Supernumerary body part Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠠⠄⠀⠀⡠⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⠀⣀⡠⠄⢳⡐⢦⢶⣶⣿⣤⡒⠀⢢⡎⢤⣀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠀⠀⢳⣞⠱⠀⢸⣷⣎⣹⡿⣿⣷⣿⣿⣾⣇⣮⡅⢠⠇⠀⠀⢡
⠀⠀⠀⠀⠀⢀⠀⠉⢹⠦⡀⠀⠀⠀⣿⣷⣤⠿⣄⡈⠙⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⠋⣠⣆⣀
⠀⠀⠀⠀⠀⢂⠀⠠⠀⠀⡙⣦⣤⡼⠏⠈⡁⠀⠸⠿⢀⣾⣿⣿⣼⣿⣿⣿⣟⠛⣿⣿⣷⣴⠋⢤⠀⠀⡀
⠀⠀⠀⡠⠒⠙⠣⢤⣤⣴⡡⠃⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣽⣿⣿⣿⣿⣿⣿⣷⣝⢻⢿⣿⡀⠈⠀⣀⡿⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⠟⣿⣯⢻⣿⣿⣿⣿⣿⡜⢻⣷⡶⠊⣅⠀⠈⢆
⠀⢀⠔⠳⠠⠤⢀⡄⠠⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⠘⠛⣱⡟⠃⠈⠹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠀⣀⣎
⠀⡜⠀⠀⢀⠀⠈⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⠟⠃⠀⠀⠀⣌⣿⣿⣿⣿⣿⣿⣿⣤⣴⠞⠉⠈⢳⡀
⠀⠀⠀⠙⠃⠀⠀⢘⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⠁⠸⢄⡄⠀⣇
⠀⠲⠶⠤⣤⣴⣶⠎⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⣿⣿⣿⣿⣯⣃⡀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠙⠛⠹⠿⠭⠥⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡀⠀⢻⢀⣼⣷⠀⢠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣨⡇⠀⠀⢀⢤⣿⠀⣿⣆⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣶⠀⢀⡀⢈⠩⣷⠀⠂⠀⠀⣿⠀⣿⣮⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣤⠟⡇⢸⠃⠈⠀⣿⡄⠀⠀⠀⣿⢠⡇⠊⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⣦⢸⡄⠄⠀⡇⢸⠀⠀⠀⣿⡇⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢻⣼⠋⠁⠈⡇⣾⠀⠀⠀⣿⣇⢀⡀⠂⢻⣸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣿⠀⠀⠀⣿⣧⠈⠀⡶⢾⣿⠈⠘⠀⣿⣿⢡⣿⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⠀⣾⣿⠀⢻⡷⢸⣿⠀⠀⠙⢹⣧⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣼⣿⠀⠂⠂⢸⣿⠀⠀⠀⢹⣿⠀⣿⣤⣸⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⡿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠞⢿⣿⡂⠀⠀⢹⣿⠀⠀⠀⣾⣿⣾⣿⣧⣿⣿⠀⠀⡃⢰⣿⡀⠀⠀⠀⠀⣼⡇⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⡇⢀⣀⣸⣿⠀⠀⠀⣿⣿⢿⣿⣿⢿⣿⣷⡶⣿⣿⣿⡇⠀⠀⠀⢀⣿⢄⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⢸⣿⡇⢸⣿⣿⣿⠃⠀⠁⣿⡿⠀⠈⠙⠋⠻⣿⠀⠃⢻⣿⣷⠀⠀⢀⣾⠟⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢰⣿⣾⣿⡇⢾⣿⣿⠟⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⢘⡿⣿⡄⠀⣾⠏⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⡿⢿⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣧⣼⠏⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⢀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀
⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠐⠀⠀⠂⠒⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⡴⢳⣆⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡄⠀⠀⠆⣻⡾⣡⡞⣿⣦⣀⢻⡇⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡝⢦⣤⣶⣼⣗⣻⡋⢿⡟⠋⣸⡇⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⠈⣢⡝⠛⠛⠛⠿⡇⠉⠀⠈⠉⠛⢉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠟⠗⠉⠉⠉⣉⠁⠀⠛⠁⠀⡇⠀⠉⠉⠘⠛⠓⠶⠒⠶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⣐⠒⠒⠒⢳⠶⠶⠦⣦⣶⣷⠏⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⡟⣭⠉⠉⠉⣉⣀⠙⠂⠀⠀⡇⠀⠀⠞⠛⠘⠿⠤⢄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠘⠡⣤⡟⠀⠔⠀⠠⣾⠃⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡂⠉⠐⠚⡷⠋⠀⣹⣛⠀⠀⣶⡤⢌⡑⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢰⠃⠔⢷⣌⣋⣭⣄⠙⡄⠀⠉⠐⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡆⠀⠀⠀⣿⡇⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⠹⠿⠀⠀⠀⢠⡿⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣷⣿⣥⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠛⠛⠙⢉⡽⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢲⢲⠀⠀⠀⠀⠀⠀⠀⡀⠠⢖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⡿⠀⠀⠀⠀⠀⠀⠈⢀⡶⠓⠀⠀⠀⠀⡄⠳⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⣤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣵⡛⠈⠐⠲⠤⠀⠀⠀⠀⠀⠄⠀⢻⣤⠖⠒⣖⠀⠀⠀⣃⡀⠀⡀⠀⢠⡟⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢧⡀⠈⠲⢤⠴⠛⠀⠀⠈⠵⠂⡄⠈⠀⠀⠈⠉⠉⠅⠀⣠⡟⠀⠀⠀⠀⠰⠂⢋⣥⠀⠀⠠⡀⠤⠴⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠂⠤⠀⢀⣀⠀⠀⠀⠀⢹⣄⡀⠀⠃⠀⠉⠣⠀⠀⠀⠄⠠⣄⢀⠀⠀⠀⠀⠀⠈⠈⠀⠄⠀⢀⡠⠖⠉⠀⠀⢀⠊⠀⠀⠀⢈⡎
⠀⠀⠀⠀⠀⠀⠀⠸⢗⠣⢠⣤⣤⣀⢀⣠⠤⠁⠐⠃⠙⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⢝⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠀⠀⠀⠀⠈⡄⢀⣴⠖⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⢤⣠⡄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠰⠛⠉⢳⡀⠀⠀⠀⠀⡁⢸⠁
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⣠⡀⢀⣤⠔⣶⡶⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⢀⠐⠾⠷⠀⠀⠀⢀⠆⠈⢦⡀
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠘⠛⡻⠋⠑⠦⠶⠎⠄⠤⠤⠤⠄⠀⠀⠠⠤⠤⠤⠄⠀⠀⣠⣄⡙⣃⠏⣸⠇⠀⣀⣀⡠⠔⢋⡀
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣴⣴⡀⣀⣀⣰⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣀⠘⡇⠀⠍⣩⠴⡋⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⡆⡉⠉⣤⠎⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⡸⠀⢀⡰⠂⠀⠉⠳⣁⢘⣉⢁⣬⠚⢿⣑⡌
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⡴⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢸⠀⣰⠃⠀⠄⠀⠊⠀⠀⠁⠀⠠⢤⠤⣤⣀⣀⣹⡴⠴⣦⠬⠌⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠒⣏⠀⠀⠀⠀⠀⠀⠀⣶⣀⡴⠟
⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡐⠚⠉⢀⣤⣴⣠⣤⣠⣤⠶⢦⠤⠤⢀⠀⠀⠀⠀⠀⣡⣬⣽⠿⢀⣀⡀⠴⣄⠀⠈⠂⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⡤⠾⠉⠉⠉⠀⠀⠀⠀⢀⠋⠀⣠⡾⣧⡖⠁⢠⡄⠀⠈⣻⣠⠀⢰⣿⠞⠛⠑⠠⢆⣄⡈⠛⢦⣄
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠎⣁⣈⣩⣤⠟⠁⠀⢠⠃⢀⣼⠟⠚⠛⠋⢹⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⢦⠀⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠉⠀⠀⠀⠀⠀⡔⢁⣶⠿⠋⠀⠀⠀⠀⠀⣄⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠨⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⢶⢿⡏⠀⠀⠀⠀⠀⠀⡾⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⠚⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⠂⠛⠒⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⠀⠀⠂⡀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠠⠄⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠐⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢤⠔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⢤⡴⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠔⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠐⢠⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⢀⡠⠊

⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣇⣘⣿⡏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢠⠂⠘⡃⠀⠀⠀⢸⡇⠰⣿⡇⠀⠀⠀⠀⠹⠀⠠⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡚⢁⡀⣹⡠⠀⠀⢸⡏⢹⣿⡇⠀⠀⠠⣈⡅⢀⡙⣄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢤⢠⡇⠈⠧⡜⢡⡦⠀⢨⡇⢸⣿⣇⠀⠠⣴⠈⣦⠎⠁⢹⢀⡼⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⣆⠯⣇⠀⠘⣓⡟⠀⣄⡾⠁⣹⣿⡿⢦⡂⠘⠯⣛⠂⠀⡟⢋⣴⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠩⠟⠛⢧⠘⣤⠏⣹⡤⠛⢁⣴⣾⣿⣿⣶⣤⠙⠳⢴⡁⢻⡜⢃⠿⠛⠻⣈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠋⢀⢄⣤⣦⡘⣟⣁⣠⣼⡿⠟⠉⠀⠀⠙⠻⣿⣦⡄⣉⠻⣡⡾⢤⡀⠀⠓⠤
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡁⠀⢸⡧⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠷⡺⡀⠀⢉⣄⠀⢀⠀⠀⠄
⠀⠀⠀⠀⡀⠠⣘⡦⢀⡱⣄⣨⠗⣨⠷⠞⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⠷⠮⣑⢯⣄⡴⢃⣀⣼⡠
⠀⠀⠀⠀⠀⠀⠉⣉⣉⡟⠋⣩⣾⠁⣠⣹⡟⠀⣰⠂⣀⠀⠀⠀⠀⠀⠀⡈⠘⡄⠀⢿⣇⡄⢘⣮⣍⠙⢯⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠞⢵⠋⠴⠟⠹⠉⢴⣾⡿⠓⢚⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠓⠛⢿⢾⡄⠙⠏⠳⠆⠹⡮⢓
⠀⠀⠀⠀⠀⠀⠁⠈⠀⢠⣀⣐⠴⢋⣸⠀⢐⡇⠰⠄⡀⠀⠀⠀⠀⠀⠀⡀⠰⠆⡽⠠⢸⡎⠹⡦⣆⣠⡄⠀⠡
⠀⠀⠀⠀⠀⠂⠴⡴⠚⢹⣿⡜⠀⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣤⠈⢱⢿⠍⠓⡦⠆⡑
⠀⠀⠀⠀⠀⠀⣀⣀⣰⠙⠃⠘⢛⣷⡟⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡜⢷⣋⡙⠂⠙⠑⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠘⢀⣀⣀⣹⢋⢇⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⡟⠙⣅⣀⣠⡀⠃⠁
⠀⠀⠀⠀⠀⢀⡀⢀⣠⠴⠻⣹⡇⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠀⡿⣜⠓⢦⡄⣀
⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣾⠁⢹⡆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⢺⠇⢹⣦⡀⡀⠈⠁
⠀⠀⠀⠀⠀⠘⠈⠉⢱⠃⠘⠀⠀⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠹⠀⠘⠃⢹⡄⠉⠉
⠀⠀⠀⠀⠀⠀⢀⢴⠞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⠄⠀⠀⢀⣄⣠⠀⠉⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⣀⢀⠀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣀⡀⠀⠀⠀⢀⡄⠀⢀⣀⣰⣿⣘⡧⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠸⠚⠀⣈⣻⢙⣅
⠀⠈⠀⠀⡀⠀⢉⡁⠀⠈⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠈⠁⠂⢴⢏⡄⡤⢂⣵⣾⣿⡇⠀⠀⠀⣀⠠⠒⠉⠀⢀⡀⠀⠀⠀⠀⠀⠠⠤⠘⢷⣞⡗
⠀⠰⠆⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⣄⠀⠀⠀⠀⢸⣷⡟⠛⠀⠈⠙⠿⣷⡦⠀⠊⠁⣀⠤⠖⠚⠁⠀⠀⠀⠀⠀⠘⠋⠛⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠠⠤⢀⣀⡈⠁⠂⠀⣹⡁⣀⣒⠓⢂⠀⣄⣫⠤⠒⠈⠁⠀⠀⢀⣀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠀⠤⢤⠄⢀⣀⣈⣁⠐⢷⢠⡍⣹⢀⡄⠋⡠⢼⡃⠀⠒⠀⠉⠁⠀⠀⠀⢀⣀⡀⠀⠀⢀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢏⠀⠧⠰⠗⠔⠀⢊⠷⠸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⡶⠶⠖⠖⠰⠄⢑⠤⠤⣤⠤⢤⡤⣒⣈⣃⣤⠤⠤⢴⠔⠙⢠⣤⡬⠤⣾⠁⢳⠀⠀⠒⠒⠒⠂⠂⢐⣀⠐⣂⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⢀⡠⠇⠘⠂⠀⡏⠀⠤⣠⡤⠞⠃⠀⠀⠀⠈⢀⡀⠀⠦⢀⡠⣿⠀⠛⡀⠀⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⡀⠁⢦⠀⠀⠠⠀⡙⠦⠘⠁⠀⠐⠀⣀⡈⠙⠆⠀⢈⡱⢆⠈⢴⣤⠄⡤⢄⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠈⠁⠀⠁⠀⠁⢈⠀⠀⠀⠈⠁⠀⢚⠀⠀⠀⢠⠏⠀⠀⠀⢀⣝⣹⠠⠀⠈⡗⣔⡈⠀⠈⠀⢼⠡⢒⡆⠀⠙⠒⠈⠁⠀⠀⠈⠀⠠⠤⠠⠄⠆
⠀⠐⠒⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡿⢦⡄⠉⢀⠀⢀⡀⠉⠐⠈⠓⠂⠀⠈⠀⠁⢀⡀⠀⢸⡟⢹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠰⡇⠀⡸⠀⢸⡅⢾⡇⢰⡔⠒⠲⠄⠀⠀⢺⣿⡄⠘⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⢀⡸⠀⡇⠀⢸⡇⢙⣳⡶⡟⢉⣀⠀⠀⡀⢸⠠⡅⠀⢳⠸⡧⣁⠀⠀⠀⠀⠀⠀⠀⣐⠀⠒
