Christmas card Emojis & Symbols

⣿⣿⣯⣶⣿⣿⣿⢟⣽⣿⣧⣿⣫⢾⡿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣫⣿⣿⣾⣟⣵⣿⣿⣿⣿⣫⣿⣿⣼⣿⡷
⣯⣾⣿⣩⣿⣷⣽⣿⡻⣿⡅⣿⢿⣼⣿⣎⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣟⣿⣿⢱⣿⣯⣿⣿⣼⣿⣟⣿⣮⢸⣿⣯
⣟⣽⣿⠿⣿⣯⣻⣾⢿⣿⡇⣵⣿⠻⣿⡟⣫⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⢫⣿⡟⢿⣿⣙⣵⡿⣿⣿⣨⣾⡟
⣿⢿⣿⣿⣿⡿⣿⣿⣶⣝⡿⣿⣿⠙⠊⠛⠛⠇⢸⣿⡿⠟⠿⣿⣿⠀⠟⠓⠊⠛⠻⣿⡿⣷⣽⢿⣿⣷⣮⣿⢿⣿⡏
⣷⣿⣿⣶⣟⢿⣿⣿⣼⣿⣇⡿⠡⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣷⣮⡻⣾⣿⣥⣿⣿⣼⣯⣶
⣮⣾⡿⢛⣿⣷⣽⣿⠿⢛⠁⠀⠰⠀⠀⠀⠀⠀⠀⣤⢄⢰⢄⡄⡀⠀⠀⠀⠀⠀⠰⠀⠁⢉⠻⣼⣿⡿⣿⣿⢸⣿⢇
⡿⣻⣿⢿⣿⣟⢿⣵⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⣾⣿⡻⢻⣿⡯
⣷⢿⣿⣾⣿⡿⣿⣿⣦⠀⠀⠀⢠⡏⣄⣧⡟⡀⣄⠀⡒⡀⢠⡄⢀⡀⠀⢀⡀⢀⠀⠀⠀⠀⢰⢾⢿⣧⣿⣿⠼⣻⣾
⣷⣿⣿⣭⣿⣿⣾⣿⡙⠃⠀⠀⠸⣇⠀⢸⡏⣇⢸⣸⣇⣽⢼⣠⡇⡏⡇⣏⣆⣼⡆⠀⠀⠀⢹⣶⣿⣏⣻⣿⣾⣿⡧
⣟⣽⣿⣛⣿⣯⣿⣿⢳⡄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⣿⡟⣿⣿⢹⡿⣋
⡻⣻⣿⣿⣷⣝⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⢻⣿⡿
⣿⣿⣯⣴⣿⣿⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠉⠉⠀⠓⠐⠃⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⢸⡿⣫⣿⣿⣼⣿⡵
⣯⣾⣿⣹⣿⣷⣽⣿⠳⣬⡀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⣈⣴⣸⣿⡟⣿⣮⢸⣿⣏
⣟⣹⣿⠿⣿⣯⣻⣿⢿⣿⡇⣥⡐⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠠⠠⠀⠀⠀⠀⠀⠀⣸⡿⢿⣿⣙⣽⡿⣿⣿⣩⣾⡟
⣿⢿⣿⣿⣿⡿⣿⣿⣶⣝⡿⣿⣿⢤⣤⡠⣴⡆⢸⣿⣷⣦⣶⣿⣿⠀⣶⣤⣤⣠⢼⣿⡿⣷⣽⢿⣿⣷⣿⣿⢿⣿⡯
⣷⣿⣯⣶⣟⢿⣿⣿⣼⣿⣇⣿⣿⣾⣿⢷⣿⡃⢸⣿⣿⣿⣿⣿⣿⠀⣿⣯⣽⢿⣸⣿⣷⣯⡻⣾⣿⣧⣿⣿⣸⣿⣶
⣮⣾⣿⢛⣿⣷⣽⣿⢿⣿⡇⣿⡿⢸⣿⣯⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣾⡿⣿⣷⣱⣿⢟⣿⣿⣼⣿⡿⣿⣿⢸⣿⢏
⡿⣻⣿⢿⣿⣟⢿⣵⢶⣟⠿⣫⣾⢹⣯⡻⣿⡅⢸⣿⣿⣿⣿⣿⣿⠀⣿⣶⣿⣿⢟⣷⡿⣿⣿⢻⣯⣾⣯⡻⢻⣿⡯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣶⣿⣾⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣬⣦⣶⣾⣿⣿⠀⠀⠀⠀⠓⠀⠀⠤⠀⠀⠀⠀⢸⣿⣿⣿⣘⣿⣫⡛⣿⣿⣿
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣶⣼⣽⣿⣿⠀⠀⠀⠰⠂⠠⢄⠥⠀⠀⠀⠀⢸⣿⣿⣿⣹⣍⡟⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⣿⣿⣥⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣿⣷⣷⣶⠀⠀⠀⢸⣿⣿⣿⣏⣀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣭⣿⣿⠀⠀⠀⠛⠹⠿⠟⠛⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⢸⣿⢿⢿⣾⣿⣿⣿⣽⣿⣿⠀⠀⠀⠀⠀⢰⢿⡀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣯⣿⣿⣿⡇⠀⠀⠀⠀⣁⠉⣀⠈
⠀⠀⢸⣿⣾⣿⣾⣿⣿⣽⣽⣿⣿⠀⠀⠀⠀⢨⣛⣿⡿⠀⠀⠀⠀⣿⣯⣿⣿⣭⣟⣿⣽⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣧⣄⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠉⠀⠀⢿⠗⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⡀⠀⣀⡿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣭⣩⣸⣿⡇⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⢸⣿⣿⣉⣿⣿⣿⣿⣛⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡇
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠑⠀⠐⠀⠀⠠⠐⠃⡌⢁⠁⠉⠀⠓⠀⠀⠂⠐⠀⠐⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢤⡈⠔⣤⠄⢠⠲⠄⡶⠗⢛⡖⣶⢲⡶⡞⠁⠀⢤⡌⠌⣠⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠠⠀⡔⢲⣌⠷⢺⡄⠘⠈⠉⠙⡏⠖⣿⠋⠩⠶⡄⠠⠀⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣰⡀⠃⠤⠤⠁⣶⣡⣋⡡⠏⣧⣤⡶⠧⡚⡖⣶⡼⡳⢠⠶⠄⠀⠄⠤⣃⣼⠄⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣈⣹⢷⣤⣤⣤⡥⣸⡖⠐⢒⠛⢟⠟⣩⢕⡗⠪⣵⠿⠛⠗⠶⣶⣶⣾⡫⢅⣤⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣧⣀⠈⠓⠶⠃⣤⡾⠂⠀⠈⠁⠈⠉⠁⠀⢀⣴⣿⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢸⣿⣿⣦⠀⠀⠑⡀⣒⡩⠘⠃⠠⠠⠁⠀⢠⣿⣿⣿⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣀⣠⣄⠁⣭⣥⣚⡁⠈⢢⣦⣤⣼⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡟⠁⢻⡀⠀⠀⠀⠀⠀⣸⠁⠈⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣯⠉⠉⠉⠉⠀⠀⠟⢁⠀⢸⠃⠀⠀⠀⠀⠀⣸⡀⠀⠈⠃⠀⣀⠈⠉⠉⢩⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⣰⢶⠟⠛⠛⠉⠀⠀⠟⠀⢀⣀⣤⣤⡄⣟⣀⣠⣧⣜⢩⡙⣋⢧⣤⣤⣤⣈⠉⠀⠀⠀⠀⠉⠉⠉⢻⢻⡄
⠀⠀⠀⠀⠀⢀⣏⡞⢀⢐⡀⠀⠀⠀⡠⣾⠋⠁⠀⠀⠉⠛⡿⢿⣿⢿⢟⡻⡞⠉⠀⠀⠀⠉⢻⡦⡀⠀⠀⢠⠈⠀⠈⣧⣧⡀
⠀⠀⠀⠀⡴⠏⠛⠀⠀⠂⠁⠀⠠⠊⣴⠉⠀⢀⣀⠀⠀⠀⠒⠾⡿⠾⢯⣽⠃⠀⠀⢀⣀⠀⠈⣵⡈⣢⠀⠀⠀⠀⠀⠈⠋⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢱⠀⠌⠁⠀⢿⣿⠇⠀⠀⠆⠀⡀⠀⠘⢹⠆⠀⠀⢿⣿⠇⠀⢹⠀⠇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠈⠢⡀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⡣⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⣇⠀⠀⠀⠀⡀⠈⠀⠀⠈⠃⢀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠃⠀⠀⡇⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⣉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠠⠄⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⣀⠀⠀⠀⢀⡀⢰⠁⢨⠶⠲⡀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠈⠲⡄⠀⠀⢈⠀⡸⠦⣄⢇⠀⡇⡀⣀
⠀⠀⠀⢀⠞⠁⠀⢀⢆⠰⠆⢁⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⠐⠠⠊⠄⠀⠈⢯⣉⣷⠧⠮
⠛⠗⢢⣼⠀⠀⠀⠀⠘⢆⠑⢤⢤⠗⠀⡀⠀⠠⠠⣦⣤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠀⠀⢀⠀⠀⢸⠋⢳
⠀⠠⠌⣿⣧⡀⠘⢀⠀⠀⣡⢄⡸⠂⠀⠀⠐⠤⠀⠀⠀⠀⠀⢀⠀⢐⠇⠀⠀⣀⠀⠀⡈⠃⠀⠁⠀⡂⠎⣈⡘
⣬⡶⠚⣉⠓⠿⠂⠠⠉⢊⡁⠀⠀⡀⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⣀⣴⠀⢀⠀⠀⠀⠀⣈⠀⠀⠰⣊⢁⣢⠔⠁
⠦⠤⠊⠀⢃⡸⠀⠀⢾⢧⣟⠃⠀⠀⠀⠀⠐⡆⠀⢀⣄⠀⠀⠀⢀⠀⠐⠨⠀⠀⢴⡀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⣠⢣⡀⠀⠪⣖⡪⠂⠀⠂⠀⠀⣷⡇⠀⢸⡇⠀⠀⠀⢀⠐⠂⠁⠀⠀⠀⠉⠢⠤⣀⣀⣠⠵⡖
⠷⢄⣀⠔⠁⠀⠀⠉⢲⠇⠀⠀⠀⠀⡄⢀⣿⠀⠀⣾⠁⢀⡖⣦⢸⠄⡆⠘⡇⣰⠀⡖⠀⣴⠀⣀⠀⣀⠈⠢⣀
⠀⢀⢂⠀⠐⠀⠀⠀⠾⠀⣀⢤⠀⡀⢀⡾⢻⡀⡰⡇⠀⢾⡞⢁⠄⢸⡇⠘⠑⣏⢨⡇⣠⡇⠀⠀⠀⠀⠀⣀⠀⠉⠓
⢄⠁⠈⠂⠄⠀⠐⠀⢠⠞⠁⠀⠀⢸⡏⡎⢠⡄⠀⠉⢠⠀⠑⠒⠦⣶⣭⣀⣠⠟⡵⠚⣹⠉⠙⠆⣀⠀⢀⠇⠉⢉⠝
⡎⢀⡄⡁⠀⠀⠀⠀⡿⠀⡸⠀⢀⣼⢹⡥⣄⠃⣾⠀⣶⠀⣏⠇⢰⠇⠀⡶⢐⡞⢻⢞⡟⠉⣶⢸⣌⠃⠦⠔⡀⠈⠁
⠔⠁⠇⠀⠐⠂⠀⠐⣷⠈⠁⠀⠉⠀⡾⢀⠞⠀⡇⠰⡇⠀⡹⣆⢹⠀⣼⠁⣸⠀⠏⠸⣇⡠⣇⠠⠻⡆⠀⡔⢁
⠀⢀⠜⠂⠐⠒⠒⠠⢘⠦⠀⠀⠀⠀⠀⠸⠀⠀⣀⣀⠀⠘⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⢩⠃
⠀⠂⡠⣄⡀⠀⠀⠀⠀⠘⢂⠀⠀⠀⠀⠂⠀⠘⢿⣽⣇⡀⠀⠐⠂⠀⡀⠊⠒⡆⠀⠄⠀⠀⠸⠍⠀⠨⠇⠰⠥⠤⠔
⣀⠞⠀⢡⠈⡆⠀⠀⠀⠀⠈⠱⡄⠀⠀⠄⠀⢀⡿⢻⠻⡟⣆⠀⠀⡀⠈⠥⠒⠂⠀⠠⠀⠀⠐⠀⠈⠀⠒⢄
⡵⠂⠤⠼⠀⡧⠒⠂⠀⠀⠒⠈⡄⠀⠀⠀⠀⣼⠷⣶⢞⢻⣍⣇⠀⠀⠀⠀⠠⠄⠀⠀⢀⠁⠤⠀⣀⠀⠄⠀⢣
⣁⠤⠤⣀⢘⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠛⡛⠋⠉⢡⡈⢂⠀⠀⠀⡀⢠⣤⣄⠸⡀⠀⢘⡰⠃⠀⠀⡸⣠⣾
⠧⠶⠔⣄⠸⣦⠀⠀⠀⠀⠂⠡⠀⠀⠀⢄⠀⠇⠀⠀⠐⠀⠊⠀⡘⠀⠰⣴⡗⢿⣿⠯⠳⣷⣄⡈⠈⣀⣀⣼⣿⣯⣿
⠤⠒⡲⢞⡓⠼⠛⠛⠯⠤⠞⠀⠀⢀⣑⣦⣬⣼⣦⣤⣀⣈⣤⣾⠧⢄⣰⠻⡟⠙⠛⠿⢿⡿⠿⣿⣿⣿⣷⣿⣟⠻⡁
⢀⡞⠀⠸⡄⠀⠀⠀⠀⠀⠀⠰⣄⠐⠼⠤⣦⢋⣍⠉⠉⠉⣏⠁⠀⠀⠉⢲⣄⠀⢀⢔⠟⠀⠀⣤⡠⡀⠉⢻⣿⣿⣾
⢸⠀⠀⠀⠹⠤⡀⠀⠰⢿⣿⠀⢈⡗⠀⣰⡗⠋⠀⠀⠀⢀⠇⣆⠀⠀⢐⣴⠃⠀⢇⣸⢀⣀⣤⠑⠔⢁⠀⠀⠹⣿⣉
⢺⠀⠀⠀⠀⠀⠀⠉⠳⠌⠶⠂⠹⠀⡜⠓⡦⠤⠤⠭⠥⠜⠀⠘⠒⠈⠁⠈⡇⠀⠀⢸⠀⠀⠀⠀⡀⠈⠑⠤⠤⠻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠡⢦⠀⣀⣀⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠁⢀⣤⡄⠐⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣰⡛⢮⠆⡴⠋⠉⠑⢾⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣿⣿⣿⣧⣄⠁⡀
⠀⠀⠀⠀⠀⢼⠀⡟⢤⡀⠀⢀⡠⣊⠘⢾⠂⡼⠁⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠤⠀⠛⠻⣿⡟⠉⡁⠐⠄
⠀⠀⠀⠀⠀⠈⠻⣞⠂⠓⢢⣱⢄⠙⠗⣠⠞⠀⢀⡤⠖⠚⠉⠀⢠⡆⠀⠀⢀⠀⢀⠀⠠⡎⢹⡄⠀⠀⠀⠐⠁⠐⢠⢈⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢈⣂⡤⠔⠋⢀⡴⠚⠁⡀⠀⣀⢀⡐⣆⠡⠈⠙⠃⠂⠁⠇⢠⡨⠀⣇⣀⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢺⠠⠒⠐⢿⠰⠉⣀⣡⠤⠖⠒⠚⠉⠉⠉⢒⠲⠦⣄⣹⠀⠀⠀⢉⡽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠁⢘⣠⠔⠋⠁⠀⠀⠀⣀⣀⢀⠀⠀⠸⣶⣚⣉⣁⣀⣀⣀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠹⣴⣫⣄⡆⠀⠀⡠⠚⠉⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⠴⠋⠀⢀⡞⠁⠀⠀⠀⠀⢀⡤⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢤⡄⠀⢀⠔⠁⠀⠀⢀⣼⣤⣀⣀⠠⠤⠤⢾⡐⢇⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠀⠀⢀⡤⢿⡄⠀⠀⠀⠀⠠⠴⣎⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠈⠁⠀⠀⠀⢀⡟⠀⣼⠀⠀⠀⠀⠀⠄⠀⢹⢀⡀⢻⠀⠀⠀⠀⠐⣏⠳⣄⠀⠀⡖⢢
⠀⠀⠀⠀⠀⣤⡀⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣻⡀⡠⠒⣤⡖⠒⢄⡿⣦⠀⠞⢦⡀⠀⠀⠀⠈⠑⢿⣦⠀⣧⡌
⠀⠀⠀⠀⠀⠻⣧⣇⣾⠟⠀⠀⠀⠀⠀⠀⠀⢸⠩⠁⡏⠹⠁⡀⢋⠑⢌⡈⣱⠟⠀⠂⢠⡇⠀⠀⣖⠐⠒⢢⣌⡣⣽⣧⢶⡄⣿⡗
⠀⠀⠀⠀⣤⣄⢸⢉⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⣳⠆⠈⠀⠈⠀⠀⠆⠀⠠⠀⠂⠄⠐⠀⣱⡀⠀⠀⠉⠉⣉⠈⣛⠻⣿⡜⢠⠃⣠⣄
⠀⠀⠀⠀⠙⠛⢿⠟⠋⣀⠀⠀⠀⠀⠀⠀⡠⠋⢁⡄⠀⠀⠂⠠⠀⢀⠀⡀⠁⠐⠀⠠⢘⠁⠙⣦⠀⠀⠘⢿⡛⢭⣷⣿⡤⠋⡞⣁⠎
⠀⠀⠀⢰⣾⣦⠘⢀⣾⡿⠀⠀⠀⠀⠀⡸⠁⣠⠈⠣⡄⠀⠀⡀⠀⢀⠀⠀⢈⠀⠁⡻⠁⢀⡀⠸⠀⠀⠀⣠⣝⡦⣄⡈⣧⡾⠛⡡⠖⢹⠇
⠀⠀⠀⠀⠉⠛⠛⡟⠉⣠⡀⠀⠀⠀⢠⠃⣼⠃⠀⠀⠳⠦⠆⡀⠈⠀⠄⢀⠀⣰⠎⠁⠀⠈⣷⠄⡆⠀⢀⡳⢄⡉⢢⡈⢫⣠⣮⡠⢐⠁
⠀⠀⠀⠀⠺⣿⣆⣃⣾⡿⠃⠀⠀⠀⣾⣾⡧⣄⣀⣀⣀⣀⣓⡤⣺⠂⡤⣊⣈⠁⠤⠐⠂⢉⢽⣝⡃⠀⢿⣿⢆⢍⡚⠿⡾⠉⠁⣰⠿⠇
⠀⠀⠀⠀⠀⣠⣈⠹⡁⣠⣦⡄⠀⠀⠉⢸⠓⠒⠒⠤⠤⠭⠭⠽⢭⠽⠇⠀⠒⠒⠂⠉⠉⠀⠘⡇⠀⠀⠀⢧⠸⡄⢱⡀⣇⡤⠚⠁
⠀⠀⠀⠀⠈⠻⠿⣷⣵⡿⠋⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡛⢮⡳⡈⠻⣿⣡⡔⠚⣉⠆
⠀⠀⠀⠀⠀⠀⢀⣤⣌⣃⣸⡿⠃⠀⠀⠈⠑⠒⠲⠶⠦⠤⣤⣤⣼⣧⣤⣤⣤⣤⡶⠶⠶⠶⠚⠃⠀⠀⠑⢄⣹⣎⠳⠋⢉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⢿⣏⣠⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠛⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⣸⠀⠀⣾⠀⠀⡇⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠠⠳⠠⠳
⠀⠀⠀⢈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠈⠀⡀
⠀⠀⢠⡙⠴⠀⠠⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣣⣿⣿⣿⣿⡥⠘
⠀⠀⠀⠀⠀⠀⢼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠉⠀⡀
⠀⠀⠀⠐⠀⠀⠘⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⡆⠂⠂⠀⠀⠀⠀⠀⠀⠀⡇⡤⢳⠀⡇⡇⡀⡀
⠀⠀⠀⠀⠀⠠⡄⣤⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠁⠀⠁
⠀⠀⠀⠀⠳⡀⡇⡿⡇⠀⢡⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠐⠆⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠁⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣆⠀⡆⠀⣰⠂⢠⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⠃⠀⠀⠀⠀⠀⢀⡄⠀⠐⠐⠀⠀⡄
⠀⠀⠀⡿⠀⡷⠀⡟⠀⢸⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠠⠀⠀⠀⠀⠀⠀⢀⠆⠀⡄⠀⢠⠄⠀⢰
⠀⠀⠀⠁⠀⠁⠈⠁⡀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠸⢸⠀⠀⠀⡃
⠀⠀⠀⠀⡇⢰⡇⠀⢱⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⣠⢸⠂⠀⠀⡆
⠀⠀⠀⠀⡇⠸⣇⠀⠈⠇⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡌⣄⠀⠐⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢸⡆⣾⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⠈⠀⠈⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠼⠀⠹⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⡄⣰⠀⡀⠀⢀⠂⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠆⠀⠠
⠀⠀⠄⠄⠤⠠⠠⠀⠄⠄⠠⠀⠄⠄⠄⠀⠀⠈⠀⠁⢁⠈⠀⠑⠈⠉⠘⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠦⠐⡓⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠛⠀⠀⠀⡀⠀⠀⢉⠀⢀⠀⠀⠂⡈⠂⠤⠉⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠈⠄⠄⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢿⢿⠁⠀⠈⠀⠀⠀⠀⠀⢀⠉⠁⠀⠘⠀⠀⡄⠀⠀⠐⠇⠀⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⠀⠀⠀⢀⣢⡥⠄⣀⣛⡀⠀⠓⠃⣀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠄⠒⢀⠈⠁⡀⠘⠈⠀⠀⠁⠀⠀⠈⠁⠋⢡⠀⢠⡆⠀⠀⠀⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡴
⠀⠀⠀⠀⠀⠤⠀⠐⠀⠀⠂⠀⣈⠀⠀⠀⠀⠀⠀⠄⠠⠂⡀⠐⠀⠤⠀⠀⠀⢀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣐⠀⠂⠂⢀⠠⠀⠐⠀⠂⠀⠀⠈⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠘⣀⢀⠀⠀⠀⡀⢀⠀⠢⠀⢀⠀⠤⠀⠀⠇⠀⠐⠠⠀⡀⠀⡆⠄⢰⠀⠀⠀⠄⠀⠀⢠⠀⠀⡂⠄⠀⡀⠀⡀⠀⠁⡀
⠀⠀⠀⠨⠀⢀⠉⡿⠄⠀⠀⠐⠿⠒⠀⠀⠉⠀⠂⡀⠀⠐⠀⠈⠀⠀⠄⠀⠀⠁⠀⠀⣀⠂⠀⡀⠂⠀⠃⠀⠀⠤⠒⠂
⠀⠀⠀⠀⡀⠉⡁⠀⠰⠀⠀⠀⡀⠀⠐⢤⠀⠀⠀⠁⡆⢸⠀⠄⠀⠀⡀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠨⢳⠆⠀⠀⠀⠈⠁
⠀⠀⠀⢠⢿⡀⠀⠀⠀⠀⠀⠀⠁⢀⠉⠲⠀⢠⠸⠂⠀⠘⠂⠀⠒⠀⣦⠀⠐⠃⠂⠁⠀⠀⠀⠤⠀⠠⣄⡀⢀⡴⠀⠠⠂⠀⠀⠂
⠀⠀⠀⠈⠚⠁⠀⠀⣀⠀⠀⠀⠀⠨⠤⠖⠀⠀⡼⠁⠀⢀⠀⠀⠀⠀⢋⢰⣄⣄⢄⡀⠀⢤⠀⠀⠒⠀⠈⠁⢙⠀⠀⣁⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠤⠀⢡⠀⠠⡀⢊⣐⡀⠐⠂⡤⠀⢀⠳⠀⠀⡈⠀⠠⠈⠀⡄⠈⠁⠃⠂⠈⡇⠀⠀⢊⠀⠐⠈⢉⣇⠀⠀⣠⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠠⠄⢡⡀⢀⠀⠁⠀⠐⠂⠢⠠⣁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠂⠈⠀⠀⠀⠈⠷⣤⣀⡀⢠⡤⠾⠋
⠀⠀⠀⠀⠀⠐⢵⠀⠀⠆⡇⠈⢐⡂⠀⠓⠀⠀⠀⠀⠀⠀⠐⠀⢂⠠⠀⢀⢂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⠆⠈⠀⠶⣤⠀⢀⡀⢀⣄⣀⡀⠰⠦⢤⠄⠀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠐⠂⢀⡀⠀⠛⠅⠀⠈⠀⢠⢀⠄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠈⡈⣀⠄⠤⡀⢠⠀⠀⠀⠀⠀⠋⠁⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⢠⠄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠖⠆⠀⢀⠠⠀⠈⠁⡀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣥⠾⠶⠇⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠶⠶⠀⠠⠁⠀⡀⠀⠀⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠈⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠶⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⢻⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡦
⠀⠀⠀⠀⠀⠀⢀⠎⠉⠀⠀⠀⠀⠀⡠⢛⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡦⠀⠀⠀⢠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠒⢲⡴⡖⠂⠀⠀⠀⢠⡾⣿⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⢞⡓⠄⠀⠀⢛⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠐⠋⣴⣿⣎⠓⠄⠀⠀⠀⣼⣿⣿⡄⠀⠀⠰⢊⡟⠆⠀⢀⣾⣿⣿⡄⠀⠀⠀⢠⣧⠀⠀⠀⠀⠶⠾⣿⣿⡿⠶⠂
⠀⠀⠀⠀⠀⠀⢠⡤⣤⠴⠀⠂⠀⢠⠞⣩⣿⡻⢦⡀⠀⠀⣸⣿⣿⣿⣷⡀⠀⠰⢋⡝⠦⠀⠀⣠⣿⣧⡀⠀⠀⠀⣾⢹⡆⠀⠀⢀⣠⣾⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⡴⢻⣿⢷⣄⠉⠀⣰⣿⣿⣿⣿⣿⣧⠀⠔⢁⡍⠶⠀⣼⣿⣿⡿⠿⠂⠀⣸⣟⣸⣿⠀⠀⠀⣀⣼⣿⣿⣦⣤⡤
⠀⠀⠀⢀⢀⡀⡀⡀⡀⠀⠀⠀⠀⠈⣠⠟⡿⢧⠈⠓⢀⣾⣿⣷⣿⣿⣿⣿⣇⠠⠞⡻⠄⠀⢀⣼⣿⣷⡀⠀⢀⣿⣻⢨⣿⣇⠐⠛⠛⢻⣿⣿⣷⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠊⠁⠀⡇⠀⠑⠀⠾⠿⠿⠿⢟⠿⠿⠿⠿⠀⠀⡇⠀⠀⣾⣿⣿⣿⣿⠄⣼⣻⣿⣼⣿⣿⡄⣠⣴⡿⢿⡿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠚⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠸⠇⠀⠀⠉

