Ring Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣛⣫⣭⣭⣭⣝⣓⠶⣄⡀⠀⠀⠀⢀⡾⠁⣿⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡵⠛⠉⠀⠀⠀⠀⠀⠉⠻⢮⡻⣦⠀⣠⠟⠀⠀⢸⣦⡴⠚⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣷⠋⣀⡤⠞⠋⣿⠉⢛⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠿⠭⣤⣤⣤⣼⣶⣯⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡷⢦⣄⠀⠀⠈⣧⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠞⣋⣩⣭⣭⣭⣍⣛⠶⣤⡀⠀⠀⠀⠀⣼⢱⡇⠀⠈⠛⠶⣄⣹⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠶⢻⡙⣦⠀⠀⠀⠈⠉⠛⢦⡙⢦⡀⢀⣴⢣⡟⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡾⠁⠀⠀⠻⣮⡳⢦⣀⡀⠀⠀⠀⢙⣦⠻⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⢰⡇⡾⠁⠀⠀⠀⠀⠈⠙⠲⢬⣝⣛⣛⣛⣭⡽⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡛⢦⣄⣀⣀⣀⣀⣀⣤⠶⢋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣭⣭⣭⣭⡭⠶⠚⠋




⣏⠉⣹⣛⣛⣟⣛⣻⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣛⣿⣿⣛⣛⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠄⠀⠤⢀⣶⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣫⣭⡶⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣵⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣤⣶⡆⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢶⣿⣿⢇⣿⣮⡻⣿⢹⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣝⠿⠸⠿⢟⣛⡀⣬⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢚⣫⣭⣵⣶⡘⣿⣿⣿⠇⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⡌⢿⣿⠀⣿⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣄⠛⠀⡟⠁⣼⣿⣿⣿⣶⣦⣄⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣦⣄⠙⠂⠀⠁⠈⠉⠛⠛⠻⠿⢋⣵⣾⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢟⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⢀⣤⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣈⣋⣭⣵⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠺⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠀⠐⠳⠂⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡈⠒⠠⠀⠄⠂⢉⡳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢖⠛⡟⠀⣀⣤⣤⣤⣤⣄⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⡩⢔⠊⡣⢸⢃⡾⢹⣯⡿⣗⠿⠍⡉⠩⡗⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣋⠕⠚⠟⠀⠀⠠⠀⢸⣸⣧⠀⢋⠂⠘⢯⡽⠿⣶⠷⡸⢿⣧⠉⠲⢄⡢⢀⠀⠀⠀⠀⠀⠀⠈⠣⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠵⠊⠀⠀⠀⠀⠀⠀⠐⣪⣼⣇⢯⡂⢡⠡⡁⠀⠻⣴⠋⡰⠉⡸⣏⣇⠀⠀⠉⠳⢍⡢⠀⠀⠀⠀⠀⠑⢝⢎⢷⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠹⣌⢇⠳⠁⠀⣠⠞⠁⠀⠀⢀⢄⡿⡿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠢⢪⣣⣳⡀
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠙⣦⣁⡤⠚⠁⠀⠀⠀⠀⠀⠲⣱⣇⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⣄⠀⣀⠑⡱⣵⢷
⠀⠀⠀⠀⢸⢰⠀⠀⢀⣀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⠶⣋⠥⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡪⢝⠢⣤⣀⠀⠀⠀⠀⠀⠙⢮⢳⡙⢎⢮⣽⣾⡆
⠀⠀⠀⠀⢿⠈⢦⡐⠧⣣⠟⠁⠀⠀⠀⠀⣀⡠⠴⣚⡩⡖⠋⠉⠂⠀⠀⠀⠀⠀⠀⣠⢚⠠⠞⢉⣆⠀⠠⠉⠟⠶⣤⣀⠀⠀⠈⣳⡙⠈⣿⣿⣸⠁
⠀⠀⠀⠀⠸⣆⠀⠉⠒⠷⠤⠴⠖⠒⠋⠉⠠⠀⠈⠁⠉⠰⢕⠉⠀⠀⠀⠀⢀⣤⣮⡡⢔⠉⣒⠀⠀⠀⠀⠀⠀⠀⠀⠊⠩⠙⠛⠹⡏⠈⣼⢣⡏
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠈⠙⠳⣦⣄⡠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡷⣳⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠀⠀⢀⠤⠞⣱⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠖⠒⠛⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣁⣸⣷⣏⣹⣾⣇⣈⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⢦⡀⢹⡆⢰⠏⢀⡴⢋⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⢦⡙⢦⣿⣿⡴⢋⡴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠙⢦⡙⢋⡴⠋⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠙⠋⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣤⠶⠚⠛⠛⠛⠛⠓⠶⣤⣀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣝⡲⢦⣤⣀⣀⣀⣀⣤⠴⢖⣋⡵⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠒⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠟⠻⡛⣛⡻⠟⠛⠛⠿⣟⣛⢛⡿⠛⠿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⡏⠀⢀⣠⠤⠒⡟⠓⢤⡀⠀⠀⢀⡠⠞⠻⡲⠤⣄⡀⠀⢹⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⠒⠻⣾⠿⠷⠶⢾⡒⠒⠒⠚⢳⡞⠛⠒⠒⠒⡿⠶⠾⠿⣷⠟⢒⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣕⢄⠙⣧⡀⠀⠀⢳⠀⠀⠀⡼⢣⠀⠀⠀⡼⠁⠀⢠⡾⠋⡠⣫⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣷⣌⢿⡢⡀⠈⢇⠀⠀⡇⢸⠀⠀⣰⠃⢀⢔⡟⣡⣾⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣵⡝⢦⡈⣆⠀⡇⢸⠀⢰⢃⡴⣣⣮⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠙⢼⣼⠇⠸⣧⡷⠋⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠳⣦⡹⡄⢀⢏⣤⠞⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⢈⣻⣷⣾⣟⡁⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣤⡶⠞⠛⠋⠉⠉⠉⠉⠙⠛⠳⢶⣤⡀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣋⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣛⢶⣤⣄⣀⠀⠀⠀⠀⣀⣀⣤⠶⣛⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠿⣿⣿⣿⣿⣿⣿⠿⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠡⡀⠀⠀⢀⠜⠣⡀⠀⠀⢀⠜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠘⢆⠠⠊⠀⠀⠱⡀⡰⠃⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠈⠟⠀⠀⠀⠀⠻⠁⠀⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠘⡀⠀⠀⢀⠃⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⡜⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠘⠒⠚⠒⠒⠓⠒⠃⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⡠⠔⠒⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠒⠢⢄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠄⣀⣀⣀⣀⣀⣀⠠⠤⠐⠊⠁


⣀⣀⣀⡀⡀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠁⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⢀⣠⣶⢻⡄⠀⠀⠀⢀⡠⠴⣚⣛⣛⡃⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⣰⡾⢳⡼⠙⠋⠀⠀⢀⡨⠔⠒⠉⡸⠹⢹⢙⢷⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡔⢠⠤⣿⡌⠛⠁⠀⠀⡠⠔⠊⣁⣀⠠⠤⠴⠥⢠⣃⣈⡏⠀⠇
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠻⠿⣏⠓⠃⠀⠀⣀⠔⠋⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠄⣀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⡜⡠⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢸⢰⣿⢏⠀⠀⠀⠀⠀⠀⡨⣩⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢁⠀⠀⠀⠀⡠⣢⠊⠀⠀⠀⠀⠀⠠⢺⡈⢏⢣⡳⢄⠀⠀⡠⢊⠜⠁⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⡾⡄⠀⠀⡰⡵⠁⠀⢀⡠⠔⠂⠀⠀⠀⢣⠈⠀⠙⠦⡙⠮⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠱⡁⠀⠸⣱⢷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠘⢄⠁⡯⣍⣉⣁⠠⠀⠀⠀⠀⠀⢀⢀⡤⠚⠑⠤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠤⠤⣄⣀⡤⢾⣤⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡘⠒⠂⠀⠀⠀⠀⢀⡰⠌⠊⠁⠀⠀⠀⠀⠀⠉⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣋⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠄⣀⣀⣀⣀⣀⡀⠼⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠛⠉⠀⠀⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⢀⣤⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠿⠟⠛⠿⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠃⢠⡟⠁⠀⠀⠀⣽⣷⣄⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⠇⠀⠀⠀⢸⡏⠈⠛⣦⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣿⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⣿⡄⠀⠀⢸⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣷
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠘⣷⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣦⣌⠙⠶⢦⣤⣤⣤⣴⠾⠛⠁⣠⠟⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣤⣤⣤⡴⠶⠛⠁⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⠃⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⣠⣤⠴⠟⠁⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⣀⣀⣀⣀⣀⣠⣤⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣤⣤⣤⣤⣀⣀⣀⢤⣀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⢻⣭⡽⢟⡋⠉⠀⠘⠃⠘⢃⡀⠉⢡⡲⠜⢳⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡷⣺⣄⠋⠁⣴⡷⡻⣿⢻⢿⢧⣶⣄⠀⡦⠙⠆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⠀⠀⣈⠚⢿⣎⠁⠙⠄⠀⠎⢀⣸⠿⠟⣄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠀⠞⢋⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⡙⠓⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⡿⠗⠾⠀⠀⠔⠚⢻⠆⠀⢻⡟⠒⠀⠀⠓⡈⠻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠀⠈⠀⣿⣦⡀⠀⠘⠀⠀⠀⠁⢀⣴⠋⣄⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣆⠲⡄⣇⡤⠝⠂⢶⣆⠒⢒⠆⡉⠛⢳⢟⡡⣾⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢖⣻⠽⡷⠻⣯⢧⡘⠂⠘⠁⢀⣉⠰⠧⣀⠁⢀⡠⠞⢾⣁⣹⠯⣝⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⣴⣿⡁⢀⣷⠒⢬⡶⠌⣿⡒⠶⠶⠶⡶⠶⣶⣿⡟⠃⠐⠚⣽⡫⢄⣽⡟⢦⡈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣠⣶⣿⣿⠿⣟⡛⢧⡀⠀⢰⡿⠿⡓⠒⠊⠛⠟⠿⢡⠇⠀⣠⢾⣩⣿⣻⣿⣧⣼⡇⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⡿⠟⠿⠛⠋⠉⠁⠀⠠⢷⡀⠈⢇⡘⢀⣴⣶⡄⢘⣠⠋⢀⣼⠁⠀⠀⠀⠀⠈⠉⠋⠙⠦⣄⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢠⣞⣡⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⢀⣠⣀⣽⣾⣷⣄⠘⠃⠻⠿⠭⠃⠘⣁⣴⡛⠚⠿⠷⠤⠄⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⢨⢧
⠀⠀⠀⠀⠀⣠⣯⠱⣧⠞⠁⠀⠀⠀⠀⠠⢤⣒⣿⣿⡿⠷⠾⠟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠲⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠳⣼⡏⢇
⠀⠀⠀⠀⡰⣳⣿⠈⠁⠀⠀⢀⣠⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⠘⣿⢸
⠀⠀⠀⢠⢃⡿⠃⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⡞⠘⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣼⠂
⠀⠀⠀⡇⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠇⠀⣰⡏
⠀⠀⠀⢹⡄⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣿⣿⣿⠏⠀⡰⠋
⠀⠀⠀⠀⠳⣄⠀⠹⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⡶⠚⠋⡿⡇⢿⣿⠟⢁⡴⠊
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣿⣿⠈⡏⡟⡛⡶⢦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠖⠛⢻⠉⠈⡄⡇⢐⣆⡧⠗⣋⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡙⠻⠦⣇⡇⡇⢸⠀⢸⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠤⢷⣛⡭⠵⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠦⢭⣍⣛⣒⣾⣤⠤⣤⣤⣤⣤⣤⠤⢠⣤⣤⣶⣶⣶⠾⠭⠗⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⡰⠊⠱⠟⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣲⣾⣭⡗⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠁⢄⣤⡴⠶⠄⠀⣀⠀⡠⠒⡩⠄⠒⠀⠒⠠⢍⠲⢄⠀⠀⠀⠀⠈⠛⢟⢛⣫⣅⣀⣀⡀⠀⣠⣾⠟⠉⠉⠀⠀⠉⠙⢿⣦⡀
⠀⠀⠀⠀⡠⢊⡡⠐⠒⠀⠒⠠⣍⠳⠎⠀⠀⠀⠀⠀⠀⠀⠘⢌⢢⠀⠀⠀⠀⣠⣶⠿⠛⠉⠉⠙⠛⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠌⡰⠁⠀⠀⠀⠀⠀⢀⢠⠑⠌⢂⠀⠀⠀⠀⠀⠀⠀⠈⠈⡆⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⢠⣿⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠘⢠⠁⠀⠀⠀⠀⠀⠀⠸⢠⠀⠘⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢄⢣⠀⠃⠇⠀⠀⠀⠀⠀⠀⡐⡠⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⣿⠇⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠈⡄⢂⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⠼⣀⡀⠀⠀⣀⠠⢊⠔⠁⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⣄⣀⣀⣀⣀⣤⡾⠋
⠀⠀⠀⠈⠢⡑⠄⣀⡀⠀⢀⡀⠄⣊⠜⠑⠂⠠⠤⠄⠐⠊⠁⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⢀⣠⣾⠟⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠂⠠⠤⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠻⣄⠀⠀⣠⠟⠉⠙⠷⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣤⣤⣤⣼⣷⣾⣧⣤⣤⣤⢤⠬⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠙⣿⡀⠀⠀⠀⠀⣿⠀⢻⡄⠀⠀⠀⢀⣴⠏⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠘⢿⡄⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⣾⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⢿⡄⠀⠀⢿⡄⠈⣿⠀⢀⣾⠏⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡿⠶⠶⢼⣷⣤⣿⣦⣼⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣀⡀⣀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠋⠀⠀⠀⣠⣤⡶⠶⠛⠛⠻⠶⣦⣄⡀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠙⠻⢿⣷⣶⣶⣶⣶⣶⣤⣤⣴⣶⣿⠿⠛⠁⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⠈⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣦⣤⣤⣤⣤⣤⣴⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣴⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠿⠛⠉⠀⢀⣀⣤⡤⠤⠤⠶⠿⠓⠒⠀⠒⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⢀⣠⣤⠶⠛⣉⣡⣤⡤⠤⠀⠀⠄⠤⠤⠤⠤⠀⠀⠀⠀⠈⠐⠲⢤⣀⠉⠙⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⢠⡿⢉⣠⠶⠋⣁⡀⠀⠀⠐⠚⠛⠲⠶⠖⠂⠀⠀⠀⢀⣁⡀⠀⠀⠀⠙⠳⢦⣄⠀⠙⠻⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⣠⡟⢠⠏⠀⢀⠤⠀⣀⣀⣀⣤⣤⠤⠤⢤⣤⣴⣦⣤⣄⣀⠈⠉⠛⢦⣄⠓⢶⣄⠙⠳⣄⠀⠈⠻⣿⣿⣷⡄
⠀⠀⠀⣼⣷⣦⣾⣿⠟⠀⠀⢰⠏⣰⠋⢀⠖⠁⢀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠙⢦⣄⠀⠉⢳⣄⠉⠳⡄⠘⢇⡀⠀⠈⢻⣿⣿⠆
⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⣠⠏⣰⠃⠀⠉⢀⣴⠟⠉⠀⠉⠉⠉⠀⠉⠉⠉⠉⠉⠙⠳⠶⣬⣷⣀⠈⠙⢦⡀⠙⢷⡄⠈⠀⠀⠱⣦⠀⠈⣿⣧⣄⡀
⠀⠀⢿⣿⣿⣿⡏⠀⠀⡼⠋⢰⠃⢠⠆⢰⠟⠁⣠⠞⠛⠛⠉⣉⡉⠉⠙⠛⠛⠒⠲⠦⠤⡀⠀⠉⠻⢦⡀⠙⢧⡀⠡⡀⠀⠀⡀⠘⢷⠀⠘⢿⣿⣿⣦
⠀⠀⠈⣿⣿⡟⠀⠀⡜⠁⢠⠃⣰⠏⣰⠃⣠⡾⠃⠐⠚⢛⡉⢉⣉⠉⠛⠛⠛⠒⠒⠀⠀⠀⡈⠑⠠⡀⠙⢦⡀⠹⡄⠀⠀⠀⠹⡄⠀⢲⠀⠈⣿⣿⣿
⠀⣠⣶⣿⡟⠀⢀⡞⠀⣰⠃⠠⠋⣼⠃⣴⠏⡴⠃⡴⠊⢁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢆⠀⢳⡄⠹⡆⠀⡄⠀⠻⠀⠸⡆⠰⣿⣿⣿
⢰⣿⣿⣿⠃⠀⡾⠁⠀⢁⡾⠁⡼⠁⣰⠋⠈⠀⠀⢠⠞⢁⣤⠄⠀⠀⠀⠀⢀⣈⣛⠓⠦⠀⠙⠀⠀⣆⠈⢧⠀⢷⠀⠙⣆⠀⡶⠀⡇⠀⠇⠀⣿⣿⠉
⣾⣿⣿⣿⠀⢸⠃⢠⠀⡾⠁⣰⠃⣰⢇⡄⢠⠂⡐⠁⡴⠋⣀⣴⠟⠀⢠⠖⠋⣀⡀⠀⢄⠀⢢⠀⠀⠘⣦⠈⠇⠘⡄⠀⠀⠀⡁⠀⡇⠀⡇⠀⣹⣿⣦
⣿⣿⣿⡇⠀⡎⠀⠀⠀⠁⣴⠏⢠⠏⡼⢠⡏⢠⠀⡘⠁⠀⡉⢹⡂⠀⢀⠀⢀⣨⡙⣦⠈⢣⡀⢃⠀⠀⠀⢀⡆⢰⡇⠀⡆⠀⡇⠀⠀⠀⡇⠀⢹⣿⣿
⠹⣿⣿⡇⠀⠁⠘⠀⠀⠀⣿⠀⠀⡈⢠⡿⢡⠇⣴⠀⠀⣸⠃⠾⠀⠀⠈⠆⢿⣀⡿⠈⠀⠘⣇⠈⡇⠀⠀⢹⡇⢸⡇⢰⡇⠀⡇⠀⠀⠀⠁⠀⣸⣿⣿
⠀⢿⣿⡇⠀⡇⠀⠀⠁⠀⡟⢀⠐⡇⢸⠁⠈⢠⠏⣰⠀⣿⠂⠀⠀⠀⠀⢀⡈⠉⣡⠀⠀⢠⡿⠀⠓⢀⠀⢺⠁⠈⠀⢸⠇⠀⠁⠀⡀⠀⠀⠀⢹⣿⣿
⢰⣿⣿⡇⠀⣷⠀⠡⠀⠀⠀⡘⠀⢇⠈⡆⠀⠀⢦⠙⠀⠙⢧⣀⠀⠙⢶⣤⣀⣈⠡⠴⠀⠈⠁⠀⠀⣸⠀⡜⠀⡐⠀⣿⠀⡿⠀⢀⠃⡀⠀⠀⢸⡿⠋
⠸⣿⣿⡇⠀⠸⡄⠀⢇⢠⠀⠹⣄⠘⣆⠸⡀⢀⠘⣧⠀⠱⢄⠙⢷⡐⠦⢤⡭⠀⠀⠂⢀⣠⠎⢀⡰⠃⠴⠀⣰⠁⢰⡇⢀⠂⠀⣸⠀⡷⠀⠀⣸⣿⣆
⠀⣿⣿⣷⡄⠀⢳⡀⠈⢎⢧⠀⠈⢧⠘⠀⠻⡌⠣⠈⠷⣄⠀⠑⠀⠉⠓⠲⠦⠤⠶⠟⠉⠀⡰⢋⣤⠄⠀⣴⠃⢠⡾⠃⠌⠀⢠⡇⢰⠇⠀⢠⣿⣿⣿
⠀⢿⣿⣿⣿⡀⠀⠱⡀⠰⡄⠁⡀⠈⠁⠀⠻⣦⣄⠙⢦⣌⡙⠓⠢⠄⠁⠀⢀⣀⣤⠶⠀⣠⠞⠋⠀⣠⠞⢁⡴⠋⡀⠘⠀⣠⠀⢠⡟⣰⠀⣾⣿⣿
⠀⠘⢿⣿⣿⣷⡀⠀⠹⣆⠑⣄⠳⣄⠙⠦⣄⡀⠙⠳⠄⢀⠉⠁⠐⠀⠀⠉⠉⢉⣠⠤⠚⠁⢀⡤⠚⢁⡠⠊⠀⠨⠀⢀⡼⠋⢀⡞⠀⠋⢰⣿⣿⡿⠃
⠀⠀⠀⢀⣿⣿⣷⡀⠀⠈⢧⠙⢧⠈⠳⢤⣌⣙⠛⠶⠦⢤⣀⠉⠛⠶⠶⠂⠉⠉⠀⣠⡶⠟⠉⢀⡴⠋⠀⠀⠀⢀⡴⠋⠀⡠⠟⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠘⢿⣿⣿⣷⣄⠀⠀⢠⡈⠳⣦⣀⠈⠙⠿⠶⠶⣤⣤⣀⡉⠓⠒⠒⠒⠒⠊⠁⠀⠴⠚⢁⡤⠞⠁⣠⡴⠋⢀⣴⠟⠁⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠑⠀⢍⣉⠙⠳⠦⣤⣄⡀⠀⠈⠉⠓⠛⠒⠶⠖⠚⠉⠐⠠⠀⠉⣠⡴⠟⠁⠀⠀⠛⠁⣠⣶⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⡀⠀⠈⠙⠲⠶⣤⣬⣉⡙⠒⠲⠦⢤⣤⣤⣀⣀⣠⣤⠴⠞⠛⢉⣠⠖⠁⠀⢀⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣦⣄⡀⠀⠀⠀⠉⠛⠻⠷⠦⠤⠤⠤⠤⠬⠥⠤⠠⠒⠋⠉⢀⣠⣴⣶⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⢿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠟⠉⠉⠛⠿⠛⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡾⣿⡶⣐⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⢹⣿⣆⣼⣻⠿⣿⣾⢡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠀⢻⡿⣿⣷⣿⣿⣿⣶⡯⣘⣛⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⢿⢹⣿⣿⡿⣫⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠸⢸⣿⣫⡾⠛⣵⡾⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣶⣤⣤⣀⠘⠁⠁⠀⢸⣿⠀⠸⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠻⢿⢟⣵⣾⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣶⣤⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣻⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠘⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡷⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣾⣿⡿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣶⣶⣤⣤⣶⣶⣾⣿⣿⡿⠟⠉⠘⠿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⡄⡰⠙⠄⠰⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠘⡇⠀⣸⠃⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣄⢱⣠⠇⣠⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣀⣽⣷⣿⣾⣅⣀⠉⠻⣦⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣴⠟⠉⠀⠀⠀⠀⢈⣽⠿⠛⠉⠁⠀⣀⣀⡀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡾⠁⠀⠀⠀⠀⠀⣴⠟⠁⢀⣤⣶⠛⢻⡍⠉⠙⠛⠶⣤⡀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⣴⠏⠁⢹⠀⢸⡇⠀⠀⠀⠀⠈⠻⣦⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠸⡇⠀⠀⠀⠀⣸⠇⠀⣼⠃⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠻⣄⠀⠀⠀⣿⠀⢠⡏⠀⠀⣴⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⠷⣤⣀⣿⣀⣘⣧⡴⠞⠁⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠉⠉⠉⠉⢁⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣟⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠙⠶⢤⣤⣄⣠⣤⡴⠾⠋⠁⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⡇⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⢈⣀⣀⣁⣀⣈⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠠⡮⢵⠮⠬⢶⠭⣧⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⢙⢦⣣⢀⣣⢞⠁⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⣁⠤⠙⠟⠡⢄⡉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⡄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡜⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⣉⣀⣈⣉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡴⠁⡴⠊⠁⡇⢸⠀⠀⠉⠲⣄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢧⠀⠀⠀⠀⠀⡘⢀⠎⠀⠀⢀⠇⠸⠀⠀⠀⠀⠈⢆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠣⡀⠀⠀⠀⡇⢸⠀⠀⡠⠊⡠⠃⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠓⠢⠤⠧⠼⠒⢋⡠⠞⠁⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠲⡖⠺⡉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠲⠤⣀⣀⣀⡠⠴⠊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠤⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡤⣤⢤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣥⣴⣯⣤⣧⣤⣽⣦⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡹⣄⠀⡇⠀⡼⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣦⣧⣼⣷⣋⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠦⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⡴⠖⠚⠛⠛⠛⠓⠶⢤⣀⠀⠈⠛⢦⡤⠶⠚⠛⠛⠛⠓⠲⢤⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⠏⠈⢳⣦⡀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠟⠈⢳⡄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⡏⠀⠀⠀⢻⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠈⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡆⠀⠀⠀⢸⠇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳⡄⠀⢠⠏⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣶⡋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠛⠦⣄⣀⠀⠀⠀⠀⢀⣀⣿⣄⠀⠀⠙⠶⣏⣀⡀⠀⠀⠀⢀⣀⣠⠴⠋⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⢈⣻⢦⣄⠀⠀⠉⠉⠉⠛⠉⠉⠉⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⣤⠤⠴⠶⠚⠋⠁⠀⠈⠙⠓⠶⠦⠤⠤⠤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠻⣄⠀⠀⣠⠟⠉⠙⠷⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣤⣤⣤⣼⣷⣾⣧⣤⣤⣤⣤⠬⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⣿⡀⠀⠀⠀⠀⣿⠀⢻⡄⠀⠀⠀⢀⣼⠏⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢿⡄⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣾⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⢿⡄⠀⠀⢿⡄⠈⣿⠀⢀⣾⡏⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⡿⠶⠶⢼⣷⣤⣿⣦⣾⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣀⣀⣀⣀⠀⣀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⣀⣤⡶⠶⠛⠛⠻⠶⣦⣄⡀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠙⠻⢿⣿⣶⣶⣶⣶⣶⣤⣤⣴⣶⣿⠿⠋⠁⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠉⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⢀⣀⣤⡒⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⣤⠲⣼⣟⣨⠷⠀⠀⠀⣀⠤⠒⣋⡩⢭⢥⡤⣭⣄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⢀⣠⢼⣿⢘⠷⠇⠉⠁⠀⠀⢀⡡⠔⠚⠉⠀⢀⡞⡸⢡⠃⡟⡷⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡴⠀⡴⢼⢷⡄⠙⠁⠀⠀⠀⣀⠴⠊⠁⣀⣀⣠⣤⣤⣯⠤⣠⣏⣐⣸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢯⣿⣏⠻⠾⠁⠀⠀⠀⢀⡴⠊⠁⢀⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⡠⠚⠁⠀⠀⣰⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠇⣪⣵⠶⠒⠒⠒⠒⠒⠒⠒⠢⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⠄⡰⠋⠀⠀⠀⠀⠀⠀⢸⢸⢻⣟⢆⠀⠀⠀⠀⠀⠀⢀⢔⡴⠃⠀⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⡰⣡⠎⠀⠀⠀⠀⠀⢀⡠⠂⢻⡈⠸⡘⢦⡑⢄⠀⠀⡠⢞⡥⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠸⡾⡄⠀⠀⢀⡜⡵⠃⠀⠀⢀⣠⠴⠒⠁⠀⠀⠀⢣⠀⠁⠀⠑⢄⡙⠲⣵⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢣⠛⠀⠀⡜⣸⣥⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠳⡀⠐⢀⡯⠭⠥⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⢈⣦⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⣯⣓⠒⠒⠚⠁⠀⠀⠀⠀⢀⠠⣒⠴⠊⠁⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠙⠓⣒⣚⣭⢾⢿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠉⠉⠁⠀⠀⢐⣀⠬⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠛⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠄⣀⣀⣀⡠⠤⠜⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢸⣿⡿⣣⡄⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣽⣛⡃⠫⢾⣿⣧⢻⠟⣡⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢸⣷⣶⡾⣠⣜⣛⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢸⣿⠟⣰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⣤⣴⣶⣿⣿⣧⠈⢿⠘⠋⣼⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠛⠛⠋⠉⠙⠛⠛⠿⢿⣿⣿⣿⣷⣌⡻⠟⠛⠉⠉⠁⠈⠀⠈⢁⣴⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣙⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⡀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣮⣭⣥⣀⣀⣤⣤⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣬⣙⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣶⣶⡄⣲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠘⣿⣮⢁⣺⠿⣿⣾⡍⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡄⢹⣷⣿⣿⣸⣿⣿⣶⣭⡟⠻⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢸⡇⣿⣿⣿⡿⢋⣼⣿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⡇⣿⣿⢟⣴⠟⣯⣶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠟⠑⠉⠀⣼⡟⠀⢴⣆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠛⠋⠉⠉⠉⠙⠛⠻⠿⣿⣿⡷⣀⣴⣾⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣷⣦⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⡿⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣦⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠺⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢟⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⢟⣵⣦⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿⠟⠉⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣿⡛⢿⣿⣿⢟⣋⣙⣇⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢽⣛⡿⢿⣵⣿⣖⣛⣫⡭⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢿⡏⢻⣿⣿⠋⣽⡟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣜⣿⣿⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠛⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠻⣿⣿⣷⡄⠘⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⠄⠘⢿⣿⣿⡄⠘⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⡀⠘⣿⣿⣷⡀⠀⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⡀⠘⢿⣿⣿⣦⡈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⠀⠀⢀⣀⣠⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉

⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠋
⠀⠀⢠⣦⣦⠀⠀⠀⠀⠀⢴⣴⡄⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⢴⣴⡦⠀⠀⠀⠀⠀⣴⣴⠄⠀⠀⠀⠀⢠⣦⡦⠀⠀⠀⠀⠀⣴⣴⡄⠀⠀⠀⠀⢠⣦⣦
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠙⠁
⣶⠀⠀⠀⠀⠀⢰⣶⡦⠀⠀⠀⠀⠀⣶⣶⠆⠀⠀⠀⠀⠰⣶⡶⠀⠀⠀⠀⠀⢶⣶⠆⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⢴⣶⡆⠀⠀⠀⠀⠠⣶⣶
⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠰⣶⡶⠀⠀⠀⠀⠀⢶⣶⠆⠀⠀⠀⠀⠐⣶⡶⠀⠀⠀⠀⠀⢲⣶⡆⠀⠀⠀⠀⠀⢶⣶⠂⠀⠀⠀⠀⠰⣶⡶⠀⠀⠀⠀⠀⢶⣶⠆⠀⠀⠀⠀⠰⣶⡶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⡿⠀⠀⠀⠀⠀⠸⣿⠗⠀⠀⠀⠀⠀⢿⡿⠂⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠐⢿⡿⠀⠀⠀⠀⠀⠺⣿⠇⠀⠀⠀⠀⠐⢿⡿

⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⠘⢿⠟
⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠟⠁⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠻⠿⠁⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠈⠻⠟
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠈⠻⠋
⣤⡀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⢀⣤⣤
⠋⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠙⠋
⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠛⠃
⣴⠀⠀⠀⠀⠀⢠⣦⡦⠀⠀⠀⠀⠀⣴⣴⡄⠀⠀⠀⠀⢠⣦⣦⠀⠀⠀⠀⠀⣴⣴⡄⠀⠀⠀⠀⢠⣦⣦⠀⠀⠀⠀⠀⢴⣤⡄⠀⠀⠀⠀⠠⣦⣤
⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⡀
⠀⠀⠀⣀⠀⣐⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⠀⠿⣷⠖⠛⠓⢦⣀⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣿⣾⣧⡀
⠀⢠⠞⠁⠁⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠞⡯⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⢻⡀
⠀⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣸⡴⠃⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⢠⠇
⠀⠈⠳⢤⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠆⠀⠀⠀⠀⠀⠦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⣭⠖⠒⠲⣄⠀⠀⠀⠀⠀⣠⠴⠒⠿⡇
⠀⠀⠀⠀⠀⠁⠒⠆⠐⠀⠀⠀⢯⠀⣀⠠⠤⢀⡀⠀⠀⢰⡇⠀⠀⢸⠋⡇⠀⠀⠘⡇⠀⢠⠞⠙⡟⢢⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢷⣀⣀⣈⣷⣃⣀⣀⡼⠁⠀⢿⠀⠀⠱⢤⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠀⠈⠓⠶⠖⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠟⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣆⡀⠀⠀⠀⠀⠀⠀⣾⣯⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⢸⣿⠁⠀⣰⣿⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣗⠀⠀⠀⠀⢸⣿⠂⢰⣿⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣦⡀⣤⣤⣼⢿⢿⣿⣿⣿
⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠆⠲⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠁⠀⠀⠉⠉⢡⡄⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠀⠀⠀⠀⠠⣾⣿⣿⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣙⠷⣀⠀⠀⠉⠉⠉⣰⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠹⠷⣼⣦⣠⣄⣠⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣶⠗⠀⠀⠀⠙⣷⣶⡍⠍⠉⠛⠻⠿⠿⠟⠋⠁
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣠⣆⣶⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣠⣆⢸⡄
⣿⣧⣤⠀⠀⠀⠀⠀⠀⢠⡿⣦⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣷⡄
⣿⣿⣿⣷⡀⠀⠀⠀⣠⡿⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠆
⣿⣿⣿⣿⣿⣦⣤⣄⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢣⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣶⣾⣥⣤⣤⣄⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣠⣤⣴⣶⠶⠶⠿⠛⠛⣿⣿⣍⠉⠉⠁⢹⣿⢷⣄⠀⠀⠀⠀⣠⣶⠟⠁
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠋⢩⣿⣿⣿⡀⠀⠀⠀⣼⡟⣿⡟⢷⣄⠀⠘⣿⠀⠙⢷⣄⠰⠿⠋
⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⢀⣴⡟⢡⣿⠀⣰⡟⣿⡿⢹⣧⠀⠀⣰⡟⠀⢸⣷⠀⠙⢿⣦⣿⡇⠀⣀⣹⣷⡄
⠀⠀⠀⠈⠛⠷⠀⠀⠀⣠⡾⠋⠀⢸⣯⣾⠋⢠⣿⠃⠀⢻⡆⣰⡟⠀⠀⠀⣿⣆⣤⣤⣾⣿⡿⠿⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣀⣀⣠⣿⣿⣥⣤⣾⣿⣤⣤⣼⣿⣿⠶⠶⠾⠿⣿⡟⠛⠉⢩⡿⠁⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣍⠉⠉⢿⣏⡉⠉⠉⢻⣧⠀⠀⠸⣧⠀⠀⠀⠀⣿⠀⠀⢠⡿⠁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠙⢷⣄⠀⠀⢻⣧⠀⠀⣿⡀⠀⠀⢸⡏⠀⢠⣿⠃⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠙⣷⣄⠀⢻⣧⠀⢸⡇⠀⠀⣾⠇⢀⣿⢃⣾⠋⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⠀⠈⠛⢷⣌⠻⣦⡀⢻⣇⠘⣿⠀⢠⣿⢀⣾⣳⡿⠁⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⣮⣿⣦⣻⣆⢿⡆⣸⣇⣾⣿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣧⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⣿⣯⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⢋⣉⣉⣥⣤⣤⣤⣤⣄⣉⡉⠉⠛⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢉⣠⣴⠶⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠷⣦⣤⡀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⢁⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⣠⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡉⠙⠻⠷⢶⣶⣤⣤⣤⣤⣤⣴⣶⠾⠿⠛⠉⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣦⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠛⠋⠉

⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣿⣦⣀⣾⣿⣧
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏
⠀⠀⢠⣶⣶⣄⠀⢀⡀⠀⠀⠘⢿⣿⡿⠁⠀⠀⠀⣠⡴⣆⠀⣴⣶⡄⠀⠹⣿⣿⡿⠃⠀⠀⣴⣶⣦⠀⢀⣀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⣴⣶
⠀⠀⢸⣿⣿⣿⣾⣿⡿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣿⣷⣼⣾⣿⣿⠁⠀⠀⠈⠉⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⢀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢰⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡄⠀⣰⣾⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃
⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⢀⣠⡄⠀⣠⣤⡀⠀⠀⠈⠿⡿⠋⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⣠⣤
⠀⠀⢸⣿⣿⣿⣴⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣹⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣦⡄⠀⠀⠉⠋⠀⠀⠀⠀⢠⣀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⠀⠀⠈⠋⠁⠀⠀⠀⢀⣄⠀⠀⣠⣴⣤⠀⠀⠀⠀⠈
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣧⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣴⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋
⠀⠀⢀⣶⣶⣆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⣴⣶⡄⠀⠀⠀⠀⠁⠀⠀⠀⣴⣶⣦⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⢸⣿⣿⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠈⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⣶⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣴⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋
⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣤⣤⡄⠀⠀⠀⠈⠁⠀⠀⠀⣤⣦⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⢸⣿⣿⣿⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣽⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠈⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠘⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⣄⡀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣠⣿⣿⣿
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟
⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⡀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣀⣠
⠀⠀⢸⣿⣿⣿⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣳⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣠⣾⣿⣷
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋
⠀⠀⢠⣾⣿⣦⢀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣴⡻⣦⢀⣶⣶⡄⠀⠀⠈⠉⠀⠀⠀⠀⣶⣿⣷⡀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣶⣿
⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⢀⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⣴⣿⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠴⠤⣄⢴⠀⢓⣦⡙⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢚⣕⣈⠽⣢⣿⠿⢋⠼⠛⠀⠀⡠⠀⠀⠠⢠⣞⢳⠘⡇⡁⠀⣏⣈
⠀⠀⠀⠀⠀⢀⠎⠰⠟⣫⣴⣿⠍⢄⣊⡀⠁⠀⠀⢀⡇⠀⡼⣤⠺⡿⠋⣘⢬⣷⢰⠃
⠀⠀⠀⠀⣠⠀⡔⠀⣰⣏⡽⠁⠰⢟⣉⠄⠀⢀⠀⠈⣇⠀⠀⣝⢆⣶⣾⠁⠢⠇⡄⠀⢼⣷⡘⡀
⠀⠀⠀⠀⣷⢸⢰⣼⣿⡏⠀⠀⠀⠈⢵⡇⠀⣿⡀⠀⠙⢦⡀⠈⢟⠃⠁⠀⠠⠾⠓⠲⣄⢣⢰⡇
⠀⠀⠀⠀⣻⣴⢏⠳⣾⢠⢣⡶⣿⡆⡈⣆⠀⣿⠛⣦⠀⣄⣉⣤⣾⣷⠁⠀⠂⣀⣀⠀⠈⠄⣿⡇
⠀⠀⠀⠀⢣⢨⣤⣰⣇⠺⡘⠿⠸⡡⡱⣿⡆⢻⣟⣺⢟⠖⠀⠀⠻⣟⢄⠀⠀⠙⡿⠁⠀⢰⡼
⠀⠀⠀⠀⠈⣷⡊⠽⣭⠗⠄⠀⠀⢀⡼⠁⠀⠈⢫⡿⡏⠰⠛⡷⠀⢸⣨⡻⠷⠒⢈⣠⣠⡟⠁
⠀⠀⠀⠀⠀⠘⢷⠞⣥⠠⣩⣄⢰⠏⠄⣤⡌⢢⠀⠑⣿⢄⣀⠐⣶⡟⡳⢖⠒⠲⢚⣛⡯
⠀⠀⠀⠀⠀⠀⠈⠁⠙⢠⣷⡿⢸⡀⠄⠙⠇⢸⢠⠄⠈⠳⢦⡉⢻⡾⡐⣇⣾⢂⡽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣟⡗⠐⠷⣄⢀⣀⣰⣿⣾⠆⢘⠢⠴⠞⠜⠿⢲⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢶⢶⠎⠁⠠⡀⢿⡤⠄⠀⢀⣠⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⡐⢿⡤⢁⠸⡝⢛⣤⢘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢼⡏⠣⠶⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣟⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢤⣀⡤⠴⢶⡶⣶⡶⠤⣤⠒⢤⡔⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡛⡞⢩⡦⢀⣀⣤⣄⣬⣤⢀⠐⠏⡋⢹⠾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡘⠓⠛⢂⣤⡝⠏⢮⠃⢋⠏⠙⣷⣦⡟⠀⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡴⠂⠈⠛⠂⠀⠁⠈⠠⠒⠉⠁⠻⢦⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣶⠤⡇⠀⠠⠴⢶⠀⠀⣶⠦⠄⠀⠰⠐⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠀⠂⢸⣦⡀⠀⠈⠀⠀⠈⠀⣠⠞⢀⠁⢀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢑⣔⢤⢿⡤⠝⠒⢶⡄⠖⠲⢈⠛⠓⣞⡡⣮⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⡯⠽⡟⠻⡼⢄⡃⠈⠁⢀⣭⠰⠧⠀⣀⡤⠞⢾⣡⡽⠭⣝⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⣼⣿⣄⣀⣷⠁⠾⠓⣾⣏⡛⠓⢒⣖⣺⣿⢿⠉⠈⢋⣿⣄⣠⣿⠙⣆⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⢀⣴⣾⣿⣿⡯⠭⠍⢣⠀⠀⣿⢛⠉⠙⡈⠉⠝⢃⠇⢀⡴⠛⠒⠛⠹⢿⡶⣟⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⣠⡴⠟⠉⠉⠁⠀⠀⠀⠀⢜⣷⡀⠹⡆⣰⣿⢿⡆⡴⠋⣠⣞⣦⣄⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠈⢦
⠀⠀⠀⠀⠀⣰⠫⣞⣠⠞⠁⠀⠀⠀⠀⣀⣤⣴⣾⣾⣿⣿⣿⣤⠤⠬⠭⠭⠶⠶⠾⠶⠦⢬⣭⣍⣁⡀⠀⠀⠀⠀⠀⠀⠑⣄⢧⢣
⠀⠀⠀⠀⡰⣿⡁⠋⠀⠀⠀⢀⣠⡤⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠦⣄⡀⠀⠀⠈⢻⡎⡆
⠀⠀⠀⡰⢸⠟⠁⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠈⠁⡇
⠀⠀⠀⡇⠛⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢠⡗
⠀⠀⠸⡇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠁⢀⡾⠁
⠀⠀⠀⢧⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⡿⠋⢠⠞⠁
⠀⠀⠀⠈⠳⡀⠈⢿⣿⣿⣶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⡏⢩⢸⢿⢸⡿⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡙⠻⣿⣸⢸⠈⢹⢙⠒⢶⠶⠶⠶⠤⠤⠤⠶⠶⠶⠒⠚⠋⠉⠉⢸⠀⠀⡀⣇⠼⢔⣫⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣍⣛⠲⠼⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⣒⠻⠽⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠚⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠛⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠻⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⣿⠏
⢠⣾⣶⣴⣤⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣿⠂
⠀⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠋⠀⠀⠀⢠⣶⣦⣾⣿⡆⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤⣴⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣤⣴⣶⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠻⣿⣿⡿⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿
⠀⠀⠀⢾⣿⡟⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⢰⣶⣿⡿⠀⠀⠙⠿⣿⠏⠀⠀⠀⠀⠀⢿⡿⠗⠀⠀⠀⠀⠉⠀⠀⠀⢰⣿⣿⣧⣤⣤⣄⠀⠈⠻⣿⣿⣿⠏
⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠉⠁
⢸⣿⣿⣶⣷⡆⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⢿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣷⣶⣦⠀⠀⠸⣿⠿⠛⠉⠀⠀⠀⠀⢀⣀
⠀⠻⣿⡿⠟⠁⠀⠀⢻⡿⠟⠁⠀⢠⣶⣦⣠⣤⡀⠀⠀⣠⣶⣶⣿⣿⣿⡄⠀⠈⢿⣿⠟⠋⠀⠀⠀⠀⢻⣿⠿⠋⠀⠀⠀⠀⠀⠀⢀⣤⣶⡄⠀⠈⠛⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⣠⣤⡀
⠀⠀⢀⣀⠀⠀⣿⣿⣦⣴⣶⡄⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠈⠙⠻⠿⠿⠀⠀⢀⡀⠀⠀⠀⠀⣠⣶⣤⣾⣷⠄⠀⠀⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷
⠀⠘⠿⠟⠀⠀⠹⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠠⣶⣶⣷⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠹⢿⣿⣿⡟⠀⠀⠀⠈⠛⠿⠁⠀⠀⠀⣠⣤⡀⠈⠋⠉
⢀⣀⣀⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠄
⠸⣿⣿⠟⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣶⣄⣠⣀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢠⣿⣿⣷⣤⣤⡀⠀⠈⠛⠋⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠀⠀⠘⢿⣿⣿⠟⠀⠀⣰⣾⣶⣿⣿⡟⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⢿⣿⠟⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢴⣾⡗
⠀⢀⣀⣠⣤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠛⠿⠿⣿⠃⠀⠀⠘⢿⣿⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠿⠿⠛⠋⠀⠀⢀⣀⣴⣶⡀⠈
⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⣴⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰⣿⣶⣤⣤⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠈⠛⠁⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⢠⣴⣿⠄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣶⡄⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣠⣤⡀⣀⣄⡀
⠀⠀⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⣴⣶⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⠶⣖⣒⠶⠖⠲⢶⣒⢲⠶⠶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢛⣇⣠⣄⣒⡟⠑⢦⣀⣀⡤⠊⢹⣒⣠⣄⣸⡻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⣦⡙⣏⠉⠉⢣⠀⠀⢸⡇⠀⠀⡸⠉⠉⣽⢋⣴⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣌⢷⢄⠈⢆⠀⠸⠃⠀⢰⠁⣠⣞⣱⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡑⢜⡄⡇⢸⢠⡣⢊⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣮⡻⡇⢸⢟⣴⢿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⢀⣙⣷⣾⣋⣀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡤⠖⠛⠉⠉⠀⠀⠉⠉⠛⠲⢤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣟⣦⣄⣀⠀⠀⠀⠀⣀⣠⣴⣻⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁










⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣷⣶⣶⣶⣶⣶⣬⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣤⣴⣤⣄⣀⣀⠀⠀⣸⣿⣿⣿⣿⣿⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠐⢣⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠋⠈⠛⠉⠛⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⢿⠛⠻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠉

⠀⢀⡀⠀⠀⠉⠛⠃⠀⠀⠠⢦⡄⠀⠈⠓⢤⣀⡔⠁⠀⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢳⡀⠀⡜
⠀⡏⠈⠖⠋⢳⠀⠀⠀⠀⠀⠀⠙⠢⡠⠤⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⡏⠉⠓⠋⠉⡧⠀⣰⣶⣾⣿⣿⡄⠀⠀⠀⠱⠜
⠀⣇⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠘⠖⠒⢢⠀⠀⠀⠀⣠⣤⣀⣴⣦⡀⠀⠳⡀⠀⠀⡴⠁⠀⠘⢿⣿⢿⣿⠁⠀⠀⣀
⠀⠸⡄⡰⠋⠀⠀⠀⠀⡠⢄⡤⡄⠀⢃⠀⠀⠀⢀⡌⠀⠀⠀⠀⢿⣾⣿⣧⡿⠁⠀⠀⠙⢦⠎⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⡞⠈⢷⠞⠛⣦
⠀⠀⠙⠁⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠈⣆⠠⠒⠁⠱⢦⠀⠀⠀⠘⢿⣿⡟⠁⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠏
⣴⣶⣤⣠⣴⣦⣄⠀⠀⢠⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⠀⠱⠊⠀⡇⠀⢐⡟⠀⢧⡴⢷⡀⠀⠀⠸⡄⠀⡴⠋
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠈⠂⠔⠢⡔⢢⡐⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡐⠀⠀⢸⡇⠀⠘⠁⠀⡇⠀⠀⠀⠳⡞⠁
⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢆⠀⠀⡐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⠀⠀⠀⠀⢣⠀⠀⢀⣨⠃
⠀⠈⠻⠋⠁⠀⠀⠀⣀⣀⣴⣷⡀⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⣠⣤⣦⡄⠀⠀⠀⠀⠙⠚⠋⠁⠀⢀⣤⣤⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣽⠟⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠉⠙⢦⢀⡤⠤⢤⡀⠀⠈⠻⣿⠇⠀⠀⠀⠀⢠⣴⣦⡀⣠⣄⠀⠀⠀⢸⣿⣾⣿⣿⡿⠃⠀⠀⠀⠠⠊⠉⣦⠤⡄⠀⠀⠈⠛⢿⠟⠁⠀⢀
⠀⠀⠈⠋⠀⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣿⣿⣿⢻⠇⠀⠀⠀⢿⣿⣿⡟⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡀⠀⠀⠀⠐⠋⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⡿⠋⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠀⠀⢣⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠙⢦⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⡔⠉⠈⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀
⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢱⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⡀⢀⠃⠀⠱⠄⠒⢢⠀⢀⣄⣄⡀⠀⠀⡜⠀⠀⢘⡔⠋⠉⢢
⠀⢀⣀⣀⣤⣄⠀⠀⠀⠣⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡃⠘⠋⠀⡇⠀⠀⠙⠋⠉⡀⠀⠀⠀⢉⠝⠁⠈⠋⠉⠁⠀⠀⡇⠀⠀⣎⡇⠀⠀⡜
⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢧⠀⠀⡰⠁⠀⠀⠀⠀⠀⠱⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠀⢀⠞
⠀⠈⢯⣟⣳⡿⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠀⠀⠀⠳⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⠔⠁
⣦⡀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡶⣶⠄⠀⠀⠀⠀⠱⠚⠁⠀⠀⠀⣠
⣿⣷⠀⠀⠀⠀⠀⣴⣿⣿⣷⣴⣿⣿⣿⣷⠀⠀⠀⢰⠋⠀⠙⣶⠋⠉⠉⢳⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿
⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⡀⠀⠀⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠳⣄⠀⠀⣠⠞⠁⠀⢀⣠⢤⣀⢀⣠⠤⠤⣄⠀⠀⣴⣶⣆⢀⣴⣶⡆⠀⠉⠻⢿⣿
⠁⠀⠀⠀⠀⣠⣄⡀⠀⠈⠻⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠃⠀⠀⡴⡏⠀⠀⠈⠋⠀⠀⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈
⠀⠀⣴⣶⣼⣿⣇⣷⠀⠀⠀⠀⠀⢀⣀⠀⢠⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⢧⢧⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⣿⣿⣿⣿⠏
⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡃⠀⠙⠃⠠⠔⢸⠀⠀⣤⣄⢀⣤⡄⠈⢧⡁⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⢻⣿⣿⠏⠀⣠⠒⢦⠞⠉⢢
⠀⠀⠈⠻⢟⡿⣿⡏⠀⠀⠀⠀⠀⠱⡈⠁⠀⠀⠀⡼⠀⠀⣿⣿⣿⣿⠇⠀⠀⠙⢦⡀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢹⠀⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠒⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠂⠀⠴⠚⠭⡛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠐⠢⡄⠠⠂⠉⡛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡚⣻⠁⢀⣠⣤⢤⣤⣀⣀⡀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠨⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣫⠵⠒⢝⢠⡇⣴⠋⣿⡿⣯⣿⣿⣋⠉⣗⣶⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⡭⠒⠿⠁⠈⠀⠔⢸⡗⣿⡐⡙⠉⠈⢷⣤⠤⣧⡖⡾⡿⣿⡉⠲⢤⡢⢄⠀⠀⠀⠀⠀⠀⠂⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡫⠚⠁⠀⠀⠀⠀⠀⠠⣚⣩⣧⢻⡇⠡⡱⡁⠠⠹⣦⠞⢌⠜⠠⣿⣷⠀⠀⠉⠳⢭⡢⢀⠀⠀⠀⠀⢄⢈⢻⡳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠀⠀⠀⠀⠀⠀⢖⣬⠾⠋⠁⠘⣎⢧⢲⡅⠀⢀⡴⠃⠀⠁⠀⡘⣼⣻⠀⠀⠀⠀⠀⠙⠲⣕⢄⠀⠀⠀⠰⡱⡵⡽⣆
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠘⢦⡃⣠⠔⠉⠀⠀⠀⠀⠀⠪⣣⣿⡀⠀⠀⠀⠀⠀⠀⠈⠳⣔⢄⠀⢉⠈⢜⢿⡼⡄
⠀⠀⠀⠀⢸⢱⠁⠀⠀⡀⡤⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⡫⠔⠀⠀⠀⠀⠀⠀⠀⢰⠏⡈⢝⡶⣤⡀⠀⠀⠀⠀⠀⠈⢳⡳⡔⢳⣺⣿⣇⣧
⠀⠀⠀⠀⢸⠈⢦⠰⢫⣪⠞⠁⠀⠀⠀⠀⢀⣀⡤⠖⣋⠵⡖⠉⠂⠀⠀⠀⠀⠀⠀⢀⠔⠡⠼⢟⣟⠁⠀⠙⠓⠶⣤⣀⠀⠀⠀⢹⣜⠎⢻⣿⡇⡏
⠀⠀⠀⠀⠸⣆⠀⠑⠲⠿⢤⡤⠤⠶⠒⠋⠉⠀⠀⠉⠒⠙⠢⡒⠀⠀⠀⠀⠀⣠⣴⡣⢔⠍⣒⡎⠈⠁⠀⠀⠀⠀⠐⠻⠿⣿⣿⣿⣿⣶⣾⣿⣾⠃
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠈⠙⠳⣮⣜⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣯⣅⡀⠀⠀⠀⠀⠀⣀⡤⢟⢟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠤⠤⠤⠴⠶⠶⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠦⠶⠖⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠿⠏⠹⠿⣿⣿⢿⣿⣿⡟⠁⢀⠀⠀⠀⢀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⣹⡇⢀⣥⣶⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣯⣿⡿⢛⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⣼⣿⣿⣶⣿⣷⣿⣿⣷⣿⣷⡻⣿⣿⣿⣇⣷⣏⡡⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣟⣯⣽⣿⣿⣾⣏⢸⣏⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⢸⣿⣷⡼⣻⠆⣦
⠀⠀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡫⣉⢶⣠⡑⣈⣿⣿⣿⣿⣿⣿⣿⣟⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣑⣮⣿⣿⣿⣟⣋⣩⣝⢿⣿⣿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣵⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣿⣟⣿⡿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣙⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣻⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣋⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢲⣤⡿⠇⠶⣈⣭⣿⣿⣷⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣧⠜⢩⡿⠁⣈⣤⣼⣷⣿⣿⣿⣿⣻⣷⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢿⣵⣫⡰⡔⢃⣷⣾⣿⡟⣻⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⣯⡷⣽⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠷⠐⣫⣍⠓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣈⣩⣤⣾⣭⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⠿⢿⣏⠙⠿⠛⣛⠿⢿⣻⠿⢿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣦⡛⠂⠀⠀⠀⠀⠈⠉⠚⠉⣛⣯⡭⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣞⣻⡑⢫⣿⣿⢢⡕⢀⠀⠤⢳⣴⡞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⣣⣙⣝⣣⠔⣤⣀⣺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣷⣾⣿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⢫⣿⡟⢯⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣭⣿⣿⣿⣿⣿⣿⣥⡿⢯⣱⠏⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡾⠿⣮⣡⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀
⠀⠀⢀⠔⠈⠁⠉⠲⠎⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠉⠲⠎⠀⠀⠈
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠉⠒⠤⢤⣀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠒⢤⡜⠁⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠑⠠⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠠⠄⠤⣀⡰⠋⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀⡰⠋⠉⠢
⠀⠀⢠⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠁⠀⠀⡠
⠀⠀⠈⠦⣀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠉⠉⠲⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⢠⠖⠒⢄
⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠙⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢤⣀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇⢿⣿⡶⣀⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣛⣻⠘⢫⣾⣿⡜⣿⢟⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣷⣶⣯⠅⣕⣛⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⢋⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⢀⣤⣴⣶⣿⣿⣆⠹⡇⠘⢡⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠟⠛⠛⠋⠛⠛⠛⠻⢿⣿⣿⣿⣷⣌⠻⠛⠛⠉⠉⠀⠀⠀⠉⢠⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣝⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠘⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡺⠿⣿⠏⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⣮⠍⠛⠛⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⢿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠶⠶⠾⠶⠶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⣼⠃
⠀⠀⠀⠹⣦⠀⠀⠀⢰⣿⣿⠟⠛⢿⣿⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⠗⣰⠏
⠀⠀⠀⠀⠙⣧⡀⠀⠈⢿⣯⡀⠀⣸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⠋
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠻⢿⣦⣶⣾⣿⣿⣿⣿⣿⣿⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠙⠻⠿⠿⡿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠁⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣤⣤⣤⣤⡶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⠆⠀⠀⠀⠀⠀⠀⠊⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠴⢁⣾⡟⠀⠁⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠱
⠀⠀⠀⠀⠀⠀⠸⣁⢀⡗⠀⡄⢀⣾⠊⡅⠀⠀⠀⠀⠂⠀⠀⣀⣀⠀⡀⠀⠀⠄⠀⠁⢸⡿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢿⠶⠀⠀⣂
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣷⡤⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠸⠀⠀⠀⠈⢀
⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠻⣷⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⢠⠇⠀⠀⠀⠀⢇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠴⢦⡄⠀⠀⡀⠀⠀⠀⠁⠀⠀⢦⣤⣴⠾⣇⡠⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠓⠶⠇⠀⣈⣉⣙⠳⡄⠀⠀⠀⢸⣯⠤⠖⠋
⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡔⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠒⠚
⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⢀⣤⡾⠂⠈⠛⠂⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⢂⠀⠈⠁
⠀⠀⠀⢯⣀⣸⠂⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠛⠁⠈⢻⣆⠀⠀⠡⡀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣝⡳⠶⢂⠀⠀⠀⠀⠈⣯⢿⣀⠀⢀⡼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠙⠉⠀⠀⠀⠀⠀⠙⢧⣍⣛⣩⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡋⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠋⠁⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠛⠋⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠈⠛⠋⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠉
⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣿⣿⣷
⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠛⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠃
⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠋⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠈⠙⠋⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠈⠛⠋⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠈⠛⠉⠀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁
⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠋⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣀⠀⠀⣀⡀⠀⠀⠈⠛⠉⠀⠀⠀⣀⡀⠀⢀⣀⡀⠀⠀⠈⠛⠁⠀⠀⠀⣀⡀⠀⢀⣀⡀⠀⠀⠈
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁
⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠋⠀⠀⠀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠙⠁⠀⠀⠀⣀⣀⠀⢀⣀⡀⠀⠀⠈⠙⠁⠀⠀⢀⣀⣀⠀⢀⣀⡀⠀⠀⠈
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁
⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣴⣶⣶⡀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶
⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡍⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠠⠴⠚⠛⣿⣿⠿⠿⠗⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡀⣀⣠⠂⠀⠀⣰⠇⠀⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣔⣀⠀⠀⠀⠀⠀⢼⠉⠉⠈⠈⠛⣹⣿⣿⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡐⢁⣴⣿⡯⠕⠁⠀⠀⠀⢤⢤⣿⣶⡶⠀⣀⣘⣿⣥⣤⣤⣤⣤⣤⣄⣀⣀⣀⣸⣿⣿⣿⣷⣤⡀⢢
⠀⠀⠀⠀⠀⡌⢰⣿⣿⠋⠀⠀⠀⢀⡀⢀⣄⠈⡷⡟⢀⣴⣍⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⠿⠃⠀⠃
⠀⠀⠀⠀⢸⠀⣿⣿⠃⠀⠀⠀⠀⠡⠒⢻⣿⣷⣤⣠⣠⣿⣷⠈⠉⠐⠀⠠⢀⣀⣀⣀⡀⢀⣀⣀⡠⠀⠀⠉⣀⡀
⠀⠀⠀⠀⠀⢀⠀⢥⣤⣤⡀⠀⠀⠀⣀⣴⡿⠟⠉⠿⣿⡿⣿⣿⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠑⠺⠻⠿⠷⠴⠖⠛⠉⠁⠀⠀⠀⠀⠈⠓⠿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠦⠤⠤⠤⠤⠼⠿⠿⠿⠛⠋⠁

⣰⠋⠀⠀⠀⢿⡄⠙⠋⣸⠏⠀⠀⠀⠙⠳⢦⡼⠁⠀⠸⣧⠈⠋⠀⢸⠇⠀⠀⠈⠻⢦⣤⠞⣖⠦⢰⠾⣧⠈⠋⠀⣸⢃⠀⢀⡻
⠀⠀⠀⠀⠀⠈⠿⣤⠜⠁⣴⡏⢷⣶⠚⣷⠀⠀⠀⠀⠀⠻⣦⡀⡰⠋⠀⣾⠉⢷⣤⠖⣤⡀⠘⠦⠊⠀⠹⣧⣀⡴⠁⢯⡙⠃⡗
⠰⡄⣴⢶⣤⡖⣶⠀⠀⠀⠸⣧⡀⠀⣰⠃⠀⠀⣴⠖⣦⣠⢬⣍⠁⠀⠀⢻⣄⠈⠁⣰⠟⠀⢴⢢⡔⢲⠀⠈⢉⣀⠀⠀⠑⢺⡤
⡰⠁⢻⣆⠉⢰⠏⣴⢦⡔⣦⠘⠳⠾⢡⠦⡤⣤⢻⡇⠉⠁⢰⡟⠀⢀⠀⠀⠙⠶⠚⠁⠀⠀⠈⠣⣀⠏⠀⠀⢸⠈⠛⠁⡴⠀⠳
⠀⠀⠀⠙⠳⠋⠀⠘⢦⣠⠃⠀⠀⠀⠈⠳⡴⠃⠀⠙⠦⠴⠏⠀⣾⠉⢻⡶⢹⣷⣠⣀⣀⣠⡶⣦⣴⢺⡆⠀⠸⣀⡤⠚⠁
⡀⠀⠀⠀⢾⠱⠆⣳⠀⢠⢦⣤⢲⡀⣠⡖⢶⣤⠴⣦⠀⠀⠀⠀⠹⣧⠀⣠⠞⠁⢳⣉⡜⠘⣷⡈⢁⡾⠁⢀⣬⣁⠀⣀⡀⠀⢀
⠿⠞⢹⣷⠈⠳⠔⠁⠀⠈⠣⣁⡜⠀⠸⣧⠈⠋⣰⠏⢠⢤⡎⢷⠀⠈⢛⣁⠀⡀⠀⣉⠀⠀⠈⠛⠚⠁⠀⢿⡄⠹⠟⢈⡟⠀⣿
⡀⢀⡼⠃⢠⡖⢶⡴⠊⢻⡄⠀⠀⠀⠀⠙⠧⠞⠁⠀⠈⠓⠤⡼⠀⠀⢳⡈⢁⠏⢸⠁⠛⠉⣸⢀⣄⠀⢀⠘⢷⣄⢀⡞⣀⠀⠘
⠉⠉⠁⠀⠸⣧⠀⠀⢀⡟⠀⣰⡖⢶⣤⠴⣶⢀⡴⠶⠶⠉⣿⠀⣠⣤⣀⣙⣋⠀⠸⢀⡤⠞⠁⠘⣎⠋⡸⠁⠀⠙⠋⠘⣏⠃⡽
⣀⣄⠀⠀⠀⠙⠷⠴⠋⠀⠀⢹⣧⠈⠁⢠⠏⠈⢿⡄⠀⢠⡟⠀⣿⡈⠉⠀⣽⠇⢀⣁⣀⠀⣀⣀⠈⠋⢁⡀⠀⠀⠀⠀⠈⣛⣀
⢡⡟⢠⡖⠦⢰⠄⠀⡄⢦⠠⡄⠙⢷⣴⠋⠀⠀⠈⠛⠢⠏⠀⠀⠘⢷⡤⠞⠁⠀⣿⡀⠹⡿⠁⣿⠃⢸⡏⠹⣧⠖⣶⠀⠀⣿⡙
⡞⠁⠀⠳⣤⠎⠀⠀⠳⣄⡜⠁⠀⠀⢀⡀⠀⣠⣤⡀⢶⠳⠖⣆⡠⣤⣀⠤⣦⡀⠘⢷⡀⣀⠔⠃⠀⠘⣷⡀⠀⣰⠏⠀⠀⠘⢷
⣀⢀⣠⣤⠀⠀⢠⡴⣦⣀⣀⣀⠀⢸⡏⠹⡿⠃⢸⠇⠈⠳⠎⢸⡇⠈⠉⠀⣼⠁⠀⠀⢉⣁⡀⠀⠀⠀⠘⠷⠞⢁⣀
⠙⠋⠀⣿⠀⠀⢸⣇⠘⠋⢀⣿⠀⠈⢿⣄⡀⡰⠋⢰⠲⠄⡦⠈⠻⣦⣀⡜⠁⠀⠀⠀⣿⠀⢻⠖⢹⡷⠀⠀⠀⣿⡙⣷⠎⣿
⣦⡀⣰⢃⠤⣄⢠⠻⢦⣠⡾⠃⣀⠀⠀⠈⠉⠀⠀⠈⠳⡔⠁⠀⠀⢈⡭⣤⣠⡖⢲⡆⠹⣇⠀⣠⠞⠁⠀⠀⠀⠘⣷⣀⡼⠃
⠈⠘⠁⠘⢧⢀⡜⠀⠀⠀⠀⣿⠙⠗⠋⢹⡆⠀⣴⢶⣤⡴⠲⣦⠀⠸⣧⡈⠛⠀⣸⠃⠀⠙⠛⠁⢰⡏⢻⣦⠖⢳⡎⠉⠀⣀
⢹⡆⢀⡤⣤⣁⡤⢶⡄⠀⠀⠹⣧⣀⡴⠛⠀⠀⣿⡀⠉⠀⢠⠏⠀⠀⠈⠻⠦⡴⠃⠐⡏⠷⠊⡆⠸⣧⠀⠀⢀⡾⠁⠀⣿⠙⠟
⠋⠀⢸⣇⠈⠋⠀⣸⠇⠀⢀⡤⣬⣉⣠⣤⠀⠀⠘⢿⣄⡴⠃⠀⢠⣾⠛⣦⣀⣀⡀⠀⠙⠦⠞⠀⠀⠹⢧⠔⠋⠀⠀⠀⠙⣧⣀
⡀⠀⠀⠻⣦⣀⡴⣃⡀⢀⢸⣇⠘⠛⢀⡿⠀⠀⠀⠀⢀⡀⠀⠀⠘⣿⠀⠙⠉⣰⡟⠀⠀⠀⢠⣶⠳⣦⡴⠲⣦⠰⡏⠳⢰⠎⠁
⠘⠞⠉⡶⠈⠉⠀⢧⡉⡸⠀⠻⣦⣀⡞⠁⢰⡞⠷⠊⠉⣷⠀⠀⠀⠘⢧⠴⠚⣡⣤⡀⠀⠀⠘⣿⠀⠉⠀⣰⠏⠀⠳⠤⠋
⢀⡤⠚⣡⢤⣄⢀⡤⣭⠀⠀⠀⠈⣽⢦⡠⢼⣷⡀⠀⣸⢃⡤⣦⡴⠒⣶⠀⠀⣿⠈⠛⠁⣹⡆⠹⣧⡤⠞⠁⢰⡖⢶⣄⡤⣤
⠁⠀⠀⣿⡀⠹⠟⢀⡿⣠⣄⢀⣀⠘⣌⢁⠎⠈⠛⠴⠃⠸⣧⠀⠀⢠⡟⠀⠀⠘⣧⡤⠞⠋⠀⣖⠦⢤⡄⠀⢸⣇⠘⠛⣠⠟
⣀⠤⣦⠘⢷⣄⡤⠊⠁⠹⡌⢃⡞⢀⣬⣅⣠⠖⣶⡄⠀⠀⠙⠷⢠⠟⠀⠀⠀⢀⣤⣄⣠⠔⣶⠙⡦⠊⠀⠀⠀⠻⠦⠚⠁⣠⠶
⠋⢠⠏⠀⠀⣠⠴⣤⣀⢤⣍⠋⠀⢻⣆⠈⠋⠀⡿⠀⠀⣀⣤⣀⡠⢤⡄⠀⠀⢿⡄⠙⠛⠀⡿⠀⠀⢀⡤⣤⣀⡔⢶⡄⠀⢻⡇

⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠢⡀⠀⠀⠀⡠⠂⠉⠁⠢⡄⠀⠀⠀⡠⠒⠉⠉⠒⢄⠀⠀⠀⢀⠔⠈⠉⠐⢤⠀⠀⠀⢀⠔⠊⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠘⠀⠓⠀⢸⠀⠀⢸⠀⠘⠃⠚⠀⠈⠀⠀⢠⠁⠐⠁⠘⠀⠀⡆⠀⠀⠃⠀⠁⠐⠑⠀⠃⠀⠀⡎⢀⡉⠀⢉⡀⢱
⠀⠀⠀⠀⠀⠀⠰⣄⠓⠤⠒⢁⠎⠀⠀⠘⢄⠒⠉⠉⢂⡜⠀⠀⠀⢦⡘⠢⠔⠊⡰⠁⠀⠀⠣⡀⠀⠠⠚⡠⠃⠀⠀⠱⣈⠢⣉⠔⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠀⠀⠁⠒⠂⠁⠀⠀⠀⠀⠀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠈⠑⠒⠈⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁


⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⢤⡀⠀⠀⠀⡠⢔⠒⢒⠢⡀⠀⠀⠀⢀⠔⠒⢒⡢⣄⠀⠀⠀⢀⠤⠒⠒⠂⢄⠀⠀⠀⢀⡤⣒⠒⣒⢤⡀
⠀⠀⠀⠀⠀⠀⡌⠘⠭⠂⠯⠇⢱⠀⠀⢰⠁⠡⠄⢤⠅⢘⡀⠀⢠⠁⠈⠆⠠⠀⠈⡆⠀⠀⠃⠀⠧⠠⢇⡀⠃⠀⠀⡌⠀⡧⠀⢼⠀⢡
⠀⠀⠀⠀⠀⠀⠰⡀⠰⢀⠖⢀⡜⠀⠀⠘⣄⠀⠀⡷⠨⣉⡕⠀⠈⢆⠀⠒⠒⠄⡠⠃⠀⠀⢣⡀⠤⠂⠌⣡⠆⠀⠀⠡⡀⢎⠀⡱⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠈⠑⠒⠒⠉⠈⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠈⠓⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠐⠢⢄⠀⠀⠀⠀⢀⠴⠂⠐⠤⡀⠀⠀⠀⢀⠤⠐⠒⠤⡀⠀⠀⠀⠀⡤⠄⠂⠦⣘⢣⡄⠀⠀⡠⠔⠒⠠⢄
⠀⠀⠀⠀⠀⠀⡘⠁⠐⠄⠶⠀⢱⠀⠀⢰⠁⢨⡀⢨⠁⠙⠀⠀⢠⠃⠈⠄⠠⠁⠈⡄⠀⠀⡎⠀⠤⠀⠤⠈⠆⠀⠀⡜⠀⠰⠀⠆⠀⢣
⠀⠀⠀⠀⠀⠀⢡⡀⠲⣤⡔⠀⡜⠀⠀⠸⡀⠠⠒⠢⠀⡠⠀⠀⠈⣄⠀⡎⢱⡄⢠⠃⠀⠀⢆⠐⢄⣀⠴⢀⠆⠀⠀⢡⡀⢈⠶⡆⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠬⠐⠊⠀⠀⠀⠀⠈⠒⠠⠔⠊⠀⠀⠀⠀⠈⠑⠠⠄⠚⠁⠀⠀⠀⠀⠑⠂⠄⠒⠁⠀⠀⠀⠀⠙⠂⠤⠐⠋


⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⣀⠀⠀⠀⠀⢀⠤⠄⠠⢄⠀⠀⠀⠀⢀⡠⠄⠠⢄⡀⠀⠀⠀⠀⡠⠤⠀⠤⡀⠀⠀⠀⠀⣠⠄⠒⠠⣄
⠀⠀⠀⠀⠀⠀⣰⠗⠂⣤⠐⠒⣧⠀⠀⢠⠃⠠⠄⠤⠀⠱⠀⠀⢀⠊⠀⠄⠠⠀⠙⡄⠀⠀⠎⠐⣂⢀⡒⠈⠄⠀⠀⡜⠁⢠⠀⡄⠈⢣
⠀⠀⠀⠀⠀⠀⢣⠉⠉⠀⣩⠍⡸⠀⠀⠸⡀⠠⠤⠤⠀⢠⠀⠀⠘⡀⢳⠒⠒⡞⢀⠇⠀⠀⣆⠀⢄⣀⠄⢀⠄⠀⠀⢣⠀⢀⠤⠄⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠤⠚⠀⠀⠀⠀⠑⠢⠄⠤⠒⠁⠀⠀⠀⠈⠢⠍⠩⠔⠁⠀⠀⠀⠈⠒⠤⠀⠄⠊⠀⠀⠀⠀⠑⠠⠤⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣴⣄⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣠⣶⣶⣦⡤⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⠿⣿⡿⢀⣀⣠⣾⣿⣿⣿⣦⠀⠀⢀⣠⣄⣰⣿⣷⣄⣾⣿⣿⣿⣿⢯⣸⠇⠀⠀⣠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⣠⣽⣿⣿⡢⣿⣿⣿⣷⣴⣿⣿⠂⢻⣿⣿⣿⣿⣇⢹⣿⢋⣶⣿⣦⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣾⡛⣿⣟⣿⣿⣟⢳⣾⢿⣿⣿⠿⢿⣿⣯⣾⣿⣿⣟⡿⠿⢿⣿⣛⣀⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡾⢻⡷⢷⣾⣿⡿⢻⡿⣷⣿⣿⣿⣶⣤⣽⠿⠟⣿⣿⡷⣿⣷⣾⣿⣿⣿⣿⡇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣯⣿⣽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠧⠘⠿⣫⡿⢛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢓⣀⣾⣋⣅⣠⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢋⣴⣾⣯⣿⣟⠹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠯⠓⠋⡎⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣭⢀⡀⠀⢰⣿⣾⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣀⣁⡀⠉⠀⠀⠀⣼⡿⡎⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⣿⣿⣿⡿⠀⠀⠰⢿⢳⡜⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⢀⣽⣿⡟⢀⣴⣕⠀⠰⠋
⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⣿⣿⣿⣧⣤⣄⣀
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⢉⣲⠂

⠀⠀⠀⣀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠂⠓⠶⣄
⠀⠀⢾⣿⣿⣟⣿⣷⣿⡇⠀⠀⠀⠀⠀⣠⠖⠛⢦⡀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⣀⣠⡚⠛⣀⣀
⠀⠀⠈⣿⣿⡟⢿⣿⡟⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠿⠻⠚⠋⠚⠹⠧
⠀⠀⠀⠘⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⠋⠀⠀⠀⠀⠀⠈⠳⢤⣤⢤⠶⠋

⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢰⡟⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⣤⠀⠀⠀⠀⠀⠀⣴⡾⣧⡀⣠⣶⣤⠀⠀⠀⠀⠀⣔⣉⣟⣙⣏⣑⣄
⠀⠀⠘⡇⡠⠒⢄⣸⠁⠀⠀⠀⠀⠀⠀⢠⡀⢀⡉⠀⠀⠀⠀⠀⠀⢹⣧⢹⣷⣿⢡⣿⠀⠀⠀⠀⠀⠳⡘⡄⠀⡜⡠⠋
⠀⠀⣾⣥⣤⣤⣤⣬⣧⠀⠀⠀⠀⠀⠀⢸⢱⡞⡇⠀⠀⠀⠀⠀⠀⠀⠻⣾⢿⣯⣾⠏⠀⠀⠀⠀⠀⠀⠈⢺⣼⠊
⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⡖⢲⠦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣰⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠉⠉⢲⢃⠀⠀⠀⠀⠀⠐⠀⡙⠻⢋⣯⠽⠀⠀⠀⠀⢠⣴⣶⣶⢰⣶⣶⣤
⠀⠀⢧⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢘⠀⠀⢸⠨⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣙⠋⠀⠀⠀⠀⠸⣿⣶⡶⡶⣶⣾⡿
⠀⠀⠈⠳⢴⣿⠴⠚⠀⠀⠀⠀⠀⠀⠘⢍⡖⢰⣉⠋⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢹⠁⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠑⠠⡦⠔⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢼⣯⡛⢿⣿⣿⣄⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⡰⠃
⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣽⡟⠀⠀⠀⠀⠀⠀⠈⣡⣿⡍
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠠⠀⠠⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⢥⣤⣥⡀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠤⠀⠤
⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠠⠊⠂⠀⠀⢐⠒⠚⠀⠀⠀⠜⠋⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⡔⠀⣴⢀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠉⠉⠀⠉⠉⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛
⠀⠀⠠⠀⠠⠀⠀⣁⠀⡀⠀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⣀⡀⠀⠀⢿⠀⠀⠀⠀⠸⠇⠀⠀⣀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠀⠈⢀⠀⠀⣠⠉⣉⠑⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠤⢀
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡂⠀⠀⠀⠻⠃⠻⠃⠀⠀⠐⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⣒⠀⠀⠀⣐⠀⢀⡂⠀⠀⢄
⠀⠀⠠⠀⠠⠀⠀⠀⠀⡄⠠⡄⠀⠀⠺⣷⠖⢶⡿⠀⠀⠀⠄⠀⠄⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡄⠀⡄
⠀⠀⠀⠒⠐⠀⠀⠀⠀⠁⠆⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠲⠶⠂
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠉⡀⠀⠐⡐⠆⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠄⠀⠀⠐⠛⠻⠟⠓⠀⠀⠐⡄⠀⡠⠠
⠀⠘⢗⡂⣚⠇⠀⠀⠀⠀⠀⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠠⣬⣴⣤⣥⠀⠀⢆⠴⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠘⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠄
⠀⠑⠠⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⡇
⠁⠀⠀⠀⠀⠀⠀⠀⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾
⢀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠋⠉⠍⠉⠉⠉⠍⠙⢿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠉⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢀⠄⠒⠈⠀⠀⠀⢀⠔⢁⡤⠒⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠔⠁⠀⠀⠀⠀⠀⢀⡎⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡌⠀⠀⠀⠀⠀⠀⠀⣼⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⢄⠀⠀⡄⠀⠁⠀⠀⠀⢠⠟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠌⠀⠀⠠⠖⠉⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠑⢦⣄⡀⠀⠀⠀⣀⣠⠔⠋⡠⠊⠀⠀⠠⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣉⠉⠉⠉⢉⠠⠐⠈




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠛⠛⠛⠲⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢸
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢳⡀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⢿⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⣶⠀⢧⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣼⠁⢀⡟⠀⢳⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡼⠀⣇⠘⣆⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣄⡤⠖⢦⢠⠇⠀⣼⠁⠀⠈⢧⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡼⠁⠀⠸⡆⠘⣆⠀⣼⠃
⠀⠀⠀⠀⠀⣴⣶⣿⣭⣥⣄⠀⠀⠛⠀⢠⠇⠀⠀⠀⠈⢧⣀⣈⣉⡉⠉⠉⠉⠙⣧⠀⣰⠁⠀⣼⠁⠀⠀⠀⠹⡄⠸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢳⡀⠘⣶⠃⠀⢠⠃⠀⠀⠀⠀⢀⣿⣤⣬⣭⣿⣿⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣰⠃⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠇⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠶⠶⠿⠿⠶⢶⣤⡈⠀⠀⢠⣤⣤⡤⠴⠾⣿⠀⠀⠀⢀⣤⠾⡟⠛⠛⠓⠲⠶⠶⠶⠦
⠀⣀⣀⣤⡴⠶⠚⠋⠉⢹⣄⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦⣠⡞⣇⠀⠀⠀⠀⢸⠀⠀⣾⢋⣠⣸
⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⠀⠀⠀⠀⢸⣤⡴⠛⠉⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⡆⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⢠⡟⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢸
⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⢸⡇⠀⠀⠀⠐⠹⣏⣹⡦⠖⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⢠⣤⣤⣼
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠛⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣧⠀⠀⠀⠀⠀⠀⣠⠴⢻⠀⠀⠀⠀⠀⠀⠘⠲⡞⠁
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡜⢀⠀⢻⠀⠀⠀⠀⣠⣾⡇⠀⢸⠀⠀⠀⠀⠀⠀⣶⣾⣧⣤
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣿⡀⢸⠀⢀⡴⠟⠁⢸⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠙⠒⠢⠤⠤⠤⠤⠖⠋⠁⢀⡾⢻⡇⣸⠾⠋⠀⠀⠀⢸⡅⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢿⡤⠤⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠁⡶⢦⣤⣄⣀⣀⣀⣀⣤⠶⠋⠀⠸⣿⠏⠀⠀⠀⠀⠀⢸⡇⠀⣸⣀⣀⣀⣤⠤⠤⠴⠶⢶⡾
⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣧⣞⣉⣁⣀⣀⣀⣀⣀⣀⣴⠋

⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠠⣾⡆⠀⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⠀⠀⢾⣄
⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢠⣶⡇⠀⠀⠒⢫⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠆
⣀⡀⠤⠞⠉⠀⠀⠀⠀⠀⠸⣿⠧⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⢘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⡇
⠘⠣⣄⠀⠀⠀⠀⠀⢀⡠⣤⣄⣣⣤⡋⠤⣺⠟⠁⠀⠀⠀⠀⠀⢀⡥⣫⣥⡀⠀⠀⠀⠀⠀⠀⡎⠛⠈⣻⣿⠇
⠀⡀⠈⣇⠀⠀⢀⠾⠋⠉⣩⡍⠉⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠏⣻⣷⠀⠀⠀⠀⠀⠀⠉⠂⠈⠉⠁⠀⠀⠀⠀⢰
⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀
⠀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⡇⠻⠏
⡆⣷⠟⠀⠀⠀⠀⠀⠀⣀⣤⡼⣽⡷⡤⣤⠀⠀⠀⠀⠀⠀⣀⢓⠤⡄⠀⠀⣤⠶⣦⡀⠀⠀⠀⠀⠣⣷⢸⡄⣤⡄⣠⡀⣼
⠉⠀⠀⠀⠀⠀⠀⠀⣤⣿⣹⣶⣿⣷⣏⠼⢷⡄⠀⠀⠀⠀⠳⡄⡀⠁⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠀⠈⠚⠳⠟⠧⠿⠶⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣇⢡⣏⣿⣿⣷⣿⡿⠁⠃⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⣄⡀⠀⠀⠀⠀⠀⠙⡏⢳⡟⡟⣿⣿⣿⣇⣉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠀⢀⢀⣤
⣿⣵⠀⠀⠀⠀⠀⠘⢧⡴⣻⣿⣮⣿⠏⢶⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢵⡆⡳⠄⠀⠀⡿⢾⣿
⠽⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠌⣳⡇⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⠻⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠄
⠘⢷⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿
⠀⠀⠉⠉⢷⣄⠀⠀⠀⠀⠐⢄⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣅⠈⠀⢀⣿⣅⠈⠀⠀⠀⠀⠀⠀⢀⠰⠂
⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⢸⣾⣿⠄⠀⢸⣾⣿⠀⠀⠀⠀⠀⠀⠀⣺⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⢩⠋⠸⡢⠘⠁⠀⠀⠘⠀⠀⠀⠈⠀⠠
⢋⢓⢦⣠⠖⠲⢦⠀⣿⣾⡿⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠙⠜⠉⠀⠀⠀⠀⠀⠄⠀⣀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡖
⡂⠁⡠⡒⠁⡠⡐⡇⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠀⢸
⠐⠕⡈⠐⠅⣈⠜⠁⠀⠀⠐⠀⠀⠀⠖⠉⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⡌⢢
⠲⣄⢊⡤⠞⣁⣤⡴⡆⠀⠠⡀⠀⠀⠀⠀⠀⣠⠁⠀⠔⠐⠶⠉⠱⠀⠂⠄⠀⠀⠀⠀⠀⠂⠀⠔⢁⠀⠀⠀⠀⠀⠐⠀⡠⠁
⠀⠈⠋⠀⠸⣿⣼⣫⢧⠀⠀⠩⢄⡀⠄⠂⠉⠀⣀⡀⢣⡸⣿⢃⠎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠆⠀⠀⠀⠀⠈⢎⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠞⠁⠀⠀⠂⢠⠢⠉⠐⠆⡈⠉⠆⠙⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢠⠀⠈⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡀⠀⢄⡀⠐⠐⢄⠀⠀⠘⡊⠀⠅⠨⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠆⠀⠈⠤⢀⡀⠤⠊
⠀⡀⠀⢠⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠈⠑⢤⣈⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⢤⢦⣤⣀⣄
⢊⣌⢂⠀⠐⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⡰⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢹⣹⣒⡄⠈⢯⢾⣏⡾⠂⠀⠀⠀⠀⢀
⡿⣿⡆⠅⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡘⠷⡀⠀⠀⠀⠀⠀⠀⠀⢧⠍⠀⠀⠸⣾⠀⠘⠋⠁⠀⠀⠀⠀⠀⢠⠂
⠘⠠⠠⠂⠀⠀⠀⠀⡄⠂⠁⠢⡀⠀⣀⣀⡀⠈⠣⡹⠓⠀⠀⠀⠀⠀⢀⠀⠀⢘⡻⠀⠀⠀⠈⠿⡴⢠⡀⠀⠀⠀⠀⠀⠠⠈⠂
⠀⠀⠀⠀⠂⠀⠂⢸⠀⠀⠄⠀⠀⡈⠻⣿⡇⠀⣠⣄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠦⡇⠀⠀⠀⠀⠀⠀⢹⠴
⠀⠀⡀⠀⡀⠈⡀⠬⠆⠀⠀⠀⠂⠁⠀⠈⢀⣴⢋⠈⡙⡲⢄⠀⠀⠀⠀⠀⠀⠙⠰⢫⡆⡤⢤⢠⠰⣉⠏⠀⠂
⡴⠛⠂⡄⠀⠊⠀⠀⠉⠀⠀⠀⠀⢠⠀⢠⠞⡑⠁⠀⠪⠊⠀⢕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠁⠀⢀⡀⠀⠀⠀⡲⠶⢦
⠀⠀⠀⣼⠀⡀⠀⠀⠀⠀⠀⠀⠤⣸⠀⡇⠈⡪⠂⢕⢕⠄⡠⣠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⡀⠀⠀⠀⣆
⠀⢀⣴⠋⠀⠈⢐⠀⠀⠒⠀⠉⠉⠀⠀⠱⢔⣔⡤⠋⠑⠒⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠊⠀⡐⠁⠀⢀⠠⠂⠀⠀⠀⠀⣰⠊⠉⠑⠶⠀⠀⠀⢰⣈
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁⡀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡎
⠐⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⠟⠓⣄⡇⠀⠘⡄⠀⣀⠠⠠⠐⠂⠀⠀⠀⠑⠠⠤⢀⣀⢀⡜⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠈⢫⢉⡓⠀⠊⠁⠀⡠⠖⣆⢢⣀⣤⢠⣄⠀⠀⠀⠈⠉⠐
⡈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠀⡀⠀⠀⠀⠳⠇⠄⡍⢨⢘⠨⠚⡃⠀⠀⠀⠄⠠
⣾⣇⣐⠀⢀⣀⣀⣀⢀⠀⢀⠀⣀⡂⠁⡒⠁⢀⣄⡀⠀⣚⣃⣀⢒⣈⣰⡐⣑⠏⡀⢀⡀⠀⠀⢈⣀⡀⠀⠀⢀⢀⡀⠀⠀⣐⣀
⣿⣿⠛⢿⠛⡟⣿⠟⣿⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣭⡉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠀⣀⣤⣶⢿⣿⣿⣿⣿⣿⣿⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠈⠉⠀⠀⠀⠻⠛⠋⠉⡙⠁⡐⠀⢨⡆⠀⢀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⢐⣀⣄⣤⣤⣞⣀⣀⢔⣀⣾⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⣏⣠⠀⢀⠀⠂⠀⠀⠠⣞⠁⠀⠀⠀⠘⠛⣈⣿⣿⣿⣒⠤⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠉⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⣾⣻⣟⡵⠊⠀⠀⠀⠀⢀⣄⣀⣷⣦⣴⠎⠀⠀⠘⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣦⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠞⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⡀⢾⣏⣽⠃⠀⣤⠛⢿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⡆⠀⣧
⠀⠀⠀⠀⠀⠀⡎⢀⣿⣿⡿⠃⠀⠀⠀⠀⡠⠔⢁⣤⣾⠦⡠⠋⠏⠀⢿⣿⡗⠻⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠉⠀⡀
⠀⠀⠀⠀⠀⠰⡁⠘⢿⣿⡁⠀⠀⠀⠀⠀⠐⠊⠉⠸⢿⣿⣿⣶⣶⣶⣾⣿⣿⣁⠀⠀⠉⠁⠐⠒⠶⠤⢤⣤⣤⠀⠤⠤⠤⠴⠖⠀⠀⠀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠡⡀⠀⣤⣤⣴⣆⠀⠀⠀⠀⠀⣀⣾⣿⠿⠋⠉⠻⣿⣿⡿⣿⣿⡟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢼⣿⣿⣿⣦⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠈⠛⢧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⣀⣀⡀⠀⠀⡀⢀⣾⣿⣿⣟⣛⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⢠⣤⣤⣄⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾
⠀⠰⣿⣿⣉⣭⣭⣭⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠉
⠀⠀⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⢰⣿⣻⣟⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⢀⣀⠤⠂
⠀⠀⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⡤⠔⠚⠉
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠠⠾⠟⠒⠒⠚⠋⠉
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀⣾⣿⣿⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⡔⣶⣢⡴⣶⣢⡄⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀⢀⣼⣿⣿⡟
⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⡄⣿⣿⢠⣿⠏⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠋⠁
⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠙⢷⢻⣏⡿⠋⠀⠀⠀⠀⠀⢰⡒⠿⠿⣶⣬⣟⣛⣛⠛⣛⣫⠤⠖⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣉⠣⢤⣈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠆
⢀⣀⣠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠺⠭⠭⠭⠤⠤⠤⢤⡀
⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡆⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⣴⠋
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣯⡀⣸⡇⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⢀⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣦
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠃⠀⠀⠀⠀⠀⢸⣾⣷⡄⠀⠀⣼⣿⡇⠀⢀⣿⣿⠀⠀⠀⣤⣾⣷⣄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣆⠀⠀⠀⠀⠀⢸⣿⣿⠄⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⡿⠋⠀⠈⠻⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⣾⣿⡇⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢸⠁⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⠘⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠹⡿⠃⠀⠀⢿⡿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⣿

⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠛⠁⡀⠀⠀⠀⠀⠁⠾⠟⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠑⠉⢢⠾⠝⠳⠀⠀⢸⠁⠀⣴⠟⢹⠀⠀⠀⠊⠁⠰⡾⡻⠂⠀⠀⠈⠈⠀⠀⠁⠀⠀⠴⣊⡥⠄⣋⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣋⣈⣹⡰⠃⠀⠀⠈⢦⣼⢏⣔⠟⠀⠀⠀⠀⣆⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢩⡴⠞⣋⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠎⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢞⡅
⠀⢀⠀⠀⢀⠀⠀⠀⢀⣒⣀⢄⣀⡠⠶⠀⠀⣾⣿⣦⣠⣴⣶⡆⠀⠀⣀⣤⣄⢀⣠⣀
⠀⠀⠐⠀⢀⠀⠀⠀⠘⣻⠿⠂⠀⢈⠻⠁⠀⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠙⠋⠉⣻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡉⠁⠀⠀⠀⠈⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠐⠀⠀⡀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣴⣿⣿⡆⠀⠀⣶⠄⢀⣬⠓
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⠂
⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠁
⢶⣄⡠⠶⣦⡀⠀⢰⡶⣆⣠⣶⣶⡆⠀⠠⠚⠒⣤⠚⠙⠲⡀⠀⠀⡂⠀⠀⣒⣆⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⡴⣀⣄⠀⠀⠀⠀⠀⠀⢴
⠸⣌⢛⡿⣸⡇⠀⢫⣿⠻⢿⣃⣽⠇⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢴⡔⠄⠀⢾⠟⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠛⠠⡾⠷⡵⠂⠂⠀⠀⢸
⢿⣱⢟⡵⠋⠀⠀⠈⠻⡺⢼⠹⠋⠀⠀⠀⠀⠀⢄⠀⠄⠁⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⡀⠃⠀⠀⠀⠀⠀⢀⣃⠘
⠀⠁⠋⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣄⡀⢀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣤⣤⠄⠀⢀⡀⠆⢀⢀⡀⠄⠀⣠⣖⣦⢀⣤⢆⣀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⢠
⣿⣷⣿⣿⣿⠀⠀⠀⢆⢡⠀⠀⠀⠀⠀⠀⠘⣿⡿⢈⢽⣼⣿⠂⠠⡄⣥⠰⠀⠶⠀⠀⠟⣯⡴⣩⡽⢯⡓⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿
⢿⣿⣿⡟⠁⠀⠀⠀⠉⢀⠀⠀⠀⠘⠀⠀⠀⠙⣿⣻⣿⠟⠁⠀⠀⠀⠀⣬⡁⠃⠀⠀⠈⠑⢟⣯⣶⠟⠁⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈
⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠲⢤⡞⠉⢳⠀⠀⠠⢯⣅⢂⡔⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⣤⠞⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠐⣤⠂⠐⠄⠀⠀⢒
⠀⡾⠁⢀⠃⠀⠀⠈⣄⣀⠠⠶⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⠏⠀⣠⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⠀⠀⠸
⠓⢤⡄⠀⠀⠀⠀⠀⠈⠲⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⠟⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠂
⡀⡀⢀⠤⣀⠀⠀⢀⡤⣤⡈⢁⣤⣄⠀⠀⢐⠀⠀⠠⠄⠀⠈⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⢀⡀⡁⠀⢀⣤⡄⠀⠠⠀⠠⡀⢀⡠⠀⠀⠀⣀
⢙⠐⡣⢪⢴⡆⠀⢸⡳⣯⣿⣫⢚⣿⠇⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⣦⡼⠻⣛⠀⠆⠅⢀⠈⠤⡐⠺⠀⠀⠆
⡈⡆⠌⢫⠀⠀⠀⠀⢳⣿⣿⣿⣿⠏⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣷⣿⢻⠭⠏⠀⠀⠂⣅⠐⢀⡄⠀⠀⠀⠘
⠈⠤⠖⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⡷⠀⠀⠀⠀⠀⠀⠙⠇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡷⣤⡞⢁⣴⠀⠀⢀⣤⣄⡄⢀⣠⣤⡀⢠⣬⡶⢎⠀⡨⠴⢄⡄⠀⠒⣤⢚⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣆⠁⠠⠴⠀⠀⠰⣗⢉⣻⣾⣿⣬⠇⠘⣁⠤⢂⣠⣴⣿⠃⠳⣀⠀⠀⠀⡑⡔⠀⠀⠂

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣶⠀⠀⠀⠀⠀⣴⣶⡤




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠶⠶⠀⢀⣤⠀⠀⣤⣤⠀⠀⠀⠀⢠⣶⠀⢰⣦⠀⢠⣾⡆⠀⣠⣶⡀⢀⣀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⡄⢰⣶⣦⡄⣶⣶⡀
⠀⠀⠀⠀⠐⠃⠀⠚⠛⠃⠀⠉⠁⠀⠀⠀⠀⠐⠒⠀⠀⠙⠗⠀⠉⠛⠀⠟⠛⠛⠘⠛⠛⠐⠟⠓⠀⠁⠐⠀⠘⠋⠀⠈⠙⠛⠀⠙⠛⠁
⠀⠀⠀⠀⠀⣠⠀⠀⣤⡀⠠⠂⠂⠀⡆⣠⠀⢀⣀⠀⠠⢎⡆⠀⣼⡿⠁⢰⣏⠀⢠⣾⣧⢀⣤⣄⠀⠠⣷⡆⠀⢶⠂⠠⠶⣶⡄⣀⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠉⠀⢀⣉⠉⠀⠈⢉⡀⠀⠈⣁⠀⠈⠁⠀⠈⠁⠀⢀⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉
⠀⠀⠀⠀⠲⠶⠄⠀⠶⠆⠈⢦⠀⠈⠋⣤⡄⢿⡿⠇⠘⣿⣿⠸⢿⣦⠀⠀⢼⡄⠙⣧⡄⠀⣤⣴⠀⢤⣤⠀⠰⣤⠀⠀⠀⡍⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⣠⣤⡀⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⡀⠀⣤⠀⢀⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⢀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠘⠷⠄⠙⠛⠃⠘⠿⠟⠀⠿⠿⠃⠈⠻⠆⠺⠄⠶⠀⢿⠿⠀⠶⠿⠃⠘⠿⠆⠈⠛⠷⠀⠀⠾⠁⢴⡶⠦⠀⠾⠆⠀⠿⠼⠇
⠀⠀⠀⠀⣶⣶⠆⢀⡀⡀⠀⠘⠀⠀⠰⠗⠀⡀⠀⢀⠠⣦⡤⠀⢠⡄⠀⣴⣄⠀⠀⠀⠀⠀⢐⡄⠀⢰⣦⠀⠀⠀⠀⠀⣰⡄⠀⢰⣶
⠀⠀⠀⠀⠛⠛⠃⠘⠋⠁⢀⠺⠇⠀⠀⠆⠀⠛⠛⠋⠀⠈⠁⠀⠘⠃⠀⠈⠛⠓⠀⠀⠀⠀⠘⠃⠀⠈⠋⠀⠘⠀⠀⠘⠛⠛⠀⠘⠛
⠀⠀⠀⠀⢨⡄⠀⢰⣴⡆⠈⣿⠇⠀⠀⠀⠀⠐⠒⠀⠺⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣤⡄⠀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⣤⠆⠀⣀⡀⠀⠀⠀⠀⠀⣤⠀⢀⡀⡀⠀⢠⡄
⠀⠀⠀⠀⠶⠆⠀⠀⠉⠀⠰⠆⠶⠀⠀⠀⠀⠉⠟⠁⠈⠉⠉⠀⠙⠋⠀⠊⠈⠀⠰⠀⠆⠀⠚⠃⠀⠀⠀⠀⠀⠿⠂⠀⠀⠓⠀⠐⠶
⠀⠀⠀⠀⢠⣤⠀⣄⢀⡄⠀⠶⠀⠀⠰⠀⠀⣶⢀⡄⣠⠶⣦⢠⡶⢶⡄⢰⣶⠄⠠⠶⠆⠀⡴⣦⠀⠰⠶⠀⣰⠶⣆⢰⡶⡶
⠀⠀⠀⠀⠈⠁⠀⠈⠛⠁⠘⠶⠛⠀⠐⠀⠀⠈⠐⠁⠻⠴⠿⠸⠧⠾⠃⠀⠟⠀⠰⠶⠆⠀⠟⠻⠀⠰⠶⠆⠰⠶⠂⠰⠷⠿⠂
⠀⠀⠀⠀⠚⠛⠀⠀⠖⠀⢰⣿⣧⠀⣿⣿⠀⣴⠴⡦⠠⣴⠟⢰⣦⣤⡄⢠⣦⠀⢰⣾⠗⠀⣀⡀⠀⢷⣶⡇⢀⣰⣿⠐⠷⣄⠠⣿⣿⡆
⠀⠀⠀⠀⣙⣙⠁⢀⣀⡀⠈⢘⠋⠀⠈⠁⠀⠀⣀⠀⠀⠉⠁⠀⠀⡉⠁⠀⠀⡀⢀⠉⠀⢀⣀⣀⠀⠈⠉⠁⠈⠋⠁⠀⠀⠉⠀⣉⣉
⠀⠀⠀⠰⣿⣿⠇⢸⣿⡿⠀⠉⠁⠠⣾⣷⠄⢀⣿⠀⣾⠿⣿⢠⣦⣴⡄⣠⠞⠁⢬⣤⣴⠈⠉⠉⠀⠳⡷⠀⣴⣶⠀⠀⣤⡆⠰⣿⣿⡇
⠀⠀⠀⢀⣬⣥⡄⢀⣭⡀⣠⣤⣤⠀⣤⣤⡀⠈⠉⠀⢀⣤⣄⢠⣭⣭⠀⣥⠀⠀⠈⣀⠁⠀⣠⣤⠀⠀⠀⠀⣨⠁⡀⠀⠀⣠⠀⠈⠁
⠀⠀⠀⠀⠻⠿⠃⠸⣿⠃⠙⠛⠛⠘⠿⠿⠓⠾⠿⠗⠛⠻⠟⠸⣿⡿⠆⠙⢦⡄⠐⠿⠧⠘⣿⡿⠀⣤⣶⠄⠹⠾⠇⠀⠚⠁
⠀⠀⠀⠐⣦⡀⠀⣶⣶⡦
⠀⠀⠀⠀⠀⠑⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⡆⠀⠀⠠⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠄⠀⠛⠓⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠀⠀⠈⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⢀⡀⢠⠀⠀⠀⡀⢀⠀⢠⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⢠⡀⢀⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⢀
⠀⠀⠀⠘⠷⠾⠃⠳⠶⠋⠸⠶⠞⠀⠷⠶⠃⠲⠶⠆⠙⠶⠟⠸⠗⠿⠀⠷⠲⠋⠸⠶⠖⠑⠶⠦⠘⠷⠶⠃⠲⠶⠞⠘⠶⠞⠃⠶⠶⠂
⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⢤⠄⠀⢀⡀⠀⣴⣤⡄⠀⠀⠀⠀⢚⡧⠀⠘⠛⠀⢠⣒⡆⠀⣴⣦⠀⠚⠃⠀⠀⠛⠀⢠⡶⢶⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠉⠀⠀⠋⠀⠈⠁

⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠈⠀⣀⣀⠀⠀⠠⡀⠀⢄⠀⠤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠐⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠉⠀⢀⠀⠴⣖⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠉⢦⡀⠁⠄⠀⠀⠀⠀⢸⡄⢀⣀⠀⠰⣀⠛⣋⡰⠆⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡈⠀⢀⠀⠀⠀⢿⡏⠁⠀⠾⠋⠈⢹⢳⣤⡡⠁⠋⠠⠀⠀⡀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠂⡀⠀⠀⣿⠀⠀⠀⠀⠀⠃⠼⣿⠀⠀⠀⠀⠀⣀⠔⠿⠀⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣕⠀⣰⣏⠀⠠⠀⠀⠀⠠⢰⣧⡾⠒⢔⣉⣤⣤⡤⠿⠇⢶⣀⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠏⠸⠛⠔⡂⠐⠀⠀⠄⡾⣣⣴⣿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠙⢦⣒⣀⠺⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣣⡀⢲⠀⠱⠊⠀⢡⣶⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⡤⡄⢂⠀⡠⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠂⢠⣆⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢺⡼⠃⣴⣹⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⡄⣸⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⠈⢿⣿⣟⣙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠠⣿⢿⠆⣿⣿⠃⠈⠀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣳⣦⠀⠀⠀⠀⡄⠀⠆⠀⠠⠁⠂
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣼⠁⠁⠀⢸⣾⣿⣿⡏⠀⠀⠀⠘⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠐⠆⠀⠀⢠⣦⠞⠉
⠀⠀⠀⣷⠀⠀⠀⠀⠘⣾⣻⣿⡂⣠⣷⣿⣿⣽⣿⡂⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣻
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣼⣯⣿⣵⣶⣿⠿⠛⣿⣟⠀⠀⠀⢀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⢀⠾⠘⠄
⠀⠀⠀⠀⠀⢡⠺⣿⣿⠟⣷⢛⣿⣿⣿⡯⣴⢻⡟⠶⠄⡀⠈⢀⠠⠀⠀⠈⣿⣿⠉⠋⠉⠁⢹⡿⢿⣿⣿⠃
⠀⠀⠐⡀⢀⡤⣾⣿⠏⣼⣵⠧⣀⣈⡝⠻⣧⣾⢁⣷⠿⢂⣷⣮⠀⠀⠀⢰⣿⣟⠀⠀⣤⠦⣼⣷⠘⢟⢟⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠒⠋⢱⣾⣟⣙⣶⣦⣿⣿⣿⣿⠨⢸⡏⣄⠿⣭⣳⠀⠉⡤⠄⠀⠝⠀⠀⠀⠀⠈⠁⠀⠨⡏⠀⠀⠀⠀⠈
⢀⠰⢠⡄⠀⠀⠀⠋⢈⣷⣬⣙⣿⣯⣿⠋⣉⢻⣤⡄⠜⠁⠻⠟⡄⣠⡞⣿⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠘⢯⣍⣛⣛⣭⠼⠃⠀⡀⠈⠁⠀⠀⠁⠀⠁⢹⠟⢹⠀⠀⠔⡀⠀⠀⠀⠀⠀⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⠏⠯⣇⠺⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠙⢿⠠⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡄⠆⠀⠃⠀⠴⠀⢿⠈⠀⠀⠀⠀⠇⠀⠀⠀⠀⢰⠈⠀⠀⡄⠅
⠀⠀⠀⠀⡀⠴⠟⠂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠆⢠⠈⠀⠀⡄
⠀⠀⠀⠀⠉

⠀⠀⠀⢐⠐⡆⠀⠀⠀⠀⡄⠀⢄⠀⠀⠀⠀⢸⢜⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⡖⠉⢲⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⡸⠯⡀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⢀⠋⢁⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⡴⠿⠶⠿⢶⠀⠀⢀⡿⠿⣧⠀⠀⠀⣠⣼⣯⣄⠀⠀⢰⡟⠛⠛⠛⣷⠀⠀⢬⣀⣨⡄⠀⠀⠀⣿⣿⠀⠀⠀⠀⡼⣛⣛⣛⢷⡀
⠀⠀⡇⠀⠀⠀⢸⠀⢠⡏⠀⠀⠀⢳⠀⢠⡏⣿⣿⡟⣇⠀⢸⡇⠀⠀⠀⣿⠂⢀⡞⠉⠉⣦⠀⢀⡖⠋⠙⢢⡀⠀⢸⢱⣿⣿⣿⣷⢳
⠀⠀⣇⣀⣀⣀⣸⠀⠀⢣⣄⣀⣠⠎⠀⠀⢿⣿⣿⣷⠇⠀⠘⢧⣦⣶⣦⠟⠀⠀⢳⣶⣶⡋⠀⠸⡄⠀⠀⢠⠇⠀⠸⣜⢿⣿⡿⢏⡏
⠀⠀⠀⣞⠉⣰⠀⠀⠀⢠⡛⠛⢓⠀⠀⠀⠀⡽⢫⠁⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⢸⠀⢨⠁⠀⠀⠉⣶⣾⠁⠀⠀⠀⠙⣿⣶⣾⡟
⠀⠀⠀⢰⠒⡆⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠧⠆⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠐⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠛⠛⠛⠁
⠀⠀⢠⡴⠖⠐⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⠀⠀⡴⢢⠀⠀⠀⢀⢄⠀⠀⢀⢄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣠⣄
⠀⢸⡃⠀⠀⠀⠀⠈⣇⠀⠀⣰⣯⣆⠀⠀⠀⣈⣏⠀⠀⣨⣏⠀⠀⠀⣈⣎⡀⠀⣈⣎⡀⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⢠⣿⣿⣇
⠀⠸⣆⠀⠀⠀⠀⢀⡿⠀⢰⠁⠀⠈⢧⠀⢰⣿⣫⣗⢰⣯⣫⣗⠀⢸⣟⢙⣿⢸⣿⢙⣿⠀⠀⢠⣻⣽⣿⣝⢆⠀⣰⠋⠁⠉⠙⣆
⠀⠀⠉⠷⠶⠴⠶⠏⠁⠀⠘⢦⣀⡤⠋⠀⠈⠻⠾⠃⠈⠻⠞⠁⠀⠈⠙⠛⠋⠀⠙⠛⠋⠀⠀⢠⣿⣿⣿⡿⡀⠀⢿⡀⠀⠀⢀⡿⠃
⠀⠀⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠄⠤⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡾⠁⠀⠈⢻⣶⣶⡿⠁
⠀⠀⠈⢆⠀⠀⠀⠀⡼⠁⠀⢀⡌⠀⠀⠀⠈⢆⡀⠀⠀⢱⡀⠀⠀⢀⡼⠀⠀⣠⡴⠶⠶⠶⣤⡀⠀⠛⠛⠁⠀⠀⠀⠀⢿⡿⠃
⠀⠀⢴⠆⡑⠠⠐⣊⢴⡄⠀⢈⡅⠀⠀⠀⠀⢸⡅⠀⠀⠀⠋⠒⢖⠋⠁⠀⠀⠙⠛⠶⠶⠞⠛⠁
⠀⠀⠀⠈⠃⣙⡇⠙⠀⠀⠀⠀⠱⢀⡀⣀⠐⠆⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠈⠀⢀⡲⠃⠀⠈⣇⠀⠀⠀⠀⣾⡆⢠⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣸⡆⠀⢨⡇⠀⠀⠀⢸⠇⠀⠀⢠⣾⡆⢠⣿⡄⠀⠀⠀⡤⣀⣀⣀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢉⡧⠀⠈⢇⡀⠀⢀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠎⠉
⠀⠀⠀⠀⠀⣼⣢⠀⠀⠀⢠⠀⠈⠉⠉⠀⠀⠀⠀⠳⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠤⣀⠀⠀⠀⠀⡠⠊⠁⠈⢢
⠀⢀⣤⠀⠀⠛⠛⠀⠀⠀⣬⠀⠀⣰⣾⣿⣷⡄⠀⠀⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⡆⠀⠀⢰⠁⠀⠀⠀⠀⢣
⠀⠀⢽⡆⠀⠀⠀⠀⠀⣴⠇⠀⠀⠳⣌⠙⣿⡗⠀⠀⠀⠀⠀⠀⣶⣿⣷⠀⠀⠀⣣⣤⢀⣀⢠⣤⠃⠀⠀⠘⠷⣦⣤⣴⡾⠛
⠀⠀⠀⠘⠟⠲⣷⠘⠿⠂⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠈⠉⠙⠟⠉⠋⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠷⢶⣄⠀⠀⠀⢀⡀⢀⣠⣶⣾⠿⠿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠴⠺⠿⣶⣦⡈⢳⣠⣄⣾⣷⠿⣋⡱⢆⣀⡀⡤⡙⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⣴⣿⣿⣷⡌⣿⣷⡟⡿⣫⠟⣁⣀⣙⠁⣿⣿⠿⠚⣣⣿⡟⣾
⠀⠀⠀⠀⠀⠀⠰⠞⣿⣄⠻⣿⠏⢹⡇⣾⠟⡡⠞⣡⣭⣝⠻⠿⢃⡙⢿⣿⣿⡿⢟⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠏⡙⠻⠶⠞⡋⠈⢡⣀⣈⡛⠷⠈⣭⣶⣶⣄⠙⣷⣶⣶⣶⣦⣟⢛
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡟⣠⢠⡖⣮⣽⣷⣦⣝⣿⣷⣦⡻⠟⢛⡻⣧⠘⣫⣴⣶⣦⣝⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣛⠀⣾⡗⢙⣋⣿⢿⣿⣿⣿⣿⣿⣔⣿⣿⢈⣾⡟⢩⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⣏⠙⢃⣼⣿⠿⢏⣭⣬⣍⡹⣅⣼⡟⣠⣴⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢎⣻⡿⠿⢟⣫⢰⣿⣉⡉⠻⣷⡝⣿⢛⣭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢛⣽⡏⡘⢿⣿⠇⠀⢹⡇⡏⣭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢡⢿⣶⢀⣜⢃⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣶⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠻⣄⠀⠀⣠⠟⠉⠙⠷⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣤⣤⣤⣼⣷⣾⣧⣤⣤⣤⣤⠬⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠙⣿⡀⠀⠀⠀⠀⣿⠀⢻⡄⠀⠀⠀⢀⣼⠏⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠘⢿⡄⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⣾⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⢿⡄⠀⠀⢿⡄⠈⣿⠀⢀⣾⡏⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⡿⠶⠶⢼⣷⣤⣿⣦⣼⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣀⣀⣀⣀⠀⣀⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⣀⣤⡶⠶⠛⠛⠻⠶⣦⣄⡀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣋⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠙⠻⢿⣿⣶⣶⣶⣶⣶⣤⣤⣴⣶⣿⠿⠋⠁⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠤⡀⠀⠀⠄⠀⠀⢀⠀⠀⠀⢀⠀⡠⠂
⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁⡜⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠠⠤⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⠁⠂⠠⠀⠁⠀⠂⢆⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⣤⠁⡀
⠀⠐⠋⠀⠀⠠⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠈⢀
⠀⠀⠠⣀⡠⠆⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⢀⣀⠀⠀⠀⠤⠀⠈⠓⢛⢁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀⠀⡁⠀⠀⠀⠉⠑⠀⠳⠋⠀⠀⠀⠀⠀⠀⠀⠘⠚
⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⣁⠀⣀⠑⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠉⠀⠀⡀⠂⠀⠀⠋⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠂⠀⠀⢄⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣓⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠲⢤⣀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠋⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢰⡿⠿⠓⢤⣀⡤⠚⠛⠳⠄⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠐⢄⢠⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢇⠀⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢠⣻⡆⠀⢸⢨⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢳⠂⠀⠀⠱⡀⠀⠛⠃⢀⠞⢄⠉⠀⢀⠜⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⠀⣠⠚⠛⢖⠋⠉⡧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠐⠒⠂⠁⠀⠀⠈⠉⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢠⡇⢀⡇⢢⡀⢈⠖⠊⢻
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣼⣠⣿⡌⠀⠈⡽⠤⠔⢾
⠀⠀⠀⠀⢀⣽⡄⠀⠀⠀⠀⠀⠀⢛⠢⢄⣀⠀⠀⠀⠀⠀⣀⡠⠤⢲⡋⠁⠀⠀⠀⠀⠀⠀⣰⠋⠁⠈⠉⠓⠶⠷⠶⠶⠋
⠀⠀⠀⢠⡾⠃⠹⣄⡤⠦⠤⣤⠖⠲⣦⣸⠀⠉⢹⠋⠉⠁⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⠉⠁⠀⠀⢣⡙⢤⡀⢸⠀⠀⠀⢸⡀⠀⢀⡽⠻⠀⠀⠀⠀⣴⠃
⠀⠀⢻⣤⣠⣤⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠙⠒⠤⠤⠼⠇⠒⠉⠀⠀⠀⠀⢀⣼⣡⡄
⠀⠀⠀⠈⠉⠉⠉⠳⢦⡀⠀⠀⢀⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⢹⣧⠀⠀⠀⢀⡤⠴⠶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠘⣿⢀⡤⡾⠁⠀⠠⠊⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡹⠀⠀⠀⠀⠈⠙⢲⣤⣀⡀⠀⣀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⡿⠋⠂⡡⠒⣡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⣠⠖⠁⠈⢰⣧⣄⠀⠀⠀⢠⣿⠃⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢊⡴⠞⠁
⠀⠀⠀⣀⡴⠚⡡⠄⠀⢀⠞⠈⠛⠿⣶⣄⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢈⡴⠋
⢀⡴⠚⢉⠄⠊⢠⣂⠵⠃⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⢿⣀⡔⢁⣀⡤⠛⠁

⠀⠀⠀⠀⣠⣴⡿⠿⠿⢿⣿⣆⠉⠿⣿⣿⡾⣿⣶⡋⠀⣿⣿⣿⢞⡋⣠⣶⣿⣿⢟⡀⠀⠀⠀⢀⣀
⢀⣤⣴⣶⣾⣷⣶⣶⣄⠀⢻⣿⠄⣼⣿⣿⡇⣾⣿⢧⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⢁⣤⣶⣿⣿⣶⣶⣶
⣿⣿⣿⠿⠟⠻⠿⣿⣿⣿⣼⣿⢠⣿⣿⣿⣾⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⠿⠟⠋
⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⣺⣿⣿⣿⣿⢨⣿⣿⣿⣿⣸⣿⣿⠹⣿⡇⣿⣿⣿⠟⠋⣁⣤⣶⣶⣶⡿
⢀⣠⣤⣤⣄⣀⠀⢾⣿⣏⣛⣉⠉⡛⢻⣏⣍⡈⢙⠿⢹⣿⠨⣿⠉⣱⠉⢝⣭⣿⣿⠀⠀⣿⡟⠋⣉⣠⣤
⡿⠿⠿⠿⠿⢿⣷⣿⣿⣿⣏⠻⣿⣷⣾⣿⣿⢉⠛⡒⠛⢛⣯⠉⣻⣿⣇⣿⣿⣿⣿⠆⣴⣿⣥⣿⠿⠿⠿
⣠⣴⣶⣶⣤⡀⠙⣿⣿⣿⣿⣦⣄⣉⣛⠛⢻⣿⡇⠁⢰⠿⠏⠶⠿⢋⡾⢛⣴⣿⣿⣿⣿⣿⣯⣤⣄⣀⣤
⠟⠛⠉⠛⢿⣿⣆⣿⣟⢿⣿⣿⣿⣿⣿⣷⣀⣭⣶⣶⣿⣿⣿⣿⣷⣶⣤⣝⡛⠿⣿⣿⣿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣜⣿⣶⣮⣭⡉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢋⠷⠿⠿⠿⢿⣷⣶
⣶⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣠⣶⣿⣿⣿⣿
⠉⣠⣤⣶⣾⣽⣿⣯⣿⣛⣛⣭⡀⣾⣿⡿⠿⠛⠛⠛⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣯⣾⣿⣿⣷⣿⣶⣶
⢾⠿⠛⠋⠙⢿⣿⣿⣿⣿⣿⡟⠀⣿⣿⠃⠐⢿⡿⡻⢿⣿⣿⣿⡶⣶⣶⣤⢸⣿⣿⣿⠉⠋⣑⣤⣦⣤⣈
⢄⣴⣶⣿⣶⣄⡈⢙⣻⣿⢉⣶⣶⣿⣏⣤⣤⣼⣿⡿⢾⣧⣾⣿⣵⣿⠃⠀⠘⣿⣿⡇⣴⣿⡿⠿⠿⠿⣿
⣿⠟⠋⠉⠙⠿⣿⣿⣿⠃⢸⢹⣿⣿⣇⠛⠛⠓⠺⢿⡌⠠⣿⠿⢽⣿⣶⣶⣀⣿⣿⠑⠋⣵⠾⠿⢷⣶⣶
⢁⣴⣶⣿⣿⣶⣶⣿⣿⠇⠘⢳⡙⣷⣠⣷⠒⠠⡖⠊⠗⠰⢦⣀⠀⢀⣽⠁⣼⣿⡏⡏⢀⣥⣴⣶⣶⣦⣬
⣿⡿⢛⣋⣽⣿⣿⣿⣥⣤⣾⠏⠻⣿⡟⠁⠀⠀⠀⠀⢾⡇⡄⢀⣠⣤⡿⠄⣸⠟⣡⠓⣛⣋⣉⣉⣉⠛⢿
⣿⣵⣿⡿⠟⠛⠛⠛⠛⢛⣁⠀⠠⣿⣧⠀⠀⠀⠘⢶⣤⣴⠆⠀⠀⠘⣷⣼⡷⠛⢁⣈⣭⣭⣭⣛⡻⢿⣶
⣿⠿⣉⣠⣤⣤⣄⣀⣴⣿⣿⠋⢸⣿⣿⣧⡃⠾⠿⢶⣶⣾⣷⡆⢀⢀⣼⡿⣠⣾⠿⠿⠿⠿⠿⣿⣿⣦⡙
⢫⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⠠⣿⣿⣿⣷⡅⠸⣶⣶⡖⠀⣀⣴⣿⣿⡇⠰⠾⠿⠿⢿⣶⣦⣈⢿⣿⣷
⣿⣿⢋⣴⣿⡿⠟⠛⠉⠁⣠⡴⣾⡿⠻⣿⣿⣿⣦⣤⣥⣤⣾⣿⣿⣿⣿⠃⠀⠉⠉⠛⠳⢿⣿⣿⣦⡻⣿
⠙⣷⣿⠟⠁⠀⠀⣠⡴⠛⠉⠀⣿⠣⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⠻⢶⣤⣀⠀⠀⠀⠈⠹⣿⣿⣮
⣼⡟⠁⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⠀⠀⠈⠙⠻⢶⣄⡐⠮⠊⠹⢯
⣢⣴⠿⠿⣷⣦⡀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣰⣿⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⢙⣿⡶⠶⠶
⠋⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⣿⠀⠀⣿⡿⠀⣀⣴⡶⠟⠄⠀⠀⠀⠀⠀⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣷⣄⠀⢿⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠛⠙⠙⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄⠄⠀⣀
⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⠀⠀⣤⠄⠀⠀⢤⣄⠀⠀⠀⠀⠀⡎⠀⠰⢤⠄⡠⡤⠆⠀⢰⠄
⠀⠀⠀⠻⢢⣄⢀⡀⣤⡼⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠉⠒⡤⠤⠤⠤⢤⣦⡖⠋
⠀⠀⠀⠀⠈⠁⠻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⢀⠀⠀⠆⡀
⠀⠀⠀⢠⠉⣩⣤⣍⡉⢄⠀⠀⠀⠀⠀⢀⡴⢋⡟⠉⠉⢿⠑⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠁⢸⣿⣻⣿⣷⠈⡄⠀⠀⠀⠐⢿⠶⢾⠒⠲⠒⢒⠗⢶⡷⠀⠀⠀⠀⠀⠀⡅⠀⠀⢨⡄
⠀⠀⠀⢠⠘⠿⣿⡿⠋⡸⠀⠀⠀⠀⠀⠀⠙⢌⢧⠀⢀⠞⡴⠋⠀⠀⠀⠀⠀⠀⣾⢹⡀⢀⡞⢷
⠀⠀⠀⠀⠲⠂⠀⡐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠃⠘⠿⠿⠂


⠀⠀⢀⠰⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⣄⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢡⣦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣧⡼⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠑⢤⡤⠊⠁
⠀⠀⠀⠈⣿⠀⢀⢘⠛⠻⠃⡄⠀⠀⠀⠀⢨⠿⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⡶⡶⡀
⠀⠀⠀⠀⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣟⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣂⠀⠀⠀⠀⠀⠀⠀⣴⡆⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡜⣦⠇
⠀⠀⠀⢈⣌⣀⢡⣁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠛⣿⠋⠁⠀⠈⠙⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⣴⣯⠿⠿⣇⣤
⠀⠀⡼⠋⠀⠈⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢀⡞⢹⡇⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠊⠀⠀⠀⠀⠹⣃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢸⣠⠞⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢷⡀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢧⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠓⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠌⠶⡀⠀⠀⠀⠀⠀⠀⡸⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⠀⠀⠀⠀⠀⠂⠁⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⣝⣣⠴⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣿⡖
⠀⠀⠀⠀⠀⠀⠢⢄⣀⣀⣀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢠⡟⢷⡀⠀⠀⠈⢷⠀⠀⠀⣀⡴⢶⡾⣅⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠃⠤⠤⠐⠛⠓⠒⢄⠀⠀⠀⠸⡇⠀⠀⠀⠸⡇⣸⠃⠀⠀⠀⣸⠀⠀⢰⠋⠀⠘⣇⠈⣷⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠄⠐⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣠⡿⢯⣄⣀⣠⡴⠋⠀⠀⠸⡄⠀⠀⠈⢳⡾⠴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣿⠿⠟⠋⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢀⣉⣭⣥⣤⣁⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢈⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡄⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣠⣶⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠀⡁⢘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠞⠛⠿⣿⡿⠟⢋⡠⠤⠤⠤⠤⢤⣠⠇⢸⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣰⡿⠁⠀⠀⡼⠋⢀⠞⢁⣴⣾⣿⣿⣿⣷⡏⠠⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢹⠄⠀⡰⠊⠀⢠⣿⡇⠾⠛⠐⠶⠿⢿⠟⢀⠀⠀⠀⢸⣿⣉⠢⢄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣄⠊⠀⠀⠀⣾⣿⣧⠰⡀⠀⠀⡠⠀⠐⠻⡄⠀⣤⣿⣿⠿⠷⢄⡑⠢⡀
⠀⠀⠀⠀⢠⠃⡰⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⠉⠙⢧⠡⣀⠈⠀⠀⠀⢀⠄⡰⠁⠉⠻⣧⡀⠀⠙⢆⠈⢆
⠀⠀⠀⠀⡜⢰⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠠⡘⢶⣇⠀⠀⣸⠊⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⡇⢸⡀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⡀⠀⠈⠂⠙⠿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠰
⠀⠀⠀⠀⢧⠈⢧⣤⡇⠀⠀⠀⣀⠠⠐⢊⣠⡾⠿⣷⣄⠀⠀⠈⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⡀⠇
⠀⠀⠀⠀⠀⠳⣄⣈⣉⣉⣉⣡⣤⣶⣾⣟⠁⠀⠀⣸⠿⠓⢄⣀⡀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠈⣧⡾⠁⠆
⠀⠀⠀⠀⠀⠀⠈⠻⢭⣻⣿⣿⣿⣿⣿⣿⠿⠖⠋⠁⠀⠀⠀⠉⠳⢦⡀⠀⠀⠈⠓⢶⣤⣄⣀⣀⣀⣀⣉⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣾⣿⣿⣿⣿⣟⣫⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⣸⡿⠛⠛⠃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠒⠂⠠⠄⠀⢀⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⠛⠁⠀⢠⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠚⠡⠆⠀⠀⠀⡦⣀⠀⠀⠀⠀⢀⠈⠧⡀⠀⠀⢰⣿⡟⣷⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⢂⠣⡀⡱⡆⠀⠙⠿⠚⢚⠉⠀⡄⠀⠤⠬⠄⠀⡘⣀⠀⠘⠿⠿⠽⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢹⠉⠅⠁⠁⠀⠀⠀⢀⡈⠀⠀⠤⠀⠨⠁⠀⠘⠂⡀⠁⣿⣷⣶⣦⣤⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠤⠄⠀⠘⠀⡀⠀⠃⠁⣿⣿⣿⣿⡇
⠀⠐⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⡒⢉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠃⠀⠀⠀⡇⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠏⠀⠀⠀⠀⠁⣬⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠁⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