Feeling Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡿⠋⢧⠀⡜⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⠀⣶⠇⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠂⠀⠀⠀⠐⠀⠸⢃⣿⡗⠀⠀⢸⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⠶⣆⠀⡀⢠⣠⡞⠁⠀⠀⠀⣾⣿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠠⠋⠀⠀⠀⣹⣿⡀⢀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣦⣴⣿⣶⣿⣿⣭⣵⣿⣧⢀⣾⣯⣿⣇⠀⢀⡤⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⡬⣽⣿⣯⣿⣿⣾⢟⢉⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣿⣿⣿⣧⣤⣤⠶⠍⠛⣦⣤⣀⣸⠟⢫⣿⣿⣶⡇⠀⠀⠀⠀⠀⠃⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣸⣿⣽⣿⣋⣴⣶⣶⣶⣬⠙⠛⠿⠷⠶⢿⣿⠿⣿⠖⠀⠀⣠⣆⣀⡠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣷⣶⣤⣀⠰⠟⠛⢀⣽⣶⣶⣾⣿⡿⠋⠀⠂⢰⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣆⣙⣿⡟⢹⣿⣿⣿⣿⣿⣿⡮⣽⣿⣿⣯⡌⠙⠛⣿⣿⣿⠛⠋⢁⣀⣶⡤⠂⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢟⣽⣿⡿⢃⣾⣿⣿⣿⣿⣿⢿⡄⠀⠀⠉⠛⠛⠿⣿⣿⣯⣤⣴⣶⣿⣿⣿⣀⣠⣴⠴⠾⣻⠿⠟⠁
⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡙⣦⡀⡀⠐⠲⢾⣿⣿⣿⣿⣿⣿⠛⠿⠿⣿⡟⠛⠀⢀⡤
⠀⠀⠀⠀⢀⣼⡿⠛⢻⣿⣿⣿⣿⣿⠿⢻⠧⠘⢿⣿⣿⣷⣿⣾⣷⣤⣾⣿⣯⢁⣿⠿⢿⠛⠙⠆⠀⠈⠁⠠
⠀⠀⠀⠀⣄⣸⡇⠀⣾⣿⣿⣿⣿⠃⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⣤⣾⡅
⠀⠀⠀⠘⣿⣿⣷⣿⣿⡿⠛⢛⣿⡒⢤⣤⣼⣿⠿⣿⣿⣿⣿⡿⣁⠛⢻⠋⠙⠿⣦⣄⣀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠈⠛⠻⠿⠋⠀⠀⠀⠀⠀⣰⣾⣿⣦⣤⣾⠿⠿⢿⣷⣼⣇⠀⠀⠀⠀⠈⠙⠳⢯⣽⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⠚⢛⢿⠛⠻⠦⣿⠿⡟⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠋⠀⠀⠀⢸⣿⠄⢱⠘⣆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⢸⠂⢹
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣉⡁⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠄⡀⠀⠀⠀⢀⣀⠀⢠⣶⠆⠲⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠴⠂⣠⣴⠤⠤⠐⢭⠃⡘⠛⠛⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠒⠁⢀⣴⣶⣦⣄⡉⠑⠫⣄⢠⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠴⠀⣴⣿⣿⣿⣿⣿⣿⣷⣦⣄⠁⡈⢲⡀⢠⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡄⢄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠈⠛⣆⠂⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠆⠀⣴⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧⢁⠘⡃⠐⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡗⢀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⡿⢈⢾⣱⡄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⠇⠌⠰⢿⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⡇⣼⢛⡉⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⠀⡏⠀⢂⠀⢸⣿⡇⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⡇⢻⡸⠟⠃⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⢸⣿⣇⠀⠉⡇⠻⣫⣿⣧⠀⠘⣿⣿⣿⣿⣿⠘⢯⣧⣀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠇⢸⣿⣿⣷⣿⣷⠀⠉⠉⠁⠀⣀⣿⣿⣿⣿⡟⣰⢷⣾⣿⡿⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢂⠘⣿⣦⣿⣿⣿⣷⣴⣴⣶⣿⣿⣿⣿⣿⡟⠀⢻⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠚⠉⠹⣿⣿⣿⣿⣿⠿⢣⡾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠿⠟⠛⠋⠁⠀⠨⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢖⢻⡟⡄⠀⠀⠀⠀⠀⡰⡩⣺⠟⢉⡉⠻⣔⣵⡀⠀⠀⠀⠀⠀⠠⢻⣟⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⣴⠁⠀⠀⠀⠀⠀⢡⣿⣁⣀⠀⠀⣀⣈⣿⡕⠀⠀⠀⠀⠀⠀⣤⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠰⠀⠀⠀⠀⠀⢸⠉⠏⠐⠂⠀⠀⠐⠂⠹⠉⡇⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢇⠀⠀⠀⠀⢨⠉⠆⠀⢀⣀⣆⡀⠀⠰⠉⡇⠀⠀⠀⠀⠸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠤⢀⣀⣀⢸⠀⠈⢆⡙⠥⠬⢋⡰⠁⠀⡃⣀⣀⡀⠤⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣅⡩⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⡠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀⣀⠀⠀⢀⣀⣀⠤⠓⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠉⡁⢉⠉⠀⠀⢸⠀⠀⠀⢸⢩⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⣦⣼⠒⠒⠒⢺⠀⠀⠀⢨⣼⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠀⠀⠀⠘⠀⠀⠀⡏⠹⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⡏⠀⡸⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠢⠄⠈⣫⡑⠤⠀⢑⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠹⠟⢛⣉⡉⡷⢶⣖⣶⡒⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⡁⢶⣦⣱⡼⣿⡼⠯⡟⠾⢻⣱⡆⠘⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣐⣦⠹⠆⢃⡬⠥⢠⡤⠄⡴⠆⣤⢩⡄⣶⡀⣿⠀⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣇⠙⣄⣀⣤⣸⠗⢂⡿⠆⣸⠷⢰⡇⢸⡇⣿⣇⡿⣠⠆⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠗⠀⣠⠿⠿⠁⠉⠛⠲⠿⣤⣻⡄⠿⡤⠸⠀⠇⠹⢇⣿⢘⡧⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⠐⢧⡄⠀⠀⣠⣄⠀⢀⡀⠈⠉⠙⣛⡛⠛⡛⠛⠉⣍⠋⢀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣀⡏⠀⣞⠁⣠⠀⢀⣿⠀⢠⡟⠉⣅⠀⢹⡆⠀⣿⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣷⠀⠈⠙⠛⠛⠉⣠⣄⠈⠛⠛⠛⠛⠋⠀⢰⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠙⠓⠦⣤⣄⣀⣁⣀⣠⣤⠴⠖⠋⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠙⠛⠛⠛⠀⠸⣇⠀⠈⢳⡄⠀⠀⠀⠀⠀⣠⣴⡞⣻⠟⠞⠋⣰⠏⠀⠒⠒⠒⠒⠒⠒⠛⠛⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠾⠃⠉⠉⠉⠉⠁⣿⠈⠉⢉⣠⠴⠛⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠉⠉⠉⠈⠉⠈⠙⠛⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡿⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿
⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣄⡀⠀⣀⢀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡧⣆⣼⠟⢋⣉⣉⣛⣿⣷⡄⣀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⡙⢿⣴⠋⠉⠉⢿⡟⣿⣷⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⢿⡀⢀⡠⢀⠀⠀⠹⣦⡄⠀⠀⠀⠀⠀⠀⠀⠡⢟⡌⢳⣦⣠⡤⢈⠃⠛⣉⣿⣿⠇⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠣⠄⠱⢤⠀⠀⠐⠂⠠⢀⣱⣤⣦⣤⣀⡀⠀⡀⠀⠀⠹⣆⠩⣁⠀⠀⠀⢘⡻⠟⢇⣴⠊⠻⡷⣎⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠘⠳⢦⣀⠒⠀⠛⠛⠻⢆⣨⢍⣻⣯⣵⢮⡔⠀⢸⡆⢙⠙⠆⠀⠀⠀⠀⠈⠊⠁⣼⢷⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢵⣚⣉⣙⠒⠈⠉⣠⡁⠉⣙⣡⠤⢦⣙⢿⡀⢻⡀⣴⣄⣀⢀⡠⢤⣾⠇⠸⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡉⠛⣍⣻⢋⠫⠂⢀⣤⠭⢽⣶⣿⣷⡅⣈⣧⣿⣫⣭⡴⢶⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣴⣿⣿⡇⢻⣿⣿⣿⣶⣿⣠⣦⣄⣺⣿⠿⣿⣛⣿⣿⣿⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⡟⣯⣽⣿⡇⢹⡟⣉⠛⢍⡻⡷⢦⣩⢃⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⡿⢀⣿⣿⠃⢱⠈⢿⡈⣙⣛⣻⣄⣠⡿⡾⣧⠀⠀⢠⣖⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠸⣿⡻⠗⠶⣵⣤⣙⢲⣏⣯⣭⡁⠀⠸⢿⣇⢀⣴⣦⠐⢘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣴⣦⡝⠿⠷⣦⠈⣉⣉⣉⡛⠛⠿⠦⡀⠀⠘⢿⡿⣇⣶⠈
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡀⠀⠀⠀⠐⠋⠀⠀⠀⠸⣿⣿⣿⣖⠲⠾⢋⣉⣽⣿⣿⣄⣄⠀⢢⣀⠀⠀⢻⡿⣿
⠀⠀⠀⠀⠀⠐⠉⣩⣿⡶⢷⡄⠀⠀⢠⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⡿⠛⠻⣯⣼⣿⣻⣿⡆⢀⠹⡎⣷⣌⢷⡿⠆
⠀⠀⠀⠀⣼⠆⣼⡿⠉⠑⠈⢧⠀⠀⠘⣧⡀⠸⠿⣿⣿⡿⠃⢿⠋⠻⣦⠀⠠⡀⠈⠹⣿⢣⠀⣞⢣⡌⢿⡆⠈⠀⣀⣤
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢠⠎⠀⠀⠀⠘⠳⠤⠾⠛⠉⠀⠀⣿⠀⢡⠘⣦⠀⠘⣆⠀⢘⠈⡇⢸⡇⠶⠸⣇⣠⢰⣦⡤
⠀⠀⠀⠀⠀⣄⢸⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡤⠔⠁⣼⣇⠆⢹⡄⠀⠿⠀⡌⣸⡇⠘⠃⠀⠘⠛⠿⠿⠏
⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣡⣼⠟⠃⢀⣴⡿⣟⣠⠛⠃⠀⠀⣼⣵⣿⠻⠄⣰⠃
⠀⠀⠀⠀⠀⢀⠈⠳⣄⠀⠀⠀⠀⠀⣀⠰⢻⢋⡁⠀⢐⣺⣭⢿⣿⡿⠉⠀⠀⠀⣴⡏⢻⡏⢶⡻⣁⠤⣀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠙⣿⣿⣾⡿⠋⡠⢒⡩⠄⣢⣥⣴⠶⠒⢁⡤⢊⠉⠁⠘⣧⠈⣧⠼⠃⠐⠢⠀⠀⡴⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣀⣀⣠⡤⢠⣮⡜⣡⣾⣿⠟⠋⢀⡤⣴⢋⡼⠋⠀⠀⠀⠘⢷⡄⢀⠂⠀⠀⠀⠀⠀⢠⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⠁⢸⣿⡴⢫⡞⠁⠀⠶⣾⣰⣇⠸⢁⡆⠀⠀⠀⠀⠀⠑⠈⠛⠒⠚⠃⠈⠊⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⣿⡵⠀⠀⠀⢹⣿⣦⠳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡇⠀⠀⠀⢹⡿⡽⣤⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢷⣄⠀⡜⠀⢠
⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⡀⠊⢿⣝⣳⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⣠⢴⢖⣒⣾⣒⡒⣤⣄⡀
⠀⠀⠀⢀⡴⣏⠔⠚⡇⠀⡇⠀⠸⠓⠪⣹⢦⡀
⠀⠀⢀⣞⡴⠁⠓⢄⠘⡄⡇⢠⠃⡠⠊⠈⢢⣱⡄
⠀⠀⣼⢸⠉⠐⠢⢬⣳⣽⣷⣯⣎⠤⠔⠊⠉⡇⢷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡟⢛⠒⠒⠒⣒⣾⣿⣀⣿⢗⡒⠒⠒⠒⡛⣻⠀⠀⠀⠀⠀⢠⠿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⢻⡸⡄⠒⠉⡠⠊⡽⣿⣯⣳⣌⠉⠒⢠⢧⡏⠀⣀⣤⣤⠦⠇⠤⠸⠵⠤⢀⠀⠀⠀⢰⢑⡃⠀⡀
⠀⣠⣌⣷⡙⣤⣎⣤⡼⢸⣿⣏⣆⣈⣳⣤⢋⠞⢀⡏⣁⣸⣹⡏⠹⠉⢻⠘⢳⡁⠤⠰⠲⢸⠼⣱⠀⢺⢽⠂
⠀⠀⠀⠁⣙⣦⣝⣺⠅⣼⣿⣯⡸⡟⣫⠴⢋⣠⡾⣟⡩⢁⡼⢹⣧⣀⣌⠀⢸⢿⢲⣾⣾⡶⣖⡿⠀⠎⠀⠃⠀⠀⡤⢀⣢⡄
⠀⠀⠀⣀⣤⣤⡈⢉⣞⣓⣷⣾⣿⣿⣿⣾⣯⣥⣤⡼⡟⢻⠛⠙⣇⠚⣸⠃⠐⠂⠙⢋⣛⠉⠁⣠⣤⣤⣤⣤⣤⣅⣐⣻⣀⣂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠒⠒⠒⣻⣿⣿⣿⣿⣦⣽⣶⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣙⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⡿⠁⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣠⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣹⣿⣿⣿⣿⣿⣿⣿⣷⣯⣖⣦⣀⣠⠴⠿⠦⣤⣧⣦⣤⣤⣈⢻⣧⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣠⣴⣾⣿⣿⣿⣿⣟⡏⠀⢀⣨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠘⠛⣫⣿⣿⣿⣿⣿⣿⣤⣀⣙⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢓⠒⠛⠁⠇⠀⠉⠉⠿⢿⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡹⣿⣿⡏⠀⠁⠉⠉⠀⢀⣴⣿⠁⠀⠀⠀⠀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡇⡀⠀⠀⠀⢠⣾⠋⠘⠻⣶⣴⣶⣿⣿⡄⣆⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣙⣿⣶⣸⣰⢀⣾⡏⣠⣤⣤⣿⣿⣿⣟⣻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⣿⣿⣿⡏⣿⣏⣉⡉⢨⠉⡏⣟⣿⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣿⣿⣿⠿⠛⣿⣟⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⠉⢿⣤⣀⣈⣉⣀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣆⠈⠘⠛⠙⣁⣾⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠛⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠛⠋⠁

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠛⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠉⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢷⣦⣀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣁⣉⣉⡠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡈⠳⣦⣴⣦⣄⠀⠀⠀⠉⠛⠛⠋⠉⢁⣴⣤⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠝⢿⣿⣿⣿⣦⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠄⠀⠀⢸⡴⠊⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠠⠀⠀⠀⢀⡞⠀⠈⠀⠀⢸⡀⠀⠀⠠⡀⠀⠀⠀⡦⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣾⣤⢤⣾⣄⣀⠀⠀⠀⣸⠁⠀⠀⠀⠱⡄⠀⠀⠇⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⡋⠰⠇⠞⠚⠙⠉⡙⢷⣎⡁⠀⠀⠢⢄⠀⢈⡠⠄⠒⠒⠢⢄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡦⡀⠠⠀⢠⠋⢠⠀⠀⠂⠀⠀⢱⠀⠤
