Fungi Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣖⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⢾⣿⣿⣟⣻⣿⣿⣿⣿⣿⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣭⣧⠿⡼⣿⣾⣯⣿⣽⣟⣻⣟⡧⡭⢀⡂⠲⣄
⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⢿⣿⣾⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣧⣭⣀⢥⡀⠀⠀⠀⠀⠀⢀⠄
⠐⠒⠀⠈⢠⣤⣤⢐⠀⠀⠀⠀⠉⠓⢷⣾⣯⣿⠿⢻⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡄⠀⠀⢀⡜
⠀⠀⠀⠨⢩⠿⡉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣯⠀⠸⠟⠿⠿⢿⣯⣛⣿⢽⠿⠿⠋⠀⠀⢀⠊
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠲⠿⣳⠤⢄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⢦⣎⣩⢉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⢸⣿⠿⡞⠙⣠⡤⣴⢶⠶⡶⢶⡤⣀⣀⡀
⠠⠂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⡐⠃⢹⣤⣼⣾⡞⢻⣶⣿⣵⣯⣫⠳⣄⢀⣀⣀⣀⡀
⠀⠀⢀⡠⠤⠤⠤⡀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⣷⣶⣿⣟⠿⣄⠄⢀⣀⣈⣽⣟⣛⣿⣿⣿⣷⣿⣿⣟⢷⡀
⠀⠀⢘⠀⢤⣀⡠⠭⣭⣥⠏⢳⠀⠀⠀⠐⠀⠀⣿⢿⡀⢀⣿⣿⣿⠿⠿⠛⢿⣾⣯⣼⣷⣿⣽⣏⣿⣷⣶⣾⣿⣿⣖⣿
⠀⠀⠈⢳⣤⣤⣄⠀⣁⣤⢤⡾⠀⠀⠁⠈⢲⢳⣍⢎⡇⢸⣿⣯⣳⣾⣵⣜⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣿⣯⣾⠟
⠀⠀⠀⠀⠁⢿⡏⣷⣿⣠⠂⠀⠀⣄⡀⣰⢌⣿⣿⢹⠁⠀⠈⠙⠛⠿⢿⣿⣻⣿⣿⣿⣿⣿⣽⡿⠟⠋⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣤⣙⣌⠓⠿⣀⣿⣇⢸⡙⣄⣿⣿⣿⢸⠂⠀⣟⢲⡀⠀⠀⠈⠻⣿⣧⠁⢿⡿⠋
⠀⠀⣠⠖⠋⣉⣁⣉⠙⡆⠨⣿⣿⣺⡇⣿⠧⣼⣿⣿⠄⠀⣷⣾⣧⣠⠒⣀⣀⣿⡿⠀⠘⣶⣖⣘⣂⣠⣄⣀⡀
⠀⢰⠃⣴⣿⡟⠋⡍⢡⣷⡀⠱⣿⣿⠁⣿⡆⣿⠿⡿⠀⠀⣿⣿⢾⣟⣃⣿⣿⣿⡧⠀⣴⣯⣭⣩⣭⣽⣿⣍⣿⣷⣦⡀
⠀⠈⠣⠻⠛⢷⠀⢷⢿⢻⢳⡀⠈⠻⡿⡉⠙⣻⣷⢲⠁⢀⣻⣾⣼⣘⣻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⣷
⠀⠀⠀⠀⢰⢮⡇⠸⣮⠁⠀⢟⡢⢤⣸⣅⢠⣿⣤⠿⣰⣿⣿⣻⠃⠙⣿⣿⣟⡟⠀⠀⢸⣿⣿⣿⡿⠏⠫⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠂⠈⠈⡇⠀⢻⡀⢀⣾⡟⠓⣬⢡⣾⡟⠙⠠⡝⢿⣿⡿⢿⣿⣿⣽⣿⣷⣀⢀⣾⣥⣷⣿⣿⠂⠀⣟⠛⣋⢉⠉⠐
⠀⠀⠀⠀⣀⣶⢿⡀⢀⢃⢈⣿⣧⡀⠰⣿⠟⠗⣚⣃⣓⣞⡏⠀⠘⢿⣿⣿⣿⣾⣏⣽⣿⣿⣟⠿⣇⡠⣼⠟⠿⠠⠮
⠀⠀⠀⠀⣈⠁⠐⠉⠉⣤⣤⣼⣿⡿⠋⠙⠻⠛⠁⣮⣙⣿⣇⡀⢀⣸⡿⣿⣧⣉⣭⣍⠉⠀⠀⠀⠐⠀⠀⣀⢀⡈⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⣾⣿⡇⠀⠀⠀⣠⣶⣿⣿⣿⣿⣯⣿⣿⡿⠟⠶⠬⠄⠤⠤⠄⠀⠀⠀⠈⣀⢀
⠀⠀⠀⠀⠐⠀⠀⠚⠁⠀⢱⣿⣧⡣⣄⣠⢶⣟⣿⣿⣿⠟⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⡻⠯⠿⠛⠑⠏⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⠛⠛⢈⣉⣉⠙⠻⢻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣌⣛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⡏⢭⡘⠇⡵⢘⡟⢿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⢿⣿⣢⣿⣧⣦⢠⣶⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠿⠿⠿⠿⣿⣿⣿⣿⢱⢈⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⣠⣪⣥⣬⣿⣷⣶⣦⣄⡀⠀⠀⠈⠙⠛⢻⠀⠸⡏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀
⠀⠀⣀⣤⣾⡿⠛⢛⣉⣙⣛⣉⣉⣛⡻⠶⡄⠀⠀⠀⠸⡃⠀⡇⠀⠀⠀⠀⡀⠀⣠⣤⣴⣶⣿⣿⣿⣿⣶⣌⣦⡀
⠀⣬⡾⣻⣭⣴⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⠆⠀⠀⠀⢠⠁⢰⣿⠀⠀⠠⣿⣥⣶⣶⣶⣶⣽⣦⣒⣶⣲⣶⠮⢥⣙⣄
⠀⢋⣴⣿⣿⣿⣿⣶⣷⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇
⠀⠈⠛⠛⠋⠉⠀⠀⠈⢛⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡄⠀⠀⠸⠿⠿⠿⠛⠛⠿⣝⡿⢙⣭⣉⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⡁⠸⠛⠛⠻⠿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⣿⣝⣛⣿⣿⣶⣄⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠸⡄⠀⠀⠀⠀⣸⠄⢀⣿⣿⣯⣽⣯⡩⣿⣿⣶⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠅
⠀⠀⡆⠄⡀⠀⠀⠀⠅⠂⠀⢷⣦⢸⡆⠀⠀⠀⢺⣋⢸⡟⣿⣿⡇⢸⡇⠉⠉⠉⠉⠀⢀⣿⠐⡆⠀⠀⠘⡷⠇
⠀⠀⠴⠂⠁⣤⠀⠀⠉⠀⣀⠈⠻⣦⠣⡀⠀⠈⣉⣇⣼⡇⠉⠉⡇⢸⢁⠀⢼⡀⠀⠀⣾⡇⢰⠀⠀⠸⡷⠷
⠀⠀⠀⠶⡶⠇⠀⡀⠑⢒⠀⡄⠀⠹⡇⠑⣥⠀⠤⡇⢸⡇⠀⠀⡇⣿⡘⢈⣻⠂⠀⣾⠏⢀⡆⠀⠲⣦⣧⠀⡀⠐
⠀⠀⠀⠀⠠⠤⠉⠃⡀⢈⡀⣴⠀⠀⣹⣤⣌⠳⣄⣏⣹⡇⣸⣠⣧⡟⡇⠐⠷⢦⣾⠏⠈⠈⠀⡄⣠⡅⠀⠶⠀⠵
⠀⠀⠀⠆⠀⢾⡅⢀⣞⡀⠓⢀⣤⣈⢿⡶⢾⡦⡿⣿⣿⡥⢸⣿⣻⢃⡍⢃⣤⡾⢡⠆⣴⢠⠰⢿⡏⠑⡇⡠⡀⡀
⠀⠀⠛⢀⡖⣀⢩⣽⡞⠗⠈⢴⡌⠀⣾⠁⣸⡧⠿⣿⣿⣿⣼⡿⢇⡾⣻⣿⡿⣱⡇⢱⣏⢨⣀⡈⠯⢧⠨⡤⠈⠉
⠀⠀⠀⠀⠈⢉⠆⢊⠸⠘⣤⠘⣋⣷⣿⠀⣿⡀⠀⣻⠟⢻⣿⣴⣾⣿⣿⣿⠁⠉⢀⣿⣿⣿⣾⣵⣄⣼⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⣀⣾⣦⣥⣾⣿⣿⣿⣛⣿⡋⠻⣿⣠⡾⠁⠈⣹⣿⣿⡇⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⠌⠐
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢧⣿⣧⠶⠷⡿⣿⣿⠷⣾⣿⠻⢿⣿⣿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠋⠛⠛⠋⠑⠒⠿⠃⠙⠲⠦⠂⠈⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⡤⣘⣁⡀⠀⢄⡤⡞⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⡈⢀⠋⠐⠋⢣⡭⣤⡾⠁⢀⣀⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠀⢶⡰⣤⠅⠃⣦⡗⢀⢂⡤⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⢉⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠙⠄⠃⠸⣧⠘⠢⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⢻⡶⠓⠆⠀⠋⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠺⠃⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⡄⠀⡀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠈⠂⠀⠀⠀⠀⣼⣅⠂⠀⠼⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠁⠰⠀⠀⠀⠀⣀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠚⡤⠀⠄⠀⠀⡀⣠⠄⢸⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡇⢀⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠⠞⠠⢶⣦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⣈⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠒⠀⢠⠀⠀⠀⡀⠊⠀⠀⠐⠂⠈⠀⠀⠀⠀⠑⠀⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢒⡩⠝⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣄⣠⣴⣴⣟⣿⣿⣿⣿⡟⣿⢿⣿⣆⠀⠀⠀⠀⠀⣀⠤⠄⠂⣀⡄⡒⠙⡡⠒⠁⠀⠀⠄⠤⠤⡀⠈⠃
⠀⠀⢠⣶⣿⣿⣾⣯⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣹⡟⠀⠀⠀⢠⠎⠁⠀⠈⠁⠛⠁⡠⠊⠀⠶⠀⠀⠀⠀⠀⡀⠘⠂
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡶⠀⣀⡀⡀⠀⠀⠂
⠀⠀⠸⣿⣿⢿⣿⣿⣿⣿⣻⣿⣷⣬⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠈⢅⠀⢤⠀⠀⠀⠀⠔⢆⠀⠙⠏⠀⠄⠀⠀⠂