⠀⢀⠀⢠⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⢰⡄⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠻⣿⣿⣿⢿⣿⣿⢿⣿
⠀⠸⣤⣀⣠⡼⠁⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⢀⣰⡄⠀⠀⠈⠋⠀⠀⠺⣤⣈⣤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡻⣿⣧⣿⣿⣿⣿
⠀⠀⠈⣿⣏⠀⠀⢰⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢁⡀⠲⠀⠀⠄⠀⠀⢨⣿⡇⠀⠀⣰⡰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠛⢀⡀⠿⢿⣶⣾⣿
⠀⠀⠘⠛⠛⠀⠀⠙⠋⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⢠⠀⠀⠘⠛⠋⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠛⣿⣿
⠀⠸⠆⠀⣀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⢻⣞⠁⠙
⠀⠀⠀⠀⠁⠀⠀⢦⣄⠜⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠀⠀⠀⠀⠈⢧⣀⡾⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠄⠀⠐⠒⠒⢻⣾⠚⠒⠒⠂⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠀⠐⠒⠒⢃⡯⢾⠖⠤⠀⠀⠛⠃
⠀⠀⠀⠀⠄⠀⣠⠋⠁⠳⡀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠁⢸⣿⣿⣿⣿⣟⡻⣟⡿⣻⣟⠿⡿⣿⢟⣿⣿⣟⢿⣟⣿⣻⣛⡻⣟⢿⣻⣿⣿⣿⣿⠀⠐⢦⣴⠋
⠀⠀⠰⠀⢀⠀⠀⣶⡇⠀⠀⢸⣯⣽⣿⣿⣿⣟⣗⣇⣾⣷⣹⣼⣿⣛⣧⣾⣶⣻⣗⣹⣿⣿⣇⣷⣼⣚⣿⣿⣯⣿⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢢⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⠀⠸⠿⠿⠿⠿⠿⡿⢿⠿⠿⡿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠦⠌⡂⡊⢪⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠉⠋⠀⢠⡂⠐⠂⣢⠀⠀⢈⠄⡅⢀⠂⠈⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠠⣷⡌⠀⠀⣆⡈⢀⡴⠀⠀⣩⢐⠔⠱⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠙⣷⡞⠁⠀⠀⠀⠀⠀⠈⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣿⡏⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠘⠃⠀⠀⠀⠀⠀⠠⠄⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⣀⠘⠛⠁
⠀⢠⠀⠀⢠⡆⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⣸⡦⠀⠀⣄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⡌
⠀⠈⢻⣶⠋⠁⣰⡔⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⠀⠀⠈⢳⣾⠋⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢠⣼⠄⠀⠀⠀⡀⢀
⠀⠀⠸⠿⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠈⠉⠀⠀⠙⠀⠀⠀⠀⠀⠺⠿⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠁⠢⠈⠀⠈⠉⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣀⡀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠛⠛⠛⠛⠛⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠉⠉⠓⠒⣦⡀
⠀⠀⠀⠀⠀⢰⡿⢭⣁⣀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡤⠶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠉⠉⠓⠲⣯⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠉⣸⠋⠉⠉⢻⡈⠙⠒⠦⣤⣀⣀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡷⢤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠖⠚⠫⣤⠞⠀⠀⢠⠏⠀⠀⠀⠀⢱⡄⠀⠀⠙⣏⠉⠉⠓⠒⠚⠁⠀⠀⠀⠀⣀⣸⠇⠀⢀⣈⡭⢿
⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢀⡞⠲⠦⣤⠴⠚⠋⢻⡀⠀⠀⠈⢧⡀⠀⠀⠀⣀⣠⠴⠖⢛⣩⡤⠖⠛⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⡶⠤⣤⣀⣀⡼⠁⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠳⡴⠚⣉⣡⡴⠖⠛⠉⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣟⣀⣤⠀⠀⣼⣁⡀⠀⠀⡇⠀⠀⣀⣠⣼⣧⠀⠀⣤⣀⠹⡏⠉⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠁⢸⢀⡼⠁⠈⠉⣿⠒⠷⠚⠋⠉⠀⠉⢹⢧⣸⠁⠈⠛⠃⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣀⣤⠼⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣿⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⣀⣸⢀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡶⠚⠒⠦⣄⠀⠙⢦⠀⠀⢀⡴⠋⢀⡤⠖⠛⠉⠙⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠈⠳⡀⠀⢳⣀⡞⠀⣰⠋⠀⠀⠀⠀⠀⢸⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢇⠀⠀⠀⠀⠀⠹⡄⠀⠙⠀⡼⠁⠀⠀⠀⠀⢀⡴⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⢤⣀⠀⠀⠀⢹⡀⠀⢰⠇⠀⠀⢀⣠⠖⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠉⠙⠓⠲⠶⠇⠀⠼⠴⠒⠋⠉⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣲⠆⠀⠀⣠⣄⠀⠀⢴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⣾⠀⠈⣧⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢰⠏⣿⠀⠀⡏⢧⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⠏⠀⡟⠀⠀⡇⠈⣇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢶⣠⠏⠀⠀⡇⠀⠀⡇⠀⠘⣆⡾⢦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡇⠀⠀⡇⠀⠀⠹⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⡇⠀⠀⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠊⠉⡦⠞⠉⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠢⢼⡀⠀⠳⠦⠀⣗⡶⠚⠁⠀⢈⣦⠚⠉⠙⢦
⠀⠀⠀⠀⠀⣴⡒⠒⠋⢹⢀⠀⠀⠀⣀⣙⣧⣦⡀⣰⣿⢽⠇⠀⠀⣼⣧⣖⣲⣦⡜
⠀⠀⠀⠀⠀⢻⣿⣦⠽⠯⣿⣷⢠⢸⣿⣿⣿⣿⢳⡿⣀⣾⡼⠗⠛⠛⠛⢯⣯⡿⠗⠊⠲⡄
⠀⠀⠀⠀⠀⢠⣟⠁⡠⣠⣾⢿⣿⣮⣯⣿⣿⣻⣾⣮⣿⣿⣿⡆⠀⠀⡀⠈⣟⣭⣁⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣯⣳⣮⠇⠈⣛⣓⣛⣛⣉⣿⣿⣿⣿⣿⣿⣵⣡⣼⣖⣥⣾⣿⣿⣿⣞⣁
⢀⡠⠔⠒⠩⠶⢿⣭⣽⠀⣾⢽⠋⠉⠀⠀⢀⣽⣿⢿⣿⣿⡄⠀⠀⢹⣿⠛⠟⠀⢀⣠⢴⢟⡇
⡎⠉⠑⠒⠤⣄⣀⠭⠛⠷⡶⢾⣇⡆⡄⡀⢸⣭⢠⣿⣿⣿⣵⣵⣴⣼⠤⠉⠓⢿⣍⠴⠣⢺⡇
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣵⡛⠻⠛⠳⠾⠿⠛⠃⠁⣀⣤⡼⣻⡀⠀⠠⠆⠘⡗⠱⢠⣸⡇
⢧⣀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠈⠉⠒⠢⠤⣠⡴⠒⢻⡿⢋⢬⣺⡇⠀⠣⠢⠀⣿⣵⣿⡏
⠀⠹⠛⢶⣦⣤⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢺⡺⣲⢟⢕⠅⣮⣾⡇⢀⡤⠔⢀⡯⡈⢺⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠓⠂⠀⣧⣀⡀⠀⠀⠀⢸⡾⣀⢀⣶⣾⢞⣿⠇⠈⠀⠀⢸⣟⠶⣹
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠉⠙⠛⠒⢦⣼⢶⣾⣿⢟⣕⣕⣿⠀⠀⠀⠀⠈⣿⢈⣾⡀
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣗⡹⡾⣫⢺⢝⢕⣿⠀⠐⠒⡁⠀⣷⢋⣾⣿
⠀⣛⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢅⣾⡫⡣⣢⡗⣿⠀⠀⢇⠄⠀⣯⣿⣿⠒⠂⠀⠀⠈⢁
⠀⣏⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⣕⢿⣪⡪⡫⣪⣿⠀⠀⠁⢁⣠⣿⡋⢠⠄⠉⠁⠀⡠⠄
⠀⠀⠉⠛⠲⢤⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢕⢕⢟⢝⢋⣾⣿⢀⣤⡶⠛⠉⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠚⠷⣦⣤⢸⡃⠀⠀⠀⠀⣿⣱⢷⢕⣵⠿⠿⠭⠋⠱⠖⠤⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠶⢤⣀⣿⡥⠟⠉⠩⠥⠄⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠁⡀⢀⠀⣀⢿⡛⢀⠀⢠⣀⠀⢀⣀⢀⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢫⡁⣦⠸⡆⣿⢸⡇⣿⠘⠃⠛⠰⣛⣬⡙⠀⡄⢉⠀⠖⠐⠦⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣥⠻⣆⢳⣼⡷⠟⠛⠛⠛⠁⠀⠀⠀⢴⣿⡍⠛⠀⣅⠈⠀⠐⠁⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣀⠳⣌⣳⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠘⠃⡀⠀⠀⠀⠈⣁⠘⢃
⠀⠀⠀⠀⠀⠀⠀⢠⠏⢤⣍⣳⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠴⠀⠂⣀⠃⣈⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠏⠘⠲⢬⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢶⣄⣁⡀⠃⠁⠀⠠
⠀⠀⠀⠀⠀⠀⣸⢠⣍⣛⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠄⠧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⠰⠤⠬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⣛⡂⣤
⠀⠀⠀⠀⠀⠀⣿⠐⠒⣚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢭⣭⡅⣿⠁
⠀⠀⠀⠀⠀⠀⢹⠈⡉⠉⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠲⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠱⠄⡀⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⠍⠁⡴
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠚⣫⡼⢷⣀⡀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠤⠀⠀⣛⡀⠀⠪⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣾⠫⠄⣁⠉⠐⠄⠀⣿⡓⡇⡀⢁⠀⣮⠄⡀⠼⠁⠀⠑⠀⠀⠀⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠱⠄⠲⣤⡀⠀⠴⡏⠀⠠⠙⡷⢒⠀⠉⠴⣀⠂⠐⡔⠐⠀⠘⢄⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂⠀⠉⠠⠐⠀⠀⣀⣀⡰⠟⠊⡖⠐⣀⠈⠃⠀⠁⠰⠃⠀⠀⢀⠀⡐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢴⡶⣤⡴⠧⠀⠂⠉⠀⠉⠙⠿⢔⣚⣳⠀⠈⠁⠴⠁⠙⡄⠀⠉⢐⡎⠀⠉⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⢘⣯⠀⠀⠆⡭⠈⢈⡡⡀⢀⡀⠀⠂⠔⠁⠀⠁⠪⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠈⠀⠀⠀⠀⠂⠀⠈⠈⠀⠰⠀⢰⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⢂⡀⠀⠁⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢢⡡⠺⣍⠄⠢⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙
⠀⡀⠐⢤⠁⠐⠁⠀⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆

⠀⠀⠀⠠⡠⠀⠠⠀⠀⠀⠀⠀⠴⠰⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠸⡃⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡄⢀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑
⠠⣀⡠⠃⠐⠋⠀⠀⠀⠀⠐⠀⠠⠀⠠⢐⠀⠀⠀⢀⠀⠐⣀⠖⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠊⠤⠖⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡁⣲⠜⡑⡤⠃⠀⠀⠀⠀⠀⡠⠲⠢⠀⠀⠀⠀⠘⠄⠸⠻⠖⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣰⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣄
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠿⠿⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣵⣶⣶⣾⣿⣿⣶⣶⣶⣭⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣭⣭⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣭⣭⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⢉⣿⣿⡉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⢄⣀⣀⣀⣀⣀⣀⣈⣻⣟⣁⣀⣀⣀⣀⣀⣀⡠⠤⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠋⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⣠⣤⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⣇⠀⠀⠀⡀⠠⠐⠂⢐⠄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠪⣁⣀⣉⠉⠁⠀⠀⠀⢸⡀⠀⠀⢀⡟⠀⠀⠀⠀⠐⠒⢊⣁⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠚⠋⣹⠟⠛⠻⢿⡛⠛⠛⠻⢿⣉⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⠞⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠈⠳⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡞⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣙⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣯⡤⢤⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣿⠀⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠈⣧⣼⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⡁⢀⠀⣀⠀⠀⠀⠘⠇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠻⣆
⠀⢀⣀⣠⡶⠶⠛⠋⠁⠀⠀⠰⠶⠀⠂⠀⠀⠠⢀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠁⠀⠀⢠⣤⠁⠒⠀⠹⣦
⢀⡾⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠊⠁⠀⣸⡆⠀⠀⣤⠀⡀⠀⠀⠀⠁⡀⠈⠁⠂⢠⡌⠀⠀⠂⠙⢷⣄⡀
⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠮⡀⠀⢈⠀⡽⣷⠀⠀⠀⢴⠄⡉⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⣤⣀⡀
⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢹⠀⠈⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠒⠂⠈⠉⠉⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠔⠐⠒⠒⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠊⠉⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⠀⠀⠀⢢⡀⠀⠀⠀⡠⠃⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣶⠟⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣄⡀⢠⣤⣤⣤⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⣀⣀⣀⣁⣀⣀⣀⣀⣠⡤⠖
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣤⠀⠀⠈⢢⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣘⠷⣄⡀⠀⠳⡌⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠒⠉⠀⢈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠉⠙⠻⢿⣦⣀⠙⠻⣀⣀⣀⠀⠀⡠⠋⡠⠊⠀⠀⠀⣀⠞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣀⠀⠀⠙⠲⠶⢤⠤⠭⠝⢺⠉⠈⠋⣟⠩⠴⠛⠒⠊⠉⠉⠉⠉⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠉⠀⣀⠈⠀⠐⢢⡤⠤⣤⣤⣤⣼⣿⡀⢸⡇⠉⠒⣌⢛⠛⠛⠛⠛⠃⠐⠀⣸⣀⣩⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠆⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠙⠛⠛⠙⢤⣤⡘⣦⠈⠙⠛⠲⠆⠀⣼⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⢄⣀⡀⠀⠀⠀⠀⢀⣨⡇⠸⡇⠉⠉⠓⠒⠢⣼