⠀⠀⠀⠀⠀⠀⠺⢣⢊⡇⠶⠀⠀⠈⠀⠀⠄⠀⠈⠁⠀⠠⠦⡿⣿⣜⡄⠀⢸⡀⢫⡀⠀⢀⡀⠀⡸⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠋⠹⠘⠅⠀⠀⣹⣦⢍⣉⣉⠤⠚⢁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠓⠤⣄⣀⡠⠴⠊⢀⠴⠃⠀⠀⠀⠀⠀⣀⠜⠃⠀⡀⠀⢄⠀⠈⠑
⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣉⡶⠢⠤⠤⠤⠒⠊⢷⠤⣀⠀⢀⠀⡠⠊⠁⠀⠀⠀⡇⠀⠸⡄
⠀⡴⠉⡩⢅⠈⣇⠀⠀⢤⣴⠊⢁⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⢹⠿⠋⠀⠀⠀⠀⠀⣈⠥⢄
⠀⢣⣼⣁⣄⡇⠈⡇⠀⠈⠉⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣹⠚⠁⠀⠀⠀⠀⢀⠤⢺⠔⠈⠚⡍⢱
⠀⢮⡤⢽⣉⡜⠉⣀⠀⠀⠀⠀⣀⣀⠀⡇⠠⠤⠄⡀⠀⣤⣄⣀⣀⡀⠀⠀⠘⠤⣌⠪⣶⡷⠙⡇
⠀⠀⠀⠰⣒⣾⣾⠽⠒⠊⠉⠉⠀⢠⠶⠃⠀⠀⠀⠳⠦⠀⠀⠀⠀⠈⠉⠁⠒⠦⢼⣷⣋⠑⠊⠁
⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡄
⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠐⠁⠐⠱⢠⠂⠀⠀⠀⠠⠀⠐⠀⠆⠀⠤⠆⠈⠇⢣⠂⠧⠀⠀⠖⠀⠀⠀⠴⠠⠀⠀⠏⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣶⣿⣷⣶⣶⣶⣶⣶⣿⣿⢖⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿⠿⠿⠿⠋⠀⠀⠀⠀⠒⠁⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠠⠔⠂⠀⠀⢠⣿⣿⣿⠋⠁⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠤⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠛⠸⠀⡃
⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠀⠀⠐⠒⠊⡁⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢀⠀⢠⠄⠁⠀⠈⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠸⠀⠀⠀⠊⢀⠀⠀⡀⠔⠂⠐⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠡⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠄⠒⠉⢀⠠⢠⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠢⠀⠂⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠈⢀⠕⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠒⠁

⣶⣶⣶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⢶⣶⠶⢶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⠀⠀⡀⠀⢀⡿⡆⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠸⣷⣄⠀⢳⣰⡇⢠⣸⠁⣷⠇⢸⠁⣤⣴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠢⡈⠲⣿⡙⢶⡀⢻⣇⠀⡏⠀⠙⠀⣿⠀⠁⣾⠞⣰⢁⣤⢔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠢⣌⡃⠙⢦⡈⡦⠈⢅⡄⠀⢧⠉⢸⠀⠀⡤⠀⠀⢰⠆⠏⢀⠀⠚⠁⣽⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣥⣀⣬⠿⠶⠱⣦⠙⠆⠀⠉⠀⠠⠄⠈⠀⠇⠀⠸⠀⠈⡠⠀⠋⢠⠀⢘⡿⠋⣀⠄
⠀⠀⠀⠀⢤⣀⠀⠀⠒⠮⣿⡌⢍⠁⠠⡀⠈⠀⢢⡀⠁⠀⠀⠄⠀⢀⠀⠂⠀⡀⢀⠁⢠⠀⠁⠀⠠⠴⠯⠗⣶⠃
⠀⠀⠀⠀⠀⠈⠙⠳⠦⠍⠒⠦⣄⠁⠀⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡀⠀⣾⡴⠀⠀⠀⠀⠀⠬⣭⣤⠄
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠀⠠⡀⠀⠀⠙⠂⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⣿⠳⠀⠻⠁⠀⠀⠀⠀⠐⢻⠯
⠀⠀⠀⣤⣤⣽⠿⠧⣤⠀⠒⠚⠛⠂⠈⠁⣀⠀⠀⠀⠈⠀⠀⠀⠠⣤⣬⣆⠈⠛⠀⠀⢀⣄⠀⠀⢠⣀⢸⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠀⢉⠿⠒⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠋⠥⡀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠰⠶⣤⣤⡄⠐⣋⡀⠀⡴⠀⠠⠀⠀⠀⠒⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠰⠇⡇⠀⢞⢣⢺⠇
⠀⠀⠀⠀⠀⠐⠋⠉⠐⠚⠉⡉⠁⠊⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⠀⠀⢀⣀⣀⡀⠈⠒⠃⢀⠬⠛⠛⠛⠻⢶⡞⢻⣦
⠀⠀⠀⠀⠒⠚⣋⣴⡾⠀⠚⠇⠀⠚⠀⡄⠀⠆⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠤⠚⢩⣟⣥⡶⢀⣠⣀⣄⡀⡁⠰⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠠⠟⠋⢰⡷⠛⢛⠁⠀⠉⠀⠀⣄⣀⣀⣀⡀⣶⠀⠀⠀⠀⠀⠉⠉⠽⣦⣠⣶⣭⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣤⣤⡼⠶⠏⠈⠉⠉⠁⠙⠳⠶⠶⢶⣦⣀⣠⣤⣾⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠔⠂⠂⡀⠀⠀⠀⠀⢀⡴⢟⡍⢈⣠⣤⣾⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠁⠐⡀
⠀⠀⡐⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⠏⠀⠀⡼⠋⠀⠀⠙⢿⣷⡌⢧⡀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠲⢠⡾⠁⠀⠀⠀⠀⠀⠹⣿⡞⢷⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠤⡀⠀⣀⠀⠉⢀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⡇⠀⠀⠀⠀⠀⠒⢄⣀⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠤⣧⠀⠀⠀⠀⢸⣷⣄⠀⠀⢹⠇⣿⢠⠐⠂⡀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⣼⢿⡆⠙⢦⡉⢰⡇⠸⣀⡠⠃⠀⠀⠀⠁⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⢷⡄⠀⠀⡏⠘⣿⡄⢈⡇⠸⡇⠀⠈⠀⠀⠀⠀⠁⠀⠐⠂⡀
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠘⢧⣠⣄⠹⣄⣀⡇⣿⣿⣷⣼⠁⠀⢷⠀⠀⠘⠦⢀⠀⠁⠀⠀⠀⠁⠂⠦⣄⡀
⠀⠀⠀⠀⢋⠀⢈⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣮⣉⣓⣏⣩⣴⡏⠀⠀⠸⡄⢠⠓⠋⢒⡄⠀⢉⣀⠐⠀⠀⠀⠀⡉⡠
⠀⠀⢀⢷⠀⢹⠁⠀⠀⠀⡐⠈⠀⢀⣼⡠⢾⡿⣿⣯⣷⠉⡼⠁⠀⠀⠀⢳⡀⠀⠀⠀⡇⠉⢀⠘⠋⠉⠳⠒⠦⠄⠥
⠐⢀⡞⠈⣧⢸⢈⣆⠈⢇⠜⠀⠄⠿⠿⣗⣿⣩⢻⡷⣟⡿⠁⠀⠀⠀⠀⠀⢻⡀⠀⠠⡇⠀⠈⠀⠀⠀⢠⡉⠁⠀⠑
⢀⡼⠀⠀⠘⣿⠹⠘⠤⠀⡁⣴⠯⢨⡧⢈⡟⢪⣻⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠁⠀⠀⠐⠀⠀⠘⠃⠓⠦⠂
⢹⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡆⠈⡷⢚⡛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⢽⡦⠄⢒⠀⠀⠀⠀⠐⢀⡀
⠘⢦⣀⣀⣠⠎⠀⠉⠀⠀⠀⠐⡄⠀⣀⣠⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠟⠛⠔⢁⠀⠀⠂⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠄⠀⠠⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠤⣤⣴⣤⣶⣶⡿⢶⣶⣶⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠴⠿⠿⠿⠿⠿⠷⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢚⠉⢑⠰⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠉⠁⠒⠒⠀⠤⠄⠠⣧⠬⠄⠚⡂⡤⢏⡗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠄⠐⠂⠒⣿⢸⠀⠀⠠⠒⢧⡟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣍⢠⠱⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠉⠢⠀⠀⡊⠀⠀⢻⠀⠀⢀⠄⠀⠀⡌⠁⢱⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣀⠙⣑⠀⣇⠄⢌⠁⡟⠀⠁⠉⢋⡠⠠⠀⢱⢐⡑⢁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠴⠀⠀⠯⢔⠖⠉⠀⠀⠀⠀⠎⢐⣓⠤⠎⠈⠐⡂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠹⢀⢀⢠⡎⠀⠀⠀⠀⠀⠀⡄⠑⢄⢑⠀⠀⠘⠈⠢⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⡀⠀⠀⠀⢀⠔⠁⡘⠀⢫⠃⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣉⣔⠉⢁⡀⢠⠃⡰⠃⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠊⠀⠀⠘⠓⠁⠈⡡⠊⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⡈⠢⠔⠁⢰⠀⠀⡔⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠓⢄⣀⡘⢀⡊⣀⠴⠶⣒⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠣⠘⠁⠁⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⡩⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⡤⢷
⠀⠀⠀⠀⠀⠀⠀⠪⡑⢄⣈⣦⡀⡸⠀⠀⠑⢤⣁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⢘⠻⣅⠀⠀⠀⠈⠀⠉⠒⠦⡤⠌⠡⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡦⡀⠀⠦⢄⠀⢀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢕⢝⣶⡙⠀⠉⡟⠛⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠛⠀⠀⠀⢱⣀⣀⣀⣀⣀⣐⣦⣤⣤⣤⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠉⠋⠉⠉⠉⠋⠛⠋⠉⠀⠈⠋⠉⠉⠭⠟⠓⠒⠓⠒⠘⠓⠢⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⢄⡀⡴⠉⠀⠀⠀⠀⠈⢢⡀⣠⠴⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠈⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠋⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⢠⠞⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣀⢀⣠⠾⣄⠀⠀⢀⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⡴⠋⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⣀
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠠⡤⠔⠋⠀⣰⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⡼⠉⠀⣠⣙⢆⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⠛⢷⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⢇⠀⠸⢿⠟⣸⡠⠒⠉⠉⠑⢦⣸⠀⢸⣿⡷⠀⡟⠂⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⠓⠤⠤⠚⢹⠁⠀⠀⠀⠀⠀⡏⠣⣄⣀⣠⠜⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠶⡏⠀⠀⠀⢀⡴⠒⠒⠒⠺⢆⠀⠀⠀⠀⣠⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢉⠓⢒⠋⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠘⣆⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠖⠚⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⢿⡟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⢁⣀⡀⠀⠀⠀⠀⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣼⣿⣿⣿⣿⡍⣡⣶⣿⣿⣿⡆⠀⠀⠔⠁⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⠇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣿⣿⡏⢉⣡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢦⣬⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣏⠀⢹⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡿⠛⠟

⠀⠔⠀⢠⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀
⡆⣧⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣇⠔⢩⠆
⠇⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠀⠰⠊⣰⠁
⠈⡄⠀⠀⠀⠉⠛⢠⠄⠷⠀⠀⠀⠀⠀⣤⣀⠀⠉⠂⠀⣀⣀⡀⣀⠀⠀⠀⠀⠐⠧⣀⠙⠀⠀⠀⠀⠠⢀
⠀⠘⠦⣀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠃⠀⠉⠀⠠⠋⠀⢓⠀⠀⠀⠀⠀⢸⠀⠉⠦⠀⢠⠒⠂
⠀⠀⠀⠈⠃⠘⣇⠀⠀⠀⠀⠀⠀⢀⠰⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⡦⠀⠀⠈⠳⠀⢀⡠⠋
⠀⠀⠀⠀⢸⠀⠹⡄⠀⠀⠀⠀⢀⡹⠖⠒⠉⠉⠉⠁⠒⠦⣀⠀⠀⠀⠰⡁⠀⠀⠀⡔⠁⢠⡎
⠀⠀⠀⠀⠀⣧⠀⢳⠀⠀⣀⠔⠉⠀⣀⠀⠀⠀⢀⣀⠀⠀⠈⠲⡀⠀⠀⠘⠀⢠⠞⠀⢠⠟
⠀⠀⠀⠀⠀⢀⣄⣠⣧⡴⠁⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⡄⠀⠱⠀⠀⢤⣠⠋⠀⢀⠎
⠀⠀⠀⠀⠀⢯⠁⠀⢸⠁⠀⠀⠀⠀⠈⠀⠀⢀⣀⠤⠚⢩⡆⠀⠀⣛⡀⣸⠃⠀⢠⠎
⠀⠀⠀⠀⠀⠘⣇⠀⡆⠀⠀⠀⣿⡏⠉⠉⠉⢁⣀⣤⣶⡿⠁⠀⢀⠀⢙⠟⢦⣴⠋
⠀⠀⠀⠀⠀⠀⢹⡀⢡⠀⠀⠀⢻⣿⣶⣾⣿⢋⣉⢀⡼⠃⠀⠀⢨⡟⠁⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⢧⠈⣄⠀⠀⠀⠙⢿⠁⢀⢠⡤⠊⠀⠀⠀⣠⠏⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠓⠶⠶⠶⠶⠖⠋⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠡⡤⠒⠒⠒⠒⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠉⠙⠓⠒⠲⠤⡶⠤⠼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠤⠤⠄⢾
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠉⠻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠣⣀⣀⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠒⠲⠶⠖⠊⠻⣤⣤⣀⣀⣀⣀⣀⣠⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠁⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠧⠴⡇⠀⠀⠀⠀⢀⠞⠀⢊
⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⡇⠀⠀⠤⠒⠁⠀⠀⣸
⠀⠰⢂⠀⠀⢋⠀⠀⠀⠀⣀⠦⠚⠣⠁⠀⠀⠀⠀⠆⢨
⠀⠉⠂⠠⣭⣭⡥⠍⠁⠎⠑⠀⠀⠀⠨⡆⠠⠄⠚⠠⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠛⠿⠿⣿⣿⣿⡟⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡈⠛⢿⣷⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⠉⠙⠻⠆⠀⠈⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⢷⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⡀⠈⠳⣦⡀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣆⠀⠻⣦⡀⠙⣿⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣦⠀⢻⣿⣄⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡛⣃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⡠⠔⣤⠞⢡⡇⠸⢦⠙⠦⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢠⣾⢼⠧⡧⣷⢰⠀⠀⠀⠀⠀⠀⣠⢸⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣿⣽⣯⣴⣴⣍⣴⣴⣥⣶⣴⣥⣦⣴⣿⣯⣽⣶⣿⠁
