Preto e branco Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⣠⣾⠟⠉⣿⡇⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⣉⣴⣿⡇⠀⠀⢀⣼⠟⠁⠀⢸⣿⣃⣠⣴⣶⠿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠾⠛⠛⠁⢸⣧⠀⢠⣿⠋⠀⠀⣠⣼⡿⠛⠋⠉⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⣿⣤⡿⠁⠀⢀⣼⠟⠁⠀⠀⠀⢀⣴⣿⣿⣾⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢻⣿⠁⠀⣴⡿⠃⠀⠀⢀⣠⣴⡿⠛⠛⠛⠛⠻⣿⡇⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣶⣤⣬⣿⣞⣿⣦⣾⣿⣶⣦⣶⡾⠛⠋⠁⠀⠀⠀⠀⢀⣴⡿⠁⠀⣀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⣿⣿⡋⠀⠈⠉⠙⠿⣿⣄⠀⠀⢀⣀⣤⣶⡿⠛⠁⠀⣼⠋⠛⣆⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠰⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢻⣿⣧⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠟⠛⣁⠀⠀⠀⠀⠻⣄⣴⠇⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣶⣶⣶⣶⣶⠿⠟⠋⢀⣾⡟⠹⣧⡀⠀⠀⠀⠀⠀⢻⣷⡀⠀⡾⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⣿⣿⣦⡀⠀⠀⠀⠈⣿⣧⠀⢷⣄⣠⡿⠀⠀⠀⠀⢀⣤⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢻⣿⡟⢿⣦⣀⠀⠀⢸⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⢻⡀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣀⣴⣾⢿⡏⠀⠀⠀⠀⠀⢸⣿⠇⠀⠉⠻⣷⣤⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠞⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⣤⣤⣶⣾⠿⠛⠉⠀⣾⠁⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣸⡟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⣰⠿⣦⠀⠀⠀⠀⠀⠀⣼⠋⢹⡆⠀⠀⠀⡀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⣿⠏⣴⠶⢶⡄⠀⢿⣄⣸⡇⠀⠀⠀⠀⠀⢿⣄⣼⠇⠀⠀⣾⠛⣦⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣠⡶⣤⡀⠀⣿⡇⢀⣴⡿⠟⠁⠈⢿⣤⣼⠇⠀⠀⠉⠉⠀⢠⡶⠶⣤⠀⠀⠉⠁⠀⠀⠀⣇⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⡏⠀⠈⣷⠀⢸⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠿⣦⡼⠃⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢸⡏⠙⣷⠀⠀⠀⠀⢀⡞⠻⣆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⢸⡇⠈⣷⠀⠀⠀⠀⠀⣿⡁⢀⣿⠀⠀⢸⣏⣈⣿⠀⠀⠀⠸⣧⣤⡟⠀⠀⠀⠀⠈⢷⣤⡟⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠈⠻⠶⠟⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢷⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣴⠇⠀⣠⣤⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣷⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣦⣄⣀⠀⠀⠈⠛⠛⠛⠁⠀⣴⠟⢶⡄⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣷⣶⣤⣤⣄⣀⣀⢿⣤⣠⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣿⣿⣶⣶⣶⣶⣿⣭⣽⣷⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢹⡇⠀⠀⡄⠀⣿⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠄⠀⠀⠀⢻⣧⠀⠀⠀⠈⣿⠀⡀⠃⠀⣿⠀⢰⡟⠀⠂⢀⡄⠀⠂⠀⠀⠄⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠠⡄⠀⠀⠀⠂⠀⠀⢻⣧⣨⣥⠴⣿⠴⠅⠀⣰⡇⣢⡟⠰⢄⣴⠟⠠⠀⠀⢠⠆⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⣾⣦⠠⢀⠀⠀⣠⡴⠞⣛⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⣄⡁⠀⢠⠄⣠⡾⠋⠀⠀⡴⢀⣴⠆
⠀⠀⠀⠀⠀⠀⣤⠀⠈⠁⠀⠀⡀⠈⠻⣷⣤⡶⢛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠟⠉⠻⣟⠓⠿⠿⣿⣷⣶⣟⡉⠀⡄⢈⣽⣡⡾⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢤⠀⣍⢀⣴⢟⣥⣾⣿⣿⡿⠿⣿⠿⠿⢿⣶⣿⣿⣿⣏⠠⠒⢀⣀⠹⣆⠀⠀⠀⢩⠽⢿⣿⣷⣶⡿⠛⠁⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠠⢀⠉⠻⢶⣤⣴⣿⣿⣿⣿⡟⠹⢿⡇⣼⣯⡀⠀⣠⣿⣿⣿⣿⣿⣧⠉⣝⣀⡀⢿⡄⠀⠀⠀⢚⡭⠭⠝⠻⣿⣛⣶⣶⡾⠟⠋
⠀⠀⠀⠙⢦⡀⠀⡈⠓⣠⣾⣿⣿⣿⠟⠉⢸⡇⣸⣿⠆⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⢨⣭⡽⠅⢸⡇⠀⠀⠀⠖⢚⣩⠭⠀⠀⢉⠻⢦⣄⡀⠠
⠀⠀⠀⠀⠀⠙⢓⣿⣿⢟⣿⠿⠋⠀⠀⠀⠈⣿⠑⣶⣞⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠦⣄⡒⠀⣾⠁⠀⠀⠀⠘⣉⣠⠤⠀⣴⠋⠀⢀⡈⠉⠓⠲⠶
⠀⠀⠀⣀⣥⣴⠿⣫⡶⠛⣁⣀⣀⠀⠀⠀⠀⢻⣇⠛⡿⣻⢿⠳⢽⡛⠛⠟⠛⣉⢼⡪⣕⠲⣼⣼⡟⠀⠀⠀⠐⠋⣁⡤⣠⡞⢁⣴⣴⡟⠛⠛⠓⠲⠶
⠀⠀⣠⣭⡥⠶⠛⢱⣶⣦⣤⣀⡈⢉⡀⠀⠀⠀⠻⣦⡸⠃⡾⡰⣿⣧⣆⢷⢹⣯⡳⣽⡈⢟⣿⡟⠁⠀⠀⣀⠴⢊⣡⡾⢋⣴⣿⣿⣿⣿⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠐⠲⢿⣻⢿⣿⢿⣷⣯⣷⣤⡀⠀⠈⠻⣦⣄⡁⠋⠿⠸⡎⠃⠁⣙⣬⣴⡿⠋⠀⠀⠀⣘⣵⣾⠟⣩⣴⣿⣿⣟⡂⠉⠀⠉⠓⠾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢮⣝⡾⣿⣾⣽⣳⠦⣤⣲⣤⣍⣛⠿⠷⠾⠿⠿⠿⠟⠛⠁⠀⠀⢀⣠⡴⢟⣫⡵⢞⣿⣿⣿⠌⠻⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣭⣙⠻⢿⣶⣭⣝⣛⡻⠿⣿⣷⣶⣶⣶⣖⣚⣒⣿⣯⡽⢶⣛⣫⣥⣾⠟⠛⢿⣿⡓⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣉⡙⠛⠻⠿⣿⣷⣾⣾⣽⣯⣭⣷⣶⣶⣿⡿⣿⡟⠉⠀⠈⠀⠀⢳⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⠀⢸⠀⣧⡏⢸⡇⢹⡅⠀⠈⠇⡇⢿⣷⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠁⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡞⠀⢰⠃⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⠁⢠⡏⠀⣠⠋⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⢦⡇⢠⡏⠀⡾⠀⢰⠏⠀⣸⢡⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣩⠵⠒⠀⢸⡇⢸⡇⢰⡇⠀⣾⠀⢠⡇⠀⠠⢌⣙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠸⣿⣶⣻⣭⣤⣼⣧⣾⣧⣾⣇⣶⣿⣤⣼⣅⣀⡐⠲⢬⣝⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢀⣠⡾⠃⣽⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣌⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣀⣀⣤⡤⠴⠶⢚⣫⣵⠾⢡⣭⣽⣲⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠸⣷⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠶⠾⢛⣛⣭⡶⠟⠠⣀⠀⣽⡇⢀⡇
⠀⠀⠀⠀⠀⢀⣠⠴⢚⡩⠁⠀⠀⢻⡦⣍⡁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠶⠛⣛⣫⣵⠾⢡⣤⢶⣫⣿⢎⡼⣁
⠀⠀⠀⠀⣴⢟⣵⢞⡥⢚⡡⠄⠀⠈⢿⡦⠄⡀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠛⣛⣛⣭⣵⡶⢣⣾⡾⢟⣫⡵⠋⠀⣍⠳⣄
⠀⠀⠀⢸⡇⣾⠃⣾⠀⣟⠀⣎⠀⠀⠀⠻⣗⠀⠀⠀⠀⠀⠀⠀⠉⢉⣛⣋⣉⣭⣭⠷⡾⣣⣾⣷⣿⣿⣿⡄⢦⡙⡌⡇⢹⡄
⠀⠀⠀⠘⢧⡘⢷⣌⠳⣬⡳⢬⣑⠀⠀⠀⠙⢿⣤⣀⡀⠀⠀⠀⠠⠤⠤⠤⣶⣶⣾⣿⣿⣿⣷⢻⣧⢻⡇⣿⢸⠇⠇⢠⡞
⠀⠀⠀⠀⠈⣙⣦⣝⡳⢦⣍⣛⠒⠤⠀⠀⠀⠀⠈⠛⠿⢿⣿⣛⣛⣛⣿⣽⠿⠿⣛⣫⣵⠿⣣⡿⢃⠞⠑⠁⣊⣤⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠐⠚⠛⢓⣶⣭⣭⣓⣂⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣒⣒⣚⣛⣻⣭⠽⠶⢛⣋⣠⣥⣴⣶⣿⣿⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠼⠿⠿⠿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠯⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⣉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠥⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢠⣤⣤⣄⣈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣬⣉⡙⠻⢿⣶⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⠿⠛⠉⠁⠻⢿⣿⣿⣿⡿⠛⠃⠀⠀⠈⠉⠙⢛⣿⣿⣿⣶⣬⣙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢏⣾⣿⣋⣁⣐⣒⡤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣥⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣦⠉
⠀⠀⠀⠀⠀⠀⠸⣿⣿⢷⡸⣿⣆⠻⡌⠹⣿⣿⣷⣄⠀⣰⣿⣿⣿⡆⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣟⢷⣍⠙⠿⢦⡃⠀⠙⠘⠻⣿⣧⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣟⣳⣕⠀⠀⠀⠑⢄⠐⢄⠀⢹⣿⣇⠘⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣶⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡮⡳⢄⠀⠠⡀⠀⠈⠳⣄⢿⣿⣆⠈⠙⠛⠋⠀⠀⠀⠀⠙⠻⠿⠇⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣮⡀⠁⢀⠈⠓⢷⣮⣎⠻⣿⣿⣦⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡳⣦⣀⠀⢤⣕⣻⠛⠿⣾⣿⣿⣿⣦⣀⠙⠲⠤⣤⣤⡄⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠳⠶⣬⣿⣟⣛⣛⣿⣿⣿⣿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣤⣄⣀⠀⡀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢻⣿⣻⡟⠀⠀⡟⡇⠀⡞⡀⣿⣷⣶⡄⠀⠀⠶⠻⠿⢿⣷⣍⢦⣿⣿⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠘⢠⠀⣿⠇⠟⠀⠀⢸⠁⡆⠀⠃⡏⢻⣿⣿⣛⠒⠐⠚⣛⣿⣿⣿⣿⣇⣿⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡟⠀⠀⠀⠀⢀⠀⡇⠀⠀⣷⠸⣿⣿⡛⠙⠀⢀⠀⠤⠶⢾⣿⣿⣿⠏⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣿⠃⠀⠀⠀⠀⠘⠀⠇⠀⠀⢹⣆⢻⣿⣾⣷⣤⣲⣿⣿⣿⣿⡿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠻⣿⡛⠛⠛⠿⢛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡘⢿⣄⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢿⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⢻⣆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⣧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⢹⣆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣿⣆⠱⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢄⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀⣠⣤⣤⣹⡄⠹⣿⣷⣌⠀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣿⠈⢸⡧⠶⠚⠛⠛⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠛⢷⡀⠙⢿⣿⣿⡄⠀⠀⠀⠀⢀⣼⣿⡿⠃⠀⣼⣥⡤⣤⣄⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠒⠛⠛⠛⠛⢷⣦⣄⣉⣛⣁⡀⢀⣀⡤⠤⠭⣭⣤⣀⡰⠋⠀⠀⠀⠚⠛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣀⡀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠴⠒
⠀⠀⢧⠀⠈⢦⣳⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣒⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣘⡁⠠
⠀⠀⠈⠑⠦⢄⠳⣅⢧⣀⠀⠀⠀⠀⠀⡠⢒⣭⢆⡼⣯⠬⣛⢆⡰⣍⡒⢄⠀⠀⠀⠀⠰⣿⡏⠈⣋⠕⣢⣄
⠀⠀⠀⠀⠤⣒⢬⡿⡀⠿⠇⠀⠀⠀⠘⠊⠯⠔⠉⠀⠹⣿⡏⠀⠙⠢⠽⠑⠃⠀⠀⠀⠀⣩⠁⢹⠀⢻⠺⠿⠂
⠀⠀⠀⠨⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈

⠀⢰⠂⠄⡀⠀⠀⣀⠀⠀⢀⠄⠐⠄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⢢⠤⠬⣴⣾⣛⣶⡬⠤⠤⠊⠀⠀⠀⠀⠁⠀⠀⣺⡭⣿⠀⠀⠈⠀⠀⠀⠀⠯⣀⡩⠥⢰⣾⣻⣷⠆⠭⢁⣈⠕
⠀⠀⠀⠁⠁⠨⢩⣭⠹⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢠⣿⣒⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠡⢫⣭⡙⡙⠉⠉
⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠐⠦⣄⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤
⠀⠀⢤⣀⣈⠓⠜⠻⠔⠋⣀⣀⠄⠀⠀⠀⢀⡤⠒⠢⠴⢹⡤⠴⠢⣄⠀⠀⠀⠀⠠⡖⠒⠢⠄⣀⠏⣆⡠⠔⠒⢲
⠀⠀⠀⠀⠈⠉⣦⣧⡈⠉⠀⠀⠀⠀⠀⠀⠘⠧⣀⡤⣖⣋⣀⢀⣀⠜⠀⠀⠀⠀⠀⠉⠲⠆⠤⠲⠍⠷⠤⠰⠖⠉
⠀⠀⠀⠀⠀⠀⢒⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡍
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠈⠑⢮⣱⣴⣩⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠲⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢱⣤⠊⢀⣀
⠀⠀⠀⠉⢓⡶⡿⢿⡷⣖⠋⠁⠀⠀⠀⠀⠀⠀⡠⠖⣏⣉⣱⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉⣀⣴⡶⠶⢶⣦⣀⠉
⠀⠀⠀⠈⠁⠀⢛⣛⠃⠀⠉⠁⠀⠀⠀⠀⢠⠊⢀⡤⢷⠒⡾⢄⡀⠱⡄⠀⠀⠀⠀⠀⠟⠛⠁⢛⣛⠓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣲⠾⠛⠉⠉⠉⠉⠙⠳⢶⡒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠉⠙⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠯⠤⠶⠶⠒⠛⠁⠀⣀⣀⣄⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠲⠶⠿⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⠖⠛⠉⠁⣀⣠⠇⠀⠀⠀⡏⠉⡇⠀⠀⠀⠀⠀⠀⢀⡴⠲⣄⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠛⠉⠁⠀⠀⠀⠀⢠⡇⠀⡇⠀⠀⠀⢀⡤⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⣴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢯⣀⡴⠚⠁⠀⠀⢀⡶⠲⡄⠀⠀⣿⠘⢧
⠀⠀⠀⠀⠀⡼⠁⢸⠇⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡄⠀⢸⡀⠈⢧
⠀⠀⠀⠀⣸⠁⠀⡾⠀⠀⠀⠀⠐⣇⣀⣀⣤⣤⣤⠼⠃⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡄⠀⠳⣄⠈⣇
⠀⠀⠀⢠⠇⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉⠉⠙⠒⠛⠉⠉⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠏⠀⠀⠈⠳⣼⡄
⠀⠀⠀⣾⡴⠋⠀⢰⡋⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡟⠁⠀⠀⠀⢳⡀⠘⣆⠀⠀⠀⠀⠀⣼⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠃⠀⠀⠀⠀⠀⣀⠴⠛⢲⠀⠀⠀⠸⡇
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠹⡄⠈⣧⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡠⠞⠁⣠⠴⠋⠀⠀⠀⢠⠇
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠸⣄⡤⠞⠁⠀⠀⠀⠀⢠⢾
⠀⠀⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢸
⠀⠀⠀⣧⠘⣆⠀⠀⠀⠀⢀⡜⠁⣸⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡼⠀⠀⠀⠀⠀⠀⣠⠶⣄⢰⠃⠀⣾
⠀⠀⠀⢸⠀⠘⣇⠀⠀⢀⡞⠀⣴⠃⠀⠀⠀⠸⣌⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠋⠀⠀⠀⠀⠀⠀⣰⠃⢀⡏⢸⡄⠀⡇
⠀⠀⠀⠈⡇⠀⢸⠀⠀⣏⣀⡼⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣀⡀⠀⠀⠀⠀⢠⠏⢀⡞⠀⠀⣧⣸⠁
