Earth science Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠄⠈⢆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣊⣹⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⣠⠂⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⣦
⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟
⠀⠂⠿⠿⣿⣿⣶⣤⣤⣤⣄⣀⣀⣀⣀⡀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣴⣾⣻⣟⣿⠇
⠀⢀⣆⣀⣻⣟⣻⠏⠛⠉⠋⠙⠛⠛⠛⠛⠻⣿⠿⣿⡿⣿⣷⣾⡿⢷⡿⣿⠟⣿⣿⣿⡿⢿⣿⠛⠙⠉⠉⠉⠈⠻⣭⠋⣿⠀⢻⡄
⢠⣾⣿⣿⡿⣯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢎⣹⠛⢛⣻⡷⣤⡄⢿⠿⠃⠀⢉⡟⠀⠀⠀⠀⠀⠀⢀⠏⠀⠿⣷⣿⡅
⠈⢿⣿⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣾⡿⡻⣿⡟⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⣿⣿⣷⠁
⠀⢺⣿⡿⣯⠉⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣡⣿⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡰⠃⢀⡠⠾⣿⣧⠤⠋
⠀⠀⠉⠃⠐⢿⡆⠹⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⣿⡏⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣠⠈⠇⢰⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡶⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⣿⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠡⠈⢋⠻⣦⡤⣄⠀⠀⠀⠀⠀⠈⠱⠻⠇⠀⠀⢠⠖⠀⠀⠀⠀⠤⠈⢷⣤⠚⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⣤⡾⠷⣤⠙⢷⡆⣰⡆⣶⡆⢠⡜⠁⠠⢶⣶⣶⡆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡒⣼⣶⢋⣟⣽⡇⢿⣧⢿⢿⢢⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⡀⢿⣟⡟⣭⡗⠈⣍⣈⡾⢻⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡿⣃⠉⣇⣈⣭⣠⣿⣿⡁⢹⡬⢭⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡚⠛⡛⠋⢽⡛⠙⠿⠽⢿⠻⠷⠿⠁⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠙⠛⠋⢹⠃⠀⢠⣄⠀⠀⠀⣀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠶⠆⢠⣤⠤⣤⠀⠀⠀⠞⡇⠀⠘⢷⣤⣤⣽⡿⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢿⣄⡤⢠⣅⣠⣋⠂⠀⢸⡇⣸⠂⠀⣈⡻⣿⣬⣥⣬⣿⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢿⡃⣾⡏⠉⠙⡿⢻⣿⣀⣻⣾⣾⢿⣿⣿⣿⣿⡄⠁⠤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣄⠠⣧⣬⡃⠈⠁⠀⢮⣿⠶⢿⣼⢿⣿⠛⣈⣿⣅⠉⠉⠀⠀⢠⣉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡷⡿⠉⣧⣴⡦⠶⠿⢿⡀⣷⠋⠛⣿⡌⠉⠁⠉⠳⣦⡀⠠⠌⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⣿⣉⣻⣶⡇⢠⡟⠉⠀⠀⠀⣿⡟⢨⡆⠀⠻⣿⡏⢀⡴⠀⠀⠐⠖⠈⣧
⠀⠀⠀⣀⣀⠀⠀⣴⣿⣷⣿⣟⣉⣸⣿⡿⠃⠰⣷⠄⠈⠛⠀⠀⢸⠀⠀⣿⠀⠀⢐⣀⢠⠀⠀⠀⡭⣶⣀⣠⡤⠄⢒⣀⠀⠀⡀
⠀⢀⡀⣹⣿⣿⣿⣽⣿⣿⡿⠏⠁⠀⠀⢠⣎⠀⠈⢃⣀⣄⡀⠀⢈⣄⠀⣿⣦⠴⠞⠛⠉⢁⣤⡾⠗⠘⠛⠉⠐⠂⠛⠉⣀⣴⠛
⠾⢿⡟⠉⣽⠿⠿⠿⠛⠛⠛⠂⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⠉⠠⠄⠐⠊⠋⠀⢀⣤⣶⣿⣿⡿⠦⠄⠒⠒⠒⠚⠛⠛⠟⢻⣏⠀⣶⣦
⠀⠀⠻⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⣠⣄⣀⣤⣀⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⢉⢋⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣠⣿⣄⣧⣿⢻⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣈⣙⣣⣄⣿⣿⣿⣥⣤⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⡶⠄⢦⠰⠶⠄⠀⠀⠀⠀⠀⠠⢤⠀⠆
⠀⠄⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡖⡀
⡇⠀⠀⠀⠀⠱⠄⢸⠶⢶⣶⣦⡀⠀⢴⣎⠻⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠑⠊⠁
⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠿⣿⣦⣄⡈⠂⠀⠀⠌⠀⠀⠀⠀⠀⠀⡠⠐⠠⣰⣶⣴⢠⠀⣀⡀⢀
⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⠀⠘⠳⣮⣹⡇⠀⢀⢀⣀⣀⣀⣀⢀⠊⠈⠔⢀⠍⠀⠿⣶⠟⣛⠷⢔⢈⠢
⠀⢀⡾⠁⠀⢰⣿⣿⣷⣄⠀⠀⠀⠀⠈⠙⠃⠀⠈⠊⠉⠁⠉⠉⠀⠈⠈⢠⠇⠀⠀⢳⣏⠐⠀⠹⠀⠁
⢀⣿⡇⠀⠀⣾⠟⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢌⡃⡀⠀⣸⠁⠑⠀⢰⠀⡀⡀
⢸⣿⣧⡐⠀⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠘⠢⣹⣮⡿⠛⣷⣆⣺⡡⢂⠔⡠
⠀⢿⣿⣿⣦⢿⣧⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣬⠁⠀⠀⠛⠀⠙⠛⠀⠐
⠀⠀⠙⠿⣿⣾⣿⠿⠟⠁⠀⠀⡀⠀⢀⣀⡠⡦⠀⠀⠸⣿⣧⣁⠄⠀⢸⣿⣷
⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠠⠂⠁⠚⠋⠀⠀⢉⣠⣶⡇⠹⢿⣿⡄⠀⣠⡿⠃
⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣠⣴⣶⣿⣿⣿⣿⡿⡇⠀⠀⠉⠉⠉⠅⠄⠐
⠀⡂⠀⠀⠀⠈⠻⣿⣿⣿⣿⡛⠛⠛⠛⠛⣛⣟⡍⢤⡇⠸⠀⠀⠀⠀⠀⠀⠈⠍⠀⡀⢀⣤⢐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⡀⢀⣾⣿⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⢀⣘⣿⣿⢸⣿⣿⣿⡀
⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⢾⣿⣿⡟⢸⣿⣿⣿⣷
⢀⣠⣾⣿⣿⣧⣄⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁
⠀⠀⠈⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠛⠛⠛⠛⠛⠿⣿⠀⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⣉⣉⣀⣾⣿⣿⣶⣶⣶⣶⠀⣿⣦⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠻⠛⠁⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣶⣶⣶⣾⣿⠿⠿⠋⢠⣤⣤⣤⣄⣁⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢰⣿⣷⣶⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⣉⠉⠛⣿⡇⢰⣶⣶⣿⣿⣿⣿⣿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣠⡏⠠⣾⠿⠟⠃⣰⡿⠟⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠀⣿⣷⣄⣠⣌⣀⣈⣉⣉⣁⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣼⠏⢠⣾⡿⠓⠀⠤⣤⣶⡿⠋⣠⡶⠟⢋⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠶⠾⠿⠿⢿⣿⣿⠁⣤⣤⣤⣤⡈⠻⠿⠿⠁
⠀⠀⠀⠀⢀⣾⣿⠀⣿⣿⠀⢾⣷⣦⡈⠻⡇⢸⣿⡅⠸⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⡆⢸⣿⣿⡀⠙⠛⢿⣿⣷⡶
⠀⠀⠀⠀⣸⣿⣿⣆⠈⣿⣦⣀⡉⢉⣉⣤⣷⡈⠻⣿⣷⣄⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⣿⣿⣿⡿⠀⣿⠟⠋⣡⡄⢹⣿⣿⣿⣦⡈⠻⣿⣧⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⢸⡿⠋⣠⡾⠋⣠⣾⣿⡇⠸⣿⣿⣿⣿⣷⣄⠙⣿⣿⣿
⠀⠀⠀⠀⢀⣤⣾⡟⢠⣾⠁⣤⣄⡉⢢⣈⠙⢿⣿⣿⠿⠂⣸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠻⣿⣇⠘⣿⠀⠟⠋⣠⡼⠟⠃⣠⡟⢁⣴⣾⣿⠋⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠑⠲⠿⠟⢀⣶⣿⡟⢠⣾⡿⠟⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣾⠿⠿⠀⠚⠉⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⣠⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⣡⣤⡶⠟⠋⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣦⣄⣉⡉⠙⠛⠛⠋⢉⣉⣉⣠⣤⡴⠶⠟⠛⢉⣠⣤⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠋⣉⣉⣤⣤⣶⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠙⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠄⠀⡀⢠⠀⢤⠀⠈⣁⣶⠤⢄⡀⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠐⢚⢀⠛⠁⠀⡀⢇⠈⠁⠂⠌⡑⠠⡀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠁⠁⠀⠀⠀⡴⠒⠋⠘⠚⢤⡰⠈⠀⡀⡆⠀⠀⠀⣘⡑⢌⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⠓⠤⡀⢀⠖⠀⣰⡌⡼⠉⡅⢀⢀⡔⠁⠀⠀⠱⢄⠳⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣀⠟⠀⠈⠓⠋⠀⠀⠲⠱⠢⠀⣷⡈⢂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⢀⣤⣀⣀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠁⠰⡃⣶⡈⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⡰⠃⠀⠀⠈⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⠔⢀⠀⢀⣘⢷⣧⠈⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⡢⠀⠀⢼⣉⠀⠀⠀⠀⠀⠀⠀⡠⢂⢄⠔⠁⠀⠑⠰⡏⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣣⠀⠀⠓⠉⠳⡀⠀⠀⡀⢔⠨⡂⠔⠁⡀⡠⢂⠞⠉⠄⠀⠀⠀⡂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠉⠦⢀⠀⠀⠀⡡⢔⠩⢐⠁⡊⠐⡡⡪⡊⠀⢥⠀⠀⠀⣸⠋⢭⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠑⢮⡔⠨⢀⠨⡐⢅⡡⢊⠔⡀⠈⣑⣤⢾⠡⡤⠶⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⢹⠤⣀⠠⢂⠡⢀⠀⠪⠀⡴⠋⠀⠀⠀⠀⠀⢀⠁⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⢄⣐⠂⠅⢰⠃⠀⠀⠀⠀⠀⠀⢰⠃⠈⡀⠨⠑⠨⡐⠀⠈⢪⡅⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠐⠈⠑⠪⠁⠀⠀⠀⠀⠀⡰⢂⠢⠉⡀⠄⡀⠄⡊⠄⠪⡆⠀⠀⡠⠄⠡⠀⡀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡁⠄⠐⡆⠀⠀⢀⡠⠒⠁⢀⠢⠁⠀⠁⡂⠈⣂⠅⡢⠈⢝⡆⠍⠐⢀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⣞⠀⡰⠊⠂⠠⠀⠂⠠⠐⡀⠤⠀⠠⠀⠁⠨⠐⣼⠤⢋⠔⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢈⣿⠲⠦⣤⢀⣀⠀⣐⡀⠄⣂⣡⡀⠱⠖⣈⠡⠂⠁⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣂⠀⠈⠁⠒⠂⠤⠠⠉⠤⠤⠤⠄⠐⠂⠉⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⡤⢤⣄⣀⣀⣀⡀⠠⠤⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⣤⣗⡛⠛⠍⠁⢢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠴⢁⡀⠀⣀⣀⠉⡓⠐⡀⢐⠀⡈⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠐⠀⠐⠀⠉⠈⠉⠒⠲⠤⠤⠭⠯⢭⠍⣭⣭⡼⣭⠯⠷⠛⠄⠂⠄⠀⠔⠐⠋⠐⠁⠋⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⣀⠀⠀⣀⣄⣀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡖⠀⠀⠀⠡⠖⢢⣀⠁⢀⡴⠐⠈⠀⠀⠀⠀⠉⠉⣁⣀⡳⠀⠒⠠⠄⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠓⠒⠂⠀⠐⠦⢄⠠⡴⣀⠀⠀⡐⠉⠉⠁⢄⡲⠄⣤⣀⡔⠃⠀⠀⠀⠀⠀⠀⠘⠋⠁⢌⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⢀⡇⠀⠈⠒⡇⠀⢠⠁⠀⠈⠉⠈⢢⠂⡂⠀⢀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠑⢤⣀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠜⠀⠀⠀⠀⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠎
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠛⢻⡳⠒⣶⡆⣤⠤⣩⣤⠬⣥⣤⣤⣤⣤⣤⣤⡤⢤⢤⣤⢤⢤⠰⠶⠂⠉⠻⠟⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠉⠛⢮⣁⣈⡱⢬⠉⠀⠀⠘⢨⠁⠂⠜⠁⠀⠀⠀⡈⢃⣦⣄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠠⡤⣤⣄⠈⣠⣤⣴⣴⣶⣾⡴⠗⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠘⠉⠀⠀⠀⠀⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⣶⣦⣲⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢹⠊⠁⠀⢀⣀⣀⣧⣀⣀⡈⠉⠓⠻⢶⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠹⡄⣠⠔⠉⠀⠀⡇⠀⠀⠈⠙⠲⢄⡀⠙⠪⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢀⣠⠤⠤⢌⣀⣠⠤⢄⠀⣧⣀⡀⠀⠀⠀⠀⠈⠢⡀⠘⢟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠀⠉⠑⠢⣄⠀⠀⠀⠈⢦⠈⢫⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠈⠳⣄⠀⠀⠀⢣⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⢆⠀⠀⠈⡆⠀⠀⠈⡆⠸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⡇⢰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠓⠤⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⠀⠀⡜⠀⠀⢀⠇⢸⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣀⡴⠋⠀⠀⡰⠁⠀⠀⡘⠀⡞⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⠉⠁⠀⠀⢀⠔⠁⠀⠀⡰⠁⡸⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠞⢹⠀⣀⡠⠔⠁⠀⠀⠀⡔⢁⠜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡀⠀⠉⠙⠚⠉⠀⠀⢸⠉⠁⠀⠀⠀⢀⠄⢊⠔⣫⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣉⠉⠳⢤⣤⣤⣤⣀⣀⣸⣀⣀⠤⠐⣊⠥⢊⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⣀⣀⣀⣻⣐⣒⣨⠥⠒⠚⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠒⠲⠒⠒⠓

⠤⣒⡦⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀⠀⣀⣄⣤⡀⣰⠓⢖⠟⢁⠔⣊⢭⡙⢶⣤⠀⠀⡠⠃⢆⠀⣿⡆
⠈⣉⡈⠀⢲⠐⡆⢲⠐⢮⣁⠀⠀⠀⣈⣽⠊⠙⠿⣿⣧⣀⠠⠂⠇⠜⠰⠚⢎⠊⣯⣇⠸⣷⣤⣶⢿⣋⣃
⠐⢧⠗⠀⠘⠀⠇⠘⢈⡳⠮⣍⣒⣫⠥⠾⡁⢀⠀⡄⢛⠓⠤⠀⢆⢣⡐⢴⢆⢦⠉⡎⠱⣌⣿⣭⡜⢢⡏
⢒⣒⢐⣒⢒⣒⢒⣒⠀⠉⠒⠤⣉⠤⠒⠉⠱⡪⠀⠈⠀⡒⠀⠀⠈⠢⢌⣹⣵⠯⠋⠀⠀⢢⠭⡂⠈⠉⠁
⠁⠀⠁⢀⡁⠀⠁⠀⠁⢀⠤⣴⣶⡤⡀⠀⢑⠂⠀⠀⡀⢚⠁⢀⠀⠀⠀⠄⠀⢀⣴⠀⠀⠈⠉⠂⠀⣤⠠
⠰⠢⢄⠛⢛⡀⢘⠀⠀⠁⢠⣦⠀⠀⣸⡆⠀⠘⠈⠌⠒⠀⡀⢢⡇⠤⠄⠁⢠⡻⠛⢆⠀⢴⣘⠆⠀⠾⠰
⠒⠎⢐⡇⣯⠐⠤⢷⠠⡀⡾⠟⠂⠘⣿⠇⠀⠀⠀⠀⠀⢌⠙⠶⡙⠃⠐⠠⡍⣠⡀⡸⠀⠤⠥⠤⠤⠤⠤
⣮⣈⡧⠝⡈⠂⠋⠁⠀⠘⠢⠄⠠⠟⠋⠀⣠⢔⠤⣍⠆⢀⠑⠤⠘⠀⠀⠀⣼⣧⣿⡄⠀⠋⠙⠋⠙⠊⠙
⠀⠤⠀⠀⢑⢀⠀⠀⡀⠔⠈⠒⠒⣄⠀⣠⠈⠀⢀⡄⠀⠈⠵⠀⠀⠀⣤⣤⡄⠀⠀⢀⡤⠤⢄⡀⠿⠗⣀
⢀⡀⣀⠀⢠⣄⠁⡨⣀⣀⠀⢀⣀⡤⠟⠂⡃⠀⣶⠈⡄⠀⢏⠗⠤⠄⠈⠀⠁⢀⣀⠸⠀⠰⠒⢸⢆⡎⠠
⠈⠃⠚⢱⠇⠀⠁⠘⠀⣠⢩⢇⠀⠀⠀⠘⠵⠀⢈⣍⡠⢴⠄⠀⠀⠀⠀⣀⣤⣄⠉⠑⢦⡊⢒⣋⠖⠹⠤
⡠⠀⠀⠛⠒⠀⠀⠀⠀⠐⠃⠚⠀⠀⠀⣀⠀⡀⠃⠚⠋⣀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⠛⠋⠀⠘⠃⠠
⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁⠴⢲⣴⡶⠄⡿⠀⣯⣀⠀⣾⢹⢷⠂⣬⣴⡢⣴⢤⣰⠢⣴⡢
⡇⠀⠀⠀⠀⠀⠀⠀⠻⠀⠺⠼⠛⠃⠀⠧⠖⠇⠸⠀⠨⡷⠘⠒⠻⠻⠄⠿⠸⠛⠒⠻⠄
⣬⣄⢠⠶⠖⡶⠂⡰⢂⠠⣆⢶⣶⡀⠀⠀⠐⡖⢦⠀⢲⣶⣶⣶⣦⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠂⠈
⠻⢿⣿⣄⠠⠐⡌⠰⢡⠖⡧⠣⢻⣵⠀⢰⠦⠷⡚⠀⢸⣿⢳⣿⣿⣿⡇⠀⢻⠛⠀⢰⣷⠀⠀⣀⠤⠒⠂
⡀⣀⢙⡛⠢⠄⢡⠰⡈⠢⡞⡄⡌⡿⠀⠀⡄⠀⠌⡃⠀⠉⠉⠻⠿⠛⠀⣠⠂⠘⣢⣸⢿⠀⣉⠦⣄⡀⣀
⠅⠀⠢⣲⠂⠀⠀⠑⠌⣙⣯⠮⠝⠁⠀⠀⠘⠂⠒⠁⠀⠀⢂⠈⠀⠄⠸⠻⢿⡿⠟⣇⢦⣆⠤⣛⠒⠬⠥
⡀⠀⢀⢢⡃⠀⠀⠀⠀⢠⠀⠀⣠⠆⠀⢰⣉⡇⠀⠀⠀⠰⢠⠒⠠⠐⠀⠑⠺⠗⠚⠭⠿⠜⠀⠀⠉⠒⠒
⠂⠤⠐⠂⣀⠨⣠⠠⠂⠂⠀⣎⠿⠧⠀⠀⠀⠀⠀⢻⠘⡏⢻⠀⠀⡀⡀⠀⢠⣄⡀⢀⡀⠀⠠⠂⡿⠟⠟
⠀⠀⠀⠠⠉⠳⣎⠛⠑⠀⣼⢁⣄⢠⠃⠘⠦⠋⠀⠘⠀⠃⠘⠀⡊⠀⣉⡓⣌⢩⠔⠂⢄⠀⠁⢰⣿⡀⢀
⡂⢌⠰⠀⡀⠠⠘⠃⠀⠀⢠⣿⣽⣧⠀⡪⢑⡪⢅⣊⢝⣊⢍⠠⡅⣀⠅⣠⢯⡼⡀⢐⡸⠂⢢⠚⠋⠁⢀
⠀⣤⠀⠈⠊⠄⠀⠀⣠⣤⣄⠀⠀⠀⡠⠤⢤⡀⠺⠷⢀⠐⠀⠀⠘⠖⠤⠗⠊⢤⠀⠀⠀⠀⠀⠁⡒⠐⠋
⢸⠀⡃⠀⠪⡸⠢⡤⠀⠁⠉⠀⣀⠀⠆⠀⡖⠠⡧⣸⠁⣈⣢⠀⠀⠀⠘⠀⠀⠐⠑⠄⠀⠔⠐⠀⠈⠉⠱
⢀⣉⣀⢤⡄⠀⠀⠀⠀⢀⣤⣤⠉⠙⢦⣖⠑⣎⡱⠙⠦⠔⠊⠀⠀⠠⣄⠠⣄⡠⠂⢌⢢⡢⠴⠄⣤⠤⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⠯⣤⣀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠈⠻⣆⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⣼⠃⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣥⣤⠴⠟⠁⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠛⢉⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡟⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⢦⣤⠞⠁⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⢀⣤⠤⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⣤⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣄⡞⠁⡠⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡈⠻⣄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠈⣿⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠳⠦⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠉⠙⠓⠶⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢠⠔⣡⡶⠛⠉⠉⠀⠉⠉⠳⣦⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢰⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠒⠂⠐⠒⢁⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠶⠛⠉⠁⠀⠀⠀⠉⠳⣌⠑⣄⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉

⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢤⣴⠖⠛⠛⠓⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠛⣉⣉⣥⣤⣤⣴⣷⣦⠶⠶⠶⠶⠶⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⣉⣤⠴⠞⣻⣽⣿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⣡⡴⠛⢉⣠⠶⣛⡥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣡⡴⠛⠁⣠⡴⠋⣡⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠉⠀⣠⠞⠋⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⢠⡞⠁⢀⡴⠋⠀⢀⡼⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⢠⡏⠀⢀⡾⠁⠀⢀⣾⠃⠀⠀⠀⠀⠙⠶⣤⡀⠀⢀⣤⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⠉⠀⠀⠀⢀⣠⡶⠶⠛⠻⣆
⠀⢠⡟⠀⢀⡾⠁⠀⢠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⢀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⠷⣤⡿⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠹⣆
⠀⡾⠀⢀⡾⠀⠀⢠⡟⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠘⠷⣦⡴⢦⣄⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣀⠀⢹
⢰⡇⠀⣼⠁⠀⠀⣾⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⢷⡄⠀⠀⠀⠙⠳⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣤⣀⠀⢰⣋⣽⠀⠈
⣸⠀⢀⡏⠀⠀⢠⡟⠀⠀⠹⣦⡀⣀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠃⠀⠀⠀⠀⠀⠘⣧⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢿⣄⠀⠉
⡟⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠛⢦⡀⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡗
⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡾⠁⠘⢧⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠃
⡇⠀⢸⡇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣼
⣇⠀⠘⣇⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⢰⡏
⢹⡀⠀⢿⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡟
⠈⢷⠀⠘⣇⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠙⠳⢦⡾⠁
⠀⠈⢷⡀⠘⣦⠀⠀⠀⠻⣄⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠳⣄⠘⢧⡀⠀⠀⠙⣦⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠈⢳⣄⠙⢦⡀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⣤⣤⡤⠶⠒⠚⠋⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠳⣄⠉⠳⣤⡀⠈⠳⣮⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡙⠳⢦⣄⡙⠳⢦⣄⣀⠀⠀⠀⢀⣀⣀⣰⡏⢰⣏⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣬⣉⡛⠳⠾⠿⠿⠷⣾⣿⣿⣭⣭⣤⣤⢼⣿⣶⡶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠶⠶⠶⠶⢯⣤⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⣿⠀⠀⠀⢹⡗⠒⠒⠲⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠙⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣉⠙⠓⠲⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⠶⠖⠚⠛⣉⣥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⡾⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⡀
⠀⠀⠀⠀⢸⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⢿⣿⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣀⣀⣠⣀⣤⣦⣶⣿⡇
⠀⠀⠀⠀⢀⣄⣘⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠿⢷⢾⣿⢾⣿⣶⡷⢶⣾⣷⢿⣿⢿⡿⣿⡿⠛⠛⠛⠉⠉⠩⡟⢹⡎⢻⡄
⠀⠀⠀⢀⣾⣿⡿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣩⠟⣛⣻⣷⣤⠘⠿⠋⠀⢹⡇⠀⠀⠀⠀⠀⢠⠋⠸⣷⣿⡅
⠀⠀⠀⠀⢹⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠛⣿⠋⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⡰⠁⠀⣀⣿⠿⢶
⠀⠀⠀⠀⠸⢿⠹⢧⣀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣴⡏⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠈⠁⣴⠚⢻⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠒⠈⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠡⡐⠲⢤⣄⣀⠀⠀⠀⠀⠀⡙⢣⣿⡇⠀⠀⢠⠃⠀⠀⢀⠀⢆⣀⣠⢾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⣁⣽⣦⡀⢶⣤⢠⡌⣧⡭⣤⠐⢁⣤⣤⣤⡀⠄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠠⣼⡌⣻⣿⡇⣿⡧⣿⠖⣴⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⢘⣿⡿⢩⣧⢨⣁⣩⡾⣿⣿⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣟⣅⣠⣤⣵⣤⣽⣿⢀⣗⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠉⣍⣈⣭⣝⣇⣛⠏⠙⡟⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⡀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢧⡀⠀⢷⣤⣀⣴⠆⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢉⠘⠋⢸⠃⠀⢠⡬⣳⠀⠈⠻⣿⠛⢋⢻⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⢻⡏⣼⡟⠛⢷⡴⣾⡃⢿⣴⣴⣷⣿⣿⣿⡈⢃⣀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣬⡇⠙⠁⠠⣾⣶⡿⣼⣿⡿⢁⣹⣏⠘⠋⠀⠀⣈⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⠉⣣⣤⠶⠾⠿⣆⢴⠛⠙⣷⡈⠋⠈⠻⢦⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣯⣭⣿⣿⣠⡟⢀⡀⠀⢸⡿⠑⢇⠀⠻⣿⠁⢜⠃⠀⠘⠃⠸⢆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⣴⣦⣀⢴⣿⣷⣿⣯⠤⠾⠿⠋⠘⢿⠀⠈⠁⠀⠸⠀⠀⣇⠀⢀⣦⢤⡀⠀⣀⠟⣦⠶⠒⣂⣥⠀⡀⣤
