Pupil Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠔⠉⠁⠀⠀⠀⠉⢁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠠⠐⠀⠀⠀⠀⠀⠀⠒⠀⠤⢀⡀⠈⠢⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⡠⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⢠⠃⠀⠀⠀⠀⣀⠈⠀⠄⣀⠈⠁⠒⠂⠤⠤⠤⠀⠈⠒⢤⠀⠀⠀⠀⠀⠀⣀⠀⡇⣿⢠⡖⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠀⠀⠂⠀⠠⠁⠀⡰⠉⠓⠤⣀⠀⠈⠀⠂⠀⠤⠄⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⢡⠙⠁⠀⠀⠐⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠁⢀⠜⠁⠀⠀⠀⠀⢉⣵⡲⢖⡲⣖⢶⡶⢿⣁⡼⠀⠀⠀⠀⠀⠀⠀⡼⠤⣀⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢀⡴⠯⠓⠫⡦⣀⠤⣤⣶⠉⢴⠆⠁⡟⠉⠀⢨⠜⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢩⣉⡶⠶⠿⣿⡀⠛⠂⡁⡟⠉⠤⠈⠫⠶⠶⢽⡋⠀⠀⡼⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠸⡀⠀⠈⢛⡓⠛⠋⠀⠀⠤⠖⠀⠀⠀⠮⠨⠂⡰⠁⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠐⠥⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣽⠒⢮⡉⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⣈⡷⠢⠄⣀⣀⣀⣀⣀⡀⠤⣴⠒⢁⡝⠀⠀⢑⠄⠱⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌⣏⠀⠀⠀⠈⠣⣄⠀⠫⣀⠀⣤⠀⠜⣀⣀⠔⠛⡀⠀⠇⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⢷⠀⠉⠐⣦⢄⣠⠤⡑⢤⡍⠀⠈⡖⠀⡉⠀⠀⠀⢡⠀⢰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡆⠀⠀⠃⠀⠀⠀⠲⢤⡙⢄⡸⢁⠞⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢁⠀⠀⡈⠀⠀⠀⠀⠀⠈⢦⠡⠎⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⢻⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠁⢸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠍⡀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠿⠀⡘⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⠀⠄⠀⠤⠶⠆⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⢇⠳⠞⠓⠃⠀⠈⠉⠉⠀⠀⠀⡀⠷⣠⣄⡀⠤⠴⠆⣤⠾⠖⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⢀⡇⠀⠈⠑⡟⠒⠒⠒⠂⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⠳⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⢀⠀⡀⡎⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠉⠉⠲⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢻⡌⠣⠇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠓⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣟⡉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣠⠆⢀⡴⠀⠀⣠⠂⠀⠀⢀⠀⠀⠀⠈⠓⠦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣷⣶⣯⣤⣴⣞⣁⣠⣤⠖⠋⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⠶⠞⠋⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⢯⣁⣀⣀⣀⣠⠴⠊⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠻⣟⣋⣉⣀⣤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣸⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢸⡟⢦⣻⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣠⣴⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣉⣛⣿⣿⣿⣿⣿⣿⠿⠿⠿⢏⠓⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⣾⡇⠀⡀⠀⣾⠁⢀⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣿⣇⢰⡇⣼⣿⣇⡘⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢦⢹⣿⣿⣿⣿⣿⣿⣿⡷⠃⡐⡄⠀⠀⠀⣰⠞⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡏⠀⢀⣤⣾⠋⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⢶⣌⠳⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⣀⣈⠀⠀⢴⣾⣿⣷⣾⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⢶⣶⣾⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⢿⣿⣷⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⡀⠈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠉⠙⠛⠶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣷⡈⠙⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠈⢁⣼⡿⠛⠛⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣧⡹⠀⠈⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⢠⠞⠋⠀⠈⠀⡸⠹⣼⣿⣿⣿⣿⠿⣿⡿⠀⡇⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⣹⠏⡿⣿⡿⠀⣿⣧⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⢰⠏⠀⢹⡿⠇⠀⢻⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡏⠀⠀⡟⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀⣤⣤⣶⣾⣷⣶⣶⣶⣤⣴⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢦⣄⣀⣴⣿⣿⣿⣿⠿⣿⠿⠛⠛⢿⣻⣿⣿⣿⣻⠟⠛⠛⣿⠿⠿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢀⣤⡶⣿⠛⠛⠉⠁⠀⠀⢠⣇⣀⣀⣙⣾⣿⣿⣿⡟⠁⠀⠁⣀⣸⡆⠀⠀⠀⠈⠉⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠢⣄⠀⠀⠀⠘⡏⠉⠉⣥⢿⣿⣿⣿⣧⡀⠀⠀⠈⢹⠇⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⣄⣙⣆⠀⣠⠞⢻⠿⡿⢿⡛⢦⣀⣠⣏⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣶⣾⣶⣷⣾⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠉⠉⠁

⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠟⠉⠁⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠔⢤⡀⠛⢿⣿⣿⣿⣿⣿
⠉⠁⠀⠀⢀⣀⣠⣶⣿⣿⡿⣿⣿⣿⣿⣿⡋⣉⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⣷⣯⣽⣿⣷⣌⢻⣿⣿⣿
⠀⠀⣀⣴⣾⣿⣿⣿⡿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣷⣙⣿⡿
⠀⠤⠼⠟⠛⠻⢿⠋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⠙⠋⠁⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠁⠀⠀⠀⠸⠿⠿⠿⠿⣿⡟⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣿⡇⢀⡆⣸⣿⡀⡰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡀⢼⣿⣧⣾⣧⣿⣿⣷⢁⠃⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣷⣄⢳⣼⣿⣿⣿⣿⣿⣿⣿⢃⣠⡖⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⠻⣟⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⣿⣧⣴⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⢀⣽⣷⣝⣳⡌⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⣤⣽⠿⠋⣹⣿⣷⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠩⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣿⣿⣷⣼⣿⣿⣿⣤⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⣶⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠏⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢀⣿⣦⠄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣁⣀⠙⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠳⢄⣀⠀⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠴⠛⣹⣿⠿⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⡇⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⣼⠟⠁⠐⠀⣸⢟⣹⣿⣿⣿⣿⣿⣿⡧⠹⠎⠹⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⠟⠈⣿⡟⣿⣿⡟⠘⣿⡇⠘⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡟⠐⢹⡿⠁⠀⣿⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡎⠀⢀⡟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⣿⣿⣿⡇⡞⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣻⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣹⣿⣿⣿⣿⠇⣿⢛⡿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⢿⣿⣿⣿⣻⣿⢹⡟⣹⣿⣿⣿⣿⢏⣾⣿⣿⣱⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⡿⣟⠿⣼⣿⢹⡾⢧⡟⣸⠇⢁⡟⠹⣳⣿⢾⣿⣿⣿⣿⡟⡿⣫⣿⣿⣦
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠗⠀⠀⣙⣇⣼⣆⣸⠇⡴⠂⠸⠇⣸⠄⢸⠃⣠⡿⠃⠈⣹⡿⢿⢏⣾⣿⣿⣯⣿⣿⣷⣄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⣀⡀⢀⣿⠘⠃⢿⣯⣼⠀⠀⠀⠀⠉⠄⢸⠾⢿⠀⠀⠀⡴⢡⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⣿⠿⢿⣿⣿⠖⠂⠈⠈⢻⡿⣧⣀⣀⠀⠀⡀⢠⣶⣦⠂⠀⠴⡳⢻⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⡀⣼⣷⠠⠤⠛⠀⠀⢀⣀⡘⡇⠈⡙⡿⢿⠿⠟⠃⠉⣷⠀⢊⠀⠀⡹⢫⢟⠍⢉⡴⣟⣿⣿⣿⢿⣿⣷
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠘⣿⡏⠒⠂⠒⠳⢶⣖⠺⡿⣇⣿⣠⣷⣼⣷⢢⣂⡀⠀⠻⣶⣤⠀⠈⢀⠔⣡⣶⣡⣾⡿⣿⣿⣿⣿⣿⣿⣇
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠛⠁⢠⣿⡁⠀⠀⣳⣄⢡⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⢈⡁⠀⠀⠀⢀⣡⠟⠚⣁⠾⠿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⡿⢿⠿⣾⣿⡛⠂⢺⣟⠔⢶⣦⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡁⠘⢷⡤⠀⠀⠀⣤⡶⠿⢕⣂⡨⢡⣞⡿⢟⣻⣿
⣾⣿⣿⣿⣿⣿⣿⣷⣶⣷⠶⠶⠀⠈⣿⣷⣼⣿⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣏⠀⠛⠃⢤⣄⠀⠛⣰⠿⢿⣛⣯⣽⣷⣻⣭⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠒⠀⠀⠈⠻⢿⣿⣹⣷⣾⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣯⣭⣭⡁⠀⠀⣼⠃⠈⠁⠐⠢⠭⢥⣴⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣏⠍⠉⠉⢋⣀⣀⠀⠀⠒⢊⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⢳⡄⠒⠲⠞⠛⣷⣾⣶⣒⣾⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣯⡀⠀⠀⢤⡍⠑⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡌⠉⠀⠀⠀⠀⠀⢹⣆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣖⢉⣼⡟⠀⢀⣴⣿⠃⢀⠖⠋⢹⠟⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⠟⠋⠀⠀⢀⠜⠳⠾⠿⠏⠠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣵⡾⠋⢄⣤⢿⣿⣷⠊⢀⣀⣀⣀⠞⣱⠏⢿⣿⠛⣿⡿⡆⠁⠘⣦⡄⠀⢰⡆⠀⠀⠀⡠⢤⣀⣈⣛⠽⣿⣿⣿⣿⣿⣿⡿
⠀⠸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣋⣴⡿⣿⣿⣿⡿⢿⣟⣫⠖⠋⠀⠀⠁⠀⠀⠀⠁⠀⢸⠈⠙⢳⢦⣧⠀⠠⡢⢌⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⡫⢂⣴⣿⡅⡀⠈⡛⠵⠿⢧⣤⠀⠀⠀⢀⣶⣦⣤⣼⠀⠀⠀⣉⣝⠀⠀⢄⣀⣈⣓⢮⣽⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣎⣴⣿⣟⡟⣴⣣⠘⠀⠀⠀⠈⢷⢿⣿⣴⣿⠃⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⡴⠀⡀⢠⠀⠈⢫⣵⣿⣷⣶⡆⣀⣦⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢿⣿⣥⣴⣇⣾⢠⢠⣾⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣵⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣽⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠒⠚⠛⠓⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡔⠀⠉⠓⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⣠⣤⣤⠶⠟⠛⡛⠛⠛⣛⠛⠻⠷⣶⣾⣄⣈⠚⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⠂⠀⠀⠀⠀⠀⠁⠀⠐⢄⢐⠫⡛⠿⣿⣯⣿⢶⡒⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⡀⠀⠀⠀⠐⠀⢀⣴⠀⠁⠐⣷⡄⠀⠂⢀⠡⡀⠈⠻⣿⣿⣷⣮⠥⠐⠀⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⣿⣷⣤⣾⣿⣧⠀⠘⠘⠀⠎⠀⠀⠈⠻⣿⣿⣶⣒⠉⣀
⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠆⠇⠘⠿⠛⠛⠛⠟⠛⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠂⢀⡀
⠀⠀⣠⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⡀⠈⣄⠈⠂⣠⢆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠿⠛⠋
⠀⠀⠻⠶⠿⠿⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠀⠠⠄⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠠⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠑⠲⠤⣄⣀⠀⠀⠀⠒⢀⠠⠀⣄⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠖⠒⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠦⠤⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⡀
⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣯⣴⣥⣤⣴⣾⣋⣀⣠⡀⣀⡄⠀⠀⠀⠀⠀⠀⡠⠀⡠
⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⣀⣤⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⡛⢛⣿⡟⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣖⣯⠔⠋
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣾⡿⠟⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣋⡁
⠀⠀⠀⠀⠀⠈⠉⣩⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠯⠉
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣅
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢄⣀⡈⠙⠛⠻⠿⠿⠟⠛⢉⣀⡠⠴⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣾⣩⠴⠖⢺⣿⡟⢲⠲⡶⠦⣄⣸⡇⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡎⣉⣇⢠⣿⡄⣰⣈⣽⣿⣇⣸⣼⣧⢇⣴⣿⣻⢶⣿⣻⡦⣠⡶⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡭⠊⣷⣍⣻⣮⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣾⣿⣿⣥⣉⠶⢄⣴⠏⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢲⣼⣏⢹⣶⢾⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠞⢳⣴⡾⠃⣰⣞⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⡱⢯⣸⣿⣿⣿⣿⡿⠋⣡⣾⡿⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⣖⡵⠓⢠⡇⢠⠁
⠀⠀⠀⠀⠄⢘⢎⣠⣾⣽⣿⣿⡟⠉⠀⣾⣿⠋⠀⠀⣠⣴⣾⣿⣿⣷⣦⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣛⣪⣿⣯⣯⣏⣴⠏⡀
⠀⠀⠀⠀⠰⣧⣾⣹⡿⠏⣾⡿⠀⠀⢸⣿⡇⠀⠀⣸⣿⠟⠁⠀⠀⠈⠙⢿⣷⡄⠀⠀⢻⣿⠢⡙⢿⣿⣿⣿⣿⣶⣿⣯⣵⠞
⠀⠀⠀⢥⣴⢏⣽⠋⠁⠀⣿⣇⠀⠀⢸⣿⣇⠀⠀⠹⣿⣶⠾⣿⣷⠀⠀⠈⣿⣷⠀⠀⢸⣿⡇⡄⠀⠙⢻⣿⣿⣯⣽⣋⣡⣤⠄
⠀⠀⣠⣿⣯⡟⢱⠃⠀⠀⢹⣿⡄⠀⠀⢻⣿⣦⡀⠀⠀⠀⣀⣾⣿⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠀⠀⢠⣿⣏⡻⡟⠟⣛⡋⠁
⠀⠾⠿⠋⡛⡀⠙⠆⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⢿⣿⣿⣿⡿⠟⠁⠀⢀⣼⣿⠏⠀⢀⣿⠏⠀⠀⢀⣾⡛⠺⠿⢦⣼⠏
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠤⢀⡀⠙⢿⣷⣦⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠃⢀⣠⠞⠁⠀⢠⡾⠿⣿⣽⡆⡸⢨⡟
⠀⠀⠀⠀⠨⣭⡁⠜⠙⢲⠤⣌⣉⠑⠛⠻⠿⢿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠒⢉⢀⣠⣤⠞⣿⠈⠃⠘⠈⢻⠃⣾⠁
⠀⠀⠀⠀⠀⠠⠬⢝⡲⠅⠀⢠⠝⠉⡿⡶⠶⡤⣴⣦⣤⣤⣤⣠⣤⣤⣤⡤⡶⠛⢙⡓⣀⠙⣀⠁⠀⠀⡠⠃⣼⠃
⠀⠀⠀⠀⠀⠟⠛⠳⣯⡒⠤⣂⠀⠀⡐⠀⠀⠀⠁⠀⣼⠀⠀⠀⠒⠘⡆⠀⠌⢢⠈⠁⠈⠂⠙⠀⡀⠔⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠌⠑⠲⢅⣀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠛⡋⠉⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣏⣹⣭⣼⣾⣷⣾⣿⣿⣿⣯⣶⣋⣴⡾⠄⢀⡀
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣠⠖⠛⠋⢩⣭⠝⢉⢉⡉⠣⣽⡉⠉⠛⠻⢿⣿⣿⣯⡤
⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⢀⣴⠋⠀⠀⠀⠀⣾⡇⢰⠕⠉⢞⠄⣿⡇⠀⠀⠀⠀⣽⠟⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⣿⣆⠀⠀⠀⠀⠘⠟⢄⠑⠛⢉⡰⢻⠃⠀⠀⠠⢚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⢛⠒⠤⠤⢠⣈⣺⣿⣿⡭⠧⠤⢒⣂⡥⠶⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠋⢰⠒⠖⠶⠖⠖⠻⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣰⡆⠀⠀⠀⠀⢀⣀⡀⠀⢆⢲⣖⠐⢦⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣿⠁⢀⠀⠐⠻⢶⣄⠙⠣⠀⠵⠙⣦⠀⠻⣎⠻⢢⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⣠⣶⣶⡿⢿⠄⠀⠉⠅⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠇⡀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⢀⡇⢠⣀⠀⠀⢀⠀⢀⡀⠀⣿⣆⠘⡟
⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⢦⣀⣜⡶⠜⠛⠚⠲⢏⡢⠀⠑⣾⣷⠀⢹⣿⣴⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠐⢅⠴⠀⠀⠀⠀⣰⣶⡿⠛⠁⢸⣿⣷⣂⣶⣦⠙⢦⡀⢿⣏⣀⣤⡋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠾⡋⠀⠀⠀⢐⠻⣿⣿⣿⢟⠀⡀⠈⠢⡑⠛⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⠀⠀⠔⠁⠚⠋⠂⠠⠞⠁⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠒⠂⠨⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣧⣶⣼⣦⣧⣄⣼⣠⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⡿⣻⣿⣷⣿⡿⠿⠿⠿⠾⠿⣿⣯⣟⣷⣻⣦⣾
⠀⠀⠀⠀⠀⠀⠀⢀⡤⡺⢟⣽⣿⡿⠛⣉⡠⠤⠒⠒⠒⠒⡶⢦⣤⣉⠛⠿⣿⣝⣷⢞⡀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⢮⣾⠟⢁⡴⠋⠁⠀⠀⠀⠀⠀⡾⠀⢨⣶⡎⠙⢦⣌⠙⢷⣾⡿⣁⡀
⠀⠀⠀⠀⠀⠀⡠⠣⢬⡿⢁⡴⠋⠀⢀⣠⠤⠶⠦⠤⣄⣇⠀⠸⣿⣿⠇⢀⡏⢳⡀⠻⣿⣿⡦⠂
⠀⠀⠀⠀⠀⠀⠤⠬⠟⣠⠏⠀⠀⣰⠫⠀⠀⢀⠀⠀⠀⠙⢦⡀⠀⠀⢀⠜⠁⠀⠹⡄⠘⣿⠿⠶⠂
⠀⠀⠀⠀⠀⠀⢀⣠⡞⡇⠀⠀⠀⡇⠁⢀⣾⣿⣟⣀⡄⠀⠘⡍⠉⠉⠁⠀⠀⠀⠀⡷⠀⠈⠛⠁
⠀⠀⠀⠀⠀⡔⠉⠀⠑⢵⣄⠀⠀⢣⠀⠈⠿⣿⣿⡿⠂⠀⢠⠇⠀⠀⠀⠀⢀⡠⢞⣴⠂
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠛⠛⢿⣒⣲⠷⠤⣤⣈⣀⣀⢀⡠⠊⠀⣀⡠⠤⢒⣩⣴⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠲⢉⡀⠉⠛⢓⠶⠒⠬⢬⣭⣥⣤⢤⠤⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠂⠬⢀⡀⠀⠈⠈⠀⠈⠂⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠂⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⡀⠑⢦⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⢄⡀⠐⢦⣄⡑⠶⣬⣓⣶⣤⣽⣿⡿⠟⠋⢉⠔⠈⠀⠀⠀⠀⠀⠉⠛⠗⠢⢄
⠀⠀⠀⠙⠳⢶⣭⣟⣿⣾⣿⡿⠟⠋⠁⠀⠀⠠⠁⠀⠀⢀⣠⣤⣤⡀⠀⠀⠈⢢⠀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠉⠹⣏⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⡆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠒⣒⣚⣷⣤⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⡆⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠐⠉⠭⠿⠛⣛⣿⣦⣄⠀⠀⠡⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣭⡭⠿⢿⣦⣄⡑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠘⠀⠀⠀⠀⠀⠀⣀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⢉⣻⣿⢿⣿⣶⣶⣤⣴⣦⣭⣤⣤⣴⣶⡶⠾⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠛⠛⠉⠉⠓⠒⠢⠤⣄⣈⠉⠑⠒⠤⣍⡉⠛⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠿⠗⠒⠒⠒⠒⠒⠲⠶⠶⠦⣤⣤⣀⡀⠈⠉⠓⠦⣄⠀⠉⠲⢄⡀⠉⠻⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢁⣠⠴⠖⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠉⡙⠒⠦⣄⡀⠙⠲⣄⠀⠙⢦⡀⠈⢢⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣀⠴⠋⠀⢀⡠⠴⠒⠉⠉⠉⠉⠉⠉⠁⠒⠒⠤⠤⢄⣀⠉⠐⠠⢉⠓⢤⡀⠑⢤⡀⠹⣄⠀⠱⠘⣆
⠀⠀⠀⠀⠀⠀⣴⠋⢀⠞⠁⢀⡠⠒⠉⣀⠤⠒⠂⣉⣩⣥⡄⠀⠉⠉⠉⠁⠒⠒⠀⠭⠕⡂⠤⣉⠂⠍⠢⣄⠑⣄⠘⣆⠀⢇⢹⡄
⠀⠀⠀⠀⢀⡜⢁⡴⢋⡠⠖⠁⣠⠔⢋⠤⠐⣊⡭⢖⣫⠥⠐⠀⢈⣀⣈⡭⠭⠭⠥⢄⣒⣂⠠⢀⡹⠂⠀⡈⠳⡌⢦⠘⡄⢸⠈⡇
⠀⠀⠀⠀⠀⠠⠊⢠⠊⢠⠀⣊⢔⡎⢁⡤⢚⠥⠚⣉⡤⠖⣚⣩⣥⣶⣶⣶⣶⣶⣶⣶⠦⣄⣉⠙⠛⠀⠀⠀⡄⠙⣎⢧⢱⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡞⡵⣫⢖⡡⠚⣡⠔⢋⣥⣶⣿⣿⣿⣿⣿⣿⣿⡟⢿⢿⣯⣛⠶⣮⣙⠢⣄⠀⠀⠀⠀⠸⡞⡄⡆⣇⠁
⠀⠀⠀⠀⠀⠀⠠⡹⡌⢊⠓⡪⠖⣉⠔⠋⣠⢞⣽⣿⣿⢿⠏⠹⣿⣿⣿⣿⣿⡀⢠⢿⠙⠳⣬⡓⢽⡢⡙⠆⠀⠀⠀⢻⢳⠧⠉
⠀⠀⠀⠀⠀⠀⠀⠑⣜⢦⣉⡠⠞⠁⠐⣋⣴⣿⣿⠟⠃⢸⠽⡄⠻⣿⣿⣿⠿⢁⠮⣸⠀⠀⠀⠹⢦⡙⢎⠣⡀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⡶⡿⠿⡟⢿⠁⠀⠀⠘⣎⡱⠄⡀⡀⢀⠀⠊⣠⠏⠀⠀⠀⠀⠀⢷⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⢠⡀⡱⡈⠳⣤⣀⠀⠈⠛⠦⣬⣥⣤⠴⠚⠁⢀⣀⣀⣀⣠⣀⣘⣁⣈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡤⠀⠢⡑⢤⣮⡓⠦⢭⣝⣓⣶⣶⣶⣶⠶⠶⠛⠛⢋⡉⠠⠤⠀⠤⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠈⠈⠁⠀⠤⠤⠐⡂⠩⠤⠐⠌⠓⠚⠛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣧⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣛⣋⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠉⠉⢉⣀⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠐⣛⣛⡃⠀⣉⡭⢭⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⠄⠉⠉⠀⠀⢧⡀⠀⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⠀⠀⠀⢀⠀⣨⢇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⣿⠟⢿⠟⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⠀⡠⠔⠁⠀⠀⠀⡤⠐⠀⢆⣀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠁⢱⠐⢄⠀⣀⠄⠘⢆⠀⠈⡄⣀⡀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠌⢀⡆⠀⠀⠀⣱⠀⣹⠇⠀⠀⠀⠣⠀⢛⠀⠈⠂⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠉⠀⡗⠐⠒⠂⢀⠛⠂⢬⣿⣶⣶⣿⣿⠊⠁⠀⠀⣀⠔⠊⠙⠁⠀⠀⠀⠀⠙⠢⣀⠈⠢⡀
⠀⠀⠀⠀⠀⡠⠂⠁⠉⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠈⠀⣤⠦
⠀⠀⠀⠀⠰⠀⠀⠐⠒⠃⣀⠇⠒⠒⠀⠹⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢤⣼⣗⠳
⠀⠀⠀⠀⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠛
⠀⠀⠀⢰⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⡄
⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⡤⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⣿⣿
⠀⢀⠌⠀⢀⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸
⣴⢟⣷⠢⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈
⠀⠞⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠀⠀⠀⠘⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠀⠀⠀⠀⠂⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡌⠀⠆⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡙⠒⠤⢀⡤⡄⠀⢰⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⠉⠢⢀⡘⠀⠋⠐⡆⠀⠀⠀⠀⢰⠉⠗⠈⠣⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠸⡀⣰⠀⠀⡇⠀⠀⠀⠀⣸⠀⢰⠀⠀⢠⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠑⠠⣃⠀⠀⠀⣠⠉⠁⠚⠋⠁⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡶⠊⠁
⣶⣶⣶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⢿⡍⢸⠉⣾⣿⣿⣿⠻⢻⣿⣿⣿⡿⢟⢋⢽⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢦⣾⣷⡘⡾⣿⣿⣿⣿⡌⣭⣿⣯⣿⣵⣿⡏⣾⣿⢷⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣏⡑⣿⡾⣷⠽⢿⣧⣽⣿⣿⠿⠻⠛⠛⠿⣿⣿⣿⢚⣿⡿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢰⣿⡟⣯⣛⣛⡛⠎⠓⣦⣼⣿⣿⣿⣧⣆⣀⣀⣠⣾⣿⣿⡜⣟⣵⡿⡿⠟⠿⣿⣿⡄
⠀⠀⢠⣿⣿⣶⡾⢻⣯⣿⢰⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢋⣉⣠⣮⣶⣾⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡯⢹⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠟⠯⣹⠛⣿⣿⣿⡇
⠀⠀⣼⣿⣌⠿⠿⣾⣷⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣡⡿⠿⣿⣧
⠀⠀⢿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⡛⢒⣽⡯
⠀⠀⢸⣿⣏⣿⣶⢟⡂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⢳⣾⢿⡮⠵⢿⣿⡇
⠀⠀⠀⢿⣿⣿⡩⣈⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣥⣾⣎⢷⣿⢿⠦⣽⣿⠁
⠀⠀⠀⠘⣿⣿⣷⣞⢹⣻⣿⣿⣿⡖⣿⣍⡹⡿⣿⠿⣛⡛⢛⠁⠧⣀⠋⠈⠉⢓⣭⣳⣷⣿⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣯⣿⡿⣾⡺⣿⣿⡏⢻⡢⡘⣧⣜⠤⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣿⡟⣼⡇⢙⠣⡄⠹⡂⡋⠀⠈⣇⢉⡈⠍⠛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣻⣻⣅⣿⣏⢸⠀⠑⡆⠣⢱⣄⢠⣸⣶⣿⣶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣯⣛⣴⣿⣤⣷⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⢿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣷
⠀⣼⣿⢿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⢸⣿⠇⣼⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠹⣿⣷
⣿⣿⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠻⣿⡆
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⣆⠀⠻⠟⠷⣦
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣨⣿⠀⠀⠈⠀⣹⡇
⢹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⢹⣟
⠈⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⡉⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠁⠀⠀⠀⣸⠇⠀⠀⠀⢿⣤⡾⠿⣩⣿⠋⠉⣿⠷⣦⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢸⣿⠀⠀⢀⣿⠀⢀⡟⠀⢸⡇
⠀⠀⠀⠀⢠⡿⢁⣤⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣀⣠⣤⡴⠒⢿⡁⠀⠀⠀⠀⠀⠀⣿⢿⡆⠉⠙⠻⠗⠚⠛⢲⣿⠁
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣉⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⣸⡏⠈⠻⣦⠀⠀⠀⢠⣤⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣤⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡤⠴⠚⠋⠉⠀⠀⠀⠀⣴⠟⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⡾⠛⠛⠷⣦⣼⡇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠃⠀⢀⠀⠀⣿⠃⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣯⡟⠀⠈⠻⣯⡛⠳⠶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⠾⣛⣋⣵⠟⠁⠀⠀⠀⠳⣤⣿⣀⣀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⢰⡟⠁⠙⢷⣤⡾⠀⠘⣷⠀⠀⠀⢈⣿⠋⠉⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢰⡟⠛⠶⣄⣠⠿⠃⠀⠈⠻⣦⣄⢀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠃⠀⣀⡀⠀⠀⠀⠉⠛⠋⠛⢶⣄⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠟⠋⠻⢶⣄⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣤⡾⠋⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠉⠉⠉⠙⠻⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⣶⡦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠛⠉⠀⠀⣸⡇⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡾⠛⠋⠉⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠻⢷⠞⢳⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠧⠤⢀⣀⠀⠛⠿⢴⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠙⠷⠶⣤⣅⣈⡛⠒⠾⠄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣼⠟⠷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢶⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⡿⠁⠀⢠⠋⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣍⣿⠀⡰⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⡼⢁⡼⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠋⣰⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠋⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿
⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡆
⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠋⠉⠉⠉⠙⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠚⠉⠉⠉⠀⠀⠈⠉⠉⠑⢶⡉⠀⠀⠉⠉⠲⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠴⠶⠿⠿⠭⠭⢭⣉⡉⠙⠒⠳⠤⣄⡀⠀⠀⢱⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠉⠉⠉⠉⠓⠒⠦⢄⣀⠉⠙⠲⢤⣀⠀⠈⠙⠢⣄⣇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠑⠢⣄⠀⠀⠉⠲⢤⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠈⢧⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡙⠢⢄⡀⠀⠈⠓⢤⡀⠹⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠈⢳⣄⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠭⣷⣦⣄⡀⠙⢢⣳
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⠏⠀⠀⠀⣹⠷⣄⡀⠀⠀⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠿⢿⢧
⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⠋⠀⠀⢀⡴⠁⠀⠈⠛⠷⣦⣤⣀⡀⠀⠈⠉⠒⠢⠤⢤⣀⣀⣀⣀⣀⣀⣠⠤⠴⢶⣿⣦⡤⠿⡆⠳⡀
⠀⠀⢀⡴⠃⠀⠀⣠⠞⠁⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠉⠓⠪⠭⣝⣒⡒⠒⠶⠶⠒⠒⠒⣛⣉⡥⠤⣶⣚⣉⣉⡤⠴⠾⣤⠀⠈⠓⠒⠉
⢠⣔⠋⠀⣠⠴⠊⠁⢀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢤⡀⠉⠉⠉⠉⠉⠉⠉⠀⠈⠙⠒⠦⠤⠤⢤⡤⢴⠒⠁⠀⠀⠀⠀⢀
⠀⠈⠙⢻⢲⡖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⡧⣄⣀⣀⣠⠔⠋
⠀⠀⢠⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠤⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⡄
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡹
⠀⠀⠙⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡖⠀⠀⠀⠀⠀⠀⠀⢀⡽⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣤⣈⣓⣒⣒⣒⣋⣠⡤⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠳⡼⣟⢿⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠁⣠⣄⣀⡤⠬⡟⠳⣄⣹⡈⠩⣿⠗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠆⠀⠉⣤⡀⠘⠁⣹⠀⠱⣄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⡄⠈⠛⣇⠐⠖⠀⢀⣠⢄⠀⢰⠁⢀⡠⠚⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⠏⡇⠀⠀⠘⣆⠀⠘⢿⣀⡼⢠⡾⠚⡏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠶⠿⢧⠀⣀⡴⢾⢓⡶⣶⣲⢶⢋⠿⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡋⠁⣀⡤⣼⣿⠿⢳⡿⢯⡠⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⣈⡧⣄⣀⣠⣴⠟⣠⠴⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⠛⠻⡖⠚⠁⠀⢨⠿⠟⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠋⠂⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠦⢄⣀⠀⠀⠈⠳⢶⠇⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣼⠟⠒⣶⠦⣄⠙⠲⠤⣄⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠞⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⡴⠟⢦⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢳⠀⠀⠈⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⣆⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠉⠀⠈⠴⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠂⠤⢄⣐⠒⢄⠀⠀⠀⠀⢠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠞⠁⢀⡀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠉⡀⠀⠀⠀⠀⠈⠁⠀⠈⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠸⠷⠆⠀⢀⠸⡷⠆⠀⠀⠀⠙⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠤⠀⠀⠠⣀⡠⠂⠀⠀⠀⠀⠀⢠⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢳⡆⠈⢁⣀⠐⣰⡆⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠠⠬⠿⠛⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢤⡤⣤⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⡀⠆⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠢
⠐⠄⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⠀⠄
⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠢⣄⠀⠀⢀⡋⠁⡁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠤⠇⠀⠀⠸⠁⠀⠂⠀⠀⠀⡂⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡇⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⣀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠐⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢛⠀⠀⢄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢘⠃⠉⠀⠀⠈⠙⠐⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡸⡇⠀⠀⠀⠀⣀⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠂⠀⠠⠄⠀⠀⠼⡂⠀⠀⣴⠀⠀⠀⢀⣀⠬⠓⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠤⠤⠤⠤⠤⠴⠃⠀⠸⢄⣀⣀⣀⣀⣀⡠⠄⠂
