Striking Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⢰⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⡇⠀⠀⠸⣧⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⣰⡇⠀⣰⡿⠀⠀⠀⠀⢻⣆⠀⠸⣧⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠞⣰⡟⠀⣼⣿⠃⠀⠀⠀⠀⠈⣿⣧⡀⠹⣧⠙⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣼⣿⢁⣰⡟⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠹⣧⡈⢿⣷⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⡇⣸⡟⠀⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⠀⢻⣧⢸⣿⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⣿⣟⣷⡿⡁⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢀⢿⣿⡟⣿⣿⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠹⣿⣿⣿⠁⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠈⣿⣷⣿⡟⣿⣿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢸⣿⡆⢿⣿⡏⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⢸⣿⣿⠁⣿⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⡆⢿⣔⣾⡿⣿⣿⣿⣿⣿⣿⣇⡄⠀⠀⠀⠀⠀⠀⢴⣽⣿⣿⣿⣿⣿⣿⢿⣯⣸⡿⢡⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⢻⣿⣎⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣡⣿⡟⣸⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡻⣧⡻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣴⢟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡽⣟⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡶⣻⣧⣤⣤⣼⡟⢾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡿⣫⢟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣮⣷⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠟⠁⢲⠂⢊⠉⠙⠏⠉⡙⢄⡞⠈⠻⣿⣿⣿⣿⣿⣿⣿⡯⣹⣿⣞⣵⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣽⣮⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡆⠀⠈⠀⠆⠈⢃⡘⠁⠰⠀⠁⠀⢰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡵⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢷⣿⣿⣯⣙⣿⣿⣿⣿⣿⡇⠞⣀⢠⠀⠀⠀⡀⠈⠁⢀⠀⠀⠀⠀⣀⠇⢸⣿⣿⣿⣿⣿⣫⣼⣿⣿⣾⣯⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣗⣾⣿⣿⣭⣿⣿⣿⣿⣿⡇⢒⣽⣾⡆⠀⢠⣍⣦⣴⣫⡄⠀⢰⣗⣿⡒⢸⣿⣿⣿⣿⣿⣿⣿⣿⢶⣚⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⠯⢿⣻⣿⣿⣽⣿⣿⣿⣿⣷⣿⡟⢁⣠⣤⣤⡌⢻⡟⢡⣴⣤⣄⡌⢻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⡿⠽⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣂⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⡞⢷⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣷⣿⣿⣿⣆⣐⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣾⣿⣿⣿⣾⣛⡿⣿⣿⠸⣿⣿⣿⣿⡿⣤⣤⢿⣿⣿⣿⣿⠇⣿⣿⢿⣟⣵⣿⣿⣿⣷⣿⣟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣭⣿⣿⣿⣿⣿⣿⣿⡿⠦⠛⢋⣁⣛⣱⣿⣿⣌⣋⣊⡙⠛⠴⢿⣿⣿⣿⣿⣿⣿⣿⣭⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣩⣽⣿⣿⣿⣿⡷⣶⣿⣿⣿⡟⢿⣿⣿⡟⢛⣿⣿⢿⣶⣾⣿⣿⣿⣿⣿⣍⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠻⣿⡿⠃⣿⠇⣿⣾⣿⣴⣂⣐⣣⣿⣷⣿⠘⣿⠃⠻⣿⠻⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⣿⠿⢾⡭⡧⡿⢿⣿⠷⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠇⢀⡰⠖⠛⠋⡉⠉⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢸⣷⡴⣺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⠄⠀⠀⠀⠸⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⠕⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⠀⠀⠉⠑⠦⣄⠀⠀⠀⠀⠀⣿⡿⠿⠌⠁⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⢷⣤⡀⠀⠀⠈⠳⣤⣄⡀⣠⠟⢣⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠙⠻⢦⣤⣷⠀⠀⢻⣿⡁⠀⠘⣧⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡏⠛⠀⠀⠸⣿⡇⠀⠀⠙⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⢀⡿⡇⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠹⣼⢱⣇⣀⣄⣴⣿⣦⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢄⡰⢃⣾⣿⣿⣿⣿⣿⢿⡗⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢸⣧⡴⢊⣴⣿⣿⣿⣯⣾⡿⡿⠈⠛⢦⣀⠀⠉⢢⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⢰⣾⣿⣿⢟⡵⠋⣰⠃⠀⠀⠀⠙⠳⢤⣧⣯⡞⡩⠖⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢠⣿⣧⣼⡿⠿⠿⠶⣿⣷⣾⡁⠀⠀⠀⠀⠀⠀⠀⠉⢩⡽⠷⣶⣶⣦⣰⢶⣶⣤⣤⣤⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠛⣤⣴⣾⣿⣿⣿⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣌⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠳⠍⠓⠉⠂⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣳⠀⠀⠀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⠀⠀⠀⠀⠀⣤⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣹⣷⣤⡀⠀⣰⣿⣄⡀⢀⣈⣀⣀⣤⣴⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⠄⢀⡤⠀⠀⢀⠸
⠀⠀⣠⣤⣤⣶⢶⣾⡿⠿⠻⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠁⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠹⣿⣿⠋⠈⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠐⢄
⠀⠀⠀⠀⠸⣿⠀⠀⣿⣟⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣩⣤⣬⣷⣄⠀⠀⠀⠀⠀⣼⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⢻⣄⣴⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⡍⠉⠉⠀⠈⠳⠄⢰⣄⠀⢈⣷⠀⣤⠀⣤⠀⠀⣠⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠻⣿⡿⠋⠉⠛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣼⡟⠀⣿⡏⠀⣿⣦⡀⢀⣼⠃⠀⠀⣴⠛
⠀⠀⠀⠀⢼⠟⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⡟⠁⠙⠓⠀⠀⠀⠀⠀⣈⣿⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣷⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣿⣆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⣿⣽⣿⡿⠿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣷⣾⣿⣿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣾⣿⠁⠀⢾⣿⣿⡿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢾⣿⡿⡻⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣬⣶⡄⠀⢀⣾⣿⣿⣧⠀⠀⠀⢀⣿⡏⠁⠈⠙⣿⣿⣿⣿⣿⠇⠀⠉⠉⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣟⠁⠀⠈⢿⣿⣿⣿⣷⣶⣶⣿⣿⣧⣀⣀⣠⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣀⣀⠀⠀⣀⣀⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⠟⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠸⠗⠰⠆⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠏⠻⢿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢷⣾⣿⣿⣷⠾⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢡⣤⣾⣿⣿⣿⣿⣷⣤⡌⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣹⣿⡟⠀⠙⠛⠿⣿⣿⣿⣿⠿⠛⠋⠈⢿⣿⣏⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⡟⢀⡀⠀⠀⣀⣐⣿⣿⣂⣀⠀⠀⢀⡀⢻⠙⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠸⠁⣿⣿⡷⣦⣄⣈⣼⣧⣁⣠⣴⢾⣿⣿⠈⠏⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣆⠀⣨⣿⣶⡿⡿⢿⡿⢿⠿⣶⣿⣅⠀⣰⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢂⣲⣸⣏⣗⡑⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏⢹⠪⣹⣿⣿⢏⡑⠏⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣿⣧⠎⠀⣸⣿⣿⣿⣿⣷⠀⠸⣜⣿⣿⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣧⠀⣿⣿⡏⢠⡴⢿⣿⣿⣿⣿⣿⢧⣄⢹⣿⣿⠀⣼⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡀⠙⣿⣷⡀⢺⣿⣿⣿⣿⣿⣿⡗⢀⣾⣿⠏⠀⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢿⣿⣿⡧⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⡿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠼⢻⠿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣠⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠞⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣭⣙⣛⣿⣝⡳⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠙⠻⢽⣻⣷⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣟⡷⣦⣤⡙⢷⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡍⠻⣷⣿⣎⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣐⣷⣿⣿⣿⣍⡉⠳⡀⠀⠀⠀⠀⠙⢿⣿⣿⡿⣿⣿⡟⢿⡀⢮⢻⣿⣞⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠈⠋⠛⣷⣿⣿⣍⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⠊⣿⣌⣷⣿⣿⣿⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠀⢀⠤⣴⣿⣿⣟⣯⠟⠋⠣⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣽⣿⣿⣷⢬⡻⣿⣿⡇⢻⣇⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⠉⢀⣸⢋⡉⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣧⣸⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⡀⢦⠀⣀⠑⠈⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠷⣤⠴⢿⡗⠈⠀⠩⣶⣄⠀⠀⠶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡁⢿⠟⠹⣿⡞⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣄⠀⠀⠀⠀⠣⡀⠀⠘⠯⠛⠲⣦⡄⠀⠀⠙⠯⠿⠻⠿⠁⠈⣿⣰⣶⣄⠀⢹⡧⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣷⣴⡄⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠅⡹⢿⠇⡜⣱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣥⣭⠟⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢄⣀⡉⡀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠟⠻⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⢷⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⡴⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⠛⡿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⡆⣠⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⢿⣿⣿⠿⢿⣿⣿⣆⠀⠀⢣⠇⠀⠀⠀⠘⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣾⠏⠀⠀⠘⠋⠙⣿⣆⠀⢠⡁⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣆⠀⠀⢀⡾⠿⣿⡆⢸⠃⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⠆⠀⡈⣿⠆⠙⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⡾⣧⣾⣇⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢻⣿⣷⣃⣿⣿⡇⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣠⣾⣿⣿⣿⡍⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣤⣄⡀⠀⠀⠐⠚⠵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣤⣾⣿⣿⣿⣿⣿⣿⡼⣿⣏⣿⣿⣿⢦⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⣄⣙⣿⣿⣿⣷⣦⣀⠀⣠⢄⡠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⠁⢿⣿⡿⣧⣾⣿⣿⠏⣿⠚⣷⡶⡾⢿⠷⠈⠙⠻⢿⢿⣿⣿⡿⣋⣽⣿⠿⢿⢿⣿⣦⡰⣫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢾⡿⠁⠀⣿⣿⣿⠀⠈⠙⠟⠀⢀⠀⡄⠓⠀⠀⣠⣾⡿⣡⣾⣿⣿⣱⣈⣳⣾⣶⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣿⡀⠈⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠈⠁⠀⢀⣠⣶⣿⣿⣿⣽⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⢰⡟⠁⣤⡶⠆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠀⠀⠀⠀⠀⢀⡔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢦⠀⠻⣌⣆⠀⠀⠀⠀⢀⠣⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠑⣾⠦⢄⡟⠉⠑⢄⠀⠸⡁⣼⠟⠓⢄⠀⢀⡤⢀⠊⡱⠠
⠀⠀⠀⠀⠀⠀⠀⡊⠑⢆⠀⠀⢸⠀⠀⠁⠀⠀⠀⢧⡰⠛⡇⠀⠀⢘⣧⠥⠒⠻⠏⡀⠒⠐⢒⡦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⣆⠀⠈⠚⠀⠡⣣⡀⠀⢸⠃⠀⢱⡀⢀⣸⡏⠀⢠⢰⠀⠻⠤⠀⠊
⠀⠀⠀⠀⠀⢠⣀⠀⠈⠛⢿⠓⠢⣄⡈⠢⠒⠥⠴⢾⣀⣀⡌⠈⠻⠿⠧⡴⠗⠁⠀⣂⣤⠖⠋⣱
⠀⠀⠀⠀⠠⢄⠀⠀⠤⠀⢗⠀⠀⠀⠈⢢⠀⣠⣾⣟⡏⣀⠀⠙⢦⢀⠔⠡⠂⠀⣀⣹⡟⠉⢉⣀⠔⢂⣉⡆
⠀⠀⢠⡖⠓⠲⠭⣒⣄⠠⢼⣷⡀⠀⠀⠀⢷⣿⣿⢫⡌⠛⠀⠀⠀⢇⠀⢠⡴⠋⠁⠀⠀⠀⣸⡟⢉⠉⠁
⠀⠀⠀⠉⠛⡛⠒⠛⠻⡀⠀⠀⠑⠦⣀⢀⣨⣿⣿⣿⡀⠀⠀⠀⠀⡸⠒⣻⣀⡀⣀⣒⠤⠊⠽⠚⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠓⠛⠲⠤⠤⠤⠼⠯⠽⢟⢮⣛⠿⣶⣶⣶⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠑⠒⠒⠂⠀⠒⠛⠚⠻⠿⠿⠿⠀⠁⢈⣉⠿⠯⣭⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠶⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠠⠀⠀⠡⠄⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣦⡀⣾⡖⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣷⠙⠃⣉⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠒⢒⣒⣻⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡀⠊⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⡿⠟⠁⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢦⣀⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣠⣤⣤⣀⣰⠖⢄⢀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⣴⡦⣄⣨⠟⡲⢴⣿⣿⣿⣿⣿⣿⣷⡄⢨⠏⠀⠀⠀⠐⠄⠀⢤⣤
⠀⠀⠀⠀⠀⠈⣉⠀⠠⠤⠬⢶⣹⡂⡀⡉⢙⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠈⠉⠀⠂⠠⠄⢲⡿⠒⠲⣅⡩⢽⣿⣿⣿⣿⣿⣿⣿⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠛⠙⠻⠿⠟⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠈⢙⣛⠛⠷⢦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢁⡀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⡄⢰⡤⠐⠓⠀⠀⠀⠀⠀⠀⡾⢠⠂⠀⠀⠀⢀⣼⠅⠀⣴⠌⣇
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣄⠀⡄⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⣀⡀⢠⣾⡿⠃⢨⠃⠀⠀⠑⠀⢤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠾⣿⣿⣿⣿⣾⡯⠀⠀⣠
⠀⠀⠀⢶⣤⣤⡼⣿⣿⣿⣿⠟⠋⠀⠙⣥⣤⣴⠆⠀⣀⣠⣤⣶⣶⣽⣦⡀⠉⠙⠿⠿⠗⣠⣴⣥⣴⣶⣦⣤⣄
⠀⠀⠈⠛⠻⣿⣷⣿⣿⣿⣶⣶⣶⣶⣶⣿⡿⠛⠓⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣀⣠⣤⣤⣀⣀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣾⣿⣶⣄⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⣿⣿⣿⣿⣷⣦⡀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⠙⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠉⠻⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠙⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠉⠉⠉⠁⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⠥⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠑⣤⡴⢣⣀⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠰⠶⠶⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠉⠀⠑⣠⠴⣒⣿⠀⢠⠆⠀⠲⡈⠉⢳⠋⠀⠀⢀⣶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠐⠋⠀⠀⠈⢹⠀⠀⣸⠃⢠⡅⠀⠀⠊⠀⠀⠙⠋⠀⠀⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⡀⠀⢳⡀⠀⠀⢀⡛⠶⢶⡯⡐⠓⡜⡝⠁⠀⠀⠀⢀⠀⣀⢦⡁
⠀⠀⠀⠘⢦⡀⢀⡞⡵⠃⠀⠉⠀⠀⠹⣦⡀⡈⠉⠛⠛⠎⠀⣼⠁⠀⣠⠴⠚⠉⠁⢀⡨⢷⣰⡄
⠀⠀⠀⠀⠀⠈⠘⠻⠁⠀⠀⠀⠀⠀⠀⠹⣷⣅⢰⡀⢀⠀⣼⣏⡴⠈⠀⢀⣠⡴⠞⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢰⠹⣿⣿⣧⣿⣼⣿⣿⣡⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡛⢶⣦⣄⣲⣾⡯⠙⠓⠯⣽⡻⣿⣥⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢤⣤⣤⣍⣿⣯⣿⣄⣀⣀⣀⣀⡹⡽⣿⣶⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢹⣿⣿⣿⣿⣿⣿⣿⢄⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⣿⣿⣿⣿⣿⠟⠜⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⡿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢁⣰⣿⠟⠉⠀⠀⠈⠳⢾⣿⣿⣿⢯⠁⠀⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣾⣿⣿⡁⠀⠀⣀⣀⣀⡀⠀⠈⣷⠦⣀⠀⡀⢀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⡀⠀⢋⣥⣤⣄⠀⠀⠳⣶⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⢹⣿⡟⠻⠁⠀⠈⠙⠁⠀⠀⠀⠀⠻⠱⡍⢋⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⣌⣠⠄⠀⣸⣿⡇⢠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⣡⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⢀⠀⢸⣿⣙⣿⣶⣾⣿⣿⣷⢠⡤⠤⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡄⢍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⣼⠀⠀⠉⣻⠿⠿⣿⣿⣿⣿⣇⠓⠛⠀⠀⠀⠀⠀⢀⠀⠉⣿⣿⣿⠟⢸⣿⡿⡷⠀⠀⠀⠀⠀⡧⢸⢸
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣷⣿⢸⢠⣧⣾⣿⣶⣦⣤⠈⣿⣿⣿⣦⣤⣤⣤⣴⡶⠞⠁⠀⠀⣿⡏⢀⢰⡿⠁⢀⢧⠱⡄⠀⢀⡄⡇⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢳⣎⠹⣿⠘⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸⣷⣿⣿⣷⣠⣺⣿⣆⣁⠀⢸⢻⠃⣿⠏⣰⡞
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣻⡀⣿⣿⠛⠿⣿⣿⣟⣭⣀⡀⠙⢋⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣇⠀⣏⡿⢀⣯⣤⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣎⣿⣽⣿⣿⠋⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣴⢞⣉⠴⠊⠉⠉⣛⣿⣿⣿⣄⣿⡇⣿⣯⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢿⣘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠀⠀⣀⣾⣿⣿⡿⠛⠁⠀⠀⠀⠉⠉⠉⠉⠻⣿⣿⣴⢃⡿⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣎⢻⣯⣠⣿⣇⣈⣉⠙⢛⣿⠃⠈⢿⣿⣦⡶⢟⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠟⣛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠻⡿⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣬⣻⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣠⣾⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠿⠟⠋⠁⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⠀⠙⠻⠿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣙⣦⠤⣺⣍⠛⣦⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠂⠀⢀⣤⠞⣩⣷⠤⣴⣋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣿⡛⢡⣬⣛⠶⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⠶⣛⣥⣌⢛⣿⡷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠽⡷⣯⣽⣿⣿⣿⡿⣿⣷⣾⣿⠿⣿⣾⣿⣯⣽⢾⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠍⢻⣿⣆⢠⣠⣆⡄⢠⣿⡟⠉⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡛⢛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠉⠉⠁⠀⠀⠀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣯⠴⠖⠒⠒⠂⠀⠄⠉⠉⠉⠙⠒⠦⣤⣹⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⠀⣰⠃⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠻⠿⢿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⢠⡀⠀⠀⣴⠲⣦⡀⠀⢢⠀⠀⠀⠀⠀⠈⠓⠻⣿⣷⣯⡷⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⠈⢿⣦⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣍⣳⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣷⣾⣧⠀⠈⢿⣦⠀⠀⠹⣄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣏⣣⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠸⣿⡿⣿⡄⠀⠀⠻⣷⡀⠀⠙⣧⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣹⢷⣝⢦
