Gratitude Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢾⡿⣿⡛⣷⣶⣶⣄⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡏⡡⢞⣭⣭⡦⢽⣷⠈⢍⣯⢹⣷⡄⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡔⠁⣈⣴⣯⠵⠂⡼⢂⢝⢿⣝⢻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⣸⣿⠿⠟⠀⠀⠟⢍⠈⣿⢻⣿⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⣸⢻⣷⡿⢁⣲⡆⠀⢸⣴⡋⢤⣿⣗⣘⣟⣺⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡠⢤⣐⠿⠷⢝⡤⠤⡚⠁⠈⠛⠓⢠⣟⡕⢐⢸⡟⠁⠀⠙⠯⣽⣿⣿⣿⠛⢿⡇
⠀⠀⠀⠀⠀⠑⠄⢀⣀⡴⠊⢫⡱⡈⠑⢦⠃⠀⢸⡟⠇⣽⡎⠀⠀⠢⢀⡔⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡼⢋⠠⠚⠀⠑⢄⠀⠀⠑⢄⠈⠇⢺⠉⠱⢂⠡⠐⠉⠀⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡸⡑⠁⠀⠀⢂⡀⠙⡇⠀⠀⠈⢢⠀⣀⡠⠤⠜⠀⠀⠀⠀⠈⠘⡿⠏
⠀⠀⠀⠀⠀⡐⡕⡁⠀⠁⠀⠀⠁⠤⡈⡀⠀⠀⠀⠹⡌⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠁⠹⠀⠃⠀⠀⠀⠀⡈⠳⡄⠀⠀⠀⠱⢈⡤⢒⠿⠋⠙⢹⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⡇⠀⠀⢧⠰⠀⠀⠀⠀⠐⠄⢰⡀⠀⠀⢀⣰⠀⠀⠀⠁⢊⡜⠀⠀⠀⠀⠀⠠⣪⠎
⠀⠀⠀⠀⡘⠀⠇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡱⠂⠁⠣⡈⠢⡀⠀⡠⠊⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⣆⢀⡇⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠱⡀⠗⠉⠀⠐⠒⠒⠒⠒⢒⠋⠀⠀⠰⠀⡦
⠀⠀⠀⢰⡀⡼⠀⠀⠀⢡⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢄⠀⠈⡦⠤⠠⠀⠀⠄⠀⠠⠀⠰⠃
⠀⠀⠀⢸⠄⡇⠀⠀⠀⠈⢂⠘⢄⠀⠀⠀⠀⠐⠀⢀⡔⠂⠀⠀⠀⡥⠤⠑⣄⣀⣀⣀⣀⣀⣀⣠⣀⣀⣠
⢠⠤⣤⣿⢀⣇⣀⣀⡠⠄⠀⠳⡀⠈⠐⠀⠂⠉⠙⠳⣦⡀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⣧⠰⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠈⠛⠢⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣤⠀⢠⣤⣄⠀⣀⣀⡀⠀⠀⣴⣤⡄⣤⣤⡀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⡆⠀⢀⣿⣿⠀⣾⣿⣿⠘⣿⣿⣿⡀⠀⣿⣿⠇⣿⣿⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡇⠀⢸⣿⣿⠀⣿⣿⣿⡇⣿⣿⣿⣷⡀⣿⣿⢠⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢸⣿⣿⣤⣼⣿⣿⢰⣿⣿⣿⣷⢸⣿⣿⣿⣷⣿⣿⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠿⢿⣿⣿⢸⣿⡿⣿⣿⡜⣿⣿⢿⣿⣿⣿⢸⣿⣿⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠀⢸⣿⣿⣾⣿⣿⣿⣿⣧⣿⣿⡎⣿⣿⣿⢸⣿⡏⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠀⠘⠛⣛⠛⠛⠉⠙⠛⠛⠛⠛⠁⠈⠉⠉⠘⢛⡃⠀⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣆⠀⠀⢠⣿⣿⢂⣴⣿⣿⣷⣄⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⣾⣿⠇⣾⣿⡟⠉⢿⣿⡆⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⡟⠀⣿⣿⠇⠀⢸⣿⡇⢿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⣿⣿⡀⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⢿⣿⣇⠀⣾⣿⡇⢸⣿⣧⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠘⣿⣿⣾⣿⡿⠁⠘⣿⣿⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠈⠙⠛⠛⠁



⣤⣄⣄⣤⣤⣄⣀⣀⣀⣄⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣠⣠⣀⣤⣠⣤⣄⣠⣤
⣿⣯⣭⣿⣯⣽⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣯⣽⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠛⠿⣶⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣠⣴⠶⠟⠛⠛⠻⢷⣆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣿⠀⠀⡀⠀⠀⢀⣿⠀⣠⣤⡀⣰⡇⠀⠀⣰⣷⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢀⣤⣾⡏⣼⠏⠈⣷⢻⡏⠀⣰⠇⣿⡀⠀⣸⡏⠰
⠀⠀⠀⠀⠀⠂⢀⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⡇⠀⢸⣧⣤⠶⠾⠛⠉⣽⠇⣿⣀⣴⡇⢸⡇⢠⡏⠀⣿⡇⠀⣿⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡇⠀⣸⡿⣦⣄⠀⠀⠀⢿⡄⠈⢩⡟⠀⢸⣧⡞⠀⠀⢹⡇⢠⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣿⡧⠞⢹⡇⠈⠙⢷⣤⡀⠈⠳⠞⠋⠀⠀⠸⣿⠁⠀⠀⢸⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⣠⡾⠁⡴⣿⡇⠀⢸⡇⠀⠀⠀⠉⠻⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣴⠾⠋⠀⢸⢁⣿⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣀⣈⣛⣁⣠⣤⣤⣤⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣴⡶⢾⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⢛⣉⣉⣉⡛⠻⠿⠰⠿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣷⣿⣿⣁
⠀⠀⠀⠀⠀⣘⣿⣿⡿⠟⣋⡛⢿⣿⣿⡁⣿⣿⡿⣿⣿⣿⡆⣰⣿⣿⢛⣿⣶⡆⣶⣾⣟⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⡿⠿⢿⣿⣿⡇
⠀⠀⠀⢠⣾⣻⣿⡟⢡⣾⣿⡿⢸⡟⣍⣳⣬⣩⣴⣿⢿⣿⠀⣿⣿⡿⢻⣿⡿⢠⣿⣿⢻⣿⡿⢹⣿⣿⢋⡉⢰⣿⠏⢠⡾⣸⣿⣿⡀
⠀⠀⠀⠈⢙⣿⣿⢠⣿⣿⣿⡃⢸⡧⠱⢿⡿⢋⡛⣿⣿⡟⢰⣿⣿⡇⣼⣿⠃⣼⣿⡏⢸⡿⠁⣾⣿⢃⣾⠇⣾⡟⢀⣩⣴⣿⣿⣿⠇
⠀⠀⠀⢰⣿⣿⡇⢸⣿⠟⣽⡇⢚⣵⠏⡼⢰⡿⢁⣿⡿⠃⣼⣿⣿⠁⣿⣯⠀⣿⣿⠃⡿⡃⢰⣿⡏⢸⢏⢠⣿⠇⢸⣿⠟⣼⣿⣇
⠀⠀⠀⠈⢻⣿⣿⣬⣥⣾⢟⠁⣿⡟⣸⠇⠟⣵⠸⠟⣡⢀⣿⣿⡟⠸⢏⣿⢸⣿⡟⡀⣱⡇⣼⡟⣤⣩⣾⡘⢋⣾⣦⣥⣾⢿⣿⡿
⠀⠀⠀⠀⢾⣿⣿⣿⣿⢣⡿⢰⣿⣷⣤⣿⣿⣿⣿⣿⣿⢸⣿⡟⣼⣷⣿⣿⣌⣋⣾⣿⣿⣷⣉⣼⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⢀⣼⣿⣿⣟⢾⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣋⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⠿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⡟⠉⢻⡟⠿⢿⠿⠻⢿⠻⠿⠻⠻⠟⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⢐⡿⠟⡒⠀⡀⡄⡐⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢴⠀⣖⣩⡖⣦⠁⠡⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⣨⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠠⠀⡄
⢼⡀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠁⠀⠀⠠⡆⢀⠔⠀⠀⠀⠀⠀⠋⠛⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡖⠠⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠉⠀⣠⡤⠀⢼⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⠛⠿⠄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠐⠦⠀⠀⠈⠉⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠰⡉⠑⢦⣀⡤⠁⠢⠂⠠
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⡄⠋⠀⠹⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣨⣟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢖⣲⡞⠒⣶⠖⢶⣟⢒⡖⣖⢒⡖⣶⢲⠒⢶⠖⣒⠀⠀⠀⣠⢀
⡀⠀⠀⠀⣀⢀⠀⠀⢂⠀⠀⠠⣌⣩⣇⣐⣟⣤⣸⣿⣸⣇⣿⣸⣧⣉⣼⣀⣼⣄⣚⠀⠠⠵⣞⣿⠃
⠀⠀⠀⠀⣿⡿⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠
⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡌⡀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠧⠀⠄⠁
⠠⠀⠀⠀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠁⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠈⣶⣖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⣐⠋⣀⠦⡄⣇⠃⣖⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠠⣈⠀⠈⠹⢡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⠀⠀⠀⠀⠀⣀⠤
⣰⠄⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠿⠿⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠅
⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠕⠀⠀⠀⢈⣺⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣷⣦⡀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠰⡄⠸⠀⣰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠀⠁⠀⠀⠀⠀⠀⠐⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠀⢈⡿⣇⠀⢸⣷⡄⢹⡏⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣼⠧⢿⡄⢸⡇⠹⣾⡇⠀⠘⠋⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠈⣿⡆⠀⠀⣴⡆⠀⣰⣦⠀⠀⠀⠀⢠⣶⠄⠈⠉⠀⠉⠉⠉⠉⠀⠈⠁⠀⣴⡿⠉⢳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⣿⠀⢾⣿⠷⠶⣿⡷⠆⢾⡆⠰⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠴⢾⣿⠒⢶⣿⡧⠀⢸⣿⠀⢸⣿⠇⢀⣾⡆⠀⣿⡇⠀⣰⣿⠀⠀⣾⡆⠀⣴⡿⠶⣾⣿⠋⢁⣶⡾⢲⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢸⣿⡇⠀⣿⣟⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣰⣿⡇⢸⣿⡇⠀⣿⣿⠀⢸⣿⣀⡼⠀⡼
⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⢿⣧⡴⢻⣿⣀⡾⣿⣀⣸⢿⣇⡴⢿⣇⣠⠿⠿⠶⠋⣿⣇⣸⢿⣧⡼⢻⣿⠀⡼⢿⣥⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠈⠉⠃⠀⠀⠀⣠⠖⠲⣦⠐⣶⠒⠶⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⡇⠀⠀⠈⠉⠉⣉⠀⠀⠀⣿⠀⠀⣿⠀⣿⠚⠂⠀⠀⠀⢈⠉⠉⠁⠀⢰⣷⢀⡀
⠀⠀⠀⢀⣤⣴⣶⣤⡀⠀⠈⠉⠁⠀⠀⠀⠐⠋⡡⠂⠀⠀⠈⠓⠚⠋⠐⠛⠂⠀⠀⠀⠀⢦⡉⠃⠀⠀⠸⣿⠟⠃⠀⣀⣀⣀
⠀⠀⣰⣿⡟⠉⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⢀⡀⠀⢠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⢸⠄
⠀⢀⣿⣿⠀⠐⠚⢡⣤⣴⣷⣄⣀⡀⠀⠀⣀⣀⠀⣀⠀⠘⣿⡟⠃⢺⠟⠀⢻⣿⠛⠀⣀⠀⠀⢀⡀⠀⠀⢀⠀⢸⣿⣃⡠⠊
⠀⠸⣿⣿⠀⠀⠀⢸⣿⠈⢹⠋⣿⡇⠀⣼⡿⠉⢹⣿⡇⢀⣿⡇⠀⣿⡇⠀⣿⡟⠀⢸⣿⠁⠀⣿⡇⠀⣾⡿⠛⢻⣿⠁⢀⣾⡟⢹⡆⢠⡄
⠀⠀⢿⣿⣄⠀⢀⣾⣿⠀⣸⠀⣿⡇⠀⣿⡇⠀⣾⣿⡄⢘⣿⡃⠀⣿⡇⠀⣿⡇⠀⢺⣿⠀⣰⣿⡇⠀⣿⡇⠀⣾⣿⠀⢸⣿⣤⠞⠁⡼
⠀⠀⠀⠙⠛⠛⣋⣽⣿⠔⠁⠀⠘⠿⠶⠛⠻⠛⠀⠿⠧⠞⢿⣧⡴⠻⠷⠴⠻⣷⣤⠟⠛⠛⠁⢻⣷⡤⠿⠷⠞⠹⣿⣄⡼⠻⠷⠤⠞⠁
⠀⠀⠀⠀⣰⠋⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢧⣀⣼⡿⠁⠀⠀⢠⣤⡀⣷⢠⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣴⠆⣀⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⣭⡿⠢⣿⡦⢽⣧⣹⣷⡄⠄⠀⠀⠀⠀⢀⣀⣴⣟⣰⣟⣠⣿⠧⢿⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠱⠟⠱⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡎⠿⡆⠻⠷⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡟⠀⣀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⢸⣿⠀⠀⡀⠀⠀⢼⡟⠀⢀⡀⠀⠀⣿⠆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡆⠀⢰⡿⠀⢠⡄⠀⢰⣆⠀⠀⣿⠁⠀⢸⣿⢀⣠⡟⠀⠀⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⢠⣿⠀⠀⢸⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⢼⣿⢸⡏⠀⠀⠀⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠘⢷⡠⠀⠈⠁⠸⣿⠀⠀⠀⠙⠀⠸⣇⠀⠀⠉⠀⢸⡇⠀⠀⠀⠉⠈⢿⡀⠀⠀⢿⡇⠀⠀⠙⠀⠈⢿⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃




⣶⣶⡾⠶⠾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⡀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠈⣿⠙⣛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣄⠀⣠⡶⠀⠀⢿⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⣤⣄⠀⣇⢀⡾⠃⣼⠋⠙⠃⠀⠀⣿
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⡇⢸⣇⣴⠞⢶⡀⢠⡟⠛⠻⣷⢸⡿⠁⠀⢻⡀⣿⡿⣧⠀⠘⠷⠶⢦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠃⠀⢸⡇⣿⠀⠀⠀⣿⠀⣷⠀⠀⢸⡇⣿⠀⠘⣧⠸⣇⣀⣼⠇⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⠀⠀⠀⣧⠸⣦⣀⣠⣿⡆⢿⠀⠀⠸⠇⠻⠂⠀⠙⠁⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠉⠀⠀⠉⣁⣈⣁⣤⣤⡤⠶⠶⠶⠒⣒⣛⣛⣛⡛⣛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣤⡴⠶⠚⠛⠋⠉⣉⡁⠀⠀⠀⠀⣠⡴⠛⣉⠉⠉⠉⠙⢦⣛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠚⠋⠁⠀⠀⢀⣤⣾⣗⠾⠋⡇⠀⢠⡏⣻⠋⠀⠀⣉⡀⠋⠀⠀⠈⣧⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠀⠀⠙⠦⣄⠉⡏⠠⡄⠀⠉⠁⠀⠀⠀⠀⣿⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠚⠛⠲⢤⣀⡀⠀⢳⡄⠻⢦⣤⣠⡴⠋⠀⢠⣿⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⠀⣶⣠⡇⢀⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢰⣆⢸⢰⡆⢸⡇⢸⡇⡰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡏⣿⡈⢿⡇⣸⡇⠘⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘⠛⠀⠉⠀⠳⠊⠉⠀⠿⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠂⢀⡔⠒⠒⠤⢤⣷⠀⣠⡄⣿⠀⠀⠀⠰⢷⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢎⠻⠶⣄⠋⣤⢦⡄⢰⡏⢉⡭⢾⣿⠀⢀⠀⢀⠀⢀⣤⣿⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⣸⣦⢡⡟⢸⡏⣼⡇⢸⡇⢸⡇⢸⡇⠀⣿⢀⣿⠀⣿⢀⣿⠀⣿⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠟⣷⠸⣧⠎⠛⠘⠧⠋⠓⠚⠧⠞⠷⠖⠙⠋⠻⠴⠛⠊⠻⡴⠛⠒⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⣁⣰⠏⠀⠀⠀⠀⢀⣤⠀⢀⡶⠂⠀⠀⠀⣠⠦⠀⣀⡀⢀⡠⠤⠄⡀⠀⢰⡆⠀⢠⣦
⠀⠀⠀⠀⠀⣼⡇⠀⡀⠀⡀⣠⣄⠀⠀⠀⣀⡀⠈⣓⠀⣾⠃⠁⣀⠀⢰⡇⠀⠰⣧⡀⠘⠗⣀⡀⡀⠀⣿⠃⠀⣐⠁
⠀⠀⠀⠀⠀⣿⡇⢰⣿⠀⣷⣏⡨⠀⠀⣼⠁⢇⣾⠩⢀⡿⠀⣾⠁⡁⢸⡗⢉⣷⠘⡷⢀⣾⠃⣽⡟⢠⡿⢀⣾⣉⠇
⠀⠀⠀⠀⠀⠹⣧⠆⢻⣇⡟⢿⡤⠂⠀⢿⣄⡨⢿⣠⠎⢷⣠⢿⣅⡠⢻⣷⣫⡏⠘⣧⡸⢿⠞⢹⡇⡸⢧⡸⣏⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⢿⠃
⠀⠀⠀⠀⠀⣄⣠⣾⡿⠟⢠⣶⡄⠀⠸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣿⠃⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣼⣧⠋
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⠃⢸⣿⣀⠴⠛⠒⣿⠇⠀⣰⡞⢀⡶⠀⣼⡇⠀⢠⡾⡋⠗⣸⡇⣠⢾⡇⢠⡾⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣠⠏⠀⣸⣿⠁⠀⠀⠀⣿⠀⣰⣿⡴⠻⣇⡼⠹⣷⡀⡼⣧⣀⠜⣿⣟⠁⢻⡧⠛⠧⠾⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⣿⠀⠀⠀⠀⢿⡴⠃⠀⠀⠀⠉⠀⠀⠈⠛⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠹⣦⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠏
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⣈⣭⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣾⢣⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⢀⢴⣦⠀⢀⣤⢤⣀⡀⢀⡀⡔⢲⠀⣿⡜⣸⢃⣿⠀⣠⠀⠀⣤⠀⢸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣰⣿⠃⢸⡿⠀⣾⠇⣠⣿⠁⣼⡷⠀⣿⢸⣿⡶⠃⢸⡇⣠⣿⠀⣰⡏⢀⣿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠁⠛⠀⢸⡇⠀⡿⠖⠁⠻⠞⠹⠃⠀⡏⢸⡏⠱⡀⣼⡟⢆⠙⠚⠈⠓⠋⠘⢷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⢺⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⠜⠀⠀⠀⢰⣏⡴⠁⠀⠀⠀⢰⡏⠈⢁⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⣠⢴⣄⣀⣿⠏⠀⣰⠟⢛⡇⡼⣷⡀⣿⠩⠀⣴⡟⡹⠀⣴⡟⠉⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⣿⡜⡡⢺⡏⠸⣿⠀⠀⣿⡉⠉⡴⠁⠘⣧⠹⣦⡀⣿⡁⣠⠼⣿⣅⠜⠘⣿⡀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⢁⡞⠀⠀⠻⣦⠴⠉⠛⠋⠀⡞⠀⢹⡄⠈⢿⡌⠉⠁⠀⠉⠁⠀⠀⠙⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡾⠣⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡄⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠺⠳⢾⡆⠰⠛⡠
⠀⠀⠀⠀⠲⡀⠘⡀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⣴⣀⢀⡀⠀⠰⠈
⠀⢰⠀⠀⡀⠑⠄⠀⠄⣀⠀⠀⠀⢀⠄⢀⠔⠀⠀⠈⠑⠒⠀⡀⠉⠉⠉⠆⠀⡀
⠀⠀⠑⢄⠈⠢⣈⡄⠘⠋⠀⢰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠶⠈⠀⢠⠄⠉⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢦⡀⠂⡀
⠀⠀⠘⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢙⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠉⠀⠀⠊⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢂⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠶⠀⠉⠉⠉⠛⠻⠶⣿⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢠⠀⠀⢀⡠⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠋⠀⣼⣷⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⠘⢷⠒⠂⠐⠒⠒⠊⠉⠁⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢻⠀⠀⠀⢧⣤⣄⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⡷⠀⣀⡀⠀⠀⣀⡀⠐⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⡀⠠⠿⢻⡟⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠓⠴⠂⣀⣀⣀⠴⠋⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠐⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠷⣤⣤⣀⣀⣠⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣤⣤⣠⣤⣤⠴⡾⠋⣀⠀⠷⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡅⠀⠀⡇⠀⠈⠁⠶⠂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⠀⠘⠀⢆⣤⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⢛⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠛⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣄⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣤⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⠶⠞⠁⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⠞⠉⠀⠻⣆⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡴⢾⠁⢠⣼⣁⣀⣀⣀⣀⠀⡴⢶⡞⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢇⠀⠁⠀⠀⠀⠀⠀⠀⠄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠡⣀⠀⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢦⣤⣤⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⠁⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⢀⠀⠀⡁⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠙⢻⣿⣿⣧⡤⢈⣀⢀⠀⣀⡴⠠⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣄⠁⠀⣶⡎⡉⣿⣯⠙⣾⣾⣷⡾⠿⣷⣾⣧⡆⠀⢤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣿⣿⣿⣷⣶⠾⣧⢴⠭⣍⣤⡟⢯⠙⣿⣿⣿⠌⣿⣅⣨⢎⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣫⠛⢿⡟⣎⣷⠈⢆⡹⣅⢀⠱⠊⠉⣈⣝⣉⡴⠿⠐⣾⢸⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠄⠙⠲⢬⡆⢰⠀⠜⢿⣧⣠⡙⢾⡇⣄⣈⡀⣋⢍⢸⣿⣶⣅⣋⣽⣔⠧⠊⠙⣼⠀⠦⣄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠳⢴⠄⠙⠋⣙⠂⠂⠠⣌⣉⣳⠶⣟⣫⣭⣝⣿⠯⣼⣿⣿⣿⠿⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠄⠠⠄⢸⠋⠰⣦⣿⣿⣿⣷⡻⠿⢻⡏⣴⣿⣿⣿⣷⣏⣠⢁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠀⣄⢀⣀
⠀⠀⠀⠀⠰⠀⠀⣌⠀⠠⢿⣿⣿⣿⡿⠳⠖⢸⡃⢿⣿⣿⣿⣿⡿⢽⣄⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠟⠁⠀⠀⣿⠈⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⡀⠈⠀⠛⣔⣂⣉⢻⣜⠙⠾⣉⣤⣂⣰⣿⣿⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⠏⠀⠀⣠⠖⠀⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠃⠀⠙⠓⠚⢿⣉⠁⡨⠉⡙⠲⠛⢩⣯⣄⠈⠙⠻⢿⡆⠀⠀⠀⠀⣶⡴⠞⠛⠛⠉⡄⠀⠾⠏⠀⠀⠀⣠⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⡉⠉⠁⠚⢿⣧⣤⠞⠃⠀⠀⠀⠉⠀⠀⠀⠀⢄⠻⣤⡀⣀⣰⣟⢁⣙⣉⠀⠀⠁⠘⠋⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⣶⡎⠀⢀⡄⠀⠀⠀⢲⡄⠀⠈⠻⠙⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠉⠁⢀⣤⡄⠐⠐⠀⢀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠘⠦⣰⣿⣇⡀⠀⠀⠀⣷⡄⠀⠀⠀⣇⣼⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠀⢀⡀⣾⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠂⠀⠘⠋⠀⠀⠀⠀⠹⢿⣿⡲⡆⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣾⠁⠋⠀⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣠⠄⠚⠂⠀⢺⡆⠐⠃⠀⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⠀⠀⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠙⠊⠙⠃⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡴⠛⠉⢻⠓⡶⠤⠄⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠐⠇⠙⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠱⠃⠀⠀⠀⠀⠳⠤⢀⡏⣸⠟⠀⠀⠀⠀⠀⠀⢀⣿⠃⣰⡿⠁⠀⢀⣾⠟⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡂⠀⠀⠀⠀⠀⠀⣸⢡⡿⡶⣠⣾⡖⣰⣷⡗⣼⡟⢡⣟⣴⣧⡖⡼⠃⠀⠀⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢘⣘⣛⣉⣉⣛⣉⣘⣋⣉⠝⢻⡇⢈⢉⣉⣙⡁⠀⠀⠀⠀⠀⠀⢀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣤⣄⠀⣠⡉⢭⣭⠅⣤⠄⣨⡥⢠⠄⣠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⣇⣸⠇⣿⣧⢠⡇⠀⣇⠀⣿⡀⢸⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⡇⣿⣇⢰⡏⣿⠈⡇⠀⡟⠁⣿⠁⢸⡀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠙⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣋⠘⠚⠃⡘⠀⠃⠀⠛⠒⠙⠀⠈⠛⢃⣘⠓⠀⠀⠀⠀⠀⠀⠀⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡸⠀⢠⠎⡹⢀⠤⠤⠤⣄⡀⠀⠀⣠⠋⡰⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡀⠀⠀⠀⠀⠀⠀⡼⠛⠳⠀⣏⠔⡁⠈⣀⡀⣀⢀⡉⠉⣛⠟⠉⠀⠀⠀⠀⠀⠀⢀⠀⡳⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠁⠀⠀⠀⠀⠀⢠⠗⠒⡲⢻⠁⣰⣏⢜⣧⢴⣧⠾⣣⠾⢿⠀⣠⠀⠀⠀⠀⠀⠀⠈⢁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠘⠒⠚⠁⠈⠛⠁⠀⠈⠁⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠀⠞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢦⠠⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠢⣞⠀⠀⠀⠀⢀⡀⠀⠶⠀⠀⠀⡠⠻⠆⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠛⠀⢉⠉⠒⠂⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣤⡄⣠⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣿⣶⣿⣿⡿⢛⢿⣿⡿⢾⣿⣇⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⡿⣫⣿⣿⡐⣿⣇⣿⠰⣷⣿⣿⣿⣧⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣝⣻⢿⣷⣤⣬⣽⣥⣬⣙⣻⡿⣿⣿⣥⣦
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⣫⣿⣯⢻⣿⣿⣿⣿⠿⣿⣟⢹⡿⣺⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢸⣿⢱⡿⢟⣡⣿⣿⣿⣿⠇⣿⣼⡟⡿⣿⣿⡏⣭⣽⣤⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣀⣤⡀
⠀⠀⠀⠀⠀⢠⣼⣿⣿⠃⣭⡅⢶⢿⣿⣿⣿⡻⡿⣿⠀⡷⣿⠘⣿⣿⣿⠁⣻⣾⣿⣤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⣻⣿⣿⠀⣿⡇⣾⢠⢋⣾⢿⡇⡟⢸⠀⢿⣟⠰⡇⣿⢸⡇⣿⣿⣿⣷⣤⣤⣤⡀⠈⣿⣿⣿⣿⣿⣿⡿⠛⣾⠅
⠀⠀⠀⠀⠀⠸⢿⣿⣿⠀⣿⡇⣿⣼⡘⠟⠸⣡⣧⣠⣤⣷⣧⢠⣼⣶⡦⣦⡘⠿⣼⣿⣿⣿⣿⠿⣧⡘⣿⣿⣿⣿⡟⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⣿⣷⣿⣿⣿⣿⣿⣿⢿⣿⢡⣮⣿⡀⢿⠿⠃⣿⣿⣿⠏⣾⣿⢻⣟⢸⣾⣷⣼⣿⣿⣿⣇⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣈⣿⣿⣿⣿⣿⡏⣰⣿⣿⡜⣻⣿⠟⣙⣿⢰⣿⠿⣿⢠⣿⣿⣮⣿⠀⣾⣿⢫⡝⣿⣿⣿⡀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣿⣿⣿⣿⣿⣿⣿⣆⠻⠿⢸⣿⠏⣬⢸⡟⣿⢸⡇⢾⣮⢠⣽⡇⢿⠈⢧⠹⣿⣾⣧⣿⣿⣿⣷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠁⢀⣽⣿⢟⣏⢰⣿⢈⣫⣶⣶⣦⣘⣵⣬⣽⠀⣿⣿⠶⣛⢾⣷⣶⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⡿⣵⣿⠇⣾⢱⡇⣿⡿⢡⣽⣿⠿⣿⠿⣇⢛⣿⡸⣿⡇⣿⣿⢿⣟⣿⣛⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣾⡇⠸⣣⣿⠇⣺⣾⣯⠈⣇⢈⣿⢛⡝⣿⢾⠂⣾⣿⣿⣿⣿⡿⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⡿⣟⣛⣛⣿⣿⣭⡅⢘⣭⠙⠀⡟⢵⣽⣆⠸⣆⢣⠸⣾⠃⣾⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣏⢾⣿⣿⡟⣿⣿⣿⣇⣾⣛⣰⠀⣇⣛⡯⣿⠀⣿⢸⣿⣾⣷⣶⣦⡹⣻⣿⢿⣛⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣩⣭⣿⣿⣿⣮⣭⣭⣷⣿⣿⣿⠿⠿⢿⣿⣶⣿⣟⣷⣉⣬⣵⣿⣿⡿⣫⣭⣭⢿⣿⡧⣿⠿⣿⡞⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣵⣾⣿⣿⣶⣦⣬⣍⣙⠛⠻⠿⠿⠟⠫⠾⠿⠟⣫⣾⣿⣿⣶⣾⡿⣱⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⣿⣿⣿⣿⣿⣿⣷⣛⣛⣿⣿⣿⣿⣿⣻⣿⣟⡛⠓⠠⠶⣶⣶⣾⣿⣿⣿⣿⡿⠟⣋⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣎⣛⣛⣉⣤⣾⣿⣷⣶⣦⣤⣭⣭⣥⣤⣶⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⠿⠿⠿⠿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⢹⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⣿⣧⡀⠀⠀⣠⡴⠿⢿⡿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠻⡆⠀⠉⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⣄⠀⣴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠁⣼⠀⡇⡌⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠼⣅⠀⡇⠀⡶⠂⢿⡀⡸⡇⢀⣴⠂⢰⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢈⡇⠣⠀⠙⠒⡠⢯⢠⢾⠁⠙⠒⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠁⠀⠀⠀⠘⠤⠎⠧⠜




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣙⣭⣍⣹⣽⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢀⠒⢄⠀⠂⣤⠔⠢⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡴⠋⡤⠯⠀⠀⣻⡄⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠋⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠷
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⡰⠀⠀⢀⡔⢀⠀⠰⠁⠔⠀⢠⠇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢇⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠐⠀⠈⠀⠃⠀⡆⠐⠀⢠⠇⠀⠘⢰⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈
⠀⠀⠐⢦⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠐⣄⣣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⡉⠉⠀⠠⠠
⠀⠀⠈⠁⣴⣦⡄⠀⢀⡤⢀⣠
⠀⠀⠀⡀⠈⠉⢁⡔⠻⠇⡉⠣⡄⣁⠉⠁
⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣾⠀⠀⣼⠃⠀⠀⠀⠀⢸⣏⡏⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠠⣿⠀⢰⡇⢀⠀⠀⠀⣀⠈⠓⠓⠋⠀⢰⠄⠀⣿⠀⢀⠀⠀⣿⠀⢀⠀⢀⣦⠀⢸⡇
⠀⠀⠀⠀⠀⡟⠀⢸⡇⢸⡇⠀⣼⠁⣰⡀⢰⣆⠀⣾⠀⠀⣿⣤⡾⠀⢰⡏⠀⣾⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⣧⠀⠘⠇⢸⡇⠀⠻⠄⢻⡀⠘⠛⠀⡿⠀⠈⠿⢹⡄⠀⠸⡇⠀⠹⠄⠸⡇⠀⢹⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠃




⣿⡿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⠀⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣇⠀⢸⣿⡇⢸⣿⡇⠀⢀⣾⡿⠁⢸⣿⣿⣿⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⢀⣿⡿⣿⣇⠀⠀⢸⣿⣿⡀⢸⣿⡇⢸⣿⡇⢀⣾⡿⠁⠀⢸⣿⡇⠀⠀⣺⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⢸⣿⡇⣿⣿⠀⠀⢸⣿⣿⡇⢸⣿⡇⢸⣿⣧⣾⡿⠁⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣄⣿⣿⠀⠀⢸⣿⠃⢿⣿⠀⠀⢸⣿⣿⣿⢸⣿⡇⢸⣿⣿⡿⠁⠀⠀⠀⢸⣿⣧⣀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠿⣿⣿⠀⠀⣼⣿⠀⢸⣿⡄⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⡅⠀⠀⠀⠀⢸⣿⡿⠟⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⣿⣿⣿⡇⢸⣿⣿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣷⠀⢸⣿⡇⢹⣿⣿⡇⢸⣿⡏⢻⣿⡄⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿⠀⢀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⣿⣿⣿⡇⢸⣿⡇⠀⢻⣿⡄⠀⢸⣿⡇⠀⠀⢻⣿⣆⣠⣿⡿⠀⣿⣿⣴⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠿⠿⠀⠿⠿⠀⠸⠿⠃⠀⠀⠿⠿⣤⠼⠿⢷⣿⣿⠿⠇⠸⠿⠇⠀⠀⠻⠿⠀⠸⣿⣷⠄⠀⠀⠛⠿⠿⠛⠁⠀⠿⢿⣿⡿⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⡟⠛⠛⠻⠂⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢠⡞
⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣰⣿⠇⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣼⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠏
⢠⣾⡿⠉⠉⢹⣿⣷⣄⠀⢻⣿⠶⣿⣿⡟⠀⠀⣠⣾⡿⠉⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠋⢻⡄⠀⣿⣿⠁⠀⢠⣿⣿⠀⢰⣿⣷⠀⠀⣿⣿⡇
⣿⣿⠇⠀⠀⣾⣿⣿⠀⠀⣾⠀⢀⣿⣿⠁⠀⣰⣿⡟⠀⢰⣿⣿⡇⠀⢀⣿⣿⠀⠀⠀⠀⣼⣿⡷⠶⠾⠃⣸⣿⡿⠀⢀⣼⣿⡇⠀⣾⣿⣿⠀⠀⢿⣿⡇⠀⠀⢀
⠹⣿⣆⣀⣼⢿⣿⡿⠀⣸⠇⠀⢸⣿⣿⠀⣴⣿⣿⣇⣠⠟⢻⣿⣇⣠⡾⢿⣿⡄⠀⢀⣴⢿⣿⡇⠀⣀⡼⣿⣿⡟⠛⠋⢿⣿⣇⣼⠛⣿⣿⠀⣠⡿⣿⣧⠀⢠⡾
⠀⠈⠙⠋⠁⣸⣿⣧⠾⠃⠀⠀⠀⠻⠿⠛⠁⠈⠛⠛⠁⠀⠈⠻⠿⠋⠀⠈⠻⠿⠿⠋⠁⠈⠻⠿⠞⠋⠀⣿⣿⣿⡄⠀⠘⠿⠟⠁⠀⠹⠿⠿⠋⠀⠙⠻⠿⠋
⠀⠀⣀⡴⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢿⡀
⠀⣴⠋⠀⢀⣿⣿⣤⣤⣤⣤⣀⠀⠀⠀⣤⣤⠀⠀⠀⣠⣤⣤⣄⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⢀⣀⣤⣄⡀⣿⣿⣤⣼⣧⣤⠀⢠⣤⣤⣤⣤⣀
⢸⡇⠀⠀⣸⣿⠃⣿⣿⠿⠿⣿⣷⠀⠀⣿⣿⠆⠀⠀⣿⣿⡿⠿⠀⢠⣿⣿⠿⢿⣿⡆⠀⢠⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠀⢸⣿⡿⠿⢿⣿⣷
⠘⢧⣀⣰⡿⠃⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⡄⠀⠀⣿⣿⡇⠀⠀⣾⣿⡇⠀⠸⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⠻⣿⣿⡟⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠉⠁⠀⠀⠀⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⢹⣿⣧⠀⠰⠿⠿⠀⢸⣿⣧⠀⠸⠿⠿⠀⣼⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡿⠀⠀⣿⣿⡇⠀⠀⣿⣿⣇⣀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⣿⣿⣇⣀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣦⠀⠀⣿⣿⡇⠀⠀⣿⣿⡿⠿⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⣿⣿⡿⠿⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⣴⣶⡆⠀⢸⣿⣧⠀⢴⣶⡆⠀⢸⣿⣧⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⢿⣿⡇⠀⢸⣿⣿⠀⢸⣿⡇⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⠃⠀⣿⣿⣷⣦⠀⣿⣿⣷⣦⠀⠘⣿⣿⣶⣾⣿⠇⠀⠘⣿⣿⣶⣾⣿⠇⠀⣿⣿⣷⣶⠀⢸⣿⣷⣶⣾⣿⡿
⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣤⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣥⣤⣤⣤⣬⣿⣿⣿⣥⣤⣤⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠿⢷⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⠀⠀⠀⠘⡆⠀⠀⢀⡠⠄⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠓⡄
⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⠋⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⢰
⠀⠀⢀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠇⠀⠀⣠⣶⣦⡀⠀⢸⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⡇
⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠤⠤⠒⠋⠀⠀⠀⠀⠘⢄⣾⣿⣿⠟⠃⠀⡸⠀⠀⠀⢸⣿⡿⠀⠀⠀⢰⠁
⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣾⣿⡿⢅⡀⠀⢀⠃⠀⠀⠀⣿⣿⡇⠀⠀⢠⠃
⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⣿⡀⢸⣿⠏⠀⠀⠈⢳⠮⣀⠀⠀⢀⣿⣿⠁⠀⡰⠃
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢿⣿⡇⢸⡏⠀⠀⢀⠔⠁⠀⣀⣱⣄⢸⣿⣿⢀⠜⠁
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠔⢁⠀⠀⣸⣿⡇⠈⠓⠔⠚⠁⠀⣴⣿⡿⠉⠹⢸⣿⣿⠁
⠀⠀⠀⠀⠉⠒⠤⠤⠐⢊⡥⠀⢿⣄⠀⣿⣿⠃⠀⠀⠀⠀⠀⠰⣿⣿⠁⠀⣀⠼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣖⡌⠛⠷⠀⣉⢠⣿⣿⠀⠀⠀⠀⠀⡀⠀⡙⠻⠖⠊⠁⠀⣿⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠠⣦⡈⡛⠓⡡⠖⠉⠀⣼⣿⠇⠀⠀⠀⣶⢪⢿⣤⢿⡀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢸
⠀⠀⠀⢀⣔⡦⢌⢛⡵⠋⠀⠀⠀⢰⣿⡟⠀⣦⢘⣷⡝⠋⣀⠤⠄⣤⠀⠀⠀⠀⠀⠹⣿⡄⠀⡸
⠀⠀⠀⠀⠙⠛⢃⠎⠀⠀⠀⠀⣰⣿⠟⠀⠀⠈⠉⠼⠁⠈⡠⠒⠉⠀⠀⢀⣀⠀⠀⠀⠙⠻⠖⠁
⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠮⠤⠄⠒⠊⠉⠁


⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠧
⠀⠀⠀⠀⠀⠀⣠⣄⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⣀⣴⣾⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣾⡿
⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣶⣄⠀⠀⠀⠀⢀⣿⡏
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣾⡿⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⢿⣿⣧⠀⠀⢀⣾⣿⠃
⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⠃
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣶⠞⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣾⣿⠃
⠀⠀⠀⠀⠒⠛⠩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣻⣿⢿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⣧⣯⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠛⠛⠿⠿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣿⣿⠇⠀⠀⠀⠀⠀⢀⣩⣽⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⢀⣴⣾⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢁⡀⢀⣴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⢀⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢯⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣥⡀⠀⢠⠀⠀⠀⡠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡟⣃⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⡄⠀⣿⠛⡿⣦⡀⠠⠚⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣷⡀⠀⢀⠀⠈⣩⣾⣻⢌⡵⠶⡆⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣐⣴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠢⣐⠳⣶⣟⢁⣱⣿⣳⡐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⠁⠈⠹⣦⣤⡤⠶⠶⠶⠶⠶⠒⠖⠀⠐⢴⣄⠀⠉⢩⠙⣤⣼⠇⢣⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠘⢧⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣵⡄⠑⠄⡇⢘⣻⡗⠷⢄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠃⠀⠀⠀⣸⡿⠀⠀⢀⢿⢀⠉⠉⡱⠞⠻⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⣦⣤⡀⠈⡿⠶⠒⠙⢦⣀⠀⢰⣿⣟⡀⢀⠂⠀⢸⡦⡀⠈⠢⡀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠒⠚⠂⠀⠀⠐⠀⢾⣿⠇⢳⡀⠇⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⡟⢰⠂⠀⢷⠀⠀⠀⠈⠑⠆⠀⠈⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢳⢠⠀⠀⠀⠀⠀⠀⠙⡆⣻⣿⣶⣿⠀⠀⢠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠈⣀⠄
⣠⢤⠶⠀⢐⡂⠀⠐⠒⠓⠲⠎⠀⠀⠈⠘⠃⣠⠖⠐⠛⠃⣴⠦⡤⠿⠔⠠⢖⠒⠲⠶⣴⣷⣿⣿⣿⣿⡤⣀⡀⣄⣀⠀⠀⠀⣠⣤⣷⣤⣀⣈⠂
⠀⠀⠀⠀⠀⠉⠠⠐⠒⠈⠀⠀⠈⠈⠁⠀⠀⠣⠐⠰⠄⠀⢀⠉⠀⢀⠀⣀⠀⠀⠀⠀⠤⠤⡀⠀⠘⠁⠀⠈⠛⠁⠀⢀⠢⡤⣁⠉⠉⠀⠀⢉⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠄⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢺⢸⡆⣧⢸⡶⣿⡁⠀⢀⣠⡴⠂⠀⠀⠀⠀⠀⠔⠉⠀⢀⣀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢾⡏⠛⠙⠟⠁⠈⠛⠛⠋⠁⠀⠀⠀⢰⡟⠆⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢠⣿⡇⢀⣷⡤⡏⠀⠀⢠⣤⡀⢀⣤⡄⢺⣧⡁⢰⡷⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠶⠖⠛⠛⠉⠁⠀⣟⣼⠁⢩⡟⠀⣷⣠⡀⣾⠘⡇⢸⡇⢤⢸⣷⣇⡠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠉⠁⠀⠘⣇⡠⢻⡈⢧⠝⠗⠙⠋⠛⠈⠁⠁⠀⠸⠿⠁⠀⡤⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⠀⠀⠈⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠀⠀⠀⠳⠴⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡙⢦⣠⠤⠖⠒⠒⢒⡛⠓⠒⠲⣄⡀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⡠⠀⠀⠐⢤⡀⠀⠀⠈⠳⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠹⣗⡨⡥⣄⠀⠀⢠⠞⠁⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⢷⠀⠀⠀⢳⡀⠀⠀⠀⠐⣶⠀⠀⠀⣄⠀⡀
⠀⠀⠄⡜⠘⡄⠈⠎⡇⡰⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣠⠔⢊⠉⠉⢍⠳⣦
⠀⠇⠀⡇⠀⠇⢀⠼⠁⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⢀⡾⠁⢰⢁⡇⠀⣷⠀⠀⠡⠘⣧
⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠞⠁⠀⢸⡘⣧⠀⣿⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣠⠞⠃⠀⠀⠀⠀⢳⣜⢸⠟⣤⣤⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⡤⠔⠋⠉⠓⠲⠦⠤⠴⠞⠷⠶⠚⠋⠁⠀⠀⣠⠒⠉⠉⠁⠉⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣵⣶⣌⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣟⣻⣿⣿⣿⣿⣿⣷⡀⢙⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣿⣽⣿⣿⡗⡸⡿⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡯⠉⠀⠀⠀⠀⠀⠉⠻⣿⣿⠇⢹⢳⣇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣠⣤⣤⡀⠘⣿⣯⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣟⡂⠀⠀⠉⠙⠛⠷⠆⠀⢹⣿⣿⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣣⣄⠠
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣯⠳
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⢤⠀⠀⠀⢰⣴⣿⡿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠠⠹⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡠⠐⢹⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢫⡑⠒⠒⠈⠁⠀⠀⢸⣿⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢮⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠨⠂⠠⠀⠬⣔⡀⡀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠍⠳⣦⡄
⠀⠀⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠉⠀⠀⠘⡆⠀⠀⠀⢾⣿⠿⣿⡀⠀⠀⠹⣿⡀
⠀⠀⠀⢄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢸⡟⣀⡘⣿⠠⠴⠂⠙⠣
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠀⠟⠋⠋⠀⠈
⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠘⠀⠀⠀⠰⠶⠄
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣸⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡏⠀⠀⠀⠀⠀⢰⡇⠀⣠⠀⠀⠀⡟⢠⡞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣼⠁⢀⡟⣠⡄⢀⡷⠋⠀⠀⣠⠀⡼⢿⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⡟⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⢰⡏⠀⢸⡿⠃⠙⠛⣇⠀⠀⣴⣟⣰⠃⣸⡤⠀⠀⠀⠀⠀⢻⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣶⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟⠀⠀⠘⠃⠀⠀⠀⠹⢤⠼⠃⣈⡵⠋⣽⠀⠀⠀⠀⣶⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡿⠀⠀⠀⠀⢿⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠂⠀⣿⡄⠀⢻⣄⣀⡴⠋⠀⠀⠀⡗⢲⡄⢆⡰⠀⠀⣰⠏⠀⠀⠀⣸⠇⠀⢠⣿⠀⠘⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⣸⠃⠀⢰⡏⠀⠀⠀⢠⠟⠀⠀⢸⣏⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠶⢶⡄⠀⢸⣿⡏⠙⢻⣦⠀⣠⣤⣤⣤⣍⡉⠉⠀⢀⣤⣴⠦⣤⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠸⠃⠀⢸⣿⡇⠀⢸⣿⢸⣿⣿⠁⠀⠈⣿⡆⠀⣾⣿⠁⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⣾⣿⣀⣠⣿⠇⢸⣿⣿⠀⠀⠀⣿⣷⢀⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠠⠴⣶⠆⣿⣿⣿⡏⠁⠀⢸⣿⣿⠀⠀⠀⣿⣧⢸⣿⣿⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⠀⣿⣿⢿⣷⡀⠀⢸⣿⣿⣤⣤⢤⣿⣿⢸⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣾⣿⠀⣿⡏⠈⢿⣧⠀⢸⣿⡇⠀⠀⢀⣿⡟⢸⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⣿⣿⢸⣿⡇⠀⠈⢿⣷⣸⣿⣧⠀⠀⢸⣿⡇⢸⣿⡇⠀⢀⡇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣿⡟⠘⠛⠃⠀⠀⠈⠉⠁⠛⠁⠀⠀⠈⠛⠁⠈⠛⠛⠛⠉⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⣀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡠⣶⡚⠉⠰⢾⣏⡁⠠⣾⣉⠠⣾⡁⠾⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠙⠓⠀⠀⠈⠙⠀⠀⠉⠁⠈⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢠⡀⠀⠀⢀⡀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⠄⣀⡧⢐⠠⠁⠀⠊⠀⠁⠀⠀⠈⠀⠑⠂⠈⠄⡂⢼⣀⠠⠠⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⢚⣡⠖⡀⠂⠀⠀⠀⣠⡴⣤⡀⠀⠀⠀⠀⠀⠐⢀⠲⣌⡓⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣫⠚⠛⠉⠀⠀⣤⡄⣤⡀⣼⠇⠀⢹⣥⡀⣠⡄⠀⢠⡄⠀⠀⠀⠉⠛⠓⣝⣆
⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠀⠀⠀⠀⣿⡇⣿⣷⣿⠀⠀⣿⣻⡇⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⢀⢨⢿