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣲⡶⠖⠶⠶⠤⢤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣻⣿⣿⣹⣉⡤⠬⠭⠤⠮⠼⠟⠛⣂⠸⣟⣲⣤⠤⣄
⠀⠀⢀⣴⣶⣿⣿⣿⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠚⠛⠛⠫⠑⣢⣤⣄⡀
⢀⣔⠋⢹⠷⢻⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠸⣿⣿⣶⡿⠛⠁
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⡀⣀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡲⡗⢏⢫⣑⣿⠂⠻⠙⢿⣿⠷⢦⣤⡄⣑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡆⠙⠋⢘⣛⣿⣿⣿⣿⣿⣟⢠⣧⠀⠈⠙⠷⣤⣬⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢸⣛⠫⣼⣿⣿⣿⣿⣿⣿⣿⣗⣞⢸⠀⠀⠀⠀⠈⠹⢶⣦⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣽⣿⣿⣿⣿⣿⣿⣿⠿⠟⣤⣼⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⠷⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⣤⣰⢿⠿⣿⢿⡟⠅⢰⡐⡦⠃⠀⠀⠀⠀⠀⠀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠾⣰⢏⣃⣚⣨⠴⠊⠀⠀⠀⢀⣠⡴⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⢁⣀⣀⠀⠀⠤⣐⠨⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣂⣒⣐⣒⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⣠⠁⠈⠀⠘⠀⣨⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢰⠓⣆⠀⠀⠀⠀⠡⠘⠀⠑⠴⠃⢀⠝⢁⠜⠀⠀⠀⠀⠠⠂⡆⢀⠀⠀⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠆⢤⠈⠝⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠡⠊⠀⠀⠀⠀⠀⠔⠘⢄⢣⡀⠰⡀⠀⠪⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⢁⠞⡠⠐⡲⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⡤⠐⠒⡚⠲⣄⠙⢄⠀⠈⠠⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠊⠀⠐⠁⠀⠀⠘⡀⠀⡠⠄⠜⠀⠀⢀⡐⠂⠂⠙⠀⠈⠆⠀⠀⣀⠴⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠩⢧⠑⠀⠂⠀⢀⠐⠀⠀⠉⠀⠀⠸⠷⠿⠁
⠀⠀⠀⠂⠀⠂⢁⡀⠀⠀⠀⠈⠹⣿⣒⡀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠁⠀⠀⠈⠒⠉⠀⠀⠀⠠⡔⠁⡀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠌⠁⠘⠃⠀⠀⢄⡀⠀⣀⣀⣀⣈⣤⣄⣉⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠃⠀⢀⢀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠘⠀⡀⠁⢠⣾⣿⠭⠯⠿⠥⠿⠧⠷⠧⠽⠯⠽⠯⠿⠭⢷⣭⣭⡟⠻⢖⡶⠂⠀⠀⠤⠔⠁⠐⠒⡄
⠀⠀⠀⠀⠀⢀⣠⡴⢚⣋⣽⡵⡄⡘⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢤⠀⠀⠀⠀⣤⣠⣶⣄⡀
⠀⠀⠀⢀⡴⢫⡴⠖⠛⠉⡀⠐⠀⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⡀⠼⠟⠏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠈⠙⢮⣽⣦
⠀⠀⠀⡼⣶⠋⠀⠀⠀⠀⠀⠀⠀⠤⠤⠰⠀⠄⠰⠀⠀⠀⠞⠀⢀⠀⠤⢠⣵⡰⠂⠢⠒⣉⠆⠀⠀⢀⢀⠀⠀⠀⠠⠂⠁⠤⢀⠈⣯⢘⡇
⠀⠀⢰⡗⡇⠀⠀⠀⠀⡴⢁⡀⢸⣄⣀⢴⠂⠀⠐⡄⠀⠈⠃⠠⡤⠤⠀⢋⡽⠃⠀⠀⠉⢀⢄⡁⠀⠘⠄⡠⠐⠃⠒⠂⠰⠤⢁⡴⢉⡝⡇
⠀⠀⢸⠻⣿⣦⣄⡀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠚⠁⠀⠀⡀⡠⠌⣡⢩⠕⠊⡠⠦⠚⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣫⡴⢋⡔⠁
⠀⠀⠸⣄⡈⠉⠛⠻⠿⢶⣶⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠧⠤⠊⡔⢉⠔⠊⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⡶⣾⣻⡯⠟⢛⣡⡶⠋
⠀⠀⠀⣸⠉⠉⣶⠶⠦⠤⠤⢉⣉⡛⠛⠓⠛⠻⠿⠿⠿⢷⣶⣶⣿⣿⣖⣲⣒⣖⣲⣻⡿⠿⠿⠿⠷⠒⢛⣛⣋⣩⡥⠴⢺⡏⠀⢹⡀
⠀⠀⡼⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓⠒⡶⠒⢶⡖⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⢳⠀⠀⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⠒⠒⠲⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⢷⠀⠀⠀⠀⠀⠠⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⠀⣿
⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠙⠉⠉⠉⠀⠀⠀⠀⠸⡆⠀⠀⠀⠸⡷⢤⣀⡀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⡟⠀⠀⠀⠀⠀⠰⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠙⢦⣈⠉⠛⠛⠛⣛⡽⠋⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣠⣤⣤⣤⡸⣷⡀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠁⠀⢹⣷⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡏⢻⣆⠀⢀⣾⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠙⠛⠛⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢁⣴⠶⢶⣦⠀⠀⠀⠀⠀⠀⢹⠉⣭⠀⠀⣶⠄⣴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⢸⡄⣿⣤⣶⣿⡀⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⣿⠀⢠⣄⠀⣼⡇⣠⡾⠛⠻⣦⠀⣟⢀⣀⣼⡟⢰⡇⠀⠀⠀⣿⠀⠀⠹⣿⡄⣾
⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣷⣿⢡⡟⠀⠀⠀⣿⠃⣿⣿⣟⠉⠀⢸⣿⠀⠀⠀⣿⠀⠀⣠⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⡿⠈⢿⣤⣤⣾⠏⠀⣿⠀⠻⣷⠄⢸⣿⣤⣤⡀⣿⣴⡾⠟⠃⠀⡛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⢀⣀⠉⠉⠉⠉⠉⠉⠉⢁⡀⠒⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠑
⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂
⠀⠀⠀⠀⢀⠊⡠⠁⠀⠀⠀⡠⠀⠀⠀⠂⠀⢀⡀⠤⠚⠛⠉⠁⠀⡀⠀⠀⠀⣀⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⢄⠀⠁⡀⠡⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⢺⡆⠀⡀⠀⠀⠀⠀⠀⠁⠢⣄⡀⠁⠂⢄⡀⠁⠢⢤⠄⡀⠀⠀⠐⠀⠐⡀⠡
⠀⠀⠀⠀⢠⠁⠀⠀⠠⠀⠀⠀⠀⠀⠔⠁⠀⠀⠹⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠠⠀⢡
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⠀⠠⠀⠀⢈⠳⢄⡀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠐⠀⠠⠴⠞⢛⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠔⠀⢀⠔⠁⠀⠀⠐⠡⠀⡐⠀⠠⠤⠠⠤⠤⠉⠙⠛⠋⣠⣶⣊⣽⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠦⣠⡀⠊⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⡀⢈⡛⠀⠀⠀⠂⠀⠀⠐⠋⠀⠤⢠⡤⢠⠃⠀⠀⡀⠀⠀⠀⠀⢁
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘
⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⠄
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠁⠄⠶
⡇⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠚
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠖⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠈
⠈⡄⠀⠈⠁⡀⠁⠀⠉⠈⠁⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠠⠀⠀⠉⠀⠈⠉⠉⠀⢀⠈⢂⢀
⠘⠵⠀⠁⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾
⠀⢖⣫⠀⠀⠰⠀⠀⠀⣀⣀⡀⠀⠤⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⢀⣀⣀⣀⡀⠀⡈⠀⠀⠜⣘
⠀⠈⠒⠒⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠒
⠀⠀⠀⠀⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠾⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠰⢏⣉⣉⣩⣉⠉⣑⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠘⠃⠀⠀⠀⠀⠈⠉⠁⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⢒⣀⣤⣤⣀⡐⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⡠⣿⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡖⠋⢁⣀⣤⢿⣶⠶⠶⠤⠤⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⡶⠋⢉⣽⣿⣿⠷⠶⠶⠿⣷⣤⣀⠀⣇⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣴⣟⣁⣴⣿⠟⠉⠀⠀⡠⢄⠀⠘⣧⠈⠙⠻⣦⡀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢠⣿⡿⠃⢀⣀⣀⠀⢇⡸⠀⠀⣿⡇⠀⠀⠀⠙⢦⣄⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⣿⣿⠃⣰⣿⣿⣿⣷⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠑⢦⣄⣠⣿⠄⣾⣿⡏⠀⢻⣿⣿⣿⣿⠃⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⣿⡇⠀⠀⠙⠛⠛⠁⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⢼⡇⣼⣿⠹⣿⡄⠀⠀⠀⠀⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣡⠄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⢸⡀⠈⠙⠓⠲⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢳⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠃⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠉⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣽⣦⣀⣀⣀⣠⣤⣴⠾⠿⠛⠋⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡦⣤⠄⠊⠁⠀⠀⠀⠀⠀⠀⡖⠊⠉⠁⠀⠈⠉⠉⠘⠣⣀⠀⠀⢀⡆⢑⣦
⠀⠀⠀⠀⠀⠀⢀⢳⣬⡷⠧⣀⡗⠒⠒⠒⠒⠒⠒⡖⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠷⠺⠧⠺⣤⣟
⠀⠀⠀⠀⠀⠀⠈⠛⢇⠀⠀⠉⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠘⢄⢀⠀⠰⠆⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠆⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡄⠌⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠓⡀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠈⠑⠤⣀⣀⠤⠊⠀⠀⠀⠀⠀⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠀⠀⠉⠉⠘⡆⠐⠒⠒⢅⠀⠀⠀⠀⠈⡢⠤⠤⠆⠒⠂⢙⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠊⠁⠀⠀⠀⠀⠀⢸⡄⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠀⠀⣀⣀⠐⠂⣤⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠠⠆
⠀⢀⠆⠈⠒⢤⣀⠀⠀⣠⢄⠀⠀⠀⠘⡦⣀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠐
⠀⡌⢀⠋⣦⣁⡈⠍⠃⠉⠒⠈⠲⠦⢠⠀⠀⠉⢰⠂⠂⠈⠒⠒⠒⠒⠲⠉
⠰⢀⡴⠋⠀⠈⠉⠓⠤⠀⡀⠀⠀⠀⠀⠀⠀⢀⠊⠘⠂⠐⠃⠀⠀⢀⠃
⠁⠉⠀⠀⠀⠀⠀⢐⡀⠔⠉⠈⠐⠠⠄⣀⣀⠬⢀⡀⠀⠀⠀⠀⠀⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⠿⠿⠿⠶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠟⠋⢉⣡⠶⠛⠉⠉⠉⠉⠉⠛⠲⣤⡀⠀⠈⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢶⣶⣶⣤⡀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣀⡀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⣧
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠉⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣤⠴⠚⣁⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠶⠮⠿⠿⠿⠗⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡠⠞⠑⠦⣄⡤⠖⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⠏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⡆⠀⠀⠀⣺⠀⠀⠀⢀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⡤⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⠳⢤⣀⡤⠖⠲⣄⡴⠚⠲⠦⣀⡤⠶⠦⢤⡈⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣡⠖⠓⠲⠦⢥⣀⡔⠛⡩⠶⡞⡷⢾⡙⠳⣤⠔⠒⠛⠲⠦⢤⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡾⠁⠀⠀⠀⠀⠀⢸⡀⠀⠠⠄⡿⡇⠤⠀⠀⡸⠀⠀⠀⠀⠀⠀⠑⣾⣷
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠾⠓⠻⠒⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⣠⢴⡳⣴⠋⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⣄⢀⣀
⢸⡙⠆⠁⠀⠀⡽⢦⣀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⡇⠘⠋⠼⣻⢤
⠳⣄⡀⠀⣠⠞⠁⠀⠈⠓⢮⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡡⠖⠉⢧⠀⠀⠀⠁⠾⡄
⠀⠀⠙⠫⣅⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡾⠗⠉⠀⠀⠀⠈⠓⠤⣠⠴⠊⠁
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠈⠻⢽⡒⠲⠤⠤⠤⠤⠤⠤⢒⣒⣯⠔⠋⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⣼⠉⠒⠲⠶⠖⠒⠚⠋⠉⠀⢳⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⢘⠞⠋⠛⠓⢦⡀
⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇
⢰⠁⠈⠁⠷⠒⡄⢀⠀⠀⠀⠀⠀⠙⠦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⢰⣧⡀⠀⠀⠀⠓⢩⠃⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠺⠥⢤⡀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠻⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠒⠒⠢⠤⠤⢤⠤⠴⠒⠒⠒⠒⠒⢹⠗⠒⠒⡏⠉⢻⣦⡀⡰⠁⠷⠋⠿⠖⣦
⠀⠀⠀⠀⠀⠀⣀⣴⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢳⠀⠀⣿⡟⡇
⠀⠀⠀⠀⠀⡜⠁⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⢸⠀⠀⡀⡇⢱⣰⠲⣄⡶⢤⠟
⠀⠀⠀⠀⠀⢇⠀⠀⢀⡇⠀⠀⠀⠀⠙⢦⣀⣀⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣒⢶⣿⠇⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠈⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠶⣶⣶⡶⠋⠉⠉⠁⠙⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣠⣤⡀⠀⠀⠀⠀⠀⢠⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠙⣦⣞⠺⠃⠀⠀⡽⢠⠒⠢⠴⣤⡜⠀⠀⠀⣿⣶⡶⠒⠲⢤⡀⢀⣤⣠⠤⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⣿⠀⠀⠸⡙⠋⠀⠀⠀⣤⢼⡗⠀⠀⠀⠛⣿⣷⡖⠢⠗⠘⣷⡄⠀⠀⠘⡶⠇⠀⠀⢸⡀
⠀⠀⠀⠀⣠⣴⣟⣀⠀⣀⣸⡄⠀⠀⡽⢦⡀⠠⣖⡾⠛⠻⢦⡀⠀⢀⡟⠈⠁⠀⠀⠀⠘⡟⠶⣤⠤⠇⠀⠀⠠⣞⠉
⠀⠀⠐⠷⠾⣃⣄⣉⣉⣭⣿⠛⢤⣼⡁⠀⠉⣳⡞⠁⠀⠀⢀⡽⣀⣼⡤⢄⢀⣤⣠⣿⣦⣇⣀⡬⠶⣄⠀⠰⣟⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣠⡀
⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⣿⡿⠟⠛⠛⠛⠛⠛⠆⠀⣰⣿⠿⠿⢿⣿⢿⣿⠓⠀⠀⠀⢸⣿⡿⠻⠛⠻⠿⢿⣿⣿⠄
⠀⠀⠀⠀⠰⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠸⠋⠂⣿⣄⠀⠀⠘⠛⠀⠀⡀⢸⡻⠉⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠐⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠈⠙⠻⢗⠆⢶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢌⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠇⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⢀⠷⠄
⠀⠀⠀⠀⠉⠁⠀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠀⠀⠀⠀⠴⠃⠃⠀⠀⠀⠀⠀⠙⠓⠂⠈⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣿⠀⠀⠀⢻⠀⢠⣾⠂⠀⢠⣿⣧
⠀⠀⠀⣠⠴⠋⣉⣐⣦⣀⢦⡈⠳⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⣹⣦⣀⣀⣸⣀⣸⣿⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⣶⣄