⠀⠀⠀⠀⢹⡀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⣀⣀⣠⣤⣄⣀⣠⠴⢚⡏⠀⠀⢿⡀⠙⢦⣄⠀⠀⠸⣄⡼⠁⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠁⠀⠀⠀⠉⠑⠒⠋⠀⠀⠀⠀⠙⠢⣄⠈⠳⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠈⢿⡀⠀⣠⠤⠤⠤⠶⠶⠶⡄⠀⠀⠀⠀⢀⣠⠖⢲⡀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳⠀⠀⠀⠀⠈⠳⠶⠃⠀⠀⢀⣠⠴⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠳⣤⠤⠤⠤⠤⠴⠋⠀⠀⢀⡴⠋⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⡶⠦⠤⣤⣀⡀⠈⠉⠀⠀⠀⠀⡞⠙⠲⢤⣀⠀⠀⠀⠻⠤⠟⠀⠀⠀⠀⠀⠀⠀⣸⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠉⠓⢦⣀⠀⠀⠀⠙⠳⢤⣀⠈⠙⢦⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣄⣈⠓⢦⣀⠀⠀⠀⠈⠙⠓⠋⢀⡴⠋⠁⢀⣀⡬⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠤⠤⣤⡤⠶⠿⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢟⡁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⡆⠀⠀⠀⢸⡟⣽⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⢧⢙⢧⡀⠀⠀⣾⣷⣇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠤⠀⢀⠀⠀⢉⣤⡤⡗⡷⣶⠁⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⢃⢳⡙⢦⡀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⡿⠻⢤⠱⣇⣾⡿⢶⡛⠛⠒⠦⣄⠈⢧⡀⠀⠀⠸⡎⣧⠈⣱⢿⠇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⢿⣙⡳⠾⠿⢛⡶⣞⡛⠛⠲⢤⡈⠳⢴⣿⣦⣀⡀⠳⡸⣾⡿⣜⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⡿⣿⣿⣶⣤⣾⡏⠻⣄⠀⠀⠀⠙⣆⠠⠭⠽⠋⠁⠀⠑⢾⣵⣇⡆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⣾⠋⣰⠏⠉⣹⣿⠨⠀⠀⠈⢳⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡙⢻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠃⠀⡿⢠⠀⠀⡟⢆⠀⠀⠀⠀⢹⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⣧⠉⠀⠀⢳⠀⠑⠤⡀⠀⠀⢻⡄⢸⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⠀⠀⢿⠀⠀⠀⠘⡆⠀⠀⠈⠑⢼⣾⣷⠿⣦⡀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢀⠘⡄⠀⠸⡆⠀⠀⠀⣧⠀⠀⠀⠀⠈⣿⡟⠀⠙⣿⡄⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣫⡤⠿⡄⠀⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢹⡄⢸⡀⠀⢀⠞⢒⣓⠚⠒⢻⠆⠀⠀⠀⠀⠀⢻⣆⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠀⠀⣷⢸⡇⢠⡾⠃⣡⠤⠤⢤⡞⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⣿⣿⣷⣿⢳⣟⡁⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⢻⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⢈⡿⡛⣥⣎⣹⣧⠶⠶⠶⠦⠤⣄⡈⠳⡶⠤⣄⣀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⢀⣼⠆⣽⣿⡿⣾⣏⠀⠀⠀⠀⠀⣀⣨⣷⣿⡆⠀⠈⠙⠲⢤⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣿⠟⣿⣿⡏⠛⠯⣉⡉⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⠀⠀⠀⠁⠀⢿⠈⢷⡄⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢧⠀⠉⠢⣄⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠑⢄⠀⠀⠉⠒⠦⣀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠈⢒⠰⡦⣤⣠⡉⠲⣦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣢⠃⠀⠈⠙⠳⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠄⠀⠠⠀⠄⠀⠄⠄⠤⠄⠀⠤⠀⠀⠠⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠤⠤⠶⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠋⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣆
⠀⠀⠀⠀⠀⣰⠇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣼⠣⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠈⡇
⠀⠀⠀⠀⡿⠀⣶⢿⡄⢠⣴⠶⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠦⠤⠠⠀⢠⡟⣠⡆⢠
⠀⠀⠀⠀⣧⣴⣯⡾⠁⠀⢀⣠⣤⣀⣠⣤⣄⡀⣤⣤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⠀⠀⠄⠀⢿⣏⠀⣿
⠀⠀⠀⠀⣇⣀⡟⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠂⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡄⠐⡈⢳⡀⠻
⠀⠀⠀⠀⣿⣿⠃⢇⣾⣿⣿⣿⢟⡿⡿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣿⡿⢿⠿⣿⡿⣿⣿⣿⣿⣄⣵⠟⡇
⠀⠀⠀⠀⢹⡿⢘⢰⣿⣿⢷⣶⠖⠀⠐⢨⣥⣤⣭⣿⠀⠀⠀⠀⠀⠀⠀⠛⠑⣾⣿⣧⣴⡎⠀⠈⠓⢤⣿⢿⣿⡇⠘⠀⣿⡇
⠀⠀⠀⠀⢈⡇⡆⢸⣧⡟⢆⢻⡄⠀⢀⡀⠈⠻⣿⡇⠀⠀⣠⣤⣄⣴⣤⠀⠀⢹⣿⣿⣿⡇⠀⠠⠐⠀⢸⣿⠻⣷⠀⠀⢹⡇
⠀⠀⠀⠀⣼⢠⢳⡘⢿⣮⣤⠈⢳⣬⣠⣈⣠⡤⠞⠁⠀⣼⣿⣿⣿⣿⣿⣦⡀⡀⠻⢯⣤⣤⣄⡶⢶⣆⢸⣧⣾⠋⠀⠀⠘⣧
⠀⠀⠀⢰⣇⣾⠁⠀⠀⠉⠙⠛⠛⠉⠋⠉⠀⠀⡴⠀⢸⣿⣿⣿⣿⢿⣿⣿⡟⢠⠂⠤⠀⠈⠉⠙⢛⠛⠛⠛⠁⣠⠂⠀⠀⣸⠃
⠀⠀⠀⢸⡟⣇⠀⠀⠀⡜⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣼⣿⣿⡋⣿⡟⢿⣿⣧⡀⠃⠀⠀⠐⣿⠑⠀⠈⠛⢢⡀⢸⡇⠀⠀⣸
⠀⠀⠀⠀⠹⣎⠣⣶⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣟⣙⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⢻⣄⣴⡟
⠀⠀⠀⠀⠀⠈⠳⢾⣧⣤⣤⣄⠐⠿⣶⣤⡀⢠⣤⠈⠛⢿⣿⣿⡇⠻⢿⡿⠟⠽⢠⠀⠀⣠⣶⣶⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆⠈⢿⣿⢋⡷⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠈⠀⢸⠋⢁⣠⣷⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⠉⣿⣿⣿⣦⢦⣿⢮⡇⡠⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡄⣾⣿⣿⣿⣿⣟⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠸⣿⣿⢿⣾⣿⣾⣧⣇⢀⣧⣀⣇⣀⣻⣀⣸⣤⣠⣇⣤⣀⣀⣹⠟⠛⡿⠙⠫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⢿⢿⣆⠉⡿⣽⢌⠀⡿⠀⢿⠁⠈⡿⠀⢹⠃⠸⡇⢸⠏⣿⡇⢀⣼⡇⣶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡻⠈⠙⢷⣷⣤⣾⠀⡇⢀⣾⢀⣰⣇⠀⣨⣀⠀⣇⢘⣷⣿⡥⢾⡽⠡⡇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠈⠸⠙⠛⣿⠻⡿⢿⡟⢿⠿⡿⢻⡿⢿⡿⠏⣿⠀⠀⠀⡀⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠀⠳⡤⢻⣀⣻⡨⣿⣸⣁⠇⣸⣧⣹⣧⡙⠀⠀⠀⠀⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⣴⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⣀⣁⣀⣀⡀⠀⠀⠀⠀⠀⠘⣁⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠯⠭⣶⠶⠶⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠖⢋⠥⠔⠲⣾⣿⠿⠛⢛⡛⠻⢶⣿⣶⡖⠐⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⢉⡉⠁⠀⠶⠃⠀⡤⠀⢀⣀⣤⠂⠀⣀⡀⠚⣿⣭⠀⣀⣰⠛⠻⠓⠄
⠀⠀⠀⠀⠀⠀⢀⠴⠞⢁⡴⠚⢉⣉⠙⢿⣀⡀⠀⠸⠷⠂⠀⣨⢹⣄⠉⠉⠀⢐⣿⠶⠟⠋⠉⠀⠀⢈⣿⣿⡶⠤⡀
⠀⠀⠀⠀⠀⣠⡖⣠⠞⢛⠅⢀⡼⠋⠀⠀⣩⡿⠓⠶⠦⣴⠛⢁⡼⠛⣧⡀⣼⠋⢠⣤⠶⠞⠛⣿⡛⠛⠁⢀⣉⣶⣾⣆
⠀⠀⠀⠀⠰⠟⣰⠟⠁⡾⠃⣾⠁⢠⠖⣋⣥⣞⠁⠀⠀⡈⠀⣿⠀⠴⠈⡱⠏⠀⣼⠇⢀⣈⠳⡞⣷⠀⢰⡟⢙⠟⢻⡉⢢
⠀⠀⠀⠀⣾⠔⢻⡀⠊⠀⢠⠋⠀⢼⠀⠁⠀⣈⣀⡆⠀⢉⣶⠏⠀⠀⣰⠇⠀⠀⣿⣄⣾⠋⠀⢀⣿⡀⣸⠁⢸⡆⠈⠻⣦⣧
⠀⠀⠀⠀⠙⠀⠠⣑⣒⣀⣁⢀⠀⠸⢷⣤⣄⡀⠈⢷⠀⠘⡏⠀⣄⠀⠁⢀⣿⣄⣸⣿⠙⠛⠿⢿⣿⡷⠿⢃⡞⠁⢀⣀⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠠⢈⡁⡈⠉⠉⢉⣠⣄⣈⣉⢀⣀⣾⣷⠶⠒⠖⡛⠿⠿⠋⠁⠈⠙⣿⣤⣤⣄⣀⡉⣀⣴⡏⠀⡾⢻⣛⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢉⠉⠀⢦⣤⣸⠟⠁⠀⣀⣠⠤⣀⣀⠀⢀⡄⠀⢰⣏⣀⣀⣽⣿⡿⠉⠹⣧⣤⣴⠀⣿⣿⣿⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⣴⠶⠞⠋⠀⠀⠀⠈⠙⣿⡋⠛⠻⠿⠛⠀⠉⢉⣁⣤⣼⣿⣏⡵⠶⢿⣿⣿⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⡇⠀⠀⠀⣀⣄⠀⠐⡄⠀⠻⣦⠀⠀⢰⣴⣾⠉⢉⣩⠍⠻⢯⡀⠀⠾⠟⡻⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣹⣷⠈⢆⡀⠸⢅⣀⢸⣧⡤⠾⠛⢀⣻⠛⠉⠠⢷⡀⠷⣤⣄⡠⢷⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣀⡤⢖⣶⣾⣷⣦⣤⣤⣠⣝⣢⣀⣀⡸⣿⠀⠀⣀⠉⣀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣧⣤⣿⣿⣿⣛⣛⣻⠿⣿⣽⣏⣻⣾⣿⣿⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⣿⣿⣿⣿⡿⠿⢿⣿⠛⢿⣿⣛⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠙⠻⢿⣿⣟⣶⣿⣿⡿⠋





⣿⢿⠿⠿⢿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⢿⣄⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣾⡏⠀⢀⠄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠤⠀⠘⣿⣷⣾⣿⠉⠁⠈⠉⢹⣿⣿⣿⣀⠀⠀⢀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠢⡀⠀⣠⣾⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣄⠀⡀⠬⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠘⡂⢁⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣾⣿⡿⣿⣿⠛⣿⡇⠀⠀⠀⠀⠀⣤⡟⢻⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣙⢿⣾⡿⢿⠀⠀⠀⠀⢸⠿⣿⣿⢿⣰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠚⠉⠠⡄⢠⡤⠈⠙⠛⠓⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠉⡋⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠉⠀⠀⠀⣠⡤⠔⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠁⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠓⠃⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⠦⢤⣀⠀⠀⠀⢀⣠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠈⠉⠳⢶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠦⠤⣄⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣾⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⡽⠉⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣀⡀⠀⣀⣀⣼⠿⠋⠀⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣽⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣯⣭⣼⣷⣷⣬⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣿⣿⣷⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⡀⠀⠀⠀⡠⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⣀⣀⢤⡤⠤⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢮⡻⣆⡠⣫⠮⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠻⣦⠀⢀⣀⣀⣀⣠
⠀⠀⠀⠠⡆⢀⡎⢣⠀⢸⠀⣀⠀⠀⠀⠀⠀⣀⣤⢶⣶⣿⣿⣿⣛⣭⣉⡐⣦⡀⠀⠀⠀⣀⡸⠖⠤⡘⣄⠀⠘⡷⠋⠉⠉⢁⠎⠀⠘⢱⡤⠤⠄
⠀⠀⠀⠀⠇⣞⣠⣄⠳⣌⠰⣁⣉⠉⡭⠽⠀⠙⠗⢉⢔⠞⡿⢯⡙⣞⠀⠈⠃⠀⢀⠔⣉⠤⠒⢢⠞⠀⠀⠉⠇⢰⠒⠒⠒⠁
⠀⠀⠀⢸⣾⣿⠋⣼⣷⡽⡆⠈⠁⠀⠀⠀⠀⠀⠀⣧⣢⠊⠉⠁⢱⠂⠳⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣀⣀⡸⠀⢸⡀⣀⠀⠀⠀⠠⢤
⠀⠀⠀⢹⢜⢿⣸⣿⡿⠁⡇⢀⡄⡸⢷⠿⣟⣀⠀⠁⠀⠀⠀⢀⠴⣏⡒⢫⠈⢓⣠⡀⠀⠀⢠⠂⡎⡏⢯⡇⣀⣨⠼⡹⢱⢄⠀⠀⠐⠃⠈⠂
⠀⠀⠀⠘⡌⠳⡀⡸⠁⢠⠃⢨⣿⣘⢼⡿⡿⣏⠀⠀⠀⡀⡰⢋⠥⠓⠄⢛⣔⣐⣶⣘⡤⢄⡀⠀⠉⠚⠻⠿⠽⠿⠻⠒⠁⠀⠃
⠀⠀⠀⠀⡇⠀⠈⠁⠀⢸⠀⢀⡙⢿⣿⡿⣿⠿⠁⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠐⠢⠔⠂⠐⠒⠀⠀⣀⣀⠀⠒⠈⠓⣊⡓⢉⣀⡀⡀⠀⠉
⠀⠀⠀⠐⠓⠒⠒⠒⠒⠚⠀⠸⣷⣴⣯⣄⢀⠠⠐⢀⣁⣠⣤⠤⠤⠤⠤⠤⠤⣤⣄⣀⡀⠐⠠⠀⠀⠑⠋⠀⣀⣈⣉⢀⡈⣁⡀⠁⠀⢄
⠀⠀⠀⠀⢷⣶⡗⠀⠀⠀⠀⣰⠟⠉⠉⢀⡥⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⠀⢀⠜⠁⠙⠲⠶⠖⠋⠙⢆⠀⠺⡀
⠀⠀⠀⠀⠈⠉⠀⢀⢀⠀⠰⠃⠀⠀⠀⠀⣠⣤⣤⡀⠀⣶⠶⣦⠀⢀⣶⣆⠀⠶⢶⡆⠰⠆⢸⡇⠀⢀⠎⠀⢦⠀⠀⠖⣤⠀⢰⠀⢣
⠀⠀⠀⠀⠚⠐⠂⠛⠻⠺⠰⣀⠀⠀⠀⠀⠙⣿⣾⣇⠀⣿⣴⠟⠀⣼⣧⣿⠀⢠⡾⠁⢸⡇⢸⡇⠀⠈⠛⠲⢾⡆⠀⠈⠉⠁⣿⠴⠛
⠀⠀⠀⠀⠀⠀⢰⢣⠀⢀⡀⠀⠀⠀⠀⠠⣴⣿⣨⡿⠀⣿⠻⣦⠀⠿⠀⠸⠃⠻⠿⠂⠘⠃⠸⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿
⠀⠀⠰⠀⠀⢴⠀⡿⠀⢸⡷⠀⠀⠀⡤⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣼⠀⠰⡶⡲
⠀⠀⠀⠀⠀⣸⣁⣀⣀⣘⣁⡀⠀⠀⠀⠀⠀⠐⣟⠳⣆⢠⠟⣆⢰⠛⡆⢹⠛⣆⣾⠀⢸⣏⣁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠿⢶⣿⣿⣿⣿⠯⠿⡾⢷⡆⠀⡀⢀⣇⣠⠇⢻⡤⠏⠻⠾⢁⣼⡴⠃⠿⠤⠸⠧⠴⢀⡀⠀⠀⠀⠀⢸⠉⠤⠤⠦⠶⠠⠴⠆
⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠐⣲⠷⠛⠁⠐⢝⠢⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⡫⠂⠀⢰⢶⡀⠀⠀⠀⠀⡀
⠀⠀⡔⠉⠈⠳⢄⣀⣀⠤⠚⠁⠉⢢⠀⠀⠀⠁⠠⠍⡓⠲⠤⠤⢤⣤⣤⡤⠤⠴⠒⡛⠩⠀⠀⠀⠀⠀⠛⠘⠁⠀⠀⠀⣀⠛⠁
⠀⠀⠘⠢⢄⣀⣀⣀⣀⣒⣃⡠⠔⠋⠀⠾⠿⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⢀⡉⠀⠀⢰⠤⣤
⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⣀⣀⠀⠀⣠⠤⡄⠀⠀⠀⣀⣤⣴⠾⠕⣊⣡⣤⡶⠿⠃⠀⡶⠂⣠⢊⠬⠿⣿⠿⠯⣈⠑⣄⠙⣆⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠉⢁⣤⡄⠀⠙⠊⠀⠀⠈⠉⠀⠰⠃⢀⣛⣃⣀⣀⣘⣛⠁⠀⠀⠀⠀⣸⠀⣸⣿⡇⠀⠀⣣⡀⠀⠀⣿⣿⣆⠻⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⠷⠒⠒⠒⠒⠒⠚⢻⣦⠀⡀⠀⠋⠀⠿⠟⠉⠐⣶⣿⣿⣶⠊⠙⢻⢿⠀⠀⠀⠡⠟⠁
⠀⠀⠀⣴⠖⠂⣠⢀⡄⠘⡟⢠⣤⠀⢳⡴⠂⢹⠪⣽⣒⣒⣒⣒⣲⢮⡟⢫⠀⠁⠀⠀⠀⢃⣴⣤⡠⠽⠿⠿⢇⠀⢀⡇⡘
⠀⠀⠀⣿⠒⢢⠿⠾⠁⣰⠇⠈⠛⠀⠀⣇⠀⢸⣠⠋⡆⢀⠞⢧⢀⡞⠱⣾⠀⠀⠀⠀⠀⠈⠻⣿⠗⠀⠤⠄⣸⣿⣿⠞⠴⠄⠀⣀⠞⢆⣀
⠀⠀⠀⠈⠙⠉⠀⠀⠛⠁⠀⠠⠀⠀⠄⠁⠀⣿⣧⣤⣼⣮⣤⣤⣿⣤⣤⣾⠇⢸⠒⠂⠦⠠⠤⠌⠑⠒⠀⠀⠚⠋⠁⠀⠀⠀⠀⢱⡤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⢶⣦⣤⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣿⢣⣴⣾⣷⣾⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⢟⣭⣿⡇⣿⠸⣿⣍⢽⣭⣽⣭⣽⢯⣝⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡸⣿⡟⢷⣿⣷⡙⣿⣷⡙⣿⣤⣝⢷⡙⣷⡼⣿⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢣⡹⣷⣌⠻⣿⣿⣌⢻⣿⡬⣟⠿⡎⣷⢹⣷⢹⣿⣎⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣷⣿⣿⣷⣦⣙⠿⣷⣽⣿⣮⡻⣶⣯⣽⣿⡌⢻⣿⢸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣌⡛⠿⣿⣿⣿⣿⣷⢮⣍⡉⢿⣿⣦⡙⢿⣋⣴⣿⡿⢸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡘⣿⣷⣦⣍⠛⢿⣿⣷⣮⣝⠻⢿⣿⣿⣦⡻⣿⣿⢇⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠞⠻⣶⣬⣿⣿⣧⠸⣿⣿⣿⣿⣷⣦⣍⡻⢿⣿⣶⣦⣭⣝⣻⣝⣣⣾⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⠿⠿⠿⠛⠛⢋⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣃⣤⣴⣶⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠙⢿⣿⢃⣿⣿⣿⣿⣿⣿⣿⠟⣩⣴⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⡦⠉⢼⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣍⡛⠿⢿⡟⢡⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠛⣉⣥⣶⣶⡶⠀⣾⣶⠀⢶⣶⣶⣄⡀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣄⠀⠀⠀⠀⣼⠃⠀⠀⢻⣿⣿⠟⠀⠀⣿⠇⠀⠀⠉⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣄⠀⠀⠉⠀⠀⠀⢸⣿⠃⠀⢀⡼⠃⠀⢰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠸⣿⣷⣶⣤⡀⠀⠈⢿⡁⠀⠀⠀⠀⢈⣿⠿⠛⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣇⠀⠀⢻⣿⣿⣿⣿⣦⠀⢸⡟⠃⠀⣀⡴⠟⠁⢠⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⢿⣟⠁⠀⠀⠀⢸⣿⡆⠀⠈⢿⣿⣿⣿⣿⠀⣼⣡⣴⠾⠛⠓⠒⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠹⣦⡀⠀⠀⣠⣿⡇⠀⠀⠀⢹⡿⠟⣿⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣈⣷⡄⠀⠘⣿⡇⠀⠀⠀⣼⣃⡾⢋
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠻⣦⡀⢿⡇⠀⠀⢀⣿⡿⠟⠁
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⢷⣼⡇⠀⢠⡾⠋
⠒⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣰⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⢀⣠⡴⠞⠁⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⣴⡾⠋⢁⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣼⠏⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⢿⣦⡀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠙⣷⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⣠⡾⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣴⠟⠁⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡴⠋⣁⣠⣤⣶⣶⣾⣿⣿⣟⣛⣛⣛⣿⣛⣛⣛⣛⣛⡛⠲⢶⣤⣄⣀⠐⠶⣄⠙⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⡿⠀⣿⠉⠙⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠉⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⢀⣿⠶⢶⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠇⠀⠀⢹⡆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⡿⠀⠀⢠⡿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⢉⣾⠁⢀⣴⠟⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⠷⠞⠋⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⣄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⢠⡿⠋⣽⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⢿⣍