⠀⠀⠀⢀⣤⣬⠾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠘⠓⠀⠠⠤⠾⠦⠆⠀⠃⢸⠟⠛⠉⣁⣤⣬⣟⣉⠀⣁⣀⣤⣤⣄⣴⡼⠇⡀⡀
⠀⠀⠀⠀⠙⢶⣶⡯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣴⣾⣿⠿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠁⠳⣦⣿⠋
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣾⠛⠟⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣽⡌⣯⡿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠠⠿⠿⠿⠤⠄

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⣙⢻⡋⣉⠛⣛⣟⠋⣿⡿⣻⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⠶⠆⢀⣀⠀⣀⡀⣤⡀⣦⡀⠀⠀⡴⠦⠀⣀⠀⡒⢀⣀⠀⣀⡀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⣟⣉⢰⣏⣿⢸⠋⢠⣏⢰⡏⣿⠀⢀⣙⡦⣾⣁⢨⡇⣿⣚⢱⠇⣿⢾⣁⢰⣟⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢠⣴⢈⡁⠐⠢⡑⠄⠀⠈⢂⠞⠁⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⡴⣷⡄⠀⠀⠀⠀⠀⢠⣇⡀⠠⣿⣽⠒⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠆
⠀⠀⠀⠠⠄⠀⡟⠊⣄⠀⠀⠀⠀⠀⠀⠉⠀⢹⣿⣀⠄⡘⢀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⣄⠀⢁⠁⠂⠀⠀⡀⠀⠂⠀⠀⠈⠀⠄⠙⠂⠈⡠⢊⠜⠀⠀⠀⠀⠀⠀⠺⢁⡀⠂
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠈⠂⠝⠬⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠈⢀⠴⠲⢂⠁⠈⠀⠀⠠⠀⠀⠀⠀⠑⠀⡀⠁⠉⠁⠀⠀⠀⠢
⠀⠐⠐⠐⠀⠀⢡⣰⣮⡰⢈⢂⡁⠈⠀⠀⠀⠀⠐⠀⠀⠤⠤⡀⡀⣓⢀⣀⢠⠤⠖⠚⠃⠀⠐⣰⠢
⠀⠀⠀⠀⠀⠠⡶⠚⠋⠀⠰⢀⣁⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⠉⠁⠀⠀⠤⠀⠈⠁
⠀⠀⠲⠤⠉⠈⠀⠐⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠐⠀⠄⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠨⡤⠇
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣄⢀⠄⠀⠀⠀⡄⠠⡌⠐⠂⠀⠂⠀⠀⠀⢀⠔⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠁⠈⠀⠀⠀⠀⢲⢈⡀⠀⠀⠀⠀⢀⠉⠂⣠⡴⠔⠒⢉⡀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠈⠀⠂⠁⠀⢀⡁⠀⠀⠀⠉⠀⠔⠀⠳⣀
⠀⠀⠎⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠄⠀⠀⠀⢀⣀⠀⠀⠨⠁⠀⠀⠸⠀⠅⠁⠄⠨⠉⠑⢦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠈⠁⠀⠀⢡⠅⠀⠀⠟⣤⡄⠢⢳⠄⠐⢲⠀⠀⡇⡘⠀⠀⠀⠐⠀⠀⡆⠀⠜⠁⢈⠛
⠀⡄⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⠀⠀⠀⠈⠉⠢⠀⠤⠈⠀⠈⠃⡁⣿⠁⠀⠈⢀⠀⠂⠀⠀⠀⠀⢠⠀⡌⠀⠠⠆
⠀⠀⠀⠉⠀⠈⠀⠀⢀⠀⠠⠀⠓⡠⠚⠂⠀⠁⠀⠐⠠⠀⠀⠀⢠⠘⡻⠄⠌⣘⡓⠢⠀⠀⠀⠀⢄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠁⠀⣤⠈⠀⢨⢀⠃⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢎⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡠⠤⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠨⠂⠠⠂
⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠁
⠀⠀⠈⠀⠤⠀⠠⠂⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡀⠐⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢎
⠀⠀⠀⠠⠀⣀⠄⠠⡀⠀⠀⠈⠐⠁⢀⠊⠁⠀⠀⠀⢠⠀⠀⣄⠀⠘⠁⡀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡀⠈⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⠁⠀⠀⠂⠀⢸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡁⠀⠈⠀⠀⠁⣀
⠀⠀⣀⠀⠀⠀⠄⢀⠀⠀⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠣⠀⢆⠀⠀⠄
⠀⠀⠀⠀⠂⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠠⡤⠉
⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡎⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⢀⠀⠐⠀⠀⡀⢀⡄
⠀⣤⣶⣶⡀⢁⠰⡦⠀⠀⠀⢀⠠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠁⠀⠀⡀⠀⠀⠀⠐
⠀⣿⣹⠟⠀⠻⠃⢤⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠄⠈⠀⠀⠰⠆⠠⠄⢠
⠀⠻⠿⠀⠀⠀⣀⡀⠉⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠰⠖⠂⠀⠀⠀⠀⠃⠈⠂⠤⠤⠐⠈⠑⠒⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠊

⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⠐⠀⠀⠈⠓⠀⠀⠀⠘⠒⠂
⠀⠀⠀⠀⠀⢀⠀⡤⠤⠀⡀⠠⢀⡀⠀⠀⠀⡀⠀⠂⠈⢆⠐⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠁⠈⠀⠀⠀⠀⠇⠈⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠈⠀⠈⠁⠀⠄⠐⠦⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠦⠄⢀⠠⡄⠀⠀⠀⠏⠀⠀⠀⠀⠂⠀⠀⢏⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⢀⡀⢀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢳⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠉⠃⠀⠀⠠⠀⠢⠤⠀⢀⡀⠠⠄⣀⠀⠀⡀⠀⡀⠉⠀⠀⠀⡀⠠⠄⠀⠈⠀⠀⠉⠀⠀⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠠⠆⠀⠀⠐⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠠⠁⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⢢⡄⠄⠀⢀⠀⣤⣀⠀⠀⢀⡀⠀⠈⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠘⠛⠚⢿⡑⠶⠞⠵⠂

⣷⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⠀⠀⡠⠄⠀⡄⠄⡄⠀⠀⠀⠀⠄⠀⠀⡀⢀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠄⡀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠼⡗
⠀⠀⠀⠁⠈⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣿⣶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠠⠀⠤⡀⠄⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⠄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠀⠂⠐⠂⠉⠀⠆⠘⠀⠂⠐⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠂⠀⠂⠀⠈⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠐⣀⠀⢇⡂⠘⠉⠀⠀⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠐⠖
⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠐⠛⣁⠀⠀⠀⠁⠀⢠⣦⡄⠉⠀⢸⢀⠀⠈⢠⣤⡁⠀⠀⠀⠈⡀⠐⠀⠀⠆⡀⠀⠉⠃⢠⡴⠂⢤⡔
⠀⠀⠀⠀⠀⠀⠲⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠆⣀⡃⢰⠈⢈⢀⠀⠀⠀⠘⠃⠀⠼⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠺⠂⠠⣤⢀⡀
⠀⠀⠀⠀⠨⠂⢀⡼⣷⡀⠀⠑⠰⠀⠀⠀⢀⠈⢠⣸⣆⣀⠀⡵⢸⠀⠄⢄⠦⠀⠀⠀⠀⠀⢀⡄⠀⠀⡄⡆⠀⠸⠞⠀⠘⠃⢠⣿⣷⣿⠀⠳
⠀⠀⠀⢀⠀⠀⠈⢏⣀⣠⡀⡂⠀⣀⡆⢠⣾⣌⡄⣘⣿⣹⣶⡇⣬⣰⡤⠘⣫⣅⣄⣴⣠⣦⣤⡷⡖⠀⣽⣷⣗⣴⣴⢶⣔⡄⠀⣩⢸⣿⡇⣡⡦⡄⡀
⠀⠀⠀⠈⠾⠰⠿⠾⠷⠿⠷⠿⠿⠿⠧⠿⠿⠿⠷⠿⠿⠽⠿⠷⠿⠿⠿⠿⠿⠿⠏⠭⠿⠿⠿⠷⠿⠻⠟⠿⠿⠿⠿⠼⠿⠧⠧⠿⠿⠿⠿⠿⠧⠧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡿⠋⠉⠹⣿⣿⣿⣿⣶⣦⣄⣠⣴⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣿⡄⢀⡴⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣤⣤⣴⣾⣿⣿⣿⣿⣷⣤⠀⠀⢸⡀⠐⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⣴⣿⣿⣿⣿⣿⡿⠿⠻⢿⣿⣿⣷⡀⢸⡅⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢛⡻⠋⠱⠀⠀⠀⠀⢹⣿⣿⡇⠘⣇⠀⢀⠀⢁⡈⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⢳⠀⠀⠀⠀⠀⠀⣸⢯⣿⡇⠀⠻⠂⣸⣤⡶⠀⠀⣘⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣶⠋⣿⣿⠉⠇⠀⣶⣾⡋⠀⢠⠚⡿⠧⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠘⠃⠀⠀⠀⣀⠠⠀⠘⠷⠀⠀⣤⡾⢋⣼⣿⠃⠘⠆⠐⣻⠿⠃⠀⣸⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⣤⣀⢻⠃⣠⣾⣿⣿⡆⣦⢀⡄⢻⡗⠀⠀⢉⣩⢠⡾⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠉⢹⡀⠉⢹⣷⡾⠟⠉⠻⢿⣿⡀⢹⣤⠴⡃⣰⣇⣈⣧⣿⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠀⠀⠀⠂⠀⣾⠇⠀⠋⠁⠀⠀⠀⠀⠀⠈⠀⠈⢧⡴⠇⢿⣀⣩⡶⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢾⣆⠀⠀⠀⠀⠀⣀⣄⡀⢀⣿⣇⣀⣀⣹⣍⣀⣩⡿⠋
⠀⠀⠀⠀⠀⠀⢠⣼⣧⠀⢄⠀⠀⠘⠀⠀⠀⢀⣀⣴⣿⡄⠀⠀⢀⣾⣿⣿⢷⡟⠋⠙⠋⠹⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢾⡇⠀⠀⠀⢂⠀⠂⣙⣿⣿⡇⠀⠀⣼⣿⣿⡉⠀⠀⠼⠄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣷⡀⠱⢣⣏⣤⣾⡿⣿⣿⠁⠀⠀⣿⣿⣏⠁⠈⠀⢤⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣛⣿⡗⣿⣫⣽⣿⣿⡿⠃⠀⠀⠀⣿⣿⡿⠇⠀⢸⠘⢖⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠸⣿⣿⣵⠀⣾⣇⣸⣗⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡐⠞⠷⢶⣤⣀⣠⣤⣤⣴⣿⣿⡿⠿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠋⠀⡀⢻⠠⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣆⣀⠐⢡⣀⣤⣼⠏⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⠀⢿⣧⠀⠀⠠⠄⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠤⣄⠀⠀⠀⠀⠀⡠⠔⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠔⠒⢒⡬⠖⠒⠒⠒⠾⣵⠒⠒⢄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠳⡄⢨⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⢠⣤⡀⠀⣀⠀⢠⣤⡀⢹⠋⠀⢸⣳⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠚⠁⡜⠈⡇⠈⠛⠁⢸⠀⠀⣠⡏⢞⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠹⣤⠃⣠⠖⢳⠊⠀⠀⡇⠙⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡑⠒⠒⣾⠁⣀⠏⠀⠀⠀⠳⢤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⢺⣀⡵⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢸⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠋⠉⢢⡀⢸⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⢦⡀⠀⠀⠀⠈⠉⠒⠒⠲⡄⠀⠉⠒⠲⠤⠒⠊⣡⠜⢢⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠘⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠳⡀⢀⠀⠀⢀⡴⠊⠁⢀⡜⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠢⡄⠀⠉⢦⠀⠀⠀⠀⠀⠈⠛⠀⡴⠋⠀⠀⡴⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠇⠠⡡⠰⡹⠀⠠⠀⡎⠐⠁⢺⠰⠈⠉⢢⠂⠀⠰⡱⢘⠅⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠆⢰⠶⠔⠂⠢⠖⠶⠀⠆⠐⠢⠰⠰⠴⠄⠶⠆⠆⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣴⢂⣼⡄⢀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣿⣿⣿⠂⠉⣽⡦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢰⣄⠀⠻⡿⣷⠞⠻⠋⠁⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠷⣬⣉⣽⣦⣤⣽⣿⣿⡿⢳⣤⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⢤⣶⣿⡛⣿⣤⡽⣿⣿⣷⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⢸⣿⣿⣿⣾⠟⠃⠀⠙⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣭⣷⠀⠠⠾⠀⠸⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣧⡿⠆⠀⠀⣀⡀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣷⠃⠀⠀⠀⣠⣿⡧
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣋⢙⢻⢿⣉⠉
⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣶⣶⡔⣉⣹⠋⠉⠉⢹⠈⠛⢛⠀⣤⠶⣶⡀
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⢂⠐⡻⠖⡏⠀⠀⠀⢸⡗⠦⠈⠉⢙⠧⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠹⣾⢠⣶⠃⠀⠀⠀⢸⡗⠀⢛⡼⢋⣡⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣼⣷⠘⠈⢻⠁⠀⠀⠀⠀⠗⠀⠈⣴⣟⠁⠀⢸⡀
⠀⠀⠀⠃⢀⡀⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡌⢃⠐⡟⠀⠀⠀⠀⠀⣧⢃⣴⣿⡟⠀⠀⠈⡇⠀⠀⢀⣦
⠀⠀⠀⢀⠞⡉⠀⠢⠀⠀⠀⢀⣿⣿⣿⣿⡇⠈⡎⠇⠀⠀⠀⠀⠀⠿⣼⠋⡉⢡⡀⠀⠀⠀⢠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⣻⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢼⣿⡆⠀⠀⢸⣿⣿⣧
⠀⠀⠠⠀⠇⢀⡁⢤⣀⣾⣿⣭⣉⣿⣿⣿⡇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠚⠀⠈⠳⣬⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣄⣀⣀⢀⣀⣀⣄⣠⣶⣾⣿⣿⡇⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠰⠆⠀⣤⣀⣸⣿⣿⣿⢿⣿⣿⣿⣿⡟⠻⢿⣽⣿⡿⢻⣿⠿⠟⡻⢿⠫⠙⢿⠁⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠿⣿⣿⣿⣷⣾⣧⣴⣿⣧⣿⡿⠀⠀⠀⠀⠈⠁⠈⠀⠨⠁⠀⠀⠀⠀⢈⣀⠴⡴⣿⣿⣿⣿⡇
⠀⠀⠀⣄⡉⠹⣻⣿⣿⣧⣼⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣠⣘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢲⠿⣿⣿⣿⣿⣿⣧⣾⣧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢤⣌⡛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠸⣻⣮⠿⣿⣿⣧⣹⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠚⠈⠁⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢶⣥⣿⣄⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠆⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⢚⡿⢾⣿⣿⡟⢿⣇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡄⢣⠙⣿⡃
⠀⠀⠀⠀⠀⠈⠶⣿⣿⣿⣵⡿⠉⠁⣴⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⢺⣯⠸⠋⠁⠀⣸⡇
⠀⠀⠀⠀⠀⢀⣶⡴⢻⣿⣿⣷⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠿⠂⠉⠀⢀⣀⣼⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠙⠀⢡⡘⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⢷⣁⠌⣾⢿⣿⣿⣿⣿⣿⢽⡾⡏⢻⣿⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⠈⠻⣿⣿⣿⣿⣿⠈⠁⠃⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣻⡿⠃⠀⠀⠀⠈⢿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠄⠂⠐⠦⠀⢠⡆⠀⠀⠀⠀⠀⢀⡜⢈⢉⢣⠀⠀⡈⠀⢰⣦⠀⠀⠀⠂⠉⢾⣧⡀
⠀⢀⠀⠠⣤⠀⠲⣤⠷⠶⠉⠀⠀⠀⠀⠀⠘⠃⢠⣤⣽⣆⠀⣏⡈⢀⣿⣄⠀⠐⠶⠀⠙⢿⣿⡀
⠀⠈⠀⢰⠳⢦⡜⠙⣦⠀⠀⢠⡇⠀⠀⠀⠀⠀⠠⠋⠀⠘⠂⢻⠃⠪⠈⠌⠦⠀⠘⠀⠀⡀⠀⠻⢧
⠀⠀⠀⣤⣄⣤⠗⠴⠛⢀⠀⢸⠁⠀⠀⠀⠀⣆⡰⠀⠀⠀⠀⡆⠢⠀⠀⠈⠺⢦⡀⠐⡿⠀⢰⡗⠈⣷⡀
⠀⠀⠰⢤⠈⣁⣘⠒⠴⡁⢠⠞⠀⠀⠀⠀⠀⢣⡆⠀⠀⠀⠀⠘⢶⣄⠀⠃⠀⢤⠅⡎⠁⣀⣼⠇⠀⢹⣿⡄
⠀⠀⢤⣀⡉⠀⢀⡄⣰⡗⠋⠀⠀⠀⠀⣠⠏⠉⠑⣚⠿⣿⡛⠛⡛⢌⠉⠛⠀⣿⣶⠿⠛⠉⠀⠀⠀⠘⣸⡟
⠀⠀⠀⠀⠇⠀⢾⠁⢙⣇⠀⠀⠀⠠⠤⠀⠀⠀⠀⢤⢀⠏⠀⠀⠀⠀⠀⣠⠞⠀⢸⡆⠀⠀⢀⡤⢶⢿⣿⡇
⠀⠀⠠⠀⠀⠀⢸⠀⠿⣿⡄⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⢳⠀⢹⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⣀⠀⠀⠱⢸⣿⡇⠀⠐⣙⠉⠁⠈⢉⡿⠋⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠞⠁⠀⠙⠃⢠⣼⠇⠀⠘⠈⠀⠀⢀⡞⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⢤⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⢀⠄⢟⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠁⢹⡇
⠀⠘⢠⣄⡀⠀⠀⠐⠿⣻⠀⠀⠀⠀⠈⡹⠚⠀⠈⠄⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⡄⠀⠀⠈⡇
⠀⠀⠑⠒⠻⡳⡒⢾⣉⡀⠀⠀⠀⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣠⣤⣤⣴⠿⠛⠁⠀⢀⡀⣇
⠀⠀⠀⠀⠀⠀⠅⠚⠛⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠒⠒⠐⢒⣃⣙⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠃⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠚⠉⠉⠛⢦⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢸⠃⠀⠀⠀⠀⣠⠞⠉⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢠⡴⠚⠛⠃⠀⠀⠀⠀⠀⠀⠸⣦⣍⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⢀⡴⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣴⠃⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠉⠙⠳⡴⠖⠒⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡼⠋⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣦⠀⢸⡇⠀⠀⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠶⣠⠞⠉⠽⢤⠀⠀⢀⣀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢘⣧⢸⠃⠠⣄⣿⠀⠀⢸⡇
⠀⠀⠀⠀⡴⠚⠋⠁⢠⡤⠶⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⠘⢧⡟⠀⠀⠈⣿⠀⠀⠀⢳⡀
⠀⠀⠀⡼⠁⠀⠃⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⣯⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷
⠀⠀⠀⠳⢤⢤⡄⠀⠀⠀⠉⠉⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⠐⡇⠀⠀⠀⠀⣿⠀⣠⠆⠀⣸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠶⠆⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣿⠀⢻⡀⠀⠀⠀⣿⠞⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠂⣿⠰⠶⠻⢦⣀⠀⣿⢀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡿⠀⡖⠘⢦⣄⣀⠀⣠⠶⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀⣀⣀⣀⣈⡁⣿⠈⠁
⠀⠀⠀⠀⠰⠯⠥⠞⠁⠰⠯⠭⠥⠤⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠉⠉⠉⠁⠀⠀⠉⠁⠟⠘⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠠⣤⠀⣤⡄⠠⡤⠘⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣾⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣻⣿⣿⡿⣟⣿⣿⣿⢋⣉⣻⠟⠉⠛⢛⡟⠉⠙⢛⠛⠻⣟⠛⠿⢿⣟⠛⠻⣷⣄⡀
⠀⠀⠀⠀⠀⢾⠿⡿⠿⣿⠿⠿⢿⣿⣷⡿⠿⣿⣷⣶⣷⣿⣶⣶⣶⣿⣽⣶⣶⣬⣦⣤⣤⣵⣤⣤⣽⣦⣤⣤⣬⣿⣦
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⣄⢀⠀⠹⣄⠀⠀⠹⡅⠈⠈⠀⠀⡇⢸⢹⠀⠠⢀⠋⠀⢀⡼⣋⠵⣊⠝⠋⠁⠀⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠⡉⠻⠿⣆⠀⠀⢱⠄⠀⠂⠀⡇⠀⡈⠀⠀⡰⠢⢐⣟⡪⠕⠊⠁⢀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢄⠀⠈⠛⢦⡌⠂⠀⡃⠀⣧⠀⡆⣠⣾⢱⢾⠟⠉⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣿⢤⣿⣾⡿⠟⠋⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⢨⡝⡿⣍⠀⠀⢀⠂⠀⣠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⡀⡾⢲⠷⣿⡀⠠⠁⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣿⣾⣳⣎⣿⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣖⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⠤⣤⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠙⠉⠃⠁⠀⠱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠄⠀⠀⠙⠋⠙⠊⠣⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣿⣿⣷⣸⣿⣿⣿⢾⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢿⣿⣿⣿⣿⡻⢿⣿⠷⠜⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠷⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⢲
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠶⢆
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣯⡽⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⣔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣴⣶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣶⣾⣶⣶⣴⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠚⠩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠐⠒⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠠⣤⣀⣙⣿⣿⣿⣶⠦⣠⣾⣯⢿⣿⣷⡖⢲⡶⢶⣦⣤⣤⡀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⣙⣿⣿⡿⢋⣵⣾⣿⣿⣿⢸⣿⣿⡇⢸⡸⢇⡿⠉⢉⠿⠋
⠀⠀⠀⠀⠀⣀⣭⣭⣽⡿⢃⣴⣿⣿⣿⣿⣿⠏⡾⠟⠀⠁⢄⠉⠉⠀⠀⠛⠒⠶
⠀⠀⠀⠀⣀⢠⣴⣾⠋⣰⣿⣿⣿⣿⣿⣿⣿⣦⠈⠃⠀⠀⠀⢹⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⣀⣬⣿⢃⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⠀⠁⠠⠄⢦⣑⣲⠖⠁
⠀⠀⠀⠈⢀⣹⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⠓⠦⠀⠀⠀⠉
⠀⠀⠀⠀⢙⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠐⢻⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⣀⣶⣶⠊⠉
⠀⠀⠀⠰⢿⡃⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡟⢀⣿⣾⡿⠟⠁
⠀⠀⠀⠀⠞⡇⣿⣿⣿⣿⣿⣿⣗⠫⠌⠿⢶⣾⠟⠋
