Comedian Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡠⠴⡆⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡆⠙⠋⠁⢸⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠄⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠉⣉⡉⠉⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⢋⣀⣤⣤⣤⣀⡐⠶⣤⣍⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⢫⠿⡤⠶⣿⣯⣿⣿⣿⣿⣿⣿⣷⣿⡻⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠋⣡⠶⠿⠿⠿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⣶⡿⠋⠉⠉⠉⠻⣿⣿⣿⣿⣿⣯⠈⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢟⣠⡤⠶⣞⣛⣛⣒⡦⠌⠛⢷⣄⠀⠀⢀⠀⠀⠀⠀⢸⠐⣿⣿⣶⣶⣄⣴⣮⣿⣽⣿⣿⣿⣿⣿⣷⣬⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⣥⣾⡿⢛⣹⣿⣿⣿⡷⣶⡌⢻⠀⠀⠀⠀⠀⠀⠀⢸⡔⠋⡿⢿⣿⠟⠈⠻⣛⡟⣿⢱⣿⣿⣏⣿⡟⣴⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣗⠀⣠⠶⠒⠶⠄⠀⣿⣾⣿⢟⣽⢋⣴⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢴⣶⣻⣿⣇⠀⠀⠀⠀⠀⠈⣿⡿⠟⢋⣠⣾⣏⡀⠀⠀⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⣄⢀⣤⠞⣫⡿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠲⠋⢀⣾⣿⣿⡷⡄⠀⠀⠀⠀⠈⠘⢿⣿⣿⣿⣿
⣿⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢤⣨⠟⣋⣴⠞⣩⣴⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣹⣿⠀⢱⡀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿
⣿⣷⣤⣿⣿⣿⣿⣿⡿⠟⠃⠀⣀⡴⠃⠏⣡⡞⢋⣴⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⡏⠀⠀⢧⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿
⣷⣬⣍⣉⣽⣛⣁⣤⡴⠂⠀⠀⠀⠀⠀⠀⢩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢁⣀⣀⣀⣿⣿⣿⣿⣿⣦⠀⢘⣧⠀⠀⠀⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠋⠀⢆⡦⠀⠀⠀⠀⢾⣼⡟⠿⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣄⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿
⢭⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣾⣀⣀⠀⢈⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠷⣷⣶⣶⣾⣿⠻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿
⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡏⠀⢸⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⣈⣹⣿⠘⣇⣀⣹⣾⣿⣿⣿⣷⣤⣴⣿⣾⣿⣿⣿⣿⣿
⣧⠙⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⠀⠈⠻⠿⠀⠻⣿⣿⣿⣿⣿⣿⡏⠉⠙⠻⢿⣿⣿⢷⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿
⣿⣆⠀⠀⣠⣴⠉⡿⠄⠹⢿⣿⣿⣿⡗⢠⣤⣤⡄⠀⣻⣿⡏⣿⣿⢿⠿⢷⣶⣷⣶⣯⣩⣾⣿⣩⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠿⣧
⣿⣿⣷⣾⣿⣿⠈⡇⢀⣄⡘⠿⠟⠋⠀⠀⠙⢿⣿⣿⣿⣿⣇⣿⣿⣧⡀⢀⡉⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⡟⠁⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣷⣶⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠉⠻⢿⣿⣤⣼⣿⣿⣯⣿⣿⣶⣶⣿⣿⣿⣷⡀⠀⠘⠃
⣿⡏⠉⠉⠻⣿⡇⠀⠈⠉⠙⣿⠿⣿⣦⣀⠀⠀⣹⣿⣿⣿⣿⠿⣷⣴⣤⣦⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠤
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣷⣶⣿⣿⣿⡻⢿⣦⣴⣿⡿⠿⠟⣛⣛⣉⣭⡉⠉⠈⠹⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠀⣆⡠⠙⠻⠏⠀⠰⠿⠿⠿⠛⠛⠁⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⣷⣷⣄⠀⠀⠀⣶⣶⣿⣿⠿⠇⠀⣀⣤⣶⣶⡿⣿⣿⣿⣿⡄⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣾⡟⠋⣯⠀⣾⣿⠀⢾⢉⣿⣿⣿⣦⠀⣸⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⢀⣿⣿⡀⢸⠘⢿⣿⣿⣿⣷⣿⣿⣭⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⢸⣿⡿⠛⢾⠀⣼⣿⣿⣿⣿⡟⣿⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣥⢷⣼⣸⣿⣿⣿⣿⣿⡏⣿⣟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠰⠄⢼⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣙⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈
⠀⠀⠀⣀⠀⠀⠀⠸⣿⣶⣷⠀⢠⣄⠀⠈⠉⠀⠀⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⣼⣧⣀⠉⠢⠀⠀⠁⠀⠈⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣴⣿⣷⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠉⣶⣲⣿⣿⡧⠀⣾⠟⠻⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣟⣛⣟⠃⠉⠋⠁⣸⡏⡔⠒⣄⡼⠛⣍⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢼⣧⠓⠖⠃⠑⠶⢃⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣀⣀⣀⣈⣓⠤⢤⠤⠔⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣽⠇⠀⠈⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠸⣿⣦⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣻⠇⠀⢿⣛⣻⠇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡄⠀⣤⣿⣧⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢹⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠘⠛⠛⠛⠓⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠸⣤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢦⣤⣿⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢀⣤⢠⣶⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⣿⡇⠀⢸⣿⢸⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢰⣿⡀⣴⢶⣦⢸⣿⢻⡇⢻⡇⣿⡇⠀⢸⣿⢸⣿⢸⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣬⡉⢹⣿⠈⣭⢾⣿⢸⣿⢸⡇⣿⣇⣿⡇⠀⢈⠿⠿⠋⢸⡿⠉
⠀⠀⠀⠀⠀⠀⠀⣬⢻⣿⢸⣿⢸⣿⣼⣿⠸⠟⠘⠃⠈⠋⠉⠀⠀⣼⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠈⠛⠁⠉⠀⠀⠀⢀⣀⣤⢀⣴⡆⣰⢛⡏⣸⡷⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣴⢻⣦⣾⢩⠏⣾⠼⠿⠞⠛⠋⠉⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠛⠋⠀⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠓⠶⠋⠉⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠈⢷⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠃⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠈⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠊⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⣠⠴⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠊⢸⠁⢠⠚⠉⢧⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⡀⠀⠀⠀⢸⠦⠽⠦⠘⣆⣹⣀⠀⠈⠲⠤⠖⠋⢻⣀⡀⠸⠤⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠉⠀⣀⡴⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠠⠜⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⢹⢠⣾⠏⠙⠳⠦⣄⠀⠀⠀⢀⡤⠖⠚⠉⠛⠲⣌⠓⡶⠶⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠒⠀⠀⡀⢸⣽⢋⣀⣤⣤⣤⡼⠃⠀⠰⣅⣤⣤⠤⠤⢤⣄⣸⢸⠇⠀⠠⣀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⡏⠉⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⢸⠀⡄⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣀⣧⣀⡇⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⢸⡼⠓⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⢻⠀⠀⠀⠰⠤⠤⢄⣀⣀⠤⠤⠼⠃⠀⠀⠀⠀⡿⠈⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠾⡇⠀⢀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠀⠀⠀⠀⠨⢍⣥⠀⠀⠀⠀⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⡏⠉⠓⠲⠶⠶⠶⠚⠋⠁⣸⠒⠒⣒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠠⢤⣇⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠚⠉⣉⣀⣤⠬⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠉⠛⠓⠲⢦⣍⡲⢤⡀⠀⣠⣖⣭⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢰⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢶⡶⢤⡄⠀⠀⠀⠀⠀⠀⢸⡷⠈⠀⢈⡠⣾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠳⠘⢳⣶⡀⠀⠀⠀⠀⠈⣴⣷⡶⣿⡿⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠠⣟⠷⠄⠀⠀⠀⠀⠋⠉⠤⠄⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⣠⡀⠀⠀⣀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⣠⠇⠘⢷⣤⠏⠁⠀⠀⠠⠠⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣾⣿⠇⠀⢻⣗⣿⡛⠠⠶⠦⢴⡧⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠄⣿⡦⠀⣸⣿⣟⢳⣄⠈⣄⡆⢹⣧⡉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣡⣾⣿⡟⢻⠈⠛⠛⣃⡹⠾⣟⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣤⡖⠈⠙⢁⠰⣯
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⣛⠋⠲⣿⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣋⣘⣻⣿⣿⡗⠢⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠟⣀⣠⢶⣤⣀⣀⣀⣼⡟⠧⢀⠀⣠⣸⠆⣀⣠⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⢿⡟⠀⠁⠀⠀⠉⠻⣷⠀⠀⠀⠛⠿⣶⣽⠉⢶⢂⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣠⡄⢉⡟⠀⢣⣼⣿⣶⣦⢀⠾⠙⢂⡀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⠇⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⡉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣸⠀⢀⣿⡿⣿⣿⣿⣿⡄⢰⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣤⣾⣿⠁⠸⣿⣿⣿⣷⣾⣿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣟⡙⢻⣿⠏⠀⠀⠙⢿⣿⡿⢻⣯⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠀⠀⠐⠟⠛⣿⣿⣿⣿⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠛⠻⠏⠀⠀⠀⠀⠀⠀⣿⣿⡋⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣤⣠⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⢉⣉⡛⡗⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣰⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⢀⡈⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⣤⣹⣿⡀⠇⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠿⠃⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣶⣄⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣴⣬⡋⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠍⠻⢿⣯⣤⣙⣂⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣼⣿⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⡇
⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣴⡶⠰⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣴⣿⡀⠀⣤⠀⠀⠀⠀⢀⣠⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣼⣿⣿⣿⣿⡀⠶⠂⠀⢀⣼⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⣶⣠⣾⣿⣿⣿⣿⣷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠛⠙⠁⠘⢁⡞⠛⢿⣿⠿⣿⡏⢻⠇⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢹⡒⠂⡷⠀⠸⠧⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⢀⣄⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢸⠃⠀⠀⠀⠀⠹⠦⠴⠟⠁⠀⠀⠀⠀⣀⡀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⣼⢿⠀⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠈⡇⢠⣴⡇⠀⢀⣴⠟⠛⢷⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠈⣯⡀⠀⣸⡄⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⡄⢸⡇⢸⡏⠀⠀⠀⠙⠻⠟⠉⠀⢠⡀⢠⣀⠀⢸⡀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠃⠸⢃⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣻⠇⢸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡞⠀⠀⣠⣴⢦⣤⡀⠀⠀⠢⡈⠏⠷⠄⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⢰⡄⠀⠰⣿⠁⠀⢹⡗⠀⠀⠀⠈⠂⣦⡀⢿⡆⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠛⢳⡶⠟⠃⠀⠀⠀⠀⠀⠈⠃⢸⡇⠀⠿⣁⠼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢠⣴⠟⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡞⠁⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⣼⠇⠀⠀⠘⠻⣷⣄⡀⠀⠀⠀⠀⠀⢸⣷⣌⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⢀⣿⠀⠀⠀⣀⠀⠈⠛⢿⣄⠢⣄⠀⠀⠈⣯⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣾⡏⠀⠀⠈⢛⠀⠀⠀⠀⠙⢷⣌⠳⣤⡀⢸⡿⠗⠀⣔⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⣿⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣬⣻⣾⠀⠠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢠⡾⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠘⣷⢌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⢻⡜⣷⣮⠳⡄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣥⡀⠀⠀⠀⠀⢀⠀⣠⠀⢀⠐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠛⠋⠉⠁⠒⢒⣒⣦⣴⣿⣿⠿⠿⢧⡀⠀⣠⣶⡉⠀⠀⠀⠈⢱⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⢶⣄⡐⢂⣈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣱⠖⢾⣏⣉⣉⣉⣉⣉⣉⣉⣁⣍⣉⣉⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠠⠀⠠⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣄⢠⣤⣤⣾⣿⣿⣶⣶⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣁⠀⣭⣽⣿⣿⡿⠉⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣇⣹⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣏⣤⠔⠀⠈⢿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⣀⣍⢻⣿⣿⣿⣿⣿⣿⢀⢡⡃⡘⢿⣿⢿⣿⡀⣦⢀⡄⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⢷⣄⣤⡀⠄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⣤⠇⠉⠈⠀⣴⣿⣿⡎⠁⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠂⠸⠉⠉⣿⣤⢰⣄⣿⣿⣿⣿⣿⡿⢉⡇⠀⣿⣄⠀⡘⠁⠀⠀⠘⣿⣿⣿⣻⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⡉⢀⣾⣖⣠⢆⣀⠞⣠⣿⣿⡿⠛⠋⢹⠇⠀⠀⢻⣎⠻⣿⡄⢻⡀⢀⡄⢸⢿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢎⢥⡀⠙⠛⠁⠈⠉⣴⣿⣿⣧⣿⣤⠀⠀⠀⠀⣴⣾⠁⠀⠘⠃⣸⡇⠀⢁⣀⠈⠻⣿⣾⠏⠉
⠀⠀⠀⠀⠀⠀⠸⣆⢓⠀⠂⢊⣸⣹⣿⣿⠿⢋⣿⣶⠃⢀⠏⠀⠿⣿⣶⡄⠀⠀⣿⣿⣶⠋⠀⠀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⢃⣜⢛⣧⠀⠀⢹⣿⡆⠀⠘⠿⠟⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣿⣿⣿⣹⣿⣯⣾⣿⣴⡟⠀⠀⠀⣿⣅
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣒⠘⠻⢿⣿⣿⣿⣿⣿⡿⠏⠀⠀⣠⣬⠍⠁⠀⠤⠀⣶⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣀⣹⢻⣞⡃⣠⡝⢹⣿⣿⣿⣿⣿⣾⣿⠛⠻⠀⣼⠂⣠⡄⡴⢧⣼⣿⣧⣴⢶⣿⣿⡟⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠙⠿⠟⠿⣿⣿⡄⠀⠟⣿⣿⠟⢿⠿⣿⣷⣾⣟⣡⡴⢟⣿⣿⢽⡿⣯⡟⠛⣿⣿⠙⠁
⠀⠀⠙⠄⠀⠀⠀⠀⢀⣄⣙⣶⣿⣿⣷⡄⠀⡜⢿⣶⣷⣦⣄⠖⢰⣿⢓⣼⣾⣷⣶⣶⠶⢁⣀⣤⣴⡴
⠀⠀⠀⠈⢷⣦⡀⠸⣿⣿⡟⠻⣿⣿⣿⣷⡄⢸⣸⣿⣿⣿⣿⠶⣪⣿⣿⠀⠒⢽⣿⣿⡀⢹⣿⣿⣿⣁⠀⣰⣶⡆
⠀⠀⢠⣤⣌⣻⣿⣶⣾⣿⣷⣤⣤⣿⣿⣛⢿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⣄⠀⣠⣽⣿⡟⣿⣿⣿⣿⡅⠀⠀⠀⠁
⠀⢺⣿⣿⣿⣿⣿⠻⠛⢿⣿⣿⣿⢛⣯⡀⠜⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡷⠈⠀⠋⠓⠻⢿⣿⣿⠏
⠀⢸⣿⣿⠿⠻⣷⣀⣠⣄⡿⣿⣷⢘⣷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⢸⣿⣶⡄⠀⠀⠀⣀⢀⡀⢀⣀
⠀⢠⣤⣤⡿⠀⣈⣿⡍⢀⠋⠙⠃⢹⣎⠉⠀⠙⠻⡟⡛⠋⠙⢿⣿⣿⣿⣶⣿⠿⠃⠀⠐⠶⠋⣼⠿⠿⠁
⠀⠀⣋⠉⢸⡧⢘⣿⡟⠉⠐⢤⡀⠙⣄⢀⠀⠠⣾⡿⠁⠀⠀⢠⣌⡟⠉⠉⠁⠀⠀⠀⠀⠀⠈⠈
⠀⠈⠿⠛⣼⡇⣿⣏⠉⠛⠁⠀⠀⢰⣿⣧⡷⣀⠀⢠⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠐⢿⡧⠛⣿⠷⣄⣀⢀⣾⡗⠿⣟⠚⢛⡷⠦⣤⡄⣲⣿⣿⣧⡄⠀⠀⠀⠀⡀⠀⢀⡠⠀⠀⣠⣾⣿⣿⠿⠿⢶⣦
⠀⠰⣶⣦⣄⣀⣀⣠⣤⡘⢛⣛⠉⣀⡴⣿⣧⣼⣷⣶⡆⣼⣿⣿⣿⣿⡟⢦⣶⠇⠀⠈⠛⠉⠀⠀⣾⣿⡿⠋⠁
⠀⠀⠈⠻⠿⠋⠀⣈⣿⣿⣿⣿⣿⣯⡈⠻⣫⣍⣿⣿⣅⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⢤⣼⣿⠁
⠀⠀⠀⠀⢀⣠⣟⡻⣿⣧⣄⣠⣍⠻⣷⡄⠈⠛⠙⢻⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠈⠋⠉
⠀⢰⠿⠟⠛⠉⠉⠁⠉⠙⠛⠛⠉⠛⠈⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⡀⢙⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡅⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠶⠘⠋
⠀⠘⢷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠈⠛⠛⠛⢿⣿⡭⣏⣭⣻⡷⣶⡀⢰⣾⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄
⠀⠀⠛⠒⠻⠿⠿⠛⠟⠛⠛⠉⠚⠻⠗⠘⠛⠿⠓⠀⠀⠀⠀⠀⠤⠐⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⢀⣀⠀⠈⠻⣦⠀⠀⠀⡠⠒⠤⣄⡀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⣾⠫⡉⠙⢆⠀⠈⢷⠀⣰⠁⠁⠀⠀⠈⢉⠖⠁⠂⠈⠳⣄
⠀⠀⠀⠀⠀⣴⠋⢀⡴⠶⣄⠀⠀⠀⢻⡀⢹⣤⣬⠄⠀⠘⣧⠃⠒⠀⠀⢀⡴⠋⠒⠀⠀⠀⠀⢈⡱⠤⠤⡀
⠀⠀⠀⠀⢰⡇⠀⡟⠠⡄⠸⡆⠀⠀⠀⠑⠟⣁⠤⠀⠀⠀⠸⣆⠀⠀⡠⠊⠀⠀⣀⡠⠤⠒⠉⠥⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠸⡇⠀⠃⣠⡷⠶⠇⠀⠀⠀⠀⠘⣴⠿⠛⠀⠀⠀⠘⢦⣊⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⡀
⠀⠀⠀⠀⠀⢻⡀⠈⠁⠠⢒⣓⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡡⢠⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣢⠞⠀⣼⡏⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠄⠀⠠⠀⠉⠑⠊⣀⣴⣿⠇⠀⣰⣿⠁⠀⠀⢀⣀⣀⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢠⠊⠉⢣⡀⢇⡀⠒⠲⠲⣶⣶⣿⡿⠟⠁⣠⡾⠃⠇⢀⡴⠚⠉⠉⠀⡉⠉⠛⠶⣦⣄⣀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⠭⣓⠦⢤⣤⣤⣤⡴⠶⠛⠉⠀⡸⢠⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⣠⢳⡟⠀⠀⠠⣤⣭⣤⠄⠀⠀⠀⠀⠀⡆⠀⠀⠙⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡍⠐⠂⠐⠊⢁⡜⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⣀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣠⠎⠀⠀⠀⠚⠛⠷⠆⠀⠀⠀⠀⠈⠳⣌⡀⠀⠀⠀⣾
⠀⠀⢠⠊⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣴⠒⠂⠀⠀⠀⠀⠀⠀⠀⣤⣐⢤⠁⠀⠀⣸⠃
⠀⠀⠘⢆⠀⢸⡀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⣼⠃
⠀⠀⠀⠈⢣⡀⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⡄⠀⣀⣀⣀⣀⡐⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⢱⠘⠁⣤⣶⣶⣭⣝⠲⢄⡀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⣀⠀⢀⡠⠔⠉⠀⠀⠀⢨⡎⠒⠤⢤⣉⠙⠻⢿⣆⠙⢦⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⡩⣖⠉⠀⠀⠀⠠⠆⡴⠁⢸⠀⠀⠀⠈⠙⠲⣄⣉⣀⡼⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠀⠈⠑⢤⡀⠀⢠⠎⠀⠀⠀⢣⠀⠀⠀⠀⠀⣀⠬⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠁⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠊⠉⠉⠑⢆⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⡄⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠲⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠖⡏⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠄⠀⢳⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠀⠀⠀⠀⠀⠳⣄⠀⣰⠋⠀⠀⠀⢳⡠⠤⣄⡠⡞⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⣧⠀⠀⠀⠀⠀⣀⠜⠁⡏⠀⠀⠀⠀⢸⠀⠀⢸⡇⢧⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠙⠒⠒⠒⠋⠁⠀⣠⠷⡀⠀⠀⣰⢻⡙⠒⣏⡀⠈⢧⡀⠀⠀⢀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠈⠉⠉⠁⠀⢣⡀⠀⠙⠢⣄⠉⠓⠚⠉⣤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢉⡠⠚⡡⢲⠟⢏⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⢐⠎⠳⣄⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠒⠻⠤⢤⠀⠀⠀⠀⠦⣀⠱⣄⠘⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡤⠒⠁⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣀⣈⣙⣚⠂⠘⣆⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠂⠅⠘⣄⡀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⢻⡳⢾⣂⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣫⠔⡢⢀⡤⠚⠒⠧⣍⡓⢦⣀⠀⠀⠀⠀⢀⡠⠤⢀⡴⠻⡿⠋⠉⠉⠑⠒⠒⠛⠉⠉⠉⠉⢹⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣻⠎⡡⠋⣠⠋⠀⠀⠀⠀⠀⠙⠲⢾⣙⠢⣄⡀⢹⠆⠐⠾⣤⣼⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡉⠛⠣⣀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⢠⠊⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠤⠀⠀⠉⠈⢳⡄⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠀⠎⠀⠀⣠⠘⣆⠀⠀⠀⠀⠀⡴⠉⢮⣛⡶⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡰⠁⢣⠸⣄⠀⠀⣀⡼⢁⡄⢈⣅⣀⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣧⠉⠛⢉⡨⢤⡀⠓⠻⣤⣀⣀⡦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡫⠕⡾⠁⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠸⠖⠋⠁⠀⠀⠓⠒⠚⡇⢘⠑⠲⢤⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⡱⢋⢤⠀⠇⢀⡾⣠⡛⠒⠤⢤⣀⣀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⢳⠏⡸⢸⢠⣧⣜⠉⡱⠛⢶⡶⣥⣬⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠋⠀⡗⣸⣷⣮⣉⡟⠓⡲⢶⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉
⠀⠀⠀⠀⠀⠀⠸⢤⡤⠞⠁⠀⠀⠿⠛⠉⠒⢯⣽⡿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠉⠉⢳⡒⠇⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠄⠀⠠⠤⠤⠤⠭⠭⠯⠭⠗

