Ladybug Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣤⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢆⣀⠀⠀⠀⠁⡂⢄⠀⠀⠀⡠⣀⠀⠀⠀⠀⣀⡶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠇⠀⠀⠀⠙⢨⠶⠶⢾⡐⠋⠀⠀⠀⢰⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢀⠀⠀⠐⣿⣀⣀⡠⡿⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⢠⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣿⣾⣷⠀⡈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠂⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣠⡟⠈⣋⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣀⣙⠈⢷⣔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⢀⣾⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢰⣿⣿⣧⡀⠻⣦⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⣾⣿⣿⣿⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠙⠿⣇
⠀⠀⠀⠀⠀⢀⣀⣤⠾⠟⠁⠀⠀⢠⠿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠻⢻⡿⠛⢿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠿⠟⠀⠀⠀⠈⠻⠶⣄⣀⡀
⠀⠀⠀⠀⢴⠏⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠈⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⣷⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣷⡀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⡄⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠇⠀⣰⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠸⣧⠀⠀⠻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⣿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣰⣾⠿⢷⣶⣶⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⣰⣾⣟⠻⢫⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣷⡄⢸⣿⡟⠙⠛⠉⢸⠛⠛⠂⠙⠛⠛⢫⠉⣿⣿⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣷⣾⣷⣾⣷⣼⡟⣰⣷⣶⣷⣶⡿⣰⣿⡿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⢶⣶⣶⣶⣶⣤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣽⣯⣯⡿⠽⢁⠁⠈⢀⣠⣄⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⡀⠀⠈⠿⡿⠿⠃⡀⠤⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡷⣄⠀⠀⠀⠁⠐⠚⠺⢟⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠒⠉⠻⠿⣿⢿⣿⢗⠏⠉⠁⠙⠿⣿⡨⣗⢦⡀⠀⠀⠀⠘⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠤⠛⠉⠈⢀⢀⣤⣄⠀⠄⢀⠙⢿⣾⣿⣦⡀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡠⣢⠀⠀⣀⠂⡈⢀⣾⣿⣿⣆⡉⣈⣭⣵⣮⣿⣿⣿⣶⣾⠆⠻⢛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣿⣦⣻⣪⡥⣀⠠⢿⣿⣿⢟⣻⣿⣿⡻⢏⣼⣿⣻⣿⣿⣦⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣿⣿⣍⡵⣿⣯⡂⠀⠀⠙⡹⣋⣿⣻⢾⣿⣿⡿⠛⠛⠀⣼⣿⡟⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⡯⣾⣿⣿⡦⡄⠀⠀⠰⢛⣽⣿⣟⣻⡯⣷⡂⢀⣼⣿⣿⠃⢀⣙⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⣎⣉⢭⣟⣆⢅⣤⣴⣵⡻⢫⣾⣾⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣉⠦⠶⠖
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠙⠿⢷⣮⣿⣿⣿⠿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⢻⣿⠋⠁⢸⣿⣯⣥⡄⠼⢻⡟⠻⡟⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠁⠀⠀⠀⠈⠁⠞⠀⠀⠀⠓⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣩⣭⣥⡄⣤⣤⣭⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠷⣤⣀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⣀⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⣠⣆⠀⠀⣸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣄⠀⢀⣴⣄⠐⠛⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡿⠋⠙⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣧⡀⠀⣠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣼⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠰⠶⠶⠶⢶⣄
⠀⠀⠀⠀⢰⡟⠛⠛⠛⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡇⢸⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣇⢸⣿⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢃⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠙⠳⣄⠀⠀⠀⠀⣠⠞⠋⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⡆⠀⠘⡆⠀⠀⢰⠃⠀⢰⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣀⣿⣤⣤⣿⣀⠀⠈⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⢻⠛⢻⣿⣿⣿⡉⢻⡿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⡿⠿⠻⠿⣾⢷⣞⣿⣿⣈⣙⣶⣿⣷⠞⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢳⣯⣤⡴⠲⡄⠈⢧⢸⣿⣿⣏⣡⠟⡟⢯⡙⠛⢯⡉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣿⡿⠁⡇⠻⣿⠟⠓⣎⣿⣿⣿⣿⠃⣰⣧⠈⣿⢶⣤⠷⠶⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⡟⠙⢦⣧⠞⠹⣦⣴⣏⢹⣿⣿⡿⢶⣋⡼⢦⢿⡞⢰⠋⣳⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣈⡿⠙⠦⣤⣇⡴⠚⢻⣇⣸⣿⣿⣿⣿⣠⣿⠷⢾⣾⠀⠘⢦⣥⣴⡋⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⡗⠦⣄⣈⣇⣠⡤⠾⡇⠀⣿⣿⣿⣿⠀⣇⡴⠋⣸⣏⣹⡞⣡⢤⡙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣾⡷⠤⣄⣉⣏⣠⡤⠴⣷⣾⣿⣏⢹⣿⡶⠛⠛⢒⡟⠛⢷⡇⢧⣒⣣⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣄⣀⠈⡏⢀⣠⣾⡇⣽⣿⢻⣼⣿⣿⠛⢒⣾⣷⡤⠞⠹⣴⠋⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⢿⡘⣯⠟⠛⣯⠏⣼⣶⣿⡿⠯⣄⢹⣿⣶⠞⢧⣸⣇⠀⠀⠘⡗⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣸⡄⠀⣿⣴⣿⣿⣿⢤⣀⣈⣙⣿⣿⣦⣰⣇⣸⣷⠋⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣸⣿⣿⣿⡿⠛⣋⣽⠟⠁⡼⠹⣿⣿⣭⡟⣻⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠿⠿⢿⣯⣽⣾⣀⣰⣧⡴⠿⠻⢿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠋⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⢀⣀⣀⣀⣀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠋⠉⠉⠙⠛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣀⣠⣤⣴⠶⣶⡶⢶⣦⣤⣄⡀⠀⠀⣿
⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠟⠋⠉⠀⢀⣤⠾⢦⣄⠀⠈⠉⠛⠷⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣾⠋⠀⢀⡴⠛⠉⠙⢳⡄⠀⠀⠀⣿⡇⠀⠀⠀⣰⠞⠉⠉⠳⣆⠀⠈⢻⣷⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⡟⣧⠀⠀⠀⣏⠀⠀⠀⠀⣹⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠳⣤⣀⣤⠾⠁⠀⠀⢸⡇⣿⠀⠀⠀⠙⢦⣄⣠⡴⠋⠀⠀⠀⠈⣿⡀
⠀⠀⠀⢀⣠⣤⣤⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣀⡀
⢀⣤⣾⠿⠛⠋⠉⢻⣿⠀⠀⢀⡴⠚⠋⠙⠳⣄⠀⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⢀⡴⠚⠛⠛⠲⣄⠀⠀⢸⣿⠉⠉⠙⠻⢿⣶⣄
⠿⠟⠁⠀⠀⠀⠀⢸⣧⠀⠀⣼⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡆⠀⢀⣿⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣆⠀⠀⠀⢀⣼⠃⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠹⣄⠀⠀⠀⢀⣼⠃⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠈⠛⠒⠒⠋⠁⠀⠀⠀⢰⡇⠀⠀⢿⠀⠀⠀⠀⠈⠓⠶⠖⠋⠁⠀⠀⣼⠇
⠀⠀⠀⠀⢀⣴⣶⣿⢿⣷⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⢸⠇⠀⠀⢸⡆⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⢰⣿⣿⣷⣦⣄
⠀⠀⠀⣴⣿⠋⠁⠀⠀⠹⣧⠀⠀⢰⠏⠀⠀⠈⣷⠀⠀⣿⠀⠀⠀⠈⣇⠀⢰⠏⠁⠀⠙⣦⠀⠀⢠⡿⠁⠀⠀⠉⠻⣷⡄
⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠙⣷⡀⠸⣆⡀⠀⣠⡟⠀⢀⡏⠀⠀⠀⠀⢿⠀⠸⣆⠀⠀⢀⡟⠀⣰⡟⠁⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠉⠋⠉⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠉⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣾⣧⣄⣀⣀⣀⣀⣠⣽⡶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⢰⡞⣉⡉⢠⣦⣀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡀⡀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⡿⠀⠸⠟⠁⠊⣉⠁⢠⡈⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣁⣱⣇⣀⣤⠎
⠀⠀⠀⠀⢀⣴⣿⡅⢀⣤⡘⠿⠃⣼⠿⠈⢛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⠋⠉⠉⠁⠢⢄⣀
⠀⠀⠀⠀⠈⠉⠻⠤⠈⠟⠃⣴⡿⠛⠿⣿⣶⡧⠀⠀⠀⠀⠀⠀⢹⣿⠁⡀⠀⠀⠍⣿⣆⠙⠻⢶⠈⠉⠉
⠀⠀⠀⠀⠀⢀⣀⡠⠖⠀⣻⡋⠐⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠉⠀⠙⠂⠈⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠲⠋⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣔⢠⣶⡄⠀⠀⠀⢦⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⢀⠼⠿⠗⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠍⠉⠀⠒⠴⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⢀⣴⡤⢀
⠀⠀⠀⡀⠀⠀⣠⣤⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠈⠙⠁⠀⠱⣄
⠀⠀⠀⣷⠀⠀⠿⠇⠠⢄⠀⢿⣿⣿⡞⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣄⡀⠀⣴⣦⠀⢀⡀⠐⣿⣆
⠀⠀⠀⠿⣈⠐⠂⣄⠀⠀⠀⠀⣻⣿⣿⠀⣂⣠⠂⠀⠀⠀⠀⢀⠰⠆⠉⢿⣿⣶⠟⢋⠑⢈⣋⣄⢌⠻
⠀⠀⠀⠘⢻⣿⣤⡛⢁⣄⣀⣼⣿⡿⢻⣗⣩⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⣶⣤⣬⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⠋⠉⠙⠳⠾⢻⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠙⠛⢿⣏⢉⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⣿⣅⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⢀⣀⣠⠀⠐⢆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠒⠆
⠀⠀⣰⣷⣾⣿⣿⣿⣯⡉⠶⢤⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠘⠿⠟⠀⠀⠘⠶
⠀⠀⠀⢭⠿⢿⣿⣿⡟⣏⠀⡀⠈⢙⡂⢁⠀⠂⠀⠀⠀⠀⠀⠀⢳⣿⣄⠀⣴⣶⡅⠀⠀⢀⣰⣯
⠀⠀⠀⠁⠀⡯⢻⣿⣿⣿⡑⠯⡀⠥⡆⠀⠀⠀⣄⠀⠀⣡⠀⠀⢠⠀⠀⠀⠘⠛⠁⠀⠀⢸⣷⡏⢁⣦⠔
⠀⠀⠀⠀⠘⠀⠈⢼⣿⣿⠇⠀⠀⠉⠁⠀⠀⠀⠒⠈⠉⠁⠀⠀⠘⢧⠀⠸⠦⠤⠀⣠⣤⠀⢀⣀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣄⣀⡙⢻⠟⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠢⣄⣤⣤⡤⢄⡀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⣀⣤⣤⣤⣄⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣻⡶⠾⣻⣿⣶⣾⠃⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣀⠰⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⣎⣀⣴⡏⢿⠻⡿⠟⣿⢻⣧⡠⣵⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣥⣾⣴⣄⣼⣷⣾⣿⣿⣿⠮⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠬⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠟⠛⠒⣒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⡐⢼⠀⠀⠀⠀⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠟⠙⠀⢀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡿⣑⠀⠀⠅⠀⠀⠀⠀⠐⠽⠂⢸⡥⠵⡀⠀⠂⠀⠀⠀⢊⡸⣷
⠀⠀⣀⣤⣀⣀⠀⣿⡇⣿⠅⢀⠀⡀⠀⢀⡲⡾⣿⠀⢃⢣⠅⠀⠀⠘⠀⠀⠀⢾⡿⢻⡇⢀⣀⣠⣤⡀
⠀⠀⠀⠀⠈⠙⠷⠿⠃⠉⠀⠀⢐⣦⣤⣼⢹⣿⣿⠠⣰⣆⢀⣠⣤⣤⣌⣠⠠⠀⠁⢸⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣟⠠⢸⣿⣿⣿⣿⣿⣿⣿⢂⠅⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠠⠹⢿⣿⣿⣿⣿⢿⣿⢹⢾⢿⢿⣿⡿⣿⠿⠏⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢬⠅⠈⡅⣮⢿⠘⣻⣿⠐⡀⢀⠉⣈⡘⡀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⡀⠠⠀⠀⡤⢈⡁⠄⣀⡿⠀⠀⢀⢖⠔⠰⣀⣤⡄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡽⣿⣶⡀⠀⢒⢽⡅⢳⢾⠃⠀⢴⢨⠘⠂⣿⣿⣿⢧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡈⠻⡋⣄⠁⠨⠂⠈⠰⢀⠄⠣⡕⠀⡀⠝⡩⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠃⠀⠈⠲⢄⣿⢰⢰⡆⠈⠇⡳⠘⡣⠕⠁⠀⠘⢷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠅⠒⠀⠂⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⢀⠀⡀⣠⣀⣀⣠⣀⣀⣠⡀
⠀⠀⠀⢸⣰⢷⢧⠇⡇⣿⠿⡿⢷⡆

⠀⠀⠀⠀⠀⠀⢀⡔⠚⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢦⣤⡄⠀⠀⠀⠀⠀⢀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⣼⡄⢠⣦⠈⢙⣤⡄⠀⠀⠀⠀⠀⠀⠀⠴⢷⣿⣖⢻⡗⠀⠀⠀⠀⠀⠀⠀⡈⢸⣷⢈⡀
⠀⠀⠀⠀⠀⠞⠉⢳⣬⡋⢼⡛⣿⣿⠂⠀⠀⠀⠀⢠⣂⣐⠟⣻⣿⣿⡷⠀⠀⠀⠀⠀⠀⣌⢩⣿⣿⣎⢁⡄
⠀⠀⠀⠀⠀⠀⠠⠎⠉⠀⠸⡀⠉⠃⡃⠀⠀⠀⠀⠀⣰⣷⠏⠙⠩⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠃⠁
⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⣻⣆⡔⠁
