Smug face Emojis & Symbols

⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠤⠤⣤⣤⣤⣤⣾⣯⣿⣦⣀⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠲⠶⠤⢤⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠻⠦⢤⣀
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣀⡀⠀⣠⠞⠁⠀⢀⣠⡶⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡟⠆⠀⣸⠇
⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣘⣓⡥⠴⣷⠶⠖⠛⠋⠁⣠⣴⣿⣿⡿⠈⠀⠙⠁⠹⠋⠹⡙⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣛⣉⣡⣤⣴⣶⣇⠀⠀⠀⠀⣼⡿⠟⣩⡤⢤⡀⠀⠀⢰⡿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣭⣭⡿⢿⣿⡇⠀⠀⠀⠈⠀⠘⣡⣶⠛⠷⠀⠀⠘⣯⠑⠽⡄⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⢸⡇⠀⣤⢤⣌⠉⠙⢿⣿⣟⠁⠀⠀⠀⠀⠀⠘⢉⣉⡤⠀⠀⢀⡴⠛⠳⡄⢱⠀⠀⢀⣤⠖⠛⣩⣍⣿⣦
⠀⠀⠀⠀⠈⠳⣦⣀⠀⠈⣷⠀⠸⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢋⡤⠀⣠⠇⠀⣧⡖⢋⣠⠶⠛⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠁⠈⠀⢀⣀⣤⠤⠞⠛⠛⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣉⣁⣀⡧⠀⠀⠀⠀⠀⠀⢰⠟⠒⠒⠒⠒⠒⠛⠉⠉⠀⠀⢀⣀⣀⡀⠒⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠙⢻⠁⠀⠐⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⠀⠀⠀⢤⠀⠀⠋⢀⡀⠉⠳⣤⠀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡆⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⡀⠀⠀⡀⠀⢸⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠾⢧⣤⠤⢷⣄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠂⠀⠘⢀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⣾⣿⠶⣼⣿⡷⣦⣄⣀⣀⠀⠛⠀⠀⠈⠀⢀⣀⣀⣤⣤⡶⠟⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢿⣿⣄⣻⣿⡅⠘⣿⣯⣉⣽⣿⡿⠛⢿⣿⣯⠉⣩⣿⣿⠃⠀⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡘⣿⣿⡉⠙⣿⣿⠉⣿⣿⡀⣿⣿⡀⠈⣿⣿⠃⣿⣿⣿⠄⢀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠛⠛⠛⠉⠉⠉⠙⠛⠛⠛⠓⠶⠿⠿⠶⠾⠿⠿⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠉⠙⠳⠦⣤⠴⠞⠋⠉⠉⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣴⡾⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠉⠙⠂⠘⠋⠉⠉⠙⣷
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣤⠶⠲⠶⣤⡀⠀⠀⢠⡟⠀⡼⢁⣴⠆⣠⠆⢠⣀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣀⣤⣾⣯⣤⣀⠀⠀⠉⠒⠂⢸⡇⠀⣇⡞⢹⣼⠃⠀⠀⠙⣧⠀⢀⡿
⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⣾⠟⠉⢹⡏⠙⠛⠛⠷⠶⠤⠄⠈⠀⠀⠘⠃⠸⡇⠀⠀⠀⠀⢹⡆⢸⡇
⠀⠀⣠⠶⠋⠁⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⡇
⠀⣼⠃⠀⠀⠀⠀⠀⢠⡿⠀⠀⢠⡿⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⠀⣿⠈⢧
⠀⣿⡄⠀⠀⠤⠤⠖⠋⠀⠀⣰⠟⠁⠀⢴⠿⠛⠛⠛⠛⠗⠀⠀⠀⠀⢀⡄⠛⠉⠀⠈⠙⠀⣿⣶⣾⠗
⠀⠘⣿⡀⠀⠈⠓⠶⠶⠶⣛⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠐⣶⣤⣤⡶⢞⣥⠀⠠⠬⠇⠘⢶⣤⣤⣤⣤⡾⠀⠀⠀⠀⢻⣤⣙⠷⣶⣶⡾⠃⠸⣇
⠀⠀⣾⠃⠰⢦⣭⣴⡾⢟⣿⠀⠀⠀⠀⠀⢤⣬⣉⣉⣩⠄⠀⠀⠀⢀⣀⠀⢙⣷⠀⠶⠚⠁⠀⠹⣆
⠀⢰⡏⠀⠈⠳⢶⣶⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⣾⠏⠀⠀⢀⡀⠀⠀⢿⡆
⠀⠸⣧⡀⠀⠀⠀⠹⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣇⠀⠀⢸⣿
⠀⠀⠈⠻⣦⣄⡀⠀⣿⢻⡟⠁⠀⠀⠀⠀⠀⠀⠿⠛⠉⠉⠛⠲⠶⠦⠤⠴⠶⠜⠋⠁⠀⠈⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠻⣿⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢾⣧⡀⠳⠖⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠋⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⢡⠟⠀⠀⢠⠋⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠙⣷⡹⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡯⢠⡏⠀⠀⢠⠁⠀⠀⠀⡞⠀⠀⣇⠀⠀⢀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢷⡜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⠀⠀⢠⠃⠀⠀⠀⡸⡇⠀⠀⢰⠀⠀⢸⠇⠀⠀⢤⠀⠀⢇⠀⠀⠈⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡎⣰⠀⢸⢀⠃⢱⠀⠀⠀⠳⡀⢠⠱⡀⠀⠸⣆⠀⠈⡄⠀⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡇⠀⣸⢠⠻⣄⠘⣎⡴⠖⠳⣄⠀⠀⠑⢸⡔⠛⠦⠤⠥⡕⢄⣷⠀⠀⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠇⠀⣿⠋⠀⠈⠙⠉⠃⠀⠀⠀⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⠇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠸⠀⡇⢀⣠⣤⣤⣤⣤⣓⡢⠀⠀⠀⠀⠠⢔⣢⣤⣤⣤⣤⣀⠀⡇⡸⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⠀⠀⠀⢇⡋⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⢳⠁⠀⠀⢸⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢸⠀⢷
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢱⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⡀⣠⡤⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⢆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠣⡀⠀⠀⠸⡶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣶⠃⠀⠀⠀⡰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠈⠢⠄⣀⣻⣀⣤⡭⠉⠉⠛⠛⠛⠛⠛⠉⠉⢉⡀⠀⡟⠀⢀⡠⠊⠀⠀⠀⢰⣦⡇
⠀⠀⠀⠀⠀⠀⠘⣇⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠉⠉⠁⠀⠀⠀⠀⣠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠙⠦⣤⣀⣀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⡤⠔⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠎⠛⠛⠛⠛⠳⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠾⠉⠉⠉⠀⣀⣀⠙⠳⠶⢀⣄⡀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⠛⠀⢀⡠⠞⠁⠤⠈⠉⠀⠀⠈⠉⠙⠳⢤⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡌⠀⡐⠋⠠⠂⠀⠀⣀⣐⢦⣀⣀⣉⣉⠁⠈⠹⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠖⢁⠈⣀⣠⡶⠛⠻⡋⠐⠻⠟⠛⠛⠛⠛⠻⠶⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣀⡼⠶⠋⣁⠟⡡⠄⠉⠠⣀⠁⠤⠄⣈⡉⠛⠳⠶⠶⠶⠶⠿⣟⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⠋⣉⠃⠀⢼⡿⠋⠁⠐⡤⠂⠀⠀⠐⠂⠀⠀⠀⠠⠤⠀⠤⠤⠖⢉⡼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡁⠸⠀⡟⢀⠘⡎⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠇⠀⠀⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⡆
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠘⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⡼⢆
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡧⢸⣦
⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⢧⡛⣽⢧⡀
⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⢸⣇⢻⡎⣷⡄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢿⣆⢻⣌⢿⣟⣿⣆
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣋⡷⠘⡀⠀⠀⠀⠀⠀⢿⣌⢿⡌⢿⣎⢿⣯⢿⣦
⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⡀⠰⠆⠈⠀⠀⠈⠠⡄⠀⠀⠀⠀⠻⣮⣿⣆⢻⣆⠿⣾⠋⣦
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣈⠿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣆⣿⣲⣧⠾⠿⣤⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⣿⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢿⣽⡿⠋⠀⠀⠀⠀⠙⣦
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣆⠟⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠈
⠙⣿⣄⠏⠹⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠘⣿⢿⣤⠶⠿⢿⣧⣿⣤⣀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣀⣁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠁⠀⡠⠠⠄⠉⠙⠀⠐⠂⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠘⠀⠀⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠷⠶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠠⠀⠀⠈⠑⠂⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⡀⠀⠀⠀⠀⠀⠀⠐⠛⢤⡄⢀⡀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⡇⠀⠀⠀⠀⠀⢸⠃⠢⣄⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠶⣤⣤⠀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠔⠛⠁⠉
⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠸⠆⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⡄⠀⠀⠀⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠘⣄⠀⠀⠀⠐⣀⠀⠀⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠃⠀⠀⠀⣴⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠈⠛⠓⠂⠀⠉⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠄⠀⣠⡶⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⢀⣰⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣉⣦⣤⠁⣤⡤⠶⣦⠔⢢⡄⢰⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠉⠉⢠⡙⢦⣌⡁⢄⠛⠳⣶⠠⣤⠀⢾⣷⣤⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠉⢿⠂⠰⣶⣤⠶⢈⣷⡄⢻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣹⠷⡄⠲⣤⡇⠸⣦⢿⣇⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣷⣄⡀⢿⣴⣦⢄⣘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣿⣿⠟⣀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⢷⡾⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡵⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⣿⣄⠈⠳⠶⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⠿⣹⠟⠁⠀⢀⡜⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠈⢻⣌⢻⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣰⠏⠀⠀⢀⠎⠀⠀⠀⢠⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠹⣧⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢰⡟⠀⠀⢀⡎⠀⠀⠀⠀⣾⠀⠀⠀⡇⠀⠀⢰⡆⠀⠀⠀⠀⠀⠱⡀⠀⠀⠹⣷⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⠃⠀⠀⡞⠀⠀⢠⠀⡼⢹⠀⠀⠀⠹⡀⠀⠸⢇⠀⠀⠘⣆⠀⠀⢣⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⠀⠀⢸⠁⣏⠀⢸⡀⡇⠈⢆⠀⠀⠀⠱⡄⢸⢈⣆⠀⠀⢹⢦⠀⠘⡆⠀⠀⢸⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⠀⡇⡰⠙⢄⡀⣧⡗⠛⠉⠳⣄⡀⠀⠈⣺⡯⠉⠛⠲⠶⠷⠝⠦⣷⠀⠀⡸⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⢸⡞⠁⠀⠀⠉⠁⠙⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⡇⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⡇⢸⠀⣀⣤⣤⣤⣤⣤⣝⡒⠄⠀⠀⠀⠀⠠⠴⣒⣤⣤⣤⣤⣤⣀⠀⡇⣸⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡄⠀⠀⠹⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⢷⠃⠀⠀⠀⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢳⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⣆⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠙⠿⠭⢭⣭⠭⠭⠟⠋⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡜⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⢀⡄⠀⠀⠀⠈⠳⣄⠀⠀⠘⣏⠛⠓⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⡏⠀⠀⠀⢀⠞⠀⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠸⣇⠀⠀⠀⠀⠀⠀⠉⠒⠤⠼⠶⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣰⣇⣀⠤⠚⠁⠀⠀⠀⢀⣾⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢛⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠴⠶⠶⠶⠶⠶⠆⠀⠀⠀⠠⠤⠤⠤⠤⣤⠀⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⠚⠛⠛⢿⣟⠛⠀⠀⠀⠠⡟⠛⠻⠿⠛⠒⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠠⡤⢤⣤⠤⠤⠤⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⣀⣸⣿⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⡛⠒⠛⠷⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⠿⠿⠿⠛⠉⠀⠀⠀⡉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣞⠁⠀⠀⠉⠳⣄⠀⠀⠉⠉⣿⣟⣛⣛⣋⣉⣉⣭⡟⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⣾⢇
⠀⠀⠀⠀⠀⢀⣤⡶⠋⠛⠶⣄⡀⠀⠀⠑⣄⠀⠀⣿⣧⣭⣭⣭⣭⣭⠿⣧⠀⠀⠀⠀⢀⡾⠋⠀⢀⣴⠟⠁⠈⠳⡄
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠙⡶⣄⡀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡾⠁⠀⣴⡿⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⢿⣆⠀⠀⢿⡇⠀⠀⢀⡞⠀⠀⣿⠀⠀⠰⠋⠀⢀⣾⣿⡇⡇⠀⠀⠀⠀⠈⣧
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡏⠀⢹⡆⠀⢸⡇⠀⠀⠘⠁⠀⢸⣿⠀⠀⠀⠀⢠⡾⠉⢸⡇⡇⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠀⠀⠀⣷⠀⠘⡟⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢠⡿⠁⠀⠸⣧⠹⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠋⠀⠀⠀⠀⠀⠀⠈⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠈⠙⠛⣿⣟⠻⠷⠶⠶⠶⠶⠶⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⢴⣶⠆⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⣠⡞⠀⠸⡷⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠉⠉⠉⠙⠛⠛⠛⠛⢉⣤⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠹⣦⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣷⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⣠⡶⠚⠓⠻⢿⣄⠀⠀⠀⠀⠀⢀⣤⣴⠶⠾⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⢀⣾⠃⠀⠀⢀⣀⣀⡙⣧⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢸⣟⠉⣛⣏⣀⣈⣏⣀⣹⣧⠀⠀⣿⠶⢴⣶⠶⠶⢶⣶⡤⠶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠈⢿⡈⠉⠉⠉⠁⠈⠉⢈⡿⠀⠀⣿⡤⠴⠷⢤⣤⣤⣦⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠈⣷⣦⣀⠀⠀⢀⣤⠟⠁⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⢰⣿⠁⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙⠻⣦⣤⣀⣀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⠶⠶⠛⠛⠿⣷⣄
⠸⣿⣧⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠈⠉⠉⠉⢹⣿⠉⠉⠉⠛⠷⠶⢦⣤⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠦⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠀⣰⡿⠁
⠀⠀⢠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣤⣴⡾⠟⠁
⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⠁
⠀⠀⠻⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⠖⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⢀⣤⣾⣿⣀⣀⣀⣀⡠⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠸⣿⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠉⠻⣷⡶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⣠⣶⡾⠿⠿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡾⢿⣦⣄
⣾⡟⠀⠀⠀⠀⠀⠀⠙⢧⣀⣀⣠⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠈⢻⣧
⣿⡇⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢹⣷⣀⣠⣤⣴⣾⠿⠋⠁⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⠿⠋
⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⢀⣤⣴⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣶⣾⣿⣷⣯⣾⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⣿⡿⢿⡿⢿⣿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠠⠶⠖⠒⠶⢶⣶⠖⠂⠀⠀⢛⣛⣿⣿⣿⡃⢸⡟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠛⠉⠻⠿⠋⠉⠁⠀⠀⠀⠀⠈⠛⠁⠀⠁⢸⣯⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⠀⢀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠄⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⢻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⠛⠱⢦⣄⡀⠈⠛⢿⣦⣀⣀⣀⣀⣀⣀⣠⣤⣤⣾⠟⠀⠀⠀⢰⡆⠀⠸⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠙⠛⢿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⣰⠟⠁⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠙⠳⣆⠀⠀⠀⠀⠀⠹⣷⡀⠀⢸⣏⣙⣋⣩⡏⠀⠀⠀⠀⣠⡾⠋⠀⣠⡞⠁⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠹⣧⠀⣸⠉⠉⠁⢸⡇⠀⠀⢀⡾⠋⠀⣠⠞⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢹⡆⣿⠀⠀⠀⣾⡇⠀⣠⡟⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⣧⣿⠀⠀⠀⣿⠀⣴⠋⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⣿⣿⠀⠀⠀⡿⣸⠏⠀⢀⡾⠁⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢻⣆
⠀⠀⠀⢰⡿⠃⠀⠀⢰⣦⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⡇⠀⠀⠀⣷⡏⠀⠀⣼⠁⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠘⣿
⠀⠀⢀⣿⠁⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⡇⠀⠀⠀⣿⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⣿⡆