⠀⠀⠀⠀⠠⢿⢸⣿⣿⣿⣿⣿⡷⠁⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢁⡘⡃⢿⣿⣿⣿⣿⣮⡖⠀⠀⠀⠀⠀⡰⠛⠉⠁⠉⠉⠙⠻⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⣷⡈⠞⣿⣿⣿⣿⣿⣾⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠘⢿⣦⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠈⢻⢸⣞⠻⣿⣿⣿⣿⣟⡄⠠⣴⣿⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⡞⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢻⣜⢿⣿⣿⣿⣿⣄⠈⠁⠀⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⣡⣽⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⢋⡑⣾⠛⢿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢤⣌⣉⡻⠿⡿⣿⣿⣷⣶⣶⣾⣿⡾⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠦⣤⠴⠒⠀⢲⡶⢄⠀⠯⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⢀⣤⡀⠀⠀⢸⡇⣸⠃⠀⠀⣠⡔⢲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠉⠙⠛⠛⠉⠀⠀⠀⠘⠧⠞⠉⣖⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⡆⠀⠀⠀⠀⠀⠀⠀⠰⠆⠠⣀⠀⠀⠀⠀⠀⠈⠓⠶⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡀⠀⢠⡇⠀⣀⣴⠄⠀⣀⣤⣤⣤⣀⠈⠙⢦⡀⠠⣀⠴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠈⣧⣎⢀⣠⠞⠀⠈⡽⠹⢠⠾⣵⣧⣼⣬⣯⠷⣄⠀⢳⠐⣿⠀⠀⠴⠓⢒⡶⠆
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⢠⡇⠀⣯⣤⣯⣤⣼⣤⣼⣤⣼⠀⠘⡇⠸⠄⠀⠀⢰⡋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣣⣀⣜⠀⡀⣇⠀⢳⣀⣿⣀⣼⣀⣼⣀⡼⣀⢸⠃⠀⠈⢓⣶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡾⠭⣤⠭⣶⠃⠘⣆⠀⠙⢬⣧⣽⣰⡧⠟⠁⣿⡯⠆⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⡖⣇⠁⣿⠈⣸⢲⠀⠈⠓⢤⡀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⣦⢤⡖
⠀⠀⠀⠀⠀⠀⠀⠁⣼⠓⠿⠚⣧⠈⠀⢀⠤⢄⡀⢈⣁⣀⠀⣠⠤⣄⢀⡜⠉⣽⣹⣏⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⠞⠉⣉⣩⣉⠉⠓⣄⡿⠀⠐⠛⢳⣰⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡞⣵⠒⣺⢳⡀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢧⣈⣻⣁⣸⠃⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⢄⡀
⠀⡠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠢⡐⢦⡙⠓⠢⠰⣎⠀⠀⠀⠀⠀⠠⠑⢂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⠀⠀⢁⡍⠑⣦⡄⢠⠀⠀⣀⣀⣤⣄⣀
⡈⣼⣿⡎⣿⡅⠀⠀⠀⠀⠀⡀⠀⢠⠈⣿⡄⠀⡁⢻⡀⠀⠀⠀⠀⠀⣷⠈⣗⡢⣬⡀⡀⣀⣿⣀⣄⣀⢀⣿⣆⡀⢘⣿⠄⢻⡇⠀⢣⡐⣿⡄⠀⠉⣿⡷⡄
⠇⣿⣿⡇⣽⣧⣿⣠⣤⣣⣸⣷⣆⣿⣷⣿⠇⠀⣆⣼⣅⠀⡀⢀⠠⣦⣿⡀⣾⣵⡿⠷⢛⣉⠩⠍⠁⠀⠈⠉⠉⠛⠻⣿⣧⣿⣷⣶⣺⡇⣣⡇⣷⣤⣾⣷
⠈⠪⠿⠕⠋⠉⠉⠉⠉⠛⠻⠿⠿⣿⣿⣏⣄⣼⣿⣿⣿⣖⣠⡾⠾⣛⣛⣭⣿⢛⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠛⢿⣿⣧⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠛⣿⣿⣿⣿⢏⣡⠖⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢷⣄⠊⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠙⠿⣿⢸⡃⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠋⢸⠀⠀⡴⣾⠟⣩⢉⡀⢠⣉⠙⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠉⠉⠃⠲⢤⠀⠸⠒⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡓⢤⡀⢀⡁⠸⠀⡾⠐⠀⢠⡎⢐⣄⡄⢈⠵⡄⠹⣎⠋⣁⣀⣤⣾⡟⠁⠀⡠⢖⣾⢰⣶⢀⠀⠑⡄⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢲⠌⡑⡆⢸⣼⣷⡂⠀⡿⠤⣾⣥⣻⡤⠤⢭⠀⣿⣿⣿⣿⣿⡿⠀⠀⠼⠛⢛⡟⢻⣄⠴⠆⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⣆⣤⠑⠂⢞⣿⣇⠀⢒⡒⠘⢿⠛⠝⢢⣿⠀⡟⠀⠀⠀⠸⣷⠀⠀⢲⠂⠈⣿⠿⠟⠲⠂⠀⠷⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠉⠛⠛⠛⣛⣿⣆⠈⠙⠦⠿⠤⡿⠟⢁⣔⣀⣀⣀⣤⣤⣿⣦⡀⠈⠲⠆⣀⡄⠔⠁⢰⡖⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠶⠦⠤⠤⠤⠶⠟⠻⠿⠿⢿⠿⠿⠟⠛⠛⠶⣄⡠⢄⣀⣠⣴⣾⣿⣿⣿⣿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠛⠀⠀⠻⠀⣤⡄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢀⠘⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠈⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣟⣻⣿⡑⠂⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠋⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⢀⣿⠿⠗⠀⠀⠀⠀⢠⣄⣤⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣍⣬⣴⠿⡀
⠀⠀⠀⠀⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠋⡤⠀⠀⠙⠉⠺⠿⣶⡄⠀⢠⢂⣴⠿⠟⠿⠻⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⣽⠃⠂
⠀⠀⠀⠀⠿⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢱⣿⣿⢶⡶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠀⠀⠀⠀⠀⠐⣞⠃⢈⠂⠁⢄⠀⠀⠀⠀⣼⠶⠘⠿⢷⣆⣤
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠀⠀⠀⠠⠄⡸⠖⠀⣠⠀⠀⠀⠀⠈⣯⠀⢸⡀⠀⠀⣄⠀⠀⠀⢸⠁⠀⢰⠈⠏
⠀⠀⠀⠄⠐⠎⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠙⣶⡔⣿⠀⠀⠀⠀⠛⡔⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⡤⠀⠁⠀⠀⣦⠀⠀⠀⠀⠀⠐⠻⠟⠃
⠀⠀⠀⠀⠀⠀⡀⠐⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⠘⠀⡄⠀⣠⠀⢀
⠀⠀⠀⢠⠆⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⣦⣼⡇⠀⠀⠀⠀⡐
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⣞⡉⡿⢿⣦⡠⠞⠃
⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠑⠲⡀⠀⠀⠀⠈⠋⠀⠀⢼⢿⣿⣷⢾⠆⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣀⡖⣀⠀⠀⠀⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⢯⣿⣽⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠛⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣦⠀⠀⠈⠀⢀⠀⠀⣤⠟
⠀⠴⠂⠀⠀⠈⠙⡆⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢨⡆⠀⠈⠀⠀⠲⣟⢷⣤⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠁⠀⠀⠀⡘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡄⠀⠀⠀⠀⠈⠉⠈⠂
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⡌⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢰⠉⠢⠀⠀⠀⠀⣀⠀⠀⠀⠃⢈⣹⣿⣿⣿⣋⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡋⡵⡆⠀⠀⠈⠟⠹⣿⡟⠋⠀⠀⠐⠁⠀⢀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⢀⠀⢸⣴⠃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠳⠖⠀⠀⠀⠞⠀⠀⠀⠐⠀⠀⠀⢀⡀⠠⢴⣮⣿⣿⣿⣆⠀⠠⠄⠀⠸⡄⠀⠃⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠘⠟⠻⠻⠟⠁⠢⠄⠀⠀⠀⠁⠀⢀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⣵⡆⠁⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠒⠀⠀⠠⣀⡍⠀⠀⠠⠀⠀⡆⠀⠀⡀⠀⣼⡥⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢁⣠⠶⢚⣿⠏⠀⠀⠀⠀⠀⡆⠀⢀⡴⠀⠀⠀⠿⠀⠀⠀⠀⠀⢰⠇⠠⠎⠀⠀⠙⠁⠠⣤⠟⠏
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⣀⡶⠋⣠⠴⠋⠁⠀⠀⠀⠀⠀⠄⢿⣿⣾⡄⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⡟⠁⠀⢀⠀⣰⣿⣖⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠉⠀⠒⠛⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⢠⣿⣀⣤⠶⠶⠶⠛⠋⠉⠉⠀⠉⠉⠳⣤⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⢀⣾⠟⠿⠿⠀⠀⠘⠟⣁⡀⢠⠀⠘⠛⣩⣿⣷⡀⠰⠋⠉⢩⣿⡄⠀⠀⡿⡿⠟⠂⠀⠀⠟⠉⣃
⠀⠀⠈⠿⠀⠀⠀⡤⠀⠀⠀⢻⡿⠀⠀⠄⠰⢷⣿⠋⠀⠀⣿⠆⠈⢹⠇⠀⠀⢁⣄⠀⠀⠸⣿⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠈⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣠⣄⣄⣶⢠⣾⣴⡾⣿⣆⠀⣤⣶⠖⠀⠀⠀⣠⣴⣷⡖⠀⠀⢠⣿⢁⢀⣠⣄⢰⣿⣤⣶⣽⣆
⠀⠀⠀⣐⣿⣿⣿⡟⠘⣿⣿⡄⣨⡿⠀⢻⣏⡈⠁⣶⠃⢻⡈⣹⣷⣆⠀⠸⣿⡿⠻⣿⡟⠀⠁⠉⠁⢸⡟
⠀⠀⠀⠈⢙⣛⠋⠁⠀⠈⢛⣛⠋⠀⠀⠈⠙⣓⠊⠁⠀⠀⠉⣛⡛⠁⠀⠀⠈⢃⣀⠈⠀⠀⠀⢀⣐⠉
⠀⠀⠀⣴⠟⠿⢿⠀⠀⠐⠟⣋⠀⢠⠀⠐⠛⣫⣿⣷⡀⠰⠟⠉⢩⣷⡄⠀⠀⣿⣿⠟⠆⠀⠀⠿⠛⡃
⠀⠀⠈⡿⠀⠀⠀⡠⠀⠀⠀⢻⡷⠀⠀⠄⢰⣿⣿⠋⠀⠀⣷⠆⠈⢻⠇⠀⠀⢁⣄⠀⠀⠸⣿⣶⣶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⢌⡢⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡐⠊⠀⢤⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⠻⠯⠷⡀⢀⣀⡀
⠀⠀⠀⢀⠖⡂⠉⠉⠉⢰⣀⡄⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠐⠻⢷
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢿⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠶⠿⢖⢀⠸⠷
⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡋⠀⠀⠀⡐⢠⡄⡁
⠀⠀⠀⠘⡈⠀⠀⠀⡖⠁⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠐⠢⠀⠏⣷⠀⠀⢑⠀⠉⢁⠔⢆
⠀⠀⠀⡆⣇⠀⠀⢀⣷⠀⠾⢂⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢩⢡⠀⠈⠇⠐⠀⠁⠈⠆
⠀⠀⢀⠵⢸⡸⠜⠡⠜⠄⠐⠿⣷⠀⠳⣶⠀⠀⠀⠀⠀⣠⠀⠀⠅⠈⢸⣰⠷⣕
⠀⠀⠄⠈⠈⢿⣦⣄⡀⢀⢰⠒⠲⣷⠶⠚⢃⡀⣀⣠⠾⠀⠀⠀⢀⠀⢨⡇⢸⡀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠃⠀⣿⣽⠜⠉⢸⠀⡇⠀⠀⠀⠁⠀⠉⠘⡈⠀⠀⠀⠀⠀⠀⠀⣁⠀⡄⠠⣤⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⡟⡀⠀⠀⠸⠀⢡⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⡧⢴⢁⠀⡇
⠀⠀⠀⠀⠀⠂⢰⠃⠰⠸⡀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢀⡠⣌⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣼⣅⡀⠀⠀⢀⣠⣤⣤⣾⡶⣼⣷⡤⢠⣌⠉⠈⠁⢰⡾⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠉⠀⠈⠉⠉⡙⠓⢶⠟⠉⠀⠄⠄⠍⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⢶⣄⡒⠖⠒⣀⠶⠒⠒⠒⠦⠤⠤⢤⡤⣤⣀⡀
⠀⠀⠀⠀⢀⠄⠶⠥⠈⠀⢤⡀⢨⡄⠉⠁⡘⠒⠒⡀⣤⣀⡀⡰⠊⠉⠀⠀⠈⠈⠿⣆⡐
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢧⠈⡕⠀⠑⢲⠀⡸⠁⠈⠋⠙⠶⢐⡀⢀⠀⠀⠀⢈⠰
⠀⠀⠀⠀⠑⢦⣤⣀⣀⣄⠀⠀⠀⠙⠁⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠟⠻⣝⠿⡇⠛⠿⡟⠒⣲⠚⠛⠛⣻⢟⡋⣉⡱⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠧⣃⣤⡄⢃⣀⣤⣤⣴⣤⠷⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠙⠆⠀⠀⠹⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⢠⠶⡄⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠂⠀⠂⠐⠀⠀⠀⢠⠋⠀⠹⣄⣭⣀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠹⡄⠀⠀⠈⠀⠁⠄⠁⠄⠀⠀⠀⠀⢻⡀⠴⢃⡟⠀⠈⢳⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣂⡏⠀⠀⠐⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠉⠛⡙⢤⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⢢⡀⢹⠉⠀⣠⠤⣦⠀⠀⠁⣀⣤⢤⣤⣤⣀⠀⠀⠚⠁⠀⣤⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣙⣶⡃⢸⠀⣾⢡⠀⣼⠀⣠⠮⣵⣇⣸⣇⣹⣬⠷⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠀⣸⠳⠚⠁⢰⡇⠀⣟⠀⢸⠃⠀⣧⠀⢹⠀⢸⠓⠦⣄⣀⣀⣀⡤⠖⢻⠂
⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⣤⣽⣾⣥⣤⣤⠀⠸⡏⠉⣏⠉⢹⡏⠉⡿⠉⢹⠁⢸⣀⡴⠋⠀⠀⠈⠙⢆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⠻⡖⢺⡗⢲⠗⣶⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠋⢩⠍⠩⠉⣧⠀⠀⠈⠓⠻⠾⠷⠟⠋⠁⠀⠐⢷⠦⢤⣈⣀⣀⣀⣠⡤⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠈⠓⠦⣤⣿⣬⣯⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢶⡶⣄⠀⠀⠀⣹⡟⠁⠀⢈⣧⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡵⠚⣧⠞⣇⠀⠀⠀⣇⠀⠀⠉⠁⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣋⣹⠆⠀⠘⢀⡀⠀⠀⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣉⣻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡚⠛⢋⣭⣉⠙⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⡏⠀⠘⡆⠈⢦⡀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢴⡯⠷⣄⠀⣜⣡⠴⠚⠁⠀⠀⠀⠀⡤⠴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠲⠤⣄⣀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠱⣌⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣤⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠙⢿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠙⠦⠤⣄⡀⠀⠀⠀⠀⢸⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⠲⣄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⡜⠁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠙⣄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣀⣀⣠⣤⡤⠶⠚⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠤⢮⡁⠀⠈⠡⡍⠁⠉⢩⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⠀⠠⢸⠗⡞⠉⠐⠲⣤⣜⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⠒⣁⠀⣣⣀⣈⡰⠃⢏⡠⠔⣛⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠑⠬⣋⡀⠀⠀⣀⣀⠤⠴⠨⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡿⡕⠢⢀⠀⡀⠆⠀⠛⠋⠟⡡⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⢤⠄⠀⠈⠀⣸⠐⠀⠀⠁⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡗⠠⠀⠠⢰⠇⠀⠀⠧⠘⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠊⠀⠀⠀⢀⠌⠀⠀⠀⡰⠊⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⢰⡊⠁⠢⡀⡠⠊⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇⠈⠙⡦⠤⢒⠁⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⢤⠒⠈⠠⠀⠔⠀⠀⠈⡂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠒⠠⠄⡼⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠣⣀⠀⠀⢀⡗⠂⠤⠀⣀⠀⠀⠈⠆⠠⠂⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂⠀⠁⠀⠀⢠⠈⠉⠈⠣⣀⡀⠀⠀⣀⡩⠟⠀⡆⠀⠀⠀⢠⠀⠌⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠄⠊⡐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⢀⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣟⢀⠈⣻⠛⠁⣩⡤⠀⠒⠒⡆⠀⣀⣄⠄⠈⢀⣀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠒⠚⠣⠂⠀⡾⠓⡌⠁⣀⠀⡟⠡⣄⣀⠰⠑⠰⣿⣿⢷⣠⣿⣦⡀⠁⢠⣤⢸⠋⠉⠃
⠀⠀⠀⠀⠀⡄⠀⠀⣤⡴⠛⡀⠄⠀⠀⠀⠛⠃⠠⠃⠈⠀⠀⠓⠐⡏⣀⠀⢳⠂⣹⡇⠀⠈⠛⠛⠡⠂⠀⢃⠈⠀⠳⡇
⠀⠀⠀⠀⠀⡀⠀⣀⡻⠃⠂⠀⠾⠁⢈⡿⠤⠀⠠⢠⠀⠀⠃⠀⠀⠃⠌⠂⠈⠆⠸⠗⠀⢺⡇⢈⡴⢤⣀⣐⣦⣁⣤⣴⠀⢀⡀
⠀⠀⠀⠀⠇⠀⠈⠀⡇⠀⠈⠀⠀⠀⡾⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⠊⠀⠀⠀⠀⠄⢈⡛⠝⢀⢸⣶⣿⠁
⠀⠀⠀⠀⠲⡀⠀⠀⣟⢣⡗⠀⢠⡆⠰⣗⢲⠀⠒⠇⠲⠘⠇⢰⠂⡇⢴⠂⣲⡄⠉⠘⠄⠀⣄⠀⡀⢠⠄⠸⠀⠦⠀⢼⢟⡊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠀⢿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⣀⣀⣠⣤⣤⣤⣄⣈⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢶⡾⠿⠟⢛⡋⠉⠉⣉⣷⣍⠛⢛⣛⣿⣿⣶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠘⠀⠀⠡⠷⣾⣏⠁⠀⠀⠘⣿⣿⠿⢷⣿⣿⡟⣶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⣀⣰⠆⢹⡿⠀⠀⠀⢰⣿⠿⣾⣿⡿⠛⠿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⠋⠀⠀⠀⠀⢰⣾⡿⠋⣧⣀⣿⡇⢀⣠⣴⣟⣡⠛⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⠋⠀⠀⠀⢀⡄⠀⠀⠟⠁⠀⠛⣿⣿⣿⠛⠿⠿⡟⠹⠧⣿⣧⢀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠸⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢤⡤⠾⣿⣧⣤⡄⢀⣀⣀⡴⢉⢻⣿⠇⠛⠁⠀⡴⠀⢀⠀⠀⢧⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠏⣠⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣠⣄⣫⣴⡶⠿⠇⠈⠁⠈⢻⣿⣷⣾⡔⠁⡀⠀⠀⠈⠁⠐⣯⡾⠆⠀⠹⠄
⠀⠀⠀⠀⠀⢀⠀⡄⣰⣿⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⠉⠀⠀⠀⠀⠐⣀⠀⠀⠙⣿⣿⠿⠀⢀⣴⣴⡮⣄⠀⣀⢀⣠⠀⠀⢀⣠⣴⡔
⠀⠀⠀⠀⠀⠘⠀⢀⣿⣿⡆⠀⣴⣶⠶⣄⢰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣴⣿⣇⡀⢀⣿⡷⠿⢷⣾⣷⣿⣿⠿⠂⠀⠀⠙⠙⣷
⠀⠀⠀⠀⠀⠀⠀⢼⡧⣿⡇⢰⡿⢻⡷⠿⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⣩⣿⣿⠋⠁⠀⠀⠀⠘⠛⠿⠀⠀⠀⠙⢦⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣦⡁⠼⣿⣿⡎⠉⠻⣿⢓⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠆⠈⠀⢻⠛⣿⣿⠀⣿⣨⣿⣿⣶⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢸⡯
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣀⠹⢿⣷⣤⡉⠀⠉⠹⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⣸⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠉⠁⣌⣷⠏⠀⣠⠶⠄⠰⠈⠙⠻⣄⠀⠀⠀⠀⢠⡟⠻⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣶⣾⠏⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣎⠦⡀⠀⠘⣇⠰⠉⠀⠀⠀⠀⠀⠀⠀⠻⢄⡀⠀⠀⠀⠀⠀⠻⠿⣠⣀⣀⣤⣤⣶⣦⡶⢼⠿⠟⠁⠀⠂⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠈⠳⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠙⢯⣿⡿⠀⠀⠀⠉⠉⠙⠿⣿⣿⣿⣯⡀⣴⠂⠀⠀⡀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠙⠻⣦⡀⠠⣄⣀⠀⠀⠈⠀⠀⠀⠀⠀⣈⣿⣶⡦⠀⠀⠀⠀⠀⠾⠛⣿⣿⡇⠷⣆⠀⢀⣽⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⣘⠀⠀⠀⠈⣿⠀⣿⠛⢿⠂⡀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠚⠁⠀⠀⠀⣆⣹⡏⠀⠀⠉⣠⣞⡟⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠲⣌⡕⡇⠀⠀⠻⡋⠀⣤⣤⡶⠿⠿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠀⢀⡴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡭⣷⡔⠀⠀⠀⢠⣴⣿⡿⠏⠀⠀⣀⠀⢠⠤⠒⡒⠉⠉⠁⣩⣿⣧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⠳⣼⣿⣦⡀⢾⣿⣿⣿⣷⣶⣼⣭⣤⣤⣤⣤⣬⣤⣀⣶⣶⣿⡉⠁⠋⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠾⣿⣿⡿⠟⠛⠋⠉⠀⠉⠀⢈⣀⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡶⠛⠛⠓⠲⠤⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠸⡇⠀⠰⠒⠀⢀⠀⠉⠓⠦⣄⡈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡗⠂
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡞⠉⠉⠓⠲⠄⠀⠉⠲⢄⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢳⡀⢰⡟⠒⠦⣄⡀⠀⠀⠉⢲⣄⣤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠱⡀⢳⡄⠀⠀⠉⢓⡤⠞⠉⠀⠀⠀⣀⡀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢦⡀⣠⠏⠀⠀⠠⠖⠋⠉⠉⠉⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢷⡀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠈⡇⠀⠀⠀⠐⢿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠈⠳⣦⡈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⠈⣇⠙⢶⣌⠳⣄⠙⢦⣈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⡷⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠳⣌⠙⢦⡉⠳⣄⡑⠂⠀⠀⠀⠀⢀⡾⠀⠈⠳⡄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠛⢦⡈⠀⠈⠙⢦⣄⠀⠀⣠⠞⠑⣄⠀⠀⠘⢦⡀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠙⠳⣄⡀⠀⠈⠛⠾⣅⡀⠀⠈⢧⠀⢦⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠛⠦⣄⠀⠀⠈⠙⠲⠦⠼⠇⠈⢧⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠶⣄⡀⠐⠢⢤⣄⣀⡼⠀⣀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠙⠳⢤⣀⡀⠀⠀⠰⠛⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠤⠀⠀⠀⠀⠀⠈⠓⠀⠀⠈⠉⠓⠶⠤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠏⠁⣤⣈⡙⢆⣈⣙⣿⠘⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢁⣰⣾⣿⡿⠟⠂⠉⢿⣿⡏⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢻⣫⣾⠟⣋⣠⣴⣿⣦⡀⣹⡟⢑⣥⣼⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣨⣿⣿⠋⠉⠙⠚⠿⠕⠺⠿⢷⣦⢭⣵⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⡇⠰⠛⣛⡓⠀⠀⢀⢘⣯⣉⡓⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣙⣿⠀⠀⠺⠻⠟⠊⠀⢸⠛⠛⠋⠁⠸⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⡄⠀⠀⠀⠀⠁⣀⢸⣄⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⠀⢰⠐⠀⣀⡍⢭⣀⠌⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠈⢍⡭⠇⠀⢀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⡉⠳⢦⣀⡀⣀⣀⣴⣿⠇⢹⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣾⠉⠛⠈⠷⣤⡀⠉⠙⠛⢉⡽⠋⠀⠸⠿⣿⠉⣿⣷⢦⣄⣀
⠀⠀⠀⠀⢀⣀⡴⣿⡉⡇⣧⠸⣆⠀⠀⠀⠈⠙⢳⢖⠶⣾⠁⠀⠀⠀⠀⣿⠀⢳⣿⣸⣿⣿⣭⠷⣄
