Distaste Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠚⠋⠁⠀⠀⠀⠀⠙⠷⠘⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣆⠀⠀⠀⣤⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠙⢷⣄⣀⡙⢷⣄⡀⠈⠻⣦⣀⣀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠇⠀⣀⣤⠍⠁⠀⠈⠙⠛⢤⣤⣉⠉⢷⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡸⠏⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⠾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢿⣠⡟⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠀⠀⠘⠋⠹⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣴⣖⡾⣯⡉⠛⠲⢤⡀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⣿⠻⢦⣄⣀⠀⠀⠀⣀⣠⠶⠛⡏⠙⠲⢦⣄⠀⠀⠀⠀⠀⢀⣠⠶⠛⢉⣩⠿⣒⣶⣤⡀
⠀⠀⢾⣻⣶⣮⡝⠃⠙⣧⡀⠀⠙⣦⠀⠀⠀⣰⠟⠁⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⡇⠀⠀⠀⠙⢧⡀⠀⠀⣰⠟⠁⠀⣰⠏⠁⢻⣽⣶⣾⣷
⠀⠀⠘⠳⢭⣉⡃⠀⠀⠈⠉⠀⠀⠘⡇⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⣧⠀⢰⡏⠀⠀⠈⠉⠀⠀⠘⣋⣭⠾⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣄⠀⠀⠀⠹⣆⣾⠀⠀⢠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡆⠀⠀⢹⣄⡞⠁⠀⠀⣠⡤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠹⣇⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⢨⡟⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠹⣶⠶⢾⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡷⠶⢶⡟⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠙⠀⢸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡇⠀⠛⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⡴⠟⠙⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠃⠙⠶⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣄⣠⡀⢠⣴⣿⡉⡉⠁⠂⠄⠀⢀⡀⢀⣴⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⣹⣿⢿⣿⡿⣿⣿⠛⣿⢶⣄⣀⣶⣾⣷⡟⡟⣿⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣗⡇⣿⢸⡛⣟⡿⣿⢸⣯⢸⣿⣯⣽⢼⢠⡇⣷⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣇⠿⣿⡿⡟⠉⢛⣦⣧⣼⣿⣷⣾⣿⣾⣷⡁⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠋⠀⠈⠀⠀⠰⣇⣾⣿⣻⡿⠋⠀⠙⠉⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⢩⡁⠀⠀⠀⠀⠀⠀⢠⡄⠉⠛⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⣴⠳⠄⠀⠀⠀⡀⠠⠚⢦⡀⠀⠀⠀⠉⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢠⡀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⢰⣿⠁⠀⣄⢀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠃⠀⠀⠱⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⣀⣿⢀⡼⣯⠀⠀⠐⠀⠀⢠⡆⠀⠀⠀⠀⠀⢿⣧⢠⠀⠀⠀⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡺⠀⠀⠀⠘⠁⣠⠉⢘⣾⡇⠀⠀⢲⣤⠶⠿⠀⠀⡀⠀⠀⠀⢸⣿⠿⠀⢀⣀⠇⠀⠀⠀⠹⣦⣰
⠀⠀⠀⠀⠀⠀⠀⢰⣻⠄⠀⠀⣾⠦⢀⣉⣈⣸⣿⣧⠀⡄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⢸⣿⣶⠀⡀⠉⠀⠀⠀⠀⠀⣃⡹
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⡼⠉⢹⢿⣿⣿⠀⠳⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⢸⣿⣏⠑⠒⠂⠁⠀⠀⠀⠠⢿⣳
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⠁⠀⠀⠐⣿⣿⠀⠀⠐⢷⣦⡠⣤⣼⣿⡆⠀⠀⢸⢻⡇⠀⠀⠀⠀⠀⠀⡐⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⠀⠀⠀⢳⠹⠅⠀⢄⢈⢳⣿⣿⠿⠛⠏⢱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣧⡶⣄⠄⠀⢸⡀⠀⠀⠸⡇⠀⠀⠉⡼⠛⡉⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠠⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣟⡿⡻⢠⣤⣼⡇⠀⢠⣶⣷⡖⠀⠠⠤⠛⣡⢂⠀⣵⠇⠀⠈⠖⠀⣤⣄⣤⠀⢸⡄⠠⣷⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣽⣾⣟⣛⢟⣶⣛⣾⡋⠀⠰⠀⢠⣼⣁⡾⠀⠏⡀⡄⠰⣀⡀⡜⢿⣧⡀⢸⡇⢑⢈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣽⣇⡀⢀⢈⣽⣟⣁⡠⠂⠀⡧⣿⣿⣾⣷⠌⢻⣟⢷⠿⣧⣘⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⢻⣵⣬⣭⣷⠠⠀⢩⣿⣿⣿⣿⣯⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⢉⣉⡉⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠸⠰⠎⠹⠒⠏⠊⢰⠟⢾⠲⠉⠯⢦⠷⠎⠋⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⢄⣀⠄
⠀⠀⡠⠔⠲⠋⠀⠀⠀⠀⠀⠀⠀⠐
⡐⠁⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆
⠱⡀⠀⡀⠀⠀⠘⢄⠀⠐⠀⠐⠀⣦⠀⠉⢰
⢰⠀⡐⠃⢀⠄⠢⡈⡆⠀⢠⠔⠁⢠⣇⣀⡄⠀⠠⠄⠀⠀⠒⠂⠀⠤⢀
⠈⠲⠀⠑⠁⠀⠀⢡⣼⠀⢁⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡀⠈⡉⠛⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠈⠁⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠒⠐⠀⠠⠐⠒
⠀⠀⠀⠀⠀⠐⠐⠀⡀⠀⣀⣀⣈⣠⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠒⠀⠀⣀⠈⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢁⣠⡤⢄⠱⡀
⠀⢀⡄⠀⠀⠀⠀⡇⠀⠀⠸⣁⡀⠀⠀⠡⡈⠲⠄⢀⡠
⠀⠀⠙⠲⢭⡑⠂⠧⢀⡀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠐⠀⡰
⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠉⠑⠂⠠⠠⠀⠀⠐⣤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢈⠓⠲⠤⣀⠀⠀⠀⠀⢀⠎⠣⣹⢲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡐⠢⢄⡜⠀⡀⣈⣄⢷⡣⡀⠀⠀⠀⠀⠀⠄⣉⣈⠵⠎⠀⠀⠀⠑⢲⢖
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢦⠠⡄⠾⠏⠋⠉⠉⠀⢳⠑⣄⢀⠀⠀⡄⢼⠂⠁⠀⠀⠀⠀⠀⠀⣨⠎
⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠳⡌⠢⣀⠀⢀⠤⣜⢂⢣⣏⡀⢠⢤⠜⠈⠳⢶⣔⣒⠣⣤⣯⠞⡁⠇
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠙⠃⣸⠀⢸⠨⡤⠭⢤⠈⠌⡉⠉⡩⠒⠛⠉⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠑⠆⠒⠒⠉⠉⠈⠒⠒⠒⠤⠷⠾⠿⠹⠳⠶⠰⠖⠖⢒⠁⠀⠀⠁
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠟⠛⠍⠁⡀
⠀⠀⠀⠀⣠⠄⠐⢶⠤⣤⣤⣤⠄⠀⠀⠠⠀⠠⢄⣠⣤⢱⡶⠂⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⡠⠤⠖⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠀⢸⠀⢀⠴⠒⠒⠒⠲⠤⢄⣀⣀⣀⣀⡴⢻⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡸⠁⡇⠀⠘⣄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠃⠀⠀⢹⡀⠀⠀⠒⠂⠀⠀⠀⠐⠒⠀⠀⢸⠁⠀⠀⡀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢧⠀⠀⢠⠖⡻⡽⡕⢢⠀⠀⢀⠏⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡵⣳⢤⡀⠀⠀⠈⠣⡀⠈⠓⡇⠁⡼⠋⠀⣠⠊⠀⠀⠀⢰⠁⢾
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠈⠓⠢⠬⠭⠤⠖⠊⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⣲⠒⠒⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘





⡀
⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⣆⠀⡆⢢⡰⠐⠂⡐⡄⣶⡀⠀⠀⣆⠀⠠⣀⠀⡆⢰⡰⠰⠀⠆⣦⠂⣆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣤⣤⣤⠀⠀⠀⠐⣤⣤⣤⣤⣤⣤⠀⣆⠀⢀⠄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⣿⣿⣿⣿⣶⠀⣀⣰⣿⡟⠛⠉⠛⢻⣇⠙⠀⠙⠀⣿⠏⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⠟⠋⠛⢿⣿⣿
⣿⣿⣿⠛⠁⠀⠘⢻⣿⡇⠀⠀⠀⢠⠿⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⣿⠟⠁⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿
⠸⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⣶⡆⠐⠶⠾⠅⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣀⣀⣀⣀⣠⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣴⣄⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢻⠀⠀⣸⠁⠀⠀⢀⣤⡤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⠀⣿⠀⢀⡞⠉⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠶⠶⠶⠖⢻⠁⡿⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠃⠀⠀⠸⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠇⠀⠀⠀⢻⡇⠀⠀⠀⠠⠴⠆⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠘⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⡄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢰⠋⣴⠲⡖⢦⡈⡧⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠟⣷⣀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠳⣿⠀⠁⢸⠖⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⣼⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠵⠚⠑⠾⠗⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠘⠦⠴⠋⢀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⠟⠛⠛⠛⠛⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⡖⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠐
⠓⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⠤⣀⠀⡀⢤⡤⠀⣄⠀⢠⠤⡀⢀⠀⠀⡤⠄⠀⢀⠤⠀⢤⡤⢀⠤⣄⠀⡠⢄⡀⡀⢀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⡇⠀⢸⢨⡇⢸⠀⢰⠩⡆⢸⠒⡇⢸⠀⠀⡗⠂⠀⠈⠢⡄⢐⡇⢸⠆⢎⢸⠀⠀⡗⡰⢧⠀⡗⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣶⣧⣼⣧⣾⣶⣷⣤⣶⣾⣶⣧⣾⣶⣶⣷⣶⣤⣶⣶⣧⣴⣧⣼⣦⣾⣦⣷⣾⣤⣷⣴⣷⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⢿⣷⣆⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⡿⠿⠛⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣖⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣴⠋⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢳⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢻⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠀⠀⠀⢀⣠⡴⢻⡟⣿⠀⣿⣿⠷⣤⣀⠀⠀⠀⣀⣤⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢯⣉⠙⠻⣿⡏⠁⣰⠟⠀⣿⠀⣷⠘⣧⡈⣽⣿⠛⢋⣩⠟⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠙⠳⠶⠤⠶⠟⠁⠀⢀⡏⠀⣿⠀⠈⠻⠦⠤⠶⠛⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⣠⠾⠁⠀⠀⠀⠙⢧⣄⣀⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢋⣡⡤⡴⠶⠒⠒⠶⢦⢬⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠉⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠉⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣤⣄⣀⣇⣀⣀⣸⣤⣤⣤⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠉

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

⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡗⠂⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⣀⠈⠈⡄⠈⠛⠿⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣯⣄⣦⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⠉⣍⣶⣿⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⡙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠘⢹⣿⡇⢼⡏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣇⣠⡞⢠⠀⠀⠀⢻⣿⣿⣿⡄⣄⠀⠀⢸⣿⣀⡀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢋⣴⣿⣧⠀⣀⡀⢻⣿⣿⢧⣿⣷⣿⣿⣿⠛⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⣿⣿⣷⣯⣭⣿⢇⣾⠿⠁⢸⣿⣿⣿⣿⣿⣇⢸⣿⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣋⡛⠻⠟⠑⠯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣯⣾⣷⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣁⣿⣿⣇⣠⣆⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣉⣛⣛⣻⣿⣗⣀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⣰⣶⡾⠿⠿⠿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇⣴⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠛⠓⠀⠀⠈⠁⠀⠀⠀⠈⠙⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠉⠉⠉⠉⠉⠉⠙⠙⠋⠉⠉⠛⠋

⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠒⠋⠉⠰⡞⠛⠉⠁⠀⠀⠀⢡⠋⠀⠁⠀⠀⢄⠀⡾⢀⠸⡄⠀⠀⣽⡧⢀⠅
⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠀⠀⣀⠀⠀⢈⠑⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣰⡷⣽⡇⢸⢀⡇⢀⡾⠟⢀
⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢀⣾⠿⢰⠏⠀⠀⣀⡀⠀⠀⠸⡿⢿⣶⡶⣿⠿⢸⡟⠓⣁⠯⡴⠛⠁⢠⠎
⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⣴⢟⣥⣶⣿⣂⠠⡀⢿⡧⠀⠀⠀⣠⡖⠀⣨⡷⠚⢋⠤⣡⠛⡆⢁⠀⠀⡞
⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⢣⡞⠉⠋⠉⠏⠀⠀⠘⡗⠂⣠⣤⣶⠶⠞⠉⣠⣴⣺⡎⠸⡄⣠⣸⣯⣙⡁
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠘⣇⢰⢲⠁⠀⠀⠀⠀⣷⠘⠁⢸⣾⣆⣐⣺⣿⣍⣋⡛⢾⣿⣿⣿⣿⣏⡇
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⣽⣇⡏⠀⠀⠀⠀⢰⣿⡀⠈⠛⠻⣿⢿⣍⣿⣿⣿⣃⣽⣟⣿⣿⠿⠋
⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⢿⣧⣇⠀⠀⠀⠀⢿⣿⡿⣀⣀⠤⣻⣷⣽⣶⣟⠻⢿⡟⣹⡵⡇⠀⠀⠀⠀⠠⢤⣤⣀⣀⣀⣀⣀
⡇⡀⣰⣄⣸⣿⣿⣿⣿⣿⣿⣿⣟⣆⡙⠳⠀⢠⠂⢸⣿⣯⣯⣵⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢝⡦⣤⣤⣤⣤⣤⡤⠤⠤⠄
⣇⣈⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡖⠀⠇⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢸⠞⢧⠢⡽⠷⢦⣤⣤⣄⣀⣀⣀⣄⣠
⡏⣉⣽⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠻⣿⣧⣄⡀⣇⢇⠈⠻⣿⣿⣍⡛⡛⠿⠭⠤⠤⠶⠖⢞⢡⣴⣾⡆⠀⠀⠀⣿⠟⠛⢻⣿⠻⢿⣭
⡟⢻⣿⣿⣿⣿⣧⣿⣿⠖⣿⢽⣇⠀⠹⣿⣿⣿⣿⡈⠂⠀⠈⠻⣿⣿⣿⣿⣶⣶⣶⣧⣿⣿⣿⣿⠿⣧⠴⣀⣠⠇⠀⠀⠀⣹⠛⠻⢿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⠄⢙⠶⠾⠿⣄⢿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣹⣿⣿⣿⣿⡿⠿⠿⠟⠛⠁⢠⢟⣶⣏⣈⡷⣤⣀⡸⣍⡓⠂

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠠⠀⠄⠀⠀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠸⠈⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⠀⠀⠀⢦⡀⠐⠈⠀⠘⢦⠀⠀⢀⡠
⠀⠀⠀⠤⢄⣀⡇⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡸⠀⠀⠀⡖⠁⠀⠀⠀⣀⠴⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⡦⠤⠏⠀⠀⢀⡀⠀⠨⢀⡀⠀⠀⠈⠂⠠⡀⠀⠘⠦⡠⠔⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⢀⠄⠁⠀⠀⠁⢀⣀⠇⡠⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡁⠈⠢⠂⠀⠉⠂⠁⠀⠀⢀⡠⠔⠊⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⠉⠉⡂⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⠷⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⡀⡀⠀⠀⠁⠂⠀⠂⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠳⠐⢆⠄⠁
⠀⠀⠀⠤⠐⠒⡆⠀⠀⠀⠁⠀⣱⠀⠀⠀⠀⠀⠀⠀⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆
⠀⠀⠀⠀⠀⠀⠃⠀⢀⣀⠠⠄⠊⠀⠀⠀⠔⠈⠁⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⣀⠠⠘⠉⠁⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⣀⡒⠀⠀⠀⣰⡄⠤⠬⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠄⠂⠈⠀⢴⠀⠀⠀⠀⣰⣞⡣⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠠⠺⣀⡠⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⠀⠀⠤⠤⠤⠄⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⢶⣾⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⢻⣿⠁⠀⠀⠀⠀⠸⠿⣯⠏⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⣀⣿⣿⣟⣿⣿⣯⣤⣤⡙⡀⠀⠀⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠶⠿⣿⢿⣿⢟⣿⣿⣿⡿⣿⣿⣟⠀⠄⢀⣼⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠉⣹⢀⡀⣸⢿⣿⣿⣿⣿⣿⣧⠒⢛⣿⣝⣶⣿⠸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣿⠀⠀⢹⡆⣠⣿⣿⢻⡾⣿⣿⣟⠃⠐⢿⣿⣿⣿⡟⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠈⢻⡅⢹⣧⣿⣿⣿⣿⣿⠙⠻⣿⣤⣴⣿⣷⣿⣿⣿⣷⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢺⣿⣦⣿⣿⣿⣿⣿⣿⣾⡷⣉⠛⠿⣿⣿⣿⣿⣿⣿⠀⠆⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣭⣍⠛⢿⣿⣶⣦⠘⣏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣉⣻⣆⡿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠉⠀⠛⠛⠛⠋⢹⡿⠛⣿⣿⠉⣿⣿⣿⡇⣸⡏⢸⡇⣿⣽⣿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠁⣸⣿⣿⣆⢹⣿⣿⢣⣿⠃⢸⣧⡧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠈⢧⠀⠀⠀⠀⢾⣿⣿⣿⢹⣮⣿⣿⣿⣿⣇⠈⣿⣣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣷⡤⣄⢨⡛⠂⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣀⣷⣤⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠷⣦⡙⣏⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠇⠘⠛⠻⠙⠋⠕⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡼⠁⠹⢷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣿⣿⡆
⠀⠀⠀⠀⢀⡾⠋⠛⠳⠀⠀⠈⠛⠛⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⡇⠹⣿⡇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣾⣇⣼⠟
⠀⠀⠀⠀⠀⠘⠷⣶⢀⣠⠂⠀⠀⠀⠈⣩⣿⣿⣿⣿⡿⢷⠀⠀⠀⠉⠻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⢨⣰⡏⣄⠸⠿⠃⡴⠋⠉⣿⣿⣿⡁⠈⠀⡀⠀⠀⠀⠙⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⠀⠀⡜⠁⠀⢠⣿⣿⣿⣧⣴⠀⠁⠀⢠⡄⠀⠘⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣠⣿⡿⠻⣿⣿⣿⠀⢠⠀⣸⣿⣇⣀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⢉⠧⢶⣿⢀⡇⠈⠛⠻⢿⣦⣤⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣟⣛⣻⣯⣙⣛⣟⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣯⣭⣿⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢠⣀⠀⠀⢠⡀⠀⠀⠀⣀⣶⠤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣤⠘⢦⡴⠊⠁⠠⠟⠉⢺⡁⠀⣀⡶⠟⠉⢀⡠⠒⠒⠤⡀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠘⠐⢓⠚⣓⠐⣟⠀⠀⠀⠀⢀⡤⠞⢿⣾⠋⠀⠀⢠⠃⠐⢠⠀⠀⠈⢳⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⠄⠻⠉⣆⠀⠀⠂⠈⢀⡤⠾⡿⠀⠀⠀⠁⠀⠀⣰⣀⠀⢀⣰⢷⡀⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⠀⠀⢀⣀⠀⠁⠚⣲⠿⢷⣄⠀⠀⠀⠀⣀⣴⣀⡴⠾⠤⠤⢾⣿⣿⣿⢿⣿⡄⠑⡀⡷⠀⡬⡀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⢹⣷⣀⣀⣽⢶⠒⣨⣶⣈⣤⢿⣷⣄⠀⠁⠀⣿⣏⠉⢈⣀⣴⣄⣿⡉⢁⠀⠈⠙⣦⣼⡧⠤⠴⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⣀⢽⣿⡄⣼⣿⣿⣿⣿⣿⣀⣀⠽⢧⣙⣰⠛⢻⣷⣼⣿⣻⣅⣠⣈⠟⣠⡄⠀⠹⢧⡴⠀⡀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡑⢰⣧⣤⣈⠻⠛⠙⠛⠉⠉⠉⡀⠠⠛⢿⡟⠛⢿⠛⠛⠻⣏⣭⡉⠃⠈⠁⠀⠀⠀⢠⡌⠁⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣀⣠⠀⠐⠛⢙⡆⠀⠀⠀⠀⠘⡀⠀⠀⠉⠓⠲⢾⣷⢄⢲⡄⠀⠀⠘⢻⡄⠀⠀⠖⠀⠀⠀⠀⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⡁⢀⣆⠀⠁⠀⠀⠀⠀⢈⣴⣶⣶⣶⣦⣄⠈⢿⡄⠀⡃⠰⠀⠂⡼⡇⠀⠀⠀⠀⠀⠀⠈⠁⡈⠀⠈
⠀⠀⠀⠀⠀⡐⠂⠀⠰⠤⠀⠀⠁⢠⣾⣿⣿⡆⢆⠈⢿⣿⣿⣿⣿⣿⠂⠀⣿⣤⣙⠘⠃⣷⣾⠇⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠉⠨⢋⠴⠂⠀⠀⢰⡌⠛⣻⠿⢳⣶⡄⠈⠳⢿⠿⠃⡼⠀⠂⢠⢈⡅⠒⢠⣾⠃⠀⡆⠀⠀⠀⢀⡃⢀⠁⠂⠁⣰
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣼⣇⠰⡁⢀⡘⢻⡿⣦⡴⠏⡠⠙⠑⠦⠀⣼⠉⠀⣷⠉⡏⠀⣼⣶⡦⠀⠀⣌⣀⠀⠁⠀⡰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠉⠀⠘⣿⣇⡊⠀⠋⠙⠛⠁⠂⠄⡄⠇⣸⣿⡹⠀⠘⣿⡎⠀⢠⣿⠟⠀⢀⣼⡿⠋⠀⢠⣾⡀⠁
⠀⠀⠀⠀⢸⠀⠀⣀⢰⣿⣆⠀⠀⠀⠀⠀⢸⣤⣘⣄⠁⢦⠐⠖⠞⢋⠐⡿⠁⠀⠀⠎⠀⢀⣿⠃⠀⢠⡿⠋⠀⠀⣴⣿⠟⠃⠀⣠⠎
⠀⠀⠀⠀⠸⡄⠀⡉⠙⣾⠉⢄⢠⣈⣰⢲⣾⡀⠈⠈⠀⢀⡦⠾⠊⣡⠌⠀⠠⠄⠈⠀⠀⠘⠁⠀⠀⠊⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠱⡔⠻⢃⠙⢴⠃⠀⠀⢮⢀⣼⠿⣞⠋⣀⣉⡃⠤⠂⠁⠀⣰⠃⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⣴⠖⠚⠓⠛⠀⠀⠀⢀⣁⠀⠑⡄⠀⠀⠀⠀⠀⠄⠀⣿⠏⣀⠴⠲⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⡀⣇⣀⠀⠄⠀⣠⡀⠀⠈⣡⣴⠀⠣⣓⢲⣶⡆⢠⠀⣷⠋⠔⠁⣤⣴⡶⢶⣶⣾⡷⠃⠀⠀⢀⣼⣅
⠀⠀⠀⠀⠀⠀⠁⢀⢃⣀⠤⠤⡁⠀⠘⠛⠉⠀⠀⢃⣀⣉⡑⠈⠷⠖⢋⣠⡿⠋⠀⢀⣠⡉⠉⠰⠔⠂⣠⠊⠈⢹⣧⠀⠰⠁
⠀⠀⠀⠀⠀⢀⠘⠱⠈⠁⣴⣾⠇⠀⠀⠀⠀⠀⠈⠉⠱⣾⡝⢷⣶⠚⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠿⠧⢤⡴⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⣀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠀⠀⢀⠐⠘⠀⠀⣿⣯⡌⣤⣤⠾⢿⣆⠀⣠⣤⣤⣀⣀⣀⣠⣤⣴⣶⡟⠉⢻⡷⣦⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣾⣦⡀⠀⠁⠀⠀⢀⣴⣿⣿⣡⣿⣉⠞⢿⡿⡻⢿⣿⠉⢿⡏⠀⠀⢿⣿⠟⠁⢀⡾⠤⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⢿⣿⣷⡾⠿⢿⡟⡉⣿⣷⣾⣟⣿⢻⣷⠀⢻⡅⢺⡇⠀⠀⢸⠁⢀⣤⠞⢀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠔⠉⠀⠀⣀⣀⣘⣿⣧⣤⢿⡟⣿⣟⣟⣛⣿⣤⣦⣍⠁⠂⢀⡾⠖⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⢀⡀⠀⡀⠸⣿⣋⡼⢷⣟⡏⣹⣿⠿⡛⠛⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠈⠛⣿⣿⣬⠿⡏⠁⠀⠁⠀⢓⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢷⡀⠈⠳⡖⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢻⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠋⠉⠉⠁⠀⢐⣤⢴⣤⣀⠀⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢀⣯⣀⡴⣤⡈⠃⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿⣤⡍⢡⢛⣋⡇⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡇⢿⡼⢤⣌⣿⡇⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢼⡇⢀⣸⣇⣸⣿⣦⣿⣿⢁⠈⡤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡏⡿⢯⡻⣿⣿⠇⢇⠎⠞⣵⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡙⠾⣿⣼⣯⣾⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⡎⣿⣿⣦⡀⠙⢿⡿⣯⣲⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣷⢸⣿⣞⢿⣆⠀⡀⢈⠉⠻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣿⣿⠀⣿⡄⣿⣿⣆⠹⣷⣼⣶⣷⣄⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡅⣿⣿⢹⠀⠘⣧⢹⣿⣿⣦⡘⢧⠉⡛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣇⡟⣿⠈⢠⠀⠹⡆⠻⣿⣿⡳⣄⠱⣌⠃⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣾⣿⡇⢿⠀⣿⢰⠀⢻⣆⠹⣿⣿⣌⢓⢬⡛⢸⡇⠘⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣾⣿⢹⡇⠀⠀⣿⠸⣧⠀⢻⣷⡙⢏⢻⣿⣝⢦⢻⣧⠀⢹⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⡟⣿⢸⡇⠀⢸⣿⠀⣿⣆⠀⢻⣟⠦⠀⢿⣿⣮⢻⣿⠀⣼⡇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣿⠘⠁⠀⢸⣿⡄⠘⣿⡄⢠⠙⣦⠀⠈⣿⠈⠋⠃⣴⡟⢇⢻⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⠀⣿⣿⣇⠀⢹⣿⡀⢳⡄⠀⠀⣿⠀⢠⣾⢿⠁⠀⠈⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢱⠀⠀⠀⣿⡏⠹⡆⠀⢹⣧⠀⢻⣆⠀⠁⢀⣿⣷⡈⠀⠀⠀⠸⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⣆⠀⢻⡄⠀⠹⣷⣄⠘⢻⡿⣷⡄⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠘⠂⠀⢧⠙⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣏⢽⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠄⡤⠤⣶⣶⣶⣶⣄⣀⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣶⣲⡆⠈⠙⣿⣿⣿⣿⠛⣍⣩⣽⣉⣝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡽⠃⠀⠀⠁⠙⣿⠿⢷⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⡜⠃⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣍⠐⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⣿⠀⠉⠛⠿⠀⡀⠀⠀⠀⠀⠰⠿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢀⣴⢣⢼⡇⠀⠠⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⣤⠀⣽⣿⠟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠃⠈⢻⡌⣡⠺⣿⡿⠗⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠐⠉⠀⡰⠧⡄⣦
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣴⣿⠀⣢⣾⣏⡛⠁⠠⠁⠰⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢘⠀⠑⢾⠆⠐⣑⣿
⠀⠀⠀⠀⢠⠃⠀⠀⢠⡞⠁⡟⣰⣿⣏⠛⠁⠶⠤⠄⠠⣴⡂⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢤⣼⣶⡶⠚⡏⠸⡟⠉
⠀⠀⠀⠀⠸⠀⠀⡰⠋⠀⠀⡃⣿⣦⡍⠀⠚⠒⠒⠂⠀⠄⠈⠈⠉⠀⠀⠀⠈⠉⠉⠉⠉⠛⠃⠂⠼⠀⠀⠀⠀⡼⠏⠀⢀⡆
⠀⠀⠀⠀⣠⠤⠚⡁⠀⠀⢸⢠⡿⠶⠤⡀⣛⣗⠀⡇⠀⡑⠦⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⡆⠀⢸⠁
⠀⠀⠀⠸⢆⠄⠠⠀⠀⠀⡸⢸⣿⡷⣡⠭⠈⡉⠀⠀⢆⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⡾
⠀⠀⠀⠰⠼⠀⡅⠀⠀⠀⡇⠰⣶⢤⣙⣤⡠⣌⠲⣀⠀⡅⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢹⡇
⠀⠀⠀⠀⣧⠄⣀⠈⠀⢰⠇⠰⣿⡙⠷⠈⠋⠈⠑⠚⠁⠋⢁⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡀⠀⠄
⠀⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⠐⣿⡁⢜⠠⠀⠀⠀⠤⣐⠺⢿⠛⠉⠀⠚⠴⠦⠀⠀⡌⠀⠀⢀⣠⠄⠋⠀⠰⣿⡈⠁
⠀⠀⠉⠁⢸⡀⡇⠀⠀⡄⢸⣦⣌⠳⠠⠀⠁⣿⣿⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⣁⣴⡖⢀⡰⣄⠀⠗
⠀⠀⠀⠀⠘⡇⢀⠀⠀⠇⠘⣿⣿⣷⡀⠀⠀⣼⣿⣟⠀⠐⠀⠀⠀⠀⠀⠀⣁⣤⡀⠶⠚⠛⠿⠁⠈⠉⣤⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠹⠀⡀⢸⠀⠀⢸⡿⠓⢄⡈⠀⠘⡷⠛⠀⣠⣦⡤⠴⠒⠋⠉⠁⠀⠀⢀⣠⣤⣶⣶⣦⣼⣿⠁⣜⣷⣾⠃⠀⠃
⠀⠀⢀⣤⠶⣷⠂⠙⣿⢠⣾⣿⡷⠀⢸⣧⣠⣆⡐⠆⠘⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢸⡇⠀⠀⠈⠉
⠀⣄⠀⠀⠀⠈⠆⠐⠋⠈⠛⢿⣧⡀⣠⣈⠉⠉⠉⠲⠲⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⣿⣷⡀⢤⣀⠀⠀⡄⠀⠀⠀⠀⠈⠉⠙⠻⣿⡶⢤⣄⡀⠀⠈⠹⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀
⠀⣿⣻⣧⠻⠿⠘⣿⢆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⠉⠙⠒⠦⢤⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢠⠃⠁
⠀⣿⣟⢿⠉⠁⢸⡇⢸⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢇⠀⣀⠑⠎⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢀⡆
⠀⢽⣾⣧⡀⠀⢸⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⡀⠃⡈⠉⠒⠃⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠈⠉⢿⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡠⢖⣩⠔⠉⣤⠶⠶⠀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠒⢹⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣸⠀⢀⠴⣊⡬⠞⡋⠄⠒⠉⠹⣆⠠⠀⡀⠀⡷⡄⠴⡀⠀⢀⣦⡀
⠀⠻⣿⣿⣇⢠⠇⢀⣷⠀⠀⠀⠸⠧⠀⢀⡀⢀⣠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡠⠺⢿⣿⣆⠈⡄⠉⠛⠻⠷⠶⢦⠤
⠀⠀⠀⠈⠉⢸⠀⣿⣄⣀⡀⠠⠀⢀⡚⢁⣄⠉⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠦⢭⣝⣧⣈⠠⠄
⠀⠐⢤⡀⠀⡇⠀⠸⣿⣿⡏⠘⠛⠿⣇⠀⠈⡏⠀⢈⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⠔⠂⠠
⠀⠀⠀⠙⠲⠀⠀⠀⠀⠙⠻⢶⣤⣤⣠⣦⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠷⠶⣧⠤⠦

⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠛⠛⠛⠛⠻⣶⣠⣤⠶⠶⠶⠶⢶⣦⣤⣀
⠀⠀⠀⠀⢀⡤⠶⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣄⡀
⠀⠀⣾⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇
⠀⠹⣦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠈⠙⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⢶⣶⠶⠟⠳⠦⣤⣤⣄⣀⣀⣀⣀⣤⡶⠟⠋⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢶⣤⣄⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣦⣀⣄⢀⣀⠀⠀⠀⢀⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠿⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣻⡄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⢸⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⡀⠀⠀⠹⣿⡇⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣹⣿⣿⣿⣿⣶⡄⠀⢠⣾⣿⣿⣿⣿⢿⣁⠀⠀⠀⣿⣧⠀⠀⢰⣁⠀⠈⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⡜⢁⡤⠚⢛⢿⡿⠋⠀⠈⠻⠿⢻⡛⣦⡀⢙⡆⠀⠈⢁⣮⣽⣖⣉⣄⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠙⣦⣀⣀⣜⣉⣭⡙⠆⠀⠀⢣⠘⣄⣳⣿⠞⡇⠀⠀⠀⢸⣄⠘⠻⠟⠁⣠⡇⠀⠀⠈⢃⡰⠃⠀⠈⠓⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⢀⡠⠔⣎⠹⠀⢃⡀⠀⠀⠀⠓⠶⠿⠶⠞⠀⠀⣀⣀⡀⠙⠧⢤⠖⠓⠋⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣄⠀⣈⡗⠀⠈⠓⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠠⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠁⠐⠒⠒⠀⠨⠛⠛⢦⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⣀⡀⣀⣀⡀⣀⣀⣀⡀⣀⣀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣛⣛⣛⣛⣓⣒⣒⠒⢒⣒⣚⣋⣉⣁⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⠱⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣷⢿⡄⠀⢸⣿⣯⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⢹⡆⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣿⣿⣿⣿⡏⣾⠁⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣸⣿⣻⡟⣿⠇⣯⢀⣼⣿⣿⠟⠻⣿⣷⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡏⣿⣿⣿⡿⠀⢻⡟⠻⣿⠁⠀⠀⠈⢳⣿⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠘⣿⢀⢿⣿⣿⡇⡷⢸⡇⠇⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⡂⣿⢸⢸⣿⡿⢸⡇⣾⢿⠀⡆⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠃⢸⡟⢸⣿⣧⣿⠃⣿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣧⠀⠈⡗⠘⣿⢹⡿⢰⡿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⠀⡇⡆⣿⣸⠇⣾⡇⡆⢸⡇⠀⡿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⢸⠀⡇⡇⣿⠛⣸⣿⠇⡇⢸⡇⠀⣷⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⢸⠀⡇⡇⣿⡄⣿⣿⢠⡇⢾⣧⠘⡟⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⢸⠆⠃⡇⣿⠀⣿⡟⢸⡇⣼⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠙⠃⠀⠀⠀⠀⠀⢠⣿⠀⣿⢸⡃⠂⡇⣿⢰⣿⡇⢸⡇⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠆
⠀⠀⠀⠀⠀⠀⠒⠲⠔⠐⠒⠒⠲⠛⢢⡆⠀⠀⢠⣿⠛⠀⣿⢸⡇⠁⡇⣿⢸⣿⠁⣾⣿⣿⣇⡄⣿⣄⠀⠀⠀⠀⢲⠛⠔⠂⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠐⠶⠀⠀⠂⠀⣨⢏⣾⢰⡇⢸⡇⡄⡇⣿⢸⣿⠀⣿⣿⣿⣏⣽⣻⠏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒
⠀⠀⠀⠀⠀⠓⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠁⣸⣧⢸⡇⡇⡇⢻⢸⣿⠀⣿⡿⣻⣿⢸⠷⠀⠀⠀⠀⠀⠀⠀⠛⡤⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠛⠂⠀⣼⠏⠀⠀⠀⠛⣿⢸⣧⡇⡇⣸⠸⣿⠀⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢲⡆⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠛⠛⠈⢙⣿⢇⠀⠀⠀⠀⠀⡹⣸⣏⡇⡇⢹⠀⣿⠀⣿⡇⣿⣿⠀⠴⠆⠀⢀⠀⢠⡀⠀⠀⠀⠉⠀⢘⡁
⠀⠀⠀⢀⠀⠛⠒⠒⠒⠋⠁⠀⠀⣴⢻⠏⠀⠀⣀⠀⠀⢠⡀⣿⡀⡇⡇⢸⠀⣿⠀⢹⣷⢸⡿⡇⠀⣠⠈⠛⢀⣤⠉⢁⠄⠀⠒⠞⢋⣡⡖⢀
⠀⠀⠀⡄⠀⠊⠠⠴⢶⠶⣤⡤⢾⣷⣾⠀⠐⠚⠛⢁⡀⢸⡇⣿⢰⠇⡇⣼⠀⣿⡇⢸⣿⡘⣧⣿⠀⠀⠒⠒⣿⣀⣀⣀⠐⠲⠶⢶⡞⠉⢛⣉
⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⣠⠀⠸⢿⡿⠀⠀⠀⠀⠀⢀⣾⢹⣿⢸⠀⡇⣿⠀⣿⡇⠈⣿⡇⢻⣾⣷⡀⠴⠊⠉⠙⠳⣿⡿⠷⠶⠚⠛⢱⠿⠛