⠀⠀⠀⠀⠀⠀⠐⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠐⡞⠉⢠⠇⠀⣟⠳⢦⣤⣤⡴⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠜⣀⡤⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡞⠀⠀⡿⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡞⠀⠀⠀⡇⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢰⠧⣀⠀⡄⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣀⣠⡀⠀⣼⠀⠈⠳⡇⠀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠹⣠⠇⠀⠀⠀⡇⠀⠀⠁⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡇⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⡀⠀⠈⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⢰⠀⠀⠀⡇⠀⠀⠘⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠁⢸⠀⠀⠀⡇⠀⠀⠀⠘⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠜⠀⠀⢸⠀⠀⠀⠇⠀⣀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⠤⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⢤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⡀⠀⡇⣀⠤⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢹⣟⡛⣻⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠖⣶⠃⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣦⣄⣥⣾⣽⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠈⠉⠉⠱⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⢀
⠀⠀⢠⣄⣨⣗⣦⣾⠂⣾⣶⠶⡾⢷⠀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄
⠀⠀⠈⢙⣿⣿⣯⣥⠀⠋⠰⡶⠀⠛⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢠
⠀⣻⣲⡶⠟⠁⢀⠀⠀⡎⠁⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⡻
⠈⠉⠉⠀⡄⠀⠀⠀⣼⡀⢦⡀⠘⢇⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠐⡆⢉⡇⢮⣵⡀⡾⠴⠀⠀⠀⣦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⣴
⠀⠀⠀⠀⠀⠀⣶⣴⡟⠉⠃⢈⡟⠀⠀⠈⠁⠠⢿⣦⠀⢃⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠄⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣤⠴⠋⠀⠈⣹⣶⣟⠁⢠⣄⠀⠀⣠⣤⣿⡆⠀⢇⠀⠀⠀⠄⠀⣞⠀⠀⠀⠀⠀⠀⠀⢲⡇⠀⠘⠣⠶⣗⣀⢸⠿⠂⠀⠀⠘⠒
⠀⠀⠀⠀⠀⢤⠼⠷⣶⠉⣿⢻⣷⣿⡻⢷⠟⠁⠀⢻⣿⠀⢀⣁⠀⠂⢀⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⠷⠀⠀⠈⣵⡒⠒
⠀⠀⠀⠀⠀⣤⠄⠈⠙⣤⣨⣼⣿⣇⠀⠀⠀⠀⠈⠩⢹⣷⢾⣇⢀⣠⠟⠛⠒⠂⠀⠀⣀⠀⠈⠑⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣭⡥⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠛⣿⡝⢫⡔⠀⠀⢠⡀⠀⠛⠀⠂⣿⡿⠁⢶⢃⣀⠀⠈⠉⢻⢠⡇⣰⠖⠛⠉⠁⣡⠄⢺⡇⠀⠀⠀⠀⠻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⣿⣿⣷⣆⠀⠀⠀⠀⠁⠀⠀⠀⢸⠿⣶⢦⡞⠻⠿⣿⠷⣤⣴⣾⣿⠣⠀⠀⠁⣾⣿⠀⢿⡆⠁⢀⣴⠂⠀⠀⠀⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣉⠛⡄⠀⠠⠦⠼⠀⠀⠀⠉⣰⣧⣀⠥⠤⢄⣠⡼⠉⠛⣿⡌⠷⠁⣠⡚⠁⢀⡀⠀⣀⣀⠀⠋⠐⠶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷⡀⠀⠀⢠⡇⠀⠀⠀⠀⣿⡏⡄⠐⠐⣎⠹⣿⣷⣄⡀⢐⣘⣁⠉⠈⠛⠃⠀⢸⢇⡈⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⣠⣤⣦⢻⣶⣶⣄⠐⠙⠳⠄⠢⡄⠙⢢⣿⣾⠁⡀⠀⠘⢏⢢⣶⣦⣠⡀⠘⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠋⠻⣿⢎⣿⣿⡏⠃⣀⠴⣷⡦⢀⢶⠟⣙⠁⠛⠁⠀⢠⢤⣾⣟⣂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠘⠛⣠⠀⠀⠈⠻⡀⠈⠃⢠⠆⠀⠀⡨⢟⣿⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠉⠉⠒⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⡀⣠⠀⠄⠀⢀⣀⣀
⠀⠀⡤⠀⠄⠀⠐⠐⠀⠢⠐⠀⠐⠀⠂⠢⠂⣄⠀⠀⢠⡄⡀⠀⢀⠀⠒⠁⠈⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠈⠒⠀⠤
⠀⠀⡁⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠂⠀⠀⠀⡐⠁⠀⠀⠑⠄⠀⡀⢀⠀⠀⠀⠀⠀⡃⠀⠠⡅⠀⠀⠀⡠⠊⠁⠀⠢⡀⠈⠀⠀⠀⠀⢄
⠀⠀⠕⠀⠀⢐⠀⠰⣶⣄⠀⠰⠊⠀⢀⠉⠳⢀⠀⠀⠨⠀⠈⡀⠀⠀⠨⠀⠀⢰⣤⡀⠃⢀⣤⡶⠂⠀⢐
⠀⠀⢐⠀⠀⠀⠡⠀⠙⠻⣷⣠⣾⠿⠋⠀⢀⢇⠑⠄⡂⣸⣄⠐⢢⣀⡀⠣⢀⠀⠙⢿⣴⠿⠋⠁⠀⡠⠂⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠈⠀⠄⣴⡏⠀⠠⠠⠖⠁⠀⣁⠌⢀⠻⠋⠀⠀⠀⠉⠋⠉⠉⠀⠀⢹⣿⡄⠐⢄⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠒⠀⠀⠀⢠⠀⢸⡿⠃⠀⡆⡀⠈⠉⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠨⢀⠀⠉⠃⢀⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠈⡀⠀⠀⠀⠠⣀⢀⠀⠤⠁⢁⡄⠀⠀⠀⠰⡀⠀⢈⠀⠠⠃⠀⠀⠀⠀⢻⠀⠑⠐⠂⠁⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢀⠼⠀⠀⢀⠀⠀⠁⠲⠿⠟⠀⡠⠂⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠤⠠⠤⠄⢐⡁⠐⠀⠀⠈⠀⣕⢀⢈⣙⢀⢈⣴⡀⠀⠀⠀⣀⣌⠢⢄⣀⡀⠀⠔⠁⠄
⠁⠀⠀⠚⠛⠀⠀⠀⠀⢀⣐⠚⢿⡟⠣⠐⠒⠂⢺⡿⠠⢘⢋⠘⠘⠟⠁⠉⠉⠀⡙⠿⠛⠂⠀⠀⠀⠀⢀⡐⠛⠁
⠀⠀⠀⠀⢡⠆⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠀⠂⠀⣤⣦⢂⠀⠐⠄⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠙⠀⡔
⠀⠀⠀⠔⠀⠀⠀⢀⠔⠀⠀⠢⡀⠘⡆⠀⠀⠀⠀⡜⠀⠈⡉⠊⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠐⠀⠀⠀⠀⠅⠀⠀⣀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⢀⠔⠀⠁⠀⢀⠀⠀⠀⠀⠢
⠀⠀⠁⠀⠀⠀⠀⠱⡀⠀⢻⣷⡀⠊⠠⠄⢌⠑⠲⠤⠀⢀⣦⡀⢀⡤⠶⠤⠀⡀⠑⠁⢀⣠⡤⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠄⠈⠁⣀⣼⠷⣶⣄⠀⠀⠈⡀⠀⠐⣸⣟⠜⠁⠀⠀⠐⠉⠉⢀⣠⡿⠛⠀⢠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⢠⣾⠿⠋⠀⠈⠙⠓⠀⠀⠔⠀⢈⠁⡆⠀⠀⠀⠌⠀⢀⣴⠿⠋⢿⣦⡈⠁⠠
⠀⠀⡀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠈⠂⠄⡤⠴⠂⠀⠀⠈⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⡤⠈⠻⢧⠀⠀⢀
⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠂⡀⠀⠀⢠⠂⠀⠀⠀⢐
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠛⠈⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⡐
⠀⠀⠐⠀⠀⠄⠀⠠⢀⠀⠄⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣐⣤⣤⣤⣤⣤⣄⣀⣀⣛⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⢍⠀⠀⠑⢶⠒⠒⠤⣤⠎⠁⠀⠀⡨⠝⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣴⠞⠓⠤⣑⠢⢀⠀⠑⢤⡤⠁⠀⢀⠤⣊⠠⠐⠓⣄⡀
⠀⠀⠀⣀⡠⠤⠒⠊⠉⠁⠀⢸⠁⠀⠈⠑⠂⠭⣙⣛⣿⣈⣑⣾⣛⣉⠡⠐⠂⠁⠀⠀⡩⠉⠒⠒⠤⢄⣀
⠀⠀⠀⡏⠉⠒⠢⢤⣀⠀⢀⣀⠬⢒⣂⡤⠝⠛⠭⠑⠋⡹⢌⠙⠊⠉⠛⠫⠍⣛⠪⢍⣀⠀⣀⠤⠒⠊⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢙⡻⠖⠊⠉⠀⠀⠀⢀⡠⠔⠊⠀⠀⠉⠒⠤⡀⠀⠀⠀⠉⠲⣎⡏⠀⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠖⠯⢄⣀⠀⠀⠀⠀⢀⣀⠼⠢⡀⠀⠀⠀⠈⠣⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠙⠒⢢⠞⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠈⠉⠒⡖⠊⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⣼⠤⠒⡋
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣠⣋⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⣢⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠜⠁⠈⡏⠒⠤⢄⡀⠀⠀⠀⡇⠀⠀⢀⡠⠔⠚⡇⠀⠐⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠀⠀⠀⡇⠀⠀⠀⠈⠉⠒⠤⡧⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠢⠤⣀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠢⢄⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⣇⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣤⡄
⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣞⢷⣤⣾⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⣿⣄⢸⡏⣀
⠀⠀⠀⠀⢠⡏⣶⣶⣿⣿⣶⣷⢸⡆⠀⠀⠀⠀⠀⡟⣸⣿⣿⣄⠈⡇⠀⠀⠀⠀⠀⠀⣼⠁⢀⣀⣈⣻⣷⣅⠘⣧
⠀⠀⠀⠀⢸⢱⣿⣿⣿⣿⣿⣿⡆⡇⠀⠀⠀⠀⢠⣧⣿⣿⣿⣿⣦⢷⠀⠀⠀⠀⠀⢸⢇⣄⣘⣛⣻⣿⣿⣿⣧⢸⡆⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⡸⣿⣿⣿⣿⣿⣿⢇⡇⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⡏⣇⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⡇⣶⣤⡌⡇
⠀⠀⠀⠀⠀⢧⡹⣿⣿⣿⣿⢟⡼⠁⠀⠀⠀⠸⡏⣿⣿⣿⣿⣿⣿⠇⡿⠀⠀⠀⠀⠀⢷⡻⣿⣿⣿⣿⣿⣿⢏⡾⠀⠀⠀⠀⢸⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣭⣭⡶⠛⠀⠀⠀⠀⠀⠀⠙⢮⣛⡻⠿⣟⣫⠞⠁⠀⠀⠀⠀⠀⠈⠻⢮⣝⣛⣛⣫⡵⠏⠀⠀⠀⠀⠀⠸⡼⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣻⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡿⣣⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣶⠄⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣧⡇
⠀⠀⣀⠀⢀⣠⣀⡀⠀⢴⣦⡌⣧⡼⢋⡉⠉⠉⠁⠀⠀⢠⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣬⣜⣿⣥⣾⢃⡀⠀⠀⠀⠀⡆⣿⣿⣿⣿⢠
⠀⠀⠈⠛⢎⢻⣿⣿⣷⣶⣤⣿⢇⡴⠋⠁⠀⠀⠀⠀⠀⣼⢀⣤⣄⠀⠘⡇⠀⠀⠀⠀⡏⣴⣷⣶⣤⣥⣿⣿⣿⣿⣧⢱⠀⠀⠀⠀⡅⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠳⡙⣿⣿⣿⣿⣯⠎⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⢧⡻⢿⣿⣿⣿⣿⣿⣿⡿⢏⡼⠀⠀⠀⠀⣇⣿⣿⣿⣿⡾
⠀⠀⠀⠀⠀⠀⠙⣎⠿⠟⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠙⠲⣍⡻⢿⣿⢟⣩⠖⠋⠀⠀⠀⠀⠀⢸⡸⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⣿⣿⣿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⣮⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣛⣛⣋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠓⠒

⠂⠀⣀⡀⠀⠐⠀⢀⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀⠘⠂⠀⣀⡀⠀⠐⠀⠁⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀
⠀⢸⠉⠀⠘⠀⠀⡴⠪⣥⠆⠀⠀⠉⡁⠀⠀⢰⠦⣴⡟⢡⠂⠀⢸⠋⠀⠘⠀⠀⡴⠪⣅⠆⠄⠀⠈⡁⠀⠀⢰⠦⣴⡞⢡
⠀⠀⠀⣀⠀⠀⡏⠁⠃⠀⠀⠀⠀⠀⠠⠺⠀⡹⢟⣿⢹⠆⠀⠀⠘⠀⢀⠀⠀⡎⠀⠂⠐⠀⠀⠀⠀⠠⠺⠂⡹⢟⣿⢹⠆
⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⠤⠀⠀⢠⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⢤⠀⡀
⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿⣿⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿
⠙⠀⠀⠀⠀⠀⡠⠀⣀⡀⠀⠉⠀⠀⡀⠄⠀⠘⠃⠀⣀⡘⠋⠙⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠉⠀⠀⠀⠄⠀⠘⠃⠀⣀⡘
⠀⠀⠀⠂⢡⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠐⠦⠀⠈⠛⠃⠀⠀⠀⠂⠢⢤⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠰⠤⠀⠀⠛⠃
⠈⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡛⠑⣀⠀⠀⠀⠀⠈⠀⠐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡟⠃⣀
⣢⣀⡀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣈⣀⠱⣶⣇⡀⠉⠀⠀⠀⠰⣢⣀⡀⠀⠀⠀⠀⣍⠀⠀⠀⠀⣘⣀⠱⣴⣆⡀⠁
⢀⠁⠀⢸⣧⠀⠀⢻⣷⡶⠾⠛⠁⠀⠀⠈⠉⠀⣠⣶⣶⣄⠀⢀⠁⠀⢸⣧⠀⠀⢻⣶⡶⠾⠛⠁⠀⠀⠉⠉⠁⣠⣶⣶⣤
⠘⣄⣦⣼⠟⠀⠀⠘⠉⠀⣤⠀⣀⠤⠴⣿⠀⢠⠘⠋⠑⡿⠀⠘⣄⣦⡼⠟⠀⠀⠘⠁⠀⣤⠀⣀⠤⠴⡟⠀⢠⠙⠋⠑⡿
⡀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⢸⣅⠀⢀⣬⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⣄⠀⣀⣼
⠀⢠⡖⠒⢠⠀⠀⢐⡲⠀⡀⠀⠛⠿⠯⠉⠀⢀⡀⢀⣡⠆⡀⠀⢠⡶⠂⢠⠀⠀⢀⡲⠀⡀⠀⠛⠿⠏⠉⠀⢀⡀⢀⣡⠆
⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠁⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘⠀⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠀⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘
⠀⢀⡀⠰⡤⠀⠣⠀⠀⠀⠐⢀⠀⠀⠉⣨⠄⠊⣑⠛⠘⠁⠀⠀⢀⡀⠰⡤⠀⠡⠀⠀⠀⠐⢀⠠⠀⠈⣨⠄⠊⣑⠛⠘⠁
⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠋⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦⣸⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠊⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦
⢯⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠦⠀⠐⠈⠀⠀⢱⡄⠀⠐⢻⡿⢯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠦⠀⠑⠀⠀⠀⢱⡤⠀⠐⢻
⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆⠀⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆
⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢍⠀⠀⠀⠀⠀⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢁
⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡆⠀⠦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡃⠀⠶
⠚⠖⠂⢠⡄⠀⠀⣷⣀⣀⣤⣤⠖⠒⠈⠿⠷⠂⠀⣀⣀⠀⠈⠚⠒⠂⢠⡄⠀⠀⣷⣄⣀⣤⣤⠖⠒⠈⠿⠿⠂⠀⣀⣀
⢰⠀⡀⢸⡿⠀⠀⢸⠿⠋⠉⠀⠀⠀⢀⣤⠀⠀⢺⡿⢻⣿⠀⢰⠀⡀⢸⡿⠀⠀⢸⠟⠋⠁⠀⠀⠀⢀⣤⠀⠀⢺⡿⠻⣿
⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠁⠉⠫⠀⠘⠀⠀⠀⠋⠀⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠉⠉⠻⠀⠘⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⡀⡀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠷⣦⡀⠀⢀⣼⣿⣿⠗⣀⣀⣀⣀⠀⠀⠀⠰⠷⡞⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡅⠀⠀⠀⠈⢻⡄⣾⢟⣩⠖⠛⠉⠉⠉⠉⣹⡷⠂⠀⠐⡿⢀⣠⠞⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣿⡇⣦⠀⠀⠀⠀⡧⢫⠞⠁⠀⣀⣤⠄⠀⣼⠏⣤⠀⠀⢀⡗⣉⠀⠀⣼⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣇⣿⣧⢰⠀⠀⠃⠀⢀⢦⣾⣫⣤⡄⣴⠋⢠⣿⡆⢀⢈⡾⠃⠀⡰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠻⣿⡆⣇⠀⠀⣄⣾⣾⡿⠟⠁⠊⠀⠀⢸⣿⣿⣼⣨⣾⠁⠈⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠂⠀⠈⢃⡿⠴⠚⣿⡏⠉⠀⠀⠀⠈⠙⠓⢮⣻⠇⣻⣿⣷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⡴⠦⣬⣶⣏⣀⣤⣾⡿⢁⡶⠒⠈⠡⢤⣀⣀⢰⣽⣾⡿⠀⠀⣴⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣤⣀⠤⣬⣽⡿⢿⠟⢃⣤⣭⣍⠉⠉⢠⣛⡳⠦⣤⣤⣶⣾⣿⣿⣷⠟⣠⣤⣤⠸⣷⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⣻⣿⣱⢟⣵⢸⠁⠀⠀⢸⣿⣿⡷⠀⠀⢈⣩⣤⢶⣿⡽⣿⡻⣿⣧⡿⠀⠀⢀⣿⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⡿⣿⡈⠀⠀⣸⠘⠿⣟⣥⠀⠀⢉⠛⠧⣽⣦⡀⠀⠈⠊⣿⡇⠀⡄⣾⢏⣴⡿⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡏⢏⣿⣬⣁⣀⣠⠏⠀⣀⣭⠷⠾⢿⡜⣷⡀⠀⠙⠿⣧⠱⡄⡏⢠⣾⠱⠏⢚⣵⡾⠿⠟⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣉⣉⠁⠈⠳⢿⣿⣵⢋⡴⠞⢩⣶⠾⠿⢸⡇⣿⣷⠀⠀⢀⣈⠃⠚⣵⠛⣇⣀⣚⡛⠛⠛⠉⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣍⠉⠉⠉⠙⠓⢦⡀⠈⢀⡞⣴⣧⡞⠁⠀⠀⢸⡇⣿⣿⠀⣴⣿⣿⠇⡀⠻⣤⣿⡟⢷⣤⣴⣆⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣇⣿⡀⣤⡠⣀⠀⢀⠁⠀⠀⣸⣿⢋⠔⠀⠀⠀⣼⢇⣿⢃⣠⣿⠟⢡⣾⡇⣷⢬⡙⠛⠶⡀⠉⠉⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡻⡸⣷⣿⣷⣌⢳⡌⣆⠀⠀⠿⣵⡿⠀⠀⢀⣼⣫⡾⣣⣿⡿⠉⡏⡿⠛⠈⠁⢀⡇⢸⣷⡍⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣙⢮⣟⣻⣿⣳⡿⢋⡴⠖⠶⣦⡙⣦⠖⠋⠚⢩⣌⣛⠋⠀⠀⣧⠀⠀⠀⣠⡞⣷⢈⢷⣍⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣶⢯⡀⢌⣉⣛⡛⠃⣿⡀⠀⠀⢸⣧⢐⠚⠛⠻⠿⢓⣮⣝⢶⣄⠹⠷⠾⠟⠉⢸⣿⣯⡳⣮⣳⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣿⣿⢟⡿⣫⣶⣿⣿⡟⣡⣆⠹⣿⣶⣶⣿⠇⣼⣿⠶⠦⠬⠭⠍⠙⠓⠹⣆⠀⠀⠀⠀⠀⠙⢿⣿⡎⠙⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢫⡿⣁⣠⣤⠞⠩⠞⠁⣸⡷⢠⣭⣭⢡⡈⠻⢦⣄⡀⠀⠀⠀⠀⣀⡴⣛⣤⣶⡟⠀⠀⠀⢀⠙⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡿⠋⠁⠀⠀⠀⣰⡿⢁⡿⠉⣿⢸⣿⣆⢀⢯⣝⣛⣛⣿⠍⠀⠾⠿⣟⣻⣃⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠻⠇⠀⠀⢀⣠⡾⣟⣵⢺⡇⠀⠸⠸⡿⣿⡌⡇⠿⠿⠛⠁⠀⠂⠉⠉⠉⢛⣿⡟⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⢸⣿⣿⡸⣇⠀⠀⠀⠃⢹⢧⡇⠐⢶⣤⣤⣤⣤⣴⣶⣾⠟⠋⠀⣠⡟⣴⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠙⢆⠀⠀⠀⠞⣾⠁⠀⠀⠛⠿⣦⣉⠉⠉⠀⠀⣀⡴⠋⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠳⣦⣠⣾⠃⠀⠀⠀⠀⠁⠒⠤⠤⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠉⠉⠉