⠀⠠⢀⠀⠈⠀⠀⢈⠀⠀⠀⠀⠀⠀⣏⠘⠃⠀⣧⠀⠘⢷⣾⡿⣧⡼⢿⣷⠶⡞⠃⢸⠀⠹⣄⠘⡇⢇⠀⠈⠉⠒⠂⠠⠄⠀⠭⠤⠄⠠⠀⠴
⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⣴⠇⢠⡖⢻⡉⠀⠀⠀⣿⢉⣀⠈⠀⠸⠂⢸⡏⠀⢹⠾⢢⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠒⠒⢘⣗⠂⠈⡇⢰⠇⠀⠠⠀⠉⠉⠉⠉⠀⠀⠀⠸⣇⠀⢸⠀⠉⠠⢄⡈⠀⠒⠂⠤
⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠉⠀⢠⠇⢁⠁⠀⣸⠀⠀⠀⠀⣷⡄⠀⠀⠀⠘⠀⠀⠘⢇⠀⠈⠀⠀⠀⠀⠁⠂⢄⡀
⠀⠠⢄⢁⡀⠀⣀⣄⠀⠀⠀⠀⡀⠄⠈⠀⢰⠀⠾⠁⣡⡿⠀⠙⠆⣆⣈⣿⢆⡀⡔⠀⢾⡰⠒⣼⠄⠁⠀⠀⠐⠠⡁⠤⡀⠀⡀⡀
⠀⠐⠀⠒⢀⠀⠀⠒⡀⠄⠂⠁⠀⠀⠀⡠⠋⠀⠀⠶⡺⠄⠳⠙⠀⠋⣉⠒⠀⠁⠀⠀⢸⠉⠁⢸⢀⠇⣳⣄⡀⠀⠀⠀⢄⠑⠒⠀⠒⠒⠂⢀
⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢚⣳⠆⡀⠀⠀⢿⣿⢛⣧⠰⡯⠿⠋⠛⠋⢵⡤⠜⠀⠀⠀⠋⢺⣦⠇⠈⠁⠠⢄⠀⠈⠄⠠⠄⠄⢤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⡀⠄⠈⠙⠃⠀⠀⢀⣼⡿⠻⣇⠂⠀⠀⣤⢼⠷⢦⠅⠸⢆⡀⠀⠀⠀⠁⠀⠘⠒⠤⣀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⡀⠔⠊⠀⢈⣇⠐⡸⢀⣝⠀⠠⡜⠀⠼⠆⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡤⠒⠁⠀⠀⠀⠀⠸⡟⠀⠀⡌⠻⣤⠀⠸⡇⠀⠀⡆⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠰⢂⠈⠀⠔⡧⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠐⠒⠂⠀⠒⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠍⠐⠋⠥⠤⠤⢷⠐⠀⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡒⠒⠂⠀⠲⠒
⠀⢉⠈⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡾⠆⠀⠰⠀⡀⠀⠀⢸⣮⠁⣯⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠠⠌⠁⠠⣬
⠀⠘⠁⠀⠀⠂⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⣸⠧⠈⠀⡈⠐⠁⣶⢆⢸⣿⠄⢠⠄⡈⠉⠁⠐⠒⠒⠢⠤⠤⣄⣀⡀⠃⠀⠘⠁
⠀⠈⠉⠉⠉⠉⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⣁⡀⠄⠀⣿⠀⠀⢀⠁⠀⠈⠁⢊⣽⠴⠀⠐⠇⠀⣠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠁⠉⠳⡄⠀⠁⠀⠀⠂⠀⠀⠀⠰⠀⢹⣷⠀⠀⣤⠎⠀⠀⠀⡡⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠁⠂⠄⡀⢠⡇⢰⡇⠀⠈⠀⠀⠃⠀⠤⠀⢸⡽⡄⢸⡇⢠⠀⠀⠀⢀⠀⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⢀⠀⠘⠅⢒⣸⠃⠀⠀⠀⠀⠀⠀⡄⠀⢸⡀⣧⣀⣧⠴⣜⢂⠉⠀⠠⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠀⠈⠀⠀⠀⠐⠀⠀⠀⢠⢻⡿⢃⠈⢻⡉⢁⣀⣀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡟⠀⠃⢠⡶⠋⠈⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠉⠡⢁⠀⡠⠄⢠⡄
⡄⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠂⠐⠉⡅
⠘⢄⠀⠠⠋⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠸⡀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⣄⠈
⠀⠀⠀⠀⠁⠂⠀⠀⠞⠀⠀⠠⣀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠣⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⢂⠀⠀⠀⠀⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠴⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⢰⠂⠀⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠤⠀⠀⠀⠀⠠⠤⠤⢄⣀⡠⠆⠀⠀⠠⣀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣷⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣤⣀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⢶⠛⠛⡟⠛⠛⠿⡶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⠈⠀⠈⠀⠀⠀⠀⠁⠤⠜⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠙⠓⠲⢤⡀⠀⠀⠀⠐⡁⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉⠈⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢀⠤⢌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⢿⣄⠤⢤⡀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠁⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠁⠀⣀⡀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠈⣿⠀⠀⠧⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣘⣅⠀⡣⠀⠀⢀⠈⠻⣦⡀⠀⠀⠸⣇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⠐⠜⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⣤⣤⣄⣀⠀⠈⠙⠻⢤⡀⠀⠀⠉⠁⠈⢿⡄⠀⠀⢻⡖⠑⢦⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠃⠈⠻⣆
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣠⠆⠀⠉⠳⣦⡀⠀⠀⠹⣦⠀⠀⠀⠀⠸⣿⡀⠀⠈⣷⠀⠈⡠⢄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠘⢷⡀
⠀⠀⠀⢠⣟⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣴⠟⠀⠀⠀⡄⠀⣿⡇⠀⠀⢹⡆⠀⢃⣀⠇⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠘⢻⡄
⠀⠀⠀⣼⠁⠁⠉⠀⠀⣠⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⣿⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠴⠦⡄⢿⡀
⠀⠀⠀⢿⠀⠀⠀⠀⢰⡏⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⢸⡇
⠀⠀⠀⠸⡇⠎⠁⡄⠀⢧⣀⠀⠀⠀⠀⠉⠻⣶⣤⣀⣠⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⡀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠈⠹⣆⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⠂⠀⢘⠙⢲⢤⣤⣤⡤⠞⠋⠀⡠⠒⣴⠏
⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠑⠋⠀⢀⡀⠀⠀⠀⠀⢘⡾⠋
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⡀⠀⠁⠀⣁⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠙⢦⣿⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⢠⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠁⠀⡖⠂⠀⢀⠖⡄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠉⠀⠀⠀⠀⠱⡄⠀⠀⠐⠓⣄⠳⠯⡄⡎⠖⢃⡜⠂⠀⠀⢀⡖⠁⠀⠀⠐⠈⢀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠀⠀⠈⠣⣆⠀⣈⣎⠓⠶⠆⠶⠞⢻⡎⠀⣄⡶⠋⢀⠀⠈⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⣄⡼⢦⡼⠋⠀⠠⣤⡤⠀⠈⢻⣴⠾⡀⣄⣁⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⣴⠇⠀⠉⠀⠀⠀⠈⠁⠀⣧⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣄⣊⢙⡆⠀⠤⠀⠙⠀⠠⠄⢀⣟⢉⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠠⠀⠠⠠⠤⠄⠐⠈⠁⠀⡩⣏⣙⠦⣀⣀⣀⣀⣀⡠⢎⣙⠯⡀⠀⠉⠐⠀⠤⠤⠀⠀⠤⠌⠠⠆⢀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⢁⣾⠿⠋⢻⠛⠻⢷⡄⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠢⣄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡱⣤
⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠀⠐⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⣢⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⠀⠀⡀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣀⠀⠀⠀⢀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⢂⠀⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠠⠄⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⠁⠀⠀⠘⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠂⣄⠀⠀⠀⠀⢠⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠆⠀⠀⠀⢀⠀⠈⢄⠐⠂⠐⠂
⠀⠀⠀⠀⠀⠀⢄⠻⡇⠀⠀⠀⡸⡿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠔⠀⠀⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠐⠂⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⡀⠀⠀⡀⠀⠐⠀⠀⠄⡀⠐⠄⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⠀⠰⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠅⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣈⠒⡑⣂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⡿⠇⠿⠿⠅⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣰⠀⠠⡄⡄⢀⠗⢃⡀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡉⠋⠐⢠⡀⠀⠀⠀⠀⠀⢀⡀⠛⠊⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠖⠖⠀⠀⢇⣁⠀⠀⠀⠀⠀⡐⡱⠀⠀⠀⠀⠀⠐⠴⠀⠀⠀⠀⠀⠊⡁⠀⢀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠁⠀⠁⠀⠀⢒⡄⠀⢸⠀⠀⠀⠀⠀⠁⠀⢀⠊⠁⠙⡀⠀⢀⡇⠀⠀⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠸⠁⠀⠀⠀⠀⠀⠨⢴⠀⢈⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠁⠀⠂⣠⠀⠀⠀⢈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠀⠀⡇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡇⠀⠡⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⡊⡂⠀⠀⠒⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠉⠁⠒⠪⠀⠀⣰⠃⠀⠀⠀⠀⢆⡀⠀⠉⠒⠊⠉⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡡⢄⡀⠀⠀⡠⢎⡘⠀⠀⠀⠀⠀⠐⣈⢄⠀⠀⠀⠀⡠⢆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠅⠨⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⡇⢈⠀⡡⠌⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡙⢦⡀⡶⢖⠶⡄⣠⠞⣡⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣧⠙⢡⣾⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣄⣎⡻⢈⠸⢻⣇⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠋⠉⠀⢄⡨⠋⡄⢲⠀⡌⢫⣁⠀⠈⠉⠛⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⢀⣀⠄⠀⣅⠂⡡⢬⢄⠁⢈⡂⠀⢄⣀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠟⠁⠀⢸⠄⣸⡄⢛⢠⣧⠁⡆⠀⠀⠙⢧⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⣏⠺⢵⢿⣇⠕⢫⠇⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠘⢦⡀⠉⢄⡼⠋⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣟⣿⣻⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣿⣟⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢾⣿⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣻⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡉⣸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⢦⣥⣾⡇⠠⢀⠾⠟⠋⡤⢁⢴⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⢤⣴⡿⠀⠈⢁⡠⢄⠀⠀⡵⢦⢸⣾⢢⣀⣀⡀⠀⢀⡀⢀⠀⠔⢀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⢹⣏⢀⣚⠿⠗⠂⢰⡍⠀⠺⡃⠀⡇⢈⣿⣻⣿⣿⣿⣶⠃⠈⢀⣀⡠⣠⣾⣿⣿⡀⠀⠀⢀⡔⡋⠙⣿⣿⣀⡿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⡁⠀⠀⠀⠘⣿⠶⢊⡇⠀⣇⣾⣷⣿⣿⡟⣹⣿⣴⣷⣿⣾⣿⣿⣿⣿⣿⠃⣠⡖⣡⢊⣴⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⢙⡏⠀⠢⡀⠀⢹⣿⣼⣳⣆⣙⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡁⠈⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⢀⠤⠫⢀⣀⠉⣈⣶⢺⣦⣟⡩⢷⣷⣶⣬⣙⣿⣿⣿⣿⣿⣿⣿⡿⣛⠁⢾⣭⣶⣶⠏
⠀⠀⠀⠀⠀⠀⣠⡄⡸⡟⠁⠀⠀⠀⠉⢻⣟⣻⣾⣩⣟⣙⣼⣿⣧⣄⣽⣿⣎⢻⣿⣿⣿⡛⢚⡵⣣⡼⠁
⠀⠀⠀⠀⠀⢐⡁⠀⠟⠀⠀⠀⠀⢀⣠⣴⣿⣿⣏⣿⣿⣿⠿⣷⣿⣿⣿⣿⣿⣏⡏⣸⣿⣿⣿⣿⣯⢶⣖⡀
⠀⠀⠀⠀⣠⠈⢾⠗⠀⠀⠀⢀⡠⠾⢛⣿⣧⡟⣛⠛⢿⣿⣷⣿⡏⣿⣜⣿⣯⡿⢋⣌⣽⣿⡿⣿⡃⢿⣍⠿⠓⠄
⠀⢀⣤⣄⣈⠘⠃⠀⠀⠀⣴⡉⠀⠔⣲⣿⣿⣿⣿⣿⣶⣤⣈⠻⣧⣼⣿⣿⠿⡓⣼⢿⣿⡯⣥⠿⠓⠈⠛⠆⣾⣿⢷⡄
⠐⠩⠷⠋⠀⠀⠀⠀⣰⠊⠐⢹⡆⠀⠛⠗⣿⣿⣿⣿⣟⣿⣿⣦⢾⡿⠛⠧⢸⣿⣯⠉⠙⠮⢴⣦⠀⠀⠀⠀⠙⣿⡟⠁
⠈⠁⠀⠀⠀⠀⠀⣮⡀⠀⣴⠟⡠⠾⢿⣷⠈⠭⢾⣳⣿⣤⣽⣿⣿⣻⣿⣷⡌⠙⠛⠦⣄⠀⣤⢾⢀⠀⠀⠀⠀⠈⢣⣾⣣
⠀⠀⠀⠀⠀⣰⢧⢠⣽⠆⠁⢾⣷⠄⠈⠀⠀⠀⠀⣁⡙⢛⣿⠏⠁⠉⢿⡿⠁⠀⠀⠀⠀⢈⣷⣼⣄⡀⠀⠀⠀⠀⢸⠿⠟⠆
⠀⠀⠀⠀⣴⡡⠀⣸⠋⢀⢦⡄⠁⠀⠀⠀⠀⠀⠰⠛⠗⢼⣿⢫⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠛⢀⠉⠇⠀⠀⠀⠀⠈⢤⣶⣶⢄
⠀⠀⠀⢠⠃⠁⢸⠃⠀⢈⡛⠀⠀⠀⠀⠀⠀⠀⠱⢴⡆⢨⡟⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⢿⡆⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⡐⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠁⢸⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⢸⠃
