Hard hat Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⣭⣽⣷⣶⣾⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡭⢒⡭⠞⢋⡵⣋⡵⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠖⣁⡴⠋⢀⡴⢋⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠜⢁⡴⠃⠀⢠⠟⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠋⣰⠋⠀⠀⣰⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢣⠃⡼⠁⠀⠀⢰⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢇⢇⡾⠁⠀⠀⢠⡏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣸⡼⠁⠀⠀⠀⡾⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⣿⠃⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⣿⣿⣿⢃⡏⠀⠀⠀⠀⡾⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿
⣿⣿⣿⣿⣿⣿⢸⠃⠀⠀⠀⢠⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⣿⣿⣿⣿⣿⢏⣼⠀⠀⠀⠀⢸⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢯⡻
⣿⣿⡿⣫⠴⢿⣏⡀⠀⠀⠀⢸⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠒⠉⠀⠀⢀⡽
⠟⣡⠞⠁⠀⠀⠙⠻⢿⣶⣦⡿⠋⠁⠀⠈⠉⠁⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣤⢶⣫⣼
⣌⢧⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠦⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠶⢖⣚⣫⣭⣵⣾⣿⣿⣿
⣿⣷⣮⣝⣓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣶⣭⣭⣛⣒⠶⠤⣤⣄⣀⣀⣀⣀⣀⣠⡤⢖⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠫⣍⠉⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⡆⠀⠀⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢲⡄⠤⠞⠃⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣠⣤⠶⠶⠶⠶⠶⢶⣶⠶⠃
⠀⠀⠀⠀⢠⣾⣷⣅⠂⠀⠀⠀⠀⠀⠀⠀⠘⠛⡟⡝⠗⠀⠘⠛⠃⠛⢹⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠖⣀⡐⢸⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣧⣾⣿⣧⣅⠢⡀⠀⠀⠀⠀⢀⣨⡇⢣⣀⠀⠀⠐⠀⣾⣟⠛⢣⠄⠀⠀⣀
⠀⠀⠀⣴⣦⣿⣿⣿⣌⡛⠿⣳⢌⡑⢤⡤⠖⡾⠀⢷⠀⠈⠙⠓⢲⡞⠻⣿⣷⣤⢁⠖⡠⠀⠈⢱⡀
⠀⠀⢠⡇⠀⠀⢀⢉⣹⡿⣦⣌⠙⠿⣦⡬⠖⠛⠙⣎⢳⣀⠀⠀⡾⠁⠀⢻⣿⣧⣼⣠⢀⠛⠧⡀⣇⡀
⠀⠀⣸⠀⣠⠴⠛⠉⠀⠀⣀⣉⡽⠖⠁⠀⠀⠀⢀⡚⢷⣿⣯⠟⣡⣶⣶⣾⣿⣿⣿⣿⣿⣆⠀⠈⠓⢍⣳⠲⢤⣄⡀
⠀⢀⣷⠯⠵⠒⠚⠛⠛⠉⢹⠋⠀⠀⠀⠀⣀⡤⠤⠤⠬⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣄⢠⣾⣿⡇⠀⠂⠉⠳⢦
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⡶⠶⠞⠻⣶⣄⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴
⣿⠀⠀⠀⠀⠀⠀⠀⢰⣈⣉⣿⡟⠀⠀⠀⠀⠈⢻⡳⣦⣾⣿⣿⡟⢿⣿⣿⣿⡿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠘⢷⣄⠐⠲⠶⢶⣾⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⣾⠹⣮⡻⡛⠋⠙⢦⡙⣿⠿⣷⠶⢿⣄⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟
⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⣿⡇⠀⠀⠀⡠⠖⠁⠀⢹⣿⣌⠢⡀⠀⠈⠁⠀⠈⢧⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠼⢱⢿⢷⣌⠁⠀⠀⢀⡀⠈⣇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡤⠀⣠⠀⠀⠀⢀⡞⠈⢿⡀⡴⢠⠇⢸⣶⠿⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢠⡇⢠⡏⢀⡆⠀⠈⠀⠀⣿⡇⣧⠈⢶⣾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣼⢃⣾⠇⣼⡇⠀⠀⣧⢀⣿⣿⡈⠀⠀⠙⠿⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣟⣀⣀⣼⣿⣿⣿⣿⣿⣦⣔⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢠⢀⠈⡉⠉⠉⠉⠉⣉⣽⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⣿⣿⣿⣷⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣴⠶⠚⠋⠉⠉⠁⠉⠉⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⠴⠋⠀⢀⣠⠤⠶⣖⡿⠿⠛⠛⠛⠛⠒⠻⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠯⢒⡽⢃⡠⠔⠊⢁⡤⠒⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠚⠚⠉⠀⢠⣞⠡⠤⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⠦⣌⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣴⢚⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣠⠤⠶⠒⠛⠉⠛⠦⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠸⣟⡢⢄⡀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠒⠢⠤⠤⠍⣉⣙⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠈⠙⠲⢮⣕⣲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠤⣤⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡄⣳
⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣭⣝⣒⡶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠲⠦⠤⠤⢄⣀⡿⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣸⣁⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠾⣭⣽⣒⣒⡶⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣷⣶⣾⣭⣭⣽⣒⣒⣲⡶⠶⠤⠤⢤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣶⣶⠶⠶⠶⠶⣾⠭⠭⠭⠥⠴⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣟⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢲⡖⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⢣⠀⠐⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⣷⠀⢠⢖⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀⣀⣀⢸⣷⡡⠀⢊⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠤⠖⠚⣉⣁⣠⡤⠤⠤⠤⠴⡶⢦⣤⡬⣽⡶⠏⠓⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣭⣶⣶⠋⢩⡟⣿⣷⡖⡀⢰⣶⣿⣻⣎⣿⡇⠘⢠⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⡟⢻⡏⠀⠋⠀⠀⠐⠁⠀⢸⠀⠰⠂⠙⡟⢡⠀⡘⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣧⠀⠀⠀⣀⠄⢄⣇⢨⣦⠀⠀⢠⡇⠈⢦⡘⢦⡀⣈⠁⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡏⢧⡀⠰⢋⡀⢀⠀⣈⣀⣹⢰⣟⠇⠀⠀⢳⣀⣀⣩⠉⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡁⠀⠈⠙⠒⣒⣒⣪⠀⢸⡟⠀⠀⠀⠀⢩⠁⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⣃⣄⠀⠀⠭⠉⠉⠁⢀⣿⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣾⡇⢰⡉⠪⣦⡀⠀⢀⡀⢀⣼⢧⡀⠀⠀⠀⠀⢠⢾⡄⠀⠀⣀⣠⢴⡍⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⡙⣧⣌⢧⠀⠈⠙⠛⠿⠿⢻⡟⢲⠛⢳⡄⠀⠀⢸⣌⣭⣭⣭⣏⣹⡤⢟⡫⠄
⠀⠀⠀⠀⠀⠀⢠⠒⠺⢸⣿⣠⢞⠛⣿⣄⠀⠀⠀⠀⢰⠃⣇⡾⠛⣿⣇⣀⠀⢸⣌⢧⠸⣅⣽⠴⣟⣛⡒
⠀⠀⠀⠀⠀⠀⠀⠐⠚⣾⡼⣋⢈⣁⣘⣌⢷⡀⠀⠀⣸⠞⠋⠙⡃⢹⡁⣏⡑⢬⠹⣽⠚⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⢀⡇⠠⡦⢠⠷⡀⡼⢹⡃⢉⢈⣉⡉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣍⣉⡉⠉⠛⠲⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣩⣤⣀⣀⠀⠀⠈⠉⠛⢦⣄⠀⠈⠙⠦⣍⠲⣄
⠀⠀⠀⠀⠀⠀⣀⣊⡀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣄⡱⣄
⠀⠀⠀⠀⠔⠈⠁⠀⠉⠉⠒⠦⣀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠙⣧⠀⠀⠀⠈⠳⣌⠳⡑⢄
⠀⠀⢀⠄⠀⠀⠐⠀⢄⠀⠀⠀⠈⠳⡀⠀⠀⠀⠈⢳⡄⠀⠀⠸⣇⠀⠀⠀⠀⠈⢧⡙⣄⠳⡀
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢻⡄⠀⠀⠀⠀⠀⢳⡈⢆⢱⡀
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⣇⠀⠀⠀⠀⠀⠀⢣⠀⠀⢧
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡄⠀⢸⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡔⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⡟⠦⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠑⠦⣄
⠈⠲⢄⡉⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠈⠑⠲⠤⣀⡉⠙⠓⠲⠦⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣧⡀⠀⡇⠀⣀⣀⣤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠭⢖⣢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠫⠷⣶⣤⣄⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣠⡤⢤⣴⣶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠚⠒⠒⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠋⣡⣤⣤⣤⠀⠀⠀⠈⠙⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⢸⡟⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢻⣧
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⣸⣿⣄⣤⣤⣤⣤⣤⣤⣿⣷⣴⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣾⣿⣤⣤⣤⣤⣤⣤⣠⣿⣇⣀⣀⠀⠀⠀⣿⡇
⠀⠀⢀⣴⡾⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⢷⣦⡀
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣿⣆⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⣰⣿
⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⣿⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣿⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⢀⠔⡠⠋⢁⠔⠓⡤⡀
⠀⠀⠀⠀⢀⠃⡐⠀⠠⠁⢀⠎⠀⠐⡀
⠀⠀⠀⠀⠀⢠⠁⠀⠃⠀⠘⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⣠⣤⠴⣶⣶⠶⠛⠛⠛⠛⠻⢦⣄
⠀⠀⠀⠠⠶⣀⠀⠀⠀⠀⠀⠀⣀⠤⡡⠀⠀⢀⣤⠾⠋⠉⣠⠞⠋⠀⠀⠀⠀⣀⡴⠞⠋⠉⠻⢦⣄
⠀⠀⠀⠣⣀⠈⠁⠐⠁⠁⣀⠄⠂⠀⠀⢀⣴⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⠟⠳⣦
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⢠⡾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰⡏⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡄⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠹⢦⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⢧⣄⣀⡀⠀⠀⢸⡇⣠⣤⣤⣤⣤⣤⣤⠴⠶⠶⠛⠛⠉⣀⣠⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⣤⣤⣤⣤⣀⣀⣠⣤⣤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⣒⠒⠒⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣩⣤⡤⠀⠂⡀⠈⠉⠒⠄⡀⠀⠉⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠉⠐⠢⣁⠢⡁⠀⡈⠂⡀⠀⠀⠨⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠐⡄⠀⣰⣼⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠐⡈⢂⢸⡎⠰⠋⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠠⠘⡀⠇⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⢡⢰⢰⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⢨⠀⠰⠸⠀⡈⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢿⣀⠀⠉⠙⠒⠒⠤⠤⠄⢀⣄⡀⠀⠀⠀⣀⠀⠠⠾⠾⠚⠚⠛⠢⠷⠧⠐⠚⠋⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠚⠋⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⢀⢀⣀⠀⣀⢀⣀⡀⡀⣀⢀⡀⡀⣀⡀⡀⡀⢀⢀⣀⠀⠀⣀⢀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⠜⠙⠛⠨⠃⠃⠘⠃⠳⠋⠣⠀⠃⠛⠣⠋⠛⠃⠘⠙⠻⠐⠆⠚⠊⠚⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢆⠀⠀⠀⠀⠄⢀⠄⡠⡐⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠤⠤⠄⠀⢉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠤⢄⣀⡀⠀⠀⢀⡠⠖⠋⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⣤⣤⣤⣤⣤⣀⢀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠉⠙⡞⠉⠀⠀⠀⣰⠋⠀⢀⣀⠤⠔⣒⣩⠭⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠑⠂⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⠁⠀⠈⠁⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣼⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⢸⣧
⠀⠀⠀⣀⣤⠶⠛⠉⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢰⢺⠿⠛⠋⠉⢸⠀⠀⣸
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⡟⠛⠉⠉⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⣸⣾⠀⠀⣀⣠⠼⠒⠉⣠⡾
⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣤⠤⠿⠛⠚⠉⢁⣀⣤⠶⠛⠉
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠷⠖⠒⠒⠒⠒⠛⠋⠉⠉⠉⠀⠀⣀⣀⣤⡴⠶⠛⠋⠉
⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡞⠀⠀⠀⠑⡆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠂⠃⠀⡇⠀⠀⠀⠀⡇⠀⠃⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠠⠆⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡦⣄⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠒⠁⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠑⢂⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢈⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠣⣀⠀⠀⡠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⢰⠊⣇⠔⠀⢰⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⣀⣀⣀⠠⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠐⢄⡏⢺⡀