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠛⠛⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠘⣿⣿⡀⠀⠈⠉⠳⣦⡀⠀⠀⠀⠀⠀⢀⣼⠛⣿⣿⠃⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣄⣀⡀⣹⣿⠷⢤⣀⡀⢀⣼⣿⣦⠀⠀⠀⢠⡟⠁⢀⣿⠿⣄⣀⣤⣶⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠉⠉⢛⣻⣧⣀⠀⠀⠙⠻⣿⡿⠉⢷⡀⢠⣿⣷⣴⠋⠀⠀⢸⠿⠛⠋⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢿⣿⣿⣿⡿⠛⠲⢤⣄⣈⢧⡀⠈⢷⡞⢹⡟⣁⡠⢤⣶⣇⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⢤⣄⣉⠙⢻⡦⢸⡇⢿⣿⣅⣠⡴⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠿⣿⣿⠉⠋⠉⠉⠉⢻⠋⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⡇⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠦⣤⣤⣤⣤⡼⠷⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠹⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣆⡞⣱⣜⢀⢠⠄⠠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣥⠓⡁⠀⠀⡄⠈⠁⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠃⠀⣸⣷⠀⠘⣿⣃⡄⠀⠈⡇⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢆⠐⡜⠿⠿⡣⡔⠈⠛⠓⠀⢠⡤⠠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡉⠆⠀⡇⢱⠾⣿⡞⢱⣶⣴⡦⠤⢀⠀⡁⠀⣨⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠈⠀⠑⢊⠔⠛⠥⢜⠛⠁⠀⠀⠘⢛⡰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⢀⣤⣸⠃⠀⠤⢀⠀⠈⠙⠛⠓⠤⠀⠈⠉⡑⠨⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⡀⡶⠒⠀⠀⠀⠀⠄⠊⠀⠆⠈⠉⣂⡄⠀⠀⠔⢫⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠘⠀⠇⠀⠀⠀⠀⠌⠀⠀⣤⠀⢶⠀⣾⠆⣿⣮⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡄⠀⠀⠀⠀⡘⢠⡌⢃⡀⣠⣄⣰⣶⡄⣬⡅⠻⠟⣠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠌⠃⡀⠀⠀⢰⠃⠀⠀⠈⠁⠾⢡⣾⡍⡤⣟⠃⠾⢆⡉⣭⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡀⠀⣦⣀⡀⡜⢠⣶⡴⠿⠶⣿⣎⡉⢾⣽⠎⣷⠹⣮⡡⢖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠇⣤⣾⠤⠀⣹⠃⡉⢀⡄⠆⠀⠺⢟⡀⣴⣷⡎⡇⠘⠛⠪⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠌⢩⡳⠀⠟⣉⠄⠖⡐⣂⠀⠀⣙⠋⣠⣭⡀⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠈⢁⣜⢛⡀⠛⢶⡆⠛⢣⣬⡛⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢻⣝⡋⣴⣦⢰⣷⢌⣻⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠚⠁⠀⠑⠋⠁

⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠶⠖⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⢀⡤⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣽⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⠀⡀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠠⠏⣼⠏⠀⠀⣼⣁⡄⠀⠃⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣿⡇⠀⠀⠀⣽⠆⠀⠐⡁⠄⠈⠀⠀⠈⣼⡇⠀⠀⠀⠁⠀⠀⠀⡉⣰⠼⠚⠀⠀⢀⣈⣿⣄⣀
⠀⠀⢀⣾⡗⠀⢀⡴⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢲⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠨⠉⠁⠉⢹⡿
⠀⠀⠀⠉⠀⠀⢙⢠⣛⣇⠀⠀⠀⠀⠀⠀⢰⡄⠀⠸⠀⠀⠀⢼⣿⡀⢈⠀⣸⣿⡇⠀⠀⠀⠠⠀⠀⢠⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⣟⣿⣿⠀⠐⣦⢀⢰⣶⣶⡆⠀⠀⠄⠀⠀⠈⣿⡇⠈⠀⠉⢹⡇⠀⠴⠀⢰⠀⣠⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣴⠄⠈⣏⡉⢛⣿⠁⠀⠀⠘⠸⢻⠿⠃⠰⠈⠁⠀⠀⠀⢹⡏⠀⠀⠀⢸⡇⠀⠶⣤⠞⠀⣿⢺⣿⣿⣷⠇⠀⠀⠀⠂⠀⠀⢄
⠀⠰⢦⣖⠂⠀⢀⠀⢸⣿⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢀⣄⣼⡇⢀⢀⣀⣸⡇⠀⠀⠁⠀⠀⢸⡄⣿⣿⣿⠀⣀⣠⣦⣄⣀⣀⣈⣳⣄⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠀⠀⢀⡀⣨⣿⠇⣠⠀⠀⠀⢀⠀⠀⠠⣀⠀⢨⠸⣿⣿⣧⠸⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⠇⢿⣿⣿⠀⠻⣯⣽⣭⣽⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠛⠋⠉⠁⠀⡸⠉⠙⢮⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢣⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⣆⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠢⣄⠈⠳⣄⠈⢣⡀⠀⠈⣿⠀⠀⢀⢀⡠⠔⠋⠁⠀⠀⠀⠈⠉⢉⣹⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣢⡤⠤⠤⣀⣀⡈⠑⠦⣌⡑⠀⠙⢦⡀⢸⡷⡆⠉⡙⢄⣀⣀⣀⣤⠤⠤⠒⠊⠉⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠯⣭⣉⣀⣀⣀⠀⠀⠀⠉⠉⠓⠒⠩⣵⣶⡤⡟⠒⢇⢸⡄⡈⠌⢧⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠓⠛⠛⠩⠭⠥⠤⢄⣀⠤⠭⠙⠛⠛⣛⣋⠭⠟⠒⠺⠧⠖⠻⢄⡇⢠⠀⢨⡭⣙⣒⠒⠒⠒⠶⣾⠭⠭⠭⠭⠚⠛⠒⢲
⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⡎⠉⠉⠉⠙⠒⠒⠒⠒⡶⠒⢺⠁⠘⠀⢸⠉⠉⢹⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡆⠀⠸⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠁⠀⠀⡇⠀⢸⡇⠀⠀⠨⡇⠀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠉⠉⠉⠉⠁⠒⠚⡇⠀⠀⠀⡧⠤⠤⢄⣀⣀⣀⣀⣀⣇⠀⣿⠀⠀⠀⠀⡧⠤⢼⡇⠀⠀⠀⡗⠒⠈⠉⠉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⣿⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠛⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⢡⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢀⡀⠀⢸⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠊⠁⠈⠑⠚⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠻⠤⣄⣀⠼⠤⠄⣤⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣄⠠⠤⠴⠧⠀⠠⠤⠛⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣶⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿⣗⠀⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠁⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⣶⣾⣶⠤⠦⠤⣤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⣿⣿⠿⠟⣉⣉⣉⠉⠛⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣿⣿⣿⣿⣿⡋⠀⠀⠀⢦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⢿⣟⣛⣫⣭⣭⣭⣥⠤⠤⣄⣘⣃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡀⠒⠂⠈⢽⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣐⣿⣿⣌⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣟⣽⣶⣿⡿⠋⠉⠈⠻⣿⣿⣿⣿⣦⡄⠄⠈⢹⣿⣿⣿⣟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⣿⣿⣿⣿⣧⡀⠀⠀⢀⣿⣿⣿⣿⣿⠶⠀⢰⣿⣿⠿⣟⡥⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠗⣴⣶⣿⣿⣿⣿⣿⣿⣛⣭⣿⠿⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠏⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢤⣾⠿⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢉⣤⣀⣠⡾⠟⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣁⣠⣤⠔⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠲⠾⣿⣿⠟⢫⡉⣁⢀⣠⡞
⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⣿⣿⠿⠋⢙⠋⠉⢁
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣸⣿⣷⠾⠿⠛⠟⠉⢁
⠀⠀⠀⡤⠚⠟⠀⠀⢀⣤⣾⣿⣿⠿⣿⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣽⣥⣿⡿⠛⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠋⠀⠀⠀⣷⠆
⠀⠀⠀⠀⠀⣠⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⣯⠁
⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⣤⠆⠰⣬⣿⣧⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣅⣀⣀⣨⣿⣿⢻⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡿⠋⣹⠝⢻⣿⣿⡟⢫⣏⠛⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠁
⠀⠀⣦⣤⡀⠀⠀⠀⠀⠀⠸⠌⠳⣴⣦⠀⠀⠀⠀⠀⠀⢸⣷⣸⡣⢼⣿⣿⠼⡳⢟⠤⣿⣿⡧⢜⣇⣾⡇⠀⡀⠀⠀⠀⠀⣴⣦⠞⠡⠇⠀⠀⠀⠀⠀⢀⣤⣴
⠀⠀⠘⣝⢿⣦⠹⢷⣦⠀⣀⠀⠲⠚⢿⣠⡄⠀⠀⢸⡇⠀⣿⢫⡷⣄⡹⢇⡤⣦⣴⣤⡸⢏⣀⢿⡝⣿⠀⢺⡇⠀⠀⢠⣄⡿⠓⠖⠀⡀⠀⣴⡾⠏⣴⡿⣩⠃
⠀⣀⣠⣬⣵⡻⣆⣣⡙⣧⠸⢿⣦⠀⢖⢿⡀⢀⠀⠘⢟⣰⣿⣿⠿⡏⠛⡋⢿⣦⢧⡿⢈⠏⢹⠿⣿⣿⣆⡻⠃⠀⡀⢀⡿⡢⠀⣴⠿⠇⣼⢋⣜⣠⢟⣾⣥⣄⣀
⠚⠯⠭⠔⠉⠄⣽⣿⠛⣩⣦⣧⣽⡇⠀⢠⡙⣎⡇⠀⢠⠟⢩⣶⣶⠁⣄⣤⡙⠃⣼⢋⣤⣠⠈⣶⣶⡍⠻⡄⠀⢸⣱⢏⡄⠀⢸⣯⣾⣴⣝⠛⣿⣍⠠⠉⠠⠭⠽
⠀⠀⠀⠀⢸⠾⠋⠁⣀⣴⡾⠋⠄⣹⣶⠤⢉⡙⡇⠀⠏⠀⠰⠟⣥⣷⡙⠃⠀⠶⠶⠀⠘⢋⣶⣬⠻⠆⠀⠹⠀⢸⢋⡉⠠⣶⣏⠠⠹⢷⣦⣀⠈⠙⠷⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠛⠉⠀⠀⠀⠉⠛⠼⠀⠀⠀⠀⠟⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠻⠀⠀⠀⠀⠷⠟⠉⠀⠀⠈⠉⠛⠃⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠰⢰⢁⡇⡠⠾⣡⠰⠀⡆⠀⠀⠀⠀⡇⢠⠄⠐⠁⡀⢀⠃⠂⠀⠇⡔⡶⢰⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠸⠋⠁⠇⠁⠀⢀⠃⠀⠀⠀⠀⠁⠈⠀⠀⠁⠁⠈⠀⠀⠀⠁⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡀⠀⠰⢤⣄⣀⠘⠲⢦⣄⡀⢠⠄⡾⠹⡀⠀⠀⠀⣿⣠⠄⠀⠀⡀⢀⠀⠀⢠⣄⣿⠀⠀⠀⢀⠋⢓⠀⠀⠀⣠⡴⠞⠃⣀⣤⡤⠆⠀⢀⡀
⠀⠀⠀⠀⠸⠻⣶⣄⠀⢉⣻⣧⣬⡾⠋⢡⡤⣾⡇⠀⢧⠀⣙⣷⣍⠛⣼⣧⡈⣙⠉⢁⣼⣧⠙⣡⣾⣃⠀⡼⠀⢸⣷⢤⡌⠙⢶⣥⣼⣟⣉⠄⣡⣶⠟⠇
⠚⢯⣭⣭⣦⣁⣼⢿⠷⣽⠋⢏⣿⠇⣀⢠⡜⢣⠇⠀⣨⢷⣦⣝⣋⡄⠁⡀⢼⡅⣸⣧⢁⠈⢠⢛⣫⣴⣾⣅⠀⠘⡜⢣⡄⣀⠸⣿⡹⠙⣭⠾⡿⣧⣈⣴⣭⣭⡽
⠀⠀⠀⡠⢚⣿⠁⢋⣼⠇⠼⠿⠋⠀⢹⣿⡀⠀⠀⢰⡷⠘⣿⢿⡿⡷⠖⢦⣿⣹⣎⣿⡴⠳⢾⣿⡿⣿⠃⢾⡆⠀⠀⢀⣿⡏⠀⠙⠿⠧⠸⣧⡙⠈⣿⡒⢄
⠀⠀⣸⣵⡿⠃⠾⠛⠁⠀⠀⠀⠛⣹⢿⠈⠁⠀⠀⠸⠃⢀⣿⣷⣟⢀⣾⣧⣈⢃⡙⣁⣼⣷⡀⢻⡾⣻⡄⠘⠇⠀⠀⠈⠁⡾⣏⠛⠀⠀⠀⠈⠛⠗⠘⢿⣮⣇
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠸⣤⡟⠘⠃⠀⠀⠀⠀⠀⠀⠸⠟⢸⣆⠹⢿⡟⢩⣾⣳⡍⢻⣿⠏⣰⡇⠻⠃⠀⠀⠀⠀⠀⠀⠘⠃⢻⣤⠇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡷⣾⣷⣿⣿⠿⣿⣾⣷⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⠀⣴⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠸⣿⡍⠁⠈⢩⣿⠏⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⣀⣀⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠋⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡋⠉⠉⠙⢢⡄⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢺⠉⢦⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡄⠀⠙⣄⠀⠀⢳⠔⠒⠒⠛⠛⠛⠋⠉⠑⠢⣤⠏⠀⠀⠀⣀⣠⡬⣦⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢷⠚⠉⠉⢦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⣴⠚⠉⠀⠀⠘⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠙⣦⡈⢧⡀⠀⠈⢷⡀⠸⡀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠈⢓⣦⡀⠀⠀⢹⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀⠈⢷⠀⢷⣄⠀⠀⠀⣰⠋⣠⠶⢋⣠⠴⠞⠉⠉⠙⢷⣄⠀⣧⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣝⢦⡘⣇⣨⡿⠷⢤⣜⣡⣾⠿⠚⠉⠀⠀⣀⣠⣤⠶⠚⠛⠣⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⣎⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠏⠀⠀⣾⡋⠉⢀⣀⣤⡶⢞⣛⣉⣡⣤⣤⣤⡴⠶⢋⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠩⣯⡀⠀⠀⣏⠉⠙⠛⠛⠛⠋⠉⢉⣉⣀⣠⡤⠤⠶⠚⠋⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠑⢦⡀⢀⣠⣤⠴⠖⠛⠉⠀⠀⠀⠀⣈⣽⠿⣟⠉⠉⠉⠉⠉⠻⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⢙⡏⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠠⣀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢀⣈⣿⣦⠤⠒⠉⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠑⢤⡀⢸⡇⠀⠀⢀⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⣦⡴⠞⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⣷⣄⠀⢸⡇⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠈⢷⣼⠇⠀⠀⠀⠀⠈⠢⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣠⣿⠴⠖⠚⠋⠉⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠷⣄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠞⠛⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠀⠀⢀⡿⠀⠀⠑⠢⢄⡀⠀⣿⠀⠀⠀⣀⡠⠴⠚⠉⢹⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠙⢦⣷⠖⠚⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣼⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡀⠀⣀⣀⣤⡤⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀⣠⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⡿⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡇⢀⡴⠋⠁
⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⢉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡆
⠀⠀⠀⡀⢘⠀⠁⠁⠈⠈⠁⠁⠈⠈⠀⠁⠉⠈⠀⠁⠁⠀⠈⠀⠁⠈⠈⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠁⢡⠀⡇
⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇
⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡇
⠀⠀⠀⡇⠐⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢐⠀⡇
⠀⠀⠀⡇⢨⠀⠀⠀⡚⠀⡆⡔⡖⡰⢠⠴⣠⠔⢰⠀⠀⠀⢠⠇⢠⠀⠂⠈⢀⢠⠀⠘⢠⢰⢠⠠⡀⠀⠠⠀⡇
⠀⠀⠀⡇⢐⠀⠀⠀⠃⠀⠁⠀⠀⡏⠁⠏⠁⠀⠘⠀⠀⠀⠈⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⢘⠀⡇
⠀⠀⠀⡇⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇
⢀⣾⡄⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡤⢰⡄⢤⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇⣼⣆
⢾⣿⣥⡷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⣧⣽⣿
⡄⠿⢛⣷⣞⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣶⣟⠻⠇
⠃⠴⡿⣿⣶⡀⠘⠻⣤⣶⠀⠀⠀⠀⢀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⠀⠀⠀⠀⢰⣦⡼⠛⠂⣰⣾⡿⡷⠄
⠂⠀⠀⢉⣉⣴⠷⢆⠀⠀⣀⡀⣴⡀⣿⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⡇⣰⡄⣀⡈⠀⢀⠶⢷⣌⣉⠁
⣖⡍⠉⢠⣼⡇⢀⢴⣷⡈⠻⠻⣿⡇⠹⡃⣩⣤⣤⡀⠀⠀⠀⠀⣠⣤⣬⡁⡻⠁⣿⡿⠻⠋⣰⣧⢄⠀⣿⣤⠈⠉⣗
⣤⢱⡴⢛⠳⡾⣍⣸⡇⠱⡄⠀⠈⢀⢺⣽⣿⣭⡀⠀⠀⠀⠰⣦⣠⣿⣿⣿⣽⣂⠈⠀⠀⡰⠁⣿⣀⠽⡶⢛⠷⣴⢩
⣯⠀⠙⣷⢿⣯⣂⢽⣇⣠⡜⣄⢼⣭⠿⠁⠈⡋⢈⣂⠀⠀⣶⡙⣿⠿⠋⠉⠉⢤⣽⢄⡜⣢⣄⣿⢥⣢⣿⢷⡟⠁⢨
⠻⣶⢚⣇⣨⡝⠉⠉⢱⡦⠛⣿⡻⣿⣶⣶⠈⣟⢿⣴⠹⣢⠘⡧⠾⠛⠡⣴⣶⡟⠟⢿⠟⠣⣶⠉⠉⠙⣥⣀⣟⢲⡾
⢀⡼⣿⠙⣧⠾⢶⣿⣿⣿⣦⣼⣿⡟⣿⡟⠃⢰⣀⣹⠷⠋⠙⣏⢯⡫⣖⣻⣾⣛⠿⣾⠀⢸⣿⣿⣷⠾⢦⡟⢹⡿⣄
⢻⡞⢓⣮⡗⠀⠺⣿⣿⡟⠿⣛⣿⣟⣳⡤⢦⣄⣾⠇⢻⣶⣿⣧⠀⢙⣿⣯⠋⠀⠹⡟⠲⠞⣿⣿⡿⠂⠘⣯⣖⠛⣿
⠻⠿⣟⣻⣿⣆⣻⠋⠉⢀⣰⠛⢩⡿⠋⠀⣹⣋⣊⣧⡀⣹⠻⡿⠚⢧⡟⠁⠀⣄⠀⠙⢲⣀⠈⠋⢻⣃⣾⣿⣛⣿⠿
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠿⠶⠶⠶⣶⣤⣄⡀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢀⣴⠟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠐⣿⠖⠒⠺⢿⣓⠒⠒⠶⠶⠶⠤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣀⡀⠀⢹⣧
⠀⠀⠀⠀⠙⠿⣦⣄⡀⠈⠉⠑⠲⠤⢄⣀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⡀⠀⠉⠙⠒⠲⠤⠤⣄⣀⡀⠀⠀⣾⠀⣿⠙⠓⠶⠦⣤⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣤⣤⣄⣀⡀⠀⠉⣽⠟⠋⠉⠛⠲⣤⣤⣤⣀⣈⣉⣛⣲⣤⣀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣿⠁⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠁⠀⠈⢙⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⡀⣀

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠈⠹⣆⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠃⠀⠀⢦⠈⢿⣤⣤⣇⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⠇⠀⠀⠘⣺⣷⣾⠁⠀⢨⡗⠋⣉⡁⠀⠀⠸⣆⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢈⡇⠀⠀⠀⠘⣁⣴⣇⡀⠀⡾⢿⡿⠭⠀⠀⢠⡾⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⠶⠶⠒⠛⣩⠏⡏⠉⢻⣷⡀⠃⠀⠀⢀⣼⠋⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⢀⡇⠀⠸⡏⢳⡄⠀⠀⢨⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⣻⣗⣀⡴⠋⢻⡀⢀⣹⡄⠙⠳⠶⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⠳⣾⡏⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⢠⣶⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⡿⢿⣿⣧
⠀⠀⢠⣿⣿⡏⠉⠙⠛⠻⠿⢿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⡿⠿⠿⠛⠉⠉⠀⠀⣿⣿⡄
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⡿⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠈⣿
⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿
⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿
⣿⣿⡇⠀⣴⣌⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣁⣾⠆⢀⣿
⣿⣿⡇⠀⢹⣿⣿⣶⣤⣌⡉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣉⣤⣶⣾⣿⣿⠀⢨⣿
⢻⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠘⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿
⠀⣿⣿⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿
⠀⢹⣿⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⡏
⠀⠈⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⡿⠁
⠀⠀⠸⣿⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⠏
⠀⠀⠀⠀⠹⣿⣧⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠘⣿⣷⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⡀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢀⣤⣾⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⣤⡀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣀⣤⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⣿⠇⠀⠈⣿⣿⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⣴⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠙⢳⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⣰⡏⠀⢸⣿⣸⣆⠘⣿⡇⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡸⠃
⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠹⣿⣤⣈⣿⣿⣟⡸⠟⠋⠉⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣷⣾⡁
⠀⠀⠀⠀⠀⠈⠻⠿⣷⡶⠶⠶⣻⣿⠙⠛⣋⡶⠀⡀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⣰⡾⢻⡟⣥⠀⠀⠹⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣶⣶⠀⢰⡅
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⢀⣾⣿⠃⣾⡟⢿⣷⡀⠀⣠⣵⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠈⠛⠤⡀⢿⣿⣇⣸⣿⠃⢸⣟⣵⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠈⣧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠿⣿⣿⠀⢘⣾⣿⠋⠙⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⡛⠛⣋⣉⣉⡉⠙⠛⣋⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠛⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠀⠀⣼⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣟⡀⣠⣾⣧⢀⣿⡇⢀⣶⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡙⠻⠷⣯⣿⣿⠿⢷⣿⣿⣷⣶⣶⠶
⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⣤⣾⡿⢿⡾⢿⣿⣿⡿⠂⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡅
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠻⠟⠋⣴⡿⢿⡄⠈⠙⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣷⣶⣦⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣧⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢙⣻⣿⣿⣿⣿⣿⣋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣾⣿⡂⠀⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠦⣤⣤⣽⣯⣭⠯⠼⠿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⢿⣿⣷⣶⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⢿⣤⣤⣤⣤⣶⣶⠾⠿⢿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠸⠛⠉⠓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣀⢹⠀⢰⣿⠇⠀⣠⡾⠁⣠⠤⢾⡇⢿⣷⣤⣀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢧⢸⡟⠀⢠⣿⣇⡞⠀⠀⠀⠀⠳⣯⣝⣛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣇⠀⠘⣿⣿⡇⠀⠀⢀⣀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠦⠤⠄⠈⠳⣄⠘⢿⣇⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣵⣶⠾⠿⠟⠛⠛⠓⠈⠻⢦⣙⢦⡹⣏
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠙⢷⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠛⠛⠛⠛⠶⣤⡙⢿⡄⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⣟⠻⣿⠀⠀⠀⠀⠀⠀⠻⣎⢧⠀⢠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡟⠛⠛⢁⣿⠀⢺⡄⠀⠲⢿⣷⣦⡀⢹⡌⠀⣾⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠸⢿⡆⠈⣷⠀⠀⠀⠀⠉⢻⡀⡇⢸⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⣀⡶⢹⣴⡏⠀⠀⠀⠀⠀⠀⠃⡇⣸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠟⠃⡼⠋⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⢋⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡼⠛⣩⣤⣴⡶⠾⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣴⠆⠀⣴⣠⡶⠀⠀⣠⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠖⠢⡜⠳⣟⣻⣆⡘⠫⢉⣴⣶⠿⢉⡥⢠⠖⣴⠞⠀⡀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣬⣾⣧⡼⣿⣿⣦⡜⣿⢤⠀⠺⣶⡽⠛⠠⡞⢿⡉⢿⡉⠳⠆⠹⣦⢠⡀⠀⠊⠃⡀
⠀⠀⢸⡏⠈⠉⠉⠉⠈⠈⣹⣿⣄⣩⢏⣀⣟⣷⣤⣦⣜⠃⣀⠀⠀⠀⠀⠈⠀⠀⠴⠶⠶⠻⣄⣹⠰⣦⠉⣣⠄
⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣍⣯⣹⢿⣾⣿⡗⣣⣄⡙⠀⣯⠀⣠⣞⢁⡖⠋⣠⠞⠀⡴⣶⠋⢙⣶⢿⡃⢙⡠⡄
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣟⡀⣬⣧⣀⡷⡟⢉⠍⢰⣆⣘⢶⣂⠈⣿⡙⢟⠛⢢⡝⠋⠁⠉⠉⠠⠶⠛⠂⢹⡄
⣿⠀⢸⡇⠀⠀⠀⠀⠀⢀⣲⣏⠹⠿⡿⠋⣱⣯⣁⡀⠺⢛⣭⡶⠹⢦⡬⠉⠈⠛⠃⠙⠲⠆⠀⠀⠀⠠⣤⣄⡈⣿⢠⡀⠀⠀⣧⠄
⣿⠀⢸⡇⠀⠀⠀⠀⠰⠿⠯⠿⠓⠊⠱⣞⢯⣻⠃⠰⣮⣦⡙⠛⠓⠘⠁⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢉⣧⡈⡇⡀⢀⣣⡄
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠙⠈⠛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡉⢹⣄⣿⠀⣽⠄
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⢉⣉⣇⢀⣣⡄⠀⠀⠀⠀⣠
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣧⡶⣀⠈⢠⡏⣠⠀⠀⡐⣦⣿⢀⣀
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣗⡌⢱⢿⡦⢠⣄⣀⠠⣾⢷⡌⢅⣿⣥
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣰⡏⢠⡟⢠⡶⣤⢀⣤⢰⣄⡤⢰⣄⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢉⣠⠀⢠⠐⣿⣀⢾⡗⠤⡀⢰⣈⠉
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢈⡟⠉⡿⢡⡟⢰⣇⣾⣡⡟⡾⣡⠟⣼⢁⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡝⣿⠶⠒⣼⣟⣾⣊⣿⡔⠳⢾⣻⡿⡟
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠉⠀⠀⠀⢸⠃⠀⢸⠃⠀⢀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⠃⣴⣿⣄⡩⣁⡹⣀⣼⣷⡄⣻⡜
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢲⣆⠆⣀⡀⣠⢤⣆⢀⣠⢇⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢿⡿⠏⢱⡿⢷⡉⠿⣿⠏⢱⠂
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⠭⠚⠙⠁⠏⠋⠏⠋⠟⠏⠟⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⠯⢶⣿⣏⣿⣷⡮⢧⣴⣿⡆
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠸⣿⠀⢸⡇⠀⠘⠻⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⡇
⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⣿⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣼⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣶⣶⣶⣶⣶⣶⣴⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣟⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣩⣼⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⢿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢡⣾⠟⠉⠀⠀⠻⣜⢧⠀⠀⢀⣤⠄⠀⠀⢀⣴⡿⠋⠀⠀⢀⣴⡿⡟⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⠟⠁⠀⠀⠀⠀⠀⠹⡌⣆⠀⠘⠿⠃⠀⣠⡾⠋⠀⢀⣠⡶⠟⠁⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠏⣀⣠⣤⣀⡀⠀⠀⠀⢻⡜⡄⠀⠀⣀⣼⣯⣄⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣴⡿⠟⠛⠛⠛⠿⣷⣄⠀⠀⠹⣿⣴⠿⢛⣛⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠸⡿
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⢷⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⡼⠱⣿⠿⠿⢿⡏⠀⠀⢠⣤⣤⣄⣀⣠⣤⣤⣤⣤⣴⣿⣤⣀
⠀⠀⠀⢸⠿⢿⣿⣿⣯⣤⣤⣬⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣿⣧⣷⣶⣶⢶⣾⠟⠃⢰⣶⣿⣿⠿⢿⣿⣿⡟⢛⣛⣛⣋⣁⣬⣿⣿⣷⣶⣶⡄
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⢛⣿⡿⠛⣫⡿⢻⡿⠟⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⢻⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣼⠏⢠⡟⠁⠀⠀⠀⠀⣌⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⡾⠁⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣿⣿⣶⣶⣶⣦⣤⣤⣤⣄⣰⠟⠀⠀⠀⡼⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣤⣤⣤⣶⣶⣿
⠀⠀⠀⣤⠀⠀⣿⠉⠉⠙⠛⠛⠛⣿⠋⠀⢀⠀⠀⠁⠀⣼⣥⣀⣀⣀⠀⠀⠀⠀⢹⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢻⠋⠉
⠀⠀⠀⠿⠀⠀⡏⠀⠀⠀⠀⠀⣼⢁⣠⣾⠿⠀⠀⠀⢰⠏⠙⠛⠿⠿⣿⣿⣶⣦⣼⣤⣶⣿⣿⠿⢧⠀⠀⠀⣤⡀⠀⠀⢠⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⡾⠋⠀⠀⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠋⠉⠁⠀⠀⠸⡀⠀⣼⠋⠻⡄⠀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢳⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⢰⠇⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣶⣶⣤⣤⣀⣀⠀⣿⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣿⣿⡿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⣀⣤⣶⣶⣶⣾⣿
⠀⠀⢀⣴⣾⣶⣤⡀⠸⣿⠟⠛⠻⡇⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⡇
⠀⢀⣿⣿⣿⣿⣿⣿⠦⣹⡀⢀⣀⣷⠁⠀⠈⣿⡿⠟⠁⠀⠀⠀⠀⢀⠔⠂⠐⠄⠀⠒⠒⠠⡀⠀⠙⠿⠁⡿⠟⣹⣿⣿⣿⡿⠁
⠀⠈⢿⣿⣿⣿⡿⠁⠀⢨⣿⠟⠛⠛⢿⣦⣴⣏⣠⣤⣶⣶⣦⡄⠀⢇⠀⠀⢀⢠⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠾⠿⠿⠿⠋⠁
⠀⠀⠀⠉⠉⠛⠛⢦⣤⣿⡇⠀⠀⠀⠀⣿⡇⠀⠘⣿⣿⣿⡿⠋⠐⠠⠀⢰⣼⠼⠰⣧⡎⢀⠡⠐⠢⠀⠀⠀⠠⢶⣶⣶⣤⣄⣀
⠀⠀⣠⣴⣶⣿⡿⠁⠀⠻⣷⣀⠀⢀⣰⣿⣥⣠⣼⣿⣿⣿⡇⠀⠀⣀⣲⡎⣴⣾⣷⣌⢰⡖⠂⠀⠀⠐⠀⠀⠈⠛⠿⠿⠿⠟⠋⠁
⠀⠀⣿⣿⣿⣿⣿⣄⣀⣴⠟⠛⠻⣿⠋⠀⠈⣧⡈⠉⠙⠛⠛⠂⠀⠀⠘⣀⢻⣿⣿⠟⣸⢷⣤⣤⣤⣾⣤⡀
⠀⠀⢿⣿⣿⣿⣿⠟⢁⣾⡀⠀⢀⡛⣤⣠⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠋⠝⢃⢬⣴⣾⠫⡑⠀⠉⢻⣿⣿⣷⡄
⠀⠀⠈⠙⠛⠋⠁⢀⣾⣿⣿⣶⣿⡇⢹⣿⣿⣿⣿⣿⣿⠀⠀⠠⠀⠀⢀⠂⠀⠂⠘⢹⣧⡀⠀⢀⠼⣦⣄⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡴⠋⠻⣿⣿⣿⠿⠃⠀⠀⠀⠀⠉⠁⠨⠀⠀⠀⢀⣿⣷⣶⠁⠀⢄⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢸⣿⣷
⠀⠀⠀⣠⠖⠲⢄⠘⡿⢿⣿⣿⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⣿⣿⠀⠀⠈⠻⣦⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋
⠀⠀⢸⠁⠀⠀⠀⠑⣿⠖⠊⡉⠓⢦⡀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠹⣷⠀⠙⣷⡉⠓⠶⣤⣤⣀⣈⣉⣭⡥⠆
⠀⠀⠘⠦⣄⣀⠀⡼⢁⣶⣿⣿⣿⣆⠙⡗⠚⠉⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⣠⠀⠀⢻⡇⠀⠘⣧
⠀⠀⠀⣠⠤⠤⠛⡇⠸⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣤⠀⣸⠃⠀⠀⢻
⠀⠀⢸⠁⠀⠀⠀⠘⣄⠙⠻⠿⠛⢁⡼⢟⡒⠢⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠘⣄⠀⢀⡤⢊⠏⠙⠒⢲⣖⠉⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⢀⣀⡀
⠀⠀⠀⠈⠉⠉⠀⡎⠀⠀⠀⢸⠸⡄⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣋⢰⣆⡙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⢀⡜⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢃⣿⡿⣷⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⢌⡉⠉⠀⠀⠐⣼⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⡟⠱⣿⠃
⠀⠀⠀⠀⢀⣾⣿⣿⡿⢠⣾⠀⣿⣦⡘⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⣧⠈
⠀⠀⠀⠀⣼⡿⠟⠋⢠⣿⣿⣧⣿⣿⣿⡄⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⣾⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⣿⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋


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

⠀⠀⡄⠀⠀⠀⡇⠸⣿⣷⡌⢢⡁⠠⣄⣠⣿⣀⣩⠀⠘⢎⢣⡻⣄⠠⢿⠆⠀⠀⢀⡏⣾⢘⠀⠀⠀⠈⠓⡄⢀
⠀⠀⠳⣤⠿⠂⠀⠀⣿⡇⢿⡄⢹⡀⢸⣧⣴⡟⠁⠘⠓⠈⣦⢳⡼⣦⠀⠀⠘⠀⠸⡇⣿⠛⠀⠠⠤⣀⣴⣻⠈⢇⡠
⠀⠀⠀⠐⠃⠀⠀⠀⣿⣷⠘⣿⠀⣧⠈⠰⣶⢍⡶⠀⠀⠀⠀⣬⡁⣩⡀⠀⠀⣤⣄⣦⢿⡜⣧⡀⠐⠚⠛⠁⠀⠈
⠀⠀⠯⠇⣈⠀⠀⢀⣿⣿⠀⡌⠁⠈⠂⢰⣿⡆⠙⠚⠢⠞⠦⠔⠀⠙⠇⠠⠀⢿⡁⠙⠟⠛⠨⠁⣀⡤⠦⠀⠀⠀⠈⠁
⠀⠀⠐⠾⣠⠀⠰⢀⠉⠛⠸⣿⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠛⠚⠉⠁⣇⠀⠰⠀⠘⠂
⠀⠀⢰⡀⡧⢳⡎⡨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠙⠒⠂⣤⠾⢃
⠀⠀⢇⢟⡧⢈⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠔⢂⠀⠈
⠀⠀⣰⡁⠄⡎⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣷⠀⠶
⠀⠀⢩⠁⢠⣇⡀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⡈⠀⠈⡄
⠀⠀⠀⠀⠀⠻⣮⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢩⠯⠂⢤⠀⠀⠐
⠀⠀⣧⠀⠀⠷⠌⠻⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣓⡌⠀⠱⠄
⠀⢠⣷⣦⢀⠀⠀⠀⢀⠄⠀⣦⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⠋⠐⡎⠁⢀⠀⠀⣤⠀⠦⠶
⠀⠘⢧⡀⠸⠆⣠⣀⢂⠔⠉⠑⠋⠀⢀⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠋⠘⠂⠀⠀⠀⠠⠟⠀⠹⠀⠟⠛⠂
⠀⠀⠰⠟⠄⠂⢿⠨⣸⠀⠀⠀⡴⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⠻⠷⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠘⠊⠀⣀⢸⡆⣿⠀⠀⣒⠎⢉⠘⠳⢀⣀⠀⠈⠁⢂⠀⠀⠈⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⢠⠞⠂
⠀⠀⣷⠿⡦⢉⡼⡰⠋⠀⠀⠌⠀⢿⠿⠆⠉⠤⠁⠀⢀⡀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⠀⠘⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠉⠀⢠⠷⠀⠀⠀⠀⢨⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢺⡗⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⢀⣀⡠⠀⠀⠀⣡⣿⡍⠀⠀⠀⠀⢠⡀⠀⠀⠀⢽⠅⠀⠠⢿⡟⠀⠀⠀⠀⠀⠀⠈⠁⠠⣼⣷⠄
⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠘⣡⣿⣌⠢⠀⠀⢠⣿⣷⡀⠀⠀⢺⡒⠀⢠⣿⣿⡄⠀⠀⢠⣆⠀⠀⠀⠴⣿⣿⠶⠆
⠀⠀⠀⠀⠀⠖⠒⠂⠀⠀⠔⣩⣿⡝⠢⠀⢀⣿⣿⣿⣧⠀⢐⠱⣝⠀⢠⣾⣿⣄⠀⠀⣾⣹⡀⠀⠀⠾⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⣿⡝⠢⠀⣼⣿⣿⣿⣿⣇⢁⡼⣌⠁⠉⣽⣯⠉⠀⣸⣷⢺⣇⠠⠴⠾⣿⣿⣟⣉⣀
⠀⠀⠈⠁⠀⠀⡇⠀⠀⠀⠚⠁⢺⠈⠂⠸⠿⠿⢿⠿⠿⠿⠀⢸⠀⠀⣾⣿⡿⠷⢠⣿⣷⣼⣿⡄⣠⡾⢿⡿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠘⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⢠⣐⠀⠀⠀⠀⢀⠀⠠⠀⡤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠤⠀⠀⠄
⠀⠃⠐⠀⢈⢈⠀⠈⠀⠀⠐⠄⠋⠈⢀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣞⣉⡛⠀⠘⡅⠀⠀⡀⠀⠀⠀⠐⠀⡀⠀⠀⣀⠀⢄⠀⠀⠄⠄⠀⠀⠀⢈⠀⠆
⠀⠠⠀⠀⠀⢂⠀⠀⠀⠀⡀⠠⡀⠡⢤⠄⠀⣀⠀⠉⠀⠀⠀⠀⠈⣴⠧⢈⠙⡃⠀⠈⠀⢀⠀⠀⠀⠠⠈⡄⠀⢠⠂⠁⠀⡠⠰⡄⠀⠀⣀⢀⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠉⡄⠀⢀⠀⠀⠀⠀⠀⠀⠐⠮⡈⠐⢩⠀⠘⠀⠁⠀⠀⣀⠀⠀⡀⠀⡀⠀⠆⠁⣘⠃⡀⠁⡘⠀⡅⠀⠁⡄⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡀⠈⡀⠈⠀⠠⠀⠀⠀⣀⢀⡀⠀⠇⠀⠄⡀⠘⡀⠀⠠⠐⠀⠁⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠂⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠅⠐⠀⠀⣀⠀⠀⠉⢀⠀⠀⠀⠦⠠⢠⠄⠀⠀⠂⠀⠠⠈⠀⠀⠀⢁⠀⠀⡠⠀⡀
⠀⠁⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣇⠎⠀⢀⠀⠀⠠⠀⡀⠀⠀⡀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠈⣁⣀⣠⣀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠈⠄⢀⡀⠐⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢄⡠⣦⠀⠄⠢⠁⠀⠀⠤⠄⠸⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠄
⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠁⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢲⠀⢲⠀⡠⢤⡀⢀⣀⣀⠀⣀⣀⡀⢀⡀⠀⡀⠀⠀⢶⠖⢲⠀⣝⠀⣀⣀⡀⠴⣷⠄⢹⣀⡀⠀⢀⡀⣿⣠⢤⡄⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠛⢻⠀⡔⠚⣗⢸⠋⢙⡇⣿⠉⣻⠈⢧⣠⠏⠀⠀⠘⠗⠛⡄⣸⠀⣿⠉⠉⠀⡇⠀⢸⠉⢹⠴⠋⢉⡏⣰⠒⢿⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠀⠛⠀⠈⠃⠙⠋⠙⢸⡟⠋⠀⣿⠛⠁⢠⣨⠏⠀⠀⠀⠘⠓⠀⠁⠛⠁⠛⠀⠀⠀⠙⠃⠙⠀⠘⠃⠑⠋⠓⠈⠛⠉⠃⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⢰⡦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠊⠀⠀⢀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠠⠈⠀⠀⠆⠑⠀⢀⠀⠀⠀⠀⠈⠐⠀⠀⠠⠀⠀⠀⠈
⠀⠀⢂⠀⠀⠤⠐⠀⠀⠀⠀⠀⠋⠰⢄⠀⠙⠀⠀⠀⠀⠂⢀⠀⠀⡀⠀⠀⠒⠒⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⢰⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⡀⠀⠁⠐⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠄⠀⠀⠀⠀⠢⠑⠤⢀⣀⡀⠐⠁
⠀⠃⠀⠀⣉⢘⠁⠀⣴⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢁⡂⠁⠀⠐⢓⡈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠆⠀⠈⠀⠠⠄⠀⠠⠀⠀⠀⠀⠈⠉⢢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⡀⡀⢀⢀⠄⡀⠈⠄⠂⠀⠀⠠⡀⢀⠀⠐
⠐⠄⠡⠀⠄⠀⠂⠀⠀⠀⠀⠀⢃⠀⠀⠃⡸⠀⠀⠀⠀⠤⠐⠒⠀⠈⢡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢦⠔⠇
⠀⠀⣤⠀⠀⠀⠀⡠⠀⠁⢀⠀⠀⡱⠂⠀⠀⠀⣀⠀⡈⠂⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀
⠀⣀⠀⠀⠀⠀⠀⢰⠀⢀⡅⠂⡰⠃⢀⠀⢀⠹⢥⠌⠀⠀⡁⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠂⠐⢀⠀⠀⠀⠄⢠
⠈⠧⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠴⠠⠰⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⡆⠀⠀⢰⠟⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⣤⡔⢶⡟⢻⡶⢢⣤⠐⠛
⠀⠀⠀⠀⠀⡀⢀⡤⢤⡀⠀⢤⣤⠀⣸⠉⠓⠇⡘⠀⠀⢃⡸⠊⠉⣇⠀⣤⡦⠀⢀⡤⢤⡀⢀
⠀⠀⠀⠚⡴⣒⡟⠸⣉⡝⠀⠀⠀⢴⠓⣣⠀⠀⢹⣤⢄⡏⠀⢀⣜⠚⡦⠀⠀⠀⢫⣉⡇⢻⣒⢤⠓
⠀⠀⠈⣸⠼⢤⣿⡤⠂⢺⡁⣴⣄⢎⡁⢀⣉⠲⢎⣿⣿⡡⠖⣉⡀⢈⡱⣠⣦⠌⡗⠔⢤⣿⡤⠧⣇⠁
⠀⢠⡎⣠⠶⢤⠀⠙⣦⡀⠀⠙⠛⢠⣊⠁⠊⠑⠒⢹⡋⠒⠊⠑⠈⣑⡄⠛⠋⠁⢀⣴⠋⠀⡤⠶⣄⢱⡄
⠀⠘⢧⣑⣒⡼⠁⠀⠘⢷⣄⠀⠀⣏⢠⠉⣲⣶⡒⡿⢿⢒⣶⣖⠉⡄⣹⠀⠀⣠⡾⠃⠀⠈⢧⣒⣊⡼⠃
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠻⣦⡄⠈⢩⠎⢘⣫⠞⠁⠈⠳⣝⡃⠱⡍⠁⢠⣴⠟⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡛⢮⠉⠛⣻⠛⡉⠀⠀⠀⠀⠀⠀⢙⠛⣟⠛⠉⡵⢛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠦⠞⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⡀⠳⠴⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠈⠉⠉⠁⠠⠤⠤⠤⣉⣉⡁⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣤⣤⣤⠤⠶⠋⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⢻⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠖⠛⠉⢀⡐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡿⠿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠶⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀⠀⢰⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣦⡟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡁⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⡴⢻⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⢷⣄⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⠛⠂⠀⠀⣿⠇⠀⣀⡴⠋⠀⢸⣿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠉⠛⠷⢶⣥⡀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠒⢛⣁⠴⠞⠉⠀⠀⢀⣾⢋⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣⡈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⡴⠟⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠉⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠋⢉⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠻⢯⡉⠉⢉⡽⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡤⢠⡄⠀⠁⢀⡀⢀⢄⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢸⣄⠈⢀⣰⠏⠀⠳⢦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⠲⠖⣶⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠴⠦⣄⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⢀⡠⠞⠙⠶⣤⣤⣤⣤⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢿⠉⠀⣻⠏⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣣⣠⣇⣤⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣢⠔⢊⣩⠟⡇⢸⠙⢦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⣠⡎⠀⠀⡇⢸⠀⠀⠳⣄⡽⠆
⠀⠀⠀⠀⡰⠶⢍⡔⠛⠛⠋⣷⢺⠛⢿⣇⣀⣀⡇⢸⣀⣀⣀⣹⠁
⠀⠀⠀⡞⠁⠘⣤⣇⠀⠀⠀⣿⡞⠀⢀⣿⠁⠈⡇⢸⠈⠉⢹⡇⠀⣀⣤⣤⠴⢦⡀
⠀⠀⠀⠘⢦⡠⠮⠿⢄⣤⠼⠟⢦⣢⣞⣿⠐⠀⡇⢸⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⣧⠴⠲⢦⣀
⠀⢠⠴⠖⡾⠁⠀⠀⡼⠳⡄⠀⠈⢯⠉⠉⢻⠂⡇⢸⣤⠞⠋⠰⡂⡆⠀⠀⠉⠀⡿⢠⠃⠀⠉⢷
⠀⢸⡀⢰⡷⠺⡆⣼⠁⠀⢷⠀⠀⠘⣆⡀⣼⡉⡇⢸⠾⡄⠀⠀⠱⣽⠀⠀⠀⠀⣿⠃⠀⠀⢀⡟⠜
⠀⢆⢻⠛⠛⠛⠛⣿⠀⠀⠸⡇⢠⣄⣿⢹⡇⠈⡇⢸⠀⢹⣄⣠⠤⠼⢧⣀⡤⠼⠓⢦⣀⣴⣏⣀⣀⣀⡀
⠀⠀⢸⡆⠀⠀⠀⣿⠀⠀⠀⣷⡏⠈⠛⢸⡉⠀⣧⡟⠉⠉⡽⠁⠀⢠⠞⠁⠑⣄⠀⠀⠹⣆⣀⣀⣀⣀⡇
⠀⠀⠸⡇⠀⡀⠀⣿⠀⠀⠀⡏⠀⠀⠀⢸⠀⠀⡏⢳⡶⢶⡷⠲⣄⡞⠛⠛⠛⠛⣆⢠⢤⣸⡏⠉⢹⠋
⠀⠀⠀⣇⠀⠀⠀⢻⠀⠀⠀⡇⡄⠀⠂⣸⠐⠀⡇⣸⠸⡄⠀⠀⠈⠁⠀⠁⠀⠀⠹⠋⢀⢈⣋⣀⡟
⠀⠀⠀⠹⠤⠤⠤⢿⣤⣤⣤⠷⠶⠶⠖⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⡀
⣿⣿⣿⣿⠟⠻⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠦⢤⣶⣿⣷⣤⣤⣤⠀⢰⡄
⣿⣿⣿⣿⡇⠀⠸⡗⢭⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⣿
⣿⣿⣿⣿⡇⠀⠀⠳⠀⠈⠲⡀⡾⠀⠀⣄⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡟⡀⠁⠈⠉⠉⠁⠺⠇⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠡⠈⠒⢲⣤⡶⠶⠂⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠁⠁⠀⠉⢣⣀⡤⠴⣄⠈⢹⡿⠿⠛⠪⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⠛⠻⣿⡇⠀⠀⠂⠀⠒⣷⣄⣀⣀⡈⠁⠙⡄⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⢰⣿⣿⣷⣶⣦⡰⠊⠀⢻⣿⣿⣿⣷⣶⠀⠀⢠⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣤⠤⠝⢿⣿⣿⡿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡏⠙⣆⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⡀⠈⢱⢊⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠉⢻⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⢀⣨⡉⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⠉⠉⠉⠉⠈⠉⠉⠁⠉⠀⠉⠈⡉⠁⠀⠀⢀⣷⣿⣿⣷⣄⠀⢀⠄⠉⠦⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠛⣿⣿⣿⣿⠏⠉⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⢀⠀⣠⠄⣤⠔⠛⠛⠛⠻⠂⣀⣀⣴⠛⠑⠆⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⡀⠀⢀⠀⠀⢀⠈⠁⠈⠀⡁⠈⠀⠁⠀⠈⠓⠄⢶⢴⣶⡶⢏⡟⢏⠛⢛⠯⣓⡀⢸⣿⠛⠛
⠀⠀⠀⠀⣿⡇⠰⠁⣰⢃⡣⡄⠆⠌⠼⠐⠄⠬⠔⠆⢦⠆⠀⠀⣾⠈⣷⢸⠻⠖⠲⠦⠰⠆⠨⢿⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠲⣼⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠁⠻⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⣀⡀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⡴⠀⠀⠀⣿⣿⣿⣦⣄⣤⢾⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣠⠔⠁⠀⠀⠀⠈⠪⣟⣻⣛⡯⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀⠀⠀⠀⠀⠀⢀⣀⣼⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⣷⣤⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⣠⠎⠀⠀⠀⠀⠀⠻⣻⠿⢿⡷⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇
⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢩⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡜⠀⠀⠀⠀⠀⠀⠀⢪⣛⣫⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠒⠂⠀⠐⠒⠀⠀⠀⠀⠀⠀⠒⠚⠛⠒⠂⠀⠀⠀⠀⠀⠐⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢄⡀⠀⠀⠀⡴⠲⠀⠀⠠⠔⢄⠀⠀⠀⢀⡠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⠁⣄⠈⠁⠀⠈⠀⠀⠀⠈⠃⣤⢈⠅⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⣘⡂⡑⠉⠁⠀⢦⢻⣤⣟⡤⠀⠈⠁⢊⢔⣁⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢧⠙⠃⠀⢱⠤⡀⠀⠁⡁⠉⠀⢀⢤⡊⠀⠙⠃⡼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠳⠄⢤⠀⠀⢳⣽⣷⢢⢣⡰⣾⣿⡎⠀⢀⡤⠈⠕⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢺⠂⠀⠁⠀⠈⣋⠡⣾⣿⣷⠈⣙⠁⠀⠊⠀⠐⡇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠂⠀⠀⠢⠤⣌⡉⢪⢉⡟⢉⣠⠤⠖⠀⡀⠐⡁⢶⢆⣀⠀⠠⢴⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠤⢄⠈⠛⠞⠈⠗⠛⠀⡠⠤⣠⠄⢾⣾⣦⠸⠾⠃⠻⠃⠶⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⠶⠖⣦⢈⠁⠀⠀⠀⢈⡁⠀⠀⠐⢿⠆⠉⢡⣶⡄⠾⠇⠸⣃⠷⡐⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⣇⠈⠀⠈⠈⠃⠀⠁⠀⠻⠃⠀⢾⣻⡆⣉⠥⠤⣬⡉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠒⠱⠓⢆⠀⠀⠀⠀⣉⡧⠁⠀⠀⠀⠀⠀⡛⠀⠰⣿⠆⣽⣶⢱⣿⣷⢈⣭⡙⠖⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡷⠀⣘⡀⠀⠈⠉⠃⠀⡠⠘⣫⠀⠀⠀⠀⢠⠀⢈⡁⠰⠷⠸⠇⣨⡋⢀⣍⠁⠸⠿⢏⣠⡈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠂⠒⠀⣀⣠⡔⣭⡍⢀⠀⠰⠀⢸⠁⢸⣇⠀⠀⠀⠀⠀⢣⠈⠁⠀⠀⣴⢦⠻⠟⠙⠟⠱⣿⠆⠘⠷⠋⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⢓⢀⣀⠙⠁⣋⡀⠂⠀⣰⠆⠻⣜⢦⡘⠋⠀⠀⠀⠀⠀⢣⠀⠀⣤⣝⠋⠀⠦⠀⠀⠀⠀⣿⠆⢾⣿⠆⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣵⠄⠀⢀⠀⢀⣬⡄⠁⠀⠀⢈⣷⠀⠀⣀⠀⠀⠀⠀⠀⠳⡀⡙⠋⡀⢙⡃⠛⢠⡄⢾⡶⢠⣶⣦⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣀⡈⠁⠀⠀⠀⠀⠈⢻⣄⢤⣌⡋⠀⠀⣀⡴⠆⠐⠽⣤⠇⢈⢩⣦⢀⣤⡀⣴⢦⠉⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠦⠞⠭⠐⠈⠁⠀⡘⠻⠄⠀⣠⡀⠉⢓⠂⢅⢄⡀⠉⢱⣾⣣⠄⠉⠀⠓⠐⣶⣜⣛⣡⣭⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⠈⠁⠈⠁⠀⠠⠀⠁⠐⣷⠀⢿⡇⣴⡏⠃⠸⣮⣙⣂⣛⣫⠀⠿⠶⣄⡀⠀⠈⠉⠙⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠈⠹⣿⢃⣤⣙⠆⢀⡉⡃⠀⠁⠈⠓⠄⠰⣄⢈⣉⣈⠀⠛⢲⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢘⣧⠀⢒⡺⠿⠀⡐⢀⢠⡿⢸⢧⢀⡖⠖⠘⡯⢋⣩⢉⡴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠈⠉⠀⡼⠟⠂⠰⠀⢂⠀⠁⠘⠋⠀⠀⠀⣠⠦⠟⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⢈⡀⠰⢂⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⠬⠒⣬⠬⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⡼⠁
