Hahaha Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠚⠉⠀⠀⢠⡤⠤⠤⠤⠀⠀⣿⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠆⠀⢻⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣤⠀⠀⠄⣿⡅⠀⢸⠀⣾⠇⠀⠀⠀⠰⠀⢸⣧⠀⢸
⠀⠀⣤⠄⠀⢤⢹⡇⠀⠈⠐⠋⢁⡀⠀⠀⡀⠀⣿⡀⠀⢸⢸⡇⠀⠸⠀⣿⠀⠀⣿⠀⠀⠁⠈⣿⣀⣘
⠀⠀⢿⡆⠀⢸⢸⡇⠀⠀⡇⠀⣼⡇⠀⠀⠈⡀⢿⡇⠀⠘⣸⣧⠀⠀⡆⣿⠀⠀⠛⠃⠀⠈⠀⣿⡉⠉⢡⠂
⠀⠀⢸⡇⠀⠀⡘⣿⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⢸⡇⠀⠀⠉⠁⠀⠀⢻⡟⠀⠀⣀⣤⠀⠀⠁⣽⣧⣀⣸
⠀⠀⢸⣧⠀⠀⠉⠁⠀⠀⢠⠀⣿⠀⠀⣦⠀⠀⢘⣿⠀⠀⢰⣶⡄⠀⢸⡇⠀⠀⡏⢻⣇⣀⣀⡍⠉⠉⠉
⠀⠀⠀⣿⠀⠀⢰⣶⡆⠀⠸⢀⣿⠀⠀⠿⠀⠀⠸⣿⡀⠀⠸⢸⡇⠀⠘⣧⣤⣤⡇⠈⠉⠉⠉⠁
⠀⠀⠀⢿⡆⠀⠘⢸⣇⠀⠀⣼⡇⠀⠀⠀⣀⠀⠀⢿⡇⠀⠈⡸⣷⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣇⠀⠀⡍⣿⣤⣤⣿⡇⠀⠀⡟⣿⡄⠀⠘⠟⠛⠛⠃⠉⠉⢉⣁⣀⣠⣤⣤⣶⠶⠶⠋
⠀⠀⠀⠈⠛⠛⠛⠃⣿⡉⠉⣹⣇⣀⣰⢁⡘⠟⠛⠛⠃⢀⡶⠶⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡶⠽⠿⠛⠋⠙⠿⣶⣄⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢂⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠈⣷⢰⠒⠶⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠛⣧⣀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⢐⢽⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⠀⠈⠋⠁⠀⠀⠀⠖⡶⠶⠶⢶⡆⠀⣸⠘⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⣀⢀⣀⡀⡀⠀⠀⠁⠀⠀⠀⢾⡀⠇⡰⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠀⠠⣠⣤⣤⢠⡇⡏⠉⠑⢁⠀⢰⠀⠀⡀⠀⠸⡅⢠⡇⡆⠀⠃
⠀⠀⢠⢠⣤⣤⢤⡇⠋⠉⠹⢱⠴⠚⠉⠀⠀⠀⢀⠀⢠⠀⠀⠀⡇⡁⠃⠀⠠⠾⢀⠰⠀⠀⣧⠀⠀⢳⣌⢿⢰⠀⢀⡆
⠀⠀⠀⢨⠀⠀⡇⡇⠇⠀⠀⣴⠀⠀⢧⠋⠉⠉⠙⣇⢸⢠⠀⠀⠃⡅⢀⠀⠀⡄⡸⡀⠀⠀⣀⡆⠀⠐⣸⠙⢠⠁⠫⣥⣤⡀
⠀⠀⢰⠘⠀⠀⠇⢻⢠⠀⠀⠱⠀⢀⠈⠀⠀⠀⠀⢮⠀⡜⠀⠀⠨⠹⠔⠀⠀⠃⡇⠇⠀⠀⠀⠀⠀⠀⢱⢎⠀⠀⠀⠁⠇
⠀⠀⠀⡈⠀⠀⢰⡻⠼⠀⠀⢠⡆⠸⢘⠀⠀⡆⠀⠘⣠⡄⡆⠀⠀⠀⣀⠀⠀⢰⠃⠄⠀⢰⢂⡌⠆⠀⠈⣼⠤⠁⠀⣡⠁
⠀⠀⠀⠇⡄⠀⠀⠀⢀⠀⠀⢰⠀⠠⠘⠀⠀⠹⠀⠀⢃⣧⠀⠀⠀⡇⡆⠇⠀⠀⠀⠀⠀⢸⠀⠰⣐⠒⠒⣒⣃
⠀⠀⠀⢰⠁⠀⠀⡏⡄⡅⠀⠈⣀⡀⡄⠀⠰⠄⠂⠀⠈⡾⢰⠀⠀⢠⣧⢰⠀⠀⡆⡓⠀⠈⠄⠀⠀⠀⠀⢨⢑
⠀⠀⠀⠐⠘⠀⠀⢁⣽⢠⠀⠀⠋⡇⠇⠀⠀⣀⢀⠀⠀⢡⠈⡄⠀⢰⠸⡀⠆⠄⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡄⡀⠀⢘⢹⡎⠄⠀⠴⢁⠀⠀⢨⠐⢂⠇⠀⠀⡢⡄⠀⠀⠚⠈⠀⠀⢀⣁⠀⠤⠤⠄⠒⣒⡈⠩⠔
⠀⠀⠀⠀⠱⠭⠭⠭⠼⡙⡁⠀⠈⢸⣀⣀⡸⡇⠈⠬⠤⠥⠥⠇⠠⠒⣂⡀⠩⠡⠤⠐⠒⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⠤⠼⠒⢒⣂⠀⠋⡑⠢⣀⠀⠀⡰⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠑⠢⡑⠒⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣾⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣼⣿⠁⠀⢹⣿⡏⣿⡟⠉⠉⠉⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⠀⠀⢸⣿⡇⣿⠀⠀⣴⡄⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣼⣿⣿⠀⠀⢠⣤⡄⣿⠀⠀⣉⡁⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⢸⣿⡇⣿⠀⠀⣿⣇⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠰⣿⣿⣷⢿⣶⡾⠟⠻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⢸⣿⡇⠀⣿⠁⣴⡆⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⠉⠁⠀⡇⠀⠛⠃⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠂⠀⢰⣶⡆⠀⡇⢰⣶⣶⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⣀⣼⣿⣇⣸⣧⣸⣿⣿⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⢻⣿⠃⠈⢿⡟⠉⠀⠀⠈⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠸⠿⠀⠀⢸⠀⠀⣴⣷⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢰⣶⠀⠀⢸⠀⠀⣀⣀⡀⠀⢸⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⣸⣿⡀⠀⣼⠀⢀⣿⣿⡇⠀⢸⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣇⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⢿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⠀⠸⣿⣿⡟⠀⢀⠀⠻⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡆⠀⠻⠿⠃⠀⢹⣿⠇⠀⠉⠀⠀⠈⢿⡏⠙⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠿⣿⣿⡀⠀⣴⣶⡄⠀⢿⠀⢠⣿⣿⣿⣦⣾⣿⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠉⠙⣿⣿⣿⠀⠰⠆⠀⠈⠻⣧⠀⢸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠉⢻⣿⡷⠀⠈⢿⣿⠀⢀⣤⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⠁⣠⣴⡀⠈⢿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠀⠹⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠉⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠉⠉⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣠⣂⠀⠀⠀⣀⣀⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢳⠀⣤⣀⡀⠀⠒⠒⠋⠀⠀⡀⠀⠢⠎⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡩⠛⠺⣏⠀⠀⣈⠁⠀⠀⠙⢯⡹⠛⠛⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢰⠘⣗⠤⣀⠀⠠⡀⡷⡈⠙⠲⠄⡎⢰⢀⢹⠀⠉⠳⢦⣼⠳⡄
⠀⠀⠀⠀⠈⢦⠊⠧⣽⠒⠆⠙⡆⠱⡇⣈⣃⡀⠀⠏⢿⠈⢀⡁⣦⡀⢠⡏⠳⢼⣦
⠀⠀⠀⠀⠀⠀⠳⠉⠃⡀⢠⢻⠓⠢⢀⠘⣆⡜⠁⠓⠺⠀⢸⠙⢮⡙⢾⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢫⠈⢣⡠⠳⡸⡄⠀⢰⡀⣰⣸⠔⠊⠙⠦⡙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢵⠀⠑⢳⢤⣘⣯⠀⢺⠀⡀⠀⠰⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢈⡷⡀⠸⣇⡼⠛⡷⢬⣸⡇⡀⠠⣧⣸⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⡉⠀⠘⢆⢻⠀⠀⠈⠀⠈⣯⠀⠀⢻⡏⠘⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠈⢃⡄⠀⠀⢸⣽⢀⣄⢸⠀⢰⡀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣶⡀⠈⣇⢀⣤⠞⣟⢸⢻⠈⢸⡠⡇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡘⡆⢻⢱⡀⣿⠉⠀⠀⠸⡮⠜⡇⣽⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢾⠁⠣⣿⠀⠀⠀⠐⠁⠀⡇⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢻⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣳⡀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠈⡼⡇⡞⠛⠛⢻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠙⣷⣄⣠⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡘⠀⠈⠛⠉⠀⠀⠀⠀⠔⣲⠶⠶⠶⢶⢦⠀⢃⡇⢁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⡞⡄⢸⢸⢸⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⢁⣀⣀⣀⣀⡀⡏⡿⠿⠿⠏⡇⠀⡇⡇⠀⠀⠀⠀⢸⢳⠘⣼⡸⡀⠀⢸⢿
⠀⠀⠀⠀⠀⢠⢄⣴⣶⣶⠲⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⢸⢸⠛⠛⢻⢹⡇⡇⠀⠀⢸⠃⢠⢁⠀⠀⢸⡆⠀⠀⣇⡆⡏⡇⡇⠀⢸⢸
⡖⡶⠶⠿⡟⣾⠈⠀⠀⠀⢸⡟⠋⠁⡤⣤⣤⣤⣤⢤⡀⢸⢸⠀⠀⠘⢸⢿⢰⠀⠀⢈⢸⢸⢸⠀⠀⢸⢱⠀⠀⠈⡀⣄⣧⢸⣀⣸⢸
⡇⡇⠀⠀⢰⢡⡇⡇⠀⠀⡇⡇⠀⢠⢁⠃⠀⠀⠀⢳⣇⠈⡄⡄⠀⠀⡏⣼⠸⠀⠀⠸⢸⡸⢸⠀⠀⢸⣈⡆⠀⠀⢃⢇⢩⢰⠒⠚⢢⢷⡤
⢃⢣⠀⠀⢸⢸⠇⡇⠀⠀⢡⠇⠀⢸⢸⠀⠀⠀⠀⠈⡞⡄⡇⡇⠀⠀⠣⠻⠴⠃⠀⠀⡇⡷⠙⠀⠀⠀⠀⠀⠀⠀⠈⡞⣼⢸⠀⠀⢸⢸
⢸⢸⠀⠀⠘⡸⢟⣰⠀⠀⢸⢰⠀⢰⢸⠀⠀⣤⠀⠀⢹⢣⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⢃⠃⡇⠀⠀⡴⢒⡒⡆⠀⠀⢱⢻⣈⣓⣒⠚⡸
⠘⡌⡄⠀⠀⠉⠁⠀⠀⠀⠸⢼⠀⠘⡄⠀⠀⡟⡆⠀⠀⠏⣿⢸⠀⠀⠀⡖⣉⢳⠀⠀⢸⠀⡇⠀⠀⡇⡇⢱⠸⣀⣀⣀⡟⡆⠀⠀⠉⠁
⠀⡇⡇⠀⠀⢠⠤⠄⡄⠀⠀⡇⡆⡇⡇⠀⠀⡇⢱⠀⠀⠸⣹⡜⡂⠀⠀⠇⣿⠸⠀⠀⠈⡠⣄⣀⣀⠇⡇⠀⠓⠒⠒⠒⢶⠗
⠀⢃⢡⠀⠀⢸⢸⡇⡇⠀⠀⢱⢣⡇⡇⠀⠀⠛⠉⠀⠀⠀⢧⠃⡇⠀⠀⢹⢻⡇⡇⠀⠀⣇⡖⠒⠒⠒⠃⠀⠀⠀⠀⠀⢸⢸
⠀⠸⠸⠀⠀⠀⡏⣷⢸⠀⠀⢸⢸⢣⠁⠀⠀⣀⡤⢤⠀⠀⠈⡆⢸⣀⣀⣸⢸⢣⣑⣒⣒⣋⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾
⠀⠀⡄⡄⠀⠀⢡⣿⡌⠦⠤⠤⠇⢰⠀⠀⢀⡡⠒⡌⡆⠀⠀⢱⢰⠂⠒⠒⠒⠃⠀⠀⠀⠀⣀⣀⡠⠤⠤⠐⠒⠒⣉⣉⡡⠞
⠀⠀⢣⣉⣉⠉⠉⢴⠱⡒⠒⠒⢲⢸⠀⠀⢸⢿⠀⢱⣈⣉⣉⣉⣸⠄⢠⠤⠖⠒⢒⣉⣉⡩⠤⠤⠖⠒⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠱⡀⢀⣺⣈⣉⣩⣭⡼⠒⠦⡀⠀⠀⠀⠀⢠⢃⠖⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠥⠤⠤⠒⠒⠒⠈⠉⠢⢌⠑⢤⡀⡰⢣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⣡⠃
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⣾⡟⠻⠿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠆⢰⣶⣾⣿⣷⣶⣶⣾⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⠉⠉⠀⣼⣿⠀⠀⠀⠘⠛⣿⣿⠛⠉⠉⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠿⠿⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣤⣤⣴⣾⣿⣷⣶⡆⠠⣶⣶⣿⣿⣾⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠛⠋⣿⣿⠀⠀⠀⠿⠿⣿⣿⠉⠉⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⣴⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣇⣀⣀⣀⣸⣿⣇⣀⠀⢀⣀⣀⣿⣇⣀⣉⣽⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⢿⣿⠟⠛⠃⢸⣿⣿⣿⣿⠿⠿⣿⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡇⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡏⠈⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⡏⠀⠈⢻⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠙⣿⣿⡀⠈⣿⣿⠇⠀⠦⠀⠙⢿⣷⠤⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠿⣿⣿⣧⠀⠈⢉⣀⠀⠸⣿⠀⢀⣤⣤⣄⢀⣿⣄⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⠀⠀⡀⠈⠻⣿⡆⠀⢿⣿⣆⣀⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⢿⣿⣧⡀⠘⣿⣿⠀⠀⠁⣀⣀⠈⣻⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠛⠋⢁⡀⠘⢿⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⠀⠸⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠙⠛⠿⠿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⡿⠿⠛⠛⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣦
⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⣴⣿⣿⡇⠀⠀⢀⣴⣾⣿⣿⡆⠀⠀⣠⣴⡄⠀⣴⣶⣆⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⡇⢸⣿⣿⣿⠀⢰⣿⣿⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⡇⠀⣸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⣿⣿⣿⡏⢀⣿⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⠃⢸⣿⣿⣿⡀⢸⣿⣿⠋⠉⣿⣿⡇⣸⣿⣿⠏⠉⠉⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡿⢠⣿⣿⠏⠀⠈⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡇⣿⣿⠏⠀⠀⠀⠹⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⣿⣿⡟⠀⠀⠀⠙⠿⠿⠁⠸⠿⠃⠀⠀⢻⣿⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠇⠀⠙⢿⣿⠃⣈⣉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⡄⢠⣶⣶⠀⠀⢠⣴⣾⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠙⠛⠋⢠⣿⣿⠀⢠⣿⣿⣿⠀⠀⢠⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⣼⣿⣿⠀⠀⣾⣿⣿⣿⣿⡇⠀⣀⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣤⣼⣿⣿⣿⡄⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣥⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⡇⣼⣿⣿⣏⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡟⠀⣾⣿⣿⠿⣿⣿⣿⠛⠋⠁⠀⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣧⣿⣿⡟⠉⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡿⢸⣿⣿⡇⢸⣿⣿⠃⠀⣿⣿⣿⡄⠀⠀⠀⢰⣿⡧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣿⡟⢿⠟⠀⠀⠹⣿⣿⣿⠀⠀⠀⠘⣿⣿⠇⢸⣿⣿⠃⠀⠉⠁⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠘⠿⠟

⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⡀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀
⢰⣶⢸⠻⡜⡂⠘⠹⠘⢻⠘⠀⢰⣶⢸⠿⡜⠂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠹⠘⢻⠘⠀⢰⣶⢸⠻⡜⠂⠘⠻⠘⢻⠘
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠆⠀⠀⠀⠀⢦⠀⢠⢷⣆⣼⢀⠆⠀⠀⠀⠀⣦⠀⢰⢷⣆⣴⢀⠆⠀⠀⠀⠀⢦⠀⢰⢷⣆⣼⢀⠆⠀⠀⠀⠀⣦⠀⢰⢧⢆⣼
⠀⣰⣼⣼⢷⢘⡀⠈⠘⠉⠛⠘⠀⣰⣼⣼⢷⢘⡀⠀⠘⠉⠟⠘⠀⣰⣴⣼⢷⢘⡀⠀⠘⠉⠛⠚⠀⣰⣼⢼⢷⢘⡀⠀⠘⠉⠛
⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁
⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⡀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀
⢰⣶⢸⠻⡜⠂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠿⡜⡂⠘⠻⠘⢻⠘⠀⢰⣶⣸⠻⡜⠂⠘⠻⠘⢻⠘
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠆⠀⠀⠀⠀⢦⠀⢰⢧⢆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢰⢧⣆⣶⢀⠆⠀⠀⠀⠀⢦⠀⢰⢧⣆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢰⢧⣆⣴
⠀⣰⣴⣼⢷⢘⡀⠀⠘⠉⠟⠘⠀⣠⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣰⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣰⣼⣼⢷⢘⡀⠈⠘⠉⠛
⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁
⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⡀⡄⢀
⢰⣶⣸⠻⡜⠂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠹⠘⢻⠘
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠆⠀⠀⠀⠀⢦⠀⢰⢧⣆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢰⢷⣆⣶⢀⠆⠀⠀⠀⠀⣦⠀⢰⢷⢆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢰⢧⣆⣶
⠀⣴⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣠⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣰⣴⣼⢷⢘⡀⠀⠘⠉⠛⠚⠀⣴⣼⣼⢷⢘⡀⠀⠘⠉⠛
⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁
⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⡀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀⠀⠀⠀⢀⡀⢀⠀⢀⣀⠀⡄⢀
⢰⣶⢸⠿⡜⡂⠘⠹⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠻⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠹⠘⢻⠘⠀⢰⣶⢸⠻⡜⡂⠘⠹⠘⢻⠘
⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠆⠀⠀⠀⠀⢤⠀⢰⢧⣆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢰⢷⣆⣴⢀⠆⠀⠀⠀⠀⣤⠀⢰⢧⣆⣴⢀⠆⠀⠀⠀⠀⣦⠀⢠⢧⣆⣴
⠀⣴⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣠⣴⣼⢷⢘⡀⠀⠘⠉⠛⠚⠀⣴⣼⣼⢷⢘⡀⠀⠘⠉⠛⠘⠀⣴⣼⣼⢷⢘⡀⠀⠘⠉⠛
⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠘⠁

⣿⣅⠀⣰⣿⡄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣾⣶⡀⠀⢀⣿⣿⣷⣄
⣿⣿⢿⣿⣿⠁⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣷⣦⣴⣿⣿⡇⠀⢸⣿⣿⠿⣿⣇⠀⠀⠀⠀⠀⣾⣇⠀⢰⣷⡀⠀⠀⣠⣦
⠟⠀⠀⣼⡏⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠘⠹⣿⠀⢸⣿⠀⠀⠈⠛⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⠁⠀⣼⣿⣿⡄
⠀⠀⠀⠉⠀⠰⠟⠁⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⢹⣿⠀⣸⡿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠁⠀⠈⠟⠀⠀⢰⣦⡀
⢠⣤⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⣠⡄
⣾⣿⠃⠀⣰⣿⣿⡀⠀⠀⠀⠀⢰⣷⣤⣿⠀⢠⣶⡀⠀⠀⠀⠀⣠⡄⠀⢰⣦⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢈⣿⣿⣆⠀⢸⣿⡿⠛⢿⣇⠀⠀⣿⣷
⢻⣿⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠸⠏⠉⡿⢠⡿⠿⡧⠀⠀⠀⠀⢸⣷⣶⣿⣿⡇⠠⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠟⣿⡄⠀⣿⠂⠀⠀⠈⠀⢰⣿⡟
⠘⠋⠀⠿⠃⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠸⣿⡏⠀⢹⠇⠀⡿⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣿⣿⠀⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠀⠀⣾⣿⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣿⣷⠀⠀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⢸⣿⠀⢰⣿⠋⠁⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡗⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⢿⡿⠀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣿⣿⡇⠀⠀⢠⣴⣿⡄⠀⠈⠻⠃⠀⠸⠇⠀⡿⠁⠀⢻⡇
⠀⠀⣀⠀⠀⠀⢰⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣾⣿⡿⠏⠿⣿⣿⠀⠀⣠⣿⡟⣿⣇
⠀⠘⣿⣷⡀⠀⢸⣿⣽⣿⣿⡄⠀⠀⠀⢸⣿⣄⣠⣿⣷⠀⠀⣾⣿⣇⠀⠀⠀⠀⢿⠿⠁⠀⢀⣿⡇⠀⢸⣿⣿⡿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣾⢿⣿⣧⠀⠸⣿⠋⠀⠙⠿⠀⠀⠀⢸⣿⡿⠛⢿⡇⠀⣼⣿⡾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡿⠋⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣟⠀⠈⣿⠆⠀⠻⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠸⠇⠀⡿⠁⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠻⠟⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠀⠀⢀⣰⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣴⡄⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⣾⣿⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢻⣿⠃⢀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⡏⠀⣸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠙⣿⡏⠀⣠⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠾⠃⢀⣾⠿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⢸⡇⠀⣿⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠿⠁⣰⡿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠟
⠀⢀⠀⠀⠀⠠⣿⣿⣶⣄⡀
⠀⣿⣿⡆⠀⠀⣿⣷⣿⢿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡞⠀⠀⢠⣶⡄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣴⣾⣿⣿⡄⠀⢻⣿⠁⠀⠙⠓⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣾⣿⡟⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⢰⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⡇⠉⠘⣿⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠙⢿⣿⠇⠀⣠⣿⣿⣽⣿⣆⠀⠀⠀⠀⠀⣾⣇⣀⣠⣿⣿⡆⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠹
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⣸⣿⠀⢀⣿⡟⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠻⢻⣷⠀⢸⣿⠉⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠟⠀⠀⠛
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠒⠛⣄⣇⡀⠙⢆⠀⢀⣠⣿⡷⣶⣀⣤⠟⠀⠸⡉⠀⠀⠀⣿⢆⢀⣠⡤⢤⡝⢤⣤⢼⡟⠉⠉⣿⡓⠊⠉⢹⠀⠈⠀⠈⣷⣄⠀⠀⠈⢧⠀⠀⠀⢇⢸⡄⠀⠀⢷
⣴⡏⠭⠟⣿⣄⠈⠓⢻⠉⠀⠀⠈⢿⢦⡀⠀⠀⢳⡀⠀⠀⢻⠈⣿⠀⠀⠀⡟⢦⡀⠈⡇⠀⠀⡇⠹⡄⠀⡎⠀⠀⣄⠀⠘⡎⢳⠀⠀⠈⡆⠀⠀⢸⣴⡧⠴⣆
⠀⡇⠀⠀⢻⠘⣆⠀⢸⠀⠀⣦⠀⠈⢇⠱⡀⠀⠀⢧⠀⠀⢸⡀⣿⠀⠀⠀⢷⠈⡇⠀⡇⠀⠀⢹⠀⢷⠀⡇⠀⠀⡟⣆⠀⢹⡈⢇⠀⠀⢸⡀⠀⢸⡄⡇⠀⢏⢣
⠀⣧⠀⠀⢸⠀⢸⠀⡏⠀⠀⡏⢣⡀⠘⡆⢳⠀⠀⠸⡄⠀⠈⡇⣿⡀⠀⠀⢸⣀⣧⠤⠇⠀⠀⢸⠀⢸⠀⡇⠀⠀⠓⠉⠀⠀⢳⠘⡆⠀⠀⢧⣠⢼⣧⡇⠀⢸
⠤⠗⠀⠀⠸⡄⢸⡄⡇⠀⠀⠉⠉⠁⠀⠸⡈⢇⠀⠀⢳⣤⠴⢳⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠸⡅⡇⠀⠀⢀⣀⣄⠀⠈⢧⠸⡄⠀⠈⢧⣈⡦⠗⠀⠸⣄
⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⣀⡠⢤⠀⠀⢳⡘⡆⠀⠀⠳⣀⠤⠟⢷⠀⠀⢰⡖⠒⠚⢳⠀⠀⠀⢇⠀⣧⡇⠀⠀⣿⡀⠘⡄⠀⠈⢇⢳⠀⠀⢴⠛⡟⢷⡀
⠉⢹⠀⠀⠀⢳⠀⢷⡇⠀⠀⡿⣄⠈⣇⠀⠀⢣⠸⡄⠀⢰⠛⠉⢳⣼⠀⠀⠈⡝⢦⣀⣸⡄⠀⠀⢸⠀⢹⡇⠀⠀⡇⢸⠋⢻⠀⠀⠈⢇⢧⠀⠘⣤⢼⢾⣱⠠⣖
⣄⣨⡆⠀⠀⠸⡀⢸⡇⠀⠀⣷⢸⡏⠹⡀⠀⠀⢣⢳⠀⠈⣧⠴⠺⣽⡇⠀⠀⡇⠀⡇⠀⡇⠀⠀⠘⡇⢸⡇⠀⠀⡇⢸⡅⠈⣧⣀⡠⠼⢾⡄⠀⠱⣘⣆⡿⠀⡏
⡇⠀⣇⠀⠀⠀⢇⢸⡇⠀⠀⣿⢸⡇⠀⣧⡤⠤⠜⢿⡇⠀⠹⣄⣠⠿⡇⠀⠀⢸⠀⣇⠀⢹⣀⣀⣠⢽⡘⡧⠒⠒⠳⣸⠃⠀⠙⢦⣀⡠⠴⠃⠀⠀⠉⡇⠀⠀⢻
⢳⠀⢸⣀⣠⡤⠼⣄⣷⠒⠒⠻⣬⡇⠀⠈⢣⣤⡤⠴⠛⠀⠀⠀⠀⠀⠹⡖⠒⠚⢧⣻⠀⢈⢣⡀⠀⠀⣙⡿⢦⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸
⣸⠀⠈⠳⡀⠀⢀⣈⣿⣧⣤⠤⠾⢳⡄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠙⢦⣠⡶⠿⠋⠉⢷⡙⠉⠉⠉⢣⠀⠁⠀⢸⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠉⠉
⡽⠖⠒⠛⣿⡉⠉⠀⠈⢧⠀⠀⠀⢸⠱⡖⠚⠉⢹⣆⠀⠀⢻⠀⠀⢸⠓⡄⠀⢠⡇⠀⠀⠀⠘⣏⢦⡀⠀⠈⢧⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⣀⣀⡠⣤⠈⠳⠤
⠀⠀⠀⠀⠘⡗⢄⠀⠀⠘⣆⠀⠀⠸⡀⡇⠀⠀⢸⡈⢳⠀⢸⠀⠀⠘⡆⢸⡄⢸⠁⠀⢸⣆⠀⠘⡄⢧⠀⠀⠘⡄⠀⠀⣧⣾⠒⠒⢳⡄⠀⠀⣯⠀⠀⢸⢣⡀
⠀⠀⣺⡄⠀⠹⡈⣇⠀⠀⠸⡄⠀⠀⡇⣷⠀⠀⠀⡇⢸⠀⢸⠀⠀⠀⡇⠀⡇⢸⠀⠀⢸⡨⠦⠀⢹⡈⣆⠀⠀⢱⠀⠀⣸⣿⠀⠀⢸⠘⢦⠀⢹⠀⠀⢸⡄⢱⡀
⠀⠀⣯⣸⡄⠀⢣⠘⡄⠀⠀⢧⠀⠀⢣⢹⠀⠀⠀⠓⠚⠛⠉⠀⠀⠀⢧⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⠀⠀⢷⠋⠉⣿⡄⠀⠀⡆⢸⠀⢸⠀⠀⠈⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠹⡀⠀⠈⣦⠒⠙⢾⡀⠀⠀⢀⣀⣠⣤⡀⠀⠀⢸⠀⢹⣸⠀⠀⢠⡖⠚⡆⠀⠈⢧⠸⡀⠀⠈⣳⣚⣯⡀⠀⠀⠷⠞⠒⠋⠀⠀⠀⡇⠀⣇
⠀⠀⢠⠴⠲⡄⠀⠈⣆⢳⠀⠀⠘⢦⣖⣚⡇⠀⠀⢻⣄⠀⠀⡇⠀⠀⠈⡇⠘⣿⠀⠀⢸⠛⣤⠼⡄⠀⠀⢧⢳⠀⠀⣇⠀⢸⡷⡀⠀⠀⣀⣀⣀⠀⠀⠀⢱⠀⢻
⠀⠀⢸⠳⣄⣱⠀⠀⠘⣆⢧⠀⠐⡏⠁⠘⣷⠀⠀⢸⠈⢷⠒⢻⠀⠀⠀⢣⠀⣿⠀⠀⢸⠀⣿⠀⢣⠀⠀⠈⣇⠇⠀⠸⡋⠙⣷⡇⠀⣿⡉⠀⠈⡇⠀⠀⠸⡄⢸
⠀⠀⢸⠀⡇⠀⢇⠀⠀⠘⣞⡆⠀⢻⡒⠉⢿⡇⠀⠘⡇⢸⠀⠘⡀⠀⠀⠸⡄⢿⣀⣀⣸⡀⣿⠀⠘⣗⠒⠉⠉⢻⡀⠀⠱⠒⣿⠁⠀⢿⠘⣶⣦⣷⡀⠀⠀⣇⠈
⠀⠀⣸⠀⡇⠀⠸⡖⠒⠚⠙⢷⠀⠀⠷⠤⠺⣇⣀⣀⣧⢸⣄⣀⣷⡒⠒⠉⠳⣾⣇⠀⠀⠱⣏⠀⠀⠈⠓⠚⢉⣉⡀⠤⣄⠀⢸⠀⠀⢾⡋⢹⠀⢸⡷⡄⠀⢸⡀
⡉⠉⠉⢣⡇⠀⠀⠙⠦⠖⠚⣋⣀⡤⠤⣄⠀⠙⣄⠀⢞⢻⡏⠀⢘⣷⣤⡤⢴⡞⠈⠳⢺⠛⠉⠙⣧⡀⠀⠀⡏⠀⠀⠀⠘⡷⣼⡀⢀⣈⣇⢸⡀⠀⣧⢿⠖⠚⠣
⡙⠦⠖⢺⠋⠉⢙⣦⠀⠀⠀⡏⠀⠀⠀⠸⡧⡀⠈⠓⠙⣏⠁⠀⠀⡇⢱⠀⠈⡟⢄⠀⠸⡀⠀⠀⡇⢳⡀⠀⡇⠀⢠⡄⠀⠹⡌⢿⡁⠀⠸⣦⡇⠀⢹⢾⣀⣠⡤
⡓⣄⠀⢸⠀⠀⠀⡇⢳⡀⠀⡇⠀⢠⡄⠀⢹⡘⢆⠀⠀⠸⡄⠀⠀⣷⢸⠀⠀⡇⢸⡇⠀⡇⠀⠀⣇⠀⣇⠀⡇⠀⢸⠛⡄⠀⢳⠈⡟⠒⠒⢻⠁⠀⢸⢸⡇
⡇⢸⠇⢸⡇⠀⠀⣇⠀⡇⢠⡇⠀⢸⠳⡄⠀⢣⠘⡄⠀⠀⢷⠀⠀⢸⢸⠀⠀⢱⣨⣧⠤⠇⠀⠀⢸⠀⢹⢠⡇⠀⠘⠓⠛⠀⠈⢧⠸⡄⠀⠈⣇⣠⠴⣧⣇
⢣⣸⣤⠼⠇⠀⠀⢿⠀⢳⢸⡇⠀⠘⠓⠋⠀⠈⢆⠹⡀⠀⠘⣆⣠⠼⣾⡄⠀⠈⠀⠀⠀⠀⠀⠀⢸⡀⢸⢸⡇⠀⠀⣀⣠⡄⠀⠈⣆⢱⠀⠀⠘⢆⣀⡬⢿
⠈⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⠀⠀⣀⣀⡀⠀⠘⣆⢳⠀⠀⠘⣄⣀⣿⠇⠀⢰⡖⠒⠒⢶⠀⠀⠈⡇⠈⣿⠀⠀⠈⣧⡀⠸⡀⠀⠘⣆⢧⠀⠀⡼⠓⠒⣾
⢰⡒⠒⠚⣦⠀⠀⠘⡇⠈⣾⠀⠀⠘⣧⡀⢹⡀⠀⠘⡄⢧⠀⠀⡞⠓⢻⣦⠀⢸⠛⢄⣀⣸⠀⠀⠀⢷⠀⣿⠀⠀⠀⡇⢹⠋⢧⠀⠀⠘⡎⢆⠀⢸⡤⠤⢿⣧
⢸⠓⣄⣀⣹⠀⠀⠀⢇⠀⣿⠀⠀⠀⡇⢹⠋⢇⠀⠀⠘⡌⣆⠀⢹⣤⠼⣏⣧⠈⡇⢸⡇⠸⡆⠀⠀⠸⡄⢹⠄⠀⠀⡇⢸⠀⠘⣆⣀⣤⠼⣼⡄⠀⢣⣀⣈⣿
⠸⡇⢸⠁⠸⡄⠀⠀⠸⡀⢻⠀⠀⠀⡇⢸⠀⢸⣄⣀⡠⠾⣽⡀⠈⢧⣀⣿⠇⠀⢱⠈⡇⠀⣧⣀⣀⣠⢧⢸⡦⠖⠒⢧⣸⠀⠀⠙⣄⣀⣠⣼⣧⡀⠀⠁⠀⢸⡀
⠀⣧⠈⡄⠀⣇⣀⣀⡠⢧⢸⡶⠶⠚⢧⣸⠀⠀⠹⣄⣀⡤⣼⠃⠀⠀⠉⢳⡖⠒⠛⢆⡇⠀⠘⢇⠀⠀⠀⢹⣿⣤⣤⣦⡿⠀⠀⠀⢹⠉⠁⠀⠀⡟⡄⠀⠀⠈⡇

⣿⣅⠀⣰⣿⡄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣾⣶⡀⠀⢀⣿⣿⣷⣄
⣿⣿⢿⣿⣿⠁⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣷⣦⣴⣿⣿⡇⠀⢸⣿⣿⠿⣿⣇⠀⠀⠀⠀⠀⣾⣇⠀⢰⣷⡀⠀⠀⣠⣦
⠟⠀⠀⣼⡏⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠘⠹⣿⠀⢸⣿⠀⠀⠈⠛⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⠁⠀⣼⣿⣿⡄
⠀⠀⠀⠉⠀⠰⠟⠁⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⢹⣿⠀⣸⡿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠁⠀⠈⠟⠀⠀⢰⣦⡀
⢠⣤⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⣠⡄
⣾⣿⠃⠀⣰⣿⣿⡀⠀⠀⠀⠀⢰⣷⣤⣿⠀⢠⣶⡀⠀⠀⠀⠀⣠⡄⠀⢰⣦⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢈⣿⣿⣆⠀⢸⣿⡿⠛⢿⣇⠀⠀⣿⣷
⢻⣿⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠸⠏⠉⡿⢠⡿⠿⡧⠀⠀⠀⠀⢸⣷⣶⣿⣿⡇⠠⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠟⣿⡄⠀⣿⠂⠀⠀⠈⠀⢰⣿⡟
⠘⠋⠀⠿⠃⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠸⣿⡏⠀⢹⠇⠀⡿⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠘⠃⠀⠈⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣿⣿⠀⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠀⠀⣾⣿⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣿⣷⠀⠀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⢸⣿⠀⢰⣿⠋⠁⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡗⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⢿⡿⠀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣿⣿⡇⠀⠀⢠⣴⣿⡄⠀⠈⠻⠃⠀⠸⠇⠀⡿⠁⠀⢻⡇
⠀⠀⣀⠀⠀⠀⢰⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣾⣿⡿⠏⠿⣿⣿⠀⠀⣠⣿⡟⣿⣇
⠀⠘⣿⣷⡀⠀⢸⣿⣽⣿⣿⡄⠀⠀⠀⢸⣿⣄⣠⣿⣷⠀⠀⣾⣿⣇⠀⠀⠀⠀⢿⠿⠁⠀⢀⣿⡇⠀⢸⣿⣿⡿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣾⢿⣿⣧⠀⠸⣿⠋⠀⠙⠿⠀⠀⠀⢸⣿⡿⠛⢿⡇⠀⣼⣿⡾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡿⠋⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣟⠀⠈⣿⠆⠀⠻⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠸⠇⠀⡿⠁⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠻⠟⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣴⣿⣿⠀⠀⢀⣰⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣴⡄⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⣾⣿⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢻⣿⠃⢀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⡏⠀⣸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠙⣿⡏⠀⣠⣿⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠾⠃⢀⣾⠿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⢸⡇⠀⣿⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠿⠁⣰⡿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠟
⠀⢀⠀⠀⠀⠠⣿⣿⣶⣄⡀
⠀⣿⣿⡆⠀⠀⣿⣷⣿⢿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡞⠀⠀⢠⣶⡄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣴⣾⣿⣿⡄⠀⢻⣿⠁⠀⠙⠓⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣾⣿⡟⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⢰⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⡇⠉⠘⣿⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠙⢿⣿⠇⠀⣠⣿⣿⣽⣿⣆⠀⠀⠀⠀⠀⣾⣇⣀⣠⣿⣿⡆⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠹
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⣸⣿⠀⢀⣿⡟⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠻⢻⣷⠀⢸⣿⠉⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠟⠀⠀⠛
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣶⣠⣤⡶⠾⠛⠉⠁⠀⠀⢹⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⠾⠟⢻⣏⠀⢀⠀⠂⠀⠀⠀⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⣧⠀⣀⡠⣴⣶⠿⠟⠋⠉⢿⡄⠀⠈⣿⡄⠁⢠⣄⠀⠀⠀⠀⣟⣷⣇
⠀⠀⠀⠀⠀⠀⢘⣷⡤⣰⣿⡿⠒⢩⠝⠀⠀⠀⠀⠀⠸⣷⠀⠀⣹⣧⠀⣾⣿⣆⠀⠀⠀⢿⣿⣿
⠀⠀⠀⣀⣀⠬⠓⢻⣏⠀⢸⣿⡠⢾⣧⡀⠀⢀⣠⠄⠀⢻⣟⠉⠉⣿⡀⣿⡿⣿⣆⠀⠀⢸⣿⣿
⠀⠀⠀⢻⡄⠀⠀⢨⣿⡖⠺⣿⡇⠘⣿⣿⣄⠉⠀⠀⠀⠈⣿⡆⠀⢻⡇⠿⠃⠘⣿⠆⠀⢸⣿⣿⣀
⠀⠀⠀⠀⢿⡀⢀⠠⢺⣷⠀⢸⣿⡀⢿⡟⠻⣧⡀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣠⡤⠴⠒⠚⠋⠉
⠀⠀⠀⠀⠈⢿⡀⠀⠀⣿⡇⠀⣿⡇⠸⠇⠀⠉⠁⠀⢀⣀⣤⣤⣶⠖⢿⡉⠁
⠀⠀⠀⠀⠀⠈⣷⡀⠀⠘⠋⠀⠈⢀⣀⣤⣤⡶⠾⠟⠛⠉⠉⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠘⣧⣠⣤⣶⣾⣿⣿⣽⣥⣥⣥⣤⡤⠤⠤⠤⢴⣶⡶⡶⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢉⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁



⣿⡏⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣠⣂⠀⠀⠀⣀⣀⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢳⠀⣤⣀⡀⠀⠒⠒⠋⠀⠀⡀⠀⠢⠎⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡩⠛⠺⣏⠀⠀⣈⠁⠀⠀⠙⢯⡹⠛⠛⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢰⠘⣗⠤⣀⠀⠠⡀⡷⡈⠙⠲⠄⡎⢰⢀⢹⠀⠉⠳⢦⣼⠳⡄
⠀⠀⠀⠀⠈⢦⠊⠧⣽⠒⠆⠙⡆⠱⡇⣈⣃⡀⠀⠏⢿⠈⢀⡁⣦⡀⢠⡏⠳⢼⣦
⠀⠀⠀⠀⠀⠀⠳⠉⠃⡀⢠⢻⠓⠢⢀⠘⣆⡜⠁⠓⠺⠀⢸⠙⢮⡙⢾⡀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢫⠈⢣⡠⠳⡸⡄⠀⢰⡀⣰⣸⠔⠊⠙⠦⡙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢵⠀⠑⢳⢤⣘⣯⠀⢺⠀⡀⠀⠰⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢈⡷⡀⠸⣇⡼⠛⡷⢬⣸⡇⡀⠠⣧⣸⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⡉⠀⠘⢆⢻⠀⠀⠈⠀⠈⣯⠀⠀⢻⡏⠘⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠈⢃⡄⠀⠀⢸⣽⢀⣄⢸⠀⢰⡀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣶⡀⠈⣇⢀⣤⠞⣟⢸⢻⠈⢸⡠⡇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡘⡆⢻⢱⡀⣿⠉⠀⠀⠸⡮⠜⡇⣽⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢾⠁⠣⣿⠀⠀⠀⠐⠁⠀⡇⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⡀
⠀⠀⠀⢠⣿⣿⡆⠀⢰⣿⣿⡇⠀⢀⣴⣿⣿⣿⡆⠀⢠⣶⡆⢀⣶⣿⡆⠀⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣇⠀⣿⣿⣇⣸⣿⣿⡇⢠⣿⣿⣏⣻⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣧⣴⣿⣿⣿⠀⢰⣿⣿⠛⣿⣿⣿⠀⣿⣿⡿⠿⣿⣿⠇⣾⣿⡿⠛⢻⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⡏⢙⣿⣿⣿⢀⣿⣿⠃⠀⣿⣿⣿⠀⣿⣿⠇⢠⣿⣿⢸⣿⡿⠁⠀⠈⢿⣿⣿⠇⠀⠀⠀⠀⠀⢠⣶⣾⡗
⠀⠀⠀⢸⣿⣿⡇⠘⣿⣿⣿⢼⣿⠏⠀⠀⠘⢿⠿⠁⢿⠟⠀⠀⢿⡿⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠃⢀⠻⠿⢇⣀⣁⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⢀⣤⣤⢀⣶⣶⠀⠀⣴⣾⣿⣿⡆⠀⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠈⠉⠁⢸⣿⣷⠀⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⣸⣿⣿⢸⣿⣿⠀⢰⣿⣿⣿⣿⣇⣠⣤⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣾⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⠿⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣟⠉⣿⣿⣿⣠⣿⣿⠟⣿⣿⣿⡇⠀⠀⠀⣿⣿⡏⣿⣿⡏⢰⣿⣿⠋⣿⣿⣿⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⣿⣿⣿⣿⣿⠏⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⠃⣿⣿⡇⠈⠛⠁⠀⠻⠿⠿⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠻⠿⠿⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠙⠁⠀⠻⠿

⢈⣳⣥⡀⠱⣄⣠⡬⠟⢶⣤⠜⠀⠸⡁⠀⠀⡟⣄⠤⠄⢮⡢⠤⢿⠉⠉⣿⡉⠉⡇⠀⠀⠘⡧⡀⠀⠘⡄⠀⠘⡈⡆⠀⠘
⠺⠂⠉⡝⢆⠀⢸⠀⢀⠀⢹⠢⡀⠀⢣⠀⠀⢃⢹⠀⠀⢸⠑⡄⢸⠀⠀⡇⢱⠀⡇⠀⣷⡀⠰⡘⡄⠀⢰⠀⠀⡗⡏⠹⢆
⠀⡆⠀⢰⠈⡆⠀⠀⢸⢆⠀⢃⢣⠀⠈⡆⠀⢸⢸⡄⠀⠈⣀⣇⡸⠀⠀⠇⠸⡀⡇⠀⠃⠓⠀⢣⢱⡀⠀⢇⣀⣷⡇⠀⡆
⠔⠃⠀⢸⠀⡇⡇⠀⠈⠉⠀⠘⡆⢆⠀⠸⡤⠚⢾⡇⠀⠀⠁⠀⠀⠀⠀⢰⠀⡇⠃⠀⣀⠤⡀⠀⢣⢣⠀⠈⢆⣥⠗⠀⠧
⠤⡀⠀⠀⡆⢠⡇⠀⢠⡖⢢⠀⠘⡌⡄⠀⢙⡖⢎⢱⠀⠀⣯⡉⠉⡇⠀⠘⡄⢿⠀⠀⠙⢦⢵⠀⠈⢆⢆⠀⢏⢩⣹⡄⢀
⡀⣁⠀⠀⢇⢸⡇⠀⢸⠱⡞⡇⠀⠘⡼⡀⠸⣀⠨⣿⡀⠀⢇⢹⠊⠇⠀⠀⠇⢸⠀⠀⠀⢸⠀⢇⢀⣈⣾⡀⠘⡍⣍⡇⢸
⡇⠸⠀⠀⠸⡀⡇⠀⢸⠸⡇⢸⡤⠤⠺⣧⠀⠣⡀⠽⡇⠀⢸⠀⡄⢸⣀⡀⠼⣼⡆⠒⠲⣸⠀⠘⢆⡀⠬⠃⠀⠈⢏⠀⠸
⠇⠀⣧⠐⠒⠣⣿⠉⠉⢲⡃⠀⠑⠒⠒⠉⠀⠀⠀⠀⠹⡉⠉⢣⣇⡠⡷⣀⡀⠼⡟⠶⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀
⣾⡀⠬⡦⠔⠒⠻⡛⠋⠉⠹⣄⠤⠴⣄⠀⠀⡆⠉⢹⢆⠈⠚⠋⠁⠀⠹⢦⡀⠀⠱⡀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡈⠣⣀
⠁⠀⠀⠹⡢⡀⠀⠱⡀⠀⢸⢸⠀⠀⢉⢳⠀⠁⠀⠘⠈⠇⢰⠀⠀⣆⠀⢣⢡⠀⠀⢃⠀⠘⣼⠆⠒⣦⠀⠀⡏⠁⢹⢆
⠀⢸⢇⠀⢣⠱⠀⠀⢃⠀⠠⣼⠀⠀⢸⣸⣀⡼⠀⠀⡇⢰⢸⠀⠀⠆⠃⠀⢆⢃⠀⠘⣄⣀⣿⡀⠀⠁⢱⠀⡇⠀⢸⠀⠇
⠀⠘⠒⠃⠀⢆⢣⠀⠘⣄⡠⢾⡄⠀⠀⠁⠀⠀⠀⠀⠃⠸⣸⠀⠀⣀⢤⠀⠘⡌⠆⠀⠙⣄⣼⠇⠀⢸⣸⡀⠄⠀⠀⡆⢸
⠀⠀⡠⠤⠀⠈⠆⢆⠀⠙⢤⠼⡇⠀⠰⣏⠉⢹⠀⠀⢸⠀⣿⠀⠀⡗⣄⣧⠀⠘⡜⡄⠀⡏⠙⣷⡀⠀⠀⠀⡀⠀⠀⢇⠸
⠀⠀⡗⡤⢇⠀⠈⡜⡄⠐⡏⢀⣿⡀⠀⡆⢳⠘⡆⠀⠈⡄⢿⠀⠀⡇⡇⠘⡄⠀⣸⣴⠀⢸⡉⣿⡇⠸⣏⠀⢸⠀⠀⢸
⠀⠀⡇⡇⠘⣄⣀⡸⣵⠀⠹⡁⣹⡇⠀⢡⢸⠀⢡⣀⡀⢥⣸⠤⠔⢧⡇⠀⠹⣅⡠⠼⠃⠀⠉⢹⠀⢈⡤⢿⠛⣦⠀⠈⡄
⠐⠒⢣⡇⠀⠙⡄⠤⠜⢃⣀⣉⠀⢳⡉⢹⢾⡒⠉⣧⣀⣀⡽⠳⠤⡴⠓⢲⡀⠀⠀⠊⠉⠉⢷⣸⠀⠈⣧⢸⠀⢷⢡⠤⠷
⡑⠄⢲⠋⠉⣷⡀⠀⢸⠁⠀⠈⢷⣄⠙⠚⣏⠁⠀⡏⠣⠀⡟⢄⠀⡇⠀⢸⠙⡄⢸⠀⢠⡄⠈⡎⢯⡉⠙⣾⡀⢸⢾⡀⠤
⡙⣆⠈⡀⠀⢁⠱⡀⠈⠀⢰⡀⠘⡌⡆⠀⠘⡄⠀⢹⢀⠀⢀⢸⠀⡇⠀⠈⡀⢇⢸⠀⢸⣹⡄⠸⡘⡍⠉⢹⠀⠀⣼⡇
⡇⣇⣀⡇⠀⢸⠀⡇⡆⠀⠸⠼⠀⠰⡘⡄⠀⢡⠀⣸⣼⠀⠘⠚⠉⠁⠀⠀⡇⢸⢸⠀⠀⠀⠀⠀⠱⡸⡀⠀⢳⠉⢹⡇
⠉⠉⠀⠀⠀⠘⡀⠇⡇⠀⢀⣀⡀⠀⢡⠱⠀⠈⢣⣀⡿⠀⢠⡔⠒⢲⠀⠀⢃⠈⣾⠀⠐⣏⠙⡄⠀⠱⢱⠀⠠⠟⠓⣦
⢲⡈⠉⠇⠀⠀⡇⢸⡇⠀⢸⢦⣸⡄⠀⢣⢣⠀⢰⠉⢻⣆⠸⡙⣤⠼⡀⠀⠸⡀⣿⠀⠀⡆⡏⢱⠀⠀⢡⣃⠀⢧⠒⢻⡆
⢰⠙⡖⢺⠀⠀⢰⠘⡇⠀⢸⠀⠁⢡⠀⠀⣫⡆⠀⠗⢹⣿⠀⡇⢠⠀⡇⠀⠀⣇⢻⣀⣠⡇⡇⠀⢷⠒⠉⣻⡄⠈⠒⠚⡇
⠈⡄⡇⠈⣄⣀⡈⡆⣧⠤⠼⣀⠀⠈⢏⣉⣀⡿⠀⠈⠙⡥⠀⠲⣸⠀⠹⡈⠉⠈⣾⣦⣠⣼⠇⠀⠀⡟⠋⠉⢹⢆⠀⠀⢡

⠙⠛⠃⠀⠀⠀⠚⠃⠛⠀⠀⣁⢀⠀⠀⠀⠘⠋⢙⣃⠘⠛⠛⠃⠀⠀⠀⠛⠃⠛⠀⠰⣃⣀⠀⠀⠀⠘⠋⢙⣃
⠈⠀⠀⠳⠃⠀⠇⠖⠰⠛⠧⠀⣬⠀⡄⠀⠀⠀⢸⠉⠀⠀⠀⠘⠳⠃⠀⠇⠖⠰⠛⠧⠀⣬⠀⡄⠀⠀⠀⢸⠉
⠶⠀⠀⠂⠂⠀⠀⠂⠀⢀⠀⠀⡿⢰⣧⠀⣤⡄⠀⠾⠦⠶⠀⠀⢀⠂⠀⠀⠓⠀⢀⠀⠀⢿⢰⣧⠀⣤⡄⠈⠾
⠠⠴⠀⠀⠀⠀⠀⢠⠏⣿⠃⢠⡄⡀⢠⣤⡀⠀⡦⠰⠆⠠⠴⠀⠀⠀⠀⠀⢠⠏⣿⡓⢠⡄⡀⢠⣤⡀⠀⠶⡸
⠀⠀⠀⠀⠀⢠⣤⠠⠧⢫⡄⠘⠋⠿⠺⠽⠇⠀⠲⠆⠠⠀⠀⠀⠀⠀⠠⢤⠠⠯⢫⡄⠘⠋⠿⠺⠽⠇⠀⠲⠆
⠀⣀⡀⠐⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣴⣦⠀⠀⠒⠀⢀⡀⠐⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣴⣦
⠃⣉⢁⡄⠀⠀⠀⣤⠀⢀⠀⣀⢠⡄⢿⣿⠿⣿⢸⣣⡯⠃⣉⢁⡄⠀⠀⠀⣤⠀⢀⠀⣀⢠⡄⢿⣿⡿⣿⢸⣣
⡀⣈⠈⢨⣴⣴⠀⠇⠀⠀⠀⢉⣈⡁⢀⡄⣠⢠⠈⠁⢉⡁⣉⠀⢨⣴⣴⠀⠄⠀⠀⠀⢉⣈⡁⣀⡄⣠⠀⠈⠁
⠀⠀⠀⠈⠙⠙⠁⠈⠑⡃⣀⢀⠈⠀⠈⠁⠛⠐⠑⠮⠀⠀⠀⠀⠈⠛⠙⠁⠈⠑⡃⣀⢀⠈⠀⠈⠃⠛⠀⠑⠮
⣠⣠⡄⡀⠀⠀⠀⠀⠀⣷⣷⡼⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⡄⡀⠀⠀⠀⠀⠀⣷⣿⡾⡇
⢿⣿⡇⠉⠁⠁⣶⡆⣶⠀⢈⡄⠀⠀⠀⠀⠰⡦⢶⠄⢹⣿⣿⡇⠈⠁⠁⣶⣆⣶⠀⢈⡄⠀⠀⠀⠀⠰⡦⢶⡄
⠠⠀⠀⣄⡄⠀⡄⣀⢀⣤⡄⠒⠐⠀⠀⠀⠀⠀⢰⠶⠀⠠⠄⢠⣄⡄⠀⡄⣄⢀⣤⡌⠒⠐⠀⠀⠀⠀⠀⢰⠶
⣀⠀⠀⡈⠀⠀⡁⡁⠈⠀⠉⠀⣿⢀⡇⠀⠀⠀⠘⣠⡀⣀⠀⠀⠈⠀⠀⠁⣅⠈⠀⠉⠀⣿⠀⡇⠀⠀⠀⠸⣠
⠛⢀⠀⠐⠀⠀⠀⢀⡤⣴⡄⠀⠋⠚⠛⠀⠛⠃⡀⢉⡉⠛⢀⠀⠐⠀⠀⠀⠀⡤⣴⡄⠀⠋⠘⠛⠀⠛⠃⣀⢨
⠈⠉⠀⠀⠀⠀⠀⠰⡅⡿⠂⢸⡧⣦⣸⢿⡆⠀⣋⡈⠁⠈⠉⠀⠀⠀⠀⠀⠐⡅⡿⠂⢸⡧⣦⣼⢿⡇⠀⡉⡈
⠀⠀⠀⢀⡀⡈⠙⠈⠉⠘⠃⠀⠀⠉⠈⣉⢁⡀⠀⠁⠈⠁⠀⠀⢀⡀⡈⠉⠈⠉⠘⠃⠀⠀⠉⠈⢉⢁⡀⠁⠁
⡄⠶⠆⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⡄⡤⡄⠴⠆⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⡄
⠀⠲⠰⠃⢀⢀⠀⡟⠀⠂⠀⠶⠸⠇⠙⠛⠉⠛⠸⠞⠯⠄⠶⠐⠃⢀⢀⠀⡟⠀⠒⠀⠶⠸⠇⠙⠛⠋⠛⠸⠞
⠂⠂⠀⠸⢿⢿⠀⠡⢄⡄⠀⠐⠶⠂⠰⡇⣾⢸⢄⣠⠐⠂⠂⠀⠸⢿⢿⠄⠡⢄⡄⠀⠐⠲⠂⠲⡇⡾⠀⢄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣖⢰⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣶⣰⡄⠀⠀⠀⠀⠄⠀⠉
⣾⣾⡇⠦⠄⠄⡀⡀⣀⠛⠻⠋⠃⠀⠀⠀⢀⡀⣈⡀⢴⣾⣾⡇⠠⠄⠄⡀⡀⣀⠛⠹⠋⠃⠀⠀⠀⢀⡀⣀⡀

