Pod Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠀⢄⠹⠄⠑⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠠⠀⠀⠋⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⠀⠀⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠧⡀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣤⣤⣤⣄⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⢿⡿⡿⣧⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁
⠀⠀⢦⡀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣲⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠴⣤⣀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠚⠒⠄⣀⡀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣀⣀⡀⠠⠏⠉⠓⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⡟⣿⣦⣰⣷⣶⣤⣄⣁⠢⠭⣒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠰⠁⢳⠏⢰⡿⡿⠋⠁⠙⠻⡧⠀⠀⠉⠢⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠈⠀⠸⠁⠃⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠣⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠲⠶⠶⡾⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠘⡛⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢐⢥⡰⡄⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠄⠀⠀⠀⠀⠀⠄⠠⠒⠁⣘⣄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⠀⠀⡆⠀⣠⣼⠀⠐⡭⠀⠏⡻⢷⡀⢀⢀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣜⣡⡾⢯⠸⡄⠀⢲⠀⠀⠃⠈⠙⠻⠿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠋⠙⡄⢳⣆⡀⢹⠠⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠙⣧⣼⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠀⠈⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠈⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢄⡀⠀⠲⠀⢰⠀⠈⠀⠁⠐⠀⠀⠈⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⠻⠿⣦⠒⠀⠈⠀⡀⢠⠀⠠⠀⠀⠀⠀⠠⠤⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠓⢓⠩⠋⠐⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⡠⠀⠉⠐⢀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣿⣮⣭⣭⣉⡁⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢂⣠⣤⣤⣤⣀⣐⣛⠻⠿⢿⣿⣿⣶⣾⡎⠀⠑⣄⡼⠚⠛⠻⡄⣤⣾⣿⣿⣿⠟⠓⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡴⠛⠙⠙⢫⠽⣛⠟⣛⣾⡿⢷⣶⣦⣬⣝⣻⠄⠠⠾⠿⠄⠀⣴⣶⣸⣤⠼⠋⠉⠁⠆⠀⡐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⡀⡤⢀⣀⡒⣶⡞⡯⣮⡙⡳⣯⣻⢯⣿⢟⣵⠇⠀⠀⣤⣄⢀⠉⢉⢜⡁⠦⢤⣀⣘⣀⠀⠥⠳⢄⡀
⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⣿⣷⣶⣟⣿⣦⣶⣿⣿⣮⣼⣿⣿⡿⢋⠐⠉⠉⠙⣦⠹⠿⠟⣿⢟⣵⣿⣿⣿⣷⢤⣍⡁⢀⠀⠀⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠉⠉⠛⠟⠿⡿⠟⣩⣾⡃⠁⠀⣠⣶⣅⢡⣴⠞⣣⣿⣯⡿⠈⠽⣳⢿⠟⣙⣳⣄⢀⡠⡦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠉⢀⠘⢻⡿⠋⡼⢋⣶⣿⣿⣿⡿⣷⠊⠃⢈⣺⣿⣦⠉⠲⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⣤⣀⡤⠀⠀⠙⢻⣃⣤⡀⢀⣠⣾⣦⣤⣤⠴⢊⠔⠛⠻⠋⠀⠀⠈⠉⣾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠪⣵⣶⠂⢀⡄⠈⢻⣄⡄⢀⣀⣴⢻⢿⣷⣾⠟⢋⠼⠋⠥⠂⠠⠰⠒⠒⠼⣷⠇⡂⠀⡈⠉⠉⠉⠋⢀⡠⠄⠀⠉⠀⠄
⠀⠀⠀⠀⠐⠁⠀⠈⠛⢆⠛⠼⣷⣪⣿⡯⣿⡿⠟⣸⣷⣶⡶⠞⠙⠀⠀⠀⠀⠀⠀⠀⢀⣵⠹⠻⠂⠈⠉⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠯⠉⠛⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⡋⢠⣴⣄⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠃⠀⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣰⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡴⠂⠀⢢⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⢿⠋⠲⣤⣤⡀⠀⡴⠁⠀⡀⠠⡾⠃⠀⠀⢼⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠁⠀⢸⠀⠀⠈⢧⠈⢺⡀⠀⠀⠠⣤⡗⠀⠀⠠⣜⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣿⠀⠀⠈⣧⠀⠀⠘⣧⠀⠹⡆⠀⠠⣼⠄⠀⠀⢀⡜⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⢹⠀⠀⠀⢹⡆⠀⠀⠹⣆⠀⠱⡄⣤⡧⠀⠀⠀⢀⡟⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⠸⡇⠀⠀⠀⢷⠀⠀⠀⢹⡆⠀⠘⡈⡀⠀⠀⠀⢙⠋⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢬⢸⡇⠀⣷⠀⠀⠀⢸⡄⠀⠀⠈⣿⠀⠀⢂⠀⠀⠀⠘⠿⠓⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡈⡇⠀⠘⡇⠀⠀⠀⣷⠀⠀⠀⢹⣆⠀⠈⡆⠀⠀⢴⠗⠀⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⡆⠀⢻⡀⠀⠀⠸⡆⠀⠀⠀⢻⡄⠀⠸⠀⢤⡴⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢳⡀⠈⣷⠀⠀⠀⢿⡄⠀⠀⠘⣇⠀⢸⢀⡰⠆⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⣧⠀⠘⣷⠀⠀⠘⣷⡄⠀⠀⣿⠀⠈⠨⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⣆⠀⠘⣧⡀⠀⠈⢷⡀⠀⢿⠀⠠⡄⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣆⠀⠘⢷⡄⠀⠈⢷⠀⠸⡆⢸⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣄⡀⠻⣆⠀⠈⢇⠀⣿⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣤⡈⢳⡄⠈⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣦⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⡀⢠⡞⢉⠉⠉⡝⠛⢶⣭⣻⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣶⣶⣿⠾⢆⡂⠉⢀⠁⠀⠀⠨⠻⣦⡤⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠁⢮⠃⠀⣀⣀⢉⣈⣃⠐⢦⣤⡀⠈⠙⢦⡤⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠘⡇⠀⠀⡎⣤⡾⠛⢻⠟⠙⣉⣙⡿⠿⠿⠷⡀⠀⠱⣮⡀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⡗⠘⣦⣸⡏⢀⣀⠼⢷⡏⠀⠀⠀⢀⣠⠴⠿⣦⡀⠈⠣⣎⠉⣛⣧⡀
⠀⠀⠀⠀⠀⠀⣾⠙⠆⠀⣻⠀⣿⣿⡷⠋⠀⠀⠀⢘⣆⡰⠊⠀⠀⠀⠀⠨⡿⣄⠀⠈⠳⢄⠈⠛⣦
⠀⠀⠀⠀⠀⠀⡿⠁⠠⠀⠸⡄⢸⡟⠀⠀⠀⡠⠔⠊⡙⠳⣤⣄⠀⠀⣀⠤⠔⠛⣷⡀⢀⠀⢱⡆⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⢀⢱⡌⢷⣄⡴⠊⠀⠀⠀⠀⠀⢈⣷⣷⡞⠁⠀⠀⢀⣩⣿⣄⠀⠁⠙⣆⡈⢳⡀
⠀⠀⠀⠀⠀⠀⢷⠃⠀⠢⠄⠀⠰⡄⢻⡇⢀⠀⢀⡠⠔⠒⠛⠛⠿⣄⣀⡠⠒⠉⠁⠀⠛⣆⠀⠀⠈⢷⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠸⡎⠂⠀⠀⠠⠄⡹⡄⠻⣏⡴⠁⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣸⣆⠀⠀⡈⣇⡀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⠀⠀⠀⠀⠘⣆⠹⣗⠀⠀⠀⢀⣠⠴⠒⢉⠛⠛⢶⣆⠔⠋⠉⠉⠻⡄⠀⠁⢰⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣀⠀⠀⠂⠀⠎⠀⠘⢇⣴⠞⠋⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⡠⠴⢿⡄⠀⠀⢳⠊⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡯⢀⠀⠂⠀⠀⠀⠑⡈⢷⡄⠀⠀⢀⡠⠔⠒⠂⠛⠚⢦⡴⠃⠀⠀⠀⢹⣆⠀⠀⣇⠘⢛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠃⠀⠁⠀⠀⠀⢄⠹⣦⠟⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣠⠔⠚⠻⣆⠀⠸⡖⢀⠰⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠃⠀⠀⠀⠀⠀⠉⠂⠘⢷⡇⠀⢀⡤⠔⠊⠉⠉⠉⠳⡾⠁⠀⠀⠀⢈⣧⠀⠹⣄⡀⠑⠫⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡑⠀⠀⠀⠀⠀⠀⢂⠀⢳⡖⠉⠁⠀⠂⠀⠀⢀⡠⣧⣀⠀⢀⡤⠊⣼⣧⠀⡈⢷⡠⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠓⠐⠦⠀⠋⡀⢢⡀⠹⣄⠀⢀⣀⠤⠚⠁⠀⠀⢹⡖⠁⠀⠀⠀⣻⣇⠀⠀⢷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⠀⠀⠀⠀⠀⠳⠀⠘⣿⠋⠁⠀⠀⠀⠀⠀⣸⣀⣀⢀⠔⠋⠡⠙⡄⠐⠘⣇⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⢁⠀⠀⠠⠀⠀⠈⢷⡀⢀⡀⠄⠊⠁⠀⠉⠉⡇⠀⠀⠀⢠⢿⡀⠀⠘⡇⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠐⠀⠀⠈⢀⠈⢻⡟⢀⠀⠀⠀⠀⣀⡴⠯⠀⣀⠔⠉⠠⡇⠈⠀⣷⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠁⠐⠀⠀⠀⠀⢣⠀⠹⣆⠠⠄⠒⠉⠀⠀⢀⠋⠀⠀⠀⠀⡗⠀⣒⣼⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡐⠀⠀⠀⠀⠘⠳⡀⠈⢳⣄⠀⠀⠀⠀⢸⠄⠐⠲⡴⠊⡇⠀⠁⣹⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⡀⠀⠀⠙⢄⠀⠙⠶⣶⠒⠊⠁⠀⠀⡔⠀⢠⣇⠀⠄⢹⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠄⠀⠂⠳⣄⠀⠈⠓⠤⣤⣀⣄⣀⣠⣼⣿⡀⢀⠘⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠁⠦⣀⠠⡀⠉⠛⣽⣧⡬⠿⢷⠨⠄⡸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣆⡀⠀⠈⠑⠤⣀⠄⠀⠙⠧⣤⡼⠇⠈⣇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢤⣀⠈⠂⣄⠀⠀⠉⢻⣆⢠⡥⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠳⠾⣧⣀⠄⠀⢿⣮⠛⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣥⣭⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⢀⡔⠀⠐⣦⣭⠍⣶⡶⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠤⠤⢤⡎⠀⠀⠀⣷⣄⠀⠰⠞⠚⢏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡠⠤⠀⣰⠀⠀⠀⢈⣿⡤⠀⣴⠞⢁⣴⣎⡀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠼⢀⡼⠀⠀⠀⠈⣳⣤⣤⠾⢋⣴⠏⢼⠁⠀⠀⣻⡿⡷⠞⠃
⠀⠀⠀⠀⠀⠀⢀⠀⢠⠏⠀⠀⢈⣳⣤⣤⡤⠾⣛⣋⣁⠀⢀⡠⠚⢷⡤⠤⠴⣿⠃
⠀⠀⠀⠀⢀⡔⠁⠈⢻⣷⠒⠒⠉⠉⠀⠀⠙⠉⠀⡡⠞⠋⠁⠀⠀⡌⠄⠀⠀⠀⡄
⠀⠀⠀⢠⢞⡫⠿⠚⠛⠉⠀⣠⣀⣀⠤⠤⣾⠀⠀⠀⢠⠀⠀⠀⠀⢿⣦⣤⣤⣬⠇
⠀⠀⡠⠛⠯⠶⠤⠴⠒⠒⠋⠉⠁⠀⠀⠀⣿⣤⣄⣴⣾⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢟⠟⣡⢾⢀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⡞⠉⠁⠀⠷⢽⡴⠇⡞⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣾⠀⠀⠀⠀⠀⣾⢠⠀⠙⣄⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠥⢊⣈⣤⡀⠀⣀⡴⢳⡇⡔⠉⠀⠑⠢⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣥⠊⠁⠀⠀⠈⠹⡰⢁⣎⢹⠀⠀⠀⠀⠀⠈⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢃⡃⠀⠀⠀⠀⠀⢀⠇⡎⢸⡈⡄⠀⠀⠀⠀⠀⡇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡡⠔⣉⣧⡀⠀⠀⠀⣠⢊⠜⠀⠀⡀⠘⡦⢄⣀⣠⡜⠴⣌⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡖⠉⠀⠀⠉⠲⠒⠘⠁⠎⠀⠀⠀⠡⡀⠘⡄⢠⠊⠀⠀⠈⠑⠠⣉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡼⠀⠀⠀⠀⠀⠀⣇⠆⡸⠀⠀⠀⠀⠀⠈⡆⠈⢮⡀⠀⠀⠀⠀⠀⢘⠣⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠔⠁⠘⣄⠀⠀⠀⢀⠔⢁⠔⠁⠀⠀⠀⠀⠀⠀⠉⠂⠤⡈⠐⢄⠀⠀⢀⠌⠀⢱⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠔⠋⠀⠈⠑⢶⠒⡏⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⡖⠲⠋⠁⠈⠙⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⠀⠀⠀⠀⠀⡨⠊⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠱⡇⠀⠀⠀⠀⠡⡘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣜⠀⠀⠀⠀⡜⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡈⠢⡀⠀⠀⠀⢘⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢞⡡⠊⠈⠢⢄⣠⠜⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠤⣤⢴⠋⠉⢃⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠰⢟⣁⡠⠔⠂⢉⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⡑⢄⠀⠈⢆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡑⠢⢴⣛⠢⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉




⣿⢿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⡋⠙⠒⠒⠶⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠊⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠠⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠒⡚⠉⠁⠉⠉⣀⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠄⣬⡀⠀⠀⠀⠀⣀⡤⠖⠙⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⠁⠀⠌⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡖⠛⠛⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠍⠤⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠉⡐⠄⡀⠀⠀⠊⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣷⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠐⡁⡀⠈⠢⡀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠳⣄⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠄⠀⠈⡤⠀⠀⡰⠁⠀⠀⠕⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣧⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠳⡄⠀⠀⠀⠀⠀⢠⡁⠄⢸⣧⠀⠀⠀⠉⢶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠢⠤⣀⡀⠀⢀⠀⠒⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⡅⠀⢀⢯⢳⡀⠀⠖⠁⠀⠈⠙⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⢳⡙⢄⠅⠀⢀⠀⠀⠀⠀⠉⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⢁⠜⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢌⠳⡠⣀⠂⠀⠀⢀⠄⠈⢁⠉⣵⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢳⡙⢮⡉⠒⠂⠉⠀⠰⠁⠀⠀⠀⠀⠙⢍⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢻⡄⠉⠓⢤⣀⠀⢰⠀⠀⠀⢀⠀⠀⠀⢡⠀⢉⡷⠶⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡻⣄⠀⠀⠈⠑⢦⡑⠢⣀⠀⠒⠀⣀⠎⢰⠁⠀⠀⠀⠀⠀⠘⢷⠒⠦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⠲⣄⠀⠀⠀⠀⠈⢧⠀⠀⠀⡀⠀⠀⠁⠈⠳⣄⠀⠀⠀⠈⠓⠦⣭⣉⣉⣁⠀⠘⢄⠀⠀⠀⡀⠀⢀⡾⠈⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠄⢸⠂⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠦⣥⣐⣒⣐⣶⣫⠵⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⢂⣠⠟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠂⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠂⢀⡀⠀⠀⠁⠨⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⡀⠀⠀⣈⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⠀⠐⠐⠈⠉⠉⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⣷⠘⠆⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠠⣧⠀⠀⠀⠀⠀⣠⣤⣤⣤⣹⠦⣀⣠⣥⠽⠶⠶⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠰⢩⡉⢲⠨⣧⠀⣻⠿⠛⠛⠋⠉⠉⠁⠱⣇⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠆⠀⠀⠈⠀⠻⣆⠨⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣽⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠠⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠸⡿⣯⡳⠤⠀⠀⠀⢀⡢⠄⠓⠚⠍⣎⢧⠀⠀⠒⠒⣂⣉⡹⠴⠥⠔⠒⣒⣈⣉⠭⠭⢋⠄⠀⡀
⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠠⣀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡭⠭⠥⠴⢒⠒⠒⠋⠛⠁⢄⠀⠀⠀⣀⠀⠀⢊⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣮⣙⠢⠀⠀⠀⠐⠀⢀⠄⢳⠀⠀⣞⡳⣄⠀⠀⠀⠀⠀⠑⡄⠐⠃⠀⠀⠀⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣦⡀⡤⠀⢀⠀⠄⠈⠀⠀⠫⣷⣬⡓⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠸⠂⠀⠈⠻⣿⡆⠀⣾⠥⢨⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠁⠀⠈⠓⠚⠃⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⠀⠀⠀⡠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⠈⠂⠀⠀⠂⠚⠰⠰⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠳⠬⢒⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠻⠶⣂⢀⠀⠀⠀⠀⠀⠀⣇⠈⢙⣿⠃⠉⣱⠀⠀⠀⠠⣦⠀⣔⡀⠒⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣈⣉⣰⣤⢆⠀⠀⠀⢀⡟⠛⠻⣿⠛⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠘⠻⠀⠀⠀⠸⣿⣷⣾⡟⠿⠻⠿⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠤⣤⣲⣤⣯⣽⣦⣠⣖⣀⡀⠀⠀⠀⠰⠟⠟⠋⠀⠀⠀⠢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢀⠁⠈⢋⣀
⠀⠀⠀⠀⠻⠟⡶⠶⠦⠤⣤⣤⡄⠬⢦⣵⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⢀⣄⠠⢀⠠⠐⠀⢀⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠚⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠐⠛⠲⢿⣤⣴⣶⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠂⠀⠀⠦⠄⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡄⠉⠉⠉⣔⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠬⠠⠔⢀⡂⢀⡀⠀⠀⢀⡀⠀⠈⣠⠍⠐⠢⣴⡺⠟⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣀⢠⡀⡂⡠⠀⠎⠁⢉⠀⡓⢂⣀⣐⠦⠶⠛⠀⠐⠻⠯⢔⣠⡄⣾⠔⠉⠀⠀⠀⠀⠀⠦⠤⡀
⠀⠘⢇⡀⠀⠑⢌⢡⠄⠢⣓⣠⡵⠶⠚⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢐⡠⠄⠀⠀⠈⢣⡼⣈⠑
⠀⠀⠈⠋⣨⣀⣠⣷⠶⠛⠋⠁⠀⠀⠀⠀⢰⣴⠁⡄⠁⠀⠀⠰⠒⠂⠬⠃⠀⠀⠀⠀⣤⣀⣀⡎⠀⢀⡒⣸⢸⠆⣬⢠⣸⠾⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠈⠀⠀⠀⠁⠋⠁

⡛⢛⡛⠛⠛⠳⠮⣭⣁⠚⠣⢤⡐⢣⡷⣿⣧⢄⣤⣼⣯⣀⡧⢸⡆⠀⣠⠾⢳⠀⠀⢀⣀⠀⠀⣀⡴⠋⢀⡤⢞
⠿⣿⣭⠗⢂⣀⠀⠀⠙⠻⠦⠄⣉⡁⠐⢧⠀⠀⠈⣿⠋⢹⡇⡼⠀⠀⠛⠀⣠⣴⣿⣿⡿⠿⣷⣿⣻⣾⣿⠟⠁
⣆⡟⢿⣿⣯⣭⡵⢦⠀⠀⠀⠀⢿⢸⣽⣏⠉⠒⢻⠟⣷⢟⣴⡵⠀⠀⣠⣾⡟⠋⠀⢀⣀⠀⠈⣿⣧⣿⣡⣀⣰
⣿⣧⡆⠸⢿⣷⣶⣛⣛⣦⠀⠀⠈⣇⣽⣟⠓⢺⢿⠖⣡⡾⠙⠀⢀⣼⣿⣶⣒⡾⠿⣟⣿⠿⣦⣿⢧⡟⠈⠉⠉
⣷⢿⣧⣆⣸⠉⡿⣿⣿⣤⠷⡄⠀⠹⣾⣟⣻⡟⣷⠋⠋⠀⠀⢠⡿⢿⢧⣏⣴⡷⢒⡛⠁⢀⣿⢫⠏⠀⠀⠀⡀
⣿⠿⣿⣿⢿⣶⡀⡟⢹⣿⣶⣻⡀⠀⠘⠛⣿⠿⠋⠀⠀⠀⠀⣿⠛⣧⡾⠥⠞⢹⣡⠟⣠⠞⣵⠟⢀⣤⠴⣉⢀