⠀⠀⡄⠤⢀⠀⡜⣰⠂⠄⠀⠀⠀⠀⠿⠇⠿⠙⠿⠀⠰⠿⠙⣷⠿⠷⠾⠺⠿⠾⠀⠀⠀⠀⠠⠐⣆⢳⠀⠀⠤⢠
⠀⠂⡖⡕⢀⣸⢱⢣⠡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠙⠷⠿⠀⠀⢀⣀⠀⠀⠀⠀⠉⠌⡜⡎⣇⡠⢪⢲⠐
⠀⠀⠀⣧⡆⡇⣼⠏⣁⣀⣀⢀⡀⣠⢀⣄⣀⡀⣰⠏⣠⣟⢀⣤⣄⠀⢀⣤⡙⣩⣷⡀⣀⣠⣠⣌⠹⣧⢸⢰⣼
⠀⠀⠀⢿⣷⡷⠃⠀⠈⣻⠉⢸⣧⣿⢸⡏⢸⣿⣿⢸⣟⢋⣈⣿⣏⠀⣿⡏⣃⢾⡿⣿⡟⢸⡷⠆⠀⠘⢾⣾⡿
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠛⠂⠘⠃⢿⡘⣛⠛⠋⠛⠈⠛⠛⠋⠛⠋⠀⠙⠛⠛⠛⠃⠛⠃⠘⠛⠃⠀⠀⣸⣿
⠀⠀⠀⠀⠹⡇⢓⠢⢔⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⡢⠔⡂⢸⠏
⠀⠀⠀⠀⠤⢿⣸⣸⡞⠀⠀⠀⣶⣶⡆⣶⠄⣶⠀⣶⣦⢰⣦⢸⡏⣶⣴⡆⣶⣶⠀⠀⠀⠀⠀⢳⣇⢇⡿⠤
⠀⠀⠀⠀⠂⠣⣧⢻⡁⠀⠀⠀⠀⣯⠀⣿⠶⣿⢸⣯⣿⢸⣿⣿⡇⣿⣿⠀⣽⣿⡀⠀⠀⠀⠀⢈⡟⣼⠜⠉
⠀⠀⠀⠀⠀⡄⣌⣷⣳⡀⠰⡀⠌⡛⠀⠛⠁⣿⢚⡃⢹⡞⢣⡙⠃⠛⠘⣧⣙⡋⢀⠠⢀⠆⢀⣞⣼⣡⢠
⠀⠀⠀⠀⠀⠓⠊⠣⡙⢏⠙⢎⠐⢞⠀⠀⠀⠙⠿⠏⠀⠙⠛⠁⠀⠀⠀⠈⠛⠁⡳⠂⡱⠋⡹⢋⠔⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⢤⡙⠾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⢋⡤⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠓⠤⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣉⠤⠞⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢠⡞⣉⡉⠉⣿⡏⠉⣴⢦⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⣼⠁⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄
⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⣿⡇⣸⣿⠌⠀⠀⠀⠀⡀⠀⠀⠀⠓⠲⣿⠊⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⣰⡟⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣷⢊⣿⢰⡟⣹⡿⢰⣿⡼⣻⡇⣿⣞⡝⠀⣼⡏⡼⣼⡇⣰⣿⢠⣿⠕⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣿⡇⠹⡇⢸⡟⢸⠿⢿⡇⣸⣿⢀⣿⢣⡿⢻⣆⢠⣿⠋⢀⣿⡷⢻⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠘⡌⠋⣠⡿⠀⠀⠀⠸⡧⠇⠀⠈⠛⠁⠉⠸⣿⢰⠆⠀⡟⢾⣿⠉⢧⠀⠀⠘⠿⠾⣷⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⡾⠙⠄⠀⢀⣴⢦⣠⢠⢶⣦⣮⢧⣠⢶⡇⣀⢿⡄⣸⢀⡞⡝⠀⠀⠙⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣜⡀⠀⣼⡷⠃⣀⣈⠋⣋⡋⠘⡙⠋⣃⣀⠀⢉⣁⣼⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⢹⣦⣿⠀⣸⣏⡜⢰⡇⠇⣾⠙⢰⣏⡜⣰⡟⢈⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡴⢻⣿⢿⣀⣿⣇⢀⠼⣷⠞⢿⡖⢻⣏⢀⣿⣧⠞⢿⡇⠈⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠾⠃⠀⠉⠀⠉⠁⡎⢹⠦⣟⡧⠀⠉⠁⠀⠀⠀⠈⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⡞⠉⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⢀⡾⠁⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⣼⣿⠀⠀⠀⠀⡀⠀⠐⠁
⠀⠀⠸⠁⠀⣠⠏⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠈⠐⠊⢡⣄⠀⠀⠀⣀⣀⠀⠀⠀⠂⢸⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣴⠖⢄⣼⡷⠀⢠⣿⠟⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠈⣿⡄⠀⠸⡏⢉⠇⠀⣶⣶⣿⣿⣿⣿⡇⠀⠀⣀⠀⣿⡇⠀⠀⣿⠃⠀⢸⣿⠁⠀⣼⡟⠀⠀⢀⢸⣿
⠀⠀⠀⢀⠀⠀⢽⣧⠀⠀⠙⠣⣀⠀⠙⠉⠉⣿⣇⠀⠀⢰⡿⠁⠀⢿⡇⠀⠀⠻⠤⠐⢸⡏⠀⡀⢻⡁⠀⠀⠂⢸⣿
⠀⠀⢈⠀⠀⠀⢸⣿⠀⠀⢀⣼⡿⠀⠀⠀⠀⣿⣿⠀⠀⢻⣇⠀⠀⠘⢿⡀⠀⠄⠀⠀⠀⠙⠊⠀⠀⠑⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠀⠀⢸⣿⠀⢀⣿⠏⠀⠀⢠⠀⠀⢹⣿⡄⠀⠈⠿⣦⡠⠂⠈⠑⠊⠀⠄⠊⠁⠀⠈⠑⢄⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⠀⠀⡠⠂⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣀⠀⣁⠀⣀⡠⠀⣤⠀⣠⠄⢠⣖⠀⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⢀⣿⠃⠀⢸⣿⠆⣿⡀⠸⠅⠀⠼⠆⠪⠟⠘⠇⠀⠛⠚⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠉⠉⠉⠓⠦⣴⡇
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣴⠄
⠀⠀⠀⠀⠀⢠⣿⢁⡉⠠⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢶⡆⢠⠄⣠⠄⠀⠶⣤⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣤⣤⠀⣿⠀⢸⣿⡈⣼⡇
⠀⠀⠀⠀⠀⢸⣿⣇⠉⡆⠀⠀⠀⠀⠀⠀⢀⡄⣀⣿⡟⠀⣿⠀⠀⣿⠃⢿⣇⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠻⣿⣇⠈⠁⢀⢢⡲⣄⢠⢸⢀⣽⠟⠀⠀⠛⠀⠀⠙⠀⠀⠙⠓⠒⠒
⠀⠀⠀⡄⠀⠀⠀⢈⠻⢿⣦⣤⡷⢻⠠⠀⠗⡞⠁⠀⡀⢠⠀⣤⣤⠄⢤
⠀⠀⠀⢠⡀⠀⠈⠄⢀⠀⣀⢩⠱⠈⡚⠐⡨⠄⠀⠀⠀⠸⠀⣿⣿⠀⠸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⠀⠀⡔⢒⠏⡷⠈⠉⠃⠇⠘⡶⢥⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢱⠎⡡⢊⠃⠁⠀⠀⢉⣎⡡⠤⣀⣧⠀⣀⠀⠀⠀⣿⣿⠀⠀⣿⡧⢶⡆⠀⠀⠆⠘⢲⣄⢠⣦⢴⣶⠀⣿⡇⠠⠀⢀⡄⢠
⠀⠀⡀⢀⣇⠀⠀⢈⠀⠀⠀⠀⠀⠉⠀⢸⡿⢧⠊⠀⠖⠀⠀⣿⣿⠀⠀⣿⡇⢸⡇⣠⡞⠋⠁⢸⣿⢸⣿⢸⣿⠀⣿⣿⣇⠀⠘⢿⣄
⠀⠀⠙⠈⠁⠙⠐⠌⠀⠀⠀⠀⠌⣀⡤⠈⠀⠌⠁⠠⡀⠀⠀⣿⣿⠀⠀⣿⡇⢸⡇⣿⡀⠀⠀⢸⣿⢸⣿⢸⣿⠀⣿⡇⢿⣆⢀⠀⢻
⠒⠂⠑⠀⠀⢆⣀⡹⠘⠀⠀⠐⢀⢙⠀⠀⡠⠆⠀⢉⠁⠀⠒⣿⣿⠀⠀⠛⠃⠘⠃⠈⠛⠓⠒⠘⠛⠘⠛⠈⠛⠀⠛⠃⠈⠛⠈⠃⠊
⠀⠀⢜⣴⠄⠀⣑⡃⠄⢀⡀⠀⠠⠧⠤⠠⡀⠀⠀⠀⠠⠐⠁⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠂⠒⠂⠀⡀⠀⢀⡀
⠀⠀⠘⣏⢒⣀⣉⣀⣈⠄⠰⠔⠃⠀⠀⢹⠌⠀⠞⢦⡴⠓⠂⢂⡏⠱⡒⢬⣉⣉⣀⣐⣂⠀⠴⠶⣤⣴⠿⠟⣾⡿⠲⠎⢛⠿⠙⠉
⠀⠀⠀⠀⠀⠈⠫⠄⠐⢄⠀⠀⠀⠀⢀⡶⠐⠦⠔⠛⠒⣦⠠⢬⠀⠀⢀⢈⠿⢭⣁⡲⠵⣿⣵⣾⠦⣴⢚⢟⡶⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣉⣇⡘⢤⠘⣭⠤⠘⠲⠶⠟⠛⠂⠒⠈⢽⣮⣁⣀⠚⠉⠉⢳⡾⠥⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢄⡀⢠⠊⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠲⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⠥⠐⠀⠁⠈⢀⠀⠁⠉⠐⠂⠌⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠊⠀⠀⠀⢳⠀⠀⢸⠀⠀⢀⡇⠀⠀⢈⠐⢌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡠⠂⠀⠀⠱⣄⠀⠈⢃⣀⣈⣄⣀⣚⠀⠀⡰⠃⠀⠀⠑⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡐⠀⠐⢦⣀⠀⢀⠕⠊⠁⠀⠀⢠⠀⠀⠁⠺⢅⠀⣠⠔⠁⠈⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠠⢀⣀⠀⠀⠈⡞⠁⠀⠀⢀⠄⠂⣈⡉⠀⠂⢄⠀⠱⡁⠀⢀⣀⠈⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠃⠀⠈⠉⠋⠎⠀⠀⢀⠄⠀⡠⠊⠀⠀⠄⡀⠀⠑⡀⠹⠙⠉⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡃⠤⠤⠤⡞⠀⠀⠀⠆⠀⠼⠀⠠⡀⠀⠀⠈⠢⡀⠁⠀⣦⠤⠤⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⢀⠀⠀⠀⡈⠀⠐⠀⠢⠀⠌⠀⠌⢀⢈⠇⠀⠀⢁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⠐⠚⠃⠀⠠⢠⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⠘⠀⠃⠀⡈⠙⠒⢀⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢂⡐⠀⠀⠀⡄⠀⡈⠀⠀⠀⡀⠁⠐⠁⠰⠁⡌⠀⠀⣅⠀⠀⠌⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠁⠀⠀⠀⠇⠀⡏⠢⡀⠀⠀⠒⢊⡕⠁⠐⠀⠀⠐⠀⢁⠎⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠐⠀⠐⠈⡀⠀⠃⠀⠀⣁⠔⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⢡⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣾⢡⠁⠀⠀⠀⠀⠀⢈⣌⡴⠅⠀⠀⠀⠀⠀⠀⢜⠦⢈⢢⡈⠀⠹⣦
⠀⠀⠀⠀⠀⠀⣸⡇⠚⠀⠐⠀⠄⠀⠐⠋⠀⠀⠉⠀⠐⠒⠒⠀⠈⠀⠀⠀⠀⠈⠑⢄⠘⣧
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢂⣿
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⠆

⣿⠛⣻⣿⢻⢟⡽⡻⢯⢻⡹⡙⣟⣿⣯⣿⣟⣟⡿⠿⢯⡻⡿⡟⡟⣯⣏⡟⡟⡽⣫⠿⢯⡻⡝⡟⣟⢿⣿⡛⢻
⣿⣷⢿⣿⣾⣾⣾⣿⢶⢷⣷⣷⣷⣶⣶⣶⣶⣽⣷⣿⣷⣷⣷⣷⣷⢾⠾⡷⡷⣷⢷⢿⠷⡷⠿⣾⣾⣾⣿⣷⢿
⣟⢮⠏⣿⠀⠀⠀⠙⠓⣦⣴⣿⣿⣿⣿⣿⡿⠿⠿⠷⠿⣿⣶⣯⣽⠓⢦⣠⡀⣤⡠⣀⣤⠖⠋⠉⠀⢸⣏⢷⣺
⣯⣿⢸⣿⠀⠀⠀⠀⠈⢨⣿⠋⠙⠛⣿⣿⣧⣄⡀⠀⠀⠀⠙⢿⣿⣶⢁⠽⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⡿⢸⢷
⣿⠿⠾⣿⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠉⢹⣿⢿⣷⣶⣤⣷⣿⣿⣿⣫⡴⢻⣿⠏⠀⠀⠀⠀⠀⠀⢸⣟⣚⣻
⣿⡭⠭⣿⠀⠀⠀⠠⣾⠃⠀⠀⠀⠀⠀⢠⠋⠇⠀⠁⢻⣿⡿⠛⠉⠉⠙⢿⣏⣿⡁⠀⠀⠀⠀⠀⠀⢸⣗⡪⢽
⡿⣪⠭⣿⡄⠀⢀⣼⠟⣄⠀⠀⠀⠀⡠⠃⠀⡅⠀⠀⢸⣭⡤⠒⠻⠛⢛⣻⣿⣯⣿⣙⠒⠦⣄⠀⠀⣸⡷⢭⣻
⡿⡇⢎⣿⣿⣾⠋⣿⠀⠈⠑⠒⠐⠊⠀⠀⠀⢣⠀⠀⢼⣩⡽⠟⠋⠉⢩⣏⡿⢿⣿⣿⣴⠀⠀⠙⣾⣿⡯⢇⣿
⣿⢬⣓⣿⡿⠀⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣿⡭⠀⠀⠀⠀⣼⣿⠉⠀⠈⢿⣷⡎⠀⠀⠈⣿⡯⠭⢾
⣯⣝⣒⣿⠃⠀⣉⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⢻⣿⠄⠀⠀⢸⣯⣭⢽
⣿⠶⠴⣿⠛⠓⠶⣿⣷⣆⡀⠀⠀⢀⣀⣀⣀⣠⣤⣿⡧⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠘⣿⣧⠴⠖⢾⡗⠺⢿
⣿⡫⠭⣿⠀⠀⠀⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣷⣏⢰⡄⢠⣿⡏⠀⠀⠀⣤⣨⣶⡟⢧⠀⠀⢾⡯⠭⣽
⣿⣚⡭⣿⠀⠀⠀⡰⣿⣿⣽⣿⣿⣿⡏⠁⠀⠀⢈⡟⣿⣅⠊⣦⢘⣿⣇⠀⠀⠀⠈⠙⠛⣿⡄⠂⠀⢸⡯⢭⣻
⣿⡇⣎⣿⠀⠀⠀⠰⣿⣟⣻⣿⣿⣅⣻⢦⣤⣤⣿⡗⢻⠟⢺⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⠄⠀⢸⣯⠇⣿
⣿⣽⢏⣿⠀⠀⠀⠈⣿⣭⣼⢠⡷⣦⣬⣿⣿⣿⣿⡇⢸⠀⠈⠃⣄⣹⠁⠀⠀⠀⠀⠀⣠⣿⣯⠃⠀⣻⣏⢿⣿
⣿⡿⣼⣿⣦⢀⣴⢾⠉⡄⠇⠀⣀⠀⡀⢨⠿⣿⣿⠇⠘⠀⠀⠀⢻⡇⢠⣄⣀⠀⠀⠀⣾⡿⢷⣄⢀⣾⣿⣼⣿
⣿⢿⢹⣿⣿⠟⢀⡝⠀⢳⡀⠀⡿⠰⢀⠋⡼⣿⡏⠀⠀⠀⠀⠀⢸⠁⢸⣿⣿⣿⣶⢞⡟⡱⠀⠈⢿⣿⡿⣸⢿
⣿⡚⢧⣿⡏⠀⠀⠳⡤⢬⣤⣀⢼⠃⠌⡜⣵⣿⢁⣀⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿⡎⠘⠐⠀⠀⠀⠀⢿⡧⠭⣽
⣿⣒⣛⣿⣅⣀⠀⠀⠈⠀⠀⢸⡘⡜⣼⣸⣿⠏⠉⠉⠁⠀⠀⢠⠇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣹⡯⢽⣻
⣿⡣⣖⣿⠁⠉⠙⠳⢤⡀⢠⣼⣿⣿⣾⣿⡃⢀⣤⠔⠒⠀⠀⢘⣷⣶⣿⣿⣿⣯⣧⠀⣀⠴⠚⠋⠉⣽⣯⡷⣻
⣿⣝⣒⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠇⠹⣿⡇⠀⠀⠀⠀⠀⢙⡿⠻⣿⣿⣿⣿⠞⠁⠀⠀⠀⠀⣿⣗⣚⣽
⣿⣒⣶⣿⠀⠀⠀⠀⠀⢀⠄⢿⣿⣿⡋⠀⠀⠹⣧⡀⠀⠀⠀⣠⡞⠁⠊⡹⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣗⣒⣻
⣿⣶⠉⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠧⢈⠀⠀⠀⢹⣌⠈⠁⣩⡿⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⣾⡷⠰⣾
⣿⣒⣭⣿⡄⠀⠀⣠⡴⠒⠛⠋⠙⠛⠲⠦⣤⡀⢀⢻⡉⢉⣿⡧⣁⡤⠶⠒⠛⠛⠒⠲⠤⣄⠀⠀⠀⣿⡯⢭⣟
⣿⣖⣒⣿⣷⣴⣯⣅⣤⣄⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣿⣯⣕⣾