⠀⠀⠀⠈⠻⠿⣿⣿⢿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣤⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠉⠙⢳⣦⣤⠀⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣽⣟⣻⢿⣿⣾⣿⣿⣷⣶⠖⢫⣵⣤⡀⠀⠀⠀⠀⠀⠀⠼⡚⠑⠃⡖⢤⡀⠉⠀⠀⠀⠝⣭⠿⠟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⠠⠀⡀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣿⣽⣭⡿⠟⠋⠀⠀⠀⠀⠀⠀⣦⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠁⢉⡙⠉⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⡄⠅⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠆⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⡀⠀⠠⢀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣿⢿⣿⣿⣿⣿⣷⡧⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣸⡇⢰⡀⢠⠀⠀⠁⣝⣄⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠃⠿⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠨⣛⣾⣷⣿⣶⡀⣰⠟⠉⠀⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣠⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⡿⠿⡿⠛⢿⠛⢿⢻⠻⠿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢛⡁⠀⠤⠠⣬⣔⣵⣄⣦⣄⣨⢞⠁⠂⡀⠉⠙⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣀⣈⣅⢀⣙⣿⢉⢬⠠⢉⣿⣿⣶⣟⣻⣶⣶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣛⠂⢀⠀⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⣶⡴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⡀⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡞⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⡼⠁⠠⢀⣾⠀⠀⢸⡄⣠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣷⠃⣼⠀⡞⡏⠀⢰⠿⡷⢁⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⢿⣿⣄⡿⣾⢡⣵⣧⣷⣾⠗⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠈⠟⠃⠛⠸⠏⠟⠁⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⢐⣘⡆⠐⢀⠀⠀⡀⢠⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢊⠒⠆⡩⣉⠇⠀⠁⠔⠒⠙⠁
⠀⠀⠀⠀⠀⠠⠀⡠⠀⠀⡒⠀⠥⠜⠛⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠢⢠⠀⣤⡄⠰⠚⠂⠀⠀⢀⣤⠤⠔⠊
⠀⠀⠀⠀⠁⢀⡄⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⢀⣤⣈⡦⠄⢀⠁⠄⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢦⡀⠁⠀⢤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠒⠠⠤⡷⠂⠠⠌⣠⠂⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣋⣝⢡⠒⣡⠬⠦⣀⡀⡽⠦⢀⠨⡁⠠⠴⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠀⠀⠀⠿⠒⠓⠒⠾⠄⠽⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠊⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣎⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠛⠿⠁⢙⠆⢠⣴⣥⠶⢀⣠⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠀⠁⠀⠀⠀⠘⠋⡫⣁⡄⠘⠃⠀⠀⠀⠀⠘⠃⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡀⢉⣅⣈⣉⢀⡴⠈⡿⢿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡀⠿⣥⠬⣉⣀⣠⡟⠛⢏⡫⠋⠀⠀⡀⠀⢀⡀⠀⠀⠠⠈⠆⠲⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠲⢢⠀⡉⢈⠡⠌⠀⠀⡰⠈⢀⣄⡀⠀⠀⠀⣀⣀⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⠀⣄⡀⠀⠀⠀⠀⠋⠀⠀⢀⠐⠛⣦⡒⢚⡛⠈⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