⠀⠀⢀⡠⠔⠊⠹⠆⢻⠀⠸⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣌⠰⠀⢸⠃⠾⠉⠐⠠⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠈⠐⠚⠃⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠛⠒⠈⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠁⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠈⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠋⠉⠉⠉⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⡶⠶⠟⠛⢛⣛⣛⣉⣉⣉⣉⣉⣛⡛⠛⠛⠳⠶⢶⣦⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠉⣠⣶⠶⠿⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠷⠶⢦⣤⡉⠙⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⡀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⠀⠀⠾⠿⠿⠿⠿⠿⠄⢤⡄⠰⣶⣶⣶⣶⡆⢸⡿⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⡇⠀⠀⣠⣴⣶⡏⠀⠀⠘⣿⡀⠓⣶⣶⡤⠄⢸⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⣿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⢹⡇⠀⠈⠉⠀⠀⣿⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢹⣿⠀⠀⠀⢠⡾⠃⣤⣤⣤⣼⡇⠚⢶⣄⠀⠀⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⣿⠀⣦⠀⠋⠀⣀⣀⣈⣉⣉⣀⣀⡀⠀⣠⢠⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⡄⠀⠻⣯⡉⠉⠉⠉⢉⣽⠟⢠⡟⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡇⠀⠀⠈⠛⠷⠶⠶⠟⠁⠀⣼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣿⠀⠀⠀⠀⠚⠛⠛⠂⠀⢀⣿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣿⡿⠋⠉⠉⠉⠉⢹⣿⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⡞⠁⣃⣤⣤⣶⣤⣤⣀⠈⢱⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⣿⡟⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⢿⡏⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢉⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣧⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣤⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧⣤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⢻⣿⣿⣿⣿⣿⣏⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠃⠛⠛⠿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠻⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠾⠿⠿⠟⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠑⣀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠉⢤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⣿⠓⠦⠀⣀⠀⠀⠀⠀⠀⢀⡀⠴⠚⣿⠤⠆⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠢⠄⠀⠠⠄⠠⠄⠒⠂⠁

⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⢻⣖⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⣴⠶⠶⠶⠶⣦⣄⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢔⣸⠟⠃⠀⣀⣀⠀⠘⠻⢇⠠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣡⣿⠏⣠⣾⣿⣿⣿⣿⣷⣄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⢿⣿⣿⣿⣿⢿⣿⣿⡾⣿⣿⣿⣿⡿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⣿⡿⠿⠿⠿⠿⠿⠷⠿⠿⢿⣿⣿⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣗⡤⠀⠠⠤⠤⠄⠀⠠⠀⠔⣺⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣵⢟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣮⡳⣄
⠀⠀⠀⠀⠀⠀⣠⢞⣴⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣮⡳⣄
⠀⠀⠀⠀⠀⠚⠉⢹⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡏⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢰⠃⠀⠈⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣉⡚⢿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣍⣻⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢭⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡑⠲⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⠖⠒⣛⡒⠲⠶⠷⠶⠶⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠠⢌⣙⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡼⠁⣴⠟⠉⠙⠳⣄⠐⢄⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠐⠲⢬⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣇⠀⣯⠀⠀⣠⣴⣾⣷⣄⠳⣄⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣬⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢦⣹⣶⣿⣿⣿⣿⣿⣿⣷⣬⣷⣄⡀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠘⢟⠛⢻⡿⡟⠛⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣷⠙⢶⣿⣿⣿⣿⣿⣿⣿⠇⢸⡀⠀⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢛⠾⣷⣦⠝⠢⠍⠉⠛⠛⠁⠀⠈⢷⡀⢠⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⣿⡿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡏⠹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢹⠉⠀⠉⠁⠉⠛⢿⣶⣤⣤⣶⣾⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣄⣀⣀⣤⢴⢟⣹⠿⣿⣿⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠅⠅⣠⣤⡴⢊⠟⣡⣾⡟⠻⠈⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣤⣥⣾⠟⠧⠉⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣇⣧⠷⠘⠋⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣉⡁⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡈⠉⢁⡉⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠁⠀⠉⠀⠉⠁⠉⠁⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⢿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣾⢩⡅⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣋⠹⢿⣰⣿⣏⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠛⠶⠶⠚⠿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠖⣤⠆⠲⠶⠾⠳⢶⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣶⢤⡔⢤⣶⣿⡆⠖⠭⣆⡠⡔⡴⠊⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⢛⣥⠚⠛⠛⢁⣴⡿⠛⢛⡧⢫⢔⣾⣿⣀⠔⠒⠲⡤⢄
⠀⠀⠀⣠⠴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠈⢑⣄⠈⢠⣤⢶⠏⣿⣶⠶⢭⣘⣷⣛⣿⡟⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⢸⣣⣾⣻⣾⢣⡀⠀⠀⠀⠀⢀⠔⠊⡝⠿⠤⠋⢓⣧⣿⣿⡆⢺⢻⠷⣛⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦
⠀⠀⠈⢻⣋⣿⡿⣿⡟⠢⢄⡠⠊⠁⢀⠐⠔⣶⣴⡖⠚⠋⣺⢿⡿⣿⣧⣾⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭
⠀⠀⠀⠀⠈⠙⢿⣿⣷⣀⣠⡔⣂⣤⣼⣏⣌⠉⢹⣻⣀⣴⢻⣆⣼⣿⣃⣶⣹⣗⣤⠀⠀⠀⣀⣀⡀⠄⡴⡶⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠦⡏⡶⠺⣽⣏⣳⣉⢶⠿⠟⣩⣽⣿⣿⠉⣉⣽⣿⣿⠀⠐⠟⠉⠀⢀⣤⠎⠁⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣧⣤⣾⣿⣿⣿⣾⣗⣾⢻⣿⣯⣛⣋⣠⣿⢿⠟⠁⠀⣠⣤⣄⡈⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣗⣱⠦⣶⢮⡖⣲⡆⠩⠭⣝⠛⣿⣿⣿⠛⠃⠀⠀⣀⣰⣿⢏⠉⠻⣆⠀⠀⠀⣹
⠀⠀⠀⠀⣀⠤⣀⣾⢫⡍⣩⡭⡭⡍⣶⣻⣿⣿⣿⣿⡿⠧⠤⢙⣛⣫⣧⣀⣀⠴⠒⠉⢨⡿⢸⠂⢣⡸⡆⠀⠐⡇
⠀⠀⢀⣴⣏⠁⣹⡿⠈⠃⡇⢷⢳⢷⡿⠌⢳⣶⣶⣶⣶⠾⣟⢿⣿⣿⣿⣿⡏⠀⠀⠀⠈⣀⢸⠀⠸⡅⣷⠀⡸
⠀⠀⡜⢹⣿⡿⣿⡇⠀⢸⢻⠸⡸⢹⣧⡤⣻⣿⠃⢰⠇⠀⠈⣿⣿⣿⢿⡿⢷⢀⢀⡀⢨⣿⡸⣶⣴⣇⡟⠀⡅
⠀⠀⠇⠸⣋⣡⡇⡀⠀⡸⡎⡇⡇⣾⣿⣾⣿⡿⠀⢈⠀⠀⢸⣼⣵⣾⣿⢷⣿⣿⡌⠳⠌⢿⠧⠃⠛⠉⠃⠠⡅
⠀⠀⠸⡆⣿⣿⣿⡅⠀⡇⣧⠳⣸⣿⣤⡭⠭⠿⣤⢾⣇⢈⠽⠋⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠘⡈⠛⠁⠙⣤⢻⣨⣾⣋⣤⣶⣾⣿⣾⣿⡆⠉⠀⠀⣀⣀⡠⠼⠓⠒⠐⡒⠂⠔⠒⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠛⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠟⠋⠁⠀⠀⠀⣷⠓⢾
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⣀⠤⠒⠀⠈⠉⠉⠁⠐⢄
⠀⠀⠀⢠⠊⡠⠐⠈⠁⢀⡀⠀⠈⠑⢄⠑⡄
⠀⠀⢀⠁⠎⠀⡀⠈⠠⠀⠀⢄⠐⡀⠀⢀⠘⡀
⠀⠀⠈⠀⠀⢰⠀⠈⠀⠀⠀⠀⠆⢰
⠀⠀⠸⠀⢂⠀⢆⠐⠄⠀⡀⠄⢀⠂
⠀⠀⠀⠑⠀⡀⠀⢢⠈⢃⠠⠒⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠉⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠤⠴⠉⠉⠉⠧⠀⠄⠤⠏⠉⠉⢱⠤⠠⠤⠞⠉⠉⠹⠆
⠀⠀⡀⠀⠀⠀⠀⠀⠇⠤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠢⡤⢤⣀⠦⠀⠀⠠⢴⠤⠄⡼⠄⠀⠀⠴⠤⠄⣰⠄
⠀⠀⠀⡄⠀⠀⢀⠠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄
⠀⢀⠀⡀⠀⠀⠈⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱
⠠⠀⠀⢐⠀⢀⢰⠂⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠈⠃⠀⠀⠀⠀⢡
⠈⠁⠐⠜⠃⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡘⠀⢀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣄⠀⠀⠂⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢨⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠒⠒⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄⠤⠤⠤⡤⠀⠠⠄⠤⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⡄⠀⠀⠀⠀⣆⢀⣰⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⢀⢤⢔⡁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⠀⠀⠀⡃⠀⣼⠀⠀⠀⠀⢸⠀⢠
⠀⠀⠀⠀⡰⣩⠊⠁⢑⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉
⡀⠠⠀⠤⠿⠧⠀⡔⠀⠠⠈⢢⠀⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢤⠤⢷⢶⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⡸⡆⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⡇⠀⠀⠀⢤⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⢾⢾⠶⠶⠶⠶⠿⢶⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠛⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡅⡆⠀⠀⠲⣗⢸⣿⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⣧⡶⠿⠶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢃⢰⠀⢀⣀⢒⢸⣿⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠶⠆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡸⡄⠀⠤⣤⢸⣿⠀⠀⠀⠀⠀⢸⣠⠤⠤⢤⣤⠤⢤⠤⢤⣼⠀⠀⠀⠀⠀⠀⠀⣿⡇⠒⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠇⣇⠀⠀⠭⢸⣿⠀⠀⠀⠀⠀⠈⢿⢦⡀⠈⠚⣤⡘⢃⢞⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣲⣤⣼⣧⣬⣤⣴⣫⠞⠀⠀⠀⠀⠀⠀⠈⠓⢿⣶⣦⣿⣾⠵⠋⠀⠀⠀⠀⠀⠀⢠⣾⣻⣧⣤⣴⣿⣳⣄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⠶⠖⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣉⣉⣉⣉⣉⣉⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠒⠒⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠉⠉⠉⠉⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠤⠤⠤⠞⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠤⠤⢤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣎⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠈⣉⣿⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢀⣐⣳⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢀⡚⣧⢿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠤⠼⡜⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⣏⡷⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠠⠬⢿⣹⣧⠀⠀⠀⠀⠀⠀⠈⣯⢧⠀⠀⠀⢶⡦⢰⣦⠀⠀⠀⠀⠀⣠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣹⠀⠀⠀⠀⠀⠀⠀⠈⢯⡳⠤⣀⣀⣀⣀⣀⣀⣀⣀⡠⠔⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣃⣠⣄⣄⣀⣤⡼⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⠦⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠉⢳⣄⠀⠀⠀⠄⠀⠙⢦
⠀⣠⡄⡰⠀⠀⠀⠀⠀⠈⠉⠓⣄⣿⡏⠀⡀⠀⠀⠀⠀⢣⡀⠈⠱⡄
⠼⢁⠂⠌⣆⠂⠀⠀⠀⠀⠀⠀⠈⢳⡧⠀⠁⠄⠀⠀⠀⢸⠀⡡⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣇⠠⢀⠀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠑⣮⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢢⡠⠄⠒⠀⣀⠀⣀⡀⡄⢨
⢹⣬⣐⢈⢐⢀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠀⠠⠔⠊⠀⢀⣀⢀⠀⠀⠀⠀⡦⠤⠄⠀⠀⠐⡀⣀⠴⠸⠉⠁⠐
⠘⣿⣿⣷⣬⣸⡞⣡⣶⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣴⣦⡌⠢⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈
⠀⠘⢿⣿⣿⣿⡄⠄⠹⠫⢳⠈⡉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⠀⠄⠖⢌⡆⢁
⠀⠀⠀⠉⠉⠁⢣⠘⣈⠊⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⣆⠣⡑⠁⣻⢨
⠀⠀⠀⠀⠀⠀⠀⠳⢬⣐⣊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣍⣒⡡⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⢖⣁⣀⣠⡤⠤⠤⣀⡂⠒⠤⠿⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠉⠁⣠⢴⣿⠁⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡠⢾⣧⡟⠁⠀⠐⠛⠲⠦⠀⠀⢀⣀⡀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠂⠙⠁⠀⠀⠀⠉⠛⠀⠀⣴⣯⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠠⠄⠀⣀⡀⣈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠳⡀⠀⠀⠀⠠⠤⠈⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠎⠑⣄⠀⠙⠢⣤⣄⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠋⠀⠠⠀⠀⠑⠄⡀⠀⠈⠿⠏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⡠⠖⠚⠋⠉⠁⠉⠘⠒⢆⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠉⠁⠀⠀⠀⠁⡀⠀⠀⢈⣶⣾⣦⡀⠀⠉⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠔⢈⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⢀