⠀⠀⠀⠀⠀⠀⡜⢸⡇⢊⣆⢿⢸⡄⡄⠀⠀⠀⡗⣾⠇⢟⢸⡟⣗⠚⠀⠀⠀⠀⠀⠀⢿⢿⡿⡿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣻⡿⣿⢿⣿⡿⡿⢿
⠀⠀⠀⠀⠀⢠⢃⡿⢀⣼⣿⣌⠁⢧⠠⠀⠀⡰⢡⡟⢀⣾⣧⡑⠹⡌⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠈⠀⠚⠛⠛⠛⠂⠈⠓⠁⠘⠁⠁⠀⠛⠛⠛⠛⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⣿⡿⠟⣿⣿⣿⣸⣿⣿⣿⠀⠀⠀⢀⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⣿⣿⣿⣧⡀⣿⣿⣿⡿⠋⠉⣛⣴⡦⣤⣾⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⡀⢀⣀⣤⣾⣤⣀⣀⠀⢀⡶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠯⠀⠘⠏⠀⠋⡙⠛⠁⠀⢰⣿⠸⢃⣿⠛⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡾⢿⠉⠁⠀⢹⠁⠀⠉⢛⡾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡟⢠⣶⡆⣦⣿⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠠⠤⣄⣀⡼⠋⠀⢈⣷⡒⠋⢻⠋⠓⢲⣞⠀⠈⠳⣄⣀⡤⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣷⣾⣿⣷⣿⣿⣀⣀⣄⣀⣀⣀⣠⡀
⠀⠀⠀⠀⣤⣤⣬⡟⠳⢤⣴⠁⠈⢳⡀⣸⠀⣠⠏⠀⢑⣤⠶⠛⣧⣤⣤⠀⠀⠀⠀⠀⢲⣼⣦⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⣷⣼⣧⣾⣦⣿⣴⠆
⠀⠀⠀⠀⠈⠉⣼⠁⠀⡸⠉⠙⠦⢄⣱⠼⢔⢉⡤⠞⠉⠸⡄⠀⢹⡀
⠀⠀⠀⠶⠶⠶⣿⠶⠒⡷⠒⠒⢒⠒⣇⠀⢀⢗⢒⠒⠒⠒⡗⠒⢺⡗⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⣶⣶⠆⢹⡄⠀⠸⣀⡤⡮⣬⡖⢩⠑⣮⠾⡶⢤⣰⠁⠀⣼⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⡧
⠀⠀⠀⠀⠀⠀⠀⣳⡴⠚⠙⢤⣍⡟⠀⢺⠀⠘⣯⢥⡴⠉⠓⣶⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⢻⠁
⠀⠀⠀⠀⠀⠒⠋⠁⠹⣦⡀⣜⡺⠓⠦⣾⠤⠖⠻⣫⡄⣠⠞⠁⠈⠉⠁⠀⠀⠀⢀⡠⣠⢎⡏⣧⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⢸⡆
⠀⠀⠀⠀⠲⠶⠾⠀⠀⠈⣹⣢⣧⣄⣀⣸⣀⣀⣤⣗⣻⠁⠀⠀⠘⠿⠟⠀⠀⠘⠛⠛⠛⠛⣿⡟⠛⡟⠛⠋⠀⠀⠀⠀⡠⠊⠀⠀⠀⡀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡯⡝⠀⠀⠀⢹⠉⠀⠀⠸⣖⡷⠄⠀⠀⠀⠀⠀⣠⡄⣴⠀⣤⠀⣿⣇⠀⢳⡀⢀⣄⠀⢤⣾⣁⡀⠀⠀⠀⡇⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢽⠃⠀⠀⠀⠉⠁⠀⠀⠀⢯⢵⡖⠀⠀⠀⠀⠀⠙⠃⣿⡀⡷⢀⣿⡟⡶⠈⢿⣄⠛⠁⠈⠻⣿⣮⣭⣍⣒⠃⠀⣄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡟⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⣏⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢚⣿⣇⠙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠿⠿⠿⠿⠿⠿⠦⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⢿⠛⡻⠿⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⠀⠀⠀⢠⣞⣙⡷⠤⠤⠶⠾⠾⠶⠦⠤⣤⣤⣒⠙⢦⠀⠀⠀⢀⡔⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡼⣧⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠿⣄⡀⢀⡜⣀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⡴⠚⢶⠀⠀⢀⣠⣀⠀⠀⠀⠠⠊⠉⠇⠀⠀⠀⠈⠙⢿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠸⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⢳⡄⠀⡹⠃⠙⣦
⠀⠀⣀⡀⠀⠀⠀⠐⣧⠚⠁⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠞⠀⠚⠁⣠⠖⢷⡒⠋⢻⠆
⠀⣼⠁⢹⣀⠀⠀⡜⠁⠀⠸⣥⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋
⠀⠘⠷⠶⢽⣷⡞⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡌⢷⡀⠀⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡸⠀⠀⠀⠀⠀⠀⠒⢉⡀⢀⡿⠘⣧⢴⣊⣥⣤⣄⡏
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣷⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢹⡏⠁⠀⠀⠈
⠀⠀⠀⠠⣯⡇⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣿⣿⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠋⢹⠆
⢀⣀⣀⣀⣙⠁⠀⠀⢳⡈⠉⠛⢶⣶⣦⠤⣤⠄⠀⠈⠈⠙⠒⠦⠤⠤⠤⠤⠖⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣦⠀⠀⠉⣿⢉⣀
⣎⡀⢈⠥⢼⡆⠀⠀⠀⠳⣄⡀⢈⣛⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠀⠀⠐⢿⣀⢹⠆
⠈⠉⠀⢀⠔⡇⠀⠀⠘⠦⣌⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠘⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠸⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⣏⠈⢧⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠜⠀⠀⢠⡀⢹⡶⢤⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⡔⠹⡌⠙⠛⠀⠀⢠⡔⠹⣆⢀⡹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢃⡟⠀⠀⢧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠳⠤⠿⣆⠀⠀⠀⠀⢹⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⠳⢤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣦⠰⠽⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠛⣧⠰⠋⠀⠀⠀⢀⡟⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⢹⠀⣠⠜⣻⣦⡀⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠀⣠⡞⢥⡀⠸⡆⠀⠀⠀⠀⢸⠀⣤⡀⢰⡈⡇
⠀⠀⠀⠀⠀⣴⠊⠉⢀⡞⠁⠀⢙⡶⣄⡀⠀⠻⠖⠋⠀⠀⠀⠀⢸⣄⣸⠄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⢹⡄⠙⡆⢿⠀⠀⠀⠀⢸⡼⠁⠹⣄⡿⡗
⠀⠀⠀⠀⠀⠘⠳⠶⢞⠁⠀⠀⠸⡇⠀⠉⢳⠶⣦⣤⣤⣠⣀⣀⣀⣉⣡⣤⣤⣤⠴⠶⡖⡟⠧⠙⢦⡀⠀⠈⠛⠊⠁⢘⡀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⡞⢀⡎⠀⠀⠀⠀⣷⠀⡴⠋⣼⠁⠀⣰⠁⠜⠁⠹⢡⠀⠘⠒⠂⠀⠀⡇⠙⢶⡄⠀⠙⢦⡀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠇⠈⢶⠲⡄⠀⠀⢻⡘⠷⠤⠟⠀⣰⠃⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠙⠳⡶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣆⢸⣇⡸⠀⠀⠘⣧⠀⠀⢀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣄⠹⠦⣬⣷⡼⠟⠀⠀⠀⣽⣂⠀⠈⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡵⠞⠛⠶⠲⠦⠟⣠⠼⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠇⠀⠀⠤⠍⡁⠶⠂⠴⣤⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢼⡟⣥⠶⣄⣀⡴⠒⠦⣤⡤⠛⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠭⠔⠒⢦⣀⣠⠔⠒⢠⡀⠀⢀⡨⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠿⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢧⣽⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶
⣿⣿⣇⣀⣀⣀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




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

⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⠦⢤⢄⡀
⠀⠀⠀⠀⢀⡴⣫⠽⡖⠋⢹⠉⠓⡾⢭⡛⣤⡀
⠀⠀⠀⣴⢫⠞⠣⡀⠘⡄⢸⠀⡸⠀⣠⠞⢣⠱⡄
⠀⠀⢰⢓⠗⠤⣀⠘⢦⣹⣸⣰⣣⠞⢁⡠⠔⢳⢹⡄
⠀⠀⣿⣼⠀⠀⠀⠩⢵⣿⠋⢻⣷⠯⠁⠀⠀⢸⡤⡇⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢹⢸⡀⣀⠤⠒⣪⢿⣿⣿⢯⡓⠢⢄⣀⢰⢒⠇⠀⠀⠀⠀⡠⡞⠈⢗⢀⠀⠀⠀⠀⠀⠀⣔⡕⠂
⠀⠀⠘⣎⢫⠀⢠⠞⠁⡎⣿⣿⢗⠛⢦⡀⢠⢏⡟⢀⡴⠚⠿⢯⡬⣥⠭⡬⢭⢥⠕⠀⢀⣀⢸⢴⢂⠀⣰⣤⡀
⠀⠲⠦⠿⢧⡟⢿⡞⡟⢀⢿⣿⠾⡶⣼⠟⣣⠞⠀⣸⣤⢶⠗⣾⢦⠅⠀⠳⠀⢸⣦⣈⢨⣨⢘⣋⣿⠀⢰⠋⢏
⠀⠀⠀⠀⠐⠘⠳⢼⣑⣾⣺⣛⣶⣽⣴⣏⣥⠶⠟⠙⣷⣇⣶⣣⣾⡖⣖⢲⡀⢸⡛⢿⣯⣿⠿⠿⠋⠀⠁⠀⠘⠀⢠⠙⣣⡜⢚
⠤⠤⠤⠴⠶⠶⠦⠤⠾⠤⠽⠽⠿⠷⠿⠿⠷⠶⠾⠿⠿⠧⠬⠤⠼⠧⠥⠾⠥⠤⠥⠤⠤⠴⠤⠤⠴⠷⠶⠶⠶⠾⠾⠶⠷⠷⠾⠤⠤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠯⠉⠉⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠒⠒⠒⠒⠲⢤⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠀⣀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣍⠂⠲⠤⠖⠒⢋⡍⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢿⣷⣶⣤⣶⣶⡟⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⢠⡏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢄⡀⠈⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠓⠲⠤⠤⠤⢴⠞⠫⢄⠀⠀⠉⠲⡀⠀⠀⠀⠙⢦
⠀⠀⢀⠤⠐⠒⠒⠒⠤⢄⣀⣀⣀⣠⡤⡾⠁⠀⠀⣀⣠⠴⠚⠁⠀⠀⢠⠇⠀⠀⠀⠙⣆⠀⠀⠀⠀⢳⡄
⡖⢶⠡⣆⣀⡹⢤⣼⣃⡀⢠⠤⠤⠤⠴⠷⠒⠚⠙⠍⠀⠀⠀⠀⢀⡴⠋⢀⠠⠂⠀⠀⠘⡆⠀⠀⠤⠊⢹
⠓⠒⠶⠮⠭⠥⠤⢤⣀⠉⠙⢧⡔⠁⢀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠂⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡗⠺⢿⡹⣄⣀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠈⡗⢬⣃⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⡇⢰⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡇⠀⠀⠀⢧⠀⠈⠀⠀⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⡸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣸⢁⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡷⠀⠀⠀⠀⠈⣧⣄⠀⠀⠤⠤⠤⠤⠤⠶⠒⠉⠀⠀⢀⡴⢧⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡞⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣠⣤⠖⠋⠀⠀⠈⠉⠛⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⣇⣇⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⡜⠁⣜⣆⠀⢀⡞⢁⣤⡀⢻⠒⠾⠦⢤⠼⠤⠼⠶⠤⣤⣄⣀⣀⠤⠤⢄⡈⠉⠉
⠀⠀⠀⠀⠀⢧⠀⠛⢉⡗⢺⠀⢸⡠⠃⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡔⢢⠀⢷
⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠘⣆⠀⠀⣠⠏⠀⠈⠉⠉⠉⠉⠉⠙⠒⢶⠒⢿⠀⠳⠁⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠉⠉⠉⠙⠓⠒⠤⢀⠀⠀⠀⠀⠀⣀⣰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢡⠀⠀⢀⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢿⠀⠀⢸⣿⠃⠈⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣤⢊⡥⠖⠒⠀⠸⠀⠀⠀⢹⠀⠐⠒⠦⣝⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢰⣍⠀⠀⠠⠔⠒⠋⣡⡎⠀⠀⣀⣠⠄⢳⠂⠀⠀⠈⠀⢠⣀⡀⠀⠁⠘⣿⢿⢿⣿⣦⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⣏⠙⠲⠄⠀⠐⡋⠆⠁⠀⠀⠀⠀⠀⡴⠋⠀⠐⢧⡀⠀⠀⠀⠀⠀⠀⠁⡿⠄⣿⣿⣿⣷⣤⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⡎⢣⣀⠀⠀⣀⣑⠄⠀⠀⠀⠀⠀⢲⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⡿⣁⠈⠳⣄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⠀⠈⠑⣶⡴⠛⠀⠐⠲⣤⣀⡀⣸⠀⠀⢀⣀⡜⢣⡀⠀⢀⣀⣤⠶⠒⠘⠿⣿⡿⠛⠓⠚⠓⢶⠾⠟⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣀⣀⣠⠞⠁⠀⠀⢀⡾⢥⣾⣿⣷⣦⣴⠋⠈⢣⡈⠙⣟⠉⠙⣮⠀⢀⡤⠞⠁⠀⠀⠀⣠⣴⠭⠽⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⢠⠀⣸⣿⠀⠀⠀⠀⠀⠀⠞⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠙⠲⣌⡳⢀⠈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⣸⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⢸⠋⠑⢺⠦⠤⠤⠤⠴⠒⠠⠤⠤⠤⠄⠒⠛⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢠⣧⣾⣿⣿⡧⢤⣤⣀⣤⡤⠶⠊⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⠤⠖⠒⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡃⠘⢦⡀⠀⣀⠾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠐⠀⠈⠉⠀⡄⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠐⠂⠀⠀⠀⣀⠠⠀⠀⠀⠳⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠋⢫⠀⠀⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣰⠏⢃⠀⢀⠜⠁⠀⠀⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣃⣀⣀⠀⠀⠀⡰⠁⠀⠀⠑⠁⠀⢀⠁⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡋⠀⠀⠀⠀⢀⣠⠤⢄⠀⡞⠁⠁⠀⠙⣆⡜⣁⣄⠀⠀⢀⠀⠂⠀⠁⠐⠠⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⣸⠀⠀⠀⡽⣇⠀⠀⠀⠀⡼⡏⠀⠀⢱⠊⠀⠀⠀⡠⠖⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠒⠊⠀⠈⠓⠤⠤⠚⠁⠑⠤⠤⠊⠑⢤⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠖⠁⠀⠀⠈⠳⣄⢀⡠⠞⠁⠀⠀⠈⢦⠀⠀⡠⠤⢄⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠁⠀⠀⠉⠒⠶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢶⠆⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠾⠂⠀⠀⠀⠀⠶⠄⠀⠀⠰⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠙⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡇⢱⡀⠀⣴⡄