⣼⡿⠲⣤⣄⣠⣴⢛⡀⠹⠂⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢇⡻⠩⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠏⠀⠀⠑⠈⠉⠻⢛⡛⠛⠛⠛⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣄⣀⡀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠋⠁⠌⠛⢷⣤⡄⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡿⠁⠀⠀⠀⠀⠀⣀⡠⠄⠋⢉⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢀⣀⣰⡄⠀⠢⡈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⢀⡤⢖⣋⣵⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠿⣆⡀⠀⣰⣿⣾⡿⢋⢉⡉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⠴⠲⠶⠿⠿⢿⣶⣿⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⣠⠬⠿⠿⣿⠏⡇⠀⠀⣿⣿⣩⠥⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠒⠠⠃⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠁⠀⠹⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠆⠑⢧⠘⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡇⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠃⠀⠛⠿⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡏⠁⣀⠀⠀⠘⠢⢀⠀⠀⠁⠀⠀⠀⠀⡄⠀⣰⡟⠁⠀⠀⠀⠀⠈⠂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠻⠓⣿⠙⢣⣀⣱⣶⣄⣲⣀⠀⠀⢠⢸⡿⢀⠰⠄⠂⠀⠀⠄⣤⣷⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠈⡟⠛⠁⠈⠉⠁⠀⠀⠸⡿⢟⠁⡀⣀⣤⡦⢄⡀⡀⡽⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠝⣿⣿⠅⠀⢸⣿⣫⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡇⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣧⠀⠀⠀⠀⢰⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢸⡄⠀⠀⠀⢸⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠐⠒⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡶⠙⢻⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠻⠟⠀⠀⠀⠀⠀⣠⣤⣤⣴⡆⠀⠀⠀⠀⠐⠊⠁⠉⠁⣀⠐⠂⢀⠀⠀⠁⠋⠙⠀⠀⠀⠁⠀⣀⣀⣀⣀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡎⢴⡉⢹⡿⠓⠐⠢⡀⠐⠖⠾⠿⠟⢻⠿⠯⠛⠛⠃⠀⠀⠀⠀⠊⠃⠀⠐⠉⠉⠉⠛⢹⠋⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠙⠋⠂⠀⠀⠀⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣄⢀
⠀⠀⠀⠀⣤⡀⣠⡄⡀⠀⠀⠀⠀⠀⢷⣴⠖⣀⣈⠀⠂⠀⠀⠀⠀⡀⣸⡄⠀⠀⠀⠀⠀⠀⠀⣤⢠⡆⠀⠀⠀⣄⡀⠀⣰⠨⠜⡇
⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠈⡝⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⢠⣤⣄⡀
⠀⠀⠀⠀⣠⠴⢒⣶⣦⣄⠀⠀⣤⣤⣤⡤⣶⠶⠶⠶⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠋⠉⠉⠀⢀⠀⠀⠐⠶⠄⠂⠀⠀⢀⡆
⠀⠀⠀⠀⣉⠒⢚⡿⠝⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠙⠲⢶⣤⣶⣤⣤⠜⠉
⠀⣤⡚⠽⠝⠋⢉⡃⣠⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠟⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠈⢉⣻⠃
⠀⠈⠁⠀⢣⠀⢰⡆⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⠇⢀⣀
