Chickpea Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⣀⠠⠄⠋⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄⣰⡃⠀⢠⠾⠁⠀⠀⠀⠀⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣿⢸⣄⣿⣤⠀⠀⠀⠀⠀⠘⣇⡿⠤⣄⡀⢀⣀⣤⡶⠚⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢂⠀⢠⣀⣿⣏⢿⡸⣾⠜⡀⠀⠀⢐⡷⣹⠇⣤⣼⡻⠉⠁⠉⠃⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠿⢶⠶⠿⣾⡛⢉⠁⠙⣦⣕⠉⠀⠹⠦⢞⣩⣶⠋⢰⠏⡩⢀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡠⡾⣆⠀⠀⠀⠪⣻⣯⡀⠘⠀⢸⠋⠛⠒⣿⢛⣿⡶⠷⠿⣧⣧⡊⠀⠀⠀⠀⢀⢸⡆⠀⠀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠀⠳⡰⣄⢀⠘⣿⣶⠂⠀⢀⠀⢀⣠⡿⠋⠁⣠⠀⡄⠀⡹⠛⢶⣇⠀⠀⠈⠸⠇⠰⠆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢁⣟⠄⠈⢸⠀⢸⣏⠀⠀⠀⠜⣧⠀⠀⠀⠀⠀⢘⣿⢠⢸⠀⡔⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠆⢀⠀⠀⠀⠀⠀⡀⣿⠀⣼⡏⠀⣀⣿⡀⣾⡏⡆⠀⠀⠀⠙⠀⠀⢠⠀⠸⡇⣹⠋⢉⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠨⠣⣠⣠⡀⠠⠀⣩⠾⠷⣤⣈⣙⣋⣥⢾⡋⡇⠀⠀⠀⠀⢰⣦⣾⣇⡘⢠⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⠚⠋⠁⠀⠀⠀⠉⠉⠉⠀⠀⠳⣌⠂⠀⡀⡿⢸⢿⣿⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣡⣾⡤⠴⠞⠉







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠄⠀⠒⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠴⠤⢄⣀⠀⠀⠀⠀⢂⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠁⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⡀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠰⡆⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⢹⡼⡇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠒⠦⠤⠀⠄⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⣀⡀⢀⣀⣀⡤⠶⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠠⠀⠀⠀⠘⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⠀⢉⡤⠚⠁
⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠺⠿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠛⠛⠛⠛⠛⠉⢩⡍⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠂⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⢀⠀⠀⠀⠔⢦⠀⠀⠀⠈⠈⠄⠀⠀⠀⠀⡀⡀⠀⡤⠄⠀⢀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠔⠊⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⠦⠄⢀⠤⠄⠈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⢮⠠⠊
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠄⠀⠀⣠⡆⠁⠀⠀⠀⠈⢣
⠀⠔⠋⠀⢀⠄⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠂⠄⠂⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠐⡄⠀⠀⠀⠀⠑⠀⠉⠀⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⡰
⠐⠀⠀⠀⠱⠄⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⠀⡀⠁⠈⠀⠀⡠⠀⠀⠰⠂⠀⠀⠈⠀⠀⠀⢰⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠳⢌⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠘⣀⡂⠐⠀⠠
⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠀⠀⠉⠰⡀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠐⠁
⠀⠟⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠀⠉⠈⠁⠈⠁⠄⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⢀
⠀⠢⡀⠈⠁⠀⡠⠆⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢄⢸⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁⠂⠄⠀⠀⢀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠑⠀⠀⡄⠀⠈⠑⠄⠀⠠⢈⠜
⠀⠀⠀⠑⢆⠀⠀⠀⢠⣴⠀⠊⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠀⡆⠀⢠⠉⠀⠀⠀⢰⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠑⢀⠀⠀⢀⣀⡥⠞⠀⠀⣀⠄⠒⠐⠀⠤⡀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠈⡀⠀⠣⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠁⠄⠐⠂⠀⠀⠀⠀⠀⠀⠄
⠄⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠘⠢⠀⣀⠁⠬⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠓⢄⡀⠀⠀⠊
⠀⠈⠂⠐⠢⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠤⢼⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⢄⠀⠐⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠴⡚⠓⠢⢄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⡠⠁⢠⡁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⡀⠀⢀⠀⠀⠀⠀⠠⠀⠁⠀⠉
⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠠⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠁⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠒⠒⠁⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠤
⣐⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣐⣀⣀⣀⣔⣀⣀⣀⣀⣀⣀⣀⣠⣘⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⡀⢠⠖⢢⢄⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⣾⠀⡤⠈⣰⠼⠷⠒⠛⠉⠉⠉⠑⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠠⡎⢀⢆⡘⠖⠋⠁
⠀⠀⢀⡀⠰⢤⣶⡎⡀⣀⡂⠀⠂⡈⠀⠀⠀⠀⠀⢀⠈⢠⡆⢈⠀⢓⠞⠋
⠀⠀⠀⠀⠂⠄⠀⠐⣉⠙⠓⢦⣤⡇⠀⠀⠀⠑⢠⡀⠀⡄⠻⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠂⠐⠀⠀⠘⠳⣄⠀⢰⠀⡀⠱⣄⣧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠁⠀⠁⠀⠈⢷⡄⠀⠹⡄⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠋⠈⢳⠐⠾⣟⠄⠒⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠃⠀⠀⢠⣬⣩⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⡁⠁⠀⢀⡞⠁⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠹⣦⣤⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⢀⡤⠔⠒⠲⢤⣤⣄
⠀⢀⡞⡡⠊⠀⠀⠀⡨⢷⠞
⠀⢸⣧⡇⠀⠀⣠⠞⠀⢸
⠀⠀⠙⢷⣶⠞⠁⠀⢀⡜⠀⠀⠀⠀⣀⡤⠠⣤⠐⡄⠀⠀⠀⠀⠀⠀⣎⣌⡉⠒⠒⢤⡀
⠀⠀⠀⠈⠻⠶⠤⠴⠋⠀⠀⠀⣠⠎⠀⠀⠀⠘⠒⢣⡀⠀⠀⠀⢀⠞⠁⠘⠇⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⢰⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣠⠞⠀⠀⠀⠀⢰⠃⠀⠀⠀⠐⡄⠀⠀⠀⠀⢁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠴⣾⣶⣶⠿⠃⠀⠀⠀⠀⠀⠈⠑⠠⠴⠞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢈⠢⡀⠀⠀⡜⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠀⣀⡉⠈⠂⠣⡘⢦⠀⠀⢈⠀⢷⡸⠄⠀⢀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣍⡶⠝⠓⠚⡿⣻⡀⠘⠀⠸⠰⠀⢰⠂⣆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣨⠄⠈⠋⠈⡹⢿⣾⡈⠀⠈⠀⢸⠅⡆⠀⠤⠚⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠦⠐⠡⠊⣡⡀⠐⣫⠙⠢⡀⠁⠘⣠⠇⠎⠀⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣌⣩⠤⠜⡡⠄⠐⠘⣻⣿⠁⢠⠀⡏⠀⡠⢁⠄⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡤⠖⠃⠀⡠⠈⠳⣤⣄⣤⠔⢰⠃⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠴⢒⡩⠒⠒⣉⡸⣿⢄⠀⡎⢠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠐⡟⠉⣠⠂⠈⠳⣶⣿⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⡨⠋⢁⠀⠀⢺⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡜⠁⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡀⠻⠀⠀⠐⠖⢡⡦⠀⠀⠒⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠃⣽⡥⠄⣀⠀⡇⠀⠀⠻⡝⣡⠄⠐⠂⠈⡄⠀⠙⣦
⠀⠀⢀⠤⠂⠀⡀⠀⠐⠀⠀⠀⠤⠤⣤⠀⠀⢀⠜⠒⢄⣴⠃⣰⣄⣀⠀⡴⠁⠀⢀⣾⠠⠤⢄⠀⠀⠀⠀⣀⠄⢀⡀
⠀⠀⣮⠁⠀⠀⢨⠀⠘⣢⣠⠀⠀⠀⠚⣇⣈⢙⠀⠀⠀⢹⠎⠀⠀⠙⡏⣹⣄⣤⡟⠠⠆⠉⠳⣁⠀⠀⣴⠁⠀⠀⠈⡆
⠀⠀⢹⣆⣀⣠⡷⠃⠈⠻⢡⡀⠀⢀⣸⣿⣿⣮⡁⠚⣁⡜⢄⣀⣀⣠⣺⠏⠀⠈⠳⣄⡀⠀⠀⣯⠀⠀⢫⡧⡀⠀⣰⣯
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠈⠉⠙⠙⠙⠛⠛⠛⠊⠁⠀⠀⢤⣀⠈⠙⠛⠛⠉⠀⠀⠈⠁⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠢⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢠⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠁⢸⣧⠴⢦⡄⠀⣦⠀⣠⡶⠶⠀⣿⠀⣠⠦⠀⣦⡴⢦⡄⠀⣠⠶⢦⡀⠀⠴⠶⣦⠀⣠⡶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⣿⡾⣏⠀⠀⣿⠀⠈⣿⢰⣿⠶⠾⠇⢠⡴⠶⣿⠀⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠆⠸⠇⠀⠸⠇⠀⠿⠀⠙⠶⠶⠀⠿⠀⠙⠷⠀⣿⠶⠾⠋⠀⠻⠶⠶⠆⠘⠷⠶⠻⠀⠲⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⠂⢸⠁⡄⢉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢠⣄⠐⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⡠⠂⡔⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠒⠂⠀⠈⠙⢎⠁⢀⠁⠀⠒⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡑⠀⠐⠀⠢⣄⠀⢈⣠⠑⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣰⡏⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢀⠀⠈⡗⢶⢄⡀⠀⠐⠛⠘⠉⠉⠀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠀⠀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢚⠀⠀⠱⠈⠢⢀⠀⠀⢂⡤⠔⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠈⢦⡑⠒⠋⠁⣘
⠀⠀⠀⠀⠀⡄⠀⠀⢷⡄⢀⡾⢀⠀⠀⠀⠢⠤⣁⡄⣮⡤⠴⠚⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⣂⢤⡤⠚⠁⠀⠀⠈⠁⠀⠁⠈
⠀⠀⠀⠀⠀⠈⠧⠀⠀⢹⣼⡅⠈⠠⡀⠀⠀⠀⠈⢰⠀⢀⠀⠘⠀⢀⡤⠤⠃⠄⠀⠀⠀⠀⠀⣾⠀⠁⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠽⡃⠀⠀⠀⠂⠀⠀⠐⠘⠀⠂⠀⣀⢤⡏⠀⠀⢀⡎⠐⠀⠀⠉⢸⠋⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠱⠀⠌⠢⣤⠄⠤⢿⠸⠋⠉⠁⢠⣎⠣⡴⠒⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣷⠴⢚⠁⢈⣡⣰⢸⠀⠀⠀⠋⠈⠁⠙⠒⠒⠃⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⢴⣆⣠⠴⠊⠀⠸⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢽⡣⡀⠀⠀⡘⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡆⠀⠀⠀⢰⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠻⠕⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡘⠠⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠈⠲⣄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⠋⠁⠀⠑⢆⠀⠀⠀⠀⠀⠣⠄⣀⡀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠁⠘⢀⣀⣀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⢃⡄⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣛⡴⠂⠀⠀⠀⠀⠲⣜⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⠊⠀⠀⠀⣈⡰⣌⣁⡀⠀⠀⠀⠀⠀⠀⣀⠸⠤⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠠⣄⠱⠶⠤⠀⠀⠀⠀⠄⣐⠀⠈⢀⠁⡄⡠⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡄⢠⠀⠀⠀⠘⠁⠀⠀⠀⠠⠠⠴⠄⠀⠙⠐⠀⠤⠀⠰⠄⠙⠀⠄⠀⠀⠀⠀⠀⠀⣄⣇⡀
⠀⠀⠀⠀⠀⠈⢡⠟⣸⢂⡗⡼⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡯⣭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠊⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢷⣌⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣶⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡌⢳⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠟⢿⢿⢹⡿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠩⠀⠃⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠁⠀⠉⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡠
⠀⠀⠀⠀⠀⠐⣶⠧⣸⡀⡇⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡜⣠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠞⠉⡴⠋⡼⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⡼⠁⣠⠖⠚⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣴⢫⣖⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠃⠘⢧⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠒⠶⢦⡴⢂⣴⠆⠄⠀⠀⢦⣐⡒⡆⢲⠀⢀⡀⠈
⠀⠀⠀⠀⠐⠛⠞⠿⠙⠃⠈⠀⡄⠀⠀⠀⠀⠈⠑⢛⠓⠄⡀⠈⢽⡗⠄⠀⣙⠲⡎⢟⠛⠀⠀⠀⠘⠀⢠⡟⡠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠉⢉⣭⠓⠠⠀⠀⠀⠀⠀⠈⠁⠀⠣⠀⠀⣄⣹⣿⣯⠀⣸⢸⣱⡏⣴⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢚⣛⡣⠀⠀⠘⠁⠇⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⢲⠌⣀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠁⠀⠀⠀⠐⢀⠐⢤⡀⣀⡀⠀⢇⣠⡌⠁⢰⠄⡳⠂⠀⠀⠀⠀⠀⠀⢠⠂⢁⣧⣦⠀⢀⡗⣠⠠⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⡀⠸⠀⠀⠰⣀⠈⢲⠏⣠⠀⠣⠀⠀⠀⠑⠚⠉⠂⠁⠀⠀⠀⠀⠀⠀⠁⠍⠉⢧⠈⠂⢰⠿⠴⣥⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠪⠅⠈⠆⠀⠀⠀⠀⠀⠀⠀⡔⢀⡄⠀⠀⠀⢇⠀⠀⢘⣫⠃⡄⡇⠀⠂⠀⢸⡆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢷⡇⢀⢀⠇⠀⠀⠂⠐⠂⠁⠀⠈⠱⠀⣸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠘⢀⣨⠀⠀⠀⠀⡐⠀⢤⠉⢦⡀⠀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠦⡄⠃⠀⠀⠀⣠⠔⢓⠦⡜⠢⠈⠃⢱⡄⠀⠀⠀⠀⠀⠀⠀⠂⢈⠁⠒⡄⠀⠀⠀⠀⢀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⡆⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⡁⠀⠼⠀⢸⣏⢀⡄⠠⠭⡀⢈⠄⠀⠀⠀⠀⡀⠘⡦⢄⠉⠀⠀⠀⢰⠃⠀⢀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠉⡀⢀⠮⡤⠀⠀⠀⠀⠀⡖⠀⠀⢹⣄⡯⠕⠋⠛⠿⠀⠐⠀⠘⣯⣾⣢⡀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠘⣄⠀⠘⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢠⠈⠈⡛⣦⡴⢋⠉⠛⢶⠋⠉⠳⣠⠀⣐⡣⠃⠝⠁⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣲⣄⣀⠙⣳⠀⠈⢢⡀⢨⣧⣄⠀⢸⡿⣤⡴⠛⠄⡈⣠⣠⡦⠛⠃⠢⠀⠀⠀⠀⠀⡀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⠆⠙⣆⠀⣨⠝⠲⢿⣶⣪⣴⠀⠀⠈⢓⢁⢠⡤⢬⣇⠀⣀⠽⢿⡇⠀⠀⢰⠀⠀⠀⠀⢸⠀⠈⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣶⠠⢿⣴⣇⡀⢠⠌⢉⡉⠙⠆⢀⡔⠺⠿⢏⠀⠀⠉⡟⠡⢄⡀⠸⠤⠀⠟⠁⠀⠀⠀⠈⠋⠴⠥⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠠⣇⠀⠀⠙⣦⢸⡁⠀⢠⠄⠀⡆⠆⢀⠦⡀⠁⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠀⠛⠀⠣⣄⣴⠀⠞⠉⠉⠀⠀⠈⠀⠁




⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢉⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠹⠂⠄⠠⠊⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⠀⠀⠀⣴⠄⢤⠞⠳⠊⠈⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⣀⠠⢾⠀⠘⠀⣰⠄⠤⠓⢄⠀⡴⠓⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠚⠉⠉⠎⠁⠀⠀⡗⠐⠊⣇⠀⠀⡠⠺⡄⠠⠀⠔⠃⡈⠁⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⣦⡀⠤⢼⢶⠟⠘⠓⠠⡞⠋⠀⠀⠉⡆⢈⡠⠈⢌⠋⠱⣀⣠⡞⠂⠐⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⠚⢄⡈⠷⠄⠰⡋⠀⠀⡰⠿⡀⠀⡀⠐⠛⢤⡀⣀⡔⠁⠀⠈⢀⡀⡀⡴⠁⠐⡄⢱⣠⡰⠿⠭⣖⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⣠⡅⠀⠀⢀⠗⠀⢎⠀⡆⠀⠛⠀⠀⠠⢀⠟⠛⠃⢀⡜⠤⡊⠀⡀⠴⢄⣠⠗⠛⠻⡄⠀⠀⣸⣧⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠱⠃⠀⢸⣾⠋⠉⢻⡶⠶⡇⠀⠀⢸⠗⠑⢰⠋⠐⢢⡶⠛⠐⠦⡀⡎⠀⠀⢠⠋⠀⠀⢰⡇⢠⢰⢀⣷⡤⠸⡟⢂⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣴⣾⠉⠀⢀⡌⡀⠀⢈⡖⠙⠃⡀⠤⢠⡆⢈⣠⣅⢀⣠⠀⡘⠓⠀⡴⡊⠁⠁⠄⠀⠙⢶⠋⠙⢧⡇⠀⣡⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⡛⠶⣿⣄⣀⣠⡎⠀⣔⠀⣴⠂⡈⠀⠉⠙⡇⠀⠀⢜⡄⡔⠔⠀⠈⢹⣤⡀⠀⢀⣼⣀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠈⠉⠓⠓⠾⠷⣿⣹⣤⣄⣀⣠⠾⣅⣀⣠⡼⢳⣧⣄⣤⣴⠯⠿⠟⠛⡿⣫⣵⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠰⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠄⢀⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣀⣀⠀⢄⠀⠀⢀⢀⢀⣰⣧⣾⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠾⠿⠿⠿⠿⠟⠛⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠉⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠠⡏⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡍
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠤⠺⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢰⠇⠐
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⠃⢰⠁⢸⠀⠂⠀⡌⠀⠀⠀⠀⠀⠐⠚⠟⠛⡲⣦⣸⣃⣾⠀⢃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢾⣽⣤⠎⠀⢀⠜⠀⠀⠀⠀⠀⠠⡶⠶⠤⣄⡀⢄⠙⢻⡽⠷⠚⠳⠶⠿⣶⡄⠀⠀⠀⠀⠀⡠⠀⠠⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣄⠈⢒⢥⢆⣠⡐⠒⠂⡄⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠘⡍⣑⡀⠌⠉⢦⢈⠉⠉⠂⢀⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⣦⣤⣤⣄⣠⠘⣧⢁⠄⢠⠖⢤⠦⣤⣀⠡⠘⡾⠶⠞⠛⠻⢷⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣠⠤⠿⣿⠿⡷⣿⣿⣇⣘⠘⣎⠷⣄⡀⢂⡘⠀⢀⡀⠈⢁⣀⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠄⠈⠂⣸⡜⠇⠀⠈⡀⢡⢉⣛⣰⣧⡀⠈⠙⠻⢡⣤⣤⠶⠴⠿⠊⠀⢀⠀⠀⠀⣀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠉⠴⠚⠉⠀⠀⠘⢆⠀⡇⡸⠀⢈⡇⠸⡷⢔⠀⠈⡆⢀⠄⠀⠀⠀⠀⡴⠟⢂⠀⠈⢠⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠁⠲⠛⠣⣸⠃⠀⠰⡇⠕⠁⠎⠀⠀⣀⠠⠤⢀⢈⠰⡀⢈⡘⢀⡠⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠦⡸⠗⠳⢾⠙⠀⠀⣸⡙⢀⣳⣯⣔⠡⠄⠀⠒⠒⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⢳⣆⠠⢿⣏⡁⠈⡛⠏⠙⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠸⡆⠀⠀⠀⢜⣃⣁⠀⠀⠙⢷⠈⢮⣀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠐⢄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣣⡀⡄⠀⠀⢀⠟⣛⠑⠮⡑⢤⣀⣲⡄⠈⠉⠙
⠀⠀⠀⠀⠀⠀⢀⢻⠀⠀⣠⠇⠉⠙⠯⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡇⠀⢨⠅⠈⠁⡖⠁⠀⠑⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⡌⡇⠀⠀⡿⣧⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣠⡐⡴⠦⣴⠀⢸
⠀⠀⠀⠀⠀⠀⠃⡆⠀⠴⠃⢋⠄⠀⠀⠀⠱⠀⠀⠀⠀⠀⣠⠁⠀⠀⠈⠣⠈⡀
⠀⠀⠀⠀⠀⠀⠸⣱⠀⠀⠀⠈⠚⢀⠀⡀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠲⠄⢠⣖⠊⠁⠐⠤
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢱⡀⠀⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣠⠤⠞⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠈⡥⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⣢⠜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢤⡀⠀⠀⠀⢠⠖⠛⠉⠉⣣
⠀⠀⡀⠀⠐⠠⢀⠀⠌⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠇⠀⠀⠀⢠⡟⢸⡆
⠀⠀⠡⡈⠳⣄⠈⠐⡁⡀⠰⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⣧⣤⣀⣀⣤⣾⠃⠀⠀⠀⣁⠀⢀⡠⠟⠁⣼⠇
⠀⠀⠀⠈⠁⠁⠒⠣⣤⣠⣀⠀⢠⠂⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠖⠚⠁⠀⠀⠀⠀⠠⠀⡬⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠩⣍⠉⠩⠿⣁⠀⠆⢀⡇⠰
⠀⠀⠀⠀⠁⠢⠤⠀⠂⠀⣈⣶⣦⡈⡰⠀⠀⠄⡊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⢁⣠⠶
⠀⠀⠀⠀⠀⢀⡠⠤⢤⠤⠀⠬⠻⡀⠀⠰⠀⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠉⠓⠡
⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠱⠂⠀⠱⡀⠀⢠⠞⢰⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣟⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⢉⣉⡉⢐⠷⣤⠤⠔⠣⠆⠉⠅⠈⠂⡐⢂⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⢀⡀⡈⠍⢣⠀⠠⢁⡔⢤⠛⡐⠠⠆⡄⠄⠀⣰⣤⠥⠶⠁⠄⠃⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠢⠤⢄⣧⣄⡊⢀⠀⢀⠈⠠⢴⣴⡶⠚⠤⢑⠄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠒⢢⠖⠂⠀⣿⠏⠀⠀⠣⡈⣦⣠⡞⠁⠡⣐⢂⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⠀⠄⠈⢸⠀⣌⠡⡔⢀⡽⡫⠀⢀⡈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠠⠤⡀⠈⡀⠙⢆⢡⡞⠁⠙⠲⠄⠍⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠄⣤⣈⣠⡼⣿⢀⣈⠁⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠒⢋⠀⠀⠘⣿⠇⠀⢉⠳⠒⢒⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠖⠊⢦⠏⠁⠀⣼⣠⠄⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡏⠀⠀⠀⡿⡅⠀⠠⢤⠤⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠈⠀⠀⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠂⢄⣐⣤⠏⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣟⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣻⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡆⠠⠂⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠲⢊⠶⠐⠃⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠉⠤⢠⠆⠒⠉⠀⠉⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠃⠠⢦⡥⠞⠀⠐⠁⠀⠀⢹⣠⠋⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⣈⢢⡄⠀⠜⠐⠀⠀⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠉⠀⢀⠀⠀⠀⠏⠁⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡎⠉⠘⠀⠀⢰⣄⠠⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⢠⢀⡀⠋⢒⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢳⣌⢳⠎⠡⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢫⡧⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡜⠂⡀⠀⠀⢁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡎
⠀⠀⠀⠀⠀⢀⡀⢀⠀⠓⠀⠊⠀⠀⠉⠀⠈⢀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⣤⠀⠀⠐⠴⠐⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠄⠀⢠⠤⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠺⡀⠔⠛⠒⠒⠣⡀⠀⠀⠀⠤⡀⠈⠏⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠈⡀⠀⢀⡠⠁⢀⣀⡤⠴⢶⣶⡶⢶⣤⣤⣧⡤⢄⠀⠠⡈⠛⢢⣄
⠀⠀⠀⠀⠀⢠⣋⣖⠈⠉⠀⣨⣴⠟⠉⠀⠀⠀⢘⣏⣗⣚⡛⠛⠛⠙⠾⣗⡂⢆⠀⢎⠙⣦
⠀⠀⠀⠀⢀⡎⠀⠀⠀⣰⣾⡿⠁⠀⡀⠀⠀⡂⠭⠗⠂⠀⠀⠈⢐⠠⣴⣾⣷⡌⠒⠚⠀⠙⡆
⠀⠀⠀⠀⢘⠦⣀⡄⣰⡿⡡⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢮⡇⢹⡀⠸⠒⢶⢿
⠀⠀⠀⠀⢸⠀⠁⢠⡿⣾⠇⠀⠀⠀⠀⠀⠔⣡⠖⠛⠛⠛⢶⣄⠀⠀⠀⡈⣇⠀⣇⠀⠖⠺⢤⡇
⠀⠀⠀⠀⠸⡄⠀⢸⡗⢣⡀⠄⠤⠤⠀⠄⢠⡏⠀⠀⠀⠀⠀⢿⠀⣇⣀⡧⣿⠀⢿⠀⠢⡀⢈⡇
⠀⠀⠀⠀⠀⠈⠉⠚⠿⢭⡟⠉⠛⠉⡻⠛⠉⢁⠀⠀⠀⠀⢀⠼⠀⠈⠉⠃⡿⠿⢾⠀⡐⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⠤⢄⠀⣰⠗⡀⠀⢐⡗⠀⠀⠀⠀⣸⢻⣧⡿⣀⣈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠳⣌⠪⣽⠃⠄⢀⠉⡉⢹⣆⢀⠔⣱⣿⣿⡿⠤⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠑⢤⣉⡓⠚⠒⠛⢉⣉⡴⠊⣿⡿⢛⠚⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣌⠙⡿⠿⠉⡯⢋⣽⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢦⡤⣤⠔⡡⠻⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠽⠏⠀⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⢠⣄⢠⡩⠓⢲⠶⠆⠀⠀⠀⢀⠒⠈⠁⠒⠁⠀⠀⠀⠀⡶⠲⡶⣶⢶⡂
⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⡔⢢⣶⠄⠀⠀⠀⠀⠠⡤⠀⠠⡤


⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⡒⡂⢀⣀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠁⢘⠆⠀⠀⠀⠀⠀⢀⠐⢀⠬⠈⠀⠀⠀⠀⠀⠀⠉⣠⣮⣃⠀⠀⠀⠀⠀⠄⡀⠀⡈⡄
⠀⠀⠀⠡⢀⡈⠆⠀⠀⠀⠀⠀⠸⠴⠄⠀⠐⠊⠀⠀⠀⠀⠀⠀⠏⠂⠇⠀⠀⠀⠀⠀⠀⠁⡠⠀⠁


⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⢲⡆
⠀⠀⠀⠰⣍⣹⠂⠀⠀⠀⠀⢀⡰⢃⠘⠿⠃⠸⠀⠀⠀⠰⣶⠶⣢⣶⠶⢾⠀⠀⠀⠀⣸⠀⢦⡆⢸⡇
⠀⠀⠴⠆⠾⠷⠰⠦⠀⠀⠀⠀⠙⠙⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⠤⢼⡇⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⡤⣀⢶⠆⣤⣤⡀⠀⠀⠻⣥⡦⠶⠶⢶⡄⠀⠀⠀⠀⠀⣴⣦⣶⡄
⠀⠀⣿⣿⣿⡑⠀⠀⠀⠀⠀⠀⡔⣿⡇⠀⠘⢽⡇⠀⠀⠀⢸⠟⣤⠛⢠⢇⡀⠀⠀⠀⣘⣼⣛⣻⣌⡂
⠀⠀⢿⠿⢿⠇⠀⠀⠀⠀⠀⠐⠙⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠿⠴⢷⡆⠀⠀⠀⠛⠛⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠴⠒⠒⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡏⠙⠉⠀⠀⠀⠀⣶⢲⠏⢹⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⠢⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠉⠓⠒⠒⡻⠛⠉⠉⠉⠉⠑⢮⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⠤⠤⣄⡀⠀⢱⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⢁⡴⡶⠄⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⣭⣯⠇⢠⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠈⠳⠤⠥⠴⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠠⠃⠔⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠁⠠⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠝⣯⢻⠝⣯⠿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡟⣲⢻⡞⣳⠛⣶⠻⡦⡤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣉⢏⣹⣏⡹⣉⣹⣏⠑⡞⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡿⠶⠷⠾⠷⠾⠶⠿⠴⣯⠯⡨⢏⡹⡇⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⢸⡇⢰⣲⡂⠀⠀⣖⣖⠀⣿⠛⠛⠚⠛⢻⣐⣲⣲⣒⣇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣈⣀⡀⣠⢀⡀⣀
⠀⠀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠚⠂⠀⠀⠛⠒⠀⣿⢸⣿⢽⠀⢼⡐⢲⢣⠃⡜⠉⠉⠉⠉⠀⠉⠀⢄⡀⠀⢤⡀⠀⠠⣀⠀⠢
⣀⡜⠆⠰⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣛⡋⢹⡋⣛⣳⠀⣯⠀⢀⠀⠀⢹⠬⠾⠷⠧⢼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠙⢦⣀⠈⠳⣄
⠈⠁⢀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠚⢻⡇⠀⣉⣏⣹⣉⣙⣹⠀⡿⠀⣖⣾⠄⣿⠀⣀⠀⠺⠠⠀⠀⠀⠀⠰⠒⢀⡀⠀⠀⠘⢦⠀⠙⣦⡀⠘
⠀⠈⠉⠉⠉⠉⠛⠦⣤⡴⠒⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠁⠀⡇⠀⠈⠀⠀⣿⠘⠙⠃⠀⢀⠀⡀⠉⠄⠀⠀⠀⠂⠀⠀⠀⠈⢳⡀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⢀⣀⠀⠀⠼⠷⠲⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠛⠚⠒⠒⠒⠚⠶⠶⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠴⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠈⢍⠜⢧⣄⣀⠘⣷⠀⢹
⠀⠀⠀⠀⢀⣠⡴⠁⠀⠀⠈⢳⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠦⠮⠦⠾⠄⠀⠉⠙⠣⢈
⠀⣀⠴⠚⠉⠀⡇⠀⢀⡀⠀⢸⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈
⠊⠁⠀⡞⠉⠙⠧⠀⠫⠠⠇⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣛⣲⡆⠠⠀⠄
⠀⠀⣀⣷⡀⠀⠀⢀⡤⠤⡄⠀⠀⠠⠾⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⢄
⠀⡼⠁⢀⣤⣤⡀⠈⠓⠚⠁⠀⡤⠴⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⢷⠀⠈⠢⠜⠃⠀⠀⢀⠤⢄⠈⠀⠁⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠑⢤⡀⠀⠀⡀⣀⠘⠤⠜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠸⡄⠀⠘⢍⠹⠃⠀⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⣀⣀⣀⣄⣀⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠤⠄⠂⠁
⠀⠀⢀⣄⠙⢦⠤⠤⣤⠶⠋⠀⢉⡉⠥⡤⠀⣠⡀⠀⠀⠀⡀⢀⠄⠀⠀⡽⠠⣦⠄
⠀⠛⠐⠗⠈⠁⠀⠀⠀⠀⡆⠀⠘⡃⠚⡟⠈⠀⠀⠏⠈⠠⠇⢨⠀⠙⠰⠇⠘⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣀⣤⣠⢀⠀⣠⠄⢠⢄⣄⡤⣠⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⣀⡀⣀⢄⣄⣀⢠⠀⣈⠄⣀⣀⢠⢠⣠⣄⣀⡠⣠⠀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣶⣶⣷⣶⣾⣶⣶⡆⠀⠀⠀⠠⣤⢀⣹⣿⣅⠠⣤⠀⠀⠀⠀⠀⣦⣤⣤⣀⣯⠗
⠀⠀⠀⠀⢸⣿⣿⣏⡲⢽⣿⣿⡇⠀⠀⠀⣾⣿⣾⣏⡩⣿⣷⣿⣦⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣷⣭⣼⣿⣿⡇⠀⠀⠀⠛⠛⣿⣷⣵⣿⣿⠛⠋⠀⠀⠈⠙⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣤⣶⣳⣦⣤⣀⠀⠀⠀⠀⠀⣀⣈⡋⢀⣄⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⢘⡟
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⠸⣿⣸⡇⠘⠚⠁⠀⠀⠀⠀⠀⢘⣟⢰⣷⢸⣿
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⢶⣌⡉⠤⣶⣦⣀⡀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿
⠀⠀⠀⠀⣴⣿⣶⣿⣾⣷⣾⣷⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠇⠀⠀⠀⠀⠾⠿⠾⠿⠾⠷⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠂
⠀⠀⠀⠀⢰⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢤⡀⣠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡇
⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣶⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⠃
⠀⠀⠀⠀⡀⠒⡒⢀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⢿⣿⡿⠛⠻⠛⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠰⠖⠞⠦⠢⠔⠀⠶⠂⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠃⠂⠃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣍⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⣛⣋⣉⡭⣯⡭⣍⡽⣛⣻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⢽⡂⣿⢹⡿⠋⣿⡿⣨⡾⠟⡾⠿⣿⣇⣶⣭⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡫⣪⣾⡿⣧⢿⡸⢹⣿⡟⢱⣿⣿⡶⡏⣾⣥⢘⣛⣯⣿⢧⠝⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣽⡶⣛⣴⣼⣿⣌⣳⠜⣿⣏⣷⣭⡷⣻⢧⣿⣭⣿⣧⢉⣶⣷⡾⣤⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣟⡚⢛⡿⢻⣥⡭⣷⣟⣻⢯⣽⣾⣥⣷⣶⣶⣶⣯⣽⣿⣯⣵⣶⢶⣦⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣝⣻⠾⢖⣻⡭⠶⢛⣭⣥⣿⣿⢹⣿⣿⢆⣭⣭⢛⣌⣻⡽⠿⣽⣃⣿⡃⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣝⢿⣿⣾⠟⡿⣏⣿⣿⢟⣭⡏⣭⣿⣭⣝⡻⣿⣿⣿⣿⢸⡟⣮⢿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⢰⣿⢏⣋⡿⣻⡽⣋⣻⠟⠿⢿⣬⣟⣋⣙⡾⢷⣝⢷⣮⢥⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠿⣾⣿⣷⢽⠷⠿⠿⢿⣿⣟⣎⣻⣿⣿⠿⠿⠟⡼⣫⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠰⡀⢠⡀⠀⢀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣤⡨⢉⣱⢀⠂⣀⠿⠶⠮⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣄⠀⠀⠘⡟⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⡇⠀⠀⠴⢇⡆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠁⣼⣿⣿⡏⠀⢸⣿⣿⣿⣿⣇⣀⣘⣇⠀⠀⠀⠘⡇⠐⠚⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣰⡀⠀⣿⣿⣿⡇⠀⢸⣏⣿⣿⢿⡏⠁⠀⠀⣀⡤⠶⠾⠧⣄⠀⢸⡌⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⡀⠉⣛⣛⠿⢿⡇⠀⠘⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡇⢾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿⠉⠐⠀⢭⠓⠺⡇⠀⢸⣿⣿⡇⢸⣇⣀⣀⣤⣠⠀⠀⠀⠀⢀⣀⣼⠃⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣟⠟⢀⠄⠀⡀⠀⠈⠂⠈⢆⣸⣿⣿⡇⢸⡏⠀⠀⠀⠈⠀⠀⠰⠶⠾⠉⠀⠠⡏⠀⣷