⠀⠀⠀⠀⠤⠄⠁⠤⣤⠤⠤⠆⠀⣸⣇⠀⠀⠀⠀⠀⣼⡟⣸⡇⢸⢠⡇⣿⠀⣿⣷⠀⢿⣿⠈⣷⣹⣷⡀⠀⠀⠀⠀⠛⠓⠀⠀⢀⣠⣧⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⣶⣜⣣⡏⠀⢀⣀⣠⣾⡿⢁⣿⠁⡏⢸⠁⣿⠀⣿⣿⡀⢸⣿⣇⠸⣷⡹⣷⣔⠀⡀⠀⠀⠀⠈⠻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣿⡏⠈⠁⠀⠸⢿⣿⣿⣟⣡⣾⡟⢸⠃⣾⢀⣿⠀⣿⣿⣇⠀⢿⣿⣆⢹⣷⣽⣿⣷⡤⠆⠀⠉⠐⠦⣾⣶⡤⠐⠊
⠀⠀⠀⠀⠀⠄⠀⠀⠄⢸⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⣠⠇⢰⡇⢸⣿⡄⢿⣿⣿⡄⠘⣿⣿⣾⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢠⣿⠏⠠⣿⣶⣏⣴⡿⢀⣾⣿⡇⣸⣿⣿⣿⣤⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠀⠀⢠⣿⠃⢠⣿⠏⠉⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡿⠁⣰⣿⠃⠀⠀⠰⠟⣿⣿⠛⠛⢻⣿⣿⣿⡁⠈⢿⣦⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣴⢤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣯⠟⢀⣼⡿⠃⠀⢀⣶⣤⣤⣿⣿⣦⣾⣿⣿⣿⣿⣷⣄⠀⠹⣿⡄⠀⠈⢻⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡿⠃⢐⣏⡹⠀⠀⠀⢸⣿⣿⣿⢻⡏⠀⣤⣈⡿⣿⣿⣿⣿⠀⠀⠘⢏⣵⠀⠀⠈⢹⣛⣦
⠀⠀⢀⢀⡤⠀⠀⠀⣔⠛⠿⠉⠀⣰⣛⡏⠁⣀⠀⠂⠈⢁⣽⡿⠼⠷⠠⣿⢻⡀⠀⠨⠀⣀⣤⡄⠀⠀⠹⡛⣆⠀⠠⠈⠩⡟⢢⡀⠀⠉⠛⠀⠐
⠀⠀⠈⡜⠀⢀⠴⣾⠟⠋⠀⢀⠶⣿⠉⠀⣾⣿⣷⣦⣰⣾⣿⣄⣠⣤⣴⣾⣿⠟⢲⣾⣿⡿⠋⢀⣶⣦⡀⠈⠿⠛⣆⠀⠀⠀⠉⢻⠟⠀⡀
⠀⠀⢀⣤⣤⡿⠖⠋⠀⠀⣠⣾⠟⠂⠀⣠⠟⠻⠟⠋⣿⣿⣿⣿⣿⡦⣀⣸⣿⣀⣀⣼⣿⣿⣿⣿⡿⠟⠓⠀⣄⠚⢿⣤⡀⠈⠁⠈⠀⠶⢿⣤⡄
⠀⠀⣾⡀⠋⠀⢐⠂⠀⢾⣤⠋⠀⠀⣴⣯⣤⣄⣠⣴⡿⠟⣿⣿⣍⣀⠹⣿⣏⢉⡛⢻⣿⣿⣿⣧⣤⣤⢀⣀⣤⣷⣞⢀⣼⡀⠀⠐⠀⠀⠈⠳⣀⡀
⠀⠀⠈⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣔⣶⣒⣒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠋⣿⣏⡉⡺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⣿⠀⠈⣹⡏⠀⠀⢀⣀⣤⣴⡶⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣧⣴⣿⣿⡄⠘⠻⠶⠶⠾⣋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣀⣀⣠⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⢻⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⢿⠈⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⣛⠻⠿⢿⣭⣽⣭⣿⠣⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠻⡉⠑⡀⣤⣤⣤⣤⣷⠀⠀⠀⠠⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠈⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣛⡁⠀⢀⣿⣶⣿⣿⣿⣿⣿⣿⢰⣷⣦⡀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣆⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣾⣷⣸⡟⠻⣦⡀⠀⠀⠀⠀⠀⡜⢠⡀⣽⣦⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣆⠀⠀⠀⣼⢁⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄⠈⣻⠓⠾⠁⣠⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣷⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⡀⠀⢀⢀⠄⣀⣄⢀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⣙⠶⣿⠃⢾⡿⠫⢈⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⡇⠶⣄⠈⢁⠀⠚⢣⣿⣿⣿⣿⠉⠛⠿⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠻⢢⣼⣿⣯⡐⠒⣸⠏⢝⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣴⣂⣱⣾⣾⣿⠀⠀⠤⠀⠙⠃⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣤⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⡇⠹⢿⡇⠹⠈⡏⠏⠿⠿⠙⠿⠿⠿⠙⠏⠁⠈⠁⢠⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⢒⣊⣉⣩⣭⣭⣥⣤⣤⣤⣤⣤⣭⣭⣍⣉⡁⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⣉⣭⣤⡶⠶⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣊⣩⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠶⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣈⠉⠓⠢⣤⣀
⠀⠀⠀⠀⣤⣤⣴⡞⣩⣴⠾⠛⠉⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠤⠭⢷⣦⡄
⠀⠀⠀⢺⣏⢿⣿⣷⣉⠀⠀⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠻⡾⢿⣿⠉⢷⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣟⣾⣿⣿⠟⢋⣤⣶⣶⣶⣦⣌⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⡿⢛⣡⣾⠟⠉⠀⠀⠀⠈⠙⢿⣦⣌⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⡆⠀⠀⠀⢀⣿⣿⢏⣴⣿⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⡀⠀⠀⢸⣿⢡⣾⣿⡟⢰⣿⣇⣰⣤⡄⠀⣤⣆⣸⣿⡧⢻⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣷⠀⠀⢸⠉⣾⣿⣿⣧⠈⠻⠿⠿⠟⠀⢀⠛⠿⠿⠿⠁⣼⣿⣿⡈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣧⠀⢸⠀⣿⡟⣿⣿⣷⣄⡀⠀⢠⣿⣿⡆⠀⢀⣀⣼⣿⣿⣿⡇⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⡆⠸⡇⢸⣿⢹⣿⣿⣿⢸⡀⠀⠁⠈⠁⣰⣯⣸⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢿⣿⡄⢱⠈⣿⡌⣿⣿⣿⠛⢿⣧⣤⣄⣥⡿⠋⣿⣿⣿⣿⣿⠇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⢿⣷⣈⣧⠘⣧⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣋⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡜⣿⣿⣿⣆⠹⣧⠹⣿⣽⣿⣶⣾⣷⣶⣾⣿⡿⢛⣡⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠹⣿⣿⣿⣦⠙⣇⠹⣿⣮⡻⣿⣿⣿⡟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣷⡈⢧⡘⢿⣿⣿⣿⡿⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣦⣳⣌⠻⣿⣿⣷⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣩⣭⣭⣽⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⢛⣛⣛⣧⡼⣿⣿⠿⣿⣿⣿⣿⣷⣄⡹⣿⣿⣿⣧⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⣴⣿⣿⣿⣿⣿⣧⢻⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢠⣿⣿⣿⡿⠿⠛⠟⣩⣌⠿⠛⢠⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢠⣶⡿⠀⢉⣠⡴⣾⠿⢁⢸⣿⣿⣿⣿⣿⣿⣏⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣏⠀⠈⢋⠉⢠⠴⣶⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢿⣿⣿⣤⣄⣀⠀⢒⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠸⣿⣿⣿⣿⣿⡿⣱⣷⠹⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣿⡇⣿⣿⣧⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⣿⣿⡿⠂⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣸⣿⠁⠘⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿⠃⠀⠀⠁⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⡼⠃⠀⠀⠀⠀⠠⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⢿⡿⢿⠿⢿⠿⠋⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢻⠟⠛⠻⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿
⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⠀⠀⣴⣿⣿⡟⠁⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⢿⣿⣿⣷⣆⢸⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣇⢠⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⠿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠻⢿⣿⣿⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⠀⠀⣨⠟⠉⠀⠀⠀⠀⠀⠀⠀⠒⠌⠙⢿⣿⣿⣿⣿⣿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠙⠒⢶⠀⠀⠀⠀⠀⢰⣾⣶⠈⠀⠀⠙⣿⡿⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠻⠇⢠⠇⠀⠀⠀⠀⠈⠈⠛⠋⠀⠀⣀⡠⡼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⢀⡤⠋⠀⡀⢀⣀⣀⠀⣀⠀⠉⠉⠁⠀⠀⢸⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⢠⣤⠚⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠹⠿⠇⠙⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠐⡀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠁⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡆⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡤⠤⠖⠋⠁⠀⠀⠹⣄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡞⠀⣀⡀⠀⠀⠀⢀⣀⠈⢧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡤⡼⠁⢛⣛⠛⢦⠀⠀⠘⣛⠛⢺⡠⢤⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⢻⠀⠀⠁⠀⢸⠀⠀⠀⠈⠀⢸⠁⣸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠛⣆⠀⠀⠀⠙⠒⠀⠀⠀⠀⢸⠗⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣠⠼⢆⠀⠀⠀⠛⠀⠀⠀⣠⠯⣀⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢷⡚⠉⠀⠀⠀⠙⠦⣤⣤⣤⠴⠚⠁⠀⠀⠉⢓⡶⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠉⠓⠦⢤⣀⣀⡤⣤⣤⡤⣤⣀⣀⡠⠴⠚⠉⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢳⡘⣞⢷⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⣣⡑⣌⢢⠳⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡄⠀⠀⡞⠀⠀⠹⣝⠎⠀⠀⠀⠘⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⢧
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡼⠀⡠⢼⣁⠀⠀⣠⠼⠓⢤⡀⠀⠀⣹⠤⡀⠀⢧⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡷⠊⠀⠀⠈⠙⡿⠀⠀⠀⠀⠸⣟⠉⠀⠀⠈⠲⣜⡆⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⢤⣀⠀⠀⣠⠇⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⣧⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⣤⠤⠤⣼⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠴⠧⣤⠤⠧⢤⡿⣧⡤⠼⠤⠤⢤⡿⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏⠀⠀⢸⡇⢸⠀⠀⠀⠀⠈⣇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⢸⠁⠀⠀⡼⠁⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢾⣀⢀⡼⠁⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠙⣆⣀⣹⠤⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣒⣒⣠⣼⣧⣼⣤⣤⣤⣀⣒⣒⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣄⡀⠀⠀⠀⠀⣀⡀⣠⣀⡀⠀⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠓⠃⠁⠋⠰⠀⠀⠀⠰⠘⠈⠐⠁⡃⠀⠀⠀⠆⠃⠁⠚⢨⠀⠀⠀⢀⠙⠉⠁⠃⠀⠀⠀⠀⠀⠋⠈⠘⠀⠂⠀⠀⠀⠐⠃⠁⠋⢸
⠀⠀⠀⠀⣀⠀⡄⠠⡄⢀⡀⠀⢀⣀⠀⠀⢈⠀⣃⠀⠀⢀⡄⣀⠀⡀⢸⡀⠀⠀⣸⢀⡀⢀⡀⣄⠀⠀⢀⡁⠀⠀⠀⢀⣀⠀⠀⣀⠀⡀⠀⡀⢈⡀
⠀⠀⠀⠀⠒⡆⠀⠀⠀⡖⠂⠀⠐⠚⠀⠀⠂⢀⠒⠁⠀⠑⠃⠀⠀⠀⡘⠊⠀⠈⠚⡀⠀⠂⠀⠓⠁⠀⠐⢂⠀⠀⠀⠘⠚⠀⠀⠒⠃⠀⠀⠀⡖⠂
⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠀⠀⠀⠀⠑⠀⠐⠁⠀⠀⠀⠀⠈⠒⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠑⠀⠐⠁⠀⠀⠀⠀⠈⠂⠀⠊

⠀⠀⠀⠀⠀⢔⡖⠆⡶⢂⠀⠀⠀⠀⢲⢲⢒⠖⡄⠀⠀⠀⡐⡖⠖⣲⠠⠀⠀⠀⠀⢴⠶⠖⡖⠀⠀⠀⠀⠀⡶⠲⢲⠒⡀⠀⠀⠀⣐⡖⠆⡶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠄⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣒⡀⠓⠀⠂⣐⡂⠀⠰⣲⠀⠀⠐⠀⣒⠄⠀⢔⡇⠀⠀⠀⢸⡢⠀⠠⣲⠀⠂⠐⠂⣗⠄⠀⢔⡂⠀⣀⠀⢰⡲⠀⠀⣒⡄⠂⠀⡂⣐⡂
⠀⠀⠀⠀⠀⠣⣀⡈⡠⠀⠀⠀⠀⠀⢄⣀⢀⠄⠀⠀⠀⠀⠠⣀⣀⡠⠂⠀⠀⠀⠀⠃⣀⠁⡠⠀⠀⠀⠀⠐⢄⣀⢀⠄⠀⠀⠀⠀⠠⡚⡈⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠈⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠉⠑⠢⡀
⠀⠀⡰⠁⢀⠤⣄⠀⠀⣠⢤⡀⠘⣆⠀⠀⠀⠀⠀⡰⠃⠀⣀⡀⠀⠀⢀⣀⠀⠘⢆⠀⠀⠀⠀⠀⣰⠃⢠⠀⢀⠀⠀⣄⠀⡀⠈⢆
⠀⢠⡇⠀⠈⠀⠈⠀⠀⠁⠀⠁⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠛⠃⠀⠀⠘⠛⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⢸⡄
⠀⠘⡇⠀⠀⢄⠀⠀⠀⠀⡤⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⢸⠃
⠀⠀⠱⡀⠀⠀⠙⠒⠒⠋⠀⠀⢠⠏⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠹⡄⠀⠊⠀⠀⠀⠀⠙⠀⢀⠎
⠀⠀⠀⠈⠢⢄⣀⣀⢀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⡀⢀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⣀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡖⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠛⠷⣄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠻⢿⡶⣠⣄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⢿⣧⣾⡇⠈⣩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢻⣿⣿⣿⢨⣥⠦⢤⠈⣽⡏⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣷⢸⣿⣿⣿⠈⠁⠀⠀⢠⢿⡇⢃⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⢹⣾⡿⢻⣷⣄⡀⠀⢠⡿⢸⣿⡏⢣⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⢷⣿⣷⢸⣿⣿⣥⣾⡟⣧⣼⣿⣿⢀⡄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣥⡘⣿⣿⡌⣿⡇⢻⣿⢀⣿⣿⣿⣯⣾⡷⢯⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣓⣀⣤⣶⣻⡗⢻⣧⣾⣿⣼⣿⣿⣿⣿⣧⣤⣀⡉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠖⣿⣦⡉⠙⠿⢿⣿⠟⡿⡀⢠⣿⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠘⢹⣿⣶⣦⢀⢈⣉⠄⣰⢯⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣝⣮⣌⣭⣿⣿⣾⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⢹⣿⣿⣿⣿⣿⣿⠀⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⣿⣿⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⢻⣷⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢻⣈⣧⣀⣿⣏⣻⢸⣿⣿⣿⣿⣿⡿⠀⢸⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠘⣿⠿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⠁⢠⡄⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡅⣸⡧⣼⣿⣏⢿⡇⣿⣿⣿⣿⣿⠃⢠⠟⠃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⠿⠟⣱⣿⣇⠘⢿⣿⣿⣧⣿⣿⣿⣿⣿⢖⣁⡤⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⢿⣷⣶⣶⣿⣿⣭⣿⡆⠉⢻⣿⣿⡿⠿⢿⣿⣿⣿⣯⣤⣴⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠉⠉⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡠⠤⠔⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠉⠀⠉⠀⠀⠂⠀⢀⠈⠐⠀⢀⠀⠈⠐⠀⣆⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡈⠀⠄⠀⠀⠑⠠⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠘⢆⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢐⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⠈⠳⣄⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢤⠤
⠀⠀⠀⠠⠁⠀⠀⠠⠂⠀⠀⡠⠁⠀⠙⠢⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⣀⣀⡀⠠⢴⣢⠄⠴⠀⠡⡀
⠀⢀⠔⠁⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠤⠀⠒⠀⠉⠀⠀⠒⠠⠀⠈
⠐⠡⣤⠐⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢃⠀⠀⠀⠀⡀
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠐⠀⠈
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠲⢼
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡇⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠤⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠞⠚⠀⠋⠛⠋⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠔⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⢹⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠇⠀⠀⢠⣿⠆⢸⠋⠉⠊⠀⠀⠀⠀⠈⠙⠲⣄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⢸⡿⢀⠜⠀⠀⠀⠀⣀⠀⣀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣠⣥⣤⣤⡼⢿⣷⣦⡤⠤⠔⠊⢉⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠟⠁⡾⠵⠜⠓⢀⢘⠐⢢⡟⡆⣀⣀⠴⡅⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠒⡞⠺⠶⠷⠯⠴⠿⠶⠖⠛⠉⡎⠉⠀⠀⠀⠙⣄⠀⣀⠤⠒⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠈⣑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠕⠀⠀⡠⠊⠈⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⣈⠷⣀⣀⠀⠀⠀⠀⣀⣴⡊⠥⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠋⠉⠉⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⠢⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠃⠀⠈⠒⠄⣹⡢⣄⠸⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⣴⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⣃⡳⣄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡤⠒⠉⢹⣮⣷⠂⠀⠀⠀⠀⠰⣖⣷⡏⠀⠈⢻⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢣⠀⠀⠈⠛⠋⠀⠀⡖⠀⠀⠀⠙⠋⠁⠀⠀⡜⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⡄⠀⠀⠀⠀⣠⣶⣦⣶⣶⣄⠀⠀⠀⠀⣰⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠙⣦⡀⠀⠀⠓⠒⠂⠒⠒⠚⠀⠀⢀⡴⠃⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⢻⣕⣤⣀⡀⣀⡀⣀⣀⣤⣾⡏⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠘⣯⣽⡇⠀⠀⠀⠀⢸⣯⣽⠃⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢫⠁⡇⠀⠀⠀⠀⣸⠈⡍⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠡⠀⠀⠀⠀⡆⠈⠑⠢⠔⠊⠀⢰⠀⠀⠀⢀⠎⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡜⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢠⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢰⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⡄⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡁⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢘⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣖⡀⣀⣄⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⡀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⡿⢀⡇⠀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣿⡇⠈⠀⠀⣿⣿⣿⣻⣿⡆
⠀⠀⠀⠀⣸⡄⠀⠸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡟⢢⢀⠀⢀⣠⣤⣠⣀⠀⡤⠘⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠄⢠⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⢀⣸⣷⣾⣸⣦⣾⣿⣿⣿⣿⡄⡀⠀⠀⠀⢠⣄⠀⠀⠐⠂⠤⢀⢸⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠰⣚⣛⣙⣿⣻⣿⠿⢿⣥⣌⣉⠹⣿⣟⢲⣀⣀⠠⠙⠿⣦⡄⠀⠀⠀⠸⣿⣿⣧⣣⡿⠿⠛⠛⠛⠛⠿⣿⣿⣧
⠀⣿⠉⡭⠩⣿⣿⣴⣾⠿⢿⣿⣿⣿⣿⣿⠟⠻⢶⡗⢦⣼⣿⡆⠀⠀⠀⢹⣿⣿⣿⠀⣠⣤⠆⠲⣴⣤⣄⣘⣻
⢨⣿⠀⠃⠀⠈⠙⠋⢁⡀⠀⣸⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⢸⣿⣿⣿⣾⠋⠀⠀⢰⣿⣿⢿⡿⠿
⢸⣿⠀⠒⠀⢀⡒⠀⠒⠀⠀⢻⠒⠒⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣇⠀⢀⠄⣾⣿⣿⣾⣿⣿
⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢸⣿⣿⣿⢻⣄⡌⠀⣿⣿⣿⣿⣿⣿
⡿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡂⠂⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⠀⠀⢸⣿⣿⣿⣿⣿⣳⣠⣿⣿⣿⣿⣿⣿
⠘⡿⠀⠁⠀⢠⠄⠀⠀⠀⠀⢼⠅⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠺⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⡃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠸⣿⣿⣿⣷⣶⣶⣶⣿⣩⣭⣽⣿⣿
⢸⣿⠀⢀⣀⣁⣀⣀⣀⣀⣠⣿⣍⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿
⠸⣿⣀⣻⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⣿⣿⣟
⠀⠉⠛⠛⠻⠿⠿⠿⠿⣿⠯⢤⣤⣼⣿⣿⣿⣿⣿⣛⣯⣭⣿⣿⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⡿⢿⠿⣯⣿⡉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣭⣭⣿⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿
⣿⣿⣿⣿⣿⠁⠀⢻⣿⣿⣿⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿
⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠘⣿
⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⣿⣿⣿⣿⣿⡄⠀⣀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣠⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⡴⠋⠀⠀⠸⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⠉⠉⠉⠉⢱⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠙⠛⠛⠛⠋⠈⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠓⠒⠒⠒⠋⠉⠉⠙⠒⠢⢤⣀⣀⡀⠀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣤⣶⣶⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣀⣀⠤⠒⠊⠉⠉⠉⠉⠙⢢
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⡀⠀⠀⠀⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠙⢿⣿⣿⣿⡛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠙⠢⡀⠀⠀⢰⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢹
⠀⠀⠀⠀⣶⣶⣤⣴⣾⣿⣷⣶⣶⠀⢠⣶⣶⣦⣤⣀⣀⣤⣤⡀⠀⢠⠖⠒⣦⠖⠋⠀⠙⠒⠒⡄⠀⢠⠔⠒⠢⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠠⡇⠀⡸⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠛⠛⠋⠛⠛⠿⠿⠿⠏⠀⠻⠿⢿⣿⣿⠿⠟⠿⠿⠃⠀⠈⠒⠒⠋⠒⠒⠤⢄⡀⠼⠀⠀⢱⣀⣀⡀⠀⣀⡼⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣭⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣭⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣀
⠀⠀⠀⠀⠀⣠⣾⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣼⣿⣷⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣬⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢟⣭⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣀⣠⣤⣄⣀⣤⣤⣴⣤⡑⢂⣀⡘⠛⠃⠀⠀⠀⠀⠀⣀⠀⠀⠐⠓⢰⡶⠶
⠀⠀⠀⠀⠀⠀⠀⢄⡀⢤⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠿⢋⡴⠫⠁⠀⠀⣱⢀⣄⣒⣈⣨⣷⣾⣿⣁⡴⠊⠀⠀⠠
⠀⠀⠀⣠⡤⠲⠋⠀⠀⣀⣨⣿⣿⣋⣤⣾⣿⡿⠿⠛⢉⣱⡖⣉⣀⠔⠲⠔⠚⣛⡋⠉⠉⠉⠉⣉⣩⡍⠙⠋⠐⠲⠤⢤⣤⡀
⠀⠀⠀⠉⠀⢀⡠⠶⢛⣿⣿⣿⡿⠿⠛⠉⣁⣴⡶⠞⠛⠉⢈⡉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠉⠀⠐⠲⠄⠠⣬⡈⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠠⠴⠾⣿⠟⣉⣤⡴⠾⠛⠋⢀⣀⣠⡴⠟⣠⣤⡖⣚⣷⣶⣶⣶⣾⣦⣶⣤⣤⣴⡾⣇⢀⡀⠀⣀⣀⠀⠀⠀⠀⠱⠀⠤⣤⣼⡷
⠀⠀⠀⠀⠀⠀⣀⠐⣠⣴⣿⡿⠋⣑⣡⡴⡾⠃⠀⢀⣀⣈⣩⣥⣴⣿⣿⡿⢿⡟⠛⠟⢀⠟⠁⠀⡀⠀⠈⠙⠛⠋⠙⠦⣤⣀⡂⣠⣿⣧⢰⣿
⠀⠀⠀⠀⠀⢀⢠⣾⣿⣿⠃⣠⣾⣿⣯⠔⠀⢀⣀⣬⡿⣛⣻⣯⣥⣤⣶⣶⣶⣾⣿⡇⣾⠂⠕⠀⠉⠀⠓⠲⠄⠀⠀⠀⠀⠉⠉⡁⣿⡿⠬⣿⡇
⠀⠀⠀⠀⠠⢮⣾⣿⣿⣷⣿⣿⣿⠏⠄⠀⠀⣈⢉⡽⣿⣿⣿⣿⣿⣿⠟⠿⠛⠻⠿⡛⠛⠒⢲⣤⣀⠀⠀⠀⠀⣀⠀⡀⠠⡀⠀⠀⢿⡅⢀⡟⠁
⠀⠀⠀⠀⣴⠿⣿⣿⣿⡿⢻⣿⡟⡰⠀⢠⡤⠟⣋⣴⣿⣵⣾⣿⡿⠟⣷⠞⠃⢉⣀⣨⣤⣄⡉⠙⠢⣝⣲⣦⡀⠉⠀⠀⠀⠀⠀⠀⣼⠁⢨⡇
⠀⠀⠀⣠⠃⢘⣿⣿⣿⣵⣿⣿⢀⠆⠀⢀⠴⠊⣽⣿⣿⡿⢟⣭⠄⢀⡤⠶⠟⢉⣉⠭⠥⠍⣛⠛⠢⣌⠳⣌⠙⢦⡀⠀⠀⣀⡀⣀⡽⣶⣿⠇⣀
⠀⡰⠛⠁⣠⡿⣫⣿⣿⣿⣿⣟⢸⠄⠀⢀⣴⣾⣿⣿⡟⣱⠏⢠⠞⠋⣠⠶⣞⠛⠛⠓⠶⢿⡏⠀⠦⣌⡑⢌⠓⠀⢧⠘⢠⣍⣹⢿⣣⣼⡇⢠⣿
⠀⠀⣠⠞⠃⣸⠟⣹⣿⣿⣿⣧⡌⠀⠔⠋⠉⠻⠛⢻⠀⢉⡤⠀⣠⣾⠉⢶⡌⠛⡘⠂⢤⠼⣧⠸⣄⠸⣧⠘⣆⠀⡞⠀⢠⣤⠀⢸⡇⡤⢂⣾⣿⠇
⠀⠈⠁⠰⠚⣁⣼⣿⣿⣿⣿⡏⡑⠀⠀⠀⠀⢀⣴⡇⠃⣸⠃⣰⠏⣙⡀⠛⡶⢲⣿⡿⢀⡠⢿⡄⠙⣇⠈⣇⠻⣸⣧⠀⠈⠃⠀⢸⡃⠀⢾⡧⠘
⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⣙⡈⠳⠂⠀⠠⠾⣿⢡⡗⢠⡟⣰⣿⣦⣬⣅⡘⠀⠛⠉⠐⠈⠁⠂⠳⣀⠉⠠⣿⠀⠁⡾⠀⠀⢠⠀⣿⠃⣿⡯⠓
⠀⠀⠀⠐⠋⢀⣻⣭⣶⣾⣿⣿⡟⣧⠀⠀⢠⠏⣼⡀⢾⣠⣿⣿⣿⣿⣿⡇⢀⡄⠐⠂⠀⠀⠀⠀⢿⣆⣐⢸⠦⡸⠂⠀⢀⡸⠀⡿⢄⣸⠁
⠀⠀⠀⠀⣺⣡⣿⣿⣿⠟⢋⣿⣇⡹⢇⢈⠃⣤⢹⡇⣾⣿⠙⠻⠿⠿⠋⣉⡀⠀⣾⣿⣿⣷⣤⡛⣸⡟⡌⣸⣤⡏⢰⣶⠈⠁⣸⠁⢀⡏⠀⣦
⠀⠀⣠⡞⠁⢹⣿⣿⣿⣧⣾⡿⣿⣶⣼⠏⣸⢃⣾⡇⣿⠁⢤⣦⡴⣠⣾⣿⣏⠀⢿⣿⣿⣿⣿⣟⡟⡸⢀⣿⣼⠁⡀⠀⠀⣀⣯⡀⣼⠁⠀⠘⠁
⠀⠐⠋⢀⡴⠛⣿⣿⣿⣿⠛⣿⢿⣿⠏⡼⠃⣼⠿⣧⢿⣷⠈⠁⣾⣿⣿⣿⡅⠛⡀⠛⠻⠿⠋⣹⡑⠀⣼⣿⡿⠀⣠⡄⠚⣿⡟⢀⣿
⠀⠀⠀⡿⠶⠾⢻⣿⣿⠇⣴⣿⣿⢃⡼⠀⣼⠁⠀⣿⣿⣿⣄⣃⡬⠁⠛⠋⠁⠚⠋⠀⠀⣋⣸⡃⢁⣐⠉⣾⢧⠚⠛⠛⣻⡏⠀⣾⡃
⠀⠀⠀⣠⠴⠾⢛⣿⣿⣿⣿⡟⣡⡞⢡⣼⡇⠀⠛⢿⣿⣏⡘⠛⠷⣶⣼⣄⣴⡿⠋⢓⣤⡞⡱⠁⣰⢋⣴⡟⠈⠻⠁⣰⡏⠀⢸⡿⢃⡀⠿⢷⡀
⠀⠀⠘⢡⣶⣾⣿⣿⣿⡿⢋⣼⠏⣠⣿⡿⠀⡄⢀⣀⣷⢌⢙⠓⡾⣴⣬⡷⠇⢘⣿⣿⠟⢈⣡⣾⡡⢛⠹⢦⣤⣄⣴⡟⠀⣰⠏⠀⠀⠀⠀⢠⡇
⠀⠀⠰⠟⠩⣿⣿⡿⠛⣡⡞⠁⣠⣿⣿⠁⠈⢀⣤⣤⢇⠀⠘⠓⡷⡶⠶⠟⣱⡿⢋⣥⡶⣿⠿⢉⡔⢯⣤⠚⠁⣠⠟⠀⣠⡟⢠⡄⠀⡴⢀⡹
⠀⠀⠀⠀⠀⣹⠟⠁⣴⠏⠐⢳⡟⢿⣿⣴⡶⢿⡏⠹⣷⣶⣤⣀⣀⣄⡠⡴⠏⠀⢈⣽⠟⡡⠖⠁⠄⠀⠉⣶⡼⠃⠀⣰⢋⣤⡟⠀⠰⠓⠉
⠀⠀⠀⢰⣿⢁⣠⡞⠁⢠⣶⡿⢻⣿⠛⢡⣶⣾⣧⡘⢻⠿⠛⣿⣏⣥⡤⠴⠞⢋⣩⣴⣾⣤⣶⡀⠀⣠⣾⡟⠂⣤⠞⠁⠸⣧⣤⢀⣀
⠀⠀⠀⠈⢡⡞⠋⠀⢀⣾⡏⠀⢸⠏⢸⣿⡁⢸⣏⣃⡞⠛⣛⣷⡤⠶⠖⢿⣟⢋⣷⣦⡼⠟⣿⣁⣹⠟⠁⢀⣼⡋⠀⢀⣀⡘⠻⠂⠸⢤⣆
⠀⠀⠀⢰⠏⠀⠀⢐⣾⠃⠀⣴⠋⢀⡼⢯⢿⣿⠏⢀⡿⣿⡛⢁⣞⢋⣐⣿⣿⣦⢉⣭⡶⣻⣿⣿⠏⠀⣠⣿⡟⠻⢶⣄⠉⠉⠑⢦⣄⢀⣈⡄
⠀⠀⠀⠀⢇⣠⣦⣾⠇⠀⢾⡿⠀⢼⡇⣰⣼⡟⡷⠘⣷⣽⣷⣮⠿⠿⠟⢋⣠⣤⣿⣽⣷⠿⢻⣇⡀⣰⣿⠛⠝⣤⣄⠙⣆⠀⠂⠸⣿⠉⠈⡟
⠀⠀⠀⠀⠀⣼⠿⠁⠀⢐⣿⠃⠀⣿⠀⣿⠜⢿⣧⣀⣿⡟⣷⣤⣶⠖⣾⠏⠛⣩⢮⣟⠫⣭⣓⢾⣍⣽⢻⣿⠀⣿⣿⠆⢸⡆⠀⠀⠈⠓⠊⠁
⠀⠀⠀⠀⣼⠁⢠⠶⠈⣤⡿⠀⢸⣏⠀⣿⣧⣼⢻⣿⣿⢿⢾⣏⣸⣆⣁⣤⠞⣩⣾⡟⢒⡾⠷⢆⢜⡥⣿⣧⠀⣿⡟⢸⡿⠀⣶⣂⠶⠶⠶⡄⠁
⠀⠰⠀⢘⡇⡀⠀⠶⣿⡟⠀⠀⣾⡟⠀⡿⣿⣿⠀⢿⣧⣼⣤⣿⣿⣿⠟⢁⣴⣿⣧⣼⠛⠻⣧⣠⢻⡅⣿⣿⠀⠈⢿⣿⣀⠀⠈⠁⠀⠠⣴⠇⣠⡤⡀
⠀⢠⠀⠨⣝⣳⣃⣴⡿⠁⠀⠀⣿⠍⠀⣿⡌⣿⠄⠘⣯⣿⣿⠃⠀⠉⠉⠛⠛⠻⣿⣯⠤⠀⣽⣿⢸⡇⠸⣿⣿⣄⡀⠈⠛⢉⠑⠒⠢⢤⣀⡀
⠀⢀⠀⢀⡾⡿⢉⣿⠃⠀⠈⢰⣿⠀⠀⣿⠀⣿⡆⠘⣿⡿⠻⠶⣦⣄⣀⡀⠀⢠⣿⠟⠀⢰⣿⣿⡏⣧⢰⣿⣏⢮⠙⣿⣷⣦⡻⢦⣤⠄⠈⢹⣷⣦
⠀⠀⠀⡸⢀⠀⢸⣗⣂⣤⣀⣸⡟⠃⠀⡯⠀⣸⡇⠀⠘⣷⡆⢀⡠⣻⣿⣿⣿⣿⠛⠀⠀⣸⣿⣿⣧⣸⡼⣿⣿⢸⡆⣿⠙⣻⣷⡀⠉⢻⣿⣿⣏⠉
⠀⠀⠀⡇⠀⠀⢿⡿⣿⣿⠇⣸⠇⠸⠀⣿⣉⣿⡏⠀⠀⠙⣷⣌⢷⠘⠟⠻⢿⣿⠀⠀⢀⣿⣿⣿⣿⣷⡶⠞⠃⣰⢇⡿⢀⡇⢸⣿⣄⡀⠛⢻⣿⡷
⠀⠐⠈⣷⠆⠄⡘⢠⣿⣿⣶⠋⢀⣀⠀⢼⡟⣻⣋⡀⠀⢀⣻⣟⠀⠌⠘⡄⠀⣏⠀⠀⣼⢿⣿⣟⠉⣀⣠⣴⣾⠏⠼⠁⣸⢗⠸⣿⡿⣿⣦⠀⠈
⠀⠠⠀⠁⠺⢴⣇⣼⡏⣿⡟⢀⡾⠋⣹⣿⣇⠘⣿⡇⠀⠈⣿⡿⢶⠆⡀⠀⠀⣿⠀⢀⡇⢸⣿⣿⡿⠋⠉⠙⠀⡠⠞⠃⠉⠑⢦⣀⣀⡠⢿⣻
⠀⠰⠀⠀⠀⣼⣿⣿⣵⡏⠀⠁⠀⠐⠿⣿⣥⣽⡿⠛⠀⠀⠈⠺⠧⣤⠑⠊⠀⠀⠉⠉⠀⠛⠙⠺⠶⠦⠔⠿⠋⠀⠀⠀⠀⠀⠀⠙⠛⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣫⡥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢻⠆⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⣵⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⣿⣱⣷⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣿⣿⣿⠟⡘⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⣧⣿⣟⣯⡷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⣿⣿⣿⣿⣯⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⣿⣿⣿⣿⣿⢳⣿⡏⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣿⣿⣿⣿⡟⢿⣯⢿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⠟⢸⣿⣧⢸⣿⣿⠙⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⣿⣯⢸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠈⠉⠙⣿⣯⣼⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣾⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⣿⣿⣿⣭⡼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣭⣭⣍⣀⠾⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣾⣿⣿⣿⣷⣶⣮⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⣛⡉⢙⣛⣛⡛⠛⢭⣭⣬⣿⣿⣿⣿⡿⠟⢿⣿⣟⣿⡿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⣽⣶⣿⣿⣿⣿⣿⣿⣿⣤⣶⣶⣌⣉⣿⣿⣉⣡⣶⣶⣄⣿⣿⣿⣿⣿⣿⣶⣾⣭⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⠟⠿⠾⢻⣿⠏⢹⣿⡿⠳⠿⠿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡆⠀⠻⣄⡀⠀⣸⣛⠁⠈⣛⣇⠀⢀⣠⡟⠀⢸⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⡄⠀⠈⠿⠛⠁⠀⠈⠳⠖⠁⠀⠈⠛⣿⡁⠀⢠⣽⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡎⠀⢀⡇⢀⣴⠞⠛⠿⠿⠛⠳⣤⡀⢸⡀⠀⢰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⣿⣿⣿⣿⢠⠀⠀⡇⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢸⠃⠀⡄⣿⣿⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡽⣿⣿⣿⣿⣎⠀⢀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡀⠀⢸⣿⣿⣿⣿⣟⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠨⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢆⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣻⣿⣿⣧⢳⠀⡀⠀⠀⠀⠀⡘⠁⠀⠀⢀⠀⡾⣼⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣮⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⣷⣿⣿⣿⡏⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⢿⣿⡄⠀⡄⠘⠄⠀⠀⢀⢿⣿⢿⣿⠎⠀⠀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⠟⠻⣆⢘⣆⢀⡎⢠⡟⠻⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢹⠏⠡⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡀⢠⡿⠁⠀⢸⣧⣤⣴⣶⣶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠉⠻⠿⠃⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡄⠀⠀⠀⢻⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⣷⣄⠀⠀⠀⣸⠏⠳⣦⣄⣀⣀⣀⣴⠟⠳⣦⣀⣀⣀⣰⠟⠻⢦⣄⣀⡼⠛⠶⢶⣶⠟⠃
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣤⣼⠏⠛⠛⠋⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣽⡿⣀⣀⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢸⣿
⠀⠀⠀⠀⠀⣰⡟⠁⣾⠉⢉⣙⣧⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⣄
⠀⠀⠀⢀⣼⠏⠀⠀⠙⠷⣤⣍⣩⣷⣀⠀⠀⢀⡏⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠀⣠⡾⠁⠀⠀⠈⢿⣧
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠋⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠈⢻⣦
⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠈⠙⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠛⠶⢤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠶⠾⠛⠉⠀⠀⢸⣷⣄⣀⣀⣾
⢻⣇⠀⠀⠀⢀⣴⡾⠻⣯⠀⠀⠀⠀⠀⢀⣴⣟⠀⠀⢠⣾⣤⣶⡟⠁⠀⠉⠉⠉⣩⣿⣟⠛⠋⠉⠙⠛⣿⣉⣉⣴⣄⠀⠀⠀⠀⣀⣴⡿⠁⠙⠛⠋⠁
⠈⠻⢷⣶⣶⠟⠋⠀⠀⠻⣦⡀⠀⠀⣠⡾⠋⠛⢻⣿⠟⠉⠁⠀⠙⢷⣤⣠⣴⠟⠋⢸⠉⢷⣤⡀⣠⡾⠛⠻⣿⡍⠙⠻⠷⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠿⠋⠀⠀⣠⡟⠁⠀⢀⣀⠀⣀⣀⣈⣉⠀⢀⣀⣤⣤⣤⣈⠛⠋⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣹⡟⣉⠉⣉⣹⡿⠿⠿⢦⣤⡈⣹⣷⣤⠶⠞⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠙⢿⣯⡽⠋⠀⠀⠀⠀⠀⢈⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣠⣴⠾⠛⠙⠛⠻⠶⠶⠶⠶⠶⠶⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣀⣀⣀⡀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⢀⣀⣀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⢻⡿⠛⠛⠛⠛⠉⠛⠻⠟⠛⢿⡟⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠶⠶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣀⣈⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠛⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⢾⣿⣿⣿⣿⡃⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡧⠈⣿⣿⣿⣟⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣸⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⣠⣴⣾⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⣿⣿⣿⣿⣿⣄⠘⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣷⡀⠙⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⡈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠹⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⣀⣀⣀
⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣭⠉⡁⠀⡀
⠀⠸⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠰⠄⠀⠈⠁⠼⠿⠆
⠀⢀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⠉⠙⠛⠛⢋⡀⠀⠀⠶
⠀⢠⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀
⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠊⠀⠀⣠⠀⢀⠴⢀⣴⣤⣀⡉⠓⠀⠀⠀⠘⢷⢶⠖⠐⣊⡉⠴⠒⠉