⠀⠀⢠⣾⣋⣵⣶⡏⡇⢿⢹⡀⢹⣆⠀⠀⢀⣤⣾⣜⣄⣵⣷⣦⣄⡀⢀⡟⠀⢸⢸⣿⣿⣿⣿⣿⡟⣷⡀
⠀⣰⣿⡄⡇⣿⣿⣿⢡⠀⠀⠃⠀⢹⣷⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠘⣿⣿⣿⣿⣿⡇⣿⣿
⠀⣿⣿⣷⢹⢹⣿⣿⡞⡆⠀⠀⠀⠀⠹⣿⣿⣿⠉⣿⣿⡟⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣻
⠀⣹⣿⣿⣏⣏⣿⣿⣷⢳⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⣿⡇⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿
⢠⡿⠸⣿⣿⣸⡼⣿⣿⣏⣇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⢧⡿⢻
⢸⡿⠶⣦⣿⡇⡇⣿⣿⣿⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣧⣿⠼⠁⢿
⣿⣷⣌⣶⣿⣿⢱⢹⣿⣿⣿⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⣿⣿⡇⣿⣿⢀⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⡀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠤⢆⡉⠉⠁⠩⡉⠈⠈⢉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠠⢈⠗⡖⠃⠒⠲⣤⣌⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⠒⡁⠀⣣⣀⣈⡠⠃⢍⡠⠔⣛⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠆⠀⠀⠈⠑⠬⣋⡀⠀⠀⣀⣀⠠⠴⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⡑⠦⢀⠀⠀⠆⠀⠛⠀⠟⣡⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⢤⠄⠀⠈⠀⢸⠠⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡖⠀⠀⠠⢠⠆⠀⠀⠇⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠀⠀⠐⠄⠀⠀⠀⡠⠊⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⢠⡊⠁⠢⡀⢀⠌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠆⠈⠘⡤⠤⠔⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⠀⠐⠀⠠⠤⠄⢤⠐⠊⠠⠀⠐⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠒⠠⠄⡼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⡗⠠⠄⣀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠈⠣⣀⠀⠀⠀⢁⡁⠟⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠤⠀⠠⠊⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣟⣿⣛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣦⣄⣹⣦⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠹⠻⠿⣽⣿⣿⣷⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡁⠀⣀⣤⣤⣤⣼⣿⣹⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⢁⣌⣴⣿⠟⠛⡉⠉⢽⠿⣿⣿⣿⣿⣿⠿⡶
⢹⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣾⣿⢋⣀⡨⠟⡲⠶⠾⢿⣿⣿⣿⣿⡆
⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠻⠛⠀⡀⠂⠀⠀⢀⠀⣀⣀⣴⣾⡟⣿⣄
⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣘⡁⠠⠖⣳⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠙⠛⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟⠋⠀⠀⠀⠀⢸⠂⢼⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠃⠀⠀⢀⠀⣼⠁⠿⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡀
⠀⠀⠀⠁⠀⠹⠿⣿⣿⣿⣿⣧⣿⣤⠀⠀⠀⣀⣨⣿⡟⢠⣖⡀⠂⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠠⠿⠗⢻⣿⣯⠀⠀⠀⢬⣹⣿⠃⠀⣯⢽⠂⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⠔⠁⠀⠀⠀⠉⠁⠀⠀⢠⣾⣿⡟⣘⠀⣿⢠⡆⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⣶⣿⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡇⣼⢁⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠐⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠁⠏⠉⣻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡇⠀⠐⠀⢉⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠐⢦⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠘⠀⠀⣯⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⣼⣷⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠀⠀⣽⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣽⣿⣿⡏⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣿⣿⡇⠀⠀⢸⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡇⠀⠀⠀⠀⠾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⡀⠀⢀⣴⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡁⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡁⠀⠈⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠃⠀⠀⣘⣻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣄⣻⣿⣿⡇⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠙⠋⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠚⢁⣤⣾⠗⠚⠛⠛⠻⢿⣿⣿⣯⣙⢟⣻⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣶⠿⠠⠾⠟⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⠶⣬⠰⡄⠀⠻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⣡⡾⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⡾⠏⢷⣄⠀⠉⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠡⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠟⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡎⠳⡀
⠀⠀⠀⠀⠀⢠⡿⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠠⣤⡼⠟⠋⠀⢠⡙⡄
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠔⢺⠐⠖⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣦⢠⠀⠈⠈⢻⡄
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⣷
⠀⠀⠀⣸⣿⢡⡀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⣠⠆⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⢠⣿⢻⣇
⠀⠀⢀⣿⡏⠘⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢀⡀⠈⢠⣶⡀⠀⠀⠀⣿⣿⣷⣤⣀⣼⣿⣇⣿
⠀⠀⢸⡿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠋⢀⡤⣬⠓⠲⣆⠀⣶⡿⢷⡄⠠⠙⣷⡤⠀⠀⠈⠿⣟⠙⠛⠛⣿⣿⢿
⠀⠀⢸⣇⠠⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣥⢤⣙⢻⣆⢤⡄⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣿⢀⠀⢰⣿⣿⣾⡆
⠀⠀⢸⣿⡰⣷⡀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠾⣯⣽⣿⣾⣿⣶⣿⡇⠀⠀⠀⠰⣦⣤⡦⣴⣿⣿⣮⠀⢸⣿⣿⣿
⠀⠀⠸⣿⣧⣹⣷⣄⣹⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠉⠉⠉⠁⠠⣤⡀⠀⠀⠀⠉⠀⢸⣿⣿⣿⣧⡻⣻⣿⣿
⠀⠀⠀⣿⣿⣝⠿⠿⢿⣦⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠘
⠀⠀⠀⢸⣿⣿⡧⡖⠊⠙⠳⠨⣿⣄⡀⠀⠀⠀⢀⣀⣠⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠷⣴⠟⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢿⣿⢱⠆⡀⠀⣠⠈⠻⣿⣿⣶⣤⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡟⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⡿⣷⣿⣻⠛⠆⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣧⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡻⠤⠓⠀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⠛⠺⠗⠒⠲⠈⠻⢿⣿⣿⣿⣿⣿⣷⡱⢶⣦⣀⡢⠤⣤⣀⣠⠠⠀⢀⣠⠦⠶⣿⣿⡁⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⡏⠋⢰⡟⠁⠀⠀⢻⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠿⠟⠀⠀⠀⠀⡜⢷⡴⣿⠟⣱⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡘⣿⢷⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠁⣼⣫⣿⣯⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡻⡿⠂⠈⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⣠⢞⣷⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣀⣼⣷⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⠿⠟⠛⠛⠻⠿⡿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠀⣠⡴⢤⡀⣠⡴⣤⡀⢠⡤⠦⡄⢤⡀⣠⡄⣠⡤⢤⡄⠤⣤⡤⠄⣤⠤⠄⢠⣤⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠀⣿⠀⠀⠸⣿⠀⢸⡇⠘⠳⢶⡄⠈⢿⡟⠀⠙⠷⢶⡄⠀⣿⠀⠀⣿⠒⠂⢸⡟⣷⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠉⠛⠛⠁⠙⠛⠋⠁⠘⠛⠛⠁⠀⠘⠃⠀⠙⠛⠋⠁⠀⠛⠀⠀⠛⠛⠃⠘⠃⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠫⠝⡇⠀⠸⣋⡣⠀⠀⠀⠠⠂⡰⠁⣤⠀⢀⠀⠀⠀⠀⠐⠑⢨⢂⣀⣀⠀⠀⠀⠀⣠⠈⠫⡭⢙⠀⡄⠀⠀⠀⠀⠀⢻⣀⣞
⠀⠀⠀⠀⠀⠘⠁⠊⡡⡒⡄⠁⠘⠀⠀⠀⢀⠀⠠⠆⠀⠄⢁⠀⠀⠀⢠⡜⡃⠰⡀⠓⢐⣤⠀⠀⢐⢠⠜⡒⠒⠨⢥⣀⠀⠀⠀⡆⡛⡿⠵⢹⠟⠈
⠀⠀⠀⠀⠀⢰⠠⠀⡈⠒⠡⠀⠠⠀⠀⠀⠀⠂⢀⠀⢀⠈⠀⠂⠀⠀⡅⠀⢈⠢⠀⢀⡋⠀⠀⠀⠘⡀⠖⣱⢔⡃⡖⡀⠀⠀⠀⠃⡁⠷⠭⠾⢆⡰⠂
⠀⠀⠀⠀⠀⠘⠳⠤⠇⠀⠸⠎⠈⠀⠀⠀⠈⠂⠈⠓⠋⠄⠂⠀⠀⠀⠈⠃⠀⠀⠀⠯⠏⠀⠀⠀⠀⠉⠠⢄⣀⠤⠐⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠄

⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⠠⠀⠀⠀⡠⠀⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠓⠈⢠
⠀⠀⠀⠀⠀⠀⡌⠉⠤⠀⠁⠊⢀⠀⠀⠀⡂⡉⢰⠒⠑⠊⠛⡄⠀⠀⠐⠀⠒⠂⠂⠀⠂⠂⠀⠀⠐⡀⠘⠛⡤⠀⡐⠀⠀⠀⠀⡐⠗⠀⠘⠤⠄⠰⡆
⠀⠀⠀⠀⠀⠈⢓⠲⠀⡌⡔⡀⠘⠀⠀⠀⠀⢀⢦⢛⠽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣸⠀⣿⡞⠀⠀⠀⢇⠠⠆⠀⢀⢴⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠰⡬⠏⠉⠀⠀⠀⠀⠀⠈⢧⣣⡜⠆⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠚⠔⠘⠉⠉⠄⠐⠀⠀⠀⠈⠠⠤⣀⡈⠳⠂

⠀⠀⠀⠀⠀⠐⢀⠤⠀⠀⡀⠂⠠⠀⠀⠀⡔⠀⢀⠀⠢⢀⢀⠀⠀⠀⢀⢀⠢⠄⠄⠄⠀⢠⠀⠀⠀⠄⢄⠐⠀⠄⠐⠀⠀⠀⠀⢀⠀⣠⣴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠡⠀⠔⠒⠬⠁⠁⠀⠀⠀⠀⠈⠠⠐⠠⠁⠀⡁⠀⠀⠁⠀⠁⠒⠒⠉⠍⠁⠀⠀⠀⠼⠤⠒⠒⢌⠸⠅⠀⠀⠀⠀⠀⠈⣀⡃⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠒⡤⠊⠀⠀⠀⠀⠀⠠⠆⠐⢴⠔⡘⢠⠄⠀⠀⢆⡆⠀⢠⡔⣠⢶⣆⠀⠀⠀⠁⠈⠣⡬⠊⠀⠈⠀⠀⠀⡀⠀⡘⢿⠗⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⠈⠉⠀⠈⠁⠉⠀⠁⠀⠀⠀⠀⠐⠈⠃⠐⠀⠀⠀⠀⠀⠀⠅⡀⠀⢐⣰⣆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠄⣈⢣⡟⠾⡁⢄⠀⠀⠀⠠⠀⢤⢬⣤⠄⠠⠀⠀⠀⢠⡴⢉⣀⡁⠈⢢⡄⠀⠀⠀⠀⡄⡤⠥⡄⡄⡀⠀⠀⠀⠀⠀⣖⣹⣸⡃
⠀⠀⠀⠀⠀⠠⡤⣭⢃⠜⣑⢷⢤⠀⠀⠀⠀⠶⠚⡧⠈⠰⠇⠀⠀⠀⢰⡀⠀⠋⠨⠀⢀⡅⠀⠀⠈⠀⣸⡀⣗⣸⡀⠠⠀⠀⠀⠀⢀⣈⢹⡭⣀
⠀⠀⠀⠀⠀⠀⡄⠹⠮⠵⠄⡇⠰⠀⠀⠀⠀⢀⠀⢀⠠⢁⠀⠀⠀⠀⠀⠘⠔⡤⢤⠂⠃⠀⠀⠀⠀⠀⡁⠈⠎⠁⠀⠀⠀⠀⠀⠀⠇⡸⢫⡁⠈⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡦⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡚⠐⠀⠐⠐⠓⣄⠀⠀⠀⠀⠀⣀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⠀⠀⢿⠢⣀⣤
⠀⠀⠀⠀⠀⠀⠼⢰⣾⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠈⣑⡂⠉⣉⠁⢰⠀⠀⠀⠀⠀⡛⠁⠲⣆⣀⢴⠄⣶⡄⠀⠀⢀⠴⠚⢿⡉⠁
⠀⠀⠀⠀⠀⠀⢷⠎⠑⠀⠁⠀⠀⢰⣶⡦⢀⠀⠀⠀⠀⠀⠀⠘⡤⠠⠄⣀⣀⠀⠠⡼⠀⠀⠀⠀⠀⡸⠶⡚⣴⡎⣲⣠⠊⠀⠀⠀⢸⣌⡒⣼⠇
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠇⢀⠀⠀⠀⢇⠰⣶⠉⠢⣀⠀⣀⠀⠀⠈⠲⢔⡀⠀⡠⠎⠀⠀⠀⠀⠀⠸⣀⣀⡯⠶⢴⡃⠈⢂⠀⢠⡖⠛⢿⡉⠁
⠀⠀⠀⢩⠙⡁⠀⠀⡘⢀⠆⠀⠀⠀⠈⢢⠀⠀⠀⠈⢦⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⠎⠈⠁⠁⠀⠀⠉⠊⠲⠇
⠀⠀⠀⠀⢢⣤⣷⣷⣴⡅⠀⠀⠀⠀⠀⠀⠟⢤⣀⣠⡾⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡄⠘⠛⠉⠉⠉⣁⣰⠀⠀⠀⠀⠀⢃⡆⠈⠫⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠘
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠢⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠐⠒⠒⠠⡀⠐⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡲⠶⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡲⠁⠀⠠⢀⠈⢻⢆⠀⠀⠀⠀⠀⠀⠀⡇⠒⠀⡏⠀⠀⠀⠀⠀⠂⠃⠀⣀⣠⢠⠀⠀⡆⡀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⡔⠁⠔
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠰⠁⠀⢀⠻⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⡇⠀⠀⠀⠀⠀⠀⠘⠄⢸⠁⠁⠀⡔⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⣀⠀⣀⣠⠺⠁⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⡰⡀⠀⠀⠀⠀⠐⠁⢀⠈⠀⠐⢰⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⠉⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⡠⢉⠏⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣿⣿⣿⣷⡑⡀⠀⠀⠀⠀⠀⠘⠁⢀⡀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⡜⢠⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⡐⡀⠀⠀⠀⠀⠈⠓⠂⠀⠊
⠀⠀⠀⠀⠀⠸⣀⠏⠀⠀⠀⣄⣂⡡⠔⢇⠀⠀⠀⠠⣛⠛⣛⢛⠛⣻⡛⢛⠓⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠔⠉⠀⠶⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠈⠱⠄⠀⠸⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢸⣄⡀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠒⠀⠋⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠛⠓⢦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠀⠛⢟⠻⠋⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠐⡎⠉⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣤⣾⡍⠉⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄
⠀⠀⠀⠀⢀⠄⠁⠀⠈⠚⡝⠁⠀⠀⠀⠀⠀⡜⠀⠈⢻⣿⠁⢀⣀⡀⢁⠀⠀⠀⠀⠠⣂⣽⠩⠽⠀⠸⡀⠀⠀⠀⠀⠅⢬⡑⢄⠀⠀⢀⠔⢊⡥⠂⢀⠄
⠀⠀⠀⢠⢁⡀⠄⠀⠤⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⢹⠀⢡⠀⠒⠀⢡⠀⠀⠀⠀⠐⢌⠁⠾⣀⠠⢣⡟⠀⢀⠤⠊
⠀⠀⠀⠀⢆⠢⠀⣀⡀⠔⠜⠀⠀⠀⠀⠀⠀⠄⠻⣿⡿⠁⠀⠀⠉⢡⠃⠀⠀⠀⠀⠀⠀⢂⠀⢃⠰⣷⠄⠆⠀⠀⠀⠀⠀⠁⠀⠙⡷⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠁⠂⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠁⡀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⡶⠿⡀⢀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠁⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢏⣿⢰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣿⣿⡿⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠷⠀⢸⠀⠀⣴⣾⡇⠀⠀⠀⣠⢆⣴⢰⣦⠰⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢦⢸⠀⠀⡫⠾⠃⠀⢀⣶⡆⣴⣭⢠⣭⡆⢶⣶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣠⠞⠀⠀⠀⠀⢘⣛⡃⣛⣉⢀⣉⣁⢘⣛⡁⠀⠸⣿⡷⣀⣤⣤⣤⡐⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠈⠀⠀⠀⠈⢿⠧⠻⠿⠘⠿⠏⠾⡿⠁⠀⠀⢋⣾⣿⣿⣿⣿⣿⣦⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠻⠎⢿⢸⡟⠱⠓⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣯⡽⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣭⣍⢛⣛⢛⣭⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠈⠉⠁⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⢰⣤⠀⠀⠀⠀⠂⣿⣿⡿⢟⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢵⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠀⠀⠀⠀⠛⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⢀⡀⠂⠁⠈⠁⠀⠀⠀⠂⠀⠀⠀⢀⠠⠀⠛⢆⠠⣮⠄⡀⠸⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⡍⠐⣂⠀⠀⠀⠀⠀⣠⡄⠀⠄⠬⠓⢄⢀⣰⠀⠀⠚⠀⠀⠀⠉⠁⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠠⣿⣷⣀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣠⣶⣶⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⢰⣻⣀⡙⣏⡤⣠⣄⣤⣄⡴⣄⡤⡤⢦⡄⢠⠀⠺⠓⠞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠋⠂⠋⠈⠓⠋⠃⠃⠓⠋⠃⠋⠓⠋⠃⠃⠘⠑⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⡤⡆⠀⠲⠆⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⣀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣤⠄⠀⠀⠠⠀⡀⠘⡣⠀⠀⠻⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⠟⠒⠀⣠⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⡋⢐⠡⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢆⣀⠀⡀⠀⠈⠻⣶⣤⢀⡾⠀⠐⣆⡀⠀⠀⠀⠀⠀⠀⠈⡑⠉⢠⡘⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⡟⣩⠟⠀⠀⠀⠛⡛⠩⢂⠀⠀⠘⣿⠆⠀⠀⠀⠀⠀⠘⢁⡄⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠉⠀⠀⠂⠁⠀⠁⠈⠡⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡈⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠃⠀⠈⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣬⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠰⣶⡿⢧⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⢰⣏⣐⣿⣿⣿⣷⣾⣿⣿⡄⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠹⢿⣏⠉⢽⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢂⣠⣿⣿⣧⣀⠀⠀⠀⠀⠀⢀⣄⠹⣮⣽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⡇⠀⠀⣠⣆⣇⠀⠻⡿⢋⣼⣿⣿⣶⣤⡄⣴⣿⣿⣷⣾⣹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣷⡜⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁⣹⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣭⣻⢿⣿⣿⠀⣿⣿⣿⢟⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⡆
⠀⠀⠀⠀⣼⣿⣿⣿⣟⣷⣿⣿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⠛⣻⣿⣿⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣠⣹⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠻⡏⠀⣠⣼⣯⠹⣿⠟⠛⠛⠛⠿⡟⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⣀⡟⠛⠛⠻⠿⡿⠿⠿⢿⣣⡿⠿⠛⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠻⠶⣷⣤⣭⡟⠻⠴⠟⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠃⠁⠀⢻⣏⠀⢀⣿⠀⠀⠀⢉⣿⣶⡖⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⠋⠹⠀⠀⠀⠀⢿⣿⡀⠀⢻⡀⠀⠀⢀⣀⣈⠙⠇⢼⣷⠄⠀⠃⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣓⡀⠀⢀⣠⠸⣶⣦⡜⣩⣷⣸⣏⠃⢀⡔⡏⠉⢋⠆⠀⠘⣁⣄⠀⢀⣷⡜⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠐⠿⠿⠂⠚⠑⢻⡀⣄⡠⠿⢿⣿⡿⣦⣩⡀⠀⣠⣿⣧⣤⣼⣿⣿⣷⣟⠁⠀⠀⠁⣀⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠿⣦⣀⣻⣃⡠⠞⠉⠁⠀⠀⠀⠀⠀⠉⠀⠘⠛⠳⣽⠆⢈⢿⡍⠻⣿⣿⠧⢤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠁⠀⠀⠀⢠⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣯⣇⠼⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠸⠷⠚⠉⠀⢀⣠⣴⣇⣤⣀⠀⠀⠀⠀⠘⣭⣙⢦⣾⠿⠛⢇
⠀⠀⠀⠀⠀⠰⢛⣻⣷⣶⡶⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠏⢀⡀⠀⠈⠀⠀⠀⢁⡴⢌⣀⣀⠐⠆⠱⠄
⠀⠀⠀⣤⡀⠀⢘⣿⣿⣭⡤⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠐⠀⢀⠤⠓⠒⠛⠋⠉⠓⠀⡀⠰⠆
⠀⠀⠀⠈⠃⠀⣄⣙⣫⠽⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⣤⠦⣄⠈⠁⠀⠀⠀⠀⠀⢰⠀⢠⡴
⠀⠀⠀⠀⣀⡀⠁⢰⣶⢆⣸⠿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⡯⠐⠙⠓⠈
⠀⠀⠀⠀⠈⠁⠀⡈⢻⣿⣿⣄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⣞⠀⠀⠀⠀⠀⢀⡼⠄⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢠⣇⡀⠸⠿⢿⠷⡷⡄⠀⠀⠀⠻⠀⠀⠘⠀⠀⠀⠀⠻⡄⠀⠀⠀⢀⣼⣷⠚⢻⣟⣡⡶⠉
⠀⠀⠀⠀⠀⠀⠋⢯⡀⠀⠺⢺⣧⣨⠟⣦⣤⡤⠽⢾⣿⣶⣇⡀⠀⡀⠀⠀⠀⣆⠳⠚⢿⠹⣷⠿⠛⢉⡀⣴
⠀⠀⠀⠀⠀⠀⢷⣄⠀⠐⠢⡶⢿⢿⣿⣧⣈⣷⢲⣬⠡⠤⣀⣋⢀⣀⣗⡴⠊⠀⠀⠠⠆⠀⠸⡄⠀⠡⠙⠛
