Sticks Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡱⡒⢼⣿⢐⢱⣶⢶⣲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣑⠹⡏⠞⡁⣨⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⢀⠶⣉⡳⠒⣚⡙⠐⠂⣛⠩⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⣳⣄⡈⢑⠔⣆⠙⢿⢋⢓⢂⣾⢰⡤⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⡨⡻⣏⢦⠟⡩⢍⣩⠟⣽⡇⢸⡷⢄⣯⣑⠲⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣙⠉⠩⡙⠧⢓⡛⣫⠿⠿⠿⣤⡉⠛⠾⢿⣖⡈⠉⠛⠿⣷⣦⣭⣳⠶⠤⣀
⠀⠀⢀⣀⣀⡒⠒⡠⠔⠲⠖⠒⠙⡿⢛⠭⣠⠄⡄⡻⣿⣷⣿⡾⠛⣢⣦⣀⠈⠙⠷⣢⣄⡀⠉⠻⣧⣾⣿⣎⡦
⠀⠤⠤⠤⠤⠀⢸⣰⢿⣷⢀⣤⣞⣠⣵⣾⢱⢡⣿⣍⢪⣿⣯⣶⣾⣿⣿⣮⡑⠤⡀⠈⠑⠫⣖⣶⡮⣉⣭⡳⣯⠭⠤
⠀⠀⠛⠋⠁⣀⣘⣼⣜⣋⣰⣿⣿⣿⣿⣿⣂⣾⣛⡛⠓⠬⠿⢛⣛⣛⣿⣿⣿⡗⢰⣿⣿⡷⣽⣿⠸⣿⢿⢟⡴⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠂⠀⠀⠀⠀⠀⠺⠷⠶⠶⠤⠤⠤⠤⠭⠭⠭⠭⠽⠷⠾⠷⠿⠿⠿⠿⠷⠶⠯⠭⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⣛⣛⣛⣛⠛⠛⠛⠛⠓⠒⢀⣀⣀⣀⣀⣐⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠒⠚⠉⠁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⠉⠁⠀⠀⠀⣀⣀⢤⣴⣶⡄⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠊⠉⠀⠀⠀⣀⣀⣤⣴⡴⠶⣿⡏⠻⣷⣦⡻⠿⠛⠒⠀⠀⠀⠀⢈⣲⡀
⠀⠀⠀⠀⠀⠀⢴⡉⠀⠀⠀⠀⠠⣤⣶⣦⡙⣷⣝⣿⡄⠻⡷⠀⠈⠛⠀⠈⠀⠀⢀⣀⡠⠤⠒⠒⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠗⢄⠀⠀⠀⠀⠙⢿⡿⠿⠋⠉⠉⠁⠀⠀⢀⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠘⡄⠑⣄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⢸⠀⠱⡀⠈⣱⣤⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⢴⣿⣋⣿⣿⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠋⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡄⣶⠾⣿⠋⠉⣿⠁⠿⠟⠈⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠀⠀⢰⣿⣿⣇⣿⣧⡟⠀⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠈⢆⡇⠀⠀⠀⡇⠀⠀⠀⠸⠿⠟⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⡇⠀⠀⠀⠀⢀⣀⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠤⠔⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁⠈⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⢏⣿⡿⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⡼⠀⢠⣄⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠿⣶⡆⠉⠀⢰⠶⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣈⠉⢻⣾⣿⡏⣋⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡃⡶⠶⠞⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⣾⢩⣭⣥⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣟⢠⡟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⡅⠸⡏⣭⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡾⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢩⣤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣟⣛⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡉⠛⠿⣿⡍⠛⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠙⠂⠀⠀⠉⠻⠄⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡏
⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇
⠘⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⢀⣾⡿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠻⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠙⢿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⠟
⠀⠀⠀⠈⠉⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⡿⠿⢿⣿⣿⠿⢿⣿⡟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣾⣿⣧⣀⠀⠀⢀⣀⣼⣿⣷⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠉⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠛⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠋⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⢸⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡇⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣴⣾⡟⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣿⣦⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠁⢀⠀⠀⠀⠀⠀⠀⡀⢀⢀⢀⡀⢀
⠀⠀⠀⠫⠀⠓⠈⠹⠀⠁⠇⠈⠃⠈⠈⠸⠁⠋⠀⠡⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠛⢦⡀⣾⠁⠀⡴⠋⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣉⣉⣻⣷⣦⣷⣿⣶⠟⢓⣺⡶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠥⡤⠶⠿⢿⠿⠿⢿⣿⣿⣷⣾⣻⢶⣄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⡄⠀⠀⡠⠞⢁⡤⠞⠁⠔⠊⠉⠛⢿⣦⣙⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⢓⠩⠒⡥⠊⠀⣊⠴⠚⠁⠀⠐⢀⠔⠋⠀⠀⢉⣿⡿⢯⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣽⣋⠁⠀⠁⠀⢀⢔⣶⣍⣀⡀⠂⠀⠀⠒⠁⡀⠀⢀⡴⠋⢺⣿⠈⢿⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢩⣴⡶⠶⠮⠳⣄⣠⠞⠁⣾⠁⠀⡈⠙⢳⣄⠀⠀⠀⠄⣰⠋⠀⠀⠀⣿⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢿⣤⠂⠀⠀⢸⡇⢀⠠⣈⠛⠛⢠⠖⠀⢻⡆⠐⡠⡾⠋⠀⠀⠀⠀⢻⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢮⡻⢶⣤⣴⣞⡵⣣⠞⠑⠁⠀⡼⢧⣀⣼⣧⠊⠀⠀⠀⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⢷⣭⢄⣉⠟⠩⠞⠛⠛⠻⣦⣂⡔⢁⣞⣵⠋⠀⠀⠀⠀⡠⢂⣼⣵⠟
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡶⣧⡹⠋⣁⠀⣴⠟⠛⢻⢦⡈⢿⡄⣸⡁⠈⢳⡄⠂⠀⠊⡠⠋
⠀⠀⠀⠀⠀⠀⠸⣇⠸⡇⠘⠃⢺⡅⣠⣿⣠⠀⠀⣸⠇⣼⢏⣴⡓⠒⠀⣧⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠦⣀⣼⡟⠁⠙⠿⣶⣬⣤⡾⠏⢸⠀⢻⣀⢴⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣟⡥⠞⠁

⠀⠀⠰⣿⣷⡀
⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⢻⣿⣿⣀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢀⣤⣿⣿⣿⡏⠉⠉⠀⠀⠀⠈⠙⢿⣄
⠀⢀⣴⠟⠁⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⢀⡿⠹⣦
⢰⡟⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⣧
⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠹⣧
⢹⣇⠉⠛⠶⢦⣤⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣀⣀⣤⣴⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶
⣿⣿⣿⡟⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡛⢦⡀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⠀⢠⡏⠙⠀⡧⠀⣼⠋⢠⡞⠧⠙⡆
⠀⠀⠀⠀⠀⠀⣏⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡽⣯⠻⡄⠹⣦⡀⠙⠒⢛⣁⣀⣿⡀⠈⠳⠦⠾⢣⡴⠖⠶⢦⡀
⠀⠀⢸⢉⢧⣀⣿⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⢧⣬⡴⠃⢀⣨⠿⠛⠋⠉⠉⠉⠉⠙⠓⠶⣄⡀⡏⠀⣴⠶⣆⣿
⠀⠀⠈⠋⡼⠩⠀⠈⡇⠀⠀⣠⠖⢋⣉⠙⢦⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⢀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠀⠹⢦⣥⠟⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠙⠦⢤⡞⠁⢀⠞⠁⠀⢾⡿⠃⡜⠀⠀⠀⠀⠀⣀⣀⡀⠉⠉⢉⡟⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢷⡀⠀⢀⣀⡀⠀⠀⠀⠃
⠀⢀⣀⠤⠤⠤⣄⣷⣠⠏⠀⠀⠀⠀⠀⠚⠓⢢⠀⠀⢠⡞⣩⠎⢹⡆⠀⣼⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⢷⣴⠋⣉⣉⢳⡄
⡔⣩⣤⡀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠐⠿⠟⣠⠇⠀⢸⡄⠘⠓⠛⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⢼⠃⣷
⢇⡻⠟⣃⡄⠀⠀⠀⠈⣿⠑⠒⠒⠒⠒⠒⠊⠁⠀⠀⠈⠳⢤⣤⡤⠶⠋⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⢸⠇⠈⠳⠤⠞⠃⠀⠄
⠀⠁⠉⡏⢰⣿⣷⢀⡞⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠲⠶⢤⣤⣄⣀⣀⣠⣤⠴⠶⠛⠋⠀⣠⠏
⠀⠀⠀⠘⠶⠭⠴⠋⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⢰⡟⠓⠶⣤⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠲⠤⣤⣤⣤⣤⡤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠙⠇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠃⠀⠀⠀⠀⠀⠀⢠⠎⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⢠⠇⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣤⣤⡤⠴⠖⠋⠁⠀⠀⠀⠀⡞⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠖⠒⠛⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠛⠋⠀⠰⣿⣷⠀⢀⣶⣄⠀⢀⣀⠀⠀⢠⣦⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠘⠛⠁⠀⠻⠟⠁⠀⠙⠛⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠒⠲⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠛⠛⠲⢦⣿⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⣀⡀⠀⠀⢀⣽⠀⠀⠀⢸⣁⠀⣀⣀⡤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⢄⡀⢀⣤⣴⠟⡀⠊⣹⣿⣧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡐⠂⡀⠀⠀⢉⡉⠀⠀⠀⣠⡟⣟⢯⠄⢛⣋⢩⠿⠇⠀⠀⣀⡽⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⣿⣿⣇⣠⡠⠌⠑⠂⠀⠀⢀⣠⣽⠾⢿⡽⣋⣀⡤⠤⠖⡚⠉⠁
⠀⠀⠀⠀⢀⢀⣀⠘⢿⣛⣿⡏⣉⡁⣠⣤⣦⣴⣾⣿⣿⣿⡥⢄⣲⡾⣧⣉⡥⠴⠚⠉
⠀⠀⠀⠀⠀⢺⣿⡾⣿⣿⣿⣿⣿⣿⡟⣩⡭⠿⠿⠛⠿⠛⡉⣄⣠⡾⠛⠉⡠⣀⣤⢴⠂
⠀⠀⠀⠀⠀⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⣀⣀⣤⣤⠶⣿⣽⡿⠷⢤⡹⠞⠿⢃⣼⣏⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⠿⠛⠋⠁⠀⠀⢿⣿⣻⣿⣷⣥⣼⣥⡌⠝⡻⣿⣶⣮⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢾⣿⣿⣿⣿⣩⢗⣿⣦⣈⡞⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⡷⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⢤⣤⣤⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⡠⠖⠉⠀⢀⡼⣿⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⣀⡴⠋⠀⣿⠀⠀⢠⣟⠁⠀⠀⠀⠀⠈⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡠⠚⠁⠀⢀⡴⣿⠀⠀⢸⡏⠉⠛⠛⣺⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⠀⠀⢀⡴⠋⠀⣿⠀⠀⢸⣇⡤⠖⠋⠁⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠖⠉⠀⠀⢠⣿⠀⠀⢸⡏⠀⢀⡠⠔⠋⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⢀⠔⠡⣿⠀⠀⢸⣃⠖⠉⠀⠀⠀⡠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡠⠞⠁⠀⢈⣿⠀⠀⢸⠇⠀⢀⡠⠔⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠋⠀⠀⠀⣠⠾⣿⠀⠀⢸⣀⠔⠋⠀⠀⠀⡠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⡠⠞⠁⢸⣿⠀⠀⢸⠏⠀⠀⣀⠔⠊⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠋⠀⠀⠀⣰⣿⠀⠀⢸⠀⡠⠚⠁⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⠀⠀⢀⠴⠊⢸⣿⠀⠀⢸⠎⠀⠀⢀⡠⠒⠁⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠖⠁⠀⠀⢸⡿⠀⠀⢸⠀⡠⠚⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⣠⠔⢹⡇⠀⠀⢸⠎⠀⠀⢀⡠⠔⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠴⠊⠁⠀⢸⡇⠀⠀⢸⠀⡠⠒⠉⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠁⠀⠀⠀⡠⢺⡇⠀⠀⢸⠎⠀⠀⠀⣀⠴⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡠⠔⠉⠀⢸⡇⠀⠀⣾⠀⣠⠔⠊⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠉⠀⠀⠀⣠⢼⡇⠀⠀⣿⠜⠁⠀⠀⠀⣀⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⣠⠔⠋⠀⢸⡇⠀⠀⣿⠀⠀⣠⠔⠊⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠊⠀⠀⠀⠀⢸⡇⠀⠀⣿⡴⠋⠀⠀⠀⢀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠤⠶⠾⠛⠁⠀⠀⡟⠀⠀⢀⡴⠒⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠔⠁⠀⠀⢀⢚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⢀⣀⡀⠀⠀⠀⠀⢹⠦⡀⠀⠐⠢
⠀⠀⠀⠀⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠤⠤⠤⢄⣠⣍⣱⣴⡿⢃⠋⣐⡚⣂⠐⠒⠂⠀⠠⣤⠤⠤⡤⠤⡤⡀
⠀⠀⠀⠀⠀⢃⣠⣀⡀⠄⡀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠰⣉⠙⡇⠢
⠀⠀⠀⠀⠀⡆⠀⠀⠉⠉⠙⠛⠷⠒⠢⠬⠤⣴⣿⢸⣥⣤⣄⣀⣄⣀⣀⣀⣀⢀⣀⠀⢀⡀⣀⡀⢿⣣⡟⠀⠓⢒⠢
⠀⠀⠀⠀⠀⠈⠙⠒⠷⠶⣤⣀⡀⣀⠀⠀⢀⠀⡁⠈⠉⠉⠙⠛⠛⠛⠛⠻⠻⠟⠻⠿⠿⢕⣛⣽⢤⣥⠖⢧⣤⠀⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠧⢅⣧⣤⣰⣤⡤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢩⣌⡿⠼⠘⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠙⠓⠢⠶⠤⣷⣒⣀⣀⠙⠯⢒⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣤⠆⣦
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠛⠁⣄
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⡛⢀⣄⢀⣄⠀⢠⡄⣀⣄⠀⢀⣠⣀⣀⠀⣠⡀⣠⡄⢀⣄⠀⢀⣄⣀⠀⣠⡀⣠⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⢸⣿⠏⣸⡇⢸⣿⠃⣿⠀⣼⠇⢹⣿⠀⣿⣷⢻⣷⠎⣿⢀⣿⠈⢺⡗⣿⡵⢹⡿
⠀⠀⠀⠀⠀⠈⣿⡀⢀⣤⢸⡇⢸⡿⠀⣿⠁⣼⡟⠀⣿⠀⣿⠀⣸⡇⠀⣿⡇⢸⡿⠀⣿⠘⣿⠀⣸⠇⣿⠃⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠈⠓⠙⠃⠀⠉⠋⠙⠁⠀⠙⠒⠙⠓⠈⠛⠀⠙⠀⠘⠃⠀⠙⠂⠉⠂⠋⠀⠛⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡀⡀⠀⠠⠔⠒⠛⠉⠉⢉⣹⡦⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠰⠀⠀⠀⠂⠠⠀⡀⠀⠀⠀⠀⠣⠀⣀⠤⠤⠖⠈⠡⠀⠀⠀⠀⠀⠀⠠⠔⠚⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⢠⡴⠖⠀⠓⢈⠀⠀⠀⢐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣀⡤⠶⠖⠚⡋⢉⠀⢈⢂⠀⠀⢀⣑⠦⠆⡈⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠈⠐⣑⣬⠶⢕⠮⡁⡀⠀⣩⠭⢆⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⡤⠶⡒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⢂⢂⠀⠉⠂⠹⠝⠆⠉⠻⢣⣀⡀⡤⠴⣲⡶⠛⠁⠀⢡⠀⠈⣦
⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⢃⢣⠀⠀⣀⠰⠅⠖⠚⠂⢣⣠⠰⠋⠁⠀⠀⠀⢘⢸⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠐⡌⣆⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡎⡟⠁⠈⠀⠀⣈⡴⠞⢻⢷⣦⢀⢀⡀⢀⡠⠂⡡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠴⢸⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣒⠬⠒⠉⠀⢰⢸⠀⣠⠰⠚⠁⢰⠀⠀⠸⠄⠋⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣙⠤⢀⠼⠆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⡋⠥⠒⠉⠀⠀⠀⠀⢀⣸⣸⠋⠁⠀⠀⠀⢸⡄⠀⠃⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⣀⠴⠂⠠⡉⠣⣸⡩⠞⡆⠀⠀⠀⠀⣀⡤⢖⡫⠕⠊⠁⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⣿⠁⠀⢐⠀⠀⢸⡥⢄⠘
⠀⠀⢀⡠⠔⠋⠁⠀⠀⠐⢬⣐⠢⢒⣼⣁⣠⠴⢒⡩⠕⠊⠁⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⢀⣿⢀⣠⢣⠔⠚⠋
⠠⠀⠉⠢⡄⠀⠀⠀⢀⣤⠾⠛⠉⠉⢁⡀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⣸⠯⠉⠁
⠁⠀⠀⠁⢸⠀⠤⠒⠉⠀⠀⠀⢀⠔⡩⠁⠁⢔⡢⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⢀⡠⠶⠟⠁
⢰⡀⠂⠒⠁⢈⢁⡔⢄⠀⠀⠀⢸⠀⠀⠀⡒⡀⠀⠀⡀⣀⡴⠟⠁⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠹⣄⠀⢆⠀⠂⠀⡆⣧⠀⢀⡼⣆⠀⠀⣦⠛⠁⣠⠇⡁⠀⠀⠀⠀⣠⠔⠋⠁
⠀⠀⠈⠳⢄⠙⠳⠚⣠⠯⠐⠁⠀⠈⠢⡀⠀⠉⢹⡑⣲⢰⠀⠠⠔⠋
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣭⡵⠋
⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣶⠾⢛⡄⠀⣠⠖⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⣤⣀
⠀⠀⠛⢉⡉⠀⠀⣾⠗⠚⠛⠒⢀⠀⢀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⣿⡖⢿⣿⣾⣧
⠀⢀⣴⣿⣷⣧⢸⡏⠀⢸⠀⠀⠘⣧⠀⠻⠄⠀⠀⠠⠄⡀⠀⠀⠀⠀⠐⠃⣰⣿⣭⣿⣆⢻⣿⣿⡀
⢀⣾⠟⡿⢻⣿⠈⠁⠀⣾⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⠀⠀⠸⣯⣀⣸⣯⢿⣼⣿⣿⡇
⠞⠁⠜⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣦⣀⣼⣿⡍⣿⡎⢹⡇⢸⣇
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠛⠿⢮⣿⣿⣧⠀⣿⠈⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣔⣤⡴⣾⢿⣿⠀⣿⣶⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠁⡼⠃⠘⣿⠀⠉⣿⣿⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⡄⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⣿⣿⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡴⢙⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠀⢹⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⡇⢸⣿⡿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⣧⠈⣿⡇⣿⠀⡼⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡟⢀⣿⣷⢟⣴⢿⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠆⠀⣟⠸⣿⣾⠿⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⠀⣿⣿⡆⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢹⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠃⠸⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⣿⠀⠀⢸⡀⠀⣿⡇⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⢀⡔⢸⡇⠀⢿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠈⡇⠀⢸⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⡠⣿⠀⢸⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⡏⢠⡾⠁⢸⡀⢸⣿⡎⣿⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡿⠁⠀⠘⡇⢸⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⢸⡏⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣷⠈⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠸⠆

⠀⠀⠹⣄⠀⠀⠉⠳⠶⠌⢷⠂⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠰⣿⠂⢀⠀⠀⢀⡰⠟⠀⠀⢀⠀⠃⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠈⠍⠉
⡀⠀⡶⣽⣅⠈⠉⠀⠀⠀⠀⠀⠀⡸⢦⠶⠀⠀⠀⠀⠀⣠⠔
⠙⢿⡯⠉⠃⠀⠀⠀⠀⠀⠀⣀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣠⡗⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢿⡟⠀⢀⣆⣄⠀⠀⠀⠀⠀⡛⠃⠀⠀⠀⠀⢀⣸⣿⡿⠂⢀⡤⠄⢠⣾⡓⠋⠀⢀⣀
⠀⠀⠀⠀⠈⣷⠀⠨⡧⢘⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠭⠀⠐⡀⠐⢿⣿⡶⠕⢾⡿⢦⠈⢭⡅
⠀⠀⠀⢀⡀⢸⡄⢰⣷⠋⠑⠀⠀⠀⣧⣠⠃⠀⠀⣠⣤⣤⠊⢀⣤⠔⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀
⠀⠀⡀⠀⡙⢿⣧⣾⣷⠎⠀⠀⠠⣄⣘⡵⠂⠣⠤⣏⠻⣯⣴⠛⢉⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠙
⠀⠀⠸⠆⢱⣺⣿⠛⣿⠖⠁⠀⠀⠘⣧⠆⠀⠀⠀⠈⠑⠈⠧⠀⠈⠳⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⣫⣭⣿⣿⡰⣿⡤⠀⠀⠐⠃⠉
⠀⠀⠀⠘⠁⣿⠻⣿⠛⡀⠀⡀
⠀⠀⠀⠀⢰⡿⠀⣿⣄⠜⠹⡇⠀⠀⣠⡀
⠀⠀⠀⠀⣸⡇⠘⣿⡏⠀⠀⡇⠀⠀⡿⠁
⠀⠀⠀⠀⣿⡇⠀⣿⣿⡀⢰⣷⠀⢰⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⣠⠷⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⣀⠤⠒⠉⠀⣠⣴⠋⡀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⠳⢦⣈⡱⠖⠊⠁⢀⡠⠂⡡⠀⡤⠋⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠤⠒⠁⠀⢀⠤⠒⠁⡠⠊⠀⠊⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⢀⠔⠊⠀⠀⡠⠊⢀⠀⠀⢀⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠡⣀⣠⠔⠚⠉⠀⠀⡀⠈⠀⠔⠁⠀⠠⠦⣀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⢇⢫⠭⡕⡄⠀⡠⠊⠀⡀⠀⠀⠀⣔⠁⠀⠀⠨⣓⠆⣔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⠙⠬⣷⠏⣤⠞⢁⣠⡊⠁⠀⡠⠊⠈⠑⠀⢈⠸⣵⡼⢉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠅⠊⢁⢸⣎⢰⣢⢀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠝⠣⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠰⠐⠂⠄⠠⠀⠄⠆⠲⠠⠆⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣿⠀⠀⠐⠋⠉⠉⠉⠙⠛⠛⠛⠛⠒⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠖⠚⠛⣉⣽⠇
⠀⢹⣆⣘⡒⠂⠠⠀⠀⠈⠉⠒⠒⠒⠒⠦⠀⠈⠉⠓⠂⠀⠀⠀⠀⡀⢀⡀⠠⠤⣤⢄⣀⠀⠙⠻⠶⠶⠤⠤⠤⡤⠤⠤⠶⠶⠶⠶⣆
⠀⠀⠈⠉⠉⠉⠛⠛⠛⠒⠲⠶⠦⣤⣄⣀⠀⠀⠀⠀⢠⡦⠤⣤⣤⣤⣤⣭⣉⣓⣓⣒⣛⣁⣀⣀⣙⠛⠒⠀⠐⠀⠀⠀⢶⣶⣶⡚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣤⣀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠲⠶⢦⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠄⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⠀⢀⠈⣂⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡈⢣⡙⡀⢀⡀⠀⠀⠠⠤⠄⠖⠖⠒⠚⠙⠁⠀⠉⡄
⠀⠀⠠⢐⠒⠒⠒⠒⠒⠒⠊⠲⠀⠀⢨⠗⢏⠁⠀⠀⠐⠄⠀⠠⠤⠤⠶⠶⠀⠀⢀⣀⣱⠄⠄⠀⠀⠀⡀⠄
⠀⠀⠰⣇⣭⡄⠀⠀⠐⠒⠒⠀⠀⠀⢐⣂⣈⠰⡀⠀⢀⠛⠤⠄⠀⠀⠀⠀⠀⠀⣀⡍⠄⠀⡀⠀⠀⣪
⠀⢀⠤⢤⠠⢤⣴⢾⣿⠶⠒⠚⠉⠉⠉⠁⠀⠃⠱⢄⡈⠀⡈⡀⠀⡶⠀⠀⠀⠀⢑⠤⡉⠢⣀⣀⣾⣧⠄
⠀⠀⡇⢹⡆⢸⡁⠡⡈⣷⠀⠀⠀⠀⠀⠠⣔⡀⠀⠀⠨⢦⠐⠁⣪⡓⠀⠈⢶⡯⢭⣭⣭⣭⣽⣿⢿⣿⣷⣦⢄
⠀⠀⠓⠛⠀⠈⠷⣬⡴⠿⠶⠒⠚⠛⠛⠉⠙⠦⣍⡵⠂⢸⣯⣾⢂⠀⠰⡄⠀⠉⠛⢛⠉⠉⢹⡏⢂⠙⠄⠀⠐
⣀⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠒⡟⢀⢴⣿⠇⡚⣆⢁⠁⠀⠀⠀⠁⣠⣄⡤⠼⡠⠈⠑⠂
⠸⢿⣤⣤⣤⣤⣤⣤⡤⠤⣶⠛⠋⠉⠉⠀⠀⠀⠋⠁⠈⣿⣠⣇⠊⣨⠀⣀⡤⠖⠛⠉⠀⠀⠀⠘⠞
⠀⠀⠀⠀⠀⠀⠀⢸⢱⠓⢮⡳⡀⠀⠀⣀⣠⠤⠖⣚⢯⣽⣫⠤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣆⠀⠳⠷⢒⣉⣥⣴⣶⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣶⣤⣴⣖⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⣗⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣷⣦⡍⢿⣿⣽⣷⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣮⣶⣿⣿⣿⣿⣿⣦⠿⠟⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣲⣦⣴⣶⡄
⠀⠀⠀⣘⣿⢾⡟⣫⣾⣿⢟⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠟⠻⣟⣿⣛⣭⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠒⠶⠝⠚⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣤⣺⣿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠿⠿⢻⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠈⢀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠁⠀⢀⡀⠀⠀⠾⠏⠀⣸⠟⠃
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢠⡏⣿⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⢹⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡟
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣦⠀⢘⣿⠀⠀⠀⡀
⠀⠀⠠⠄⠀⠀⢤⣤⣀⡀⠳⣼⣤⣾⣋⣈⣴⠞⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣿⣷⣾⣯⡍⠩⢅⠀⠀⠀⠀⠘⣷⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣶⠇
⠀⠀⠐⠂⠁⠀⠠⠶⠟⠋⡙⡸⢿⠈⠟⠓⠞⠓⠐⠀⠀⠀⠈⠻⣿⡿⡿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠃⠈⢣⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⣺⣇⣴⣄⡀⣀⠆⡀⠂⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢴⣺⣸⣟⣿⣿⡿⢿⡧⠠⣗⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⢴⣷⣀⢲⣟⠷⢰⣤⣤⠯⠸⠛⠁⠈⠀⠀⠀⠀⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⢯⣷⣶⣿⣯⣾⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡙⠿⢛⣿⣿⣿⡗⣾⠖⠀⣀⣀⣢⣠⣗⣶⡆⠽⢿⠿⠿⠻⠙⢿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⢿⣿⡬⡉⠜⢃⢚⣿⠎⠙⠉⠙⠋⠙⠛⠋⠐⠉⠃⠃⠀⠉⢷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣽⣿⣾⣧⠤⡔⠛⠁⢠⢂⡀⡀⠀⠀⠀⠀⠠⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⣽⠟⠋⠝⠛⠁⠀⠀⠀⣨⣾⣷⣷⣧⣤⣷⣼⣷⣤
⠀⠀⠀⠀⠀⠀⢈⣛⣍⢟⣷⣶⣴⡄⠀⠀⠀⠀⢰⠬⢏⣝⠻⡛⠿⣿⡿⣽⡛⠛⠛⢀⣀⠀⠄
⠀⠀⠀⠀⠰⣾⣿⣿⣻⣧⣿⣿⣿⣿⣾⣷⣿⣿⣿⡃⠀⠀⠀⠀⠀⠈⠁⠈⠷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠛⡛⠿⢿⣽⡏⢙⡚⠻⣯⣿⣿⣿⢿⣤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠁⠈⠕⢫⠀⠁⠀⠉⢦⣿⣼⣿⣿⣿⣿⣿⣿⣧⣰⣀⣨⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⡟⣿⣿⣿⣉⡉⣡⣁⣨⣉⣉⡙⣉⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⡏⣾⣿⠓⠦⣶⣴⠿⣿⣟⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡇⣿⠓⣿⡗⠈⠀⠏⠘⠂⠉⠧⢻⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠝⠟⠉⠁⠿⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡂⢆⡦⢀⠈⠤⠒⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠁⠠⠀⠀⠀⠀⢸⡷⢺⣷⠒⠒⠘⠉⠀⠠⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠁⠐⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⡎⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⡁⡴⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠑⠀⠂⠈⢪⣧⠠⢤⣤⢀
⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡃⠀⠀⠀⠀⠈⠁⠀⠡⠞⡻⠽⠦⣄
⠀⡀⠀⠀⠀⠀⢈⣓⡄⠀⠀⠀⠈⠐⠀⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡡⡤⠀⠀⠀⠀⠀⠀⠀⡏⠳⠛⢀⠀⢀⢀⢠⠀⠀⠀⢈⣴⣶⢾⡽⡆
⠀⣧⠁⠀⢠⡎⠩⣿⣅⢆⠀⠀⠄⡀⠀⠀⠀⠈⠙⠻⠥⣀⠀⠀⠀⢰⡏⡀⠀⢀⣀⡠⠤⠒⠋⠉⠞⣋⣬⡤⠼⠞⣟⠷⢤⣀⠘⠁⣹⠏⡧⡇
⡎⡈⢶⡀⢰⣌⠀⠀⠙⡌⠀⠀⠀⠌⠑⠀⠀⡀⠀⠀⠀⠀⠉⠑⠂⠼⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡐⢮⣉⡉⢀⡰⠃
⠑⠄⠀⠙⠂⠉⠀⡢⠎⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠉
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠬⠛⠛⠉⠉⠉⠉⠉⠉⠛⠒⠒⠢⠬⣥⣀⣐⡂⠀⠤⢀⣀⣀⣀⡀⠐⣢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣓⣒⣒⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡜⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠟⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢠⠌⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣴⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡔⠀⠀⢠⡎⠀⠀⠀⣠⣂⠁⢀⡀⠴⠋⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣠⠎⠀⠀⠀⡞⠀⠀⠀⠘⠏⣿⣿⡭⣿⣦⠀⢀⡠⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣴⠃⠀⠀⢠⠞⠀⠀⠀⠀⠀⢶⡈⠹⠤⢋⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣸⠃⠀⠀⢀⡺⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠛⠉⠀⠀⠀⠀⡠⠠⡀⠀⠀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⠁⠀⠀⠀⡵⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄⡠⡠⠁⡠⠊⢁⠴⠋⠀⢙
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡄⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⡇⠀⣠⠀⠀⡠⠂⠜⡰⡡⠊⣠⠞⠁⠀⡠⠒⣁⠤⠂⠐⡀
⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⡸⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⣰⠃⠀⠔⢀⢀⡼⠃⡠⠊⠀⠀⣠⠾⠀⡉⢀⣠⠴⠒⣄
⠀⠀⠀⠀⠜⣡⣤⡆⢀⡞⠁⠀⠀⠀⡼⠀⠀⢀⡠⠴⠒⠲⠦⣄⠀⢸⢰⣣⠟⠠⠂⡀⠁⣨⠔⠋⠀⢀⡠⠊⠁⣠⠴⢚⡩⠔⠚⠉⢁⠄
⠀⠀⠀⠀⢦⡃⢡⠿⣯⠶⣾⡀⡀⡜⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣼⣃⢼⠿⠋⠁⠀⢀⠔⣫⡴⣒⡭⠖⠂⠁⣀⠤⠚⠉⠁
⠀⠀⠀⠀⠈⠫⠘⠙⡋⢀⡸⣡⠏⠀⠀⠀⠀⠳⠤⣤⡤⠤⠤⠴⠶⢿⣿⠛⣙⣷⣿⡀⠀⣀⣴⣾⠿⠛⠋⠁⣀⡤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⢀⣿⠊⠀⠈⢧⣹⣞⠋⠁⠀⢀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⢀⠔⠉⢿⠀⠀⣠⢿⡅⢸⡧⠔⠊⡁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠂⢀⡠⠊⠀⠀⠀⢀⡴⢊⡠⠼⠟⠋⠁⠀⠀⢧⠀⢀⢣⡄⡆⠀⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠙⣿⣿⣇⣴⣯⡀⡄⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣄⡈⣎⢠⡏⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡎⠛⣿⡛⠿⣿⣷⢠⣶⡿⢻⡣⠀⠀⠀⠀⠀⠀⣀⡠⠒⡀⠘⣳⣿⣿⣿⠋⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠺⣯⢹⣤⠿⢟⣿⠋⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠚⠟⡾⠟⣿⢿⡟⠻⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⡶⠾⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⡟⢼⡿⢭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠱⣼


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠁⣛⣁⠀⢀⣀⡉⠉⢖⡒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⡤⠀⡠⠊⠁⠀⠀⠀⠀⠀⢈⠉⠢⣄⠙⢄⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢰⠊⢠⠎⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠈⢦⠀⠍⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡤⠊⠀⡎⠀⠀⢠⠖⡚⠉⣻⢼⠛⡆⠀⠀⠀⠘⡆⠘⣄⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣸⠈⠀⡀⠀⠀⠀⢃⠉⠀⠀⠀⡞⠀⠀⠀⠨⠀⡆⢠⠊⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⠥⡐⠀⡇⠀⠀⠀⠘⠀⠀⠀⢰⠁⠀⠀⠠⠁⡸⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⣀⠜⢀⠘⡄⠀⠈⠐⡧⠀⠠⡆⠀⢀⠀⠃⡰⠁⠀⢀⡼⠆⠠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⠀⠀⠀⠈⠉⠀⠣⠀⡈⠢⢄⠀⠁⠠⢰⠀⠊⣈⡤⠊⠠⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠙⠆⠸⡀⠀⠀⠀⠀⠀⠐⠃⠉⠁⢠⡕⠙⢲⣾⠿⡉⠤⢄⣠⠖⠒⢼⠀⠀⠀⠀⠀⠀⠀⢀⠆⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⡠⠂⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠃⣈⣡⠾⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠌⢀⠆
⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣬⠋⢉⣩⣤⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠘⢀⠎
⠀⠀⠀⠀⠀⢈⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠄⠀⠀⠀⠀⠉⠻⣛⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠸⠈⡠⠋
⠀⠀⠀⠀⠀⠀⠑⠒⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣅⠠⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠠⠈⡠⠞
⠀⠀⠀⠀⠀⢠⠊⠉⠐⡄⠀⠀⠀⢹⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠥⢌⣊⠂⠔⠠⢂⠂⠄⡄⡰⠐⠀⠇⢈⡠⠄⠊
⠀⠀⠀⠀⠀⠈⠓⠒⠘⠀⠀⠀⠀⠘⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠠⠦⠀⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠬⠃⣀⡀⢰⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⡜⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢴⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣉⠒⢅⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⢩⢐⠅⡨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠫⢔⡨⣚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡪⣐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠎⣻⠅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢿⢻⡂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡘⡟⠸⠃⣽⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⢄⡠⢀⠄⡠⢂⠠⣂⢔⡠⢐⠠⢂⠄⡠⢄⡠⣀⠄⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢼⣅⣞⠷⡼⢧⣀⣷⢄⡠⣀⢄⣠⡤⠤⠀⠄⠄⠀⠀⠀⠁⠈⠀⠁⠀⠀⠉⠀⠁⠈⠁⠈⠀⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣤⠄⠤⣀⣈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣀⠐⠄⣤⣴⣤⡿⠃⠀⠉⠉⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⠤⠤⠤⠄⠀⠐⠒⠒⠊⠉⠁⠀⠀⠀⢸⡷⣿⢿⣋⣁⣬⠭⠿⠓⢙⣿⡏⠁⠀⠀⠂⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣤⢐⠂⢀⠂⢠⠤⠒⠂⣉⡀⠀⠠⠞⠛⣅⢹⣎⣻⢤⣤⠶⠶⠐⠋⠀⠁⣀⣄⡤⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣛⣿⣿⣿⣨⣤⠤⠖⠂⠊⠉⠁⠀⠀⢀⣠⣤⣶⠿⠿⠛⢿⡟⣎⣤⣤⡶⠾⠒⠛⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⢿⣟⠛⠉⠉⠁⢀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣅⢤⡀⢰⣶⣿⡧⡉⢉⣁⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠐⢰⣿⣟⣠⣔⣿⣾⣿⣿⣧⣷⣿⣶⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠴⠚⠉⣉⣰⣷⡿⠛⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣅⣴⠾⠟⠋⠉⠉⠉⠉⠉⠁⣠⣰⣶⣮⠿⠟⣀⠠⣠⢊⣵⠴⢲⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⣟⣯⢿⣿⣿⣿⣯⡀⠀⢀⣀⡠⣔⣤⣴⢾⣿⣭⣷⡶⠶⢦⣄⠙⡇⠻⣿⠋⣠⣷⣏⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⠼⠟⠉⠁⠀⠘⣿⣿⣾⣿⠦⣶⣩⢁⣀⣚⠂⣘⠿⢿⣿⣿⣷⣍⣁⢂⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⡩⡛⠵⣒⣿⣿⣾⣿⡿⢿⠘⣧⡌⠑⠪⡙⢿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢨⠙⠻⣿⣿⣿⣿⣿⣼⢈⣼⣈⣿⣿⣧⣄⡻⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⣶⣿⣿⣿⣿⣵⣿⣽⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠿⠛

⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⣀⣧⠀⠀⠀⢳⡇⠀⠀⠀⠀⢻⣧⣄⠀⠈⢿⡧⠤⠀⠀⠀⠀⣸⠀⠀⠀⢠⣿⡄⠀⠀⠀⣤⣯⡄⠀⠀⢠⣿⢆⠀⠀⠀⣠⣧⠗⠀⠀⠀⣬⣥⡀
⠀⠀⠀⢠⠟⠀⠀⠀⣾⣭⠁⠀⠠⠖⣹⠇⠈⠀⠀⢸⡇⠀⠀⠀⠀⣨⠿⠀⠀⠀⠘⣿⡟⠀⠀⠀⢹⢻⠁⠀⠀⢈⣿⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢿⢧
⠀⠀⠀⠈⠂⠀⠀⠀⠚⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠘⠀⠀⠀⠈⠀⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠈
⠀⠀⠀⣀⠀⠀⠀⣄⢠⡄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⢀⡄⠀⠀⡀⢠⡀⣀⠀⠀⣠⣀⢄⠀⠀⢠⡀
⠀⠀⢀⣬⣷⡤⠀⠈⢻⡿⠁⠀⠀⢩⣿⠯⠤⠀⠀⣻⣠⠄⠀⠀⠀⣾⠴⠂⠀⠐⠒⣶⡖⠒⠀⠀⢸⣿⡇⠀⠀⠈⣷⡞⠁⠀⠀⠑⣷⠎⠀⠀⢠⢻⣯⠃
⠀⠘⢫⠏⠀⠳⠀⢠⠋⠙⠤⠄⠀⡼⠙⡆⠀⠀⠀⠻⠚⠃⠀⠀⠈⢻⠆⠀⠀⠀⢀⡏⢇⠀⠀⠀⢸⢻⠁⠀⠀⠀⡏⢻⠀⠀⠀⠀⠋⡇⠀⠀⠀⡞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁
⠀⠀⢰⠠⠄⠀⠀⠀⣠⣄⠖⠀⢀⡀⢶⣀⡀⠀⠀⢀⣶⠀⠀⠀⢰⡀⡄⠀⠀⠀⠀⣶⣠⠀⠀⠀⠀⣦⠀⠀⠀⢀⣠⡆⢀⠀⠀⠀⡶⠀⠀⠀⠀⣦⡄
⠀⠀⠀⣿⣿⠀⠀⠀⣱⢿⠷⠀⠀⠉⢹⣿⠈⠀⠀⣿⣿⡑⠀⠀⠈⢹⡟⠃⠀⠀⠌⣿⠃⠀⠀⠀⢯⣿⡿⠀⠈⠉⣿⡋⠁⠀⠀⠸⣿⡇⠀⠀⠀⢹⣿
⠀⠀⠀⡗⡇⠀⠀⢠⠇⠈⠆⠀⠀⠀⠹⠇⠀⠀⠀⠈⠇⠃⠀⠀⠀⢸⢻⠀⠀⠀⠰⠿⠄⠀⠀⠀⠸⠸⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠟⠀⠀⠀⠀⢸⡏

⠀⠀⢽⡦⠞⠀⠀⠀⣬⡷⠂⠀⢀⡀⡆⠀⠀⠀⠀⠰⣿⡄⠀⠀⣀⣻⡴⠄⠀⠐⠦⣼⡴⠋⠀⢀⣿⣄⡀⠀⠀⢀⢴⣿⡄⠀⠠⢤⣯⠤⠄⠀⠀⠦⣼⡇
⠀⠀⢸⡟⠙⠄⠀⠐⠙⡗⢆⠀⠈⡽⢷⣾⣍⠀⠀⡴⢻⠈⠀⠀⠀⡴⢷⠀⠀⠀⢀⡿⢧⠀⠀⠘⣿⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⣿⠀⠀⠀⠰⠛⢿
⠀⠀⠸⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠙⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠈⠓⠀⠘⠉⠀⠀⠳⠀⠀⠛⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣀⢠
⠀⢀⣀⣣⣞⠀⠀⠀⡸⣿⡁⠀⠀⢀⣾⣦⡀⠀⠠⠴⣿⠖⠂⠀⠑⢮⡥⠤⠀⠀⢘⣶⣃⡀⠀⠀⢡⡿⡀⠀⠀⠐⢹⠧⠖⠀⢠⢾⣟⠁⠀⠀⠒⠒⣿⠋
⠀⠀⠀⣠⢻⡇⠀⠀⢠⢾⠁⠀⠀⢠⠟⢳⠀⠀⠀⢠⠛⢦⠀⠀⢀⡼⠳⡄⠀⠀⡟⢧⠀⠀⠀⠀⢹⢦⠀⠀⠀⢀⡟⢻⠀⠀⠈⠈⡏⢳⡀⠀⠀⠀⣿⡄
⠀⠀⠀⠘⠀⠑⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠘⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠛⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⠀⠀⠉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢒⣩⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠲⢭⣲⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⣚⡩⠴⠒⠋⠁⠸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠋⠙⠲⠤⣀⠀⠀⠙⠒⠭⣕⣢⢤⣀⣀⣠⠴⢒⣋⠥⠔⠚⠉⠁⡀⡠⢠⣶⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠜⠁⠈⠙⠢⢄⡀⠀⠀⠉⠒⠤⣄⣀⣠⠬⡷⠚⡇⡖⠊⠉⡀⠀⢀⠰⡖⠘⡅⢿⠄⠋⠂⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⠓⠤⣀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⡇⠀⠀⢸⠀⡇⡇⠀⢸⣽⠸⣸⠀⠃⠀⠀⢀⡠⠴⢚⣩⠴⠃
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⢸⠓⢦⡇⠀⠀⢮⠀⡇⡇⠀⠸⠎⠀⢀⣀⠤⠒⣋⠥⠖⠋⠁
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⡆⣠⠃⢠⠏⡇⠀⠀⢸⠀⡇⢇⣀⡤⠔⣊⡭⠔⠚⠉
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡴⡱⠃⢠⠏⠀⡇⠀⠀⢸⠀⣇⡨⠵⠚⠉
⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡜⡰⠁⢠⣇⡀⠀⡇⣀⡤⠚⠉⠁
⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡞⡴⠁⣰⠃⠀⠉⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⣰⠃⢀⠾⢜⡁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠓⠒⠋⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣤⡀⠀⠀⠀⠀⠀⣾⠁⠀⢶⠸⣇
⠀⠀⠀⠀⠀⠀⢠⡿⣁⠀⠀⠈⢻⣦⣀⠀⠀⠀⢹⡆⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠸⣇⠹⣦⡀⠀⠈⢁⡿⠷⣤⡀⢸⡇⠀⢸⡇⢹⡆⠀⠀⠀⣠⡞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡻⠦⠀⠈⠀⠐⠟⣻⣾⡇⠀⠀⣿⢸⡇⠀⠀⠺⣯⡀⢀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣜⠻⠂⠀⠐⠋⠘⣿⠀⠀⠙⠀⣿⠀⠀⠸⠌⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⣿⣀⡀⠀⠀⢸⡟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣤⡀⠀⠀⠀⠉⠳⣿⡏⢁⡀⠀⠘⣧⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⠁⠀⠀⠀⠀⠀⢸⡟⠋⠉⠀⠀⣿⡀⠀⢀⣠⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⣿⡲⣄⠀⠀⠀⢸⣷⠞⠛⠀⠓⢹⡇⢀⣾⠁⠰⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣿⠋⠀⠀⠀⢸⣧⣠⡤⠀⠀⠸⣿⢾⣧⠲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡳⡄⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠑⠃⠀⠀⠀⣿⢀⣤⠶⠶⢦⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⡴⠾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠈⠳⠆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⣶⣤⣄⣀⠀⠈⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⢤⣤⣀⡈⠑⠒⠤⢄⡀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠄⠄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠉⠢⠤⣄⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠠⠀⡀
⠀⠀⠀⠀⠀⢀⣦⣵⡁⣀⣀⡤⠖⠈⠀⠀⠈⠻⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⣻⣿⣿⣛⠉⠁⠀⠀⠀⡀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⣌⣿⡇
⠀⠀⠀⠀⠀⠁⡻⣿⡿⣀⠀⠀⠂⠁⠀⠀⠀⡰⠦⣤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⡞⢋⣀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠸⠇⠀⠙⠓⢾⡉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠠⢤⣤⢤⣿⣆⡹⣶⡟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡀⢠⣴⣺⡆
⠀⠀⠀⠀⠀⢀⣀⣰⣤⣭⣿⣿⣶⣯⡴⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣷⡉⣀⣵⣟⣀⣸⡗⠆⣲⣶
⠀⠀⠀⠀⣀⡬⡛⠉⢉⣻⡿⢿⣿⣝⠗⢶⡟⠁⠀⠀⠀⠠⡶⡶⢶⣶⣿⣿⡿⣿⣿⣿⣧⣯⣷⠃
⠀⠀⠀⠀⠙⠥⣤⣶⡟⠉⠀⣿⠛⠈⢳⡀⠈⠣⡀⠀⠀⠀⠘⢶⣾⠟⠁⡲⣿⠛⢿⡍⠧⢀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢀⢠⣿⠀⠀⢈⠷⠦⠒⠃⠀⠀⠀⠀⠀⣾⢖⣊⣴⡇⡄⡸⡻⠦⠤⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠁⠉⠓⠬⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⢦⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡋⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠟⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣼⣿⡇⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣇⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠧⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠂⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⢀⣐⣶⣤⣤⡟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⠀⠐⠒⠚⠛⠉⠉⢩⠏⠹⣦⡀
⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠻⣆
⠀⠉⠒⣄⠀⠷⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠈⠳⣀⠆⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣆⣠⡂⠀⢠⡟
⠀⠀⠀⣀⣤⡶⠟⠙⠿⣦⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡟⢻⣿⡰⡎
⠀⠀⠉⠁⠉⠀⠀⠇⠀⠈⣿⣿⣿⠀⣸⣿⣿⢿⣿⣿⡇⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⣿⣿⠀⣿⣿⣿⠘⣿⣿⡇⢠⣾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⣿⣿⣿⣴⣿⣿⡟⢀⣿⣿⣇⡿⠃
⠀⠀⠀⠀⠐⠒⣲⡶⠞⢾⣿⣿⣧⣼⣿⣿⡇⢸⣿⣿⡿⠃
⠀⠀⠀⠠⠶⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢀⠁⠈⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠈⠀⠀⣿⣿⡇⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠟⣛⣛⣛⣛⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣴⠟⢉⣴⠞⢛⣭⠙⠛⣶⡌⢻⣷⣀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⣠⡾⠋⣩⡴⠒⣶⣬⠙⢷⡄⠙⠿⣿⣿⣿⣿
⠀⠀⢀⣿⢿⣄⠘⠿⣦⣈⣛⣒⣛⣋⣠⡾⢟⣿⣷⠀⠀⠀⢸⣿⣿⣿⠀⡀⢻⣧⣀⠛⠷⠤⣭⠤⠶⠟⢁⡤⠀⠘⣿⣿⣿
⠀⠀⠘⣿⡀⠙⠷⣦⣤⣍⣉⣉⣉⣉⣥⣶⠟⢁⣿⠀⠀⠀⢸⣿⣿⣷⠈⣿⣦⣈⠙⠛⠷⠶⠶⠶⠶⠛⢋⣠⣾⠀⣿⣿⣿
⠀⠀⠀⠙⢷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⣠⣾⠋⠀⠀⠀⢸⣿⣿⣿⣦⡈⠿⢿⣿⣿⣶⣶⣶⣶⣶⣿⣿⠿⢁⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠻⠷⢶⣦⣤⣤⣤⣶⠶⠟⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣀⣈⡛⠛⠛⠛⢛⣛⣉⣥⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠈⡉⠙⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⣴⣬⣍⣛⠻⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠿⣿⡿⣖⣦⣬⣉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠉⠛⠛⠿⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣺⠿⠛⢉⣤⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣩⣴⣶⣿⠃
⠀⠀⠀⠀⠀⣠⣴⠿⠛⢹⣿⠸⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣈⡴⣞⣯⣿⣿⣿⣿⡇
⠀⠀⣰⣴⠿⠋⠁⠀⠀⢸⣿⡌⣿⣿⣿⣿⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡯⣗⡿⢟⣯⣿⣿⣿⣿⣿⢶⣤⣄⣀
⢠⣾⡋⠁⠀⠀⠀⠀⠀⠈⢿⣿⣦⣤⣛⠿⠭⢿⣗⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢐⣻⣽⣺⢽⣿⣿⣿⡏⣽⣿⠀⠈⠉⠛⠻⢶⣦⡤⡀
⠻⠿⢷⣤⣀⡀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣷⣦⣭⣛⠛⠿⠧⣤⣀⠀⠀⠀⠀⠀⢀⣇⠨⣟⡿⣺⢟⡿⣻⠿⣱⣿⢏⣀⣀⡀⠀⠀⢀⣿⡟⢈⣤⣀⡀
⠀⠀⠀⠈⠙⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣤⣌⡑⠢⠀⠀⠀⢸⣿⠈⣷⠿⣺⡵⢛⣥⣾⢟⣥⣿⣿⣿⣿⣿⣶⠟⣋⣴⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⠻⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣄⣸⣿⡄⠑⣛⣥⣾⠟⣫⣴⣿⣿⣿⣿⣿⠿⢋⣤⣾⣿⣿⣛⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠻⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⡿⠟⣩⣴⣿⣿⣿⣿⣿⡿⢛⣡⣶⣿⣿⣛⡷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠐⠒⠿⣿⣿⣿⣿⡿⠟⣩⣶⣿⣿⢿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⣀⣀⣠⣤⣶⣷⣿⠟⣩⣶⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠟⣫⣴⣿⣿⣿⣿⡷⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⣚⠇⠀⡀
⠀⠀⡀⠀⠀⡀⠖⠀⠀⠀⠖⠰⠀⠀⠀⠀⠀⠐⠆⠀⠀⠠⠀⠁⠀⠀⠠⠜⡒⠒⠀⠀⠀⠀⠀⠀⠀⡀⠦⡀⢀⠃
⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠂⠀⠀⠀⠀⠀⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠀⠀⠛⠀⠂⠀⠀⠀⠀⠈⠀⠉⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠈⠀⠀⠀⠁⠉⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢀⠠⠀⠠⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠁⠈⠁⠈⠃⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠈⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⢄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⡀⠀⠀⠄⠀⠀⠊
⠀⠀⠄⠜⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠱⠰⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⢠
⠀⢀⠐⠂⠀⠀⠐⠁⠀⠀⠒⠈⠀⠀⠀⡀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠄⠀⠀⠀⠖⠐⠀⠀⠂⠲⠀⠀⠀⠀⠀⠀⠰⠓⠶⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠐⠈⠁⠀⣶
⠀⠀⠂⢠⡄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠠⠄⠐⠀⢀⠀⠈⠀⢚⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠢⠀⣈⠁⠈⠁
⠀⠀⠀⠀⠉⠂⠠⠐⠤⠀⠀⠁⠀⠀⠀⠀⠂⠀⠂⠨⠜⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⢂⠀⡁⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡄⢀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠒⠁⠐⠅⠀⠀⠀⠀⠐⠀⠀⠍
⠀⠀⠀⣠⠤⣄⡀⠀⠀⣀⡤⢤⡀⠀⠀⠀⢀⠀⣀⠀⠀⡀⢀⠀⠀⠀⠀⢠⡀⡀⣀⠀⠀⠀⠀⠀⠀⣠⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⠞⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⢰⣣⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡴⠀⠀⠀⠀⠀⠈⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠁⢰⣶⣄⠀⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠻⠟⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⣀⣠⠄⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⣤⣀⣀⣀⣠⣤⣤⠤⠴⠖⠚⠋⠉⣀⠴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣶⡈⠃⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣤⣤⠤⠴⠖⠋⠁⠀⡠⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠹⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢻⡀⠙⣦
⠀⠀⠀⠀⠒⠲⠶⠤⠤⠤⢤⣤⣀⣠⠟⠳⢦⣤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣹⠄⢭⣽⣷⣞⣹⣶⡦⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠀⠙⠲⢶⠚⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠚⠻⣿⠏⠛⠓⠒⠒⠶⢶⣠⠴⢚⣉⣀⣀⣀
⠀⠀⠀⠶⣎⣉⠉⠀⠤⣄⠀⠀⠈⠑⠠⠤⠀⠀⠀⠐⠤⢄⣀⠀⠀⠀⣠⠤⠤⠤⣴⣉⣁⣀⣠⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠒⠒⠶⠦⠤
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣍⣈⠁⠀⠀⠤⠄⠀⡉⠁⠒⠠⠀⠀⠉⢘⢧⣄⣀⣠⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣄⣈⠙⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⠀⠀⠀⠀⢸⠆⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠷⣤⣤⣿⡄⠀⠀⠀⢻⠀⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⣠⣶⣿⣀⣀⣠⣽⡆⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⢙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢘⣿⣿⣿⠉⠉⠁⢸⣿⡿⠿⢷⡄⠀⠀⠀⢸⡗⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⣿⠏⠀⣿⠀⠘⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡄⠋⠀⣀⣿⠰⠀⠀⢸⣧⠾⢷⡄⠀⠀⠀⢠⡏⠀⢀⡏⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⣴⣿⣭⣿⠀⠀⠀⢸⣧⠀⠀⢻⣆⠀⠀⢸⡇⠀⢸⡇⠀⣀⣤⣾⣫⣥⣤⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣇⠀⠀⣠⣿⠀⠀⠀⢸⣟⣧⠀⠀⢹⣧⡀⣾⠀⠀⢸⣷⣿⡿⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⣠⣴⠖⠀⠀⠀⠀⠸⣿⠿⠟⠹⡟⠀⣰⣫⣿⠀⠀⠀⢸⡿⣿⡀⠀⠀⢻⣿⣟⠀⣠⣾⡉⠉⠛⣛⣷⠶⠶⢽⣷⡀
⠀⠀⠀⠀⠘⠛⠛⠛⠷⣭⣳⡄⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠿⠋⣿⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠙⣿⣹⢻⣟⣿⣴⠿⣿⡶⠶⢾⣿⡛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⣰⣿⣿⣿⣏⠁⠀⠀⠀⠙⠳⢦⣌⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣸⡄⠀⠀⣿⠀⠀⠀⢿⡇⠀⢀⣤⡾⠟⣿⠁⠀⣾⠉⠛⢦⣄⣀⣠⡤⠶⠾⠿⢿⣿
⠀⢀⣤⣶⣿⣿⣿⣾⣟⠛⠛⠋⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⢸⣷⣾⠟⠉⠀⠀⡿⠀⠀⡿⠀⠀⠀⡿⣿⠻⣿⣿⡛⠛⠉⠁
⠀⢛⣭⠾⣻⣿⣿⡟⠙⠛⠛⠶⣶⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⣿⠛⠛⠛⢻⡿⠁⠀⠀⠀⢸⡇⠤⢸⡇⠀⠀⢸⡇⢹⡄⠻⣎⠙⢦⡀
⣰⣿⡷⠛⢡⣏⣼⣃⣤⣤⣶⠛⠉⠁⠀⠈⠙⠳⣄⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠰⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⢸⠇⠀⠀⠀⣷⢸⡇⠀⠘⠷⣌⠻⣄
⠉⠁⠀⠀⣿⡿⢛⣿⡏⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢸⠇⠀⠀⣿⠀⠀⣿⠰⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠸⣾⡇⠀⠀⠀⠙⠳⣿
⠀⠀⠀⣰⣟⡴⢻⡿⡇⢸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⡄⠀⣀⣸⣦⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠛⠋⠀⣸⣷⡇⠈⢷⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠛⠋⢹⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣰⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⣠⣾⡿⣿⠿⠶⢶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣧⠀⣿⠀⠀⠰⢸⣿⣯⣅⠀⣿⠀⠀⢸⡷⠛⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠁⠀⠀⣿⠀⣿⠀⠀⠀⢸⡟⠷⢬⣿⣿⠀⠀⣼⣡⡴⢞⣛⣩⣭⣽⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⣀⠀⠀⢸⣷⣄⠀⣸⡇⠀⠀⣿⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢹⡀⣟⠉⠀⠀⢸⣛⡿⠟⢹⡇⠀⠀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠃⠀⢸⣧⣿⠀⠀⠀⢸⠏⠀⠀⣸⠃⠀⠸⣿⣦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⢀⣀⣸⡀⠀⠀⣿⢀⠀⢀⡏⠈⠛⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⢸⣿⡿⠿⠟⠛⣻⠇⠀⠀⣿⠈⠀⢸⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣤⣤⣤⣿⡇⠀⠚⠂⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢠⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⣿⣂⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠃⠀⠀⠀⣿⠀⠀⢀⡿⠛⠛⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⣿⢄⡀⠀⠀⣿⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠘⠂⠀⠀⣿⠀⠀⠸⠇⠀⠈⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⡀⠀⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⢤⣿⡴⠂⠀⣠⣿⣄⠀⠀⢻⣿⠁⠀⢸⣿⣄⠀⠀⣿⡟⠀⠀⣠⣼⣧⡀⠀⠈⣧⡼⠁⠀⢀⣼⡟
⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⡏⠀⠀⢸⢿⠀⠀⢸⣿⠋⠀⠀⣿⡇⠀⠀⠁⣿⡇⠀⠀⠀⣿⣧⠀⠀⠈⣿⡇
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠘⠙⠀⠀⠘⠛⠀⠀⠀⠛⠃⠀⠀⠀⠉⠃⠀⠀⠀⠉⠉⠂⠀⠀⠛⠃

