Bible Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⢀⣰⣟⡷⣠⠅⣤⡠⠀⣤⣀⠀⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠙⠛⠙⠊⠙⢈⠉⠁⠁⠋⠀⠀⠀⠈⣡⡾⣿⣟⡴⢠⢄⡄⢀
⠀⠀⠀⠀⠀⡠⢜⣶⡤⠊⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠐⠀⠉⠝⡉⠉⠈⠀⠁⠎⠰⠅⡴⠊⠁⣠⣦
⠀⠀⠀⠀⠤⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢜⠟⠏⠤⣩⣷
⠀⠀⠀⠀⣠⠞⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡳⣣⣯⢪⣾⣿⠟
⠀⢀⣠⡿⠁⠀⠀⠀⠉⠈⠋⠛⠛⠛⠳⠖⠒⠦⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠑⠁⣵⣿⡷⠃
⣴⣿⣿⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢠⣶⣦⢤⠀⢈⠷⠒⠒⠲⢶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣼⣍⡼⣠⣾⣿⠟⣵⡄⡀
⠀⠉⠉⠉⠋⠛⠛⠛⠻⠿⠿⢿⢏⣾⡟⣡⣿⣶⣿⣦⢀⣀⠀⠀⠀⠉⠻⠗⢈⢵⣒⣦⠤⣲⡟⣩⠇⣴⣿⣽⣻⡞⢃⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢠⣿⡻⣿⣥⣿⣻⣿⣿⣿⣶⣦⣄⣀⠉⠈⠀⠈⠹⡏⣼⢯⣾⣯⠞⠉⠉⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠬⢡⡟⠉⠛⠛⠛⠋⠁⠀⠈⠉⠙⠓⠾⢯⣟⣳⡶⣤⣤⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠯⣯⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠄⠒⠒⠒⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢰⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⡀⠡⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡆⠘⠀⢄⡀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡀⠀⢰⠂⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠄⠀⠥⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠀⠐⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣁⣒⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢣⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣥⣴⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁

⢾⣿⡿⢉⠋⠩⠭⠩⠿⠉⠉⠛⠛⢿⣛⣿⣿⠛⠛⢻⣿⣿⡿⢿⡟⢛⣿⠛⡉⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⡄⢀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠉⠙⢇⠒⢠⡤⠀⠛⠶⡾⢿⡿⠻⠟⡃⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠈⠁⠀⠀⠁⠀⠉⠐⠀⠀⠀⠈⠿⠿⡿⢿⡿⠿⢿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠴⣤⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠒⠂⠐⠛⠃⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⢸⣇
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣦⡀⠀⣀⣀⣀⡘⠀⠛⠟⢿⣄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⣷⣯⣿⣿⠁⣀⣁⡴⣾⣿⣧⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⠛⣹⣿⣼⣿⣿⣿⣶⣿⣯⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠐⠒⠒⠒⠒⠚⠒⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠹⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⢀⢀
⠄⠀⠀⠤⠀⢄⣀⠈⢡⠀⠀⠀⢠⠀⠀⢸⣯⣿⣿⣇⢸⠀⢹⣿⠀⠇⢻⡿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⠿
⢀⡀⣠⣘⣿⡻⣿⠻⣽⡏⠁⣼⠟⣡⣴⣿⣿⣿⣿⡁⠀⡇⢸⡏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⠐⠈⠁⢀⡀⠶⠚⠛⠉⠀⠀⠀⣴
⣏⡉⣿⣿⣿⠅⢈⠷⠚⠲⠀⣤⣾⣿⣿⣿⣿⣿⣿⣇⣀⣿⡾⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠂⢁⣴⣶⣿⠿⣿⣶⠂⠨⠿
⣟⡊⠿⢿⣧⣤⠀⣄⠗⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡧⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⠴⠟⠛⠉⠁⠐⣀⣀⣠⡄⢒
⣿⣧⠄⠈⠛⣿⠆⡍⠹⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣧⢨⣷⢈⣿⣿⣿⣷⣿⣿⣿⠯⠀⠀⠠⠤⠐⠀⠀⠀⠀⠀⣀⠴⡛
⡟⠁⠀⠀⠀⠹⠗⢹⠇⣴⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⠙⣿⡀⢻⣽⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠀⠀⠠⣽⣤⣤⣛⣻⣾
⡴⡗⣱⠄⠀⠀⠂⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠂⠀⠹⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡒⢤⣤⠤⠀⣤⣤⠄⣈⣿⣿
⣿⣦⠈⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠘⠹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣼⣾⣭⣀⣨⡭⢽
⣿⡁⠀⠀⠀⠀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⠠⠀⠀⠀⢀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡆⢠⠀⠄⢠⣈⢅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠐⠤⠄⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⠓⣋⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⣀⣉⡛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⢀⠒⠂⠀⠀⢈⣉⣀⣤⣶⣶⣌⣛⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣯⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣻⡿⣻⢿⣿⣝⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣿⣻⣿⣿⣿⣿⣿⣿⠿⠻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠋⠁⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣠⣤⣴⣾⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠖⢒⣒⡚⠉⣩⠭⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⡉⠀⠀⢐⠆⠈⠉⠉⠀⠀⠀⠀⠀⠙⠲⢄⠊⢧
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠐⡀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠰⣾⡀⢀⡠⢴⣄⣌⠀⠀⠀⠀⠀⠀⠀⡴⠒⠢⠀⠀⠀⠀⠀⠢⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣷⢪⠊⠁⣨⣆⠀⠀⠀⠀⠀⢀⡠⠜⠀⠀⠉⠉⡀⠀⠀⠀⠡⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢀⠪⠔⠉⢂⠀⠀⠀⠀⠀⠂⢀⠠⠀⠀⢖⠁⠀⠀⠀⠀⠐⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢆⠀⠀⠀⠀⠀⠐⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⠀⠀⣡⡀⠀⠀⠀⠀⠀⠀⠑⠒⠈⠀⠀⠀⠀⠀⠀⠀⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⢊⢅⡽⡔⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡱⡫⠊⠁⢈⢄⣈⡁⣐⠆⠠⠤⠤⠤⠤⠤⠀⠰⣍⠀⠘⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡁⣠⠀⢴⡷⠒⠐⠒⠒⢀⣽⣿⣿⣶⣒⣂⣠⣤⡥⠍⠩⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠜⣼⣭⣩⣭⣱⡶⠒⢠⡤⠤⠄⢀⣀⣤⣀⣶⣀⣛⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡭⠬⠭⠷⠶⠖⠚⠛⠛⠊⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣤⣴⣿⣯⠀⠤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣰⠶⣦⡄⠀⣴⣦⠀⣠⣶⣦⠀⣠⣶⠶⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⣶⣠⠴⠀⠀⠀⢸⣿⡇⢀⣴⣟⠀⣿⡇⢰⣿⡟⣴⠛⣿⠏⣼⣿⠃⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠛⠘⠃⠀⠀⠀⣾⣿⠃⢸⡟⠙⢶⣿⠧⣾⣿⡟⠁⠀⠈⢰⣿⠇⠀⣰⣿⡏⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠸⣷⣤⠿⠃⠰⣿⡟⠀⠀⠀⠀⢸⣿⣤⡾⢻⣿⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⢓⣾⡿⢀⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣉⣉⣀⣀⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠁⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠀⠀⣤⣤⣤⡤⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⠘⠃⠸⠃⠀⠛⠘⠘⠃⠹⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣆⣾⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠾⠟⢛⣿⡟⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠷⠟⢻⣿⡟⣹⣿⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣿⣆⠀⠀⣸⡿⠀⣸⣿⢠⢾⡿⢢⣾⢋⡷⢠⣿⢢⢾⣿⠂⢀⣴⡟⠉⡇⠀⣾⡿⢀⣿⠏⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⢻⣿⡀⢠⣿⠃⢠⣿⣿⠃⠈⠁⣿⡟⠛⢁⣿⡿⠁⣿⠏⢀⣾⡟⠀⣼⡿⢰⣿⠇⢸⣿⠚⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣾⡿⠁⢸⣏⡴⢻⡿⠁⠀⠀⠀⢿⣤⡴⢺⣿⠁⠸⣿⡤⠛⣿⣁⢼⣿⠇⣾⡟⠀⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠁⣼⡿⠀⣿⡇⢀⡍⠀⠀⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠈⠓⠛⠁⠀⠀⠈⠳⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠤⣄⡤⢿⠀⢠⡤⡤⣤⢤⡀⡄⠀⠀⠀⢠⡄⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠘⠰⠋⠓⠚⠀⠘⠃⠃⠛⠈⠷⠀⠀⠀⠀⠸⣆⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠙⣦⠀⢠⣶⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⠀⣾⣿⠁⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠉⠁⢸⣿⠃⣠⣤⡀⢀⣬⡁⠀⣠⣤⢄⠀⣾⡿⠀⠀⣤⣤⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠈⣿⣷⡀⠀⣿⣏⡜⣿⡿⠁⣸⣿⠁⢰⣿⣡⠞⢰⣿⠇⢀⣾⡏⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⣿⡇⢸⣿⡟⢠⣿⠃⣠⣿⠇⣠⣿⡏⠀⢀⣾⡟⢀⣼⡟⠀⣠⣿⠃⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣤⣿⠟⠀⢸⡟⠀⠈⠿⠞⠉⢿⠾⠃⠹⠷⠞⠋⠻⡷⠏⠙⠷⠞⠙⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢤⣠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠎⠙⣧⣤⣿⢸⡇⣟⢻⢹⡇⠀⣸⡃⣜⢻⠌⡁⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⣀⠀⠉⠉⠛⠲⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣴⠋⠀⠀⢀⡀⠀⠀⠀⣰⠏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⡾⠁⠀⠀⣴⠋⠉⠛⠶⠶⠏⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⠏⠀⠀⠀⠀⠛⠶⢤⣤⡀⠀⠀⠀⠀⠈⠉⠙⠛⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⢏⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⠶⠤⣄⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⢃⠎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣰⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢁⠎⠀⠀⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⠀⢀⡼⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⡀⢀⡾⢡⠎
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠠⠃⡀⢀⡞⣡⠏
⠀⠀⠀⠀⠀⠀⡼⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠁⡜⢠⡟⣰⠏
⠀⠀⠀⠀⠀⣸⢡⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠜⢠⠟⣰⠃
