Garbage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⠬⢭⢆⡀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢰⠁⣸⠇⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠒⢴⠃⠐⠛⡀⠀⠀⣠⢽⡉⠋⠘⡆⡌⢀⣿⡰⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠈⡄⠀⠀⡇⣀⣜⢣⠈⠱⢤⣀⡷⠦⢾⡿⡱⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⡀⠳⣄⢀⣷⣖⡩⢿⡃⠈⠂⠁⢀⡾⠉⡇⠀⠸⠫⠛⠓⣬⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢁⣀⡀⠀⡈⠉⢯⠁⠀⠀⣇⠀⠀⠀⡿⠀⠀⠘⠒⠛⠒⢲⣎⣸⠀⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠉⠏⠀⠠⠋⠉⢉⣩⣴⣾⣛⣒⣚⣒⣒⣒⣐⣩⠭⠭⠖⠚⢲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢄⣀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡀⣶⠶⢸⠳⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠧⣀⠀⠈⡙⠊⠀⠀⠰⠶⣶⠗⢻⣟⣻⢍⠆⣎⣦⣼⠹⣹⣿⣽⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠢⢄⣙⢋⣿⣯⠄⠀⠀⠀⣿⠀⠸⠋⠉⠓⠃⠋⠀⠀⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⠢⡄⠀⠀⠤⠤⣄⠀⠀⠀⠀⠤⠤⠄⡀⠀⠀⠠⠤⠤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⢤⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠀⠰⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠦⣠⣇⣀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠀⢱⠒⡜⢾⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⣀⠤⣄⣼⠖⢠⡀⢰⠀⠇⠀⠉⠢⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣄⣀⣀⣠⣤⠤⢤⣀
⠀⠀⠀⠠⡎⠉⠉⣑⡀⠈⠉⠉⠑⠀⠀⢸⣄⣀⣀⡠⠵⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⣧⠀⠀⡏⠉⠉⠁⠀⢨⠛⠫⡉⠉⠁
⠀⠀⠀⠀⠳⣤⡼⠷⠈⠉⠒⠊⠉⠑⠢⣼⣼⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢃⡿⢓⠞⠓⠶⠬⣉⣉⣉⣦⣄⡈⠑⡄
⠀⠀⠀⣀⡜⢀⠀⣠⠀⠀⠸⢆⠀⠀⡠⠈⡟⢾⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠈⣧⣈⡒⠢⠤⡤⠋⠉⠽⠤⣇⡈⠉⠓⠢
⡴⠚⢁⡤⠴⠋⠙⣆⣡⡀⠀⠀⢱⠀⠀⠈⠀⣸⢤⣬⣉⠉⠀⠀⠀⠐⠒⠒⠚⠁⠀⠀⣉⣉⣤⠴⠋⠀⠉⠙⠒⠠⠤⠤⠴⠦⠤⢬⡷⠦⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⠶⠶⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⣭⠴⢶⣒⣲⣶⣶⣐⣂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⡶⠿⠛⠛⠋⠉⣉⣉⣉⣙⣛⣛⣻⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣭⣶⠶⠾⠟⠛⠛⠛⠛⠋⠉⣉⡉⠙⠛⠛⠛⠿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢏⠉⠀⢀⣠⣀⣀⣠⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣄⣀⣂⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣡⣤⣶⠾⠿⣿⣿⣿⣿⢿⣿⣽⠿⣷⣟⡝⣡⣋⣽⢿⣟⢿⣿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣏⠘⣛⣿⣶⣿⣿⡟⣇⣁⢹⣏⣷⡿⠛⣟⠛⢻⣷⣴⣶⣶⣿⣶⣥⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢃⢎⣸⣶⣿⣭⣶⣀⣙⣿⣥⣹⣾⣏⠹⣯⣵⢩⠙⢁⣦⣿⡿⢛⣋⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⢿⣿⠄⠀⠈⣉⣉⡉⠛⠛⣛⣻⣧⣬⣭⣽⣿⣿⡿⠛⠛⠟⡟⢻⡯⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢁⣠⣼⣿⠁⠿⢸⡿⣌⣿⠀⣸⡏⠙⣏⠀⢠⡖⢶⡄⠂⣰⢶⣦⠀⢸⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣯⣤⣤⢸⡇⠀⣿⠀⣿⠇⠀⣿⠀⢸⠀⢸⣷⠀⣿⢸⣿⣶⠾⠿⠟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⠶⠿⢸⡇⠀⣿⠀⣿⠀⠀⣿⠀⢸⠀⢸⣿⠀⣿⠸⣿⠛⣷⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢹⠀⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⣺⡇⠀⣿⠀⣿⢠⡄⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣿⡟⠀⡏⠀⣿⠀⠀⡇⠀⢸⠈⢸⣿⠀⣾⠄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠀⣿⡇⠀⡇⠀⣿⠀⠀⡇⠙⣿⠀⢸⣿⠀⣿⠀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡇⠀⣿⠠⢀⡇⠀⢿⠀⢘⣿⠀⣿⢰⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣄⡀⠈⠛⠋⠀⠀⢿⣀⣸⠃⠀⠸⣦⣼⠟⣀⡈⠋⠁⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⣽⠲⢽⣷⡦⣤⣝⣒⣀⣉⠀⠀⣀⣀⣀⣀⣬⡭⠵⣖⣻⢵⢛⣿⠀⠀⠀⢀⣤⣤⡀
⠀⠄⠀⠀⠀⢰⣿⣿⢶⠾⣤⣙⢷⣄⡒⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⢶⠶⠶⠚⠛⠉⣀⣬⣼⢏⣴⡾⢿⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣷⣮⡠⢎⠽⠿⣯⡛⠻⠿⢷⣦⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⡶⠶⢟⣛⠉⢿⣿⣛⣛⡹⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠜⢿⡔⢡⢖⣽⣫⣻⡆⠓⢀⣠⣤⣀⠈⠉⠉⣩⣿⣷⣶⣶⡆⢾⣿⣿⡗⠀⣍⢛⠛⡛⠛⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣾⣮⣵⠟⠀⠀⢿⣿⣻⣿⠆⠈⠉⠻⢷⣤⣤⡾⠇⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⠀⠀⠀⠀⠀⠠⠄⠀⠐⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣏⣤⣤⣴⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠛⣩⣼⡿⢋⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⠛⡻⠏⠉⠀⠈⢀⣿⣿⡿⠿⠧⢤⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣗⣚⣿⣀⠀⠀⠀⠀⠐⣫⡉⠁⡀⠀⠀⠀⢙⠩⢍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠤⠤⠿⠟⠉⢿⡄⠀⠀⢀⣠⣞⢲⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⣿⣆⠀⡾⠟⠉⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣭⣿⣿⣿⣏⣀⡀⢠⣤⠀⣼⣿⣻⣷⡀⢶⣅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣻⣿⣿⣿⣿⣽⣿⠀⢻⠀⢹⡇⠀⢻⡁⢸⡇⢿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣷⣿⡄⢸⡀⢸⡇⠀⣸⠁⣿⡅⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⢸⡇⠀⣿⠀⣿⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⡧⢸⡄⢸⡇⠀⣿⠀⣿⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢸⡇⠸⣇⠀⣿⢀⣿⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣿⠀⣿⠀⣿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡗⠀⣿⠀⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠞⣹⢟⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⠀⢹⠀⣿⠰⡟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠀⠈⠀⠛⢸⡷⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣭⣿⣛⡟⠚⠀⠴⢀⢭⣥⡶⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣿⡂⢸⡗⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣽⣿⣿⣿⣿⣧⣤⣤⣤⣽⣷⡾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⢿⡀⢸⠉⠉⠻⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢠⠀⠀⠆⡀⢉⣭⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⢾⣤⣇⣞⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣍⣀⠠⠍⠛⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣄⣿⠿⢦⣬⣙⣺⣯⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⡿⢛⣉⡭⢟⣽⠛⡟⠛⣏⠙⡏⠻⢿⣭⡛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⡴⠚⠉⢁⣴⠟⠡⢄⢳⡀⠀⠠⠈⠀⠀⠈⠙⢷⣄⠙⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠋⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠄⠳⠀⠀⠀⠘⢦⡀⠀⠀⢀⡈⠂⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⠆⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠨⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠂⠀⠀⠘⠷⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠟⠋⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠦⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⢀⣴⠟⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⢋⡁⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⠋⠉⠄⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠰⢄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⡆⠀⠀⠀⣴⠟⠹⣇
⠀⡀⠀⠀⠀⠀⠀⣴⠋⠁⢿⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠻⣧
⠀⠀⠀⠀⠀⢾⣁⣀⣤⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⢀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠷⠶⢦⣤⣭⣿⣶⣦⣤⣤⣀⡀⠀⣀⣈⡁⠀⣀⣀⣀⣈⣁⣀⣤⣤⠶⢾⣯⣥⣤⣴⣾⣤⣤⣤⣤⣤⣤⣦⣼⡷
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⡠⣤⠴⢁⣉⣑⡲⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣭⣍⣉⡀⡄⣀⣔⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣩⣿⡛⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠁⠈⠁⠀⠉⠑⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢰⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠸⠉⠃⠀⢠⣿⣿⣧⣀⠀⠀⢀⣠⠤⠒⠈⠉⠉⢻⣇
⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⣿⣿⣿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⢿⡇⢀⠖⡒⠤⢀⡀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣸⠗⠊⠀⢃⠀⠀⠈⠁⠀⠉⠉⠉⠉⢰⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡮⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢰⡀⠈⠑⠀⠾⠤⠤⠤⠄⠀⠀⢤⣶⠃⢠⠃
