Nerds Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢠⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠈⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠰⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠂⠰⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠀⠀⠀⠀⠠⠀⠄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣠⣤⣤⣤⠶⠶⠶⠖⠛⠛⢉⣁⣠⡄⠀⠀⠀⠀⠀⠄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣭⣥⣤⡤⠴⠶⠶⠶⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠶⠶⠦⣤⡀⠀⠀⢴⠶⣚⣛⣋⣉⣛⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠶⢦⣍⠀⢠⣶⠟⠉⠉⠉⠉⠉⠛⢶⡄⠀⣠⡾⠋⢀⣤⣶⣄⠀⣼⠇
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⡀⠀⢹⡆⣿⠁⣤⡄⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⠀⣾⠀⠈⠋⢠⡿
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⠃⠀⣸⡇⠻⣆⡉⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠈⠛⢂⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⠴⠞⣿⠃⠀⢈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠀⣼⠛⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣼⡇⠀⠀⣰⠇⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠷⣤⣾⠏⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣀⣀⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢀⣤⣾⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣀⠀⠈⠛⣷⣦⣤⣴⣾⠿⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠙⠛⠷⣾⠟⠋⠉⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣄⠀⢰⣿⡆⠀⠀⠀⠀⣀⣠⣴⠾⠛⠋⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⢋⠙⠻⣷⣾⣉⣿⣤⣶⣾⠿⠛⢻⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣀⣴⣿⡿⠁⢠⣌⣷⣴⡟⠉⠉⣿⣋⣀⣀⠀⠀⣼⠇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢠⣾⣿⣿⣿⣇⠀⠀⢀⣹⣿⠀⠀⣰⣿⣍⣉⠀⠀⢰⡟⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠷⠶⣿⣥⣌⣉⠁⣠⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣰⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠶⠶⠶⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠛⠒⠤⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⣖⣤⢤⣦⣴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠏⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡺⠙⠓⡿⠿⡦⡶⠿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⡷⡄⠀⠑⠚⢅⠓⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⢈⣩⢀⢃⠹⣤⡀⠀⠒⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⠀⢻⢘⢈⣳⣼⣿⡗⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠘⣆⡎⢶⠟⢿⣿⣿⣾⠇⠀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣄⠀⢢⣿⠃⠀⠀⠀⠘⣿⠃⠀⠀⢰⣇⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠈⠃⠁⢀⢹⣶⡀⠀⠀⣰⡿⡐⠓⠒⣾⡿⠼⣠⢄⡇⠀⠀⠀⠀⢀⣀⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢶⣶⣶⣯⣿⣿⣷⣤⡞⡃⣵⠃⣀⡜⠻⣗⣫⡗⠉⠈⢲⣶⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣷⣿⠋⠉⠉⣿⣿⣿⣿⣾⣿⣷⣿⣿⣷⠀⠀⠀⠑⢆⠀⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣒⠶⣵⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣀⣰⣿⣿⣷⣝⣶⠶⠶⢂⣤⣐⣲⣦⣴⣶⣤⣤⣄⡑⢆⣻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠵⠀⠀⠀⠈⠙⣷⣶⣭⣝⣛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣵⣟⣵⡿⢠⠀⠀⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠇⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣦⣴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠛⠛⠛⠿⢶⣦⣄⡀⢸⡟⠀⠀⠀⠐⠲⣆⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣧⣀⣀⣀⣀⣴⠏⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠷⣶⣄⣀⣀⣀⣠⣤⣤⣴⡿⠻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠾⠛⠛⠻⠷⠶⠶⣶⠏⡼⠁⠀⠀⠹⣏⠉⢉⣁⣀⣾⣻⠃⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠶⣶⠶⠶⢶⣦⣤⣤⡏⣰⠃⠀⠀⠀⠀⣿⠟⠋⠉⠙⣷⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠛⢿⡀⠀⣼⠃⠀⢸⠁⣿⠀⠀⠀⠀⠀⡿⢟⣛⣛⣻⣧⣇⠀⠀⢀⣿⣀⣀⡀⠀⢀⣤⣀⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡿⠘⢧⡀⡏⠀⠀⡆⠀⣿⠀⠀⠀⠀⢀⠇⠀⠉⠉⠉⠉⠉⠙⢛⣛⣻⣿⡿⠋⠀⢸⡏⣿⡇⢻⠀⣿
⠀⣴⣦⣴⢶⡾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣧⠀⠀⢷⠀⣿⠀⠀⠀⠀⡸⠀⠀⠀⢀⣤⣴⣾⠛⢋⣿⠉⠀⠀⠀⣾⠛⢷⠘⠇⠸⠀⠛
⢀⣿⠸⡇⢸⡇⢸⣷⠶⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⠸⣆⠀⢀⣴⠃⠀⠀⠀⠀⠀⠻⢾⣶⠟⠁⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣀
⣿⡿⠆⠀⠀⠀⠛⠁⣼⠟⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⢁⡴⠶⢤⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⠛⠉⠹
⠘⢷⣄⣀⣀⣠⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠞⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⣿⠛⠉⠉⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿
⠀⣿⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀
⠀⢻⡆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣇⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⢘⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡿⢦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣿⡀⠀⠀⠀⠀⠘⠷⣦⣄⣀⣠⣤⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⠿⣿⣉⡉⠉⠉⠁⠀⠀⢀⣸⠏⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠲⠶⠶⠶⢶⣦⣤⣤⣤⣤⡴⠛⠁⣿⠀⠀⠉⠙⠳⠶⣤⣤⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣶⠖⠶⠶⠶⠶⠶⠶⠶⣤⣤⣴⠋⠀⠀⠀⠹⠶⠶⠶⠶⠶⠶⠒⠶⣿⡿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⣸⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣷⡶⠟⠛⠛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠙⠛⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣦⣤⣤⣤⣤⣤⣶⣶⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠶⠶⠶⠿⠿⠿⠿⠿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣆
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣭⣭⣭⣭⣭⣭⠍⠙⠻⣷⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⢻⣿⡿⠛⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⠈⠛⠋⠀⠀⠈⠉⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣿⠟⠛⠉⠀⠀⠖⠛⠛⠛⢿⣿⡿
⠀⠀⠀⠻⠋⠀⠠⠦⠀⠀⠀⠠⠦⠄⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠤⠦⠀⠀⠀⠰⠤⠀⠀⢿⠃
⠀⠀⠐⣾⠛⠛⠛⠛⢶⣤⣤⠖⠛⠛⠛⢻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠛⠛⠛⠒⢦⣤⣴⠖⠚⠛⠛⢷⡶
⠀⠀⠀⢸⡀⠀⠘⠗⣸⠃⢹⡀⠳⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⠇⣸⠁⢻⡘⠗⠀⠀⢸⠇
⠀⠀⠀⠀⠙⠓⠒⠋⠁⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠙⠒⠒⠚⠉
⡤⠒⠀⣖⠀⢐⠀⢠⡀⢰⣆⣰⢰⡒⠀⡴⡀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡄⠀⢀⠀⢰⣀⡄⢰⡂⢀⢶⠀⡶⠲⡄
⠑⠀⠂⠃⠀⠀⠀⠈⠑⠘⠃⠙⠸⠄⠸⠃⠃⠺⠔⠃⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠁⠀⠀⠀⠸⠁⠃⠸⠄⠘⠙⠇⠳⠴⠃



⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠖
⠀⠀⠀⠀⣤⣀⠀⢰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢀⣤⣶⣯⣶⣿⡿⠋⠁
⠀⠀⠀⠀⢿⣿⣿⣶⣿⣿⣿⣎⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣤⣈⣻⣿⣿⣦⣍⣉⠉⠁⠀⠈⠉⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣽⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⣿⣿⣿⣿⣛⣛⣛⣛⣉⣤⣴⣶⣄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⠆
⠀⠀⠀⣿⣿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⠽⠿⠿⠿⢿⣿⣟⣛⠛⠛⠛⣿⣿⠃
⠀⠀⠀⢸⡟⠀⢀⣀⠀⠀⠀⢀⣀⡀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⡿⠁
⠀⠀⢠⣬⡥⠶⠤⠤⣄⡀⣀⣤⠤⠤⠦⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣥⠤⠤⠤⣤⡀⢀⣠⠤⠴⠶⢤⣥⡄
⠀⠀⠀⢹⠀⠀⢰⡦⢸⡟⢻⠠⣦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣦⠀⡟⢻⡇⢶⡄⠀⠀⡯
⠀⠀⠀⠘⠶⠤⠤⠴⠚⠀⠈⠳⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠞⠁⠀⠓⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢠⡤⠀⣠⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⠀⢀⠀⢀⠀⠀⡄⢠⠀⡤⠀⢠⡀⢀⡤⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢻⢸⣃⢠⠧⢧⢸⣀⠇⠀⠀⠀⠀⠀⠀⠸⣀⡆⢸⡀⢸⣀⠰⠛⠄⡟⢸⠀⣇⣀⠶⢷⢸⣄⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⣻⣭⣟⠳⢶⣚⣛⣛⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣋⣵⣶⣿⣿⣿⡿⠿⣿⣟⡛⣿⣿⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣋⣭⣶⣿⣿⣿⣷⣿⣿⠋⠀⠀⠈⠙⠿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣴⣿⢋⣽⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⣿⣧⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⣾⣿⢻⣿⣾⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢻⡜⢷⡲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠟⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⣠⣤⡶⠶⠖⣂⣠⣤⡶⣿⠟⠛⡛⠙⣯⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣼⣿⣿⣿⣿⣿⣿⣿⠏⣴⡿⠟⣡⣴⠶⠛⠉⢹⡄⠀⢻⡿⢀⠙⣆⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⣿⣿⣿⡿⢿⣿⣿⣿⠁⠀⢡⣴⠟⣿⣦⡄⠀⠀⠀⣷⠀⠀⠙⣅⣀⣠⠟⢧⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⣶⣿⣿⣿⣇⣠⣴⣶⣿⠀⢻⡛⠁⠀⠀⢀⡟⠀⠀⠀⠈⢫⡄⠀⠀⠙⢷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡝⣿⠛⣉⡉⢻⣿⡟⠛⠉⠁⠀⠳⣄⣉⣀⡤⠶⠛⢠⡖⠀⠀⠀⠀⠘⢆⠀⠀⠀⢻⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡇⠸⠁⠈⡆⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠼⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣷⡀⠀⠀⠙⠃⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠂⠀⢘⠃⠀⠀⣸⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⡿⠻⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⢀⠆⠀⠀⢀⡿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠮⠖⠛⠳⠾⢭⣝⢶⣄⠀⠀⠀⠀⠀⠈⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢷⣝⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠋⠀⠀⢰⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣆⠈⠉⠑⠒⠢⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⣠⡼⣳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣹⡇⠀⠀⠀⠀⢀⣤⠾⣋⣛⡳⢶⡶⢞⣫⡵⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣚⣛⣭⣾⠃⠐⢤⡀⠀⠀⠈⡛⢦⡻⣌⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣴⣾⣿⣿⣿⣿⣿⣷⡀⠀⣉⣒⣤⡔⣣⣬⡷⣬⣛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡘⢻⣿⣿⣿⣿⡶⣿⣿⣷⣮⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⡿⠟⠙⠿⠃⠸⣿⣿⣿⣿⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢠⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⡜⣆
⠀⠀⠀⠀⠀⠀⣰⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣷⡹⣆
⠀⠀⠀⠀⠀⣰⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠘⠘⢸⣿⣯⣿⣿⣿⣷⡹⣆
⠀⠀⠀⠀⣰⢣⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡇⢾⣿⣿⣿⣿⣿⣿⣷⡹⡄
⠀⠀⠀⢰⣇⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⣌⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠄⢡⢘⣿⣿⣿⣿⣿⣿⣿⣷⠹⡄
⠀⠀⠀⢺⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡳⣝⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⢳⡀
⠀⠀⠀⠀⠙⣮⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢷⡀
⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⡟⠛⡮⢿⣽⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡻⢿⣿⣿⡿⡁⠀⠙⠓⠒⠒⠒⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣭⡻⣧⡼⢦⡀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠳⣝⡳⢤⣄⡀⢠⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡌⣿⣦⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⢟⣄⢿⣿⣿⣿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠮⣭⣟⣛⣛⣟⣣⠞⠹⣌⣛⣻⣭⡭⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠌⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢳⡾⠋⠉⠛⢶⡀⠀⠀⠀⠐⠊⠀⢠⡙⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢐⣾⡇⠆⠀⠀⢸⣇⣀⠀⢰⢁⠄⠠⠠⠉⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠳⣤⣤⡴⠛⠃⠉⠛⠿⢊⡈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⣈⠂⠀⠀⠀⢀⠠⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⠀⠀⠄⠂⠈⣀⠋⠀⡀⠁⠈⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠃⠀⠰⠂⠀⠘⠂⠈⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠤⠀⠀⠠⠄⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⡀⢹⠄⠀⢀⠠⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⣘⢏⠂
⠀⠀⠀⠀⠸⡐⠉⠖⠐⠀⠀⢀⠂⠄⠀⠀⠀⠀⡰⠀⠎⢸⡀⠈⡵⡚⣢
⠀⠀⠀⠀⠀⠡⢠⡀⠈⠀⠀⠀⠂⠀⠀⡶⡀⢁⠸⠂⡉⠼⠀⣴⢎⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠻⠟⣼⠀⠘⢑⠄⠀⠀⠈⡀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡴⠂⠠⢀⣇⣀⢀⢤⣦⡔⠻⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠊⠀⠀⢀⠀⠀⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⣤⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡈⢿⣿⡀⢛⢿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⠀⠐⠛⠒⠚⡉⠉⣡⢈⠓⢓⠒⠽⠭⣗⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠉⠐⠐⠁⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠖⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣆⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣛⠧⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠐⡄⠀⠘⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⡀⠀⢸⣄⠀⠀⣴⣀⢀⡼⠦⠴⢳⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢣⡀⢀⣟⡲⠶⢉⡙⠒⠊⠈⠑⠊⢁⠼⡋⠛⣗⢦⣎⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⢠⣎⣉⣁⡋⠙⠦⡀⠙⢦⢀⠤⣤⣤⡞⢦⡈⠣⡈⠢⣭⠀⢳⡞⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠳⢖⠒⠒⠸⡟⢦⡀⠙⢦⣸⠗⠊⠉⠉⠱⡀⠙⢆⠈⢢⠎⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⡀⠀⠀⠀⠙⠦⣝⣤⡠⠏⠠⡤⣤⠤⡆⠈⠓⠒⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⣄⠀⠀⢀⣤⣤⣶⣞⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠛⢛⣛⡛⠛⠛⠛⠛⠓⢒⣒⠒⢠⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⣉⣭⣥⣤⣭⣽⣯⠥⣀⢠⠠⠜⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣩⠭⠥⠤⠤⠬⠭⠭⠭⠿⠛⠿⠿⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⢅⣠⠁⡤⠤⠤⠤⢤⣤⣤⣭⣭⣭⣭⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⠽⠛⠛⢛⣛⣋⣉⣩⠭⠭⠭⠭⠭⢭⠽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⢷⣤⣄⡒⢒⠂⠀⠀⠀⠀⣐⣛⣓⣒⢂⣾⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠘⠡⠋⣿⣔⣒⣒⣐⣒⡲⢤⣤⣤⣾⠻⠘⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⠚⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠲⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠤⣤⠤⠤⠤⣤⣤⣤⢤⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⣀⣀⣻⠀⢘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⠤⠬⠭⠟⠻⠿⠶⠖⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⢀⣀⣀⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣿⣿⣿⣿⡿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣭⣭⣭⡻⣿⣿⣷⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠛⠻⠿⠋⠁⠀⠀⠙⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣧⣀⡀⠀⠀⠀⠀⠀⣀⣠⣄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡿⣩⢿⣿⣷⣶⣶⣶⣿⣿⣯⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣷⡀⠁⠀⣩⣯⠈⢧⡁⠀⠀⣠⢿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⣍⠛⠛⢩⣏⠀⣀⠙⠛⠋⣡⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢀⠙⣿⡿⣋⠀⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣧⡈⠩⣭⠭⠍⠉⠁⣠⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢿⣿⣷⣄⣀⣀⣀⣠⣾⣿⢇⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣤⣶⣿⣿⣿⢿⣆⠙⠿⣿⣿⣿⣿⡿⠟⢁⡾⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢿⣿⡿⣿⣿⣿⣿⡆⠀⠀⣠⣌⠹⣿⢡⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠷⣽⣿⣿⣿⣧⠀⠰⠿⣿⣿⣿⡿⠿⢷⡀⢸⣿⣿⣿⣯⠾⡫⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠙⠻⢿⣿⣶⡀⠀⢹⣿⣿⡁⠀⠘⣷⣿⡿⠿⡛⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣘⣛⣛⣃⡀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣟⣻⠿⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⠛⠉⠁⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣀⣠⣶⣶⠦⠤⠀⠀⠀⠀⠀⠀⠤⠴⣶⣶⣄⣀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣟⣉⣹⣯⢿⣿⣶⣤⣀⣀⣀⣀⣤⣶⣿⡿⣽⣯⣉⣿⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣏⢻⠐⠁⣿⣿⠱⣽⣿⠿⠿⣿⣯⠎⣿⣿⠌⠂⡟⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⣿⠈⢧⣀⣈⣡⡾⢻⡿⡄⢀⢿⡟⢷⣌⣁⣀⡼⠁⣿⣩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣀⡀⠈⣉⣡⣴⠟⣁⠇⠸⣌⠻⣦⣈⣉⠁⢀⣀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣭⣿⠛⠛⠛⠉⠁⠸⣽⣄⣠⣯⠇⠈⠉⠛⠛⠛⣿⣭⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠟⠃⠉⠉⠉⠉⠉⠉⠘⠻⡄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⡀⠀⠀⠉⠛⠛⠉⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠛⠛⠻⠿⠛⠿⠿⠿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣟⠀⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠆⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠁⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⢀⣀⣿⣧⠀⠀⠀⠙⠛⢿⡷⣦⠀⢀⠀⠀⢠⡞⠃⠀⠀⠀⢸⣿⠟⠒⣦
⠀⠀⠀⠀⠀⠙⠉⠹⣯⠀⠀⡄⠀⠀⢀⣤⡄⣴⡀⠀⢠⣼⣿⡷⠀⠀⠆⠀⠿⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢀⣀⣨⣟⣡⡿⠁⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠷⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⠠⠄⠀⠀⠀⠀⠠⠄⠁⠀⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠛⠛⢀⠀⠀⠈⠠⠤⠐⠀⠀⠀⠀⠉⠀⠘⠛⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣄⠘⠛⠋⠀⠀⠀⠀⠘⠀⠀⢰⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢠⡄
⠀⠀⠀⠀⣠⣾⠟⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢣⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠁⣀⡀
⠀⠀⠀⣴⡿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠘⢷⣄
⠀⠀⣴⡟⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⢀⡿⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⣿
⠀⣿⠇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻
⣀⠟⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣶⠶⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⠐⠛⠛⠛⠛⠒⠒⠒⠛⠷⠒⠒⠛⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⣶⣶⠄⢀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠙⠉⠀⠀⠀⠀⡿⠁⠀⠐⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠆⠀⠰⠀⢀⣄⠘⠃⣀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠈⠁⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⠈⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣀⣀⣀⣀⣀⣀⡴⠆⠉⠁⠀⠀⠀⠀⢀⣀⣴⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⢀⣀⣀⣀⣀⣤⣴⣶⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣤⣤⣴⣦⣤⣤⣤⣦⣤⣤⣶⣶⣴⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠉⠉⣉⡉⠉⠛⠋⣀⣀⣁⣀⣀⣀⣀⣀⠀⢀⡀⠀⠀⠈⠁⠀⣿⣧⣌⣡⣤⣬⣭⣩⣤⣤⣤⣤⣤⣭⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢶⣄⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠙⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠻⠶⣦⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠙⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⡏⠉⠙⠛⠻⠶⠶⣤⣴⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣰⡟⣀⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⡀⠀⠘⣷⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣼⢏⣾⠋⢁⣀⣀⠈⠙⣷⡀⠀⠀⠀⠀⢠⡾⠋⢀⣀⣀⠈⠻⣧⠀⣿⣴⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⣾⠇⢠⡿⣭⣝⣷⡀⢸⣧⣤⣤⣤⣤⣿⠁⢰⣟⣭⡝⣷⠀⢸⣷⠟⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢰⡟⠃⢻⡇⠈⢷⣤⣤⠟⠀⣸⡏⠀⠀⣶⠀⢿⡄⠘⢷⣤⣴⠟⠀⣸⡇⠰⣦⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠛⠀⠀⠻⣦⣀⡀⢀⣀⣴⠟⠀⠀⠀⢿⡄⠈⠻⣦⣀⡀⢀⣠⣴⠏⠀⠀⠾⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⢀⣼⠇⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀⣤⣤⣤⣶⡆⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠈⢿⡄⢸⣿⣹⣏⣹⡇⢀⣾⠃⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠛⢶⣭⣉⣉⣩⣵⠟⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣀⠈⠉⣉⢉⣀⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉



⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣶⠤⠶⣤⢤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠏⠉⠉⠀⠩⠶⣶⠶⠮⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠰⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡀⠀⢠⠷⡄⠀⡰⠑⢤⡀⣠⠛⢆⠀⠀⠀⠀⣿⠀⡀
⠀⠀⠀⠀⠀⢸⠀⣀⣤⠀⠀⠀⡏⠙⠶⢃⣀⣙⡶⡁⣀⣀⣈⣁⣀⣈⠳⢴⡆⠀⣿⡏⢹⠴⢆
⠀⠀⠀⠀⠀⠈⠋⠀⣏⣀⣀⣼⠗⡫⠭⠭⠓⠒⣦⢨⠤⢤⡇⡔⠚⠩⠭⣭⢳⣤⣏⣁⡄⠰⡚
⠀⠀⠀⠀⠀⢀⢠⡴⠿⠛⢿⠻⡎⡄⢀⡀⠀⠀⢸⢸⠀⠀⡇⡇⣀⠀⠀⣿⢸⣟⢯⡀⠓⠊⠁
⠀⠀⠀⠀⠀⠀⢸⠰⡟⠓⠘⠋⢳⡱⣈⣁⣀⣠⠟⣸⣤⣤⣣⠳⣍⣡⣴⢏⡼⠙⠋⡇
⠀⠀⠀⢠⠀⠀⢸⠀⠁⠀⠀⠀⠀⠙⠒⠭⠤⠐⠊⠁⠀⠀⢹⡕⠒⠒⠒⠋⠀⠈⢀⠇⠀⠀⠀⢀
⠀⠀⠀⢸⠀⡄⡀⠓⠤⠤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⢰⠢⠴⠊⠀⠀⢀⠄⢸
⠀⠀⠀⢸⠀⢀⣉⡖⢲⡆⠀⢸⡀⠀⠀⠀⠾⣄⠀⠀⠀⠀⢀⣠⠔⠀⠀⣼⠀⡤⡀⠀⠀⢀⡀⢸
⠀⠀⠀⠸⠀⢸⣉⡁⠘⠛⡇⠀⢷⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢠⠗⣖⢋⣉⠇⠀⠑
⠀⠀⠀⢰⠀⠀⠸⢧⣼⠉⠁⠀⠀⠙⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠘⠚⠰⣂⣈⡅
⠀⠀⠀⠘⠀⠠⠀⢠⠤⢦⡀⠀⠀⠀⠀⠀⢹⢦⡀⠀⠈⠉⠀⣴⢯⡀⡠⢤⠀⠀⠀⠀⠏⢁⣹
⠀⠀⠀⢰⠀⠀⡤⠼⠗⠚⠋⢧⠀⢒⡆⠀⡏⠀⠙⠓⢶⣶⠚⠁⠀⣯⡅⢈⣳⠀⠒⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠹⠴⢶⡶⢾⠋⠀⠀⡴⠚⠳⢤⣀⣀⣾⠙⡷⠤⠞⠉⠙⠻⡄⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠓⠊⠀⠀⢸⡇⠀⠀⠄⠀⠄⢸⠒⡇⠀⠀⢠⠀⠀⣇⠀⠸⡖⡃⠀⠀⢸
⠀⠀⠀⠘⠄⠀⠀⠄⠤⠀⠄⠄⠠⠸⠇⠰⠤⠤⠀⠤⠸⠲⠧⠤⠀⠴⠴⠀⠿⠠⠄⠀⠄⠀⠠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⣀⣼⣧⣴⣿⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⠉⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠉⠀⠀⠀⠀⠈⠀⠀⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣀⣀⣤⣤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⣟⣿⣷⣶⣿⡟⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠉⠀⠉⠁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣿⠀⠀⠀⠀⢸⡇⠀⠗⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣬⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡙⠒⠀⠀⠀⠀⠀⠀⣠⣵⣤⣤⣤⣶⣴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⡀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣶⣤⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠟⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⢀⠀⠀⠀⠀⠹⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡦
⣴⣿⣶⣎⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠉⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠀⠀⣀⣀⣤⣶⣿⣿⠿⠋
⠀⠀⠀⠈⠻⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠁⠘⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣾⣷⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠳⢦⡀⠀⣖⣁⠤⠐⠒⠒⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⢀⡠⢿⡎⠉⠀⠀⠀⠀⠀⠀⠀⢠⣕⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠤⠚⠁⠀⠈⠇⠀⣠⡶⠛⠛⠛⠛⠳⣤⡘⠃⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢀⠔⠀⠐⠏⠀⠀⡾⢀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣯⣀⠀⠀⣀⡀⠘⠉⢾⠁⣠⠖⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢉⡭⠽⠥⠀⠀⠀⢰⠀⠈⠉⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠀⡄⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⣴⠋⠙⢷⣄⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⠴⠀⠁⠀⠀⠀⠀⠙⠳⠶⠤⠶⠞⠋⣠⠖⠀⠀⠈⣻⡦⠤⠤⣤⡒⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣮⠏⠀⠀⠀⠀⣀⣤⠤⢤⡀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠂⠠⠤⠤⢤⠤⠒⢞⠉⠑⡴⠋⠀⠀⢀⣀⠀⠀⠉⠁⠈⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⢣⠀⠈⣡⠈⠁⠀⠴⠊⠁⠀⠁⠐⠂⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠲⠀⠀⠀⠀⠀⡐⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⡞⢲⣂⠉⢋⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠃⠚⢅⠀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣉⢊⠉⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠻⠿⠬⢷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⢠⡶⠖⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⢠⢤⣄⠀⢀⣀⡀⢰⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢾⣄⠀⢸⣸⠁⠀⠀⠤⠁⠈⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢂⡀⠒⢻⡀⠊⠉⠁⠀⠀⠀⡀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠭⠭⠭⠥⡃⠀⠀⠀⠀⠀⢀⡠⢞⠻⣦⠀⡀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠒⠒⠒⡿⠀⠀⢀⡠⠔⢉⡰⠇⡄⢹⡇⠃⠀⠀⠘⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⠍⡽⠧⠔⠊⠉⠉⠲⠍⠭⠭⠕⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠄⠀⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣩⣵⣶⠂⠀⢚⣉⣋⣉⡉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠿⣿⣿⣿⣷⣾⢿⣿⣿⣿⣿⣦⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⣀⠉⣿⠉⠀⠀⠀⠈⠙⢿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⠀⣠⣤⣀⣀⠀⠀⠀⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣭⡻⣿⠿⣶⣶⣾⣷⣾⣿⣿⣷⣶⣾⣿⠿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢸⡁⠙⠁⢤⡈⠙⣿⠬⠟⢩⡟⠘⣿⣇⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣠⣦⠀⢽⣟⠀⠈⠿⠟⢋⣀⡀⣻⡟⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⢻⠋⢆⠀⠀⠀⠀⢀⠀⢀⣘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⣜⣿⣧⠱⣌⢶⣄⡀⠀⠀⠀⠹⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣶⣿⣿⡜⣿⣷⡈⠓⢽⣿⣶⣤⣀⣀⣴⠟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣿⣄⠀⠙⠻⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣆⠀⡀⠀⣽⣯⢹⣿⣿⣿⣿⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣎⠋⣸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠛⠛⠛⠉⠁