⠀⠀⠀⠀⠈⣦⣾⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠁⠸⠁⠻⠿
⠀⠀⠀⠀⠈⠛⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⣤⣿⠄⠀⠀⠀⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠑⢶⠁⢐⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⣤⣾⡟⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣎⡇⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣀⠀⠀⠀⠠⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⠏⠁⠀⠘⠛⠀⠀⠀⠀⠀⣠⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⠀⠀⠀⠙⠒⠦⠤⣄⡀⣀
⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠚⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠐⠦⣄
⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⡦⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⢀⣸⡧
⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠞⣉⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣛⣿⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⢤⡴⠶⠒⠒⡨⠛⣉⡠⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣼⣿⣮⣬⣉⣙⣭⡉⠉⠉⠉⣹⡙⠄⣀⣴⣤⣶⡮⠶⠶⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢻⣿⠟⠛⠃⠐⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⠀⢸⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣼⠻⠳⠦⠆⢶⡊⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⣿⣷⣾⣤⠶⠚⠛⢧⠀⠀⠀⢠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡀⡇⠀⣿⡿⠃⠀⠀⠀⠀⠘⡆⠀⠀⡟⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠹⢄⢳⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠹⠀⢸⡇⣠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡎⣾⡙⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⢠⡇⢰⠁⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧⡿⣷⣸⣿⣦⠀⠀⠀⠀⡇⠀⠀⠀⣿⢸⣀⡾⠊⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠑⣯⣿⡏⣿⠀⣀⠀⢠⠀⠀⠐⠀⢻⡞⢠⠇⢰⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⢧⣿⡇⣿⠂⡇⠀⡆⠀⠀⠃⠀⣿⡇⡾⢠⡟⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣾⣿⡗⣿⣰⠀⢸⠀⡄⠀⢸⠀⣿⣿⣵⡾⢋⣉⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⡄⢸⡀⣷⢰⣸⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠷⠛⠛⠛⠉⠉

⠀⠀⠀⠉⠻⣟⠓⢛⡟⠉⠀⠀⠀⠀⠀⢸⠿⣶⢒⡛⢻⡛⢙⣧⢰⡘⣇⠀⠀⠀⠄⠤⠀⢠⠶⢚⡋⡉⠀⠀⣠⠀⠛⠀⢃⢈⠆⢠⡈⠻⣆
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⣻⢸⣷⡞⡇⡎⠘⠜⣧⠸⣆⠀⠐⣦⣴⠀⠈⠒⢬⡙⢡⣄⡆⢨⡄⠱⡀⠘⠋⠀⡈⢁⡄⠈⠛⢲⡄
⠀⠀⠀⠀⠀⢰⠟⢻⠀⠀⠀⠀⠀⠀⢀⡟⢀⠸⡎⠟⣷⢀⡇⢶⢠⡙⣧⠹⣆⠀⠈⠁⠀⠀⠀⠀⠛⠶⢦⣄⡀⠹⠶⠟⠀⠆⠈⠃⢀⠙⠦⠴⠀⠻⢄
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⣼⡷⣼⠀⢿⡀⣿⠘⣿⠸⣧⢷⣘⢧⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣶⣾⠋⠛⠓⠒⠒⠉⠉⠑⠋⠁