⠘⠛⠻⠃⢀⠀⠀⡀⠸⠻⠸⠿⠄⠀⢸⠂⠀⠐⠀⠀⠀⠀⠛⠛⠙⠛⠀⠸⠛⠛⠻⠃⢀⠂⢀⠀⠸⠻⠰⠿⠄⠀⢺⠂⠀⠐⠀⠀⠀⠀⠛⠛⠑⠛
⠀⠆⠀⢠⣤⣴⠀⠀⣤⢠⡄⢀⡶⣦⡀⠛⠐⠃⠀⠀⠀⠀⠀⠂⠐⡟⡖⠀⠂⡄⠀⢰⣶⣰⠀⠀⣤⢠⡄⢀⡶⣦⡈⠙⠓⠓⠀⠀⠀⠀⠀⠀⠐⡟⠓
⠀⠀⠀⠀⠈⠁⠀⠀⠋⢘⠀⠛⠀⠈⠁⠀⣇⡇⢸⡇⠀⠀⠀⠀⠰⠇⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⡋⢘⠀⠙⠀⠈⠁⠀⣇⡇⢸⡇⠀⠀⠀⠀⠰⠇⡄
⠾⠆⠀⠀⢁⠈⠀⠀⠃⠘⠐⠒⢀⣀⠀⠀⡟⡇⣾⢷⠀⢶⣶⠀⠀⠚⠛⠳⠾⠆⠀⠀⢁⠈⠀⠈⠃⠘⠑⠐⢀⡀⠀⠀⡟⡇⣾⢧⠀⢶⣶⠀⠀⠚⠛
⢠⣀⡤⠀⠀⠀⠀⠈⠀⠀⣸⡏⣿⢻⠆⢠⡀⠁⠀⢀⣀⡀⠀⠀⣦⢀⣶⡀⢠⣀⡄⠀⠀⠀⠀⠈⠀⠀⣺⡏⣽⢻⡇⢠⡀⠀⠁⢈⣀⣀⠀⠀⣤⢄⢶
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⡆⡿⠃⠀⢸⡷⢲⣄⣼⠓⣿⠀⠀⣉⣈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢳⡆⡿⠃⠃⢸⡷⢲⣆⣼⠛⣿⠀⠀⣁⣈
⠁⠀⡀⠀⢸⠀⠀⠘⠓⠃⠉⠉⠘⠛⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠉⠉⠀⠘⠁⠀⠀⠀⢸⠀⠀⠘⠓⠃⠉⠉⠘⠛⠂⠀⠀⠈⠁⠈⠉⠁⠀⠀⠉⠉⢀
⠁⢀⣉⡀⢸⢱⣎⡇⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⢠⣦⣾⣷⣾⡇⠀⠀⠈⠉⠁⢀⣈⡀⢸⢱⣎⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣦⣾⣧⣾⡇⠀⠀⠈
⠇⠈⠛⠃⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣷⢸⡟⣸⣏⠇⠘⠛⠃⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣿⢨⡟⣸
⠄⠘⠛⠘⠙⠀⣈⣀⠀⢸⠙⠀⠙⠀⠸⠿⠰⠛⠄⠉⠉⠉⠈⢉⠐⠛⠊⠶⠆⠘⠛⠘⠙⠀⣀⣀⠀⢸⠙⠈⠙⠀⠸⠿⠸⠛⠄⠉⠉⠉⠈⢉⠘⠛⠘
⠶⠰⠄⠀⠀⣿⣿⣿⡂⠈⣤⡀⡄⠀⠀⠰⢤⡖⠀⣧⡇⢰⣇⢸⣄⠀⡀⠰⠶⠰⠆⠀⠀⣿⣿⣿⡄⠈⣤⡀⡀⠀⠀⠰⢤⡖⠀⣧⣷⢰⣇⢸⣄⠀⡀
⠂⠐⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⣉⢠⡀⢠⠀⠀⠀⠁⠁⠈⠉⠘⠈⠓⠂⠀⠂⠐⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⣋⢠⡀⢠⠀⠀⠀⠁⠉⠈⠉⠘⠈⠓⠂
⢀⡀⣀⡀⢀⠀⠀⠈⠀⠀⠀⠀⣿⢼⣇⣟⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⢼⣇⣟⣇⠀⠀⠀⠀⠀⢀⠁⢀
⣼⣿⣿⡇⠛⠲⠴⠂⢀⣠⠀⣤⠙⠘⠛⠁⠙⠀⠀⠀⠀⠀⣤⡀⣀⣀⠀⣿⣼⣿⣿⡇⠘⠶⠴⠄⢀⢠⠀⣤⠙⠈⠛⠁⠙⠀⠀⠀⠀⠀⣤⠀⣀⡀
⠙⠛⠻⠃⢀⠒⠀⡀⠼⠻⠰⠿⠄⠀⠸⠃⠀⠐⠀⠀⠀⠀⠛⠋⠙⠛⠀⠘⠛⠛⠻⠃⢀⡂⢀⡀⠼⠻⠸⠿⠄⠀⢺⠓⠀⠐⠀⠀⠀⠀⠛⠋⠑⠛
⠂⠆⠀⢠⣤⣴⠀⠀⣤⢠⡄⢀⡴⣦⡀⠉⠐⠓⠀⠀⠠⠀⠀⠂⠐⡟⡗⠠⠂⡄⠀⢰⣶⣰⠀⠀⣄⢠⡄⢀⡴⣦⡈⠙⠑⠃⠀⠀⠀⠀⠀⠀⠐⡟⡓
⠀⠀⠀⠀⠈⠀⠀⢀⠋⢈⠀⠛⠀⠈⠁⠀⣇⡇⢸⡇⠀⠀⠀⠀⠰⠇⡄⠀⠀⠀⠀⠀⠈⠀⠀⢀⡋⢘⠀⠛⠀⠈⠁⠀⣇⡇⢸⡇⠀⠀⠀⠀⠰⠇⡄
⠾⠆⠀⠀⢃⡈⠀⠀⠁⠘⠐⠒⢀⣀⠀⠀⡟⡇⣾⣷⠀⣶⣶⠀⠀⠚⠛⠳⠾⠆⠀⠀⢁⡈⠀⠀⠁⠘⠑⠒⢀⡀⠀⠀⡟⡇⣾⣧⠀⣶⣶⠀⠀⠘⠛
⢠⣀⡄⠀⠀⠀⠀⠈⠀⠀⢺⡏⣿⢻⡄⢀⡁⠀⠀⢈⣀⣀⠀⠀⣦⢀⣶⡀⢠⣀⡄⠀⠀⠀⠀⠀⠀⠀⣺⡏⢿⠻⡇⢀⡀⠀⠁⢈⣀⣀⠀⠀⣤⢀⢶
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢳⡆⡿⠃⠀⢸⡷⢲⣄⣼⠓⣿⠀⠀⣁⣈⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢳⡆⡿⠃⠀⢸⡷⢲⣄⣼⠓⣿⠀⠀⣁⣈
⠁⠀⡀⠀⢸⠀⠀⠘⠓⠃⠉⠉⠘⠛⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠉⠉⠀⠘⠁⠀⡀⠀⢸⠀⠀⠘⠓⠃⠉⠉⠘⠛⠀⠀⠀⠈⠁⠉⠉⠉⠀⠀⠉⠉
⠀⢀⣉⡀⢸⢱⣎⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣦⣾⣧⣾⡇⠀⠀⠈⠉⠁⢀⣈⡀⢸⢱⣎⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣦⣾⣧⣾⣇⠀⠀⠈
⠇⠘⠛⠃⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣷⢸⡟⣸⣏⠇⠘⠛⠃⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⣿⢸⡟⣸
⠀⠘⠛⠘⠙⠀⣀⣀⠀⢸⠙⠀⠉⠀⠸⠿⠸⠻⠄⠉⠉⠉⠈⢉⠘⠛⠈⠶⠆⠘⠛⠘⠙⠀⣀⣀⠀⢸⠙⠀⠙⠀⠸⠿⠸⠛⠄⠉⠉⠉⠈⢉⠐⠛⠘
⠶⠴⠄⠀⠈⣿⣿⣿⡄⢈⣤⡀⡀⠀⠀⠰⢦⡖⠀⣷⡇⢸⣇⢸⣄⠀⡀⠰⠶⠰⠄⠀⠀⣿⣿⣿⡄⠈⣤⡀⡀⠀⠀⠰⢦⡆⠀⣧⣿⢸⣇⢸⣄⠀⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⣋⢀⡀⢠⠀⠀⠀⠁⠁⠉⠉⠘⠈⠓⠃⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠈⠙⣋⢀⡀⢠⠀⠀⠀⠁⠉⠈⠉⠘⠈⠓⠂
⢀⡈⣀⡀⢀⠀⠀⠌⠀⠀⠀⠀⣿⢼⡇⣟⣇⠀⠀⠀⠀⠀⢈⠁⢀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿⢼⣇⣟⣇⠀⠀⠀⠀⠀⢈
⣼⣿⣿⡇⠛⠲⠴⠄⢠⢠⠀⣤⠙⠈⠟⠁⠙⠀⠀⠀⠀⠀⣤⠀⣀⣀⠀⣿⣼⣿⣿⡇⠘⠲⠰⠄⢀⢠⠀⣄⠙⠈⠛⠁⠙⠀⠀⠀⠀⠀⣤⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⢠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⢾⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠁⠀⠀⠀⣴⠂⠒⠒⠂⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⣀⠀⠀⡀⠀⣿⠀⠀⠀⠘⠀⠸⣿⠀⠘
⠀⠀⠀⠀⠀⠀⣤⠤⠤⡄⣀⡤⠖⠋⠁⠀⠀⣶⠀⠀⢂⢿⡄⠀⢠⠀⣿⠀⢀⡀⠀⠁⠀⣿⡀⠀⡆
⠀⠀⣿⠀⠀⢃⣿⠀⠀⢨⠁⢀⡦⠀⠀⠠⠀⢻⡇⠀⢸⢸⡇⠀⠸⢸⡇⠀⢰⣧⠀⠈⠄⢹⡧⠤⠇
⠀⠀⣿⡄⠀⠠⢹⡇⠀⢸⠀⢸⡇⠀⠀⠀⢁⢸⡇⠀⠀⠼⠇⠀⠀⣼⡇⠀⠀⠀⠀⠀⠠⢸⡇⠀⢸
⠀⠀⢸⡇⠀⠘⠜⠃⠀⠘⠀⢸⡇⠀⢀⠀⠈⡸⣷⠀⠀⢀⣀⠀⠀⣿⡇⠀⢰⢾⣧⠀⠀⠼⠿⠷⠾
⠀⠀⢸⣧⠀⠀⣀⣀⠀⠀⡇⣸⡇⠀⢸⡄⠀⠐⣿⠀⠀⠙⣿⡄⠀⢸⡀⣀⣸⠀⠿⠶⠶⠾
⠀⠀⠀⣿⠀⠀⠋⣿⠀⠀⠀⣿⠀⠀⠈⠁⠀⠀⣿⡇⠀⠰⢸⡇⠀⠘⠛⠛⠋⠀⠀⠀⠀⠈
⠀⠀⠀⢿⡆⠀⠸⢹⡇⠀⢸⣿⠀⠀⡴⢶⡄⠀⠸⣷⣶⣶⠎⠛⠛⠛⠃⠀⢀⣀⣀⣠⣤⡴
⠀⠀⠀⠸⠷⠶⠾⢬⡟⠛⢛⣿⠀⠀⡇⠸⡷⠶⠶⠆⢀⣤⣤⡶⠶⠾⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣼⣿⠿⠟⠓⢷⣦⣀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣸⣿⠿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣿⣿⣶⣤⡄⢰⣶⣶⣿⣿⣤⣤⣼⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⠛⠛⠋⠉⣿⣿⠉⠉⠀⠸⠿⢿⣿⡿⠟⠛⢻⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣸⣿⣇⣀⡀⠀⣀⣀⣿⣿⣤⣤⣬⣿⣧⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⢿⣿⡿⠛⠃⠀⣿⣿⣿⣿⡿⠿⠛⣿⣿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⠉⠀⠀⠀⣿⣿⠀⠀⠀⠉⠉⣿⣿⡄⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠿⠛⠃⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣤⣤⣤⣤⣿⣿⣤⣤⡀⢠⣤⣤⣿⣧⣤⣤⣼⣿⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠿⠿⠿⠛⣿⣿⠛⠋⠁⢸⡿⢿⣿⡿⠿⠛⢻⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⢿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣻⣧⣤⣶⡾⠟⠛⠉⡁⠔⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣶⣾⠿⠿⣿⡏⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠘⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⣀⣠⠴⣖⣿⠽⠚⢻⣿⠀⠀⠀⢻⣷⠀⠔⠁⢀⠀⠀⠀⠀⠀⠀⣆⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢸⣿⡤⢶⣚⡭⢷⠚⠉⠁⠀⠀⠀⠈⣿⣇⠀⠀⠸⣿⡇⠀⢀⣿⣦⡀⠀⠀⠀⠀⣿⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⣖⣪⠽⣿⣟⠉⠀⣲⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⣠⣿⣷⠀⢸⣿⣿⣷⡀⠀⠀⠀⢸⣼⣿⣇
⠀⠀⠀⠀⢀⣐⡬⠴⠒⢻⣿⡀⠀⢹⣿⣄⠜⣿⣷⡀⠀⠀⣀⣠⡤⠀⠀⢿⣿⠉⠁⠸⣿⡆⢸⣿⡿⣿⣷⡀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠈⣿⣧⡴⣾⣿⣧⠀⢹⣿⣿⣆⠀⠉⠁⠀⠠⠁⠘⣿⣇⠀⠀⣿⣷⢸⡿⠃⠘⣿⣷⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠈⢹⣿⡎⠀⢹⣿⡆⠘⣿⣽⣿⣷⡀⠀⠐⠀⠀⠀⠹⣿⠀⠀⠘⠁⠀⠀⠀⠀⠈⠁⣀⣠⣤⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠘⣷⡀⠔⠊⠀⢿⣧⠀⠈⣿⣷⠀⢻⣧⠀⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠸⣿⡆⠀⢹⣿⡆⠘⠋⠀⠀⡀⠀⠀⢀⣀⣤⣤⣶⣾⠿⠟⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠻⠋⠀⠀⠉⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣀⣠⣤⣶⡾⡿⡻⢋⢝⡕⠀⠂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⣿⢟⢭⡫⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⢿⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣣⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠸⡸⡏⣴⠛⠿⠿⡏⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠳⡷⡀⠀⣀⣤⣰⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⣿⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⠘⢿⠞⠛⠉⠀⠀⠀⣠⠤⢤⣤⣤⣤⠤⡀⠀⡇⣧⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠉⠉⠉⠉⢳⢣⠀⢱⢻⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⣀⣀⣠⣀⣀⣀⡀⠀⡇⡇⠀⠀⠀⠀⠘⡞⡄⢸⢸⠘⡄⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠟⠋⠁⠀⡤⢤⣤⣤⣤⢤⡇⢸⠉⠉⠙⡎⡇⠀⠇⡇⠀⠀⡀⠀⠀⢰⢱⠈⡎⡇⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢺⢩⠭⠿⠿⡝⣆⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⡇⡏⠉⠉⢹⢸⢹⢸⠀⠀⠀⡇⡇⢸⢠⠃⠀⠀⣿⠀⠀⠀⠏⠇⡇⣧⢸⠀⠀⡇⣇
⡏⡭⠛⠋⢻⢹⢿⢸⠀⠀⠀⠀⡟⠋⠁⢠⢤⣤⣤⣤⣤⡄⡄⠀⡇⢃⠀⠀⠈⣼⣾⠘⡄⠀⠀⢀⡇⢸⢸⠀⠀⠀⡇⣇⠀⠀⠀⡐⣤⣸⠈⣄⣀⡧⢿
⣧⢱⠀⠀⠀⡇⣼⠸⡆⠀⠀⢰⢡⠀⠀⢸⢸⠀⠀⠀⠀⠹⣹⡀⢱⢸⠀⠀⠀⡇⡟⡇⡇⠀⠀⢸⢸⡘⢸⠀⠀⠀⣇⣹⠀⠀⠀⢃⢧⢸⠰⡒⠒⠒⣎⣶⡦
⢹⢸⠀⠀⠀⡇⡇⡆⡇⠀⠀⢸⢸⠀⠀⡆⣾⠀⠀⠀⠀⠀⢧⢃⢸⠸⡀⠀⠀⢱⢱⡧⠗⠀⠀⢸⣼⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡌⡆⡇⠀⠀⢱⢸
⢸⢸⡀⠀⠀⢻⢡⡇⢃⠀⠀⠸⣸⡀⠀⡄⡇⠀⠀⢠⠀⠀⠘⡼⣈⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⡇⠀⠀⢠⠤⠴⢲⡀⠀⠀⢳⢻⡇⠳⠤⠤⠞⡸
⠘⡄⡇⠀⠀⠘⠒⠉⠁⠀⠀⠀⡏⡇⠀⡇⡇⠀⠀⡿⡆⠀⠀⢱⢻⡇⣇⠀⠀⠀⡤⠦⠲⡀⠀⠀⢇⠀⡇⠀⠀⢸⢸⠙⡆⢇⠀⠀⠀⣎⡏⠉⠉⠑⠒⠃
⠀⡇⢇⠀⠀⠀⠀⢀⣀⠀⠀⠀⢣⢇⢠⠁⡇⠀⠀⡇⢣⠀⠀⠈⡎⣧⢸⠀⠀⠀⠇⣿⡆⡇⠀⠀⢸⢠⠃⠀⠀⢸⢸⠀⢹⣈⣉⣉⣉⣉⣸
⠀⢱⢸⠀⠀⠀⢏⣥⡄⡇⠀⠀⢸⢸⢸⢸⠀⠀⠀⣧⠬⠆⠀⠀⢸⡸⠘⡄⠀⠀⢸⢹⡇⢱⠀⠀⠈⡎⣓⣒⣒⣋⡼⠀⠀⠀⠀⠀⠀⢸⢸
⠀⢸⠸⡆⠀⠀⢸⢸⡇⢣⠀⠀⠸⡾⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣇⠀⠀⠸⡎⣷⠸⡀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⡄
⠀⠈⡆⠇⠀⠀⠘⡸⣿⢸⠀⠀⠀⡇⠏⠘⠀⠀⠀⡔⠒⠒⡆⠀⠀⠸⡄⠹⠤⠤⠤⠇⡏⠦⠭⠭⠥⠤⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠃⡇
⠀⠀⢣⢰⣀⣀⣀⡇⡿⡆⠳⠶⠶⠚⠀⡇⠀⠀⢰⡄⠋⢧⢹⡀⠀⠀⢣⢺⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠂⣈⣉⣁⡤⠤⠴⠚
⠀⠀⠸⠬⠭⠭⠭⠥⢷⠙⡍⠉⠉⠉⡇⡇⠀⠀⢸⢳⠀⠘⢤⠭⠭⠭⠭⠼⠆⢠⠖⠒⠚⢉⣉⣉⡤⠤⠤⠒⠒⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠹⡀⠀⣸⣇⣉⣉⣉⣭⡼⠒⠢⣀⠀⠀⠀⠀⠀⣰⢃⡖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣈⣉⣉⠤⠤⠴⠒⠒⠊⠉⠲⣄⠑⠦⣄⠀⣰⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠙⢡⠎
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣀⡄⣶⣦⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠿⠳⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⣀⡴⠞⠛⢉⡉⠉⠉⠛⠳⣦⡀⠀⠀⠀⣦⣶⢸⣿⡆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡴⣫⣶⠟⠁⠀⣴⣤⣠⡌⠻⣦⠀⠀⠈⠈⠈⠀⡀⢀⠀⣰
⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠈⠀⠋⣴⡟⠁⠀⠀⠀⠉⣱⠏⠹⣆⢻⡇⠀⠀⠀⢸⡿⡧⡿⡇⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⡶⢀⣠⠞⠁⠀⠀⣿⢸⡟⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃⠀⠀⠀⢻⣧⠈⠱⠛⠁⠀⠀⣀⣴⣿⡿⠁⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣈⣙⣛⣛⣭⣾⣿⣿⡆⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⡿⣿⠁⠘⢻⣿⣿⣶⡄⣸⡟⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢹⡇⠀⣼⣿⣿⣿⣿⠟⠁⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣼⣷⡾⠛⠶⠻⢿⣿⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠤⠤⢤⣶⠶⢾⡶⠶⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⠀⠸⣿⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣾⣿⡏⠀⠀⢻⣿⣿⠉⣿⡿⠋⠉⠀⠉⠛⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⣿⣿⠇⠀⠀⢸⣿⣿⠀⣿⠁⠀⠀⣤⠀⠀⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⠀⠀⠀⠛⠛⠛⠀⡿⠀⠀⠸⠿⠇⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⠀⠀⠀⢰⣶⣶⠀⡇⠀⠀⢠⣶⡆⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠀⣇⠀⠀⢸⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⣄⣤⣾⣿⣿⣤⣿⣦⣤⣾⡿⢿⣷⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⢻⣿⣿⠃⠈⣿⡟⠁⢀⡀⠈⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⣿⠀⢠⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣤⠀⢠⡇⠀⣴⣶⣶⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⠀⢸⡇⠀⣿⣿⣿⠀⢘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⢶⣶⣿⡿⠿⢶⣾⣿⣶⠿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠈⣿⣿⡇⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⠇⠀⠀⣿⠀⠀⠀⣾⣷⡀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣀⣀⡀⠀⠀⣏⠀⠀⠀⠉⠉⠁⠀⠀⢸⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⣿⠀⠀⢰⣿⣿⡇⠀⠀⢸⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⢠⣿⣿⣆⡀⣀⣿⣄⣀⣼⣿⣿⣇⠀⠀⣸⣿⠏⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣏⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⣤⣤⣦⠀⠀⣰⣿⣿⣦⡄⠀⠀⠀⠀⠀⢀⣤⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⢸⣿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⣶⣿⣿⣿⡇⣿⣿⣿⣿⣼⣿⣿⣿⣿⣧⡀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣄⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⠹⠿⠿⠋⠛⢛⣿⣶⣤⡀⠀⠈⠻⣿⡿⠋⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣄⠀⠈⠀⠀⠀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣮⠿⠛⠛⣿⣿⣿⡆⠀⠀⣿⣿⣿⠿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⣿⣿⣿⣧⠀⠀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡿⠿⠋⠙⢿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡄⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⢈⣩⣤⣄⡀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠉⢁⠀⠀⠀⣴⣶⣿⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⣾⣿⣿⡇⠀⠀⣿⣿⣿⣇⠀⠀⢀⣾⣿⣿⡿⠁⠙⣿⣿⣿⣷⡀⠸⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣀⣤⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠉⣿⣿⣿⡇⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⣠⡀⠀⠀⣰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠐⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠐⣿⣿⣿⣧⠀⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢀⣿⣿⣿⣿⢀⣿⣿⣿⣟⢻⣿⣿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⢿⣿⣿⣿⢾⣿⣿⣿⠛⠛⠛⠋⠻⣿⣿⣿⣷⠄⠀⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠈⠛⠛⠉⠀⢁⣽⣿⣿⣶⣄⡀⠀⠙⠿⠟⠁⠀⠀⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⣠⣤⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣎⠛⠋⠉⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⠻⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠿⠋⠙⠻⣿⣿⣿⠃⠀⠀⠀⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⢉⣡⣤⡀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡁⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⡀⠀⠀⠙⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠙⠋⠉⠀⠀⢀⣤⣴⣦⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠀⠀⢠⣶⣶⣧⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⣿⡟⠙⢿⣿⣿⣿⣦⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⣈⣿⣿⣿⣷⠀⠀⣰⣿⣿⣿⣿⣶⣶⣮⣿⣿⣿⣿⣧⠀⠻⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡇⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡅⠀⢸⣿⣿⣿⣇⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⡀⢠⢦⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢷⠸⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⣤⡶⠶⠿⠿⠿⠷⢶⣤⡀⠀⠀⠈⠀⢠⢠⡄⣴⣦⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡾⢋⡵⢀⣤⠶⠀⠀⣀⠀⠈⠻⣷⡀⠀⠀⠸⠟⠇⠏⠙⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠋⣴⢋⣴⠟⠁⠀⠀⠰⠿⢣⡾⣦⡘⣿⡄⠀⠀⠀⠀⣤⣰⢠⣶⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⣾⡏⠀⠀⠀⠀⠀⣰⠟⠀⠘⣷⢸⣧⣀⠀⠀⠀⠻⠙⠛⠙⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇⠺⡿⢁⣴⠞⠁⠀⠀⠀⣿⣸⡏⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠸⣷⡄⠘⠉⣀⣀⣀⣀⣤⣾⣿⣿⠁⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣌⣉⣩⣭⣷⣿⣿⣿⣇⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⠿⣯⠀⠘⢻⣿⣿⣿⣆⢀⣿⠁⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⣿⡄⠀⣼⣿⣿⣿⣿⡿⠃⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣿⣧⡾⠿⠷⠿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣇⣤⣤⣤⣤⣤⣭⣥⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⠀⠀⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⠇⣿⢡⠖⠶⠶⡖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡟⣷⡀⠀⣀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢰⢸⢸⠀⠀⠀⡇⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠀⠘⠿⠟⠋⠁⠀⠀⢠⠤⣤⣤⣤⣤⢤⠀⠸⢸⡸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⠀⠀⢃⢇⠀⡏⡇⡄⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⡤⢠⣤⣤⣄⠤⠀⠈⠘⠀⠀⠀⠀⠈⡼⠀⠇⡇⠃⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⣠⣴⠶⠋⠉⠀⠀⢰⢲⠶⠶⠶⡒⡇⢸⠀⠀⠘⢰⠀⡀⡆⠀⠀⣄⠀⠀⢱⢇⢰⢿⠸⠀⠀⠇
⠀⠀⠀⡄⣤⣤⣤⡤⣼⠐⠃⠈⠘⡞⣶⠶⠛⠉⠀⠀⠀⠀⢀⠀⠈⡘⠀⠀⠀⡇⣳⠘⠀⠀⢰⣿⠀⡃⡇⠀⠀⡿⡄⠀⠈⡜⡌⢸⡄⡆⠀⢀⡇
⠀⠀⠀⡇⡇⠀⠀⢃⢸⠀⡇⠀⠀⡁⡇⠀⠀⡏⡞⠛⠛⠛⢊⣇⠀⡇⡅⠀⠀⠃⢹⠀⡆⠀⠀⡄⡀⡇⠁⠀⠀⡇⢇⠀⠀⠰⢸⠺⠃⡕⠒⢋⣡⣀⡀
⠀⠀⠀⠇⠇⠀⠀⢸⠸⡆⠇⠀⠀⠏⡇⠀⢀⠁⠃⠀⠀⠀⠘⡘⡀⠃⠇⠀⠀⢸⢻⡇⡃⠀⠀⠇⣷⢸⠀⠀⠀⠃⠉⠀⠀⠀⢃⡆⡇⠃⠀⠀⡇⡏⠉
⠀⠀⠀⠰⢠⠀⠀⠘⣠⡇⢰⠀⠀⢰⢡⠀⢸⢰⠀⠀⢀⠀⠀⢱⢣⠇⢰⠀⠀⠈⠁⠁⠀⠀⠀⢰⢽⢸⠀⠀⠀⣀⠀⢄⠀⠀⠨⢱⡇⠠⢀⣀⡅⡇
⠀⠀⠀⢰⠸⠀⠀⠀⠓⠓⠋⠀⠀⠸⠸⠀⠘⢸⠀⠀⢸⡄⠀⠀⠇⣷⠸⠀⠀⠀⣠⠤⢤⠀⠀⠘⡀⠘⠀⠀⢸⢰⠲⠈⡄⠀⠀⢢⢏⠀⠒⠒⠒⠃
⠀⠀⠀⠈⡀⡆⠀⠀⠀⠀⣀⡀⠀⠀⡇⡆⡄⠘⠀⠀⠀⢡⠀⠀⠸⣸⡄⡇⠀⠀⠇⢾⠘⡀⠀⠀⡇⠇⠀⠀⢸⢸⠀⢃⣙⣘⣛⣋⣨⡄
⠀⠀⠀⠀⡇⠃⠀⠀⢸⢡⡄⡇⠀⠀⢁⢃⡇⡆⠀⠀⠦⠸⠂⠀⠀⢧⠇⢁⠀⠀⢸⢸⡇⠇⠀⠀⢱⢙⡂⠀⢉⡼⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⠀⢠⢸⠀⠀⠘⠜⡷⢡⠀⠀⢸⢹⡇⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠀⠀⠈⡌⡇⠰⡀⣀⣸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇
⠀⠀⠀⠀⠘⠘⡄⠀⠀⡇⣿⠘⡀⠀⠀⡜⢃⠁⠀⠀⠎⣩⡙⡆⠀⠀⢡⢈⡓⢒⣒⣃⡞⠓⠒⠒⠒⠚⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⢃⡄
⠀⠀⠀⠀⠀⡇⠧⠤⠤⠴⢸⢧⣉⠉⠉⣁⢸⠀⠀⠀⣇⠀⢡⠱⠤⠤⠤⠎⡄⠀⢀⡀⣀⣀⠠⠤⠄⠒⠂⠈⣉⣁⠥⠤⠄⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠹⡌⢆⠀⠀⠸⠸⠤⠤⠤⢻⣄⡈⠃⠚⠛⠛⠋⠁⡰⢉⠥⠤⠐⠒⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠒⠒⠚⠛⣂⣀⡀⠬⠤⣈⠓⢄⡀⠀⠀⡜⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⡉⠲⠜⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠁

⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀
⠸⠧⠛⠇⠁⠈⠉⠉⠋⠀⠸⠧⠛⠇⠁⠈⠉⠙⠋⠀⠸⠧⠛⠇⠁⠈⠉⠉⠋⠀⠸⠧⠛⠃⠁⠈⠉⠙⠋
⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀
⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻
⠀⠉⠋⠈⠋⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠋⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃
⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀
⠸⠧⠋⠃⠁⠈⠉⠉⠋⠀⠸⠧⠛⠇⠁⠈⠉⠙⠋⠀⠸⠧⠟⠇⠁⠈⠉⠙⠋⠀⠸⠧⠋⠇⠁⠈⠉⠉⠋
⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀
⠀⣄⣄⣦⠸⠀⠋⠟⢻⠘⠀⣄⣄⣦⠸⠀⠋⠟⠻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⠻
⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃
⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢠⣠⣠⢀⡀⠀⠀⣀⠀⡀⢠⣠⣠⢀
⠸⠧⠛⠇⠁⠈⠉⠉⠋⠀⠸⠷⠋⠇⠁⠈⠉⠉⠋⠀⠸⠷⠟⠇⠁⠈⠉⠙⠋⠀⠸⠧⠛⠇⠁⠈⠉⠉⠋
⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⢀⠀⢠⣀⢀
⠀⣄⣄⣦⠸⠀⠋⠟⠻⠸⠀⣄⣄⣦⠸⠀⠋⠟⠻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻
⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠋⠀⠀⠀⠀⠀⠀⠉⠋⠈⠋⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃
⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀⡀⠀⠀⣀⠀⡀⢀⣠⣠⢀
⠸⠧⠛⠇⠁⠈⠉⠉⠋⠀⠸⠧⠛⠇⠁⠈⠉⠙⠋⠀⠸⠷⠏⠇⠁⠈⠉⠙⠋⠀⠸⠧⠋⠇⠁⠈⠉⠉⠋
⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀⡀⡀⠀⠀⠀⠀⠀⢠⣀⢀
⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⠻⠸⠀⣄⣄⣦⠸⠀⠋⠟⢻⠸⠀⣄⣄⣦⠸⠀⠋⠟⠻
⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃⠀⠀⠀⠀⠀⠀⠉⠋⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⡀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⡇⢀⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠘⣿⣿⣧⢸⣿⣿⡿⣿⣿⣷⡀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⢿⣿⡿⠻⢿⣿⣙⠋⠈⠻⣿⣿⠇⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⢀⡀⠀⠀⣾⣿⣿⣷⣦⡀⠈⠁⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣤⣶⣞⠛⠋⣿⣿⣿⠀⠀⣿⣿⡟⢿⣿⣿⣶⣄⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣀⣀⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⡿⣿⣿⣷⡆⠀⠀⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡟⠉⠁⠀⠈⠛⢛⣁⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⠁⠘⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⣉⣀⠀⠀⣶⣾⣧⠀⠀⠀⣼⣿⣿⢿⣿⣿⣦⠀⢠⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⣿⣿⣿⠀⠀⣼⣿⣿⣯⣤⣽⣿⣿⣷⡀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠙⣿⣿⣇⠿⠿⠃⠀⠀⠀⠀⠀⠹⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⢿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⡀⠀⠀⠀⠀⢀⣴⡟⠁⠙⣷⣄⠀⠀⠀⠀⠀⣀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠙⠻⢷⣤⣀⣠⣾⠏⠀⠀⠀⠈⢿⣦⣀⣠⣶⡿⠛⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠉⠛⠟⠁⠀⠀⠀⠀⠀⠀⠙⠟⠋⠁⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣴⣶⣶⠶⠾⠿⠿⠟⠃⢸⣿⠀⠀⣿⡇⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠶⢶⣶⣶
⠈⢿⣦⠀⠀⠀⠀⠀⠀⢸⣿⠿⠾⣿⡇⣰⣿⣙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠻⣷⡀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣷⡿⠛⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠁
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⡇⠀⢀⣾⣆⠀⠀⠀⣸⡿⠀⠀⠀⠀⠈⢿⣦
⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⡇⢀⣾⡟⣻⣆⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠻⣷⡀
⣾⣿⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇⣼⡟⠛⠛⣿⡄⣼⡏⠀⠀⣀⣀⣀⣤⣤⣤⣼⣿
⠀⠀⠈⠉⠉⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋⠋⠀⠀⠀⠈⠛⠛⠀⠀⢠⣿⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣀⣴⡾⠟⠻⣧⡀⠀⠀⠀⠀⣠⡿⠛⠿⣶⣄⡀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠿⠛⠁⠀⠀⠀⠙⣿⣄⠀⢀⣼⡟⠁⠀⠀⠀⠙⠻⢷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣾⠋
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠙⠛⠃⠀⠀⠀⠚⠃⠛⠀⠀⣃⣀⠀⠀⠀⠘⠋⢙⣃⠘⠛⠛⠃⠈⠀⠀⠛⠃⠛⠀⠰⣃⣀⠀⠀⠀⠘⠋⢙⣃
⠈⠁⠐⠳⠃⠀⠇⠖⠰⠛⠧⠀⣬⠀⡄⠀⠀⠀⢸⠉⠀⠈⠁⠘⠻⠃⠀⠇⠗⠰⠛⠧⠀⣬⠀⡄⠀⠀⠀⢸⠉
⠶⠀⠀⢂⠂⠀⠀⠂⠀⢀⠀⠀⡿⢰⣷⠀⣤⡄⠀⠶⠦⠶⠀⠀⢂⠂⠀⠀⠓⠀⢀⠀⠀⣿⢰⣷⠀⣤⡄⠈⠾
⠠⠴⠀⠀⠀⠀⠀⢨⠏⣿⡃⢠⡄⡀⢠⣤⡀⠀⡦⡴⠆⠠⠴⠀⠀⠀⠀⠀⢠⠏⣿⡓⢠⡄⡀⢠⣤⡀⠀⠶⡸
⠀⠀⠀⠀⠀⢠⣤⠠⠯⢫⡄⠘⠋⠿⠻⠽⠇⠀⠶⠆⠠⠄⠀⠀⠀⠀⠠⣤⠠⠿⢫⡅⠘⠋⠿⠻⠽⠇⠀⠖⠆
⠀⢐⡀⠘⣦⡂⠀⠀⠀⠀⠀⠂⠀⠀⣤⣴⣴⣦⠀⠀⠒⠀⢐⡀⠐⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣴⣦
⠃⣉⢁⡄⠀⠀⠀⣤⠀⣀⠀⣀⢠⡄⢿⣿⡿⣿⢸⣣⡯⠇⣉⢁⡄⠀⠀⠀⣤⠀⣀⠀⣀⢠⡄⢿⣿⡿⣿⢸⣣
⡁⣈⠈⢠⣴⣴⠀⠇⠀⠀⠀⢉⣈⡁⣀⡄⣠⢠⠈⠁⢉⡁⣈⠀⢨⣴⣴⠀⠇⠀⠀⠀⢉⣈⡁⣀⡄⣠⢠⠈⠁
⠄⠀⠀⠈⠛⠙⠁⠈⠑⣃⣀⢀⠉⠀⠉⠓⠛⠘⠑⠾⠀⠄⠀⠀⠈⠛⠙⠁⠈⠑⣃⣀⢀⠉⠀⠉⠃⠛⠀⠑⠾
⣠⣠⡄⡀⠀⠀⠀⠀⠀⣿⣷⡼⡇⠀⠀⠀⠀⠀⠠⠀⢀⣠⣠⡄⡀⠀⠈⠀⠀⠀⣿⣿⡾⡇⠀⠀⠀⠀⠅
⢿⣿⡇⠁⠉⠁⣴⣆⣶⡀⢈⡄⠀⠀⠀⠀⢸⡦⢶⠄⢹⣿⣿⡇⠈⠉⠁⣲⣆⣶⠀⢈⡄⠀⠀⠀⠀⢰⡦⢶⡆
⠠⠄⢀⣄⡄⠀⡄⣀⢀⣤⡄⠒⠒⠀⠀⠀⠀⠀⢰⠶⠀⠤⠄⢠⣤⡄⠀⡄⣀⢀⣤⡌⠒⠒⠀⠀⠀⠀⠀⢰⠶
⣀⠀⠀⡈⠀⠀⡁⣁⠈⠀⠉⠀⣿⢀⣇⠀⠀⠀⠘⣠⡀⣀⠀⠀⡈⠀⠀⠁⣁⠈⠀⠉⠀⣿⢀⡇⠀⠀⠀⠸⣠
⠛⢀⠀⠐⠀⠀⠀⠀⡤⣴⡄⠀⠋⠚⠛⠀⠛⠃⡀⢉⡉⠛⢀⠀⠐⠀⠀⠀⠠⡤⣴⣄⠀⠛⠚⠛⠀⠛⠃⣀⢉
⠈⠉⠀⠀⠀⠀⠀⠸⡅⡿⠃⢸⡧⣦⣼⢿⡆⠀⣋⡈⠁⠈⠉⠀⠀⠀⠀⠀⠸⣅⡿⠊⢸⡧⣦⣼⢿⡇⠀⣉⡈
⠁⢀⠀⢀⡀⡈⠙⠈⠉⠘⠃⡀⠀⠉⠈⣉⢁⡀⠈⠁⣈⠁⢀⡀⢠⡀⡈⠙⠈⠉⠘⠃⠀⠀⠉⠈⣉⢁⡀⠁⠁
⡄⠶⠆⠀⠛⠂⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⡄⡤⡄⠶⠆⠂⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⡄
⠅⠲⠰⠃⢀⢀⠀⡟⠀⠒⠀⠶⠸⠇⠙⠛⠉⢛⠸⠞⠯⠅⠲⠰⠃⢀⢀⠀⡟⠀⠒⠀⠶⠸⠇⠙⠛⠋⠛⠸⠞
⠒⠒⠀⠸⢿⢿⠀⠡⢄⡄⠀⠐⠶⠂⡶⣇⣾⢸⢄⣠⠐⠂⠒⠀⠸⢿⢿⠄⠡⢄⡄⠀⠐⠶⠂⡶⡇⣾⠸⢄⣠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣶⣖⣰⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣖⣶⣰⡄⠀⠀⠀⠀⠄⠀⠉
⣾⣾⡇⠦⠤⠌⣀⡀⣀⠛⠹⠋⠃⠀⠀⠀⢠⡁⣈⡀⢴⣾⣾⡇⠢⠄⠄⣀⡁⣀⠛⠻⠋⠃⠀⠀⠀⢠⡁⣈⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⢰⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠑⡄
⠀⠀⠀⠈⠳⠤⠤⠤⠶⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠞⠁
⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⢿⣿⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠤⡀
⠀⠘⠶⠖⠛⢀⣀⣤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⡾⠀⠐⠓⠒⠤⠃
⠀⠀⠀⣠⠾⢿⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠘⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠉⠙⠛⠲⠶⠦⠤⠤⣤⣤⣤⣤⣤⠤⠤⠶⣶⡶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠓⠒⠲⠦⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⣤⡴⠚⠉⠙⠳⣦⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣰⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣤⠾⠋⢠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⡤⠴⠶⠚⠛⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠄⠀⠀⠄⠀⠀⠀⠄⠂⠤⠤⠂⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣠⣾⡿⠛⠉⠁⠀⠀⠈⠙⠻⣿⡿⠿⠿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠿⠿⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢸⣿⡄⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⣷⠀⠀⣿⡿⣷⡀⠀⢿⠃⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⢠⣤⡀⠀⠈⣿⣦⡶⢿⣇⢠⣿⠿⠛⢿⡆⠾⠂⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣤⡀⠀⢸⣿⢻⣷⣄⠸⣿⠀⠘⠿⠈⠋⠀⠀⠀⠀⠀⠀⢰⣾⠟⠁
⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢴⣦⠀⣹⣷⡄⢸⣿⠛⠋⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢻⣿⠋⠙⣿⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠻⣿⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣠⣾⡿⠛⠻⢷⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆
⠀⠀⠀⢰⠋⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⢲
⠀⠀⠀⠈⠳⠤⠤⠤⠦⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠴⠋
⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣻⡿⠿⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡷⢶⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠤⢄
⠀⠘⠲⠒⠊⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⢰⠇⠀⠈⠒⠢⠚
⠀⠀⠀⣠⠞⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠘⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⣶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠙⠒⠒⠲⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⠀⠠⢤⠤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⠴⠖⠒⠒⠶⢤⡖⠋⠉⠉⠳⣦⣴⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡼⠁⠀⠀⠀⠀⠀⠀⠙⠀⢀⣠⡴⠟⠁⣸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⠤⠤⠴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢖⣶⣟⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣤⣀⣀⣀⣀⣤⠴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠋⠉⠑⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢉⠙⢦⡀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣝⡲⣄⠀⠀⠀⠀⠀⢀⡴⣂⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⢆⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠟⠋⠉⠑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠈⠁⠈⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣄⣈⣉⠉⠛⠲⢦⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠂⠈⢧⠀⠀⠀⠀⠀⠀⠸⣍⠙⠻⣿⠉⠉⠉⠉⡿⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣀⣠⠞⢳⡀⠀⠀⠀⠀⠀⠘⠦⣄⣘⡆⢀⣠⠞⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣤⡤⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠉⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉
⠀⠀⠀⠀⠀⣤⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠶⠶
⠀⠀⠀⠀⠀⠀⠉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⠶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠶⠆⠀⠀⠀⠈⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡶⠶⠛⠛⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣀⣀
⠀⣴⠟⠉⠙⠳⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡴⠞⠛⠉⠙⠛⠳⠶⢤⣄⣀⣀⠀⠀⣠⣾⣋⠀⢀⡾⠋⠙⣧
⢸⡇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠳⣟⠀⢀⣴⠏
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤⣤⣤⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⢻⡶⠟⠁
⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠁⠀⠀⢀⣾⣇
⠀⠀⠉⠛⠳⠶⠶⠶⠤⠤⠤⠶⠶⠶⠛⠛⠋⠉⠁⠙⠳⢦⣤⣤⣤⣤⡤⠶⠛⠁⣾⠁⠀⠀⠀⠀⠚⠋⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⠼⠷⣤⣤⣀⣤⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠛⠉⠉⠛⠛⠛⠻⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⢀⡠⠄⠤⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⠞⠁⢀⣀⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢠⠊⠁⢀⣀⣀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⠃⢠⣾⣿⠏⠉⠁⠀⠀⠀⢡⠀⠀⠀⢀⠃⢀⣾⣿⠏⠉⠙⢦⠀⠀⢡⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣧⣀⣀⡷⠀⠀⢨⠀⠀⠀⢨⠀⢸⣿⣿⣧⣀⢀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢃⠘⢿⣿⣿⣿⡿⠃⠀⢀⠎⠀⠀⠀⠀⢆⠈⠻⣿⣿⣿⡿⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠉⠁⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠑⢄⡀⠈⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⡆⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠀⠀⠸⡄⠀⢀⠇⠀⠀⢠⣿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣤⣤⣤⣤⠴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣤⣤⣤⣤⣤⣤⣶⡶⠾⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠨⠋⠁⠀⢀⡀⠤⠀⠀⡠⠊⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠤⢄⠀⠀⠀⠈⠀⠤⠒⠁⠀⠀⣀⡀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠋⠉⠉⠂⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⢰⠁⠀⡀⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢡
⠀⠀⠀⠀⡠⠀⠃⠀⠀⠀⢸⣀⠀⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⡇⠀⠘⠀⠀⠀⠀⠠⠔⠒⠅
⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠘⡆⠀⠀⠠⢖⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⠄⠀⠀⢠⡟⠀⠘
⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠸⣷⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⣿⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⠀⢻⣷⣶⣦⣭⣩⣍⣉⣉⣩⣍⣁⣠⣦⣶⣿⠇⠀⠀⠀⡐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠻⣿⡿⠿⠛⠛⢿⠿⠿⠟⠿⢿⣿⡿⠋⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⣀⠀⠀⠈⠉⠀⠂⠈⠉⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠤⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⣦⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠉⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⢹⣿⡏⢀⠈⢿⣿⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⡀⠛⢃⠀⢿⠇⢀⣠⣄⢹⣍⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠛⣿⣿⠀⠠⠈⠻⣧⠈⣿⣧⣼⣶⣾⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠙⠿⠂⠘⢿⠀⢠⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠐⢾⣧⣨⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠿⠿⢿⣿⠿⠟⠋⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠶⠶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⠶⠶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⣸⡿⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢿⡿⠂⠀⠀⠀⠀⠈⣿⡆⠀⠀⣾⡏⠀⠀⠀⠀⠀⠺⣿⠇⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢻⣧⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣾⡏⠀⠈⣿⡆⠀⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠙⠻⢷⣄⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠈⢿⣆⠀⠀⠀⠘⠿⣦⣄⣀⠀⢀⣀⣠⣴⠟⠁⠀⠀⠀⣼⡿⠀⠀⠀⠸⣿⡄⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢀⣾⠏
⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠙⢿⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣶⣤⣤⣶⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣦⣤⣴⣶⣶⠿⠟⠉