⠀⠀⢸⡟⠀⠀⠀⠀⢾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠿⠃
⠀⢀⡿⠃⠀⠀⠀⠀⠘⢛⣿⡿⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠈⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣀⡀
⠠⢶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⡶
⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⡾⠿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣾⣿⡀⠀⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⢿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡷⠦⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢟⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠂⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠌⠀⠀⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠢
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡉⠙⠲⠢⢄⡀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⡈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣞⣺⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡛⣿⣝⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⢯⠛⠛⠛⠛⠳⠀⠉⠉⠃⠈⠁⠀⠉⠉⠀⠉⠀⠀⠀⣁⣘⠋⠉⠉⠉⡉⠉⠉⠉⠛⠉⡁⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⣉⣄⣀⣀⣀⣀⣀⡀⠀⡀⠀⢀⡐⠀⠀⠀⠀⢀⣈⠤⣦⣠⣤⣶⣾⣿⣿⣶⣾⣿⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⠆⢶⣴⣿⣋⣐⣹⣟⣿⣿⣿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⡉⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⡿⠁⠀⠀⠀⢀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠙⠛⠛⠏⠙⠛⠋⢸⠟⠛⠛⠿⢿⠿⡿⠟⠛⠛⠛⠞⠋⠙⠙⠋⠉⠉⠉⠉⠉⠛⠛⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⢸⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢸⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⢹⣿⣿⣿⣿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⣺⠃⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⡀⠀⠀⢀⣀⣤⣾⠿⠋⠀⠀⠀⠀⠀⡰⠋⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠘⣿⣿⢻⡿⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⣿⡿⠛⡥⢄⣀⠀⢠⣀⣠⡞⠀⡄⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠘⠛⢇⠙⠀⠈⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠘⢿⢯⣤⣼⠀⠀⠉⠁⠀⠈⠹⡷⠀⠁⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⢄⠀⠘⠧⠠⢀⣈⢰⡂⢔⣶⡲⢤⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⣷⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⠉⠐⠾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣤⣄⠀⣀⣀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⠁⠉⠉⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣤⣤⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣤⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⢻⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠀⣴⣿⣿⣉⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⠘⢃⣴⣿⣿⣿⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣀⣀⣰⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⢿⣟⢿⣿⢟⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠙⢷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣴⡾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣴⣿⡧⠀⠀⠀⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡧⠀⠦⢤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⣿⣿⣿⣿⣷⣶⣾⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣟⠉⠉⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣿⡿⢿⣦⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣰⡿⠃⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⡿⠿⢿⡿⠿⠿⠿⠿⠿⠿⣿⠁
⣿⡇⣤⢸⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⡀⠀⠀⠀⠀⠀⢠⣿
⢸⡟⢻⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⣾⡿
⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⢠⡿⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣿⠃⢰⣿⠀⣿⠁⠀⠀⠀⠀⠀⣾⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡟⠀⠀⠀⠀⠀⢀⣾⠇⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣾⠁⠀⠀⠀⠀⢠⣾⡿⠀⠀⢸⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢰⡏⠀⠀⠀⠀⠀⢈⣿⠃⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢰⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡸⠃⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠁⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢸⣿⠀⠇⠀⠀⠀⠀⠀⣿⠇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡘⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣼⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⢠⡿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣼⡇
⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡏⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠇⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⡟
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡏⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⣠⣿⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣇⣀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠷⠖⠚⣋⡙
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣧⣤⠴⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⡿⠛⠛⣹⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⢏⣙⠻⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠤⠈⢀⣼⠟⠁
⠀⢿⠇⠀⠀⠀⠀⠀⢀⣾⠁⠀⢉⣴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⣴⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣷⡶⠀⠀⠈⢻⣿⡿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠉⠀⢻⡆⣹⣿⣿⣿⠇⢀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢉⡡⠋⢹⣿⠏⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠏⠛⠀⠀⠀⣀⡤⠂⣀⣤⠆⠔⠓⢚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠘⠿⠶⠗⠒⠺⣿⡯⠐⠋⠁⠀⣀⣤⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠐⢀⠄⠀⠀⠀⠀⣦⣀⣩⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⠀⠄⠀⠀⠈⢿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢿⡿⠋⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠉⠉⠉⣉⡵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠂⠀⠀⠀⣀⠠⠤⠤⠤⠦⠤⠤⠤⢤⣀⣀⠀⠀⠀⢀⣀⣀⣤⠬⠭⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠔⢩⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⢄⠀⠀⣷⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⡆⢀⡇⠀⠀⣀⣀⣀⣈⣙⡛⠀⠀⠒⠋⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠤⠬⣍⠓⠾⠀⠀⠀⢸⠁⠀⠛⠀⢸⠍⣹⠈⠉⠉⠉⢣⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠚⠲⢤⡀⣠⠞⠦⣅⣀⠀⢀⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠙⡇⠈⣹⢧⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⣀⠤⠤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⢱⢤⡀⠀⠀⠀⠀⠀⠁⠀⢤⣀⣀⣀⡼⠀⣸⠁
⠀⠀⠀⠀⠀⢀⡠⠖⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡴⠁⢀⠏⠓⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢏⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⣠⣋⣀⣘⣁⣰⡭⠭⠟⠒⠂⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⢦⣀⡤⡔⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣹⢃⣀⢿⡄⠀⠀⠀⠉⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡴⡊⠁⢀⡔⠐⠒⠒⠙⢷⠋⠀⠀⠀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠥⠞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣤⢤⡤⠴⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⣸⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⣾⣤⣤⣽⡤⢤⣷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠭⠟⠚⠉⠑⠺⠿⠾⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢦⡤⠚⠋⠉⠛⠲⠖⠋⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣠⠴⠒⠚⠋⠉⠉⠙⠛⠒⠯⡀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠛⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠁
⢀⣞⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢰⠒⣄⠀⠀⣾⠛⠒⣦⠈⠛⠶⣤⣄⣀⣀⣀⣤⡴⠞⠛⠷⠶⣦⣤⣤⡴⠶⠋⠁
⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠘⠛⠁⠀⠀⠻⣤⠞⠋⠀⠀⠀⠀⠈⠉⠉
⢸⠇⠀⠀⠀⢄⠀⢀⡀⠠⠄⠤⠶⠦⠒⠒⡶⠞⠃⢻⡄
⠘⣇⠀⠀⠀⠀⢱⠀⠀⠀⣠⡞⠆⠀⠀⠀⠁⠛⠀⠀⢻⡄
⠀⠘⡆⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⡀⠀⠠⡀⢳⡄
⠀⠀⢷⠀⠀⠀⠀⠆⠀⠀⢠⡀⠀⠀⠘⡆⠀⢣⡀⢀⠇⠀⢳⡀
⠀⠀⢸⡆⠀⠀⠀⢰⠀⠀⠀⠳⣄⣀⡴⠃⠀⠀⠙⠉⠀⠀⠀⢷
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢠⠞⠉⠁⠐⠤⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⣿⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠀⣿⠉⠁
⠀⠀⠘⢦⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣀⣀⠀⠀⡀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⣀⣀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢦⣤⠄⢀⣀⡀⠀⠀⠀⠀⠀⢀⡼⠷⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢉⡽⠿⣤⣀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⡀⠀⠀⢀⣴⠏⠀⠀⠀⠉⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣳⣶⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣾⣦⣾⣷⢹⡆⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣫⠿⠛⠭⠴⠶⢾⣛⣛⣵⣫⣿⣼⣷⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢟⣵⣶⣿⠿⠭⣽⣭⣝⣛⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡿⠙⠂⠀⢀⣢⣤⣤⣾⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣴⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⡇⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠷⣀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠿⣿⣿⣿⣿⣿⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠭⡛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢳⣦⣀⣀⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣆⢻⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⡟⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆⠋⠥⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠙⣰⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠐⠂⠀⡄⢳⢄⠀⠀⠀⠀⠀⢼⠂⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⡴⠣⠁⣴⡇⠀⠀⠐⠀⠀⠠⣈⠞⠈⣆⠀⢸⢂⣼⣿⣿⣬⣀⣀
⠀⢀⠀⢰⡅⠘⡇⠈⠀⠀⠀⠀⣸⠀⢠⣾⡿⣿⡇⠀⠀⠀⠀⠀⡼⠃⠀⡰⢁⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⣸⠜⣽⠟⣿⠻⣿⣿
⠀⢸⣧⡘⠀⢀⠙⡀⠀⠀⠀⠀⡏⣰⣿⠟⠁⣿⣇⣤⣶⣾⣿⣿⡷⣤⡐⠀⣸⢒⠠⠀⠀⠀⡔⡇⠀⠀⢀⡄⠀⣿⣼⡿⢀⡏⠀⠑⠘
⠀⠸⣿⣧⠀⡀⠀⠸⡀⠀⠀⠀⣳⣿⠏⠀⣠⣟⣻⣿⣿⢫⣬⣄⢩⣷⡟⣆⡧⠈⠀⠀⢀⠎⠀⠀⠀⠀⢸⣷⣴⣿⣿⠃⢸⡏
⠀⠀⠸⣿⡄⠀⠀⠉⢿⡄⠠⠀⣿⡟⠀⠘⣿⠐⣿⣿⣿⡈⣙⡋⠠⣭⠠⣿⡱⠀⠀⢠⠏⠀⠀⠀⠀⠰⠇⢀⡻⣿⠁⠀⣾⠁⠀⢡
⠀⠀⠁⣿⣧⠀⠀⠀⣸⣷⠀⠀⣿⡇⠀⠀⠙⣦⣽⣿⣿⣿⣿⣦⣠⠄⠂⢺⠄⠀⢠⠏⠀⠀⠀⠀⠀⣽⣾⣥⠔⠋⠀⢰⣿⠙⢶⢾⡇
⠀⠀⠀⢹⡏⣷⡀⠀⢸⣿⡅⢠⣿⡇⠀⣐⣶⣿⠿⠿⢿⣿⡟⠻⣿⣿⣿⡿⠀⣠⠋⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⢀⣿⣿⡆⠀⢠
⠀⠀⠀⢸⡇⣿⡇⠀⢰⣿⠃⠸⣿⠀⣸⣿⣿⣿⣶⣶⣿⣿⣿⣦⣤⣤⣿⣀⣰⠡⠴⣶⣾⣾⣿⣿⣿⣿⠏⠰⠀⢠⣾⡿⣷⠈⠀⠀⣻
⠀⠀⠀⣸⣷⣿⠂⠀⢸⣿⠀⢀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣩⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⢏⠅⠀⠀⢀⣾⡿⠁⠈⠀⢀⣼⣿
⠀⠀⢀⡟⣿⣿⠀⢀⣾⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣼⣿⠿⢡⠟⠁⠀⠀⣾⣿⠃⠀⠀⢠⣿⡿⢹
⠀⠀⢸⠁⣿⡏⣠⣾⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠒⠶⣶⣾⡿⠛⢀⡏⠀⠀⠀⣼⣿⠏⠀⠀⢠⣿⡟⡰⠿
⠀⢀⡏⠀⣿⣷⣿⢧⠤⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢀⠀⡇⠀⠀⣴⣿⠏⠀⠀⣴⣿⠏⠀⠎
⠀⠸⠧⠄⣿⣿⡇⠘⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆⢀⠂⣴⣶⣦⠀⠀⠀⠈⣃⠀⢠⡇⠀⣰⣿⠏⠀⣠⣾⡿⠃⠀⠁
⠀⠈⠀⠀⣿⣿⣿⠀⠹⡄⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣿⣧⢤⣾⣿⣿⣿⣿⣄⠀⠀⠑⢄⢸⠀⣴⣿⠏⢀⣾⣿⠟⠁
⠀⠰⠤⢀⣿⣿⣿⡄⠀⢳⠀⠀⠀⢿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⢾⣿⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⡠⠚
⠀⢀⠀⢸⣿⣿⡿⠸⢄⠀⢇⠀⠀⠘⣿⣿⣿⣧⣠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢷⣆⡀⠀⢱⣿⡿⠃⠀⠀⠀⠀⠀⢀⢞⠱⡀
⠀⠀⢀⣿⣿⣿⡇⠀⠀⢵⢼⡀⡈⠀⠙⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣦⣤⣤⠻⠿⢶⣶⣿⠛⠻⢶⣖⡂⠀⢀⡿⡘⡄⢳
⠀⠀⣿⣿⣿⣿⢇⠀⠀⠀⢧⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠻⢿⣿⣷⡀⠀⠀⠙⢧⡀⠀⠈⠉⠋⣿⠀⣡⠘⢉
⠀⢸⣿⣿⣿⡇⠘⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣟⠿⠿⣿⣿⢿⣷⣦⠀⠈⢻⣿⣿⣶⣄⣀⡀⠙⣷⣄⠀⣸⡿⣍⠀⠀⠈
⠀⣿⣿⣿⣿⠁⠀⠈⢧⡀⠀⠀⠀⢀⠀⠀⠀⡠⠞⠤⣈⡁⠀⠀⣰⣿⡇⠀⠀⣿⣷⡆⠈⣧⠙⢿⣿⣿⣿⣶⣌⣿⣿⣿⠃⠈⠳⢤
⠀⣿⣿⣿⡿⠀⠀⠀⠈⢐⢄⠀⠀⠀⠘⢦⡞⠁⠀⠂⠈⠃⢠⣾⠟⣼⠈⠃⠀⣿⣿⣷⡀⣷⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⡝
⠀⢻⣿⣿⠀⠢⠀⠀⠀⠈⢢⠣⡀⢀⡴⠋⠀⠢⠀⠀⠄⣴⣿⡋⢸⠏⠒⠄⠀⠿⣿⣿⣧⢹⠀⠘⠁⠀⡈⣻⣿⣿⣿⣿⣿⣿⣷⣄⣱
⠀⣿⣿⣯⠀⢀⠀⡀⠀⠀⠀⢣⡸⠎⠀⠀⠀⠀⠀⢈⣼⠟⠁⣇⣾⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠂⣀⠞⠠⠃⠎⠳⣌⢉⠉⢻⠿⠿⢟
⠀⣾⣿⣿⠀⠈⠀⠛⠐⢂⠄⡟⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠠⢸⠇⠀⠀⠄⢀⡠⣾⣻⣿⣿⡁⠌⠡⠀⠀⠘⠀⠀⠟⣦⠹⣤⣀⢃⡘
⠀⢻⣿⣿⠀⠀⠀⢠⠜⠇⡄⠀⠀⠀⠀⠐⠀⢰⣿⠃⠀⠀⢀⡏⠀⢀⠤⠚⣡⠞⠁⣸⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠁⡷⡀⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣟⣛⣛⣛⡿⢯⡿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⣽⣷⣶⣶⣶⣶⣾⣷⣯⣭⣍⡙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠊⣬⡍⠻⣿⣿⠀⠀⠀⠀⠠⠖⠿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣿⣿⣿⣿⣿⡏⢰⠈⣼⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⠀⢧⡘⠟⠷⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢤⡴⠟⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠙⠳⠂⠀⠀⠠⡀⠀⠀⠀⠀⢀⣴⣿⠷⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠰⣄⠀⠀⠈⠒⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣀⣀⠀⠀⠀⠀⠈⢷⣦⣄⣀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⢲⣤⣀⠀⠈⠙⠓⠲⠦⣤⣀⡙⣿⣿⣿⠃⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠚⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⢁⣤⣉⠛⠃⠀⢀⣾⠀⠈⠻⢿⣶⣤⣄⣀⠀⠀⠀⠉⠛⢿⣧⣶⣇⠀⠀⠀⠀⠀⠀⠸⢟⣉⣤⡈⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠃⣾⣿⣿⣿⣷⣦⣼⣿⡀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣠⣦⡀⢻⡇⢈⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡸⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣌⣙⣿⣿⣿⣿⣿⣮⣷⣿⣾⣛⣛⣻⣿⣟⣛⣻⣛⣛⣛⣃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠲⠤⠤⠖⠒⠒⠛⠛⠛⠛⠋⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⢸⣧⠴⠖⠛⠲⣄⠀⣄⣻⠉⠓⠲⠿⡍⠓⠲⠦⠿⠏⠛⠲⢦⣄⣀⠀⢦⣤⣀⠀⠑⠦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠢⢾⡀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⡞⠠⠌⠉⠓⠛⠀⠉⠓⠦⣌⣷
⠀⠀⠀⠈⠛⣟⣛⣒⣒⠒⠒⠠⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠶⢦⠀⠉⠁⠀⠀⠀⣀⣀⠀⢀⠴⠒⠓⣄⢠⠞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⡠⠔⣊⣁⣀⣀⡤⠤⣤⠤⣤⣤⡬⠍⠛⠛⢛⣿⡿⠶⠶⠶⠦⠈⠧⣄⣀⣀⣠⡄⠀⢠⡴⢋⡽⢩⠅⢰⠋⠀⠀⠀⡜⠈⣦⠄⠀⠀⣸