⠀⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠠⣿⣧⡏⢸⢸⣇⡟⣠⢟⣼⣧⣠⣽⡷⠟⠁⡠⠶⠒⠶⠒⠦⣄⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠈⠿⠷⠷⢾⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠘⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⢻⠀⠀⠀⠀⢸⡇⠂⠠⣤⡄⠀⠀⠀⣷⠀⠀⠀⠀⣼⣿⡇⢠⣀⠀⡀⠀⣱⠀⣇⡀⡄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡸⣿⡇⠀⠀⠀⠈⠳⠶⢤⣿⣀⠀⢠⡞⠁⠀⠀⠀⢠⣿⣿⠁⠘⠛⠛⠛⠛⢉⠋⠉⠉
⠀⠀⠄⠠⠀⠀⠀⠈⠲⠾⠃⠀⣰⣿⣷⣆⠀⠀⠀⠀⠀⣿⡧⠿⢿⡀⠀⠀⠀⠀⣰⡿⠁⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠉⣁
⠀⢔⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⢧⡉⢉⡼⣣⠀⠀⠀⠀⡏⢠⣦⣤⣷⡀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠠⡀⢠⠤⠔⠀⠀⠀⠀⠀⣿⠀⠋⠍⠀⠸⡆⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣰⡟⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠈⠁⢔⡀⠀⠀⠀⠀⢠⣇⣩⠖⠦⠦⠴⢹⡀⠀⠀⣿⣿⠿⢉⣿⠙⡇⠀⣿⡇⠀⠀⠀⢀⣠⣶⣿⠏⢀⣾⠃⠀⠀⠀⠀⡀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠦⣤⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⠀⢱⡀⠀⢹⡇⡄⣿⣿⣦⣷⠀⠙⠃⠀⢀⣴⣿⣿⠟⠁⢀⡾⠃⠀⠀⠀⣤⣴⣷⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠦⢴⣦⣤⠖⠛⠈⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡿⣁⢂⣴⠟⠁⠀⠀⡠⢿⣿⠟⠿⠿⢛⣷⡆
⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⢸⠉⠈⡀⠀⠀⠀⠀⠀⠘⣿⣇⣉⣉⡿⠀⠀⠀⠸⣿⣿⣷⡴⠟⠁⠀⠀⠀⠈⠢⠤⡤⣤⢤⢄⡈⠐⠛⡄
⠀⠀⠀⠀⠘⣾⠱⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢀⣰⠉⠁⠁⠁
⠀⠀⠀⠀⡄⢊⠈⠀⢄⠀⠀⠀⠀⠀⡇⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠶⣶⠲⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿
⠀⠀⠀⣘⠊⠉⠉⢈⣠⠇⠀⠀⠀⠀⢡⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠷⢏⣦⠸⠟⢡⡍⠛⠓⢄⠀⠀⠀⠀⠀⠀⣸⡧⡟⠀⠀⠀⢀⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠈⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠂⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢁⣀⣤⣤⡴⢴⣶⠶⢤⣤⣄⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣞⣋⣯⢦⢺⡇⠀⣿⡴⣌⢿⣿⣿⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⠻⢿⣷⣯⣼⡇⠀⢸⣿⠽⠿⣛⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠈⠙⠒⠲⠶⢾⠃⠀⢸⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠏⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣾⠀⠀⠀⠀⣠⣴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠉⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢰⠇⠀⢀⡏⠀⣤⡶⢛⠉⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢾⣿⣯⣿⠉⣿⣻⣳⣦⠀⣷⠀⠀⠀⢀⡟⠀⠀⣼⠁⢸⡇⠁⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠬⡏⠀⣯⣭⡭⠥⠞⠁⠀⠀⢀⡞⠀⠀⢠⠇⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡟⠀⠀⠀⠀⠀⢨⡟⢳⡶⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⠁⠀⠀⠀⠀⣠⠏⠀⡼⠀⠀⢀⣀⡴⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠀⠀⠀⣴⠃⠀⣼⠁⣴⠚⠋⠉⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠀⢳⡀⠀⢳⡄⠀⠀⢰⡇⠀⠀⢸⠁⠀⠀⠀⡼⠁⠀⣼⠃⠀⠻⣜⢯⣹⡗⢢⣄⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠛⠶⣄⢳⡀⠀⠙⣆⠀⢸⡇⠀⠀⣼⠀⠀⠀⣸⠃⠀⣸⠃⠀⠀⠀⠈⢱⠟⢀⣟⣺⢟⣠⠟