⠀⠀⢰⣶⣴⣶⢠⣤⣄⣤⣤⣴⣦⣠⣤⣤⣄⢰⡦⣤⣤⣤⣄⢀⣤⡄⣠⣤⡄⣠⣤⡀⣠⣄
⠀⠀⢸⣿⣿⣿⣹⣿⡇⣸⣿⢸⡿⣿⣿⡇⣿⢸⡇⣿⢸⡇⣿⢨⣿⣿⣿⣼⡇⣿⢿⣏⣻⣧⡀
⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠁⠉⠈⠁⠉⠈⠋⠉⠘⠟⠃⠉⠋⠁⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠈⠙⢷⡄⠀⢠⡿⠋⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠈⣿⡀⣾⠁⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⡆⠀⣿⠁⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣿⣇⠀⣿⠀⠀⢀⣿⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢻⣿⠀⣿⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠘⣿⠀⣿⠀⠀⣿⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡇⠀⣿⠀⠀⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⣠⣤⣤⣽⣤⣄⣁⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠁⠀⠀⠈⠉⠙⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠉⠙⠻⢷⣄⠀⠀⠀⢀⣴⠞⠋⠉⠉⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⣗⠐⠒⣾⠃⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⠇⠀⠀⠀⣿⠀⢸⡏⠀⠀⠀⠰⠄⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠀⢀⣠⣶⠟⠁⠀⠈⠻⣦⡀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠒⣄⣤⢤⣤⡴⢻⣆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⠘⠀⠀⠸⠂⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠁




⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠀⠁⠀⠀⠁⡀
⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀
⠰⡂⠙⠀⠀⣶⠋⠙⢦⠊⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣠⣠⣄⡂⠀⠱⡀⠀⠀⡀
⠛⡀⠴⠤⡄⣿⠁⠀⢨⣇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣽⠍⢹⣾⠁⢀⢸⣿⠇⠁⠀⠀⠈⠀⠴⢃⡀
⠀⠀⢀⠀⡛⠉⠓⠒⠋⠈⠩⠅⠀⠁⠀⠀⠁⠀⠀⠀⠀⠹⠦⠞⠘⠷⠤⠟⡀⠊⢶⡄⠀⠠⠐⠁⠁
⠀⠀⠛⠁⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠁⢀⣈⣠⣈⡀⣤⡤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⢀⣿⡉⠀⠸⡇⣿⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠃⠀⠀⠁⠀⠄⣼⣌⠳⠤⠼⠁⠘⠲⠚⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠈⡀⠐⢄⠀⠀⠐⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠂⡤⠀⡀⠀⠀⢿⢠⡄⣄⠀⠀⠀⢀⣰⡀⠈⡀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢀⠘⠈
⠀⠀⠀⠀⠀⠀⢀⠂⢠⣶⡀⢘⡀⠢⠀⠁⠰⢊⡝⠓⠆⠀⠀⠠⠎⠰⠆⠉⢀⠀⠀⠀⠀⠀⠔⠉⠲⡃⢇⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠉⠉⠀⢈⣁⡀⠨⠀⠀⠀⠀⠈⢠⣀⠌⢨⠒⠖⠂⠘⢕⠀⠀⠀⠀⠂⠀⠀⠀⠆⠱⠶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⡔⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡷
⠀⠀⠀⠐⠀⠀⠻⠁⠤⠤⠀⠀⠰⠒⠀⠟⠠⠐⠒⣤⢒⣄⣀⡄⠀⠀⢀⡀⣢⠤⠠⠤⣄⣀⡀⠀⠀⢈⡉⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣴⣶⣶⣿⣷⣶⣶⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣄⣀⣀⣼⣿⣿⣄⣀⣀⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣴⠻⡆⢸⡟⠃⢸⡟⠃⢸⣠⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⢰⡆⢸⡗⠂⢸⠗⠂⢸⢿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⢻⣿⣿⡇⠀⠀⠀⠀⠹⠾⠃⠸⠷⠆⠸⠷⠆⠸⠂⠳⠀⠀⠀⠀⢸⣿⣿⡟⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠀⣈⣿⡇⣴⢦⡀⡴⢤⢠⡄⡴⢤⢠⡶⠄⢠⡶⣤⠀⣴⠠⣄⡤⢸⣿⣁⠀⠀⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⡇⣹⠞⠃⡷⣞⢸⡇⡇⢸⢸⡗⠀⢸⡇⣿⢰⣿⡇⢹⠃⢸⣿⣿⣧⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠁⠉⠈⠁⠉⠉⠈⠉⠁⠈⠙⠁⠈⠀⠁⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣿⣿⣿⣯⣤⣤⣤⣤⣼⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣭⣭⣭⡭⢍⣙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠙⠛⠁⠀⠛⠁⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠟⠛⠉⠠⠶⠿⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣘⣃⣀⣒⡀⠀⢐⣒⣀⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⣐⣒⠀⠀⢐⣂⣀⣻⡁
⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⡤⢹⠻⡏⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠠⡌⢹⢺⡡⡄⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠁⠀⠑⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⠄⣶⢴⢸⠁⣾⡄⡏⡆⠀⠀⠀⠀⠀⠀⡀⠂⠀⠄⠠⠀⡦⣶⣾⢀⣶⣰⣓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆⣠⣤⣶⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⠿⣿⣯⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣷⣶⠤⢀⣀⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣶⣶⣿⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⣿⣿⣿⡻⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠤⠀⠀⠀⠤⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠⠀⠀⠀⠤⠄⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢺⠛⠛⣙⢲⢶⡞⡋⠋⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠋⢛⠓⡶⣶⣋⠋⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠭⠞⠀⠳⠥⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠬⠴⠁⠙⠬⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⢰⠀⣤⡀⡄⡄⠀⠀⠀⠀⢀⠀⠀⡀⢠⠀⣀⢠⣀⣤⡆⢠⣆⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠀⠉⠁⠓⠁⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠁⠉⠃⠈⠉⠘⠈