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢰⡏⠀⣼⢃⣼⠷⠿⠒⠓⣲⣦⡄⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣿⠀⣼⢻⣿⡴⠶⠞⠛⣛⣩⣿⣧⣤⡤⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠹⣾⣧⣴⣿⣟⠛⠛⠋⠉⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠮⣭⣄⣙⣛⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⡀⣿⣄⣤⣶⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿⣤⣭⣽⣋⣩⣽⣟⣻⡛⢿⣿⣷⣦⣄⡀⠀⢠⠈⠀⠀⠀⣀⣀⠀⡀
⠀⠀⠀⠀⣷⡝⢿⣿⣟⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣿⣿⣿⡶⢃⣼⣠⡴⠟⠀⠀⠀⢌⣳⡀
⠀⠀⠀⠀⠿⡇⠐⠨⠝⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣡⡶⣿⣿⣿⣷⠋⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣛⣛⡛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠽⠆⠄⠖⢿⣿⣿⠍⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢁⣿⠋⠉⠀⠀⠀⠀⠠⠀⠘
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⣹⠏⠀⠀⠀⠀⣠⠔⢀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣏⣀⣀⡤⠖⢋⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⠿⠋⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡶⠟⠋⠑⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠟⠛⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡤⠃⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣀⡠⠤⠤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠔⠋⠀⠀⠀⣠⠶⠦⡌⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⡾⠃⠀⠀⠀⠀⠀⡏⠀⣄⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣾⠃⢀⣴⠶⠶⣄⠀⠹⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⢸⢣⡎⠳⠈⡇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⣧⠘⣎⠳⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠻⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠠⠃⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣶⠋⠉⢀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠂⠘⠋⠀⠀⣀⣀⣀⣀⡤⠤⠤⠴⠒⢋⣠⠾⢻⣷⣒⡓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠚⠉⠀⠀⣸⠋⠉⠙⡇⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠐⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⡄⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⠴⠟⠁⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⣀⡈⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⣭⣉⣉⣁⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⣀⣤⡥⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠤⠖⠂⠀⠀⢀⠀⠀⠀⠀⠀⢀⣞⣡⠄⢻⣿⡣⡁⠀⠘⣿⣂⠀⠀⠀⠐⠂⠘
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠒⠀⠀⠰⢆⡀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠰⠔⡅⠞⠀⠀⠀⢠⠏⠁
⠀⠀⠀⠀⠀⢠⠀⠡⠀⠀⠀⠊⠀⠀⢳⡄⠀⠀⠀⠀⠀⢩⢠⠈⠤⠓⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠐⠁⠂⠄⠁⠀⡞⡶⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠐⣄⣀⣆⣀⠔⠀⠀⠀⡀
⠀⠀⠀⠀⠈⡀⠀⠐⡀⠀⠂⠀⡀⠠⠁⠖⠂⠀⠀⠰⢤⣄⡬⢐⠴⠚⠘⠀⠄⠀⠁⠹⠁⣄
⠀⠀⠀⠀⡀⠁⢀⠀⠁⠐⠀⡀⠀⠀⠁⠀⠀⠀⢛⠄⡐⢨⢉⠁⠀⠂⠀⠃⠀⡀⠁⠀⠁⢀
⠀⠀⠀⠀⠠⠂⢀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠁⠀⢛⠈⠀⠂⠨⠀⠀⠁⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⡉⠉⠀⠠⠀⠠⠀⠈⠀⠃⠠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠃⠘⠀⡄⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠐⠊⡚⠂⠠⠔⡄⠀⠀⠀⣀
⠀⠀⠀⠁⠀⠐⠒⠀⠛⠋⠋⠙⠀⠁⠀⠀⠀⠀⠀⢀⠰⣶⡆⢀⠀⠀⠉⠉⠲⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⠀⠀⠀⠗⠀⠰⠊⠆⠀⠠⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⡿⡄⠶⣀⣟⠦⠒⠄⣀⠃⡁⠆⠀⠀⢀⠀⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢀⡀⢾⠀⠘⠢⡡⢙⠘⠈⢀⠠⠠⠘⠁⢶⠘⠛⢿⡗⡄⢠⢾⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡁⠀⠐⡀⠀⠑⡈⠀⠀⠀⠠⠆⠀⠐⢀⢐⣀⢐⢇⡷⠀⣌⠘⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢘⠀⡀⡀⢠⠁⠀⠀⠀⠀⠀⠀⢋⠀⠔⡀⠂⠚⠆⠃⠄⣅⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠉⠀⠀⠘⠈⣀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⢤⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠓⠢⠀⠐⡄⠐⠀⢄⣃⡡⢆⡈⡀⢀⡠⠂⠀⠀⠀⢶⡾⠁
⠀⠀⠀⠀⠀⡟⢉⢴⠂⠡⠀⠃⣰⠀⠀⠈⠑⠛⠭⠽⠁⠀⠀⠀⠁⠁⠙⢋⠉⠀⠀⠀⠀⠀⠰⠈⣀⡶⢣⠋⠈⢁
⠀⠀⠀⠀⠈⠀⠀⠨⠀⠆⢐⠔⠋⡱⢆⠑⠚⢫⡡⡱⡀⠀⠀⠀⠈⡐⠀⠂⠀⠀⠀⠀⡀⢠⣾⡆⠀⠂⠘⠠⠄⡄⠆
⠀⠀⠀⠀⠀⢀⢠⣬⢐⡆⣀⡀⠀⠀⠡⠀⢸⡜⠇⣕⠀⠀⠀⠀⠀⠁⠠⠈⠈⠀⠀⠚⡀⠩⢈⠓⠀⡈⠃⠍⢀⡁
⠀⠀⠀⠀⠀⠉⠀⠑⠺⠛⡟⢁⢄⡤⣿⡷⠖⡚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠊⠂⠡⠁⢁⡐⠋
⠀⠀⠀⠀⠄⠀⢂⠁⠀⠀⠉⢉⠋⡅⢠⢀⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠄⠄⣠⠤⠊⠁
⠀⠀⠀⠀⠆⠀⠀⡀⢈⢀⠠⢀⠐⠀⡀⡄⠌⡀⠄⠀⠀⠀⠀⠀⠀⠀⢂⠄⢀⣄⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⡂⠆⠀⠠⠠⠀⠄⠂⠀⠀⠂⠕⠂⠀⠀⠀⠀⠀⠄⠂⠀⢙⠻⠯
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠈⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢐⠿⢷⡀
⠀⠀⠀⠠⠔⠚⠁⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢤⣕⣦⠾⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⡀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢤⠀⡀⠀⠔⠉⠩⠐⠀⡁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡈⠀⠀⠢⠀⠉
⠀⠀⠀⠀⠀⠠⢠⣰⣤⣉⣤⣀⠀⠀⢀⢁⡈⠀⡂⠅⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠆⠘⡈⠩⠁⠀⠀⣄⠁⠄⢁⠐⠀⢄⠃⠀⠀⠀⠀⠀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠐⠀⡂⠃⠀⡄⡁⠀⠀⠁⠀⠀⠂⠈⡁⠃⠀⠀⠀⠀⠀⠑⠀⠉⠉⠀⠀⠖⠘⠲⣶⣦
⠀⠀⠀⠀⠀⠀⡈⢀⠁⡈⢀⢀⠁⠀⠀⠀⠀⠀⠀⠂⡁⠀⠀⠀⠀⠀⠀⡀⠂⠄⠀⠀⠀⠂⠀⠞⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢂⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠐⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢷⡄⠀⠀⠰⠛⠐
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠈⠓⠈
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣷⣤⣿⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠂⠀⠀⠄⠀⢀⠀⡔⢲⠲⣶⢲⡖⣲⢰⡂⢸⠃⡶⡶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠰⠶⠖⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠑⠚⠚⠃⠀⠀⠘⠚⠒⠘⠀⠉⠁⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡁
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⡄⠀⠀⠀⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠁⠒⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣒⣾⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠆
⠀⠀⢀⠀⣀⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠘⠛⠛⠃⠀⣀⠘⢂⡀
⠀⠀⠀⠀⠤⠤⠴⡆⠀⠀⠀⠀⢀⣄⣄⣤⡤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⣀⣀⠀⣀⣀
⠀⠀⠀⢰⣶⣶⡆⠃⠀⠀⠀⠀⣀⣠⣤⣤⡀⠸⡄⠀⠀⠀⢰⣶⠀⠈⠛⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⢠⣀⣀⡄⠀⠀⠀⠛⠛⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠀⠉⠀⠸⠿⠀⠀⠀⠀⠸⠿⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⡀⢠⠄⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠑⣼⠀⠁⠾⢟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠁⠁⠐⣿⠖⠂⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⣀⣀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠈⠈⠛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠄⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢦⠖⡀⠊⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠛⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣈⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠁⠂⠀⠋⠐⠄⡀⠀⠀⠋⢀⡠⠐⠈⠐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠀⠀⠀⠁⠀⢀⡀⠄⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠒⠀⠠⢄⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠸
⠀⠀⠀⠀⠀⢂⠀⠀⠐⣄⡔⠀⠀⠀⠀⠐⠌⠳⣄⡐⠀⡀
⠀⠀⠀⠀⠀⠛⠉⠁⠁⠙⠳⠃⠀⠀⠀⠀⢀⠴⠛⠫⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠤⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠴⣻⠃⢀⣠⡤⠶⠚⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠙⡗⠀⠀⣠⠞⠁⣰⣧⠶⠋⠁⠀⠀⣀⣼⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⢰⠃⠀⣠⠟⣡⠄⢀⣠⠴⠛⠉⠀⠀⠈⠉⠛⢳⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣷⠶⣾⣷⢻⣠⠏⠀⡴⢃⡞⢁⣴⠿⠶⠶⠒⠒⠒⠶⠶⠶⣲⠿⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣹⣎⢷⡀⣸⢡⡞⢀⣤⠖⠒⠂⢀⣀⣀⣠⣤⡶⠟⠉⠀⠙⠁⠀⢻⡆
⠀⠀⠀⠀⠀⣠⡴⠖⠛⣋⣁⣹⣆⠻⣄⢾⡶⠋⣠⠴⠖⠛⠉⠉⠉⣤⠀⠀⢰⣄⠀⠀⢠⣄⠀⢻⡄
⠀⠀⠀⢠⡞⠁⠠⠚⠋⣉⣁⡤⣨⠟⠋⠊⠀⠀⢹⠀⢠⡄⠀⠀⠀⠙⠃⠀⠀⠉⠀⠀⠈⠟⠀⢸⣧
⠀⠀⢀⠏⢀⣠⣴⡞⢋⡿⢋⡜⠁⢠⡇⠀⠀⠀⣿⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⢀⣾⡾⠋⢡⡟⢠⡟⠀⠎⠀⣰⠛⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣤⡀⠀⠸⡇⠀⣿
⢠⠞⠋⠀⠀⣾⠀⣾⠁⠀⢀⡼⠁⠀⠹⣦⡀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢠⡏⠀⢻⡀⠀⣾⠁⠀⢰⣆⠈⠛⣦⡈⡇⠀⠀⣶⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡀⠀⢿⡇
⠀⠀⠀⠀⢸⡇⠀⠈⣇⢸⠇⠀⠀⠀⠉⠀⠀⠈⠻⣿⠀⠀⠈⠁⠀⠀⠀⢸⣷⠀⠀⠀⠀⠸⡷⠀⠸⣷
⠀⠀⠀⠀⢸⣧⢲⡄⢸⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣿⡄⠃⠀⡇⠀⢠⡀⠀⠀⠘⠃⠀⠀⠀⢀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⢹⣇⢸
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⢠⣀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⡀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣉⡀⠀⠀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠹⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣶⣤⣀⡀⠀⠀⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣦⣄⡀⠀⠀⠀⠀⠻⣦⠀⠀⠰⣦⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠀⠀⣀⣤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠄⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⢄⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠦⣍⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⣸⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠔⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠉⠀⠀⢀⢠⠃
⠀⠀⠀⠈⢇⠀⣈⣉⠐⠠⣀⢮⠢⡐⠄⠀⠀⠀⠀⠀⠙⠓⠲⢦⡴⢒⡵⠟⠁⠀⠀⠀⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠑⠄⡀⠁⠲⣈⣧⠦⠬⠱⡾⣽⡴⡦⡄⣀⣰⣯⣥⣔⣉⡀⡀⠠⢀⡂⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣉⣗⣾⡿⣿⠬⡯⢑⢔⡉⣀⠛⣧⣶⡖⡖⠒⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠤⠠⡿⠿⠹⣞⣛⢯⡉⣯⣽⣿⣘⢣⢻⣯⢏⣿⣾⣤⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠜⠀⢘⢐⣝⡭⠨⣽⢛⠖⢚⣿⣓⠾⠏⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠂⠈⠑⠋⠉⠒⠃⠘⠯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠛⠃⠚⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢐⣾⡯⣹⢟⣛⡿⣭⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠍⠐⡖⡠⠤⢄⠰⡒⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣭⣹⣿⣛⣿⣫⣽⣿⡇⢰⣾⣟⣿⣷⣯⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠂⠚⠪⠀⠠⠀⠱⠒⠀⠘⡟⠉⠀⠄⠈⠑⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣛⣿⣿⢻⣿⡟⢻⣿⣿⡇⢸⣷⣽⡿⣿⣿⣽⡦⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠰⠀⠀⣤⠀⠀⡄⠀⠰⠀⡇⠈⠃⢀⠈⢂⠆⢝⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⢿⣿⡿⣿⣿⢻⣿⢡⢿⣯⣿⡿⢿⣾⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠄⠀⡄⠀⢀⡀⠀⢠⠀⣸⢁⠀⠂⠀⡀⠉⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣻⣿⢿⣿⣟⣿⡿⠃⣿⣿⣿⣿⣿⢿⣷⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠠⠀⢀⠀⢠⠀⢀⣴⡃⠀⠀⠀⠀⢀⠀⣁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠿⠋⠀⠀⠹⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⣀⣴⣾⣿⣷⡀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⡄⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⠃⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⠏⣸⠇⠀⢰⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⡞⢈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣾⠁⠀⡼⠁⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⠏⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠖⠚⢻⡀⠈⠁⣿⠀⠈⠀⠀⠀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣋⣥⣤⠴⠶⠶⠒⠈⠻⠦⡘⠙⢧⡘⠛⠳⠶⠶⠤⢤⣀⠉⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣀⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡁
⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠶⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⢸⣿⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⠀⠀⠀⠀⠀⣹
⠀⠈⢿⣦⡀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⠿⣶⣄⣈⠙⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣤⣬⣉⣉⡙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠁⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⢢⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡒⡄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣊⣸⣿⣾⣶⡟⠀⠀⠀⠀⢀⣀⣽⣾⣾⣟⣀⡀⠀⠀⠀⡤⢻⣿⣿⣾⣞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢉⣽⠟⣿⠻⡉⢣⠀⠀⠈⣿⡋⠝⣿⣟⡋⠉⢢⠀⠀⢸⠈⠛⣿⢿⠛⠗⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠋⠀⠁⠀⠈⡇⠀⠸⡄⠀⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠙⣄⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠑⠦⠤⡠⠴⠋⠀⠀⠀⠀⠀⠈⠲⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⢀⠄⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⢐⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢀⣇⣠⡀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣿⣤⡴⠀⠀⠀⠀⠀⠀⠀⢼⣾⣷⡤⣻⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣷⣾⣿⠋⠉⢧⡀⠀⠀⢠⠎⠀⠀⢮⠀⠙⠇⠱⡄⠀⠀⠀⠀⠒⠚⠻⢿⣿⣿⣷⣧⣄