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣿⣧⠹⡗⢄⠀⠀⠙⣷⡄⠀⠈⣧⠀⠹⡄⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠹⣿⣿⣟⣧⣷⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⡸⢹⣿⡀⢹⡇⢀⠀⠀⠹⣿⡄⠀⠈⣧⠀⢹⡀⠀⠀⢰⡟⠉⢻⣄⠀⠀⠀⠘⣬⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡗⠈⣿⣧⠀⢻⡄⠀⠀⠀⢹⣿⡀⠀⢹⡄⠀⢳⠀⠀⣻⡇⠀⠘⣿⣆⠀⠀⠀⢻⣏⠻⣿⡞⢷⡀
⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢰⠃⠀⢻⣿⡀⠀⢻⣆⠀⠀⠈⣿⣧⠀⠀⣧⠀⠘⣇⠀⢹⣇⠀⠀⢹⣿⢆⠀⠀⠀⠿⡆⠹⣿⣼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡌⠀⠀⠸⣿⣧⠀⠀⢻⡄⠀⠀⢹⣿⡀⠀⢻⠀⠀⢻⠀⣸⡏⠀⢠⡾⣿⣷⡄⠀⠀⠀⢽⡀⠹⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢿⣿⣆⠀⠀⢻⡀⠀⢸⣿⠇⠀⢸⠀⠀⢸⠀⢸⡇⢀⡿⠁⣿⠟⣿⡄⠀⠀⠘⣿⡀⢿⢋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠈⣿⣿⠀⠀⠘⡇⠀⢸⣿⠀⠀⡿⠀⠀⢸⠀⢸⡇⣾⠃⢀⣿⠀⢸⣧⠀⠀⠀⢸⣧⢸⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢀⣿⡏⠀⢀⠇⠀⠀⢸⠀⣼⣿⡏⠀⢸⡏⠀⣸⡿⠀⠀⠀⢸⢹⡆⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢻⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣤⣿⡟⠀⠀⡜⠀⠀⠀⡌⢀⣿⣿⠁⠀⣿⠇⠀⣿⠇⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣇⣠⡞⠉⠙⠋⠉⠳⢄⠈⠉⠙⠉⠀⢀⡜⠀⠀⠀⢰⠁⣸⡿⠃⠀⢸⡿⠀⣸⣿⠀⠀⠀⠀⠀⣿⠛⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⡠⠋⠀⠀⠀⠠⢃⣼⣿⠃⠀⣠⣿⠃⢀⣿⠇⠀⠀⠀⠀⠀⠸⡇⢁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠄⠀⠐⣶⠟⢡⡏⠀⢀⣿⠃⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⣇⠸⣿⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⣾⠁⠀⣼⡟⠀⣼⡿⠀⠀⠀⠀⠀⢀⡆⢀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⡆⠈⠙⠶⢶⣤⣀⣀⣀⣤⣠⣴⠖⠛⠉⠀⠀⠀⣸⠃⠀⠀⣿⣁⣼⠟⠀⠀⠀⠀⠀⡀⡾⠀⢸⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⢳⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣴⡶⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣨⠿⠋⠀⠀⠀⣿⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⡉⠛⠦⢤⣀⣀⡀⠀⠀⠀⠈⠙⠲⠾⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠊⠁⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢿⣷⢤⠀⠀⠀⠈⠉⠛⠓⠶⠤⢤⣤⣀⣀⣀⣀⣀⣤⠤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⡟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡺⠿⠷⣬⣙⠛⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠐⠂⠉⠓⠦⣄⣀⣉⣙⠳⠦⢄⣀⣀⠀⢀⣀⣀⣠⡤⠤⢒⣋⠶⠶⠶⠾⠿⠛⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣘⣛⣠⣭⣥⣤⣤⣴⣚⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠻⠿⠛⡿⠿⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⣠⡤⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣠⠞⠛⠘⠁⠀⠉⢙⣮⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠁⠀⠀⠀⠀⠈⡀⠀⣨⣧⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⡄⢠⡿⠟⠀⠀⠀⠀⠀⠀⠐⠀⢀⣸⠇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⣼⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⢰⡾⣃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⢤⡼⣫⣴⠟⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⢄⡽⣫⡾⠛⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠀⢰⣪⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⢄⡰⠅⠀⠙⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣲⠿⣛⡽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⡼⠃⠀⣘⣽⢯⣀⣀⣀⣀⡤⠤⠔⣒⠯⢑⣊⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⣴⠞⠋⠛⠛⠛⣿⠓⠀⣴⣿⣯⣄⣀⣀⣀⣠⣤⣤⡴⠶⠿⠛⠛⢛⡛⠉⠛⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣀⠀⢀⡟⠐⠰⣿⡿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠉⢈⣉⣁⣀⣀⣉⢙⡆⠀⠘⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⡄⠘⣿⡇⠈⠻⣷⣖⡒⠒⠚⠛⠛⠉⠉⠉⣉⣉⣩⡿⠛⠀⢀⢜⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣿⣿⣄⠀⠀⠩⠝⢛⣛⣛⡻⠿⠿⣿⠟⠉⠉⠈⣁⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣦⣤⣄⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣄⠀⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⣶⣶⢸⣿⣇⣰⣿⡟⠀⣿⣿⣥⣤⠁⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⣾⣿⣿⣿⣦⠀⢿⣿⡀⢿⣿⣿⣿⣷⣤⣿⣿⠛⠛⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠛⢱⣿⣿⣏⣨⣿⣿⡀⢸⣿⣇⠘⣿⣿⡈⠻⣿⣿⣿⣶⣶⠆⢰⣶⡆
⠀⠀⠀⠀⠀⢠⣾⡶⠀⠻⠛⢿⣿⣆⠀⠋⠙⣿⣿⠛⢿⣿⣶⣿⠿⠀⠙⣉⠀⠑⡆⠀⠀⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣾⣿⣀⣀⣀⣀⡈⢻⣿⣆⠰⡄⠘⣿⡷⠀⠙⠁⠀⠀⠀⠀⠸⣆⠀⢇⠀⢀⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⢿⣿⡆⢿⡿⠇⢱⠀⢈⡀⢀⣨⠤⡦⣄⠀⠇⠀⠈⡶⠾⠷⣌⠀⢠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠇⠀⠀⠀⠀⠣⣠⠟⠋⠀⠘⢶⣿⠀⡀⠈⠀⡇⠀⠀⠈⠳⡌⢀⠞⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡴⠃⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⡎⠀⢀⢎⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡸⣥⠀⠀⢀⣠⠞⡁⠂⠀⢸⢀⡞⠁⢈⠀⠀⠀⠀⣷⢠⢂⠜⢀⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢦⡀⠀⢠⠶⠚⢡⠼⠓⠛⠉⡄⠀⢇⠐⣀⠄⣸⣁⢠⠞⡀⢀⣶⣶⣿⢠⠊⡰⠃⣠⠎⠀⠀⢠⠞⠁⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠳⢤⡀⣙⠢⠤⢤⣄⣣⡀⢣⠀⠂⠸⠀⠘⡴⠁⠀⠙⠋⠁⡄⠙⠦⠭⠘⠡⠃⠔⢁⠜⠁⠀⣠⠖⠁⣠⠔⠁⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡉⢻⡃⠀⠀⠀⠈⠙⠢⣁⠀⠆⢃⢰⠁⠀⠀⠀⠀⢀⡇⢠⠁⡠⢀⠀⢢⡀⠁⠀⣠⠞⢁⡤⠊⡡⢄⠎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢳⡀⠀⠀⠀⠀⠀⠈⠆⠈⠀⠇⠀⠀⠀⣀⢀⡜⠀⠃⠰⠁⢬⠀⢀⡇⣠⠞⣡⠔⢉⠰⠊⠀⡼⠀⣠⠞⣁⡴⠂
⠀⠀⠀⠀⠀⠀⠛⠲⠦⣄⣀⠀⠈⠳⣄⠀⠀⣴⣻⠖⡧⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢸⣀⢸⠀⠁⢀⣠⠘⠁⢀⡴⠾⣷⢴⠟⠋⠁
⠀⠀⠀⠀⠙⠛⠒⠲⠤⠤⣉⣉⡒⠀⠌⠑⠒⠻⠷⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠙⠘⣆⠀⠉⠁⢀⡼⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠬⠍⠐⣲⡆⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⡆⠤⠖⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠒⠀⠀⠐⣦⣸⡿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⣇⠀⢸⠁⠀⠀⠀⠀⢠⡧⠤⢤⣖⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠤⠤⠄⠀⠀⠘⣿⣇⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⣤⣄⡇⠀⢸⣷⣤⡀⢀⡴⠋
⠀⠀⣀⣀⣀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣑⡟⢀⣀⠀⠙⠒⢒⡫⠤⠓⠒⠀⡉⠉⠙⠙
⣠⠚⠉⠉⠀⣀⣨⠼⣀⣀⣀⣀⣀⡠⠴⠚⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢌⡙⢋⡥⠚⠁⠀⠀⠀⣶⣠⢹⡀
⠀⢄⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢀⡀⠤⢒⡠⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⣿⡥⠤⠤⢔⡊⠀⠀⠀⠀⠀⠀⠘⣿⠟⣀⠓⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠈⣠⠄⠀⠁⡠⠄⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⡏⠸⣤⡤⠖⠋⠉⢒⡒⠲⠖⠒⠚⠻⢤⣄⡀⠉⠉⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⢀⠔⡠⠂⠀⣠⠄⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠳⣤⡼⠦⣄⡀⠢⡀⠀⠀⠈⠑⠤⣈⠐⠠⢄⡀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠤⢂⠴⠊⠀⢀⠔⠁⠀⠀⠀⢠⢀⠀⠀⡀⠀⠀⠀⡄⠠⠀⡏⠀⠀⠀⠉⠳⣌⠓⢄⠀⠀⠀⠀⠉⢂⣀⠈⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠞⠁⠀⢀⡰⠃⠀⡠⢠⠂⢠⠃⡌⠀⠀⡇⠉⠀⠀⠺⠀⠂⢱⠀⠀⠀⠀⠀⠘⢏⠢⠑⠀⣀⠴⠚⠉⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⡠⠊⠀⢀⡔⠰⠃⠀⠀⡸⠀⠀⢰⢀⠠⠀⠀⢠⠇⠀⡌⢧⠀⠀⠀⠀⠀⠈⡄⣠⠞⠁⠀⠀⠀⠀⠀⠀⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠀⠀⠀⠀⢠⠃⠀⠀⠘⢸⠀⠀⠀⠈⠸⡀⠀⠈⠳⣄⣴⣿⠟⢊⡿⠁⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢉⣙⡹⢟⣰⡄⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠞⠉⠀⢀⣤⠿⠛⠒⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣦⣤⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢲⣀⠤⠃⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠙
⠀⠀⠀⠀⠀⣠⣠⣁⡀⢠⡾⢉⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠁⠀⠀⣄
⠀⠀⠐⠲⣶⡏⠈⠉⢰⠄⢱⠞⠙⣷⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡄⠀⠺⠛⢔⡟⠀⢉
⠀⠠⠤⣤⣾⣿⣦⠀⡄⠀⠟⠀⢠⣿⣦⡤⠤⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣴⠀⠘⠀⠀⠙
⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⢀⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠰⠆⢀⡙
⠀⠀⠉⠉⢿⣿⣿⣿⣿⣷⣶⣿⠿⣧⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⢁⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠉⠀⢸⠋⠙⡿⠉⢻⡀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢿⣆⠠⣶⣿⡿⢋⣤⠀⢠⣦⠀⠀⠀⠀⠀⣀⡴⢞⣋⣭⣭⣉⣻⣶⠄⣀⢀⢀
⠀⠀⠀⠀⠀⡀⢡⣦⣄⠸⠃⠘⢏⣀⠛⠁⠀⠀⠀⢀⡴⢫⡶⠋⢹⣿⣼⣿⡿⣿⠂⠸⠿⣿⠏⠹⢝⡢⣄
⠀⠀⠀⠀⢠⣴⣾⡻⣿⣿⣿⢩⣿⣿⡅⠀⠀⠀⠀⡾⢱⠏⠀⠀⠈⠿⠛⠛⠁⣉⣶⣾⣿⣿⣿⣶⡄⠙⣝
⠀⠀⠀⠀⢿⣿⣿⣷⣾⢹⣿⣿⣿⣶⠄⠀⠀⠀⠀⡇⣾⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣸
⠀⠀⠀⠀⠹⣿⣿⡿⣫⣶⣿⠆⠁⠄⠀⠀⠀⠀⠀⣧⢹⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡠⢫
⠀⠀⠀⠀⠀⠈⠙⠛⠟⠋⠅⠐⠂⠀⠀⠀⠀⠀⠀⠘⣦⠻⣄⠀⠘⣿⣿⣿⣿⣿⣿⣚⣒⣒⡚⢖⡯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣙⠳⠿⠿⠿⠁⣋⡄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠋⠁
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣟⠛⣛⣛⡓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣿⣿⣿⣿⡿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣏⣹⣤⣷⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣛⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡛⢓⣚⡓⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⣿⠉⡁⢀⣀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠷⠟⠛⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⢿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠚⠓⠚⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣾⣿⡀⠀⠀⢀⡄⢠⡆⣸⠇⠀⠀⠀⠀⠸⣇⢠⣆⢠⡀⠀⠀⢀⣿⣷⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣿⣿⣧⠀⣤⡞⣱⣿⣧⢏⣠⠀⠀⠀⠀⣄⠻⣼⣿⣎⢷⣦⠀⣼⣿⣿⡄⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣾⣿⣿⡟⣱⡿⣽⣿⠟⣵⣿⣿⠀⠀⠀⠀⣿⣷⣮⡻⣿⣯⣿⣎⢻⣿⣿⡇⣿⣿⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⡿⣱⢰⣿⣿⣿⣿⣿⣿⣿⢇⡆⠀⠀⣤⣹⣿⣿⣿⣿⣿⣾⣿⡆⣦⢻⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡿⢡⡆⢻⣿⣿⣿⣿⡿⢟⣿⣿⠇⡄⢀⢻⣿⣿⡿⣿⣿⣿⣿⣿⡿⢹⡈⢿⣿⣳⣿⡏
⠀⠀⠀⠀⠀⠀⢰⡜⣿⣿⣿⣇⠸⡇⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⢸⠇⣸⣷⣿⣿⢣⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣮⣿⣿⡀⡿⠿⣜⡮⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠿⡿⣿⣿⣿⡿⣽⣿⢵⣣⠿⣿⢁⣿⡿⣡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠛⣿⡐⢬⣳⣿⣿⣮⡿⢻⣁⡀⠀⠀⠀⠀⠀⣈⡟⠿⣹⣿⣿⣏⡵⢂⣿⠚⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢹⣦⣙⢿⣿⠦⢿⡟⢶⣼⣿⣯⡿⣼⣤⣼⣓⢆⡀⢀⡰⣈⣯⣤⣧⣿⣽⣿⡿⡶⣹⡯⠼⣿⡿⣋⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣠⣿⣟⣻⣿⣿⣿⣿⠷⠿⠛⠙⢧⡼⠋⢚⠿⠾⣿⣿⣿⣿⣿⣻⣿⣄⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣝⠻⢷⣬⣽⣿⣷⣿⢷⣾⣿⣿⣷⡾⢷⣿⣿⣿⣷⡾⣿⣾⣿⣯⣥⡾⣟⣻⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣛⣿⣿⣌⣛⣿⣶⣿⣿⣿⠾⣿⣿⣿⡿⣷⣾⢿⣿⣿⣿⠷⣿⣿⣿⣿⣿⣋⣠⣿⣿⣛⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣏⡻⠿⣿⣿⣿⣿⣷⣶⣦⣬⣼⣿⣿⣧⣽⣶⣶⣾⣿⣿⣿⣿⠿⢟⣹⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⣻⣶⣛⣿⣿⣿⣻⡏⠙⠁⠋⠋⣸⣟⣿⣿⣿⣛⣷⣟⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣿⣿⣬⡏⢻⣿⣿⣿⣷⡯⡽⠧⢷⣿⣿⣿⣿⡟⣿⣥⣿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠟⢷⣿⠙⣷⣻⣾⣷⡀⢀⣾⡷⢯⣾⠙⣿⡾⠻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠻⣏⢻⣿⣧⠻⡇⣼⢏⣼⣿⠏⣿⠟⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⠃⣿⡇⠀⠀⢸⣿⠸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠟⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢸⣧⣴⠏⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠈⣿⣏⣀⡀⠀⠀⢀⣴⣿⣷⡶⠀⠀⠀⢰⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠸⠷⣦⣄⡀⠀⠀⠀⠈⠁⣿⠁⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⣶⣤⣠⡇⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠸⣿⠟⠁⠀⠀⠀⠀⣴⡾⠁⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠿⣷⠆
⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⡿⢃⡀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⣠⠆⠀⠀⢿⣿⣟⣡⡄
⠀⠀⠀⠀⠀⠸⠿⣿⡟⠀⠀⠀⣼⣿⣷⡾⠀⠀⠀⣼⣿⣿⣶⠆⠀⠀⠀⣠⣾⣃⡀⠀⠀⠘⢉⣿⠟
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠈⠁⣿⣿⠃⠀⠀⠀⠚⠛⣿⡿⠁⠀⠀⢀⣾⣧⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⢈⣼⡟
⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠂
⠀⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣠⣾⣤⠀⠀⠀⠿⢻⣿⠟⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠾⢿⣿⠇
⠀⠀⠀⠀⠰⠿⣿⣥⡤⠀⠀⠀⠚⢹⣿⠃⠀⠀⠀⠀⠾⣿⡶⠀⠀⠀⠀⠈⢁⡟⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢸⡏⠛⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢼⡆⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⠁⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠞


⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣥⠂⠀⠀⠀⠀⠀⠀⢚⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣀
⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠈⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠺⣿⡖⠀⠀⠀⠀⠀⠀⠋⣿⠃
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣸⣦⣄⣀⠜⠀⠀⠀⢠⠖⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣧⣤⣼⡀⠀⠀⠀⠀⠀⣰⣯⣙⡻⠃⠀⢀⣀⣀⠃⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⢀⣤⣾⡄⠀⠀⠀⠀⠀⢰⠋⠉⠉⢣⠀⠀⠀⡠⠊⠉⠛⢿⡇⠀⢠⠋⠀⠈⢿⣶⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣡⣼⣦⡀⠀⠀⢀⠇⠀⠀⠀⠀⠣⡠⠊⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣶⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠀⠈⠑⠢⡜⠀⠀⠀⢀⣀⣠⣀⣀⣀⠀⠀⠀⢠⢀⡤⠺⠿⢿⡟⢙⠁
⡠⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡼⠋⠀⣾⣿⣶⡆⢸⠀⠀⠀⠀⠀⠀⣠⠒⠉⠈⠙
⢇⠀⠀⠀⣸⣦⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⢁⠸⠤⠤⠤⣤⣶⠻⣿⡆⠀⢀⡴
⠀⠑⠢⠴⣿⢋⣼⡿⠒⠒⠂⠤⠵⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡧⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠒⠂⠀⠀⠉⠉⠉⠉⠓⠒⣄⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠐⠂⠀⠀⠈⠘⡤⠶⠶⠬⠥⠀⠀⢀⣐⡂⠉⠩⠛⡇⠀⠀⠀⠀⠢⠘
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢛⣉⣉⣈⡉⠉⣉⠁⠨⠕⢲⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠲⠄⠤⠒⠒⠒⠒⠮⡁
⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢐⣀⣈⣉⣭⠉⠰⢒⠎⠀⠀⠀⠀⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣃⣀⣙⣛⣭⣍⣭⡕⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣐⣒⣒⠲⠤⡜⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠠⠄⣀⡀⠀⠀⠀⠈⠖⠒⠲⠰⠟⣍⣀⠤⠤⠒⠉⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⣛⠛⠛⠛⢡⠀⠀⠀⠀⠀⡒⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠐⠒⠠⠠⠤⠤⠤⠀⠀⡻⠶⠶⢶⣷⠊⠁⠀⠀⠀⠀⠀⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡭⠭⠖⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣿⠁⢠⠊⠙⣦⣏⡷⠀⣟⠹⡆⢰⠉⠙⡆⡴⠋⢻⡆⡞⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡼⠦⠴⠃⠟⠳⠖⠿⠘⠷⠜⠦⠞⠅⠳⡤⠞⠇⠳⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠂⠀⢠⡄⠀⠀⠀⠠⣤⠀⢸⠄⢀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⢸⡇⠀⠀⣤⣤⠀⠀⠀⣾⠁⢠⣤⡄⠀⠀⣼⠃⢀⣤⣄⠀⢻⡀⣶⡀⠀⠀⢀⣤⣄⠀⠀⠀⢰⡆⠀⠀⢠⣤⡄⠀⣿
⠀⠀⠀⠈⠻⣦⣀⣛⣛⡀⠀⠀⠙⣦⣘⣛⣃⠀⠀⠘⢷⣈⣛⣋⣰⠟⠀⠈⢷⣤⣀⣘⣛⣋⣀⣀⣤⡟⠀⠀⠀⣘⣛⣃⣴⠏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣷⡀⠀⠈⣿⣿⣿⡏⠀⠀⠀⠀⠈⠹⣿⣿⣿⡟⠉⠁⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣏⡿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢺⡉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⠿⣿⡿⠇⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠘⠷⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡆⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠸⠇⠸⠇⠀⠀⠀⠀⠿⠀⠿⠀⠀⠀⠀⠿⠃⠻⠇⠀⠀⠀⠀⠀⠀⠿⠃⠻⠇⠀⠀⠀⠀⠀⠀⠿⠀⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠉⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡸⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣀⣀⣀⣀⡼⠁⠀⠀⠉⠛⠿⠿⢿⣿⣿⡿⠿⠿⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣫⣟⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣭⣭⣽⣿⣯⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢖⣚⣫⣭⠿⠶⠟⣛⣛⣋⡩⠭⠭⠭⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⠶⢾⣭⣽⣿⣿⣿
⠀⠀⠀⢀⣠⣴⠶⠛⠉⠥⠖⠚⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠭⠭⠽⣿⣿⣿⣿⡿
⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣬⣤⣭⣥⣤⣴⡿⠟⠋
⣾⡏⠀⠀⠀⠀⢀⣠⣤⣶⠾⠛⠉⠁⣀⣀⡤⠴⢒⣂⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠛⠋⠁
⠙⢷⣦⣤⣴⣾⡿⠛⢉⣠⣤⠶⣚⣫⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉
⠀⠀⢮⣭⣙⣛⣛⣛⣻⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣟⣻⠿⠯⠭⠭⠤⠤⠴⠶⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⣭⣭⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⣛⣛⡿⠷⠶⠶⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⢮⣭⣭⣭⣛⣛⣓⣒⣒⣒⣂⣀⣀⣠⣤⣤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣶⠶⠭⠭⢭⣭⣭⣭⠭⠥⠤⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣩⣭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠭⠍⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡭⠭⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠟⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣻⡴
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣭⣿⣯⣽⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠀⠀⠀⠀⠀⢿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡂⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢀⡅⠐⣼⠀⡀⠤⠀⠀⠘⣷⠇
⠀⠀⠀⠀⢀⠀⢠⣇⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣻⠀⠀⠀⠀⡀⣄⢸⢺⠀⠀⠀⠀⠀⠀⢸⡇⠤⢿⠀⣧⠈⡆⡀⡰⡞
⠀⠀⠀⡼⣤⡣⡼⣿⢷⠀⡀⣄⠀⠀⠀⠀⢠⣿⡎⣻⠀⠀⠀⣼⣶⢪⠊⡛⠀⠀⠀⠀⢀⠀⢊⠁⣿⠁⠀⣿⣃⣀⠗⡡
⠀⠀⠘⣿⠀⡙⢿⡻⡦⢤⢿⠈⡁⠀⠀⠀⠀⡻⣯⡩⠇⠀⠀⣿⠢⢤⡔⠁⠀⠀⠀⠀⣼⠤⣾⣄⣸⠀⡀⣷⡼⣯⠂⢻⡄
⠀⠀⠀⢸⣇⢣⠈⣴⣿⡉⢿⠀⡇⠀⠀⠀⠄⣗⣿⣷⠀⠀⠀⡿⠃⠢⡅⠀⠀⠀⠀⠀⠈⣷⠦⢮⠀⠈⢷⣈⣄⣸⣏⣉⡃
⠀⠀⠀⠘⣸⠴⠣⠃⡀⠃⠈⠙⠃⠀⠀⠀⡁⡷⠀⠋⠀⠀⠀⢧⡤⢤⡇⠀⠀⠀⠀⠀⢁⣭⡤⣈⠃⠀⢹⣏⣻⣈⡏⠈⡏
⠀⠀⠀⠀⠘⠄⠈⣤⡳⠀⠀⠀⠀⠀⠀⠀⠉⠃⢸⡀⢂⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⢦⢰⠀⠀⠀⠀⣏⡀⠸⡇⠀⡇
⠀⠀⠀⠀⡀⢰⣴⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⢘⡇⡰⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⡄⡎⠀⠀⠀⢠⢸⢡⢰⡆⠘
⠀⠀⠀⠀⡕⣹⣻⣻⡿⠂⠀⠀⠀⠀⠀⠀⠀⢇⢸⣴⡁⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⠀⠀⡃⢧⡀⠀⠀⢸⠸⡼⠀⠇⠠⠄
⠀⠀⠀⠀⠙⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠳⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠁⠀⠀⠀⠀⢀⡔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠳⡄⠀⢪⣣⠘⡇⠀⠀⠀⠀⠀⠀⠀⢧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠢⣀⣻⡁⠀⠙⣿⠿⢦⡀⠀⠀⢠⣤⠒⢢⣦⣾⣄⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠈⣿⠛⠓⢦⡇⠀⠀⠉⢢⡀⠀⢸⣠⣿⠋⠁⠈⠳⡄⢀⡤⠒⢠⡃⡠⠋⠀⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠙⢆⠀⢄⠀⢸⢀⠀⠈⢀⡀⠀⠀⠀⢳⣠⠞⠹⡇⠀⠀⠀⠀⣿⣤⠴⠒⠛⡟⢁⡔⠚⠉⠉⢙⣯⠠⠜
⠀⠀⠀⠀⠀⠀⠙⢶⠄⠈⢧⠀⠑⠈⢎⠆⠀⢇⢣⡀⠀⠀⠈⡏⠀⠀⢳⡀⠀⠀⢸⣿⠁⠀⢀⠄⡇⠘⣿⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢽⣦⣿⣧⣅⡀⠀⠐⠀⡀⢣⠑⠀⢀⣀⡇⠀⠀⠠⠓⢤⣾⢿⣟⠀⣠⢞⠼⠁⠀⣀⠀⢀⣤⡤⢤⡀
⠀⠀⠀⠀⠒⠦⠄⠀⠀⠀⡹⠋⠀⠉⠓⠢⣄⠈⠁⠈⠚⣋⣽⣧⡴⠶⠦⣀⠀⠉⠃⠙⠓⡊⠡⠤⠬⡥⣴⣿⣟⣁⣠⠴⠁⠀⢀⣀
⠀⠀⠀⠠⣄⠀⠀⠐⠤⠀⠳⡇⠀⠀⠀⠀⠈⢣⠠⠀⣪⣿⡟⣿⠀⣀⠀⠀⠙⣗⢀⠖⠡⠖⠊⠀⢀⣀⣸⣏⠁⣀⣀⣀⠔⠊⢁⣠⡿
⢠⡔⠒⠲⠤⣉⡒⢄⠀⣀⣀⣿⢦⠀⠀⠀⠀⠀⢷⣾⣿⣿⢯⡁⠘⠛⠀⠀⠀⠈⣇⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢹⠿⠓⠋⠉⠉
⠃⠳⢶⣶⣦⣤⣹⣿⡟⠀⠀⠉⠑⢥⡀⠀⠀⠀⠘⣿⣿⣿⣼⠋⠀⠀⠀⠀⠀⠀⢸⣄⣠⡏⠁⠀⠀⢀⣀⠀⣀⠔⢑⣫⠴⠒
⠀⠀⠀⠒⢒⣀⣀⣀⣙⣦⡠⡀⠀⠀⠈⠓⢤⣠⡶⡿⣿⢿⣿⣷⣄⠀⠀⠀⢐⣠⡜⠀⠐⠷⠤⠤⠤⠴⠖⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠲⠤⠴⠖⠚⠒⠒⠿⢝⠧⣝⡻⠿⣿⣷⣾⣿⣿⣁⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠩⠟⠁⠒⠒⠒⠂⠀⠒⠒⠿⠶⠭⠽⠿⠽⠾⠵⠂⠀⠀⠈⠩⢭⣭⣭⣍⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢶⣿⣟⣻⣟⣫⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣴⡿⠟⢛⣿⠟⠻⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡾⠃⣾⣿⣠⣿⡿⣶⣟⣑⢂⣶⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠋⢰⡿⠀⣸⣿⣿⣇⣸⣷⣼⣿⣿⢿⣿⣿⠟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⡀⢀⣏⣠⣼⠇⢠⣿⣿⣿⣿⣿⡿⢿⣏⠉⠚⣿⣧⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠂⠉⠉⠻⣿⡟⠉⡉⠉⠹⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⡟⣼⣿⣤⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠀⠀⠀⠀⠀⢰⣿⣇⠀⠃⠀⢀⣿⠋⠀⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢷⡀⠘⣯⣀⣠⠈⣶⡐⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⣿⡏⠀⠀⠈⠠⡈⠻⣟⠀⠀⢛⠂⠈⠙⠛⢿⣿⣿⣿⣿⡿⠛⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡸⢫⡆⠀⠀⠀⠀⢓⡀⠉⣶⣤⡈⠠⢿⣦⣦⣤⣤⣤⣸⠟⢳⠆⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠐⡄⠀⠀⢻⣄⠀⠀⢠⣇⣾⠃⠀⠀⠀⠀⠈⢳⠀⣿⡟⠉⠒⢂⢤⡌⢻⣋⣿⣿⣃⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡁⠀⠀⣻⣯⠀⢰⠾⠟⣿⡆⠀⠀⣼⠂⠀⠀⠂⢿⣿⡾⢦⣽⣦⣀⣠⣿⠉⠉⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⢿⡿⠀⠸⢀⣼⣿⡇⠘⠈⠁⠀⠀⠀⠀⠈⠻⣿⣶⣦⣾⣿⡿⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢻⡇⠀⡠⢾⣿⣿⠁⠈⠰⠆⠠⠀⢀⣀⠀⠀⠈⠉⠻⣟⣿⠇⠀⠀⠀⠀⠀⣤⠿⣤⠃
⠀⠀⠀⠀⠀⡴⠋⠀⢠⠀⠀⠀⠀⠀⢘⠀⠀⠀⠸⠀⠀⠊⢀⣴⣩⠁⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⢃⡰⢀⠀⠀⠀⣾⠋⣰⠿⠆
⠀⠀⠀⢀⠘⠀⠀⠀⠈⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⣴⡟⠼⢇⣀⠰⠦⠈⡄⠀⢀⡠⠟⠁⠀⠀⣼⣿⠉⢆⣱⠆⠀⠀⠘⠋⠀⠛⠀⠰
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⢋⠀⠛⠁⠸⢍⣀⠶⠂⢀⠀⠀⣀⠀⣀⠀⣸⣿⠁⠰⠞⠁⣤⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠁⣴⠀⠈⣂⢤⣟⡐⠊⠁⢠⣤⡈⣀⡀⢉⡴⠀⣼⡿⠋⢠⡾⢧⣾⠏⠀⣤⠀⢀⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠘⠁⣤⣈⠩⠔⢋⡀⢠⡾⠋⠛⡁⠉⠹⣧⠀⢰⡿⠇⢰⡿⠑⠏⠁⠀⣸⠁⠀⡾⠀⠀⠀⠀⢰⣄
⡇⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠿⢷⡋⠀⠸⣧⠰⠀⡖⠉⠠⠀⠈⠀⣯⣅⠀⠀⠻⣇⢰⣿⡇⠘⢣⡶⠋⠀⠀⢀⡀⢨⡏⠀⠘⠃⢀⡜⠀⢠⠀⠙
⠹⣆⠠⠄⠀⠀⠀⠀⠀⢀⡄⠀⠈⠙⢲⣄⡀⠀⠀⠸⠟⠀⢠⣠⣦⠙⢿⣄⡄⠀⠘⣿⣟⠻⣶⣇⠀⠀⠀⠀⠀⣧⡄⠀⠂⡄⠀⠀⠀⠀⠋⢀⣤
⠀⠙⢦⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠐⠆⠙⠉⠁⠀⣠⣿⣿⣷⠀⠈⢿⠀⠀⠛⢤⡀⠀⠀⠀⢹⠁⠰⠀⠀⠀⠀⠀⠀⢀⡘⣿
⠀⠀⠀⠙⠻⢠⡶⠏⠁⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⢀⣠⠴⠋⠁⠠⠊⢠⣿⠻⣷⡀⠠⠀⠃⠰⢤⢈⠃⠀⠀⠀⠀⣶⠀⠁⠀⠀⠀⠀⠀⠀⠑⣸
⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⢀⠉⠀⠀⢀⣀⠀⠛⢣⠀⠘⢿⣀⢀⣤⣶⡶⠋⠀⠀⠀⠀⠀⠀⡜⣡⡄⠀⢀⣀⣀⠀⠀⡏
⠀⠀⠀⠀⠀⡌⠙⣷⣐⣄⣀⡴⢈⡵⠲⣤⣦⠀⠀⠀⠀⠀⢠⡈⠻⣆⡀⠀⢿⣠⠂⡉⢙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠓⠒⠉⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⡇⠀⠹⣯⠉⣥⣤⣹⣷⡴⢧⡜⠣⣤⡀⠀⢷⡈⠳⣄⠈⠁⠀⠘⠃⠀⢹⠋⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠒⠀⠙⠉⠉⡀⠊⢁⡔⠈⢻⡶⠀⠱⠖⠁⠀⠂⣤⡎⢡⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⣠⣦⣴⢶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡞⢀⡴⠋⣰⣾⠋⠀⢀⣄⠀⠀⣀⠞⣣⣴⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠋⠀⠀⢰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡁⡀⢠⡄⠈⠵⠋⠀⣼⠛⠁⠀⠀⠀⠙⢀⣰⠟⢠⣿⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠱⠀⠀⢁⡄⠀⣠⡟⠁⠀⠀⠀⠀⠀⢠⠼⣧⡴⠟⢙⣠⣀⢰⡞⠃⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠃⠘⠟⣥⣟⣿⡿⠋⠠⢹⡀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⡀⠛⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠙⣿⡿⣷⣟⣶⠊⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢷⣤⠀⢹⠀⢠⣤⡁
⠀⠀⠀⠀⣤⠀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠛⠟⣛⢀⣾⠋⣾⣿⡟⣥⢋⡰⠲⠄⠀⠀⣷⠀⠀⠀⢀⣠⣴⣦⣾⣭⡆⢸⣂⣀⡙⠇
⠀⠀⠀⠀⡉⠀⠀⢰⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢭⠉⠀⠀⠉⠻⠿⠟⠀⠑⠂⢀⣠⡝⠒⠈⡻⠿⠿⠛⢿⣭⡽⠷⠚⠉⠉
⠀⠀⠀⢨⠁⠀⠀⠀⢱⠀⠀⠀⠀⢉⣤⡀⣀⣠⣤⣀⣀⣀⣀⣤⣼⣤⣤⡀⠀⠀⠀⣀⣀⣤⡙⣿⣷⡾⠟⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠘⣆⠀⣾⣿⣷⣭⣼⣍⣠⣍⣈⣉⣉⣉⣩⣤⣄⣉⣁⣠⣶⣦⣭⣿⣿⡿⠿⠟⢻⡆⠀⠀⠀⠒⠀⠀⠒⠐⠚⠂⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠙⠋⠉⠉⠉⠛⠉⠛⠛⠋⠙⠋⢻⠶⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢆⡠⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠳⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⠀⠀⠀⠀⢀⡀⢄⡴⢤⣚⠿⠋
⠀⠀⠀⠀⠀⡸⠇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⣀⠀⣤⣶⣾⣾⣧⡽⠿⠛⠊⠁
⠀⠀⠄⠀⢠⡅⢰⠀⠁⠀⠀⠀⠄⠀⠉⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡿⠚⠿⠋⠉⠉
⠀⠀⢀⡀⠸⣷⡄⠐⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣤⡄⢠⣄⠛
⠀⢠⠀⠘⠆⠁⣿⠀⠀⠀⠐⣀⠇⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢶⡀⠀⠀⢰⡿⣼⣷⣷⣀⠻⣇
⠀⠸⠀⠀⢠⡆⠀⣹⣦⠀⠀⠈⡄⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⢸⣷⣿⡿⣿⢿⣶⣉⡙
⠀⢸⡆⠀⠀⣸⢧⡛⣷⣄⠀⢀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠰⠆⠀⠀⠀⠀⠀⣰⠀⠀⠀⢻⣿⣾⠷⠚⠛⠋⠛⠀⣀
⠀⢸⠀⠀⠀⣠⡾⠁⠘⠿⡠⠈⠁⠀⠀⢀⣦⣜⣻⣧⣀⡰⢀⣴⣤⣀⣤⣈⣿⠀⠀⠀⠚⠋⢀⣤⣶⣶⠿⠟⠋⠉⠀⠂
⠀⢸⠇⠀⠀⠟⠳⡀⠀⠠⠃⠀⠀⢀⠻⣤⡈⠉⠉⠉⠉⠁⠈⠀⠉⠙⠛⠛⠛⠃⣴⡄⠀⠾⠿⠋⠉⠁⠀⣀⡀⣴⣿⣀⠀⡀
⠀⠈⠀⡀⠀⠀⠀⢡⡌⠀⠀⣶⡄⠺⠧⣞⠉⠓⠒⠲⠄⠀⠀⣤⣤⠤⣤⣤⠀⠀⠠⠶⠆⠠⠄⠀⠀⠈⠈⢁⡘⢿⣿⣿⣴⡇
⠀⢀⠐⢧⠀⠀⠀⠳⠀⢳⠄⠛⠀⣀⠀⠙⠀⠀⠐⠂⠀⢀⣀⠀⠀⣀⣠⣄⣄⣀⣀⠀⣀⠀⣠⣤⣤⣈⠑⢾⣷⣾⣿⡿⢧⡇
⠀⠃⠀⠀⢠⣄⠀⠀⠚⠋⠁⠀⠀⠻⠀⢠⣿⠐⣤⣦⣌⠈⢁⣤⠶⠟⠻⡉⢿⣉⣿⣿⣿⣡⠿⣿⣷⢿⡄⠀⢹⡏⣿⣎⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠖⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀⣘⡒⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠽⠀⠀⢄⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡆⠀⠈⠁⠐⠒⢾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡌⠀⠀⠀⢘⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠛⣷⣶⡤⣄⣰⣿⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣄⣀⡤⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣕⡋⠀⢰⣿⠿⠿⠿⣿⣿⣯⢈⠃
⠀⠀⠀⠀⠀⠢⢢⡀⠀⢸⣽⣿⣿⡟⡿⠮⠀⠀⠀⠀⢰⣀⣤⣆⡠⠚⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⢿⣏⢀
⠀⠀⠀⠀⠀⠾⠺⣿⠿⡾⣿⡾⢉⠁⠀⠀⠂⠀⠀⠀⠀⠉⠡⠈⠉⠑⠢⢤⡀⠀⠰⠐⠀⠀⠀⠀⠘⠀⣯⠓⠰⠃
⠀⠀⠀⠀⠰⣆⣿⣦⢸⣳⣿⡟⣺⡄⢀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢋⣤⡐⠛
⠀⠀⠀⠀⠀⠞⠃⠉⢉⣿⠚⡇⡿⠟⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠁⠀⠀⢐⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡈⠀⢀
⠀⠀⠀⠀⣍⡂⠀⢀⠞⡇⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠰⣿⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⣧⠀⡘⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠶⡿⢷⡀
⠀⠀⠀⠀⡼⡟⢶⠃⣢⣤⠄⠀⠀⣀⣀⣠⣴⣶⠎⠱⠴⠀⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋⠁⠒⠛⠛⠓⠾⠯⠁⠀⠡⠀⢀⡌⠃
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡰⢠⣆⠤⢢⠄⠴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣤⣤⣤⣠⣤⣤⣠⣤⣄⣤⣠⣤⣤⣤⣤⣤⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⣉⡉⠉⣉⣉⡉⠉⣉⣉⠉⢉⠉⢉⠉⢉⡉⠉⣉⣉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣯⣅⠈⢹⡟⠁⢸⣿⣿⡇⢸⡇⢸⣷⠟⠁⢸⣿⣍⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⣽⡷⠀⠸⡇⠀⠸⡿⠿⠄⢸⠇⢸⠟⠷⠄⠸⣿⡤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⢿⡄⠀⠀⣠⡾⠟⠻⢷⣄⠀⠀⣠⡿⠁⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠈⢻⣆⠀⣿⠀⠀⠀⠀⣿⠀⣰⡟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠻⡀⣿⡀⠀⠀⢠⡿⢀⠏⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠘⠿⣦⣴⠟⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣆⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣴⡾⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢰⡿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⡄⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⠾⠃⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠷⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⣤⡶⢶⣤⣀⠀⠀⠀⢸⣧⠀⢰⡆⠀⣾⠇⠀⠀⠀⣀⣤⡶⣶⣤⡶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁⠀⠀⠉⠀⠀⠈⢹⣧⠀⠀⠈⣿⠀⢸⡇⠀⣿⠀⠀⠀⣾⡏⠁⠀⠀⠉⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡄⢸⡇⢰⡟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠈⠁⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣏⣙⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠛⠛⠋⠙⣿⣿⣿⡻⣖⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠵⣦⣤⣤⠆⢹⣿⣿⣷⣌⡲⣄⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣀⣦⢠⡟⣿⣿⣿⣿⣿⣿⣧⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠤⣤⡖⠒⢦⣴⠿⣾⣿⣿⡾⢷⡿⠾⠛⠛⣿⣿⣿⡷⠊⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⢦⠙⢦⡈⢣⣀⣼⢿⠦⠈⣿⠁⣠⢾⣿⢤⣤⠞⠋⢉⡏⢀⡚⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⣄⣱⣶⠋⢳⠞⠀⣀⣉⠈⢤⠔⠛⢶⠉⣿⣶⠋⡰⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣈⣹⡟⣸⡀⢸⠀⠀⠁⣸⠀⠀⠀⠀⠀⠀⡿⢁⡜⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣽⠏⣸⣿⡷⣾⡉⢉⡛⠟⠉⠀⡈⠉⣿⠟⣠⣿⣍⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⣿⣿⣿⣿⣧⣸⣇⠉⠢⣼⠀⠠⢴⣾⣏⣴⣟⠿⠳⢭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣽⠿⣥⣤⣭⣭⣭⣙⣷⣟⠶⠞⣻⡯⣿⣿⣻⡄⢰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣯⣝⣳⣞⣿⡁⠄⠹⣧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⢻⡿⠿⠛⡻⣿⣿⣻⡿⠟⢉⣠⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣷⣾⡿⡓⢈⣿⣿⣿⠿⠛⠋⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡉⠻⣿⣹⣯⣤⡜⠃⠀⢸⠃⠀⠀⡠⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠉⠛⣿⠓⠞⠉⡇⠀⠀⠸⡇⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⢶⣇⠀⠀⠀⡇⠀⠀⠀⣷⠄⠀⠀⠀⠈⠛⡆



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣀⢀⣠⡄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡽⠀⠀⠀⣸⠀⠈⠉⢰⠃⠘⠛⠀⠀⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⡀⠐⠋⠤⡄⠀⢀⣈⣣⡀⠀⠀⠀⣧⣼⠃⡔⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⢉⣽⣦⡀⠀⠰⡀⠀⠀⠸⡶⠃⢠⠀⣀⠀⠀⣰⠧⣿⠊⠀⢶⠋⠈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣴⣯⢫⡾⠓⠦⣤⣥⣤⣀⡀⠀⠀⡌⠘⠿⠃⣰⠃⠀⢸⠀⠀⠐⠒⠓⠆
⠀⠀⠀⠀⠀⢀⣠⠜⠳⢤⡄⠉⠻⣅⠀⠀⠀⠈⠉⠙⠻⢷⣤⡃⠀⢀⡜⡏⠀⠀⠸⡇⠀⡠⢀⡥⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢹⣀⣄⣮⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠘⠀⡇⠀⠀⣸⠇⠈⣠⠎⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢠⣴⣮⠅⢀⡉⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢷⣀⣠⠏⠀⣰⠃⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠋⠙⠀⠀⠈⠉⠑⠒⠄⠹⣧⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⣠⠩⠅⠀⣰⣇⠀⠀⠀⠀⣠⠞⢁⡀
⠀⠀⠀⠀⣠⠒⠢⣄⣀⠀⠀⠀⠺⠷⠀⠈⠻⢿⣶⣤⣤⣤⣶⣿⣺⢯⡇⠤⠄⣀⡞⠹⣽⣶⣤⣠⠞⠁⣀⡰⣆
⠀⠀⠀⠀⠈⠲⠤⣾⣩⡟⠒⠒⠒⠒⠲⠦⣄⡀⠈⢉⠛⠛⠿⠿⠗⢋⣠⡴⠞⠻⣻⣦⣀⣩⠟⠁⠀⠀⣹⣀⣰⠋
⠀⠀⠀⠀⠀⢀⣠⠤⠌⠙⢦⣀⠀⠀⠀⠀⠈⠙⢶⣄⠀⡔⠘⣠⠖⠋⠁⠀⠀⠀⠙⠻⢿⠏⠈⠉⠉⠐⠂⠀⠉
⠀⠀⠀⠀⠀⢴⡆⠀⣠⣄⠀⠙⠷⣦⣤⣀⣀⣀⣼⡿⡼⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡰⡄⠀⠈⠉⠉⠉⠉⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠑⣠⣤⠘⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠚⠁⠙⢒⡦⠀⠀⠀⡐⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⢠⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣻⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⡀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⣤⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⢿⣧⣀⣀⣀⣠⣴⠾⢛⠁⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⢸⣧⣠⡾⠁⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⣿⣟⠀⠀⠀⠀⠀⣠⣾⣯⣤⠆⠀⠀⠀⣼⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⣾⣧⣄⡀⠀⠀⠀⠀⠈⠉⣹⠃⠀⠀⠈⠉⣸⡿⠃⠀⠀⠀⢠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣤⣀⣀⡏⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⣿⣷⠂⠀⠀⠀⠀⠀⣤⡾⠁⠀⠀⠀⠀⣿⠈⠙⠁⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢠⣿⣯⠄
⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⡴⠃

⠀⠀⠀⠀⠀⠀⣤⣶⡶⠀⠀⠀⠀⠀⣰⣶⠖⠀⠀⠀⠀⣰⣦⣤⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠⣤⣴⡶
⠀⠀⠀⠀⠀⠀⣿⣿⣴⡖⠀⠀⠀⣰⣿⣡⣤⠀⠀⠀⣰⣿⣿⣏⡄⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠉⣿⠏⠀⠀⠀⠘⠛⣻⡿⠃⠀⠀⠰⠛⣿⣿⡿⠁⠀⠀⣠⣾⣿⣷⡶⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠞⣿⡿⠃
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣶⡿⠃
⠀⠀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⣿⣿⣏⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡀
⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⣠⣾⣷⡖⠀⠀⠀⠛⢹⣿⠋⠀⠀⠀⠀⠀⣼⢤⡄⠀⠀⠀⠘⠛⣿⡿⠁
⠀⠀⠀⠀⠐⠛⢻⣷⣶⠂⠀⠀⠈⠉⣽⡟⠀⠀⠀⠀⠀⠟⣿⡟⠁⠀⠀⠀⣀⣀⡼⠁⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⡏⠙⠃⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣿⡿⠂⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⢰⡿⠂
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⣷⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⡻⣿⣻⣿⠃⠀⠘⠻⣿⠀⠀⠀⢸⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⢴⣴⣷⣀⡀⠠⠁⢸⡿⣼⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡖⠋⠉⠉⢽⣄⢈⣿⣿⡐⠝⢟⣻⣥⣿⣾⣿⣿⣦⣿⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⣲⣦⣭⡄⠈⠉⣿⣶⢿⣿⣿⣿⡏⣿⡿⣿⣿⣿⣏⡉⠉⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⣻⡿⠓⢶⣤⣠⣆⣯⣝⣛⢿⢿⠇⣿⣿⣿⠛⠿⣿⡟⣷⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⢀⣿⠛⢿⣿⣿⣿⣿⣿⢶⣶⠷⠈⠑⢲⣩⣽⣇⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣛⣿⣿⣷⣿⣟⡉⠙⠁⣠⣾⣤⣶⣤⣾⣷⣿⣽⠵⣃⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣿⣿⣿⠻⣿⣟⢻⣯⣛⢿⣿⣿⣿⣿⣏⠉⠁⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣼⡿⠛⠉⢉⣿⣿⣿⣃⣠⣾⣿⡏⣿⣮⡛⠿⢟⣿⡿⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢰⣿⣿⣿⣿⡿⣿⣿⣿⡷⠿⣿⣿⣿⣿⡉⠙⠂⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣝⠛⠋⣰⣿⣿⡇⠀⢠⣾⣿⣿⣋⡛⠓⠀⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣾⣿⣿⣿⣿⠇⣿⠀⢀⣼⠿⣿⠋⠉⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣿⣿⡟⣸⠇⢾⠉⠀⠚⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡭⠴⠛⠋⠀⠁⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢿⣿⣿⣿⣿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠙⢆⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⣀⣤⡄⠀⠀⢠⣾⣶⣤⡀⠈⢳⡀⠀⠀⢎⠀⠀⠱⣈⠀⢀⠔⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣰⠟⠉⠉⠉⠀⠀⠀⠀⠀⠙⠧⠀⠀⢷⠀⠀⠀⡑⠀⠀⠀⠀⠁⠀⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠁⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠐⢤⠀⠀⠀⠀⠀⢠⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⠀⠀⣤⡄⠀⠄⠀⣤⡄⠀⠀⠀⠀⠀⣷⠀⠀⡰⠁⠠⠂⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠈⠁⠀⠀⢀⡀⠀⢹⡔⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠉⠀⢸⣇⡠⠂
⠀⠀⡀⠄⠤⢄⠀⡠⠒⠉⣁⣼⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⣼⠉
⢀⠴⠒⠒⠀⠀⠙⡤⠊⠉⠀⠀⣧⠀⠈⣷⣶⣄⣀⣀⡀⠀⢀⣀⣀⣤⣶⣾⡟⠀⠀⡟
⡇⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⢹⠀⠀⠘⢿⠿⠛⠛⠻⠛⠛⠛⠻⣿⣿⡟⠀⠀⢠⠇