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣙⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⡛⠻⣿⣶⣤⣤⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣀⢀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣛⡻⠿⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠻⠿⠿⢿⣿⠿⠿⢛⣫⣽⣿⣿⣿⠿⠿⢛⣩⣵⣿⣿⣿⣿⣿⡝⣿⣿⣿⣷⡄
⠀⠀⢀⣾⣿⡟⢁⣿⣿⠃⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡟⢿⣿⣆
⠀⢀⣾⣿⠏⢀⣾⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢛⣻⣿⣿⣿⡿⠟⢹⣿⣿⣿⣿⣿⣧⠈⢿⣿⣆
⠀⣼⣿⡏⠀⠸⠋⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠸⣿⣿⣿⡿⠁⠉⠀⠈⣿⣿⡄
⢰⣿⣿⠀⠀⠀⠀⠀⢿⣿⡟⠁⠀⣀⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠙⣿⣿⠇⠀⠀⠀⠀⢸⣿⣷
⣼⣿⡇⠀⠀⢀⣀⡀⠘⣿⣿⣀⣬⣤⣴⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣄⣸⣿⡏⠀⣀⣀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⣿⣯⡉⢳⣿⣿⣿⠟⢉⡽⠋⣭⣿⣯⡛⢿⣦⣠⣤⣴⣤⣄⣠⣾⠟⣫⣿⣯⡍⠻⣍⠛⢿⣿⣿⣷⠋⣩⣿⡇⠀⠀⣿⣿
⣿⣿⡇⠀⠀⢹⡌⣹⣿⣿⣿⡏⠀⢿⡀⠘⢿⣿⣿⠇⠀⢹⣿⠉⠀⠈⢹⣿⠁⠀⢿⣿⣿⠟⠀⣸⠇⠈⣿⣿⣿⣿⡉⡼⠁⠀⠀⣿⣿
⢿⣿⡇⠀⠀⠀⠡⡹⠛⢿⢿⣇⠀⠀⠁⠀⠀⠙⠓⠂⠀⢸⣿⠀⠀⠀⢸⣷⠀⠀⠀⠈⠑⠂⠐⠁⠀⢠⣿⢿⠟⠻⡱⠁⠀⠀⠀⣿⣿
⢸⣿⣷⠀⠀⠀⠀⢻⣄⠀⣨⣿⣷⣄⡀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⣀⣴⣿⣯⡀⢀⡼⠁⠀⠀⠀⢰⣿⣿
⠈⣿⣿⡆⠀⠀⠀⠀⠙⠛⠋⠘⣯⠙⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠙⠃⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⢩⡏⠈⠛⠛⠁⠀⠀⠀⠀⣾⣿⠇
⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠛⣛⠛⠛⠉⠀⠀⠀⠀⠀⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣦⣄⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⣴⣾⣋⣁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠙⣿⣿⣦⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠳⢦⣄⡀⠀⠀⣀⣤⠶⢻⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠙⠓⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠠⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠻⢿⡿⠛⠀⠀⠂⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠉⠀⠰⢦⡰⠃⡀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⢁⠀⠴⡂⢰⠃⡀⠀⠔⠀⠄⠀⠀⠰⠀⠀⠠⡄⡂⠀⠠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠈⠁⠀⠙⠢⡑⠊⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠈⢳⣿⠀⠀⠀⢠⠄⠀⣀⠄⡀⠀⠷⠟⠀⠀⢂⡀⠀⠘⠑⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠂⠀⣴⡆⠀⠀⠈⠀⠀⠀⠘⠈⢿⡄⠀⠄⢹⣤⢚⣁⡀⢁⠀⠢⢰⡀⠀⢢⣰⣀⠀⠈⠉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠋⠉⠰⢉⠳⠀⠱⠦⢄⣀⣈⣧⠠⢁⡾⠁⠀⠙⠁⠜⢁⠀⠀⢀⣤⡔⠊⠁⣀⡀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣄⠀⠀⠀⠄⠀⠀⠸⣆⣾⠀⠀⠀⠰⠀⢠⠞⢻⣴⡏⠀⠀⠀⢸⠭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⢢⠰⣄⣀⠇⠀⠀⠀⠀⡀⠀⠹⣇⠀⡀⠀⠙⠁⢨⠨⠸⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠷⠤⠴⠶⢶⡶⠶⠿⣶⣤⣻⣦⠁⠐⠀⠀⠩⢛⠓⢾⡇⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠔⠀⢀⠞⠁⢄⠀⠀⠀⡬⠙⢿⣷⡀⠀⠀⣐⡙⠁⢈⡇⢠⠀⠵⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⡀⠰⣠⠇⠈⢉⡠⠃⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠕⠀⠄⢀⠂⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⢋⢁⠁⢠⠿⢶⠶⠛⠁⠑⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠐⠃⢀⡀⠒⡀⠤⢢⣀⣀⣠⣄⣀⡛⢸⣿⣆⣡⡾⠃⢀⠄⠠⣎⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⢐⠚⠉⠉⠉⠀⠈⠙⠻⢿⣿⡿⠋⠀⠐⠊⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠏⠑⢀⠰⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣄⡰⠾⠂⠈⢠⠰⡴⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠁⠀⣤⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡤⠔⠉⠁⠉⠀⠈⠐⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢭⣀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⢈⢳⡔⠃⠁⠀⠀⠀⠀⢱⡠⡀⠀⠀⠀⠀⠀⠀⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⣊⠁⢀⡧⣠⠴⠚⠛⠛⠳⢦⡹⠈⠐⢄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡔⠁⠈⠁⢀⡾⣰⠃⣀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⢒⣶⣞⠁⠀⢿⠘⠓⠀⠀⠀⠀⠀⣼⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠘⠀⠀⠘⢦⡀⠀⠀⠀⢀⡴⠋⠳⣄⡐⠁⠀⠀⠀⠀⣀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠐⠀⠀⠀⠀⠀⠉⠓⠒⠋⣁⡼⠁⠀⠈⠛⠒⠒⠢⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⣀⢀⣀⡀⣤⠐⢎⡝⠁⠀⠀⠀⠘⠯⠝⠂⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠙⡄⢈⡶⠋⢀⠔⠊⠉⠐⠒⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠉⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⡧⢠⠀⠴⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⠃⠚⡂⡘⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣀⣍⣍⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⢨⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⢄⣀⣀⣀⡀⡸⣹⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠲⣄⣐⣰⡁⠀⠠⠥⠉⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁⢹⡏⠀⠀⠀⠀⠐⣠⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠩⠭⢽⡇⠀⠀⡀⠔⣩⢂⠘⡇⠆⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡭⢭⡽⠇⠒⠉⢑⠞⢭⠹⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠳⠤⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⣤⠀⣥⣴⠀⢷⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠙⣀⠀⠪⠠⡁⠄⢽⠗⣶⠊⠑⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠉⠄⢬⠀⠀⠀⢤⡋⢘⡤⣿⡀⣀⣿⡜⠃⠠⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠞⠀⠰⠂⢠⡄⠆⣈⠀⠝⣿⡤⠘⣌⠓⢼⡀⠀⣶⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠉⠀⢖⠀⠀⠀⢬⡀⢼⣷⣰⠧⢿⢸⡆⢻⣀⠉⣸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠜⡳⢀⡀⠁⣀⠉⠁⠉⠻⣾⣿⠫⣄⠛⠔⢣⠸⢾⢤⠀⢸⡇⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢰⠻⠆⢋⡦⢡⠈⠉⠠⠜⠀⠀⢹⣿⡅⠘⣦⡄⣽⠀⠚⠀⠓⠈⠁⠋⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⢴⢛⠆⠀⠈⠁⠼⠪⠄⠒⢠⠸⣶⣷⠎⠹⠇⣾⠀⢁⡼⠉⢧⠹⡄⡁⢹⣞
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⡃⢠⣠⡈⠙⢰⡌⠘⠂⢃⡀⠙⢀⢸⣿⡿⠀⣠⢆⠀⣲⡞⣀⣀⠸⡆⡿⡗⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⡈⠛⡁⠰⠈⣤⡁⢀⡒⠀⠀⣀⢸⣿⣷⣿⣷⢻⡀⣹⠀⢸⣇⣽⣓⠲⠧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢦⠚⡰⠚⠁⠒⡈⡀⠩⠠⠀⠙⢿⣿⣿⣯⠀⢷⣛⣿⠠⣸⠇⠳⢤⣖⣲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠩⠀⠙⠃⠄⠗⠀⠀⠸⣿⣿⡄⠀⠐⠜⠻⠴⠛⣾⣉⡉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠖⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⡶⣿⣿⣿⣧⣿⣿⣿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡂⢰⡆⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠘⠣⠂⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠣⠀⠉⠶⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡶⣲⠖⢒⣚⠗⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⠀⠀⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠓⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠀⠀⠘⠁⠀⠰⠶⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⣀⠀⠀⠀⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠋
⠀⠀⠀⠠⠄⠀⠁⠴⠀⠸⠀⠀⠀⠀⠀⠺⠦⡀⣤⠄⢈⡋⠀⠀⠀⠀⢸⠂⠂⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠰⠀⡼⠉⠀⠒⣦⠀⠀⠀⠀⠈
⠀⠀⣄⠀⠀⢀⠀⠁⠄⠀⠄⠀⠙⠀⠀⠀⠀⠄⡀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠘⠃⠀⠀⠀⠈⠉⠂⠀⠀⠀⠣⠀⠂⠊⠀⠊⠁⠰⠂⠐⣠⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠐⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠈⢎⣉⡠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⢦⣄⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⣻⣭⣿⣿⣿⣦⣠⣤⣶⣾⣿⣿⣿⣿⣾⣿⣷⣤⣴⣿⣿⣿⣭⣟⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⢟⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣷⡽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣠⡶⠟⠛⢛⣻⣿⣶⣄⣀⣀⣀⣀⣠⣶⣿⣟⡛⠛⠻⣶⣄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⠀⠀⢰⡟⣩⣯⢿⢻⡿⠉⠉⣿⡟⡾⣽⣍⢻⣆⠀⠀⢻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⣴⣿⣷⣍⡥⠋⣼⠃⠀⠀⠘⣧⠙⢬⣭⣾⣿⣦⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⡈⠛⠿⠟⠋⢠⣾⢃⣤⣤⣤⡄⡙⣷⡀⠙⠻⠿⠛⢁⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⠷⠶⠶⠾⠛⠁⠙⠿⣿⣿⠿⠛⠈⠛⠷⠶⠶⠾⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⡄⠂⠀⠀⢸⡇⠀⠀⠐⢀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣀⠤⣀⠀⠈⠒⠒⠒⠻⠟⠒⠒⠒⠁⠀⣀⠤⣂⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢿⣒⠶⣄⣀⠀⠀⣀⣠⠶⢚⣭⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣬⣽⣯⣭⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠛⠁⠀⠀⠀⠙⠻⢿⣿⠃

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


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


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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⠛⠛⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡾⠀⠀⠀⠀⡀⠠⠖⢛⠒⠀⠁⠈⠐⠂⠀⠀⢀⣄⡀⠉⠂⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠣⠀⠀⠈⠀⠀⡀⠀⠀⠀⠄⠀⠰⠆⠘⠀⡄⢸⡇⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠚⠀⠀⠀⠀⠱⠄⣀⠀⠀⡖⠀⢸⡇⠀⠀⠀⢨⠟⠀⣀⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡇⠀⢸⡇⠀⠀⠀⠘⣆⠀⣇⠀⠀⠉⠙⠛⠒⠶⠶⠶⠶⠶⠚⠛⠉
⠀⠀⠀⡇⡆⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⠀⠀⣾⠁⠀⠀⠀⠀⡸⢃⡼
⠀⠀⠀⣧⡅⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⣿⠀⠀⠐⠚⠋⠉⠉
⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⢰⡏
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡴⠞⠳⠀⠀⠀⠀⠀⡏⢄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠀⣿⠳⠦⣄⡀
⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠢⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠁⠀⠀⠀⠉⠀⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠈⣧⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⢠⠃⠀⡿
⠀⠀⢹⡄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⡇
⣤⣤⣤⣿⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣦⣤⣤⣤⣤⣤⣤⣧⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣿⣿

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

⠀⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣄
⠀⠀⠀⢸⠃⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠸⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⠎
⠀⠀⣀⣴⠫⣀⠝⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣦⠉⣌⢡⣦⣄⣀⠀⠀⠀⠀⠀⣀⡠⡿⢕⡪⢷⢄⣀⠀⠀⠀⠀⠀⣀⣤⣴⡌⣻⡁⣴⣤⣄⡀
⡔⠉⠀⠈⠋⣿⠛⠁⠀⠉⢢⠀⠀⠀⣾⣿⣿⣿⣆⣿⣸⣿⣿⣿⡇⠀⠀⢠⠋⠀⠀⠳⢺⡗⠇⠀⠀⠑⡄⠀⠀⣼⣿⣿⣿⡇⣼⣀⣿⣿⣿⣿
⡇⠀⠀⠀⠠⣿⠆⠀⠀⠀⠸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦
⠀⠀⠀⣸⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣀⣀⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸
⠀⠀⠀⠙⣄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⠉⠙⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢀⡞
⠀⢀⣀⣰⣯⠉⣹⣆⣄⡀⠀⠀⠀⠀⢀⢠⣤⠋⠒⠊⠣⣤⢀⡀⠀⠀⠀⠀⠀⣀⣰⣏⠐⢉⣧⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⠃⠉⠁⢳⣤⣀
⢠⣿⣿⣿⣾⣿⣷⣿⣿⣿⡆⠀⠀⢰⠁⢸⣿⣄⣀⣀⣼⣿⠀⢹⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡜⠁⢸⣿⣷⣶⣶⣿⣿⠀⠑
⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠸⠀⠸⠿⠿⠿⠿⠿⠿⠄⠸⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠇⠀⠾⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⢠⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣦⡀
⠀⠀⠀⢸⠉⠛⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠉⢻⡇
⠀⠀⠀⠸⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠒⠁⡸⠂⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣰⠇
⠀⢀⣠⣴⣏⠒⣫⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣷⠄⢴⡁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣍⣩⣿⣶⣄⣀⠀⠀⠀⠀⠀⣀⡤⡾⣷⣦⣾⣿⣦⣄⡀
⣰⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠀⠀⢠⠔⠊⠉⣧⢿⣥⠋⠙⠒⢀⠀⠀⠘⢻⣿⠛⠿⠟⠿⠿⠿⡿⢿⣧⠀⠀⢠⡞⠉⠀⠹⡼⣿⣿⡼⠁⠈⠙⣆
⠹⠛⢻⣿⣿⣿⣿⣿⡟⠛⠻⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢾⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠿⠀⠀⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸

⠀⠀⢀⣠⠖⠒⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠒⢦
⠀⠀⢸⣧⡀⡀⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⡇
⠀⠀⠀⠳⣀⣉⡀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠋⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⡀⢰⠃
⠀⠀⣀⣠⡟⠿⢛⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣑⣤⣚⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⢟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⢿⣿⣦⣄⡀
⣴⣿⣿⣿⣿⣶⣿⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣟⠉⢹⡿⠿⢿⣧⠀⠀⢠⣴⣾⣷⡻⣮⡴⢏⣿⣷⣦⡀⠀⠀⢠⣾⡿⠿⠿⣿⡆⣿⠿⠿⠿⣿⣦
⣿⣷⣶⣶⣶⣿⣶⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣧⣽⣯⣼⡇⣿⣬⣿⣥⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠒⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠗⠁⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⡜⡆
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠦⢸⣇
⠀⠀⠀⠀⠀⢼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠂⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢢⢈⣽⡿
⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠇⠀⡤⡆⠁⠄⠠⠀⠀⠠⠐⠀⠀⠀⠐⠀⠀⠀⠐⡂⢸⠈⡈⣯⡇
⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣚⣉⣍⣭⡅⠀⠀⠀⠡⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⢸⡏⡿⣾⡂
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠨⠀⠀⠀⠀⢥⡤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⣆⢷⢘⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡃⣨⢸⡏
⠀⠀⠀⠀⢸⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⡅⣧⡇⠁
⠀⠀⠀⠀⢸⠘⢣⣖⣒⣒⣢⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢄⣄⣔⡈⡇⣷⠃
⠀⠀⠀⠀⠀⢇⣉⠉⠉⠉⠉⠙⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠿⠿⠛⣛⡃⡛⠉⠘⠋⠘⢑⣱⣿
⠀⠀⠀⠀⠀⠀⢀⣈⠙⠩⠮⠁⠀⠀⠀⠀⠀⠀⠒⣊⡹⠿⡿⣶⣶⣾⢟⡿⢝⡱⠶⠀⠒⠪⠍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠐⠛⠒⠉⠲⠴⠶⠦⠤⠤⠤⠶⠖⠒⠲⠾⠷⢴⣿⣹⣽⡾⠿⠛⠒⠛⠓⠒⠂⠂⠄⠰⠰⠦⠶⠾⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣲⣬⣭⣤⣤⣶⣶⣿⣿⣿⣿⡿⢿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣛⣻⡶⠶⣖⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣻⡿⣟⣉⣉⣭⣤⣐⣒⣒⣒⡒⠒⠒⣚⣛⣛⣭⠀⢸⣿⣷⣾⣶⣿⣿⣿⣷⣶⣯⣭⣞⣻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣵⡿⢽⣷⣿⡿⠿⠛⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣉⣧⠀⢺⣿⣿⣿⣿⣿⣿⣟⣿⠿⢿⣿⣿⣷⣶⣜⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣳⡿⣫⣾⠿⣛⣥⣶⠾⢟⣛⡿⠿⠿⠿⠛⠛⠛⠻⠿⠿⣿⠀⣸⣛⣉⣉⣉⣉⡛⠿⠿⣻⣿⣶⣾⣳⢌⣻⢿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⠟⣡⣾⡿⣯⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢮⣵⠾⣿⣯⢿⣷⡹⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⢫⣾⣟⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣝⢷⣍⢷⣜⢯⢿⡆
⠀⠀⠀⠀⠀⡴⠟⢿⣯⣿⣯⣼⣥⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣶⣿⣧⣿⣯⣷⠖⠉⠳⣆
⠀⠀⠀⠀⢸⠃⣤⣄⢿⣿⣿⣾⣷⣦⣤⣤⣤⣭⣤⣬⣿⣯⣤⣽⣿⣿⣿⣿⡆⠀⠀⣼⡟⢉⣍⣩⣭⣥⣤⣤⣤⣤⡶⡶⠶⠶⣷⡆⣿⣯⡿⢿⣴⣿⠄⡿
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢹⡟⣿⡏⠀⠉⠉⢉⣉⣤⣐⣠⣀⠀⠀⠀⢸⡿⣿⠇⠀⠀⣿⣷⣿⠉⠉⣡⣼⣯⣭⣤⣒⠀⠀⠀⠀⢸⡃⣿⡟⠁⣾⣿⣿⡴⠃
⠀⠀⠀⠀⠀⠙⣯⠉⠀⢸⣷⣿⡄⠀⢀⣴⡟⠉⢹⣍⡹⢿⣆⠀⠀⢸⣿⣻⡄⠀⠀⣿⡿⣿⠀⣼⣿⣁⣨⣿⣉⣿⣷⡀⠀⠀⣸⠁⢸⠀⢀⠈⣻⠏
⠀⠀⠀⠀⠀⠀⠹⣦⣴⢸⣿⣿⡀⠀⡎⢿⡿⣾⣿⣿⡋⢸⣿⢆⠀⢸⡇⠿⠷⣶⣶⣿⡇⣿⢰⡟⣿⡛⣿⣿⡏⣸⣿⣹⠀⠀⣿⢁⣿⠀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⣿⣿⡇⠘⠷⣼⣿⣌⣉⣉⣠⡾⠏⠉⠀⢸⣧⣶⠾⠿⠻⣿⣿⡇⠈⠉⠻⢿⣤⣤⣶⠿⠟⠁⠀⠀⣿⢶⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠈⣿⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣀⣸⣿⣿⠶⣤⡶⢿⣿⣷⣤⣴⣶⠶⢶⡶⠶⠶⠶⠶⠶⠾⡟⠸⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣟⣹⡧⠀⠀⠀⢸⣧⣤⣤⣦⣤⣤⣤⣤⣤⣦⣦⣤⣶⣿⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣿⠿⠿⠿⠿⠟⢿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠙⢿⠛⣿⠶⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⡴⢶⡾⠛⣙⠿⣦⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢹⡇⢻⡄⠀⢸⡏⣻⣿⣿⣿⣟⢹⡿⠀⣸⡇⢰⡟⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣧⢸⡇⠀⠸⠿⠋⢉⡟⠀⠉⠻⠇⠀⣾⠁⣼⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⡟⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢡⣿⡀⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⣿⣧⣾⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢻⡇⣿⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⢸⠉⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⣿⣿⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠢⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣄⠀⠀⢿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣖⠀⠀⠀⠀⠉⠙⢶⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⠳⣄⠀⠀⠀⣀⢿⢻⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣯⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠤⠼⠷⣄⣠⢹⡆⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢿⣉⣉⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠉⠈⠉⠙⠛⠛⠛⠛⠶⠶⠶⠶⠶⠚⠋⠁⣸⣿⣷⣜⣛⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡋⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠐⢶⣤⡶⠞⠀⠀⠀⠀⠀⣰⠏⠀⠙⠓⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡏⡀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⡇⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣼⠁⣷⣀⣀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠿⠿⠿⠿⡿⠀⢿⣿⣿⣧⣿⣷⡄


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣍⣿⣋⣙⣋⣛⣉⣤⣉⣙⣟⣋⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣭⣯⣭⣬⣭⣿⣯⣭⣭⣭⣯⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠉⠀⠉⠍⢩⣷⠀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⢀⠒⠒⠘⠯⢰⠂⠀⠀⠀⢀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡷⠆⣄⠔⢒⣖⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠱⢀⠀⠄⡐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡶⡄⠀⠀⠀⠞⠓⠈⠀⠀⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠷⠃⠀⠐⡄⠀⠀⠀⠀⠀⠀⠋⠛⣦⠀⠀⠇⠀⠀⠀⣀⣀⠠⢀⠀⢐⠸⡀⠀⡌⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠆⡀⡀⠈⣧⢤⣴⡞⣻⠟⠛⣛⣛⡿⠀⠓⣀⡀⠀⠀⠉⠉⠀⠐⠲⠇⠄⢁⣈⠐⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠀⠀⠐⠂⠀⠁⠁⠄⠀⠈⠁⠳⡤⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢱⢀⠀⡀⢋⠙⢁⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢋⠠⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⡧⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⢉⣉⣛
⠀⠀⠀⠐⣒⣀⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀⡇⠀⣤⠟⡀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⡆⠀⠒⠀⠀⢠
⠀⠀⠀⠸⠤⠄⠀⠠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠠⠀⠠⠧⠀⣀⠐⠈⠠⣐⣀⡀⣀⠚⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢁⢀⢀⡀⡀⠐⡄⠀⠀⠀⠀⠈⠀⠀⠐⢺⣂⣀⡀⠉⡆⠧⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠁⠈⠀⠐⡂⠀⠀⠀⠀⠀⢀⠀⠀⠈⡅⡀⠉⠹⠷⣆⠀⠀⢠⠤⣨⠍⢁⣀⡀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣄⣠⣀⣀⣀⡥⣶⠒⠃⠁⠘⠄⢀⣤⣶⠤⣄⠹⠔⢒⣦⡀⡀⡀⠀⠀⠈⠋⡃⠀⢀⣆⡿⠴⠟⠁⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁⣤⣤⣤⣄⠀⢀⠧⣯⡀⠀⠀⠀⠀⠀⣆⠘⠩⠿⠶⣔⣄⣰⣏⢠⡾⠇⠰⣶⢀⡠⣤⣀⣈⣁⠀⠄⢰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⠛⠟⠃⠀⠨⠢⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠈⡙⠱⠒⠈⠀⢨⠤⢈⣉⠛⠛⠁⠀⠀⠸
⠀⠀⠀⠀⡀⠰⠀⠴⠀⡆⠀⢸⣍⣉⠁⠠⠀⠉⠠⣾⣄⡀⠀⣈⣁⠭⠿⠃⠀⠀⠀⠀⠀⢶⠍⢉⠛⢛⢙⡉⡙
⠀⠀⠀⠀⠉⠁⠈⠈⠉⠁⠀⠀⠠⣤⣄⠀⠀⠀⠀⢠⠀⠆⡞⠟⠁⡐⣉⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠹⣆⠀⠀⢀⠈⠈⠞⢡⠀⠌⠈⡹⠡⠀⠈⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⣬⠂⠀⡇⣠⠀⠁⠠⠤⠀⠄⠄⠀⠉⠁⠔⠀⠈⠄⠀⢀⡺⡓⠀⢡⢠⠀⠀⡀⠀⠀⠘⠃⠀⠀⠀⣀⡇⠀⠀⡀⠀⠀⠄
⠀⠸⠿⠴⠀⠀⠀⠉⠙⠓⠢⢄⡀⠘⠁⢤⣤⠤⠒⠪⠤⠔⠋⠐⡠⠇⣸⡆⢠⡀⢠⠀⢐⠀⠀⠀⠀⠀⠀⠈⠉⠀⢚⣣⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠲⠖⠂⠀⠈⠀⠀⠈⠉⠉⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⢀⣁⠬⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠐⠢⠀⠢⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⣄⠀⠈⠐⠀⠘⠓⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡁⢐⡄⠤⠀⠀⠙⣄⣄⠘⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠄⡠⠄⠀⠀⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣴⣤⣤⣦⣤⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠿⢿⠿⣿⣿⣿⣿⠿⡿⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠋⠀⠀⠀⠈⠘⠿⠿⢃⠁⠀⠀⠀⠙⢿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢛⣶⢉⣾⣿⣥⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣬⣿⣷⡉⣶⣛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡋⣇⣰⡾⢛⣿⡿⣿⡿⣷⣾⣯⣽⣳⣾⢻⣿⢿⣿⡛⢷⣆⣹⢙⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡛⣽⣿⣻⡇⠈⠻⢿⠶⣾⣿⣿⣷⣾⣿⣿⣷⠷⠿⠛⠁⢸⣿⣿⣏⢛⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⣀⣒⠶⡿⢛⡉⠈⠀⢉⡻⢿⢶⣐⣀⣼⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⢢⠴⡁⠘⠿⠷⠾⠿⠃⢈⠦⡐⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢻⣿⣿⣿⣿⣿⣧⣒⣛⣩⣭⣭⣭⣭⣭⣭⣭⣛⣒⣼⣿⣿⣿⣿⣿⡟⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⣿⣿⣿⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠜⣿⡿⢣⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠟⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠻⠿⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠾⠿⠛⠛⠛⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣶⠀⢰⠏⠁⠉⣧⠀⢰⡖⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢹⡇⢸⡄⠀⠀⣼⠀⣾⠇⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠈⣿⠛⠙⠒⠚⠉⢻⡿⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣇⠀⠀⠀⠀⣾⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣈⣿⣀⣀⣀⣰⣟⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⣛⣛⣛⣛⣛⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⣿⡟⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⠶⠿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠿⠶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⣿⡇⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣄⣀⡈⠻⣷⣿⣷⡿⠋⣀⣀⣤⣤⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠶⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠛⠳⡟⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡄⠀⠤⠬⠠⡀⠀⠀⢠⠿⠚⠛⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠁
⠀⠀⠀⢀⣴⣄⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁
⠀⠀⣼⢾⡁⡉⠉⠁⠀⠀⠔⠉⡁⠀⠀⠀⢀⠉⠀⠀⠀⠀⠉⠒⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣿⠀⠀⠀⠈⠙⠁⠀⠀⠀⠁⢠⠀⠀⠈⠁⠀⠀⠸⡄⠀⠀⢀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠸⣆⠀⠀⠀⠈⢃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⢡⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⠕
⠀⠀⠀⠈⠳⠶⢶⣀⠈⠑⢄⡀⠀⠒⠒⠒⠊⠁⣠⢴⣥⠴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠁⠄⡀⢀⠠⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠐⠀⣿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⠇⠀⠀⠹⠶⣤⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠈⠀⢀⠀⠀⠀⠀⢱⠀⠙⠦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⢛⡔
⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡆⠀⠀⠼⠧⣤⣀⣀⣤⡤⠶⠖⠛⠉⡀⠈⠀⠁⠱⡄
⠀⠀⣠⡾⠋⠀⠢⠀⡀⠀⠀⠄⠀⠀⠈⠠⠁⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⠀⣀⡤⠖⠃
⢀⡴⠋⠀⠀⢀⡴⠋⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⣄⣀⣀⣠⣤⠔⠀⠀⠀⠉⠋⠉⠁
⣿⠁⠀⠀⡴⠋⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠈⠉⠉⠩
⠘⣧⠀⠀⠹⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠀⠂⠀⠀⢠⡔⠋⠉⠉⠉⠉⠳⡄
⠀⠈⢷⡀⠀⠘⢦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠹⣦⠀⠈⢧⡤⠶⠧⢀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⣀⣀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠹⣤⠤⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣮⣉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⢣⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⡄⠀⠀⢀⣀⠤⠒⠒⠉⠉⢱⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣠⡔⠒⠒⠒⠒⠒⠒⢺⣿⣷⢶⠊⠉⠀⠀⠀⠀⠀⠀⢸⡽⣤⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠓⢄⠀⠀⠙⢶⣿⣇⣀⣀⣀⣤⣾⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⠀⠀⣹⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜⠓⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡷⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡎⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣸⡿⣿⡇
⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⠶⠯⠓⠒⠒⠚⠒⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⡖⠒⢲⠶⣶⣾⣿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠾⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⣭⣶⣍⣓⣫⣭⣭⣝⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣒⣫⣶⡿⣻⣿⡿⠛⠻⢿⣭⡹⣿⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⣥⡶⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠻⣿⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣵⣿⡏⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⢿⡼⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣾⣿⣿⣿⣿⣿⢿⠋⢀⣀⣀⣠⡀⠀⢀⣼⣥⣼⡧⢷⡎⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⣡⣿⣿⣿⣿⣿⣿⡿⢁⣴⡾⠟⣉⣥⡴⠾⣟⠉⢹⣧⡄⠰⡀⢻⡼⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠛⣩⡴⣿⣉⠀⠀⠀⢹⡄⠀⠹⣆⠀⠈⡾⣇⠧⣀
⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⣿⣿⣥⣾⣿⣿⣀⣀⣤⣿⡀⢻⠿⠃⠀⠀⢸⠇⠀⠀⠈⢯⡉⠀⠈⠳⣬⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡝⣿⠟⣉⡙⢿⣿⠟⠛⠉⠉⠳⣀⣉⣀⣤⠴⠋⣰⠀⠀⠀⠀⠙⣄⠀⠀⠈⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡇⠸⠁⢁⠈⠛⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠒⠀⠀⠀⠀⠼⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣷⡀⠀⠀⠙⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠘⠃⠀⠀⡾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣏⢾⠟⣙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⣠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠙⠓⠾⣝⢷⣄⢄⠀⠀⠀⠀⠈⠂⠒⠂⠀⠀⠀⠀⠀⡀⠀⠀⢠⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢧⡑⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⢻⡄⠀⠀⠉⠉⠛⠓⣶⣤⣄⣀⣀⣀⡤⢖⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⣋⡞⠃⡀⠀⠀⠀⠰⣯⣱⡖⠾⢭⣭⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣭⣷⣾⣿⣿⣦⠀⠙⠦⡄⣀⡤⢈⣳⣝⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠺⣿⣿⣶⣿⣯⣽⣿⣮⣝⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⣿⡟⠻⠿⠈⣿⣿⣿⣷⡹⡄
⠀⠀⠀⠀⠀⠀⢀⡼⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡹⡄
⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣹⡀
⠀⠀⠀⠀⢀⢎⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣷⣿⣿⣿⣧⢳⡀
⠀⠀⠀⢀⡎⣾⣿⣿⣿⣿⣿⣿⠟⡵⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⠰⠸⣿⣿⣿⣿⣿⣿⣧⢳⡀
⠀⠀⠀⡜⣸⣿⣿⣿⣿⣿⣿⣿⣮⡳⣌⠳⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡀⣿⣿⣿⣿⣿⣿⣿⣧⢳
⠀⠀⠀⢳⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣝⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣎⢧
⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⢧⡹⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢇
⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⠛⠡⣾⣝⣛⣣⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⡻⣿⢁⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⢿⣿⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠭⠶⣝⠦⣄⡀⠀⢀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⢟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣍⢳⣄⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⡻⢿⣿⣿⣿⣿⣿⣿⡿⣣⢹⣿⣿⣿⣿⣿⣿⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⣝⣛⣛⣛⡼⠙⢧⣛⣯⠭⠷⠒⠒⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢒⣢⣭⣒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡦⢽⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⡿⣿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣟⢁⣤⣀⠀⠀⠀⢿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠙⠉⡟⢿⠛⡷⣴⣿⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⡂⠠⠘⠒⠊⠀⠲⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢫⡇⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣖⣿⣿⠀⠀⠀⠈⠳⣄⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⣹⣿⣿⡆⢠⣆⠀⠀⡟⠉⠁⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢃⣿⣿⣿⣇⠈⠱⣻⡁⡰⣾⣷⣾⣷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⢿⣿⣶⣶⡶⣮⣿⡀⠀⠸⡀⠀⢿⣿⣯⡲⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡿⡌⣾⣿⣿⣏⢾⣿⣿⣧⠀⠀⠀⠰⣾⣿⣿⣿⢆⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡈⣿⣴⣿⣿⣿⣿⣆⠻⣿⣿⡆⠀⠀⢀⣿⣿⣿⢃⡽⣛⣒⣒⣶⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡇⢻⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⡄⡀⣾⣿⣿⣡⢯⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣥⣾⣧⣬⣿⣿⣿⣿⣾⣿⣿⣿⡾⢿⣷⡀⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠭⣛⣭⡅⣰⣯⣽⣿⣿⠿⠋⠀⢔⠨⡙⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣫⣾⣿⣿⠏⠀⠀⠀⠀⣈⠂⢁⣴⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣶⣾⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠿⡿⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢦⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠳⢶⠋⠀⠀⠀⠀⠹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣀⣉⣉⣑⣤⠀⠀⠀⢠⡷⠾⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠑
⠀⠀⠀⠀⣠⡆⠀⢠⡟⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠢⢤⡊⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄
⠀⠀⢰⣴⡏⡛⠛⠋⠀⠀⢀⠜⠓⠀⠀⠀⠀⠀⠓⠀⠀⠁⠉⠒⠦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈
⠀⠀⣿⠈⠁⠀⠐⡖⡆⠀⠀⠀⠘⠀⡀⠀⠀⠚⠀⠀⠀⢳⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⡆⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠁⢂⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠖⠀⠀⠀⡀⠅⠀⠐
⠀⠀⠀⠹⢦⣤⣠⡤⠐⠳⢄⠀⠀⠠⢄⣠⡤⠔⠀⢀⣠⣂⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠰⠱⢀⡀⠀⡠⢈⠔
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠾⠗⠂⣤⠤⠤⠤⣤⠒⠉⠉⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠰⠴⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣧⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⡟⠋⠀⠀⠀⢨⠉⡏⠻⢦⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢹⡀⠀⠉⣻⣆⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠀⠐⠳⠺⣄
⠀⠀⠀⢀⡴⠟⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠈⣇⠀⡀⠉⠉⠛⠲⠖⠚⠛⠉⠉⠀⡠⠐⠀⠀⠀⠀⣸
⠀⢀⡴⠋⠀⠀⠀⣠⡤⣄⠀⡈⠀⠀⠀⠀⠃⠀⠀⠀⢸⡶⣤⡀⠀⠀⠀⠀⠀⠀⡀⠐⠑⠳⣄⣤⠴⠒⠋⠁
⣠⠞⠁⠀⢀⣠⠞⠁⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⠶⡦⠴⢶⠞⠉⠂⠀⡀
⢻⡀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⠀⠀⠀⠈⠁⠀⣀⣀⣤⣄⣀⣀
⠀⢻⡄⠀⠈⢷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⣄⣉⣉⣉⣡⠏⠁⠀⠀⠀⠀⠀⠹⣌⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀
⠀⠀⠹⣦⠀⠀⠹⣄⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠳⣄⠀⠙⠶⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣀⣙⣦⣀⣠⣤⣀⣤⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠧⢤⡄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⣀⣀⣀⣀⣀⣀⣸⣣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⢀⣴⣾⡿⠿⠿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⢶⣦⣌⣻⣦⡿⠋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣿⠿⠛⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⢀⣠⠄⠀⠀⠈⠉⣹⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣼⣿⣿⠋⠀⠠⢟⡥⠶⠲⠦⣀⡞⠉⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢰⣿⡿⣿⡄⠀⣰⡏⠀⠀⠀⠀⠈⠀⡤⠀⠀⢨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢸⣿⣰⣿⣇⢀⣤⡀⠀⢀⡞⠀⠔⠁⠀⠀⠡⠀⠀⠀⠀⠀⢂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣼⡟⠛⠻⠿⠿⠛⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⢀⢈⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⡀⠂⠀⢀⠀⠀⠀⠀⠀⠐⠀⠂⠘⢀⣠⠀⠀⠀⠀⠠⠂⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢱⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠈⠦⠁⡀⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⡠⠦⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⡀⠈⠔⠀⠀⠁⠢⠀⠀⠀⠀⣠⣴⣦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⠊⠀⠀⠀⠀⠆⠀⠀⠀⢰⣿⣋⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠛⠛⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠎⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⡐⠀⠀⠈⢀⠀⠐⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⢀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⢀⣀⣀⣀⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠙⠛⠋⠻⣿⣿⡉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣷⡦⠀⢸⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠊⠀⠨⣿⣿⣿⣿⣿⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠈⢿⣿⣿⣿⠏⢵⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⡶⢿⣿⣆⠀⠘⠟⡠⣤⣄⠀⠀⢀⣿⡏⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣃⣋⣙⠙⣃⢀⣸⣴⠶⢾⣤⠴⠿⠁⡀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢯⠉⠉⣻⠏⠹⠇⡄⠀⠀⠀⣵⢷⠒⣤⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠃⠀⠻⠄⡠⠓⣕⣀⣀⣠⠿⠆⠐⢛⡐⠤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣡⣀⣀⣠⡄⠭⠁⠀⠀⢠⡶⣡⠤⢅⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠠⢀⣟⢡⣶⡤⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⣀⣤⠴⡀⠀⣼⣿⣦⠟⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡋⠀⠐⠀⠀⢻⡁⠉⠀⠐⢃⠙⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⢿⡿⢣⣿⣿⣿⣿⡇⠀⠀⠀⠀⢙⣷⠀⠀⠀⠑⢤⣄⠀⠉⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣨⠃⠈⠻⣿⠿⠋⠀⠀⠀⠀⣰⠟⠀⠱⡀⠀⣠⠂⠙⢦⡀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣄⠀⠀⠀⡐⠂⠀⣼⠢⢁⠒⣂⠁⣴⠋⠀⠀⣴⣿⡴⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠘⡀⣿⢀⡄⠈⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠊⠀⠰⢻⡿⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠸⡆⠃⢻⠸⠇⠀⠀⠀⢸⠃⠀⠀⠀⡴⠃⠀⠀⢀⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠄⣿⡄⢸⢀⣀⠀⠀⠀⣾⠀⠀⠀⡼⠁⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡟⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠶⠿⠿⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠱⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠦⣤⣴⣶⣶⡶⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠶⣶⣖⠛⠂
⠀⠀⠀⠀⠀⠀⡠⣟⡂⢤⣀⠀⠀⠈⠙⠓⠦⢤⣀⠀⢹⣯⣄⣀⣀⣀⣠⡤⠤⠶⣒⣚⣩⣭⣽⡿⠷⠶⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢐⠀⠀⠉⠂⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠚⠿⢿⣿⣿⠷⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⣘⣇⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⣌⠑⠢⢄⡀⠀⠀⠀⢀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢶⣒⣛⣯⣽⠅
⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣈⠑⠢⢀⣸⣿⣖⣀⣀⣀⣤⣤⣶⣖⣻⣽⣿⡶⠾⠛⠛⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡄⢈⣙⣻⣭⡽⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠂⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⣿⣧⠄⠀⠀⠀⢀⣀⣀⡤⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡷⠶⠟⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣽⣦⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡴⠖⠛⠿⣄⣀⢀⡼⠷⠶⣄⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡋⣿⡄⠀⠙⠃⡟⠉⢯⠘⠟⠀⢈⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠉⠓⠒⢾⣀⠀⣘⡷⠶⠆⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣀⣋⣏⣹⣀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⣈⣝⣛⡋⠁⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⣄⡤⠀⠁⠄⠅⣠⠶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠈⠙⠒⠚⠒⠋⠁⠀⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣞⠁⣧⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠈⠙⠒⠒⠒⠒⠚⠉⠁⢀⠜⠁⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⠶⠾⠿⠟⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣙⡛⢶⣝⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣹⣿⣿⣝⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣷⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣭⣛⣓⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣿⣶⣶⣿⣿⣷⣬⣭⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⣥⣿⣿⣿⣶⣶⣾⣶⣶⣶⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣟⣹⣇⣀⣠⣿⣥⠬⢭⣿⡿⣷⣦⣤⣀⣀⣀⣀⣤⣴⣾⢿⣯⡭⢥⣬⣿⣄⣀⣸⣟⣻⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⢿⡇⢸⠁⣴⡿⣿⡝⣎⢻⣿⣿⣿⣿⣿⣿⡟⣱⢫⣿⢿⣦⠈⡇⢸⡿⢻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣇⠘⣇⠈⠀⠙⠿⠿⢃⣨⣿⣿⡏⠀⠀⢹⣿⣿⣇⡈⠻⠿⠟⠀⠁⣸⠃⢸⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣰⣿⠀⠘⠷⣤⣀⣀⣩⡿⠟⣽⣿⢻⠀⠀⡞⣿⣧⠹⢿⣍⣀⣀⣠⡾⠋⠀⣿⣇⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠉⠉⠁⣠⣾⡿⠃⣸⠀⠀⣇⠘⢿⣷⣄⠈⠉⠉⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⣿⣶⣶⣶⣶⣾⠿⠟⠋⢠⠞⡏⠀⠀⢸⠳⣄⠙⠻⠿⣷⣶⣶⣶⣶⣾⠿⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⣿⡄⠉⠉⠀⠀⠀⠀⠀⠘⠷⢿⣤⣤⡾⠾⠋⠀⠀⠀⠀⠀⠈⠁⢀⣿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⣼⡧⠤⠴⠶⠾⠿⠿⠷⠶⠦⠤⢼⣷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠘⠋⠀⠀⠀⢄⣀⠀⠀⣀⡠⠀⠀⠀⠈⠳⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⣀⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣀⣀⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠋⠉⠉