⣿⠋⢻⣿⣏⢯⡻⠝⣹⣹⢹⢣⣭⣛⣛⣛⣭⣼⡛⠼⣹⢩⢏⢏⣏⣽⢹⣸⡹⡹⣍⠧⣫⢏⡇⡏⡏⣿⣿⠋⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⡇⡟⣰⠃⠀⠀⠀⠀⢀⡠⠖⠚⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⢠⣈⠳⡄⠀⢿⡷⠁⠀⠀⠀⢀⡴⢋⡤⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣙⣻⣄⢸⠁⠀⠀⠀⢠⣮⢖⣉⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⢻⠀⠀⠀⢠⠟⢋⡩⠴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠙⢦⠀⠀⢸⡇⠀⣰⣋⣴⣟⣚⠋⠉⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠛⠉⠛⡒⠦⢤⣀⠘⣆⠀⡄⢧⠀⠈⡇⢰⠟⠁⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠉⠓⠢⢬⡓⢬⡳⣜⣾⡀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⣬⣙⣿⣆⠙⢧⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢾⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡴⠊⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠛⠛⠓⠲⢤⡀⠀⠀⠀⠀⠙⣄⣷⠀⠀⠀⠀⠀⢠⣞⣤⢞⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠉⠓⠢⣌⣣⡀⠀⠀⠀⠸⣿⠀⠀⠀⣠⠴⠛⠋⠉⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠤⢽⣷⠀⠀⠀⠀⡇⠀⣤⡾⠵⠶⠶⢿⡿⠯⠤⠈⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢰⣧⠞⠁⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⣒⡦⡀⢳⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣤⣤⣭⡿⣄⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⡟⡰⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣾⠋⠿⣀⠀⠀⣉⠉⣿⠓⠦⣄⣠⣿⠜⠁⢀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢾⡏⠀⠸⣇⣴⢻⡟⢰⡏⠀⡴⠆⣿⠛⠶⠞⠋⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⠤⣿⠧⠤⠖⠀⠉⢸⣇⣾⠇⢸⡇⣸⡟⣰⡇⢀⡆⠀⣿⠀⡀⠀⠀⠀⣤⡄⠀⣶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢰⣿⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢿⡤⠊⠉⣽⡇⠹⠧⢼⡇⡐⢧⠔⠁⠀⠀⠀⠉⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣁⡼⠋⠀⠀⣀⠀⣰⠞⡆⠀⠀⠀⠀⠀⠻⠃⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⣀⢣⠑⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢠⠂⠀⠀⠀⡴⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠃⣿⡟⢹⡇⡴⢲⢤⣄⡔⢹⡇⢸⣇⡴⠀⣿⠀⠀⠀⢰⣇⠆⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢸⣿⠀⠛⠃⢾⣇⡗⢿⣀⣿⡇⣼⡇⢸⣿⡁⢰⡟⢰⡇⠀⢸⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡟⠁⠀⠀⠀⢿⣄⡴⠁⠈⠉⠀⠀⠀⠀⠀⢿⣇⢾⡇⠳⢼⡇⠸⠦⢯⣸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⡇⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠛⠀⠀⠀⠈⠉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⡇⠠⢶⡀⠀⠀⠀⠀⢀⣼⢃⡜⠀⣴⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢹⡇⠉⠀⠀⠀⠀⠀⣾⡷⠋⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢸⠇⠀⠀⠀⠀⠀⣠⣿⣀⣀⣠⣿⠀⣴⢖⠆⢀⡀⢀⠶⣦⠀⢠⣾⣫⢇⣤⢦⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣰⢻⣿⠁⣿⢹⡏⢸⡟⠁⣰⠛⣿⠋⠀⣿⡇⢿⡏⠀⣸⣏⡴⣿⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠛⠸⠟⢸⣇⠟⠷⠖⠁⢰⢻⡇⢮⣽⠃⠈⠛⠋⠀⠀⠀⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠻⣧⢀⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⣼⠟⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⠀⠙⢿⣿⢻⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡟⣿⡿⠋⠀⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠈⢻⠈⣿⣜⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣣⣿⠁⡟⠁⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠫⢄⠀⠀⠣⠘⣿⣷⣮⡙⠛⠛⠿⠿⠿⠟⠛⠛⢋⣵⣾⣿⠃⠜⠀⠀⣠⠝⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣷⡄⠀⠀⠀⠁⠈⢿⣿⣝⡳⢦⡀⠀⠀⢀⡤⢞⣫⣿⡿⠁⠀⠀⠀⠀⢠⣾⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠛⠿⣿⣌⢦⠀⠀⢤⣽⣮⣭⡀⠙⠀⠐⠋⢀⣩⣵⣯⡄⠀⠀⡴⣡⣿⠿⠛⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣶⣬⠌⢑⣷⡄⠀⠉⠻⢿⣿⡄⠀⠀⢨⣿⡿⠟⠉⠀⢠⣾⠊⢀⣥⣶⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣉⢛⣷⠄⠈⣽⠻⠦⠀⠀⠈⠋⢻⡄⢠⡟⠙⠁⠀⠀⠴⠟⣯⠁⠠⣾⡛⣋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⠋⠉⠴⢿⡂⠈⣿⡘⣇⣀⣠⠀⠀⠀⢹⡏⠀⠀⠀⣄⣀⣸⢃⣿⠁⢐⡿⠦⠉⠙⠛⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⠀⠀⣻⣥⠄⠛⢿⣮⣭⣵⣿⣦⣴⠟⠻⣦⣴⣿⣮⣭⣵⡾⠟⠠⣼⣟⠀⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣩⣿⣷⣀⣈⡀⣰⠏⢠⢿⡟⠀⠀⢻⡿⡄⢻⣄⢀⣁⣀⣿⣿⣍⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠁⡞⠁⠀⠉⠙⢻⠏⠀⠀⠀⠀⢻⠀⠀⡟⠀⠀⠀⠀⠹⡟⠋⠉⠀⠀⢣⠈⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡇⢀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠀⡀⢸⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣿⣶⡇⠀⢰⡷⢾⣇⣀⣄⠀⠐⠂⠀⣠⣀⣸⡷⢾⡆⠀⢸⣦⡷⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠟⣷⠀⢸⣇⠈⠛⠙⣿⣦⠀⠀⣴⣿⠏⠛⠁⣸⡇⠀⣿⠿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⣧⠸⣿⣦⣧⠀⢻⣿⣧⣼⣿⡟⠀⣼⣴⣿⠇⣼⠏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠏⠻⣧⡈⢿⡌⢠⡿⢁⣼⠟⠹⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠇⠸⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠊⣴⠀⠀⠎⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠎⠦⠀⠀⠀⠀⠀⠀⡌⢀⠀⠸⠄⠄⠀⠁⠃⠀⠓⠰⠀⠄
⠀⠀⠀⡀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡂
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⢀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢀⡻⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠠⣅⡚⠝⠄
⠀⠐⠊⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡄⠀⠊⠀⣾⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠻⢟⠿⠻⠟⢺⠃⠀⠀⠀⢘⢀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡠⠰⡅⠀⠀⠀⠀⡠⠴⠀⠀⠀⠀⠀⠀⢡⠟⠓⠻⠓⠚⡝⣾⠾⠓⠟⢲⡟⠀⠀⠌⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠘⠔⠁⢐⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣄⡀⡀⠀⠀⠀⠈⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢨⠄⠀⠀⠀⠁⠀⠀⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠃⠄⠀⠠⠂⠀⠀⠐
⠀⠀⠀⠀⢀⠄⠀⠚⠀⠡⠀⠀⠀⠠⠄⠈⠁⠉⠣⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠
⠀⠀⠂⠂⠀⠀⠀⠐⡃⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠒⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⡀⢲⠀⠀⠀⠐⠐⠊⠐⠀⠀⠁⠊⠈⠤⢀⢐⠁⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠸⠤⡇⡀⢠⢌⣠⠉⠀⠀⠀⠀⠀⠈⠐
⠊⠀⠀⠈⠁⠀⠀⠁⠄⡄⠁⡉⠀⣂⡀⠀⠀⡀⠐⠘⠠⡁⠠⢀⡆⡄⠄⠶⠦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠈⡨⠡⠥⢅⢀⡤⠡⠑⢂⡚⠺⢤⠉⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠌⠄⢠⠞⠠⠄⠀⢡⠀⠀⢀⠴⢄⡑⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢾⠄⠀⠀⡀⠊⠀⠐⣔⠂⠐⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠙⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠊⠈⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠐⡖⢠⠀⡆⡴⠀⠀⠀⠀⠸⡀⢀⠀⠀⠉⠀⠀⢀⣀⣠⣔⡒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⡇⠈⣆⠁⡗⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⣠⠞⠉⠀⠣⣀⣨⠆⠀⠐⡀⠁⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠀⡧⠀⠿⠀⣇⡀⠀⣴⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⢄⠀⠀⠊⠄
⠀⠀⠀⢀⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄⠁⠒⠒⠒⠊⠁
⠀⠀⠀⢸⣷⣀⣸⡇⠀⠀⣠⡤⠀⢰⣢⢶⠆⣾⣧⠋⢠⠛⣶⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⢂⡄
⠀⠀⠠⠚⣿⠁⢸⡇⣼⡇⣿⣼⣧⠾⠃⠺⠴⡿⠻⠤⢠⢲⡿⠄⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠆⠀⠚
⠀⠀⠀⠀⢻⡀⢸⣷⠻⡷⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇⠀⠀⠀⠠⠈⢀⠀⡀
⠀⠀⠀⠀⠈⢧⠎⠋⠀⠀⣴⣷⣷⣦⡄⠀⠀⠀⠀⠀⡀⡆⡇⣿⠀⠏⢸⢸⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⣷⢻⠃⣿⠀⢰⢸⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⢶⠒⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠘⠀⠛⠂⠈⠘⠈⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢹⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⡕⠀⣀⢠⡹⣶⡀⢂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⣀⢻⡄⢻⡌⡿⠾⠛⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠐⠉⠀⠈⠀⠀⠀⠀⣉⡀⠀⠀⠀⢀⡀⠀⣤⢀⡄⠀⠈⢹⡇⢸⡿⠀⣿⡌⠻⠋
⠀⠀⠀⠀⠀⠃⠀⡄⠘⠄⠀⠀⠀⣼⣫⣶⡀⠀⠀⡿⣸⡇⣿⠞⢃⡟⣵⣆⢧⡼⠧⠞⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂
⠀⠀⠀⠀⠀⠡⠀⠀⠠⠀⠲⡄⠀⠈⠁⠉⠀⠀⠀⠛⣹⠇⠛⠀⠈⠛⠉⠉⠀⠀⡀⣀⢀⡀⠀⡀⢀⡀⣀⡀⢀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠬⠁⠀⠀⠀⠀⢸⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣸⢸⡀⠀⢇⢸⡞⢸
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠁⠘⠀⡖⠠⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⢸⠀⡸⢼⠈⢇⢸⡄
⠀⠀⠀⠀⠀⠰⠃⠘⠁⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣷⡄⠉⠈⠈⠃⠁⠈⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⢀⣤⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⣾⣿⠿⠿⠿⠿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⢣⣶⣤⡀⢲⣤⣤⣶⣶⣿⣿⣷⡦
⠀⠀⣩⣤⣴⣤⣤⣀⠀⠈⢉⠟⢻⣷⣶⣶⣶⣤⣄⠀⠀⠀⢠⣴⣜⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⣼⣿⡿⠟⠻⠿⣿⣿⣶⣧⣤⣄⣙⣿⣿⣿⣿⣿⣷⡄⢠⣿⣿⣿⠈⠻⢿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢸⡟⠁⠀⠀⣀⣀⡀⢠⠋⠻⣿⣿⣿⣿⣷⣦⡀⠈⠙⠃⠘⣿⣿⠟⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣤⣾⣿⣿⣿⣿⣿⣤⣤⣶⣯⣭⡙⢻⣿⣷⠀⠀⠀⠀⠙⠉⠀⠀⠀⣀⡤⠄⠀⠹⣿⣿⡍⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠘⠉⠁⢈⣉⣉⠀⢧⠈⠉⠛⠻⢿⣿⡄⠹⢟⣀⣀⣀⣀⣀⢀⣀⡴⠋⠁⠀⠀⠀⠀⠈⠉⠀⢹⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢀⣾⣿⣿⣿⣿⢿⣾⣷⣶⣄⣨⣿⣾⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣷⣶⡦⣀⡀⠀⠀⠀⠀⠈⢻⣿⡿⢿⣿⣿⣷
⠀⠀⠚⠁⣠⣤⣬⣁⣸⡉⣻⣶⣯⣽⣿⣿⣿⣸⣿⣿⣿⣷⣿⣿⣿⣿⠟⣯⣿⣿⣿⣷⣤⣀⠀⠀⠀⢀⣤⣤⣽⣟⠋
⠀⠀⠀⣾⠟⠛⣛⡛⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣷⣄⠀⠹⢿⣿⣿⣿⣷⠄