⠀⠀⠀⠀⠀⠀⢀⡿⡧⠦⡄⠀⠈⠄⠀⠀⠀⢾⡄⠀⢹⡇⢸⡇⠀⠀⠀⢀⡀⣠⠇⠀⠀⣀⣀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠎⠀⠀⡀⠀⠀⠈⣿⠀⠈⡇⢸⣇⠀⢀⣀⡰⠛⠁⠀⠀⠉⠁⠈⠙⡄⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠂⠉⠐⠀⠀⠀⠨⡄⠀⠀⢸⡟⠉⠉⠉⠻⢄⣀⠀⣀⣀⠆⠀⣸⠇⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⢰⢶⣺⡆⢐⣀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡄⠀⠈⠉⡉⠁⠀⢸⡏⠐⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢸⠿⣿⡇⠀⠁⠀⢀⠀⡀⢸⡇⠀⠀⣸⠋⠓⠚⠛⠙⠓⠒⠋⠙⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣘⠲⢼⠁⠀⠀⠀⣾⣾⣿⣾⡇⢀⣴⠋⠀⣠⠷⣤⠞⠉⠙⢦⡀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⡖⣦⣉⠞⢀⡾⠀⢰⡅⣹⣿⣿⡷⠛⠓⠶⠚⠁⠀⣸⠇⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣹⣶⡞⣥⣷⣿⣿⣿⣿⣿⡇⠀⣀⣰⣦⡴⠟⠋⠀⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣿⣽⣿⣿⣿⣿⠃⣟⠚⠋⣉⠀⢀⣴⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠈⠒⠶⠟⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⣴⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠁⠈⠉⢁⣘⣿⣿⣿⡟⠛⠀⠀⠀⠀⢀⡀⢀⣾⡿⠛⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⡇⢸⡀⠀⣠⣤⠘⠓⢸⣿⣶⣿⡶⠆⠁⡀
⠀⠀⠀⠀⡠⠐⠀⠈⠀⠀⠀⠀⢀⡀⠀⣀⡀⢀⣿⣿⣿⣿⣄⣀⣀⣿⣿⣿⣿⣧⣾⣄⠀⠙⣿⣷⣤⣤⣿⣿⣷⣶⣶⣗⣀⠄
⠀⠀⢰⣏⠀⣾⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⡟⣿⣥⣶⣿⣿⣿⣿⣿⣥⣤⣀⣿⣿⢰⠀⣿⣿⣯⣯⠰⢀⣴⣲⠖⡃
⠀⠸⠿⠯⣭⠄⠶⢷⡨⠉⠀⠀⠀⠀⠉⠉⠀⠉⢙⡛⠛⠛⠋⡛⡛⠋⠉⠙⠋⠉⠙⠋⠛⢛⠉⡉⠀⠀⠉⠉⠙⢃⠀⠁⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢀⣂⡀⢁⣀⠀⠠⣀⢀⢠⣠⠄⡁⣄⡀⡡⣀⣄⠤⠀⠀⠀⠀⢠⠀⢀⠀⠀⡈⡀⠄⠄⢀⠀⠀⠀⠠⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠘⡧⣥⠙⢠⠶⠁⢹⣇⡛⡒⣍⣳⡇⢠⣰⡗⢁⡺⣔⡧⠀⢰⠉⠋⢃⠛⣀⠇⢀⣴⢏⠉⡻⠰⢏⡔⢩⣬⣧⣘⡀⠁⢐⡄⢷⡃⢂⣀
⠀⠀⠠⣿⢸⢡⣞⠞⢠⡈⢻⡄⣰⣍⡏⢸⠊⢹⡇⠪⡽⢙⡿⣶⡆⠙⢰⢢⠟⠟⢠⠟⢙⣶⣋⡁⠜⣿⣌⡹⠄⢰⡴⣄⢠⢼⠁⢻⠿⠉⡉
⠀⠀⠐⣿⢸⡷⣻⠨⣄⣽⢿⡀⣷⣌⡷⠲⡗⢸⣧⣸⡵⠌⣃⢤⣹⡣⡆⣼⢖⠃⢰⡖⠁⠹⢘⢹⣇⣉⣮⠁⢈⡑⣚⣉⠙⣀⢠⡼⠳⢒
⠀⠀⠤⣯⣸⡇⣻⢰⡇⠀⣿⠂⠻⢮⣧⣈⣟⢼⡅⣽⣟⠈⡗⣘⢩⡅⠜⣩⣻⡇⢸⡇⠾⢧⣄⢨⣿⣡⢯⡿⠸⡷⣩⣯⡵⠈⠟⠶⠃⡤
⠀⠀⠠⠿⢤⣀⣹⡈⣁⠃⣿⠐⠶⢓⡷⠼⠇⢠⡇⣺⣁⣘⣇⠐⢱⣍⡎⠉⢹⡃⢜⡃⠶⡫⠅⢼⣯⠘⢡⣿⣸⡟⣩⣟⡛⢘⡟⠸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠚⠓⠦⠤⢤⣧⣤⣄⣘⡇⠈⠁⢸⣟⠀⠉⠛⢘⡱⠈⠥⠀⢸⣟⠀⣀⣷⣸⡏⣰⠏⠋⢸⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠓⠶⠦⠤⢼⣅⡀⣀⠀⠰⡿⠀⠀⠁⢸⡏⠁⠑⠒⢺⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠓⠲⠰⠦⠬⠧⣀⡀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠉⠉⠀⠀⠉⠉⠙⠻⢿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠺⠿⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠙⣿⡷⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⢰⣿⣧⠀⢸⡇⠀⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⣧⣶⣿⠟⠀⠀⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉
⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⢿⣿⣷⣦⣄⣀
⣴⣿⠛⠿⢿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣶⣤⣀
⢻⣿⣤⣀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣷⣶⣦⣤⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡛⢿
⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣷⣶⣶⣶⣶⣶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠈⢁⣾
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⡿⠿⠿⣿⣿⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠻⣿⣶⣤⣾⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡴⠋⠁⢀⠀⠉⠳⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣼⠀⢠⡞⠉⠙⠂⣰⠛⠉⢀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⢦⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠟⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣄⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⢀⣠⠛⣦⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠈⠉⠀⠀⠀⠀⢸⠀⠀⢠⡄⠀⠀⣤⠀⡇⠀⠀⠈⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠀⢸⠀⢀⣟⣹⡄⠀⣿⠀⡇⠀⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⣄⣀⡴⠛⠛⠀⢸⠀⠾⠁⠀⠳⠀⠿⠀⡇⠀⠛⠛⢦⣀⣠⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⣀⣀⠀⠸⠦⠤⠤⠤⠤⠤⠤⠴⠇⠀⣀⣀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠉⠉⠀⢀⡄⠀⠀⢀⡀⠀⠀⢠⡀⠀⠉⠉⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠀⠀⠀⢀⣸⠇⠀⠀⢸⡇⠀⠀⠸⣇⡀⠀⠀⠀⢿⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣦⠀⠀⠈⠳⠞⠁⠀⠀⣴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⡄⠀⠶⢦⡀⢰⠆⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⠳⠶⠗⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠘⠃⠈⠳⠆⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠛⠶⠀⠀⠀⢶⠲⣦⠀⠻⣤⣤⠄⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⣴⠛⠓⠀⠀⠀⠀⡞⠃⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡤⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡦⢀⡤⠶⡄⠀⠛⠀⠀⢰⡖⠘⠁
⠀⡶⠶⣤⡀⠀⠈⢷⣤⡀⠀⠀⡀⠉⠀⢦⣤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢿⣀⡳⠆⠀⢂
⠀⠀⠀⠈⠀⠀⠀⠀⠸⠟⢲⡌⣷⠖⣧⡀⠖⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣡⡶⠀⠈
⠀⠀⠀⠀⠀⢀⣀⢰⣆⣀⣦⠀⠹⢤⠅⢙⡧⠀⠁⠀⠀⠀⠀⢴⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠘⢧⣀⡉⣭⡁⢠⡀⠀⢹⡇⠀⠀⠀⢀⡀⠀⢀⣨⣽⣾⠃⠀⠀⠀⢀⣀⡀
⠀⠀⢀⣀⡸⠛⠛⣯⠀⠈⠉⣉⣥⡀⠛⠁⠀⠀⣀⣨⠏⣤⠙⢃⠀⠋⠀⠀⠀⠀⣸⡍⠹⠃
⠀⠀⠙⠉⠙⡷⠀⠀⠀⠀⠸⡯⠀⠀⢻⣅⡀⠀⠉⠁⠀⠉⠛⠛⠀⠀⠀⠀⠀⢾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣰⠞⡛⠀⠀⠀⢠⡴⠒⡦⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢦⡀⠀⠀⠀⠀⣀⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠠⣦⣤⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣶⡂⢈⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣤⣤⡀⠈⠙⠛⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⢋⡴⠛⠓⠛⠉⠙⠃⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⣿⣤⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠉⠀⠘⠃⠾⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠉⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⣿⣿⣧⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠛⠛⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠋⠙⠁⢠⣿⣿⣿⡟⠀⠀⠀⣀⣤⣴⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⢿⣿⣿⣿⣄⣸⣿⣿⣿⡿⠋⠀⣠⣶⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⠛⠛⠛⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⡇⢸⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⢸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠄

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⢄⣀⠀⠒⠀⢄
⣀⣀⣀⣀⡀⢀⡀⠤⠤⠤⠤⠄⠀⠀⠁⠒⠒⠒⠒⠒⠒⠐⠒⠒⠂⠙⠛⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠂⠈⠙⠉⠉
⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠁⠉⠁⠂⠀⠁⠀⠀⠢⠄
⠀⠀⠀⣴⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠤⠶⠚⠛⠃⠑⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⠀⠰⠶⣤⠠⠴⠚⠈⠉⠁⠀⠈⠁⠈⠉⠂⠀⠀⠀⠀⡈⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠤⠤⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠛⣛⡋⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠄⠤⠒⠒⠂⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠰⣾⣿⠟⠀⣿⣿⣿⡄⠀⠀⠤⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠠⠀⠛⠛⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠄⠀⡠⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠠⠀⡤⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⡀⠀⢠⠄⠄
⠀⠀⠤⠀⣇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⢀⡀⡀⠀⢰⡀⠀⠀⠀⠀⣨⡇⠓⠀⠘⠄⠀⠀⠀⡄⠐⠠⠀⠏
⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⢺⡀⢀⡞⠀⠀⠀⠀⣀⠀⠉⠀⠠⠤⠞⠀⠀⢀⠀⠀⠘⠲⠄⠲⠃⠐⠀⠐⣆⠀⠀⠀⡀
⠀⢉⢁⠀⠁⠐⢶⣤⡤⠱⡾⠃⢈⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠐⠀⠐⠌⠁⡴⠊⠀⠀⠰⠉⠐⢢⠀⠈⠓⠴⠴⠁
⠀⢠⠴⠠⣄⠀⠀⠠⠀⠈⢠⠂⠀⠙⠉⡥⢀⠁⠀⡏⠈⠈⠙⢦⡐⠂⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠘⠀⠁⠀⠑⠈⠀⠀⠀⠈⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠹⠄⡀⢀⠄⢀⡀⠀⠊⢀⡠⠶⡄⠙⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠈⠠⣄⠙⠒⠰⠊⠀⠠⠤⣄⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠐⠓⠋⠀⠉⠙⡀⠀⠀⠀⠐⠐⡄
⠀⡀⠀⠀⢀⠁⢀⠀⠀⢀⠏⠀⠂⠀⠀⠀⠀⠀⡍⢀⠤⡄⠀⠀⣆⠀⠀⠀⠀⠲⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠃
⠀⠂⠸⠀⢄⠄⠀⠉⠀⠁⠤⠠⡁⠐⠦⠄⠀⠚⠀⠀⠀⢰⠀⡁⠈⠁⣀⣀⡠⠘⠃⠠⠴⠒⠀⠀⠈⠤⠴⠂⠀⡀⠅
⠀⠀⠀⠀⠴⠒⠒⡀⢀⠀⠀⠀⢉⠀⠀⠀⠀⡀⠀⢀⠤⠈⠀⠊⠀⡀⣀⢤⡀⠀⡠⠤⢀⠀⠀⠂⠄⠀⠀⠄⠀⠐
⠀⢰⠇⠀⠀⠀⠀⠃⠐⠀⢀⠀⠋⠐⠓⡀⠀⠀⠀⠒⢲⡀⠀⠀⠀⠀⢀⠠⠛⠄⠃⠀⠀⠱⠀⠀⢠⠀⠀⠀⠀⠒⣄
⠀⠘⡠⠂⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠄⠁⢀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⣰⠀⠀⠈⠳⠀⣀⣈⠌⠃
⠀⠈⠳⠖⠂⠁⠰⣀⣀⠀⡴⠃⡀⠀⠀⢢⠀⠉⠉⠈⢀⠀⠢⢤⠤⠀⠀⠋⢀⢀⣀⠀⠈⠁⠚⠀⠠⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠒⣄⠀⠀⠀⠀⠈⠀⠒⠁⠀⢠⠀⠀⠀⠰⠁⠀⠀⢠⠁⠀⠏⠀⠈⢠⠀⠀⠂⠜⠂⢀⠖⠳⠀⠁
⠀⠧⢄⠀⠀⢀⡌⠃⠀⠀⢀⠀⠴⠢⠀⠠⠀⠚⠀⠀⠀⠘⢄⡠⠀⠘⠀⠀⢂⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠄
⠀⢀⠀⠀⠋⠉⡄⠀⠀⠀⡎⠀⠀⠀⠀⠒⡀⠀⣀⢈⠃⠀⠄⠀⠀⠀⠇⠀⠠⠀⢠⡐⠉⠀⠈⠲⠀⠀⠀⣠⠀⠀⠁
⠀⠁⠀⠀⠀⠀⠐⠄⠚⠀⠲⠤⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠈⠒⠘⠁⠈⠓⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⡠⡀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡭⠟⢉⣁⠀⣠⠖⠦⣝⡎⠑⠢⡄⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢢⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⠐⠈⠲⣤⣯⣄⡀⠇⢀⣀⡘⠃⠀⠀⢹⡀⠒⠚⠒⠦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠠⣄⠀⠀⠀⠀⠉⠛⢬⡉⠙⢳⣄⣀⣀⣀⠀⣸⠀⠀⡈⠀⠂⡀⠀⠀⡔⠐⠂⠄⢴⠀⠈⢠⡅⠀⣴⠟⠁
⠀⠀⠀⠀⠀⡠⠊⠢⢤⠀⠀⠘⣆⡤⠒⠀⠀⠀⠰⠋⠙⡆⢹⡄⠀⠈⠻⠇⢄⠀⣻⠔⠛⠃⠀⢠⠀⠀⠀⠀⠀⠂⠄⠒⠀⠈⠛⠠⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠤⣤⡀⠀⢀⡏⠀⡞⠀⠀⠀⠀⠒⢤⡀⠀⠇⠀⢀⠤⠤⠼⣇⠛⠓⢦⠀⠈⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢧⡀⠀⠀⡝⠀⠸⡅⠈⢧⡀⠀⠠⠚⠓⠄⠙⡄⠀⠀⣀⡀⠐⣄⠈⢳⡀⢈⣃⠀⠀⢀⠄⠈⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢃⠀⠳⢄⡀⣧⠀⠀⠃⠀⠀⠀⠀⣰⠢⠀⠣⠀⡇⠀⣋⣀⡉⠁⢸⠇⢈⡇⠁⠘⠛⠈⠠⠄⠀⢀⠄⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢁⠐⢦⡀⠘⢦⡀⠀⠀⠀⠈⡉⠁⢠⠗⢀⠈⠑⠛⠁⠀⠙⠚⠋⢉⣀⣤⡴⢊⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⠿⠛⠿⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠹⠄⠀⠈⢉⣀⣀⡀⣇⡄⠺⢀⠀⠈⠐⠒⡖⠒⢀⣎⡀⢈⠀⠠⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⠀⠀⠀⠉⡹⠷⠐⠤⡈⠢⡀⠀⠙⢆⠀⠩⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠠⠀⡜⢂⠘⡂⠈⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠔⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⡠⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣷⣶⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠈⣰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢄⠈⠀⢜⣦⡀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡂⢀⠀⠻⠓⢀⠀⠀⢀⡅⠀⠗⢶⣤⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⠂⠀⢠⠂⠀⢸⠈⠈⠀⠀⠉⠃⢀⠠⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠈⠀⠀⠉⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠋⡀⢰⠂⠀⠀⣱⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣂⠘⠂⠀⢀⢞⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⡀⢥⣀⡃⢠⠁⠂⠀⠀⠀⠀⠀⠀⠀⣠⠇⠁⠀⠀⢀⠠⡀⡠⠀⠀⠌⠐⡊⢁⡐⠤⠂
⠀⠀⠀⠀⠈⢀⠀⠰⠂⠀⠀⠁⠂⠁⠀⠁⣤⠧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡀⠀⠀⠀⡄⠀⠆⠐⠏⠐⠀⠀⢄⡄⠀⠔
⠀⠀⠀⠀⠢⠀⡇⠀⠐⠀⠀⠀⠀⠀⠄⠄⡄⢢⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢌⠘⠁⣀⠀⠐⠀⠀⠀⠂⠁⠐⠁⠄
⠀⠀⠀⠀⠀⠢⠑⠀⠀⢈⠀⠆⠈⠉⠀⡈⠁⠔⠂⠀⠀⠀⠀⠀⠰⠛⠓⢦⠏⠀⠀⠀⠀⠄⠀⠀⠀⠁⠰⠀⡂⠀⠠⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠒⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠦⣤⣄⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠄⠀⠀⠿⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠦⠀⠆⢠⡄⠉⠃⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠶⢈⠴⠂⠁⠀⠀⠀⠉⢠⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⢸⠀⠀⢀⠄⠂⠀⠀⣛⠰⠏⠄⠀⠀⠀⠙⢦⡀⠀⠤⠀⢷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠘⠄⠀⠀⠀⠀⡎⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣠⠄⠀⠀⠔⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠚⠁⠀⠒⠀⣶⠀⠉
⠀⠀⠀⠀⠀⣀⠔⠊⠉⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠚⠀⣼⠃
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⠔⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠉⠁⠀⠈⣳
⠀⠀⠀⢀⠴⠲⠄⡴⠚⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠐⢦⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣄⡤⠤⠤⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠱
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣀⣀⣤⡦⠺⣉⡀⠀⠀⠀⠀⣀⡤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠲⣶⡿⡑⢀⠈⡛⢷⣶⣖⣺⢍⠅⠀⠈⣉⣀⣀⣀⣠⣤⣴⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠕⠦⠭⠛⠛⠊⠁⠀⠀⠀⠻⠛⠛⠛⠛⣋⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠙⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢀⣠⠴⠞⠓⠚⠓⠲⡄⠀⠀⣠⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡤⠞⢋⣁⣀⣀⡀⠀⠀⢀⡏⠀⢸⠷⣄⣰⢯⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣧⠀⠀⢀⣴⠇⠀⠀⠀⠀⢀⡤⠞⠋⣉⣤⣶⠋⠁⢤⣄⠀⣹⣀⡴⠋⠀⣰⠋⣠⠟⠁⠀⠈⣆
⠀⠀⠀⢲⠒⣶⣦⣌⡣⣑⣻⡏⠀⣰⣿⠧⢰⠀⠀⠀⢠⡏⠀⠀⢸⡇⠌⠋⠀⠀⢈⡁⠀⢻⠁⠀⠀⡴⠁⣰⣯⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⢧⡀⠙⢿⣿⡄⢹⡇⢰⣿⣿⠀⡌⠀⠀⠀⠘⢧⣀⣀⣼⠛⠀⠀⠀⠀⠁⠬⣿⠋⠀⢀⡜⠁⣼⠃⢹⣄⣀⠀⠀⠀⣘⣧