⠀⢀⣠⡤⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣄⠔⠉⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⠔⢡⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡊⡄
⢀⠃⢀⠀⠐⢄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢂⠀⠀⣿⣿⣿⡆⠀⠀⢀⠄⠀⠀⠀⠃⠀⡄⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣏⢰⠰⡀
⡘⠀⠀⠱⠀⠀⠱⣄⠀⠀⠀⠀⢦⣀⡀⠀⠀⠀⠀⠢⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⢀⡴⣿⠁⠘⡆
⡇⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠱⣄⠈⠁⠀⠀⠀⠀⠀⠑⣼⣿⣿⣿⡀⠀⠀⠀⠘⠀⠀⠀⣰⢃⢠⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⢰⠃⠀⠀⠏⠐⣉⡠⢴
⡇⠈⠲⠤⠄⠀⠀⠀⠀⢲⡀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⡴⠁⠈⠙⠢⡀⠀⠀⢠⡞⠉⠀⠀⠈⠳⣄⣠⠆⠀⠀⠀⢀⠈⡁⠄⠚
⢧⠀⠀⠀⢤⣀⠒⠄⡁⠢⣷⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠆⣰⣋⣀⡀⠀⠀⠀⠀⢈⡷⢶⣚⣛⣣⣤⡄⡟
⢸⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⡀⠀⡶⠐⠐⠒⠀⠘⣋⣉⣭⣭⣭⣭⠄⠀⢠⠞⢁⣀⣮⣽⠫⢇⡜
⠀⡆⠀⠀⠀⠀⠀⠲⠤⣤⡼⡇⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⠇⡀⠀⡇⢀⡴⠒⣶⡖⠚⠛⠛⠛⠻⢿⣶⡁⠀⠠⣧⣴⡏⠀⣽⠋⣀⡈⣵⠇
⠀⢿⡄⠀⠀⠀⣀⡤⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠰⠊⠠⠼⠗⠊⣸⢀⠾⣇⠀⠀⠀⠀⠀⠀⠉⠛⠷⠂⠛⣛⡻⣵⣞⠛⠉⠉⠉⡳⣤⡀
⠀⠈⣷⠀⢠⡞⡫⠒⠈⠉⠙⣶⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠠⠂⣤⠋⠀⠹⣄⠀⠀⠀⠀⠀⣀⠤⢲⡿⠛⢁⠀⠈⡹⢿⣷⣄⠀⠀⠀⠝⠶⣄
⠀⠀⢹⡀⠏⠀⢀⠌⠀⠒⣫⠥⠒⠓⠦⠤⠤⠤⠤⠤⠤⡤⡤⠚⠁⠀⠀⠀⠀⠀⠄⣼⠀⠀⢀⠘⠳⣷⣦⣞⣉⣤⠚⠩⠚⠋⠁⢀⠔⣡⠞⠉⠙⢿⣦⣄⠀⠀⠂
⠀⠀⠘⣿⠀⢰⠞⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞⠁⠀⠀⠀⠀⠈⠻⢿⣄⡀
⠀⠀⠀⢻⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⣀⣀⡴⠒⠒⠒⠚⠻⣎⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢊⡵⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣
⠀⠀⠀⠈⢧⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠿⠋⠀⠀⠀⠀⠀⠀⠈⠓⢾⣁⡄⠀⠀⠐⠂⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠙⠂⠈⠓⠻⠶⢤⣄⣀⡀⠀⠀⡤⠄⡄⠀⢧⣠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣷⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣿⢼⠀⢠⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠉⠈⠈⢈⠈⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠃
⠀⠀⠀⠰⣄⠀⠠⠰⢌⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠈⠈⠂⠀⠘⠁⠈⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⢀⡀⠀⠀⠀⠆⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠁⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠟⠂⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣧
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠗⠠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡞⠀⢠⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡟
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⠈⠧⠀⢠⠇⠀⠀⠀⠀⢀⠀⠀⢔⡫⠞
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠴⡐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⡀⠀⢀⠴⠒⠀⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠘⡆⠀⠸⡆⠀⠀⠀⠀⠉⠈⠂⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⡀⠀⢠⠇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⡿⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⢀⣴⣾⣿⣷⣤⣤⣤⣤⣤⣴⣤⣦⣶⣶⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠂⠠⠀⠀⡂⠀⠀⠤⠀⡂⠀⡇⠀⠂⠀⡆⠀⠂⢠⠀⠀⠀⠀⠀⠀⢰⠀⠰
⠀⠻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣯⡀⠀⠐⠃⠐⠐⠀⠂⠂⠂⠐⠀⠒⠀⠐⠂⠂⠒⠃⠐⠃⠘⠐⠂⠂⠒⠀⠂⠐⠂⠚
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢂⡀⠀⢀⡐⢄⠀⠀⠠⢀⡁⠀⠀⣁⠢⠀⠀⢀⠔⠈⢠⣄⠈⠂⡀⠀⠀⡄⠊⠀⠀⢈⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠠⡀
⠀⠈⢀⣼⣶⣒⣾⣧⡀⠄⠠⢡⣿⣷⣀⣰⣿⣧⠀⠀⠀⠀⣠⣼⣿⣆⡀⠈⠀⡐⠀⠀⣴⠟⣦⡀⠀⠂⢀⣀⡤⠀⠤⠄⠀⢄⣡
⠀⢀⠈⠈⠧⣶⠼⠀⠁⠀⠐⠀⠀⠀⣽⣅⠀⠀⠀⠀⡀⢸⣟⠁⠋⢿⡇⠀⠀⢡⠀⢠⠿⠾⠋⠁⠀⡄⠈⠹⣀⣀⠞⢆⣀⠼⠘
⠀⠀⠁⢐⠚⠛⠓⡂⠐⠀⠀⠐⠄⡘⠛⠛⠀⠔⠁⠀⠈⠨⡉⠉⠉⠉⠡⠂⠀⠀⠲⠼⠀⠀⠀⡀⠈⠀⠀⠑⠄⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢀⠀⠁⠀⠀⠀⠈⢠⠀⠀⡐⠁⢰⣆⢠⠎⠐⠀⠀⠠⠊⣼⣷⣠⣶⣶⡄⠀⢀⠀⠀⠀⣦⠀⠀⠢⠀⠀⠔⠁⠐⣺⡀⠀⠀⡀
⠀⠀⢠⣶⣤⣶⣤⣤⣄⠇⠰⢰⣧⠀⡁⠀⣤⡄⠐⠀⣤⣶⣿⣿⣿⣿⡿⠃⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠠⠀⠀⠐⠀⠁
⠀⠠⡴⣿⣿⡿⢿⣿⠟⠀⠈⠀⠉⠤⠀⣠⣀⠀⡐⠀⠟⢿⠿⠟⠛⠉⠀⠀⠀⠃⠀⠀⠀⣇⠀⠀⢀⠂⠈⡀⠀⣶⣷⣦⡄⠀⡀
⠀⠀⠐⠠⠀⠀⠀⠄⠀⠀⠀⠈⠢⠇⠈⠉⠩⠀⠀⠀⠀⠀⠄⠀⠀⠠⠐⠁⠀⠀⠑⠠⠴⠿⠦⠄⠁⠀⠀⠈⠠⠿⣿⡿⠇⠊

⠀⠀⡠⢢⣶⣿⣷⡄⠢⠀⠀⢠⣾⣿⠀⠀⠈⠂⡀⠀⢀⢀⡵⠒⠒⠢⣔⠄⠀⠀⣤⠀⠀⠀⠀⠑⢀⠀⠀⡠⠂⠀⠀⠤⠀⠢⡀
⠀⠀⢀⣼⣏⣩⣹⣣⡄⠁⢀⠁⠀⠠⠀⣀⠀⠀⠀⠀⢠⡏⢰⣿⣿⡟⠛⡆⠀⡈⠹⣿⣦⣄⠀⠀⠀⠀⠀⠂⢸⣷⣦⠀⠀⠀⢠
⠀⠠⢀⣠⠽⣿⡿⣄⡀⡀⠈⡀⢠⣶⣶⣿⣿⣶⡶⠀⠘⣧⡴⣿⣿⡇⢠⢇⠀⢁⠀⠈⠙⠛⠳⠀⠀⠀⠀⠀⣼⣿⣿⣶⠀⠉
⠀⠀⠙⠅⢀⠀⢀⠨⠋⠀⠀⠐⠀⢙⠛⢛⠛⠉⠁⠀⠈⠪⠓⠦⠤⠒⠅⠀⠀⠀⠀⠄⡀⠀⢀⠲⠊⠀⠀⠑⢾⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠐⠂⢩⣷⣶⣄⠀⠀⠀⠐⠊⣇⠀⠗⠢⡀⠀⠀⠄⠂⠈⣩⠁⠢⡀⠀⠀⠀⠊⢀⠀⠈⠐⢄⠀⡀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠾⠿⠿⠋⠃⠀⠛⠲⣀⡼⠦⣄⣀⣀⠀⠈⣰⣆⠀⣿⠀⣿⠀⡀⠀⠀⠀⣤⠷⣏⠀⠀⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⣼⣁⡃⣸⡋⢉⠀⢸⣿⣿⣿⣿⣧⣿⠀⠃⢀⠀⢸⡄⠒⠨⠅⠀⠀⠀⠀⠀⠄
⠀⠀⠻⠛⠛⠒⠛⢃⠐⠁⠀⠘⠣⡎⠁⠉⢻⠄⠃⠀⠈⠻⣿⣿⣿⣿⡿⠊⠀⠀⠢⢀⢷⣤⡶⠡⠊⠄⠀⠀⠀⠀⠄⠠⠄⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠠⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣄⠀⠀⠀⣀⣴⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⣛⣧
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣦⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣀⣀⡀
⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡷⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢛⣿⣿⣿⣧⠀⠀⠘⣿⣿⡏⠀⠉⢙⣿⣿⣿⣿⣿⣟⠋⠁⠀⣿⣿⡿⠀⠀⢀⣿⣿⣿⣟⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄⡀⠈⠻⢃⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠘⠛⠀⣀⣴⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢀⣀⡤⠴⠛⣛⣭⣭⣴⣶⣶⡶⠶⠶⠤⠀⠀⠈⠙⢿⣿⣿⠟⠉⠀⠀⠠⠤⠦⣤⣶⣦⣤⣭⣭⣛⡻⠦⣄⣀
⠀⠀⠀⠀⠀⢀⡼⠿⢛⣭⣥⣤⣤⣤⠶⠖⠒⠀⠀⠀⢈⣿⡅⠀⠀⠀⠒⠲⠤⣤⣤⣄⣭⣍⡛⠿⢿⣄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡾⢟⣿⣿⣛⣭⠤⠖⠒⣉⣠⡤⠶⠛⠉⠙⠳⢦⣄⣀⠐⠲⠤⣭⣛⣿⣿⡿⢷⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⢁⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢤⡈⠉⠛⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣭⣮⣭⣯⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠷⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⢠⠾⠛⠀⠀⠀⠀⠀⢠⠂⠓⢳⣝⢄⠀⠀⣀⠀⠀⢀⠔⠁⠁⠠
⠀⠀⠀⠀⠀⠀⠄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠘⠧⡀⠀⢻⠁⠀⠠⢀⠂⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⣀⢀⠀⠀⠀⠀⠀⠾⠓⠀⠘⠀⠃⠀⠀⡄⠀⠀⠈⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⢀⡀⠌⠀⠀⠀⠈⠀⠀⠀⡀⠀⢀⠀⠀⠀⠘⢦⠀⢀⣀⡐⠒⠊⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⡨⠎⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⡀⠀⣒⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠇⠀⠘⠰⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⢰⣾⡄⠀⠀⠀⠀⠀⢀⡰⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠠⠀⠀⠡⡀⠁⠀⠀⠻⡇⠀⠀⠀⠀⠐⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣬⠘⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢢⡀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠘⠁⠀⠠⡄⠈⠀⠀⡀⠁⠀⠀⠈⠀⠀⠒⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠀⢀⡤⠒⠈⠉⠀⠉⠛⠒⠠⢄⠀⠀⠀⠀⠈⠀⠄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡴⠃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⣤⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⢠
⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⠀⢀⡿⠁⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠸⣄⣀⣤⣤⠀⡴⠆⠀⠀⠀⠰⠁⠀⠀⠀⣐⡠⠔⠀⠀⠀⡇⠀⠈⠀⠀⠙⠀⠀⠘⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⢀⡔⠁⠀⠀⡐⠀⠁⠀⠄⠀⠀⢸⠁⠀⠀⠘⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⡟⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠐
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣀
⠀⠀⠁⠀⠀⠀⠉⠓⠿⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠙⠁
⠀⠀⠀⠁⠀⠑⠀⠠⠀⠀⠀⠐⠀⣠⡿⠃⠀⢠⡐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢀⠈⠓⠪⣍⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⠢⢄⣠⠋⠀⠀⠘⠁⠙⢲⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣾
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣁⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⢿⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⠛⢦⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠛⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⡛⢶⣌⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠷⣤⠾⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠉⣻⡆⠈⢛⣶⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠙⠛⠋⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣴⠏⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⡆⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣠⣤⣄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠃⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠻⢦⠟⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠶⢤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠛⠉⠁⠀⢰⡟⢷⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣠⡿⠶⠶⠚⠛⠛⠓⠶⠶⠶⢦⣤⣤⣤⠶⠶⠛⠋⠉⢷⡀⠀⢸⣏⠛⠛⣻⣿⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢸⡇
⠀⠀⠀⠀⣰⡟⠀⠀⣴⠶⣦⠀⠀⣶⠛⣶⠀⠀⣠⣤⣤⡀⠀⠀⠀⠈⢿⡄⠀⢻⡄⠀⢿⣼⠟⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣰⠏⠀⠀⠀⠙⠷⠋⠀⠀⠘⠛⠃⠀⢰⣏⠀⠀⣿⠀⠀⠀⠀⠈⢻⡄⠀⠻⣦⠀⠀⠀⠀⠀⠸⣇⣀⣠⡟⠀⠀⠀⠀⢀⣰⠟
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠈⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠞⠛⠉


⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡟⠛⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⡇
⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣦⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⢘⠿⣿⣟⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⢀⡠⠃⠀⠆⠹⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠉⠋⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢧
⠀⢠⠊⠀⠀⠿⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⢀⣴⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⢧
⠀⣶⣶⣶⣦⣄⡀⠀⢀⣰⡆⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠲⣖⠛⠒⠦⠞⢧
⠀⠸⣿⣿⣤⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⡶⠀⠀⠀⣾⣿
⠀⠀⠈⠻⢿⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣤⣤⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⢿⡿⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡒⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡟⢻⣿
⠀⠀⠉⣸⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣿⣟⣻
⠀⠠⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣂⣀⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣟⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣠⠊⠱⣄⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢢⡀
⠳⣄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣀⣬⡵⢮⣧⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⣄⠀⣶⠃
⠀⠈⠑⡄⡴⠋⢶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠠⡎⠉⡳⡏⠰⠆⢹⢞⠈⣱⠆⠀⠀⠀⠀⠀⠀⣀⣧⣀⣺⠋⠀⣙⡍⠁
⠀⠀⠀⡇⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠑⢟⣲⣖⡛⠊⢏⠁⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠈⡧⠤⠟⠁
⠀⠀⠀⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⡄⢠⠛⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠐⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠈⠑⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠁⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡷⠀⠀⠔⠣⠉⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠈⠦⠆⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⠸⢟⣛⠚⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠈⡄
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠁⠀⠀⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⣽⣽

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠶⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⢠⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠁⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⢰⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣢⠀⠀⠀⠀⢀⣠⠖⢁⣤⠖⠃⠀⠀⠀⠀⠀⠈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠈⠉⠉⠒⠤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡟⡩⠳⠀⣀⣴⣯⠵⠚⠉⡗⠒⠒⠒⠒⢲⠒⢲⠐⠏⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠉⠀⠒⠤⠀⣐⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡵⢀⣴⡾⠟⠋⠁⠀⠀⠐⢟⠓⠒⠒⠒⢲⡓⠒⢚⠟⢠
⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠷⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⡀⠤⠂⣠⡞
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠳⡀⠀⠀⠀⢀⠼⠞⠒⢿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢴⠶⠞⠉
⠀⠀⠀⣀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠆⠘⡄⠀⠀⡎⠀⠀⡀⣼⠱⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠣⣸⢄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠘⠏⠁⠀⠀⠈⠦⣐⠃⠀⢰⠋⡟⢀⡾⠁⣀⣀⣿⣿⣿⠀⠀⠀⠀⠉⠙⠉⢹⠈⠁
⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⢀⢤⠙⠀⡤⠃⠉⠉⣼⣅⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣸⣿⡇⣿⣇⡀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣆⠠⠋⠀⠀⠇⠀⡀⠀⠀⢰⣿⢿⣾⣿⣻⣿⣿⣿⠀⠀⠀⢀⣼⡇⠀⢸⡇
⢠⣿⣟⣷⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠣⡀⠀⣀⠠⠚⠁⠀⠀⠀⠀⠈⣈⠖⠦⠀⣿⣿⡀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⠟⡀⠀⢸⡇
⢸⣿⣯⣿⣿⣿⣿⡜⠆⠀⠀⠀⠀⠀⢺⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣾⣏⡼⡇⠀⣸⣧
⣘⣛⣛⣛⣽⣿⣿⣇⠀⠳⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠴⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⢲⣿⠟⠒⠒⠾⠒⢻⣿⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠳⣄⠀⠀⠀⠀⢀⣀⠠⢰⡋⠀⡄⠀⠀⢀⠀⠀⣠⣤⡤⠤⠤⠠⠀⠀⠄⠀⠾⠃⠀⠀⣰⠰⠄⢾⣿
⠀⠠⠤⠤⠴⡌⡃⠀⢻⠀⠀⠀⠀⠑⠀⠄⠒⠉⠀⠀⠈⠁⠀⠇⠀⠐⠘⣀⡀⣡⣎⢰⣶⠌⢻⣏⣠⣤⠀⣠⣀⣠⠆⠈⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⡇⡃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣐⣈⣻⣡⣭⡷⠶⠻⠋⠋⣿⣽⠃⠈⣩⡹⣇⠀⣹⡇⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠇⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣯⡿⢩⠀⣄⡄⠀⠀⡀⠀⣿⣿⡆⠒⡇⢠⠙⠛⣻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⣇⣼⡄⣽⡷⠘⣤⡃⡀⢸⣛⣇⡞⠁⠈⢢⣴⣿⣿⣟⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⢚⣿⡏⣿⡇⣼⡷⢶⣿⣿⣿⠿⠿⢿⡗⢲⣶⣼⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⣀⣠⠤⠄⠒⠊⠉⠀⠀⠀⠀⣠⣼⣿⣿⣿⣷⡿⠟⠛⠉⠁⠀⠀⠀⠘⣟⠀⣿⠃⠠⠦⠄⢳⠈⠋
⠀⠀⠀⠀⣀⣀⠾⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣻⣤⣤⣤⣤⡇
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠛⢛⣉⠉⣀⣠⡤⠤⠶⠶⠤⣤⣄⣀⢉⣉⡛⠛⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⢋⣡⣴⣶⣿⣿⡏⡞⠁⠀⠀⠀⠀⢸⣆⠀⠀⠉⢷⢹⣿⣿⣶⣦⣌⡙⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⢋⣤⠾⠟⠛⠿⠿⣿⣿⢹⣇⣀⣠⣤⣤⣤⣤⣭⣤⣤⣀⣸⡎⣿⣿⣿⣿⣿⣿⣷⣤⡙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⣡⡞⠋⠀⠀⠀⠀⠀⠰⣂⣩⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣝⣛⡿⢿⣿⣿⣿⣿⣿⣷⣌⠛⢿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⠟⣠⣾⠏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣶⢶⣶⠤⠤⠤⠤⠤⠤⠤⠶⣶⡶⣶⣾⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣷⣄⠻⣿⣦⡀
⠀⠀⠀⠀⣠⣿⡿⢁⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⣻⢿⡿⣦⣾⢡⣾⣿⣿⣿⣿⣿⣿⣷⡌⢷⣶⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣄
⠀⠀⠀⣰⣿⠟⣠⣿⣿⣿⣷⠀⣿⣟⣿⣟⣿⠀⠀⣿⢸⣿⣿⢻⣦⣙⣛⣛⣛⣛⣛⣛⣋⣴⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣆
⠀⠀⣰⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣤⣤⣿⢸⣿⣿⡟⣁⣶⣤⣾⣤⣤⣷⣴⣦⣝⢻⣿⣿⡇⣿⣿⣯⣹⣿⣻⣿⣿⣿⣿⡿⣿⢿⣿⣆⠹⣿⣆
⠀⢠⣿⡏⢰⣿⣿⣿⣿⣿⢽⣿⣿⡿⣿⡿⠿⢟⣛⣵⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣮⣛⣛⠼⣿⢾⣿⣿⣿⣿⣯⣿⣽⣿⣿⡆⢹⣿⡆
⠀⣾⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣡⣮⣵⣏⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣮⣝⡿⠻⣿⣿⣿⣻⣿⣿⣿⡀⢿⣷
⢸⣿⡇⣸⣿⣿⣿⣿⣿⣿⡚⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠠⠞⢿⣿⣿⡿⠳⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣷⣿⣾⣿⣿⣿⣇⢸⣿
⣼⣿⠀⣿⣿⣿⣿⣾⣿⡿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣿⣿⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠯⣿⢽⣿⣿⢻⣿⠀⣿
⣿⣿⢠⡿⣿⣿⣿⢽⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣏⣿⣿⣿⡆⣿
⣿⣿⢸⣿⣿⣿⣿⣽⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣯⣿⡇⣿
⣿⣿⠀⣷⣿⣿⣿⣻⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⠁⣿
⢸⣿⡀⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⢙⠟⠿⠿⣿⣿⣿⠟⠉⠀⠀⠀⠠⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢀⣿
⠘⣿⣇⢸⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⡇⣸⣿
⠀⢻⣿⡄⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣶⣼⢡⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⢀⣿⡟
⠀⠈⢿⣷⡘⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢃⣾⡿⠁
⠀⠀⠘⣿⣷⣬⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣥⣼⣿⠃

⣾⣿⣿⣿⣿⠀⢸⠀⢰⠐⠞⠻⠖⠀⣤⠀⣿⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⡾⠀⠘⠛⠛⠛⠛⠃⢠⣆
⣿⠟⠛⠛⠛⠂⢸⠀⢸⠀⠀⠁⠀⠀⡇⠀⣿⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⢸⣷
⢯⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠰⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠙⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⢸⣿⠀⠀⠀⡟⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠟⠀⠄⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⡒⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⣿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢻⡏⣤⣶⣟⢧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡇⣸⡧⢿⣿⣿⣴⠇
⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣉⣉⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⣆
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣦⣤⣄⠀⠙⠃⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠈⢿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠘⢿⣦⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⣀⣠⡼⠿⣟⠋⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⣿⡿⠋⠉⠀⠀⠈⠉⠻⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣾⣿⠿⠋⣀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣿⣤⣀⣀⣀⣀⣠⡴⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⠾⠿⠿⠿⠿⠷⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠴⠀⠀⠉⠒⠂⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠀⠀⠀⣸⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⣤⣴⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠿⠂⠀⠀⠀⠀⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠃⠀⠸⡿⠿⠿⠿⠛⠁

⠀⣁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠄⠀⢸⢹⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⣴
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⢢⠀⣜⣇⠄⡇⡄
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠋⠉⠑⠲⠁
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢰⠋⢙⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⠘⢃⠃⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢠⠞⠀⠀⢠⠄⠀⠀⠀⠀⠘⠒⠋⠀⠀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢀⡠⠔⠊⠁⠀⠀⠀⠐⠀⠀⠀⡠⠚⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀⠀⠀⠀⠀⠓⠋⠁⠀⠀⠀⠀⠀⣠⠔⢁⡐⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⠀⠀⠤⠤⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⠠⡀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢸⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠘⠛⠋⡅⠀⠀⠀⠘⡀⠠⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠷⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢱⠀⠀⠀⢀⣃⠀⠀⡨⠉⠉⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣁⡀⠤⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠆⠀⠀⠀⢿⠀⠀⡀⠀⣤⣀⡰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⡀⠀⠀⠀⠠⢴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠐⣚⣆⣀⣀⠀⢸⢶⣭⣀⣀⣠⡊⣽⡖⠰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢀⡟⠁⠀⠀⠐⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠔⣠⣄⠐⠈⣦⣼⣽⡍⡻⣳⡶⠂⠑⠻⠛⢻⠟⠛⢃⣻⣿⣯⠀⠀⠀⠀⠀⠑⠶⣿⣿⣾⠁⠀⠈⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⢙⡃⢀⣄⣼⡿⠿⣿⣧⣿⣳⣠⣿⣿⣶⣶⣟⣷⣿⣿⣿⣯⠙⠀⠀⠀⠀⠀⠀⠘⣿⣎⢦⡀⠀⠀⠀⢠⣇⣀
⡀⠀⠀⠀⠀⣀⡀⠀⢄⡹⠶⢨⠃⡈⠉⠁⣷⣶⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⣄⠀⢠⣤⡄⠀⣆⣼⣿⣷⣯⣤⣴⣶⠿⠏⠉
⠈⠀⠀⠀⠀⢹⣿⣿⣫⣥⣰⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠘⣿⢾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣴⡀⠀⠀⠈⠉⠹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣶⡀⠼⣿⡛⣿⣿⣿⣿⣿⣿⣿⡿⣧
⣰⣾⡁⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⡇⠙⡇⢈⡿⣿⣿⣿⣿⣿⣿⢧⣿⢀⣠
⢿⡿⣇⠀⠀⢰⣷⣄⣀⠈⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢺⡇⠐⢫⣁⠙⣿⣿⣿⣿⣿⣯⣬⡉⠻⡛
⣤⣷⣿⠦⡀⠻⠿⠿⠿⠦⠹⠛⠛⠛⠻⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠺⣿⠈⡏⠚⣽⢿⣿⣿⣿⣿⡿⢿⣿⣾⣷⢀⣀
⠉⠉⠉⠀⠀⣰⠸⠶⢤⣤⢀⣤⣤⣴⣶⣾⣷⣾⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⢸⠻⣀⠿⠠⢟⠉⠑⣻⣿⣿⣿⣿⣧⢀⡛⠙⣁⣘⣿
⠀⠀⠀⠀⠐⠿⠷⣆⠈⢉⣉⣉⣉⣭⣤⣤⣴⣾⣿⣭⣿⣿⣷⣶⣤⣉⣿⣿⣿⣿⣷⣮⣼⣶⣿⠈⠶⠍⡲⠼⢿⣿⣿⣯⣾⣿⣿⣿⢿⣿⠭
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣽⣿⢀⡤⢀⠨⡆⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣿⠀⢮⡁⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢿⣿⡏⢀⡤⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⡇⠰⡞⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⡔⣂⠌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢌⠁⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠉⠉⠉⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡀⠀⠁⠂⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⡠⠀⠺⠠⠤⠀⠀⠀⠀⠂⠂⠉⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠘⠆⠔⠀⠂⠀⠀⠀⠠⠂⠀⠲⠀⠘⣲⠎⢥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠲⣆⠀⣶⡄⠀⣀⡀⠀⣶⡄⢀⠠⠊⠀⠀⡘⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠈⠐⢨⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠈⠉⠁⠀⠉⠁⠈⠉⠉⡃⠀⠀⠀⡄⠈⠢⡀⠀⠈⠀⠀⠀⠀⢀⢔⠕⠄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡆⠀⠀⠄⠂⠁⢢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣀⠁⡀⢁⠀⣁⠈⢀⠀⡀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢋⡃⠀⠀⠀⡠⠁⠈⠀⠀⠀⠀⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢈⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣉⡀⠀⠀⠀⠐⡀⠀⠀⠈⠈⠢⣀⡌⠀⠀⠁⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠍⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⠀⠈⡀⠀⡐⠀⠀⠀⢀⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢰⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠡⠀⠡⡀⠀⠀⠀⢀⠒⠂⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠛⢶⠇⡇⠀⠀⠀⡖⠈⠀⠉⠐⠀⠀⠀⠸⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⢀⠘⠀⠄
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⣀⣀⣀⣀⠀⢢⠀⠀⠢⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠂
⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠧⠄⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣾⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠉⠉⠉⢙⣿⡇⠘⡦⣉⠁⠒⠢⡀