⠚⠋⠉⠉⠉⢻⡅⢠⠖⢾⠉⠹⡇⠀⣵⣔⣚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢻⡉⣵⢻⠀⢾⠀⠀⠀⡰⠃⠀⠈⢟⠛⣛⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠳⠜⠢⡀⢣⠀⡇⠐⢋⣁⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⣸⣿⠀⠀⠑⠒⠊⠁⢀⠀⠀⠨⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠉⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠳⣼⡄⠀⠀⠀⠐⢦⠉⠄⠀⢀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣇⢠⣤⣤⣀⣀⡀⣀⡰⢛⣶⡤⢾⣿⡆⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣙⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠛⠋⠉⠉⠹⣄⣈⣉⢙⡄⠀⠀⠉⠀⠁⠸⡅⢀⣼⠉⠹⣆⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⡟⠉⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣏⡃⠀⠀⠀⠀⠀⠀⠘⠲⢋⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢇⠀⠀⠀⠀⠀⠀⣎⠁⠀⠘⠛⠊⠁⠀⠀⠀⠀⠀⡀⠈⢧⣀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⡾⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⢀⣨⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⢶⢲⠊⡿⠀⠀⠀⠈⠓⠦⣄⣀⣀⡀⠀⣰⠁⣧⠔⣹⢿⣿⣤⡀⠀⣴⢮⣉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠁⠀⠀⠀⠀⠀⢰⠃⡏⠉⢩⠏⠁⠀⠉⠀⠀⠀⠙⠦⣙⢺⢁⣤⡨⢏⡁⣹⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣯⣄⡀⠀⠀⠀⠀⢿⠾⠗⠂⢺⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢰⡶⣄⣹⠁⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡜⠻⠏⡟⡿⡲⠶⡾⢶⣊⠻⠟⠻⠇⠉⢳⠀⣀⡠⠤⠒⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⢀⣃⡤⢖⡽⠿⣧⡀⠈⢑⣲⣭⣛⣓⠶⢲⡏⣟⡓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣴⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠥⣶⣛⡃⢂⣴⡧⠯⢬⢤⠖⠒⢺⠻⡉⠁⠀⠈⠉⡿⠿⣷⣶⣶⣤⣤⣍⣉⣉⠒⠒⠲⠦⠤⢴⠋⣠⣾⠟⣋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡸⠀⢸⠉⡏⠻⣳⠀⢸⠾⡆⠀⠀⡇⢧⠀⠀⠀⠀⢹⣤⣤⣤⣌⣉⣉⡛⠛⠻⠿⠿⢿⣶⣶⣶⠾⣋⡥⠊⢹⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠇⠀⢸⠰⡇⠀⠉⠀⠈⠉⠁⠀⠀⢳⠼⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢤⣄⣼⡧⠚⠁⠀⠀⢸⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢀⠀⠐⣦⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢒⣴⣎⠁⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⡇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠔⠁⠀⠈⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣽⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠛⠲⣄⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⡱
⠀⠀⠀⠀⠀⠀⠀⢅⠦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⣀⣀⡤⠔⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣏⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣮⣤⣤⣤⣴⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣯⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⡞⣽⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡸⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⣢⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⡇⠀⠶⠶⢤⡀⠀⢀⡤⠶⠖⠀⢀⠹⣿⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠇⠸⣶⣤⡤⠀⠀⠀⢤⣤⣶⠇⢸⣷⣿⣼⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⢏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠜⣡⣫⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠃⠀⣠⣴⣾⣿⣿⡟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠀⠀⣸⣿⣿⣿⢟⣵⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⡿⣫⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣨⣾⣿⢟⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⣿⣿⣿⣿⣿⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⡦⠐⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠈⢻⣿⢿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⠀⠀⠀⡀⠐⡈⣋⠂⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣶⣄⠀⠀⠀⠀⢨⣴⣿⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠭⠿⡿⠟⠀⠀⠀⠀⠸⢿⡿⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣴⣶⠆⠀⠀⠀⠀⠀⠘⢧⣁⣙⣛⣉⣵⠋⠀⠀⠀⠀⠀⠐⣶⣦⣌⢲⣀
⠀⠀⠀⠀⢦⡴⣶⣶⡄⠈⠸⠛⠛⠁⠀⠀⡀⠀⠀⠀⠀⠈⢿⣻⢟⡿⠁⠀⠀⠀⠀⢢⠀⠀⠈⠛⠛⠃⠉⢠⣴⣶⢦⣴
⠀⠀⠀⠀⠈⠾⠿⡿⢷⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⡾⢿⠿⠷⠃
⠀⠀⠀⠀⠀⠈⡖⢄⢸⣧⣴⡶⠂⢀⠀⠀⠸⢷⣦⡀⠀⢀⣀⢀⡀⢀⡀⠀⢀⣴⡾⠏⡀⠀⡀⠐⢶⣦⣼⡇⣀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡬⣷⣟⡀⠀⠘⡦⡄⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠘⡏⢀⢴⡃⠀⠀⣙⣶⠥⣼⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣤⣌⣿⣁⡴⠊⠀⢻⠀⠀⢈⠌⠉⠻⡿⠋⠁⡀⠀⠀⡟⠀⠙⢦⣈⣿⣁⣤⡿⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠈⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠏⠀⠀⠀⠻⢿⣿⣿⡆⢰⣿⣿⣿⠟⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣠⡐⢬⡮⣾⣿⣿⡿⣿⣾⣷⢿⣷⣷⣧⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠻⠿⠛⠀⠈⠋⠙⠋⠁⠈⠈⠉⠈⠙⠛⠛⠸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⡄⠐⢮⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⡄⠀⢳⡸⣄⣀⣀⡀⠀⠀⠀⠀⠁⢀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠉⢻⣿⣿⣿⣦⡀⠳⣤⠥⣍⡉⠳⡀⠀⠀⠀⠂⡐⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠹⣿⣿⡏⠻⢶⣄⡀⠈⠻⢦⣈⠓⠀⠀⠀⠀⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠈⠉⡁⠀⠀⣿⣿⣶⣤⣄⣀⣀⣀⠀⠀⣀⣀⣀⣿⣿⡟⢳⡄
⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⢀⣤⡀⠀⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠉⠉⠿⠟⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⣼⡇⠀⡄⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⣿⠁⢰⠇⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⢶⡀⠀⣀⣈⠙⠄⠀⠀⠀⠀⠀⠀⢰⣾⣿⠋
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣡⡄⠀⠚⣩⠍⢳⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠾⠋⠀⣀⠴⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢲⡤⢖⣚⣥⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠈⣷⠀⠀⣀⣴⠟⠁⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠘⠷⠾⠛⠁⣀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣤⣀⣀⣠⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠚⠛⣻⣿⠿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⡁⢰⣿⣶⣭⣛⠿⣿⣳⣾⣟⢿⣿⣿⣷⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢣⡾⠃⣀⠻⣿⣿⣿⣽⣲⣭⡳⢯⡻⣮⣻⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠟⠀⠀⠸⣿⣮⣙⠿⣿⣿⣟⣷⠦⣝⣮⣳⣝⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠙⠻⣽⣛⡶⢍⠙⢻⣿⣶⣮⣝⡻⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⣛⡲⠀⠀⠀⠴⣛⡛⠛⠳⢦⣝⠻⣿⣿⣿⣷⣌⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⠛⢿⠆⠀⠀⠐⠛⠿⠻⠖⠀⠉⢷⡈⢻⣟⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡙⡄⠙⣯⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣄⢱⠰⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⣴⣿⣇⠇⠁⣧⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⠐⠒⠀⠀⠀⠀⣀⡴⢺⣿⣿⢇⡶⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣤⣤⡔⠊⠁⠀⠸⠟⢁⡾⠁⢰⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣾⣿⣿⣿⣿⣿⡇⠀⠀⢀⡴⣻⠟⢡⠋⢛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣾⣿⣿⣿⣿⡿⠟⣀⡤⢞⣽⢞⣵⢯⣾⣌⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣾⣿⣿⢿⣫⠭⣖⣫⣵⣾⣿⣿⢟⣵⢟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣫⣵⣿⣿⣿⣿⣿⢟⡵⢟⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⢰⣿⣿⣿⠟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠿⠃⠀⠙⠛

⠀⠀⢀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⢀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠣⢀⠀⠀⢀⡀⠀⠀⠀⣸⣤⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣶⣿⣶⣼⣿⣷⣧⣾⣼⣿⣷⢂⣤⡀⠀⠀⠀⠀⡠⠔⠋⠀⢠⠖⠋
⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢤⡈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⡤⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤⣞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠠⠤⠤⠴⠒⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠚⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣱⣿⣿⣿⠻⠿⠋⠉⠿⢻⣿⠃⣼⣿⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢈⣩⣽⡒⠀⠘⠋⠀⢘⣩⣤⣼⢿⣿⣿⠛⠁⠀⠀⠀⣀⠀⢈⡀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠈⠁⠛⠋⠀⠀⠀⠀⠉⠀⠛⠉⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡤⠄⠀⠀⠠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⣄⡑⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⣈⣤⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣴⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣤⣄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠉⣨⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠻⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⣠⡟⠁⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠰⣿⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢻⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⠞⠉⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠟⠁⢀⡀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠿⣦⡀⠀⠘⣿⣤⣤⣤⣤⣄
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠉⠉⠉⠉⠻⢿⣤⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⣇⢠⣤⣤⣤⣼⣿⠀⠀⢸⣿⡍⠉⠉⠿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠈⢿⠋⠁⢈⣿⠀⠀⠀⢻⣧⠘⠀⢰⣿
⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⢀⣀⢀⣴⣶⢾⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠈⢷⣤⣀⢀⣴⡾⠁⠀⠀⠀⠘⣿⣆⣠⣾⠟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⣀⣠⣶⠶⣾⣿⣿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⣧⡀⠀⠠⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠁
⠀⠀⠀⠀⡈⢿⣷⠠⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠁⠀⠀⠈⠛⠃⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠸⠷⣸⣿⣦⣴⠄⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢻⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣶⡿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣤⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⢰⣿⢋⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣀⡄⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⣸⢿⡆⠀⠀⠀⣾⡏
⠀⠀⠀⢸⣿⡘⠇⣀⡰⠄⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⣾⣶⡾⠟⠋⠀⠀⠀⣼⠏⠀⡇⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠹⣷⣤⣉⣠⣤⣾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠛⠿⠦⣤⣤⣤⣤⣤⣤⣤⠿⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⠰⠖⠋⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠿⢿⣆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠻⠿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠛⠛⠉⠀⢀⣀⣠⣤⠛⢿⣶⣶⣶⣦⣤⣤⣴⣶⣶⣿⣿⣿⡇⠘⣶⣄⣀⡀⠉⠛⠛⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⣀⣠⣴⣶⡾⠿⠛⣿⡇⠀⠈⢈⠻⣿⣟⠛⢿⣁⢀⣠⡄⢻⡏⠰⠀⠹⣿⠟⢿⣷⣶⣤⣀⣀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣶⣿⠟⠋⠉⠀⠀⠀⠀⣿⡇⠀⠀⠹⠀⠈⢻⣦⣀⠋⠉⠙⣿⡟⢀⡸⠃⠀⢻⣧⠀⠀⠈⠛⠛⠿⣷⡆⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⣰⡟⠑⠎⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢰⣿⠃⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠘⢦⣀⠀⠹⢿⡟⠁⡠⠊⠀⠀⠀⠀⢹⣿⠀⠀⠀⢠⣿⠃⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣤⣾⣿⡟⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣿⣦⣤⣀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠀⠉⠛⠛⠛⢷⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠛⠋⠉⢹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⢹⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⠀⢸⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣶⠄⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢶⣶⣿⣿⣙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠉⠀⠀⠀⠀⠀⠀⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠟⠛⢷⣦⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠉⠻⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣤⣤⣼⠟⢿⣏⠉⣉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣉⣿⣧⣤⣤⣤⣤⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⢿⣷⡉⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⠀⠀⠀⣀⢀⣀⡀⢀⣠⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⠀⠘⡿⠋⢹⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣿⣤⣤⣴⣿⣤⣤⣤⣿⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⡿⠿⠿⠿⠿⣿⣿⣿⣿⠋⠉⠉⢉⣏⠉⠉⠉⢹⣿⣿⣿⣿⠿⠿⠿⠿⠿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⢻⣄⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⢸⡇⠀⠀⠀⠈⠻⠛⠛⠋⠀⠀⠀⠀⠀⣠⡟⠋⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⠀⠀⠀⣙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⣏⡀⠀⠀⠀⠀⢀⣀⡀⣀⠀⣀⠀⢠⣿⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣤⣤⣤⠶⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⠛⠛⠉⠀⠈⠱⢆⡀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠛⠋⠉⠀⠀⢠⡞⠓⢤⡄⡀⠀⠈⠳⣀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣿⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⣙⠂⠀⠀⠙⣆⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠻⠀⠀⠀⠀⠰⣇⣀⠀⠀⢀⠻⠀⠀⠀⠀⠘⣧⣀⣀⣠⣤⣴⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠰⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠈⠉⠲⠦⠾⠀⢀⣤⣴⠶⠿⠛⠉⠉⠉⠁⠀⠀⠀⠉⠻⠶⣦⣄
⠀⠀⠀⠀⠀⣼⡇⠀⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣈⣙⣷⡄
⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢧⡀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⠛⠉⠉⠉⠹⣧⠈⠉⠉⠙⠋
⠀⠀⠀⠀⠘⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⢧⣴⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⢠⠞⠉⢙⡦⠀⠙⣷
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠰⣆⡰⠋⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠘⣷⡀⠈⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠒⠶⣄⠀⢸⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣾⣦⣤⣶⣿⣤⣄⣀⣀⣀⣀⠀⠀⢀⣴⡟⠀⠀⠀⠀⢠⠀⠀⠀⢸⡧⣤⣤⣤⣿⠀⠀⢿⡏⠉⠙
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣹⠏⠉⠛⠛⠛⠛⠛⣉⣿⡄⠀⠀⠀⢠⡄⠀⠀⠘⣦⡈⠁⢠⡾⠀⠀⠈⣿⠁⣠
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣇⣀⠠⠶⠶⣶⡟⠃⠀⠈⣷⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⢻⡟⠋