⠀⠀⠀⠐⢮⢲⣧⡰⠀⠲⠀⠀⠐⠃⡸⠋⠀⠀⠀⠀⠀⢀⡤⡤⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡂⠠⠷⠀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠳⣾⣦⣷⡞⠋⠁⠀⠀⠀⠀⠠⠀⠘⠛⢧⣰⠆⠘⡀⠀⠀⠀⠀⠊⣸⠃⣤⠀⢴⠈⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣋⣤⡍⠀⠀⠀⠀⠀⢾⣤⣞⣀⣐⣂⣽⣷⡾⠁⠀⠀⠀⠀⠀⢯⢷⡄⢀⡀⠖⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣶⣶⠶⣶⠶⣶⣶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⢿⠷⣶⣾⣿⣴⣾⣿⣿⠶⠶⣤⡀⢀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡤⠖⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⡋⢻⣿⣿⣿⣧⣀⠀⠀⢹⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣾⡏⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠸⣿⣿⣿⡿⠋⣿⣿⣿⣿⣧⣀⣤⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠛⣿⣿⡆⢀⣽⡞⣿⠗⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣰⡿⠋⠀⢀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡟⠿⣧⠉⠁⢠⣿⣿⣿⡿⠀⠉⢻⡟⠁⠈⠛⠷⠆⠖⠚⠁
⠀⠀⠀⠀⣰⣿⠀⠀⠀⠸⣿⣿⠟⠁⠀⠀⠀⡀⠄⡒⠁⣿⣿⣿⣯⣿⠟⠁⠐⠙⢷⡀⠀⠀⠀⠉⠁⠀⢀⣾⠃
⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⡩⠒⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⢤⣀⣈⣖⣿⠏
⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⣀⡤⢚⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠛⠿⠿⠋
⠀⠀⠀⢸⣿⠣⠀⠀⣠⢞⡭⠚⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⠷⢶⣤⠤⠶⠶⠚⠓⢦
⠀⠀⠀⠘⣿⠇⣠⣾⠕⠃⠀⠀⠀⠀⢸⣿⢿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣴⣾⣿⡆⠀⠀⠘⠳⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠹⣠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⡼⠟⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠻⣿⣦⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⣠⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⢀⡀⢠⣴⣴⠿⠋⠁⠀⠀⠉⠛⠷⣦⣤⡀⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣟⣤⣤⣶⣴⡶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣤⣔⣫⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⠀⠀⠀⢠⡶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠢⣄⠀⠀⠀⣴⡞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⢔⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣦⡀⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⡿⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠠⠋⠈⠀⠀⠠⣠⣴⣶⣿⣿⣷⣶⣦⡀⠀⠀⠉⠀⠈⠒⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⠀⠁⠀⠀⠐⢻⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠨⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⠛⢻⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢡⢽⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣦⡀
⣤⠴⢾⠿⠛⠛⠋⠉⣿⢿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠈⢿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⣿⣿⣇⣓⠀⠀⠀⢠⣿⠇⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⣿⠈⠻⢧⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠿⣿⣿⣿⣟⣻⠀⠀⠠⣿⡄⠀⢰⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⣷⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣀⣅⡀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢯⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⠻⣿⣿⣿⣿⣤⠀⢀⣁⣴⣦⣿⣿⣦⡀⠀⢄⢠⣿⣿⣿⡿⠋⣵⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠙⠧⡁⠀⠀⠀⠙⠹⠿⣿⠛⠛⠛⠋⠉⠉⢛⡽⠞⠁⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠃⠀⠀⠀⠈⠒⢤⣀⠀⠀⠀⣿⡂⠀⠀⣠⡴⠚⠉⠀⠀⠀⠻⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠞⠉⠚⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⣄
⣶⣶⡶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠲⢤⡉⠿⠖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⣠⠖⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢶⠶⡦⠄⣀⡀
⠀⠀⡀⠀⢀⣶⠀⠈⠳⣄⣀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣀⣤⡞⠁⠀⠻⠿⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣧⣶⣾⣿⣦⡀⡠
⠀⠀⡟⠀⠈⠁⠀⠀⠀⠿⣿⣤⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⡛⠋⠀⠀⠀⢀⡀⠀⠳⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⣤⠀⠀⣤⣿⣿⣁⠸⢿⡧⠤
⢀⣰⠓⡄⢰⣦⠈⠀⠀⣰⣥⣨⣿⣿⡳⡄⠀⠀⠀⠀⠀⠀⢠⢿⣟⢀⣠⣿⣦⠀⠀⠀⢿⠇⣀⡄⠀⠀⠀⠀⠀⣴⡆⠀⠀⠙⢋⣠⣶⠛⠿⠏⢻⠷⡄
⠈⠁⠀⠈⠂⠡⣤⣀⣴⣿⡟⢻⣿⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠛⢿⣗⣤⡤⠲⠞⠉⠁⠀⠀⠀⠀⠀⠙⣱⣤⢦⣾⠛⣱⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣰⠋⠁⠻⢬⡉⢉⠛⠁⠉⠉⠀⠀⠀⠈⠉⠉⠻⠿⠿⣟⡲⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠊⠈⠰⠇
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠶⣄⡀⠀⠈⣡⣤⣔⡁⠀⠀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠈⢷⣠⣨⣿⣿⣿⣅⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡞⠁⠀⠈⠙⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠈⠑⢤⣾⣿⣿⣷⣤⠰⠄⢱⡄
⠀⠀⠀⡀⠛⠛⣳⣶⡞⠛⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡠⠿⣟⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣴⣿⠈⢻⣿⣿⡿⠁⣶⣦⠈⠙
⠀⢠⣿⡆⠀⣀⠈⠟⠁⡀⠁⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣤⣬⣧⡀⠀⠘⢿⠇⢀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠙⠁⠀⠈⣿⣿⠁⠀⠈⠋
⣠⠏⢹⠁⢸⣿⡇⡇⢸⣿⠆⠈⠻⠹⢄⠀⠀⠀⠀⠤⢀⡟⣻⣿⣿⠟⠙⢿⣷⣤⣄⠒⠚⠛⢳⡀⠀⠀⠀⠀⠀⠱⡘⠿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠠⡿⡬
⠀⠀⣼⠄⠀⠉⠀⣷⠀⠁⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⢶⠿⠹⣿⡟⠃⣴⡈⠀⠈⠉⠙⣧⠀⠀⠉⠙⠁⠀⠀⠀⠀⠘⠦⡀⢀⣴⣞⡋⢛⣷⣤⣀⠠⠊
⠀⢀⡇⠘⢶⡀⠀⣿⠀⠀⣴⠟⡟⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢘⡄⡸⠃⠀⠀⠀⠀⠘⠷⠦⠄⠀⠀⠀⠀⠀⠀⠠⠴⠛⢿⠛⢿⣷⣿⡿⢻⢿⣃
⠐⠋⠀⠀⠀⠑⠂⠚⠐⠒⠁⠀⠹⡄⠀⠀⠀⠀⠋⠁⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⣿⡿⣋⠿⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠳⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠀⠀⠀⠀⠠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣀⣀⣀⣠⣀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡾⠟⠛⠋⠉⠉⠲⢫⣾⡟⢲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣏⣹⣄⠀⠀⡀⢠⠤⠐⠈⠁⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠳⣄⣈⡙⠛⠿⢿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⢲⣤⣀⠙⠻⣿⣿⣿⣿⣿⣿⡇⠁⠙⠾⣇⠀⠀⠀⣸⡟⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡼⡿⠿⠿⠦⢄⡉⠻⢿⣿⣿⡇⠀⠀⠀⢻⡄⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣫⣾⣿⡄⠀⠀⠀⠉⠳⣄⠙⠻⣷⣄⠀⠀⢸⡇⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣫⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⢄⠈⢿⣷⣦⠈⣇⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣉⣁⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠉⠛⠁⠀⢠⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣴⣾⣷⣦⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀
⠀⠀⢀⣠⣴⣿⠃⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠶⠦
⢀⣾⣿⡟⠉⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠋⠃⠙⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣤⠀⠈⢳⡀⠀⠀⡴⠋⠀⢠⣶⣆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⢻⡄⣼⠁⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⣽⡅⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣠⣾⣷⡄⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⣴⣿⣶⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⣷⡄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣤⣦⡄⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣾⣿⣦⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠿⠿⠋⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠛⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⡾⠉⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠹⡆⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠸⢿⠟⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⡀⠀⠀⢠⣤⡀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⢀⠀⠀⢿⡿⠀⠀⠈⠛⠁⢸⣶⣿⣶⡴⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⡀⠀⠀⣠⣄⢀⣾⣿⣿⣿⠷⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣔⣿⢿⣿⣿⣿⣿⠏⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣮⣟⢿⡟⢿⣾⢏⣿⣶⡄⠀⠀⠀⣀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⣿⣿⣿⣿⢧⣾⣿⠿⣿⣝⠅⠀⠀⠸⡿⠇⠀⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣌⣏⣿⢶⣽⢛⣾⣯⣟⡿⢿⠁⠀⠈⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢦⣽⣿⣾⣝⣛⣾⠇⠀⠀⠀⠀⠀⠀⠈⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⢒⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛


⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⣀⡠⠔⠂
⠀⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⠀⢀⣦⣤⣼⣆⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢦⣸⣡⣿⣿⣿⣧⣼⣤⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⠛⣿⣟⠋⠉⢀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡀⠀⠀⠀⢾⣿⠆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠿⠗⠀⠘⠿⠿⠃⠀⠻⠟⠀⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢠⣶⣿⣿⡟⠛⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠤⠤⠖⠋⠀⠀⠀⠀⠀⢀⣤⡀⠀⣴⣤⠀⠀⠀⠀⠈⠑⠢⠤⠤⠀⠀⠀⠀⣠⣽⣾⣿⡿⠋⢻⠃⡀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡔⠛⠀⠀⠀⠈⠛⠁⠀⠙⠉⠀⠀⠀⠛⢢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢣⠀⠉⠀⣽⠃⠈⢲⣤⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣤⠀⡐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠉⢦⡀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠙⠛⠀⠀⡆⠀⠙⢋⠄⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠄⠀⠁⠀⠀⠀⠀⠈⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠋⣭⣉⠛⣓⠲⢤⡀
⠀⠀⠀⠀⠈⠻⣿⡳⠶⠶⣤⡤⠤⠴⠶⠖⠛⢉⣁⠀⣉⣵⠶⢾⠛⠋⠉⠙⠛⡻⣶⣄⠀⠀⢀⡴⠤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⡶⢤⣄⣀⣀⣐⣤⣤⡴⠞⢋⠟⠀⡰⠃⠀⣠⡴⠚⠉⠉⠉⠛⢷⣴⡏⠁⠀⢬⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡀⠀⠔⠈⠀⠐⠁⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⢶⣻⣿⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⢀⣀⣼⣇⣠⡟⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠛⠆⠀⠉⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⠀⠐⢿⣶⣶⣿⣵⣾⣷⣦⣠⣾⣷⣮⣿⣶⣶⡾⠂⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠂⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠳⠶⣴⢀⣰⣿⣿⣿⠀⠀⠀⠀⠀⢐⡄⣦⠴⠞⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡘⡌⠙⣿⣿⡋⠀⠀⠀⠀⣴⠋⢁⢃⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠾⣿⠿⠦⢽⣿⠀⠀⠀⠀⣸⡿⠴⠿⣿⠷⠶⠤⣤⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⣰⠟⣶⠦⣤⣪⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣕⣦⠴⣶⠻⣆⠀⠀⣀⡀
⠀⠀⠀⠀⠉⠈⢯⣝⡳⡏⢠⡏⢸⠇⣿⣿⣿⡇⢀⡤⠀⠀⠀⠀⣦⡀⢸⣿⣿⣿⠸⡇⢹⡆⢹⠞⣫⡭⠅⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠟⠀⣿⠀⠿⠿⠛⠀⠘⠁⠀⠀⠀⣸⣿⠃⠀⠙⠿⠿⠀⢿⠀⠻⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣤⣾⣿⠀⠀⣾⣷⣤⣀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⡗⢤⡶⢶⡄⠲⢶⣿⣿⣿⡿⠟⠃⠀⠀⠘⠻⢿⣿⣿⣿⡶⠒⢀⡖⢶⡤⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⢻⢸⡇⢠⡀⠙⠋⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠙⠋⢀⣄⢸⣧⡾⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠺⣷⠘⣿⢆⠀⢀⣾⡿⢿⡿⢿⣿⡿⢷⠀⠀⡴⠸⠇⣾⠏⠀⣴⠂
⠀⠀⠀⠀⠀⠀⢀⣤⠄⠚⡓⠀⣿⠇⠸⣿⣧⠈⠻⡶⣿⠛⠂⣿⡇⠘⠀⣾⣿⠀⣼⣿⠀⠘⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠟⠂⠉⠉⠈⠉⠀⢠⡈⢉⣀⣀⡀⠉⠀⡀⠉⣁⢀⣀⡉⢁⡐⠁⠉⠁⠈⠉⠛⠂⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠻⣿⣿⣷⠀⠃⣾⣿⣿⠟⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⠛⠛⠰⠆⠛⠛⣡⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⢧⣤⡀⢠⡔⣫⢍⡀⢿⣿⣿⡿⠋⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⣴⣶⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣍⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⣤⣏⢀⣼⣿⣿⣿⣷⠀⣠⣴⣶⣤⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡀⠙⠳⣶⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠐⠦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡶⠶⠶⠦⢤⣤⣀⡙⠻⠟⠁⠠⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⠁⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠘⠎⣇