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣴⠞⠛⠛⠛⢛⣿⡟⠻⠿⢿⣷⣶⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣴⣿⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠘⡉⠙⠛⠛⠋⠉⢁⠃⠀⠀⠀⠀⠈⠉⠉⠉⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⢀⣼⠂
⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⢒⡊⠉⡑⠒⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣶⡿⢿
⠀⠀⠀⠀⡇⠀⠀⠀⠈⡄⠈⢉⣡⣤⠄⠀⣷⣦⣤⣤⣤⣤⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠀⢀⣠⣧
⠀⠀⠀⠀⠉⠐⠒⠒⢲⣁⠀⠀⠀⠀⠀⣀⣿⠿⠿⠿⠿⠿⠿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠑⢄⠀⣀⠔⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠢⡀⠐⠄⠀⠄⠇⠀⠠⠂⢀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢈⠀⠚⠒⠠⣈⠦⠶⢁⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠥⠞⡛⠂⠂⠺⢄⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠃⠁⠀⠄⠀⠀⠈⢂⠀⠈⠂⠌⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠢⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠈⠢⡀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣀⣰⣁⠔⠁⠀⠀⠀⠀⢀⠅⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠈⠢⠄⠐⠂⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠂⡁⠀⡆⢀⠀⡀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢂⠂⢠⡘⣗⣀⠀⡀⠀⠁⠄⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠚⡷⠞⡀⠁⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡤⠶⢄⠀⢀⠴⠦⡄⠀⡠⠶⢤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⢸⡔⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡏⢣⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠈⡆⠀⠀⠈⢻⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠀⠹⡀⠀⠀⠀⢣⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠀⠀⢧⠀⠀⠀⠈⡆⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣧⠀⢸⡄⢸⠀⠀⡇⢰⡇⠀⣾⠀⢸⡇⠀⠀⠘⡆⠀⠀⠀⢳⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢿⠀⠘⡇⢸⠀⠀⡇⢸⠃⠀⡿⠀⢸⠁⠀⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣀⣠⠃⠸⣄⣀⠇⠸⣄⣀⠇⠀⢸⠀⠀⠀⠀⠀⢇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⠘⣆⡼⠁

⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠈
⠀⠀⡅⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⡤⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠁⠀⠀⠎⣾⡇⡋⢸⡇⣿⣷⠋
⠀⠀⢠⠀⠠⢘⣉⣁⠛⠉⠉⣉⣁⣋⠄⠰
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀
⠀⠀⡄⠓⡄⠀⠀⡷⠂⠀⠑⣦⠀⢀⡔⠃⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈
⠀⠀⡇⠀⣇⠀⠀⠳⠤⣔⡤⠏⠀⢘⡇⠀⡇⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠣⠤⠋⠁⠐⠀⠐⠒⠀⠐⠀⠉⠦⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⢿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣏⢠⡄⠀⢁⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣟⠉⠙⣿⣿⣿⣟⠉⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠐⠿⣿⠋⠉⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣶⣿⣦⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣶⣶⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢶⣛⣩⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡗⠒⣶⡖⢲⣶⠖⢶⣶⠒⢶⡖⠒⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠙⡇⠀⣿⡇⢸⣿⠀⢸⣿⠀⣼⡇⠀⣿⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣿⡇⢸⣿⠀⢸⣿⠀⣿⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⣷⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣍⣛⡛⠻⠷⠾⠟⣚⣋⣭⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠯⢉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠒⠒⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⡄⡶⡶⢶⢤⡤⣤⡤⣄⡀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⠀⠀⠀⠀⠀⠈⠇⠀⣿⠀⠀⠀⠹⡄⠀⢻⣍⣉⣙⣛⣋⠉⠉⠛⢷⡀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠰⠀⠀⠀⠃⠀⠀⠀⣹⠀⠀⣷⡄⠹⡄⠈⢷⠤⠤⢤⣤⣤⣤⣤⣤⣷⡀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⠢⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠹⡄⢻⡀⠈⢧⣤⣤⣤⣤⣤⣤⣤⣬⣷⠀⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠓⠒⠒⠒⠒⠒⠒⠤⠶⠶⣿⠀⢈⡇⠀⣷⠀⢧⠀⠘⣿⣿⠙⠿⠿⠿⢿⣿⣿⣧⠀⠸⣧⠀⠀⠀⠀⣴⣶⡶⠲⠶⠶⣶⡶⣖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⣶⠒⡶⢰⢤⠤⠴⣶⠴⢶⣿⠀⢸⡇⠀⢸⡄⠈⠀⠀⢰⡼⡄⣦⠴⢄⣼⣿⣿⣿⡄⠀⣿⡆⠀⠀⢸⠓⣒⠂⠘⠫⠬⣅⣇⠘⣧
⠀⠀⠀⠀⠀⠀⣤⢞⣉⡠⡏⠀⣿⠀⠀⠀⡄⠀⠀⢻⢰⠸⡟⠀⠘⡇⠀⣸⠃⡄⢰⠀⠀⣧⠻⣏⠸⢣⡽⣿⣿⣿⣇⠀⣸⣷⣀⣾⣏⡧⣿⣛⣿⣭⠭⠽⠾⢻⣷
⠀⠀⠀⠀⣤⡄⣿⠁⣠⣤⣿⠀⡿⠀⠀⠸⢠⠀⠀⠀⠀⠀⡇⠀⠀⡇⢀⡟⢰⢟⠀⡄⠀⢹⠀⣯⣀⣈⠑⢿⡋⢙⠿⢴⣿⣉⣿⠟⠈⢷⣿⠀⠀⢀⣴⣶⡀⢸⡟
⠀⠀⠀⠀⣿⣿⣿⡀⣿⣽⡟⠀⣿⣀⣀⣀⠀⠀⡀⠀⠀⢠⡇⠀⠈⡇⣸⠃⡿⢸⡆⢃⠀⠘⣾⠁⠋⣉⣿⠙⢷⣿⣶⣾⠏⢹⡇⠀⠀⠀⡏⠛⣆⣾⣾⣟⣁⡼⢧
⠀⠀⠀⠀⣩⠶⢿⡏⣿⠟⡇⠀⠈⠉⢉⣉⠉⣉⣉⣉⢈⣙⡇⠀⢸⡇⡏⢠⡇⢸⡇⢸⠀⠀⢻⠶⠞⢉⠀⣠⣾⣯⣽⡏⠀⣸⡇⠀⠀⠀⠙⢧⡈⠉⠉⠉⠉⢲⡞
⠀⠀⢠⡞⠉⠉⠉⢻⠇⠀⠀⠀⣿⠙⡌⡡⠀⠈⠘⠙⠘⠏⡇⠀⢸⣷⠃⡼⢀⣸⡇⢰⠀⠀⢸⣶⣧⣤⣾⣿⣥⣽⣿⡇⠀⠹⣷⠀⠀⠀⠀⢘⡧⠀⠀⠀⢀⣽
⠀⢀⡏⠐⠂⠁⠀⢘⠀⠀⡆⠀⠸⠇⠁⠃⠀⠀⠀⠀⠀⠀⡇⠀⠸⠟⠀⠗⠉⡁⡄⢸⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⢘⣿⠀⠀⠀⠀⢸⣷⠤⠤⠤⣾⣿
⠐⢿⡇⠀⢀⣤⡄⢸⠀⢠⡇⠀⠀⠀⠀⠀⣠⠴⠒⠒⢒⣠⣧⠀⠀⠀⠴⠾⠧⣤⡇⣸⠀⠀⠀⣇⣆⣔⣀⣀⣠⣾⣿⡇⠀⢮⣿⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⣿
⠀⣸⠀⢰⣿⣿⣿⢸⣄⣼⣧⣠⣤⣤⣤⣤⡇⢤⢀⣀⣿⣡⣽⣄⣤⣤⣤⣤⣄⣀⣀⣽⣤⣤⣤⣭⣭⣭⣭⣭⣉⣉⣉⣁⣀⣸⣿⠀⠀⠀⠀⣾⡰⢻⡟⣿⠛⣿
⣿⣙⣷⣿⢋⣹⣿⣿⣿⣿⢡⡍⣿⣿⢯⡹⣿⣾⣿⣾⣷⣾⣿⣿⣩⢻⣿⢫⡙⣿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣬⣭⣭⠭⢽⡇⠀⠀⠀⠀⢰⠀⣸⡇⢸⡇⢿⡀
⠀⠀⠈⢿⣎⣽⣿⠃⠘⣿⣌⣵⣿⣿⣯⣡⣧⣿⠉⠁⠀⠀⢹⣿⣬⣼⣿⣜⣥⣿⣾⠃⠀⠘⢿⣿⣿⣿⣷⣿⣿⣿⢯⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠸⡇⢸⡇
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠛⠛⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠋⠁⠙⠛⠛⠋⠁⠀⠀⠀⠈⠙⠋⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⣸⠇⠀⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠹⠆⠩⠙⢳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣰⣀⢠⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⣉⠝⠳⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣷⣼⣟⣶⢯⣽⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⠭⠒⢉⡴⢃⢣⠈⠀⠃⠉⠛⢦⣙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠘⠀⠀⢀⡼⠉⠀⠀⠀⠃⠀⠀⠲⢄⠀⠀⡀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢀⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠂⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⠻⠦⠤⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⣠⠾⡇
⠀⠀⠀⠀⢀⡼⠉⢧⠀⠀⠀⠙⠦⣄⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠁⠀⢹
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠐⠋⠉⢧
⠀⠀⠀⠈⠛⠛⠳⠦⢤⣬⣷⣦⣤⣤⣀⠀⢀⣐⠀⢀⣀⣐⣀⣀⣤⠴⣞⣩⣤⣄⣀⣤⣤⣤⣬⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⠢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣠⡂⡩⣅⡂⢀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠭⢭⡽⢦⡀⠀⢄⡐
⠀⠀⠀⠀⠀⢙⣿⣽⠋⣿⠢⢨⢠⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢷⣦⣦⠤⣂⣦⣶⣿⡟⣧⣌⡀
⠀⠀⠀⠀⠀⣾⣧⢸⠀⢸⡄⡌⡆⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠛⠻⠿⠾⠟⠛⠋⠙⠋⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠘⣿⠚⡇⠀⣇⡠⢿⣉⡄⢀
⠀⠀⡠⢦⢜⠙⢳⠏⢿⠼⡾⢭⣻⡍⣷⡾⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⠛⠓⣦⠀⠀⡀
⠀⠘⠟⠿⠚⠾⠿⠿⠏⠉⠉⠾⠿⠿⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢀⣃⡈⠄⢠⢠⢋⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⢷⢿⣿⣳⢰⡞⣟⡀⠡⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠹⠾⢯⣇⡃⠈⢉⣿⣿⡷⣷⣽⣷⠬⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠈⠿⠿⠿⠙⠉⠉⠿⠿⠚⠁⠈⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠀⠰⢶⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⠀⡐⡈⣷⠀⠉⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹⣾⣾⣧⠁⠿⣤⣆⣀⣥⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄
⠀⣰⠀⠀⢀⠀⡤⠖⠛⠟⣛⣛⡶⣭⣽⡟⢫⣿⡿⢶⣷⣦⣤⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⢡⣶⣷⣄⠀⠠