⠀⠀⠀⠈⠓⠦⣄⣩⡟⠋⣽⣫⠽⠋⠉⠉⠉⠉⠉⢹⣿⡴⢺⣿⠀⠀⠀⡼⠀⠈⣿⠀⠀⡞⠀⣿⣧⠴⠋⠉⠉⠉⠛⠛⠉⠉⢹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠘⣿⣄⣀⡼⠁⠀⠀⠙⠓⠚⢁⠴⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠈⠉⠛⠋⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠋⠀⠀⢀⣼⠃⠔⢶⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠖⠀⠀⠀⢀⡴⠋⠈⢿⠀⠈⣧⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢧⡴⠆⢀⣾⠉⣩⡗⠀⠈⠓⠾⠹⣆⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⠏⢰⡶⠲⣄⠀⠀⠀⣠⣶⣿⣆⠘⢄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡀⡞⣆⠀⠀⠀⠀⠘⠧⢤⠾⠀⠈⠓⠐⠋⣠⠤⡄⠛⠛⠛⠘⢢⣬⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠘⢆⠹⠇⢿⠀⠀⠀⠀⠀⠀⠘⢆⣄⠀⠀⠠⣤⢀⣀⡀⢤⡄⠀⠀⢠⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠃⠀⠀⠀⠘⠦⢂⡴⠀⠀⠀⠀⠈⠙⢦⣀⠀⠑⠂⠀⠐⠋⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣤⣤⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⣀⡴⠒⠒⢿⣏⡉⠃⠀⠉⠛⣲⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢧⠀⣀⣠⠾⠁⠀⠀⠀⠀⠙⠯⠟⠷⠾⠛⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠛⢿⣷⡀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠁⠀⠹⣷⢀⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⣤⠴⠖⠚⠛⣿⢤⡤⠤⡤⣤⣤⣤⣤⣤⣤⣤⣍⣹⣧⠴⣺⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡊⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣷⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠀⢸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠞⠁⠀⠉⢹⠉⡏⠉⠀⠰⣄⠀⡏⠈⣧⢀⡴⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠈⠳⣇⠀⢸⡅⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⠀⠀⠀⠸⠈⣼⣇⠀⢀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢣⢸⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⢧⠀⠀⠀⠀⠀⠈⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢦⠤⠤⣤⣤⣴⠁⠘⣤⣤⣤⡤⠤⠤⢾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⣀⣴⡇⠀⠀⠀⠀⠀⠀⢣⣤⡤⠴⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣉⠓⢤⠤⠤⡆⠀⠀⠀⠀⠀⠀⠘⡖⠤⡴⢊⣩⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢫⡙⣌⣊⠤⠃⠀⣴⠶⡇⠀⠀⠀⠀⠀⠀⠀⣷⣌⢢⡁⢿⣀⢜⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡂⢄⣠⡄⠤⠤⠴⠖⠚⢛⣒⡇⠀⠀⠀⠀⠀⠀⠀⠿⠻⢦⣥⡀⢈⠉⠀⠀⣡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠆⠭⠭⠥⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⣴⣿⣿⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⣿⣦⡀⠀⢀⣿⠟⣿⡆⢀⣼⠏⠸⣿⠀⢠⡾⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⢿⡇⠙⣿⣦⣾⣿⣶⣿⣿⣿⣿⣶⣤⣭⣴⡟⠁⢸⡏⠀⢀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢷⣄⣠⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣿⣧⣾⠟⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⢸⣧⣼⠿⠋⢁⣠⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠈⠛⢿⣿⣿⣀⣤⡶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣦⣿⠟⣁⣴⠾⠛⣽⡿⠋⠁⢀⣤⣶⣶⣿⣷⣦⣍⠉⠙⢿⣿⣿⣶⣦⣄⡙⢿⣿⣅⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣥⡾⠋⠀⠀⢰⡟⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢹⣿⠀⠈⠙⢿⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠉⠀⠀⠀⠀⣿⠃⠀⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣷⠀⢠⣿⠀⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⣿⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣄⠀⠀⠀⢿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⡏⠀⠀⠀⢀⣼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠋⣨⣿⣿⣿⣷⣄⠀⠘⢿⣆⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣼⠟⠀⠀⣀⣴⠿⠋⣸⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢻⣿⣍⠻⣷⣤⣈⠻⣷⣦⣤⣉⣛⡛⠛⣛⣉⣤⣶⠟⣁⣠⣴⠾⠛⢁⣠⡾⠟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠋⣿⣷⣤⣉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⢀⣠⣴⡿⠋⠐⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⢹⡿⢿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣾⠟⠋⠙⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠉⠙⣿⠋⠛⢿⡟⠉⠻⣟⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠉

⢠⠄⠀⠀⠀⠀⠀⠐⠂⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠐
⠀⠀⠆⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠬⠃⠀⠐⠁⠐⠀⠄
⠀⠀⠈⠀⠠⠀⠂⢀⠀⢰⡄⠀⠀⠀⠀⣀⢀⠒⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⢈⡂⠀⠀⠀⠀⠀⢁⡀
⠀⠀⣀⣄⠀⢢⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠸⢐⠂⠀⠀⠀⠀⠀⠀⠈⠤⠐⡀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠑⠂⠓⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠠⠌⡈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠒⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠟⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠠⠂⠒⠀⡄⠀⠁⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⢰⠒⠂⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢐⠉⠐⠀⠀⠀⠀⠠⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠠⠄⣄⠀⠀⠀⠤⠐⠀⠠⠀⠠⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⠃⠀⡐⠁⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠃
⠀⠀⠀⠀⡀⠀⣀⠁⠄⠀⠀⠉⠀⢀⠄⠀⠀⠰⠀⠀⠘⡀⠀⢀⡀⠤⠒⠇⠀⠀⡄⣀⠈⠀⠀⠀⢡⠀⠀⡰⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠑⠀⠂⠁⠀⡄⠀⠀⠀⠑⠂⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠃⠒⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⢀⠀⡄⠒⢤⢀⠀⠀⠀⠀⠀⢐⣰
⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⢉⠣⡀⠇
⠀⠐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠐⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠐⡄⠂⡀⠀⡡⢄⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠈⠀⠐⠁⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠑⠰⢉⡈⠄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⢀
⠀⠀⠀⠀⠒⣤⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡎⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠐⠀⠒⠁
⠀⠀⢄⠈⢢⠦⣔⠁⡰⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⠄
⠀⠠⠌⣠⠝⠰⢂⣈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⢀⡠⠣⠀⡀
⠀⠀⠀⠠⠄⠓⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⢢⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⢁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⠐⢄⠀⠁⠀⠀⠀⠀⠢⠀⠀⠠⠊⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠆⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠡⠀⠐⠈⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⠀⠀⠐⢠⠉⢒⢀⠆⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠔⠠⠰⣅⠀⠀⠀⠀⡐⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⡀⠄
⠀⠀⣀⡀⠁⢠⠄⠐⠢⢀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡡⠊⠀⠀⠀⠀⠀⠈⠁⢀⠀⠒⠀⠀⡐⠀⠀⠀⠀⠀⡠⡅⠙⢠⠁⠀⡃⠜⠠
⠀⠈⠄⠈⠖⠀⠀⠀⠀⠈⢃⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⢀⠂⠉⠁⠢⠀⠀⠈⠀⠀⠀⠜⠀⠀⠀⠀⠀⡀⠴⠐⠦⡀⠆⠬⠁⣀⠈⠰
⠀⢀⠦⠠⢆⠀⠤⠀⠴⠀⡈⠀⠢⠈⠀⠀⠔⠁⠀⠂⠁⠀⢀⠃⠀⠐⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠆⠠⠤⢄⡀⡀⢅⡀⠈⠱⠄
⠀⠈⠰⠁⢀⠱⠤⢤⡄⠊⢋⠐⡆⠀⠀⠀⠈⠢⠀⣀⡀⠀⠊⠡⡀⠀⠀⠈⠁⠀⠀⢀⠴⣀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠚⢀⡠⠎⠀⣧⠀⠑⡈⠄
⠀⠀⠀⠀⠁⠼⡀⣤⠈⠦⠜⠀⠀⠀⠀⠥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⢂⡠⠂⡐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣻⣟⣿⣛⣿⣿

⠀⠀⠀⠀⢀⡠⠴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⢾⡀⢿⠇⢨⠇⠀⠀⠂⠀⠰⠂⠀⠘⠀⠀⠀⡄⠀⢠⡄⠈⢢⠀⠀⠰⣆⠰⢀⡳⠪⠂
⠀⠀⠀⠀⠀⠙⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠁⠁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⣒⠳⢆⠀⠀⠀⣠⣴⠶⢂⡠⣀⠀⠀⠀⠀⢀⠀⡀⡀⢀⠀⠀⠀⠠⢀⣀⢐⡠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣏⡴⠋⠀⠀⣾⣛⡃⣿⠏⣋⣻⡇⠀⠀⠈⡧⣥⣧⢤⣤⢤⠂⠀⠀⢣⡼⠜⡣⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡠⡴⠽⠋⠀⠀⠀⠀⠣⣙⠽⠯⢊⠜⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⢀⠄⠀⡀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠆⢆⠀⠀⠉⠁
⠀⠀⠀⠀⠘⢦⠌⠛⣰⠞⠀⠀⢠⠚⢡⡜⢢⠀⠀⠀⠀⢺⡏⠰⠌⣫⡆⠀⠀⠀⠀⢠⠔⣺⡗⠦⡄
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠄⠐⠃⠀⠀⠀⠀⠡⠂⡄⠆⠱⠈⠀⠀⠀⠀⠈⢢⠌⠥⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡤⣄⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠐⡏⢸⠽⢹⠄⠿⠀⠀⠠⣏⠀⢠⡄⢀⣻⠆⠀⠀⢠⠀⠠⠀⠀⢰⠀⠀⠀⠀⢖⠁⢴⡆⠁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡈⠙⠉⢀⡑⠀⠀⠀⠈⠙⠀⠀⠁⠂⠀⠀⠀⠈⠀⠀⠂⠠⠃⠀⠀⠀⠀⠈⠓⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠄⢠⣦⠀⣄⠀⢀⣲⣈⣻⡿⢼⣿⣡⡦⠀⠀⠀⠀⠀⠀⢐⣔⣀⡀
⠀⠀⠀⠀⠀⠐⢄⠻⠣⢀⠆⠀⠀⠀⠀⠘⠓⠃⠀⠀⠀⠛⣯⣩⢾⠟⡉⡽⠁⠀⠀⠀⠠⠀⡖⢹⠭⢢⢈⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠉⠢⢿⣷⠿⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠻





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣼⣿⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣼⣿⠀⣿⣿⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣼⣿⣿⣿⣿⠁
⠀⠀⠀⣤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢶⡄⢻⣷⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠸⣿⣾⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⢸⣿⠀⠀⣤⠄⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁
⢲⣄⣻⣿⣿⣿⣿⣷⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡆⠀⣴⡆⢠⡆⢀⡄⠀⠘⣿⣆⣸⣿⡄⣰⣿⠀⣸⣿⣿⡆⠈⣿⣿⣿⣿⣿⣿⠃
⠀⠻⣿⣿⣿⣿⣿⣿⣧⣴⣾⡟⠁⠀⠀⠀⠀⢸⣧⣼⣿⣧⣿⡇⣾⡇⣀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠂⠀⠀⠀⠘⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠢⣄⢠⠖⠉⠙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠀⠀⠀⢸⠃⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⢸⠀⠀⢀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠙⣆⠀⠀⠉⣸⢆⠀⠈⠁⣠⠛⠒⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠁⠀⠀⠀⠀⠈⠙⠒⠛⠁⠈⠙⠒⠋⠁⠀⠀⡃
⠀⠀⠀⢠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⢺⠀⠀⠀⠀⠀⠀⢠⠋⠹
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢸⠀⠀⠀⠀⠀⠀⢸⠀⠠⡤⢄
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠀⢸⠀⠀⠀⠀⠀⠀⣸⠖⠚⢆⡜⠉
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠔⠲⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⡃⠀⠤⠎⠉⢸
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⣠⠤⠚⡇⠀⠀⠀⠀⠀⠀⠀⣹⣿⡏⠀⢸⠀⠀⠀⠀⠀⠀⠙⢦⡂⢀⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠼⠥⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠦⢄⣀⣀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⡠⢄⣀⡠⠖⠋⢷⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⢢⡤⠌⠀⠀⠀⠀⠳⠤⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡔⠁⢀⡴⡇⠀⠘⠙⠢⠔⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠉⠒⠉⠀⢇⣀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡏⠉⠉⠉⠉⠉⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⣀⡀⠘⡄⠀⢀⠇⠀⢀⣀⣀⣤⣤⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠈⠙⠛⠛⠛⢹⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠐⠄⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠒⣎⡉⠁⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣼⠗⠀⠀⢰⡟⠒⠒⠒⠃⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠲⠤⣤⠋⠉⠉⠙⠃

⠀⠀⠐⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⠀⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡉⠙⠳⢦⠀⢠⠶⠛⠋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠵⣶⡶⠀⠀⢴⣶⠶⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠀⠀⠰⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠰⠋⠺⠇⠀⠀⠀⠀⠀⠀⠿⠂⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⡤⠀⣀⣤⠀⠀⠀⠀⠀⢠⣄⢀⠀⣄
⠀⠀⠀⠀⠀⠃⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠋⠀⠀⠀⠀⠀⠠⢤⡤⣤⡤⠤⠄⠠⠤⠤⣤⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣴⣿⣿⣿⣿⣾⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠛⠿⠿⠻⣿⡿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢛⣡⠀⠀⠀⠀⠀⢀⣄⠀⠙⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⢀⣶⡆⠀⠀⠀⠀⠠⣿⡇⠀⠀⠉⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⠀⣀⣠⠉⠀⠀⠀⡠⠁⠀⠀⠌⠒⢜⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⠛⠉⠉⡠⠄⠀⠁⠀⣀⠄⠊⠀⣀⠩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡒⣹⡆⠒⠈⠉⣧⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠈⠀⠀⠙⡿⠻⠃⠀⢀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠄⠒⠁⠀⣻⣄⢠⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⣦⢀⠠⠒⠀⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡿⠸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢀⡄⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢾⣳⣿⡿⠁⠀⠀⠘⣿⣿⡭⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢉⠝⠀⠀⠀⠀⠀⠘⠿⠟⠋⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊
⣿⣿⣿⣿⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣿⣿⣻⣟⣿⣟⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠿⣷⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠁⠀⢰⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⡄⠀⣠⣥⣌⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠘⡿⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡁⠉⠁⠀⠀⠙⠉⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠣⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⣄⣧⡤⠴⣾⣿⠄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢰⡾⠏⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢤⣴⠟⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⠟⠛⢿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣸⣿⠟⣿⠁⠀⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡁⣿⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⣿⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣯⠀⠀⢸⣿⠉⢸⣇⣼⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣽⣆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣼⡇⠀⠟⣸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⣿⠉⠀⠀⠀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢿⡇⠀⡀⣿⣿⣿⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠾⣿⣼⠀⠀⢀⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⡀⠛⠃⢸⠇⣿⣿⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⣿⣿⠀⠀⣼⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⠀⣻⣿⠇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣠⠘⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⡏⣿⡙⢿⣿⣿⣿⡆⠀⣿⠀⣿⣿⡄⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣷⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣈⣁⣸⣿⣿⣿⣿⡀⠃⠀⢿⣿⣵⣤⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣟⡛⠛⣻⣿⣿⣿⣿⣷⡀⠀⠸⢿⣿⣿⣿⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣥⣤⣥⣭⣉⣹⣧⠀⠀⠀⠹⣿⣿⡋⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⡀⢻⣿⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢿⠀⠀⠻⠌⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣴⡆⠀⠀⢀⠀⠀⢢⡰⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡁⠀⠀⠀⢻⣇⣸⡆⠈⠣⣀⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢸⣿⡇⠁⠀⠀⠀⢸⣿⡷⣉⣤⣾⡏⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠀⠛⢿⣿⡿⠹⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⠿⠛⡽⠋⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠙⠃⡠⠋⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⡄⠀⠀⠀⢠⠞⠀⠄⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢀⠀⣀⣀⠀⠀⢀⠀⢀⣀⣀⠀⡀⠀⠀⢀⡀⠀⠀⠉⠉⠁⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣿⣆⠀⢀⣿⣿⣿⠉⣻⡆⣿⡇⠉⣉⠉⠀⣿⢀⣾⠋⠙⢻⣆⣶⠀⠀⣿⣾⡋⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣧⢸⣿⢻⣤⡿⠻⠏⠛⠛⣻⡄⣻⡁⠀⣿⠀⠀⣿⠸⣧⡀⠃⣸⡟⣿⡀⢀⣿⠈⠛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠙⠛⠛⠀⠙⠁⠀⠀⠀⠛⠋⠑⠛⠛⠀⠛⠀⠘⠛⠃⠈⠛⠛⠋⠀⠈⠛⠛⠁⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⠀⠀⠐⠪⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣛⣭⣽⣶⣶⣯⣥⠤⠤⢤⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡵⠻⠋⠙⠉⢀⣤⣀⠀⠀⠀⠀⠀⠈⣯⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠀⠀⠀⢤⠀⠠⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠴⠄⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡣⠁⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠏⠉⠀⠀⣀⠴⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢇⠀⠀⡀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠄⠀⠀⠀⠀⢤⠤⠖⠊⠁⣠⠇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡐⡄⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⢦⠔⠒⡩⡾⠀⠀⠀⠀⢀⡞⠀⠀⠀⡠⢊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⣠⠏⠀⣀⢤⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣀⡤⠔⡟⠓⠒⢉⡡⠴⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⣀⡠⠔⠋⠑⠤⠤⠃⢋⠉