⠀⠀⢤⣄⣀⠛⠶⠤⠬⠷⣾⠃⢸⣟⣩⠔⡁⠀⠀⠀⠀⢀⣭⣽⠏⠀⠀⢳⣤⣀⣀⣈⡏⠀⢠⠞⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠸⡌⢹⣿⣷⣦⠀⠀⣟⠟⢩⣾⡿⢫⠃⠀⠀⠀⣰⠏⠀⣿⡀⠀⠀⢸⣿⠟⠉⣏⠉⣻⠋⢀⣾⣄⡀
⠀⠀⠀⢳⡈⠩⣿⣿⡇⢸⣹⢠⣿⢟⣵⣧⡶⠾⢿⠛⣿⣀⣀⣈⠛⡶⠶⡿⢣⣤⡤⣽⡰⠃⢠⡞⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠉⠓⠒⠚⢳⣇⡟⠉⢈⣴⠟⠁⠀⠀⢸⡎⠛⢻⡉⠙⣧⢻⡸⣱⠋⣸⣛⠛⠳⢤⡟⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⢠⡟⠁⠀⠀⠀⠀⠘⡇⠀⠘⡇⠀⠈⠊⠃⠃⣴⢯⣉⠙⠂⠀⠹⣆⠀⠀⢀⣀⣀⣬⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⡿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⣴⠿⢤⣈⠁⠀⠀⠀⢻⣤⣞⠋⠉⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⠀⢠⢀⡿⠀⠀⣿⣶⡶⠶⢶⡿⠶⣤⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⢤⣤⣤⣿⡆
⠀⠀⠀⣀⣤⠞⠛⣻⠿⠿⠾⠟⠓⢦⡀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⣠⡿⠁⢀⡾⢿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡀
⣴⠞⠋⣽⢶⣤⣾⣥⣄⢀⣤⡀⠀⠀⢻⣄⡀⣿⡟⠀⠀⠀⠀⠀⣰⠟⠀⢀⡾⠃⠀⠉⠙⠛⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠻⣤⣤⣷⠀⠘⣿⡀⠈⣏⠈⢹⡟⠻⣼⠏⢻⣿⠁⠀⠀⠀⢀⣼⠏⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⢻⡆⠀⠙⡇⠀⢻⠀⠸⡇⠀⠹⠀⣼⡟⣀⣤⣤⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠘⠳⢦⣤⣿⣄⣼⣄⣀⣧⣀⣠⣾⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣿⣶⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡅⢀⠨⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠴⠀⠁⣨⠠⠄⢀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⢁⢀⢀⡅⢈⡡⠀⠠⠐⠀⠀⠈⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠹⡀⢠⠟⠀⠨⡇⠀⠀⠹⢀⠀⠀⠃⠐⠀⠀⢀⠖⠈⠀⣀⣰⢄⡄⣀
⠀⠀⠀⠀⠀⢀⡴⠺⠀⠀⣠⡁⠀⠠⠆⢠⠀⠀⢑⠀⠀⢀⠟⠋⢀⠠⠄⡀⠰⠄⡴⠚⠻⠀⠀⣹⢍⢱
⠀⠀⠀⠀⠀⢸⡈⠀⣸⠏⠀⠛⠛⢃⡀⡰⠃⠀⠉⠀⠚⠁⠈⢢⡄⠀⡠⠁⠀⠘⢇⣢⢴⡷⠈⠏⣴⡞
⠀⠀⠀⠀⠀⠀⢻⣿⡿⠤⣴⡁⢀⡜⢁⠀⣠⠁⠃⠐⠗⠀⠀⡆⡠⠈⠁⢀⡀⠀⢀⣇⣀⣷⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠉⠁⠒⠺⠿⠵⣤⣠⣤⣧⣀⣤⣓⣧⢤⠴⠟⠟⠚⣛⣭⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣄⣶⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠿⠾⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⡄⢀⣀⠐⠒⠛⠛⠦⢄⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢸⡰⠋⠉⢳⠀⠙⠲⣄⡴⠊⠀⠐⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠠⢈⡤⠤⠄⠀⣏⠀⠀⠈⣁⡤⠤⢤⠴⠊⠉⠉⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⡼⣧
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⠀⠈⣀⣀⣀⡾⠃⡀⢼⠁⠀⠀⠀⢤⣀⢸⠁⠀⠀⠴⠒⡀⠀⠀⠘⢦⣇⣠⡷⠃
⠀⠀⠀⠀⠀⠈⡀⢀⡆⠀⢘⢦⠞⠉⠀⢸⠃⢸⠁⠈⠂⠀⠀⠀⢠⡈⠻⡄⠀⠀⣠⠖⠈⢀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠐⠉⠉⣚⣇⢰⡁⣀⣀⠀⠀⠸⠀⢀⡔⠊⡀⠀⠀⠀⠀⡇⠀⢹⠀⠈⡇⠀⠀⢠⠆⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡞⠁⢀⠞⠉⡁⠀⠁⠀⠀⠀⡞⠀⢮⠀⠀⠀⠀⠊⠀⠀⠘⠀⠀⣿⢀⠴⠃⠠⡀⠀⠀⠀⠀⢀⠀⠤
⠀⠀⠀⠀⠀⢈⣀⡀⡟⠀⣼⠁⠔⠋⠙⠂⠀⣇⠀⠘⣆⠈⠲⠤⠤⠀⠀⠀⠀⣠⠏⢀⡴⠋⠐⢫⣉⣉⣡⣤⣤⣶⣶
⠀⠀⠀⠀⠀⡇⠀⠀⢧⠀⠘⠦⢠⠞⠉⠉⠓⠺⠆⠀⠈⣆⠀⠐⡄⠀⠀⠐⠊⠁⠀⠚⠀⠐⠀⠨⠋⠉⠉⢉⣭⡅⣀
⠀⠀⠀⠀⠀⠑⠀⠢⠤⠵⠄⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⢠⡧⠖⠛⠩⠴⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡦⠄⠀⠀⢀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠠⡀⠺⠇⠀⣸⠃⠀⠀⢀⠈⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⠐⢒⡂⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠠⠈⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠆⠒⠒⢒⣤⠤⠴⠶⠦⣤⣤⣤⣄⣀⣀⣀⣉⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠊⠁

⠀⠀⠀⠀⠀⡤⠤⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠤⠤⠀⠤⠤⣠⠤⠤⠄⠠⠤⠤⠄⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠂⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡜⠁⠀⠀⠀⠈⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠂⡀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠠⠰⢐⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⡀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠄⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠌⡓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠨⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⠀⠈⠋⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠾⢀⡄⢀⠠⠄⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⠠⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠠⢆⠌⢰⠈⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠴⢠⡞⠀⠀⠀⢠⠐⠉⠁⠀⢀⡄⠀⠀⢠⡁⢀⠁⠃⡀⠑⡒⠙⡁⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠈⡟⡇⠀⠀⠀⠂⠠⠔⠤⡎⠉⠁⠲⢠⠂⠀⠃⡄⠠⢦⠐⠲⠀⠀⠀⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢠⢹⡆⠀⠀⣷⣇⠀⠀⠀⠀⡀⠠⠐⢀⣙⢠⠂⢈⠑⠀⠊⢀⡁⢸⠀⠀⠀⠀⢠⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡆⠁⠸⣄⣀⡼⡃⠜⣧⠀⠀⠀⠔⠢⠈⢀⡠⢄⢀⡘⠄⢀⡁⠐⢅⡸⠀⡀⠀⢰⠋⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠱⠈⠀⠁⠸⡄⠀⠀⠀⢀⠁⠉⣀⠀⢀⠝⠠⢎⠁⠀⠈⠀⠀⠐⣤⡜⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠻⣄⠀⠀⠈⠂⡀⠀⠀⠚⡀⠀⠀⠑⠆⠀⠀⢠⠞⢡⡰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠁⢒⠤⠠⠊⢀⡴⠋⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⠙⠲⢤⡀⠀⣰⡄⠀⣀⣄⣀⠔⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠫⠁⠳⠤⠞⠃⠐⠐⣀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⢀⡴⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣴⣹⢀⣠⣤⣀⣤⣤⡾⠁⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⡇⣨⣿⡝⠟⢁⣴⣿⠀⢻⡇⢰⣿⣿⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠻⣆⣼⣿⣤⠷⣿⠿⣿⣇⡾⢧⣘⡿⠛⠿⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣱⠞⣓⣦⠖⡛⢲⣴⠒⠲⣤⠶⣤⡴⢶⠗⢦⣀⡶⠲⣤⡖⢲⣾⠟⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡇⢠⡿⣾⠀⠟⣼⣿⠀⡄⢻⠀⣜⠇⣿⠀⣦⢹⡇⢰⣮⡆⢸⣿⠀⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡿⣼⡀⣷⠈⣯⣰⣇⢸⡇⣿⣀⣿⣀⣿⣨⣇⣸⠇⢧⣸⣧⣼⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠙⠉⠛⣩⠽⠺⣯⣩⣁⣭⣁⣩⢽⡏⢉⡏⠀⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠨⢳⡏⠰⣣⡟⠀⣿⡔⠏⢀⣦⡟⠜⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣻⡿⢧⣐⣫⠾⣆⣛⡽⣇⢰⣿⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠐⢂⣀⡀
⠀⠀⠀⠀⠀⠠⣄⠉⠁⡗⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀
⠀⠀⠀⠀⠀⠘⠁⢸⡲⣤⠀⠀⠀⢉⣀⡁⠀⠀⠀⠂⢀⣴⢿⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠀⠀⠀⠀⠯⠤⠐⠆
⣋⣠⣄⣶⣀⣀⣀⢸⡇⠰⠄⠀⢀⣌⣡⢁⡀⠀⠐⡀⠙⠁⠈⠉⠀⠻⠆⠀⠀⠀⠀⠀⠀⢠⡄⠐⠀⠀⠈
⠞⠋⠉⠉⠉⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠉⣀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⡤⠀⣤⣤⣤⣬⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠿⠿⠷⣶⣶⣦⣤⣀⣠⡤⠶⠇⠀⠀⠀⠀⢰⠄⠉⠉⠉⢦⡀
⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀⡤⠴⠟⠉⠀⠀⠀⠄⠀⢿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⣀⣀⠀⠄⣐⠀⠀⣤⠠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⢀⠀⠠⠂⢘⠈⠉⠰⢶⠿⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠀⡆⠤⠀⠀⠋⡉⠤⠄⡒⠒⠛⡀⠈⠈⠉
⠀⠀⠒⠈⢀⡁⠤⠐⠀⠉⠗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠤⢒⠫⠴⠓⠂⢁⣀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣿⣄
⠄⠀⠀⠠⠀⠔⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣴⣿⣿⣧⡀
⠤⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣺⣿⣿⣷⣿⣾⣤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⢀⣼⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⢿⣿⣿⠟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠋⠉⠙⠋⠉⢻⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣶⣶⣶⣾⣶⣦⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⠀⢸⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⣶⣶⣶⣶⣶⣿⣿⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⢿⡴⠢⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠘⣶⢿⣯⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢁⡶⠢⠀⠀⠀⠀⠎⠈⡆⠀⠀⢹⣇⣽⣯⣍⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠾⣟⢀⡴⠖⠈⠷⠄⠀⣸⠁⠀⣠⠞⠻⠶⣬⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠾⢿⢁⣴⠶⠆⠀⠀⢿⣥⠊⢀⡀⠀⠘⢿⡄⠀⢿⣄⡀⢺⡇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣶⠟⢏⣡⡶⠂⢀⡟⠀⠀⢀⠻⣆⡀⢈⣿⠄⢸⣯⠱⣿⣶⡟⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠀⠰⡿⠉⠳⣄⣼⡆⠶⣤⡈⠠⣿⣥⡴⠿⣿⣉⠻⣷⡈⠙⢷
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⠻⣦⡀⡌⢻⡀⠀⣰⡆⠀⠀⠀⠀⠀⠐⠱⠌⠀⠀⢹⣧⡔⠙⠻⣦⠀⠈⢿⡇⠈⠇
⠀⠀⠀⠀⠀⠀⣠⣴⣝⣿⡧⢸⣿⡀⣿⡇⠀⣭⡷⠀⣴⠀⠀⠀⠀⡾⠀⢀⠔⠃⠉⠻⣦⠀⠹⡇⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠉⠀⣾⡛⠀⣾⠇⣠⣿⡗⠤⣿⠆⢉⣾⠀⠀⢀⢰⣦⡴⣣⠒⠑⢢⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠀⣼⠇⣠⣾⠏⠑⣾⡗⢺⣯⡅⢅⡹⠋⠀⠀⠀⣼⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⣾⠏⠙⠪⠁⠀⠖⠀⠀⠀⠀⢻⠋⠉⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠐⠃⠉⠀⢀⡀⠀⠀⢄⡴⠇⠀⢰⡇⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣗⣹⠅⠀⣀⠙⠀⢀⣶⣟⠀⢀⣠⣭⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⢠⠆⠀⠀⠘⠀⠸⣿⣥⢤⣿⡋⠒⠓⣿⠁⠀⣸⡟⢥⣼⡿⠗⢀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠀⠛⡀⠀⠀⢰⡦⠺⣿⡄⢻⣿⣿⡆⠀⣿⣷⠀⢿⡷⡰⣿⡍⢲⣾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣙⢿⣷⡄⠀⢹⡇⠀⢨⠐⠘⣿⣶⢸⣿⣿⡅⠀⠸⣅⠀⢸⣷⠁⠻⡇⠘⢻⡘⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢷⣾⣯⢿⡷⢺⣷⠠⣿⢅⣀⣀⡇⡄⠈⢻⣸⣭⣿⠇⠀⠀⠘⠀⠀⠋⠀⠈⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⣿⣿⣿⢿⣼⡿⢰⣿⠀⠀⠉⢢⠀⠀⢹⠁⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠋⠁⠛⠀⠘⠁⠀⠁⣀⣀⠀⠈⡤⠀⢰⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡉⠟⢳⣏⠑⡧⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣝⣛⠿⣷⠘⠷⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⣿⣟⠻⡏⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠛⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣶⡀⢀⣴⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⣤⣠⠀⣤⣄⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⢀⡠⠀⠀⠀⡀⠀⣄⡠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢘⣋⣀⣀⣄⣀⠀⠀⠁⠠⠀⠀⠐⠊⠉⠈⠉⠀⣴⣤⠶⣷⡆⠀⠀⠈⠀⠁⠘⠀⠍⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⠀⣤⠖⢀⡐⠀⠐⠒⠀⠀⠉⢠⡀
⠀⠈⢏⠉⠀⠈⠉⣷⡀⠀⡀⠀⠂⣀⡠⠀⣀⠀⠀⠈⠛⠟⠛⠁⠤⡄⠀⠐⣖⠀⠀⠀⠀⡀⠀⣀⠀⠀⡖⠀⠘⠃⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⢠⡀⠀⠀⢀⣐⠉⠀⠀⠀⠀⠀⠀⠐⠋⢨⠤⠀⡀⢀⡄⠀⢆⣀⠗⣀⣠⠇⠀⠀⠀⡼⠀⠀⠯⣄⠈⠉⠘⠛⠂⠀⠀⠀⠀⢰⠘⠢⠛⠉⠹⠆⠈⠓
⠀⠉⠀⢤⠞⠋⠉⠻⡆⠀⢠⠀⢠⡖⠒⢢⡀⠀⠈⠀⠴⠁⠀⢠⠀⠀⠉⠁⠀⠀⠀⠐⠳⣶⡀⠀⠀⠀⠀⣀⠤⡄⠀⠀⠀⠃⠈⠋⣀⡀⠞⢉⡆
⠀⢀⣀⠀⠑⠠⢤⠾⠃⠀⠀⠀⢸⡂⣀⠠⠃⢿⡄⠀⠀⠀⠀⣤⣀⠀⠀⠀⠉⠀⢀⣀⠀⠀⠀⠀⣠⠀⠀⡏⠀⠀⢿⠀⠀⠂⠀⡞⠈⠁⣶⡀⠤⢀
⠀⠀⣈⠃⠀⢰⣦⠀⣀⣀⡄⠀⠀⠉⠉⠀⠀⠀⠀⠋⠉⠳⠀⠙⠋⠐⠀⠀⠀⣾⠉⠉⢳⠀⢰⡿⢿⠁⠀⠑⠗⠛⠋⠀⠐⠄⠀⠣⡤⠤⠞⠀⠐⠄
⠀⠈⠠⠇⠐⠀⠁⠀⠛⠃⠘⠳⠀⠀⠀⠈⠀⢾⣤⠐⠊⣱⠀⠀⠀⠀⣴⠷⠄⡻⣤⣀⢰⠀⠸⣷⡞⡆⢠⣒⠀⡀⠀⠀⠳⡠⠀⠀⣤⡆⠀⠰⡘
⠀⠀⠀⠐⠢⠄⠀⠀⣠⠶⣄⠀⠀⠀⠀⠠⠂⠀⢀⣄⠀⠀⡤⣀⠀⠀⣿⠀⠀⠈⠀⠈⠉⠀⢠⠐⢄⠀⠀⠩⣀⢹⠄⠀⠓⠄⡄⠈⠷⠇⠀⠲⠌⢀⣤
⠀⠀⠀⢀⠑⠀⠀⠀⢠⡂⠀⠀⠀⠀⠀⢆⠀⠀⢿⡇⠀⠀⡐⠀⠀⡀⠘⢧⣄⣠⠄⢿⣈⠇⠘⠒⠀⠀⡀⠠⠈⠈⠀⠀⠐⠤⠀⠀⠀⠀⢀⠁⠀⠸⡿
⠀⠀⠐⢡⠁⠀⢀⡴⠀⠹⠆⠀⠀⠀⠰⡆⢀⠀⠀⠀⠀⠶⡙⠀⠀⡄⢤⡀⠙⠁⠀⢀⠁⢀⡄⡐⡄⠐⠸⠀⠀⢀⡀⠀⠐⠤⠂⠀⠀⠀⠀⠀⡀⣀⣤
⠀⠀⠈⠆⠀⠀⠈⠁⢲⡄⠀⢀⠒⠀⢠⣄⠁⠀⠀⠀⠀⠢⠁⠀⠀⠳⠄⠁⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡅⠀⠰⠄⡰⠆⠀⠀⠀⠘⢿⣧⣾
⠀⠀⠀⠀⠀⠀⠰⢊⡄⠁⠀⣀⣉⠀⠄⡀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡆⠀⢀⡀⠀⠀⠀⢀⠠⠶⠒⠀⠀⠉⠀⣠⠀⠀⠀⠀⣴⠁⢀⡶⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⠈⠇⠀⠀⢿⡀⠃⠀⠉⠃⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡅⣴⠋⠉⢷⠄⠀⢰⡧⠀⢠⡔⠂⠀⠀⠈⠋⠀⠘⠀⠈⠓⡋⠀⠀⢌⠓
⠀⢾⡗⠀⠀⠀⠀⠃⠀⣠⠒⠒⣀⠀⠀⠙⠲⠁⠀⠀⢠⣾⡃⠀⣿⣀⠀⠀⠁⢷⢀⡠⠎⠀⠀⠀⠀⠀⠀⠃⠐⠖⠛⠀⢀⡀⠀⠀⠀⠘⠷⠤⠴⠛⠁
⠀⣠⢧⠀⠀⠎⠀⠀⠀⢗⢀⣀⣺⠇⠀⠲⠄⡶⠇⠀⠀⠙⠁⠈⢉⠉⠁⢔⣐⠘⠉⢰⠔⠓⢦⢀⡴⠲⢀⡀⠀⠀⣷⠁⠉⡉⣀⣀⠀⠀⢠⡀⠀⢀⡤
⠀⢙⠈⠀⣠⠀⠀⠀⢀⡈⠉⠉⠁⣄⠀⠀⠀⠀⠀⢰⠋⠁⢐⡦⠈⠁⠠⠀⠀⠀⠀⠸⣤⡄⠊⠸⠀⡀⣄⡿⠀⠀⠀⠀⠀⠓⠈⢉⢁⡀⠘⠛⠠⡏
⠀⠈⠇⠀⣁⡀⢿⣶⣉⣿⠇⠰⢤⢸⠆⠀⠀⠰⠀⠘⠶⠶⡋⠀⠀⡀⠀⠈⠀⡤⠆⠀⣄⠀⠀⠀⠙⠉⠉⠠⢶⠀⠀⢲⡆⠀⠀⠀⠁⠠⣠⠀⠀⠱⠖
⠀⠀⠄⠐⠦⠄⠀⠉⠉⠁⢀⠀⠀⠀⢀⣴⣾⠄⠀⠀⠰⣮⣀⣀⣠⡿⠂⣀⣄⠀⠘⠁⠠⠀⢰⠀⢠⠀⠰⢄⡸⠃⣀⡼⠁⠀⠀⠀⡀⠀⠈⠀⢀⠀⠄
⠀⠈⠁⠀⠀⠠⠆⠌⠆⠈⠘⠀⣠⡄⢸⣏⣽⡀⠀⡠⠄⠀⠉⠉⠁⠀⣼⡟⣏⠀⠀⣤⠀⠀⠁⠠⠊⠁⠀⠄⠀⠀⠀⠀⠀⢾⣶⣛⣿⡗⢀⣤⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣉⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣉⣀⣈⣁⣀⣀⣀
⣿⣻⣛⣟⣟⣫⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣯⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡆⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠓⠶⣤⠛⠀⡸⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠇⠈⣶⡋⠀⡤⣤⣿⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠈⢳⡀⠸⣐⠼⠻⢷⣾⠁⠠⠀⠙⢶⣣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠺⢤⡀⢠⡼⠻⢿⣀⡤⠒⠲⡅⢀⠀⠐⡤⡸⢙⠷⠋⠑⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠰⣴⢝⠋⠰⣀⠀⠈⣆⠀⠀⠈⢺⣤⠄⢒⣠⡎⡀⠐⠀⠀⡆⣬⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⢤⡜⠫⠑⢠⣄⣛⣄⠀⠘⡇⢀⠞⠓⢦⣀⠾⠛⠛⠻⣅⠈⠂⠤⣾⣿⡟⠁⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⢹⠀⢠⠖⠉⠉⠻⣿⣥⣿⡁⠀⠠⠀⢩⣀⣸⠗⠓⢾⣄⣠⠖⠛⠻⢇⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⡼⡠⠺⢶⡿⢤⠀⡰⠀⠠⣀⠈⠱⣀⡠⠃⠛⠛⢧⠀⠀⠀⢼⠃⠐⢦⡀⠈⡷⠤⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠸⣇⠀⠀⠈⢷⡄⣯⠀⠀⢠⠐⠈⣇⠴⠀⡸⢤⠈⠈⢈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠾⠦⠜⠁⠱⣤⢄⣴⠀⠘⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⢠⠮⢰⣌⠙⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣟⠐⣾⠇⠤⠛⢺⡉⢷⣖⠲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⢐⠉⣠⣈⣎⡀⡑⠒⠉⠻⢷⢠⡟⡀⣱⠌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠪⠑⢌⠀⣊⢣⡸⠁⣿⠋⢐⡤⠊⠠⡀⣱⡆⣬⠈⠢⣀⢁⣑⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⣼⣁⡀⠐⠗⢈⠁⠄⡛⠉⠁⠬⠁⣠⢒⠶⠫⢘⡿⢊⣼⣟⠣⠜⣏⡠⣎⠐⠂⠠⢀⣀
⠀⠀⠀⠀⣠⣶⡿⠛⠅⣀⠽⠟⢁⣜⠂⢁⡨⣷⡌⡪⠑⠄⢈⡿⠋⣠⢊⠇⡯⣥⢤⠈⢳⣀⡀⢵⣬⠘⠋⣱⣶⣎⣑⣄
⠀⠀⠀⠠⣃⣀⠴⣾⠟⠈⠇⡸⠣⢖⡆⢹⡒⣍⠻⢗⠠⢃⠶⡱⣴⠟⡙⠉⣡⠏⢸⡔⠀⡐⠒⠬⡳⡶⠋⢙⢤⠄⠰⠊