⠈⠒⠟⠹⠀⡆⠀⢂⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠘⠤⠔⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⣿⡿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣳⣎⣭⡿⢉⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡷⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠛⠿⣿⡿⣿⠱⣤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡟⠓⠄⣀⡔⠉⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠖⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣭⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⣾⣿⡟⠁⠀⠀⣸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⢱⣿⣿⠁⠀⠀⢠⡿⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣬⣻⣏⠀⠀⠠⢏⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣿⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣀⣠⣼⠿⠻⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠛⠃⣀⣴⣿⣶⠦⠀⠐⠚⠛⠋⢁⣀⣀⣝⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠿⣿⣿⣿⣿⢿⣿⡆⠸⠋⠉⠉⠀⠀⠀⠀⢀⠤⠒⠉⡝⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⣿⣿⣿⣄⠈⠙⠼⠿⣦⣹⣧⡀⠀⠀⠀⣀⣤⠒⠉⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⠆⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣼⠀⡞⠻⠟⢿⣿⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⣧⡷⢨⣿⣿⣿⡏⠀⠀⠀⢀⣤⡾⣿⠀⠀⠀⢠⣤⣤⣤⢤⣀⣠⣤⠦⡄
⠀⠀⠀⠀⢄⡾⠏⡑⢺⣽⣶⣿⠁⠀⠀⢠⡾⠋⢰⡏⠀⠀⠀⠀⠉⠉⠁⠠⠿⠁⠛⠛⠛⠛⠛⠛⠿⠷⠶⠶⠦⠦⠤⠤⠄
⠀⠀⠀⠀⠁⠀⢸⠇⣙⣿⣿⠇⠀⠀⢀⠚⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣴⣾⣴⣤⡤
⠀⠀⠀⠀⠀⠀⠸⣀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢯⣆⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠹⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣟⣔⣤⠄⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⢨⣬⡿⠀⠀⠀⠀⠀⠀⠀⡠⢿⣿⣾⣷⠶⡀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⣺⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡟⣉⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢯⣼⣧⡀⠂⠀⣠⠎⠀⠀⠀⣺⢿⡿⢫⣿⣿⣿⡿⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡮⠛⠉
⠀⠀⠀⠀⠀⠀⣀⢾⠷⠿⡿⢷⣾⣋⡀⠀⢀⠤⠐⠂⠀⠈⣷⢿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣤⣄⣤⡤
⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣾⣿⣿⡿⠿⠯⠥
⠀⠀⠀⢘⣁⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⣿⡿⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠁⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠀⠀⠀⠀⢀⣤⣾⣿⡟⠀⠀⠈⣿⣿
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠋⠀⠀⣿⣿
⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿
⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣾⣿⣿⠏⢿⣿⣿⣶⣤⣄⣤⡤⠄⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⠇
⣿⣿⣿⠀⠀⠻⢿⣿⣿⣯⣀⠀⠀⣀⣿⣿⣯⠀⢤⣄⣀⣀⣀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⡀⠀⠀⢠⣿⡟
⣿⣿⣿⡄⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣦⣄⣈⣉⣙⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⠀⣸⣿⣿⣿⠿⢉⣠⣴⣿⠟
⢸⣿⣿⡇⠀⣠⡖⠀⠀⠀⠀⣤⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⣭⣉⣉⣤⣾⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠸⣿⣿⣧⠀⣿⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⠉⠉⢹⣿⡏⠀⠀⠀⠉⠛⠛⠿⠿⠛⠋⠁
⠀⢻⣿⣿⣷⣿⣄⣀⢀⣀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠴⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣞⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⡀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢐⠖⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠜⢀⠔⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠀⠚⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⠋⠀⠀⠀⠀⠀⡐⠁⢀⣝⠁⠀⠀⠀⠀⠀⠘⡩⠃⠀⣾⠆⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢮⠎⠀⠀⠀⠀⠀⢀⣔⣀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠼⡴⡣⠊⠁⠀⠀⠀⠠⢈⡃⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⡠⠊⢠⠮⡤⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⡔⣁⣀⡉⢀⠜⠁
⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡀⡠⠊⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⠁⢀⢌⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠠⠃⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠾⣥
⠀⠀⠀⠀⠀⡰⠁⠀⢾⠖⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠈⠉⢉⡩⠛⠁⠀⠀⠀⠀⠀⠀⠰⠂⡲⠀⢠⠞⠁
⠀⠀⠀⠀⠴⠐⡞⡴⠋⠀⠀⠀⠀⠀⠀⠾⠐⠈⡩⠋⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠔⠁
⠀⠀⠀⠀⢀⡾⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠴⠊⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣸⠔⠁⠀⠀⠀⠀⠐⡅⠤⠤⡤
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠒⠁⠀⠀⠀⠀⠀⡘⠀⠑⣄⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣄⣀⣀⣀⠀⡴⠁
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠘⡢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠾⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠰⣶⠞⠋⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠻⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣠⡾⠇⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢸⠉⠀⡠⠀⠀⢀⣀⣀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⡤⢊⠷⠀⢰⣿⡿⠿⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣠⠟⣷⣡⣠⡦⣸⣿⠀⢀⡴⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⠋⠀⢿⣿⣿⣿⡇⠛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⠀⡉⠀⠐⠞⠁⢰⣦⣾⣿⣿⣿⣿⣿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠄⠰⠿⡆⣽⣿⣾⡶⣀⣤⣀⣴⣿⡠⠴⠚⡉⠤⠄⠠⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣿⡄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⢟⣩⡽⠒⢊⡀⠤⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⣿⡿⢿⣿⡄⠀⠀⣿⣿⣿⣿⢿⣼⣴⠟⠁⢠⣴⡟⣁⣤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢸⠏⠃⠀⠀⠿⣿⣿⣷⣿⣽⣯⣵⣤⣿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠾⣿⣿⣿⣿⣿⡘⠟⢻⠽⢟⡟⠉⠀⠄⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢁⠸⣏⣨⣿⣿⣿⣷⣴⣶⣴⣯⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⣿⣾⣇⣤⣌⣹⣭⣥⣬⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡭⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⢋⣡⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢋⢰⣿⣿⣿⣿⣿⢿⣿⣿⢫⣽⣿⣿⣓⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⡀⠀⠀⣀⣠⣾⣿⡀⢱⣾⣷⣿⣿⣿⣿⡵⠟⠙⠻⣆⣀⣤⠿⠛⢿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣌⠙⣷⣄⠐⠾⠿⠋⠉⠁⠸⣿⣿⠋⠙⠁⠁⢀⣠⣾⡿⠿⠛⣓⣶⣞⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠈⠛⠿⣦⣄⠀⠀⠀⠀⠘⣻⣤⣴⡶⠟⠋⠁⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡤⠔⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠳⣄⠀⠀⠀⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠋⠉⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⢀⣀⡀⠀⠀⠄⠀⠙⠀⠀⠀⠀⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⡀⢹⡆⠀⠀⠀⡾⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠁⠀⠀⠀⢀⣠⠖⠋⠁⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⠁⠐⢁⡜⠀⠀⠀⢀⡤⠊⢀⠂
⠀⠀⠀⠀⠀⠀⠜⠁⢀⡴⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡴⠋⠀⢀⠤⠖⠉⣀⠴⠁⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⢀⡠⠞⠋⠀⠀⠈⠀⣀⠤⠚⠁⠀⢀⠀⠐⢠⡞
⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠖⠚⠁⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⠶⠊⢀⣤⠖⠋⠄
⠀⠀⢀⠞⠁⠀⠀⠀⠳⡄⠀⠀⠐⠶⠲⠶⠖⠒⠶⠒⠊⠉⠁⢀⣀⣤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣤⠶⠊⠉⠀⢀⡜
⠀⠀⢸⡀⠀⠀⠀⠀⠄⠀⠉⣀⣒⣀⣀⣀⣀⣀⡤⡤⠤⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠂⠉⠁⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠻⢆⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢀⣀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠁⠀⠛⠲⢄⡰
⠀⠀⠀⠀⠀⠉⠛⠛⠒⠶⠶⠶⠶⠒⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⢷
⠀⠀⠀⠀⠀⠉⠑⠖⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⠔⠒⠈⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠖⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢉⣁⣠⠤⠒⢂⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠶⠚⢛⣉⠩⠤⠒⠂⢁⣀⠴⠛⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠊⠉⠀⠀⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠄⠐⠒⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠤⠤⠀⠐⠒⠒⠈⠉⠉⠀⠀⠀⢀⣀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡒⠂⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠉⠉⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠛⠁⠈⠉⠓⠶⠤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠤⢤⣤⣤⣤⠴⠶⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠙⢢⣄⡀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣠
⠀⠀⠀⠀⠀⢠⠊⢀⡾⠃⠈⠓⠤⢄⡈⠉⠒⠒⠲⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⣀⣀⡴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠳⣼⡁⠀⠀⠀⠀⠀⠁⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢹⡋⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣤⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠠⠤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠲⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣶⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣙⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀⢒⣠⣤⡶⠏⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢀⣀⡀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⣀⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠛⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⢦⣄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣀⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠐⠒⠂⠀⠀⠀⠀⠠⠴⢞⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠂⠀⠀⠀⠀⢀⣠⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⠈⡀⠀⠀⣙⠇
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣴⠋
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⡆⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠾⣷⣿⣿⣿⣿⣏⠑⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡟⠓⠎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⡟⢿⣿⡿⢾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢈⣿⣿⣿⣿⣿⣇⠈⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠠⣾⣿⣿⣿⣿⣿⣿⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡛⠀⡵
⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠋
⠀⠀⠀⠀⠀⠀⢸⠁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣤⣤⣤⣄⣀⣠⣤⣤⣼⡟⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣮⡳⣄⠀⠀⠀⠀⠀⠀⠠⡿⠿⠿⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢧⠀⠉⠛⠿⣿⣿⣷⣵⣦⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢧⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣦⣤⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⣧⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣏⡉⢉⣉⣉⣱⠶⣿⣶⣾⣿⣿⣿⣶⡦
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠉⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡲⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢎⣴⣾⠟⠿⢦⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⠋⠀⠢⠀⣈⠚⠁⠈⡙⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⡂⠖⠲⠁⠀⢿⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠠⠀⠠⢀⠀⠂⠀⣼⠃⠹⠁⢲⡏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡰⠒⠒⠀⠄⡃⠀⢢⠐⣅⠀⣧⡸⣅⡀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠍⢀⠄⠀⠀⡴⢹⠋⠀⢈⠉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡕⠀⠀⠀⠱⠂⠁⠈⠀⠄⠤⢠⠋⠶⡌⠀⢀⢴⠆⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡈⡀⠰⠀⠀⠀⠨⠀⡐⠆⠰⢀⠀⡺⠀⠀⠀⠄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠀⠀⢀⠀⠀⠀⠀⠀⣷⣤⣽⡀⠀⠀⢀⠀⠀⢨⣤⡤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⡃⢦⡸⠄⠀⣂⣀⠀⢉⠈⡻⡇⠀⠀⠀⠀⠀⠐⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢁⠀⠀⠁⢧⠀⠀⠀⠀⢸⡴⣿⢵⡖⣒⡲⠒⢪⣇⠼⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠂⠀⠔⣆⠀⠀⢀⠈⠳⠀⠀⠀⠈⠀⠀⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣂⠀⠀⠈⣄⠀⠀⠀⠀⠐⡆⠀⠀⣄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⣍⠂⣈⢆⠈⢀⠀⠀⢂⠀⢀⡉⠆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠁⠑⠀⠐⢤⡎⠀⠸⡇⡘⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⢱⠇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠆⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡥⠀⠹⠇⠨⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡤⠴⠆⠰⠤⢤⡤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠲⠒⠀⠐⠀⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠂⠀⠀⠁⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡉⠄⠀⠀⠀⡀⢀⣢⡀⠀⢤⣄⣀⠂⠀⠲⠤⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠋⠀⠀⡠⠖⠋⡟⠉⠉⠻⣆⠘⡗⡩⣽⢶⠉⡷⡖⠓⠲⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⡠⠪⠀⠀⠀⠘⠂⠀⠀⠘⡆⠃⢳⡈⠢⡉⠈⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⢒⣼⢆⠊⢳⠀⠁⢄⢣⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠉⠡⢈⠳⡀⠅⡠⣬⢡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⡿⠄⠀⠀⠀⠀⠀⠀⡜⢁⡆⠀⠀⠊⠀⡈⠐⠨⢈⠈⠙⢦⡀
⠀⠀⠀⠀⠀⢠⠁⠈⠀⢸⠀⠀⠠⠐⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠖⠉⠡⣤⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠲⡆⠀⠀⠀⢸⠶⠁⠀⠀⠀⠀⠀⠀⠐⠀⠴⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢇⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⢀⢠⡶⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⢸⣆⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡼⢉⣀
⠀⠀⠀⠀⡆⠘⡦⠀⠀⠀⡈⠘⢆⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⠂⠀⠀⠀⠘⢢⡀
⠀⡀⠁⠀⢀⠀⣀⣐⠚⠠⢨⡤⠌⢣⡁⢸⡄⠀⠀⠀⠀⠀⠀⣼⠈⠁⠐⠀⠀⠀⠀⡚⢀⡄
⡀⡠⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⣉⡠⠀⠨⠀⠐⠁⡤⠆
⣄⠈⠀⠀⠀⢀⣠⠤⠒⠓⠉⠃⠁⠀⠀⠸⠷⣗⢄⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠂
⠈⠀⠀⢀⠊⠁⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠉⠲⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⢀⡄⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢹⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⠠⢀⠀⠀⣸⠀⠂⠐⠆⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠲⣆⠀⠉⠑⠢⠤⠤⠴⠀⠀⠀⠓⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⠁
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⣤⠄⠀⢀⠤⠒
⠀⠀⠀⠀⠀⠀⠁⠐⠤⣄⣀⠀⠀⠀⠀⣀⢀⡐⠤⠖⠂⠈
⣿⣿⡿⢿⠾⠿⣿⡿⣷⡶⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⡀⣶⣶⣶⣶⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣶⣶⢻⣿⣤⣿⡟⠀⣿⣿⣤⡄⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⣠⣿⡿⢿⣿⠄⢹⣿⡀⣿⣿⡿⣿⣶⣿⣏⠉⡁⠀⠿⠃
⠀⠀⠀⠀⠀⢀⣄⡀⣶⣿⣿⡏⠘⠿⢿⣧⣾⣿⣦⣈⣿⡇⠘⡿⠇⢨⡛⠛⠛⠻⠃⠸⠿⠇
⠀⠀⠀⠀⢠⣿⠏⠀⠈⠀⢻⣿⣄⢠⠈⢿⣷⡈⠻⠟⠉⠉⠀⠐⡄⠀⢡⠀⢸
⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣦⢹⣿⠆⢧⠈⠋⠀⢀⣀⣄⡀⠀⠀⠘⡦⣼⣦⡂⠀⡄⠀⠌
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢾⣿⠀⠁⠀⠈⠆⣸⠞⠉⠰⢷⡷⠀⠀⠀⡇⠀⠀⠙⢎⠁⡼⠀⢀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⡞⠁⠀⠀⠀⡰⠃⠀⡇⠀⣧⣀⠀⠀⠈⣗⠀⢀⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠞⢦⣀⣠⡴⠊⡄⠇⠀⠇⡞⠁⢸⠀⠀⢀⢸⡀⢣⠃⡔⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢄⠉⠳⣄⠀⠊⠁⡔⠉⡉⢁⢰⠀⠃⢠⠞⠰⣧⠰⠃⢄⠲⠟⡿⠀⢁⠎⣠⠊⠀⢀⠔⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡙⠲⣶⠉⠙⠛⠒⢦⡑⠀⢂⢇⢠⠃⠀⠀⠀⠀⡇⢀⠉⡁⠀⣀⠀⠜⠁⢀⠔⢁⡠⠊⣠⠖⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠸⡄⠀⠀⠀⠀⠈⢣⠈⡌⣿⠀⠀⠀⠀⡰⠁⠆⡴⠡⠁⠈⡇⢀⠔⣡⠔⠩⠔⠁⡼⠀⣠⠎⣠⠄
⠀⠀⠀⠀⠀⠐⠢⠤⣀⡀⠀⠙⢄⠀⢀⣴⡳⢪⢀⣠⣴⣶⣿⣿⣿⣿⣶⣤⣀⠀⣇⢸⠀⠉⢈⡥⠊⢀⡠⠾⡧⣼⠓⠉
⠀⠀⠀⠈⠙⠓⠒⠶⠤⠌⣑⡒⠀⠑⠒⠛⢓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠋⢇⠈⠉⠀⣰⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣒⡒⠉⢒⠆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⡆⠒⣸⠁⠀⠀⠀⠀⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⣈⢳⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡷⠀⣧⡀⠀⠀⢀⡾⠓⠛⠛⠋⠉
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⠉⠁⠀⠀⠈⣿⡶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡸⠿⣷⠃⠀⠻⢿⣒⠴⢫⣤⣄⠠⠤⠤⠤
⢀⠴⠚⠛⠋⢙⣛⣀⠀⠀⠀⣀⣠⠴⠂⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡐⠲⢄⡤⠒⠉⠀⠀⣴⡹⡄
⠩⣄⣠⠴⠚⠉⠀⠀⠉⠉⠉⣁⠤⠐⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡧⠤⠬⣴⡋⠀⠀⠀⠀⠀⢻⡿⢃⣒⠲⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⣀⠤⠂⢁⠠⡂⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢹⠉⠧⠤⠖⠋⠉⢐⠒⠒⡒⠛⠣⠄⣀⡉⠉⠙⠒
