠽⠺⢁⢂⣥⡑⢹⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠉⢸⠐⡣⢶⡨⢊⣢⣖⢔⢄⡌⡘⣙⡱⡰⣡⣴⡱⣱⠳⡗⠃⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⠁⢸⡀⠏⠁⢳⢃⠎⢫⠊⡘⠱⡜⡰⠈⢣⡝⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠋⠀⠀⠛⠀⠀⠘⠃⠀⠀⠁
⠀⠀⠀⠀⢤⢔⠔⡠⢂⠔⡠⢄⡠⢂⠔⢂⠄⡠⠀⡠⢀⠀⢀⠄⣀⠀⡀⢀⠀⡠⠀⡠⠀⡀⡠⢀⠄⡠⢀⠄⡠⢀⠀⡀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠈⠈⠁⠊⠐⠁⠒⠁⠊⠡⠊⡠⢊⠔⡡⢊⡡⢊⡴⣈⢔⡡⢊⡠⢊⠔⡨⠊⠔⠡⢊⠔⡡⢊⠔⡡⢊⠞⠕⠱⠡⠊⠊

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⢀⣈⣋⣀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⠟⠛⠛⠛⢿⣦⡉
⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⢀⣀⣀⢠⣿⠃⠀⠀⠀⠀⠀⢻⣷⢀⣀⣀⡀
⠀⠀⠸⠿⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⢸⣿⠀⠀⠀⠈⠉⠉⠘⣿⡆⠀⠀⠀⠀⠀⣸⡿⠈⠙⠉⠁
⠀⠀⢀⣀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣙⢿⣦⣤⣤⣤⣾⠟⣁
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⠀⢸⣿⠀⠀⠀⠀⠀⢰⡿⠋⠀⠈⢉⣍⠉⠁⠈⠻⣷
⠀⠀⢸⣿⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⢸⣿
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⠀⣼⣿⡀
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⣾⡟⠹⣷⡄
⠀⠀⢸⣿⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⢛⣿⡏⠀⠀⠹⣿⡄
⠀⠀⢸⣿⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣼⣿⡀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⢿⣶⣶⡾⠟⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⡟⠀⠀⠀⢀⣾⡏
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠘⠿⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠿⠟⠀⢸⣿⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢠⣿⠏⠀⠀⠀⢠⣿⠏
⠀⠀⢸⣿⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠸⠿⢿⣷⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⣻⡧⠀⠀⠀⠀⣹⣷⠀⠀⠀⠀⢹⣷
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠋
⢀⣤⣼⣿⣦⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣧⡀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⠙⢦⣀⡾⠁⠹⣆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡿⢧⡀⠀⠘⣦⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⠀⠀⠙⢦⡀⠸⣇⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⣇⠀⠀⠀⠀⠙⢶⣟⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠘⢦⣄⣀⣀⣤⠞⠙⢧⣄⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠁⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠖⠛⠋
⠀⠀⠀⠀⠀⢀⣤⠶⠲⠦⣤⣤⡤⠶⠲⠦⣤⣤⡤⠶⠲⠦⣤⣠⣤⠶⠶⠶⣤⣄⣤⠴⠖⠶⢤⣄⣠⡤⠶⠲⢦⣄
⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠴⠶⠶⠶⠶⠶⠦⠀⠀⠀⠤⠦⠶⠶⠤⠦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠒⠒⠶⠶⠶⠶⠶⠶⠶⠂⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠻⣆⠀⢀⡴⠋⠳⣄⠀⢠⡾⠛⢷⡀⠀⢠⡞⠙⢧⡀⠀⢠⡞⠙⢷⡀⠀⢀⡼⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠓⠛⠁⠀⠀⠙⠛⠋⠀⠀⠀⠙⠶⠋⠀⠀⠀⠛⠖⠋⠀⠀⠀⠙⠶⠛⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠂⠐⠐⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂⠒⠐⠀⠂⠐⠀⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠂⠐⠐⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂⠒⠐⠀⠂⠐⠀⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⢀⠤⠜⠓⠐⠀⠀⠀⠰⡇⣦⡔⡉⠤⠠⢾⠀⠀⠀⣀⠃⠀⠀⠈⠈⣡⠃⠀⠈⠀⢀⠤⠚⠃⠒⢀⠀⠀⠸⡑⣤⢐⠩⠄⠠⡎⠀⠀⠀⡐⠁⠀⠀⠉⢈⡔⠁
⢿⡃⠀⠀⠀⣠⣾⣄⠀⢀⠜⠁⢀⠤⠄⠀⣨⢀⡠⢴⡉⠡⠖⠊⠉⠈⠉⠉⡹⠶⣾⠁⠀⠀⠀⣠⣿⡀⠀⣠⠋⠁⢠⠤⠀⢀⡌⢀⠠⢞⠉⠰⠒⠊⠉⠉⠉⢉⡵
⠀⣱⠖⠦⠐⠋⠈⠉⠙⡿⠓⠒⠛⠓⠢⣤⠼⣏⠀⠀⠀⠀⠀⠴⠶⠦⠐⠾⠀⠀⠀⡶⠖⠆⠘⠉⠉⠉⢙⡟⠒⠒⠛⠒⢦⡤⢿⡅⠀⠀⠀⠀⡠⠶⠶⠆⠰⠮
⠨⢂⠀⣀⡴⠤⢤⣤⠜⠁⢀⣠⠄⣄⣀⣀⣠⠌⠉⠟⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠹⣀⠀⣀⡴⠤⢤⡤⠞⠀⣀⣠⠤⣄⣀⣀⣠⠈⠉⠟⠉⠀⠀⠀⣀⡀
⠛⠋⢉⣼⣄⠀⣀⣠⠀⠉⠈⢂⠀⠀⠀⠀⠴⠞⠀⢀⣀⡤⠔⠋⠉⠉⠀⠀⣨⠚⠛⠋⣁⣼⡀⠀⣀⣠⠂⠁⠙⢄⠀⠀⠀⠀⠴⠇⠀⢀⣀⡤⠒⠉⠈⠉⠂⠀⡥
⢴⡊⠀⠀⠈⠉⠁⠘⣦⠤⠐⠈⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢀⡠⠐⠒⠚⠳⠦⣴⠊⠀⠀⠉⠉⠀⢹⣄⠄⠒⠋⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣀⠤⠒⠒⠛⠲
⣀⣧⣀⡀⠀⢀⠠⢘⣻⠶⠟⠋⠀⠀⡠⠴⠦⣾⡤⠾⠿⠛⠞⠉⢿⠀⠀⠀⢀⣀⣈⣧⣀⡀⠀⢀⠤⣚⡷⠶⠚⠁⠀⢀⡠⠴⢴⣦⡴⠾⠟⠳⠞⠙⢗⠀⠀⠀⣀
⠀⠀⠀⠈⡹⡡⠞⠛⢦⡤⠖⠙⠒⠋⠀⠀⢀⡏⠀⠀⢀⡴⡀⠀⠀⠈⣻⠋⠁⠀⠀⠀⠀⠈⡟⡴⠚⠛⢤⡤⠊⠙⠒⠉⠀⠀⢰⠃⠀⠀⢀⠜⡀⠀⠀⠙⡿⠉
⢄⠀⢀⡴⠗⠁⠀⠀⣠⠗⠊⠀⢀⣀⠀⣠⠞⠀⢀⣄⢀⡈⡤⠖⢉⠚⠧⠤⣠⠄⠄⠀⢀⠼⠊⠀⠀⠀⣸⠗⠁⠀⣀⣀⢀⣠⠋⠀⢀⡆⢀⠀⡤⠚⢁⠛⠧⢄⣠
⠔⠛⡋⠀⠀⣀⡤⠚⠁⣀⠔⠈⠀⡘⠉⠛⠦⠒⠋⠉⠠⠀⣸⣤⣾⡀⠀⠀⠀⡠⠚⠛⡃⠀⠀⣀⠄⠛⢁⣠⠀⠁⢀⠘⠉⠓⠄⠚⠉⠉⡤⠀⣧⣤⣆⡀
⣠⣒⠁⠐⠈⢿⠆⠉⠀⠉⢧⠀⡸⠁⢀⠔⠀⠀⣀⡀⠤⡦⠋⠁⠀⠈⡣⠐⠔⢁⣴⣒⠥⠐⠉⡿⠊⠁⠈⠙⣆⠀⡞⠀⣠⠄⠀⠀⣀⡠⢤⠜⠉⠁⠀⠉⡦⠂⠊
⠀⠀⠀⢀⡤⠻⣀⣀⣀⣤⣊⠴⠁⣲⣁⠄⠚⠉⣡⣴⠊⠀⠀⠐⠥⠺⠤⣤⠔⠉⠀⠀⠀⢀⠔⢧⡀⣀⣠⡴⣣⠾⢐⣞⣁⠔⠚⠉⣠⣴⠃⠀⠀⢀⠕⠺⠤⣤⠞
⠀⠀⡠⠊⣀⡔⢹⠁⠀⠀⣠⠖⠋⠉⣡⠤⠒⢈⡡⠋⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⡐⠁⣀⡌⢹⠁⠀⠀⡤⠚⠉⠉⣠⠔⠂⢉⡵⠋⠀⠀⠀⠀⣀⣤⡾⠃
⢠⠞⠀⡈⠘⠁⣼⡀⠀⣠⡧⠀⠲⢔⣡⢶⣯⠥⠤⠤⢤⣀⠠⠊⠀⣠⢧⡠⢤⡄⡠⠎⠀⡁⠙⢀⣾⠀⢀⣼⡤⠒⠦⢎⡠⣶⣯⠤⠤⠤⢤⣀⠔⠈⢀⡼⢇⡠⢤
⣴⣀⣘⣀⡴⣚⣀⣉⣉⣙⣦⣐⣚⣋⣠⣮⣌⣿⣀⣀⣀⣠⢂⣐⣉⣀⣤⣠⣞⣁⣄⣀⣚⢠⡖⣋⣀⣉⣉⣻⣤⣒⣛⣉⣴⣧⣬⣟⣀⣀⣀⣸⣂⣚⣉⣀⣄⣰⣊
⣿⢿⠿⠿⢿⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣛⣿⣟⣿⣿⣻⣟⣿

⠀⠀⠀⠀⡆⢀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⡀
⠀⠀⠀⠈⡇⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⡏⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣇⢠⡇
⠀⠀⠀⠀⢻⡄⢹⡄⢻⡄⠀⠀⠀⠀⠀⠹⣆⠹⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⢸⠃⣼⠃⡷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⣷
⠀⠀⠀⠀⠈⣇⠀⣧⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⢠⡏
⠀⠀⠀⠀⠀⡏⠀⠟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢳⠀⠇



⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⡞⠀⡏⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢳⡠⡏⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼
⠀⠀⠀⠀⢷⡀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠈⢷⠈⢷⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⢯⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡄⠻
⠀⠀⠀⠀⠀⢳⠈⢷⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧⠘⠇⠀⠀⠀⠀⠀⠀⠹⢀⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⣆
⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡼⠁⣼⠀⢻⣄⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⢰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⡀