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣤⡼⡺⢼⠛⢿⣿⣇⠀⠀⠀⠀⠡⠟⢐⣾⣿⠈⡠⣶⣿⠀⠀⠀⢠⠛⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣤⣠⡓⠚⠋⠁⢻⣆⠂⠀⠀⠆⠋⠀⠰⠚⠛⠋⢙⠀⠀⠠⠀⠂⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣟⣿⣿⣷⣄⣀⠀⢹⠠⢤⡄⢠⣴⡶⠀⠀⠐⠒⠁⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠈⠇⠘⠿⠯⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⣩⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠐⡦⢈⢰⢰⣴⣆⣬⠾⢯⡀⡠⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⣄⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡄⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡷⠀⠀⠠⠚⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⣶⣿⣯⣅⣿⣿⣿⣿⣿⣿⡷⡾⡅⠀⣤⣄⡀⣼⠆⠀⠀⠀⠀⢯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢴⣿⣝⢸⣿⣯⣿⣇⠀⣿⣿⣿⣿⣿⣯⣄⣿⣭⡃⠟⠚⡇⠀⠀⠀⢀⠀⠀⢸⡅⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡘⠀⣿⠁⢸⣿⣧⣿⣿⣥⣿⣿⣿⣧⣟⡐⢤⠘⠿⢨⢉⡀⠀⡀⠆⡰⠃⠀⢀⣿⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣀⣸⡇⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⢃⠛⠈⠉⠏⠀⠃⢸⣁⣰⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠉⢹⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⣰⣶⣶⣄⠐⠀⠀⠀⢸⠀⢸⣿⣿⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠁⠀⣸⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠀⣠⣿⣿⣿⣇⡐⡀⠐⢠⣾⠤⢼⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⡧⣿⣤⠀⢹⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣤⣿⣿⣿⡿⡾⣤⣄⠈⠁⣿⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣀⣨⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠘⡇⡿⣻⣾⣿⣬⣿⣿⣿⣿⠿⠷⠿⠿⠆⢐⣾⣿⣿⣿⣿⣷⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠇⢹⣿⣿⣿⢿⣿⣿⣻⢿⡅⢼⣻⡿⣿⣟⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣟⣷⡆
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣼⣟⣿⢸⣾⠿⠛⣿⣣⣻⠾⣿⣿⣿⣷⣼⣾⣤⣵⣥⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠒⣼⣿⣿⣿⡈⣿⠉⠉⠛⠛⠛⠂⠻⠿⠚⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣸⣯⣿⣿⡿⣡⣿⣷⡀⠐⢶⡶⠖⢠⢤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣶⣿⣿⣿⣷⡼⡇
⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⢳⣿⡿⣿⣿⣿⣿⣿⢿⣿⣄⠀⠻⣿⣾⡀⢻⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠡⣬⣿⣿⣿⣿⣿⡄
⠀⢽⣾⣿⣿⣿⣿⣿⣿⣿⢏⣀⣴⣿⣿⣇⣿⡿⢿⠛⣿⣾⣿⣿⣦⠀⠹⣿⢷⠈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⣿⣷⡀⢈⣿⣿⣿⣾⡏⣿⣧⡀⠈⢾⣵⣿⣿⣷⣌⠡⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣦⣄⡀
⠈⠿⠿⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢻⣿⣿⣾⡶⠏⣿⣷⡄⠀⠉⠉⠉⠉⠙⠂⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠟⠿⠿⣋⢳⣶⡟⣿⣿⣶⣶⣾⣾⣿⣶⣶⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⠿⠻⠿⠛⠛⠛⠒
⠀⠀⠀⠀⠲⠤⠤⣀⡀⠀⠉⠉⠉⠙⠛⠛⠿⠟⠲⢿⣷⡀⢡⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⠿⣿⣿⡿⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡉⢙⣫⣭⣍⣉⠀⠀⣴⣦⣤⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠦⣤⣤⣤⣔⠛⠾⣿⣿⣿⣿⣿⣷⠰⠿⢿⠿⠇⠀⠀⠀⠀⠀⠀⣀⡤⢤⠴⠖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠭⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⢤⣠⣤⣄⣀⣄⡀⣤⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⢛⣟⣻⣟⡻⢿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⣴⠜⠀⢄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠶⢼⣿⣄⠙⠻⠓⣾⣿⣷⣄⡀
⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣿⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⡿⠶⣄⠀⠱⠟⠈⠕⣤
⠀⠀⠀⠐⣉⣙⣿⡿⣿⣿⣯⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣿⡿⠛⠃⠀⠀⠀⠀⠈⢐⣈⣷⡀
⠀⠀⠀⠀⠙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⣭⢿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠸⣿⣿⣿⣿⣿⠿⠻⠉⢻⡁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣌⠈⣷
⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠹⢻⣯⣿⣿⠀⢠⡄⠐⠲⢤⢀⣀⠀⠀⠀⠀⠀⢠⣄⢠⣗⣿⡇
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢉⡛⠛⠻⣿⣤⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⢻⣿⣷
⠀⠀⢰⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣴⣿⣿⣿⣿⣦⣄⠀⠀⠀⠠⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⠴⣿⣿⡿⠀⠀⠀⣀⡀
⠀⠀⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣶⣾⣯⣀
⠀⠀⠀⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠖⡠⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⢟⠋⠉⠃
⠀⠀⠀⠀⠀⡿⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠦⣀⣦⣴⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠈⠿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣮⣯⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⠽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣯⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠿⢿⣿⠿⠯⠀⠀⠉⠓⠀⠃⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠉⡀⠀⠀⠀⠁⢄⡀⠀⠁⠀⠀⠀⠀⠌⠀⠀⠈⠀⢤⠈⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⠀⡃⠁⠀⠀⠀⠄⠁⠐⠀⠀⠀⠊⠀⠈⠐⢄⡀⠀⠠⠚⠲⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠤⠄⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠂⠀⠀⠀⡁⢠⣉⠦⠐⠉⠀⠀⠀⣑⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⢂⢠⣌⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⣔⠐⠉⠁⢤⠀⠐⠂⢂⠤⡬⣁⠠⣰⠽⢁⡣⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠈⠀⠀⠀⠀⢀⠀⣀⣀⠉⠻⠧⣄⡀⠀⠀⠑⠤⡂⠈⠲⡈⢯⢂⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⡀⠀⠀⠀⠀⠙⠦⣄⠀⣠⠼⢆⡀⠀⠊⢊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠨⠈⠀⠀⠀⠀⢀⠈⠓⠿⣄⠀⢙⢬⡀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣀⣄⠤⠚⢁⠀⠂⠀⠀⠁⣄⠐⠓⠃⠉⠀⠄⠈⠓⠖⣓⡈⠢⡀⠠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡐⢀⣀⠀⢠⠂⢀⡀⠀⠄⢀⣠⡀⢀⠤⠂⠉⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⢼⡟⠆⢼⡢⠀⠀⠀⠀⠄⠀⠀⡀⢀
⠀⠀⠀⠀⠁⡐⢗⡀⠀⠀⠉⠀⠀⠉⠁⠘⠀⢀⣤⣤⣐⡀⠀⠀⠁⠀⠠⡀⠀⠀⠐⠀⠠⠆⠀⠈⠀⠴⣾⣧⠀⠀⠉⢺⡴⠆⠀⠠⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠑⠒⠆⠉⠀⠀⠀⠀⠀⠀⢄⡄⡀⠀⠀⣚⠀⡈⢠⡿⢽⡧⠄⠀⠢⠀⠈⠒⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠂⡀⡄⠀⠀⠀⠀⠐⠋⠀⠀⠑⠆⠀⡀⢀⠀⢸⣼⠧⠀⣳⢆⣁⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠄⠀⠀⡀⠀⠀⠈⢃⠮⣀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠘⠤⢄⣈⣘⣄⠿⠓⠛⠃⡉⢀⡂⠀⠀⠀⢄⠀⠁⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠘⠀⠀⠁⠀⣀⡐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁⠄⡤⠔⠓⠋⠉⠋⠁⠉⠀⠀⠀⠉⢢⡂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⠁⠀⠀⠑⠀⠐⠀⠀⠑⠓⠀⠉⠅⠓⠀⠀⠀⠒⠓⠤⠒⠀⠂⠨⠃⠓⠒⠒⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠤⠤⠀⠀⠤⠤⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠁⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠁⠲⠭⠝⣋⡑⠒⡢⠤⣄⠀⠀⠀⠀⠁⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠤⠀⠀⠼⠀⠀⠀⠀⠀⠉⠀⠙⠷⠒⠅⣤⣶⣶⢆⠠⣒⠌⠈⠀⠀⠀⠀⢀⣂⣀⣉⣁⣠⣤⢤⣬⣍⣁⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠒⠒⠒⠒⠂⠀⠀⠀⠛⠓⠓⠀⠀⠈⠐⠒⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠣⣈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣀⠀⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢈⡟⠋⠀⣄
⠀⠀⠀⠀⠐⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠹⡿⣷⣶⣦⣤⣄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⣀⣀⣀⣤⣠⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⢀⠀⢹⣿⠿⡿⠿⠛⠛⠛⠛⠿⠿⠿⢷⡶⢷⣶⣶⣶⣶⡶⣶⣶⣶⡴⣾⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⠩⢟⠛⣿⠉⢿⡀
⠀⠀⠀⠀⣠⣾⢿⡿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣯⣾⢋⠟⠛⣻⣷⣌⡄⢿⡷⠃⠀⢻⡇⠀⠀⠀⠀⠀⠀⡜⠃⢽⣦⣿⠁
⠀⠀⠀⠀⢹⣿⣿⣦⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣳⡿⠻⣿⡟⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡜⠀⠀⠠⣿⣿⣬⠂
⠀⠀⠀⠀⢰⣿⣏⢿⡉⠁⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣅⣀⣿⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠊⠀⣠⠾⣿⣯⠄⠋
⠀⠀⠀⠀⠀⠀⠃⠀⠩⡗⠈⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⢿⣿⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢀⠌⠵⢠⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⢢⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢾⡇⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠃⠀⢉⢛⡶⠤⣀⠀⢀⡀⠀⠘⢰⡟⠇⠀⠀⡠⠎⠀⠀⠀⠀⠀⠡⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣶⠛⠳⡄⢹⣷⣰⡇⢹⡆⢲⣎⠀⠔⠻⣶⠾⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠶⣾⣇⣽⣽⡇⠸⣿⢻⢻⢰⣿⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⣸⡿⣾⢡⣾⠀⢤⣤⡟⣹⢿⡯⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣣⣄⣴⣤⣼⣶⣶⣿⣇⣸⣆⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⣐⡒⢖⡰⠷⢞⠛⠛⢛⠋⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣄⠀⠘⢦⣄⡀⣠⠆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠉⡁⠚⠂⠘⠂⠀⢀⣌⢳⡀⠀⠹⣿⡿⠻⠟⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠙⣾⢉⣿⠶⠷⣤⣤⣼⠇⢾⣠⣀⣶⣬⣿⣾⣿⠘⠇⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠻⠈⠿⠁⢀⢼⣅⣿⢓⡾⣿⡻⠙⣿⠋⠿⠟⠀⠈⠉⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣷⣤⡟⠻⠀⣀⣀⣴⣿⡇⣏⡹⠿⣿⡀⠿⠟⠳⣦⡀⠀⠀⢹⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣦⠻⢿⣭⡇⢸⡟⠋⠀⠀⢰⡿⢞⡂⠀⢾⣷⡆⠀⣤⠀⠙⢠⠄⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣽⡿⠛⢹⣷⣿⠄⢤⡄⠀⢺⡇⠀⢱⡀⠀⢹⠃⠈⠂⠀⠠⠀⠀⠈⣴⡀⢀⣀⡈⠠
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⣿⣿⣿⡛⠓⠉⠉⣡⠀⠀⠣⠀⢠⠀⠀⠈⡅⠀⣾⣦⣀⠼⠟⠓⢀⣠⣶⠉⠽⠛⠁⠀⠲⠛⠀⣠⠷⡀
⠀⠀⠀⠀⣴⣷⠚⢻⡿⢻⠿⠿⠿⠓⠀⠀⠀⠀⡈⠁⠀⠀⠋⠛⠑⠇⣀⠉⠠⠟⠁⢀⣠⣴⣾⣿⡍⠤⠤⠤⠤⠀⠂⠠⠶⠶⡋⠀⣤⣄
⠀⠀⠀⠀⠀⠙⣷⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣀⢀⣤⣀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣽⠋⠛⣹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣼⣄⣭⣾⢋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠥⠤⠮⠿⠿⠧⠤

⠀⠀⠴⠀⠀⠀⠀⡤⢀⠄⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⢄⡀
⠀⢀⠀⠀⠠⠆⠀⠀⠀⠚⠁⠀⠀⡇⠀⠀⠀⠐⠈⠁⠂⠢⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠰⢜⡀⠀⠐⡄⠀⠀⠀⠀⠈⢲⡖⠚
⠀⠈⡁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜⠊⠋⠉⠀⠀⠀⡁⠀⠀⠀⠀⠀⠂⢂⣡⡀⠀⢀⣀⣈⣀
⠀⢶⠱⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠄⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠁⠀⢧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠀⠀⠐⠉⠀⠉⠁⠀⠀⠘⠀⠄
⠀⠀⠑⠓⠉⠉⠉⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣶⡆⣶⡆⠀⠰⠶⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠖⠉⠉⠁⠀⠀⠀⠐⡰⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⢰⠈⠀⠁⡆
⠀⠀⠀⠙⠲⣦⠄⠁⠀⠀⣀⠀⠈⠉⠀⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠈⠄⠴⠀⠀⠰⠈⠟⠃⠀⢀⠀⠀⠀⢠⣶⣶⣿⠿⠶⠂⠠⡀⠀⠄⠄⠢
⠀⠀⣠⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠀⣂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠂⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠘⠁⠀⠀⠳⠶⠞
⠀⠘⠋⢀⠄⠦⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠁⠀⠀⠀⠠⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠁⡤⠠⠠⠀⠁
⠀⠂⠀⠀⠀⡀⠀⠄⠐⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠠⠐⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡀⡀⠀⢀⢀⡀⠀⠀⠀⠠⠁⠂
⠀⠈⠐⠒⠈⠀⠀⢀⠄⢸⡇⠀⠀⠀⣠⡼⠀⠀⠀⢰⠾⠀⠁⠐⠀⠂⠐⠁⠀⠀⠀⠐⠀⢘⠇⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⠀⠀⠀⠢⢐
⠀⣀⣀⣤⣤⣤⠄⠈⠀⢸⡇⠀⠀⡀⠄⠀⠀⠀⠀⠀⠋⠐⡅⠤⠀⠀⢐⠀⠰⠑⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡄
⠀⢼⣿⣿⣿⣿⡗⠀⠀⣸⡇⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣾⣷⠀⢠⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠿⠛⠛⠛⠃⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠈⠻⠟⠀⠀⠙⠃⠀⠀⠀⠨⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⡤⣿⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠛⠋⠀⠈⣯⣠⡿⠶⡄⠀⠰⢦⣴⢛⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠙⠓⣶⠀⠀⠀⠀⠀⠈⠻⠶⠶⠃⠀⠐⢻⡇⠀⠁⣠⡼⠷⠒⠒⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣀⣤⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⡛⠉⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠠⠤⠄⢀⣀⠀⠀⠀⠉⢀⡀⠀⠀⠘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠒⠒⠚⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠉⢻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⡀⠀⠀⠀⠀⣤⢶⣿⡇⠀⢀⡤⢶⡿⢻⠛⢷⠶⣄⠀⠀⠙⣦⠀⠙⠟⠃⠀⠀⢀⡞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⡀⢻⡄⠀⠀⠀⢀⡟⠸⠇⣴⠿⢦⣿⣤⣼⣤⣬⣷⠼⠷⡄⠀⠈⣧⠀⠀⠀⠀⠠⠟⢀⡀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠘⠃⠀⠹⣄⠀⠀⢸⠁⠀⢰⡇⠀⢸⠃⠀⢸⡀⠀⢸⡄⠀⢻⠀⠀⢸⡆⠀⠀⣠⠆⠀⠘⠃⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠐⠂⠀⠀⠙⣆⠀⢸⠀⠀⢸⡟⠛⢻⠛⠛⢻⠛⠛⢻⡟⠛⢻⠀⠀⢸⡇⠀⣰⠋⠀⠀⠐⠂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⣇⠀⠈⢧⣀⣼⣧⡤⣼⡦⢤⣾⣄⣠⡟⢠⠀⣼⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠻⢦⣹⣄⢸⣃⣴⣣⡼⠋⠀⣾⣼⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣳⣤⣴⡃⢀⡀⠘⢷⣄⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠛⠋⠁⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣭⣥⣭⣿⡾⠂⠀⠀⠈⠛⠶⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡗⠲⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠂⢸⠐⠂⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⢉⡟⢙⡏⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣳⣄⣼⣠⣼⡃⠘⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠘⢦⡤⢾⡉⠉⣹⢦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠉⠁⠸⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠸⡇⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠠⠄⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢀⣶⣖⣲⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⣿⣶⣾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢻⡿⠉⠉⠛⠿⣿⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠘⠷⠶⠀⠛⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣔⣶⡀⠀⠀⠀⣸⡟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⣄⡄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣦⣀⠄⠀⢀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠂⣠⠉⠤⠠⢾⡟⠋⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅⣠⠂⠀⠠⣄⠀⢸⣿⠀⠀⡙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠂⠘⢧⡀⠀⠉⠄⠈⣿⡇⠘⠟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠘⢿⣄⡀⠈⠀⣿⣿⢠⡀⠀⢻⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢠⠞⠓⢶⣬⣿⣿⣿⣷⣿⣿⣿⡟⣀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠃⠀⠀⠁⠀⠀⣀⠉⣹⠟⠛⠋⠘⠙⠃⢹⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⢀⡞⠁⢰⠃⢤⠀⠀⠁⠀⣀⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣦⠀⣀⡴⠀⡀⠀⠀⡀⠀⢠⣿⡄⢀⡀⣿⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⠛⠀⠞⠋⠀⣀⠁⠀⢸⠛⢲⣿⣿⡟⣿⡁⠘⠟⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣻⣶⠁⢀⣴⠋⠀⠈⣿⠃⢀⣿⠀⢀⠸⣿⡆⢿⣇⢀⣀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⠋⣠⡾⠁⠀⢀⣸⣿⠀⣼⣷⠀⠨⢀⣿⡇⠼⣿⡌⠻⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣋⣠⣴⣾⠟⠁⠀⣀⣦⡀⠄⢠⣿⣿⠀⠀⠃⣿⣇⠀⣻⣿⠰⡧⠄⠈⣿⣧
⠀⠂⠀⠀⠀⠀⢀⡀⠀⢰⡀⣰⣿⣿⣯⣉⣽⣿⣿⡥⠀⣛⣨⢠⣸⠄⢂⣿⣯⣿⡇⣧⠸⢿⣿⠀⡻⣿⣆⠀⠘⢀⣿⣿⡆⠀⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡟⣿⣯⣭⣤⣉⣿⠃⠀⠀⣤⠘⠷⠀⠀⣸⣿⣽⣿⠀⠀⡀⠸⣿⡀⠙⡩⣿⣆⢠⣿⡔⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣿⣿⢯⣴⣦⣼⠟⠃⠀⠀⢀⡀⣤⠀⠀⢴⡟⠉⠝⠁⠀⠀⠤⠀⣿⡇⠀⢤⠹⣿⡄⠘⢧⣈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣟⣷⠟⢁⣀⡀⠀⠈⣮⣷⠏⠀⢰⣦⣤⢠⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠓⠹⣿⡄⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⠟⣡⣶⠿⢁⡶⠾⣛⠋⠁⠀⢠⣿⣿⠃⢸⠀⠀⠀⢠⠀⠀⣿⣧⠀⠰⠆⠀⢸⣿⣆⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣡⡾⠋⠁⠀⠈⢦⣴⢿⣀⣠⣾⡿⣿⣿⡄⠈⠀⠀⠀⠀⣶⠀⣿⣿⠀⣧⠀⠀⠸⣿⣿⣦⠸⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢋⡀⠠⠀⠀⢠⡾⠛⠇⣈⣽⣿⠇⡿⠉⣧⡆⠀⠀⠀⠈⣙⠀⣿⣿⠀⣿⢿⣄⠀⢹⣿⣿⣷⡝⣿⣇
⠀⠀⠀⠀⠘⣷⣿⣛⡟⣋⢶⣿⢧⠀⠸⠟⠉⢁⡴⣾⣿⠟⣿⠀⣧⠀⢹⡇⠀⠀⠀⠀⠃⠀⣼⣿⡀⣿⠚⢿⡶⠘⢛⣻⡙⢿⣿⣿⡄
⠀⠀⠀⠀⢀⣼⣿⡟⣧⣿⠂⣝⠂⢶⠀⠂⠀⣠⣼⠟⣹⣧⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⢩⠀⠀⠀⠀⣙⣿⣇⠈⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣟⠾⣿⣿⢛⣿⡀⠃⢀⣴⣿⢋⠀⣾⡇⢻⣦⢘⣴⣄⣉⣄⠀⠀⠁⠀⠀⠀⣿⢿⡇⢸⣀⠀⠐⠈⢿⣿⡯⢶⡌⢻⣿
⠀⠀⠀⢰⣯⡙⣿⣶⠌⠹⣿⣥⡾⣣⣿⣿⡇⠸⣧⠘⠷⢀⣿⣿⠙⠣⠉⡙⠇⡀⢀⠆⠀⠀⣿⢸⡇⠛⣿⣄⠀⢼⡆⠈⣵⣿⡇⠈⣩⡾
⠀⠀⠀⣾⣿⣷⣄⣉⣀⣘⣍⠁⣼⣟⣿⣿⡅⢦⣙⢷⣄⣸⣿⣿⣷⣀⠀⠰⠶⠟⢻⠿⠦⠀⣿⠀⡇⠀⢾⣿⣧⠄⠀⡐⠋⢉⣅⠀⢹⣧⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⠹⣧⠈⢳⣿⣿⠃⠘⢿⣾⣿⣿⣿⣿⠇⣿⣦⣀⠀⠀⠛⠂⣾⠀⣿⠀⣷⠀⠛⠾⠁⣠⠞⠁⠀⢸⣷⠀⠸⣧⣁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠻⠂⣼⣿⣿⡆⠀⠀⠙⣿⣿⣿⡿⢠⣿⡟⠙⢷⣦⣄⣀⠀⠀⣿⠀⣿⠀⣀⣤⣾⠃⠀⢀⡠⢿⣿⣤⠀⣿⣿
⠀⠀⣾⣿⣿⣿⣿⠿⠋⠈⠳⠀⠐⣿⣿⣿⣿⡄⢀⠀⠹⣿⣿⡇⣸⡟⠁⠀⠉⠉⠉⠙⠛⠶⠿⠀⠹⠟⠋⢀⣾⡇⠐⣃⠀⢸⣿⡃⠀⢿⣿⠄
⠀⠀⠙⠛⠛⠻⣧⣤⣤⣄⣀⣀⣴⣿⣿⣿⣿⣧⡀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡇⢠⠍⠀⢸⣿⠀⠀⢸⣿⢦
⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣛⣿⣿⣿⣯⣿⣭⣿⣟⣀⣤⣾⣿⣿⣶⣶⣦⣤⣀⣈⡁⢀⣀⣀⡀⠀⠀⣀⣀⣶⣿⣧⣀⣀⣠⣼⣿⡤⠴⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣿⣟⣻⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠉⠉⠙⠛⠛⢶⣾⣿⣿⡿⠿⣻⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠠⠤⠄⠀⠀⠀⠉⠀⠛⠛⠛⠛⠛⠛⠻⠶⠿⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣶⣷⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⢀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠴⡈⠘⠉⠁⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⢀⡁⠀⡀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠙⠋⢁⣰⠇⣀⠀⠀⢀⡄⢰⡇
⠀⠀⠀⠠⠀⠠⠤⡀⢀⡘⠒⠛⠋⠁⠀⠙⣿⠳⠁⠀⢸⡇
⠀⠀⠀⠀⢀⠀⢶⣛⣿⠁⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢸⡷⡄
⠀⠀⠀⠀⠀⠘⠛⢛⡉⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸⡗⢃⠀⢀⠔⡦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⣿⡄⢹⡠⠊⢃⡇⠀⠀⠘⢿⣷
⠀⠀⠀⠀⢀⣤⣿⣿⠀⠀⢠⣾⣫⡿⠁⠀⠀⠀⠀⠀⠧⢻⣿⣶⠶⠟⠀⢠⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⢹⡆
⢀⠀⠀⠀⢀⣿⣯⠀⠀⣠⡿⣽⠃⠀⠀⠀⡀⠀⠀⠀⣦⡤⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⠀⢣⡀
⠀⠀⠀⠀⣸⣿⠃⠃⣴⠛⢠⡏⠃⠸⠀⠀⠓⠀⠃⠨⠉⠘⠂⠀⠀⠓⠀⠂⠘⡄⠀⠊⠢⠀⢷
⠀⠀⠀⠀⣿⡿⠀⣼⠃⠀⡞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠆
⠀⠀⠀⢰⣿⠇⣰⡿⡎⣼⢁⠃⢠⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠘
⠀⠀⠀⢸⣿⣄⣿⢡⢠⡿⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣯⡏⡿⠀⠜⢠⡆⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢇⠀⠀⠈
⠀⠀⠀⡈⣫⣽⡭⢀⠇⠀⠀⣼⠀⠀⡠⠀⠀⠀⠀⠀⢸⠰⣀⠀⠀⠃⠀⢠⠶⢄⡀⠈⠀⠀⠀⢠⡀
⠀⠀⢠⢸⣿⢿⣧⣾⣶⡄⠀⣇⣀⣴⠛⣶⢄⡀⠀⠀⣿⡆⢋⠳⠢⣴⠠⡆⢃⠀⠀⠚⡶⠂⠃⠀⠀⢄
⠀⠀⢸⣎⠻⣷⣿⣿⣿⢏⣴⣾⠁⣿⠀⠇⠈⣿⠙⠚⣻⡇⢸⠀⠇⢛⠳⣹⡜⡆⠀⠀⡀
⠀⠀⢸⣿⣷⣦⣽⣿⡶⢸⡏⣿⠀⣿⠀⠀⠀⢿⠀⠀⢹⢻⠀⡇⠀⠈⡄⢸⣧⢰⠀⠀⢄⠀⡀
⠀⠀⣸⣿⣿⢿⡏⢸⠇⣼⠷⡏⢐⡇⠀⢠⠀⢸⠀⠀⠈⢸⡄⣿⡆⠇⠀⠀⢿⠀⠂⠀⠈⠀⠁
⠀⠀⢿⣿⣄⡀⠀⡟⠠⠋⠸⠃⢸⠷⠀⠿⠂⠈⠁⠰⠀⢸⠄⢸⠀⣀⠀⣀⣼⣦⣤⣄⣤⣤⣤⣴⣶⠖
⠀⠀⢻⣿⣿⣿⣿⣿⣷⣾⣶⣶⣷⣴⣴⣶⣴⣾⣿⣷⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠁
⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠖⠒⠒⠲⠶⠶⠶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢋⣉⣠⡤⠤⠴⠶⠖⠒⠒⠲⠶⠀⠀⣀⣉⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⠀⢋⣉⠁⠔⠒⠚⢋⣉⣉⣉⣠⣤⣶⣦⠄⠈⠉⠛⠷⠤⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⣠⣴⣾⡿⠀⠚⠋⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠉⠛⣷⣶⣦⣤⣈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠴⣾⣿⡿⢛⡭⠀⣀⣀⣠⡄⠀⠐⠒⠒⠛⠳⠿⠷⠤⣤⣾⣿⠖⠀⠒⠶⠬⢭⣉⣀⣈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⡰⠃⣰⠟⡹⣫⢖⣵⣾⣻⢟⣭⣭⡓⠒⢲⣶⣶⣤⣶⠖⢲⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣤⣭⣍⣁⣻⣧
⠀⠀⠀⠀⠀⠀⣸⠧⠾⣗⣻⣭⣟⠉⢩⢿⠇⢀⢶⡲⣔⣆⠀⡶⠛⠿⠶⠬⠓⠒⠛⠛⠛⠀⠒⠒⠪⠭⢭⣙⣁⠰⣾⣿⣿⣧
⠀⠀⠀⠀⠀⢰⠗⠚⠋⠙⡛⠉⣉⣁⣘⠊⡃⠻⠬⠛⢸⣾⢣⠆⠀⢀⣀⣤⡤⣤⣤⣤⣤⣀⣀⠀⠀⠀⣀⣆⠏⠇⠿⠃⠹⡽⣧
⠀⠀⠀⠀⠀⣎⣸⣿⣿⣿⠧⠴⢿⣿⣿⠀⠙⣒⣖⣻⣽⠇⠈⢀⣠⡽⠛⠉⠀⠉⠉⠉⠉⠉⠉⠉⠙⠓⠻⠿⢽⡛⣶⢷⡶⣷⢿⡄
⠀⠀⠀⠀⢰⠏⠉⣁⣀⣠⣤⣤⣤⣤⣩⠑⠿⠋⠉⠀⠀⠀⠖⠋⠡⠤⠶⠶⠶⠿⣿⣿⣿⡿⠋⣳⣦⣤⣀⣀⣀⣉⠘⠘⠗⢻⣿⡇
⠀⠀⠀⠀⢸⡆⠻⠛⠛⢋⣉⡩⠴⢒⡀⣠⣤⣴⣶⣶⡿⠧⠶⢖⣒⠛⠉⠀⠀⠀⠻⢓⣶⣆⣀⡤⣤⠭⣭⣤⣽⢹⠟⣿⣿⡏⢩⡇
⠀⠀⠀⠀⢸⣿⣶⣤⣿⣿⣿⣆⡈⣽⣦⣤⣭⠥⢴⠒⠒⠋⣉⣁⣀⡤⠤⠀⠤⠖⠒⠒⠒⠶⠬⠟⠛⣛⣋⣉⣻⣹⣉⣿⣷⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⢿⡿⢿⢿⡿⢩⣭⣽⠶⠞⠋⠛⠛⠋⢉⣉⣡⡤⠶⠚⢀⣤⣶⠯⣭⣭⣽⣶⣤⣄⣩⣏⣩⣍⣧⣼⣿⣾⣿⢾⠇
⠀⠀⠀⠀⠘⣿⣿⣟⢓⠀⠤⠤⠌⠓⠓⣀⡤⠴⠶⠒⠋⠉⣁⡤⠴⢒⡿⣻⣿⣿⣿⡆⣸⠖⣻⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⡿⡞⣖⢦⠲⠘⠉⠁⢀⣀⣀⣠⣤⣤⣴⣶⣤⣽⠶⣿⣿⠷⣻⡿⣣⣼⢁⣰⣿⣭⡄⠛⠓⣲⣿⣿⣿⣽⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⠛⠛⣻⣿⣿⣿⣟⣿⣿⣿⡿⢗⣀⣤⠴⣻⠏⣀⣬⣾⣿⣤⡽⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢿⣟⠻⡟⠩⠭⣭⣽⣿⠛⠛⠛⠛⣃⡀⠉⠉⠉⠁⢙⣻⣀⣼⣷⣿⡿⣿⠿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣟⣿⣿⣿⣿⣿⠿⢿⡿⢟⣽⡟⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣛⡛⠻⣾⣛⡛⠾⣿⣿⣿⣿⣿⣟⣯⣭⣶⣾⣿⣾⣿⣿⣿⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣛⣿⣽⣞⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣮⣭⣿⣿⣭⣽⣿⣿⣯⣿⣿⣽⣷⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠾⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⣦⣋⣀⡙⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠰⠆⠀⡔⠁
⠀⠀⠀⢀⡴⠁⢫⣿⣽⠉⢦⠀⠀⠀⠀⠀⠀⢦⣴⡿⠂⠀⠀⠀⠀⠀⠛⠛⢿⣿⣏⣽⣿⡄⠀⠀⠀⠐⠀⡀⢀⣾⣿⣧⠁⢀⡄⠂
⠀⠀⢀⡞⣠⡔⠈⠻⠉⠢⣌⢳⡀⠀⠀⢠⣀⣾⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠈⠠⡁
⠀⠀⠀⠉⠉⠉⠙⠛⠒⠚⠛⠚⠃⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⡿⠂⠀⠀⠀⠰⠄⠋⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠄⠂⢀⠐⠣⣀⡀⠀⠶⠀⠀⠀⢠⠁⠀⢻⣿⠇⠀⠱⡀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠠⣀⠀⡸⢄⢀⠜⠂⣆⣤⠀⠀⠀⢀⣴⣶⡦⠷⣴⣶⣦⠀⠀⠀⠀⠀⠀⢺⣝
⠀⠀⠀⠀⠀⠻⠿⠿⡿⠿⠿⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⢸⡋⠀⠀⠈⠻⡋⠀⠀⠀⠀⠀⣤⣾⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⣆⠀⡅⠀⠄⡀⠀⠀⠀⠀⣀⡙⢠⣤⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢲⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠛⠣⡀⠀⡶⠀⠶⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⡄⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠰⠳⠄⠃⠀⢰⡆⠀⠀⠀⠀⢀⣀⠼⠿⠀⠐⢷⠀⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⠀⠀⢀⡇⠀⠈⠁⠀⠀⠀⠀⠘⠋⠀⢸⡆⠀⠀⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠛⠛⢻⡀
⠀⠀⠐⠿⠿⠶⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆⢰⣶⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⡀⠀⠀⠀⠀⢀⣠⠤⠶⠤⣤⡀
⠀⠀⠀⠛⢛⡿⠛⣷⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⣄⢀⣤⣄⢠⣤⠀⠀⠀⠀⠀⣀⣨⠟⠛⢿⣷⠀⠀⠀⠀⢊⣾⡡⠄⢀⣴⡙⡀
⠀⠀⠀⠀⠾⣥⣴⣧⣤⣤⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣾⣇⣠⣼⡿⠀⠀⠀⠸⢏⠉⠙⣀⠋⠁⠉⠇
⠀⠀⠀⠀⠀⠰⣟⣁⣁⣨⣇⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠈⠈⠐⢿⣿⡿⠛⠁⠀⠀⠀⠀⢂⢠⣀⣿⣀⡴⡠⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠩⠟⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠈⠙⠂⠘⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠉⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣸⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣴⡂⣀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⣾⣿⣿⣿⡇⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠛⠉⢻⣿⣷⣦⣄⣈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣷⠀⠀⠀⠙⢿⣆⣙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⣶⣄⠀⠸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢰⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡟⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠋⠁⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⡀⠀⠀⠀⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⢾⠯⣠⣛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣪⡤⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⢫⣹⣳⣤⣤⢤⣤⣤⡶⠤⠤⠖⠚⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣀⣦⡶⠟⠛⠉⠁⣷⠣⣫⡿⡟⣠⠂⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢴⣿⠟⠋⠀⢀⡰⣶⣤⡘⣿⣿⣿⣿⠯⣠⣾⠋⠀⢀⠀⠀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠋⠁⠀⠀⠀⠀⠙⠷⠎⣸⣿⣿⣿⠋⣠⣛⢛⠚⡓⠉⠀⢀⣠⣽⡁
⠀⢀⣤⣤⣤⣤⣤⡤⠤⢤⡜⡁⢀⡀⣀⣀⣀⣠⣤⣴⣶⡿⠟⠛⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠹⠿⠃
⠀⠀⠉⠛⠳⣶⣿⣇⡈⠟⢛⠙⠛⠿⣋⣭⠉⠠⠀⠓⠁⠀⠀⠀⡀⠀⡠⠆
⠀⠀⠀⠀⣼⣷⡿⠟⠁⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠈⢀⠀⠀⠀⠀⠁⠴⠒
⠀⠀⠀⠸⠟⠋⠀⠀⠈⠀⠀⠉⠉⠀⠉⠉⠶⠤⢄⣀⣀⠀⠐⠀⣀⣀⠠⠐⡒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠀⠘⣟⠛⠛⠃⠐⠄⠀⠀⠀⣶⣄⢀⡀⠀⠀⢰⠒⠢⣤⠂⠐⡄⠀⠀⡹⠒⠊⣁⡀⠄
⠀⠀⠀⠀⠤⠀⠟⠀⠀⠀⠤⠀⠀⢀⣴⡆⠀⠑⡀⠀⢀⠜⢖⡚⠓⠓⢦⢶⢄⣉⠀⠀⠸⢀⠝
⠀⠀⠀⠀⠀⡜⠉⠀⠀⢇⠀⢰⢙⠟⠁⠰⣴⣾⠁⠀⠈⠳⢄⣗⣄⠚⠀⡿⡀⢉⠆⠰⠽⢂⠔⡳
⠀⠀⠀⠄⣸⣶⣾⣷⣿⣿⣆⠀⣿⢣⠀⠀⠬⠬⡀⠀⠀⠀⠂⢣⠈⢙⠾⠥⠏⠀⠈⢀⠔⡣⠊
⠀⠀⠀⢠⢹⣿⣿⣿⣿⣿⠏⡆⢸⠤⠧⠤⠭⠭⠽⢄⠀⠈⠃⠀⠉⠁⢀⣤⠤⡀⠶⣿⠋⠀⠊⢹⡂
⠀⠀⠀⢀⡑⠚⠉⡉⠑⠒⠈⠈⠈⠂⠤⠤⠤⠴⠐⠋⠀⠀⣤⠀⠀⣴⢋⠔⡒⢢⠱⠈⡄⠀⠃⣸⡉
⠀⠀⠀⠈⠁⠀⠀⠻⠇⠀⡀⡀⠀⠀⠀⠀⠐⠊⠢⠔⠀⠀⠈⠀⠀⣇⠸⣨⡋⡱⢡⠀⡇⠀⡄⣷
⠀⠀⠀⠐⢃⠞⠉⠳⡄⠈⠉⠉⠀⠀⠀⠀⠈⠉⠒⠒⠁⠀⠶⢦⠰⠈⠲⣀⣄⣀⠡⠊⠀⠀⠓⠋
⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⢒⣠⣀⠀⠀⢀⠀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡉⠉⠉
⠀⠀⠀⠀⣀⠀⠀⢹⡆⣰⠃⠀⠈⠧⠀⢹⠀⠀⣾⠉⠉⠀⢀⣷⠀⢀⡇⢠⠞⠉⠷⠀⢸⠖⠒⠂
⠀⠀⠀⠀⢘⠶⠤⠞⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿⠴⠶⠀⢸⡇⢧⢺⡇⣾⠀⠀⠀⠀⢸⣤⡤⠤
⠀⠀⠀⠀⠈⠀⠀⠀⢠⠹⣄⠀⣀⠄⠀⣿⠀⠂⢿⣀⣤⡀⠼⠀⠘⠾⠀⠘⢤⡤⠞⠁⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠉⠁⠀⢈⡉⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⢠⣢⢤⡠⠀⡠⣊⠅⢌⠑⡔⠂
⠀⠀⠀⠀⣦⠀⣁⠀⠀⡸⠀⢀⡈⠀⠈⠫⣀⠎⠁⠂⠮⢈⡉⠉⠁⠀⡤⠀⠠⣄⡾⠖⠁⠀⡠⢀⠇⣴⠄
⠀⠀⠀⢀⣀⣸⣓⣇⣸⣹⣀⣏⣇⣀⠀⣠⠁⠂⠀⠀⠀⠈⠁⢶⣶⠀⡙⢦⡸⠍⠀⠀⠀⠸⠷⢊⣀⠋
⠀⠀⠀⠈⣿⠗⣇⡏⢹⣽⠉⣯⣿⠋⠀⠈⢢⠀⠀⢀⡀⠀⡀⠀⠁⢀⠵⠋⠈⠱⢄⠀⠀⠃⠀⡀⢻⠗
⠀⠀⠀⠀⣿⠀⣿⡇⢸⣿⠀⢿⣿⡄⠠⠒⠁⠱⠆⠒⠂⠈⠁⠀⠤⠎⠀⠀⠀⠀⠀⡧⡄⠀⠀⣷⡷
⠀⠀⠀⢴⣿⣶⣿⣷⣶⣶⣶⣶⣿⠷⠆⠓⠁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⢹⢳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⢠⣷⡘⠀⣿⡇⠀⠀⣀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢻⣿⣤⣾⣿⣿⣿⡀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⢢⣄⠀⠀⢐⣄⣀⣠⣼⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡇⠀⢀⣴⡆⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠙⣿⣶⣶⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀⠹⠗
⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⢀⣀⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡆⢨⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣼⣥⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠁
⠀⠐⠿⠟⠛⠛⠛⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣧⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣁⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠙⠃⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠿⠛⠻⠟⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⣿⡁
⠀⠀⠀⠀⠀⡾⠛⠉⠀⠀⠀⠀⠀⠀⣰⡿⠋⠻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⢯⡈⢳⣾⣷⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⢄⠀⣰⡿⠉⠙⣿⣿⣿⣶⣿⣿⣿⣿⣿⣏⠻⣮⣿⣶⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠏⠀⠀⠐⢿⣿⡿⠻⣿⠿⠟⣿⠋⠙⠀⠈⠻⣿⣿⣿⠟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠻⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⢀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠩⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡽
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠸⣿⡉⠁⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⣠⣠⣠⣠⠀⠀⢠⡄⠀⠀⠀⠀⣾⠀⠆⠑⠍⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠘⢆⣐⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠒⢤⠀⠀⠀⠀⠀⠐⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠠⠾⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢒⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠳⣄⣀⣀⣀⣠⠴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠈⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠀⠀⠀⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠒⢿⡄⠤⠤⠔⠠⢄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠤⠔⠋⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠁⠀⠈⠙⠦⡀⠀⠀⠀⠀⢰⠩⠩⣦⠀⠀⠀⠀⠀⠀⠀⠸⣁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠉⠉⠉⠙⠛⢳⠀⠀⠀⠀⠈⠒⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⡛⠿⠆⢀⣀⢀⡀⠀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⠛⠇⠀⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠉⠉⠛⠉⣩⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⣄⣀⣀
⠀⠀⠀⠀⢰⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⠋⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣬⣭⣭⣭⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⢀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠁⠀⠈⢿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣻⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⢧⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠙⠋⠁⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⣠⣤⠀⠀⠀⠀⠀⠀⠻⣿⣧⣭⣭⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⡀⠀⠀⢀⣠⣀⣀⣀⣀⣸⣿⣿⡇⠀⠀⠀⠀⢀⣠⣌⡛⠛⠛⣋⣡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠀⠀⣿⣿⣿⡿⢋⣭⣭⣿⣿⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠙⠿⣿⣿⣿⡿⠋⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠋⠀⠀⠠⠴⠶⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠛⠛⠿⣿⠿⣷⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⣿⠟⢿⣿⣿⣿⣿⣶⡾⠶⢶⣬⠻⣿⣯⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⡿⣿⣿⣿⣿⣿⡀⠀⠀⣀⡰⣾⣿⠃⠛⢿⣶⣿⡿⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⣛⣍⣷⣿⣿⣿⣿⣿⣿⣆⣼⣿⣿⣦⣥⠀⠀⠀⠉⠛⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠘⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⣿⡟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣦⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣷⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡄⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠉⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠘⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⡀⠀⠀⠀⠀⠀⠈⣋⣀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣶⣶⣶⣿⣭⡤⠴⠶⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠁⢳⡀⠀⠀⠀⠀⢀⡾⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠚⣇⠀⠀⠀⠀⢸⡃⢠⣴⡞⣆
⠀⠀⠀⠀⠀⢠⡞⢛⡷⢄⠀⠀⠀⠀⠀⠘⣧⣿⣄⠀⠹⡀⠀⠀⠀⢸⡇⠛⢀⣱⣿
⠀⠀⠀⠀⠀⠘⡄⠀⠑⠈⢆⠀⠀⠀⠀⠀⢻⠇⢁⡶⢿⣇⠀⠀⠀⠘⡇⠀⠚⢉⣽⡇
⠀⠀⠀⠀⠀⠀⢰⡀⠂⠀⣪⣢⠀⠀⠀⠀⠘⡆⠈⠀⠀⠈⡄⠀⠀⠀⣷⣯⡀⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⢳⡸⠆⠀⠄⢣⡀⠀⠀⠀⠉⠀⠀⠀⠀⢺⡀⠀⠀⠟⠀⣰⣟⠃⢛
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠀⠀⠀⢳⡀⠀⠀⢨⠀⠀⡠⠖⠚⣇⠀⠀⡄⠰⠋⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠠⠻⣄⠀⠈⣇⢠⠀⠀⠀⢹⠀⢠⡅⠀⠓⠀⠐⠺⡆
⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡴⠎⠲⠌⢆⠀⠸⡀⣆⠀⠀⠘⡇⠘⡇⠀⡀⠠⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛
⡟⠡⠈⡓⢄⠀⠀⠀⠀⠀⠈⢶⠁⠠⠂⠂⠸⣄⢀⠃⠈⠙⠛⠋⠙⠚⠁⠊⢦⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⠂⣔⢀⡴
⠹⣤⣤⣤⣶⣷⣄⡀⠀⠀⠀⠘⣧⠀⠐⠠⠀⠘⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⢿⣄⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⡆⢀⣤⢄⡞
⠀⠈⠲⣀⠩⠉⠉⠙⠲⣄⡀⠀⠈⣧⠀⠸⣧⣄⠀⠀⠀⠟⠁⡀⠀⠀⢤⡄⣴⣂⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⡼⠣⠝⣿⠿⡿⡾
⠀⠀⠀⠈⠳⢆⠀⢰⣿⣀⢛⡶⠤⠞⠀⣄⣂⠀⠀⠀⠀⣠⣾⣿⣿⡦⠿⠃⢿⡿⢿⠀⠠⣿⣦⠀⠀⠀⠀⣸⠡⠾⠊⢍⡀⣾⠃
⠀⠀⠀⠀⠀⠀⠳⣟⠉⠀⢂⠠⢀⣴⠄⠉⠀⢠⡆⣀⡼⠟⢫⣥⣠⣀⠦⠀⢿⣵⠊⡷⢠⡷⠢⠷⣄⣀⣼⠏⠠⢘⠀⠝⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠁⠀⠀⠀⠠⡀⢀⠈⠁⠉⠀⠀⠀⢉⡿⠟⡷⣟⣶⡑⡓⢡⠟⣿⣠⠶⠈⠉⠉⡉⠐⣨⡋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡙⢢⣐⠁⠀⠀⠀⠈⢀⠀⠀⣴⣦⡀⠀⣾⠟⠃⠦⠁⠼⠃⡯⢵⠄⣆⢰⣦⡀⢰⣎⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠛⠟⢀⠐⠊⠓⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⢶⣽⡄⢀⣒⣋⢴⣓⠋⠉⠀⢺⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⣴⣾⣿⣿⠶⠀⠀⠀⠚⠀⣿⡿⠀⠀⠀⠉⠁⠨⠈⣉⠀⣹⢻⡟⠀⢩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⡟⡿⡆⢰⠀⠀⠀⠀⠈⠁⠐⠒⠀⢳⡻⣄⣿⠇⠀⢠⣄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠈⠃⢀⠀⠘⡆⠀⠀⠀⢠⣀⡀⠀⡀⠉⠂⢝⣯⢀⣶⣾⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡠⠇⠉⠀⠀⠙⠒⠀⢀⡀⢉⠛⠃⠀⡀⠀⢀⣳⣌⡙⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣐⣒⣶⣂⣈⣉⠉⠉⣭⠀⠤⢅⣀⡀⠉⠉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡠⣴⡯⣤⣀⠀⠰⠶⢿⠿⠂⠈⠀⠈⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⣤⡄⠀⠤⠤⠄⠀⠸⣿⠏⠡⠶⠶⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡌⠉⡄⢆⡶⣴⣶⡦⠀⢀⠀⠇⠐⠟⢻⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠷⢶⣼⢃⣄⢠⡄⠀⣾⢠⢆⡄⢀⣾⠸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣴⣵⣶⣿⣼⣼⣮⣤⣼⣿⣼⣼⣼⣮⣧⣶⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡶⣆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠄⠀⠀⠀⠀⢀⠠⠄⠀⣃⡃⠀⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢰⡅⠈⠘⡦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⢠⠀⡔⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⢀⠀⠀⠀⡀⠀⠂⠂⡰⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠈⠱⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢤⡔⠁⠀⠈⡠⠒⢤⠀⠀⠀⠀⡴⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠂⠀⠂⠔⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠈⠉⠉⠀⠓⠤⠞⠀⠀⠀⠀⠀⠀⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⡁⠊⠀⠀⠀⠄⠀⠇⠀⠈⠉⠉⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⢀⣀⡈⠂⠁⠀⠂⠀⢠⠤⠤⠴⠧⠤⠤⢤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⡠⠀⠀⠀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠶⢀⠆⠀⠠⣀⠀⠒⣒⠚⡛⢛⠛⠛⡉⠂⠀⢀⠂⠀⠀⠐⠀⠀⢱⣄⠀⠀⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢊⡁⠀⠊⠰⠀⠀⠂⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⢴⡦⢀⣤⣀
⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⢤⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠰⠏⢉⠉
⠀⠀⠀⠀⠤⢆⡕⠀⢀⠀⠀⠈⠋⢰⡟⣶⠀⠸⡟⠂⢀⡶⠛⢷⡀⢰⡇⠀⠀⠀⣴⠛⢳⡄⠀⣴⠛⠻⠆⠀⠻⣆⢀⡾⠁⠀⠀⠘⠒⠖⠉
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⢸⡷⣯⠀⠀⣿⠀⢸⡇⠀⢸⡇⠘⣇⠀⠀⢸⡏⠀⠈⣿⢰⡇⠀⠰⣦⠄⠀⠹⣾⠃⠀⠀⠀⡆⠀⠄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡼⠃⢠⣿⠄⠘⢷⣤⠾⠃⠀⣷⣤⣤⠈⢿⣤⡾⠃⠀⢷⣤⣴⠟⠀⠀⢸⠇⠀⠀⠀⠉⠀⠈⠂⡀⠈⠉
⠀⠀⠀⠀⠀⠤⢀⠒⠀⢃⠀⠂⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⡅⠀⠈⡢⠠⠇
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⠀⠀⠣⣀⡸⠀⢀⠜⡁⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠐⠂⠚⢃
⠀⠀⠀⠀⠀⠠⠐⠓⠄⠓⠂⠀⠈⠙⠀⠀⡠⠁⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⠀⠀⠀⡄⠀⠂⠀⠀⠀⠐⢂⠀⠈⠑⠂⠁⡀⠀⡈⠁⠉⠠
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠃⠀⠡⠀⠀⢰⡒⠀⠀⢈⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠴⠀⠀⡘⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠆⠠⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠆⠀⠀⠀⠾⡂⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠠⠄⠄⠀⠀⠀⠀⠀⠀⢰⠄⢠⠄⠀⡄⠀⡄⠠⢐⡀⢀⡀⠀⢀⠊⠀⠤⠀⠀⠀⠀⠘⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠤⠠⠀⠀⠀⠐⠚⠲⡀⠀⠀⢠⠤⠇⠀⠈⠀⢸⠢⢤⢈⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠈⠀⠂⣄⡂⢐⡀⠀⠀⠀⠀⠀⠈⠂⠢⠤⠤⠴⠋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠘⠒⠀⡀⠀⠉⠁⠀⡈⠘⠂⠀⢀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠲⠁⡚⠦⠘

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠈⠁⢠⣤⠀⠈⢁⠀⠀⡀⠀⢼⣗⢀⡐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⡄⠀⠀⢀⢠⡤⠰⠀⠴⢆⣀⡀⠀⡂⠀⢀⠒⠿⠇
⠀⠠⠇⠀⠀⢀⡀⠈⠃⠀⡁⠀⠉⠠⠈⠀⠀⠀⠀⠒⠐⡀⠀⠀⠀⠀⣄⠀⢶⠆⠒⠋⡹⣆⠀⠴⣖⣤⠤⠆⠀⣩⠍⠀⢀⠀⠀⠈⠀⠒
⠀⠀⠐⠓⠂⠈⠿⠇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠀⠀⠈⢳⣇⠈⢠⠀⠤⢈⡆⠀⠚⢊⣤⠜⠂⠀⠀⠀⠺⠽⠇⠀⠀⣤⡖
⠀⠀⠙⠂⠀⠰⠶⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠐⠂⠐⠃⠉⠰⣀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠄⠀⠂⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠏⠀⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠰
⠀⠀⠂⠀⠀⡀⠀⠀⡀⠈⠃⠃⠀⠀⠀⠀⠀⠀⠍⠈⠿⣿⣿⡷⠆⠀⠀⠀⠁⠀⡀⠀⠀⠃⠀⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷
⠀⠀⣤⣦⡀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⢀⣦⡄
⠀⣞⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠛⠿⠿⠿⠛⠁⠀⠀⠀⠋⠁