⠀⠻⡧⠀⠀⠀⠀⠀⠛⠛⠻⠁⣸⠟⠳⣶⣿⣽⣿⣗⣳⠛⠋⠀⠀⠀⠀⠀⣀⣰⣾⣗⣛⡭⢭⣹⣏⣶⣶⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⣠⣴⣴⣿⣻⠻⡹⣿⢾⣛⣻⣿⣿⣧⣷⣿⣷⣶⡄
⠀⠀⠀⠉⠒⠀⠀⠀⠀⠂⠀⠀⠀⣄⡀⢀⡀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠁⠈⠙⠈⠀⠉⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠿⠆⡄⢤⡠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡟⢛⠡⠀⠀⠀⠀⠠⣆⣶⡾⠿⠶⠶⢆⠤⣥⣰⠄⠶
⠀⠀⠀⠀⠀⠀⠠⠀⠾⣉⠒⠶⡄⠀⠀⠀⠀⠀⠧⠄⠀⠂⠀⢀⡰⠓⢠⡏⢾⣍⠐⠂⠙⠊⠾⠃⠀⠶⢠
⠀⠀⠀⠀⠀⠀⡴⡆⠩⢉⠏⣵⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠠⠄⠀⠚⠃
⠀⠀⠀⠀⠀⠀⣿⢱⠃⡜⠀⣿⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⡤⢜⠗⢁⣷⠤⣿⣦⣤⣤⢋⣀⡀⠀⠀⠀⠀⢿⡄⢰⠦⠶⡤⣧⠴⠦⣴⣶⣆
⠀⠀⣀⣌⣡⡅⠪⠼⡟⣣⣶⡿⣳⡷⢧⡞⢇⢳⠀⠀⠀⠀⠐⠟⠀⠀⠀⢩⠀⠀⠐⠀⠉
⠀⠊⠁⠈⠑⠂⠙⠾⠿⠶⠚⠒⠙⠀⠈⠑⠊⠁⠈
⣿⣿⠿⢿⡿⠿⢿⢿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠕⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠀⠈⠫⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣀⣀⠀⢀⠤⡲⠀⢀⣀⠀⠣⢀⢶⠚⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⢄⠀⣀⣀⠜⠁⠀⢀⠟⠁⠀⡗⠊⠏⠀⢱⣀⡸⣌⠢⢼⡬⢷⠀⠀⠠⡀
⠀⠀⠀⠀⠰⡀⠀⡎⠈⠪⢄⡀⠀⡠⠒⠉⠑⠢⡀⠣⣔⣓⣀⠜⡠⠤⠼⢄⠀⠀⠀⡆⠀⢰⠃⠀⢀⣄
⠀⠀⠀⠀⢠⠃⡜⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠘⠋⠉⣥⡀⠀⠀⠀⠀⠀⠱⠤⠴⠍⠉⢢⣀⠤⠞⡷⠃
⠀⠀⠀⠀⢈⠞⠓⠒⠒⠒⠤⠴⠚⠉⠉⠱⠄⠀⠠⢄⡴⠑⡵⡎⠀⠀⠀⠀⣀⡀⣀⡀⠀⡜⠁⠀⠀⢱
⠀⠀⢠⠊⠁⣠⠔⠒⠀⠀⠀⣀⠤⡀⠐⠒⡄⠀⠀⠀⠱⠖⠚⠃⠀⠀⠀⠀⣫⣉⣉⡇⠊⠉⠉⠉⠑⠟⠒⡄
⠀⠀⣇⣠⣎⠤⠴⡉⠉⢑⣦⣀⣀⣈⣒⡒⠁⠀⠀⣀⠤⠄⠒⠒⠒⠠⠤⣀⠀⠀⠈⣃⠤⠊⠑⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣴⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣿⠿⠳⠶⣿⠁⠆⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠯⠉⠀⠀⠀⠉⠀⠀⠀⠀⠛⠛⠃⠀⠈⠉⠉⠿⣫⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⠶⠤⠴⠴⠶⠤⠄⠀⠀⠀⠀⠰⠚⠛⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⡛⠙⢉⣉⡉⢉⠉⠉⠉⠉⠛⠛⠛⠳⢶⠤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠉⠛⠛⠛⠻⠷⠷⢮⣤⣧⣤⣤⣤⣴⣤⣴⣴⡶⠶⣶⠶⠶⠛⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠲⠦⢄⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣤⣭⣤⡷⣶⠞⣫⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡔⣶⣿⢀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⠉⢨⣿⠀⣼⠃⣻⠀⣿⡿⣫⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣍⠉⢋⣩⣤⣴⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢸⠀⢸⡿⠀⣿⠄⡼⠀⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⢀⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁⢰⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠃⢠⠀⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣤⠀⠅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡆⢸⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠬⠀⠀⢠⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⢸⡇⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⡧⠈⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⢡⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⡀⠛⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣀⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠒⠒⠛⠛⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣽⣽⣿⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣷⣶⣷⣾⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠶⠶⠶⠾⠶⠶⠶⠷⠶⠶⠶⠿⣶⠀⠀⢸⣿⢶⡿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠒⠖⣶⠖⠒⠒⠶⠲⡶⠒⠒⣶⠟⠀⠀⠀⠹⣾⣷⠶⠲⠒⢶⠖⠖⠒⠒⠲⡶⠒⠖⢶⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⣗⡀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣶⢶⣦⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣰⣮⠉⠀⠉⣥⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣗⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⣤⢀⣤⣽⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠀⠉⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⡆⠀⠀⠀⠀⢠⡞⣩⢭⡝⢦⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⢷⠀⢶⠀⢦⠀⢦⠀⢶⣿⡇⢸⡇⠀⠀⠀⠀⠸⣇⢷⣤⠟⣸⡅⠰⡄⠀⣆⠀⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠉⠛⠓⠚⠓⠚⠓⠚⠓⠚⠉⠳⠞⠁⠀⠀⠀⠀⠀⠙⠳⠶⠚⠛⠛⠒⠛⠒⠚⠓⠚⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠃⠀⠒⠆⠀⠀⠤⠀⠀⠠⠀⠀⠰⠒⠀⠘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢉⡀⠀⠰⠀⠀⢠⢀⡀⣀⠀⠀⣀⢀⡀⠄⠀⠀⠄⠐⢀⡉⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠁⠐⠀⠒⠠⠀⠀⠤⠀⠀⠤⠠⠀⠄⠒⠀⠂⠈⠀⠁⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠒⠀⠄⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠠⠄⠒⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠠⠀⠤⡀⠀⢉⠉⠉⡉⠈⢀⠤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⢀⡁⠀⢸⠀⠀⡀⠀⢸⣀⠠⠄⠀⠊⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⠶⠶⠶⠒⠒⠒⠶⠦⠤⠤⡼⠀⠀⡇⠈⢙⠒⠲⠤⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢝⣒⣒⣣⠉⠙⢙⠇⠀⢹⡾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠺⡗⠪⠔⡨⢉⠓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠈⠉⠓⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢑⡦⢁⡀⠬⣁⠊⠀⠀⠀⠈⠁⠐⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠳⠤⠤⠤⠥⠤⢬⢝⣒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠦⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠒⠶⢬⣍⠀⠒⠩⠍⡐⠂⠠⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⠴⠾⣯⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢒⣏⣉⣉⣙⣳⣦⡈⡇⠀⠀⠀⠀⢀⣩⠉⠒⠂⠤⡤⠄⣀⣀⣒⣒⠒⢒⠒⢒⠒⢀⣉⠭⠞
⠀⠀⠀⠀⠀⠀⠀⣾⢰⢋⠎⠁⠛⢿⠿⠿⠿⢦⢹⡇⠀⠀⠀⡞⠁⢘⢀⡤⢄⠘⣇⣀⠀⠀⢸⡁⠅⠙⠤⡀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⣅⢠⠀⣄⡀⠀⢸⠀⠀⣠⠞⣸⡇⠀⠀⢸⠀⠀⠘⠁⠀⢨⣴⠀⢹⠀⠀⣸⠀⠀⠀⠀⣿⢀⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⠦⣭⣍⣉⣋⣩⠵⠚⢩⡇⠀⠀⢨⠀⠀⠀⠀⡇⢸⡇⠀⠈⢻⡆⡯⠀⠀⠀⢠⠏⡜
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⠇⢸⠀⠀⠀⡼⠳⠃⠀⢀⢠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⠀⠀⠀⡆⡞⠀⠀⢠⡇⠀⠀⠀⣼⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⢸⢰⡇⠀⠀⡾⠀⠀⠀⢰⢿⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⢀⠏⡟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⡿⠀⠀⢸⠃⠀⠀⠀⡏⢸⠀⠀⣼⠁⠀⠀⡜⢠⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢀⡇⢰⡇⠀⠀⢸⠀⠀⠀⢠⠇⣾⢰⠁⡇⠀⠀⣰⠃⣸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠀⢸⠁⣸⠀⠀⠀⡟⠀⠀⠀⣼⠀⡟⠀⢸⡇⠀⢠⡏⠀⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⢀⡏⠀⡟⠀⠀⠀⡇⠀⠀⢠⠇⣸⠁⠀⣼⠀⣠⠟⠀⣸⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣸⠀⢰⡇⠀⠀⢀⡇⠀⢠⠏⢀⡏⠀⠀⣿⠚⠁⠀⢠⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡁⣰⠃⠀⣾⣀⡠⠊⠀⣧⠔⠋⠀⡼⠀⠀⠀⠸⣆⠀⣠⠞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠐⠂⠻⢥⣤⡤⠏⠀⠀⠀⠀⠹⣤⣀⡴⠃⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢇⡈⠑⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠴⢒⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣤⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⡥⠤⠤⠴⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣾⠿⠂⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣠⣼⡆⠀⠀⠀⠀⠀⢀⣴⣿⠿⠝⠿⣳⣴⡆
⠀⠀⠀⠀⠀⢀⣤⣷⣾⣷⣦⣤⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡟⠁⠀⠀⣌⣀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⡀⠉⠁⠀⠐⠿⠿⠟
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⣿⣿⣿⠟⣶⣧⡼⢿⣿⣿⡿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠻⣿⣿⡄⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣄⣛⣟⢁⣼⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣷⠉⠓⠀⠀⠀⠀⡄⠀⣿⣟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠿⣿⣶⡆⠀⢠⣾⣷⣶⣿⠏
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⡿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠻⡟⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠂⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⣤
⠀⠀⠀⣠⢴⣞⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⢻⣛⣛⣛⣛⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠛⣙⣿⣿⣿⠇