⠀⠀⠀⢄⣀⡠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢸⣿⠁⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠐⢺⣷⠒⠀⠀⠈⠉⣿⡟⠉⠀⠀⠀⣀⣿⣄⠀⠀⠀⢴⣾⣦
⠀⠀⠀⡸⠛⢆⠀⠀⣼⡏⠉⠉⠙⣷⠀⠀⢀⡞⠙⣆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⠿⠿⠿⠤⠀⠀⣰⠋⢣


⠀⠀⠀⢢⣶⠀⠀⠀⠀⣶⠀⠀⠀⢀⣶⠀⠀⠀⠀⣶⡴⠀⠐⣄⡆⠀⠀⠰⣶⡶⠀⠀⢠⣤⠂⠀⠀⡆⡆
⠀⠀⠀⢸⣿⡷⠀⠠⢺⣿⡷⠄⠀⢿⣿⡗⠀⠠⢾⣿⡄⠀⠀⢸⣿⠢⠀⠀⣿⡇⠀⠰⣿⣧⠀⠀⠐⣿⡇
⠀⠀⠀⢈⣿⠁⠀⠀⠸⣿⠁⠀⠀⢸⣿⠁⠀⠀⢸⣽⠁⠀⠀⢸⣽⠀⠀⠀⣏⡇⠀⠀⢹⡏⠀⠀⡰⢿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣀⣀⣖⣀⡀⠀⠀⠀⡀⣀⢀⡀⠀⠀⠐⣄⠦⡴⠀⠀⢀⠀⣶⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⣰⡰⠂
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⣿⡏⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⢀⣿⡇
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠴⠾⠿⠷⠶⠀⠀⢠⠎⠉⢣⡀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠛⠛⠓⠒⠀⠀⠀⡏⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠁⠀⣠⣶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠂⠉⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠆⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣰⣿⡏⠀⠀⣀⣤⠶⠚⠋⠁⠀⢀⣀⣤⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠦⣤⣤⣤⣀⣿⢷⣄⡲⠈⠉⡞⢻⣷⠶⣻⣿⣟⠀⢀⣠⣴⣾⡿⠛⠉⠉⠀⠒⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠰⠶⠦⠤⠀⠀⢀⡀⠠⠌⠛⢧⣻⡙⢶⣦⠀⣸⢣⠞⠉⣵⣷⣾⣿⣽⣥⣁⣀⣀⡀⠀⠀⠀⠀⢤⡄⡐⢠⣶⣄
⠀⠀⠀⠀⠀⠀⣄⣀⠙⠻⠟⠛⠁⣠⣤⣤⣤⣬⣽⣗⡺⣿⣈⡼⣟⣿⣾⣓⠛⠛⠲⠋⠀⠈⠉⠉⠙⠛⠛⠛⠷⠿⠿⠗⣚⠻⡿
⠀⠀⠀⠀⣀⠀⠈⠛⠁⠀⠀⢀⣠⣿⣿⣯⣅⣈⣍⣿⣃⠯⣿⠗⣻⣍⡉⠉⠛⠛⢻⣶⣿⣷⣶⣶⣦⣤⣄⣀⡀⣀⡀⢸⣽⣫⠇⠈⠁⠀⣂⣀⡀
⠀⠀⢠⣬⡀⠀⠀⢀⣠⡴⠞⠉⠀⠀⢘⡿⣣⠿⠿⣫⡿⣼⠏⣷⡙⠲⣝⠿⢾⠇⠀⠀⠀⠉⠉⠉⠙⢛⡛⠻⠿⢿⣿⣦⣭⣵⣦⣤⢠⡾⢿⣿⠓
⠀⠀⠿⣦⡿⢰⡚⠋⠁⢀⡠⢀⠠⠔⣫⣾⣥⡾⠿⠋⣹⡇⣿⣿⣧⡑⠘⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣾⣷⠈⣉⣀⠀⠀⠀⢾⣯⠗⠒
⠀⠀⠘⢾⣷⣿⠇⠐⣀⣴⣦⣤⣤⣤⣿⣿⣿⣿⣷⣶⣾⣷⣿⣿⣇⣉⠓⠮⡝⠋⠙⠛⠻⠶⠶⢶⣶⣶⡶⠟⢿⣿⠀⠍⠉⠉⠁⠐⠈⠉
⠀⠀⠀⠀⠁⠀⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⢿⣿⣿⠥⠈⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⡾⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢒⠖⡒⠒⠀⠴⡒⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠽⢰⠀⠏⠔⠁⠀⡵⠊⡠⠀⠀⣸⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⢀⢩⠌⢹⡐⢛⠉⠉⣶⠢⢄⡀⠀⠀⠀⠀⠀⠀⡄⡁⡶⢠⡀⡁⠰⠋⠀⠀⠠⠃⠰⠀⣰⡎⠀⠀⠂⠤⡽⢦⡀
⠀⠀⠀⠀⠀⢀⠔⠀⠒⠆⠀⠈⠴⠀⠀⠑⡬⡀⠀⡆⠉⠀⠙⢦⡀⠀⠀⢀⠎⠀⢻⠀⠀⠀⣡⠂⢠⡒⠆⠘⠀⣧⠌⢀⠄⠀⠀⠀⣼⠫⣭⣷⡄
⠀⠀⠀⢀⠔⣀⡈⠈⠃⠐⠀⢀⠃⠀⠘⠒⠄⠃⠀⠁⡠⠀⠀⠆⠱⡄⠀⠚⣀⠃⠈⠀⢸⠁⠇⠀⡼⣣⡁⢁⠔⠁⡔⠁⣠⣢⠔⠁⠀⠘⠶⠦⢹⡀
⠀⠀⠀⠎⡈⡁⠐⠠⠄⢸⠀⡏⠀⠑⠆⠀⠀⢰⠀⢡⠔⠁⢄⠴⠀⠡⢠⢦⠙⢄⠀⢃⡈⢰⢠⣼⠃⡟⡠⠊⢁⡴⠃⠉⠉⠁⠐⠶⡅⠈⠛⢺⣇⡇
⠀⠀⠈⣰⠁⠓⠤⡆⠸⣚⢀⠇⠀⡼⠀⡄⢠⠀⠃⣘⢀⢀⡎⠀⢉⣈⢘⠈⣭⡂⢴⡄⠗⠲⢸⡗⣀⡱⠅⠊⠁⡤⠆⠀⠁⣆⠐⣀⡉⢛⠒⣿⣫⣷
⠀⠀⠇⢠⣈⠐⠂⠣⡈⠒⠫⡐⢤⡃⠀⠃⡆⠀⠀⢩⠄⢸⠇⡸⠟⢣⠈⡿⠄⢰⣉⣇⠀⢀⡶⣺⢟⡠⠐⠀⠀⠰⠚⠀⠘⣝⠦⡷⢣⠀⣀⣬⣭⣼
⠀⠀⣇⠔⠀⠉⢀⠈⠙⠢⢀⠈⢢⡙⢄⠳⣄⣜⡄⣜⡀⡎⢘⠡⢄⠎⠀⠘⢂⢙⣯⠉⣿⢛⡎⣿⡽⠦⠄⠐⠒⠲⠶⠤⣤⡄⠀⠖⠒⠦⢈⣛⡢⡿
⠀⠀⣟⠀⠘⢒⠤⠨⢦⡀⠀⠝⠒⠷⠄⢡⣌⡎⢉⠀⠸⠶⣋⠁⢌⠀⠀⢰⡧⢈⠚⢀⢙⣿⢶⢿⡄⣂⠀⠀⠀⠐⠪⡽⠂⠈⢳⠒⠤⢾⡶⠶⢶⡇
⠀⠀⠸⡀⠘⠋⣀⠀⡤⠬⡖⠦⢄⡰⠖⢠⢡⠛⢮⣿⣲⠂⠩⠀⠀⠑⠒⠊⣀⠁⣀⡠⠊⠈⠀⡇⠰⣌⠑⢤⡠⢤⢀⡜⠤⠁⠀⢩⣴⠒⠁⢲⠞
⠀⠀⠀⢱⡀⡀⠤⠀⠐⠂⡀⠖⠃⢀⠀⢳⠀⠕⠉⠏⣿⠀⣴⣷⣀⠴⠚⠛⠣⣀⠀⠤⠶⣧⣄⠀⠀⠀⢰⠊⡙⢦⡀⠀⠀⠀⠀⠸⠶⠞⣻⠋
⠀⠀⠀⠀⠱⡀⠄⣀⣀⡄⡤⠀⡜⢁⡴⠃⢢⠊⣭⠂⡿⠶⢋⣾⠁⠀⠀⠀⠤⠈⢢⡀⠀⠻⡙⣦⠘⠀⢀⠀⡇⠨⠠⣌⠉⢛⢁⠐⢂⠔⠁
⠀⠀⠀⠀⠀⠘⠪⣩⠃⠀⡠⠊⢉⣠⠖⠀⠇⠸⠃⢀⡡⠒⠈⣏⡀⠀⠀⠀⠃⠀⠀⠙⠮⡀⠹⢸⡇⠀⡘⢠⠟⢄⡤⠻⠉⢀⠤⠚⡀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⢿⣴⣉⢀⡀⠄⠄⠐⠋⠁⠀⠀⠀⠙⠳⣦⠀⡽⢽⠀⠀⣀⣀⣀⣀⣴⠿⢀⣑⣉⣩⠤⠔⠒⠉⠀⠀⠀⠀⢨⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣴⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠖⠚⠉⠉⠉⠀⠀⣀⣶⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠤⠤⠒⠒⠒⠋⠉⠉⠁⠀⡀⠀⣀⡀⠤⠤⠤⠀⣒⣋⣭⣭⣵⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠖⠊⠉⠉⠁⠀⠀⠀⡀⣀⣀⣀⢠⡄⠤⠤⠐⢒⣂⣊⣉⣩⣅⣶⠴⠶⠾⠿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢗⢛⡛⠓⣢⣉⣉⣩⣥⣴⣦⠤⠤⠴⠞⠛⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⡀⢀⣀⣠⣤⣤⣤⣤⣶⣲⣶⣖⠠
⠀⠀⠀⠀⠀⠀⠀⣀⣐⡀⣉⢡⣀⣠⣤⣤⣤⡈⣤⡭⠤⠤⡴⠶⠤⠴⠶⠀⠚⠋⠉⠉⠉⠀⠉⠉⠰⠳⢡⣄⢫⣷
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⢈⣠⡸⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠈⠾⣦⣄⣰⣀⣀⣀⣠⣄⡀⣠⣄⣀⣀⣀⣀⣀⣂⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⠿⠿⠿⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⠕⠖⠠⠄⠉⠥⠄⢀⣀⣔⣒⠂⠀⠤⠤⠄⠀⠀⠀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢇⣴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠛⠛⠓⠒⠒⠶⠶⠦⠤⠚⠾⠽⠭⠤⠭⠿⢄⢀⠀⣲⣶⣄
⠀⠀⠀⠀⠀⠀⠘⠏⠄⣐⣂⣀⡀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠿⠿⠿⠾⠿⠿⠧⠤⢤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣀⣲⣦⣤⣤⣤⣤⣴⣴⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⠉⠉⡉⠁⠀⡇⠀⡏⠀⠀⢸⡏⣡⡀
⠀⠀⠀⢀⣀⠄⠀⠠⠅⠛⣿⡆⣴⣏⡀⠀⠃⠀⡇⠀⠀⢸⡇⢃⡈⠳⢤⡀
⠀⠀⠖⠛⠉⠀⠀⠀⠀⢀⣿⠆⣈⠉⠹⠷⢲⡶⣧⠀⠀⢸⡇⠀⠀⡄⠏⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⣿⣴⡦⠰⡟⢀⣼⠀⡷⣾⣧⢰⠞⠁⡀⠀⠚⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡿⠋⡟⠳⢀⢢⣠⢸⣧⠐⠛⠉⠉⠼⠠⢨⢄⠄⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⣷⠠⠷⡀⠘⠳⢟⠃⠻⠠⣤⣄⡷⣴⣧⡾⠃⠀⡆
⠀⠀⠀⠀⠀⢠⡴⢾⠃⢀⣿⣀⣠⡇⠀⠀⡾⠀⢠⡀⠈⠻⡇⠘⠟⠁⠀⢀⠃
⠀⠀⠀⠰⢻⠀⡗⠈⠀⠈⣷⠅⠀⠀⠀⠀⠁⠀⢆⠘⣄⠘⡇⠀⠀⠀⠀⢈⢀⣇⡇
⠀⠀⠀⡄⢀⠀⡇⠀⠋⠢⣿⠦⣤⣤⡀⠀⡀⠀⠌⣣⢼⣤⣇⠀⢢⡀⠀⠉⢹⣿⠁⠒
⠀⠀⡜⠀⠘⠀⢉⠀⠀⠀⠺⡀⠈⢿⣤⢉⡟⢻⠏⠽⢶⠻⡇⠀⠀⠳⠀⢣⡜⠍⢰⡀⢀⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠃⠌⠀⠘⣦⣴⢃⠌⠀⢹⡇⠀⠀⠀⠠⣌⠻⠾⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⡀⠆⠀⡇⢹⠮⡀⠀⠀⡟⠳⠊⠈⠷⣌⠑⠂⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣏⣀⡗⠃⠀⡇⢸⡆⠀⠀⢀⣇⣸⠖⠀⠀⣀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠹⠘⣇⠀⠀⡝⢨⡇⠀⠀⠀⡇⢈⣀⢴⡉⠛⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⣨⠉⠀⠀⡇⠀⡇⠀⠀⢰⡇⡏⢠⠀⢳⡈⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡇⡇⠙⣆⠀⠀⡇⠀⣧⡆⠀⣩⠀⠁⠀⠀⠀⠣⠀⠙⠇
⠀⠀⠀⠈⠀⢀⠀⠈⠉⢳⡇⠀⠀⠀⠀⠀⣇⣴⣯⠄⠊⡡⠄⣴⣤⡤
⠀⠀⠀⠀⣊⠤⠂⠀⡠⠘⡇⠀⠆⠀⠀⠀⣯⡜⠃⢿⠋⠐⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣧⣠⣄⡄⠀⠀⣿⣯⡀⠀⠙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠁⡁⠀⠀⣿⢀⡇⠀⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡀⠀⠀⠀⠀⣿⢨⡇⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⠀⠀⠹⡆⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠉⠀⠀⢸⣧⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡇⠀⠀⢀⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⠆⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⣼⡄⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢸⡇⠀⣿⡇⡇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣆
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⣿⣿⣿⠟
⠀⠀⠀⠀⠈⢴⣶⣄⠀⢀⣠⣾⣿⣿⣷⣄⡀⠀⣠⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀⠀⢀⣽⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⢿⡟⠋⠸⢟⣿⣿⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣧⡄⢴⣿⡿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣿⠟⠀⠀⠀⠀⠀⠈⠛⠛⢹⣿⣿⣿⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠈⠋⠉⢋⣻⣿⣿⣿⣿⣿⡙⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⢠⣶⣶⣶⣶⣾⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⠛⣻⣿⣿⣟⣛⣥⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠞⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⣿⣿⣿⣿⡿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⢸⣿⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠛⠛⠛⠛⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠁⠈⠁⠀⠀⠒⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣦⠀⠀⣠⣴⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣛⣁⢀⠔⣠⣶⣧⡤⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⣭⣛⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⠿⠿⠿⠿⠿⣃⣀⣘⣿⣏⣀⣀⣀⣀⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣛⣿⣿⣾⣍⣻⣛⣛⣁⣀
⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣷⣷⣿⣿⣶⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⢉⣭⣿⣿⣿⣿⣿⣭⡉⠉⠉