⠀⠀⠀⠀⠀⠀⢀⣠⢿⣿⡾⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠙⠦⣀⠈⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⣏⠈⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠳⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠙⢶⣾⣿⣿⡿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣽⢿⡃⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠛⠁⠸⡜⣷⡄⠀⠘⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣵⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⣶⣤⣤⣴⣶⡶⠾⠟⠛⠉⠈⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⢩⣿⣿⣿⣿⣿⣭⣭⣉⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠬⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠿⠶⠦⣴⡊⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⢦⣈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢰⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠛⠙⠛⡿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠈⠐⢦⡎⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⠋⠀⠀⠀⠀⠀⢀⣾⣿⡧⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢼
⠀⢀⡀⣀⡀⠀⠀⢀⡞⠋⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⡇
⠀⠙⠛⠉⠙⠃⢦⡈⡧⡄⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡸
⠀⠀⠀⠀⠀⠀⠀⠉⣳⡴⠀⠀⠀⠀⠀⢶⡉⠀⠀⢀⠀⠀⠀⢀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡷⢄⠈⢐⠀⠀⢸⡇⠀⠀⠈⠢⠀⡠⢂⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢌⠑⣄⣀⣀⣾⣄⣀⣀⣤⣼⡟⣾⠉⠻⡿⣦⡢⢀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠋⠁⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⡧⠀⠉⢉⡏⠉⠻⣀⠻⡍⠣⣰⣇⠴⣼⠆⢣⠈⢛⡷⣯⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡶⣶⠟⢛⠋⠧⣿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠄⣾⠇⠀⠀⠈⠉⠛⣧⡷⢛⡼⠃⠀⠀⠰⣯⢵⡒⠈⠉⠭⣉⡉⠉⠉⠉⠙⢦⣀⣉⠴⠊⠀⠀⠀⠀⠑⠄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⢀⡴⠚⣡⠔⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠄⣀⢽⡳⡄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣢⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡄⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣹⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠐⢠⡟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⡢⢄⣀⠀⠀⠀⠘⢑⣠⠔⠓⢀⠀⣀⠴⠀⠀⠀⠁⡺⡞⣳⠄
⠀⠈⠢⠴⠈⠇⠀⠀⢀⡄⠀⠀⢰⠀⠀⠀⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠓⠻⣙⣻⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠐⢀⠇⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠠⠭⣦⡄
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠡⡀⠀⠄⠘⠠⠒⠂⢄⠀⠀⢰⡀⠣⡬⢺⡀⠀⠀⢈⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠐⡔⠀⠀⠀⠀⠙⠋⠁⠀⠑⠀⠀⠀⠀⠀⠐⢤⡤⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠷⠂

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠠⣤⡁⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠦⠐⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠊⠁⠒⠦⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠄⠀⠀⠀⠀⠀⠐⠳⠤⢴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠚⠈⠁

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣀⣤⡀⠀⡀⢀⡆⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⢸⡹⠁⠀⠀⠔⠋⠠⠄⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠈⢈⢃⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠁⠂
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠉⠳⣄⠀⠀⠀⠀⡴⠛⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡾⠁⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠂⠀⠀⠀⠀⢸⡄⣸⠃⠀⠀⠀⠀⠰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣷⣿⣧⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠁⠀⠀⢸⡇⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡴⢋⣤⣀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣠⠤⣄⠉⠲⣄⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⡰⠋⠀⣇⠀⢸⠇⠀⠀⠀⠀⠀⢸⠀⠀⢰⠋⠀⠀⠀⢳⠀⠈⢣⡀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡾⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⣀⠞⠀⠀⠀⢱⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠖⠒⢢⡀⠀⣿⠀⠀⠀⠀⠉⠉⠁⠀⠀⣠⣄⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⣠⣤⡀⠀⠸⣄⠀⣸⠃⠀⣿⠀⠀⠀⠀⣀⡀⠀⠀⡞⠀⠀⢹⣿⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠀⢠⠏⠀⠀⠈⡆⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⡞⠁⠙⡆⠀⠘⠦⠤⠊⡼⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠸⡄⠘⢦⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⡼⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣄⠀⠀⠀⢰⠃⠀⠈⡇⠀⠀⡇⠀⠀⠀⢰⠁⠀⠈⡇⠀⢀⡞⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠈⠢⣀⠀⠈⠳⠤⠔⠁⠀⢸⡇⠀⠀⠀⠈⠳⠤⠞⠁⡠⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠼⠧⠤⠤⠖⠒⠉⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⡴⠋⢡⣴⣶⣷⣾⠿⡿⣿⣄⣬⣭⣓⠲⢤⣀
⠀⠀⠀⠀⢀⡾⢰⣿⣿⣷⣿⣽⣿⣿⣟⣿⣿⣿⣿⣵⣿⣷⣮⡕⠢⣀
⠀⠀⠀⠀⣼⣿⡈⠿⠿⣿⣿⣿⣾⣿⣿⣏⠀⢀⣽⣟⣽⣿⢿⣿⣷⡈⠳⡀
⠀⠀⠀⢠⣿⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⡛⠻⠻⢾⣯⣿⡃⢳
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣬⡙⠛⠹⣿⣿⣿⣿⠍⢁⣺⣿⣆⡅⠀⠈⠙⢧⡼
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣬⣿⣿⠟⠀⠀⡀⠀⢀⣱⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠀⠀⠀⣠⣶⣿⣿⣿⣿⣴⠖⠀⣉⣻⣶⣃
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣞⣂⣭⠀⣴⣿⣿⣿⣿⣿⣿⣿⣇⠠⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠙⢊⣙⡛⢿⣿⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠁⠈⠛⣋⠈⠀⡿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⡀⣠⣏⠐⠀⠁⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⡿⠯⠉⠻⢿⣿⣿⣿⣧⠚⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠈⠻⣿⣽⣾⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠋⠉⢉⠉⠙⠛⠙⢿⣿⣗⡀⢀⠀⠀⠻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣆⠈⠀⡀⡀⠀⠁⣠⣌⢻⣿⣿⣿⢏⡙⢂⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣤⣴⣾⣿⣅⢶⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⡔⡞⠓⣶⡄⠀⠀⠀⠀⠀⢀⠎⠀⢱⠀⠀⠀⠀⠀⠀⠀⡴⠉⠙⡄⠀⠀⠀⠀⠀⢀⣔⣲⠛⠛⣷⡀
⠀⢰⠁⠈⠈⠉⠁⠘⡄⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠱⡀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠀⢱
⠀⠘⡄⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠣⡀⠀⢀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⢸⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡜
⠀⠀⠈⠢⠤⠤⠴⠁⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠈⠚⠙⠚⠉⠁⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⢀⣴⠾⠷⢢⡄⠀⠀⠀⠀⡠⠊⠉⠾⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⡨⠛⢦⡀⠀⠀⠀⠀⣶⣿⣿⣿⡅
⠀⠀⣞⡂⡅⢘⠹⣸⡅⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⠀⠀⣠⡾⡿⣷⣄
⠀⠈⣖⠅⡁⢀⢒⢼⠆⠀⠀⠘⣄⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⣀⢀⠤⠊⠀⢀⠇⠀⠀⠀⠀⣟⣝⢾⡳⣿
⠀⠀⠛⡯⣄⣀⡧⠋⠀⠀⠀⠀⠈⠑⠲⠤⠴⠊⠀⠀⠀⠀⠘⣅⣀⣀⣨⠴⠋⠀⠀⠀⠀⠀⠹⣵⣳⣻⠏
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⡰⠋⠙⠉⠉⢆⠀⠀⠀⠀⠀⢀⡠⢤⣀⡀⠀⠀⠀⠀⠀⢀⡠⠦⣧⣤⡀⠀⠀⠀⠀⡴⢋⠁⠈⢉⢙⢦
⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⠀⠀⠀⠈⡆⠀⠀⠀⢀⡞⠃⡐⠀⠁⢹⡀⠀⠀⠸⠐⠀⠀⠀⢀⠐⠈⡇
⠀⠀⠱⡀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⠇⠀⠀⠀⢸⡇⠉⠁⠀⠸⢸⠁⠀⠀⢠⠀⠠⠀⠀⠀⠠⢠⡇
⠀⠀⠀⡇⠀⢀⡰⠃⠀⠀⠀⠀⠀⠈⠓⠒⠒⠁⠀⠀⠀⠀⠈⠻⣴⣀⣀⣠⠎⠀⠀⠀⠈⠢⣁⡀⢀⣀⡴⠏
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠸⠃⢢⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣄⢴⣶⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀
⠀⠀⠀⠀⠈⡂⠈⠆⠀⠀⠀⠀⡠⠚⢉⠃⣭⠷⣄⠀⠀⠀⣀⡴⠹⠁⠐⢁⠤⡏⠀⠀⠀⠀⢠⣖⣟⣲⣋⡇
⠀⠀⠀⠀⢀⡎⠀⡁⠀⠀⠀⢸⠷⠓⢀⠗⠀⠐⢮⡆⠀⠀⠇⣁⠤⢲⠀⠀⡾⠁⠀⠀⠀⡤⢯⠒⢯⠤⣧⢼
⠀⠀⡠⠒⠁⢀⠔⠁⠀⠀⠀⠈⢄⢛⠨⠆⢠⡉⡴⠀⠀⠀⢇⠀⠀⠎⢒⠞⠋⠀⠀⠀⠀⢳⠞⡶⢻⣲⠏⠁
⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠒⠉⠀⠀⠀⠀⠀⠉⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣄⡀⠀⠀⠀⢀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣀⠀⠀⣰⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣏⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⠻⣿⣿⣿⣿⣟⣉⣽⣿⣿⠿⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢴⣿⣿⣿⠇⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡟⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠛⠿⠿⠿⠿⠿⠿⠛⠳⣀⣀⣤⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⢰⣿⣿⠃⡰⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⡎⢀⣤⣶⣤⡀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣾⣿⠇⢠⠁⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠻⢿⣿⠟⠁⢰⠁⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤
⠀⠀⠀⠀⣰⣿⠟⢀⡇⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⠿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣷⡀
⠀⠀⠠⠾⠛⠁⠀⢸⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠃⠀⢠⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⣴⣶⣶⡄⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣿⣿⠿⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⡟⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠻⣷
⠀⠀⠀⠀⣀⣼⣿⣿⠀⢣⡀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⣼⣿⣿⡇⠀⢸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⣠⣶⣿⡿⠿⠛⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⣠⣾⣿⣷⡃
⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠻⢿⣿⣿⣿⣿⡿⢧⡀⠀⠀⣴⣿⣿⠷⠚⠁⠀⠀⣿⣿⠇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣏
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿

⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⣀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⢿⣿⠆
⠀⠀⠀⠈⢣⡀⠀⠀⢀⣧⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⢱⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠙⠻⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠃⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⣠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⣤⣶⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦
⢸⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠂⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠹⠛⠃⠀⠀⠀⠀⣠⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢳⠂⠀⠀⠀⠀⠀⠀⠡⡀⢲⡀⣰⢀⡅⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⠢⣑⣘⣜⡐⠉⠴⠂
⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢒⣽⢿⢿⠒⣒⠒⠂
⠀⠀⠀⠀⠀⠢⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡰⢃⠂⡇⠑⡌⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣠⡄⠀⠀⣏⠰⢿⣇⠀⠀⠠⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⠉⢓⠋⠁⠐⠀⠀⠀⣿⣿⠏⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠂⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠀⠉⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠛⠀⠀⠀⢀⡍⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢸⡀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⣰⠀⠀⢸⡇⠀⠀⠈⢸
⠀⠀⠀⠀⢶⣦⡂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢧⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢪⠉⠀⠄⢸⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠇⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣀⣭⡆⠀⡄⠂⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠂⠈⣄⡀⠀⢸⡇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣶⠆⠀⠀⠀⠠⡝⢻⡇⠀⠀⠆⠣⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⢸⡇⠀⢀⠄⢠⣤
⠀⠀⠠⠀⠐⠠⠀⢀⡀⠀⠀⠀⠘⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠀⠀⣆⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠹⣄⣿⡇⢀⡎⠀⠀⠙
⠀⠀⠀⠂⠀⠀⠁⢸⠃⠀⠒⡠⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠻⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠐⠚⠐⠂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠂⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⡤⠊⠁⠠⠊⠁⠀⠀⢄⠀⠀⠀⠀⡀⠁⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠔⠋⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⣠⡀⠀⣾⡀⠀⠀⡆⠀⢀⠄⣿⡖⢡⣶⠇⡀⠁⡀⠀⠀⠀⣾⡏⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠹⠗⠀⠉⠁⠀⠀⡀⠀⠀⠀⢉⠙⢊⠛⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠴⠶⠟⠀⠠⠃⠀⠰⡀⠘⠿⠟⠠⠈⠠⢀⠂⠀⠀⠀⠙⠿⠓
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢀⠄⠠⠠⠄⠐⠐⠀⠀⠀⡀⠀⠐⠀⠀⠀⣀⠈⠀⠠
⠀⠀⠀⠐⠀⠀⠀⠘⣤⡄⠘⢻⣦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⢀⣦⡄⢀⣀
⠀⠀⠀⠰⠀⠀⠀⠀⣿⠿⠀⠈⠉⠀⠀⠀⢠⠀⠀⠀⠀⠐⣿⡆⠀⣿⠇⠀⠀⠀⠀⠀⠾⠿⠁⣼⡿
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⣶⠆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠛⠁⠀⠁⠀⠀⡄⢰⠀⠀⣀⣀⣀⠈⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠴⠾⠛⢀⠜⠀⠀⡄⠀⠈⠛⠉⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠁⠀⠀⠀⠈⠂⡀⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣄⠀⠀⠀⠳⠤⣀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⠀⠀⠀⢰⡌⣄⣀⣠⣊⡷⠀⠀⠀⣶⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⠀⠀⣾⣻⣿⣏⣷⠄⠀⢄⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣴⣇⣠⣠⣿⣟⣿⣿⣿⣤⣤⣼⣦⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣞⡍⠐⠁⠙⣇⣈⣙⣿⣿⣿⣉⢡⡴⠋⠀⠀⠻⢷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢠⣿⡏⠀⠀⠀⠀⢈⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣸⣿⣇⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢠⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡄⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣾⣿⣿⣿⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⡀⠈⢻⣧⣀
⠀⠀⠀⠀⢀⣠⡾⠟⠁⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⠙⡿⠉⠹⠏⠋⠀⠀⣀⡀⠀⠀⠈⢹⠟⡇⠀⠀⠈⠛⠻⠶⣤⡀
⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⢸⡇⢸⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⣾⣿⣿⣿⣿⣿⣦⠀⠀⠐⠀⣼⣿⣿⣿⣿⣿⣿⠀⢠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠘⡤⠀⢘⣿⣿⣿⣿⣿⣿⢠⠀⢨⠀⢿⣿⣿⣿⣿⡿⠋⠀⠌⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⢿⣇⠣⠀⠀⠙⠻⠿⠿⠟⠋⠸⠀⢸⠀⠀⠙⠻⠋⠙⠁⠀⢀⠃⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣮⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡄⡞⢁⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠈⢻⣿⣿⣿⣿⣦⡀⠀⠀⢸⢠⢸⡀⠀⢀⣾⣿⣿⣿⣿⡟⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠙⢿⣿⣿⣿⣿⣀⠀⢴⢸⠈⠄⠀⣾⣿⣿⣿⡿⠋⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠉⠻⢿⡻⠿⣷⣯⢸⡈⡇⣴⣬⣯⠝⠋⠀⠀⠰⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠉⠑⠒⠿⠚⠳⠛⠛⠉⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⢤⣀⣤⣴⣶⣤⣤⣀⡤⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢠⠖⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⡶⠒⢦⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣴⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠙⢧⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⡼⠁⠀⣠⣤⣄⠀⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠀⠀⣴⣶⣦⡀⠀⢻⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢰⠃⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⠂⠀⠀⠀⣿⣿⣿⣷⠀⠀⢷⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡿⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠸⡇⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠛⠻⠍
⠀⠀⠀⠀⠚⠩⠏⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣄⠀⢸⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⠀⢻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⢸⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⢰⠏⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣧⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠘⣧⠀⢰⣿⣿⣷⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣾⣦⠀⠀⠀⢰⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⢦⠈⢿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⢠⠏⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠈⢳⡀⠈⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠁⠀⠁⣠⠏⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⣀⣾⣄⠀⠀⠀⣀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠟⠛⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠁⠀⠀⠑⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡆⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⠀⠀⠀⠀⠐⣶⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠐⠒⠤⠄⠀⠒⢢⠽⠛⠉⠙⠁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠈⠉
⠀⠀⠀⠀⠃⠀⠀⠀⠑⢦⡀⠀⠀⠀⠎⠑⢶⣿⣤⣤⣤⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠚⠀⢶⣬⣇⠘⣡⠚⠛⠚⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡶⢾⡀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⠀⠀⣠⢻⡮⠿⢩⣄⡤⣤⣈⡿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢄⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⢀⢀⡼⠃⡀⢶⣶⡌⢀⠌⠉⠀⠈⠁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⡀⢲⣶⡀⠊⠱⠐⠄⠑
⠀⠀⠀⠀⠀⣀⣀⡀⠲⠶⠛⠻⢥⡀⠒⢤⠌⠣⠛⢀⣀⣀⣀⠈⠻⠇⠠⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠠⢀⣴⠷⢿⣡⠾⠀⠐⠰⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⣼⡼⢠⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⣤⣤⣤⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠎⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠒⠂⠀⠈⠁⠾⠀⠐⠀⠀⢀⣰⡆⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢷⡀⠀⠐⠠⢄⣠⡀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠶⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢀⠐⡖⢀⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⠀⠒⠢⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡐⠛⠚⠏⠩⣿⣆⡤⠘⣋⣈⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠁⠈⠁⠀⠀⢀⡀⢘⣷⣞⢣⣾⡯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠙⣯⡀⢀⣼⠓⠀⣐⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⢀⣠⡀⢀⣥⡿⡟⠧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢡⡤⢤⣗⢆⣜⠁⠀⢈⡉⣦⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⢀⠀⠀⡤⠄⢂⣈⢁⣨⡧⠄⠟⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠺⠿⠀⠀⠀⣏⠀⠱⣶⡘⠈⠀⢸⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡉⠲⡶⣀⡴⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠑⢂⠀⢀⠦⠤⠐⠁⠀⠀⠈⠁⠀⢐⠈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠐⢶⠃⣾⡿⣄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠁⠀⠋⠙⠢⢀⡀⠀⠀⠀⠤⠛⠂⠀⠄⠀⠀⠀⠐
⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣉⡻⠿⠿⠿⠿⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣤⣄⡀⠀⠀⠀⢀⣤⣤⡤⣤⡀⠀⠀⠀⢀⣠⣤⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡀⣀⣰⣿⡼⠧⠸⠧⣿⣦⣀⢀⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⠛⠁⣿⠂⢀⡀⠐⣿⠈⠛⠳⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⣤⣤⣴⣿⢤⣾⣷⡐⣿⣦⣄⣤⣼⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠁⠀⠉⠛⠻⠿⣿⣧⣹⣏⣼⣿⠿⠟⠛⠉⠀⠈⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣤⣤⣴⣮⣩⣯⣿⠏⢹⣿⣿⣯⣷⣦⣤⣤⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠸⡏⠉⢻⢿⠛⠋⠉⠀⠀⠈⠙⠛⠛⡟⠉⣸⠃⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⡀⢸⣶⣶⣶⣶⡆⢴⣶⣾⣿⣿⡇⢀⣿⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⡿⢸⣏⠁⠀⠐⡄⠀⢁⠈⢈⣿⡇⢾⡏⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣶⠇⢸⣿⣿⣿⣿⡃⢘⣿⣿⣿⣿⡇⠸⣷⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡾⠚⢀⣼⢋⠉⠉⠙⠋⠈⠋⠙⠋⣛⣧⣀⠛⢷⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣁⣈⡾⠿⣿⣾⣤⣤⣤⣦⣴⣤⣿⣿⡿⢿⣗⣌⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠋⠀⠐⡟⠉⠉⠙⢹⡟⠛⠙⠛⣻⠃⠀⠙⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⢧⡀⠀⢿⣶⣶⣤⣸⣇⣴⣶⣶⡿⠀⢀⡼⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⢷⡄⠘⣏⠙⠛⢻⡏⠋⠉⣻⠃⢀⡾⠁⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⣻⡀⠹⣿⣿⣿⣿⣾⣿⠏⢀⣿⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⣺⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣗⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⠿⠋⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠬⠋⠐⢂⡀⠀⣀⣠⣆⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⣰⠋⠁⠀⠀⣤⣤⣤⣤⣶⣿⣦⠀⠀⠀⠀⢀⡀⠀⠙⢻⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠄⠀⠀⣼⣇⠀⠀⠀⠙⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠊⣿⣶⡾⣿⣿⠿⣿⠽⠯⠅⢠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⣿⣗⣀⣈⣀⡄⠀⠁
⠀⠀⠀⠀⠀⢀⠒⠛⣻⣷⣶⣿⡏⠑⠛⠙⠙⠉⡉⡀⣴⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⠿⣿⣿⣿⡉⣽⣏⣁⣤⡀
⠀⠀⠀⠀⠀⠐⣤⣴⣿⣻⣿⣿⣷⡀⠀⣤⠶⠾⢻⣿⣿⣶⣤⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣸⣿⡙⠗⠿⣿⣿⣯⡗⠂⣀
⠀⠀⠀⠀⠀⠀⠘⢧⠚⢧⣾⠟⢻⣙⣶⣯⢻⣼⣿⣿⣿⣿⢷⣄⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠈⠹⡍⠀⣄⣉⣹⣿⣿⣿⣶⣿⡷⠄
⠀⠀⠀⠀⢨⣵⣾⣾⣶⣿⣿⣿⢙⣿⣿⣿⣿⣿⣿⣿⡋⢀⣤⣭⡟⠻⣿⣷⣄⣀⣀⣴⡄⠀⢁⣀⣥⣤⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠿⣿⣿⠿⠿⠟⠛⠛⠿⣿⣿⣿⣿⣾⠿⠿⠛⠛⠛⠛⠛⠒⠻⠿⠿⠿⠟⠻⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠⡄⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⣻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠙⠁⠀⠀⠀⠀⠋⠠⡥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡤⠄
⠀⠀⠻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠀⢀⣀⣀⠀⡀⠓⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⣀⠀⠈⠰⠦
⠀⠀⠰⡀⠀⢠⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠨⣿⣿⣿⡿⠿⠟⠻⢿⣿⣦⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠃
⣦⠀⠴⠄⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠀⢀⣄⣀⠄⠀⠙⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣿⠙⠒⣦⠀⡘⠀⠀⣬⠀⠈⠆⠀⠀⠠⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠉⠁⠀⠀⠀⠜⠢⠀⠰⠀⠀⠀⠐⠀⢠⠀⠀⠐⠀⠀⠤⢀⠆⠠
⢻⢀⠀⠈⠀⠃⠀⠀⣽⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣷⣺⣶⠃⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠇⠀⠀⠀⠄⠀⢀⠀⠀⠰⠤⠇⡉
⠀⢯⣠⡀⠈⠁⠀⠀⣽⠀⠈⣠⡀⠀⠀⠀⠀⠀⠀⠘⣿⡽⢀⡠⡶⢀⠀⠊⠍⠀⠀⠀⣔⠀⢠⣰⣿⡀⠀⠀⠀⠠⣄⠠⠇⣀⣄⠘⠃
⠃⠉⢀⠙⠄⠰⡆⠀⢺⣁⣼⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⣇⠬⠴⠃⠸⢂⠀⠀⢀⡞⢡⠘⠚⠀⣏⡣⠀⠀⠀⠁⠀⠈⠶⣌⠉⠀⠀⢴
⠀⣰⠏⢳⡀⠀⠆⠀⣽⠛⣿⣧⠀⡀⠀⠀⠀⠀⠀⡘⢀⠃⠒⠚⢃⣸⠏⣰⣤⣻⠂⣻⠀⢦⢠⣿⠃⠀⣄⠀⠀⠀⠀⠀⠈⠚⢁⠊
⠀⠁⠀⠀⣷⢀⣶⠄⣿⠀⡟⢹⣏⠁⠀⠀⢀⠄⣾⠇⣼⠆⠀⠀⢎⢈⠓⠈⠠⣨⡁⠚⣥⠜⠮⠝⠂⠀⠈⢢⠄⡀⠀⣀⠀⣀⠈⠐⠆
⡴⡄⠀⠓⠮⠋⠦⠀⣼⠰⠁⠙⣿⠳⡄⢀⠌⢠⡟⠀⠀⠓⢦⡇⡠⠈⡀⠀⠀⠀⠁⣞⣾⣷⣄⣠⠀⠐⢀⡰⢠⢴⠐⠣⠠⠘
⠀⢸⣦⣆⣠⡠⡾⠿⣿⠦⠃⢨⣟⠀⢶⠀⢰⡾⡶⠄⡄⠤⣭⢜⣓⢺⡏⠉⢀⡠⠀⠘⣺⣏⡀⡦⠀⢀⣉⡀⣼⡜⡙⠃⠀⠁⠀⠶⡀
⠆⠈⣥⣄⠐⢛⣿⣀⣿⡀⠐⠢⡌⠀⡈⠰⢦⣞⠁⢠⠓⠂⠀⠀⠀⢾⣁⣠⡜⠱⢾⡠⠝⠛⠛⣰⠆⡙⠛⠁⠛⠁⠒⣤⡳⢄⣤⣤⠙
⠻⢎⣿⣿⣡⣬⣭⡙⣿⢛⠀⠀⠸⠀⣷⣶⣾⣿⣷⣷⣶⠌⢀⠀⢴⡟⡱⠾⢆⣟⣻⣀⣗⠘⠋⠝⢨⡅⠀⠀⢀⣄⣀⠬⠁⣹⠘⠓⠶
⡖⠺⣟⡽⠿⢋⣙⣫⠁⡀⠀⢤⡄⢨⣯⣾⠽⠿⢿⡿⠀⠒⠤⠀⠘⣝⠿⢛⠈⢻⠿⢿⠏⠉⢰⠓⠃⢩⠑⢂⣼⠾⢻⣤⣺⣭⠶⣄
⠉⠴⣿⣂⢶⡄⢀⣤⢞⣿⣿⠶⣲⢹⣏⡈⢥⣣⠛⣟⣀⣤⡗⢠⣦⣯⡟⣁⢀⠘⠸⡏⠃⢀⠘⡁⠻⢻⠄⠛⢣⢛⣜⣷⢷⢿⡙⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡶⢾⣿⣷⡷⢶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣀⡤⢾⣿⣷⣾⣿⣿⣷⣾⣿⡷⢤⣀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣠⣶⣿⠋⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣶⣄⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣧⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⢶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠉⠉⠉⠙⢛⣿⣿⣿⣿⡛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⣿⠁⠀⣴⣿⣿⡆⠀⠀⠀⠀⠈⠛⠻⢿⡿⠟⠛⠁⠀⠀⠀⠀⢰⣿⣿⣦⠀⠈⣿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢰⡟⠀⠰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠆⠀⢻⡆⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡇⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⢸⡇⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠏⠀⠀⢸⡇⠀⠀⠹⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⢿⡄⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⣶⣶⠀⠀⢠⡿⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠻⣦⠀⠻⢿⣿⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⠟⠀⣴⠟⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⣠⡾⠋⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⣾⣷⡶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠠⣄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⢻⡿⠂
⠀⠀⠀⠈⢣⡀⠀⣀⣵⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⠋⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠈⢿⡿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣾⣿⣦⡄⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁
⠀⠀⢀⣀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀
⢠⣾⡿⠿⠿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠈⠉⠀⠀⠀⠀⠀⢀⣠⣶⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣴⣾⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣲⡦⠀⠀⠀⠀⠀⢀⣀⣠⠈⣐⠖⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣋⣿⣥⡿⠯⠀⠐⣀⣰⣦⣤⣈⣿⡍⠈⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⢠⣼⣿⣶⣶⣶⡆⠀⠀⠾⠿⠿⠋⠋⣰⣤⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⣀⡄⣠⣀⡉⠛⠋⡟⠉⠉⠀⠀⣺⣿⣿⡻⣟⣷⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⣮⣿⣿⣿⣿⣾⣿⣷⡀⠀⠀⠀⠀⣲⣿⣿⢿⣷⣿⣿⣿⣿⡁
⠀⠀⠀⠐⣰⣿⣛⣷⣽⣿⡆⠀⠃⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⠷⠀⠀⠀⠀⠀⢀⣾⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⠂⠀⠀⣩⣿⣯⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢀⢿⣿⣶⣿⣷⣿⣿⣶⡄⠀⠀⠰⢹⣻⣿⡏⢈⣿⣯⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⣿⡿⠷⣟⡽⠛⣯⣿⡏⠋⠀⠀⢸⡣⢿⣿⣿⣿⠿⣿⣻⣿⣷⡆
⠀⠀⠀⡾⣿⣯⣝⣏⣯⣿⡿⠀⡇⠀⠀⠉⠛⣿⡆⡆⢀⣽⣷⣾⠿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡹⣀⠀⢸⠀⠀⠿⣿⡇⠀⠀⠀⢸⠈⠀⠙⢹⡇⠀⠈⠉⠋⠉
⠀⠀⠀⠙⢿⠟⢸⡁⠀⠉⡿⢿⣿⣶⠀⠀⠀⠉⠁⡇⢸⠙⣿⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⠻⢹⣿⡇⢸⠀⠀⠀⠀⠀⠀⣀⣤⣼⣄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅⢩⣉⣿⠀⠀⠀⠀⠀⡛⠈⠓⠛⠓⢠⣶⣿⣿⣿⣿⣷⣶⡀⠛⠹⠿⠦⢾⠤⠤⠤⠤⠤⠠⠨⠹⣿⣿⡇⠀⢨
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠓⠛⠿⠷⣾⣿⣿⢷⣶⢿⣤⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡤⣿⣶⣿⣷⣿⣦⠀⢰⡴⣿⣿⣀⣀⣸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⢀⡀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⣃⣶⣶⣽⣲⣶⣿⣯⡿⣟⡡⠊⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⠁⡰⠄⣿⣿⣿⣿⣿⣦⣽⣿⣿⣍⣉⣿⣾⡯⣥⣴⣶⡶⣶⣶⣶⣿⣿⣷⣾⣶⣄
⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠳⣀⠉⢻⣩⠽⣉⡿⠬⠏⣀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷
⠀⠀⠀⠀⣿⣟⣿⣿⣿⠟⠛⠛⠛⢛⣛⣩⣭⣵⣾⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣶⣒⣻⣥⣾⣿⣿⣿⣿⣿⣷⣿⣿⢥⣟⣛⣻⣿⢿⣏⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠸⠯⠿⠿⠿⠤⢤⣴⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⡟⣻⢿⣛⣿⣹⡽⣯⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⣿⣶⣦⣌⠉⠙⢻⡇⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡛⢚⣿⣿⣿⣿⣷⣿⣯⣿⣿⢿⣿⣿⣿⣿⣟⣿⡄⢹⣿⣿⣿⣟⣿⣿⣿⢿⡆⠀⠀⠁⠙⠛⠛⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⡆⢹⣿⣿⣿⠷⠮⣿⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⢿⣿⣯⣽⣿⣿⡽⠟⠋⠈⠓⠛⠛⠛⠛⠛⠒⠒⠒⠚⠛⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⣀⣤⣴⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠋⡀
⠀⠀⠀⠀⣼⣯⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣿⣿⡇⠀⠀⠀⠀⠀⡼⠉⠉⠉⠉⠈⢳
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣥⣾⣿⡿⠁⠀⠀⠀⠀⠀⢳⠀⠐⠂⠂⠀⡸
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠞⠁


⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡇⠀⢀⡀⠀⠀⠀⠀⠀⢰⣦⣀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣤⠂⠁⠀⠀⠀⠀⠀⣴⣿⣷⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⠛⣆⣽⡆
⠀⠀⠀⢼⣿⣿⡟⡻⠿⠿⠇⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⡿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⢋⣁⠐⢢⠀⠀⠀⠀⠀⢐⠻⣤⣺⣾⣿⡿
⠀⠀⠀⠀⠙⠻⠿⢷⡟⣶⠲⣯⠆⠀⠀⠀⠘⢿⣿⣿⣶⠖⡴⣦⣠⠆⠀⠀⠀⠀⠙⠻⠟⢦⣤⣭⣴⡿⠀⠀⠀⠀⠀⠈⠛⠿⢿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠈⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢻⣿⣷⠏⣀⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢼⣿⣿⡿⣟⣿⣿⣀⣤⡀⠀⠀⠀⠀⠀⣾⣿⠛⢿⣿⡄⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣠⣠⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣿⣿⠛⠁⠀⠀⠀⠀⠀⣿⣧⣀⣰⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣙⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⢾⣿⣿⣷⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⣾⢟⣽⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢧
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣿⣇⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣼⣾⣦
⠀⠀⠀⠀⠀⠀⠻⣟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⠽⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠙⠳⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣤⣤⣤⣤⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⠀⠀⠈⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡶⠟⠛⠉⢉⣹⣏⡉⠉⠛⠻⢶⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⢰⠏⠁⠈⠹⡆⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⣰⠇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⠟⠀⠀⡴⠚⠛⠳⣦⠀⠀⠀⣹⣿⠁⠀⠀⣠⠞⠛⠛⢦⡀⠀⠻⣶⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢸⡇⠀⠀⠀⢸⠆⠀⠀⣿⣿⠀⠀⠀⡇⠀⠀⠀⢈⡇⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠳⢤⣤⡴⠋⠀⠀⠀⡿⢸⠀⠀⠀⠙⠦⣤⡤⠞⠁⠀⠀⠀⢿⡄
⠀⠀⢀⣠⣤⣤⣤⣼⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡇⢸⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⣧⣴⣶⣦⣤⡀
⣠⣾⠿⠛⠉⠉⠉⣿⠀⠀⢀⡞⠋⠉⠉⠙⣦⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢳⡄⠀⠀⣿⠁⠀⠉⠙⠿⣷
⠙⠁⠀⠀⠀⠀⠀⣿⠀⠀⢸⡁⠀⠀⠀⠀⣸⠀⠀⠀⢸⠃⠀⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⡷⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠳⣄⣀⣀⡴⠋⠀⠀⠀⣾⠀⠀⢿⠀⠀⠀⠙⢦⣀⣀⣠⡼⠃⠀⢸⡏
⠀⠀⠀⠀⠀⣀⣤⣼⣷⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡟⠀⠀⢸⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣾⣧⣄⣀
⠀⠀⠀⣠⣾⠟⠋⠉⠹⣧⠀⠀⢀⡶⠛⠛⢲⡄⠀⢰⡇⠀⠀⠈⡇⠀⢠⡶⠛⠓⢦⡀⠀⠀⣼⠟⠉⠛⠻⣷⣄
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠹⣧⠀⢸⡄⠀⠀⢀⡇⠀⣸⠁⠀⠀⠀⣿⠀⢺⡀⠀⠀⢸⡇⠀⣼⠏⠀⠀⠀⠀⠈⢿⣧
⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⠒⠒⠋⠀⠀⡟⠀⠀⠀⠀⢸⡆⠈⠛⠶⠖⠋⣠⡾⠃⠀⠀⠀⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⢸⠇⠀⠀⠀⠀⠈⣇⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣿⣄⣀⣀⣀⣀⣠⣽⡶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠲⣄⠀⠀⠀⠀⣠⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠈⢧⠀⢀⡞⠁⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠘⡆⣼⠁⠀⠀⠀⠰⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠞⠛⢻⠟⠿⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠉⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣈⠙⠢⣀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⠶⣄⡀⠀⣠⠊⢰⠉⢙⡆⠀⠀⠀⠀⢸⠀⠀⡴⠉⠈⠙⣆⠈⠣⡀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠁⠀⠀⠙⠉⠀⠀⠀⠀⠀⢸⠀⠀⠳⣀⠀⣠⠏⠀⠀⠱⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⠊⠙⣦⠀⢸⠀⠀⠀⠈⠉⠀⠀⣠⠤⢄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠒⠢⡄⠘⠦⠤⠋⠀⣼⠀⠀⢀⡤⢤⡀⠀⣇⠀⣨⢿⠒⠶⠶⠦⢤
⠀⠀⠀⠀⠀⠶⠖⠛⠻⡄⠸⡁⠀⢀⡽⠀⠀⠀⠀⠀⣿⠀⠀⠘⢤⡤⠃⠀⠈⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠉⠉⠀⢀⣀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠀⢠⡏⠀⢹⠄⠀⣿⠀⠀⠀⡎⠀⠙⡆⠀⡰⠃⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠈⠢⣀⠀⠑⠒⠊⠀⠀⡇⠀⠀⠀⠙⠒⠚⣠⠞⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⢸⡇⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠀⢀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣇⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠀⠙⠻⣿⣿⣿⣿⣿⣁⣰⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣿⣿⣿⡟⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠟⠁⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⢧⠀⠀⣀⣠⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⢀⠇⠀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⡜⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠀⡜⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠛⠛⠛⢦⣄⡀
⠀⠀⠀⠀⠀⢠⣿⣿⠁⠰⠁⠀⣼⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⢰⠁⠸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣿⡿⠃⢀⠃⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣷⡀
⠀⠀⠀⠞⠛⠉⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⠀⠀⠀⠠⣿⣿⣿⣿⣿⡆⠀⠀⠀⣼⣿⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⢃⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⠀⠀⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠙
⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⢧⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣤⣤⣄⡜
⢀⣴⡿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠳⣀⣀⣀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠴⠊⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⣿⣿⣿⣿⡿⠿⡀⠀⠀⢠⣾⣿⡿⠿⠊⠁⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠙⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣛⣛⣽⣷⠿⣙⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣬⣴⣶⣿⠟⣻⣻⡻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣾⡿⢿⣿⣿⣾⣿⡇⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢿⣿⣿⣿⣿⣿⣷⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣤⢤⡀⠙⢧⣼⣧⣴⣒⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣟⣁⠀⠉⠛⠛⠛⠋⠉⢀⣿⣷⠓⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠠⡀⠀⣀⣄⣠⣾⡃⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡟⠛⠿⠙⠇⣿⣿⡿⠻⡟⣷⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⢸⠀⡇⣴⣾⠀⠀⠸⢣⢷⣿⣿⡛⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠲⢾⠰⠇⠋⠈⠀⠀⠀⠀⠈⠘⢻⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣽⣿⣿⣿⣿⣿⣿⡽⣷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣽⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⣿⣿⣿⣿⣿⣿⣧⣛⢀⠀⠁⠀⠐⢄⠀⣠⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣮⣻⣿⣿⣿⣿⣿⣿⣾⣆⣤⣾⣷⣮⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣷⣿⣿⣟⢉⣉⣉⣉⣉⣭⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠟⠿⠛⠿⠟⠛⠿⠿⠓⠂





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠒⠐⠒⠚⢐⣶⣬⣭⣧⠠⠔⠈⠉⠉⠉⠀⠀⠀⢠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠁⠀⠀⡀⠀⠀⠀⢀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢰⣾⡅⠀⡦⡄⡦⣤⣴⠀⠀⢡⣶⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏⣛⠁⢰⣷⣷⣷⡿⣿⣆⠀⢸⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠿⠇⣠⣿⣤⠼⠛⠂⢈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⠀⢻⣷⡀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡿⢨⣬⠉⢴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠄⠉⢪⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⣯⠇⠀⠻⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠈⡴⠋⠀⠀⢰⣮⡆⣠⡄⠀⣴⣦⠀⠀⠀⣴⣦⠀⢠⣤⢠⣏⣦⠀⠀⠐⡌⢣⡀
⠀⠀⠀⠀⠀⠀⠶⠦⠁⠀⠀⠀⠈⠏⠄⠻⠃⠀⠹⣿⠃⠀⠀⣿⡟⠀⠀⠋⠈⡧⣿⠀⠀⠀⠈⠎⠣⡄
⠀⠀⠀⠀⠀⣀⠓⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣀⣌⡇⢄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⢰⣷⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠀⠇⣿⣾⠇⠀⢠⣤⢸⠀⣴⡀⢰⠛⢃⣹⠛⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠹⣿⢦⣤⣸⡛⣼⣀⣼⣱⣼⣬⣭⠝⠁⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠟⢿⣷⣿⢻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠻⠝⠉⠈⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣀⣀⣀⣤⣶⣶⣶⣶⣤⣀⣀⣠⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣿⡿⠋⠉⠀⠀⠉⠙⢿⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣤⣤⣄⣀⣤⣤⣶⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣠⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣄⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⠋⠀⠈⠙⠛⠿⠿⠿⣿⣿⣾⣿⠿⠿⠿⠛⠋⠁⠀⠙⢿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣀⣄⣀⠀⠀⣿⣿⠀⠀⣀⣠⣀⡀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⣰⣿⠟⠛⢿⣧⠀⣿⣿⠀⣼⡿⠛⠻⣿⣆⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢻⣿⣄⣀⣾⡿⠀⣿⣿⠀⢿⣧⣀⣠⣿⡏⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠙⠛⠛⠋⠀⠀⣿⣿⠀⠀⠙⠛⠛⠋⠀⢀⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠀⢠⣾⡿⠿⣷⣆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣾⠿⢿⣷⡄⠀⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⢹⣿⠀⢿⣿⡀⠀⣼⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣇⠀⢀⣽⡿⠀⣿⡏⠉⠻⣿⣆
⠀⠀⢠⣶⣦⣤⣄⣼⣿⠃⠀⠀⠸⣿⡆⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⢸⣿⡇⠀⠀⠙⣿⣷⣤⣤⣶⣶⡆
⠀⠀⠀⠈⠉⠉⠛⠛⠃⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⣿⣿⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⢀⣹⣏⠁⠀⠀⠀⠈⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣧⠀⠀⠀⣰⣿⠛⠛⢿⣧⠀⣿⣿⠀⣼⡿⠛⠻⣿⣆⠀⠀⠀⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢿⣧⡀⠀⢻⣿⣄⣠⣾⡿⠀⣿⣿⠀⢿⣧⣄⣠⣿⡟⠀⢀⣾⡿⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠻⣿⣦⡀⠉⠛⠛⠋⠀⠀⣿⣿⠀⠀⠙⠛⠛⠉⢀⣴⣿⠟⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⠀⠀⠀⣿⣿⠀⠀⠀⣀⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣿⣿⣶⣾⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡀
⠀⠀⠀⠀⠀⢾⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⢠⠟⢡⡀⠀⠀⡀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢀⠀⠀⠀⢀⣾⡄⠀⣰⠇⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣜⠀⠀⢀⠹⣿⣁⠀⣿⠀⢀⣼⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⢿⡄⠱⡄⠀⣿⣿⢠⡟⢀⣾⠇⣸⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠛⣿⣦⠀⠀⢿⣿⡇⢠⠟⢠⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣆⠻⡇⠀⠀⢰⣿⡰⠇⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣄⡲⣦⣈⣑⠉⢿⣄⠂⠀⣸⣿⡇⢀⣴⡿⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢳⣌⠛⣷⣤⡟⠀⠠⠋⣸⣷⣾⠟⣣⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⣄⣈⣻⠿⣿⣮⡛⠀⠀⢠⣿⡟⠁⣾⡏⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣿⣆⠸⠉⢿⣆⠀⣹⠏⢀⣴⣃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣽⣿⣇⠀⠘⠛⣿⣁⣰⡝⠛⠉⣡⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⠀⠀⠛⣿⡁⣠⠾⣿⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡛⣿⣷⣀⠰⠛⣿⣤⢎⣹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣻⣧⣉⣫⣿⣦⣾⣿⡛⠾⠛⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣿⣯⣭⢉⣿⠉⣿⣷⣄⣹⣷⣶⢢⣽⣏⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣴⠋⢻⡄⣼⡟⠙⢷⣿⠁⢈⡿⠟⢿⣀⢈⡿⠻⡝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⣿⠃⠁⣸⠷⣿⠀⠁⢰⡿⠤⣾⡇⠘⠀⣿⣬⣇⠈⢘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠈⣧⣼⡇⠠⢸⣇⣠⡟⠀⠀⠈⣷⣀⣶⠁⠀⢻⣄⣸⠃⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠹⡇⣶⢃⠘⣷⠀⣼⠃⡉⢳⡄⠀⢠⡟⠉⠹⣧⠀⣸⠏⢹⡆⢰⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡍⣠⠿⢿⠀⢰⡿⠾⣿⠀⠁⢺⠷⠶⣿⠀⠘⠐⣿⠴⣷⠀⠀⣿⣿⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣷⡏⠂⣼⣤⣿⠀⠂⢹⣦⣤⡿⠀⠂⢸⣧⣤⣾⡃⠠⢸⣇⣸⡇⢸⣇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⣢⡟⡀⢹⡄⢠⣞⠁⠈⣷⣀⣀⣼⠃⠀⠙⣵⠂⡼⠋⠹⢇⢰⡉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢀⡿⠙⣇⠀⣼⠛⠛⣯⠀⠀⣾⠋⠉⠿⠀⠀⢀⡿⠲⣷⠀⢀⡾⢷⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣼⡇⠃⣽⣴⣏⠐⠂⢸⣦⣼⣏⠈⠁⢸⣧⣤⣾⠀⠀⢸⣇⣸⠇⢸⣇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣻⣱⡏⡄⢹⣂⣠⡏⢠⠀⢿⣅⣠⣾⠃⡀⢹⡆⠀⡜⢉⢹⡆⡸⢋⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⠙⣇⠀⣼⠋⠙⣧⠀⠀⣾⠋⠉⢿⣀⠀⢰⠟⠺⣧⠀⢸⡿⣷⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣏⠁⣿⡤⣿⠀⠀⢸⣧⣼⡇⠀⠀⢸⣧⣤⡿⠀⠀⣿⣤⣿⠐⣽⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢸⣂⡛⠄⢸⣆⣐⡟⠀⠀⢹⡄⣰⡎⠡⠈⢷⡀⣰⠏⠀⢿⡀⢏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠉⣿⠄⣨⠋⡉⢿⡀⠀⣼⠋⠙⣧⠀⢠⡟⠉⢻⡄⢀⡞⠻⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣬⠗⠻⣆⠁⣸⠗⠚⢧⠀⢐⡿⠒⢿⡆⠀⣼⠷⣾⢈⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⡻⠖⢿⡀⠁⣸⠷⣿⡀⠀⣸⡷⢴⡏⠀⣨⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣷⣀⢈⣿⠴⣯⠀⣨⣷⣼⣇⠀⣠⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⣿⡿⢿⣿⣿⠿⠿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⠀⠀⣴⣿⣷⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠻⠿⠟⠁⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⢠⣶⣶⡄⠀⠀⠀⣠⣶⣿⣿⣷⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⣴⣶⣶⡀⠀⠉⠛⠿⡿⠃⠀⠘⠿⠿⠃⠀⠀⣼⣿⠟⠉⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠤⡈⡀⠀⠀⢸⣿⣿⣿⡗⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠈⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⣴⣲⠀⠀⠿⠟⠈⣀
⠀⠀⠀⠀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⢸⣿⣿⣿⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠶⠆⠀⠌⠀⠈
⠀⠀⠂⠁⠀⠀⠀⠀⠠⠀⠀⢀⣶⣶⣦⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠁⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠈⠀⠁
⠠⠠⠤⠄⠀⠀⠠⠀⡀⢀⠀⠘⢿⣿⡿⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠈⠙⠳⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠐⠀⠀⠀⠐⠀⠄⠀⠀⠄⠀⠰⡀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦⡀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢄⠂⠀⠄⠐⠂⠀⠀⠒⠐⠂⠉⠈⠑⠀⠀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠧⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⣼⠗⠁⠰⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠟⠁⠀⠀⢀⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣶⡾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠘⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⢸⣶⣾⡶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠈⠁⠂⠀⠀⢀⠀⠀⠀⠀⠄⠠⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣞⡛⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⠙⣻⢿⣝⣛⣯⡛⠩⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣷⠶⣾⣱⣛⣮⣉⣎⣿⣻⢿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠠⣾⡇⢉⢿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣾⢼⡃⠀⠀⣴⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡾⣣⣄⠀⣸⣻⣿⡔⠀⢻⣿⣿⣿⣿⣿⣿⢡⣿⠃⣼⣿⣤⡾⢯⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣫⢞⣾⣿⡏⣿⣯⠀⠈⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⡿⡇⠸⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣛⣻⣾⣿⠇⠀⢰⣿⣿⣿⣿⣿⣷⠸⣻⡄⢩⡵⣌⠢⢍⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡉⠛⠛⠛⠛⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⠹⣹⣆⡘⠣⠿⢸⡟⣧⢀⣀⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣳⡶⣦⣼⣿⣿⣶⣦⣤⣤⣴⢞⡽⠋⠁⣿⢻⢻⡇⠈⢙⢷⣈⡚⠽⣖⣶⣤⢤⣬⡝⣷⣻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢯⣿⣿⣿⣿⣿⣿⣿⣷⡋⢀⣴⣾⡟⢸⠘⣿⣷⣄⠠⢻⣿⣷⣮⣭⣥⣥⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠖⣭⣛⢿⣿⣿⢫⠀⣾⣿⣿⠃⢸⠀⢻⣿⣿⡆⢢⣿⣿⡿⢛⣻⣭⡿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠲⣄⢻⣧⠹⣿⣿⠘⣿⣿⠏⠀⣸⡄⠘⣿⣿⠇⡘⣿⢫⣾⠿⠋⢁⣠⣷⢢⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⣿⡙⣷⢸⡇⣚⡇⣿⣿⡆⢹⡟⠀⠀⣺⡇⠀⠸⣷⣀⣿⠇⣿⡏⠀⣴⣻⡷⠧⢿⣽⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠿⠚⢡⡟⣰⡇⣺⠇⣿⣿⣿⡿⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⡇⣿⡅⠀⣿⣿⡜⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢋⡴⢏⣼⣿⣿⡿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠙⣿⣷⠘⣷⡀⠹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣴⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠈⢿⣷⣌⠻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠠⢸⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⡀⠀⠀⠀⣾⣿⣿⣧⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⣸⠟⠛⠛⠻⣎⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⢰⠏⠀⠀⠀⠀⠹⡇⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⣠⡷⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣀⠤⢚⠀⠀⢉⣯⣮⣻⣿⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣤⣯⡀⠀⠀⣀⢢⠎⢈⢿⡥⣻⣏⡟
⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡛⢛⡟⢈⡀⠀⢘⣶⣽⣿⠇
⠀⠀⠀⠀⠀⠀⣠⠇⠀⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⣒⣻⣤⠟⠁⠠⢖⣽⣟⡟
⠀⠀⠀⠀⠀⠀⣼⠀⠀⡓⢀⡂⠀⠀⠄⢰⣿⣿⣯⣿⣿⣛⢿⠋⠀⡙⣿⢿⣿⠿⠟⠋⠀⠀⠀⢀⡃⣉⢎⡠
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠓⠂⠀⠈⣐⢢⣟⢿⣿⡿⣿⣭⣤⡀⠀⠀⣘⠀⡽⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠟
⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠁⠐⠒⠤⢸⣿⣿⣿⣿⣿⠻⠋⠀⠀⢼⣿⢣⣾⡏⢳⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣯⣦⣦⣶⣲⢄⣤⣿⠛⣿⡄⢻⡀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⢀⣾⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢻⡇⢀⡩⠟⣿⣧⠀⠄⠀⠰⠊
⠀⠀⠀⠀⠀⠀⢈⡅⠀⢀⣼⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠒⣾⡏⠀⠀⠀⠀⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢛⠙⠡⠂⠀⡧⢸⠏⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢾⠗⣻⡆⠖⠚⠉⣿⠀⠀⠀⣀⡐⠾
⠀⠀⠀⠀⠀⠀⣸⣷⠀⣿⣿⣿⣿⣿⢿⣞⣋⠝⠁⠈⠀⠀⠀⠀⠀⢀⡤⠚⢁
⠀⠀⠀⠀⠀⣰⣿⡏⢀⣿⣿⣿⣿⣿⣯⣍⣥⡤⠤⡴⠐⠂⠀⠔⠃
⠀⠀⠀⠀⢠⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣀⣅⡤⠤⠦⠴⠃
⠀⠀⠀⠀⣾⣿⣟⠀⣿⣿⣿⣿⡿⠓⠫⠅⡀
⠀⠀⠀⢸⣿⢿⠍⢰⣿⣿⡿⠋⢁⡐⠲⠤
⠀⠀⠀⣿⣿⣏⣠⠟⣩⠊⡀⠂⠀⠈⠑
⠀⠀⢐⢿⡿⠊⠁⡐⠁
⠀⠀⢈⡛⠁⠀⠐
⠀⠀⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⢀⣀⡤⢤⣀⡀⠀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠀⠀⠀⠀⠙⣏⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⢀⣸⣀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣠⠞⠁⠀⠉⠉⠙⠋⠉⠉⠀⠈⠳⣄⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠉⠙⠲⠤⠤⣤⣀⣀⣤⠤⠴⠒⠋⠉⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⣀⡀⠀⢸⡇⠀⢀⣀⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠋⠀⢹⠄⢸⡇⠠⡏⠀⠙⡆⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠓⠖⠋⠀⢸⡇⠀⠙⠲⠚⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢼⠀⢀⡔⠒⢦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡴⠒⢢⡀⠀⡧⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⢸⡀⠘⢆⣀⡼⠁⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⡰⠃⢀⡇⠀⠙⢦
⠀⠀⠈⠙⠓⠚⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⢠⠋⠉⢳⡀⢸⡇⢀⡞⠉⠹⡄⠀⠀⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠘⢦⡀⠈⠧⠤⠞⠀⢸⡇⠀⠳⠤⠜⠁⢀⡴⠃⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠾⠧⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣾⣿⣿⣿⡗⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⢿⣿⣿⣿⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡄⠀⠀⠀⠀⢀⣾⣷⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣤⠀⠀⠀⢾⣿⢿⣄⠀⠙⠇⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠾⠟⢀⠀⠀⠈⠁⠈⢙⢄⠀⠀⠁⠀⠀⠄⠱⢤⡀⠀⣤⡀⠀⢸⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⣀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆⠈⢀⣾⣿⣿⠀⢸⠀⠀⢀⡟
⠀⣀⡀⣶⣾⣿⠷⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠘⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣨⡿⢿⣿⣿⠃⢸⠀⠀⠀⠁
⠀⠀⠂⣿⣿⣿⡟⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⢱⡹⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡇⠐⠈⠙⡏⠀⢸
⠀⠀⠀⠹⡟⠑⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⣷⠀⠰⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⠀⠀⡇⠀⣀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⢸⡆⠀⠀⣙⣷⠀⠀⠀⠀⠀⠰⠀⠀⢸⡇⢆⠀⡃⠀⢾⣀⠀⠀⠀⠀⢀⠾⠁⣹⢿⡆
⠀⢸⡄⠀⣇⣰⣿⡆⠀⠀⠈⢉⡛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣧⣾⣿⠁⣷⠀⢀⠿⠀⣴⠀⢾⠀⠀⠀⠀⠀⠈⣧⠘⡆⢿⠀⢼⠈⠳⡄⠀⠀⠀⠀⠀⡉
⠀⠈⣧⣸⡿⣿⣿⣇⣠⣴⣤⡾⠃⠀⠀⠀⠐⠴⣾⣿⢿⠟⣿⡛⠉⣧⠸⣆⡡⠐⢠⡇⠀⠈⠀⢀⣶⡚⠀⠀⣿⠀⢹⣼⡀⣺⠐⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⢿⡇⢤⣿⡿⣿⣭⢿⡇⠀⠀⠀⡆⠀⠀⠈⢳⣄⠀⠘⡇⠀⠸⡆⣿⠅⠀⣼⠃⠀⢐⠀⠈⠀⠁⠀⠀⢹⡇⠈⣿⡇⣿⠀⡇⢀⠻⡆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢸⠀⣇⢸⣯⣤⣴⣾⣿⣧⣤⣤⢦⣇⢀⣠⠀⠘⣿⣦⡀⣿⠀⠀⢿⢸⡄⠀⡟⠀⠀⢸⠀⠀⢠⠆⠀⠐⣼⡷⣤⣼⣿⣿⠀⣿⡎⠀⢻⡀⠀⠀⣠⣾⠇
⠀⠀⣾⡄⣿⡾⢻⣿⡛⢿⣿⣿⣦⡁⣾⣿⣿⣿⣿⡀⢿⣿⣷⣿⠀⠀⠸⡿⣇⢸⡇⠀⠀⡼⠀⠀⡼⠀⣠⠀⢸⣷⡀⠈⣿⣿⠀⢻⠋⡄⠈⣧⡀⠀⢻⣿
⠀⠀⡟⡇⡏⡇⠈⡏⣇⠘⣍⠙⢿⣛⠹⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⣷⣿⢸⠇⠀⢠⢑⠰⣾⣧⡾⠋⠀⣼⡏⢳⢰⣿⣿⢀⣸⡀⢣⠀⢻⠀⠀⣼⡏⠄
⠀⢰⡧⣇⣷⣿⣠⣷⢸⡆⠘⣧⠸⡇⠀⣿⠿⣿⣿⠇⢸⣿⡿⢿⡇⢰⠀⣿⣼⣾⠀⣀⣼⣸⣶⣿⣉⢷⡀⡼⢸⡇⢨⣿⠀⣿⢻⣿⣆⡸⡆⢸⡇⣷⣿⠀⠸
⠀⣸⠀⣯⡿⣿⡉⣿⡀⣷⣿⣿⣧⣿⣴⣿⣷⡏⣿⠀⠸⣯⡇⠘⡇⢸⠀⢸⣿⣿⣿⣿⡟⣿⣿⠛⣻⠛⢰⠃⠘⡇⢀⣿⡄⣿⢸⡇⡇⠙⣧⡈⣿⡇⣿⠀⠀⡇
⠀⣿⠀⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⡀⠀⣿⡇⠀⡇⢸⠀⢸⣿⣿⣿⠈⣷⢸⡏⣰⠃⠀⡿⠀⢀⣧⠦⣿⣇⣿⣾⡇⡇⠀⢹⢻⣿⡗⡏⡁⠀⢹
⠀⣿⠀⢼⣧⣾⡟⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⢸⡇⢰⡿⡇⠀⡇⢸⠂⢸⡿⣟⣷⣤⣇⣸⢧⡏⢀⢰⣇⡴⢛⡟⠀⣿⣿⣿⣿⣧⡇⠀⢸⣈⣿⣧⡏⣿⠀⢸⡀
⠀⣿⠀⣾⣿⠨⡇⢸⡿⣿⣿⣿⣿⣿⣿⣿⡇⢣⣸⡇⡼⣷⣷⠀⣧⣼⡇⢸⠆⣿⣿⣿⢻⣿⣾⣷⣾⣾⠟⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⠴⢻⡅⣿⣿⡇⢹⠀⠸⡇
⠀⣿⣴⣿⣿⠀⡇⢸⡇⢻⣿⣿⠟⢹⡿⣿⡷⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⢸⠀⡿⠹⣿⢸⣿⣿⣿⣸⡿⠶⡀⢸⠃⢀⠹⣿⣿⣿⡿⣷⡄⢸⡇⣿⣿⡇⢸⡇⠀⡇
⠀⢿⡟⢹⣿⣆⡇⢸⣷⣾⣿⣿⣧⣾⣧⣿⠇⢸⢻⣿⡏⡟⠀⡏⢿⢻⠈⣿⠀⡇⠀⣿⣼⣿⣿⣿⣿⡇⠀⠀⣿⠀⢸⢀⣿⣿⣿⡇⢹⡇⢸⡇⣿⣿⡇⠘⡇⠀⡇
⠀⣿⠃⠈⣿⣟⣷⣿⣿⠿⣿⠙⡟⢿⡏⣿⠀⢸⣾⣿⠀⡇⡆⣇⣿⢸⠀⣿⡆⡇⠀⣿⡿⣿⠏⡟⢸⣇⣀⣀⣿⠀⣼⡞⢿⣿⣯⡇⠀⠀⠈⡇⣿⣿⣧⠀⡇⠀⣿
⠀⣿⠀⢸⣿⡿⡏⠈⡿⣧⣸⠀⡇⢸⡇⣿⠀⢸⣿⣿⠀⢧⠃⢻⣿⠸⠀⣻⣿⡇⠀⢻⡇⡇⢸⣿⣿⣿⡉⢩⣿⠀⣿⠇⢸⣿⡏⠁⠀⠀⠀⡇⣿⣿⢹⠀⡇⠀⣿
⠀⢸⠀⢸⣿⡇⡇⣦⡇⠀⢹⠀⡇⠀⡇⣿⠀⢸⡟⣿⠀⢸⡀⢸⣿⠀⠀⣿⣿⡇⠀⢸⠁⡇⣼⠋⢸⠈⠛⢸⣿⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⡇⠛⣿⢸⠀⣷⠀⣿

⠀⠀⠀⢀⣠⡀
⠀⠀⣴⣿⣿⣿⣷⣄⠀⠀⢀⣶⣷⣶⣿⣿⣆⠀⠀⠀⣀⠀⣠⣶⣦⣠⡄⠀⠀⣠⣠⣄⣴⣽⣦⠀⠀⠀⠀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣷⣆⣀
⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢻⣿⣿⣿⡏⠀⠀⠀⠀⠙⠋⠉
⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠉⠛⠛⠛⠛⠙⠉⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢀⣴⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⠀⠀⠀⠀⣠⣤⣷⣴⣰
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⣿⣿⣿⠤⠀⠀⠀⠂⠀⠀⠄
⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠻⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⢿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠀⢀⠈⣡⣀⣤⣤⣤⣀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣇⣠⣄⡀⠀⣰⣿⣿⣿⣷⡀⠀⣼⠻⢿⣿⣟⡀⠀⠒⣿⣾⣿⣿⡓
⠀⢰⣏⢾⣯⣿⣿⣿⣿⣿⡂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠛⠛⠛⠛⠛⠃⠀⢻⣷⣤⣿⡿⠀⠀⠈⠉⠻
⠀⠈⢿⣷⣿⣿⣿⣿⡿⠟⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣤⣤⣤⡄⠀⠀⠈⢿⣿⡯⠁
⠀⠀⠀⠈⡉⣩⠉⣁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠟⠋⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣀⣀⡀
⠀⠀⢐⣾⣿⣿⣯⣿⣒⠀⠀⠀⢠⣴⣶⣷⣾⡤⠀⠀⠀⣀⣀⣈⣤⣤⡀⠀⢀⣾⢿⣹⣟⢿⣷⡀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠾⠿⠿⠿⠿⠿⠿⠆⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⣰⣿⣿⣿⣿⣿⣧⡀⢸⣿⣎⢷⣿⢨⣿⡇⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⢻⣿⣿⣿⣿⣿⡿⠁⠈⠻⣿⣷⣍⣾⣿⠁
⠀⠀⠈⢻⣿⣿⣿⣿⠋⠀⠀⠀⠈⢻⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠙⠛⠛⠁⠀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⡤⠈⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀⠀⠀⠀⠀⣰⢒⡀⣀⡠⣄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⠐⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⢺⠒⢄⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⣑⣤⣵⢂⡀⠀⠀⠀⠀⠀⡹⠺⣄⠀⠀⠀⠀⠀⡱⠤⠵⣋⡜⠀⠀⠀⠀⠀⠀⠀⢸⠆⠑⡄
⠀⠀⠀⠀⠀⢎⠆⠀⠉⠁⠈⢣⠀⠀⠀⡸⠀⠀⠘⡄⠀⠀⠀⢸⠰⠀⠂⠂⢸⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⡜⡁⠀⠀⠀⠙⡄⠀⠀⠈⢎⢘⢀⢁⠆⠀⠀⣀⠠⠴⠚⢁⡠⠊⢀⠎
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⣠⠃⠀⠀⢇⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠒⠊⠀⠀⠘⢥⣉⣉⣉⣁⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠉⠉⠁⠀⠀⠀⠀⠑⠒⠒⠲⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢢⠀⠀⠀⠀⠀⢀⡤⠤⠼⡤⢮⡀
⠀⠀⠀⠀⠀⢀⣴⠍⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⢀⡤⢴⡺⠯⢷⡼⠤⠤⡀⠀⠀⡏⡎⠀⠀⠘⡄⠈⡆
⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢰⠚⢄⡠⢇⠀⣰⡿⠶⡖⠁⠀⠀⡗⠂⠀⠀⠐⠁⢀⡇
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣟⡩⠻⢄⠀⡏⠉⣧⠠⠏⠀⠀⠀⠘⠦⣀⣀⣀⠴⠊
⠀⠀⠀⠀⠀⠈⠳⠤⣀⣀⠤⠖⠁⠀⠀⠀⠸⠤⠷⠤⠚⠐⠉⠉⠁⠀⢠⢤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⡌⠁⡈⠀⠄⢀⠀⢁⠀⠂⡈⡸⢰
⠀⠀⠀⠀⠀⠀⢸⠦⡸⠙⣤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣦⡀⠂⠠⢈⠀⡄⠐⠀⡰⣡⠃
⠀⠀⠀⠀⠀⠀⢸⠳⢽⢊⡤⢻⠀⠀⠀⠀⡴⠊⠁⣶⣀⠤⢄⣀⠀⠀⠀⠀⠀⠙⠪⢕⡲⠤⠤⠤⣒⡭⠚⠁
⠀⠀⠀⠀⠀⠀⣠⠓⠻⣟⠙⣧⡀⠀⠀⠈⠓⠒⢩⣋⠀⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣎⠈⣦⡊⠈⡱⠧⣵⠀⠀⠀⠀⠀⡇⡃⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀⠴⢒⠩⠭⣑⠒⠤⢄
⠀⠀⠀⠀⠀⡟⠻⣄⣩⠾⢄⣰⢹⠀⠀⠀⠀⠀⠳⣁⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⢳⠶⣊⠑⢢⠼⣉⠏⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠁⠀⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠑⠶⣋⡱⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠄⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠙⠀⠉⠉⢈⢽⢯⡭⢵⡰⠚⡈⠉⠁⠀⠀⠀⢀⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⢰⣿⡇⠀⡖⣆⣖⣶⣦⠀⠀⣼⣷⣘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢸⠀⣸⡿⠻⣿⠟⢿⣇⠀⠒⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠀⠉⠓⠒⠛⠒⠊⠉⠀⣤⡄⠐⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠄⠀⠈⢿⡆⠀⠀⠀⠐⠀⠀⠀⠸⡿⠃⢺⣦⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢺⠟⠃⠀⣀⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠀⠀⠑⡙⢆
⠀⠀⠀⠀⠀⠀⣀⠰⠁⠀⠀⠀⣿⠛⣼⡇⠀⢺⣿⡄⠀⠀⣿⣷⠀⠸⡇⠿⢻⡇⠀⠀⠑⣤⠧
⠀⠀⠀⠀⠀⢠⡍⠁⠀⠀⠀⠀⢀⡄⠈⠀⠀⠈⠛⠀⠀⠀⠛⠁⠀⠀⠀⢸⣯⡇⠀⠀⠀⠈⠀⠑⢄⡀
⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⢧⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣧⡃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠋⢺⢴⣦⣆⠀⠀⣀⢸⠀⣀⠀⠀⣾⢀⣾⡏⠙⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⣿⣅⢀⢨⠿⣸⠀⡻⢢⣰⣽⣿⠛⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢾⣿⢧⣷⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⠛⠛⠙⠉⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠙⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⢙⡆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣞⢙⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠶⠒⠒⠲⠾⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤⣤⣼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠉⢡⠃⢀⣀⠤⠔⠚⠉⢹⡏⠉⠓⠢⠤⣀⡀⠘⡌⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⢀⣴⠞⠋⠀⠈⠂⠒⣊⠵⠚⣉⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣉⠓⠮⣑⠒⠂⠁⠀⠙⠳⣦⡀
⠀⠀⠀⣠⡟⠁⣀⠄⠀⢠⣴⠊⠁⠰⠋⠀⠀⠀⠑⠄⠀⣿⣿⠀⢠⠊⠀⠀⠀⠉⢆⠈⠑⢦⡄⠀⠠⣀⠈⢻⣄
⠀⠀⣰⠏⠀⡐⠁⢀⡴⠋⠈⢂⠀⠇⠀⠀⠀⠀⠀⡘⢠⡟⢻⡆⢂⠀⠀⠀⠀⠀⠸⠀⡐⠁⠉⢢⡀⠈⢃⠀⠹⣦
⠀⢰⡏⠀⠀⢣⡰⠋⠀⠀⠀⡜⠀⠈⠢⢀⣀⡠⠔⠁⣼⠃⠘⣷⠈⠢⢄⣀⡀⠔⠁⠀⢡⠀⠀⠀⠙⢆⡘⠀⠀⢹⡆
⢀⣿⠀⠀⢀⡜⠑⠢⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠊⢳⡀⠀⠀⣿
⢸⡇⠀⢀⠎⠀⠀⡠⠔⠒⠂⢄⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡠⠔⠒⠢⢄⠀⠀⠱⡀⠀⢸
⣼⡃⢀⠎⠀⠀⡘⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢣⠀⠀⠱⡀⢈
⣿⠀⡜⠑⢆⠀⠡⡀⠀⠀⠀⢀⡌⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢡⡀⠀⠀⠀⢀⡌⠀⡰⠊⢣
⣿⣸⠁⠀⢈⠀⠀⠉⠒⠀⠒⠉⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠙⠒⠀⠒⠉⠀⠀⡁⠀⠀⣇
⣿⡇⠀⡠⠊⠀⠀⢀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠑⢄⡀⢸
⢻⣧⣤⡶⠶⠶⠛⠛⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠛⠛⠶⠶⢶⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣤⣤⣤⣤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⡤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⠺⣿⣿⣿⠿⠿⠶⠶⠀⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢰⣾⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡧⢤⣴⠒⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣶⣆⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠑⢦⣸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢠⠄⡀⠀⡸⠁⠄⠀⠀⠀⠀⠀⠀⠀⠹⡋⠉⠉⡟⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣀⣀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⣴⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡸
⠀⣶⡶⠶⢤⣀⠀⡏⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠈⠃⣯⢁⠀⠀⠀⠐⢄⠄⠀⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠿⢛⠌
⠀⠀⠀⠀⠀⠀⠀⠘⡧⣀⠂⠄⠀⠈⡇⠀⠀⠐⠄⠀⡠⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡈⠢⣁⢀⣰⣇⣀⣀⣀⣸⠮⡖⠹⣿⣦⡦⡀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠁⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡼⡇⠈⠉⣻⠉⠙⢅⡻⡍⢯⢸⣠⢦⡧⠱⠈⢛⡶⣥⣆⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⡶⢛⠏⠯⣗⠳⢄
⠀⠀⠀⠀⠀⠀⠈⠉⠁⡠⣠⠟⠀⠀⠀⠉⢛⣧⠖⣡⠎⠀⠀⠐⠿⢭⣁⠘⠉⠍⣉⡉⠉⠁⠙⢦⣈⡵⠊⠀⠀⠀⠀⠑⠂⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣠⠖⢋⡴⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠠⢘⣟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢟⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠺⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡞⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⣦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⡶⠀⠈⠄⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣯⡵⣤⣴⡾⡅
⠀⠀⢀⠀⠀⠀⠀⠠⠂⣰⠿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡣⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢹⡟⠀⠀⢀⠀⠀⠀⠀⢼⣧⠀⠀⠋⠀⢉⠙⠀⡅
⠀⠀⢈⠀⠀⣼⠀⡆⣼⣇⢾⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠀⡄⠻⡇⠀⠀⠀⠀⢰⠀⣀⣿⣿⣦⠀⠀⠀⡏⣠⠈⡇
⠀⠀⠀⢳⠠⢿⣆⣴⠿⠻⣿⣿⡿⠿⣿⣿⢟⠾⠛⣿⣿⣿⣿⡄⠀⠀⠘⠀⢀⡀⠀⠀⣦⠀⠀⠀⠀⡀⡇⢘⡇⠀⠀⠀⣠⣿⣦⣿⣿⡛⠿⣷⡆⠀⠈⠀⠀⡁
⠀⠀⢀⡸⣷⣸⣿⣧⢿⣾⣿⣿⣧⣤⣽⣿⡟⠿⢿⣿⣿⡿⢿⣷⡄⠀⢨⠀⡌⠀⠀⠀⣿⠀⢀⢄⠀⢻⠁⣯⡇⠀⠀⣲⣿⡟⣅⢱⣀⣠⡀⢈⠻⠀⠈⠿⠁⠑
⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⡤⠀⢀⡏⠉⢉⣻⣿⣶⣶⣤⣄⡀⠀⠊⠉⣿⡄⠫⠘⢨⠋⣷⣏⣧⡼⣴⣿⣟⡙⢻⣶⣾⣻⣆⣿⣾⣂⠠⠄⠀⠁
⠀⠀⠈⡈⣿⣻⣿⣿⣿⣿⣿⣿⣿⠿⢹⣿⣟⣻⣦⣶⠐⢲⣾⣿⣿⣿⣯⣿⣿⣷⣶⣶⣿⣺⣦⡶⣾⢻⣿⣿⡿⣇⣦⣼⣇⠀⢀⣹⡉⣿⣿⣿⡿⣿⠀⡀
⠀⠀⠀⠁⢿⣾⣿⣿⠛⣿⣿⣿⣿⣶⣿⣯⡽⣝⣿⣿⣄⡈⣿⣿⣿⣿⡿⢻⣿⣿⢿⣿⣿⢿⢿⡿⠿⢾⣿⣻⣿⣿⠿⣿⡿⣴⣾⡿⢃⡷⡯⣝⣯⣝⣻⣿
⠀⠀⠀⠒⣼⡿⣟⣼⣷⣿⣿⡥⠸⣏⠉⢙⣿⢾⣯⡇⢸⣈⡿⢿⣷⣶⡟⣾⣷⠷⣶⣿⣿⣾⣟⣿⣅⣸⣿⠟⠻⣾⠞⣣⣿⠟⣻⣿⠿⢷⣾⣭⣽⣾⣟⣿⠀⠈⠃
⢠⡦⠀⠀⠀⠿⣡⣾⡟⣛⣻⡵⢶⣿⣿⣿⠿⡷⡼⣻⣿⣱⣆⣲⢿⣿⣻⣿⠛⢹⣿⣿⣿⣿⣧⣼⣎⣿⣿⣽⣇⢷⡼⣯⣧⠂⠉⢘⣿⣏⣣⣿⣿⡿⢷⣟⠀⢷⠂
⠈⠡⣀⡆⠘⣰⡧⢿⠛⣡⡿⣏⢛⡟⣇⣁⡃⣠⠜⢡⡀⣽⣩⢟⣘⠻⣿⣏⢈⣈⣻⣿⣿⣿⣿⣯⣿⢿⡿⣿⣏⠻⣿⡛⣿⡄⠀⣀⣿⣿⣿⢛⡿⣧⣾⣿⠀⠨
⠀⠙⢻⡁⡼⠁⠀⣂⡼⣿⣤⢫⢿⣿⣈⡯⢷⣅⣮⢠⡿⠇⣹⣤⣾⢕⣿⣿⣳⣿⡟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣭⣹⣿⣿⣟⣻⣿⣿⣿⠻⣟⣹⡿⢿⣽⡿⡇⢸
⠀⠀⠀⢣⣠⡆⣼⢿⣷⣟⣿⠰⢚⡽⢿⡿⢦⡋⡵⠾⠂⢶⣧⢹⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⣼⣷⣿⡙⠿⣿⢷⣾⣷⣿⣿⢻⣧⣼⣻⣿⣫⣠⣿⡇⢠
⠀⠀⠀⠀⠧⢸⢧⣸⣿⣾⣧⣷⣤⣾⡏⣳⢼⣿⣷⣿⣿⣿⣿⣸⣿⠟⡻⠛⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⠒⠤⣤⡌⢶⣿⣿⣯⣻⣏⣈⣇⣹⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠘⣾⣾⢿⣿⣏⣉⣉⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⡼⣀⣰⣿⡿⢻⣿⠿⣼⠁⠉⢊⡁⠀⠻⠿⠷⡶⣿⣿⡏⢹⣿⣧⣿⢟⡿⠏⡹⣷⣶⡇⠉⡹⠘
⠀⡞⣁⡉⡑⣾⣇⣘⣿⣿⣿⣿⣷⣽⠹⣿⣿⣿⣿⣿⣿⣿⡗⣣⡿⢻⣿⣷⣾⣿⣦⡼⣀⣸⠋⣁⣸⣠⣄⣿⠀⣨⣾⣿⣾⣿⣿⠿⣏⡁⠘⣤⣌⣷⡏⢿⡦⠖⠋
⠈⠋⠀⠈⠹⣿⣦⡉⣿⣿⣿⣿⣟⠿⣦⡼⣿⣿⣿⣿⣟⣿⣾⣿⣦⣿⣿⣟⣻⣿⣿⣿⣿⣿⢿⡟⡄⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⢍⣢⠈⠁⣿⢷⣷⣤
⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣿⠋⠀⢠⣿⣿⢶⣽⣿⣿⣿⣿⣿⣿⢡⣟⣽⣿⣻⣏⣭⣿⡗⢻⣾⡦⢶⣿⡻⡏⢛⠋⣿⢙⠀⣿⡏⣲⣾⣿⣿⣿⢶⣟⢌⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣻⡟⣿⢣⡄⣰⣿⣿⣿⣞⣿⣿⣿⣿⣯⣻⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣵⣾⣿⡅⢠⡊⠀⣿⣭⣿⣿⣽⡁⠩⠏⢙⠀⠀⢹⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢻⡇⠹⣿⠸⡰⠟⠃⠟⠻⡟⢿⣿⡿⡿⠟⣿⡿⣿⠻⠿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⣷⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠻⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠉⠀⠙⠁⠉⠉⠈⠉⠈⠉⠀⠀⠀⠈⠃⠋⠙⠙⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠙⠛⠻⢶⣄⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣇⣀⡀⠀⠀⠙⣿⠁⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣙⡛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠻⠷⣦⣄⠁⠀⠀⣿⣧⣴⡶⠟⠛⠛⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠛⠉⠉⠉⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠈⠻⣷⡀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠠⣽⣷⣂⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠈⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⢷⣤⣿⢿⣿⠃⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠤⣾⣥⣤⣴⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⢀⣀⣀⣠⡤⠄⢀⣀⣤⡤⢶⣶⣾⣿⠡⢖⣿⡷⠛⠋⠉⠉⠉⠛⠻⠷⣦⣤⣀⣀⣀⣤⣤⠶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣭⣭⣭⣤⠶⢛⣋⣡⣤⡶⠛⢉⡾⢁⣴⣿⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣿⣿⣷⡾⢿⣿⠟⠋⢀⣴⣿⠃⣼⠿⣿⣿⣜⠿⣵⣶⡭⠖⠀⠀⠚⠛⠛⠿⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠉⠀⣴⠟⠁⢀⣴⠿⢻⡏⢠⡿⠀⠈⠙⠿⣷⣶⣶⣶⣶⣾⠿⠟⠛⠋⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡿⠋⠀⠀⢀⣾⠋⢀⣴⠟⠁⠀⢸⡇⢸⡇
⠀⠀⠀⠀⣠⣴⡾⠿⢛⠋⠙⣿⡿⠿⠿⣶⣦⡾⠁⣠⡟⠁⠀⠀⠀⢸⡇⢸⡇
⠀⠀⢠⣾⡿⠋⠀⠀⠸⣦⣘⣫⡤⠀⠀⠀⡿⠁⣰⣏⠀⠀⠀⠀⠀⢘⡇⢸⣇⣀⣀⣀
⠀⢠⣿⠏⠀⡀⠀⠀⠀⠈⠙⠉⠀⠀⠀⢸⠃⣰⠟⣿⣧⠀⢀⣤⣾⣿⣿⡀⢿⠿⠿⢿⣿⣷⣦⣄⡀
⠀⢸⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡟⠀⢸⣿⣴⣿⡿⠋⠁⠈⢧⡈⣦⡴⠀⠀⠈⠉⠛⢿⣦⡀
⠀⢸⣿⡀⠻⣇⠀⠀⠀⢀⣀⣤⣤⣶⣶⡇⣾⣤⣤⣿⣿⣿⠏⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠈⢿⣷⡄⠉⠀⣠⣾⡿⠟⠋⢁⡀⢸⡇⡏⠀⠈⠙⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠻⣿⣶⣾⠟⠉⠀⠀⠀⠀⠻⣾⣧⣧⡶⠃⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿
⠀⠀⠀⠀⢨⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣸⣿
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢠⣿⠏
⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠰⠾⠛⠉⢀⣴⣿⠏
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⢀⣿⣿⣤⣄⣀⣀⣀⣀⣀⣤⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠀⣠⣾⡿⠁⠉⠛⠛⠻⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣉⡛⠛⠿⠿⠛⠋⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡛⠙⠻⣧⣄⣤⣀⠤⠂⠘⢇⠀⠀⠀⢿⣶⣦⠀⢀⣼⣾⡇⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⠁
⠀⠀⠀⠀⠀⣠⡌⡵⠇⠀⠘⣿⣿⣮⣿⡦⢻⣁⡘⢷⣦⣠⣤⠟⠃⣼⠟⠛⠛⠁⠘⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠖⠃
⠀⠀⠀⣰⣰⣿⠟⠃⠀⠀⠀⠉⠀⠈⡀⠀⠚⠋⠁⠈⢋⡫⡀⠀⠾⣝⣴⣾⠿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠨⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⢠⣶⡄⠀⠀⣸⡁⢀⣷⣦⣶⣿⣿⣿⣿⣀⠀⢿⣷⣶⡦⠀⢰⣁⠆⠀⢀⠰⡲⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠄⢂⢀⡚⠮⢿⣦⣀⠀⠨⠉⠛⣷⣶⣿⠃⡄⢀⣽⣥⣿⣿⡿⣻⣷⣄⣼⣿⢀⣬⡻⠽⢯⠄⡀⣀⠉⠀⠤⣄⠀⠄⠀⡀
⠀⠀⠀⢀⡀⢠⠼⠮⠛⠀⢨⣧⢈⣠⢀⣀⡁⠨⢁⣍⢓⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣷⣿⠋⢀⣟⢁⣤⣙⣿⣿⣧⣖⣶⣫⢀⡐⢠
⠀⠀⠀⠀⣽⠀⡐⠒⠶⠀⠀⣿⡽⢿⣿⣿⡦⣠⣾⣀⣷⣿⡿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢿⡿⠻⣛⣻⣿⣿⣾⣷⣮⣠⢡⡏⠄⠁
⠀⠀⡄⠀⡉⠀⡇⠀⠀⠀⢀⣿⠀⢸⣿⠿⠇⢁⠛⠛⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⡟⠀⣿⡏⠘⠙⣿⣿⣿⠛⣻⣉⠿⠦⡤⠄
⠀⢀⡇⢀⣡⢨⡇⠀⠀⠀⢸⣿⡇⣸⡏⣠⣶⣾⣤⣀⣽⣿⣿⣿⣿⡿⢿⣿⣿⣿⣻⢿⣿⣷⣶⣿⣇⢀⣴⣿⣿⣿⣾⣿⣷⣿⡟⠐⠀⣀
⠀⠈⡇⠠⡦⢸⣷⣴⣾⢳⣿⣿⣇⣸⣷⣿⣿⣧⡿⢯⣽⣿⣿⣿⣾⣽⡿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⣷⣾⡿⠿⠿⠋
⠀⢘⣇⣀⠀⢸⡇⠉⠁⠶⠖⢺⣇⢸⡷⣾⣛⣗⣬⣵⣚⣿⣿⣿⡿⢿⣿⡏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣿⣿⢋⣼⡥⠚⠲⠁
⠀⠺⣿⣿⢀⠘⡏⠀⠀⠀⢠⣻⣗⣸⣇⢉⣉⣿⠛⠉⠙⢛⣿⣿⣿⣛⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠄
⠐⢶⣝⡧⠊⠈⠍⠀⠀⠶⡈⢻⣿⣽⣇⠀⠀⣟⢀⣠⣤⠄⠩⠤⣽⣿⣿⣄⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡘⢴⠞⠀⠀⠀⠀⢀⣘⣾⡿⠿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⣉⣩⣽⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄
⠀⠷⣿⠇⠀⠀⠀⠀⠀⠈⠹⠿⣶⠂⡐⠀⠀⠿⠃⠀⠀⠀⢀⣀⣶⠤⠭⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⠁⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠀⠀⠢⠲⠂⠀⠀⠀⠘⠋⠃⠀⠀⠀⠁⠤⠈⠉⠉⠈⠉⠹⠤⠀⠉⠙⠿⠦⠁⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⣀⡀⠀⠀⢀⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠙⣿⣿⣿⠀⢠⣿⡿⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠛⠉⣰⣿⡟⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⣿⣿⣷⣶⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⢉⣩⣭⡉⠻⢿⣿⣿⣿⠟⢋⣩⣭⣉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⣼⣿⣿⣿⠁⣰⣿⣇⠀⣹⣦⠈⢿⣿⠃⢠⣿⣏⠀⣸⣧⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⢿⣿⣿⣿⣿⣿⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠈⠿⣿⣿⠿⠃⣠⣿⣿⣧⠈⠻⢿⣿⠿⠋⢀⣾⣿
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣷⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣶⣤⣤⣤⣶⣿⣿⡿
⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣍⣛⡛⢛⣛⣫⣥⣾⣿⣿⡟⠁
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⢀⠇⠀⠀⢀⣤⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠈⠙⠻⠿⠿⣿⠿⠿⠟⠛⠉⠘⡄
⣸⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠓⠒⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠒⠒⠋⠉

⠙⢹⡏⠉⠏⠀⠀⠀⠀⠀⠙⠁⢰⣬⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⠃⠊⠉
⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⡼⠗⣹⣿⣿⣿⣿⣿⣻⣻⣧⣾⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣏⠳⠰⠆
⠀⢹⡀⠀⠘⠙⢿⡶⠄⣴⢰⣯⣦⣿⣋⡼⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⢶⣀⠁
⠀⠈⠁⢀⣶⣶⡏⢿⣻⢋⣉⣙⣿⣯⣿⣠⣾⣿⣿⣿⣷⣿⣇⣀⠙⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠂⣿⡍⡆
⠀⠀⠀⠙⢣⡙⠗⢽⡉⠇⠈⢻⠟⠛⣟⡿⡟⠛⠛⠛⠻⣿⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠉⠈⠹⣀⠑
⢀⠀⣍⣽⡇⠀⠀⢠⣉⣓⠶⣶⡶⣟⡽⡀⢀⠀⠀⠀⣤⣧⣀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠦⣶⣾⡍⠤
⠀⠛⣿⣿⣿⣿⣧⡶⢦⠋⠹⣿⣿⣿⢀⣁⠀⠒⠂⠙⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⢉⣸⡿⠉⠐⡀
⠀⠀⣇⢹⣷⣿⣿⣷⡆⢳⣿⣿⣷⣿⢟⡉⠤⣿⣖⣀⣀⣀⣬⣷⣿⣿⣿⢿⠚⣻⣿⣾⣿⣿⣯⣼⡷⡆⠀⠀⠁
⠀⡄⢽⢹⡇⢻⣿⣿⡑⠀⠑⡻⣿⣿⡟⢱⢴⢾⠿⠧⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠷⠃⡅
⠀⠻⣿⢺⡧⣞⣿⣿⣅⣤⣶⣦⣿⣿⣻⣿⣻⣤⣴⣶⣶⣿⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡀⡇
⠀⠀⢿⣿⣇⠈⠋⣿⡛⢋⣀⢀⡩⣍⠛⢟⣛⣿⣿⣥⣿⣿⣿⣋⣵⠾⠟⠛⡉⣛⣽⣿⣿⣿⣿⣽⠃
⠀⠀⠂⣼⣿⣿⢇⣾⣇⠀⠙⠻⠻⠛⠾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠲⠭⢍⠁⠌⠉⢉⣻⣭⣀⠄
⠀⠀⠀⠈⠋⠁⠀⠁⠈⠀⠄⠀⠀⠀⠀⠁⢠⣿⣿⣿⡿⠀⠀⢀⠀⠀⢀⡈⡉⠛⣏⣠⣄⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠙⠷⠧⢾⣯⣻⣿⡟⣷⢶⡄⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠊⠛⢋⣛⡯⣓⣹⣦⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠉⠛⠛⠻⣯⠁⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢠⠐⠂⠁⠁⠀⢀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢰⠋⠀⠑⣄⠁⠲⠄⠀⠀⠀⢀⣠⣤⡁⠄⠀⠀⠀⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠈⢣⡀⠀⠀⠀⢠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠈⠻⠿⠛⠁⠀⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠢⠀⢇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠈⢄⣀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠙⠋⠅⠁
⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠐⠀⠀⠠⠚⠁⠠⣀⠀⠀⠀⡀⠄⢤⣤⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠓⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⠏⠉⡡⠌⠁⠀⠀⢠⣾⣿⣿⡿
⠀⠀⠀⠠⠖⠁⠀⠂⠀⠀⠀⠀⠀⠈⠉⠀⠁⠁⠀⠀⠀⠀⡴⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠢⠐⠀⠀⠀⠀⢀⡀⠐
⡀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢨⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡤⣀⣀⣀⡀⠀⠀⢀⡐⠠⠒⠚⠉⠀⠀⠈⠉⠙⢃⠀⠿⠂⠠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠴⠤⠤⠤⠆⠀⠈⠑⢢⣤⣠⡄⣀⡀⠀⠀⠀⠠⡀⢀⠘⣧⡀⠀⠘⠐⠉
⠀⠀⠀⠀⠤⡄⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠈⠛⠚⠋⠁⠀⠀⠀⠛⠉⠑
⣀⣀⠐⠃⠉⣀⣤⣠⣤⣰⣤⣴⠜⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠂
⣄⠴⠿⠒⠛⠋⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢄⠤⠰⣄⠀⣤⡴⠤
⠀⠀⠀⠀⠀⣀⡄⣠⠀⠘⠃⢀⡀⠀⡀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙⠠⠦⠄⠈⠉⠁
⠀⠀⠀⠘⢳⠀⣤⣀⠠⠄⠀⠀⠀⠀⠀⠀⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠄
⠀⠀⠀⡿⠿⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⢠⡄
⠂⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠶⠶⣶⣦⡀⠀⢠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠈⠁⣠⣴⠞⠋⡋⠀⠀⠀⠀⠷⣤⡀⠛⣇⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⠮
⠀⠉⠀⠹⣆⠀⠀⠙⠏⢠⡜⢀⡄⠀⠀⣵⡀⢙⣿⣆⠸⠋⠓⣤⢻⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠚
⡄⢠⣤⠈⣿⣄⠐⣷⣦⣼⠀⣼⡇⠀⠀⠻⣖⠄⢚⣿⠡⡎⠇⢠⣾⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