⠀⠀⠀⠱⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢹⣿⡿⣿⣿⣟⢹⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⡀⠀⠀⠉⠉⠙⠋⢀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢼⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣈⡀⠀⢀⣴⣿⣴⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠝⠋⠛⢛⠟⠋⡇⣀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⡷⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠀⠺⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⣿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠋⠁⠀⠀⠀⠀⠀⣼⠂⠀⠉⠉⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿


⠀⠀⠀⣤⣶⣾⣿⣿⣶⣶⣄⠀⠀⠀⠀⢀⣠⣤⣤⣦⣵⣤⣤⣀⠀⠀⠀⠀⠀⣴⠶⠶⠚⠛⠛⠛⠛⠷⠶⢦⡄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢈⢛⣟⣛⣛⣛⣛⣛⢛⡁⠀⠀⠀⠀⠸⠶⢶⣶⣤⣤⣴⣤⡶⠶⠾
⠀⠀⠀⢸⣿⣿⢛⠛⣿⣿⠀⠀⠀⠀⢏⣻⠐⡇⢸⡇⢸⠀⣿⢰⣏⡇⠀⠀⠀⠠⣴⣄⣄⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠠⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⢸⣿⠉⣿⠿⢙⣿⠀⠀⠀⠀⠀⢹⠈⡇⢸⡇⢸⠀⣿⢸⡇⠀⠀⠀⠀⠀⣀⡀⠈⠉⠉⠉⠨⠀⣀⠀⠀⠀⠀⠀⢹⣶⣾⢻⣶⣾⣻⣷⣾⡟⣷⣶⡏
⠀⠀⠀⢸⣿⣷⣾⣶⣿⣿⠀⠀⠀⠀⠀⢸⡀⣷⢸⡇⢸⠀⣿⢸⠇⠀⠀⠀⠀⠀⢉⡋⠛⠛⠓⠛⠛⠉⣉⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⢹⣿⣿⡇⣿⣿⡇
⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⢸⣇⣿⢸⡇⣸⠀⣿⣸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣶⣶⣾⣿⡏⠀⠀⠀⠀⠀⠶⠿⠿⠾⠿⠿⠾⠿⠿⠷⠿⠿⠶
⠀⠀⠸⠟⠿⠿⠿⠿⠿⠟⠿⠀⠀⠀⠀⠀⠺⠭⠽⡧⡿⠭⠽⠒⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⠿⠿⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⢠⠤⠖⠒⠛⠉⠉⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⡏⠉⠉⠀⠀⠀⣤⡀⠀⣿⣿⠀⠀⢀⣤⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⢸⣇⠀⣿⣇⠀⣿⣿⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⣿⣿⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣶⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣫⣵⣾⣿⣿⣦⣶⣿⠟⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠶⠶⠒⠚⠋⠁⠀⠹⣜⣻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣤⠀⣄⡄⡄⢶⢠⣄⢰⠆⣄⡴⢄⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠃⠉⠃⠃⠘⠘⠙⠘⠀⠋⠓⠋⠋⠙


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣿⡟⢉⣤⣤⣤⣶⣶⠆⣿⣷⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠉⢩⡟⢋⣀⠾⠀⠀⠀⠈⠻⠦⣿⢹⣶⡌⠉⠉⠉⠙⠛⠻⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠷⠶⠶⠶⠶⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⠙⠻⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡶⠾⠛⠉⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠛⢦⣀⠀⠀⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⢀⡠⠔⢋⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢳⣦⣤⣉⡙⠲⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠒⢒⣂⣩⣤⣶⣾⠋⠙⢻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠈⠉⠛⠻⠛⠋⠛⢛⣶⣶⠊⠉⠉⠉⠩⠭⠭⠭⠭⠭⠭⠥⠄⠀⠐⠒⠓⠒⠀⠈⠉⠈⠙⢧⡀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡶⠶⠶⠶⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢿⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢰⡄⠀⠀⢿⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣻⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠸⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡄⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⢰⣿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⡇⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡿⠀⠀⢸⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⠘⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⢸⡇⠀⠀⣾⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⣿⡄⠀⣿⡆⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣼⡇⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢻⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢰⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣿⠀⢹⡇⠀⠀⠀⣿⠄⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⡀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⣿⠀⠀⠀⣿⠇⠀⠀⣿⡃⢀⣸⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣤⣤⣿⣇⣀⣀⣀⣿⣅⣀⣠⣿⣧⣴⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣒⣶⣖⣒⣶⣖⣖⣲⣄⠀⠀⣤⣤⣤⣤⣤⣴⣖⣒⣒⣒⣒⣒⣒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠇⠀⠘⣯⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠈⢿⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠀⠀⠉⠀⢰⡇⠀⠀⠀⠀⠈⣇⠀⠀⠁⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠠⣶⢲⣦⠀⠀⢸⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⣿⠀⠀⠰⣷⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠸⠛⠁⠀⣸⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⡇⠀⠀⠀⠀⠀⠀⠀⣏⣤⡀⠀⠀⠀⠀⢸⣧⣄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⣧⡀⢀⠀⡀⢀⠀⣀⣿⠀⡇⠀⠀⠀⠀⡏⡔⢲⡽⡆⢀⠀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡼⠻⣧⣼⣤⣷⣬⣦⡽⠻⣤⠇⠀⠀⠀⠀⠳⣽⣯⣼⣧⣬⣦⣼⣦⡽⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁
⠀⠀⠀⠀⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣏⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡃
⠀⠀⢸⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣄⣄⣠⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠉⣈⠙⣿⡥⠔⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁
⠀⠀⢸⣿⢫⡍⣭⠉⠉⣍⣤⢀⣤⢠⣤⡄⠘⠟⠘⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⢆⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⣻⡆
⠀⠀⢸⣿⣀⣁⣀⣀⣠⣏⣫⡬⠯⠬⠥⠧⠤⠤⠤⠤⢼⣿⠀⠀⠀⠀⠀⣀⠐⢸⠇⣣⢺⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣘⡙⡇⢠⡄⠸
⠀⠀⢸⣿⢬⣤⣤⣤⣤⣤⣶⣶⣶⠶⢶⣶⣶⣲⣶⣶⠛⣿⠀⠀⠀⠀⠀⠺⠇⠀⢀⠭⠻⠧⠒⠋⠀⠀⠀⠀⠀⠀⠠⢺⠋⠁⠔⢿
⠀⠀⠸⣿⢸⣋⣉⣉⣉⣁⣨⣭⣿⡄⢈⣭⡭⠭⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣷⠀⠀⠀⠀⠠⠂⢀⣾⠀⠀⢀⠘⠁
⠀⠀⠀⣿⣼⣶⣤⣤⣤⣤⣴⣶⣦⣴⢱⠠⡔⢶⣾⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠉⠀⠠⠀⠀⣤⡴⠿⠃⠀⠀⠼⠲
⠀⠀⠀⣿⣿⡿⠿⠻⠛⠿⠿⠿⠿⠿⠀⠀⢰⠈⠻⣿⣧⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠸
⠀⠀⠀⡿⠿⠇⠀⠀⠀⠀⠠⠞⢣⡐⣠⣦⡜⢁⣤⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡤⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡇⢸⣀⣀⣘⣋⠁⠀⠀⠀⠳⢉⡾⠞⠉⢅⣽⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡋⠉⢁⠈⠁
⠀⠀⠀⣿⣈⣿⣹⣯⣉⣱⡀⠀⠀⠀⠸⡥⠘⣇⠀⢿⣈⣼⠇⠀⠀⠀⠀⠀⠀⢰⡆⠤⠬⠭⢱⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⣿⣿⣿⡏⠀⠀⠘⢦⡀⠀⢀⡙⠀⠙⣦⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⠨⡎⠁⠀⠀⠀⠀⠀⠀⠁⢴
⠀⠀⠀⠀⠿⣿⣿⠇⠀⠀⠀⠀⠑⠂⢸⠁⠀⠀⠿⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡿⣀⠀⢤⡄⠀⠀⠀⠀⠀⠀⢧⠈⠀⡠⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡀⠇⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⠘⣁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡹⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⢣
⠀⠀⠀⠀⠀⢰⣿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⢡⠀⠀⠀⠀⠀⠀⠀⡈⠀⢨⠏⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠐⠋⠀⠘⠃
⠀⠀⠀⠀⢀⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣈⡡⠤⠄⠀⠀⠀⠀⠀⠀⢰⣼⣷⣤⣼⠂
⠀⠀⠀⠀⠸⣿⣿⡏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣖⠓⠾⣿⢿⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀⡿⠟⣽⠖⢪⡿⣿⡂⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⡉⠘⠃
⠀⠀⠀⠀⢠⠃⢸⢁⠀⠀⣄⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠂⠀⡈⠁⠀⠀⣸⠃⠀⠀⣰⡟⠁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠐⠏⠀⠈⠋⠀⠀⠿⠆⠀⠘⠁⠀⠀⠀⠀⠀⠘⠒⠀⠉⠀⠀⠀⢏⡀⠀⣰⣿⠁⢠⠓⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢶⡒⠒⠶⣶⡶⠞⠋⠉⠉⠉⠙⠓⠶⠦⣤⣤⠶⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⠈⠷⠀⠀⠋⠀⠀⠰⡄⠀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠀⣿⣷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣟⠳⢤⡈⠛⢦⣀⣴⢦⣤⡦⠶⣶⠿⣤⣄⡀⠀⠀⠠⠀⣠⠞⠀⢀⢠⡿⠉⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠉⢳⡦⠤⣤⣀⣿⠠⣞⠁⢤⡀⠸⣄⠀⢨⡟⠻⣟⠛⢾⡗⠶⢦⣾⠈⠁⠀⣠⡟⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⣴⠋⢤⠀⢀⡴⠋⠀⠀⠈⠉⢻⣤⡼⠷⢦⣍⣀⣈⣀⣼⣃⣀⢌⣉⣈⣉⣷⣀⣹⣟⣄⣠⣯⠀⠀⠄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⣤⠀⠀⠀⠀⣴⠟⠙⢧⢀⡀⠀⠉⠉⠉⠉⡿⠙⠻⠿⠿⢯⡍⠉⠉⠉⠀⠀⠻⡄⠀⠀⠀⠀⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣀⡴⠋⠀⠀⠀⠀⠚⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠘⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠻⣆⠀⠀⠄⣤⠀⠀⡀⠀⣷⠀⠀⠣⠂⣤⠀⠀⠀⠀⠠⠀⠈⢧⠀⠀⣄⠀⠀⠀⠀⡆⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⡄⡇⠀⣈⣤⡤⣼⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⠞⣧⠀⣀⣤⠴⠦⣄⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⠟⠋⠉⠀⠀⠈⠛⠛⠛⠛⠓⠒⠒⠖⠖⠒⠋⠀⠀⠈⠉⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠰⡂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣼⢃⠀⠘⠉⠓⠶⡄⠀⠀⠀⢸⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣟⠂⠀⠀⠀⠁⠀⢹⡄⠀⠀⠀⠻⣟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡇⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠙⠲⢦⡈⠂⠀⣠⡟⠀⠀⠀⠀⠀⠈⠳⢮⣟⡲⠦⢤⣤⣤⣤⣀⣀⣀⣠⣤⠴⠖⣛⣭⠞⠋⠀⣠⢶⠛⢻⠉⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠀⣠⡴⠞⢦⡀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠤⠤⠤⠤⠶⠶⠚⠋⠉⠀⠀⠀⠀⣏⠈⠆⡀⠀⣈⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⡈⢿⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣤⣷⣋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡳⣤⣙⠀⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣍⣛⣻⡶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣿⣯⣭⣭⣭⣭⠙⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠉⠉⠠⠾⠛⠀⠀⠀⠀⠀⠀⠛⠧⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣤⣤⣤⣀⣐⣒⣒⣒⣒⣀⣀⣤⣤⣤⣶⣾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠟⣒⣲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣙⣛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠾⠿⠛⢛⣉⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⢻⣿⠛⠻⢷⣶⠶⠶⢶⣾⠟⠛⣿⡟⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡏⠀⠀⣿⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡄⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⣤⣼⣿⣀⣀⣿⣧⣤⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⠎⠁⠒⢀⣨⠳⡀⠀⠀⠀⠀⣀⠾⠧⡚⠀⠀⠀⠀⢰⠁⠀⠀⢸⠋⡍⡆⠀⠀⠀⡣⠟⣤⢴⡖⠒⡄
⠀⢸⣀⣠⣧⣊⠋⠁⡇⠀⠀⠀⠰⣇⡀⠀⣡⡄⠀⠀⠀⢸⠀⠀⠀⢸⣦⡡⠃⠀⠀⠈⢣⠚⠩⢄⣦⣾⠇
⠀⠈⢦⡋⠏⠀⢀⡰⠁⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣶⣶⡿⠟⠁
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠟⠉⠉⠻⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠠⢤⣶⡡⠤⣬⣷⡀⠀⠀⠀⢰⡲⠒⢢⣀
⠀⠀⠀⢸⠘⠃⡇⠀⠀⠀⠀⠀⠀⢘⠙⠿⡇⠀⠀⠀⠀⠀⢨⠛⡄⠀⣿⣿⡇⠀⠀⠀⣦⣤⣶⣎⣸⣶
⠀⠀⠀⠘⣤⣤⠃⠀⠀⠀⠀⠀⠰⣀⠁⠈⣀⠆⠀⠀⠀⠀⠀⠀⠑⢄⣿⠟⠀⠀⠀⠀⠻⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠰⠆⣁⠀⠀⠀⠀⠀⠀⢰⠒⠒⡂⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢸⣁⡤⣤⣤⣤⣤⡀
⠀⢰⣩⣵⣶⣾⣿⣿⡆⠀⠀⠀⠀⢸⣶⣤⡇⠀⠀⠀⠀⢰⣈⣉⡀⢀⣉⣑⡆⠀⠀⠀⠉⡇⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⣿⠿⠿⡇⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠈⠍⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠻⠿⠿⡿
⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⢿⠞⠃⠙⠀⠀⠀⠀⠀⠀⠀⡔⢦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣟⠀⠈⢇
⠀⠠⣤⣤⡤⢤⣤⣤⠄⠀⠀⠀⣀⣰⢓⢢⢠⠔⠀⠀⠀⠀⠀⢨⠟⠛⡅⠀⠀⠀⠀⠀⠉⠙⠒⠼⣆⣀
⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠹⠤⠤⠥⠎⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣍⠁
⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣂⣐⠒⠒⣢⢱⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡂⠁⠀⡠⠀⠀⡠⠂⠀⠀⠐⢄⠀⠠⢀⠀⠀⢂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢛⠁⠒⠂⠀⠀⠀⠤⠤⠬⠤⠤⠤⠤⠤⠤⠥⠤⠤⠀⠀⠐⠒⠒⠈⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠠⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠄⡀⠀⠀⣀⡀⠀⠀⠐⠀⠐⠂⠀⠀⣀⣀⠀⠀⢀⠤⢤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠂⢤⠀⠀⠀⡇⠀⠈⠀⠀⠇⠀⢸⠀⠀⠆⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⡤⠐⢂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠈⠲⠀⡀⠀⡇⠀⠀⠀⠀⠄⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠖⠁⡘⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⢤⡀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠁⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡇⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⡇⠀⡁⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠤⠇⠀⠀⡇⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠠⠄⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⣁⠒⠤⠀⠁⠒⠁⠀⠀⠄⠸⠀⠀⠈⠒⠁⡀⠤⠒⣈⠤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢈⡑⠂⠀⠠⠌⠉⠉⠁⠈⠉⠉⠥⠄⠀⠐⢈⠡⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣤⡄⠀⠀⣠⣄⠀⠀⣠⣄⠀⠀⢠⣤⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢿⡿⠀⠀⢿⡿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⡻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣋⣵⣾⣿⣿⣷⣮⣙⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣩⣶⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣶⣍⡻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣮⡛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣭⣻⣿⣿⣿⣿⣿⢸⡟⣭⣭⡝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣃⣛⣛⣛⣛⣛⣛⣛⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣭⣭⣭⣭⣭⣭⣭⣭⢸⡇⣿⡿⠷⠭⠭⠭⡍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⡿⠿⠿⠿⢛⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣝⣛⣛⣛⣛⡛⢸⣇⣛⣛⣃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤

⠀⠀⠀⠀⣠⣴⣦⣤⣤⣤⣤⣾⣻⣶⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠘⣿⡀⠀⣀⣀⣉⣩⣭⣭⣩⣉⣉⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⢹⣿⠉⣿⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⠾⠿⠿⠷⢶⣤⡀
⠀⠀⣠⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣀
⠀⢸⣿⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⠷⣦⣀
⠀⢸⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠈⠛⠛⠛⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠷⢶⣄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠶⠾⠟⠋⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⠿⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⢶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠹⠘⠃⠛⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢛⠚⡛⢛⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠹⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠛⠙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠻⣧⠀⠀⠀⢀⣠⣤⣄⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⢀⣰⠟⠋⠈⢻⣷⠟⠋⠻⣧⠀⠀⠀⣴⠟⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣄⠀⠀⠀⠀⢀⣴⢟⣡⣤⣤⣴⣿⡁⠀⢀⣴⣯⡀⠀⢸⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⣠⡾⠛⠻⠟⣉⣀⡀⠀⠉⠻⠿⠛⠿⣯⣿⡆⠀⠻⣦⣄⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢠⣴⢿⣤⢶⣤⣿⠋⠉⠻⣦⣀⣴⠶⢶⣄⠹⠿⢷⣤⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣸⣿⣿⠁⣾⡇⠀⠈⠻⣦⡀⠀⠈⠻⣧⡀⠀⠙⢷⣄⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢿⣅⣿⠆⢈⣿⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠈⣿⠁⢰⣿⠁⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣤⣿
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢀⣿⡀⠀⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠺⠋⠻⢶⣤⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡈⠻⣦⡀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣋⣿⣦⣌⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠟⠋⣾⠋⠉⠉⠉⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣤⣤⠀⢾⣿⣿⡆⠀⠀⢠⣤⡄⣾⣿⣿⡇
⠀⠀⠀⠀⡀⠙⣻⡁⠀⠀⠀⠀⠀⠘⠛⣃⡀⠀⠀⠀⣀⣛⣛⡀⠺⢛⠛⠃⠀⠀⣘⣛⣃⠛⠛⠛⠃⠀⢰⣿⡦
⠀⠀⣟⠀⠙⠒⣿⣿⠀⠀⢀⡏⢀⣠⡴⣿⣷⠀⠀⢰⣿⣿⣿⡟⠓⠛⠁⠀⠀⠀⣿⣿⣟⠻⠿⠂⠀⠀⠀⠉⣷⣦
⠀⢹⣿⣿⣿⡏⣽⡿⠀⠀⣿⣿⣿⣿⡏⣿⡿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣠⣾⣿⣿⠀⠤⡀
⠀⢨⠉⠉⠉⠀⣿⡇⠀⠀⢬⠉⡍⠁⠀⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⣿⡟⠛⠋⠀⠀⡇
⠀⠀⠁⠁⠀⠀⣿⡇⠀⠀⠈⠉⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢿⡗⡀⠀⠈⢲⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣹⣿⠷⠞⠛⣦⡀⠀⠀⠀⢀⣤⣿⢟⣠⡀
⠀⠀⠐⢦⡀⢀⣤⣄⡀⠀⠀⠀⠈⠛⠛⣻⠛⠛⠃⠀⢀⣄⡀⠀⠀⠀⠀⢰⣿⡟⠛⠛⣿⣤⡈⣿⠀⠀⠀⣾⣿⠉⠉⢁⡼
⠀⠀⠀⠈⢳⣞⣿⣋⣻⠄⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⣹⣿⡃⠀⠀⠀⠀⠘⣿⡇⠀⠀⢿⣿⠇⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠻⠒⠛⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣿⣟⣿⠀⠀⠀⠀⣸⡟⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⣿⠃⠀⠙⠃
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣶⣶⠄⠀⠀⣿⣿⡏⠁⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠛⠙⠋⠀⠀⠀⣠⣶⣄⣠⣴⣄⠀⠀⠀⠀⠀⣴⣦⣀⣴⣤⡀
⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣽⣷⣿⣭⣷⡄⠀⠀⣠⣽⣯⣹⣿⣯⣻⡄
⠀⠀⠀⠀⢀⣼⠛⠃⠀⠀⠀⠀⠀⠀⢀⣬⣛⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠙⢛⠛⠉⠂⠀⢰⣿⣿⣿⡟⠻⡛⠋⠑
⠀⠀⠀⠀⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣭⠿⢦⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣶⣶⣶⡆⠀⠸⢿⣿⣿⣧⣶⣶⣶⣦
⠀⠀⠀⠀⠻⢿⠿⣿⡄⠀⠀⠀⠀⠀⠻⠿⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉
⠀⠀⠀⣿⣿⣷⠀⠸⡷⠀⠀⠀⠀⠀⣿⣿⣷⠸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡉⠉⠛⠛⠷⢶⣦⣴⣟⣉⠁⠈⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⢶⣦⣤⣀⣈⠉⠙⠛⠻⠶⣶⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⡉⠉⠛⠛⠷⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢶⣦⣤⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⡀⠀⠀⣀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⠀⠀⣿⠀⠀⣿⡇⠀⣸⡇⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿⡀⠀⣿⡇⠀⣿⡇⠀⣼⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⣿⡇⠀⣿⡇⠀⣿⠃⠀⣿⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢸⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⣭⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⡗⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⢸⡇⠀⣿⡇⢀⣿⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠀⠸⠗⠀⠿⠇⠸⠿⠀⠾⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⣠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡄
⠀⠀⠀⠀⠀⢈⡷⠢⣔⠺⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡼⠀⢹⠇⠀⠀⣣
⠀⠀⠀⢀⡔⢡⡬⢋⣟⡏⠉⠀⠀⠀⠀⡤⠤⠖⡶⡏⣿⢻⢹⠒⠒⠚⣏⠀⢱⣂⣸⡀⠀⢸⠋⠑⠢⡀
⠀⠀⠈⠁⠀⢸⠀⠀⠓⣿⠀⠀⠀⠀⠀⣸⠀⠀⡇⣇⢹⢸⢸⠀⢀⣀⢸⡀⠀⠀⣠⠜⠋⠛⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠣⡀⠀⠀⠀⡏⠀⠀⠇⠸⠘⠘⠸⣤⢋⠞⢠⠇⢠⠞⠁⠀⠀⠀⣠⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⡀⠀⠀⠳⡀⠀⢸⠀⠀⠀⠀⢠⠏⠀⡾⠀⣞⠀⠀⠀⣀⡜⠀⠿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⠛⠻⠀⠀⠀⢀⠟⠋⠉⠙⠀⠸⣄⠀⠀⠛⠇⢴⡦⡆⠀⢀⠇
⠀⠀⠀⠀⠀⠘⠢⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡗⠀⡀⠤⠄
⠀⠀⠀⠀⠀⢰⠁⠈⢉⣛⠳⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣓⣧⡀
⠀⠀⠀⠀⡄⠺⡏⡩⠭⠭⢭⣓⠲⢬⣆⠀⠀⠀⠀⡔⢋⣩⠭⡽⡪⠓⢄⠀⠀⢀⢜⣛⣫⠉⣟⡢
⠀⠀⠀⠀⡏⠈⠉⠀⠀⠀⠀⠀⠉⠑⢲⡟⠀⠀⢸⠁⣼⣠⠾⣎⠀⡠⠎⠀⠀⢸⢸⡅⠀⠀⢳⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠏⠀⠛⢁⡴⡡⠊⠀⠀⠀⠀⢸⠘⡇⠀⠀⢸
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠏⠀⠀⡴⠯⠊⠀⠀⠀⠀⠀⠀⡸⠀⢇⣀⣀⣸
⠀⠀⠀⠀⠈⠒⠦⠤⣄⣀⡤⠤⠴⠒⠋⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⣀⡀⠁⠒⠀⠐⣒⣒⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣥⣶⣾⣿⣿⣿⣶⡶⢾⣿⡿⠿⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠟⠉⠉⣴⣿⣯⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣏⣹⣏⠐⠉⠀⠀⢀⡴⠟⠛⠉⠉⠉⢣⠾⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣶⣆⣀⣰⣖⡺⣦⡀⠀⠀⠀⣠⠤⢕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣷⠶⠒⠒⠒⠻⣷⡀⢠⡾⠯⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣿⣿⣿⣿⢀⣀⢀⣤⠀⣿⣿⣾⡄⣴⣆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⣙⣿⣿⣿⣯⣿⡇⢹⡆⢹⡇⠀⣿⠀⣿⢸⡏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣧⢸⡇⢸⡇⠀⣿⢀⣿⢸⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣛⣿⢸⡇⢸⡇⠀⡟⢸⣯⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⣸⡇⠀⡇⠀⡇⢸⣯⡼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣿⣿⣿⣿⣿⣿⡶⡇⠀⣇⠀⡇⢸⡧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣫⣽⣿⣿⣿⣿⣿⣿⣿⡇⣷⠀⣿⠀⣷⢸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠔⣿⢿⢿⣿⣿⣿⣿⣿⣿⣽⣧⣿⠀⢛⠀⣿⢼⣿⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢡⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠀⠗⣸⠎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠍⠀⠄⣀⢨⣼⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣏⠀⠀⠘⣿⠀⣻⣐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣷⣶⣶⣶⡿⠾⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣗⡠⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠋⠈⠩⢛⡶⣻⣇⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⡷⠁⠀⣀⡸⢟⣯⡿⠏⣿⣼⡟⢦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣭⡀⠀⣀⣴⣷⣾⡿⠫⠀⣰⡿⢻⣷⣿⣧⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⡼⢿⣿⣯⣿⡵⡟⢿⠏⣴⣶⣿⣷⡾⠏⠱⣿⣿⣭⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡴⣟⣳⣟⣻⡏⢱⢼⣿⢳⡿⣷⣷⣿⣿⣿⣿⢃⣼⣤⣼⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢴⡟⢾⡿⠃⣿⣙⠷⣼⢻⣿⣿⣱⣷⣾⣭⣚⣿⣿⣷⣿⣿⣦⡤⢸⡿⠿⢿⣿⠯⡀⢠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠏⠙⢿⣤⣇⡄⠠⣶⣿⣭⣟⣛⣿⣏⣭⣴⣨⣝⣻⡿⢝⣿⣿⢁⣀⣽⣇⠀⣾⠇⠀⠀⠀⣧⡼⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣛⡀⠀⣠⠞⢹⣿⡇⣿⣿⡿⢷⣶⣭⣿⡿⡌⠉⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣶⣤⣣⣿⠿⣴⣤⣭⠃
⠀⠀⠀⢠⣤⣿⣿⣿⣯⡋⢀⣨⡟⢳⣹⣿⣿⢿⣿⣷⣿⣿⡇⠀⠛⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣇⠂⠈⠿⣦⡀
⠀⠿⠻⠟⠿⠶⠶⠿⠟⠛⠛⠿⠗⠛⠛⠛⠁⠘⠙⠉⠉⠙⠈⠛⠐⠋⠛⠉⠻⠷⠾⠟⠻⠗⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠿⠿⠿⣿⣶⣦
⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⡿⢁⣿⣿⣿⡇⢸⣿⣿⣧⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⠃⣼⣿⣿⣿⡇⢸⣿⣿⣿⣇⠸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠋⠀⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠁⠀⠴⠛⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢿⣦⣤⠀⠀⠀⠀⢀⣤⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠿⣿⣿⣿⣿⣿⣿⣿⡶⢶⠶⠾⣿⠻⡿⢦⠴⢾⣿⣿⣿⣿⠿⠿⠶⠶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣤⠖⢚⣉⠩⠄⣐⠲⠤⣙⠫⠍⢛⡈⠥⠒⢉⠥⠂⣠⠊⠀⢻⠀⠘⣦⠉⠒⢌⡉⠛⢤⣐⠊⠭⣀⡒⠂⠬⣍⣙⠓⠦⠴⣤⣄
⠀⠀⠀⠀⣼⣿⡚⣉⠤⠐⡀⠠⠀⠂⣀⠤⠐⠉⢀⡠⠂⠁⢀⡔⠁⠀⠀⠘⠀⠀⠈⢷⡀⠀⠙⢦⡀⠈⠙⠲⢬⡉⠛⠲⢦⣈⠉⠓⠤⣨⣿⣷⢀
⠀⢀⣴⠀⢟⠛⠻⠷⣶⣯⣤⣤⣖⡉⠀⠀⡠⠔⠁⠀⠀⢠⠎⠀⠀⠀⠀⡖⠀⠀⠀⠀⠳⡀⠀⠀⠘⢲⣄⠀⠀⠉⠓⣤⣀⣀⣭⣶⣶⣿⢿⡿⠀⣧
⠀⢺⣿⠶⣬⣍⣁⠐⠀⠈⠉⠙⠛⠛⠿⠿⢷⣶⣶⣦⣴⣧⣤⣀⣀⣀⣀⣥⣀⣀⣀⣀⣀⣵⣤⣤⣤⣤⣿⣷⣶⠿⠿⠛⠛⠋⠉⠍⠰⠶⣋⣥⣾⢿⣆⡀
⠀⠈⢿⣀⠀⠈⠉⠙⠓⠒⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⢶⠶⠟⠛⠋⠏⣡⣮⣹⡇
⠀⠀⠀⠙⢳⣶⣦⣤⣄⣀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⠠⣤⣤⣤⡤⠤⠴⠶⠚⠛⠛⠛⠙⠉⠁⠀⠀⣀⣤⣦⣾⠿⠛⠟⠋
⠀⠀⠀⠀⠘⣿⠀⠀⠙⠉⠛⠛⠶⠶⢤⣤⣤⡾⠲⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠿⠟⣿⡟⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⢿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠹⡇⣿⠇
⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣼⣿
⠀⠀⠀⠀⠀⠨⡟⠑⣶⠆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡨⡴⢾⡟⢹⣿
⠀⠀⠀⠀⠀⠀⡿⠀⣷⠀⢰⡟⠁⢱⣤⡶⠓⣆⠀⣠⡤⢄⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣠⡤⡀⢀⣴⠒⢢⣰⠏⢈⣷⠀⢹⡇⣸⡇
⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⢸⣧⠀⢨⣿⡀⠀⢘⢸⡏⠀⠀⡄⣼⠋⠁⢰⡆⣸⠋⠁⢈⣦⣿⠁⠀⢹⣼⡏⠀⢸⣿⠀⢸⣏⠀⣾⠁⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⢿⡆⠸⣿⠀⠀⣿⡇⠀⢸⣸⡇⠀⠀⣿⠉⠀⠀⠘⣿⣷⠀⠀⠀⣿⡏⠀⠀⢸⣿⠀⠀⢻⡿⠀⢨⡷⠀⣿⠀⡿⠃