⠀⠀⠀⠀⠀⠀⢷⣬⣿⣿⣿⣿⡟⢻⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⣿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢢⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⠀⠀⠀⠀⣀⣴⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣿⣿⣿⣿⣿⠏⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⣴⠁⢀⠃⠀⣀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣠⡥⠤⠚⠈⠁⠀⢀⣠⠤⠒⠛⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠛⠚⠧⢝⣦⡄⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠉⠁⠀⠀⠀⠀⢀⣀⠤⠿⣍⠁⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⡀⡤⣔⠶⣆⠁⢀⠀⠊⠀⢀⠔⠉
⠀⠀⠀⠀⠀⢀⡴⠛⠻⠶⣦⡀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⡀⢄⣪⠗⡏⠙⢆⡠⠻⣇⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⡠⠀⢠⠀⢹⣷⣤⣤⣶⠋⠁⠀⠀⠀⢀⠤⢒⠭⠒⠁⠀⠀⣣⠔⢋⠀⠀⣈⠋
⠀⠀⠀⠀⣂⡺⡏⠀⠐⠋⠀⢸⠋⠀⠀⢠⠠⣀⣠⠔⣊⠕⠊⠁⠀⠀⢀⡴⠊⠑⠀⢈⠰⠊
⠀⠀⠀⠌⠁⠟⢸⣕⠠⢀⣰⡇⠀⢀⠈⠀⢈⣩⣖⠋⠀⠀⠀⠀⣠⠖⠉⠀⠀⢀⠼⢯⡀
⠀⠀⠠⠀⠄⠀⣼⣧⢔⡩⠒⣇⠀⠀⠁⡄⡠⢄⠈⠱⣄⣤⡴⠞⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠓⣁⠾⣫⢶⠟⠦⢀⣼⢦⡀⠀⢧⣀⡰⠁⢰⡿⠋⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠉⠁⠘⢠⠏⢶⠀⠀⠈⢣⡈⣰⡶⠠⠖⠚⠻⡄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠐⠊⠀⠀⡀⡿⡯⠴⠴⠌⠀⠀⣷⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠒⠁⢸⠀⠚⠃⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⢤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⢀⠀⠠⠀⢀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⡄⠡⢀⣀⠜⢀⠎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠄⠀⠀⠘⢄⠈⠒⠤⠤⠔⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢣⠀⢆⡀⣀⠆⢠⠑⢄⠀⢱⠀⠀⠀⠀⢸⠀⠀⠀⠀⡔⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠤⠀⠠⠔⠁⡠⠃⠀⢸⠀⠐⠂⠀⠘⢀⠔⢡⠀⢆⡀⡀⠎⢠⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡎⠀⠀⠀⢸⠀⡆⠀⠀⢨⠀⠣⡀⠑⠤⠀⠠⠔⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⢄⠀⡇⠀⠀⠀⠘⠤⠃⠀⠤⠆⠀⠀⠈⡆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠀⠸⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠀⠠⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠀⠀⣼⣡⠄⠀⠀⠀⣠⡇⠀⡰⡿⣏⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠏⣤⣼⠀⠀⣿⠋⢁⣦⣄⠀⠈⢀⡞⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⡏⣀⣈⠉⠁⠀⠀⠏⢠⡮⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠐⡟⠘⠟⠁⠀⢀⡟⠀⠀⢲⢲⣤⣌⡻⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢨⡅⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⣸⡇⢸⡇⡀⡴⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⢻⡇⣾⣇⡳⣤⠀⡟⣀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⢿⠘⠉⠋⣸⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢦⢨⡇⣿⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣼⡇⣿⠐⠶⢹⡇⠀⢠⠺⣉⣰⠠⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡇⣿⠀⠰⣿⢰⠿⠢⠀⠒⠋⣀⡠⣮⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠄⢰⠃⠈⠀⢀⠔⠀⠀⢻⠛⠜⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⡿⠀⠀⠐⠁⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⡇⠀⠊⠀⡸⠻⠻⡄⢠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⢊⠉⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠉⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⠈⢣⡀⠀⠀⠈⠐⢄⡀⠤⠄⠒⠀⠂⠁⠀⠀⠀⠀⠀⡠⠄⠠⠖⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢷⡠⠀⠀⠙⢎⡀⠀⠀⠀⠀⠠⡀⣀⠀⠀⠀⠐⠈⡀⠀⠀⠀⠀⠀⣠⣤⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⢖⡐⠀⠉⠀⠀⠀⠀⠈⢦⡀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⢠⣤⠶⠟⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠎⡄⣺⡿⣶⡄⠄⢀⣀⣤⡄⠄⠒⠚⠙⢄⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠐⠶⠒⠋⠁⠀⣠⠔⠚⠉⡀⢰
⠀⠀⠀⠀⠀⠈⠀⠀⠸⡰⠿⢃⢞⣛⠫⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⢀⣀⣛⣂⠡⡀⠈⢛⡇⢩⠟⣷⠀⠀⠀⢀⣀⡤⠴⠒⠋⠳⣄⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠐⢄⡀
⡈⠐⠙⠋⢤⡴⠿⠣⣌⠐⠄⠉⠓⢞⣡⡥⠶⢎⠉⠁⠀⠀⠀⠀⠀⠈⣣⡂⠀⠀⠀⠙⢮⡀
⠀⠀⠓⣾⠏⠔⣹⣗⢸⠅⡍⢉⡽⢋⣤⠖⣶⣦⡈⠄⠀⣀⡀⠔⠀⠁⠀⠈⠪⡲⣄⠀⠀⠙⢦⡀
⠀⠀⠠⠔⠃⠢⠀⠤⢋⣜⣀⠠⠁⣾⢨⡀⠠⠾⠇⡾⢋⠁⠀⠀⠀⠀⠀⢀⡠⠞⠪⡁⠀⠀⠐⠙⢦⡀⠀⠀⠀⠀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠹⡄⢱⠦⡄⠫⢟⡇⠠⠀⠀⣀⠤⠚⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⠲⣤⣀⠀⢀⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣤⡽⠿⢁⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⢬⣿⣏⢨⡻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣔⡁⣯⣉⣏⡷⣶⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠻⠿⠯⠔⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠺⠿⠿⠿⠿⠿⠿⠟⠛⢛⠛⠓
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢰⣿⠋⠿⠉⢹⣿⠃
⠀⣸⣿⣠⣾⣰⣷⡆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠇
⠀⣽⣿⣿⣿⣿⢻⡄⢸⠀⢸⠇⠀⠰⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠘⡿⢹⠉⠹⠛⠀⠃⠈⡆⣸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠇
⠀⡇⠈⠀⠀⠀⠀⠀⠀⣧⣿⠌⠀⣀⡠⠀⢠⡏
⠀⡇⠀⠀⠀⠀⠀⠀⡄⢹⣿⡖⠉⠉⠀⠀⣾⡅⠀⠀⠀⠀⠁
⠀⡇⠀⠀⠀⠀⠀⠀⢿⠀⠙⡌⠳⣄⠀⣸⣿⠅⠀⠀⠀⡀
⠀⡇⡄⠀⠀⠀⡆⠀⠸⣶⠀⢻⠀⠈⢳⣿⣿⠀⢀⡀⢰⠃
⠀⣇⡇⠀⠀⠀⡇⠀⠀⠿⠀⠈⠄⠀⣾⣿⣿⣦⣿⣧⡿⠀⠀⠀⠀⠀⡼
⠀⡏⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⠇⢠⠎
⣰⣇⣷⣆⣶⣿⡇⠀⢠⠀⠀⠀⠀⣿⡟⠉⠛⠻⠃⠿⠀⠀⠀⠀⠀⡟⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁
⢹⣿⣿⣿⣿⣿⠱⢀⡇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠠⡇⠀⠀⠀⠀⣰⠀⠈⠀⠀⠀⢀⡀
⢸⣿⢿⣿⡿⣿⢀⣄⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⠀⢀⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠒
⢸⣏⠈⠋⠀⡇⢸⣿⠀⢀⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⣠⡶⢋⠁⠀⠰⠋⠀⠀⠀⢠⣄⠀⠀⣀⡄⢐⣥⣀⡈
⢸⡏⠀⠀⠀⡅⠀⣿⡞⠁⢠⢠⡟⠀⠀⠀⠀⠋⢀⡼⠋⡴⠀⡀⢀⡆⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠁⠀⠉⠙⠒⠤
⠈⡇⠀⠀⢰⡇⡀⠹⠀⢀⠋⣼⠁⠀⠀⠀⠀⠀⠊⣡⠊⢀⠌⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢸⡇⠀⠀⠀⡇⠁⠄⢠⡎⢰⡇⠔⠀⡘⢀⡆⠀⡰⠁⣠⠎⠀⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠄
⢸⡇⠀⠀⠀⣇⠸⣧⠟⢠⣿⣷⣹⣿⣿⣿⠃⡔⢁⠀⠁⠀⡼⠁⠀⠀⠀⠀⠱⠀⠀⠀⡴⠃
⢸⡇⠀⠀⠀⣹⠀⢹⠀⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢀⡜⣡⠀⢀⠀⠀⠀⠀⢀⡴⠚
⢸⣤⢠⢹⡀⣿⡇⠈⠀⣿⠿⠃⠀⣿⣿⡏⠀⠀⠀⠀⠊⢰⠇⢀⠋⠀⠀⣠⠴⠋
⢾⣿⣿⣶⣿⣯⠀⢸⣴⡏⠀⠀⠈⢯⣿⠁⠀⢀⠆⠀⠀⣾⣠⠉⠀⠠⠺⡟⢿⣿⠦⠤⣤⡐⣦⣤⣤⣤⣀⡀
⢰⣿⡟⠿⣿⣿⠃⠸⡏⠁⠀⠀⠀⢸⠇⠀⠀⡀⠀⠀⠀⣹⡅⠂⠁⠀⠀⠹⡄⠀⠑⠠
⠀⡏⠁⠀⠀⠀⡀⢠⠇⠀⠀⠀⠀⣼⠀⠀⡔⠀⢀⠠⠂⠀⠀⣠⠖⠀⠀⠀⠙⢦
⠀⢃⠀⠀⠀⠀⢃⠘⠀⠀⠀⠀⠀⡏⢀⡞⢀⠶⢋⡀⠐⠘⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⢔⡡⠴⠋⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⣴⡿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠲⠤⢤⣀⡀
⠀⠀⠀⠀⣀⡤⠒⠋⠁⢀⡰⠚⠛⠒⠒⠚⠛⠛⠛⠻⠯⠭⠭⠭⠭⣝⣒⣒⣤⣤⣀⠀⠉⠙⠲⠤⣀
⠀⢀⡤⠊⠁⠀⢀⠴⠚⢩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⠦⡀⠀⠈⠑⢤⡀
⢠⠊⠀⠀⢀⠜⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣈⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠣⡀⠀⠀⠑⡄
⡇⠀⠀⠀⡎⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢱⠀⠀⠀⢸
⢇⠀⠀⠀⢣⠀⠀⠀⣗⠒⠒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⡠⠞⣼⠀⠀⠀⡜⠀⠀⠀⡸
⠈⢢⡀⠀⠀⠳⢄⠀⢫⠻⢍⣉⣛⣒⣒⣒⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⢺⠀⡠⠎⠀⠀⢀⡔⠁
⠀⠀⠉⠢⣄⠀⠀⠙⠲⢭⣒⡲⠦⠤⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣛⡭⠗⠋⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⡀⠉⠉⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠉⠉⢀⣀⡤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⣿⣿⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠁⠛⠛⠋⠙⠿⣿⠟⢻⣿⣿⢿⣿⠋⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠛⣷⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⠟⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠛⠛⠛⠋⠁⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⣤⡀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠉⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⣿⡿⠟⠉⠈⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉
⠀⠀⠀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁
⣴⣾⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣶⣤⡀
⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠇
⠈⠻⣿⣿⠃