⣿⡇⡄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⣿⡟⠙⢿⣿⣿⣿⣹⡏⠀⠀⠀⢸⣿
⣿⡇⣧⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣅⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⠁⠀⢺⣿⣿⣿⡌⠃⠀⠀⠀⢸⣿
⣿⡇⣿⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⣾⡟⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⢸⣿⢸⣿⣿⡄⠀⣾⣿⣿⣿⢳⡇⠀⣠⠄⢸⣿
⣿⡇⣿⠀⠀⠀⢠⡞⠃⠀⠘⣦⠀⢀⣿⡇⢀⣤⣴⣶⣶⣶⣶⣴⣶⣿⣿⣦⡀⢸⣿⢸⣿⣿⡇⢀⣿⣿⣿⣿⣾⡇⢼⣷⡆⠀⣿
⣿⡇⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⣿⡷⢠⣿⣿⣿⡿⠿⠟⠛⠛⠻⠛⠿⢻⡎⣿⢸⠿⠿⠀⣿⣿⣿⣿⡿⣿⣷⣦⣹⣷⡆⠿
⣿⡇⣿⠀⠰⠖⣿⠀⠀⠀⠀⢸⣿⣠⣿⣯⣘⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣘⠀⠀⠘⢿⣿⣿⣿⡿⠛⠛⢿⢿⡿⠟⣤
⣿⣿⣿⢠⠀⢦⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣀⣠⠀⠠⣾⠀⣷⣿⣿⠀⠀⠀⠀⢻⠛⠿⣏⠀⣀⣀⣈⣤⣤⣴
⣿⣿⣿⡞⠁⠻⢿⣿⡶⠶⠶⣿⣿⣿⢿⣿⣿⡛⢻⠂⠀⠀⠘⠛⠋⠀⠀⠀⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢣⣽⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠐⢿⡀⠀⠀⠈⢿⠁⠚⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⢶⣤⣀⣼⣯⣿⡿⢿⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠛⠻⢿⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⡿⣿⣿⣿⠫
⢿⢹⣿⣆⠀⠀⠀⠘⣧⠀⠀⠀⠱⣀⣼⣿⠏⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⡷⠋⠉⠉
⢸⣸⣿⣿⠂⡀⠀⠀⠈⠀⠀⠀⣰⡿⣿⡟⠀⠀⠀⠈⢿⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁
⢸⣿⣿⣿⠀⠹⣄⠀⠀⠀⢀⣾⣿⡁⣿⡇⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠓⠶⠖⠂⠀⠤⠄
⣿⣿⣿⡏⠀⠀⠈⠳⡦⠶⠟⠛⣩⡇⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⠁⠀⠀⠀⠀⣦⣄⣼⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡿⠿⠶⠶⡶⠶⠶⠶⠿⡿⣿⣿⣶⣿⣷⣶⠶⠶⢶⡶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⠶⢶⢶⣶⣶
⣿⡇⠰⠀⠀⠉⠀⣴⠆⣀⢀⣿⣿⣿⣿⣿⡏⢀⣤⡯⠀⠀⡗⠀⣸⣿⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢰⣿⣿⣿⣿
⣿⡃⠀⢀⡀⠀⣰⡿⢠⡏⣼⣿⣿⣿⣿⣿⠃⣠⣾⠁⠀⡀⠁⢰⣿⡏⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣸⣿⣿⣿⣿
⣿⠄⣠⢟⣿⢠⣿⡇⣾⠁⣿⣿⣿⣿⣿⣿⢀⣿⣾⣇⣀⣿⡄⣼⣿⠀⠀⠀⣀⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡿⢠⣿⣿⣏⡿⣿
⣿⣼⣿⣿⣿⣼⣿⣻⠃⣰⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⢿⣶⣿⣿⣠⣶⣾⣿⣿⣿⣿⡆⡀⠋⠀⠒⢛⣯⣿⣇⣾⣿⣟⡜⣰⣿
⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠉⣻⣿⣿⣟⣿⠉⠛⠉⣿⣿⣿⣟⠙⢻⣿⣿⠛⠉⠉⠉⠉⠉⠉⠉⢻⣿
⣿⣿⣿⡜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣸⡟⢸⣿⣿⣿⣿⣦⢸⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠃⡿⣿⡇⢸⣿⣿⣿⣿⣿⣬⣷⣿⡿⠿⠆⠀⠂⣄⡀⠀⢸⣿
⣿⡿⠿⠛⠿⠟⠋⠉⠁⠀⡛⠑⠋⠈⠉⢸⣿⡇⠀⠀⠀⢸⣿⡟⠀⡇⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⢦⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⢄⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⣿⣿⠁⢸⣿⣿⣿⣿⣿⡿⢏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠦⣧
⠀⠀⠀⠀⠀⠀⠀⢐⣶⡶⣶⣶⠀⠀⠈⢸⣿⣧⠀⠀⠀⢸⣏⠁⠀⠙⠿⠀⢸⣿⣿⣿⣿⣿⡾⢋⡗⠀⠀⠀⠀⢀⠰⠀⢀⡁⣿
⠀⠀⢀⡀⠀⠀⠀⡈⣽⢀⣻⣿⢆⠀⠈⢸⣿⣿⠀⠀⠀⢸⣿⢀⠀⠘⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸
⠀⠀⠀⠐⠂⠰⠂⠴⠁⠀⠉⠋⠘⢁⣴⣾⣯⣿⡆⠀⠀⠀⠻⠄⠀⢸⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣦⣴⣶⣦⣤⣶⣶⣧⡀⠀⠀⢰⣄⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⠸⠾⠇⠀⣿⣿⣿⣿⣿⣿⡿⣇⡀⠀⠀⠀⠀⠀⣠⣤⡄⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⡇⢸⡿⣿⣿⣿⣿⣇⣦⣀⢸⣿⣀⢁⠀⣸⣿⣿⣿⢿⣿⡽⣧⠈⢣⣄⠀⠀⠀⠀⠃⠀⠀⣿⣿
⣿⣿⠛⢻⠿⠟⢿⠿⠛⣿⣿⠛⠁⠀⠀⣸⣿⣿⣿⣿⠿⠯⡿⣿⣴⣿⢠⣿⣿⣟⠹⣿⣉⡷⣾⣳⣄⡙⠷⠶⠦⠀⠀⢠⣆⣿⣿
⠉⠁⢀⡀⠀⠒⣦⣠⣂⣿⡇⢀⣀⣀⣨⣿⠿⠛⠀⠉⠁⠁⠈⢁⣈⣠⣶⣿⣿⣿⠀⠀⠉⠁⠉⠉⠀⣉⣀⡀⠀⠀⣀⢼⣿⣿⣿
⢀⠁⠘⠇⠃⠀⠀⣬⣬⠛⣿⡿⠿⣿⣿⣿⠀⢀⣠⣤⣤⣿⣿⣯⣿⢿⣿⣿⣿⣿⣌⣁⣠⣤⣴⣶⣫⡤⠛⠛⠗⠸⣿⣿⣿⣿⣿
⠈⠀⠀⠀⠀⠀⠤⠀⠀⢀⣬⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉
⡀⠀⠀⠐⡦⠄⠀⠀⠑⠛⡉⣹⣍⣭⣿⣿⢿⣿⣿⣿⠿⡿⠿⣿⣿⠿⠟⠿⠿⣟⠉⠉⠉⠁
⣡⣴⣶⣦⣿⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣄⠐⠏⠁⠀⠀⠀⢀⣁⢠⠀⠀⠠⣞

⠀⠀⠀⠀⢰⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶
⠀⠀⠀⢠⣶⣿⣶⡌⢢⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣠⡄⣠⡀⣤⣀
⠀⠀⠀⢸⣿⣿⣿⡇⠨⠁⠀⠀⠀⠀⠈⣿⡿⠟⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⠀⢿⣏⢺⣛⣻⢊⣽⠇
⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢩⣇⠀⠀⠀⠀⢸⣿⠟⡃⠘⢰⠐⠐⡛⢿⡏⠀⠀⠙⣿⣾⣾⡟⠁
⠀⠀⠀⠀⠸⠿⠇⠀⣼⡄⠀⠀⠀⠀⠀⢲⠟⠀⠀⠀⠀⠸⣯⣍⣀⣩⣀⣨⣁⣀⣭⡧⠀⠀⠀⢿⣿⣿⠇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣾⢟⣵⡄⠀⠀⠀⠀⠀⠀⢀⡴⡚⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠛⠋⠿⣀⡀⠀⠀⠀⠀⡞⠀⠜⠀⠚⠄⠸⡆⠀⠀⠀⠖⠲⠂⠀⠀⠀⠀⠀⠀⢰⡃⠛⠁⣧⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⢳⡘⢿⣿⣿⠟⣸⠃⠀⠀⠀⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⣇⠙⠺⣝⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠙⠒⠶⠖⠊⠁⠀⠀⠀⠘⠀⠀⠃⠰⠷⠆⠀⠀⠀⠀⠈⠓⠶⠛⠋
⠀⠀⠀⡤⠴⢄⣀
⠀⠀⠀⠀⠀⣼⣿⣍⠙⠛⠀⠀⠀⣶⣶⢤⣶⣦⣴⣶⡆⠀⠀⠀⣠⠴⠖⠶⢤⡀
⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠐⠿⣿⣺⠿⣗⣿⢿⣺⡿⠗⠀⡼⠁⢄⠄⢄⢀⡹⡄⠀⢀⣠⡀⢠⣶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⣼⢹⡇⠀⠀⠀⢀⣤⣿⣻⣽⣟⣿⣽⣻⣧⣄⠀⢧⠰⣶⣤⣶⡶⢠⠇⠀⠸⣿⡇⣿⣟⣿⡇⣿⣿
⠀⠀⠀⠀⠀⢻⢸⡇⠀⠀⠀⠀⠉⣿⣾⢽⣷⣿⣽⣾⡏⠁⠀⠈⠳⢬⣉⣩⠴⠋⠀⢠⣤⣿⢢⣴⣿⣷⣤⢺⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⡀
⠀⠀⣀⣀⣀⣀⣠⣶⣶⣶⣶⡄⠀⢀⣬⣿⣥⡙⢄⠀⠀⠀⠀⠀⠀⢉⣤⣌⠁
⠀⠀⡇⣀⠀⣀⢸⣟⣻⣟⣻⡇⠀⢸⣿⣿⣿⡇⢘⠃⠀⠀⠀⠀⠀⠘⣿⠟
⠀⠀⡇⣍⣀⡩⢸⣿⣛⣛⣿⠁⠀⠈⢹⣿⡏⠁⢸⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⢺⠉⣇⡞⣭⡙⣏⢹⠃
⠀⠀⠰⢄⣉⡠⠎⠙⠛⠛⠋⠀⠀⠀⢸⢿⡇⠀⣈⡀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⢼⣀⣛⣥⣛⣡⣟⣘⣻⠆