⠀⠀⠀⠀⠀⠀⢸⠆⢸⡇⠀⣿⠀⢰⣿⡇⠀⠈⢹⡇⠀⠀⣿⢠⠀⠀⢸⣿⡏⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢺⡟⠀⢾⠇⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣿⠀⠀⣽⣧⠀⢸⣸⡇⠀⠀⣿⠰⠀⠀⠀⣽⡇⠀⠀⢠⣿⡇⠀⠀⢸⣿⠀⠀⢾⡇⠀⢼⡄⠀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠈⣷⠀⣿⠀⠈⣿⡟⠀⠀⢸⡇⠀⠀⣿⢰⠀⠀⠀⣿⡇⠀⠀⠘⣿⡇⠀⠀⢸⣿⠀⢰⡾⠇⠀⢸⡇⢰⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⣦⠀⣿⠀⣿⡆⠀⢿⡇⠀⠸⢿⡇⠀⠀⣿⣿⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⡐⡏⠀⠀⣧⡀⠀⣿⠁⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢹⡇⠀⢸⡇⠀⠀⢸⣧⠀⠀⣿⣾⠀⠀⠀⣿⣿⠀⠀⣸⣿⡇⠀⠀⡇⡁⠀⠀⣿⠁⢠⣿⠀⣸⠁⣼
⠀⠀⠀⠀⠀⠀⠀⢹⠀⢿⡄⢸⣇⠀⣸⡇⠀⠀⣿⣿⠀⠀⠹⣿⠀⠀⢀⣿⡏⠀⠀⢹⣿⡇⠀⠀⣿⡇⠀⠰⣿⠀⠘⣿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡇⢸⡟⠀⢻⣷⠀⠀⣾⡿⠀⠀⢸⣿⠀⠀⠈⢻⡇⠀⠀⠀⣿⠇⠀⠰⣿⡇⠀⢰⣿⠀⢰⡿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⡇⠘⣧⠀⢸⣿⠀⠀⠘⣇⠀⠀⣼⣿⠀⠀⢠⣿⡇⠀⠀⢠⣿⡀⠀⠀⣿⡅⠀⢸⡟⠀⢸⡇⢀⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⣿⠀⢸⣿⡀⠀⢼⣧⠀⠀⢙⣿⠀⠀⢀⡾⡇⠀⠀⣿⣿⠃⠀⢠⣿⠁⠀⣾⡇⠀⣸⡇⢸⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⣿⠀⢸⣿⡇⠀⠈⣿⠀⠀⣿⣿⠀⠀⠈⢻⡇⡀⠀⣿⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⠃⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⣿⠀⠀⣿⡅⠀⠛⣿⠀⠀⠸⣿⠀⠀⠀⣾⡇⠁⠀⢻⣿⠀⠀⣸⣿⠀⢀⣿⡇⠀⣿⠀⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⣿⠀⢻⡃⠀⣿⡇⠀⢠⣿⠀⠀⢰⣿⠀⠀⢠⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠈⣿⡃⠸⣿⠀⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢻⡇⢸⡇⠀⣿⡅⠀⢸⣿⠀⠀⠘⣿⠀⠀⠀⢸⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⢠⣿⠀⢸⡟⢀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⣿⣇⠀⢸⣿⠃⠀⢸⣿⡄⠀⢀⡼⡟⠀⠀⣿⣿⠀⠀⣿⡏⠀⢸⣿⠀⢸⡇⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⢸⡇⠀⣿⡏⠀⠘⣿⡄⠀⢸⣿⡇⠀⠀⣽⠇⠀⠀⣿⣟⠀⠀⣿⡇⠀⣸⣿⠀⢸⡇⢸⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⢸⣿⠀⢿⡇⠀⢀⣿⡇⠀⠘⣿⡇⠀⠰⢿⡇⠀⠀⣿⣿⠀⢠⣿⡇⠀⣽⡿⠀⣼⠁⣼⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢸⣷⠀⠈⣿⡇⠀⢸⣿⡇⠀⢀⣽⡇⠀⠀⣿⣿⠀⢘⣿⡀⠀⣿⡇⠀⣿⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⠀⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⠈⢻⡇⠀⠀⣿⡇⠀⠘⣿⡇⠀⣿⡇⠀⣿⠀⡯⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⡄⣿⠀⢸⣿⠀⠀⣿⡇⠀⢠⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⢠⣿⠁⠀⣿⠇⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠃⢾⡄⣨⣿⠀⠀⣿⡇⠀⠀⣿⡁⠀⢀⣼⡇⠀⠀⣿⡇⠀⢸⣿⠀⢠⣿⡀⢠⡿⢛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡈⠛⠓⠠⢾⣧⣀⡐⣿⡇⠀⠈⣻⡇⠀⠀⣿⣇⠀⣸⠿⠒⠚⠋⣉⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⢠⠀⠀⠀⠀⠉⠁⠀⠘⠋⠉⠀⠈⠉⠁⠀⢀⣠⡴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠂⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⢱⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠙⣉⠑⣦⡀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⢦⡀⠰⡄⠈⠷⠄⣤⡴⢧⠙⠚
⠀⠀⠀⠀⠀⠀⠀⡆⠱⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⡈⠀⠀⠀⠀⠀⠘⣧⡀⣟⠀⢟⢡⠀⢀⡀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⠑⠀⠐⠁⠀⠀⠐⠄⢀⣐⢀⡶⠃⠀⠀⠐⠙⠉⠀⠈⠉⠉⠁⠈⠀⠀⠀⠈⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠂⠥⠀⠀⠀⠐⠀⠀⠁⠀⠁⠒⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠐⡀⠉⡄⠀⠀⠀⡀⢙
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢁⢀⣷⠀⡀⠀⡤⠞⠛⠛⠓⡦⣠⡞⠉⢂⡀⠂⠤⠌⡀⠔⠀⠀⡀
⠀⠀⠀⠄⠐⠒⠢⣀⠀⠲⠚⣧⠀⠀⠀⠀⠀⣾⢃⣾⠃⡜⡴⣫⡾⠀⡾⢱⡿⢰⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⠑⠀⠀⠀⠐⠈⠓⢉⠳⣊⢈⣉⠙⠄⢪⠗⣸⠁⠻⠴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠈⠈⢠
⠀⠀⢦⡀⠈⠀⠀⠀⠂⠀⠠⠴⠖⡲⠁⠀⠄⢠⣟⣴⠯⣞⠾⢴⣇⡾⢢⢟⡶⠀⡦⢄⡀⠀⠀⢁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢚⠖⠁⠐⠫⡊⣦⣡⠏⢀⡤⠤⠮⠥⣭⣁⡈⠋⢁⡴⢋⡽⠀⠀⡈⠁⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⣸⣏⣤⠀⣡⢤⡤⢠⣄⣤⡄⣬⢭⡟⣩⢤⠀⠠⠀⠉⠀⠄⠀⠀⠀⠀⠁⣈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢱⠃⠾⠵⣟⡱⠿⢫⣏⠼⠇⣾⢀⣣⠈⠀⡀⠃⠀⠀⠀⠁⠀⠀⢀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠰⠂⠀⠄⠁⠀⠀⠙⠈⠁⠀⠀⠡⣄⢀⠀⠀⠀⠀⠀⢀⠪⡄
⠀⠀⠀⠀⢰⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⢀⡄⠀⠀⣀⣀⡀⠀⠀⠀⠠⠀⠀⠀⡀⠸⡀⠀⣄⠀⠀⢠⠀⠐⠂⠈⠢⠁
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⠉⠠⠶⠛⠀⠀⡀⠉⠉⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠡⠚⣳⠤⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠌⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⢤⣤⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⣑⡏⠉⠔⠚⠢⠠⠄⢀⠀⠀⠀⠈⠠⠤⠭⠻⢷⣥⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠘⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠉⠉⠉⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⣀⡀⠄⠤⠤⡤⠄⠈⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡠⠀⠀⡠⠔⠂⠉⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⡃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠄⠀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠈⢄⠈⢆⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠠⠁⠼⠒⠠⠄⠀⠀⠀⠈⢂⠉⠐⠂⠀⠐⢂⠀⠀⠀⢀⠀⠀⠀⠀⠠⠂⡀⠄⠂⠁⠀⠀⠀⠀⠀⠲⠑⢌⠀⠀⠀⠘
⠀⠀⠀⠀⠸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠠⢆⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⢠⣀⠀⠀⠀⠈⡴⠀⠀⡆⠀⠀⠀⢠⣤⡀⠀⠀⠀⣀⠀⠀⠀⣄⣤⠀⠀⠀⢰
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠈⠒⠁⠀⠀⠈⠁⠀⠀⠀⠈⠈⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠌⠈⠁⠀⠈⡘⡄⠆⠀⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⢧⢠⣀⠀⠀⠀⠀⠈⠒⠈⠀⠀⠀⠀⢀⠠⠊⢸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢤⠄⠀⢠⠤⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠁⠒⠂⠤⡤⠀⠀⠤⠄⠐⠂⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠀⠓⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⠀⡀⡠⠔⠀⠀⠀⠈⠒⠠⢀⡇⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⠨⠨⣠⠀⠰⠳⠿⠟⠁⠰⠴⠐⠴⠆⠀⠷⠀⠘⣆
⠀⠀⠀⠀⢰⠱⠀⠀⢠⠀⠤⢦⠀⠀⠀⠀⠀⡀⠀⠀⡏⠀⣚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⢄⠀⠀⢤⢤⣤⣽⠀⠀⠰⡟⡇
⠀⠀⠀⠠⠈⠈⠢⠀⠈⠘⡁⠀⠡⢀⠀⠀⣀⠀⢀⠜⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢯⠀⠀⠤⣀⠤⣤⡤⢤⡠⠆⢀⠾⠀⠀⢀⠖⠛⠳⢆
⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠒⠤⠄⣈⡁⣍⠈⣀⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⡓⠢⠠⣀⣉⢨⡁⣁⡠⠄⡂⠀⠀⠀⢸⠀⠀⠀⠨
⠀⠀⠀⠅⠀⠀⠀⠇⠀⠀⠠⠀⠀⠤⠤⠤⠤⠤⠤⠀⠨⠄⠀⠀⠀⠀⠠⠍⠡⠄⠀⠀⠀⠀⠇⠀⠀⠤⠤⠤⠤⠤⠄⠀⠥⠀⠀⠀⠸⠀⠀⠀⠠
⠀⠀⠁⠐⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠀⠠⠤⠤⠤⠤⠀⠀⠐⠒⠂⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠉⣿⠉⠉⠉⢻⡟⠛⠛⠛⡟⠉⠉⠉⣿⠉⠀⠀⢸⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⣷⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⣼⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠿⣿⣶⣦⣤⣼⣄⣀⣀⣸⣇⣀⣀⣸⣇⣀⣀⣠⣧⣤⣴⣶⣿⠿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠈⠉⢹⡟⠛⠛⢻⡟⠛⠛⢻⡟⠛⠛⢻⡏⠉⠁⢰⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⡇⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣿⣀⠀⠀⣧⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣸⠀⠀⣀⣾⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢻⠟⠿⠿⣿⣶⣶⣶⣷⣴⣶⣾⣷⣶⣶⣿⠿⠿⠛⡟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⢿⠀⠀⠈⡇⠀⠀⢸⠁⠀⠀⣿⠀⠀⢰⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣧⣀⣀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣏⢀⣀⣼⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⣢⣄⣤⣤⣤⣶⣶⣶⣷⡿⣿⣿⠿⢿⣿⣷⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣈⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣭⡧⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⢸⣿⣾⣿⣿⢏⣿⣟⡛⣻⣿⠿⠟⣿⣿⣿⠿⣿⡿⠿⠿⣿⡿⠿⠛⢿⣟⣫⣿⣽⣭⣶⠚⢯⣿