⠀⠀⠀⠀⠒⠀⠀⠇⠀⠀⢠⣀⠀⣀⣐⣒⣯⢀⣀⠤⠖⡛⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡀⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠈⠙⠲⠤⣈⡙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠉⠉⠉⠉⠃⠤⣀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢱⡒⣦⠤⠤⠆⣈⣁⠤
⠀⠀⡀⠀⠀⠀⠀⠀⡠⢊⡤⠒⠙⢧⠤⠴⠖⠒⠶⠄⣉⠉⠛⠒⠶⠤⠴⣂⣀⠀⠀⠀⣸⢁⠇
⠠⡀⡀⠀⠠⢄⣀⠜⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢶⡀⠀⠀⢠⡏⣀⠠⢐⡣⠊
⠀⠑⢌⠢⡀⠀⠈⠓⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡼⠒⠒⠊⠁
⠀⠀⠀⠑⠮⡓⠄⡰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⢰⠛⠱⡀⠀⠀⠀⠀⠀⠔⠀⢢
⠀⢀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⢸⠀⠀⢱⠀⠀⠀⠀⡘⠀⠀⢸
⠀⢸⢁⠀⠠⢌⠳⡄⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠉⠑⠠⡀⠸⡴⣚⠧⢀⠼⣴⢤⣸⡀⠀⠀⢀⠃⠀⠀⡌
⠀⠈⢮⡈⠁⠀⣹⣼⣆⠀⠀⠰⠃⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆⠹⡀⢀⠀⣀⠈⠓⠺⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠑⣤⠀⣩⡅⠈⢆⠀⡅⠀⡀⠀⣠⠔⠢⣄⠀⠈⠀⠘⡀⠈⢦⠀⠀⠀⠉⢀⡇⠀⠀⡧⠂⠀⡎
⠀⠀⠀⠀⠈⠻⡻⣠⣤⣼⡄⡇⠀⠁⣰⠃⠀⠀⠈⡆⠀⠄⠀⡇⠀⠀⢏⠀⠀⠀⠨⠇⠀⢸⣣⡠⣰⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠿⠟⢃⢸⠀⠀⠜⢦⡀⠀⡰⠁⠂⠆⢠⠁⠀⠀⢸⣤⣤⣍⣸⠀⠀⣇⣈⣍⡏
⢀⡤⡀⠀⠀⠀⠀⢸⡄⢒⢻⠀⢃⠀⠠⠀⠈⠉⠀⠠⠂⢀⠎⠀⠀⠀⠤⠸⠿⠿⠇⠀⠘⠯⠿⡾⠁
⣏⠀⠈⢂⠀⠀⠀⡸⠁⠀⠉⠀⠀⠳⢄⠀⠀⠀⠀⣀⠔⠁⠀⣀⠴⠚⠫⢲⡵⠃⠀⢀⣤⢬⡉
⠱⡀⠀⠀⡱⣲⢾⡅⠀⢠⠁⠀⠀⠀⠀⠉⠉⠈⢉⠔⠒⠊⠉⠀⠀⠀⣠⠞⠀⠀⢰⠁⢿⡄⢸
⠀⠑⠤⢀⡇⢘⠂⢉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⠔⠊⠀⠀⠀⠀⠈⠢⠿⠧⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣤⣤⣀⣀⢤⢀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⡟⢃⣴⡿⢿⣏⣽⠿⣤⣈⣢⣉⠀⢈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣿⣳⣿⣷⣿⣯⣿⣾⣿⣿⣿⣾⣿⣷⣶⣆⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣯⣽⣯⣥⡤⠶⠤⣄⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡸⣿⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣷⣶⣾⣭⣬⣝⣒⠶⡂⠠⠀⡀⢤⣾⣽⣶⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠋⠁⠈⠀⠀⠀⠀⠉⠉⠉⠛⢙⠻⢟⣿⣿⣶⣭⣒⠮⢕⢌⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣸⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣰⣥⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣅⣈⠡⠿⣿⣿⣿⣿⣦⣿⣿⣷⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣽⡟⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡝⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⡿⣿⡿⣛⣿⣿⣿⣿⡿⣻⣭⣬⣍⡻⣿⣿⣿⣿⣿⠉⠇⠀⠉⠛⠟⠻⠿⢿⣿⣿⣯⣛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⢠⣿⡔⢿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠻⢻⣿⢷⣮⣽⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠘⣿⣿⣮⣿⣿⣿⣇⣿⣿⣿⣾⣿⣿⣣⣿⣿⣿⣿⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⡿⣿⣿⠻⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠻⡯⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣯⣝⡛⠛⣛⣵⣿⣿⣿⣿⡿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠰⢦⣾
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⣤⣀⠠⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡟⣿⣿⣿⡿⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡟⡓⠞⢋⣡⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠙⣿⣿⣿⣿⣿⣿⡧⢿⡿⡇⣿⣿⣻⡷⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠒⠋⠉⠲⣌⣬⢏⢁⢸⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣼⣧⣿⣿⣟⣋⣀⣀⢠⢤⠤⠖⠲⠀⠉⠁⠂⣉⠴⠉⡆⣴⠽⠛⢃⡇
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⡛⠙⠋⠉⠉⠹⠭⠋⠿⠉⠉⠟⠟⣿⠙⢛⠩⠄⠃⠀⠀⠀⠀⠀⠀⡠⠅⢈⠉⠘⡋⠊⣽⡿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣿⣿⣕⡲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠁⠐⣋⡼
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⠙⠒⣦⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡛⢻⣥⣴⡩⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢶⣬⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⢀⠐⠤⠄⠀⡀⠀⠀⠉⠭⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣛⡻⢶⣟⣫⣵⠶⠀⣀⠠⡵⠆⢀⡁⠀⠀⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠉⠽⠗⠒⠤⠀⠀⠐⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠙⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠠⠐⠀⠉⠀⠉⠁⠙⠓⠒⠠⢄⠀⠀⠀⠀⠀⡀⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠱⢦⣤⠖⠁⠀⢂⠀⡠⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠠⠔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠞⢋⠟⠀⠀⡆⠈⠛⠓⠶⠶⠦⠤⠤⠤⠤⢴⡆⠀⠀⢴⡏⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⡤⢺⠀⠀⠀⡏⠐⠀⠤⠀⠀⢀⡀⠀⠀⢤⡾⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠞⠳⣿⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠳⢤⣀⣉⣉⣁⡴⠚⠁⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠄⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⡁⠛
⢠⡀⠴⢊⢃⣀⡠⠄⠀⠀⠀⠠⢄⣀⡀⠒⠂⠤⣄⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⣀⠄⢀⠼⣄
⠀⠀⠰⠪⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⢌⣉⠒⠠⠤⣉⣓⣦⣤⣄⣀⣀⣀⣀⣀⣤⡴⠞⢉⣀⡀⠠⠤⠐⣒⣩⠤⠚⠁⠀⠈⠳⢤⠀⣰⠀⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣌⢀⡝⠩⣉⣩⡽⢋⢉⣉⡀⠠⠤⠤⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡁⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠌⠀⠀⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡆⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢲⡟⠀⢌⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢌⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠶⣤⣤⣤⣄⣀⣀⣤⡤⠾⠋⡁⢄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠠⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡃⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣴⣿⣿⣿⣷⡄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠹⢿⣿⣿⠿⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠻⣦⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡶⠿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠀⠀⠘⠷⣦
⠀⠀⠀⠀⠀⠐⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢠⡾⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢶⣤⣀⠀⠀⠀⠸⣷⣄⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣈⠛⠻⢶⣤⣀⣈⣉⣭⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣈⠉⠛⠋⠁⠙⣿⠛⠻⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠙⠛⠻⣷⠀⠀⢿⡷⢶⣶⡄⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠿⠛⠻⣧⡀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸⣇⠀⠈⢿⣆⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣤⣬⡿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⣿⠀⠀⠀⠻⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣠⣾⢿⣧⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣾⡟⠁⠀⠹⣧⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢷⣄⠀⠀⠘⢷⣄⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠻⣧⡀⠀⠀⠙⠻⣦⡈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠇⠀⠀⠀⠀⠈⠻⣶⣼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⡤⠰⠂⠉⠉⠁⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠼⠁⡀⠀⠀⠀⠀⠠⢰⠁
⠀⠀⠀⠀⢠⠂⠢⢀⠈⠓⠂⠈⠀⠀⣎⠉⡇⠀⠠⠀⠠⠸⠃⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⠑⠶⠛⠄⠀⠀⠀⠀⠀⠘⢝⡀⠀⢀⡀⣁⡈⠀⠀⠀⢀⠝⠲⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢂⠈⢂⠀⠀⠀⠀⠀⢀⠉⠠⣄⠀⠜⠀⠀⠀⢠⠂⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠀⠐⠂⡚⠁⠑⠂⠁⠻⠐⠂⠀⠒⠁⡐⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉
⠀⠀⠀⠇⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢲⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠒⠒⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠈⢂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆⡨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢼⠀⠀⠀⠀⠀⠘⣠⣽⣿⠵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣧⣼
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⢤⣤⡀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣄⣠⠤⠄⣀⣀
⠀⠀⠔⡀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠐⠠
⠀⠓⠄⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠊⠐⠃
⠀⠀⠀⠀⠋⠉⠀⠋⠉⠒⠒⠒⠂⢤⠤⢤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠊⠉⠑⠀⠉⢃
⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈
⢇⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠀⠉⠉⠉⠉⠉⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⣴⣀⣀⣀⣀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⠃⣄⡠⠔⠂⠽⠃⢀⡀⠀⠀⠀⠀⠉⠉⡀⢒⡠⠃⣀⠀⠀⠀⠀⢠
⠀⠀⣠⡄⣰⠏⠀⠀⢀⠖⣡⣶⣀⡁⢀⠀⠀⢠⣃⣶⣧⣌⣱⡀⠉⢲⡄⠀⢸
⠀⢸⠀⠀⣿⠀⠀⠀⠰⠒⠋⠉⠉⠉⠉⠀⠀⠈⡉⠀⠀⠀⠈⠁⠀⠈⣇
⠀⠈⢳⠒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠀⠀⠀⠀⠀⠀⠀⡿⢤⡀
⠀⠀⠘⡄⠈⢧⠀⠀⠀⠀⠀⠀⢦⢀⡀⠀⠀⠀⣀⠤⠆⠀⠀⠀⠀⢸⠇⠀⠹⠂
⠀⠀⠀⠙⢆⡈⠳⣄⠀⠀⠀⠀⠀⠳⣌⣉⠉⣁⣠⠞⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀⠠⠐⡲⠀⡀
⠀⠀⠀⠀⠀⠙⠲⠾⠷⢤⣀⡀⠀⠀⠀⠉⢩⠉⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠂⠀⢑⣀⠂⠀⠮⠔⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢶⠶⠤⡶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠑⠀⠀⠤⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⡤⠜⠀⠀⠳⢤⡤⠤⠄⣀⣀⠀⠀⠀⠀⠀⢀⡠⠜⠂⠀⠀⠀⠀⠀⢠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⠢⠤⠤⠔⠋⠀⠀⠀⠀⠀⢹⡇⠈⠉⠉⠁⠀⠀⠀⣀⠀⠀⠀⣠⢄⣉⡰
⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢈⣇⣀⣀⣀⣠⡤⠴⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣘⠀⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠁⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡘⠁⠀⠀⢄⢸⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⢼⠇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠆⠀⠀⠀⢀⠞⢳⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⠓⠦⡸⣄⠤⠛⠊⠀⠀⠀⠘⠀⠈⣇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠀⠇⠀⠀⢹⠀⠀⠠⠤⠆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇
⠀⣀⠴⠒⠭⡁⠀⠉⠉⠉⠉⢩⠛⢧⠀⠀⠀⠀⠀⡟⠁⣉⡉⠭⠥⠈⢱⡄
⡜⠁⠀⠀⠀⠈⢠⡀⠀⠀⠀⠈⣲⠾⡇⠀⠀⠀⠀⡗⠉⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠳⠶⠤⠤⠤⠤⠤⠧⠤⠤⠤⠬⢥⠤⠇⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣀⣀⣀⣤⠟
⣿⣿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⢤⣾⠿⢻⢿⣯⡂⠀⠀⠀⣠⠔⣻⢟⣿⣇⠀⠀⠀⠀⠠⢶⣾⣿⣿⣦⠀⠀⠀⢲⡿⠛⢉⣽⣻⢆⠀⠀⠀⠀⢠⢾⣿⡗⢲⡄
⠀⠀⠀⠘⢧⣄⡘⣾⡿⠃⢀⡀⠀⠙⠲⠼⡿⠿⠃⠀⠄⠀⠂⠀⠈⢿⣷⣯⡿⠇⠀⠀⠘⢷⣀⢻⣿⣿⠀⠇⠀⠐⠂⠨⠿⠿⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⣄⡀⠀⠀⠀⠀⢀⠤⡒⡒⢄
⠀⠀⠀⢠⡀⠉⡏⢆⠦⠀⠀⠀⠀⠠⠖⢸⣏⣷⠂⡀⠀⠀⠀⠀⠸⣿⡿⠀⢙⠢⠀⠀⣀⡔⣿⣿⣿⢛⣶⡄⠀⠀⡄⠸⣿⣿⠆⢤
⠀⠀⠀⠀⠉⠂⠙⠋⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢭⠉⠉⠓⠋⠁⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠤
⠀⠀⠀⢀⡄⢾⡿⡶⢀⡑⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⣀⢠⣴⡶⢒⡶⠔⠂⠀⠀⠀⠢⣄⣰⣶⣤⠀⠀⠀⠀⠤⡄⢰⣾⣶
⠀⠀⠀⠀⡉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠙⠛⠛⠂⠐
⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⣤⢰⣤⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠠⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠐⠀⠘⠷⣏⠤⠄⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠁⠀⠈⠉⠉⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠨⡧⢴⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣏⢉⡿⠀⠀⠲⡏⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣙⣃⣀⣀⣀⣋⣙⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡤⢤⣤⣤⣤⣤⡤⡤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣯⣭⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣭⣭⣭⣭⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣌⠋⠉⠛⣰⣦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠴⡿⢧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⢩⡟⠈⢿⡁⠈⡽⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠺⠇⠀⠸⠛⠓⠷⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⡇⠀⠀⣼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⣄⡀⠀⣀⣴⢧⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡼⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⢠⠏⠈⢻⠀⡀⡜⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⡼⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠃⠀⠘⠋⠙⠓⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠰⣖⠋⠙⣶⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣽⠴⢬⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡤⣤⣤⣤⡤⣤⢤⣤⣤⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡤⠞⠉⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⣀⡤⠞⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⠴⠒⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⣀⣀⡤⠴⠖⠛⠉⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⣀⣀⣀⣀⣀⣠⣼⡤⠤⠶⠒⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡏⠉⠉⠉⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠐⠒⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡠⠒⠈⠉⣀⠀⢀⣀⠄⣀⡀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠚⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠚⠀⡤⠤⠚⠁⠉⠁⠀⠀⠀⠉⠲⣄⠀⠈⠉⠢⡀