⠀⠀⠀⠀⠀⠀⠈⢁⣾⠙⣿⠈⠺⠇⠀⠀⢳⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡬⠁⠀⠀⠀⠀⠈⠻⡿⡿⠥
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠋⠀⠀⠀⠀⠀⠈⡇⠀⠈⠳⣄⠀⠀⠀⠀⢀⠜⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⡆⠀⠀⡜⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⣠⠖⠚⠁⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠣⣀⣀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⠀⣀⡀⣠⣤⢠⣤⠤⠤⠤⣤⢤⣤⣤⣌⣀⠀⠀⠀⠀⢰⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂⠉⠀⠄⠁⢠⡤⠀⠀⠀⠀⠀⡶⠿⠿⠟⠿⣿⣿⣿⣶⣧⣄⡈⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠦⠀⠀⠀⠀⢠⠀⠰⣖⣀⣠⣤⣤⡶⠂⠀⢀⠀⠉⠉⠙⠃⣭⠉⠻⣿⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢦⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢁⠀⠁⢙⠂⠔⠚⢛⣻⣦⣀⠀⠙⢀⣀⣿⣿⠇⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡙⠒⠤⣀⣀⠀⠀⠀⠛⠁⠈⣉⠑⢀⡀⠀⠀⠶⠿⠿⡶⢶⡯⠟⣁⡴⢳⣾⠃⢀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢶⣦⣮⣍⠁⠀⠀⠀⠀⠠⠶⠿⠷⠤⠰⠿⠒⢛⣍⣡⠀⠀⣿⣿⣰⣿⡟⠉⠉⠻⣖⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⣰⢶⠗⠀⠈⠉⠛⡀⠘⠃⣿⢿⣿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣼⣏⣀⣠⣴⠿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣅⣉⣉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠊⠙⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣟⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠠⠃⠀⠀⠀⠀⠀⠴⣻⣷⣵⣆⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣰⠽⣿⡿⣋⡬⣿⠀⠈⠀⡈⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⡁⢹⣧⣠⣹⣿⣆⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⢀⣩⠾⣿⡾⣥⣾⣷⠤⡀⠀⠐⠀⠘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⣹⣭⣿⡋⠥⣺⠏⣻⣷⣷⣦⣄⣀⡀⠀⣀⠀⡂⢀⣉⣀⣴⣾⣿⣱⢊⣽⣓⣺⣿⣾⡷⠏⠀⠁⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢛⣡⠀⢀⠉⠉⠛⠻⠿⠿⢿⣿⣿⠿⣿⡿⠿⠿⣿⣻⢍⣫⡙⣌⠛⠉⠀⠀⠀⠀⢀⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣥⠬⠋⠀⣴⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠋⠛⠉⠀⠀⢀⣥⢴⢊⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣀⣀⣀⣀⠠⣄⣀⠀⡀⠤⣶⢎⢃⠈⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠂⠀⠂⠁⠁⠀⠀⠀⠘⠓⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⣄⣆⠀⠀⠀⠀⠀⠐⠀⢀⡖⢶⡟⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣈⠙⠒⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡏⢠⢇⠁⡿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣋⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠯⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣳⡀⣀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠀⠀⠀⠀⢀⡄⢸⣧⣽⣯⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠻⠿⠿⠿⠿⠟⠻⠿⢿⣿⣛⣛⣫⣵⣾⡿⠟⠒⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⠇⠂⢀⣤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠃⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⢁⢿⣾⣿⠿⢛⣭⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠙⢃⡴⢣⣤⣤⣬⣁⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣨⠿⣻⣠⣹⣷⣻⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢨⡇⠸⣿⣿⣿⣿⠿⠓⢲⣤⣴⣶⣿⣿⣿⢭⣾⣷⣿⣿⣿⣿⡿⠛⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢳⣂⠁⠄⠙⢿⣿⠯⠀⠀⠀⣿⣿⠿⠿⠷⠟⠛⠛⠛⠛⠛⣋⠉⠁⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡪⣤⡄⠀⠙⠿⠷⢾⣿⣏⣙⠛⠲⠶⣤⣄⣀⠀⠒⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣄⣀⣀⠀⠀⠀⠉⠈⠃⠮⣍⡛⠛⠂⠠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡿⡟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣾⣧⣧⣷⣼⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠸⡄⢂⠀⠀⢀⣤⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⢀⠀⣠⠤⠢⠤⢤⣤⣀⠀⠀⡀⠀⠤⢐⡄⣷⠴⠂⡀⠩⠍⠅⣬⠵⠉⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢸⠗⢊⠀⢀⡟⢀⡀⠀⢤⣕⡾⡤⢂⠀⠂⠳⠞⠀⠀⠉⠉⠉⠛⣍⠀⢲⡬⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡰⠀⠊⠀⢀⠴⠳⠏⠀⠁⠠⣠⠜⠈⠀⢀⣤⡆⢀⠀⡀⠉⠀⢳⡔⠲⠤⠐⣦⢄⢺⡆⢢
⠀⠀⠀⠐⣁⣀⣠⣀⣉⣐⣂⠠⢬⣀⠀⠀⠀⠀⠻⢄⠚⠀⠀⢵⠂⢳⠒⠒⠊⠁⡘⣀⠴⠊⠑⢄⢀⡜⠠⣧⠀⠀⣠⠠⡁⣲⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠋⠓⣬⠆⠀⠀⠀⠀⠀⠙⣆⠀⠁⣵⡇⢰⠇⠀⠀⠉⠁⠰⡆⠀⣀⡈⠀⠀⠂⠸⠎⢈⢌⢄⣣⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⢀⠀⠀⢀⡈⢛⣛⠙⠒⠛⠀⠀⠨⡇⠀⠀⠀⠲⡅⢠⣄⠀⠀⢉⡃⠀⣤⠀⠠⡆⠑⡡⡁⢠⣮⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡐⣀⠀⠀⠀⢸⡋⠑⠀⠀⠒⠷⠆⢣⠀⢶⠀⠀⠀⠀⠉⢀⠀⠸⠂⠀⠈⡴⠀⠠⠪⡺⡶⠀⡛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⠀⢁⠏⠦⣀⠀⢸⠁⠠⣦⠀⠀⠀⠀⢨⣆⠈⠀⠀⠸⠄⠀⣘⠧⠀⣸⣦⠀⠉⢤⣮⢢⣪⣤⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡞⠜⠆⠨⢢⡾⠲⠀⠀⠀⠘⠷⠀⠈⠉⠦⡀⠀⡤⠀⠀⢿⠀⠀⢀⣉⠀⣠⣈⣉⡨⣋⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢷⡆⠲⣤⠀⠀⠀⠲⡦⠀⠀⠀⠀⣤⡜⠛⠢⣈⠁⢀⡄⢀⣄⠈⠹⠀⢉⣉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢄⣤⣀⣠⣀⢀⠀⠀⣤⣀⠉⠀⠀⡻⠎⡱⢪⣅⠀⠙⠀⠀⢠⡄⡝⣨⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⢉⣀⣤⣀⠈⠉⠂⠀⠚⠷⢀⣪⠽⠝⠋⠑⠒⠤⢀⣂⠨⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠐⠠⢀⠀⠀⠄⡀⠔⠀⠀⠁⡈⢀⠈⠂⢀⠀⠀⠄⡈⠐⠀⠀⠅⡈⢐⠠⢀⠀⡂⢀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠁⠂⠐⠀⠀⠁⠈⠐⠈⠐⠉⠀⠠⠈⠐⠀⠀⠁⠀⠀⠀⠊⠀⠀⠐⠀⠈⠐⠈⠀⠁⠂⠀⠀⠀⠀⠀⠈⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⡿⣷⣄⠀⠈⣿⣿⣦⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣜⣿⣿⣿⣧⠀⢸⣟⠹⡇⠀⠀⢠⣶⣿⣥⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣉⠳⡄⡇⢀⡇⣠⣶⣿⠏⡽⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣤⣶⣤⣀⣀⣈⡻⣌⣻⣿⠿⢿⠷⠧⣤⣿⣿⣯⣄⠄⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⢿⡿⠋⠙⣛⢻⡟⣿⡽⢻⣿⣿⢚⣼⠟⠻⠦⣾⠿⠭⣬⣹⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⢀⡄⢟⣾⣿⡌⣄⢠⣶⠺⣿⣧⣼⠿⣴⡟⣶⣾⣇⢠⣶⣉⣽⠤⡈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡦⠀⡼⢣⠗⣿⣿⠏⠿⣿⠼⣿⣧⣿⡏⢿⣶⣿⣿⣿⣯⢹⡇⣍⣻⣧⢤⣝⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠃⡀⠙⢻⣶⡼⠉⡸⣿⣿⣶⣿⣏⢿⣿⣿⣿⢿⣿⣸⣿⣿⣇⡙⢋⢻⡟⣏⢻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢁⡘⠻⢸⢧⡝⣿⣞⠟⣆⣿⣶⣾⣿⣿⣿⠿⣿⣾⣿⣿⣿⢿⣏⣿⣘⣿⣯⣿⡜⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⠀⢸⣇⢀⣌⠻⢃⡟⢯⡛⡿⡛⢿⣿⠿⣿⣿⣿⣿⣿⣿⠛⣿⣼⣿⣽⣿⣱⣿⣏⣻⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣄⢱⣦⡸⣇⣈⡓⢾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣻⣷⣽⣿⣛⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⢻⢦⣝⠋⢘⠧⠿⢰⢦⢘⣿⣶⣌⣿⣿⣡⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢺⡻⠇⣿⠵⡶⢹⢟⣤⣘⢻⣿⣯⣹⣿⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⠿⣆⢈⡛⢿⠄⡈⢷⣽⣿⠿⢿⣿⣿⣗⣎⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡿⠆⠁⠻⣿⠆⣤⣿⡷⢹⣿⣿⣯⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡜⠼⣦⢶⡦⡍⠛⢿⣟⣿⡟⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢾⣏⣿⣇⠸⣿⢿⡽⣘⣿⣿⠛⣿⣿⣬⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⠹⣿⣙⡾⠎⠉⣴⣯⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠿⣝⠆⡻⠷⢮⣿⣯⣭⢿⣿⣿⣿⣿⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡸⠦⠻⢆⠻⣧⡹⣿⠿⢿⣿⣭⣿⣿⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡘⢟⣿⢿⣧⣼⣻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣟⠻⣽⣿⣝⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⢾⡷⠶⢶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠠⢶⡾⣷⠦⠀⠀⠀⠈⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣤⣤⣴⡖⠀⠀⠀⠉⠀⠀⠁⠀⠀⠐⣦⣤⣤⣤⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢰⡏⠀⢀⣠⣾⠀⠀⠀⢸⣦⡀⠀⠈⣷⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⠶⠛⠁⠸⣧⡀⣠⡿⠀⠙⠳⠾⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠘⠆⠀⠀⠰⠇⠀⣈⠙⢉⡀⠀⠷⠀⠀⠀⠾⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡄⠀⠀⠀⠋⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠉⠀⠀⣀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢠⡀⠀⠉⠀⢠⠀⠈⠁⠀⣠⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⠁⠀⢠⠀⠈⠀⢠⡀⠀⠉⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠈⠀⢀⠀⠈⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢶⣄⠀⢠⡟⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣷⣿⠁⠀⠀⣿⣴⠞⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⣀⣘⣿⠀⠀⣸⠟⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠋⠉⠉⠛⠛⠛⠛⠷⠶⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡄⠀⠀⢀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠁⠀⠀⠀⠀⠀⢰⡆⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣤⣄⣼⣿⣿⣿⣆⣠⣴⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣦⣀⠀⢠⣿⣧⠀⢀⣠⡆⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣬⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣍⡛⠛⠿⠿⠿⠿⠿⠟⠛⣋⣩⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡟⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣴⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣾⣿⣿⣯⣼⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣧⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⠏⢻⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⡏⢻⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣴⣿⣿⣿⠛⣿⣿⣿⣤⣿⣿⣿⣿⣾⣿⣿⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣟⣸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣷⣼⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣷⣼⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠹⣿⣿⣿⣾⣿⣿⡿⠛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣇⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢨⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠙⠋⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡱⡦
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⡶⠶⣾⣿⠎⠂⡂
⠀⠀⡄⠂⢈⠉⣴⣶⠀⣠⣷⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⠒⠒⠀⡠⠉
⠀⠚⠀⣀⡿⣠⡀⠘⢛⠛⣿⣾⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⣀⣀⣰⢆
⠀⢳⣼⣿⣡⣿⠁⠀⣿⢸⠃⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡶⠛⠉⠉⢀⡼⠁
⠀⠀⠙⠛⠿⢧⠀⡶⢉⣾⠁⢸⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣰⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠶⣤⣬⣥⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠐⠒⠀⠀⠂⠀⠄⠀⠰⣿⡷⠐⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢸⡇⣬⣥⣿⣖⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣿⣧⣠⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣶⣀⣠⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣾⣿⣿⣿⠿⠿⠿⠛⣋⣥⣾⣿⣿⣿⡷⠶⠶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠂⠂⠀⠀⠀⣤⣤⣤⣤⣿⣦⣶⣶⣶⣿⣿⣿⣿⡿⢻⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣻⣿⣿⡿⣿⣿⣿⣻⣿⠗⠊⠀⠐⢀⣠⣴⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢿⡿⠋⢸⡟⣠⣤⣶⣾⣿⣿⡿⠟⡧
⠀⠀⠀⠀⢀⣤⣶⠞⠋⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣹⡗⣿⡖⢀⡽⢹⣯⣿⡿⠛⠉⡁⠤⢲⣯⠓⢠⡀
⠀⠀⠀⣰⡟⠻⠧⠂⢤⡴⠃⣀⣠⣀⣀⣘⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⡟⠃⢀⣵⣿⡿⡿⠀⢚⠻⣶⣼⠟⣷⠘⡳⣿⡀
⠀⠀⢰⣿⠃⢀⠀⢫⣜⠁⢰⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠰⠀⢀⢺⣿⣿⣿⣿⣿⣿⠺⣫⡿⢯⣤⣴⣿⣶⢻⡎⣿⣿⣸⣷⣾⢿⣮⣿
⠀⠀⢾⣿⣧⠀⡀⢸⡇⠀⠸⣿⣿⣿⣿⣿⣿⣯⠻⡿⠁⠙⢷⣤⣄⣀⣗⣿⣿⣿⣿⣿⣿⣣⢬⣽⣿⣼⣿⣿⣿⣻⣾⣿⣿⣿⡿⠋⠉⠁⠀⠀⣿
⠀⠀⠀⠻⣷⣄⣀⠛⡻⠀⠀⠻⣜⢻⣟⠛⢿⡿⠆⠀⠀⠀⠀⢻⣻⣟⣽⣿⣟⣿⣟⣛⣩⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⣈⠊⣉⣄⠁⠐⠂⠀⠈⠻⣿⣿⡷⠶⠂⠀⠀⠀⠀⢸⣿⣿⡛⠻⠟⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⠳⣝⠻⣯⣘⡀⢀⠀⠀⠈⠻⣿⣿⣶⣶⣶⣾⣿⣷⠈⠫⠝⠻⠶⣤⣤⣄⣀⠀⠀⠀⠀⣀⠤⠄⠚⠋⠁⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣍⣛⠛⡁⠀⠀⠀⠀⠈⠉⠛⠙⠛⠛⠛⠛⠶⠦⣄⠂⢤⡉⠛⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠘⢶⣄⡉⠛⠗⠫⠉⠀⠀⣴⡶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠶⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠷⠶⠦⠤⠤⠤⠼






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣷⣿⢛⠻⡏⣺⠀⡾⢛⢻⠟⡛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣼⣯⣷⣼⣵⣧⣿⣬⣷⣌⣿⣦⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⣇⣀⣠⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢲⡄⠀⠀⠀⠀⠀⡎⠹⠇⣀⣀⠀⠀⠀⠀⣠⣞⣤⣼⣷⣶⣿⣿⣷⣼⡿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠡⡖⠛⠈⠀⣾⠟⠃⣸⠆⠝⠀⠀⠀⠀⢙⣻⣽⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⠀⠀⢄⠀⠉⠂⠀⠋⡤⠒⠀⠀⠠⣴⣼⣿⣾⣿⣿⣿⡻⣿⣯⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⢷⣽⣱⣿⢟⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠦⢷⣾⣿⣿⣿⣏⡿⠏⣻⣿⣻⣿⡇⢇