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⢐⠕⠈⢀⠔⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠋⢳⡶⠓⠢⢤⡓⢄⠀⠀⠈⠒⠬⡑⠢⠤⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡠⠚⠁⠀⡔⠁⠀⠀⠂⡰⠰⠀⢠⠠⠀⠀⡆⠡⠸⡀⠀⠀⠀⠙⢦⢑⠄⠀⠀⢀⣠⠵⠒⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⡠⠊⠀⡀⡡⠃⠀⢀⠃⠀⡈⠀⡀⠀⢸⠀⠁⢧⠀⠀⠀⠀⠈⠆⠁⣠⠖⠉⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠜⠁⠁⠀⠀⡎⠀⠀⠇⡆⠄⠀⠀⢣⠈⠈⠳⣄⣠⣶⡶⢺⡞⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆⠀⠀⠈⢙⣛⣾⣏⣄⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⣠⠼⠛⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠉⢱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠃⠀⠀⠀⢊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⠀⠀⢠⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠓⠛⣶⢀⠀⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣤⣦⣀⣄⣀⣀⠀⠄⠁⠸⢣⣄⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⣿⣏⠉⠉⠙⣦⠁⠀⠋⠋⠻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⣿⡏⢀⡀⢽⣇⠀⣴⠛⠀⠙⡻⣿⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠄⠨⣿⣿⡟⢻⣷⠈⠙⠘⠋⠘⡯⢨⢥⣄⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠒⣲⣿⡏⢇⠀⠐⢧⠆⠈⠁⠃⠀⠀⠀⠀⠀⠀⠁⠘⢷⣟⢀⠀⠀⢈⣀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠋⠁⠘⣿⣮⡿⣾⣦⣠⣨⠒⠀⠀⠀⠀⠀⠀⠀⣀⢶⠾⠿⠓⠷⠷⠯⢠⠀⠈⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠞⠋⢻⡮⢤⡄⠊⠃⢤⠀⠀⠈⠧⣠⣤⠀⠀⠀⠀⢀⡿⠛⠋⠀⠀⠀⠀⡜⣦⣀⣥⠰⢤⡀⠈⠢⢄⠀⠀⢀
⠀⠀⠀⠀⣠⣤⣤⣀⠔⠁⠀⣠⠀⠐⠉⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⢣⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣰⠋⠉⠀⠀⠀⠈⠒⢄⡀⠈⢺⠋
⠀⠀⠀⠀⠉⢻⣧⡁⠀⢐⡶⠄⠀⠀⢀⡄⡠⠂⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠁⢀⡤⠆⠀⠀⠀⠀⢬⡉⠀⢈⡷⠄
⠀⠀⠀⠀⣠⠊⠙⠛⠓⠚⠃⠀⠀⣀⠝⣡⠊⠀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣹⠓⠂⠀⠀⠀⠀⠉⢁⠟⠀⠈⡀
⠀⠀⠀⠀⠙⠲⠒⠶⠶⠀⡤⠀⠰⢥⠤⣀⠉⠈⠉⠁⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣀⣀⢀⣠⠶⠛⠲⡧⠀⠀⠴⠤⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⠄⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠠⠄⠀⠀⠀⢀⠀⠁⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡀⠀⠐⠂⠀⠀⠀⠈⠁⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⢈⠀⣀⠀⠁⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⡌⠀⠀⠀⠀⠀⢈⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠀⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⡁
⠀⡁⠀⠀⠀⠀⠠⠐⠀⠠⠀⠀⠀⠀⡰⠒⠲⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠐⠁⠀⢀⣴⠞⠉
⠀⢇⠀⠀⠀⠀⠈⠂⠒⢰⡆⠀⡠⠘⠁⠀⠀⠉⢁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢢⠄⠀⠀⠀⠀⠀⠀⣏⠉⠉
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⣀⠴⢀⠀⠈⣱
⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⢀⠚⠛⠊⠉
⠀⠀⠀⠀⠀⠌⠀⠀⣀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢀
⠀⠀⠀⠀⠌⠀⠀⢠⡞⠈⠑⢀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠈⠀⠀⠱⠏⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⡌⠀⢀⠜⠀⠁⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⡒⠂⠀⠶⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠈⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⠂⠀⠀⠀⠀⠀⠀⡠⠊
⡰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⠀⠀⠀⠀⠀⡠⠊
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠉⠉⢻⠂⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠐⠠

⠀⠀⠀⠀⠀⢀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⢀⢔⠆⡻⠁⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠠⠊⠄⠂⠀⠐⡌⢂⠀⠀⠀⠀⠀⠳⢣⢝⡤⣒⡤⡩⣪⣂⠀⠀⠀⠀⠀⠀⡂⠄⠀⠈⢠⠠
⠀⠀⠀⠀⠈⢀⠀⠀⠀⠸⢈⠀⠀⠀⠀⠀⢏⠇⣨⡫⠪⣎⣶⠡⢹⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⢌⠢⠄⠠⠔⠁⠎⠀⠀⠀⠀⢪⣪⣦⢉⣚⡡⢡⣆⡾⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠓⠶⢊⡲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠿⢷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣠⠖⢋⣉⣉⡙⠲⣄
⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠠⠀⠀⠀⢰⠋⣴⡾⠟⠻⢦⡙⣿⡄⠀⠀⠀⢀⣾⡿⠉⠀⠀⠈⠻⣿⣆⠀⠀⠀⣼⢁⡾⣫⠖⠲⢍⢷⡘⣇
⠀⠀⠀⠀⠀⠸⡏⠀⢈⣇⡀⠀⠀⠀⠀⠀⢸⡟⢰⠃⠀⢨⠇⣿⡇⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⡇⢸⡱⡇⠀⠀⢸⢸⠇⡿
⠀⠀⠀⠀⠀⠈⠛⠳⠻⠑⠀⢀⠀⠀⠀⠀⠸⣿⡘⢷⣤⣤⡾⠟⣠⠇⠀⠀⠘⣿⣧⡀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠱⡈⢧⡙⠮⠬⠷⣋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣴⠾⠋⠀⠀⠀⠀⠈⠻⢿⣿⡿⡿⠛⠁⠀⠀⠀⠀⠀⠑⠄⠉⠛⠛⠛⠉

⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠻⢷⣦⡀⠀⠀⠀⠀⣠⡲⣝⣛⣛⣦⣢⢄
⠀⠀⠀⢠⣿⣿⡽⠛⠻⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢡⡖⣫⠭⢍⠲⡘⣷⡀⠀⠀⢰⢯⡿⡯⣒⣒⣟⣟⣝⢆
⠀⠀⠀⣿⣿⢿⠀⠀⠈⣮⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⢸⠀⠀⠀⡃⡗⢸⡇⠀⠀⢬⢺⣸⣸⡥⣼⣹⢹⢸⣺
⠀⠀⠀⢘⣿⣿⣗⣶⣖⣵⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣌⠢⢔⡮⠞⣡⡿⠀⠀⠀⠸⣷⣻⣿⠯⠝⣿⣿⣟⡎
⠀⠀⠀⠀⠙⠻⡿⠿⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⠈⠛⠶⠶⠛⠉⠀⠀⠀⠀⠀⠈⠓⠯⠟⡯⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡱⠊⠉⠐⢝⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣶⣾⣷⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⡀⠀⠀⠈⡆⣾⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⡏⠀⠈⢹⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣝⡢⠤⢚⣔⣿⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⢿⣷⣾⠟⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠾⣥⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠤⣬⠗
⠀⠀⠀⠀⠀⠀⠀⢱⠌⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⠡⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢗⠐⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠒⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠕⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠣⢵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⡸⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⢆⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠧⠴⠟⢉⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠳⠧⠼⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠣⠤⠶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣶⠖⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣿⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⢟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⠻⣜⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠟⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠹⣿⣧⠀⠀⠀⠀⣼⣿⠏⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⣿⣧⡀⢀⣾⣿⠃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢿⣷⣿⡿⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣼⣿⠏⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⡔⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⡴⠃⠙⢦⡙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⣁⠔⠁⠀⠀⠀⠀⠙⠢⣈⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⣋⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠿⣿⣶⣄⡀
⠀⠀⢤⣤⣤⣴⣶⠿⢟⣛⡭⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⣛⠻⠷⣶⣤⣤⣄⡤
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡿⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣽⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣤⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣠⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢷⣦⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠀⠑⢄⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠴⢷⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡟⠀⠀⠀⠀⡏⠀⠀⠀⣀⠴⡪⠁⠀⠀⠉⡇⠀⠀⠀⣀⢤⣤⣀
⢀⣴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡇⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢹⣀⡀⢀⡼⠁⠀⡠⢚⣵⠟⠀⠀⠀⠀⠀⡇⠀⢠⠾⠟⠛⠉⠛
⣿⣯⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⡛⠛⢻⠁⢠⢞⣾⡿⠁⠀⠀⠀⠀⠀⠀⡇⢀⠏⠀⠀⠀⠀⣰
⠹⣿⡄⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡼⠻⠿⢿⢠⢣⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⣷⣄⣀⣠⠞⠁
⠀⠈⠙⠢⢤⣴⠟⠁⣵⡤⣤⣀⣀⡀⢹⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⢎⠀⠀⠀⠈⢯⣾⡟⠀⠀⠀⠀⠀⠀⢀⣀⠴⠻⣧⣈⡝⠋
⠀⠀⠀⠀⠀⠀⠙⢾⠋⠀⠀⠀⠀⠈⠉⠙⠛⠧⣀⠀⠀⠀⠀⠀⡰⣱⠏⠀⠀⠀⠀⠘⡿⠀⠀⠀⢀⡠⢔⣪⠽⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡼⣽⡏⠀⠀⠀⠀⠀⠀⢸⡀⣠⢖⣭⡾⠟⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⣟⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⡇⣿⣇⣀⣀⣀⣀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠿⣷⣤⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣝⡻⣷⡾⢫⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣯⡈⣧⡀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⡄⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢹⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣉⣛⣛⣻⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⢀⣴⣾⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⣟⢹⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡙⢂⠀⠀⠀⠀⠀⣀⣈⣷⠀⠈⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠓⢢⠑⢄⠀⠀⠀⠈⣹⠿⣿⠞⠁⠈⢈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢱⣤⠔⠋⠀⢺⠏⠀⠀⢀⡼⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢮⣿⣀⣤⣴⡟⠀⢠⣠⣿⣷⣄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡁⠀⢸⣿⣿⣿⣿⡆⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⢀⡔⠁⣀⠴⠟⠁⠸⣿⣿⣿⣷⣠⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⢊⡴⠊⠀⠀⠀⠀⢀⣴⣾⣿⡏⣿⠇⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⡎⠀⣸⡥⠀⠂⠀⠀⣠⣿⣿⣿⣿⢱⣿⣦⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⢀⡀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⢣⡿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⡦⢬⣭⡝⣵⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⠞⣲⠀⠀⠀⠀⣴⣿⣿⣿⣿⢏⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⣼⣿⣿⣿⣿⢧⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠙⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⣼⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡁⣿⡿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠑⢤⣀⠀⠀⠀⠀⢿⣿⣿⠿⣱⡇⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣱⣿⡿⠿⠛⠿⢿⣿⣿⣷⣤⣀⣠⠝⣳⠀⠀⠀⢸⣾⣿⣿⡇⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠓⠋⠀⠀⠀⠀⠸⢿⣿⣿⠇⠀⣶⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⡿⠿⠿⢿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣿⢿⠿⣿⢿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣿⡆⣿⣿⡇⢸⣿⣷⢀⣤⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠻⡗⠒⠶⠦⢤⣤⡾⠁⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⢸⣿⣿⠀⠈⢷⣤⡤⠴⠶⠒⢺⠟
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣉⣭⣥⣬⣉⣃⡘⢿⡗⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⠿⢠⣾⣶⣤⣦⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡄⠀⠀⣾⣿⣿⣿⡃⠿⣿⣷⣾⣿⠟⣫⣽⣿⠀⠀⠀⠀⢠⣞⣀
⠀⠀⠀⠀⠀⠲⢿⡉⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡌⣿⣿⣏⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢉⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿





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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠁⠋⠀⠗⠀⠀⠀⠀⠠⠚⠍⢉⡜⠀⠀⠀⠀⠀⠀⠀⠢⠉⡠⡌⣀⠀⣐
⠀⠀⠀⠀⠀⠀⠀⡸⡆⠁⣀⠀⠀⠀⣀⣠⣶⡻⢋⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢀⣀⣈⡁⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⣠⠤⠴⠞⢛⠛⢡⠦⠤⠔⠂⠀⠀⢀⣀⠀⠐⠀⠀⠀⠁⠼⣇⡷⠀⡀
⠀⠀⠀⠀⠀⠀⠇⠸⣷⣄⠀⠒⠐⢆⣭⠄⣉⠆⠈⠀⠀⠀⠞⠁⠒⠁⠀⠀⠀⠀⠀⢡⢤⠿⠇⠀⠹⡄
⠀⠀⠀⠀⢠⢞⠖⠂⠿⠅⢟⠃⠀⠁⠀⠉⠁⠀⠀⠀⠀⠐⣀⠀⠀⠀⢀⠄⠈⢉⡁⡀⣀⣀⣀⠀⠀⠘⣄⡀
⠀⠀⠀⠀⢘⣮⡆⠀⡇⠀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢯⢴⣎⡉⠙⢦
⠀⠀⠀⠀⢸⢡⡇⠠⡀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠛⠁
⠀⠀⠀⠀⢸⠘⠙⠀⡈⠢⠤⣤⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣏⠉⢐⣒⣲⣭
⠀⠀⠀⠀⠘⢂⣤⣤⣤⡤⠀⢸⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢈⡿⣾⡿⠟⠱⠧⢄⡀
⠀⠀⠀⠀⢀⠞⠠⠈⠀⠠⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠶⣀⡤⠀⠈⠑⠒⠫⠤⢬⣓⡄⢀
⠀⠀⣀⠠⡉⠀⢀⠀⠀⠀⠀⠀⠘⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀
⠀⠀⠈⠑⠒⠦⠄⠀⠒⠒⠤⠤⣤⣬⣌⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠷⠶⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠓⠒⠲⠆

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⡄⣤⢤⣤⣤⢤⠀⠀⢤⣤⣤⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣾⣽⣼⣧⣭⣬⣤⣿⣼⣥⣧⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢹⣿⡉⡏⠀⢹⣿⡯⣶⣿⣾⣿⡩⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⠙⠋⣼⡏⠉⢉⣩⣉⣙⠛⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⣿⠇⢀⣜⡿⠁⣴⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⢰⣿⡿⠇⢀⣹⣟⣁⡉⣿⣿⣴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣾⣿⣰⣿⣿⣾⣿⣌⠁⣰⣧⣾⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⡄⠘⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢡⣿⣿⣇⠈⠛⠛⠋⠀⢿⣿⣿⣿⣏⣿⣿⣿⡶⣀⣘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠐⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠜⢋⣿⣯⣿⣾⣿⣤⣄⡀⠘⢻⣿⣿⣿⣿⣿⣿⡿⣀⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢨⣿⣿⣯⠡⢹⣧⣀⣸⣿⡟⠛⢉⣽⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⡇⠈⣿⣿⣿⣶⣬⢻⣿⣿⣿⣷⣶⣿⣿⡧⣺⠋⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⣿⣭⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⡿⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡛⢿⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠂⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⢰⣀⣼⣿⣿⣿⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⠿⠿⠿⢿⢿⢿⣿⠿⠒⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⢹⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣿⣿⠂⣿⣿⢿⣿⢿⣠⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠋⡇⠀⣿⣿⣿⣿⣿⣹⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⠿⡟⠿⣹⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣽⣿⣦⣤⡼⣿⣶⣧⣤⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠿⠟⢸⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⠿⠛⡿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠘⠟⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠈⠁⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠐⠀⠀⡀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡀⠈⠂⡀⠀⠀⠄⠘⠁⠀⠰⠀⠒⠀⠀⠄⠠⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⣤⠚
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⡂⠤⡆⠀⢀⡀⠀⠀⢀⠈⡀⠀⠀⠆⡀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠉⢂⠠⠈⠀⠂⠀⠀⠙⡆
⠀⠀⠀⠀⢀⡅⠀⠉⠐⠀⠠⢀⠀⠐⠐⠃⠠⠀⠈⠄⢥⠀⠀⠠⠄⠀⡴⠀⠦⠤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭
⠀⠀⠀⠀⣂⡃⠄⠀⠀⠀⠀⠀⠀⠀⢍⠐⠒⢶⡀⡀⠀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⡰⡦⢴⢧⠄⠤⠄⠀⠀⠀⠀⠀⠀⠳⡆
⠀⠀⠀⠀⠲⠠⡀⠀⠀⠀⠐⢡⠀⡠⠀⠈⠈⠀⢀⠀⢁⡄⠀⠀⠀⠀⠀⠠⣖⡌⠧⠆⠤⠰⠈⠉⠀⣔⠁⠂⠀⡀⠀⠀⠀⢧
⠀⠀⠀⠀⡀⠀⠀⠁⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠺⡌⠢⠀⠀⠀⠀⠲⢌⠠⠀⠠⠌⠂⢬⡨⠂⠈⠉⠓⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⢠⣶⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⠀⠈⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⣀⣀⡀⠄⠠⠐⠂⠀⠀⠀⣄⡠⠀⠀⠠⢀⡶⡆⠀⠓⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠬⠀⣀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣘⡨⢆⣀⣠⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠰⢤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⡆⣾⣶⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣿⣿⣿⡿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡯⣭⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢟⣥⡉⢉⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⢏⠔⢋⣩⣿⣿⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠁⠸⢱⣾⣿⣿⣿⣿⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠛⠀⢒⣼⣷⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⣲⠷⣿⣽⣻⣿⣿⣿⣿⢣⡏⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣿⣿⣿⣿⣟⣵⣿⡇⠀⠈⠻⣦⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⣥⣾⠿⠛⠛⠛⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⢿⣿⣿⣘⢧⣀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣵⢞⣷⣦⡀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣿⡿⣛⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣯⣽⣯⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣯⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡄⢀⡴⠶⠦⡀