⠀⠀⠀⠀⠀⠙⠧⠈⠣⠴⠶⠖⠲⠉⣥⣾⠋⣉⠂⠠⣼⣆⣔⢰⠅⠈⢀⠝⠀⠀⣹⣷⠚⣉⠦⡀⢈⣣⣶⠅⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢺⣿⣷⠗⡉⠌⠻⠀⠁⠦⠾⡉⠛⠊⠴⣎⡨⢶⣆⡴⠋⡨⢋⣀⡤⢳⡈⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠚⠀⠰⢈⢀⠠⠔⢉⡑⠒⠰⠋⢀⣶⡼⠐⠚⠛⠃⠀⠐⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠴⠟⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⡁⢈⣙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡘⠃⠘⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠇⠸⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣆⣰⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⣀⠀⠀⠀⠈⢹⡏⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣇⣠⣼⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠷⣦⣼⡟⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⢸⣿⣿⡟⠉⠉⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣉⣉⣉⣉⣉⣉⡙⠛⠟⠿⠟⠿⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠐⣶⣶⣶⣆⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠟⠛⢋⣠⣶⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢳⠁⠀⢀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⣀⡇⠀⠀⠯⢈⣦⠁⢀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⢤⠀⡀⣠⢄⠈⢠⠘⢀⠈⠃⢄
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⠤⠒⠒⠀⡐⠁⡠⠁⡠⠁⡠⠃⠀⢎⡵⠀⠱⡄⡤⠠⠆⠀⠀⠐⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⣄⢣⠀⠀⠀⠀⢈⣘⠁⢀⠁⢠⠧⠤⠞⠀⠀⢗⡀⢐⡀⢀⠂⢄⠀⣀⣀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢰⠘⡀⠀⠀⠀⢀⠠⠐⠋⠉⠫⢀⡀⠈⠘⠒⣅⠀⠀⣈⠏⡔⠚⡆⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣸⠀⣇⠀⠀⠀⠠⠠⠄⠒⢌⠀⡀⠅⠴⠃⢐⠀⠣⣀⠧⢀⡄⠀⡇⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⢆⠀⠀⠀⠠⠐⠖⠈⠀⢀⠤⣆⠀⠘⠤⡀⠈⠓⠀⢇⢀⡇⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠐⡀⠂⢘⠁⠀⠈⠈⢹⠄⣄⠋⠀⠀⡴⠋⠀⠀⢳⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠑⢄⠈⠉⠁⠡⡬⠁⠀⠀⠑⢄⠌⠀⠀⠀⠀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠃⠀⠀⠀⠈⠦⣀⡀⠀⠀⢡⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⡀⠀⠀⠈⢜⠒⠒⠁⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⠀⢀⠗⠄⣀⣀⢼⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠑⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣤⣠⠶⣾⣛⣿⡵⣾⣳⣦⣴⠲⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡷⣫⣿⢾⡿⢾⣧⣾⢿⣴⣿⣾⣭⣯⣽⣷⣬⣭⣶⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢺⣾⣩⢟⡿⢿⣷⡶⢿⣥⣿⣿⣿⣴⣿⣿⣋⣻⣷⣿⣽⣮⡿⢻⣥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣰⣷⢊⣽⣤⣷⠾⡿⠻⣿⠶⣾⣉⣭⣿⣿⣿⣭⡟⣛⣿⣿⣯⣿⣼⡿⣿⣽⢷⣓⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣻⡟⣩⡟⣉⡿⣭⣷⣾⡿⢿⡧⢾⣿⡿⣟⣁⣾⣾⣾⡟⢿⠿⢿⣻⡏⢻⠿⣿⢼⣖⢿⣭⠧
⠀⠀⠀⠀⠀⠀⣞⣭⢿⡿⣏⣽⢿⣿⣿⢱⣿⣴⣿⣴⣾⣵⣾⠛⣏⣽⢧⣶⢿⡿⣷⣼⣭⣹⡟⣻⣿⣿⣿⣿⣽⠶⠾⣘
⠀⠀⠀⠀⠀⢠⡸⣯⠿⣾⣿⠟⣼⡿⣼⣟⣻⣭⣿⣍⣿⣧⣾⠿⢿⣴⣾⣿⣶⣿⢿⣷⣿⣟⣿⣛⡿⣷⣿⣿⣿⡛⣿⢋⡆
⠀⠀⠀⠀⠀⠈⣿⣌⠺⣧⡾⢾⣿⡿⣿⣙⣏⣼⣿⣏⣿⣾⢿⣶⣿⣧⣾⣧⣶⣿⣽⣯⣍⣷⣶⢿⡷⡿⠟⣤⣾⢟⣵⡿
⠀⠀⠀⠀⠀⠀⠸⡿⠷⣮⣙⠶⣿⣷⣾⡿⣿⣭⢿⣥⣽⣷⣾⢿⣶⡿⢿⣧⣼⢻⣋⣽⣾⣋⣳⣾⣶⠷⣚⣭⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠙⠲⠭⣝⣛⠿⠵⢶⣿⣾⣿⣉⣻⣷⣷⣿⣋⣿⣯⡿⠦⠿⢚⣫⡥⠖⠻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠂⠈⠁⠀⠀⠀⠀⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⢿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⡷⣤⣀⠀⠀⠀⠀⠀⠀⢄⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣨⣽⣿⣷⣌⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⢉⣡⣿⣿⣿⣿⣿⡆⡀⢀⢀⣄⢀⣀⣰⠶⣤⡆⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠻⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⠚⣻⣿⣏⡶⣿⣿⣻⣶⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠈⠉⠋⠛⠻⠛⠟⠻⠶⠿⠉⠉⠋⠀⠙⣷⣟⣻⣯⣭⣿⣛⣿⣿⣽⣧⡿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠶⠀⠘⠛⠃⠈⢹⣿⣿⣿⣿⣯⣽⠷⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⠀⠀⠀⠀⣧⣄⠀⠻⠟⠉⣀⣀⠈⠉⠉⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⣤⡆⠀⠙⢋⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡏⠉⠉⠉⠉⠉⠁⠀⠈⠀⠒⠤⢤⣄⠤⠄⣐⠒⠀⢨⡈⠍⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⡳⡄⠀⠈⠆⢠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠤⠴⠤⣀⠀⠀⠀⠀⠣⡤⠤⠤⣤⢄⠀⠈⠻⡀⢀⢞⠘
⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⠈⠢⡤⠨⠟⣛⣢⣄⠈⢣⠬⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⡛⠓⠦⠯⠹⠾⠷⠞⠀⢉⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢍⣒⠨⢿⣍⣡⠶⠒⠀⢲⠂⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢐⡒⠂⡍⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢆⣀⣘⣄⡔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠍⡙⠃⠀⠀⠋⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣱⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠎⢀⡠⠤⢤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⡏⠀⢎⡀⢹⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠳⣄⣒⣁⡼⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠈⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠉⠒⠒⠒⠒⠒⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⣤⠞⢀⣀⡋⠀⠊⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⣀⠟⠀⠚⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⡾⠀⠠⠘⠂⠠⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠈⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠈⠀⠹⠷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⣶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀⣠⣴⡾⣿⣦⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢠⡾⠉⠙⠙⡄⠀⠀⠀⠀⠀⠒⠲⣾⣿⣿⣿⣟⡀⠀⠈⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠈⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡀⠙⣿⣿⣿⣿⣿⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠁⠀⠑⢌⠛⠳⢤⡇⠀⠀⠀⢀⡾⠉⠁⠉⢠⣌⠙⠓⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⢷⣦⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠱⡼⠶⠷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⢁⢴⣀⡠⡠⡅⠀⠀⠈⢁⣄⣁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣿⣻⣛⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⠀⠘⠇⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢀⠤⡐⢐⡐⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⢢⠋⠀⠘⢧⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⢆⡀⢀⡃⠀⠠⢰⡀⠀⠀⣠⠇⠀⠀⢀⢦⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠁⢚⡍⢙⠒⠁⢠⡟⠠⢈⠁⠁⠨⠭⠷⢾⣄⠀⠈⠺⠦⡾⠋⠀⠀⠆⠿⠀⣴⠀⡀
⠀⠀⠀⠀⠀⡌⣤⠀⣹⠀⣠⠀⡠⠊⠱⣣⠞⠙⢦⠐⠒⢄⣠⠀⠀⠠⠀⠃⠀⠀⡦⠀⠶⣀⠈⠰⠁⣄
⠀⠀⠀⠀⠀⢧⠻⢊⡡⠤⠄⢀⡇⠀⠈⣙⡤⠤⢈⠔⠋⠉⠁⢀⠀⠀⠀⠀⠀⣀⠀⣠⠀⠈⢷⠀⠾⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠉⠀⢠⠀⢀⣠⣤⣼⠋⡠⢼⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡈⠀⠙⠀⢀⡾⠀⢶⡈⠁
⠀⠀⠀⠀⢀⣀⡀⡆⠀⡈⠲⠉⠀⠀⡇⠘⠀⢀⠀⠀⠀⠐⢄⠈⢆⢤⣤⡤⠀⠁⢸⠆⠚⠉⣀⠸⣅⠉
⠀⠀⠀⠀⠀⠀⡤⠷⢸⡥⠤⠤⠀⠀⠃⢀⠎⢡⠀⠀⠀⢀⡼⠀⢸⠀⠈⡆⠀⣠⠏⠀⣶⠁⠽⠈⠁
⠀⠀⠀⠀⠀⠜⠀⢰⠋⢠⠄⣀⡤⣄⠀⢌⠀⢸⡀⢀⡀⠈⠀⠀⠀⠀⢀⡇⠐⣡⠄⠂
⠀⠀⠀⠀⢰⠒⠀⢿⠀⢻⡀⢀⠤⠤⠤⣸⡄⠀⢳⡀⠉⡉⠀⠀⠀⠤⠊⠀⡜⠁
⠀⠀⠀⠀⠘⠄⢤⣈⣂⡀⠀⠈⠀⠀⠀⠀⣁⠀⠀⡇⠀⣸⣠⠴⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠂⠀⠉⢸⡇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠈⠂⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠧⠅⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠇


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

⠀⠀⣀⠀⡀⠀⠀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⣀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣸⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠷⠶⣿⣿⣿⣵⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣶⣍⡑⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⡿⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⠋⠇⢸⣿⠌⠻⣝⣝⠉⠒⡄⣀⡀⢀⣠⣤⣬⣵⠀⠀⠀⠀⠀⣻⣻⠃⠁
⠀⠰⠖⠂⠀⠀⠒⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣋⡙⠿⢠⣿⣯⣿⠶⠆⣀⣴⣼⣴⣄⣀⠠⠤⠤⠶⣶⣶⡆
⠀⠀⠶⠶⠶⠶⢦⣠⣤⣄⡍⠿⠿⢝⣻⣟⣻⣿⣿⢸⣿⣃⢸⣿⣿⣿⣏⣽⣦⣶⣼⣿⡏⠉⠉⢹⣿⣄⣀⣤⣤⣿⣿⣛⣭⠁⠐⠒⠂⠐⠒⠒
⠀⢠⣤⣤⣤⣤⣤⣤⣬⣤⣖⣲⣿⣿⣿⠿⠿⠟⠛⠘⠛⠋⣻⣟⣛⠛⠋⠙⠛⠛⠘⠛⠃⠀⠀⠚⠿⠿⠿⠛⢿⣭⣷⣴⣾⣴⣶⣶⣤⣤⣠⣤⡀
⠀⠸⠿⠿⠟⠛⠛⠉⢉⡩⠁⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⢙⣿⣏⣤⣤⣰⡤⣴⣤⠴⣶⣤⣤⠦⢀⣀⣀⡀⣀⣛⣻⠿⠿⠿⡿⠿⠟⠻⠿⠿⠇
⠀⠀⣀⣿⣇⣤⣜⣅⣀⣦⣤⣏⠙⣿⣟⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣟⣱⡇⠢⣦⣿⣿⠃⣠⠈⢻⣿⣤⠈⠛⢛⣦⣤⣤⠂⠀⠀⠀⡀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣶⣀⡀⠀⠀⡀⠀⠀⠈⠙⠿⣿⣿⣿⣷⢀⡟⡿⠿⣢⣿⡀⢸⣿⠿⣆⠀⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⡀
⠀⠠⣶⣾⣿⣽⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡷⠗⠖⠺⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣇⣹⢇⣿⣿⠗⣾⣶⣾⣿⡶⣶⣾⣿⣾⣶⣶⣶⣶⣿⣿⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣨⣯⣗⣀⣐⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣻⣻⣛⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⢠⠂⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠢⢀⡀⠄⠀⠁⠀⠀⠀⡀⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⢨⡠⠄
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⢀⡤⠀⠀⠀⠀⠀⠱
⠤⠘⠁⠀⡠⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠠⠁
⠀⣀⠄⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠈
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠁⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠐
⠀⠑⠠⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠆⠀⠀⠀⠀⢄
⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢁⠀⠀⠀⠞⠁⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠘⡅⠀⠀⠀⠀⠠⠀⡸⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠀⠀⠀⢠⠀⠀⠃⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠐⠀⠤⠒⠀⠉⠀⠀⡰⠀⢀⣀⠀⠈⠂⠀⠀⠀⠂
⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⡀⠈⠁⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠐⠰⢦⠀⠀⠀⠀⠠
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠈⠂⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡤⠤⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠁⠐⠀⠀⠈
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠈⠀⠀⠓⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠁⠉⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡠⠤⠕⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⢀⠰⠋
⡿⢿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣿⠿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⢀⣄⠀⡄⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠘⢢⡡⡇⣆⢿⢆⣇⡏⡶⣴⣠⢂⢀⡉⠉⠀⠒⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⠠⣇⠲⢰⣶⣌⢪⡳⣟⣾⢮⢰⣦⢦⡎⡅⣵⣵⢫⠾⣡⣶⣶⠶⢀⢀⣀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡄⣒⢹⣇⢿⣜⣓⣈⣁⠈⠛⠉⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠈⠁⠲⠶⠾⣫⠍⡉⡉⣉⠩⠤⠖⢀
⠀⠀⠀⠀⠀⠀⡀⠒⢶⢘⣷⠘⣷⣭⣥⡤⠤⠔⠮⠙⠤⠤⠀⠀⢸⣿⣿⣿⡿⠀⠀⢘⡡⣤⠔⢒⣒⣒⡈⠛⠛⠛⠡⢤⣤⣤⡤
⠀⠀⠀⠀⠀⢤⡉⢁⠐⠌⠉⢓⢉⠉⡉⠢⢐⠤⠀⠐⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⣞⣊⢀⡨⡓⠙⢊⠭⠭⠭⠭⣛⠶⠄⠀⠀⠁
⠀⠀⠀⠀⠐⠂⡙⠮⣭⢤⢤⣤⣤⣤⣤⣬⡁⠐⣠⡶⣒⢀⣀⣀⣀⣤⣀⣄⣀⣀⣀⡠⣤⡘⠥⠠⠐⠌⠩⠭⢍⢭⣐⠭⠭⠭⢕⠲⠖⠶
⠀⠀⠀⠀⠈⠤⠂⢀⡄⣐⢶⣤⠤⣒⢖⣶⣶⣛⠵⣪⢕⢋⠻⡿⣿⣿⢹⣿⣿⡿⡻⣿⣜⣛⣛⣛⡲⠌⠛⢛⡪⢝⣈⡉⣍⡛⢦⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠐⢴⣿⡅⢿⣄⡿⢂⠵⣂⣘⣒⣑⢋⠠⣢⠕⣩⠾⣻⢇⣸⡀⣄⡳⠜⠖⠒⢒⣒⠮⣛⣛⢳⣦⢹⡆⢿⣇⢿⣿⠸⣷⣬⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢫⢎⡔⣀⡀⠲⠕⣫⡴⢛⡵⢞⠝⣩⣿⠻⣎⢻⣶⣶⣶⢸⣿⡷⠌⣿⢸⣿⣦⢹⣦⢹⣶⣬⣍⠂⣍⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⢸⡟⢰⣶⠞⣡⡞⢩⣴⠇⣾⣿⣿⠇⣿⡘⢻⡟⣩⢸⣿⠑⣿⣿⠀⣿⣿⠈⣿⣆⠉⢛⡛⢷⣈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠁⢿⣿⠰⣿⡀⠾⣿⡇⠙⣿⡇⢸⣿⣧⣿⠁⡿⢈⠺⢿⣿⠿⠛⠿⠿⠐⠟⠿⠿⣿⠁⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠸⠿⠄⡹⡿⣄⠿⠟⠀⠋⠉⠉⠉⠉⠛⢾⡃⣹⡷⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⢀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⡍⠉⠉⢹⠉⠉⠉⡏⠉⠉⡿⠉⠉⢉⡟⠉⠉⠉⡹⠋⠉⠉⠉⠉⡩⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠹⡄⠀⠘⡇⠀⠀⡇⠀⢠⡇⠀⢀⡞⠀⠀⢀⠞⠁⠀⠀⢀⡴⢊⡂⠤⠀⠀⠀⣀⠤⠀⠀⠋
⡇⠀⢹⡄⠀⢻⠀⢀⡇⠀⣸⠀⠀⡾⠀⠀⣰⠋⠀⠀⣀⠔⠉⠀⢀⡀⠠⠀⠀⠀⠛⠁⠀⠀⠀⠀⣀⠠
⡷⣄⠀⢳⡀⠸⡇⢸⡇⢀⡟⠀⡼⠁⢀⡞⠁⠀⣠⠞⠁⠀⣄⢐⣵⡄⠀⠀⠀⠀⠀⣀⡤⠔⠂⠁
⡇⠘⢧⡀⢷⡀⣷⢸⡇⣸⠃⣼⠣⠴⠏⢀⡴⠞⠁⠀⠀⠀⢀⠀⠘⠃⣀⡤⠴⠚⠉
⡟⢷⣌⠻⣌⢷⠹⠞⠇⠿⣼⣧⡴⠂⣿⣿⣷⣶⣶⣶⣤⣤⣽⡵⠶⠌⡏⠀⣀⣀
⡟⢶⣍⡳⠉⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣇⣝⣙⡿⠻⢿⣿⢻⠃⣰⣧⠤⢶⣽⣿⣷⡄
⣟⡳⠮⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⢸⣿⠿⣿⣹⡙⠿⢿⠸⣦⡟⡀⠈⣿⣿⣿⣿⣻⠉⠉⠉⠀⠀⠀⠀⠀⡀
⣯⣭⡅⠀⠀⠀⠀⠀⠀⠒⣆⠀⢠⣤⠾⠏⢼⡿⡛⣾⣿⣿⣿⣿⡟⢇⣠⣿⡟⠀⣿⢋⣀⣀⠀⠀⠀⢀⠞⠁
⡷⠶⣖⡀⠀⠀⠀⠀⠠⡷⢾⠤⠄⠸⠦⠀⠸⣆⠢⣸⢿⣿⡏⠉⢁⣆⠙⢀⡋⣰⠏⠀⠀⡩⠋⠀⠀⠀⠰⠈