⠀⠀⠀⠀⠀⠀⠀⢠⢖⣡⣒⣲⣶⣥⣒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠾⠟⠁⠃⣿⡢⠐⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠿⠾⠗⠏⣀⢱⣕⣄⣤⣄⠠⠤⠄⠐⢒⠒⠂⠀⢐⣂⢴⣦⢤⣤⡀⠀⠀⠀⢸⡇⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡈⡀⠴⣶⣶⣯⣭⣭⠅⠀⢀⣀⠀⠄⠀⠠⠀⠀⣠⣾⣿⠉⠁⢀⠀⠘⠇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⢴⣶⣾⣍⡉⠻⠛⠿⢷⣾⣥⠀⣂⡄⣴⠿⣯⣯⣏⠀⢀⣾⣿⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠾⡝⡟⠃⠘⠐⠒⠂⠀⢽⡇⡿⡿⠁⠨⣼⣿⠿⠀⢸⣿⣿⠀⠀⢠⢶⣶⣾⣴⣦⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣸⣼⠀⠠⢴⢻⠊⣥⡤⠭⠖⠀⠒⠏⣿⡗⠀⢀⣸⣿⡏⠀⠀⠘⡆⠄⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠲⣯⣹⣿⢩⣥⣼⠽⣮⡀⠀⠀⠀⠀⠀⡇⣾⣶⠆⠀⢙⣾⡇⠀⠀⠀⡁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⣿⠤⠄⠠⡇⠀⠙⢩⣶⣤⣀⠖⡆⣽⠿⣤⠄⠀⢤⡆⠀⠀⠀⣇⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡿⠀⠀⠀⣥⠄⠠⠠⢝⡻⢷⣶⣶⠏⠉⠀⠀⠀⠀⢿⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣀⣠⣤⣴⣛⣔⠙⣻⡇⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡟⠁⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⡀⣶⡶⠷⠀⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⣾⣯⠄⠠⢤⣿
⠀⠀⠀⠀⠀⠀⠀⢃⠨⣈⢀⡁⠀⠀⠘⠀⠠⠀⠀⠀⠀⠠⠄⠉⠀⢰⣏⡀⢁⠀⠀⢀⣸⠁⢲⣧⡤⠀⠀⠀⢀⡶⠦⠐⢻⣿⣷⣶⣿⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠻⣿⣦⣰⡀⢢⠀⠀⠁⢄⡐⠃⠀⠀⣧⣾⣿⠆⠀⠆⠀⣾⠇⣴⠋⠀⠀⠀⠀⣠⠟⠀⠀⠀⠠⠈⠫⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠒⠲⢽⣿⣿⣤⣷⡄⢄⠀⠃⠀⠺⣷⣿⡟⢡⠞⠁⠰⣼⣯⠺⡅⠀⠀⢀⣠⠞⠁⠀⠀⡀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢈⣿⣿⣿⣿⣾⡆⢀⢦⠀⣽⣿⠖⢁⡔⡀⢠⣿⡿⡳⡔⡶⠚⣉⢀⡀⣒⢠⡵⠀⡤⠀⣀⡤⠁⠈⠀⠲⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠈⠀⠈⠻⢿⣿⣿⣷⠀⢿⣧⣿⡏⣴⠋⢰⣗⣿⣻⡟⢛⢀⣹⣿⣿⢿⣿⣥⣾⣷⣴⣿⠟⣉⣴⣞⣡⣶⣤⡂⠔⠁
⠀⠀⠀⠀⠀⠀⣠⢴⣴⣶⣿⣄⠀⠙⠓⠒⠦⢬⣿⣿⣆⣦⣻⣿⣾⠃⣰⢋⣿⡛⢿⣇⣠⣽⣿⣟⣥⣸⡿⢟⣿⣿⣿⣷⣾⣿⣿⣿⣿⣛⠛⡲
⠀⠀⠀⠀⠀⢣⠠⠌⣛⠿⢿⣿⣷⣄⠈⢁⣀⣀⣉⣻⣿⣮⣿⣿⣿⣟⣵⣿⡟⢳⣿⣯⣏⣉⣽⠷⣮⣵⣾⣿⣯⣥⣿⣿⣿⣿⣿⣷⣿⡏⠈
⠀⠀⠀⠀⠀⠘⡥⠂⠈⠛⢿⣿⣿⣿⣷⣦⣤⣤⣿⡿⠿⠿⢿⣿⣿⠿⠿⠋⢁⡟⢉⠈⣉⠉⠀⠐⠂⠁⡈⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⢀⣀⣻⣿⣿⡿⣿⡿⣹⢟⣾⡛⢳⣶⡤⠀⠀⠀⢸⠀⠈⠀⠁⠀⠘⠀⢠⠀⠁⣿⣿⡏⣭⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣬⣬⣭⣵⣿⣿⣿⣼⡇⣿⢸⣷⣿⣷⣿⣿⡶⢶⡆⣾⡀⠆⠰⠀⠃⠀⠄⠀⠰⠀⣿⣿⠷⠈⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⠷⠽⠿⢷⣼⣿⣿⣿⣷⢟⡞⠉⠀⢰⠏⡷⡈⢀⠰⠀⠶⠀⠒⣀⣾⣿⣿⣼⣷⣿⣶⣫⣴⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠛⠛⡟⠙⣩⡟⡟⠟⢿⡗⠂⠀⡿⢠⣝⡻⣮⣄⠒⠀⠂⣠⣿⣿⢿⡿⠋⣿⠓⠸⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠆⠀⠴⠀⠉⠛⠉⠠⠾⢦⣼⣽⡟⢸⡇⣏⢽⣿⣛⢿⢶⣥⣾⣿⡿⠛⠋⠉⠲⠃⠀⠰⡀⠀⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠴⠀⢠⠀⠀⠆⠀⠰⠀⢠⣿⡿⠀⠾⣦⣴⣿⣿⣷⢶⣾⣯⣞⠋⣡⠄⠃⠘⠂⠀⠓⠀⡀⠐⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠄⢠⠀⢀⡄⢠⠀⢀⣼⣿⠿⢒⡿⠛⣋⣿⡛⠉⣹⣟⠛⠳⣶⣽⡄⢒⠀⣄⠀⢤⠀⠈⠀⢀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⡀⠀⠀⠰⠂⣠⣿⠶⢼⠓⠒⢫⠁⠀⠈⡒⠋⠈⢙⣷⣾⣏⠙⢮⡁⠄⣀⠀⣀⠘⠆⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣥⣦⣿⣴⣾⠟⠋⠄⡞⠂⢠⠀⠀⡆⠀⠁⠀⠰⠄⠈⣿⣿⠀⠀⠉⠓⠴⣦⣤⣠⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠀⠀⡰⢠⣧⡤⠈⠀⠀⠀⠰⠄⠀⡆⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡟⠘⣇⠙⠀⠚⠀⠀⡄⠀⠀⢠⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡞⠀⠀⠘⣆⣈⠂⠰⠆⠀⠀⠆⠈⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡞⠀⠀⠀⠀⠈⠻⡀⠰⠀⠛⠀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣴⠟⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⣤⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠠⠘⠂⠘⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡠⠢⢁⢠⠖⢀⠋⠀⡠⠊⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣁⠰⠃⢈⠀⡨⠂⠄⠀⠀⡰⠳⠄⠒⠒⠒⠐⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠚⠾⢤⡀⠨⡞⠠⠈⢹⠁⢈⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠁⠀⠀⡀⠀⠀⠐⠈⠑⢗⠔⠈⠉⠺⠀⠀⠀⠐⠆⠈⠈⠁⠉⠀⠂⢀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠖⠀⠀⠁⡀⠀⣀⠴⠦⠚⢦⠀⢠⣱⣦⡞⠂⣠⠓⡒⠀⠀⢀⢀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠁⠰⡂⠲⣿⡿⢇⠂⠀⢨⠄⠼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠐⠀⠀⠀⠂⠀⠄⠀⢠⠐⠧⠰⢠⡀⡀⠉⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠂⠀⠉⠆⠒⠀⠀⢠⠊⠀⡀⠀⠉⠈⠄⢀⡀⠔⠄⡄⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠘⠃⡀⢀⡐⠀⠙⠀⠀⠀⠀⡀⠀⢠⠀⠈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠀⠈⠀⠀⠰⡆⠀⠁⠀⠀⠀⠘⣷⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⢠⠄⠐⡃⢀⠀⠴⡀⠐⠀⠀⠈⠀⠘⡄⠀⠄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡃⠐⠂⠛⠁⠊⠁⠀⠈⠄⠀⠂⠀⠀⠀⠘⠄⠠⠈⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠵⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠆⠉⠈⡈⠉⣁⠉⠀⠀⠀⠀⣼⣿⣿⠿⣽⣶⣶⣶⣮⣉⠙⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠴⠀⠀⠀⠀⢀⠀⢲⣂⣀⣤⣤⣶⠂⠀⡀⠀⠁⠘⠛⣩⠉⠿⣿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⣸⠃⠀⠀⠀⠀⠀⢈⡀⠁⣙⡂⠐⠛⢿⢷⣄⡀⠈⠃⣬⣿⡿⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢌⣙⠀⠤⣄⣀⠀⠈⠉⠀⠩⠁⣠⡀⠀⢐⠛⢛⣛⡻⠞⠋⣤⡞⢾⣏⣀⣄⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣶⣶⠄⠀⠀⠀⡀⠐⠓⠛⠓⠂⣙⡋⠩⣷⣴⡀⢠⡿⣧⣿⡟⠁⠀⠙⣯⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠁⠀⠀⠉⠄⠈⢸⠏⣿⠇⠀⠀⣠⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠶⠶⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠷⠟⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡴⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⣶⣿⡿⢟⣳⡄⠈⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⣿⣉⣻⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⠀⣛⡾⣿⣟⣩⣿⣥⡀⠀⠱⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣯⣽⡟⠉⣻⠏⣿⣾⣧⣤⣀⣀⠀⣀⢀⠀⣀⣁⣠⣶⣿⣫⢊⡽⣒⣿⣿⣷⠞⠀⠀⠀⢘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣥⡀⣌⠉⢙⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣟⢍⣯⣱⣝⠋⠁⠀⠀⠀⢀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠰⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠉⢁⠀⢀⡮⢶⡋⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠤⠦⠤⠤⠤⣄⣢⣤⣄⣤⡐⠿⠯⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠰⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣊⣉⠭⡤⠤⢤⠤⠬⢭⣙⡒⠦⣀⠀⠻⠷⣀⣀
⠀⠀⠀⠀⠀⡰⣡⣶⣋⣭⣥⣴⣶⣶⣶⣶⣧⣔⠀⠈⠑⠪⡻⣄⠀⠀⠈⠑⣄⣀⣠⠄⠀⠤⠄⠠⢄
⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣺⠗⠀⠀⠀⡹⣏⣹⠦⠤⠚⡁⠀⣀⡴⠂⠈⠀⠠⠊
⡴⢊⡭⠭⣙⣿⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢀⣠⠤⣚⣹⡟⠋⢀⠄⣸⣷⣚⡉⠉⠀⠀⣠⣆
⣇⢸⡀⢀⡼⣿⣿⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⣍⣴⣿⣿⣿⢧⣤⡷⢞⣉⠁⠀⠀⠀⣴⣉⣠⣄⣙⡢⢄
⠘⢦⡙⠯⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⠀⣌⢧⡀⠭⠀⢠⢶⡉⠻⠀⠉⠈⠉⠂⢘
⠀⣼⠙⡷⠦⣽⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⣌⠙⣿⣻⣿⣿⠇⢀⡿⣧⢿⣯⣭⣓⢦⣉⣀⣠⠤⠤⠶⠋⠉
⠀⢿⠠⠂⠀⠀⠙⢦⡀⣠⣄⣀⣄⣠⣀⣦⣿⣼⣿⣿⣿⣿⣶⣄⣠⡿⣻⣧⣩⣿⠟⠁
⠀⠘⢆⠠⠀⠀⠀⠀⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠤⠆⣱⣿⣍⣧⣽⣀⡀
⠀⠀⠈⠓⢤⡀⠀⢀⡀⠀⠀⠈⠙⠛⠿⠟⠿⠿⠟⠛⠉⠉⢀⣠⣾⣿⠛⢽⣿⣭⣿⢿⡆
⠀⠀⠀⠀⠀⠈⠓⠲⠤⣍⣁⣀⣀⣀⢀⣀⣀⣀⣠⣤⣶⣞⣩⣿⣿⣿⢀⡘⠿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠉⠉⠉⠈⠿⣯⠭⠶⣿⢿⣝⢧⣀⣾⣯⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣼⠏⠳⠬⡅⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢁⠈⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠐⠻⣾⣷⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣴⣌⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⡝⠉⡉⠁⠀⡀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠹⠟⢃⣀⣀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⠀⠀⠘⢀⠄⣒⠉⠉⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⡤⠀⠈⠀⠐⠄⢀⠀⠄⠘⢀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠂⠀⠀⠀⠀⠂⠀⡄⢀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈⠀⡄⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡂⢐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠃⡄
⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⣄⠀⣠⣠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢄
⠀⠀⠀⠀⠀⢀⡤⠁⠀⠈⢉⠽⠿⠯⢡⣶⣲⡔⠛⠱⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⡀
⠀⠀⠀⠀⢀⠀⡀⠀⠄⢠⠞⠅⠀⠄⠁⠞⠁⠀⠐⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈
⠀⠀⠀⠀⠚⠀⡄⠀⠉⠏⣀⠠⠀⠀⠀⠀⡄⠀⠀⠘⠀⠁⠒⠀⠀⠀⢀⡀⢀⡀⠀⢀⠀⠀⠁⠀⠀⢀⠁
⠀⠀⠀⢀⠀⠀⠀⠁⠐⢣⠀⠀⠘⢀⠀⠁⢀⠀⠂⠀⠁⠘⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠖⠋⠀⠒⠀⣀
⠀⠀⠀⠘⠀⠀⠐⠀⠂⠸⠅⠘⠀⠀⠀⠀⠈⠀⠐⠀⢠⠘⠁⠠⢀⠀⠃⠀⠈⠀⠂⠀⠂⢛⡐⠀⠚⠀⠀⠆⠀⡄⢰⡄
⠀⠀⠀⠃⠈⠠⠀⢀⠀⠠⠀⢦⡀⠀⠀⠀⠘⠀⠰⠀⣀⠀⠀⠀⠈⠀⠀⢀⠀⠀⡀⠀⠒⠀⠀⠠⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠐⡊⠀⠐⠀⠀⠄⠠⠋⠀⠀⠠⡄⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢃⠀⢰⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⡄⣇⠎⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠹⡏⠀⣠⠖⠉⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡠⠤⠤⠤⠤⠤⠤⠤⠀⡅⠀⠀⠀⠀⢠⠈⠁⠀⠀⢀⠔⠁⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠚⠛⠉⢉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡲⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠉⠀⠀⠀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠱⢄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠙⠢⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡠⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠈⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠊⠁⠉⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣠⠀⢀⠀⠀⠄⠂⡁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⣎⡂⠀⠀⠁⠀⠈⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⡗⠌⠁⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⢔⡨⠁⠒⠊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠕⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠠⠁⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡤⠁⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠐⠰⣀⡇⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠑⠊⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠀⠀⠀⢀⡀⠄⠂
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣿⣦⣭⣯⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣶⣶⣿⣿⣶⣶⣾⣷⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠒⢂⡄⠠⠌⠀⠤⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⡤⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠒⠒⠶⠤⣀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡞⠉⠀⠀⣠⢀⢤⣤⣤⣤⣤⣀⠠⠘⠃⡥⠀⠶⠂⠀⢠⠀⢀⡀⠘⢯⣤⡑⢄⢀⠄⠀⠀⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⢀⣶⣿⣿⡀⢛⣽⣿⣿⣿⣟⠱⠀⠠⡄⠀⡘⠟⠀⢄⣲⣼⡏⢠⡀⣿⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣈⠻⣿⣿⣿⣿⣶⡀⡼⠟⡿⣧⠀⠀⠶⠀⢖⣨⣴⣿⣿⣯⣿⣿⣼⣷⠟⠁⠀⢀⡤⡀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠻⣷⣦⣌⣉⠛⠻⠿⠿⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠐⠒⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣼⡆⠈⡍⡛⠛⠛⠻⠷⠶⣶⣤⣤⠬⣭⣥⠤⠤⠄⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⡀⢠⣶⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⢄⡂⢄
⠀⠀⠀⠀⠈⠔⣽⣿⢟⣯⣽⢿⣿⣿⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⡑⢄
⠀⠀⠀⠂⠠⠀⠾⣁⣈⣿⣛⣫⡿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣆
⠀⠀⠸⠀⠸⠀⠀⢿⣿⣛⣿⣿⣯⣷⡙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⣿⡀⠂⢸⣿⡿⡟⢿⣶⠶⠤⣤⣤⣤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡿⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣤⣶⣦⣼⣿⣿⣷⣶⡟⠋⢑⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣄⣠⣤⠀⠀⠀⠀⢠⡞⣡⠊
⠀⠀⠀⠀⠀⠀⠐⠈⠂⢀⡀⠀⠈⠻⣿⣿⣿⡆⠀⠛⠻⠓⠚⠉⠀⠀⠀⠀⠀⠀⠘⠿⢯⠷⠀⠀⠘⠻⠿⠿⠶⠈⠀⢄⡤⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣢⠴⣊⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⠒⠛⠉⠅⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠙⠛⠿⣿⣷⠀⠈⣷⣷⣶⡶⠶⠶⠶⠲⠟⠋⠁⠋⠙⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⡐⠁⠀⠀⠀⣀⠈
⠀⠀⠀⠈⢻⠒⠂⢿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⣀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⣤⡄⠀⠀⢸⠏⠈⡓⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣾⣿⣿⡟⠉⠀⠀⠀⠀⠀⣅⠠⠄⠀⣐⣀⣨⣥⠤⠶⠖⠚
⠀⠀⢶⣶⣾⡇⠀⣷⣤⣤⣤⠀⠀⢾⠃⢠⠄⠀⢧⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠐⡞⠛⠉⠉⠁
⠈⠄⠀⣿⣽⡇⠀⢫⣭⢹⠉⢀⡦⢾⢧⣮⡀⢀⡠⠿⠟⠃⠇⠀⠀⠀⠀⠀⠀⠀⡇
⡐⡀⠀⢻⢻⡇⠀⢸⣿⢸⠀⠄⣾⣬⠾⣽⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢰⡇
⠲⠀⠀⠀⢸⡇⠀⢸⢼⢸⠀⠀⠸⡽⠀⠛⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⡇⠀⠀⡠⠧⠤⠤⠤⠤⠶⠖⠒⠒⠒⠒
