Zzz Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⡄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⣟⣀⡀⣨⡿⠀⢀⣛⣂⣀⠈⠀⠈⢀⡠⣀
⠀⠀⠀⠀⢠⠊⠑⢆⠀⣀⡠⠤⠖⠒⢋⡽⠋⠀⠀⠾⠷⠖⠂⠀⠀⠀⠀⠉⠉⠐⠂⠀⡸
⠀⠀⠀⠀⠀⢑⠀⠈⠉⡀⠀⠀⠀⠰⠿⠶⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡆⠀⠀⡈⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⡀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡜⠑⠒⠦⠤⢤⣤⣤⠤⠤⠤⠤⠀⠒⠒⠉⠉⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢦⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠉⠉⠉⣀⡐⢂⠟⠁⠀⠀⠀⠀⠀⠀⠐⢀⣩⡅⠀⠀⠠⢰⠄⠀⠩⠍⣉⡙⠒⠠⠀⠀⠀⠀⠀⠀⡼⠁⠀⣻⠃⠀⢹⡀
⠀⠀⠀⠀⠀⢠⠆⢈⡜⣵⣯⢤⡤⡀⠀⠀⠀⠀⠐⢈⡵⠋⣀⠀⠀⣞⣁⣀⠀⣠⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠾⠷⠒⠀⠈⡇
⠀⠀⠀⠀⠰⠟⢂⡾⠴⠂⣀⡉⠁⡇⠀⠀⠀⠀⠀⠟⠛⣉⠡⠤⢤⣄⣀⣀⣀⣀⣈⣉⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣺⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠤⢒⣫⠭⠗⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⢼⠀⠀⠸⠷⠒⠀⠀⣧
⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠀⠐⣩⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣘⡆⠀⠰⠷⠖⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⠖⠒⠲⠶⠦⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠐⣻⠟⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠈⡆⠘⠷⠒⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⣸⠊⢸⢻⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⣻⠃⠀⢉⡽⠃⠀⣩⠟⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣰⠋
⠀⠀⠀⢀⠀⠀⢠⠥⡁⠨⢍⠀⠀⠀⠀⢳⡀⠀⠀⢀⣼⡥⠄⠀⠾⠤⠄⠐⠷⠆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⢿⠑⢦⡀⡇⢱⠈⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⢸⠘⡓⢍⡛⢸⡄⠇⡀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⡀⢀⠠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⠒⢒⡒⠒⣂
⠀⠀⠀⠈⠀⠇⠀⠙⠦⠇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠭⠠⠾⠅⢠⣏
⠀⠀⠀⠀⢀⣀⠀⠰⡓⢦⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠋
⠀⠀⠀⠀⠸⡬⣕⠦⣡⠘⡄⢿⡀
⠀⠀⠀⠀⠀⢣⠹⡑⢦⡀⢣⠸⡆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢁⡤⣍⠺⠀⢧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⢂⠶⠀⠀⠐⢪⡇
⠀⠀⠀⠀⢻⢒⡤⣄⠳⡘⡆⠀⠸⡆⠀⠀⠀⠀⠀⣇⠀⠀⠐⠒⡸⠃⠀⢀⡾⠂⠀⢈⠟⠀⠀⣿⠀⠀⠀⠀⠘⠛⠲⣤⠀⠀⡜⡇
⠀⠀⠀⠀⠈⢏⠉⠓⠭⣃⠸⡄⠀⢳⠀⠀⠀⠀⠀⣿⠀⠀⠀⡴⠁⠀⠀⠛⠛⠂⠐⠧⢤⠄⠀⣿⠀⠀⠀⢰⠀⢠⣾⡥⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⡤⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠸⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠄⠀⠀⣾⡄⠀⠀⣠⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠤⢤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⢿⠟⠀⠀⠀⠳⣑⣀⣚⠓⠒⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣀
⠀⠀⠀⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠻⠿⢶⣦⣄⠀⠀⣀⣤⣤⣴⣶⣶⣦⡄⡼⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠇⠈⡄⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⠛⠉⠁⠀⠀⠀⠉⢁⡤⢊⣥⡀
⠀⠀⠀⢠⡶⣄⠸⡀⢰⠶⠚⠛⠋⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠑⢪⡹⡗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣤⡄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⢰⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠟⣿⠁⠀⠀⠀⢀⣠⡶⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠭⠭⠕⠒⠒⢡⣾⠃⢀⣠⡴⠾⢻⣿⠃⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⣰⣿⠃⢀⣴⡿⠃⠀⠀⠀⠀⠀⣠⢾⣿⣄
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣼⡿⠁⣴⣿⣋⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⢿⣷⡀
⠀⠀⠀⠀⠀⠻⣶⣄⡤⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⢠⣾⠟⠀⠀⠙⠛⠛⠛⠿⠿⠿⠇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠿⠿⠾⠿⠿⠿⢿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡶⠶⠶⠶⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢛⣤⠄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⡀⠴⠀⠙⠋⠀⠀⠈⠉⠉⠉⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣴⠾⠟⣁⣀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣙⡻⣶⣄⢰⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣀⣀⣀⣀⣠⣤⣶⣿⠿⠋⣸⣿⠘⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⡴⠁⠈⣇⠀⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠉⠛⠛⠉⢩⡏⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣾⣥⣤⠶⢿⣦⢼⣧⣤⣤⣤⣴⠾⢿⣋⠀⠀⠀⣀⣠⣤⠾⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠘⣿⠄⠈⢿⣄⠀⠀⠀⠀⠉⠉⠉⠉⢯⣁⣀⣀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢸⠃⠀⠐⣺⠇⠀⢙⡿⠀⢈⣽⠟⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣨⡿⠋⠀⠀⠈⢙⡷⠂⠀⠐⠲⢦⣤⣀
⠀⠀⠀⠸⣧⠀⠼⠧⠶⠀⠿⠶⠂⠘⠛⠃⠋⠃⠁⣰⠇⠀⠀⠀⢠⣾⣋⣀⡀⠀⣀⡴⠋⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠉⠛⠶⢦⣤⡀
⠀⠀⠀⠀⠈⠛⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠛⠛⠲⠶⠀⢠⣶⣋⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠆⠀⢴⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⡇⢀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⡶⣄⣼⠃⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⡶⠋⠀⠀⣾
⠀⠀⠀⠀⠈⠀⠈⠋⠘⢹⠟⠃⠀⠀⠀⠀⠀⢀⡀⠀⠴⢻⠇⢉⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠖⠚⠛⠛⠛⠒⠶⠦⣤
⠀⠀⠀⠀⣿⢷⣄⣾⠀⢸⠀⠀⠀⠀⠀⠀⢠⣾⠃⢀⡄⣿⡴⠋⠀⢀⡼⠃⠀⠀⠀⢀⣤⠖⠛⠉⢁⡀⠀⠶⣶⠆⠐⢛⣷⠂⠀⠀⣰⠏
⠀⠀⠀⠀⠋⠀⠙⠃⠀⢸⠀⠀⠀⠀⠀⠀⠋⣟⣰⠟⠀⠛⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠙⢳⡄⢀⣿⣁⡀⠼⠷⠶⠰⠿⠥⣤⠀⠀⢻⣄
⠀⠀⠀⠀⣶⣤⡀⠀⣿⢸⡆⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢈⣉⣉⣠⣤⣤⣤⠤⠤⠶⠶⠶⠤⢤⣽⡆
⠀⠀⠀⠀⠏⠀⠙⠷⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠁
⠀⠀⠀⠀⣀⠀⠀⠀⣀⠸⡇⠀⠀⠀⠀⢠⠴⠋⠁
⠀⠀⠀⠀⣿⠳⣄⠀⣿⠀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢹⠀⠈⠳⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⣒⢲⡆⠀⠀⠀⠀⠀⠀⠀⠠⠴⢶⣶⠆⠐⠲⢶⡶⠀⠐⠛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⣿⣿⣿⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡟⠀⠀⠀⠉⠁⣾⡟⣼⣿⠿⠃⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠘⠛⠛⠛⠋⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠸⢾⠷⣫⢾⡃⢀⣤⡀⠀⣼⣿⣬⠽⠖⠋
⠀⠀⠀⠀⠀⢠⣾⣷⢆⡇⠀⠀⠀⢀⣿⡷⣫⠵⣫⡾⠃⠀⠈⣀⣤⡴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠁⣾⢸⡥⢖⠀⠀⣾⣫⠾⠃⢊⣁⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⢉⣩⡽⠟⠋⠀⠀⠉⣹⠟⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣶⡾⠟⠁⢀⡈⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡿⢁⣤⣶⣋⣁⣀⣀⠀⠀⣴⣟⣁⣀⡛⠛⣹⠟
⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣈⣉⣉⠉⠉⠉⠉⠋⠀⠉⠉⠉⠉⣡⣾⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢴⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠀⢀⡾⠃⠀⠐⠒⣺⡟⠀⠀⠴⠯⠄⠀⠀⠘⠀⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⣶⠟⠛⠋⠉⣩⣿⣋⣉⠉⠉⠉⠉⠛⠒⠒⠲⠤⠤⣄⡜⠉⠉⢳
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⣆⣀⠤⠴⠒⠋⠉⠁⠀⣠⡾⠁⠀⠀⢀⣀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋
⠀⠀⠀⠀⠀⠀⠀⢙⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠾⠿⠖⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡐⠈⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⡁⠊⠀⠜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⠁⢀⠊⠀⡠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡦⠁⠀⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⣄⠊⠀⡠⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠐⣀⠜⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠹⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡖⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠚⠉⠁⠀⠀⠓⠦⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡠⠏⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⠖⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡤⢀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣋⣉⣉⡉⢉⣿⣿⣿⠋⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠰⠿⠿⠿⠟⣿⣿⣿⠋⣠⣿⣿⣟⣁⣀⣀⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣥⣤⣤⣤⣤⣴⣶⣿⡟⠁⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⡶⠒⠲⠤⣤⠚⠉⠉⠉⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠋⠀⠠⠀⠒⠄⠀⠀⠈⠣⢀⠀⣠⠔⡃⠀⣄⠠⢄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠰⢶⡷⠆⠀⠀⠀⠀⠀⠀⠒⡰⢃⣠⡄⠝⠂⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢀⠀⠀⠈⠦⠃⠀⠀⠀⠀⠀⢸⠩⠖⠒⠃⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠂⠐⠒⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⡀⠀⠘⠀⠀⠀⢀⣤⡒⠐⠀⠀⠀⠀⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⡖⠂⠀⠀⠀⠈⡱⢱⣧⡴⠀⠀⠀⠁⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠈⠁⠁⠒⠀⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠷⠄⠀⠀⢀⣠⠶⠚⠉⢉⡇⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⡩⠴⠒⣻⢀⡞⠀⠀⠑⠀⠀⠀⣀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠃⣸⠁⠀⠀⠀⠀⠀⠀⠈⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡇⠀⠀⠀⠐⢄⡀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣤⡄⠀⠂⠦⣄⡀⠀⡆⣾⠀⣀⡤⠖⠋⢳⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢄⠀⠀⠀⠈⠃⠟⣋⣥⠴⡶⠛⠉⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡑⠀⠈⠀⠀⠀⠑⠉⠁⠀⠀⢀⡂⠀⣀⡠⠞⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣦⡀⠁⠀⠢⠄⠀⠀⢀⡱⠒⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣌⠓⠉⠑⠒⠲⠶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠤⠤⠤⠖⠒⣊⣩⣤⠀⠃⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⠂⠉⠉⢁⠘⠛⠲⠶⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣤⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡦⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢲⡞⠀⠀⠀⢉⡽⠋⠀⠀⠉⠒⣲⡦⠀⠉⠁⠀⠀⠀⠀⠀⠀⣸⠁⠀⣠⠞⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣹⠃⢀⡞⠵⠿⠖⣾⠇⡆⠀⠀⠀⠀⠀⠀⢠⡾⣋⣠⠄⠀⠐⠯⠤⠤⠄⠰⠾⣏⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⠛⠛⣉⠀⠀⢻
⠀⠀⠀⠀⠀⠞⠓⠃⡾⠖⠋⣀⣤⠅⢀⡇⠀⠀⠀⠀⠀⠀⠛⢋⣁⠤⠴⠖⣶⣦⡤⠤⠤⠤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢈⣽⠟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⢒⣊⡭⠵⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⢼⠃⠀⠀⠿⠷⠒⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠒⢚⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⡀⡆⠀⠀⣴⣯⣤⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢻⠀⠀⠠⢤⣤⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢠⡄⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣠⣞⣁⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠀⡷⡇⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⡀⠀⢀⣤⡀⠀⠀⠀⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⠉⠉⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡬⡍⠀⡇⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⣠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣀⡴⠋
⠀⠀⠀⠀⣀⠀⠀⢠⡶⣅⠀⡴⠅⠀⠀⠀⠀⠸⣆⠀⠀⠀⢀⣼⣥⠴⠀⠰⠷⠶⠆⠀⠛⠒⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠸⡏⠳⣄⠀⣇⢸⡄⠃⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⡇⢳⢦⡑⠾⠈⡇⢸⠘⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠃⢈⡆⠈⠳⣄⣿⠀⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⡇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⢀⡞⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣉⠀⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣠⣭⣭⣥⣼⡏
⠀⠀⠀⠀⠀⡖⠦⣀⠀⢧⠈⡇⠸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠸⡰⣮⣑⠾⠆⢹⠀⣻
⠀⠀⠀⠀⠀⠀⢣⠹⡈⠳⣄⠈⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⣌⠙⠟⠈⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⢦⠀⠀⠀⠀⠀⠀⠒⣰⠗⠀⠀⠐⢢⢹