⢀⢸⠹⠀⢀⠀⠀⠀⠀⠀⢀⣶⠀⣿⠘⡄⣇⠀⢳⡀⡇⠹⡀
⢸⢺⡄⢳⠟⠀⠀⠀⠀⠀⢸⠁⠆⣿⠀⠘⢿⠀⠀⢻⡇⠀⠱⡀⡷
⢀⢧⠁⠀⠀⢀⠀⠀⠀⡷⣸⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠡⢠
⠈⢿⣆⠀⠀⠿⡇⠀⢸⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠹⣦⡀⣷⡄
⠀⢈⢆⠀⠀⠀⢹⣠⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⢹⣄
⠀⠀⠻⡷⠀⠀⠀⠿⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡙⠃
⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢻⡄⠀⣱⣆⣤
⠀⠀⠀⠈⠻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡠⠙⠛⠁⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣰⡆⠀⢠⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠺⣧⣬⣀⣀⣈⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠋⠁⣷⠀⣾⣇⢸⡇⠀⠀⠀⠀⠀⠘⣆⠀⣿⡄⠀⡠⢨⡏⠉⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⡀⡇⠀⠀⠀⠀⠀⠀⢹⡀⣿⢱⡈⣷⡷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡇⠀⣇⢷⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⢣⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣼⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⡀⠘⣇⢀⢀⠀⠀⠀⠀⠀⢸⣧⢀⢀⠀⢻⡄⢀⢀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠂⠒⠐⠐⠀⠀⠠⠀⠐⠈⠈⠈⠀⠀⠀⠀⠀⠈⠀⠒⢜⡄⠀⠀⠀⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢀⠀⡄⣀⠄⠀⠈⠉⠍⠻⠥⢐⢄⠀⠀⠀⠀⠁⠉⠀⠀⠶⠞⢗⣦⠬⠓⠂⡀
⠀⠇⠀⠀⣆⡀⢰⠚⡏⠀⣀⠠⠴⡔⠠⢄⠀⠸⡠⢤⣼⠀⢀⡤⠤⢤⠤⢠⠀⢸⠿⠀⣰⠠
⡀⠀⠠⠀⠯⠓⠂⠠⠯⠄⠃⠀⠈⠻⠀⠀⠃⠨⠂⠀⠿⠠⠉⠀⠠⠛⠄⠀⠇⠔⠁⠀⠓⠂⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠂⠙⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠠

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢸⡃
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⣾⢀⠀⣀⠿⠟⠛⣶⡀
⠀⠀⠀⠀⠀⢸⣧⢸⣺⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠤⠞⣿⠈⢻⣿⣶⣶⣤⠾⠃⢠⡄
⠀⠀⢈⡇⠀⠀⢿⣦⡀⢰⣷⣄⡄⠀⢀⣾⠃⠀⣶⠆
⠀⠀⠘⠓⠀⠀⠀⠙⠻⣾⣿⣻⣷⢀⡸⡎⠀⣼⠋⢀⡀
⠀⠀⠂⠀⣀⢀⠐⠀⠚⠉⢉⡉⠙⠉⠈⠿⣿⣧⣀⣼⠓⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⡘⣇⡀⠀⢀⡤⠛⠓⠒⠀⠀⢀⣽⣿⣿⣷⣠⣶⠎⠀⠀⠀⠀⠙⡧
⠀⠀⠀⢸⣿⣻⣷⡞⠉⠀⠀⠀⠀⠀⠀⠿⡁⣿⣿⡟⢠⡄⠀⠀⠐⠸⣟⡶⠁⠀⣆
⠀⠀⠀⣿⣿⣿⣿⣁⣤⣶⡄⠀⠀⠀⠀⢲⡟⣸⣿⡿⢿⡹⣆⠀⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⣄
⠀⠀⠀⢿⣿⣿⣿⣼⣿⡉⢿⡄⠀⠀⠀⢸⣧⣿⣿⣿⠈⢁⢹⡄⠀⠂⠀⠀⠀⢰⣏⢀⡶⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠹⠀⢿⡆⢠⣴⣿⣿⣿⣿⣿⣷⣴⣿⣇⠀⠀⠈⠀⢠⡿⢁⣾⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣾⡟⢨⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣆⡀⠀⠀⢹⡇⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣬⣿⣿⣿⣿⣿⡿⣻⣿⣴⠆⠀⣸⠇⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠂⣀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⢧⣈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣥⣤⣾⣃⡼⢾⣧⡀⠀⠀⠁⠀⠀⣼⡟⡶⠃⢀⠐⠁
⢀⡀⠀⠀⠘⣷⡟⠀⠀⠀⣾⣩⣿⣿⣿⣿⣩⣽⣿⢉⣿⠛⣿⣿⣿⣿⣿⣿⣛⣧⡄⠙⣿⣶⣤⣤⣤⣴⡿⠏⠠⠊
⠈⣿⡀⠀⠰⡟⣻⣦⡑⣶⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣠⡇⢻⣧⢈⡙⠓⠀⠀⠀⠀⣠⠾⠃
⠀⠈⠿⣦⣄⣸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠟⠛⠛⠛⣿⡿⣿⣿⣿⣿⣿⣯⣿⣦⣿⢧⡄⠀⠀⠀⢀⠀⠊⠁
⠀⠀⠀⠠⢙⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣾⣯⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿
⠀⠀⠁⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡀
⠀⠐⠿⠿⣿⣿⣧⣿⡇⠀⢀⣤⣶⣶⣶⣶⣦⣶⡀⠀⢠⣾⣿⠿⠛⢿⣿⣶⣤⣿⣿⣿⣧⣼⣷⠛
⠀⠀⠀⠀⠈⢹⣿⣿⣿⣾⣿⠿⠋⠀⠀⠀⠙⣿⡟⠀⢸⣿⠉⠀⢀⡐⣋⣿⣿⣿⣿⣿⣿⣿⠯
⠀⠀⠀⠀⠀⣸⠁⣿⣿⣿⣿⣤⣤⣤⣤⣶⣾⣿⡄⠀⠸⠿⠿⠿⠿⠿⠟⠛⠋⠈⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠇⢰⣿⣿⠇⠙⠻⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠈⠿⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠚⠳⠚⠁⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⢀⣰⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⣷⣶⡋⢻⡿⠛⠋⠉⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠸⣿⣿⣿⣧⡀⠀⣀⡴⠂⠼⠃⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣵⣾⣯⣥⣭⡿⠛⢻⠟⠋⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣤⠤⠄⠀⠀⠈⣷⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⠃⠁

⠀⠀⠀⠀⠀⠀⢀⠄⠠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣶⣀⠀⡤
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣱⢤⡀⠤⠄⣀⡀⠤⠤⡄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠟⢫⡄⠶⠀⣜⠻⣅⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⡅⠉⠁⠀⠉⡧⢎⠉⠓⠀⠀⠁⠀⠓⠀⠀⠀⠀⠐⢺⠋⡒⠤⣱⣶⣾⠤⢒⠙⡖⠂
⠀⠀⠀⠀⠀⠀⣿⠀⢀⣷⣀⣂⠀⠀⣇⣀⢀⠒⠀⠀⠀⠀⢠⢀⡀⠀⢀⣬⣹⡉⢋⣩⣿⣿⣿⣍⡙⠉⢯⣥
⠀⠀⡔⠀⡋⡉⢹⠀⢀⣀⣈⣳⣄⣀⡤⠤⠀⡄⠀⠈⣀⡀⡀⣤⡂⠀⠀⠀⣭⠻⣉⣿⢧⣦⠸⣷⢉⠟⠤
⠀⠀⠉⠈⣣⡟⢛⠈⢨⡅⠀⢀⡀⠀⠁⣶⣄⣤⣤⣼⣿⣦⣤⣀⣁⣆⣀⣀⣻⢀⣬⣿⠒⢿⠖⢻⡅⠀⠙⠁
⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣶⣟⣟⠉⠏⢹⠁⠉⢉⡍⣿⡏⠙⣉⢩⣿⣿⣷⡌⠘⣿⢿⠻⣿⣤⣾⣿⣴⣷⣄
⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣙⣷⣿⣇⣍⣉⣡⣿⣯⣿⣁⣼⣿⣿⣿⣯⣽⣿⣼⣯⣿⣷⣯⣿⣿⣿⣿⡆
⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⡠⠐⢢⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢼⠤⣀⣠
⠀⠀⠀⠀⠀⠀⡇⠀⢸⢀⣨⠓⢆⡒⢀⣤⡤⠀⠀⡄⢀⡀⠀⠀⠀⠀⠠⢤⡜⢉⢦⢺⠂⡴⡉⣣⡤
⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢹⠂⠀⣈⠀⠀⠀⠀⠉⠀⠀⠀⠀⣉⣟⣈⣑⣠⣿⣾⣔⣉⣀⣏⡁
⠀⠀⠀⢀⡀⠀⡧⠠⢼⠡⢴⠀⠀⠸⠤⠤⠯⠁⣘⠃⠀⡶⠠⡤⠄⠰⠦⣭⣈⣤⣾⢿⢿⣷⢤⣀⠗⠶
⠀⠀⠇⢀⡀⣄⣇⡀⢼⠤⠬⠷⠦⢶⠒⠀⠋⠀⢀⣤⠊⠇⠶⠅⠀⠀⢐⡏⠳⢼⡟⢺⡆⢻⡤⠊⣁⡀
⠀⠀⠀⢀⣾⣅⡁⣠⣾⡀⢀⣤⣀⣠⢼⡧⠷⣶⣾⣿⣿⣷⡶⠤⣤⣦⠈⡇⣒⣿⣯⣹⡏⢉⣷⡀⣀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⣄⣸⡇⠠⠤⠆⣿⡧⣤⡆⢸⣿⣿⣿⠠⣽⡏⣏⢿⣷⣯⣿⣾⣿⣷⡀
⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⢾⣿⣷⠷⣶⣾⣿⣷⣷⣶⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⣿⣋⣌⣿⣿⣿⣿⣿⣿⣷⣾⣷⣮⣽⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣭⣿⢿⣿⣿⡘⣿⡄
⠀⠀⠀⠀⣾⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠛⠉⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣷⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣋⣠⣼⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠿⠿⣿⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣷⠉⣰⡿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣻⣿⣿⠀⣿⣟⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣤⢻⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡔⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣤⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⢽⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⣿⣧⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⣿⣿⣿⣿⡌⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠻⣿⣷⢻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠹⢿⣿⣿⣷⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⢸⣏⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣷⣤⣭⣿⣿⣿⣜⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣸⣿⡏⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣶⡀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢀⣼⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡌⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠃⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡆⠀⠀⣿⡏⠈⠀⢸⣿⡏⠛⣾⠦⠤⢤⣴⠤⠔⠋⠀⣿⠀⠁⣿⡟⠻⠛⣿⣿⣿⣿⣿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⠸⢿⣇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⣿⡄⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡅⠀⣾⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄⣿⣄⣾⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡃⣄⠹⠿⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠇⣰⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡿⢻⡟⣿⢸⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⣿⣿⣿⣿⣸⣿⣿⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⠿⢿⣿⣼⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠈⠉⠋⠉⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⡀⠀⡀⠈⠃⠀⠀⠀⠀⠈⠙⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄⠀⠀⠀⠀⠀⣰⠟⢀⣿⣯⣾⣽⡧⠀⠀⠀⢀⠈⠀⠀⠀⢋⠀⠁⠀⠀⠀⠀⠀⠀⣤⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⢿⣯⢋⡼⠃⠀⠀⠀⠛⠃⠀⠴⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢠⡄⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠂⠐⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠨⡆⠀⠀⠀⠀⠀⠛⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⡀⠀⠀⠀⠀⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠐⠒⠒⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠛⠉⠀⠰⠊⢁⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢂⡇⢠⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⠃⠀⣈⠀⢀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠨⣉⢢⣴⡷⠁⠀⣠⡞⠉⠁⠀⣸⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣢⠌⠁⠁⠀⢠⠋⠈⠋⠀⢀⠴⣻⡆⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⢆⡤⠤⠶⠶⠶⠯⠅⠀⠀⠀⠀⠈⠑⠒⠚⢋⠉⠤⣉⣁⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣇⣾⠀⢴⣊⣛⠺⢏⣖⣦⣀⠀⠀⠀⠀⠀⣀⠼⢄⠀⠀⠉⠉⠒⠢⠤⢤⣀⡀
⠄⠀⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠖⠊⠁⡸⠀⠀⡇⠀⠁⠀⠙⠂⠭⣱⡊⠉⠉⠳⣄⡤⠒⠉⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⢻⠃⠉⠉⠑⠄⠀⢀⣏⠁⠀⠀⠀⠀⠀⠓⠒⠄⡇⠀⠀⠀⣀⣀⡤⠤⠌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣂⣀⣀⣠⠔
⠀⠀⠈⡄⠀⡠⠶⠓⠊⠁⠈⠣⡀⠀⠀⠀⠀⠤⠀⠈⠑⠒⠿⣏⠉⠀⢤⡄⣀⡠⢀
⠀⠀⠀⠑⢄⢰⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠀⠑⣀⡠⢶⣿⣿⠤⢾⡶⠼⡀⠵⡴⢳⠀⢀⠼⣥⣼
⠀⠀⠀⠀⠀⠉⠒⠂⠠⠤⠄⣀⣀⠀⠀⠈⢉⠒⠒⠩⠽⣁⠀⠛⠙⠒⠂⠭⣁⠸⡱⢳⠆⠇⠔⡍⢠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠥⡀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠙⠒⠉⠈⠂⡸⠙⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠑⢄⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣀⡴⣾⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠉⠉⠉⠉⠋⠙⠓⠛⠛⠛⠒⠠

⡀⠀⡀
⣇⡀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠓⣤⣾
⣿⣿⠙⣆⣄⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⡀⢱⣾⣿⣿
⣿⣿⣿⣿⠿⡉⣀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⢏⢢⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠺⣥⣷⣿⣁⢀⡴⠂⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⢡⢈⡢⣄⣾⣆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡾⣿⣂⣼⣤⣸⡎⡹⠀⢰⡄⣠⠄⠀⣄⢄⠰⡀⣦⠀⡀⢲⣀⠙⢄⣠⣷⣏⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣤⣷⣿⣴⣿⣿⣴⣄⣼⢸⣾⣶⣧⢿⣴⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡎⡞⡕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣡⡞⢰⣇⠹⣆⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠁⣾⡟⠀⢸⣿⠀⢹⣷⡄⠙⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠐⡿⠓⠠⣾⣿⡆⠘⠛⠓⠀⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡂⢀⠀⣿⣺⡇⣠⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠿⢿⠚⠂⣿⣺⡇⠁⡿⠿⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣀⣆⣀⣴⣿⣾⣷⣄⣀⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄



⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⣶⠶⠶⣶⡶⢶⢶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣠⠤⠖⠦⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⢳⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠆⠀⠀⠀⠀⢰⡷⣌⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⣰⡿⠠⠎⠉⠛⠛⠛⢛⣛⠛⢻⡿⣅⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⣾⠟⢀⡤⠞⠛⡷⣄⢀⡞⠉⠙⠢⣷⣌⠳⠀⠘⢦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠰⠖⣫⣿⠃⠀⣾⠀⠀⠺⠿⢺⢿⣇⢿⠗⠀⣹⣿⢧⡀⠀⠀⠙⢦
⠈⠉⠁⠀⣰⠃⠀⠀⠀⠀⠀⢸⠟⠻⠀⠐⠮⣷⣶⠶⠞⠉⠀⠙⢷⣶⣶⡯⢾⣄⣳⡀⠀⠀⠈⣧⠀⠀⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡼⠃⠀⠀⠀⣫⡏⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢦⣌⠳⣄⡀⠀⢻⡘⠿⣧⡤⠀⢀⣤⣴⠶⢶⠶⠶⡶⠶⣶⢦⣄⠙⣄⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢓⣒⠒⠛⡟⢠⣾⢻⡀⣸⣄⣸⣄⣀⣇⣠⣿⠤⣿⣇⣿⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⣾⣇⢘⣿⢟⡿⠋⡿⢸⠄⣿⠙⢧⡙⢦⠘⡏⢿⠻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢹⡇⠸⠁⠰⠃⠙⠀⢿⢧⡈⠁⠘⠃⠛⢸⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣟⣎⢧⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⡾⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢻⡏⣨⡷⠂⠀⠀⠀⢀⡟⠀⢿⡀⠀⠀⠘⢷⡖⠒⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠴⠒⠛⠛⣻⠟⠁⠀⠀⠀⢠⡶⠁⠀⠀⠀⠙⣆⠀⠀⠀⠙⢿⡲⣬⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡴⢖⣻⠟⠁⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠙⠘⣯
⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⠀⠈⠁⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷⣦⣀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣤⠴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣀⢨⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡅⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡄⠠⡀⠁
⠀⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢁⡠⠜⠀⠀⠀⠠⠁⡰⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠂⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠇⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⢀⠀⡈⠀⠀⠀⠀⠀⠀⢰⠠⢀⠌⠀⠀⠀⠀⠀⠀⠀⡇⡠⠂
⠐⠀⠁⠀⠌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⡇⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠉
⢠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⢤⠊
⠀⠀⠐⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠒⣆⠁⠀⢀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠒⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣾⡏⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⣼⣿⣿⣿⢷⢁⣄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⣻⢿⣿⣿⣿⢸⣿⣯⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡈⢦⡀⠰⣆⣰⠏⢸⣿⣿⣿⣿⣿⡁⢷⢀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣌⠓⢄⣀⠳⡀⢸⡟⣿⣿⡀⣿⣿⣿⣿⣿⣿⡿⣼⣿⣇⣿⣀⣄⣸⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣌⡳⣶⣿⣤⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣦⡙⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣇⡶⢣⠀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⢈⡓⢤⣰⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⣻⠋⠁⣠⠆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⢞⡡⠶
⠀⠀⠀⠀⣄⠀⠈⠹⣦⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⣤⠄⠀⣴
⠀⠀⠀⠀⠈⠓⢤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡷⠤⠴⠃
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠲⠏⠉⠉⠍⠐⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⡀⠀⠀⠗⠀⠀⠂⠀⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣠⣀⣰⣆⣀⣰⣆⣨⡀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡊⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠳⣄⠀⠉⠀⠀⠀⠀⠀⠈⠉⢀⣠⠞⠑⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠈⠳⣤⣀⠀⢀⣀⣤⠴⠚⠋⠀⠀⠀⠸⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⣠⠞⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⢸⠀⢀⠞⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⡘⢠⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⡇⢸⣿⡿⠛⠛⠋⠙⢿⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⢻⠸⡿⠶⠶⠶⠶⠶⠾⡿⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠘⡄⠻⣄⠀⠀⠀⠀⡴⠃⡀⠀⠀⠘⡆⠀⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠗⠋⠀⠀⠀⢀⠇⠀⢠⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⡴⠃⠀⠀⠚⠀⠀⡾⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠉⠉⠈⠳⣄⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⠀⠀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⢆⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⠶⠤⠶⢻⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠈⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠔⠉⠀⢀⡞
⠀⡠⠒⠂⠒⠒⠒⠀⠀⢼⠁⠀⠀⠀⠀⣇⢸⠀⠀⢰⡄⠀⠀⠠⠶⠂⠀⠀⠘⡄⠀⠀⣧⠀⠀⠀⠀⠀⠸⡦⠤⠖⠛⠉⣉⠱⡄
⡜⢀⠎⠀⡰⠒⠁⠀⢠⡏⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠹⡄⠀⡟⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⡄⠈⢧⡱
⠙⠺⢤⣤⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⡀
⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⣠⠤⠤⠏⠀⠀⠀⠀⢸⣦⣤⣤⣤⣷⠤⠀⠀⠀⠀⠈⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡴⠋⡀⠀⠀⢀⠀⠀⠀⢀⠈⠳⡄⠀⡞⠁⠀⢰⠂⠀⠀⠀⡀⠀⠀⡄⠘⣆⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠈⠛⠛⠚⠛⠛⠙⠦⢼⣃⠀⠀⢸⡀⠀⠀⢸⣄⣀⣿⠀⠳⠤⠴⢿⣄⣀⣀⣀⣇⣀⣀⡽⠒⠋⠙⠛⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠒⠒⠒⠤⠤⣄⣀
⠀⠀⠀⠀⠀⣤⢀⠴⠊⠉⠀⢀⠠⠔⠒⠉⠉⠉⠉⠉⠀⠈⠙⢦⡀⣀⠀⢀
⠀⠀⣔⠻⠁⠘⠁⣀⠠⠀⠀⠀⠀⢀⠠⠤⠒⠒⠋⠉⠀⣠⡤⠖⣋⢵⡴⠃
⠀⠀⢈⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⢉⡵⠞⣇⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⣲⠶⠓⠒⠒⠲⠤⣄⠀⠀⢀⣠⣤⣄⡀
⠀⢠⠏⠐⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⢀⣀⣠⠴⠞⠉⠀⠰⣹⡀⠀⠀⣠⠔⡶⢊⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⢦⣋⣤⠂⠈⠙⣆
⠀⡞⢠⠁⢠⠃⠀⠀⠀⢠⠀⠀⠒⠒⠚⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⠃⢠⡞⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⡝⣆⢉⡀⢸
⢠⡇⢸⠀⠈⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢹⢀⣿⢠⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⢀⡽⣼⡀⠀⣸
⠈⡇⠘⠆⡴⢁⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣸⢸⠺⢀⠧⣄⣀⣤⣀⡀⠈⠉⠒⠤⣀⠀⠉⠒⠦⣤⡴⢞⠅⣧⡤⠖⠁
⠀⣿⠀⢰⢡⡏⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⢀⠇⠈⢇⣸⠠⠀⢀⡄⠀⠉⠙⢦⡀⠀⠈⠳⡀⠀⠀⠀⠐⠋⡴⡇
⠀⠸⣇⠀⠈⠣⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⢇⡞⠀⠀⠀⢿⢀⠄⠀⠁⠀⠀⠀⠀⠓⢸⡍⢣⠁⠀⠀⢀⠀⡰⢣⡇
⠀⠀⠹⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣀⠴⣫⠋⠀⠀⠀⠀⢸⡘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠶⠎⠀⠀⠀⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⢬⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣡⠞⢁⡀⠀⠀⠀⠀⠀⢳⡤⠤⠞⠁⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠤⣄⡀⠀⣠⣤⠤⠶⠚⠉⠀⠀⠙⠃⠀⠀⠀⢀⣄⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠘⡎⠀⢒⣈⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢽⡀⠾⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠀⠈⠉⠓⡆⠀⡠⠒⠲⠿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⠀⠉⠢⡀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⠀⠀⠀⠀⡠⠂⠑⠒⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢄⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠈⡄⢇⡀⠀⠀⠀⠈⠛⠿⠃⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠂⠀⠀⠀⠀⠀⢠⠀⠈⠑⠒⠤⢤⠖⢦⣤⠤⠖⠋⠹⠀⠀⠀⠀⠀⢸⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢏⣢⠀⢀⡠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠈⠉⡩⡅⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡤⡆⠀⠀⣠⣴⢺⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠊⠀⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣈⠙⠒⠛⠂⠛⠋⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠙⢓⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣊⣑⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠭⠃⡔⢚⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣠⡄
⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⠿⠀⣡⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⢠⢞⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡶⢤⠄⠀⠙⠃⠀⠀⠀⠀⢰⣾⡲⡆⠀⢰⠀⠀⠀⠀⠀⣀⡀⣰⣿⣫⣦⡄⣤
⠀⠀⣼⣧⠀⠓⢢⣼⡀⠀⠀⠀⠀⠛⣿⡇⠀⠀⣽⠤⠀⠀⠀⠀⠀⢸⣿⣷⡿⣿⣿⣗⠀⠀⠀⠀⠽⣟⠿⢽⣷⡷⠛
⠀⢰⠇⠸⡇⠰⠚⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣵⣶⣿⣶⣦⠀⠀⠀⠀⠀⠈⠉⣠⣶⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⡄⠒⠐⠃⠒⢀⠀⠀⠀⠀⠀⡠⠖⠛⠷⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠞⠀⠁⣀⣄⠈⠄⢳⠀⠀⠀⠀⢀⡀⣀⣤⡄⠀⠙⠂⠀⠀⠀⠀⣠⣤⣾⣿⣿⣦⣤⠀⠀⠀⡉⠀⠈⢂⡠⠤⡀⠀⡠⣀
⠀⣇⠠⢀⡝⢻⡀⠌⣘⠀⠀⠀⠚⠸⠡⣭⣿⣟⡱⢆⡀⠀⠀⠀⠀⠉⠉⠋⣿⡏⠋⠉⠀⠀⠀⠀⠀⠀⠁⢡⠀⠑⠊⠀⢸
⠀⠈⢦⣜⠀⠀⣣⡰⠁⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠠⠴⠘⣿⡇⠳⠶⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠐⠁
⠀⠀⠎⠀⠈⠁⠈⠱⠀⠀⠀⠀⠐⠻⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠟⠂⠀⠀⠶⠶⠶⠶⠶⠦⠶⠶⠶⠶
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⣠
⠀⠀⣯⣅⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣾⡶⣹⣧⣤⡀
⠀⠀⡟⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⢨⡇⠀⠀⠀⠀⠘⡀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢛⣵⣿⡫⢝⠻⠁
⠀⠀⣧⣤⣤⣤⣤⣴⠀⠀⠀⠀⠀⣰⡏⠓⠤⠀⢀⢸⡇⠀⠀⠀⠀⢸⣿⣿⠷⠢⣤⣄⠀⠀⠀⠀⢠⣯⣽⡥⠊⠁
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡴⠋⣷⠀⠀⢀⣼⣽⣇⡀⠀⠀⢰⣾⣿⣷⣶⣶⣾⣷⡶⠀⠀⠀⠘⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠛⠛⢛⡟⠛⠛⠁
⣀⠠⡤⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⢀⣄⢀⣀⣸⣿⡟⠃⠀⠀⠀⠀⠀⠀⠈⠃
⠈⢛⣿⣉⣉⣀⣀⣀⣀⠀⠀⠈⢩⡿⢧⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠘⣿⣟⠡⠉⣿⡇⠀⠀⠀⠀⠀⢠⡄⢀⣤⣠⣤
⠀⠀⣸⢷⡀⠀⠀⠀⠀⠀⠀⣀⡞⢷⠀⣉⣀⣤⣤⣤⣿⣿⡇⠀⠀⠀⠈⠛⢲⡞⠋⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠈⠃⠀⠓⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠊⠐⠙⠋⠐⠑⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠸⣿⣿⠀⢀⣠⠆⠀⣼⣿⣿⣷⠀⠀⠘⢦⡀⠸⣿⣿⠀⣰⠋⠀⠀⢰⣿⣿⠿⠞⠀⡀⠀⠀⣿⣿⣿
⠀⢀⣀⣤⣶⣿⣿⡟⠋⠀⠀⣇⣘⣻⣿⣯⣀⣀⡀⠀⠹⣷⣿⣿⡿⠁⠀⢀⣠⣾⣿⣧⡀⠀⠀⢧⣠⣴⣾⣿⣶⣄⢀⡤
⠀⠈⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠉⣿⣿⡇⠈⠉⠁⠀⠀⢸⣿⣿⡇⠀⠉⠉⠁⣿⣿⣏⠉⠓⠆⠀⠀⢸⣿⣿⡏⠉⠉
⠀⠀⢴⣄⣼⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⠂⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⢿⣧
⠀⠀⠀⠈⠉⠁⢹⡇⠀⠀⠀⠀⠀⡿⣠⠞⠁⠀⠀⠀⠀⢸⠏⠸⡇⠀⠀⠀⠀⢧⠸⢦⠀⠀⠀⠀⠀⣿⠃⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠐⠇⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠷⠀⠀⠀⠀⠨⠃⠀⠙⠀⠀⠀⠐⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣉⠉⣉⢉⣉⣉⣉⣉⣉⣉⣉⢉⣉⣉⣉⡉⣉⣉⠉⣉⡉⣉⠉⣉⢉⣉⡉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠿⠧⠿⠿⠟⠾⠻⠿⠾⠧⠿⠿⠧⠯⠼⠻⠇⠿⠀⠿⠹⠿⠦⠿⠿⠿⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠖⠂⠒⠖⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣿⠆⠀⠀⠀⠀⠀⠀⢶⡿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⡠⣾⢸⡶⡄
⠀⠀⠀⠀⢀⣀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠈⠀⠀⠀⢺⣿⣿⠆⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠃⣿⢸⣇⡇
⠀⠀⠀⠀⠈⣿⡿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠙⡃⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⢙⣼⠉⣶⡄
⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣐⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠈⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣤⡀⠀⠀⡏⠀⠘⡇⠀⠀⣠⠄⠳⠆⠀⠰⢠⠈⢦⡀
⠀⠀⠀⢀⢠⡄⡀⢀⣠⢊⡏⢿⡲⣄⠀⢀⣠⣄⡀⠀⢀⣠⣤⣄⡉⠒⢾⠧⠄⠤⢿⠖⠊⠁⠀⠀⠀⠀⠇⠘⠀⠀⠙⠦⣤⠒⠒⠒⠒⠲⡄
⠀⠀⠀⢹⠙⠋⡇⢸⡷⠟⣓⣛⠳⣾⣇⡉⢹⡏⠉⣈⣭⠉⣿⡍⣩⠁⡎⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠸⠛⠯⣉⣉⡭⡷⠒⠒⠢⣄
⠐⣛⠂⣸⣖⣶⣇⣾⣿⣿⣻⣟⣿⣿⣏⠐⣨⣇⡐⢁⣿⣼⣿⣤⣿⣜⣒⣿⣺⣶⣗⣘⣀⣴⣶⣤⣾⣤⣿⢾⡗⠊⠉⠓⣶⠒⠊⠉⠉⠉⣛⠒⠒⠋⢁⢺⠲
⠀⠀⠀⠘⠉⠉⠃⠹⠽⠧⡉⠁⠭⠻⠁⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠍⠩⠉⠍⠉⠉⠩⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠀⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠁
⠀⠀⠀⠐⠀⠀⠠⠆⠀⠆⠆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⡦
⠀⠠⢤⠀⠀⠀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠠⠊⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠈⢢⢀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠑⠂⠤⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⢄⠉⠀⠂⠀⠂⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⠹⢅⠈⠁⠈⡁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠤⠀⢢⡀


⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⡀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠈⠁
⠀⠀⡠⠄⠀⠉⠉⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡑⣠⣤⣤
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠄⠀⠀⠀⠀⠤⠄⠀⡈⠻⣿⠟⢡⡙⠻⠛⠀⠀⠐⠒
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠂⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢠⣄
⠀⠀⠀⢤⣤⣤⣀⡀⠀⠀⠀⣀⣠⣤⠶⡦⠀⠀⠀⠀⠀⣠⠴⠞⠋⢠⠜⠓⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠐⡀⠈⠛⠛⠛⠿⡛⠻⠁⠀⢀⠇⠀⠀⠀⠀⠸⡀⠐⠲⠂⠀⡀⠒⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⣠⡀
⠀⠀⠀⠀⠉⠂⠀⠐⠊⠀⠈⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠁⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⢀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⡀⠄⢰⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈
⠀⠀⠀⠀⡰⠯⠤⠤⠘⠀⠔⠒⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠇⠄⠄⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷
⠀⠀⠀⠈

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠋⠌⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⢠⡟⠛⡆⢀⠈⢩⠆⠀⠀⠀⠀⠀⠀⠀⡋⠉⢙⠀⠀⠂⠀⠀⡀
⠀⣀⣰⣆⠀⠀⠀⠀⠰⡍⠬⠆⠀⠀⠀⠀⣸⠉⠉⠇⠶⠀⢺⣦⣠⠁⣾⣦⣾⡀⠀⢀⠤⣄⠀⠀⢸⡷⢀⡾⠦⠀⠀⠚⠉⠙⡆
⠀⣿⣿⣿⣷⠀⠁⡆⠂⣠⢀⣦⣤⠀⠀⣾⣿⣣⣴⠁⡧⢠⣼⣿⣯⠄⡁⢸⣿⡇⠀⠠⠀⢈⡄⠐⠇⠀⢀⣠⡗⢀⡠⢶⡄⢠⡇⠀⠀⠀⣄
⠀⢸⣿⣿⠏⠀⠀⠀⣼⣧⣿⣿⣿⣷⠀⣿⣿⣿⡏⠀⣧⣌⣛⠙⠃⠀⠘⠈⣿⣦⠀⠴⣧⣼⡃⠀⠀⠺⣿⣧⠈⠀⢡⣾⣷⣿⣿⡀
⠀⣼⣿⣿⠀⡇⠀⠦⢻⣿⣿⣿⣿⠱⠶⣿⣿⡇⠀⣿⣿⣿⣩⣵⣤⡀⠀⢀⠹⠛⣁⣶⣾⣿⡇⠀⠀⢐⢟⡟⣳⣄⣸⣿⣿⣿⡙⠁
⠀⣿⣿⣿⣤⡇⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣶⠇⢨⣭⣭⣿⣽⣿⡿⠀⠀⢡⡄⣺⡿⣿⣿⣟⠑⠀⠀⢼⣿⣿⣿⣿⣯⣭⣭⣉⣥
⠀⢹⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⢛⣿⣿⢠⣿⡿⣻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⡇⠀⢸⣿⡟⣿⣿⣿⡇⢸⣿⣧⣿⡟⠁⢿⣿⣿⣿⡄⠀⠀⣸⣿⣿⠀⠙⣿⣿⣧⠀⠀⢸⣿⠁⢸⣿⣿⣿⢿⣿⡁
⠀⢸⣿⣿⣿⡇⠀⢸⣿⡇⢹⣿⣿⡆⠘⣿⡏⢿⣇⠀⠀⠻⣿⣿⣿⡀⣼⡟⣿⡇⠀⣸⣿⡿⠟⠀⠀⢸⣿⠀⣸⡿⠘⣿⡇⣿⡇
⠀⢸⣿⣿⡟⠀⠀⠈⣿⣧⠘⣿⣿⠅⠀⣿⡇⠘⠿⠄⠀⢀⢸⣿⡜⢿⡿⠀⠛⣀⣠⣿⡏⢀⠀⠀⠀⠸⠿⠀⠿⠃⠀⡘⠿⣿⡇
⢀⣾⣿⣿⠀⠀⠀⠀⠾⣿⠃⢿⣿⡄⠀⠉⣑⡀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠄⠀⠈⢄⣘⡀⠀⠀⠀⠀⠐⠀⠠⡀⠀⠀⠀⠀⠘⠛
⢨⣿⣿⡟⠀⠀⠀⠁⠈⠀⠀⣌⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢸⡟⠁
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⢋⣿⣿⣿⣿⣿⣿⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠐⢰⢶⣏⣰⣷⣿⢿⡇⠀⠀⠀⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣦⣘⣻⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣟⣿⣿⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣯⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⢿⣿⣿⣿⣿⡻⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡿⢿⢿⣯⣿⣇⣛⣶⡏⠂⠈⢻⣿⣿⣿⣿⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣃⣴⣿⡦⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣮⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⡴⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠉⠤⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢚⠇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⠆⠃⠀⠀⠀⢸⣿⠻⣿⣦⣀⠀⢸⠀⡈⠀⣠⡆⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⠻⢿⣿⣶⣶⣶⡾⢿⣿⣤⣤⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⢠⣿⡇⠀⢰⡖⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠴⠒⠚⠛⠷⢤⣀⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢹⣷⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡞⠉⢷⡾⡁⢀⣤⣤⣄⡀⢠⣟⢹⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣁⣀⣠⡾⠿⠶⠶⢤⣸⣿⣝⢦⡀⠀⢠⡇⠀⠈⢿⣢⡟⢩⡴⠛⠛⠉⠉⣿⣇⠀⢸⣄⣀
⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⢀⣀⠀⠀⠈⢻⣿⠀⠹⣆⣸⠁⠀⠀⠘⣿⣧⡀⣷⣠⡤⠀⠀⡿⣿⠖⠚⣇⡀
⠀⠀⢸⣃⢈⠄⠀⠀⠀⠀⢀⣿⣿⢤⡾⠛⠛⠻⣦⠀⠀⢻⣔⡀⠈⠛⠀⢠⡆⠀⠸⣎⠉⠛⠉⠀⢀⡼⢃⣽⣀⣤⣽⠛⠛⠓⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣼⣿⣿⣿⡁⠀⠀⠀⠘⡇⠀⢸⡯⣿⣄⠀⠀⢸⢿⡄⠀⢹⣶⣶⣶⠾⠛⣽⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⢷⣄⣀⣠⠾⠁⠀⣸⣿⣿⠙⢷⣤⡿⠀⠿⠛⠛⠁⠀⢄⠀⠀⠹⣄⠀⠀⠀⠀⡐⠀⠆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠛⣿⣿⣷⠈⠉⠁⠀⢀⣴⣿⠿⠟⠃⠈⠉⠀⠀⠀⠀⠀⢀⠵⣎⠻⣍⠉⠉⠁⠀⠀⠀⠀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠞⠁⠙⢻⡿⠶⠶⠿⠛⠙⢿⣆⠀⠀⠀⠀⣠⡤⠤⠤⣄⢸⠀⠘⢦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠞⠁⣀⣠⣄⡀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠖⠀⠀⠀⠀⠀⠀⣿⢰⠏⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⠝⠀⠀⠀⠀⠀⢹⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠴⡆⣶⡆⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡔⢦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠐⡎⠗⠃⠃⠟⠃⠃⠷⠆⠀⠀⠀⠀⠀⠀⠀⠸⢯⠆⠀⢮⡵⠀⠰⠿⠆⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⢐⠁⠢
⠀⠀⠀⠀⠀⠀⡟⠚⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⣸⠈⣅
⠀⠀⠀⠀⠀⠀⣥⣤⣤⣤⣤⣦⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡶⠆⠺⣆⠀⠀⠀⠀⠀⠀⠈⢷⣻⣄⢠⠇⠀⠸⡆⢀⣤⢾⡆
⠀⠀⠀⠀⠀⢀⣿⣷⣤⣤⣥⣤⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⠀⠀⠀⠢⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡟⢀⠀⣴⣿⣿⡼⠛⠁
⠀⠀⠀⠀⠀⣾⠰⡦⢹⠉⠉⠉⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣤⣤⣤⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠈⠓⠖⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠶⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣀⣴⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣀⣤⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾⢻⣾⠳⠟⣶⡟⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠦⣽⢺⣿⡿⢏⣥⣾⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡇⢸⣿⢸⡆⣿⡇⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣸⡏⣼⣧⢻⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣑⣶⣾⣟⣛⣭⡵⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡿⣷⣿⣿⣾⢿⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢄⢀⣤⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠁⠛⠛⠈⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⣭⣇
⠀⠀⠀⠀⠀⠀⣠⣴⠿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⣤⣴⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠲⠶⡆
⠀⠀⠀⠀⠀⠸⠿⠗⠚⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠀⠀⠀⢰⠟⠻⡆⠀⠀⠀⢛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡝⠷⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠾⠛⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢠⠃⠀⠀⠐⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣮⡀⢀⡀⢀⣵⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢛⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠉⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠾⣏⠐⠀⠀⢀⣴⣶⠒⠂⠀⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠐⣤⣽⣦⣄⡼⢁⣹⣆⠐⠂⠀⢀⡀⠈⠑⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢊⣤⠤⠀⣨⣿⣿⡇⠸⠟⣹⠿⣧⡀⢨⡇⠀⠀⠉⣄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⣁⠀⠀⠲⠀⠈⢹⣿⣄⡸⠏⡀⠀⢽⠛⣦⢤⣀⣄⠈⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠸⢃⠔⠋⠀⠀⠀⠀⠀⣴⣏⣾⢋⣿⢟⣿⣶⣦⡸⢿⡼⢻⣽⢾⣆⠀⠀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⠃⠀⠀⠀⣠⠤⠔⠚⠉⢿⠏⠘⠛⠟⣣⣾⣋⣸⠿⣷⣸⣟⠛⠹⣞⠃⢈⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⠀⠀⡀⠀⣠⣄⣼⣿⡿⢛⠻⢾⣿⣼⢿⣄⣰⢟⣿⠻⣃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⠛⣁⠀⠰⠀⣤⢤⣿⡟⠉⠐⢥⡴⠆⢙⠱⡟⢻⣯⡿⢋⡇⠙⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⢀⣀⡀⠉⢀⠀⢠⠄⡼⠟⡀⠀⠀⠐⠃⠤⠙⠀⠸⢿⣿⡄⡾⢿⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣿⣿⣧⠔⠀⠠⠂⣠⣀⣤⣤⣤⣤⣄⣙⣇⠸⣿⡿⠁⠈⣧⢀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣿⣿⣿⣿⣿⣷⣮⡄⠁⣴⣿⣿⣿⣿⣿⣿⣷⣿⣟⡀⠻⣟⠀⠓⡿⢠⣀⠈⡇
⠀⠀⠀⠀⢀⣀⠠⠖⣐⣼⠟⣡⣿⣿⣿⣿⠿⣿⣿⣿⣿⣆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣯⢰⣦⡇⠬⠋⠀⡇