⡷⠞⠫⠶⠀⠀⠀⠀⠤⡭⠠⣤⠤⠤⢤⣤⣘⣹⣼⣾⣬⣟⣀⣀⣠⣩⣇⣸⡁⢛⣂⣐⣀⣀⣀⣠⣤⣤⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠦⠕⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠻⣿⣿⣿⣦⡀⠀⠀⠀⣾⣿⡟⣡⣙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⢠⣤⡈⢻⣿⣿⣷⡀⠀⠀⣿⣿⡷⢦⠌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣀⣉⠙⢻⣿⣿⣿⡇⠀⠀⠙⢿⣿⣦⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠙⠿⠃⣠⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠉⠁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣷⣟⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣱⢏⡽⣿⣿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣵⣜⢱⡾⣳⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣌⢩⣹⡽⠿⣿⣻⡖⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢧⢋⣠⢨⣰⠻⠻⠶⠿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣯⡛⠷⢢⣤⢘⣃⡛⣻⡗⣽⣾⢹⣽⣛⣿⢿⣿⡿⠟⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠁⠑⠓⠹⠴⣨⡍⣹⣏⢯⣿⡿⣼⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⢱⠀⠊⠄⣨⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⢒⠆⣶⠆⠐⠁⣀⠋⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠤⠚⣁⡀⢠⡈⠡⠔⢃⠄⢺⠁⠀⠀⢀⠀⠀⠀⠀⡀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⣠⠎⠑⢄⢠⠁⣠⠄⠀⣠⠀⠐⡀⠐⣄⠀⡤⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⡋⠁⠄⠀⠁⢀⠀⠁⠀⢳⡞⡈⣀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠉⠠⣈⢠⣎⡀⠠⠋⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡬⠙⠀⠠⡀⠀⠐⠂⠼⢋⣁⡀⠀⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠤⠔⠐⠤⠄⠃⠀⢢⡀⠀⠀⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣅⣠⠄⠀⠀⠀⠈⣿⣀⢁⣀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⢘⣦⣠⡀⠀⢈⠩⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠙⣆⡈⢳⡴⠁⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠝⠀⠀⠀⠀⠂⠉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠿⠿⠿⠿⠿⠛⠛⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢿⣽⡀⠀⢲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠞⠻⣿⣿⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠗⠛⠃⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢰⢤⠀⠀⠀⠈⡟⠒⡾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⡼⠇⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡼⠃⡀⠴⠀⠀⢡⠤⠦⠄⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⠈⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠈⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢇⡀⠀⠀⠀⠢⠖⠐⠓⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⠉⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢻⡇⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⠶⢤⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢸⡇⠀⣀⠤⠀⠀⠀⠀⣹⠟⠀⠀⠀⠈⠙⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⡞⠁⠀⠀⠀⠀⢰⡏⠀⠀⠐⢤⡀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⣦⡀⠀⠀⠀⠘⣧⡉⠙⠀⠁⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⠙⠶⢤⣤⠴⠟⠙⢳⡖⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢈⡟⠳⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣠⡾⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠋⠁⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⡀⠀⠀⠀⣀⣀⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣻⣉⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣭⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢃⡄⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣀⠄
⠀⠀⠀⠓⢜⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⡏
⠀⠀⠀⣀⡐⣶⣷⡂⠀⠀⣴⠾⢤⣀⣀⠈⠀⠒⠀⢀⣠⠄⠊⢸⣿⠋⡹⢣⡀
⠀⠀⠀⠈⠁⠀⠠⢤⡖⢀⣻⢲⡶⢻⣿⣟⣦⣤⣞⣉⠤⠀⠘⣼⠃⣠⠇⣸⡏⢉⣳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠉⢲⣼⣾⣥⣬⣇⣈⣚⣉⢣⠀⠀⠉⢡⢇⣼⠃⠀⢻⣇⢈⡟⠻⡗⠛
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡼⣻⡟⡏⡟⠙⠟⡿⢻⠿⡿⣿⣷⣾⣿⣾⣿⣼⣾⣿⣾⣷⣤⣿⠰⠦
⠀⠀⠀⠴⣶⣶⣦⣬⣯⣀⣐⣧⣶⣿⣿⣷⣷⣮⣤⣄⠧⣇⡅⡥⡁⡻⡿⠟⢟⣟⣷⠛⢿⣶⠄⠤
⠀⠀⠀⠀⠀⠀⠨⣭⣍⣀⣀⣀⣈⡉⠉⠙⠛⠻⠷⠿⢾⣿⣛⣽⢓⠛⢧⠀⢘⠁⠈⡄⠈⠉⠁
⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣶⣶⣶⣶⣴⠦⠶⠶⠦⠤⠤⠤⢤⡭⣭⣝⣛⣳⡆⢀⣀⠀⠄
⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⡆⠄⠀⣤⢢⠰⡂⢰⢲⠂⢠⠖⡄⡖⠆⢸⠀⠀⠀⠀⡉⠔⠢⡔⠒⣤⡖⡄⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠁⠈⠁⠈⠊⠈⠊⠀⠈⠀⠈⠐⠁⠁⠀⠈⠁⠀⠀⠀⠁⠈⠊⠈⠊⠀⠁⠁⠑⠁

⠀⠀⠀⠀⠀⠀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠤⣤⣼⣷⠄⠀⠀⠀⠀⠀⢠⣶⢳⣦⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠿⠀⠿⠘⠟⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⡻⣫⣷⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣶⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠿⠼⠏⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣝⢫⣽⡵⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣽⠆⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠻⣦⡀⠀⠀⠀⠀⠀⠰⣿⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⢿⣽⠿⣅⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⣧⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠻⠟⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⢶⣦⡀
⠀⠀⠀⠀⣿⠛⠋⢸⡷⠙⠛⢿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣥⣿⣿⠁⠀⠀⠀⠀⠀⠀⠲⣦⡄⣴⡾⢿⣶⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣵⠂⠈⠛⠂
⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠟⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠲⠒⠛⠓⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⣴⣶⣄⠘⠁⠀⠀⠀⠀⠀⠀⣶⢀⢀⣿⣉⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠰⠗⠮⣭⠧⠾⠄
⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⢶⢢⠀⠀⠀⠀⠀⠀⠤⢠⣿⡟⠿⣿⣷⠠⠤⠀⠀⠀⠀⠀⣿⣙⣹⡿⠿⠿⡿⡷⣷⠀⠀⠀⠀⠀⠀⠸⢷⣻⣿⣿⣾⡃
⠀⠀⠀⠀⣀⣤⣿⣧⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠤⢼⣿⠶⠷⠶⡶⠞⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿
⠀⠀⠀⠀⣧⣼⠿⡿⢿⢿⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡼⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣭⢽⡀
⠀⠀⠀⠀⢀⣴⣧⡒⢿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣄⣎⣀⣇⢰⣲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣶⡵⠋
⠀⠀⠀⢠⣾⣿⣿⡇⣠⣴⣇⡻⡀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠟⠛⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣾⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣶⡔⢂⢶⣶⡄
⠀⠀⠀⠘⠟⠝⠶⠭⢿⣿⣿⣯⠏⠀⠀⠀⠀⠀⠀⢸⣷⠏⠈⢿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⢸⣿⣿⣿⡏⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣍⣾⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡗⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠿⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣠⣾⡇⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡿⠇⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣦⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢀⣴⣿⣃⣼⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⣾⣧⣤⣄
⠀⠀⠀⢾⣿⣿⣿⠯⡹⢹⡦⠄⠃
⠀⠀⠀⠀⠀⠀⢹⣄⠁⡀
⠀⠀⠀⠀⡀⠄⣟⠫⣥⡏⢲⢀⠀⠞⡓⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣠⡀
⠀⠀⠀⡞⠀⡀⠉⠑⢿⠋⡉⠀⠑⡞⠀⠀⠁⠀⠀⠠⠄⠀⠀⠔⠡⠦⠙⢿⣿⣿⣷⡄
⠀⠀⠀⣧⠀⠹⣦⢶⣌⣢⢵⣂⠘⠛⣰⣄⣠⡴⠀⠀⠀⠀⠀⠠⣦⢰⡆⢰⣀⣶⣿⣯⡀
⠀⠀⠀⢹⣷⣫⡽⣟⠛⠋⠻⠂⠰⢾⣿⢿⠟⠃⠀⠀⠀⠀⠀⠒⠒⠒⠂⠚⠛⠛⠛⠛⠛⠲
⠀⠀⠀⠀⣿⣿⣤⡈⠑⠤⣤⠶⠤⣤⢀⣸⠖
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣤⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣾⢿⣻⠄⠀⡠⠀⢸
⠀⠀⠀⠀⢸⣿⡯⠀⠀⢰⡇⠀⡜
⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⠃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⣋⠉⠉⠉⢉⡛⢷⣦⡀⠀⠀⠀⠀⠀⣿⣉⣽⣏⣹⣏⣹⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣧⣀⣉⣀⣀⣀⣈⣁⣀⣿⢿⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⣴⠶⣦⠀⣠⡄⢀⣴⢶⣄⠀⣿⠟⣿⡇⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡇⠀⠿⣶⠟⠀⢻⠇⠘⢷⡾⠟⠀⣿⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣇⢀⣤⣤⠀⣠⠀⢠⡄⢠⣤⣤⠀⣿⠉⠉⣀⣴⠶⠶⢶⣤⣀⣠⡶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⣠⣤⣤⣤⣤⡄⠀⠙⣿⡏⠀⢰⡟⠁⠀⠀⠀⠙⠿⠋⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣿⠉⠉⠉⢹⣇⣠⡾⠋⠀⠀⢹⡇⠀⠀⢠⣿⠛⣷⠀⣿⡇⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠷⠶⠶⠾⢿⡟⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⠻⣿⠀⣿⡇⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠿⠿⢿⣄⠀⠀⠀⠀⣼⡿⠶⠾⠶⠶⣦⡀⠈⠻⣦⡀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢋⣴⢶⣄⣀⣀⣈⣿⣄⣀⣼⠿⣦⣀⣴⠟⠙⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⣿⡶⠟⠙⠛⠛⣻⡟⠙⠉⠀⠈⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣷⣄⠸⠷⠶⠶⠶⠶⠶⠶⠿⢃⣴⡟⠛⠛⠛⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⠟⠀⠻⢷⣶⣶⣶⣶⣶⣶⣶⣶⡾⠛⠿⣦⣤⣤⣤⣤⣤⣤⣾⠿⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣿⠀⠀⣻⣇⠀⠈⠛⢷⣦⡀⠀⠉⠉⠉⠉⠉⠉⠻⠿⠟⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⣻⡷⠿⠛⠛⠉⠀⢾⣏⣻⡇⠀⠀⠀⠈⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⣉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢀⢤⡔⠂⠤⣄⠀⠀⠀⠉⠒⢤⡀⠀⠀⠳⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡼⣁⣀⠼⠂⠀⡤⣱⡀⠀⠀⠀⠀⠈⠲⡄⠀⠀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢀⡼⠤⠤⠥⠤⠤⠯⠤⠬⠧⡄⠀⠀⠀⠀⠀⠈⢦⣸⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠳⡙⠛⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠉⡇⠀⡔⠉⠲⣤⠶⠶⢆⡔⠋⠲⡄⢸⠉⡇⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠒⡇⠀⠳⠤⠔⢹⠀⠀⠨⠳⠤⠴⠁⢸⠖⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡹⢯⠀⠀⡨⠯⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣑⣒⣒⣒⣒⣊⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⠇⣀⣸⠀⠀⠀⢀⢀⡀⣀⡀⠀⠀⢸⣀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡞⡸⠁⢹⠀⠀⠀⣮⢧⢷⡻⠀⠀⠀⡸⠀⢣⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⡏⠀⡇⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠒⠁⠀⠸⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢸⠀⠀⠀⠀⠀⢠⢏⡙⡄⠀⠀⠋⢹⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡌⠙⠒⢄⣎⡎⣱⢱⡀⠀⣤⣼⠀⢀⡔⠋⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠟⠀⠀⡼⣨⣭⣥⡀⢧⠀⣿⣿⠀⠸⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠸⠷⠁⠀⠀⠘⠾⠇⠳⠟⠀⠀⠀⠀⠠⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢢⠀⠀⢄⣔⣤⢰⠂⠴⢦⢂⢠⣒⡄⡴⠆⢸⠄⠀⠀⠀⠈⡠⠒⢤⠒⢄⣴⢢⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠈⠀⠉⠀⠡⠀⠐⠁⠀⠈⠨⠈⠢⠀⠁⠀⠈⠃⠀⠀⠀⠀⠈⠤⠉⠠⠀⠉⠈⠐⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⣴⣾⣯⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣨⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⡦⢻⣿⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⣜⣿⣷⢘⣿⣟⣤⡀
⠀⠀⠀⠀⠀⠀⠛⣿⡟⡃⣹⣇⠀⠀⠀⠀⠀⠀⠀⡴⣶⣿⠈⡏⢹⣷⡶⡄⠀⠀⠀⠀⠀⠙⠻⠟⠻⣶⡋⢁⡾⠗⠀⠀⠀⠀⠀⠛⣿⢽⣿⣿⣿⢹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣟⠆⣻⡿⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⡇⢸⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠛⠙⠋⠋⠛⠙⠋⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⡆⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠐⠒⠶⣆⣀⣠⣤⣴⣳⠄
⠀⠀⠀⠀⢀⣟⠁⢿⣿⣿⣿⠇⢹⣇⠀⠀⠀⠀⠀⠀⠛⠛⣠⣤⡘⠛⠃⠀⠀⠀⠀⠀⢀⣀⣤⡟⠛⠈⢠⣾⡏⠀⠀⠀⠀⠀⠀⣸⣿⣻⣀⣄⣸⣻⣿
⠀⠀⠀⠀⠯⠈⠇⠘⠿⡿⠟⠀⠇⠨⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⣷⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣥⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋⢙⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⡦⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢠⡄
⠀⠀⠀⠀⠀⣀⣤⣤⣄⢠⡄⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣼⣯⢿⢣⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣛⡿⠃⠀⣴⢶⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣵⣶⠒⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠾⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣧⣽⣿⢿⣹⡿⣘⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠾⠀⠰⠗⠀⠺⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⣿⢟⣿⡟⢯⣾⠏
⠀⠀⠀⠀⠀⠷⠿⠿⠟⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠸⠿⠗⠾⠿⠒⠿⠿⠀⠀⠀⠀⠀⠀⠸⠶⠾⠶⠗⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⠸⠚⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⢺⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢬⣿⠂⠀⠀⠀⠀⠀⠀⠀⣰⢖⣧⣾⢻⣦⡓⢶⡀⠀⠀⠀⠀⠀⠀⠸⣿⣍⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣬⣟⠛⣩⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⠿⠨⣟⣵⣿⡇⠀⠀⠀⠀⠀⣤⣀⠈⠉⢁⣠⣤⣄⡀⠀⠀⠀⠀⠀⠉⠀⢿⣿⣨⣿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣆⢦⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⢿⣯⣾⣿⣮⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣜⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠖⠀⣿⣿⡇⠐⠄
⠀⠀⠀⠀⠀⠀⠙⠋⠙⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣤⡄⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⣠⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠔⠋⠙⠁⠀⣴⣿⠿⢋⠞⠀⣠⡤⢛⡽⣠⠄⡀⠉⠻⠂⠀⠀⠀⠀⠀⠀⠄⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⣰⠎⢀⣤⣿⡿⠁⠐⢁⣴⠟⠋⣴⣯⣾⡿⠛⠛⢁⣤⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⡼⣯⡤⢾⡿⠋⣤⣤⡶⠊⠁⠀⠈⠉⠉⠁⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠸⠟⠋⠀⠛⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⢶⣦⣄⣠⣤⡀⢠⣷⣿⣿⣿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠏⠀⠀⠈⠉⣻⣿⢻⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠋⠉⠛⠉⠉⠉

⣪⣷⣴⡖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣆⡀⠀⠀⠀⠀⠀⣠⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠂⡄⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢬⣿⠀⢀⡴⠃⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠜⠁⠀⣠⠀⠄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠦⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣠⣖⣐⠀⢁⣠⣴⢧⡤⢤⡀⠀⡀⣠⣢⢀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢠⣷⣿⠟⠋⠉⠙⣶⣤⣤⣤⣌⣵⣷⣾⡏⠀⠘⠻⣿⣿⣇⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠁⠀⢠⢀⣴⡿⠋⠉⠈⢻⡅⠀⠠⠱⡀⠀⠐⠹⢿⣿⠄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢳⡴⠟⠉⠀⠀⠀⡠⠿⣷⠀⠀⠀⠘⡄⠀⠂⠰⢝⡻⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⣨⠞⠋⠀⠀⢰⣶⡰⠋⠀⠀⢸⣷⡀⠀⠠⢈⠆⠀⠀⠈⠁⠐⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠁⠀⢀⣦⡤⠡⠋⠀⢀⣤⣴⣏⠀⠹⣄⠀⠀⠲⣔⢤⣀⡀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣶⣾⠗⠁⠀⣀⡜⠛⠁⢀⣠⠀⠀⢈⢦⡀⠀⠻⠿⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢠⣴⠶⢫⠤⠶⠒⠉⠁⠃⠄⠈⡥⠷⠦⠀⠒