⠀⠀⠀⠀⠀⠀⣴⠞⠓⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣾⠀⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⣶⡇⠀⢸⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣠⣤⣀
⠀⢀⠀⠀⠀⠀⠀⢳⣤⣴⣾⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣏⢈⣉⣉⡇⠀⠈⣷⢰⣶⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⡆
⣾⠁⠈⠱⣄⠀⠀⠈⣇⣥⣾⣧⣠⣴⣾⣿⣿⠿⠿⣿⣿⣷⣾⠿⠸⠿⠿⢧⠀⢸⣷⢰⣦⣿⠀⠀⠀⠀⠀⠸⡆⠀⢠⡇⠀⠀⢀⣀
⠙⢄⠀⠀⠙⣆⠀⢀⡟⢈⣵⣿⡿⠿⢿⣿⣇⠀⠀⣽⣿⣿⠏⠀⠀⠀⠀⠀⢇⡾⠉⠈⠉⠉⢇⠀⠀⠀⠀⠀⣧⣤⣼⠁⣠⠞⠉⠉⠛
⠀⠀⠙⢦⡼⢟⣧⣼⢣⣾⣿⣿⡀⠀⣠⣿⡿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸⡧⣿⣿⠞⠁⠀⠀⠀⣠
⠀⠀⠀⠀⢻⡞⠋⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⣄⣀⣤⣿⡶⣋⢹⣶⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠈⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢿⡷⠟⡏
⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠜⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠹⣜⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣸⠀⠀⢠⠊⠀⠀⠀⡿
⢀⠔⠒⠚⠛⠒⠒⠺⣿⡟⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡟⢀⡴⠁⠀⠀⠀⢰⠇
⣾⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⠟⠁⠘⡄⠀⠀⣠⠏
⠘⠷⣤⣤⠤⠶⠒⠚⠋⠉⠉⠛⠳⢶⣤⣄⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⡿⠷⡀⠀⠀⠀⣰⠃⠀⠀⠀⠙⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⢿⣿⣿⣿⡟⠋⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣟⣶⣬⣻⠛⠟⢁⠠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⣹⡿⠀⠀⠙⠃⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠐⣿⡟⣆⠈⢑⡞⠉
⠀⠀⠀⠀⠀⣠⠖⠛⠁⠉⠒⢄⡀⠹⠳⣾⣻⣯⡿⠆
⠀⠀⠀⠀⣼⡃⠀⠄⠀⠈⠃⠄⡈⠐⠀⠈⣷⣤
⠀⠀⠀⠀⠿⢀⣀⠀⠀⠀⢀⡄⢻⡀⠀⠀⢿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠙⠢⠄
⠀⠀⠀⢠⠖⣉⣥⣤⣄⢻⡄⢙⡟⠃⠀⠀⢠⠃⠰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⢾⡿⠓⠀⠀⠀⠀⢿⡦⠀⠀⠀⠀⠠⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠻⠿⣦⣄⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠳⢄⡀
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⣀⠀⣈⠓⠄⠀⠀⠀⣠⢟⣿⠏
⠀⠀⠀⠀⠐⠋⢈⠟⣷⠢⢄⠀⠀⠀⠀⠀⠀⠀⠁⠲⠽⠄⠄⠓⢦⢞⣡⠟⠁
⠀⠀⠀⠀⠀⢀⣾⠀⠁⣷⡘⠙⠒⠦⠤⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣰⡏⠀⠂⠄
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣛⠳⣦⠔⠋⠙⠦⣐⠶⠷⡆
⠀⠀⠀⠀⠀⢿⡿⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⢾⡿⠿⠿⠛⢿⡿⠃⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣶⢊⣛⡛⣛⣧⡔⢸⣤⠀⣿⠶⠶⠋
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠹⠎⢹⣧⠉⣠⡷⠈⠀⡀⡆
⠀⠀⠀⠀⠀⠘⠦⠤⣀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡇⢹⣁⠀⢰⣄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠃⠀⠂⠴⠀⠼⠿⠢⡀⢻⣷⣄⣵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⠀⠧⠀⠀⠀⠀⠠⢤⡀⣀⠐⢦⣠⣟⣛⣙⣳⣴⣴⣦⣄⣤⣴⣴⣿⠾⣃⢩⢏⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣷⣿⣿⣉⣉⣉⣩⡝⣋⢩⣉⣿⣷⣾⡇⢘⣤⢸⣾⡇⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⠿⣿⣿⡄⠀⠀⠀⠀⠈⠙⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢿⣿⣯⡛⠘⢸⢸⡿⠃⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣭⣥⣶⠶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⢸⠛⢻⠀⣾⣿⣿⣿⣿⣧⠹⣷⡽⡇⠀⡀⠈⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠸⣯⣤⣭⡉⢹⠻⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⡌⠀⠃⠛⢿⡿⣿⢆⢠⠀⢇⣴⡀⠇⢀⣼⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠘⡇⠠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣄⢧⠇⠀⢰⠀⠈⠿⣎⠰⣹⣷⣿⢿⡇⠀⠠⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡯⠥⢼⡇⣸⣿⣿⣿⣿⠀⠀⠀⠀⣀⡤⠄⢿⡘⣤⠀⠀⢀⢂⡀⠙⣆⠹⣿⣿⢆⡇⠀⠰⡇⢠⣿
⠀⠀⠀⠀⠀⢀⠄⣸⣿⣿⣼⣷⣍⣛⠻⠛⠟⠉⡙⢠⡅⠀⡀⠀⠀⠳⣹⠄⠀⠀⠀⢳⡄⠈⠀⢻⠸⡞⡇⡇⠘⢃⢸⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣿⢿⡿⠼⢿⣷⣴⣶⣦⣤⡶⠖⠃⢤⡄⢠⠑⣄⠀⠀⠀⢁⢿⡄⡀⠘⢂⠘⡇⢡⠀⠂⠸⢸⡒⠖⠀⠄
⠀⠀⠀⠀⠀⠀⢿⣯⣄⣉⡀⠸⠇⠀⠰⠋⢀⢽⠗⠉⢳⣶⣾⡷⠠⣄⠣⠈⠷⠀⠀⠈⠀⠀⠰⡠⠈⠓⢰⠀⣀⢀⣾⣯⣿⣛⠃
⠀⠀⠀⠀⠀⠀⠀⣉⡬⠿⠛⢿⣷⣶⣶⡶⠷⠟⠁⠀⢠⣭⣿⠃⠙⣦⣖⣠⠀⠀⠤⢠⣤⣤⣤⡴⢷⣤⣤⡼⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠎⠉⠩⠉⠁⠀⠀⠀⣠⠾⠩⠿⠿⠿⣿⢈⡃⠘⠛⢛⡿⠷⠿⠿⠿⠿⠟⠛⠻⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⡿⠟⠛⠩⠛⣿⡘⡋⡈⠛⢻⣏⣉⡘⢹⠘⡿⠬⠛⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠄⣤⣄⣠⠤⠄⠘⡟⠋⠋⠛⠛⡏⠉⠉⠛⢻⣃⡀⢀⡿⠤⡞⣻⣿⡟⠢⠉⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠙⠀⡄⣈⣧⠀⠀⠀⠀⣀⠤⠄⠄⠐⠇⠀⠀⠀⠀⠀⡉⠛⠁⠀⠀⢻⣤⣀⠀⠀⠁⢀⣼⠟⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡁⠀⠀⠀⠀⠈⠁⡀⠔⠒⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⡆⠀⠀⠈⠉⠒⣵⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠒⠀⠀⠀⠀⢠⠎⠀⠀⠀⠠⠊⠀⠀⠀⠀⠤⠒⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⡿⠌⠊⠙⡧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠖⠓⢤⣉⠉⠀⠲⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⡅⣿⡿⠀⠀⢸⠁⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠉⠙⠛⠂⠀⡀⡀⠂⠀⠀⠀⠀⠀⠀⠻⠋⠁
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣿⣧⣀⡀⣸⠀⠀⠀⠠⢀⡀⠸⠓⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⡯⠉⡉⠉⢙⠛⠒⠢⠤⠄⢀⡉⠁⠐⠠⠄⣀⠉⠓⠀⢀⠀⡠⠒⠊⠀⠀⠐⠁⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠄⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠢⠍⠀⠀⠀⠀⢀⠄⠐⠉⠀⠐⠒⠂⠀⠒⠒⠒⠂⠀⠂⠈⠈⠉⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⣉⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠒⠒⠦⠤⣄⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠛⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⣁⠒⠒⠚⠉⢀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⣿⣿⣿⣿⣿⣯⣿⣿⢬⣦⣀⣠⣩⣦⠀⠀⠀⣀⣀⡤⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠃⠀⠒⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⢋⠉⠈⠘⣷⠛⠉⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠀⢀
⣿⣿⣿⣿⣿⣿⣽⣟⣹⣿⣽⢷⣀⣶⡿⠟⠀⢀⡠⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠘⠲⠶⠲⠶
⣿⣿⣷⣴⣿⣿⣿⣿⡧⡿⢿⣧⠉⠁⢀⣴⠚⠉⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐
⢿⣿⣿⣿⣿⣿⣿⣿⣳⡿⣛⠟⢰⣿⠟⠋⠁⠢⠀⣠⡶⠉⠀⠀⠀⠀⡘⠀⢣⠧⠄⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠐⡀
⣬⣿⣿⣿⣿⣿⣿⣿⣯⡿⣅⣶⢟⠛⠷⣢⡀⠀⠀⠉⠠⣀⠀⠀⢀⣼⣀⣀⡆⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⡤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢫⠖⠀⣠⡿⠇⢄⠀⠀⠀⡙⠲⣾⠟⠀⠀⠀⠉⠈⠃⠀⠀⠀⠀⡏⠉⠉⠀⠀⠀⠀⠀⠤⠒⣏⠀⠀⠈⣆⣀⣠⡴⠞
⣿⣿⣿⣿⣿⣿⣿⣿⢛⣔⣿⣵⡳⣬⣾⠟⠁⠀⠀⠉⠲⠞⠀⣰⡏⠀⠀⠀⠀⠀⠀⢤⢄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⢧⣤⡾⢿⣿⠡⠄⡠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠉⠐⣿⣤⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⢉⡶⠒⠒⠒⠈⠉⣄⣀⣶⡴⠶⡋⣼⠀⠀⣠⣷⣎⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣸⣿⣿⣖⣛⠿⡶⢰⡤⣾⣤⣤⣄⣠⡷⠀⠀⠀⠀⠀⢿⣄⣀⢤⣍⣷⣼⣦⡶⢟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣿⣮⣶⣷⣠⣦⣬⣭⠀⠀⠀⠀⢀⠀⢼⣿⣽⣾⣿⣿⣿⣿⣿⣈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣁⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⢠⢟⠇⠀⠀⠀⣤⣙⠛⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⡇⢳⡐⠶⢶⠏⢸⠀⠀⠀⠀⠈⢩⠖⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⡇⠀⠛⠶⠎⠀⢸⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣡⡶⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡀⠈⠳⢧⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⣤⣤⠤⠤⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⠀⠀⠀⠀⢀⡴⠊⠁⠀⣤⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠓⣴⣿⣿⣿⣿⡏⠀⠈⢳⡀⠀⠀⠀⠀⠀⠻⡦⠉⠀⢀⡠⠀⠀⠀⠀⠀⠀⠹⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⣤⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⠶⣶⣶⡶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⣰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⣠⢾⣿⡷⠤⠄⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢾⣿⣿⣀
⠀⠀⠀⠀⠀⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣷⣶⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣌⠉
⠀⠀⠀⠀⣰⠟⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢹⡆
⠀⠀⢀⡾⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠻⠶⣄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠸⠧
⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⢠⠴⣾⣿⣿⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⠴⠖⠺⣿⣷⣦⣠⡏⠀⠀⠀⠀⠀⠀⢀⡀⢒⣿⣷
⠀⠘⠁⣼⣿⡟⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⣿
⠀⠀⢀⣿⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠋⠻⣧⡀
⠀⣰⠟⠁⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠉⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠈⢧
⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢀⣰⣷⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⢀⡆
⠀⣰⠟⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⣿⡏
⠚⠁⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⠁⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣷⢶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢸⡇
⠀⣸⠃⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠴⣾⠟⠁⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣀⣤⡤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⠤⣍⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣡⠴⠒⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢗⡒⠲⢦⡿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⣋⣩⣽⣄⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠙⠳⡇
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠁⠀⠀⠘⢦⡀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⣼⡧⠶⠶⢦⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⢀⣠⠞⢡⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣴⠟⠁⠠⢒⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣡⠶⠋⣰⠏⣿⠢⣄⡀⠀⠀⠀⠀⢸⡄⢀⡾⠃⠀⠀⠃⣾⡿⠁⠀⡠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡖⢛⣡⠾⠃⠀⠈⠙⠶⣤⣀⠀⠸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⢈⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠞⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠦⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⣦⣤⣴⣶⡿⠋⠈⠙⠳⢦⣤⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣾⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠋⠁⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⢟⣣⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠛⠁⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣫⣭⣶⣿⣿⣿⣿⠿⣫⣵⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣄⣀⠴⠾⠿⠿⠿⠿⠿⣟⣛⣛⣛⣭⣭⣵⣶⣾⣿⣿⣿⣿⣿⠿⣛⣭⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣭⣵⣾⣿⣿⣿⣿⡿⠟⠋⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟⣛⣛⣫⣭⣵⣶⣶⣿⣿⣿⣿⡿⠿⢛⣛⣭⠴⣒⣥⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣒⠒⠲⠶⠿⠿⢿⠿⢿⠿⠿⠿⠿⠿⣛⣛⣫⣭⣭⣶⡶⠿⣛⣭⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⣵⣶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⢛⣛⣛⣩⣭⣶⣾⣿⣿⡿⠿⠿⢛⣋⠉⣡⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣶⣶⣶⣶⡿⣟⣯⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣛⣿⣭⣿⣾⣿⡿⠿⣟⢋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣷⣶⣾⣿⣿⣭⣭⣽⣿⠶⠶⣚⣯⣷⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢿⣿⣿⣯⣿⣶⣶⣿⣿⣿⢿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⣶⣶⣿⣿⣿⠿⢶⣖⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣚⣛⣿⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡞⠛⠀⠀⠀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣓⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠠⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣖⠰⠲⠒⠒⠒⠛⠋⠉⠉⠉⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡛⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⠙⢉⣿⠋⠀⠀⠀⢀⠀⠀⣴⠉⠆⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⢁⣀⣠⠤⠾⠾⠒⠒⣿⠀⠐⠀⠈⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠠⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢰⠘⢀⡾⢿⠛⠁⣷⠒⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠥⡜⡤⢻⣦⣌⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠄⠈⢻⣿⣶⠿⡥⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⢸⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠘⣰⡏⢿⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠂⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣹⠀⣇⢲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠿⣾⠁⠀⣿⢦⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⣸⠃⠀⠀⣿⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣸⣉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣗⣾⠁⠆⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡀⢀⣯⣩⡿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⣾⠁⢀⡇⠀⠀⠀⠀⠀⠀⢀⣤⢤⣶⠟⠊⠉⠉⠁⠀⠉⠉⢃⣀⣾⣿⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠙⠉⢰⡇⠀⠀⢀⠀⣀⣀⢻⣿⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⢿⣿⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠉⢈⣿⡀⠆⠀⠀⠀⣀⣀⣀⣀⡄⠀⠀⠸⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⣀⠀⠀⣠⡦⢰⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶
⠰⠐⠇⠙⢙⣿⣻⢏⠈⡄⠀⢠⠀⡀⣀⠈⠈⠀⠀⠀⠀⠀⢀⣀⠀⠐⠆⠀⡀⠀⣠⠀⢀⠀⠃⠀⠀⠹⠿
⢠⣴⣿⣶⣿⡏⠀⢧⠶⣧⠀⢘⡟⢱⣯⠅⣿⡳⠀⠀⠀⠀⠈⠙⠁⢀⡦⠞⠛⠉⠉⠙⠻⢤⣴⣚⣶⡄
⠘⠉⣉⠙⠻⠷⠒⠆⠀⠸⠀⠈⠁⠘⢳⡘⠮⠇⠀⢀⠀⠀⠀⠀⠐⡞⢀⠀⠀⠀⢀⠃⡀⢀⢈⡿⢁⠙⠆
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⣇⠀⠤⣦⣷⣨⣤⣷⣧⣾⡵⡏
⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠘⢦⣐⣚⣭⠭⡙⠛⣫⠭⠕⠃
⠀⠀⠈⠕⠂⠶⠠⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠤⡀⠀⠀⠀⠸⠀⠁⠀⠡⠀⠀⠀⢰
⠐⠽⠆⠀⢂⠁⠀⠀⠀⠈⠨⢉⠀⠻⠀⠀⠠⠁⠈⠀⣀⣤⡤⠀⣄⡠⠇
⠀⠀⠁⣰⠜⡿⢬⡀⠀⠀⠀⣁⠈⠃⠀⠀⠁⢰⠾⠛⠋⠉⢀⡶⠳⠶⠾⠶⠄
⠀⠰⣀⠻⣗⡀⠀⠀⠑⠀⣠⣄⢀⠀⠀⠀⠁⠀⡀⠠⢰⠺⠁⠉⠂⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣤⣄⢆
⠀⠀⠁⠠⠄⠑⠀⠤⠒⢚⠙⢋⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣵⡿⣿⡆⡿⣿⣿⣯
⢠⣶⣶⣶⣶⡖⠊⠁⠛⠃⢓⠃⠈⠐⠆⠀⠀⠀⢀⡤⣤⡂⠀⠀⠀⠀⠀⣶⠀⠀⠈⣉⠛⣍⠳⠷⣯⡽⢻
⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣶⣴⠤⡶⠀⣿⢾⣦⠡⣿⢰⡧⡄⠀⠈⣶⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡆⠀⠀⠀⠀⡠⡈⣄⣙⣋⠤⡛⢛⣀⠒⠂⣒⠘⠛⢐⣻⣝⣳⠇⢀⣴⣿⣿
⠸⠒⠀⠀⢀⣠⠝⠻⣿⣿⣿⠕⢾⣶⣶⣶⣷⣎⡺⠊⠙⠲⠮⢭⣤⣉⠉⠉⣉⣉⣁⣭⣭⡵⢾⣿⢿⣿⡿
⠀⠀⠀⠐⠐⠀⢲⡀⠀⠀⢀⡖⠀⠀⠀⠀⠈⠈⠈⠈⠀⠀⠈⠉⣩⣉⣧⣮⣭⣿⡅⡉⢩⠉⠉⠉⠉⠈⠘
⠀⢠⢿⣤⣏⣠⡀⢁⣄⢀⣀⡀⡀⣀⡴⣦⣸⢳⡟⠟⠦⣤⢠⣠⠄⢻⣿⣿⣿⣿⣧⣿⣇⣤⡖⢀⡀
⢠⠟⠈⠿⠻⠼⠃⠿⠿⠿⠼⠛⠯⢋⣛⡏⡟⠚⠀⠀⠀⠀⠀⢀⣠⣿⣿⢿⡛⣛⢿⣋⣯⣹⣙⣫⡀
⢀⠢⠤⠂⣀⣄⣀⣄⣀⢤⣴⣶⣽⡿⠔⠓⠀⠀⠀⠀⠐⠀⠀⠚⠛⠛⣿⣶⣿⣿⢿⡿⢿⡿⢯⠙⠃
⠾⠿⠧⠀⠠⡥⠿⠇⠀⠀⠉⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠽⠿⠿⠿⠿⠼⠇⠸⠟

⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠚⠛⢧
⢰⡿⣟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀
⣾⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⢷⡀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠙⢦⡀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠙⢦⡀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠻⣄⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢀⡴⠞⢯⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⠳⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣦⡀
⠀⠀⠀⠀⠈⠙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⠀⠀⠙⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⢄⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⡇
⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠑⢦⡀⠀⠀⠀⠙⢦⠀⠀⠀⢀⣴⠋⠀⠀⠸⣧
⠀⢀⡏⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠈⠢⣄⠀⠀⠈⢷⡀⣠⠞⠁⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⢸⡗⠂⢒⣸⠇⠀⠀⠀⠐⢿⠶⠤⢤⣹⣦⡀⠀⠈⠑⢦⡀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠟⣇⣀⡤⠤⠒⠚⠋⣩⢾⡇
⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢀⠔⠁⡾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣸⠃
⠀⠀⠀⣿⠛⠭⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢰⠇
⠀⠀⠀⢸⣄⡀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⣀⣤⠶⠋
⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠈⠓⠦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠞⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⡠⠤⢦⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡠⠔⠊⠉⠀⠀⠀⢸⡀⠀⠀⠀⣠⠎⡏⠳⣄⠀⠀⠀⠀⣩⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⣀⣤⡤⠶⠖⠚⠛⢻⡇⠀⢀⠜⠁⠀⡇⠀⠈⠢⡀⠀⠀⣿⠛⠛⠛⠛⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⢸⡇⣠⠏⠀⠀⣠⢧⣄⠀⠀⠈⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⣿⠁⠀⣠⠞⠁⠀⠈⠓⢦⡀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⡞⠋⣩⡿⠀⠀⠀⠀⠀⠀⢸⣤⠞⠁⠀⠀⠀⠀⠀⠀⠉⠳⣼⠇⠀⠀⠀⠀⠀⠀⢿⠢⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡔⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠒⠒⠲⠂⠈⠁⠈⠉⠉⠀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠐⠖⠛⠋⠉⢀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠐⠂⠈⠀⠀⠐⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⣩⠠⠀⠀⠀⠀⠀⠀⠄⠀⢲⣄⠀⠀⢠⣤⣄⠀⠁⠀⠀⠀⠶⣤⡄⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⢠⠞⠁⠀⠀⠀⣀⣤⠶⢿⠿⠛⠿⠿⣷⡀⠀⢿⡈⡜⢛⣲⣄⡂⠰⣤⢤⣴⢦⢤⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠳⡀⠀⠀⠀⠈⢳⡀⢸⡇⣧⠙⣿⠉⣇⡰⢏⡟⠀⠀⠀⢳⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠐⠀⠀⠀⢠⠞⠱⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⢿⡀⠠⠘⣧⠈⠃⢄⠀⠈⡀⠀⠀⠀⢸⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⣷⡀⢇⠹⡧⡀⠊⠢⠀⠳⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠾⠃⠹⣌⠀⠑⠌⠀⠀⠙⢆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠳⠀⠈⠑⣠⠜⠳⣄⠒⠤⡓⢾⣄⣶⡌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⡀⠀⢸⠀⠀⠆⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢠⠆⠀⠀⠀⠇⠀⠀⠇⠉⠢⢌⡁⠀⠀⢙⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⢇⠀⢸⠀⠀⠀⠀⠀⠄⠀⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠛⢤⣠⡴⠛⣆⠀⣸⠀⠀⠀⠀⠀⠁⠀⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⢄⢸⠀⠀⠀⠀⠁⠀⢸⡇⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠛⠀⠀⠞⠢⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⠁⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⢆⠄⡀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⣹⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠒⠀⠈⠉⢉⡿⣧⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡶⠏⠤⠤⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠄⠀⢾⡆⡀⠀⠄⠀⠀⠀⠘⣧⣀⣤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⠃⠀⠀⡀⠀⠀⠘⢢⡄
⠀⠀⠀⠖⠠⠀⠀⠄⠘⡀⢡⡤⢤⠀⡘⠂⠠⡘⢿⡙⠇⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠁⠶⠀⢀⡀⠀⠉⠁⠀⠁⢠⠏
⠀⠐⠁⠀⠘⠀⠶⠀⠰⠗⠶⠄⠀⠉⠱⠚⠓⠋⠀⢹⣔⠂⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⢠⣄⠀⠀⠃⣀⠐⠀⢀⣄⠀⠼⠆
⡜⢀⡔⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠆⠀⠰⠀⠀⡍⠀⣤⠶⠗
⡅⠀⠓⠂⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠒⠒⠂⠀⠠⠌⣹⣦⡷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢁⠠⠁⠀⠈⠀⠠⠤⠀⠀⢰
⠱⡆⢀⣀⠉⠀⣠⠴⠚⠋⠁⠀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡔⠀⣀⡾
⠀⠙⠠⠀⠀⡾⠁⡀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢖⡀⠀⣀⠀⠀⠀⠀⠻⣦⣀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⢰⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇⠠⡝⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠒⠚⠒⢦⣀⡀⠈⠁⠶⠂⠀⠀⠸⠆⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠰⠂⠀⢀⡄⠈⠀⠀⠀⠀⠀⠙⢾⣄⠀⠀⠀⠉⠚⠢⠤⠶⠤⠞⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠰⠄⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⢀⠀⠤⠄⠐⠚
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⡤
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠷⢀⣀⡄⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠁⠀⡄⠀⣈⣁⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⣤⣀⣀⣀⣀⣀⣰⡶⠶⠐⠛⠋⠉
⣿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⣿⡛⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢑⠀⢿⡿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⣴⡟⠁⠈⢙⢢⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⣠⠠⠬⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⣾⠟⢀⡞⢧⠀⠿⣷⡤⣀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣿⠻⣦⣸⠓⢀⣔⡒⢿⡀⠀⠡⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣛⡛⢥⡀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠀⠙⢧⣶⠋⠻⣿⣿⡓⠂⢁⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡁⠉⡀⠀⣿⣿⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠞⠋⠫⠀⠀⢙⡏⠙⣆⠈⡆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⣿⣦⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿⡖⠦⠞
⠀⠀⠀⠀⠀⣀⠀⢀⣠⣿⢿⠃⠀⠘⢿⢸⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂⠀⠀⡟⠀⢀⣿⡇
⠀⠀⠀⠀⠀⢡⠀⣸⣿⣿⡟⠀⣄⡀⢸⡼⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠃⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⡁⣿⠉⠀⣦⡶⠶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⠀⢰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢑⢿⡀⠀⠀⠓⣀⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣿⣿⣿⣇
⠀⠀⠠⣤⠤⠤⣶⣾⣧⣤⠤⣤⡼⡮⣽⣿⣿⣷⣞⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣴⣿⣿⢿⢈⡾⠁⢸⡌⡏⣿⣿⡏⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢠⠇⢠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣯⣌⣼⡃⠀⠈⠛⡇⣿⣿⣿⣤⣿⡇⠀⠀⠀⠀⠀⠐⣠⠀⢠⠃⠀⣿⣧⡐⠀⡘⢀⣾⠏⠙⢿⣿⣿⣿⣷⣄⠐
⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠁⣿⡆⠐⠁⠉⠻⣿⡥⣾⡇⠀⠀⠀⠀⠀⣸⣿⡶⠘⣧⠀⣿⣿⡁⠀⠂⠉⠉⠈⣧⠀⠉⢻⣧⣤⢹⣿⣀
⠀⠀⠀⠀⠘⠿⣿⡟⣡⢤⣼⠟⠀⠀⢠⠀⠘⢿⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡇⠙⣿⣧⠀⣀⠀⠀⠀⣿⠷⣦⠀⢿⣷⣔⣽⣿⣤
⠀⠀⠀⠀⠀⢀⠞⠀⠁⣾⡟⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢿⣿⠀⠀⠀⠀⠊⠀⠀⠈⢳⡈⢻⣿⠈⠁⠁
⠀⠀⠀⠀⢠⠋⠀⠐⣼⡿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠶⠖⠀⢣⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢿⡄
⠀⠀⠀⠀⣀⠀⠀⠼⡿⠀⠀⢀⡀⠻⠶⣿⣶⠔⠱⠀⠈⢿⠓⣤⣤⣴⣶⡞⠛⠃⢁⣀⠀⡞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⢷⡀
⠀⠀⠀⠀⠁⠀⠀⣠⣿⣤⣶⣯⣤⡤⠀⠈⢤⠴⠾⠻⣷⡰⠼⠢⢈⠙⣳⠀⠂⠚⢉⡠⠴⣀⣹⣤⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⢀⠈⣳⣄⣤
⠀⠀⠀⠀⠀⣀⣸⠿⠋⠉⠀⢻⢿⣶⣶⠀⢀⣿⣤⡂⢻⣿⣶⣀⠀⠀⠀⠀⠀⠰⣄⡠⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⣹⣿⠟⠁
⠀⠸⠒⠐⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠩⠟⠿⠿⠉⠫⠽⠀⠌⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠓⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡴⠖⠛⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣉⠙⠳⣼⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠖⠛⣉⣉⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣉⣉⡙⠳⣿
⠀⣠⣶⠿⠛⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠛⣉⣩⣭⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠉⠳⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋⠉⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⢠⣟⣁⣀⣀⣀⠀⠀⠀⠀⢻⡀
⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢈⣻⣴⡶⠿⠛⠛⠋⠉⠉⠙⠛⠛⠷⢶⣼⣇
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⣠⡾⠃⣸⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡾⠋⠀⢠⠖⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⣁⣠⠞⠁⣠⡏⢹⡦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⣼⠟⠀⠀⠀⢇⢸⣿⡿⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣠⣤⠾⠋⣠⡟⠁⠈⠛⠶⣤⣀⠀⠀⠀⠈⣿⠀⣾⠏⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⠋⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢠⠊⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠸⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⣀⣀⣀⣠⣴⣿⠟⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠉
⣀⢀⣀⡀⣀⣀⡀⣀⣀⢀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⠛⠛⢻⠛⣋⣁⣀⣀⣀⣀⣉⣛⠛⠷⣶⣶⣶⣶⣶⣶⠟⠛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠒⠓⠒⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⡗⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠄⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⣿⣿⣿⠏⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⠀⠀⣻⣿⣷⣶⣿⣿⣿⡿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⣿⣿⣷⡿⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠏⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠿⠿⠓



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠆⢚⡅⣠⠟⠋⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⣶⣞⣡⠾⠁⠀⠀⠀⠀⠙⠿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣫⣤⣀⣵⣛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢄⣀⣀⣬⢟⣥⣘⠻⠄⣉⡉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠋⢠⣿⣥⣙⠶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠈⢦⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣉⣉⣉⡉⠒⠤⣄⠀⢀⡠⠤⠤⢄⠀⠀⠀⠀⣀⡴⢟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣆⠀⠀⠀⠀⠀⠄⠀⠀⠈⢆⠀⠀⠀⠉⠛⠲⡾⢋⡉⢱⠂⠙⠲⣑⣀⣴⣞⠁⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣃⠀⠀⠀⠀⠀⠀⠀⠑⡀⢫⣭⣍⣉⠭⠿⢋⣨⣵⡾⢉⣉⣽⣿⠿⠿⠯⠉⠉⠉⠐⠒⠒⢖⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠁⠀⠀⠀⠀⢀⡠⢀⠤⠀⠈⢆⠰⠤⣀⡤⠴⠚⠋⠁⢈⣿⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣆⡀⠤⠀⠀⡀⣸⠀⠛⠃⣸⠒⠒⠚⠒⠒⠒⣲⠶⠛⠓⠒⠒⠒⠂⠐⠂⠀⠀⠀⠀⠈⠉⠩⠉⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣦⣦⣧⡾⢰⣄⢠⣿⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⠿⣷⣿⣿⣿⣿⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⢀⡀⠴⢾⣲⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⢺⣷⣿⡄⢥⢠⠈⡀⠀⣤⢐⡀⡤⢈⡡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡸⠿⢿⢿⡤⣬⣷⠟⠀⠛⠸⣷⣡⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⠗⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠙⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣿⡇⠀⢀⣤⣶⣾⡟⠙⠉⢹⡛⢻⣶⣶⣤⣂⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡶⠶⣾⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣷⣦⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣤⡤⠖⠚⠛⠛⠽⠿⠛⠈⠉⠉⠻⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠻⢦⣶⣶⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣸⠟⠉⠀⣄⣤⣶⣶⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⡈⠉⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⠋⠀⠀⢀⡄⢿⣿⣿⠿⠿⣫⡾⡇⠀⠈⢀⣤⡾⣿⣾⣿⣿⣷⡄⠀⠀⠈⠣⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠀⠀⠀⠀⠈⠟⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠿⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠻⠟⣳⢶⣦
⠀⠀⠀⠀⠀⠀⢀⡾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣛⣭⡾⠇⠀⠀⠀⠀⠀⠐⣾⢹⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣇⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢰⢻⣧⠀⠀⠠⠀⠀⣴⡀⠀⠁⠀⠀⠀⠞⣿⣿⣿⣶⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⡘⣿⡷
⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡾⣿⠂⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⠾⠃
⠀⠀⠀⠀⠀⠘⢿⡦⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⠀⠀⠐⣻⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠀⠀⢿⣿⣿⣿⣄⡀⠈⠛⠋⣭⣙⡫⣿⠋⠀⠀⠀⠀⢐⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠰⠀⠙⠻⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⢠⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣴⣂⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣈⠻⣦⠀⠁⠀⠀⠀⠀⠀⠸⠿⠯⠍⠀⠁⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⢶⣶⣤⣤⣟⣻⣧⣠⣼⣏⣡⣀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠿⣯⡽⠿⠁⠀⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⠄⠀⠀⠀⣷⠈⠛⠋⢹⠃⠸⠿⠁⠀⣠⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⣚⡁⠀⠀⠀⠹⣄⠀⠀⠀⠀⣏⢀⡟⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⢶⡄⠀⠀⠱⡀⠀⠈⠙⣆⣰⠚⠛⠋⠁⠀⠀⠀⣟⣻⠧⠎⢀⡠⠟⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣾⠟⣛⣦⡀⠀⠳⠀⠀⠀⠘⠁⠀⡄⢰⣶⡄⢀⡼⠉⢹⡀⠀⠀⢳⣠⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⠻⢯⣼⠋⠁⠉⠛⠻⠿⢶⣶⣤⣀⠠⡇⠀⠉⢠⣿⠁⠀⠀⣇⠀⠀⠀⡠⠀⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠲⡚⠁⠙⡶⠂⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⣠⠃⡇⠀⠀⠀⣿⠀⡴⢊⡴⠋⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⡀⡷⠖⠾⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⡇⠀⠀⢰⡏⠀⢠⠞⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢾⣿⡖⠀⠠⢄⣀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠹⢦⡴⠟⠀⢠⠏⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠉⢙⡂⠘⢿⣦⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠤⠤⠒⠀⣠⣿⡀⠀⠀⠀⢀⡴⠋⠠⠤⠄
⠀⠀⠀⠀⢠⡖⠒⠲⣤⣀⠀⠀⠀⠹⠟⠁⠀⠙⠿⢿⣷⣷⣾⣶⣿⣟⡽⣻⡇⠲⠀⣀⣾⠉⠫⣿⣷⣦⡴⠋⠀⣀⡴⢦⣀
⠀⠀⠀⠀⠀⠙⠶⢾⣏⣼⡟⠉⠛⠛⠛⠛⠶⣤⣀⠀⠉⡉⠛⠛⠿⠷⠞⢋⣠⡴⠛⠹⣋⣷⣄⣀⡽⢋⠀⠀⠀⣽⣀⣰⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠴⠉⠻⢦⣀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢠⠆⢈⣤⠞⠋⠁⠀⠀⠀⠈⠛⢿⣿⠁⠈⠉⠑⠒⠀⠠⠌
⠀⠀⠀⠀⠀⢠⣤⠀⢀⣠⣀⠀⠙⢷⣦⣤⣀⣀⣀⣠⣿⣇⠏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣠⡀⠈⠉⠉⠛⠛⠛⢉⢎⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠙⠆⣠⡀⠸⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠴⠀⠳⠤⣤⠀⠀⠀⠀⠎⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⢾⠁⠀⠀⠀⠈⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠋⠙⠺⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠙⢄⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⢸⣷⡀⠀⠀⠀⠀⢀⣤⣾⠟⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣶⠿⠟⠉⠀⠄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢰⣄⣦
⠀⠀⢸⡋⣼⠟⠀⠀⠀⠀⠀⠄
⠀⠀⣿⣇⣿⠁⠉⠉⠑⠳⣴⣮⣭⣅⣤
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣟⣙⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠛⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣷⣿⠏⠰⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢴⣿⠉⠉⠐⢤⣤⣴⣄⣤⠤⣞⣿⢟⣿⣿⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⣻⣿⡟⠻⠓⠈⠳⣜⢃⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠀⢻⡔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⠀⠀⠀⢠⣾⡇⠀⠀⠀⢀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣿⣿⣿⣷⣴⣶⡿⢻⣷⣀⣀⣠⣿⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣟⠧⠀⠀⠒⠓⠀⠭⠉⢻⠋⣿⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⡗⢀⣀⠚⠉⠀⢀⠀⠀⠀⠘⢿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣬⡛⠀⠀⠐⠉⢻⣧⠃⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡲⢽⡁⠀⣨⣏⠁⢘⡏⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⣬⣘⠃⠀⠀⣏⣡⢼⡇⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⡿⠭⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠐⢦⡈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠘⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠸⠇⣾⢿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠻⠷⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡟⡁⠉⠉

⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⡠⣾⠀⢻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠎⠀⡷⠀⠸⢀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠁⠀⢀⡽⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠑⣤
⠀⠀⠀⠠⠉⠈⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠆⠙⠦⣀⣀
⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣯⣷
⠀⠀⠀⠀⠀⣰⠇⠙⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢧⡀
⠀⠀⠀⠀⠀⡏⢳⡀⠑⢄⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢰⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣧⠀⠀⠈⡇
⠀⠀⠀⠠⠚⢅⡞⠀⠀⠀⢡⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠼⠀⣿⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⢣⡀⠀⠀⠃
⠀⠀⠀⠆⠀⢸⡷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠀⠀⢀⡇⠸⡙⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣷
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡇⠀⠀⠀⠀⣏⠀⠀⠑⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⢨⡵⢏⠟⠳⢄⢀
⠀⠀⠀⠀⠀⠘⢪⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⡋⠀⠀⠀⠠⡎⠹⣖⡆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣇⠸⡀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠎⠀⣟⢈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠣⡀⠀⢠⡏⠀⢀⡼⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢰⠇⠀⠸⡂⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢠⠏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠘⡄⠀⡏⠀⠀⣸⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠀⢀⠞
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠁⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠒⠀⠁⠈⠈⣁⢀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠂⠤⣀
⠀⠀⢀⠤⠒⠉⠀⠀⠠⠀⠒⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠄⣀⠀⠀⠀⠑⢦
⠀⠰⠁⠀⠀⠀⠀⢀⠠⠄⡒⠠⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠁
⠀⠰⡀⠀⢀⠔⢈⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠂⠤⢀⡀⠀⠀⠀⠀⡠⠃
⠀⠀⠉⠢⣌⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣢⡴⠕⠊