⠀⠀⠀⠀⠀⡇⡀⢰⢿⡉⠛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⠞⣠⠏⣼⠃
⠀⠀⠀⠀⠀⡇⠃⢺⢸⡀⠁⠒⠤⣄⣉⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⠎⣰⠏⡼⠃
⠀⠀⠀⠀⠀⡇⠀⢸⣿⢀⡈⠁⠒⠀⠀⠉⠐⠀⠈⠉⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⠎⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠘⡄⠀⠿⣤⣀⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⢀⣾⢃⠔⢠⠏⣼⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠉⠛⠶⢤⣀⠀⠀⠀⣶⠲⠤⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⣠⡿⢃⠞⢠⠃⡼⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠀⠈⠉⠒⠄⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡁⠒⢿⢡⠊⢠⢃⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠢⢄⣈⠁⢻⠃⠠⢃⡾⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡇⠈⠑⠢⢤⣀⡀⠀⠀⠉⢸⡤⢁⡞⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣤⡄⠀⢀⡞⠁⠈⠐⠢⢄⣀⠉⠓⠦⣤⣘⣧⠟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠉⢸⡇⢀⡞⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡉⠉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠰⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⡀⠀⣄⠀⢠⣤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡾⠇⢸⢴⢠⠛⠀⡔⠀⢀⡦⣤⣄
⠀⠀⠀⢀⡀⠐⠀⠀⠀⠈⢸⠧⠀⠀⠀⡇⠀⢚⢸⣼⣄⣠⠇⠄⠀⡇⣏⠙⠀⣠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣟⣷⡯⠋⣙⠆⡘⠀⢀⣰⠹⠃⢀⡾⠁⢼⠦⡄⠁⠀⠀⠀⠀⢰⠀⠀⠁⠄⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠣⠞⠃⣠⠟⢿⡞⠀⣼⣿⠆⠀⢀⣾⣿⠃⢠⠟⠀⢀⡞⣀⠘⠀⠀⠀⠀⢀⡼⠁⠀⠆⡄⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡿⢠⡞⠀⣸⠋⣿⡭⢤⣚⣿⠇⢠⣿⣦⠵⠋⠠⣼⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⢤⠀⡀⠁⠴⢁⡀
⠀⠀⠀⠂⡠⠀⣀⣤⡄⢂⡄⠀⡾⢱⣾⠃⢰⣯⣼⠃⠀⢠⣯⡟⢠⡏⠉⠁⠄⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⢀⡾⢿⠖
⠀⠀⠀⠀⠀⣾⠁⡴⠀⣼⠁⢸⣧⡞⠁⠀⣾⡿⠁⠀⠀⢸⣿⠁⣿⣷⠈⣧⣠⠀⣰⠈⣧⡾⠀⢀⡦⠋⢠⡞⠻⡞⢁⡾⠀⢸⡿⠟⠁
⠀⠀⠀⠀⣼⢁⣼⠃⢰⠇⠀⢸⠛⠃⠀⢠⡟⣠⣆⣦⠀⡟⠀⢸⣿⠇⣠⣿⠇⣴⡟⢀⡿⠃⢠⣿⠃⢀⡟⠀⠀⠁⣼⠃⠀⠈⣇⢠⡀
⠀⠀⠀⠳⢬⡟⠩⠀⣿⢀⣆⢸⣄⣬⠀⢸⣷⣧⠭⠟⢠⣹⣦⡼⠛⣤⣽⠛⠶⣿⢵⣻⠁⠄⠿⠟⢠⣀⣻⣴⠂⣸⣏⢿⣄⠀⠘⡏⠳⣱⡆
⠀⠀⠀⠴⠲⠀⠀⠱⠾⠛⠁⠙⢙⠁⠀⢸⠏⣽⣾⢰⢗⢸⠻⢃⣿⣳⣟⣳⠷⣇⡸⣧⠀⠒⣄⡀⠀⠈⠁⠀⣰⠟⠁⣾⠁⠀⠀⢹⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠡⠦⣥⠴⠛⡼⡨⣼⠾⣯⠎⢁⣠⣤⡭⣽⣻⡇⠀⣼⣷⠏⢀⡴⠃⣰⠏⠤⠀⢿⠀⠐⠀⠀⣷
⠀⠀⠀⠀⣀⡀⢆⠐⡡⠚⠻⠋⢀⣴⣗⡖⠀⡴⢻⡞⠳⣯⠄⢣⡀⣿⠄⠠⠄⠰⡟⠀⣭⣿⡛⠂⢀⣠⡾⠃⠀⠀⠀⠈⢷⣁⣀⣼⣿⠂
⠀⠀⠀⠀⠁⠀⠈⢸⠓⡠⠁⢠⣾⠟⢃⠀⣸⢁⣾⠀⢼⢧⠀⠀⠈⠉⠀⣾⢳⡶⠁⢰⣿⣿⣿⡟⠋⠉⠈⠉⠀⠀⠀⢀⠀⠈⠙⠉
⠀⠀⠀⢠⡀⠀⢲⠓⠛⠀⣴⠛⠄⠀⠀⢀⣷⣞⡀⢷⢾⡟⠀⠐⠢⡄⠀⢹⡟⠂⠀⣾⠏⠉⠩⠤⣄⠈⠛⠀⠀⣴⢐⣿⠽⣎⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠐⠋⠀⠀⢰⡏⠀⠀⠀⠀⢸⣷⠛⠁⣸⣿⠇⣷⡏⢀⡿⢄⠀⠻⡇⠀⣿⢠⡆⠻⠆⣸⡗⢀⡀⠘⣗⢨⡇⢀⡎⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⢦⡄⠀⢸⠉⠀⠀⡴⠀⠸⡟⢆⣸⣿⡏⢠⣿⡅⢸⣇⡾⢣⠈⢻⣄⢻⣈⡀⠀⣴⡯⠆⡄⠈⠀⠉⠙⠛⠋⠀⠠⡀
⠀⠀⠀⠀⠀⢀⠀⣥⡀⠸⡄⠋⠄⠀⠤⠲⡇⢸⣿⢛⣣⣘⣿⣛⣾⠛⢱⡏⡄⢸⡏⡵⡖⠒⣏⠃⢹⣼⡏
⠀⠀⠀⠀⠀⠈⠡⣏⠽⡳⠒⢦⠟⣧⠀⠘⣇⢘⣻⡻⡅⡝⢾⢇⣤⢙⡞⠓⠁⣸⠳⠱⣍⠀⢻⡐⠛⠡⠘⠁⠀⠀⡸⠠⠐⠑⠦⣤
⠀⢀⠀⢀⠀⠀⡴⠁⣠⢡⠆⣸⠶⠃⠀⠤⠿⠿⠿⠏⠱⠏⢋⠃⢟⡼⠶⢻⠿⠛⣓⢦⣍⢗⠀⠋⠀⠈⠀⠐⠀⠀⠌
⠀⠉⠀⠀⠀⠄⠀⠀⠉⠉⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⡆⠂⠀⣀⠈⠱⢿⣿⣰⣧⠳⠁⠟⠫⠀⠀⠀⠐⠠⠀⠖⢀⠰⠘⠃⠉
⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡌⠀⠠⠀⠀⠀⠀⠀⠓⠲⠀⠀⢈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠚⠋⠁
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠈⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣟⣝⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠉⣀⢀⡄⠈⢉⠑⠒⡴⠂⠉⠀⠀⢀⡀⠀⠉⠑⠲⠤⣄⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠶⠶⠒⢋⠉⣠⣤⡐⢺⠃⠈⢣⠢⣀⡼⠋⠀⡇⠚⠛⠉⡉⠘⠈⠉⠘⠛⡆⠤⢠⢈⣉⣉⡉⡇⣄⡀⣀⡀
⠀⠀⠀⠀⢠⠤⣤⣿⣿⠀⢻⡟⠙⡏⡘⠁⠀⢻⣼⠆⠀⡀⠁⠉⠀⠀⠀⠆⠀⠙⠛⠘⠀⢀⠀⢼⠻⣇⠀⡁⠀⠹⠉⡋⡇⣿⣇⣿⡇
⠀⠀⠀⠀⠀⠀⣯⠛⣿⠀⢸⢳⠶⡇⠡⣠⣠⡾⡿⠷⠾⠇⠰⠠⠀⠀⠀⡀⢠⠰⠶⠃⠘⠚⠲⠎⠶⠊⣀⣀⣀⢸⠉⠃⣶⣧⠙⣿⠄
⠀⠀⠀⠀⢰⠀⣧⢞⣻⠀⣠⣬⣴⠷⠀⠈⠉⠀⠀⠀⠀⠉⢛⣃⠰⠶⠀⠀⢰⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠦⡾⣧⢈⠰⣿⡃
⠀⠀⠀⠀⢸⠀⣧⡾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠸⣄⣿⡇
⠀⠀⠀⠀⢠⠀⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⢻
⠀⠀⠀⠀⢸⠀⡟⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢸⢁
⠀⠀⠀⠀⠸⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠨⠀⡇⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢸⢨⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸
⠀⠀⠀⠀⠀⡄⢿⠀⢸⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠀⠀⠈⠁⠂⠀⢸⠰⠃⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢸
⠀⠀⠀⠀⠀⡇⢸⡇⢸⡆⠄⠀⠀⠠⢤⠀⢀⣠⠴⠒⣋⠉⠉⠿⠯⢽⣷⡶⠶⡉⠻⠙⠁⠀⠀⠀⠉⠓⠦⢤⣠⣔⠀⠐⢠⣿⣆⢸⢸
⠀⠀⠀⠀⠀⡇⢸⣿⡿⢷⣤⣤⣤⡶⣶⡾⠛⠁⣠⣞⣛⣁⣀⡀⢁⠔⡡⠚⢆⠃⠀⠀⣀⣴⣀⣀⡄⠀⠀⠀⠉⠛⠛⠛⣿⡄⢿⣼⠈
⠀⠀⠀⠀⠀⡇⢸⠏⡤⠞⣡⠴⢁⣈⡭⠴⠚⠋⠉⠉⠀⠀⠈⠉⣙⣞⣁⠀⣈⣷⣞⡉⠀⠀⠈⠉⠙⠳⢦⣄⣀⢀⢀⣠⡈⣀⣨⠻⠂
⠀⠀⠀⠀⠀⡇⠘⠛⠒⣛⣒⣋⣉⠤⠤⣄⣐⡂⣖⠶⠤⠭⠙⠒⠒⠚⠛⠉⠑⠒⠀⠐⠒⠒⠛⠉⠠⠬⠔⠚⠿⠛⣻⣿⣟⣻⣭⣄⣤
⠀⠀⠀⠀⠀⠙⠯⠝⠛⠚⠚⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⡐⠉⢁⡀⠐⢀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠰⣯⠅⠀⠀⠀⠉⢡⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⡢⠀⠀⠆⠀⠈⢧⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠧⠄⠒⠂⠐⠒⢲⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠂⠀⣤⣄⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⡤⠠⡄⠄⠐⠋⠙⡄⠈⢄⠣⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⣴⠃⠀⠀⠃⠀⠀⠀⢂⠀⣀⠀⠀⠀⢀⡎⠀⠀⠀⡆⠀⠀⠀⠀⠈⢦⠠⠤⠐⢶
⠉⠂⠤⠤⠴⠺⢄⠀⠀⠀⠀⠀⠀⠀⡰⠢⢤⣄⣀⣤⠔⢏⠠⠀⠂⠀⠄⠐⢈⢁⡈⠀⣀⠈⠀⠂⣼⠀⠀⠆⠀⠰⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠡
⠀⠀⠀⠐⠀⠀⠀⠱⠦⢤⣀⣀⣤⠞⠁⠀⡁⡀⡤⡜⡆⠚⢷⠐⠖⢠⣀⠯⠭⠀⠀⠂⡄⠀⠀⠠⡇⠀⠀⢠⠀⠀⢇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀
⠀⠀⠈⠀⠀⠴⠋⣀⠁⠤⣀⠀⠀⡉⠻⢅⡁⠈⠁⠀⠁⠀⠨⠡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢆⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠂⠈⠑⠦⢀⡀⠀⠐⠂⡀⠐⠜⢂⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠀⠈⢆⠀⠘⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⠀⠀⠃⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢸⠀⠁⠀⠀⠈⢧⡀⠘⣄⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡜⢇⠀⢀⠀⠀⠀⠑⠄⠈⢢⡀⠀⠘⣄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡇⠈⢆⠀⠂⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⢐⠂⠤⠼
⠀⠀⠀⠢⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢀⠀⠐⢀⠇⠀⠈⠣⡀⠐⠄⠀⠀⠀⢀⣀⠠⠁⠀⠠⡁⠀⠀⠁⠀⠀⠘
⠀⠀⠠⣀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⣄⠀⠀⠀⠀⠀⠈⠘⡘⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠁⠀⢀⢊⢀⢀⠀⡰⠁⢅
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠊⠼⡈⠠⠉⠒⠸⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⣤⡀⢀⣤⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠁⠀⠀⠀⠀⠀⢹⠏⠀⠠⠠⠄⢀⠉⠙⠲⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣀⣠⣤⡴⣶⢛⣍⠩⠭⢉⡍⡛⢦⣼⣤⡶⣋⠭⠉⠀⠬⢍⣛⠳⠶⢤⣄⣀⠀⠀⠀⢻⣾⣇
⠀⠀⠀⠀⢠⣿⡿⢗⡚⢛⢫⡹⠵⢚⣋⣉⣥⠤⠤⠤⢤⣈⢑⣵⣿⣿⣯⣒⣨⣭⢭⣭⣥⣖⣙⣑⣖⠤⠭⡙⣛⣓⠲⢿⣿⣆
⠀⠀⠀⢠⣿⣟⣒⣂⣨⣭⣴⣒⣋⣭⣤⣤⣶⣶⣶⣾⣿⣿⣿⡶⠭⠭⢶⣿⣿⣿⣷⣶⣶⣶⣶⣯⣿⣿⣷⣶⣤⣤⣤⣬⣭⠿⣆
⠀⠀⠠⣿⣷⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠘⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠛⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠦⠤⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⡀
⢠⡤⢤⣀⣀⣀⣀⣠⣤⠤⠤⠼⢿⣥⠤⠄⠀⠈⠙⠿⣿⣇⠀⠀⠀⠀⣾⣿⡿⠉⠁⠀⠤⢤⣾⠟⠦⠴⢶⣶⠤⠤⠤⠴⠒⣊⠟
⠀⠈⢲⣤⣄⣀⣀⠈⠙⢿⣖⣒⡀⠉⠻⢿⣍⣁⡤⠀⣈⠏⠀⠀⠀⠀⣿⠁⢀⣀⣈⣻⠿⢋⣁⣀⣚⡿⠟⠁⠀⠤⠴⢒⠟
⠀⠀⠀⠈⠲⣦⠤⠤⠄⠀⠈⠙⠿⢧⣤⣤⢀⣉⣉⡽⠋⠀⠀⠀⠀⠀⠈⠻⣖⡛⠉⠠⣤⣤⡽⠟⠁⠀⠀⠒⢒⡲⠟⠃
⠀⠀⠀⠀⠀⠀⠉⠳⠶⣒⣒⣀⣀⣀⡨⠭⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⠭⣁⣀⣀⣀⡈⠭⠟⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠊⣏⣸⢀⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⣹⡍⢹⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠭⠤⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡐⢀⡠⠾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣴⣿⣷⣦⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢫⠟⠱⢽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠔⠀⠀⠀⠄⠂⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⠅⠉⠁⠀⠀⠀⠀⠀⠊⠁⡝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⠅⠁⣂⣀⠀⠀⢀⢀⣠⣄⡀⠐⠀⠈⠀⠀⣨⣙⢝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠕⠑⠂⠀⠀⠀⠀⠀⣠⣶⡫⠟⢧⣄⠀⠀⠀⠀⠀⠀⠀⠐⢙⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠕⠋⠀⠀⠀⠀⠀⠰⣤⡾⠫⠊⠀⠀⠑⢝⢷⣄⠀⠀⠀⠒⠀⠀⠀⠙⢿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⠁⠀⠀⠈⡀⠀⠉⣩⡾⡡⠊⠀⠀⠀⠀⠀⠂⠑⢝⢷⣅⠀⠀⠀⠀⠀⠀⠀⠗⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⣴⣶⡤⠘⠛⢺⡧⠭⠥⠍⠁⠀⠀⠀⠀⢽⡿⠛⠁⠀⣢⣖⠀⠐⠂⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢹⣽⣟⠀⠀⢸⣟⠀⢠⡤⠤⠬⠥⣄⠀⣿⡇⠀⠀⢘⣿⣿⣃⠀⠁⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠠⠴⢺⣿⡷⠀⠀⢹⡷⠀⢸⡅⢠⡄⡂⡇⠀⣾⡇⠀⠀⠰⣿⣿⠂⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠂⣀⣸⣿⣟⠀⠀⢸⣗⠀⠘⡄⢸⡇⡇⡇⢀⣻⡇⠀⠀⢺⣿⣿⡂⠐⠆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠠⣾⣿⣯⠀⢀⢸⡯⠀⠠⠀⠸⠇⡈⡧⠄⣽⡇⠀⠠⢴⣿⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡂⠀⠀⠛⠛⠛⠂⠀⢸⡷⠀⢰⡇⠐⠐⠆⡇⠀⣿⡇⠀⠀⠘⠉⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣁⡀⠈⡉⠉⣉⣁⣹⣹⡶⠈⠉⢉⠩⠿⢽⣿⣏⣏⣀⣁⣉⣀⣉⣉⣀⣁⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⢠⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣆
⣿⠃⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢻
⢿⡀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣾
⠈⢷⡀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠾⢻
⠀⠈⣿⣄⡀⠀⠀⠀⠀⠙⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢿⡌⠙⠦⣀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢸⡇⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⢻⡇
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⠉⠰⢦⣀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠈⢻⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣀⣤⣾⠇⠀⠀⣴⠟⠱⣦⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣾⠟⠉⠁⠀⢀⡾⠃⠀⠀⠙⠳⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⣸⠃⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠈⢷⣄⠀⢻⡄⠀⢀⣠⠞⠳⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠄⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⢹⡇⢸⡇⠀⡉⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢠⣤⠆⠀⠀⠁⠾⠁⠎⣴⠇⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠙⠛⠻⠶⣤⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⡀⠀⢠⡿⠀⠂⢺⡟⢷⣶⡀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠉⠛⠋⠑⠀⠤⢸⣷⠀⠘⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠻⢦⡀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠙⠶⠤⣤⠴⠶⠶⠶⠛⠛⠛⠿⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⢶⣀
⠀⣠⣴⣀⠀⠀⠀⠀⠀⠀⢄⣀⣢⣄⡻⢍⣉⣺⠝⠲⠖⠂⡈⠁⠠⢴⣂⣠
⢠⡏⠀⠉⠓⢦⣀⠀⠀⠀⠀⠉⠓⠂⠉⣁⣀⡠⢄⡲⢯⣵⠮⣕⡢⢤⠬⠗⠚⠉
⣾⡊⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠉⠳⢾⣛⣲⡯⠓⠊⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡖
⠀⠉⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠖⠛⠉⠁
⠀⠀⠀⠀⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠖⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⣶⡾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⡀⠀⠀⠀⠀⠀⠐⢸⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠖⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⣀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀⣿⡧⠀⠀⠀⣀⣠⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣧⣴⠲⠯⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢳⠎⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢶⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡖⡇⠸⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⡗⠆
⠀⠀⠀⠀⠀⠀⢀⡄⠱⡀⢃⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠃⡰⠁⣄
⠀⠀⠀⠀⠀⠀⠘⡙⢄⠑⡄⢦⠸⡄⠀⠀⠀⠀⠀⣠⢶⠒⢄⠀⠀⠀⠀⠀⠀⢀⡼⢠⠆⠔⢁⠜⡙
⠀⠀⠀⠀⠀⠀⠐⡙⠦⣝⡢⠀⠁⠈⠢⣤⣀⠀⠈⠓⠄⠀⢸⠀⠀⠀⢀⣠⡤⠊⠀⠁⠠⢒⡥⠞⡕
⠀⠀⠀⠀⠀⠀⠀⢾⡓⠦⠬⠀⠀⠀⠀⠀⠈⢆⠀⢀⡄⠀⠀⡄⠀⢀⠎⠀⠀⠀⠀⠀⠨⠤⠖⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⢉⣉⡁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣉⣉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⣁⡤⠊⡰⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠰⡈⠢⣄⡉⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣠⠞⢡⠎⣰⡃⣇⣠⡀⠀⠀⣰⣀⡇⣳⡘⢆⠙⢦⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠙⠋⠁⣀⡠⠊⠁⠀⠘⠉⠢⣀⡀⠉⠛⠉⠉⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠃⠀⠀⠀⢐⣉⣁⠠⠀⠀⠀⠀⠀⠀⠀⠠⣀⡉⣑⠀⠀⠀⠀⢣⢢
⠀⠀⠀⠀⠀⠀⠀⢠⢧⠇⠀⠀⠀⠀⠀⢅⡠⠔⠂⡠⢀⠀⣀⠠⡈⠓⢤⣄⠅⠀⠀⠀⠀⠀⢧⢧
⠀⠀⠀⠀⠀⠀⢠⣏⡞⠀⠀⠀⠀⠀⠀⠘⢦⣤⠊⠀⢸⠀⢸⠀⠈⢢⣤⠞⠀⠀⠀⠀⠀⠀⠘⡎⣧
⠀⠀⠀⠀⠀⢀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠻⢄⠼⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣆
⠀⠀⠀⠀⢀⡾⣽⠁⠀⠀⠀⠀⢀⣀⠠⠤⠶⠐⠦⢤⡀⢸⠀⣠⠤⠶⠶⠦⠤⣄⣀⠀⠀⠀⠀⠀⠱⡹⡆
⠀⠀⠀⠀⣼⣳⠇⠤⠤⠐⠚⠉⠉⠀⠀⠀⢀⣀⡀⠀⢈⣻⡊⠀⠀⣀⣀⡀⠀⠀⠈⠉⠑⠒⠠⠤⢤⣷⣻⡄
⠀⠀⠀⣼⣿⣁⣀⣀⣀⣤⣤⣴⣶⣦⣭⣭⣭⣿⣿⣷⣍⣀⣈⣵⣿⣿⣿⣭⣭⣽⣶⣶⣤⣤⣄⣀⣀⣀⣹⣿⡄
⠀⠀⢸⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠉⠛⠛⠛⠋⠁⠈⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⣀⣈⣁⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢄⠐⢦⡀⠀⢰⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⠀⢧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠳⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢷⡴⠖⠋⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⢀⡐⠒⠋⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠛⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠀⣶⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠓⠒⢲⠀⠘
⠀⠀⠀⣤⠤⠶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⣤⣀
⠀⠀⠀⠤⣤⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣈⠉
⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠀⠰⡄
⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠉⠉⠧⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠉⠙⢦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⣴⣭⢍⠉⣉⣉⡀⠉⣹⣿⡤⠤⠤⢴⣶⣖⣒⣲⣶⣶⠶⢶⣶⣶⣶⣶⣶⠶⠎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣈⣒⣻⣿⠿⠛⣫⣿⣿⣷⣖⣶⡿⣿⡿⠟⠛⢛⢂⣀⣐⣛⣿⣿⣿⣿⣿⣿⡿⡟⢁⠀⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠏⠍⢙⠋⠉⣛⣿⣛⡿⡿⠿⠛⣋⣍⣀⠋⡄⣁⡸⣿⣻⣿⣿⣿⣿⣿⣿⡿⠻⠊⠀⠺⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⠥⠉⠒⢒⡼⢿⣿⣿⣟⡹⠔⠛⣛⣙⣙⣂⣀⢒⣀⢻⣿⡍⡿⣫⣿⣿⣿⣿⠛⢶⡄⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢶⣿⣿⢶⡷⣽⡍⠦⢺⣿⣭⢿⣿⣟⣛⣒⣀⣀⣐⣒⢲⣤⣝⢻⣳⣽⣶⣒⡻⣿⣿⠟⠃⠀⠀⢃⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢟⣥⣒⡈⠅⣀⣴⡞⣻⣭⡈⠉⢙⡿⣶⣿⣿⡯⣭⣭⣒⣋⣻⣯⣡⣊⠂⣀⣐⣲⣨⡾⠻⣶⡆⠀⠀⠀⠈⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣽⣿⣏⣉⣛⡉⣵⣶⣾⣻⡿⠫⡅⠀⠀⠀⠀⠀⠀⠀⡆⠀⢪⠤
⠀⠀⠀⠤⠶⣿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣭⣽⣶⣷⣾⣷⣷⣶⣿⠋⠺⠷⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⡁⠸⠋
⠀⠀⠀⠉⢻⣿⡛⠛⠒⠒⠛⠛⠛⠻⠾⠶⠞⠷⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣿⣿⣶⡀⢻⠁⠀⠠⠀⠀⠇⣦⡃⠀⠠⠈
⠀⠀⠀⢀⣾⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣧⢰⠀⠀⢰⠀⠀⠄⣷⡆⠂
⠀⠀⠀⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠰⠗⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡆⠀⠀⢄⡀
⠀⠀⠀⣠⣾⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠤⣄⣀⢤⡯⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠠⣤⠀⠋
⠀⠰⠛⡟⠛⠛⠛⠛⠛⠛⠛⢓⣒⠒⣰⠋⠀⠀⡀⣴⢿⣮⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠆⡏⣾⣿⣶⣆⣀⣀⡀
⠀⠀⢀⡴⠞⠛⠛⢦⡀⢠⡾⠉⠉⠳⠊⢀⣠⠶⠋⠁⢹⣿⣿⠿⠿⠿⠷⢶⣶⣶⣦⣔⡐⠒⠒⠒⠶⠽⣿⣾⠿⠿⠯⠚
⠀⢀⡿⠀⠀⠘⠀⠈⠗⢸⠧⠐⣊⠤⣤⡉⠀⠀⠀⠀⠈⡇⠙⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣤⡀
⠰⣸⣇⣀⣤⣤⣶⣾⣟⣠⡔⠉⠀⠀⢽⣷⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⢹⣿⣿⠿⠟⠋⠉⠀⣾⠹⠇⠀⠀⠀⠀⠙⠷⣦⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄
⠸⡟⢻⣧⡀⠀⠀⣰⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠛⠉⢲⣄
⣤⣤⡤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⡿⡡⢄⣒⡟⣋⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠀⠀⠒⢒⣒⠒⠒⢲
⠉⠛⠿⢭⣭⣭⣀⡀⢀⡀⠀⢀⡉⠉⠒⢦⡀⠀⠀⣰⢡⡎⠀⠀⢀⠌⠹⡍⢩⡕⠀⠀⠀⣠⠖⣉⡤⢠⠤⠀⣀⠆⠀⠤⠶⠶⠒⠋⠁
⠀⠺⢶⡂⠤⠤⠀⠀⠠⣜⡃⠀⢱⠮⠈⠀⡹⠀⠠⢤⠆⣷⠚⠀⠀⠀⢀⣿⣆⣱⣄⠀⠀⢧⡀⠈⢚⠷⠄⠀⠽⡃⠀⠀⢘⣚⠭⠚⠁
⠀⡤⠤⢤⠉⠀⠂⠀⢀⢀⡀⢰⡖⠀⠀⡟⠁⠀⠀⢻⡶⣿⣧⣄⣀⡠⠈⣿⠉⢥⣿⡄⠀⠀⢹⠀⢈⣻⠖⠐⢊⠁⠈⠁⣀⡠⠤⣶⠆
⠀⠀⠁⠐⠂⠀⠀⠠⡜⠒⠒⢊⣽⠅⡀⢳⡀⠀⠀⠈⢧⣜⢻⣿⠉⠀⠀⠹⣿⣷⣿⡅⠀⢀⡎⢀⣌⡵⣀⡉⠠⠒⠒⠒⠒⠈⠉
⠀⠀⠠⢖⡛⠓⠒⢂⡰⢦⠄⠀⢀⠻⢥⠄⠙⢦⣀⡔⢻⠾⡋⠁⠀⠀⠀⣀⣼⠋⠙⢳⡴⠋⢀⣤⠃⡖⠉⠤⡔⠃⣀⣀⠠⠭⠿⠂
⠀⠀⠀⠀⢤⠒⣊⠉⠀⠀⠾⡋⠀⡀⠈⠈⣤⣀⡟⠀⡞⠉⠀⠙⠱⡶⠋⠀⠃⠀⠀⠈⢷⡴⠽⠉⠀⢤⣀⡩⠭⠄⡈⢉⣙⡶⠂
⠀⠀⠀⠀⠀⢀⣔⣊⠄⠀⢀⢉⡀⠉⢫⡄⡈⣿⠁⣸⡇⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠘⣧⣴⠴⠖⠨⡀⢤⣀⠀⢄⡢⣄
⠀⠀⠀⠀⠰⣏⣭⠖⠀⡠⠄⠘⠋⣠⣇⡔⣻⢇⠆⣿⢣⠀⡀⠀⢀⠀⠀⢄⠀⢀⣰⠀⢰⠸⣏⣷⣼⣇⠙⠲⢬⠙⣒⠊⢿⠗
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣒⣩⠔⠒⣹⢿⡿⠛⠤⣋⢻⣦⣁⡿⣄⣁⣤⣄⡀⣱⣰⣿⠀⠀⣀⣨⣿⡟⣌⠲⣤⣐⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠝⠈⣱⡮⠂⣠⢾⣷⠀⠀⣾⣿⡿⠋⠉⠓⡷⠲⡞⠹⡛⢿⣿⣤⠋⠀⠀⣬⣷⣼⡢⣤⣄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡾⠗⡿⠁⠠⡑⣼⣿⡇⠀⠀⠀⠀⠀⢣⡆⠃⢸⣿⣿⠀⢁⡎⠀⠹⣿⣿⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⢠⣷⡿⢸⡇⡇⠀⠀⠀⠀⠈⢀⠄⠘⢿⣿⣧⣄⣥⡀⠀⠹⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡞⠀⣴⣿⣿⠇⢸⠃⢹⡀⠀⠢⢀⣒⠩⠀⠀⠻⣿⣿⣿⣿⣿⣆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣾⠻⣿⣿⠀⢸⡀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣫⣷⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢦⡟⣤⠻⣿⠀⢸⡇⢀⢻⡙⠷⢀⡄⠔⠀⠀⠀⣿⣿⣿⡿⢱⣿⢿⣷⣼⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⢠⣿⡇⣿⠀⢸⣷⠈⡜⣿⣶⡞⠀⠀⠀⠀⠀⣿⣿⣿⠇⠸⣿⣾⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⣷⠘⣿⡄⢹⡆⢸⡇⢇⢳⡘⣿⠀⠀⠀⠀⠀⢰⣿⣿⡿⣧⠀⠀⣿⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣿⣦⠈⣴⣿⣧⠈⣿⠘⣎⣷⠈⠁⠀⠀⠀⠀⢸⣿⣿⠃⣿⣧⠀⡿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⣧⠀⣿⣿⣿⠀⢿⣇⠹⣾⡁⡞⠀⠀⠀⠀⣾⣿⡇⢰⣿⡟⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠙⢿⣿⣧⠸⣿⡄⢻⣷⡇⠀⠀⠀⠀⠻⣿⣿⣦⠟⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣸⣿⣆⢻⣷⠆⢿⡿⡀⢻⡟⠀⠀⠀⠀⣤⣿⠟⢡⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⠛⠋⣾⣿⡎⡘⣿⠣⠀⢳⡰⡄⠘⢰⡿⣿⣦⡀⠈⠷⠖⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⠋⠀⣠⣿⠻⣿⡀⠸⣧⠁⠀⠻⣌⠀⠉⢀⣿⢿⣿⠆⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠽⢒⣬⡅⢰⡌⢙⣀⢻⣧⠀⠹⣧⠀⠀⠈⠳⠶⠿⢏⣞⣿⡀⢽⣀⣡⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⣟⠸⣆⢻⡀⣿⢿⣧⣿⣇⠀⠙⣧⠀⠀⠄⠀⠙⠛⢭⣾⣿⣄⣰⠌⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠉⢩⣿⠟⣸⣿⣿⣿⣆⠀⢌⢧⢰⣶⠶⣠⠖⠒⣮⠻⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣁⢸⡋⣿⡿⣻⠿⣎⢦⠁⠀⠽⠟⣡⠴⠂⠬⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠉⠰⠷⣽⡄⠠⡖⠈⠓⠀⢀⠴⠢⠤⣰⣉⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠦⢤⣤⣤⡤⠤⠶⠒⠒⠶⢤⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⢀⡇⣠⠆⡠⢄⢠⠂⠀⢀⠀⢹⠖⢫⠥⡄⠀⢠⠉⠉⡙⠒⠒⠒⠒⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠘⠀⠘⠀⠧⠜⠸⠀⢧⠃⢠⠏⠀⣟⣙⡇⡇⣸⠉⡆⡇⣴⣒⡄⣼⣿⡟⢿⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⢠⠏⢀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠑⠒⣼⣿⣟⢮⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢠⣾⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⣰⣿⣯⣄⣁⣀⣀⣀⣀⣀⣔⣀⣀⡀⠀⠀⠀⣰⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣧⣶⣶⣶⣶⣦⣤⣈⡀⠀⠀⢀⣶⣴⣿⣿⣿⣿⣿⣿⠏
⣠⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⠏
⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⢿⣟⠉⠉⠙⢿⣿⣿⣿⣶⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠉⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⢿⣦⣤⡀⠀⠀⠀⠀⣤⠖⠀⠛⠛⢀⡀⠀⠀⠀⣀⠀⡼⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣻⣿⣽⣿⣇⠀⠀⠐⣿⠈⠊⠀⠀⠀⢄⠀⡀⡀⣬⣮⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣾⣟⣿⣦⢂⠀⠿⠒⡀⠶⠀⢱⡔⢀⠓⣾⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣝⣿⣿⣷⡿⣶⣌⡃⠀⢠⢂⣀⣣⠒⢫⣿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⠋⠁⡀⠈⠐⠒⠁⢀⠙⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣰⡾⠆⠀⡚⠋⢀⡿⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢡⠀⠂⣀⣤⠛⢛⠡⢄⠲⠢⠶⢇⡀⠀⠈⠉⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡔⣾⡿⣿⣡⠏⠉⠓⣿⣁⡅⠐⢯⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⡇⠀⠀⢻⣻⡀⠀⠸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠘⠻⢧⡀⠀⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠁⠀⠙⡃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠴⣿⣿⣿⣿⠛⠛⣛⣿⣶⣤⡀⠀⠀⠀⠀⢠⡀⣭⣄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⡖⠀⠀⠸⣑⠀⣆⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⡯⠇⠀⠀⠀⡴⠸⠛⢶⡈
⠀⠀⠀⣄⡈⠀⠀⠀⠀⢠⡇⡀⠀⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⢀⢑⠀⠀⠐⣠⡆
⠀⠀⠀⢈⠀⠀⠀⠀⢀⣾⣿⠁⣀⠂⠀⠀⢸⡀⠀⠀⠀⠀⢀⣀⣤⣶⣶⡤⣀⠀⠀⠀⠈⠈⣳⡶⣶⠋⢰
⠀⠀⠀⠸⠦⠄⠀⠀⠰⠿⠟⠛⠉⠛⠒⠋⠀⠁⠀⠀⣴⣾⣿⢿⣿⣿⣿⣿⣾⣯⣀⠀⢀⣰⠁⡀⠈⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⡾⢿⣿⣿⣿⣿⣿⠇⠀⠸⡥⣈⣿⠣
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠞⠋⠀⠀⠀⢤⣀⠀⢨⡿⣬⣿⣿⣿⣿⣿⣿⢿⣿⣽⣿⣷⡆⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠛⠽⣿⢧⣄⣀⡤⢒⡽⠀⠈⠙⠿⣿⣿⣿⣿⣧⡿⢿⠟⠛⠋⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⠗⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⢀⣴⣿⣶⢲⣦⠤⣄⣰⣦⣤⡀⠀⠐⡾⠁⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡏⢈⡛⠘⢀⣉⣉⣿⡇⢰⠤⡤⢉⣻⡿⢹⡇⠀⠀⠀⠀⣴⡏⠏
⠀⠀⠀⠀⠈⠘⠒⠂⠀⠀⠀⠀⠘⣿⠀⠛⣤⠿⠽⠾⢿⣿⡇⢾⠀⣁⣘⡿⣿⣺⣷⠀⠀⠀⣀⣹⠋
⠀⠀⠀⠀⢡⣤⣄⣀⣠⠀⠀⠀⢰⣇⣄⣋⣩⡤⠶⣶⣾⣿⣷⣤⣤⣉⣉⣿⣿⣿⣿⠀⠀⢼⠛⣻⡆
⠀⠀⠀⠀⠉⠁⠉⠉⠉⠀⠀⠀⠿⠿⠿⠿⠿⠷⠴⠾⠟⠙⠻⠿⠿⠿⠷⠾⠿⠧⠼⠀⠀⠈⠉⠉

⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠛⠉⠉⠉⠉⠉⠙⠻⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠄⠊⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣶⠂⠁⠠⣤⡄⠀⠀⣤⡤⠀⠁⠐⣤⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠉⠉⠙⠿⣦⣀⠀⠀⠐⠀⠀⢀⣠⡾⠟⠉⠉⠉⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⠈⠻⡿⠃⠀⠂⢺⡿⠋⠀⠀⠀⠀⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣅⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⢰⠇⠀⠀⣼⡄⠀⠀⢧⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣤⠀⡞⠀⠀⠀⠿⠇⠀⠀⠘⡆⢠⣤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⡀⠀⠀⠐⠒⠒⠀⠀⠀⣀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣩⣬⡉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣉⣽⣿⣿⣿⣌⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢾⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠫⣿⣿⣿⣿⡏⢶⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⢶⣄⠀⠀⠀⠀⣿⣾⣥⣽⣿⣿⣃⣾⡇⠀⠀⠀⢠⣴⣞⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⢠⣾⣷⡄⡝⠛⠛⠛⠛⠛⠿⢽⣿⣿⣿⡿⠟⠛⠛⢿⣿⡿⠟⣽⣷⣦⡘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣃⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡔⢻⣿⣿⠋⢋⣉⣉⣉⣉⣉⠉⢹⣿⣿⣿⡏⢉⣉⣩⣿⣿⣯⣍⢿⣿⡿⢑⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣤⠾⠉⠉⠉⠉⢹⠀⢸⣿⣿⣿⡇⢸⡇⠀⠀⠀⠙⢿⣷⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣯⣿⣶⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢀⢸⣿⣿⣿⢇⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⣶⣿⣿⣿⣿⣷⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠿⢿⣿⣿⣿⡿⠛⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠈⠻⠿⠟⢉⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠖⠋


⣿⠿⠿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣛⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠄⣀⣀⠀⠀⠀⣀⠤⠖⠊⠉⠓⠦⢤⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⢋⣠⠤⠤⠄⢀⡀⠉⠉⠉⢀⣀⠤⠖⠒⠒⠤⢄⣀⣀⡀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⡤⠴⠤⢄⣀⠈⠉⠉⢁⣠⠴⠖⠒⠲⠤⣄⣀⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠃⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠛⠁⠀⠀⡀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢻⡛⠶⣄⣀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠘⠿⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⢃⣀⣠⣬⠿⠻⢯⣉⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢻⡟⠙⠛⠋⠁⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢹⣿⣿⣿⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⣴⣦⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠙⢧⣀⣠⠴⠤⣤⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣿⠰⣿⡿⠋⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣿⣿⣿⣧⠀⣀⡤⠞⠙⡇⠀⠀⠀⠙⢦⡀⠀⣀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢼⡟⠋⣹⣿⡏⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢁⠀⠀⠀⠀⣇⠀⠀⣀⠼⣗⠉⠀⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⣶⣷⡄⠀⢹⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠎⠀⠀⠀⠀⠈⠓⠉⠀⠀⢹⡀⠀⠸⡇⠀⠀⠀⠀⠛⢷⣄⡀⠸⠿⠃⢠⡟⠀⠈⠳⢀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⢻⣤⠴⠛⠁⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣶⡟⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⠀⠈⠁⠀⣠⣴⠊⠁⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠒⠒⠤⣤⡀⠀⠀⠀⢸⡆⢠⡤⠤⠤⠤⠤⠤⢼⣿⣿⣇⣀⡀⡀⠀⠀⠀⢹
⠤⠄⠀⠀⠀⠉⣹⡟⠉⢹⣿⠉⣀⣴⠛⠁⠀⠀⠀⠈⠉⠉⠲⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢰⡤⠤⠦⢰⣦⣶⠤⣿⣿⡧⢄⣀⣀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⣿⡁⠀⢀⣿⠞⠋⠈⠉⠙⠲⢦⣀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠠⠖⠒⢤⠀⠀⠈⡇⠀⠀⢀⡿⠀⢠⣶⣿⡿⣿⣿⣿⣿⠃⠀⠀⢹⠄⠀⣰⠃
⠀⠀⠀⠀⢀⡾⣟⣠⠖⠋⢻⡄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡇⠀⠀⢸⡇⠀⠈⣿⠃⠀⠈⢿⣿⡿⠀⠀⠀⢸⡀⢠⠏
⠀⠀⠀⠀⣿⣴⡟⢡⣶⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⠳⢦⣀⠀⠀⢀⣤⠞⠀⠀⣸⠇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢈⣿⠃⠀⠀⠀⡯⢀⠎
⠀⠀⢀⡴⠛⣿⡇⠘⣿⣿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠉⠉⠉⠀⠀⣀⠔⠃⠀⠀⠸⣏⠀⠀⠀⠀⣷⠀⠀⢸⡟⠀⠀⠀⠀⣃⡌
⢀⡼⠋⠀⠀⠘⣿⣦⣌⡁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠉⠓⠦⠤⡤⠤⠔⠚⣉⠀⠀⠀⢀⡀⠈⠲⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡟
⢸⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠂⠀⠀⢀⣀⣀⣀⠤⠤⠶⠶⠒⠓⠚⠛⠓⠃⠈⠉⠉⠉⢉⡉⡛⠲⢦⣇⡀⢸⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⠁
⢸⠀⠀⠚⠈⠈⢀⣿⣿⣿⣿⡿⢷⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠄⢠⠎⡪⠀⠀⠀⠈⠓⢾⠀⠀⣴⠟⠁
⢸⠀⠀⢠⡞⢉⣼⣿⣿⣿⣿⡇⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⠋⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⢸⠀⠀⡾⠀⢺⣿⠋⠉⢹⣿⠇⠀⠀⠈⠙⠛⠶⠶⠤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢀⣀⡤⣾⠟⠀⠀⠀⠀⠀⠘⠛
⠸⡀⣼⠃⠀⠈⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⢻⣿⣿⣿⡿⠛⠛⠛⠛⠉⠁⠀⠋
⠀⣧⡏⠀⠀⠀⠸⣧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⣿⠁⠀⠀⠀⠀⢿⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠃
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠛⣩⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣉⣉⣏⣨⣉⣩⣝⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣄⡀⢤⣶⠄⠠⢶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣤⡄⠀⠀⠀⠀⠀⣠⣤⢀⠀⠀⠀⠀⠀⠋⠙⢿⣿⣧⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣶⠈⡷⣤⡴⣾⣿⣿⠂⠀⠀⠀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡃⠀⠀⠀⠀⠁⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠛⠉⠉⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣷⡀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⣠⣴⣾⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣿⣷⣾⣿⣦⡀⠀⢸⣿⣿⣿⡇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢈⣯⡀⠀⠉⠉⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⣿⣿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠈⢹⣿⣿⢏⡍⠀⠀⡀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⢸⣟⡈⠯⣷⠆⣿⣧⢈⣽⣇⣄⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⢨⣭⣦⡀⣠⣶⣿⣿⣾⣿⠃⢀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⣁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣅⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⠤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢰⣶⡆⠈⠉⣠⡧⠖⠒⠒⠲⢤⡀⢀⣀⡤⠻⣄⣀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄
⠀⠀⠀⠀⠀⢸⣿⡇⢀⠔⠁⠀⠀⠀⠀⠀⠈⡇⠈⠉⠓⣴⠋⠉⠀⢸⠄⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡀⠀⠀⣿⠀⠀⢀⠄
⠀⢰⣶⣶⣶⣿⣿⣷⢂⣶⣶⡆⠀⠀⠀⠀⠜⠁⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⢀⣀⣈⠀⣁⣀⡀⠀⢸⠄⠀⠘⢻⠟⠓⠒⣿⠒⠚⠿⡟
⠀⣈⠭⣉⠉⢹⣿⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠛⠋⢉⡿⠀⠈⠉⠙⠛⢾⡆⠀⠀⠀⠀⠀⢀⣿⡀
⡎⠀⠀⠀⠉⢸⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢰⣲⣶⠀⠀⠉⠿⠉
⠑⠤⠴⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⣴⡻⠃⣿⣿⣿⠘⢿⣦⠀⠀⠀⠀⠘⠛⠛⠙⠛⠛⠂⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣀⣸⣿⣏⣀⣤⣥⣥⣤⠀⠀⠀⠀⢀⠀⠀⢀⣤⣤⣤⣄⣸⣟⣟⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠂⠂⠀⠸⣿⣿⣿⡿⢾⡿⡷⢾⣿⣿⣧⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠰⣲⣄⠀⣼⣿⣷⠀⢀⣖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠛⠷⣿⣿⣿⠸⠝⠁⢀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠤⡀⠀⢸⣿⣿⠀⠀⢀⣀⣀⡇⢧⣀
⠀⣤⣤⣀⣀⡽⣿⣿⣿⢹⣀⣀⣤⡄⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⠀⠀⠈⠓⠒⡆⡖⠒
⠀⣿⣿⣿⣷⣶⣿⣿⣿⣶⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⡆⠀⠀⠸⣽⡿⠀⠀⠀⠀⠀⡇⡇
⠀⣿⣿⣿⡿⠿⣿⣿⣿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠘⠃⠓⡗⠃⠓⠀⠀⠀⠪⢭⡀⡨⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠛⠋⠉⠉⣷⣿⣿⣿⣬⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠃⠀⠀⠀⡟⠃⠀⢘⡗⡇⠀⠀⢰⡏⢱⠦⣠⣶⣤⣶
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⡀⡀⠀⠁⢸⠀⢐⢀⠀⠀⠀⠀⠀⠯⠁⠀⠈⠭⠁⠀⠀⠠⠿⠶⠤⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣠⠄⠸⣤⣤⣤⣄⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣠⡀⢀⣄⣀⣀⡼⠀⠀⠀⠀⠀⢦⣸⣿⢀⡴⠀⠀⠀⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿
⠀⠀⢀⣬⡓⣹⣿⠱⣧⡀⠀⠀⣀⣰⣆⡈⠀⠄⠄⠀⠀⢸⠈⠠⠠⠀⠀⢀⠀⢰⡩⣯⠭⣿⠬⣿⡍⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⠛⠿⡟⢻⣿⢛⠿⠟⠃⣀⣉⣹⣏⣁⣀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣆⢠⣄⠀⠉⠼⢻⣿⠘⠧⠉⠁⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠘⠋⢾⣿⠎⠓⠀⠈⠉⢉⣹⡏⠉⠉⠀⠀⠀⠀⢸⠀⠀⠠⢠⡤⢠⡤⠠⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⢹⡟⠂⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣤⣴⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⢻⣿⣿⣿⣿⢛⣊⢩⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣿⡘⢿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⡇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢸⣧⠀⠴⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⢺⢿⣿⡿⣿⣿⣿⣷⠟⠋⢩⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣵⡶⢶⣶⡶⠶⣶⣶⣴⣼⣶⣭⢥⠤⠤⠤⣤⣤⡛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢨⣿⡿⠟⠋⠀⣀⣠⣤⡾⠖
⠀⠀⠀⠀⠀⠀⠿⠴⠷⠿⠛⠛⠛⠛⡜⠻⠿⡿⠿⠿⠘⠶⠶⠶⠲⠢⠴⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣟⣡⣶⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠠⣄⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣲⣮⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⡠⠖⢋⡡⠤⠾⠿⢷⡀⠀⠀⠀⠀⠀⠀⢀⣗⣴⣟⣲⣦⣤⣀⡀⠀⠈⠀⠀⣀⡤⠴⢒⡓⠋⣁⡀
⠀⠀⠀⠀⢀⣿⣿⠦⠤⠰⡽⢛⣿⣻⠗⢄⡔⠁⣀⣤⣤⣶⣖⣒⣒⣶⣧⡀⠀⠀⠀⠀⠀⣌⣐⣊⣁⣀⣻⣿⣶⣯⡑⢴⣰⣟⣟⣛⠛⠒⠒⠂⠼⠿
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠈⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠢⡀⠀⠀⠀⠀⠠⠊⢀⠄⠂⠀⠀⠐⠠⡀⠑⠄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠒⢄⡀⠀⠃⠠⠃⠀⠀⡠⢄⠀⠀⠈⠄⠘⠀⢀⡠⠒⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠚⠀⠃⠰⠒⠈⠀⠀⠁⠒⠆⠸⠀⠓⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⡀⠀⠀⠀⠀⢠⠁⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡄⠀⠀⠀⠘⢄⡀⠀⠀⠘⠤⣀⣀⠔⠁⠀⠀⢀⡠⠃⠀⠀⠀⢠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡨⠋⠀⠀⠀⢀⡀⠀⠀⠀⠙⢅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⢀⠌⠀⠀⠰⠁⠀⠰⠆⠀⠈⠆⠀⠀⠣⡀⠀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠘⠂⡤⢤⡴⠤⠀⠁⠀⠀⠀⠱⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡆⡰⢆⢰⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠋⠀⠀⠙⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡤⠤⠀⣀⣠⠀⠀⣄⣀⠀⠤⢠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠐⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠄⠀⠀⠀⠂⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀⠀⠀⠀⠀⢀⠔⣒⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡤⣶⣶⣷⡄⠀⢠⡷⠞⣉⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⡔⠁⢰⡾⠇⠈⠀⢠⠀⠀⠈⠒⠉⠀⠑⣄⠀⠀⠘⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⠁⡌⠀⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡟⣀⡷⠀⣀⡈⢆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⡉⢀⠁⠀⠀⠀⠈⠘⢆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠂⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⢸⠐⠂⠀⢈⡔⡀⠐⠃⠈⠒⠂⠩⠄⠀⠀⡰⠀⠀⠀⠀⡸⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠈⢆⠀⢀⠃⠀⢠⢀⡰⠂⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠃⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⡀⠀⠀⠑⠤⢕⣒⣺⣉⡀⠀⠐⠈⠀⣠⠞⠀⠀⠀⠀⡀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠢⢄⣀⡤⠀⠀⠀⣀⣸⠦⠤⠖⠋⠀⠀⠀⣀⠔⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⢀⣀⠤⠒⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠒⠒⢐⠊⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⡇⠀⠀⠀⠀⠀⠂⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⠑⠃⠏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠒⠒⠀⠀⡤⠤⠴⡄⠀⠀⠀⠀⡗
⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⡁⠀⢠⠇⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠤⠤⣄⡀⠁⠀⠈⢠⣀⣠⠤⢤⠒⠒⣒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣀⠠⠤⠄⠀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠈⠐
⡇⠀⠀⠀⠀⠠⡄⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠉⠓⠚⠀⠀⠁⠈⠀⠀⢤⣄
⡇⠀⠀⠀⠀⠃⠀⠈⠑⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠁⢀⠉⠁⠈⡀
⡇⠀⠐⢡⡤⠀⠠⠄⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠂⠀⠀⠀⣈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠂⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⠔⠎⠀⠈⠓⠒⠒⠒⠋⠀⠀⠀⠓⠒⠒⠒⠊⠀⠀⠷⠦⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣷⣦⡄
⠀⠀⠀⠱⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⠦⠔⠉⠉⠉⠉⢻⠀⠀⠀⡏⠉⠉⠉⠙⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠉⠉⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠀⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⡒⢢⠤⠤⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⣒⣩⠭⠓⡚⢁⡿⣅⣐⡠⡤⠀⠀⠘⡀⠀⣠⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⢖⡶⣿⣿⠿⠟⣛⣯⣥⣖⣊⣭⠖⣩⢒⡕⣩⢿⣍⡣⠦⡜⠀⣰⣿⣿⣿⣿⣿⡧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣖⡺⠿⢖⣊⠍⠍⣈⡥⠴⢒⣻⠵⢚⡽⠛⣻⠖⠿⢕⣿⣞⡵⢋⣴⢻⣆⣇⠀⠸⣯⡽⢟⣩⣭⡶⠟⠁
⠀⠀⠀⠀⠀⠘⣺⠓⠉⠠⠔⣊⣉⠤⠔⣋⠁⠀⡟⠉⡀⣚⠁⠔⠋⡄⢚⡰⢋⡄⠛⣞⣥⠞⡁⠛⣾⠒⢘⣻⣿⣿⣿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠁⠀⠈⠀⠀⠀⠁⠁⠈⠈⠀⠁⠀⠈⢓⠀⣈⠐⠲⢚⣧⣦⠷⣿⣿⣥⣷⣰⢋⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢊⡶⢣⡿⠟⣩⠟⣻⣝⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⢀⠀⠠⣤⣤⣴⣄⢀⠀⡁⢋⠔⠣⢞⣴⠣⢠⡅⡻⣴⠆⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⣉⠿⡫⠕⠊⠀⠁⠀⣽⣿⡹⠖⡷⢶⠟⢛⠉⣭⣶⣛⠛⠉⢟⣂⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢛⠩⠀⠀⠀⠀⠀⠀⢀⣠⡤⢛⠄⢁⠀⠂⠀⠀⣼⠷⢦⣀⣈⣶⠶⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡿⢋⢀⠅⢀⠌⠞⡜⠒⢞⠀⢺⠏⡈⠡⠖⣸⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡽⠈⣰⡡⠔⠻⣖⠁⠀⢆⣀⣶⠊⠋⢁⡤⣸⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠌⠀⠁⠀⠩⠁⠀⠀⠀⠀⠀⡠⢎⡞⠡⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⢀⣤⠌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠢⢔⣴⡎⠁⡡⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠐⠤⢀⡀⠀⢀⡀⠠⠀⠴⠟⠁⠀⡀⢠⠊⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠈⠉⠒⠤⠀⠈⢙⢛⣂⢀⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠀⠀⠀⢀⠀⠀⠠⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠒⠒⡄⢉⠛⠛⠉⠀⣀⠀⠀⠀⠀⠀⠀⣀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢆⡐⠉⠀⠐⠂⠀⢀⡀⠠⠄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢀⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⡌⠳⠀⠤⣀⣙⢳⡄⠀⠀⠀⠀⠀⢠⠖⠦⠀⠀⠀⠀⠀⣠⠞⠁⣀⠀⣴⠖⣡⠄
⠀⠀⠀⠀⠐⢯⠝⠳⠶⢤⣼⣿⣝⠻⣦⡀⠀⠀⠀⠸⢤⠐⠂⠀⠀⡠⣾⠛⠋⠹⣇⠤⠔⣛⣠⡤⠂
⠀⠀⠀⠀⠐⠒⠻⡿⢛⣓⣿⡯⣽⠁⠈⢌⠂⠀⢠⠎⠀⠀⢳⠀⠈⡰⠃⠈⠓⠘⢓⣲⡯⢭⠼⠢