⠀⠀⠀⢧⡀⠘⠷⣄⠈⠙⠆⠀⠀⠀⠀⢻⡄⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠰⡜⣆⠳⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡷
⠀⠀⠀⠈⠻⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢳⣌⢷⡙⣆⠇⠀⠀⠀⠀⠀⠀⢸⡀⠙⣆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣌⡛⠿⣿⣿⠟⢃⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⣿⣏⣿⡟⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣿⣽⣿⣰⣿⣿⢛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣜⣣⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢠⠄
⠀⠀⠀⠀⠀⢦⠀⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠀⢻⡀
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠻⣦⠙⢷
⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡟⣸⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡞⠁⡼⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠜⠁⠜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠡⡎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡇⠀⢆
⠀⠀⠀⠀⠀⢸⡄⢠⡏⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⢋⡵⠲⣄⠀⠀⠀⠀⠀⠀⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣼
⠀⠀⠀⠀⠀⠘⣇⠸⡇⢰⡏⠀⠀⠀⠀⠀⠀⣠⠶⣄⠈⢳⠈⢯⠒⠃⠀⠀⠀⠀⠀⠀⢰⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠧⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠳⠘⠁⠀⠀⠀⠀⠀⠀⠙⠾⠈⠇⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄
⠀⠀⠀⠀⠀⡏⠀⣸⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡇⢀⡄
⠀⠀⠀⠀⠀⢻⡄⠁⠀⣸⠀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠏⣠⠞
⠀⠀⠀⠀⠀⠀⢹⠄⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⡇⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⣱⢸⠀⠀⠀⠀⣱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣋⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣦⣤⣀⠀⢀⣾⡿⠃⠀⠀⠈⠻⣿⣶⣤⡀⣠⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠋⠉⠉⠉⠉⠉⠛⢿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣴⡶⢶⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠁⠀⢈⡗⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⢸⡇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣷⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠆⡆⢴⠠⠀⠀⠤⠤⢠⢆⠃⠠⠀⠠⠲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓
⣦⠤⠦⠤⠤⠤⢤⡤⢤⡀⠖⠒⠶⠤⠦⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠒
⠀⠶⠶⠶⠦⠠⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠦⠤⠤⠄⠀⠀⡀⠠⠄⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⢀⠀⣤⣤⣄⣀⣀⣀⣀⣀⡀
⣦⣿⡿⠛⠿⠯⠿⠿⣿⣿⣿⡟⠛⠒⠒⠛⠙⠛⠉⠉⠉⠉⠉⠛⠛⠉⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠍⠩⠛⠛⠓⠀⠛⠉⠛⠛⠛⠛⠛
⣿⣿⣿⡿⠶⠶⠶⢶⡶⣖⣀⣠⣀⣀⠀⠀⠀⠀⠤⠄⠀⣠⣤⣀⣀⡀⢀⡀⠀⠄⠀⠠⡄⠀⠠⠄⠀⠤⠤⠄⠀⢀⡀⠤⠠⠤⠤⣤⠄⢀⣀⡀⣀⣀⣀⣀⣀
⢠⣤⣤⣄⣀⣀⣴⠔⠀⠀⠀⠤⠤⣤⣤⣤⣠⣤⣤⣄⣠⣄⣠⣄⣀⣀⡀⣀⣀⠀⣀⣀⡚⠛⡛⠚⠛⠒⠒⠂⠲⠤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠬⠷
⠀⠀⠀⣀⣈⣱⣶⠶⠶⠶⠶⠶⠆⠀⠐⠶⠄⠀⠠⠩⣭⣴⣶⡦⢤⣭⣯⣍⣭⣉⣉⣉⣙⣛⣛⣛⠛⠓⠒⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠘
⠀⠀⠶⠿⠿⠟⠉⠉⢉⡙⠙⠋⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⢀⠀⠀⠈⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣿⡟⠒⠶⠦⢤⡄⠀⠈⣄⡁⠀⠀⠀⠀⠀⠄⠀⠀⣤⠀⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠐⠂⠈
⠐⠒⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⠄⠄⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠠⠤⠒⠀⠠⠤⠤⠤⠤⢤⠀⠒⠒⠒⠀⠠⠤⠄⢠⣄⠀⠀⠲⠴⠀⠠
⠤⠄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣐⣒⣶⣶⣦⣤⣄⣹⡟⠛⠙⠀⣐⣒⣲⣶⣶⣶⣶⣦⡴⠀⣀⣠⣄⣘⣓⠀⠠⠬⢤⣭⠙
⠀⣀⣀⣀⣀⣀⣀⣲⣤⣤⣤⣤⣤⣤⣤⡶⢷⣾⣿⣻⡿⢛⠛⠛⠶⠶⠶⠶⢤⣤⣄⣀⡛⠛⠛⠛⠿⠃⠠⢤⣤⣤⣤⣍⣉⣛⣉⠉⠙⠚⠛⠻⠗⠒⠒⠂
⣭⣭⣭⣍⣉⣙⣛⡛⠿⠿⠶⠶⣶⠿⣿⣿⣶⣾⣯⣭⣭⣭⠽⠽⠿⠐⠒⠶⠶⠶⠾⢯⣭⠉⠉⠙⠿⠶⠶⠶⠶⠶⠾⠍⠉⠉⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⢀
⠆⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⣄⣤⣤⠤⠶⠶⠷⠿⣿⣿⣟⣛⣛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⢀⡠⠔⠒⣉⣙⣒⣊⣉⣉⡒⠂⠄⠠⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠉⠉⠀⠙⢄
⠀⠀⠀⡑⠤⢤⣍⣀⣀⣀⣀⣤⠤⠔⠁⠀⠀⠀⠤⠴⠮⢍⣁⣒⣈⣩⣴⣦⠤⠄⠀⠀⠀⠀⠀⣠⡔⠐⠂⠒⠒⢤⡀
⠀⠀⠀⠘⠢⣄⡀⠀⠉⠁⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠈⠓⠒⠁⠀⡄⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠

⠀⠀⠀⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠄⡀
⠀⠀⠀⠐⠒⠤⢬⣥⣄⣀⣉⣑⣢⣤⠤⠀⠀⠀⠈⠀⣀⣀⣀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠿⠿⠶⣭⣤⠤⠀⠀⠀⠀⠀⠖⠚⠡⠾⠯⠋⠆⠑⠢⠄
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⡀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣀⡤⣄⡀
⠀⠀⠄⢤⣀⣀⣈⠡⠤⠴⠒⠒⠒⢰⠟⠀⠀⠀⠀⡠⠊⠀⣀⣀⣀⣀⠀⠑⡄⠀⠀⠀⢠⣐⣊⣁⣀⣄⣀⣀⣤⣀⣉⣒⣠⡄
⠀⠀⠀⠀⠈⠑⠒⠲⠶⠶⠶⠖⠛⢁⠄⠀⠀⠀⢰⣁⣴⢮⣤⣤⣤⣤⣽⣦⣈⡄⠀⠀⠀⢍⠛⠦⢤⣄⣀⣀⣤⠤⠖⢛⠅
⠀⠀⠀⠀⠀⠀⠂⠄⣀⣀⣀⡀⠄⠊⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⢀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠔⠊⠉⠉⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠲⠶⠖⠒⠦⣄
⠀⠀⠀⠀⢀⣴⣋⣉⣉⣉⣉⣭⣦⣄⠀⠀⠀⠀⠢⣤⣄⠠⠥⠤⠒⠒⠒⢀⡾⠃⠀⠀⠀⠀⠁⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⣾⣿⣿⡿⠟⠡⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢫⢏⣼⡿⡵⠁⠀⠀⠀⠀⣼⣿⢛⠶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⢋⡜⡱⢣⢎⣼⠑⠁⠀⠀⠀⠀⢸⡟⣡⢋⠞⡴⣩⢛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡴⣡⢋⠜⡴⢣⢎⡜⣦⣄⠀⠀⠀⠀⠘⢿⣥⢋⠞⡴⣡⢋⠞⡽⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠞⡴⣡⢋⠞⡴⣡⢋⠜⡵⢻⡇⠀⠀⠀⠀⠀⡿⢫⢞⡼⣡⢋⠞⡴⣡⠟⣭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡡⢫⠞⡴⣡⢋⠞⡴⣡⢋⠜⡵⢡⠇⠀⠀⠀⠀⡼⡱⢣⢎⡜⡡⢫⠞⡴⣻⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡜⡱⢫⠞⡼⣡⢋⠞⡴⣡⢋⠞⣴⡏⠀⠀⠀⠀⡍⢿⣱⣧⣎⣜⡱⢋⣮⡴⢿⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⡜⡱⢣⢎⡼⡡⢫⠞⡴⣡⢋⠞⣽⠀⡀⠀⠀⠀⠉⠒⠈⠀⠈⠙⠻⠟⠋⠀⢸⢸⣯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡜⡱⢣⢎⡜⡱⢣⢎⡴⡡⢋⠞⡼⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡵⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡜⡱⢣⢎⡜⡱⢣⢎⡜⡱⢋⠎⣴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢞⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⡞⡱⣡⢎⡜⡱⢣⢎⡜⡱⢣⢎⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣫⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣞⡰⣡⢎⣾⣵⠷⠾⣾⡱⢣⢎⡜⡹⢻⢟⡶⡶⢦⡀⠀⠀⠀⠀⢀⣀⣀⠀⢸⢸⡵⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡈⠙⠛⠛⠉⠀⠀⢠⡘⢿⣧⣎⡜⡱⢣⢎⡼⡱⢋⠟⣧⣄⣀⣴⡿⣩⢋⠗⣿⢸⢟⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠉⠲⠮⠍⠛⠻⣧⢎⡜⡱⢫⢞⡼⣩⢋⠞⡴⣡⢫⣾⠇⣼⣣⣿⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⡱⢣⢎⡜⣱⢋⢞⡽⠳⠟⢋⠾⣿⣿⣿⡿⣜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⢣⢎⡔⡱⢣⢾⡄⣶⠰⠿⣾⠁⠀⠈⢾⢾⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠐⣧⢚⡟⡱⢣⢎⡜⡱⣣⢾⣾⣿⡰⣷⡿⣦⣠⣤⡶⣶⠆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⣀⣀⡈⠛⠿⣽⣂⡀⠀⠀⢹⢏⠞⡱⣣⢊⣜⣵⡗⠋⠁⠀⠳⣜⠿⣽⣽⣵⠵⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠭⢍⠉⠚⠛⠛⠛⠛⠛⣛⡋⠉⠉⠉⢉⡥⠭⢍⣉⠙⠓⠚⠓⠲⠶⠖⠚⠛⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠲⠤⠄⠀⢉⣉⣉⣉⡤⠀⠀⣀⠀⢀⣤⣤⣀⡀⠉⠉⠓⠒⠊⠁⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣀⣀⡀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡾⠉⠀⠙⣧⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠆⠀⢀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣇⣀⣀⣀⣸⠀⢸⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡷⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠇⠀⣿⣿⣿⣿⣿⠀⠸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠆⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣼⣿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠘⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣄⡀⠉⠉⠉⠉⠉⣀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⢀⣴⣾⡦
⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣴⣶⣶⣶⣶⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣶⣶⣶⣶⣦
⠉⠉⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣤⣶⡆⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣾⠋⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣧⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣧⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⡿⠃⠀⠀⣿⡇⢠⡠⠆⢤⡠⠠⣄⠔⠢⡀⢸⡇⠀⠀⠹⣷⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⣠⡿⠁⠀⠀⠀⣿⡇⢄⣠⠄⢄⡠⠦⣀⠔⠤⡀⢸⡇⠀⠀⠀⠹⣷⣀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⣿⡇⢄⣀⠤⢄⣠⠠⣀⠐⢤⠀⣼⡇⠀⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠁⠀⢈⡴⣦⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠀⠃⢸⡇⣻⠀⡉⢰⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⠀⡔⢸⡇⣽⠈⢇⠰⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢡⠀⠇⢸⡇⣻⠀⡔⢠⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠀⡌⢸⡇⢾⠀⢇⠸⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢃⠀⡇⢸⡇⢻⠀⡬⢀⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠘⢸⡇⣿⠐⡆⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠎⠀⡁⢸⡇⣹⠀⢙⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⠀⠀⣸⡇⢿⡄⠀⠈⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣶⠟⠁⠘⢷⣦⣴⡾⠋
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣹⣼⠺⡎⣏⣱⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡺⡶⢤⡐⢾⣾⡟⣼⣈⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡎⢅⠢⡻⣮⡂⢳⣿⡿⠟⣩⣾⣿⣝⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⣿⣮⣡⣄⣘⣿⣿⠟⣡⣾⣿⣿⣿⡛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣿⣭⠿⣛⣥⣾⣿⣿⣿⣟⣎⣳⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⢷⣽⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣛⣯⠮⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠲⠶⠳⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠙⠃⣠⠾⡾⣋⢨⢈⣤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣘⣡⣄⣠⡄⠉⠉⠻⡅⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢃⠀⠈⠙⠳⠃⠦⡀⠄⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠰⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠤⠒⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠸⢄⣀⡴⠋⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⡴⠃⠀⠀⠐⠀⠀⠘⠁⠀⠀⠀⡠⠜⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⡴⣶⠟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⣠⠆⠑⣻⡿⠡⡀⠀⣜⠳⠒⠶⠶⠦⠤⠀⢀⣰⠇⢰⠀⠀⠀⠂⠀⠀⠈⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣶⢡⡁⠀⠀⠁⠀⠀⠀⠙⢣⡤⣀⠀⠀⠁⠀⠀⠀⠉⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠟⢦⡀⠄⠀⠠⠀⢳
⣀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢠⡉⠢⡀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⢤⡀⠄⠁⣠⡾⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠑⢄
⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠉⢦⡈⢂⡀⠀⠀⠀⣀⠐⡏⠀⠀⠀⢈⣵⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠈⡇⠀⠀⠐
⠀⢸⡟⠿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠱⡄⠀⠱⣄⣳⣄⠴⣋⡤⠖⠁⣀⣤⠾⠋⣁⣴⠋⠈⠁⠀⠈⠀⠀⠈⠀⠀⠤⡴⠀⠀⠀⠀⠀⡼⠃
⠀⢸⡇⠀⠈⠛⢷⣄⠀⠘⡄⡆⠀⠀⠘⣆⠀⣀⠤⠒⠉⠁⣠⣴⠞⠛⠛⠉⠉⠉⠈⠳⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⢀⠀⢈⡼
⠀⢸⡇⠀⠀⠀⠀⠙⠳⣦⡁⠸⠆⠀⠀⠈⠋⠁⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⢀⡇⠀⡀⠠⢏⣀⠀⠀⣠⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣴⠾⠋⠁⠀⠀⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣸⠟⠳⣏⢰⠆⠀⡸⠖⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⢹⠀⠀⠀⠀⠲⣼⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁⠀⢀⣿⠁⢀⡴⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⢀⣀⣤⣶⢮⣭⣤⣤⣤⣤⡤⢶⣿⠈⠉⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠼⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⡄⠀⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠹⡄⠀⠀⠀⠀⠀⢳⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣸⡇⠀⠀⠹⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠘⣿
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣿⣀⣤⡴⠿⠛⠉⠉⢻⡏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣠⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢀⡼⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠞⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠀⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⣷
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⢿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⠲⢤⣀⣀⣀⣀⣀⣤⡶⠀⠀⢀⡇⠀⠀⠀⠀⠐⠛⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠤⣀⣉⣉⣩⠵⠋⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡠⠒⠀⠁⠀⠀⠀⠈⠙⠳⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠄⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠀⠀⠀⠂⠐⠂⠀⠀⠀⢄⣀⣀⠀⠀⣻⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢀⠤⠆⠶⠤⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠘⡇⠀⠂⠀⠀⠀⠈⠂⠁⠀⠀⠀⡀⠀⠀⣎⠇⠀⠀⠐⠁⠐⠄⠂⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⢄⡀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠼⠟⠀⠐⠀⠂⠀⠀⢀⠠⠈⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠉⠒⠂⠀⠐⠂⠁⠀⢀⣼⠝⠁⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣤⣀⣀⣀⣀⣤⡖⠟⠁⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠁⠈⠀⠀⠢⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠡⠀⠄⠂⠀⡀⡔⠔⡨⡤⢄⠀⢉⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⣫⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠔⠈⠠⠄⢁⠀⢠⠆⠪⢔⠊⡢⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⢎⣁⡀⠈⠢⣄⠀⠀⠀⢀⠆⣀⠐⠈⠊⠐⠀⡊⠔⠪⠒⠡⢺⠀⠀⠀⠉⠢⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡜⠂⡙⢤⡈⠣⡀⢀⠎⢈⠡⣰⠀⡀⠀⠠⢀⠅⡢⠐⣀⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⠀⠘⡄⠦⡢⠑⠶⠀⠐⠀⢸⠀⢒⠁⡠⢀⠡⠀⠈⠂⠅⠀⠀⢀⠊⠈⠔⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠂⠕⢀⠌⠀⢸⠀⠁⠁⡊⠐⡈⠐⠈⡠⠙⣄⡴⠕⠊⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⢀⠀⢁⠀⠀⠀⠈⡀⠅⠠⢀⢀⡀⢈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠄⡂⢐⠠⢂⠡⠂⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⢀⠦⠈⠁⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠅⠈⠘⠀⡄⠠⠆⡈⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡂⠀⠀⠂⡇⠀⠁⠀⠠⢀⠀⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠂⠀⠀⠠⠀⢊⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸⠀⠀⠈⠀⠀⠀⠊⠄⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠄⠄⠸⠀⠀⠀⠇⠀⠊⢀⠄⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣔⠁⠀⠀⠀⠐⠃⡖⢀⠄⠠⢠⠀⠐⠁⠊⠂⠂⠈⣓⠀⡀⢄⠠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠃⠒⠐⠂⠊⠁⠈⠐⠈⠘⠁⠚⠛⠋⠘⠁⠊⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⣄⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢂⠀⠀⠁⠂⢂⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠐⢤⡏⠈⠡⠂⠠⡂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢠⡄⠁⠀⠀⠀⠀⠈⠀⣹⠀⠀⡀⠘⠀⠀⠀⠀⠀⣀⠀⢈⡘
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⡀⠤⠀⠀⠀⢐⠀⠀⡄⠀⢠⡤⠂⠉⠁⠀⠀⠉⠀⠔⠄⠀⠀⡢⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠂⠐⡜⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣵⡴⡄⠈⠀⠈⠀⢣⠀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠀⡈⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠘⢰⠓⠄⠊⠀⠍⠀⡀⠀⠒⠀⠆⡗⢢⠔⡠⠂⠠⢈⡒⠡
⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠃⡀⢄⡀⠒⠀⠀⢠⡟⡜⡀⠀⢠⠄⣡⠫⠐⠍⠀⠴⢆⠖⠁⠀⠠⣈⡬⠀⡛⠠⠂⢁⠰⠀⡜⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⡢⠀⠠⠠⠀⢢⣄⣀⣠⠋⢉⠀⠉⡒⢡⠚⣋⠤⣄⠈⢤⠲⠧⠤⠤⣤⣈⣼⢡⠨⢥⠈⡁⠡⠤⠂⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⡋⠙⠂⠀⠁⡤⣄⣀⣈⣸⠆⠀⠉⢈⣉⣀⢀⠀⠐⡀⠀⣢⠂⢀⢹⣴⠃⣾⠷⠧⣔⡁⠁⠐⡀
⠀⠀⠀⠐⠐⠀⡀⠀⠀⠈⠀⠙⠢⡄⢀⣾⡁⠶⠛⡦⠀⢀⡀⠴⠊⠈⠙⠲⠖⠁⠛⠊⢈⡇⠀⠀⡀⠀⠍⣀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠀⠁⠀⠁⠀⠁⠈⣑⡄⢠⣖⣟⣁⣀⠨⣁⡴⠖⡀⢉⡐⠂⠁⢀⠠⢀⠀⠀⠐⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠂⠄⠀⠈⠘⠀⠂⢐⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠄⠉⢁⠄⠴⠿⠍⠒⠛⠀⢀⠀⡐⠀⠂⠀⠀⠉⢉⡉⢀⢀
⠀⠀⠘⢀⠀⡨⠘⣯⠁⢂⠀⠀⡠⣞⠀⠁⠀⠀⢐⠃⣼⣫⡝⢀⡀⠀⠔⠀⠑⡄⢁⠀⠆
⠀⠀⠀⠘⠐⠻⢿⡁⠀⠈⠉⢉⠐⠀⠀⠀⠀⢓⠡⠊⠁⠀⠀⠀⠀⠀⠀⠈⢀⠁⠨⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⣇⠀⠀⠀⠃⠛⠀⠄⠐⠈⠁⠀⠰
⠀⠀⠠⠀⠀⠀⢀⢈⡷⡂⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠠⣀⠼⢋⢛⢂⠀⡀⠀⠁⠀⠤⠔⠸⠄⢐
⠀⠀⠀⠀⠚⢿⠨⠐⠂⡐⠂⠀⠄⠀⠀⠀⠀⣀⠒⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠚⣉⠉⠀⠀⠀⠈⠉⠻⢿⠲⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡅⠀⢀⠜⢹⡀⡞⠦⣄⡀⠀⠀⣀⠀⠀⠉⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⠗⠛⠛⠂⠻⠇⠀⠐⠛⢿⡓⠻⡆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⡾⢶⡄⠀⠀⢀⣤⣤⡀⠀⠀⢹⠀⠀⠀⢸⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⡻⠷⡖⡶⣆⠀⠀⠀⠀⠙⠀⠀⢸⢀⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠖⣫⠔⠶⡙⣦⣳⣹⡿⡇⠀⠀⠀⠀⠀⠀⠘⣫⠄⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇⢰⡧⠚⠛⠋⢻⠙⠛⢉⡇⠀⠀⠀⠀⠀⠀⣠⣁⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡇⠀⠀⢻⣴⣳⣤⣤⣤⠾⢶⣛⠉⠀⠀⠀⠀⣀⣤⡞⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⣇⠀⠀⠈⢻⡇⠀⠀⠘⠀⢴⡛⠙⡟⠛⢃⡬⠟⠉⣻⡉⢉⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢦⣀⡴⠋⠀⠀⠀⡀⠀⢠⡇⢰⣧⠔⠋⠀⢰⡏⠉⠀⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠋⢀⣀⣠⠤⢴⠇⠀⠀⢻⡼⠁⣀⡤⢾⠯⣅⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠛⠋⠉⠁⠀⠀⣼⠀⠀⠀⣼⠗⠋⠁⢀⡟⠲⠦⣌⡓⡆⣀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⡇⠶⠀⠀⢸⡇⠀⠀⠀⢹⠇⣼⠳⣄⣤⠦⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠈⠛⠶⠤⠴⠏⢰⣧⠴⢯⡉⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠶⠀⠀⠀⠀⠀⠀⢀⡤⢿⡁⠀⠀⠙⣆⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⣫⠴⠒⠛⣆⠀⠀⠘⡆⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢠⡄⠀⠀⠀⠀⠻⣷⣒⣛⣩⡌⣇⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⢶⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⣠⣶⡦⠀⠀⠠⠄⠠⠤⠠⠀⠤⠠⠄⠀⠄⠆⠀⠀⠄⠀⢠⣦⡠⢀⣤⣀⣀⡀⡀