⠀⠀⠀⠀⠀⠀⢀⡘⢿⡆⠀⠀⠀⠀⠀⢻⡍⠀⠀⠀⠉⠁⠀⠀⣠⠋⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⠋⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⢀⣰⠟⠀⠀⣸⡇⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠷⠿⠷⠶⠟⠁⠀⢀⡼⠀⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠋⠙⣷⠀⠀⠀⠀⠀⠀⠟⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠏⠁⠀⠃⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⡧⠀⠀⠉⠒⠂⢤⣤⣤⣤⣤⡤⠺⠋⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠈⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠾⢿⣷⣶⣤⣄⣀⣴⠟⠻⣦⣴⣾⣟⠉⠙⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣶⣿⣿⣠⣤⣾⠿⣦⡉⠉⢹⡇⠀⠀⢸⣷⠟⠉⢿⣦⣾⢿⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢛⡷⠾⠛⠋⠀⠙⠋⠀⣿⠀⠀⠉⠛⠛⣿⣆⣴⡾⣷⡀⠀⠀⣿⠁⠀⠀⠙⠹⣶⠛⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠻⣦⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⢸⡇⢨⣿⣇⠘⢿⡀⠀⢹⣧⠀⠀⠀⢀⡾⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⢧⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⣸⠛⠻⣧⠸⡇⠀⠈⢿⡄⠀⠀⣸⠇⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠛⠛⠲⣦⣼⡟⠀⠀⠀⠀⢸⣧⠟⢀⡆⠈⠻⣷⠀⠀⢸⣷⠚⠛⠋⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠘⠉⠀⢈⡃⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣷⣦⣤⠀⠀⣿⣇⣀⣀⣀⣀⣀⣠⡶⠛⠛⠳⣦⣀⣀⣀⣠⣿⢀⣠⣴⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠉⠀⢰⣿⡉⠉⠉⠉⠉⠉⠛⢧⣤⣤⣤⠞⠉⠉⠉⢹⡇⠀⠉⠉⠙⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠋⠻⣧⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⠶⣶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠉⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠿⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⢶⣶⠶⣿⠶⢶⣶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⣿⠖⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣀⣯⣀⣠⣭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠿⠿⠿⢿⣿⣿⡟⠉⠉⣿⠉⠉⢙⣿⣿⣿⠟⠛⠳⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⣆⠀⠀⠀⠈⠉⠉⠁⠀⠀⡿⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⠞⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢤⡶⢦⣴⠻⣦⡄⢤⡄⢴⡄⠰⣦⠀⣿⢶⡆⠤⠴⠂⣴⠖⢶⠦⠿⠂⣤⠠⢦⡝⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠒⠛⠓⠚⠛⠚⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡛⢦⣀⣀⣀⣀⣠⣞⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠀⠉⠉⠉⠉⠉⠘⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠿⠽⠆⢠⡄⠘⠉⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠠⣤⠼⠧⡤⠄⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣴⠀⠀⠀⠀⣧⠖⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⠀⡴⠚⠛⠛⠋⠉⣿⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡼⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⣼⠈⢳⡀⠀⠀⠀⠀⠀⠀⣷⢰⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠐⣿⣀⡀⠈⠳⡄
⠀⠀⠀⠀⠐⠟⢃⣠⣴⣿⣴⣶⣿⠟⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠓⢿⣤⣶⡴⠿⠖⠲⠶⠶⠶⠖⠋⠁⠈⡿⠀⠀⡷
⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⢹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣌⡓⠿⢶⣤⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡿⠟⢛⣉⣭⠥⣤⣶⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠟⣡⠴⢚⣩⠴⠒⠉⣁⢤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢟⣵⢞⣡⠖⠋⠀⢀⠔⣨⠶⣛⣭⡽⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣵⣟⣵⣟⡅⠀⡠⠂⢠⢞⡵⡿⠋⣁⢤⣭⣭⣿⣿⣿⣿⡍⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⡴⠊⠀⣴⣿⢟⣠⡶⠋⢀⣴⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣥⢖⣡⣾⣿⣿⣿⡟⣡⣾⣿⣿⣿⡿⠿⠟⠁⠰⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡼⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⢰⣮⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠙⣽⢏⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⢟⣯⣿⣿⣿⣿⣿⣷⣝⣿⡿⣻⣯⡤⣶⣦⣤⣄⡀⠀⠀⠀⣀⣼⣿⡏⠠⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢿⣿⣿⣿⣿⣿⣿⣿⣫⠞⢽⣷⣶⣿⣶⣯⠛⠃⠀⢠⣾⣿⣿⣿⣇⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⠀⡄⠐⠿⡯⠤⠁⠀⠀⠀⣿⣿⣽⣿⣿⣿⣬⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⡟⠿⣿⣿⣿⠛⢾⣿⣧⠆⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⣇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠿⣿⣿⣿⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⢄⣠⣿⣿⣿⣏⢷⣬⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠛⣻⣿⣿⣛⣛⣛⣽⣿⣿⣧⠀⢀⡾⠁⠀⠀⠀⠀⢀⣀⣈⣿⣿⣿⣿⡌⢀⣐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣌⣛⣿⣿⣿⣿⣵⣾⣿⣿⣷⣏⡀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣱⣎⣉⣡⢟⣿⣿⠟⠉⠉⠻⣿⣿⣶⣄⡀⠀⠀⠉⢙⣿⣥⣥⣭⣥⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⡿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣧⣸⣿⣾⣶⣼⣷⣾⣶⣷⣾⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣥⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣭⣿⣿⣷⣽⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢰⣿⡶⠿⠛⠛⠛⠛⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢠⡎⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⠈⢿⡄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠈⠁⠀⠐⠒⠢⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⡴⠋⠀⠈⢿⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢋⣤⣤⣄⠀⠈⢷⡀⢀⡀
⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠲⢦⣄⠀⠀⠀⠀⠀⡀⠀⠀⢰⡏⢀⡀⣈⣷⠀⠘⣿⠛⠻⢷
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⠀⡀⠀⠀⠘⣏⠘⠋⢀⡸⠀⠀⢻⡆⠀⣸
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⠀⣀⣀⣰⣶⡖⠙⠉⢻⠀⠀⠀⠀⢿⡀⠀⠀⠈⠛⠄⠉⠀⠀⠀⠈⣿⡾⠋
⠀⠀⢀⠸⣧⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠙⠁⠀⠀⣾⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⢻⢿⡟⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢻⡆⠀⠀⠀⠀⠀⠈⠛⠓⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣤⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⡟⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⢀⣤⠟⠀⢀⡜⠃⠀⢀⣾⠁
⠀⠀⠸⣧⡐⠀⢀⣠⣶⡄⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⢀⠞⠁⠄⠀⣼⠇
⠀⠀⠀⠈⠛⠿⠛⠉⠀⢻⡄⠀⠀⠀⠀⠀⡏⠳⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠇⠀⠀⠈⠉⠒⠶⠶⠐⠒⠒⠊⠁⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠶⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠶⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⠋⣁⣬⡟⠷⣶⣤⣤⣤⣤⣴⠞⠋⠀⢶⡄⣾⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡤⣚⣲⡿⠛⠋⠻⠛⠉⣿⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⣿⠋⠉⠙⠛⡵⣞⠢⣄
⠀⠀⠀⠀⠀⠀⢻⣄⠻⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⡿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⢧⡀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⠶⢦⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠶⠾⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣤⣤⡀⢀⣿⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⢠⣤⣴⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡟⠀⠀⠉⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠒⠛⠋⠹⣧⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠒⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⢀⣀⡀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⡏⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡇⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⢻⡟⠻⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⡏⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢹⡏⠉⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣾⡿⠻⢻⡿⠺⠿⣷⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⣍⠉⠁⠀⠙⠛⠛⠀⠀⢸⡇⠀⠀⠛⠛⠛⠀⠀⠀⢈⠟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⠤⣤⣤⠿⣤⠤⢤⠤⢤⠤⢤⡤⢼⣧⡴⠦⠴⠤⠴⠤⠤⡤⠾⠦⠤⠤⡬⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠿⠿⠦⠾⠶⠾⠶⠾⠶⠴⠷⠼⠷⠶⠶⠶⠶⠶⠶⠴⠷⠶⠶⠶⠶⠿⠞⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣾⣿
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣶⣶⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠶⠶⠛⠋⠛⠓⢦⣤⠶⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣀⡤⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠶⣤⡀
⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣰⠊⠉⠉⠉⢦⣀⣀⣀⣀⣀⣀
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠤⠊⠉⠁⠀⠀⠈⠉⠙⠳⢤⡀
⠀⠀⠙⠲⠶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⡄
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⢀⣀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⣀⣠⣤⡄⠀⠀⢀⣀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠀⠀⠉⠙⠓⠒⠒⠒⠛⠋⠁⠀⠸⣇⣀⢀⣿⠀⠀⠻⣄⡇⠀⠀⢨⠏⠀⠲⡒⠂⠀⠤⠤⠤⢀⡀⠤⠆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠁⠀⠀⠀⠘⢦⠀⠀⠐⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣁⢈⢦⠀⣐⢉⠈⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⢀⡸⠀⢗⠛⠀⠃⡿⠀⠀⠀⠀⠄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⠠⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡄⢀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠒⠁⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠉⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠿⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⠞⢧⡉⠉⠁⠀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠙⢦⡀⣀⡤⠶⠚⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣗⣒⠭⠉⡝⢿⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠠⠀⠁⠀⢠⣽⣼⡇⠀⣠⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠁⠀⠀⡀⢠⣾⠽⠾⠙⠪⠛⠹⠧⠶⣄⡘⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⢀⣠⠖⠋⠁⠩⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⢘⢆⠈⣃⣀⣴⠆⠀⠀⠀⠤⠤⣤⣄⣀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⢕⡅⣈⣭⡽⠁⠀⠀⠀⢤⢶⡿⠿⠉⠀⢰⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡑⠂⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠈⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⢢⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠱⣄⡀⠀⠀⠀⠈⠙⠿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣶⣤⣀⣀⣀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠻⢿⣿⣿⢿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⢋⣪⣤⣤⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⠛⠇⠀⠀⠀⠠⢐⣺⣽⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠙⠿⣿⣶⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠙⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⣿⣇
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⣼⣿⠟⠉⠁⠀⠀⠀⢻⣿
⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣠⡆⠸⣿⡇
⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⡙⠻⣶⣤⣀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣾⣿⡇⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷⡀⠈⡙⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠊⠁⠀⠀⠀⣀⣴⡿⠟⠁⠈⣿⡀⠘⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠉⠛⠿⣦⠹⣧⠀⠀⠀⠀⣀⣴⡾⢿⣉⡀⠀⢠⠀⠸⣧⠀⢹⣿⡄
⠀⠀⠸⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⡀⠀⣿⡇⠈⠀⢠⣴⣾⠟⠋⢤⣬⢿⣿⣷⡏⠀⣀⣹⣧⠀⢻⣿
⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⣿⡆⣿⣇⠀⠀⣾⣿⠀⠀⠀⠀⠉⠂⠋⠉⠓⠀⠈⢿⣧⣀⣾⡟
⠀⠀⠀⢸⣿⡆⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣷⡀⠀⠀⣙⣛⠁⢸⣿⠀⢀⣿⡏⠐⢦⡀⠀⢀⣰⣶⠦⠄⠀⠀⠀⠻⣿⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠹⣿⣄⠐⣿⠙⣧⢠⣿⠀⣸⣿⠇⠀⠈⢻⣶⡿⠻⣇⢠⣶⣄⠀⠀⠀⢻⣿⡀
⠀⣤⣶⠿⠟⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⠻⣷⣿⠀⣿⣿⠏⢰⣿⡟⠀⠀⠀⠸⠟⠁⠀⠈⣃⣈⡉⣷⣄⠀⢸⣿⡇
⢸⡿⢀⣤⣤⣤⣤⣤⣴⣿⣿⣧⣄⡀⠀⠀⠀⣿⡄⠘⠛⠀⣿⠇⣠⣿⡟⠀⠀⠀⠀⢀⣠⣴⣞⣛⣉⣍⣉⣻⣿⣷⣼⡿
⢸⢇⣾⠟⠛⠛⠉⠉⠉⢙⣻⣿⣿⣿⣷⣦⣄⠈⠿⣶⣶⡾⠋⡴⠟⣿⣷⠀⠀⠀⠀⠉⠉⠙⠓⠿⢬⣍⣋⣙⣹⢿⣧
⣼⡿⠃⠀⢀⣠⣴⣾⡿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣈⣉⣁⣼⣿
⠁⢀⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣤⣀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⡀⠀⠘⣦⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢿⣿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣻⣷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣆
⣿⣷⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⡇
⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠸⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠹⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡇
⠀⠀⢹⣿⣿⣿⣼⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣿⠻⡏⠀⠸⠙⣿⣿⣿⣿⣿⡟⠀⠈⢿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣃⣤⣥⣤⣤⣀⡹⣿⣿⣿⣿⡇⠀⠀⣠⣵⣶⣶⣶⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⢱⠋⠀⢸⣿⣿⣿⡗⠘⢿⣿⣿⠀⠀⠚⢹⣿⣟⣿⣿⠀⠈⡷⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠸⠟⠒⠛⠁⠀⠀⠙⠿⡄⠀⠀⠈⠉⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⠉⠉⠃⠀⡇
⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⣠⣷⠀⡇
⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣴⣿⣿⡇⡇
⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠒⠒⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣿⣿⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣯⠛⢿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠖⠛⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠉⠙⠻⣤⠀⠀⢀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⠖⠚⠒⢀⠀⠀⠀⠀⢀⣀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠇⠀⠶⠀⠀⢰⡞⡂⠀⠀⠉⠉⠁⣀⣨⠉⣯⠒⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠘⢦⠄⠊⠁⢀⡴⢋⣱⠿⠛⠚⠒⣒⣾⣿⣿⣶⡏⠙⠢⠔⠺⠟⠑⠒⢦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣟⡰⠀⢀⣴⡿⣡⣾⠁⠀⣠⣶⣉⣼⣿⡋⠃⠀⠉⠳⣤⡇⣀⢀⡀⠀⣊⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣦⣴⠋⢀⣴⣿⣿⢰⣿⠿⠗⠉⠀⠀⠀⢉⣄⠁⠀⠀⠀⠀⢹⣿⣿⣶⠀⢀⡾⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⣿⣿⡏⠃⢸⡁⠀⠀⠀⠀⠀⢠⣴⣿⣗⣴⣦⡄⠀⣾⠙⠿⠛⢁⠈⠓⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡏⠰⠀⠘⣿⠏⠀⠒⠰⠎⣧⠈⠻⣿⢰⣿⣿⣶⣄⠁⢀⣷⣦⣤⠾⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⡄⠀⢿⡁⣶⠈⣰⣿⡦⠀⠀⠀⠀⣿⡗⣸⢃⣼⣿⣿⣿⣿⣧⢼⣿⡅⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠋⠉⠉⠀⠘⢿⢿⣆⠻⠟⠁⢀⣀⣠⣤⣿⠗⣱⣾⣿⣿⣿⣿⡿⢉⣈⣉⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢶⠀⠈⠃⢻⡀⠀⢀⣌⡁⠀⠀⠙⠛⠛⠛⠟⠿⢿⣿⣧⣿⣿⣿⣧⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣇⠀⠀⠀⠀⢀⣽⠖⢲⡌⠙⠢⠉⠻⢿⣷⡀⠀⠀⠀⠀⠀⠂⣈⣥⣴⣶⣾⣿⡟⠁⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣾⣇⡀⠀⠠⠄⣿⠁⠀⠀⢿⣖⣄⣀⠀⣠⣭⠙⢀⣀⣰⣤⣶⣿⣿⣿⡿⠟⠿⣿⣿⠀⠶⠀⠀⢰⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡶⠀⠐⢻⡆⠀⠀⠀⢻⣯⣉⣉⣀⣀⡑⠾⣿⠿⠿⡟⠛⠛⠋⠀⠀⠀⢹⣿⠀⢢⣴⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠓⠀⠀⠀⠀⠉⠉⠹⣿⡁⠀⠀⢀⠈⠀⠀⣂⣁⣤⣤⣀⠠⠾⠃⢀⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠠⢦⡾⢶⡄⠀⠀⠀⠀⠀⠈⠑⠀⠀⢹⣿⣿⣯⡉⠀⠁⠉⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠛⢡⣿⣷⣶⣤⣤⣄⠀⠻⣿⣿⣿⣧⡈⡀⠀⢀⣀⡠⢾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠉⠹⠛⠉⠁⣛⠆⠸⣧⣹⣿⠻⣇⠉⠋⠙⠉⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠀⠤⠓⠀⠰⣿⠋⠱⠍⣷⣤⠠⠾⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠜⠃⠉⠙⠀⢄⡃⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠖⠒⠶⠄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⠟⠁⠉⠀⠀⠀⠀⠀⠀⠿⠋⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠋⠀⢀⣴⣶⣿⠀⠀⠀⡀⠀⢠⠖⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡤⠀⠀⠀⠀⡾⠋⠁⣀⡀⠀⠀⠳⣄⠸⠟⠈⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢟⣤⠀⠀⠀⠀⠀⢰⠟⠋⣀⡀⠀⠀⠈⠙⠳⠶⢤⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣫⠀⠀⠀⠉⠉⠉⠉⠉⣉⣠⠤⠴⠶⠤⠤⠶⠚⠉⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢟⠀⠀⠀⠀⢠⠶⠾⣭⣤⣤⣀⣀⣀⣀⣀⣀⣠⣼⠇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠻⢧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠉⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠘⠳⢦⣀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢧⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇⠈⢣⡀⠀⠉⠓⢦⣀⠘⠉⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠴⠛⠒⠓⠲⢄⢀⣀⡤⠶⠛⠉⠈⣇⠀⠀⢳⡀⠀⠀⠀⣿⣷⣄⣿⢮⣙⠶⢦⣄⣀⠀⠀⠀⠀⢀⡴⢋⣤⠞⢉⡙⣦