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⣾⣧⠀⢀⣘⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠰⠲⢒⣒⣊⡀⠐⢠⠐
⠀⠀⠀⠀⠒⢻⠿⠄⠀⠈⠀⠀⠀⢠⣇⣹⡀⣤⠍⣿⢠⣤⡤⣠⢤⡀⣦⢤⡄⣰⠞⢦⢰⡶⣦⠞⢳⣴⡆⠀⣶⠀⠘⠋⠩⠭⠶⠀⠂⠐⠄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠷⢤⠀⡾⠉⠙⡇⣙⡷⢸⠀⡿⠀⢧⣠⠇⣷⠠⡇⢻⣄⡼⠸⡇⢿⠀⢸⡟⢧⣴⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠆⠀⠀⠠⠄⠀⠰⠀⠈⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠠⠀⠀⠀⠂⠀⢀
⠀⠀⢘⣛⣛⣀⡩⡄⠀⠛⢠⠣⠌⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠋⣀⡄⠀⠀⡀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠩⠁⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣤⠈⢠⠀⠀⢸⡇⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⣰⣷⠗⠀⠀⠀⠀⠐⠀⠀⣠⠀⠘⣩⠖⠀⠀⠀⠈⠻⣶⣤⡴⠀⠀⠀⠀⠀⠐⠁⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣶⣾⣿⣿⣥⠤⡆⠀⠀⡀⠀⠀⠰⠷⠶⠚⠁⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠟⠂⣠⠆⠀⠀⠂⠀⡀⠉
⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⣻⠃⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠄⠀⢀⣠⠆⢀⣠⠄⠁⠀⣼⣿⠁
⠀⠀⣰⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣞⣶⡦⡀⠀⣼⠇⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠁⠐⠉⠁⠀⣀⡀⠿⠿⠄
⠀⠠⣿⠃⠀⢹⣿⣿⡶⣾⣷⠀⠀⠀⠀⠈⠏⢧⠇⣧⠼⢋⡀⠀⠀⠀⠸⣿⣦⣀⠀⠀⠀⠀⢀⠀⠄⠒⠉⠀⡶⠃⠀⠀⠀⠉⢁⠀⠀⠠⠄
⠀⠤⠀⠀⠀⢿⡿⠁⣠⡿⠃⠀⠀⠀⠀⠀⠈⢨⣁⠠⠔⢋⡴⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⢀⣤⠄⠀⢶⡆
⠀⠀⡀⠀⢀⠀⠀⣄⠈⠀⠀⠦⠠⠄⠘⠂⠀⠘⠂⠐⠛⠉⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⠃
⠀⢺⠿⠀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⣀⣀⣄⠀⠀⣠⡀⠀⠀⠺⡧⠂⠀⠂⡀⠀⠀⠀⢀⠀⠐
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠟⠃⠀⠃⠀⠀⠃⠀⠐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠉⠀⠀⠂⠀⠀⠈⠋⠁⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⡁⡄⠖⡆⢤⠸⠇⠟⠀⠀⠀⠛⠰⠄⠀⠀⢀⢄⢀⡠⠀⠀⠀⡰⣆⠀⠀⡔⠉⠀⠀⠀⠀⡄⠀⣀⠂
⠀⠀⢦⠀⢠⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣿⡄⡽⠉⠀⠀⠀⠘⠁⠈⠃⠸⡅⠀⠀⠀⠀⡰⠃⢰⢹⠇
⠀⠀⠀⢧⠸⠀⠀⠀⣤⠀⡴⠀⠀⠰⢶⡀⢄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠤⠒⠖⠒⠷⣆⠒⠋⠉⠀⠀⢸⢸⡀⢀⠴⣾⠂
⠀⠈⠣⡘⡇⠀⠠⡴⠁⠉⢶⠀⠀⠀⠀⠀⢀⠵⠒⠒⠒⠤⠶⠆⠾⠀⠀⢰⠋⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈⢦⣉⣁⠤⠊
⠀⠀⠀⠘⣇⠀⢠⠜⣤⣠⠾⠀⠀⢰⡆⠘⠁⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠣⡀⠀⠀⠀⣀⡴⢃⣴⠶⠆⢠⡄
⠀⠀⠀⠀⢸⢀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⡾⠋⣰⠒⠉⠉⠉⠉⢣⡉⠉⠁⠀⡎⣏⣀⢠⠃⣇⢞⠆⣠⣤⡀
⠀⠀⠠⢄⢸⡞⠀⢀⣀⣀⡀⣀⠤⠢⡀⠀⠀⠀⢀⣀⣀⣉⣀⠀⠱⡄⠀⠀⠀⠀⠀⡣⠀⠀⠀⠓⠴⠊⡧⢤⢀⡎⠀⠻⠿⠂
⠀⠀⠀⠀⢺⡇⠀⢋⡀⢀⡜⠑⠢⣤⠁⠒⠰⠚⠋⠁⠀⠈⠁⠉⠛⠛⡖⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢊⡴⠲
⠀⠀⠀⠀⢈⠂⠀⠈⠳⣞⠀⠛⢠⡎⠳⡀⠀⠀⠀⠀⠀⠐⠀⠀⣀⡰⠃⠀⠀⠀⠠⡶⡄⢰⢶⠄⢠⣤⠄⠀⢸⣿⢠
⠀⠀⠀⠀⠘⠀⠀⠉⢸⠀⢉⡲⠯⠄⠤⠃⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣧⡇⢸⣾⠀⢸⢷⠀⠀⠈⠣⢸
⠀⠀⠀⠀⠀⣀⡀⠀⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⡖⡄⠀⠀⢰⠒⣄⠀⠀⣿⠁⢸⣿⠀⢸⣼⠀⠀⠀⡀⢸
⠀⠀⠀⢠⠀⡏⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠒⠃⡇⠀⠀⡦⢙⢘⠀⠀⠟⠀⠘⠛⠀⠐⠛⠀⠀⠸⣇⣸⢠⠄
⠀⠀⠀⠀⠀⣯⠀⠈⠀⠀⠀⠀⠀⠐⡶⠶⡖⠀⠀⢸⢸⠉⢢⡄⠐⡄⠣⣉⠞⠀⠀⠀⡺⠀⠀⠀⠀⡀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠀⠀⠋⠀⠈⠁⠘⠁⠀⠀⠀⠀⠀⠾⠧⣠⣤⠔⠊⠉⢢⡔⠁⠀⠑⢦⠄
⠀⠀⠀⠀⣠⠊⢆⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⢣⠀⠀⠀⠖⢠⠄⠀⠒⣒⠒⡆⠀⠀⠀⠀⠈⠑⣤⠀⠀⠀⠀⠀⠨
⠀⠀⠀⢰⠓⠩⠾⡆⠀⠀⠀⠀⣼⣦⣶⣷⣶⣾⡆⠀⠀⠀⢸⠀⠀⠀⣿⠀⣉⠉⢉⡍⠉⠙⣆⣀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠸⡀⠀⢀⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄⠀⠄⢼⠀⠀⠀⣿⠀⣿⡇⢠⡇⠀⣠⡀⠀⠈⣭⠉⢩⣇⠀⠀⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⣠⡀⠛⠿⡿⠿⠿⠿⠿⠛⠃⠀⠀⢸⠀⠀⠀⡏⠀⡏⠀⢈⡇⠀⡟⡇⠀⠀⣿⠀⠸⢻
⠀⠀⠀⠀⠀⢀⣯⣦⡤⠜⢳⡶⠒⢻⣷⠒⠒⠀⠀⠀⠀⠐⢸⠀⠀⠀⡇⡀⡇⡀⢀⣇⠀⣇⡇⠀⠀⣿⡀⢀⣿⢀⡀⣷
⠀⠀⠀⠾⠟⠉⠀⠘⠁⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠸⠂⠉⠙⠉⠉⠉⠩⠉⠉⠉⠉⠍⠉⠩⠩⠉⠹⠉⠉⠉⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢆⡙⠉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⡀⠀⠀⢈⠃⠀⠘⠁⢀⣤⣠⠾⢓⠆⠀⠀⠀⢀⡀⠀⠄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣾⡗⡀⠀⠀⠀⣅⡰⠀⠀⡀⣰⠀⢻⠉⠑⠄⠐⠁⠰⠀⠀⢸⡿⠿⠶⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⣤⣴⣷⡛⣿⡏⣹⣄⢦⠀⣶⣿⣴⢀⣴⣇⠻⠀⠀⠛⠀⠀⠰⠂⠀⢸⣆⢻⣿⣿⡶⠴⢾⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⣾⡿⠿⠟⢿⣿⣷⣿⣿⡇⣼⣿⣿⣿⢧⡙⢒⣢⠀⠀⠒⢠⣀⣀⣠⣶⣌⣿⡗⠋⠁⠀⠀⢀⠹⣿⠄⠁
⠀⠀⠀⠀⠀⠀⠉⠀⢦⣄⢀⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣯⡘⢸⣧⣅⣀⣀⣤⣿⣹⣿⣿⣿⠿⠏⠀⠀⠀⢀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⠋⠉⠛⡛⠙⠉⠿⡿⣿⠻⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠒⠉⠉⠁⣀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠟⢁⠀⠈⡀⠑⠂⠀⠀⠀⠹⣷⣮⣽⣦⣭⣭⣿⣯⣤⡼⣿⣿⣟⣽⣿⠋⠀⠀⠀⠀⠀⢠⠀⠀⠈⢀⣤⠈⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⠀⢀⡶⠁⢙⡆⠀⣠⠀⠒⠀⣹⣿⣿⣿⣻⡿⣿⣿⣻⢿⡿⠍⢻⣿⣧⣄⠀⠀⠀⠀⠠⡇⠀⠀⠀⠈⠃⠛⠀⠁
⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠀⢩⡟⠿⠻⡟⠀⠘⠀⠛⠿⣿⣿⣿⣿⡿⢻⣄⠻⣧⢿⣿⣶⡁⠛⠻⢿⣷⡀⠀⠀⠘⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠏⠀⠈⣄⠀⢨⠙⢢⠄⠃⠀⠸⠀⢀⢦⡈⢿⠛⠛⠛⣈⠘⢟⣹⣿⣿⠀⢀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠈⠇⠀⠀⢉⡀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⣡⡄⠀⠀⠀⠘⡶⠓⠀⠀⠀⠀⡀⠉⠈⢀⣸⣛⣩⠝⣿⢏⡉⠠⣟⣧⡀⠀⠀⠀⢻⣾⣿⡆⠀⠀⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⠀⠠⣀⠀⠀⠀⠀⠘⠀⠀⣴⣿⣿⣿⣯⣾⡇⢘⠛⠛⠻⣿⣿⣶⣤⣤⣤⣹⣿⣄⠀⠀⠀⠀⠀⠂⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠁⠔⠊⠉⠀⠀⠀⠰⡛⢿⠛⠻⠟⠿⡿⠟⠂⠀⠀⠘⠋⣽⣿⣻⣿⣿⣿⣿⣤⣀⣤⡤⠤⣶⣦⣬⣵⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠀⠈⠉⣀⣴⣾⣿⢟⣃⣡⣩⣶⣶⣾⣿⣿⣿⣿⣿⡷⣦⣼⣛⢻⣿⣃⣙⣿⢿⣿⣿⠿⠉⠢⣨⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⢻⣿⣿⣿⣶⣾⣯⣍⣌⣁⣤⣤⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⡿⢦⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣏⠙⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⣛⠳⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠐⣁⢀⣄⣈⠉⠛⠋⠛⠋⣉⡁⠀⠀⠀⠈⣿⣧⠘⠏⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠛⣤⣬⣭⣭⣷⠀⠀⠀⠻⣿⣥⣀⡀⠀⠀⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⠋⠀⠀⠀⠀⠀⠘⠈⠛⠙⠁⠀⠀⡏⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⢾⡉⠛⠛⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠠⠂⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠠⠄⠀⠰⠂⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⡜
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠁⠀⠠⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠐⡀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⢀⠀⠘⠀⠀⠐⠀⠂⠀⠀⠀⠀⠉⠋⣀⡞⠈⢀⡏⢈⠫⠀⡐⢰⢠⠂⢁⡛⠁⠐⡆⠀⠀⢄⠀⠶
⠀⠐⠐⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠊⠑⠂⠀⠒⠐⠃⠊⠀⠐⠁⠒⠂⠀⡂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠉⠂⠀⠈⠛⠃⠘⠛⠘⠛⠛⠋⠙⠛⠃⠀⠛⠛⠚⠃⠀⠁⠀⠶⠀⠂⠈⠆⠀⠠⠂
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠐⠛⡀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡠⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⢸⣿⠀⠐⣿⠀⠙⠋⢁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠙⠢⠄⠀⠸⠇⠀⠀⠿⠀⢰⡄⠀⢀⠠⠁⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣰⣿⣿⣿⣿⣿⠶⣤⣠⣷⡾⠛⠋⠛⠻⠶⠿⠇⣠⠞⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣀⢸⣷⡿⠻⡿⠂⠚⠁⠀⠀⠀⠀⠀⠈⠀⠐⢸⣿⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⢯⣹⣿⣿⡿⠇⣠⣤⣄⢀⠴⠴⠄⢰⡀⠀⠀⠀⠺⢿⣿⣦⣁⣀⡽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣄⡀⢠⣿⣿⣿⠗⠊⠉⠻⠿⠿⠶⠆⠀⠈⠁⠀⠀⠀⠀⠸⡟⠛⠋⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠃⣶⣦⠀⣀⢱⣶⡬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⣿⣿⣿⣶⣿⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡷⠀⠀⣾⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠺⢩⡾⠿⣿⣿⣿⣿⡗⠀⠀⠀⢀⡿⢻⣿⣿⣿⣿⡀⠀⡀⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⡄⠺⣦⠟⣿⣿⣿⣿⣆⣀⣠⣿⣷⣿⣿⢟⠁⠉⡻⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣷⡆⠁⠀⣈⠀⠌⠍⠿⣿⡿⠏⠫⣥⣼⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⡤⠔⠂⠯⠐⠀⠀⠀⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣠⣊⠁⠀⢀⠄⠀⠒⠊⠀⠀⠀⠗⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⢠⣇⡦⣭⣄⠂⠀⣠⣤⠀⡌⡀⡠⢀⣴⡿⡇⠂⠖⠒⠶⠆⠠⠀⠀⠤⠠⡀
⠀⠀⠀⠀⠐⠒⠓⠋⠉⠀⠀⠀⠀⠀⠈⣿⡓⣮⡍⠘⠧⢉⠀⠀⠂⠀⠀⠀⠙⢾⡇⠀⠀⠀⢀⠀⠄⠀⠀⡢⠄⠈
⠀⠀⠀⠓⠒⠤⠴⠾⠏⠡⠤⠄⠀⠉⠉⠉⠙⠳⣀⣈⡀⣀⡀⠋⣩⡀⠤⣤⣄⡀⠀⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣽⣿⣿⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠙⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠠⠤⠤⡤⠤⠤⡤⠤⢀⣻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⣠⠿⣀⠢⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠠⠂⠀⠀⠀⢀⣼⠏⠀⠙⣆⠘⢤⡀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢀⠜⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠈⠳⡄⠙⠢⡀⠀⠀⠀⠀⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⠋⠀⢀⡰⢊⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⣠⠞⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢦⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡶⠁⠀⠀⠄⠀⢸⡿⠀⠀⢀⣌⠀⠀⠀⠀⠀⢸⡿⠂⠀⠀⠀⠀⢷⣼⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣄⠀⠀⠀⠀⠀⢠⠎⠀⠀⣤⡄⠀⠐⠆⠀⠀⠀⠀⠀⣠⡞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠶⣤⣄⣀⣠⣶⣶⣤⣤⡀⠶⠾⣿⠁⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⢻⣍⠁⠀⣀⣀⡀⠠⠀⣁⣴⡏⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⣽⠀⠀⠀⠀⢸⡏⠙⠢⠤⠤⠤⠴⠛⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⡸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⣄⣿⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⢰⡅⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⣸⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣄⠆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠐⢦⣀⣴⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢾⠏⢸⡆⠀⠀⠀⠀⠉⠹⠁⠀⠀⠀⠂⠀⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢻⡀⠀⠀⠀⣀⣼⠀⢸⡃⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡴⠶⠶⣴⣿⣤⠄⠀⠀⢀⠋
⠀⠀⠀⠀⠈⢣⣴⢚⡷⠶⠾⣆⢸⡁⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣹⣇⠄⠀⠀⢋⣴⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⢈⡟⣀⣀⣀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢤⣃⡠⢾⠃⣴⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⢶⣶⠟⠁⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠗⠛⣡⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢸⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠸⠆⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡠⠚⠀⠀⢀⣠⡤⠶⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣾⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⡯⡴⠶⠖⠛⠋⠀⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢻⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣁⡤⠀⠀⠀⣀⡠⡤⠚⠋⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡿⠒⠒⠤⠤⣤⡤⣤⣤⡶⠾⠋⠒⠒⠊⢿⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢏⠀⠉⠉⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠷⣤⣄⠀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢶⡀⠀⠉⠻⢦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢀⣤⣶⠿⠟⠛⠛⠛⠛⠛⠻⠿⠷⢶⣶⣤⣤⣀⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣀⣠⣤⣴⣶⠶⠿⠿⠟⠛⠛⠛⠛⠿⠷⣶⣄⡀
⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⠉⠛⠿⢶⣦⣄⡀⢀⣠⣤⣶⠿⠛⠉⠁⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢀⣨⣽⡿⢿⣿⣅⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠈⠙⠻⢷⣦⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣤⡾⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⢷⣦⡀⠀⠀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⡿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢻⣶⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠙⠿⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⡿⠋⠻⣷⣄
⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⠻⢷⣤⡀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣤⣾⠟⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣤⡾⠟⠁⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣶⣦⣄
⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⢹
⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠙⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣾
⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠉⠛⠿⣶⣤⡀⢀⣠⣴⡾⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠿⣿⣥⣀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣀⣠⣴⣶⠿⠛⠉⠁⠀⠀⠀⠉⠛⠻⢷⣦⣤⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢿⣦⣀⠀⠀⠀⠀⣴⡿⠛⠻⣦⡀⣀⣀⣤⣤⣶⢿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿
⠀⠀⠉⠛⠻⠿⠷⢾⣿⠀⠀⠀⣿⡟⠛⠋⠉⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠷⠶⠶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⢀⣠⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠅


⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⣠⡄⣐⡀⠠⠤⣤⣤⡤⢤⠄⣀⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣖⡈⠈⠁⠀⠵⠟⢲⣄⡌⠀⣠⠖⠚⠉⠀⠀⠀⠀⠈⠉⠉⣉⣀⣙⠆⠘⠛⢲⠦⢆⠄⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠚⠧⠴⠀⠀⠐⠦⣄⠀⢠⣦⠁⠀⠀⢀⠟⠋⠉⠉⠄⠠⣤⣀⣄⢀⡤⠋⠈⠀⠀⠀⠀⠀⠀⠐⠲⠏⢤⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣊⠈⠉⡢⠀⠈⠲⢴⡆⠀⢀⡆⠀⠈⠓⠋⠙⢄⡤⠠⠠⡀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠘
⠐⠀⠀⠀⠄⠀⠈⢦⡀⠀⠄⣀⠀⠁⠀⠀⠀⠀⠀⠈⢧⣸⠇⠀⠀⠀⠈⠷⠄⠞⠁⠀⠀⠄⠀⠀⠈⠂⠒⠒⠀⠊⠁⠀⠀⠀⠀⠊⣉⡤⠂⠀⠀⠀⠀⠈⡂
⠀⠀⠀⢤⠖⠀⠀⠀⠉⠳⢶⣤⣤⣀⣒⣀⡀⠀⠀⠀⡤⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠉⠻⣼⣯⠿⡯⠽⢯⡙⢾⣻⡇⠲⠶⡶⢶⠖⠲⢲⡶⠞⠛⡟⠹⢻⡍⠛⠏⠙⠋⢠⡄⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠳⢤⣤⣤⣙⢾⠀⡇⠀⠀⠁⣼⠀⠁⠟⠀⠀⠀⡁⢰⡅⢡⣿⣦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠒⡤⢴⣶⡄⢠⣤⣆⣴⡶⣾⣿⣧⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠈⠚⠉⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣠⡱⠿⠃⠷⣰⣿⣿⣷⣤⡜⢦⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠒⢙⣱⠆⢳⣤⣍⣙⡿⢹⣽⢺⡟⡾⣮⢰⡞⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣹⡬⠟⠣⡤⣷⣙⢛⡀⣠⡙⠻⣾⣶⣄⡰⡛⣋⣹⠟⡶⣖⣫⣾⣧⣴⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠶⢹⣿⣿⡞⣷⣖⡴⣿⢽⣴⣿⣎⢇⣼⣄⣡⣬⢓⣾⣿⡟⣿⣿⢿⠿⣿⢛⣸⢘⡀
⠀⠀⠀⠀⠀⠀⢸⣦⠡⠤⣄⢬⣉⠋⡵⢶⣸⡇⣠⣼⢎⡛⢡⣆⢿⣯⣿⣁⣼⠛⢻⣯⣷⣷⣼⣄⠁⢿⢿⣼⣿⢀
⠀⠀⠀⠀⠀⣀⡳⣿⠦⣸⣏⣿⣼⡟⢿⠜⠻⡯⣶⠤⡶⠧⣞⣷⢜⡙⡛⢽⣿⠛⢹⣿⢹⣦⣶⣟⠆⣿⣻⢛⢃⣙⣠⡄
⠀⠀⠀⠀⢀⡻⡕⡇⣴⢦⠸⡿⢟⡲⢖⣾⣷⣿⢿⡿⣁⣷⣿⣷⢏⢿⣮⣾⣯⢶⣼⣟⣸⣯⣽⣉⡋⣿⢻⣏⢩⣭⡛⢊⡧⠄
⠀⠀⠀⢤⡌⣵⠑⡾⢾⡞⣶⣿⣏⣻⡿⠎⣵⣶⢧⣯⣻⣭⡥⣍⣸⠼⡿⠷⠿⣈⣍⣉⠻⠶⢾⣛⣩⣹⣘⣹⡟⢈⣭⣿⣧⣼⡇
⠀⠐⢲⣢⣗⣭⣿⣥⠸⢾⣿⣷⡄⠆⡿⢰⣯⣬⠻⠿⣇⠿⢘⣯⣉⣛⡿⣰⣆⣿⣿⡏⡷⣷⣍⣩⣿⢡⣌⢻⣿⣻⣍⣤⣤⡔⣣⡀
⠀⠰⣆⣽⢅⡋⠏⢶⣿⢰⣶⣶⣶⢷⣴⣼⡙⣡⣖⠷⣶⣶⡏⣬⣭⣭⡭⡙⢁⣼⣯⣀⣷⣿⣦⣝⠋⡼⢡⣻⣿⣷⠙⡛⠉⣅⣿⡅
⠀⣼⣿⣿⣿⣿⣧⡻⢡⢼⣙⣗⢡⣾⡩⣹⡶⠿⡿⢣⣾⣷⡾⣾⣍⢹⣄⢿⣶⣭⣽⠶⢅⣻⣿⢇⣴⡧⢼⣷⣉⣴⡾⢛⣧⣬⣽⣦⡀
⠀⠛⠛⣻⣟⣽⡻⣿⣌⣿⣿⡼⡛⢎⣷⡬⣛⡿⣦⠚⣇⣸⣙⣹⡟⢧⠿⣿⣿⣳⣙⢿⣶⡴⣋⣽⣹⠟⣞⣹⡢⢻⠧⠾⢫⡿⣛⣹⡋
⠀⠀⠀⠀⠀⢰⢆⡾⠹⠿⣩⣿⣿⣿⣄⡶⢫⢸⣿⠏⢟⠻⠛⣷⣾⣿⣿⣮⣿⢻⣿⣇⣶⣺⠿⣿⡿⣇⣦⣽⣿⣿⣭⣍⣋⡛⣿⣾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⢽⠿⠟⠛⣊⣛⣻⣈⣏⡒⣷⣆⡞⣭⣛⢻⣻⣯⣹⠾⣷⡋⣛⢡⡞⣯⣽⡤⠉⣿⣛⠿⢻⡆⣿⢧⣛⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠛⠛⠋⠉⠈⠀⠘⢳⣼⣷⣴⣣⡾⠉⠂⠉⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢂⠀⠀⠀⠀⢀⠀⢀⡀⠀⢀⣀⢀⠀⠀⠀⣀⣠⢞⣁⡒⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡘⠦⢉⣿⣞⡿⠞⠒⠛⠍⠀⠀⠀⠀⠀⠀⠤⠂⡀⠀⠀⠀⠀⠀⠙⠚⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡭⠭⠵⠶⠺⣷⡲⠛⠘⠁⠶⠒⠀⠀⠀⠂⠀⠀⠀⠃⠀⢨⣀
⠀⠀⠀⠀⠀⡄⡄⠀⢠⡝⣡⠚⣉⡙⠳⣾⡀⠾⠛⢢⢠⡗⣿⡇⣆⠐⠓⠁⠀⢬⡄⣈⣋⣀⣀⣠⣤⣿⠁
⠀⠀⠀⠀⠀⢸⠧⡿⣘⡇⠁⠘⠙⠛⣤⡎⠘⠂⡄⢸⡿⢣⣾⢎⠰⠔⠣⣷⠀⠀⢰⡄⣿⠟⢻⠛⣱⣯
⠀⠀⠀⠀⢠⠃⠐⢰⠟⣿⠀⣦⡠⣾⣍⢁⣀⣄⡙⣧⠀⢸⣿⠿⠓⣞⠒⠆⢀⣤⣍⡮⠉⠀⠀⣠⡏⠙⡆
⠀⠀⠀⠀⠀⠳⠄⢸⡖⡛⢸⠿⠛⢉⣀⣈⡀⠀⠀⠁⠐⠉⠉⠚⣇⡷⡜⢉⢇⣤⡜⠀⠀⢀⣴⡟⠤⠈
⠀⠀⠀⠀⠀⠠⠂⠐⠼⣏⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⡈⣷⡾⣵⣾⣧⡀⢀⣴⠮⣭⣅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠉⠘⣿⣿⣿⢿⣿⣿⣿⣯⣿⡿⣯⣿⣿⡿⢿⡿⣿⣿⣿⠿⣿⣟⣿⣄⡒⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣿⣿⣿⣏⠸⠛⠻⣿⣯⣿⣬⣿⡿⢿⣿⡇⠘⢛⣽⣹⡿⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠿⠿⠟⠀⠀⠹⠍⠻⠿⢿⣽⣿⣿⣯⣤⠘⣿⣿⠁⢻⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣋⣃⠀⠐⢛⡉⠑⠂⠀⠀⠙⡿⠿⣟⣿⡉⠀⣨⣿⣆⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠀⠀⠉⠘⠓⠀⠀⠀⠀⣿⣿⣯⣿⣿⡚⠿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣿⣿⣿⣽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠤⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡾⡿⠉⡀⠁⠈⠉⠉⠉⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡙⠙⠒⠶⣿⣷⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⣈⣹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⠗⠀⣻⡟⠃⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡏⢼⣿⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠶⣾⣿⣇⡀⣰⣿⣿⣿⣌⣩⣽⣿⣿⣿⡿⠋⠀⣀⠀⣼⠟⠚⠃⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠘⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣠⡀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣔⣈⡿⠃⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣰⠿⢿⣧⣱⡀⡄⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⢿⡲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠛⠿⠗⠿⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⠺⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣍⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣻⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⣿⣿⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡸⠛⣹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣥⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠀⠀⠀⠀⢀⡶⠶⠄⠀⠀⠀⠀⠀⠀⠠⢄⣤⣧⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⢀⣠⣄⠀⢸⡇⠀⠀⠷⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠯⣟⣿⣅⡀⠼⢿⣿⣷⣿⣻⡆⢶⣗⡀⠀⠀⠀⠀⠁⢉⡏⣇⡥⠇⡛⡧