⠷⠄⠀⠠⢸⡇⠀⢸⠑⢸⠞⠀⠠⣸⣀⣴⣶⡄⠀⣰⣶⡆⠈⠉⠀⣤⣧⣤⣠⡤⠤⠤⠤⢤⣤⣤⣄⣠⣤⣤⣤
⣷⡗⠀⠀⢸⡇⠀⣸⠓⢸⠘⠛⢛⣻⡛⣿⡛⠗⢠⣿⣿⣿⣷⣿⣿⣿⡯⣷⣾⣷⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣳
⣿⡇⠀⠀⢸⡇⠀⣸⠏⢸⢠⣾⣯⣭⣿⣿⡿⠿⢿⣿⡿⣿⢹⣿⣿⡟⠃⠞⠻⠸⠿⣿⣿⡏⠿⢿⢿⣿⣯⣿⣸
⣿⡇⠀⠀⠸⡇⠀⠀⠠⢤⣤⣭⣉⣭⣭⣴⣶⣶⢸⣿⣿⣟⣿⣿⣿⢳⡖⣶⡦⢰⣴⣾⣿⡇⣒⣐⣀⠛⠯⠅⠈
⣿⣧⠤⣴⣶⣟⠀⡀⢾⣿⣿⣮⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⢸⡇⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇
⣿⣋⢁⣛⣻⣏⡁⡅⢸⠿⠿⠿⠿⠛⠛⠛⣛⣛⢸⣟⣿⣿⣿⣿⣿⢸⠿⢿⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇
⣙⣖⣴⣻⣿⡃⠀⡇⢠⡶⠶⢶⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⡿⣿⣷⣶⡆⣷⣿⣿⣿⡇⠿⢿⣿⣿⣿⡇
⣯⣷⣭⣵⡾⡿⠛⡇⢙⣺⣿⣿⣯⣟⣻⣽⡿⠷⡀⡿⠝⠻⣿⠿⡿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣶⣶⣤⣭⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣦⡀⠀⠀⠀⢀⣀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⣿⣿⣿⣦⠀⣰⣿⡿⢿⣟⣡⣴⢃⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣶⣶⣿⣿⣷⣶⣶⣽⡻⣿⡇⣿⣵⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣟⣿⣿⣿⣿⣿⠮⠱⠿⠿⠿⠿⠿⠶⠿⠯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠁⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡾⠋⡆⠀⣾⢋⣠⠀⠀⢰⡟⣠⡆⠀⢰⣿⢀⡀⢠⣾⣦⢈⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠧⠛⠁⠀⠙⠎⠋⠀⠀⠈⠻⠝⠁⠀⠀⢿⣮⣃⣼⣿⣿⣮⣁⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⡆⡀⢠⡞⣠⠀⠀⢀⣾⠃⡄⠀⢀⣾⢁⡄⢸⣿⣿⣁⡌⣿⣿⣉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠟⠃⠘⠷⠛⠀⠀⠈⠿⡼⠃⠀⠈⣿⣜⣣⣾⣿⣿⣝⣡⣿⣿⣛⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⢀⠀⠀⣰⡟⣠⠀⠀⢠⡿⣀⡜⣿⣿⣏⣠⢹⣿⡟⢉⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠹⣧⡛⠀⣀⠘⢷⣙⣡⣿⣿⣎⣁⣼⣿⡇⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⣾⢀⡆⠀⢸⡏⣴⠀⣿⣿⢡⡌⣿⣿⣋⡙⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⠷⠁⠀⠘⢷⣬⣴⡿⢿⣮⣴⠿⢷⣉⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣿⠾⢸⣿⣿⠲⢃⣿⣏⠖⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⣩⢻⣷⡟⣡⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣭⣼⣿⣿⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢉⣉⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠘⣇⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀
⠀⠀⠀⣿⠳⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⠀⡿⣄⡀⢠⠇⠀⢻⡄⠀⣠⡶⠛⠉⠉⠈⠉⠉⠻⢦⡀
⠀⠀⠀⢻⡀⠀⠉⠙⠒⢦⡴⠚⠉⠀⠤⢄⣈⢹⠂⠈⠉⠛⠒⠶⢬⣧⡾⠃⠀⠀⠀⢀⠔⠀⢀⣤⠞⠋
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠾⠓⠲⢦⣀⠀⠀⠈⠙⢧⡀⠀⠢⣄⠀⠀⠈⢷⡀⠀⣠⠔⠁⢀⣴⣏⠙⠶⣤⣄⣠⣤⣤⡴
⠰⣤⣤⣤⣤⣭⠿⠶⠶⠒⠂⣈⣠⣽⢷⣤⣀⣀⠀⢙⣆⠀⠈⢳⡄⠀⠀⢹⡏⣁⣠⡴⠿⣤⣬⣭⣤⣤⣤⣤⠶⠋
⠀⠙⠳⠦⣤⣀⣀⣀⣀⣉⣉⣁⡤⠴⠚⠉⠉⠙⠛⠛⠻⣦⣀⣀⣿⣀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⣠⠞⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠘⣧
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⡴⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠁
⣿⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⢻⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⢠⠀⣼
⠘⣧⠀⠈⠀⠀⢠⠀⠀⠄⠀⠀⠀⡆⡆⠀⠀⠀⠘⣦⠇⠀⠐⠀⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⡖⠀⠀⡀⠀⣸⡏
⠀⠹⣇⠀⠀⢀⠀⠀⠀⠊⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⢰⡟
⠀⠀⠙⣧⠀⠸⠀⠀⠘⣢⠀⠀⠀⢆⠀⠀⠀⢰⠰⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡄⡀⠀⡴⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠈⠚⠀⠠⡤⡀⠉⠀⠀⠀⠡⠁⠀⠀⠳⠼⠀⣀⠀⠁⡀⠀⣠⠄⠀⠤⢪⡟⠁
⠀⠀⠀⠀⠀⠻⣆⠀⠢⠀⠀⠈⠤⡀⠀⠀⠠⠜⠀⠀⠤⣄⠀⠀⠀⡤⠔⠠⠀⠀⠀⠀⠀⠈⠀⡀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⡀⠀⠚⠁⠀⠐⠠⡀⠀⠀⠀⠉⠀⠀⠘⠀⠁⠠⠚⠀⠀⢀⡒⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠁⠀⠀⠄⠐⠀⠀⠁⠀⠘⢁⠆⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠔⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠢⡀⠀⠀⠄⠀⠉⠠⠀⠀⢀⣀⡀⠀⠁⠀⠀⠂⠀⢠⠖⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠐⠂⠄⠀⣀⡀⠀⠁⠀⠠⠄⢀⡄⠀⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡐⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡀⠁⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠈⠁⠀⢀⣀⠁⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣈⣓⣀⣠⣴⠾⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡻⠚⠂⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠴⣴⠄⠠⢀⣠⡴⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢄⡆⠁⠀⢈⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢛⠢⡀⠉⡟⡋⠋⢀⡀⠤⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠸⡇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⡀⡇⠇⠀⢀⡄⠀⠀⢀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠎⢠⠉⡔⠋⢠⠟⡰⠞⠃⢀⠏
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡐⠡⠀⠀⠀⠀⠁⠀⠙⠀⣈⠀⠸⣆⢀⡞⠠⠄⠀⠀⠂⠀⠐⠂⠀⠀⠠⠀⠇⠠⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⠀⡷⠁⠈⠀⠀⠁⠀⠐⠄⠀⠁⠀⠄⠈⠏⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠂⠀⠐⠒⠀⠠⠄⠀⠦⠀⠘⠃⢀⢀⠊⡄⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠂⠀⢀⢺⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⣀⠀⢀⡀⠀⠒⠀⠈⠆⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢛⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⢘⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢁⠀⠐⠃⠀⠐⠒⠀⠈⠁⠀⢀⠀⡎⠁⠀⠀⠠⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡣⡀⢤⠀⠀⣠⠀⠀⠒⠀⠈⠀⡇⠐⠂⠀⢀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡂⠀⠀⠀⡀⠀⠠⠀⠀⠹⡇⠀⣀⠀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠚⣇⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣟⡿⠋⡁⠄⠀⣠⡀⠈⠁⠒⠬⢝⡺⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠕⣿⡷⠉⢂⠕⠊⠀⠀⠈⣃⠐⠀⠀⠀⠐⢍⠲⣝⢷⣌⠷⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠁⣾⠏⡄⡔⠁⠠⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠈⠎⢿⣦⣤⠖⢘
⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⢸⣼⠘⠨⠀⠀⠀⠁⡀⠀⠀⣴⠄⡀⢁⠀⠀⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⢿⢀⠀⠐⡀⡀⠀⠘⠦⠄⣀⣄⠀⠈⠀⠀⠸⠀⠀⠀⠀⡇⣿⣥⣀⡀⠀⠈⠀⠒⠄
⠀⠀⠀⠀⢰⠃⠁⠀⠀⠀⢻⠸⡈⠌⢆⢳⣕⢄⡀⠀⠀⢈⠃⡴⣋⢔⠵⠁⠀⠀⡸⢸⢹⣿⠟⠻⠉⢗⡶⠦⠤⠃
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠈⣧⠱⡀⠀⠀⠙⠻⢟⣓⠶⣤⡁⣴⡿⠃⠁⠀⢀⠜⡱⢣⣿⠛⠃⠀⠀⠈⣧
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣷⣙⢦⡈⠲⢄⡀⠀⠉⠢⢱⠇⠀⠀⠀⠔⣡⢞⣵⣿⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⢍⡲⠤⣌⣁⣀⣀⣀⣀⣠⠤⢖⣫⠴⣻⠞⠁⠀⠀⠀⠂⠀⠀⡟
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢂⠈⠻⡷⣍⣙⡒⠲⠾⠭⠶⠶⠒⢛⣉⡴⠞⠁⠀⡀⠠⠀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⡣⡈⠮⡛⠻⠷⠶⠶⠶⠖⠛⠋⠉⠀⠀⠠⠊⠀⢀⠌⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⡪⢀⠕⡡⠊⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⡀⠀⢀⠀⠈⠀⠀⠀⠂⠉⢀⠁⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠄⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣗⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣖⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠯⠭⣿⣿⠿⠭⠭⠷⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⠠⡄⢀⠀⠀⢒⠒⠒⠯⣷⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣬⣜⠋⠀⣤⣉⡻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⠋⢘⣿⣿⠄⡀
⠀⠀⠀⠀⢸⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣭⣿⡿⢠⣇⡴⠞⠉⠙⣦
⠀⠀⠀⠀⠘⣿⣮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⣼⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣭⣟⣛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣻⡍⡋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣇⠁⢤⣀⢀⠀⠀⠀⠀⠀⢀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠿⠿⠦⠧⠤⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣰⠟⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣴⡿⠟⠋⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⣤⣻⣆⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⡾⠁⠀⠀⠞⠉⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠈⠉⠛⠷⠀⠀⠀⠀⢈⣻⣄⣀⣾⣃⣀⡀⠀⠀⠀⠀⣠⡾⢋⣤⡶⠛⣿⣦⠶⠞⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣀⣠⣤⣶⡾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣾⣿⣾⣏⡁⠀⠚⠉⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠉⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣶⣶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠿⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⣠⠞⠉⠛⢾⠛⠛⠀⠀⠀⣤⣤⣤⡄⠀⠀⠘⠿⠿⣡⡤⠤⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢠⠀⠀⢿⡀⠀⠀⣸⠃⠀⠀⠀⠀⣿⠿⢿⡇⠀⠀⠀⠀⢰⡇⠀⠀⠈⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⠀⠀⠀⠙⠓⠛⠁⠀⢠⡄⠀⠀⠘⠦⠾⠁⠀⠀⢀⡄⠈⠳⣤⣤⠼⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⣦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⢠⡆⠀⠀⠀⠀⠐⢋⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⢀⡀⠰⡆⠀⠀⠀⠀⠀⢀⣧⣴⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠟⠛⠉
⠀⠀⠀⠀⣀⡀⣀⠰⢠⣯⠔⠊⠁⠀⠈⠙⢥⠀⠀⠀⠀⡀⢸⣉
⠀⠀⠀⠀⠙⢛⡻⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀⠁⠀⠂⠀⠹⡶⡄
⠀⠀⠀⠀⡴⢶⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⡀⠄⢀⠀⠰⡿⠷⠦⡀
⠀⠀⠀⠀⠑⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⡄⠀⠀⠀⢀⡙⢣⣛⡀
⠀⠀⠀⠀⠈⠉⠈⣇⠀⠀⠐⡀⠀⠀⠀⠄⠀⠀⠀⠀⠘⡀⠁⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡘⠢⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡐⠀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠚⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠔⠊⠁⠂⣀⢀⣀⠔⡰⠁⠀⢀⠔⠀⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢰⠁⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠯⣁⡈⠀⠀⠀⠀⠈⠁⠠⠤⠀⠀⠀⠸⣀⡂⠔⠩⠄⢐⡂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⣤⣤⣉⣉⣀⣂⡀⠄⠀⠂⠼⠿⠿⠟⠓⠠⠖⢀⣀⣀⣀⣀⡺⠾⣃⠿⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢽⠙⠛⠻⠏⠿⠛⠛⠛⠒⠶⠒⠲⣶⣶⣶⣶⣾⡿⣿⣿⣿⣿⠛⣿⠃⢠⠔⠊⠁⠀⣀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣒⢀⣩⣭⠿⠿⣟⣛⠀⢑⡁⢸⠃⣠⠞⠉⠉⠉⢳⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⢀⡲⢒⣚⣿⡿⠽⢖⣀⠡⡂⢻⠞⠀⠀⠀⠀⠀⠀⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣤⠀⠠⠴⠶⣒⣻⡭⠄⣈⠁⣿⠀⠀⠀⠀⠀⠀⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠒⣂⣀⣉⣭⡭⠤⠖⢀⠥⠀⡇⠀⠀⠀⠀⠀⣠⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠬⠍⠭⠴⠶⣒⣛⡁⠐⠂⠀⣇⢀⡀⠤⠒⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⢀⣚⣛⣭⡭⠵⠀⢋⠄⢠⡷⠂⢀⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠠⠴⠶⣒⣚⣫⡅⢐⠀⣼⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠈⣉⣭⣭⣵⠖⠤⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣷⣍⠠⠀⠀⠀⠀⠀⣀⢀⣀⠐⣀⣐⣚⣿⣿⡿⠯⢈⣥⣏⣀⣀⣀⣀⣀
⠀⠀⠀⣀⣤⠴⠖⠒⠚⠹⠏⠉⠀⠀⠠⠤⠤⠿⣗⣦⣔⠀⠀⠀⠐⠀⢛⡀⣺⣿⣿⣿⣯⣧⣴⣿⠿⠿⠿⢿⣯⣍⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⢠⡀⠀⠀⢎⠀⠀⠀⠀⠀⠰⠀⠀⠈⠣⠄⣀⣀⣀⣀⣀⣉⣍⣩⣶⣶⣶⣤⡦⠞⠁⣀⡠⠀⠀⡀⠀⠈⠉⣙⠆⠀⣠⢈⣹⣹⡇
⠀⠀⠀⠙⠳⢭⣛⡒⠶⠦⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢉⠉⠉⠉⣉⣉⣡⣄⣀⣀⣀⣰⣶⣶⣶⣦⣤⣤⠤⠤⠶⠖⢛⣫⡽⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠦⣤⣤⣤⣀⣈⣉⣉⣉⡉⠉⠉⠉⠩⠉⠉⠩⠭⠉⠉⠩⠉⠉⣉⣉⣉⣉⣀⣤⣤⣤⠴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣤⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢠⠃⠀⠀⠀⠀⠀⠀⣤⢀⠀⢀⡄
⣠⠜⠁⠀⠀⠀⠀⠀⢀⠤⠮⢁⠙⠁⣇⡀⠀⠀⠀⠸⡀⠂⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⡤⠦⠉⠙⠁⣇⡀⠀⠀⠀⠸⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⢪⠀⠀⠀⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⢺⠀⠀⠀⠀⠀⠑⠒
⣀⢀⢤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢠⣄⣀⢀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢠
⢈⠁⠰⡄⠀⠀⠀⠀⢇⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠼⠬⢈⠁⠰⡄⠀⠀⠀⠀⢇⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠼
⠀⠈⠙⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠶⣠⣀⡠⡆⠀⠀⠀⠀⢁⠀⠀⠂⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠶⣠⣀⡠⡆⠀⠀⠀⠀⢁
⠔⠋⠀⠀⠀⠀⠀⠀⡠⠒⠓⠠⢈⡀⢣⠄⠀⠀⠀⠈⡄⢀⣀⠔⠋⠀⠀⠀⠀⠀⠀⡠⠒⠓⠠⢈⡀⢣⠄⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⢹⠀⠀⠀⠀⠀⠀⠁
⢤⡠⠒⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰⢦⢤⡠⢒⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰
⠠⠄⣘⡖⠀⠀⠀⠀⠣⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⢀⠔⠚⠒⠤⠄⣘⡆⠀⠀⠀⠀⠢⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⢀⠔⠘
⠀⠠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⠀⠂⠠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⡛⠴⢤⠜⡇⠀⠀⠀⠀⢡⠀⠀⠁⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⡛⠴⢤⠜⡇⠀⠀⠀⠀⢠