⠀⠀⠀⠀⠀⠀⠀⡞⠁⣠⠴⠒⠒⠒⠺⡋⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⣄⠀⠀⣿⣴⡛⢿⠀⠀⠀⠀⠨⣍⡛⢦⡀⠀⢾⢁⡾⢁⡴⠋⢁⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢳⠈⠁⡴⠖⠚⠓⠒⢿⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠘⢧⣸⢹⢋⣷⣼⡆⠀⠀⠀⠀⠈⠻⣮⣿⣦⠈⠻⢧⡟⢠⡾⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡤⠶⠒⠒⠢⣼⠇⠀⠀⠀⠰⢶⡦⠬⣗⣦⣄⣀⠘⣯⢸⡿⠃⣼⡇⠀⠀⠀⠀⠀⠀⢻⣄⡈⠳⣤⡇⠙⡏⠀⢻⡝⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣴⠃⢻⡀⠀⠒⢶⣾⣤⡟⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠉⠁⠘⣿⠁⣼⠟⣧⠀⠀⠀⠀⠀⠀⠸⡇⠙⣦⠈⠈⠳⠈⢲⡞⠀⢠⡟
⠀⠀⠀⠀⠀⣸⠃⠀⠈⢷⡀⠀⠀⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠘⣾⡏⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣄⠀⠀⠀⢸⡇⠀⢸⠃
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⣧⡾⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣆⠀⠈⠳⢠⡟
⠀⠀⠀⢀⡟⠀⠀⣀⡴⣶⢼⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠘⣧⠘⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣼⢿⣀⡀⠀⣸
⠀⠀⠀⢘⣥⠶⠟⠉⠀⠘⣄⠳⣄⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣤⣉⣙⠛⠋
⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⢣⡈⠓⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠙⢲⡀⠀⣰⠞⠛⠛⠋⠙⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣔⣩⠤⣤⠀⠀⠀⠹⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠛⢉⡴⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⢀⢈⡗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠛⠁⠀⠀⠉⠓⠦⣄⣀⠐⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡉⠳⣄
⠀⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⠮⣷⣄
⠀⣠⠶⠛⢉⢉⠙⠲⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠠⡀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠙⠂
⢰⡇⠀⠀⠉⠘⢳⠀⠹⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⢹⡀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⢰⣤⣀⠀⠙⢷⡀
⠈⠛⠲⠤⠤⠤⢤⡄⠀⠀⠀⣀⣤⣴⠲⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⣀⣀⠘⣆⠙⠳⢦⣽⣏⠓⢦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣾⣿⣿⣇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⢀⣀⣉⣉⣉⠀⠸⡆⠀⠀⠀⠉⠀⠀⠈⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠀⠻⣇⣹⣾⠀⢹⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⡴⠚⠉⠉⠀⠈⠉⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢧⣈⡷⡁⢸⢦⡀⠀⠘⣆⡀⠀⢠⢀⣿⠘⠀⠀⠀⠀⢀⣠⠞⠀⠀⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡄⠀⠀⠙⢦⣼⠧⣄⣹⢦⡀⠀⠉⠓⢶⡛⠙⠲⠴⠶⠖⠛⠉⠀⠀⢀⣠⠼⠛⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠻⣦⡀⠀⠀⠻⣦⣼⠳⣦⣽⡷⣤⣜⡂⢃⠀⠤⠆⣀⠀⢠⠢⠀⢀⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠀⠐⣿⠻⢦⡐⠈⠀⠙⠓⢾⣟⠛⢮⣿⠛⠶⣦⣌⣀⣁⠀⠈⠈⠀⢈⠉⠉⠙⡏⠁
⠀⠀⠀⠀⣠⠞⠀⠀⢰⠃⠀⠀⠸⣆⠀⠙⠦⣀⠀⠀⠀⠈⠙⠺⢤⣉⣳⠟⠤⢤⣼⣉⣉⡟⠉⠈⠀⠀⠀⣿
⠀⠀⠀⡴⠃⠀⠠⠽⡟⠀⠀⠀⠀⠙⢆⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠉⠙⠒⠾⠧⢤⣯⡀⠀⠀⠀⠀⣰⠇
⠀⠀⡼⠁⠀⠀⠀⣸⠃⢀⣠⣤⣶⣚⡿⠃⠀⠀⠀⠀⡼⠻⡝⠛⠲⠦⢤⣤⣀⣀⣀⣀⣈⣀⣀⣠⡤⠞⠋
⠀⡼⠁⠀⠀⠀⠀⣯⠶⢋⣿⡿⠟⠛⢦⣀⣠⠖⣲⠴⠗⠒⠛⠒⠶⡞⣽⣶⠀⠀⠈⠈⠉⠉⠁
⡼⠁⠀⠀⠀⠀⠀⠛⠋⠉⠀⠙⠦⣄⡀⠉⠉⠈⡉⠓⠶⣄⠀⠀⠀⡗⢙⠿⠶⣦
⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠠⡈⠓⢦⡼⠀⠀⠀⠈⠋⠙⢺⠃
⠀⠈⢳⡈⠙⠓⠶⠦⣤⣄⣀⣀⣀⠀⣼⢈⠳⣄⣸⠦⡼⠃⠀⠀⠀⠀⡴⠒⠻⡆
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢉⣉⠉⣿⢸⡀⠀⠁⠀⠀⠀⠀⠀⢰⡋⠙⢶⡞⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⠀⠀⠀⠿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠐⣺⡛⠛⠻⣄
⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⢄⠀⠀⠀⠀⠀⠹⡟⠲⠶⠋
⠀⠀⠀⠀⣼⢷⣶⢤⣤⣀⣀⣀⢀⣀⣠⣏⣀⣙⣦⣤⣀⣀⢀⣀⡴⢃
⠀⠀⠀⠀⠙⠶⢍⣛⠳⠤⣬⣭⣏⣁⣀⣀⣭⢿⣩⣃⣈⣉⡿⠁⢀⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣸⣿⣧⠀⠀⠀⠀⣀⣈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡁
⠀⠀⠀⠀⢀⣾⠟⢉⡉⠙⢻⣿⠀⣤⣞⣋⠠⠤⠤⣤⣹⣿⣧⡀⠀⠀⠀⠀⣴⣿⠟⣉⣁⣉⣛⢶⣄⠀⠀⠀⢠⣾⠛⢉⣉⡛⣦
⠀⠀⠀⠀⣼⠃⠀⢀⣹⡆⠸⠟⢸⡏⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⣾⡿⠟⠛⠉⠉⠀⠀⠀⢛⡇⠀⠀⠀⠁⠀⣾⡟⠀⣼
⠀⠀⠀⠀⣿⡀⠀⢾⣯⣀⣠⡀⠈⢛⠲⠴⠤⠤⠤⠤⣼⣿⣿⠇⠀⠀⠀⣿⣦⣤⠀⢀⣀⣀⡠⠴⡟⠁⠀⢀⣀⣀⣀⣿⣷⢀⡿
⠀⠀⠀⠀⠸⠇⢀⣼⡿⠟⠋⢁⠀⠀⠘⠒⠂⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⣹⡿⠟⠛⠀⠩⠥⠄⠚⠀⠀⠀⠀⠉⠛⠿⣿⣧⡼⠃
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⣀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣾⠋⠀⠐⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠺⠶⠿⣦⣤⣠⣾⡿⠉⠛⠋⠀⠀⠀⠀⠀⣠⡶⠛⠉⠈⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⢹⡇⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣼⣷⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠼⠛⠉⠙⠛⠻⠷⢶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠋⠉⠀⢠⡿⠁⠀⠈⠙⠻⣦⠀⠀⢻
⠀⠀⠀⣠⣴⠞⠋⠉⠉⠉⣀⣤⡶⠾⠶⣦⡀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⡅⠀⢀⣀⣤⣄⣼⡇⠀⠘⣷⣤⣄⡀
⠀⢀⡾⠋⠀⠀⠀⠀⠀⢾⣏⣁⡀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦
⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⢹
⣿⠃⢀⣴⠇⠀⣠⣤⣄⠀⠀⢀⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⣤⠖⢻⡇⠀⠀⠰⣆⠀⠈⣿⡿
⣿⠀⢸⡏⠀⠀⡏⠀⠙⠷⠾⠛⠋⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⠸⠀⠀⠀⠀⣿⠀⣰⡟
⠸⣧⣼⡇⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⡆⠀⢺⡆⠀⠀⣠⡿⠞⠋
⠀⠀⠈⠙⠶⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣤⣿⡧⠀⠈⠻⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠄⠐⠒⠒⠶⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠉⢹⡿⠉⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠉⠙⠙⠛⠛⠛⠋⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⣀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠉⠙⠛⠿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣤⣤⣤⣶⣶⡶⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠶⠶⠞⠧⠴⠒⠒⠒⠒⠶⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣈⣧⠰⡆⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡴⠂⠒⢤⠀⠀⠐⠒⠒⠒⠀⠈⠉⠀⠋⠁⠈⠃⡐⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠐⠀⠀⠾⢤⣤⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠿⠿⠋⠇⠈⠦⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡠⠀⠧⠀⠀⠢⠍⠩⠔⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⠂⠁⠀⢀⠀⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠃⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣠⠊⣤⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⣈⢂⡀⠀⠀⠀⠀⠈⠉⠀⠒⠀⠒⠒⠀⠀⠀⠀⠀⣠⡾⠋⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⣄⠀⠀⠀⠀⣶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠖⣻⠀⠀⠀⢀⡏⢹⡦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠀⠀⠈⠀⢈⠁⠠⠀⠀⠀⠆⠀⠀⠈⠉⠙⠛⠛⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢢⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣤⣤⣤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠴⣿
⠀⣤⣤⣤⣾⠁⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋⢹⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠳⠶⣤⣀⡀⢰⠐⠀⢀⣺
⢀⡿⠆⠀⡀⢡⡀⢀⣤⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠛⠳⣤⣿⡿
⢿⣖⡡⡀⢠⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠻⠶⠻⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣗⠀⠀⠈⠉⠉⠛⠀⠈⠀⠉⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠧⠤⠤⠤⠤⡴⠀⠀⠤⠜⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠤⠤⠤⠶⠶⠶⠷⠶⠶⠶⠤⢤⣤⣿
⣿⣿⡿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣦⡶⠿⣷⣴⣾⣿⣄⣀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠙⠛⠻⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠧⡀⢀⣀⡠⠤⠄⠀⠚
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠐⢶⡾⠋⠉⠉⠑⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠶⢶⣇⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣄⣀⣀⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠛⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⣿⡀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡀⠀⠀⢀⠈⠻⢿⣿⠟⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⢶⣶⣶⡶⠚⡇⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠘⣧⡀⠀⠀⠀⠀⢊⠁⠀⠉⠀⠀⠀⠀⠑⠶⠀⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠉⠉⢀⡴⠁⠀⠈⠷⣀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠈⢷⡄⠀⠀⠀⠈⠣⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠄⣸⠆⠀⠀⣠⡿⠁
⠀⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠚⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠻⢶⡴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢀⣶⣦⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠛⠛⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣶⣤⠄⠀⠀⠀⣀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠻⠿⠀⠀⠀⠀⢿⠯⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠿⠷⠀⠰⠾⠶⠶⠶⠾⠟⠀⣸⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠚⠳⣤⡀⠀⠀⠀⡁⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⢹⣷
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠙⣦⣀⣼⣿⣉⡀⠀⠀⠈⠉⠉⠙⠶⠦⣀⡀⠀⣠⣤⣄⣤⣴⠛⠉⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣼⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠛⠦⣀⠀⠀⠀⠀⠀⠀⠘⢿⣏⠁⠀⠀⠉⣛⣤⣤⡦⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠀⠙⢦⣄⡿⠞⢿⡯⠉⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠿⣶⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠴⠚⠛⠉⠉⠛⢦⠄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢶⣶⠿⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣀⣀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡄⠀⠒⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠿⠷⠀⠀⠒⠲⠶⠶⠠⣷⡦⠐⠒⠂⠉⠉⠉⠉⠉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣤⣠⣤⣤⣤⣤⣀⣀⣤⣼⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠿⣷⣄⠀⣀⣴⣾⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏⣽⣷⣤⣹⡍⣼⠋⢉⣉⠷⣆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠒⣀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠟⠀⣸⠃⣿⣺⣿⡆⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⡟⠁⠀⠀⣀⠀⠀⠀⠀⠈⠷⠶⠶⠾⢷⣦⣿⣏⣉⣀⡤⠟⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠀⠀⠀⠘⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⢘⣿⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠁⠀⠀⠀⣠⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢰⡿⢸⡇⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⡠⢿⡄⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⣼⠁⠈⢻⣦⡀⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⢸⡏⠈⢦⡔⠁⠈⠻⣦⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⢿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢠⡄⠈⢿⣤⠊⠑⢦⣄⡀⠈⢙⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⡀⢷⡀⠈⠛⠷⠶⠶⠶⠿⠛⠋⠁⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠻⣤⡙⠶⣤⣀⡀⠀⠀⠀⠀⢀⡴⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠠⣿⣧⣤⣌⣛⠶⣤⣀⠀⠀⠀⢀⣴⠋⠀⠀⡀⠀⡈⣧⣤⣤⣤⣶⠶⠿⠛⠃⢠⠇
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠉⠙⠛⠿⣯⣝⠓⣶⠞⠁⠀⢀⠀⢻⡀⡂⢹⡀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠘⠿⠶⣶⣶⠶⠶⠶⢶⣤⣀⠙⣿⠋⠀⠀⢀⣼⡇⠸⡇⢷⣼⠛⠛⠛⠛⠛⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣤⡶⠋⡘⢣⠀⣷⡞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣘⢉⡽⠀⢠⠃⠈⡷⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣠⡏⠀⣼⣁⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⢿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⠈⠉⠉⠉⣹⠋⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣷⡤⠹⣧⢀⡿⠀⠠⣾⣧⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⠶
⣤⣤⣀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣿⠶⠟⠋⠁⢀⣼
⢳⡀⠉⠛⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠃
⠀⠻⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⠾⠛⠻⠿⠶⣶⡶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⠟⠀⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⣴⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣀⡒⣾⣿⣿⣿⣿⣿⠛⠁⢿⣄⢡⡽⠆⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣄⠀⣀⡤⠞⠳⣄⠀⠀⠀⠀⢀⡠⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠈⠛⠁⠀⠀⠀⠈⠙⠲⠤⠖⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⠀⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠙⠓⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼⠋⠀⠀⠀⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠿⢟⣛⣛⣓⣒⣲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⣾⣿⠟⣫⣵⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⠏⣴⣾⣿⡿⣛⣯⠷⣞⣻⣿⣿⣿⣿⠋⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡿⢣⣾⣿⢟⡵⢟⣩⣶⡿⢟⣻⣿⣿⡿⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⢃⣿⠏⣱⢋⣴⡿⢛⣵⣾⣿⣿⡿⠛⠀⠀⠀⠒⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣼⡟⠐⣱⡿⢋⣴⣿⣿⡟⢉⠁⠀⠀⠀⠀⠰⢿⢧⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⠃⢰⡿⣱⣿⢟⣿⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⠀⣿⣱⡟⣱⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣿⣷⠹⠀⡏⡟⣼⡟⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⣰⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢿⣧⠀⠃⡇⣿⡁⠀⠀⠀⠀⣠⣾⣿⣶⣤⣀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣿⣧⡀⠁⢻⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡝⢄⠀⠻⣧⡀⠀⠸⣿⣿⣿⣿⡿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⡁⠀⠙⢷⣄⠀⠀⠉⠉⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣒⡋⣤⣴⣏⣉⣩⣄⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠋⠉⢀⣙⠁⠘⢿⣿⣿⣿⣷⣶⣬⣿⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠾⠐⢵⣿⣿⣿⣦⣄⠋⣉⣯⠙⢿⣿⣿⣦⣄⣉
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣟⡲⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⣰⠃⠴⣣⡄⠀⠀⠀⠀⠀⠉⠛⠟⠋⠛⠁⣹⣿⡿⢻⣿⢟⣹⣆
⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠟⠃⠠⣿⣿⣫⠝⡄
⠀⠀⠀⠀⠀⢹⠈⢿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡀⢸⣿⣯⣤
⠀⠀⠀⠀⠀⠄⠆⢾⠇⠀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⢸⣿⡽⠁
⠀⠀⠀⠀⠀⠘⢂⠈⠀⠘⠉⢈⣩⣽⣟⠿⠟⡀⠰⣶⣾⠿⠛⢛⠻⠈⠟⣠⡄
⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠟⠛⠛⢛⣽⠙⢣⠀⣼⣿⠛⠛⢛⣿⠧⠀⣶⠀⡁⠀⠀⠀⣀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⣿⣿⣆⠀⠩⣽⣇⠀⢏⡜⠁⠀⠀⣮⣀⠀⠤⠤⠤⠄⠴⠶⠶⠶⠟⠒⠑
⠀⠀⠀⠀⠀⠀⠑⠴⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⣿⣿⣿⡄⠀⣠⡇⢀⡸⠁⠀⣀⠴⣿⣿⡿⡿⠾⠖⠂⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⠟⠉⠐⠢⠦⣤⣾⣭⣝⠻⣿⣿⡇⡌⠀⡠⢊⠴⠚⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠠⡶⠶⠤⣤⣤⣤⣭⡍⣻⡇⢸⡟⢰⠀⠀⡇⡏⠀⠀⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⢤⣀⠂⢀⣸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡳⡀⠀⠀⠀⠀⢤⠠⣤⣼⣾⠋⢡⡿⢡⡆⠀⠀⢣⡙⠄⡀⣿⣿⡇⡇⠀⠀⢀⡀⠀⠀⠈⠛⡟⠻⠍
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠘⢆⠀⠀⠀⠈⠉⠉⠛⠁⢠⣟⣑⣸⣿⡀⠀⠀⠉⠒⢄⣿⣿⡇⠇⠀⠄⣉⠩⠀⢀⣀⣠⣧⡀⠀⣸
⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠀⠈⠳⣄⠀⠀⠀⠀⢰⣶⡯⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⣿⣿⣷⡆⠀⢀⣙⠟⠛⠙⡟⠉⠉⠉⢻⠁
⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠻⣶⣿⣷⡾⠋⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⣀⡙⢿⣿⣇⡀⢉⣁⣣⣤⡀⣤⡄⠛⠉⣉⠢
⠉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⡿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⣿⣿⣿⣿⣾⡛⠓⠒⠚⣿⠻⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢿⡇⠀⢿⢇⡙⢄⣹⣄⡟⡵⠊
⠀⠛⢉⣩⣾⣿⣿⣿⣿⣿⣿⡇⠀⡠⠊⣨⣿⡇⠙⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠴⠃⠀⠈⣠⡿⢫⠎
⠀⠀⠉⠉⢰⣿⣿⣿⣿⣿⣿⣷⠈⠀⢰⣿⣿⣿⣆⠘⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣀⣾⠟⠴⠃
⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣤⣤⣽⣿⣿⣿⣿⣤⣬⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣷⣶⣶⣶⣶⣤⣀
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠛⣛⣿⣿⣆
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⣠⣤⣾⣿⣿⣿⣿⣇
⠀⢠⣿⣿⡿⠛⠻⢛⣩⣤⣄⣐⣾⣿⣿⣍⣿⣿⣿⠟⢿⣿⣦
⠀⢸⣿⣿⠇⢠⣽⣏⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⢻⣿
⠀⢸⣿⣷⣤⡦⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠸⣿⣿⣿⣶⣷⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⣫⣭⣮⣿⣿⠋⠀⠀⣠⣶⣶⣶⣶⠶⠀⠀⠀⠰⠿⠛⡿
⠀⢸⣸⢩⠟⠩⠏⠀⠀⠀⠁⠴⣶⡖⠦⠀⠆⠀⠘⣻⡟⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠈⢿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⢶⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠇
⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣆⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⡀⡙⠁⠀⠀⠀⠀⠀⢀⡴⠃⠘⠖⠂⠠⠾⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡼⠁
⠀⠀⢀⡾⢷⣳⡀⢀⠀⠀⢠⡄⠛⠶⢶⡶⠶⠶⠶⠞⢀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⣤⡼⢿⣦⣤⣤⣤⣤⡤⠤
⢀⠴⠋⠀⠈⢧⠳⣄⠳⣄⠀⢳⡀⠀⠀⣉⣉⣉⡀⠀⡼⠀⠀⠀⠀⢀⡤⠞⠉⠉⠉⠑⢆⠀⠀⠀⠁⠀⠈⠉⠉⠁⣰
⠀⠀⠀⠀⠀⠈⢧⠈⠳⣌⠳⣦⣀⠀⠀⠀⠉⠉⠀⢰⠃⠀⠀⡴⠞⠉⠀⣀⡤⠾⠛⠃⢸⡃⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠳⣌⠻⣿⣶⣤⣤⣤⠴⠟⠀⠀⢠⡇⠀⣠⠀⠀⠀⢀⣠⠴⠾⣽⣒⣒⣒⣒⣒⣲⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠳⣮⣟⡀⢸⠀⠀⠀⠀⠀⣸⠀⠸⠁⠀⠀⠀⠉⠀⣀⣤⡿⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢠⠴⣿⣿⣷⣸⠀⠀⠀⠀⢀⡇⠀⣆⡄⢀⠀⠀⠰⠿⠛⠋⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⢹⣿⣿⣹⠀⠀⠀⠀⣼⣸⠀⠻⠁⠘⠒⠁⠀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣠⢶⣷⡇⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⢀⡟⠀⠘⠁⠀⠙⣟⢦⠀⠀⡀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠯⠳⢾⡶⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣟⣻⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡄⡀⠠⠖⣂⡄⢴⡦⢀⡐⠲⠄⢀⢰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⣠⡞⣫⠭⢥⣬⣤⣶⠶⠷⣥⡼⣻⣉⢷⡞⣉⣟⢧⣬⠾⠶⣶⣤⣥⡬⠭⣝⢳⣆
⠀⠀⠀⠀⠀⠀⢻⡸⣽⣧⡾⠫⢬⣽⡿⠿⢯⣥⡭⢞⣼⣧⡳⢯⣬⡽⠷⢿⣯⡥⠝⢷⣼⣦⢇⡟
⠀⠀⠀⠀⠀⠀⢠⡦⢠⡟⢨⠖⠀⠀⠂⠁⠀⠉⠛⠋⠁⠈⠙⠛⠉⠀⠈⠐⠀⠀⠲⡅⢻⡄⢴⡄
⠀⠀⠀⠀⠀⠀⡾⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠇⡇⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠥⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟⠬
⠀⠀⠀⠀⠀⠀⠀⣙⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢩⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⡍
⠀⠀⠀⠀⠀⠀⠀⡒⣼⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣧⢒
⠀⠀⠀⠀⠀⠀⡆⡃⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⢸⢰
⠀⠀⠀⠀⠀⠀⢳⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠘⠗⠘⣧⢘⠦⠀⠀⡄⡀⠀⣀⣤⣄⡀⢀⣠⣤⣀⠀⢀⠠⠀⠀⠴⡃⣼⠃⠺⠃
⠀⠀⠀⠀⠀⠀⣼⢱⠻⡟⢷⣔⢚⣻⣷⢶⣞⠛⣓⢮⢻⡟⡵⣚⢛⣳⡶⣾⣟⡓⣢⡾⢻⣟⡎⣧
⠀⠀⠀⠀⠀⠀⠹⢧⣝⣒⡚⢛⠛⠿⠶⠶⡛⢳⣽⣉⡾⢧⣉⣯⡞⢛⠶⠶⠿⠛⠛⢓⣒⣫⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠇⠁⠐⠦⠌⠁⠺⠗⠊⠡⠴⠂⠈⠸⠖⠚⠃

⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠻⠿
⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣶⣖⣒⣒⠚⠻⠷⣶⣶⣒⠶⠶⣶⣶⣮⣍⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠈⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣶⢦⣬⣉⡓⡤⡀⢰⣿⠏⣴⡿⠛⢋⣉⠉⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣅⣀
⠀⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠇⣼⣿⣿⣿⣿⣿⣷⣾⣿⣿⣼⣿⣥⣴⣾⣿⠿⠿⠷⣎⢳⣄⠀⠀⠀⠀⠀⠸⣿⠹⣿⠿⠌⣁
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⣟⣿⣿⣿⣿⣿⡿⠻⠮⡉⠙⠛⠛⠻⣿⣿⣿⡦⠀⠘⣷⠌⠳⡄⠀⠀⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣿⣿⣿⡿⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠙⠻⣄⠢⢐⣿⠀⡄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡼⢻⣿⣿⡟⢸⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⡿⢳⣾⣿⠀⠘⣧⠈⢻⣆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠈⣿⡘⠱⡆⠀⢀⣠⣤⣄⣸⡇⠀⠀⠿⠿⠿⢶⣶⣧⣼⢿⣿⡇⠀⢻⣆⠀⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠃⣤⢐⡞⠉⠀⠀⠈⣿⠁⠀⠀⠀⣀⣀⣀⡿⣽⠻⣎⣿⣧⠀⠀⢿⡆⢁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣦⡀⢀⣾⠁⠙⢏⣿⣿⣣⣴⠋⠀⠀⠔⢫⠝⠁⠸⡻⣇⠀⠙⢿⣿⡀⠀⠈⣿⣿⢸⡇
⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⢡⣿⠁⠀⠀⠀⠀⣩⠟⠁⠀⠀⠀⠀⠘⢶⣶⣤⣁⠘⠇⠀⢸⣿⡿⣄⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⡀⠀⠀⣾⡏⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠞⠿⢦⣀⠀⣿⣿⡿⢳⣄⠽⣿⡇
⠀⡇⠀⠀⠀⠀⠀⢀⣾⣧⠈⡄⢰⣿⠃⣤⠀⠘⣿⣿⣷⣦⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢝⣦⣿⡇
⠀⡇⠀⠀⠀⠀⣠⣿⡿⠏⠀⠃⢸⡿⠀⠘⢦⣀⠈⠉⠈⠛⠋⠛⠉⠀⠀⠀⠀⣂⣶⣶⣤⠀⠀⣼⣿⣿⠃⣻⣯⠿⣿⣧
⠀⠀⠀⠀⢀⡴⣫⠎⠀⠈⡄⠀⣾⡇⠀⠀⠀⠈⠙⠓⠒⠐⠀⠀⠒⠒⠚⠛⠋⠉⠀⠀⠀⠀⣰⣿⣿⠇⠐⣩⣿⡆⢻⡟
⠀⠀⠀⠀⣨⠔⠁⠀⠀⠀⠐⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣨⠖⣹⣿⡁⣀⣴⣏⣿⠟⣾⠇
⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠊⢁⣾⠛⣸⣿⡏⢱⡿⢋⣼⠟
⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢠⣟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⠋⠀⠀⢀⣴⡟⠁⠀⣿⣿⢰⡟⠁⣿⡏⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⣦⣄⣀⡄⣠⣴⣿⣿⣿⣥⡴⠶⠶⠶⠿⠋⠀⠀⠀⣿⣿⡶⠁⡀⢹⣷⣄⠀⠀⠀⠀⢠⠇
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⢲⣦⣄⣿⡏⣦⠀⠀⠀⡾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢹⣷⡀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡀⢸⣍⣿⣿⠃⠄⠀⢸⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⡾⠁⢹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢀⠼⣿⣹⡙⢻⣾⣿⢦⣄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣴⢿⣿⣦⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠀⢹⣯⣷⠈⢟⣠⣶⡀⣿⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣷⣾⣧⠀⠀⠀⠀⢠⠖⠋⠀⠀⣿⠀⠀⢀⣻⡟⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⡟⠀⣀⣼⢻⡿⣿⣿⣿⣿⣻⠇⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⡇⣾⣿⣿⣿⣟⣿⣿⣧⡞⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⣷⢸⣿⣣⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠿⣿⡿⣿⣿⣿⣿⠟⣭⣿⣿⢿⣿⡟⠻⡙⠓⢦⡀⠀⢤⠼⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠷
⡀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡟⢉⢀⠀⣼⣿⣿⡟⢋⣥⣾⡿⠋⣙⣾⣿⣿⡄⠈⢂⣀⣿⠀⠀⠀⠀⢸⡇⣾⣻⣿⣿⣿⣿⡿⠋⣡⡇⡆
⣝⠀⠀⠀⠀⠀⠀⠀⠠⣦⡽⠗⠋⠠⣴⣿⣡⢾⣻⠿⠋⡀⠐⡿⠋⢹⣿⣿⡄⠀⠹⢿⡀⠀⠀⠀⣿⢃⣿⣿⣿⣿⠟⢉⣴⣾⣿⣵⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣟⣿⣤⣌⣀⠀⠀⠀⠠⠿⢿⣿⣦⡀⠘⣷⢀⠀⠀⢹⣸⣿⣿⣛⣥⣾⣿⣿⣿⣿⢻⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⢶⡿⠉⠀⠀⠀⠄⠀⠀⠹⣮⣳⣦⡙⣄⠱⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⢀⣀⠀⠀⠈⠻⢦⡙⢿⣷⣄⣿⣣⠾⠛⠛⢿⣿⠿⠛⠁⠀⡼⠁
⠀⠀⣀⢀⣀⡤⠒⠒⠀⠀⠀⢀⣀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢴⡀⠉⠀⠈⠛⢿⣄⠀⠀⠀⠛⠀⠀⠀⠀⠀⢻⡄
⠀⠰⠀⠀⡼⢡⣌⡴⠚⠉⠉⠉⠁⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿

⠀⠀⢠⣾⣷⢶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣷⠄
⠀⠀⢺⣯⡏⠀⠀⠉⠙⠛⠿⣶⣤⣄⡀⠀⠀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⣀⣠⣶⣴⢇⣶⣃⣀⣀⠀⣀⣠⣴⡶⠿⠛⠉⠁⠀⢀⣽⡟
⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣦⣽⣿⣿⣿⣿⣷⣶⣿⣶⣷⣿⣸⣷⣿⢱⣿⣿⣿⣳⣿⣿⣿⣿⣷⡾⠟⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠘⢯⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠻⢿⣿⠛⠉⠉⠛⠛⠟⠻⠇⠛⠋⢿⡿⠿⠻⠟⠻⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠾⣡⡟⠁
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣥⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣦⣦⣶⣤⣤⣤⣄⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠈⠛⠛⠉⠉⣉⣉⣉⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠐⣾⣿⡄⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠉⠉⠙⠻⣿⡄⠀⠀⠀⠀⠀⣰⠏⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⣿⡟⠀⠀⢀⣄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢠⣄⠀⠀⣻⣷⠀⠀⠀⠀⢰⣿⣾⡋
⠀⠀⠀⠀⠀⠀⣤⣿⠛⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠉⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣤⣤⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣴⣾⣿⣿⡏⠀⠀⠀⠀⠀⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⣶⠽⣷⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢋⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣯⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣰⢿⣧⠀⠈⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣻⠟⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣤⣰⣿⠿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢋⣠⣼⣧⡀⣠⣿⡃
⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠉⠙⠋⠀⠈⠙⣿⠃⠀⠉⠙⠛⠷⢦⣤⣄⣀⣀⣤⣤⣄⣤⣾⠟⠋⢠⣶⠟⠋⠁⠈⣿⠉⠈⠙⣶
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠷⣤⣀⡀⠀⠀⠀⠈⠉⠈⠛⠛⠛⠛⠉⠀⠀⠀⢸⣧⠀⠀⣶⣴⡿⠂⠀⢀⡏
⠀⠀⢰⠟⠀⠀⠀⠀⣴⣾⡿⠀⠀⠀⠀⠀⠀⣶⡶⠀⠀⠉⠛⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠁⠀⠀⢹⡗⠀⢠⡾⠋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⣴⣄⠀⠀⠹⠿⣦⣄⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⢀⣤⣀⣈⠳⣶⠊⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣧⡀⠀⠀⠈⠙⠶⡄⢠⡶⠟⠋⠀⠀⢀⣠⡿⠋⠉⠉⠙⢿⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢠⣀⣀⣀⣠⡄⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠈⠈⠁⠐⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢲⡀⠀⠠⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⠶⢺⠋⠀⠀⣤⡤⠄⠀⣀⠄⠆⣐⣠⢂⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡘⠀⠀⠀⢳⣦⠤⠄⢀⡠⠔⢫⠸⠒⠁⠀⠀⠈⣗⢈⡆
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⠾⠤⢤⢀⠞⠉⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠿⠶⠷⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠴⠖⠒⠒⠒⠒⠒⠶⠦⢼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠔⠚⠛⡿⠻⠿⠿⢿⣿⣿⣶⣦⣤⣤⣀⣰⠀⢀⣠⣤⣶⣿⠿⠟⠛⠉⠁⣧⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠡⠿⠟⠉⠉⠉⠉⣿⠀⠈⠟⠿⠿⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠢⢀⡀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⡤⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢱⡀⢨⡗⠒⠶⠤⣤⣤⣤⠤⡤⣤⠤⠐⠒⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣧⠔⣆⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣄⣀⡀⠀⠀⣷⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠀⢰⣷⠀⠀⢀⣿⠁⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢠⡇⠀⢠⠎⢸⠀⢀⡼⠉⠀⠀⢠⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⣤⣾⣿⠖⠁⠀⠈⢷⣿⠃⠀⠀⣠⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠙⠓⠚⢋⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣀⣠⣤⣴⣶⡄⣀⣠⣤⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⢩⣾⣿⣿⡟⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠙⠻⠿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠉⠉⠀⣴⡲⠄⠀⠀⠀⣾⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠚⢇⠀⠀⠑⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣨⠟⣛⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠤⠞⠀⠐⣦⡀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠰⡇⠘⠉⠓⠀⠀⠀⠀⠀⠀⠀⡠⠚⠣⠀⠀⠠⠒⢊⠁⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣖⡒⢒⣶⣾⣿⡧⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣞⡿⣿⣿⣄⡹⣿⣷⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⣷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣤⣈⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⢰⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠋⠉⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐⡀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠁⠀⢀⣀⡀⠀⠀⠀⠐⠙⠛⠛⠉⠀⠂⠀⡇⠘⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀⠀⠀⠁⠐⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⡄⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡾⠿⠧⠀⡘⠀⢀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣎⣄⠀⠀⠈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠂⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡉⣻⣿⣷⡲⣤⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣛⣳⣮⠑⣼⣷⣄⣾⣷⠀⣠⡀⠀⣥⠀⠀⢀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣭⣷⡸⢟⣙⣿⣋⣻⣃⢿⣇⣠⣿⠀⣾⠛⣹⣗⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣭⣵⣷⣙⠷⠶⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⣦⣿⣿⣿⣋⠀⠀⣘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣷⡟⣫⣾⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢐⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣯⠄⢀⣤⣶⣶⣾⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠻⣿⣿⣿
⠀⠀⠀⡇⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣛⣋⣠⣤⣾⣿⣿⡿⠋⠁⢀⣾⣆⠀⠙⢿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⡿⣿⣧⠀⠈⣿
⠀⠀⠀⢸⠄⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⠿⠛⠋⠀⠀⠹⣧⠀⢸
⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇⠀⠀⡎⢧⣄⠀⠀⠀⠀⢻⡀⢈
⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣭⣍⣙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢹⣦⡀⠀⢸⠁⢸
⠀⠀⠀⠀⠙⠦⡈⠉⠩⢿⣽⣧⠈⠉⢉⡿⠒⣲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣼⣿⡇⠀⠏⠀⣼
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠫⢽⣇⡴⠋⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠀⠀⠻⠿⠋⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠛⠉⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⣆⠀⠀⠀⢀⣤⣾⣿⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠙⠻⠿⣿⠿⠟⠉
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⣄⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⢸⠟⠒⣶⣿⣁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠔⡛⠛⠁⣀⣀⣤⣶⡿⠃⠀⢉⡓⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠳⠦⡤⠶⢺⣴⣿⣿⣿⡿⠛⠉⢀⢤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣉⡉⢀⣀⠔⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⠴⠋⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢻⣿⣿⣿⣿⣿⣿⣷⣦⣄⣤⣀⣤⣴⣿⣼⡆⠀⠀⢀⣴⣿⡏⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣴⣿⣿⣿⡇⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣾⣿⣿⣿⣿⡷⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⡟⠋⠁⠈⠉⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣈⡁⣼⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡶⢀⣬⣿⣶⣄⣀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡿⣧⣿⣿⣿⣭⡉⠉⠀⠀⠀⢠⣄⣠⣤⣤⣤⣤⣤⣤⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⡀⠈⠉⠀⠉⠻⣿⠻⣿⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣟⣧⣸⣿⣿⣿⣿⣿⠇⣶⣦⡀⠰⣾⠛⢣⡄⢻⣷⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡈⢙⣿⣿⣿⣿⣷⣾⡿⠧⠀⠘⠛⠛⠀⠹⠇⠘⠿⣾⣿⣿⣿⡏⠙⠻⡟
⠀⠀⠀⠀⠀⠀⠸⣷⣦⣀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣄⡀⠀⠒⢤⡤⠀⠀⠀⠀⠀⠈⠛⢿⣿⣉⡄
⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣷⣿⣿⣿⣦⡈⣉⣩⣇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠉⠀⠙⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢉⣿⣷⣶⣦⣀⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣹⣿⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠈⠻⣿⡟⢿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠉⠁⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣀⡛⠉⠀⢠⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⣀⣠⠶⣶⡶⢿⠿⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠹⣿⣀⠀⠀⠸⠋⣹⡿⢿⣿⣿⣿⡆⠀⠟⠋⠛⠒⣾⣷⣿⡷⠶⠆⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠈⡉⠀⠀⠀⣼⡿⠇⠀⣽⣿⠛⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⣼⣷⠀⢀⣀⠸⣿⡀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠋⣻⣟⣀⣈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⣿⣿⣤⣽⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⠋⣿⠿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠁⠈⠉⠁⠈⠉⠁⠀⠀⠀⠀⠉



⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣷⣶⣶⣄⠀⠀⠀⠀⠀⣴⡄⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⡀⡶⣤⣄⠀⠀⢀
⣿⣿⣿⣿⠁⠀⣸⣿⠿⠻⣿⣿⠿⠻⣿⣿⣷⠆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣿⣿⣧⡀⢀⣀
⣿⣿⠿⣿⣶⣀⠈⠁⠀⢤⣄⣡⣄⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣶⣿⠟⠻⠋⠐⣶⡾⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⣉⠀⠂⢠⡄⠻⢷⡄⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡏
⣿⣿⣠⡏⢀⣴⡆⠀⢀⣤⣄⣘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣷⡸⠿⠉⠛⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⡀⠻⠏⣸⠁
⣶⣟⣿⣿⣿⣿⠙⣿⣧⣀⡀⠀⣀⣀⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⠟⠀⠈⢻⡄
⡌⠛⢿⣿⣿⣷⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠀⠀⠈⠻⣿⣿⣿⡴⠃⢠⡾
⣀⠀⠀⠙⢿⣿⣿⠋⢿⣿⣷⣶⣿⣿⣿⡿⠡⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇⢀⣾⠇
⠙⠣⡄⠀⠀⢙⣿⣿⠟⠿⠿⠻⢿⠿⠿⢇⣀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⡀⠀⠀⠂⠀⠈⠹⠟⠀⣠⣤⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠛
⣹⣶⣀⠀⠀⢠⡄⠀⠘⣭⣉⠿⠟⠀⢀⠀⠀⢀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣤⣾⣗⣀⡀
⣿⣹⣿⣧⡀⠈⠋⢀⣈⡉⠟⠀⣴⠀⠀⣠⣶⣾⡿⠿⢿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠻⢿⣿⣿⣿⣍⣿⡿⠷⠄
⣿⣿⣿⣿⣁⡀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠘⠋⢀⣴⣿⣿⣿⣿⣟⠿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⢻⣿⣦⣾⣶⣶⣶⣿⣏⣀⣿⣿⣁⣤⣤
⣿⣿⣿⣿⣿⣿⠆⠀⠘⣿⣷⣄⠀⣰⠟⠀⠀⣼⠃⠀⠀⣤⣌⢻⣶⣿⣿⣿⣿⣿⠿⠛⠀⢘⣛⠧⠈⠉⠉⠀⠀⠀⣁⣸⣿⣋⢼⣿⣿⣷⠶
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⠿⠂⠁⠀⠀⡆⠀⢰⣾⣿⣿⡟⢀⠉⠀⠹⣿⣟⣿⣶⠿⠇⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⡉⣿⣿⣼⠟⣯⣄⣤
⣿⣿⣿⣿⣿⠿⠟⢰⡆⠀⢀⣠⠀⠀⠀⠀⣀⡀⢾⣿⣿⠟⢁⡼⠀⢸⣿⣿⣿⠋⠉⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⢿⣿⡟
⣿⣿⣿⣿⣿⡖⠒⣿⣷⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⣽⣧⣴⣾⠀⠀⢠⣿⣿⣤⣶⣾⣷⣶⣿⣿⣿⣿⣭⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⢛⣉⡀
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣾⣿⣿⣄⠀⠀⣸⣷⡀⠘⣿⣿⣁⣀⡂⢠⣿⡿⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿
⠛⠋⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠋⠀⠁⠙⠛⠁⠀⠛⠛⠛⠛⠁⠀⠉⠉⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠾⢦⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢧⣶⣿⣶⣤⣄⡀⠉⠀⠻⠿⠃⠀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⡟⠋⠹⢷⣄⣰⣶⣶⣾⣿⣿⣷⢆⣀⠸⠷⣨⣧⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠇⣀⣽⣿⣿⣯⡝⠛⣛⡻⠿⢿⣧⠸⢿⡏⠛
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠛⣿⡛⢿⣁⣠⡿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⢁⡀⠀⠀⠀⠀⠀⠻⢿⣿⣦⣶⣲⣶⣾⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠻⢿⠿⠿⣿⣿⣿⣌⣿⣿⣾⣿⣿⣷⣿⣿⣿⠛⢿⣋⣉⣛⣤⣤⣌⠁⠀⢀⡀⣀⠀⠀⣠⣶⡛⠛⠛⠉⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣆⠻⣿⣿⣿⣿⣿⣷⡄⣈⣿⠋⠀⠀⠉⠙⠳⠀⠀⠀⠀⠀⠹⣿⡟⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⡟⣿⣿⠛⠛⣏⣉⣷⣏⣿⡿⢿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⡿⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣤⠞⢻⣿⠟⠁⡾⠟⢩⣿⣿⣿⣿⣾⡿⠛⠿⣿⣿⡿⠿⠉⣿⣿⣿⣿⡗⠀⠀⠀⣿⣿⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠉⢰⣿⡿⣷⣿⣗⣶⣮⣼⡿⣿⣁⠁⣀⣴⣾⣿⣿⡀⠀⠀⢿⡿⢿⣿⠟⠀⠀⠛⢻⣿⡇⠚⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣦⣤⣀⣀⣵⣿⣿⣿⣿⣿⣟⣿⠿⢿⣿⣿⣛⣿⣿⠿⠿⠿⢀⠀⣠⠀⠀⠸⠗⠀⠀⠈⠙⣷⠀⠀⠙
⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠈⢿⣿⣿⣷⣶⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠙⠛⠒⢾⣿⡆⢠⣦
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣄⡸⣿⣿⣿⣿⣿⡀⠀⠀⣿⡿⠛⠛⢿⣿⠋⠹⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣦⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣷⣤⣀⣀⡀⠀⢠⣤⣤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠙⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠉⠉⠹⣿⠟⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠏⣀⡀⠘⢢⡴⣿⣦⡀⠀⢠⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⢻⡶⢤⣄⣀⠀⠀⠀⠀⢀⡀⢰⣄⣠⣴⣴⣿⣿⣴⣤⣿⡆⡀⣰⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠸⣿⡄⠈⠉⠙⠛⠳⢦⡈⠿⠛⠋⠁⠙⠟⠈⠀⠀⠉⠉⠉⠃⠹⠿⠿⠿⣶⣿⣀⣠⣤⡶⠟⠛⠉⢁⣾
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢯⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⢺⡿⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣦⣶⡀⠀⠀⠠⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠇⠀⠀⢘⣧⣠⣼⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢘⣿⡏⠀⠀⠀⢺⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠿⠿⠇⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠙⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⡄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠙⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠱⢶⣶⣄⣤⣠⣀⡀⣀⣠⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⠿⢿⣏⣉⣽⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢭⣿⣤⣼⡿⠏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣄⣀⣀⣀⣀⣀⣤⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢿⡿⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡾⢻⠃⠀⣠⡤⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠓⠶⢦⡴⠋⠙⠶⣄⡴⠚⣡⠞⢀⣯⡴⠋⢸⠀⣿
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣠⠏⠀⠀⠀⣠⡟⠁⡰⠋⠀⡼⠋⠀⣠⠞⠛⠛⠳⢤⡀
⠀⠀⠀⠀⢀⡞⠁⠀⠀⣰⠏⠀⠀⢀⡼⠋⠀⢠⢃⣀⣼⠁⠠⠀⠻⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣜⠁⠘⡄⣰⠏⠀⠀⢠⣾⢢⠀⣀⠾⠛⠉⠈⠙⠒⠶⢤⣈⢷⡀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢩⠀⢰⢠⡟⠀⠀⢠⣿⠜⣠⠞⠁⠀⢠⣾⣿⣷⡄⢸⡷⠟⣸⠃⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢀⣸⠦⣈⡿⠀⠀⢀⠧⣘⡿⠁⠀⠀⠀⠀⠀⠈⠙⠃⠀⠉⠉⠁⠀⠀⢰⣾⠗⠀⠀⠹⣆
⠀⠀⣴⠋⠀⠀⠈⡇⠀⢀⡜⢀⡿⠁⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠐⢦⡀⠹⣤⠾⢻
⠀⠀⣇⠀⠀⠀⠀⢷⡀⣼⡧⢸⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⢦⠀⠀⠀⢀⠀⠀⠿⠲⠖⢚⡃⠀⢿⡄⢸
⠀⠀⠙⢦⣄⣀⣠⠞⠉⢿⢃⣼⣇⠀⠀⠀⣸⠁⣀⣀⣤⡆⠘⢻⡇⠀⠀⢻⡄⠀⠈⠃⠤⠊⠀⠀⠘⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠀⢹⡄⠀⠀⠸⡇⠀⠀⠉⠁⠀⣸⠃⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠹⣄⠀⠀⠙⠦⣀⣀⣠⠾⠋⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣀⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣠⡿⠀⢠⠇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣭⣤⡼⢧⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠈⠙⠋⠛⠁⠀⡰⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠁⠉⠒⠤⠄⣀⣀⣀⡀⠴⠋⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⢉⣙⡷⢶⣤⣤⣄⣤⠶⠋⢡⣌⣹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢒⣬⠟⠛⠙⠛⠙⣯⠀⠀⠀⠀⠨⡁⠀⠀⠀⢿⡋⠉⠙⠛⡵⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠳⣄⠀⠀⠀⢰⡟⠀⠀⠀⠋⠀⠀⠃⠀⠀⠘⣇⠀⠀⢠⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⣤⣀⢸⡇⠀⠀⠀⡀⡀⠀⠐⠄⠀⠀⢹⣄⣤⡟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⣿⠄⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠘⡇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠶⠄⠀⣿⣤⣤⣤⣤⣼⡛⠉⢙⣦⣤⣤⣤⡏⠠⠶⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⣀⣀⡴⢿⡄⡀⠀⠀⠈⠉⠛⠉⠁⠀⢀⣼⣅⣀⣀⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡄⠀⠀⢀⡀⠀⢀⣴⠞⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⣄⣤⣀⣀⣨⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⡿⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣀⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣏⣉⣭⣉⣩⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⠛⠛⠛⠿⠿⠇⠀⠉⠀⠈⡿⠿⠏⠙⠛⢲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⡸⢄⡀⣀⢀⡀⢠⡀⣤⣀⢀⡀⢀⡀⣀⣠⡇⡀⣀⡹⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⠓⠓⢚⢓⡚⢓⣚⣒⣓⣚⢓⣛⣛⣓⣒⣛⣒⣒⡓⣒⠚⠛⠓⣋
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠶⠶⠒⠛⠻⣦⡶⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡔⠋⠉⠉⠙⣆⠀⢀⣀⣀⣀⡀
⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⡾⠀⠀⠀⢄⡀⠟⠉⠉⠉⠉⠉⠉⠙⠒⢦⡀
⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⡀
⢸⡇⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢳⠤⠤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⠏⠀⠴⠘⠂⠀⠀⢠⠌⠉⠉⠁⡩⠋⠀⠀⠀⣸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡠⠤⢏⠀⠀⢀⠤⠤⣄⠀⠈⠙⠂⠀⠀⠃⠀⠀⠀⣼⠃
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⠏⠀⢸⠲⠀⠎⣇⢠⠧⡤⠤⠤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢈⡦⠤⠤⠃⠈⢧⣀⠈⣉⡟⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠋⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⡟
⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣠⠶⠶⠴⠶⠟⠳⣤⣀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣈⢳⣄⠀⠀⠀⠀⠯⠤⣤⢀⣀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠧⠒⠈⢁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⢀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠒⠂⠉⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠛⢧⣀⢀⣀⣀⣀⠀⠤⠤⠄⠖⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⡤⠞⠛⢦⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⠴⠖⠋⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣤⣀⠲⣄⠀⣠⠖⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡯⢿⣿⣻⢿⡿⣷⣽⠿⠖⠛⠛⢿⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣮⣝⡻⡾⠟⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⢿⣿⣿⣿⠿⠗⠿⠋⠁⠁⠀⠀⠀⠀⠀⣀⠀⢸⡆⣀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡤⡞⡏⡇
⠀⠀⠀⠀⠀⢀⣿⡿⣟⣫⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿⣋⣤⣽⡆⠀⠀⠀⢀⣴⣿⣆⣷⠟⠁
⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⢸⡏⠀⢘⡇⠀⣀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠡⢿⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣷⠖⠀⠀⠀⠀⠈⣷⠀⢸⣧⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢟⣧⠀⢀⡤⠶⠒⠋⣉⣿⣦⣄⡀⠀⢼⣤⣤⣄⠀⠀⠀⠀⢹⣦⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⡀⠀⠀⠀⠀⠉⠀⣠⡼⠋⠀⠀⢀⣤⣼⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⢠⡿⠒⠶⢮⣷⡄⠀⠀⠀⢠⠞⠙⢧⣐⣦⣀⣸⡯⠶⠖⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠻⣆⠀⠀⣏⣀⣤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⣠⣤⣿⣴⡟⠛⠛⠛⠛⣿⠛⢛⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠘⣧⡀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⠀⠀⠈⡇⠀⠀⠀⠀⣿⣠⢼⣄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⣿⠀⠀⠀⡇⠀⠀⢀⣤⡿⠿⢷⣬⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⡇⠀⢰⡚⠦⢬⡙⢢⠘⠇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠶⠶⢤⣤⢞⡽⠚⣋⣉⡽⠀⠀⡇⠀⠈⠉⠳⣄⠱⠀⢂⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⢰⡟⢫⡴⠿⠛⠛⠦⣄⣀⣛⣻⡟⠛⣻⠟⠐⢊⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⠛⢻⡇⠁⠠⠶⠶⢾⣓⠋⣿⠿⢿⠷⡾⠃⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠀⠈⠉⠙⢳⣼⡇⠀⣠⣤⡤⣤⣈⣷⡟⠀⢻⡘⣇⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡅⢀⣌⡛⠓⠒⣾⣾⠃⠀⠀⠙⣎⣧⡬⠷⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢠⡀⠀⠀⠀⠀⢸⠇⠀⣿⠏⢹⣿⣁⣿⠀⠀⢆⠀⠈⢿⣇⣀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣧⠀⠀⠀⣠⡾⠀⠀⣿⠀⢸⣿⣿⡿⠀⠀⠈⢷⡀⠀⣿⠁⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣆⣴⠞⠙⣧⣠⡴⠋⠀⠀⢹⣿⣧⠀⠀⠀⠀⢳⡀⢹⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣠⡾⠋⠙⣆⠀⠸⡇⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⢳⡸⠆⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⣠⠞⠁⠀⠀⠀⣹⡶⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣥⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠻⢟⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠜⢀⢌⠛⠁⣤⠄⠀⠈⠛⠻⢫⣶⡍⢋⣥⡀⠀⠃⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠡⣎⣹⣧⠀⠀⠉⣷⡄⠀⠀⢠⡦⠀⠀⠀⣡⣄⣠⠈⣀⠀⣿⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢋⡴⣦⢸⣿⣿⣀⣤⣶⠖⠤⠀⠀⠹⣿⣿⣄⣬⡀⠻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡙⠃⠘⠟⠛⣿⣿⠟⢴⡶⠠⠐⠀⠘⢷⠘⠛⠁⣩⣤⠀⡀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣽⡥⠀⠀⣀⡀⠈⠁⠀⠀⠀⠐⠠⠀⠁⢀⣀⡀⢠⣬⣯⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠠⢶⡿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠠⣾⡿⢿⡶⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢷⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠛⠣⡀⠀⠀⠀⠀⢀⣤⣤⡤⢤⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⢻⣄⠀⠀⠀⠙⠏⡴⠂⠿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣆⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠘⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠰⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⣿⣿⠟⠋⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠂⠀⢸⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣷⣄⠀⠀⠉⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠉⢋⡄
⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠉⢿⡻⠏⠆⠀⠀⣠⣶⣿⠁⠀⠐⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⡀⠑⠀⠁⠀⢺⣿⣿⣿⣿⣶⠤⠍⠀⠀⠻⣿⣶⣦⡙⣶⠀⢀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⢀⠰⢴⣶⣶⣶⣾⡄⠀⠁⠀⠀⠙⠋⠀⠀⠉⠀⠠⢀⠀⠀⠀⠉⠉⠙⠀⠀⣿⣿⣿⣝⡋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣽⣿⡿⠿⡇⠀⠀⣠⣀⠠⢰⠃⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢋⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠰⠻⠇⠀⠀⠤⣄⡂⣛⣯⣴⣿⠀⠂⠀⠀⠀⣾⠀⠀⠀⠀⠓⢶⣶⣾⣧⣰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠙⢿⣿⡻⠇⠀⠐⠀⣆⠘⢉⣼⠀⠀⠉⠻⣾⣧⣤⣿⡿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢰⠀⠘⠀⡿⠙⢤⣀⠘⢿⣏⣀⣼⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⠀⡀⣀⣠⣴⣾⡴⠀⠀⠀⠇⠀⠀⠉⠙⢋⣩⣿⠋⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⡷⠀⠘⠀⠀⠀⠀⠀⢀⡀⠋⠁⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡿⠀⠀⠀⠈⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣝⣦⠤⠴⠖⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠍⠑⠒⠢⠖⣩⣶⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⢸⣿⠟⢃⡭⢵⡖⠚⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡖⠒⠒⠲⢄⡙⠻⠿⡿⢸