⠀⠀⢠⠛⠓⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⠀⡼⣻⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠺
⠀⠀⣠⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⡿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢱⡄
⠀⢰⢿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠛⠁
⠀⠰⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡣
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⣻⣉⣝⣛⣟⢋⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠿⠛⣛⣛⣛⣛⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢛⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣉⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢋⣴⣿⣿⠟⢋⣩⣤⣴⣶⣶⣤⣬⣉⡛⠿⣿⣷⣦⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⣴⣿⣿⣋⣤⣾⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣷⣌⣻⣿⣿⣄⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⣾⣿⣿⣿⣿⣿⠟⣋⣤⣶⣾⣿⣿⣷⣶⣬⡙⢿⣿⣿⣿⣿⣿⣆⢹⣿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣭⣍⣙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⡀
⠀⠀⠀⠀⢀⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣾⣿⡿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⠟⣠⡌⢿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢁⣼⣿⣿⣦⠉⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⡿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣥⣬⣭⢹⣿⣿⣿⣿⣿⣿⠋⣭⣤⣼⣿⣿⠟⣡⣿⣿⣿⣿⣿⡏
⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡀⣿⣿⡿⢛⣡⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣉⢸⣿⣿⣿⣿⣿⣿⠀⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢸⣿⣿⣿⣿⣿⣿⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⢸⣿⣿⣿⣿⣿⣿⠄⣿⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣥⣿⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⡖⣶⢰⣖⡂⣶⢲⡄⡴⠒⢢⣖⡒⢰⠒⡦⢲⡖⢲⡆⣴⠒⣦⢰⣦⢰⡆
⣀⣀⠀⣀⠀⢀⢀⣏⣉⣸⣯⣅⣿⢹⣆⢷⣤⣘⣯⣅⣸⣋⡁⢸⣇⣘⣇⢻⣤⡟⣸⡏⣿⡇⢀⠀⣀⡀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠐⢀⣠⡄⠤⠴⠤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⣠⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠈⠀⠂⠂
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣂⢄⡀⡀⠀⢄⡀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣾⣍⣹⣿⣿⣆⣱⡈⣦⠀⢉⣳⣤⠀⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠄⠂⢠⣾⣿⣵⠿⠻⣿⣿⣿⠛⣹⣽⣿⣿⣛⣿⣏⣽⣉⡀⢠⣶⢿⣿⠛⢦⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡴⠟⠉⠁⠀⠀⠀⠀⢰⢏⣽⣿⡿⠹⣿⣍⡀⣀⡀⣸⣟⠉⠁⣸⠂⠀⠀⠈⢣⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣺⣿⠏⠀⠀⠈⠙⠙⠋⠙⢏⢻⣆⠀⠁⠀⠀⠀⠀⠀⠙⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃

⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⡦⢤⣄⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⢹⠆⠀⢸⠏⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠀⠀⣀⣠⢴⢾⠀⠀⠈⠳⠄⣀⣀⣀⣀⣀⡤⣤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⢠⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⠈⠁⠀⠈⠀⠀⣀⡴⠖⠉⠉⠀⢀⣠⢴⠇⠀⠀⠀⠀⠢⢤⣀⣀⣀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡆⠀⠀⣻⡷⡚⠁⠀⠀⠀⠒⠉⡉⠠⠁⠀⠀⠀⠀⠀⠀⠀⣈⢝⡟⠀⠀⠀⠈
⠀⠀⠈⠛⡓⠦⢤⣤⣇⣀⣈⣷⣼⣿⣄⣴⠏⠀⠈⠾⡉⠉⠉⠉⢀⡄⠀⠀⠀⠐⠦⣄⡠⠔⠉⠀⠘⠁
⠀⠀⠀⠀⠘⣆⠀⣀⣙⣿⣿⣿⣿⣿⣿⣯⡤⢴⠶⠓⠛⢯⣙⣉⠉⣄⠀⢱⣄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠾⢛⣿⣿⣿⣿⣿⣿⣿⠉⠁⠘⣗⠲⡶⢧⣤⣯⣭⣶⣪⣍⣀⣄⠀⠀⠰⡄⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠋⠁⠀⠸⣿⣿⣿⣿⣿⡿⠙⣎⡆⠀⡸⠀⢈⠹⣽⠉⣤⢀⡁⢠⣜⠁⠈⠉⠉⠛⠛⠒⠒⠒⠒⠬⠷⠤⣄⣀⣀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢸⠃⠀⣸⣷⡀⠀⠀⠈⣷⣁⠄⠈⠋⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣾⠐⠁⠀⠀⠉⠀⠀⠀⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢀⡼⠁⣀⡤⠒⠉⣿⠉⠢⠀⠀⠀⠀⠘⣦⣠⠄⠀⠀⢈⡾⠁⠀⠀⠀⠀⠀⠀⠠⠤⣾⣇
⠀⠀⠀⠶⠿⠾⠯⠭⣄⠀⠁⡗⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠒⢤⣀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠇⠀⠀⠀⠘⡄⠀⠀⢀⠠⠤⠤⠤⣄⡀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣌⡓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠋⣰⠆⠀⠀⠀⠀⠀⠈⠙⢿⠃⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⣞⣡⠀⠀⠀⠀⠀⠀⠀⣀⠈⢧⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢫⣶⣿⣿⣶⣮⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡫⠝⠉⢉⣀⡈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡠⢊⣠⣤⣀⠈⠑⢂⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⠀⣼⣿⣿⣿⣷⣦⣀⣂⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⣉⠁⠀⠀⣈⠑⠳⡀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⡠⠊⢁⣠⣤⣤⣤⡈⠢⡉⠢⣤⣤⣈⡛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠁⢠⣿⣿⣿⣿⣿⣿⣦⣌⠀⢀⠀⠀⠀⠀⠒⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢘⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⣉⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠈⠄⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⢻⣯⡉⠛⠋⢉⡀⠒⠃⢀⣁⡈⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣯⣉⣉⣤⣴⣶⣿⣿⣿⣿⣷⡈⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠊⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡫⠙⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡐⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠻⡦⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢱⢡⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡷⡀⠀⠀⠀⢠⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣱⠀⠀⠀⣾⢾⡃⠀⠀⠀⢀⣀⠀⠀⠀⢀⠏⡹⠀⠀⣼⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣤⣀⠘⢦⠳⡀⢠⡾⠋⠉⠙⢦⢀⣞⠜⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⢷⣹⣿⣠⣤⠤⠤⠬⠿⣎⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠡⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⠛⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡏⢀⡏⠎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡏⠱⡌⣄⠈⢿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢸⢻⡀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡞⠀⠀⢻⠹⠂⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⣇⣣⠀⠀⠙⢷⣄⠀⠀⣠⠞⠀⠀⠀⢸⣸⡄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢯⡣⡀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⡟⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⢉⣉⣭⠤⢬⣭⣉⣙⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⢀⣴⣪⣅⣀⠀⠀⠀⠀⠀⠉⠙⠲⢌⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣠⡞⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⣀⣀⡤⠀⠀⠀⣠⣶⠾⠿⠶⢦⡀⠀⢀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠙⠛⠉⠉⠀⡄⠀⣼⡟⠀⠀⠀⠀⠀⠃⠀⠀⣧⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⣀⠄⠀⢸⠀⣿⠀⠀⠀⢀⠄⡀⠀⠀⠀⣿⢀⠴⢤⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠠⣠⠼⣰⠏⠀⠀⣰⠋⣀⠈⠐⠀⢰⡟⠈⠀⡼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠁⠀⠀⣼⠁⣀⠀⠁⠀⣰⡟⠀⢀⡼⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⢰⠇⡀⠀⠁⢀⣾⠏⠀⢀⡏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠈⠁⣰⡿⠃⠀⠀⠘⠄⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡁⣰⡟⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢠⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠚⠁

⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠦⣄
⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠘⢦⣀
⠀⣾⡁⠀⠠⠶⣏⠖⠀⠀⠀⠀⠱⡝⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⣤⣤⠤⠤⠤⠴⠖⢲⡆
⠀⠙⠿⣵⣒⣠⡄⠀⠀⠀⢀⣠⠜⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡀⣤⢤⣠⣀⣠⣤⣤⠤⠴⠚⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢿⣆⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢾⡶⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢄⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣷⢂⠀⠀⠘⡗⠒⠲⠶⠶⠤⢤⣤⣄⣀⡀⠈⠻⣳⣄⡀⠉⠛⠒⠶⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⢸⢺⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣬⡛⠯⣗⡦⢤⣀⠀⠀⠀⠈⠉⠛⠲⢤⡀
⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠘⡟⠆⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠉⠙⠚⠯⢷⣶⢤⣀⠀⠀⠀⠙⠳⣄
⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⣿⢃⠀⠀⠀⠈⢿⣭⣏⣙⣒⠲⢤⣄⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠈⠙⠾⡳⡄⠀⠀⠀⠘⢷⡀
⠀⣸⠃⠀⠀⠀⠀⠀⠀⣠⣎⡵⠛⠙⣯⡆⠀⠀⠀⠀⠉⠳⣤⣉⣉⣛⣮⣳⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠈⣧
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⡏⠀⠀⠀⠘⣯⣆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⠀⣯⣇⠀⠀⠀⠀⠀⠀⢧⣇⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣾
⠀⢻⡸⣆⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀⢀⣠⡴⠋⠁⠀⠀⠀⠀⠀⢰⡏
⠀⠈⢻⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢦⣤⣤⣀⣀⣀⣀⡉⠉⠉⠉⠀⠀⣀⣀⣀⣠⣤⡤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠙⢦⡑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠙⠳⣍⡳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡖⣪⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣭⣙⡖⠶⢤⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠶⣖⣋⣭⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⠷⢤⣿⣭⣽⣭⣽⣯⣭⣯⣭⣭⣭⡥⠿⠶⠾⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠞⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡁⠀⠀⠀⠀⠀⠀⠀⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠙⣷⠶⢦⣤⠴⢶⡟⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢿⡀⠀⢻⣄⡀⢀⣠⡾⠃⠀⣼⠃⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⣤⢶⣶⣦⡍⠳⣆⠀⠀⠀⠈⠻⣦⣀⠉⠉⠉⠉⢀⣠⠾⠃⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣼⢳⡏⠙⣧⢻⡄⠘⣧⠀⠀⠀⠀⠀⠉⠛⢻⡟⠛⠋⠁⠀⠀⠀⠀⣼⠃⠀⣀⣤⡴⢶⣤⣄⡀⠀⢹⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢻⡆⢀⣾⠃⠀⢹⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡏⢰⡞⢹⡏⣾⢻⡎⣿⠙⣷⡄⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⠋⣴⠟⠁⠀⠀⣼⠛⠶⢦⣄⣀⠀⢀⣸⣧⡀⠀⢀⣠⣤⠶⠞⣷⠈⠙⠾⣷⣝⣛⣵⡿⠟⠁⢠⡟
⠀⠀⠀⠀⠀⠈⢷⣄⠀⢾⡀⣠⡏⠀⠀⢀⣼⠋⠀⠀⠀⠈⠉⠛⣿⠁⠈⢻⡟⠋⠁⠀⠀⠀⠙⣧⡀⠀⠀⠈⠁⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣭⣭⣤⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⣠⡿⠷⠶⠿⣄⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⢀⣴⠋⠀⠀⠀⠀⠙⢷⣄⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠁⢀⡶⠀⠀⠙⠳⣟⠁⠀⠀⠀⠀⠀⠀⠀⣹⠟⠋⣡⡴⣶⣦⣌⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣿⣿⣉⣿⣿⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠈⠉⠁⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢰⡏⠀⣀⣀⠋⢻⡆⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⢠⣤⡄⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠛⠛⠋⠙⠓⠛⠁⣀⡾⠃⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⡿⠲⠶⠖⢿⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⡴⠞⠋

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

⠀⠀⠀⠀⠀⠀⠈⢉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠙⢦⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⢣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠛⠉⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠉⠙⢦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠰⠄⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⢿⣿⣦⡀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠐⠶⠀⠠⠞⠢⠋⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠀⠋⠉⠉
⠀⠀⠀⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠐⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠛⠁⠀⠀⠘⠛⠋⠉⠁⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣠⡤⢶⣶⣶⢄⠀⠀⠀⠀⠀⢀⢤⣶⣶⡴⣤⡀⠀⠀⠐⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⢦⣘⠛⠃⠀⠐⠀⠀⠀⠐⠁⠈⠛⠛⣠⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠈⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣷⣶⣴⣶⣶⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣶⢖⣲⠟⠒⣿⣷⣶⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⡟⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⣁⣻⣯⣾⣿⡿⠿⠟⠛⠛⠛⠛⠿⠿⣿⣛⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⡣⢸⣿⣿⠟⠋⠀⠀⠸⣿⣷⣦⣤⡀⠀⠀⠀⠈⠉⠛⠨⣹⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢹⢑⣿⣿⠇⠀⠀⠀⠀⠐⠷⣽⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠈⠙⣼⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠨⢻⣿⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⠀⠒⠔⠀⠀⠁⠘⢧⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣗⡀⠸⢿⣽⣦⣄⡀⠺⢿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅⠨⢚⣉⣉⣿⣟⣟⡛⠻⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⣄⣠⣴⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣂⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡮⠻⣷⣶⣤⣀⠙⢿⣯⣛⢿⣿⣿⣿⡿⠷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡼⣫⣿⠟⠋⠙⠛⢀⣷⣾⣾⣲⣬⢹⣆⠙⣿⣿⣿⣿⣶⣽⣏⠰⣾⠋⠿⠛
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⣦⣴⣤⡤⣤⣴⣶⠶⣲⡐⢇⡛⠴⡿⠃⢀⣤⣶⣶⣶⣮⠉⠛⠾⠿⣿⣮⡇⢸⣿⣿⡚⠛⠻⣿⣷⣄
⠀⠀⠀⢀⣴⠿⣿⣷⣟⣤⣿⣿⣿⣿⣿⣿⣿⡟⢾⠟⡑⠻⡟⠈⣤⣰⡿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣷⡔⡁⠉⣻⣿⣷⡄
⠀⢀⣴⣿⣯⣾⣿⢿⣿⣿⣝⣿⠿⠋⠉⠙⠛⠿⣿⣷⣬⡿⣷⣶⠿⣟⣷⡿⠶⠶⠦⠂⠀⢢⠄⠀⠀⠀⠀⣾⡏⠀⠘⢿⣿⡇⠀⠹⣿⣿⣆
⢀⣾⣿⣷⣿⡟⢁⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣿⣿⣍⣴⠄⠀⠀⠀⠀⠀⠀⢀⣠⣼⣾⡟⠁⠀⠀⠀⠉⢀⡠⡄⣼⣿⣿⡄
⢸⣿⣻⣿⡿⠀⢸⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⡄⣿⣬⣻⠝⠀⣀⣀⣤⣴⣶⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢻⡟⢀⣿⣿⣿⣧
⢸⣿⣿⡟⣷⠀⠀⢿⣿⣿⣹⢷⣤⣀⣀⡀⠀⠀⠀⠀⢻⢻⣿⣿⣿⡿⣻⣩⢼⣿⡿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣳⣿⣿⣿⡟
⠸⣿⣩⣷⣿⣧⡀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣖⣲⠿⢿⣿⠿⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣴⣿⣿⣿⣿⠃
⠀⠹⣧⣿⣿⢹⣿⣷⣦⣄⣀⣈⣉⣉⣙⣛⣛⣻⣿⣿⡿⣷⣯⣹⢿⣿⣿⣾⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣇⣿⣿⣿⣿⠃
⠀⠀⠙⢿⣾⣿⣾⣮⡽⣿⣿⣿⣿⣿⣟⣛⡻⣟⣽⣭⣿⡇⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠛⠿⢯⣿⣿⣻⣿⣷⣽⣿⢿⣟⣿⣽⡿⢯⠞⠉⢸⣿⣼⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣙⣉⡉⠉⠉⠀⠀⠈⠀⣠⣿⣿⠏⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠉⠛⠛⠿⠿⣿⣷⣶⣶⣾⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠲⠚⠓⠀⠉⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠔⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⣠⡴⡶⣛⣳⣶⡦⡀⠀⠀⠀⠀⠀⠐⡯⢊⣁⠀⣀⠂⠉⠓⣕⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠘⠀⠳⠥⠵⢧⡊⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡉⠁⠉⣉⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢰⡡⠀⠠⠤⢱
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣦⠐⡄
⠀⠀⠀⠐⡥⠒⠐⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡖⠒⠛⠁⠀⠀⠀⠀⠀⠀⢠⠠
⠀⠀⠀⠘⢧⢍⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠠⠤⢀⠀⠂
⠀⠀⠀⠀⣤⠞⠝⡞⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠠⠗⠂⠀⠲⠀⠀⠀⠀⠀⠀⠈⢼⣛⢧⡀⣀⡀
⠀⠀⠀⣀⠉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⡂⣀⣀⠠⢌⠂⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠈
⠀⠀⢀⠔⢤⡠⠖⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⢖⡖⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⣾⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊⣶⠞⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⢻⡀⠀⠀⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢀⡠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠠⣠⠾⡄
⠀⠀⠀⢳⢸⠊⡇⡼⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⡶⠴⠒⠊⠉⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠘⠋⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠓⠒⠒⠒⠁
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⠶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠠⠤⠄⠂⡀⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠆⠄⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠠⠄⡶⢀⡋⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⣀⠠⠊⠁⡜⠁
⠀⠀⢀⡔⠀⠠⢦⡀⠉⠉⠑⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠀⠀⠀⠀⠄⠈⣩⢁⣠⣴⣤⡎⢀⢂⡀⠀⠱⡀
⠀⠀⠀⣀⠈⠱⡄⠀⠑⠦⣄⡀⠀⡀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠃⠀⠉⠀⣠⠊⠙⠁⢢⡀⢀⠀⠐⠊⠲⣄
⠀⠀⠀⡀⢁⠀⠈⠢⠀⠀⠀⢉⠠⣤⡀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠠⠁⠀⠈⢾⠀⠁⠀⠀⡴⠊⠁⠈⠉⠂
⠀⠀⠀⡇⠀⠅⠀⠀⠁⢄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡔⠀⠢⡱⡅
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡒⠒⠒⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⢄⠀⠀⡇⠀⠄⡀⢀⢀⡀
⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢴⠒⠮⠀⠀⢊⣀⡈⢆⣠⠀⠠⡤⠒⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⢈⣅⠀⠿⡿⣗⠀⢠⣄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣤⠥⠚⠉⠁⠀⢠⡶⠞⠲⡁⢀⠀⠀⠀⠲⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠡⡌⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠅
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡉⢀
⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠈⢆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⡀⠀⠀⠋⠈⠀⢠
⠀⠀⠀⠀⢀⠀⠘⡄⠀⣀⠀⠀⢡⡢⠀⢀⠼⠁⠀⠁⠤⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢤⡀⢳⢠
⠀⠀⠀⠀⠄⠀⠀⢰⠀⠈⠈⠀⢂⠱⢦⠁⠔⠂⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⠒⠒⠂⡈⢸⣗⠘
⠀⠀⠀⠀⠀⠤⠄⣀⠀⠤⠄⠀⠀⠈⠀⠙⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⠁⠀⠀⠀⠀⣀⠤⠈⠉⡴⢉⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠑⡄⠀⠀⠔⠁⠀⡀⠤⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠈⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⡀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠁⠀⠀⠀⠀⠀⡼⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠠⠠⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠂⠁⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣄⠀⣤⣆⠀⠀⠀⠀⣤⡄⠀⣠⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣦⠈⠟⢃⠀⠀⠘⠻⠁⣼⣷⠌
⠀⠀⢀⢠⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢈⣁⡇⠸⣟⣀⣀⣻⠇⢨⣌⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠈⠛⠂⠉⠉⠘⠿⢯⣽⣶⣄⡀⢸⣿⣧⣶⣸⣿⣿⣇⣶⣾⣿⠇⢀⣤⣾⣯⠽⠿⠚⠉⠉⠘⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣅⣱⡀⣿⣽⣿⣿⣿⣿⣿⣿⣽⣿⢠⣎⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠰⣶⡍⠿⠛⠸⠿⢷⢿⡿⢟⣡⣘⠻⢿⣿⡽⠿⠇⠛⠿⢹⣶⠦⣄⡀
⠀⠀⠀⠀⠀⡠⢾⡿⠉⠁⠁⠀⠀⠀⠀⢀⣸⣧⣰⣿⣯⣽⣿⣂⣼⣅⠀⠀⠀⠀⠀⠉⠈⠙⢿⡷⢄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠻⠿⡿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣋⢭⣾⡿⠛⠀⠘⢇⠻⠿⣿⣷⡭⣙⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠃⣾⣿⣷⡇⠀⣾⡇⠀⠘⣿⣿⣷⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡔⠰⣿⣿⠱⠠⠀⠙⣿⠀⠠⠈⣿⣿⠀⢂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢿⣯⡦⠀⡀⢀⣿⠀⠀⢰⣿⡿⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⢿⣿⣦⡀⠉⠛⢁⣴⣿⡿⠁⠀⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠙⠻⢷⣾⣿⡿⠟⠋⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣟⣿⣟⣛⣡⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠏⠙⢋⣹⣶⣿⣿⠿⣿⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⠟⠁⣠⣀⣾⣿⣿⣿⡛⢿⡶⠈⠛⢀⣴⡶⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣿⠇⢤⢆⣉⣿⣿⣯⣴⣿⠛⠁⢠⣤⡼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠿⠋⢠⡀⠘⢻⣧⠉⠉⠉⢉⡘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢀⣀⣬⣿⣦⠀⠻⢇⣠⣾⣿⡇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⠀⣸⣁⣀⣴⣿⣿⣿⠛⠁⠈⠃⠜⠛⢻⣤⠿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣍⣉⣿⠟⢿⡿⣿⣿⣿⣿⡿⣿⣇⠀⠀⠀⠀⠀⠈⠀⠀⢀⣈⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⣿⣿⣿⣦⣤⣄⡀⠉⠉⠻⣿⣟⢢⣄⠀⠀⠀⣠⡴⠖⠋⢉⠉⢁⠉⣞⢋⣤⣖⣀
⠀⠀⠀⠀⢠⣤⣤⣄⣀⠰⠏⠱⣾⢛⣿⡛⠻⣿⣷⡞⠣⣄⡠⢿⣧⡈⢳⣶⣿⣏⠀⠀⠐⠀⢀⣀⡾⠟⠛⠃⠙⢳⡶