⠀⠀⠀⠀⠐⡗⠦⢄⡀⠱⡄⢣⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠒⣲⠂⠀⠀⢲⠖⠀⠀⢙⡷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠿⠶⣂⣀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠹⡙⡓⠮⣕⠻⠈⢧⠀⠀⢳⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⡼⠁⠀⠀⠰⠧⠦⠀⣴⣏⣀⡀⠀⢸⡇⠀⠀⠀⢠⠀⠀⣠⠞⢁⡀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠙⠲⢼⡆⠀⠈⠃⠀⠀⠀⠀⢸⣿⠀⠀⢀⣼⣡⡤⠴⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⢇⠀⠀⠀⢸⢸⠀⠉⠉⢩⡤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⣿⡎⠀⠀⠀⣿⢸⠀⢀⣴⣋⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⠙⠚⠝⠁⠀⠀⠀⠘⠦⠥⠤⢭⣭⣤⣤⡴⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⢿⣿⡿⠁⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡤⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠛⠛⠛⣿⣿⠟⠁⠀⣶⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠼⠵⠒⣒⣁⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⢀⣾⣿⣋⣀⡀⠀⠀⢀⣾⣿⣃⣀⠀⢀⡀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠀⢠⣿⣿⡿⠿⠿⠿⠀⠀⠾⠿⠿⠛⠋⠀⠙⠋⠈⠁⠈⠋


⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠦⠶⠶⠖⠒⠒⢒⡦⠀⠀⢸⠀⠸⠦⠶⠶⠖⠒⠒⢒⡦⠀⠀⢸⠀⠀⠷⠶⠶⠖⠒⠒⠒⣶⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠛⠓⠛⠒⠚⠒⢲⠀⢸⠁⠀⠀⠀⠛⠓⠒⠒⠒⠚⢲⠀⠀⡏⠀⠀⠀⠘⠛⠒⠓⠓⠓⠚⡆
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣼⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣼⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣤⣀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠄⠀⠀⠀⠀⠀⠠⠤⣤⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠋⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣿⠋⠀⠐⢒⣶⠇⠀⣠⡾⠋⠀⠀⠀⠀⣀⡤⠤⠶⠶⠶⠦⢤⣄⡀
⠀⠀⠀⠀⣀⠴⠋⢀⣀⡀⡰⠋⠀⠀⠀⡀⠀⠀⠀⢀⣾⣿⣥⣶⣄⣴⡿⠃⢰⣾⣿⣤⣤⠄⠀⣰⠊⠁⠀⢠⠀⠀⢀⡄⠀⠀⡍⢳⡀
⠀⠀⠀⠘⠛⠋⠉⠉⢡⣞⣥⠤⠴⢀⠞⠁⠀⠀⠀⠈⠋⠉⠁⠀⢾⡿⠿⠿⠗⠀⠀⠀⠀⠀⠀⡇⠀⢀⣴⣁⡀⢰⣏⡤⠠⣎⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⠖⠒⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⠀⠀⠀⠀⠀⠀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⠟⠀⣶⣶⣶⣶⣶⡶⠂⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡠⢒⠜⢁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠉⠉⣹⣿⠟⠁⠀⠻⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠧⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣥⣤⣤⠀⠀⣼⣿⣿⣶⣦⡄⠀⢠⣿⣿⣶⡶⠀⣶⠄⠶⠀⣶⠄⠀⠀⠀⠀⠈⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠋⠉⠉⠉⠉⠀⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⢤⡄⠀⠀⡇⠀⠧⠤⠤⠤⠤⠤⢤⡄⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⣤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⣠⠚
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⡇⠀⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⣷⠀⢸⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠳⠴⠶⠶⠶⠶⠶⠶⠶⠶⠇⠀⠳⠤⠶⠶⠶⠶⠶⠶⠶⠶⠏⠀⠸⠦⠶⠶⠶⠶⠶⠶⠶⠶⠞

⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣾⡿⠃⠀⣀⣀⡀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⡀⢀⣴⠟⢀⣴⣟⣁⠀⠀⠀⡠⠖⠋⠉⠉⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠴⠯⠴⠖⠒⣣⠎⣀⣀⠀⡨⠂⠀⠀⠀⠻⠟⠛⠛⣽⣿⣧⣤⡍⠉⠉⠉⠉⠀⡞⠀⠀⢠⠎⠀⢠⠞⠀⣠⠟⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⢡⣎⡤⠤⠠⠤⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠛⠉⠁⠈⠉⠁⠉⢉⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠛⠋⠉⠉⠛⠛⢿⣿⣿⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠋⣹⣿⠟⠀⠀⠀⠛⣻⣿⠟⠀⠀⠀⠛⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣁⣀⠀⠀⢀⣴⣿⣃⣀⠀⠀⢀⣴⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠀⠀⠈⠛⠛⠛⠛⠀⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⢀⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣤⣀⣀⣀⣀⣀⣤⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡄⠙⢿⣿⣿⣶⣤⣤⣀⣀⣤⣤⣶⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠹⠿⠏
⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢤⡀
⠀⠀⠀⠀⠀⠒⠂⠉⣱⠂⠀⠀⠀⠀⠀⠀⣐⡀⠉⠉⢠⠉⠐⢒⡂⠠⠄⠀⠀⠀⠀⢀⠎⠀⡶⠀⢳
⠀⠀⠀⠀⡴⠀⡽⡴⢥⡤⠀⠀⠀⠀⠀⣨⣋⡀⠀⠶⢅⠀⢠⣖⡋⠀⠀⠀⠀⠀⠀⠸⠀⠾⠒⠂⠘⡆
⠀⠀⠀⠘⠉⢚⣋⢤⣄⡠⠆⠀⠀⠀⠈⠩⠁⠐⠒⠶⠐⠒⠒⠂⠤⠄⠀⠀⠀⠀⠀⡇⠀⢨⣟⡁⠀⡇
⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠨⢥⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣠⠀⠰⠷⠂⠀⡇
⠀⠀⠀⠀⠀⠀⣄⢀⡀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠚⠒⠒⠒⠲⠤⢄⡀⠀⠀⠀⠀⠀⠈⡄⢀⡝⠁⢀⡇
⠀⠀⠀⠀⠀⠠⡆⢸⡇⠀⠀⢠⠊⠀⠀⠀⢤⠄⠠⢤⠄⠀⡤⠀⠀⠈⠳⡀⠀⠀⠀⠀⠱⡈⠉⢀⡜
⠀⠀⢀⠀⠀⡬⡃⠸⠁⠀⠀⠸⡄⠀⠀⣠⣫⠄⠰⠧⠄⠼⠥⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠑⠒⠋
⠀⠀⢸⢑⠦⣰⢰⢀⠀⠀⠀⠀⠈⠢⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠘⢈⡙⠪⣘⠀⣠⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⡄⡄⠈⠉⠀⠀⠀⠀⠀⠀⣠⠆⢠⠎⢩⠄
⠀⠀⠀⢀⠀⠀⡬⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠙⠁
⠀⠀⠀⠸⢬⣂⢥⢸⠸⡄
⠀⠀⠀⠀⠃⠎⠓⢄⡆⣇⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠒⢲⠶⠤⠤⡄
⠀⠀⠀⠤⢄⡀⢏⢢⠀⠸⡀⠀⠀⠀⢰⠀⠀⠠⢤⡄⠀⢠⠄⠀⢠⠄⠈⡇⠀⠀⠀⠰⠯⣀⠀⢸⡇
⠀⠀⠀⠘⡍⠘⠭⣃⢃⠀⢣⠀⠀⠀⢸⡀⠀⢀⠎⠀⠀⠛⠂⠰⠧⠄⠀⡇⠀⠀⠀⠀⣴⠥⠀⠀⡇
⠀⠀⠀⠀⠐⠐⠀⠀⠉⠀⠀⠀⠀⠀⠸⣇⡀⠛⠉⠁⠀⠀⠀⠀⠀⢀⣰⡟⠀⠀⡿⡂⢀⣜⡁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠓⠒⠒⠐⠒⠁

⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⣤⣤⣀⠀⠀⠀⣀⣀⣠⣤⣀⡀⡘⠉⣱⠆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⡟⠛⠉⠉⠉⠙⢃⡠⣪⣅
⠀⠀⠀⡖⢦⡐⡀⠴⠖⠛⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⢿⣆
⠀⠀⠀⠈⣰⠝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠒⠒⠶⣶⡶⠀⠀⠀⠈⣿
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠤⠤⠴⠶⣚⡛⢋⡽⠁⠀⢀⣠⣴⡿⠀⠀⣠⡾⠋⠀⠀⠀⠀⣸⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⢠⡾⠃⠀⠊⠉⣡⡿⠁⣠⣾⠟⠀⠀⠀⠀⢀⣴⡋
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣴⡟⢁⣴⣟⡁⠀⠀⠀⠀⠀⠁⠉⢻⣧⡀
⠀⠀⠀⠀⠹⣦⣤⠄⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢀⣾⠏⠀⠈⠛⠛⠛⠿⠿⠟⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⣾⠏⠀⣀⠀⠀⠛⠛⠛⠛⠛⠻⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣾⠿⠛⡋⠭⠭⠭⢙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣘⣩⡶⠂
⠀⠀⠀⠀⠸⡇⠀⢀⡀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣭⣛⠉⣥⠖⠒
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠀⣀⣀⣤⣴⡿⠟⣿⡎⣷⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⡔⠀⢳⡀⢄⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⠛⠛⠋⡿⠁⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣾⠥⠶⠻⣷⠾⢧⡤⠴⠶⠞⠻⣅⣀⣀⣠⡤⠶⠛⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠿⠂⠈⠻⠀⠀⠀⠀⠀⠀⠀⠹⠦⡤⢖⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠈⠛⣿⣧⣀⣴⡶⠿⠛⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠛⠛⠛⠿⠶⣴⡿⠁⠀⠀⠀⠀⠀⠈⢻⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⡷⠀⠀⠾⠿⢿⣶⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠰⠿⠟⠛⠉⠉⣀⣴⠟⠁⠀⠀⣠⣶⡿⠋⠀⠀⠾⢿⣶⣤⡀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⢠⣾⣿⣋⣀⣀⠀⠀⢀⣤⣾⠟⠁⠸⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣤⣀⠀⠀⢀⣴⡿⣋⣤⣤⣶⣶⠘⠻⠛⠛⠛⠛⢀⣴⡿⠛⠁⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⠀⠀⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢠⣴⣿⣿⣤⣤⣄⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠿⠿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠛⠉⢿⣿⣷⣤⣤⣴⣿⣦⣄⣀⣀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⣀⣀⠉⠛⠛⠛⠋⠁⠉⠛⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣄⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟

⠀⠀⠀⠀⠀⢰⣶⣶⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠆
⠀⠀⠀⠀⠀⠈⠉⠉⣩⣿⠋⠀⠿⠿⢿⣿⡿⠃⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⣡⣣⠄⠐
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⢠⣿⣟⠀⠀⠈⠉⣹⣿⣃⡀⠀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠠⠨⠭⠭⠌⠈
⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠷⠀⠰⠿⠿⠿⠿⠇⠀⠰⠿⠿⠛⠃⠈⠛⠈⠁⠙⠁


⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⢰⠒⠒⠒⠒⠒⠒⠒⠒⢒⡆⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⡶
⠀⠀⠀⠀⠀⢸⣄⠤⠤⠤⠤⢤⡄⠀⢸⠁⢸⣤⡤⠤⠤⠤⢤⡄⠀⢸⠁⠀⣇⡤⠤⠤⠤⠤⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠜⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠜⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⡠⠊⠁⢀⡴⠋
⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣔⣁⣀⣀⣀⠀⢀⠔⠋⠀⢀⣔⣁⣀⣀⣀⠀⠀⡠⠊⠀⠀⣴⣋⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⠁⠀⢲⡶⠂⣠⡴⠋⠀⠀⠀⣀⡤⠤⠶⠶⠶⠤⣄⡀
⠀⠀⠀⣠⣖⣁⣠⠤⡠⠊⠀⠀⢠⠀⠀⠀⢠⣿⣿⠶⢶⣴⣟⣁⠾⠿⠶⠶⠀⢠⠎⠁⠀⡴⠀⢀⠄⠀⢠⠍⢣
⠀⠀⠀⠉⠁⠀⠀⠾⠗⠒⢛⡴⣁⡀⡀⡀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠸⡄⠀⠾⠔⠂⠛⠒⠐⠓⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣤⣤⣠⣤⡤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⠇⠀⠀⠘⠿⠿⣿⣿⠇⠀⠀⠸⠿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣶⣶⡆⠀⠀⢰⣿⣿⣶⣶⡆⠀⠀⢸⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣆⠈⠻⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣶⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⣶⠂⠀⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⡾⢁⣀⣀⣉⣹⣟⣀⣀⣞⠓⠂⠀⠀⠁⠀⠈⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠒⠢⣄⠀⠀⠀⠀⢀⣀⣠⠤⠶⢒⣾⠟⠉⠉⠉⣤⣟⣡⣤⠀⠀⠀⠉⠉⠉⠛⠒⠒⠤⠴⠁⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠘⡤⠴⠒⠋⠉⠁⠀⠀⣴⠟⢁⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠠⢄⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠁⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣨⠊⠀⠔⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢁⢀⠊⠀⡠⠊⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⡃⠀⠔⠀⢀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠐⡎⠀⡐⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠰⠎⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⠋⢦⣀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣠⠏⠈⠉⠓⠒⠶⠦⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠦⠤⣤⣄⣀
⠀⠀⠀⢠⡶⠋⠁⣀⠀⣀⣀⠀⢀⣀⡀⠉⠛⢦⡀⠀⠀⠀⠀⠀⢤⣤⣄⠀⠀⢀⣀
⠀⠀⠀⣿⠀⠀⣩⠟⠀⣨⡟⡁⢠⣞⡁⣀⡀⠀⢿⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡽⠋⠀⠀⠙⠓⢶⡤⠀⠀⠀⣀⣀
⠀⠀⠀⠘⢦⣀⠛⠛⠃⠉⠉⠉⠈⠉⠀⠀⠀⣠⠟⠀⠀⠀⠐⠿⠶⠶⠄⢀⣴⣋⡀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠉⢙⣳⡦
⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⡀⡀⠀⠠⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠛⠷⣤⣄⡀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠
⠀⠀⠀⠀⢠⣄⢀⡆⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣧⡾⠂⠀⣿
⠀⠀⠀⠀⠋⠈⠻⠁⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⡟⠘⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠶⠶⠦⠤⣤⣄⡀
⠀⠀⠀⠀⡶⣄⢠⡆⢸⡇⠀⠀⠀⠀⠀⣠⣾⠀⣀⢸⣧⠞⠁⠀⣰⠏⠀⠀⠀⢀⡤⠶⠚⠋⠉⠁⠤⣤⠄⠐⢶⡶⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠃⠈⠛⠀⢸⡇⠀⠀⠀⠀⠠⢻⣇⡴⠋⠘⠃⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣦⠀⣸⣃⡀⢴⠯⠤⠰⠿⣤⡄⠀⠸⣇
⠀⠀⠀⢰⡶⣄⠀⢸⣾⡇⠀⠀⠀⠀⠀⢸⠟⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣉⣉⣥⣤⣤⠤⠤⠤⠶⠶⠤⠤⣼⣧
⠀⠀⠀⠈⠃⠈⠙⠾⠇⡇⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉
⠀⠀⠀⢠⣄⠀⠀⢠⡀⣷⠀⠀⠀⠀⡠⠚⠉
⠀⠀⠀⠘⡏⠳⣄⣸⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠇⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⣫⣭⢹⠃⠀⠀⠀⠀⠀⠀⠀⠒⣻⠟⠃⠈⢉⡽⠟⠀⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠰⠿⢛⣿⢯⣏⣤⠀⠀⠀⠀⠀⠀⢠⣞⣁⣤⡄⢰⣯⣤⣤⡄⢼⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣾⣽⣿⠁⠀⠀⠀⢀⣾⠏⢿⣛⡯⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⣴⣚⢹⠀⠀⠀⠉⣵⣻⡷⣠⢞⣷⠀⠘⠛⠋⣉⣠⡤
⠀⠀⠀⠀⠀⠻⢻⡇⣏⣀⠀⠀⣴⢟⣵⠎⠔⠋⣁⣠⠴⠖⠛⠉⠀⠀⠀⠀⠀⢀⡄⠀⠐⠒⣒⣶⠗⠀⠀⠴⢶⡶