⣿⣿⣿⣿⣿⣿⣮⣍⡛⠛⢋⣩⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠑⠶⣦⣤⡶⠖⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⣀
⣿⣿⣿⣿⣿⣭⣖⣒⣂⣀⣀⣒⣻⣯⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠒⠯⠿⠿⠿⠿⠿⠄⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢅⣀⣀⣠⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⣰⠀⠀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆
⠀⡌⢀⠀⠀⠀⡚⠛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠆
⠀⠈⠻⠤⣿⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠃
⠀⠀⠀⠐⠐⠂⠒⠰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣇⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡠⢼⣧⣄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠠⢱⠀⠀⠀⡽⣧⣄⡄⠀⠀⠀⠀⠀⠀⠀⢰⡠⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢳
⠀⣶⣄⣀⣀⣀⣀⣀⣰⣿⡀⠀⠀⠀⠀⠀⠁⢸⠃⣠⠀⡡⠘⢧⢻⡆⠀⠀⠀⠀⠀⠀⠀⠱⣦⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⢶⣒⠺⠯⠭⠭⠭⠽⣿⣶⣿⠀⠀⠀⠀⠀⠀⠈⠐⣿⠐⣟⢂⡘⠞⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠈⠉⠙⢻⠉⡼⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⣻⡄⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⠀⢆
⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡹⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢡⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⢢
⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳
⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹
⠀⠠⠤⣒⣗⠀⣼⣶⣤⢤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠙⣢⡀⠀⠀⠀⠀⠰⣇⣛⣷⣗⡓⢓⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠚⠂⠀⢨⡿⣧⡶⠾⠛⠁⠀⠀⠀⠀⠀⠘⠷⠦⠄⢠⣧⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣴⠶⢾⡷⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠁⠀⠀⠀⢸⣿⣿⣿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣴⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠆⣌⣀⣤⡒⣉⣤⠀⠀⠀⠀⠀⣀⣤⣴⠶⠛⠛⠉⠈⠡⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢙⠀⣰⠂⢭⣭⠄⢾⣿⡿⠀⢀⠠⠒⡩⠞⠁⠀⠀⠉⠐⣤⠖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⢅⡘⣵⣶⠏⡈⣉⠀⠨⠀⣤⠊⣀⠐⢠⠄⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡌⢍⣴⡾⣵⣷⡌⠿⣿⡗⠊⡜⠡⣀⡈⠁⠀⢀⣠⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡜⢿⡟⠁⠻⣿⠀⠠⠊⠁⡜⠀⠀⠀⠀⠀⢀⠎⢣⡆
⠀⠀⠀⠀⠀⠀⡐⠒⠊⠩⡑⡌⠰⢮⡄⠀⠀⠈⢤⢠⣾⣇⠀⠀⠀⢀⠂⠀⣸⠁⠀⠀⠀⠀⡀⣃⠰⠉⠛⠦⠄
⠀⠀⠀⠀⠀⢠⠈⠐⠋⡂⠋⠀⠐⢂⠒⡇⠀⢀⢠⡀⢁⡾⣆⠀⠀⡌⠀⠀⠇⠀⠀⣀⠴⢊⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢂⡀⠤⠂⢂⡀⠸⠢⠰⡉⠷⢌⣥⣉⢼⣿⣷⢴⣇⠄⣘⡦⠴⡎⠱⠾⠿⢁⣈⠛⠇
⠀⢀⣀⣸⣿⣇⠀⡘⠢⢐⠀⠁⠀⠀⠀⠀⠙⠛⠉⠁⠈⠂⣔⢿⣷⢹⣤⣺⠃⣠⣾⣷⣆⣁⢻⣿⣇⠚⠈⠈⠁⠢⡀
⠐⠻⠿⢿⣫⣥⣀⠐⠀⡙⠀⡈⣁⡠⠆⡀⠀⠀⠀⠀⠀⠀⠛⠗⣠⣤⣽⢹⡦⠌⣿⣯⣾⣿⡧⠉⢉⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠿⠿⢻⣿⡗⠀⠈⠀⣰⢋⠁⢉⣈⠉⠐⠂⢄⠀⠀⠀⢠⠉⠛⠛⠙⣿⡀⣿⣿⠝⡻⠇⠀⠀⢤⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠎⠉⠀⣀⡂⢠⠁⢧⡄⡩⢉⠃⠀⠀⠀⠑⡀⠀⠈⠀⠀⠀⠀⠃⠳⠈⡡⠰⠃⠀⠀⠀⢸⡿⢂⣄⠀⢀⣶⡇
⠀⠀⠀⠀⢀⠠⡁⡂⠲⡆⠘⠀⠀⡸⠂⠻⠃⠀⠀⠀⠀⠄⠀⡈⠀⠀⠀⠀⠘⡀⡸⡁⠆⠀⠀⠀⣠⣤⡆⣿⣿⠀⢿⡿⠃
⠀⠀⠀⠀⠀⠄⠈⡀⠉⡘⠠⠪⣀⢀⡄⠅⠐⠆⠠⣨⠉⠀⠀⡁⠀⠀⢀⠀⠄⢳⢘⠌⠒⢀⢀⣺⠿⣛⠠⢌⣁⡊
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠈⠈⠁⠀⠂⣗⠀⡀⠀⠈⣀⠀⠠⠀⠐⢂⠁⡀⠐⡙⢁⠈⠣⡆⠺⣿⣷⣤⡐⢿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠰⠀⠀⠀⠈⠠⠈⣀⠁⡀⠀⠀⠀⠄⢀⡠⠁⢊⡄⠈⣀⣘⣦⣄⠄⠀⠀⠉⠁⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠈⢀⠀⠀⠀⠑⠁⠀⢀⠀⠞⠀⠀⢂⠀⠀⠚⠿⠿⠟⡁⣷⣿⡄⣈⡀⠛⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠌⡀⠉⢳⠀⡀⠀⢀⠀⠂⢀⣴⣇⠙⠻⡁⣿⣿⡇⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⠒⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⣼⣿⠃⠈⢀⠀⢛⢿⣷⠘⠻⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢰⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠉⠂⠀⢀⣀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠤⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⣿⣿⡂⢀⠀⢠⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⢀⡈⣢⠈⠙⣿⣿⠻⢿⣿⣿⣿⣿⣷⢈⣿⣿⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣽⡆⠀⠁⣤⣹⣿⢄⣎⣿⣿⣷⣾⠿⣿⣶⣤⠟⡾⠻⡍⠙⣀⣄⡤⠖⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣟⣁⡀⡁⠼⢿⣷⣌⠛⢋⣿⡟⠀⠈⠻⣷⣤⣄⠀⠙⢤⠛⢇⠰⣢⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⣻⣻⣷⣴⡄⡈⢙⡀⠻⣿⠁⠀⠀⠀⢹⡿⡿⠛⠻⠀⢆⠀⣴⡟⢩⡀⠡⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⣎⢀⡾⢾⡟⢹⣿⣶⡁⡈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⠿⣿⣿⡶⢤⣀⣠⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠰⢛⣀⣤⣾⠗⢞⡿⠄⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⡍⢉⣿⠀⡿⣿⣿⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠷⠶⣷⠀⠼⣿⣼⡷⠚⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣙⣯⡿⣿⣧⢸⡟⠛⠒
⠀⠀⠀⠀⠀⠰⠇⠀⣀⣀⠀⠐⡸⡟⠁⠐⡆⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡟⠙⢿⣿⠿⠠⠁⠒⠲⠄
⠀⠀⠀⠀⠀⠘⠁⠈⣭⢛⣧⢠⢑⣿⠷⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡘⢸⣇⣄⣀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⡏⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡏⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠐⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⣿⣶⠀⢰⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⠀⠀⠙⠃⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠽⠏⢻⢿⠊⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠉⡟⠛⢳⡀⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⣿⣦⢾⣾⣿⠛⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⢾⣙⣿⡗⠀⣸⣟⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡸⣿⣆⠘⠙⢛⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣏⠉⠀⢁⣽⡟⠦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠹⢦⡸⣿⡄⠐⠁⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠍⠀⠺⡗⠆⠀⠀⠀⠀⠀⣠⣿⠦⣤⣷⠀⠀⣹⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠙⠉⠈⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠜⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢨⡿⠂⠩⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡬⠁⠺⣯⠀⡄
⠀⠀⠀⠀⠀⡀⡀⡋⢀⠛⣶⣌⡔⣋⣙⡛⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠞⣟⣉⡻⣎⣴⡞⢃⠈⡃⡀⡀
⠀⠀⠀⠘⣿⠟⠻⠋⡘⢿⣿⡿⠇⣸⣿⡇⢸⣿⣏⣈⣨⣉⠨⣀⠳⣌⠈⠁⠈⠈⠁⣽⢛⣡⢉⣨⣅⣿⣿⠀⣿⡿⡀⠿⣿⣳⠟⡈⠻⠛⢿⡟⠁
⠀⠀⠀⠈⠹⠿⠂⣦⡼⢨⣉⣡⣾⣷⣦⡾⠉⠛⢻⠿⡿⢯⡵⢫⠿⠋⠀⠀⠀⠀⠈⠻⢯⠻⢯⠿⣿⢿⠛⠻⣦⣶⢻⡦⣈⣩⠸⣤⡆⠺⠿⠉
⠀⠀⠀⠀⠀⠀⠺⠿⣧⠸⢿⣷⣸⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡠⣿⠿⢠⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣶⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⡆⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠏⢻⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠻⣿⡧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠃⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣅⠱⡟⡅⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⠀⢰⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⡷⢃⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠋⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣦⠼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢤⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡋⡼⣧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠿⡌⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠅⣼⣷⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣿⡄⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢆⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠛⠻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠛⠃⢼
⠀⠀⠀⠀⠀⠀⢶⣶⡟⢰⣿⡯⢙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⠉⣿⣷⠘⡷⣶⠆
⠀⠀⠀⢠⣰⡶⠄⠟⢻⢼⣍⡹⢿⡿⠛⣷⣄⣤⣼⣿⣿⣿⡳⣌⣷⣤⡀⠀⠀⠀⣀⣴⣟⡴⣿⣿⣿⣿⣦⣴⡛⠻⣿⠿⣉⣽⠼⠛⠇⠴⣶⣠
⠀⠀⠀⣠⣾⡦⣶⣄⢡⣾⣿⣒⡄⣹⣿⡇⢸⢿⠍⠭⠹⠁⢜⢡⠶⢧⢠⠄⢤⢠⠄⣵⢦⡉⡤⢹⠉⡿⢿⠀⣿⣿⡁⣶⣿⣿⣧⢀⣴⡦⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⡈⠤⠿⠩⠷⣭⣥⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣧⣩⡵⠏⠝⠧⠌⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⡷⠄⣐⠛⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢛⡂⠴⣏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠙⠀⠀⠀⠀⠀⡔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⡄
⠀⠀⠀⠈⠁⠀⠀⠀⢠⠁⠀⠐⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠤⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⢏⡋⠀⠀⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠂⠀⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠉⠓⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠠⠀⠈⡇⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡇⠀⡠⠔⠀⠂⠀⠳⢤⣀⣀⠄⢹⠀⠀⠀⠀⠀⡌⣀⡤⢂⠀⡀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢰⡇⡠⠸⠏⠀⢠⠀⠀⠀⠠⢄⣉⡷⢦
⠀⠀⢻⣸⡇⠀⠺⠀⠀⠀⠈⢳⣦⠀⡀⠀⠀⠀⠀⠀⠀⢯⠀⢧⣘⠀⠐⢀⠄⢀⡜⣇⠀⠀⠀⠀⠀⣧⠈⠀⠒⠒⠒⠒⠒⠒⠚⠉⢻⡆⡾
⠀⠀⠀⠻⣯⣒⠀⠀⠠⠤⠒⢛⣡⠞⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠒⠒⠒⠛⠉⡸⡟⠀⠀⠀⠀⠀⠹⣆⠀⠴⡎⠀⢀⢄⠀⠀⢰⠋⣰⠇
⠀⠀⠀⠀⠀⠉⡽⢿⠒⡚⢯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣀⣀⣀⣀⡀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢐⣫⠞⠃
⠀⠀⠀⠀⠀⠀⠃⢀⠌⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⡝⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣁
⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⣉⡀
⠀⠀⠀⡄⠈⠀⠀⠀⢸⠱⢒⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠋⠀⠀⠀⠀⠉⠉⡒⢶⣤
⠀⠀⠀⠘⠶⣤⣀⣀⣀⣉⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠂⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣄⣀⠀⠀⠀⠀⠓⢚⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣷⡀
⠀⠀⢠⠟⠋⠉⠉⠛⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⣩⠤⠶⢦⡈⢳⡀
⠀⠀⡟⠀⢰⡏⠉⠉⠛⠲⠦⢤⣍⡓⠦⣄⡀⠀⠀⠀⠀⣀⡤⠶⠿⠷⠶⠦⣄⠀⠀⠀⢀⣠⠴⠚⣉⣤⠴⠚⠉⠀⠀⠀⣠⠇⠀⡇
⠀⠀⢳⣄⠀⠙⠲⠤⣄⣀⣀⣀⣀⣉⣙⣛⡻⣶⣤⣀⢰⡏⠀⠀⠀⠀⠀⠀⢸⣇⣠⣶⣿⢗⡚⣉⣁⣀⣀⣀⣀⣤⠴⠚⣁⣠⣞⣁⣀⣀⣀
⡟⠛⠛⠉⠛⠲⠦⠤⢤⣤⣭⣭⠭⠭⢭⡿⠛⠛⠋⠉⢉⣻⣶⠦⠤⠤⠶⠖⠛⢿⣉⠛⠛⠛⠛⠛⠛⠿⠿⠟⠗⣶⠛⠋⠉⣻⠉⢻⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠋⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠋⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⠀⠀⢀⣿⡀⣸⣀⣀⣀
⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠉⣿⠉⠉⠉⣿⠉⢻⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣄⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⢤⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣸⣀⣠⠤
⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣇⣀⢀⣀⣀⡀⣸⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣿⣀⣤⠴⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠶⠤⣤⣀⡀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣤⢤⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢦⣀⡀⠀⠀⠀⠀⠀⢀⣸⣧⣼⣅⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣀⡤⢖⡿⠛⠉⠀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠋⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠛⠒⠦⢤⣄⣀⣀⣀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠛⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⣤⢤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠛⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠉⢣⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⡐⠀⠙⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡅⠁⠀⠈⠃⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠰⢆⠀⠠⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣨⠔⠚⣁⢐⡉⠀⠀⢼⢯⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠲⠊⠀⠀⢀⠀⠠⠀⣡⣿⣀⠃⠱⢦⠄⠀⠰⡦⢠⣀⠶⠘⠃⠀⠀⠀⠀⠀⠀⢱⠈⡅⠐⣪⠴⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⣠⠿⠁⠀⠈⡧⠀⠀⠁⠀⠀⠀⢈⠀⠀⠀⠐⢀⠄⢁⠀⠄⠀⠀⢁⠐⢁⠑⠄⢠⡀⡠⢎
⠀⠀⠀⠀⡀⠶⡄⠀⠀⡀⠄⠒⠛⠀⠰⣶⠦⠕⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠠⣈⠀⣁⠀⠀⠄⠀⠊⠀⠈⢡⠌⠠⠁
⠀⠠⡤⠤⣤⠄⠀⠀⣠⢤⣠⠇⠉⣡⠀⢀⣥⠃⠀⠀⠀⠀⠀⠀⠆⠄⠁⠀⠥⠀⠂⠀⠀⠀⠄⠀⠀⠠⠈⠁⠀⠀⠁
⠀⠁⡀⠀⠀⢠⡄⡈⠂⠌⢄⠀⠈⠉⠀⠉⠙⠂⠀⠀⢤⣠⠤⠔⠀⠀⠀⠀⠀⠑⠂⡄⢀⠀⢐⠄⠀⠀⠄