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢉⠑⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⠐⠃⠀⠀⠀⠀⠀⠂⠀⠠⢄⡀⠀⠀⠈⠑⠑⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠚⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⡤⠄⠒⠈⠁⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎⢡⣀⣀⣀⣀⣤⡤⢰⠊⢉⠅⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠰⠤⠒⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠂⠀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⢀⠴⠥⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡆⠀⠀⠀⠀⠐⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢠⡒⠒⢣⠤⠀⡀⠀⠀⠀⠀⠀⠛⠛⠉⠡⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠇
⠀⠀⠀⠩⣉⡩⠄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠤⠤⠤⠤⠤⢀⣀⣀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠦⣀⡠⠛
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣠⣠⣄⢤⣄⣄⠀⠈⠣⡄⢀⡠
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠶⠢⠀⠉⠃⠑⡿⢆⠙⠀⠁⠀⠁⠈⠿⢆⠀⠘⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠳⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⢁⡀⠀⠀⠀⠠⠈⠀⠉⠉⠀⠠⢄⡀⢀
⠀⣀⠔⠊⠃⠀⠀⠀⣀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⠤⢀⡄⠄⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢠
⠘⠂⠀⠛⠐⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠞⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⣀⣤⣴⡶⠶⠿⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣤⣴⣶⣶⣿⣿⣿⣿⣧⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣯⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⣀⣀⣀⣿⣿⣫⠷⠋⣿⣿⣍⠙⠛⠳⣿⡄⢰⣿⣿⣿⣿⣶⣶⣦⣶⣯⣥⣤⣤⡀
⠀⣴⠋⠁⠀⠀⢉⣿⣯⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠈⢿⣦⣄⣀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣭⣄⣈⢿⣿⡿⠋⠉⠻⣿⡏⠀⣀⣀⣀⣀⡀
⠀⠀⠉⠙⠛⠛⠻⠿⣿⣿⣿⡏⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠛⠉⠁⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⣿⣤⣴⣾⣿⣿⣷⡾⠁⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠈⠻⢿⣿⣿⣿⡿⠃⠀⠈⢿⣿⡇⠀⠉⠙⠛⠛⠉⣽⣿⣿⢯⡾⠋
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣤⣭⣯⣍⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⣠⣼⣿⠟⢱⡿⠁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⢀⣽⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣇⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠋⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠔⠃⠉⠘⠁⢄⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠬⠥⠄⢴⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⠤⠀⠀⠀⠀⠉⠀⠒⠒⠂⠀⠀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠔⠀⠐⠐⠀⡜⠁⠈⠀⢸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⠴⠶⠍⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⠀⠀⠠⣶⣿⣤⠀⠀⠀⣸⠀⠀⡔⣐⠂⣇⠀⠐⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡑⠀⠀⠚⠉⠂⠠⠔⠁⠀⠀⠊⢁⠂⣀⡀⢁⠉⠚⠊⠊
⠀⠀⢀⠤⣤⣄⠸⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⠴⡂⠠
⠀⠀⠇⢠⣿⣿⡆⠈⠒⠢⠤⠄⣀⣀⣀⣀⡀⠀⡤⢴⠒⠻⡀⠀⠀⠀⠠⣨⠂⠒⠁
⠀⠀⠰⡟⢻⠋⢢⠤⠀⠠⠄⠂⢩⠈⠀⣀⠄⡈⠠⠃⠐⢄⠀⠀⠜⢡⡴⢠⠃
⠀⠀⡀⠣⡄⣇⡴⠀⢀⡠⡴⠂⢡⠴⠊⠀⠀⠃⠀⠀⠀⠘⠑⠠⠖⣮⠀⠋
⠀⠀⠀⠀⠠⠄⠉⠈⢀⠊⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠃
⠀⠀⠀⠀⠈⠁⠀⠰⣁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠀⠂⠄⠀⢀⣀⣀⡀⠀⠀⠠⠂⠁⢠⠇
⠤⠀⠠⠤⠆⣠⠂⠰⠷⠄⠀⠙⡓⠂⠀⠀⠀⢀⣠⣴⡗⠀⡄⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠛⠉⠒⠒⠦⠤⠤⠀⠀⢂⣀⡷⠟⠧⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠒⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣦⡀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⣠⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡏⠙⠿⠿⠿⠟⠁⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣇⣠⣴⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⣿⣿⣿⣿⣿⣿⣿⣷⣀⢀
⠀⠀⠘⠿⠟⠋⣹⣿⣿⣿⣿⣿⡿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⣾⡿⣿⣿⣿⣿⣿⣏⠉⠛⠿⠃
⠀⠀⠀⠀⠀⣾⣿⠏⣹⣿⠉⢿⡆⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣿⠉⣿⣿⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠉⠁⠀⢿⡿⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⢻⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣷
⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣴⣦⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠉⢛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣗⠀⠸⠹⣃⠤⠴⢶⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢽⡿⠒⠖⠂⠀⠀⠀⠀⣿⠋⠈⠑⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⢱⣤⣤⣀⣀⡴⢿⡶⢿⡄⣰⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⢸⣧⣤⣇⠘⠋⠉⠁⠀⠀⠀⠹⣯⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡿⢿⣿⣿⣷⣿⡄⡄⠑⠀⠁⠆⠀⠲⠾⠿⡜⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠷⠋⠛⢿⢻⣷⠹⣶⣶⡿⠀⠀⣦⠀⠀⠠⣉⠀⡇⠀⢀⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠺⠇⠘⣷⣦⣔⣁⡴⠆⠀⠂⠀⠘⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠐⢠⠀⠀⢸⡄⠀⠀⢀⡀⠀⠀⣿⣟⣿⣿⣿⣦⣀⠀⠀⠀⢀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡐⣄⡀⠠⣿⣧⣀⣀⣼⡛⠽⠿⠿⣿⣿⡄⠀⠀⠈⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⡸⡿⢿⣿⣉⠀⣀⣀⣘⠛⠿⠿⣴⣤⡆⣀⠟⣿⣧⣀⠂⠀⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠣⣤⠀⠀⠀⠀⠀⣀⣒⠚⣽⣿⡛⣭⣭⠹⠿⠛⣿⣾⣯⣴⠁⠐⠛⠻⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠑⠂⠀⠀⠀⠐⣀⠨⠿⢿⣿⣿⣧⣹⣿⡆⠐⠐⣿⣿⣿⣷⣧⠀⡀⠰⣧⡂
⠀⠀⠀⠀⠀⠀⣐⠂⢠⡘⣷⣤⣀⣀⢠⡆⠀⠀⣤⡄⢀⣽⣿⣷⣽⣿⣿⡿⠁⠀⠀⣠⡵⢿⠰⠃⠀⠘⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢸⠯⣽⣷⠀⠀⠈⢻⣿⣿⣷⡄⠈⢩⣁⠈⠛⠛⢿⣿⣿⣿⣤⣶⣦⣰⡟⠁⠀⠀⠀⠀⢷⣀⠀⣼⣿⣦
⠀⠀⠀⠀⠀⠸⢿⣾⣿⣗⡀⢃⣘⣿⣿⣿⡿⣳⡄⠻⠇⢤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣧⡼⠛⡀⠀⡆⠐⢛⣶⠌⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠺⢾⡟⢿⣿⣿⡟⠻⠋⢐⠻⠶⣤⣤⣤⣈⣉⣛⣿⣿⣿⣯⣿⣿⣿⡴⢋⡤⠂⣤⣶⢿⡁⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⣧⣄⠀⠘⣷⡐⠀⠈⠉⠙⠛⠿⠿⠛⠛⠋⠁⠈⢿⣿⣯⡴⠏⠏⢡⣶⡁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣌⣄⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣠⣴⠈⢻⣿⣧⠀⣧⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡂⢨⣭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠻⣿⢸⣿⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠈⣻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣻⣿⣿⣆⣿⣿⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⢟⣫⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⣿⡟⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⣠⣟⡧⠀⠀⠀⠀⠀⢀⣛⡠⠀⠀⠀⠀⠀⣴⣶⡷⣤⣴⣶⣤
⠀⠀⡏⠀⢸⠇⠀⢹⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⢿⣿⠳⠁⠀⠀⠀⠀⢻⣾⣛⣿⣿⣿⣿
⠀⠀⠿⠻⠾⠷⠞⠿⠄⠀⠀⠀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠳⠾⠋


⠀⠀⠸⠿⠿⠿⠿⢿⠀⠀⠀⠀⢠⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⣤⣀⣶⡆⣤⡄
⠀⠀⡄⠀⢰⡇⠀⢘⡄⠀⠀⠀⣿⡿⠿⠿⢿⣿⠀⠀⠀⠀⠀⣀⡹⢋⣀⠀⠀⠀⠀⣤⣭⣤⣙⣡⣭⣥
⠀⠀⢰⣶⣶⣷⣶⣶⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⠀⠀⠀⠉⠙⠿⠿⠿⠏⠉


⠀⠀⢶⠄⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣀⡀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢸⠦⠀⠀⠀⠀⠀⠀⠉⠅⢠⠀⠀⠀⠀⠀⣸⣅⣼⣍⣷⣽⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠂⢐⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉


⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡠⢚⠉⠉⠲⡄⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣶⣶⣾⣿⣶⡆⠀⠀⠀⠀⡇⠈⠀⠈⠁⢸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡿⡿⢿⣿⢿⢿⠇⠀⠀⠀⢿⡿⢿⣿⣿⠇⠀⠀⠀⠀⠑⢌⣐⣊⠵⠃⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋

⠀⠀⣀⠠⠴⠆⠀⠐⠀⠂⠄⢀
⠴⢥⠴⠖⠒⡀⠀⠀⠀⠀⠀⢀⠜
⠀⠞⠂⠽⠍⠀⠒⠀⢠⠀⡰⠟⢄
⠀⠀⠀⠀⠀⠀⢎⠀⠈⡆⠀⠀⡈
⠀⠀⠀⠀⠀⠀⣄⠔⠁⢺⡀⠀⢇⠠⡠⠂⣄⠠⠤⠤⠤⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⣦⠀⢨⡀⣸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⡛⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠂⠑⡀⠘⡗⠀⠙⠆⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢹⡀⠀⠈⠂⠀⠈⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⣛⡦⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠹⣏⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠢⠦⡀⠀⠀⠀⠀⠀⠸⣦⣠⣴⣗⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡗⠀⠟⠓⠠⣀
⠀⠀⠀⠀⠀⢀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢿⠂⢢⠀⠀⠀⠑
⠀⠀⠀⠀⡦⠜⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡄⠀⢃⠀⡀⠀⠀⠀⠀⠀⠠⢴⠔⢠⠓⠈⣦⠤⡀
⠀⠀⢠⠎⠐⡻⠢⡴⠂⠢⠀⠀⠀⠀⢀⠀⣠⣍⠉⢉⡷⣺⡗⣢⣤⣄⣤⣞⢇⠀⠛⣂⠽⢛⠠⣠⠜
⠀⠀⠀⠀⠀⢘⠂⠣⡠⠀⡋⠈⣯⠬⠭⠏⠇⣇⣕⢻⢄⡙⡝⢷⢿⠯⡐⠢⠊⠒⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢛⣈⠦⠜⠷⠯⠄⠀⠀⠀⢠⢿⠊⠏⢷⢘⡛⢿⢛⣀⠐⡣⠀⠀⠀⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠠⠒⢄⠇⢠⠔⢀⠀⠀⠀⠀⠀⠞⣂⡈⠀⠆⠰⠰⡏⡿⣊⣦⠔⡏⠀⠀⠀⠀⠀⠈⠔⡀
⠀⠀⠀⢠⢁⠋⠃⠀⠀⠉⠣⡱⡀⠀⠀⠀⠀⠃⠉⠙⠐⠒⠾⡼⠃⡉⠘⠲⡀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠘⠁⠀⠣⡀⠀⠀⠠⡑⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠼⣭⣀⠁⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠈⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠱⠚⡌⠁⠁⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⢹⢻⠢⣅⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡢⠄⠐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠠⢇⣀⣤⣤⡤⣰⠢⠆⠀⠀⠀⠀⠙⠛⠃⠘⢶⡵⣆⣄⣀⣤⣀⡀⠀⠀⠀⠀⠰⡻⠋⠆⠉⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⠏⠛⠪⠬⠤⠴⠶⠶⠶⠾⠶⠶⠿⠥⠭⡿⢭⣽⠦⠄⠼⡆⠀⠀⣼⣀⡁⠠⢔⡠⠊⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⢀⠊⣄⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⠐⣂⠀⠄⡰⠂⠒⠊⠈⠿⠿⠖⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠈⠀⠤⢀⠱⠁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣷⣶⡶⡐⠀⠉⠂⡤⠀⡶⣄⠀⠀⠀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠛⠿⢛⡴⡁⠀⠀⠀⠧⣀⠀⠈⠉⠀⠘⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠐⠀⠋⠀⠀⠋⠀⡜⢀⠠⠃⠀⠈⢈⠗⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠤⠤⠚⡀⠁⠒⠀⠤⠀⠂⠀⢀⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⠀⠀⠀⢆⠉⠐⠀⠀⠔⠁⠀⠀⠠⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠀⠀⠚⠒⠄⠈⠢⣀⡠⠀⠀⠀⠀⠀⢠⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠛⠢⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⡠⠘⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠐⠉⠀⠀⠀⢀⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⢀⠊⠀⠀⠀⠀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⢀⠤⠄⡀⠈⠐⠤⣀⣀⠠⠤⠐⠀⠀⠀⠁⠀⠀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠁⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠄⠀⠀⠀⠀⠀⠀⢐⠑⠁⢀⠄⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠀⠈⠙⠄⠀⠀⠀⠒⠒⠒⠁⢀⣔⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⡦⢄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⢀⠔⠈⠀⠠⠈⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠸⠀⢆⠀⠄⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠻⠒⡽⠃⠈⠂
⠀⠀⠀⡠⠂⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠤⠤⠤⢄⣀⠈⠁⠀⠀⠀⠄⠦⠤⠤⠤⠀⠀⡀
⠀⠀⢀⡁⠀⠀⠀⠂⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠐⠠⡀⠀⠈⠑⠢⣄
⠀⠀⢼⠀⠀⠀⠂⠐⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢦
⠀⠀⠘⡀⠀⠀⠀⠈⢼⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⡈⠍⠛⠀⠀⠈⠀⠁⠈⠉⡉⠉⢉⠉⠉⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠳⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠄⠈⠢⠠⠤⠂⠡⠔⠠⠀⠠⠀⠀⢄⡀⠐⠢⠤⠀⡀⡀⢤⣶⠀⠀⠀⣀⠤⠄⠒⠈⢀
⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⡃⠀⢀⢬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠈⠁⠈
⠄⠀⠤⠁⠀⠐⠂⣁⣀⡀⠀⠀⠀⠁⠊⠁⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠈⠍⠉⢀⡀⢀⣐⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡠⠘⠣⣀⢀⣈⡍⠁⠈⠀⠀⠀⠀⠀⢰⠂⠐⠒⠒⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⡔⠋⡀⢀⠀⡬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠂⠈⠒⠑⠊⠀⠀⠣⠘⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡨⠄
⠀⠀⢀⠂⠤⠀⣀⡀⠀⠘⠀⡠⠔⣊⠤⠊⠀⠀⠀⠀⡀⠀⡀⠀⠐⠉⠉⠉⢨⡂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠈⠑⠒⠦⠬⣉⣹⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠐⣐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠜⠀⠰⠤⢀⣉⠐⠒⠠⠤⠤⣸⡄⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⢣⢸⢇⠎⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣯⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣟⡿⡾⢾⠶⣟⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡾⢽⣭⣟⣹⡏⢹⣅⡼⡆⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣏⣹⠓⡿⢤⣯⡼⣯⡽⢧⡤⢷⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢧⠀⠀⣸⠦⣯⣹⣿⣾⡟⠚⣟⠶⢻⠒⠛⡏⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⣟⢺⠧⢼⡇⠙⣏⢛⣏⠓⣻⡓⢚⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⣍⣽⠓⢺⡇⠀⢹⣏⣿⣙⣹⣏⣉⣼⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣻⡀⢼⠢⣿⣝⣽⠇⠀⠈⡯⢾⣍⣉⣿⣭⢼⣾⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠻⡇⣾⣉⡿⢤⡟⠀⠀⠀⡷⢺⡯⡭⢿⠦⠾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡷⢭⣽⣾⠁⠀⠀⢰⣟⣹⡗⠶⣿⡓⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡖⢿⡏⠀⠀⠀⢸⣧⣤⣏⣋⣹⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢿⣹⠇⠀⠀⠀⢸⠶⠾⣧⣉⡽⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣓⢚⣷⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡇⠀⠀⠀⠸⣯⡟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢳⣄⠀⠀⠘⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣷⣤⣼⡧⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢀⡜⠛⣿⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⢠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢀⣴⠋⠘⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠂⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣤⣶⣶⣶⣷⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡿⣿⡿⣻⣿⣦⣄⣀⣀⣀⣀⣀⣤⣶⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⡿⠿⢿⣿⣖⣀⣀⣤⣼⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⡇⣿⣧⠑⣽⣦⣈⣉⣙⣛⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⣿⡛⠟⣉⣽⣻⣿⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠹⣏⠻⣶⣦⣭⣭⣽⡿⢟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⣷⡻⠛⢋⡽⠋⢠⡾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠇⠀⠈⠀⠀⠈⣉⣉⣥⠴⣊⣽⡿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠃⠘⢷⡄⠉⠀⣰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢩⣤⣾⡿⠋⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡄⠀⠀⠀⠀⡇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣸⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣠⠟⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢰⡏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡿⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢠⣿⠃⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠿⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⢿⣿⣧⠻⣿⠿⠆


⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⡀
⠀⠀⠀⡔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠔⠒⡄
⠀⠀⠀⠳⠤⠤⠴⠄⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠴⠃
⠀⡄⠂⢀⠀⠀⠀⠀⠀⠀⠴⠛⠻⠟⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⣿⠷⢦⡀⠀⠀⠀⠀⡀⠀⣀⠀⠂⡄
⠀⠈⠉⣠⣴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⠉
⠀⠀⠞⠉⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠙⠓⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠤⠤⠤⣄⣀⣀⣀⣈⣉⡉⣉⣉⣉⣁⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⠔⠋⠉⠉⠙⠻⠍⠀⠀⢙⡷⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢤⣀⣀⣀⣀⣤⡤⠶⠚⠋⠀⠘⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⡿⣟⠲⣄⠀⡈⠢⣄⠀⠀⠀⠀⠀⠀⠈⣉⣽⢿⣰⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠤⣩⡇⢿⡘⡜⢦⠈⠢⣈⠳⢄⠀⠘⣻⡶⢫⣵⣿⣿⠟⠵⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡶⠟⣫⡵⠞⠁⢸⡄⠀⢻⡌⢿⡀⠈⢹⣖⡛⢲⣿⣰⣿⣿⠟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣴⣿⣋⡀⠀⠀⢀⣻⡄⠀⠳⣄⣉⣒⠈⣹⡷⢫⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⣿⣿⣷⡮⠀⠀⠩⢲⣿⣶⣦⣤⣭⣿⣿⣿⢃⣾⣿⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢡⣿⣆⠀⣀⣀⠀⠀⠀⠀⢸⢃⣽⣿⠃⣼⡟⣿⣿⡜⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣎⠻⠍⠉⠀⢀⣰⡦⢺⡿⠁⣸⣿⢱⣿⣇⠇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⣜⢿⣿⣦⣀⣤⣴⡿⢋⣴⠏⠀⢰⣿⢿⠘⣿⣏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢻⠘⢷⣿⣿⣿⢹⢏⣴⣻⠃⠀⢠⣿⡷⠚⢣⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⢰⡄⢣⡀⢨⡿⣡⣏⣾⣳⠃⠀⢠⣿⣿⠁⢠⠞⢹⡟⡳⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⢀⡿⣿⠀⠇⢀⣴⢿⣿⣿⠃⠀⢀⣾⡟⢸⠀⠀⠀⠀⣷⠘⣎⠳⣍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣿⣠⠾⢋⣘⣸⣿⠃⠀⢀⣾⣿⠁⡼⠀⠀⠀⠀⢸⣧⡈⢳⢬⡳⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠟⣉⡤⠖⠋⠉⣹⢇⠄⢀⣾⡿⠻⠷⣧⣀⣀⣠⣤⣾⣿⣷⣄⠑⢌⠚⢦⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣓⠟⠁⠀⠀⠀⣰⢃⠎⣰⣿⡿⠁⠀⠈⠲⣝⢿⣿⣿⣿⣿⣿⣾⡳⣤⣙⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠞⠁⠀⠀⠀⠀⣰⢏⡞⣰⣿⡿⠁⠀⠀⠀⠀⠙⣦⠹⣿⣿⣿⣿⡿⣟⢦⡹⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⡀⠀⠀⠀⠀⣰⢏⡞⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡆⢿⣿⣿⡟⠁⢻⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⠀⠀⠀⠀⣰⢏⣞⣼⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣗⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⢏⣾⣾⢏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⣴⢯⡟⣾⢃⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠈⠁⠈⠉