⠀⠀⠀⠀⠀⠀⣴⠁⠸⣿⣿⣦⡑⠀⢀⡀⠈⢹⡿⠯⣆⠉⠛⠛⠫⠁⠀⠀⠀⠀⠀⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣿⣿⠇⢰⢿⣽⣶⣾⣿⣷⡇⠠⣁⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠾⣿⣿⠅⡀⠈⣸⣿⣿⣿⣿⡿⠁⠀⠀⠓⠀⠐⢢⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣘⠂⠀⠀⡜⠁⠀⠁⠀⣿⡆⠋⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⣤⣄⠀⢠⢖⣹⠀⡀⠀⠄⠀⠀⣄⠀⠁⠀⢀⣐⠓⠄⠘⠂⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠈⢹⡟⡀⠀⠸⠋⣠⠽⡖⢀⢊⡄⣴⣹⣺⠃⣿⢿⣷⠀⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣾⣷⡁⠀⣀⣾⣧⠀⠀⠀⠘⠈⢻⣿⣿⣠⣿⣦⠹⣧⡇⠘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⠀⢿⠃⠀⠁⠀⠘⠀⠀⢸⣿⣿⣿⠻⣟⠀⣿⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢄⡀⢰⣄⠀⠀⠀⣿⣿⡇⠙⠟⠙⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⣸⠏⡋⠀⠀⢠⠸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⠀⠀⣿⠇⠈⠀⠀⢈⠢⡟⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣀⣼⠏⠀⢀⡀⠀⣸⠃⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⠀⠀⠈⣠⣞⡰⣇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⠋⠀⠀⠙⠿⠆⢹⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠣⠽⠯⢿⣛⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠉⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⡀⠑⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣖⠤⣀⣀⠀⣀⡠⢂⣦⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⣠⣾⡿⢛⢛⣛⣛⠚⢻⣿⣿⢋⢭⣭⠭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡠⠬⢍⠛⢿⣿⣧⣀⣈⣛⣀⣁⣼⣿⣿⣄⣬⣭⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠊⠈⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⡛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠊⠈⠒⠁⠈⠒⠁⠈⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣀⣀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⣀⣀⣀⣀⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣴⣾⣿⣷⣦⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠛⠋⠛⠛⠛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⣶⣶⣶⣶⣤⡀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣼⣿⣿⣿⣆⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠉⢻⣄⣹⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⣿⡀⠀⠀⠀⠀⠀⢹⣿⣿⠛⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠪⡀⠀⠀⠀⢸⣿⣿⣿⡿⠛⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⠀⠀⠀⠈⠳⣰⢌⡀⡿⠿⠋⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠀⢀⠀⠀⠀⠈⠲⡀⡈⠔⠁⠀⠠⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠸⠴⠀⡖⠒⠃⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡐⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢎⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠈⠒⠀⠠⠰⣦⠐⠀⠁⠀⠀⠀⠑⠄⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⢻⣧⡘⠃⠀⠀⢄⠀⠈⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣉⠭⢿⡿⡀⠀⠀⠻⣷⣄⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣆⣀⣀⡀⠀⠄⠀⠀⠀⠙⠻⢷⣤⡀⠀⠠⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡠⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⡣⠊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷⡔⠋⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣀⠠⠚⠀⠀⠘⠦⣻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡛⠀⠰⣿⠀⠀⠀⠀⣾⡇⠈⠿⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣄⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⣀⣾⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠳⡀⠀⣠⣶⣶⣦⡀⢀⡜⠁⣸⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠙⠒⠬⢷⣿⠟⠛⣿⡿⠷⠒⠊⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⣿⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢿⣿⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣰⠋⢻⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠻⠃⠀⠘⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠘⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣆⡀⠀⠀⢀⣇⡀⠀⠀⣼⡀⠀⢠⣇⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢻⢦⡼⠓⠒⠊⠀⠉⠓⠒⠋⠀⠙⠒⠚⠁⠈⠉⠉⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡥⠤⣄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠳⠬⠽⢦⣀⣠⠇⠀⠀⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢦⡀⢀⡼⠁⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠂⠤⠐⠋⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠀⠀⠀⠀⢀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠴⠋⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡴⣿⠀⠀⠀⢀⡄⠀⠀⠈⣉⣰⡖⠒⠶⠶⠤⢤⣴⠶⠒⠚⠋⠉⡀⠀⣸⣷⣶⣏⣤⠤⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠋⠀⠘⢦⣀⣠⠾⠛⠛⠛⢻⡏⠀⢉⣹⠶⢶⣋⠉⠈⣳⠶⠶⠒⠚⠛⠋⠉⠉⢹⠀⣈⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁⡀⠀⠀⠀⠀⣀⣀⣠⡼⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠲⢤⣤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⡟⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣸⣇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⢩⡟⠛⠛⠛⠛⠻⡍⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠦⢤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣉⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣬⣉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⣁⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⣤⣄⡀⠉⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣡⠞⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠈⠉⠛⠦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡵⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣼⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠻⢿⣿⣿⣿⣿⣦⠀⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢠⣤⣤⣤⣤⠀⠀⣴⣶⣤⣤⡄⠈⣿⣿⡇⣼⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⠇⢸⣿⣿⠁⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠉⠛⠋⠀⡴⢦⠈⠙⠋⠁⣠⣿⣿⡏⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⣠⡄⢠⡄⣰⣄⣠⣿⣿⣿⡟⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠙⠋⣹⣿⣿⣿⠏⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣇⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣯⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠧⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠁⠑⢦⠈⠹⡕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠈⣿⠄⠀⣼⡏⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠐⠲⢤⠤⠲⢄⠴⠚⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⠠⠤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡠⠤⠀⠒⠀⠀⠀⠁⠒⠒⠒⠂⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡏⠒⠶⠤⠤⠤⠤⠤⠄⠠⠀⠀⠒⡈⣇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠓⠲⠶⠶⠶⠶⠂⠁⠀⡼⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢓⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠤⠒⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣨⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢁⣄⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡼⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡿⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣠⠁⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⡄⠀⠀⠀⠀⢱⠈⠳⢄⡀⠀⠀⠀⠀⠀⣀⡀⠈⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢫⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⢠⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠰⠷⠂⠀⠘⠀⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⢀⢤⡀⢤⡖⠚⢀⠉⡆⠉
⠀⠀⠀⠀⠀⡎⠐⠐⠐⢀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⡸⠀⠀⠀⠀⠀⢧⡀⠀⡘⠀⠀⠀⢠⡀⠊⠄⣟⠡⠖⣂⢀⠀⠐⠢
⠀⠀⠀⢠⠴⠀⠀⠀⠀⠀⠑⠢⢤⣀⡈⠀⠃⢠⡄⠐⢀⡀⠤⠤⣼⡜⠀⠀⠀⠀⠀⠀⠀⠙⡀⣇⠄⠐⠒⠢⠉⠂⠁⠀⠀⠠⡅⣂⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠈⠀⠀⠁⠀⣈⠁⠒⠐⠒⠒⢀⣤⣈⣀⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠚⠛⠋⠁⠀⢠⣴⣄⣍⣥⣤⠄⠦⠝⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠴⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠲⠶⠤⣤⣀⠀⣠⠎⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⢧⣀⣤⠤⠖⠒⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠤⣤⠄⠀⠀⣀⣤⠤⠖⠒⠚⠛⠉⠓⠒⢒⣤⣄⠉⠃⠀⠀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⠞⠉⠀⢰⣃⡤⠒⠉⠁⠀⠀⠀⠀⠀⣀⡤⠖⣋⣉⡁⠀⠀⠀⠈⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⣠⠀⠀⠀⠀⠀⠀⡰⠋⡽⢀⡴⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⢀⡴⠚⠛⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⡴⠉⡆⠀⠀⠀⠀⢰⠁⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠞⢠⣦⣤⡇⠀⠀⠂⡀⠊⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠂⠤⠀⠤⠒⠀⠉⣉⠈⠉⠂⠀⠈⠐⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠐⠀⠋⠉⠉⢳⠀⠀⢣⡈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡓⢄⠀⠁⡀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡄⣼⣶⣶⣾⣿⠀⢠⠔⠛⠀⠀⠀⠀⠘⠢⡄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⡠⠊⠉⠀⠀⠙⢦⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢁⣁⣀⣀⣀⣸⡇⢿⡠⢺⠃⠀⠀⠀⠀⠻⣅⠀⠀⠀⠀⠐⠋⢀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⢸⠿⠿⠿⠿⢿⡇⠈⠁⡾⠀⠀⠀⣀⠀⠀⠸⡆⠀⠀⠀⠠⠊⠀⠈⠺⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⢸⣤⣴⣶⣶⣾⡇⠀⠀⠛⢛⣉⣉⣉⣧⣀⣠⠇⠀⠀⠀⡃⠀⠀⢴⡆⢹⠓⣶⡤⢤⣀⣀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡘⠉⠙⠋⠉⠙⡇⠀⢠⡞⠉⢀⡤⠤⣀⡈⠉⠀⠀⠀⠀⠁⠀⠀⠀⢀⣾⡀⠉⠀⠀⠀⢉⡿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢱⣴⣶⣶⣶⣾⡇⠀⣾⠀⠀⠉⠓⠚⠋⠁⢀⠀⠀⠀⠀⠀⠁⠒⢲⠿⠃⠙⠿⠒⠖⠛⣋⣀⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⢸⠉⠉⠉⠉⠉⡇⠀⠈⠳⠦⠤⠤⠶⠒⠋⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠀⠀⢨⠇⠀⣸⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⢸⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠓⠒⠒⠊⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⡜⣅⣀⣀⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⣁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣇⣿⠟⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠐⠂⠑⠬⠔⠁⠀⠀⠀⠀⡠⠷⣾⣧⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠸⡸⣶⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢄⠀⢀⠌⢳⡒⢋⠭⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠙⠓⠶⠂
⠀⠀⠀⠀⠀⠀⠀⢳⡹⣉⣡⣼⠃⠀⠀⠀⠀⠀⠀⣀⣀⣤⣏⣀⡄⠙⠁⠀⠐⣷⡄⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢡⡹⠿⠿⠠⠴⠤⠤⠶⠤⢾⡁⣜⢰⡇⡼⠃⠶⠴⠶⠤⠤⢤⣤⣄⣗⡛⢶⠒⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⣀⡤⠞⠉⢀⠀⣀⡀⢀⡔⠉⠇⠀⠀⠀⢀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠞⠀⠃⠘⠓⠂⠠⠤⣀⣀⠀⠀⡘
⠀⠀⣀⠄⠊⠁⠀⠀⣀⣹⣯⡀⠀⠎⠀⠀⢓⠇⠘⠋⠈⡟⠂⠛⠒⠒⠒⠂⠀⠤⠤⠄⢠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠤⣄⣀
⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⣀⣡⠀⢆⠀⠀⠉⠀⠀⢀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣄⣇⠀⠀⠀⠀⣀⠀⠉⢀⡰⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢄⣀⠀⠀⢀⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⣯⠤⠤⢠⣄⣈⣉⠉⠉⠉⠉⣡⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠈⠑⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣄⣀⣸⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠀⠀⠠⠐⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠆⠐⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠻⣷⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣿⡟⠁⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢷⣾⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣶⡾⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣈⡙⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⢋⣁⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣾⣶⠇⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢿⡄⢘⣦⢹⡇⢠⣼⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡙⠦⣵⣀⢹⣿⡄⢸⣿⣿⣏⠀⠀⠀⣸⣇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⣦⡀⠉⠁⠰⣦⣿⣧⣻⣴⠉⣴⠀⠀⠀⢀⡈⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⠀⠀⠻⣿⡇⠀⢀⡀⠛⢿⡆⣿⠀⠀⣿⠃⠀⠀⠸⣷⢋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⡀⠀⣷⡜⢃⠀⣾⣷⣾⣿⠛⣿⠰⣷⣆⣶⠀⢀⡴⠛⠛⠁⠀⠀⢀⣶⡀⢸⡇
⠀⠀⠀⠀⠀⣀⡦⠀⠀⠉⠉⠿⣀⠈⠳⣿⡀⣿⠁⠈⢻⠀⢻⠀⣹⣟⣻⠴⣿⣅⡀⠀⢀⣿⡄⢙⡾⠁⠸⠛⢁⡄
⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣼⡆⠈⠛⣠⣬⣧⡾⠒⠛⠛⡦⢈⡴⠿⠋⠀⠀⠀⠉⠃⢀⠀⡿⢀⡾⠁⢸⣷⠿⠟⠁
⠀⠀⠀⠹⢶⡄⢣⢸⡄⠻⣿⣠⣾⠁⠀⢠⣾⢷⡀⠀⠀⠱⣌⠁⡾⠀⠀⡀⠀⠀⠀⠈⠿⣀⢀⣰⣎⣉⠈⠙⠁
⠀⠀⠀⠀⢀⣀⠘⡇⣴⡄⠙⣿⢁⣄⠀⣤⠁⠘⣿⡇⠀⠀⢻⣀⣠⣤⠀⡇⠀⢀⠀⣶⢰⠟⠉⠁⣽⠈
⠀⣤⠀⠀⠀⢹⡶⢿⣿⡀⠀⢡⡈⢻⣰⣟⠀⠀⠙⠳⡄⡶⢸⠋⠁⠀⢸⡇⠀⠸⠂⠹⠃⠀⠘⣷⡟
⠀⠻⣷⠀⢀⠀⠀⠀⠀⠿⢷⠈⣧⡤⣽⣯⣄⠀⠀⠀⢈⣧⣨⠀⢰⣄⠸⠀⠀⢸⣿⠀⣼⠶⢾⣍⢁⣇⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠘⠿⡧⢘⡁⢀⠀⠀⢀⣤⣶⣶⠀⡏⣹⢹⢲⡄⠰⡟⠋⣿⠀⠀⠻⣧⣴⡶⠟⠉⢷⡄⢠⢀⣼⠟⠿⠁⣀⠀⣰⡄⢼⣇⣿⢁
⣸⡇⠰⠿⠛⠲⣿⡄⢀⣌⠀⠀⠈⣧⣁⣼⣸⡿⢿⡀⠀⠀⣿⠀⠉⣙⣇⠾⠶⣦⡄⢈⣁⡾⠛⠋⠉⠀⣠⣏⠘⣟⡡⣾⠻⠿⠏
⢀⡑⣄⠀⠀⠀⠉⠑⢢⣿⣆⠠⣦⠀⠟⢻⡏⠀⠀⠀⣹⣶⡍⣠⡄⣼⡁⢀⣤⢴⣿⠊⠉⢀⠀⠀⢀⠀⣼⣿⠉⢉⠳⡾
⠀⠙⠻⠷⡄⢿⡄⠀⠀⠈⢁⡼⠿⢦⡘⢇⣱⣶⠶⠾⠟⢸⡇⣿⠧⠞⠋⣈⠀⠰⠟⠀⠀⠿⠆⠀⠸⡷⠉⠀⠀⣾⠋
⠀⢤⣤⣦⡈⢻⣅⡀⠀⠀⠀⠀⢾⣠⠑⢾⡇⠈⠳⣄⠀⠸⡷⢤⡖⠀⠀⣹⡶⢠⡏⠀⠈⠹⢷⣤⠶⠽⠃⡠⠛⠛
⠀⠀⠈⠁⠀⠀⢸⣟⣆⣀⡀⢶⣈⠛⠀⣠⣿⡄⠀⢈⣧⣴⠀⢘⠁⣀⠀⠉⢷⡌⠀⠀⠀⡦⡴⠃⢠⠄⠊⣀⣠⣄⣠⠄⠀⣤
⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠑⠾⣿⡀⠘⠃⠀⠸⡶⠛⠁⢽⡀⢂⣤⠻⠶⠶⠄⢁⣠⣄⣴⣿⠖⠈⣉⠀⣖⠛⠋⣤⠓⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢢⡄⢻⡁⣲⡶⢺⢿⡀⠀⣾⠛⠁⠀⠸⣗⢠⠘⣋⡶⠏⠘⠃⠀⣼⠇⠀⠛⣷⠀⠘⣧⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣻⡎⣖⠀⢹⣾⡟⠛⠲⠀⠀⢱⠀⣿⠀⠰⠟⢷⣾⡧⢾⣉⣤⣶⣤⡤⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠉⣉⣷⣿⣦⠀⠘⠇⣴⡟⡇⠀⠈⢆⡿⠀⣠⠤⠚⣡⡔⣉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⠙⢢⣼⡟⠁⢀⣠⡆⠘⣿⠚⠓⠾⣉⠛⠻⠉⢀⡤⠾⢶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣴⡀⠀⠻⡀⠀⣿⠀⠀⢠⣏⣀⡴⠚⠁⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⠘⢄⡁⠀⠀⠞⠁⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡶⣶⠾⢿⠀⣼⣴⡾⠟⠛⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢻⡉⠀⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡷⣦⣤⣀⢹⡄⣿⠒⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠓⠿⠿⣿⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⠅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⣿⣿⣿⣿⣏⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠀⠀⢀⢀⡷⢠⣼⣿⣿⣿⣿⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣾⣶⣦⣄⣒⣒⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⢿⣟⣿⣿⣿⠿⢿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠊⠙⣿⠛⠛⠛⢫⣿⣿⡇⠀⣼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠣⠄⢀⣈⣓⠈⠳⣠⣴⣾⣿⣿⣯⣿⣷⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠚⠒⢀⣈⣭⣛⠷⡹⣿⣯⡙⠿⠭⠿⢿⣿⣦⣄⡙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠊⠀⢀⣈⠄⠂⢀⣡⡶⢟⣫⡥⠀⢀⢀⠻⣯⠈⠄⠀⠀⠀⠀⠈⠙⠻⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⠃⠀⣠⠞⠋⣀⣴⢞⣫⡵⠞⣫⡿⣧⣿⣾⣿⡆⢹⡆⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢶⡶⠔⠁⡠⠞⠋⠊⠈⠁⣠⣾⣫⣾⣿⣿⣿⡟⢿⡄⢳⡀⢆⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠺⢷⡔⠈⠀⠀⢀⣤⣴⣾⣿⡿⢫⣾⠇⣿⣿⣿⠘⣿⡀⣷⠀⠙⢮⡓⢤⡀⠀⠀⠀⠀⠀⠀⠠⠭⠃⡄⠀⠀⠀⢠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠴⠿⣿⣇⡙⢿⣿⠿⠟⠋⣭⣴⣿⡟⢸⣿⣿⢹⣇⢹⣧⠘⡄⢢⡀⣹⠞⠃⠵⠤⡀⠀⠀⢢⣄⣴⣖⣷⣤⣀⡁⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠐⡲⠞⢿⣟⣿⠦⠤⣤⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⠈⣿⠀⠘⡄⠻⠃⠀⢀⡀⠀⠤⠚⢛⣀⠀⠀⠀⠀⠉⠤⠶⢛⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡆⠙⢿⣾⣟⣧⣀⣠⣾⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣠⣾⣷⣾⣿⠉⠁⠘⠂⢡⣠⣤⣴⠾⠛⠉⠓⠒⠠⠟⠃⠬⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡴⠸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⡿⣧⣤⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢯⢼⡇⣿⣿⣿⣿⣧⣟⢻⣿⣿⡿⠿⣿⣭⣽⡿⢴⠟⠔⢷⣹⡇⠻⢿⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣽⡿⣶⣿⣿⣿⣿⡟⡌⢸⡟⣽⣿⣿⡟⣿⡿⣿⣴⡦⠄⠶⠋⠑⠈⠁
⠀⠀⠀⢀⠀⠀⡐⠀⠀⡼⠀⣿⣧⣿⣿⣿⣿⣿⡅⠀⢸⢃⣿⣿⣿⢱⡜⢷⠘⢿⣿⣆⠀⠀⠢
⠀⠀⠀⠀⢀⠀⢈⡦⠊⢀⣠⣿⠃⢻⣿⣿⣿⣿⠁⡇⣿⢸⣿⣿⣿⠈⣿⣦⡀⠀⢻⣿⣷⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠀⠠⡀
⠀⠀⠀⠈⠂⣄⠉⣠⣾⣿⡟⠁⠀⣼⣿⣿⣿⡇⢀⣇⠀⢸⣿⣿⣿⣇⢻⣿⢷⡼⣆⠻⣿⣦⡙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⣀
⠀⠀⠀⣐⡤⠜⣹⣩⡟⣫⠀⠀⠀⠹⣿⣿⣿⣧⣼⣯⠀⠰⣿⣿⡞⣿⣦⡙⠧⢻⣜⠆⠹⡻⣿⣶⣄⡙⠃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠋⠀⠠
⠀⠀⠀⠈⠉⠛⠋⠉⠁⡉⠀⠀⣠⣮⣙⣿⡏⢻⣿⣿⣧⠀⢻⣿⣧⠹⣿⣿⣦⢤⣛⣦⡀⠱⡈⢻⠛⣿⡷⠦⣌⡁⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢉⢻⣿⣟⢷⣸⣿⣿⣿⣷⡌⢿⣿⣶⣿⣿⣷⣶⣮⡙⠻⣦⠰⡸⠂⡿⢻⣆⠀⠠⣀⡀⠀⠀⠀⠀⠠⠖⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣮⡻⣿⣄⢘⣿⣿⣿⣿⣿⡜⣿⣏⣿⣿⠟⣥⣝⢿⣷⡈⢧⠹⣶⣄⠀⣹⡶⡀⠀⠉⠓⠦⠀⠀⠀⠀⢒⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠙⡙⠃⡽⠋⣴⢿⡿⢛⣭⣿⣿⣿⣿⢻⣧⣼⣷⡜⢎⢿⣷⡈⣇⠹⠻⣮⢿⣧⡿⠄⠈⠑⢦⣀⠀⢤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠛⠀⠸⣦⣉⣤⣤⣾⣿⣿⣿⣿⣿⣿⡙⢻⣿⣿⡌⢮⣿⣷⣹⣦⢳⣬⣿⣿⣇⠸⣆⢢⡀⠉⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⡄⠘⠛⠿⠿⣿⣻⣿⣿⣿⣯⣾⠃⠀⠈⠙⠛⠦⣜⣈⣛⣛⣳⡛⠛⡧⣻⣆⣿⣦⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⡿⠄⠀⠓⠒⠊⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠁⠀⠉⠀⠀⠈⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠄⢀⠀⠈⠀⠈⠄⠀⠀⠀⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠊⢤⢀⡌⠀⠐⣊⠀⠀⠰⠄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⡀⠀⠋⠀⠁⠠⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⢃⠀⠀⠀⠐⠄⠀⠀⠀⡀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠴⣀⠀⣠⡀⠠⠄⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢯⣤⠧⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⡀⠀⠀⠀⠀⠀⢀⡀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁⠈⠀⠐⠀⠀⠀⠀⠀⠊⠈⠰⢦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡜⠀⠡⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠤⣒⠼⠟⠛⠛⠻⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠈⠉⠛⠅⠢⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⣀⣀⣠⣤⣤⣷⣵⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⣾⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⡾⠛⣿⣿⣿⢿⠇⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⢟⠙⠿⠿⠟⠉⠛⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠱⣄⡀⠉⣀⣞⡤⣺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡕⠮⣦⣤⣤⣤⣒⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠛⠛⠋⣁⢼⣿⠙⠋⡇⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡶⣄⡈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣴⠿⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠟⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠨⠀⠘⡀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣀⣀⣀⣤⣤⣤⣤⣄⠤⠤⠀⠐⠀⠂⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠉⠉⠀⠀⠀⠀⠁⠒⠲⠬⣕⣒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠊⠉⠀⠀⠀⠀⢀⣀⡠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠍⡐⠀⠀⠐⠈⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠤⣴⣆⠀⠀⠀⠈⠛⠋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣿⣶⣄⡀⠀⠀⠉⠲⠆⣀⣀⣀⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⣼⠿⠿⠛⠚⠛⠿⢶⣤⣴⠂⢀⣉⠉⠈⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣧⣤⡀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡝⠛⠷⢶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣴⢆⠀⠀⠀⣠⣄⣀⣀⣀⠀⠀⢀⣼⣧⡈⠙⣦⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣼⣧⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⠻⣷⣄⠙⢧⡀⠈⠉⠉⠁⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠙⢷⣦⣱⣄⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⣷⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⢻⣏⠉⠉⠉⠻⣿⣿⡛⣩⡿⠋⣿⠉⠉⠉⠙⢿⣿⣯⠉⣿⡏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⣤⣄⣀⣈⣭⣿⠟⠁⠀⠈⠻⣦⣤⣄⣈⣩⣭⡿⠋⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠈⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢰⠟⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⣤⣄
⠀⠀⠀⠀⠸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠹⠶⠶⣿⡇⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣶⣶⣤⣤⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣦⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢶⣶⣦⣤⣤⣀⣀⣀⣀⣄⣀⣠⣤⣤⣶⣶⣶⣶⡶⠶⠶⠶⠟⠛⠛⠋⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡜⢫⡀⠀⠀⠀⣿⠉⠀⠀⠀⢹⡏⠁⠈⣿⠃⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⣷
⠀⠀⠀⠀⠀⢠⡎⠀⢸⠁⠀⠀⣠⠋⠀⠀⠀⠀⠈⡇⠀⢠⡟⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⣿⠇⠈⣿
⠀⠀⠀⠀⢀⡜⠁⢰⡃⠀⠀⠀⣿⠀⣠⡄⠀⠀⠘⠗⠀⣿⠇⠀⠀⠀⣤⣄⣸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠘⠃⠀⠸⠇⠀⠀⠀⡿⠻⣯⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⢀⣼⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣷
⠀⠀⠀⢀⣇⣀⣴⣦⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⣸⣿⠃⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠤⠤⠤⢾⣿⠀⠀⢸⣿
⠀⣴⠊⣹⠟⠋⢀⣴⡿⠿⣦⠀⠀⠀⠀⠹⣧⠀⠀⣿⡏⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⢀⣠⣴⢶⣾⣿⠀⠀⠈⣿⣀⠀⡀
⠀⣹⠾⠁⠀⣰⡞⠁⢀⣴⣿⡀⠀⠀⠀⢀⣸⣧⣤⣿⣧⣠⠴⠾⠿⠶⠶⠶⠶⠶⣶⣮⣉⣀⣀⣸⣿⠀⠀⠀⣿⣏⣀⣁⣤
⣤⠟⠀⢀⣼⠋⠀⢰⠏⠁⢈⣿⠖⠾⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣤⣤⡶⠿⠋⠉⠉⠉⠉⠙⠿⠀⠀⠀⢹⣯⢈⡉⠉⠿
⠻⣤⣰⣿⠁⠀⡸⠃⠀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⠛⣿⡄
⠀⢸⡿⠀⢠⡼⠋⠀⣴⣿⣦⣤⣶⣤⣶⡶⠞⠛⠛⢻⡏⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⠶⠶⣿⣧⠈⠢⣄⣀⣹
⠀⠸⣷⣄⢘⣷⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⣦⡼⠛⠃
⠀⠀⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⣱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⡇
⠀⠀⠀⠀⠸⣷⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⠿⠿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠿⣿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⢻⡟⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣤⠼⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠖⠊⠉⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣀⡀⠀⠀⠉⠻⠿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣤⠞⠛⠉⠉⠀⣠⣴⣿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⡻⠿⣦⣤⡀⠀⠀⠀⠉⠙⠛⠛⢦⣤⣤
⠀⢠⣶⣤⡘⠋⣀⣀⣀⢤⣶⡾⠛⠋⠀⠘⠿⠰⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠈⠛⠛⠛⠲⠶⠶⠶⠶⠄⠛⠛
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠛⢻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⣿⠀⠀⠀⠀⠄⠄⢺⣿
⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠠⡗⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠖⠁⠀⠀⠀⠠⢌⣳⢺⣿
⣿⣟⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠋⠉⠙⢛
⣿⣿⣟⠀⢴⣷⢤⡤⠔⠂⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣰⡆⠀⢀⢀⠀⠀⠹⣿⠇⠀⣀⢀⣿
⣿⣿⣿⠀⠨⣏⢪⡀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⢀⣼⣷⡀⠀⠙⠀⡀⠀⣼⣿
⠟⢻⣧⠀⠈⡟⠁⠛⢄⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡴⢦⣬⣍⣻⡷⠀⣠⣼⡇⢰⣿⣿
⠀⠀⢈⠆⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⡟⠻⡟⠁⠀⢛⠇⠀⢤⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠂⢀⠾⢿⠅⠐⠘⢦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠉⠀⠁⠀⠀⠀⠁⡜⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢠⣴⣴⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⡀⠀⠿⢿⠁⠈⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣡⣤⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠒⠛⠛⠓⠒⠤⠌⠉⠛⠛⢗⠒⠒⠲⠶⠦⣤⣀⡀⣼⠁⢀⢀⣴⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠁⣼⠿⠋⠑⢦⣀⠀⠀⠠⡀⡄⠀⠑⣄⠀⠀⠀⠀⠈⠙⠣⠶⣾⣿⠇⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠐⢀⠔⠋⠁⣸⡣⠀⠀⠀⠀⠉⠟⠦⢀⠈⠻⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡿⠑⢦⡀⠀⠀⣠⡿⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⡇⠀⠀⢀⡠⢔⣦⣀⠼⢷⡄⠈⠀⠀⠀⢳⡄⠀⠀⠀⠀⣼⠁⠀⠀⠙⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⢯⡅⠀⠐⢿⣿⡯⠃⢰⣿⣆⠀⠀⠀⠈⣿⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢻⠟⢱⣦⠈⣉⠁⣠⡯⢻⣿⡄⢠⡆⠀⢹⠏⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣶⣆⠀⠀⠀⢈⣽⣿⣷⠹⣿⡇⢨⡇⣰⠘⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡸⣿⣿⣧⣶⡞⠀⢠⠟⠉⠀⣿⢣⠟⢁⡿⠀⢸⣿⣾⣷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣞⢿⣿⣿⠡⠄⠂⠀⣰⡟⠁⠀⠀⠘⠃⢠⡿⣿⡟⠻⡄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣕⣈⠻⣇⣠⣀⣼⠏⡔⠀⡄⠀⠀⢠⣿⡟⢻⡇⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣿⣽⣮⣝⠿⡟⣸⡇⠀⣿⠀⠀⣿⠋⠀⢸⠃⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠇⠁⣿⡇⢹⣶⡇⣿⡇⠀⡔⠀⣼⠇⠀⠀⠘⠀⡆⠀⠁⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⢰⠟⢠⠸⡆⠀⠀⠀⡫⢷⡞⡓⠀⢛⣃⡰⢁⣼⢿⠃⢠⠀⠀⢠⡇⠀⡀⡄⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⢗⣠⠎⠀⠀⠊⡞⠁⢀⢠⣾⡑⢿⣿⣿⣤⣿⣄⣺⣾⡇⠘⣸⣧⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡭⠖⠉⠇⠀⢀⠀⢀⡱⡀⣰⡟⢦⣙⡶⡹⣿⡿⢿⣿⠿⣿⣿⣾⡿⢿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⢀⠀⠀⠀⢸⠀⢸⠹⠿⠧⢙⢳⣞⠓⠁⡌⠃⠀⢀⡄⠀⠉⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⣧⣾⣗⠄⣠⡿⣦⡟⣦⠀⠐⠀⠙⠀⠀⠰⡇⠀⠀⠈⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠁⠉⠛⠉⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣷⡤⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⢠⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢉⣼⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠿⠛⠃⠀⠀⠀⠀⢠⣾⣶⡄⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠻⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡾
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠋⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣴⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠔⢲⡆⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡽⠁
⠀⠀⠀⠀⣿⢦⠀⠀⠀⣿⡏⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠋⠀⠀⢠⣟⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⣾⠂⠀⠀⠹⢿⣤⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠁⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠙⠛⠶⠦⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⠾⠷⠒⠶⠶⣦⣄
⠀⠀⠀⢀⣤⡶⠟⠋⠉⠁⠈⠙⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠦⢤⣤⣤⠤⠤⠴⠖⠒⠉⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⢹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠹⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣷⣦⣤⣀⣀⣀⣤⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉⠉⠙⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣿⣻⣻⣻⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣷⣶⣦⡀⠇⣠⢒⣤⣤⣤⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡄⠐⢁⣠⣤⣴⣿⣿⣿⣿⣾⣿⣿⣿⣶⣶⣧⣤⡀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣠⣶⣿⣿⣿⠿⢟⠛⠛⢿⣿⣿⣿⡿⢿⣿⣷⣬⡙⢷⡄
⠀⡀⠀⠀⠀⠀⣀⠠⠤⠒⠒⣼⠟⣽⣿⡿⢋⡴⠛⠈⠀⠀⢿⣿⠹⣿⠀⠺⣟⢿⣿⣦⠄⣀
⠀⣆⠈⠉⠃⠁⠀⠀⠀⠀⠰⠋⣼⣿⡿⢱⠈⠒⢁⣤⣤⠀⠸⠏⠀⠏⣤⣌⠹⠈⢻⣿⣧⡀⠉⠓⠢⢄⡀
⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠈⠀⠰⢛⡭⡲⠢⡀⠀⠀⠀⠹⡿⡷⠀⠀⣿⣿⣧⠀⠀⠀⠀⠈⠓⠦
⠀⠀⠈⢳⠄⠀⠀⠀⠀⢀⡀⢸⣿⣿⡇⠀⠀⢠⡇⠀⠆⠀⠀⠀⢀⡀⠀⠁⠁⠀⢠⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢀⡞⠀⠀⠀⠀⠔⡜⣡⡼⠿⣿⠁⠀⠀⠸⡄⠀⠀⠉⠁⠀⠈⠁⢀⠆⠀⠀⣼⣿⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣼⠁⠀⠀⠀⠀⠈⡾⠁⢨⣀⠀⠀⠀⠀⠐⠨⠥⠶⠃⡀⠈⡷⠺⠗⠊⠀⠐⢿⣛⣋⢳⠈⠇⠀⠀⠀⠀⠀⡄⠀⠈⡇
⠀⢠⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⠟⢸⢸⠇⠁⠀⠀⠀⠀⣼⠁⠀⠀⡇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠹⣄⡀⢀⣹⡀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞⠀⠀⠀⣀⣴⠋⢀⣠⠞⠀⠀⠀⠀⠀⣰⡏⠀⠀⢠⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⢈⠉⠉⢉⡱⢤⡒⠔⠖⠋⠁⠀⠀⣀⣠⣴⣶⡋⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⢠⢿⠁⠀⠀⣸⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⢤⣦⠭⠒⣈⡥⠶⠛⠛⢿⡏⠀⣰⠋⠉⠙⢿⡇⠀⠈⠑⠢⠄⡀⠀⠀⠀⢀⢇⠇⠀⠀⠀⡿
⠀⢻⠀⠀⠀⠀⠀⠐⠙⠖⣚⠥⠒⠉⠀⡵⠃⢠⣿⣎⡀⣰⣵⣿⠉⠐⠛⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣸⡞⠀⠀⠀⣸⠇
⠀⢸⡇⠀⠀⠀⠀⠀⠈⢉⡀⠀⠀⡠⢊⠜⣱⣿⡟⢸⣷⡇⢸⣿⣇⠀⡄⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣰⠏
⠀⠈⣷⠀⠀⢠⠒⠒⡎⠀⠈⠂⠘⢀⠏⣰⣿⡿⠀⣸⣿⡇⠀⢿⣿⡄⠘⣆⠀⠹⣄⠀⠀⠀⠀⠀⢸⠁⠀⠀⣰⠏
⠀⠀⢹⣧⣤⣾⠀⠸⡇⠀⢦⠀⢸⡛⠒⠛⠛⠧⢄⣿⣿⣧⠤⠞⠛⠛⠒⠛⠲⠦⠽⣄⡀⠀⠀⠀⠈⠀⠀⣼⠏
⠠⠒⠉⠀⠀⠈⠦⣀⠿⠄⠼⠢⠜⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⢀⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣁⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠉⠉⠉⣙⣢⣀⠀⠀⠈⠿⠋⠀⠀⠙⠿⠋⠀⠀⢀⣴⣚⣋⡉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠙⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣦⣀⡀⠀⠀⠀⢻⠀⣾⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⡴⠋⡀⠀⠀⡘⠷⣄⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⣥⠴⠚⠁⠀⠀⠈⠳⢮⣭⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⢄⣀⣠⠞⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⠲⢤⣀⣀⠄⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠳⠶⢤⣷⣤⣀⣀⣀⣀⣀⣀⣼⣤⡴⠶⠚⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⠉⠉⠉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠛⢶⣤⡀⠀⢀⣠⠞⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁


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

⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⢀⣴⣶⡜⠻⠿⢣⣾⣷⡎⢻⣟⠁⢼⣿⡇⠀⢀⣀⣠⣤⣾⣿⣦⣶⣶⣿⣿⣶⣶⣤⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⠛⠛⠃⣾⣿⣦⠙⠋⠁⣿⣿⣿⡇⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠻⣿⠿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣤⣤⢠⣤⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡶⠄⠀⠀⠀⠀⣀⣀⣀⣉⠉⠉⣁⣀⣉⡉⠉⢩⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠀⠻⠿⠿⠿⠿⠀⠸⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣤⣴⣒⣲⢶⣶⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠂⠤⡀
⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⢠⠶⠀⠠⠀⠀⠂⠀⢄⠈⠑⠒⠀⠀⠠⠄⣀
⠀⠀⠀⠠⠤⠤⠶⣦⡤⠙⠿⠿⠿⠿⠛⠛⠙⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⡀⠀⠁⢌⠑⢄⡀⠀⠀⠀⠀⠀⠘⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣾⠁⠀⠈⠳⢦⠀⠀⠀⣸⠀⠀⣷⠀⠆⢧⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠆⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⣿⠀⠀⠀⠀⠀⢀⡴⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢰⡟⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠙⠦⠔⠋⠁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⣼⣟⠀⠘⠮⣴⠄⠀⠀⢰⡀⠀⣷⠐⡈⣧⠇⠀⠀⠀⢀⣴⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠈⠀⠀⠀⣟⠁⠘⣿⣷⠀⠈⠀⠀⠀⣠⠿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⢠⣿⠇⠀⠀⠀⠀⠈⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠏⠀⠀⠀⠲⣶⡦⠞⠋⠁⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣿⡇⢀⠰⢤⡀⠀⠀⠀⢠⠀⡀⣧⢠⡖⣆⠀⠀⠀⠀⢀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣿⣿⡀⠀⠙⠢⠉⠀⠀⡀⡞⢀⡷⣿⠀⢧⠙⠀⠀⠀⣰⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠈⠀⠀⠀⠀⠀⢠⣾⠇⠸⣿⡿⣄⠀⠀⠀⠠⠞⠉⢸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠢⣄⣠⡴⠟⠃⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⢸⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⡀⠀⡠⡀⠀⠤⠀⠤⠐⠁⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡁⠀⠛⠀⠀⠀⠰⠄⠀⡀⠴⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⠠⡀⠀⠀⡀⢄⠀⢰⢏⣠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⠀⠀⢀⠄⠀⢈⠱⠖⢓⠷⢖⣣⡀⡄⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⢐⡤⠐⠁⡠⠊⡡⠒⠉⠇⠀⠈⠄⠀⠈⢢⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣈⠉⠐⠺⡀⡌⠀⠀⠘⠓⠆⠀⠞⠀⠀⠀⢂⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⠐⡦⡈⠉⠀⠂⡄⠀⠰⠖⠬⠀⠀⠀⠈⡄⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢁⠈⢢⠀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢄⠀⠀⠀⠑⢌⠓⠓⠏⣐⠂⠠⠄⢀⣀⡀⠠⠜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠠⡵⠂⠄⡀⠈⠁⠐⠒⠀⠀⠐⠒⠒⠀⠈⠉⢉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠉⠁⠀⠐⠒⠒⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡐⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠾⠿⠿⠿⠿⠿⠿⠷⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣦⡀⢠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⢠⣴⣶⣶⣼⣿⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣦⣄⡈⠙⠃⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣦⡀
⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠀⣿⠀⠀⢰⡆⠀⠀⢹⣷
⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⣿
⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢰⡇⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⠀⠀⢸⡿⣆⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⢸⡇⢹⡄⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⣹⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⣼⠁⠀⠀⣾⠁⠀⣷
⢿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡄⠀⢀⡴⠋⢸⡇⠀⠀⠀⣰⠟⡇⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⡾⠁⠀⠀⠀⢠⠿⣄⣠⡼⠃⠀⠀⢸⡄⢸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⠛⠋⠀⠀⠘⢧⣤⡴⠟⠁⠀⢷⣤⣤⡴⢿⡀⠀⠀⠀⣠⢿⡇⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡅⣼
⠈⣿⡆⠀⣠⣤⣤⣆⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠛⠶⠶⠛⠁⠈⠳⣤⡴⠾⠋⠀⢀⣀⣀⠀⠀⠀⠀⠘⠁⣿
⠀⢻⣷⠘⣇⠀⢠⣉⠳⣤⣼⠇⠀⠀⠀⠀⠀⠀⢀⣉⣙⣲⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣏⣉⣉⣀⣀⡀⠀⠀⠀⣾⡿
⠀⠘⣿⡀⠙⢶⣄⣹⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⣉⣭⠿⠛⠁⠀⢀⡤⣤⡶⢶⣴⠶⣤⡶⣤⠀⠀⠈⠻⢦⣉⠉⠉⠉⠁⠀⠀⠀⢻⡇
⠀⠀⣿⡇⠀⠀⠈⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⣸⡏
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⣾⡟⠀⠀⢀⡀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⢸⣿⠁⠀⣰⠟⠀⠀⠀⣶⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠋
⠈⠛⠷⠿⠻⣧⣤⣤⣴⣿⣤⣀⣀⣷⣄⣀⡀⠉⠙⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠙⠛⠛⠿⠾⣷⣤⣤⣍⣉⣿⠷⣦⡀⠀⠀⠤⢤⣤⣤⣤⣤⣤⡶⠶⠟⠛⣩⡁⢀⣶⣄⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠿⢷⡀⠈⠛⢦⣄⣀⣀⣤⠞⠁⠙⢷⣤⣴⣿⡾⠛⠻⠟⠋⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢻⡏⠀⠀⠀⠀⠙⣦⡀⣠⠞⠻⣯⡀⢀⣠⡶⠋⠉⠙⣿⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢻⡄⢀⣤⠀⠀⠈⠛⠉⠀⠀⠈⠛⠋⠁⠴⢶⣄⢠⡟⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⡀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢠⣿⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠉⢻⣧⢠⣄⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠈⣿⠧⣀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠀⠀⣾⣋⡄⢹⣇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢀⣰⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠙⠷⠾⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠿⠛⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣈⣉⣉⣿⠟⠉⠉⠉⠉⠉⠉⠉⣿⣭⣭⣤⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠉⢉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣶⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣦⣴⡾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠃⠠⠊⠐⠗⣪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠲⠤⡈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠁⠀⠀⠀⢀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠄⠀⠀⠀⣠⣭⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⡸⠁⠀⢪⣌⢱⠀⢴⣁⠅⠀⢣⢠⣄⠘⠛⠛⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠐⢀⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠁⠀⠇⢈⢱⠎⢸⡁⠀⢔⣀⣀⠀⢸⠀⠑⠃⠐⠶⠆⠀⠀⠀⠀⡠⠂⠀⢀⠘⠈⣠⠎⣀⠀⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⡲⠘⠀⠀⠉⢿⡊⠁⠀⠈⢳⢸⠀⠀⠀⠀⣀⣀⠄⠀⡠⠊⠀⢠⠰⠀⠁⠔⠁⢊⡡⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠔⣡⠊⠉⠳⢊⡆⠀⠀⠠⠂⠄⢂⡤⠊⠀⢠⢸⠘⣠⠖⠀⠀⠀⠉⢀⡜⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠉⠁⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⠀⠐⠈⠉⠐⠒⠀⠁⢀⡀⠸⠸⢀⠜⠁⠀⠀⠀⠀⠀⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠒⣋⣠⠤⠒⡹⢀⠊⡰⠁⡔⢀⠔⢀⠔⣠⠆⠀⢠⢂⠆⠇⣁⠔⠁
⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⢄⠖⡺⢉⢝⡴⡣⢂⠜⡰⢃⠜⡠⠊⡠⢋⡰⢋⡴⠁⡠⠂⡀⢨⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠠⡙⠲⠤⣁⡊⡔⠡⠊⠚⠐⢃⠞⡰⢃⠜⣡⠎⡠⠊⡠⠋⡠⠊⡠⢰⠁
⠀⠀⠀⠀⠀⠀⠲⠬⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⢈⡔⣡⠞⣡⠊⣡⠎⡡⠊⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠔⣡⠞⣡⠞⡡⠊⡠⣊⠔⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡡⠊⡡⢊⡵⢋⠔⡫⢚⡁⢠⠏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢈⡴⢊⠔⢁⢔⡥⢊⠔⢉⠀⡇⠀⢸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣉⠔⣡⠖⠁⢌⠐⡡⠒⡡⠄⢸⡀⠸⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠎⠀⠀⢘⡡⢊⠴⠋⡇⢸⠈⠰⣋⠴⢊⠀⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢠⠀⠀⠀⡩⠔⢁⠔⠀⡅⡈⠀⡨⠔⣊⠡⠂⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⡄⢸⠀⠀⠀⠴⢊⡥⠒⠀⡿⠀⠀⢒⡩⠔⢊⡁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⠁⠸⡆⠀⠀⠚⣡⠔⣃⠄⢱⠀⢈⡥⠖⣊⠥⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢧⠀⠀⠈⠴⣊⠔⡂⠈⢇⠀⢒⣉⠤⣒⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠊⠔⠪⠶⠀⠘⠀⠀⠀⠈⢁⡰⠳⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡠⠤⠐⣀⠄⢳⠀⠀⠀⠀⠀⠀⠀⠙⠻⡭⢛⠥⣂⠤⠒⡩⠄⢊⡡⠔⣂⠤⢒⡢⠔⡀⠤⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠔⠊⠥⢒⠶⢶⡤⠤⢤⣤⠴⢐⡣⠐⣈⠤⠒⠁⠔⠊⡡⠖⠊⠀⠊⠁⠀⠉⠀⠈⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠛⠉⠉⠩⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⡀⠀⠀⣾⣿⣿⣦⠘⡏⣿⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣿⣿⣷⠀⠀⠻⣿⣿⡿⠀⠃⣹⡇⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⠿⠃⢰⣤⡄⢨⣭⢀⡎⢐⣿⣷⠀⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡇⠘⠛⠃⠀⢡⣿⣿⡿⢻⣿⣷⡀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⢰⠘⣿⣿⠇⣼⣿⣿⠟⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠉⠛⠀⠐⠉⢉⣾⠛⢃⠀⠀⠀⠀⠃⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣇⠀⢀⣀⣀⣴⣿⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄⠉⠁⠙⠏⠻⢿⣿⣿⣷⣿⠃⡰⠈⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠈⠁⠀⠀⠐⣄⠀⠉⠛⣿⣿⠀⠀⠀⠀⢠⠾⠄⠄⠠⠤⠄⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢿⡄⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠤⠤⠦⠤⢿⣄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠈⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠄⡠⣠⣤⣤⣘⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡓⣷⣿⠋⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠘⠀⠀⠈⠀⢀⣝⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠁⣤⣷⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢹⡎⠉⠉⠉⣠⣾⡛⠉⠂⠀⠀⠀⢄⠠⠀⠃⠁⠀⢠⠆⢰⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⢈⡇⠀⡐⠀⢠⡏⢀⡇⠀⣿⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠉⠀⠀⠀⠀⡟⠿⠇⣶⠀⠀⠈⠀⠸⠀⠀⢿⣄⣀⣈⠹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⠀⠀⠈⠛⠛⠧⠀⢀⠀⠀⠀⠀⠁⠀⠀⠸⡇⢸⣷⠀⣶⠂⣠⣾⣿⣱⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠁⠈⠽⠀⡟⢰⠏⠀⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠒⠒⠚⠃⠀⠀⠈⠓⠒⠒⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⢀⡀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠹⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠁⠀⣸⠀⠀⠈⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡀⠀⠀⣠⠀⠀⠀⠉⠀⠀⣄⠀⠀⢠⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠉⠀⠘⠋⠙⠃⠀⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣝⡳⠦⠤⠄⠠⠤⠴⢞⣫⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⢋⣿⢧⣀⠀⠀⠀⠀⣀⡼⣿⣹⣿⣿⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣾⣿⠀⠉⣹⣿⣿⣏⠉⠀⣿⣿⣾⣿⣿⣿⣿⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣿⣿⣿⣿⣿⣯⣿⣿⣀⡼⣿⣽⣟⣿⢧⣀⣿⢻⣭⣿⣿⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⣿⢻⣿⣿⢿⡏⠀⢸⣯⣽⡇⠀⢹⣿⣿⣿⣿⣿⣧⣿⣻⣿⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣿⣹⣿⣿⡿⣿⣿⣿⣿⣿⣷⠀⣸⡿⢻⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⢹⡏⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣏⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠽⠿⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣤⣤⣤⡄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿
⠀⠀⣿⠟⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠛⠛⠛⠃⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢳⡄⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠳⣴⠏⠀⠸⠟⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠷⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣘⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣄⡀
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣤⠀⠀⠀⣆⠀⠀⠸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠀⠀⠀⢻⠀⠀⠀⣿
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠀⠀⣾⡀⠀⠀⠀⣼⡀⠀⠀⣼⠀⠀⣀⡿
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠘⣦⣀⢹⡦⠤⠤⠋⠉⠓⠦⠤⠾⠋⠙⠲⠤⠖⠋⠙⠒⠒⠋⠉⣿⠋
⠀⠀⠀⠀⢈⣽⢆⣀⠀⠀⠀⠀⢹⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⣠⠟⠱⣏⠨⠽⣄⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣿⠀⠸⣧⡀
⠀⠀⣴⠏⠀⠀⠈⠙⠛⠛⢳⠶⠞⠁⠀⠀⢹⡀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠚⠁⠀⠀⠈⣷
⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⢰⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠃⠀⠀⠀⠀⠀⠀⢀⡾⢷⣄
⣿⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⢀⡀⢹
⣿⠀⠀⠀⠀⣠⣾⢀⣀⣀⠀⢀⡀⠀⠀⣽⠛⠒⠦⠤⠤⠤⣤⣤⣤⣤⣤⣤⠴⠶⠒⢋⡵⢛⡷⠞⣫⡿⣾
⠙⢦⣤⣴⠞⢱⣟⠛⢧⡉⠉⠚⢷⡦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⢳⣿⣁⣀⣤⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢾⣍⣙⣤⣤⠤⠴⠾⢧⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣷⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠉⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⡟⠛⠓⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⢻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠉⠉⠉⠉⡏⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⣀⣀⣿⣀⣀⣸⣇⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠂⠈⠉⠉⠉⠁⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠄⠂⠀⠀⠀⠀⠀⠀⠀⠒⠠⡤⠂⠈⠁⠀⠐⠢⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡤⠄⢐⡒⠒⠒⠂⠀⠤⠬⢄⡀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⠋⠀⠀⠀⠤⢀⡀⠈⠁⠒⠀⠄⡀⠀⠀⠁⠒⠠⣀⠀⢃⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀⠀⠉⠂⢄⡀⠀⠀⠉⠚⢄⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢘⢻⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠈⠑⠤⡀⠀⠀⠈⠒⠄⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢣⡀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⢈⡑⠠⣀⠀⠀⠈⠢⢄⠈⡄
⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡃⠀⠀⠑⣄⠀⠀⠈⠑⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠭⠒⣤⣀⣀⡑⠜⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡐⠀⠀⠀⠀⡌⠛⢦⣀⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠼⠛⢳⠑⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⠜⠀⠀⠀⢀⠜⠀⠀⠀⠉⠛⢔⡢⠄⣀⡀⠀⠀⠉⠁⠒⠠⠤⠄⢀⣀⡀⣀⣀⣠⠤⠄⠒⣛⡴⡢⠔⠃⡆⠑⢄
⠀⠀⠀⡰⠁⠀⠀⢀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠌⢁⣐⠒⠂⠀⠀⠀⠒⠒⢈⣁⡠⠤⠤⠂⣉⣁⠀⠤⠄⠪⢄⠀⠀⠁⠂⠉⠉
⢀⠔⠊⠀⢀⠠⠒⠁⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⢀⠤⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠁⠒⠂⠤⠄⠀⢠⠤⢤⠒⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠙⠯⠥⡤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡦⣀⠀⠀⢀⡠⠔⠁
⠀⠀⢀⠌⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠⠀⠉⠉
⠀⠀⠊⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢡
⠀⠀⠣⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠊
⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠈⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠠⠀⢀⣀⣒⣀⣀⣂⡁⠀⠠⠤⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣄⠀⢀⠤⠄⠶⠒⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡁⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣶⣦⣤⣀⡀⢳⠒⠉⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣿⣿⣿⣿⠿⠟⠋⠹⡿⠛⠋⢸⣿⡿⠋⣸⣿⣿⣿⡟⠉⠀⠀⠈⠙⠻⢿⣿⣿⢑⠢⠤⠤⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣿⣿⡍⠁⠀⢤⣤⠴⠞⠁⠀⠰⠟⠋⠀⠐⠛⠋⠉⠀⠙⠳⠦⣤⠄⠀⠀⢸⣿⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡿⠃⠀⠀⠀⠰⢶⣲⡶⠆⠀⠀⠀⢰⠂⠀⠀⠀⠰⢶⣶⠶⠀⠀⠀⠀⠘⢿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠭⠥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣤⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠈⠓⠢⠔⠊⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⠀⠀⠀⠀⣀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡏⠈⠉⠉⠀⣭⣤⣤⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠈⣿⣿⣿⣷⣄⠀⠀⣠⣿⣿⣿⡟⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣅⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠲⠏⠀⠟⠚⠋⠉⠉⠙⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⢠⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⢸⡀⠀⢸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢲⢄⡷⠀⠚⠉⠒⠶⠚⠉⠒⠒⠊⠙⠒⠋⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡗⢢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠙⠒⠛⡤⠔⠀⠀⠘⢄⠀⠀⠆⠀⠀⠀⢀⡈⠂⠐⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡴⠂⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡰⣟⠀⠀⠀⣤⠀⣸⠛⠒⣦⠀⠀⠤⢤⣤⣴⣒⡚⠉⢀⣸⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⠘⠦⡤⠞⠉⠉⠁⣀⡾⠉⢆⢀⡴⠋⠀⢙⢧⡉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢉⡇⠀⠈⠉⠀⠀⠀⣏⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⣁⣠⣴⠊⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⢤⠴⠶⣶⠶⠶⠶⣤⠤⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠚⢻⡀⠀⢠⡏⠓⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣧⠀⢸⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⡆⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡉⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣛⣛⣓⣂⡀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡟⢿⣦⣤⣀⣹⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⢸⣿⣿⣿⣆⣄⢁⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⣹⣿⠈⣿⣿⣿⡋⠙⣿⣷⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠄⣰⣿⡿⠀⣿⣿⠻⢿⣿⢻⣿⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠗⠀⢹⣿⣧⡀⣿⣝⣶⣿⣿⢸⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⡴⠲⣤⣿⣿⣿⠛⣿⣿⣷⣿⣿⡼⣿⣿⡦⢀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠔⣛⣛⢟⢻⣿⣻⣿⡟⢿⣿⣿⣷⣿⣿⣶⣿⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠋⠴⢋⣴⠟⣱⠏⢡⣾⣿⠁⠈⣹⣿⣿⣿⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣾⣿⣿⣥⣾⡿⢠⣿⠟⣛⠓⢶⣿⣿⢏⡵⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⢟⣾⣿⠂⢸⣿⣯⣾⣃⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡏⠀⢰⣿⣿⣿⠃⢰⣿⣿⣿⣿⣭⡾⢶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠀⠈⣿⣿⡿⢃⡄⣾⣿⣿⣿⣿⣿⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⣿⣿⠃⠀⢸⣿⡿⢡⡾⢱⣿⣿⣿⣿⣿⣿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⣿⣿⣿⣿⣯⠀⠀⣿⣿⣷⠟⠁⣼⣿⣿⣿⣿⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⡀⢰⣿⡿⢃⣀⣠⣿⣿⣿⣿⣿⣿⣇⣰⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣿⣿⣿⣿⣿⣿⣠⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣿⡗⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣷⠻⣿⣿⣿⣯⣔⣛⢻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧⣤⣤⠤⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠁

⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿
⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢸⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣽⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠙⠃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣞⠀⠀⠀⠀⣀⣤⣄⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠵⠄⠀⠀⠀⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠖⠛⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢠⠖⠛⠒⠚⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠳⣄⠹⣤⣀⣀⣀⡴⠚⣁⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡄⠀⠸⡆⠀⠉⠉⠁⢦⡀⣈⠳⡄⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⠀⠀⢹⡀⠀⠀⠀⠈⣇⠸⡆⠹⡆⠀⠀⢸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⡼⢧⠀⠀⠷⠤⠤⠤⠤⠤⠤⢤⡀⢹⠀⠀⢸⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣀⠀⢀⡴⠋⠀⠘⢦⣀⣀⣀⡀⢀⣀⣀⣀⣠⡇⣀⣀⡀⣸⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⢳⡋⠀⠀⠀⠀⠀⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣉⣉⣉⣀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠼⠆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠨⠀⠸⠰⠁⢘⠀⠀⠘⠀⠈⠀⠈⠸⠂⠂⠀⠒⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⣠⡶⣿⡿⢿⣿⣿⣶⣦⡀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠹⣷⡘⣿⡌⠻⣯⡻⣿⡻⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⡘⣿⡄⠹⣷⡘⢿⣎⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠸⣷⠀⠹⣷⠈⢻⣆⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻⡇⠀⢻⣇⠀⢿⣆⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠈⣿⡀⠈⣿⡄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠿⠆⠀⠻⠇⠀⠹⠇⠀⠸⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠒⠛⠛⠓⠲⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢛⣩⣤⣶⣶⣾⠛⠋⠉⠀⠀⠉⠙⢛⣿⣶⣶⣤⣍⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣴⣾⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⣿⣿⣿⣦⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣿⣿⣿⣿⣿⣷⣦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⣾⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⢰⡸⣿⣿⣿⣿⡯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢽⣿⣿⣿⣿⡟⡄⣠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⡆⣧⢿⣿⣗⠪⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⣺⣿⣿⢱⠃⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣇⢸⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⡟⢸⣿⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣿⠀⣿⣿⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⢿⣿⢿⡇⢸⡏⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⠀⣿⣞⠻⠿⠶⠯⣥⣤⣀⠀⢰⡏⣆⠄⠀⠀⡀⠀⠀⡄⣾⣆⠀⢀⣠⣤⡭⠷⠾⠿⣫⣿⠀⣿⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣿⠀⣿⡿⠟⠒⠀⠀⠀⠀⠠⣄⣼⡁⠽⢰⢸⣠⡇⣿⣦⠐⠇⣿⣠⡔⠀⠀⠀⠐⠒⠚⠿⣿⡀⣿⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣿⠠⢋⣶⠀⠀⣀⣀⠀⠉⠉⠙⠻⠷⣦⣾⣿⣿⣿⣿⣿⣶⠾⠟⠛⠉⠉⠁⣀⣀⠀⠀⣰⣌⠇⢻⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢰⣿⣧⣾⣿⣿⣿⣿⣿⣶⣄⡈⡑⢄⡹⣿⣿⣿⣿⡟⣡⢄⡁⣠⣴⣾⣿⣿⣿⣿⣿⣮⣻⣦⠘⣿⠃
⠀⢀⠀⡀⠀⠀⢀⠀⣀⠀⢠⣏⣀⢨⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣝⣷⣽⣿⣽⣿⣽⣯⣵⣫⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣏⣀⡙⣦⣀⢀⠀⠀⠀⡀⢀⡀
⠀⠈⠈⠁⠈⠁⠈⠉⠉⠀⢹⣙⣿⡏⡛⢙⣿⣿⣿⣀⣘⣻⣿⣿⣿⣿⣿⣿⣋⠈⠉⢛⣿⣿⣿⣿⣿⣿⡛⠄⣘⣿⣿⣿⡟⠙⠋⣿⢇⣿⠁⠈⠈⠀⠀⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⠇⣤⣰⡌⣿⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢰⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡄⠛⠈⠲⠶⠿⢿⣛⣛⣿⣿⣟⣛⣣⣿⠏⣸⣿⣿⣷⢘⣿⣮⣛⣛⡿⣿⣟⣛⡻⠿⠷⠖⡃⠘⢁⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⣿⠀⠀⢽⣿⣭⡁⠀⠉⠉⢻⣿⣿⣿⣿⢃⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⠋⠉⠁⠀⣭⣿⣿⠇⠀⢸⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⣿⢇⣾⣿⣿⣿⣿⣿⡌⣿⡍⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⣹⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡷⣦⣦⣤⣶⣲⠦⢆⣀⡀⠀⠀⢸⡜⣿⣿⣿⢿⣿⣿⢇⡟⠀⠀⠀⣀⡐⡤⣶⡶⣤⣤⣶⢾⣻⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣻⡿⣿⣿⣛⣥⣾⣿⣿⣿⣷⣆⠈⠙⠘⠿⢻⡟⠿⠏⠚⠁⢠⣾⣿⣿⣿⣿⣮⣛⣻⣿⡿⣿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⢿⠉⠉⠁⣿⢿⣿⢽⡇⠀⠀⢀⠘⠇⡀⠀⠀⢠⣿⣿⡿⣿⡇⠛⢻⣿⢿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠁⢹⡿⣷⣼⣷⣆⣼⣬⡀⠀⣧⣹⣤⣼⣿⣴⣿⣿⠇⠀⠈⠈⠀⠀⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⡏⢿⢻⠙⢿⢳⡶⢻⠋⡟⢹⠙⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣼⣆⡘⠀⡇⢸⣀⣧⣼⡾⠻⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣾⣿⣿⠟⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠉⠋⠘⠉⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠶⠶⣶⡿⢛⡛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢨⣿⠤⠄⠀⠀⠀⠀⠀⠀⠀⣷⠙⠋⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠾⠿⣶⠟⠋⠀⠀⠀⠛⠛⠉⠉⠉⠙⣷⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⣯⠀⠀⠀⢰⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣼⡟⠀⠀⠀⢸⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⠀⠘⠿⠶⢶⠶⠶⢶⡖⠀⠀⣰⣾⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡠⣶⠜⣇⣀⣀⣀⠀⠀⠀⢀⣈⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠾⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⢀⡾⠋⠁⣤⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡻⣿⣿⣿⣿⣿⡆⠀⣠⠴⠋⠀⠀⠀⠓⢤⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠑⢻⣿⣿⣿⣿⢿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⣹⣻⣿⣷⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠃⠉⠙⢻⣿⣿⣿⣟⢿⣿⡿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢦⣄⡀⠀⣠⡤⠂⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢺⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠇⠁⠑⠺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⣿⢿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠟⠀⠉⠨⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