⠛⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡁⠀⢱⠀⠀⠃
⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠛⠛⠛⠋⠀⡰⢄⡑⡀⣘⣤
⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠉⡻⣾⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⡡⢋⠜⢂⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠃⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠛⡵⠃⢀⠌⠈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡅⠀⣀⣀⣀⣀⣀⣀⡀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⡰⢁⡀⠀⠔⡡⣅⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡈⢆⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⠊⠠⢊⣼⡣⠗⠁⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠧⠠⣤⡥⠤⠦⠈⣄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⡐⠁⠀⠙⠊⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣄⡀⠀⢸⡂⠀⣀⣤⣾⡆⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⡝⠿⢿⣧⣸⣥⣾⠿⠟⠉⠈⡄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⠀⣿⣿⣿⠇⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⠁⠀⠈⠻⣿⡿⠟⠀⠀⠀⠀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣾⣍⠁⠀⠀⠀⢐⠠⠄⠐⠒⠒⠒⠐⠚⠀⠢⢄⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠀⢀⠔⠁⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠁⢾⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠏⢠⠀⠀⡌⠀⠀⠠⠊⠁⠀⠀⢀⠇⠀⠈⠲⡀⠀⠈⠃⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠐⠚⠐⠂⠀⡄⡇⠀⠀⡄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⠀⠰⠒⠒⠒⠒⠛⠛⠛⠒⠒⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣇⣀⣀⡧⠤⠤⠤⣴⣁⡤⠤⠄⠔⢦⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡇⠀⠀⠀⠓⠐⠃⠀⠀⠀⢰⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡇⠀⠀⠀⣶⣶⣶⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠋⣛⠛⢛⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠂⠀⢰⠂⠀⠀⠀⠒⠀⠀⠚⠁⠀⠀⢸⠀⠀⠐⠃⠀⠀⠉⠃⠀⠀⠛⠈⠉⠉⠉⠉⠉⠉⠋⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠄⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡼⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠩⠩⠍⢁⠒⠠⢬
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠁⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠖⠀⣁⠤⠐⠀⠀⠀⠀⠀⠬⡀⢅⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠃⠈⠐⠲⡄⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢠⠔⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠁⡠⠊⠉⠑⢄⠈⣤⠜⠀⠀⠀⠀⠱⠄⠠⠂⠀⠀⠀⠀⠢⠀⠔⠁⠀⠀⢀⠊⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢃⣶⣶⣶⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠉⠉⠉⠀⠀⡌⡉⡍⡉⡉⢉⡉⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠉⠀⠁⠉⠉⠉⣉⠙⠛⠋⠭⠝⠋⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣥⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⣿⠿⠟⠛⠛⠃⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠸⠆⠀⠀⠀⢶⠀⠀⠀⠀⠀⠓⠛
⠀⠀⠀⠀⠀⠀⢂⠀⠈⠄⠀⠀⠀⣠⣤⣤⣶⣶⣶⣦⣤⣀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣃⢀⣾⣿⣿⣿⣿⣿⣿⠉⠁⠀⣠⣿⣿⣿⣷⣶⡄⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠘⣁⣈⡉⠉⠉⠁⡏⠉⠉⣩⣿⣿⠿⢏⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠏⠉⠉⠉⠓⠛⠛⠛⠻⡿⠁⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⣪⡿⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⢀⡀⠊⠉⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠒⠿⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢤⡦⣀
⠀⠀⠀⠸⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠤⠒⢻
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠃⢱⠀⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢄⠀⠀⠀⢀⡜
⠀⠀⠀⠀⢐⢲⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠋⠑⠊⠁⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠚⡀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠠⠦⠄⠀⠀⢄⢀⡀
⠀⠀⠀⠓⠀⠀⠠⠐⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⣑⠀⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠄⢤⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣄⣄⠀⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠈⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠓⠄⠄⠀⠤⠋⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠒⠒⠂⠀⠑⠢⠼
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢚⡿⠉⠁⠀⠉⠙⢯⠻⡟⠻⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⢷⡘⣆⠀⠀⠉⠓⢭⡓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡆⠀⠀⠀⠀⠉⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠞⡟⣰⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢳⠀⠀⠀⠀⠀⠀⠈⠳⣌⢧⡀
⠀⠀⠀⠀⠀⡴⠋⢰⢇⡏⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣇⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳
⠀⠀⠀⢀⡞⠁⠀⢸⢸⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⡿⠛⠒⠒⠒⣶⠦⣄⠀⠀⠘⡎⣇
⠀⠀⠀⡾⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⣇⠀⠀⠀⠀⢸⡀⠘⡆⠀⠀⢹⣸⡀
⠀⠀⢰⠃⠀⠀⠀⠘⡇⢷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠈⣿⡇
⠀⠀⣼⠀⠀⠀⠀⠀⢹⡘⡆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠧⡤⠤⠾⠷⣶⠲⡖⠒⠛⠒⠃⠀⠀⠀⣿⣇
⠀⠀⣿⠀⠀⠀⠀⠀⠈⡇⢹⡀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠿⠶⠓⠒⠒⠒⠒⠦⣄⡀⣸⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠸⡄⠀⠀⢀⣠⠤⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣸
⠀⠀⢹⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣀⡽⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠘⡆⠀⠀⠀⠀⠀⢹⡀⢿⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⢤⣄⣙⠢⣄
⠀⠀⠀⢻⣀⣀⣠⣤⡤⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢉⣠⡤⠴⠖⠒⠶⣤⢬⣙⠻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠘⣆⠈⠳⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢧⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⢀⣠⢞⡥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡠⠤⠤⠤⠤⠤⠠⠤⠤⠄⠢⠀⠀⠀⠀⠀⠠⡆⠂⠀⠒⠒⠒⠀⠀⠄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢄⠀⠘⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠦⠔⣦⠀⠀⢰⠔⠒⠾⣄⠀⡊⠀⠀⡽⠐⠢⢀
⠀⠀⢀⠠⠐⠁⠡⠀⠀⠄⠀⠄⠰⢐⢸⠀⠀⠀⠸⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠚⠀⠀⠻⣄⠀⠀⠀⠘⠀⠀⠂⠀⠀⠄⠀⢠⠈⠢⡼⠁⠀⢀⡞⠉⠢⡀
⠀⠀⢣⠀⠀⠀⠀⢃⠀⠀⠀⠈⠡⣘⠾⡁⠋⠀⠡⢉⠀⢡⡤⠤⠤⠤⣤⣤⣶⣖⣾⢣⡈⠉⠋⠁⣸⠁⠀⠀⠉⠀⠀⠉⠀⠀⠃⠀⢸⠀⠠⠉⠐⡤⢞⡀⠀⠠⠁
⠀⠀⠀⠡⠀⠀⠀⠈⢆⠀⠀⠐⢒⠯⢟⡹⢖⠒⠒⡞⠒⢺⠿⠿⠿⣿⣟⡟⢛⣛⣛⠒⠙⢦⠀⠐⡗⠒⠒⠒⠒⠀⠀⠒⠒⠒⠃⠀⠸⠠⠁⠀⡰⠁⠀⠀⠉
⠀⠀⠀⠀⠑⡀⠀⠀⠘⠃⠀⠀⠈⠀⢠⠃⠨⠀⠀⢠⡀⠘⡋⠉⠉⠉⠉⠉⠉⢉⠝⠀⠀⠸⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡐
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⡘⠀⡈⢡⠀⢇⠀⢀⣣⡀⠀⠀⠉⠐⠀⠊⠀⠀⠀⠀⠀⠀⢧⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡐
⠀⠀⠀⠀⠀⠀⠈⢢⣰⡄⠀⢀⠀⢠⠀⠀⠆⠈⠆⢸⡥⣷⠀⠀⠀⠰⠶⢶⠀⠀⠀⠀⠀⠀⠸⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠈⠢⡄⠀⠀⠈⠄⠠⠀⠁⠘⡄⠀⠀⠰⠒⠚⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆⠀⠀⠀⠀⠀⠀⡔⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠄⠀⡆⡀⠘⠀⠀⠀⢘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡀⠀⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠊⠒⠐⠒⠀⠃⠃⠐⠒⠒⠀⠚⠒⠃⠂⠀⠀⠀⠀⠀⠀⠀⠓⠂⠘⠂⠒⠒⠀⠚⠀⠒⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠊⠋⠀⠀⠀⠈⠋⠂⠌⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠉⠉⠉⠉⠉⠉⠉⡩⠋⠁⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠲⠤⠤⠤⠤⣄⣀⣀⣀⣀⣤⠤⠤⠤⠴⠒⠒⠐⠚⠉⠉

⡀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⢿⢟⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣟⣿⣿⣛⣿⣟⣛⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁⠁⠀⠈⠈⠩⢿⣿⣿⣿⣿⠀⣾⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣻⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣿⣿⠀⣿⣿⣿⠿⣿⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⣿⡟⣷⣿⣿⠿⠿⠯⠀⠘⢷⣦⣭⣙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠘⠛⠀⣥⣤⠄⢴⣿⣷⣿⣿⢿⣿⣿⣿
⣿⣿⣿⡿⣡⠆⢀⣿⣿⣿⣿⠷⠁⢦⠀⠰⠀⢀⣿⣿⣿⡛⠛⣻⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⡄⢰⣿⢀⣿⣿⣐⠱⣿⣿⣿⣿⡿⣹⣿⣿
⣿⣿⢣⣾⡟⠀⣾⣿⣿⣿⣿⡇⠀⡳⣶⢓⠶⣟⣻⣿⣿⡇⠠⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠃⢀⣉⣾⣿⣯⣧⡴⣿⣿⣿⣿⣇⣿⣿⣿
⣿⣡⣿⣿⠇⢰⣿⣿⣿⣿⣿⡇⠀⣿⠿⠞⠓⠛⢋⣁⣤⡄⠐⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌⠿⢿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠏⡾⣾⣿⣿⡿⠛⣛⡁⠀⣤⢶⣶⣿⣿⢹⡟⢷⡧⠠⢽⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣾⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⢆⠀⣆⣄⢌⡍⡛⢻⢾⡆⡇⠀⠈⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠰⣈⣃⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⣹⡎⠀⢻⣿⢿⢾⣆⠄⠀⠈⠀⠀⠀⠀⢠⣮⣧⡤⠯⠏⠙⠛⠀⢈⣄⠀⠈⢛⢻⠏⠫⠿⠉⠙⣿⣿⣿⣿⣿⣿⣧⣻
⣿⣟⣵⣿⣿⣿⣿⣿⣿⣹⣿⡆⠀⣿⣿⣿⣿⣿⣧⠒⠄⠀⠀⠀⠀⡟⠛⠀⣄⢤⡄⢀⠀⠀⠀⡿⢦⠀⠀⢰⣦⢰⡶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠇⠀⣿⣿⡏⠙⣿⣏⡅⠀⠀⠀⡐⣼⣅⣴⣀⠁⠉⡁⠀⢤⣆⡀⠹⣿⣦⡀⢰⡄⠸⠕⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⡃⠀⣿⣿⡷⠀⢛⠛⠂⠀⢀⢌⡜⣡⣌⠃⡩⣷⡶⠂⠀⡽⣿⣁⡀⠛⡼⢷⡄⠀⠐⠈⠌⠩⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣥⣠⠏⠟⠃⢀⡉⠁⠀⢠⣿⣿⠁⣿⡿⢸⣆⣿⣿⣦⡀⣿⣋⣙⣷⠠⣞⣾⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡗⠀⠀⢸⡉⠙⣧⣉⣡⡾⣽⣿⣿⣿⣿⡿⢿⣿⡶⡦⠠⢾⣿⡿⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿
⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣏⣯⣿⣿⣶⡛⣷⣶⣶⣾⠿⠃⠀⠀⣀⣘⣿⣿⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠉⠹⠟⠟⣻⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⠴⣏⣽⣿⣿⣟⣿⣿⣾⣾⣿⠏⠀⠀⠀⠀⠃⠘⠿⠍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣝⣻⣿⣽⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡠⠊⠀⠀⠈⠁⠢⣄⡀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠀⠀⠀⠀⠔⠂⠈⠁⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢜⠐⠈⠀⠀⠀⠠⡄⠀⠰⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠚⠀⠁⠀⠀⠀⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠀⠀⠀⠀⠂⠀⡘⣷⡀
⠀⠀⠀⠀⢃⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⡔⠁⠓⠄⢀
⠀⠀⠀⠀⢀⡧⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⡀⠀⠐⠉⠀⠈⠀⠀⠀⠈⠁⠐⢄⡀
⠀⠀⠀⡀⢌⢳⠠⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠑⡄
⠀⠀⠀⢳⠨⠆⠛⠄⢠⠀⠀⠀⠀⠀⠀⠀⡰⡁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠑⠀⡄⠄⠊⠱⡀⠀⠀⠀⠀⡰⢱⠁⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠉⠢⡀⠀⢠⠁⢸⢠⠀⠀⠀⠀⡆⢸⠘⡄⠉⠀⠀⢀⣀⣤⠇⢸⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠑⣄⠃⠀⢸⠈⠂⠀⠀⠈⡄⢸⠀⡇⢨⡁⠀⠀⠀⢈⡅⠈⡅⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠘⠂⠀⢸⠀⡄⠀⠀⠀⠇⢸⠀⡇⠀⠇⠀⠀⠀⠀⠃⠀⢱⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠠⠊⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠸⠷⠰⢦⣬⣿⣯⣦⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠥⢀⠀⢰⡿⠻⠺⣿⠛⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⣶⣿⢡⣤⡄⣬⣟⢧⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣷⡶⣶⣶⡄⠛⣟⣛⣿⣿⣿⣿⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣿⣿⣿⢸⣿⡇⣿⡏⠀⠱⣜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⢸⡇⣿⡆⣿⠀⣫⡉⠛⠛⢛⡛⠓⠃⠀⠀⠀⠀⠀⢸⣶⡖⢀⣠⡤⠴⣸⠟⠛⠛⠓⠘⠇⠀⠀⠈⣖⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠚⠐⠹⢿⠇⠈⠛⠁⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠰⠿⣿⣇⠜⠠⣴⡟⢹⣇⡛⣧⠾⠛⠻⡶⢶⣾⡋
⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠙⢧⣶⣾⡿⢿⣻⣯⣥⣭⡆⠌⣷⢰⡀⢹⣽⠄
⠀⠀⠀⠀⠀⠀⣿⡄⠉⣉⣒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⠛⠁⠀⠋⠉⢿⣧⣼⠏⠀⠀⠙⠷⠌⠡⠂