⣿⢿⠿⡿⢿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣛⣻⣿⣛⣛⣻⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠛⣿⣿⣿⣿⡿⣿⣿⣿⡟⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠉⠛⣻⣛⣽⡋⠀⠈⣿⡟⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⢀⣘⠛⣉⡀⠀⠀⠈⠉⠀⣠⣤⣤⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣼⡿⠛⠋⠙⠳⣀⠀⠀⠈⢁⡀⢤⣄⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣥⣾⠗⠘⣶⣿⣀⠀⠙⠋⠀⠐⠈⠁⢻⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡟⠁⠉⠀⠈⠉⣋⣩⣀⣠⡀⠀⠀⠀⠀⣸⣅⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⢀⡀⠀⠸⠟⠻⠟⠁⠀⠀⡀⢤⡠⣿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣦⠈⠉⠂⠀⣀⣤⣄⣤⠤⣤⣉⠘⣇⢹⡇⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣛⣿⣿⣿⡇⠠⣾⣿⣏⣀⣀⣀⣤⣬⣿⠓⠗⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡆⠹⣟⠛⠉⠉⠉⠀⠉⠁⠀⠀⢀⣷⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡄⠙⠀⠚⠋⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣙⠿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⡉⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡠⠞⠹⣿⣯⢹⡆⣿⠸⠏⣷
⠀⠀⠀⠀⠀⢀⣾⣯⡀⣿⣿⠟⠛⢛⡏⢩⡿⠤⠀⠈⠿⣿⣧⣴⣶⣴⣦⠶⠋⠀⠀⠀⠁⠘⠀⠁⢹⣿⣿⠋⠉⠉⠁⠈⠉⢉⣉
⠀⠀⠀⠀⢀⣼⣯⠈⢿⣿⡏⠀⠀⠀⠁⡼⡃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢸⡿⠁⠀⠂⠀⠀⠀⠠⠷
⠀⠀⠀⣠⣾⠃⠀⢀⢼⣿⡇⢠⡀⣀⣠⠀⠁⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣿⠀⠀⠀⡟⠁
⠀⠀⠘⣇⣤⠀⠀⠈⣾⣿⢼⣿⣷⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣤⣶⣞⠁⡀
⠀⠀⣈⣥⠾⣦⣴⡿⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⠟⢠⣽⣆
⣾⣿⠦⣴⣶⠾⠛⠓⢤⣿⣿⣿⣿⡿⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣀⣀⣠⣠⣦⣼⣿⣯⣀⣶⣿⡿
⢹⣿⣾⣝⣶⣦⡄⠀⠀⢈⠟⠻⣿⠁⠈⠀⠑⢄⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠐⡄⢪⣾⡟⣼⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠙⠿⣿⣿⣿⣿⣿⣶⠃⠀⠀⠈⠀⠀⠀⠀⡈⠁⠒⠀⠈⠉⠶⢦⣿⣿⠷⠾⠛⠙⣿⡟⠛⠿⠿⠃⠀⠙⠻⠿⠿⠟⠉
⠀⠀⠀⠈⠙⠛⠿⣿⣟⢀⡄⠀⠀⠀⠀⠀⢠⠹⠄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢸⣧⣀⣠⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣤⣤⣶⣬⣆⣀⣀⣀⠀⠀⠀⠐⡤⠂⠀⠀⠀⢀⡈⠛⠛⣭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣇⠉⠉⠁⠐⠒⠒⠘⠀⠂⠐⠈⠉⠁⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⣿⣇⠙⠁⠀⠀⠀⠀⠀⠀⢈⣤⣤⣠⡴⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣯⠀⢀⡀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠠⢀⡆⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⣣⣼⠀⠀⠀⠀⢸⣿⣿⣿⣿⠢⡉⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡧⠁⣹⣷⠀⠀⢀⢨⣿⣿⣿⣿⣷⡉⠂⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡶⢻⡟⠀⠀⢸⡎⠀⠀⠀⢻⣿⣿⡆⠀⠀⠶⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⡄⣿⠀⠀⠀⠇⠀⠀⠀⠘⣿⣟⢿⣶⡁⡦⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣿⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡍⢙⠆⠀⡀⠀⠀⠀⠀⠸⣿⡄⠁⠀⠀⠘⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡟⠑⠂⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡦⠀⠀⠀⠀⠀⠈⠓⠘⠒⢀⣀⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣻⣿⣿⣟⣀⣀⣀⣀⣀
⣿⣿⢋⣉⡿⠟⠿⠿⠿⠟⠻⠟⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠛⠛⠛⠙⠛⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⠴⠖⠒⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⣤⣤⣤⣤⠶⢚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠄⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠞⠉⠉⠢⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠲⣄⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣐⡲⢦⡀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠱⡈⣧⡏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠕⠀⠀⠀⠀⠀⢸⠗⠛⠓⠆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠐⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⠆⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠶⠾⣇⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠛⠒⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⣷⣶⠶⠤⠤⠤⣽⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠋⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⢯⠀⠀⠀⢀⡞⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠖⠋⠙⢦⡀⠀⠀⠀⠀⠀⠈⠳⠦⣼⣧⣤⠶⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠶⠶⠒⠚⠋⠁