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢶⢶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⣠⣾⠿⠛⢿⣦⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⢰⣿⠃⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣤⣤⣼⣿⣶⡄⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠉⠉⢹⣿⢁⣤⣿⣿⣤⣤⣤⣶⣿⣿⡿⠗⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠉⣿⣯⠉⠉⠁⠀⣿⡇⠀⠀⠀⠀⠸⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡿⠀⠀⢹⠇⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⡇⠀⠀⢀⣴⣿⠿⣿⣆⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣧⣀⣀⣠⣿⣧⡀⠀⣼⡿⠁⠀⠈⣿⣇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠛⠛⢛⣿⡏⢁⣠⣿⣧⣤⣤⣤⣽⣿⣤⣶⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠈⠛⣿⡏⠉⠉⠉⢹⣿⡏⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠘⠟⠁⠀⠀⠈⠁⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣼⣿⠀⠀⠀⢰⣿⠋⠀⠈⢻⣿⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣿⣿⣤⣤⣤⣾⣿⣶⡆⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠉⠉⠉⣿⡯⠀⣠⣴⣿⣥⣤⣤⣤⣴⣿⣷⣾⠆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡏⠀⠈⢹⣿⠉⠉⠉⠉⢹⣿⡇⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⠇⠀⠀⠘⠿⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠿⠷⢶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⣶⡟⠁⠀⠀⠀⠀⠀⠉⠉⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠛⢿⣿⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⡀⠀⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⢿⣿⡍⠻⠂⢹⡇⠀⣀⠻⡆⣿⣿⣿⠃⠀⠀⠀⠀⣀⣿⠃⠀⠀⠀⢀⡀⠀⢸⣆⣸⣇⢰⣯⠿⣦⢹⠄⠀⢠⡿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠻⣿⡇⠀⠙⢷⡐⣿⣤⣧⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣼⠟⠁⠀⠀⣤⠀⢸⣟⣶⣄⢿⠀⠛⠘⠁⠀⠀⠀⠀⣶⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠛⣁⠘⣇⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⣦⠾⣷⡸⠏⠀⠉⠀⠀⠀⠀⡀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠹⠆⠈⠀⠀⠀⠀⢠⣄⠀⠈⢿⣟⠋⠉
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠿⠿⡿⠿⠟⠁⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣀⣤⣤⣤⣤⣤⡶⠟⠉⠛⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉

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

⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢲⡄⠀⣀⠀⠀⡠⠤⢰⠤⡀⢀⣀⢀⣠⠄⠀⡀
⠀⠀⠀⠸⡇⠐⠄⠀⢀⠀⠀⠀⠆⠈⠀⠀⠀⠀⢸⠇⠈⠃⣚⠁⠸⠑⠌⠋⠃⠢⢀⠸⠉⢢
⠀⠀⠀⠀⡇⢀⠠⢄⠀⢀⠤⣀⡀⠈⢁⣄⡰⠀⣾⠀⠀⢀⡀⠁⠀⠀⢰⣄⣠⠄
⠀⠀⠀⢰⡇⠀⣲⠆⣇⠀⡴⠖⢸⠀⠸⢟⡁⠴⠃⠀⣳⢻⠖⠂⠀⠐⠚⣿⠿⠦⠀⠙
⠀⡀⠐⠾⠃⠀⠠⠒⠋⠰⠃⠠⠞⠀⣤⠞⠀⢱⠆⠀⠀⠀⠀⠀⠻⠀⠀⠃⠀⠀⠀⠠⠀⠂⠀⠁
⠀⠁⠀⠀⠀⡂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⡀⠀⠀⠒⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠇⡧⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⡀⠀⠀⠀⠀⢀⣀⡄⠀⢀⢠⣤⠄⣤⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠍⠀⠀⠈⣷⠤⠀⠴⠘⠉⠲⠃⠙
⠀⠀⠠⠄⠒⠚⠀⠀⠀⠀⠁⠀⡀⠀⠀⡀⠀⢠⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠀⠠⠀⢠⡀⠆⣆⡆⠀⠀⠰⠤⠈⠄⡁⠀⢠⡀⠰⠀⠊⠁⠀⢐⠂⢡⠀⠆⠀⠀⢐⠀⠀⠀⢀⣩⡁
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠈⠀⡄⠀⠀⠘⠀⠀⠀⠀⣀⣀
⠠⡄⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠅⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⢠⠀⢠⠀⠠⠄⠀⠀⠐⠠⠄⠀⠀⠐⠂⠁
⠀⠔⠠⠤⢀⡆⢰⢶⠆⠀⠀⢲⢄⢰⡀⣉⠈⠁⣏⠆⠀⠀⠀⢀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣤⡀⡇⢰
⠀⠃⠐⠇⠘⠉⠸⠉⠢⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⢠⡀⠀⠀⠁⠀⠂⠀⠀⣀⠀⠀⠱⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠁⣠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠗⠂⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠄⡀⠀⢠⠊⠀⠀⠈⠏⠗⠒⠀⠀⠀⠀⠠⠁⠀⠄⠀⠀⠈⠀⠠⠦⠐⠠⠀⠣⠄
⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠀⢀⠀⠀⠤
⠀⢀⢠⠀⠀⠂⢰⠀⠀⠀⡀⠀⠀⠀⠀⠐⡶⠇⠀⡄⠗⠃⠇⠘⠒⠸⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠄⠈⠀⠤⠒⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡎⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⡀⠀⠀⠀⠁⣠⣶⡷⠛⠉⠉⠀⠀⠀⠀⠘⠈⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⣇⢀
⠀⠤⡀⠀⢂⡞⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠋⠻⢿⣦⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⣤⣤⡀⠀⠸⠀⢰⠀⠀⢀⣼⡇⢸⣿⣿
⢀⣧⡹⣶⡿⠃⠀⠀⡀⠀⣠⣿⣿⡷⠾⠿⣶⣄⠀⢀⠉⠃⠀⠀⢘⠀⠀⠀⠀⢀⠐⣰⠂⣼⣿⣿⣇⠀⡣⢠⡟⣧⠀⣿⣿⡇⣾⣿⣿
⠈⢫⣿⣿⣅⡀⠀⣰⣿⣾⣿⣿⠃⠀⠀⠀⠈⣻⣧⡷⠆⠀⠀⠀⢸⣿⣯⣤⣤⣬⣿⡿⠀⣿⣿⣿⡏⢈⠁⣸⠋⣿⡀⢹⠉⢱⣌⡹⣏
⠀⣼⠋⠈⣿⣿⣶⠟⢙⣿⢿⣿⣤⣀⡐⢧⣶⣿⠟⠁⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⣈⠉⡉⠻⠀⢀⡟⠀⢿⣿⢨⠀⣿⣿⣿⣿
⠐⣿⠀⠀⣹⠞⠁⢀⣾⡏⠀⠛⠿⠿⢿⠚⠁⠀⠀⠀⠀⠀⠀⣦⢼⣿⣿⣿⣿⣿⣿⣿⢸⠟⣿⠉⡇⢈⣾⡲⠄⣌⣿⠀⣸⡜⠸⣷⠃
⠀⠉⠀⡴⠃⠀⠠⠚⠋⠉⢻⡏⠓⠄⠤⠀⢠⣤⣄⠀⠀⠀⠀⣿⢼⣿⣿⣿⣿⣿⣿⡇⣸⡆⣘⠀⡁⢸⣏⣄⠀⠺⣿⣶⣿⣿⠐⢸
⠀⢀⠜⠀⠀⣀⣠⡤⠶⣶⠟⠃⠀⠀⠀⠀⠀⣽⣿⣿⣿⣷⣀⣱⣏⣻⢿⣿⣉⣿⢿⣿⣿⣷⠛⠀⠁⢸⡿⡛⠁⠢⣜⣿⣿⡏⠀⠈⠃
⠀⠈⠠⣾⣻⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣦⣽⢿⣿⣽⣿⣿⣗⢠⣿⣿⠁⣼⣿⣿⣿⣆⣤⡀⠙⣁⠘⢆⢠⡝⠾⣹⠇⠀⠐
⠰⠄⠀⠙⢿⣿⠟⡟⠁⠟⠀⠋⠀⠉⠩⣉⣉⣉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⡳⠌⠉⠸⠒⢠⠀⠉⣜⠐⢶⡴
⠐⠀⠀⠀⡈⢻⡀⣀⣀⣤⣤⣤⣦⡒⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣦⠀⠸⠀⢿⡿⠏⡁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣋⣽⣿⣿⠿⠿⣿⣿⣿⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⢉⠙⠛⢿⣄⢂⠠⣈⠁⣾⢇
⠀⠀⠀⠀⠀⠀⢸⡟⠙⢻⣿⢋⣹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠿⢿⣿⣷⣿⣷⣶⣿⣿⡄⠀⠀⠁⠘
⠀⠀⠀⠀⠀⢀⣼⣇⣠⣟⣿⣨⣽⣿⣿⣿⣿⡭⢵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⢿⡿⣾⡿⡟⢙⣻⣷⣿⣿⣿⣷⣮⣉⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡈⣽⣿⣿⣍⡙⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⣷⠟⠙⣿⡿⣿⣿⣿⣿⣿⣿⣶⣯⣤⣤⣽⣿⣿⣭⣬⣭⣥⣴⣶⣾⣿⣿⣿⣿⣿⠿⠷⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠇⠀⢻⣟⠀⠀⠀⢠⣿⣿⣿⣿⣟⣯⣽⣯⠙⠉⠉⢩⠍⠉⠉⠉⠻⠏⠙⢻⣿⣿⣿⣿⣿⠀⢀⣿⡏⠙⣿⡄
⠀⠀⠀⠀⠀⢀⠈⠀⡖⢈⣿⣶⣶⣾⣿⣿⣿⣿⣿⢛⠈⣿⣷⣀⣀⣀⣠⣤⡀⠀⠀⠀⠀⠈⠀⣼⣿⣿⣿⡟⢇⢰⣿⣷⡄⢸⡀
⠀⠀⠀⠀⠀⠸⣶⣆⠀⢸⣿⠟⠋⢉⣿⣿⣿⣿⣿⠸⣧⢩⡏⠉⠉⠉⢹⣿⠉⠉⠁⠀⠀⠀⣠⣿⣿⣿⣿⡇⡀⠀⠉⠀⡀⠀⠘
⠀⠀⠀⠀⠀⣷⣿⡇⠀⢹⣷⣶⢶⡟⢩⣿⣿⡿⠉⠂⠛⠛⠦⠄⣠⣤⣿⣿⣦⣤⠤⠴⠦⡄⠛⠛⣿⣿⣿⡇⣇⣀⡀⠀⢡⣀⣴⣇⡄
⠀⠀⠀⠀⠀⣸⣿⠁⣰⣿⡟⣼⠏⠀⣼⣿⣿⡇⠀⠀⣷⠀⠀⢸⡏⠈⠉⠉⠉⣻⠀⠀⠀⣧⢀⠀⣿⣿⣿⣇⢻⣼⣿⣻⡟⠉⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣿⢀⣿⣟⣼⠋⠀⢸⣿⣿⣿⡇⠀⠀⡏⠘⠋⠸⡇⠘⠲⠶⠃⣻⠀⠀⠀⣿⠁⢀⣿⣿⣿⣿⡌⣿⣿⣧⣧⠀⣿⣧⠃
⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣇⠀⢀⣿⣿⣿⣿⣿⣡⣴⣧⣴⣦⣶⣿⣦⣤⣤⣤⣿⣤⣤⣴⣿⣬⡉⣿⣿⣿⢿⣷⡘⣿⣿⣿⢰⣿⣧
⠀⠀⠀⠀⠀⠀⢿⡈⠟⢻⣿⡄⣼⣿⣿⣿⣿⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠹⣿⣿⡬⣿⣿⣾⣿⣿⣼⣿⡿
⠀⢠⠀⡀⠀⠀⢸⡇⣷⣌⣻⣷⣿⣿⣿⣿⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢻⣿⣷⣫⠙⠻⣿⢿⠻⢿⠁
⠀⡀⠀⠀⢀⡀⠀⣷⣿⣿⣮⢻⣿⣿⡿⠃⠀⠀⣰⠎⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠐⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣤⡇⠀⠉
⣼⣃⣰⣿⣿⣿⣸⣿⣿⠃⣽⣿⢏⣿⠃⠀⢀⡾⠁⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⢿⡟⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⢏⣾⠇⠀⢀⣿⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣼⣥⣇
⣿⣿⣿⣿⡿⢣⣿⢿⣿⣿⡏⣾⡟⠀⠀⣾⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⣿⣶⣦
⣿⣿⠟⠁⢠⢋⣿⠟⠟⠛⣸⣿⠃⠀⠀⠀⠀⠀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠠⠀⠈⢿⣿⣿⣿⣿⡿⣿⣿
⣿⣿⡆⣰⠃⣼⠏⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠸⢿⣿⣿⣿⣷⣿⣿
⣿⣿⣿⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠹⡿⣿⣯⢿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣎⠁⢻
⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⢀⣀⣿⣷⣤⡄⢨⣴⣬