⠀⠀⠀⢀⠔⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠄⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡇
⠀⠀⢠⠂⢀⡤⠚⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣰⠃⡀⠀⠀⣀⣀⣀⣀⣀⠀⠸⡄⠀⠀⠀⢸⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠰⡅
⠀⠀⣃⡤⠋⠀⠀⠀⠀⠀⢃⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⢠⠃⢠⠗⠊⠉⠁⠀⠀⠀⠉⠉⠓⢧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⡀
⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀⠀⠀⡏⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⢉⠏
⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠈⢑⢶⠁⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢠⠏⠥⠊
⠀⠀⠀⠘⢄⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜⠋⠀⠦⠀⠀⢣⠀⠀⡀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠳⡿⢦⡀⠁⠂⠤⠄⢒⡁⠀⠀⢀⡴⠋
⠰⠷⣀⠀⠈⠳⢄⡀⠉⠂⠤⠤⠔⠋⠀⢀⡴⠋⠀⣢⣴⠈⢄⠀⣀⣳⣀⠈⠀⠐⠁⠘⢧⣀⡀⠀⠀⠀⠀⢀⡼⠁⠀⠉⠓⠦⢤⡤⠥⡤⡖⠚⠁
⠀⠀⠈⠉⠒⠑⠂⡬⠓⢖⠒⠒⢲⠒⣍⠥⠤⡴⠋⠀⠀⡀⠀⠉⠁⡀⠈⡱⠶⠤⠤⡤⢤⡚⢉⡉⠛⠛⠉⣩⡄⠀⠀⡄⠀⠲⠃⢇⠀⡀⠈⢷⠒⢲⣀⣀⡀
⠀⠀⠀⠀⠈⠒⡞⠁⠀⠀⠁⠀⠁⠀⠘⢖⠀⠂⠀⠀⠾⠏⠉⠉⠹⡀⢠⠃⠀⠀⠐⠁⠘⡄⠀⡹⠒⠋⢡⡤⠑⠚⠉⢁⠠⠁⠀⠈⠉⠀⠀⠀⠳⡞⠀⠈⠉⠙⠟
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠀⠉⠁⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⡘⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⢀⣀⡧⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢶⡉⠉⠁⠢⠤⣄⣈⠝⢏⠉⠉⠉⠉⠉⣇⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠘⢳⠒⠀⠀⠈⠉⠉⠉⠉⠉⢉⡩⠷⣶⠂
⠀⢀⣠⣽⠄⠀⠀⠀⠀⠀⠀⠈⢠⠟⠉⠉⠉⠀⠀⠀⠀⠀⠉⠒⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠬⠇⠀⠀⠀⠀⠼⣤⠤⠤⠤⠤⢵⠖⠒⠋⠉⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡒⠒⠚⠉⠉⠀⠱⣄⠀⣠⠂⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠙⠲⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡤⠀⠀⠀⠀⢳⡿⠀⠸⠿⠀⠀⠀⠀⠐⠒⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠁⠤⠄⠉⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⢤⡀⠀⠀⠀⠀⣠⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠙⣆⠀⠀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠹⡾⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢀⣀⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⠀⠀⠀⠀⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠀⡀⠀⠀⢀⡀⣀⠀⢡⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣠⣠⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⣣⡞⠀⡿⢚⡇⢨⠔⠁⠀⢺⡆⠘⠀⠀⠀⠀⠀⢰⠀⠀⡠⠀⠀⡴⡿⠻⡷⡀⠀⢢⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡟⠁⢀⢀⠀⠈⠀⠀⠀⣀⠀⠈⢹⠈⠀⠀⠀⠀⠀⢸⠀⠈⡠⢖⡊⠀⠀⠀⠀⢘⠢⢤⡑⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⢠⠒⠀⠀⠀⠀⢀⢂⠀⠀⡜⠑⠂⠀⠀⠀⡠⢎⠊⠉⠀⠖⣢⠀⠀⠀⠀⢐⠺⠀⠉⠑⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠸⠀⠈⠚⠀⠀⠤⠀⠀⠛⠀⠀⠃⡠⠀⠀⠀⡜⠀⠈⠦⣄⠀⠈⠋⠀⠀⠄⠀⠘⠋⠀⢠⡠⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⠀⠀⢀⠀⠀⠀⡀⠀⠀⡘⠋⠀⠀⠀⠀⣳⠃⠀⡀⡸⡀⠀⠀⡀⠀⠀⠀⡀⠀⢀⠿⠠⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⡉⠀⢁⡠⠄⠊⠀⠀⠀⠀⠀⠶⠅⣀⠠⠞⠁⠘⠦⣀⠀⠁⠀⠁⣀⠤⠋⠀⠑⠤⢀⠨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⠈⠖⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠽⠉⠉⠙⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠂⠀⠀⠠⠆⠀⢙⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢱⠀⠀⠀⠀⠀⠀⡘⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⡆⠀⠀⠀⠀⢠⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢃⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠸⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡜⡄⠤⣘⡷⠐⠒⠒⠒⠒⠒⠒⠒⠒⢲⡷⢂⠀⡱⠀⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠉⠁⠀⠀⠈⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢤⡤⠀⣀⡼⣄⣀⣀⣀⠠⠔⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠏⡏⠉⠉⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢍⠀⢈⡁⣈⠁⣈⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢅⡀⢉⡀⣌⠀⣨⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠀⠀⡁⣄⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠀⠀⡄⡀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⣀⡠⠖⠉⠁⠉⠀⠦⣠⣐⠂⠛⠀⠀⠀⠀⠀⠀⠀⠳⠐⣂⡠⠔⠉⠀⠉⠑⠤⣠⣖⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠂⠒⠚⠛⠛⠉⠛⠛⠉⠙⠓⠛⠛⠿⡿⢭⣿⣷⣶⣖⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠋⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣠⣲⣶⣶⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠉⠉⠉⠉⠙⠛⠛⠿⢿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⢀⡠⠤⢔⣒⣀⣀⣀⣒⣒⠒⠶⣦⣄⡈⢻⣿⣿⣯⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣴⣿⣿⣁⡴⢚⡥⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣍⡻⢿⣿⣿⣿⡿⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣰⣿⣿⣿⡿⣫⣶⠋⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⡇⠁⠐⠙⢿⣦⡙⢿⠟⠉⠈⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠻⠿⣫⢾⡿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠀⠰⠁⠀⠀⠐⢀⣼⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⠩⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡠⡁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠙⠂
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢊⠁⠘⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠃⠸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠤⢀⠄⠄⠤⢀⠘⠢⠤⠤⠔⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⣀⡄⠇⢣⠀⠆⠐⠀⠘⠀⠂⠄⠐⢢⣀⠀⠀⠀⠀⠀⡠⠋⠉⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠊⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠃⠁⡲⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠁⠀⠀⠀⠀⠉⢢⣄
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠄
⠀⠀⠀⠀⠀⢰⠑⠓⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡈⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡙⠃
⠀⠀⠀⠀⠈⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠑⠦⠄⡀⠤⠚⠁⠀⠀⢀⡠⠊⠀⢆⠀⠀⠀⠀⡠⠀⠀⠀⣠⡣⠀⠃
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⠀⠀⣀⠞⠀⠀⢈⠒⠤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠊⢀⠂⢀⠀⠉⠐⠐⠉⠀⠀⣠⠔⠡
⠀⠀⠀⠀⠀⠉⠁⡠⡀⠀⠈⠁⠉⠀⠀⣀⠕⠁⠆⠀⢈⣋⠉⠉⠰⢚⣛⡉⠀⠀⠆⠐⠷⠴⠏⠐⠒⠤⠐⠂⠉⡰⠄⠀⣰⠄
⠀⠈⠃⠀⠀⠀⠀⠐⠒⠭⡐⣂⡐⠶⠷⠃⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠐⠒⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠂⠀⡀
⠀⠀⠀⠀⠀⣔⡊⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠢
⠀⠀⠀⠀⠀⢰⡥⠄⠐⠻⠤⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠀⠒⠠⠨⠀⠀⠀⠀⠒⠃
⣶⣖⢲⡶⣶⣶⠶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠀⠁⠐⠒⠒⠈⡉⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⠊⡠⠔⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢢⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢀⠀⠀⠐⠈⡠⠐⢀⠐⠢⡀⠠⡀⠀⠈⠂⠀⠣⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠠⠀⠀⢁⠠⠒⠁⠀⠀⠈⠢⣀⠀⢄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⠀⠀⠀⠀⠀⢡⠤⢤⠀⠒⠀⠉⢀⡀⠀⠀⠀⠀⠀⢀⡀⠁⠂⠌⠁⣀⣀⣄⠀⠀⠀⠐⠰⠌⠸⠡⢗⠝
⠀⠀⠀⠀⠀⠀⣘⡑⠆⠏⢀⡠⡄⠀⠀⠀⠀⡇⠊⣺⠀⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⣜⠗⠉⠀⠀⠀⡰⢄⠀⠀⠀⢈⠝
⠀⠀⠀⠀⠀⠀⠩⡁⠀⠀⠀⢰⠀⠀⠀⠀⠀⢱⠀⡌⢃⠀⠀⢆⡰⠀⠀⠐⠤⠔⠀⠀⠀⠅⠅⠀⣔⠀⠀⠠⢄⠔⠊⠀⠀⠑⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢢⠤⠐⠊⠢⡀⠀⠀⠀⡮⠀⢨⠞⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⡙⢆⠀⠈⢆⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠑⠂⠴⠁⢠⠂⢤⠂⡎⠁⠂⢤⠄⣀⠀⠀⡠⠄⡶⠁⠀⢡⠈⠂⠀⠙⡀⠠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢠⠦⣠⠎⠀⠈⡀⠈⢲⠄⣠⠣⣈⠒⠒⣁⠜⣄⡠⠐⢩⠀⢸⠀⠀⠈⣖⠡⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡊⠁⠚⠀⠀⠀⡇⠀⢸⠀⠘⢄⠙⡀⢠⠋⡠⠃⠀⠀⠘⠀⢨⡀⠔⠉⠀⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⠓⡖⠀⢰⠀⠀⡄⠀⠀⠀⠑⡜⠣⠊⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠰⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⢀⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢣⠎⠤⠀⠀⣀⣀⣀⣀⣇⣁⡀⠀⠀⠤⠤⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⢄⣀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⣀⡠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠁⡠⠉⠹⠁⠐⠒⡖⠐⡆⠘⠉⢹⠀⢡⠈⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡰⠁⡐⠁⢠⠃⢀⠇⠀⡇⠀⡇⠀⡆⠈⠀⠀⠆⠈⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠌⢀⡜⠀⢠⠃⠀⡘⠀⢠⠁⠀⠃⠀⡇⠀⡇⠀⠘⡀⠈⢆⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⠁⢠⠊⠀⡰⠁⠀⢠⠁⠀⢸⠀⢰⠀⠀⡇⠀⢠⠀⠀⢃⠀⠈⢣⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⠧⢀⡜⠀⠀⢠⠃⠀⠀⡆⠀⢸⠀⠀⡇⠀⠘⡄⠀⠈⡆⠀⣀⠱⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠈⠁⡲⠃⠤⠀⢠⣁⣀⣈⣀⣀⣇⡀⠀⠧⡄⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡏⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣡⣄⡠⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠂⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⡀⢀⡱

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣴⣧⣼⣼⣷⣾⣼⣼⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣿⣾⣿⣿⣶⣶⣶⣾

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⡀⠀⣿⣿⡇⠙⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡾⠛⠉⠀⠉⠻⣿⣷⠀⣿⣿⡇⠀⠀⠈⠛⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⡇⢠⡟⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢰⠛⣿⠀⠀⡇⣾⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠄⡆⢸⣏⣜⢸⠀⣿⠀⠀⡇⢻⡆⠀⠀⠀⠀⢀⣼⣿⠃⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⡤⠸⣇⡸⢻⠀⢿⠀⠀⡇⠈⠻⣦⣤⣤⣴⡿⠟⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣤⣄⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣀⣀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠟⠛⠋⢉⣉⠉⠀⠤⠄⣀⠈⠙⠛⠿⢿⣿⣷⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⡀⠘⠃⠤⡀⠀⠉⠛⢆⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⠻⡗⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⣀⠓⠤⣀⠀⠈⠃⢤⡀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⠀⠹⣆⠀⠙⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⠐⣤⣄⡀⠈⠢⢌⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡌⠀⠀⠈⡷⣄⡀⠀⠈⠓⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠾⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠞⠀⠀⢀⡔⠁⠈⠙⠷⣤⣄⣀⡀⠀⠐⠓⠠⠤⠄⠀⢀⡀⣀⣀⣀⠠⠤⢴⣶⣄⠴⣦⠘⢿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠙⠁⠴⠖⠒⣀⣀⣀⣈⣉⣉⢉⣩⡥⠶⠶⠊⠁⠤⠔⢣⡄⠀⠙⠛⠉
⣿⣿⣟⣁⣀⠤⠒⢉⣀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤⠀⠉⠁⠀⠀⠉⠁⠀⠉⠐⠒⠀⠀⠠⡄⠖⠂⠀⠀⠀⠀⢀
⣿⣿⣿⣿⡟⢲⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⣷⣤⣤⣤⣾⣿
⣿⣿⣿⡟⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣧⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠒⠒⠀⠀⣉⡤⢤⡍⠀⠐⠒⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⡟⠒⠂⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠐⠺⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠉⠉⠦⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠴⠙⠿⣿⣿⣿⣿⣿
⡿⠋⠁⠀⠀⠀⠀⠀⠙⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⠄⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿
⣷⣶⠇⢀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠰⣤⣼⣿⣿
⣿⣧⣴⡟⢀⡴⠀⣰⣤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢠⣄⠀⣤⠀⢻⣄⣸⣿⣿⣿
⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⡟⠙⠁⠀⠠⠤⠤⠤⠄⠀⠀⣀⠀⠀⠤⠤⠤⠄⠀⠚⠛⠀⢹⣷⣾⣿⣿⣦⣼⣷⣤⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣀⣀⣀⢀⣀⣀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠒⢿⣩⣭⣭⣭⣭⣍⡛⠁⣿⠉⢻⣉⣭⣭⣭⣭⣭⡿⠗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠛⠃⠒⠒⠒⠒⠈⠋⣠⡟⣄⠙⠓⠐⠒⠒⠚⠛⠛⠂⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣂⣀⣀⣀⣀⣒⣠⣵⣾⣮⣑⣂⣀⣀⣀⣀⣀⣒⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠂⠈⠉⠉⠉⠄⠂⠈⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⢀⡀⠀⠠⠤⠀⠀⣀⡀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⡀⢀⠃
⠀⠀⠀⠀⠀⠀⣸⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⢆
⠀⠀⠀⠀⠀⣰⠃⠀⣬⠁⣤⠀⠀⠀⠀⠀⢀⡈⠉⢉⡁⠀⠀⠀⢀⠀⠈⢀
⠀⠀⠀⠀⠀⡟⠀⣎⣠⣤⡤⣷⠄⠀⠀⣰⠧⣤⣄⣀⣈⡀⠀⠀⠰⣀⢀⠈
⠀⠀⠀⠀⢰⡇⠀⠫⡈⠻⢿⠆⡀⠀⠀⢠⡀⠻⣿⣧⡿⠀⠀⠀⠀⠀⡈⠢
⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⣭⣀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⣠⠤⠂
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠒⢒⠒⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⢶⠒⠉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⡀⠀⠀⠀⠰⣉⠀⠀⠔⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡤⠀⣀⠀⠴⠁
⠀⠀⠀⠀⠀⢀⡜⠀⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢦⠀⢀⠘⢧⡐⠁
⠀⠀⠀⠀⠀⡾⠀⢠⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⡄⠀⠀⠳⠎⠀⠀⡱