⠀⠀⠀⠀⠀⠀⣾⣯⣭⡶⠆⠀⠑⣉⠤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠁⣠⡴⠚⠉⠀⠀⠀⣠⡶⠋⠠⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠐⠟⠋⠁⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁⢛⣛⡛⠛⠛⠛⠒⠀⠛⠛⠉⠉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠻⣿⣦⡀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠶⢶⣦⣄⣰⡟⠁⠀⠀⠀⠈⠻⣷⠟⠋⠁⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣶⠀⠀⠾⠿⣶⣦⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠾⠛⠋⠉⢁⣴⡟⠁⠀⠀⣠⣾⠟⠁⠀⠾⠿⣶⣄⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢠⣾⣿⣤⣤⡀⠀⣠⣴⠿⠃⠀⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⣠⣾⣯⣴⣶⠿⠟⠈⠉⠉⠉⠉⣠⣾⠟⠁⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⢿⣶⣦⡄⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⣿⣷⣤⣀⣠⣶⣦⣀⠀⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣶⠿⠟⠋⠁⠀⣀⣈⠙⠻⠿⠛⠉⠙⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄
⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠈⢿⣿⡄
⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣴⣿⣿⣋⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⣿⣿⣿⠟⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⣿⣟⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢸⣿⣇⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢿⣿⣿⠟⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠸⣿⣷⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣶⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⢙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠉
⠀⣼⣿⠇⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⡿⠟⠋⠁
⢰⣿⣿⣤⣴⣶⣿⣿⠿⠟⠛⠉⠁⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⢿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⠄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢉⣉⠉⢩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡿⠃⣰⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠋⣠⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⣿⣧⣤⣤⣴⠆⢠⣿⣿⣿⠟⠀⠸⠟⠛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⠘⠛⠛⠛⠛⠋⢻⣿⣿⠟⢁⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣤⣤⣤⣤⣶⣶⣶⣾⡿⠋⠀⠀⠉⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⡶⠒⠲⠤⣤⠚⠉⠉⠉⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠋⠀⠠⠀⠒⠄⠀⠀⠈⠣⢀⠀⣠⠔⡃⠀⣄⠠⢄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠰⢶⡷⠆⠀⠀⠀⠀⠀⠀⠒⡰⢃⣠⡄⠝⠂⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢀⠀⠀⠈⠦⠃⠀⠀⠀⠀⠀⢸⠩⠖⠒⠃⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠂⠐⠒⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⡀⠀⠘⠀⠀⠀⢀⣤⡒⠐⠀⠀⠀⠀⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⡖⠂⠀⠀⠀⠈⡱⢱⣧⡴⠀⠀⠀⠁⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠈⠁⠁⠒⠀⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠷⠄⠀⠀⢀⣠⠶⠚⠉⢉⡇⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⡩⠴⠒⣻⢀⡞⠀⠀⠑⠀⠀⠀⣀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠃⣸⠁⠀⠀⠀⠀⠀⠀⠈⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡇⠀⠀⠀⠐⢄⡀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣤⡄⠀⠂⠦⣄⡀⠀⡆⣾⠀⣀⡤⠖⠋⢳⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢄⠀⠀⠀⠈⠃⠟⣋⣥⠴⡶⠛⠉⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡑⠀⠈⠀⠀⠀⠑⠉⠁⠀⠀⢀⡂⠀⣀⡠⠞⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣦⡀⠁⠀⠢⠄⠀⠀⢀⡱⠒⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣌⠓⠉⠑⠒⠲⠶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⠿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠹⣿⣆
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⣻⣿⠟⠁⠀⠀⠀⠀⢹⣿⡄
⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣷
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣠⣾⣟⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⡿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠘⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢿⣿⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⠀⢠⣾⣿⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣿⡆⠀⠀⠀⠀⠀⠈⠉⣩⣿⡿⠋⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⢻⣷⡀⠀⠀⠀⠀⣠⣾⣿⣯⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠈⢿⣷⡀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⣾⡿⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠿⠋⠁
⣸⣿⣧⣴⣶⣿⡿⠿⠛⠉⠁⠈⠙⠛⠿⠿⣿⣿⣷⣶⣶⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⠄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢉⣉⠉⢩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡿⠃⣰⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠋⣠⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⣰⣿⣿⣿⣿⣿⣧⣤⣤⣴⠆⢠⣿⣿⣿⠟⠀⠸⠟⠛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⠘⠛⠛⠛⠛⠋⢻⣿⣿⠟⢁⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣤⣤⣤⣤⣶⣶⣶⣾⡿⠋⠀⠀⠉⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠚⠛⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⠛⠉⠙⠲⣎⠁⠀⠀⠀⣀⠀⢀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⣀⣈⠂⠀⠀⠀⠃⠀⣐⡊⡅⠀⠢⢍⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠈⠙⢫⡑⡀⠀⠀⠀⠀⢈⠼⣑⣒⡆⠀⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⣠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠒⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠤⠀⠀⠀⠐⡠⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢱⡤⠄⠀⠀⠀⠛⡟⣔⣀⠀⠀⠀⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠑⠀⠀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠒⠄⠀⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠄⠀⠀⢀⣠⠖⠚⠉⣩⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠭⠖⠒⡷⢰⠇⠀⠉⠂⠀⠀⣄⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠑⠢⢄⠀⠀⢀⢁⡟⠀⠀⠀⠀⠀⠀⠈⢷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣈⣐⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⠀⠀⠑⠲⠄⢠⠀⣏⡠⠴⠋⣉⣸⠆⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠐⠀⠀⠀⠈⣀⡡⠶⠚⠙⠉⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡁⠀⠄⡀⡀⠀⠀⠀⠀⣀⡒⢀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⡲⠄⣀⣈⣁⣠⣤⠖⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠒⠶⠦⣤⣀
⠀⠀⢀⡞⠉⠀⣤⠀⠤⣤⠀⢤⡦⠀⠉⠳⡄⠀⠀⠀⠀⠒⣶⠆⠀⠀⢤⣤⠀⠀⣀⣀
⠀⠀⠘⣇⠀⣴⣧⠄⠾⠥⠄⠿⠆⠒⠂⠀⡿⠀⠀⠀⣤⣞⣁⠀⠀⣠⠞⠁⠀⠀⠀⣩⠟⠃⠀⠀⠒⠦⣤⣀
⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⠀⠀⣀⣀⠴⠚⠁⠀⠀⠀⠀⠉⠉⠀⠘⠓⠲⠦⠀⣴⣏⡁⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠰⠿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠂⠄
⠀⠀⠀⠀⣤⡀⡤⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡴⠂⠀⡟
⠀⠀⠀⠈⠁⠙⠁⢻⡟⠂⠀⠀⠀⠀⢀⠀⠠⢺⠃⡋⠀⠀⣼⠃⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠖⠒⠒⠲⠶⠤⣄
⠀⠀⠀⢸⠳⣤⡇⢸⡇⠀⠀⠀⠀⣰⡟⢀⡄⣿⠞⠁⢀⡼⠁⠀⠀⠠⣴⡚⠉⢁⡄⠀⢲⠆⠀⣹⠟⠀⠀⣴⠃
⠀⠀⠀⢀⠀⠈⢠⣸⡇⠀⠀⠀⠀⢹⣷⠋⠀⠁⢀⡴⠋⠀⠀⠀⠀⠀⠈⣿⠀⡾⠴⠄⠛⠛⠀⠛⠛⢂⠀⠙⣦
⠀⠀⠀⠿⠙⢦⣼⣿⡇⠀⠀⠀⠀⠘⠁⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠸⠧⠶⠖⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠁
⠀⠀⠀⣀⠀⠀⢈⠀⡇⠀⠀⠀⢀⠤⠚⠉
⠀⠀⠀⢹⠳⣄⢸⡇⠇
⠀⠀⠀⠘⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣒⣭⣽⠃⠀⠀⠀⠀⠀⠀⠒⣻⠟⠀⠘⣻⠟⠀⠉⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠺⠛⣿⣿⣯⣤⠀⠀⠀⠀⠀⢰⣾⣥⠤⠀⢾⠥⠤⠄⠾⠥⡤
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢢⣾⢿⣿⠀⢀⡀⠀⣼⣏⣻⠿⠗
⠀⠀⠀⠀⢠⣾⡷⡼⠀⠀⠀⣰⡿⣻⢞⣵⠟⠀⢉⣡⡤⠶⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⢰⡇⢷⣒⡀⠰⠿⠞⣁⡩⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⣠⡞⠀⢀⣭⠞⠋⠁⠀⢈⡽⢋⣀⣀⣀
⠀⠀⠀⠀⠀⣾⠿⠛⠉⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡟⠰⠾⠿⠶⠶⠤⡄⠰⠿⠶⠶⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀
⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠹⣿⣇
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⣿⣋⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⡿⠃⠈⠻⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡄⠀⣠⣾⣿⣯⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⢋⣿⣿⠟⠁⠀⠙⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⢿⣿⡀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠋⠁
⠀⣾⣿⠁⠀⠀⢀⣀⣤⣴⣶⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠛⠉
⣸⣿⣧⣴⣶⣿⡿⠿⠛⠋⠉⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠛⢿⣿⣦⠀⢀⣠⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣦⣤⡀⢰⡿⠃⠀⠀⠀⠀⠀⠙⢿⣷⡿⠛⠉⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠿⢿⣿⠏⠀⠀⠙⠛⣻⣿⡿⠀⠀⣀⣀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⢀⣴⡿⠃⠀⠀⢀⣠⣾⠿⠋⠀⠀⠘⠛⠻⢿⣶⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢰⣿⣿⣥⣤⣤⡀⠀⠀⣠⣶⡿⠋⠀⠘⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣦⣄⡀⠀⠀⣴⣿⣋⣤⣤⣶⣶⡶⠈⠛⠛⠛⠛⠉⢁⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⠀⠀⣾⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣤⣤⣄⣀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠻⠿⢿⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠙⣿⣿⣶⣄⣀⣠⣼⣷⣄⡀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣶⣶⣶⠿⠛⠋⠀⠀⠀⠈⠛⠿⢿⣿⠿⠛⠙⠻⠿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢰⠟⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⠉⠁⠀⠀⠀⠉⠙⠻⢿⣿⣿⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡄⠀⠀⢠⣶⣶⣶⣶⡆⠀⠀⠠⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣁⣀⡀⠀⠀⢠⣾⣿⣃⣀⡀⠀⠀⢀⣴⣿⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠈⠻⢿⣿⣿⣷⣶⣶⣤⣤⣤⣶⣶⣿⣿⣿⡿⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢰⠃⠉⣹⠆
⠀⠀⠀⠀⠀⠀⢸⠀⢂⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣤⣶⡿⠟⠛⠛⠛⠛⠻⠇⠏⡴⢟⠁
⠀⠀⠀⠀⣀⡀⠈⡄⠀⢂⣀⣠⣤⡤⠤⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠰⠋⠐⢿⣷⡀
⠀⠀⠀⠀⢫⡙⢦⣰⡀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⢨⡶⠙⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⠛⢻⣿⠇⠀⠀⠀⠀⢈⣿⠃
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⠴⠶⠾⢛⡛⣩⡟⠀⠀⠀⢀⣠⣴⣿⠁⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠉⠉⢁⣴⡟⠀⠠⠴⠶⠛⣫⣿⠃⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⣴⣿⠃⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠤⠾⣿⣦⡀
⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣾⡿⠁⣴⣿⣏⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠙⢿⣦⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⢠⣾⠟⠀⠀⠉⠛⠛⠛⠻⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠰⣿⣿⠾⠾⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣴⣶⡶⠶⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⣤
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣠⠀⠹⠟⠉⠁⠐⠒⠒⠒⠒⠒⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢶⣬⡉⠁⣴⠞⠛⠓
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡷⢹⣿⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⠆⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠿⠶⠶⠶⠿⠿⢿⡟⠋⠁⠀⣼⡟⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⢠⡟⠀⢀⣹⣆⠘⣆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⠷⠟⠛⠙⣿⡛⠛⢿⡿⠿⠟⠛⠋⠙⠦⣤⣤⣤⣤⡶⠾⠛⠉⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢻⠗⠀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⠤⠖⣛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⣀⠤⠶⠶⠖⠛⠛⠛⠛⠓⠲⢤⣀⠴⠖⢛⡛⠉⠉⠛⠓⠦⣄⠀⣀⡤⠶⠶⠦⣤⡀
⠀⠀⣠⢠⠞⠁⠀⠀⠀⢰⠃⢐⠀⡴⠣⠀⠀⠀⠀⠀⢸⡇⠀⢠⡴⠲⠖⢤⠉⠁⠀⣠⠀⠀⠀⠙⢦
⠀⢰⠃⡟⠀⠀⠀⠀⠀⢻⠉⠉⣾⣤⣤⣤⠀⢠⣤⢤⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣼
⠀⠘⣀⠷⠀⠀⡴⢢⠀⠸⠶⠆⡷⠰⠀⢼⣄⣸⣀⡀⢸⠀⠀⠸⡖⠂⠀⡐⠀⠀⣾⠤⢼⠀⠀⠀⠀⠙⢦
⢀⡞⠁⠀⠀⢸⡓⠚⣃⣤⠴⠶⣧⠶⠶⠞⠀⣀⠀⠈⡉⠉⠙⠒⡇⠀⠀⠀⠀⠀⢸⠀⣰⠀⠀⠀⠀⠀⢸
⣾⠀⠀⠀⠀⢸⡋⡏⠉⠀⠀⠀⢸⡆⠐⠃⠀⠻⠀⠘⠇⠀⠾⠀⡇⠀⠀⣦⡀⣀⢸⡂⠍⠀⠀⠀⠀⣠⠞
⣿⠀⠀⠀⠀⠀⢣⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠇⠀⠀⠇⠘⠉⠛⠉⠶⠛⠆⠀⠀⠈⡄
⠘⢧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⣠⠖⠲⣦⠔⠒⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠈⢟⠂⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡿⠃⠀⠙⠀⠠⣶⣶⠄⠙⣆⠀⠀⠀⠀⠂⠀⣠⠾⠁
⠀⠀⠈⠓⠦⣤⠀⠀⠀⠀⠀⠀⠘⠃⠀⢠⢞⡹⠏⠀⣠⣤⠄⠀⠀⠀⠒⢺⠭⣽⡢⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⡀⠀⠀⠀⢸⡈⠀⠀⠀⢤⡀⠀⠀⠀⢠⡄⠀⢀⠟⠁⠠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠛⠲⠤⠤⣴⠁⠀⢠⡄⠀⠀⢠⠎⢤⠀⠙⢀⡾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢀⣸⠀⠀⢠⡀⠀⠉⠋⠀⠀⠀⣴⠋⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣴⠞⠛⣛⣶⣬⣦⡀⠀⠉⠒⠊⠀⣀⡴⠛⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⡶⠞⣉⠁⠠⣀⠙⠖⢶⠶⡶⠛⠙⠦⠔⠁⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣸⡿⠋⠀⠐⢌⡆⢢⡀⠀⢱⡾⡶⠁⠒⠒⠀⣷⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠏⡠⣴⡶⠂⠈⢥⣀⠹⣄⡸⠾⢅⡠⠊⢀⣮⠖⢹⣦⡀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠘⣄⣛⠁⠀⠀⣴⣡⣌⢉⣀⠤⣌⣂⣧⡔⠉⠉⠓⢦⣅⣁⠀⣀⡎⠀⢀⣀⣀
⠀⠘⢭⡭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⡭⠭⢭⠭⠭⠭⡭⢭⡭⢭⣭⢭⠭⢭⠭⣭⠭⣭⡭⣭⠽⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⣤⣶⠴⠀⣀⠆⠀⠄⡀⠔⠂⠠⢀⡤⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠋⠉⢉⡉⠉⣻⠉⠈⠙⣿⠀⠒⠁⠀⢀⠤⠚⠁⡀⠀⣸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⢀⠖⠉⠀⢀⡏⠀⠀⣰⣿⠀⠀⠀⠋⠁⠐⠀⠈⠀⠀⣹
⠀⠀⢸⡇⠀⠀⠀⢀⡠⠂⡠⠂⢀⢽⠀⠀⠀⠀⣼⠁⠀⢠⡏⣾⠀⠤⠊⠀⢀⠄⠀⠀⠀⡀⣿
⠀⠀⢸⡇⠀⠀⠀⠁⠀⠈⠀⣀⠀⢺⠀⠀⠀⣰⠇⠀⣀⡞⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿
⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣤⣶⣹⢻⡟⢿⣥⡼⠟⠷⣶⠆⣿⣀⣀⣀⣀⣀⣀⣠⣀⣀⣠⡿
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣎⠀⠀⠀⠀⢠⠤⠖⠃⡀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠂⠀⠀⠀⠁⠐⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠰⠀⠰⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠠⡂⠀⠀⠀⠀⠐⠁⠀⢀⡱⠄⠀⠔⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣄⣀⠀⣀⡤⠒⠀⠀⠀⠀⠀⠠⣀⢸⡀⠀⠀⠀⠀⠀⠀⢀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⡄⠵⣌⠉⠁⠀⠀⢀⡠⠀⠀⠀⠀⠈⠉⠙⢧⡂⠀⠀⠀⠀⢀⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⠲⠶⠷⠶⠶⠚⢯⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠸⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠦⣄⠀⠀⠀⢰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠆⠀⢀⠃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⠀⠀⡌⠀⠀⠀⠀⠹⡀⠀⠀⠀⣀⣀⣀⣀⢻⠀⠁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢠⠁⠀⠀⠀⠀⠀⠑⣄⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢠⣠⡀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡇⠳⣄
⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠂⠀⣀⡀⣀⣀⣀⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢸⠁⠀⠀⠀⠀⢸⠃⡸⠋⠉⠉⠓⢤⣀⠉⠂⠀⣷⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣀⣠⠬⠬⣤⣄⣀⣀⠀⠀⠙⠓⢤⡀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠟⠒⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠴⡺⠕⣊⠭⢔⣒⠀⢀⣠⠴⠖⠒⠐⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠡⠴⠩⠶⠭⠩⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠤⢤⣤⣀⣀⣀⡴⠋⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠲⡄⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠄⡔⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⡖⠲⠒⠒⠂⠐⠂⠒⠀⠐⢺⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⠀⢰⠀⠃⠀⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⢰⠀⠃⠀⠀⠀⢰⠀⡄⠀⠀⡈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⠈⡀⢰⠀⠀⠀⠀⠀⢁⠀⠀⠀⡇⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⠀⠀⡇⢸⠀⠀⠀⠀⡇⠸⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠧⠤⠔⠃⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⢤⡈⢇
⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠖⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⠒⠒⢲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣄⡀
⠀⡀⠄⠀⠤⠤⠤⠤⠤⠤⠀⠀⠀⣀⣀⣀⣒⣒⣒⣒⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠴⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣽⣿⣿⠿⠋⠁⠀⠀⠈⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠃⠀⠀⠀⠀⠀⡀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠁⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠙
⠀⣤⠄⠤⠤⠄⠀⠀⠐⠐⠒⠒⠒⠒⠒⠒⢺⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠹⠿⠋⠀⠀⠀⠀⢐⡞⠀⠀⠀⠀⠀⠀⠧⠆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠒⢄⠀⠘⣼⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠒⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢣⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⠀⢇⠀⠀⠀⢡
⠀⠀⢳⡀⠀⡆⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⡄⠀⡄⠀⡆
⠀⠀⠈⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠁⠀⠀⠀⠚⢤⣸⠀⠀⠠
⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠄⣀
⠀⠀⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠒⠒⠒⠊⠉⢀⣀⠠⠤⠒⠉⠀⢀⣀⠠⠤⠔⠒⠃⠉⠉⠉⠉⠉⠉⠙⠂⠤⢀⡀
⠀⠀⠀⢰⠥⢤⣔⣧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⣀⡠⠄⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤
⠀⠀⠀⢠⠂⠁⠀⠘⠀⠀⠀⠀⢀⠄⣀⠴⠊⠁⠀⢀⡠⠔⠊⠉
⠀⠀⠀⠘⠀⢰⠁⠀⣀⠤⠒⢉⠴⠊⠁⢀⡠⠔⠊⠁
⠀⠀⠀⢀⠀⡆⠀⡸⠁⢀⠔⠁⠀⢀⠔⠉
⠀⠀⠀⠈⠀⡇⠀⡇⢀⠇⠀⠀⡰⠃
⠀⠀⠀⠉⠀⠇⢀⠀⡌⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠞⠀⠉⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡈
⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡷⠆⠀⠤⠀⠠⠂⠀⠀⠀⠀⡰⠒⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠘
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸⠀⠀⢀⠄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠃
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⢠⠑⠤⢀⠌
⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠁⠂⠀⠀⠄⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡟
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⠀⠀⠀⢀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠆⠂⠒⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⢀⠴⠊⠉⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢂⠀⠀⠚⠂⣀⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⠒⠢⣄⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢲⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⢀⡼⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢂⠀⠆⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠁⠀⠠⠂⠁⠈⢰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠒⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠙⠢⢅⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠜⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠤⠤⠤⠤⠀⠒⠀⠀⠀⠀⠈⠓⠤⠤⠄⠀⣀⣀⡀⠠⠤⠤⠬⠦⠤⣤⣤⣄⣀⣈⣉⣀⣀⣀⣠⣤⠤⠤⠤⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣄⡀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢻⡟⢻⣧⣬⣳⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠓⣹⣿⣿⣦⣼⣿⣟⠅⢿⣿⣽⡿⠿⣿⣿⣦⡀⠀⠀⠀⠠
⠀⠀⠀⢀⣀⡄⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠚⢻⣿⣿⣿⣿⣿⣷⣭⣦⠀⠀⢦⣤
⠀⠀⢀⡏⠀⣰⡿⢿⣿⣿⣿⣿⣿⣿⡛⢿⡆⠀⢄⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢶⠀⢼⠟
⠀⠀⠈⠏⠀⠀⢠⣿⣿⣿⣟⣿⣿⣿⣵⣿⡹⢶⣍⢁⡤⣺⣿⠿⣭⣿⣿⣿⡟⣹⡄⠀⣾⣆⠀⠈⣶
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣟⠻⢿⣿⣿⣷⠈⣀⠈⠀⠿⣽⣷⣿⣿⣿⣿⣾⣿⣿⣾⠻⢿⣦⡀⠹⣶
⠀⠀⠀⠀⠘⢿⣀⣿⣿⣿⣿⣿⣷⣾⣿⡿⠋⠰⠉⠀⠀⣾⣵⡿⠛⢻⣿⣿⣿⣿⡿⠿⠀⠀⠈⠀⠀⠀⣆
⠀⠀⠀⣀⣀⡀⣠⣿⣿⣟⣛⣋⣿⡿⠋⠀⣰⠀⠀⠀⠀⣿⣿⠃⠀⢠⡿⣻⣟⣿⡟⠀⠂⢀⠀⠀⠀⠐⣿
⠠⣶⣾⣿⣿⣿⣿⣿⣅⠩⣿⣿⠏⠀⠀⠀⣻⡷⢲⡇⠀⠸⡏⢀⣴⣷⣶⢟⡿⣱⡏⠀⠀⠀⠁⡀⠀⠀⠁
⢱⣤⣿⡟⢻⡟⣅⠀⣸⣷⣮⣻⣦⡠⠶⣾⣿⠅⠀⠁⠀⠀⢿⣿⣿⡏⠐⠈⣰⡿⠃⠀⠀⠀⣦⢰⡀
⣿⢻⡻⢿⣮⣧⣼⣷⡿⠴⣿⣾⢿⣿⣯⣶⣿⣷⠀⠀⠀⠀⠠⢿⣿⣧⡆⣰⣿⡇⠀⠀⠀⢠⢹⡾⣧
⠀⠈⢲⣷⣿⠿⠉⢹⣿⠷⡈⣿⠞⠿⣯⠘⣿⣏⠀⠀⠀⠀⠀⠀⠙⢿⢰⣿⣿⢷⡀⠀⠀⡿⠀⣿⣿
⠀⢀⣠⣭⡄⢠⣿⣿⡿⠗⠐⠋⠀⠀⠀⠈⠙⠛⢀⣀⣀⣀⣀⣠⢴⣧⣿⣿⣿⠀⣷⡀⡇⡴⠀⣿⡟
⠀⠀⢲⣟⠃⠈⠸⣿⣧⡀⠀⠀⡀⠀⠔⠒⠈⢀⣨⣤⣥⣶⣿⣿⡌⡿⣿⣿⣿⣴⣸⢰⢁⣧⣰⣿⠁
⠀⢀⣾⡉⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡸⣍⢿⣿⣿⣿⣼⣾⣿⣿⠇
⠀⠘⢿⠇⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢻⣿⣿⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣭⣉⡉⠛⠳⠦⠌⠻⠿⠟⠛⣿⣿⣿⣿⣿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⠶⠶⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⡄⠀⠀⠀⠆⠀⠈⡀⠀⠀⢼⣷⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠿⠃⠿⠿⠿⠿⣆⠀⠀⠰⡀⠀⠁⠀⠀⣾⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠈⠀⢀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣧⡀⠀⠱⡀⠠⠂⢀⣭⣭⣏⣉⣉⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠀⣰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣦⡤⠊⠀⣴⣿⡉⠀⠙⢿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⢀⣠⠚⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡟⠠⠂⢸⡿⠃⢤⡄⠀⠀⠹⢿⣷⡄⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢰⣏⣉⠀⣩⣭⣭⣭⣭⡁⣩⣭⣭⣭⣭⣭⡸⠀⠀⡈⠀⠀⠠⠄⠀⠀⢰⡀⡘⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⡆⠀⣀⣀⣀⣀⣿⣿⡇⣿⣿⠿⠿⠿⠃⠛⣻⠿⠿⣿⣿⡇⠀⢀⠀⠀⠤⠀⠀⠀⠐⠀⢐⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠃⠉⠁⠀⠀⠀⠉⠉⠙⠓⠢⢄⡀⠀⠀⠀⠉⢬⠀⠘⢁⡿⠙⠁⢆⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠉⠳⡄⠀⠀⠀⠚⠀⠘⢇⣠⣤⣈⣦⣤⠀⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢻⠵⠶⠷⣖⣛⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⢃⠀⣀⣉⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⠄⠀⠀⠀⠀⠀⠑⡅⠐⠴⠀⠀⠉⠙⠝⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠛⢥⢀⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢄
⠀⠀⠀⠀⠀⠀⣀⡠⠤⣲⣶⣾⣿⣿⣿⣿⣿⣿⣦⣤⠤⠤⠔⠒⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠔⠁⢀⡧⠀⠑⢄
⠀⠀⠀⣠⠔⠋⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠓⣄⣀⣱
⠀⡰⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣯⠁⠀⠀⢣⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠙⢽⠂⠀⠤⠤⠤⢀⣀⣀⣀⣀⣀⣀
⠀⠇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠳⠟⢬⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠤⠤⠤⠤⠤⡤⠀⢀⠄⢉
⢸⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⠻⠀⠀⠈⠡⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠠⠊⣀⡔⠁
⠘⡀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠄⠀⠀⠀⠀⠀⠀⢠⠊⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢁⡤⡞⠁⠇
⠀⢃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⡴⣿⠟⠛⠚
⠀⠈⢆⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⠢⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⣴⣾⡿⠚⠁
⠀⠀⠀⠳⣄⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀⠀⠀⠘⠒⠂⠀⠀⠀⠀⢀⠎⠈⠀⠙⡄⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣠⣖⢿⡿⠋
⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠙⠻⠿⠿⡷⠤⣀⡀⠀⠀⣀⣀⣀⣤⡊⣀⡀⠀⣀⡇⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠸⠉⠉⠉⠒⠒⠒⠒⡶⠁⠁⠸⡡⢈⠿⠥⣀⡰⠃⠀⠀⠀⠘⠛⠥⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠆⠠⡇⠀⠀⠀⢠⠉⠁⠐⠂⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠕⢢⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⣤⣦⢧⡀⠀⠀⠈⢷⡶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠐⠂⢠⠂⠁⣀⣼⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡜⠙⠛⠿⣷⠀⠀⠀⡸⣽⣟⣿⣾⣿⣿⣿⣿⣶⣶⠦⣤⠤⢨⣤⣞⣿⣾⡄⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠁⠀⢤⣸⠳⠿⢿⣿⣟⣿⣿⣿⣟⣿⠀⢀⣟⡿⠟⠁⠱⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠤⠤⠤⠐⠃⠀⠀⠀⠀⠉⠉⠙⠋⢿⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠋⠉⠉⠉⠉⠑⠒⠦⣄⠀⠀⠀⠀⢀⣀⣠⡤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⣤⣀
⠀⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠈⠳⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠉⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣠⠷⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣠⡴⠋
⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢹⣦⣀⣀⣀⣀⣠⡤⠞⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⡿⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠈⠳⠦⣤⣴⣋⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⢶⣶⣿⣿⡿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢹⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋⠁⣀⡈⠲⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⠶⠊⣞⠉⠉⢱⢀⠖⠂⢢⢀⠀⢢⠀⠀⠊⣿⣿⡟⠹⠀⠀⠙⠋⠁⠈⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉⠈⠓⠖⠊⠀⠉⠈⠀⠀⢸⣿⣿⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠓⠲⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠽⠆⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣚⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣾⣿⣧⠀⠀⠈⠻⠀⢿⣆⠀⠀⠀⠀⠀⠀⢀⠔⠙⣿⣿⣿⣿⣿⡿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠹⣿⣿⣇⠀⠀⠀⠀⠘⣿⣿⠖⠒⠀⠀⠠⠀⠀⠈⠻⣿⠉⠻⣾⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠈⠷⣦⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢠⡶⠀⠀⠈⢆⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣄⣀⣀⣀⣀⣤⡟⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠢⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⡇⠠⠀⠀⠀⠀⠐⢄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡁⠀⠱⣄⠀⠀⠀⠀⠈⠢⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠑⢌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠙⣿⣿⣿⣿⡿⠋⠀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢁⣀⣤⠀⢠⣿⣿⣿⠉⠀⣠⣾⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⣼⣿⣿⣿⣷⣾⣿⣿⣿⠀⠀⢹⣿⣿⣿⠟⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢠⣿⣿⣿⠿⣿⣿⣿⣿⣿⡀⠀⢸⣿⠟⠁⢀⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⣿⣿⠇⠀⠚⠋⠉⣀⣤⣼⣿⣿⣿⣿⡇⠀⠈⠁⢀⣴⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣶⣶⠖⠀⣰⣿⣿⣆⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⠏⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠠⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠙⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀⠀⠀⠙⠛⠿⠿⠛⠋⠁⠀⠹⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢣⠀⠀⠀⠈⠓⢤⡀⠀⢠⠮⣉⠉⠲⠈⠂⠀⠀⠀⠈⢿⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⣏⡒⠳⢄⡀⠀⠀⠀⠉⠒⠼⢄⡘⣗⠾⣇⠒⠐⠠⡀⠄⠀⢵⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⣤⣉⡷⠋⠓⠢⠤⠤⢤⡤⠶⠈⢿⠈⠉⠑⢢⡴⠧⠄⠒⣠⣟⠒⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣠⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠁⠀⠀⣠⣾⡋⠈⠒⢦⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠈⠱⠖⠊⠉⢉⠭⠽⠶⠮⠴⢯⡷⠋⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠘⣛⡢⡽⣆
⠀⠀⠀⠀⠀⢀⠤⠯⠀⠀⠀⠀⠀⠀⡠⢶⣀⡤⠚⢆⠀⠀⢺⡷⠞⠢⠤⢤⣀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠀⠀⣀⠔⠋⠢⠬⠡⠞⠘⠧⠴⠶⠦⢤⡀
⠀⣀⠤⠒⠒⠳⢤⠄⠀⠀⠀⠀⢀⠎⢀⡞⠈⠳⡀⣀⠵⣄⠀⠀⠀⠀⠀⠀⢱⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⢀⡠⢞⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢤⠞⠀⡠⡄⠿⣤⠊⠉⠑⢄⠀⠀⠀⠀⠸⠤⠤⢤⣀⣉⠑⠒⠤⠤⢴⣿⢢⡈⢦⢣⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠘
⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠅⢲⠎⢀⡇⠀⠘⠦⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠛⠛⠒⠚⠫⠳⡈⠉⠛⠉⠁
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣄⡀⠀⠀⠀⠀⡼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⡍⠁⠀⠀⠀⠈⠃⠒⠂⣟⣳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢢⣤⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠁⠐⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢤⣀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠉⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⢀⣠⠞⠣⢄⣀⡠⠞⠁⠉⠁⠒⠄⠋⠢⣄⠀⠀⠀⠀⠀⠀⣀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢛⢩⡿⠛⠀⠀⠀⣠⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡘⢂⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⡿⠀⡙⠿⠃⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠋⠀⠀⠀⠀⠘⣿⣿⠿⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⢠⣄⢀⣄⣀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠡⠶⠷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠈⠀⣻⠷⡀⠀⠀⠸⢉⣿⣟⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠐⢿⡀⠙⠃⠀⠀⢀⣀⠈⢻⣏⠙⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⣶⡀⣠⣶⣦⠀⠼⣿⢿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⣿⣩⣤⡀⠀⠛⠘⠟⢹⣤⣴
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⡿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡇⠀⠻⠿⠃⠀⠀⠀⠴⠿⡞⠟
⣿⣷⣶⣤⣤⣄⡀⠿⢃⣀⣾⣿⣿⣿⣿⣿⣿⡀⣀⣀⣿⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢠⣼⡷⢂⣶⣿⠿
⣿⣿⣿⣿⣛⣯⣽⣷⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣶⣶⣦⣤⣼⣷⣍⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣟⣛⡿⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⡿⠉⠸⣿⠿⠿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠲⠟⣿⣿⣭⣿⣿⣿⣤⠀⠉⠘⠿⢛⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣯⣿⡓⣲⣦⣾⣿⣿⣅⡝⠻⣿⣿⣿⡦⣀⣘⣃⣟⠛
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⢿⣾⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣋⣩⣿⣿⣟⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣀⠀⠄⠐⠀⠀⠀⡀⠀⠀⢀⠛⠁⢂⠀⠀⠀⠤⠜⠉⠀⠀⠀⠈⠀⠀⠂⠀⠉⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠞
⡔⠈⠁⢱⠀⠀⢲⠂⠀⠀⠁⡄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣌⡀⠀⠀⣄⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠐⠄⣀⣀⣠⢴⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⢨⠁⠈⠉⠃⠀⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠘⠁⠐⠀⠀⠈⠛⠁⠠⠖⠀⢀⠀⠀⠠⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠃⠈⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠢⡀⠀⢄⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⠈⠁
⠀⠀⠀⡤⠯⠤⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡓⠀⠂⠀⠂⠀⠀⠆⠀⠀⠃⠐⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠰⠀⠀⢀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⣤⡃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣀⠀⣄⠀⢠⠀⠀⠀⠘⠛⠀⠀⠀⠲⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣶⡀
⠈⠀⠰⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠙⠿⠓
⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠃⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⣶⠄⢰⡶⠀⠀⠀⠘⢃⣶⣦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠟⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠈⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⢘⣷⣆⠀⠀⠀⠀⠐⠀⠀⠀⣈⣁
⠀⠀⢠⣀⠀⠀⠀⠘⡂⠀⠀⢀⡠⣤⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⣴⣶⣾⠿⠛⢻⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠛⠉⠁⠀⢀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦
⠀⠀⠀⠀⣀⡴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣤⡶⠾⠛⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠛⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⠋⠉⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⣯⣽⢉⢙
⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠼⡶⡀⡹⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⣠⣆⣆⢤⣰⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠿⠉⠃⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢠⠀⠀⢠⣤⣤⣄⣄⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⠈⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠













⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣈⣀⣀⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣏⢯⠀⢭⠽
⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠠⣸⠃⠀⠤⠀⠴
⠀⠀⠀⠀⠙⣄⠤⢤⠤⠬⠀⠀⠀⠄⡀⣰⠀⠖⠈⠩⡑⢄
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠢⢐⠒⠂⠀⠡⠂⠀⠀⠀⠀⠘⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⢹⠱⡄⠘⡆⢹⡀⠉⠀⠀⠘⠆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡆⠘⢶⡁⠘⢏⠉⠉⠉⠉⣉⣉⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⣠⠎⠳⣦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠚⠁⠀⢠⠁⠀⠀⠀⠀⠉⠉⠉⠑⠒⠲⠤⡄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣀⢤⠄⡸⠤⠤⠤⠤⠄⠀⠠⡀⠀⠠⠃⢠⠇⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠊⠀⠀⠀⠀⠤⠴⠒⢀⠔⠀⢠⡈⠒⡀⢀⣞⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠦⠤⠤⠤⠴⠄⢰⠀⠀⡞⠀⠀⠀⢳⡀⠈⠡⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠇⠀⠀⠀⠀⠹⡄⠀⠙⢆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠙⢆⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠈⠂⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⣀⠲⡄⣈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠒⢲⠚⡁⠀⢰⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣰⡄⠙⠳⠶⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣠⠄⡀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣄⠀⠀⠀⠀⠘⠁⠏⠲⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠰⠈⠈⠡⡀⠀⠀⠀⠀⠤⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠁⠀⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡃⠀⠉⠀⠀⠠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠠⠂⠈⠁⠀⠀⢠⡶⢃⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣼⠈⠡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠢⢀⠀⠀⠀⠰⡶⠟⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡆⢠⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⣀⠀⠀⠐⠀⠀⠰⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠘⢀⠀⠀⢀⣤⣠⠜⠀⠀⠀⠀⠀⠈⠋⠀⠀⢮⠁⠀⣄⠐⠄⠀⠀⠀⠀⠀⠄⣢⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠀⠉⠀⢆⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠴⠆⠀⠞⠠⢀⡡⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣷⣴⡶⠉⠉⢀⠀⠀⢹⠀⠀⠀⢢⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⣿⣿⣿⣄⣀⣶⡌⠀⠀⣘⡄⠀⠰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢸⣿⣿⣿⣿⣿⡏⠠⢶⣾⣿⡇⠀⠀⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣿⣿⣿⣿⡿⠋⠀⠀⣼⣿⣿⣧⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⡀⣀⣴⣿⣿⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠴⠛⠛⠙⠻⠟⠛⠛⢻⣿⣿⣿⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠑⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡋⠀⣤⡀⠠⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⡿⠁⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢀⣸⣿⣦⣥⣈⣉⣛⣻⣿⣟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣷⢀⣀⣴⣧⣬⣿⣿⣭⣿⣿⣿⣯⣽⡵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠸⠻⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣠⣄⢠⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠉⢈⠉⠈⡉⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠛⠃⠛⠜⠛⠜⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⡀⠀⠒⠒⠒⠲⢟⠯⣙⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⣯⣶⣷⠀⠀⠀⠐⣿⣿⣿⣿⡄⠓⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢻⣿⢯⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣀⣈⠿⢛⣻⣿⣯⣄⢹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣼⣆⣠⠀⠀⠀⠉⠙⠿⢿⣿⠁⠐⠠⠦⢤⣾⣏⣽⡟⢛⡽⡿⢋⣴⢿⣿⣿⡿⠻⠋⠿⠿⠛⠿⢿⣿⣿⡿⢛⡿⣧
⠀⠀⠀⠀⠀⠀⢰⠟⠻⣿⡛⢿⡓⢆⠀⢀⠀⠀⠈⠻⣿⣶⣄⠀⢠⡼⠋⢁⣾⣫⢞⡴⣿⠟⣿⠃⢃⢁⡀⡀⠀⠀⠀⠉⣡⣶⠋⡴⢇⠀⠻⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⣯⣙⣮⣗⡁⠀⠀⠀⠀⣘⣿⣿⠟⠁⠀⠒⠈⣡⠴⠛⢁⠙⣧⣸⡇⢀⠻⠟⠺⡿⠛⠛⠋⠩⣤⣤⣌⣈⣵⠆⢱
⠀⠀⠀⠀⠀⢠⡿⠿⠻⠟⢉⢿⣿⣿⣿⣗⡦⡀⠀⣤⡿⠛⠁⣯⠀⠀⠀⠈⢠⣷⠴⠈⠓⠈⠙⠃⠀⠀⠀⡀⢹⣤⠤⣀⠀⠀⠀⠀⠀⠁⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡅⠦⠀⠀⠀⠀⠙⠛⢻⣿⣿⣾⣾⣿⣴⣞⣁⣄⠀⠒⢸⣴⣿⣷⣤⢴⡷⣶⣶⣦⣦⣠⣀⣀⠀⠉⠓⠦⢉⡲⠖⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠘⠓⠒⠚⠚⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠾⠶⢦⠷⠾⠰⠷⠶⠮⠷⠰⠷⠶⠶⠲⠆⠾⠃⠶⠗⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠓⠦⡀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢘⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠸⣄⠀⠀⠀⠈⠳⢄⡀⠰⠓⢤⡀⠈⠆⠀⠀⠀⠀⠀⠹⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡗⠬⢍⣲⣄⡀⠀⠀⠀⠉⠚⠣⠌⣽⠛⡧⣁⠀⠐⡄⠂⠀⠘⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡷⠖⡶⠋⠀⠈⠉⠒⠒⠒⠒⠛⠀⠱⣄⣀⣀⠱⠚⠓⠀⢉⡴⠯⣁⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠀⠤⢄⣀⠀⢀⣀⣀⣀⣀⠀⡠⠊⠽⡔⡦⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⠿⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⣰⠒⠂⠙⣻⣈⣱⠟⠀⠀⠀⠀⠈⠑⠢⠄⢀⣀⣀⣀⡀⠀⠄⠀⠀⠀⣀⠀⠤⡜⢎⢳⡀
⠀⠀⠀⠀⠀⡰⠒⠛⠀⠀⠀⠀⠀⠠⠒⢹⢶⡎⠉⠣⡀⠀⠙⠛⠋⠉⠁⠒⢦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠉⠒⠐⠁⠀⠛⠉⠉⠙⠳⢦⡀
⢀⡴⠊⠉⠁⠉⠐⠃⠀⠀⠀⠀⣰⠁⢠⠃⠀⠙⣦⠐⣋⠦⡀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⣠⠞⠣⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡵⠃⣠⠒⡆⠉⢿⡁⠀⠈⠲⣄⠀⠀⠀⠈⠉⠐⠒⠂⠤⢤⣉⣉⡒⠛⣶⣙⠆⢱⡼⣄⠀⢀⣀⠠⠖
⠓⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡹⠁⢠⠁⠀⠀⠑⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠉⠈⠉⢀⠩⡌⠄
⠀⠀⠓⠤⡤⠀⠀⠀⠀⠀⡀⠀⠀⣰⡁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⣀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠣⢼⣁⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠽⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠄⠀⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠑⠶⠤⠤⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⢀⠤⠴⠀⢀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠖⠋⠈⠐⠒⠚⠁⠀⠀⠀⠀⠁⠀⠉⠲⠤⣀⡀⢀⣀⡠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡏⠚⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⡻⠿⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⢿⣿⡟⠁⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⢀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠂⠑⠛⠘⠀⠃⠀⠃⠂⠃⠋⠞⠁⠐⠃⠀⠋⠘⠁⠐⠑⠃⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠢⠀⠀⠀⢠⠄⢡⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠂⢄⡀⠀⠀⠀⠉⠀⠀⠀⠁⠈⠢⢀⠠⠐⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠄⢐⡀⡰⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢡⡀⢨⠒⠲⡤⢀⠀⠠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠉⠀⠀⠂⠐⠁⠀⠑⡀⠄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡁⡂⠈⠁⠀⠈⠉⠉⠉⠉⠁⣠⠀⠀⠂⠁⢂⠀⢑
⠀⠀⠀⠀⠀⠠⠀⠠⡀⠀⢀⠠⠠⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠄⠀⠀⡑⠀⠘⡀⠀⠑⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠇⢀⠃⠎⠁⠀⠀⠈⡲⠤⠀⠀⠀⠀⠀⠀⡠⠇⠀⠄⠀⢈⠆⠀⠁⠀⢨⡆⢈⠃⢃
⠀⠀⠀⠀⡏⠉⠉⠉⠉⠈⢆⠣⠀⠀⠀⢀⠁⠀⢱⣠⠤⠐⠊⠁⠀⢀⠠⠈⠀⠈⠠⠀⢠⠀⠁⢀⠂⠀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⢧⠸⣇⠀⠀⡥⠤⠔⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠏⢀⠂⠀⡐⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠂⠌⠀⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⠀⠀⠈⢂⡨⡅⠁⠀⠀⠀⠀⠀⠀⠀⢲⠀⢂⠈⢉⢁⢈⡢
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠢⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠑⢆⢑⠄⠀⠀⠀⠀⠀⠀⠰⠀⢛⠀⠀⠄

⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⣾⣿⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠛
⠀⠀⠀⠙⠿⣿⣿⣿⣶⡶⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡋⣡⡀⠀⠀⠀⠀⢀⣴⣶⣾⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢁⠀⠀⠀⡐⢿⣿⣿⣿⡿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣿⣿⠋⠐⠀⠀⠀⠠⠎⠛⠛⠋⠀⠀⠀⠀⠀⢠⣦⣶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠈⠰⠁⠉⢀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣻⣿⣿⣿⣿⣭⡙⠀⠀⠀⠀⣀⣤⡀⠀⢀⣤⣀
⠀⠀⣠⣾⣿⣿⣿⣴⡦⠀⠀⠀⠀⢰⣿⣴⣿⣆⣾⣧⣼⣧⣸⣆⣼⣇⣼⡷⠀⠁⠈⠙⠛⠛⢉⠛⠁⠀⠀⠀⣾⡟⢻⣿⣶⣿⣿⣿⣧
⠀⠀⠸⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⣼⣇⣿⣇⣿⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠻⣿⣿⣿⡟
⠀⠀⠐⠆⠇⠹⠰⠀⠀⠀⠀⠀⠀⠸⠿⠏⠿⠇⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠩⠁⠀⠨⠍⠍
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠲⠤⠤⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⢀⣴⣶⣿⣯⣹⣟⢷⣦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⣤⣀⣀⣴⣿⣿⣿⠳⠋⠉⠉⠀⠛⠛⠿⢿⣄⠒⢉⣀⣀⡀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣿⣷⣼⣿⣿⡏⠉⠀⠀⢠⡀⠀⠀⠈⠻⣷⣾⣿⠀⢀⣀⠤
⠀⠀⠀⢀⣠⣤⣤⣄⣠⠔⠊⠉⠉⠺⣄⣴⣴⣿⣿⣿⣿⠷⠇⠀⠀⢲⡀⠙⠀⠀⠀⠀⢸⡷⠟⠊⠉
⠀⠀⠀⠡⡀⢀⠘⠻⢿⣿⣶⣄⠀⠀⠈⢻⢿⣿⠿⣯⠗⠀⠀⠀⠀⠀⠀⢠⠄⠒⢄⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⢻⣷⣦⣀⠌⠙⠷⣿⣷⣄⠀⣸⡀⠌⡀⠀⢸⡀⠀⠀⣦⡀⠠⠄⠘⢦⠄⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⠻⢿⣷⣥⠄⠘⠿⣝⣿⣼⠁⠀⠁⠀⠀⠱⣄⠀⠈⢋⡤⣔⠔⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠹⢷⣦⡤⠀⠙⠛⢷⣄⡄⠈⠻⠿⣿⣌⠀⢠⣴⣾⠟⠋⠉⠀⡠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠙⢿⣧⠂⠀⠀⠙⠻⠥⣀⣀⠀⠉⠓⠬⣉⠀⢀⣠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣂⡤⠀⠀⠀⠉⠻⣷⣤⣠⡀⠉⠛⣤⡀⠐⠤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣊⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⣀⠄⠀⠘⢿⣿⣦⡉⢀⡘⢿⣦⡀⠀⠉⠓⢤⠀⠀⢠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠉⠛⠷⠤⠄⠀⠹⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠈⢻⣿⣄⠀⠀⠀⣴⣶⢀⣴⣦⣄⠀⠰⢶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢀⣀⣤⣤⡹⣿⣧⠀⠀⢹⣿⣼⠟⠉⣿⣠⡄⢀⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡿⠿⠛⠋⠉⠻⣷⡘⣿⠀⠀⢀⣿⣿⣤⣤⠏⣿⡿⠛⣿⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠙⣿⣿⠀⠀⠀⠀⣿⡇⢸⣷⣤⣾⣿⡇⠀⣡⣾⣿⠀⢸⣿⣾⠟⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠘⢿⣦⣤⣴⡾⣋⣤⡿⠟⠛⠛⢿⣿⣾⣋⣸⣿⣄⣼⣿⣿⣀⣠⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣠⣄⣈⣭⣿⡿⠋⠁⠀⣀⣠⣤⣄⣀⣉⡙⠛⠛⠋⠉⠙⣿⡛⠛⠁⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠉⢀⣤⣶⣿⣿⣿⡿⢹⣿⣿⣿⣷⣦⡀⠀⠀⠹⣷⡀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⣠⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⢹⣷⡀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⡛⠿⠛⢿⣿⣧⡀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⣰⣿⣿⣿⣿⠿⠿⢿⣿⣿⠟⠛⣙⣛⣿⣷⣶⣽⣿⣷⠀⠘⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⣿⣷⣶⣶⣤⡿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣿⠟⣩⣴⣾⣿⣾⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⣿⣿⠿⠿⣿⣿⣿⣿⠿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣹⣿⣿⣿⣿⠀⠀⠹⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢻⣿⠆⣰⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⡄⠀⠀⠸⣿⣿⠙⢿⣷⡄
⠀⠀⠀⢸⣿⡄⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⡇⢘⣯⣼⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧⣀⣷⣿⣿⣿⣿⡀⠀⠀⣴⣿⠟⠀⠀⣿⣿
⠀⢀⣤⣼⣿⠇⠘⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⣧⣼⡟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⡇⠀⣸⡿⠋⠀⠀⠀⣿⡟
⠀⠈⠙⠉⠀⠀⠀⢈⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⠁⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣿⣿⣿⣿⣿⡇⣰⡿⠁⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠚⠿⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣧⣾⣿⣿⣿⣿⣿⡇⠹⣿⣦⠀⠀⠀⠀⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⣿⣿⡿⠿⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢾⣶⣠⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠉⠙⠿⣷⣦⣀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⣀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣏⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠟⠇⠀⠠⣄⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠅⠻⣿⣿⣿⣿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡖⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠛⠋⠛⠷⣶⡀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⢶⣄⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣶⠟⠉⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣷⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⠈⠹⣆⣀
⠀⠀⠀⠀⢀⣴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣟⠶⣄
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢀⣸⠟⠛⠛⣁⠈⢿⣦⣌⣷
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⡍⠙⢋⣁⡀⠀⢺⣿⠇⠈⣿⡌⠉
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢻⣧⠀⠹⠿⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣠⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣴⡾⠟⠁
⠀⠀⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠈⣙⣿⡿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣸⡟
⠀⠀⢀⣤⡾⠋⠁⣠⡿⠟⣳⡦⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣀⣠⣾⣿⠷⠿⣿⠋
⠀⣰⡟⠁⠀⣰⡾⠋⠀⣼⠏⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠹⣇⠀⠻⣦⠀⠙⣷⡀
⠀⠹⣧⣤⡾⠋⠀⢀⡾⠛⠻⢶⣶⡶⠾⠋⠙⢷⣄⣀⣀⣀⣤⡾⠋⠀⠙⠷⢶⠶⠶⠿⣆⠀⠹⣧⠀⠘⣷
⠀⠀⠀⠈⢿⣄⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣹⠷⠾⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⢀⡶⠴⢄⠀⠀⠀⠈⣼⣿⡿⠙⣿⠟⠁⣓⢒⣠⢶⣟⣻⣶⠈⢀⣽⣟⢣⠐⣿⣻⢞⢦⠠⠄⠀⣀⠀⠀⠀⠀⠀⢀
⠠⠀⠀⠀⠙⡿⢖⡼⢠⠀⢘⡄⣠⣼⣧⣼⠊⢉⠉⢹⡿⣟⣾⡞⢞⡇⠂⠘⠻⠛⢺⣧⢹⡋⠈⡣⠿⡄⢀⣾⣿⣱⠀⠀⠀⠈
⠀⠀⠀⠄⠀⠀⡢⢀⠀⠂⠙⢘⣛⡿⣓⢿⠖⠋⠄⢠⣿⣹⣻⠝⠛⠊⠘⠿⠃⠀⠀⡙⣾⠟⠑⠀⠀⠳⡀⠈⠛⠉⠀⠠⢀⣤⠤
⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⢰⠉⡱⣏⠈⢲⡤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠂⠺⡟⠀⢿⣿⠄⠀⠈⢦⠉⠉⡄⠀⢀⠀⢸⢙⣼⣺
⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠢⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣷⣾⣿⣀⡀⡑⣾⣧⠈⣇⠀⠀⠀⢸⡿⣶⠱
⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⡟⠋⠩⣉⠁⠡⡓⠨⢻⠄⢻⣄⡀⠀⠀⠈⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢒⡶⣾⣗⠺⠉⠁⢐⠟⣠⡿⣻⠏⠀⡀⠣⠕
⠀⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠶⣾⡿⢿⣩⣧⠀⡰⠺⠛⠈⠉⠊
⠀⡁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⡀⠀⠀⠀⠀⠀⠀⢸⣾⢤⣤⡏⢰⠛⠋⠀⠒⠒⠁
⠀⠀⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣀⡀⠠⢢⢈⢷⠺⠁⠀⠐⠁⠀⠰⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠒⠂⠀⠴⣄⠀⠀⠀⠀⠐⠁⠀⢜⡥⠟⠲⡢⠃⠀⣘⠀⠀⡤⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠠⢶⡠⡉⠻⠻⢆⣃⣂⣤⡎⠂⠃⠀⠐⢀⡰⠏⡄⠐⠛⠑⠃
⠀⠀⠀⠀⠀⢀⠀⠄⡀⠃⢆⠘⢀⡆⢀⠈⢂⣰⡀⠛⠁⣁⠆⠂⠈⠂⢠⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡖⠄
⠀⠄⠀⠀⠂⠘⠓⢴⣴⣤⠠⠙⢶⠟⢅⢶⢁⠈⠀⠀⠀⣹⠸⠶⠾⢣⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠡⠊
⠀⠀⠀⠀⠀⠀⠐⡾⢋⠠⠔⣖⠨⢦⣈⡿⠀⠀⠀⠒⠠⢅⡔⡦⣴⠿⢻⠀⠀⠀⢐⡇⡀⠀⠀⠀⠀⠒⡰⢀
⠀⠀⠀⠀⠀⢠⠊⡰⠉⠂⠤⠚⣪⠀⣹⣷⡀⣀⣻⡀⢀⢹⡆⣠⡃⠠⢸⠀⠀⢠⢾⣯⠂⠀⠀⢸⠀⠠
⠘⠒⠃⠀⠀⠁⠄⠁⠀⠀⢀⣈⣠⠔⡏⡷⣽⢋⣿⠇⠈⠽⠻⡉⠀⢀⣈⡄⢠⡧⠃⠀⠀⠀⠛⡜⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢄⡠⠒⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠃⠀⠀⠀⠀⠘⠃⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠠⠎⠀⠈⠖⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⢢⠀⠀⠀⠀⡀⠀⠐⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠠
⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠂⠀⠀⠀⠁
⠀⠀⠊⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡤⢄⠀⣀⣠⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⠋⠉⠛⢢⡴⠋⠉⠛⢢⣀⡔⠘⠃⠠⠀⠈⠉⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡞⡉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠀⠀⠀⠀⡰⠤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠆⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⢀⣜⣣⡀⣤⠜⢉⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠉⠐⠈⢢⡈⠀⢀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⠴⠃⠀⠀⠀⠀⠈⠐⠁
⠀⠀⣀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⣀⡠⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜
⠀⠀⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⢔⢒⣉⣡⠤⠒⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⢠⠎⣠⠛⠒⠊⢣⣀⣠⡴⢆⣀⣠⠟⠦⠴⠎⠙⠁⠈⠓⠤⣉⡒⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠳⡀⠀⠀⠉⠃⠀⠀⠘⣀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠂⠀⠀⠀⡠⢂⠀⠀⠀⠀⣀⡇⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡠⠄⢂⡩⠔⡇⢸
⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠤⠒⣉⠤⡇⢠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠖⠊⠉⠀⠀⠑⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠸⠆⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠐⣉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣁⣀⣀⣀⣀⣀⡀⣀⣀⠀⡀⣀⡀⢀⠀⣀⣀⣀⢀⡀⣀⡀⣀⠀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠒⠂⠴⠒⠒⠚⠉⠙⠒⠒⠢⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠣⣀⣀⣀⠀⡀⠀⡀⠀⠀⡀⢀⠀⣀⣀⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠈⠉⠈⠉⠙⠒⠊⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠒⠠⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢎⡁⠀⠀⢀⡨⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⠀⠀⠘⠒⠒⠂⠒⠒⠉⠉⠉⠀⠀⠀⠀⢀⣈⠑⡄
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡤⠀⠀⠀⠈⠺⡢⠃
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢅⣀⣀⣀⡴⠁⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⡜⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠖⢦⠀⠀⠈⠣⡀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⡼⠁⠀⡜⠁⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠒⠒⠚⠃⠀⡇
⣠⣤⠎⠉⠀⠀⢸⣀⣀⣀⣀⣈⣑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⣶⠂⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢠⡾⢁⣀⣀⣋⣹⣟⣀⣀⡞⠓⠂⠀⠀⠁⠀⠈⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠒⠢⣄⠀⠀⠀⠀⢀⣀⣠⠤⠶⢒⣾⠟⠉⠉⠁⣠⣟⣡⣤⠀⠀⠀⠉⠉⠉⠑⠒⠲⠤⠴⠁⠀⢸⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠘⡤⠴⠒⠋⠉⠁⠀⠀⣰⠟⠁⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠⣄⠀⠀⠀⠀⠘⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⠃⡨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⠊⠀⡔⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⢀⠊⠀⡠⠋⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⡃⠀⠔⠁⢀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠐⡎⠀⡰⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠠⡎⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠋⢦⣀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣠⠏⠈⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡾⠋⢸⡟⠀⠀⢀⣠⣴⠾⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⣴⠟⠀⠀⣿⣁⣤⣾⠟⠋⠁⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢀⡟⠀⠀⣠⡿⠋⠀⠀⠐⢿⡿⠋⠀⠀⠀⠀⣼⡟⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⠀⠀⣼⠇⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠿⠛⠛⠉⠉⠉⣠⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇⠀⠘⠛⠀⠀⣀⣀⠤⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⡄⢻⡄⠀⣿⣿⠀⠀⠀⠀⠉⠉⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠀⠀⠀⠀⠀⠀⢿⣯⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣌⠻⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠁⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣙⣛⣻⣶⣶⣦
⠀⠀⠀⠀⠘⠛⠳⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠶⣬⣭⣍⣉⡁
⠀⠀⠀⠀⠀⠒⠒⠶⢶⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⣠⣀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦
⠀⠀⠀⢀⣾⡿⠟⠻⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠴⢶⣶⠶⣶⣶⣶⠶⠆⢤⣄⣀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣼⠉⠙⠓⠶
⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣩⣵⡾⠛⠀⠀⢩⡛⠛⠓⠲⣾⡄⠀⠀⠀⢀⣼⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠙⢦⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠋⢻⣀⠀⠀⠀⢸⡟⠒⠒⠂⠉⠁⠀⠀⠀⠘⠛⠛⠛⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⡆⠀⢀⣴⣾⣿⣿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣸⣦⣤⣤⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣧⠀⣾⠟⠉⠀⠁⠀⢹⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⡴⠏⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢿⣦⣀⠀⠀⣠⣾⣿⠿⠋⡀⠀⣀⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢧⣍⡙⠛⢛⣋⣩⣤⠶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⠤⢤⣭⣭⣭⣥⣤⣤⣍⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣁⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢈⡿⠈⠉⠉⠉⠉⠉⠉⠙⠛⢻⡟⢻⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠙⠓⢺⠗⠲⠒⠒⠒⠒⠶⠶⠤⠤⣼⠗⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡟⣿⢂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠤⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠃⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⠷⠶⠶⠶⠶⠶⠤⠤⠤⠶⠦⢾⡀⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠖⠋⠉⠁⠀⠈⠀⠙⢦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⡀⠀⠀⣀⣀⠄⠀⠀⢀⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠀⠚⠓⢲⢴⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠀⢠⣤⡀⠉⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠬⠟⠁⠰⢦⠈⠀⢀⣄⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠛⠒⠙⠦⣄⣠⣀⠘⢻⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⠏⠐⠯⠽⠟⠙⡃⡀⣄⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣏⠳⣤⣀⣀⣀⡼⡱⢡⣻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⠈⡝⠉⠉⢀⣢⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠾⣛⢹⠻⢤⣶⡷⠄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠒⠒⡹⢛⣷⣿⣿⣿⣿⢹⠤⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⠤⠄⣥⡌⣵⣿⣿⣿⣿⣻⣷⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⡿
⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠸⠿⣿⠿⠛⠛⣉⣀⣈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠿⣿⣏⣠⣬⡽⠗⠀⠹⣿⣿⣿⡿⠟⠉⠀⠀⠉⠛⢿⣿⣼⠇
⠀⠀⠀⢠⡇⢸⠑⠒⠶⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀⣤⡴⠶⠛⣿⣿
⠀⠀⠀⢸⠀⡾⠀⠘⠠⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠉⠀⠀⢸⡀⠀⢻⣿
⠀⠀⠀⣿⢠⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣿
⠀⠀⢀⡏⢸⠃⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢇⢿
⠀⠀⢸⠇⠸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡸⡆
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣇⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠉⠐⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠂⠉⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠈⠀⠁⠀⠀⠀⠁⢢⡀⠁⠈⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⡀⠀⠀⠀⡄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠈⠀⠀⠈⠀⠈
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠻⠇⠀⠀⠀⠘⠃⠈⠀⠤⠀⠄⠂
⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠄⠘⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡤⠠⡄⠂⠀⠀⠆⠀⠀⠄⠀⠤⠆⢀⠴⡀⠀⠤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠐⠀⠀⠂⠀⠈⠀⠂⠀⠀⠃⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠐⠠⢀⠓⠀⠀⠀⠄⠃⠀⠀⢀⠀⠈⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⢀⠦⠄
⠀⠀⠀⠀⠀⠀⠂⡀⠀⡀⠀⠚⠀⠆⣀⠘⠀⢀⠀⠤⠀⣀⠹⠀⠀⠊⠠⠀⠀⡀⠀⠰⠀⠈
⠀⠀⠀⠀⠐⠀⠀⠉⠀⢐⠀⠈⢠⠤⠀⠁⠀⠀⠠⠤⢎⡠⠀⡄⠠⠄⠀⠂⠀⢃⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⣠⠤⣄⢶⣖⡶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡔⠒⠲⠁⢔⡚⠄⢀⡂⠠⢀⡜⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣸⠁⢀⠈⠃⠉⠀⣠⢞⣀⡀⢠⡔⠖⢚⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡤⠈⠀⠁⠀⠠⠀⠀⠛⠛⣉⠘⡥⠖⠈⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡉⠀⠀⡀⠀⠀⠂⠀⠀⠔⠀⠀⠀⠀⠈⠢⡀⠗⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠄⡀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠐⠀⠀⠉⠀⠈⡹
⠀⠀⠀⠄⠡⠄⠀⠀⠀⠀⠀⠀⡃⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⢀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠁⠀⠀⣄⢠⠀⣙⠢⡖⢇⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢰⠻⣀⣀⣼⠦⣒⡠⠉⠐⠊⠩⠀⢡⠀⠳⣀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆⢰⠁⢃⠘⡀⠀⠀⠀⠀⠀⠀⣧⠸⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡌⠀⢸⡀⢇⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢽⠁⠀⠀⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠻⠐⠁⠜⢒⣲⢌⠑⣆⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡀⠂⠀⠄⠲⣅⠀⠐⠀⠷⠊⠁⢢⠀⠀⠀⠀⠀⠀⠐⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠑⠦⢡⣙⡴⠛⠾⠽⣧⠀⢀⠀⠐⠆⠸⢋⡀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡤⠀⡠⠁⠐⠂⠀⣀⡀⠀⠀⢄⠀⠀⠈⠙⠸⠳⢦
⠀⠀⡀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠔⠊⠐⠂⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡼⠃⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠀⠠⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠈⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢣⣐⠠⡆⠀⠀⠁⠀⠀⠢⠀⠀⠄⠉⠐⢦⠠⠊⠀⠀⠀⠀⠀⠀⠀⠠⠅⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡴⠁⡌⢠⠳⠬⠔⢣⠀⠄⣠⡀⠀⢀⢦⡀⠜⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡼⠊⠀⢠⠁⡸⠀⠀⠀⠀⠁⠈⢠⠉⡒⢣⠀⡆⠀⠀⠀⠀⠈⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠮⡇⠀⠀⠀⠄⠀⠀⡘⢀⠃⠀⠆⠰
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡽⠀⠀⠸⣴⣇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣄⠀⠀⠀⠀⠀⢠⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣼⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢸⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⣿⡄⠀⠀⠻⣿⣦⣄⣀⣀⣀⣠⣴⡿⠇⠀⠀⠸⢿⣦⣄⣀⣀⣀⣠⣴⣿⠟⠀⠀⢠⣿⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⣸⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⣀⣀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⡞⢁⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡗⣘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⢾⡗⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢣⡇⣼⣇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡿⠇⡍⢃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⡀⠛⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⠁⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢔⡯⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⠈⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⠓⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢳⣍⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠍⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠫⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣙⣋⣠⣂⡀⠀⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⣤⣴⣦⣼⣿⡟⣿⡙⣟⠃⠀⢠⣴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⡿⣿⡀⣠⠌⢿⠰⡆⠀⢸⣿⣿⡿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⠟⠓⠻⠖⠿⠛⠇⠀⢺⡿⣿⣿⣿⣾⡖⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣧⣤⣤⡤⡄⠤⢸⣄⣄⠀⢠⣿⣿⣿⣿⣿⠘⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣿⣿⣷⣶⡖⠆⠐⠚⠛⠛⠀⠪⣿⣿⣿⣿⡟⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣏⢡⠆⡰⠆⠀⠀⠀⠀⠀⠹⠿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣄⣸⣷⣶⣤⣼⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣋⣹⣇⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠀⠠⠆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠠⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⡠⠔⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠇⡀⠄⠀⠿⠀⠀⠄⣄⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣈⠁⠀⠀⠐⠀⠉⠒⠘⠀⠀⢄⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣁⢦⢄⠀⠀⣠⠂⠄⢷⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠂⠀⠠⠋⠀⠡⠈⠀⡃⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠛⠒⠄⠁⠀⠀⠀⠀⡠⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢀⣀⣀⠠⠔⠘⠠⢀⠀⠀⠀⠁⠀⢀⡀⠔⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠤⠄⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠂⠠⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⠀⠈⠳⡀
⠀⠀⠀⠀⠌⠘⠤⣼⠚⢉⣐⠀⠀⡰⢚⣌⠙⢤⠀⠁⠈
⠀⠀⠀⢀⡀⡃⠀⢇⠀⠈⠋⢈⠀⠃⠈⠛⠁⢠⠀⠀⠀⠀⢄
⠀⠀⠀⠠⠄⠆⠀⠈⠢⠀⠤⢊⡀⠑⠂⠤⠒⠁⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⢠⢂⡄
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠁⠈⠀⠀⠈⠉⠁⠀⠀⢀⡞⠌⠂
⠀⠀⠀⠀⠀⠰⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠾⡾⠁
⡔⡆⠀⠀⠀⠀⢰⠀⡸⠂⠄⠠⡤⠠⢤⠴⠖⢚⠁⢀⠞⡁⠀⠔⠦
⢃⡀⢀⠀⠀⢀⠀⢠⠁⢀⣤⠤⠀⠀⠀⢤⣄⠀⠡⠈⠈⠰⠄
⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⡀⠀⢠⢎⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡡⠂
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠈⠉⠁⠀⣰⠃⠀⠐⣲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠁⠀⠙⠉⠀⠙⠦⠤⢄⠀⠀⠉⠉⠀⠰⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⣀⣤⡤⠖⠲⢤⣄⣀⠀⠈⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⣸⡁⢀⣀⣀⣀⣀⣀⣀⠀⢈⣆⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡎⠀⢸⠋⠠⡄⠀⠀⣤⠀⢹⠁⠈⢧⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠓⠒⢻⡀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠊⠀⠐⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢧⠀⠀⡀⢀⠀⢀⡞⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠑⠤⣄⣠⠴⠋⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⢙⡦⢤⠖⢢⠤⢴⠉⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⡇⢸⠀⢸⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠑⠊⠀⠀⠑⠋⠀⠙⠊


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

⠀⢰⡃⠠⠀⢀
⢀⡈⠙⠂⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣄
⠀⠀⣠⣞⣿⣿⣻⣄⠀⠀⠀⠀⠀⢨⢹⢣⡿⣿⣷⣹⢻⠀⠀⠀⠀⠀⠀⣿⣤⣦⣄⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡅⠀⠀⢀⣭⣷⡄
⠀⠈⢹⣿⡟⢻⣿⡏⠁⠀⠀⠀⠀⢸⣿⣿⣆⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣟⣻⣿⣿⣿⣗⡀⠀⠀⠀⠀⢀⡤⠝⣿⣷⡛⠛⣻⣿⠟⠤⣀
⠀⠀⠸⠿⠃⠘⠿⠇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠾⠀⠀⠀⠀⠀⠀⠿⠋⠉⠉⠉⠉⠉⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠸⠿⠋

⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠋
⠀⠀⠀⣠⣤⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⢰⣿⡛⠉⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠙⠁⠀⠀⣀⡀
⠀⢀⣴⣶⣿⣶⣦⡀⠀⠀⠀⠀⠸⠙⠻⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠏⠻⢷⣶⣇⡀⢓⣢⣄⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀
⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠻⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠚⠉⠉⠁⠀⠀⠀⠠⠖⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⣴⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣰⣦⣄
⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠋⣿⣿⡏⠙⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢳⣀⣄⣰⠃
⠀⠈⣭⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿⠃
⠀⢠⣤⣧⡄⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡶⠊⠀⠈⠲⣾⣿⣿⡆⠀⠀⠀⠀⢸⣇⣀⣸⣿⣶⣄⣀⣀⣀
⠀⠸⠿⠿⢃⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⡿⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠈⠿⣿⠇⠀⠀⠀⠀⠸⡟⠛⠋⢹⢿⠏⠉⠙⠿
⠀⠀⠀⠀⠙⠁


⣴⣶⣦⠀⠀⠀⣶⣆⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢂⣀⣀⠶⣶
⠁⢻⣿⣇⣀⣼⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⢩⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣁⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡟⠛⢛⠛⠛⣛⣛⡛⣿
⠀⠀⠙⠿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣩⣿⢤⢤⣼⠇⠀⠀⠀⠀⠹⣦⣋⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠄⠤⠄⣿
⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠾⠙⠛⠋⠹⠆⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⠛⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⠃⠀⢀⣄⠀⠿⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣗⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠈⡛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⣤⣤⢶⣶⣧
⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⡦⠀⠀⠀⠃⠀⠀⠉⢀⠈⠻⠇⠘⢿⡏
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢀⣄⣴⣤⣤⣄⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠛⠁⠀⢠⣤⣤⣴⣤⣄⠠⣤⣶⣶⣶⣶⢰⡷⡆
⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠁⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⣭⣿⣿⣿⣿⡀⣞⣿⣿⣿⣷⠀⠁
⠀⠀⠀⠀⠀⢐⣈⣿⠙⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⡼⣿⡛⢿⡛⠛⠁⠈⠻⠋⠋⠁
⠀⠀⠰⣤⣴⣿⣿⣿⣷⣴⣶⠀⠀⠀⠀⠀⣠⠀⢸⠀⠀⠀⠀⠀⡆⠙⢧⠈⠁⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠈⣩⣿⣿⣿⣿⠿⠷⣶⣷⠆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣦⣤⡄
⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⢀⣀⠀⠈⠛⣻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠺⣿⣿⡟⠉⠁⠆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⣹⣿⢹⣿⠛⠀⢰⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⣦⣏⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠛⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⢠⠀⢀⢀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠘⠛⠻⡟⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻
⠀⠀⠀⠈⠱⣾⣥⣄⠀⢠⣤⣤⣤⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢿⣿⠙⡇⠀⢹⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⡶⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧⣀⠠⣄⣀⣼⣦⢤⡄⠀⠀⠀⠈⠙⣿⣿⡆⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣌⣉⣹⣶⡽⠦⠠⡴⠿⠿⠋⣉⠁⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠁⣈⠟⠀⣴⣧⣀⠐⠟⣻⡟⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣡⣼⣿⠀⠀⠈⠙⠻⠃⠀⠻⠷⠜⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⢣⡿⢃⣿⠀⣴⡖⠀⠰⢷⣤⣴⡶⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⢘⣷⣿⣿⣿⣥⣴⣶⣦⣠⣿⢻⣤⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠛⢿⣿⠿⠋⢉⣠⣼⣿⣿⣿⣿⣿⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠁⣤⠉⠀⠙⠉⢹⡎⠁⠈⠉⠀⢨⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⠀⣿⡆⠀⠀⢰⡿⢀⣠⡄⠀⢃⣾⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠘⣀⠈⢁⣀⠀⣼⣦⠈⣻⣄⣀⣸⡏⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠈⣁⠀⠉⠀⠈⠁⡀⠉⠉⠁⠉⢹⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⡆⣿⡗⠲⠀⠿⣴⣿⣴⠆⠀⠃⢨⡅⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠠⣿⡉⠱⠦⠀⠸⣄⠀⠹⣤⣤⠶⣆⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠼⡄⠀⠀⠀⠀⠀⣦⠀⠀⡀⢀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⣤⡿⢤⠈⠀⢿⠤⣶⠔⠂⠀⠻⠷⠄⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠤⠴⣤⡤⠀⠤⠔⠦⠤⣶⠶⠦⠤⣤⣴⣶⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⢰⡄⢸⣄⣀⠠⢶⣤⠀⡀⢠⣤⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠺⡏⠙⣿⡃⠀⠚⠉⠀⠉⣿⠛⢀⣌⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⣰⠇⠀⣸⣿⠂⣴⡿⠆⠸⣿⣶⠀⢸⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⢷⠀⠻⠿⠈⠛⠛⠦⠜⠻⠟⠀⢸⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⡾⣄⣐⠀⣴⣦⣴⣷⣤⣦⣼⣦⣼⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡀⣤⡀⣿⣤⣼⠀⣀⣠⡤⠀⣤⣀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⢙⣷⣿⣿⣟⣶⠟⢿⣶⢠⣶⡎⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠻⣿⡿⣿⣿⠋⠀⠈⣡⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣿⣷⣿⠇⠀⠀⠘⣿⠿⠛⣿⡿⠇⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠘⠿⢢⣶⡤⠾⡇⠐⢻⡜⠻⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣽⣿⣿⣿⡀⠲⣿⣿⠀⠀⣿⣶⣆⠸⠇⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢳⣾⣿⣯⠀⠀⣻⣿⡟⣿⡶⢿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⠿⣿⠟⣿⣌⣿⠿⣿⣿⢿⣿⠴⣿⡥⠤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣶⣿⣷⣌⣼⠛⣷⡬⠿⠿⠟⠠⠟⠀⠀⠛⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠻⠿⠿⠿⠛⠿⠆⠈⠇⠀⣶⡆⣀⣴⣶⣄⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⢠⣤⡄⢰⣤⣀⡿⢤⣶⡿⢀⡟⠁⣼⣟⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠉⠁⣾⡟⠿⣄⠀⣹⡷⢸⡿⠾⣿⣿⣿⡿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀⢼⡆⣹⡿⢤⣥⠀⠿⠓⠺⠿⠾⠟⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣼⣤⣿⣤⣤⣄⣤⣤⡀⣶⣤⣤⣤⣼⣯⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⠀⣽⠀⠀⢀⢈⣇⣸⣿⣀⡈⠋⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠄⢰⣿⣿⣿⡟⢱⣏⡁⠶⠉⠁⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠉⠋⠁⠜⠁⠙⠛⠀⠸⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⣀⡴⢃⣠⠀⣰⠆⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆⠹⠏⠀⠘⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⣿⡀⣤⣀⣼⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠈⠁⣀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣆⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠟⠛⠻⣿⣿⠏⠀⠛⠻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢻⣿⣏⣁⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣦⣠⣾⡟⢠⡄⠰⣄⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣹⣷⡆⠀⢠⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀⠚⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⢀⠰⢦⡾⢾⣏⣀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠿⠀⢸⡇⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡍⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠹⣿⣿⣿⣷⠀⠀⢀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠘⢀⣴⢀⣿⣿⣿⣽⣿⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠀⠼⠁⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣹⣶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣙⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠀⠈⠆⠀⣾⣿⣿⣿⣿⣿⣿⣿⡁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠀⠀⠙⢿⠿⣿⣿⣿⡿⠋⢩⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠉⠐⠀⠀⠀⠀⠈⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠷⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⠟⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⠤⣤⣀⠄⠀⠀⠁⠀⠀⠀⠸⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠻⠿⠿⡿⠿⠿⠟⢻⣿⣿⡿⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣁⡀⠀⠸⣿⣤⣤⣄⡴⠖⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠋⠀⠀⠙⠉⠉⠙⠿⠿⠷⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⡄⠀⠀⠀⢀⡀⠀⢀⣀⡀