⠀⢀⡴⢈⣩⣵⣶⣿⣿⡟⠀⠈⠿⠛⡉⢁⣾⣿⣿⣿⣿⣿⠄⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⠘⢛⣴⡄⣠⢰⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣧⣾⣿⣿⣿⣿⣿⣿⣀⣩⣽⣦⣤⣍⣉⣉⣉⣉⣡⣀⣼⣿⣿⣿⣿⣿⣷⣿⣶⣾⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⢉⣙⣢⡖⣉⣁⣉⣑⣶⣉⣡⡍⠻⢦⣥⣤⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣹
⠀⠸⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠟⡉⣨⣛⣭⣿⣿⣷⣿⣿⣿⡟⣩⠦⠿⠏⠉⢠⡟⢥⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠘⠻⣿⣿⠛⠿⣿⣿⢿⣿⡇⠉⠐⣶⣠⣼⣿⢋⠳⣤⡴⠋⠋⠙⢋⣴⠆⠀⢀⡎⢀⡿⢏⡱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢱⠴⠁⣿⣿⡦⣄⣿⣿⣾⡟⢹⠀⠀⠀⢩⣽⣮⠀⢿⠛⣿⠃⠀⣰⢿⡇⢀⠀⢸⠃⢨⠴⠋⡑⣯⣿⢾⣿⣿⣿⣿
⠀⠀⠘⣇⣌⡻⣿⡄⡏⣿⣿⣿⡇⠀⣇⢈⡀⠠⣏⠳⡀⠀⢱⡇⠀⣰⣿⣿⠄⢸⠀⣾⠀⢸⣣⠆⢀⠊⢉⠣⣿⡟⣿⠃
⠀⠀⠀⢹⡎⣷⣿⡟⢃⣽⣿⣿⠇⠀⢿⠀⡇⠀⠟⢿⡅⢶⢈⠁⡄⣿⣩⢿⠀⠘⠠⡇⠀⠈⣥⠊⠿⢠⣎⠐⣻⣡⡏
⠀⠀⠀⠀⢳⡌⢿⠟⠛⡼⢟⠼⠀⠀⠞⠀⠁⢠⣇⣰⡄⠀⠀⠀⡆⣿⣽⣾⠂⠀⢸⡇⠀⢰⡆⣠⡇⡀⠼⢸⣿⠏
⠀⠀⠀⠀⠀⠹⣄⠀⠤⣑⣄⠘⡄⠀⣷⠀⠀⢈⡟⢹⡧⢶⠀⠀⠀⢿⣉⢻⡁⠀⣼⢀⡄⠰⠃⢼⠇⢻⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣖⡈⢱⣏⠳⡀⠨⠀⠠⠄⢿⡎⠓⠀⠘⠡⠠⡞⢛⡺⠁⢸⠟⡞⢠⣧⢴⣏⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⢒⣤⠹⢆⠀⠈⢣⡶⠇⠰⠆⢾⡟⣾⠃⢠⡆⠀⠀⣺⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠚⠷⠒⠾⣿⠿⠿⣾⠿⣿⡶⠲⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠆⢀⠁⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡴⠶⣿⢿⢃⣹⠷⢆⣙⣿⢦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣤⡰⣾⣿⣿⡻⣏⠿⢿⣿⡿⣷⣬⣧⣰⣿⣄⣠⣸⣥⣥⣤⣾⢿⡟⢿⡹⢿⣭⢿⡗⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⣾⣿⣯⣤⣄⣸⣥⣮⣿⣿⣾⣿⣺⣷⣾⣷⠖⢿⣤⢻⣝⠒⢃⣀⣥⣈⣹⣿⣷⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠫⠿⣿⣿⣏⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢀⠀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣿⣿⠿⠛⠁⠀⠀⠀⠈⠻⢿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠟⠋⠉⠉⠉⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠒⠉⠉⠙⠛⠛⠿⡶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⣿⠗⠆⠀⠀⠀⠈⡄⠀⡎⠀⠀⠀⠀⣠⡶⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠙⠿⠷⠃⠀⠀⠀⢰⠀⠀⢁⠀⠀⠀⠀⠙⠿⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠻⠶⠶⠶⠴⠮⠀⠀⠀⠀⠀⠀⠉⠦⠤⠶⠶⠶⠶⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⢶⣦⣤⣤⣤⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡤⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠻⠿⠧⠤⠤⠤⠄⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⣤⡀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠒⣶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠥⠀⠀⣈⡹⣿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣿⢱⢀⠇⣠⠠⡉⠂⢶⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⢁⣀⣀⡁⠀⠙⠑⠁⣾⣴⣷⣿⣃⡤⠖⠲⢶⣶⣦⣤⡀⠀⠀⢼⣹⣿⣏⢹⠷⢦⣶⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣬⣭⣵⣶⠖⠀⠙⡿⣷⠭⡉⢁⠀⠀⣀⠘⠛⠻⢿⣦⠖⠛⢏⠉⠩⠷⠦⠀⣿⣛⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠼⡇⠀⢻⣀⣩⠜⠁⠀⢠⠞⠁⠹⠶⡄⠂⠀⣀⣝⢳⡄⠀⣰⢣⣎⠱⢸⠇⢠⡒⢢⠀⣻⢛⡟⣏⣿⡃
⠀⠀⠀⠀⠀⠀⠀⣇⠀⣳⡀⠀⠀⠀⠀⠀⡰⠃⢸⡒⠢⣀⠁⠀⠘⢟⠸⠀⠁⢾⠇⠈⠉⡖⡁⠀⠘⠓⠁⠀⢹⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠓⢦⡀⠀⣰⠁⢠⠿⢿⣷⣌⠳⣀⣦⠀⠁⠀⣠⣾⠐⣯⠤⠤⠤⠤⢬⡶⠶⠂⠈⠋⡿⠛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠂⡏⠀⢺⣀⣉⣉⡽⠛⠋⠉⠁⠀⠀⠙⢿⠀⠸⣿⡿⢛⣻⠏⠀⠀⠀⠀⢠⡏⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢦⢻⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠄⠰⠋⣸⣆⠀⠳⡭⠔⠁⠀⠀⠀⠀⣠⡟⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠖⠋⢹⣹⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡠⠤⢚⠀⠀⠀⠀⠈⠁⠀⠉⠳⢦⡀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣾⡅⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣤⣤⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠟⢷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠗⠛⠁

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄⠢⡀
⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⡴⠴⠀⠂⡀⠀⣼⠀⠀⠀⠀⠈⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠚⠀⠐⢂⠀⢀⠀⠀⠀⠄⠰⠀⠀⠑⠀⡠⠀⠀⠀⠀⠠⠀⢀⠀⠖⠀⠎⠀⡠⠀⠀⣄
⠀⠐⢐⠀⠀⡀⢀⠀⠀⠄⠀⡀⠀⠈⠀⠤⠀⠁⠀⠄⠐⠀⠁⠐⠇⠊⠀⢅⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠂⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠑⢀⠀⠘⠃⡄⠀⠀⠄⠢⠀⠀⠀⠀⢀⡄⠀⠀⣀⣤⢀⠀⢀⠘
⠀⠠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⢀⡀⠐⠀⠀⡁⢀⣀⡄⠤⠀⠀⠀⠀⠁⠀⠀⠀⢀⢀
⠀⠁⠀⠀⠀⠄⠄⢀⠀⡎⠀⠠⠀⠀⠗⠤⠤⠊⠉⠀⠈⠑⠁⠐⠿⡧⠀⠀⠀⠉⢈⣁⣀⠀⠀⡠⠊⠀⠀⠀⠰⠀⠲⡇⠐
⠀⠀⠤⡀⠀⠀⡀⠀⠀⢀⠀⠀⠁⠀⠄⠀⢀⠐⠀⠔⠀⠀⠄⠀⠀⠀⢠⢰⠀⣴⣿⣿⣟⣷⡀⢀⡀⠀⠈⠀⠀⡀⠀⠐
⠀⠁⠈⡤⠀⠀⠮⠀⠃⠀⠀⠀⡄⢠⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠀⠻⣿⣿⣿⡿⠁⠠⢅⠀⠀⠀⠄⠀⡀
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠓⠀⠀⠄⡘⠀⠀⠀⢀⠡⠄⠐⢁
⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠰⠀⣀⠔⠂⠤⠤⠊⠉⠀⠀⠄
⠀⡃⠄⠀⠀⢀⠄⠊⠌⢀⡟⢀⠟⠛⠉⢹⡇⠀⠀⠀⠄⠁⠠⠀⠂⠀⠀⠈⠐⠂⠁⠀⠀⣐⡄⠀⠐⠂⠀⠂⠀⠘⢆⠉⣀
⠀⡌⠃⠀⠀⠀⠀⠀⠱⠈⢷⣤⣤⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠊⠀⠎
⠈⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠦⠀⡀⠀⠀⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⠃⠂⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠂⠠⠈⠀⢠⢠⠀⣈⣀⣀
⠀⠀⠀⡀⠈⠀⠀⠀⠒⠆⢀⠀⠙⠀⠂⠀⠀⠀⠁⠁⠀⠀⠀⠀⠔⢀⢀⠀⢈⣠⣶⠄⠀⠀⠀⠀⠀⠀⠈⠁⣼⡟⠛⢻⣷
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣻⣏⣤⢸⣠⡦⢰⡖⢠⠰⡄⠻⣿⣶⣿⠟
⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⡆⠀⠀⠐⠀⢀⠀⠀⣀⠈⠀⡐⠀⠀⠀⠀⠄⢸⠿⢯⡜⢣⡿⢠⠟⠀⠄⠐⠀⠀⠋
⠀⣀⠁⠤⠤⡀⠀⠐⠆⠀⠀⠀⡇⠸⣾⠇⠀⠀⠙⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⡀⠀⠐⠀⠈⠀⠀⠀⠃
⠀⠔⠀⠀⠀⠣⠈⠀⣀⠀⠀⠀⢃⠀⠈⠀⠀⢀⢽⠀⡆⡀⠀⠀⠐⣿⠗⠀⠀⠀⠀⠃⠀⠀⠠⠊⠀⠀⠃⠈⠀⠀⠀⠀⠂
⠀⢀⡀⣀⠀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠰⠃⢀⠉⠈⠁⢉⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⢀⠀⠀⠀⠀⠠
⠀⠈⣡⢉⡌⠃⠀⠈⠀⠠⠄⠀⠔⠐⠀⠀⠀⠀⠀⣦⠄⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⣠⠂⢠⠆
⠀⠀⠀⢠⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⣄⠀⠉⠁⡄⠊
⠀⠄⠀⠀⠣⡀⠄⣀⠀⠀⢀⠀⡠⠀⠀⠈⠒⠀⠈⠉⠙⠆⢀
⠀⠀⠢⢈⠒⠵⠈⢁⡀⠆⠀⠘⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠸
⠀⠀⠄⠈⠁⡀⠐⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⡀⠂
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢹⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢂⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⣤⣤⣹⣄⣡⡀⠀⠤⠤⠤⠐⠒⠊⠉⠀⠀⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢂⠀⠀⠀⣀⠴⠃⠀⠉⠀⠀⠀⠀⡈⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢤⡤⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⢣⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠐⢀⠀⠀⠀⢀⠀⣠⡤⠖⠋⢠⣾⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢰⠀⠈⢋⡀⠈⠉⠁⠀⣀⣤⣾⣿⣿⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡦⢀⡀⠀⠀⠀⠠⠐⠁⠀⠀⠀⢸⠆⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡸
⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠙⢿⣿⠋⠀⠁⠀⠟⣻⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠑⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠒⠡⠤⠤⠚⡁⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⣏⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣀⣀⣀⠤⠶⢋⠀⠎⢉⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡳⠀⢀⡅⠀⠁⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠄⢀⣀⠀⠠⠖⠋⠀⠀⢁⢡⡞⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠙⡀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢦⣀⢀⠜⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠠⠐⣊⡤⠾⣍⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢉⣴⠃⣀⡞⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠘⣆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢆⠉⠉⠉⠉⠉⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠢⣀⠀⠀⠀⣰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⡙⠛⠛⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⡿⠿⠀⠀⠀⠰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡄⠀⠀⣶⠶⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠰⠶⢤⡀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠁⣀⠔⠀⠀⠀⡀
⢀⠁⠀⠀⠀⠈⠆⢀⠀⠀⠈⠄⠀⠀⠶⡿⣶⣤⣄⣀⣤⣀⣤⣴⣾⠟⠛⢫⠉⠀⡇⠀⠀⠀⣄⠸⢠⡄⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠈⠃⠛
⠸⡀⠀⢿⠂⠀⡇⠸⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠋⠙⢍⠁⠈⠁⠀⢀⠎⠀⠀⢇⠀⠀⠘⠿⢠⠘⠏⠀⠀⡌⠀⠀⠠⠀⠀⠀⣠⠀⡀⠀⠀⠀⢠
⠀⠣⡀⠀⠀⡠⠃⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠐⠂⠁⠀⠀⠀⠈⠢⡀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠠⠀⠀⠛⠌⠁⠀⠀⠐
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⡀⠀⠀⠠⠞⠉⠁⠀⠀⠀⠀⠁
⠀⢀⣠⡴⠛⠉⢠⡐⠉⠳⢤⣀⠀⠀⠀⠤⠴⢤⣤⣀⠀⠀⠌⠀⠀⠀⠁⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠊⠀⠀⠀⠀⠋⠐
⠐⣏⠁⢠⣤⠀⠀⡇⢤⡄⠀⠀⡍⠀⠀⠀⠀⠀⠈⠹⣷⢸⠀⢠⣄⠀⠀⠁⠀⠀⠆⠀⠀⢀⣶⡆⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⢸
⠀⢩⠄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠇⠀⠀⠰⠦⠦⢌⣦⠈⠀⢆⠸⠟⠀⢀⠂⠀⠀⠡⡀⠀⠀⢈⡁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃
⠀⠀⠢⣐⠀⢀⠖⠉⠠⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠙⠂⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠘⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠞⠃⣀⠉⠙⠷⢤⣀
⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⢡⡴⣦⣤⠘⠁⣴⢷⠶⠐⡀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢰⡆⢶⠀⠀⠀⢸⠁
⠀⠠⠀⠀⠀⢀⡆⠀⠀⠀⠠⡶⠀⠀⠀⠀⡇⣠⣽⣷⡅⢀⠀⣙⣿⣿⠆⡇⠀⠀⡤⠀⠠⠄⠀⡄⠠⠄⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⣠⠀⠀⠀⠀⠌
⠀⠀⠃⠀⠀⢀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⡀⠉⠋⢁⠊⠡⠉⠈⠁⡐⠁⠀⠀⠐⠀⠀⠀⡸⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠰⢤
⠀⢠⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢠⠀⠀⠉⠭⢙⠆⠀⠀⠀⢈⡥⠴⠂⣄⠀⢀⠬⠍⣇⠀⠀⠀⠀⢀⠴⠀⠰⠄
⠀⠉⠀⠀⠀⠀⠀⠀⠺⠿⠻⠷⠆⠀⠀⠀⡌⢀⢒⡓⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣤⣶⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠄
⡆⠀⠀⠲⠶⠀⢐⠀⠀⠶⠶⠒⠀⡆⠀⠀⡇⠉⠙⠉⠀⠄⡆⢰⠋⠀⣰⠆⠀⠈⡇⠀⠀⠘⠿⢸⠂⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠈⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⠴⠚⠀⠀⠀⠹⡄⠬⡭⠔⣴⣎⠛⠯⢚⡟⠀⠀⠀⢳⡄⠀⠀⢀⡼⢄⡀⢀⡴⠇⠀⠀⠀⢃⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠂⠰⠋⠀⠈⠙⠉⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠛⠛⠉⠙⠛⠻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠻⠟⠛⠉⠁⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠶⠶⠶⠶⠶⣼⣆⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⢀⣼⣥⠶⠶⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⣽⣿⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣷⠀⠀⠀⠈⠛⠋⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⢿⣖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⡿⠿⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⠛⠛⠛⠋⠉⣉⠥⢊⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⣀⣀⠤⠤⠶⠚⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉

⡶⣶⢶⠶⠶⠶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠁⢀⣀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠘⣟⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠫⠟⠀⠀⢴⠾⠓⠀⠠⠄⢸⡇⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠛⠃⡕⣧⠀⡇⠀⠀⠘⢿⣿⣫⣭⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⢻⣟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠂⡀⠈⠛⠋⣠⣼⣯⠻⠏⠊⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠏⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⣐⠈⠙⠛⠛⠀⠀⢀⣴⣶⣶⣤⣀⣂⠀⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢈⣀⡤⠀⠲⠿⣟⠉⢻⣿⣿⣿⡀⠘⢿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣋⣝⠀⢴⣷⣾⣿⡷⠀⣿⣿⣿⣧⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠂⠈⠋⢛⣫⣤⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠰⠂⠀⠾⠛⠛⠛⠉⠉⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⠀⠀⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣄⡀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⢹⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⣄⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢰⠸⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣀⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠁⢀⣱⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠠⠂⠀⠀⠄⠀⠐⠠⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠐⠀⠀⠤⠀⠠⢄⠀⠀⠀⢀⡀⢀⠀⠈⢆⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠠⠀⡇⠀⠇⠀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠚⠃⠐⠄⠀⠔⠻⠎⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠤⠠⠄⣠⠄⠀⠠⢤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢰⠀⠀⠈⠂⡀⠀⠀⣀⠜⠁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠙⠉⠛⠁⠀⠚⠃⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠐⣤⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣠⠈⠀⠰⢀⣄⠒⠈⢧⡀⠀⠀⣤⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢱⡤⠂⠛⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠠⡤⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠀⠁⠆⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠂⠉⠁⠀⡄⠀⡃⠀⠠⡆⠀⠀⢀⡀⠀⠈⠁⣴⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠜⠀⢃⠀⠀⠈⠀⢀⡀⠀⠠⡄⠋⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠤⠔⠊⠀⠀⠀⠈⠄⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠁⣸⠇⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⡀⢈⡀⡾⠀⠸⠀⠀⢀⣀⠰⠈⠁⠀⠀⠀⠀⠇⠀⢠⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠠⣤⠒⠉⠀⠀⠃⠀⠀⢆⡰⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡠⠤⠤⢀⠀⠀⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠀⠀⡇⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠸⠀⠈⠢⣀⠀⣀⡴⠀⠀⠀⠀⠀⡀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡰⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣟
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⣀⣀⠀⠈⠐⠂⠐⠀⠀⠀⠀⠉⠐⠒⣶⡄⠒⠀⡛⠀⢀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠐⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠀⠁⠀⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀
⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡱⠆⢾⣋⣉⣉⣉⣉⣉⣉⣉⣙⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣤⡀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⠀⠀⢀⡀⣾⠇⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⠀⠀⢿⡄⠛⣳⣄⣀⣀⠀⠀⡿⢠⠀⠀⠀⣀⢀⣤⣻⢁⡼⠀⠀⠀⠀⠀⢿⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠀⠀⠀⠀⠈⢻⣆⠻⠙⠛⢉⣶⣤⡇⢸⣤⣴⠻⠟⠉⣿⠁⣼⢃⣷⡀⠀⠀⠀⣴⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠀⢤⠀⢿⣦⡀⠀⠀⠿⡙⠁⣿⡏⣹⡄⢀⣀⣻⠾⠁⠈⣭⠀⢲⣄⡸⢈⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣷⠆⠀⠀⠙⠀⠀⠈⠛⠿⣷⣤⣿⠀⠀⣧⣿⣷⠞⠋⠁⠀⠀⠀⠁⠀⠀⣼⠃⣾⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⡇⢻⣶⢰⠄⣴⠀⠀⠀⠀⠀⠈⠉⢹⠀⠺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣴⠇⢰⡏⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣷⠈⢿⡎⠀⠘⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡘⠃⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠸⣇⠀⣀⣤⡴⠶⠶⠟⠛⢻⡿⢢⡾⢾⣿⠋⡉⠉⠙⠛⠲⠶⣴⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣿⣿⣋⠀⢀⡀⠀⠀⡔⢠⣄⡀⣤⠈⣥⠀⣴⡎⣠⠀⠐⠆⠘⠛⢶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠡⠄⠻⡄⠋⠀⠀⠀⢸⣆⠉⣩⡀⠀⢿⡀⢈⡄⠉⠣⢁⠀⠈⠑⠢⡀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⡀⠀⣀⣀⣄⠀⣤⣄⣄⡘⠆⠀⠀⡿⠃⠀⠼⠟⢀⣤⡶⠤⢄⠵⣦⣤⣤⡈⠳⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣦⣶⣶⣶⡠⣿⣵⣷⣿⣶⣿⣷⣾⣷⣿⣁⢀⠀⢀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢷⡈⢇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⡛⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⢁⣿⣿⣤⣿⣿⣽⣿⣿⣿⠈⢛⠛⠛⠻⠻⠿⢿⣿⡿⣿⠿⠘⠀⠈⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠹⡀⠀⠹⠛⠛⠉⠉⠀⠀⠀⡰⣾⣦⡄⢾⣷⢛⣏⢸⡿⢿⣿⠛⢀⡄⠹⣶⣶⠤⠀⢀⣁⠀⠀⢶⠟⢀⣰⡿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣸⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⣾⠓⣾⠏⣿⠀⢸⡇⠀⣿⠁⢘⣿⠀⢾⡇⣀⠀⠉⠱⢶⡄⠛⠰⠘⢿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣠⡿⠋⠀⠻⠶⠀⢢⠐⠇⢀⣀⡾⢷⡀⠸⣇⣸⣧⣸⡇⢀⣿⣤⣼⡇⢠⡾⠁⣿⡄⣼⢣⡌⠀⢉⡟⡀⠈⠛⢿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠰⠞⣠⣶⣤⡀⢸⠀⣄⣌⠛⢇⣼⣿⣶⣿⣿⣿⣿⡀⢨⣿⣿⣿⣷⣾⣿⣾⡁⣼⣇⠘⠀⢤⣿⣟⣹⣿⠀⢠⡆
⠀⠀⠀⠀⠀⠀⢸⡁⠐⣧⢀⣿⣧⣿⣿⣾⢀⣏⠹⣷⣿⣿⣿⣿⣿⣏⡙⣿⡇⢸⣿⠋⣹⣿⣿⣿⣿⣿⡁⣿⢇⣼⣿⣿⣿⣿⣿⡆⢰⣿⢀⡀
⠀⠀⠀⠀⠀⢤⡀⣿⣤⣨⣿⣿⣿⣿⣿⣿⣦⣙⣴⣿⣿⣿⣿⣿⠋⠀⢉⣿⠇⠈⣿⣾⡉⠈⠹⣿⣿⢿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡧⢜⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⡛⢿⣿⣿⣿⡿⢿⣿⣿⡟⣼⣿⡅⠈⢛⣥⣶⣿⣟⠁⣶⣾⣧⢸⣿⣿⣟⠉⢁⣼⣿⣻⣿⣿⡿⢻⣿⣟⣫⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣌⠙⣿⡅⠸⣿⣿⣿⣿⣇⣿⣷⠾⠛⠉⣿⣿⣿⢫⣽⢿⣿⣿⣯⣉⡛⣿⣿⣿⣿⣿⣿⡇⢰⣿⠿⠿⠇⢾⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠚⠛⠛⣻⣻⣧⠀⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣡⣿⣿⣀⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⡀⣼⣿⣶⣶⢤⡴⠍⣀⡙⢦⡀
⠀⠀⠀⠀⣠⣿⠷⠶⠂⡿⠛⢻⣿⣿⡆⠀⣿⣿⣻⣿⣿⣿⣿⣿⢿⠉⢻⠀⢠⠄⢈⡏⠉⡿⠋⢿⣿⣿⣿⣿⣿⠋⠄⣿⣿⢿⡟⢨⡄⠀⠀⠁⠻⣷
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠐⣶⠟⣡⣿⣿⡀⠹⣿⣯⣿⠛⠋⠙⣇⣸⣧⡼⢦⣼⠧⣾⣧⣼⡇⢰⣏⣿⠛⣿⠻⡿⠀⣼⢿⣿⠈⠱⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠐⠊⠳⣈⠿⣧⠀⠹⡁⠙⣦⣼⠿⠋⠆⠁⠀⠀⢻⡀⢿⠀⢸⠿⣟⢻⡿⠿⣧⡴⢡⠞⠹⠀⣟⡜⠃⠲⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢻⠂⢹⡇⠀⠉⡄⠉⠃⠀⠀⡐⠀⡄⠀⢼⡇⠀⢀⠃⠀⠃⠀⢠⡇⢸⡇⠀⣤⣤⠀⢻⡇⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⠣⠔⠀⢠⣦⠀⢠⣼⠛⠀⠃⠀⠀⠀⣼⠈⢣⡀⣠⣧⡞⢠⡿⠷⠀⠹⠿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠀⠸⠋⠀⠀⠀⣠⣤⡄⠄⠐⠀⠹⠀⠀⠀⠛⠹⡿⠀⣠⡄⢰⡗⠠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠀⠀⠐⠛⠗⠀⢁⣀⣠⣤⡤⠴⠶⣶⣦⣤⣤⣤⠀⠸⠉⠀⠀⡠⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⣀⣀⣤⠶⠛⠉⢿⣿⣷⠀⣾⣿⣿⡟⠉⢹⡇⠶⢶⣿⣁⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠰⠟⠋⠀⠀⣠⡶⢛⡿⣿⡿⠂⢻⣿⠉⡏⠑⢦⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠉⢀⣶⠐⠉⢻⠃⣿⠁⠸⣷⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⠁⠀⢀⣿⠀⣿⣄⠀⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠐⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡖⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣄⢀⣴⣾⣿⣿⣶⣄
⠀⣠⣶⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡏⣶
⠀⠀⠀⠀⢠⣤⣤⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⠞⠷
⠀⠀⠀⠀⠀⠐⢻⣿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠂⠉⠈⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠁⠀⢀⠀⠤⠤⣴⣚⣃⡒⠀⠀⠀⠀⠀⡀⣤⣶⡾⣿⣟⣙⠳⣀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠃⠀⡺⠟⠒⠒⠒⠛⢿⣉⠀⠢⡄⠀⠐⢋⡜⠩⠀⠃⢰⠈⠆⠑⣌⢳⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢧⡀⣀⡢⢂⠎⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⠃⠀⢀⡞⠠⠀⠀⡀⢸⠀⠠⠀⠈⡆⢷⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⡎⠀⠀⠀⠀⠀⠀⠀⡰⠄⣄⠀⠱⠀⠀⠃⠘⠀⠀⠁⠀⠀⠀⠀⠀⠂⢸⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡁⠀⠀⠀⣠⣾⡃⠜⠁⠀⠈⢷⠀⢃⠰⡇⠀⠤⠄⠀⠀⠠⠄⠀⠀⠀⡞⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⣀⢀⠀⠀⠇⢸⠐⠷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠔⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣤⣀⠒⠛⠃⠀⠀⠉⠀⢀⠜⢀⠆⠀⠀⠈⠐⢠⣤⣀⣤⡄⠖⠁⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡸⠟⠁⠀⠀⠀⠀⢀⠠⠴⡿⠦⢄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⡖⠀⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠆⠙⠲⣄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⠉⡄⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⣤⠀⠱⠀⠀⠀⡈⢢⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⡇⠀⠱⡀⣸⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡘⠁⠀⢰⠀⠀⠀⠁⠀⠀⡹⢻⠀⠀⠀⠀⠀⠀⡔⠋⠙⢦⠘⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠁⣠⠶⠚⠤⠀⠀⠰⠴⠊⠀⢸⠀⠀⠀⠀⠀⠀⠱⠤⡤⠚⠀⠀⡅⠈⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⡠⠃⠀⠉⡄⣀⠴⠁⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣹⡄⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠁⠀⢨⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⣏⣀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇⠀⢸⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡦⠀⠀⠃⠀⢸⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠉⠉⢹⠋⠉⠁⠀⠀⠀⢸⠀⠀⢸⠀⠇⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⢠⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⠀⠀⠀⠀⠀⣼⠤⡀⠀⠀⠀⢀⠁⠀⠀⡇⠀⠸⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠀⠀⡇⠀⢅⠀⠀⠀⠘⢲⡖⠶⠃⠀⠐⠖⢰⡎⡁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠐⠁⣇⠀⣦⡀⠀⠀⠸⠱⣄⣀⣥⣤⣄⣉⣈⣉⣠⣤⣤⣤⣀⠁⠀⠀⠲⠄⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⡀

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