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠈⠁⠈⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡠⠀⠀⡀⠀⠀⢱⢤⠒⠀⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠉⠀⠈⠑⠒⠠⠀⣸⠃⠄⠰⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⢷⠈⠂⠃⠊⢈⠡⠃⢀⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡤⢀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠆⢀⠴⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⣤⠕⠐⠠⡀⣀⡀⣀⠀⠠⠛⣦⣨⢆⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠈⠀⠀⠀⠀⠣⠘⢸⠔⠃⠀⠀⠀⠑⠄⠄⠀⢠⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠰⡀⠀⢁⡀⠀⠀⠀⡖⠠⠀⠀⠀⠀⣧⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠃⠀⠀⠀⣣
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠀⠀⠉⠂⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⡤⠄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢥⡴⠡⠐⢃

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠣⠀⠀⠻⠋⢀⠀⢀⠀⠀⠀⠀⠀⣤⣾⣷⣤⡀⠀⠀⠀⣌⠀⠀⠀⠸⠟⠃⢰⠀⠀⠀⡀
⠀⢀⠠⣑⠦⠥⠯⢔⡨⠀⠐⠀⠀⠀⢠⣴⣶⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠁⠀⠀⢠⣿⣿⣿⣶⣧⣆
⠀⠀⠑⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠟⠍⠩⠯⠩⠻⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⢾⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣗⣛⡯
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠀⠀⠀⠀⠘⠋⠁⠈⠁⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⣿⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⠦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣯⣟⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠿⠿⠿⢀⣃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⠧⢼⠤⢤⢤⠧⢼⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣷⣿⣛⣿
⠀⠀⠀⢺⣿⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⢞⡒⣖⢞⡲⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠿
⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠀⠀⠀⠀⢧⡤⠶⠶⠶⠶⢲⠖⠛⠛⠛⠛⠛⠛⠁⠀⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⣰⠛⣆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠦⡴⠁⠀⠈⢦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠈⠙⠛⠛⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠙⣛⣟⢻⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡅⠸⠢⠉⢹⠾⠼⠛⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⣿⣿⣿⣿⣿⣝⠻⣿⣆⠀⢀⡤⠀⢀⣴⣆⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⣶⠖⠀⣀⢀⣨⣭⣿⣿⣩⣿⣿⣿⠧⠔⠀⠀⣿⣿⣿⣿⣶⣤⡀⢸⡿⠷⡦⣤⠰⢿⣿⠟⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠉⣹⣿⣆⣾⠟⠛⢿⣶⣿⠼⣿⡿⠟⠒⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣶⣴⣷⣿⣦⣴⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⡛⠾⠿⣷⠀⠰⡾⢳⡿⠿⠋⠄⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⡓⠚⠛⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠲⡄⠀⠀⠀⠀⠀⠀⠁⠀⢰⡶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠤⠤⡀⠀⡀⠀⠀⠀⠀⠀⠀⣤⣌⣩⣤⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡄⠈⣿⡿⣿⣿⡇⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣼⣿⣿⣿⣿⣷⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⣰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⠸⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⡟⠻⠃⠀⠀⢛⡏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣷⣦⣄⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⢰⠟⠣⢤⣤⣤⡤⠟⠻⡖⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣟⣀⣀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⣀⣀⣻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡏⣛⢻⣿⣿⣿⠿⢿⣿⣿⣿⡟⣛⢽⣿⣯⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⣸⣿⣿⣧⣽⣵⣿⣿⡏⠀⠀⢹⣿⣿⣮⣯⣼⣿⣿⣇⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠸⣟⢿⣿⣿⣿⣿⣿⢟⣿⣡⣤⣤⣄⣻⡻⣿⣿⣿⣿⣿⡿⣻⠇⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠹⣶⣽⣿⣿⣿⡷⠋⢿⣿⣷⣾⣿⡿⠙⠾⣿⣿⣿⣯⣾⠏⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡡⠀⠀⠀⠛⠛⠻⡇⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⢸⠛⠛⠛⠁⠀⠀⢀⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⡄⠀⠀⠀⠀⢙⠶⠤⠤⠰⠾⠿⠆⠤⠤⠶⡋⠀⠀⠀⠀⢠⣶⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣦⠶⡀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⠶⣴⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣷⣶⣾⣶⣿⣦⣤⣄⣠⣤⣴⣿⣶⣷⣶⣾⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣽⣿⣿⣯⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡉⢆⢺⡒⠶⢌⣪⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⠢⣝⣋⣾⣥⡿⠟⠋⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣽⡿⢋⣩⡁⠀⠀⣤⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣾⣿⡆⠈⠉⠀⠀⠀⠉⠁⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠉⣧⠞⢳⡶⠋⡿
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣄⠀⠠⣤⡤⠀⣰⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠃⠀⣰⠏⠀⡾⠁⣸⠃⣤⣶⡄⣠⠶⣦⠴⡆
⠀⠀⠀⠀⠐⢿⣯⣱⣿⣿⣿⣿⣿⣿⣷⡄⠄⣤⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⣼⠁⢰⡇⠸⣁⡾⢳⠏⠰⠃⢸⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⢦⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠹⣿⣿⣿⣿⣿⢿⣿⣿⠏⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢈⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠻⢿⣿⣿⣿⡟⠁⠀⣤⠀⠀⠸⡀⠀⠀⠀⠀⣸⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⢷⡀⠈⠙⠿⠋⠀⢀⡼⠙⣆⠀⠀⢻⡀⠀⣠⡾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⢻⠀⠀⠴⠀⠀⢨⡇⠀⠸⡄⠀⢀⣷⡞⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⠤⣾⡀⠀⠰⠂⠀⠈⢧⠀⠀⣹⣶⠟⠉⢳⡄⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠈⠙⠓⢦⡀⣀⢠⣞⣦⡾⠋⢱⡀⠀⠀⠹⣄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢄⣀⡀⠀⡸⠁⠀⢸⡿⠛⣆⠀⠀⢳⡄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠛⣻⣦⣤⠿⠏⠀⠈⢦⡀⠀⠱⡀⠀⠀⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⡾⢿⣿⣿⠀⠀⠀⠀⠀⢹⣤⠀⠙⠶⠞⠁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢋⣀⣸⣿⣿⣦⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⢀⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⠿⠛⠛⡿⠛⡿⠛⠻⡟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⢰⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣷⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣾⠇⢸⣿⠟⣿⣌⣳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠈⠛⠀⠙⠻⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣄⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⢞⣷⣹⣮⣽⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠈⢯⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠬⡅⠀⢰⠀⠀⠀⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣘⣓⠀⠀⡄⠀⠆⠖⠀⢠⣤⣒⡀⠀⠀⠀⠀⢀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠶⠀⠋⠉⠉⠉⠉⠉⠉⠉⠁⠘⡀⠀⠀⠀⠄⡞⠀⢸⣽⠛⠒⡦⣄⠂⠠⠂⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠀⠀⠠⠈⠓⠋⠘⠦⠤⠇⡿⢓⢀⡀
⠀⠀⠀⠀⠀⠀⠥⠤⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⡀⠡⠊⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⠀⠂⠀⠀⠀⠀⣆⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠤⡅⠀⠀⠀⠀⠀⡀⠀⠏⠀⠠⠠
⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠈⡡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠄⢀⠀⠀⠀⠰⡀⠀⠘⣀⠈⠄
⠑⠒⠀⢀⠀⠀⠄⠀⠠⠤⠄⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠤⠄⠠⠀⠤⠀⠀⠀⠀⠠⢔⢀⡂⡀⠀⠃
⠀⠀⠀⠀⠀⠀⣃⠀⠤⠄⠂⠬⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢃⠀⠄⠠⠘⠐⠌⢠⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠐⠈⠖⠫⠀⠂⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠰⠶⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢾⠀⠀⣀⡄⣀⣀⣀⣀⣠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⣀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠁⠠⠁⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠴⠀⠺⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡡⠀⠄⠀⣔⣭⣄⡁⡀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣳⡀⢀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢠⠉⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠠⠼⠀⠀⠓⠲⡖⣦⠀⠊⣴⠆⢀⠀⡔⡸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⠂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⣀⢉⠗⢦⠀⠁⠀⡠⠋⠀⠆⠀⢀⡀⠀⡀⠀⠄⠀⠀⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠐⠁⠀⠀⠀⠀⠧⠆⠀⡀⠀⠀⢈⠙⠒⠊⣐⣢⡞⣒⢂⣀⢈⡤⢦⡍⠀⠈⠁⠀⢰⠖⢺⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⢀⠀⠀⣤⠀⢀⠰⣾⡷⠄⢐⠉⠙⣇⣭⣒⣩⡀⠉⠁⠁⠀⠀⡔⠂⠘⠢⠜⠳⣆⢱⡦⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⠑⢄⠀⠀⠛⠀⠃⢀⡀⠀⠠⢻⡍⠀⠀⣿⠂⠀⠀⠀⠀⠐⡇⠀⠢⢰⣦⣄⡀⢠⠀⠈⠀⠀⡂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢈⡉⠐⠀⣠⢁⠄⠤⠀⠀⠙⢆⢀⣏⡴⠶⢗⠀⠄⠀⠀⠀⠆⠈⠛⠋⠁⢀⡖⡚⠂⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣀⠈⢉⡀⣲⢤⣾⡀⢘⣿⡋⠀⢀⣨⠟⠂⢀⡀⠀⠈⠊⡰⠀⡄⠜⠀⠀⠠⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠥⠤⣀⠴⠛⠃⠣⠀⠀⢹⡟⠁⠉⠛⣠⣥⣄⠀⢀⣉⣀⡀⠀⠀⠀⠏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠀⢀⣠⠖⠚⣧⡀⠀⠀⠀⠖⢮⡇⢠⠤⠤⡻⠿⠟⢀⡀⣸⠋⠙⣀⣠⡞⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠈⢰⠊⠉⠓⢤⣾⡇⠀⢀⡇⢈⠂⠀⡧⠀⠀⠣⢸⢹⡞⠉⠁⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠿⣿⡇⠀⠀⠀⠀⠘⠛⠛⠛⣶⣾⣇⠹⠶⣶⣷⠶⠶⣤⠾⠋⠙⠦⠆⠁⠐⠁⠈⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⣠⠆⠄⠠⠈⢿⣿⣦⣼⠇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣒⣦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⢼⣷⣦⣌⡙⠂⠄⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈⠭⠍⠁⠀⠀⠀⠐⠀⠀⠀⠠⢄⠀⣀⣀⢀⡠⠼⠛⠛⠿⣧⣤⠀⠄⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠐⠂⠀⠈⢉⣀⡠⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠀⠉⢉⣉⣁⣀⣀⣀⣀⣐⣒⣀⣒⣀⣤⣤⣤⣿⣗⣋⣐⣿⡿⠦⠁⠀⠀⠀⠉⠩⠭⠭⠭⠭⠥⠄⠙⠛⠛⠓⣒⠨⠇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⡒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠤⠒⠉⠉⠉⠉⠀⠀⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢉⠼⠃⠼⠿⠦⠤⠤⠤⣀⡀
⠀⠀⣠⢎⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣀⠲⣒⠠⠤⢤⣤⣀⠀⠀⠀⡠⠤⠰⠆⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠞⠓⠄
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠛⠃⠀⠀⠉⠉⠀⠀⠉⠀⠙⠐⠊⠀⠉⠉⠀⠀⠉⠳⠾⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠀⢴⡮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣕⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢂
⠀⠰⠃⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠃
⢠⢡⠆⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⡿⣹⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⢻⡆⠀⠀⠀⠀⠀⢸⣱⠂⢀⣴⠟⢋⡉⠉⠉⠻⢷⣶⠚⠷⠙⣄⢷⡀⠀⠀⣀⣈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠻⡇⠀⠀⣀⢀⠄⣸⣿⢀⡾⠡⣄⡔⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⢛⣼⠖⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⠀⢇⠀⢠⢫⡆⠛⠁⠸⣸⡇⠈⠉⡄⠀⠀⠀⠠⡶⢸⣿⠶⠶⣦⡟⠁⠀⠀⠀⠀⢖⠙⡜⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⢰⡀⠀⠀⠀⠀⠀⠀⠀⡿⡇
⠀⠀⠘⡄⣻⠘⠁⠀⠀⠀⠙⢿⣌⠉⠁⠀⠀⠀⠀⢀⣼⠇⢤⡀⣿⠁⠸⠃⠀⠀⠀⠘⡒⡋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⡎⠉⠐⣴⣰⡇⠀⠀⠀⣸⠛⠃
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⣤⣤⡶⠟⠱⠆⡶⠙⢿⣇⠀⠀⠀⠀⢀⣜⣩⢃⣾⣅⠀⠀⠀⠀⠀⠀⣰⢀⡎⠛⠀⠀⠀⠀⠀⠋⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠙⢃⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⣀⠀⢀⣀⣴⠞⠁⠛⢉⣉⠑⠲⣄⠔⣽⠞⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢧⣤⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⡛⠉⠁⠀⠀⠀⠉⠉⠙⠦⠈⡆⠁⠀⠀⠀⠀⠀⠀⠀⠰⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠑⠚⢻⠒⣢⡤⠤⣤⣤⣤⠄⠘⠆⠀⢀⠤⠊⠐⠤⠀⠤⠜⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠸⠿⠿⠛⠉⠉⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⢽⣶⣶⡖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡋⣻⣄⡼⠧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣠⣔⡀⠉⠀⠀⠱⢤⠞⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠴⠥⣄⣉⡉⠓⠲⠤⣀⢸⢀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡠⢄⣀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣈⠙⡧⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⣶⣦⣄⡀⠀⠀⠀⠀⠀⠙⢆⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠻⠿⣿⣦⣄⡀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠢⠤⢷⠬⢙⡿⣗⠢⣀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢫⣙⡜⢦⣀⣸⡀⠀⠀⢣⣄⣈⠳⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠂⠀⠀⠀⠀⠯⢕⣲⠋⠎⠭⠛⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠄⠠⢄⠀⢀⠤⠀⠠⢄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠊⠉⢨⠀⠀⠀⠀⢱⠃⠀⠀⠀⢀⠍⠉⠲⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢼⢆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠈⠢⢄⡀⠀⠀⠀⡀⢸⠀⡄⠀⠀⠀⢠⠠⠔⠈⠑⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⢰⠀⠀⡄⠀⠀⠁⠀⠀⠀⠃⢸⠀⠁⠀⠀⠀⠊⠀⠀⡀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠸⡀⠀⣈⢀⡀⠀⠴⠂⠀⠀⢸⠀⠀⠀⠐⠤⠀⢰⢀⡁⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠑⡌⠁⠀⠁⠀⠀⠀⠎⠠⢼⡦⠄⠢⠀⠀⠀⠉⠀⠈⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⢀⡀⠀⣠⠋⠳⡀⠀⢀⠀⠀⣠⠄⡰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⠕⠊⠁⠀⠀⠈⠒⠤⠤⠤⠔⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠤⣒⣉⠡⠤⠄⠒⠒⠢⢤⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⡴⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡖⠒⠒⠒⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋

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

⠀⠀⠀⡠⡎⢹⢉⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡤⣤⡀⣶⠀⠀⠀⠀⠀⠀⣀⣤⠶⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⢸⠐⢌⢸⢀⡴⢸⠀⠀⠀⠀⢰⡎⠉⢱⠰⡟⠉⢳⠀⠀⠀⠀⠀⡇⡀⢀⠀⡇⣿⠀⠀⠀⠀⠀⣏⣛⠻⠿⠛⣩⠤⠀⠀⠀⠀⠀⣦⡀⠀⡷⣏
⠀⠀⠸⡀⡜⢺⠚⣀⡨⠀⠀⠀⠀⠀⠳⠒⠁⠀⠙⠒⠏⠀⠀⠀⠀⠀⡇⠀⠐⠀⡇⢿⠀⠀⠀⠀⠀⣗⠦⠬⣍⠥⢒⡨⠀⠀⠀⠀⠀⣿⠙⣴⡳⠾⡆
⠀⠀⠀⠘⠢⠼⠬⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠃⠉⠀⠀⠀⠀⠀⠈⠉⠑⠒⠊⠁⠀⠀⠀⠀⠀⠀⠿⠦⠣⠿⠦⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⡞⣭⣭⣭⡝⡆⠀⠀⠀⠀⠀⢸⠁⠤⠬⠈⡇⠀⠀⠀⠀⠀⠀⠈⠤⡤⡤⠤⠁⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⡇⠭⠭⠍⡅⡇⠀⠀⠀⠀⠀⢸⠨⠭⠭⡡⠣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠍⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠈⢸⠉⣿⣿
⠀⠀⠀⡇⠅⠥⠅⡇⡇⠀⠀⠀⠀⠀⢸⣬⣭⣭⣦⣴⡀⠀⠀⠀⠀⠀⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣜⣙⡉⠉⠍⣉⣣⠀⠀⠀⠀⠀⠀⠘⢛⠛⠛
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠘⠛⠓⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠓⠒⠚⠛⠛

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣾⢟⣭⢻⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⠒⠲⢄⠀⠀⠀⠀⠰⣶⣶⣶⣶⣷⣶⠆⠀⠀⠀⠀⠤⣤⠤⠀⠤⢴⠤⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢿⣮⣭⣼⣿⡆⠀⠀⠀⠀⢠⣿⣻⣿⣿⡆⢀⡨⠀⠀⠀⠀⢐⠀⢀⡖⢲⡀⠀⡇⠀⠀⠀⠀⠀⠛⡻⣿⠿⠛⠀⠀⠀⠀⠀⠀⢰⠖⠻⣿⣷⣦
⠀⠀⠀⠈⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠙⠟⠛⠋⠀⠉⠀⠀⠀⠀⠀⠨⣀⣀⣓⣚⣀⣀⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⣌⡥⠷⣿⣿
⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢰⠖⢲⣶⣴⠶⣆⠀⠀⠀⠀⠀⠀⣀⡀⡰⢒⣁⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⣿
⠀⠀⢸⡖⢾⣿⣧⣤⡟⠀⠀⠀⠀⠀⣾⠛⣻⢿⣿⣝⣦⠀⠀⠀⠀⠀⣿⡟⠒⠓⢺⣿⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠸⣿⡿⠋⠉⠙⢾⣿⠀⠀⠀⠀⠀⠿⠷⠤⠤⠾⠟⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠸⣿⡿⠈⠛⠋

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

⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠈⢿⡞⢃⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣿⣿⠧⣤⣤⢿⣿⣦⡀
⠀⠀⠀⠀⠈⢳⡈⢠⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⡙⠛⠋⠹⢻⣿⣆
⠀⠤⠀⠠⠄⡈⣷⠀⢀⠐⣾⠿⡇⡃⠀⠠⡀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣤⣾⣿⣿⠃
⠀⠙⠂⢴⣇⣀⠀⠀⢨⡧⣽⡀⣘⡇⠀⠀⢹⣦⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⡄⠀⠀⠀⢀⡄
⠀⠀⠀⢀⡀⡹⠂⠀⠸⠀⡛⢻⣿⣿⣄⠀⠠⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⢀⣠⡾⠃
⠀⠀⠀⢿⡄⢀⣀⠀⠀⠀⠀⠸⣿⣿⣿⢦⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣼⣿⣿⣿⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⣤⢀⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣅⡀⠀⠀⠀⠈⣧⠘
⠀⠀⣠⣴⣿⣿⣇⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⡜⠷⢊⠻⣿⣷⡈⣿⡿⣿⠛⣿⣿⣿⣄⠈⠩⠙⢿⣿⡿⢿⣦⣤⠀⠀⠈⠢⡈
⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣼⣶⢸⣿⣿⣿⣿⣿⣷⣾⣷⡈⣻⣷⠄⠀⠄⠻⣆⠀⠀⠀⢀⡀⠀⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣏⠀⣿⡟⢃⠀⠐⡄⠈⢳⣦⣶⣻⣿⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣷⣿⣷⣿⣧⡀⠀⠀⢻⣄⠀⢹⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣷⡞⠈⠛⠀⢿⣿⡉⠀⠁
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠰⣦⡟⢉⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣧⡔⣺⣿⣦⡼⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⢧⠀⠀⠀⠀⠀⣧⡄⠀⠀⢧⠀⠀⠘⠷⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⢆⠀⠈⠳⢄⠀⠀⢀⡾
⠀⠀⠸⣿⡆⠀⠀⠀⣿⣶⣄⠰⢀⣴⣿⣧⣄⣀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠏⣸⠛⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢣⡀⡿⠁
⣄⡀⠀⢈⡀⠀⠀⠀⠘⣿⣿⣦⣄⣀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣦⢠⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢸⡿⠃
⠈⠙⠻⠶⣿⡶⢀⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⡸⠃
⠶⢊⡁⠀⠙⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣯⠀⠘⠁⠀⣼⠃⠀⠀⠀⢀⣼⠇
⠀⢹⣧⣀⣄⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣄⡀⠀⣈⠀⠘⠀⠀⠀⣽⠉⡄
⠀⣿⣿⣿⣥⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣄⣴⠿⠷⠀⠀⠀⠘⠉⠰⠋
⠾⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠤⡀
⡀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⢛⢿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⣠
⠳⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣧⠈⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⣿⣷⣿⣿⣯⣹⡇⠀⠘⠻⢉⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠂⢶⡇
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣯⢂⣤⢿⣿⣿⣿⣿⣿⣿⠙⢀⠀⢺⣿⣿⣿⡿⠱⡌⠉⣦⠂⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠙⠏⠀⠀⠀⠁
⢻⣿⠟⣯⣿⡟⢻⡴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢹⣿⣿⣾⠷⣿⣧⣄⠃⠘⢲⡀⠙⢿⠿⣿⣿⣿⣿⣿⣿⣿⡟⠓⢤⡄
⣿⣿⣿⣿⣿⣶⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⢠⣽⣿⣿⣿⢿⣿⣿⣿⣶⣾⣤⠀⣀⠀⠈⠻⣿⣿⣿⣿⣿⡇⠀⠀⣷⠀⠀⠀⣂
⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⢑⣢⣴⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣿⣷⣦⣠⣹⣿⣿⣿⣿⣿⣄⠀⢸⣆⠀⠈⠙⣧
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣏⢿⣿⣿⣿⣿⣻⡆⠀⣹⠆⠀⠀⢸⡇
⢻⣿⣿⣀⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣄⣽⣀⡉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠉⡁⣼⣿⣿⣿⡙⠻⣷⣤⣽⣦⣶⣴⣿⠃
⣿⣿⣿⣧⢻⣿⣾⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⡏⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠈⠻⣿⣄⡀⠁
⣿⣿⣿⣧⠀⠈⠉⠀⣠⣄⠙⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣇⣤⣼⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⢿⡿⣣⣿⣿⣿⣿⣟⡻⣼⠷⣶⣄⡈⠀⠁⡾⠁
⣿⣿⣿⢹⡆⢰⣤⣿⣿⣿⣷⣾⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⡏⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⡯⣩⣤⣾⠟⠿⢿⣿⣿⣿⡙⢿⣦⣄⡸⣿⠇
⣿⣿⣿⠀⣿⣄⠛⠿⣿⣿⣿⣿⣿⣿⣟⡛⣿⣿⣿⣿⣿⣿⣿⡶⠖⢻⣿⣿⣿⣿⣿⣿⣿⣽⢿⣟⣋⣻⣇⣶⣶⣿⣿⣿⡟⣡⣶⡿⢣⣾⣆⡈
⣿⣿⣿⡇⣿⡿⣷⣀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣶⣿⣷⣾⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣶⣇⠻⠄⠀⠈⠛⠻
⣿⣿⣿⣁⣀⠀⠈⠿⢿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⡎
⠿⠿⠿⠿⠿⠿⠿⠶⠿⡿⢿⡿⠿⠿⠿⠿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠋⠈⢻⡏⠻⠷⠾⠏⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡀
⠀⠀⢀⠱⣀⣠⣄⣤⠔⣂⠥⠤⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⡈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣤⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣓⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠒⠓⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣟⣛⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⣀⣤⣶⣄⡀⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⢽⣽⢽⠤⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠞⠻⠻⠛⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣛⣿⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣩⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣶⣶⣦⣶⡾⠿⡇
⢀⡀⠀⣀⣾⣿⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣡⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣼⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠋⠉⠉⠙⠛⠳⠌⠙⠋⠡⠞⠛⠋⠉⠉⠙⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠛⣹⡿⢩⢉⣽⡟⡛⢿⣶⣶⣶⣶⠿⢻⡿⢉⠉⣽⡿⡙⢛⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣰⡿⢡⢃⣾⡿⡰⢠⡟⣿⡟⢻⣿⢠⣿⢃⠎⣼⡿⡱⢁⡞⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢡⠃⣾⡿⡵⢡⡟⣰⡿⠁⠈⢿⣿⠃⠎⣼⡿⠱⢁⡞⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠾⠿⠷⠶⠾⠿⠋⠁⠀⠀⠈⠙⠿⠶⠿⠷⠶⠾⠷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⠀⠀⢸⣿⣿⣧⣤⣤⡄⠀⠀⣿⣿⣷⣾⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⡿⠿⠟⠀⠀⢸⣿⣿⡿⠿⠻⠃⠀⠀⣿⣿⡿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣤⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣉⠙⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⢿⣿⣶⣤⣈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣁⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣷⣶⣶⣦⣤⣀⣉⠙⠻⢷⣦⡈⠳⡄⠀⠀⠀⠀⠀⢀⡞⢋⣤⡾⠟⠋⣁⣈⠻⢿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠛⠛⠉⠉⠛⠛⠛⠛⠛⠿⠿⢶⣦⣌⡙⠦⠈⠀⠀⠀⠀⠀⠉⡴⢛⣡⣴⡶⠿⠿⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣠⣤⣤⣶⣶⣄⠻⣿⣿⣿⣿⣿⣇⣀⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⣤⣴⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⠟⠘⢿⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⠻⠟⢛⣡⣾⡇⠀⠙⠛⠛⠋⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃⠈⢿⣿⡿⠃⠀⠛⠉⠉⠉⠁

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