⡿⠿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⢰⣿⣿⣆⣶⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠘⣻⣟⣻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⠛⠛⢩⣤⣤⣤⣤⣤⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠖⠶⢦⣴⠶⠾⣷⣤⠶⠶⣶⠛⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣀⡀⠀⠀⠛⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⢨⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣛⣿⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⣠⡴⠶⠟⠛⠛⠋⠉⠉⠉⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡶⠿⠟⢛⣿⡟⣉⣉⣉⢻⣯⡉⠉⠙⠛⠿⣷⣤⡀⠀⠀⢸⣇⠀⠀⠻⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡿⠛⠉⠁⠀⠀⠀⢸⡧⢠⣤⣨⣿⣿⣽⡇⠀⠀⠀⠀⠈⠙⣿⣆⠀⠛⢩⡇⠀⠀⠀⠉⠻⣏⠉⠉⠉⠉⢩⣿⠟⠁
⠀⠀⠀⠀⠀⣰⡾⠋⠉⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠈⠷⣤⣀⣀⣀⣤⠿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣠⠞⠁⠀⠀⠀⠀⠀⠹⣇⠀⠀⣰⡟⠁
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⡿
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣼⣿⣶⣦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⣄⠀⠀⠚⠉⠀⠈⢿⣷⠀⠀⠀⠀⠀⠈⣇⢸⡁
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣿⠀⣷⡀⢹⡆⠀⢀⣄⣀⣀⣾⡟⠀⠀⠀⠀⢀⣴⠟⠋⠛⢷⣆
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⣀⣤⣾⣷⣶⢾⣿⡟⠛⠋⠁⠀⠀⠀⠀⢠⡟⠀⡿⡇⠀⣿⠀⠈⠛⢻⣿⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠛⠉⠀⠀⠀⠀⢹⣧⠀⣴⠒⠶⠶⠶⠶⠋⢀⣼⢢⡇⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⣿⠀⣀
⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢷⡀⢦⣄⣀⣀⣴⠾⢅⣸⠃⣸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⣿⣷⣿⣿
⡀⠹⣧⡀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢦⣉⠩⣭⣦⡴⠞⣁⣴⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠿⠛⣹⡿⠁
⠻⢷⣬⣹⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠛⠶⠶⠶⠟⢫⡟⠈⢻⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⣷⣴⣾⠟⠁
⠀⢶⣬⡽⢿⣿⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢻⡆⠀⢰⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣤⣄⣀⣀⣀⣀⣀⣠⣿⣿⠟⠲⢦⣄⣠⡴⠶⢾⣧⡴⠖⠀⠀⠘⣿⣀⣸⣿⣤⣤⣤⣶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⣹⡿⣿⠀⠀⣀⣬⡏⢀⠀⢠⣿⣀⡀⠀⠀⠀⣿⠋⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⡀⠈⠁⢈⣷⣬⣭⣿⠋⢉⡿⠿⢶⣾⡟⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠘⣷⡀⣠⡾⢻⣤⢻⡉⡖⣻⠃⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠛⠋⠀⠀⢻⣸⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣸⣇⣀⣤⣤⣤⣤⣤⣤⣶⣶⣶⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣿⠋⠉⠉⠉⠉⠉⠉⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠦⣤⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡧⠴⠶⢶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣰⣀⣠⣤⡌⠙⡆⠀⠀⠀⠀⢠⠟⢠⣤⣀⣀⣇⣀⣼⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣽⣭⣥⣤⣤⣤⣤⣼⣷⣿⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢖⣶
⠀⠀⠀⠀⠀⠀⣿⡀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⢀⣠⣬⣿⣦⣝⣶⣦⣴⣶⣾⣿⣿⣶⣾⣿⣿⣿⣯⣤⣤⣤⣀⣀
⢀⠀⢀⡴⢋⣴⣶⣿⣿⣿⠿⠛⠉⠁⢠⡿⠋⠀⠀⠀⠈⠻⣧⠀⠈⠉⠉⠛⠛⠷⢶⣦⣤⣀⡀
⠙⠶⠋⣰⡿⣻⣿⣿⠟⠁⠀⠀⠀⠀⣿⡇⣾⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⣀⣀⣴⠟⣱⢿⣿⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⣠⠼⠁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢰⣿⠏⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠉⠉⠉⢻⡿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠻⣿⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠛⠋⠻⣦⣀⠀⠀⠀⣠⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡄⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⢈⣿⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⡐⡱⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⢴⣦⣈⠏⡴⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⢰⣿⠁⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠹⢿⣦⡀⠀⠀⠀⣠⣿⠋⠊
⠀⠀⠀⣸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣷⣶⣿⡿⣷⣷⣤⡄
⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣟⣾⣷⣯⣭⡶
⠀⠀⢠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢠⢊⠟⠿⣾⠏⠉
⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿
⠀⣀⣨⡍⠉⠉⣿⠋⠉⠉⠉⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠁
⠀⠻⣿⣯⣤⣤⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⢉⣡⣤⣤⡤⠤⠤⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢰⡏⠁⠀⡄⠀⢀⣀⡄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣤⣀⠀⠀⢰⠀⠈⣻⡆
⣀⣀⣀⣈⣻⣶⣶⣷⣚⣋⣉⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣏⣙⣛⣷⣾⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠻⠛⠉⠉⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠛⠂⠀⡀⠴⠆⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⣇
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡁⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠙⠦⣤⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡄⠀⢀⣀⣀⣀⠀⣤⣄⡀
⠀⠈⠉⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠙⣟⠁⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢰⣿⡆⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢀⠀⠀⢼⣿⣿⣿⣿⡄⢸⣿⣇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡏⠀⠀⠷⣄⣀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⣿⡟⠀⢸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠃⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠤⠶⠶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠈⠑⠦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⢱⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠜⠁⠀⠀⠀⠀⠠⠤⠂⣡⠶⣄⠀⠀⠀⠀⠀⠀⡖⠲⡄⢀⢀⡀⠈⠧⢤⡆
⠀⠀⠀⠀⠀⠀⠈⠦⣄⣀⠀⠀⠀⡤⠠⠀⠔⠁⠀⠇⠀⠘⠀⠀⡏⠉⡏⠈⠀⠀⠓⢸⡡⠎⠀⢠⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢎⣁⣀⡀⠀⢦⣀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⣵⠀⠀⠀⠀⠀⢱⠀⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢒⡶⢀⣀⣀⡱⢄⡀⠀⠀⠀⠀⠀⢰⠈⣿⠀⠀⠀⠀⠀⢰⠗⠒⠒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⣩⠗⠦⣄⡀⠀⠀⠑⠂⠀⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠉⠀⠈⠉⠉⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⡼⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣳⠀⠀⠀⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣏⢘⠄⠀⠀⠀⠀⢌⢙⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣀⣤⣄⣀⣠⣄⣠⣄⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠇⠀⠀⠀⠀⠀⠀⠧⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆
⠀⠀⢰⡉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣱
⠀⢀⠀⠉⠉⠉⠉⠀⠀⢀⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠁⢀
⠀⠣⠄⠒⣀⣤⡄⠀⠀⠉⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠙⠂⠀⠀⠀⢰⠀⠐⠂⠤⠃
⠀⠀⡴⠛⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠉⠓⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⠒⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⣀⠤⠤⣄⣠⠤⠚⠶⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡎⠀⠀⠀⠀⠈⢂⣀⣤⠞⠉⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠉⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣏
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠋⠉⠁
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡄
⠀⢀⣠⣼⣿⣿⣿⣿⣿⠿⠿⠇
⢴⣿⣿⣿⣿⣿⡿⠿⠉
⢸⣿⣿⠟⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢑⣰⠇
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠂⡀⢀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠠⡀⠈⢧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⡞⠑⠠⣀
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⣰⡁⠀⡉⡆⠀⢘⠀⠁⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⣀⠀⣢⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠿⠟⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠴⠄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠉⠁⠉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⡀⠀⠀⠀⠀⠀⢀⡸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠒⠐⢶⠺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠘⢂⣴⠀⠀⠀⠀⠀⣀⣠⢄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⠀⠘⠣⠀⠀⠀⠀⡶⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠙⠃⠀⠀⠀⠀⠀⢴⣆⠠⠦⠄⣀⡀⠈⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠆⠀⠀⠀⠀⢸⠟⢿⣶⡄⢠⠤⣶⠄⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠇⠘⣄⠀⠀⠀⠀⠳⡄⢸⡀⠀⣠⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠰⡶⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⡛⠛⠛⠻⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣀⣀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡗⠲⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠠⢤⠄⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠚⠁
⠀⠀⠀⢀⠜⠀⠙⢦⡀⣀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡰⠋⠀⢀⠴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡜⠁⠀⣴⠎⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⢀⣠⣼⣇⢀⣠⡤⠀⠀⠀⠀⡆
⡆⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠠⠈⠀⠉⠙⣿⣿⣷⠆⠀⠀⠀⠁
⠷⠠⠤⠠⠤⠤⠤⠀⠀⠰⡎⠉⠉⠉⠀⠀⠈⠁⠀⠼⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠠⠤⢤⡤⠤⠁⠸⠤⣤⠤⠤⠤⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣾⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⣿⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡌⠓⣤⡀⠀⠀⠀⠀⠀⢰⣦⣀⣠⣾⣿⣿⠃⠀⢸⠓⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡔⠉⠙⢠⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⢶⣶⣎⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠋⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣰⣿⣿⠟⢻⣷⠀⠀⠀⠀⣼⡟⠻⣿⣿⣇⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⡏⠀⠀⢻⡇⠀⠀⢸⡟⠀⠀⢹⣿⣿⠟⠛⠋⠉⠉⠉⠉⠙⠻⢿⣷⣤⡀
⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠸⣧⠀⠀⢸⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⣠⣿⣶⣶⣿⣅⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⡟⠁⠀⠀⠈⢻⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⣠⣾⠿⠿⠿⢿⣿⣿⡆⠀⠀⠀⠀⠈⠻⢷⣦⣤⡾⠟⠁⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣆⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠈⠙⠿⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⠀⠿⠋⠁⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⢸⣿
⠀⣿⣇⣠⣤⣤⣿⠃⠀⠀⠀⠀⣰⣷⣠⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣄⣼⣆⠀⠀⠀⠀⠈⣿⣤⣤⣄⣸⣿
⣰⣿⠛⠉⠉⠉⢻⣷⠀⠀⠀⢠⣿⡿⠋⢀⣀⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⠿⣶⣶⣤⣄⡀⠉⢿⣿⡆⠀⠀⠀⣼⡟⠉⠉⠉⠹⣿
⣿⣧⠀⠀⠀⠀⠀⣿⠃⠀⠀⣸⣿⣷⣴⡿⠛⠉⠀⠀⠐⠲⢖⣛⣛⡒⠖⠂⠀⠀⠉⠛⢿⣶⣾⣿⣧⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸
⠘⣿⣦⣀⣀⣠⣾⠟⠀⠀⢀⣿⣿⣿⠋⣀⣤⣤⣤⣀⡀⠀⠉⠉⠉⠉⠀⢀⣀⣤⣤⣄⣀⠈⣿⣿⣿⡀⠀⠀⠻⣷⣄⣀⣀⣴⣿
⠀⠈⠉⢻⣿⡏⠀⠀⠀⠀⣸⣿⣿⡇⠈⠉⠉⠉⡙⠻⣿⣶⣤⠂⠘⢦⣶⣿⠿⢋⠉⠉⠉⠁⢸⣿⣿⣇⠀⠀⠀⠀⢹⣿⡟⠉⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⠙⣿⣷⠀⠀⠶⠿⠻⢿⣿⣿⣷⠀⠀⣾⣿⣿⡿⠿⠿⠶⠀⠀⣾⣿⠏⢿⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢸⣿⡅⠀⠀⠀⢸⡇⢠⣿⠛⠁⠀⠀⠈⠙⠛⠉⠀⣿⠀⠀⣿⠀⠉⠛⠋⠁⠀⠀⠈⠙⣿⣄⢸⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣷⣿⡇⠀⠀⠀⠀⠀⠀⢀⡴⠶⣿⠀⠀⣽⠶⢦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⢿⣇⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⣰⣿⢀⣴⠇⠀⠀⠸⣦⡀⣿⣆⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣦⣄⠐⠾⠛⠉⠓⠹⣇⡀⢀⣰⠏⠚⠋⠛⠷⠂⣠⣴⣾⣿⣿⣿⡇⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢻⣷⣤⡴⠾⢿⣿⣿⣿⡉⠙⠧⠀⡀⠀⠀⠀⠈⠻⠟⠃⠀⠀⠀⢀⠀⠰⠛⠉⣿⣿⣿⡿⠷⢶⣤⣾⡟
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⣿⣿⡇⠀⠀⢰⣇⣠⣴⠶⠶⠶⠶⠶⠶⣦⣄⣸⡆⠀⠀⢸⣿⣿⠁⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢀⣾⣿⣿⡀⠀⢼⡟⠉⢀⣤⣶⣶⣶⣶⣤⡀⠉⢻⣧⠀⢀⣿⣿⣿⡀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣽⣿⣶⣶⣶⠿⠟⠀⢻⣷⡀⠘⠷⠖⠻⠛⠉⠉⠉⠉⠛⠟⠳⠾⠃⢀⣾⡟⠀⠻⠿⣶⣤⣶⣿⣿⡀
⠀⠀⠀⣠⣶⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⢈⣻⣷⣶⣄
⠀⠀⠀⢻⣿⣿⠿⠿⠿⣷⣶⣶⣤⣤⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⣤⣤⣶⣶⣾⡿⠿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⢹⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠻⢿⣶⣤⣤⣤⣤⣶⡿⠟⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣀⣀⣀⠀⠈⠉⠉⠉⠉⠁⠀⣀⣀⣀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣶⣾⣿⠿⠿⠿⠿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣿⠿⠿⠿⠿⣿⣷⣶⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣠⠄⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠍⣶⠐⠆⠀⠀⠘⡄
⡔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⡅
⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢿⠆⠂⠀⢴⠀⠃
⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢈⠣⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠈⠤⡀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠄⠀⡀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡄⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠑⠒⠒⠀⠋⠭⠀⡁⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠉⢉⡓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣎⠉⠙⢻⠀⠙⣇⠀⠀⠑⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠳⣄⠈⢻⠙⠋⠀⠀⢸⡄⠀⢀⡀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠘⠰⣄⣸⠖⠛⠀⠀⢀⡔⠋⠀⣤⣈⠫⠑⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠓⢦⡀⢀⣠⡤⠶⠶⠯⣅⠀⢠⠀⠀⡜⢀⠞⠙⠿⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀⠀⢸⡉⠑⠒⠢⣄⣿⣋⡁⠀⠀⠀⠀⢸⠀⠀⢀⡾⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠺⡍⠉⢉⣙⣓⣦⡴⠋⠀⠀⣸⣓⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠙⠚⠁⠀⢀⡈⠉⠀⠀⠀⡿⠛⢉⣁⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠠⢀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⡼⢭⡟⠂⠀⠀⠀⠞⠋⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⣠⡎⠑⣾⡇⠀⠀⠀⣶⠠⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⢈⣿⣿⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠫⣄⣸⠋⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠠⠚⠉⢳⡠⠞⠹⣀⠟⡿⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠏⡼⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠁⠀⠀⠀⠀⠀⠈⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⡤⠴⠺⡇⠀⠀⠀⠀⠀⠀⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡰⣎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⣖⡛⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠖⠚⠋⠁⢀⡠⠴⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⠜⠻⡄⣠⠞⡉⠉⠉⠉⠁⠀⠀⢀⣀⡤⠖⠊⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⢡⡖⠒⠺⢥⡀⢰⡀⢀⣀⡤⠔⠚⠉⣁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠖⠛⠫⠽⠆⢹⠆⠉⠉⠀⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢈⡿⠥⢛⣀⠀⠘⠠⠤⠤⠔⠒⠚⣿⣦⡀⠀⠀⠀⢠⠆⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠲⠤⣀⡘⠦⠖⠋⠈⣇⣀⠤⠤⠤⠤⡤⣤⡸⠿⠿⠦⣄⣠⠋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⢆⠀⡞⢩⠟⠒⠲⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣁⣀⡤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠉⠀⠀⠈⢻⡁⡎⣀⣀⣠⣤⣤⠤⠤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⢻⢳⠒⠤⣤⣤⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠘⠋⠉⠈⠀⠀⢀⠠⢽⡦⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠴⠴⠄⠤⠄⠛⠛⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣭⣽⣶⡿⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣄⢠⡄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⣼⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⣀⣀⣠⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠉⠉⣩⣀⣤⣾⣿⢿⡋⠁⠈⠁⠀⠀⠉⠀⠈⢹⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⣿⣿⠋⠩⠀⠀⠀⠀⢴⣤⣿⣥⣔⡶⡄⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠈⣷⡀⠀⠀⠀⠐⣴⣿⣿⣿⣿⣿⣿⣞⡆⠀⠀⠀⢠⡄⠀⠲⠿⠿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣠⢫⣢⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⢿⣛⢿⣵⡀⠠⢤⠙⢿⣿⣿⣿⠿⢿⢏⡀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⡿⡆⠀⠀⠀⠈⡻⠿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⠦⠀⠀⠤⠀⠨⣵⡭⢿⣟⠷⠆⠀⠀⠀⠀⠉⠀⠀⢦⠀⠀⠀⠀⠛⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣤⣄⡉⠀⠐⠁⠀⣀⣀⠀⠀⡀⠀⠀⠀⠘⡂⠰⠆⠀⠀⠀⠀⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿⢁⠀⣤⣛⠀⠉⢹⣧⠘⠃⠀⠀⠙⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣓⣯⣼⣿⣿⡇⢲⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢿⣿⣧⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠛⠉⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣿⣿⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠘⢿⣿⡷⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣨⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠉⢾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⡏⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣀⣀⣄⣉⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣽⣷⠶⣦⣄⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠉⠉⣙⣿⣷⣶⣦⣭⣁⠀⠈⠙⠯⠙⢦⡀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠿⣿⣿⣿⣶⣤⣀⠀⠀⠙⣶⡿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠿⢷⣦⣀
⠀⠀⠀⠀⠀⣀⣠⣤⠿⠏⠉⠉⠉⠙⠛⠛⠿⠛⢿⣷⣤⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⣠⢝⣛⣡⣤⣶⣶⣶⣶⣶⣶⣶⣤⣄⠀⠈⢻⣿⣦⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠼⠛⠉⡽⠟⠋⠟⠉⣚⣯⣭⣭⣟⣿⣿⣿⣦⡀⠙⣿⣷⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⣀⣀⣿⡟⠛⠛⠛⠛⠂⠀⠀⠉⠉⢉⣿⣿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⣷⣦⣀⢻⣿⡀
⠀⠀⠀⢀⡼⢟⣋⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄
⠀⠀⣰⠿⠛⢻⠟⠛⣿⠿⠛⠻⠿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣄⣀⣀
⠀⣀⣴⣾⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣷⣦⣀
⣰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢀⡤⠶⠶⣶⣄⡀⠀⠀⠀⠀⢀⣠⣶⠶⠶⢤⡀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⠀⠀⠠⣤⣶⣾⣿⣿⣿⠃⠘⣿⣿⣿⣷⣶⣤⠄⠀⠀⣿⣿⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠙⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⡿⠁⣼⣿⣿⠀⠀⠀⠀⠀⠠⠼⠛⠏⠀⠀⠸⠛⠧⠄⠀⠀⠀⠀⠀⣿⣿⣧⠈⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⠿⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣧⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⣼⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣤⣹⣿⣿⣷⠀⣀⣀⣤⠟⠉⠁⠀⠀⠈⠉⠻⣤⣀⣀⠀⣾⣿⣿⣏⣤⡀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⢿⣼⠀⣿⣿⣿⣿⡀⠈⠛⢛⣷⣾⣷⣀⣀⣾⣷⣾⡛⠛⠁⢀⣿⣿⣿⣿⠀⣧⡿⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣅⠸⡄⠿⢀⣿⣿⣿⣿⣧⣶⡿⠿⣿⣟⣛⣩⣍⣛⣻⣿⠿⢿⣶⣼⣿⣿⣿⣿⠀⠿⢀⠇⣨⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠶⠛⠶⠟⢻⣿⣿⣿⣿⣿⠀⠰⣦⣬⣉⣉⣉⣉⣥⣴⠆⠀⣿⣿⣿⣿⣿⡏⠳⠶⠟⠶⣿⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡝⡆⣀⡤⠶⠛⣿⣿⣿⣿⣿⣆⠀⠉⠀⢸⣿⣿⡇⠀⠉⠀⣰⣿⣿⣿⣿⣿⠛⠲⢦⣀⢰⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡿⣅⠀⢀⡤⠞⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡟⠳⢦⣀⠀⣩⢿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣽⣿⡋⢀⣴⠞⠉⢙⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡋⠉⠳⣦⡀⢙⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣎⣷⣟⡁⠀⣴⠟⠁⣰⡏⠉⣿⢿⣿⣿⡿⢿⠉⢹⡆⠈⠻⣆⠀⢈⣻⣾⣱⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣯⣽⣾⣯⣄⢠⡟⠀⢸⡇⠀⢻⡟⠀⢸⡇⠀⢻⡄⣠⣼⣷⣫⣽⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣮⣿⣛⡶⣼⣥⣤⣸⣇⣤⣬⣧⣶⣛⣿⣵⣾⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣶⣷⣿⣿⣶⣶⣿⣿⣿⡿⠿⠿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣯⣽⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣾⣤⣦⣤⡀
⠀⠀⠀⠀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⢻⣿⣿⣿⣿⠁⠈⠈⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⣿⣿⡟⠐⢤⡀⣤⠤⠂⠀⠛⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠘⣿⡇⠀⠘⠀⠹⠂⠀⠀⠀⠈⢹⡿⣻⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠤⠂⠙⡄⠀⠀⠀⣸⣱⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢀⠾⠶⢻⣕⣶⠖⠀⠀⠀⣠⣿⡿⠛⠋⠁
⠀⠀⠀⠀⡠⠆⢙⡿⠶⣼⣿⡷⠃⠀⣠⣾⣿⣿⣿⣶⣶⣶⠒⠂
⠀⠀⢀⡴⠓⣲⣾⣷⣤⠌⣿⣥⣴⢎⣙⠋⣿⢿⡿⠟⢁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣰⣿⣿⣿⣾⡟⠀⡧⠈⠉⠀⠙⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠊⡹⠀⣠⠊⠇
⠀⢠⣾⣿⡿⣿⠿⠿⠁⠀⡼⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠃⣴⠁⡎⠀⡠⢲⠂
⣴⣿⡿⠋⠙⢤⡀⠀⠤⠰⣇⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠸⣷⠇⡘⣠⠎⡰⠃
⠉⠉⠠⠀⠀⢸⠷⣀⣀⣤⣿⠀⠀⠀⠐⣦⣄⠀⠈⠉⠒⠒⠀⠀⠘⣧⠀⠀⠀⠰⠀⡀⢄⠀⢠⠃⠀⢀⠀⠛⢁⠌⣀⡠
⠀⠀⠀⢢⡀⠘⠀⠀⡉⠺⢿⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢁⠱⡄⠑⠎⠀⣰⣄⠒⠀⠀⢉⡩⠖⠋

⠀⠀⠀⠸⡄⢰⠀⠀⢨⣁⣾⡆⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⡟⣿⡀⠀⠀⠀⢆⠸⡀⠀⠀⢹⡿⠀⠀⣰⠃
⠀⠀⠀⠀⡇⢸⠀⡄⢄⡈⠀⢧⣀⣀⡀⠤⢞⡤⠀⠀⠀⠀⠀⢀⡼⠀⠘⣧⠀⠀⠔⠛⣲⢧⠀⢀⠘⠁⢀⡜⠁
⠀⠀⠀⠀⡇⠸⠀⠘⠂⠀⠀⢸⠧⠤⠤⠀⠁⠀⠐⠂⢐⠶⠾⣿⠃⠀⠀⠘⡇⠀⠀⠀⠀⠄⠀⢈⣷⡖⠉
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⣾⣷⠶⠂⠀⠀⠀⠀⠀⠀⠠⠚⣿⡀⠀⠀⠀⠘⢦⣤⣠⣬⣷⡽⠟⠉
⠀⠀⠀⠀⠀⠈⠳⠤⣀⡴⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⢀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨⢀⠌⠃⠀⠀⠀⠁⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⡀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢄⠁⠀⠠⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠐⠀⢀⠔⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠀⠁⠀⠀⠂⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⠋⠀⡠⠊⠈⢐⡄⠀⢄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⠀⠀⡀⠀⢸⠄⠈⠜⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⠀⠀⠠⠄⠀⠀⠀⠄⠿⢀⣆⢠⠎⠀⠀⠀⡜⠀⠀⠀⡅⡀
⠀⠐⠀⠀⢃⠀⠈⢆⠀⠀⢀⠀⠀⡟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⡀⠀⢀⡌⠀⠀⢀⡜⠀⣸⠂⠐⠐⠒⠂
⠀⠀⠀⠀⠀⠐⠤⠸⠦⣀⠜⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠉⠉⠁⠑⠤⠤⠋⠔⠚⢁⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣶⣾⣿⣼⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⢠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠫⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠿⣟⠻⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⡄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠉⠈⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠲⠦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡤⠤⣄⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠞⠳⢦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⡞⠉⠀⠀⠀⠀⠈⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠉⠙⠓

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣼⡇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣿⠀⣠⡀⠀⠀⢠⡄⡆⢀⡄
⠸⡇⢿⡄⢸⣿⡄⠀⠀⠀⠀⠀⠸⡟⣿⢸⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⡏⢠⣿⡇⠀⠀⢸⢷⡇⢸⣿
⠰⣿⢾⣇⢸⡟⣿⡀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡿⠀⡾⢻⡇⠀⠀⠈⠘⠃⠟⠘
⠀⠙⠘⠟⠀⠃⠀⠀⣀⢠⡄⢀⠀⠀⠀⠀⠀⠐⣀⠤⠔⠗⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⠁⣼⡇⠀⠀⠀⠀⠊⠁⠄⠀⠀⣀⠠⠄⠒⠠⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⣧⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠇⡿⢠⡟⡧⠀⠀⠀⢀⡀⠀⣠⠐⣉⠀⠀⠉⠀⠂⠀⣀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠀⠈⡏⢿⠀⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢅⠚⠰⠈⠀⠲⡄⠀⠀⠈⠒⠀⡀⠀⠀⠀⢀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠈⠛⣆⠟⠀⠀⠀⢀⣀⡀⠙⠠⡀⠈⠁⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⠄⠁⢀⠀⠀⠀⢉⣭⣍⡉⠀⠈⠢⠀⠐⡄⠈⢁⠛⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠑⠄⣀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠱⡀⠀⡀⠀⢣⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢠⠯⡁⠈⣄⡈⠨⠀⡇⠀⠀⠀⢠⣿⣷⣦⣤⣉⣉⣩⡽⠀⠀⠀⢀⢠⠔⡠⠸⠀⠇⠀⠀⡇⠀⢸⠄
⠀⠀⠀⠀⠀⠋⠐⣈⠔⠈⠀⣠⠀⢻⡀⠀⠀⣿⣿⠿⢿⡿⣿⣿⣿⡇⠀⠀⠀⠈⠚⠠⠁⠨⡇⡆⠀⠀⡇⠀⠋⠂
⠀⠀⠀⠀⠀⠑⢅⠁⠀⠀⢴⡃⠀⠀⠳⡀⠀⠻⣿⣤⣀⣀⣼⣿⠟⠀⠀⠀⠀⠀⠀⡠⠒⠢⠈⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠣⠐⠈⠙⢄⠀⠀⠈⠢⡀⠈⠉⠙⠉⠉⠀⠀⠀⠀⣀⣀⡴⢄⣀⣀⣀⡤⠓⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⢳⠒⠒⠈⠉⠒⠲⠤⠀⠀⣀⠖⠚⠛⠩⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠂⠈⠉⡋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠤⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠰⠉⠉⠀⠉⢴⠂⠉⠀⠈⠀⠅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⠀⣀⠀⠀⢀⣀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣀⣀⣀⣀⣀⣀⣀⣉⣀⡸