⠀⠻⣿⡆⠀⠀⠠⠠⠀⠀⠀⠠⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠻⠐⠟⠱⠿⠯⠿⠿⠶⢶⣤⣆⣀
⠀⠀⠈⠀⠀⠈⢰⢲⣠⢦⣀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠆⠈⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⣆⣰⣀⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢛⠈⠁⠁⠁⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠟⠻⠘⠃⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡤⠀⠀⠄⠈⠀⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡶⠩⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠒⠞⠀⠀⠀⠠⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠋⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠗⠒⣀⠀⠀⠀⠀⢀⣾⣄⠄⠀⣀⠀⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢤⡤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠈⠀⠈⠉⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣒⡒⠖⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⡌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⠀⠒⠒⠠⠤⣀⠀⠀⠀⠀⠸⠀⠀⠙⢄⠀⢠⣚⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⣲⣦⣤⣤⣤⣀⣀⠀⠀⠀⠈⠑⠤⡀⠸⠀⠀⠀⠈⡆⠉⣠⡴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⣻⠿⠛⠛⢿⣿⡶⢦⣄⣀⠈⠑⣄⠀⠀⠀⢡⠀⠀⠈⠓⠊
⠀⠀⠀⠀⠀⠀⣄⠀⢸⡆⢀⡆⠀⠀⠀⢸⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⢠⢝⡛⠛⢢⣌⢆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠙⠂⠈⠁⠘⠃⠀⠀⠀⢸⠀⠀⠞⢹⠄⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⢿⣿⣆⢠⠁
⠀⡤⠒⠒⠒⠀⠀⠠⠤⢤⣤⣤⣤⣤⣄⣬⡀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠁⠱⠀⠀⠀⢸⣿⡟⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠻⠙⢿⣿⣿⣟⠲⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠁⢸⣿⠀⠰⠒⣹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠊
⠈⠢⠤⠤⠤⠄⢀⣀⣠⣿⣄⠀⠀⠀⠀⣾⡯⠊⠉⠋⠥⣀⣀⣀⣀⣀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢢⠀⠀⠀⢡⡤⠴⡖⠛⣿⣧⣀⣀⣠⣤⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⠀⠀⢳⡈⠸⢠⣼⠿⣛⣛⢍⠩⡍⠙⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⡀⠀⠀⠀⠙⠛⠉⣷⢋⠰⠀⠀⣀⠈⠂⣼⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⡄⠀⠀⠀⠀⠀⢸⡎⡇⣀⣈⢙⣡⡾⢋⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣶⢶⠒⡛⣷⣭⣵⠾⣛⡉⠊⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⠙⠉⠁⠀⠀⠀⠁⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠀⠸⡞⠁⢀⡂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠
⢀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠃⠀⠉⢀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢂⠊⠀⠠⡆⠀⠀⠁⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⢁⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⡤⠀⢀⠀⠀⠀⠠⠂⢀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⢀⡆⠀⠀⠀⠀⠁⠀⡔⠀⡀⠀⠀⠀⠀⠒⡂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠀⠀⠁⠀⠤⠀⡀⢀⠀⠐⠀⡀⠈⠀⠆⠀⠀⠀⠀⢀⠀⡐⠤⠀⠀⠀⠢⠀⠀⠀⠀⡀⠀⠀⠊⠄⡀
⠴⠂⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠁⡀⠀⣀⠄⠀⢀⠂⠀⠀⡀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠐
⠀⠀⠀⢨⡇⠀⠀⡀⠐⠤⡁⠀⠀⠀⠀⠀⠆⠒⠊⢀⠀⢀⠘⠂⠀⠐⠆⠀⢁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⢠⡔⠈⠁⠀⠀⠁⠀⠲⣉⠀⠈⠘⠀⠀⠀⠐⠒⢄⠀⠀⠀⡠⠀⠂⠀⠔⠀⠀⠉⠀⠂⠀⠀⠀⠒⠒⠀⠒⠘⠂⠤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⠀⠘⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⡔⠀⠀⢠⠄⢈⠐⠀⢀⠀⠘⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣇⣾⡇⠀⣠⠏⠀⠀⠀⠀⠀⠐⠊⠁⠔⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣄⡀⢀⠀⠀⠀⠀⠀⡀⢠⣿⡿⠃⢸⣏⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊
⠀⠀⠀⠀⠀⠼⢳⣞⠀⣤⣀⣠⣾⣷⢋⣵⣦⡴⠚⠋⠀⠀⠀⠀⠀⠀⠉⢱⣄⠀⠀⠀⠀⠀⠘⠁⠀⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⣿⠟⠻⢣⣿⡟⠉⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠖
⠀⢠⣤⡄⠀⣼⣿⣿⡏⠙⣿⡙⣷⣿⣏⣤⣾⣿⠀⣀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠉⠓⢺⣯⠭⠛⠂⠿⠇⠸⠯⠿⠟⠛⠋⠿⢛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠒⠒⠉⠤⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢀⣙⣷⡑⢿⡀⣆⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠈⠙⠟⢮⡁⣺⣿⣿⠀⠀⠄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⣆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠃⠈⢻⣿⣿⣾⢠⡿⠂⠀⠀⠀⠀⠀⠀⠘⡀⠐⡗⡀⠀⠀⠀⠀⠀⠀⡀⣜⢷⣿⣿⡁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣎⣧⠀⠠⣿⣿⡇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠐⣼⡷⡄⣠⣀⢀⣿⣿⠗⣾⠟⣁⡀
⠀⠀⠀⣤⡤⣠⣀⣠⣶⣿⢿⣿⣷⣄⣿⣿⡄⢻⣿⣷⣤⣐⡤⣄⠀⠀⢠⣤⡄⢻⣿⣿⣍⢫⡿⠟⢱⡷⣸⣯⣞⣁⣠⠀⢀⠤
⠀⠀⠀⠙⠺⠭⠍⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⢙⣛⡋⠁⠉⠀⠀⠀⠉⠓⠋⠉⠉⠛⠛⠛⠛⠛⠛⠋⠭⠭⠛⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀⡤⠴⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠒⠒⠊⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⠤⠔⠓⠢⡀
⠀⠀⠀⠲⡌⠭⠤⠤⠤⠴⠶⠶⠤⠤⠤⠤⠭⡕⠂⠁⠀⠀⠀⠈⠀⣀⠠⣀⣀⣀⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⠀⠀⠀⡀⠀⠈⢆
⠀⠀⠀⠠⡈⠓⠲⠦⠤⣤⣤⣤⣤⠤⠶⠒⢋⡀⠀⠀⠀⠐⠒⠚⠛⠲⠤⠤⠭⢥⣤⣴⠿⣷⠖⠒⠂⠀⠀⠀⠀⠀⣀⣴⣊⣀⣉⣉⣉⣉⣓⣤⡀
⠀⠀⠀⠀⠈⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠐⠢⣀⠀⠀⠀⠈⠓⠒⠁⠀⠠⠀⠀⠀⠀⠀⠊⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡈⠉⠓⠦⠤⠤⠴⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠑⠒⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠢⠤⠖⠒⠢⡀
⠀⠀⠀⠀⡈⠉⠓⠶⠶⠶⠤⠤⠤⢤⠭⠵⠶⠦⠂⠀⠀⠀⠈⠀⢤⣄⣠⣤⣀⣀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⣀⠀⠀⠀⣀⠀⠈⠢
⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠀⠀⠀⡀⠀⠉⠑⠲⠶⠿⠷⢶⣭⣤⡤⠄⠀⠀⠀⠀⠀⠰⠒⠛⠳⠶⠯⠭⠵⠶⠛⠲⠦⠄
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠒⠒⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠒⠋⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠲⠶⠖⠒⠲⢤⡀
⠀⠀⠰⠂⢤⣤⠤⠬⠤⠴⠖⠒⠛⠋⠉⠉⣩⠟⠀⠀⠀⠀⠀⣠⠎⠀⢀⣤⣀⣀⣠⣄⠀⠈⢦⠀⠀⠀⠀⠠⣤⣉⣠⣤⡤⠤⣤⣤⣤⠤⢤⣤⣌⣁⣤⠤
⠀⠀⠀⠀⠀⠀⠉⠒⠲⠶⠶⠶⠶⠶⠞⠛⠁⡐⠀⠀⠀⠀⢰⢁⣠⣶⣧⣤⣤⣤⣤⣤⣽⣦⣀⢣⠀⠀⠀⠀⢈⠙⠳⠦⣤⣀⣀⣀⣀⣠⣤⠴⠞⢋⠅
⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠉⠁⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠁⢀⡀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠛⠒⠒⠛⠛⠓⠦⡀
⠀⠁⠀⠀⠀⢀⣤⣞⣉⣉⣉⣉⣉⣉⣱⣦⡀⠀⠀⠀⠀⠀⠦⣤⣤⠄⠪⣥⠤⠴⠶⠒⠒⠒⣸⡟⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠐⠛⠛⠻⠿⢿⣿⣿⡿⠿⠛⠛⠛⠂⠀⠀⠀⠀⠀⠈⡙⠿⣶⣶⣶⣶⣶⣶⣶⡿⠋⡄⠀⠀⠀⠀⠐⠒⠚⠛⠻⠿⠿⡿⠿⠟⠛⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⠉⠉⠛⠉⠉⢀⠜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣤⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣼⡿⠁⠈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⣾⡿⠁⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠟⠋⠉⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠛⢉⣙⣛⡒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢠⡀⢀⣴⣿⠟⠁⣀⣴⣿⣿⡿⠟⢛⣉⣡⣤⠤⠤⠀⠈⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣷⣿⡿⠁⢠⣾⣿⣿⠟⣉⣴⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢰⡀⢹⡟⠀⣰⣿⣿⠟⣡⣾⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡆⠘⣧⠀⠀⣰⣿⣿⠋⣰⣿⡿⠿⠿⠿⠿⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢸⣿⠀⣿⠃⢰⣿⣿⠇⠀⠉⣀⣀⣤⣤⣤⣤⣀⣀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡞⣿⠀⠀⠀⡔⠀⠀⠀⢠⡀
⠀⠀⠀⣼⡟⢠⣿⠀⣼⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣷⣦⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣾⣿⠸⣰⣆⢸⣇⠀⠀⢠⣿⣷
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⡖⢀⣴⣿⣿⣿⠀⣿⣿⣦⣿⡀⣶⠘⡿⠁
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⡆⣽⣿⣿⣿⣷⣿⡇⠀⠀⡄
⠀⠀⠀⢿⣿⠈⣿⡀⢸⣿⣿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⣿⣿⣿⣼⣴⣿
⠀⠀⠀⢸⣿⡄⢻⣇⠀⢿⣿⡇⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢼⡔⢋⡡⠖⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⠘⣿⣿⣿⣿⣿⣿⡿⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣧⠘⣿⡌⠘⣿⣿⡄⠹⣿⣆⢤⣤⡤⠶⠶⠚⣛⣩⡥⠎⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⠉⠿⣿⠿⠋⠀⠀⠀⣸⣿⣿⡿⠋
⠀⠀⠀⠀⠹⣿⣆⠘⣿⣆⠘⢿⣿⣄⠙⣿⣧⡐⠒⠚⠛⠋⠉⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣭⣤⣤
⠀⠀⠀⠀⠀⢻⣿⣆⠘⢿⣦⡈⠻⣿⣧⡈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⠤⠤⠤⠤⠀⠈⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣧⡈⠻⣿⣦⡈⠻⣿⣦⣈⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣛⠿⢷⣭⣛⡦⠄⠀⠀⠙⠻⢦⣤⡀⠀⠀⣀⣴⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣌⠙⢿⣦⣈⠛⠿⣷⣦⣄⡉⠛⠻⠷⣶⣤⣤⣄⣛⣛⣶⣾⣯⣭⣛⣛⣛⡤⠤⠐⢀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣌⡙⠛⠦⣌⣙⠛⠿⢷⣶⣤⣤⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⢁⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⡿⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣿⣧⣴⡷⠀⠀⠀⠀⠀⢀⣴⣾⣶⣄⠀⠀⠀⠀⠀⠰⣷⣤⣿⣧⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠋⠀⠀⠀⠀⠀⣠⣿⠟⠁⠙⢿⣧⡀⠀⠀⠀⠀⠈⠻⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⢸⣿⡀⣿⣧⡀⠀⠀⠀⣼⣿⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠈⢿⣧⠀⠀⠀⢻⣷⣌⠛⠁⠀⢀⣴⣿⠃⠀⠀⢰⣿⠏⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⢸⣿⡇⠀⠀⠀⠙⠻⠿⣿⠿⠿⠋⠁⠀⠀⠀⣿⣏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⢀⣀⣠⣤⣤⣀⡀⠀⠀⣀⣠⣤⣤⣀⣀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠋⠉⠛⠛⠿⠿⠿⠛⠛⠉⠛⠛⠿⠿⠿⠟⠛⠋⠙⠛⠻⠿⠿⠟⠛⠋⠉⠛⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⢀⣀⣤⣤⣤⣀⣀⠀⢀⣀⣠⣤⣤⣄⣀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠋⠉⠙⠛⠿⠿⠿⠛⠛⠉⠙⠛⠿⠿⠿⠟⠛⠉⠙⠛⠻⠿⠿⠛⠛⠉⠉⠛⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⢿⣧⠀⠀⠀⣰⣿⠋⢻⣷⡀⠀⠀⢠⣿⠏⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⢀⣸⣿⠀⠀⠀⣿⣧⡀⣠⣿⡇⠀⠀⢸⣿⡀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⠙⠿⠿⠛⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⣠⣴⣄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠉⠙⣿⣿⡟⠉⠉⢻⣿⣿⠋⠉⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠀⠀⢀⣀⣀⡀⠙⠿⠟⢀⣀⣀⣀⠀⠀⠀⣿⣿⣿⠀⠀⠐⣿⣿⣇⣀⣀⣸⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠀⠀⢿⣿⣿⣿⠂⠀⠀⢿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠉⠁⠀⣴⣶⣦⠀⠉⠉⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇
⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠘⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⠈⠉⠁⣴⣶⣶⣤⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⢀⣀⡀⠻⠿⠿⠟⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⣶⣿⣿⣦⠀⠉⠀⣴⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠛⠛⠛⠋⢀⣤⣀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢀⠀⠀⠀⢻⣿⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢃
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠓⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢋⠹⣿⡟⣕⠒⣪⢻⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣴⢦⡀⢠⡞⣭⡝⣦⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡟⣡⣿⣷⡘⢧⣙⠶⣋⣼⣿⡇⠀⢸⡇⠀⠀⠀⢀⡾⠁⠈⢳⡌⠷⣭⡵⠏⠀⣿
⠀⠀⠀⢸⣿⣿⣿⠏⣴⢋⣼⣿⣿⣌⢻⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⣠⠟⣰⠇⠀⠀⠹⣆⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⡏⢸⠃⣾⣿⣿⣿⣿⡆⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡟⢸⠃⠀⠀⠀⠀⢹⡆⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣷⠸⣧⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⢷⡘⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣷⣌⡛⠻⠛⣋⣥⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠈⠳⢦⣤⣤⡤⠞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡟⣻⢻⠿⣿⢿⢛⠟⢻⢻⣛⣿⢟⡟⢻⡇⠀⢸⣧⣄⣤⣤⡤⣤⣤⣠⣄⣠⢠⢤⣠⣤⣤⣿