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡤⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⣄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣴⣶⣦⡀⢷⠀⠀⠀⠀⠀⡾⢀⣤⣶⣤⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⡇⣾⣿⣿⣿⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⢿⣿⡿⢫⠏⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣨⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⢬⣉⣁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣶⠶⠦⢤⡤⠤⢤⡤⠴⠶⡶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿⠛⠛⣿⠿⠿⢿⠿⠿⢿⡿⠿⠿⡟⠛⠛⣿⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⢤⣀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⡴⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⣶⣤⠀⠀⠀⠀⠀⠈⢷⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⢷⡀
⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣽⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣍⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⢀⣤⣶⠀⢠⠾⠛⠋⠉⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠛⠻⢦⠀⢀⣀⠀⠀⠀⠀⢸
⣿⠀⢀⡴⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠳⣤⡀⠀⢸
⣿⠀⣿⠀⠀⠀⢘⡇⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⣿⠀⠀⠈⢻⡆⢸
⢹⡆⠙⢦⣤⣤⠾⠃⠀⣿⠈⠉⠛⠳⠶⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠛⠋⠉⣸⠇⠀⢿⣀⠀⠀⣰⠇⣾
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠙⠛⠛⠁⢰⡏
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⣿⠲⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⣾⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣤⣤⣄⣀⣤⣤⣤⡀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡟⠀⠀⠈⠉⠁⠀⠈⣷⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣓⣶⣶⣶⣶⣶⣶⣶⣖⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠉⠀⠀⠀⢀⣀⡀⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⢚⣓⡀⠀⠀⢨⣭⠉⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢸⣿⣷⠀⠀⢸⣿⡆⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⢀⣄⠀⠸⣿⡏⠀⠀⠸⣿⠇⠀⠀⣨⣽⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠸⢰⣄⡀⠀⠀⠀⠀⠀⠁⣀⣤⣾⡟⠉⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠻⣿⡷⣶⢶⡶⢾⠟⢿⣽⡿⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠘⠷⣿⣴⣧⣼⣷⣾⠟⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠉⠁⠀⠉⠀⢀⣤⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⠿⠟⠛⢿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣄⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡆⠀⢠⣄⠀⠀⣴⠀⣾⡆⢀⣾⣿⠀⢰⡆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣼⣧⠀⣿⡇⢀⡏⢀⣿⢿⡄⢀⣿⠀⣿⡇⣾⣏⣿⣄⣾⣧⣼⣧⣤⣀
⠀⠀⠀⠀⠀⠰⣦⣆⣶⡄⢸⣿⡿⠷⣿⠀⣿⡿⠿⣷⢿⣿⠿⣿⠿⣿⡿⣿⣿⡿⠿⢻⣏⣿⣿⢹⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠃⡏⠁⠙⠘⠹⠇⠀⠸⡇⣿⡇⠀⣿⠀⠀⠀⢿⡆⣿⡇⣿⡟⠀⠀⢸⣿⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢿⠀⠀⠀⠀⠁⠙⠃⠹⠃⠀⠀⠀⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣲⠶⢦⠈⠀⠠⡤⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⣘⡍⠖⠢⢤⠈⠉⢥⣤⣤⣄⣈⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠀⠀⠀⠀⣕⠶⢯⡈⢱⢀⠀⢆⠠⢤⣀⠹⣏⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠼⡜⡯⣍⠒⡌⢦⠏⠢⠀⠉⢼⢾⡜⠚⢀⡀⠉⡿⢬⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠒⠋⠠⣰⠲⠬⠀⢻⣩⠓⠲⠼⠀⠙⠪⡛⢾⡃⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠳⠂⢢⡜⢆⢧⠀⢰⣀⣡⣰⠖⠉⠲⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠙⡦⢤⣷⡀⣿⣀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⡴⠳⡄⢿⠜⠉⠗⠢⢿⠤⠀⢷⡾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠺⡇⠀⠀⠀⣼⡇⠀⢸⠃⠘⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⡇⠀⣀⣽⢆⣿⠘⡄⢸⢸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⢳⢸⡗⡀⣿⠊⠁⠘⣿⡩⡆⡟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢱⡏⠑⣿⠀⠀⠀⠋⠀⡇⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠸⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⠀⠀⢸⠠⠄⣸⠀⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⠓⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠳⡼⠛⠂⠀⠀⠘⠀⢀⣏⡼⠤⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡷⢤⡀
⠀⠀⠀⠀⣀⢠⠟⠲⢄⠙⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⢀⢰⠀⠀⡀⠀⠙⢦⠀⠁
⠀⠀⠀⠀⠈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠠⠀⠣⠈⠇⠳⠀⠀⠀⢸⠀⠀⠘⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⢿⠀⠀⢀⣠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠃⣀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠧⠀⠈⠈⠀⠁⠀⠈⠐⠂⠀⠀⢀⢰⡀⢄⠦⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⡄⠀⠂⠈⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⠉⠂⠀⠈⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⢀⡀⠐⡄⠱⠓⠄⠀⠀⠀⠀⠀⡀⠀⠸⠀⠁⠜⠃⠀⠘⠈⠀⠛⠀⠂⠀⠠⠆⣴
⠀⠀⠀⠀⠀⠀⠈⢯⣱⠦⠀⠀⠃⠀⠈⠀⠀⡀⠀⡀⠐⡄⢡⡕⠑⠄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠐⣄⡀⠁⠀⠀⢀⠀⠀⢀⡰⢄⠢⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢴⠄
⠀⠀⠀⠀⠘⢾⡁⠙⠀⠀⢀⠈⡢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⢀⠈⠀⠠⠣⠄⣠
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠊⠁⠀⢀⢰⡄⠀⠄⢰⠠⢦⠀⠘⠈⠃⠟⠄⠁⠸⠈⠁⠜⠂⠀⠀⠈⠘⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠔⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠄⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⢀⣴⣶⣄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⣿⣿⣇⡇⠀⠀⠀⠀⠐⠒⠋⠉⠙⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠻⠿⢫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢿⣿⣦⣤⣉⣉⠉⠛⠓⠒⠒⠒⢛⣋⣉⣩⣥⣼⣿⠏⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠈⠙⠻⠿⢿⣿⣿⠿⠿⠛⠉⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣿⣆⠀⠉⡒⠤⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⣿⣿⡄⠀⢸⣿⣶⣶⣾⠀⠀⠀⣠⠴⠂⠐⢦⡀
⠀⠀⠀⠀⠀⢠⠊⠒⣒⣒⣒⣒⣒⣤⡀⠀⠀⠀⡜⢰⣿⣿⣿⣧⠀⠸⣿⣿⣿⡏⠀⣠⠞⠁⣠⣀⣀⠀⣷⡀
⠀⠀⠀⠀⠀⢁⣠⡀⠀⠙⠿⣿⣿⣿⣿⣦⡀⢠⠃⠈⢿⣿⣿⣿⣦⠀⢻⣿⣿⢃⡜⢁⣴⣿⣿⣿⣿⣷⣌⢿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⢿⣿⣿⣿⣿⣿⡞⠀⣠⣾⣿⣿⣿⣿⣷⡌⢿⣿⣯⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢷⡀
⠀⠀⠀⠀⣨⣿⡿⠋⠉⡆⠀⠈⢻⣿⣿⣿⣿⠁⠀⠀⠠⣾⣿⣿⣿⣿⡷⠠⠿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⡘⣷
⠀⠀⠀⠀⣿⠏⠀⠀⠀⠑⣄⠀⠀⠸⢿⡿⠋⠀⠀⠀⠐⠺⠿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⡇⡟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣶⣦⣬⣷⣤⣤⣴⣿⡿⢟⣿⡟⠀⠀⠀⠀⠀⠀⠘⠛⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⡤⢒⣭⠷⠒⠒⠀⠀⠈⠙⠛⠛⠛⠛⠛⠿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⣁⣤⠒⠋⠉⠈⠀⣈⣉⠙⣻⣿⣿⣶⣄⡙⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣴⣿⣿⡿⠦⠄⠀⠀⠀⠩⠴⠛⣻⣿⣿⣿⣿⣿⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢿⡏⢉⣶⠷⣶⣤⡀⠀⢸⣟⣿⣿⣿⣿⣿⣿⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣤⠁⠘⠣⠌⠉⣿⣿⣿⠈⢿⣿⣥⡾⣿⡿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⠉⠙⠀⢸⣿⡄⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣹⡆⠀⠀⠀⠀⣀⣤⠀⠀⢸⣿⣿⣄⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡀⢀⠀⡾⠉⠙⠶⣤⣼⠿⢟⣻⣧⣿⣿⣧⣶⣶⣶⣾⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣇⠸⡄⢿⣿⣟⢓⠒⠒⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠰⡌⠋⣿⠈⣻⡟⠉⢹⣿⣿⣿⣿⣿⣿⠯⣉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠆⠈⢢⠸⠃⠀⢸⣿⣿⣿⣿⣿⡯⣉⠐⠊⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⡄⠀⢀⣿⣿⣿⡿⢟⠪⡉⠂⠁⠀⠀⠐⠄⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⡒⢛⣡⣿⠟⠀⠀⠑⠈⠀⠀⠀⠀⠢⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡵⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣤⠐⡃⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡤⠒⠒⠦⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠒⠦⠤⢤⡀
⠀⠀⠀⠀⢀⡞⠁⢀⠀⡤⣴⡀⠆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣀⡄⣦⣰⠀⡇
⠀⠀⠀⠀⡏⢠⠂⠧⠟⠑⠁⠁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠡⡰⡼⠓⠋⠉⠀⣀⠞⠁
⠀⠀⠀⠀⠳⣄⣩⣄⡤⠖⠮⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⣀⠀⠀⠱⢄⣡⣄⡠⠖⠢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⠀⡀⣠⢠⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣰⡼⡹⠙⠛⠈⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠸⣄⠈⢛⠁⣠⢄⣰⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣠⠴⠲⠊⠁⠈⢉⠉⠳⡄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠴⠲⠞⠁⠀⢉⠉⠓⡄
⠀⠀⠀⠀⡠⠇⣀⣠⣴⣶⠽⠞⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⠀⡖⢲⢧⠼⠄⢠⠇
⠀⠀⠀⠀⡇⠀⠣⠟⠁⣀⠠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠒⠉⢁⠀⣤⠔⠃
⠀⠀⠀⠀⠈⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉⠉⠁

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



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣶⣠⣤⡶⠾⠛⠋⠁⠀⠀⢻⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⡶⠟⢻⣏⠁⠀⠠⠊⠀⠀⠀⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⣧⠀⣀⣠⢤⣶⠿⠟⠋⠉⣿⡄⠀⠈⣿⡔⠁⢠⡄⠀⠀⠀⠀⣿⣷⣇
⠀⠀⠀⠀⠀⠀⢘⣷⡤⣔⣿⡿⠛⢫⠝⠀⠀⠀⠀⠀⠸⣷⠀⠀⣹⣧⠀⣾⣿⣆⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢀⣐⠤⠗⢻⣏⠀⢸⣷⡠⢾⣧⡀⠀⢀⣠⡄⠀⢻⣟⠋⠉⣿⡀⣿⣿⣿⣆⠀⠀⢸⣿⣿
⠀⠀⠀⢻⡄⠀⠀⢈⣿⡶⠾⣿⡇⠸⣿⣿⣄⠉⠀⠀⠀⠈⣿⡄⠀⢻⡇⠿⠃⠘⣿⠆⠀⢸⣿⣿⣀
⠀⠀⠀⠀⢿⡀⢀⠠⢺⣷⠀⢸⣿⠀⣿⡞⢻⣧⡀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣀⣠⡤⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠘⢿⡀⠀⠀⣿⡆⠀⣿⡇⠸⠇⠀⠙⠁⠀⠀⣀⣠⣤⣴⠖⢿⡉⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⠋⠀⠈⢁⣀⣠⣤⣶⠶⠟⠛⠋⠉⠁⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠘⣧⣀⣤⣴⣶⣿⣿⣿⣽⣵⣤⣤⣤⣤⡤⠤⢤⣤⣶⣶⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠊

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣦⣄⡀⠀⠀⢻⣿⠀⠀⠉⠻⣿⣷⡄⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠈⠉⠛⠿⣿⣶⣄⣿⣿⠀⠀⠀⠀⠈⠻⣿⣄⢠⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠈⢻⠁⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⠀⠀⠘⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⣰⡿⠋⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠙⣿⣷⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠻⠇⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠸⠟⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢸⣿⠗
⠀⠀⠀⠀⠀⠠⢿⣿⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⢹⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠈⠙⠻⠶⣶⣶⣶⡶⠿⠛⠁⠀⠀⠀⠀⢀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣛⣉⣀⣀⣀⣀⡀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠋⠁⢀⣀⣤⠶⠞⠒⠈⠉⠉⠉⠉⣉⡉⠉⠉⠉⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⡙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠛⠳⣆⠀⠀⠀⠀⠸⠀⠀⠀⣠⣴⡿⠟⠛⠛⠛⠛⠿⠿⣦⣍⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣴⣿⣋⡴⠞⠋⠙⠛⠒⠀⠀⠀⠉⠓⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣠⣴⡿⠟⠛⣉⣉⣙⣛⣿⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⠹⣿⣆
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢠⣾⡿⠋⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠁⠀⠀⠘⢿⣧
⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣧
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⠐⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⠟⠋⠉⠙⠻⢶⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠻⣿⠹⣿⠀⠙⢿⡀⣿⣿⣿⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⣄⠰⣆⠀⣿⣦⡀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⡏⠙⢿⡆⢡⣄⣻⣰⣶⣴⣷⣿⡿⠋⠀⠀⠀⢀⣴⠟⠁⠀⡀⠀⣾⣦⡀⢻⡟⢻⠼⠏⠉⠃⠉⢀⣴⠟
⠀⠀⠀⠀⢰⣿⣿⡏⠻⠆⠹⡇⣠⣦⣽⣾⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣾⠃⠀⢰⣄⣿⣆⢸⠛⠙⠊⠁⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣆⣻⣶⣿⣿⣿⣿⣿⣿⣿⣏⠙⠉⠁⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠻⠄⠙⠀⠀⠀⠀⣀⠀⠀⢿⡒⠛⠋
⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠿⣿⣿⠿⠟⠁⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣠⣤⣄⣠⣤⠾⠋⠛⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠋⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠙⠛⠛⠻⠿⢶⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠋⠀⠀⣠⡴⠛⠉⠁⠀⠀⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠈⠉⠛⢦⣄⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠈⡄⠀⠀⢠⠃⠀⠀⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⠁⠀⢹⣿⣿⣧⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⣸⡏⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣷⣴⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⢿⣷⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⡤⠤⠖⠋⠁⠀⠀⠀⡴⠋⠳⠄⠀⠀⠀⠈⠙⠲⠤⢤⣤⠤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⡶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢖⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣔⠋⠍⣙⠓⠒⠒⠲⠒⠒⢒⡚⠋⠉⠙⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⣀⣉⣀⣀⣀⣤⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢤⣤⣀⡀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣴⣾⡿⠛⠷⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣤⣴⣶⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠶⠶⠒⠛⠛⠉⠉

⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣧
⠀⠀⠀⢠⣿⣿⡆⠀⣰⣿⣿⡇⠀⢀⣾⣿⣿⣿⡀⠀⣴⣾⡆⢰⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⠁⠀⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⡇⠀⣿⣿⣇⣼⣿⣿⠀⣸⣿⣿⣇⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣶⣶⣿⣿⣿⠀⣸⣿⡟⠹⣿⣿⣧⠀⣿⣿⠟⢻⣿⣿⢠⣿⣿⠟⠛⢻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡏⢹⣿⣿⣿⢠⣿⣿⠁⠀⣿⣿⣿⠀⣿⣿⠀⢸⣿⣿⣾⣿⠏⠀⠀⠘⢿⣿⡿⠀⠀⠀⠀⠀⠀⣴⣾⣿⠇
⠀⠀⠀⢸⣿⣿⡇⠸⣿⣿⡯⣿⣿⠇⠀⠀⠘⠿⠟⠀⠿⠋⠀⠈⢿⠏⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⠁⣀⡙⠿⣃⣤⣄⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⢠⣴⣦⢰⣿⣇⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠉⠁⣸⣿⡇⠀⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣿⡇⣼⣿⡏⠀⣼⣿⣿⣿⣿⣤⣤⣤⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⡿⠿⠃⠀⠻⠿⠿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⣿⣿⣿⣸⣿⡿⠛⣿⣿⣿⡇⠀⠀⢸⣿⣿⠋⣿⣿⡇⣼⣿⡟⠀⣿⣿⡇⠀⠀⠀⣰⣶
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠿⡿⠃⠀⠹⣿⣿⡇⠀⠀⢸⣿⡿⢸⣿⣿⠀⠈⠉⠀⠀⠻⠿⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠙⠟⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡿⠉⢹⡇⣼⡟⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡟⢦⣄⠀⠀⣸⡇⠀⣿⣿⢏⣤⣀⣀⠹⣧⡀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣾⢧⣤⣮⠙⠀⠿⠶⠶⠟⠁⣶⠾⠯⣭⡅⠈⠁⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠏⢸⣉⣹⡇⠀⠀⠀⠀⠀⢸⣧⣤⣀⣾⠃⠀⠀⣠⡍
⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣉⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣦⣀⠀⠀⠀⢰
⠀⠀⠀⣴⠶⠻⣆⢸⡇⠀⠈⣙⠻⠶⠀⠀⠀⠀⢰⠟⠛⢋⡩⠍⡉⠉⠉⠁⠠⠀⢄⠀⠈⢿⣿⣿⣷⣄⠀⢸
⠀⠀⠀⢹⡆⠀⣿⣿⠇⣾⣛⣯⣥⡀⠀⠀⠀⠀⡎⠀⠀⠀⡤⠤⡀⠀⠀⠀⡠⠖⢦⡀⠀⠀⠙⠿⣿⣿⣆⠈⢀⣀⣀
⠀⠀⠀⠸⣷⣀⣸⡇⠘⣿⠉⢀⣹⡇⠀⠀⠀⣠⠇⠀⣠⠤⠤⣀⠀⠀⠀⠀⢀⣠⠤⠤⣄⠀⠀⠀⢸⣿⣿⡾⠿⢿⣿⣧
⠀⠀⠀⠀⠛⠋⠉⠀⠀⠻⠟⠛⠉⠁⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠉⠁⠒⠋⠁⠀⠀⠀⠘⠄⠀⠀⣼⣿⡟⠀⠀⠀⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢣⡀⠘⡀⠀⣀⡀⠠⠶⢖⢀⡀⠀⠀⠀⡸⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠦⣈⡛⠦⠤⢄⣀⣀⡠⠬⠷⠖⠊⣀⣀⡴⠋⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣰⠀⠉⠓⠲⡖⠀⠀⢀⡻⠟⠋⠉⠉⢀⠈⠳⡀⢠⡘⠦⠀⣼⠁⣹⣣⡾⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢼⡇⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⢸⢱⠀⣹⠀⢣⠀⢼⣧⣰⠟⣫⡦⣤⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡠⠃⠀⢈⣄⡀⢠⡄⢰⣏⠙⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡞⣰⣷⣋⢳⡀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠒⢲⠶⠶⠶⠒⠒⠒⠒⠒⠒⣟⠋⢻⠃⠀⠀⠀⠉⠁⣞⣻⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠞⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠁⣀⣤⡶⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣟⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣟⡛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠘⢷⡀⠀⠀⠸⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⢉⣼⠏⠁⠀⠀⠀⣿⣇⠀⠀⣼⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢷⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢸⡇⠙⣧⣀⣿⠀⠻⣄⠀⠸⣷⣄⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠙⣷⣄⢸⡇⠀⢰⣦⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣴⠶⣶⠀⢲⣼⡇⠀⠈⠙⠛⠀⠀⠙⢷⣄⣻⡝⢦⣄⠹⣿⣦⣄⡀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣿⠀⠈⣟⠃⠀⠀⣀⣀⣀⣀⠀⠀⠈⠛⠛⠀⠙⣷⣿⣶⣿⣿⣿⡶⠶⣶⣤⣼⡷⢠⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⣿⠀⢴⡿⠿⠿⠿⣿⡷⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣿⣿⣦⢸⡇⠀⣀⣬⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⣻⣿⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⣧⡾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣷⣿⠀⠀⠾⠋⢿⣿⡟⠙⠂⠀⣶⠀⠀⠐⠛⠹⣯⣿⠙⠿⠀⠀⢸⣯⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠀⠀⢰⣶⣶⣶⣶⣶⣶⡏⢉⣿⣶⣶⣶⡶⢶⡶⠀⠀⢸⡇⠉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠀⠀⣿⣷⣦⣤⣬⣉⣉⣀⣀⣉⣩⣤⣤⠴⢾⣿⠆⢀⣿⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢹⣷⣤⡀⠉⠉⠉⠉⠁⠀⠀⢤⡴⣿⠇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠙⠲⠦⢤⣤⣤⣤⣤⣤⠤⠶⠚⠉⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣴⠤⣤⡀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⢶⡇⠀⠸⣷⣶⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⢠⡀⠀⠀⢁⠘⠟⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠿⣆⣰⠟⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⢟⣋⢛⠛⠉⠀⠀⠀⠀⠔⠒⠒⠒⠂⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢿⠿⣺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⢪⣭⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⠀⠀⢰⣶⣦⠤⢤⣤⣀⣀⣠⡤⠤⠴⣲⣶⠆⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠟⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠛⠻⠛⠻⣿⠿⠃⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣠⣶⣾⡿⣿⡟⣷⠀⠀⠀⣠⠒⠚⢁⣀⠉⡍⡆⠀⠀⢀⣴⣶⣿⣿⠿⣻⣦⠀⠀⠀⡔⠒⠋⢀⢈⠩⠱⡄
⠀⠀⠀⣰⡟⣏⣢⣬⣶⡿⠏⠀⠀⢰⠡⣔⠽⠓⠋⣁⡴⠁⠀⠀⣾⢟⣏⣥⣶⣾⡿⠁⠀⠀⡞⢠⠴⠖⠉⢉⣁⠞
⠀⠀⠀⠘⠛⠛⠛⠉⠉⠀⠀⠀⠀⠈⠢⠔⠤⠚⠒⠃⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠑⠤⠄⠔⠓⠚

⠀⠀⠀⠀⠀⣀⣀⣤⣤⣠⣀⠀⠀⠀⠀⣀⣀⠔⠲⠤⢤⡀⠀⠀⠀⠀⣀⣠⣤⣄⣤⡀⠀⠀⠀⢀⣀⡠⠖⠢⠤⢄
⠀⠀⠀⢀⣾⣿⠿⢛⢝⣽⡿⠀⠀⢀⡞⢀⣀⣤⣶⠖⢀⡇⠀⠀⣠⣿⣿⣛⠛⢏⣿⡗⠀⠀⣠⠏⠀⡤⣴⡀⠂⢨⠂
⠀⠀⠀⢺⣿⣯⣾⡿⣿⠛⠁⠀⠀⢸⡀⠳⠋⣀⣀⡄⠊⠀⠀⠀⣿⣷⣭⣶⢿⡟⠋⠀⠀⠀⢇⠀⠓⠉⣁⣠⠐⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠆⠀⠀⠀⠀⠠⣶⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⢈⠀⣷⡘⣤⠃⠀⠀⠀⢴⣶⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠞⠰⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⡇⢀⠀⠀⠀⢴⣯⠄⠻⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢈⣾⡀⠂⠀⠀⠀⠀⠀⠀⢻⣼⡇⢸⣷⣬⣄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⢹⣷⡀⢀⠀⠀⠀⠀⠴⢾⣿⣷⣴⣻⣿⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠸⣿⣏⣿⣄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⢹⣿⢃⠀⠀⠠⡶⣤⡀⡀⢐⠀⠂⢀⠀⠀⢠⣤⣌⠊⣠⠄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠉⡿⣿⣿⣎⢄⠐⠀⣰⣿⣿⣿⣿⣿⡌⢹⣇⣄⠀⠠⣦⠀⠻⠆⡉⠀⣠⡌⠀⠀⢿⣿⢿⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠂⠀⠠⣿⣿⣿⣾⣴⣿⣿⡿⠿⠿⠿⠟⠷⣶⠾⣻⡟⣰⣿⡿⣃⠀⠛⠙⢿⣿⠀⠸⣀⣷⡶⠐⠦⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⡄⠀⠀⠀⠉⠲⣿⣿⣠⣿⣷⠀⠀⠚⠁⠠⣦⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠈⢛⣽⣿⣿⣿⡌⠻⣿⣿⣏⠁⠀⠀⢀⠀⢄⠀⠹⣿⣿⣿⣿⡿⡄⠀⠀⠀⠨⣤⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣹⢇⣿⣿⣿⣿⣿⠀⠀⠘⣿⣿⣆⠈⠲⣿⠀⢀⣷⣄⠹⣿⣿⣿⣿⣄⣄⠀⢠⠜⠇⠀⡄⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⣿⣿⡿⣿⣿⣿⣇⠀⠀⠈⠻⠮⢂⣀⠀⣳⣼⣿⣿⣆⢻⡇⠹⣷⠙⢿⣇⠀⠀⠀⢻⡉⢀⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠄⣼⣿⣿⠁⢀⣿⣿⡏⠀⠀⠀⠀⠀⠐⠈⣛⣽⣿⣿⣿⣿⡌⠃⠀⠛⠀⣻⡏⠆⠀⣀⠊⡀⠀⠀⠀⠀⠚⣃⣀⣠
⠀⠀⠀⠀⠀⠀⠀⡼⠏⠂⣴⣿⣿⣿⢀⡼⡟⢿⠁⠀⠀⠀⠀⠀⠀⣸⣿⠿⠝⠋⢻⣿⢷⡀⠀⠀⢠⣁⠀⠀⣶⣿⣦⡁⢀⣀⣄⣀⠰⠿⠛⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣷⡄⠀⢸⠋⠙⠁⠻⠛⠋⠁⠀⠉⣻⣉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⡇⣿⡟⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⡀⣸⠿⠃⠀⠀⢀⡀⠀⠀⠀⠁⠈⠃⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣧⢱⣌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⣿⡗⣟⣴⣶⣶⡖⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢽⣿⣿⣿⣾⡌⢹⡁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⠃⣾⣿⣿⣸⣿⣿⡏⠙⠁⠀⠀⠀⠀⠀⠈⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣰⣿⣿⠛⠛⣱⡄⢀⠀⠀⠀⠀⣠⣾⠇⣠⡞⢰⣿⣿⢧⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢼⣹⡇⠐⣿⣿⡁⠄⠹⢿⣿⣶⠉⢿⣶⣿⣡⢼⣿⢫⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠸⣇⠛⢷⣄⠈⠁⠀⠀⢀⢋⣿⡏⠀⠀⣿⣿⢃⡼⣣⣿⡟⠀⠀⠀⠀⠀⣠⣴⣾⣿⡟⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣷⢤⡀⠀⠀⠀⠀⠈⠸⣿⣿⠀⡀⣼⣧⢋⣾⡿⠋⠀⠀⠀⣀⣠⣾⣿⣿⣿⣷⡅⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠘⢢⠄⠀⠀⠀⠀⠀⡇⠙⣽⡻⣧⠙⣤⣼⠏⠀⣠⣴⣶⢿⠇⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠀⠃⠀⠀⠀⠀⠐⠲⢦⠄⠿⢷⣦⡀⢻⡝⢠⣾⣿⣿⣧⠄⢠⣿⣿⡿⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⢶⡄⢀⣠⡞⠢⡄⠛⠁⣿⠒⢻⣿⣿⣯⣍⠀⢸⡿⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣀⠘⠀⣠⣞⣿⣿⠀⠀⠻⣤⢀⡟⠀⠀⠹⣏⠁⠈⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣶⡏⠙⠀⢀⣀⠀⠀⢠⣾⠃⠀⠀⠀⢨⣭⣿⣿⡇⠀⠀⠀⠀⡃⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⡠⠛⠋⢀⣴⠿⠁⠀⠀⠀⠀⡙⠛⠃⠈⠀⠀⠀⠀⠀⢃⣈⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⡀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣀⡠⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡄⠀⠀⡀⠀⠀⠀⠀⡘⢆⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⢀⡀⠀⣨⠀⠈⠑⠊⠙⠤⠒⠂⠒⠁⠀⠑⠂⠐⠢⣀⡠⠔⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠑⠤⢀⣀⠜⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⡀⠀⠀⣀⡠⢤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠻⣐⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠖
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡆⠉⡆⠀⠀⡆⠉⡆⠀⠀⠰⠀⠀⠢⠀⠀⠀⠂⠉⠂⠀⠀⠋⠑⠂⠀⠀⠰⠉⠀⢢⠀⠀⠐⠂⠁⡆⠸⢄⡀
⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠇⠀⣇⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⡄⠈⡆⠀⠀⡆⠀⠇⠀⡊
⠀⠀⠀⠀⣀⣠⢃⠀⠀⠀⡃⠀⣀⣀⣈⡀⠀⡇⠀⡀⠀⡄⢀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⡎⢡⠀⠰⠀⠀⠃⠰⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⡇⠀⡇⠀⠀⡇⠀⡇⢀⠁⣀⡀⡀⡀⠀⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⢃⠀⡼⠾⡄⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠥⠤⠇⠀⠀⠧⠤⠇⠦⠤⠃⠀⠀⠘⠤⠬⠄⠥⠤⠇⠀⠀⠤⠤⠀⠤⠤⠃⠀⠀⠸⠤⠬⠄⠧⠤⠇⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡔⣀⠠⠐⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠒⠚
⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⣰⡠⠖⠈⠱⠀⡠⠄⠤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⠒⠒⠎⠀⠑⠋⠉⠒⢄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠏⠀⠀⠀⠓⠉⠀⠘⡄⠰⠋⠉⠢⡴⠋⠀⠘⢄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠿⠿⢿⣶⣤⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⠟⠛⠛⠛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣾⣦⡀⠀⢹⣟⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⣸⣿⡀⢰⣿⣽⣷⣄⢘⣏⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⣰⣾⠿⠋⠀⠀⠀⠀⠀⠀⣿⣶⣄⠀⢻⣟⠋⢻⣧⠸⠏⠉⠈⠛⠈⠉⠀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⣀⠀⠸⣷⡀⠀⣿⣾⠿⢷⡌⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣧⡶⠿⣷⡄⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠻⣷⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢛⡛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠈⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⣶⣤⣤⣬⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⡾⠿⢿⣶⣶⡾⠿⠛⠁⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⣈⣦⡀⠀⠀⠁⠒⠂⣁⡠⠤⠒⠺⡇⠀⠀⠀⠛⠶⠃⠀⠀⠀⠈⠛⠁⠀⠀⠸⠀⠀⠀⠉⠉
⠂⠉⠐⠢⢸⠀⠀⠀⠀⣿⡋⣻⠃⠀⠀⢸⠿⣿⡏⠐⠢⢤⡴⠊⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣶⡄⣀⣀⠀⠀⠀⣀⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠶⠟⠃⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⠟⠉⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠈
⠉⠉⠉⠙⠛⠣⢄⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣨⣦⣄⣀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠙⠦⣄⣀⡤⠔⠁⠀⠀⡇⠀⠀⠀⢸⣿⠿⣿⠀⠀⠀⢸⠿⢿⣷⠸⠀⠀⠀⣠⣶⣄⠀⠀⠀⢀⣤⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡠⠄⠒⠒⠒⠒⠒⡇⠀⠀⠀⠈⠻⠶⠋⠀⠀⠀⠘⠶⠿⠃⡇⠀⠀⠀⢿⣋⡻⠀⠀⠀⠉⡜⠁
⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠸⡋⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⠉⠀⠀⠀⠀⡘⠀⠀⣠⣤⣤⣤
⠁⠀⠀⢿⡋⣻⣿⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣤⣄⢣⡀⠀⠀⠀⡠⠚⠉⠈⠉⠓⢄⠀⠀⠈⢆⠀⠀⠀⢢⡀⠀⠀⠀⡇⠀⠀⢻⣿⣟⢙
⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⡇⣠⣤⡀⠀⠀⢸⡟⢻⡇⠑⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠤⠚⠓⠀⠀⠀⠙⠲⣐⠈⡇⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⠀⣿⡋⣻⠀⠀⠀⠙⣫⠔⠊⠁⠀⠀⠀⠀⠈⠉⠒⢤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢷⡀
⣀⠀⣀⡠⠞⠁⠀⠀⠀⡰⠃⠤⣈⠉⠁⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠑
⠀⠈⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠈⠢⡀⢠⠃⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠻⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠢⣀
⢤⡤⣤⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡿⠿⡇⠀⠀⠀⣶⣿⣷⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠲⡶⠖⠀⠀⠀⣇⡀⠈⠑⢲
⠛⠁⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⠘⠳⠞⠁⠀⠀⠈⢧⣸⡿⠀⠀⠀⢀⡦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢳⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠘⠛⠃⠀⠀⢣⠀⣠⠤⠒⠈⠉⠉⠐⠂⠤⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⣰⣉⣛⣃⣀
⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⣠⠧⠤⠤⠄⣀⠀⠀⢀⠴⠒⠉⠀⠀⠀⠀⠀⠉⠑
⠈⢢⠜⢆⠀⠀⠈⠢⠄⣀⠠⠔⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠞⠁⠀⠀⠀⠀⠀⠉⠻⣅
⢠⠋⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⡀⠤⢺⠀⠀⢼⣿⡿⢿⡆⠀⠠⣿⠿⣿⣿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠃⠀⠀⠀⣠⣶⣄⠈⠉⠉⢩⣿⣷⣆⢸⠀⠀⠈⠻⠷⠞⠁⠀⠀⠙⠶⠿⠋⠀⠀⢀⣀⠀⠀⠀⢰⣿⢿⡆⠀⠀⠀⢻⣿⣆⠀⠀⠀⣿⠿⣿
⠀⠀⠀⠐⣿⡟⢻⠀⠀⠀⠸⣍⣹⡟⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢷⠀⠀⠀⠳⠼⠃⠀⠀⠀⢸⣄⡟⠀⠀⠀⠙⠶⠟
⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠈⠉⠀⠀⠑⣄⠀⠀⠈⠢⢄⣀⣀⠤⠊⠀⠀⠀⡔⠙⠳⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⣀⣀
⣣⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⡸⠳⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠺⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⡠⠁⠰⣾⣮⣭⣭⣶⡾⠃
⠈⠣⡀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⡐⠀⠀⠀⣨⣍⠐⠒⠒⠈⣉⡀⠀⠀⠘⡀⢀⠤⠊⠉⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠉⠉⡩⠟⠉
⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠇⠀⠀⢸⣿⢻⡇⠀⠀⠘⠉⠉⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣠⠊
⠀⠀⠀⢀⠀⠈⠉⡳⠒⠒⠋⠉⠁⠀⠀⡄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢈⠃
⠀⠀⠒⠛⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⣄⠀⠀⠀⢀⠄⠀⠀⠀⡇⠀⠀⢰⣿⠿⣷⠀⠀⠀⡟⢻⣿⠀⠀⠘⠀⠀⠠⣄⣀⡀