⠀⠀⠀⠁⣰⡿⠛⢋⡿⠿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣻⣿⣷⢺⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣟⣥⣻⣿⣮⡉⠉
⠀⠀⠀⠀⠋⣠⣴⢸⣿⣷⣾⣾⣿⣖⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⡟⠀⠻⣿⣯⣿⣿⣿⣽⣿⣿⣿⣷⡿⡿⣿⣿⣭⣿⣿⣿⣿⣷⣝⣿⣿⣯⣿⣿⣿⣿⣿⠙⠻⠏
⠀⠀⢀⠤⠸⡿⠀⠀⠀⣾⣭⣷⣍⣿⣷⣭⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠄⢠⡤⠈⠀⢠⣿⣿⣿⣿⣮⣹⡿⣿⣿⢿⣿⣿⠟⠻⣟⣿⡿⢂⢙⣿⣿⣿⣿⣿⣷⣿⣻⠿⠃
⠀⠀⠠⠀⠀⢀⡠⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣿⠁⣻⠿⡛⠀⢟⣾⣿⣾⣙⣿⢿⣿⠉⠀⠀⠀⣤
⠀⠔⠈⠉⠉⠀⠀⠀⣿⣿⡋⢿⣿⣿⣿⣿⣿⣿⢻⠟⢻⠀⢟⣼⡇⢰⠏⣨⠿⣿⠟⣡⣶⡏⠀⠀⠀⣰⣿⣦⡀
⠀⠰⠋⢀⠀⠀⠀⢰⣿⠟⢡⠘⣿⡟⢿⡏⣽⠻⠄⣾⠘⠀⣈⢿⡄⣟⢰⡋⡼⠃⣼⣿⣿⠁⠀⢀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣵⣶⡈⢷⡘⡇⠈⠷⣈⣠⣼⣿⣿⣶⣿⣿⣷⣿⣿⣷⡆⢾⣻⣿⠃⠀⢠⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⣷⡘⢷⡀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠿⠟⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣬⣿⣿⡏⣾⢱⡟⠗⣿⣿⡝⡝⣾⡫⣿⢾⣿⣿⠟⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠇
⠀⢀⣴⣾⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⣿⣸⣧⣶⣿⣿⣟⣧⣿⣇⢿⣿⣿⡏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⢟⣷⣶⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠿⣿⣿⣿⣿⣿⣿⠿⠟⠂⠀⠀⠀⢉⣿⣿⣿⢿⡿⠃
⠀⠉⠉⠀⢸⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⢓⣢⡼⣿⣿⠿⣅⣠⣦⣀⠀⠀⠀⠈⠻⠟⠃
⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠙⢿⣟⠛⠛⠛⠛⠿⠿⠛⣋⣵⣾⠟⠁⡀⠈⣉⡉⠉
⠀⢀⡴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠿⠟⠛⠉⠀⠰⣤⣉⣉⡉⠳⠖⠀⠀⢠⠀⣠⣶⣶⡄
⠀⣾⠁⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡠⠘⢦⣀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣦
⠀⠹⡀⠀⠀⡟⠀⢸⡆⠀⣀⣀⡀⠀⡄⠀⣄⠀⢰⣀⠀⠛⠶⡶⣶⠎⣧⡀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡿⠾⢿⡓
⠀⠀⠙⠦⢤⣷⡴⠟⢠⣿⣿⣿⡏⠛⢿⠦⠌⠳⡤⠉⠛⣻⠟⢰⣿⠀⠈⠉⠀⠀⠀⢰⣿⣿⡟⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠙⠀⠛⠛⣿⣿⡇⠀⠉⠀⠀⢾⡁⠀⠐⣿⡀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠹⠻⠇⢸⠟⣿⠉⠀⠀⠀⠀⠀⠀⠀⠱

⠀⠀⠀⢠⢢⢢⣠⢄⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⢦⡤⣀⣠⡄⣦
⠀⠀⠀⠈⠁⠉⠛⠚⠀⠀⠀⠀⠀⠀⠀⡶⢴⢴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⠋⠛⠃
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠸⠐⠂⠇⠀⠀⠀⠀⠀⠀⠀⢠⣄⡄⢤⡤
⠀⠀⠀⢰⢢⣰⢶⣴⡆⠀⠀⠀⠀⠀⠀⠈⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⠐⠒⠘⠑⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠘⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡆⠀⢀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠘⢇⠀⠇⠀⠀⠀⠀⠀⠓⠋⠐⠊⠋⠀⠀⠀⠀⠀⣶⠀⡆⠀⢶⠀⢰⠚⠀⠀⠀⠀⠀⠀⡄⢠⠀⣰
⠀⠀⢀⠀⡀⢀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢰⢰⢸⠀⠀⠀⠀⠀⠀⠿⠄⠇⠀⠸⠀⠸⠤⠀⠀⠀⠀⠀⠀⢃⠈⠈⠘
⠀⠀⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠆⠰⠀⠄⠀⠀⠀⠀⠀⠀⠈⠮⠆⠇⠥
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠄⠀⠆⠆⠆⠦⠀⠀⠀⠀⠀⠀⢰⠖⠰⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀
⠀⠀⠀⣀⢀⡘⢁⢀⠀⡀⠀⠀⠀⠀⠀⡶⡄⡆⠀⠀⠀⠀⠀⠀⠀⡐⠂⣀⣀⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠈⡇
⠀⠀⢸⡬⢫⡼⣜⡞⣾⠁⠀⠀⠀⠀⢄⢁⠁⡁⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠈⡁⣈⠁
⠀⠀⠈⠀⠀⠁⠉⠁⠁⠀⠀⠀⠀⠀⣈⢉⡉⡀⢈⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⢀⠄
⠀⠀⠀⠤⠀⠠⠀⠄⣤⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠁⠀⠀⠀⠀⠀⠘⠀⠊⠉⠓⠁

⡀⠀⡀
⣇⡀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠓⣤⣾
⣿⣿⠙⣆⣄⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⡀⢱⣾⣿⣿
⣿⣿⣿⣿⠿⡉⣀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⢏⢢⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠺⣥⣷⣿⣁⢀⡴⠂⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⢡⢈⡢⣄⣾⣆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡾⣿⣂⣼⣤⣸⡎⡹⠀⢰⡄⣠⠄⠀⣄⢄⠰⡀⣦⠀⡀⢲⣀⠙⢄⣠⣷⣏⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣤⣷⣿⣴⣿⣿⣴⣄⣼⢸⣾⣶⣧⢿⣴⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠂⠀⢀⡤⠀⠀⠀⠀⣿⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⠄⠀⠀⠈⢸⡇⢸⡟⢠⡄⢀⣤⡄⠀⡀⣠⣦⢸⡧⣡⠀⣼⡇⢀⡀⣀⠀⣿⡄⠀⠀⠀⠀⢴⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠠⠤⠄⣾⠀⡸⡿⣾⠡⣿⣽⣇⣼⡟⢸⡇⡼⣿⣏⠀⣿⡇⢼⣃⣿⢀⢿⠀⡐⠤⠄⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠔⠁⠀⠙⠃⠀⠈⠋⠀⠀⠘⠋⠀⠉⠙⠞⠺⣧⠈⠁⠳⠂⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢠⠀⠠⠤⠄⡀⠀⠀⢀⡄⠀⠤⠤⡤⠄⠤⠤⠠⠄⠬⠄⠤⠠⢤⠀⠀⠠⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡘⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇⠀⢠⢠⠏⠀⢰⠀⢀⠂⠀⠀⡇⠀⠀⠉⠁⠀⡇⠉⠀⠀⡸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠻⠄⠰⠤⠀⠥⠠⠅⠀⠈⠴⠀⠀⠀⠀⠀⠤⠠⠀⠧⠀⠀⠀⠐⠦⠤⠀⠀⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡄⢠⢠⢐⠂
⠀⠀⠀⠀⠀⠀⠑⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠀⠐⠂⠹⡄⡇⢾⢾⠸⠇⠂⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠊
⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠦⣰⠂⠀⠀⢀⠀⠀⠀⠐⣆⠤⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠓⢂⣤⠀⠈⠀⡍⠨⢭⡉⢦⡀⠀⠈⠐⠚⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡆⠐
⠀⠀⠀⠀⠀⠈⠙⠓⠂⠁⠀⠀⠀⠀⠀⢰⣿⡌⠀⠀⣿⠁⠃⠀⠀⠘⠆⠀⢙⣳⡇⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⡀⠀⠈⠐⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠃⣠⠄⢻⣿⡁⣄⢀⣿⡜⢀⣶⠂⢰⣇⢠⣖⠀⠃⣴⠢⢀⣴⢾⣿⠀⠐⠠⣄⠘⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠻⢠⣿⠏⢿⡄⠸⢿⡴⠋⣿⣯⢻⣖⠉⢿⡤⠊⠻⠺⣿⡰⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠘⠷⠃⠀⠀⠀⠛⠋⠘⠁⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⡿⠉⠈⠙⠶⣄⡀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⢼⣿⣷⣄⡀⠐⣿⣿⣷⣤⣾⣿⣿⣤⠀⠀⠀⠀⠀⢀⣤⣾⣿⡯⢄
⣿⣿⣿⣿⣿⡟⠀⠀⢀⠀⠀⡤⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣴⣿⣿⣿⣿⢤⡀⠀⡀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠀⣠⣿⣿⣿⣿⣿⣿⡋
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⡜⠃⣰⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠈⠛⣛⣿⣾⣿⣻⣿⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣙⠋⠉
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠛⠁
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⢿⠎⢻⣿⣿⣿⣿⠏⢿⣯⢉⡀
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣤⣀⣴⣿⣿⣿⣿⠁⠀⠸⣿⣦
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⡄
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠋⠈⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢻⣧
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠁⠀⢿⣦⣄⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖⠚⠉⠛⣷⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡒⠀⠈
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢈⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠉⠙⢷⣦⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠙⠉⣼⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⢸⡇⠁
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡍⢷⣤⡀⠀⠀⠉
⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⣽⡋⢰⡿⠁⠈⣿⡇⢦⡄
⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣾⠀⠈⠀⠀⠀⠃
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⡅
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠁
⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣍⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣾⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠁⠈⠉⠀⠀⠀⠈⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⢱⠀⠀⠔⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⢠⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣠⠔⣣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⢦⠀⣀⡄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡄⠀⠀⠀⠀⠀⢤⠐⠚⠉⠀⡀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢂⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠘⢶⠒⠒⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠨⠖⠋⠀⠀⠀⠀⣠⠟⠀⢀⡠⡄⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⣀⠀⠀⠀⠀⣀⣠⠴⡿⠥⠒⠒⢉⣀⠟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢍⠀⣠⠃⡆⠒⠉⠉
⠀⠀⠀⠀⢠⡄⠀⠒⡆⠀⠀⠀⠀⠀⠀⣀⣀⡠⡆⠀⠀⢉⣀⠴⠆⠀⠉⠀⠀⢳
⠀⠀⠀⠀⠀⠤⠀⠁⢁⣀⡠⠜⠒⠒⠒⠒⠒⠚⠳⠒⠊⠉⢀⡞⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡬⢌⣉⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⢤⣄⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠼⣄⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣯⠙⡆⠴⠖⠂⠤⠤⠥
⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠊⡠⠚⠉⠹⡦⠴⠖⠒⠒⠒⠤⢀⠉⠉⠛⠒⠲⠶⠶⣂⣀⠀⠀⠀⠀⡏⣸⠁
⠠⡀⡀⠀⠠⢄⡀⢀⠎⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣇⠀⠀⠀⢀⡏⢀⣀⠤⢚⡴⠃
⠀⠑⢌⠲⡀⠀⠈⠙⠦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣼⠢⠤⠒⠊⠁
⠀⠀⠀⠑⢌⡒⢄⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣭⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⠿⠿⣦⠀⠀⢀⣴⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⣸⣿⠀⢀⣾⣿⣣⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣶⣦⡀⢀⣤⣶⣶⣿⠟
⠀⠀⠀⣼⣿⠟⠁⣤⠀⠀⠀⣼⣿⠏⠀⣾⣿⡿⠋⣀⣀⠀⢀⣴⣶⣄⣼⡟⢉⣿⣾⡿⢋⡼⢣⣾⡟⢁⣾⠏⣠⣶⣄
⠀⠀⠀⣿⠃⠀⣰⣿⠀⢠⣾⡿⠁⠀⣸⣿⡏⢀⣾⣿⢻⣷⣿⠏⣠⠏⢻⣧⠞⣼⣿⠗⠋⢠⣿⡿⢠⣾⡟⣰⣿⣿⡟
⠀⠀⠀⠀⠀⣰⣿⣯⣴⡿⠛⣷⣦⡞⣿⣿⢀⣾⡿⣡⠟⢁⢻⣮⣵⠖⠀⣻⣿⢻⣏⢀⣴⡿⠻⣿⠏⣿⡇⢈⣿⠟
⠀⠀⠀⠀⣰⣿⣿⡿⠋⢀⣴⣿⣿⢇⣿⡇⣸⣿⠋⢁⣴⠏⣴⣿⣧⠀⣼⢻⣿⠄⠛⠟⠋⠀⠀⠀⠀⠙⢷⠿⠋
⠀⠀⠀⢠⣿⡿⠁⠀⣰⢟⣿⣿⡟⢸⣿⣱⢿⣿⣠⡿⠃⢸⣯⣿⡟⠀⠿⣾⠟⠀⣀⣠⣤⣶⠶⠶⠄
⠀⠀⠀⠸⣿⡇⢀⣾⢋⣾⣿⡟⠀⠈⠻⠋⠀⠉⠉⠀⠀⠈⠛⠛⢁⣀⣤⣶⣾⠿⠟⠋⠉
⠀⠀⠀⠀⠙⢡⣿⢁⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⢾⣷⣿⡿⠃⠀⠐⢧⣤⣴⣶⣾⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢼⠃⢠⠏⠀⠀⠀⠀⢀⣤⠤⠴⠶⠶⠦⠤⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠈⠉⠁⠀⣠⡶⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⣀⠀⣿⡀⣸⢁⣠⣴⠇⠀⠀⠀⣠⣤⢤⣸⡿⠀⠀⢀⣴⠞⠲⡄⠀⢀⣴⠞⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠈⠙⠛⠉⣽⡏⠀⠀⠀⣰⡟⠁⢠⣿⡇⠀⠀⣾⡏⠀⠸⠁⢀⣾⣏⣀⡴⠋⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢠⣿⠀⠀⢀⡼⢿⣇⣰⠋⣿⡇⢀⡼⣿⡇⠀⠀⢠⠞⣿⡏⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠒⠋⢸⣿⠀⠀⠀⠀⠘⣿⣄⡤⠊⠀⠈⠉⠁⠀⠹⠷⠞⠀⠘⠷⣤⠴⠋⠀⠙⠷⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⠙⠓⠶⢤⣾⣯⣀⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢸⡟⠳⡄⢀⡾⠻⡆⠐⣷⠀⢸⡄⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⠀⠀⢸⡇⠀⣿⢸⣇⣠⠇⢸⠃⠀⢿⠀⡟⣇⢸⡇⠀⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣾⣷⠀⠀⠀⠀⠈⠙⠻⠁⠀⠀⢸⡇⠀⡿⠸⡏⠁⠀⢸⡄⠀⣼⢰⡇⢹⣼⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⢻⣿⣿⠀⠀⠀⣤⡤⡄⠀⠀⠀⠀⠈⢧⣰⠇⢸⡇⠀⠀⠈⢷⣴⠏⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⢿⣿⠀⠀⢸⣯⡶⡇⠀⣀⡴⠀⠀⠀⠀⣀⣀⣠⣾⠂⠀⠀⣠⡤⢦⡀⠀⠀⣠⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣰⠇⢸⣿⠀⠀⣨⠟⠻⠓⢻⣿⠁⠀⠀⢀⣾⠋⠀⣿⡟⠀⠀⣸⡿⠁⠠⠇⠀⣸⡿⢀⣠⠟⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⠀⣿⣿⣠⡾⠋⠀⠀⠀⣼⡇⠀⠀⢠⢾⣿⢀⡞⣿⡇⠀⣰⢿⡇⠀⠀⠀⡴⣿⡏⠉⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠉⠀⠀⠀⠀⠀⢿⣇⣀⡴⠋⠈⠛⠋⠀⠸⢷⡴⠃⠘⣷⣄⡠⠞⠀⠹⢷⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢻⣿⡏⠠⡀⢄⢀⠠⡠⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣸⣿⠇⠀⠘⣬⣦⣵⣼⣬⣶⣦⣶⣤⣤⣤⣤⣤⣠⣤⣤⣤⣀⣀⣀⣀⣠⣤⣀⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⣿⡟⠀⢀⡴⣡⣋⢞⣴⡣⠋⠁⠉⠀⠄⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⠟⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣠⣿⡟⠀⠀⠈⠀⠁⠁⠈⠉⠀⠀⠀⠀⠀⡰⡻⣛⣻⡿⠛⠓⠛⠛⠛⠛⠛⠛⠓⠒⠓⠛⠒⢻⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⣼⡇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠰⣶⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣷⣀⣀⣻⣀⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠟⠻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⣻⣿⡇⢸⣿⣷⠈⣿⣿⡏⣿⣿⡿⠿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠻⣿⠃⢸⣿⣿⠀⠻⠟⠁⣿⣿⡧⠀⢸⣿⣿⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠛⠟⠁⠀⠸⣿⡿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⡀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢸⠃⠀⣧⠀⣿
⠀⠀⣀⣠⡤⠴⠶⣶⣶⣿⣿⣿⡿⠿⢿⣿⡿⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⠀⠀⣿⣧⠀⣾⣿⣿⣿⣷⡀⢹⣿⣿⣿⡏⠙⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢰⣿⣿⣇⠈⠛⢸⣿⣿⣷⠀⠙⢻⣿⣷⠀⢸⣿⣿⠀⠉⣹⣿⣷⣹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⣿⠏⠀⠀⠘⢿⡿⠃⠀⠀⢸⣿⣿⠃⢸⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠋⠀⠀⠻⠿⠟⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣿⠏
⠀⠀⠀⠀⠀⢀⠠⠤⠰⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠶⠖⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠶⠾⠿⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⣿⣷⣶⣶

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣿⣯⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠈⡀⣶⠠⠟⠀⢻⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣇⠀⠀⢸⣿⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠻⢌⠃⠀⠀⣷⡼⣷⣄⡀⠀⡦⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈⠱⢦⠀⠈⣷⠈⠀⠻⡀⠰⢀⡆⢰⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠸⣦⡀⠄⠀⠀⠀⠃⠀⠀⢃⡐⣻⣿⣤⣝⢁⡄⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⡀⠀⢀⠈⣷⣯⣿⡿⡍⠧⣉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡀⠀⠀⠀⠀⠀⠛⢦⣙⢿⣦⢫⡘⣿⡄⠙⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⡀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⠀⠀⠀⠰⠾⠯⣙⣶⠿⠋⠙⠿⠀⠀⠀⠀⠀⠀⠀⡰⢻⠟⢀⣀⠀⠀⢀⡀⢀⣾⡟⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣛⡛⠛⠉⠒⠂⢠⠆⣡⡴⠋⣠⠀⢰⡉⠂⠠⠤⠤⠤⠀⠠⠚⠁⣡⡴⠛⣉⡼⢓⣀⠄⢺⡿⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⠦⠄⠀⣸⣿⣿⠆⠛⠁⡤⠀⠻⢦⣄⣤⣄⡠⠶⠞⠛⠛⢉⣴⠟⣩⠞⠈⠁⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠶⠾⠟⠹⣿⠁⠀⣰⣿⠣⠀⡀⢚⣛⣋⣡⣤⣤⣄⠀⠀⠋⠀⢀⠰⠾⠿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠀⠙⠀⠀⢻⣿⣄⠀⣘⠷⢦⡄⠀⠈⠉⣉⣉⣳⢶⣤⣀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠙⡸⠇⠀⠙⡲⣤⠠⣴⠒⠾⡋⠙⠻⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣈⣹⢷⠻⣿⢢⣾⣷⣙⠳⠀⠻⢆⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣻⠀⢻⣿⢿⡴⡟⣸⡌⢻⣻⣄⡀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠶⠞⠀⠻⣆⢻⣮⠻⠃⡿⢠⢄⠻⢮⣭⡁⠘⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠖⠀⠘⠀⠀⠘⢆⡙⠳⡀⢳⣿⢸⣦⡀⠙⠛⠷⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣤⠈⢳⡄⢹⣾⣿⠈⢻⣿⡄⢠⠒⠝⠣⠀⠈⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣀⣴⣿⡿⠀⠀⠁⠀⠻⣿⣷⡜⢦⡙⠂⠉⠂⢴⡀⠀⠀⠀⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡁⠀⣠⢀⡆⠀⠀⠀⠙⡿⣿⡄⠙⣦⡀⢠⡄⠱⣄⠀⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣴⢶⣾⣷⠟⣡⣾⠃⣰⠀⢠⡀⠻⣿⣷⠀⠹⣿⡌⠳⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡍⠀⢸⣿⡏⢰⣿⠃⡼⠛⣇⠈⢷⣀⠙⣿⣷⡀⠙⢿⣦⡀⠀⠐⠒⢉⣡⣤⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢼⡇⢨⠁⠀⣿⣷⡀⢻⣷⡈⠻⣷⣄⡀⠙⠿⠶⠶⠾⠛⠛⠉⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⠀⠀⠀⠘⢿⣇⠈⠻⣿⣄⡈⠙⢿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⢳⡄⢀⠹⣷⣄⠈⠙⠛⠳⢶⣍⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣧⠈⠻⠿⠿⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣌⠒⢶⣤⣄⠉⠻⠿⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠂⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠈⠃





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠄⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠐⠦
⠀⠀⠀⣶⡤⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⣾⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣽⢷⡀
⠀⠀⢠⢽⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡟⠷⡄⠂⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⣸⡇⣸⡇
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠉⣠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⣽⣿⢿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠈⣿⣷⠀⠀⠀⠀⠀⠸⢿⣿⣷⣶⡀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣦
⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠾⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣾⡏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⣼⣿⣿⣿⢷⢁⣄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⣻⢿⣿⣿⣿⢸⣿⣯⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡈⢦⡀⠰⣆⣰⠏⢸⣿⣿⣿⣿⣿⡁⢷⢀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣌⠓⢄⣀⠳⡀⢸⡟⣿⣿⡀⣿⣿⣿⣿⣿⣿⡿⣼⣿⣇⣿⣀⣄⣸⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣌⡳⣶⣿⣤⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣦⡙⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣇⡶⢣⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⢈⡓⢤⣰⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⣻⠋⠁⣠⠆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⢞⡡⠶
⠀⠀⠀⠀⣄⠀⠈⠹⣦⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⣤⠄⠀⣴
⠀⠀⠀⠀⠈⠓⢤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡷⠤⠴⠃
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