⠀⠀⠀⠀⠸⠀⠀⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣦⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠘⢦⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠙⠢⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⠄⠘⡞⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⠋⠀⠹⣆⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢰⡏⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣞⣁⣀⣀⣀⡀⣨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⠤⠐⠶⠋
⠀⠀⢀⠠⢖⠫⣌⢩⠉⠁⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡀⠆
⢀⠄⠉⠐⠄⠁⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠊⠀⠀⠐⠂⠄
⡄⠀⠀⠀⠀⠀⢀⡢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠤⠀⠀⠀⠀⠈
⡑⣂⣀⣐⠒⢊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⢂⣀⣄⣀⣤⣃⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣤⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠉⠚⠋⠉⠉⠉⠛⠋⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣻⠿⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠽⠋⠉⠁⠀⠀⠀⠀⡀⣀⣀⣦⣤⣤⣀⣰⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⡟⣯⣿⣧⣿⣿⣿⣿⣿⣿⣽⣓⣴⢏⣠⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡻⣄⣿⣿⣿⣿⣷⣿⣿⣿⣾⣾⢯⣿⡷⣾⣷⣿⣥⠔⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣿⡿⠋⢹⣏⣨⣿⣿⣶⣤⣝⢿⠀⠈⠙⠳⢟⣿⣷⣾⡯⢖⣡⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠟⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢹⡛⢿⣿⡿⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⡏⠀⠀⠀⠀⠀⠙⠿⣷⣿⣶⠿⠋⠀⠀⠀⠀⣀⣰⣯⣛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠚⢂⡴⣠⢒⡀⠤⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣏⠙⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠙⠁⠘⢱⡾⡇⣶⢰⡟⠌⠙⠿⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠛⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⣷⢤⡿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡞⠉⡶⠂⠀⠰⣞⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⣀⡀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠋⠳⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠚⠒⠓⠓⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢧⣄⡀⣀⣤⠟⣦⣄⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣀⡟⠁⠙⣇⣀⢰⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠃⠀⠀⠋⠉⠙⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⢧⣀⡀⣀⣠⠟⢆⣀⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣠⡟⠁⠙⣆⣀⣰⠏⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠃⠀⠈⠛⠉⠙⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣶⠚⠉⢳⡶⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢨⣧⠤⣼⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡏⠉⠁⠉⠀⠉⠉⠁⠉⠈⣽⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇
⠷⠴⠶⠶⠶⠶⠶⠶⠶⠴⠞⠀⠀⠀⠀⠺⠶⠶⠶⠶⠶⠶⠶⠶⠤⠾⠁⠀⠀⠀⠀⠀⠻⠶⠦⠴⠶⠶⠶⠶⠦⠴⠾⠃⠀⠀⠀⠀⠀⠷⠶⠦⠶⠶⠶⠶⠶⠶⠶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⡄⠀⠀⢠⣄⠀⢠⡤⠤⣤⡀⠀⣤⠀⠤⢤⡤⠤⠀⣤⠀⠀⣠⠤⢤⣄⠀⠀⣤⠀⠀⢠⡄⢀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⣧⠀⠀⡜⢻⡄⢠⡎⣿⠀⢸⣧⣤⣼⡁⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⣼⠁⠀⠀⠸⡇⠀⡏⢳⡄⢸⡇⠘⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠙⣧⠀⡇⠀⢷⡞⠀⣿⠀⢸⣇⠀⣨⠇⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠹⣆⠀⢀⡼⠃⠠⡇⠀⠙⣾⡇⠀⠀⠈⣹⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣶⣷⣶⣾⣶⣶⣾⣶⣶⣿⣿⣷⣶⣶⣿⣶⣶⣾⣷⣶⣶⣿⣶⣶⣾⣿⣿⣶⣶⣶⣷⣶⣶⣾⣷⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠒⠲⠶⢤⣄⢠⠴⠋⠉⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠀⣉⡷⠖⠲⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓⠶⣄⠀⠀⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠞⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣄⡀⠀⠀⠀⢷⡼
⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠓⠦⣤⣀⣀⣀⣠⣤⠤⠖⠋⠀⠀⠀⠀⠙⢤⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣾⠁⠀⠙⢛⡗⠀⠀⠀⠘⢧⡜⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⡀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢼⡛⡆⠀⠀⠀⠀⢀⡴⠞⠛⠉⢷
⠾⣍⠀⠀⠀⠘⠛⣻⠀⠀⠀⢠⣇⢀⣠⠖⠊⠉⠉⠉⠁⠀⠀⠀⠀⠘⢉⣀⠀⢀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠇⠀⠀⠀⢰⠋⠀⠀⠀⠀⣾
⢰⣟⠃⠀⠀⠀⠰⣧⢴⡄⠀⢯⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⣀⣠⠞⠁
⠀⣉⡹⠆⠀⠀⣀⡤⠞⠁⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⠛⠉⠉
⠀⠙⠶⠶⠚⠋⠻⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣆⠀⠀⣀⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⣀⡀⣤⠦⡄⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡖⢶⢀⣤⡀
⠀⠀⢠⠾⣯⠙⠟⠀⢹⣄⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⣠⣤⡏⠀⠘⠋⡴⠛
⠀⢰⡏⠳⠈⠀⠀⠀⢸⠍⠉⠓⠶⢤⣀⡀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⣀⣠⠤⠖⠛⠉⠁⠀⣧⠀⠀⠀⠀⠚
⠀⢰⣟⠃⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠙⠓⠶⣤⣙⠳⢦⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠛⣁⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⢈
⠀⠀⠉⠛⠒⢦⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⡧⢤⣉⣉⠉⠙⠉⠉⠉⠉⠉⣀⣼⡖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣠⡤⠶⠶⠶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢸⡀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠛⠋⢉⣹⡇
⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢾⡥⣤⠀⠀⠀⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠀⠀⠀⠀⠀⠀⠀⢰⠋⡀⠀⠀⠀⢶⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⣿⠉⠉⠙⢿⠈⠉⠉⠉⠉⠛⠓⠒⠒⢶⠖⠒⠒⠒⠊⠉⠉⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⢫⡟⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠿⠀⠀⢰⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⢦⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠒⣼⡷⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣾⣀⣤⡤⠤⠤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣄
⠀⠀⠀⠀⣀⣤⡀⢀⡀⠀⠀⣀⣤⡤⣤⡶⣤⣼⡇⠀⠀⠀⢿⣀⣠⣀⣠⠦⣄⠰⡏⠀⠈⠁⢿⡶⢧⢠⡞⠉⠳⣆
⠀⠀⠀⢸⠁⠀⠉⠉⣇⣠⠞⠁⠀⠀⢀⣹⠟⢩⣇⣠⢤⣠⣴⣿⣧⠈⠀⠀⢨⣧⡟⠀⠀⠀⠀⠀⢼⡿⢳⠄⠀⠈⣷⠲⠤⢤⡀
⠀⠀⢠⣞⡀⠀⠀⠀⠉⡏⠀⠀⠀⢀⣼⡏⠀⠈⠉⠀⠺⠶⣿⡇⠀⠀⠀⠀⣨⣿⣧⡟⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣿⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢨⡝⠀⠀⠀⠀⠻⠴⣶⠒⠋⣽⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⣠⡜⠛⠉⠉⢻⣉⠷⢀⣠⠞⠻⡄⠀⠀⢰⣯⣀⣀⣀⣀⡽⠿⢴⡆
⠀⠀⠀⠀⠉⣻⡦⠀⢀⣤⣾⣁⣀⣀⣿⣼⣿⡄⣠⣦⠀⣠⣤⣾⣄⡀⢿⣀⠀⠀⠀⠀⠹⣟⡉⠀⠀⠀⣹⣤⡴⠛⣿⣷⡄⠀⢀⣠⣀⣻⣿⠿⠃
⣀⣀⣀⣀⣀⣛⣒⣛⣉⣓⣀⣈⣉⣀⣀⣀⣀⣙⣋⣈⣛⣀⣀⣉⣈⣉⣉⣉⣓⣦⣤⣤⣚⣉⣙⣓⣒⣋⣁⣘⣁⣀⣉⣉⣉⣉⣉⣛⣁⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠓⠲⢤⡀⠀⠀⠀⢀⡞⠉⠀⠈⠙⢦⠀⠀⠀⠀⣠⠎⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢧⡀⠀⠀⠀⠀⠈⢇⣰⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⠀⠀⠀⠀⢠⣻⡆⠀⣜⣦⠀⠀⠀⠀⢀⣀⠈⣇⠈⠳⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⡇⢸⠁⠘⡏⠀⠀⠀⠀⠈⠛⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⢰⠋⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠸⠶⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⢳⡀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠉⠩⠭⠉⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀
⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠄⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠙⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠇⠀⠀⡆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⡀⠒⠈⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣨⣴⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣥⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢤⣢⣾⣿⣿⠿⠛⠋⣁⣀⣤⣄⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢿⣿⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⣿⡇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠠⠀⠀⠙⢦⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡼⠉⠁⠰⢆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢡⠀⠀⠀⠘⠀⢠⣄⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠤⠀⠀⠀⠀⠈⣀⡀⢀⡀⠤⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⡀⠀⢋⣦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠙⠃⠤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠧⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠰⠀⠈⢰⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⠀⢀⠃⠀⠀⢡⡈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠘⠡⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠂⠐⠒⠒⠂⠈⠉⡅⠀⠠⠀⡀⠁⠲⡀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠊⣥⢀⠀⠡⡀⠀⣠⣤⡄⠀⠀⢠⠁⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⢤⠀⠀⠀⠀⠙⡄⢀⠄⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⣀⠇⡀⠀⠀⠘⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠁⠁⣶⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠒⠒⣷⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⠟⠉⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣀⡀⠀⠀⠀⢀⣀⣀⣠⣤⡶⠾⠛⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡅⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡎⠈⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣀⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠄⠁⠈⠠⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠆⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠃⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡉⢀⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠄⢀⠎⠀⠀⠀⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣤⣤⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣆⣥⣤⣹⣆⣴⣄⣄⣶⣨⣴⣾⣁⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠰⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀
⡄⣦⠀⣸⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⡀⠜⢹⠄
⠇⠈⠀⠘⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⢶⠁⡠⠃
⠐⡀⠀⠈⠉⠉⠳⠋⣀⢹⠄⠀⠀⠀⠀⠀⣀⠀⠈⠁⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠶⣍⠈⠻⠀⠀⠀⠁⠠
⠀⠙⢄⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠹⡌⠲⠀⠀⠒⠈⠉⡵⠈⢱⠀⠀⠀⠀⠀⠀⡇⠀⠢⡄⠀⢠⠦⠌
⠀⠀⠀⠙⠦⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⢤⡀⠀⠀⠱⣄⠀⠀⣠⠎
⠀⠀⠀⠀⠀⡆⠸⡇⠀⠀⠀⠀⠀⠐⣮⣤⡄⠐⠒⠲⠆⣤⣀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⡴⠁⢀⠛⠁
⠀⠀⠀⠀⠀⢣⠀⢻⡀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠱⡄⠀⢀⠎⠀⢠⡎
⠀⠀⠀⠀⠀⠘⡀⠈⢣⠀⡴⠃⠀⠀⠘⠃⠀⠀⠀⠈⠒⠀⠀⠀⠀⢳⡀⠀⢀⠀⣠⠋⠀⢀⡟
⠀⠀⠀⠀⠀⢰⠟⠚⠋⡟⠁⠀⠀⠀⠀⠐⠈⠑⠀⠀⢀⣠⠖⡃⠀⠀⢡⠀⢸⣷⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠈⡆⠀⢠⠃⠀⠀⢠⣤⡤⠤⠤⠤⠔⠒⠉⢀⣼⡇⠀⠀⠘⠢⣼⡃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⢹⡄⠸⠀⠀⠀⢸⣿⣆⣀⣀⣠⡤⠶⠛⢿⡟⠀⠀⠠⣤⡰⠋⠈⠳⣅
⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠻⣿⣿⠿⠏⠐⠒⣰⠞⠀⠀⠀⣰⠇⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣄⠀⠀⠀⠈⠓⠠⠤⠐⠊⠁⠀⠀⢀⡴⠋⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠸⣤⠴⠶⠶⠦⠄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠉⠑⠒⠲⠶⠤⢤⣤⣤⣬⠆
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠐⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⠖⠊⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⣀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠉⠉⡉⠁⠀⠙⠛⢳⠖⠲⠶⠶⠦⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡷⠀⠀⠀⠀⠸⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⠁⠀⠀⠀⠀⠀⣀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⢻⠄⠀⠀⠀⢀⡴⠉⠀⢠
⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⠀⠸⣆⠀⡄⠒⠁⠀⠀⢀⢼
⠀⣘⠡⠀⣀⣉⣄⢀⠀⠀⣀⠜⠒⠉⠃⡀⠀⠀⠀⡀⠃⣰
⠀⠈⠓⠠⠤⠴⠤⠤⠀⠸⢅⠂⠀⠀⠀⠰⠆⠤⠄⡙⠠⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠢⠢⣐⡀⠜⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠰⢀⠐⠀⠤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢰⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⢀⠀⡍⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠊⢀⠈⠊⠡⠘⢄⠀⠠⠀⠀⠔⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣶⡡⠔⠊⡠⠖⠁⠀⠀⠀⠑⢄⠳⣔⠀⠀⡀⠀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠄⠀⠀⠁⠁⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠤⠇⠁⠀⠀⠀⠀⠐⠢⠠⢤⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠉⠀⠘⠒⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣏⣛⣟⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⣤⣀⣠⠶⠛⠋⠉⠉⠉⡿⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⠤⢼⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢀⡞⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣏⡳⠦⢼⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣭⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢤⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣀⠀⠀⢹⣆⠀⠀⠀⠀⠐⢦⣀⠀⠈⠳⣦⡀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣄⠀⠘⢇⠙⠶⢤⣄⣀⣀⠈⠳⣦⡀⠘⣿⡆⣿⠻⣦⣤⣀⡀⠐⢤⡰⣄⠀⠀⣆⢹⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠙⢧⣐⣮⡓⠄⠀⠀⠀⠈⠉⠑⠈⢻⡄⣿⣿⡟⠰⣌⠻⣍⠙⠓⢦⡹⣿⣆⢸⣿⢈⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡄⠀⠉⣻⣮⣥⡴⠶⣤⡄⠀⠀⠀⢀⣿⠿⠋⠀⢀⣀⣁⣉⣃⠀⠀⠻⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠱⡡⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠙⠢⠀⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠙⠲⣄⣀⠀⠀⠱⡀⠀⢠⠴⢒⣻⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡒⠦⣄⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢳⡄⢀⡇⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⣼⣿⠋⢳⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠸⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⠁⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠻⠭⣭⣉⣩⠭⠟⠋⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⡇⠀⠉⠓⠶⢤⣤⡤⠶⠛⢹⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠸⣖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡄⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣼⢉⡷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢰⢃⡞⠁⠀⠀⠈⠙⠓⠶⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢀⡞⠀⠀⠀⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠙⢄⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠉⠉⠀⠈⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠁⠀⠈⠉⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⡂⠤⡀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⢀⠤⢒⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣯⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣿⣿⣿⣿⡍⠙⣿⣿⣿⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣷⣦⣭⣭⣭⡥⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣬⣭⣭⣭⣶⣿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣬⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣸⡇⢙⣾⠘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠪⠀⡗⢣⠞⣿⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠈⠁⢸⠀⠀⡄⣸⡏⣼⡆⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡄⠀⠀⠀⢀⡀⠀⠀⡏⢰⡂⣸⣿⡇⣾⡇⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣌⣆⣡⠀⠀⠳⠉⠀⠀⠸⠀⢾⠁⡿⣿⣟⣿⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠘⠓⠤⠆⠀⢀⣾⢂⣇⣿⣿⣿⣧⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⢾⣞⠸⠛⡟⠛⢻⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠰⣄⢘⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡾⠟⠻⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⠀⣶⣿⢷⣿⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢠⣘⣿⣿⣿⣷⡟⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠄⢜⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⢄⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢉⡟⣛⠿⣿⡿⢿⣿⣤⣆⣈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⡆⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠉⠛⠿⣷⡍⠛⠿⢿⣿⡻⠿⣿⣿⢿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣁⡀⠀⠉⠁⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢁⠴⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠆⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⠉⠉⠂⠀⠀⠀⠀⠀⠚⠉⠉⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⣀⣀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣴⣯⠭⢭⣝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠭⠽⣷⣄
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣆
⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠠⠴⣞⠉⠀⠈⢓⣶⠤
⠀⠀⠀⠀⢀⢜⣿⡿⢿⣶⣭⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⣵⣾⠿⣿⣟⠆⠀⠀⠀⠀⠀⠀⠱⡄⣰⠋
⠀⠀⠀⠀⢀⣼⣹⣳⣾⣿⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣾⣧⡿⣿⣶⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⢾⠀⠀⠀⠉⠉⠛⠛⠯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠖⠚⠋⠉⠁⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠶⠒⠛⠉⠁

⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠇⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠙
⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⡀⢀⡤⠶⠋⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣀⡤⠔⣺⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣍⠙⠓⠒⠒⠲⠶⠶⠶⠶⠒⠒⠚⠋⠉⣠⢴⣾⣿⣿⠋⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠲⠞⠲⠶⣄⣀⣀⣀⣀⡶⠒⠛⠋⠁⠀⠙⡿⠁⠀⠀⠀⣀⣤⠏⢀⡴⠾⠷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⡼⢡⠏⢠⠏⣠⠶⠾⠷⠶⢭⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⡥⣏⣰⣋⣰⠃⣀⣀⣠⠶⢬⣷⣷
⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⠀⣀⠀⠀⠀⠀⣀⣀⡀⣼⠋⠀⠀⠀⠀⠀⠀⣧⡌⠀⠉⠧⠖⠁⠈⠁⣠⣄⡀⠙⡇
⠀⠠⢶⡋⠈⡷⠆⠀⠀⠀⠀⠀⠀⠈⠻⣍⡉⠛⠙⠛⠛⠉⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⢳⠞
⠀⠀⠀⠹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢀⣀⣤⡎⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠉⠛⠲⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣄⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢉⣁⠈⠀⢸⡿⠛⠛⠥⠄⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣠⠒⢄⠀⠀⠀⠀⡖⠒⢦⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⢳⣾⣿⢂⡤⠶⣄⠳⠿⠟⠀⠿⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡜⠛⠀⠈⢀⠀⠀⠚⠁⠀⣀⠀⠀⠀⠋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠴⢆⡀⠀⠉⠑⠢⠤⠤⠒⠉⢀⣠⠋⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⢶⠤⠤⢤⠖⠚⠉⣋⠹⠁⡸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣄⣀⣬⡷⣄⢎⠉⠧⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠉⠉⠉⠀⠈⢪⢳⠄⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢑⢣⠴⠒⠀⠈⢱⢠⢜⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⡞⢸⡀⠀⠀⠀⢸⠸⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢰⠃⠀⠉⠉⠉⠉⠉⠀⢹⣀⣄⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢤⣀⣀⣤⠤⠖⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠈⣛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⣄⣀⣸⣀⣀⣮⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠉⠁⠉⡹⠉⠙⠋⠀⠀⢱
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣒⣒⣒⣋⣈⣓⣢⣴⣶⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⢀⡀⢠⢄⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀
⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣿⣿⣷⣦⣤⣤⣤⣄⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠠⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠂⠃⡜⢁⠣
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣉⡙⠛⠛⠻⣿⠃⠘⠀⠀⠀⢁
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢱⠀⠀⢀⠂
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠕⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⢸⠀⠌⠀⠀⡄
⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠒⠰⠌⠢⡠⠂⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⡕
⠀⣉⣓⠂⠀⠁⢀⡀⠀⠀⠈⢿⣿⠋⠛⠿⠀⠀⠀⠀⠀⠀⠐⢀⣤⣴⣾⠇⠀⠀⠘
⠘⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⣀⡀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠠⠄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⣹⠒⠀⠒⠒⠒⣶⣶⣶⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⠏⠀⠀⠀⠀⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠖⠂⠉⠉⠉⠉⠉⠉⠁⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡄⠀⣀⣤⡤⠶⠶⠾⠿⢷⣶⢤⣄⡀⢸⡇⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⠀⣀⣴⡿⠾⠛⠉⠀⠀⢀⣴⠶⠶⣆⠙⠀⠀⠙⠻⠿⢦⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢀⠟⠉⠈⠉⡴⠞⠉⠳⣆⠀⡾⠀⠀⢀⡈⣧⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡼⠀⠀⠀⢰⠁⢀⣴⠖⣿⢸⡇⠀⣴⡟⢉⣿⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢰⣾
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡇⠀⠀⠀⢸⡀⢸⣿⣶⡿⠈⣧⡀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣇⣠⢤⣾⠉
⢀⡴⢾⠁⢻⠳⢄⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠙⠦⣿⢋⣡⠀⠈⠙⠛⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡆
⢸⡇⠸⡄⠀⡇⠈⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡾⠃
⡏⢱⠀⢷⡀⠰⠀⢱⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠷⠦⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⢣⠈⠇⠈⠁⠀⠀⢸⡶⠒⠒⣄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠈⣧⠀⠀⣠⡤⠤⣼⠅⢠⠔⠛⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠞⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠘⣆⠈⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁
⠀⠀⠘⢧⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⡟⠋⠉⢻⣶⣄⡀
⠀⠀⠀⠀⠙⠶⣦⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡻⠉⠉⢛⡙⠛⠉⠉⠉⢀⣼⠃⠀⠀⠀⢻⡄⠙⢶⣄
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣧⠀⠀⠀⠀⠀⢠⠞⠉⠀⢰⠇⠀⠀⠈⠳⢦⣤⣤⠴⠟⠁⠀⠀⠀⠀⠈⣿⡀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢹⡄⠀⣀⣤⠶⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢱⠖⠋⠀⣸⡇⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⣀⣿⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣀⡤⠞⠙⠷⠶⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⠙⠻⣆⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠘⣧⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠜⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡰⠃⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡀⠹⣀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣀⣀⣠⣤⣤⣤⣤⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⢻⡏⠉⠀⠀⣿⠟⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢤⣄⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣽⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⣶⡶⠶⢤⣤⣤⣤⣼⠀⠀⠀⠸⠶⣶⠶⠶⠟⢻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣠⡞⠁⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⣀⣹⠋⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠒⠒⠈⠉⠉⢉⣩⡭⠿⠷⣤⣀⣀⣿⠀⠀⠀⠀⠀⠀⠈⡆⠈⠉⠉⠙⠆⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡙⠳⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠀⠀⠀⠀⠀⠀⠀⣿⡤⠤⠤⠤⠴⠴⠒⠚⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⣀⣈⣉⣉⣉⣉⣉⣉⣉⡰⠟⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⠤⣤⡴⠶⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠅⢰⡖⢖⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣴⢶⣄⠐⢘⠁⢻⠪⠢⡢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣣⣿⣟⠫⡫⡉⡕⠕⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣆⢈⣝⣋⣁⡧⢑⠄⢌⢊⠪⡢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡦⠀⠀⠀⣤⣤⠀⠀⠃⣠⡇⢸⡦⡑⡔⢌⢆⠢⠠⡑⢔⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡾⢶⠋⣿⣁⣴⠆⠈⠷⣿⠤⡀⣴⢦⠤⡼⡑⢜⢌⠪⡢⡑⢕⢌⠢⣑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠋⢀⢎⣸⡟⣛⣁⠀⠀⠀⢇⣃⠇⡯⠐⢕⢌⢌⠢⡑⢕⢌⠪⡢⡑⡕⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⣿⣿⠿⠟⣿⡆⠀⢰⡖⢗⢖⢋⠪⡢⡑⡕⢌⢨⠢⡢⡑⢌⢎⠪⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⣷⠀⠀⢸⣷⣒⣸⡢⡡⡑⡑⠕⢌⢌⠪⡢⡐⢑⢌⠪⠢⡑⡕⢜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢸⡑⡝⢍⢌⢎⠪⡪⡢⡢⡁⢕⢌⢊⠢⡰⡑⢔⢌⢌⠪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢡⣤⢤⣤⢼⢌⢪⡪⡣⡓⡕⢄⢈⢮⠪⡢⡀⡡⡑⠌⢌⠢⠢⡱⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢸⡧⡣⡪⠢⡳⡑⡕⢕⢌⢎⠢⡡⡑⠕⢌⢌⢎⢊⢊⠢⡡⡑⡜⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⡲⡾⢻⠮⣟⣕⢜⢌⢪⠪⡢⡱⡑⡕⢌⢎⢢⢢⢣⠣⡱⡱⡑⡌⢌⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠊⠘⠓⠑⠑⠑⠑⠑⠓⠑⠑⠜⠘⠌⠊⠂⠃⠁⠁⠁⠑⠑⠐⠈⠂⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡟⢻⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣄⡀⠀⠀⣿⠁⠀⠈⣷⠀⠀⢀⣴⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⡻⣦⣀⠙⢷⣤⡾⠋⣠⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡉⠁⠀⠀⠀⠈⢉⣴⠟⠁⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⢿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠰⣶⡿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⣀⣀⣠⣤⠀⠀⠀⢿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⡶⣶⣶⣿⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠶⠾⠛⠛⠃⢸⡧⣿⡇⣿⣿⣗⠘⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣭⣁⠀⠀⠀⠀⠀⢀⣀⢸⡏⣿⡇⣿⣿⡿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⡶⠟⠋⠁⢸⣇⣿⡇⣿⣿⣿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣴⡟⠀⠛⠛⠀⠘⠛⠃⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⡏⢙⣷⣤⡀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠟⠁⠈⠛⢷⣤⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠛⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠐⠄⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⣆⠂
⠀⠀⠀⠀⠀⠀⢠⢀⢂⡔⡀⠀⠀⠀⠀⠀⠈⠖⢰⠈⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢄⠰⡆⠀⠀⠀⠍⠒⠀⠁⢊⣌
⠀⠀⠀⠀⠀⠀⢤⡜⠈⠃⠇⠂⠀⠀⠀⠀⠀⡈⠨⡄⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⢠⠇⢁⠀⠀⠀⠀⠓⢄⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠇⠀⢈⣄⠀⠑⠘⠀⠀⠈⠐⠀⠀⠀⠀⠀⠁⡠⠣⣀⠀⠀⠆⠁⠀⠀⢀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠈⠐⠄⡀⠀⡘⠀⡠⠋⢈⣡⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠺⡓⠶⠚⠢⡀⠘⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢉⠁⢀⠓⠒⡅⠠⢀⠀⠙⠆⢲⡀⢒⠆⠉⠃⠀⢀⡄⠀⠁⠀⠑⢄⡠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠮⢑⡆⠀⠀⠀⠀⢸⠁⠘⠀⢦⠀⠀⡖⠁⡃⠈⠀⢀⠀⠀⠀⡀⠘⢆⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⡤⣀⠀⠀⡀⠀⠘⠀⠀⠑⠀⢠⡌⡠⠊⠀⠀⠀⢸⠀⠐⠈⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⣁⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡁⠀⠀⠀⠈⠉⠉⠁⡍⠉⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⢁⠖⢠⠆⢰⠀⠦⠀⡦⠰⠀⠒⠀⠏⠡⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠊⢠⠊⢀⠆⠀⡆⠀⡄⠀⠇⠀⡀⠀⠀⠐⡀⠡⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⠁⡠⠃⢀⠎⠀⠰⠀⠀⡇⢀⠀⠀⡇⠀⠀⠀⢡⠀⠱⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠜⢀⡔⠁⢠⠊⠀⢠⠁⠀⢰⠀⢸⠀⠀⡇⠀⠂⠀⠀⠆⠀⠑⡄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡮⢀⡔⠁⠀⢀⠆⠀⠀⡘⠀⠀⠀⠀⡇⠀⢰⠀⠀⠰⠀⠀⡈⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⡠⠞⠀⠤⠤⠇⣀⣀⣀⣀⡇⠠⠤⠦⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⡤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