⠀⠀⠀⠀⠀⠀⣿⣇⡸⢿⡉⠙⠒⠛⠛⠛⠓⢄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠘⠋⠛⠓⠲⠦⣤⣸⣇⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⣸⣿⡗⠁⠉⠀⠀⠀⠀⠀⠈⢿⣷⡆⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣽⠛⠛⠛⢧⡀⠀⠒⢄
⠀⠀⠀⠀⠀⣿⢹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⣎⠻⣶⣿⣿⣿⣷⣶⣦⡆⠀⠀⠀⠀⣤⣭⡍⣈⣉⣀⣬⣯⡄⠀⢴
⠀⠀⠀⠀⠀⠚⣛⡿⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣟⣀⣐⣉⣁⠀⠀⣆⣽⡿⠿⢿⣿⢋⡍⢻⠿⠃
⠀⠀⠀⠀⢀⣿⠉⣰⠏⡀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⢿⡿⡿⢦⠀⠀⠲⠾⠤⣄⣉⣿⣿⣯⡌⢰⣶⣶⣶⣿⣿⣿⣿⡿⠆
⠀⠀⠀⢀⣙⣹⡀⠀⠀⠠⠰⠀⠀⠸⣿⣿⣿⣿⠿⢿⣿⣯⣧⣧⣴⠷⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠘⠛⠛⠉⠉⠁
⠀⠀⠀⠸⢿⣿⣿⡄⡀⠀⣁⡀⠴⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣿⣽⣯⣯⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠟⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠎⣡⠾⡇⠀⠀⠀⠀⠀⡀⠀⠛⠓⢠⣴⠲⡄⣠⡤⣤⡤⣀
⠘⠿⠷⢶⡃⠀⠀⠀⠀⠀⠀⣀⡀⠐⠒⠒⢚⡧⣽⠉⢩⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠙⣄⣠⢀⠠⠴⠶⣿⠀⠀⠀⠀⣽⣿⡟⠀⢸⠀⠐⠀⠀⠀⠀⠀⡆⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⢻⠀⠀⠀⠤⣬⣭⠅⠀⢸⡤⡀⠀⠀⠀⠀⠀⡇⠀⠼⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢛⠀⠀⠀⠀⢼⡏⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠠⠞⠀⠀⠀⠀⠀⠒⠀⠠⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡼⠀⠀⠀⠀⢽⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡧⣠⣀⢠⡀⠨⣇⣀⣄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢧⠶⠂⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁
⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⢀⣌⠀⣥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸
⠀⠀⠀⡜⠀⠀⠀⠀⠧⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⣤⠄
⠀⠀⣰⠆⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⠈⠀⠙⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣾⠀⠀⠀⠀⠀⡂⢧⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠐⣸⠇⠀⠀⠀⠀⠀⠘⢺⡄⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤
⢇⡼⠀⠀⠀⠀⠀⠀⠀⠇⠵⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠨⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣬⣇⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠉⠉⠉⠐⠤⡀
⠀⠀⠀⠀⠀⡠⠊⠁⡠⠄⠒⠈⠁⠀⠈⠁⠒⢄⠈⠳⡀
⠀⠀⠀⠀⡜⢀⠔⠁⠀⡀⠤⠀⠒⠂⠠⢄⠀⠀⠑⡄⠘⡄
⠀⠀⠀⡜⠀⠏⠀⢠⠊⢀⠔⠈⠉⠉⠢⡀⠱⡄⠀⠘⡀⠰
⠀⠀⠀⡇⠀⠀⠀⡎⠀⠎⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⢁⠀⠱⡀⠰⡄⠀⠀⠀⡠⠃⢠⠃⠀⠀⢀⠀⠃
⠀⠀⠀⠈⢆⠀⠀⢄⡑⣄⠈⢦⠒⠉⣀⠔⠁⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠑⠢⠄⣀⠘⡄⠀⡗⠉⠀⠀⠀⠀⠀⠀⠘⡀⠈⠄⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⡤⠤⠤⢤⡀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠒⠲⠻⠀⠀⠀⠘⡓⠐⠒⠐⠚⠃⠀⠀⠀⡟⠒⠐⠒⠚⠃⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⢤⣤⣤⣀⣰⠤⠀⠀⠀⠤⣧⣤⣄⣠⡧⠄⠀⠀⠤⢼⣤⣄⣀⡮⠄
⠀⠀⠊⠀⡄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠇⠀⠀⠀⠲⡉⠠⠆⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⠺⡄
⠀⠀⠤⠠⠄⠀⠀⠀⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠠⡇⠀⠀⠀⠀⠀⠸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⡇⠀⠀⢈⠁⠀⠤⢘⣋⠒⠂⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢣⠀⠈⠀⠀⠀⠀⠀⠀⠢⡀⠠
⠘⠉⡀⠂⡜⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⡀⠂⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡎⠀⠠⠄⠀⠀⠐⠀⠊⠢
⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠂⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠂⠐⡄⠀⠀⠀⠀⡀⠀⠀⠐⡠⠀⠀⠀⠀⢀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠐⢧⡀⠠⠀⠀⠀⠤⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⠀⢰⠄⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠬⠄⠀⠩⠀⠤⠬⠥⠤⠤⠤⠤⠬⠤⠤⢤⡥⠤⠤⠤⠤⠤⡥⠤⠤⣬⠤⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄⠀⣾⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠀⠆⠀⠀⠀⠀⠀⠿⠃⠀⠸⠀⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠂⠀⠠⠃⠀⠀⠀⠀⠀⡇⠀⠂⡇⠀⠀⠀⠀⠀⢰⠂⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡤⢒⣪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⠆⠀⠀⠀⠀⠀⣥⠀⣀⠇⠀⠀⠀⠀⠀⢸⡀⠀⣄
⠀⠀⠀⠀⠀⣰⠞⡼⠋⠀⠙⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⣀⣿⣼⡁⠀⡠⠈⠉⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⢄⠁⠀⢸⠇⠀⠀⠀⢠⣀⠤⠄⠀⠓
⠀⠈⠳⣄⡀⠢⣌⡐⠂⣿⡇⠀⣀⣀⢈⠁⠀⠀⠀⠈⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠑⠒⠮⠭⠭⠿⣋⣁⡉⠉⠀⠀⠀⠀⠀⢃⣀⠈⣻⣶⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠉⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⠀⠰⠄⠀⠀⠀⡐⡀⣖⡉⡀⠀⠀⠀⠀⢀⠄⠂⡭⠉⠀⡄⠀⠀⢢
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠥⠀⠉⠙⠂⠀⠀⠃⠀⠀⣅⠀⠀⠛⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠄⠤⠀⠀⠀⠀⠀⠀⡈⠁⠀⢀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡄⠈⠁⠂⠀⠀⡂
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠤⠤⠠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣔⠁⠀⡐⠉⠢⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢙⡀⠀⠅⢀
⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠶⠮⣤⣉⣸⣧⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠸⠥⠙⠻⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⡀⢲⡔⠀⢄⣀⢴⣀⠉⠉⣭⠥⠬⠭⠥⠤⡆⠀⠒⠀⠀⠀⠘⡇
⠀⠀⢘⡁⢀⢀⣀⣀⣄⠼⠆⠀⠐⠀⠀⠒⠆⠀⠀⠂⠐⠒⠂⠀⠒⠀⠀⠷⠖⠀⡀
⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣀⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠈⠂⡄⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠒⠁⠀⠑⠒⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢢⡤⢲⠀⠀⠀⣠⠀⡴⠐⢤
⠀⠀⢀⠀⠀⣀⡀⡀⢀⢀⣀⡀⢠⡄⠀⠀⠀⠀⠀⢀⠻⡆⠀⠀⠀⣠⠠⢥⡴⠤⡄⠀⠀⠀⠈⡀⠀⠀⠀⢻⠀⠀⠓⠎⣁⣀⣀⠁⠶⠃⠀⢸⠇
⠀⣀⢨⠂⠊⠔⠌⠂⠑⢐⠂⠀⠀⠠⠄⠀⠀⠀⠶⠦⠠⠄⠀⠀⠀⠸⠀⢐⠂⠀⠧⠀⠀⠀⠸⠇⠀⠀⠀⠈⠳⠖⠂⠀⠹⠷⠞⠀⠀⠒⠶⠋
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠠⠉⠉⠄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠠⠂⠤⠤⠐⠕⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠉⠉⠅⠀⠀⠀⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠆⠀⠀⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠉⣀⠀⠁⠂⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠊⠀⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢑⠂⠉⠀⠀⠀⠤⠉⠒⠉⢁⡀⡇⠀⠀⠀⠀⠀⢀⡅⢀⡰⢘⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠰⠀⠀⠀⢠⠒⠀⠀⠀⡇⠀⠀⠀⠀⡠⠁⠈⠐⠖⠁⠈⠀⠂⠄⠈⠂⡀
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠑⢕⠄⡀⠀⢀⡠⠀⡀⢐⠅⠀⠀⡇⠀⠀⠀⣔⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⠄⠈⠄
⠀⠀⠀⠀⠡⠀⠀⠠⠃⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠗⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠶⠚⠠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⢒⠓⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡐
⠀⠀⠀⢀⠀⠀⠘⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠠⣤⠙⠠⡀⠀⡠⠊⠁⠀⠈⠀⡤⠊⢀⠔
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠄⡀⠙⠀⠀⠀⠀⡄⠀⠈⡔⠁
⠀⠐⠂⣀⠀⠀⠈⠕⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠀⠠⠀⠀⠉⠀⠀⠈⠀⠁⠀⠀⡇
⡀⠄⡀⠈⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠇
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠔⠚⠛⠲⠦⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡖⠂⠐⠲⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢠⡀⠀⢿⣿⣿⣿⣿⣿⣿⡇⢸⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠶⢦⡼⠟⠃⠘⢿⣿⣋⣩⠽⠟⠃⢠⣿⣿⡄⠈⠻⠯⢍⣙⣿⡿⠃⠘⠻⢡⡴⠶⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢛⡉⠍⢾⣷⡤⡄⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣽⣿⠀⠀⠀⣠⠀⠀⠀⠀⣠⢤⣾⡧⠉⢉⡛⢧⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢈⡔⠁⠀⠀⠀⠈⢻⣯⣂⣀⡠⣤⣀⡀⠀⠀⠛⠟⠻⠋⠀⠀⢀⡠⣤⣄⣀⣐⣵⡟⠁⠀⠀⠀⠈⢢⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢸⡅⠀⠀⠀⢀⣄⣺⡎⠉⡹⢦⣄⡼⢧⡤⣤⢤⢄⣠⡤⡤⢤⡼⢄⣠⡴⢏⠉⢸⣗⣠⣀⠀⠀⠀⢘⡇⡁
⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⡶⠿⠛⢋⣉⣹⡟⠀⠦⠤⣀⡑⠲⣧⣄⣀⡘⠁⣀⠠⠼⠖⢋⣀⡤⠔⠀⠁⠉⠉⠙⠛⠿⣶⣿⠇⣷⣤⡄⣀
⠀⠀⠶⠶⣿⢻⣏⣹⣤⣿⣴⠶⠛⠛⠻⢻⣗⣊⣩⡵⠶⠟⠛⠉⠉⣉⡠⠤⠴⢒⡚⠭⠵⠶⠮⣍⣑⣢⣌⠂⠽⠿⠿⠛⠁⣀⣿⣻⡟⣿⠶⡖
⠀⠀⢰⣶⣿⠿⣿⢻⣏⣻⣏⢠⣶⣾⣦⡿⣍⣁⣀⡤⠄⠖⣒⠩⠉⠐⠀⠛⠋⠩⠍⣒⠲⠤⠤⣀⣈⣩⢿⣷⣦⣴⡞⢻⣟⣹⡟⣿⠿⣿⣶⡆
⠀⠀⠀⣥⣼⡷⢿⠟⣿⠻⡟⢹⣿⣿⣿⣾⢹⣷⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢮⡟⣿⣿⣿⣿⡟⣻⠟⣹⢻⣿⢾⣧⣤
⠀⠀⠀⠉⠈⣷⠘⣇⢹⡆⢻⣽⣿⣿⣿⣿⡷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣯⣿⣯⡟⢰⡏⣸⠃⣾⠁⠉
⠀⠀⠀⠘⣀⣸⣦⣿⣶⢿⡟⢿⡉⢉⣉⣤⣤⠶⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠶⣤⣤⣉⡉⢙⡿⢻⡿⣶⣿⣴⣇⣀
⠀⠀⠀⠀⠸⡉⣹⣆⣿⡤⢿⡞⣿⡉⠡⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠈⢉⣿⢳⡿⢤⣿⣰⣏⢉⠇
⠀⠀⠀⠀⠀⠙⠉⠽⠒⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠐⠯⠉⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⠤⢤⡀⠀⠀⢀⣀⡤⣴⣂⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠤⢴⣆⡈⠉⠉⠁⠀⢤⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⠀⠀⠀⠙⣶⢤⡴⠖⠲⠤⢤⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠀⠀⢉⡴⠃⠸⡇⠀⠀⣀⠀⢹⠟⡷⠄⠀⠀⣠⣴⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⢄⠀⢀⣳⣤⣀⣁⡴⠋⠀⢇⣠⠴⠋⣨⢼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄⠀⠀⠀⠀⠙⠢⣌⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢊⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡻⣄⡀⠀⠀⠀⠀⠀⡴⠛⢳⡶⢶⠶⢶⠖⢻⠯⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣯⠹⡄⢙⠒⠦⣀⣠⠔⢣⠀⢰⠈⠁⢈⠋⣴⠋⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠗⠠⣄⡜⠀⠀⠀⠁⠀⢸⠀⡌⠀⠀⠈⢰⠹⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⣣⡀⠀⠀⠀⠀⠏⣤⠃⠀⠀⢠⡇⠀⡆⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⡏⠈⠁⠒⠒⢸⢠⠃⠀⠀⠀⠀⠇⠀⣥⣀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠆⠎⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⢰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⢠⠴⠚⠒⢄⠀⠀⠀⢠⠎⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡹⠢⠄⣨⠖⠀⠀⠀⡱⠤⠚⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠃⠀⠀⠈⡿⠤⠶⢊⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠏⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⡠⠀⠀⠀⡀⠀⠀⠀⢔⠒⠒⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠁⠂⠄⢀⢀⠃⠀⠀⠈⠉⠉⠀⠉⠉⢫⠀⠀⠀⢘⠉⠁⡛⠀⠀⠀⠀⢨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠑⠲⠤⠤⣀⡸⠂⠠⠀⡀⠀⠀⠀⠀⠀⢘⡄⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠈⠁⠀⠂⠀⠀⡇⠀⠀⡸⠀⠀⠠⡀⠀⠀⠀⠰⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠙⠦⠤⣄⣀⣀⢸⠂⠈⠉⠑⠒⠠⠤⢤⣀⣀⣀⡇⢠⠊⠀⠀⠀⠀⠈⢢⠀⢠⠃⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠇⠀⡆⠁⠀⠈⠉⠀⡆⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠀⢰⠂⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⡃⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠣⣀⠀⠀⢀⡠⠃⠠⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠐⠂⠀⠀⠠⢤⡴⠀⡀
⠀⠀⠀⠀⠀⢀⠤⠤⠴⠶⠖⠒⠀⠀⠀⠀⠀⠐⠁⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⡒⠠⠀⠀⡀⠰⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⢃⢀⠁⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⡀⠉⠓⠚⠀⠁⣨⠀⠉⠀⠀⠀⠈⠉⠳⣄
⠁⠀⠑⠠⠀⡤⠀⠀⠀⡠⠀⠀⠀⠈⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⢤⣀⠀⠈⢳⡻⣦
⠀⠐⡀⠀⠀⡁⠀⠀⡜⠀⢀⠄⠀⠀⠀⠠⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣁⠈⢙⣋⠃⣈⢁⡀⡀⠙⠏⣀⣠⡀⠀⠔⢢⣰⡒⠦
⠀⠀⢿⣷⣤⣑⣄⢠⠀⠀⡆⠀⢰⠒⠄⠀⠅⠀⠘⣦⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠸⠈⠁⠈⡟⢫⠁⠉⠀⠇⡁⠀⠰⠅⠀⠉⠙⠉⠈⠙⠉
⠀⠀⠈⠻⣿⣿⣿⡍⠀⠀⠀⡀⠀⠁⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⠐⠀⠀⠀⠀⠀⠁⠀⠔
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠠⠐⢀⠀⠈⠀⠀⠁⠀⡠