⠀⠀⠀⠀⠐⠊⠀⠁⠁⠀⠀⢠⣫⣤⣤⣇⣀⣸⡇⣀⣀⣀⡼⠀⠀⠀⠀⠀⠀⢀⢎⡀⣀⣧⣀⣀⣀⣠⣧⣤⡀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⠛⠻⢯⣁⠀⢀⠀⠀⠀⠀⠰⠾⣿⠿⠂⠀⠀⠀⠉⠁⠄⣀⣤⡾⠟⠛⠋⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠉⠛⠶⣬⣄⣀⡀⠀⠀⣿⠀⠀⣀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠙⠛⠻⠾⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣾⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡄⢀⣀⣤⠟⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡀⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠠⠸⣧⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⢻⡄⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣟⣷⠀⠀⡀⣿⣿⣷⣼⣷⣿⣿⣿⣿⣿⣿⣿⠿⠷⡶⠶⠶⢶⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠤⠤⠴⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠓⠒⠒⠋


⠀⠀⡀⢠⢀⠀⠀⠀⣀⠀⠀⡁⢀⠀⠀⡀
⠀⠀⠋⠉⠉⠈⠁⠀⠁⠈⠃⠀⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣶⡿⠿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠻⠿⢿⣶⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⠶⢶⡖⠀⠰⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⠶⢶⠂⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢱⠀⠈⢿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⠈⣆⠀⠹⣿⣿⣿⣿⣿⣿⣷⡏⠠⣾⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⣄⠈⠙⠿⠿⠿⢟⠿⠁⠀⠀⠀⠀⠈⢦⡀⠈⠻⠿⠿⠿⣻⠏⠀⠀⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⢠⡀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣀⣀⡤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠛⠉⠁⠀⠀⠴⠾⠗⠒⠒⠒⢒⣲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⠶⠀⢀⣠⡤⢖⠲⡄⠀⠀⠀⠚⠉⠉⠉⢉⣭⣶⠶⠄
⠀⠀⠀⠀⠀⢀⣠⠶⠺⣍⠀⢀⣠⠟⠛⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⠵⠦
⠀⠀⢀⡤⣞⠭⠖⠒⠒⠦⣤⣋⠤⠤⠶⠲⣶⢺⠃⠀⠀⠀⠀⠀⣠⠖⠒⠲⡶⠿⠖
⢠⠖⠉⠀⢯⡉⠉⠉⣋⡽⣇⠀⠀⠀⠀⣀⡴⠋⠀⣴⢀⡤⠀⠘⠡⠾⢋⡾⠁⠀⠀⠀⠀⡀
⣏⠀⠀⢀⠤⠽⠯⣍⠁⠀⠈⠱⠒⠉⠉⠉⠉⡤⢾⡵⢋⣡⣾⣭⣽⠿⣥⣀⣀⠀⠀⣖⢼⢹⣞⡆
⠘⠦⣀⠘⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉⠀⢀⣀⡀⠀⡿⡶⠒⠦⠬⠭⣈⠉⢸⠁
⠀⠀⣆⣿⠿⣶⡶⣶⠒⠲⣖⠒⠛⠋⠉⢡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠓⠲⠤⣄⡀⠈⠙⠃
⠀⠀⠈⠑⠚⠄⡆⢘⠒⣚⠭⣦⠀⣠⠔⠁⠱⡄⠀⠀⠀⣠⠴⠦⢄⡀⢀⣀⣠⡔⠊⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠋⠥⠟⠀⠀⠀⠙⠛⠁⠀⢀⣀⡹⣄⡀⡼⠁⠀⠀⠀⠈⠙⠷⠀⣷⡾⠛⠛⠛⠭⣭⣕⣦⡀⠀⠀⠀⠀⠀⢀⡔⡄⠀⢠⢌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⣗⡢⢽⣷⣄⡀⢀⡾⣓⠦⠤⠖⠉⠉⠛⢿⢿⣼⠁⠀⢹⡧⠐⠆⠐⢀⢠⠾⠀⠖⠶⢬⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢥⣌⣓⣬⡣⣙⣼⡇⠀⠉⠒⠦⣄⣀⣠⠞⠮⠟⠤⠴⠚⠀⠀⠁⠀⠈⠈⠲⡆⢸⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣭⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⣀⡀⠀⠀⡀⢠⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠉⠉⠁⠀⠈⠉⠋⠉⠙⠛⠋⠉⠉⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⢟⣛⣭⣭⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢵⣺⣭⣿⣿⣿⣿⣿⣿⣿⡿⢃⡿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⠿⢛⣩⡴⢟⣵⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣭⣭⣭⣥⣶⠶⢟⣛⣭⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠙⠿⣿⣿⢿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣮⣝⡻⢿⣭⣭⣍⣉⣭⣥⣶⢟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠂⠀⠀⠀⠙⢮⡙⢶⣌⠻⣿⣿⣿⣿⣿⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⡀⠀⠂⠰⣶⣶⠆⠀⠀⠀⠈⣻⡄⢹⣷⡌⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠃⢸⣿⣿⡘⣿⣿⣿⠟⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⢸⣧⣿⠟⠁⠀⠐⠐⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣿⣦⠀⠀⠀⠀⢶⣦⡄⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⢸⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠛⢿⣇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⡀⢻⠀⠀⠀⠀⠀⠀⠀⠚⠛⢟⣿⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡟⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢣⣿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠴⠖⠛⠷⠀⠀⠀⠀⠀⠀⢸⣼⣷⠀⠀⠀⠀⠀⣿⡆⠀⠀⠾⠛⠲⠦⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣶⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣿⣿⣿⣿⠃⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠿⠀⣠⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣷⣦⣜⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣄⡀⠀⠀⣨⣭⣿⣿⠙⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣴⡿⠛⠉⢤⠀⠀⠉⠙⠿⣷⣴⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⢸⣿⣿⠀⠀⠀⠀⣾⣿⠁⠀⠀⣤⠀⢀⠤⢤⡀⠈⢿⣿⣿⣿⣿⠾⠿⠶⠶⣶⡶⠆⠀⠶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⢴⣿⠋⠁⠀⠀⠈⣿⡟⠀⠀⠀⢰⣿⣿⠀⠀⠀⢸⠀⠘⠶⢦⡄⠀⠈⢿⣿⣿⣿⠀⠀⢀⡇⠸⣷⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⢿⣧⠀⠀⠀⠘⣿⣿⣧⣀⠀⠚⠂⠘⠒⠐⠁⠀⠀⠘⠿⠘⢿⣷⣴⡾⠥⢤⣿⣧⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⣸⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠙⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⠄⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣿⣿⣀⣀⣀⣀⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣦⣿⣿⠉⠉⠉⠉⢉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⠀⠘⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡻⣿⣆⠀⠘⠓⠋⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠈⠛⠷⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⢸⣿⡿⠻⢿⣦⣄⡀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠏⠀⠀⠀⠀⠘⠛⠀⠀⠀⠉⠿⠿⠿⠿⠿⠿⠛⠉


⣿⣿⠿⢿⡿⠿⠿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣁⠀⠀⢂⢠⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠁⣀⣤⣤⠄⠀⠀⠀⠉⠉⠉⠀⠈⠂⠙⠘⠀⠀⠀⢉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣦⣄⠀⠀⠀⢨⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⠀⠀⠀⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⠀⢇⡀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⣿⣦⣄⣀⣬⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠿⣭⡣⣹⣯⢁⣶⡆⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⡄⠿⣹⣿⣛⢿⢸⣇⢀⣿⠀⣶⢛⣳⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣵⣧⣛⣶⣾⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣿⣶⣤⡶⣿⣄⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉⠛⠿⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢱⡀⠀⠀⠀⠀⠀⠀⠀⠤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀⣀⣀⣤⠶⠶⠶⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡑⠶⢶⡖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⣿⠁⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⡸⠛⠉⠉⠛⢦⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⠋⢶⣶⣶⣶⡶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣡⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢹⡇⠀⠀⠀⠀⠀⢠⠴⠂⠴⡇⠀⠀⠀⠢⣀⡉⠛⠋⣁⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠙⠻⠿⠟⠁⡹⠂⠈⢦⡀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠻⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣤⣤⠞⠁⠀⠀⠈⠙⢶⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠙⣦⡀⠀⠀⠀⠈⠛⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣠⠞⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⣤⡀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠈⠛⠷⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄⣴⠿⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣶⣶⠒⠂⠀⠀⣤⣤⣤⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠛⠛⠛⠀⠀⠀⠀⡿⠧⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠿⠛⠳⠀⠙⠿⠗⠶⠶⠛⠛⠀⣼⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠙⠛⢦⣄⠀⠀⠀⢘⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣤⣤⡤⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠙⢷⣄⢀⣾⣏⣉⠀⠀⠀⠀⠉⠛⠓⠶⢦⣀⡀⠀⢠⣀⣀⣀⣤⡞⠋⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠉⠙⢿⡟⠉⠁⠈⠉⢓⣢⣤⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢺⡶⢿⣿⠋⠁⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠿⣿⡶⢿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠶⠶⠋⠉⠶⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠛⠓⠀⢤⣤⣤⣶⡾⠿⠛⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠃⠒⠀⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⣀⣀⣀⣿⣧⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⠓⠀⠒⠒⠛⠛⠖⠰⣷⡦⠐⠛⠃⠉⠉⠉⠉⠁⠀⠀⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⢀⢀⣤⣤⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢣⢠⡉⡈⡄⡀⡖⣶⠦⣴⢿⡶⢠⢷⣿⠁⠀⠀⠀⠀⠀⠀⡀⣾⢿⢸⡏⡟⢻⢸⣿⡿⣿⢷⣦⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⡀⠈⠋⠁⠁⡀⠐⠀⡀⢠⠠⣠⡘⡀⠀⡟⠀⠀⠀⠀⠀⠀⡄⠳⠼⠸⡃⠁⠣⢸⣟⣻⣿⣿⣆⣏⣿⡄⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⠃⣜⢠⢌⠠⠀⠀⡀⠁⢁⢨⣥⣁⡁⣼⠁⠀⠀⠀⠀⠀⢰⢳⡶⡆⣯⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠁⢡⢿⢸⠇⠇⠳⠹⢸⢿⠾⣾⠛⣟⣿⠀⠀⠀⠀⠀⠀⣼⣲⡄⢇⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⠁⠆⢈⠀⠈⠀⠀⠀⡀⠠⠠⣀⢠⠽⢽⡿⠀⠀⠠⠄⡀⠀⠍⠭⠭⠭⣙⠺⣛⡛⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣧⢠⢱⡇⡞⡀⣦⡼⢠⡁⡌⡀⠳⡈⠃⠀⢀⠨⠀⠀⠀⠀⠀⠀⠀⢛⣰⣏⠻⠏⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠑⢇⢣⠋⠁⠀⠤⠀⠰⠄⠀⠀⠀⠀⠀⠐⠀⠘⠛⠂⠀⠈⠉⢉⠉⠐⢠⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠘⡄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠙⠶⣶⣦⡀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⢀⣤⣀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢀⡀⠀⢀⣀⡀⢀⣀⠀⢼⣄⢳⣷⣄⠀⠀⠈⠻⣶⠶⠽⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣦⠀⣼⠁⢻⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠐⠖⠈⣧⡀⠈⠙⣋⣭⠹⢄⣀⣥⣾⠿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⡉⠙⠻⣦⣄⠀⣴⣫⣿⡿⠛⠀⠷⠶⠎⠰⠆
⠀⠀⠀⣦⣤⣦⡀⠀⡘⣃⡀⠀⠀⠰⠂⣞⣀⣴⣯⡶⠺⠷⠄⢳⡍⠷⠀⠒⠂⠞⠃⠀⠀⠀⠀⠀⠈⠻⠿⠦⣄⣀⣈⠖⠋⠙⠻⣧⠒⡰⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠸⣿⣿⣿⣷⠟⠉⠛⠢⠤⠀⢸⣿⢿⣟⣦⡀⠀⠀⠀⠀⠉⠻⠶⠄⠰⠶⠶⠶⠤⠤⠀⠀⠀⣀⠢⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢿⣿⣿⠏⣦⠀⠀⠀⠀⠀⣾⣿⡿⣧⠿⣿⡦⣤⣤⡀⠀⣤⣤⡤⠤⠤⠀⠀⣠⠀⠀⠀⢠⡈⢄⠀⠙⠛⠀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠐⠂⠰⡖⢲⡆
⠀⠀⠀⢠⠜⠈⠁⠀⠈⠓⣄⠀⠘⢼⣿⣿⡇⢻⡠⣿⠱⣿⣿⠓⢷⡟⠀⠀⡆⠐⠀⠀⠀⠀⠀⠓⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠁⠈⠐⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⢾⣄⠒⠘⠿⠿⠃⠀⠀⢸⡦⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⠻⠋⠁⠀⠈⠓⠢⠄⠀⡀⠀⠀⠈⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠆⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣸⠀⣴⢀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⠺⣥⠞⣰⠇⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⢀⠲⣾⠃⢀⣠⣴⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿⢯⢍⡉⠉⠛⣹⣿⠿⠟⠛⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢽⢸⣤⣴⣿⣿⣿⣿⣿⢿⡏⣰⢿⣿⠛⣿⠟⢁⣴⣾⣷⣨⡇⢉⡵⠞⣻⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⢿⣽⣿⣏⣨⣉⡭⠠⡟⣤⡀⠈⡁⢸⡏⣼⣿⣿⣿⣿⣿⢧⠞⠁⠈⢩⠾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⠃⣾⣿⣼⣿⣆⡱⣾⢁⣿⣿⣿⣿⣿⢏⠃⢀⠆⣠⣴⠾⠛⠻⠆⣈⣹⣿⣿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣧⣿⣼⣿⢿⠟⠋⠁⣠⣼⡷⢛⣉⣤⣤⣶⢀⠀⠀⢹⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⡿⠏⠂⣀⣰⣶⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡛⢿⣆⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⣒⣒⣉⣩⡭⣽⣿⣿⣿⣿⣿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢿⣿⣿⡟⠙⠛⠶⣭⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⣁⠀⠙⠛⢿⣿⣿⣇
⠉⠉⠉⢉⠍⠉⢭⣭⣉⡽⠛⢻⡍⢩⡌⠁⣭⠃⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣟⠛⠛⠭⣤⣤⣬⣿⣿⣿⣿⣿
⠀⢤⡤⠞⠀⣠⣄⢀⡡⢤⢴⣦⣝⠋⣀⣴⣫⣤⡼⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣶⣆⣤⣙⣿⣿⣿⣿
⠒⠢⠒⠺⠋⠁⠉⠉⠀⠀⠈⠙⠛⠻⠿⠚⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢬⣟⡻⠿⠟⢛⣉⣴⠟⢉⣀⡉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣠⣶⣿⠟⠉⠁⠀⠙⠄⠀⠀⠙⢻⣿⣿⣿⡟⠁⡴⣿⣿⣿⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣽⣿⢿⣟⡀⠁⠀⠀⡙⣻⣿⣿⣷⣦⡤⠂⠀⠀⠀⠀⠀⠈⠉⢽⣿⡇⠈⠷⣌⠻⠙⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠉⣣⣿⠀⠀⠘⠁⠉⣩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢠⣷⡿⢀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢲⣿⣿⡆⠁⠰⠟⣰⣿⣿⣿
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⡴⠒⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠛⠁⢸⣿⣿⣄⣠⣴⣾⣿⣿⣿⣿
⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⣿⠀⠀⠀⠀⣀⡀⠜⠓⠶⠶⠟⠣⣄⠀⠈⠀⠀⠀⠀⣴⣿⣿⡿⣻⣿⣿⡿⢻⣿⡿
⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⣿⣦⣤⢤⣾⣡⣤⣴⣦⣶⠶⠛⠀⢹⠀⠀⠀⠀⠀⠀⣩⣿⡿⠁⢹⡏⠙⢷⣿⠛⠁
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣷⡟⠉⠿⡥⠴⠿⠿⠏⠙⠋⢁⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⢉⠁⢠⠃⢸⡇⠀⠀⠉⠓⢦
⠿⠟⠛⠛⠛⠛⠛⠒⠒⠀⠉⠉⠉⠉⠉⣉⣉⣀⣀⣀⣀⣀⡤⠤⠼⣷⣄⠐⣤⡶⠶⠶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⠋⠀⣾⠁⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣎⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠀⠀⠀⠀⢀⣀⠤⣄⠀⣀⣀⢠⡆⠀⣠⣶⡟⣡⠞⠁⢀⣾⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢋⣿⡖⠒⠋⠉⠀⠀⠀⢇⠀⠀⠩⢉⣤⣾⣿⣿⠞⠁⠀⣠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣀⣀⡐⢲⣶⣶⣿⣿⡿⠋⠀⠀⢀⣴⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡿⣿⢻⣄⠂⣀⡀⠘⣛⣋⡹⠿⣿⣿⣿⠿⠋⠀⠀⠀⢠⣿⡿⣹⡏⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣾⡿⢿⣿⣿⣿⣿⣿⣿⡇⠙⣳⡛⠛⠉⠉⠉⠀⠀⠀⠀⠀⡕⠀⠱⡀⠀⠀⠀⠈⠋⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⡟⣿⡇⠀⢿⣿⣿⣿⣿⣿⠀⢸⡏⠀⠀⠀⣠⡤⠔⠒⠒⠒⠛⢄⠀⠀⣼⣄⠀⠀⠀⢠⡿⠠⣀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠁⠀⠘⢿⣿⣿⣿⡏⢠⠈⠁⢀⣤⠀⠀⠀⠀⠀⠀⢀⣠⣎⣠⣾⣿⣿⣷⣤⣀⣿⠁⠀⠈⠙⠶⣤⣤⠤⠔⠂⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡀⠀⠀⠈⠛⠿⢿⡇⠘⠀⢀⠸⠂⠀⠀⠀⠠⠶⠛⠋⠁⣨⠛⢿⡉⠁⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠙⠃⠀⣿⡇⠀⠀⠀⠀⠀⡼⠁⠀⢸⡇⠀⠀⠰⠀⠀⠀⣀⣤⣴⣾⣽⡇⠀⢷⡀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⠀⠀⠀⠀⢸⣧⠀⠀⣠⣴⣿⡧⢇⠀⠸⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⡇⠀⠘⡇⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢶⣧⠱⡄⠀⠀⠻⢀⢀⡿⠻⣏⡔⠈⢆⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡟⠁⠸⠃⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⣿⣿⣿⣿⣿⠟⠛⠋⠛⠀⠀⠈⠳⣽⣦⣀⠀⠁⢸⡇⠀⠹⣎⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭⠋⠀⠀⠀⢧⠀⠀⠹⣆⠀⠀⠰⢒⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⡿⠛⠁⠀⠀⣀⣤⣶⡶⢶⣦⣄⠴⠋⠀⠀⠀⠀⠀⠈⢳⡄⠀⠈⠣⣠⡼⣿⣿⣿⣿⣧⠛⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀
⣶⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⢇⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⢿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢹⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢠⣿⣿⣿⣿⢸⣿⣿⠛⡿⠀⠀⢸⠉⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣸⣿⠋⠀⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠀⣀⣀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡿⢣⡴⠿⠛⣿⣿⣶⣶⣄⡙⣿⣿⣿⣿⡟⠇⠀⢀⡴⣾⣿⡟⣿⣿⣏⠉⠻⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⣟⠀⠀⢰⣿⣿⠼⣿⡏⠁⠈⠻⣿⣿⡇⠀⠀⠈⠀⢿⣿⣋⣻⣿⡏⠀⠀⠽⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠿⠓⠚⠛⠃⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⣰⡆
⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡇⠀⣴⣿⣿