⠀⠀⠀⠀⠞⠉⠉⠉⠛⠷⣶⡄⢀⣈⡉⣻⣷⣾⡿⢃⣶⠏⠁⣤⠛⠿⢷⣿⣿⣿⣦⣤⣤⣶⠛⠛⡠⠆⡀⠀⠀⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣤⣀⡀⠀⣾⠃⢠⣼⣄⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⠿⣿⣿⣿⣧⠀⣀⣤⣤⣾⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⡿⠿⣷⡶⠉⠀⠉⠛⢿⣿⣶⡃⠀⠀⢙⠟⠁⠀⠀⠀⢈⣿⣿⢿⡿⠿⠿⢛⣿⣤⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⠶⣄⣉⠶⠿⢀⡀⠀⠀⣿⣿⣧⡀⠀⢈⡷⠀⠀⡀⢶⣿⠟⠋⡀⢠⣴⣾⣿⠿⣯⡙⢧⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠛⢿⠋⠁⡐⠀⡈⢷⡄⢀⣿⣿⣿⣿⣶⣤⡓⢀⣬⣳⣶⡏⠲⢦⣸⣿⣿⣯⡿⠟⠈⣁⣠⡶⠟⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢠⣿⡏⢘⣷⣌⣉⡀⠀⠀⢱⡌⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠛⠀⠙⣄⠈⢻⡏⠉⢉⣹⠟⠋⠉⠁
⠀⠀⠀⠀⠀⢀⣴⠿⠟⢻⣿⠟⣹⣿⡙⢟⣛⣿⣶⡀⠀⡀⠀⠀⠛⠛⢛⠻⠟⠛⣿⠃⠀⣠⣶⣾⣿⣷⣈⠑⣶⡿⢿⣀⣠⡄
⠀⠀⠀⠀⠀⣼⠉⠀⡀⢀⣽⣿⠋⡀⠁⣾⣿⣿⣿⣷⡄⠁⠀⠀⠒⣀⠊⠠⠀⣼⡏⠀⢀⣼⣭⣿⣿⣿⣿⡿⠛⠀⠾⡙⠃
⠀⠀⠀⠀⠀⠼⠀⡤⠀⠀⣿⡁⠀⠀⢸⣿⠿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠊⠁⠀⣾⡄⠀⢈⡿⠈⣿⣿⡿⠋⠀⢀⣠⡄⠈⠓⠒⠂⣤⣄⡀
⠀⠀⠀⠀⠀⢶⠀⢴⠿⣿⣏⠀⠀⠀⠹⣿⣶⣎⣉⢿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣍⠀⠈⢡⡖⠀⠻⡧⠐⠛⠋⠀⢀⠀⢉⣴⠶⣷⠾⠿⣄
⠀⠀⠀⠀⠀⠿⠀⢠⣿⣿⣧⣤⣀⠀⣰⣿⠟⢻⣿⣿⣋⣉⠙⠛⠿⢿⣿⡿⠿⠿⠿⠇⠀⠈⠙⠳⣶⣤⡐⢢⣤⣤⣬⡛⢛⣍⡀⠀⠀⠀⠉⣅
⠀⠀⠀⠀⠀⠀⠛⡀⢹⣿⣿⠟⠿⢷⣾⣿⣿⣿⣉⣹⢿⣿⣷⣦⣤⠈⠻⣿⣿⣷⣄⣸⣿⠦⢀⡀⠀⠀⠉⣠⣤⠾⠌⠛⢿⣿⣿⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⡉⠙⠿⠆⠀⠘⢿⣻⡿⢿⣿⣿⣦⣽⣿⣿⣀⣤⡄⢻⣯⡸⢿⣌⡁⣠⠈⠙⠃⢸⣿⣿⠁⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⣤⡄⠀⠚⣿⣿⣿⣯⣍⠉⠉⢉⣼⠟⠁⢠⡌⠻⣷⣼⣿⣅⣠⡄⠀⠀⢀⣽⠿⣇⠀⠀⣠⣾⣿⣿⣿⡿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⡄⠋⢻⣿⣿⣿⢿⣷⣄⠿⠃⢠⣴⣏⣀⠀⠉⠻⣿⣿⣿⣶⣦⠀⢀⣿⠁⢹⣦⣾⣿⣿⣿⣿⣿⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣠⠀⢹⣧⣷⣄⣉⣉⠶⣿⣭⣿⣿⣿⣦⡀⠀⠈⢿⣿⣿⣿⡀⣸⠟⠀⠘⠿⠿⣿⣿⣿⣿⣇⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠲⠶⡀⠿⠻⠿⢿⣤⣿⣿⣿⣿⣿⣿⣿⣿⡇⣽⠁⠀⡀⢺⣿⣿⣿⡀⣽⠀⠐⠀⠰⠀⠏⠙⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣰⡾⢻⠟⢿⣦⣁⡄⠐⠊⠀⠈⠛⣿⣿⣿⣿⣿⣿⡟⢡⣿⡄⠀⠀⣻⣿⣿⣿⣦⣠⠦⣠⠀⣰⡀⠀⡀⠀⡏
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠹⢿⣄⣀⡀⠘⢆⡀⢌⠛⠛⠛⢦⡁⢸⠟⠀⠀⢈⣿⣿⣿⣷⣿⢃⣄⡉⠐⠋⠙⣶⢡⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣠⣴⣷⣦⣤⡟⠻⣦⣤⣷⡀⠀⠀⣤⣤⡽⠟⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠟⢀⠿⢧⣄⠀⠘⠛
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣏⠉⠛⣷⣧⣄⠉⠛⠉⠘⠋⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡷⠀⠘⠀⠀⢹⡤⠁⢸⣀
⠀⠀⠀⠀⠀⠘⣁⣴⠒⢿⣿⣿⣿⣿⢿⡀⠀⠈⣿⣿⣷⣶⣶⣤⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢀⡝⠠⣄⡹⣿
⠀⠀⠀⠀⠀⠀⢿⣴⣷⣿⣿⣿⡿⠿⠟⢧⡀⠀⣤⠀⡙⠛⢿⠛⠛⠛⠛⣹⣍⣉⣿⣿⣿⠿⣿⣭⣄⣀⠀⠀⢠⡼⠁⡄⢸⣷⠍
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠃⠀⠀⠀⠀⠘⣧⠄⢤⣇⠀⠂⠘⣶⣶⣿⣾⣿⣿⣿⡿⢟⡹⠟⠉⠙⠿⠛⠛⢶⡎⠀⠞⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⣍⠉⠀⠀⠀⠀⢀⣄⣸⣿⣶⡀⢛⡌⢁⡀⠘⠋⠙⠿⠿⠿⡟⠶⠛⠁⠀⢠⠀⠴⢾⣴⡆⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⣠⡼⠟⡁⢹⣿⣿⣿⣷⣎⣵⡄⠐⣄⠈⢡⠀⠠⡖⠋⠀⠂⡀⠀⠀⣴⠀⣾⣿⣤⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⡦⠠⣾⠏⠀⢰⣿⢻⣿⣿⣿⣿⣿⣿⣷⠀⠈⠀⠀⠀⠀⠈⠀⠛⠠⣴⠃⠀⢤⣼⣿⣿⣿⣘⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠇⠠⠇⠀⢀⡿⢻⣾⣿⣿⠟⢡⡍⢉⡿⠗⠿⣀⣄⠀⠐⠃⠀⠀⠀⢁⣄⣼⣿⣿⣿⣿⣿⡟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢂⡤⠀⠓⢨⣿⡿⠱⠀⢤⣰⡏⠀⣠⣶⣯⣿⣿⣿⡶⠛⠻⠿⣾⣿⣿⣿⠿⠿⢻⣯⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡧⠜⠀⠛⣿⡄⢉⣴⣾⣛⣟⣻⣋⣹⠿⣟⣿⣿⡁⠀⣠⣤⣠⣤⣄⣴⣤⣴⣶⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⠀⠆⠉⠻⣿⡋⠉⡉⠉⠉⠙⣻⠿⣈⣣⠘⢻⣀⣄⢁⣤⣼⣿⣿⣿⣿⢇⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⡀⠀⠺⠁⣤⣤⣾⣳⣿⣿⣿⣿⣿⡄⡌⠋⣤⠛⠁⣸⣿⣿⣿⣷⠾⠋⠈⣹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⢀⡀⠈⢤⣙⡻⠟⠻⠿⣿⣿⣿⣿⣷⠀⠠⣤⢶⣾⣿⣿⣿⣿⣿⣶⣦⣾⣏⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣱⡆⠀⣉⠁⠀⣁⠰⠉⢻⣿⣿⣿⡟⢦⠁⠜⢻⡾⠛⠻⢯⣻⡏⠙⠛⣿⠟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣌⣀⡐⣧⣤⠈⡊⢩⣿⢿⣷⡎⠀⣴⠞⠳⣤⡀⠀⣿⠟⢂⡀⠈⢿⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢿⣿⠋⠙⠁⠀⠈⠛⠀⠀⣿⡉⠳⣀⠠⣐⠀⢨⡿⢟⡋⠤⣈⠴⠄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡜⠿⣆⠀⠀⠀⠀⠀⠀⠠⣾⠁⠀⠙⢦⡘⢧⣼⡤⢜⡋⠀⠻⠶⠀⠲⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣹⣦⡀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠈⠋⢥⣤⠘⠛⠀⠈⠀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣦⣄⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣥⣶⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣶⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠣⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢂
⠀⠀⢀⠤⠒⠒⢒⡂⠒⠤⡀⠀⠀⡠⠄⠒⢒⣖⠒⠂⢄⡀⠀⢀⡠⠔⠓⠒⡒⠓⠠⢄⠀⠀⣈⠄⠚⠛⢩⡛⠓⠠⡁
⠀⠀⠑⠠⢄⣈⢛⣁⡠⠄⠊⠀⠈⠓⠤⢀⡙⢛⣀⠠⠔⠁⠀⠈⡢⠄⣀⡛⢋⡀⠤⡞⠁⠀⠑⠦⢄⣈⣛⣁⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠈⢣⠀⡞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠉⠑⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣻⣿⣇⣨⣿⣶⠀⠀⠀⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡆⠈⣓⣪⡼⡛⣩⣄⠃⢄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡼⣷⣤⣨⣀⠘⣟⢁⣼⣃⣼⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⣿⣼⣿⣿⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣶⣄⡈⠻⣿⣿⣿⣿⣿⠟⠋⢀⣠⣤⣤⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣿⣿⣿⣷⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣿⣾⢿⣿⣿⣿⢿⣿⣿⣶⣶⣾⠿⠿⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⠀⠀⣴⣿⣿⣿⣴⣾⣏⣽⢿⣶⣶⣶⣶⣄⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⢸⣿⠋⣿⡝⠛⢿⣏⣼⠂⠀⠀⠈⠉⠛⣷⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠊⠀⠀⢑⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢄⢀⠔⡥⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠘⡄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠘⠀⠐⡄
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⡀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠣⡀⠀⢀⣀⠠⠤⠀⠤⠄⣀⢀⣀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠈⠂⠀⠐⠁⠀⠀⠊


⠀⠀⠀⠀⠀⠀⠀⠀⢏⠑⠤⡀
⠀⠀⠀⠀⠄⠄⠤⠤⠤⠱⠆⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠀⠐⠢⣀
⠀⠀⢠⠘⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠀⠀⡠⠄⠒⠉⠀⠊⢠⢄⠑⠀⠀⠀⢀⠇⠀⠀⠀⠀⢠⡶⠉⢀⣀⣀⣀⣀⣀⠀⠁⢦⠄
⠀⠀⠀⢙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠄⣀⠀⠀⠢⡈⢈⡠⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠈⠀⢈⡀⠔⠁
⠀⠀⠀⠀⠸⢉⣉⣉⣉⣀⣀⣀⣀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠞⠤⠘⢆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠊⠄⠂⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⢢⠀⠔⠒⠢⠄
⠀⠀⠀⠀⢀⠔⠁⠀⠈⡤⠒⢠⠃⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⡰⠋⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠬⠒⠀⡇⠠⠂⢄⠀⢀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠔⠁⠓⠤⡞⠈⢉⡁⠂⠀⠀⠀⠀⠀⠀⠰⢉⠄⠄⠀⠀⠀⣇⠠⠀⠀⡁⠠⡑
⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⠀⠫⡦⡀⢠⡄⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⡿⠀⠀⢄⠁⠀⠅
⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⢀⡖⠃⠀⠀⠰⢄⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⡀⠀⠀⡇⠀⠀⢨⠀⠨⠐
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠪⡐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠨⠐⢀⠃⠘⠢⠅⣀⠆
⠀⠀⠀⠈⠒⠀⠀⠄⠊⠀⠀⠀⠐⠠⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠀⠑⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣄⣀⣄⣀⣠⡀⡀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⡴⠼⠞⣛⠉⠹⣿⣍⣩⣍⣻⢏⡉⠉⠓⠛⠳⠦⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠴⠚⠉⠀⠀⠀⠀⠈⠀⠀⠠⡿⠿⣿⣁⡀⠁⠀⠀⠀⠀⠀⠀⠉⠲⢤⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⢀⡅⠁⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣬⣿⣿⣿⣆⡄⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠺⡷⢸⣆⠀⣚⣿⣿⣿⣿⣟⠀⠀⣥⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣸⢷⣞⡿⣿⣿⢹⣿⣿⣷⡴⢧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣐⡺⠯⢥⣚⡿⢿⡏⠘⣟⡿⢟⣿⠷⣚⣅⠀⠀⠀⠀⣀⡠⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡚⠳⠦⢥⣌⣓⠶⢭⣓⠂⠐⢿⣚⡭⢖⣋⡭⠶⣒⣒⣒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣬⣙⣓⣒⣶⣬⣭⣙⣒⣯⣽⡁⠀⣻⣭⣵⣚⣫⣭⡖⡚⢫⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⣂⣉⣉⡉⠿⠶⢶⣊⣉⠷⢶⣇⠀⢀⣨⡷⠶⢏⣉⣲⠶⠿⣉⣁⣲⠶⠶⠦
⠀⠀⠀⠀⠀⠀⣠⣤⣤⠔⠛⠛⠛⣻⣥⣤⣤⠚⠛⢻⣤⣼⠁⠀⠀⠀⣡⣤⣼⠛⠛⢠⣤⣼⡟⠛⠓⢤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠚⠛⢛⣯⣤⣤⣤⠜⠛⠛⢛⣧⣤⣤⠛⠛⢻⡷⠂⠐⠾⣿⠛⠛⢧⣤⣤⡟⠛⠛⢠⣤⣤⣿⡛⠛⠛⠂
⠀⠀⣠⣤⣤⣤⡄⠛⠛⠛⠛⣻⣤⣤⣤⡜⠛⠛⢻⣤⣤⣜⠀⠀⠀⠀⠘⣤⣤⣼⠟⠛⠛⣤⣤⣤⣽⠛⠛⠛⠠⣤⣤⣤⣤⣄
⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠘⢿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣛⣭⣶⠾⠿⠿⢶⣾⣽⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢫⣾⣟⣭⣶⡿⠿⠿⢷⣶⣝⣿⣮⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢣⣿⣿⣿⣯⣵⡿⢿⠿⣷⣮⣿⣿⣿⣧⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⡏⣾⡟⢿⣎⣿⣿⣿⣿⣿⡞⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣧⢫⣾⣮⡣⣿⣿⣿⣿⣿⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣏⢿⣿⣿⡿⣵⣿⣿⣿⣷⡝⣿⣿⣿⢏⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣧⡻⢿⣶⡌⣿⣿⣿⣿⣴⣾⡿⣫⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣭⡁⣿⣿⣿⣿⣩⣵⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣏⡇⣿⣿⣿⣿⣟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⡀⣀⢀⡀⣀⢀⣀⡀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠧⠟⠟⠧⠸⠄⠟⠁⠇⠟⠧⠗⠟⠇


⣏⣈⣯⣏⣟⣯⣿⣹⣽⣽⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣝⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣛⣫⣿⣤⣀⣀⣤⣤⣤⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⣛⠻⠿⠿⠛⠛⠛⠉⠉⠉⠉⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀⠀⠀⣸⡏⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡏⠠⣿⣿⣿⣿⣿⣿⡿⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⠙⠻⢶⣿⣿⣷⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣿⣿⣻⣿⣸⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠈⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢹⣧⡀⠀⠀⠀⡀⡀⠀⠀⠀⣠⣿⠃⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠙⢿⣦⣀⣾⣷⣿⣆⣠⣾⠟⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⣩⣽⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣭⡉⠙⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣴⡾⠟⠹⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣶⣄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⠀⠀⢻⣆⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠙⢷⣦⣭⣭⣿⠟⠁⠀⠀⠀⠀⢸⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠤⣄⣴⣶⣶⣄⢺⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⢰⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⣀⣸⣿⣿⣿⣿⣏⣾⢛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣾⣜⡁⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡾⠉⠙⣿⣿⣿⣿⣿⡷⠟⠙⠳⡄⠀⠀⢀⣠⠴⣾⣿⣿⣿⣿⣷⣛⣁⠀⠀⠀⠀⠀⣀⣠⢴⣶⡴⡀⡼⠁⠀⠀⢀⣠⠤⠒⠒⠉⠉
⠀⠈⢀⡶⠟⣿⣿⣿⣿⣿⡷⣦⡀⠀⠙⠤⢀⡾⠀⠀⣿⣿⣿⠟⠹⣆⠀⠈⠓⠦⡀⠀⠸⠁⠀⢃⣴⣷⣿⣿⣦⣴⠾⠭⠤⠴⠂⠀⠠⠄
⠀⢀⡞⠀⠀⣽⠉⡟⠋⠛⣇⠈⢿⡄⠀⠀⠘⠁⠀⠀⠀⣠⠟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⢳
⠀⡞⠀⠀⠀⣿⠀⡇⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⢹⡆⠀⠳⡀
⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠛⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⣀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⡄⠀⠀⢰⠓⡀
⠀⠀⡰⢤⡈⣆⢰⣿⣿⣿⣆⢸⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣧⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⡇⠀⠀⡿⠀⢱⡀
⠀⠀⠁⠀⠹⣼⡸⣿⣿⣿⠏⣜⡼⠁⠀⠀⠀⠀⠀⠀⠐⠢⢤⣀⣈⢿⣿⣿⣿⠇⣀⡤⠤⢄⣀⡀⠄⠀⠀⠀⠀⢀⡿⣅⠠⣃⣄⡄⡇⢀⡽⣇
⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣾⣿⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣿⣷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢈⣻⣿⣿⣿⣿⡋⢀⣼⠂
⠀⢀⠔⠁⠀⡇⢉⠟⠋⠙⢏⠀⡇⠈⠳⡀⠀⠀⠀⠀⠀⣠⠞⢋⡽⢿⣿⣿⠿⡏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠡⣿⣿⣿⡏⢹⠛⠁
⠀⡜⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⣠⠞⠁⠀⣸⠃⠈⠀⠈⠀⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠻⣿⠟⠀⠘⡄
⠀⠁⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⠁⠀⠀⠸⠀⠀⠈⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢠⣤⣤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠊⠛⠛⢛⣻⣏⡱⠶⢉⣉⣥⢦⡈⢻⣿⠗⠰⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⠟⢋⣉⡋⠀⢰⡿⣯⣉⡛⣤⣸⣏⣵⠛⢷⡈⣰⣦⣶⣻⡅⢸⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣭⣆⢠⣿⢾⣿⣫⡞⢫⣴⠯⠻⠷⠦⠶⠿⠰⢿⣾⣯⣿⡞⣻⣝⡃⠘⠛⢋⡁⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⡱⣏⡴⠟⣿⡯⠛⠛⠁⠀⠀⠀⢀⣀⣀⣤⣤⡀⣀⣀⠉⠁⠻⢿⣟⣃⣴⣿⣥⡽⠅⢤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⡟⢃⣴⣿⣥⡶⠋⠀⠀⠀⣀⣤⠶⡿⠛⡹⣿⣏⡈⢻⣿⣶⣿⠆⢄⠀⠁⢮⡙⠷⣿⡾⠀⠎⢵⠆
⠀⠀⠀⠀⠀⠀⠀⡼⢟⡳⠿⣄⡙⣿⠟⠀⠀⠀⣠⢞⣿⡣⠚⣤⣌⣽⣧⣬⣉⣟⣿⡞⠻⠛⠛⠀⠀⠀⠁⡀⠙⣷⣶⡜⣤⣶
⠀⠀⠀⠀⠀⠀⢀⡷⠛⠙⣿⡉⣠⡟⠀⠀⢠⣾⣧⡜⢩⣿⣿⡏⢼⣿⣿⣏⡟⣉⡶⣄⠀⠀⠀⠀⠀⢠⠀⢱⠀⠈⢛⡿⢃⡼⡃
⠀⠀⠀⠀⠀⠀⢘⣿⣿⢀⣿⠠⣜⣧⠀⠀⢸⣿⣯⣿⠦⠴⠤⡈⠛⠿⣿⠿⡿⣏⠐⠧⣀⣀⡀⠀⣠⠏⠀⠠⠄⠀⢸⡟⠳⢟⡃
⠀⠀⠀⠀⠀⠀⠀⣅⠿⠶⣿⡄⠸⠿⣦⣤⢺⣟⠋⠀⠀⣄⣠⡴⠿⣆⠙⢷⣄⡈⢳⣄⡁⠙⠻⠞⠋⠀⢀⠃⠀⠀⠸⣿⢶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠠⣸⡿⠶⠶⠿⣋⣾⠻⠀⠀⢰⡾⠅⠀⠀⠀⠙⠧⠿⣷⣶⣿⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⣸⣿⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡝⣶⣶⣾⣛⠿⠀⠀⠘⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣛⣇⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠈⠀⠉⠀⠀⠀⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠜⠁⠀⠀⠀⣴⣿⣍⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠃⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣟⣣⣤⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠘⠀⠀⠀⠀⠀⣀⣠⠒⢿⡿⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣦⣾⣟⡥⢸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣰⣦⣾⣿⣿⠁⣠⣛⣿⣶⣏⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⢀⣄⣠⣤⣴⣦⡿⢾⠟⢿⣻⣿⣫⡟⣀⣻⣽⣷⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⠓⣶⣤⡄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡤⣾⡶⢦⣼⠿⠏⣉⠷⠟⣿⣆⣀⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣏⣿⠲⡿⠽⠅⢀⣶⣦⡀
⠀⠀⠀⠈⠀⣀⡴⢶⣻⣿⠷⣞⣶⣿⡹⠶⠞⠛⠛⠉⣠⡼⣿⡿⣿⣿⣿⣿⡿⠛⠛⠉⠉⠀⢀⡀⡀⢈⣈⡉⡉⠙⠛⣷⣄⣰⣶⣷⣾⣻⠟⠛⢃⡀
⠀⠀⢄⣠⣶⣟⡛⢻⣤⣷⣶⠾⠛⠉⠀⠀⠀⠀⢀⡾⢯⣤⠿⣷⣿⡟⠋⠡⣀⠠⠖⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠⢈⠹⢿⣿⡟⢻⣿⠿⣛⢿⣄
⠀⢀⣎⠉⣻⡏⢿⡦⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⣾⢻⣾⣽⣗⣴⡟⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠐⠄⢻⣇⣀⢤⣤⣙⣿⣿⡄
⠀⢸⡏⠻⠉⠟⠂⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠹⣯⣽⠻⣷⣀⠀⠀⢀⣀⣤⣤⣶⣿⠏⢉⣻⣥⠿⣽⣿⢷⡆⢤⣀⠀⠀⠐⡿⢻⣿⠿⡝⣿⡍⣅
⠀⢸⡟⠻⠿⠿⣾⠻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣶⠠⣤⡄⣾⣻⡿⠿⢋⣹⠟⠛⢻⣯⠐⣉⣭⣤⣤⠁⣽⣆⠁⣾⣿⢠⡀⠀⣽⣶⣍⣀⠸⣿⡷⢻
⠀⠸⡧⢿⢧⣾⣿⠃⢠⡸⣦⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣶⠦⣄⣾⣧⣾⢂⣤⢤⣬⣙⡋⠀⣰⣿⣽⣶⣏⣷⣾⡋⠐⠉⠛⣙⣁⣾⡟⠻⣟⢹⡄⣢⠃⡈
⠀⠀⢿⡿⣼⠟⢁⡿⣆⠳⣉⠅⢢⣄⠀⠀⠀⠀⠀⠈⠻⢾⣷⣿⡉⠷⠀⡽⢯⣾⣯⣿⠟⠛⠉⠀⠀⠀⠀⠉⠙⢷⣦⡘⢻⣿⣿⣿⡛⢻⠿⠿⣿⠃⣸⠃
⠀⠀⠈⢻⣿⣶⣍⣷⣯⣾⣻⠎⠲⡿⠷⠂⢤⣀⡀⠀⠀⠀⠉⠙⠛⠒⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣮⡝⠚⠙⢲⡿⢷⣴⣾⡿⠉⣠⠋
⠀⠀⠀⠈⠻⣿⡋⣹⣥⣘⢫⣄⢤⣷⡶⠚⢷⡜⠋⠲⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣾⣿⣷⡾⢯⣌⣷⠖⠁⢨⡅⠸⡯⠉⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠋⢻⣯⣿⣄⣉⣶⣭⡿⣇⡻⠇⢘⡳⢄⣰⡿⢻⣿⡋⣆⡸⣇⣼⣏⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣅⣼⡆⢠⢾⡿⣦⡇⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⡙⠻⢿⣯⣴⣿⣻⣇⠘⠋⣾⣏⣱⡾⠻⠷⠨⠀⣼⣿⢿⣿⣿⠯⣽⣿⣿⣿⣿⣿⣿⣧⣿⠺⡇⠻⣮⣱⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢤⢉⡉⠉⠛⠓⠿⠿⡿⠅⣴⣾⣷⣄⣼⣿⣷⣾⣿⣿⡖⣮⢹⣿⣿⠿⠛⠋⢁⣀⡜⠃⠀⣾⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠠⡄⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠙⠉⠁⠉⠀⠀⢀⡀⠤⠒⠈⢉⡿⣠⡴⢻⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠐⠂⠒⠀⠀⠀⠀⠐⠐⠒⠒⠀⠉⠁⠀⠀⠀⠠⣏⡀⢻⡗⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠖⠂⠀⠌⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣡⣄⣠⣼⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⣠⡄⠀⣴⡓⣶⣧⣶⡟⢳⡶⣄⠰⠀⠀⠀⠀⣀⡤⠞⠓⣼⢧⡾⣻⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⣾⣥⣿⣶⠟⠛⠛⠋⠀⠹⢿⠿⢿⡿⣶⠁⡰⠾⢋⣼⣿⠿⡿⠶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠻⣿⠿⢧⣤⣤⣤⣤⣤⣤⣴⣷⡿⠧⣿⡇⢸⣶⣿⣧⣭⡿⡧⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢀⣻⡶⢲⣈⣍⢳⡶⣮⣙⣿⣿⡷⢠⠿⣷⡸⠏⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡌⠛⠛⠛⢙⠻⣯⡷⠿⠉⣿⣿⡇⢾⣶⡿⡓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠛⠛⠓⠒⠂⠚⠉⠻⣥⢾⡇⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣿⠶⠞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣞⢣⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉⣧⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠒⠒⠒⠂⠈⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠀⢀⠈⠉⢻⠀⠀⠀⣿⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠀⠉⠉⠀⢹⠀⠀⠀⣏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠛⠃⠀⠀⠀⠰⣧⠀⠀⠀⡇⠀⠀⠀⠈⠁⠁⠸⠀⠀⠀⣏
⠀⠀⠀⢸⠀⠀⣠⡀⠀⠀⡇⢠⣧⢜⣿⣁⠀⠀⠀⠀⠸⡄⠀⠀⡇⠀⠉⠉⠁⣶⡄⣿⠀⠀⠀⣧
⠀⠀⠀⠸⠀⠀⡩⡰⢦⡀⢠⡈⠁⠀⢩⡉⠀⠀⠀⠀⢰⡇⠀⠀⣇⣤⣈⡃⠢⣤⡄⢿⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠀⠶⠯⠉⣾⣻⣷⣶⡿⠀⠀⠀⠀⣀⠀⠃⠀⠀⣿⣿⣿⣿⢷⠟⠐⣘⠀⠀⠀⡿⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠁⠀⠒⣰⣿⣿⣿⣿⠇⠐⠂⠀⣼⣇⠀⢺⡇⢠⠿⡿⠿⠛⠓⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⢺⣿⣧⠐⠀⠀⠀⠀⢸⡄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡠⠤
⠀⠀⠈⠒⠲⠤⢾⣛⡛⠓⠀⠀⣿⣿⠀⠀⠀⠀⠀⡈⡧⣿⣏⡀⠀⠆⠀⢀⣀⡠⠤⠶⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠻⠭⢍⣀⣀⠀⠀⠀⣀⣉⣩⠯⠿⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⣡
⠀⠀⠀⠀⠀⠀⢰⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠛
⠀⠀⠀⠀⠀⠀⢾⡇⣷⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊
⠀⠀⠀⠒⢤⣀⠼⠧⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠂⠀⠀⣀⣠⣴⣦⣁⡀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠉⠛⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠨⠘⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⠀⠠⢄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠐⢄⠀⡀
⠀⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢆⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣴⣿⠛⠛⠉⢻⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣆⠀⠠⠄⣻⣿⡇
⢀⠀⢀⢀⣀⠙⣿⣿⣶⣶⣿⣿⡟⢁⢀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠀⢀⣀⣀⣀⡀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⠟⢰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣰⣷⣌⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⢿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠙⣿⢿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣤⡿⢧⣽⢿⣛⣿⡻⣷⣦⡀⠀⠀⣿⢿⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡿⢻⣟⣻⣾⣿⣟⣷⡞⣿⣉⣯⣭⡿⣿⣿⣿⣿⣿⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⣿⣹⣿⣿⣇⠀⠀⠀⠉⠫⢿⡾⣿⣻⣷⢿⣿⣹⣷⡽⣁⣤⣤⣶⣶⣶⣾⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠈⣾⣏⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠉⠛⠛⣻⣿⣽⣿⣿⣿⣿⣿⣿⡿⢻⣿⣏⣹⣿⣿⢹⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢱⣿⣹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⢿⣿⡻⢻⣿⣍⣼⢛⠛⣿⣁⣽⢧⣀⣼⣏⣀⣻⠾⣿⡿⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣽⠿⣿⣻⣿⣻⣿⣽⡻⣿⣿⣏⡿⠛⢷⣿⣁⣸⡟⢶⣾⡯⠟⠋⠊⠉⠉⠛⠫⣿⢷⡞⠳⣶⣞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠯⣧⣬⣧⣏⣀⣽⢿⣤⡞⠙⢶⠿⠤⠜⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⡴⣟⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢻⠯⠿⢿⡿⢿⣿⡿⢲⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣦⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⢿⣷⣮⠟⢳⡟⢳⡾⠿⣤⣾⡏⣹⡟⠿⠿⣿⣶⣶⣦⣦⣤⣤⣴⣶⣾⣿⢿⣽⣿⣁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣟⣻⡶⣾⡷⠛⠙⠛⠛⠻⠻⠶⣾⣍⣻⠞⢷⣾⣮⣩⡟⢛⣙⣩⣏⢽⣦⣿⡾⣟⣁⣿⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⡟⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⣛⣶⣿⣍⣽⡟⣻⣟⣉⣿⣅⣽⢶⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣧⡽⢿⣶⣤⣤⣤⣤⣤⣤⣶⣶⡿⡿⢿⡿⠿⠿⢿⣿⣿⣉⠈⠙⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣴⣩⣉⣭⢉⡋⣿⢫⣧⣰⣶⣶⣧⣽⣷⣟⣧⣾⣽⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⢿⣽⣯⣿⣽⣿⣾⠿⠿⠙⠛⠈⠉⠉⠉⠙⢿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⢀⣠⢾⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⣷⣷⣶⣦⣯⣉⣿⣿⣛⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡟⠀⠀⠀⠈⠉⠛⠛⠻⠿⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢤⣭⠳⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣄⡀⠀⠀⠀⠀⢀⠴⢞⠋⠋⡓⣦⡀
⠀⠀⣴⠚⡟⣩⢻⠓⣦⠀⠀⠀⠀⠀⣧⢰⡋⣟⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠸⡇⠀⠀⠀⠀⠀⣴⣋⣀⣈⣀⣀⣹⡆⠀⠀⠀⡞⢲⣎⣣⣑⡲⠄⣿
⠀⠀⠈⠛⠻⠶⠞⠛⠁⠀⠀⠀⠀⠀⠘⢦⡙⠺⢣⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠙⡆⠀⠀⠀⠀⠈⠳⢤⣄⣠⠴⠋⠀⠀⠀⠀⠙⢶⣶⣮⣕⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠁⠀⠀⠀⠀⠀⠙⠛⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠸⠿⣛⡻⠇⠀⠀⠀⠀⠀⠀⠀⢱⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⢸⠁⠉⠉⠈⡇
⠀⠀⠀⢸⣿⠿⠷⡆⠀⠀⠀⠀⠀⠨⢊⡽⣿⠯⢀⠉⠀⠀⠀⠀⠀⠀⢈⣛⣋⠘⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⡄⢸⠁
⠀⠀⠀⢰⣶⣿⣷⡄⠀⠀⠀⠀⠀⠠⠜⠀⢘⠦⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠻⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⡶⣞⡀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢰⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣄⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣀
⠀⠀⠀⢉⡸⠹⠟⣥⠀⠀⠀⠀⠀⢠⣟⢛⠋⣛⢟⣆⠀⠀⠀⠀⣼⣿⣏⡦⣎⣿⣿⡄⠀⢀⣀⢠⣿⣿⣿⢸⣿⠁⡀⣀⣀⢀⡀⠀⠀⠀⣿⠀⠈⣧⢀
⠀⠀⠀⠈⣴⣿⣷⣶⡆⠀⠀⠀⠀⣿⣯⣿⠀⢿⣼⣿⠀⠀⠀⠀⠹⣿⡿⣿⡿⣿⡿⠀⠀⢊⡀⢈⣛⠛⠙⣃⣴⣾⣿⣿⣿⡏⣴⣶⡶⠞⠁⠀⣠⡏⣸
⠀⠀⠀⠀⠈⠛⠿⠟⠀⠀⠀⠀⠀⠙⠛⠁⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠤⠥⠼⠠⠤⠤⠽⠿⠿⠿⠷⠬⠤⠙⠶⠶⠞⠋⠴⠿

⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠘⠳⢦⡱⡄⠀⠀⠀⠀⣾⢠⠶⠶⡶⣦⢹⠀⠀⠀⠀⢠⣒⣭⣭⣭⣕⣆⠀⠀⠀⠀⡔⠻⢤⡀⢠⡬⠓⡆⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⢠⡴⠲⢦⣀⣤⡇⠹⡄⠀⠀⠀⣿⢸⢰⡏⣭⣥⠞⠀⠀⠀⠀⠹⠁⠀⢹⠁⠀⠛⠁⠀⠀⠀⠈⠛⠺⡄⣸⠚⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠆
⠀⠘⣆⣀⡀⠀⠀⢉⣠⠇⠀⠀⠀⢙⡏⠈⣻⠁⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⡶⠶⠂
⠤⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⡠⠚⡟⢢⡀⠀⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣸⠀⠀⠀⣐⠁⠸⡿⠀⢑⠀⠀⠀⢸⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⢦⡀⠀⠹⣧⢨⣯⢠⡟⠁⠀⣠⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡙⢦⠀⢿⡘⡇⣸⠃⢠⠞⣡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⣷⣼⣷⢴⣿⣴⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣀⡸⣿⠛⡉⠩⡙⢻⡿⣀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠾⠙⠛⠙⠉⠀⠙⠉⣹⣿⣟⡶⠰⣍⣿⣿⠋⠙⠁⠈⠉⠙⠛⠹⢆⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡾⣿⢛⡄⠀⣜⣻⡿⣾⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⡿⢃⢈⣿⣿⣸⣿⠏⢀⠻⣇⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡴⠂⢀⡾⢡⠃⣾⠉⠻⡿⠋⢺⠆⢣⠹⣆⠀⠢⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠀⠀⠀⢨⣿⠋⠀⠘⠀⠀⠀⠀⠘⠀⠈⢳⣯⠀⠀⠀⠈⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠔⠂⢌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⢤⠖⠒⢦⢤⡀⠀⠀⠀⠀⠱⣁⠨⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡤⠐⢊⠁⠀⠀⠉⢢⠀⠀⠈⠻⣿⠟⢻⠛⢻⡿⠋
⠀⠀⠀⠠⢎⡇⡇⢤⣿⠆⡅⣏⠦⠀⠀⠀⠐⡭⠀⢀⠃⠀⠀⢀⠆⠀⠀⠈⡄⠀⠀⠀⠀⢈⠝⣁⠄⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠉⠚⠦⠤⠞⠚⠁⠀⠀⠀⠀⠈⠂⢁⠎⠀⠀⠀⠊⡀⠀⠀⠀⡈⡆⠀⠀⠰⡡⠜⠑⠪⣴⢀⠔⠁⠀⠀⠀⠀⠀⠢⠈⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠑⠙⠢⠴⢚⠑⠁⠀⠀⠁⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣖⠂⠀⠀⢒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣭⣭⣭⣭⣭⣍⣭⣩⣩⣩⣍⣉⣉⣉⣉⡉⣋⢩⣎⢉⣻⢾⣭⣭⣩⣭⣩⣭⣭⣍⣭⣭⣍⣉⣉⣉⣉⡉⣙⣉⣋⣙⣋⣛⣉⣉⣉⣛⣋⣛⣉⣛⣙⣋⣙
⣾⡋⢡⣤⡎⠉⣑⠀⣷⢙⡋⠉⠛⣦⡤⠖⣽⠶⠲⢯⡁⠀⠹⡏⣥⠽⠇⣬⠻⡿⠁⣿⢠⡿⠛⠛⠉⡛⢙⡛⠛⠛⠛⣛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⡯⢿⠢⢈⣡⣽⠶⠒⢾⣥⣍⠠⢾⠈⢰⢳⣅⠪⠀⡇⠸⢦⠇⡿⠀⠀⢸⠀⢹⠈⣿⢁⣷⠒⣛⠉⠀⠀⠙⢶⣶⣶⣿⣿⣿⡿⢻⠟⢿⣿⣿⣿⣿⣿
⠻⣦⣄⡵⣏⣿⡇⢤⣿⠆⡿⣏⡷⣤⢮⣏⢱⡏⠀⢁⠃⡼⡲⢲⡇⠀⠀⠈⣄⢸⠀⠛⠄⣩⡿⣩⠄⠀⠀⠀⢸⠇⣀⡔⠱⢸⠀⢸⠀⢸⠁⡯⣥⡤⢬
⣀⣻⡿⣆⢌⠙⠿⠦⠤⠾⠟⣉⡠⣶⣾⠀⣜⠂⢁⠎⢰⠉⢰⠏⡀⠀⠀⠀⡈⡇⠠⠚⣰⣩⠔⠓⢾⣵⢀⡴⢫⡄⠙⢳⠰⡘⢧⣈⡠⠞⠰⠐⢹⣇⣾
⣿⣿⣈⡈⠁⢑⣿⣨⣬⣽⣿⠑⠛⣾⣇⣠⣛⠒⢁⢀⡿⠦⢮⠓⣙⠦⠤⢊⠑⠁⢈⢜⡋⠺⠆⣹⣦⠬⢩⡌⠛⣾⣶⠿⠅⢘⣓⣰⢤⡛⢜⣶⣸⢿⣿
⢶⠛⠛⠛⢻⠛⡿⢩⠉⣹⣿⣿⣿⣿⣿⣿⣯⣿⣵⣾⣿⣦⣸⣊⣭⣼⣣⣬⣵⣾⣶⣿⣬⣧⣶⣿⣥⣶⣮⣤⣶⣿⣯⣴⣯⣤⣾⣷⣰⣵⣮⣷⣿⣾⣼
⠿⣥⣤⠄⣠⠄⠌⢸⣾⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣠⣴⠶⠎⢘⣉⣿⣟⣛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⠿⢿⣿⣟⠁⠂⠾⢟⡋⠉⠁⠁⡈⠁⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣯⣴⠟⠉⠻⠒⠂⠀⠀⠀⠙⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠶⡿⠿⠿⠿⣷⣬⣾⣿⣿⡟⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣷⡆⢀⡴⣿⠗⡈⢿⣶⣦⣾⣶⣼⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣿⣩⣮⣩⡆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⣿⣴⠻⠞⠋⠈⠉⠙⠛⣿⣿⣯⠻⣿⣿⣿⣿⣷⣤⣀⠀⠀⣀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⢧⣿⣯⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⢨⣝⢛⣿⣿⠙⣻⣿⣿⣿⣿⣿⣿⣯⣾⡏
⠀⠀⠀⠀⠀⣿⣿⣾⣿⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣷⣾⣿⡿⢿⣿⡟⠛⠳⠟⠉
⠀⠀⠀⠀⠀⣿⣏⠻⣿⣥⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠎⠙⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣯⣜⢿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣮⡛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣎⢻⣿⣷⣌⣙⢶⣶⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣶⣾⣿⣷⣯⣷⡎⢉⣿⢛⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⣿⣿⣯⣬⣤⣽⣟⠾⢿⣿⣛⣿⣿⣿⣿⣿⣷⢿⡟⣿⣿⣿⣿⣽⣿⣿⣷⣤⣍⢸⣿⣿⡟⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡑⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢻⣿⣿⣿⣿⣿⣿⡿⣹⣿⠈⢿⣿⣿⣷⣮⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⢿⣫⣽⣿⣿⣭⣭⡭⠅⠠⠿⠟⢿⣷⣎⠲⣿⣿⣿⣿⣿⠿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠛⠿⠿⠿⠋⠐⠒⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣿⡿⣿⣿⡆⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣷⢻⣷⣿⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⡆⣿⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣻⡿⢃⣤⣴⣶⣦⣽⡿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣵⣶⠀⣿⣿⡟⣼
⠀⠀⠀⠀⠀⠀⣢⣿⡎⠙⢁⣤⡿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣫⡿⠿⢿⣬⠿⠿⢟⣶⠶⢶⡄⠰⠶⠾⣯⣴⣿⡿⠇⠀⠈⠻⣿⣿
⠀⠀⠀⢀⠀⣿⣋⣿⣷⠶⢦⣤⡔⢻⣿⣶⣿⣷⣿⡿⣷⣄⣹⣛⡿⣿⣿⣿⠿⣿⣾⣿⣶⣶⣶⣿⣿⣿⢿⣿⣿⠟⣺⣿⣯⣽⣶⣶⡀⠀⠀⠉
⠀⠀⢠⣿⡇⢿⣿⣏⣤⡾⠿⠛⣡⣼⡷⠟⠛⠛⠛⠻⠛⠻⠿⠿⢿⣿⣿⣟⣤⣿⣿⠿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣾⣿⣷⣮⠛⠛⢠⣤⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⡿⢿⣷⣦⣶⣿⣿⣿⣿⠈⠁⢹⣿⡟⣛⣻⣿⣿⡿⠟⠁
⠀⠀⣩⣾⣿⣷⣤⡀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠙⠫⣭⣤⡤⠼⣿⠿⠿⠛⠉⠁
⠀⢸⣿⣿⣿⣯⣽⣧⣽⡇
⠀⠀⣏⣿⣿⣿⣿⣿⡍⢷
⠀⠀⣹⡛⢿⣿⡟⢛⣰⠘⣷⣤⡀
⠀⠀⠘⢿⣦⣿⣿⣞⣋⣽⣿⣟⠻⡿⠷⣴⣤⣤⣤⣤⣠⣤⣤⣤⣠⣄⢠⣤⡀⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠈⢿⡿⣿⣾⣿⣿⣭⣛⠻⣿⣿⠿⣿⣤⣤⣼⣿⡿⠻⣛⣇⣸⣷⣦⣄⢺⣿⣿⣿⡄⣄⣠⣿⣿⣶⡶⢠⣄
⠀⠀⠀⠀⠀⠙⢷⣹⣿⣿⣿⣿⡿⣫⡟⣿⣧⣴⣤⣭⣿⣧⣿⣿⣭⣿⣯⣭⣽⣿⠏⠛⣡⣼⢿⣿⠿⣿⣧⣤⣾⣿⣿⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⣿⣽⣾⣭⣤⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣶⣿⣿⣯⣻⣷⣿⣿⣏⣰⣷⣿⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣛⣿⣿⢏⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡶⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠿⢌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡰⡇⢠⡟
⠀⠀⠀⠀⠀⢀⢀⣀⣤⡤⠴⠒⠲⠶⠶⠾⠶⠶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣬⠛⣦⠞⠁
⠀⠀⠀⣠⣤⡾⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠟⠃⠀⣩⡾⠋
⠀⠔⠋⠈⠛⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠭⣭⣛⣛⣛⠛⠛⠿⠟⠛⢛⣛⣉⣡⣤⡤⠞⠂⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠐⢆⠛⠋⠉⡱⠦⡄⠐⠀⠂⠀⠒⠠⠀⢀
⠀⠀⠀⢧⣌⢀⢃⠂⠀⠒⡁⣀⣠⣀⡔⢈⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠈⠁⠐⠂⠐⠉⠀⠀⠉⠉⠛⠦⣀⢠⠀⠁⡀⢀⠄⠀⠀⠀⡠⠒⡈⠁⠀⡀⠀⠀⠋⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⡒⠤⢀⠘⠀⡰⠊⠂⠀⠀⠈⠀⡀⣀⠆⡈⠀⠄⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠄⠀⢀⢐⡖⠀⠁⠈⣀⠶⠋⠉⠀⠉⠫⣄⢂⠐⠂⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠐⡀⢂⠀⣾⠃⠐⠀⢀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡀⠀⠀⢄⠖⣿⠁⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⣌⠐⢀⠁⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠄⠀⠀⢢⠋⠀⢸⡉⡀⠀⠄⢃⠀⠀⠀⠀⠀⡼⠁⠠⠀⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢒⡈⠄⠰⠀⠀⠀⠀⢇⠠⠀⠄⠀⡄⠀⠀⢠⡞⠂⠐⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠄⠐⢠⠁⠀⠀⠀⠀⠈⡠⠂⠠⠈⡰⠀⡰⡏⢁⠈⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠁⠀⠁⠰⠀⠀⠀⠀⠀⠀⣽⠈⠐⠂⢀⡴⠅⠠⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠂⠐⢀⠡⢄⣀⢀⣠⠞⡁⢂⠐⠀⡼⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠂⡐⠀⠀⠀⠠⢀⠆⣷⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⣀⠀⠀⠀⢀⣀⠠⠈⠀⠀⠹⡀⠀⠀⠒⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠦⣄⣁⠀⠀⠀⠘⠀⠐⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠢⠦⢤⣄⡀⠄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢵⣦⠀⠤⠤⠶⠚⠩⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠋⠀⠁

⣿⣏⢹⣟⠻⣿⣟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠄⠀⠀⠀⠀⠀⠀⢀⡤⠊⣉⠿⣍⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⢀⠞⠀⣻⡷⠀⢻⣇⠀⠱⡄⠀⠀⠀⠀⠀⠰⣾
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⢦⠀⠀⠀⠀⡾⠀⠸⣷⣿⠀⢸⣿⡇⠀⢻⡀⠀⠀⠀⠀⢤⡏⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠘⢴⠄⠀⠀⠀⠀⠀⠀⠈⠣⣽⠤⡀⠀⡇⠀⠰⣏⣥⠄⢁⣚⠿⠀⣸⡇⠀⣀⡄⢒⡼⠁⠀⠀⠀⠀⠀⠀⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡦⡀⠀⠀⠀⠀⠀⠈⠢⡈⠛⣏⡀⢰⣭⡿⠃⠸⢶⡄⢀⣿⠖⢋⡿⠟⠋⠀⠀⠀⠀⠀⢴⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⡐⠄⡀⠀⠀⠀⠀⠈⠑⠿⣷⣄⣠⣶⡆⣠⣶⣶⣶⠿⠚⠉⠀⠀⠀⠀⣀⣤⣀⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⠁⢤⣤⠖⠀⢲⣶⣲⣿⡟⠛⣿⠿⠙⢿⣍⡯⣭⣏⣉⣿⣏⣏⡭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⢉⡅⣾⣿⣧⢐⢻⣄⣀⣾⣟⣷⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢉⣴⢟⡴⣿⢩⡧⣝⢻⣿⣟⣪⡻⣄⢑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣭⠴⢣⠋⡼⠁⡼⠶⣃⣷⠞⡇⠙⣆⠐⢆⠙⠻⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠔⠁⢀⣮⡾⠁⠀⢓⡀⠀⠀⢰⡃⠀⠈⠱⣏⡣⠀⠈⠳⣬⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡜⠁⠀⢠⢆⡜⠁⠀⠀⠀⠃⠀⠀⠚⠀⠀⠀⠀⠈⠳⣜⢄⠀⠀⠙⠦⣕⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⠉⠀⠀⠀⢂⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡗⡀⠀⠀⠈⠳⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⠋⠀⠀⠀⢀⢾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦

⠀⠀⠀⠀⢠⠔⠚⣉⣉⠙⠲⢄
⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⢦⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣭⣭⣍⣀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢇⠈⡇⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠁⣿⡻⣿⡶⣿⡏⠙⣶⣄⡀
⠀⠀⠀⠀⠸⡤⣄⣀⣀⡀⠙⣼⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠼⣿⣶⣾⣿⠶⠞⠋
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠹⡀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⡿⠀⠀⢀⡴⠁⠸⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒
⠀⠀⠀⠀⣾⢃⡴⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠴⠠⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⡸⠀⠀⡤⠢⡄⠀⢸⠀⠀⡇⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠂⠤⠒⠉⠀⠈⡡⠒⠊⠀⠀⠀⡇⠀⢰⠀⢸⠀⠀⡇⠀⡸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢰
⠀⠀⠀⠀⣀⡤⠤⠤⡤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⡆⢇⠀⠈⡆⢸⠀⠀⡇⢸⠁⠀⢸
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢇⢸⠀⠀⡇⡍⠀⠀⡇⡎⠀⠀⡏
⠀⠀⠒⠚⠉⠉⠉⠙⠉⠉⠉⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠸⡸⠀⠀⢃⠀⠀⠀⣇⠇⠀⢀⠃
⠀⠀⠀⠒⠤⠤⠤⠤⠤⠤⠤⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣷⡇⠀⠈⠁⠀⠀⠉⠀⠀⢸⠀⠀⠀⠀⣀⡤⠤⢤
⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢠⡞⠁⠀⣀⡼
⠀⠀⢀⠔⠉⠀⠀⠉⠉⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣧⣴⠋⠀⢠⡾⠋
⠀⠀⠁⢀⠀⠀⠀⠀⡄⠀⢠⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⢀⡴⠛⠋⠉⠐⠂⠁⠒⠈⠙⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣠⡴⠂⠀⠀⠀⠀⠀⠀⢶⣤⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠐⠓⢦⡀⠀⠀⡠⠒⠒⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠒⠁⣀⣀⠀⠒⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠺⢙⠀⣄⣀⢀⠠⣯⣩⡘⠉⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠩⠈⢄⡢⠋⠁⠀⠉⠈⠤⠀⠤⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⡑⢈⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣐⢀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠤⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠄⡀⡨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠐⠊⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠁⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠒⡀⠰
⠀⠀⠀⠀⠀⢀⣤⢴⠲⡲⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢐⡠⠄⢈⡄
⠀⠀⠀⣀⡞⠣⢂⡉⠂⠉⠉⠉⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠂⣉⠠⡆
⠀⠀⢰⠋⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⢻⢷⡄⠀⠀⠀⠀⠀⠀⢀⡏⠅⠢⣐⡃
⠀⠀⣿⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⠀⠀⠀⠀⠀⠀⠸⢃⡂⡉⢨⠇
⠀⠀⣷⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⣖⢂⠐⡀⡞
⠀⠀⠘⣁⠄⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢄⠂⠀⡜
⠀⠀⠀⠡⡄⢁⠀⠗⠦⣄⡀⠀⠀⠀⠀⠀⣀⣀⡠⠔⠊⠡⢀⡑⠀⣄⠏
⠀⠀⠀⠀⠐⢂⡀⠩⢘⠐⢉⢙⠈⠛⠙⡉⠅⠀⡃⢌⠐⢈⠐⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠃⢤⡐⡀⢃⠀⠐⡘⢀⠂⠁⠄⣂⡸⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠘⠒⠃⠚⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣦⡚⠷⠅⡀⢤⣤⣬⣅⠈⠁⠐⠢⠒⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⣷⣮⣕⠢⢔⣺⣿⣷⣾⣷⡔⠬⢥⣿⣿⣧⠨⣑⢂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠷⠾⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣟⣿⠇⢑⡈⠈⡁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣋⣋⣟⣽⣻⣿⣿⣿⣿⣯⢾⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣬⣀⣠⣿⢿⣾⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣯⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣳⣟⣻⣿⣿⣿⣭⣭⣿⣿⣷⣾⣿⡇⡕⡽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣄⢘⣏⣒⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠉⠉⠛⠿⣿⡿⣿⣿⣿⣿⣿⡟⢁⠆⠓⢸⢂
⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡽⣿⣽⣿⣿⣶⣿⣿⣯⡟⣄
⠀⠀⠀⠀⣿⣟⣿⣿⣿⣯⣿⣿⢿⣩⣿⡿⢿⡾⠽⣻⡟⣽⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠱⡷⣿⣿⣿⣿⣿⣿⣿⠁⢥⡄
⠀⠀⠀⠀⠛⠫⠽⠧⠤⠤⠜⠚⠙⠋⠁⢲⣲⢶⣶⢶⣦⣬⡛⠋⣛⣿⣿⣿⣿⣿⣜⢧⠷⣒⣋⣭⣽⡷⢦⣄⡀⠹⡟⠻⠟⡿⡿⠛⠁⠈⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣑⠀⢩⣿⣿⣿⣶⡙⠿⣿⣿⣿⣿⣿⡌⣼⣿⣿⣿⣸⣿⡂⣽⣿⣦⡑⢊⣸⣶⣴⢶⢦⣤⡄⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠃⣿⢷⣽⡿⣷⢰⣟⢿⣿⣿⣿⣷⣿⣟⣿⣿⣿⠗⢒⣠⣶⣿⡏⢿⣿⣿⣿⡇⢁⣬⠟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠡⠐⠹⡆⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⢽⣿⣿⣿⣿⣇⠀⠛⢛⣿⠛⠛⠉⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⡶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠩⠿⢰⡿⢿⣿⣿⣿⣿⣿⢿⡽⢽⣿⣟⣿⣿⣿⢡⣾⣶⣾⣧⣌⡀⠐⢠⠆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⡷⡤⡀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⣿⣦⠃⡈⢿⣿⣿⣿⢃⠈⠿⣴⣿⠁⠈⣻⠀⢰
⠀⠀⠀⢀⣼⡽⢣⢛⢿⣿⣩⣿⠪⣟⣮⣿⣝⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢦⣟⠿⠟⢟⣻⣿⣿⣷⡆⠃⠾⠟⣡⣴⣦⣴⡾⠙⠛⠛⠉⠀⠂
⠀⠀⢀⡿⣿⣇⣭⡭⣩⣙⠻⢽⡛⢾⣿⣽⢟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠈⢷⠵⠔⣾⣿⣿⣿⣿⣟⣿⣿⣷⣼⣷⣿⠟⣷⣄⠀⠀⢀⠃
⠀⠀⣾⣉⣡⣿⠷⣲⡗⠂⠁⠻⠋⠀⠈⣬⣎⢫⢿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡀⠳⣦⣻⢿⡿⠟⠋⣿⣿⡿⣿⡿⠙⢧⣁⣁⡼⢀⠤⠊⠄
⠀⢐⡋⡏⠅⢒⢋⡛⠁⠀⠀⠀⢀⢄⣤⣀⡀⡀⠈⠉⣳⣻⣿⣿⣿⣿⣿⣿⣿⣦⣌⡳⢾⣦⣀⠀⠻⣷⣿⣿⠇⠀⠈⠉⣡⡴⢈⣀⠀⡅
⠀⢸⣁⠐⠁⠩⠄⠀⠀⠀⠀⢴⠊⠀⠀⠉⠳⣵⡐⣤⡄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣝⣿⠿⣶⣬⠠⠤⢦⠿⠀⠉⣤⣮⡟⠐
⠀⢸⣨⠃⠐⠤⠀⠀⠀⠀⠠⢷⠀⠀⠀⠀⠀⠀⠙⣏⠉⡄⠐⠾⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣽⡖⡶⣺⣟⠘⠠
⠀⠈⡴⠇⠀⠀⠀⠀⠀⠀⢀⠠⢢⡀⠀⠀⠀⠀⠀⠈⠻⣼⡔⢠⠀⠘⠯⠝⢿⣿⣟⣿⣿⣟⣿⣿⡿⣛⠋⢿⣗⣸⣿⠁
⠀⠀⢡⢰⠑⡄⠁⠀⠀⠀⠄⠖⣈⠛⠷⣦⣤⣤⣀⣀⣀⣈⣹⣿⣿⣧⣀⠉⠉⠹⠿⢳⣿⠿⣾⡿⢋⠔⢊⠎⠹⠋⠁⠀⡠⠀⠰
⠀⠀⠀⢺⠇⠀⠾⠀⠀⣠⣀⣠⣤⣤⣤⠦⣴⡧⣭⣭⣭⣉⣉⠉⠉⠻⢿⣷⣴⣦⣮⣀⠀⠈⠀⠀⠁⠀⠀⠀⣠⣄⠐⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢃⣸⣼⣷⣿⣿⣿⣴⣿⣇⣿⠀⣶⠄⣉⠙⠓⠦⣄⣈⡙⣛⠿⠿⡿⣷⣾⠿⠷⠶⠒⠿⢥⣄⡀⠀⠐⡀⠀⠀⠐
⠀⠀⠀⢀⠒⠄⣦⠿⠛⣍⣧⡷⠾⠿⠛⠛⠛⠻⠿⢿⣿⣶⣿⢠⡤⣀⠀⠉⠙⠿⣍⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣷⠟
⠀⠀⠰⢀⣦⠛⣀⠶⠛⠉⠑⠂⠄⠠⣆⣠⠒⡰⢠⣤⠤⠍⠛⠿⢷⣯⣐⡆⢀⡀⠀⠉⠓⠢⢤⣀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣅⡐⠀⠀⠀⠀⠈⠋⠙⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠺⢦⠀⠶⠴⠦⠶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠁⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠂⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀⢀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡉⠉⠉⠉⠉⢉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠢⡴⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡐⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡠⠄⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠦⣀⣠⠆⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠸⢁⠆⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠐⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠢⠀⠁⠂⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠃⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣿⣿⣷⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⡇⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⢸⣿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠿⢿⡇
⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣤⡀⠀⠀⢀⣤⣶⣶⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡷⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⡿⠟⠛⠁⠀⠀⠙⠛⠿⣿⣿⡇⠀⠀⠀⠀⣠⣤⣼⣿⣿⣧⣤⣄⠀⠀⠀⠀⢸⣿⣿⣿⡛⠉⠀⠀⠉⠛⣿⣿⣿⡇
⠀⠀⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠇⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⡦⣆⠆⠞⣯⡃⠀⠀⠂⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⣈⣁⠀⠉⠡⡴⢂⠀⠰⢶⣦⣄⠘⠃⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣄⣀⣤⠟⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⢶⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠐⠀⢿⡿⠀⠀⢻⣿⢿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣉⠘⣿⣷⠀⠀⡀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡷⠪⣿⠖⠁⠀⣲⣴⣾⡏⢘⣿⠟⠛⡿⠻⣿⣯⣷⣭⣶⣿⣿⣷⠿⢳⣄
⠀⠀⠀⠀⠀⠀⣄⢶⡾⢉⠛⣻⣿⢻⣿⣿⣿⣿⣿⡷⠿⣦⠿⣷⣿⠆⠹⣿⣿⠛⠉⣠⣿⡀⠐⠀⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣔⠙⣾⡇⢿⣿⣿⣿⣿⣿⣿⠟⢤⣤⡆⣾⠷⣮⣷⣾⣯⣼⢿⡛⠁⠘⣱⢁⣠⣬⡿
⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠀⠘⣿⣏⣻⠷⠦⠤⢤⣄⠹⠃⠀⠀⢹⣿⢏⠉⠙⠻⠇⠀⠀⠋⣌⡄⠈⠟
⠀⠀⠀⠀⢀⣾⣿⠟⢿⣿⣤⡤⠧⠽⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠡⢤⣄⢼⠀⠀⢀⣀⠏⢀⢠⠄
⠀⠀⠀⠀⠚⠛⢉⢰⡠⢦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⢔⣻⠿⡄⠀⠀⠘⠿⣟⣡⣿
⠀⠀⠀⢰⡇⢠⡄⡠⠄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠈⠀⠃⠀⠀⣸⣿⣻⢿⣿
⠀⠀⠀⠈⣷⠚⣽⠟⣶⠟⡄⢀⣠⣶⣾⣿⣿⣿⣿⣷⣶⣶⡑⡄⢿⣿⣤⣴⡶⠀⠀⢀⡟⣿⣶⠿⡇
⠀⠀⠀⠀⢻⡄⠑⠂⠾⣄⣿⠟⣻⣿⣿⠿⢿⣿⣿⢟⢿⣿⣧⣿⣸⠟⠹⣿⣥⣠⠐⣿⣃⣼⠗⠛
⠀⠀⠀⠀⠀⠻⣦⣀⣄⠋⠠⠀⢸⣿⣿⡿⠿⣿⡟⠃⢶⣿⣿⣿⡟⣀⣼⣿⣿⢣⣿⢹⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⠄⣤⠈⣿⣿⣿⠁⢀⡃⣿⣧⣾⣿⣿⠟⠛⠟⠿⠟⠡⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⣿⣿⠧⠴⠾⠷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢿⣷⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠻⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⢠⣿⠁
⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠴⠶⠶⢶⣶⣤⣤⡀
⠀⠀⠀⠀⠻⣿⣿⣷⠶⡶⠶⠶⠶⠶⢿⣟⣏⣹⣽⣵⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠈⠛⠿⠾⢿⣿⣾⡿⠿⠿⠿⢛⣛⡋⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⢛⠛⠛⠛⠿⠶⢶⣶⣶⣶⡶⠾⠛⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⢁⣤⣶⣶⠯⠿⠿⠿⠿⢶⣶⣴⣶⣤⢦⣄⢴⢦⣾⠿⠃
⠀⠀⠀⠀⠀⠀⣴⡟⣁⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣷⣺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢿⣿⡹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠸⣿⣿⡝⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⣛⣫⣷⣶⣿⣿⠿⠿⠟⠛⠛⠿⢿⣿⣿⣽⣷
⠀⠀⠀⠀⠀⠀⠙⣿⣷⣤⡙⠻⢷⣶⣤⣄⣀⣀⣀⣠⣤⣴⣶⠿⠟⠛⣭⣦⢶⣯⠵⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣻⢠⠀⠈⠉⠙⠛⠛⠛⠋⠉⠉⠀⣰⣟⣾⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠺⠦⢤⣅⣤⣄⣴⣤⡶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣡⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠤⢖⣊⣡⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣯⣟⣷⣶⣶⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡎⢛⣭⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢖⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