⠀⠀⠀⠀⠀⠀⠉⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢎⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⡘⠶⠶⠦⡄⠀⠈⠉⠁⠀⠀⠀⠀⠀⡠⠊⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠁⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠠⢀⡇⠀⠀⢀⠄⢲⣿⡤⢄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠱⡀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⡠⠀⠀⠈⠢⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⢠⣤⡄⠤⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡏
⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⢘⠢⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡖⠁⠀⠀⠀⠀⠱⠦⠤⠖⢏⣢⣀⠀⠀⠀⠀⡎⠀⠀⡵⠀⠀⠀⠀⢰⠁
⡰⠁⠀⠀⠐⠂⠤⠤⠀⠀⣤⠀⠤⠑⠒⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡰⠚⠓⣄⠀⡎⠀⠉⢳⠀⠀⠀⠙⠢⠾⠁⠀⠀⠀⠀⠘⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⠄⡤⠃⠀⠙⠒⠒⠁⠀⠀⠀⠀⠀⡼⠋⠉⢢
⢀⡞⠉⢳⡀⠀⠀⠀⠀⠈⠻⡁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡜
⠈⠧⡤⠞⢣⡶⠒⠒⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠤⠤⣄⠀⡏⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠠⠄⠀⠀⠄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⡼⠌⢧⡀⠀⣠⠞⠀⠀⠀⠀⡞⠉⢲⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢠⣶⡄⣠⠀⠀⠀⠀⠐⠀⠀⠀⢈⡀
⠙⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠀⠈⠀⡀⠛⠀⠄⡀⠀⠀⠻⠃
⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠁⡰⠄⡀
⠶⠋⠀⠺⡀⠀⠀⢰⠉⠙⣦⣀⠄⡴⠋⠉⠳⡄⠀⠀⠀⠈⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠌⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⠀⠀⠀⠑⠄⠀⠈⠑⠚⠁⠀⠈⢧⡀⠀⢀⡧⠀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣤⡤⢤
⠀⠡⡀⠀⠀⠈⠂⣀⡀⡀⠀⠀⠀⠀⠉⠉⠉⠀⡟⠁⠀⠙⣦⠀⢐⡎⠀⢹⠀⠀⠀⠣⡄⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠳⢄⣀⣠⠏⠀⠀⠈⠐⠁⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⢀⠃⠀⠀⠀⠀⠀⡀⠀⠐⠈⡆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⢸⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠗⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⡇
⠀⠀⠘⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢡⠀⠀⠀⡠⠂⠉⠉⠙⢢⡀⠀⠀⠀⠀⡠⢤⡜⠓⠉
⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢧⡀⠀⠀⠀⣰⠃⠀⠀⠀⠸⡀⢀⡸
⠀⠀⠀⠀⠀⠉⠒⠒⠐⠒⠈⠀⠀⠀⠀⠀⠀⢀⡖⠋⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢠⠀⠀⠀⠀⣟⠒⠒⠛⣥⢦⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⢤⡜⠷⢄⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡎⠉⠙⡄⠰⡂⠀⢀⡇
⠀⠀⠀⠀⠀⢠⠖⢠⡀⠀⠀⠀⠀⢺⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠚⠁⠀⠉⠛⠉
⠀⠀⠀⠀⠀⢃⠀⣀⠇⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠀⠂⢁⢀⠀⠠⠀⠀⠀⠀⠄⡕⠀⠀⠀⠀⡐⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠀⠄⠈⠀⠀⠀⠀⠀⡠⠀⠈⡦⠄⠀⠀⠀⠀⠀⠀⠀⡠⠂⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠐⠊⠀⠁⠀⠂⠁⠀⣀⠄⠁⡀⠀⠀⠐⠈⠀⠀⠀⠠⡆⠠⡆⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠁⢠⡠⠐⠆
⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⠠⠀⠉
⠀⠀⠀⠀⢀⡄⠂⠀⠀⡄⠀⠂⠁⠀⠀⢠⠆⠀⠐⠖⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠒⢃⠀⠀⠀⠘⠀⠀⠀⠀⠐⠃⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠙⡂⠀⠠⠀⠀⠀⠀⠴⡄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠴⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠢⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠠⣀⠀⠤⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⡀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠉⠀⠨⠭⠍⠁⠉⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠣⡦⠀⠀⠀⠀⠈⠙⠀⠀⢀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⡀⡀⠀⠀⡅⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠆⢀⠈⠁⢀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠀⣀⣐⠀⠸⠄⠀⠂⠁⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠰⠄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⠄⠈⠑⠀⠀⠰⠀⠀⠀⢐⠂⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠱⢄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠄⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠁⠈⠂⠠⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠖⠀⠀⢀⡔⠀⠀⠀⠠⠈⠇
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠄⣀⠀⡀⠀⠀⠀⠊⣀⠀⠠⡀⠘⠁⠀⢁⠰⢦⠐⠝⠶⠂⠀⠀⢀⡠⢎⠁⠀⠀⠀⠀⠐⠀⠐⠂⠂⠁⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢄⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠉⠁⠺⣄⠀⠀⠀⠀⠀⠕⠀⠀⢄⠨⠁⠀⠀⠠⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠂⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠔⠈⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡀⠀⠀⠀⠐⠃⠀⠀⢀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠊⢀⠀⠀⠀⠀⠐⠠⠂⠐⠈⠢⠀⠀⠀⠀⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠄⠀⣀⢀⣀⣠⠀⠄⠈⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠄⡀⡀⠀⠀⠙⢀⠇⠁⢀⠀⠂⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠀⣇⡀⠂⢐
⠀⠀⠀⠀⠀⢰⠁⠐⠠⠁⠀⠒⠀⠀⠠⠐⠃⠀⠈⠃⢀⣀⠨⠀⠀⠀⡀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠆⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠙⢠⠒⣆⠉⠄⠀⠀⠀⠀⠀⠈⠁⢀⠐⠒⠂⠀⢀⠄⠀⠁⠀⠀⠉⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠘⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠈⠩⢭⣍⣙⡒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⣫⣶⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢮⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢻⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⠿⢇⣾⡿⣿⣯⠈⠀⠀⠀⠀⢀⡎⢩⣧⠀⠀⠀⠀⢀⡠⣤⠀⢠⣤⣶⣾⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣷⣶⣶⣿⣿⣃⣿⣟⢠⡃⠀⠀⠀⢸⠃⢸⣿⠀⠀⠀⠀⣿⣧⢸⣦⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣯⣭⣭⣭⣿⣿⣭⣭⣽⣿⣛⣛⣙⣛⣛⡂⠀⢠⣤⣿⣟⣸⠛⠿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⡏⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣯⣭⣝⣓⠖⢿⣾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣶⡶⠶⠄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⣛⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠘⢿⠆⠖⠂⠀⠀⠀⠀⢹⣿⣿⠟⠁⢺⣄⠙⣧⣿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⣤⠽⠀⢹⣷⣤⣿⣿⡗⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⢠⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⠀⠀⢈⣴⡿⢿⣿⣿⣿⡀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣠⣿⣾⠟⠉⠀⠀⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠄⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⠟⠁⠀⠀⠀⠀⠀⢘⣿⣿⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣂⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣶⣿⣿⠿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⣿⢶⣷⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⢟⡿⢋⡡⠞⣫⢿⣫⢿⣿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠀⠀⠀⡠⢊⣵⣿⠗⠁⠀⠀⣠⢚⣱⣿⣿⣿⡏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣿⣿⣦⠀⢀⠎⣠⣿⡿⠃⣰⣦⢤⠞⣱⡿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⣿⣿⣿⣿⡟⠁⠀⣿⡿⠁⠼⠛⠣⠏⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠂⠓⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⢀⢠⡤⠤⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⠉⢙⠇
⠀⠀⠀⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠀⠀⠀⠀⠀⡰⠀⠀⠳⡀
⠀⠀⠀⠁⠀⠀⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠁⠈⠀⠁⠀⠉⠁⠀⠀⠀⣱
⠀⠤⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⡀⣰⣾⠤⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢨⠂
⠀⠀⢣⠁⠀⠀⠀⡼⠁⠀⠛⠙⣄⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠄⠀⣀⢀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⢀⣀⡤⢐⣒⠚⠷⠊⠧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲⠒⠾⠼⠟⣲⠀⠀⠀⠀⠀⠠⠶
⠀⠀⣀⠀⠀⠘⠀⡀⢰⢛⡾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠥⠦⣀⣩⡗⠒⠄
⠀⠠⠀⠀⠀⠰⢄⠀⠀⠘⡇⢉⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠰⣷⠀⠀⠐⠂⠀⠀⠀⡏⠉⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣍⣉⡗
⠀⠀⠛⠀⠀⠀⠀⠀⠢⠤⠇⠀⠀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠠
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣀⡀⠀⠄⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣒⣂⣉⣭⣭⣭⣭⣭⣭⣭⣍⣁⣒⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡛⠻⠿⠯⠝⠛⠉⠛⠋⠙⠛⠭⠹⠿⢟⣻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⢀⣠⣶⣦⣄⣀⣠⣴⣿⣷⣤⣀⣀⣤⣶⣦⣀⢈⣹⡆
⠀⠀⠀⣀⣀⡀⠤⠤⠤⠐⠒⢒⣂⣈⣉⣉⣉⣉⣩⣭⣭⣭⣥⣤⣤⣤⣤⣬⣭⣭⣭⣉⣉⣉⣉⣉⣐⣒⠒⠂⠤⠤⠤⠄⣀⣀⡀
⢠⣚⣩⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⡿⣹⢁⣽⡁⠔⠒⠒⠲⣷⢈⠼⢌⣰⡷⠒⠒⠒⠄⣹⣅⢩⡹⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣭⣑⣢
⠈⡟⠛⠿⠿⣿⣿⡟⠛⢻⣿⣿⣿⡥⡌⡮⣀⣽⣿⣿⣷⣦⠈⠙⠤⠜⠉⢠⣶⣿⣿⣿⣅⣨⡎⡤⢹⣿⣿⣿⠛⠙⣿⣿⣿⠿⠿⠛⡏
⡎⠉⡇⠀⢀⣼⣄⠑⠶⠋⢉⣏⢹⡁⢊⠀⣾⣿⣿⣿⣿⣿⣇⠐⡄⡄⢀⣿⣿⣿⣿⣿⣿⡆⢈⠚⣽⢉⣏⠉⠳⠖⠁⣼⡄⠀⠀⡏⠉
⠈⠉⠀⠀⠈⠛⠋⠀⠀⠀⠻⠿⠀⢏⠁⠀⣿⣿⣿⣿⣿⣿⣿⢠⢸⢠⢸⣿⣿⣿⣿⣿⣿⡇⠀⢉⠇⠸⠿⠃⠀⠀⠈⠛⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⠀⠀⢹⣿⣿⣿⣿⣿⡏⡙⢁⠙⡈⣿⣿⣿⣿⣿⡿⠃⠀⢨⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢊⣁⡳⣄⡙⠛⠿⠟⢋⡞⣴⣿⣷⡜⣮⠛⠿⠟⠛⣁⡴⢃⣉⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡳⠜⠡⠴⠛⣉⡥⠄⠻⢰⣿⠿⢿⣷⠘⠃⠤⣍⡙⠓⠤⠘⠴⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠉⠒⠒⠒⣾⣿⣷⣖⣑⣦⡑⠻⠓⣩⣖⣑⣶⣿⣿⡗⠲⠒⠊⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⣀⣀⣤⣴⣿⢻⣿⠙⠙⠉⡄⢰⠀⡌⠙⠙⢹⣿⠻⣷⣦⣄⣀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⡇⠘⠻⡇⡖⠤⢀⣸⢀⠰⠔⡆⡻⠛⠀⡇⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣧⡐⣤⠁⠦⠸⣿⡰⠤⠆⣡⡔⣠⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣗⢱⣴⣒⢄⢔⣲⣴⢁⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢬⣁⡜⣄⣩⠼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠑⠦⣌⣉⠒⠤⡉⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⢍⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⣒⠦⢍⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣭⣷⣦⣌⣁⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣄⣠⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡜⠘⣯⠉⣿⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠘⢧⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣴⣦⣬⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣼⣿⠏⠉⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⢤⣭⣉⣉⣉⣉⣉⣉⣛⣛⣛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠤⠬⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣒⡒⠒⠒⠢⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣭⣭⣭⣭⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⡦⠄⠤⠄⠀⠀⠀⠉⠁⣀⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣭⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠾⠿⠛⠀⠀⠀⢸⠃
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣉⣤⡤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡶⠶⠖⠛⠛⠛⠉⠉⠁

⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠈⠁⣀⣸⡿⡇
⠋⣀⠀⣄⠙⣿⣿⣿⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⢠⡶⠲⠿⠉⢷⣿
⡄⢿⣧⣿⣿⣿⡿⠋⠁⣼⣵⣶⣶⡈⠀⢀⢦⣶⡆⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠀⠀⠁⠀⡀⠀⠀⢈⣿
⣿⣿⣿⣿⣿⣏⣁⣤⣴⣿⣿⣿⣿⣷⣧⣽⣗⣀⣀⠀⠀⠀⢀⣻⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⣐⢳⣶⣤⣐⢶⣿⣿⣷⣾⣤⣿