⠀⠀⠒⠠⠍⠂⠀⠀⠀⠀⠁⠀⣀⠄⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣶⡎⢲⣿⣷⣮⠭⢗⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣖⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠈⣀⠤⠀⠀⠀⠀⠈⠛⣷⠖⢚⡹⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡃⠒⠲⡞⠀⠀⠀⠀⠀⠀⠉⢧⣄⣀⣠⠀⠀⠂⠸⢆⣀⠚⢉⠉⡀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠓⠣⠤⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠭⠛⢋⠁⢠⡡⢀⠀⠀⠀⠤⢁⡲⠄⠠⠰⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⡀⠀⠙⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⡑⠂⠐⠈⢣⠊⠒⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⣀⣀⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣀⠤⠤⠐⠒⠒⠒⠂⠀⠀⠀⠀⠐⠒⠂⠤⠤⣀⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠒⠒⣂⣉⣉⣉⣉⠀⠀⠀⠀⣀⣉⣉⣉⣐⠒⠒⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⣴⠛⠒⠀⠒⠀⠠⠄⠀⠤⠤⠀⠤⠤⠤⠤⠀⠤⠄⠠⠤⠤⠄⠠⠤⠄⠀⠒⠒⠒⠛⠀⡀
⠀⠀⣀⠘⠋⠀⠈⠉⠀⠉⠁⠈⠁⠀⠀⠂⠀⠒⠀⠒⠒⠀⠒⠂⠐⠒⠒⠀⠀⠉⠉⠀⠉⠉⠉⠁⠀⠙⠃⣀
⠀⣾⠓⠦⠤⠤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠤⠤⠴⠚⣷
⠀⠙⠲⠦⣤⣄⣀⣀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣠⣤⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡒⠒⠋⠉⠓⢦⣀⣠⣤⣀⣀⠀⠀⢀⡴⠋⠉⠉⠙⠓⠒⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠉⠢⣄⠀⠀⠀⠈⠢⡀⠀⠉⠉⡳⠋⠀⠀⠀⠀⢀⡴⠋⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢶⣶⠛⠙⠲⢄⣈⠑⠤⣀⠀⠀⠈⢦⣠⡜⠁⠀⠀⣀⡤⠖⣁⡠⠔⠊⢑⢢⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠛⠉⠁⠀⡰⠋⠉⠉⠙⠒⠦⣌⣉⠓⠺⠿⢶⣶⠖⠛⢲⣶⠶⠾⠗⠒⣉⡥⠴⠒⠋⠉⠁⠈⢻⠓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⢰⠾⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⣀⣀⣀⣀⡀⣀⣭⣵⠶⢖⣻⠦⡄⠾⣛⠲⠶⣮⡭⠤⠤⢄⣀⣀⣀⠠⠴⠋⠀⠀⠀⠀⠀⢉⣉⠷⡆
⠀⠀⠀⠀⢹⠀⠀⠈⠙⠒⠦⢤⣀⡀⠀⣀⡤⠴⠚⣉⣡⡤⠶⠒⠉⠁⠖⠚⠉⠀⡴⠓⣄⠀⠉⠁⠀⠈⠉⠒⠦⢄⣉⠓⠲⢤⣀⠀⣀⡤⠴⠚⠉⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⣳⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠉⠳⢴⠚⢻⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠺⣄⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠱⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⡟⠙⠲⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⢶⠖⠋⠉⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⣎⣠⠴⠚⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣄⠀⠀⠀⢀⡜⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢱⡀⠀⠀⠀⠀⢀⡴⠳⢤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⠞⢣⡀⣠⢾⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣄⠀⠀⣠⠎⠀⠀⠀⡏⠉⠛⠲⢤⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠴⠞⠋⢹⠀⠀⠀⠙⠁⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢣⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⣸⣇⣠⠴⠚⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠦⢄⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣬⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣸⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⢸⠀⠀⢀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣼⡴⠖⠋⠁



⣀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡤⠠⣤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠓⠄⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠋⠁⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠃⠃
⠀⠀⠉⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢳⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠄⡀⢀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢰⠺⣄⠀⠀⠀⠈⠀⠐⠃⣹⢾⠀⠀⠀⠀⠀⠧⣀⠀⠈⠀⠉⡆⠀⣹⡼
⠀⣿⡖⡁⠘⠀⠀⡢⢆⣾⠀⠀⠀⠀⠀⠀⠈⠆⠈⠙⠒⠒⠒⠒⢚⠉⠁⡾⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⣰⠿⠁
⠀⠘⢯⡉⠘⠀⠐⠀⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⢑⠦⢭⣙⣛⣉⣥⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠝⠛⠛⠻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢀⡀⠀⠀⠈⠁⠁⠐⠣⣦⡄⠀⠀⠀⠀⠀⠀⠀⡀⢄⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠂
⠈⠐⣆⡀⠀⠀⠀⠀⣀⣑⡯⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠨⠀⡈⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠑⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠤⠤⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⣧⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠱⣄⡀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⣹
⠀⠀⠀⠙⠲⢤⣀⡀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⡀⣀⣀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠀⣀⡤⠚⠁
⠀⠀⠀⣀⠴⠚⠁⠈⠉⠑⠒⠒⠒⠚⠛⠛⠓⠒⠒⠒⠊⠉⢉⡽⠃⠀⠀⠀⠀⠈⢯⡉⠑⠒⠒⠠⠤⠶⠶⠶⠮⠭⠥⠤⠤⠒⠒⠚⠯⣀
⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠖⠒⠒⠒⠲⠶⠼⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⡸
⠀⢓⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⡇
⠀⠸⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⢀⣀⡀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⠠⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠈⣦⡀
⠀⠀⠀⠀⠀⢠⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠤⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⣤⠤⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⡼⠁⠀⠀⢀⣀⠤⠔⠒⠂⠉⠉⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⠉⠉⠒⢲⡤⢄⡀⠀⠀⠀⢷
⠀⠀⢸⠇⣀⡴⠊⠉⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠈⠙⠢⣄⠈⣇
⠀⠀⣯⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣠⣄⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣤⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠱⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⡀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠤⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠔⠒⠊⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⢈⣈⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⢠⣶⡄⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠏⠙⣿⣄⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠰⣶⣄⠘⡿⢃⣴⣶⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠙⢻⣆⢠⡿⠋⢙⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡶⠛⠛⠛⠒⣾⣿⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣤⣭⣍⠁⠚⣋⡁
⠀⢸⣿⣿⣿⣿⣿⡟⠋⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣇⣀⣿⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⡑⢟⣻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⣿⣿⠂⠀⠙⠻⣿⣷
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⠒⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⢀⣀⢠⠀⠦⠒⡏⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢶⠖⢶⠊⠛⠀⠁⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠈⠙
⠀⠀⢲⡀⠘⠊⠳⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⣠⣤⠶⡀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠳⡀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢁⢐⣛⢱⠏⡠⢫⡆⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠡⠈⡽⣨⣇⢼⡁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠰⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣙⡆⠀⠀⠰⠂⠐⣲⠋⠏⠿⢡⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡔
⠀⠀⠀⠀⠀⠈⠆⠻⠛⢀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠼⣇⠀⣮⠴⠆⠉⠀⠁⢀⣀⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠰⠖⠚⠉⠁⠒⠀⠀⠠⣖⢺⣔⠖⠚⢿⡖⠹⡿⠗⢂⣬⠔⡀⠈⢄⡥⠊
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠷⠈⣤⡧⠄⣘⡥⡴⠚⠻⠁⠉⡦⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⡇⠀⠀⠀⢀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⡄⢿⡆⠀⣿⠀⣷⣖⣮⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢹⡄⣾⣄⢳⡈⣿⠳⠟⠦⠍⠈⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢳⡟⠊⠛⢳⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⣀⣀⢠⣸⣆⠀⠀⠀⠈⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠔⣧⠀⡖⣆⣶⢳⡼⣇⣷⠉⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡄⠘⣆⠟⠘⠛⣏⠃⢻⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣠⣧⣀⣀⣠⣼⣄⣀⣀⣀⣬⣄⢠⣭⣄⣀⢀⣬⣥⠀⢰⣽⣀⠀⣀⣠⠞⠦⣄⣀⢼⣥⡀
⠀⠀⠀⠀⠀⢺⣤⠠⠒⠁⠨⠥⠁⡛⠋⠙⢹⠜⠁⡇⠀⠙⠃⣿⠀⠀⡼⠀⠈⠀⠉⠀⠂⣶⠁⠄⢿⣟⣿
⠀⠀⠀⠀⠀⠈⠈⢦⣄⡐⠛⠂⡀⠟⠂⠁⠈⢷⠀⣹⠀⠄⠀⣿⢀⣰⠃⠀⠀⠀⠐⠀⠈⡿⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⣸⠀⠀⠺⠟⠟⡆⠙⠚⠛⣁⠀⠈⣆⠘⢀⣠⣤⣿⣀⡟⠘⠻⡦⠀⣀⡀⢀⡻⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⣼⢀⠀⢀⡀⠰⣁⠀⢁⣿⣿⣿⡷⢘⣸⡜⡇⠸⠇⢿⢏⠳⣄⣉⣛⠛⠁⣉⡷⠠⠀⢾⡍⣿
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠑⠁⠀⣸⣿⠿⠿⠇⢼⣆⣸⡇⠉⣿⣤⣼⠃⣯⠻⣿⣆⡀⣟⡛⠀⠀⣻⡧⠉
⠀⠀⠀⠀⠁⣾⣉⡂⢀⣄⡀⠐⢲⠏⠀⠀⠀⢀⠀⠋⠻⠇⠀⠇⠉⠀⠀⠓⠚⠟⠛⢿⣯⣛⠀⠀⣭⣷⣶⠻⠃
⠀⠀⠀⠀⠀⣼⣯⣭⣦⣉⡀⣚⡏⢀⡀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢠⣤⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠃⠨⠼⡭⠭⠥⠤⠩⣀⢡⡄⠶⡶⠲⠛⠰⠻⠎⠛⠛⠻⠛⡃⣶⡶⠀⢈⡂⠫⣩⠩⠭⢹⡬⠽⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠶⣗⣒⣺⡟⠉⣯⠈⢹⠉⡉⠙⣏⣛⣻⡿⢯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢽⣯⡭⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⢿⢷⢿⣼⣯⡇⣷⠇⠙⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⡇⡀⠀⡘⢻⢻⣉⠿⣷⡎⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡌⠁⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠁⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠠⠚⠍⠙⠷⢆⡀
⠀⠀⠀⠀⠀⡀⢀⣶⣯⢿⠴⣧⣤⡐⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢍⢂
⠀⠀⠀⠀⠸⢽⣭⣽⣿⣿⣭⣭⣭⣬⣄⣤⣀⣀⣀⣀⣀⣼⣧⣄⣀⣀⣀⣀⣠⣤⣥⣤⣭⣬⣭⣿⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⢿⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠇

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

⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠔⠉⣀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠐⠒⠒
⠀⠀⡇⠀⠊⡇⠀⣰⠴⢺⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡰⠀⠀⠉⠉⢉
⠀⠀⢴⠀⢀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⡀⢀⣠⡠⠆⣰⠁⠈⠀⠀⠀⠀⡼⠂⢤⠇⡀⢀⢀⡴⠁⠠⠄⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡰⡆⣳⡟⠸⠱⡤⠃⠀⠘⠂⠀⠀⠠⠁⠀⡞⠰⠀⢈⡏⣠⠞⡤⠒⣠⠊⠈⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⠤⢤⡄⢠⠀⡆⢸⠇⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠀⢞⡥⠂⡠⢆
⠀⠀⠀⠀⢰⣇⣀⣀⣀⠸⡤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠠⠉⠙⡖⢁⡤⠊⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠛⠋⠤
⠀⠀⠀⠀⣤⠂⠀⣠⠞⠁
⠀⠀⠀⠀⠙⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢧⡼⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠀⣀⣀⡤⢏⣼⣧⡹⢦⣀⣀⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⡀⣠⠶⣻⠉⡿⢛⣭⢴⣿⠛⠻⣿⡦⣭⡛⢿⡉⣟⠶⣄⢀⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⢥⣼⡟⣋⡽⢧⢿⢀⡄⡿⡼⢯⣙⠻⣧⡬⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣟⠀⠀⠀⡼⠃⡏⠀⠈⢻⣷⡈⣻⣿⠓⡚⠛⠛⠙⠛⢓⠚⣿⡟⢁⣾⣟⠁⠀⣹⠘⢧⠀⠀⠀⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠲⠶⣎⣀⣤⣿⡀⣠⣹⣿⠀⠈⠃⠘⠓⢀⠀⠀⣀⠚⠃⠐⠁⠀⣿⣇⡄⢀⣿⣤⣀⣱⠶⠖⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠶⣮⣷⣦⣤⡀⠙⢿⣻⣙⣿⠙⢋⡤⠀⠀⠘⣤⠉⡀⢀⠉⣤⠁⠀⠀⢤⡙⠋⣿⢋⣾⡿⠋⢀⣠⣴⣾⣥⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⠘⣿⠙⢻⣶⡀⢷⣿⠒⠁⣐⣤⣴⣒⡀⠈⠀⠀⠀⠀⠁⢀⣒⣢⣴⣂⠈⠒⣿⡾⢀⣴⡿⠋⣿⠃⣯⣶
⠀⠀⠀⠀⣀⡀⠀⠀⠘⣎⠙⠻⢶⣴⣿⣷⣼⣅⡌⠛⣥⠈⠣⣾⣩⣦⡁⠀⠀⣈⣴⣍⣷⠜⠁⣨⠟⢥⣨⣧⣼⣿⣶⠶⠟⠛⣱⠁⠀⠀⢀⣀
⠀⠀⠀⠾⠯⣍⠳⢦⣾⠁⣯⣤⣤⣷⣿⣷⣏⣼⡁⠈⠉⠓⠦⠌⠀⠘⡃⠀⠀⢸⠃⠀⠠⠴⠚⠉⠁⢈⣧⣹⣾⣯⣾⣤⣤⣽⠉⣷⡴⠞⣩⠝⠷
⠀⠀⠀⠀⠀⠈⣷⠈⡿⠛⠉⠀⣀⣨⣽⣧⣴⡏⢁⠀⠟⠁⠤⠀⠀⢸⠁⠀⠀⢈⡇⠀⢠⠤⠈⠻⠀⡈⢹⣦⣼⣯⣄⣀⡀⠉⠛⢿⠁⣾⠁
⠀⠀⠀⠀⠀⠀⣟⣿⣷⠻⣍⣷⡦⣝⣿⡑⢻⣵⣿⣥⡦⠀⡐⠒⠂⠚⡠⠒⠒⢄⠓⠐⠒⢂⠀⢴⣬⣿⣾⡟⢊⣿⣫⢵⣟⣩⠟⣾⣿⣻
⠀⠀⠀⠀⠀⠀⣟⣻⣮⣤⣈⣙⣦⣼⣿⢡⠏⠁⠸⢻⣧⣀⠁⠀⣄⠀⠀⠱⠊⠀⠀⣀⠀⠈⣀⣼⡟⠇⠈⠹⡌⣿⣧⣴⡋⣁⣤⣽⣟⣿
⠀⠀⠀⠀⠀⣸⢋⣩⣿⣿⠝⢿⡷⠋⣿⣏⣼⣶⢠⠀⠛⢛⣡⡀⠵⣆⡢⠴⠦⢔⣰⠮⢀⣄⡛⠛⠀⡄⣶⣧⣹⣿⠙⢾⡿⠫⣿⣿⣍⡙⣇
⠀⠀⠀⠀⠀⠹⣿⣷⣿⢿⡅⡟⠁⣠⠟⠛⠻⣏⣿⣶⡆⠈⢩⣷⣀⡘⠠⣤⣤⠀⢁⣀⣾⡭⠁⢰⣶⣷⣹⠟⠛⠻⣄⠈⢿⣠⡿⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⢹⣿⣾⠟⣾⣇⣆⡻⠋⠹⣷⣄⠈⣻⣿⣀⣭⠟⣩⡿⣿⣿⣿⣿⢿⣝⠿⣭⣰⣿⣟⠁⣠⣾⠏⠙⢟⣠⣸⣷⠻⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠳⡿⣶⣞⣿⣿⡍⠳⣴⠃⢹⡖⢋⣹⣿⣀⡇⣀⡷⠸⢿⡿⠇⢶⣀⢸⣀⣿⣏⡙⢲⡏⠘⣧⠞⢩⣿⣿⣳⣶⢿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡷⣽⣿⡀⣿⣀⣿⠞⣻⢹⡟⠟⣿⣍⢀⣦⠀⠀⣤⡀⣉⣿⠻⢿⡏⣟⠳⣿⣄⣿⠀⣿⣯⢿⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣼⣿⣿⢸⣥⣶⠷⣄⣿⣟⡿⣆⣇⣸⣰⢿⣻⣿⣠⠾⣦⣬⡇⣿⣿⣧⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⣙⢧⣙⠛⠛⣏⣔⣬⡳⣬⣻⣟⣥⢞⣥⣢⣹⠛⠛⣋⡼⣋⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣌⣻⡾⠯⢽⣿⣷⣼⣿⣯⣧⣾⣿⡿⠩⢿⣟⣩⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣾⠟⢿⣟⠚⠓⣻⡿⠻⣷⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣴⡋⣹⠛⢛⣏⢹⣦⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣧⣼⡿⠿⠟⠉