⠊⠁⠀⠀⠀⠀⠀⠀⡔⠈⠉⠐⠠⠄⡱⠂⠀⠀⠀⠈⢆⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⡔⠈⠉⠐⠠⠄⡱⠂⠀⠀⠀⠀⠆⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠲⠔⠩⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢸⠳⠲⠔⢉⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢸
⠐⠀⢬⡇⠀⠀⠀⠀⠑⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠐⠀⢬⡏⠀⠀⠀⠀⠑⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⢠⠊⠉
⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠄
⢀⡰⠊⠀⠀⠀⠀⠀⠀⣀⣍⠚⠲⠋⡇⠀⠀⠀⠀⠰⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⣀⣍⠊⠲⠋⡇⠀⠀⠀⠀⠰
⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠀⠀⢼⠁⠀⠀⠀⠀⠃⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠈⠐⠂⢼⠁⠀⠀⠀⠀⠃⠔
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀
⠙⠊⢠⣀⠀⠀⠀⠀⡄⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠸⣙⠙⠊⢠⣀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠸
⠈⠁⠲⡇⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠁⠲⡇⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣠⠄⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⡏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢻⣧⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠻⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⡼⠃⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠒⠒⠂⠒⠒⠒⠒⠐⠒⠒⠒⠒⠐⠒⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡖⢉⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡰⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⠤⠊⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⣀⣀⠤⠐⠁
⠀⠀⠀⠀⡠⠄⠒⠒⢈⣉⣭⣤⣬⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⣤⣤⣤⣉⡁⠒⠒⠠⢄
⠀⠀⠀⠀⠢⢄⣀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⢤⡀⠭⠭⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠭⠭⢀⡤⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠂⠀⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠛⠛⢛⡛⠻⠯⣶⣢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣻⢟⡭⠚⢁⠠⠤⠀⠠⣄⠈⣉⠉⠑⠲⠬⣙⠪⣝⢶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣩⢞⣟⡵⢋⠔⡨⠔⠊⠀⠀⢈⡫⡠⠀⠀⠀⠀⠐⠪⡙⠦⡙⢿⣮⣙⡦⠄⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡴⠋⢠⢯⠞⡀⣠⠊⠀⣠⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠈⢢⠘⢎⢫⣧⠀⣀⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠋⠀⢀⡏⡏⡜⠰⠃⠀⠀⠴⠄⠠⠀⠐⠀⠀⠀⠠⠆⠀⠁⠄⠀⠀⠀⠈⠈⡏⣿⠋⠀⢠⢧
⠀⠀⠀⠀⠀⠀⢠⡟⡰⠁⠀⠀⢸⢸⠁⠃⠃⠀⠀⠀⠀⠈⣄⠀⡀⠘⠇⢀⠆⠰⠀⠇⠀⢸⠀⠀⠀⠀⢸⢸⣂⡴⠛⠒⠧⠤⠀⣀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣾⠸⡆⡆⢀⠀⡄⢂⠀⠁⠈⠓⠠⠦⣴⡅⠊⢀⠘⢀⠁⡎⠀⠀⠀⠀⢸⢸⣿⣶⣤⣄⡀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⢸⠇⠄⠀⠀⠀⠀⢻⡆⢧⠰⡈⢆⢘⣄⡢⣀⠀⠀⠀⠀⡈⢀⡼⢋⠔⡡⢊⠀⠀⠀⢰⢃⡟⣼⣿⡋⠋⠧⠙⢿⡖⠦⠤⠜
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣷⡘⣆⠀⠀⠀⠙⠻⣶⡛⠷⢶⣤⡙⠈⣴⣵⠟⠊⠀⠀⠀⡰⢃⡞⣰⣿⠙⢃⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⣿⠰⠀⠀⠀⠀⠀⠀⡟⣷⣌⢧⡀⠐⢤⡀⠀⠈⠙⠲⢌⠻⢸⠟⠁⠀⠀⠀⡠⠞⣡⢏⣴⡿⠁⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⠲⣄⡈⠑⠢⠤⣄⠀⠐⠘⠀⠀⠀⢀⣠⡴⢋⣵⣾⡿⠃⠀⠀⠈⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡳⠮⣍⣓⠲⠦⠤⠤⠤⠤⠴⠒⣚⣩⡤⠞⣋⡴⠏⢀⢀⠂⠀⢀⠆⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⠙⠟⢷⣤⣈⡍⠛⠛⠛⠛⠛⠛⠋⢉⣠⡴⠞⠋⠀⠀⠡⢂⠄⢀⠌⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⠣⡈⠲⣍⠛⠻⠿⢷⠶⠶⠷⠛⠛⠋⠉⠀⠀⠀⡀⠜⠁⡁⢠⠊⡔⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⢑⠢⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢐⡨⢐⡡⢊⠔⣡⠎⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡐⠀⣀⠈⠁⠐⠂⠒⠂⠐⠀⡀⠤⠒⡡⠒⡡⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠨⢐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡠⢖⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢽⣳⡶⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢖⣊⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠾⢭⣭⣭⣽⣭⣭⣭⠭⠶⠖⠛⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡏⣾⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡏⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⢸⣿⣿
⣾⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣇⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⢛⣿⢋⣴⡿⣯⣿⠈⣿⣿
⣿⣞⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣾⣿⣿⡋⢹⣏⢉⣿⣿⣿⡿⢿⠿⣤⣿⣟⣿⡏⣿⠀⣿⣿
⣾⢿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣋⣼⣿⣿⣿⣿⣿⣷⠟⣽⣿⣿⣿⣿⣧⣿⠀⣿⣿
⣿⣾⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢥⣾⢏⣿⣿⣯⣿⣅⣿⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⣹⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠶⣶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣭⣬⣽⣿⣿⣿⣿⣯⣿⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠙⢿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣣⠭⠿⢟⣛⣻⣿⣻⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠙⠻⣮⡻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡟⣧⡌⢳⣾⣏⣩⣿⣿⣿⣥⡍⠋⠉⠀⠈⠉⠙⠋⢻⣿⢻⡟⠿⠿⢶⣿⣿⣿⣿⣿⡿⢿⣿⠀⣿⣿
⠀⠀⠀⠀⠈⠻⢯⣿⣿⣿⣿⣿⣿⣽⣿⣯⡿⣯⡽⠛⠛⢀⣉⣩⣽⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⡟⠘⠟⠂⠀⠒⠁⠿⣿⣿⡿⠀⠈⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⣤⣿⣿⡿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⢶⡑⠀⠒⠀⢰⠀⠀⠀⠁⠀⢠⡏⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⠄⢀⣀⣤⡄⠀⠀⠀⠀⠀⠉⠉⢹⣏⣥⣤⣤⣾⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⢸⣿⡷
⠀⠀⠀⠀⢴⣶⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⢻⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠏⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡇⠸⣿⣿
⠀⠀⠀⠀⠀⣿⠠⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⢻⣷⢸⡇⠀⣠⡄⠀⠀⠀⠀⠀⠀⠈⢁⣈⣀⣸⣷⣶⣶⣶⣶⡄⣿⡛⣾⡇⠀⣿⣿
⠀⠀⠀⠀⢀⣶⠸⠃⢸⣿⣿⣿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣇⠀⠈⠀⠀⠀⠀⠀⠘⠒⡆⢸⣿⣿⣿⡷⠒⠀⠐⣿⡆⣿⣁⣾⡇⠀⣿⣟
⠀⠀⠀⠀⠈⠛⠀⠀⠀⣿⣿⢻⡇⣿⣧⠀⠀⢠⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠰⠈⠀⡇⢸⣿⣿⣿⡇⠀⠀⠀⣿⡇⣿⠟⢹⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣧⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⡦⢸⡆⠀⠃⢸⣿⣿⣿⡇⠀⠀⠀⣿⡇⠻⠀⢀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡧⢾⣿⠀⠀⠀⠀⠀⠂⠀⠀⡇⠘⣿⣿⣤⣤⣼⡟⣤⣼⡿⠀⠀⢸⣿⡟⣿⡧⠀⠀⠀⣿⡇⣼⣿⢾⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠸⠿⠤⠶⠤⠄⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣗⣠⣄⣠⣸⣟⣿⣇⣀⣀⣀⣿⡏⣿⣏⣸⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⡶⣶⡶⡶⠒⢲⠶⡀⠶⠀⠰⠶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⠻⢻⡿⣿⣿⡿⠿⡟⢻⠃⣿⣿⢻⡇⠘⣿⡇
⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡇⢠⡶⠦⢶⠀⠠⠶⠆⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣧⣾⣿⡏⠘⠛⢻⣷⣿⣿⢸⡇⢰⣿⡇
⠀⣿⢈⠀⠒⠀⠀⠀⠘⣿⣿⠀⣇⢸⡇⠀⠘⠀⢀⣄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠐⠾⡇⣿⡿⠀⠛⡀⢸⣿⣿⣿⣿⡇⢸⣿⡇
⠈⠛⠀⠀⠀⠀⠀⠀⠀⣝⣿⠀⣿⢸⡇⠀⢀⣀⡀⠀⡆⠀⣀⣤⡀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⡤⠀⢾⣿⣿⣿⣿⡁⢸⣿⡇
⠀⢰⡔⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡷⢰⣿⣿⣿⣾⡿⢸⣿⣿⡇⠀⠈⢹⣿⣦⠠⡦⠶⠶⠶⠶⠾⠛⠛⠛⠛⠋⠀⠺⣿⣿⣿⣿⣿⡇⢸⣿⡇
⡜⢾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣀⣿⣿⡇⣾⣿⣿⣿⣿⢷⣿⣿⣿⣿⠆⠀⠘⠿⣿⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡇⢸⣿⡇
⠁⢸⡀⠀⠀⠀⠀⠀⠀⣾⣿⠏⢉⣁⡰⣾⣿⣟⣛⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠠⠀⣷⡄⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼⠇⢸⣿⡇
⣷⣾⡇⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠⣤⣤⣤⣤⠴⢄⣤⣴⣶⠚⠻⣿⠀⡀⠀⢹⣿⠊⣿⡧⠀⢢⠀⣋⣉⣉⣀⣀⣠⣤⣤⣤⣤⣤⣤⣿⠄⢘⣿⣧
⢿⡿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣷⣯⣿⣷⡾⠿⡷⠾⣟⠿⠟⠛⢋⠓⠀⠀⠀⢉⣏⠁⣼⣿⠀⡀⢘⢹⣿⡟⠛⠛⠙⠋⠉⠉⠉⠉⠉⠉⠀⠈⣿⡟
⢻⣿⣿⠄⢀⠀⠀⢀⣀⣿⣏⡛⠦⣬⣉⠶⣦⣴⡶⢦⠐⠠⠂⠉⠶⡗⠀⠀⠀⣃⠀⠸⣿⡄⢳⣿⠈⣿⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⢠⣉⣿⣏⣉⠛⠻⣀⡀⠀⣤⣬⣑⠲⣮⣿⡧⠞⣯⡥⠶⣀⠐⠂⠤⢦⣤⠀⠀⢻⡄⢀⠈⡇⠈⠟⠀⣅⢹⣆⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⡟⣿⣿⣿⡿⠶⠊⠉⠋⠡⠔⠋⠟⢁⠉⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠁⠠⠀⠀⠀⢉⣩⡇⠀⠀⡇⠉⢸⣿⣷⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⣏
⠀⣿⣿⣿⢿⣷⡀⠰⠶⣤⡂⠛⠦⣞⠉⠰⠶⣈⠙⠂⢤⣀⠈⠳⠄⢈⠀⠀⠀⢀⠘⢹⣿⣇⠀⡄⡇⠀⣸⣿⣿⣿⣦⡈⠧⣀⣀⣀⣀⡀⢰⣿⡇
⣟⣿⣿⠻⢦⣄⡙⠒⢤⣄⣉⠑⢂⣀⠈⠀⢂⡤⣈⠁⠴⢄⣀⣤⠦⣤⣬⠷⠀⣼⢿⣿⣿⣾⠀⠁⢇⠀⣿⣿⣿⣿⣿⡿⠄⢘⠋⢻⢹⡇⢸⣿⡇
⣿⠟⠛⡷⠞⠛⢉⠶⠛⠋⠁⠞⠉⣁⡬⠛⢁⡴⠛⢁⡀⠀⠀⣀⠴⠋⢡⠔⠀⢁⣼⠿⣿⣿⡀⠀⢸⠀⠉⢹⡅⢀⠀⡆⠀⠈⠀⣼⣾⡇⢸⣿⡇
⠄⠘⢁⡀⠄⠁⢀⠴⠚⢃⡠⠄⠉⣀⡤⠞⢁⣠⡄⢈⣁⢶⣞⠁⡠⠦⢀⡀⣻⠳⢦⣾⣿⣻⡇⠀⠸⠆⠠⠈⠀⣿⡀⢸⣶⢀⠀⢸⣿⡇⢸⣿⡇
⠤⣌⠁⣠⠰⢶⣄⡀⠈⠻⢦⡶⠛⠙⠣⢶⡍⢉⠉⠻⣋⣄⡈⠛⠒⢤⣀⡉⠙⠷⣦⣬⣿⡿⠀⠀⣶⠀⠀⠀⠀⢿⣇⠀⢿⢸⡀⣾⢿⡇⢸⣿⡇
⠀⠈⠉⠦⣄⡀⠈⠑⠢⢄⡀⠀⠁⠶⢤⣄⡀⠈⠻⢤⡀⠀⠝⠻⢶⣆⡹⠟⢷⣤⣄⣈⠙⣻⠁⣸⣿⡄⠘⣦⠀⠸⡟⠄⠀⢸⡅⣴⣼⡇⢸⣿⡇
⠛⠶⣀⠀⠈⠙⠲⣄⡀⠀⠙⣶⣦⣦⢀⣨⣷⠶⣤⣀⣼⠟⢲⣦⡴⢟⡹⣳⣶⠋⢀⣙⡿⠋⠀⣬⣿⡷⠀⢻⣥⡀⡏⠀⣀⠈⠃⣿⣿⡇⢸⣿⡇

⠀⠀⠀⢸⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣤⣷⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠠⠄⠀⠀⢀⣤⣤⣄⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠄⠀⠐⢠⣤⣤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣀⣤⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣦⣤⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⣾⣿⣷⣤⣤⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⠃⠀⠀⠀⠈⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣌⡀⠂
⠀⠀⠀⣼⣿⣿⣿⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡌
⠀⠀⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡂⠐⣁⡀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠊⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠈⢁⠀⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠈⡁⠀⠀⣾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡇
⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠄⠠⢀⠀⠀⠀⠀⣠⣶⣄⠀⠈⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣴⣶
⣿⣿⡆⠀⠀⠈⠻⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⠻⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢠⣾⣿⣿⣦⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣤⠠
⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⣀⣤
⠀⠀⠀⠀⠀⠀⣶⣶⣶⡾⠿⠿⠟⢿⣿⣿⠋
⠀⠀⠀⠀⢀⣤⡾⠿⢿⡷⠶⠿⠟⠛⠛⠛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⣾⠁⠀⣀⠀⠀⣠⠀⠀⠀⢠⣦⠀⠙⣿⡆⠀⠀⠀⣀⣴⠾⠛⠉⠉⠙⠻⣷⡄
⠀⠀⠀⢸⣯⠀⠰⠏⠀⠀⠃⠀⠀⠀⠈⠋⠀⠀⣿⡇⠀⢀⣴⠟⠁⢠⠶⠃⣠⡄⠀⠹⣿⣠⣶⣿⠂
⠀⠀⠀⢸⣿⠀⠀⠀⣠⠀⠀⠀⡿⠀⠲⠆⠀⢀⣿⠁⢀⡾⠁⠀⣀⡀⠴⠆⠀⣠⠀⠀⣿⡟⠸⣿⣦⣶
⠀⠀⠀⠈⢿⣧⠀⠀⠈⠀⠀⡀⠀⠀⣠⠀⢀⣾⠃⣠⣿⢡⡾⠁⢨⣤⠀⠀⣤⡄⠀⠀⣿⡇⠾⣿⣍⡁
⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠟⠀⠀⢉⣠⡿⠃⠀⣿⣇⠈⠀⢴⣦⠀⢠⣤⠀⣀⠀⢠⣿⡇⢾⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣤⣶⡿⠋⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⢶⡄⢀⡉⠀⢸⡿⠻⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣦⣤⣌⣡⡴⠛⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣀⡿⠀⠀⠀⠀⠀⠀⠘⠻⣆⠘⠁⠀⠀⢩⠟⠀⠉⠁⠀⠀⠀⠀⢘⢀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⣶⣍⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⠂⠈
⠀⠀⠀⠀⢠⡌⣀⣀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣃⠘⣆⠐
⢀⢀⣠⠖⠋⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀⠀⠀⠀⠙⢄
⠉⠉⠁⠀⢀⠀⠀⠀⢀⠴⠛⠊⠀⠳⣦⡀⠀⠀⠀⠀⠹⣤⡀⠀⠀⢀⠜⡁⠀⠀⠀⠸⠏⡐⣧⣿⠡⣴
⡱⢴⠶⠦⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠀⠀⠀⠀⠨⢭⠶⡴⠋⢄⠁⠀⠀⠀⠀⠀⠀⠡⠄⠹⣿
⡴⠃⠀⠀⠈⠱⠤⠀⠀⠀⠀⠀⠀⠀⡴⡍⠋⠑⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⣀⠀⠀⠀⢤⠁⣲⠀⡀⠀⠀⠑⢀⠀⠀⠀⠀⠀⢀
⠳⠶⠚⠉⠀⠀⣀⣼⠿⢭⡀⠀⠅⠀⠀⠀⠀⠂⢟⠀⠀⠀⠀⠲⠄⠙⠄⠀⠀⠀⠀⢤⡆⠠
⠀⠀⠀⠀⠀⢰⠏⠁⠀⠈⢠⠀⢳⡄⠀⠀⠀⠀⠘⣀⣀⣀⣀⣀⣤⡴⠃⠀⠀⠀⢀⡌⣇⠀⠃⠠⠀⠀⠀⠀⡀
⠐⠀⠀⠀⠀⠾⣂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠑⢦⣤⣴⡿⠓⠈⠁⠀⢰⠀⠈⠒
⢀⡀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⣆
⡏⢁⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠠⠀⠳⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣆⣀⡠
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣷⡀⠠⡀⢸⡅⢠⡾⣷⡞⠀⠀⠀⠀⠙⣿⠄⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀
⠀⠐⠀⠒⢟⣰⠜⠁⠀⠀⠘⢧⣀⣰⢞⠛⠏⠀⠈⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⡶⠁
⠀⠀⠀⠀⢸⣯⠰⠀⠀⠀⠀⠈⠛⠛⠛⠆⠀⠀⠀⢦⠀⠀⣀⢴⠟⠁⠀⠀⠀⠤⠺⣟
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⢠⠼⢉⡦⠤
⠛⠚⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠂⠠⠏⠀⠀⠀⠀⠂⢀⡆⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⣀⡤⠄
⣀⣀⣂⣀⣀⣀⣀⣀⣡⣀⣄⣐⣀⣉⣻⣦⣤⣝⣀⣁⣀⣀⣀⣀⣀⣑⣟⣀⣃⣐⣀⣐⣀⣀⣀⣀⣀⣟⣁⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠐⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠆⠈⠀⠈⢀⣀⣠⠔⠚⠂⢀⣀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⡀⢀⡀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠉⠈⢀⠀⠠⠀⠀⡀
⠀⠀⠀⠐⠂⠀⠀⠐⠒⠕⠁⠀⠁⠙⠀⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠛⡾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⢀⣀⠀⣠⡖⠚⠅⡀