⠀⠀⠀⠸⣧⣵⣼⣶⣿⣾⣬⣶⣼⣶⣿⣿⣮⣷⣾⠇⠀⠘⣟⣛⣛⣛⣃⣃⣛⣙⣓⣛⣚⣘⣛⣛⣛⡟
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠰⡄⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⣄⠀⠙⣄⠀⠀⠹⡄⢀⠀⡇⡀⢰⠇⡀⠀⡴⠁⠀⢀⡤⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠈⠳⢄⠈⢢⠐⢄⠱⠸⠄⡇⡇⡞⡜⢠⠎⡠⢀⡴⠋⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠠⡑⠄⠑⠌⠀⠁⠀⣇⠀⠐⡰⠃⢊⠔⠉⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⡀⠨⣑⠈⡀⠀⠠⢀⠠⠤⠤⠄⡈⠢⠀⡀⠔⣋⠤⠂⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠲⠶⠶⠤⢤⣄⣀⣀⡨⠁⠂⠁⢀⠋⡐⠁⠀⠀⠀⠀⠈⢢⠁⠀⠠⠀⠂⠉⣀⣀⣤⢄⣠⣤⣤⠖⠒⠶⠤⠶⠒⠲⠦
⠀⠀⠀⢀⣀⣀⣠⠤⠤⠤⠤⣀⡀⠀⠈⠉⠈⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡖⠉⠉
⠐⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠓⠤⢄⡀⠀⠀⠀⢣⣀⣀⣀⣀⣀⠤⠚⠂⠉⠉⠉⣡⠏⢁⣠⠀⠀⠠⠤⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⡀⠄⠂⢀⡀⠀⠀⠀⠀⢀⠀⠉⠑⠒⣍⣁⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⢁⣐⠊⢀⡀⠀⣠⠤⠒⠉⠉
⠀⠀⣀⡀⣀⡀⢀⠀⠀⡁⢈⣁⣐⢒⣤⣍⣀⣀⣈⣨⣽⣦⠄⠀⠀⠀⣀⡼⠁⠄⠐⣀⠀⠀⠀⠀⡩⠄⣲⠉⠲⡒⡢⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡼⣃⡤⠷⣬⠶⠷⠬⣁⣀⣀⣐⣠⣅⣀⣄⣈⣉⣀⣠⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⢖⡒⠒⢦⠶⠚⢛⡛⠉⠠⠤⠤⠤⠤⠤⠴⠀⠨⠉⠁⠀⠀⠈⠉⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠴⠚⠉⠉⠉⠉⠉⠉⠙⠯⢤⠀⠀⠀⠀⠤⠶⠮⠍⠤⠤⠥⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠢⠀⠠⠤⣄⡀⠀⠀⢀⣀⠀⠀⠀⣄⣀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⢅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡂
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠒⠢⢄⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠰
⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣨⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠉⠻⢶⣤⡤⠞⠉⠁⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠁⣀⡀⠀⠀⢠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⠃⡀⠀⣿⡇⠀⠀⠀⣷⡀⡀⣀⣿⢻⡆⠀⠀⣰⣷⢷⠃⡀⠀⢀⠀⢀⡀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣧⡿⢃⢃⠠⣸⣿⣦⢰⡐⡑⣷⢷⢻⣿⣿⣿⣦⣿⣿⣟⡞⡸⢡⢧⣿⢇⣾⡖⠀⣀⡄⠸⠻⣦
⠀⠀⠀⠀⣠⣾⡿⣿⣏⣷⠘⢏⣿⣽⣝⣿⣦⣣⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣯⡟⣠⣾⢯⣿⢃⣷⢹⣧
⠀⠀⢀⣼⣿⣿⣿⣿⣯⣝⣷⣶⣿⣿⢿⡿⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠟⠛⢏⠻⣿⢿⣷⣿⣵⣷⣷⣧⣞⣷⡀
⠀⠀⠈⢻⣿⣿⡿⣿⣿⣿⣿⣯⣻⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠘⣿⢻⢿⣿⢿⡿⣿⠇
⠀⠀⠀⠀⠻⣟⣼⡿⣹⣿⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣮⣿⡆⡏⡏⣿⠋
⠀⠀⠀⠀⠀⠙⢎⠱⠻⣧⣿⣿⡏⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣼⣸⣿⣿⣇⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠏⣿⢠⣿⣧⡞⢠⣄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢀⠀⡆⢀⢠⣿⣧⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠉⢻⣿⣿⣿⣿⡏⡞⣰⢀⢠⡇⣀⠀⡀⠀⢀⢰⣼⣶⣷⣼⣿⣿⡟⢻⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣿⣿⣿⣿⡇⠇⡇⠆⠈⠁⣿⡿⣿⣿⣿⢸⣿⣿⣿⠿⣛⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣤⣅⣀⣀⣀⣃⡀⢿⣾⣈⣈⣤⣼⣷⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣷⣦⣀⣾⣿⣿⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠹⡿⠿⠀⠛⠃⠉⠀⠈⠁⠈⠁⠿⣿⣿⣿⢹⣿⣿⡿⠛⠋⠀⠉⠉⠀⠙⠛⠛⠛⠿⠟⠛⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⣬⣍⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣤⣶⣾⣿⣦⣴⣶⣶⣴⣶⣶⣶⣦⢀⡀⠀⣀⠀⠀⠀⠀⢠⣄⢀⣤⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⡿⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⡟⢿⡿⣿⣿⣿⣿⣿⣿⣿⠏⠿⠟⠛⠃







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠻⣿⡿⣛⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣴⣶⣶⣦⣤⣤⣦⣶⣶⣦⣶⣦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣩⣥⣭⣙⢿⣿⣿⣿⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⢅⣾⣿⣿⣿⣿⡆⢩⣭⣭⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡜⣿⣿⣿⣿⣿⡟⠸⠿⠿⡿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣷⣜⠿⠿⠿⢟⣵⣶⣶⣶⡇⠀⠸⠿⢛⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣷⣦⣝⡻⠿⠿⠿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠈⢿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇⠀⡅⡃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣀⣀⣤⠤⢄⡀⠀⠀⣾⠑⠄⠀⠀⠀⡠⠞⡠⠚⡠⠃⠀⠼⠋⠅⠀⠀⠀⠀⠀⣸⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣨⣭⣈⠉⢢⡈⢆⠀⠉⠀⣄⠀⠀⠎⠁⠀⢀⣴⣁⠔⠂⣠⠔⠁⠀⠀⠀⠀⠀⣼⠃⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠳⠀⡇⢨⡀⠘⠀⠸⡖⠀⢀⠀⣀⣆⣻⠁⠀⢰⡟⡀⠠⢀⠀⠀⠀⠚⠁⣠⢆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠘⢷⣇⠀⣠⠴⣛⣋⡭⠭⠭⠭⣍⣛⠲⢦⣠⠈⠀⠀⠀⠀⠀⡉⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡾⣫⠴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⡳⣤⢨⠾⠭⣀⡀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠉⠱⣀⣀⠠⠁⣴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢳⡀⠀⣀⣠⠀⣁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠈⠙⠓⢸⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢳⡈⢉⠁
⠀⠀⠀⠀⠀⣠⣺⣍⣉⡉⠢⣄⣳⡀⢀⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡗⠒⠠⠐⢄⠀⠀⠀⠀⢀⣤⠶⣤⡀
⠀⠀⠀⢀⡼⠛⣁⣤⣄⡈⠢⠈⠙⠭⢽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠈⠂⠑⠤⠴⠶⠛⣡⡶⠋⠉