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

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⡤⠀⠤⣤⣤⣤⡤⡤⠀⠀⠀⠀⢀⣠⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠉⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⢀⡴⢋⠉⢉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠤⣀⣀⡜⢁⠂⠉⠋⠀⢸⠀⢸⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡰⠋⡆⠀⢩⠀⠸⠄⠀⡇⠀⢸⠀⠘⣠⠴⠒⢒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡜⠁⠀⡖⠚⠒⡆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢻⠀⠀⠀⠧⠤⢄⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⡄⠀⠀⡇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠦⣀⠉⠹⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠀⠖⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣁⣿⡄⣿⣷⣦⠤⠰⢿⣿⠃⠀⢙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢻⡆⠈⣄⡛⠀⢀⠉⢻⣿⣿⣿⣿⣿⡄⠀⠈⠁⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⣿⣿⣿⡟⠁⠀⠀⣤⡄⠃⠀⠸⠙⠋⡇⠀⠀⢅⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⢰⣿⣾⣿⠃⠀⠀⠀⢿⣷⣾⣿⣂⣠⣦⡅⠀⠀⣾⡇⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢻⠂⠀⠀⠀⠋⠙⢛⠛⠻⠟⢿⡇⠀⠀⣯⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⢿⡇⣸⠀⠀⠀⣶⣤⣤⣟⣠⣞⣁⡶⢥⠀⢠⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠁⠀⠈⠁⢿⡀⢀⢸⣿⣏⣠⠈⢙⣿⣿⣿⣿⠀⢸⡿⠀⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠃⠀⠉⠺⠆⠉⠾⠭⠀⣾⡥⣀⠈⢛⠀⣾⠃⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⠄⠻⠷⠶⣾⣯⣤⣿⣤⣼⢁⣿⣤⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⡍⠀⠀⠀⠀⠈⢿⣿⣿⣉⠀⠙⠃⢙⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⢠⡙⠿⠛⠁⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⠿⢿⣿⣦⣰⣤⡶⠟⢋⣡⠀⢀⡘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⣼⡿⠀⠀⣭⣿⣿⣿⣿⣿⣷⣦⣤⣿⡇⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⣰⣿⠇⠀⠀⢤⠘⢻⣿⣿⣿⠋⠉⠉⢸⣿⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣧⡀⠀⠀⢀⡀⠿⣿⣿⡄⢀⣀⣀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⣴⣦⣘⡇⢰⠆⣿⣿⢠⠀⣿⡇⠸⣿⣿⠃⢺⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣭⠉⡏⢠⡌⣿⡏⣼⡆⣿⡇⠀⢻⡏⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⠀⡇⢸⡇⣿⡇⣨⡄⢻⡇⣰⠈⢡⡇⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣬⣥⣤⣇⣸⣇⣿⣅⣿⣇⣸⣇⣼⣧⣼⣇⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠛⣷⠒⢿⣋⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣧⡾⣐⡿⠠⡾⢿⣿⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⣿⠋⢻⡉⢹⢛⡗⢺⣫⣹⣻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣻⣿⠟⢧⣤⠛⢷⠟⢻⠅⣟⠉⣹⡿⢿⣿⣿⣿⡿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡻⣻⣿⣊⣿⡋⢻⡿⢿⡶⠚⣇⣤⡀⢻⡤⢌⣲⣾⣿⢿⣿⣿⣧⣸⣮⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣏⣿⣀⠿⡿⠶⡞⢑⠶⢿⠁⣱⡿⢦⣼⠛⢢⠷⢶⣿⡟⣿⣿⣯⣿⠏⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠰⣥⡇⢀⠀⠀⠀⠞⠉⣳⠷⠴⢿⡗⢦⡞⣧⣀⣧⠾⢤⠽⢿⣿⡏⠻⠟⠯⠿⠏⢀⢀⠀⠀⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⢲⠾⡙⠉⠉⠙⠦⠚⠓⠚⠉⠁⠀⠀⠀⡠⠄⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡄⠀⠀⠀⠒⠁⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠚⠿⣤⠀⠀⠀⠀⠀⢸⡿⢿⢿
⠀⣀⣶⣶⣴⣶⠀⣠⣀⠀⣀⣄⣤⢀⣀⣠⣶⣿⣿⣶⣶⡆⠠⡀⠐⢶⣶⣾⣿⣿⣦⣶⣴⣶⣟⣢⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⡞⣷⣶⣆⣤⣿⣿
⠙⢻⢛⠛⠏⠉⢉⠉⠉⠩⢋⡉⠹⠉⠛⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠈⠀⠉⠁⠀⠉⠿⠟⠉⠉⠉⠉⠁⠀⠀⠿⠀⠀⠀⠀⠠⠥⠾⠿⠿⠗⠿⠿⣿⡿⠿⣿
⢠⠇⡀⠀⢠⠐⠀⠁⠀⡉⠀⠀⠁⠂⠄⠀⠀⠄⠀⠀⠀⠐⠒⠈⢉⡉⠉⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⠦⢄⡈⠉⠲⠤⣈⠓⠤
⡘⢠⠀⢸⠀⠆⠀⠡⡀⠈⠑⢄⠀⠀⠢⠀⠁⠢⠄⡀⠐⠤⡀⠀⠀⡀⠐⠀⢀⡤⠔⠀⣁⡠⠐⠀⠀⠀⠤⠀⠀⠉⠀⠄⠀⠀⠀⠁⠢⢄⡈⠑⠶⢄⡀⠙⠲
⠁⢼⠁⢠⡀⠐⡄⠀⠐⡄⠀⠀⠹⢄⠀⠀⠁⠂⡀⠈⠲⣄⠈⠓⢄⡀⠑⢄⡀⠀⠀⠉⠁⠀⠐⠂⠀⠀⠒⠀⠀⠠⠤⠒⠚⠀⣀⡀⠄⠀⡈⠐⢤⡀⠉⠳⣄
⠀⢸⠀⠈⡅⠀⠰⠀⠀⠘⣦⠀⠀⠀⠱⡄⠀⠀⠈⠢⢄⠈⠓⢤⡀⢙⠒⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⠰⠶⠦⠉⠠⠀⡤⠀⠛⠢⡀⠙⠳⣄
⠀⢸⡂⠀⢿⠀⠠⠆⠀⠀⠘⢀⣐⡂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠰⠄⠀⠀⢄⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠠⣤⣄⠈⠁⣀⣀⡀⠙⡤⠀⠨⠆
⢀⠀⠠⡄⠀⠤⠀⠀⠀⠀⢀⠀⠀⣄⠀⠤⡀⠑⠀⠀⠀⠀⢀⠙⠂⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠰⠂⠀⠰⠶⠤⠀⠀⢀⣀⡀⠈⠀⡤⠄⠀⠒⢁⡀
⣀⠀⠠⠄⠐⠦⠀⠈⠃⠀⠀⢁⣀⡀⠀⡀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠁⢠⠀⠂⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠶⠃⠀⣈
⠈⠂⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠁⠀⠀⠀⠛⠁⠀⠀⠀⠐⠻⠁⢀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠀⠀⢀⠀⠁⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠿⠚⠋⠀⠀⠰⠖⠉⠀⡀⠀⠀⠀⠈
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⡤⠄⠄⠀⠀⠀⠐⠛⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠖⠀⠀⠀⠀⠠⡞⠛⠁
⠀⠀⠀⠀⢶⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠀⣤⠖
⢀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⣀⣀⣠⡄
⠻⣿⠖⢠⡶⢀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠶⠆⠄⠀⠀⠀⠀⠀⠰⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠛⠋⠉
⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠚⠀⠀⠀⠀⠀⠀⠀⠺⠚⠐⠂⠒⠀⠀⠀⠀⠘⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠠⣤⠴⠶⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⡁⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡇⠀⠀⡅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⡄⠘⡃⠐⠋⠈⡁⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢅⡤⠀⠊⢃⡀⡀⠴⠁⠉⠉⡰⠀⠊⠀⢀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠒⡀⠀⠀⢁⣀⡄⢤⡀⣀⡼⠂⠀⠈⡈⠁⠀⠤⠠⠴⡀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠘⣄⠀⢀⠟⠂⠉⢩⡆⠀⠀⠀⣵⣀⠀⠀⠀⠇⠀⠂⠀⠑⠀⡰⠒⢈⠠⡀⠀⣃⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠉⠀⠈⡊⠀⠁⠈⣤⠀⡄⢉⠁⠲⠎⠀⠀⠈⠲⠶⠇⠀⠠⢀⠘⠀⠊⣅⠀⣸⠤⠷⡋⠀⠈⠹⡞⠑⢦
⠀⠀⠀⠀⠀⠀⠀⡞⡵⠀⠑⣤⠴⠚⢧⣤⣤⠎⠀⠀⢱⠜⠀⡴⠂⢄⣠⠴⠀⢄⠀⠐⠁⠈⢑⠆⠀⠀⣹⠁⡀⡄⢸⣀⠠⡾⠏⢂⣼
⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣴⡟⠀⠀⡘⠁⠀⢡⡴⠲⠋⢀⡀⢠⠀⠂⡍⢀⠀⠀⠱⠆⠀⣤⠘⠒⠀⠀⠘⠦⡴⠒⢿⢹⠀⢁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠳⢦⣞⣅⠁⣀⠌⠀⡆⠀⡦⢀⠃⠐⠉⡟⠀⠉⠑⠁⡠⠀⠀⠁⢠⣀⠀⠀⢠⣆⣀⣼⣿⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⡀⠉⠙⠒⠶⠾⣷⣞⣣⢌⡀⣀⡠⣇⣀⣀⡠⢷⢧⣀⣀⣴⡻⠿⠶⢞⠟⣉⣤⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⡦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠚⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠔⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣗⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣤⣀⡀⡀⡀⠀⠀⡀⣀⣰⣧⣾⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠷⠞⠛⠛⠋⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⢰⠈⠉⠉⠉⢉⡍⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠐⠈
⠀⠀⢸⠀⠀⠀⠀⠊⠘⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠃⠈⠠⠤⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⠤⠤⠬⠄⠀⠤⠀⠀⠶
⠀⠀⢸⢀⡌⠉⠉⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠇⡄
⠀⠀⢸⠀⡆⢀⠴⠀⠦⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠤⢀⠀⠀⠀⣤⣀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠘⢆⡀⠀⢀⠶⡀
⠀⠀⠸⠀⠀⠁⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⣉⣉⣁⣀⣉⣤⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠉⠁⠀⠈⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠟⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠒⠒⠒⠒⡒⡂⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠁⠀⠀⠀⠁⠈⠁⠘⠓⢠
⠀⠀⠀⢅⢹⠉⠛⠉⠀⠀⢸⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠉⠉⠁⠀⠀⢸⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂⠀⠀⢠⠆⠀⡼
⠀⠀⠀⠓⠀⠀⠐⠒⠒⠒⢚⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠄⠀⡄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⣀⣸⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠉⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⠠⢁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡡⠒⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⢀⠠⢀⠤⣠⣶⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢷⣄⠀⠀⠀⠀⣀⡀⠼⡧⢐⡒⠈⠂⠈⠀⠁⠉⠉⠛⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡒⡀⡼⠛⠓⠚⠉⠉⠉⠔⠁⡣⢌⠙⣌⣐⣄⠐⣠⠀⠖⠀⠰⢓⡩⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⢀⠺⣄⠫⠀⠀⠀⡀⠄⠦⣀⡀⡩⠚⠈⡉⠝⠀⣀⡢⠅⣒⣩⢥⡶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⢳⠠⡹⣇⡠⠶⠦⢀⣀⡤⣤⡮⠍⠭⢗⡶⠮⠽⡆⠒⠏⣊⣥⣶⣶⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠱⢤⡈⠺⢟⢺⡯⠭⠶⠒⣒⡋⠡⣈⡥⢔⣨⣕⣶⡿⢟⣿⡿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠈⠀⠙⢦⠉⣶⣦⣤⣖⣺⠭⣟⣫⣽⣻⣝⣫⣾⠟⢋⣥⣶⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡂⢴⡆⠩⣭⣔⡒⠊⣩⣽⣯⣵⣾⣿⣾⣥⣴⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢗⣿⠂⠑⣲⣾⣭⣥⡿⡷⠿⠟⠛⢋⠁⡀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠀⠀⡈⠉⠩⢈⠁⠈⠀⠀⠐⠠⡂⢀⠁⡨⢀⠅⡈⢐⠁⠁⡀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀⠀⠀⠤⣤⠖⡲⡯⡭⠏⠲⠖⢒⣲⣒⡿⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠗⢃⡀⢴⡐⠀⠠⣼⠳⠲⡝⠳⢾⡧⢖⡚⠚⠛⣡⠖⠨⠟⣛⢒⠢⢤⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⣠⢊⡥⠄⠀⠈⠀⠒⠐⠂⠀⠞⣀⣀⠀⠳⠶⠀⠸⢥⣬⠀⠃⡀⣀⡄⠤⡤⠥⠋⠁⣀⣀⣳⡄⠉⣃
⠀⠀⠀⢀⡠⢚⣁⡀⠀⡀⠀⣀⠀⠀⢳⡂⠀⢒⠀⠙⠒⠀⠀⠠⠴⠤⣤⡴⠚⢩⠿⢀⡤⠘⠀⢦⣁⣠⡀⢀⣀⠈⠫⡑⢀
⠀⡤⣚⡁⢀⠈⠉⠀⠘⠁⠀⠓⠒⠀⠀⠀⢄⡈⠙⠒⠒⣀⠐⠶⣀⠄⢀⠀⣠⡄⢀⠀⠀⠀⠀⢀⠈⢀⠤⠀⢁⡀⠄⠙⣨⠇
⠀⠘⢍⡴⠒⡴⣞⡉⢁⠙⠂⠀⢢⡥⢠⠀⡠⢄⡤⠖⢦⣓⣄⣠⡆⢀⡀⠠⢭⠈⢁⣀⣀⡀⣤⣀⣒⣃⡴⠇⠀⠠⠔⠒⣋⢤⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⢀⡌⢁⠘⠚⢋⠀⡀⠀⠀⢤⡈⢤⠴⢍⠐⠂⠻⣄⣙⣁⢓⣉⠏⠁⠀⠀⢠⡆⡦⠀⠄⠀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠗⠿⠗⠀⢠⠀⠀⠀⠉⠛⠁⠀⠒⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠈⠁








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠊⠉⠉⠉⠉⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⣉⡉⠁⠀⠀⠀⢀⣤⣄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⣾⣿⣿⣷⠀⢿⣿⣿⣷⣄⣠⣶⣿⣿⡿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⡿⣛⣥⣤⣄⡙⢿⣿⣿⢻⡿⠿⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠉⠁⣸⣿⣿⣿⣿⣿⠀⢉⣱⣿⣿⡄⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠠⣿⣿⣿⡿⠁⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⢀⣀⣀⠠⠤⠔⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠀⠀⠳⠞⠃⠀⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⣀⡤⣤⡀⠀⠀⢀⣤⢤⣀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣼⠁⢀⡀⠙⡆⢰⠋⢀⡀⠈⣧⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⠖⠋⠉⠛⢿⡀⠀⡇⢸⠀⢀⡿⠛⠉⠙⠲⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢰⡇⢰⠋⠉⠀⠀⠁⠀⡇⢸⠀⠈⠀⠀⠉⠙⡆⢸⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣨⣧⡈⠁⢤⡀⠀⠀⠀⡇⢸⠀⠀⠀⢀⡤⠈⢁⣼⣅⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼⠁⣠⠉⠀⠀⠉⠉⠉⠙⡇⢸⠋⠉⠉⠉⠀⠀⠉⣄⠈⣧⠀⠈⠷⡼⠃
⠀⠀⠀⠀⢠⡞⢶⡀⠀⢷⠀⠳⣄⠀⠀⣀⣤⠀⠀⡇⢸⠀⠀⣤⣀⠀⠀⣠⠞⠀⡾⠀⠀⢀⡇
⠀⠀⠀⠀⠈⠓⠚⠁⠀⠈⢳⠦⠤⠤⠞⠳⡄⠀⢦⡇⢸⡴⠀⢠⠞⠳⠤⠤⠴⡞⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢻⡀⠀⢠⠴⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠦⡄⠀⢀⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠓⢦⡏⠀⠀⠀⡦⠀⡇⢸⠀⢴⠀⠀⠀⢹⡴⠚⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢷⡀⠛⠚⠁⣰⠃⠘⣆⠈⠓⠛⢀⡾⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠉⠛⠚⠋⠁⠀⠀⠈⠙⠓⠛⠉⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⡀⠀⢠⡴⢦⠀⠀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠵⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠔⣢⠟
⠀⠀⡜⠃⠀⠀⣀⣀⣀⡀⠀⠀⠸⢑⠿⣄⣤⠴⡀
⠀⠠⡱⠊⣤⣈⡀⠐⠁⠈⡃⠰⠀⠀⠀⡘⣿⢹⡵⠲⣄⣀⠀⠀⠀⢀⣤⣤⡤⣤⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⢀⠓⠋⣸⣿⣿⢾⡷⠠⢏⢣⣀⢀⡼⠞⢃⠈⠹⠗⠫⠽⠄⣀⠀⠀⣭⣹⣁⠈⠄⠁⠟⠲⣴⠀⢘⣶⣀⣾⡄⡬⣇⠀⠀⠀⢀⢀⡀⣦⣆⣄
⠔⠋⠉⠁⠘⠻⠿⣿⠇⣾⡎⣊⢃⣘⣅⡈⣮⣄⠠⠄⠤⢀⣠⠖⠚⠀⠓⠒⠀⠐⠚⠁⣵⣚⣿⠋⡉⠘⡖⣿⢤⠄⠘⠄⠀⠀⠜⢳⠙⠆⠈⢈
⢁⠖⠦⣻⣿⠧⣄⠙⠀⠙⡈⠚⢚⡗⠇⠀⢘⣦⡂⠄⢀⣾⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣟⣽⠧⣤⠄⢻⡀⠓⣷⣖⡳⠀⢀⡱⣄⢲⣠⣀⠺⡷⣄⡀
⠀⣁⡀⠘⢉⠠⡄⠙⠛⢻⠇⠀⢨⠁⢸⠁⠉⡻⡇⠘⢿⣻⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⡼⠛⠟⢱⠜⠀⣨⠤⠄⣀⣀⣀⣘⣧⠂⡈⢠⡈⠃
⢖⣿⡥⣾⠿⣶⣷⣶⣶⡯⢤⠶⢿⢦⣶⠦⣾⣿⣿⣭⠟⣿⠀⢘⣦⣀⠀⠀⠀⠀⢀⠆⠀⡄⠀⡷⢁⣜⡠⠞⠛⠘⠛⠀⠦⣏⣩⡟⠏⢻⣽⣿⣟⠛⠓
⠀⠉⢠⠀⣉⡹⠿⠏⡸⢀⠚⠀⢨⠀⠀⠀⠸⠸⡇⢫⠔⣁⠔⡻⣃⠼⠛⡷⡶⠒⢺⠀⣰⡕⣯⢃⡎⢣⣱⠈⠀⠀⠀⠀⠀⠈⠉⠻⢶⡈⠉⠀⠁
⠀⠀⠀⠈⠁⠄⢀⢰⡁⠀⠀⢀⣸⠀⠀⠀⠠⡱⣿⡞⡎⠀⠀⢱⢇⠀⣐⣅⡇⠀⢸⢸⠁⠱⣹⡂⠀⢀⡕⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠜⠳⠔⠤⠝⠿⢟⠛⠛⠠⠐⣼⣇⡇⠐⠒⢾⣵⣷⣈⠉⠛⠀⢸⡞⠀⢠⣻⠣⡀⢼⣸⡂⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠈⠐⠒⠁⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢂⠈⠉⠁⠀⠀⠀⠐⢂⠉⣐⠤⠤⣸⣐⡀⠀⠰⠿⠧⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠐⢺⣏⡴⠂⠀⠀⠀⡀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠄⠀⠒⠊⠙⠉⠉⠁⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠓⠉⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⣀⠀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠃⠤⠴⡥⣤⣤⣯⢽⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢩⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣯⣿⠏⠋⠿⠻⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⡲⠀⠀⠀⠈⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⠄⠀⠀⢀⠀⠀⣀⣾⣿⣗
⠀⠀⠀⠀⠀⠀⠀⡞⠙⢺⡅⣂⣀⣾⣅⣤⣀⣄⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠵⣫⠄⠰⠀⣀⠀⠘⢮⠓