⠀⠀⢀⠃⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠠⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⡄⠋⠀⠀⠀⣀⡀⠐
⠀⠀⠈⠐⠀⠂⠠⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠀⠀⠀⠈⠀⡀⠀⠰⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠨⠁⠀⠀⠀⠠⠀⠀⠠⠀⠀⢈⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣠⣤⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠂⠀⠀⠠⠀⢈
⠀⠀⠀⠀⢠⠄⠁⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡈⠃⢿⡔⠂⠂⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠀⠀⠁⠠⠀⣀⠂
⠀⠀⢀⠠⠟⣄⣰⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⢠⠊⡟⠄⡡⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢰⠒⢛⢛⢁⢯⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢀⡀⠂⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⠂
⠀⠀⠀⠛⡖⠛⠀⠷⠀⠄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⠄⠀⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡶⣾⢿⣧⡄⡀
⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⢀⣂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢿⣷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⣀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠆⠠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡶⠞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠢⣀⣠⣄⣄⣠⠆⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣈⠌
⠀⠀⠀⠄⠀⠀⠀⠀⡝⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⣧⠜⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠁⡀
⠀⠀⠀⡀⠀⠀⣀⠀⠑⢀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠲⣿⡶⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡄⠀⠀⠱⢧⡈⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠤⠤⠤⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠆⠉⠀⠀⠠⠤⠤⠀⠀⠒⠂⠀⠀⠤⠤⠄⠀⠀⣈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠒⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠓⢀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣋⠱⠀⠈⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡩⠔⡆⠀⡖⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢡⣉⠢⢄
⠀⠀⠀⠀⠀⠀⠠⠊⠔⠩⠀⠁⠈⢆⡈⠲⢅⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣀⠤⠒⠂⠀⠠⡯⡓⠄⠱
⠀⠀⠀⠀⠀⠀⡇⡌⠀⠀⠀⠀⠀⠀⣹⠒⠠⠀⡀⢁⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠪⠊⠀⠀⠀⠀⢀⡴⠠⠈⠈⡄⡇
⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠈⠳⠒⠓⠀⠠⠤⠤⠤⠄⠀⠀⢀⡈⠤⢄⣲⡦⢤⣤⡤⠔⠋⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠠⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⣠⠴⠖⠂⣈⠥⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠌⡐⠀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠒⠉⠀⠀⠀⣀⣀⣤⣤⡤⠴⠖⠂⡁⠄⡢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⣀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⡀⠤⠤⠐⠒⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠢⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠁⠀⠈⠙⠶⠦⠤⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡄⠀⠈⢳⣄⠀⠔⠙⡀⠀⣰⠓⠚⠀⠀⢀⡶⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢙⣄⠀⠀⢻⡎⠀⠀⠡⠀⣿⠀⠀⠀⢀⡜⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠙⢷⡀⢰⡇⠀⠀⠀⠈⢿⡀⠀⠀⡾⢁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣧⣄⣘⣿⣾⡇⠀⠰⣀⠀⠀⠑⣄⠀⡟⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠈⠉⠛⢿⣧⠀⠀⠸⡀⠀⠀⠈⢢⣷⣊⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⢷⡀⠀⢳⡇⠀⠀⠈⢫⠉⠑⠀⠀⠀⠀⢀⡤⠖⠊⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⣇⡐⠀⠀⢸⡆⠀⠀⢠⣀⣰⡿⣰⡟⠀⠀⢠⠞⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢣⡈⣻⠶⠶⠦⣼⣧⡤⠤⠀⠉⠁⡴⠋⠀⡇⢀⡎⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡃⠀⠀⠀⠀⠀⠀⢀⡠⠀⢰⠁⡀⢀⡷⠋⠀⢀⡼⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⣀⣀⣀⠀⠄⠋⠀⢀⣀⣴⠗⠛⣉⣥⠾⠋⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣉⣉⠀⠀⠀⣏⠉⠉⠉⠙⠛⠶⡖⠒⠋⠉⣀⠄⢠⡀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣿⣤⣤⡄⠀⠀⠀⠈⠀⠴⠖⠁⢀⡾
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⣀⠀⠈⠉⠛⠶⣅⡂⣄⡀⠀⠀⠠⠶⠋
⠀⠀⠀⢠⡏⠀⠀⠴⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣏⠠⣄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣌⡁
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⠀⠈⠛⠛⠲⠶⠶⠶⠒⠚⠋
⠀⠀⣼⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⡇⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⢸⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⣸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⣿⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣴⠾⠋⠁
⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉
⢻⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠁
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠋⠉
⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡤⠤⠖⠒⠛⠉⠁
⣤⣤⣤⣭⣿⣶⣶⣶⣶⣶⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠿⠶⣦⣄⡀⢀⣴⠋⠁⠸⣷⡀⠀⠀⠀⢀⣤⣾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠈⠙⢾⣏⠀⠀⠀⠸⣧⠀⣠⡴⠟⠋⣁⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠈⢿⣦⠀⠀⠀⠁⠐⢤⡙⣆⠀⠀⠀⣿⠜⣩⣴⠾⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠤⣄⠀⠙⠳⣦⣀⡀⠀⠀⠈⠺⣧⣴⣠⣷⣿⠿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢷⣴⠏⠀⠀⠀⣀⣽⣿⠟⠛⣉⣙⣿⣿⣿⣷⡆⠐⠒⠄⠈⠙⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡤⣄⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠀⠀⠈⠀⣠⣾⣏⠙⠢⣍⠳⣤⡀⠀⠀⠀⣈⣹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⣯⣠⠞⠀⠀⠀⠈⠻⢿⣿⣤⣄⣀⣀⣤⣴⣾⠿⠉⣿⣆⠀⠈⠣⠈⢻⡿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠉⠁⠀⠀⠀⢀⡤⢤⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠘⢿⣷⣄⡀⠀⠀⢿⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣏⣠⠎⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⣠⡀⠙⠻⠿⠷⣶⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⠚⣦⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣟⠀⣼⠀⠀⠀⠀⣼⠁⣼⠀⠀⠀⠀⠀⢰⠒⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣾⠃⠀⠀⠀⠀⢀⡔⠒⡄⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠹⠿⠋⠀⠀⠀⠀⠀⣼⣀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠃⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠙⠛⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠋⣷⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡞⠉⣹⠀⠀⠀⠀⠀⣿⣀⡾⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡔⢦⡀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⢀⣴⠛⢙⡆⠀⠠⣾⣠⡴⢁⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡘⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠛⠀⠀⠀⠉⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠈⠉⠀⠀⣠⠖⡄⠀⠀⠀⣠⠶⢦⠀⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⢰⣧⣴⠃⠀⠀⠸⣧⡴⠋⢀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣭⣀⣀⣀⣀⣤⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠓⠶⣄⡀⠀⠀⠀⣠⠴⠒⠋⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⢰⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢀⣠⠤⠴⠖⠒⠛⠛⠋⠉⢉⣉⣉⣉⣉⣩⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣍⣛⣛⣓⣒⣲⠦⠤⢄⣀
⣿⢶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⣿⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⢸⡇
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⠤⠤⠤⠤⣄⡀
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⡀⠀⠀⠙⢦⡀
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⠋⠉⠀⠉⠳⣄⠀⠀⢳
⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈
⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀
⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⢀⣤⣤⡀⢰⡷⢰⣿⢠⣿⣴⢏⡼⠁⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡜
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣾⢷⣼⣧⣾⢃⣾⠛⠛⢧⡴⠟⠁⠀⠀⠀⠀⠀⢀⡾⠁⢀⣀⣀⡤⠴⠒⠋⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣀⣿⣠⠟⠁⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠉⠉⠁⠀⢀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠠⣿⣷⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⠤⠴⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠁⠀⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⢸⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠿⠿⠉⠓⠦⣄⣸⠀⢠⣾⣥⣤⣶⠶⠶⢶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠋⠉⠁⠀⠀⠒⠻⠯⣤⣀⠈⠉⠙⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣶⣾⣿⢿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡉⠻⢦⡀⠀⠀⠀⠙⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠈⠉⠓⠦⣷⡀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢰⡟⠀⢀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠈⢿⡄⠀⠀⠀⣠⡌⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⣿⠀⢠⡾⠛⠉⠀⠀⠀⠉⠙⣿⡗⠀⢠⡴⢿⡄⣰⠟⢿⣷⡀⠘⣷⠀⠀⠀⢿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⣼⣿⠋⠀⠀⢀⣴⣶⣧⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⣰⡟⠁⠈⢿⠏⠀⠀⠙⢿⡄⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⣼⣿⠃⠀⠀⠀⠸⠷⠋⠹⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰⠷⡄⠀⠀⣿⠀⣧⠀⠀⢰⡟⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠘⣿
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠀⠘⢷⣿⡄⠀⠈⠻⠃⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡏⠀⠀⠀⠀⣿
⣼⡿⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿
⣿⡇⠀⠀⠀⠹⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏
⠘⣿⣧⠀⠀⠀⠀⠀⠀⢠⡶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡄⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠻⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡄⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠘⢧⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡿⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⣾⠳⡄⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣦⣤⣤⣤⣴⣾⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⣤⠤⣤⣤⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⠿⠛⢋⡭⠆⣶⡤⠒⠾⡞⠋⢻⣝⣿⣿⣿⡿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠁⢠⠂⠀⠀⠀⣿⣇⠀⠀⠀⠀⣠⣿⡿⠟⠀⠈⣷⡿⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠘⡀⠀⠀⠀⠙⠛⠷⡀⣰⡾⠋⠁⠀⠀⣀⣴⠿⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠘⢿⣿⣿⡿⠇⠀⠀⠐⠂⠀⠒⠐⣂⣠⡀⠉⢀⣤⡶⠖⠛⠉⠀⠀⢀⣠⣿⣿⣿⣿⡟⣠⡀
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⡀⢢⣴⣶⣶⡤⣴⣮⡣⣾⢋⣡⣤⣴⣶⣶⣿⣿⣿⣿⣿⡿⠿⣫⣾⣿⡋⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠿⢷⣶⣤⣄⣽⣿⣷⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠐⠀⠀⢉⡟⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⣾⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠏⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠏⠠⠠⠔⠒⢉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠉⠀⠀⣀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠳⠆⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⣀⠀⠀⣀⣀⣠⣴⣦⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡴⢶⣿⢦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠀⠀⠠⢴⣾⣷⡦⠄⠀⠀⠈⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣤⡴⠂⠀⠀⠈⠀⠀⠁⠀⠀⠐⢦⣤⣄⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠁⢠⡿⠀⠀⣠⣾⡇⠀⠀⢸⣷⣄⠀⠀⢿⡄⠈⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⠶⠞⠋⠘⣧⡀⢀⣼⠃⠙⠳⠶⠟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠐⡆⠀⠀⠀⡆⠀⢈⠛⠛⡁⠀⢰⠀⠀⠀⢰⠂⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⠀⠀⠀⠹⠀⠀⠏⠀⠀⠀⣀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠙⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠋⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢀⡀⠀⠘⠀⢀⡀⠀⠃⠀⢀⡀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⠃⠀⢀⡀⠈⠁⢀⡀⠀⠘⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠈⠃⢀⡀⠘⠁⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠘⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠞⠋⠁