⣉⣉⣉⣋⣙⣛⣛⣛⣛⣿⣿⣿⣿⣿⠿⠿⠿⠿⣯⣀⡀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣬⡙⣿⣿⣴⣴⣶⣦⣤⣄⣶⣮⣟⣿⠻⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠛⠁⠀⠻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢀⣤⡆⠀⡀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣛⡖⠛⠒⢲⣛⣞⣛⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢉⣿⣿⣍⣉⣛⣿⣿⣿⣿⣿⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅⣜⣿⣿⣿⣿⣿⣿⣇⣾⠟⢋⣿⣿⣿⣿⣿⣿⣿⣿⡏⢈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⠿⠿⠿⠿⠟⠀⠀⠈⠀⠀⠀⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠉⢀⣀⣤⣤⣶⡶⠀⠀⠐⠂⠀⠂⠶⣶⣮⣭⣝⡻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣠⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣭⡻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⢤⣄⣀⡀⣤⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⠒⠚⠛⢶⠀⠀⣿⣿
⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠈⠉⢧⣀⡀⠀⠀⠀⣸⣄⣀⣀⣀⣀⠀⠀⣰⠋⠉⠙⠛⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⡀
⠀⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠚⠁⠀⠀⠈⠉⠉⠉⠉⠻⣤⣀⣀⣀⣀⣸⠓⠒⠒⠒⠒⠒⠒⠒⠻⠀⠀⠀⠀⠀⠼⠛⠋⠙⠛⣙⣀
⠀⠘⠣⢠⣤⣤⣤⣤⣤⣤⣄⠀⣠⣤⡀⣠⣤⡀⢠⡀⠀⠀⣠⣤⣤⡀⠀⣀⠀⢀⣤⣤⡀⣀⠀⠀⣀⠀⠀⠀⣠⣤⡀⢀⣤⣤⣤⣤⣄⠀⢠⡤⠄⠐⠿⠛
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠀⠛⠛⠀⠙⠛⠃⠀⠛⠛⠃⠀⠛⠛⠀⠛⠛⠛⠈⠛⠛⠀⠘⠛⠛⠛⠛⠀⠛⠛⠛⠀⠙⠛⠛⠛⠛⠛⠁⠈⠋⠁

⠀⠀⠀⠀⢠⣎⣠⠀⠀⠀⠀⠀⠀⢰⣆⣠⠀⠀⠀⠀⠀⠀⠀⣰⠗⠀⠀⠀⠀⠀⢀⣰⣟⣠⠀⠀⠀⠀⠀⠀⣠⣎⡄⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⢠⡠⠎⠀⠀⠀⠀⠀⣠⠖⠋⣹⠏⠁⠀⠀⣠⠶⠚⠉⣽⠁⠀⠀⠀⢀⣠⠤⢲⠟⠁
⣠⣤⡾⠋⠀⠀⠀⠀⠀⡀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣶⡟⠉⠀⠀⠀⠀⠀⠊⠀⠀⡴⠋⠀⠀⠀⠀⠁⠀⠀⣠⠏⠀⠀⠀⠰⠋⠀⢀⡟
⠉⠸⠃⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⡴⠒⢦⠀⠀⠀⠀⠀⠀⣠⣶⡦⠀⠀⠀⠀⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⣠⡜⠛⣧⠀⠀⠀⠀⠀⢈⡿⣿⡷⠀⠀⠀⠀⢀⣾⠿⣷
⠀⢀⣄⠀⠀⠈⢢⠀⠀⠀⠀⣨⠖⠀⠀⠀⠀⠀⣠⡞⢿⣻⠏⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠠⠈⠉⠣⣀⡾⠏⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⢷⡜⠉⠀⠀⠀⠀⠠⡾⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠨⡏⠉⠉⡅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡻⠟⠀⠀⠀⠀⠀⠀⣼⣯⠖⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠘⡟⠛⠛⡇⠀⠀⠀⣨⡷⢙⣦⠀⢠⣤⡀⠀⠀⢠⡾⢁⡼⠃⠀⠀⠀⠀⠀⢀⣴⠟⠛⠃⠀⠀⠀⠀⢀⠀⠋⠁⠀⠀⠀⢀⣤⣾⣿⠿⠋
⠀⢰⡇⠀⠀⡇⠀⠀⠀⠿⠦⠶⠿⠿⠶⠿⠀⠠⠞⠉⡴⠋⠀⠀⠀⠀⠀⢠⣴⡟⠁⠀⠀⠀⠀⠀⣠⣾⣿⣷⡦⠀⠀⠀⢠⣾⠛⠋⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠈
⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣶⣶⣶⣶⡄
⠀⠀⠙⠳⣦⣰⣷⠀⠀⠀⣾⣦⣤⣼⣿⡇⠀⠀⢸⣿⣿⣿⢿⠛⠂⠀⠀⠛⢻⣿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣴⣿⡟⠃⠀⠀⠀⠉⠉⠉⡩⠜
⣰⡖⠒⠛⣿⣿⣿⠀⠀⠰⠿⠭⢭⣿⣿⡇⠀⠀⣿⠃⠀⠀⢸⠀⠀⠀⠀⠀⣸⣏⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⢰⡆
⠙⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠻⠃⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⣴⣤⡀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠁⠀⠀⢠⣤⣤⡀⣾⠛⣷⠀⠀⠀⠀⡀⠀⣿⣻⡧⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⢼⣿⣛⣿⡷⠀⠀⠀⠽⠒⠒⠒⠒⠒⠂⠀⠀⢻⣧⣼⠃⣙⡛⠁⠀⠀⢰⣿⣿⣿⡏⠋⠁⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠛⠛⡟⠛⠃
⠴⠾⠿⠿⠿⠷⠶⠀⠀⢰⣶⣦⣶⣤⣤⡄⠀⠀⠀⠀⠀⢸⣏⣹⠆⠀⠀⠽⣿⣶⣿⠏⠀⠀⠀⠀⠀⠻⡷⠦⠄⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⣀⡀
⢼⡟⠛⢿⡆⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠀⢀⣤⣤⣄⣀⣠⣤⣤⡀⠀⣴⣶⣿⣶⣷⣶⣦⠀⢀⣀⣤⣶⣾⣿⠎⠀⠀⠀⢠⣾⣷⣿⣿⣷
⢸⣧⣥⣼⡇⣀⣀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠓⠚⠁⠀⣿⣥⣿⣿⣿⣼⣏⠀⠀⢉⣽⡿⠯⣉⠄⠀⠀⢠⣾⣿⣿⡿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⣠⠔⢲⢰⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡆⡇⠀⠀⠀⠀⠀⣼⠕⠂⢀⠀⢠⠂⠀⠈⢆⠀⠀⠀⡜⠁⠀⢸⢸⠴⣚⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢸⠀⡇⡇⠀⠀⠀⠀⡸⠁⠀⢠⢇⡴⠃⠲⡀⠀⠘⡄⠀⠀⡇⠀⠀⢸⠔⠊⠁⢀⡇
⠀⠀⠀⠀⠀⠀⡞⢳⢸⠀⢸⠀⡇⡇⠀⠀⠀⢀⠇⢀⡴⣷⢍⠀⠀⡠⣈⢦⠀⠱⡄⠀⢹⠉⡣⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣧⣸⣼⣠⣼⣄⣧⣇⣤⡀⠀⢺⠊⣏⠀⠐⠆⠀⠀⠰⠀⠀⣳⢄⣹⢀⡼⠃⠯⡖⠒⠊⠁⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣷⡀⠀⣐⣂⠀⢀⠟⡏⢀⡞⠈⠉⠉⠉⠁⠀⡤⡄⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢹⢦⡀⠀⠀⠀⠹⣄⣇⠙⡦⢍⡡⡴⠋⢠⢃⡟⠀⠀⠀⠀⠀⢀⡴⠋⠙⢢⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠭⠬⠉⡇⠀⠀⠀⠀⠈⢁⣹⡇⠀⠀⣸⣾⡓⠋⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡴⠚⠉⠓⡇⠀⣛⣛⠀⡇⠀⢀⠔⢺⠉⠉⢣⡈⢲⡞⢁⠇⠈⠙⡶⠦⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠰⡁⠀⠀⡀⠳⠤⢤⠤⠄⠇⠀⠆⠀⣸⠀⠀⢘⠑⢻⡟⠋⠃⠀⠀⡇⠀⢹⠀⠀⢸⠖⠖⠖⠒⠲⠚⠳⠲⡇
⠀⠀⠀⢀⠔⠁⠀⢄⣇⡠⠀⠘⢦⠀⠀⢰⠀⠀⢻⠀⠀⠘⠖⠑⣞⠒⠂⠀⠀⡷⠀⠈⡄⠀⢸⠀⠀⠰⣉⣱⠀⠀⠀⡇
⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢠⠇⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣧⠀⠀⡇⠀⢸⠀⠤⠤⠤⠬⠤⠄⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣇⣀⡀⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠀⠀⢳⠀⢸⣀⣉⣉⣉⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠒⠬⣷⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣟⡩⠭⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡞⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣇⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢼⣄
⠀⠀⠀⠀⠀⠀⠤⢤⡴⣫⡿⠓⠒⢾⢶⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣳⣯⣻⣦⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠺⠛⡏⠀⠀⣀⠜⠓⠚⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⢀⠴⠂⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠐⢄
⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒
⠀⠀⠀⠀⠉⣉⣽⣿⠿⡏⠁⠀⢠⠊⠉⢲⣀⡾⠋⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣽⣿⡿⠏⠁
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⢸⠀⠀⣰⠋⢧⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢁⣠⡆
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠠⡀⠂⠀⠀⠁⡐⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠟⣀⣤⣶⡇⠀⠀⠀⠀⠈⢔⢄⠀⠀⠂⠀⠀⠀⠠⡈⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠇⠀⠆⠀⠐⠀⢠⡤⠴⡆⠀⠀⠀⡟⢁⣤⣶⡇
⠀⠀⠀⠀⠀⣿⠿⠛⢁⡇⠀⠀⠀⠀⠀⠈⡀⢱⠆⠀⠀⠀⠀⠀⠙⢀⣀⣀⡐⠁⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⢐⠒⠿⠅⠀⠀⠀⣿⡿⠛⢉⣀
⠀⠀⠀⠀⠀⣃⣴⣾⡿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠀⠀⠀⠀⠀⢰⠃⢀⠃⠀⠀⠀⠀⢒⣛⣭⡅⠀⠀⠀⣏⣤⣶⡿⠇
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⠄⠀⠀⠀⠀⠀⠠⣬⣬⡇⠀⠀⠀⡿⠛⢉⣠⣦
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠜⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠩⠥⢖⡂⠀⠀⠀⣷⣾⡿⠟⠁
⠀⠀⠀⠀⡀⠋⣁⣤⣾⢇⡀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠀⠀⠀⠀⠀⠀⠀⣘⢒⣲⠇⠀⠀⣀⠛⢁⣤⣶⠗⡀
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠠⠠⢀⣀⣀⣀⣀⠉⠉⣉⣙⣉⣁⣈⠭⠂⠀⠀⠀⠀⠀⠂⠉⠭⠅⠀⠀⠠⠠⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠈⠢⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⢠⠀⠀⢠⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⡯⠐⠁⠀⠀⠀⠒⢸⣧⡆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⡟⣤⠀⠀⠀⠀⠀⠀⢠⠙⣅⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠓⠷⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣀⡱⠃⠀⡀⠀⠀⠀⠀⠀⠀⣀⢘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⡀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣥⣤⡦⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠨⠗⠂⠀⠀⠀⠘⠬⣿⡟⠦⡀⠀⠀⠀⠴⢪⣿⡧⠊⠀⠀⠀⠀⠀⠀⢀⡧⠄⢺⡇⠀⣹⠢⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣦⢀⠀⠀⣌⣠⡚⣉⠀⠀⠀⠀⠀⢀⣤⠊⠍⢹⣆⡌⠛⠛⠁⠀⣶⠋⠁⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⠀⡿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠓⠒⠒⠒⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣿⢁⡴⠶⠶⠀⠀⠀⠈⣿⡿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⢸⡿⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡄
⠀⠀⠀⠀⠀⢠⡿⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⢿⡄
⠀⠀⠀⠀⢀⣾⠃⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠘⣷
⠀⠀⠀⠀⣸⡏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⢠⣿⣀⣀⣀⣤⣼⣿⣤⣤⣤⣶⣶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⢶⠾⠷⣶⡶⣶⣤⣤⣿⣧⣤⣤⣀⣀⣿⡆
⠀⠀⣾⠟⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠻⣷⡀
⠀⢸⣏⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣹⡇
⠀⠀⠻⠷⠾⠿⠿⠿⣷⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠿⠷⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⢸⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾
⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣶⡄⠀⠀⣠⣶⡶⠶⠶⠶⣶⣤⣶⠶⠶⠶⠶⣶⣤⠀⠀⢀⣴⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠸⣇⢹⣿⣷⣝⠆⢰⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⣠⢶⣿⣿⣿⣧⠀⢛⣽⣿⣿⢃⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣿⡟⢠⣿⣿⣿⣿⣯⡓⠂⠀⠀⠀⠀⠒⣻⣿⣿⣿⣿⣇⠘⣿⣿⡏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡟⢀⣾⣿⣿⣿⣿⠿⣽⠃⠀⠀⠀⠀⢫⡽⢿⣿⣿⣿⣿⣆⠹⣿⣧⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣰⣿⣿⣿⣿⡿⢭⡛⠂⠀⠀⢀⠀⠀⠀⠚⣩⠽⣿⣿⣿⣿⣦⡘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⡿⠿⠿⠛⠋⠉⠀⣀⡀⠠⠭⡭⠭⠥⢀⣀⡀⠈⠉⠛⠻⠿⠿⣟⣊⣀
⠶⣤⢤⣤⣀⣀⣀⣺⣿⣿⣿⣿⣿⣶⣶⣶⣤⣬⣭⣽⣿⣿⣿⣿⣭⣭⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣃⣀⣀⣀⣤⣤⣴
⠀⠈⠛⠶⣬⣉⡉⠛⠓⠌⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠔⠛⠛⣉⣩⣵⠶⠋
⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣤⣤⣄⣀⣀⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣀⣀⣤⣤⣤⣴⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠿⣿⠿⠿⠿⣿⣛⠿⢿⣿⣿⣿⣿⣿⣿⠿⣻⣿⠿⠿⠿⣿⣿⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣇⠀⢰⣿⠉⢳⣌⢿⡿⣿⣿⡿⣡⠏⠙⣿⠀⠀⣸⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⠮⢥⣤⣼⣿⣾⠷⠋⣿⣿⣿⣄⣀⡽⠴⢾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣛⣿⣿⠋⠉⠀⠘⢤⡀⠀⣰⡼⠁⠀⠉⢻⣿⣿⣛⣉⣀⣀⡀
⠀⠀⠀⠀⠠⠔⠚⠋⠉⢉⣀⣠⣤⢽⣯⣭⠭⠀⠀⠀⠙⡶⠃⠀⠀⠈⢭⣩⣽⢥⣤⣀⣈⠉⠉⠙⠒⠤
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⡠⢔⣪⠿⣖⣊⡥⠀⣀⣴⣿⣦⡀⠀⣈⣑⡾⠯⣓⠤⣀⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠐⢉⠴⠋⠀⢀⠜⠙⠛⢛⣫⡇⠀⢹⣝⡛⠛⠙⢌⠀⠈⠑⢄⠁⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠃⠀⠀⠀⢿⣿⡇⠀⢹⣿⡷⠀⠀⠀⠱⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠙⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣼⣭⣼⣽⣯⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠄
⠀⡇⡁⠀⡇⢀⣀⠀⠀⣤⡦⣦⢴⠴⠤⠆⠦⡤⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠈⠄⠄⠀⠀⠀⠄
⠀⠇⠸⠂⡕⠺⠐⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⡇⠘⠀⠁⠀⠈⠁⢀⠀⠐⠀⠀⠀⠁⠐⡢⠄⠀⠀⠠
⠀⠁⠀⠀⠀⠁⠀⠁⠀⠉⠈⠀⠀⠁⠀⠈⠀⠁⠀⠀⠘⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠈⠙⠂⠀⠠⠀⠀⠂⠀⡀⠠⠀⠠⢀
⠀⠀⠀⠠⠔⠀⠀⢀⡀⠀⡅⠀⠀⠀⢉⠀⠀⠀⠀⠐⠐⡐⠬⠀⠀⠀⢀⠠⠌⡄⠠⠀⠀⠀⠀⠀⠀⠪⡀⠀⠁
⠀⠀⠈⢀⠐⠀⠂⠁⠠⠀⠀⠀⠀⠀⠀⠈⠡⠀⡀⠐⠀⢂⠔⠀⡀⠀⣋⠀⠀⠀⡚⡀⠀⢤⣈⠀⠰⠒⠦
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⠊⠀⠈⠁⠀⠀⠸⡀⠈⠦⢠⠄⠋⠀⡇⠐⠣⠑⡕⠀⠀⠀⠀⡀⢆⠆
⠀⠄⠀⠀⠠⠀⠀⡀⡀⠀⠠⠤⠁⠀⠀⠀⢀⡁⠀⡀⢀⡀⠀⠀⠀⠈⠲⠤⠤⠤⠒⠉⠁⠀⢀⠄⠄⠀⠐⠀⠀⠄
⠀⠀⠀⠀⢀⠃⠀⠀⢆⠀⢈⠈⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠁⣀⠀⠀⡠⠀⠈⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠐⠀⠀⠁⠀⠀⠡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠠⠀⠈⠃⠊⠐⠄⠀⢀⠀⠀⠠⡠
⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠡⠀⠀⠀⠀⣤⠀⠀⣴⣶⣶⡄⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠠⢲⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⣿⠀⠀⣿⠀⢸⡇⣿⣠⡷⠀⠀⠁⠀⠀⠈⠀⠀⠀⠐⠈⠀⠀⠀⠀⢀
⠀⠀⣆⢬⢹⠀⠀⢸⠀⠀⠀⠠⡀⢀⠄⠀⣿⠀⠀⣿⠛⢻⡇⣿⢻⣧⠀⠀⡗⠂⠙⠻⡀
⠀⠀⠁⠀⠀⠀⡄⠬⠀⠀⠀⠀⠁⠀⠀⠀⠿⠟⠀⠟⠀⠘⠃⠛⠛⠃⠀⠀⠁⠀⠀⠀⡇⠀⠈⠀⠀⠀⠁⠀⠄
⠠⣀⠁⠀⢀⠠⠀⠈⠀⠀⠀⠦⠄⠀⡄⠀⢸⣟⣯⣟⢻⡛⣻⡟⣿⠀⠀⠀⠀⠐⠐⠀⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠄⠀⣂⠀⡀⠀⠀⠀⠀⣁⠀⡀
⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠈⠔⠀⠀⠀⠀⠀⠈⠀⠀⠣⠀⠀⠀⠈⡅⠀⠀⠀⠀⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠈⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⡀⠀⠈⠅
⠀⠀⣿⢸⡇⣀⡀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠁⠐⢈⠤⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⣿⢻⡏⢹⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡐⡀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠁⠀⠀⠀⠈
⠀⠀⠙⠘⠃⠛⠃⠙⠛⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⣿⣿⣿⣻⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣛⣻⣿⣟⣻