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣶⣿⣿⣿⣓⣿⣗⣿⣇⣿⣟⣿⣯⣿⣿⡿⠿⣋⡿⣥⣶⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣟⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⠍⠉⠉⠉⣤⣶⣾⣿⣿⣿⣧⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠒⠒⠒⠋⠉⠉⠉⣛⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⢸⣿⣿⢹⣿⣿⣿⠛⣿⠀⢠⠠⣦⠀⠀⢰⣶⠀⠀⣿⡏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠸⣿⣿⢸⣿⣿⢸⣿⣿⣿⡄⣿⠀⠘⠀⣿⠀⠀⢸⣿⠀⠀⣿⠇⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣿⣿⢸⣿⣿⢸⣿⣿⣿⠇⣿⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⣿⢀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⢸⣿⣿⡞⣿⣿⣿⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿⠀⢠⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⡏⣿⣿⡇⣿⣿⣿⡄⣿⠀⠀⠀⣿⠀⠀⣸⡿⠀⢘⣿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⣿⣿⡇⣿⣿⣿⠃⣿⠀⠀⠀⣿⠀⠀⣿⡇⠀⢸⡟⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⡇⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⢽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⣿⠀⠀⣽⡇⠀⢸⡇⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⣷⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⣿⠀⠀⣽⡇⠀⢸⡇⢺⣏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⣿⣿⢹⣿⣇⣿⣿⣿⡇⣿⠀⠀⠀⣏⠀⠀⣿⡇⠀⣾⠁⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⢸⣿⣿⢻⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⣿⠃⠀⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿⢸⣿⣿⢸⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⣿⣼⣿⣿⢸⣿⣿⡇⢸⠀⠀⠀⡇⠀⠐⣿⠀⠀⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⡇⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⡏⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⡟⣿⣿⢸⣿⣿⡇⢸⠀⠀⠀⡇⠀⠀⣿⠀⢠⡗⢠⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⣿⡇⣿⣿⢸⣿⣿⡇⢸⠀⠀⠀⡇⠀⠀⣿⠀⢸⣇⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⣿⣯⣿⣿⣇⢸⣄⣀⠀⣇⣀⣠⡯⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠴⠦⠭⠩⠤⠥⠴⠾⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠀⠠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⢀⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⣱⠚⠢⣄⠀⠀⠑⢢⠤⠤⣄⣀⣀⣀⣀⡤⠖⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠁⠀⠠⠊⠒⢤⡀⠸⢀⠔⠁⡇⠀⠀⠀⠂⠀⠀⠀⠀⠘⣤⣤⣤⡤⢤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡴⠁⠀⠀⠀⠈⢻⠁⢀⡔⠓⠒⠒⠚⠳⡄⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⡜⢀⠎⠀⠀⠀⠀⠀⠀⠈⢦⠃⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀
⠀⠀⠀⠀⠀⡀⠀⢀⣼⠤⠤⠤⠤⠤⠴⠹⣄⠸⠀⠼⠄⠀⣀⣀⠀⠀⠀⡀⣘⢀⢀⣇⠀⠀⠀⠀⣀⢀⠀⣀⠀⣸⡀⡀
⠀⠀⠀⠀⢏⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠆
⠀⠀⠀⠀⠀⢳⠒⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣒⡒⠒⢒⡒⠒⢒⡒⠒⢒⣒⠒⢒⣒⠒⠒⣒⠒⠒⢲⠃
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⡇⢸⠀⡇⢸⠀⠈⢸⠀⢸⠀⡇⢸⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⡇⢸⠀⡅⢸⠀⠀⢸⠀⢸⠀⡇⢸⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⠀⡇⢸⠀⡇⢸⠀⠀⢸⠀⢸⠀⡇⢸⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⡇⠀⡏⢱⠒⡶⢠⠀⢶⣲⠖⢤⠤⡤⢤⠤⡤⠛⡇⡇⠀⡇⢸⠀⡇⢸⠀⠀⠈⠀⠘⠀⠁⢸⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠇⠀⡇⢸⠀⡇⢸⠀⢸⠁⠀⢸⢸⡇⢸⠀⡇⠀⡇⡇⠀⡇⢸⠀⡇⢸⠀⠰⢼⠀⢸⠚⡇⢸⢸⡇⠀⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢇⡸⠀⢗⡺⠀⢸⣠⠀⠸⣼⠇⠸⣀⠇⠀⣇⡇⠀⢇⡸⠀⢇⡸⠀⢠⣸⠀⠸⣀⠇⠸⣼⠇⢸
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣭⢬⡟⢧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢿⣦⣤⣳⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠲⡌⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡈⠀⢸⣉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣀⡷⠒⠒⠛⢧⡀
⠀⠀⠀⠀⠀⠀⣟⠉⣉⣙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣁⣀⣀⡀⢀⣀⣸⡀
⠀⠀⠀⠀⠀⢀⡌⡉⢩⠈⠛⠦⣌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠚⠲⢬⣙⠳⠆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⣠⣄⣀⡊⠀⠃⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⡇⠹⡄⠀⠈⠑⠂⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠊⠉
⠀⠀⠙⢌⠻⣟⠓⠖⠚⠛⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠻⣿⣦⣄⣠⡤⠤⣄
⠀⠀⠀⠀⠑⠮⣉⣉⣁⣠⣀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢬⡒⠤⠤⠄⠀⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠾⠒⠿⣂⡀⠸⠗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⡿⠛⢻⠷⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣟⣉⣹⣤⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠀⣞⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣤⣙⠛⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⢈⠶⠴⠾⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⢤⠾⠤⠤⠤⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣉⣉⣀⣀⠀⢀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠽⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣠⣤⠀⠘⡆⠀⠀⠀⠀⠀⠴⠖⣋⡥⠶⠤⠤⠇
⠀⠀⠀⠀⠀⠀⢠⠞⢳⠀⠙⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢯⣼⠓⠀⡇⠀⠀⠀⠀⠀⠴⠋⠁⢀⡴⢰⡆⢲⡀
⠀⠀⠀⠀⠀⠀⢈⡵⠯⠤⠤⡏⠉⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⣤⣙⠓⠤⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣠⣶⡶⢺
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠛⠁⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡄⢄⠈⠙⠲⢄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣬⠭⠤⢋⡴⠋
⠀⠀⠀⠀⠀⢹⡉⠉⠀⠉⠉⠉⠉⠉⣹⠃⠀⠀⠀⠀⢠⡤⣤⣬⣁⠈⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤⠒⠒⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠲⠲⠚⠁⠀⠀⠀⠀⠀⠀⠱⣌⠻⣉⣓⣋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠭⠤⠤⣄⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡗⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠸⣅⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⣄⠜⠉⠉⠉⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⡛⢛⡛⠛⠛⣛⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠤⠤⠤⠤⠄⠠⠤⢬⡇⠀⠀⠀⠀⠀⠀⠋⣉⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠋⢙⡥⠤⣈⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡼⠮⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠙⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠔⡖⠀⠀⠀⠀⠀⠧⠀⡴⡶⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡴⠊⠉⠈⢉⡶⠊⠀⢰⡧⠔⡞⠉⠙⡆⠀⠸⡄⢣⡀⢾⡙⣿⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠹⣠⠚⠉⠙⠁⠀⢀⡤⠯⢄⡀⠀⠸⡀⢈⣧⠀⠀⡿⠚⠉⠉⣆⠈⠉⠉⠁⢧⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠈⢦⠀⣰⠁⠀⠈⠑⠢⣄⠀⠴⠁⠀⠀⠀⠈⠳⣄⣙⣧⠤⠴⠚⠡⠒⠉⠉⠙⠢⡀⠀⠀⣼⣀⣀⡘⠀⠀⠀⢰⡲⣄
⠀⠀⠀⠀⠀⢰⠃⢰⠃⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠹⠒⠒⠣⠄⠀⢱⣠⠴⠶⠁⡟⠁
⠀⠀⠀⠀⠀⢈⡕⠛⠚⠋⠛⠓⠢⢤⠔⠊⠉⠉⠉⡣⠤⠖⠢⠤⣀⡼⠙⢎⢾⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⡼⠁⠀⠀⠀⢹
⠀⠀⠀⢠⠖⠉⠀⣀⡤⠤⠀⠀⠀⠀⠀⣀⠀⢀⠴⠦⡀⠀⠀⠀⠈⢣⡤⠬⠷⠇⠀⠀⠀⠀⠀⡟⠒⠚⢹⠡⠚⠉⠉⠉⠓⢦⡿⠤⣄
⠀⠀⢰⠃⠀⣠⠎⠁⡠⠤⠖⢦⣀⠐⠋⠉⠱⢄⣀⣀⠇⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⠀⠀⠐⠋⠉⠙⢺⠀⢀⡤⠤⢄⣀⣀⣁⡤⠼
⠀⠀⠘⠶⠾⠓⠋⠉⠑⢄⢀⡤⠜⠓⠲⠒⠒⠒⠢⢤⣀⣀⣠⠴⠒⠉⠉⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⡤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢠⣤⣀⡤⢤⠀⠀⠀⠀⠀⠙⢧⣦⡌⠁⠀⠀⠀⠀⠀⠈⠉⢳⣽⠁⠀⠀⠀⠀⠭⠈⣉⠁
⠀⠀⠀⢀⣹⠹⡧⠟⠀⠀⠀⠀⠀⠂⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠠⠇⣿⡇⠀⠀⠀⠀⠀⠈⠻


⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⢄⡀⠀⠈⢀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁⠀⠀⠀⠀⢀⣀⣀⠇⢸⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⠀⠀⠸⢿⣿⠿⠀⠀⠀⠀⠀⢞⢿⣿⠗⣸⡀⠀⠀⠀⠐⠂⠈⠠⡄⠒


⠀⠀⠀⠀⣴⣶⡶⡄⠀⠀⠀⠀⠀⢠⣐⣒⣀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⣴⡖⡄
⠀⠀⠀⠈⣐⣀⣐⡈⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣤⢸⣶⣾⠀⠀⠀⠀⠀⠀⡗⠠⡜
⠀⠀⠀⠀⠛⠿⠟⠃⠀⠀⠀⠀⠀⠘⠛⠻⠛⠀⠀⠀⠀⠀⠸⠿⠾⠾⠷⠀⠀⠀⠀⠀⠈⠙⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠹⠆⠀⠀⠀⠀⠀⢀⠀⢼⣷⡀
⠀⠀⠀⢀⣠⣄⣀⣀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣃⡇⠀⠀⠀⠀⠠⣾⡿⠶⢿⡦
⠀⠀⠀⠈⠉⠉⠋⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