⠀⠀⠀⠉⠛⠻⢷⣶⣦⠶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⣿⡳⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⠿⢿⣯⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⢩⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠂⠿⣛⠿⣿⣿⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣴⡏⡄⢳⠈⣽⣯⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⢂⢰⠩⢇⡧⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢀⠉⡁⡂⠐⠆⢧⣭⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠐⢈⢆⡡⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢼⡄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢉⣠⣫⣜⠪⡝⣿⣿⣿⣿⣿⣟⡿⣦⣀⢄⡑⠶⠀⢃⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡽⢷⣾⣇⣳⣿⢻⣿⣿⣿⣿⢿⣿⣿⣯⣻⡍⣄⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢉⢻⣿⣿⣿⣟⣿⣹⢃⡾⢿⡦⢹⣿⣿⣿⡹⡯⠠⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⢀⣀⣭⣿⣿⣯⡛⠋⠛⢿⣿⣿⣾⡿⡟⣏⣾⣿⢗⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣿⣁⠞⠁⠉⠻⣿⣶⡀⠘⣿⣿⣿⡿⢿⢿⡏⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⣧⡀⠀⠈⠀⠉⠛⣿⣄⣹⣿⣿⣿⡿⡟⡱⣀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⣻⣤⡀⠀⠀⠀⠌⠛⣿⣿⣿⣿⣧⣯⠕⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠿⣿⣯⡻⠖⠀⢀⠀⠀⠈⠋⠻⣿⣿⡟⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⢿⣷⣄⠀⠣⠀⠀⠁⠀⠀⠈⢟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠃⢀⢰⡦⣄⣀⠙⢿⣿⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢝⠪

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣿⡆⢫⠀⠀⠀⢸⠀⠀⢺⣼⡇⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⡇⠸⡖⣿⣷⠀⠑⠀⠀⠸⡄⠀⣬⢿⡃⠀⢀⣀⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢾⣇⠀⠀⢹⣿⠀⠀⠃⠰⣶⡇⢀⣿⣎⠃⠀⠀⡬⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡇⠀⢤⣤⣿⡆⠀⠀⠀⢸⡇⠈⣿⣿⠁⠀⠰⣥⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣈⠀⢸⠁⣿⡇⠀⠀⠀⠸⡇⢠⣋⣅⠀⠀⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠈⣿⠀⠈⠀⠹⡇⠀⠀⠀⠀⠇⢼⣿⡉⠀⢰⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⢸⠀⠀⢻⠀⠐⠶⢶⣿⠀⠀⠂⢲⣷⢸⣿⡇⠀⠘⢻⡟⠀⠀⠀⠠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠰⠀⢦⣀⣀⣠⡀⠀⠃⢘⣿⠀⠀⠀⠀⣿⢸⣿⡇⢠⣇⣸⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣟⠃⠀⡄⣈⣻⠀⠀⠀⠀⣿⢸⣿⠅⣸⢻⡿⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⠇⠉⡄⡋⣿⣿⢇⠀⠀⠀⠻⣿⣿⡀⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⠝⢱⡇⢸⡏⢺⡖⠀⠰⣾⣿⣿⡿⡇⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠠⣾⣧⣤⠁⢋⣴⠀⠀⣿⣷⣿⠎⣿⣿⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣀⠔⢨⣿⡿⠀⣀⠞⢀⠀⠸⣿⡏⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣾⣆⠀⡀⠀⠙⢻⣿⣷⣶⣿⣶⡞⠀⠀⢹⠓⠜⣫⣿⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣧⣠⠟⠋⣼⣿⣿⣿⣿⡇⠀⠀⣿⠆⣨⣿⣿⣿⣟⣉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠐⡾⢿⣿⠋⣿⡟⠠⠠⢿⢿⣀⣛⣿⣿⣿⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⣰⣿⣿⣄⣲⣏⡁⠈⣡⣄⠈⢻⡿⠋⠁⣼⣿⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣠⡀⠓⢤⣶⡿⠋⠉⣿⣿⣿⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣥⣢⣀⡀⠛⣁⣐⣶⣿⣿⡟⠋⠋⠉⠈⠈⠁⠉⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⠙⠉⠉⠉⠉⠁

⠀⠀⠾⠆⠀⠀⠀⢾⠆⠀⠀⠀⠲⠂⠀⠀⠀⠰⠂⠀⠀⠀⠰⠂⠀⠀⠀⠺⠆⠀⠀⠀⠰⠆⠀⠀⠀⢾⡷

⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠃⠀⠀⠀⠈⠁⠀⠀⠀⠘⠃⠀⠀⠀⠈⠁

⠀⠈⣶⡎⠀⠀⠀⣶⡆⠀⠀⠀⠠⠄⠀⠀⠀⣶⡆⠀⠀⠠⢼⠦⠀⠀⠀⠠⠄⠀⠀⠀⢴⡆⠀⠀⠀⢰⡆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠿⠿⠻⢿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠦⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣀⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠉⠉⠉⠛⠛⠶⠶⣤⣄⣀
⠀⢀⣤⠶⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣾⠉⠀⠀⠈⠉⠉⠉⠉⠉⠙⠳⢦⣄⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⣰⠟⠁⠀⠀⠀⠀⠉⠙⠻⠶⣤⣄⡀⠀⠀⠈⣻⠄⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠳⣤⣄⣀⣀⠀⠀⠀⠉⠛⢦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣾⣏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠀⠀⠀⠀⠀⠙⣷
⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⡷⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠉⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡼⠛⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⣀⣀⣀⣀⣤⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢶⣦⣤⣀⡀⠀⠸⠏⢠⡄⠀⣠⠀⠀⠈⠉⠛⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⢠⡿⠏⠉⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⣠⠿⠙⠛⠷⣤⡾⢧⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⣀⡾⠋⠀⠀⣠⠟⠀⢀⣀⣀⣀⣠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣼⣋⠀⢠⡞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣀⡀
⠀⠀⠀⠀⢀⣴⠏⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢶⣄
⠀⠀⠀⣠⡞⠁⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏
⠀⠀⣼⣏⡀⠀⠀⠀⠉⠛⠓⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠇⠀⢠⡾⠁⠉⣹⡟⠻⠶⠶⠶⠶⠶⠚⠛⠉
⠀⠀⠹⣏⠙⠛⠳⠶⣤⣄⣀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣰⠟⠀⠀⣴⠋
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣈⠙⠛⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⣠⠞⠁⠀⢀⣼⠋⠀⢠⡾⠁
⠀⠀⠀⠀⠹⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣤⣀⡀⠀⠉⠛⠶⠖⠛⠁⠀⠀⣠⡞⠁⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⡀⠀⢀⣴⠏⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⠁⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⠻⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣤⠾⠁

⠀⠀⠀⠐⣧⡀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⢀⣴⣄⢀⣴⠆
⠀⠀⠀⠀⠈⢷⣝⣋⡀⠀⠀⠀⠀⠀⢀⣬⣭⣄⠀⠀⠀⠀⠀⢀⣠⣭⣤⡀⠀⠀⠀⠀⠀⢀⣭⣥⡾⠃
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣦⠀⠀⠀⣴⢿⣿⣿⣿⢷⡄⠀⠀⢰⣟⣿⣿⣿⣻⡦⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠘⠁⣼⣿⣿⣿⡄⠛⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠛⢁⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢹⡏⢻⡏⠀⠀⠀⠀⠀⠉⣿⢹⡏⠀⠀⠀⠀⠀⠈⣿⠉⣿⠁⠀⠀⠀⠀⠈⢹⡏⣿⠋
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠘⠃⠘⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃⠙


⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⣰⣶⡀⣰⡗
⠀⠀⠀⠐⠶⠶⠶⣶⣶⣶⠶⠶⠶⠂⠀⠀⠀⠀⠻⣦⠠⣾⡆⢠⡿⠁⠀⠀⠀⠀⠀⠈⢷⣬⣭⣼⠏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣿⣿⣧⣤⣤⣤⠀⠀⠀⠀⣠⡟⠉⠉⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠈⠿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣄⠀⣜⣦⠀⢴⣤⡶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠿⠽⠆⢹⡿⣀⣱⠃⡀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣦⠀⠛⠫⠓⠂⠸⠟⠹⠗⠋⠸⠃⠀⠀⠀⠀⠀⠴⣷⡖⢰⠆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⣹⠟⠀⠀⠀⠀⠀⠠⢶⣦⡀⠀⠀⠀⠀⢿⢿⡶⠒⠉⢹⡷⠋⠀⠀⡶⣦
⠀⠀⠀⠀⣠⡎⠀⠘⣋⣁⣤⠄⠀⢔⡆⠀⠀⣠⠀⠻⠀⢀⣰⠆⠈⠦⣳⣤⣤⠞⠁⠀⠀⠀⠙⠃
⠀⠀⠀⠠⠀⢀⠀⠀⠧⠴⠀⠀⠀⣀⣤⡔⠈⠉⡠⡶⢠⡼⠁⠀⢰⠲⣄⢠⠆⢠⣤⠖⣲⣄⠺⠃⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⣶⡆⠀⠀⠐⢿⠿⠃⠘⠀⠉⠁⠀⠀⣄⡀⠀⠉⡭⣤⠀⣿⢲⣿⣿⣽⠆
⠀⠀⢀⣠⣤⣀⠀⠀⣠⠄⠒⠒⠢⡀⠠⠀⠀⠀⠀⠀⠀⠘⠧⠇⠄⢸⡴⠋⠀⠘⢮⡯⠿⠋⠀⢀⡀⠀⡄
⠀⠀⣿⣿⣾⣽⠄⣄⠁⠀⠀⠀⠀⠈⠀⢀⡀⠠⣤⠴⠃⣀⣀⡀⢀⠈⣤⠀⡶⣦⡀⢠⠀⢠⣖⠧⠕⢀⢤
⠀⠀⠹⣿⡯⠞⠀⠙⠷⣤⣄⣀⡄⠊⢀⠈⠁⣀⠠⢀⣸⢐⡤⢁⠎⠰⠂⠀⠈⠻⣿⠈⡄⠀⠉⠀⠀⠙⠊
⠀⠀⠀⢀⠀⠀⠰⠆⢰⡄⠈⠡⠀⢠⣽⣶⠐⠁⠀⣀⣄⡀⢀⠟⠀⠀⢀⡦⠄⢀⣤⣤⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣤⣶⠦⠛⢻⡟⠃⠀⢶⣻⠿⠅⠄⠀⣀⠄⣀⡢⢄⠀⣿⣶⣻⠄
⠀⠀⠀⠀⠀⠐⠻⡻⠏⠉⠉⡃⠁⢧⡼⡗⠩⡄⠀⠈⠓⠂⣠⣤⠒⡄⣥⠿⣦⢃⢹⡮⠟⠀⠔⠀⡰
⠀⠀⠀⠀⠀⢠⡄⠀⢠⣄⠸⠕⠐⣀⡈⠀⢀⠀⠀⠀⡠⠚⠉⣿⠀⡀⠍⠉⠉⠁⠄
⠀⠀⠀⠀⠀⠈⣿⡆⠀⠈⢳⡀⠀⠈⠓⡈⠁⣤⡖⠁⠀⢀⣴⡏⢀⣀⢰⡷⠒⠂⠰⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠀⠁⠀⠀⠉⠟⠐⠊⠙⠒⠛⠟⠋⠀⡑⠖⠀⠀⠁⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠊⠙⡶⠆⢀⡐⠑⠀⠀⠀⠸⣟⡆⢀⣡⡶⠃⢰⣶⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡾⠓⣷⠸⢍⡆⠀⠀⠀⠀⠀⠀⠊⠁⠀⠰⠶⣭⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠟⠁⠀⠐⠁⢼⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡝⠀⠀⠀⢀⣤⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣿⡿⠿⢻⠿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠠⠄⠀⠁⠀⠠⠀⠀⣾⣿⡿⠿⠿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠙⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⢠⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⢀⠠⠀⠀⠰⠃⠀⠀⠀⢀⠄⠀⠀⠀⠜
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡶⢶⣄⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠿⠀⠀⠈⢳⣟⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡻⣇⠀⠀⠀⠀⠀⠀⢰⣟⣿⣧⡴⠾⠛⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣏⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⣿⡀⠀⠀⠀⠀⠀⠀⠉⠁⠹⣦⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠿⠧⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡟⢦⡀⠀⠀⠀⠀⠀⠀⣀⣈⠙⠛⠛⢋⣀⡀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣧⠀⠙⢷⣄⠀⠀⣠⡼⢻⡏⠛⠛⠛⠋⠉⣿⣦⡀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⠶⠟⠀⠀⠀⠉⠛⠛⠉⠀⣾⠀⠀⠀⠀⠀⠀⢻⡈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠉⠉⠉⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡾⠿⣶⠄⠀⠠⣦⣴⡿⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣛⡀⠻⢦⣤⠀⢠⣿⣁⡈⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠉⠉⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣦⡋⠷⠀⢠⠆⠢⢈⣅⡞⠽⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡧⠈⢈⡇⠀⠰⠄⢸⠀⠀⠈⠉⠀⠀⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢥⠀⠀⠉⠁⠀⠀⠀⡏⠀⠀⠀⣀⡤⢒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢶⠀⠀⠀⠀⡀⠘⠞⢀⠡⠈⠐⠗⠋⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠁⠀⣠⠏⠀⠀⣠⠤⠀⢀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡜⠛⠸⢦⠀⠀⠀⠀⠀⠺⠀⠀⠐⡠⠊⠀⠀⣀⠈⠓⠒⠈⣀⣉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⠶⠻⡴⠷⠀⠋⢸⡰⠂⠀⠀⠀⠀⢀⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠙⣆⠀⠙⠈⠆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠑⠢⣠⢄⠀⠀⠀⠀⠛⢩⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡜⠃⠙⠁⠀⠀⠐⠠⠤⠤⡤⢤⠀⠀⠞⠁⠀⠀⠀⠀⠀⠑⠢⠴⠤⠏⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠄⣄⠀⠀⠀⠀⠀⠠⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⢃⡖⠋⠳⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⣀⢤⡀⠀⡀⠀⡞⠑⠀⠇⠀⠀⠀⠾⠋⠓⠒⢤⢄⣘⠁⢈⠋⠀⠰⠇⠁⠠⠐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣡⠞⠀⠐⠀⠸⠀⡴⠀⠀⠀⠰⠢⡀⡇⠀⢸⠀⠀⠉⠛⢓⠀⠤⣀⠀⠀⢠⠄⠐⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡁⣠⡴⢻⠇⠰⢧⡄⠀⢀⡀⢀⡙⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠸⡆⣠⡸⠙⠎⠀⠀⠀⠀⠀⢸⠃⠀⠈⠉⢢⠀⠀⠻⢝⢦⢙⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠃⠀⠀⠀⠀⠀⠀⠻⡬⠃⢀⠀⠀⡆⣀⡠⢀⡽⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢦⠦⡇⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠛⢸⡅⢸⡇⢠⠠⣄⢠⣤⢤⡄⢠⡦⠠⣤⣠⢀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⢸⡇⣼⢚⣿⠀⣿⢨⡿⢸⡇⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠀⠈⠁⠈⠁⠉⠉⠉⠀⠉⠘⠁⠈⠉⠀⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠂⢠⣉⠉⠉⠉⠉⠉⠉⣀⠒⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣮⣁⢀⢀⢂⢀⡄⣀⠩⠤⠶⠤⣠⣴⠂⠘⠃⠀⠠⠀⠀⠈⡑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡠⠒⢁⠀⢀⠀⠀⠀⣐⣀⠜⠉⠀⠀⠀⠀⠀⠀⠡⢀⡀⠀⠀⠁⠵⠖⠀⠛⢄⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⡠⠊⣀⡀⠄⠈⢸⠇⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠄⠀⠀⠀⡴⠁
⠀⢀⠌⠀⢈⡉⠀⢠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠿⡶⣤⡀⠐⠛⠂⢠⠀⠀⠀⢀⡜
⠀⠎⠀⠀⡘⠓⠰⠁⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣼⡎⠻⣦⡀⠀⠀⠀⠀⢀⠾⡀
⡘⠀⠀⠌⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⣶⠿⠿⠛⠁⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠀⠈⠓⠀⠀⠀⢠⠏⠀⢣
⡇⠰⠟⠀⠀⠀⢻⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⡋⠀⠀⢠⠀⠀⠀⣠⠇⠀⠀⢸
⡇⠀⠀⠱⡶⠀⢀⠙⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠿⠋⣁⣀⣵⠀⠀⠾⠀⣠⡶⠃⢀⡶⠀⢸
⢡⠀⠀⢐⠐⠀⠘⡄⠀⢿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠀⠀⠀⣼⡟⠀⢀⠢⠀⠀⡄
⠀⢆⠀⣸⠄⠀⢀⠘⢄⠘⣿⣏⠛⢿⣿⣷⣦⣤⣤⣾⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠁⠔⠀⢠⣾⡟⠐⣶⠒⠂⠀⣴
⠀⠈⢦⡉⠀⠈⠋⠁⠀⠑⢿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠿⠋⠀⢀⣏⡀⢀⡼⠃
⠀⠀⠀⠱⣄⠀⠀⠱⡀⠀⠀⣛⠛⠛⠭⢁⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠋⠀⣰⡿⠛⠁⠒⠉⠉⡡⠋
⠀⠀⠀⠀⠈⠑⢦⣀⠿⠀⠘⠁⠖⠠⠀⠀⠀⢈⠉⠀⠀⠀⠀⠀⠉⢉⠀⠀⠀⢀⣼⡿⠇⠈⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠈⠰⣄⠀⠚⠱⠡⠀⠀⡀⠀⠠⠋⠈⠀⢠⣾⠿⣆⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣀⣈⣀⣀⣙⣂⣠⣈⠀⣰⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠹⢿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠇⠀⠀⠀⠉⢀⡠⢊⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠂⠀⠂⠀⠀⠘⠀⠠⠐⠈⠀⠀⠀⠀⠀⠁⡸⠀⠀⠀⠀⠀⠀⠈⠉⠢⠀⠚⠋⠀⠌⡀
⠀⠀⠖⠄⢤⠀⠰⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⢀⣀⠀⠄⠊⠀⠑⠀⠄⣀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠑⡀⠀⠆⠀⠰⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⡠⠂⠀⠀⠀⠀⠤⠄⢐⠀⠀⠁⠀⠈⠀⠂⠠⢀⣀
⠀⠀⠁⠀⠀⠠⠁⠀⠀⠀⠀⠠⠁⠀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠊⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠘⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⢋⠀⡰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠒⠂⡁⠠⠒⠂
⠀⠀⠀⣀⡀⠐⠡⠀⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⡀⠀⠀⠀⠀⠀⢁⢤⡵⡊
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠊⠀⠈⠃⠀⠀⠈⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠠⡀⠐⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠐⠠⠀⠠⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢣⡈⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠉⠐⢄⠀⠀⢀⠀⠬⢶⠄⠀⠀⠀⠀⠀⠡⢥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⠂⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⢢⠊⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⢠⡏⣄⢈⡑⠀⠁⠒⡀⠉⠀⠀⠀⠀⠀⡧⠀⢀⠀⠀⠀⠀⠀⠀⡁⠒⠈⠀⠀⣴⠀⡎⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠄⢠⢀⢁⠀⠑⠄⠀⠀⠀⢈⠙⠀⠀⠠⠀⠀⠉⠀⠰⠏⠀⠀⠀⠀⠀⠉⠀⠈⠀⠁⠘⠀⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠒⡐⠀⠀⠀⠀⠀⢐⠀⠚⠈⠥⠲⠉⠘⠀⠀⣐⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠛⠉⠀⡄⠀⠀⠀⠀⠀⠠⡀⠀⠈⠙⠋⠉⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉
⠀⠀⠀⠀⠀⠀⠚⠀⣂⢀⠀⡀⠠⠹⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂⠄⠤⠀⠀⡰⢄⠀⠀⠠⠞⠀⠠⠠⠠⢀⢠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠒⠒⠊⠙⠩⠍⠉⠲⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⣒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠒⠒⡋⡩⡥⠆⠈⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠐⠦⢄⣀⡀⡇⠀⠀⠀⠀⠀⠀⠢⡄⠈⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀
⠀⢠⠒⠢⢤⣀⡙⢆⠀⢀⡀⠀⠈⠁⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠉⠷
⡤⠓⠦⣤⣀⡀⠈⠉⡛⠢⠄⣉⠀⠀⠀⠀⣀⠞⠑⠦⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠟⣋⣤⡴⠚
⠙⢶⣤⣙⠻⢿⣷⣲⡾⢥⣀⡀⢹⠛⠒⠶⠾⣷⣤⡀⠈⡕⠶⣆⣇⠀⠀⠀⢀⣐⡥⠤⠒⣚⣉⡥⠴⠺⠛⢉⠜⠁
⠀⠀⠳⣍⠛⠲⢤⣍⡛⠩⠄⣉⠓⠦⣄⣀⠀⣀⠉⣹⣦⣀⡠⠼⠟⢒⣊⣩⠥⠴⠒⠻⠉⢁⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠑⠲⠤⣀⡉⠑⠲⠤⣍⣑⡒⢚⣛⣛⣉⣩⡤⠴⠶⠚⠉⠉⠁⣀⣠⠤⠔⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣄⣈⠉⠉⠉⠉⠀⠀⣀⡈⠤⠔⠒⠋⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠭⠤⠄⠀⠀⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⡀⠤⠄⣀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠐⠂⠑
⠀⠀⠈⠀⠤⠤⠤⠤⠤⣤⣀⣠⣄⣀⣈⣀⣀⡀⠀⢀⡀⠂⡀⠐⠀⠀⢀⣀⣀⡤⠄⠐⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⣏⣡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠉⠉⠉⠉⢉⡉⠠⠍⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠐⠂⣁⣠⣿⠻⢎⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠬⠍⠛⠛⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣈⣁⣄⣤⣴⡶⠛⠛⠛⠻⠛⢾⣦⠈⣭⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠄⠀⠐⠒⣂⣀⣁⡠⠔⠦⢴⣶⣶⣿⣿⣟⣉⠅⠀⠀⠀⠀⠀⢠⡾⠙⠼⠿⠀⠑⠁
⠀⠀⠀⠀⠀⣁⣀⣀⣁⣠⢤⠤⠤⠤⠤⠶⡶⠾⢞⣿⠩⣭⡉⢇⣀⡠⢛⣨⢥⡍⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠭⠿⣆⠁⡬⠁⠀⢲⣍⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢀⣷⡘⠊⠻⣶⠶⠿⠿⣶⣄⣨⡉⢐⡒⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⡷⠀⢀⣼⡆⠀⠀⠀⠀⠀⠉⠉⠉⠁⠃⠰⠶⠤⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠁

⠀⠀⠀⠀⣄⣀⡀⠀⠠⠤⣴⣶⣾⣿⠿⢿⣷⣶⣶⣶⣤⡀
⠀⠀⡀⠀⠀⠉⣛⣿⣶⣦⣼⡀⠀⣦⢰⣿⡇⢻⣿⢿⣟⠻⢧⡀
⠀⠀⠐⠀⠘⠋⠉⠉⣩⣴⣿⣷⣦⡇⢸⡏⡁⠀⠹⠄⠉⠣
⠀⠀⠀⠀⠀⠀⠀⡴⢟⣿⣿⠿⣿⡇⠘⠉⠙⠀⣀⣠⣤⣤⡤⠤
⠀⠀⠀⢀⠀⡀⠈⠀⣼⣿⠏⠀⢿⡇⠀⢀⣰⣿⣿⡯⠭⠄
⠀⠀⠀⠸⠀⠱⠀⣸⣿⡧⣀⠀⣾⠀⣴⣿⣿⣏⠻⣷⣄
⠀⠀⠀⠀⠡⠈⣴⠟⠀⠀⠀⠀⢻⣼⣿⣿⣿⣿⠀⠀⢉⣣⣤⣤⣤⠄
⠀⠀⠀⠀⠀⢱⠛⠀⠸⠀⠀⠀⣸⣿⠏⢹⠙⣿⡇⢀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠋⣤⡄⣀⠀⠀⠀⡿⠁⠀⠀⠀⢸⡇⢼⡟⣿⣿⣧⠙⠳⡀
⠀⠀⠀⠀⠀⢀⣾⢳⣿⢿⣦⡰⠁⠀⠀⠀⠀⠈⣷⢸⣷⠘⣏⢻⣧
⠀⠀⠀⠀⠀⣿⡟⢸⣿⡘⣿⣿⡀⠀⠀⠀⠀⢠⡿⠀⡟⠀⠹⡄⠙⡄
⠀⠀⠀⠀⣸⡿⠁⢸⣿⣇⠘⣿⣧⠀⠀⠀⣄⣸⡇⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡿⠁⠀⣜⣿⡇⠀⠈⠻⡄⠀⠀⡏⠀⠆
⠀⠀⠀⡼⠁⠀⠀⡏⢻⡇⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠃⠀⠀⠀⠁⢸⢿⢀⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠶⠗
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⡗⠤⢄⠀⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠐⡄⠀⡇⠀⠀⠀⠀⠀⣴⣟⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠔⠚⠂⠀⡇⠀⡆⠀⠀⣸⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠉⠉⠹⠀⣸⣿⡟⢿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠃⡆⠀⠀⢠⣿⣿⣿⡄⠙⢧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢀⡁⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢘⣇⢠⣀⣸⡃⠀⣼⢹⣿⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠋⢩⢐⠀⠀⡇⠀⣿⠀⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⠀⢰⠢⣿⠀⠃⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⡇⠀⢸⠀⠰⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡆⢠⠀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⠄⠀⠀⠀⢈⡀⣎⣀⠆⠈⠛⠶⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠁⠀⠀⠀⠀⠐⠄⢠⣀⣀⡨⠛⠒⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹
⠀⠀⠀⠀⠀⣀⠤⠤⣄⠉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠁⠀⠀⠉⠓⠄⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠰⣄⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⠁⠶⢦⡀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣤⡰⣒⠤⠈⠐⠀⠀⠘⣿⣈⣹⠒⠤⣤⠔⠒⠒⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⢳⡀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣉⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣆⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣄⡀⠀⠉⠉⠉⠉⠙⠛⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣌⡻⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⣴⠿⠛⠉⠀⢠⣾⡿⢻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣿⣄⡀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣧⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢸⣿⡇⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⣠⣶⣿⣿⣿⣿⠃⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢸⣿⠃⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⣡⣿⣿⣧⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠸⣿⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⢠⡾⢻⣿⣟⠻⣷⣄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⣆⠀⠀⣸⡏⢸⠀⠀⠙⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠟⠀⢸⡀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣿⣾⣇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣿⣟⡀⠀⠰⠶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠟⠛⠛⣹⣿⣿⠀⠀⢿⣿⣿⣏⣼⡿⠟⠛⢿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⣠⣾⣿⠿⣿⣧⢰⣿⣿⣿⡏⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⣿⡿⠋⠰⣿⠙⣿⣿⣿⣿⠁⠀⢀⣼⠿⣿⣿⠹⣿⡏⢿⣿⣧⡀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⣿⠀⠘⢿⣿⣿⠀⣠⡾⠃⢸⣿⡇⠀⣿⡇⠀⠹⣿⣿⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣶⡏⠀⠀⢸⣿⠁⠀⢹⣿⢀⣴⠟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⣿⣿⢿⣿⣦⡀⢸⣿⠀⠀⢸⣿⠟⠁⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠙⢿⣿⣾⡏⠀⣠⡾⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠀⠀⠙⣿⣇⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⡇⠀⠀⢠⣿⡇⠀⠀⠀⠀⠶⠾⠿⠿⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣿⣿⡇⠀⠀⢸⣿⣧⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠁⠀⣠⣿⣿⡏⣸⡟⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⣰⡟⢻⣿⣿⠏⠀⠸⣿⡌⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠏⠀⣿⣿⡏⠀⠀⠀⢻⣇⠀⠀⠉⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⣿⣿⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠘⠿⠿⠿⠇⠀⠐⠿⠿

⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⣿⣀⣱⣾⣠⣤⣼⡧
⢀⣿⠀⠀⢸⡇⢀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣽⠉⢹⠏⣀⠀⣘⡁
⢸⣿⣦⠀⣼⡤⠸⡄⢀⣿⠁⠀⢸⡇⠀⠀⠀⢰⣯⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⣿⢀⠂⠀⠈⠑⢿⠇
⢘⣿⠃⠀⢿⣇⣀⣻⣾⣃⣀⣀⣼⣧⠀⠀⠀⠈⣿⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⢿⠁
⠨⣿⠀⠀⢨⡟⠛⢹⡟⠉⠉⠉⣿⡗⠀⠀⠀⠀⣿⡀⠀⠀⣿⡖⠐⢄⢀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼
⠀⣿⡀⠀⢸⡄⢀⠎⠈⠙⢦⡀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⠇⠀⠨⡻⣧⡆⠸⣿⠀⠀⠀⠀⠀⣿⡄
⠀⣿⣿⠀⢸⡇⠊⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠠⣿⠳⠀⠀⢸⡄⠒⠀⠈⠻⡀⠀⣿⠀⠀⠀⠀⠀⣾⡗
⠀⣿⡏⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠁⠀⠀⣼⡄⠀⠀⠀⠀⠸⣄⠏⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣿⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⣿⣃⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡥⠀⣼⡄⡰⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣿⣼⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡿
⠀⣿⡇⠀⣿⣿⠅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣁⡀
⠀⣿⡇⠀⣾⡯⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡅⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⠃
⠀⣿⡇⢀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡗⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⣯⣁⠀⠀⠀⠀⣿
⠀⣿⣧⠎⢹⠀⠀⠀⠀⠀⠀⠀⢸⣭⡀⠀⠀⠀⣿⣿⡀⢠⣿⡖⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢻⡀
⠀⣿⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣀⣠⣿⣹⣧⣞⣿⠤⠤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇
⠀⡿⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⣯⠛⢻⠛⢿⠷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⣁
⠀⠂⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠿⢋⠉⢇⢸⡆⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⡏
⠠⣦⡤⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠐⢱⣾⡏⠀⠈⠾⠟⠀⠀⠀⠀⠀⢀⣏⡀⠀⠀⣠⠀⢸⠇
⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⣿⠸⡄⢰⡏⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠾⠛⠋⠉⠉⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢹⡇⢸⡇⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⣿⡈⣷⠀⠀⠙⢷⣄⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠉⣻⡆⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠙⢷⣄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣾⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠙⢷⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠻⠶⠛⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⢷⣄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠇⠀⢀⣤⠾⠛⢷⡄⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠹⣦⣴⠟⠁⠀⠀⠘⢷⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢷⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠙⢷⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠾⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠒⠀⠒⡇⠀⢴⠂⠀⢠⠚⠒⣲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠤⠤⠖⢒⠖⠉⠀⠀⠠⠀⠢⣤⠼⣂⣾⡤⠴⠃⠀⠀⠀⠈⠻⠯⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⡄⡀⣀⠀⠀⠁⠀⠈⠐⠒⠤⠀⢀⡀⠀⠀⢀⡩⠛⠓⠶⠶⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⡆⠃⠸⠀⠀⡄⢀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠤⠄⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣀⣀⣀⣰⠉⠈⠉⣶⣄⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠉⠒⣶⠛⠉⠉⠉⠀⠀⠠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣠⠞⠉⠀⠀⠈⠉⠳⠄⢀⡈⠛⠿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠂⠄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⡇⠀⠄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠑⢦⣀⠀⠀⠀⠀⠀⢸⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠈⠳⢆⡆⢰⢀⠆⠐⠁⠊⠀⠀⡠⠀⠀⢀⡨⠥⠶⠶⠬⠿⣵⡢⢄⡀⠀⢸⠠⢊⠜⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⠀⠈⠛⠜⣦⡆⡀⠀⠀⢰⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢙⢷⣽⣷⣾⣀⠈⠀⠀⠡⢀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠖⠚⠋⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣼⣴⣸⠸⡔⠦⣤⡀⠈⠓⢤⡀⠀⠀⠹⣿⣷⠈⠉⣳⠦⣄⡁⣁⣈⡤⠤⠶⠒⠚⠋⠉⠉⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣝⣒⠃⠋⠀⠀⠀⠀⠈⠒⢤⣽⡷⠶⠿⠧⣤⠀⠉⠛⠦⣄⡒⠐⠒⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⢓⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠉⠚⢽⡳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⡀⠀⠈⠑⠻⢟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⣤⡉⠳⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣓⣚⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣤⣀⡄⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠈⠑
⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡼⠃⠐⠂⠀⠀⠀⠀⠐⠂⠀⢷⣼
⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⢀⣀⣀⡀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠁⣌⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠈⠻⠟⠃⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠛⢬⡓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢢⣤⣤⠀⢠⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⡇⢸⣇⠀⠀⢸⠛⠛⡟⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⠷⣿⠉⢻⣿⠉⣳⠞⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⠐⠷⣤⡀⠀⠀⠀⠀⠀⡇⠀⢀⣀⠀⠀⠈⠉⡇⠈⠙⠋⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉⠳⣄⠀⠀⠀⠀⡇⠀⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠙⢧⡀⠀⠀⡃⠀⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠹⣦⣰⡇⠀⡙⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠻⠃⠀⣿⣿⠀⠀⠀⠠⡇⣸⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠸⣟⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡇⢈⠀⠀⠀⠀⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⠤⠤⣼⠧⠼⠓⠀⠰⠖⠚⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣤⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇⢈⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢶⡶⠶⢾⡇⢰⡷⢶⡶⠶⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⢀⡗⣈⠇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣈⣛⡿⠄⠻⢶⣦⣜⣻⠿⠄
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠰⣯⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢧⡀⠀⠀⠀⠀⠀⣴⡿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢦⡀⠀⠀⠀⠙⠿⠿⢿⣷⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⠀⠀⠀⠀⠀⠸⣿⣤⣾⢇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠄⠀⠀⠀⠀⠈⠉⢡⣿⡋⣠⣶⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⠙⠿⠟⣱⣿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠘⢿⣼⡿⣡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠸⣿⣅⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⢾⣿⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣷⣽⣷⠀⢹⣿⢷⢆⣄⠀⠀⠀⠀⠀⠀⠈⢹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⢿⠃⢀⣾⢏⣴⡶⣶⠄⣀⠀⠀⠀⠹⣏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠸⣿⣤⠀⣼⣿⣴⡶⠄⠀⠈⠳⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠿⠛⣿⡀⠀⠀⠀⠀⠈⠳⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢻⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣤⣥⢿⣓⣤⠀⠒⠒⠉⠈⠉⠁⠈⠁⠀⠀⠀⣀
⠀⢀⣀⡤⠴⠀⠀⠀⠀⠀⠀⠤⠄⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠒
⡔⠋⢤⣆⣽⡷⡄⠀⠌⠺⣟⢋⣙⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡏⢹⢱⡆⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠂⠀⢀⡠
⠣⡄⡈⢩⣿⣣⡿⣶⡀⣠⡼⠘⠙⠳⢸⠀⠀⠀⢀⠀⠀⠀⢀⢂⣞⣀⡇⡿⣗⡶⠓⠊⠉⠁⠀⠀⡀⠀⠄⢀⣀⡤⠴⠋
⠀⠀⠐⢩⣶⠶⢥⡀⠑⠀⠰⢦⡠⣾⣟⣀⣀⣠⡤⠴⠒⣺⠭⠈⠋⣹⠕⠀⣨⠀⣀⠀⠠⠀⣀⣤⠔⠒⠛⣉⠄
⠀⢀⠠⣟⣶⠴⣛⠻⣆⠀⡶⣛⡁⡼⣁⣈⣉⠀⠀⠀⠀⠁⠀⠀⣴⡣⠁⠐⣸⠿⠖⠀⠉⢉⠀⠀⠀⠈⠁
⠀⢸⣷⢍⣹⣟⢿⣷⣹⢸⢸⢟⣛⠘⠛⠙⡟⣇⢀⣀⡤⠀⠀⢀⣩⠴⠒⠛⣛⡩⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢀
⠀⠀⠣⠘⡿⣁⢰⠿⣣⢾⣮⡳⠾⢦⣙⣛⡥⢇⣀⣠⠤⠒⠉⣁⠤⠔⠂⠈⠉⠀⠀⠀⠀⢀⠀⠀⠠⠐⠂⠀⠀⠀⠀⢀⠤⠄⠔⠚⢉
⠀⠀⠉⠛⠛⠅⣴⠿⣛⣛⣷⢮⢓⠒⠛⠗⠂⠉⢁⠀⠄⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠄⠀⠐⠀⠀⠀⠀⣀⣄⠤⠖⠋
⠀⠀⠀⠀⠀⠘⢿⠿⣩⡿⢷⣬⡇⢀⡠⠴⠒⠉⠀⠀⣒⠂⠀⠀⠀⠀⠀⠀⣀⣀⠠⠖⠒⠁⠈⠀⠀⠀⠀⢀⣀⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠉⠁⠀⠀⠀⠐⠀⢹⠟⣿⢄⣤⡴⠚⠂⠉⠀⠀⠀⠀⠀⢀⣠⡤⠔⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢭⡎⣶⣦⠀⠀⠀⢀⣀⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠓⠷⣫⠤⠒⠊⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠽⣿⣿⣿⣿⡿⠿⠿⠿⡷⠶
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⠀⠀⠖⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠈⠋⢠⡀⠀⠀⠀⠀⠈⣠⣦⡀⠀⠀⠀⠠⣼⡦⠀⠀⠀⠀⠀⠀⣳⣾⠀⠠
⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢨⠹⠋⠀⠀⠀⠀⢀⠈⠋⠠⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠐⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣴⣄⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢴⣷⠀⠈⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠋⠀⣀⡀⠰⠂
⠀⠀⠀⠙⠁⢀⡀⠀⠀⢀⣠⣧⡀⠐⠃⠀⠀⠠⠀⢀⡀⠻⠀⠀⠀⠀⠺⡿⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠠⠆⠀⠀⠙⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠰⠖⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⠀⠀⠀⠂⠈⡀⢀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈⠁⠀⠉⠏⠁⠀⠀⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠉⠀⢀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠢⣄⠀⢀⠀⠀⠀⠀⠀⠒⠉⠐⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣇⢀⡀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢢⡈⠙⠺⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠗⡄⠀⠀⠀⠀⢀⣠⠄⠒⠀⠀⠀⠴⠗⢀⣀⡀⠴⢂⣉⠽⣆⠠⢀
⠀⠀⠀⠀⠀⠀⢐⠀⠉⠀⠂⠌⢛⣶⣄⡀⠐⠂⢤⠔⠊⠉⠁⠀⠀⡄⠀⠀⡀⠉⢠⣾⠀⠀⠀⠀⣀⠀⠀⠀⠀⠚⠉⠁⠀⠀⢸⣗⡀⠑⣄
⠀⠀⠀⠀⠀⠀⣀⠁⠢⡀⠐⠠⠄⢀⡽⢻⣷⢄⣬⡀⠀⠀⠀⢀⡀⠃⠀⠀⠱⠄⠙⢿⣄⠀⠀⠀⣼⠀⠀⠀⣀⣀⣤⠤⠖⢚⡩⠵⠖⠛⠁
⠀⠀⠀⠀⠠⠀⠈⠉⠉⠙⠳⢶⣤⣄⣑⣀⣙⣿⡀⠉⠓⠶⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠒⠤⠿⣒⣢⣭⣭⠴⠖⠙⠉⠁
⠀⠀⠀⠀⢀⡉⠉⠉⠒⠶⠤⢄⣜⣀⣀⣭⡏⠀⠈⠀⠀⠀⠀⠨⣑⣢⡄⠀⠀⠀⠀⠀⠈⠐⠒⠛⠛⠋⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠢⠾⠞⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⢉⡈
⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⣤⣬⣝⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢘⠀⠀⠀⣀⡀⠀⠀⢀⠴⠚⠉⠛⢓⡤⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣇⡆⠀⠀⡼⢋⣠⡙⣿⡛⡌⠀⠀⠀⠀⢠⠞⠋⠉⢖⣄⠣⢤⡴⠀⡜⠁⠠⠖⠁⠀⠀⢠⠋⠀⠀⠀⠀⠁⠴⡚⢷⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠙⢷⣦⡄⡗⠾⠿⡗⢘⠳⢿⡄⠀⠀⢀⡇⠀⠀⢠⣾⣶⠃⢀⣧⢞⡁⠀⠀⣠⠀⠀⠀⡏⠀⠀⠀⠀⢀⣴⠟⢿⣸⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢌⠒⠂⠐⠁⠀⠀⠀⠠⢚⠋⢧⡀⢀⡿⠜⠃⣠⢞⠛⠋⠓⠒⠉⢁⣸⠀⠀⣇⠀⠀⠀⣀⣾⠉⠀⡰⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢉⡛⠷⢘⣋⣥⠶⢒⣚⣙⡉⠉⠉⣿⠀⠀⠘⣧⢠⡆⢹⡟⠟⠋⠀⡀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠒⠒⠲⠂⠀⢀⢭⣀⣠⠤⠦⠬⠷⠒⠒⠛⠁⠀⠀⠀⠈⠛⠻⢼⣷⣍⣰⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠴⠦⠤⠖⠋⠉⠉⠁⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠊⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⣃⠤⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠤⠚⠉⢀⠤⠂⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⢀⠤⠊⠁⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡠⠚⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣀⠔⠋⠀⠀⠀⣠⣴⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⡠⠞⠁⠀⠀⢀⡠⠞⢡⢏⠎⢆⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡠⠊⠀⠀⠀⢀⡴⠋⠀⢀⡏⡜⢀⠈⢆⠀⣀⠴⠒⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡠⠊⠀⠀⠀⢀⣴⠿⡒⠠⠤⣞⡀⡇⠀⠃⢸⡞⡡⠺⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣀⠀⠀⣠⠊⠀⠀⠀⠀⡰⠋⡞⠀⠱⡀⢤⠤⣉⠻⣜⠀⢸⠰⠑⣡⠞⢉⣉⡉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡸⡡⣢⠲⣳⣜⡁⠀⠀⠀⡠⠊⠀⡼⠀⠀⠀⢹⠦⣉⣒⡁⠈⢣⣌⡠⠞⠑⠊⠁⠐⣪⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠻⢾⣡⠒⡍⣆⢠⠞⠀⠀⣰⠁⠀⣀⠤⠛⢉⠤⠈⣉⡩⠜⠟⢫⠉⢉⠁⠚⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠉⠔⣣⠞⡿⢂⢐⢦⣇⣜⡛⠢⡤⣄⣁⣠⠟⣁⠄⢸⠀⡀⢳⡈⠣⣉⠢⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠸⣠⢱⣤⣿⡔⠐⡜⡶⠀⠀⢠⢃⠎⡱⠁⣼⢰⠙⡀⠉⠑⠂⠠⠤⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣈⣩⣜⠴⠁⠀⠀⡜⠀⢉⠠⠊⠘⡌⢆⠃⡘⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠚⠀⠉⠀⠀⠀⠀⠘⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⣴⣿⣷⣶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣤⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠘⢿⣿⣃⢻⣆⠀⠀⢀⣀⣀⣀⣀⣀⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⢷⡄⠀⠈⠳⣤⣀⡀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠀⠀⠀⣼⢃⣶⣶⡌⢷⡄⠀⠀⠀⠀⠀⢸⠧⣾⣷⣼⡇⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡻⡿⠟
⣀⣀⣀⣀⣀⣀⣘⣿⣟⣀⣠⣤⣤⣤⣼⣯⣤⣽⣿⣥⣬⣿⣤⣤⣤⣤⣤⣾⣤⣽⣯⣴⡷⢤⣤⠤⠤⠤⠀⠀⢰⣿⣿⣿⡷⣄
⠉⠉⠉⠉⠉⠉⣿⣿⣿⡏⠉⠉⠉⠉⠉⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⠏⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⢈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⣇⠀⠀⠀⢀⣤⠾⠋⠉⠀⠈⣿⠀⠀⠀⠀⠙⢃⡀
⠀⠀⠀⠀⠀⢠⡿⠁⡘⣷⡀⠀⠀⠀⠀⢠⡾⠁⡀⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⢁⣀⢻⣆⠀⠀⠈⠁⠀⢀⣠⣄⠀⣿⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⢠⡿⠱⣿⣿⠞⢷⡀⠀⠀⢀⡾⠡⣿⣿⡆⠻⣆⠀⠀⠀⠀⢠⡿⠱⣿⣿⠇⠹⣦⠀⠀⠀⠀⠘⢿⠿⠁⣿⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣬⣿⣋
⠰⠶⠖⠲⠻⠷⣶⣾⣷⣶⠾⠿⠶⠶⠾⠷⣶⣾⣿⣶⠶⠿⠶⠶⠶⠖⠚⠳⣶⣾⣿⣿⠒⠛⠛⠛⠛⠛⠛⣿⣿⣿⡟⠛⠛⠛⠛⠛⠃⣀⠀⠀⣿⣿⣿⠟⢷⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠐⣿⠛⠛⠿⣿⠋⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⣰⡟⠻⣧⠀⠀⠀⠀⠀⠀⣰⡿⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⣧⠀⠀⠀⠀⠀⠀⢀⣼⠛⠻⣧⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠸⣧⠀⠀⠀⠀⣰⡟⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠘⣧⡀⠀⠀⠀⢀⣾⠃⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠘⣷⠀⠀⢰⡟⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠘⡷⠀⠀⠀⠾⠃⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿
⣿⣿⡟⠛⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠛⠛⠛⠛⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣠⠚⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠃⠀⠀⠀⠀⠀⣈⣳⠤⠖⠚⠉⠁⠀⠀⣷⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⢀⣀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡧⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢳⠀⠀⠀⠀⣠⠏
⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁⠀⣠⠞⠀⠀⡤⠚⠁
⠀⠀⣼⠉⠀⠀⢀⣀⡤⡴⠒⠋⢩⠞⠓⢤⣀⠀⢀⣠⠤⠖⠋⠉⠀⠀⠀⠀⣠⠎⠁⠀⠀⣸⠁
⠀⠀⠈⠒⠒⠋⠉⠀⢰⠃⠀⣰⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠙⠒⠲⠶⠦⠤⠤⣄⡀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠉⠓⠢⠤⠔⠊⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣠⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣀⣴⠤⠴⠒⠚⠃⠭⠭⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠟⠋⠀⠀⠰⠂⠀⠘⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣵⠎⠀⠐⠇⠀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠞⡐⠛⠋⠀⠀⠆⣠⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⢀⡜⠀⣶⠀⠀⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡈⢇⢀⠀⣴⠦⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⣌⠁⠘⡈⠀⠠⠄⢻⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠍⢼⢃⡀⠀⠀⠁⠀⢴⠀⠸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣜⢈⡁⡀⠶⠀⠀⢀⠀⠀⢸⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣟⠘⢡⡅⠶⠀⠀⠈⠰⠻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠓⢴⠰⠤⠉⢄⠀⢢⢲⠈⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⠀⠁⠲⣝⠑⠦⠬⠥⢴⣽⣧⡰⣦⡄⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⡈⠆⢱⠜⡆⠃⠂⢴⣠⠿⠷⣝⢧⣀⣄⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠹⣜⠛⠁⣴⢛⣛⢦⡿⡀⣙⡛⠂⣩⠉⢥⣄⡐⣦⠀⣀⠀⢄⣀⢀⣤⢀⣺⠶⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⡍⣼⣿⠇⠀⣂⢠⣶⠀⠁⢆⡤⢬⣾⣢⣄⡉⢀⣨⡜⣵⡏⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⢤⡱⢮⡀⢀⠸⠟⠀⣥⢀⣌⠈⢐⢱⠂⠰⠇⠈⠑⡟⠐⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢖⢬⡖⠭⣓⠆⢤⡤⢀⠈⠀⠠⠀⠀⢸⣰⣇⣰⡾⡑⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠹⠛⡋⠉⠓⢶⡬⢬⢒⣒⢒⣓⣪⢭⡭⠆⡘⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠸⢑⠲⠂⠠⠝⠐⠀⠀⠀⢠⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣩⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