⠀⢀⠀⠀⠀⠀⠉⢉⣡⣬⡵⠿⠗⣳⡂⣈⡓⠚⠃⠀⠀⠀⠀⠓⠋⠀⡀⢠⡤⠦⣌⣁⣚⠓⠤⠀⠀⠀⠀⠀⡀
⢒⡿⠿⠿⠿⠿⠛⠛⠀⠀⢀⡴⠾⠻⣻⣿⠿⠿⡆⠀⠀⠀⢠⡷⡴⣦⣘⣙⣾⣔⠶⠏⠉⠓⠚⠛⠻⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⡀⠀⠸⠐⠀⠛⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡜⢁⣀⡠⠀⡀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠀⡈⠀⠘⠀⠄⠀⠱⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠈⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡒⠁⠉⣡⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣯⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡅⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⢻⡟⣿⣿⡟⢎⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⠶⡴⣷⣿⣿⣷⠘⢓⣴⣗⡠⡀
⠀⠀⠀⠈⠀⠠⣦⡬⣿⡇⠘⠁⠀⢀⣤⠄⠀⠦⠀⠀⠠⠀⣤⡄⠀⣦⣀⡈⠉⢉⣀⣲⣆⢠⠆⠠⠀⠀⠀⢠⣾⣿⣷⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠰⢶⡴⠿⠟⠘⣷⣤⠤⠀⠿⠷⣦⡄⡠⠀⠀⢀⠒⢶⡆⢻⣿⠆⡀⠀⠀⢼⣿⡿⣲⠦⢆⠀⠀⠀⠀⠻⠿⢋⣭⣉⡉⢻⣿⡏⣽⣿⣿⣿⠟⠟⠃
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣙⠂⢻⡇⣾⣿⡇⢏⡸⢏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠘⡇⣿⣿⡇⣿⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⣸⡇⣿⣿⡇⣿⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⣿⣿⠇⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⣸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡽⠂⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⣿⣿⡿⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⣭⡍⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⡏⠁
⠀⠀⠀⠀⠀⠀⡠⠊⢩⣿⡦⢹⢰⣾⡭⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣍⢣⣤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⡇⣠
⠀⠀⠀⡀⢀⣆⡀⡹⠋⢠⢧⣿⣤⠃⠉⢢⣠⣱⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠆⠘⠋⠀⡀⢀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠈⢹⡟⠇
⠀⠀⠀⣧⣉⣁⠲⢋⣩⡘⢠⣎⠰⢒⣮⣉⠛⣃⡉⠁⠀⢀⠀⢾⣔⠦⠄⣀⠀⠈⠣⠀⠠⠴⣲⠧⠄⡀⠈⢫⣤⣤⣼⣿⣧⣤⣼⣧⣤⣤⣽⣿⣧⣬⣵
⠀⠀⠀⡇⠘⣻⣶⢉⣹⣿⢆⣹⣷⣭⣿⡉⢿⣙⠃⠀⠀⠀⠀⣶⠞⣡⠀⠸⠆⠀⢠⠶⢦⣬⢳⣶⠀⠜⠁⠈⠧⠀⠜⠛⠂⠀⢸⡇⠀⠀⠀⠛⠷⠤⠭⠇
⠀⠀⠀⠁⠻⡏⠙⠀⠀⠀⡟⠉⠙⡇⠀⢀⡟⠛⠒⠀⠀⠀⠀⠙⠷⠁⠀⠀⢰⡆⠀⠀⠀⠢⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠀⢰⣾⠀⣸⠀⢰⣖⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠺⡄⣶⣿⡿⠟⡔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠐⣿⡆⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⣼⣿⣷⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣵⡟⢮⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣼⣯⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠟⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡶⠟⣙⣭⠵⠚⠛⠓⠲⣝⢿⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣴⡶⢟⣛⡭⠶⢚⡟⠳⠒⢲⠀⠀⠀⡠⢌⣳⣝⢿⣦⡀
⠀⠀⢠⢞⡽⠛⠉⢰⡟⠉⠁⠀⠀⢯⣀⠀⠀⠙⠶⠖⠋⠁⠀⠨⡏⠳⣝⢿⣦⡀
⠀⣴⠋⠀⠑⢄⠀⣾⠀⠀⠀⢀⡤⡤⠞⠃⠀⠀⠀⠀⢀⡤⠶⠒⠃⠀⠈⠳⣝⢿⣦⡀
⢰⡇⠀⣀⠈⠢⡱⣌⠳⣄⠀⢸⡃⠀⠀⣀⡤⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣝⢿⣦⡀
⢸⡇⢰⡏⢷⡐⢝⢮⡳⣌⠳⣄⠉⠙⠚⠁⠀⠀⠙⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢷⣝⢿⣆
⢸⡇⢸⠀⠀⠙⣦⠳⡝⢎⠳⣌⢷⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠶⡄⠀⠀⠀⠀⠙⢧⡝⢷⣄
⠈⣷⢸⡀⠀⢰⡇⠀⠈⢢⡳⣌⠣⡙⢧⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⡏⠀⠹⣷⣄
⠀⠘⢿⣇⠀⢸⡇⠀⠀⠀⠙⢮⡳⣌⠢⡙⢦⡀⠀⠀⠀⠀⠀⠙⠋⠛⠁⠀⠀⠀⠀⣀⡤⢞⡡⢔⣪⣵⠛
⠀⠀⠀⠙⢷⣼⡇⠀⠀⠀⠀⠀⠻⣎⠳⡜⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⡫⢕⣪⡵⠚⠩⢲⡏
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢈⡷⣜⠢⡙⢮⡻⠦⠦⢤⣄⡤⢖⡫⢕⣪⡵⠞⠋⠀⠀⠠⠒⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⣼⠁⠘⢷⣌⠂⠙⢦⣀⡠⢔⣪⡵⠞⠋⠁⠀⠀⠀⠀⠀⡀⠄⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⡟⠀⠀⢀⡟⠀⠀⣠⡵⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡇⠀⠀⢸⠁⠀⣾⣯⠔⢂⣡⣴⣖⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⣿⠀⢰⣿⠗⠚⣿⡄⠀⢹⡇⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠀⢸⣿⠀⠀⢸⣷⠀⢸⣷⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣹⣀⣠⣼⣿⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠻⣷⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⣤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⢸⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠘⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⣼⡄⠹⡄⠹⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡸⠁⡼⠁⣼⡄
⠀⠀⠀⠀⠀⠀⠀⢳⡙⢦⡘⢆⠹⡄⢳⡀⠀⠀⠀⠀⠀⢀⡔⢻⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⢁⠞⣠⠞⣱⠃
⠀⠀⠀⠀⠀⠀⠀⢯⠙⠦⣙⠢⢅⠈⠀⠙⠦⣄⣀⠀⠀⠉⠳⡄⠀⢨⡇⠀⠀⠀⢀⣀⡤⠞⠁⠈⠀⠥⢚⡡⠞⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠠⡗⠦⢤⣉⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡇⠀⠀⣇⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣈⣡⠤⠖⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠶⠶⠶⠒⠀⠀⠀⠀⠀⠈⠓⠶⠚⠀⠀⠀⠘⠲⠖⠋⠀⠀⠀⠀⠀⠐⠲⠶⠶⠶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡶⠖⠚⢁⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣄⠙⠒⠶⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡤⠞⠁⣠⠏⣰⠀⡆⢸⠀⡀⠀⠀⠀⠀⡄⢸⠀⡆⢰⡈⢧⡀⠙⠦⣤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⣾⣁⣴⢃⡴⣇⡼⠤⣷⡀⠀⠀⣰⡧⠼⣄⡷⣄⢳⣄⣹⡶⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠋⠉⠉⠉⠉⠉⣀⣀⡤⠖⠁⠉⠀⠈⠀⠓⠦⣄⣀⡈⠉⠉⠉⠉⠙⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠏⠀⠀⠀⠀⣎⠉⣉⣠⠴⠂⠀⠀⠀⠀⠀⠀⠲⢄⣈⡉⢉⡆⠀⠀⠀⠈⢯⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡟⠀⠀⠀⠀⠀⢸⡋⣉⣠⠔⠁⡀⠀⠀⠀⠀⡀⠑⢤⣈⠉⡻⠀⠀⠀⠀⠀⠘⣇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⡾⠀⠀⠀⠀⠀⠀⠀⠸⡍⢀⣠⠞⠀⢸⠀⢸⠀⠘⢦⣀⠈⣽⠀⠀⠀⠀⠀⠀⠀⠘⡎⣧
⠀⠀⠀⠀⠀⠀⢠⡟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢇⣀⣠⣟⠀⢘⣧⣀⣀⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣧
⠀⠀⠀⠀⠀⢀⡾⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⣆
⠀⠀⠀⠀⢀⣾⢱⠏⠀⠀⠀⠀⣀⣀⡤⠴⠖⠚⠋⠉⠉⠓⠲⣄⢸⣀⡴⠒⠋⠉⠉⠛⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⢷⢻⣆
⠀⠀⠀⠀⣼⣧⠿⠤⠴⠖⠚⠉⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣈⣿⣏⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠒⠶⠤⠼⢧⣿⡄
⠀⠀⠀⣼⣿⢃⣀⣀⣀⣀⣠⣤⣴⣶⣖⣻⣭⣭⣥⣶⣾⣿⣿⣁⣀⣀⣹⣿⣿⣶⣦⣭⣭⣽⣓⣲⣦⣤⣤⣀⣀⣀⣀⡀⢻⣿⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠿⠿⠿⠟⠋⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡯⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠋⢏⠏⠻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⣿⠀⢸⡇⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣧⣿⣤⣬⣇⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⣄⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⣯⡇⠸⣿⣿⠇⢨⣷⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠒⠉⠀⡈⣇⣄⠀⢀⣄⠈⡇⠈⠑⠪⢗⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣨⠗⠊⠀⣀⡀⠀⠀⡇⣿⣿⠀⢸⣿⠇⡇⠀⠀⢀⡀⠈⠓⢯⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⡴⠻⣿⡏⠢⡀⡇⣿⣿⡀⣸⣿⡇⡇⣠⠊⣷⣯⠓⡄⠀⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣇⣤⠀⣤⡀⡇⡇⡏⢉⣥⣍⠉⠁⡇⡇⣠⣄⢠⣄⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⣿⣿⣿⣻⡇⡇⣇⡇⡞⢸⠈⡆⢠⡇⡇⣿⣿⣟⣟⠇⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⣿⣿⣿⣿⡟⣇⣽⣇⡇⢸⠀⣇⣼⣇⡅⣿⣿⣿⣿⢲⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⣿⡿⠛⣿⣇⠇⡏⡏⡇⢸⠀⡇⠉⡏⣇⣿⣟⠻⣿⡰⠈⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡧⣼⣤⣤⣤⣤⣤⣤⣤⣧⣷⣷⣾⣶⣷⣦⣧⣤⣤⣤⣤⣤⣤⣤⡤⢧⣄

⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⣄⠀⠠⡤⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⢀⣠⡆⣶⣄⡀
⠲⠂⠀⡀⠐⠀⣈⠀⠐⠂⠀⠨⡡⠶⠶⢯⡷⠶⠶⢈⠆⠀⠰⠶⠰⠶⢾⠲⠖⠰⠖⠀⠰⠆⠭⠭⠵⡯⠭⠭⠰
⠀⠀⠀⠋⠀⠘⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⠏⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⢠⣶⣿⣿⣷⣆⠀⠀⠀⠀⣀⢀⣔⡇⣿⣠⡀⣀⠀⠀⠀⠀⡀⣤⡏⣦⡀⠀⠀⠀⠀⠀⠐⡶⣾⣷⢶⠂
⠀⠻⢿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠤⠩⡥⡅⣬⢤⠥⠤⠀⠀⠐⠒⠯⣭⡅⣭⡭⠒⠒⠀⠐⠿⠿⡿⣿⣿⢿⠿⠿
⠀⠀⠀⠛⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⡇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠊⡇⠏⠂⠀⠀⠀⠀⠀⠐⠛⢻⡟⠛⠂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⡉⡎⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⢀⡤⣠⣾⣗⣟⣿⣦⡤⣄⠀⣀⣤⠤⠤⠇⠣⠤⠤⣄⡀⢀⠔⣠⣂⣀⣿⣀⣀⣄⠂⡀⠠⠤⣀⣀⣸⣗⣀⣀⠤
⠈⠒⠙⢿⡭⣯⣿⠟⠑⠋⠀⠀⠉⠉⠉⡇⡜⠉⠉⠉⠀⠀⠁⠈⠑⠀⣿⠀⠈⠉⠀⠀⠈⠉⠁⠈⢸⡇⠁⠈⠉
⠀⠀⠀⠀⠁⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠲⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠵

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠙⠐⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉
⠀⠀⠀⠀⠈⠠⢈⠉⠑⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢄⠀⠀⠀⣰⣹⡄⠀⠀⠀⠠⠁⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠈⠂⢸⡗⠉⠌⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠳⣄⠀⠘⡎⠁⠄⠀⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢁⠄⠨⠀⠾⣄⠀⠁⠀⡀⠈⠀⢻⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢱⣄⢠⢰⠀⠄⠀⠀⠀⠱⣠⠀⡤⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠘⠹⠂⠀⡄⠀⠀⢀⣀⠀⠈⠕⢾⠄⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠚⢃⠀⠈⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠀⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠑⠄⠀⠠⠆⡄⠀⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠠⠀⠀⠀⠀⠐⠊⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⡀⠀⣴⣿⣦⠶⠛⠛⠉⠋⠉⠉⠉⠁⠐⠛⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠉⠲⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⢰⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣇⣻
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⡆⠀⡀⠀⢀⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⣿⣿⡆
⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣤⣤⢉⡀⠀⠀⠁⢀⡼⢚⣭⣭⣋⣭⣥⣍⡐⠂⠤⠤⣤⣤⣄⣠⡀⠄⠘⣿⣿⣷
⠀⠀⠀⣾⣿⣿⣁⣤⣤⠀⢀⣤⡤⠤⣤⣲⣿⡿⠿⡭⢾⣭⢭⡛⢾⣷⡀⣀⡧⡚⢋⣡⣤⣤⣭⣭⣳⣽⣷⣶⣦⣤⡤⣤⣄⣄⡀⣄⣿⣿⣿⡄
⠀⠀⢸⣿⣿⡟⠘⠛⠛⠷⠽⣗⡒⠛⣉⣥⣤⣲⣦⣤⣶⣶⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣷⣿⢿⣿⣻⡯⠿⠮⢥⡄⠈⣉⣿⣿⣧
⠀⠀⣿⣿⡟⠉⢉⢉⣩⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣷⣿⣿⣟⣛⣿⣿⡄
⠀⣸⠍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠛⠿⠿⣶⣿⡿⠿⠟⢋⣿⡏⠉⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣏⣐⣃
⠀⠙⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡄⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠀⠀⢱⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⢀⠀⠀⠀⢐⠃⠈⠀⡀⠐⢂⠒⢂⣀⡀
⠀⠀⠀⠀⠀⠀⣴⢢⣨⣓⡩⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⡀⠀⠀⠀⠀⠁⢹⡷⢤⣀⡟⢀⣶⣴⣮⢖⡋⠴⠞⠋⠁
⠀⠀⠀⠀⢀⣼⣥⣾⣉⣥⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡿⠊⠉⠉⠋⠀⡀⠀⠈⢀⣤⡀⡿⠛⠛⠛⠻⠾⢽
⠀⠀⠀⠉⠀⠀⠀⠀⣠⣤⣤⠀⠀⢸⠀⠀⠀⠀⣠⡤⢀⡀⠀⠀⠀⠐⣦⠈⠲⠂⢱⣯⡅⠁⠀⠚⠷⣶⣤⣈⡤⠴⠤
⠀⠀⠀⠀⠀⠀⢧⣠⣼⡏⢸⡰⣤⣤⡄⠀⢠⣤⡜⠀⢸⣇⣀⡀⠀⠀⠈⠳⠒⠚⠛⠉⠻⣦⣄⡀⣼⠏⡁⠉⠀⠀⠀⠀⠀⠠⠉⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠱⠘⠺⠇⢀⡤⠊⠔⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠙⠛⠧⣄⣀⣀⠈⠉⠛⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡄⠰⡀⠀⠀⠠⠤⠔⢊⢆⠞⠀⢰⠀⣀⠀⠀⢀⣎⡗⠐⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋
⠀⠀⠀⢀⣄⡄⠀⠀⡇⠁⠻⣄⡈⠲⠤⠄⠠⠼⣫⠞⢸⡆⠈⣇⠉⠀⠠⠞⠾⣇⠀⢰
⠀⠀⠀⠼⢉⠃⠀⠀⡧⠀⠀⡏⠃⢰⣶⣶⡆⠀⣏⡀⣸⠁⠀⠘⣆⢠⢠⠀⢠⡙⢧⣤⠀⠀⠀⣀⡤
⠀⠀⠀⠀⣾⡆⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠈⠁⠀⠀⠀⠀⠀⠙⠳⢴⢷⣿⠶⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⣠⢀⣄⠀⣀⣄⣢⡄
⠀⠀⠀⠀⠞⡉⠂⢧⡀⢄⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠞⢁⡉⠢⡽⠂⡄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢤⣉⠘⢻⠿⠁⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠂⠢⠀⢂⣸⠃
⠀⠀⠀⠀⠐⠛⠛⠶⠁⠀⡥⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡤⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠹⢦⣬⣤⡴⠃
⠀⠀⠀⠀⠀⡀⠀⡘⣦⣼⠃⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡀⠀⠀⠀⠀⠀⠘⣿⠾⠃
⠀⠀⠀⠀⠀⡁⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠁⠀⠀⢀⣘⢾⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢋⣥⣴⡀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⣴⡁⡐⢤⣤⣿⡷⠄⠀⢸⠀⠀⠀⠀⠀⣴⠟⡛⠲⠿⢿⣿⣷⣄⠀⢀⡠⠊⠁⠒⣈⣉⣉⣙⣃⠀⠀⠀⠀⢖⡫⠘⣖⠂
⠀⠀⠀⠀⠁⠛⠛⢃⣬⠏⠀⠀⠀⢸⠀⠀⡆⠀⣰⣿⣷⣿⣭⣭⠭⢽⣶⢭⣵⣆⣤⣴⣶⣿⣿⣿⣷⣾⣿⣧⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠸⢂⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠄⠒⠑⣆⡞⠀⠀⠨⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⢄⡠⢿⠀⠀⠀⠀⠙⡌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠀⠀⠘⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠈⢎⠂⠀⠉⠲⣄⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢧⠀⣦⡃⠀⠀⠀⠈⢣⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡀⠠⠄⢠⠜⠀⡇⠑⠐⠂⠀⠀⠀⢳⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠘⠡⠈⠢⠒⠁⠀⠀⠘⢆⠀⠀⠄⠀⠀⠀⢳⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠎⢰⢰⡸⡀⠀⠀⠀⢷⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⢸⠀⢇⢧⠀⠀⠀⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⠆⠀⠀⡇⢣⠀⠀⠀⠀⢻⣸⠀⢸⡸⡄⠀⠀⠀⣷⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠸⡀⠀⠀⠃⠘⡄⠀⠀⠀⣼⡿⠀⠀⣇⢧⠀⠀⢀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢧⠀⡠⠊⠑⢄⠀⠀⠀⣸⠃⠀⠀⢹⢸⡀⠀⣸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⢧⠳⣄⡀⠈⠀⠢⡄⠃⡀⠀⠀⠘⡎⡇⢀⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠳⣄⠢⣔⠒⠉⢀⠠⠈⠐⣄⠀⢇⢧⣼⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠳⣄⠀⠀⠀⠙⠺⢤⣄⣀⣀⣀⡀⠇⠀⢈⣼⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠈⠳⣤⡀⠀⠀⠀⠈⠀⠀⠀⣀⣠⠴⠟⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠙⠂⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⢄⠄⠠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⠀⠀⠀⠀⠀⠀⣀⡆⡆⣦⡀
⠴⠄⠀⡁⠠⠄⢈⠀⠀⠦⠀⠀⡚⣶⢶⣶⣛⡶⣶⡶⢒⠀⠀⠴⣖⢠⣴⢽⢠⠄⣲⠦⠀⠠⠄⠬⠭⠥⣧⠭⠭⠄⡦
⠀⠀⠀⠞⠆⠰⠓⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⡇⡇⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣠⣼⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⣨⡂⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾⡄⡀⠀⠀⠀⠀⠀⠀⣀⣸⣾⣀⣀
⠀⣤⣴⣿⣿⣿⣿⣿⣤⣄⠀⠀⠐⢒⣚⣚⠃⣓⣓⡒⠂⠀⠀⣀⣀⠤⠿⠇⠿⠧⣄⣀⠀⢀⡤⣤⣿⣽⣿⣽⣧⣴⣄
⠀⠈⠘⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠁⠙⢽⡏⡯⠋⠈⠁⠀⠀⠀⠀⢉⡿⡇⡿⡋⠁⠀⠀⠀⠈⠉⣿⣽⣿⣽⣏⠉⠁
⠀⠀⠀⠀⢩⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽

⠀⠀⠀⠀⢠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠂
⠀⠀⠀⢀⣄⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣠⠖⣴⣿⣏⣯⣷⣷⣄⢦⡀⠠⣴⡖⡒⠚⠀⠒⠒⢲⣦⠀⢠⠈⣴⣤⣤⣿⣤⣤⣤⠀⡄⠀⡖⣦⣤⣤⣿⣤⣤⣔⣲
⠈⠉⠀⠿⡦⡧⣾⠗⠀⠋⠀⠀⠀⠉⠉⢹⠐⠉⠉⠉⠀⠀⠀⠀⠈⠐⠀⣿⠀⠂⠉⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡿⣻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⣽⡗⢙⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠀⢸⣷⣼⣧⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢲⡒⠲⢾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⢸⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣁⠀⣸⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⡀⠀⠸⠟⣿⡿⠳⡧⢈⡟⣱⠟⢿⣿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⢜⠛⠒⠀⡷⣤⡴⢡⣦⡌⠻⣟⡛⠓⠒⠨⠁⠤⣁⣊⣉⣈⡉⡙⠫⠿⠿⠗⠶⣲⣶⣦⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⡃⢀⠀⢛⣺⣷⢀⣘⠿⣏⣀⠘⠛⠁⠀⠈⢤⣀⣊⣍⣑⣂⣀⣀⣈⣉⡉⡉⠁⠤⠄⠠⠀⣀⣀⣉⣑⣀⢀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡆⢐⡁⠉⠂⣻⡇⠐⣿⡃⢠⣶⡖⠀⠐⢻⡏⠉⣩⣿⣯⡍⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠳⠀⠘⡷⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⡟⢺⢞⠀⠀⣿⡄⠀⣿⠇⣾⣿⡯⠀⡀⢺⡄⣼⣿⣿⣿⣿⠂⠀⢱⠀⢀⣴⣾⣿⣆⠀⠀⢠⠀⢠⣴⡀⠀⠀⢀⣾⣏⡇
⠀⠀⠀⠀⠀⠀⢤⣶⠰⠿⣿⠿⡷⢸⠟⣁⠿⠿⢆⠀⣿⠀⢿⠿⣧⠀⠀⢸⡃⢻⢿⣿⡟⢿⠀⠀⢸⡄⢸⣿⣿⣿⣿⡄⠀⢺⡉⣿⣿⠇⠈⣀⢀⠛⣿⡥
⠀⠀⠀⠀⠰⠤⠀⠁⠉⠂⠀⠀⠀⠸⠿⠥⠺⠿⠟⠂⣠⡤⠤⠂⠀⠀⠀⢸⡧⠍⠀⢀⡀⢸⣿⠂⠸⠇⠿⠿⠿⠛⠟⠃⠀⠀⣗⠛⠛⠂⠨⠉⠈⠀⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠈⠛⠀⠀⠀⠀⠀⠁⠀⠀⠛⠖⠒⠒⠀⠤⠀⠙⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣻⣻⢿⣻⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠩⠞⠀⠀⠠⠀⠔⠲⠀⠠⠠⣤⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠛⠛⠻⡖⠉⠃⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣀⡀
⠀⠀⠀⠲⣀⣀⣀⠀⠀⠆⠀⠀⠒⡀⠄⠊⠂⠐⠀⢀⠀⣭⢁⣤⣒⣤⠑⡄
⠀⠀⠀⠀⠀⠁⠒⠶⠀⣀⣀⡄⢊⠀⡣⢬⣽⢿⡷⢖⣍⠤⢢⡶⠿⢎⢀⣇⡀⢄⣠⣤⣀⣀⣀⡄⣀⣀⠐⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣋⣀⠁⡘⣷⢧⣞⣿⣋⡭⠿⠶⣋⣴⣦⡾⣏⠩⢧⡌⢘⠟⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡀⠀⠒⢲⣆⠄⣒⠩⢟⣚⣽⣥⣿⡿⠿⣀⠀⠀⠀⠀⠀⡉⠁⡠⠞⠈⢀⠖⠁
⠀⠐⠂⠰⠖⢢⣭⣑⡊⠽⣶⣮⣅⡸⢿⡟⠁⢡⢦⣽⣯⣍⡉⠤⡄⣿⣎⣡⠖⡰⡄
⠀⢀⠠⠀⠀⠒⢂⡪⠭⠍⠭⣍⡡⠤⢼⣤⣀⠄⠞⢸⡻⢽⣿⣗⡴⠋⡠⠏⢰⣷⣾⠆
⠀⣤⠈⢩⠟⢊⢡⣴⣤⡴⠊⠁⠒⠀⣀⠩⠥⢀⣀⣾⣿⣿⣏⠻⢍⡉⠀⣠⣫⡟⢻
⠀⠈⠁⠈⠈⠁⢌⠙⠋⠁⠀⢀⣠⢚⣥⠤⠀⠀⢹⣿⡿⠉⠉⢓⢶⢖⡚⠁⠙⠀⠸
⠀⠀⠀⠀⠀⠀⠘⠷⢶⣶⢶⠿⠟⠋⠀⢀⡠⣤⡿⠣⢃⠀⣚⠁⣀⠾⣯⣲⣁⠄⠁
⠀⠀⠀⠀⠀⢠⠟⠠⣼⠿⡀⠀⠀⢀⣴⣽⠞⣼⠕⠁⢸⡿⣁⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⠤⣆⣠⣤⠾⡟⣦⡴⢻⡿⠃⠜⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢰⡇⣠⣴⠾⠋⠁⠀⠞⡠⠊
⠀⠀⠀⠀⠈⠁⠁⠎⠀⠠⠄⠈⠒⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣒⢒⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣤⡄⠀⠀⠀⠀⠀⠀⠀⠯⠉⠀⢈⠁⠀⠀⠀⠀⢀⠀⠉⠹⠃
⠀⠀⠀⠀⠀⠀⢀⣠⢸⣎⢷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢠⣀⠄⣠⠤⠠⠤⠒⠀⠰⠓⠐
⠀⠀⠀⠀⠀⡀⠘⠃⣀⠉⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⡇⠀⠉⠐⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠑⡀
⠀⠀⠀⠀⠀⣷⠀⢈⠀⢃⣀⡠⠀⠀⠀⠀⠀⡸⠀⠀⡟⠜⠀⠀⡄⠀⠀⢀⠙⣦⣠⠔⠂⡈⠃⠀⠁⡀
⠀⠀⠀⠀⠀⠸⢆⡈⠠⠌⡉⠠⡀⠀⠀⠀⠃⠁⠀⣀⣄⣹⣿⠀⣏⠀⠀⠈⠁⠛⣅⣀⢗⠍⠀⠀⣀⡗
⠀⠀⠀⠀⠀⢀⡴⠀⢥⠀⢩⠀⡰⠀⠀⠘⠀⠀⠊⠽⠇⠉⠛⠀⠇⠀⠀⠀⠀⠀⠘⢷⠈⢠⣦⣶⣏⣴⠀⡀
⠀⠀⠀⠀⢀⡾⠁⠀⠁⠀⠐⠀⣷⡄⠀⠰⢀⡀⡤⠃⠀⠀⠼⠀⠘⠀⡔⠀⠀⠀⠀⠀⠀⠍⠀⠀⠈⣿⢷⠃
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠔⠈⢳⣿⢁⡤⣀⣿⠷⠆⡀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣾⣿⠀⡀
⠀⠀⠁⠠⢧⠀⠀⠀⠀⢱⢄⢈⣯⠘⢰⣿⠟⢧⣤⠁⠀⠀⠚⠀⢸⡄⠀⠀⠀⠀⢀⣠⡅⠀⠀⠀⠸⣏⢹⡦⠄
⠀⠀⠀⠀⠀⠑⣄⠀⣀⠀⠀⣿⣿⣦⣿⡏⠰⡈⠇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠠⠂⠘⠟⠀⡆⠀⢸⣏⠈⠃
⠀⠀⠀⠀⠀⠀⠰⣄⣿⢃⣶⠿⢻⡿⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⣀⠂⠀⠃⠀⠈⡆⠀⢰⢀⣀⠀⠀⠀⠀⠀⠰⢶⣮⡄⠶⢾⠆⠸⡄⢠⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⡘⠁⠀⠀⡃⠀⠀⠀⠀⠀⣿⠀⠀⠲⣆⠐⢶⣶⣶⣆⣀⠀⠀⠀⠀⢀⠀⠔⠋⠈⠁⠀⢠⠈
⠀⠀⠀⠀⠀⠀⣀⡄⠀⣀⡇⠀⠀⡄⠀⠀⠈⠀⠀⠀⠂⠀⠀⠁⠀⢌⠉⠛⠓⠖⠛⣯⡤⠀⠀⠀⠀⠀⢈⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠿⠄⢀⡰⠛⢤⣤⠇⠀⠰⢶⣾⣷⣾⠿⠒⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠐⠈⠁⠐⠒⠂⠂⠉⠁
⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠑⠢⡄⠀⠀⠀⢀⡔⢁⣤⠾⠋⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⣤⣾⣿⣿⣿⣿⠆⠀⠀⢸⠿⠋⣁⣄⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠂⠀⠀⢠⠎⠀⢠⣶⡿⠀⠀⠀⠀⣠⠀⠀⠀⠉⠒⠉⠀⠀⠑⣄⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠉⠃⠄⢠⠃⠀⣴⠾⠋⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⠋⠳⣄⠀⠀⠀⠳⡀⠀⠀⠐⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⢀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠿⠋⡴⢚⠃⠉⠑⠂⠄⣱⡀⠀⠀⠌⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢦⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⣤⡀⢤⠀⠈⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⠀⢀⠄⠀⠀⠀⠀⠀⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠏⠀⠀⠀⠇⠤⠤⠄⠀⠀⢊⡡⠊⠀⠶⠀⠠⠤⠠⠴⢋⡀⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⡇⠀⡀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⡰⠛⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢰⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠳⡀⠀⢸⠁⠀⠀⣠⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⠎⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⠲⢌⣓⡒⠲⢷⡁⠀⠀⠀⠘⠀⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⠦⡀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠙⣦⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⢀⡠⠚⡇⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⠲⠶⠀⠀⠀⠀⠀⢶⠚⠛⠛⠉⠁⠀⠀⠀⠀⢀⡠⠔⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡄⠀⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⢚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⣀⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⣴⣾⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⢀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢱⣾⣿⠋⢫⡀⣿⣿⠃⡜⠀⠀⠀⠐⠑⠃⠀⠀⢀⣤⣉⣦⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢠⠊⠐⠶⠋⠀⠑⠀⠸⠄⣻⡏⢰⠁⠀⠀⡇⣠⠔⢀⣀⠀⠈⢻⠻⣿⠀⠀⢠⠃⣀⠆⢡⡐⡄
⠀⠀⠀⠀⠀⡔⠃⢶⢀⠀⠀⠀⠈⠂⠂⣬⣿⠁⡇⠀⠀⠘⢛⡇⠀⠈⠁⠴⣶⢬⣧⣿⠀⠀⡄⠠⠃⢠⡘⠁⡰
⠀⠀⠀⠀⠾⢀⠀⠀⠁⠀⠀⠀⠀⠃⠲⣠⣿⡀⡇⠀⠀⢀⣻⣇⠀⢀⡒⠀⢠⣼⡿⠃⠀⡐⢠⡄⣸⠜⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠀⠁⠐⢳⢿⣧⠘⣄⠀⠈⠩⠈⢦⣬⣤⣶⣿⡿⠂⢀⠄⣰⣼⠗⠁⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⢀⣉⠿⣭⣤⡤⠁⠀⠀⠀⠀⢙⣿⡁⣁⣠⣴⣡⣾⠏⠈⠖⢀⠀⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠈⠠⠀⠀⠊⠉⠀⠀⠀⠀⠀⠈⠲⠿⢻⢿⣿⣿⣿⡟⡀⠌⠀⠠⠀⠀⠠⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠲⠦⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣿⡿⢇⠀⠀⠃⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⡚⠀⠀⠀⠠⡔⠀⠀⠁⠀⠀⢀⠀⠀⠀⠸⠤⢈⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣤⠞⠀⢀⣣⠄⠀⠀⠀⠀⠀⠀⢰⣄⠀⠁⠀⢀⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠔⡟⠁⠀⢠⣟⠋⠀⠀⠀⠀⠀⠀⢀⣿⢸⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢸⠃⠀⣠⣿⣭⠁⠀⠀⠀⢀⡠⠴⠋⢹⠐⠄⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣴⣿⣿⡀⣀⠀⠀⠴⠋⠀⠀⠀⢸⠀⠑⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢀⣀⣶⠞⠓⠒⠒⠲⠠⠄⢠⠇⠸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠤⣀⠈⠉⠚⠳⠤⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣠⠶⣥⡀⠀⠀⠀⠀⠀⠀⠰⢦⣀⣹⠦⣝⡲⢤⣬⡽⠆⠉⠛⠉⠁⣀⠀⠐⠲⢯⣥⠤
⠀⠀⢠⡟⠈⠐⠈⠓⢦⣀⠀⠀⠀⠀⠀⠈⠙⠂⠀⢉⣀⣀⠠⣄⡒⢿⣩⡷⠬⣝⡲⢤⡤⠼⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠
⠀⠀⣾⡊⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠘⠻⢥⣾⠓⢲⣭⠓⠮⠙⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⠔⠁
⠀⠀⠘⠻⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠙⠿⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣴⣾⡿⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠰⢸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⡲⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣤⡀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⡜⣿⠁⠀⣀⣠⢴⠒⠋⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⢿⠯⠛⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠠⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠐⠀⠒⠚⠑⠀⠂⠀⠀⠀⠉⠁⠒⠂⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠠⠤⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⡇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣜⣟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠓
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣮⡿⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢾⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢢⣦⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠐⠀⠀⠌⡀⠌⠀⠄⡰⡿⠍
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠐⠠⡀⠠⠐⠄⠂⠅⠂⠄⡐⣰⣷⠏⢄
⠀⠀⠀⠈⣴⠸⢷⣶⡴⣀⠠⠄⠀⣀⠀⠁⠄⠢⢀⠄⠀⠅⡀⢀⠠⢁⠀⡁⢌⢠⠁⠾⣿⠁
⠀⠀⠀⠀⠹⡆⠀⠀⠉⠉⠛⠓⠲⠴⠦⢬⣀⡀⠒⠠⠤⠁⢀⡐⠀⠀⠔⠤⠂⠌⢎⣝⠏⠐
⠀⠀⠀⠀⠀⠢⢖⠤⢀⡀⠀⠀⠀⠀⠀⣀⡉⠛⠻⠿⡶⠶⣤⣤⣌⣁⣐⠒⢉⡔⡘⡌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠒⡠⠈⠐⠊⠐⠂⠽⠣⢶⡲⠤⠄⠀⠈⢁⠃⢀⢡⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠡⠾⡀⢭⣤⡀⣀⠀⠤⠈⣄⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡍⠏⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠛⠀⠈⠓⠲⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⣶⠀⢰⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⣠⠤⣄⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣀⡴⠋⠁⠀⠈⠑⢦⡀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢀⡤⠞⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠳⣄⡈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣠⡞⠋⠀⠀⠀⠀⣠⢞⣫⠽⠷⡶⠿⣭⣳⣦⠀⠀⠀⠈⢻⣦⡀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡴⣾⢹⡇⠀⠀⠀⠀⡾⣱⠏⠁⠀⠀⡇⠀⠀⠙⣿⣧⠀⠀⠀⢸⠉⡿⢦⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⢸⡇⣿⠒⠒⠒⠲⡗⠒⠒⠒⣿⣿⠀⠀⠀⣼⢰⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⠀⡇⠀⠀⠀⠀⢷⡹⣄⡀⠀⠀⡇⠀⢀⣴⢿⡟⠀⠀⠀⣿⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠶⣿⠀⡇⠀⠀⠀⠀⠀⠙⠲⢭⣙⣛⣛⣛⣭⠵⠋⠀⠀⠀⠀⡇⢸⠷⣌⡛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠴⠚⠉⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⠙⠳⢦⣍⡛⠶⣤⡀
⠀⠀⠀⣠⡴⠞⢉⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠈⠙⠶⣤⣙⠳⢦⣀
⣠⠶⠋⣁⣤⡞⠋⠁⠀⠀⣀⣤⣤⣄⡀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣠⣶⣶⣶⣦⡄⠀⠉⠳⢦⣌⠛⠶⣄
⠹⠶⠟⢻⠘⡇⠀⠀⢀⣾⡿⠛⡟⠻⣿⣆⠀⣿⠀⣿⠀⠀⠀⠀⢀⣤⢶⣾⣿⣿⣓⠶⣄⠀⠀⠀⠀⠀⢸⠃⡏⠀⣰⣿⠋⣹⡉⢹⣿⡆⠀⠀⢀⡿⣻⢦⣤
⠀⠀⠀⠘⡆⣿⠀⠀⢸⣿⠖⠒⡶⠒⢾⣿⠀⢹⠀⢸⡀⠀⠀⢠⡞⣱⠏⠁⠈⡇⠉⠻⣮⢷⡄⠀⠀⠀⣸⠀⡇⠀⢹⡿⠉⢹⠉⣹⣿⠇⠀⠀⣼⢣⡇
⠀⠀⠀⠀⢿⢸⡆⠀⠀⠻⢵⣤⣷⣶⡿⠋⠀⢸⡆⢸⡇⠀⠀⣿⢠⡏⠀⠀⠀⡇⠀⠀⠈⣷⢻⡄⠀⠀⣿⢸⡇⠀⠀⠙⠷⠾⠾⠟⠋⠀⠀⢰⡏⣼
⠀⠀⠀⠀⢸⡆⣇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⢸⡃⠀⠀⠀⡇⠀⠀⠀⢸⡎⡇⠀⠀⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡇
⠀⠀⠀⠀⠈⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⢸⠉⠉⠉⠉⡏⠉⠉⠉⢸⡇⣿⠀⢰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾
⠀⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⢸⡇⣼⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⣿⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⡇
⠀⠀⠀⠀⠀⠸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠇⣿⣤⣤⣤⢤⣧⠤⣤⣤⢼⡇⣿⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾
⠀⠀⠀⠀⠀⠀⣷⢸⡆⣤⠤⠤⠴⠶⠶⠶⢶⠀⣿⠀⣷⠀⢸⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⣿⠀⣿⠀⡟⣴⠶⠴⠦⡤⠤⠤⣤⠀⡿⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⣧⢻⠀⠀⠀⡇⠀⠀⢸⡆⣿⠀⣿⠀⢸⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⡇⢠⡇⠀⡇⢸⠀⠀⠀⡇⠀⠀⣿⢸⠇⡟
⠀⠀⠀⠀⠀⠀⠀⣇⢹⣸⣇⣀⣀⣧⣀⣀⣸⠇⣿⠀⣿⠀⢸⡄⣧⠤⠤⠤⢤⣧⡄⢠⢀⣸⡇⡇⢸⡇⢰⡇⢻⣀⣀⣀⣇⣀⣠⣧⡟⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡌⣧⠉⠉⠉⠉⠉⠉⠁⠀⢹⠀⣿⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⡇⢸⠇⢸⡇⠀⠉⠉⠉⠉⠉⠉⣸⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡇⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⣿⣿⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠞⠛⠋⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⡟⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣷⡄⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠛⠛⠛⢷⣻⣇⠼⠛⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⣠⣤⣤⡀⠀⠁⣠⣴⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⡦⠚⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣭⣭⠍⠙⠛⠛⠛⠛⠙⠛⠉⠛⠻⠿⠄⠀⠀⠈⠉⠙⠋⠙⠛⠛⠛⠛⢛⣿⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣟⢿⣿⣿⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠻⣏⠙⣿⣷⡟⠀⢀⡖⢀⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢻⡇⠀⢎⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠡⠂⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⣿⣿⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⣿⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣷⣶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠷⠚⠁

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