⠀⠀⠀⠀⠀⠀⠀⣿⣼⡞⣽⣿⣷⣦⣄⠹⣿⢿⠙⠷⠿⣏⠉⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣷⡧⠀⠤
⠀⠀⠀⢀⠀⠀⢰⣿⣿⠿⢿⣟⡐⣻⣿⠐⣏⡑⡀⠀⠀⢤⠄⢠⡤⠀⠀⠒⠚⠿⠿⠿⠿⠿⠧⠈
⠀⠀⠀⣶⣿⣶⣿⣟⣛⣛⠿⡷⢿⣿⣿⡆⢿⣷⣧⢀⡄⣰⡾⣁⣤⣤⡮⠤⠤⠤⠶⠖⠒⠒⠛⠛⠉⠉⡁⠋⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⣿⣯⣭⣬⣅⣉⣡⣭⡭⠤⠴⠶⠒⠒⠚⠶⠂⠀⠀⢀⠰⠿⠁
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇⠻⢿⣿⣿⣿⡿⠀⢸⠀⠀⠀⠀⠀⣠⡆⠀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⠄⠀⠀⠈⠉⠙⠛⠻⢿⣏⢹⡇⠐⠂⠀⠉⠉⠛⠲⠾⠄⠀⢀⣀⣀⠉⠀⠺⠧
⠀⠀⠀⣴⣦⣤⠝⢢⠂⠐⣄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢀⣀⣴⣤⣼⣿⣿⣿⣿⠊⠙⠀⠀⠀⣀⣠
⠀⠀⠀⠐⠉⠁⠀⠉⠁⠁⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠤⠽⢿⠿⠿⠟⠛⠛⠛⠉⠉⠁⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⢀⡠⠔⠀⡀⠀⠀⠀⠀⠣⢸⠆⠀⠀⠀⠑⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣰⠀⣁⠁⠀⢀⡴⠚⣁⡤⠒⠉⠀⠀⠀⠀⠀⣠⠏⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠇⠰⣂⡦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⢀⡞⠈⣰⠎⠐⢀⠝⣠⠞⢉⡴⠚⠉⠀⠀⠀⠈⠉⠁⡹⠚⢁⡐⠁⠀⠀⡀⣀⡀⠠⠴⣠⡤⠀⠀⠀⣹⢠⡆
⠀⠀⠀⠀⠀⠀⢀⠤⠶⡅⣸⠅⠀⠀⢰⠏⢀⣾⠗⢠⡼⠃⣴⢁⠔⠋⠉⠃⠀⠉⠀⠀⠱⠞⠁⣀⠔⠋⠀⢀⡴⠋⠉⠀⢀⡴⠞⠉⠀⢀⣤⡞⠷⠋
⠀⠀⠀⠀⠠⠖⡖⢑⡎⡃⠘⠀⠀⠀⠸⠆⠘⣷⣫⠎⠀⠀⢰⣯⡒⠦⠠⠤⠄⡶⠖⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⠝⠉⠀⠀⠀⠀⠫⡄⣿⠤⣀⡀
⠀⠀⢀⡀⢠⠚⡇⣩⣇⣬⠀⠀⠀⠀⠘⢆⠐⣟⠃⠀⠀⠀⠘⠆⣁⣀⣠⠄⠊⢁⡤⠊⠀⠀⣠⣤⠀⠲⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠠⣝⡳⢦⡀
⠀⠀⠀⠰⡿⠛⣿⣩⣏⡼⠁⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠈⡠⠞⠁⡠⠾⣁⠀⠀⠐⠋⠐⠂⠢⡀⠀⠀⠈⠉⣙⣛⣗⣂⡼⠛⠉⠀⠀⠀⠀⠀⠀⠁⠉⢳⡀
⠀⠀⠈⣠⢾⡾⢛⣯⣿⡇⠀⠀⠀⠀⠀⠀⣀⡴⣷⠀⠀⠀⠀⠀⢀⡤⡌⢶⡄⣀⡉⠲⢄⡀⠀⠀⠀⢻⢟⣻⣏⠀⠉⠉⠙⠛⠢⢤⠀⠀⠀⠀⢀⣀⣤⡔⣦⠼⠁
⠀⠀⠀⠘⠄⢀⣎⠟⠁⠀⠀⠀⠀⠀⠀⠰⠃⠠⢼⡅⠀⠀⠀⣰⢈⠀⣀⠀⠀⠀⠀⢠⣀⠉⠓⠄⡀⠀⠈⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢴⡞⢻⠭⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢠⠤⠞⠈⠀⣾⣧⠀⠀⠀⣤⠻⢟⡛⠛⠉⠉⠉⠉⠉⠛⠉⠻⣅⠘⢆⠀⠀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠪⡾⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢦⠤⣻⠴⢾⢟⣧⡄⠀⠀⠀⠈⠧⣀⡻⠾⢿⡛⠛⠻⠟⠉⠋⠉⠉⠛⣹⠀⠀⠀⠃⢀⠤⠄⠀⠀⠀⠀⡤⠴⠁⠈⠠
⠀⠀⠀⠀⢀⣀⣄⣀⡀⢈⣙⣶⡾⣭⣦⣼⡒⠁⠀⠀⠀⠀⢀⡀⠀⠉⠲⢀⡹⠦⢤⡤⠶⠶⠶⢦⡼⠋⠀⠠⠦⡄⠀⠲⠀⠀⣀⣀⣤⠶⠖⠃⠀⠀⠀⢀⢠⡄
⠀⠀⠀⢸⠿⣥⠌⡁⢛⣥⢶⡋⣠⣋⡼⠿⠛⠀⠀⡀⠀⠀⣷⡞⢣⠀⠀⠀⠉⠓⢄⡸⢷⡶⠞⠉⣠⠞⠀⠀⠀⠀⠉⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠰⠆⠩⡝
⠀⠀⠀⠀⢀⠀⠠⣟⡭⢰⣦⣍⡁⠀⠀⠀⠀⠀⠀⠀⠣⠜⢀⣦⣄⠉⡼⠖⠂⠀⣀⠝⠋⢀⣤⡞⠳⡄⠀⠀⠀⠀⣀⣀⠀⠐⠒⠒⠒⠀⠀⣈⠉⠁⠀⣲⠁
⠀⢠⣴⡓⣊⣍⣰⢿⣧⠎⡘⠋⠁⠀⠀⡄⠂⠚⣁⡸⠖⠈⣉⠁⡜⠈⠁⣀⠴⠊⢁⡠⠶⠿⠾⠿⢶⢿⡄⢠⡶⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⡋⠁
⠀⠀⠀⠁⡭⣘⡡⢺⣁⡀⠀⠀⠀⠀⠠⠃⠜⠋⢀⡀⠄⡆⠀⠈⢁⡠⠊⢁⣠⣶⣭⣤⣤⣤⡤⡤⢴⡟⠃⠘⠀⠀⠀⠀⠀⠀⠠⢶⣷⣧⣠⣄⣤⠆
⠀⠀⢰⠶⠀⣿⣧⠉⣟⠿⠂⠀⡨⠦⢠⠅⠀⡌⠀⠁⢸⠁⠀⡴⠋⣀⣴⣋⣀⣀⣀⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⢈⣫⠾⠟⠋⠁
⠀⠈⠻⡦⡿⠹⠇⠙⠀⠀⠀⠀⠐⠓⠁⠀⠀⠁⠀⠀⠀⠀⠈⢀⡞⠉⠉⠉⠉⣉⢉⣩⠝⠋⣀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⢀⢚⡇⠀⡞⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⣿⠂⠀⠈⢈⡹⠟⠉⣠⠔⢊⡁⠤⠂⣁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠳⠈⠉⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠀⠀⠀⠀⠹⠟⡷⠞⠁⡀⠀⠈⣡⠔⠹⠃⠀⠀⠀⠑⠠⠀⠀⠀⠀⢠⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⢚⠁⠀⠀⠀⠀⢀⠈⢀⠖⠛⠢⢄⡀⠉⠈⠉⠀⠀⠑⠂⡀⠀⠀⠀⠀⢠⠇⠀⠀⢠⠀⠀⠂⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠿⢊⡁⠀⠀⠀⠀⠀⠀⡰⠛⠛⠛⠾⢧⡈⠀⠀⠀⡀⠀⠀⢀⡿⣣⣦⣤⣴⠂⠀⢀⡴⠊⠳⠄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⡄⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⠀⢰⡄⠰⠦⠄⠠⠶⠽⠇⠤⣗⠈⠳⢄⠈⠆⠙⠃⠈⢇⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡷⢰⠄⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣤⡀⠀⣠⣶⡶⡤⠤⠬⡳⠤⡘⣧⠀⠀⠀⠀⠈⢆⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⢼⠀⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⢄⡈⢹⣦⣄⠀⣤⣤⣤⠤⡄⣰⢿⠀⠀⠀⠀⠀⢸⠆⣸⠇
⠀⠀⠀⠀⠀⠰⡿⠀⠀⡷⢣⠂⠀⠃⠂⠀⠀⠀⠀⠤⡀⠀⣠⣶⡀⠛⠉⠓⢥⡈⠛⣦⡀⠀⣀⡔⢋⡜⠀⠀⠀⠀⠀⣫⣾⡏
⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡈⢙⠷⢉⠀⠀⠀⠀⠀⠈⠒⢌⡩⠟⢉⡠⠋⠀⠀⣠⠴⢆⡸⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠬⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⠾⠁⠉⠀⠐⠛⠀⠀⠈⠀⠀⠙⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠐⢅⡀⠉⠢⢄
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡿⣧⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠐⡶⠆⠀⠀⠈⠳⡀
⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⢣⠀⠀⡟⠀⠀⠀⠀⠀⠘⢄
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⢰⣿⠗⠉⠀⠀⠀⠀⠀⠀⢄
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⢰⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡼⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠴⠃⠿⠃
⡇⠀⡔⠈⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⢀⡴⠉⠀⠈⢣⠀⠀⠀⠀⠀⠀⢠⠖⠠⠒⠂⠀⠀⡠⠂⡀
⡇⠀⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⡜⠁⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⣇⣼⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣠⡃⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⢦⣀⠀⠐⠁⠀⠀⠀⢠⡾⢹⡽⣆
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⡆⠀⠀⠀⠀⠀⠀⠀⢹⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠻⠤⠁
⠈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⢆⣣⡑⠢⡀⠀⢀⡀⠤⠂⢠⠇⣀⣀⣤⣴⡂⢀⠀⣀⡀⢀⣰⣤⢰
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠙⣏⣁⠠⡏⠁⠀⠀⣼⡟⢻⠋⢀⡽⠾⠦⣭⠁⣈⡏⠉⢁⡎
⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⣟⣀⣒⣒⡋⠈⠉⠶⣛⣁⠀⠀⠀⠀⠛⣒⡁⠀⠀⠞
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠓⠒⠛⠛⠻⠟⠛⠛⠃⢡⠖⢢⣠⠀⠁⠀⠞
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⠁⠀⠈⠁⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣀⣀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠄⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠤⠤⠤⢠⠀⡀⣤⠠⠄⠤⠀⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⣶⡀⢀⣽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⡀⠀⠀⠀⠀⠀⣿⡇⠂⠈⣿⠁⠀⠉⠀⠀⡀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠋⠙⠶⢤⡀⠀⢠⢿⠀⠀⠀⡿⡆⠀⣴⡀⠃⢠⢄⡤⠉⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣾⣿⣿⣿⣿⣶⣤⡀⠀⢰⣦⣼⣴⠶⠶⠶⢷⣦⣤⣿⠇⠀⣠⠧⢀⢂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢱⣿⣿⣿⣿⡿⠛⢻⣌⣻⣿⣿⣿⣋⢀⣾⠁⠀⠈⠀⠀⠈⠉⠻⠇⠀⡀⢰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⣸⣿⣿⣻⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⢠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠻⣦⣼⣿⣿⣿⠟⢀⣤⣄⣢⡀⡀⠾⠾⠦⠀⢶⡀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣄⣀⣴⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣀⣀⠀⠀⣸⣿⣿⣿⡷⠐⠋⠉⠻⢿⡷⢿⣶⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⣿⠛⠛⠛⠉⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣿⣿⠛⠓⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠘⠦⣄⣀⠀⠀⠀⢦⣬⡄⢘⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠔⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣀⣤⣾⣿⣧⢠⣶⡌⠛⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⣿⣿⣿⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣾⣿⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡟⠀⠀⣼⡿⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⢠⣺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠰⠟⣡⣾⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣏⠀⠀⠀⡀⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠉⣦⣀⣹⢿⣿⣿⣿⣿⣆⠀⠀⢀⣼⣿⣴⣿⣿⣿⠟⠛⠻⢷⣼⣭⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣷⣧⡀⡨⠟⠁⠺⠿⠛⠿⢿⣿⣶⣶⣿⣿⣿⠿⠛⢻⣴⡀⠀⠰⢊⠁⠉⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣆⡀⣀⢰⣄⡀⠉⠳⠘⢟⣿⣟⠛⠀⢳⣷⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠈⠛⠛⠁⠀⠀⠘⢿⠟⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠈⣩⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⡁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⡍⠛⠓⣒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠇⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⢸⠀⠀⠀⡐⠉⠿⠓⠀⠼⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠾
⠀⢰⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⡎⠀⠀⠐⠁⠀⠀⠀⢰⢅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠛⠉⡁⠀⠀⠀⠈⢢⣀⣀⣠⡾⡁⠀⠀⠀⠀⢠⠒⠐⢀⠀⠀⠀⠀⠸
⢠⠁⠀⠀⠠⢀⡀⠀⠀⢸⠀⢁⠠⢄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠼⠦⡀⠀⠟⠙⠒⡄⠀⠀⠘⢄⣠⠋⠀⠀⠀⠀⠀⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸⣀⣀⠇⠀⠳⠤⠔⠃⠀⠀⠀⠀⢸⠖⠂⠄
⢀⠔⠒⢆⠀⠀⠀⠀⠈⠻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡼
⠘⠦⠤⠞⣤⠄⠒⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠔⠦⡀⢰⠃⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⡀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣀⠞⠘⢦⡀⣀⠜⠁⠀⠀⠀⡎⠈⣣⠀⠀⠀⠀⠀⡔⠁⠀⠊⠀⠀⠀⠀⠀⠀⠠⡑⣄⠀⠀⢀⠠⣶⠄⢸⣀⠀⠀⠀⠂⠀⠀⣠⡄⠀⠀⠀⠘
⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠐⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠙⠁
⠀⣠⠄⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠐⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘
⠋⠁⠀⠑⡀⠀⠀⢇⣀⡽⠒⠠⡎⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠙⠤⣤⠼⢃⡤⠂⠰⡄⠀⢀⡠⠀⢄⠀⠀⠀⠒⠲⠛⢹
⠀⠈⠄⠀⠀⠀⠐⠂⠀⠀⢀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣸⠀⠀⠣⠤⠜⠀⠀⠀⢩⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⠀⠀⠀⠀⠈⠒⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠱
⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⣅⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢰⠀⠀⠀⢠⠐⠒⠒⠦⡀⠀⠀⠀⠀⢀⣀⡳⠆⠊
⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠁⠀⡁
⠀⠀⠀⠀⠈⠒⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⢠⠖⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠇⠀⠀⠈⡓⠶⠒⢫⣥⣀⡀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠢⣤⠸⠦⣤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢠⡊⠉⢳⠀⢎⠀⠀⡹
⠀⠀⠀⠀⢀⠔⠐⠄⠀⠀⠀⠀⢨⡀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠊⠀⠀⠉⠉
⠀⠀⠀⠀⠈⠄⠀⠄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠰⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄
⠀⠀⠀⠀⠀⢰⣿⣿⠏⡀⠿
⠀⠀⠀⠀⢀⢸⣿⣿⡄⠀⠰⠶⠀⠂
⠀⠀⢠⡀⠠⢈⡛⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡈⠿⣷⠀⠀⠀⠀⠀⠀⣿⠛⠿⢿⣿⡇
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⠁⠈⢈⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠟⠇
⠀⠀⠀⠀⠀⠐⢠⠄⣠⢲⣤⠖⣺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠛⠀⠹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠔⠀⡄⠀⠀⠀⡀⢖⢄⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠄⠀⡰⠀⢽⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣄⣀⡰⣶⣯⣴⠁⠀⠀⠀⠀⠊⠘⠃⠀⠀⢶⡬⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⢈⠀⠆⠀⠀⠀⠠⠀⠀⠁⠀⠀⠉⢹⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠄⠀⠘⠃⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣼⠀⠘⢀⣤⢶⣶⣶⣦⣄⠀⠀⢦⢠⣄⠀⠀⠀⠀⠀⠸⠀⠀⠴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣩⠀⣪⠟⢸⢹⡇⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠏⠅⢠⡏⢀⣿⠀⢸⢸⠁⠀⠀⢠⠀⢀⡀⠀⢀⢰⣀⠀⠀⠀⠴⣨⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⣿⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣹⣧⠀⠀⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⣾⣿⣧⣾⣿⣷⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠛⡇⠀⠀⠀⣀⠁⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠸⢇⠠⠀⠀⣞⡄
⠀⠀⠀⠀⠀⠢⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⣯⣻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡗⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣿⣿⠃⠀⠀⢿⡶⢀⢠⡿⠂⠀⠀⡀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥⠿⢷⡏⠀⢀⡴⠈⠁⠦⠀⠁⣶⡄⠄⣳⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣾⡇⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⡿⢶⡄⠪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠙⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠂

⠌⢿⢛⠛⠟⠧⠤⠏⢑⡧⣖⢻⣄⠀⠀⠀⠀⠀⠀⠁⢀⠉⠁⠨⢙⡯⢴⣄⣢⣸⣲⣿⡇⢾⠭⣡⡞⢠⣀⣤⣶⣾⡿⣿⣿
⢈⠀⠀⢠⠀⣄⠀⠀⠀⠓⠛⠿⡇⠀⠀⢀⠀⠀⠀⢀⠈⡅⠀⠀⠠⠖⢢⠷⢟⠋⢤⡀⠚⠺⣷⣾⣿⣶⣴⣽⠧⠽⣿⣿⢯
⡀⠁⠀⠚⣠⣶⡖⠲⣮⣶⣿⣶⣦⡅⠂⠠⠤⢄⣤⣜⣀⣴⣦⡤⣗⣚⠙⣿⣿⣝⠦⡄⠢⠀⠋⠸⣿⣿⣿⡷⢒⡀⣿⣣⠞
⠃⠀⣶⣾⣿⠏⣿⢟⡿⠿⢹⡏⣷⢯⡇⠀⠀⠸⢻⣿⣿⣣⠁⡀⣾⣿⣿⣿⣿⣿⣿⣦⣆⣩⠀⢀⠀⠉⠻⢷⣿⣿⣏⣿⣷
⠀⠀⣵⣤⣿⣶⣞⣽⡾⢿⣿⣧⠿⢽⠯⡧⡅⠁⠐⢹⣿⣽⣿⣶⣿⣿⠛⢹⣿⣿⣿⣿⣿⣷⣯⣢⡄⠀⠀⠠⠀⠁⠈⠁⠁
⠀⠀⠛⢿⣤⠈⠃⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣵⣶⠬⣠⣠⢴
⠀⠀⠀⠀⠀⢀⡤⢀⣴⢄⡀⠀⣿⠐⠀⠀⠈⠢⡀⠈⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡃⠁⠀⠀⡬⣉⠂⢘⡞⣄⠀⣠⣿⠒⠀⠀⠀⣤⢿⡄⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡁⢸⣿⡟
⠙⣸⠀⠀⠉⠀⠀⠀⡧⢘⡇⠀⠈⣇⠀⠀⠀⢻⣨⣾⡦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣧
⡇⠻⠀⠀⠀⢨⣀⣸⢿⣷⡘⣲⡞⢯⠀⠀⠀⠈⣦⣼⠟⠀⠀⠀⠀⠀⠀⠀⢢⠅⣹⡛⡭⠽⣷⢽⡟⣟⠿⢾⣿⣿⡟⠥⢘
⡴⡆⠀⢀⡄⠈⡟⢤⢾⣿⣷⣼⠀⢸⢶⠤⠀⢹⢿⣿⡗⠃⢠⠀⠀⠀⠀⠀⠈⠇⡀⠁⡊⢀⣩⠖⡩⠉⠀⠘⣩⢺⢿⣷⠟
⠟⢇⠀⡾⠁⠀⢿⣨⣶⣛⣿⡧⠀⢺⡌⠄⢇⠉⠀⣟⣂⠉⡏⠀⠀⠀⠀⠀⠀⢸⣤⣈⡀⢉⡹⢦⣁⡧⡄⡠⠎⢙⣿⣿⣮
⠃⢤⡦⣇⠀⠄⠊⢹⢿⣿⠟⠁⢀⡎⠀⡠⢈⣠⣾⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⣬⡃⠀⢀⡀⠀⠬⠤⠄⠀⠀⠒⠿⠿⠟⠨
⡶⢈⣳⣤⡀⣂⡴⠈⠠⢐⢀⣴⣿⡴⠈⢁⣦⣿⡍⢿⣿⣿⡄⠀⠀⠀⠀⠀⢰⡇⣁⢰⣠⠖⠒⠀⠰⠀⠀⠠⠀⡠
⣾⣿⣿⡎⣹⣍⣑⠟⠒⣡⠊⠈⢸⢆⢿⢨⠿⡍⢳⣬⣝⠾⠃⠀⠀⠀⠀⠀⡼⠃⢆⠀⢻⢷⠃⠟⣥⠌⠀⠀⠀⠀⠀⠀⡀
⣕⣾⣿⡗⣿⣾⣤⠤⠛⣏⠄⡈⠟⢦⣲⣿⡀⢿⠈⣎⣽⡟⠀⠀⠀⠀⢀⣰⢡⠀⢦⢣⠈⣦⣤⡀⢤⡀⠙⠦⣀⠉⠉⠀⢄
⠛⢛⢿⡿⠋⣳⠰⣧⡀⡘⠃⠀⢀⣼⣧⡙⣇⢸⣆⠹⣼⡇⠁⠀⠀⠀⢀⠏⣾⠀⢾⣄⣆⡈⢻⣿⣿⣯⣢⣀⠈⠁⠲⢤⣀
⣷⣶⣿⣴⣾⣿⣶⣿⣉⢥⣦⡎⠙⠓⢿⣿⣌⢸⠹⡄⠹⡤⠀⠀⢀⡀⠐⠾⢋⡃⠈⠉⠸⡘⣦⠛⠻⠛⠻⢿⣷⣤⡀⠀⠉
⠃⢀⣸⠟⠉⠁⠈⠛⠛⢿⡏⡇⠀⣠⠸⣟⣋⣳⣧⡘⢶⢇⠀⠀⠀⡀⠈⠀⢰⠀⠀⠀⠂⢁⠀⠉⠆⠙⠀⠀⠀⠤⠜⢿⣤
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢠⠟⢆⠀⠙⠂⠈⠉⠛⠳⣶⣄⣀⣀⣤⣞⣿⣠⣀⣀⣄⠙⠁⡠⠤⢤⠢⠼⢐⣀⡀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⠀⠀⠀⠲⡂⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠃⠐⠀⠀⡀⢄⠀⢀⠫⠷⡿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡤⡊⡱⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠈⢱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡴⠃⣽⢚⠕⠎⠁⠀⠂⠰⠂⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠈⠣⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡄⠀⠘⠃⠀⠀⠀⢀⠀⠀⠀⠈⠌⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡴⣖⠵⠚⣦⠦⡄⠀⡄⠛⠻⡄⠈⠄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠘⡀
⠀⠀⠀⠀⠀⠀⠐⠀⢈⡟⡡⠠⡄⠈⠋⣾⠻⣉⣌⡉⠀⠀⠘⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⢁
⠀⠀⠀⠀⠀⠃⠀⠰⠇⠀⠀⠈⠣⢄⣠⢀⠀⢀⢀⠇⠀⠀⠀⣀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠑⢦⡄⠁⢨
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⢦⡀⠀⠀⢀⣉⡛⠇⡉⠉⠙⠧⣆⠀⢀⠀⠀⠀⠘⡉⡀
⠀⠀⠀⠀⠀⠄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠱⣄⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⢻⠈⠀⠀⠀⢀⣙⢹
⠀⠀⠀⠀⠀⢀⠹⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⢷⣀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠈⠀⠹⣷⣤⡤⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠩⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠙⢻⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡈⠀⠀⠀⠀⠊⢁⠀⢁⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡱⣄⠈⠻⣆⣀⠀⠀⠀⠀⣤⠅⠀⡀⠁⠀⠀⠄⠀⠂⣉⠀⠐⠁⠀⠠⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣬⡢⠀⠹⣷⡆⠀⠀⠐⠄⣀⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠂⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠻⣧⢄⠀⠀⠀⠘⢀⠜⠀⡄⠈⠀⠀⠀⠄⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠿⣷⣤⣤⣠⡇⠐⠀⠛⠀⣀⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢉⠧⡀
⢀⣀⠀⠀⠀⠣⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠆⠉⢩⠕⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢸⠀⠀⠀⠠⣤⣄⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⡇⠀⠀⠨⠀⠈⠁⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⢸⠀⠀⠀⠀⠢⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⠉⠲⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰
⠀⡎⠀⠀⠔⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠲⠂⡠⠀⠀⠀⠰⣀⠀⢀⣰⠃⠀⠀⠀⠀⡠⠂⢄⠀⠀⠀⠀⡞
⢠⠁⠀⠠⡀⠀⠀⠀⢐⠈⠀⢄⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠭⣉⠁⡱⠓⠢⡀⠀⠀⢇⣀⡜⠀⠀⠀⠀⢅
⠁⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢇⣀⡸⠀⠑⠤⠔⠁⠀⠀⠀⢀⠖⠒⢄
⢠⠊⠑⡄⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡜
⠈⠢⠔⢫⠖⠉⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢢
⠊⠑⡄⣌⠀⠀⢀⡄⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠂⠀⠀⠀⠑⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠞⠁⠈⠢⠔⠊⠀⠀⠀⠀⠦⡠⠃⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠛⠃⠒⠀⠤⠀⠀⠀⠀⢴⠆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠄
⠖⠋⠐⠄⠀⠀⣔⠉⣢⢀⢠⠔⠉⠒⡄⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⠈⠡
⠠⠀⠀⠈⢂⠀⠈⠉⠁⠀⠈⠣⣀⡠⠇⣠⠤⢤⡀⠀⠀⡠⢀⠀⠀⢀⠤⠴⠖⡢
⠀⠑⠀⠀⠀⠑⠄⠀⠀⡀⠀⠀⠀⠀⠀⡅⠀⠀⡹⠀⠘⠄⠠⠃⠀⠀⢁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠈⠑⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡷⢤⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡁
⠀⠀⢃⠐⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⠢⠜⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⡰⠈⠉⠉⠲⡀⠀⠀⠀⡠⠤⡝⠈⠁
⠀⠀⠈⠣⢀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠱⣄⣀⣀⡴⠁⠀⠀⠀⠆⠠⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⢰⡁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡠⠾⣍⠀⡔⠓⠲⡀
⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡎⠀⠉⡇⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠜⠀⠑⠤⠚
⠀⠀⠀⠀⢂⠀⡨⠀⠀⠀⠀⠙⠒⠂