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠉⠉⠉⠛⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠭⠅⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠶⣄⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⡴⠋⢀⠠⠀⠀⠀⣤⠞⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡾⠀⡐⠁⠀⠀⠠⢴⠃⣰⠃⠀⠀⠀⠀⡀⠤⠀⠒⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⡇⡘⠀⠀⠀⠀⡇⢻⢰⠇⠀⢀⠔⠊⠁⠀⠀⠀⣀⣠⠤⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠤⠀⣀⢸⡀
⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠁⠈⣿⡠⠊⠁⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣄⣀⣀⣠⡤⠤⠤⢤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠐⠀⢸⡀⠀⡠⠐⠀⣹⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠀⢹⡆⢸⠃
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⣷⠉⠀⠀⠀⣿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡾
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡀⠀⠀⣿⣆⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣼⣰⠃
⣴⢿⠃⠀⠀⣠⠃⠀⠀⡰⠀⠀⢠⡇⠙⣦⠴⠦⢼⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠊⠁⠀⢻⡃
⣇⠈⠙⠛⠋⠁⢀⡠⠊⠀⠀⢀⡞⠀⢸⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠘⢷⡀⠀⠀⠉⠀⠀⠀⢀⣠⠟⠀⠀⠸⣆⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⣿
⠀⠀⠉⠓⠲⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣤⣶⣶⣿⣿⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠿⡿⢛⡿⠁⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠊⠉⠀⢀⣠⡞⡉⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡤⠦⣄⣀⣀⣀⣀⣀⣀⣤⠤⡶⠚⢻⡉⠉⢻⠓⠒⠒⠒⠒⠒⠉⡉⡙⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⢀⡙⠉⠉⠉⠃⠠⠤⠚⠀⠀⠈⣧⣀⣼⠤⠤⠤⠤⠶⠤⣀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢬⣅⣾⠋⠀⣀⠀⡄⠐⠒⠒⠒⡆⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠳⠤⠞⠘⠗⢻⡄⠀⠀⡰⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣤⠤⠤⣄⠀⢠⣭⠏⠉⠉⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠒⠐⣶⠈⠉⢹⡏⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⢸⣇⢀⡀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠏⠉⠙⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠃⠉⠀⠘⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣂⠀⠀⠀⢀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡗⠶⠖⣿⣬⣿⡄⠀⠀⠀⠀⠀⠀⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⠿⠃⠐⠒⠒⠀⠿⣿⣷⣤⣀⣀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣤⣴⣶⣾⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⣿⣿⣿⡆⢰⣿⡏⠉⠛⠻⠿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠉⢙⣋⣛⡛⠻⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⠃⠀⠀⠀⣀⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣦⣄⠀⢤⠀⠀⠀⠀⠀⠘⢋⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠚⠏⢿⣿⠏⠟⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠐⠚⠛⠒⠂⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣄⠀⣠⡄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣘⣦⣾⣷⣾⣿⣿⣿⣿⣿⣿⣧⣾⣃⣼⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣸⣖⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣲⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⣶⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⣿⣿⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⣿⡿⠁⠀⣀⣤⣤⣤⣄⣀⠀⠀⣀⣤⠖⠛⠉⠉⠉⠙⠻⣶⣄⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⣠⣾⣿⠟⠋⠁⠀⠀⠉⠉⠉⠀⠀⣀⣀⣀⣀⠀⠀⢹⣿⣧⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣿⣿⣿⠁⣼⣿⡿⠃⠀⣠⣴⠾⠛⠳⢶⣄⣠⠞⠉⠁⠀⠉⠻⣆⠀⠙⠛⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⡇⠀⠛⠋⠀⢠⡾⠋⠀⢀⣀⠀⠀⢻⡏⠀⠀⢠⣤⡀⠀⢻⡆⠀⠀⠀⠀⠀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠿⠃⠀⠈⣿⡀⠀⠈⠉⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣼⠛⢿⣦⣄⡀⢀⣠⣾⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⣾⣿⠁⠀⠀⠹⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⡿⠋⣏⠀⠀⠀⠀⠹⣿⡙⠳⠦⠀⠀⠀⠀⠀⠀⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢀⣠⣿⠀⠀⠀⠀⠀⢻⣏⠳⣄⠀⠀⠀⠀⢀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⡀⠀⠀⠀⠀⣼⡿⠀⣿⠀⠀⢀⠂⠀⠐⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⢷⣤⣀⣤⣾⡿⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠀⠀⠨⢙⣟⢻⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣨⣷⣾⣧⠀⠀⠀⠐⠀⠂⠀⢀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣿⣧⠀⠀⠠⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⠏⠂⠀⠀⠀⠀⠀⢀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⡄⠀⢀⠀⠂⠀⠀⠉⠀⠈⠙⠛⠳⣦⠀⠀⠀⠤⠄⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣥⣀⠁⠄⡀⠀⠀⠀⠀⠄⠀⠘⠄⠸⣷⢀⣠⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⠷⠶⠶⠾⠶⠿⠿⠟⠛⠋⠁


⠀⠀⡀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠿⠿⠟⠛⠛⠻⠿⠿⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠉⠛⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣶⣶⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⢀⣠⡾⣿⡄⠀⡼⠟⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠙⠻⢧⠀⠀⣤⣄⠀⠀⠀⠀⠀⠸
⣿⠀⠀⢀⣴⠟⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠙⢷⣦⡀
⣿⡆⢠⣿⠁⠀⠀⠀⠸⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡏⠀⠀⠈⠻⣦⡀⢰
⢹⡇⠘⣿⡀⠀⠀⠀⣰⡿⠀⢸⣟⠻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⣿⠇⠀⣾⡇⠀⠀⠀⠀⢹⡇⢸
⠸⣷⠀⠈⠻⠶⣶⠾⠛⠁⠀⠈⣿⡀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣤⣤⣤⣴⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⢰⡿⠀⠀⠘⣷⣄⣀⣀⣠⡿⠃⣾
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠉⠛⠛⠉⠀⢰⡟
⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟⠛⠻⢶⣦⡶⠟⠛⠻⣦⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢶⣶⣤⣶⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⢛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣯⣍⣈⠙⠻⠳⣄⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠿⠿⢿⣿⢦⣄⠈⠻⡉⠉⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣩⣥⣶⣶⣶⣦⣤⣀⠙⢷⣄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣫⡬⠥⠿⠟⠛⠻⠿⣷⣬⡿⠷⠗⠒⠒⠒⠲⠶⠶⣤⣄⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⣭⣤⣶⣶⣶⣶⣶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠛⠉⠛⠉⢉⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⠶⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠠⠶⢦⣄⣀⡀⣀⣀⡴⠶⠤⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⣀⣀⣀⣀⣀⣠⡞⣹⣿⠀⠀⠚⠛⣻⠿⠁⢹⢿⡛⠛⠂⠀⢸⣿⠹⣦⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⢻⡄⢿⣿⣄⠀⠀⠀⣠⡄⠀⠀⣄⠀⠀⠀⢀⣾⣿⠃⣿⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⠲⣿⣿⡇⠰⢶⣅⣤⠀⣠⣌⣷⠶⠀⣿⣿⡗⢢⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⢹⡘⢣⣿⣿⣷⣴⠾⠿⣿⣭⣽⡿⠿⢶⣼⣿⣿⣇⠛⡸⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⢉⡩⢿⣿⣿⣧⠀⠛⠲⣶⡶⠚⠃⣸⣿⣿⣿⢭⣉⠉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣍⣠⠞⢻⣿⣿⣷⣶⣾⣿⣿⣶⣾⣿⣿⣿⠛⢤⣈⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⣦⡞⠁⣰⠋⡻⢿⣿⣿⣿⣿⢻⡋⢧⡀⠑⣤⢟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⢥⣰⡃⢸⠉⢻⠁⢻⠀⣧⣠⢷⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣾⣾⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠛⠋⠉⠁




⣿⢿⠿⡿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣛⣻⣿⣿⣛⣻⣿⣿⣿⣿⢻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⡀⠀⠀⠰⠓⠀⠀⠀⠀⠀⢿⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡔⢒⣀⣶⡆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠦⣈⡿⠟⠀⠀⠀⢀⠜⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠀⣠⣤⣶⠁⠀⠀⣉⢩⠋⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠠⡎⢀⡤⠖⠁⠈⡄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠤⠈⠙⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢁⠀⠀⠀⠀⣄⣀⠀⢀⡀⠄⠀⠁⠀⢈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⢸⡀⡠⢶⣶⣌⣁⠉⠄⠀⠒⠀⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⡠⢀⣀⣸⣿⡟⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠠⠤⠤⠤⠀⠀⠒⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢨⠉⠉⠉⠉⠉⠁⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡆⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⡇⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⡄⠀⣴⣶⡝⠋⡇⠀⠀⠀⠀⠀⢀⠿⣶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⠤⠟⠛⠛⠛⠁⠀⠀⠀⠠⠖⢅⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠗

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⢿⠿⣉⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⢛⣿⣛⣛⢛⣿⣿⡛⣛⢛⣛⢛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠀⠐⠢⠔⠂⠉⠁⠀⠉⠁⣰⣧⡤⠤⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠋⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡀⡇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠀⣼⠀⠀⡴⠚⠃⠀⠀⠀⠸⠖⠦⠀⢁⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡊⠀⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠊⡽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠈⠉⠀⠀⠀⠊⠀⢡⠀⠀⠀⠀⠴⠉⠱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠐⢁⠌⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣿⠀⡄⠀⠀⠀⢠⣷⠀⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠁⢠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡎⠁⠀⠀⠄⠀⠒⠒⡤⠀⠒⢣⡶⢒⠒⣦⠁⠠⢅⠀⠻⣀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⢀⣀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠅⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠑⣦⣛⣤⣀⡠⠤⠔⠊⠁⠀⠀⠀⠀⣀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢂⠀⡅⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⡁⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛⠩⡀⠧⡀⠀⠀⠘⠈⠛⠦⣄⠀⣀⣀⡀⠀⠀⠀⢀⡀⠔⢁⠃⠀⠀⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⠀⠀⠛⠀⠈⠂⠤⢀⣀⡀⠀⠈⠣⡤⢀⠤⣤⠛⠉⣀⠠⣔⠁⠀⠀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠶⣄⠀⠱⠄⠤⢁⡴⠋⠀⠀⠈⠆⢠⠞
⠀⠀⠀⠀⠀⠀⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠈⠓⠠⠤⢖⠁⠀⠀⠀⠀⠀⠘⠁⠱⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢀⠛⢆⠀⢿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠐⠁⠀⠀⠡⡐⠗
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠢⣀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘
⠀⠀⠀⠀⠀⠀⢠⠉⠒⠄⣀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠁⢢⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⡰⠇⢈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡎⠉⠑⠆⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠉⠒⠒⠐⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠠⠐⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⢤⠔⠂⠁⠀⠉⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢀⡀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⠶⣷⠄⠀⠀⠀⠀⠀⠀⡠⠞⠀⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠨⣧⠀⠀⠀⣠⠔⠉⣀⡰⠏⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠐⣤⣄⡜⠑⣳⢞⠁⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⡀⠤⠍⠀⠨⠏⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣜⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠋⠉⠙⠛⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠍⠡⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⢤⡀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⠀⠀⡀⢀⡽⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢠⠏⠀⡔⠈⠀⠀⢀⣰⠏⢠⠏⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⣿⠀⠎⠀⠀⠀⢠⠁⣿⢠⡏⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⣀⣀⣁⣢⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⢿⡘⠀⠀⠀⠀⠸⠀⠻⣾⠀⡠⠔⠁⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢼⡆
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡄⠀⣿⠊⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⣀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⣒⡛⠛⠛⠓⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡇⠀⢻⡀⡠⠐⠉⠀⢸⠇⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠈⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡟
⠀⣀⡀⠀⠀⠀⢸⠇⠀⠀⢠⠁⠀⠀⡿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣟⡼⠁
⡼⢻⡃⠀⠀⣠⠎⠀⠀⢀⠆⠀⠀⢸⠇⠘⡶⠖⠒⠶⢧⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠋⠁⠀⠀⢻⡁
⣧⠀⠉⠛⠋⠁⠀⡀⠔⠁⠀⠀⢠⡟⠀⢸⠁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠘⢷⡀⠀⠀⠉⠁⠀⠀⠀⢀⡴⠋⠀⠀⠸⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⡇
⠀⠀⠙⠓⠶⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⢿⡟⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠥⠤⠤⠒⠉⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣾⠛⠖⣶⠤⠤⠤⣤⣤⡤⠔⠒⠓⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠉⠛⠒⠒⠚⠛⠛⠛⠉⢉⡀⠀⠀⢹⡄⠀⣇⠀⠀⣀⣀⣀⣀⠀⠸⢈⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡂⠤⢀⡶⠀⠀⠀⠀⠊⠉⠒⠀⢈⠀⠀⠀⢿⠒⠋⠉⠉⠉⠉⠉⠈⠙⠢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⣶⡃⢀⡴⢳⣀⣥⠄⡀⢀⠎⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠈⠉⠋⠀⠈⠡⢼⡁⠈⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠛⠓⠦⣄⠲⡖⠃⠀⢀⣀⣀⣀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠠⣤⡤⠄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⢼⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠉⠉⢹⡏⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⢸⡇⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠉⡄⠄⠠⢅⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠄⠐⡀⠀⠀⠖⠒⢄⠘⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢂⢨⠛⢄⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⢀⠀⠐⠠⠀⠀⠁⣈⠀⢀⠜⠀⠹⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠓⠞⠒⠄⢀⠑⠤⠄⠁⠀⠀⠀⢠⡐⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠤⠡⠄⠀⠑⠤⣀⣀⣀⡠⢐⠕⢠⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⢃⠀⠀⠀⠀⠀⠈⠒⠠⣄⠁⣭⠔⠋⠀⠃⢠⠀⢸⠀⠀⠀⠞⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠣⣄⠀⠀⠀⢀⢴⡁⠄⠉⠂⢶⢀⡀⡀⠘⣀⠀⠢⡀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁⠀⠑⢤⠀⡀⠀⠀⠀⠀⠀⠈⡄⠈⢳⡀⠀⠳⢄⡀⠀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠆⢃⠀⡘⠑⢄⠆⠀⠀⠀⠀⠀⢸⡄⠑⠈⠲⠤⠀⠈⠩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠆⣀⠄⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⠀⠀⠀⠀⢀⠂⡠⠉⢄⠀⡀⠀⠀⠀⠀⢀⡀⢼⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠒⠂⠉⠀⠀⠀⢀⠍⠈⠁⠀⠈⠉⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⠒⠒⠲⠒⠒⢺⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⣸⢠⡀⠐⡶⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠤⠤⠤⠊⠉⠒⠀⠀⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣎⢱⡀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⣁⠀⠀⠐⡉⠀⠀⠈⠁⠒⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⢄⠤⢃⣀⡄⠀⠀⠀⠀⠀⢑⠠⢀⠀⢀⠖⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣩⠁⠀⢸⡈⢧⡀⠀⢀⡰⠀⠐⠃⢀⠕⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠃⠀⠀⠀⡇⢇⡱⢦⡀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠘⣴⣍⠃⠀⠀⠀⠀⠸⡈⠓⢆⡈⠓⠠⢀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⢀⠃⣅⠀⠀⠀⠀⠀⠀⢣⡈⠢⡀⠀⠀⠀⠀⢱⠀⠀⠉⠒⢧⣠⠁⠑⠒⠂⠤⠄⠠⠤⠗
⢀⠄⠒⠉⣀⠃⠸⠒⠒⠀⠀⠠⠎⠈⠢⢈⠒⢄⠀⠀⠀⢣⠀⠀⠀⡧⠤⠭⡁⠐⠂⠂⠀⠀⠀⠀⢀⠄⢀⠂⣄
⢷⣀⡠⠂⠦⢄⠦⢀⡀⠀⠀⠀⠀⠀⠀⠘⠱⢄⠉⠢⣀⠀⢣⡀⡐⣠⠤⠀⠟⠀⠀⠀⣀⠀⡀⠀⠘⢠⣾⣿⣿⡀⠀⡀
⠀⠉⠀⠀⠀⠀⠀⠀⠉⢲⢄⠀⠄⠀⠀⠀⠀⢀⠉⠢⡀⠑⠤⡑⢶⡈⡄⠀⠀⠀⣠⠚⣤⠟⠙⢆⠀⠻⠿⠿⠛⠜⠋⡜⢴⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠘⡒⢄⠈⠑⠦⣉⠒⠂⠚⠒⠋⠀⠀⠀⠈⣆⠔⡄⠀⡀⢆⠐⠷⣿⢆
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⡀⠀⡔⢷⢯⠞⡤⠑⠢⡀⠀⠙⠲⢤⣀⠀⢰⠂⠒⢲⣹⣂⢡⠤⠂⠀⠑⢶⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢎⡹⢽⠁⠀⠀⠉⠓⢄⠀⠀⠈⠉⢚⢆⠠⠋⠀⠸⣑⣄⡠⠢⢐⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡒⠀⢄⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⡐⠉⣑⢠⡀⠀⢁⠩⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠉⠑⠶⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠋⠘⢤⠠⠙⢶⣶⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡘⠀⢀⡏⠉⠐⠒⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠋⡭⠣⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠜⡋⠖⢀⣒⠶⠆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⡀⠀⠀⠀⠀⡲⠞⡛⠶⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠙⠋⠐⠻⠾⠫⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠉⠋⠰⢿⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠫⡉⢃⠀⠠⠶⠮⡛⢛⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠻⠡⠧⠜⠀⣶⠉⡆⢀⠘⣿⢻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣶⣦⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠠⣤⣤⣤⣀⠀⠉⠈⠀⣿⣶⡿⢿⣯⣴⡚⠉⠙⠛⠿⣿⣿⣦⡀⠀⠀⣿⣼⣿⣿⣌⠟⠛⢿⣿⣧⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡷⣶⣮⣽⡷⠓⠀⠈⣹⢟⡝⢢⠀⡀⠀⠀⣀⡈⠉⠙⠻⣿⡴⢛⠙⢂⡀⠀⠞⠛⠦⢀⣿⣛⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡇⠀⠸⣇⣀⡤⠊⠀⠀⢠⠞⠁⠉⠗⢦⠒⠁⠀⣠⣌⠻⣆⠀⢀⡞⣰⡅⡑⢨⠇⢀⣖⠒⡄⠈⣿⢻⡏⣏⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣹⡄⠀⠈⠁⠀⠀⠀⣰⠃⢸⠖⠦⣄⠈⠀⠀⠺⢗⡨⠃⠙⢰⡿⠀⠈⠉⡔⡁⠀⠘⠛⠚⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠉⠳⢤⣀⠀⠀⣸⠁⢀⣾⣿⣷⣄⠑⣄⠐⡄⠀⠁⠀⠀⣠⡇⢴⣃⣤⣥⣤⣄⣀⣢⡤⠤⠄⠘⠛⡿⠛⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⠀⡇⠀⢼⠀⠒⠛⢻⣷⠾⠛⠛⠆⠀⠀⠸⣿⡇⠀⢻⣶⣶⠶⠶⡾⠋⠀⠀⠀⠀⢈⠇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⡀⣇⠀⠈⠛⠚⠋⠉⠀⠀⠀⠀⠀⡀⢀⡖⠊⣧⠀⠸⣟⠓⣂⠟⠀⠀⠀⠀⠀⢀⣿⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠱⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣠⠖⡿⣦⠀⠈⠉⠁⠀⠀⠀⠀⠀⣴⠻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠀⠀⠧⠃⠛⢦⣀⠀⠀⠀⠀⠀⡼⠁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⣠⠴⠚⢉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣯⣷⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷⠖⢟⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠑⠓⠻⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢠⡄⢠⣶⡆⠒⡀⠀⠀⠀⠀⠀⠠⠀⢲⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣷⠘⢂
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣁⡐⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⣺⢿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠛⠈⠛⠃⠙⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠋⠘⠛⠛⠘


⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠒⢠⣶⡦⢀⡀⠀⠀⠀⠀⠀⠀⣤⢰⣾⣦⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣃⠚⠀⠀⠀⠀⠀⠀⠐⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⢿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠘⠛⠃⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠛⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁⠈⢻⣿⡆⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⡿⡄⠀⠀⠹⠿⠿⠿⠿⠿⠤⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠒⠋⠁⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠠⣤⡀⠈⠙⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠛⠁⢀⠴⠦⣀⣀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁⠘⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡟⠁⠀⠀⠁⠀⢢⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠴⢿⣿⡁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠗⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⠀⣀⣀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠢⣌⣳⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣡⠟⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣠⣦⡀⠀⠀⠀⠀⠀⠀⠘⡿⠛⢿⡿⠋⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢈⡓⠚⢉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⣄⣀⠉⠉⠁⢀⣠⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠠⠤⠀⠀⠴⠆⠐⠶⠒⠲⠖⠒⠒⠒⠋⠉⠙⡷⠄⢀⡼⢱⠄⡀
⠀⠀⠀⠰⣋⣉⣡⣤⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠋⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠯⣙⠚⠀⣸⠀⠒⠠⠄⣀⠀⠀⠀⡄⠀⡇⠀⢠⠂⠀⠀⠠⠤⢤⣄⣀⠀⠀⠐⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠒⢧⡠⢧⣴⠡⠔⠒⠉⠉⠉⠉⠉⠉⠉⠂⠀⠈⢄⡠⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⣄⣤⣄⡀⠘⡄⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⠀⠀⠀⣀⠤⠞⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣵⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢀⡤⠒⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡞⠣⣀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢀⠤⠎⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢿⣿⡀⠀⢣⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠁⠀⠈⣆⣀⣀⣀⣀⢠⣆⡠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠛⠒⠢⠤⣸⡇⠀⠈⢿⣧⠀⢸⣁⣀⡀⢄⣤⡭⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢈⣿⡴⠋⠀⠀⠉⠑⠪⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣴⣾⣿⣿⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⢹⣿⣿⣿⣿⣿⣯⡉⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣦⣀⣛⣹⣿⣿⣿⡿⣿⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⡿⢿⣿⣟⢿⡿⠱⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣏⣬⣀⣠⣴⠾⢋⡀⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣭⣴⣶⣿⣥⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢸⠏⢀⡔⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠈⢠⣿⡇⢠⣾⣏⣤⡴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⡃⠀⣿⣿⣷⣿⣿⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣤⣾⡇⠸⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠛⠁⠀⠤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⣤⡀⢀⠘⣿⠟⠉⣀⡠⠀⠀⠀⣠⡿⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠁⠀⣿⣯⣄⣤⣶⣿⣭⣤⣤⣴⣾⣿⣿⡿⠃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢫⣿⣿⣿⣿⡿⠋⣀⣤⡾⢋⣤⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⢻⡇⢸⡿⢫⡿⠋⣠⣾⣿⣿⣿⠟⢁⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠃⠘⠃⣼⠁⠐⣻⣿⣿⣿⣏⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⢠⡆⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⠀⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠁⠈⠞⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠚⠛⢳⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠐⠀⠐⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⡟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⣀⣠⡍⠀⣤⣀⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠀⠈⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠡⠤⠀⠀⠤⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡧⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠛⠛⠃⣾⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡟⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⣀⠈⠀⢀⣈⡩⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣭⣭⣛⣛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢹⡟⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣶⠀⠀⢰⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⠄⠀⠄⢀⣀⠀⠀⠀⠀⠸⢯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠋⠉⠉⠁⠀⠀⠀⠀⠛⠙⠓⢶⣤⣄⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡵⠄⠀⡄⢠⣶⠾⠿⠀⠀⠀⠀⠀⠀⢠⣯⣡⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣀⠀⢲⡄⢠⠀⠀⠀⠀⠀⢀⡀⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⣷⠆⠀⠰⠆⠀⠀⠀⠀⢀⢖⣥⣶⣶⣶⣮⡂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⣠⣶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢃⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⠴⠶⢤⣾⠟⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⢡⣾⣿⣆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢠⢙⣛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠀⠀⠚⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⢀⠂⢀⡄⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢶⡆⠀⠀⠘⠿⠋⠉⠁⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣸⣷⠀⠀⠀⠀⠀⠈⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣧⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣼⣿⣶⣤⣿⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯⣰⣿⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠏⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⢻⡿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠈⠋⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣦⣄⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⣀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠊⠀⠀⠈⠀⠀⠀⠉⠀⠀⠈⣧⠠⠤⠔⠒⢒⠶⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⣉⣡⣀⣁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠠⠋⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠠⠤⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢀⠀⠐⠈⠑⢀⣴⡁⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡜⠁
⠀⢀⠔⠒⠶⢄⠀⡜⠠⢸⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡼
⠀⡌⠀⠀⠀⢠⠁⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⡤⠆⠀⠠⠋⠀⠐⠊⠈⠉⠉⠑⢤⡀⠀⠀⢰⠁
⢠⠁⠀⠀⠀⡆⠀⢘⠂⡀⠀⡀⠀⠃⠀⠀⠀⢠⠃⠈⠁⠀⠀⡇⠀⠉⠉⠉⠉⠉⠀⢹⠀⢀⠇
⡌⠀⠀⠀⢠⠃⠀⢨⣀⡡⠂⠄⠀⣂⡀⠀⠀⣸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠊⠁⠈⠉⠉⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇
⠣⢄⣀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠉⠀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⣃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠤⣇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡤⠤⠞⠁

⠀⠀⠀⠠⠾⠇⠀⠀⠠⢦⡀⠀⠀⠀⠀⠀⣰⠶⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⣀⣠⣴⢆⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⠀⠀⠀⠉⠉⠀⢀⠈⠆⢀⠀⠉⠁
⠀⢀⠀⠀⠀⠚⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢁⣌⠉⠀⠀⠄
⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈
⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠀⠶⣆
⠀⠀⠀⠒⠒⢦⠀⠰⠦⡄⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠀⣀⣄⣀⡄⣀⣄⡠
⠀⠀⠀⠀⠀⠀⠆⣀⠀⠀⡀⠀⠀⠀⠀⡈⠉⠛⠛⡾⠚⠛⠉⢀⠄⠀⠀⠀⠀⠀⢀⣹⣷⡅⡀⣹⣿⡆
⢀⡀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠹⡀⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠄⠀⠀⢉⠀⠀⠈⠉⠀⠉⠀⠀⠈⢀
⠀⠣⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⡀⠀⠄⠈⠀⠀⠀⠀⠠⡄⣀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⢠⠁⠐⠀⠐⠀⢡⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠨
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠅⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⠀⠞⠛⠀⠀⠀⢠⣤⡄
⠀⢀⠉⠛⠛⠺⠗⠻⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠒⠀⡆⠐⠒⠀⠀⠂
⠀⠀⠁⠀⠂⠁⠀⠁⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠉⠠⠌⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠤
⠀⠀⠲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⠀⠀⠘⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⠀⠀⠀⠀⠀⢀⡤⠒⠀⠐⠒⠦⢼⡆⠄⠴⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⢶⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⡏⢷⠤⠖⢊⣅⠀⠀⠀⢀⣠⠞⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠉⢿⣆⢦⣴⣶⣷⣶⣦⣂⡶⢠⡄⠀⠀⠈⠻⣷⣄⡀⠀⠈⠓⠄⠐⢈⡉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⢰⣏⡸⠛⠋⠙⣛⠻⠷⣾⣿⣿⣃⡀⠀⠀⠈⠻⣍⠐⠂⠀⢤⡼⠉⠙⢿⡆⠀⠐⠂⠀⠀⣤⣤⣤⣤⣤⡴⠿⠂
⠀⠀⠀⣾⠈⠻⣦⡀⠀⠀⠉⠒⢮⡛⢿⣿⣿⣆⠀⠀⠀⠀⠈⠐⠀⠸⡇⠀⣀⣠⡟⠀⠀⠀⠀⠀⠉⠉⠀⢀⠔
⠀⠀⡘⡛⠂⠀⠈⠻⣆⠀⠀⠀⠀⠙⢦⠙⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⢠⠁⠀⠀⠀⣀⡴⢻⡿⣆⠀⠀⠀⠈⢷⠘⣿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡍⠀⠘⠓⢶⡄
⠀⣾⣄⡤⣾⡟⠉⠀⠀⣿⢹⡄⠀⠀⠀⠀⢷⠘⣿⣿⡀⠀⠀⠀⠀⠀⣀⠞⠘⠿⢿⣿⡏⢠⣀⣀⣺⠋⠁
⢮⣿⣿⢿⣿⠀⠀⠀⠀⣿⡆⣿⠀⠀⠀⠀⠈⡄⢹⡏⠓⠀⠀⠀⠀⣿⣴⢆⠀⢀⡀⣮⠁⣼⠋⠉⠉⢦
⠀⠉⠁⠘⠋⠀⠀⠀⠀⣿⡇⢹⡆⠀⠀⠀⠀⠉⠈⣷⡀⠀⠀⠀⠀⠀⠀⠑⠀⠋⡠⠉⠉⠉⠒⢄⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡞⠇⠀⡈⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡈⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⠂⠄⡀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠐⠀⠉⠐⠂⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⢀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢢
⢀⣴⣾⣭⣒⠤⣀⠀⣾⠏⠐⠂⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣠⠟⠉⠁⠉⡙⠒⡤⣄⠀⠀⠀⠀⠀⠀⠁⠺⣓⣄
⠘⢿⣿⡿⣿⣿⣦⣝⣿⣄⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁⣤⡸⣿⣮⣻⣶⣤⣤⣄⡀⠀⠀⠀⠑⠅⠠⣀
⠀⠈⠻⣿⣎⢿⣿⣿⡟⣧⡄⢈⡥⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠈⠀⠀⠰⠀⠠⡄⢰⡞⣿⣷⣿⡌⠻⣿⣿⣷⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⢿⣷⣝⠯⣴⡉⣵⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠐⠆⠀⠲⣀⣀⣈⣿⣿⣿⣼⣻⠀⢿⣿⣿⣿⣷⣄⠀⢀
⠀⠀⠀⠀⠈⢻⡟⠲⢱⣉⣀⠈⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⡄⠁⢙⣫⣿⣍⡟⣷⣸⠀⠸⣿⣿⣿⣿⣿⣦⡼
⠀⠀⠀⠀⠀⠈⠑⢬⠮⣭⣿⡇⠀⠀⠈⠙⠛⠻⠿⠾⠶⠶⠿⠾⠶⢾⣷⡀⠀⠠⠀⠀⢰⣏⣽⣯⣯⠇⡇⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠹⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠘⠿⠿⠸⠀⠸⡀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠀⠂⣤⠀⣤⣕⣤⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣦⡾⢦⡈⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡚⣤⡗⣿⣿⣷⣦⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠺⠿⣿⣿⡄⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠙⠛⠷⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⢾⣮⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠛⡟
⠀⠀⠀⠀⠀⠙⠛⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⣿⣇⠈⣻⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣧⠙⣿⡛⢠⣿⣿⣿⠀⠀⠀⠀⠀⣱⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢻⣾⣿⣷⣿⣿⣿⣿⣇⣀⣠⣄⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣽⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣥⠀⠈⠻⠿⠿⢿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢰⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢠⣤⣤⣞⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠄⠀⠀⠀⠀⠉⠛⠻⠿⠆⠀⠀⠀⠸⠿⠿⠷⠦⠤⠤


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⠶⠶⠖⠚⠋⠉⠉⠉⢉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢤⣀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠛⠉⠁⢀⣀⣀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⢀⣠⣤⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⢠⣚⣩⠤⠶⠛⠉
⣀⠤⠤⠶⠶⠲⠦⢤⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠚⠉⠉⠉⠋⢳
⠀⣀⣠⡤⠤⠶⠖⠛⠉⠉⠁⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢷⠿⡗⠚⢛⣛⣛⣛⣛⣲⡄
⠉⠁⠀⠀⠀⠀⣀⡤⠶⠒⠒⠚⠋⣉⣀⡤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢺⠃⠈⢹⡜⣍⠉⠉⢻⡌⣧
⡤⠴⠖⠚⠋⠉⢁⣀⡤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣹⠦⡄⠀⢳⡌⠻⡇⠘⡇⢹
⣀⣀⡤⠶⠒⠋⠉⠀⠀⠀⠀⢀⡠⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡼⠁⡤⠟⠀⠀⠈⠳⢿⡟⢻⣼⣤
⠉⠀⠀⠀⠀⠀⠀⡴⠒⣋⣭⠵⢿⢸⣉⠉⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⡾⠘⠀⡼⠁⠀⠀⠀⣀⡤⠈⣧⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣻⡧⠐⣆⣩⣀⢼⣆⣇⢸⡎⣹⠀⠀⠀⠀⠀⢰⡇⠀⣸⠃⢀⡤⠖⠋⠁⠀⠀⡿⣆⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⡖⢋⣹⠄⠸⣧⢌⡙⠿⣟⢳⡄⠀⠀⠀⠘⣇⣰⢿⣤⣤⣤⣤⣤⣤⣤⣤⡟⠛⢦⡀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢁⡔⢹⣿⠀⠀⠹⣌⣛⣦⣌⠀⢙⡄⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠻⡇⠀⠀⠈⠠⣄⢱⡄
⠀⠀⠀⠀⠀⠀⠀⢧⣴⣻⡿⠞⠋⠀⣀⡤⠀⠁⢹⠈⠙⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣦⠀⠀⠀⡇⠀⠀⠀⠐⢺⠾⠇
⠀⠀⢀⣠⠖⠓⠲⢄⠈⢻⠇⠀⠖⠋⣁⣀⣤⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣆⠀⣿⢹⠀⠀⠇⡇⠀⠀⠀⠀⢸
⣠⡶⠋⡀⠀⠀⠤⠴⠀⡸⠶⠞⠛⠋⠉⠉⠁⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠀⣿⠘⡇⢸⠀⣿⠀⠀⠀⠀⡞
⠁⠀⠀⠉⠒⠊⠀⠀⡀⣠⠀⢀⡔⠚⡣⠀⠀⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⣧⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⢤⡴⠄⠀⠀⡼⢡⡏⢀⡞⠀⠀⡇⠀⢠⡇⢸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣿⠀⢻⣤⣄⣸⠀⠀⠀⢸⠁
⠊⢡⡄⠠⠦⠀⡘⠁⠸⠁⡞⠀⠀⠀⡇⠀⠾⠀⢸⡇⠈⠓⠆⣀⠀⠀⠀⠀⠀⣿⣿⡿⣿⠀⠸⡿⠿⢿⡇⠀⠀⢸
⠀⠈⠰⠠⡀⠀⡇⠀⠀⢠⡇⠀⠀⠀⣇⠀⠀⠀⠈⢧⠀⠀⠀⠈⠁⠀⠀⢀⢴⡷⠀⠀⣿⢠⡄⢷⡀⢶⣿⡄⠀⡇
⠀⣀⡀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠘⢧⠀⢀⣴⣿⢷⡀⠀⠀⠀⠀⠀⣏⠋⣁⣠⢤⠏⠀⠀⠀⠛⢶⣍⣹⠀⠱⠶⠶⠶⡶⠶
⠘⠁⠰⠖⠲⠀⣇⣀⣀⣸⡇⠰⠒⢄⣀⠈⢳⡿⠛⠁⠀⣧⡀⡄⢀⡴⠚⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⢦⡀⣀⣀⣀
⣀⣀⣀⣀⡀⠀⣿⠿⠿⠻⡇⠀⠀⠈⠉⠁⠀⠹⣤⡄⣠⡏⠄⣰⠿⢤⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠀⠀⠀⠳⡙⠛⠿
⠉⠉⠉⠉⢀⣤⡋⠀⢀⣠⡇⠉⠙⠛⠛⠓⠂⠰⢭⠴⣧⠶⠋⠀⠀⠀⠈⠛⠉⠉⠉⠓⠦⠤⣾⣀⣀⡀⢀⣀⠀⠃⠀⠀⠀⠀⠛
⠿⠿⢟⣆⠈⠛⠛⠛⠉⠙⠀⠘⠿⠿⠂⣀⣀⣀⣠⠞⢁⠀⠖⢾⣭⠭⠝⠓⠶⢤⣀⣤⠤⠤⠤⢤⣤⣀⠀⠉⠤⠶⠤⡄
⠀⠀⢤⣿⣿⣿⣿⣿⠿⠿⠗⠒⠒⠛⠉⠓⠙⠛⠛⠚⠫⠙⠂⠀⠐⠲⠲⠦⠀⠀⠉⣷⣄⣀⣀⣤⣤⣬⣽⣤⣤⣶⡾⠋⠉⠙⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣦⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢟⣻⣟⣛⣛⣛⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇⣿⣿⣿⡿⠿⠿⠘⠛⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣄⣀⣀⣀⣀⣠⣤⠴⢟⣴⣿⣿⣏⠀⠀⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢏⠉⢉⣍⠉⠀⢀⠀⠈⠻⣿⣿⠟⠀⢸⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⠀⠘⠛⢁⠀⢿⠗⠀⡔⢻⠃⠀⢀⠌⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡶⠠⠝⠁⢀⡀⠀⠷⠃⠀⢀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢟⠶⠶⠶⡷⠁⢸⠁⠀⢠⠋⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡻⣿⣿⣶⣷⣶⣮⣤⣶⣿⠀⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣼⣿⣶⣮⣝⣛⣿⣻⣛⣛⡉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⢰⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡆⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡔⠁⠀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠎⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⡆⠀⠀⠀⢀⣛⡻⢿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⣠⣶⣿⣿⣿⣿⣷⣾⢤⣯⣭⣭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠱⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠰⣄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷
⠀⠈⠻⣶⣤⣄⡻⣝⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠉⠻⣯⡻⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣽⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣣⣄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣓⣒⣒⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣯⠁⠀⠀⠐⠂⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣀⠀⠀⠀⠀⠀⠙⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⣿⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⠛⠃⠀⠀⠀⣹⣿⡿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⢠⣿⡏⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⣾⣿⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⣴⣿⠟⠁⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢀⣿⡏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⠿⠃⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⣿⠃⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⢸⣿⡏⠀⠀⢀⣿⡏⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠀⠀⠀⢀⣀⣠⣾⡿⠋
⠀⠀⠀⠀⣾⣿⠀⠀⠀⣾⣿⠃⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢁⠀⠀⣤⣾⣿⠟⠋
⠀⠀⠀⢠⣿⡟⠀⠀⢠⣿⡏⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⣿⣏
⠀⠀⠀⣾⣿⠃⠀⠀⣾⣿⠃⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⣿⡿⠀⠀⢸⣿⡏⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀
⠀⠀⣼⣿⠇⠀⠀⣼⣿⠁⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⣿⣿⠀⠀⢰⣿⡏⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⣾⡿⠃⠀⠀⢸⣿⡀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⣶⣿⣧⣤⠤⣤⣾⣿⡇⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⢿⣿⣿⣿⣄⣿⣿⠋⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠈⠻⣿⣿⣿⡿⠁⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⠿⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⢸⣿⢹⣿⡇⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠘⣿⡇⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⣻⣿⠰⣿⡏⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣀⣠⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⣤⣀⣠⣴⣿⡟
⣤⣤⣄⣀⣤⣄⣀⣀⣤⣀⣀⣀⣩⣿⣿⣿⣿⣍⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣀⣈⣽⣿⣿⣿⣿⣁⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠉⠀⠛⠋⠙⠋⠉⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠛⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣄⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠉⠻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⣤⡤⠀⠀⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠂⠀⠄⠀⠐⠷⠀⠀⠰⠄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣤⠤⠼⠛⠁⠀⠀⠀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣯⡗⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⣴⣦⣹⣧⣴⠋⣻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢿⣧⡀⠀⠀⠀⠈⠙⣿⣿⢻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⣄⠀⢀⣤⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⣾⣿⣿⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣼⣿⣧⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣁⣹⣿⣏⣉⣿⣿⣧⡀
⠀⠀⠀⣤⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⡀
⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠈⠋⠘⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠋⠛⠁⠀⠀⠈⠻⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠠⠤⠄⠀⠀⢀⡀⢀⣀⣀⣀⣀⠀⠀⢠⣤⠐⠃⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⡇⠀⠀⠀⠙⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣘⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇






⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⠂⠘⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⡿⢿⣿⡿⣿⠿⣿⡿⣏⠹⢿⢿⣿⣿⠿⠿⢿⡿⠛⣿⣿⣿⡿⠛⠛⢻⠿⢿⣿⣿⠿⠿⠿⣿⡿⠻⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣶⣦⣿⣶⣦⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⢣⠴⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠏⣴⡀⢈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢩⣈⢣⣤⡀⠻⢿⡠⣤⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠸⠸⢿⣷⡄⣸⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⡇⣨⡏⠉⣙⠦⢠⣞⠁⢙⠇⠘⠄
⠀⠀⠀⠀⠀⠀⠀⣼⣾⣄⢙⣋⣵⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡆⣿⣿⣿⡄⣾⣿⣿⣧⣄⣡
⠀⠀⠀⠀⠀⠀⣾⡿⣻⠿⡪⠙⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣱⣿⣿⡳⠿⢻⣟⠹⡿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⢀⣴⣿⣿⠛⠀⣣⠂⠀⠛⣍⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⢀⣼⣿⣷⣦⡿⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣦⣹⢻⣿⣃⣴⣿⣧⣤⣤⣭⣅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠈⠻⠿⠿⠁⠀⢠⣿⣿⡿⠃
⠀⠀⠀⠀⠙⠋⣼⣿⣻⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡷⣼⡆⢠⡤⣤⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⢿⣿⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⠛⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣄⢛⣿⣿⠬⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣦⠙⢹⠇⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣟⡋⣠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣥⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⣾⠉⠿⠇⠀⡹⢀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣶⣾⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⡆⠀⢠⡇⠀⠠⠢⢤⣄⠀⣀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⡻⣿⣿⣽⠿⠋⠻⢿⣽⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣥⣂⡄⢸⡷⠀⠀⣀⢿⣿⣿⣿⣷⠊
⠀⠀⠀⠀⠀⠀⣷⡝⠋⠀⠀⠀⠀⠀⠈⢿⣿⢧⠀⠀⠀⠀⠀⠀⢀⠀⢸⡻⠊⢿⣿⣿⡯⠄⠀⢰⠇⠈⠻⢻⡿⠛
⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠺⠓⠒⠀⠀⠀⠀⠈⠲⡃⠀⠀⠀⠉⠐⠛⣀⣠⠿⢀⡀⠀⠀⠀⠐⢤⣤
⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠘⠻⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣧⣶⣤
⠀⠀⠀⠀⠀⠿⣿⠀⠀⢰⣿⡧⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣾⣿⣷⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁⣠⣴⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⠰⣌⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣼⣿⣿⠿⢛⣥⠆⠀⣀⣠⣤⣴⣶
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⠦⣭⣭⣝⣛⣛⣛⣛⣛⣛⣛⣛⣩⣭⣭⣴⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠉⠉⣉⣉⣭⣭⣿⣿⣿⣿⣿⣻⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣾⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣤⣶⣿⣿⣿⣶⣤⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢠⣶⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⣠⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁
⣸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠸⣿⣿⡇
⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠁
⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⣦⠀⠻⠻⢶⡄
⠀⣨⡿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⡇
⣼⠏⠰⠶⢦⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠟⠁
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠻⣦⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣠⡼⠋⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⣻⣛⣿⠋⠉⠁⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣉⡛⠛⠛⢿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡀⠉⠁⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠛⠉⠛⢻⡶⢶⣤⣤⣄⣀⣀⣴⠟⠁⠀⣶⠀⠀⢸⡟⠛⠛⠻⢶
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⡀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣭⣽⠿⠟⠁⠀⢀⠀⠀⣼⡇⠀⠀⠀⠉⠉⠉⠉⠀⢀⣰⡿⠶⠶⣿⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠸⣧⣀⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⣽⠟⠛⠛⠛
⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠛⠛⠻⠿⠿⠿⠟⢿⣦⣀⣟⠉⠉⠛⣻⡏
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⡄⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⣠⣾⣇⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣄⣠⣄⣠⣀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡏⣹⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⡞⢻⣶⣶⣶⣶⣿⠇⢸⣷⣶⣶⣶⣶⠟⢱⠖⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠙⡷⠦⠿⠖⠃⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⠘⠶⠷⠴⡿⠋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠲⠶⠶⠚⠲⠶⠶⠶⠶⠒⠶⢾⣧⣾⡗⠶⠶⠶⠒⠶⠶⠖⠳⠶⠖⠞⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⡿⢿⣷⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡠⢏⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣄⠀⣸⣇⢰⣶⣟⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣆⣿⣿⣌⠻⣏⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢹⣿⣿⣷⣌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⢿⣿⣿⣿⡆⢿⣦⢲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣿⠈⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣠⣞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⡆⢸⣇⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢱⡆⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠸⢿⡇⠘⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⣀⣶⣶⣤⣼⣡⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⣾⣟⣻⠿⣿⣿⣿⠟⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠈⣿⣯⣽⣻⣿⣯⣶⣶⣽⣧⣾⣿⣆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣄⢨⣭⣛⣿⣿⣯⣝⢷⣶⣌⠹⠟⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠿⠟⣻⣿⣿⣿⣿⣿⣿⣦⡈⠁⠀⠀⠉⠙⢿⡿⠏⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⣼⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⡀⢀⣸⣇⣤⣬⣸⣩⣥⣬⣹⣆⣭⣥⣬⣠⣏⣤⣀⣠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣥⣤⣭⣽⣿⣯⣥⣤⣭⣽

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⡇
⡇
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⡄⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠀⠋⠻⠻⠀⠀⠀⣆⡟⣰⡇⣰⡆⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠻⠛⢷⠿⢇⡋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⢤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠈⡅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⡄⠀⢀⢀⡀⢠⡄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⣀⣀⣀
⡇⠀⠀⠀⠀⠳⠄⠘⠨⠇⠘⠙⠜⠋⠀⠄⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⠐⠟⠳⠞⣧⡟⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡂⠐⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠔⢊⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠆⠠⡆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠁⠀⡤⢄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡄⠀⠀⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠆⡠⠖⡆⠀⠘⡄⠑
⠀⠀⡤⠤⡙⣄⠳⣠⡏⣡⠏⠀⠀⠀⠀⠀⣴⢛⡉⠉⠉⠀⠈⠉⠉⠙⢶⣄⠀⠀⠀⠀⡴⠒⠲⣼⡈⠛⢡⣞⣀⡀⠀⢸
⠀⠀⠈⠲⢌⡛⢡⠞⠃⣇⠀⠀⠀⠀⠀⢸⠇⢹⣆⠀⠀⠀⠀⠀⠀⠙⡶⠹⣆⠀⠀⠀⠛⠉⢹⠀⠙⡆⠈⣥⠴⠃⠀⢸
⠀⠀⠮⠉⢨⠀⠋⠀⠀⡾⠀⠀⠀⠀⠀⣿⠀⢼⠈⠳⣤⣀⣀⣀⡤⠞⢱⠀⢻⡀⠀⠀⠀⠀⢸⡄⠀⢈⡠⢬⣵⠀⠀⠎
⠀⠀⠀⢰⡡⠒⠦⠴⠦⡈⠢⡀⠀⠀⣰⠛⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠓⣦⠀⠀⡰⠋⣴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢦⡀⠘⣷⠀⢸⣿⣷⣦⣤⣤⣤⣤⣶⣿⠀⠐⡶⠃⣠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣇⡴⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⠀⠀⠸⡿⢿⣿⣿⣿⣿⣿⠇⠀⠀⡏⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣄⠀⠀⢉⡽⠋⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠱⣄⠈⠙⠛⠉⢀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠢⢄⣈⡙⠓⠖⠛⠉⢀⣀⡤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⢁⡴⠒⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠘⠀⠀⡏⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⢦⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡞⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠱⣄⠀⠀⢙⡶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣴⢾⠀⢀⡴⠚⠉⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠓⢤⠞⣰⠃⠀⠉⠉⠉⠉⠉⠉⠀⠀⠘⢦⡑⠋⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣀⣠⠶⠋

⠀⠀⠀⠀⡀⠀⣠⣤⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣤⡀⢠⡀
⠀⠀⠀⠘⠟⠀⣿⣿⣏⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⣿⣿⠁⡰⠆⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⣿⣿⣿⠀⠐
⠀⠀⠀⠀⠈⣠⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣿⣿⣿⣄
⠀⠀⠀⠀⢺⡟⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⠃⠛⠛⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠿⠿⠟⠹⠃



⠀⠀⠀⠀⠀⢠⣶⣾⣶⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢶⣶⣶⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣶⣷⡇⠤
⠀⠀⠀⢀⠁⢾⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢿⡿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢨⣿⣿⣿⢀⠚
⠀⠀⠀⠈⠁⣿⣿⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣭⡌
⠀⠀⠀⠀⢸⡿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⡟⣿
⠀⠀⠀⠀⠉⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢉⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢘⣷⣤⡟⣷⠂
⠀⠀⠀⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⢿⣷⣤⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⡆⠀⣹⣿⠁⡀⢨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⣼⣷⡄⠁⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠸⠿⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠇⡀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⡁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⡇⠠⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⡇⠀⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⠃⣴⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣦⣶⣿⣷⣶⣶⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⣀⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣤⡀
⠀⠺⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣀⣀⣤⠀⠀⠘⣿⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⠓⣀⠀⢀⣴⣄
⢻⣿⣿⠆⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠉⠀⠉⠁⣿⣿⡿
⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠲⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⣶⠾⠐⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠅⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⡀⡠
⠀⢀⣀⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡁
⡐⠀⠀⠀⡁⠀⠀⠀⠀⢿⣾⡇⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠹⠁⠙⠈⠉
⡇⠀⠀⠀⠀⡄⠀⢀⠐⠿⠛⠀⠰⠃⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠠⡀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⢰⣴⣶
⠀⡱⠀⠀⠐⠒⢢⡀⠀⠀⠀⠀⠠⠤⢀⣀⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣠⣿⣾⠁⠀⠀⠘⣿⣿⡀
⠀⢈⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠐⢀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠹⠏⠀⢰⣴⣿⠟⠻⠓
⣶⣶⣶⣷⣿⣾⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣮⣵⣦⣤⣧⣼⣯⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠒⠚⠛⠉⠉⠉⠉⠉⠉⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡆
⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠖⠒⠒⠒⠦⠤⢄⣀⠘⣧
⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⠀⠈⠳⣿
⠀⣠⡾⠋⠀⠀⠀⣠⠞⠉⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣴⣯⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⡟⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠤⠶⠃⠀⢸⡇
⠈⣷⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠹⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢹⣆⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢻⣆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣹⡆⠀⠀⠀⣿
⠀⠀⠀⠀⣹⣦⣤⣤⡾⠃⠀⠀⠀⠀⠀⢿⣷⣶⣦⣶⣶⣶⣾⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠰⣯⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠁⠀⠀⣾⠁
⠀⠀⠀⠀⠙⠷⠶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣋⣀⣈⣥⠴⠋⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡯⠤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠿⠶⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠛⠋⠉⠹⣦⠀⠀⠈⡟⠛⠛⠋⠓⠒⠒⠒⠒⠊⠉⠀⡀⠉⢉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣠⡴⠷⠶⠶⠶⠶⠶⠶⠶⠦⠶⣄⡈⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡂⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢻⣿⠁⠀⠀⠀⡴⢶⡀⣀⣤⡀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠾⠿⠤⠤⠤⠞⠁⠈⠉⢁⣤⣇⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣄⣤⣄⣠⡤⣄⠀⠀⠀⢤⣤⡼⠃⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⠙⠛⠲⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠻⠷⠶⠶⠶⠶⠿⠟⠛⠛⠋⠉⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⣉⣹⣍⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⣿⠉⠉⠙⡏⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⣿⠀⠀⠀⣿⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣴⣿⠀⠀⠀⣿⣶⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⣀⣴⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⡾⠟⠉⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣠⣾⠏⠀⠀⠀⠀⣀⣀⣀⠉⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣴⡟⠁⠀⢀⣴⣾⣿⣿⣿⣿⣿⠄⠀⠈⠙⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣼⡟⠀⠀⠀⠿⠿⠛⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣛⣿⡶⣶⣶⣦⣴⣶⡶⠶⠶⢶⣶⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠻⣦⣤⣤⣤⣶⠆⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣠⡾⠛⠁⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⣀⣀⡀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⣀⣀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠾⠛⠋⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣴⠶⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢹⡷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⣿⡿⢦⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠈⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⣁⣤⣶⣿⡇⠀⠀⠀⠀⢠⣤⠀⢀⣸⡷⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠻⠶⠶⣶⣶⣶⣶⣶⡶⣾⡿⠛⠉⠀⣼⡇⠀⠀⠀⢀⣿⣏⠀⠈⠉⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣠⡟⠁⠀⣰⠟⠀⠀⠀⢀⣿⠃⠀⠀⠀⣼⡿⣿⣦⣀⣀⣠⣴⡿⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢠⡟⠀⠀⠀⠉⠀⠀⣀⣴⡿⠋⠀⠀⢀⣼⡿⠁⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣈⣷⣤⣤⣤⣶⡾⠿⠛⠉⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣶⣾⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠆⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢻⡷⣳⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣷⢿⡿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⣿⡇⣘⠾⠷⣿⢋⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢰⣿⣿⠙⠛⠳⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⣿⣿⡟⠀⠀⣠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⡇⣀⣾⣷⣿⣛⢻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠻⠿⣿⠷⣏⣼⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢷⡻⣿⠼⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⡛⣷⣿⢾⣼⡟⣻⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣏⣿⣳⣏⣛⣿⣴⢾⢯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢶⣛⣽⣻⣯⣿⣿⣿⡥⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣟⣏⣿⣿⡧⣿⢿⡞⣷⣋⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣟⣵⣯⣿⠿⣿⣶⢮⡷⡾⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⣽⣖⣻⣿⣿⣿⣿⣟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣛⣽⣿⢷⡻⠖⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢾⣿⢾⣮⣿⣿⣫⣺⠿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣟⣾⣿⣟⣸⣧⣿⣾⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡼⣿⣿⣿⣿⣿⣿⡟⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢿⣿⣿⣿⣿⣿⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⣿⣿⣿⣿⣿⣿⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣇⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣏⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠇