Angiosperm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡴⠳⠟⠛⢳⡾⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠻⡆⠀⠶⠆⠘⠷⠀⢠⠀⠀⢀⡀⢀⣬⣯⡴⠶⣤
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢴⣟⠀⠀⠀⠀⣠⣾⣧⣼⠗⣾⡟⢻⣿⡟⠀⠀⠈⣷⣤⡄⢠⣄
⠀⠀⠀⠀⠀⠀⢀⡘⢃⠀⠀⢀⡀⠀⠾⠀⣿⡷⠀⢠⡰⠿⢿⣿⣷⣾⡛⣧⣼⣿⡗⢶⡾⢶⠛⠀⠀⣀⡿⠷⠶⣦⡀
⠀⠀⠀⠀⣤⠟⠀⠀⠛⠓⠶⠏⠁⢠⡄⣤⣈⣵⣄⠘⠁⢠⣴⡏⠩⣿⣧⡿⣿⣿⣵⡾⢻⣤⣲⣦⣶⣏⠀⡄⠀⣮⣵⡄⢀⣤⣶⣤
⠀⠀⠀⣤⠏⠀⠀⠀⠀⠀⠠⣦⣀⠀⠀⣿⣿⢿⣿⡆⠈⠀⠀⣙⡶⢋⣍⣻⠟⢻⣿⣶⣿⢉⣿⡁⣶⡾⠀⣠⡾⠁⠀⠈⣿⢿⣿⣿⣷
⠀⠀⠘⣤⡀⠀⠀⠀⠀⠀⢠⣟⣹⣿⣿⠟⠁⢸⠋⠀⣠⣤⠀⠙⢷⣾⡟⣿⣶⠟⣋⣠⣴⡿⠛⣿⣿⠁⢀⣽⣧⣄⠀⣴⣿⣿⠟⠻⣿
⠀⠀⠀⠀⠙⠳⢦⡶⠦⠤⠿⠟⠛⣛⠋⠀⠀⣿⣄⠀⢈⣿⣤⡄⠛⢻⣷⣼⣏⢸⣿⠀⢹⣦⣾⡏⢈⣀⣿⡿⠟⠠⠿⢿⣿⣿⣇⣴⣿⣀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⡀⠟⠋⠀⠘⠻⣿⡄⠀⠸⠿⢿⣿⠁⢹⣷⠟⢁⣼⣇⡼⠿⣁⡀⠀⠀⠀⠀⠈⣿⣿⣿⣟⣻⣿⣶⣄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠂⠎⠁⠀⠀⠠⢾⣶⣿⣿⣦⠀⠀⣼⡿⠿⣼⣿⡞⠛⠋⠁⣀⠀⠀⠐⠂⠀⠀⠀⢸⣿⣿⣿⠻⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡌⢁⡀⠀⠷⣦⣀⣀⣀⠐⠋⢉⡿⠋⠀⠀⠘⣋⡶⠀⠄⠀⡀⠀⠀⠀⠲⣄⠀⣾⠇⣀⣴⣾⡟⢻⣿⠶⣉⡛⠘⣿
⠀⠀⠀⠀⠲⣤⡀⠀⢠⡶⠋⠁⠈⠃⠀⠰⠿⠿⠿⣿⠀⠀⠈⠁⣤⣤⣤⠾⣟⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⣿⣿⣾⣿⡿⠋⠛⠿⠛⠳⣿⣤⣆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣀⣀⣤⣾⡿⢭⡇⠀⠻⠀⠀⢀⣠⣿⣷⣤⣴⣶⠛⣿⠿⣿⡟⢷⡆⣛⠀⣀⡀⠉⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠈⠀⢸⣧⣴⠀⠀⢳⣿⣿⣿⣏⡙⠻⡅⠀⠀⠄⠹⣧⡄⠀⠙⠷⠟⠃⢀⠁⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⢤⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠘⣥⢻⣧⡼⢻⣄⠀⣦⠀⠄⠀⠙⣧⡀⠐⠂⣤⠀⠀⢁⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠰⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠄⠉⢀⣦⠈⢻⣴⣿⠀⠠⠀⠠⣌⠛⠳⠾⠇⠰⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⡀⢸⡇⠀⣼⡇⣿⠇⠀⠦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⣶⠟⣡⡿⠇⠀⠀⠀⠘⠟⣹⡇⠀⠿⢧⣍⣀⣉⡠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠉⣀⠂⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡟⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠺⣄⠀⠐⠆⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠙⣦⠀⢀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢸⣦⠈⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠹⣆⡸⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣽⢧⣎⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠶⠶⢤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠚⠉⢉⣤⠄⠒⢲⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⢀⡀⠤⡞⠛⠀⢀⡰⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⣀⠀⠊⠁⠀⣀⡤⠖⠋
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠋⠀⠀⣠⡴⠃⠀⠀⣠⡴⠛⠁
⣿⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⢀⡶⠋⠀⢀⣠⠞⠉⠀⠀⣀⠈⠁
⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⣠⠖⠉⢀⣤⠐⠊⠀⠀⣠⡴⠛⠁
⠀⠁⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠰⠚⠁⣠⠖⢉⣀⠠⠶⠞⠛⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⣤⠾⠿⠻⠒⠉⠉
⠀⠀⠀⠘⠄⠙⠄⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠓⣀⡀⠀⠀⠈⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣄⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⣴⡆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣽⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣀⣀⣤⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣀⣤⣤⣭⣥⠄⢠⠁⠀⠐⣈⢾⠟⡤
⠀⠀⠊⠉⠀⠉⠉⠁⠀⣠⣾⣿⡆⠹⠟⣏⠉⢀⣠⠄⠹⠀⠀⠀⠀⢀⠄⠀⠀⠠⠤⠀⢀
⠀⠀⣤⠀⠄⠀⠀⠀⠀⢿⣷⣿⡇⠀⠀⡀⠀⠉⠀⢀⣤⠀⠀⠐⠙⠣⡀⠀⠀⣀⡓
⠀⠀⠀⠀⠚⠇⠀⠀⠀⠈⢿⣿⠁⢙⣶⣷⠒⠀⠈⢭⡿⠁⠀⠈⠃⣀⣘⠣⣄⢈⠁⣀⣀⡀
⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠈⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣤⡈⢻⣴⣿⣿⣿⣦
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣾⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⡟⠈⠻⣿⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡠⠃⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡼⠅⢆⠐⡐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠶⠤⠧⠴⡇⠣⠤⠄⠀⠀⢸⡆⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠒⢠⠶⢷⣶⣿⡟⢢⡀⣼⡷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠟⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇⠈⡻⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣿⡏⠀⡿⠋⣿⣿⣿⣟⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⣀⠹⣿⣿⡏⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢴⡿⠀⠘⡿⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠈⠛⠀⠀⠚
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡃⠀⠀⠈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣽⣦⠀⠀⠀⠀⠃
⠀⠀⠀⠂⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠠⠙⠀⠀⠀⠀⠂⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣄⠀⠀⢴⣴⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣿⣿⣿⣦⡈⠻⠿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣯⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠛⣛⣉⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⢠⣾⣭⣿⣛⣛⣭⣤⡷⣿⣿⣿⡄⠀⢰⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣟⣩⣧⣴⢧⠶⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⡰⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣾⣣⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠠⣠⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣹⣏⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣷⣮⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠁⠀⠀⠀⠀⠰⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣿⣯⣿⡆
⠀⠀⠀⠀⠀⢀⡄⣨⣶⣾⣿⣷⣤⣤⣦⣇⠀⠀⠀⠀⠀⠘⠛⡚⠀⠀⠀⠀⠀⠀⠀⠶⡞⠀⠀⠉⠛⠉⠁
⠀⠀⠀⠀⡠⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠙⢣⡄⠀⠀⠀⠀⠀⠀⢿⣧⣖⡄
⠀⠀⠀⠰⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠠⣽⡗⠀⠀⠀⠀⠰⠈⡏⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠎⡼⠁
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡈⢰⠇
⠀⠀⠀⠀⠀⢻⡫⣙⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠈⢷⡈⠳⢄⠈⠙⠢⣄⡀⠀⠀⠀⠀⢀⡏⠀⠀⠸⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠑⢤⡀⠀⠙⠢⡀⠀⠀⢸⡇⠀⠀⡖⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢀⣠⣾⣿⣾⣷⡀⠀⠀⠙⢦⡀⠀⠈⢢⡀⢸⡇⠀⠀⣧⠀⠀⢹⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⠉⠁⠉⠁⠀⣉⠴⠒
⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⢦⣀⠀⠀⠙⢦⡀⠀⠱⠈⣧⠀⠀⣿⠀⠀⡼⠀⠀⣠⠞⠉⠀⠀⠀⢀⠤⠂⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣽⣷⣤⣀⠀⠙⢦⡀⢡⠸⡆⠀⡏⠀⢠⢃⡠⠎⠁⠀⠀⣠⠴⠊⠀⠀⡠⠚⠁
⠀⣀⡠⠤⠶⠖⠒⠒⠒⠒⠒⠲⠿⠿⣿⣷⣦⣀⠀⠀⢣⡇⢰⣃⣰⠷⠋⣀⣠⠤⠖⠋⠁⠀⢀⠴⠊
⠛⢯⣉⠓⠒⠒⠤⠤⢤⣄⣀⣀⠀⠀⠀⠈⠉⠛⠷⢶⣮⠛⠾⣭⣖⣚⣋⣉⣀⣀⣠⡤⠔⠊
⠀⠀⠈⠛⠦⣤⣀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⢒⣠⢴⣿⣻⡶⣤⣈⠉⠛⠿⢿⣛⣓⠒⠒⠒⠒⠒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠶⠶⠖⠒⢛⣉⠔⠉⢸⡇⣷⠤⢌⡙⠻⢷⣴⣀⠉⠙⠻⠯⣁⡐⠂⠀⠀⠈⠉⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⣠⠎⢀⢻⢠⢻⡈⢦⡙⠳⡢⢤⣉⠙⠳⠦⣤⣄⡈⠁⠒⠤⢄⣀⣀⣀⢨⠭⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⢀⡤⠞⠁⢀⢂⡟⢰⠘⣧⠀⠳⣄⠈⠢⣄⠉⠙⠒⠒⠛⠛⠳⣦⣤⣀⠈⠉⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⣠⠴⠋⠀⠀⡠⠃⣼⠁⢸⠀⢹⡄⠀⠘⢷⡀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣄⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠊⠁⠀⢀⡤⠊⠀⣰⠃⢀⡟⠀⡀⢻⡄⠀⠀⠙⣆⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡾⢁⠔⠉⠀⣀⡤⠞⠉⠀⠀⢰⠏⠀⣸⠁⠀⠀⠀⠻⣦⠀⠀⠈⢣⡀⠈⢇
⠀⠀⠀⠀⠀⡼⠏⣁⡤⠞⠋⠁⠀⠀⠀⠀⠀⡿⠀⢠⠃⠀⠀⡇⠀⠀⠈⢷⣄⠀⠀⠱⡀⠈⡆
⠀⠀⠀⠀⢤⠵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠀⠀⢀⠇⠀⠀⠀⠀⠉⠳⣤⡀⠘⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡄⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣈⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣰⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠈⠐⠄⢀⠔⠃⢸⠀⡒⢄⠀⠔⠈⠀⢿⣿⣿⣿⠃
⠀⠀⡀⠤⠠⠄⠤⢀⡔⠁⠙⠛⠙⢕⡄⠀⠈⠁⠀⠀⢸⠀⠃⠀⠉⠀⠀⣔⠝⠙⠛⠁⠑⣄⠠⠄⠤⠠⠄⡀
⡐⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠱
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢰⡀
⠐⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢀⠄⢸⠀⢅⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡐
⠀⠈⢢⠀⠀⠀⠀⠀⡆⠀⠀⠀⡐⠀⠀⠀⢀⠆⠡⠄⢸⠀⠄⠁⢄⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠑⢄⠀⠀⠀⢰⠀⠀⢠⠀⠀⠀⢀⠃⠄⡠⠂⢸⠀⠢⡀⠄⢃⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠑⢄⠀⠀⢇⠀⢸⠀⠀⠀⠈⠘⢠⡍⠰⣁⡱⠈⣥⠘⠈⠀⠀⠀⢸⠀⢀⠆⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⢆⠈⡀⠄⠀⢀⢠⠰⣃⠀⠀⠀⢀⡛⢠⢠⠀⠀⠄⡀⢀⠎⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢢⠐⡈⢄⠈⠄⢂⠑⠔⠠⢐⠔⢁⠂⠌⢀⠌⡐⢠⠊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣂⣑⢌⠢⡑⠨⢦⡑⠘⠀⠢⠐⣡⠮⠐⡡⣊⢔⢡⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⢚⡺⠶⠤⠈⠀⠀⠀⠈⠠⠴⠾⡚⠢⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢶⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣶⣆
⠀⠀⠀⠀⠀⢡⡄⣷⣐⣎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⢸⡏⡇⢉⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣀⠀⠃⠑⠀⠀⠀⠀⠀⠀⠀⡄⣾⣿⣿⠘⠀⠉⠁
⠀⠀⠀⠀⠠⢀⠀⠃⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠆⣦⢿⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣬⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠳⢶⠏⡤⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠀⠘⢀⡀
⠀⠀⠀⠀⠀⢙⣣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⢀⣀⣐⠂⡁⢀⣿⣆⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠿⠿
⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⠁⠀⠈⠛⢃⡇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⡎⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠉⠀⣄⣧
⠀⠠⠄⠀⠀⠀⠈⡴⢆⡰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣝⡄⣴⣠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡔⢀⠠⠜⠃⠔⠀⠂
⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣴⠁⢢⠀⢰⠶
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡈⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⣷⣷⡾⠶⠶⠿⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣾⣿⣷
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⡴⠀⠐⠓⠒⠻⢋⠉⠉⠁⠹⠀⠀⠀⠄⠀⢀⡴⠁⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢡⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣴⠎⢀⣠⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢾⡷⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣿⡥⢊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣙⣿⡲⢼⡷⡌⠀⠀⢠⡀⠀⠀⠀⠀⢻⠀⣀⢾⡿⠒⠈⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢻⡯⣉⠉⠙⠛⠇⠠⣧⠘⠃⠀⠀⠀⠀⠈⣥⣼⡿⢛⣻⡿⠓⠒⠒⣤⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡿⡘⠙⠉⢙⣳⣤⣤⣤⣘⢫⡀⠤⠀⠀⠀⢂⡴⣾⣯⠡⠤⣅⣄⣀⣀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣊⠀⠀⠀⠀⠈⠉⠙⠺⣿⣷⣷⡄⠀⠀⢠⠞⣨⣝⠀⠀⠀⠈⢿⣍⠉⠉⠑⠺⡏
⠀⠀⠀⠀⠀⠀⡰⠃⠋⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢷⣀⠀⣾⡞⣿⠟⠓⠀⠀⠀⢸⡌⢢⠀⠀⠀⢀⡀
⠀⠀⠀⢐⣿⢮⡇⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠈⣿⣧⣼⣤⡟⣹⠋⠀⠀⠀⠀⠀⠀⠰⠀⢣⠀⠀⠈⠃
⠀⠀⠀⠈⠱⠀⠑⣤⠼⠣⣄⡀⠀⠄⠀⠀⠛⠀⠰⠭⣆⣿⣿⣿⡿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠑⠀⣐⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⢿⠿⠿⢇⣾⡄⠲⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣾⡟⠁⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⠇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣶⣿⠀⢾⣡⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠻⣧⣒⠴⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⡠⡋⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⡏⠉⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡦⠢⠤⡀⠀⠀⠀⢸⠀⢸⡇⠀⡁⠀⠀⠀⢀⠤⠔⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣿⢏⠀⠀⠈⠢⣀⠔⢹⠀⢸⡇⠀⡏⠢⣀⠔⠁⠀⠀⣹⣿⣿⣿⡿⣅
⠀⠀⣀⠤⠄⠒⠒⠂⠠⠤⡔⠁⠀⠉⠉⠈⠣⡣⡀⠀⠀⠁⠀⢸⠀⢸⡇⠀⡇⠀⠈⠀⠀⢀⢜⠜⠁⠉⠉⠀⠈⢢⠤⠄⠐⠒⠒⠠⠤⣀
⢠⠊⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢡⢣⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⠀⠀⡌⡌⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢡⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣰⡌⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⠀⠀⢡⣇⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠣⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢀⠼⠀⢸⡇⠀⠧⡀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡘⡘⠀⠀⠀⢀⠔⢁⠤⠒⢹⡏⠒⠤⡈⠢⡀⠀⠀⠀⢃⢃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠸⡀⠀⠀⢠⠁⡇⠀⠀⢀⠎⡠⠃⡠⡪⢽⡯⢕⢄⠘⠄⠱⠀⠀⠀⢸⠈⡄⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢣⠀⠀⢸⠀⡆⠀⠀⠘⠀⠃⡘⠎⢠⠊⠑⡄⠡⢃⠸⠀⠇⠀⠀⢸⠀⡇⠀⠀⡼⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢆⠀⠸⠀⠇⠀⠀⠇⢸⠀⣿⠀⠀⠑⠊⠀⠀⢾⠀⡇⠀⠀⠀⠸⠀⠇⠀⡰⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠈⢆⠀⢇⠰⡀⠀⢰⠈⡆⢡⢢⠀⠀⠀⠀⡔⡌⢰⠀⡆⠀⢀⠇⡰⠀⡰⠁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠣⡈⢆⠑⢄⠀⢇⠘⣄⠑⢍⠒⠒⡩⠊⣠⠃⡘⠀⡠⠊⡴⢁⡜⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⡙⢤⡑⢄⠑⠤⣣⣈⠓⠬⠀⠸⠤⠚⣁⣜⠤⠊⡠⢊⠴⢋⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢕⠨⠒⣽⣦⣄⡉⠓⠀⠀⠀⠀⠚⢉⣠⣴⣭⠶⢕⡪⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠉⠡⡀⠀⠀⠀⠀⢀⠌⠉⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡀⢸⣦⠈⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⠠⡀⢄⠇⠀⣿⡄⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠸⣿⣿⣾⣿⣿⣿⣧⣸⠇⣀⣀⣀⠀⢀⡄⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢒⣩⣽⣿⣿⣿⣿⡿⠉⠉⠉⠀⠀⠀⢁⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠐⢛⣷⣮⣽⣿⠉⢯⠁⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠐⠛⠀⠈⠁⠀⠀⠀⠈⣅⠀⠀⣀⡔⢠⣿⡿⡡⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⢀⠈⠉⠒⢬⣃⣄⢠⣄⠀⠀⠀⣰⡟⣶⣿⣿⢂⣻⣯⡋⠠
⠀⠀⠀⠀⠀⠀⣠⡄⣠⣄⣴⣶⣾⣶⣦⣤⣤⣬⣅⣘⣿⣟⣿⣸⣿⣻⣴⡠⠟⠋⠀⠀⠀⠀⠂⠐
⠀⠀⠀⡠⢒⠉⠒⣈⣠⠤⠔⠚⠛⠛⣛⡋⠉⠿⠿⡛⠛⣛⣛⠋⠉⡉⠉⠛⠂⠙⢆
⠀⣠⠞⠫⠔⠂⠁⢀⠀⠀⠂⠠⠍⠀⠠⠲⠄⡈⠉⠉⢩⣴⣿⣿⡿⠤⠤⣤⡀⢌⠆⠑
⡰⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣙⡒⠤⣀⠉⠛⠳⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣴⣶⣶⣾⣿⣿⣿⣶⣶⣤⣤⣼⣅⣉⠛⠳⢤⣝⠢⢄⡀⠁⠄⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠴⣊⠴⠚⠉⢀⣠⠴⠚⠋⠉⠙⠛⡻⠿⠛⠛⠛⠉⡃⠀⠈⠛⠦⣌⠒⢤⣀⠈⠓⢤⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⢔⡅⠊⠁⣀⠴⠞⠉⠀⠀⢀⠀⠔⠈⠁⠠⠀⣶⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⢀⡀⠀⠀⠈⠢⣀
⠀⠀⠀⡠⡡⠁⢀⡤⠞⠁⢀⠠⠀⠒⠈⣀⠤⠐⠈⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡐⠌⠣
⠀⠀⢠⠐⢁⡴⠋⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠐⣡⢋⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣁⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐
⣶⣶⡶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠸⡂⣈⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⡸⠃⣤⣶⡟⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠤⠀⠘⠃⠀⠾⠐⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠠⠟⠀⠀⡽⠇⠀⠀⠀⠀⠀⠠⠘⢆⠀⡇⠀⠀⠊⡌⠀⢠⠀⢀⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢀⣾⡀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠓⣄⠀⡄⠈⠁⢠⠇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⡠⠁⠀⠈⠉⠳⡄⠀⠀⠀⠀⠀⠐⢄⡙⠀⠇⠀⢠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡌⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢆⠐⠈⠉
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠠⠿⠂⠀⠀⠀⠀⠀⠀⠀⠠⠘⡄
⠀⠀⠀⠀⠀⠁⠤⠀⠀⠰⣦⡦⣄⠀⠀⠀⠀⢠⡞⠂⠀⡅⠀⠀⠀⠀⠀⢰⡟⢀⣠⣴⠶⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠂⠤⣀⠁⢻⡷⣄⠀⢠⣛⠃⠀⢠⠃⠀⠀⠀⠀⠀⣟⣱⣾⠟⠀⠀⠀⠀⣠⣠⠖⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠙⠢⡘⢯⣤⠆⠉⠀⠀⣸⠀⠀⠀⠀⢀⡾⠋⠁⠄⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠂⣥⠀⠀⠀⠀⡟⠀⠀⠀⣰⣫⠆⠀⠀⢀⡠⠊⠀⡴⠋
⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠈⠓⠀⠀⠀⠈⢣⠀⠀⠀⡇⠀⠀⡞⡹⠁⢀⠴⠃⠀⠀⢀⠞
⠀⠀⠀⠋⢀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠴⡄⠀⢃⠀⢸⡇⠀⡞⣴⢁⠔⠛⠐⠂⢀⠔⠁
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⠒⠠⢀⣈⠀⡀⣆⢸⠁⣾⡼⠟⠁⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢾⡞⠉⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡏⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⣠⠀⠈⠄⠀⠀⠀⠀⡟⠀⠀⠂⠀⡴⠀⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠈⠀⢀⠚⠄⠀⠀⠀⢠⡟⠀⠀⠀⠔⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠴⠀⠀⣎⠀⠆⣄⠀⠀⢸⡆⢀⣐⠁⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠰⠀⠀⠁⠲⣄⠀⠀⠀⠀⠱⡀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣸⠅⢀⣀⠀⠴⢆⢸⣷⠟⠤⠒⠂⠁
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠁⠀⡀⠆⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢀⠙⢄⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡠⠁⠀⢀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠈⠳⣄⠀⢀⠀⠀⡇⠀⠀⠀⡴⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡈⢧⣈⢰⠀⡇⠀⢀⠞⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⢸⡇⡇⢠⡾⠀⢤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡇⣧⠞⠠⢖⡿⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡿⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣴⠋⢹⢟⣶⣖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠋⢀⣼⡟⣄⣹⡄⢙⢻⣷⣉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠚⠋⠁⠈⠃⠀⠈⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡅⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⣀⣀⠘⢿⣿⣿⣿⠇⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡙⢿⡿⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣄⣳⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⢰⣿⣿⣿⣿⣿⡿⠟⠛⠻⢿⣮⣛⢙⣿⣿⠿⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠦
⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⣀⣤⣈⢻⣿⡟⣵⣾⣿⣶⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠛⠷⠀⢸⣿⣿⡾⠿⣿⣿⡿⣿⣿⠻⣤⣿⡿⠃⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣿⡿⠟⠛⢻⣧⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡀⠀⠀⠙⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⣀⠀⠸⣧⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⡀⠀⢀⡼⠁⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣧⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠀⠀⠀⠀⡄⢰⠃⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠲⢾⣿⣿⣿⣦⣧⣿⣨⣷⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠹⠿⣹⣿⣿⣿⣀⣈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⣙⠻⣿⣻⠿⠛⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠖⠂⡻⢿⣿⣿⠛⠋⠛⠈⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠀⠀⠰⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣔⣠⣾⣿⠟⠁⠀⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠻⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢰⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠘⠟⠀⠀⣠⣤⣌⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡻⣦⡀⠀⣾⣯⣠⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⠿⠚⠁⢀⡼⠫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣼⡇⠈⠉⠉⠉⠀⠀⠀⢀⣠⣤⣶⡿⠉⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣴⡿⠋⢹⡿⠁⠀⠀⠀⠰⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡟⢿⣷⣄⠀⣠⠟⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢿⡁⠀⢠⣿⡿⠁⠀⠀⠈⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠉⠚⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠲

⣶⣶⣤⣶⣶⣦⣴⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣦⣶⣴⣶⣶⣶
⣿⣿⣏⣀⣁⣨⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠆⠲⣠
⠀⠀⠀⣐⠟⠁⠀⣤⣤⣷⣾⣠⡂⢀⠀⠀⠐⢶⣄
⠀⠀⡟⠉⢤⡟⢻⣾⣷⣦⣰⣞⢿⣷⡄⠤⠤⠀⠇
⠛⠐⠛⠷⣾⣍⣹⢻⣿⣄⣼⣿⣾⣿⣿⡠⠐⢾⢟⡂
⠀⠒⣲⠿⠉⢿⠗⠛⠉⠋⠁⠉⡙⢉⡁⠀⠢⣾⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⠒⠲⣿⣄⣤⣧⣤⣧⣀⣸⣤⡴⠾⠾⠋⠁
⠀⠀⠀⠀⠀⠉⠿⢿⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠘⠂⡍⡤⠈⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠘⠟⢲⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣐⣒⣓⣛⣒⣋⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣜⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣨⣿⡿⢿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣤⢄⣿⣼⣿⣻⣿⣿⣿⣿⣿⣿⣾⣶⣠
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢿⣍⡟⠣⣭⡵⣹⣿⡿⠛⣿⣿⣿⣿⣿⣧⣶⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣦⣴⠆⠉⣤⡹⢟⣭⢼⣿⣿⡿⠈⢍⡿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣲⣿⠁⣀⣀⡾⢣⠈⡟⣿⣿⣿⡏⠐⣿⣿⠟⣿⣿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⠛⠀⠀⣽⠓⠊⠀⢽⣿⣿⣿⡇⣈⠁⣯⠻⣿⣿⣿⣿⣻⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢔⣾⡇⠬⢛⣿⡐⠉⠔⣿⣿⣿⣿⡇⠀⠀⠉⣲⣿⣿⣿⣿⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⡒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠄⠉⣻⡅⠀⠈⣷⢀⠙⣌⣻⣿⣿⣿⠂⠀⠀⣶⡭⣽⣧⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⠁⠌⠛⠑⠀⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡙⢿⠀⣴⣿⠋⡠⣫⣿⣿⣿⣿⡇⢀⣀⣿⡿⢿⣿⣿⣿⣾⣧⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠶⣼⣿⣷⣷⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠾⡿⠿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣹⡿⠯⡁
⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠡⠐⢾⣻⣯⣿⣿⣿⣿⣿⣿⣿⡯⠓
⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⣰⣧⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⣈⠍⠓⣫⠞⡟⢻⠏⠉⠛⠉⠢⢄
⠀⠀⠀⠀⠀⠀⣠⡄⢼⣷⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⡸⠆⠀⠀⠀⠁⠀⠈⠠
⠀⠀⠀⠠⢷⡄⠉⠉⠈⢿⠙⠻⠿⠦
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⢰⢶⣦⡶⢀⡶⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡎⠛⢰⠘⣟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⣁⠘⠂⣽⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠆⢤⢈⢀⠻⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⣤⢸⣿⠀⠃⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠄⠀⢰⢘⡛⠀⡁⢼⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠃⠀⣶⡈⢹⢦⣅⢸⡇⡀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠟⠄⠈⢸⡅⢸⠃⠐⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠆⠀⠈⠀⠀⡄⢸⠀⢘⢀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣾⠀⠄⡇⠈⠁⠈⠸⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⠀⡀⠀⠀⠰⠀⢠⡌⠃⠀⡇⠀⠐⢋⡀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⢀⠄⠀⡆⡐⣆⠎⡸⠀⠀⢳⠀⠀⡀⠉⠂⠈⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⠄⡀⠀⠠⠂⢀⠈⡄⠁⠃⠀⠀⠈⢳⡁⠀⠀⡨⢢⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢰⠏⠀⠰⠀⠀⠀⠀⣀⡀⣠⡁⢈⠰⠆⣁⣬⠷⠮⣭⣔⡰⢇⣐⠀⠐⠕⢄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠔⠀⠁⡜⣱⠃⣁⠾⠛⠻⠃⡠⠤⡌⠀⠉⠳⣍⠓⠀⡨⢋⢧⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠈⢠⡆⠀⣹⠋⠀⠉⠀⢀⢀⣏⠀⠀⠈⣆⡀⡀⠋⠛⢄⠀⣈⠃⠃⠀⠀⠃⠀⠀⠸⡄
⠀⠀⠀⡟⠀⠄⠄⠀⠀⠀⣶⠁⣴⠗⠠⡤⠚⠳⡁⠀⠙⣦⠄⣴⠁⠀⠈⡦⠠⡀⠧⢈⠀⠐⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⡇⢠⣴⠂⠀⠀⢠⠍⠀⣷⠀⠀⣧⡀⣀⣦⡄⣼⠁⠀⠀⣦⢀⡾⠃⠀⣸⡀⢿⠦⠋⠳⠠⠀⠀⠀⠀⢸
⠀⠀⢠⡇⠸⣿⢘⠀⠀⠻⠄⠒⣿⠉⠈⡟⠁⠙⡇⠀⠀⡷⠠⡞⠁⠀⣷⠠⣼⠋⠉⢻⡆⠠⠻⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢸⡇⢰⣿⢸⠀⠀⠀⠀⠀⣿⣤⡤⢶⣤⠴⠛⢰⣾⣁⠀⣸⠖⠳⣁⠀⣨⡆⠀⣼⣇⠀⣤⠆⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠈⢻⠀⠀⠀⠘⠀⠐⣿⣼⡌⠻⣄⡄⢀⡃⠀⠈⣻⣇⡀⠀⣸⠚⠋⠙⡏⢻⡄⢰⡏⠃⠀⠀⠀⠀⢠⠃
⠀⠀⢸⡇⠐⣿⡆⠀⠀⠀⠀⠀⣿⣯⠀⢀⡁⠀⠀⣷⠦⢶⡏⠀⠈⡟⠉⠙⣦⠼⢇⣾⠁⢙⡁⠀⠀⠀⠀⠀⡼
⠀⠀⠈⡇⠀⢇⠁⢡⠀⠀⠀⢠⢹⣉⡏⠉⠳⣤⣼⡃⠀⣀⡧⣀⡴⠧⣀⣠⠏⠀⢸⡧⡄⡘⠁⠈⠀⠀⠀⢰⡇
⠀⠀⠀⢷⠀⠀⠀⠄⣆⠠⠀⠂⠞⣦⡷⣤⣴⡇⠀⢠⡿⠉⠀⠀⠁⠀⠈⠗⠓⢤⡞⢃⠃⠁⠔⣰⣦⠂⠀⢸⠁
⠀⠀⠀⠸⣆⠀⠀⠘⠎⠒⠐⠀⠘⢻⣇⣀⣸⠟⠂⢺⠛⠢⢦⠜⠓⠖⠺⠀⢀⣼⠗⠀⠀⠎⣤⠻⠁⠀⢀⡟
⠀⠀⠀⠀⢻⣤⠀⠀⡑⠄⠸⣄⠀⠄⠉⠻⣯⣀⡠⠟⡶⠘⠎⣀⡠⢧⣀⣸⡿⠁⠠⣠⣠⠞⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠘⠢⢥⣀⠀⠀⠀⠑⠒⠿⣤⣤⣤⡶⠴⠖⡋⠁⠐⢋⡴⠟⠃⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⢀⣀⣀⣠⣤⣤⡶⠶⠶⠷⠶⠶⠶⠤⣤⣍⣁⣀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣿⣿⡇⢤⠄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⡀⠀⠀⠀⠘⠃⠀⠀⠀⡤⠤⣀⣶⣶⡦⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⣿⣿⡇⠼⠄⠀⠀⠀⠸⢢⠆⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠩⢹⡇⠀⣿⠿⠿⠟⠛⠋⠀⠣⣰⣾⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⡷⠤⠴⠤⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⡇⠀⢠⣶⣾⢸⡀⠀⠀⢠⡎⠉⠉⠉⠉⠉⠀⠀⠀⢀⠄⢠⢰⡆⣿⡇
⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼⡇⠀⢬⣿⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠛⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣀⣀⣀⣀⣘⣃⣿⡇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠒⠒⠀⠀⠀⠀⣶⣦⣤⠤⣤⣤⢤⣤⣤⣤⣶⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⠍⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡓⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠚⠛⠛⠛⠀⠀⠀⠀⣿⣿⣶⡇⠀⠀⢀⣀⣠⣤⠀⠀⠛⠉⠀⠁⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⡶⠆
⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⣠⡟⠛⠻⢯⣀⠀⠀⠀⢸⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⣀⡊⠓⠲⢻⠀⠀⠀⢸⣷⣿⣾⣿⣿
⠀⠀⠀⠀⢈⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠀⠀⠀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠛⠛⠒⠀⢸⠤⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⢫⠀⠀⠀⠀⠀⢶⣦⣤⣴⡾⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⠻⠟⠿⠟⢃⡀⠀⠀⠀⠛⠘⠳⠁⠠⢸⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⠤⠴⠦⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡰⡿⢃⣠⠞⠉
⠀⠀⠀⢀⡠⠤⠴⠶⠦⠬⢽⡃⢸⠃
⠀⠀⠔⠁⠀⠀⠀⠀⢠⣶⠗⣉⣛⣰⣾⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠤⢾⣿⣷⣤⡀⣼⣿⣿⣿⣟⣿⣟⡿⣿⣟⣤⡀⠀⠀⣦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⢿⣶⡙⣧⣙⣿⣿⣿⣿⣿⣿⣿⣇⣛⣿⣿⡆⠀⣝⡴⠂⡈⠿⢷⢰⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⢿⣿⣾⣞⢆⣿⣯⣿⣋⢀⣠⣌⢳⣿⣿⣀⡸⠉⠉⠁⠉⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣾⣿⣪⡻⣿⣿⣿⡾⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡏⣥⠟⣡⣤⡀⠀⠀⠀⢴⣦⣿⡿⠿⢿⣶⡄⢷⠄⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣾⡿⠇⠀⠈⠁⢰⣿⣿⣿⣿⣿⢛⣻⢟⣿⣿⣿⣿⣿⣷⡆⠻⡿⠀⠀⢀⣼⣿⣿⢿⣷⣶⣿⢆⠔⢋⣛⠃⠀⢜⢡⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴⣾⣿⢒⣲⣿⣿⣿⣿⣟⣾⣿⣿⡿⠿⣛⣿⣿⣿⠇⠀⠀⠀⠀⠀⠉⣿⣿⢿⣿⣿⠋⠴⠚⠛⠒⠲⢤⡻⠃⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣫⣾⣿⡟⡮⣯⡉⠁⣿⣿⣿⢿⣿⣏⣿⡿⣿⣿⡄⠀⣀⠀⠠⢦⡄⠈⠉⣠⢡⣾⣶⣾⣶⣆⡀⠀⠀⠉⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡿⠛⣿⣿⣸⣿⠀⣴⣿⣿⣿⣿⣿⣿⣾⢩⣍⠀⠘⣋⣠⣤⣜⣃⠤⠊⢵⣷⣿⣻⣿⢃⢿⣟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡁⠀⠀⠹⣿⣿⠏⠀⠉⢛⡟⣭⡤⡙⠛⣹⣮⣿⡇⢱⡶⠏⠉⠡⡀⣴⣶⠌⣽⣿⣿⣿⣴⣿⡛⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⣾⣷⠿⠁⡇⠀⠈⢛⡟⢱⡿⢋⡟⠂⠀⣷⣬⠁⠀⠉⠉⢻⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⠂⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠁⠀⢀⡌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢰
⠀⠀⠀⡀⠂⠉⠀⠀⢠⡰⠀⠘⡄⢦⠀⠀⠡⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⡄⡤
⠀⠀⠀⡀⠀⢀⡴⠈⠂⠃⢀⡀⠘⠈⠀⢡⠁⠀⠀⢠⠦⠦⣄⠀⠁⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⡇⠈⠓⠧⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢭⡈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣆⠣⣰⠀⢐⠀⠀⢀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠀⠂⠀⠀⠀⢀⠠⡎⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⡄⠀⡀⠠⠈⠀⠀⠠⡀⠀⠀⠀⠁⠀⠀⣤⡲⠤⠤⠰⠄⠁⠘⠁
⠀⠀⠠⠂⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠒⠂⠐⠈⠟⠂⡀⠀⠔⠈⠀⠀⠦⣀
⠀⠰⠀⠀⡐⠈⠀⢄⠁⠢⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠼⠏⠀⢀⣀⠀⠀⠸
⠀⠀⠀⢰⠀⡀⢐⠄⠃⠀⢢⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡆⠀⠀⠀⠀⣠⠆⠀⠈⠀⠆⠀⢀
⠀⠇⠀⠸⡀⠈⠉⠀⡄⠀⠘⠀⠀⠀⠈⠄⠀⠋⠀⠀⠰⠴⠀⠀⠲⠤⠈⢀⠀⠀⠀⠀⠁⠀⢀⠀⠈⠀⠈⠀⠀⢐⠃⠀⠀⠀⡐
⠀⠈⢦⡀⠈⠒⠒⠈⠀⢀⠄⠀⠀⣀⢀⡈⠀⡠⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠔⠉⠀⠈⡀⠀⠀⠀⡂
⠀⠀⠀⠉⠓⠲⠤⠤⠒⠋⠠⠔⠊⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠌⠘⠁⠀⠀⠀⠀⠀⠈⠚⠓⠂
⠀⠀⠀⠒⠐⠂⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣀⢤⣄⠠⢤⣀⣀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⠝⠒⠛⣩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⢉⣉⠀⠒⠒⠒⠀⠀⠒⠚⠤⡀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡠⠊⠁⠀⠉⠒⠂⠠⠤⠤⠤⢄⡀⠘⣦⢀⣀⡤⠊⠁⠀⠀⠀⠀⠀⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢉⡏⠉⠀⡠⠚⠉⠒⠠⠤⠤⠤⠤⢀⡀⠈⢢⡝⢢⡀⠈⠑⠢⡀⠀⠀⠀⢀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠎⢹⠊⠉⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⡄⢀⡇⠀⠳⠤⢄⡀⠈⢦⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡤⠒⠚⠀⢠⢧⠀⢀⠀⢰⠀⢸⠀⢀⠃⠀⠀⠀⣀⡜⠉⠂⠤⣀⠀⠙⢄⠈⣖⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⣠⠤⠔⠉⢄⠣⡈⢢⠈⠀⠈⠀⠘⠀⡴⠁⣠⠏⠀⡀⠀⠀⠈⠢⣄⠀⡹⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠀⠀⡇⠀⢀⡀⠢⡑⠌⢲⠣⣰⠿⠚⠓⠶⢤⠞⢁⡠⠊⠀⠀⠀⠀⢀⢆⡜⠇⢸
⠀⠀⠀⠀⠀⠀⢀⠤⣄⠚⠓⠂⢷⠤⢄⣈⡒⠌⠐⠟⡞⢰⡿⠖⣶⡆⠼⣇⣡⠔⠀⠀⠀⠀⣠⠞⢏⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⢱⠀⢸⠀⠀⠐⠨⢭⣚⡷⢧⡰⢧⣴⡾⢁⡷⢟⠲⠾⠿⠶⠒⠉⠀⠀⢸⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠇⠀⡼⠀⠀⠀⠈⠭⢐⠁⢼⡑⠢⡠⣶⠋⠁⡈⠷⡶⣌⠍⠉⠉⢀⠔⠋⠀⡸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣇⠀⠀⠀⠤⠒⠥⢊⠎⡆⢀⢡⠈⡀⢱⠈⢆⠈⢢⣍⣁⠶⠇⠀⡰⠊⠀⣠⠃
⠀⠀⠀⠀⠀⠀⢀⡼⠣⡈⠢⡀⠈⠑⣄⠀⠀⢀⡴⠃⠰⠁⢸⠘⠂⠇⠀⡆⠈⠀⠀⠹⡁⢀⡤⠔⢁⡴⠊⠁
⠀⠀⠀⠀⠀⠰⢣⠀⠀⠘⠢⣈⠢⡀⠈⣑⡶⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣊⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡠⠋⠉⠉⠀⡇⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡜⠁⠀⠀⠀⠀⠱⣒⡊⠁⠀⠙⠒⠀⠒⠒⠲⠒⠉⠉⢁⣀⠴⠃
⠀⠀⠀⠀⠀⠠⣥⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⠀⠀⠀⠀⠀⠀⣠⠓⠊
⠀⠀⠀⠀⠀⠀⠳⠸⠀⠀⢀⠀⡤⣄⠞⠊⠁
⠀⠀⠀⠀⠀⠀⢶⡇⢀⠶⠊⠋
⠀⠀⠀⠀⠀⠀⠘⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠠⠀⢀⡀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣗⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠻⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⢶⠃⠀⡇⠀⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠐⢀⢐⠀⠀⠀⣴⡦⠀⠀⠸⠂⠀⠇⠀⠈⠀⠀⠀⢈⣁⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢬⠂⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠛⣁⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⢬⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠉⡋⠀⠈⠑⠓⠄⠀⠀⠀⠀⠰
⠈⠋⠉⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡔⠀⡄⠀⠀⠀⠣⠄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠳⠀⠀⠐⡄⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠐⣆⠀⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡁⠠⠤⠄⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠂⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣵⣤⢄⠐⠂
⠀⠀⠁⠀⠀⢐⣀⣠⡀⠐⠲⠶⣶⡶⠀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠋⠁⢠⠆⠀⠀⠀⢘⠀⠀⠀⢠⣤⣴
⠀⠐⠁⠀⠀⠀⠉⠁⣀⠀⠀⡄⠁⢐⡆⠀⠀⠀⠀⠀⠐⠈⠆⠶⣄⣀⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠉⠑⠦⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⠀⣴⢀⠀⢰⠁
⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠧⠠⠀⠃⠀⠀⠠⢄⡂⠐⢦⠘⠱⡀⢧⠇⢀⡠⠤
⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡓⠦⢙⣦⣁⣆⢀⡡⠀⡨
⠀⠀⠈⢀⠀⠠⡄⠁⠈⠀⠀⠀⠀⠀⡘⠖⠉⣠⡼⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠚⠛⠙⠻⢿⣿⣿⡶⠒⠛⠁
⠀⠀⠀⠈⠷⠢⣤⠀⡀⠀⠀⠀⢀⠐⢁⣦⠟⡉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⢰⣥⠴⡺⠋⢁⠞⡿⢹⢟⢣⣀⠀⠁⠁⠀⠐
⠀⠀⠀⠀⠤⠒⠄⠁⠀⠀⠀⠀⠀⣴⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄⠤⠂⠁⠐⠀⣰⠈⣥⠜⢆⠉⠁
⢄⠀⡄⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠀⠀⢠⡄
⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠎⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢐⡀⠀⠀⠀⠴⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠄⠀⠃⠀⠈⠸⠀⣴⡃⡀⠄⡠⠀⠀⠀⠀⠐⣡⠾⠀⠀⠀⠀⠐⠀⠂⡨⠿⢧⡀
⠠⠄⠀⠀⡀⠀⡐⠀⠀⠃⠀⠈⠁⠀⠊⠀⠀⠀⠀⢸⣿⣁⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⢸⠀⠀⠁⠈⠁⠀⠀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⠀⠀⠀⠀⠀⢀
⠅⠊⡀⠌⠀⠀⠠⠄⢀⠠⣈⣉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠁⣀⣒⠒⠂⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣈⠄⡤⠈⠓⠒⠂⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠊⠉⠄⠘⠉⠉⠐⡄⠀⠀⠀⠤⢀⡐⠠⠀⠠⣄⠉⣠
⠀⠀⠒⠀⠂⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠰⢄⡙⣳⣄⡀⠁⠘⠛⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⣤⠉⠛⠶⣶⣸⠀⠋⠃⡀
⣉⣯⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠐⢒⡀⠙⢛⢃⣀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡤⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡛⠛⠳⡿⡿⠋⡿⡛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⢷⠀⢡⣾⣿⢿⣑⣿⠇⠙⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⢠⣿⡟⠉⠋⢛⣭⣁⠐⢿⢟⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠋⠀⢰⣷⠿⣿⣿⡅⠀⢺⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⠋⠙⠋⣽⣋⣤⡄⠸⠇⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢟⣀⡉⠶⠻⡆⠉⠉⠉⠉⠀⠀⢤⠡⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢢⣏⠣⠄⠀⠀⠀⠈⠃⢀⠌⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣈⣡⢶⠜⠃⠀⠐⠀⠀⠀⣣⠀⢸⣗⣀⣀⣰⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠉⠙⠁⣠⠴⠀⠀⡁⣀⠀⠀⠻⢞⣡⠟⢆⠑⠍⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠋⠓⢤⣴⠃⡄⠄⠊⠛⠡⡀⠀⠀⠀⠚⢮⠙
⠀⠀⠀⠀⠀⠨⠖⠀⢀⠀⠀⠀⠀⠠⠉⠐⠀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠉⣠⠴⢶⢣⡔⠄⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠈⠑⢀⡗⡀⠀⠀⣨⠀⠀⢀⣸⣦⡉⠉⠉⡠⠊⠀⠖⠸⢠⡔⠺⠉⠙⢣⠢
⠀⠀⠀⠀⠀⣀⣰⠋⠉⠀⠀⠀⠀⠀⢀⠘⠁⠘⠃⢄⢠⠂⠀⠀⠀⠈⠱⠁⢠⠂⣴⣯⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠿⠃⠀⠀⠈⠀⠀⠀⣠⠌⠀⠀⠢⡄⠀⠀⢀⠇⠈⠀⠉⠠⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠐⠀⠀⠀⢎⠄⠀⠀⢀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠈⠰⡅⣈⠠⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠨⠃⡘⠞⢚⠠⠗⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠤⢊⠀⠂⠒⠒⠬⡑⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⢀⠔⡨⠄⠒⠈⠉⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠐⡀⠇⠀⠀⡀⠖⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⢊⠔⡩⠂⠡⠀⠀⣠⠖⠄⠀⢹⠀⡄⠀⠀⠠⡀⠀⠀⠈⠀⠀⠱⡀⢢⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡐⠁⠠⠊⠀⠀⠀⠀⠀⠣⠀⠀⠴⠉⠐⠇⠀⠀⠐⡄⠀⢁⠂⠀⠀⠀⠀⠀⠁⠡⡀⢆⢀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡈⠀⠁⠀⠀⠀⠐⠀⠘⢇⠐⠈⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠀⠶⠀⠀⠀⠀⠙⠂⠀⠀⠐⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠘⠀⠀⡛⢉⡖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢰⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡹⡷⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⡁⠀⠰⢠⠆⠀⠀⠀⠀⠕⠀⠁⠀⠀⠄⠠⠀⡀⢢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡀
⠀⠀⠰⢣⠀⠀⠀⠀⠀⠀⠀⠉⣀⠠⠾⡦⣤⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠌⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠞⠄⠉⠀⢀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠐⡀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⡈⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠑⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠒⠂⠁⠀⠀⠀⠀⠂⠂⠀⠂⠀⠄⠐⠀⣤⠀⡆⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⢀⡀⡄⠀⢰⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠊⠀⠀⠀⠉⠣⡀⠀⠡⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠶⠛⠋⠛⠋⠀⢏⠉⢣⡢⡀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⠈
⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⢜⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡢⢀⠀⣀⡀⠀⠀⠉⠢⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣴⠛⡏⠉⠷⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣤⢴⠆⠈⠁⠒⡭⠄⠀⠀⠚⠉⠀⠀⡀⠀⢀⡡⠀⠦⠌⢛⠡⠂⠀⠘⣄⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡗⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠊
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢄⠀⡀⠄⡧⠀⠀⠀⠠⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⢀⠔⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠑⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡜⠃⠀⠀⣀⠄⠈⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⠒⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠂⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⠀⡠⠁⠀⠰⠇⠐⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢣⠀⠀⢀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠸⠀⢈⠀⡔⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⢢⠀⠀⣀⡄⠻⡲⣕⣾⡰⠧⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢸⢠⠋⠀⠀⠀⠈⡩⠯⡖⠉⠁⠉⠑⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠐⠒⢤⠙⣤⣀⠴⢁⡤⠊⣀⠠⠬⢕⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠠⠷⣜⠀⡰⠻⠛⠭⡑⠄⠀⢀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠐⠳⠀⣸⠊⠀⠸⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⣆⠀⢀⡐⢤⣴⠗⠉⢉⠇⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠢⣗⡔⠁⡄⡸⠋⢰⣌⢡⠊⠉⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⢄⣀⣱⠾⣧⡀⠃⠃⢠⣜⢼⡶⠤⡴⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣍⣿⡟⠰⠀⡇⠘⣄⡥⢎⠀⢀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⡟⠲⡄⠡⣀⡿⢀⠄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⢒⡩⠤⣀⢱⠀⣹⡗⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣀⡀⠴⣋⠻⡀⠈⠉⡩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠆⠙⢄⠀⡁⢘⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠉⠑⠪⣢⡄⣴⣭⣛⣺⣾⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣒⣂⣉⠤⣏⠁⠀⡠⠤⠤⢬⠖⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡠⠁⠰⠁⠱⠀⣇⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⠀⠀⠈⢢⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣻⣟

⠀⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⡄⠀⠀⠀⡀⠀⠄⠀⠀⢠⠀⠀⡠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢠⠀⠠
⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠁⠀⠈⠀⠀⠆⠀⠃⠀⢸⠀⠠⠀⢀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄⢀⡴⣇⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣴⣾⡀⣆⣠⣦⠖⠀⢸⠀⢠⡄⢒⢫⣴⣷⣿⣅⠀⠀⠀⠀⠀⡀⢦⢎⠃
⠀⠀⢀⣼⢉⣼⣿⠟⠞⢁⣼⣾⡾⢰⠇⠀⠙⢾⠯⡽⣿⡷⢀⢴⢐⣆⡘⣿⣿⡟⡳⠇⠀⠀⠀⠀⡀⡄⠠⠤⠬⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠉⣁⠀⣸⡿⠇⣀⣒⠀⢲⡈⣩⣆⣄⣾⣴⢎⠲⢑⣠⣃⠈⠠⣤⣒⠊⠀⠆⠀⠐⠇
⠀⠀⠀⠀⠀⠀⡂⠀⡀⢡⡄⡠⠨⣱⣏⣯⠁⠀⠀⠀⣀⣶⣿⠦⢅⠀⠄⠋⢩⢤⡯⢤⠀⢉⣀⠆⠤⢤⢲⠆⠂⢀⣴⠂
⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠙⢿⡅⢌⠀⠀⠳⠀⠀⢉⠁⡆⠈⠁⠀⠘⠉⣉⠁⠀⠀⠈⠉⠁⠀⠈⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠘⠑⡞⠒⠓⠻⠘⠛⠐⡶⠀⡟⠉⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⢖⢀⣀⡂⣟⠂⡐⠁

⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠸⡆⢸⡆⠀⠀⢠⡇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈
⠀⠀⠀⢷⠸⡇⠑⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠰⠀⠄
⠀⠀⠀⠘⠀⠇⢀⡇⠈⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⢲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠳⣀⠃
⠀⠀⠀⠀⡰⠀⠀⢠⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⡇⠈⠠⡆
⠀⠀⠀⡀⢃⠀⢰⡸⡄⢻⠇⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠲⡿⢀⣀⠄
⠀⠀⠀⠇⡞⢰⠀⠷⣧⠋⡆⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣳⡀
⠀⠀⠀⢸⠃⠸⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⡿⠘⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣧⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣤⡰⠦⠀⣙⡛⠀⠀⠀⠀⠋⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣦⣃⡀
⠂⡐⠀⠈⠀⠀⠈⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⡛⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⡈
⠀⠀⠀⠀⠀⢤⢠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡞⠅
⠀⠀⠀⠀⠀⠈⢀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇⢼
⠀⠀⠀⠀⠀⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⣾⣉⣈⣆
⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠈⠁⠀⢡⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠆⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠐⠢⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢾⣶⣤⢏⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣬⣯⣤⣿⣬
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠠⢄⣀⢀
⠀⠀⠀⠀⠀⠀⢠⡖⠦⡀⠀⠀⠀⠀⠀⢰⠏⢀⣴⢱⡿⢛⡎⣑⢒⣳⣶⣾⣦⣴⢯⡠⡄
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⢶⡀⠀⠀⠀⠰⢬⢣⡄⠈⠹⢮⠑⣮⣿⣿⣿⣿⣿⣿⣦⡷⣿⠄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣾⣇⠀⠀⠀⢠⣡⢻⠙⢉⢳⣼⣀⠸⠉⠉⣿⣿⡟⠉⠻⢧⢿⣆⡴⣠⡄
⠀⠀⠀⠀⠀⠀⠠⣳⣶⣛⣿⠃⠀⠀⣀⡏⠙⠳⣄⣹⡆⠧⠬⣉⠙⠲⠣⢽⣿⣵⣶⡛⠛⣿⠃⣸⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠶⡄⠠⠀⠃⠈⣟⠻⠉⠛⣧⣰⠂⠈⠓⠀⠀⠀⠛⡻⠭⣤⣤⣿⣝⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣓⡛⣆⠉⢽⣿⣿⣧⣼⣏⣛⡉⡉⠈⠙⡷⣾⣻⣯⣽⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⡀⢱⠀⠋⢿⡡⢩⡗⠂⠤⠄⠤⠋⢹⣿⣏⠛⢙⣿⣿⣿⢯⣠⠾⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣿⣿⣦⣶⡵⣿⡾⡫⠊⠐⠀⡤⠰⠒⠺⠛⠪⡟⡛⢿⣮⣍⣙⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⢌⣻⣵⣿⡭⠴⠂⢀⣶⠆⠀⠀⡆⠀⠀⢠⠤⠶⠯⢁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣷⡖⣦⣈⣿⠁⢀⠆⢰⣿⡔⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣆⡝⠀⠘⢁⣠⣴⡎⣩⠂⠀⠀⢢⠀⡀⠀⠀⢠⡞⣹⣋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢁⣀⣀⢹⣄⡇⣾⠿⠿⠛⢻⣀⣠⠘⣦⡰⠀⡠⣰⣿⣟⣫⠏⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠚⠻⠿⣗⠷⢦⣧⣀⢀⠀⠀⠘⢧⡅⠀⠀⠀⣿⣏⣇⠀⣼⡇⢰⠟⠋⢰⡍⢀⣰⣢⣤⣴⣾⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢶⣿⢷⣌⣄⠀⠀⠣⡀⠀⠀⡆⢿⡟⠀⣹⠠⠞⣶⣯⣾⣾⣫⠭⠥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡭⠷⠿⢿⣾⣴⣤⣈⡐⢤⡇⠘⡇⠔⠁⣠⣞⠋⣙⣻⣉⣴⣟⢂⣄⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⡤⠟⠊⠉⠉⠉⠉⠙⠻⢷⣛⣿⣛⠻⠛⠷⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣤⡀
⠀⠀⢠⡆⠀⢠⡶⠀⠀⠀⣦⡄⠀⢀⣀⠀⠀⠸⡄⡏⡄⠀⠀⠀⠿⣿⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⢷⡀⣼⠟
⠀⢀⠘⡇⡴⢋⡀⠀⣀⡀⠹⠿⣾⡿⠟⠀⢀⠳⣜⢊⡴⢀⠀⢀⣠⡛⠁⠀⠀⠀⣴⣿⣷⣦⡄⠀⠀⠀⠀⠈⠳⢁⣴
⠀⠘⣧⡈⢠⠾⠁⡄⠻⣷⣄⠀⣠⣴⡦⠀⣬⢷⡈⡡⠴⢋⡄⠘⠿⣿⣿⠀⢠⣷⣹⣯⣿⣾⣽⣷⡄⠀⠀⣈⢣⡿⠋
⠀⠀⢨⠛⠆⣠⡾⠃⠀⣨⡍⡀⠛⢋⠀⢀⡙⢇⢹⣠⠶⢟⠀⠀⢀⠀⡀⠀⣼⡛⣿⣿⣿⣿⣿⣻⣥⠀⠈⣿⢿⠁
⠀⠀⠸⣦⡀⠋⠀⠀⠀⠘⠿⣧⣾⡿⠃⠀⡳⡌⣯⣶⠞⠁⠀⢾⡿⣿⣿⠀⣻⠿⣯⣿⣿⣿⣿⢿⣿⢃⣀⠈⢿⢋⣤⡄
⠀⠀⠀⠈⠳⣠⡄⠀⠀⢿⣦⣏⣴⣶⠀⠀⠳⣘⣡⠆⠀⠀⠀⢀⣀⠀⣀⠀⢿⣋⣿⡟⢻⣏⣹⣿⣦⠈⠿⡀⡸⠚⡋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠟⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢿⠿⠿⠂⠀⠉⠉⠀⠀⠻⠿⠿⠃⠀⠀⠰⢡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⣶⣄⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣤⣤⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⢸⣧⠀⠀⡀
⠀⠀⠀⢻⣿⠆⠃⣀⡀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠈⢁⣈⠛⠋⠀⠀⠀⠺⣦⡀⢹⡿⠃⠀⠀⠙⠀⣾⠇
⠀⠀⠀⠀⠉⣥⣾⣿⠇⠀⠀⠀⠘⢿⡇⣠⣴⣶⠀⣤⡄⠸⡟⢷⡶⠀⢀⣤⣤⡬⢥⡄⣀⣀⣄⠻⣶⡤⡄⠀⣰⣿⠂
⠀⠀⣀⣤⣀⠘⠋⠀⣀⣀⠀⢀⣄⠀⡀⣿⡿⠃⠀⢙⣥⡄⠀⢰⣶⡆⠈⠛⣡⡀⠀⠱⠀⢉⡋⠀⠀⠀⠬⣀⠛⢁
⠀⠀⠻⣿⡿⠿⡇⣾⣿⡿⠂⠸⣿⣷⠸⡇⢀⣴⡆⠈⠻⠓⢤⠘⠛⡁⠀⠘⢿⣧⠀⢆⣾⡿⠃⠀⢀⣤⣀⢸⠐⠁
⠀⠀⠀⠀⠀⣀⣸⠈⠉⠀⠀⠀⠉⠻⡄⣶⣿⣿⠟⠀⠀⠐⣄⢀⠞⠀⠀⠀⠀⠉⢠⠋⠁⠀⠀⠀⠀⠙⢻⠘
⠀⠀⠀⠰⣾⣿⡿⣠⣴⣆⢸⣿⣷⡄⢸⠛⠋⠁⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠈⠁⠀⠟⠛⠁⠀⠙⠛⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢀⠏⠀⠀⠀⠀⠀⢠⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠂⠀⠀⠀⠠⢴⠇⠀⢚⠒⡀⠀⡄⢁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠂⠀⣀
⠄⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⣄⡀⡠⠀⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠄⠀⡂⠀⠀⢠⡄
⠀⡀⢁⠀⠈⠀⠀⠀⠁⠀⠀⠀⡀⡄⠈⡀⠀⠀⠀⠀⠐⠄⠠⠀⠁⡀⠀⠃⡀⠀⠀⠀⠈⠀⠀⠀⠂⢤⡏⠀⠀⠀⠅⢑⡀
⣆⠀⠠⠀⠀⢀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠞⡀⠀⢀⠄⠀⠙⣇⡠⢤⠀⠃⠀⢀⣄⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠃⠐⠠⡄⠈⠀⠀⠀⠀⢆⠃⠀⠀⠀⢴⣞⣃⣀⢠⠈⠋
⠀⠰⠄⠀⣡⣤⣶⣊⣠⡎⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠃⠀⠀⠛⢛⢻⡏
⠀⠀⠓⠛⠛⢿⣿⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⣡⣶⡆⠀⠐⠛⢧⡀
⠀⠀⠀⠀⡀⠀⠛⢝⣤⣄⡀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣾⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣶⡤⠀⠀⠀⣾⣿⣿⣷⡄⠘⠃⢰⠇
⠙⠀⠀⡀⠀⠀⠀⠈⠁⠙⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠄⠈⡆⠀⠁⠰⢄⡿⣛⡻⣆⠛⠃⠀⠀⠀⠇
⡠⡴⠛⣁⠀⠀⣤⣶⠀⠀⢀⠄⠀⠊⠀⢀⠀⠄⠀⠀⠀⠀⠉⣄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣇⣧⡼⣼⡀⠄⠤
⠀⠐⠄⢛⠁⠈⠛⠁⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠉⠀⠀⠀⠀⠀⠔⠈⣀⡀⠀⠀⠈⢹⠀⠉⠓⠀⠀⠐
⠀⠀⠀⠀⠀⠢⠀⠀⠈⠀⢀⠘⠉⠛⠉⠀⠀⠀⡌⠭⠁⠀⣀⠀⠂⠀⣀⣇⠀⠀⠀⠈⢠⡄⠉⣤⡄⠀⠀⢀⠀⠀⢢⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣶⣤⢝⡄⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⢉⣿⠶⠂⢿⢿⠟⠈⠀⠀⠈⠙⢂⠈⠁⠀⠀⠀⠀⠠⠤⠁⠤⣀⣀
⠀⠀⠀⠀⠀⡁⠉⠒⠋⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠁⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⣄⠀⡏⣱⠂⠀⣤⠀⡠⠈
⢀⢴⠄⠐⢀⠃⡀⠀⢀⠞⠀⠀⢤⣀⡄⠈⠀⠀⠀⠀⠀⢀⠀⠀⢁⠀⠐⠀⣤⠀⠀⠀⢀⠀⠀⣀⠐⡒⠪⣳⠌⠀⠠⠟⠤⠤⠔
⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠅⣠⣧⠁⠀⠀⠀⠀⠀⠀⡆⢀⢰⡆⠀⠀⠀⠀⠀⣠⣬⠁⢊⢀⡰⠀⠀⠀⠀⠐⠂⠀⠆
⠀⣀⣤⣄⠀⠀⠀⠞⠀⠀⠘⣿⠄⠨⠁⠈⠁⠀⠀⠀⠀⢀⣘⡭⢀⡔⠀⠀⢠⡾⠛⠉⠉⠀⠀⠀⠀⡠⡄⡄⠀⠀⠀⠀⡞
⠠⣿⡹⠿⠀⠀⣄⡀⠀⠄⠀⢀⡄⠀⠀⠀⠀⠀⣠⠄⠀⠀⠚⠃⠘⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⣳⡹⠄⠊⠁⠀⠠⡀⠀⠠⣀⠄
⠀⠘⠇⠀⠀⠀⠀⠃⠀⠀⠦⡌⢃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠡⠀⣇
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⢹⠀⣀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠐⠊⢉⠀⠀⠀⡀⠸
⣀⣴⣿⢿⠯⣷⠃⠰⢦⠄⠀⠀⠀⢰⣨⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠈⠀⠀⠒⠀⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠄⠩⠏⠀⠀⠀⠀⠀⠁⠀⠀⣤⡤⠞⢩⣷⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⠀⢀⠀⠐⢇⠠⢀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠸⣏⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢂⡀⠀⢀⠀⠈⠀⠀⢠⠀⠀⠀⣄
⠀⠂⣀⠀⠀⠠⠄⠀⠀⠰⠠⡦⠰⡞⢀⠉⠀⠒⠤⠀⠀⡤⠤⠀⠀⠀⠀⠀⠀⠙⡀⠀⠸⠇⠀⡜⠁⠀⠀⠀⠀⠈⢡⡤⡈⠓⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠉⠡⠌⡁⠀⣐⠂⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⢁⣀⣀⠀⠀⠀⡀⢄⠀⠈⠊⠀⠐⠂⠐⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⡱⠊⠓⠠⠀⠤⠂⠁⠀⢎⠀⡠⠔⠀⢀⠀⡀⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡀⡀⡨⠂⠢⠄⠀⠁⠀⠀⠁⠑⠄⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠘⢅⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠐⠠⢄⣁⠀⢬⡄
⠀⠀⠀⠀⠀⠀⣤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠉⠁⠐⠐⠃⠂⠤⢀⢁⡀⢀⠀⠀⠀⠀⠀⠀⢁⠄⠘⠒⡀⠀⢢⡀⠠⠤⡀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠐⠂⢁⣀⡀⠸⡀⠀⠀⠐⠀⠀⠠⠂⠀⠀⠡⠌⠀⠁⠐⠃⠀⠠
⠀⠀⠀⠀⠀⢠⠈⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠲⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠁⡅
⠀⠀⠀⠀⠀⠀⡀⠠⡈⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡄⠀⠀⠀⠀⠀⡀⠤⡾⠃⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⢄⠐⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⣐⠣⡀⢀⣀⠐⠒⠊⡘⠁⠀⠸⠄⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⠈⡢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠔⠁⠔⠍⠀⠈⠉⢀⠡⠞⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠙⠋⠒⠒⠲⠖⠀⣀⣀⣀⣀⡒⠈⠅⠀⠒⠈⡠⠴⠁⠀⠦⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠂⠀⠤⠤⠤⠤⠤⠀⠐⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⡀⠀⠀⣺⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠛⠿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢸⣿⣿⡇⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠙⠉⠿⣿⢹⣿⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⣖⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⠀⣿⣿⣿⣿⣿⢸⣿⣿⡇⠸⣿⠁⢿⡿⠿⣿⠸⢹⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠛⠉⣉⣹
⢠⣶⣶⣶⣶⡀⣸⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⢸⣿⣿⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠴⠚⣩⣿⠿
⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠟⠁⠉⠉⠙⠙⠿⠀⠟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⣤⠶⠛⠉⢀⣤
⠘⣿⣿⣿⣿⠟⠙⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡶⢾⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⢚⣉⣿
⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠠⣄⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣹⣯⢛⣿⣶⡀⠀⠀⠀⠀⠀⠛⢠⡤⠶⠾⠿⠛⠙⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠶⠀⠀⠀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⣶⣷⡞⣿⣿⣿⡿⣤⡇⠀⠀⠀⢴⠖⠞⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠐⠈⢥⡤⠄⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠿⣾⡉⢀⣨⡟⣏⢿⣛⣶⢹⣿⠀⣠⠄⠀⠰⢶⣶⣤⠄⠀⣀⣀⡀⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠐⠀⠀⣰⡿⠀⠾⠛⢡⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡾⡟⢿⣿⣷⣇⠂⣬⡔⣟⡻⣄⢠⣤⣦⣤⣤⡤⠆⠀⠘⠛⠉⠀⣤
⠀⠀⠀⠀⠀⠀⣾⣦⡀⠠⠀⢀⠚⣁⣁⣀⢂⠛⠆⢁⠋⢐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⣶⣿⡯⣿⣆⢠⣿⣛⣥⣞⣛⡀⠈⠙⣿⠿⢇⣀⣀⣠⣤⣤⣤⠶
⠀⠀⠀⠀⠀⠀⣿⡟⢿⣮⣥⣷⣶⠝⠿⠁⢈⣇⣠⠆⡴⠛⢋⡀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠇⣷⣁⠲⢀⣿⡽⠜⢻⠏⠀⣈⣭⣅⠀⠒⠚⠛⢹⣿⣋⣥⣯⠅⠀⡀
⠀⠀⠀⠀⠀⠀⠋⢿⡀⠉⠉⠀⠈⠉⠀⠐⢁⡟⣇⣠⠴⠟⢃⡇⠀⠀⠀⠀⠀⣀⡠⣿⣏⠀⢹⣧⠴⢦⡝⢷⡿⣿⡳⠟⠉⠉⠛⣀⠀⠀⠀⠙⠛⠋⠡⠄⠀⠛⠉
⠀⠀⠀⠀⠀⠀⢀⡈⢁⣠⡄⠁⠀⠀⠄⣤⣾⣗⠭⠔⣂⡐⠊⠅⠀⠀⠀⣠⣴⣿⡅⣿⣿⣧⣿⡿⡤⡄⢠⣿⠤⠾⣿⣦⠀⠀⠀⢁⣀⣐⣺⡭⣥⣴⣶⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠙⢶⡿⠀⠀⠀⠀⠀⢡⣿⣷⡞⠉⣻⡿⣷⡀⡀⢰⣣⠟⠁⠘⠃⠙⠟⣛⣯⢠⣇⢀⣿⠃⠀⠀⠈⣿⠀⠀⡆⠉⠉⢩⠶⢾⣿⣿⣿⣿⡛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣠⡄⣨⡞⠛⠋⣀⡀⢈⣷⣿⡿⡿⠾⡟⠂⠘⢃⡅⣰⣿⣿⣿⣿⠻⣿⡇⠀⠀⠀⢰⣿⣿⡿⢁⡤⣴⣶⣿⣛⣻⣿⠿⢿⣿⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠟⠉⠀⠘⠋⠙⢛⣁⣼⣿⣿⡟⠙⠀⠀⢀⣼⣏⠏⠉⣽⣿⣿⣿⠀⠙⠁⠀⠀⢀⣿⣻⣿⡇⢀⣷⣴⢿⣿⡿⠋⢀⣴⣾⡿⠛⠿
⠀⠀⠀⠀⠀⢄⠀⠀⠀⢠⢟⣥⡶⣾⠿⠟⢿⣷⢦⣍⠁⠹⣿⡁⠀⠀⠀⠀⠹⣯⡆⢰⣿⣿⢿⣿⡀⠀⠀⠀⠀⠀⠀⢩⣿⠋⢉⣿⣷⣿⣟⠀⡘⢫⣿⣿⣇⡀⣀
⢂⣤⣶⣶⣶⣶⣦⣾⣣⣾⡾⠃⠀⠁⠀⠀⠈⠁⠀⣿⣄⣦⡿⠿⠶⢤⡑⡀⠀⠐⠃⣀⣤⡀⣸⣿⣧⠀⠀⠀⠀⠀⠀⢺⣿⠗⣀⣉⣼⡿⢿⣷⠋⣠⣿⣿⣿⣅⠿
⠟⢋⣠⣾⣿⣿⣿⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⢠⠞⣻⣿⠿⣧⡀⠀⠈⠙⢋⢤⠀⠀⣿⣏⣿⣷⣿⣿⣧⠀⠀⠀⠀⠀⠾⠿⠟⠛⠛⢻⠆⠈⠁⠈⠉⠁⠈⣿⣿⡟
⣾⠿⣻⣿⣿⣿⡿⢃⣿⡷⠀⠀⠀⠀⠀⠀⠐⢁⣴⠟⢹⣿⣏⠀⠀⠀⠀⡼⠈⣦⡆⣿⣿⠟⠋⡍⣙⡿⢷⠦⠀⢀⡄⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⣴⠿⠛⠋⠸⣿⠆⣴⡿⠁⠀⠀⢀⠀⢀⠄⠀⠀⣿⢳⣯⣸⡇⠀⠀⢀⣿⠁⢰⠻⡇⣿⠿⣶⡄⠀⢹⡇⢸⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣽⡝⡆⠀⠀⠀⢀⠠⠄⢤⡀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀
⠀⠀⠀⣠⠰⣈⣏⡖⢂⡀⠀⠛⠿⠿⠁⠀⠀⠠⠍⠀⠀⠀⢘⡆⠀⠀⠀⠀⣏⠀⠀⠀⣿
⠀⠀⢐⠮⡿⠉⠈⠹⡧⠣⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⠲⠤⠖⠃
⠀⠀⠐⣊⠻⣤⣠⣼⢏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠈⠑⠣⠧⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⣾⠛⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠈⣿⣄⣀⣼⠟
⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣾⡟⠉⠙⣿⡄⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⢿⣷⣤⣴⣿⠃⠀⠀⠀⣴⢯⣽⣾⣭⢷⡄⠀⠀⣤⣿⠟⠛⢿⣧⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢿⡸⣿⣿⡿⣹⠇⠀⠀⢾⣿⣄⣀⣸⣿⡆⠀⠀⠀⢀⣤⢶⣤
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠾⠞⠋⠀⠀⠀⠀⠻⠻⡿⠛⠃⠀⠀⠀⠀⠸⣧⣀⣿⠃
⠀⠀⠀⡶⠏⠋⠹⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠘⣇⠀⠀⢀⣿⠀⠀⠀⣠⡶⢦⡀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠉⠳⠗⠚⠀⠀⠀⠀⠿⣤⣼⠇⠀⠀⠀⢠⣿⡟⠁⢹⣿⡇⠀⠀⠀⠀⢀⡴⣶⣶⢦⡄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⡿⠃⠀⠀⠀⠀⣿⣽⠉⠉⢳⢹⡄
⠀⠀⠀⠀⣿⣿⠆⠀⢀⣠⣠⣄⡀⠀⠀⠀⣤⣦⣤⡀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠻⣝⡶⢶⣫⡾
⠀⠀⠀⠀⠀⠁⠀⠀⣾⡏⠉⢹⣖⠀⠀⠘⣿⣿⣿⠇⠀⠀⢠⡿⠿⣶⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⢶⡾⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⢷⣶⠟

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠄⣂⣳⣶⣲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠅⠑⢻⣽⣷⢿⣔⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠼⠛⠉⢿⠟⠋⠉⠹⠿⢿⡿⠟⠉⠛⠚⠛⠚⠛⠛⠋⠀⠙⠛⠷⣯⣑⡠⢄
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣗⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⣦⣄⣀⣀⣀⣀⣤⠆⠀⠀⡈⣿⣿⣾⣿⡄⠀⠀⠀⠀⠀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠈⠉⠙⢻⣶⣾⣿⣿⣿⣿⡟⠛⠛⠀⠈⢿⣿⣶⣶⣾⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⠞⠁⠀⠀⠀⠀⢸⣿⣿⣿⡿⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⠀⢠⠀⠀⠀⠀⠀⢸⣿⣿⣿⢆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣻⢻⠖⠊⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⢀⡴⠒⠒⠀⠻⣿⣿⡆⠀⠀⠀⠆
⠀⠀⠀⡀⠀⠀⠀⠀⣹⣿⡟⠋⢨⡿⠿⣿⡄⠀⠀⠀⠀⠘⣿⡘⠀⠀⠀⢰⡀
⠀⣰⣾⠃⠀⠀⠀⣴⣿⡗⠀⠀⢸⠀⠀⠉⣿⣿⢶⣇⣀⣠⣿⣃⣀⡀⠀⠀⠳⡀
⢰⣿⡇⠀⢀⣤⣴⣿⣿⠀⠀⠀⠀⣆⢀⣾⡿⠁⠀⠉⣿⣿⣿⠁⠉⠻⠆⠀⠀⠐⠀⠀⠀⠀⠆⡔
⠀⠿⣧⠼⣿⡇⠀⠙⢿⣷⣶⣤⣤⣿⣼⣯⣀⣀⡀⣤⣼⣿⣧⡐⠉⠙⡆⠀⠠⠀⢀⡀⠀⡜⢰⡅⡠⠆
⠀⠀⠀⠀⠹⣷⣤⣶⡿⢿⠟⢺⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣆⣤⣀⣶⣶⣿⣿⣷⡷⢴⡾⣣⡿⣷⣷⣦⡴⠂
⠀⠀⠀⠀⠈⠉⠙⠓⠒⠀⠀⠘⠉⠉⠉⠀⠈⠉⠈⠁⠉⠀⠀⠉⠛⠻⠋⠛⠛⠙⠟⠻⠻⠋⠁⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠢⠤⠤⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⢣⠀⠀⠀⣀⡠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠙⢦⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣇⠔⠊⠁⠀⠀⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷
⠀⠀⠀⠀⢠⠛⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⡀⣆⢩⣸⣖⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠇
⠀⠀⠀⢠⠃⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢈⣻⣿⡿⣿⣾⡏⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⡄
⠀⠀⠀⠸⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⣿⣿⢿⣿⢻⡷⢾⡧⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⡆
⠀⠀⠀⠀⠙⢄⠑⢦⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠻⡏⡁⠛⣛⣋⠐⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⢀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠑⠤⣈⣒⣤⠤⠒⠚⠉⠁⠀⠀⠀⠀⠁⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⣠⠇⠙⢄⡀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠤⠤⠴⠚⠁⠀⠀⠀⠉⠉⠉⠉

⠀⢹⡀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠐⠒⣀⡀⠀⠀⡐⠀⢸⡃⠀⠗⠀⠀⣱⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠸⠀⠀⢰⠂⠸⠤⡀⠉⠩⠚⠁⠐⠲⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢺⠸
⠀⢸⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠧⠈⢀⠀⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠁⠀⠀⠀⠀⠀⢣
⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡈⠀⠀⠀⠀⣼
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣀⡀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡀⣠⣧⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢲⣺⠀⠀⠃⠀⣠
⠁⠈⠹⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠄⠛⠚⠛⠉⠀⠀⠉⢆⠀⠀⠀⠀⢀⡠⠔⠁⠀⢀⠠⠋⠀⢠⠖⠘⠓
⢠⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠔⠊⠁⠀⠀⠀⠀⣱⠁⠀⠀⡇
⡉⠀⠀⠀⡟⠀⠀⢄⠠⠀⢀⢀⠀⠄⠈⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠄⡙⠀⢘⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢱⠠⡀
⣀⣀⣜⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣀⣀⡠⠬⠔⠊⠀⠀⠈⢆⡃⠀⢀⣀⠴⠁⠀⠀⠀⠀⠀⠳⣵⣄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠐⠒⠂⠠⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀
⠀⠀⠀⠀⠈⡆⠀⠈⠌⠂⡲⠄⠠⠔⢀⠀⠀⠀⠀⠐⠴⠋⠀⠀⠈⢳⡀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠐⠒⠋⠑⠒⠃⡀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠈⠀⠀⡇⠀⠀⡠⠊⢀⡃⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠐⠁⠀⢀⠃
⠀⠀⠀⠀⡠⠊⠀⢀⢀⡀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠈⢠⠄⡄⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠂⠐⠐⠈⠀⠀⠀⠈⠿⢏⠀⠜⠓⠀⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠴⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀
⠆⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠁⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠂⠀⠀⠀⠀⠀⡔⠈⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣄⣹⣥⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢢⠀⠀⠀⠀⡌⠁
⠀⠀⠀⣠⠁⠀⠀⠈⣀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠘⡨⠼⣫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠱
⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠻⠆⠄⠐⣀⠞⠀⠀⠀⢇⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠱⡀
⠐⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠀⠀⠀⠀⠀⠉⠒⠛⠒⠚⠁
⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡡⠄⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⣀⣁⢱⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⢀⣾⣿⣿⠟⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢿⣏⣠⡤⠶⢿⣿⣟⣒⠒⠶⣶⣿⣿⣿⣿⡿⠷⠂⠀⠀⠀⠀⠀⠀⠄⠈⠙⠺⠧⠀⠀⠈
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡼⣿⣥⣤⡀⠈⠻⣿⣿⣷⡀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⢐⠀⠀⠀⢹⡀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠘⢿⣆⢀⡎⠀⠈⠻⣿⣿⡄⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠵⢀⠀⠀⠀⠀⠂
⠀⠀⠀⢶⣤⣄⠀⠙⡿⠀⠀⣀⣤⣤⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠬⠥⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢻⣿⣇⠀⢡⢠⡼⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠐⢛⡁⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠉⠉⠙⠻⡏⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠀⠀⠀⠘⠡⠂
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡘⢄⠀⢀⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⡀⢢⠸⡅⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⣻⠀⠀⠀⠀⠀⠀⠐⠪⠅⠳⡗⡨⠊⡁⠀⠀⠀⠀⠀⠹⣦⡄⠸⡇⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣦⠀⠀⡼⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢅⠀⠶⡗⢐⠄⠄⠀⠀⠀⠀⠀⠀⠘⢿⡄⣇⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡆⢀⣇⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠑⢤⡷⠥⡐⢥⠀⠀⠀⠀⠀⢀⠀⠀⠻⣿⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡇⡼⠋⢀⣀⣀⣀⡀⠀⠀⠀⠀⢈⢢⡀⢐⣠⡧⠊⠀⢐⡀⠀⠀⠀⠀⠘⣿⡆⠀⢹⡇⠀⣠⣴
⠀⠀⠀⠀⠀⠀⡾⢆⠀⠀⠹⣾⣞⡥⠚⠉⠀⣠⠜⠁⠀⠀⠀⠀⠈⠂⠀⠠⣄⣧⣐⠈⠄⠊⠀⠀⠀⠀⠀⠈⠻⣄⢸⡇⢠⡿⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠳⡄⢠⠟⠛⠓⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⣬⣧⠐⠀⠈⠩⠀⠀⠀⠀⠀⡀⠀⠘⢾⣇⡞⠁
⠀⠀⠀⠀⠀⠀⢧⡀⠀⣷⠏⣀⡠⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠐⠠⣀⡇⠄⠀⠀⠰⠀⠀⠀⠀⠀⣿⣦⠀⢸⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣿⣚⣁⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠊⠉⠠⣁⠈⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⣦⢸⠃⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡐⠙⡟⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣾⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⡟⠃⠠⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀
⠀⠀⠀⠀⠀⢀⡠⢔⠂⠀⠄⠀⠀⠀⠂⠀⠀⠉⠢⠑⡀⠀⠀⠀⣠⡤⠀⠀⠀⠤⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠄⢁⠪⢀⠀⠠⡀⠉⠑⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⡩⣫⣵⣤⣀⣠⣄⡀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠐⠁⠐⠁⠀⠀⠀⠸⠀⠀⠔⠃⢆⠀⠀⢂⠀⢃⠀⠀⢈⢨⠀⠁⠊⢡⢄⣐⠟⠋⠀⠀⠀⡤⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠘⠀⠙⢄⠂⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠘⢃⠀⡀⠀⠁⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠠⠤⠀⠀⠀⡠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠆⠀⠀⠀⠘⣿⣇⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⢠⢀⡄⠀⡀⠁⠢⢐⠀⠀⠀⠀⠀⠀⠒⠒⠰⠦⠤⠤⣄⡀⠀⠀⠓⠀⠀⠀⠄
⠀⢠⢣⠀⠀⠀⠀⠀⠈⢁⠤⢾⡧⣄⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠉⠀⠀⠀⠀⠀⠀⠀⡐⠐⠏⠃⠀⢹⠁⢰⠋⢢⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⣀⣀⠠⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⣄⡀⠀⢸⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⢰⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢣⡀⠀⡄⠀⠀⠀⠀⠀⠀⢰⣂⣀⠀⢰⠀⢠⡀⡘⠀⠀⠀⠀⢀⠀⠤⠄⠄⠀⠀⠀⠈⠀⠀⠀⠀⢈
⠀⠀⢀⠔⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⡉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠮⠁⠀⡐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠸⡀
⠀⠀⢀⠀⠀⠂⢄⢐⣋⠀⠘⠢⡀⠀⠀⢠⡤⠂⠀⠐⡄⠘⠒⡄⠀⠀⢸⠉⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⢸⡄⠀⠀⠂⠋⣨⢇⣂⡀⠀⠢⡀⢤⠀⠀⣴⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢠⠑⠂⠀⠤⠴⠇⠀⠀⠈⠐⠄⠈⢀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⢸⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⣀⣀⠃
⠀⠀⠀⠑⢄⠒⠐⠚⡇⠀⠀⣀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠂⠢⠄⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠈⠓⠤⠀⢀⠺⡤⡤⠟⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⡓⠀⠀⢀⠀⠒⠉⠉⠉⠛⠃⢦⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⡀⠀⠀⠘⠛⠁⠀⣠⡦⠀⠀⠀⠀⢀⠉⠀⠀⠙⣇
⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⢠⣿⣾⡀⢀⠀⠀⠀⠀⠠⢀⠄⣿
⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠊⠁⠀⠈⣷⠊⢡⡄⠁⠀⠀⠀⠀⠄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠲⠟⠀⠠⠈⠃⠀⠀⠀⠀⠀⠙⢉⣄⣐⠂⠀⠀⠀⠈⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠐⠇
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠀⡴⠺⠓⠆
⠀⠠⡇⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠺⣀⡁
⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢱⣶⣶⡾⢿⠷⢶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠐⠀⠐
⣆⠠⠤⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠋⢀⣄⣀⠀⠈⠛⠁⠀⠀⠐
⣯⠙⠲⡄⡈⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⠉⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔
⠙⡤⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠘⢿⡿⡇⠀⢀⠀⢀⢠⠂⠀⠀⠀⠘⠀⣠⠀⠀⠀⠂⠀⠈⡚⠀⠀⡄
⠄⠗⠓⡄⢀⡄⠈⡗⢠⡌⠂⠀⠀⠀⠀⠀⠀⢿⠁⢂⠏⢠⠀⠈⠀⣠⠄⢃⠄⠳⠟⠂⠀⠀⠀⠑⣌⠱⠚⠂⢀
⠀⣰⠲⡄⠀⡅⢀⡟⢾⣇⠀⠀⠀⠀⠀⠈⠒⠠⠃⠤⠆⢰⠂⣤⡰⠇⢸⠀⡄⣸⡞⠀⡀⠀⠀⠀⠀⠙⠖⠀⠈
⠀⠁⠀⢱⢀⣤⢨⡇⢸⢻⣎⠁⠀⠀⡄⣼⠆⡷⠀⠀⢌⢙⠂⠃⢍⡁⠻⣤⠳⠝⠃⠀⠑⣤⢀⠀⢀⠀⡀⠁⠆
⠔⠄⠈⠒⠋⢗⣠⡇⢂⠉⣿⠳⠀⠄⠀⡏⠀⠈⠳⣔⡀⢈⣀⡀⠀⠀⢾⣾⡧⢤⠀⠐⣤⠒⡖⣆⡓⠀⠈
⠀⢾⠾⠔⣶⣟⢙⡟⠋⡐⠿⠀⠇⢐⢟⡟⢠⣄⡒⠊⠭⣹⠀⠀⡴⢀⠈⣽⣦⡔⡀⢠⣤⢰⡿⣄⡁⡀⠀⠘⢣
⣥⢰⣿⡦⡰⠿⢾⣷⠄⠀⢣⠐⣌⣩⣿⣤⣜⣀⠀⠀⣀⡽⢒⣟⢠⣽⠈⡄⠤⢼⠃⠆⠀⠀⠀⠀⢳⠏⣔⣿⣠
⡤⢺⢿⣷⠟⠻⡖⠃⠀⢀⡈⢸⣫⣿⣿⣿⡿⠉⢄⡀⠹⡳⡷⠆⠻⣿⣿⠗⠂⡬⠜⢓⠠⢀⣯⣾⣁⣴⣣⣄⠈
⠁⢼⣗⣤⣄⢀⣥⣺⣶⡦⣦⣻⣉⢡⣶⠚⣗⣀⡴⠀⣤⣟⣆⡁⡘⠻⡞⠂⠀⢃⠰⢺⠐⠿⡝⣜⣧⢿⢷⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣀⢀⠠⠤⣤⣈⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠈⠉⢀
⠀⠀⠀⠀⠀⠀⡐⢱⡟⠻⣄⠁⠀⠀⠋⠀⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠆⢀⡔⠋⠁⣀⠀⡀⠀⠈⢱⡄⠀⠀⠀⢰⠀⡀⠀⠁⠆⠀⠀⠀⠀⠀⠘⢷⢸⣧
⠀⠀⠀⠀⠀⢷⡇⠚⠻⠶⡥⠀⠀⠀⠀⢀⠋⠛⢉⣹⠆⠀⠀⠀⠀⢸⠀⠎⠀⠀⣴⡿⢿⡇⠀⠀⢀⡷⠀⠀⠀⢸⠀⡅⢀⠁⡆⠀⠀⠀⠀⠀⠀⢘⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠊⢀⣤⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠔⠻⡄⠀⡷⢀⣠⠾⠃⠀⠀⠀⠻⣦⠿⠶⠞⠃⠀⠀⠀⠀⢀⠴⡟⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣿⣢⠀⢰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢆⡳⣤⣀⡤⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠁⢀⣧⢹⠂
⠀⠀⠀⠀⠀⠀⣀⡴⠲⠀⠀⠀⢫⢀⡆⠀⠀⣤⣤⣤⣤⣤⣰⠀⠀⠀⠀⠀⠀⠀⠠⢃⣀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢤⡴⣿⠛⠀⠀⠀⡆⠀⣿⠃⠀⠀⠋⠈⠋⠟⠛⠽⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠢⠌⠑⢦⡀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⣷⣼⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⡿⠀⣄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂⠀⠀⠀⠀⠀⠀⠀⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡼⢿⣷⠀⠀⠀⠀⠀⠀⢀⡷⠋⠁⣤⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢠⡽⢟⡄⠀⠀⠀⠀⣠⣀⡴⠶⣶⣟⠟⠀⣦⠀⠀⠀⠀⠀⠀⠠⠇⢠⣤
⠀⠀⠀⠀⠀⠀⡠⢚⣯⡬⢺⠹⠼⠻⠀⠀⠀⠀⡉⣿⠂⢰⣿⣿⣾⡿⠛⠀⠀⠀⠀⠀⠀⢠⠀⠀⠛⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢀⡔⢁⣾⠋⣇⠀⢷⡇⠁⣆⠀⠀⠀⠷⠻⣦⣿⠋⣿⢽⣧⠀⠀⠀⠀⠀⣀⣴⠻⠗⠀⢼⣤⣄⠀⠀⠀⢸⣆⣀⣤⣤⣄⣀⣀⡀⣀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⡞⡐⢮⠀⣦⠿⣳⡎⡿⣇⣍⠀⠀⠀⠀⠙⢹⡟⢰⣧⣿⣿⡆⠀⠀⢸⣧⡈⢿⣶⣷⣏⣻⢟⡲⢷⠀⠀⠀⠙⠛⠻⠿⠛⠛⠛⠛⢻⣶⣦⣶⡾
⠀⠀⠀⢰⡏⠒⠋⠶⣿⣶⣩⠏⡠⢣⠀⠀⠀⠀⠀⠀⢸⠀⡏⠉⠀⠀⠀⠀⠀⠈⠻⠿⠟⢻⣿⣿⡿⢿⣷⡞
⠀⠀⠀⠀⠈⠛⠙⢻⣿⣿⣿⡋⠛⠚⠁⠀⠀⠀⠀⠀⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠟⣿⡇⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠛⠾⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠶⠶⠶⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠃⠀⠀⠀⠘⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄

⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠔⠐⠂⠢⡀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⡄⠀⣀⣀⠀⠈⠑⠀⠀⠐⠈⡆⢀⣠⣀⡀⠀⠁⠠⠀⢠⣤⠈⠀⠀⢀⣤⡀⠸⣏
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠠⡀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡐⠈⠁⠀⠁⢀⠀⠀⠀⠀⠀⠔⠉⠁⠉⠑⠄⠀⠀⠀⠀⠀⣤⠊⠈⠉⠑⠄⠀⠀⠀⠀⠀⠠⠊⠁⠀⠉⠢⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢈⠀⠀⠶⠆⠸⠀⠀⠀⠀⠀⠰⠀⠠⠶⠄⠀⠀⠁⠀⠀⠀⢈⠀⠀⠶⠆⠀⠀⠠⠀⠀⠀⠀⡅⠀⠀⠶⠒⠀⢹⡇
⠀⠀⠀⠀⠀⠑⢄⡀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠡⢀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⡔⠈⠀⠀⠑⠄⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⠁⢨⠈⠑⠄⠀⠀⠀⠀⠀⢀⡶⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠰⠀⠠⠶⠄⠀⡁⢀⠀⠀⠀⢈⠀⠀⠶⠖⠀⢀⠀⠅⠀⠺⠇⠀⠀⠀⠐⠒⠂⠘⣇⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠡⢀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠱⠄⡀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⡐⠈⠀⠀⠑⠄⠀⠀⠀⠀⠀⡰⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠀⠑⠄⠀⠀⠀⠀⠀⢀⡶⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⠀⠠⠦⠤⠀⡁⢀⠀⠀⠀⠀⠀⠀⠶⠦⠀⢈⠀⠀⢀⠺⠇⠈⠀⠀⠰⠒⠂⠘⣇⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠡⢀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠱⢄⡀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠢⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠟

⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣼⣿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢤⣀⠀⠀⣿⡟⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠈⠑⢞⡶⢦⣄⠉⠁⠀⠘⠁⠀⠀⣠⠀⠐⣿⡇⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⠷⠄⠀⠀⠀⠀⠀⠟⠀⠀⠹⡇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠹⠆⠀⠠⡀⢀⡇⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠶⠖⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠁⠄⠀⠙⢿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠛⠛⠻⠷⠶⠤⢀⠀⠂⠀⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣠⠒⣀⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢦⣴⣠⠄
⠀⠀⠀⠀⠀⣠⢄⡸⠀⡞⠀⠀⠀⠀⠹⣶⢸⣿⠁⠀⢠⠾⠉⢓⡔⠃⠀⠀⠠⠀⠀⠀⢠⣾⡻⢟⡾⡱⠏⠀⠟⣢⠀⢀⡴⣫⠀⠂⠀⡬⢋
⠀⠀⠀⠀⠀⢣⠀⢷⣤⠖⢰⠀⠀⠀⠀⣇⣰⡶⠃⠀⢻⠐⠡⠀⠀⠠⠀⠀⢀⠇⠀⡔⠋⠂⠁⠈⠀⠀⠀⠀⡸⠁⠀⣸⠃⢀⡀⠐⠂⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠳⢾⣿⠒⠁⠀⠀⢀⣾⡇⠀⣱⠀⠀⠈⠣⢀⠈⠀⠀⢀⣢⠎⠀⠀⠁⠉⠂⠄⠀⣀⣀⡠⠞⠁⠀⠀⢻⠋⠈⢁⡄⠒⢀⠜
⠀⠀⠀⠀⢀⡴⠒⠚⠓⠒⠒⢄⠀⢸⣏⠀⠀⡿⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⣄⠀⠀⠀⠀⠘⢘⣋⣈⡴⠛⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡐⠀⢸⣴⠇⣼⠃⠀⠀⠀⠀⣠⠄⡶⣠⣫⠦⠤⠀⠀⠀⢠⣿⣯⠻⠾⠟⠋⣧⠀⠀⠀⠀⠀⠉⣁
⠀⠀⠀⠀⠈⠓⠂⢤⣀⠔⠈⠀⠀⢸⡟⢠⠃⠀⠀⠀⠀⣜⡕⡠⠁⠈⠡⢜⠀⠈⡆⠀⢹⠃⠾⡷⢘⡿⠫⡇⠀⠀⠀⠀⣀⠤⣿⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠯⠃⠀⠀⠀⠀⢰⡿⢠⡖⣣⠀⠀⣸⠇⢰⠇⠀⠀⠁⠀⠠⠤⠀⠀⠇⠀⠀⡴⣻⠀⠀⠻⠄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⡀⢤⡠⢀⠀⠀⢸⡇⢸⣷⡇⠀⢠⡟⢀⡜⠀⠀⠀⠀⡀⠤⠂⠉⢲⣄⠀⢸⡫⣿⢣⢆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠟⢋⣴⠏⠐⣍⣽⡛⠛⡞⣁⠄⠘⢧⣿⡻⣧⣴⣿⣧⠞⠀⠀⠀⡴⢋⡤⣂⣦⠀⢸⢇⠀⠘⣿⠡⠢⠂⠉⠂⠀⠀⢠⠟
⠀⠀⠀⠀⠀⡠⢾⠂⣍⠉⠁⠀⠀⠳⡼⢿⣀⣸⣴⠀⠀⠀⠀⠙⠛⠘⠛⠁⠀⠀⠀⢸⡁⠸⣿⣇⠟⠀⡿⡸⠀⠀⠈⠑⠦⢤⣄⣤⠞⠖⠉
⠀⠀⠀⠀⣼⠁⠈⠀⢸⡀⠀⠀⠀⠀⣗⣻⡟⡾⠀⠀⠀⢀⣀⣤⡆⠀⠀⠀⠀⢀⣤⠘⡧⠀⢀⣀⣠⠞⡑⠁⠀⢀⣀⣤⣤⣌⡉⠁
⠀⠀⠀⢀⣿⠆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⡝⣋⣀⣀⠀⣀⣴⠏⡟⠀⠈⠁⠀⠀⠀⠁⠀⢠⣞⢁⠀⠀⠀⠀⠈⡆
⠀⠀⢀⣾⣿⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⣀⣀⣙⣷⣾⠿⠟⠛⠋⢁⢰⡿⠋⡸⠁⠀⠀⠀⠀⠀⢀⡀⠀⢻⡿⠂⡀⣀⣴⠊⠁⠂
⠀⠀⢸⢟⠟⣰⠀⠀⠀⠀⢸⠀⠀⢀⣶⠉⣸⠁⠀⠀⠀⠑⠄⠀⠀⢙⢻⡁⠈⠀⠀⠀⠀⠀⠀⣀⢸⡇⠠⡀⠙⠒⠀⣿⠰⣝⠔⣀⠀⣀⠀⢸
⠀⠀⠘⢯⡐⣥⠏⡆⣀⣠⠏⠀⢀⡏⢪⠀⡗⢨⠄⠀⠀⠀⢸⠀⠀⠘⠀⠉⠢⣀⠀⢀⠀⠀⣰⡇⣸⡇⠀⠱⠀⠀⠀⠈⠂⠨⠧⣀⣠⣡⠴⠋
⠀⠀⠀⠀⠉⠛⣾⡞⠋⠀⠀⠀⠘⡇⠁⢄⣧⠀⡤⠀⠀⠀⡼⠀⠀⢴⠠⣄⢼⢉⣁⣀⠇⠀⣿⡅⢿⣩⠀⠀⡇⠀⢀⡀⠀⠀⣤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⣹⣌⠐⢀⣠⠞⠁⠀⡇⣬⠀⡼⠈⠛⠚⠋⠀⠀⠸⡖⢸⣏⠆⣠⠁⠀⠀⢳⣯⠘⣯⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⠀⠀⠀⠀⠀⠉⠁⠉⣉⡁⠀⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠞⠁⠀⠀⠲⡸⢛⣛⡻⠋⡷
⠀⠀⠀⠀⠀⣴⣏⣿⠏⢻⣇⠀⠀⠀⠀⡠⠒⢸⠏⠉⢀⡶⡉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠙⢢⣠⠅⠠⡀
⠀⠀⠀⠀⡔⢻⣹⠟⢳⣸⢾⡀⠀⢀⡾⣡⠀⠈⢹⢀⡼⢁⠵⠄⠈⠋⡆⠀⣠⣾⣏⠷⠚⠁⢰⣆⠀⠀⠀⠀⢀⣾⠗⢱⣂⠾⠬⡁⣋⠸⡀
⠀⠀⠀⠘⣧⣧⣿⢄⢸⠁⠀⡇⠀⢸⢡⡟⣀⠀⢻⠋⠀⠀⢡⠀⢸⠀⡵⠀⢿⣡⠀⠀⠀⠀⠀⠒⡘⠀⠀⠀⣼⣿⣡⣼⡣⢀⣠⠍⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠿⡴⠞⠀⠴⠁⠀⠸⡏⢿⡏⠀⡞⠘⠀⠆⠈⢀⡏⢠⠇⠀⢿⠁⠁⠀⠰⢆⠀⠀⡀⠠⠀⠀⣯⠑⣶⡏⢌⢦⡄⢠⢀⠤⡇
⠀⠀⠀⠀⠀⢀⠀⠿⠀⠀⠀⠀⠀⠀⠳⣌⢷⣸⢧⢀⡼⢀⠴⢋⡴⠃⠀⠀⠀⠁⠈⢱⠘⡣⠉⠀⠁⠀⠀⠀⠹⡶⢣⡮⢯⣹⣶⣇⢎⡼
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠏⠁⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠪⠷⣞⠿⢽⣿⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⠠⡄⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠅⣸⡇⠀⠀⣠⠏⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⣿⠀⠀⠀⠀⠀⠀⡘⠀⠀⢁⡼⣽⠀⠀⣰⠋⠀⠀⢀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⡹⡀⠀⢴⡀⠀⠀⠀⠀⡱⢋⢁⡇⠀⣸⢃⡀⢀⣴⠏⢰⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢝⢆⣿⠨⣄⠘⢣⡀⡄⠀⠘⠀⠀⣸⠁⡼⢃⠐⣴⠟⠁⠀⢸⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣥⢄⠈⠳⡌⢶⠆⠀⠁⢰⣠⠇⡼⠁⣶⠞⠉⣆⠀⠀⠈⠀⠀⠙⠶⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠾⡇⠀⢷⣄⠉⢿⠀⠀⠀⡏⡿⣻⣅⡟⠁⠀⠀⠹⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⢄⡀⣀⡤⠖⠋⠁⠀⠀⣧⠀⠀⠙⢷⣄⡀⠀⠀⡰⣾⡯⠋⠀⠀⠀⠀⠀⠹⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⠫⢂⣠⠼⠂⠀⠀⠀⠟⠀⠀⢀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢰⠉⠀⠀⠀⠀⠀⢠⣌⡄⢠⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⠦⣄⡀
⠀⠀⠀⠀⣀⣠⠴⠖⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣧⣼⡯⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉⠓⠦⢄⣀
⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠤⠔⠊⠀⢀⣿⡟⡇⠀⠉⠾⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠙⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠶⠂⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠃⠀⢸⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠉⢻⣿⣧⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣴⣾⣿⠿⠿⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⠿⠿⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠋⠁⠀⠀⠀⠀⠈⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠀⣼⣿⣿⠇⠀⠀⠀⣀⣤⣶⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣦⣤⣀⣀⠀⢻⣿⠿⠓⠛⠿⣿⣠⣤⣶⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⢻⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⡇⠀⠀⠀⠀⠀⠀⢨⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⡀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠈⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠟⠋⠉⣿⣷⣶⠶⣾⣿⣷⠀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣶⣶⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢰⣿⣿⡏⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣷⣶⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣾⣿⣿⠿⠋⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⢀⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⡤⢤⠍⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠉⢈⡦⡀⠀⠀⠀⠀⠀⠂⠀⠀⠰⢀⣄⣣⡆⠐
⠀⠀⠀⠀⠀⠀⣦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣻⣧⡜
⠀⠀⠀⠀⠀⠈⢳⡊⢹⡆⠀⠀⠀⢀⡤⠀⠠⣤⠤⠤⡳⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠋
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠳⣄⠔⡨⠉⠀⠀⠀⠀⠈⣆⠀⠃⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠈⠃⠀⣀⡀⢀⡀⠀⣀⠸⠻⠀⠀⠀⠀⠀⠀⠘⠀⠀⡍⠀⠀⠀⠀⠀⠉⡱⠤⠤⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠣⠤⠏⠀⡭⠍⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡀⠀⠀⠀⠀⢀⣔⡡⠔⢈⠤⠂⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡑⢄⠐⡀⠀⡆⠀⠀⣵⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠏⢀⡠⠐⠁⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠬⢑⣄⢣⣀⠈⢸⡊⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠲⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⢻⡏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠜⢀⠀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿⠏⡃⠄⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠋⣃⢀⠀⣐⠈⠓⠐⡱⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠈⠀⣸⡶⠃⢞⠛⠤⠀⠀⠉⢳⠂⠀⠈⠳⠀⠠⡀
⠀⠀⠀⠀⠀⢀⣔⠌⠋⠀⠀⠀⡠⢒⠅⠈⢡⣀⡘⢀⠀⠀⠀⠈⠐⢃⠀⠀⠀⠀⠀⠈⠓⠢⢤⣀
⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠔⡠⠊⠀⢀⠊⠀⢡⠀⢂⠐⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄
⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⢀⠔⠲⠶⠞⠠⡀⠄⠀⠀⠀⠀⢦⢈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠠⠁⠀⠀⠀⠀⠀⠘⠌⢃⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠀⠀⠀⠀⠠⡀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠀⠀⠀⠀⠈⠢⠀⠀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢃⠀⠀⣀⡀⠠⠐⠂⠉⠰⠀⠡⡀⠀⠀⠀⠀⠀⠑⢰⠀⢆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⠀⢠⡞⠀⠀⠈⢄⠀⠀⠠⠂⠀⠈⢱⠘
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⢎⣁⠤⠒⢁⠆⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣀⣴⣻⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣐⠤⣀⠀⠀⠀⠘⠁⡤⠞⠓⠄⢀⠠⠂⠀⠀⢀⣘⣷⣻⢆
⠀⠈⠒⠦⠘⠄⠀⢀⡰⠀⠀⠘⡆⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠀⠀⠀⠀⠀⠘⠛⠮⠻⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠈⠐⠃⠀⠡⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢠⠀⢄⡀⠀⠀⠙⠄⠀⠀⠀⠀⠒⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠈⠈⡀⢁⠀⠑⠀⠀⠸⣄⠴⠋⠱⡀⠀⠀⠂⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢁⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠤⠙⠂
⠀⠉⠃⠀⠈⠉⠒⠠⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢴⣀⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠴⠒⠂⠀⠀⠀⠠⠀⡠⠃⠂

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⣀⡀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠉⠁⢉⠀⠈⢸⢳⠂⠀⢀⠴⠋⢠⠄⠀⠀⠈
⠀⠀⠀⠘⠀⠀⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣬⣥⡛⠿⠛⢳⣿⠟⠁⠀⠀⠀⠁⢠⣀⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣸⢠⢃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡟⢩⡔⠃⠗⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠺⠉⠙⣗⠂
⠀⠀⠀⠀⠠⢀⠀⠀⡈⠀⣸⡎⠈⡄⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡀⠘⢷⣶⣤⣴⠡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣌
⠀⠀⠀⠀⠀⠀⠀⠁⢕⢄⣿⣯⠔⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡾⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡏⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⣿⡟⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣼⣱⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⠆⣰⣛⢘⢻⣄⠈⠀⠀⠀⢀⡀⠀⠀⠀⠐⢾⣇⣿⣳⡓⠂
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣿⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣾⠿⣿⣿⡿⢛⠻⠷⣶⣬⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠙
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡞⠉⠁⠱⡘⠄⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⡰⠄⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡆⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠆⠸⡄⢻⣿⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢷⠍⣉⣉⣛⣻⣷⣤⣻⡇⠀⢀⠴⠖⠛⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠙⠻⣇⠀⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣤⣤⣌⣳⣦⣆⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣷⣾⡿⠟⠻⡟⠁⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠱⠀⠀⣠⡶⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣀⠀⠀⣇⣰⣯⣔⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢮⡳⣦⣼⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⣧⠀⠀⣀⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣟⠟⡉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⡾⠚⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠟⠉⣡⡴⢛⡽⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠁⡆⢰⢏⠜⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⣮⡳⣄⠀⠀⣿⣤⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠏⠹⣆⠀⠀⠀⠀⣾⡇⠀⠀⠀⣷⣄⠀⠀⠀⠀⢸⡇⢭⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢂⠀⠀⣠⢿⢿⠀⠀⢠⡿⠏⠳⢦⡀⠀⢸⣠⣾⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠘⡆⠀⢩⠈⠘⡇⠀⠸⣣⠀⠂⠌⣷⠀⠈⣿⣿⣷⡿⠟⠛⠉⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢠⣴⠃⠀⠈⠣⠀⣜⠀⠀⠉⠛⢶⣷⣣⡴⠚⠉⠀⠉⠉⠒⠤⠤⠤⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⠏⠀⠀⠀⠀⠀⣿⠀⠀⢀⡠⠼⢯⣥⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠲⢾⣷⠤⠤⠤⠤⠤⠤⣾⠖⠊⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣦⡀⠀⠀⠀⢰⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡹⣧⠀⠀⣀⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣧⡹⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣉⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡑⠶⢄⡀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡛⣷⠀⠀⠀⠈⢳⡤⡀⠀⠀⠀⠀⢉⡶⠶⠿⢢⣀⣀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⡀⠀⠀⠀⢧⠈⢧⠀⠀⠰⠋⠝⠀⢒⣁⡁⠙⠾⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠇⠀⠀⠀⠀⠙⠺⠧⠀⠀⠀⣆⠤⣀⠉⡉⠉⠲⢤⡞⠉⢀⠄⣰⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⣀⡀⡀⠀⠀⠀⢐⡚⠙⠳⡆⡠⠓⠲⣄⠀⠈⠀⣤⠄⢡⡴⠳⠚⠁⠀⠀⠀⠀⢀⠤⠔⠂⡋⢹⢾
⠀⠀⠀⠀⠀⠀⣘⣷⠺⠃⠀⠈⣇⡹⠉⢏⡁⠢⠠⠙⠳⢄⡻⢄⡀⠀⢈⠳⠔⠊⠁⣀⣠⣅⣠⠴⠖⠀⠀⡤⠚⠁⣀⠐⠈⡀⠁⢠
⠀⠀⢠⡖⡶⠘⠿⠟⠀⠀⠀⠀⠀⠀⡀⠀⠙⡄⠠⢰⠘⡁⠀⠀⠀⣤⣖⣒⣓⡠⡀⠈⡋⠈⣡⡔⢒⡶⠛⠁⠄⠄⠀⠀⠈⠠⠁⠠
⠀⠀⠈⠛⢻⠙⣆⠀⠀⠀⠀⠈⣄⣾⣋⡧⠀⠁⣴⠆⠐⣱⠁⢀⣘⠉⣉⡉⠉⢧⣉⡤⠦⠾⡀⠀⠀⠀⠢⠀⢀⡠⠂⡠⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⢨⡄⠀⠊⠄⣠⠞⢁⠀⡸⡸⡖⠿⠓⣳⢺⠹⠅⠁⠒⠤⠑⠀⣆⡠⠖⠉⠀⠘⢀⠠⠞⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢆⠀⠀⣆⠀⠰⣆⢰⠀⡈⠁⠀⠀⠈⠡⣈⣑⢚⡼⠉⠸⡇⠀⠉⠀⠀⠀⡆⡙⢂⣀⠉⠀⠐⠎⣀
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣨⡓⣀⢸⡁⡀⠻⣼⡃⠑⢄⠱⡈⢆⠀⢁⠀⡠⢀⢗⠄⠁⣸⢠⠆⢼⠸⠻⠓⠲⠤⣀⠀⠀⢀⠀⠈⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⢀⣱⣤⠈⠀⢥⠀⠂⢹⡆⠸⣝⡁⠠⠔⠠⣆⡰⢟⠍⠁⠀⠘⠠⠀⠁⠄⢀⠀⠈⠑⠦⣈⠀⡀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠮⠋⠁⠁⡨⠕⣦⠶⣦⢀⢧⠡⠐⠌⠓⠤⡴⣋⢉⣀⣃⡄⣀⡀⢀⠀⠀⠐⠂⡀⠑⠈⠀⡀⡀⠑⠌⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠤⠈⠀⠠⠀⠉⡠⣳⣥⠦⠤⠔⠈⠁⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⢀⣈⢀⠀⠐⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠝⠀⢠⡴⠠⠀⣀⢴⠗⡀⢁⡙⡆⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠀⠊⢀⠻⠐⢋⠀⠸⠊⢀⠘⠀⡇⠀⠊⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠝⣇⠀⠀⠀⠒⠀⠀⢀⡤⢺⡄⠠⡀⢳⠀⢀⠆⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠀⠒⠊⠁⠀⠀⡆⠀⠀⠈⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡀⠀⠂⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⣲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⡟⢻⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢰⣶⣶⡶⠛⠛⢷⣦⠀⠁⣀⣶⡄⠙⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢠⣴⣿⣿⣷⠈⠁⡀⠐⣦⠼⠛⠁⠸⣟⣿⣿⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢿⡷⠘⣿⣿⠈⠻⠉⣿⠣⣦⣤⡟⠻⣿⣤⠤⡈⠰⠟⠳⢄⣀⣀⡹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠚⠋⠀⠈⠁⠀⠙⠛⠀⠹⠶⠻⡶⢹⣿⡇⠀⣿⠇⣸⣿⠀⢀⡤⠈⠛⢿⣧⡜⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⣶⣦⠀⣸⣿⣆⠈⠁⢰⣦⡀⠿⠿⣆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠈⠛⣿⡟⠛⣿⣦⣴⡆⠉⢱⣶⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡜⠻⠏⠙⠃⢰⠇⣠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡕⠒⠄⠀⠺⣦⣴⠋⠉⣉⡉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⢠⣶⡿⢿⣧⣤⣾⠏⠁⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢠⣼⣏⠀⣠⣍⠛⠿⣆⠰⠀⠙⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⣀⠀⠿⠿⢻⣾⣿⣿⣧⣄⠈⠀⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⢀⣰⣦⣤⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⣿⡄⠀⢀⡄⠀⢀⣀⣠⣀⠀⠀⠊⠁⠀⠀⠀⠀⠉⠁⠀⢻⣿⣤⡘⠉⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠻⠟⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡆⣤⣤⣦⠼⠿⠿⣿⣷⣦⡄⠀⠀⠀⠀⠈⠂⠀⠀⠈⠉⣹⣿⣷⡄⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡄⡉⠁⣀⠀⠀⢀⣌⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⠀⣀⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠈⠉⠰⣆⡐⠀⠀⠀⠀⠠⠄⣀⣉⡀⠛⠋⢁⣶⡟⠀⣴⣿⣿⣆⠈⣿⣧⠀⠀⠀⣾⣇⠀⠀⠀⠀⣹⡿⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡆⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⢀⣤⣾⣿⣿⡿⠇⠠⣾⠟⠁⠀⢿⡿⠉⢛⡀⢹⣿⣇⠀⠀⠘⣟⣷⣦⠀⠀⠻⠃⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢛⡇⠀⠀⠀⠀⠀⠀⠀⢻⣧⣼⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⢻⡿⠀⠀⠈⣿⣿⣿⠀⠀⣀⡀⣠⣾⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠶⡶⣾⡆⠈⣿⡇⣿⡿⠋⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⣟⣻⡄⢀⠀⢿⣿⡿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣌⣿⣷⠀⠀⠀⠀⠙⢣⡀⣿⡇⢹⣧⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢠⣄⣀⡈⠁⢀⣴⠶⣟⡿⠇⠈⠀⠘⣿⣿⠋⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⣠⡀⠀⢠⣿⡇⣿⡇⢸⡿⠟⠃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⢻⡿⠁⠀⠈⠀⠀⣈⣤⣴⠂⠀⠀⠈⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡿⠙⢿⣿⣿⣿⣿⣿⡇⢸⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣼⣿⠆⠀⠰⠖⠀⢀⣴⣿⣿⣿⡆⠠
⢀⣠⣤⣶⠶⠂⠀⣤⡶⣿⣿⣿⣿⡅⠀⠀⠉⠻⣿⣿⠟⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⡅⠀⠀⠀⢰⣾⣿⣿⣿⠿⠿⠃
⣿⣿⣿⠁⠀⠀⠀⠈⣧⣬⣿⣿⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣴⣿⣦⣍⣹⡿⠋⠀⠀⠀⠀⠚⠻⠿⢿⣿⣿⣿⡇⠀⢸⡀
⣿⣿⣿⣤⣤⠀⠀⠀⠹⣿⣿⣿⣇⠀⢀⠀⠀⢀⣤⣄⡀⢰⠀⠙⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋⠐⠂⠀⣤⣶⣿⣿⣿⣿⣶⣬⡟⠛⠋⢀⡀⣀
⣿⣿⡿⠛⣅⠀⡤⠲⣾⣿⣿⣿⣿⡄⠀⠙⢿⣿⡿⠿⠟⢻⣀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⣿⡿⠿⠟⢀⣴⣿⣿⣇
⣿⣿⣿⡄⠙⢻⣦⢀⣄⣼⣿⣟⣻⡇⠘⣶⣾⣿⡇⠀⢰⣿⣿⡀⢀⡀⣀⣀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡌⠁⠀⣰⣿⣿⣿⡇⠀⣄
⣿⣿⣿⠀⢰⣶⠁⣸⣿⣿⣿⣿⠟⢁⠀⢹⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⢠⣦⣤⣶⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⠀⢸
⣿⣿⣿⡄⠈⢍⢘⣿⣿⣿⣿⡏⠀⠈⠻⢎⠙⢻⡇⠸⣿⣿⣿⣿⣿⣿⠿⣿⡀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡀⠀⢸⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣷⣦⠀⣿⣿⣿⠿⠿⢃⣀⠀⢀⣸⣷⣄⠀⢠⣿⣿⣿⣿⣿⣿⣀⡻⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⡇⠀⠘⠛⠻⠿⣿⠿⣿⡇
⣿⣿⣿⣿⢉⣿⣿⣿⠃⣀⣴⣿⡿⠸⠋⠁⣿⣿⣷⣾⣿⡛⠿⣿⣿⣿⣿⠃⣀⠀⠀⣄⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⢘⣿⡇⠀⣀⣠⣴⣦⠠⣄⡛⠃
⣿⣿⡟⠉⣿⣿⣿⠃⣴⣿⣿⡿⠁⠀⠀⠀⣿⣿⡿⠋⣿⣷⣶⣟⠿⠿⣏⠸⣿⡇⢰⣿⠀⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣿⣧⠾⠿⠿⠿⠃⠀⣿⡇
⣿⡟⠃⢸⣿⣿⣿⠸⣿⡟⠉⠀⠀⠀⠀⢠⣿⡏⣰⣿⣿⣿⣿⣿⣷⣤⣿⣄⣈⠀⠈⠛⣼⣿⣿⣿⣻⣿⣿⠇⣀⣿⣿⠟⠿⠄⠀⠀⠀⢠⣾⡁
⡿⠃⢀⣿⣿⣿⣿⡄⠉⠀⠀⣠⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣇⢰⣿⠏⣈⣽⣿⡟⠛⠃⣼⣿⡷⠀⠀⠀⠀⢠⣿⡆⢸⣿⡇
⠀⠀⣾⣿⣿⣿⡟⠁⠀⠀⣸⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⣄⡀⠸⣿⠰⣿⣿⣿⣟⢠⣾⣿⣿⣇⠀⠀⠀⠀⣿⡿⠁⢸⣿⡇
⠀⠀⢿⣿⣿⣿⢠⣶⣀⣸⣿⢿⣿⠿⢋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⣿⡦⢈⣤⣼⣿⣟⢻⣿⣿⣿⣿⣿⣶⣦⣤⣾⡟⠁⣠⡿⣿⡇⠀⡀
⠀⠀⠀⢻⣿⣿⡈⠿⣿⣿⢏⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠀⠈⢻⣿⣿⣿⣿⡉⢻⣿⣿⣿⣿⣿⡿⠟⠁⢰⣿⡀⢻⡇⢸⣿⠀⢠⣄⣠⣤
⠀⠀⠀⠀⠹⣿⣿⣦⣀⣤⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣘⣀⣴⣇⣴⣿⣿⣿⣷⠸⣿⣿⣿⣿⡟⠁⠀⠀⣸⣿⡇⠈⢀⣿⡇⠀⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣇⠀⢸⡿⣤⠀⣿⣿⣿⣿
⠀⢸⠃⠀⠀⠀⠀⠙⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⢻⣿⣿⣿⣿⣿⣧⣴⣿⢿⣿⣿⣿⣿⠛⢻⣿⠈⢻⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⠘⠀⠈⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣮⣿⡛⢿⣿⣿⣿⣿⣿⣿⠟⠀⣾⣿⡄⢸⣿⣿⣿⡀⠀⣸⣿⣯⣴⡆⠀⢠⡄⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠀⢸⣿⠛⠀⢈⣿⣿⡟⠁⢠⣿⣿⣿⣿⣿⣄⡼⠃⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠟⠛⠛⠁⢰⡟⠉⠀⢰⣿⣿⣯⠀⢀⣼⡟⠋⢹⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣿⠁⢠⣆⣸⣿⣿⣿⣠⣾⣿⣿⣦⠘⢿⣿⠏⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢠⣾⣿⣿⡿⠉⠙⢿⣿⣿⣿⣿⣆⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⣾⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⢰⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣸⣿⣿⣿⣤⡀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣿⣿⡈⠇⠀⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦⣤⣀⣼⡿⠋⣉⡀⢀⣤⣤⣤⣴⠶⠂⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⣶⣿⣿⣿⣷⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢳⣄⡀⢸⣷⣬⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠶⠆⠀⠀⠙⢿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣴⣦⣀⣀⣰⡇⢰⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⣿⠿⠛⠁⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠶⠿⠁⠀⣾⡇⠈⣿⡿⣿⣧⣠
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠉⠀⢀⡰⢇⣠⡟⠀⣠⣿⡇⣼⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣧⢨⣿⣿⡆⢻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⢿⣾⣁⣼⣿⣿⡃⠸⢿⣿⣇⢿⡏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢿⡏⢹⣿⣿⢻⣿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠙⠃⠘⢣⠻⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠂⠀⠒⠲⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠒⠀⠀⠒⠶⣤⣄⡀
⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄⡀
⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣦⢠⣠⣏⣀⣀
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⡀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢣
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠜⠀⠀⠋⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠞
⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢂⡠⠋⣹⣿⣯⣉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠰⠀⠀⢸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⢸⡀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣸⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⣹⣿
⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿
⢓⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡻⣌⠉⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡭⢽⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⢺⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠕⢺⣿⣄⡀
⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠩⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⡀⢀⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⠀⣀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⢀
⠒⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠉⠐⠒⠈⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣉⡐
⠀⣤⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⣤⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠐⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣒⠶⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠶⢄⣀⣒⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⢝
⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣀⣸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀
⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢉⣁⣸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣺⣿⠟⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠤⠄⠒⠢⡄⠀⠀⠀⢀⣴⠢⠀⢀⣀⠀⠀⠀⠀⢠⡄⠀⠀⡀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⢀⡦⠔⠺⢆⠀⠀⢠⠅⠀⠰⣔⣠⠈⡀⢀⣓⠀⠀⣴⠿⣿⡂⠀⠉⠿⠗⡶⠀⠀⠐⡆⠀⠀⠐⠀⠂⠀⠒⠻⡆
⠀⠀⠀⢊⣰⡟⢯⡷⣤⠀⠠⣤⡀⠀⠀⣍⡢⠀⠠⡇⠀⠀⠈⠀⠺⣦⡄⠐⠀⠀⠀⠀⠀⠀⠁⠀⠐⠠⠀⠂⡀⠀⠀⡂
⠀⠀⠀⠙⠿⠇⠘⠛⠉⢀⡄⠈⠀⢨⣉⣉⣹⠛⠁⠀⠀⢠⣴⡾⠀⢀⢱⠆⠐⠻⠷⠶⢂⠄⣠⡤⠊⠠⠄⠀⢁⡀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠬⠮⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠉⠈⠲⡜⣤⡀⢀⡜⠀⠀⠈⡀⠀⢀⠠⠐⠀⠈⠠⠓
⠀⠀⠀⠀⡠⠀⢠⡄⠀⠀⡄⠀⠀⠀⠤⠤⠆⠀⠀⢤⡀⠀⠀⠀⠀⠈⠻⠶⠋⢀⠀⠀⠀⠈⠐⠀⠐⠚⢻⠀⠑⠊
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠁⠀⠄⢀⣀⠊⠀⠀⢀⣠⢼⣥⢤⠌⠀⠀⢸⠀⠀⠀⠀⠚⠿⡄
⠀⠀⠰⣿⣷⠀⠀⠀⠔⣠⡤⠤⢌⡂⠀⠀⠀⠘⠇⠘⠆⠀⠐⠀⠤⠐⠀⠀⠈⢻⠟⠙⠾⠀⠀⠀⠸⣤⡇⠀⠈⢀⠀⠁
⠀⠀⠀⡄⠀⠀⠀⡌⡜⢡⣶⣶⠄⢹⠀⠀⠞⢠⣀⣤⡈⠀⠀⠀⡄⠀⠀⠀⠀⢈⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⢠⢸⣠⡆
⠀⠀⠀⠀⠀⣀⠀⡕⡄⠞⣇⢻⡠⠞⠀⢀⣐⣯⣶⣦⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠴⠶⠔⠃⠀⣤⠀⠈⢋⡍
⠀⠀⠀⠀⣴⣿⣧⠈⠛⢛⣇⣔⠀⠀⠀⠀⢀⡘⡛⡛⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢘⡧⠄⢑⣊
⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⣸⠋⠀⠀⢰⠀⠈⠁⣿⠁⠀⠀⠛⠲⠿⠖⠒⠶⠶⠤⠾⠀⠀⠀⠸⣿⣿⡶⠉
⠀⠀⠈⠛⣿⣿⠛⠀⠀⠀⡇⠀⠀⢴⣶⣦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⢰⣉⡅⠀⠀⠀⠀⣇⠀⠀⠀⡙⠀⠀⢀⣱⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣀⣀⡅⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞⢐⣆⣤⡸⢦⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡖⠐⠒⠾⢤⡈⡹⠖⠋⠀⠀⠀⠁⠈⠈⠀⠉⢓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠚⡲⡄⢈⠟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠁⠀⠀⠀⠐⢉⠤⠀⡀⠔⠒⠠⠀⠀⣠⠀⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠁⠀⠀⠀⠈⠑⢤⡀⠀⠀⢰⣲⠋⠀⠀⠀⣠⠖⢀⠂⠀⡀⠸⠧⠤⣔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⣀⣠⠚⠁⠀⠀⢁⣴⠋⠂⢨⡗⢚⠦⠄⠀⡀⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡟⠷⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣿⢿⣿⣷⣦⣀⡾⠛⠉⠀⠉⠉⠉⠀⠐⠠⣄⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠠⠀⢸⠐⡀⠀⠀⠀⠀⣼⡿⣧⡘⣧⣞⣩⣤⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⢀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡀⠊⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⣿⣀⣡⢽⣴⢯⡐⢆⣻⡃⠀⠀⠀⠀⠀⠀⠰⡄⢸⢄⢀⠁⠀⡀
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠊⠀⢀⡴⠀⢀⡏⢠⣀⣀⡤⠔⠚⠉⠉⣹⣿⢛⡄⠋⠀⠀⠈⢢⠀⠀⠀⢀⡀⠶⠛⢪⢀⠏⡠⣚⠁⡀
⠀⠀⠀⠈⡢⠄⠀⠒⠒⠀⠀⠁⠀⡀⠤⠚⠕⠀⢀⠞⡠⢚⠍⠉⠉⠁⠂⢀⡞⢡⣿⠋⠀⠀⣀⣠⠄⡪⠓⢏⠁⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠠⢋⡎⠀⠂⠀⠀⠀⢀⢀⡎⢀⢀⠃⠀⢀⣞⡿⣮⡟⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⠈⠀⠀⠀⠀⠀⢸⡄⠀⠂⣀⡀⠀⠀⡸⢀⠀⠸⡀⢀⡊⠐⠛⠉⠀⠀⠀⡐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⡀⠀⠂⠀⠻⣦⣀⣀⣀⣀⠴⠣⠂⠀⠀⠙⢯⡀⠀⢀⡀⠠⠐⡈⠔⠁⠀⠀⠐⠠⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂

⠀⠀⠀⠠⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢
⠀⠀⠀⠀⠀⠀⠀⣎⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡁
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⣀⣠⣤⡤⠤⠄⢀⣀⡰⡆⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⠧⡟⠋⠁⢠⠁⠀⠀⠀⠀⠱⠘⡟⠓⠶⠲⠟⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡎⠀⠀⠀⠀⠀⠀⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠡⣀⡀⠀⡀⠀⢀⢶⠘⠀⠀⠀⠀⠀⠔⠑⡎⠉⠰⠔⡆⠐⠒⠂⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡨⠃⠁⠺⡀⢁⠾⠀⠀⠀⠀⠀⡠⢂⣀⠆⠄⡔⠉⠀⠀⠀⠀⠀⠀⢈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠢⡀⠁⠀⠀⠀⢠⢄⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠠⠂⢀⠜⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠬⣆⠀⠀⢰⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠔⠀⠔⠁⡠⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢴⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢮⡊⠀⠄⠈⠀⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣧⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡢⠤⠄⠂⠽⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠡⠀⣬⡗⡾⠵⣄⠂⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣖⠁⠀⡢⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⡠⡪⠛⠉⠙⠀⠙⠉⢢⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣥⠔⠈
⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠌⠐⠀⠰⠁⠘⠂⠀⠀⠈⢣⠀⠻⡄
⠀⠀⠴⠋⠀⠀⠀⠀⠀⣸⠀⠀⠀⡈⠒⠒⠑⢄⠀⠀⠀⢁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢸⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⢉⣀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡀⠀⠉⠉⣁⠠⢼⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠁⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠶⢲⣩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠈⠔⠁

⠀⠀⢀⠠⠒⠀⠀⠀⠢⢀⠀⠀⠀⡠⠆⠤⣢⡀⠀⠀⡢⠄⣠⠄⠀⠀⠀⠏⠃⠡⠤⠈⡠⡄⢠⡀⢠⡀⠘⠑⠀⠀⠀⠠⠂⠛⠓⡆⠀⠀⠀⠀⠀⢀⣻⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⠁⣶⢐⡅⠁⠀⠀⢩⠋⠉⠐⠄⠀⠀⠈⢌⠀⠠⠅⢨⡆⠐⡂⢀⠔⡡⠀⠀⠀⠀⢢⡁⠀⠁⠇⠀⠀⠀⢠⡾⠥⠤⠔⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡣⠀⢀⠞⡡⠂⠀⠀⡄⠀⠀⠀⠨⡀⠀⠀⠀⠑⠨⢐⠒⠒⠒⡚⠅⠊⠀⠀⠀⠀⠀⡠⠀⠤⠄⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⡀⠀⠀⢠⠄⠲⠀⢄⠀⠈⠢⡀⠄⡰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠄⠀⠠⢀⠀⠀⠀⠀⠐⢄⠀⠀⠀⣠⠂⠀⠀⢀⠠⢀⣀⠠⢀⠀⠀⠀⠸⣀⡙⠦⣀⠀⠈⠀⠀⠉⡀⠀⠀⡠⠀⠀⠀⠈⠈⡀
⠀⠀⠀⠐⣧⡜⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠲⢦⣠⣐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⡪⡃
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠠⢔⣲⠒⢂⡢⠌⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢽⠿⠴⢅⡀
⠀⠀⠀⢀⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡒⠊⣉⡉⠃⠂⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠐⠀⠂⢀⠀⠀⠀⢀⠊⠁⠢⠄⠀⠁⠀⠀⠐⠄⡀⠀⠀⠀⡀⠄⠀⠀⢧⡠⢀⢋⣁⡤⠸⠀⢀⠍⠡⡀⠀⢆⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠃⠙⠢⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠍⠉⠀⠔⠁⠀⡣⢲⠣⣿⡆⠈⠢⡀⢀⠀⢀⠔⠁
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠦⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⢀⠤⠒⣾⡈⣴⠦⡀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠰⢲⢑⡐⠆⠀⠀⠀⠀⡈⠀⠀⣀⠁⠀⠀⠀⠀⣀⠀⠀⢻⡷⠤⠀⠀⠂⠐⢻⠗⠒⢯⠳⠘⡀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⢀⠃⠀⠀⢰⣄⣰⠻⢎⡜
⠀⠀⢀⠄⠊⠉⠑⠠⡀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⣠⠊⢀⠀⡲⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠃⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠸⢍⣸⠠⠤⠋
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠺⠀⠀⢌⡤⠀⠘⠁⠀⠀⠀⠀⢀⡀⠈⠂⠄⣀⣀⠠⠔⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⡠⠒⠉⠒⢄
⠀⠀⠢⢀⠀⠀⠀⡀⠊⠀⠀⠀⠊⠁⣀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢹⡇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠁⢀⠊⠀⠀⠀⠀⠀⠘⠄⠉⠀⠀⣄⠀⠀⠰⡚⠘⠳⢤⡤⢞⣰⠃⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠙⠢⠤⠤⠒⠉⢀⣠⡤⠒⡇⠀⠀⠈⢣⢀⠖⠁
⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠈⠒⠬⠶⠖⠋⠁⠀⠀⡀⣤⣲⢒⢝⣧⠼⡎⠉⠁⠀⣀⣤⣲⢪⢏⣹⡰⠚⠁⠀⠀⠀⠀⣻
⠀⠀⡤⠀⢀⣀⣀⣀⣀⡠⠄⠊⠁⠙⢯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠎⠯⣱⡣⠶⠊⠁⠀⠀⠀⢰⡦⠚⣵⡰⠧⠛⢉⡀⢀⢀
⠀⠀⠈⠒⠠⢀⡀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⠓⠘⠇⠀⠀⠀⠀⠇⠼⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠑⠀⠀⠶⠂⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠢⡙⢆⡀⠉⣉⡀⡀⠩⠀⠰⠂⠙⠀⡰⣰⠀⠠
⠔⠠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢄⡀⠀⠀⡆⠀⠀⠀⠀⠠⢈⠉⠀⠀⢨⡁⠀⠀⡙⠢⡀⠀⠀⠀⢀⡔⠔⣠⢮⡼⠁
⠐⠂⣄⠀⠀⠊⠂⡀⠀⡧⠐⢁⡠⡄⡀⠀⠀⠸⣀⣀⠀⠀⠄⠀⠡⠀⡄⢀⠁⠂⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠀⠖⠋⠑
⠀⢀⠈⠀⠃⠈⢀⢐⠀⠀⠠⠍⠀⡀⢀⠐⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠰⠆⠀⢀⠠⠔⠀⠁⠀⠤⠠⠏⠀⡁⠀⠀⠌⠀⠆⠠
⠀⠀⠀⠐⠈⠀⠁⠨⠀⠀⠀⠀⠀⠶⠅⠠⡨⠀⠀⠀⠀⠀⠠⠂⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠁⠀⠁
⠒⠂⠀⡀⢈⡁⠀⠀⡠⠂⠢⡀⠀⢀⣔⣽⠣⠁⡠⠒⠀⣈⡀⠄⠀⠀⠈⠀⠊⠁⠀⠐⠀⠀⢀⠀⠀⠤⠀⠔⠁⡇⠀⠀⠀⣀⡘
⠀⠀⡀⣉⢠⢀⠀⠐⠀⠀⠀⠃⢠⡵⠋⠄⠁⠀⠁⠆⠣⠀⠈⠁⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⡋⠀⠀⠀⠉
⠈⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠱⣞⡖⠀⡀⡀⠐⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⡀⠄⢀⠄⠀⠄⠄⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣒⠒⠂⠀⠀⠀⠀⠀⠌⡀⡟⡁⠀⠀⢈⠆⠀⡀⢁⠁⡀⠀⠀⠀⠐⠊⠀⡀⠀⡀⠀⠐⠂⠜
⠀⠌⠂⡀⠀⠡⠀⠈⠁⠀⠀⠀⠀⠀⠀⠴⠃⡇⠹⠀⠀⠀⠀⠀⠀⢁⠀⡤⢨⠀⠀⠠⣀⠀⠀⣷⡄⠀⠈⠀⠁⠀⠀⠀⠀⠀⣤
⠄⠘⢀⠀⡀⠀⠐⠀⠀⠀⠀⠀⡔⢇⠀⣃⣤⠁⠈⠠⣀⠄⠀⠀⠀⠶⠂⠀⠀⢀⠠⠄⡀⠀⠄⠀⠀⠀⠀⠈⠛⠀⠒⠒⠊⢀⠤
⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⠀⠀⠰⠀⣠⠀⠀⠀⠀⠸⠐⠂⠀⠀⠰⠅⡀⡂⢀⢔⠀⠰⠴⠃⡄⠠⠀⠀⠀⠀⠀⢀⠂
⠠⠀⠠⠄⠒⠂⢄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠌⠋⠀⠘⠁⠄⠐⠘⠂⠁⠂⠀⠀⠀⠐⠒⠅⣀⠄⠢⡀⠀⠄⢀⠈⢖⡰⠀⣀⣄
⠀⠅⠰⡂⠢⠄⠀⠅⠀⠀⠰⠀⢀⠔⠁⠀⠀⠤⠤⡞⠁⡀⠀⠀⠀⠀⠀⢃⠁⡄⠀⠠⠊⠀⠀⠀⡗⠀⢀⡌⠀⠉⠀⡀⢀⠀⠂
⠈⠀⠀⠏⠡⠂⠀⠂⠀⣄⡀⠤⠁⠠⢊⡡⡁⠐⠄⠀⠠⣧⠀⠀⠀⠀⢁⡀⢠⠀⠀⠅⠀⠀⢈⠈⠀⠀⢉⠀⠀⠀⠀⡀⠀⠐
⠀⠉⠈⠐⣀⠀⠀⠈⠀⠐⢀⠀⠀⠁⠠⠀⠈⡠⠂⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠑⠊⢀⡎⢀⡀⠃⠀⠐⠀⠒⠀⠀⠀⢀
⠄⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠉⠀⡁⠀⠀⠂⠀⠀⠀⠧⠔⠂⠀⠀⠀⠑⡀⠀⠂⠈⠈⠠⠀⡁⠀⡁⡀⠀⠀⡙⢈⠁⠀⠀⣠⣁
⢉⠂⡄⢊⠐⠁⠀⠀⠀⠀⠀⣀⠊⢀⡀⠀⢠⡠⡠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠙⡄⢁⠒⠀⠂⠈⢻⣿⠀⠀⠀⠀⠈⠐⠀⠀⠀⠒
⢀⠄⡉⠠⢎⠶⠀⠀⠀⢀⠆⠀⠐⠀⠐⠔⠐⠒⠀⢂⢀⡀⠀⠀⠒⠀⠀⣀⠀⠁⠄⠁⠀⠠⢤⠀⠇⠀⢀⠤
⢈⠀⠆⠀⠆⢺⣡⠧⠔⣸⠀⢀⠀⡀⠆⠀⠀⠀⠀⠸⠀⠉⠀⠀⠒⠉⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠈⠤⠈⢁⢆⡸
⠀⠀⠂⠂⠀⠡⠗⠆⠀⣆⠔⢠⣠⢀⠈⠂⠤⠠⠐⠁⢄⠒⢬⡀⠀⠀⣂⠀⠀⢀⢂⠀⡀⠀⠐⠒⠒⡄⠈⢀⡉⠈⠋⠀⠀⠀⢀
⠂⠄⢀⠆⠀⠀⢀⠀⠀⢀⡴⣲⢏⠂⣠⢠⡄⠐⠀⢤⡠⢀⠀⢈⠓⢊⠰⠁⠂⠡⠈⠈⠀⠀⠀⠀⢸⠀⠀⢉
⠘⠄⠋⠠⠐⠂⠀⠂⠀⠠⠟⠁⠂⠀⠀⠣⡕⣠⠊⠒⢝⡷⡒⠀⠀⠀⡀⠲⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠂⠀⠤⠀⠀⡠
⠀⠀⠁⠀⠀⠀⢀⠐⡀⠀⠂⠄⠀⠀⠀⢐⢻⣅⠅⠀⠀⠙⠙⠂⠀⠀⠁⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠛⠊⠃

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⢲⣤⣤⣤⣤⠠⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⣀⣄⡀⠀⠀⠐⡶⣶⣶⡶⠶⣶⣶⣤
⠀⢹⣧⡉⠙⠷⣷⣷⡘⡦⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⠩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⡟⠛⡄⠀⠀⢘⣶⢮⡱⣞⢀⠓⠀⠁⢂
⠀⠀⠍⠿⢗⣀⠀⠈⠉⠃⠁⣿⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠴⢒⠓⠲⠸⠄⠀⠀⢿⣿⢿⣦⣈⢣⠀⠀⣼
⠀⠀⠘⢿⣭⡽⠞⣣⡄⡀⠀⠙⠃⠀⠀⢀⣔⠀⢸⡁⣤⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⢀⠀⠀⠂⡄⠀⠀⠨⠿⣷⣎⡈⠁
⠀⠀⠀⠀⠑⠂⠁⠀⠈⠺⠷⠦⣀⠠⠀⠀⠙⠛⠎⠓⠚⠓⠲⠇⠀⠀⠀⠀⠀⠉⠀⣤⣀⢻⣄⣄⣬⡿⣿⣷⠿⠆⠀⠀⠀⠻⠯⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣅⢹⣶⣽⣟⣿⠿⠛⠀⢀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠔⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠀⡄⡏⠉⠠⠀⠢⡀
⠀⡀⠀⠖⠀⠀⠀⠀⠀⠨⢻⣤⠀⠀⠀⠀⠀⡐⢿⠇⢰⢃⠀⠀⢀⢠⢤⡄⢀⢀⠀⣤⡤⢀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢩⠀⠸⠀⠀⠰⠆
⢸⣷⠋⢰⠆⠀⠰⡄⠀⠀⠂⢽⡇⠀⠀⠀⠀⡝⣿⠀⢨⡟⠀⣰⢻⠿⣼⢣⡤⠾⠋⣁⢴⠖⠀⠀⠀⠀⠀⠀⠈⣑⠦⠀⠠⡀⠀⠀⠀⠀⠀⠀⣆
⣸⣿⡿⠁⠀⠀⣤⠘⣆⠈⣻⣲⡇⠀⠈⡐⢀⣤⡌⠀⣼⡧⣠⣿⠂⠴⠋⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⣀⠐⠀⠀⠀⠀⠀⠀⠀⣸
⢹⣛⡄⠂⠄⠀⠀⢀⠀⢠⣜⡟⠀⠀⠀⠁⠀⠙⠁⠀⠻⠁⠛⠁⠀⠠⠤⣼⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡤⡄
⠀⢿⠡⢎⡄⠀⠀⣄⢁⡰⡎⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡐⠷⠶⢶⣔⡶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⣬⣌⡀
⠀⠘⢷⣎⠄⠀⢦⢰⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠿⠳⠦⢤⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⢻⡆⢼⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⢛⠑⠀⠀⠀⢀⡀⡐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⣧⣄⣀
⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢳⠘⡅⠘⢧⢉⠀⠙⠟⠟⠈⠉⠀⠀⠀⠀⠀⢫⣄⢀⠀⠘⠀⠀⠸⣯⠿⡿⣷⣶⡆⢠⣄
⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣀⣤⣤⡶⣾⣾⣷⣦⣄⠀⠀⠙⢿⡄⡀⠀⢠⣶⣿⣄⠸⠇⢸⡟⢿⡿⣠
⠀⠀⠀⠀⠈⠳⣯⣟⣷⢮⡙⠆⠀⢀⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠋⠻⣮⠻⣿⠝⣻⠆⠀⠀⠀⠙⠁⠀⢘⡻⢿⡖⠂⠀⠘⠀⠞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟⠂⢤⣄⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣽⡷⠖⣚⣶⣦⡆⠁⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⣙⡀⠀⠈⠐⠋
⠀⠀⣴⣶⡶⠶⢷⠖⠂⠀⠀⠘⠳⣶⣿⣛⠲⡄⠀⠀⠀⢀⠴⢿⣿⢯⣭⠄⠀⠈⠙⠉⠄⠀⢀⠀⠀⠀⠀⠁⠀⠈⠻⢿⣿⣽⣥⡄⠀⠰⡀
⠀⠀⢈⣙⣛⣛⣛⡛⠃⠀⢤⡙⢷⣤⡀⠉⠙⠉⠀⠀⣠⣾⡛⣛⣫⣤⠴⢀⣀⣴⣤⡾⠂⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣾⣿⣿⣯⣿⣯⡅⢀⡐⢦⡙⠳⣽⣻⣆⠀⠀⠀⠀⠉⠛⠟⠿⣿⣿⣿⠟⣻⣿⡏⡅⣴⣟⣰⣿⠇⠀⠀⠀⠀⠀⠀⢦⣶⡆⠀⠀⣠⡄
⠀⣈⣭⣭⣽⡶⠄⠠⣌⣳⣦⡛⢷⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⢳⣿⣿⠿⠏⠀⠀⠀⠀⠀⢰⠤⢹⣙⣤⡞⣽⢟⡏
⢼⣯⣿⣿⡿⠟⠀⠸⣿⡿⣿⣟⠿⠿⠀⠀⠀⢀⣠⣤⠤⢦⠀⠠⣤⣤⣀⠀⠈⠉⠁⠁⠁⠀⠀⠀⠀⢠⣶⠀⢀⡶⣢⣾⡿⠋⣻⣿⣆⣋⡜⢳⡀
⠀⣤⡶⠟⢛⡯⢠⣷⣬⡻⢿⣿⡆⠀⢀⣤⡾⠿⡛⠣⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⣤⡌⠀⠛⠙⠳⢚⢉⣭⡸⣟⣿⡯
⠘⠻⠟⠛⣩⣧⡌⢻⣿⢿⡄⠀⠀⢠⡾⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠁⠀⠀⠀⠀⣀⠄⠤⠚⠀⠉⢀⡈⢠⣥⣈⡘⠻⣷⣾⣥⣿⣧
⠀⠀⠀⢸⣿⣿⡇⠀⠙⠻⠃⠀⣠⣏⠀⠀⠀⠀⠀⠀⢠⢂⢂⣾⣶⡾⠁⠀⠀⠀⠀⠀⠊⠁⣀⣠⠴⢤⣯⠉⠙⠿⣿⣿⡛⢦⣬⣽⣣⡌⠉
⠀⠀⠀⠘⢿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠙⠓⠀⢠⣤⣤⣷⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠢⠀⠀⢈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⠒⠒⠂⠐⣶⡄⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣌⠀⠀⠀⠑⢄⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠈⠐⣤⡜⠀⠀⠄⢠⠀⠀⣄⠀⠠⠀⠄⢻⠀⠙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠰⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠊⠀⠀⢀⠀⣀⢽⡜⠁⠀⠙⡄⢣⠐⡀⢠⠱⠀⠀⠆⠁⠆⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢄⠀⡐⠀⣾⢰⢣⠄⠒⠒⠂⢄
⠀⠀⠀⠠⡀⡀⣀⡤⠏⠈⠈⠃⠀⡄⠀⠈⡀⠀⠐⡄⠈⠀⠀⢠⡶⠸⠮⡄⠀⠀⠄⠀⠀⠀⠀⡀⠐⢤⢠⠀⠐⠐⠀⠀⢸⠇⡄⠛⠚⠃⠀⠀⠀⣀⠀⢡
⠀⠀⠀⠀⠀⠉⢭⡀⠑⠆⠀⠀⠀⠀⠀⡄⠀⠈⠂⠠⠄⠀⠀⣼⠸⣄⡞⡀⢐⡆⠀⠀⠀⠀⠀⠈⠀⠀⠰⢤⠀⢀⣀⠘⡄⣢⠀⠀⠀⠀⢀⡠⡞⠹⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢈⠉⢀⠈⢃⡀⠉⠀⠀⡀⠁⠀⠠⢔⡀⠀⠀⠀⠁⠒⠐⢄⠀⣀⣰⣀⣀⡀⢬⣀⣇⡠⠖
⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠁⡀⠀⠀⠀⠀⠀⠠⡔⠁⠈⠉⠀⠀⢄⠀⠀⠀⠐⠄⠀⢀⣀⡇⠀⡠⢠⣽⣆⠉⠁⠀⠀⠼⠂⠠⣸⡄
⠀⠀⡠⠋⠀⠀⡠⠔⠒⠲⠄⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢡⠈⠰⣄⡔⠁⠀⠀⠀⠈⠓⠶⠂
⠀⢠⠁⠀⢠⠎⢀⠤⠰⢅⠀⠑⡀⠀⠀⠀⢱⠈⠀⠀⠠⠀⠀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠘⡀⠀⠐⠛⠉⠀⠚⠛⠀⠀⢀⡤⣆⠀⠀⠘⡀
⠀⡄⠀⠀⣼⠀⡇⠐⡢⠀⠆⠀⢡⠀⠀⠀⠀⢂⠀⠀⢀⢀⠀⠀⠙⠁⠉⢆⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⣂⠀⣗⠀⠀⠀⠀⠀⢀⠼⠀⠀⡐⠀⡑⠀⠀⢰
⠀⢙⠀⠀⢧⠀⠈⠛⠁⠀⡄⠀⢸⠀⠀⠀⠀⠈⢆⠀⠘⢧⠃⠀⠀⣾⡷⠀⠑⠢⡤⣈⣈⣀⠠⠐⠀⠀⠀⠃⠁⠀⠠⠀⠇⠉⠙⠃⠀⠀⠐⠅⠀⠃⠀⠀⠼
⠀⠈⢆⠀⠀⠓⢤⣤⠤⠊⠀⠂⡸⠀⠀⠀⠀⡠⠐⠢⠀⠈⠀⠀⠘⠀⢀⡀⠀⠀⠳⡄⠀⠉⠀⠀⠀⠀⠀⠁⠶⠀⠁⠀⠀⠀⠀⠀⠀⡠⠇⠀⠐⠎⠀⠈⠂⠐⢄
⠀⠀⠀⠑⠦⣤⣀⠀⠀⠀⣁⡴⠃⠀⢀⡤⠚⠒⠈⠁⠈⠇⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠠⠄⡠⢤⠔⠒⠋⠀⠀⠀⠈⠐⣤⣤⣀⡀⠞
⠀⠀⠀⠀⠀⠀⠈⢉⠉⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠚⠉⠛⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠈⠋⠉⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠁⠒⠂⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠈⠙⠢⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⣆⠀⢠⠇⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠘⢦⡀⠘⡆⡼⢰⠃⡀⠂⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠙⢦⣹⣇⣮⠜⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⠶⠦⠭⠿⣶⠋⠀⠙⢶⠿⠛⠒⠛⠛⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠰⠖⠒⠒⠛⠋⣹⣄⠀⢀⣜⠛⢿⣛⠛⠒⠒⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠏⢈⣏⠉⣎⠢⣀⠈⠙⠂⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣠⠤⠖⠋⢁⡞⡽⠀⢸⠹⡄⠸⡄⠈⠓⢤⣀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠸⠀⠀⠀⢳⠀⠀⠀⠈⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢻⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠲⣄⠀⠀⠀⢀⡇⠀⠀⠀⠳⣄⡀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠙⠲⣌⠳⡄⠀⢸⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⠑⢾⣆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣀⡤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠚⣁⡤⠴⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣞⡤⠚⠁⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣋⣠⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠉⠓⠦⡀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⢀⡤⠒⠉
⠀⠀⠀⢀⣤⣼⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢰⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢠⣮⣤⡀
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇
⠀⠀⠀⢀⠊⠁⠀⠈⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠑
⠀⠀⠀⠨⣄⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡇⠀⢀⡌
⠀⠀⠀⠀⠹⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡉⠁⠀⡋
⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢨⣅⣀⣸⣿⣿⣿
⠀⠀⠀⠈⠉⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⢄⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢸⣿⣿⠇

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠄⠀⠘⠱⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠀⠀⠀⠓⠛⢁
⠀⠀⠀⠀⠁⠓⡀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⡚⢠
⠀⠠⠀⠈⠀⢠⢁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡉⡸⠀⡀
⠀⠀⠀⠀⠀⡤⠀⢤⠀⡻⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠐⠲⠤⡀⢀⠜
⠀⠂⠀⠀⠈⠀⠠⣀⠆⡴⢎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢀⠀⠠⠒⠁
⠀⠀⠀⠀⠨⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠈⢥⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⡀⠀⠀⢨⠟⢃⠀⠂⠤⠄⠀⠊⠐⡀⠀⠀⡤⢄⠀⠀⠀⠀⠀⡄
⠀⠀⠸⡁⠀⠀⠀⠀⣴⠂⠀⠀⠀⣉⣁⠤⠀⠀⠀⠀⠘⢅⢁⠂⢀⠄⢄⠀⣤⠘⠂⠀⠀⠀⠀⠀⢀⠢⠤
⠀⠀⢘⡇⠀⠀⠀⡶⢗⠀⠀⠀⣀⣀⠀⠀⠄⠀⠀⠀⠖⠂⠠⢤⠞⠀⢀⠈⠁⠀⣀⠄⠀⠀⠀⢤⠀⠑
⠀⠀⢸⠁⠀⠀⠢⣭⠵⠀⡀⠀⠀⠀⠁⠀⢀⠄⠀⠀⠀⠐⢃⣬⣄⠛⢆⠲⠷⠢⠀⠀⠀⠀⠠⠀⠀⢈⣤⡀
⠀⠀⡎⠀⠀⠙⠦⠶⠍⡉⠀⠀⠲⠄⢀⠠⠡⠀⠀⠀⠀⠀⠘⠀⠀⠫⠀⠆⠠⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠉⠲⢶⠾⠛⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁
⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠜⠁⣠⠤⡔⠀⢨⠀⡤⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠂⠀⠻⣦⠊⠁⢸⣶⣶⠿⢦⣃⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⠀⠀⠈⡄⠀⠊⠉⠀⠀⣿⣧⠴⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣷⢤⣀⡟⠀⠀⠀⠀⠐⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⠻⣾⣥⡤⠄⣀⠀⠴⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⡂⠀⠢⣠⠀⠀⢹⢀⣠⠄⠀⡄⠀⠀⢠⠋⠀⠀⠀⠉⠛⠶⠦⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣠⣴⣴⡿⢶⣤⡶⠃⠙⠀⡰⠀⠀⡄⠁⠀⣀⡀⠠⠒⡄
⠀⠀⠀⠀⠐⠈⠀⠘⠂⠉⠑⠾⠈⠻⡇⠀⠿⠁⠀⣴⠞⠁⠀⡾⠀⠀⡔⢸⡀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⣃⢄⡇⠀⣀⣠⠞⠉⠀⠀⢺⠁⠀⢨⡇⣼⡁⠁⢀⡇
⠀⠀⠀⢀⡤⠶⠒⠛⠢⠄⠀⣄⣨⡽⠗⢺⡗⠒⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣧⣴⡇⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠤⠤⠢⠖⠁⠀⠀⠈⢻⡄⠀⠀⠀⡈⠀⡠⠈⢀⡿⠁⠀⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢠⠁⠀⠀⢠⠟⠐⢆⣣⣤⣅⡐⠃⠀⢀⡁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣼⠠⠀⢀⡎⠀⠀⠼⢮⢙⣟⠟⢋⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠱⣜⡀⠼⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠉⠐⠦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣼⠉⠫⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣿⠀⠀⢠⡿⠉⠓⠒⠤⠤⢤⣀⢀⣀⠀⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠉⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠔⠆⡀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⢠⣀⣡⣴⡀⠀⠀⠀⡀⠈⠀⢀⡠⠀⠐⠀⠀⢺⠀⠀⢀⡀⣿⡻⠄⠀⠘⢹⣶⢤⡀⠀⠀⠨⠅⠀⠀⠈⡄⠰⠀⠀⠀⢼⡇
⠀⠀⠀⢠⠎⣠⣶⡢⣍⡀⠀⠀⠁⠀⠀⠘⠛⠛⡀⠁⠀⢚⠀⠀⠈⠓⠀⣽⣁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⡀⠀⢀⠀⠀⠀⠀⡁
⠀⠀⠀⢠⡞⣿⠀⢿⣂⡿⠀⠀⠛⠖⠐⠤⠴⠶⢤⣀⠄⠙⠀⠀⠀⢀⣤⠈⠙⢗⡘⠀⣠⣤⣀⣠⠀⠀⠀⠀⢀⠄⠁⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠆⠀⠀⠀⠒⠶⠶⠐⠂⠀⠀⠀⠀⠲⠿⢅⡀⢰⠘⠇⠀⠈⡩⠉⠉⠰⠁⠞⠚⠀⠀⡂⠀⢀⠰⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣷⡇⢀⡴⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠒
⠀⠀⠀⠀⠠⠂⠀⠰⠄⠀⠀⠄⠀⠀⠀⠀⠂⠴⠂⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠙⠓⠋⢀⠄⢀⠀⡀⡀⠀⠀⠀⠙⠉⠃⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠈⠀⠠⠀⢠⣄⠈⠀⠀⠀⢀⡴⣲⡖⢤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠛⠇
⠀⠀⠀⢿⣿⡆⠀⠀⠀⠔⢀⡀⠠⣄⡑⠀⠀⠀⠀⠘⠇⠀⠳⠀⠀⠂⠤⠤⠐⠀⠀⠀⠉⠷⠋⢹⠾⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⠀⠀⠀⠀⠆⡐⠉⣤⣤⡄⠈⣇⠀⠀⠔⠀⡀⢀⣈⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⢨⢠⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠠⠀⠸⣏⠹⡆⣠⠏⠀⠀⢠⣔⣭⣥⣉⡰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠋⠀⢠⣄⣀⠤⠀⢀⠀⠀⠙⠞⠛⠁
⠀⠀⠀⠀⠀⣰⢿⡄⠢⠣⠤⠾⢀⠉⠁⠀⠀⠘⠃⢩⡿⡿⡉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠠⣄⢀⠰⣌⠛⠀⠀⠈⠁
⠀⠀⠀⢀⣼⣗⣿⢇⠀⠀⠀⢳⣶⠂⠀⠀⡀⠀⠰⠔⢴⠤⠘⠀⠰⣅⣀⣤⣀⣀⣀⣀⠀⠀⣠⠀⠘⠃⢸⢯⢭⣄⡲⠦
⠀⠀⠠⣷⣯⣟⣾⣿⠀⠀⠀⢸⠀⠀⠀⢀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠛⠋⠁⠀⠀⠀⠈⠾⠚⠗⠁
⠀⠀⠀⠀⠸⡿⠇⠀⠀⠀⠀⡜⠀⠀⠐⠛⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢧⣠⠄⠀⠀⠀⠀⣧⠀⠀⠀⠀⡁⠀⠀⠠⣤⠄⠀⠀⠀⠀⢠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡤⠀⠀⠀⠀⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⡶⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⡀⠀⠀⠀⠶⣀⣐⣪⠴⡈⡆⠀⠀⡠⠔⠊⠉⠉⠉⠀⠒⠒⠒⡺
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢤⣄⡀⠀⠀⡇⡇⣠⠊⢀⡠⠄⣒⣂⣤⡀⠤⠄⠊
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢌⢢⠀⡇⡷⠁⣤⠏⢂⡩⠤⠤⣀⠬⠷⠦⣀
⠀⠀⠀⠀⠀⡧⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠀⠀⠀⠀⠀⠈⢆⢳⣷⢁⣺⠵⠾⡅⠀⡠⠊⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⢐⢀⡠⠔⡖⢻⠿⠿⣼⠿⣫⠔⠚⡇⡅⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠉⢆⠀⠀⠀⠀⡇⢰⠋⠀⠀⠣⣈⠒⠤⣳⣊⣹⣆⠀⠳⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠃⠀⠘⡄⠀⠀⠀⣷⡇⠀⠀⠈⠐⠤⡉⠒⠒⠒⠒⠉⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⡇⢸⠀⠀⢀⡇⠀⠀⠀⠰⡀⠀⠀⣇⡥⠔⢄⠢⢄⠀⠀⠉⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⠀⠀⠀⠀⠀⣧⠀⢀⠏⣤⡞⠊⡀⠀⠀⡡⠊⠁⠀⠀⠀⢀⣤⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⡆
⠀⠀⠀⠀⠀⡇⢸⠀⠀⢡⠀⠀⠀⠀⠀⠀⢣⣌⠮⠴⢥⣀⡀⠈⡞⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡎⢠⠁
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢑⢢⣤⡤⢊⢣⣼⡟⠊⠉⣀⣉⡡⠽⡃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡸⠀⡎
⠀⠀⠀⠀⠀⡧⠼⠤⠤⠤⠍⠲⢲⡶⡶⢋⡟⢫⡙⠃⢌⡀⠀⠀⠹⡄⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⡰⠁⡜
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠋⡠⠊⡇⠈⡍⠳⢄⡙⢆⠀⠀⡇⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⠜⢁⠜
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠚⢰⠉⠀⠀⢡⠀⡇⠀⠀⠉⠚⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠐⠐⢧⡀⠀⠈⠶⠀⠀⠀⠀⠀⢀⡔⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠉⠳⠦⣆⠀⠀⠶⠒⣉⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣡⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠠⠘⢹⣿⣿⣿⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠻⢿⡙⠲⠶⠦⡀⡀⠀⠀⠀⠀⠸⣿⠻⡟⠀⠀⣸⠏⠉⣉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣖⠲⣿⣅⡀⠀⠈⠣⠀⠀⠀⠀⠀⠹⢠⡄⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠌⠁⠀⠉⠙⠳⢄⠀⠈⢀⣄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠑⠈⠀⠁⠘⠓⠀⠀⠀⠀⠐⠶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠰⡶⠒⡆⠄⣿⠀⠀⠀⠀⠀⠀⣈⠻⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢉⠐⠠⡄⠋⢠⡶⠀⠠⠀⠀⠠⠚⠉⠛⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠠⢤⡔⠛⠀⠀⠀⠘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⠤⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠙⣎⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠀⠀⠀⢀⡾⠀⠀⢠⡆⠀⠀⠀⢻⡄⠘⣷⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⡾⠁⠀⠀⣿⠁⠀⠀⠀⠈⢻⡄⠀⠉⠛⠻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡷⠀⣴⣇⡿⠀⠀⢀⠀⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠁⠀⠀⢻⣿⠃⠀⠀⠀⠓⠘⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⢃⠀⠀⠡⣰⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠃⠀⢿⢲⡦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠸⡷⠤⢀⡒⠲⢽⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠢⡀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠉⠀⠀⠈⠀⣄⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡗⠲⡄⠈⠀⠀⠠⠌⢀⣦⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣏⡄⠀⠀⠀⠀⠀⠀⠈⡄⢰⠀⠀⠀⠀⠀⣼⡇⡀⠘⠅⠙⠤⠤⢄⠤⠌⠂⢄⣉⣿⡋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠄⠀⠀⠀⠀⠀⠰⠧⠄⢷⣄⡀⣠⡖⢠⠀⠐⢃⡀⢊⣉⢐⡘⢒⡞⣓⣉⣍⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⢀⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣯⣀⣰⠟⠙⣏⣲⢐⡶⠒⡯⢭⣯⣽⣧⣴⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⠶⢾⣶⣤⣤⣤⣄⣐⣂⣤⣦⣼⣿⡁⠀⣹⣶⡶⠟⠉⠉⠙⠋⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠙⠛⠳⢦⡀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⠛⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠟⠿⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢉⣀⣤⠤⠶⠶⠦⢤⣄⡉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠘⢷⡀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⢀⣀
⠀⠀⢀⡴⠞⠋⠉⠉⠛⠓⠢⠤⣤⣀⡉⠙⠳⠦⣤⣀⡞⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢻⣄⣤⠴⠞⠋⣉⣀⣤⠤⠒⠚⠋⠉⠉⠙⠳⢦⡀
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠦⡌⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢡⡴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢂⣌⡛⠶⣤⣤⣀⣀⣠⣤⠴⠚⣩⣔⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠟⠀⠉⠓⠶⠤⠤⠤⠤⠶⠚⠋⠁⠙⣦⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⡴⠞⠋⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⢦⣤⣤⣤⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣄⠀⠀⠀⠀⣠⠶⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣾⣷⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠋⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⣔⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡰⡩⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢑⠢⡀⠀⡁⠰⡐⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⠎⠑⢌⢢⡇⢸⢃⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⣁⡀⠀⣀⠱⡱⠸⡞⡐⠪⡓⠄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠬⡙⣷⢡⡁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠬⣠⠤⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⠳⣄⠀⣀⣠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠙⡯⠗⣚⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⢀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⣀⣤⣀⡠⠐⠂⢄⠔⠚⠚⠟⠋⠤⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣠⣶⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢑⡖⠋⠀⠀⠀⠟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣿⣿⡿⠋⠹⣓⠦⢖⣤⡭⣭⠶⣄⢄⣀⠼⡸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⡩⢮⡺⠟⢸⡏⠑⠲⡮⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢀⠤⢊⠴⠋⠈⠀⡀⣇⠀⠀⠱⡴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠤⠍⠔⠊⠁⠀⠀⠸⠴⠡⠜⠀⠀⠀⠈⠢⠂

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣟⣻⣿⣻⣟⣟

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢺⢛⠻⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡘⠂⣬⢢⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠇⣼⣈⣱⡧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⣿⣿⣷⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠪⣥⠞⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⢏⣣⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡀⠀⠀⡠⠤⠞⠁⠀⠀⠙⠫⢍⠁⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠈⠻⣴⡫⠔⠁⠀⠀⠀⠀⠀⠀⠘⡇⣃⠆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⣠⠤⠤⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢃⠀⠀⠀⠁⠀⢀⣠⠤⢤⡀⠀⢀⡠⢻⠙⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢈⠗⠒⢒⡦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⠀⠀⢦⣀⡿⠁⢈⡉⣙⠛⢄⠀⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⡰⢃⣠⠔⠋⣀⠔⠊⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡳⣄⠀⠀⡇⠀⢸⠀⠀⠁⠘⣦⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⣀⣀⣠⠞⠛⠉⣀⠴⠊⠀⠀⠀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡑⠢⢧⡀⠘⡀⠀⠀⠞⣭⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢏⠀⠀⠀⠀⠀⠀⢀⣠⠤⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠿⠦⣹⠦⣄⣀⢻⣍⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢠⢠⠀⠀⠀⡎⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢿⣷⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⠜⠃⡄⠀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⣿⠇⡃⢣⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡦⢇⠀⠀⠀⢠⠉⢶⡤⠼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠐⢉⠔⠁⢠⡚⣶⠢⣴⣷⡀⠀⠀⠐⠴⣶⣺⠁⠀⠲⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠛⠚⠀⠀⠀⢀⠤⣾⡟⠉⠘⠇⠁⠻⠀⠂⠀⢠⡘⡁⠀⠀⠀⠈⠓⠈⠐⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡎⠔⠉⠀⠀⠀⠀⣠⠞⡡⠚⠀⢰⣠⣀⡀⢀⠀⢂⠀⠐⠀⠄⠱⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀
⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⡜⢁⠜⠁⠀⢀⠊⠀⠈⠆⠀⢆⠀⠆⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠞⠁⠀⠀⠀⡸⣄⣀⣠⠦⡀⠈⠆⠘⠀⠀⠀⠀⢠⡀⠻⣦⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⢮⠃⠀⠀⠀⢠⠊⠀⠀⠉⠁⠀⠈⢢⡈⢦⠀⠀⠀⠀⠀⢇⠑⠨⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠏⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢆⠀⠀⠀⠀⠈⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡝⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⡀⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠈⢃⠀⢃⠀⠀⠀⠀⠀⠀⠀⠑⠦⡡⠀⢣⠀⠀⢸⠀⠀⠀⠂⠀⠰⠂⠀⡤⡤⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠘⡆⠂⠐⠒⠚⠉⠉⠀⠀⠀⠀⣸⠀⠀⠱⡄⠀⠀⠀⠀⠀⡠⠀⠘⢧⠈⡇⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⣀⡀⠤⠐⣊⢿⠁⠀⠀⠀⠈⢦⡀⠀⠀⠁⠀⠀⠀⠀⠓⠸⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⣤⡤⠔⠊⠀⡸⢁⠀⠀⠀⠀⠀⠈⠒⢄⣀⠀⠀⠀⠀⢠⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡉⣒⣒⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⢦⠀⠲⡀⠀⢀⡇⠀⠀⠐⠃⣠⠜⠁⠀⠀⠀⡀⠀⠈⠀⠀⠀⢆⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠅⠘⢎⠀⠉⠓⡞⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠳⡀⠀⢠⠇
⢀⠀⠀⠰⡄⠀⣠⡇⣀⠀⠀⢐⠁⠀⠀⠀⠐⣲⠀⠈⢆⠀⠀⠈⠰⠀⠘⢄⠀⣀⣀⠸⡄⠀⠈⠓⡞
⢪⠀⠀⠀⡧⠀⠆⠐⢆⠀⢒⡟⠀⠀⠀⠀⠀⠁⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠇⠀⣀⣸⡇⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣎⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡄⠐⠌⠁⠀⢣⠀⠘⡇⠁
⠀⢀⠀⡀⠸⡆⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢷⣠⠏⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⡀
⠠⠀⠈⠈⠙⢇⠀⠠⠀⠀⠀⠀⠙⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠆⠀⠀⢸⠁⠉⠁
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠠⠀⠀⠈⢷⡋⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠘⢦⡀⠀⣠⠀⠀⠀⡈
⠀⠠⡀⠀⠀⢸⡄⠀⠀⠀⠀⠆⠂⡶⠀⠳⡄⠀⠀⠀⠀⠆⠀⠈⠲⡀⠀⠀⠹⣶⠅⡀⠀⢠⠃
⠀⠀⡸⠓⠂⠘⡇⠀⠄⢦⡀⢀⠎⢀⠀⠀⠹⣄⠀⠀⠀⠰⡤⠂⠀⠀⢄⠀⠀⡈⠀⠈⠂⡌⠀⠀⠀⢀⢀
⢄⡖⠀⠑⠀⠒⡇⠁⠀⠀⢰⠃⠀⠀⠢⠐⠈⠙⣆⠀⢀⣤⡄⠀⠀⠀⠀⠱⡀⠀⠀⠀⣸⠂⠀⠀⣀⠀⠈
⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡸⠞⠉⠀⠀⢤⠀⠀⠀⠀⠘⢦⡀⣰⠏⠉⠋⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠁⠀⢀⡴⠃⠤⠄⠀⢠⠀⠀⢣⡀⠀⠀⠀⠘⣦⠛⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠛⠗⠀⠒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡦⠀⠀⠈⠻⢿⠄⠀⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⣤⣶⣆⠀⠀⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠿⠹⠋⠀⠀⠈⠁⣤
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣄⢠⠀⠀⠀⠀⢺⡷⣆⠄⠀⠀⣤⣴⡄⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⢸⣟⠀⠀⢠⠀⠀⢀⠏⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⢤⣀⠀⠈⠷⣪⠻⡀⠀⢨⠆⢀⠎⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠚⠛⠓⢄⢃⠀⣨⣤⡾⢚⣀⠀⢳⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⠄⣠⣹⣤⢉⣟⣴⣿⣇⠀⢸⣂⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠁⢱⣿⣟⣛⣻⠃⠀⠸⠿⠤⠤⣄⣀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⠙⠳⢾⣇⡘⢄⠘⣣⡈⠀⠀⠀⠀⠀⠹⠀⠀⠀⠇⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣌⡷⠀⠀⢿⢻⡌⢶⠁⠀⠐⢰⣀⠀⠀⢸⠀⠀⢰⢢⡤⠖⠁
⠀⠀⠀⠀⠀⠠⣄⠐⠚⠹⢧⡀⠀⠀⠸⣟⠁⢀⢧⠀⠀⢘⠉⠁⢀⡎⠀⠀⠙⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠱⣄⣾⠀⠁⢦⢠⠆⢻⡄⡄⠀⢣⠀⠠⡀⠀⢠⠛⠢⠤⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠼⠃⠀⠀⠀⢓⠀⠀⢻⡆⠀⠘⡀⠀⢣⣔⠁
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⣠⠟⠀⠀⢼⢷⡀⢄⣇⡴⠉⣯⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⢺⠀⢳⡀⢹⠀⠀⠘⠆⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣇⠀⢳⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠂⠀⠈⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⢀⡠⠤⢄⠀⠀⠀⠀⡇⠠⠄⠀⠈⠁⢀⡠⠈⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣳⣤⣞⠀⠀⠀⢳⠀⠀⠀⠘⢦⣀⠀⠄⠀⠀⢀⡼⠁⠀⠀⠀⢀⡴⠋⣉⣀⣠⣤⣤⣤⣭⣭⣉⡓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠈⢳⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠁⠀⠀⠀⠀⢀⡾⠞⠛⠉⠁⠈⠀⠀⠀⣀⠤⢬⣉⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡇⠀⠀⠀⢨⣧⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣀⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⢤⣤⣤⣄⣀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴
⠀⠀⠀⣰⡿⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⡫⠉⠐⠒⠈⠐⠒⠂⠈⢍⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡁⠜⣅⢀⠀⡀⢀⢀⠀⡀⣀⢂⡡⣰⠌⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣥⣿⣿⣿⡿
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢁⣤⡔⢕⢁⣬⡀⠗⣩⣍⠦⢗⣴⡁⡘⠄⢿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣷⣾⣿⣿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣟⡵⠋⠀⢙⣒⠋⠈⠓⠞⠁⠉⠛⣋⡈⠙⢶⣔⢸⠀⠀⠀⠀⠀⢀⣴⣿⣖⣋⣩⣯⣿⣷⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⣧⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⢻⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠛⠻⠟⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠋⠀⠀⠀⠀⢠⠚⡇⠀⠀⠀⣾⣷⠀⠀⠀⣤⠀⠀⠀⣾⡆⠀⠀⠀⣼⠓⡀
⠀⠀⠀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢠⠶⣬⣦⢷⠀⠀⣤⠈⠁⣠⣤⡤⠤⢤⣤⣄⠀⠠⡤⠀⢀⣯⣞⡵⢦⠀⠀⠀⠀⠀⣠⣤⣤⣴⣧⠤⣤⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣴⠉⠸⠥⢈⡏⢧⡀⠀⠀⠀⠳⣄⡀⠀⣀⣠⠏⠀⠀⠀⣠⠎⣟⠀⡽⠊⢱⡄⠀⠀⢸⠁⠀⠙⠛⠋⠁⠀⢻
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠦⣤⣠⠾⢦⡀⠉⠲⢤⣄⣀⣀⠉⠉⢉⣀⣀⣤⠴⠚⠁⢀⠼⢦⣀⣤⠞⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⠒⠛⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠓⠒⠉⠀⠀⡼⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣠⠏
⠀⠀⠉⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⡉⢱⠀⠀⢀⠜⠁⣸⠁⠀⠀⠀⠀⠀⠀⢧⣤⣤⣀⣀⣀⣀⣠⣤⣤⡿⠀⠀⠀⠀⠀⢀⣠⡤⠴⢶⣶⡯⣍⠉⠓⠦⣌⠉⠓⢦⡀
⠀⠀⠀⠀⠀⡜⠘⡆⠱⠀⣡⠖⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠉⣮⡇⠀⠀⠈⡧⠀⠀⣹
⠀⠀⠀⠀⠰⡇⠀⣀⡤⠚⠡⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣠⣠⡾⣄⣀⣄⣠⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢰⢯⠀⠀⢠⡖⠁⢀⡟⠁
⠀⠀⠀⠀⠠⡏⠉⠁⢄⠙⠆⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣺⠀⣿⣒⣲⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⣿⠀⠀⡼⠀⠀⡼
⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⢀⣸⠆⣏⡀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣸⡿⢤⡼⠧⠤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⡄⣄⣤⣤⣤⠄⡄⣦⠄⡄⠀⢤⣤⢠⡠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣷⡗⡿⢿⢧⠸⢧⡇⣿⡄⣧⢷⡴⢀⡼⣿⣿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣧⡇⢸⡟⡈⢸⢧⢿⡇⣿⢸⣦⢹⢆⢿⣵⡗
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣸⣿⢃⢿⣨⢸⠸⣯⢸⣇⢿⣯⡮⣳⢽⡗
⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⣟⣿⣽⣼⣬⣿⡄⢃⠙⣧⡫⡻⡻⣯⣛⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⣧⢷⣻⣷⠹⣿⣿⣷⡌⢷⡮⣿⡚⢿⣾⡿⣻⡟
⠀⠀⢀⡀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⣯⢻⣿⣷⡽⣿⣾⣿⢶⣙⢮⣝⠻⠿⣿⣿⡇
⠀⠀⢺⣧⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣧⣻⢿⣿⣮⡻⣿⣷⢽⡷⣾⣿⣓⣶⣦⡆
⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⣿⣿⠪⣿⣷⣮⣞⣻⣿⣛⡿⡗
⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡿⣮⣻⣦⣕⣿⣿⣿⣟⣛⣿⡇
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣍⡻⣿⣽⣿⣿⣿⣿⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⢵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣛⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⡞⣾⣧
⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠇⢹⣿
⣿⣶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠁

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣬⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⠰⠂⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⢰⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠀⠀⠁⠀⠀⠀⠀⢠⢰⢇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠠⡈⡞⣞⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠧⠀⠀⠀⢲⠰⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠆⠅
⠀⠀⠔⣉⡒⠀⠀⠀⣠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠐⡋⠁⠉⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡀⠎⠠⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠐⡠⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⡂⠀⠀⠀⠀⠠⠀⠀⡀⠆
⠀⠀⠐⠦⠴⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⡀⠀⠢⠄⠀⠀⣀⣀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⢀⡄⠀⠀⠀⠀⠈⠐⠈⠁⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⡾⡴⢬⢱⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢠⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣴⢀⡤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠆⠀⠀⠒⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢦⣀⣀⡄⠤⠋⠀⠀⠀⢠⠠⠤⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠌⢀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠑⠠⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⡾⠀⠀⠀⠀⢁⡀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠐⠘⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢠⠀⠀⠀⠀⠃⢠⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠆⠀⠀⠀⠠⠄⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠄⠀⠀⠂⡀
⠀⠀⠀⢀⠠⢶⠀⠀⠀⠀⢀⠐⣦⡀⠀⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠰⠀⠀⠀⢀⣶⣦⣤⣤⠀⠄
⠀⠀⠀⠀⠑⣌⠃⠀⠀⠀⠐⠃⠈⠀⠀⠀⠈⠉⠀⠀⠀⠰⡦⠀⠂⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠿⣛⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠁⠀⠀⠀⠀⠈⠑⢦⡔⠒⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⡀⠀⠠⡀⠀⠀⠈⠄⠀⠀⡱⡄⠀⠀⠀⠈⣉⠭⠶⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⢢⡀⠘⢄⠀⠀⠘⡄⠀⢰⠘⣄⣀⡠⠚⠁⠀⠀⠄⠀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠙⢄⠈⢆⠀⠀⢰⡀⠈⡄⢿⠀⡀⠀⠀⢀⠎⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠢⡀⠀⠈⢣⡈⠆⠀⠀⡇⠀⠇⢠⠀⡇⠀⠀⡎⠀⠀⣠⠊⢀⠎⠀⠀⡔⠁⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇⠀⠀⠀⠀⠀⠈⠢⡀⠀⠱⠈⠄⠀⠰⠀⠰⢸⠀⡅⠀⠘⠀⠀⡴⠁⣠⠃⠀⢀⠜⠀⠀⢀⣜⠎
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢤⢀⢣⠀⠀⠀⠀⠐⠂⠤⣈⠢⡀⠑⡀⠀⡄⠀⡀⣤⢠⢡⢀⠀⡀⠐⠀⡐⠁⠀⠀⠊⠀⠀⠠⠊⢸
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠑⢄⣀⠀⠀⠀⠀⠀⠙⠺⣮⣷⡤⣽⣀⡗⠉⠙⠚⡛⠈⢳⣔⣴⠤⠞⠋⠉⢉⠥⠖⠒⠒⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠉⠉⠐⠒⠂⠉⠯⣜⠒⠒⠒⠒⢸⢛⡁⠤⡀⠈⠙⢦⣀⡔⠁⠀⡆⠙⢅⠀⠀⠀⢰⠋⠀⠈⠉⠉⠁⠀⠙⠆
⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠐⠂⠠⠤⢄⣀⣀⠀⠉⠲⠤⢄⣯⡀⢀⣴⣾⢦⣄⣀⠉⠉⠱⡌⠀⠀⢸⣀⣀⠴⠃⠀⠈⠛⠛⠀⠀⠀⢀⠇
⠀⠀⠀⠀⡸⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣁⣀⡀⠠⡤⠋⡟⠁⠈⠀⢉⣙⣿⣓⢦⣇⠉⢀⡾⡒⠀⠈⠉⠉⠉⠉⠉⠑⠂⠀⠸⡀
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢀⣀⡀⢈⣉⣽⠀⠀⠣⣔⣾⡿⣯⣩⢟⠂⢠⠇⢠⠛⠀⠀⠉⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠤⠔⠒⠉⠁⠀⠀⡴⠊⠉⠾⢤⣖⣉⣦⢾⢬⡀⠀⠂⢁⡼⡖⢧⡉⢢⡐⠢⢄⡀⠀⠀⠈⠑⠢⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠀⠀⣀⠤⠒⠈⠀⡸⠀⣀⣀⠈⠛⡱⢻⠂⣼⠀⢹⠉⢹⡅⢱⠘⣆⠹⡄⠑⢤⠀⠈⠑⠢⣄⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠠⠐⠉⢀⡠⠔⠀⣠⠗⠋⠀⠈⢩⠋⠀⡇⠀⡟⠀⠈⠀⠀⠁⠘⠀⠘⡄⢸⢄⠀⠱⡄⠀⠀⠀⠉⠂⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⠊⣁⣠⠔⠊⠀⠀⠀⢀⠔⠁⠀⡜⠀⢸⠇⠀⠀⡄⠀⠀⢠⠀⠀⢱⠇⠈⢦⠀⠘⠄⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠀⡀⠔⠁⠀⠀⡜⠀⠀⢿⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡆⢢⠀⢃⣀⣀⣀⣀⣀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠇⠀⠀⢀⠁⠀⠀⢸⠀⠀⠀⣇⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠂⠀⠀⠀⡜⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠀⠀⠐⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠉⠹⠁⣀⢴⡟⠓⣆⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣐⣛⣒⣒⣲⠯⠓⢉⡶⠚⠁⠀⠀⠀⡴⠋⠁⠛⠓⠒⠒⠂⠀⠀⠀⢀⢨⡂
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠖⠙⢏⠙⢶⡁⠀⢨⠗⣒⣲⢾⣟⣛⣻⣍⡹⢦⠀⠀⢧⡄⠂⠰⠀⠄⠐⠰⠀⠀⠀⡠⠔⠁
⠀⢀⡤⢒⡉⣉⢀⠀⢀⣀⠀⣀⡈⠁⢀⠯⠍⠭⢭⣓⣍⣩⢤⠤⢦⡈⡙⢾⠀⠀⠀⠈⠉⠒⠒⠒⡎⠙⣇⠈
⠈⠓⠫⠿⠾⠯⠯⠽⠿⠟⠛⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⠀⠀⠀⠀⠀⠀⠀⢰⡤⢀⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢿⣲⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠇
⠀⠀⠀⢀⡤⠒⢲⠾⡷⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢸⣼⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢀⠀⡠⡀⢀
⠀⢠⡖⢙⠀⢠⠊⣠⠃⠀⠘⡝⠢⡀⠀⠀⠀⠀⢰⠋⠀⠐⠗⠀⠈⡆⠀⠀⠀⠀⢀⠴⠊⠉⠉⠙⢠⢩⠄⠛⢠⡉⠉⠢⡀
⠰⠁⢘⠌⢢⠀⠑⢜⢦⡀⠸⣔⡅⢳⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠎⠀⠈⠁⣘⠡⠐⣠⠄⠭⠀⠀⠀⠀⢱
⡇⣴⢣⠞⠁⢀⡠⠊⣀⠇⠀⠈⡟⣾⠀⠀⠀⣠⢃⣀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡆⡀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸
⢣⠈⣺⢦⡀⠸⣀⠸⣅⠀⠀⣎⢸⡝⠀⠀⣰⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠘⢼⣦⠎⠑⠒⠒⠀⠀⠀⢀⣀⣠⠔⠁
⠀⠳⣧⣼⠁⠀⡨⠇⡼⠀⢀⣼⠟⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠙⠓⠼⠤⡸⠤⠤⠛⠁⠀⠀⠀⠀⠘⢦⡀⡀⡀⠂⠀⠀⣀⡀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⡆
⠀⠀⣴⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠡⠠⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣰⢧⠇
⠀⠀⠹⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡻⠉⠁⠙⠋⢻⡇
⠀⠀⠀⠱⡀⢢⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡾⠿⢦⠩⠿⣶⢦⡀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠋⠰⠓⠀⠶⠄⠙⡶
⠀⠀⠀⠀⠱⠈⢇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡮⣀⠀⠸⠀⠀⢀⡱⠻⡀⡄⠀⠀⠠⣔⠃⠘⠁⠔⠀⠀⠤⠀⢠⠀⢵
⠀⢀⡤⠖⠲⢆⠘⠄⢱⣤⣀⣀⣀⠀⠀⠀⢀⣇⠁⠉⠪⡳⡇⣄⢔⠡⠐⠆⢧⠸⡄⠀⢀⡇⠀⡀⠀⠀⠤⠄⢀⡀⠀⠀⢘
⢠⠋⠀⡴⣲⡶⠷⣠⠞⣵⣲⣦⠈⠳⡄⠀⢸⣿⡀⠤⣰⠾⢀⠲⡀⠀⠀⡄⡞⠀⡇⠀⠈⡂⠀⠉⠀⠂⢀⡀⠀⠀⠚⠀⡞
⡞⠀⠀⡾⠉⢾⣠⡇⣸⠋⡇⠘⣆⠀⢹⠀⠀⢇⢎⠉⠀⠀⠂⠐⢌⠢⡔⡱⠁⡜⠀⠀⠀⡧⠐⠂⠀⠀⠀⠀⠀⠖⠀⡸⠇
⢣⠀⠀⣧⠀⠀⢳⣷⠉⣆⡇⠀⣿⡀⢸⠀⠀⠈⠣⣑⠦⠧⣀⣀⠤⣣⢞⡤⠊⠀⠀⠀⠐⠓⢄⡀⠘⠁⠀⠉⢀⣀⡖⠁
⠈⠣⣤⠗⠀⠀⠠⡼⠀⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠚⠁⠈⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣠⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠨⠄⠒⣠⣄⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢑⠃⠀⢀⠔⠂⠀⠀⠀⠉⠀⠈⣳⡄⠀⠀⠀⡀⠀⡀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡄⠀⠀⠉⠀⠨⠑⠰⠆⢠⢠⡀⠀⢀⣀⠀⢊⠤⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠀⠀⠐⢔⠶⠒⢄⠀⠀⠀⠀⢰⣇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡺⣇⢀⣸⡳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⣀⠉⠁⠒⢬⡇⠀⠀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⡡⠊⠀⠀⡆⠀⠠⠏⡙⠁⠀⠈⠳⢦⣀⣀⣀⢠⠂⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⡠⢡⣠⠮⠚⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡂⠀⡁⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⡁⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠻⡏⠏⠁⠡⡤⠴⠦⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢡⠀⠂⢙⣌⠁⠀⡑⢄⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣄⠤⠆⠀⠈⢩⡵⣉⡁⠉⠉⠛⠂⠍⣉⣁⣳⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡞⢋⠀⠀⠺⠀⠀⡟⠎⠀⠈⡄⠀⠀⠀⠀⠀⠀⠡⣹⢷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢼⠋⠀⠀⠀⠀⡄⠀⢰⠈⡦⠄⠂⡄⠀⠀⠀⠀⠀⠀⠘⡂⠙⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⡐⠡⠂⠀⠀⠀⢀⠌⢰⡀⢘⠰⢀⣀⠠⠋⠐⠤⡀⠀⠀⠀⠀⠐⡄⠐⢵⠀⠀⢸
⠀⠀⠀⠀⢠⡖⠁⠀⠀⠀⠀⣌⡰⡁⠶⠀⠀⠈⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⢹⠀⠈⢧⠀⠸
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠘⠇⠀⠈⣇
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠸⡆
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣁⡇⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⠀⠀⠀⠀⠌⢆⠀⠀⠀⠀⠀⢀⠴⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠡⠀⠀⠀⠀⠀⠀⢰⠈⢦⠀⠐⠐⠋⠀⡠⠞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠀⡀⠀⠀⢀⠆⠀⠀⠓⡤⠄⠒⠉⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⠒⠲⡈⠁⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⣰⣃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠘⠁⠀⠀⠀⢰⠀⣟⢉⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠹⣤⠀⠴⠿⠅
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠃⠀⠀⠀⢸⡈⢷⡁⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⡤⠀⠀⠀⠈⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠆
⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⡄⠀⠀⡠⠂⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⡑⣐⠎⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆⠖⠀⠀⠀⠀⣠⣿⡃⠀⠀⠀⠀⠘⣦⠁⣀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠄⠙⢋⠀⠀⢠⡆⠛⠀⠀⠀⠀⠙⡿⠟⠁⠀⠀⠀⠐⣆⡍⠉⠀⠀⠀⠀⠀⠀⠧⡀⠂⠁
⠀⠀⠀⢀⠀⠀⠀⢀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠸⡇⢴⠆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢼⣖⡤⠶⡟⠁⠀⠀⠈⠈⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⣀⠀⠀⠀⢀⠔
⠐⢤⣤⣄⡀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠢⡧⠃
⠀⠀⠙⠂⠀⢈⡢⡄⠀⠈⠀⣾⢓⡄⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⠻⣄⠑⢀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢬⡙⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⡀⠀⠀⡴⣣⠴⠣⠀⢠⡏⠂⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡴⠉⠀⢸⠀⠀⠈⠒⢼⣿⣅⠀⡼⠀⢸⢀⢃⠔⠉⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡞⠀⠀⣠⢼⣲⠂⠀⠀⢸⣍⠻⠛⣦⡀⡎⢸⣷⠖⠲⣠⠤⢄⠘⣄⠀⠀⠀⢰⢆
⠀⠀⠀⠀⠠⡇⠀⠀⠀⢸⣷⡡⠀⠀⡘⢇⠀⢀⠉⠙⡿⠋⠁⠀⢀⠏⠀⠀⠻⣄⠱⡀⠀⠈⠀⡗⢄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⢿⣿⠖⢩⣷⡮⠭⠉⢷⣦⣇⠀⠀⠀⡏⠛⠁⠀⠀⠈⡶⠃⢐⠀⢸⣷⡬⠖⠋⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠃⠀⠊⠉⢻⣷⣶⠳⠤⢍⠳⡄⢾⠀⡄⢀⣠⣶⠟⠓⠤⢄⣆⢸⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⢀⠄⠚⡝⢻⡠⠀⠀⠓⠛⢻⣿⣿⡿⠋⢻⠀⠀⠀⠀⢿⣬⡝⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⡀⠀⠉⢸⠇⠀⢀⡄⢠⠞⠍⡞⠀⠀⢸⣖⣄⣀⣴⠋⠉⠓⢦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠔⠉⠉⣽⠁⠀⠈⠛⠚⡇⠀⠀⢘⣟⠛⣿⢸⣤⠢⠈⣢⡇⠀⠀⠀⢤⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠬⠷⣄⡀⣀⣧⣸⣄⠀⣾⣷⢰⣿⢸⣤⣟⠻⡁⠀⣀⡄⠀⠀⣧⠴⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠲⢿⡟⠳⡌⢙⠟⠀⠘⣿⠿⣷⢦⣿⣦⡃⣸⣿⣄⣠⡼⠯⠥⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠱⣄⣼⠸⠀⠀⡼⡇⢠⠃⢀⡿⠀⢻⢷⡄⣀⣀⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠷⣹⠙⠻⣿⡁⠀⠾⠞⢿⣟⣉⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⢧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠃⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠑⣆⢸⡃⠀⡼⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⡞⠀⣻⠛⣴⡉⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣯⠴⠛⠳⠒⠋
⠀⠀⠀⠀⠀⡴⠲⢤⣰⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⢧⠀⣸⠿⣤⡼⠃⠀⠀⠀⠀⠀⢸⠻⡄⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣠⠤⣍⠙⢆⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⡄⠀⠀⡏⠀⠀⠀⣠⠖⢻⠃
⠀⠀⠀⢸⡀⢀⣺⠗⠋⠑⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⢳⠀⢸⠁⢀⡴⠋⠀⢠⠏
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⡀⠀⠈⣇⡏⡴⠋⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢳⡀⠀⢿⠟⠁⠀⣠⠞⠁⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠑⣄⠹⡄⠀⠀⠀⠙⠲⣾⠒⠚⠉⠀⢀⠞⠁⡾
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⠀⠸⡄⠈⠳⡽⡄⠀⠀⠀⠀⡏⠀⠀⠀⡴⠋⠀⣸⠁⠀⠀⣰⠋⣹
⠀⠀⠀⠀⢠⠤⢄⠘⢴⡾⠀⠀⢳⡀⠀⠙⣿⡀⠀⠀⢀⡇⠀⢠⠞⠁⠀⢰⠇⠀⠀⠀⡜⢻⡗⠒⡆
⠀⠀⠀⠀⠹⣄⣾⠧⢤⣷⣀⠀⠀⠙⢦⡀⠈⢷⠀⠀⢸⡇⣰⠋⠀⠀⣠⠋⠀⣠⠴⠚⢿⣩⣿⡋⠁
⠀⠀⠀⠀⠀⡴⠻⣿⠋⠁⠈⠙⠲⢤⡀⠉⠓⠚⣇⠀⢸⣴⠃⠀⢀⡴⢃⡴⣋⡥⡶⠀⠈⠯⣤⠝
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢠⡦⢤⣄⡉⠲⡀⠀⢸⡀⢸⣇⣠⠴⠋⣰⣫⠞⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠙⠲⢽⣆⠀⢧⢸⠁⠀⠀⡼⡵⠁⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠿⠂⠈⠈⠀⠀⠀⠈⠛⠉⠁