⣳⡴⠛⢯⡻⣿⣿⣧⣾⣸⠿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣄⡽⠆⣀⣤⢖⡵⠞⢁⣾⠋⠓⠶⠻⣿
⢙⣷⠖⠋⢳⣹⡿⣿⠿⠿⣶⣟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠰⣟⣿⣿⣦⣤⣀
⡉⣻⣤⠤⠾⢧⣿⡘⡇⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⣿⣿⣿⣯
⡹⣀⣷⡦⠔⠛⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣏
⣿⣿⣯⡷⡶⠚⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣼⠟⠀⠀⢉⣉
⠻⣾⡻⣷⣾⣟⣻⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣷⡦⢀⣤⡴⠊⠉⠉
⠀⠈⠛⠮⣽⣯⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣤⣤⣴⣤
⣤⣿⠷⠦⠤⣌⠙⠛⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣲⣿⣟⣻⡴⣟⣻⢿⣗⡤⠀⠀⠀⠀⢠⣎⣴⣿⠛⠋⠛⣻⣿
⣉⣀⣤⠖⣡⠏⠀⠀⠀⠀⢴⣾⣷⠄⣰⡇⣰⣟⡇⣹⡞⣷⢿⣯⡽⣻⣞⣻⣵⡄⠀⠀⠘⢧⣿⡿⣶⣶⡾⠟⠛
⣿⣿⣿⣿⠋⠀⠀⢠⣄⣠⣄⠀⠀⢰⣿⣄⢿⣩⠿⣩⡿⣿⣼⣋⣶⣧⣹⢺⡾⣏⣄⠀⠀⠀⠉⠻⣾⠏⠒⠒⣶
⠿⠋⡠⠋⠀⠀⣴⣤⡽⠛⠉⠀⠀⠘⣿⣿⣆⠙⢦⣿⡾⣧⢶⣏⣽⣏⣽⣏⣿⣻⣽⡗⢦⠀⠀⢠⣟⣉⣹⣿⣿
⠚⢩⡶⢴⣖⠂⡀⠀⣀⡄⠀⣀⠀⣾⣿⢿⣿⡷⠄⠙⠲⢿⣴⣷⡴⢾⡦⣽⣧⣽⣾⣿⠀⡇⢠⣋⣀⣼⣿⣿⣿
⢞⣿⠷⠾⣯⡽⢿⠳⡬⠛⠛⠁⠀⣾⣿⣷⣝⠻⠶⣄⠀⠀⠀⠈⠙⢿⣶⡿⠾⣿⠏⢁⣾⡇⣌⣭⣿⡿⢿⣼⡏
⣛⣿⣟⣋⣻⡶⢾⣷⣞⣡⠦⢀⠀⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⢘⣉⡽⢈⣩⠟⢻⣇⣸⣿⣶
⣹⣯⣉⣻⣧⣤⣾⣥⣬⣻⣟⣛⣡⣿⣿⣟⣉⣉⣛⣛⣻⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣀⣘⣓⣂⣉⣉⣉⣉⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠲⢏⡔⡠⢋⣝⠀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠠⡀⠀⠀⠀⠻⢧⣴⣬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠳⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⢮⣝⣒⣤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⢀⢀⡀⠱⣖⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠌⠘⢃⠵⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠬⠻⠧⢄
⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢛⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠈⠁⠤⠄⠤⠤⠶⠒⡛⠿⢿⠚⢒⣶⠾⠀⠀⠀⠀⠠⠤⡂⣄⢦⣂⠆⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢄⣤⣌⣀⡠⠀⠠⠐⣊⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⠔⢀⡐⣓⠭⠖⣋⠑⠤⠀⠀⠉⠠⠔⣲⣿⠆
⠀⠀⠀⠈⠙⠫⡂⢤⣲⡹⠬⢒⣩⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣅⣤⣦⣿⠁⢀⠄⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠠⢤⣍⣨⣤⣴⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣤⣤⣖⣾⣭⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣦⡀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣩⣭⣭⣭⣍⠛⠿⠿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣯⣽⣭⠜⣉⣤⣴⣶⣿⣿⣿⣿⠟⣋⣵⠾⠛⠛⠿⠿⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣯⡟⣡⡾⢋⣤⣾⣋⣿⡉⣳⢦⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣿⣿⢛⢉⡼⠋⣠⠞⢤⣼⣅⣀⣈⣿⠒⢳⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢋⣴⢃⡾⠁⡴⠓⠢⣴⡧⣀⣀⣈⣿⠛⠻⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢟⣵⡿⠇⣼⠃⣸⠷⠦⣴⣧⣀⣀⡀⢹⠟⠃⠲⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣱⣿⣿⡿⢰⡏⢀⡿⠤⢤⣼⣅⣀⠀⠉⡿⠓⠲⣴⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⣾⣿⣿⠟⠁⢸⡇⢸⣧⣄⣠⣏⠀⠀⠉⣿⠷⠶⢤⣼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⣿⠿⠋⠁⠀⠀⢸⡇⢸⣄⣀⣸⡋⠉⠁⠺⡿⠤⢤⣤⠏⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠸⣇⢸⣉⠈⢹⠋⠉⠓⡾⠧⣤⣠⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⠿⠛⠲⣶⣦⣄⣠⠋⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⡖⠛⣦⠴⣧⣀⠙⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠳⣤⣧⣤⠞⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠁⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡲⢦⣤⠴⣦⠤⠄⠠⠄⠐⠶⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣥⣍⡂⢈⣷⡤⠓⢂⣠⠄⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡪⣭⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢠⠏⣿⠀⠀⠉⠀⠀⠈⠣⡾⡑⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⡴⢀⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡟⠀⡴⠗⠐⠃⡀⢀⠐⡀⠀⠀⠀⠀⠀⠀⡴⠿⠛⠂⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠐⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡿⠁⢰⡃⣈⠀⠈⢀⠼⡆⠙⠈⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡀⢠⣄⠀⠀⢻⡷⠛
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠀⢰⡿⠁⠀⠘⡇⢠⡀⠀⣰⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠁⣀⣸⠦⣴⡆⠀⢘⠃⠈⠂
⠀⠀⠀⠀⠸⡄⣴⢀⣴⠏⢀⣶⠟⠁⠀⠀⠀⢧⢤⡤⠀⠺⠆⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡲⠗⢦⠃⠸⠛⠃⠀⠟⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠷⣷⣋⡤⠾⠛⠀⠀⠀⠀⠀⠀⢹⠊⢳⠀⠒⣲⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢆⡴⢁⠘⠁⠀⠀⠀⢈⠀⠀⠀⠀⠀⢁⠀⠀⠠
⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠹⣥⣏⠀⣽⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡀⠈⠀⠀⡀⢀⢩⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⡠⠀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠃⣀⢀⠀⠀⠉⠀⠀⠀⠈⠛⠺⡬⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠁⣤⡄⠤⣤⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠔⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⡧⢘⣀⡀
⠀⠀⠀⠀⠀⡈⠢⠀⠠⢀⣄⣀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⠀⣐⢁⡬⡄⢠⡀
⠀⠀⠀⠀⠀⣇⠀⢠⠟⢠⡤⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠀⡠⠀⡆
⠀⠀⠀⠀⠀⠹⣰⡟⣠⣟⣤⠞⠁⠀⠀⡀⠄⠈⠁⡄⠀⠄⠰⠀⠐⠀⠰⠄⢷⣾⡿⠦⣤⡀⢃⠀⠀⠀⠀⢰⠁⠶⢫⢤⡍⠘
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠊⠁⠁⠀⠀⠐⠀⠂⠀⠒⠈⠓⣈⣿⢟⠙⠂⢀⡄⢣⠈⣦⠀⠀⠀⠀⠒⠀⠈⠀⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠤⣠⡀⢀⡀⠀⠀⠀⠠⠤⠔⢒⣴⡿⠋⢑⠀⠀⢀⠋⠀⣸⠀⡻⠀⠈⠒⠄⠀⠀⠀⠀⠀⠈⢶⣄⢀⣴⠃
⠀⠀⠀⢠⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⠀⠴⠖⠔⠛⠟⢁⠀⠀⠀⠐⠂⠀⢠⡏⣴⠃⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠸⢺⡟⠁⠀⠀⠀⡄⠊
⠀⠀⠀⢿⠀⣷⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠀⢠⡼⠋⣴⡇⠀⠀⠀⠐⡃⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠈⣝⡄
⠀⠀⢠⡼⣄⣸⡷⠐⠛⠁⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⡤⢀⡥⢀⣸⠎⡡⠚⠛⠀⠀⠀⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⢈⣿⠋⡴⢂⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⣆⠀⠗⣉⣥⠊⠑⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠐⠊
⠀⠀⠈⠣⠠⡽⠀⠀⠂⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣉⣤⡶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠐⡄⠉⠠⠀⠠⠂⡀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠙⢉⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠐⢧⣈⣄⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠓⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠘⡀⠀⠀⠀⠃⠠⠀⠀⢠⡦⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠱⣀⠓⠾⠤⠀⠛⠁⣴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢩⠄⠀⠀⡀⠀⠃⠀⢾⠀⠀⡢⠌
⠀⠀⠀⠀⠀⠀⠙⣷⣤⠔⠻⡎⠉⠸⡆⠀⠀⢀⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢦⡭⡅⠀⠀⠀⠀⠢⠀⠰⡄⢬⠀⠀⠈⢖⡻⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣰⣀⠜⠃⠀⠀⠲⠦⣀⣤⠂⣠⠆⠀⠀⠀⠀⠀⠢⠴⠒⢂⣈⠀⠙⠒⠚⠀⠀⠀⠀⠄⢹⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠤⠤⠴⠂⠀⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠉⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⣀⣠⣶⢀⡴⣛⣥⠾⠋⠉⠉⢻⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⢻⣾⣿⣏⣡⢼⡟⠉⠳⡄⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡃⠀⠀⠘⠛⠻⢿⣿⡧⣟⠀⢰⠇⠀⠀⢀⡾⠀⠀⠀⠀⠀⢠⡶⠶⠶⢤⡴⠋⠙⠓⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡟⠀⠀⠀⢀⡴⠛⢉⢿⣰⠋⠀⡏⠀⡴⠞⠋⣤⣤⠀⠀⠀⠀⣸⠃⣴⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠀⠀⠀⠀⢾⡀⣴⠋⠦⡧⠀⠈⠧⠞⠀⠀⠀⠉⠉⠀⠀⠀⢿⣁⠐⡏⢳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠀⠀⠀⠀⣿⠃⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⢉⣙⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢀⡴⠒⢦⡀⠀⠀⢠⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠠⢶⡞⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢯⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠶⢦
⡞⠀⠀⠀⢹⡄⠀⣼⠀⢸⠃⠀⠀⠀⠀⠀⠀⣠⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠀⠀⣀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢠⡏
⡇⠀⠀⠀⠀⣇⠀⡇⠀⢸⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠎⣁⡈⢿⠷⣮⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⡇⠀⠀⠀⠀⢸⠀⡇⠀⠘⣿⠀⡴⠒⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠉⠁⠙⢦⣠⠶⠲⠖⣁⠀⢰⡛⣦⡴⢦⠀⣾⣿⠀⡇
⡇⠀⠀⠀⠀⢸⡀⡇⠀⠀⠸⣆⣧⣤⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⢇⠀⠉⢩⠇⠸⣿⣿⣿⠀⡇
⡇⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠈⠳⣼⡆⠙⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⣠⠞⠀⠀⣿⢹⡏⢸⠁
⡇⠀⠀⠀⠀⠀⡇⠘⡀⢠⠔⠢⣄⠈⠻⣄⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⣠⠖⢦⣄⣼⠁⢸
⡇⠀⠀⠀⠀⠀⢳⠀⢻⡏⠀⠀⠈⢣⠀⣸⠇⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢹⡟⠀⡾
⡇⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠈⡆⢧⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⢀⡞⣻⠀⠀⠀⠀⢸⠇⠀⡇
⢳⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⣡⠇⠀⠀⠀⠀⡿⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠹⣆⠙⢦⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⡜⣿⠏⠀⠀⠀⠀⣸⠃⠀⢰⠃
⢸⠀⠀⠀⠀⠀⠀⢧⡀⠀⢻⡀⠀⠀⠀⠀⠙⢦⡈⢇⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡴⢋⡼⠃⠀⠀⠀⠀⠀⡟⠀⢀⡼⠀⠀⠀⠀⠀⢀
⠈⡇⠀⠀⠀⠀⠀⠀⠉⠲⣜⣇⠀⠀⠀⠀⠀⠀⠙⢾⣆⢀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢋⡴⠋⠀⠀⠀⠀⠀⠀⣸⣣⠖⠉⠀⠀⠀⠀⠀⠀⢸
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣤⣤⣄⣀⣀⣠⣤⣤⠤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣿⠍⠉⠁⣿⠩⠛⠉⢸⣧⢀⠉⢠⡆⠀⠀⠀⠀⠀⠀⡷⣀⠀⠀⡇⣠⠄⠀⢸⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⡀⣿⠀⢀⣂⢸⣟⠋⡤⢸⡃⠀⠀⠀⠀⠀⢰⡟⡏⠓⠀⡇⡻⠎⠀⢸⠋⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⣿⡛⡛⠛⢹⡏⡉⠋⢹⠀⠀⠀⠀⠀⠀⢸⡏⢛⡛⠛⣿⠛⠛⠛⢱⠙⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⣿⠆⠿⠀⢹⠋⠀⠁⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⠐⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠿⠟⠷⠛⠿⠟⠻⠶⠛⠃⠀⠀⠉⠀⠀⠀⠀⢸⡷⠿⠒⠒⡔⠶⠶⠶⢦⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⣤⢸⡇⢀⡆⠀⡇⠏⠀⠀⢾⡽⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⡞⠀⠀⠀⠀⠀⢀⠄⢸⣧⡤⠤⠤⡣⢤⠬⠥⡸⠤⠬⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠁⠀⠀⣿⣿⣇⢸⣇⣷⡦⠀⡇⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠶⣀⣴⣿⣦⣤⣖⣤⣤⠶⢺⠁⠀⠀⠀⠀⠀⣧⣭⣥⣄⣮⣭⣥⣤⣣⣤⣤⣄⡽
⠀⠀⡄⠢⠤⠠⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠿⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⣷⠅⠀⢀⡇⠀⡀⠀⡇⠀⠀⠀⡃⠀⠀⠀⢀⣀⠄⠒
⠀⠀⣻⣦⣤⠴⠖⢻⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣃⣈⣀⣀⢷⣀⣀⣀⡇⠀⠀⠰⠿⣷⣄⢠⣴⡴⢶⢾⠄
⠀⠟⠁⠀⠀⠀⠀⠈⣿⣿⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇⠉⠉⠉⣷⠉⠈⠉⡇⠀⠀⠀⠀⠀⠉⠻⠝⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣍⣀⣀⣀⣀⣀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⣇⢀⣀⣈⣁⢀⠀⢀⣹⣀⣀⣀⡇⠀⠀⠀⠀⠀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⣤⣀⣠⣹⣧⡴⡶⠖⠒⠦
⠀⠀⠀⠀⠀⠀⢀⣔⣂⣷⣶⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⢮⣅⢀⣀⠠⠀⠀⠀⠀⠀⠻⣯⣉⣁⠀⠀⢈⢷⣚⣠⣤⣤⣴⡄
⠀⠀⣠⣴⣮⣤⣸⣿⠿⠻⠿⢭⣿⣿⣿⣶⣤⠆⠀⣠⣄⠀⢰⣞⠫⠀⢀⣙⠦⣔⡲⠚⠁⠛⢯⡀⣀⣤⠤⠀⠀⠀⢻⣳⣿⣿⠿⠭⠵⠞⠉⠛⠒⠀⠁
⠀⠄⠙⠻⣿⣿⣶⣿⣿⣷⣶⣾⣯⣅⡀⢈⡭⠟⣊⣭⣶⣢⠄⠙⢷⣯⡉⢀⣀⣈⡷⢦⣖⠾⠉⠈⠓⢮⣀⣠⣤⣄⠈⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣧⠀⣹⠿⣿⣿⣽⣧⣶⠊⠁⠠⠔⠒⠋⠀⠈⢻⣿⣾⣭⡻⠀⢀⣙⣷⢤⣶⣻⣷⣾⣿⣿⡄
⠀⠂⠀⠀⠀⠀⠀⠈⠛⠓⠂⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣧⣾⣿⣿⠿⠿⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀
⠀⢄⣀⣀⣀⠀⠀⢼⡄⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢳⡀
⠀⠈⠉⠳⣤⡉⢂⠀⠵⡀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠳⣖⠒⠶⠤⣄⠀⢳⡀⠸⡄
⠀⠀⠀⠀⠀⠉⡧⠤⣶⣗⡤⠘⠢⠂⣂⣀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠈⠀⠀⣧⡀⠈⡄⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⡠⢈⠁⠀⠈⠨⢿⣿⡧⠑⠳⢀⣠⡤⠖⠚⠍⠁⠀⠀⠀⠀⠀⠀⠀⠠⢄⣸⠿⠓⢉⠙⠓⢦⡸⡏⠀⠀⢀⣄⡠⡤⠴⠛⠓⠉
⠀⠀⣼⡔⠁⠀⠔⠀⠀⢸⣿⠰⣢⠀⠄⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠖⠉⢀⠀⠘⣌⡟⠢⣀⠘⠸⠀⠉⠀⡀
⠀⢰⣿⣷⠆⠀⢀⠠⠂⡉⢙⢷⡩⡾⣀⡑⢆⡀⢀⠀⠀⠀⠀⠀⠀⠀⠾⢧⡤⠞⠁⢀⡠⠼⣎⠣⡈⠛⣤⠀⠈⠁⡜
⠀⣼⣿⣿⣞⡋⠁⣨⠀⢁⡜⢾⣷⣜⠟⣵⡆⠀⠄⠁⠀⠀⠀⠀⠀⢤⡀⢉⣶⣂⠉⢀⢀⣤⠿⣧⣉⠲⣤⡉⢔⠀⠘
⠀⠀⠀⠀⠀⠀⠠⠐⣠⡈⠓⣄⣍⠻⢿⣶⣼⣥⣤⣠⠀⠀⠀⠀⠀⠞⠃⠉⠛⠑⠒⠂⠁⠀⠀⠈⠈⠮⣶⣬⣈⡓⠦⠉
⠀⠀⠀⠀⠀⠀⠐⠚⢈⣍⣱⡿⠿⠀⠀⠈⠉⠉⠙⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠀⠒⠒⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠈⣿⡟⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⣀⣠⡤⠔⣛⠛⠤⣌⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡰⠀⢀⡠⠒⠈⠁⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠈⠲⢤⡴⣫⢷⡿⠛⠛⡻⠿⣎⢣⡀⠙⢿⣶⣤⣤⣀⣤⣴⣭⣤⡴⠷⠾⠓⠈⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠐⣒⠚⡿⢫⡾⢓⡻⠃⠀⢙⠗⠄⠘⣷⡙⣄⣠⣿⠦⢌⣀⠀⠉⠢⡀⠀⠁⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠋⢤⡉⠀⣈⡟⠃⢀⣹⠳⣤⢻⠀⠈⠆⠀⢋⠀⢤⣄⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⣸⣗⡐⡬⠀⣀⣽⠛⠂⣀⣯⠱⡿⢸⠀⠀⠸⠀⠈⡆⠀⢃⢀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢤⡹⠄⠀⣤⣏⠁⠠⡜⠢⣽⠇⣾⠀⠀⠀⠀⠀⢹⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟⢦⡔⠀⢠⣮⡀⠐⣶⣉⣻⠏⣰⠙⠀⠀⠀⢰⠀⠈⡀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣟⠲⡉⠀⠶⠄⢀⡽⠦⣾⠏⣰⠃⠀⠃⠀⠀⠸⠀⠀⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣳⠄⠸⠓⢠⠞⣒⠞⣡⠾⡀⠠⠀⠈⡀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⡽⢀⡋⣠⢯⠝⣣⠞⠁⠀⠑⡄⠁⠀⢣⠀⠀⡇⠀⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣷⣬⡴⢛⡴⠚⠁⠀⠀⠀⠀⠈⢆⠀⠀⢆⠀⠀⠀⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠄⠘⢠⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⠟⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣟⣛⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠂⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠀⠈⠈⠀⠀⠀⢀⣀⠀⠢⠈⠀⠀⠀⠠⠄⠉⠑⠀⠁⠈⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠠⠊⡐⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠂⠐⠳⡄⠀⡀⠘⡀
⠀⠀⠀⠀⠠⠁⠌⠠⠂⠀⠀⠀⠊⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠱⠀⠀⢙⣈⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹⣃⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠁⠀⠀⡤⠂⠀⠀⠀⠀⠈⠀⠀⢀⣤⠦⣬⠀⠀⠀⢸⡆⠐⡈⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠁⠀⠀⠀⢠⠣⠀⠀⠈⠂⠂⠀⠁⠀⠀⠈⠑⠈⡑⢫⠋⠩⠂⠈⠙⣗⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠁⠈⠈⠙⠉⡊⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠉⢪
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡠⡀⣀⠂⠀⡁⠀⠀⠁⡀⠀⠁⠉⠡⡞
⠀⠀⠀⠀⡂⠀⠀⠀⠈⠀⠀⠂⡀⠂⠀⠀⠐⠀⠀⠀⠀⠎⠁⠉⠈⠉⠉⠠⠀⠀⠈⠑⡦⠂⠀⢠⡁
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠐⢠⠐⠈⠳⠤⠜⠱⢃⢑⣦⡌⢀⠐⠀⠱⠰⠄⠖⠄⠤⡤⠄⠀⠀⠘⣠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢀⠀⠀⠉⢏⠀⠈⡛⠋⠋⢻⣀⠀⠠⠂⠐⢂⠀⠐⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⡜⠃⠠⠔⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⢀⠀⢀⢀⠄⠀⠀⠠⠄⡀⢡⡄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⡄⠀⠂⠀⠂⠀⡄⠆⡬⢡⢀⣶⠄⠀⠀⠀⠀⠈⠀⠀⣀⠙⠦⡀⠸⠘⠀⠠⡀⠂⡦⡤⢀⠀⢠⡀