⠀⠘⢿⣯⣰⡾⠋⠀⠀⠈⠢⡀⢀⠀⢘⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡿⠚⢿⠳⠄⡑⠦⠤⠤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⢧⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⠈⢣⠀⠉⠓⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠤⠾⠂⠈⢧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⡟⢀⠽⠓⢄⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⠆⠀⠠⠤⣀⣼⣳⣌⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠃⠈⠄⠄⠀⠈⠐⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠒⠛⠓⠀⠈⢱⢮⣙⠲⠤⢤⣄⣠⣤⠤⠖⣚⡩⠖⠛⣬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣡⠒⠉⣁⣀⠀⠀⡂⠀⢀⢈⠉⠙⢲⡶⠖⠒⢚⠋⠉⢄⠀⢀⠏⢧⠀⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⠟⠁⠀⠀⠀⠀⠀⡸⠃⢀⡠⢺⡇⠀⠀⠈⠀⠈⠙⠆⠈⠀⠘⡀⢆⠈⠛⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣿⠀⠀⠀⠀⠀⠀⡉⢀⠀⠀⢠⠉⡠⠀⡡⠊⠀⠀⠰⡀⢸⡆⠀⠙⠢⢟⣶⠶⠴⠦⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶⠟⠀⠀⠀⠀⠀⠀⠓⠋⠀⠰⡃⡞⠀⣾⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣣⡀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣟⣻⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⠿⢿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⣿⣶⣤⠀⠀⠀⠀⠀⠹⣿⣆
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⠁⠀⠀⣤⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⠳⣤⡼⠁⢳⡀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡞⠛⢦⡀⢻⡄⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡁⣆⠀⠙⢦⡷⠀⠀⢰⠇⠀⠀⠲⣄⠘⠃⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠳⣬⣉⣠⠾⠛⢦⣠⠟⠀⠀⠀⠀⠈⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⢀⣀⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣭⣉⣉⣉⣭⣉⢉⣉⣭⣉⠉⣉⣭⣉⡉⣉⣩⣍⡉⢉⣩⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠬⠭⠥⠤⠬⠭⠥⠤⠬⠭⠥⠤⠬⠭⠥⠤⠤⠭⠭⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡒⠒⠒⠒⡒⠒⠒⠒⡒⠒⠒⠒⢒⠒⠒⠒⢒⠒⠒⠒⠒⡒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠳⡤⠞⠉⠳⡤⠞⠉⠳⣤⠞⠉⠳⣤⠴⠋⠙⢦⡴⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠎⢀⠀⠀⠀⢰⠀⢈⠐⢳⠂
⠀⠀⠀⠀⠐⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡠⠎⠀⠀⠀⠀⠀⠀⠓⠊⠀⢢
⠀⠀⠀⠀⠀⠀⢄⢨⣇⡓⠀⣀⠀⢀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⡄⡀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠐⠀⠀⢏⠈⡄⡂⡹⢦⢩⣷⢀⠀⠐⡆⠘⡀⢢⣄⢀⣆⣰⡀⣤⣀⠠⣃⠀⢄⡀⡀⠀⠈⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠉⠀⢸⠉⠀⠀⠰⢷⠀⠁⠣⠁⡁⠇⣼⠟⢉⠁⠀⡜⠇⠙⣷⢾⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠠⠼⣧⢸⡯⣄⡆⠈⣿⡂⠠⣀⡄⡇⠂⣡⠀⢸⣂⠀⡆⠀⡀⠠⢀⠉⠃⣽
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣃⡏⣹⣨⣍⢬⣣⠀⢠⡇⠀⠐⠃⡇⢪⢽⡰⠨⠦⠀⢧⠃⠹⠤⢘⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⢛⣙⡋⡧⠽⠚⡀⠘⡧⠀⠸⠙⢳⣠⢸⣠⠀⡆⠤⢸⣤⢰⡀⠌⣎⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣨⣭⡶⡦⢤⣤⡴⠶⠛⠉⢉⣁⣭⣭⡽⠶⠦⣤⣤⠴⠶⠛⠉⢉⣠⣤⣬⡭⠷⠦⠤⣤⣤⠴⠞⠛⠉⣁⣠⣤⣭⡭⠶⠦⠤⠤⠴⠞⠛⠉
⠀⠀⠀⠀⢀⣠⣤⣴⠦⠴⠶⠶⠶⠿⠛⠋⠁⢀⣠⣤⡴⠶⠶⠶⠶⠶⠿⠛⠋⠁⢀⣠⣤⡶⠶⠶⠶⠤⠶⠾⠟⠛⠉⠀⣀⣤⣤⡶⠶⠶⠶⠶⠾⠛
⠀⠀⠀⠀⢉⣉⣠⡤⠴⠖⠒⠒⠶⢤⣔⣒⠛⠛⢉⣁⣤⠶⠒⠒⠒⠶⢤⣐⣚⠛⠛⢉⣁⣤⠴⠒⠛⠓⠲⠦⣤⣒⠚⠛⠋⣉⣠⡤⠖⠒⠛⠒⠶⠄
⠀⠀⠀⠀⢉⣉⣀⠒⠛⠛⠻⢶⣦⣤⣈⣉⣙⣋⣉⡁⠘⠛⠛⠿⢶⣦⣤⣈⣉⣉⣉⣉⣁⡀⠉⠛⠻⠷⣶⣤⣀⣉⣙⣛⣉⣉⣀⠈⠙⠛⠿⢶⣦⣄
⠀⠀⠀⠀⣈⡉⠛⠻⠷⣦⣤⣤⣄⣉⣩⣭⣍⣉⡉⠛⠻⠷⣶⣤⣤⣄⣈⣭⣭⣭⣍⡉⠙⠻⠷⣶⣤⣤⣤⣈⣉⣭⣭⣭⣉⡉⠛⠻⠷⢶⣤⣤⣀⡀
⠀⠀⠀⠀⣈⠙⠓⠶⠤⠤⠴⠞⠫⢥⣤⣤⣄⣉⠙⠓⠶⠦⠤⠶⠒⠩⠥⢶⣤⣤⣈⡉⠛⠶⠦⠤⠤⠶⠚⠭⠤⣦⣤⣄⣉⠙⠛⠶⠶⠶⠖⠚⠩⠤
⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⢶⣦⣤⣀⡀⠉⠛⠛⠓⠲⠒⠒⠲⠶⣶⣤⣄⡀⠉⠙⠛⠳⠶⠶⠒⠒⠶⢶⣦⣤⣀⡀⠉⠙⠛⠛⠒⠒⠒⠲⠶⣶⣤
⠀⠀⠀⠀⠴⠒⠚⠛⠛⠲⢦⣤⣀⣉⣙⣛⣻⠽⠒⠛⣛⣛⠓⠶⣤⣀⣈⠉⢛⣛⡯⠕⠒⠛⣛⡛⠲⢦⣤⣀⣉⣉⣉⡩⠽⠒⢛⣛⣛⠳⠶⣤⣀⡁
⠀⠀⠀⠀⣴⠾⠛⠋⠉⣉⣀⣀⣍⣉⣉⣡⣤⣴⠾⠛⠋⠉⣉⣁⣠⣬⣍⣉⣉⣤⣤⡶⠟⠛⠉⢉⣉⣀⣤⣬⣉⣩⣤⣤⡶⠿⠛⠉⠉⣉⣁⣠⣬⡁
⠀⠀⠀⠀⢀⣤⣴⡶⠿⠛⠉⢉⣀⣤⣤⣤⣤⣀⣤⣴⠶⠿⠛⠉⢁⣀⣤⣤⠤⣤⣤⣠⣤⣴⠾⠟⠛⠉⢁⣠⣤⣤⢤⣤⣤⡤⠶⠾⠿⠛⠉⣁⣠⡤
⠀⠀⠀⠀⠠⠄⠒⠶⠶⠶⠞⠛⣉⣤⣴⠶⠶⠬⠍⠓⠲⠶⠶⠞⠛⣉⣤⣴⠶⠶⠆⠤⠐⠲⠶⠶⠶⠛⢋⣡⣤⡶⠶⠶⠬⠉⠓⠲⠶⠶⠛⢋⣩⣤
⠀⠀⠀⠀⠶⠶⠖⠒⠒⠒⠚⠛⠋⠁⣀⣠⣴⡶⠶⠚⠒⠓⠚⠛⠛⠋⠁⣀⣠⣴⡶⠾⠛⠛⠛⠒⠒⠛⠛⠉⢀⣀⣤⣶⡶⠟⠛⠛⠛⠛⠛⠛⠉⡁
⠀⠀⠀⠀⣠⣤⠶⠖⠛⢛⣉⣙⠛⠻⢉⣉⣁⣀⣤⡴⠾⠛⣋⣉⡙⠛⠫⢉⣉⣁⣀⣤⡴⠞⠛⣉⣉⣉⡛⠛⠋⣉⣉⣀⣠⣤⠶⠟⢛⣉⣉⣙⠚⠉
⠀⠀⠀⠀⠠⢤⣶⣊⣉⣉⡉⠉⠛⠷⢶⣤⣍⣩⣤⣴⣄⣉⣉⠉⠙⠛⠷⢶⣤⣭⣭⣤⣴⣬⣭⣁⡈⠉⠙⠻⠶⣤⣬⣉⣥⣤⣴⣬⣅⣀⡉⠉⠛⠗
⠀⠀⠀⠀⢤⣤⣀⡀⠉⠙⠻⠷⢶⣤⡤⠤⢤⣤⣤⣀⡈⠉⠛⠻⠷⠶⠦⠤⠴⠶⢶⣤⣄⣀⠉⠙⠛⠿⠶⡶⠤⠤⠴⠶⣦⣤⣀⣈⠉⠙⠛⠷⠶⠦
⠀⠀⠀⠀⢤⣤⣉⠛⠻⠶⠖⢂⠠⠴⠶⠶⠶⣤⣌⣉⠛⠛⠶⠒⡂⡄⠴⠖⠲⠶⢦⣤⣉⡙⠛⠶⠶⢒⣀⠤⠶⠒⠶⠶⣤⣌⣉⡛⠛⠛⢚⣂⠤⠆
⠀⠀⠀⠀⣀⣈⠉⡛⢛⡛⠛⠛⠛⠿⠶⣦⣤⣀⣈⣉⣛⣛⣛⡉⠙⠛⠛⠿⣶⣦⣄⣀⡉⠛⣛⣛⣋⠉⠙⠛⠻⠷⣶⣤⣄⣈⣉⣙⣛⣉⡉⠙⠛⠓
⠀⠀⠀⠀⠉⣉⣉⣉⣉⡙⠛⠷⢶⣤⣄⣀⠈⠉⣉⣉⣉⣉⣉⠛⠻⢶⣦⣤⣀⣈⠉⡙⠛⠋⠉⠉⠙⠛⠳⠶⣦⣤⣠⠌⢉⣉⣭⣭⣉⣙⠛⠿⢶⣦
⠀⠀⠀⠀⠛⠉⣉⣠⣬⣭⡕⡲⠦⢤⡤⠴⠖⠛⠉⣁⣠⣤⣬⠽⠶⠦⢤⣤⡤⠶⠞⠋⠀⣀⣤⣤⣤⠴⡶⠦⠤⠤⠶⠞⠛⢉⣠⣤⣤⣭⣿⡶⠦⡤
⠀⠀⠀⠀⠶⠿⠛⠉⠁⣀⣤⣤⡶⠶⠶⠶⠶⠶⠟⠛⠉⠁⣀⣠⣴⡶⠶⠶⠶⠴⠶⠾⠓⠃⠀⢀⣀⣤⣶⠶⠶⠶⠶⠶⠞⠛⠛⠉⢀⣀⣤⣶⠶⠖
⠀⠀⠀⠀⠰⢤⣞⡛⠛⠛⣉⣠⣤⠶⠞⠛⠒⠶⢤⣚⣛⠛⠋⣉⣠⣤⠶⠚⠛⠛⡲⢦⣔⣛⠛⠛⢋⣁⣠⡴⠖⠛⠛⠓⠶⢤⣝⣛⣛⣉⣁⣤⡴⠖
⠀⠀⠀⠀⠙⠓⣈⣉⣙⣋⣉⣁⣠⣴⡶⠿⠛⠛⠁⣈⣉⣉⣉⣉⣁⣤⣴⡶⠿⠛⠉⠉⣀⣉⣉⣙⣋⣉⣀⣤⣶⠾⠟⠛⠉⢀⣀⣉⣉⣉⣉⣀⣤⡦
⠀⠀⠀⠀⠾⠟⠛⢉⣉⣉⣭⣍⣉⡠⢤⣤⣴⡾⠟⠛⢉⣉⣩⣭⣍⣉⡠⢤⣤⣴⠾⠟⠋⢉⣉⣤⣭⣭⣉⣁⠤⣤⣤⣶⠾⠛⠋⢉⣁⣤⣭⣅⣀⡀
⠀⠀⠀⠀⠴⠖⠛⠫⠭⢤⣤⣤⣈⡉⠓⠶⠤⠤⠶⠚⠫⠥⢤⣤⣤⣉⠙⠓⠲⠤⠤⠶⠚⠫⠥⠴⣦⣤⣄⡉⠛⠲⠤⠤⠴⠶⠛⠭⠥⢤⣤⣤⣈⡉
⠀⠀⠀⠀⠰⠶⠶⣶⣤⣄⡀⠈⠙⠛⠷⠖⠶⠶⠶⠶⣦⣤⣀⡀⠉⠙⠛⠓⠒⠒⠒⠒⠶⢶⣤⣄⣀⠈⠙⠛⠻⠶⠒⠒⠒⠲⠶⣦⣤⣄⣀⠉⠙⠛
⠀⠀⠀⠀⠲⠶⣤⣀⣈⣙⣛⣻⠷⠞⠛⠛⠛⠲⠶⣤⣀⣉⣉⣛⣻⠷⠛⠛⠋⠛⠛⠶⣤⣄⣈⣉⣛⣛⣻⠶⠛⠛⠛⠛⠳⠶⣤⣄⣈⣉⣉⣩⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⠿⠿⢟⣛⠉⠉⠉⠛⠛⠛⠉⠉⠉⣛⡻⠿⠿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣀⣀⣠⠤⣶⣶⣶⣦⣤⣤⣴⣶⣶⡦⢤⣄⣀⡈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⡁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣭⣥⣦⣤⣀⣀⣀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠛⠀⠙⠛⠛⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⣀⣠⣼⠏⠀⠘⠖⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠛⡋⠉⣀⠀⢻⡆⠀⠀⠈⠉⠀⠸⠆⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣿⠀⢙⡀⠀⠀⢶⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⣷⠀⠀⠀⠸⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣀⠀⠀⠀⠈⠁⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢋⡀⠀⠀⠀⠀⠈⠁⠀⣀⡘⠿⣿⣿⣿⣿⣿⡿⠛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠀⠀⠀⠘⣁⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠉⠉⠉⢁⡀⠈⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠀⠀⠀⠀⢻⡄⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⡀⠀⢀⣿⣷⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣾⣷⡀⢾⣿⣿⡷⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠙⠁⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⠇⠈⠙⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣦⣄⣠⣴⣦⣄⣠⣴⣤⣄⣤⣶⣤⣀⣤⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠿⠿⠛⠛⠿⠿⠛⠻⠿⠟⠛⠻⠿⠟⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣶⣶⣿⣷⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠙⠛⠛⠋⠙⠛⠛⠉⠙⠛⠛⠉⠛⠛⠛⠉⠀⠛




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