⠀⠀⠀⠀⠘⠬⡙⠳⢦⣤⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠄⠒⠉⢃⠄
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢍⣙⣢⠮⠭⠭⠀⣀⣀⣀⣀⣈⣉⡉⠀⠁⠀⠂⠀⣀⣠⠤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⡀⠁⠡⣍⠙⡓⠛⠚⠲⠶⠴⠦⠀⠉⡛⠒⡒⠪⠭⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠐⠶⠤⠉⣉⢁⣀⠀⠀⠀⠀⠀⠉⠉⠉⣁⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡖⠭⠭⠭⠛⠛⠛⠛⠋⠭⠉⠁⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢶⣚⡀⣀⣀⣀⣀⣠⠶⠷⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠤⠄⠀⠈⠀⠈⣉⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠄⠠⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠍⠁⢌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⡦
⠀⠀⢠⣶⣦⡤⠴⠖⠋⣩⠃⡀⠁⠀⠀⢀⣠⣤⠄
⠀⠀⠘⣿⣷⡀⠀⢀⡼⠃⣰⠀⠀⠀⣠⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠈⠉⠀⢀⣿⡅⠀⠇⣀⣀⣀⢘⣿⣿⠿⢷⡖
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⢹⡻⡿⣿⡿⢿⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⡁⠊⠁⣷⠈⠧⠈⢧⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣼⣟⣧⠀⠀⡄⠀⣇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⢷⡀⢻⠀⠹⣿⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣮⣳⡘⣇⠀⣹⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⢹⣷⣴⠞⠃⠙⢦⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⡟⠉⢀⠙⠈⡵⢤⣤⣷⣾⣂⠀⠀⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⣷⣼⢦⡁⠀⠁⠀⠀⠉⠉⢙⣛⠷⣶⡿⣿⣿⣯⣤⡤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡆⡀⠙⢧⡇⠀⠀⠀⠀⢈⡹⠿⣿⣿⣿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⠌⢣⠀⠀⠃⠀⠀⣀⣠⣬⣶⣿⣿⠟⠀⠀⢀⣨⣝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡌⠀⠈⠀⠀⢠⠀⠀⣿⣿⣾⣿⣿⠇⠒⣚⣛⣛⣛⣛⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡖⠀⠀⡸⠀⠀⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠅⠠⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣇⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠹⣿⡄⠀⠳⠀⠀⢸⣿⣿⣿⡿⣿⠀⢰⣷
⠀⠀⠀⡀⢄⣀⣶⣿⠦⠂⠀⣿⣿⣿⡇⢠⣶⢽⣿⣤⡜⣀⣭⣼⣷⣿⣿⣷⡇⠀⢸⣿⣇⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠠⠐⠉⢭⣽⣿⣾⣾⣿⡿⣋⣧⠘⠿⣾⣿⢟⡌⢹⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⢿⡄⡁⠀⠀⠀⢸⠒⠊⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⣿⣶⣄⡀⢸⣿⣷⠙⠿⢿⣿⣿⣿⣿⣷⠀⠀⠈⡘⣧⣡⣤⡶⠤⠌⠈⠐⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠲⠒⢛⣛⣿⣿⡛⠛⠻⣿⣷⣦⠼⠏⢉⣛⢋⡉⢙⠉⣡⣟⡒⣶⠀⢀⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀⣀⣈⠀⠀⠠⠤⠄⠀⠚⠓⠉⠃⠀⠉⠙⠛⠛⠻⠿⠛⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣻⣟⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣟⣛⣻

⠀⠀⠀⢀⠀⠀⠀⠄⢧⠀⡄⠀⠀⡀⠀⠀⠀⠀⢰⠰⠀⠀⠀⠀⠀⢆⠀⡄⠀⠀⠀⣀⢀⡀⠀⠄⢠⠀⢀⠄⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⢘⡆⢑⠀⠀⠀⠀⠀⠀⠀⠸⠸⡆⡆⠀⡄⡄⠐⠀⠃⠀⠀⢰⢃⡞⠀⠂⠀⠈⡠⠀⣠⠊⠁⢀⡴
⠀⠀⠰⡀⠀⠀⠀⠁⠀⢣⡈⡆⠀⠀⣰⠀⠐⠀⠀⡇⡇⣷⢸⡇⠃⠀⠀⣠⠀⢀⠏⡼⠀⠀⠀⢀⡜⢡⠞⠁⢀⡴⠀⠀⠄
⠀⠀⠀⠙⢄⢀⠡⡀⠰⡜⣧⠸⠀⡀⢿⡀⠀⡆⠀⢇⣷⠈⢸⡇⢰⠀⢰⡏⠀⡜⣰⠁⠀⠀⠀⠊⠐⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠳⡙⠄⠈⠘⣧⢁⢡⢸⡇⢀⠃⢸⢸⢸⣀⣼⡇⣸⠀⣾⠁⣸⢱⡇⠀⣠⠈⠀⠀⠀⠀⠀⣀⣠⠊⠀⠀⡠
⢀⠀⠀⠀⠀⠰⡄⠙⢔⣄⠀⠘⢧⠀⡜⣷⠀⠸⢸⡎⣘⡇⢹⣷⡇⣇⡇⢆⢃⣿⠁⣤⠏⡰⠁⢀⠄⢀⠼⠋⠁⠀⠀⠊⡄
⠀⠃⠀⢠⠀⠀⠙⣆⠸⣾⡣⣜⠘⢄⠃⣿⣧⠀⣿⡇⣏⡇⢸⣿⡇⢹⡸⠈⡾⢃⣾⢃⡼⠁⣠⠞⠐⣡⡾⠃⠀⠀⠀⠀⡀
⠸⠆⠀⠀⠣⠀⠀⠘⢆⠀⢦⠸⡷⣌⠀⢻⣿⡆⠸⡇⣿⠃⠘⡇⠁⣿⣷⡼⢡⠟⣡⡿⢁⣴⣿⣶⣿⠋⠀⠀⣠⠆⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⠎⡆⢽⣮⡀⠘⣿⣷⠀⠣⢿⡄⠘⡟⠀⣿⣿⡅⡫⣺⡿⡡⣿⡿⢻⡿⡡⢊⡴⢞⠑⢉⡥⡪⠞
⠀⠀⠀⠠⡀⢠⡄⠀⠀⠸⣿⡌⠈⣿⡇⠀⠘⣿⢀⠀⠸⡇⠀⠃⠀⣿⡟⠁⡼⠛⢃⣾⠟⣴⣿⢋⣀⣬⢞⠏⠐⠋⠀⢀⡀
⠀⠀⠀⠀⠈⢷⣌⠀⡄⠀⢻⣿⡄⠘⣷⠀⢀⠸⢸⡀⠀⠃⠀⠀⠀⠋⠀⠘⠁⡠⢈⠁⣺⡟⣱⣿⠟⣡⠞⢉⣤⠴⠚⣋⠄
⢈⡓⠂⠀⠀⠀⠙⠷⣌⡀⠀⢻⣷⡀⠘⠄⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⢃⡼⢋⣼⡿⣫⣾⣫⠔⣋⣀⠴⠊⣁⠤
⠀⠉⠉⠓⠂⠄⢀⣳⣬⡻⣆⠀⠙⣗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⢋⡼⢛⣩⣶⡟⠉⠀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠉⠉⠛⠛⣛⣂⠈⠀⠈⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⣟⣋⣀
⢰⡶⠶⠶⠶⠶⠶⠤⠤⠜⠛⢁⣀⣠⠶⢋⣡⣶⠆⢀⡀⢀⡿⢂⠀⣤⡄⣀⠒⢦⡘⠚⢛⣋⣉⣉⣿⣿⣿⠿⣿⣿⣿⣿⣿
⣾⣷⡶⠶⠿⠿⢀⡀⣐⣛⣛⣓⣠⣴⣾⣿⡽⢂⣴⡟⣰⣞⣥⣣⣿⡹⢥⡌⠷⣔⢌⡻⣷⣍⠻⣯⣙⠊⠉⠉⠉⠛⣛⣻⣿
⣿⠧⣤⣄⣠⣤⣤⣤⡈⠉⠉⢉⣉⡀⡛⢉⣴⣿⡋⣼⣭⠉⠉⠛⣿⣿⣌⢻⣦⠀⠁⠉⠊⠉⠀⠈⠉⠁⠐⠻⠶⣦⣄⣙⣿
⣿⣶⣦⣬⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣭⣴⣶⣤⣤⣤⣬⣭⣭⣬⣭⣥⣶⣾⣿⣦⣾⠿⠿⠷⠶⠶⠶⠶⠶⠾⠿⠟
⣿⣿⢟⠿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⣸⣿⣿⡇⢹⠆⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠀⠀⠒⠒⠒⠛⠋⠉⠁⠙⠋⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢰⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠀⠀⠀⠉⠛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠀⠀⠀⢠⠖⠦⣀⡀
⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⣲⠓⠦⠤⠤⠤⣤⣤⣤⣤⣬⣿⣿⣦⡀⠀⠀⠀⠀⠻⣦⣀⣸⣿⠦⢀
⠀⠀⠀⠀⠀⢰⣿⣷⣤⣤⣤⣤⣴⠣⠀⠀⠀⠀⠀⠈⠋⠉⠉⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣅⡀
⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠉⠉⠛⠒⠶⠆
⠀⠀⠀⠀⢠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⠇⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢸⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠹⣿⣷⡀
⠀⠀⠀⣴⡿⠋⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠹⣿⡇
⠀⠀⢸⣧⡀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⢀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⢀⠐⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠈⠅⠀⠀⠀⠀⢈⣀⡄⠐
⠀⠀⠀⠠⠠⠄⠀⠀⡀⠀⠀⠀⠁⠀⠀⠔⠀⠉⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⡅⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠠⠠⠴⠶⠾⣶⡖⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⠉⠛⠿⣯⣙⠲⢄⡀
⠀⠀⠀⢣⠁⠀⠀⠹⣟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠙⠳⡦⣈⠑⠢⣀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠘⢎⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠈⠛⢖⢄⡉⠓⠢⢀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢧⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠈⠈⠢⢄⢀⠉⠂⢠⡄⢠
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢣⡡⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⢀⠈⠙⠛⠟⠒⢬⠗⢾⢶⣦⣀⡞⢹⡆⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠱⡐⠄⠀⠀⠀⠢⡀⢂⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢂⠀⠀⠐⠋⠀⠈⠂⠈⣹⣷⣾⣤⣀⣞⢁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢀⠀⠀⠀⠀⢄⠁⠐⠢⡀⠀⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⣺⠀⠀⠀⠏⡹⠉⠈⡅⠕⠋⠹⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠈⢆⠢⠀⠀⠀⠀⠡⡀⠠⠤⢤⡀⠀⠀⠀⢱⠂⠄⠀⠀⠀⠀⡠⠐⠀⠉⡐⠠⣤⡐⠀⠠⠊⠀⠀⢠⣤⠂⢳⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠢⠱⡀⠀⠀⠀⠈⠳⠠⣀⠓⠤⠀⡀⠙⡀⠠⠀⠎⠈⠋⠀⠀⠀⡅⠈⠀⠈⠣⠐⠁⠀⠀⡠⢊⡇⣠⢖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡘⠄⠀⠀⠀⠀⠑⠄⠑⡀⠀⠀⠈⡣⠘⠄⠐⡀⡇⠀⠀⠀⢐⠀⣄⠀⠀⣯⠤⠤⣊⠔⠁⠀⠁⠀⢈⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠌⢂⠀⠀⠀⠀⠈⠢⠈⠢⡀⠀⠇⠀⠈⢤⠃⠈⠄⠀⠀⢸⠀⠈⢂⡠⠀⠀⡠⠁⠀⠀⣠⠴⡁⢀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠡⡀⠀⠀⠀⠀⠐⢄⠀⠉⠇⠀⠀⢸⠀⠀⠈⣆⠀⢸⠀⠀⠀⣅⠔⠈⠁⠀⠀⣠⠁⠛⠍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢂⠐⡀⠀⠀⠀⠀⠀⠢⡀⠘⠀⠀⢰⡄⠀⠀⠀⠀⠈⠄⠀⠀⠘⠂⠈⠳⠤⠚⠁⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠡⠈⠄⠀⠀⠀⠀⠀⠀⢄⠈⠈⠁⠐⠄⠠⠂⠀⠀⠀⠢⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡐⠁⠀⠀⠑⠀⠀⠅⠀⠀⠀⠀⢸⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠘⣤⣤⣤⡌⠀⠀⠀⡟⠛⠛⢛⠀⠀⠀⢣⣤⣤⣤⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠶⠶⢶⠂⠀⠀⣭⣉⣩⡇⠀⠀⢀⠿⠿⠿⠿⡄⠀⠀⢸⣍⣩⣭⠀⠀⠐⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠿⠶⠾⠄⠀⠰⠉⠉⠉⠡⠀⢀⠎⠀⠀⠀⠀⠘⡄⠀⠌⠉⠉⠉⠆⠀⠠⠷⠶⠷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢡⡌⡰⠀⠀⠀⠀⠀⠰⡌⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⢸⡠⠇⠒⡒⠒⠒⠠⢄⣇⠀⠀⠀⠀⠐⠸⠆⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠰⠀⠀⠀⢀⠔⣡⠖⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠆⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⢃⠞⠁⠀⠀⠀⢠⣦⡄⠸⠶⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⡞⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠳⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠂⠀⢣⠀⠄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠠⠀⡜⠀⠐⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠂⠀⠀⠀⠀⠀⠠⠔⡫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠉

⣤⠀⠘⠉⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠂⠉
⠀⠄⠠⠄⡀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠄⢄⠀⠀⠀⠀⠐⢄
⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠄⠀⠀⠑⠠⠀⠀⡀⠀⢄
⠀⠢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢠⠇⠰
⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠑⠂⢀⠀⠀⠀⠀⠐⠂⠐⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡨⠁
⠀⠀⠀⠀⠑⠶⢒⠀⣀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠂⠄⠀⠀⠐⠂⠀⠒⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠃⢠⠄⠀⠀⠠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣒⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠤⠀⠀⠀⠀⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠀⠀⢀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⢀⣀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢱⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡧⠤⠐⠂⠂⠈⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡾⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠟⠋⠉⠻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣆
⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣧⠈⠻⣿⣿⣿⣿⣿⣿⠟⠛⢀⣸⣿
⣤⣶⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣦⣀⣠⣾⡿⢿⠃⠀⠈⠙⠛⠛⠋⠁⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡎⠁⠘⠉⢳⡀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⢰⣄⣉⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡏⠀⠉⠁⠀⣴⢿⣿⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡇⠀⠀⠀⠀⢸⡘⠏⠀⢤⠤⠀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠸⡧⢤⠶⠶⠾⠏⠉⠀⠀⣈⣄⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠉⠙⠛⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡂⠔⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⡆⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡞⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠙⢦⡀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⢦⣼⢀⠄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⢣⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⣦⣤⣁⣀⣀⢀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣴⣷⣶⣅⡉⠉⠉⠛⠛⠛⠛⢻⡋⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢳⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢉⣩⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣰⢇⡾⠀⠀⣠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣡⠏⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠗⢲⡄⠀⠰⣿⠔⢨⣡⡼⠓⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣡⠾⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣌⠀⠻⣤⣤⡤⠟⠀⣸⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠉⠀⣿⣿⡇⠠⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⠢⢤⣀⣀⣀⠤⠞⠁⠀⠈⠛⠶⠶⠶⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠄⠀⠀⠈⠛⠛⢛⠛⢆⠒⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠄⠐⠀⢀⠀⠀⠀⠈⠐⢈⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣦⣤⣌⣀⣀⣀⣀⡔⠉⠉⣁⣀⣀⣀⣀⣀⣀⠇⠈⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠉⠈⠁⠁⠉⠈⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⣀⡴⠖⠲⢦⡄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⣠⢤⡄⠀⠀⠀⢠⣤⡀⠠⣤⡀⣤⡄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⣤
⠀⠀⠀⠀⠀⣀⣤⠊⠀⠀⠆⠙⠀⠀⣀⣸⡇⢲⡀⠛⠿⣦⡘⢷⣼⣷⠿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠐⢀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⢠⠊⡉⠁⠀⠀⢀⢀⣤⣤⣶⢻⢸⡇⢈⢃⠰⣦⡌⠙⠣⣍⠀⠀⠀⠈⠷⣶⡖⠀⠀⠀⠀⠀⠀⠀⢤⣤⣶⣦⣶⣾⣷⣶⣤⣦⡀⠐
⠀⠀⢠⢺⣿⣿⣠⡶⣶⣏⡿⠇⠛⠘⣛⠋⠁⢘⠋⠉⠠⠀⠀⠀⠀⠈⠁⠐⠒⠲⠻⣿⣄⠀⠀⠀⣧⣤⠠⢤⣹⡟⢻⢿⣿⣿⣿⠿⣿⣁⣨
⠀⠀⡰⠀⣿⠿⠟⠛⠁⠀⠀⠑⠂⠁⣀⢀⣤⣌⡀⠠⠀⣀⠖⠀⠠⠄⠒⠠⢂⣁⠾⣭⣉⠁⠂⣠⣆⢉⣁⢀⣠⣤⣀⣀⣌⡶⣥⠆⡮⢹⣾
⠀⠀⠱⢄⡀⠀⣈⠁⢀⣈⣠⡀⣀⢿⠁⠀⠉⠛⠳⣄⠀⠈⠀⠤⢰⡶⠓⠆⢷⡝⠷⠈⠁⠀⠀⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⠉⠉⢋⠋⠋⣹⡟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⡆⠀⠀⠐⠀⠀⠀⢠⣿⣿⣷⣨⣭⡻⣛⡛⢡⣾⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠻⢿⣿⠆⠀⠀⠀⠀⠀⠘⠿⣟⣶⣼⣯⠁⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠹⣀⠠⠀⠳⣄⡀⠀⠀⠉⠀⢸⡿⠁
⠀⠀⣢⡤⠒⢀⠇⠁⠀⠉⡛⠒⡒⠂⠀⠀⡀⠀⠀⠀⢹⣿⠿⢃⡀⠀⠀⠀⠸⢷⣤⠆⠛⠉⠀⠂⣴⣶⠾⠀⠀⠀⠀⠹⣷⠀⠀⠀⢀
⠀⠸⣿⠀⣀⣁⣐⣀⠀⠀⠉⠈⢁⡰⠃⢀⡇⠀⠀⢸⣇⢱⡾⠛⠿⢤⣀⠴⠖⠒⠺⢄⠀⠀⠀⠀⠐⠆⢠⢴⡞⠀⠀⠀⠀⠀⠀⠒
⠀⠱⢧⣬⣭⡝⠛⠉⠚⢛⣾⣿⡾⠋⠀⢺⣿⠀⣀⣼⡟⠉⢠⣤⣶⣾⠁⠀⣾⣿⡆⠀⣹⣛⣦⠀⠀⣀⠁⠀⢻⣷⡆⢘⠟⠁
⠀⠀⠈⠛⠻⠿⠿⣿⢿⣿⡏⠙⠇⣴⡆⣼⣿⠀⣿⣿⣷⡼⣦⣽⣏⠟⣸⣶⣽⣋⣴⣦⣿⣏⣠⡟⠛⠁⠀⠀⠀⠀⠀⠜⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⢰⣶⣦⣿⡇⣿⣿⣆⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⠁⠈⠿⣷⣆⠀⠀⠀⠀⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⣿⣷⣿⣿⠃⠀⠀⠁⠀⠙⠻⣿⣯⣿⣿⡯⢹⣼⣿⣤⣇⠀⠀⠀⠈⠻⣧⣀⠀⠄⢀⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠸⠿⢿⢿⡀⠔⠀⠀⠠⠀⠀⢀⣽⣿⣿⣷⣦⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠘⣢⢉⠻⡤⠒⠁⣀⣶⣾⣅⡠⠀⠀⠠⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⣶⡖⠋⣉⣠⣤⣤⣀⣀⣓⡻⠛⠱⠂⠺⠟⠍⠀⠀⠁⠀⠠⠐⠀⠐⡀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠈⠋⠁⠉⠓⢆
⠀⠀⠀⠀⢀⣿⡧⠤⠞⠉⠀⠉⠋⠁⠀⠈⠛⣿⠆⠀⠀⠀⠀⠀⠀⡠⢀⡖⢋⠉⠽⠩⠇⠌⡆⠀⡀⢾⠁⠬⢄⣄⢦⣦⣀⡀⠈⡉⢻
⠀⠀⠀⢠⡞⣁⣤⣤⡤⣰⢈⣀⣠⣤⣤⡀⢀⡀⠑⣤⡄⢠⣷⠀⠀⣇⣸⣇⣢⣄⣩⣈⣧⣿⣏⡀⠃⠀⢿⣿⠛⠹⣧⣼⣿⣿⣿⣷⡞
⠀⢀⠀⢼⣿⣿⡁⣸⡿⣿⣸⡿⣯⣤⣼⢷⣾⣿⠟⣼⡇⢸⣿⠀⠀⠈⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⠀⠰⡸⠿⢆⠀⠈⡩⣄⡥⠛⢋
⠀⠈⠀⠈⠳⣝⡛⠋⠑⢤⠀⠀⠀⠀⠀⠈⠁⠋⠛⣹⠃⣸⣿⠀⠀⠄⠸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣀⡀⣠⠌⠹⠾⠷⠉
⠀⠀⠀⠀⠀⠀⢋⠙⢧⣀⠉⣀⣤⡉⠈⢀⣀⣰⠈⠇⢠⣿⠇⠀⠀⣠⣒⡋⣉⣉⡉⠉⠉⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣤⣉⠉⠉⠉⠉⠟⠛⢿⠟⠈⠾⠟⠋⠀⠀⠀⠀⡀⠄⢡⣶⢿⡦⠄⢀⣠⣦⢰⣀⣄⣬⡰⣰⠀⠀⡤⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠇⠀⠀⠀⠀⠀⠀⢠⡈⡀⠀⠀⢸⡀⠈⣾⣿⢻⣷⢠⣿⣯⣿⡈⣤⣾⣿⡧⢟⣹⣷⡆⢀⣤⡀⢀⣤⡄
⠀⠠⠀⠤⠏⡦⡄⢀⠀⠀⣒⠂⢀⡤⢤⠀⠀⢨⣠⡤⠀⣼⣷⣶⣾⣾⣇⠀⠼⠿⠾⠟⠀⣠⡉⢙⣣⠉⢉⡻⣷⠿⢿⣿⣴⣿⡿⣴⣿⣿⣧
⠀⠀⠀⠨⠀⠉⠐⠇⡥⠍⠀⠀⠁⠀⠀⠁⠉⠐⣘⠓⠂⠀⠀⠀⠉⣉⣉⠓⠓⠓⠔⠢⠚⠿⠿⠿⠿⠿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠴⠩⣟⣶⢀⣐⢀⣤⡀⣤⣤⣦⠀⢡⠀⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⢍⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠃⠘⣯⣿⢾⣽⢻⣬⡇⡿⢿⢸⡇⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠄⠀⠀⠀⠀⡀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠳⡤⣍⣁⠉⠑⠉⠈⠛⠒⡈⢘⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢁⡀⠈⠁⠀⠀⡀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠂⠗⠚⠛⠒⠒⠚⠛⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⢠⣤⣄⡀⢠⢠⡄⢠⠀⠐⠒⢲⡒⠒⠒⠀⡀⠀⢦⣶⣾⣿⡟⠀⠀⠀⠀⠀⠰⣤⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠢⠀⠀⠸⠀⡀
⠀⢸⣿⠿⣳⠻⠆⠸⣜⣃⣄⣀⣤⣄⣧⣴⢋⢋⠳⠾⣿⣿⡿⠁⠂⠀⠀⠀⠀⠀⠐⢀⣠⡤⢆⠀⠀⢀⣴⠀⠀⠀⠀⠀⣄⣠⡕
⠀⢸⣿⣶⡄⢬⡷⣿⣼⣿⣿⣿⣟⣻⣿⣶⣾⣷⠇⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⢢⡈⢡⣀⡠⠀⣀⠏
⠀⠘⠛⠊⠛⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠋⠛⠋

⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⡤⠀⠀⠀⠀⠀⢠⣾⣇⡀
⠀⠀⠀⠀⢠⣿⣿⣧⣤⡄⠀⠀⠀⠀⢠⣿⣿⣧⣤⣤⠄⠀⠀⠀⠀⢩⣿⣿⣿⡿⠁⠀⠀⠀⠀⠐⢻⣿⡿⠁
⠀⠀⠀⠀⠸⠿⣿⣿⡟⠀⠀⠀⠀⠠⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⣼⣿⣧⡄
⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠁⣽⣿⡟⠁⠀⠀⠀⠀⠀⠘⢻⣿⠏
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣧⡄⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠰⣿⣿⣿⣤⡀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠈⣹⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⡾⠃
⠀⠀⠀⠀⠀⠈⣻⣿⣿⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⠂⠀⠀⠀⠠⠿⣿⡷⠀⠀⠀⠀⠾⠿⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠉⣿⣿⠟⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣧⣤⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⡰⠁⠀⠀⠀⢱
⠀⢠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠁⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⢠⠇⠀⢀⣠⠄
⠀⠘⣄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡠⢃⡤⠒⠉
⠀⠀⠈⢶⡾⠟⢧⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣈⣉⡉⠐⠒⠤⡑⠂⠸⣏⠥
⠀⠀⠀⠈⢷⡾⠟⠳⣄⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣷⣦⡈⠳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣧
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠑
⠀⠀⣀⠤⠤⢜⣆⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠗⠒⠤⠤⠤⠤⣾⡝⣿⡆⠀⢠⡶
⠀⡸⠁⠀⠀⠀⠙⡧⡤⠤⠒⠒⠦⢿⣿⣿⠘⣿⢣⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠟⠛⠊⠉
⠀⠱⡀⠀⠀⠀⢠⣧⡇⠀⠀⠀⠀⠀⠻⢷⠀⠀⡼⠋⠁⠀⠀⠀⠀⠀⠀⢀⡠⠔⣠⠞⠁
⠀⠀⠉⠒⠒⠚⢻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠲⠾⢓⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣓⢤⣀⣀⠀⠀⢀⡴⠃⠀⠈⠒⠒⠒⠒⠒⠒⠒⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣷⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