⠀⠀⠀⠀⡇⠀⠀⠳⠖⢂⡤⠤⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣀⣁⣀⡀⠠⠤⣤⡰⠶⠾⢊⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⡄⠈⡧
⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⡧⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⢁⠌⡀⠀⠂⠀⡀⠁⠀⠀⠄⠀⠀⠀⡠⠊⠐⡁⠐⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⢄⣀⡀⠀⠀⠀⠀⠄⠡⠊⠔⠁⠈⠀⠡⠂⠈⠀⢀⣀⡄⡤⡶⢚⠁⢂⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠲⠒⠐⠒⠒⠒⠃⠙⠩⠀⠉⠀⠈⠀⠁

⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡀⠀⣀⠀⠀⠀⣠⢄⡄⢠⣤⣄⠀⠀⢤⣤⠀⢀⡀⠀⠀⣤⡄⢠⠤⢤⣤⣄⠀⣀⣤⡤⡤⢤⣄⡀⠀⠀⠀⠠
⠟⠀⠀⣀⡤⠤⠤⣤⣄⣀⣀⠀⠉⠉⢩⣄⣙⡷⠋⡀⢠⡏⡞⣹⣟⠀⠈⢳⣦⣞⠉⡾⢸⠁⠘⠛⠉⣰⣋⣹⣿⢿⠉⣀⣿⣯⣿⠀⣿⣹⣿⣷⣤⣀
⣠⣾⣏⠀⠀⠀⠀⠀⠀⣙⣿⣟⠛⣇⣄⠉⠉⠀⠀⠅⣰⣹⡿⣇⠈⠲⣶⣾⠋⢉⡿⢡⠞⠀⠀⠀⣰⢥⣼⣿⠛⠀⣦⣿⣿⣾⣿⡹⣿⣿⣻⣿⠉⠻⣷
⣾⠿⠿⢿⣿⣿⣶⣿⣿⣿⣿⠿⣿⠿⣿⣀⣀⣠⠀⠀⣿⡽⢧⣤⡿⣿⠥⠾⣛⣭⣒⠁⢠⣠⡿⢠⠷⣾⢿⠉⢦⣼⣿⣿⡿⠦⣼⣱⣹⢿⣇⠈⠳⡀⠈
⣿⣤⣤⣴⣾⣿⡛⢻⣭⣤⠽⠷⣟⠓⠾⡻⢿⡿⣄⣀⠘⣿⣿⣯⠟⠲⠶⠛⠑⡋⣁⣀⡈⣉⡀⢸⣻⡟⣦⣷⣿⣿⡟⡼⠇⠤⣾⣉⡇⣼⣿⣦⡀⢹⡆
⢷⡿⠫⠤⣤⣈⣿⠿⠛⠓⠢⣄⣈⣻⣶⠿⢾⣻⢏⡏⠀⠈⠉⢁⣠⠶⠟⠛⣡⣴⠾⠋⢉⠅⣹⢨⠟⣷⠿⠛⣿⢻⣇⣷⣄⣠⣿⣈⢷⣧⣿⢿⣧⠀⢳
⡞⢛⢒⣶⣾⣿⠟⠒⠲⣤⡴⠾⢿⡁⠈⢷⣀⡟⣸⣷⠀⠀⡴⠋⣁⣤⠶⠛⢋⣡⣤⣾⣿⣾⠅⠘⠊⠁⠀⠀⣡⣸⣿⠧⠤⣼⣇⡼⣾⡾⣿⣄⢻⣄
⣀⣽⣿⣿⣿⣧⣤⣤⠞⠛⠳⡀⠀⢳⣄⣨⡟⢠⢏⡟⢀⡾⣁⡾⠋⠀⠀⢠⡾⠟⢛⣿⠏⡿⠀⢀⡀⢠⡀⠀⢸⣿⣿⠦⢤⡾⣼⡿⣿⠁⠈⠪⣵⣿⣶
⣿⠿⣯⣿⢣⣄⣴⣛⣳⣤⣤⣿⣤⡾⠟⢋⣴⣧⠞⠀⡿⣿⡿⠀⠀⣠⡴⠋⠀⣠⡿⣡⠔⠁⣤⣌⣽⠟⠋⠀⠘⣿⣿⡶⣾⢶⡟⣽⠏⢀⣤⠤⠀⠉⠓
⣏⣀⣿⣧⣬⢭⣭⣭⣭⣭⣭⣥⠤⢒⡾⣿⡟⠁⠀⣤⠞⢡⣴⡾⣛⢁⣀⡤⠞⢉⠞⠁⠀⠀⠀⠁⠀⠰⠽⠿⠂⠹⣿⣟⣵⡯⠞⠁⠀⠻⠇
⠉⠉⠉⠁⠀⠈⠙⠿⣭⣙⣛⣛⣯⠥⠞⠉⠀⠀⠀⠷⠶⣥⢶⠿⠿⠿⠭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⢸⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⡓⠤⣀⠀⠀⠉⠀⠀⢀⠴⠒⠋⠁⠀⠀⠀⠀⠤⠐⣒⣩⠔⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠢⢬⣐⠀⠀⠀⢸⢀⣀⡠⠄⠀⠀⣠⠖⠖⠩⠋⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣀⣈⠀⠐⠀⠁⢠⡄⢈⣉⡥⠴⠚⢋⡟⠁⢠⡤⣀⣀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⠀⠀⠁⢸⠁⠀⠀⣰⣎⡠⠖⠛⠛⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⢧⠈⠫⡛⠲⣄⣸⡤⠶⠊⠉⠀⠀⠀⠀⠀⠀⠤⠤⠿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡀⠈⢧⢸⠗⢦⠀⠸⠥⠀⣀⣀⡀⣀⣀⣤⢤⠴⠒⠒⢩⠏⠁⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠈⠻⡀⠀⢱⢲⠒⠉⠁⠙⠉⠀⠀⣷⠋⣠⠊⠀⠃⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠎⣿⠊⠉⡗⢀⠑⢄⡈⢸⠀⠀⠀⠀⢀⣀⡴⠥⠾⠁⠐⠒⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠗⡄⠀⠠⠁⠀⠘⣤⠀⢓⠌⠓⠚⠒⠒⠉⠉⢉⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠃⢀⡀⠀⠀⢳⡜⠀⠀⠀⠀⠀⠀⠤⣴⣶⢿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠇⡀⠀⠀⢀⠀⣠
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⢉⣉⣤⠴⣲⣿⡄⠀⠀⠀⠀⠀⠰⠲⣇⣯⠿⣟⡾⢁⡆⠀⠀⠀⠀⠠⢔⡇⠂⡀⢠⠃⣉
⠀⠀⠀⠀⠠⠂⠀⡀⠔⠛⠁⣠⣴⢚⣵⠟⠀⠀⠀⢠⠊⣅⢲⣻⡭⠶⣛⢁⣴⡞⠀⠀⠀⠀⠘⠘⣿⣇⡜⢿⠬⡽⠇
⠀⠀⠀⠀⠀⣐⣉⣠⡤⠶⣟⣽⣾⠟⠁⠀⠀⢀⣠⣈⣉⣉⣭⣥⣶⣿⡿⠟⠋⠀⠀⠀⠀⠐⠾⣺⣿⡏⠛⢳⣶⠻
⠀⠀⠀⠈⠙⠛⠛⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⣴⣿⡿⠙⢓⣶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠿⠟⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⡾⠀⢀⠉⡨⠆⡋⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⣀⡒⠐⣦⣀⡀⣄⣀⣈⣉⣽⣇⡖⢿⡧⣻⢿⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠂⠈⣀⣤⠤⠴⡶⠴⠶⠒⡻⠿⢿⡓⢛⣶⡶⠿⠁
⠀⠀⣄⣁⣒⣉⣁⣀⠄⢠⠤⢖⢺⣽⢿⣓⠟⢈⣷⠋⠉⠁⠀⠀⢠⡞⢩⡷
⠀⠀⠀⠉⠛⠿⢟⠥⣂⠠⡟⣎⣳⠼⣛⢥⣴⡟⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢠⣤⣤⠀⣠⠤⠄⣤⣤⡄⢠⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠙⣓⣋⣥⣔⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⢹⢪⣿⠀⢸⡇⠈⡇⣿⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢀⡦⢻⣀⡾⠘⣿⠀⡸⣇⣰⠇⢻⣤⢾⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠤⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠠⣈⡠⠂⠀⠀⠀⠀⠀⠀⢀⠤⠤⠂⠀⠀⣀⣀⠀⡠⢀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢀⠠⠃⠀⠀⠀⠀⠀⣀⠤⠔⠈⠐⠁⠀⠀⠀⠀⠊⠀⠈⡀⠀⠅⣂⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⢁⠖⠁⠀⢀⠄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⣈⣗⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠃⠀⠀⢠⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠈⣥⣂⠡⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠠⠂⠀⠀⠀⠄⠔⢀⡆⠀⠀⠈⢏⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢲⠂⠀⠀⠀⠀⠀⢈⠀⢀⣠⣴⢢⢤⠁⢊⡂⣈⠇⠈⡭⣅⠮⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠜⠢⡈⡀⢀⡐⢁⣀⠀⠀⠀⠀⢉⠐⠉⠩⠉⠁⠒⠈⠉⣳⠭⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠈⠀⠁⠩⠍⡉⣁⡂⡠⠀⠀⢀⡀⠀⠀⠂⠁⠀⠀⠀⣈⠻⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣌⡤⣨⣌⠀⣀⡀⠀⠀⠁⡀⠀⠒⠉⠁⢾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⢄⠐⢂⠁⠠⠊⠀⠐⠀⠀⠌⠊⠁⠈⠉⠉⠄⢄⠀⡄⠑⢦⠐⠐⠠⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡕⠊⢛⠦⡴⠋⠛⣘⣹⣭⡁⢨⠂⠀⠖⡲⢀⠖⠔⠤⣤⠄⠠⠊⠚⢀⢗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠍⠊⠀⠈⠳⣅⠀⢉⠚⠉⠩⣒⣀⠁⠨⠊⠐⢂⡈⠀⠊⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⢐⠛⠂⠈⠓⠐⠀⠀⠁⠀⠀⠐⠂⡀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠐⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠒⣻⣿⣶⠃⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⣣⠀⠀⠀⠀⡋⠀⠀⠀⠐⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠆⠀⠀⠄⣄⣲⢠⢟⡀⢣⣄⠐⠁⠀⠀⠈⢒⠀⠁⠈⠻⣍⠙⢻⠻⠛⢁⠃⡆⣶⢡⡉⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠤⣀⡤⠀⠀⣈⣈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠀⠃⠃⣂⣁⡀⠉⢠⣄⣡⡤⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠚⠛⠋⠉⠉⠁⠀⠨⠭⠭⠍⠉⠉⠉⠐⠒⠒⠂⠈⠉⠉⡁⠀⢠⢁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⢆⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⡐⢁⠈⡐⡠⢂⠈⡐⠀⠠⠁⢀⠀⠊⡡⢋⡠⢊⠄⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠠⠐⠀⠈⠀⡠⢊⠔⡠⠊⠐⠡⢂⠔⡠⠁⠌⠐⠐⢀⣁⡤⠖⡈⠐⡁⢀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠒⠦⠦⠤⣤⣬⣄⠁⣈⡀⢈⣤⢤⣤⠄⡠⠶⠂⠜⠛⠉⠊⠔⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣾⣿⠁⠀⠀⠀⢰⣷⡄⠀⠀⠐⠦⢄⠴⠖⠀⠀⣀⣶⣷⡆⡀⠀⢀⡀⠘⠿⠀⠀⠠⠚⠁⠈⠁⡆
⠀⠀⠘⠸⠟⠀⠀⠀⡀⣈⢛⡁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠻⡿⠣⠀⠀⠸⠋⠠⣴⡤⠀⠀⠈⠉⢩⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢀⣴⠙⢁⣦⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⡞⢶⡆⠀⠀⣀⣀⢀⢀⡀⠀⠀⠀⣤⡅⠂⠀⢠⣤⣤
⠀⠘⡩⡄⠘⣏⠃⠀⣻⣿⣻⣿⡇⠀⠈⣿⠀⢀⡟⠀⠰⣏⣿⣸⣮⡇⠀⠈⠈⣿⡃⠌⠀⠀⣾⣿⠟⠻⠆
⠀⠀⠈⠑⠀⠁⠀⠀⠛⠀⠛⠛⠃⠀⠀⠛⠉⠛⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠀⠃
⠀⠀⠀⡐⢂⡀⠀⠀⠀⢀⡗⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡴⢰⣀⠀⠀⠀⡤⠀⢄⡀⠀⠀⢀
⠀⠀⢀⡁⣉⠀⠀⠀⠀⣾⡉⠛⠀⠀⢀⣴⣧⣀⠡⠀⠀⢹⢐⡍⠔⠂⠀⠘⠌⣮⢚⠃⠀⢠⣾⡂⢐⣿⠂
⠀⠀⠈⠻⠟⠀⠀⠀⠰⠃⠃⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠻⠛⠛⠉
⠀⠀⡀⠀⠀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀
⠀⠘⠿⣀⣘⠿⠃⠀⢰⣿⠟⠈⠀⠀⠀⣤⣿⣿⡇⠀⠀⣔⡀⠰⣄⠀⠀⣶⠒⢲⡄⠀⠀⠀⣿⢰⡆⣤
⠀⠀⠀⠧⠟⠀⠀⠀⠸⠏⠀⠁⠃⠀⠀⠸⠦⠤⠃⠀⠀⠻⠛⠛⠿⠃⠀⠻⠗⠚⠛⠿⠂⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣄⣤⣠
⠀⠀⢠⢖⡢⡄⠀⠀⠀⢸⡆⢹⠀⠀⠈⠃⠸⠀⠛⠀⠀⠒⣿⠷⠄⠀⠀⠀⢠⡾⠅⠀⠀⠀⠈⣿⢁⣀
⠀⠀⠣⣘⣃⠎⠀⠀⠀⢼⡧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⣶⠂⠀⠀⠀⠙⠀⠀⠀⠀⢰⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡴⢣⡐⢦⠀⠀⣴⣧⢀⣿⡆⠀⠀⢰⣶⡇⠀⠀⠀⡔⢉⣥⠛⡄⠀⠀⣠⣼⣶⡄⠀⢀⣾⡏⢹⣷⡄
⠀⠀⢣⡉⢃⠞⠀⠀⢣⣨⣶⣀⠇⠀⠀⠲⠌⠶⠃⠀⠀⢧⡘⢛⣠⠃⠀⠀⠟⠁⡼⠃⠀⠈⢿⣿⣺⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢸⡄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠠⠤⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉⠁⠀⠐⠄⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣠⣴⠀⠀⠀⠀⢀⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⠓⠚⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠠⠤⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠤⠀⢀
⠀⠀⠀⠀⠀⢄⣀⡀⠈⠁⠀⠀⠀⡀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢀⡀⠀⠀⠁⠒⠁⣄⡤⠄⡀⠈
⠀⠀⠀⠀⠀⠀⢆⢀⠠⣄⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠈⠋⠉⡁⠀⠀⣀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠠⠀⠂⠀⠤⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣶⣶⣶⣶⠶⠾⠿⠛⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣜⣿⣎⢿⣿⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⡴⣡⣴⣾⣧⡀⠀⠀⠀⠀⠀⠀⢿⣿⣭⣍⣙⡻⢿⣿⣿⣿⣿⣎⣿⣿⡄
⠀⠀⠀⠀⣠⣾⡿⢿⣛⣯⡷⣫⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡷⠬⠛⣻⣿⣿⣯⣿⠃
⠀⠀⠀⠰⣻⣵⣶⣿⡿⢃⣾⣿⣷⢟⣾⡿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠬⠖⢚⣋⣩⡭⣭⣭⣍⢳⣤⡴⠆
⠀⠀⠀⠘⠋⠉⠉⠉⢁⣿⣿⡾⢣⣾⣿⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⡴⢋⣤⡶⣿⠳⡛⢏⢹⡉⣏⣯⡧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢣⣿⣿⣣⣿⣿⠇⠀⠀⠀⠀⠀⡤⠞⠁⢿⣌⣧⢸⡖⣗⣻⣉⠯⠗⢊⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣼⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣦⣤⣭⣭⣤⣤⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣯⣭⣭⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⡆⢸⣏⡝⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠘⡇⡀⠀⠀⠀⠀⣠⠟⠀⣼⠃⡈⣯⢂⠉⠉⠱⠒⢂⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠖⠒⠊⠉⠁⠀⠸⠁⠀⡇⢹⡆⡑⠀⠀⠀⠈⠀⠉⠑⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠚⣿⣛⡻⠟⠋⠀⠙⠀⣠⠀⠀⠀⡆⢸⠇⠳⣤⠤⢀⠈⠀⠈⠑⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⣀⠀⠁⠀⢠⡏⣶⠀⠀⡇⢸⠀⠠⡟⢀⠀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠠⣾⠷⠋⠀⠀⣠⡜⠁⠀⠀⣿⣿⣷⠿⠀⠸⠀⣿⡄⠀⠁⠘⢦⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣿⡴⠃⠀⠀⢀⡴⠋⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠘⣿⣇⠀⠀⠀⠈⢳⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣎⡁⢠⡟⠁⠀⠀⣠⠊⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⢄⣄⡀⠻⣿⡄⠀⠀⠄⡀⢢⡀⠀⠰
⠀⠀⠀⠀⠈⠛⠶⠾⣿⡾⠿⣷⡟⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠟⢁⢈⡈⠁⠀⢐⣋⣡⠰⢿⣷⣄⢠⠀⡇⣾⢓⠦⢭⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠀⠀⠀⣶⠃⠀⠀⠀⢤⣼⠋⠠⠀⣨⠄⠀⡀⣺⠤⣼⠀⢿⣿⣿⣦⣇⣰⣄⣹⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡰⠁⠀⠀⢀⢰⣾⠃⠀⠀⠀⠴⠀⠤⢤⡟⠲⠖⠀⢸⢹⣟⢻⣿⣿⣝⡛⠻⠛⠛⢓⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣠⠁⠀⡄⠠⢄⣾⡏⠀⠀⠄⣴⠂⠀⠀⣼⠛⠓⡏⠀⠈⣼⣏⢀⠿⡄⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡉⠐⠤⠀⣠⣾⣿⠃⠀⠂⠠⠈⠀⠐⢤⣌⠉⢻⠃⠀⠀⢿⣿⣇⢰⣷⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠄⣠⣾⣿⣿⣿⠀⠀⠐⠲⠉⠀⠒⢾⡅⠉⡋⠀⠀⠀⡞⠈⠙⢾⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⠀⢉⠐⢊⠈⠀⠲⠅⣀⡸⠁⠀⡴⡼⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠋⠁⠀⡄⢨⢨⠄⠀⢀⡾⠦⢤⠃⠀⣰⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⣴⠂⠢⣼⣃⡒⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠃⠶⠗⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣶⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣉⢯⡇⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢹⣯⠻⣤⠚⠁⣠⡤⢌⣙⣦⣄⡀⠀⢀⠚⠁⠀⠀⠀⠀⠈⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢠⠋⡼⠶⠅⢠⡾⠿⣀⡂⠁⣸⠿⢿⣧⣿⣦⣠⣀⠆⠀⠀⢰⠃⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣠⡏⢠⠀⣠⡶⠋⠀⣸⠟⢁⣿⢁⣤⠏⢁⣸⠟⢻⣿⣶⣤⠾⠇⠠⢄⣰⠖⠀⠈⠳⢤⡀
⠀⠀⠀⠀⢀⠞⠁⠐⠈⠁⡇⠀⠹⣦⡿⢅⣴⠇⠀⢠⠜⠉⣩⣧⣤⠎⢁⡴⠚⠉⣹⣏⢦⠀⢂⠀⠈⣫⠓⠲⢶⣤⣄⣽⣥⣖⡫⠇
⠀⠀⠀⠀⡜⠀⠀⠀⣠⣼⠃⠀⠀⠈⣀⡼⢟⣤⡞⠀⣠⠞⠁⢀⡽⠷⢏⠀⣠⠄⠛⢻⡷⢇⠀⠳⠀⠈⡄⠀⠘⠆⠀⢨⡶⠋
⠀⠀⠀⢰⡇⠠⠆⠈⠁⢂⠀⠑⠸⣇⢻⣀⠜⠉⢱⢾⡇⠀⣰⠋⠀⢠⡿⠚⡷⢀⡤⠾⣷⡸⣧⣈⣀⠀⣰⣀⣀⡼⠒⠁
⠀⠀⠀⠸⠃⠀⠀⠀⠘⢹⠀⠀⠀⠹⡎⢿⣀⠞⠀⢀⡿⢻⣃⣠⣶⠋⠀⢠⡿⢿⣤⣤⣾⢆⡇⠈⡻⣦⣄⣡⠸⣷
⠀⠀⠀⠀⡆⠀⠀⢰⠒⠙⡷⠰⠆⣀⡻⣎⡻⣤⣴⠉⢀⠜⠛⢹⣧⡶⠋⡀⣠⣤⣯⣀⣼⢸⡇⡔⠁⠀⢳⡄⠀⢻
⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⢣⠀⠀⠈⠀⢹⡷⡈⡻⢶⣏⣀⡰⠋⣹⣦⡾⠋⢀⣤⣽⣛⣏⣿⣀⠆⠀⢠⡞⢧⠀⢸⡄
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⠇⠀⣠⡀⢨⡇⠈⠙⠢⠬⣛⠳⠶⣧⣰⠿⣾⠉⣠⡤⣿⣷⠋⠉⠳⢤⡜⠁⠀⠱⢆⡛⡄
⠀⠀⠀⠀⠀⠀⢳⡀⢀⠖⠘⡆⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠉⠒⠨⣽⣶⣿⣏⣁⣠⣿⡟⠀⠀⠀⠀⠉⠁⠀⠘⠀⠩⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⣼⠃⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢍⡛⠻⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⠙⣣⡞⠁⠀⠀⠀⡀⠐⠂⠀⢈⡷⠀⠀⠀⠀⠀⠀⠹⣄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⡿⠀⠀⠀⠀⠀⠳⣤⡶⠞⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣀⣀⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⠀⣉⠛⠶⠀⠲⠦⣤⡤⠶⢀⣴⠂⠐⣉⡤⣀⠀⢀⣤⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠙⠛⠖⠲⠶⠦⣄⣠⡄⢉⣉⣁⣌⣹⡘⣥⣠⡤⠦⠶⠚⠓⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⠖⠒⠀⠤⠔⣶⣮⠟⠚⠉⠉⠉⠀⠀⡟⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⡠⠌⠉⣴⠃⢰⠟⡳⠶⣿⠧⡤⢤⡀⠀⠀⠀⢠⠇⢀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⣀⠞⠀⠀⠀⠉⢶⡇⡌⠀⠀⣸⠀⢿⠀⢷⠀⠀⠀⡼⠀⠈⣷⠀⠈⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠀⠈⣻⠀⠀⠀⠀⢀⣾⠇⣿⠀⠘⣿⠀⣿⠀⣾⠀⠀⢀⡃⢀⡜⠀⠻⡦⣀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⢀⡿⣄⡀⠀⢀⣾⠋⠀⠟⠀⠀⢹⠃⢿⡇⠻⡄⠀⠈⣧⠋⠀⠀⠀⣿⣄⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡠⠞⢻⠀⠉⢰⡾⠁⠀⣀⠀⠀⠀⠈⠀⠘⡇⠀⠙⢄⠰⣧⠀⠀⢀⠜⠁⢹⣆⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⣦⠀⠺⠆⠀⢰⠁⠀⠀⠀⡼⠀⠀⠂⠀⠀⠈⢆⠙⡇⠠⠎⠀⠀⢀⠉⠡⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠓⣿⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡆⡇⠀⠀⠀⠀⠌⠀⠀⡶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⡠⢻⠀⢻⡀⠀⠈⡀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⢀⡇⠱⡀⠀⠀⠀⠀⠀⠀⣰⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⢀⣄⠘⣇⠀⠀⠁⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⢸⠇⠀⡇⠀⠀⠀⠀⠀⢰⠛⣧⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣼⠏⢳⣿⢆⠀⠀⢣⠀⠀⠀⠄⠀⣶⠀⠀⢀⡟⠀⠀⠈⠲⣤⠀⢀⣠⠈⠀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣫⢀⡾⠃⠈⠳⣄⠀⢠⠀⠈⠃⡠⠏⠀⢀⡞⠁⠀⠀⠀⠀⠙⠦⣸⠁⠰⠟⢿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠁⠀⠀⠀⠀⠈⢳⣈⡇⠀⢀⡵⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠡⢏⡐⡠⢃⣜⡀⢄
⠀⠀⢰⣶⠤⢄⣀⠒⠀⠀⡀⠀⠀⠀⠻⢦⣴⡬⠉⠓⠎⠴⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠈⢳⣝⢢⣄⡀⠓⠂⠀⠀⠀⠀⠀⠀⠉⠛⠓⠷⠾⠶⠴⠖⠒⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠻⣮⣝⣒⣤⢤⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⢀⢀⡈⠱⣆⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠍⠘⢃⠥⢹⣿⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠆⠀⠀⠀⠀⠈⠋⠳⠂⢤⠁⠤⠹⠯⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠘⢒⣿⢁⠰⣖⢙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠠⠶⠒⠻⠿⢟⠚⢂⣶⠾⠀⠀⠀⠀⣄⠐⣒⣖⢛⣛⠛⣓⣀⣀⣀⣀⣀⣉
⠀⠀⢤⣠⣌⣀⡠⠀⠠⠐⣋⠭⠳⡇⠋⣸⠆⠈⠀⠀⠀⠀⢔⢀⡐⣛⠭⠖⣋⠁⠬⠀⠀⠉⠠⠔⣺⢿⠄
⠀⠀⠀⠈⠙⠫⡂⢥⣲⡩⠬⢒⣉⣠⡾⠃⠀⠀⠀⠀⠠⠤⣬⣨⣥⣤⣦⣿⠁⢀⠀⠈⠁⣠⡴⣪⡵⠋
⠀⠀⠀⠀⠄⢤⣍⣨⣤⣠⣦⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣐⣾⣭⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠶⠶⣶⡾⠿⠿⣿⡶⣶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⠂⠐⠤⡉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣄⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣿⣷⣶⣿⢡⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠸⡟⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠉⠉⢱⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠾⠀⠄⠾⠸⠲

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣳⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠙⠟⡇⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠐⡂⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡷⣄⣿⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣌⡃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⠀⠀⠀⢀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⡀⢀⠄⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡉⢀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠇⠧⠸⠜⠧⠸⠄⠿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠓⠀⠛⠛⠀⠒⠛⠚⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠶⡄⣴⠄⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠒⠖⣡⣾⢟⠕⢮⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⠦⠴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠾⢿⠗⢡⢀⢠⠘⢮⠻⠷⠀⠀⠀⠀⠀⢀⡴⠚⢻⣿⡑⠤⡀⢀⢃⣽⡞⠛⢶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣏⠈⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡠⠄⠚⠀⠀⡀⠀⠀⠀⠀⢸⡅⠀⠋⢀⡗⠀⠈⠛⢺⡗⠛⠂⠀⣿⠀⠀⠀⠀⠀⣿⣿⠏⢹⣿⡶⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠁⠶⢀⠀⣀⢈⠀⢀⣤⢄⠲⠈⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁⠀⠀⠀⠀⠛⠶⠶⠞⠁⠀⠀⠀⠀⠀⣿⣿⣦⣤⣿⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⡒⠐⠂⠰⠶⠰⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠄⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⡒⠉⠉⠉⠀⠒⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠤⣸⡀⠀⠀⠀⠀⣀⠤⠖⠙⠁⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠨⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠖⠚⠛⢧⡀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠒⠀⠠⢀⡀⠰⠀⠤⠎⢀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡐⠄⡀⠀⠀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣧⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠇⠀⠈⠢⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠳⣄⠀⠀⠀⠀⠈⠑⠦⣤⣀⠀⠀⠀⠀⠰⢁⠀⠠⢀⢀⠁⠁⠁⠄⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣧⠀⢈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡄⠀⠀⠀⠀⠀⢀⠃⠀⢹⣧⠀⠀⡀⡉⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣀⡀⠀⠀⠀⠂⢂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⢯⢳⡀⠀⠄⠁⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⢣⡙⣄⠃⠀⢀⠀⠀⠀⠀⡍⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢎⢻⡪⡄⠢⠀⠀⢀⠇⠀⢁⡩⣑⡶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡙⢮⡊⠀⠀⠉⠀⡐⠁⠀⠀⠀⠀⠙⢍⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠲⢤⣀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠃⠀⢉⠷⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⡹⣄⠀⠀⠈⠑⢦⣑⠠⣄⡀⠈⠀⣠⠏⡰⠁⠀⠀⠀⠀⠀⠉⢛⠲⠦⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠓⠲⣄⠀⠀⠀⠀⠈⢧⠀⠀⠀⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠓⠦⣅⣉⣉⡀⠀⠘⢄⠀⠀⠀⠱⠘⢄⡾⠈⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠐⠀⢸⠆⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠦⣥⣀⣠⣴⣴⣫⡵⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠊⣠⠟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠤⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠂⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠀⠀⠀⠀⡀⠠⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⠀⣈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⣷⣶⣶⣶⣶⡷⢿⣾⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢷⣾⡷⠾⠿⣷⣶⣷⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⠀⣀⣠⣤⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠶⢶⣾⣭⣿⢿⣿⠛⠉⠁⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⡤⣤⣤⣤⣴⡾⠛⣩⡥⠀⠸⠏⠀⢀⣾⡿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣀⣴⠿⠋⣥⠀⢠⣿⣵⣾⣿⣿⣶⣿⡏⠀⢹⣿
⠀⠀⠀⠀⢀⣼⡿⠛⠛⣻⣿⠟⣁⠀⣸⠏⣠⣾⡿⠟⣩⣾⣿⣿⣿⣿⣤⡼⣿⣤⣤⣀
⠀⠀⠀⢠⣿⠋⠀⣠⣾⠋⠀⣼⠇⠀⣿⣾⠟⣡⣴⣿⣿⣿⣿⣿⣿⡟⢁⣀⣀⠙⢿⣿⠷⡄
⠀⠀⠀⣿⣧⣴⣿⢿⡇⠀⠀⡟⢀⣾⡟⢁⣾⣿⡟⣩⣾⡟⣹⣿⡟⣠⣿⣛⣻⣷⠈⣿⣶⡙⣆
⠀⠀⢠⡿⢋⡞⠁⣾⠀⠀⠀⢣⣿⡏⢠⣿⣿⠋⣴⣿⠟⣰⣿⡟⣰⣻⡯⢬⣽⠾⡇⢸⡇⣷⢸⡆
⠀⠀⣿⠁⠀⠀⠀⣉⣀⣤⡀⣾⡿⢀⣿⣿⠁⣼⣿⠏⣰⣿⣿⢠⣟⡿⠤⠤⣾⣛⣿⠸⡇⢻⡄⣷
⠀⠘⣧⠶⠚⠛⠛⠛⠋⠉⠀⣿⡇⢸⣿⠇⣼⣿⠃⣰⣿⣿⡇⣼⢻⡧⠶⢶⣿⣭⣽⠀⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣿⢸⣿⢃⣼⣿⣿⣿⠁⣿⣻⠴⠶⢶⡿⢤⣿⠀⠀⣾⢇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⢣⣾⣿⣿⣿⣿⠀⣟⣳⠶⠶⢾⣟⣻⠇⣸⢀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣷⣿⡿⢛⣽⡿⣿⡆⢿⣽⠖⠒⣿⢤⠏⢰⡏⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣾⠟⠋⠀⠸⣷⠘⣿⣏⣙⣿⠏⣰⣟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠙⣷⡈⠻⠞⣡⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⣰⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣶⡀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣤⡻⣇⠀⢀⡤⠴⠦⣤⡀⠀⠀⠀⠀⢀⡠⠴⠶⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢰⢷⣙⣿⠁⣠⡶⠒⠶⠽⣲⣤⡀⣰⡉⠀⠀⠀⠀⠀⢀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠰⠃⠛⠉⢀⡼⠛⣳⣶⢀⡴⠋⢹⠷⠾⣷⣖⣶⡁⠀⣠⠏⠀⠀⢀⡈⠳⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠤⠶⠓⢼⡀⣾⠏⢀⡄⠊⠁⡼⠧⣾⠁⣀⠖⢁⣸⠿⢿⣷⡉⠠⡈⠒⢯⢤⣀⣀⠙⠂⢤⣄⡠⢤⡀
⠀⠀⠀⠀⡼⠀⠀⠀⢀⡗⠀⠀⠛⠋⢹⡟⢀⡤⠊⢀⠴⠛⢹⣇⣤⠊⠀⡀⠾⣷⣹⡄⠰⡄⠘⠃⠀⠉⡟⠓⠚⣛⠟⠊
⠀⠀⠀⢸⠁⣀⠀⠚⢋⠁⠠⠄⡆⢶⠁⠸⠟⣧⣴⠋⠀⡴⠎⠀⣸⠶⣿⠀⢀⣼⣇⢳⡀⠃⠀⠀⡄⠀⢀⣤⠞⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠐⡆⠀⠀⢻⡜⣷⢀⡔⠉⢹⡶⡶⠁⣠⡼⠃⢀⣵⣾⢁⣈⣿⢨⠃⠻⣧⡘⠛⡛⣯
⠀⠀⠀⢰⠀⠀⠀⡤⠐⢧⠤⠄⠀⢳⣜⢿⣄⡤⠋⢀⠿⠛⣯⣤⠞⠈⢁⣨⣟⢉⡝⣸⠀⠸⠈⠹⣍⠁⢸
⠀⠀⠀⠈⣇⠠⠀⠀⠀⢸⡀⠀⠉⠁⢻⣧⠙⠷⣴⡁⢀⠞⢛⣧⣴⠟⠉⣉⣽⢛⣇⡟⠠⠀⠀⣤⢿⡀⠸⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢘⡇⠀⢀⡀⢨⠁⠈⠐⠠⢝⡛⠶⣾⣀⠿⣶⠋⢠⣤⣿⣽⠉⠙⠦⣄⠋⠀⠑⢦⡛⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠴⠀⣦⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⠙⠢⠽⣶⢾⣇⣁⣠⣿⡏⠀⠀⠀⠈⠉⠀⠘⠂⠩⣌⡆
⠀⠀⠀⠀⠀⠀⠈⢧⠀⣰⡋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡙⠻⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠁⣣⡞⠁⠀⠀⢀⠀⠀⠀⣀⣽⠃⠀⠀⠀⠀⠀⠙⢦⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⠀⠀⠀⠀⠀⠳⠶⠛⣩⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣄⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⢦⣄⡀⢀⣀⠀⠀⢀⠀⢀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠒⠀⠒⢤⣥⡀⣈⣙⠋⠉⠀⠻⠦⠆⠚⠏⢶⠀⠈⢉⣀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⣽⣾⡯⠶⠷⠚⠛⠙⡾⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠒⡐⠚⣡⠀⣠⡦⣥⣼⣯⣁⣀⠀⠀⠀⠀⢰⡆⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡴⠃⠀⠀⠛⣤⡇⠐⠀⠈⡇⢸⡄⢳⠀⠀⠀⡾⠀⢰⡄⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡄⠀⠺⠀⠀⠀⠀⢠⣾⠁⡇⠀⢾⡃⢸⡀⣼⠀⠀⢸⠃⢀⠜⠁⢦⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⣾⣄⡀⠀⢀⡿⠃⠠⠇⠀⠸⡇⢸⡇⠸⡄⠀⠸⡇⠋⠀⠀⢸⣦⡀⠁⠷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⣠⠔⢻⠀⠉⢰⡿⠀⠀⡀⠀⠀⠀⠃⠀⠇⠀⠙⢄⠸⣇⠀⠀⣠⠊⠈⣧⣀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⣧⠀⣻⠀⠀⠘⠀⠀⠀⢠⠇⠀⠐⠀⠀⠈⢂⠙⡆⠰⠁⠀⠀⡈⠡⠉⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣽⠉⣛⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠀⠀⠀⠘⢤⡆⠀⠀⠀⠀⠁⠀⠳⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡴⠹⠀⢹⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢠⡆⢱⡄⠀⠀⠀⠀⠀⣸⡇⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⣦⣀⣧⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡸⠁⠈⠇⠀⠀⠀⠀⢀⠇⣵⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠁⢸⡟⢢⡄⠀⠃⠀⠀⡌⠀⣸⠀⠀⢰⠇⠀⠀⠀⠹⣆⠀⣴⠃⣀⣼⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⠴⠋⠀⠀⠙⢦⡀⢦⠀⠀⠰⠁⠀⣰⠃⠀⠀⠀⠀⠀⠈⠓⠇⣀⠋⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠹⣞⡀⠀⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⡀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠂⢀⢀⡀⠲⢤⣄⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⢀⣤⣞⡿⠷⠝⣻⡧⣸⣷⣷⠦⢤⣄⠀⠀⠀⠀⠀⣴⣿⠿⠿⢳⡌⠛⢏⡨⢧⠈⣿⢿⣷⡄⠀⢻⡟⠿⣿⣶⣄⠀⢾⣷⡌⠉⠉⠋
⠀⠀⠛⠿⣿⣶⣦⣞⣵⠟⣹⢻⡿⡝⣿⣖⡶⢙⢦⡀⠀⣾⣷⣿⡿⢿⢨⠴⠘⠦⣤⡼⠁⣿⣻⠟⠀⠀⠈⠻⣿⣮⣻⣿⣆⠀⠘⠟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢻⡜⣿⣖⠉⠳⡱⡄⠀⠙⠻⠿⣿⣧⡍⠢⣁⣈⣡⣾⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣹⣿⠀⣀⣀⠀⠀⠀⢸⠃
⠀⠀⠘⢿⣷⠀⢹⣿⡆⠀⣿⢸⣷⠸⣷⠸⣿⣶⡤⠹⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣠⣤⣤⣤⣄⣀⠀⠈⠛⠛⠁⠀⠻⣽⣿⣦⣠⡏
⠀⠀⠀⠀⠀⣶⡆⠉⠁⠀⢿⢸⣿⣇⢿⣧⣿⣇⠚⢤⠇⣷⠀⢠⣶⠀⠀⠀⠛⠳⠶⢞⠍⠭⡭⠭⢙⡛⠿⣿⣷⣤⡀⠀⠀⠀⠀⠴⠛⠉⢹⠁
⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠸⣇⢿⡻⣿⣿⣿⣿⠃⠶⠂⣿⠀⠿⠃⠀⠀⣠⣿⣄⠀⣯⢀⠊⠄⠠⠀⡀⠋⢰⢝⡻⣿⣆⠀⠀⠀⠀⣠⢖⡻⣆
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣜⣿⣿⠋⠉⠉⠉⠛⠀⠀⠀⢀⣾⣿⢻⣿⣷⡈⠪⡳⣄⠀⠉⣷⠴⣀⠈⡐⠙⣿⠀⠀⢀⡾⣵⣿⣧⢻⡄
⠀⠹⣆⠀⠀⠀⣀⣀⣤⣤⡤⠤⢤⣄⣈⠉⠛⠀⠰⣶⠶⠀⠀⠀⢀⣿⢹⡏⣸⣽⣿⣇⠀⠘⠺⡿⠶⢦⣀⣇⣯⣴⣶⡼⠇⣠⢟⣾⣷⡟⢾⣿⡧
⠀⠀⢻⡟⠛⢻⣿⡟⣩⢲⢒⡖⡶⡦⣌⣙⠶⣄⠀⠀⠀⠀⠀⠀⢸⡇⣼⠇⣿⡇⣿⣿⠀⠀⠀⠀⠈⠒⠳⠿⠒⠛⠉⠀⢠⣟⣾⣝⣧⣽⣾⣿⠇
⠀⠀⠘⠇⠀⠀⢻⡄⣿⠻⢺⠟⢟⢷⣵⣽⣳⣌⠳⣄⠀⠀⠀⠀⣿⡇⣿⢀⣿⡇⣿⡿⠀⠀⣴⢲⣦⠀⣿⣶⣤⣄⠀⠀⢸⣿⣻⠿⣙⣼⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣾⡿⡾⣾⣬⣌⣙⣳⣷⡙⣦⠀⠀⠀⢻⡇⣿⢸⣿⣸⣿⠇⠀⢸⣷⡘⣿⠀⢻⣯⣿⠻⣧⠀⣿⣗⣿⢿⡻⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣽⣳⣷⠿⠷⠿⠿⠾⠿⢈⣧⠀⠀⠈⢿⣿⣸⣿⣿⡏⠀⠀⠈⢿⣧⣙⣧⠀⠛⠿⠷⠿⠀⢻⣙⣚⡯⠞⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠛⠛⠉⠁⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣀⣀⣀⡀⣀⣀⣠⣄⣲⠟⠛⠛⢿⡟⠉⢈⣹⠆⠀⠀⠀⢀⣤⣤⣴⠖⡄
⠀⠀⠀⢸⠿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢾⣿⡋⠉⠉⢹⡟⠉⢀⢻⣧⠴⠶⠶⠛⠉⠈⠉⠀⠀⠀⠀⠀⠈⠓⠶⠖⠂
⠀⠀⠀⣯⠀⢘⡇⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠈⠛⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⢢⠀⠀⠀⠀⠀⠀⡄⠀⠀⠦⡀
⠀⠀⢀⡯⠀⢸⡇⠀⠀⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠋⣠⠎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⣆⡀
⠀⠀⠈⡧⠀⠠⡇⠀⢀⡟⠈⢷⠀⠀⢠⡞⢋⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⣼⠛⢷
⠀⠀⢸⡇⠀⠀⡁⠀⠀⣳⠤⢸⡄⠀⠘⡇⢸⠅⠀⠀⠀⠀⠀⠀⡄⠠⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢹⠀⠀⠀⣏⢀⣠⠃
⠀⠀⢘⡿⠀⠶⡇⠀⠀⢁⡀⢀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⢤⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣆⠀⠀⢻⡄⡌⠀⠀⠀⡏⠀⠉⠃
⠀⠀⢸⡇⠀⢀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢿⡄⠀⠆⠀⢠⠀⠀⠀⢻⠂⢲⡄
⠀⠀⠘⡧⠀⢺⠀⠀⠀⠈⣿⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢐⡇⢻⠀⠀⠀⢸⣀⣼⠃
⠀⠀⠘⡇⠀⠘⡀⠀⠀⠀⢹⣄⢸⡇⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣤⣤⡤⠤⠠⣄⡀⠀⠀⠀⠀⠰⡇⠈⠀⠀⠀⣾⠀⢘⡇
⠀⠀⠀⠐⠂⠀⢃⠀⠀⠀⠈⣗⠀⢷⠀⠀⠀⠉⠒⢿⣿⣉⣉⠉⢹⡟⠋⣉⡛⣿⡀⠤⠤⠴⣟⣃⣤⠶⠞⠃⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣽⠂⣿
⠀⠀⠀⠀⠑⣤⢲⠀⠀⠀⠀⢹⠁⠹⡄⠀⣀⣀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠒⠂⠀⠒⠈⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡇⡰⠀⠀⠀⣿⢀⣾
⠀⠀⠀⠀⠀⠈⢾⠃⠀⠀⠀⠘⡀⠀⡅⠀⠹⣏⠉⠙⠶⣦⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀⡄⠀⠀⠸⠞⠃⠀⠀⠀⣿⣠⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡⡀⢸⠀⠀⠙⢿⣷⣄⣼⡅⣀⡈⠙⡿⠒⠒⠲⢶⣦⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾
⠀⠀⠀⠀⡤⢤⡤⠀⠀⠀⠀⠀⠀⠙⣮⡆⠀⠀⠀⠈⠛⠿⠿⠿⣿⣾⣷⣤⣦⡤⢴⣧⣀⣀⣈⠙⣷⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣴⣋⠞⠁
⠀⠀⠀⠀⠓⠤⠤⠚⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠛⠛⠛⠛⠛⠉⠁⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣋⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⢤⡼⣅⡀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠹⡱⡀⠀⠀⢠⣤⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠤⠄⠀⠀⠀⡴⠁⠱⡀⠀⡾⠈⠃⠀⠀⠀⢱⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⡠⠀⣀⠀⢞⠀⠀⠀⠱⣀⠇⠀⠈⠀⠀⠀⠀⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠢⢶⣷⣠⡄⠁⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠬⢿⠀⣀⠈⡀⠉⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢉⡖⠀⠀⠀⠀⠄⠈⡆⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠣⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠦⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠉⣰⠔⠀⠀⠀⠀⠑⡄⠀⠀⠀⠲⠏⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⢎⣠⡤⠤⢠⣤⡄⢀⠐⠀⢀⠀⠀⢀⡜⠹⣇⣉⣀⣁⣀⣀⣈⣂⣤⠴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢮⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡀⠀⠀⡀⠀⠄⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡤⠀⠀⠀⠀⠀⠀⣢⠄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠀⠉⠲⠀⠐⠚⠀⠀⠈⠀⣄⠀⠀⠀⠙⢄⠀⣿⣀⣀⡀⣤⠤⠖⣋⣵⢊⢿⣶⠀⠀⢀⠄⡑⢉⣄⣰⣈⠱
⣘⠀⠐⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠀⠈⠢⢀⠀⠀⠀⠳⡇⠀⠀⠀⢨⠄⢚⣿⣿⡷⣔⣊⢉⠒⠀⠚⡋⢟⣿⣿⡥⠄
⠙⠐⠂⠪⠛⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠁⠀⠀⠈⠉⠂⠂⠈⠀⠀⠀⠀⠀⠓⠥⣨⡀⠤⠋⠀⠈⠄⠀⠀⠙⢬⣌⣁⣩⠞
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣴⢀⡤⣀⠀⠀⡀⠀⢠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⢶⣦⣤⠞⠁⠀⠀⠀⠀⡀⠀⠄⠀⣡⠔⠉⠁⠀⠀⠈⢀⢠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣊⠅⠄⡀⢀⣿⡇⠀⠀⢀⣤⡖⡁⢀⠀⠄⢸⡍⠀⠀⠀⠀⢀⡠⣣⠎⠀⠀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⢠⢃⠊⠀⠀⠈⢳⢸⡇⠀⢀⢐⠑⠋⠀⠀⠀⣶⡄⠚⠻⢋⣤⠶⠞⣩⠞⠁⠀⣠⠖⠉⠀⠀⠈⠂
⠀⠀⠀⠀⡰⢡⡏⠀⠀⠀⠀⣿⡟⠀⢠⣆⠁⠀⠀⠀⡀⢀⣸⠃⣀⣀⡿⠶⠟⠋⠁⠀⠀⢸⡇⡀⠀⠀⠀⢠⠄
⠀⠀⢀⢜⠁⠘⡂⢀⣄⡦⢻⠟⠀⠀⠃⢺⢦⣄⣈⡶⣖⣫⡥⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠙⣫⠶⠞⠋⠁
⠀⢀⠀⡀⣤⣀⣀⢍⣹⡾⠋⠀⠀⠀⣪⣠⣬⣭⠧⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠐⢢
⠀⣈⡜⠁⠀⠘⠛⠊⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠊⠁⠀⠈⠉⠁⢘⣷⣄⣀⣀⡀⠀⠠⠂⠀⠀⠀⠀⢚
⢰⣿⠁⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⢀⡠⠀⠘⠁⠂⠀⠀⠀⠀⠀⢀⣠⠾⠋⠉⠉⠉⠁⢰⡆⢠⢀⢄⣐⡠⠖
⢘⠹⣢⢀⢀⡰⢿⡿⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠐⠊⠀⠒⢰⡤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠈⠫⠿⠖⠛⠁
⠸⡆⠀⡈⠉⣆⣼⠃⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣣⠖⠈⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄
⡌⠐⠁⠀⠋⣯⣿⡄⠀⠠⠋⠀⠀⠀⣀⠀⢀⠼⠛⠁⠀⠀⠀⣠⡶⠛⠁⠀⠀⢉⢭⢻⡶⠒⠛⠉⠀⡴⠋
⠇⠁⠀⠀⢀⣼⣸⡇⢰⠀⠀⠀⠀⠀⡵⠋⠀⠀⠀⢀⠠⠄⠬⠈⡀⠀⠀⠀⣠⣸⣾⣋⡤⠒⠐⠂⢉⠈⢆
⠀⠣⢀⢀⣜⣿⢽⡇⣼⠄⣀⣀⠴⠊⠀⠀⠀⢴⠾⠛⠉⠈⠁⢠⣝⣯⣿⣿⣿⠿⠯⠁⠒⠀⠁⠣⠉⢄⢈⠣
⠀⣤⠀⠻⢙⠉⡿⠀⠗⠚⠉⠀⠀⠀⢀⣤⡂⣆⠀⠀⠀⠀⣤⡾⣻⠿⠚⠉⢀⠀⠀⢀⠀⠀⢀⡄⠀⠈⡊⠐⡆
⠀⠈⠻⣷⣈⣼⠃⠀⠀⠀⠀⠀⣠⠚⠁⠀⠰⣙⠓⡴⢾⣿⣾⠟⠁⠀⠀⠀⠈⠆⠄⠀⠀⡠⠊⠁⠆⠀⠀⢺⡇
⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⢠⣞⠉⠀⠀⠀⠀⣾⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⢰⡀⠀⡰⠗⡃⠀⠀⠀⠀⠄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⢰⣰⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⡜⢁⠀⡠⢔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠙⣻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢾⠁⢈⡐⠑⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣊⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣦⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⢀⣀⣤⣾⣿⣷⡚⠓⠂⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⡤⠊⠁⢈⠂⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣴⡧
⠀⠀⠀⢀⠀⠉⠘⠒⠤⠀⡤⠀⡀⠴⠦⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⠀⠀⠈⠳⡄⠀⣀⡴⠋⠀⠀⠀⢀⠀⠀⠘⢛⣟⠃⠟⠙⠛⠛⢻⣿⣦⣄⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠷⠶⠶⠶⠓⠚⠉⠀⠀⠀⠀⠚⠰⣦⠀⠀⢹⡶⣤⣀⣀⢢⣄⣮⣵⣿⣷⠶⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣠⣀⣾⣿⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢿⣿⣯⡉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⣇⢁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠌⢻⢠⣏⠂⡴⢐⢸⡇⣴⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠋⠀⠙⠊⠋⠉⠉⣇⠘⢺⡗⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡾⠋⠡⠊⡀⠀⢸⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠘⢷⣶⣆⡘⠁⠀⠸⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠻⢿⣤⣴⣿⣾⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤⣰⡄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣩⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠶⠚⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢿⢻⣽⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⣀
⠑⠃⠀⡰⠁⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡞⠋⠁⡢
⠀⠀⣸⡇⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⢰⡞⠂
⠀⠀⢿⡇⠀⡞⠀⢀⡔⠀⠀⢠⡄⠀⢡⠀⣧⠀⢠⠰⡟⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠁⠀⣻⡇⢠⡇⢀⡿⠁⠀⠀⠀⠀⠘⡉⣇⣻⠀⡆⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠄⠀⣿⣇⡼⠁⣼⠇⠀⠀⠀⠀⠀⠀⢹⣿⡟⣸⣏⣴⣿⢃⡀⠀⠀⣀⣀⡤⠘⢿⣼⣸⠟⠶⣶⣶⣦⣤⣀⠐
⠀⢠⣿⡟⢁⣼⠇⠀⠄⠀⠀⠒⠀⠀⠀⠙⠷⠟⠿⠿⠛⠚⣠⣴⣾⣷⣶⣶⣆⣀⣿⡗⠇⢤⣼⡾⡿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⣀
⠀⣸⠟⡴⠟⢁⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡏⢀⣹⣧⡷⣀⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⣰⠟⣭⣷⣦⣽⣧
⢰⣿⣫⠔⠚⠋⠉⠀⠈⠉⠙⢶⡀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠒⣶⣿⣶⣿⡿⠙⢿⣮⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣘⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⡖⠔⡄⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠀⣾⣿⣿⠟⠫⠁⠈⠲⡟⢿⣿⣷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠋⠈⠉⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢀⣀⣀⡀⠀⢤⢀⣀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠀⠀⢀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣛⣋⣭⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠙⠻⣭⠭⠿⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢄⣀⠠⠄⠀⢀⣀⣀⣀⡀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡌⠉⣁⣂⡀⠀⢠⣤⣤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣯⣭⣛⠿⢷⣮⣿⡿⣥⡀⠈⠉⠀⠤⣒⣄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠙⢶⣄⠀⠀⣠⠀⠈⠙⠲⣤⡀⠀⠀⢀⣠⠀⠀⠀⠀⢻⣷⣹⣶⠙⣶⣆⣀⡽⠿⢷⣽⣷⣦⣤⣀⠀⢉⠀⠀⢀
⠀⠀⠀⣠⣮⣀⠀⠀⠀⠀⡈⠷⣮⡀⠀⠀⠀⠀⠀⢙⣶⡿⠛⠁⠀⠀⠀⠀⠘⣿⣯⠻⣿⣀⡔⠛⣲⢶⣈⡩⠿⣮⣿⣍⠉⠐⠂⠀⠀⠳⢄
⠀⠀⠀⠻⣎⡻⣷⣄⠐⠊⠀⠀⠀⠙⠷⣤⣠⣦⣇⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⢿⣧⣼⣁⡀⠛⢣⣔⠉⣻⣿⣿⣶⠶⠖⡠⠄⠊
⠀⠀⠀⠀⠈⠙⢧⣽⣷⣦⡀⠀⠀⣀⣤⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣍⡛⠻⠿⣶⣇⣨⠿⣥⡬⣿⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣗⣮⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣦⣭⣛⣓⣻⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⣤⠤⡀
⠀⠀⠀⠀⠀⠀⡀⢠⣞⡷⣑⢟⡳⢯⢤⢿⣿⣥⣛⡧⢰⣒⡀⠄
⠀⠀⠀⠀⠀⠀⢹⠂⢈⠡⠉⠙⠉⠉⠉⠙⠒⠛⠛⠃⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⡴⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠤⠶⠽⠷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⣠⣤⣴⡞⡁⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣻⣿⣿⣿⣎⣷⣿⣦⡀⠀⠀⡀⠐⣀⠍⡄
⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡂⠀⠀⠀⠀⠀⠀⠙⣟⣽⣽⣿⣿⣿⣿⣿⣿⣖⣀⣴⣪⣽⡧
⠀⠀⠀⠀⠀⠰⢧⣤⣤⣬⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢿⡿⣿⡿⢻⡁⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⢖⠿⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⣮⡟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣿⢉⣤⣤⡀⠀⠀⠀⠀⠀⠀⢘⣻⡆⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⠋⣴⢮⣷⠀⠀⠀⠀⠀⢠⣾⣏⣤⣀⣀⣀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣉⠉⢹⡇⠀⠀⠹⡆
⠀⢀⡶⠛⢷⠀⠀⢸⣇⢰⡞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠉⠉⠻⣎⠛⣷⣠⣄⣿⠀⠀⢰⠟⠳⣆
⠀⣼⠁⠀⢸⡇⠀⣾⠻⣮⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⡈⠀⠈⢹⡆⠀⣿⠀⠀⢹⡆
⠀⡟⠀⠀⢸⡇⠀⢹⡄⠀⠈⢻⡏⠛⠛⢦⡄⠀⠀⠀⠈⠛⠶⠟⣦⠀⠀⠀⣠⡞⠁⠀⠀⣸⠁⠀⣿⠀⠀⠘⡇
⢸⡇⠀⢀⡸⠇⠀⠘⣇⠀⠀⠸⣇⠀⠀⠀⠙⠛⣻⠄⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⢀⡟⠀⠀⠟⣀⠀⠀⣿
⢸⡇⢠⡟⠙⢷⡀⠀⢹⣆⠀⠀⠙⠳⣆⠀⠀⣀⡿⠀⠀⠀⠀⠀⢸⠀⢠⡟⠁⠀⠀⢀⣾⠁⠀⣰⠟⠙⣧⠀⣿
⣼⠀⠈⣧⠀⠀⠻⣆⠀⠹⣦⠀⠀⠀⠹⣦⣸⠋⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⢠⡾⠁⢀⡾⠋⠀⢠⡏⠀⢸
⣿⠀⠀⠘⣦⠀⠀⠈⠳⣄⡈⠳⣄⡀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⡴⠋⠀⠀⢠⡟⠀⠀⢸
⢿⡀⠀⠀⠈⢷⣄⠀⠀⠈⠻⣦⡈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣠⡾⠋⠀⠀⢀⣴⠏⠀⠀⠀⣸
⠈⠻⣦⡀⠀⠀⠙⣷⠀⠀⠀⠈⠻⣆⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⡾⠋⠀⠀⠀⢰⡟⠁⠀⠀⣠⡾⠋
⠀⠀⠈⠻⣆⠀⠀⠘⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠻⠀⠀⣀⣀⣀⣀⣠⣬⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣀⣀⣀⣀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⢸⡛⠛⠋⠉⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠛⠛⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣀⣤⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣄⣀⣀⣀⣀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⢠⠐⠀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⡂⣄⠀⠀⣸⠀⡈⠁⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⣀⠀⠀⠀⢀⡶⠀⠀⡄
⣀⡄⠈⣿⠀⠀⠀⠀⠀⠀⠤⠀⡰⣄⠀⠸⠇⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠈⠐⢶⠗⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⠰
⠶⠦⠛⠉⠹⣋⢛⣴⣦⣤⡆⠄⣐⠀⢣⠰⠛⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠔⠊⢁⠀⠀⡀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠙⠂⠘⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠮⠡⡀⠀⠀⠈⠉⠉⠻⠂⠀⢠⣄
⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⠄⣀⣀⡄⠔⠂⠈⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⣀⢀⣀⡀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠁⠉⠑⠖⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠰⠿⡓⠡⠆⠂⠤⣥⣖⡂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠑⠦⡖⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⡼⠉⢝⠲⠀⢀⣶⠠⠔⠀⠁⠀⠀⠆⠈⠁
⠀⠀⢠⢤⡤⣶⢏⡌⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠙⢤⠤⣶⢺⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢄⠚⣶⡄⠀⠀⠀⣆⡁⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠙⢤⡆⡀
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠌⠠⠋⠀⠉⠀⠀⣰⢃⡃⠀⠀⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠠⠀⠀⠀⠘
⣤⣤⣦⣜⣀⡛⠀⠈⠈⠀⠐⢀⠊⠀⠀⠀⢰⡾⠞⠁⠎⠀⠀⠐⠀⠀⠀⡆⠐⠀⠈⠐⠁⠒⠀⠉⠀⢀⣠⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤
⠈⠁⡉⠀⠐⠃⡤⠀⠤⠠⠞⠁⠀⠀⠀⠀⠛⠁⠈⠂⠀⠀⠀⠀⠀⠀⠍⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣀
⠠⠐⢀⠄⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣠⡀⢄⠀⠠⠁⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢰⡆⡀⠀⠘⠿⠃⠂⠀⢈
⠟⠀⢁⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠈⡴⠀⢀⡀⠀⠀⠀⠀⠀⠲⣀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⢀⡀⠢⠠⠈⠓⠠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡄⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⡂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢡⠆⠀⠀⠀⠀⠀⠘⡂⠀⠀⠠⢾⡿⠈
⠀⠀⠀⠀⡨⠣⠄⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢣⠹⠌⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠈⠀⠀⣠
⠀⠀⠀⠘⠀⠀⠂⠀⠄⠀⠀⠀⢀⣰⠆⠀⠀⠀⣀⠀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣀⡔⠁
⠈⠀⠀⠀⠀⠄⠤⠊⠈⠀⢣⡀⠠⠁⡆⣀⠖⠉⠀⠁⠀⠈⢋⡋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠔⠈⠀⢀
⠀⠈⠐⣄⠀⠀⠀⠀⠀⢈⠄⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢰⡵⠂⠉
⠺⠃⠀⠁⠐⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠠⠀⠉⠊⠉⢳⠂⠀⢄⡰⢾⠀⠀⠚
⢂⠒⠀⠀⠀⠀⠀⡈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⠺⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⡀⣀⣀⡉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡀⠀⠀⡴⣫⣠⣴⠶⠾⠛⡛⠉⠉⠉⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⡭⠟⠛⣛⣿⠦⣼⡿⠃⡏⠉⠳⠾⠿⠿⠿⢷⣶⣦⣄⡀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⠖⠋⠁⢀⡴⠚⠁⣠⢶⡿⠛⠛⢁⡴⠞⢻⡶⠲⣦⣶⢦⣈⡙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡵⠋⠀⠀⣠⠞⠁⠀⢀⡼⠁⣸⡇⠀⡴⠋⢳⣴⡟⠀⣠⠟⢹⡾⣯⣷⠤⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡼⣡⠟⠀⠀⢀⠞⠁⠀⢀⡴⠋⠀⠀⣿⡇⢸⣇⣠⠟⢹⣷⡟⠉⢠⡿⠀⣿⠿⣆⢻⠲⡄⠙⢿⣆
⠀⠀⠀⠀⠀⢠⡞⣰⠋⠀⠀⡴⠋⠀⠀⡠⠊⠀⠀⠀⢠⣿⣷⠀⣿⠁⢀⣼⠛⣿⣾⠋⢀⣼⠇⢰⡿⢿⣄⣿⢦⡀⢻⣧
⠀⠀⠀⠀⢀⡟⢠⠏⠀⢀⠞⠀⠀⢀⡞⠁⠀⠀⠀⢠⠞⣿⣿⣧⡹⢶⡟⠁⣼⠟⣿⣶⠟⠁⣠⡾⠁⢸⡏⣿⣬⣷⡀⢿⣇
⠀⠀⠀⠀⣸⠃⡾⠀⢠⠏⠀⠀⣠⠏⠀⠀⠀⠀⣠⠏⢠⢿⣷⣿⣷⡈⢳⡞⢁⣼⠏⢹⣷⣾⡏⠀⣠⡿⢡⡿⢸⡷⣇⠘⣿⡄
⠀⠀⠀⠀⡟⢠⡇⢠⠏⠀⠀⣰⠃⠀⠀⠀⠀⡰⠃⣠⠋⣨⣿⣿⣿⢳⣄⠙⣿⣁⣠⡾⠀⣿⢷⣿⣏⣴⡟⢁⡾⢀⣿⡀⣿⡇
⠀⠀⠀⠀⣧⢸⠇⡞⠀⠀⡼⠁⠀⠀⠀⢀⡜⠁⡴⠃⣰⠏⣸⢿⡿⢽⣿⣷⣌⠹⣏⠀⣠⠟⢸⡏⢹⡟⢻⡿⣷⡾⣹⡇⢸⣷
⠀⠀⠀⠀⢻⣼⢠⡇⠀⣰⠃⠀⠀⠀⡠⠋⢀⡜⢁⡼⢋⡼⠃⠈⠻⣿⣿⣿⢽⣷⣍⠙⠧⠶⣏⣤⣟⣀⣾⢁⣿⣤⠟⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠈⣿⢸⡇⣰⠃⠀⠀⣠⠞⠀⣠⢏⣴⢟⡵⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣶⢶⣶⣬⣭⣭⣩⣤⣤⣤⣤⣀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣧⠏⢀⡠⠖⢁⡤⣞⣵⡿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⠿⠿⠿⠿⢿⣶⠿⠁
⠀⠀⠀⠀⠀⠐⣿⣿⣾⠿⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣾⠻⣷⣴⡟⠗⣾⠻⣷⣤⡟⣿⠂⠀⠀⡿⢻⣧⣼⣫⡇⣴⠋⡿⢰⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠚⠛⠶⠋⠘⠷⠖⠛⠶⠋⠘⠗⠿⠀⠀⠀⠻⠾⠃⠙⠷⠞⠻⠞⠳⠚⠇⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢚⡹⠁⢈⡭⡝⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠥⢒⡏⢶⠬⢮⠂⠀⢗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠈⡄⠀⠀⠀⠀⠈⣢⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡵⢥⠁⠀⢁⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡘⠓⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠓⠲⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡞⠒⢴⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⠧⣄⣈⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠇⠀⠀⡸⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡘⠙⠒⢴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⠧⢄⣀⡌⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣎⡀⠀⡸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣸⠁⠉⢻⠃⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢠⠋⠓⢤⠎⠀⢀⠀⡎⠀⠀⠀⠀⠠⠊⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠟⠤⣀⠎⠀⢀⠌⡜⠀⠀⠀⠀⡀⡐⠄⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡿⢄⣠⠎⠀⠀⡠⠊⠀⠀⠀⠀⢆⠀⡸⠀⠀⠀⡉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠊⢁⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡮⠔⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢉⡆⠀⠀⣎⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⠀⢾⢷⢦
⠀⠀⠀⠀⠀⠀⠀⠸⠓⢒⣁⡀⠀⠀⡠⠔⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢛⣁⡀⠀⠀⢠⣤⣶⣇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢇⡀⠉⢦⠀⠘⢶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢺⣿⣿⣧⠀⠈⠻⣿⡟
⠀⠀⠀⠀⢸⠹⡀⠈⢆⠀⢹⡰⠃⠀⠀⢀⡷⠁⢰⢉⡆⠀⠀⠀⠀⠀⣷⠀⠈⢦⡀⢸⣿⠃⠀⠀⠀⡹⠁⠀⣾⠇
⠀⠀⠀⠀⠘⡄⢡⠤⡀⠑⠢⠤⠤⠤⠒⠉⣠⠤⡌⢸⠀⠀⠀⠀⠀⠀⣿⣦⣤⡀⠉⠂⠤⠤⠤⠔⠊⢀⣠⣤⣿
⠀⠀⠀⠀⠀⢣⠀⠱⡌⠓⢄⠀⠀⠀⡴⠊⢡⠊⢀⠎⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄⠀⠀⠀⢠⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⢸⡀⠀⢀⠁⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠀⠀⠀⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⣉⡱⠀⢏⣉⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡶⠀⢴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣴⡿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣿⣿⣿⣿⣿⡿⠞⠛⠻⣦⡴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣩⡿⠿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⢏⣾⣿⣿⣿⣿⣿⢋⣴⡏⢿⠤⢹⣿⣞⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⣟⣛⣷⣶⣶⣶⣯⣻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⡾⠡⣫⢞⣵⣿⣿⣿⢃⣾⡮⠬⣿⢹⢸⢹⣿⣌⣿⣿⣿⣷⡀⠀⠀⠀⢀⣴⠟⣋⠥⢖⣻⣿⣿⣿⣿⡿⠟⣋⡭⠥⠠⡤⢬⢩⡹⣆⣤⡶⠆
⠀⠀⠀⠀⣸⡇⡰⣻⣿⢟⣿⣿⡇⣾⣿⡛⣲⠗⢾⠸⢸⣿⡿⣯⣿⣿⣿⡇⠀⠀⣴⡟⢁⡤⢖⣻⣭⣿⣾⣿⠟⡉⢴⠛⢅⢹⣤⢷⠛⠏⢣⢵⢹⡅
⠀⠀⠀⠀⣿⠀⠘⣫⠷⢛⣩⣾⢸⢷⠧⢤⡽⣉⠏⠀⣼⢸⡷⣿⣯⣿⣿⡇⠀⣾⡋⠀⢒⠚⣿⣟⣛⣛⡿⢡⡎⢳⣤⢶⠚⠇⠙⡀⢡⡼⡎⡟⣸⠁
⠀⠀⠀⠀⠻⠲⠶⠒⠛⠋⠁⣿⢸⣿⡃⣚⠳⠎⡴⢲⢇⡾⠀⠹⣿⣼⣿⡇⠀⠈⠙⠷⣦⣭⣙⣛⣻⡟⡰⢦⠽⣎⠱⡀⣠⡸⡆⢻⡉⡥⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⠼⡭⢋⡼⡱⢫⡾⠁⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⣷⡚⣦⢻⡎⢻⡈⣦⡸⠞⣋⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢋⣤⣊⣨⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠭⣭⣭⣭⣭⠥⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣿⡟⡩⡿⠟⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠰⢦⣄⣀⣛⣙⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣠⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣭⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⣿⣿⣟⡽⠚⠋⠉⠀⠈⠁⠀⠀⠀⠠⣄⠀⠰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡀⡀⢐⡶⢤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠍⠐⡁⣀⣀⡥⣀⣬⣌⢻⣟⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⠴⢿⡿⠿⠿⠷⠿⠟⠿⣿⠿⡟⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣧⣠⡄⣔⣤⢰⣤⡀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠛⠊⠉⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⠃⣀⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢹⣾⠋⠃⠀⠀⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣯⢀⡤⠋⠉⠀⠀⠀⠀⠀⢠⣿⠀⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠻⡏⠀⠀⣀⠀⠀⠀⠀⢠⠏⣽⠀⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠲⠤⣤⡤⠾⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢹⣷⡊⠁⠀⠀⢀⣼⠀⠀⠀⢀⡿⢠⣿⠀⠸⠙⣧
⠀⠀⠀⠀⠀⢀⠀⢀⡿⠻⢦⣄⣄⢈⣷⡤⠒⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢟⢀⡴⠈⠊⠀⠀⢠⣴⢢⣾⠀⠀⠀⣼⠃⢴⡧⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⣿⣠⣴⠞⠻⣿⡋⠁⢀⣠⠿⣄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢼⠻⠉⠀⠀⠀⣠⡾⠛⣿⠟⠀⠀⢰⡟⠀⣿⡏⠀⡇⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⡽⣤⡀⠉⠛⢿⠙⢻⡄⠀⡇⠀⠀⠀⠀⠔⠛⢳⣦⡖⠀⠀⠀⠀⣠⠴⠃⠀⣾⠏⠀⠀⢀⡿⠁⠀⣿⠁⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣀⢀⡿⠀⠈⠿⠶⠖⠟⠛⡆⠙⠒⠁⠀⠀⠀⡆⠀⣐⡿⠋⠀⠀⠀⢀⡴⠁⠀⠀⣼⠏⠀⠀⠀⣼⠇⠀⣸⡟⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠦⣄⣀⡤⠜⠁⠀⠀⠀⠀⠀⢰⣿⡾⠋⠀⠀⠀⠀⡠⠛⠀⠀⣠⡾⠃⠀⠀⠀⢰⡟⠀⢠⡿⠀⠀⡜⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⢀⠔⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⠐⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⡰⠃⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠁⠀⡠⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣀⡀⢙⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠁⠀⠀⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠾⠖⠛⠛⠁⠀⠋⠈⡆⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣴⡏⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣼⠃⠀⠀⠀⠀⠀⡰⠃⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⣾⡟⠀⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⢠⠃⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣠⡟⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⠏⠀⠀⠀⠀⢀⠆⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣴⡄⠀⢳⠀⠀⠀⠀⠀⢀⡎⠁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣿⠇⠀⠀⠀⠀⠀⣜⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠎⠀⠀⠀⠀⠀⢸⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠻⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣀⡀⢤⢤⠧⣆⢰⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠦⣴⠶⠟⠒⠤⡆⠀⠀⠀⢱⡶⢶⠆⣺⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⣀⡠⣤⣄⣀⡜⠂⠀⠀⠀⠘⣷⣀⡀⠀⠐⣶⠅⠺⢟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⢀⠌⠁⠀⠀⠀⠹⣧⡀⠀⠀⠀⢈⣩⡾⠃⠀⣀⠁⠀⠀⠀⠓⡔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠃⡠⠒⠒⠂⠀⣬⡀⠀⠀⠀⠀⢂⣿⣿⠗⣀⣸⠿⡋⠀⡰⠿⣟⣤⠀⠀⠀⢴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣅⠀⣸⠀⠀⠀⠀⠀⠄⢱⣦⣤⣤⣴⡾⠋⣠⣢⠞⠡⡎⠀⠀⠀⠀⠻⣿⣿⣿⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠀⠀⠀⠻⣧⠀⠀⠀⠀⠒⣡⣿⣿⡿⠟⠁⠀⠊⠉⠉⣠⣷⣌⠀⠀⠀⣠⣧⣿⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣀⣾⡘⠀⠀⠀⠀⣄⣼⡷⠶⠶⣛⣋⣉⡥⠴⡶⢦⡆⢐⣶⠞⠛⠁⠹⠿⠟⠛⠛⠛⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠈⣿⠿⠷⠒⡊⠉⠉⠀⠀⠀⠈⠈⠉⠀⠀⣀⣠⠴⠛⠉⠁⠀⠀⠀⠰⢢⠖⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⢀⣜⡿⢖⣲⣬⠴⠾⠛⠁⠀⠠⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣕⡒⠲⠖⠶⢲⡆
⠀⠀⠀⠀⢀⢞⣥⠖⡉⠁⠀⠀⠀⢀⣀⣰⣒⡫⠴⠖⠛⠋⠁⣽⡃⠀⠀⠀⢀⠀⣧⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⣿⠟
⠀⠀⠀⣠⢎⣉⡋⠨⠧⠴⠖⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢿⡷⢦⣄⣠⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠛⠉

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⠀⠀⢰⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠲⣿⣿⣿⣿⢿⣦⡤⡂⠀⠀⠀⠀⠀⢀⣻⣿⣿⡷⢼⣀⡙⢛⣻⣶⠶⠆⠀⠀⣠⣶⣿⣿⣷⡮⢮⡹⣿⣿⣿⣆
⠀⠀⠀⠀⡽⣫⠕⣉⢜⣻⣯⣳⡄⠀⠀⣴⣿⣿⣿⣿⣷⣄⠉⡙⢥⣄⠻⡇⠀⠀⣸⣿⣿⣿⣿⢳⠅⢆⣳⠈⠛⠻⠟⠓
⠀⠀⠀⡼⢟⠕⢊⣤⣿⣿⠉⠉⠃⠀⠀⠉⣻⣿⣿⣿⣿⡿⢸⡠⡌⡎⢆⠀⠀⠀⠿⠿⠿⠿⣟⡎⠜⣐⡏
⠀⠀⠀⠉⠙⠚⢻⣿⠿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠁⠈⢷⡐⠈⣿⠀⠀⠀⠀⠀⠀⠀⢹⡴⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠤⠒⠂⠈⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠉⠉⠉⠉⠒⠦⣀
⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠒⢄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡯⠉⢳⡀⣱⣄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⣶⡄⠀⢸⡏⠀⠸⣿⣿⣿⣧⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢹⣾⣿⣇⠀⢹⣿⠛⣿⣿⣿⣿⣷⣤
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⣿⣿⣿⣦⣘⣿⡄⢿⣿⢋⣶⣿⠿⣷⣶⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣷⣿⢩⢻⡿⣿⣿⡧⠌⣟⣿⡟⢁⣤⠾⢻⣿⣦⣤
⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⣿⡏⣧⢸⠏⣷⡌⢹⢇⣶⡿⢸⡼⣿⣋⣴⣿⣿⠿⣿⣧⣄
⠈⡄⠀⠀⠀⠀⠀⠀⣠⢿⣿⡟⢸⡇⠇⠟⡦⡿⣸⣦⠸⢹⢳⡿⢰⣿⢹⣿⠋⢠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⡄⠀⠀⠀⠀⣼⢃⣵⣾⠘⣼⣷⣶⣮⠁⣵⠛⡾⡇⡞⢸⢃⣿⠃⣸⡏⢹⣿⣿⠿⢛⣹⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⡄⠀⢀⡼⠡⣪⢟⢕⠟⢉⠛⢛⣣⣴⡇⢸⡇⡇⠁⣿⢸⡟⣰⡿⠁⣸⡟⢳⣤⣿⣿⠿⠛⢿⣷⣄⠀⠀⠀⢀⠄
⠀⠀⠀⠘⢄⢞⡵⠉⠑⢁⡵⠾⠛⠛⢋⡽⢹⣧⢸⡇⠃⠀⡇⣿⢱⡿⠁⣴⣿⠇⣼⣿⣟⠁⢀⣤⣶⣿⣿⡿⠂⢀⠎
⠀⠀⠀⣠⣿⣫⣀⣠⣞⣥⣶⣶⣶⠿⠋⠀⣾⣿⡄⣷⠀⣸⢃⣿⣿⠄⣼⣟⡞⢸⣿⠇⢘⣿⣿⣿⣿⠟⠋⠀⡰⠃
⠀⠀⣼⣿⣿⣿⣿⣫⠥⠤⣀⡉⠛⣿⡧⠀⣿⡇⢣⣹⠀⣿⣿⣿⠏⣼⠏⠊⣰⣿⢣⠀⣨⣿⣿⠟⠁⠀⠀⠔⠁
⢀⡼⣼⣿⣿⣿⣿⣿⡿⠟⢾⣿⡆⢸⡇⠀⣿⣇⣆⣿⣰⢏⣾⠏⣼⣳⠃⣰⣿⣿⡆⢰⣿⣿⠏⠀⠀⠀⠊
⣾⣿⣿⣻⣿⣿⣿⠿⢶⣶⣤⢸⡇⡼⡰⢠⣿⣿⣿⣿⢃⡾⠃⣼⣿⠃⡰⣿⣿⡟⢀⣿⡟⡏⠀⢀⠔⠁
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡼⠑⠋⠁⠀⠀⠹⣿⠟⠁⣼⣿⠃⡔⣼⣿⡟⢀⣾⣿⢱⠁⡠⠃
⠀⠙⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⣿⠀⣸⣿⠃⣘⣾⣿⡟⠁⣼⣿⣏⡯⠊
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⣶⡟⢁⣴⣿⠫⠋⢀⣾⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠉⠉⠛⠓⠒⠛⢛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣱⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣿⣯⣽⣯⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠿⢦⡴⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠁⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⣀⠄⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⠠⠴⣿⣦⣦⣄⣠⠐⠀⣀⡤⠶⢞⢉⣻⣿⣿⡶⠮⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠛⠯⠿⡻⢻⣿⢶⣮⣭⣟⣯⠀⠴⠿⠀⣠⣦⣎⡽⠛⠉⠆⠀⠂⠁
⠀⠀⠀⠀⠀⠀⢠⣷⡄⢂⣤⡅⢺⣽⡧⢽⣿⣿⡿⣫⠿⡀⢠⣶⣤⣬⣅⣯⣓⡤⠬⠤⠀⠘⠢⡀
⠀⠀⠀⠀⠐⠠⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⢟⣫⠂⠀⢠⣜⢊⣉⡼⣵⣿⡛⣿⣯⡷⠶⣄⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠉⠉⠂⠐⢻⠟⡸⣫⣾⡿⣿⣶⡖⠃⣳⣿⣏⠛⠃
⠀⠀⠀⠀⢀⣤⣀⠤⠒⢶⡖⠀⠈⠙⣷⣦⢠⣶⠿⣶⠶⠊⠀⠉⠋⣀⠀⠀⠻⠿⢿⡿⠿⠄
⠀⠀⠀⠠⠀⠛⢯⠰⣧⣤⣷⣇⣶⣾⢹⣝⣋⡥⠎⠁⠁⠀⠉⠀⢈⢻⡤⠀⠂⠀⠀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠓⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠴⠿⢁⣤⣀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠛⠳⢦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠶⠟⠋⢀⣶⣟⣻⣦⠀⠀⠀⠀⠀⠉⠛⣷⣶⡦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⡋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢮⣙⢷⡄
⠀⠀⠀⠀⠀⣠⡶⣫⣵⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⢰⢏⣼⠋⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀
⠀⠀⠀⡰⢡⠞⠁⠀⠀⠀⠀⣿⣿⣷⣌⣳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣧⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⣌⢷⡄
⠀⢀⡾⠕⠁⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⢼⣿⣿⣟⣻⡄⠀⠀⢸⢸⡗
⠀⣿⠁⠀⠀⣀⣴⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠁⠀⠀⣾⢸⣇
⠀⡏⠀⠘⣿⣿⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⠀⠀⠀⠀⠀⠀⠈⠃⢿
⢰⠃⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⢷⠀⠀⠀⢀⣶⣆⢆⠸
⣿⣮⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢸⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⠀⠘⠦⠿⠿⠋⠉⠀⠀⣴⣻⣿⡿⣸⢸
⢿⣿⡀⠀⠀⠀⠀⣾⣿⣿⣮⣷⠀⠀⠀⠈⠛⠿⢿⣻⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢣⣿⡟
⠈⢧⣛⣦⣐⣶⣤⣜⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⣿
⠀⠈⠻⢻⡝⠿⠋⣙⠿⠛⣷⣶⣄⡀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⣠⣼⡇⣿⠋
⠀⠀⠀⠰⠍⠀⠀⠉⠀⠀⠉⠉⣿⣅⢸⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⡇⠀⠀⣨⣵⠟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠀⢟⡿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⠁⢀⣾⢟⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⢙⡷⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⢧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣧⣤⠀⠀⠄⣿⣿⣶⣿⡇⣠⣴⣶⠷⣟⣋⣉⣩⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⣿⣛⣛⣓⣚⡛⠛⣋⣈⣥⣾⣶⡿⠟⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣬⣽⣿⣿⣿⣯⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠦⠶⢶⠦⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠛⣁⡀⣀⣒⣢⣍⣠⣬⢁⣨⣿⣖⣄⣀⣠⣴⣾⠁⢀
⠀⠀⠀⠀⠀⠀⠙⢷⣦⣽⣋⣹⡟⢉⡟⠉⣥⣋⣁⣞⣡⠏⠋⠁⠀⣴⠋⠈⣛⢦⣛⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠿⠟⠛⠛⣋⣁⡀⣀⣠⣾⠁⠀⠀⢻⣄⣿⣷⣦⢄⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠠⢐⣻⣿⣿⣿⠁⣼⠙⣆⣀⣀⡀⠙⠿⢿⣮⣤⣱⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⣬⠶⣫⣿⣿⣿⣿⠇⣸⠇⢀⡏⢻⣷⣜⠒⠀⣄⠈⠉⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣫⡶⣿⣷⡿⣿⣿⣿⣿⣧⣴⣏⠀⠸⣷⡀⠻⠿⢷⣦⡈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢫⣾⣿⣻⣿⣿⣿⡿⢷⠿⢿⣻⡆⠀⣿⡦⠀⡆⠈⢿⣿⣦⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣫⣿⣿⣿⢿⡿⣽⣏⣉⡟⣿⣷⠀⠘⣿⣾⡇⠀⠈⢹⣟⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣱⣯⣿⡟⠠⣿⣛⣿⣿⡀⠀⢻⣿⣆⠀⠀⠀⣿⣷⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣳⢷⣾⣏⣑⣾⢯⣼⣿⣿⣷⡀⠀⠙⣿⣿⡆⢠⠟⣿⡍⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⣿⣯⣟⣻⠿⣄⣽⠷⣴⢿⣿⣿⣟⢻⡄⠀⠈⠹⣷⡜⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣏⣳⡾⣉⣳⣿⣿⣯⠙⠻⢷⣷⣄⠀⠀⠈⢿⣶⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⢦⣟⡶⣻⣿⣿⣿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⢿⣟⣿⡿⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠿⠛⠉⠀⠀⠻⣧⣅⡧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡍⠛⠛⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠇⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠄⠀⠀⠀⠀⠀⡯⠁⠀⠀⠀⠀⠀⠢⢀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡬⠂⠄⡼⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢄⢣
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠁⢀⠀⠑⡄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠈⠀⣂⡤⠔⠚⠒⠲⠶⣆⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠃⠀⠈⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠈⠀⠀⢂⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⡀⠤⠼⡇
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⠀⢠⠄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⡀⡀⢄⠙⠠⢐⣤⠞⠁
⠀⠀⠀⢇⢀⠂⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⣂⣒⣖⣶⡾⠿⠟⠁
⠀⠀⠀⢸⡼⠀⢀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠘⢆⠂⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢣⢀⡎⠀⣔⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠻⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣸⣐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠀⠒⡊⠉⣉⠁⠀⠄⢙⠁⠄⠈⣭⠄⠋⠀⢠⠁⠀⠀⡼⠁⠀⠀⠀⠀⢀⣠⠤⠐⠀⠀⠀⠀⠀⠓⠒⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣥⣮⠵⠬⠦⢼⡶⢞⠒⣚⠒⡓⠦⢤⣴⡁⠀⠀⠀⠰⠀⠄⠊⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⢀⡠⠤⠀⢒⠀⢦⠈⠆⠐
⠀⠀⠀⠀⠀⢤⠞⡏⠻⢡⡋⡀⠝⣦⡈⠮⡀⠎⠙⡎⢡⠐⠚⣅⢰⣍⣉⣲⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠁⠀⠂⡁⢠⠂⡅⢡
⠀⠀⠀⠀⠤⠄⠿⣗⠓⠖⠄⠷⠚⠀⠐⠐⠒⠒⠒⠒⠒⠓⠉⠉⣑⣲⠟⠉⠀⠀⢀⡠⠤⠐⠢⣤⡀⠀⠀⠀⡠⢊⠮⢁⡠⠤⠢⠈⣢⢔⡔⠁
⠀⠀⠀⡠⠔⠈⠩⠉⣻⢦⣄⣀⣀⠀⠀⠀⠀⠠⠀⢐⣲⠴⠟⠛⠉⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠔⡾⠋⠀⡰⠁⠄⠈⢃⣀⠤⠒⣉⠴⠋
⠀⠀⠚⠀⠠⠄⠐⠈⡡⠚⠄⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢿⣦⠄⠀⠀⠀⣀⡠⠔⠀⠀⠠⠅⠀⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠘⠀⠀⠹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢉⢦⡁⠒⠄⠀⠀⠀⠀⢀⠄⡽⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠥⡀⢲⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢊⣰⠏⣰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⢠⡄⠀⠞⠀⠀⣼⠥⢈⣵⠁⠀⠀⠀⠀⠀⢀⠴⠋⠀⡀⣀⡠⢀⡀⠠⠚⠁
⠀⠀⠀⠀⠀⠓⠶⠄⠈⠁⢀⡸⠏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣇⣄⠀⠀⠄⠉⠋⠉⠉⠓⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⢀⡄⣠⠀⣉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⣜⣆⡵⠖⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢠⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⢐⠀⢄⡀⡈⠃
⠀⠀⠀⠀⠘⠉⠁⠀⢀⠀⠘⠂⠀⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠄⠠⣾⡆⠀⠀⠀⠀⠀⠔⠈⠁⠀⡀⠀⡰⠁⠀⠀⠀⠀⠈⠦⢀⣀⡜⠀⢠⡜
⠀⠀⠀⠀⠀⡀⠀⠳⢆⡄⣠⢀⢠⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠰⠀⠀⠀⠀⠀⣰⢀⡴⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠈⠀⡇⠈⠙⠀⠀⠣⠀⠀⠀⣠⣀⠲⣶⣠⡆⠀⠀⢸⡄⠀⡀⠀⠀⡀⡿⠏⠀⢀⣽⠃
⠀⠀⡀⢂⣀⡠⠂⠀⠀⠀⠀⡇⠀⠐⠀⠀⠀⠀⠀⣀⢤⣌⡉⠉⠘⠃⠀⠀⢘⠓⢀⠉⠐⠆⠉⠁⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡇⡆⢸⠀⠀⠀⠀⠰⢀⠚⠉⡁⠀⠀⠁⠀⠀⢃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⠀⠀⠀⠀⣶⠿⠶⠛⠁⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⢀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠸⠒⠂⠰⠦⠶⠴⢂⡜⠀⠀⠀⢀⠠⠒⠉⡏⠀⠀⠀⠒⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⡀⠀⠀⠀⢾⠰⣿⡂⠒⠒⠒⠉⠀⡀⠔⠈⠁⠀⡀⢀⡀⠠⠂⡞⠀⠀⠀⢀⡔⠉⡷
⠀⠀⠀⠀⡀⠀⢀⢰⣀⠀⠦⡀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⠹⠃⠀⠀⠀⠀⢀⠀⣀⢦⣙⠦⠛⠀⣀⠀⠈⠀⠀⢀⡆⠁⢹⠊⠁
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠛⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣯⢛⣽⠆⠁⡀⠀⠊⠁⠀⠀⠀⠊⢀⡼⠀⠉
⠀⠀⠀⠀⠀⠀⣠⣠⠆⠀⠚⠉⠀⠀⠀⠁⢀⡀⠀⢀⠀⢤⢀⠤⡒⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠶⠮⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⡄⣤⣶⣄⣶⣧⣦⡀⠤
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢢⣠⠄⠀⠀⠀⠀⠀⠀⠓⠢⠄⣀⣀⠀⠀⠀⢄⠙⢕⣾⠖⣟⣿⠎⡾⣿⣿⣼⣻⢵⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠯⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣈⡉⠉⠙⠛⠽⠽⠷⠲⠛⢋⡵
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠂⡀⠀⠐⠀⠀⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣐⡋⣉⢉⣤⣴⣂⣧⢃⢒⡌⢱⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣿⣿⡿⠻⡿⢻⡿⢿⡿⢿⡟⢿⣿⣿⡇⠀⠀⠮⣋⡶⣮⣹⣺⠶⡟⠾⣾⣸⣷⣻⣾⡷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠸⣿⣿⣶⣷⣾⣷⣾⣷⣾⣷⣾⣿⡟⠀⠀⠀⠀⠈⠉⠑⠋⠑⠛⠓⠛⠛⠚⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠟⠁⠀⢀⡀
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡯⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢠⡿⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣠⣴⢶⣤⣤⣤⣄⣀⠀⠀⠀⡾⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⢫⣾⡿⢣⣶⡙⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⣱⣿⣿⢡⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⣼⣿⣿⣿⡜⢿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⣭⣙⡛⠟⠼⢿⡿⠛⠿⣿⣶⢠⣾⣿⠟⠛⣿⡟⡛⠿⢛⣻⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢰⣿⣶⡆⢹⣷⣦⡙⠘⣫⣶⣿⡇⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡘⢿⣿⣿⠰⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣶⣭⣥⢠⣬⣥⣾⣿⣿⣿⣿⣿⣟⡻⠟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣥⣤⣤⣤⣤⣤⢠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠏⠹⣿⣿⣿⠟⡹⣿⣿⣿⠟⡙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⢠⣶⣿⣿⣶⣬⣴⣾⣿⣶⣤⣴⣾⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠸⠿⠟⣛⣛⣛⣉⣉⢸⣿⠟⢿⣿⣿⡿⠻⣿⣿⣿⠟⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣩⣥⣶⣦⠹⣿⣿⣿⣿⣿⣿⢈⣵⣾⣦⣭⣭⣶⣷⣬⣭⣵⣾⣦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⢸⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣌⢿⣿⣿⣿⢘⣋⣵⣦⣙⣛⣫⣴⣬⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⡙⢿⣿⢸⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠒⠤⣀⣀⡜⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣠⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠦⠒⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⣴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠴⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣹⢟⡈⣟⠀⣀⣷⠆⠀⠘⢂⡁⢠⠀⠀⠀⠀⢘⠠⣈⠙⠶⣾⣿⣁⠀⠀⠉⠐⠢⣄⡙⠢⡀⠀⠀⠀⠀⠀⠑⠂⢀⣉⠻⠿⣒⠆⢸⠍⠛⠓
⠀⠀⠀⠀⢻⢣⢣⠘⣇⡀⢸⡖⠂⠀⠠⣁⠀⡄⠀⠀⠀⠈⠀⠨⡳⢶⣂⠈⠈⠁⠢⠤⣀⠀⠀⠙⢳⣜⣦
⡐⠊⠀⠀⠀⢄⢇⢧⡘⢧⡀⢦⠤⠀⠀⠽⠄⡇⠀⠀⠀⠀⠀⠈⢫⢦⡉⢢⡀⠀⠀⠀⠈⠙⠢⣤⣄⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣌⢣⠸⡤⠤⠄⢼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡑⢄⠙⠂⠀⢀⠀⠀⠀⡘⢟⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠀⠀⡰⢄⡼⠐⢫
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡙⢷⡄⠠⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢂⠤⡀⣈⠑⠛⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⡠⠂⡈⠁⠀⠀⡰⣁⡼⠟⠋⢀⣴⠏
⠀⢀⡤⡀⠀⠀⠀⠀⠀⠉⠻⢿⣦⠙⠀⡏⠀⠀⠀⠀⠀⢀⠴⠒⠢⠤⠒⢤⠀⠀⠀⠈⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠂⠈⠀⣼⠁⣰⣟⠴⠛⠉⠀⡀⣴⠟⠁
⠀⠈⣹⣷⣖⠒⠤⢀⠀⠀⠀⠀⠉⠓⢌⠃⠀⠀⠀⢠⠊⠁⢀⡤⠄⢄⡀⡀⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠠⡀⣸⣧⠖⠋⠁⠀⢀⣀⡾⠟⠁
⠀⢀⣿⣿⠛⠃⠀⠀⠉⠢⢄⠀⠀⠀⠀⠉⠀⠀⢀⡜⠀⠔⠃⢀⠂⡎⠀⡞⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡾⣠⡿⠞⢉⡀⠀⠈⣉⡡⠔⠊
⠀⠰⣟⣿⠰⠟⣠⠀⠠⠤⠀⠑⢠⠀⠀⠀⠀⠀⠘⣇⠀⣘⡀⠔⢢⠟⠀⢴⠂⠀⡗⠀⠀⠀⠀⠀⠀⣠⠀⣾⡜⠛⠁⠠⠀⢀⣠⠤⠒⠉
⠀⠀⢿⣿⣧⠘⠟⣥⣤⠀⠛⣂⡀⠀⠠⡀⠀⠀⠀⢸⠀⠈⠣⢀⣨⠆⠱⠂⠀⡌⠀⠀⠀⡀⢀⣀⡰⠟⣉⣂⣠⠤⠶⠘⠋⠉⠀⠀⢀⠀⡀⣀⡰⣦⠀⠄⠒⣎⠙
⠀⠀⠈⢻⢿⣧⠘⠿⢋⣴⡄⠉⢩⣄⡀⠑⢄⠀⠀⠀⠑⠒⢄⠀⢀⣀⠀⡰⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡠⡀⣻⠖⠋⢄⠀⢠⣤⡷⡟⢠
⠀⠀⠀⠀⢻⣻⣷⡄⢿⡿⢣⣶⡆⠈⠠⡄⠀⠳⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢰⠥⠰⠅⠀⠘⣦⡤⠟⠉⢠⠟⣰⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡘⠟⢱⡷⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⣀⠠⠄⠠⢤⠈⠁⡀⠀⠀⠀⠀⠀⡜⣠⡇⠒⠂⢤⡀⣸⡖⠉⠀⠀⡶⢋⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣾⣶⣶⣤⣤⣴⠠⠀⠀⠀⠀⠀⠀⠀⡄⠚⠠⠖⣟⢀⡀⠋⢀⡖⠃⠀⠀⠀⠀⡜⣰⠊⠁⠐⢄⣀⠟⢇⠈⣠⡼⢋⠴⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠾⢿⣿⣷⣧⠀⠀⠀⠰⣀⣀⠉⠃⢈⠒⠁⣀⣶⠖⠹⣧⡆⠀⠀⠀⡘⡰⣅⠀⠐⣤⡜⢉⡤⠾⢋⡡⢖⡥⣾⠎
⣀⣀⣤⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠈⢍⠻⡎⠙⣿⡁⠁⢻⡆⠀⣿⠀⠀⠀⡐⡐⠁⡠⠴⠞⢓⣈⣥⣶⣚⣥⣖⠩⠊⠁⠀⠀⠀⢀
⣽⣿⣿⡇⠀⠀⠑⡶⢶⣓⣠⡄⠒⠂⣀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣦⡘⢿⣄⣸⣧⣰⠃⠀⠀⠠⣀⡩⠤⠴⠶⠿⠿⠿⠿⠟⠋⠉⠀⠀⢀⠀⣒⡾⠛⠉
⠀⣼⠏⣿⠀⠀⠀⠘⠀⠉⠉⠲⣿⣏⣁⡀⠄⠀⠀⠐⠀⠤⡀⠀⠀⠀⠀⠀⠉⠛⠶⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⢠⣤⡾
⣸⡟⢰⣿⠀⠀⠀⠀⠈⠑⢦⠀⠀⠈⠙⠻⢴⣶⠖⠛⠁⠁⠈⠀⠂⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣤⣤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠚⠁⠤⠉⠙
⠿⣀⡞⡍⠀⠀⢰⠀⠀⠀⠀⠙⠦⡀⠀⢀⠀⠉⠑⠲⢿⣂⢀⣀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⢦⡀⠉⠛⠶⣤⡀⠀⠀⠀⠈⠄⠣⠀⠦⡋⠰
⠀⢻⡀⠃⠀⢀⠸⠤⠀⠀⠀⠀⠀⠈⠙⠦⢡⡄⠢⡀⠀⡈⠑⠶⢤⣤⣀⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⢯⡀⠀⠀⠙⢦⡄⠀⠘⢿⣦⡀⠀⠀⠈⠂⠈⠂⢥⣀
⠀⡠⠃⠀⠀⡎⢰⢟⡉⠄⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⢦⣀⣄⠀⠀⠈⠛⠲⢦⣀⣑⡄⠀⠀⠀⠀⠀⠸⣧⠀⠙⢦⡀⠀⠀⠙⢦⡀⠀⠡⡑⣄⠀⠀⠀⠈⠒⠠⠈
⠀⠀⠀⠀⢰⡇⣮⠿⣤⠀⠙⢒⡆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠾⠷⠶⠶⠿⠽⠯⢦⡀⠀⠀⠀⠀⢹⣷⡄⠀⠉⠲⡀⠀⠀⠙⢆⠀⢵⣹⡄
⠀⠀⠀⠀⣸⡇⢻⠀⢳⡄⠀⠐⢇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢶⣄⠀⠈⠓⠦⣄⠈⢷⡸⣿⡇
⠀⠀⠀⠀⡟⣇⢸⠃⠉⢱⡖⠂⠘⢠⢠⠁⠀⠀⠀⡤⠠⢤⠔⠂⠶⢮⣥⠤⠞⣛⢊⠒⠢⢄⣀⠀⠀⠀⠀⠀⠈⠀⡉⠷⡤⣀⠀⠚⠟⠦⠁⠈⣹⣄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⣛⣉⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣀⢀⣰⠋⠀⠀⣽⠀⠀⠀⠹⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⡤⠐⠋⣰⣆⢯⡷⠂⠀⢠⡇⡇⢀⣠⣤⣤⣤⣀⠀⢀⡚⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠆⡋⣦⠽⠆⠀⣸⢸⣶⡏⠈⢹⡄⠈⠹⢷⣤⣷⣴⣟⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣇⢸⡀⠀⢠⣇⡞⠹⣇⣀⣼⠃⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡞⠀⢹⠇⠀⡾⡸⠀⠀⠀⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⠎⠙⠅⣼⠀⢠⢷⠃⠀⢀⡤⠖⠛⠉⠋⠉⠉⠍⠁⣉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⠟⢤⠈⢰⠇⠀⡞⡞⢀⡴⠋⠀⡀⠀⠁⠀⠀⠄⣀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢠⣇⣀⡘⢀⡏⠀⢰⣷⠃⡞⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡼⠁⠀⣼⡼⢸⡇⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⡜⠓⠄⠀⣸⠃⠀⢀⣿⠇⣼⡣⠈⠀⠀⠀⠀⠀⠀⢀⣠⠜⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣰⡁⢀⠆⢠⡏⠀⠀⢸⣿⣼⡵⢦⡒⠆⠐⠒⠒⠞⠋⠁⢀⡀⠀⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⡇⠀⡾⠀⠀⠀⢸⢻⠎⠀⠀⠙⢦⣀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠂⢈⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠄⠀⠀⠀⠀⡠⠔⠙⢲⠀⠈⠀⣼⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠦⣀⡀⠀⠀⣀⣀⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠘⠷⠒⠒⠒⠒⢺⠏⣇⡀⠀⡘⠀⣠⡼⠁⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⣩⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⢸⡾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇

⠠⠱⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⡶⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠋⠃⠀⠠
⠀⠂⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠄⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢧
⠀⠀⠀⠀⠀⢣⣥⡆⠀⠀⠀⠀⠀⠀⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠀⠈⠀⠲⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⡃⡔⠀⠀⢸⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠄⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠢⢞⠄⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠘⠁⢐⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢁⡀⢀⠀⠀⠀⡀⠀⠐⠋⠘⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠠⠀⠀⠀⠀⠀⠀⠊⠉⠑⠀⠀⠈⠃⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠖⢰⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠰⠀⢀⡂⠀⠀⠀⣀⣀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⣅⠉⠀⢀⠛⡔⠀⠁⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣶⣌⣠⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠘⠃⢋⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠂⠀⠀⢠⢷⡾⠋⠁⠄⠸⡷⠆⠀⠐⠃⠀⠰⠆⠈⠁⠀⣄⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠸⢼⠇⠀⠀⠀⠀⢠⣶⣦⡀⠀⢰⠀⠀⠁⠀⠀⠀⢘⣈⠁⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠀⠀⠀⢸⡇⠈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹
⠀⠀⢠⠀⠀⠀⠀⠀⠚⡀⠁⠀⢠⡥⢿⣄⠀⢻⣶⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⢸⡀⠀⠀⠀⠀⠠⠁⡄⠀⠈⠀⠈⠙⣧⢸⡆⠀⠤⣀⢒⣾⣿⣀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀⣀⣀
⡿⠋⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠈⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⢀⣿⡥⣾⡿
⠃⠀⠀⣸⠏⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣧⠀⢀⡀⠀⠀⠰⣶⣦⣼⡄⠀⠀⠀⠀⠈⢧⠀⠈⢿⡇⠀⠃
⠀⠀⠀⢫⠾⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡏⠁⠐⠂⠀⠀⠀⠀⠈⠐⠀⠀⠀⡀⠈⢇⠀⠀⠈⢀
⠀⡠⠀⠀⠀⠀⠉⠀⣀⣀⣤⣶⠛⠛⡐⠂⠀⠀⢤⣀⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡼⠶⣶⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⠏⠉⢛⡷⣾⣛⠉⠉⠀⠀⠀⠈⠈
⠀⠀⢤⡀⢀⠠⠖⠉⠀⢀⠈⠉⠀⠀⡠⠀⢤⡀⠀⠀⠀⠁⠀⠀⠱⡍⠢⢭⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠐⠓⠀⠀⠐⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⠀⠀⡶⣧⡇⠀⠀⠀⠉⠉⠛⠛⢷⡴⠞⠁⠀⠀⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⡀⠀⠀⠸⠇⠀⠀⠀⣀⡘
⢙⣀⢀⠀⠀⠀⢀⡦⠀⠀⠀⢀⣀⣀⡦⠀⠀⢀⠀⠀⠀⠀⢀⠎⠲⣀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠐⣾⠐⠉⠳⠒
⠋⡏⣸⠧⠤⠖⠋⢀⣶⢟⡺⢯⣍⣭⡄⠀⠀⠠⠀⠀⠀⢠⡟⣗⠀⠈⠑⠂⠀⠀⠀⢻⠀⡔⠻⠆⡠⠚⠐⠟
⢀⡁⠀⠀⠀⠀⠀⢸⡟⢸⣋⣐⡤⠄⠀⠀⠀⠀⠳⡀⠀⠈⡤⠚⣀⠀⠀⠀⠀⠀⠀⢈⣾⣱⡁⢀⡷⠤⣴⠋⢰
⠘⠀⠀⠀⠀⠀⠀⣠⡿⢛⠿⡿⣿⢯⣧⣠⡀⣰⠀⡗⠀⠀⣹⡉⠱⠆⠀⠀⠀⠀⠀⢸⣬⡉⠙⣿⡀⠠⣟⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠏⠐⣾⠚⠇⠈⣷⡦⣤⣴⡇⠘⠀⠀⠀⢹⡖⠆⢛⠀⠀⠀⢀⡜⡀⠠⠿⣤⣯⣤⣟⣁⣠⣀
⢶⡄⠀⠀⠀⢀⡀⠀⠀⢩⠀⠀⠀⣿⣼⠛⢿⡸⡆⠀⠀⠀⠘⣷⠬⠋⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠉⠈⠉⠉⠉
⠉⠀⢀⡴⡞⢯⠉⠳⠄⠈⠀⠀⢀⣿⡇⠀⠰⣇⠁⠀⠀⠀⠀⠿⡂⠴⣢
⠀⠀⢸⠀⠀⠀⠀⣄⡗⠀⠀⠠⠞⡿⠁⠠⠴⠋⠀⠀⠀⠀⠀⠈⠣⠿⠟⠀⠀⠀⠀⠀⢀⡔⠛⠛⠐⠒⢠⡀
⣀⣀⣰⣛⣆⣀⣂⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣐⣒⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣸⣀⣘⣛⣀⣀⣀⣘⣄
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣒⣿⠿⠟⣻⠿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡥⠞⠋⠁⠀⣠⠞⠁⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣡⠞⠋⠀⠀⠀⣠⠞⠁⠀⠀⢸⠇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠞⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣾⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣰⠋⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢰⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡼⠁⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠁⠀⠀⢀⡾⠁⠀⠀⠀⣀⣀⣀⣀⣼⣁⣀⣿
⣾⣿⣶⣄⣠⣤⣖⣒⣛⣛⣛⠛⠛⠛⠛⠛⠷⣤⣀⣠⢟⣠⡤⠖⠛⣉⣉⣡⣤⡤⠤⠭⢭⣽⣳⣦⣄⡀⠀⠀⢀⣴⢿
⠀⠀⠉⢻⣯⣁⠀⠀⠀⠉⠉⠉⠙⠃⠰⢤⣀⣀⣹⠿⢋⣡⠤⠖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣿⣿⣛⣭⠤⠾
⠀⠀⠀⠘⣷⡈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⡼⢛⣡⠖⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠉⠁⠀⣠⣿⠋
⠀⠀⠀⠀⢹⡿⣄⠀⠉⠓⢦⣀⠀⠀⣰⢋⡴⠋⠀⠀⠀⠀⣀⣠⠤⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⢣⠏
⠀⠀⠀⠀⠀⠹⣌⠳⣄⠀⠀⠈⠁⣾⡷⠋⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣌⡓⠤⣀⣼⠟⠀⡀⠰⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣼⢏⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠚⠁⢀⡴⠋
⠀⠀⢀⣤⡶⠶⠶⠶⠶⣦⡰⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⢀⣴⣟⣤⣤⡤⣤⣤⣤⡀
⠀⠀⠘⠻⢿⣴⣶⣶⣶⣟⠁⠀⠈⠙⠳⠦⣤⣤⣤⡤⠶⠚⠋⠁⠀⠀⢀⣀⣤⠖⠋⠀⠿⣷⣤⣤⣤⣤⡾⠏
⠀⠀⠀⢠⣾⣟⠛⠉⠙⢛⣻⣶⣶⠞⠛⠛⠛⢿⡏⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠛⠿⠶⠶⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢈⡏⠈⢻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠇⣌⣴⡶⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠘⢿⣿⠀⠀⣸⡿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢯⡙⢄⡹⣿⠀⠐⠇⢀⠞⠋⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⢿⣦⢻⡀⡁⠀⠀⣀⣴⡏⠀⣠⣿⢋⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⢹⢸⡇⣿⢸⡇⣟⣧⠀⣾⢻⠁⣿⠟⠙⢻⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢨⣰⠃⡿⢸⡇⠿⢸⣇⣯⠟⣰⡌⢸⢰⣿⠀⣴⠏⢉⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣚⡽⣡⡾⠃⣼⣿⣶⣦⣨⢎⡼⢋⡇⣿⢸⡇⢸⣿⠰⠏⠈⣽⣿⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣋⡾⠋⣁⣈⡙⠻⠾⢫⠌⠈⣇⢾⢰⡟⢸⡇⣼⡿⢰⣦⡾⠛⢻⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣩⡶⠟⠋⠉⠙⠷⣦⡄⠘⡆⢻⠀⡿⢰⢸⣀⡿⢁⣾⡏⡀⣠⡿⢻⣿⣗⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣯⣿⡿⠗⠛⠛⠿⣿⣦⡀⠀⢻⡄⣧⢸⡄⢁⠈⢸⡿⢡⣾⡿⠁⣿⡏⠀⣻⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⡿⣵⠾⠿⣷⣦⣄⠀⠹⣷⡄⠸⡇⢸⠈⣧⡘⠆⣾⢣⣿⡟⠁⣼⣿⡇⡞⠉⢼⣿⠁⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⣥⣦⣄⠀⠉⠛⢷⡄⢈⡿⠀⡇⢸⡆⡎⣧⢸⠏⣾⡟⠀⣼⣿⡿⠁⣷⡤⢿⡇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⣧⠉⠛⠻⠿⣶⣤⠈⠇⣼⠃⣸⠇⣸⠀⣷⢨⠞⣼⣿⠁⣼⣿⡿⠁⣸⣿⣷⣿⡃⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣾⡇⢹⣿⠳⢒⠲⢤⣄⠸⠀⣼⣣⢴⡟⢠⡟⢠⡏⠀⣰⣿⠃⣸⣿⣿⠃⣸⣿⡇⢸⠿⣇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⣿⣛⣠⣿⣿⣿⣿⣷⣇⣿⣼⣸⡷⠿⠏⢰⡋⠶⠟⣠⣼⡟⠁⣰⣿⣿⠏⢀⣿⣿⡇⣾⢸⡏⣷⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⢀⠻⣶⠿⠟⠋⢠⣾⢟⣿⠇⢀⣾⣿⣿⠁⣿⣾⣷⡿⣧⢹⣦⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣉⣴⡿⠃⢀⣼⣿⣿⡇⢰⣿⣿⠃⣿⠸⡆⢯⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢰⣾⣿⠟⠀⢀⣾⣿⣿⠟⠀⣼⣿⡇⢰⣿⠀⢿⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⢠⣿⣿⡿⠋⠀⣴⣿⣿⠇⣾⣿⠀⢸⡆⣿⣿⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢠⣿⣿⠟⠁⢀⣾⣿⣿⡏⢰⣿⣿⠀⢸⣧⢹⣿⡇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠻⣆⠀⠀⠀⣿⣿⠋⠀⣴⣿⣿⣿⡿⠀⣼⣿⣿⠀⠀⣿⠀⣿⡷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⢿⣶⣾⡟⠁⣠⣾⣿⠟⠋⠀⢀⣴⣿⣿⣿⠀⠀⣿⡀⣿⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣄⠉⠉⠀⠠⠛⠉⠀⣀⣤⣴⣿⣿⣿⣿⡟⠀⢸⣿⡇⣿⡷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠉⠉⠛⢦⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⠀⣿⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠉⠙⠛⠛⠋⠉⠀⠀⢀⣼⣿⡏⢠⣿⡇⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⣾⣿⡧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠁⢀⣼⠿⣿⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⢻⣷⢦⣤⣤⣤⣤⣴⣾⣿⡿⠟⠛⠁⠀⣠⡾⠋⠀⣿⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⢦⣽⠿⠛⠋⠉⠉⠉⠀⣀⣀⣀⡾⠋⠀⢀⣴⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⠿⠶⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠊⢛⣿⣿⠀⠀⠀⠀⠐⢦⣤⡀
⠀⠀⠀⠀⢻⡇⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⡖⠀⠀⡀⠈⣿⣿⣦⠀⠀⠀⢰⣦⣄⣠⣾⠿⠃⠙⢷⣤⣀⣀⣸⣿⠇
⠀⠀⠀⠀⢸⡟⠀⠊⠀⠋⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⠀⠀⠀⣷⠀⢸⡟⡏⡆⠀⠀⠀⠉⠙⠉⢁⠀⣠⣤⣤⡈⠉⠉⠁⠄⠰⠲⣦
⠀⠀⠀⠀⠘⣿⣄⠀⠋⡄⠀⠁⠔⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠇⠀⠀⠀⢹⠀⢸⣏⣧⢡⠀⠀⠀⢀⡔⠀⠀⢀⣽⣿⡿⠃⠀⠀⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⡠⢀⠀⠈⠀⡔⠀⠀⠀⠀⠀⢸⠀⡿⠀⡀⠀⠀⠘⠀⢸⣿⣿⢸⠀⠀⠀⠀⠙⠻⠾⠛⠛⠉⠀⠀⠀⢰⣄⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣧⡈⠋⢠⠀⠈⠤⠂⠀⠀⠀⠀⢸⠀⠁⠀⠁⢠⠀⠀⠀⠘⣿⣽⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠘⣤⠂⠴⣡⠀⠀⠀⠈⡀⢀⡄⠀⠈⠀⠀⡇⢀⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢤⣤⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣄⡀⢶⡄⠉⠆⠀⠀⠀⢳⢸⣧⡄⠀⠀⢰⣇⣸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢚⣶⡤⣶⣿⣧⣀⣠⣽⡗⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣍⠁⢨⠄⠀⠀⢾⡇⢿⣿⣄⠀⣸⣿⣿⣿⠏⠀⠀⠀⣀⡀⠰⠷⣤⣾⠁⣠⣿⣯⣉⣉⣙⣿⣷⡈⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠒⠒⠀⠀⠈⢿⣼⣿⣿⣇⣿⣿⣿⠋⠀⠀⠀⠊⢙⣷⣶⡔⠿⣁⣈⣽⣿⡟⢻⣿⣿⢿⣯⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣄⣀⣠⣤⠄⣀⣀⣀⣤⣉⠛⠿⢿⡿⠛⠁⠀⠀⠉⠻⠶⣿⣽⣏⣻⣟⣛⡛⣿⣾⣯⣭⣽⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣾⡀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠿⠿⠿⠛⠛⠉⠉⠉⠀⠀⠀⢀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢻⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠛⠉⠉⣽⡆⠙⠋⠉⠉⣿⣆⠛⠛⠉⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠐⠂⠀⠀⠀⡠⠀⢠⠆⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠏⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠷⠶⢻⡶⠶⠦⠒⠲⣿⠲⠶⠦⠄⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠂⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⢧⠀⢀⠀⠀⠀⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣠⠴⠆⠘⠿⠶⠂⠰⠾⠿⠶⠶⠒⣇⢲⠉⢷⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⠇⠀⠀⢝⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠏⠀⢀⡴⠊⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡠⠭⣭⣭⠥⠤⠿⠃⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣰⡀⢠⠾⠁⣐⣦⡎⠀⠀⠀⣤⣤⣄⡀⠀⣍⣉⣉⣉⡋⠁⠀⠛⠩⠡⠴⠄⣾⣿⣯
⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡀⠠⣶⣼⣥⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⡇⠐⢦⣀⠤⣈⡈⠛⠓⠬⠿⣯⡉⠉⠛⠛⢓⣶⣶⠶⠶⠿⠿⠿⢯⣄⣠⣶⣾⣷
⠀⠀⢀⡿⢷⣄⢦⣄⣀⣸⣷⣾⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠢⣉⠛⠻⣍⡛⠶⣤⣀⣀⡀⠁⠛⠂⠉⠏⠀⠀⠀⠀⣠⣴⢾⣿⣿⣿⣿⡋
⠀⠀⠀⣀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠂⢄⠈⠛⢶⡛⠉⠉⠛⠛⠷⠶⣤⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⡾⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁⠢⢀⠙⠷⣤⣴⣶⣾⣿⣭⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⡿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠃⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢈⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣛⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⠤⢶⠒⠠⣰⣯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣬⠅
⠀⠀⠀⠀⠀⢀⢺⠁⠉⠂⠀⠀⠀⠀⠀⠀⠘⢲⣿⣿⠅⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂
⠀⠀⠀⠀⢠⣢⠩⠄⠀⠀⠀⠀⠀⠀⠈⠠⠄⡤⢿⠋⠉⠀⡠⠊⠀⠀⠀⠀⢀⣄⠀⠀⢸⡹
⠀⠀⠀⠀⠀⡟⠸⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠈⠀⡀⠂⠀⠀⠀⠀⠀⠀⣮⢭⡆⠀⣰⣼
⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⢖⠉⠃⠀⠀⠀⢀⣴⣤⠮⠤⠀⣀⠀⠀⠀⠀⠀⡗⠐⡇⠀⠙⠁
⠀⠀⠀⠀⠀⡇⠀⠂⠀⣰⢫⠀⢸⠀⢀⡄⢸⣿⡿⠗⠒⠲⠭⣑⢦⡄⠀⠀⠮⢩⣷
⠀⠀⠀⠀⠀⢇⠀⡁⠀⣿⠈⡇⠀⡆⢈⠀⠈⢹⢿⣿⠒⠢⣤⡨⠋⠀⠀⠀⢰⠒⣿
⠀⠀⠀⠀⠀⢸⠀⠇⢠⡏⢀⠇⠀⢱⠀⠀⡀⠀⠀⠉⠁⠤⣵⡄⠀⠀⠀⠀⠸⣋⣿
⠀⠀⠀⠀⠀⠸⡔⠀⠈⠇⢸⠀⠀⠀⢃⠀⠈⢰⠂⠀⠒⠲⠾⠇⠀⠀⠀⠀⠀⢧⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡎⠀⠀⠀⠀⠢⡀⠘⡄⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠰⡎⢰⠁⠀⠀⠀⡀⠀⢝⣾⣷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣥⠃⠀⠀⢀⡄⢀⡰⢛⣽⣿⡎⠣⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣦⠔⠉⣠⣾⣿⠉⢇⠀⠀⠀⡠⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢉⣩⠿⠟⠃⠀⢠⣥⠀⠀⠈⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡛⠓⠛⠲⢶⣦⣍⡛⠙⠛⠶⣤⣀⠀⠀⡾⢰⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠀⠈⠿⠿⣶⣄⡘⡏⠳⣤⡗⠈⣿⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠘⠛⢿⣿⣄⠘⣧⠀⢻⡇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠈⠉⠉⠛⢷⣿⣧⠈⣿⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣶⣶⣶⣶⣶⣾⣿⣶⣤⣄⠀⠀⠀⠀⠿⢿⣧⣸⣿⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⢃⠴⠈⠉⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣐⣿⡿⣿⣿⣾⠁
⠀⠀⠀⠀⠀⣠⣺⣥⡦⠀⣀⠀⢀⢀⣴⣿⣿⠿⠟⠛⠉⢉⠉⣅⣠⣤⢆⣤⣤⢄⡀⠀⣾⣙⣿
⠀⠀⠀⡤⢾⣿⣿⣿⣿⣿⣿⣷⣾⡿⠟⢉⡠⢴⣉⣴⠶⠾⣏⠉⢢⠈⢢⣸⣿⣿⠇⡘⠻⠿⣝⣣⣄⡀
⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⡿⢋⢄⣪⡴⠛⠫⢄⠀⠑⢆⠈⣧⣤⣷⢿⡽⠛⢁⡾⠁⠀⠀⠀⠈⠳⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⢟⢴⡤⠯⣅⠈⠑⣄⢀⣷⣤⣾⣿⡿⣏⡽⠋⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠳⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣜⡛⢤⣤⣤⠷⠶⡟⢛⣿⣭⡷⠼⢛⣡⣴⣾⡿⠛⠀⢀⣀⣀⡤⢄⣠⡤⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠺⣥⣄⣉⣙⣛⣛⡛⢛⣛⣛⣋⣩⣴⣶⣿⣿⣿⠟⠋⢀⣶⣉⣀⣸⣿⣿⣷⣶⠟⡻⠄⡛⠚
⠀⠀⣴⣾⣄⣛⣻⣶⡄⡩⠿⠿⡿⢿⣯⣭⣭⣬⣿⡶⠾⣿⣿⠋⢀⣠⡔⠛⠋⣿⣿⣻⠋⢸⣿⡿⣷⢤⣴⣿⠆
⠀⠀⠈⠛⠻⠿⠟⠿⣿⣵⣦⣶⣿⠀⠀⠈⠉⣼⣁⣴⣳⣷⠟⠁⠘⠿⠶⣶⣿⣿⡟⠹⣿⣿⢿⣧⣀⢀⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⣔⢋⡀⣈⣿⠀⠈⠀⠀⠙⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⡶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣴⠋⠀⠀⣷⠀⠈⠹⣄⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⣤⣼⠁⠀⠀⠀⣿⠀⠀⠀⠘⣦⡤⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢸⠋⠛⠒⠶⠶⣿⠴⠶⠒⠋⠹⡆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠺⡟⠒⠒⠒⠒⠚⣿⠒⠒⠒⠒⠒⢻⠓⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣸⣷⠖⠛⠉⠉⣍⠉⠉⠛⠲⣾⣅⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣏⠁⠸⣆⠀⠀⠀⣿⠀⠀⠀⣴⠃⠈⣻⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⣷⣄⡀⣿⢀⣠⣾⣥⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠢⠀⢀⠀⢀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢠⠴⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣱⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠤⠤⠤⠤⠤⠔⠀⠒⠀⠀⠀⠈


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

⠀⠀⠀⢀⢠⠄⠐⠶⠀⠈⢈⡠⠊⠁⠀⠐⠃⠸⠃⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢟⠏⠉⠀⠈
⠀⠀⢰⢁⡾⠀⠀⠀⣀⡼⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⡔
⠀⠀⠈⠠⢤⣀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢠⠄⠁⠀⠶⠘⢲⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⡤⠀⠀⠀⠶⠁⢀⡴⡁⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢶⣶⢶⢲⠒⡆
⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠐⡷⣮⡏⠿⠘⢰⠃⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣤⣴⡖⡖⡆⠀⠀⠀⠚⠳⠧⠆⠎
⠀⠀⠀⠾⠀⢠⠆⠀⠀⠰⣄⠀⢸⣴⡇⠀⠀⠀⠀⠤⠔⠀⠀⠀⢲⣾⡏⡇⠀⡇
⠀⠀⡀⠀⠰⠈⠀⠀⠀⠐⢂⢼⣿⣷⠯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⣰⡾⣠⠀⠀⠀⠀⡤⣡⠻⡟⠻⡕⠀⠀⠀⠀⢠⢿⣷⠀⣠⡤⣶⣶⡶⠀⠀⠀⣄⠒⣄⠀⢠⠀⠀⡆⠂
⢠⠀⠀⢐⣿⠁⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠩⠬⠱⡰⠉⠁⠀⠀⠀⠀⠈⠁⠘⢀
⢸⡄⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡽⣆⠁⠀⠀⠀⠑⣃⠀⠀⠀⠀⠀⠀⣀⡝⠀⠄⠀⣤
⠀⢳⡄⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢸⣿⠁⠝⠀⠠⠄⠀⠄⡑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃
⠇⣺⡄⣾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⡄⠚⠃⠀⠈⠠⠢⠁⡀⠒⠬⣆⠀⠄⠀⠀⠀⠶⠄⠀⠁⠈
⣇⣟⣼⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠀⠀⠀⠀⠀⠸⠾⠛⠋⠈⠙⠀⣆⠀⠀⠀⠀⠀⡀
⠘⠋⠁⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣺⡇⠀⠀⠈⠢
⣤⠶⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠈⣆⠘⡇⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⡀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠨⡇⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡯⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢆⣰⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⢀⠀⢀⠄⠀⠀⢸⠀⠀⠀⠀⠀⢰⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⡃

⠀⢀⠀⣀⠀⠀⠀⣠⣤⠀⠀⣠⡄⢀⣠⡀⠀⠀⡤⠠⠄⠀⠠⠀⢀⣠⣄⣀⡀⠀⠀⣀⣀⣀⡀
⠀⠸⠀⠉⡜⠀⠀⠙⠿⠃⠐⠿⠀⠐⠛⢋⣤⣤⣤⣶⣶⣤⣤⣄⣈⠙⠉⠉⠁⠀⠻⢿⣿⣿⡿⠂
⠀⢠⠠⠊⠀⠴⢶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡈⠁⣀⣀⠀⠀⠀⠀⡀⠄⠀⢸⢰⢺⡇
⠀⡄⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠘⠿⣿⣿⣿⣿⣿⡿⡛⠛⠛⠛⠛⢻⣿⠛⠁⠀⢤⢠⣿⠇⠀⠀⠈⠈⠈⠁
⠀⠈⠑⠀⠀⣾⣖⠀⠀⢠⣤⠀⠀⠉⠉⠀⠀⠀⢡⠄⠀⣀⠀⠀⢠⣶⣆⠀⠈⠻⠀⠀⠀⠈⠛⠀⠀⠀⠀⣶⢠⡄⡆
⠀⢰⣿⣷⡄⠙⠇⠀⠀⠘⠋⢠⠀⠀⢠⠀⠀⠀⠉⠀⠀⢉⠀⠀⠀⠉⠁⢠⣴⡶⠿⡇⠀⠀⠀⠀⣍⡀⠀⠛⠘⠃⠃
⠀⠘⣿⠀⠀⠀⠀⠘⠃⠀⡀⢸⡇⠀⠈⠉⠡⢠⢀⣤⢠⠉⠉⠀⡤⠀⢀⠘⠻⢷⣦⡇⠀⠀⠀⢸⣿⠄⠀⢠⢀⣀⡄
⠀⠀⣿⠀⠀⠀⠀⢸⣦⠀⣿⢸⣷⠀⠑⠀⠀⠀⠄⠀⠀⠀⢀⠠⠈⠀⠈⣀⣀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⢸⠸⠯⡇
⠀⠂⠻⠤⢝⣤⡄⠀⣿⣆⣿⣿⡿⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⢿⡇⠀⠐⠄⠀⠀⢰⡀
⠀⠘⢷⣲⣼⡽⠀⠀⠈⠿⣿⠟⢁⣀⡈⠉⠀⠀⡆⢰⢶⡶⠂⠀⠀⡄⠈⠀⠀⠀⣰⣿⣿⣿⣦⠀⣳⠀⠀⢸⣶⣴⡇
⠀⠀⠀⠉⣿⣇⠀⠐⠀⢀⣴⣾⣿⡟⠓⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠰⣿⡄⠀⠈⠈⠉⠁
⠀⠠⡀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⠃⠀⠀⢰⣿⣷⡄⠐⠐⠄⠀⠄⠀⠀⠀⠀⣀⣸⣿⣿⣿⣥⣄⢼⠃⠀⢀⢤⣤⡆
⠀⠀⠀⠀⠘⣿⠿⡿⢛⡉⠿⠛⠁⠀⠄⠀⠀⠉⣿⣷⣄⠂⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⢿⠿⣿⢻⡆⢀⠀⠰⠛⠹⠇
⠀⠀⠀⣶⣤⠈⠓⠁⠋⡀⠀⠀⣴⠀⠘⡶⠀⠸⠏⢹⡏⠀⠀⣿⠘⣿⡄⠙⢛⣱⢇⡇⣿⢸⣇⡀⠛⠋

⠀⠀⠀⠀⠀⠀⣠⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠛⠷⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⡿⣱⡿⠃⠀⠈⢷⡝⠷⠀⠀⠀⠀⠀⣼⣿⣿⡇⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⠀⣘⠀⣁⠀⡃⢈
⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠄⢤⣤⣤⠀⡐⠀⠀⠀⠀⠀⢸⡟⠙⢿⠁⠀⣸⣿⣿⠀⠀⠀⠀⠀⠠⠠⠤⠠⠤⠠⠤⠠⠄
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠘⠦⣈⣙⣻⠔⠁⠀⠀⠀⠀⠀⠀⢻⣦⣬⢧⡾⢋⣼⠏⠀⠀⠀⠀⠀⢈⠬⠭⠬⠏⢈⡉⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋


⠀⠀⠀⣄⡞⣉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠒⢦⡄
⠀⠀⠠⠠⠄⢤⠀⠤⢰⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠙⢆⠂⠁⠔⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠀⣠⣶⠶⠹⡆
⠀⠀⢀⣔⡆⣰⣠⣶⣆⣀⡀⠀⠀⠀⠀⠀⢡⣄⣀⠀⡈⣧⠴⡞⡃⠀⠀⠀⠀⠀⠈⠻⠷⠀⢠⡶⠆⠀⠀⠀⠀⠀⠀⢻⡀⠉⠁⠉⠁⠀⢠⡇
⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢿⡜⠛⣻⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣶⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀
⠀⠀⢠⣾⠏⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢀⣀⡴⠉⠉⠉⠉⢳⣀⡀
⠀⠀⢸⠫⠠⣤⣤⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⣿⣻⣖⢄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣽⣛⣿⣿⣻⣿⣛⣻⠁
⠀⠀⠰⣦⣤⣽⣿⣷⣷⣦⣄⠀⠀⠀⠀⠀⠀⡆⠞⠁⠆⠰⠠⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠹⠿⠋⠉⠏⠉⠻⠟
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠐⢿⣀⣃⡜⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⠂


⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⡀⢈⣤⠶⣦⣌⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⢿⣿⣭⡇
⠀⠀⠀⠁⢾⡁⣾⠍⣹⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⣤⣾⣇⡸⡿⢠⡿⠟
⠀⠀⠀⠁⢈⠻⠷⠾⢋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠖⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⢀⣤⣆⠹⠁⠀⠀⠀⠀⡀⡈⢆⠀⠀⠀⠀⠈⢤⣀⣀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠆⢀⣫⣙⡟⠀⠀⠀⠀⢀⣞⡣⠀⠀⠑⠀⠀⠀⠀⠀⠈⢫⣧⣄⡀⠙⠠
⠀⠀⠀⢠⠄⢀⡐⢠⡿⠼⡿⠀⠀⠀⠀⠀⢸⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⣱⡀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⡀⢀⠶⣼⠀⠀⠀⠀⠀⠀⠠⢻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡼⡆
⠀⠀⠀⡀⠀⢠⢷⠘⠗⣻⠀⠀⠀⠀⠀⠀⣼⡟⣷⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⢷⠀⠀⠄
⠀⠀⠀⡇⠀⠀⠀⣔⠛⣾⠀⠀⠀⠀⠀⠰⡉⠛⠡⠁⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⢞⠳⢞⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠹⡛⠉⡇⠀⠀⠀⢠⣴⢿⡖⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠃⠀⠀⠎⠙⠗⠋⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢻⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠑⢀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⡁⠈⠁⡀⠐⠈⠁
⠀⠀⠠⠛⠆⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠸⠀⠌⢲⢒⢶⣄⢀⡃⠀⠀⠀⢀⡀⠙⡄⣀⣀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⡀⠀⠘⠙⠄⠙⢧⡄⠢⠀⠀⠸⠁⠁⢘⢧⡙⠷⣔⠲⠀⠀⠀⠀⠀⠒⠠⡀
⠀⠀⠀⠂⠈⠀⠀⠈⢃⢴⠀⠀⠒⠀⠀⣂⠠⠈⠀⠈⡤⢈⠳⠀⢁⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠊⡀

⠀⠀⠀⠀⠀⠀⠸⠿⠛⠿⠂⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝
⣻⣶⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⠀⠀⠀⠀⠑⠓⠀⠀⠀⠀⢸
⠛⡻⠀⠀⠀⠀⣀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣠⣴⡄⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣵⣠⣶⣶⡶⢶⣖⠒⠀⠀⠀⠀⠀⠐⠛⠀⠺⠿⠿⠀⠀⠀⠀⠀⡏⡀⠀⠠⣧⣷⣾⣷⣾⢿⠇
⠀⠀⠰⠶⡀⠳⣿⣜⢿⣛⠛⢛⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⣼⣿⠿⣟⡆
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣳⠫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⣄⡀⠀⠀⠀⠀⠘⣯⣾⡿⠁
⡀⠀⠀⠀⠀⠀⠣⡘⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣠⡴⠾⠟⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⣀⣸⣾⠆⠀⠀⠀⢰⣶⣶
⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⣠⠀⣄⠄⠄⠀⠀⠀⡟⠉⠀⠀⠘⠃⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠠⢤⣀⣄⣠⠎⣽⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠑⢤⣀⢹⣷⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⣀⣀⣀⣺⠩⠁⣀⣬⣤⣤⣒⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠛⢷⣿⡟⠢⠀⠀⠀⡀⢀⣿⣿⣶⡎⣀⠀⢈⣀⣠⣤⣴⡿⣽⣿⢾⡷⣷⡷⠟⢿⣿⣿⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⢀⣤⣾⣿⣿⣷⣿⣶⣿⣿⣛⣿⢿⣟⠿⣯⡭⠾⠃⠀⠀⠈⠙⢻⡿⠟⠉
⣤⣄⣤⣴⣆⠀⠀⠀⠀⠀⠰⠝⠿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠐⠀⣠⠬⠟⠚⡻⠟⠀⠀⠀⢀⡠⠔⠋⠁⠀⢀⣀
⠉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⣽⣿⣀⡀⡠⠄⠒⠀⠠⠽⠖⣚⣋⣉⠠⠄⠀⠶⠿⡿⠿⠂⠀⠀⠀⠛⠛⠃
⠀⠹⣿⠏⠀⠀⠰⠶⠶⠀⠀⠀⠀⣠⠒⠛⣿⣿⠋⠐⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠬⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠟⠛⡉⠉⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⢀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⡿⠂⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣶⣶⣭⣵⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠟⠯⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⢀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠃⢈⣿⡷⠂
⠿⠟⠛⠒⠤⠀⠀⢸⡁⡠⠄⠀⠀⢠⣶⣴⣤⣷⠄⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢺⠟⠛⠁⠀⢺⠟⠁⠀⠀⢰⠤⠤⠴
⠀⠐⠂⠀⠀⠀⠀⠀⠑⠚⠃⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⡀⢠⣶⣾⣿⣯⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⢀⣠⢄⣀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠁⠀⠿⣷⣷⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣦⢰⡇⢰⡆⣴⠛⣦⢰⡞⢷⡄⣾⢻⡆⣺⠀⠀⢰⣿⡆⠛⣿⠓⢺⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢸⣧⣼⡇⣿⠀⣿⢸⡇⠈⠈⣿⢸⡇⣹⠀⠀⣸⠟⣇⠀⣿⠀⢸⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣿⢸⡇⢸⡇⢿⣀⣿⢸⣇⣼⠇⣿⣸⡇⣸⣄⣀⣿⠛⣿⠀⣿⠀⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠁⠀⠉⠀⠀⠈⠁⠀⠈⠉⠀⠈⠉⠁⠁⠀⠈⠀⠉⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠉⠉⡁⠀⠈⠉⠉⠭⠹⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠠⠁⢠⡟⠙⠦⠒⠒⠒⠶⠦⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠁⡐⡽⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠄⢀⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⡜⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⡀⠀⠌⡜⠀⠀⣴⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠩⠄⢒⠒⠒⠂⠀⠀⡀⠀⢀⢀⣀⠎⡜⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠉⢹⠁⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠂⠀⠀⠀⠀⠈⠉⠉⠉⠰⠋⠉⢹⢁⡞⠀⢀⡾⣉⡀⠉⠐⠒⠒⠲⢤⡄⣀⣸⣀⣴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⢃⡞⠀⠀⡾⠁⠀⠉⠙⠒⠂⠀⠚⠁⠀⢀⠝⢋⠤⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⣀⠠⠀⠀⢴⣠⡀⠠⣡⡞⠀⠀⣼⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢊⠅⠊⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠊⠀⠁⠀⠒⠀⢤⡇⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠘⠛⠀⠐⠂⠀⠀⠀⠀⠤⠤⠊⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠞⡄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡼⠀⣾⠶⠶⠦⣤⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣼⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⢸⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠄⠉⠀⢀⣀⠀⠀⠀⠐⠀⠄⠀⠀⣼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠚⢀⢠⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠩⣾⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠤⠤⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠄⢀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠆⠉⠉⠉⠀⠐⠠⠀⠀⠀⠀⠀⠘⠃⠀⠰⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣡⠔⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣰⠟⠁⢀⠜⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢠⠃⠀⠀⡎⠀⠀⠀⢠⡿
⠀⠠⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠢⢀⠀⠈⢆⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠸⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠀⠈⢢⠀⠸⣄⡀⡇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠘⠛⢌⡀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⣾⠆⠐⠋⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡖⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠳⢲⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢈⡆⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡉⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠠⠤⣈⡉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠊⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠇⠀⠀⠀⡆⠀⠀⠀⡀⠀⠉⠁⠉⠋⠀⠓⠶⢮⣷⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠁⠂⢤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠂⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀
⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠁
⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢀
⠀⠐⠠⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⢠⠀⠀⠈⠙⠛⠛⠋⠉⠉⠉⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