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠙⠋⠃⠉⢻⠛⠉⠉⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡥⢞⡡⠀⠝⠀⢦⠈⢧⢸⡦⣯⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⠀⡆⠀⡄⠀⠉⠀⠞⠀⣦⡘⢧⡙⢷⠹⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⢸⠇⣿⢠⡇⡐⠀⢀⣁⠄⠈⠹⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⢁⠞⣱⠟⡇⠀⠋⠀⢠⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⠄⢳⠤⠀⠀⢀⢠⠀⣠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠀⠀⠀⠀⠀⠀⢀⠉⠃⠀⡁⣰⡇⣿⠋⣴⠇⣴⠏⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⡈⠠⣁⠀⠤⢿⡁⣿⠄⢻⠀⣯⢼⠃⣶⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣷⡟⡇⠽⠀⡀⠘⢗⣻⡆⠘⠆⠛⠈⠁⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠸⠻⠋⠃⠋⠀⠐⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⣁⡄⠀⠀⠀⠀⠀⠀⠀⠤⠬⣝⢧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡜⢹⠛⠲⠀⠀⠀⠀⠀⠐⣠⠓⠮⠩⣁⠈⠳⠂⢄⠀⠀⠀⠀⠚⡻⠟⠀⠀⠂⠀⠀⠀⠀⠀⢔⣴⡅
⠀⠀⠀⠀⠀⠀⢸⣃⡿⠀⠀⠀⠀⠀⠀⢈⡰⠳⣤⡌⢓⣊⠽⣁⠰⠏⢀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⣘⡾⡽⠖
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠈⠈⠑⠄⡀⠴⠀⠁⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠀⢄
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⠠⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡆⢐⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡀⠀⠀⠆⠆⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠉⠂⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⡀⢀⠤⢤⡴⠔⠂
⠀⠀⠀⠀⠀⢀⠈⠰⣖⠨⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠐⠢⢤⣰⠊⠀⢁⠐⠐⢅⠀⠑⢔⠒⠀⠔⠠⠖⠀⠑⠀⠁⠀⠉⠁
⠀⠀⠀⠀⢇⠀⠁⠀⠀⠈⠐⠀⠀⠀⢠⡄⣀⠀⠀⠀⠐⠀⠩⣅⠀⠠⠊⠀⠀⠀⢁⢀⢠⣤⣍⡶⠖⢚⠻⠛⢛⠉⢛⠑⠛⠟⡫⠌⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⢑⣄⡠⠠⢒⠽⠐⢊⠡⠂⣁⠔⠈⡀⠔⢀⠄⢊⠠⢂⠄⠈⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠅⠀⠀⠀⠀⠀⠀⠉⠑⠒⠪⡔⠉⢹⡠⠂⢁⠔⠊⡠⠐⠁⠔⠁⢀⠄⠀⡠⠊
⠀⠀⠀⠀⠀⠀⡼⠁⠈⢢⠁⠂⢄⡀⠀⠀⠀⠀⠐⠊⢀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢰⣆⢀⢠⠀⠙⠓⢲⡅⡠⠂⠀⠠⠂⠁⡠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠈⠀⠂⠄⠐⠚⠈⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠈⠉⠍⠛⠀⠀⠨⠀⢀⣙⠨⠳⡒⠠⢪⣀⡀⢁⠠
⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠃⠀⠀⡀⠀⠀⠀⠀⠁⠐⠐⠀⢉⠁⠀⠂⡐⠀⠀⠈⠀⢀⠀⠐⠉⢔⡒⠠⡉⡑⠢⢰⡒⠬⠉⡐⢄⠉⡒
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢨⠖⠂⢙⠇⠀⠀⠀⠀⡀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⡌⡡⡚⠉⠑⡀⠠⣈⡀⠂⠨⡀⠀⠳⢔⠪⢄⡑⠬
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣀⠤⠾⡇⠠⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠜⡍⣌⡠⠐⠠⢠⠉⠳⡆⠠⡈⠉⠒⠤⡉⠢⠈⡑⠤⡈
⠀⠀⠀⠀⠀⢀⠀⡠⠐⠘⡅⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠥⠤⣁⠂⠬⣑⠒⠢⣈⠐⠎⣁⠂⠬⣐⡀⠤⡉⠐⠬⡒⠠⠁
⠀⠀⠀⠀⠂⠉⣀⢦⠐⠉⠀⠀⠀⠂⠀⠀⠀⠀⠠⠐⠀⢠⠴⠀⡤⠀⢅⠐⠬⢁⠂⠢⡉⠀⠤⣉⠢⢄⠉⠀⢀⠘⠢⡈⠀⠀⠤⠁⠠
⠀⠀⠀⢀⡰⠖⠉⠀⠀⠐⡀⠀⠀⠀⠀⠀⠠⠈⢠⢀⡴⢋⠔⠑⢄⠢⢌⠑⠒⢤⠈⠁⠺⡑⠠⣀⡑⠢⡉⠐⢄⠙⠒⠠⢈⡠⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⡔⢊⡁⠢⢀⠑⠢⡉⠑⠠⡈⠓⣄⠈⠂⠤⢌⠢⢌⡑⠠⣕⠠⡀⠀⠁⠪⡐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠄⠘⠢⠀⠑⠀⠈⠁⠢⠈⠂⠀⠑⠀⠀⠀⠡⠀⠠⠀⠈⠡⠀⠑⠀⠀⠘⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠛⡛⠉⠉⠛⠛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⢡⣄⠀⢀⡓⠒⠄⡀⠸⠂⣀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠘⠄⠨⠔⠓⠓⢢⠄⠀⠅⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⣄⡐⠀⠠⡅⠀⠀⢸⠤⡄⠀⣀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠰⠆⠀⠸⢈⡁⡥⠀⢇⡀⠉⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣬⣥⣤⣄⡀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⠃⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣄⡀⠀⠀⣠⢾⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⢀⡼⠁⠀⠀⠀⣈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢏⣾⣷⣦⣄⢠⣾⠟⣛⠟⣷⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⢯⣾⣿⣿⣿⣿⣮⣛⠀⢩⣞⣸⡇⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣝⡁⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣕⢮⡙⠻⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠙⣿⣿⣿⣿⣿⣷⡝⠳⣄⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠁⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⠦⣄⠀⠀⠀⠀⣠⢞⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⡳⣶⣶⡞⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⢸⣿⣿⣿⣿⣿⠈⠉⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠻⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡐⠆⠧⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡊⢠⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡗⣦⢠⣄⣤⡀⠀⢀⣽⣿⣿⡇⠀⣟⣋⡁⠀⠀⠀⠀⠘⣊⣗⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠁⠀⠀⣀⣀⣁⠉⠛⢤⣿⣿⣿⣿⠃⢰⣿⣿⠿⣞⣁⣀⣠⣤⣭⣾⢿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⣿⡿⣽⣳⣄⡙⢻⣿⣿⠀⣼⣿⡿⠛⠋⠉⢉⣉⣠⣤⣴⠶⠋⢩⣼⣭⣙⡂⡀
⣠⡶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡿⣘⡹⣧⣹⣿⣠⣟⣠⣤⣶⣿⣿⣿⣿⣿⣝⣽⠚⡤⣔⠀⣀⠠⠘⠉
⢿⡁⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⢟⣿⣿⣿⡟⢮⣻⣼⣿⣿⡉⠓⣯⠁⠀⠀⠉⢻⡶⠶⡶⢄⡼⠋⠁
⠀⠻⣄⣢⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠿⠻⡿⢻⡿⣧⣉⣿⣿⣯⢙⣱⠿⠹⣿⣆⠀⢀⡞⠄⣠⠄⠉
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠈⣧⠈⠻⡇⣿⠰⠈⠉⡋⣿⣷⣾⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⢀⡤⠃⠀⠘⠀⠀⡇⠉⠉⠁⠛⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⠀⢘⠙⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣠⡇⡀⠊⠀⠀⡰⠁⠀⠀⠀⢀⠁⣀⢀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣴⣓⣓⣚⣏⣉⣛⣀⣀⣀⣀⣈⣃⣃⣚⣀⣙⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⡀⠀⠀⢀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣝⣽⠇⠀⡀⠀⠀⠀⠀⢀⣿⣦⠀⢯⣇⠁⠀⣀⣿⣥⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⢐⡠⠞⣿⣴⡾⠁⠀⠀⣴⢀⠀⣿⣧⣆⡠⠟⠻⣏⠁⠙⢶⣆⠉⠻
⠀⠀⠀⠀⠀⠀⡜⠉⢀⣤⡀⢸⡆⠀⠀⠀⠀⠀⠀⠋⠞⣉⣈⣻⢷⡇⠀⢰⣿⣴⠀⢈⡽⠻⣷⣶⣆⠉⠛⠆⠀⠈
⠀⠀⠀⠀⠀⠰⢀⣤⣾⢟⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⣶⣻⡄⠈⠻⣿⡖⠃⠔⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠚⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣦⣾⡿⠶⠶⠶⢦⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠦⠀⡀⢰⡶⠄⠀⠀⠀⠀⢠⡯⠀⠀⠀⠀⠀⠀⠙⠳⠟⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣦⠠⡀⠘⠃⣸⣿⣷⡤⠘⣿⡛⠳⢦⣄⠀⠸⡇⠀⠀⠀⠀⠀⣀⠀⠀⡤⠀⠀⣠⠟⠇⢀⡺⢁⢰⣿⠛
⠀⠀⠀⠀⠀⢀⡀⢻⣦⣤⠘⢷⣄⣐⣯⡭⢭⡭⠴⣧⡤⢿⣷⠀⣨⢷⡾⡇⠀⠀⠀⠀⣾⢿⡀⠘⡇⠠⠀⣿⣡⠇⣿⣧⡏⢸⣿⠂⣀⡤⠤⣄
⠀⠀⠀⠀⠊⢄⠐⠂⡙⡻⢋⡟⠁⢼⠃⠀⠘⢀⠀⡿⠀⠀⠹⣄⠁⠀⠀⢽⡀⠀⢀⣶⣤⠀⡇⠸⢿⢂⣀⠻⠧⣴⣜⠛⠋⠋⠉⢹⡍⠛⠳⠾⠷
⠀⠀⠀⠀⠀⠀⠀⢄⠔⠁⣸⠁⠀⣋⠄⠠⡄⠂⠀⠙⠁⠀⠀⠀⠀⠀⠀⠘⣧⢀⡆⠘⣻⣦⠰⡆⠈⠉⢸⡇⠀⠐⢿⡄⠀⢾⣄⠀⣿⣲
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⡔⡞⠉⠙⢲⣇⡘⠀⠹⢿⠏⠀⠙⠟⠀⠀⠙⠃⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡟⠀⣅⠐⠀⢂⡻⣟⠆
⠀⠀⠀⠀⣤⠀⠀⢰⡂⠀⠐⣧⠠⠀⠀⠀⣾⡓⢄⠀⢰⣇⠢⠀⠈⢷⡥⠀⢸⠃⠀⠈⠀⠀⠙⠣⣌⡛⣨⡇
⠀⠀⠀⠀⢻⣦⡀⠈⣷⣴⡄⢻⣷⣳⠀⠀⢻⣷⣼⡀⠘⣿⣧⡡⠀⠨⡐⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⢔⣿⡿⢿⣿⡦⠮⣻⠷⢦⡿⠻⠶⢶⡶⠭⢿⣤⡄⠈⣿⡿⠀⠀⣑⣸
⠰⠤⠉⠀⠀⢰⡿⠁⣾⠟⠀⣼⠣⠂⣤⢋⠔⠀⣠⠌⢀⣀⡴⠖⠄⣦⠙⢿⣆
⠀⠀⠀⠀⠀⠊⠀⠰⠃⠀⠀⠻⠁⠀⠿⠀⠀⠀⠙⠀⢸⡋⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣀⢀⡀⠙⠄⠀⢸⡏⠀⠀⠀⠀⢿⠓⠀⠀⠀⠀⠀⠠⡤
⠀⠀⠀⠀⠀⣀⡶⢤⠀⠀⢠⣶⣶⣆⠀⠁⠀⡒⠶⠦⣄⣈⣷⡀⠀⠀⠀⠀⠈⠆⠀⢀⢽⣤⡀⠀⠀⠘⣾⡀⠡⠀⢀⣄⣀⠎
⠀⠀⠀⠀⠸⡏⠀⠈⠃⢠⣸⣇⢸⢿⠀⠀⠀⠠⡀⣠⢴⣾⣿⠁⠀⠀⠀⠀⠀⠀⡀⣿⣆⢻⣷⣇⣤⣤⣿⣧⣄⣀⢸⣿⠇⠠
⠀⠀⠀⠀⠀⠈⠓⠒⢦⡀⠙⢿⡇⠈⠀⠀⠀⠀⠁⣇⣴⣿⠹⠇⠀⠀⠀⠀⠀⠀⠡⢀⡏⠀⠈⡸⠁⠀⠀⡟⠀⠀⢉⠋⢛⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⡠⠈⢳⣦⡀⠀⠀⠀⠀⠻⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⢀⠀⠀⠇⡀⠀⣶⢃⠀⢰⣼⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⠉⠀⠀⠈⠳⣦⠍⠀⠀⠀⠻⡖⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⡠⡕⢿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⢠⡇⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⣉⡉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣶⣶⡄⠸⣿⡿⠒⠒⠒⠒⠺⣿⡿⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠋⢀⣀⡀⠙⠋⠓⢤⣀⣀⣀⣀⣀⣀⣴⣦⠀⠀⣾⣷⡄⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠸⣿⠟⠒⠒⢤⡀⠀⠀⠀⠀⠀⠈⠛⠛⢀⡴⢋⡋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⣀⡀⠀⠀⠀⢠⣾⣄⠀⠀⠀⢀⣠⠞⠉⠉⠀⣿⣿⠆⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠸⣿⣿⡀⠀⠀⠘⠿⠋⠀⠀⠀⢿⣿⠇⠀⠀⠀⣰⠃⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠛⠛⠛⠛⠛⠒⠢⣀⣀⠀⠀⠀⠀⣞⠁⠀⠀⠀⢼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⢲⣤⣀⠘⢿⡿⠀⠀⠀⢾⣿⡇⢠⣶⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣀⣀⠀⠀⢀⣤⡀⠀⠀⡼⠛⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠻⠿⠟⠉⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⠀⠀⡴⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠆⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⢀⡀⠀⠀⠀⠀⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⣧⣤⠀⢉⣹⡆⣿⠉⠁⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠀⠀⠉⠀⠀⠀⣧⠀⢀⡟⢸⡇⣿⠀⠀⡏⠋⣸
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⢈⡄⠠⠀⠁⢸⠀⠀⠀⣿⠀⠘⢧⣸⡇⣿⠀⠀⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠂⠱⡄⠈⠈⠁⠀⠀⠀⠀⢀⣠⠀⢀⢶⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠸⣇⠃⠈⣷⠁⣿⠀⡖⠛⢤⠄⠀⠀⡄⢸⡅⠀⠀⠀⢀⣀⡀⢸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡆⠁⠠⠁⠀⠀⠀⢿⠀⠀⡇⢠⡇⠀⣇⢀⠾⠀⡘⠙⡇⢈⣷⡎⠑⠀⣾⠀⠝⠈⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠁⠁⠉⠁⠀⠀⠀⠀⠘⣇⠀⡇⠸⣆⡰⠀⢀⣀⠉⠀⠈⣇⠌⢻⠘⢄⡠⠛⢦⣀⠔⢳⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠟⠈⣡⠄⡀⠀⣼⠉⠀⠉⠀⣠⢤⡀⠈⠇⠀⠀⠐⡖⠀⢀⡀⣑⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⣦⢻⠀⢸⡆⡇⠀⠀⠀⢰⠇⠀⣷⠀⣿⡀⣿⠀⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⢸⠶⣏⠀⣧⠀⢺⡆⣿⠓⠉⣿⢠⡇⢳⣿⠀⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠳⠴⠃⠸⠀⠘⠆⠘⠦⠞⠀⠛⠀⠀⠿⠸⠃⠈⠇⠠⠿⠀⠛⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡼⢀⠤⠒⠒⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⡴⢾⠏⢤⠱⢄⡀⠀⣠⡼⠦⣄⢀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⢀⡴⠒⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⠀⢰⡁⠀⢀⠙⠶⣞⠛⠋⠉⠛⠻⣷⢦⡉⠲⡄⠀⠀⠸⡄⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣿⠶⣼⣧⡀⠈⠀⠀⠈⠳⡀⠀⠀⠀⠈⠳⡞⠆⠸⢀⣀⢀⣸⠀⠀⠀⠀⠐⠮⣧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣸⡿⠐⣅⠀⠙⢶⣄⣀⡀⠴⠷⣄⣀⠀⠀⠀⢱⣴⡷⠛⠉⠉⠉⠉⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣿⠗⠊⠉⠙⠓⠲⠿⢽⡏⠒⣄⠀⠉⠉⠲⣿⠷⠋⠀⠄⠂⠀⠀⠐⠊⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣾⠿⡃⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⠁⠀⠈⠉⠐⠂⡿⣶⣀⣀⣀⣀⣤⡤⠾⠓⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡤⠻⠶⠶⢤⣤⣄⣀⡠⠶⠛⠲⢦⣤⢶⣤⣴⣿⡵⠛⠋⠉⠉⠳⢦⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⣇⠀⠀⠀⠀⠀⢀⣈⣎⠀⠀⠀⠀⠀⠙⠛⠋⠉⠛⣄⠀⠀⠀⠀⢠⡞⠓⠶⢤⣤⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠣⡀⢻⠓⠒⠶⢦⡶⠟⠋⠳⣄⣀⣀⣀⡼⠀⠀⠀⠀⠈⠴⣤⡴⠞⠛⡄⠀⠀⣀⠤⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⣠⠋⠁⠀⠀⣠⠟⠛⠛⠉⣦⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡠⠋⠀⠀⠀⠱⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠟⠀⠸⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢽⣧⣄⣀⢀⣴⠇⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⣼⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡎⠙⢾⣁⢴⠀⠀⠀⣠⠶⠛⠉⠉⠉⣯⠀⡆⡀⠀⠀⣠⡾⠀⠀⠀⠀⢰⣇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⢻⢶⢦⠟⠋⠀⠀⠀⠀⠀⢀⠆⠀⠳⣧⡴⠖⣿⠤⠀⠀⠀⣠⠏⠈⠙⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠒⠠⢀⠀⠀⠀⠀⠠⡀⠈⠀⢻⡇⠐⠉⠁⢀⣠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠛⠻⠔⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⠀⠄⠀⠀⠡⠀⠠⠀⣿⠒⠒⠚⠉⠀⠀⠀⣀⡤⠶⠒⠚⣓⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠢⣀⠈⠂⠀⠀⠀⠀⠀⠑⢄⣀⡟⠀⠀⠀⠀⠀⢠⠊⠁⠠⠤⠀⠈⠓⡟⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠐⠶⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠋⣠⠀⢀⠄⣠⠀⣤⢀⡀⠀⡄⢠⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠂⠐⠧⠀⠈⠓⠻⠄⠟⠀⢧⠘⠳⠸⠄⠫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣄⡀⠈⠐⠁⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⣿⣦⣙⠦⣄⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣙⠻⣷⣮⡑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠞⣡⣶⡆⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡙⠿⣷⣌⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢋⡴⠋⠀⠀⠀⠀⠀⠈⠁⠀⣄⠀⠀⠘⠀⠀⡰⠁⠀⠀⠀⠴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⠁⠀⠀⢀⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡠⠤⣀⡤⠊⠁⠀⢰⠁⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠰⠁⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠒⢤⠇⠈⠢⠀⠈⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⠤⠔⠃⠀⠈⠀⠀⠀⡇⠀⠀⠺⡅⠀⠀⠀⠐⢄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠑⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡠⠀⢀⡇⠀⠀⠀⠀⢸⠀⠘⢄⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠦⠖⠒⠒⠀⠈⠉⠣⠀⠀⠹⠌⠑⠀⠀⣤⡎⠄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠀⠀⠀⢀⠼⡁⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢤⣶⠆⠉⣁⡀⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠿⠃⠀⡀⠈⣛⠿⣖⣒⠫⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠓⠦⠍⠑⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




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