⠀⠀⠀⠀⠀⠀⣼⠟⠅⠀⠀⠀⠀⠀⣺⠇⢻⡄⠀⠈⢧⣾⢀⠀⡄⣿⠀⠀⢠⡟⠀⢠⡏⠀⠀⠀⣀⡴⠋⢠⠞⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡤⢴⡖⠋⠁⠀⠀⢻⡄⠀⠘⣿⡸⢠⢲⣻⡆⠀⣸⢷⢰⢸⡇⢀⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⣄⠀⠀⠀⠀⢷⢁⠄⢻⣏⡿⡘⣿⣷⠀⣿⣾⣾⢸⣷⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠻⣶⣶⣶⡜⣟⢄⡠⢿⡹⡱⡣⠹⡇⣿⠿⢡⣾⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣶⣶⣶⠏⢻⣃⡆⠊⣿⣴⣥⡼⠃⢹⡇⡴⠰⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢧⣥⡼⠏⠀⠀⠀⠀⠀⠙⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠏⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠉⠁⠀⠀⠉⠁⠂⠓⢤⡁⡘⠆⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢅⠀⠄⣀⠒⠾⠰⠀⣠⡀⠀⡀⡀⠀⢠⢆⠠⢠⠀⠀⣀⣤⡜⢯⣳⣷
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⠐⡶⡤⣤⣄⣀⣠⣁⡤⠀⠤⠤⠀⠠⠄⠁⠉⠠⠤⠀⠋
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⡐⠶⠐⠆⠂⠀⠀⡐⠀⠂⠀⠀⠀⠀⠀⢱⠀⠀⢠⣅⠀⠈⠉⠻⢇⣿
⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠿⣦⠗⠛⠉⠛⢾⣿⣤⡄⠈⠓⠀⠀⠀⠀⢃⠀⢸⠇⠉⠐⣄⢀⣠⣽⡇
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⣸⡃⠀⠀⠀⠀⠘⣿⣆⠐⠂⠀⢯⠀⠀⠀⢸⡄⢈⠀⠀⠀⠀⠈⠀⣿
⠀⠀⡞⠀⠀⠀⠀⠀⠀⣾⡥⣿⡇⠀⠀⠀⠀⠀⠈⢻⣄⡈⠁⠭⠄⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠛⣷⢹⡇⠀⠐⠤⠠⢄⠀⣸⣷⡥⢌⠍⠁⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢊⡚⢻⡆⠀⠀⠀⠀⠛⢻⡇⠝⠃⠀⠀⠀⡀⠃⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠘⡅⠀⣀⡀⠠⠊⠀⠀⠀⢸⠄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠢⣤⠀⠸⣼⠀⠀⠀⠀⠀⣼⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠲⢷⣦⣿⠇⣰⠀⠀⡀⢹⢧⣮⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡜⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⡤⣀⣡⣾⡜⢷⢣⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠰⣀⠏⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠁⠯⡟⡁⢣⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠠⡀⠀⠀⠀⠀⠀⠀⡘⠱⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢦⡆⠟⡧⠹
⠀⠀⠀⠀⠀⠀⡀⠸⢸⢋⠀⢘⡆⠘⡂⣨⠀⣶⢟⣸⡀⢀⠀⠘⠀⠀⡀⠀⡀⣠⢏⡛⣆⠇⢀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠾⠖⠿⣧⡇⠿⡈⡇⡟⠈⡏⠃⠎⠐⡀⢠⣰⡾⢶⡟⠉⠀⢳⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠰⠈⠃⠘⠠⠇⢾⠚⠷⠔⠋⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⡿⢿⡟⡿⠿⣋⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡴⠖⠚⠛⢋⠙⢻⡟⣉⣑⡒⢂⣶⠀⣠⡀⠀⠂⠀⡸⠏⠃⠉⠟⢛⡒⠻⠿⠧⠃
⠀⠀⢀⣠⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣾⠗⠛⠛⠛⠙⠟⠻⢿⡿⣿⡿⠃⢯⡖⠭⣀⣂⣉⣥⣤⣷⣬⣶⣿⠯⠋
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣭⣿⣿⣿⣟⢾⣢⠼⠫⠛⠉⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⢿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠶⠾⠧⢤⣀⡀⠈⣙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢷⣾⣿⣿⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠢⠠⠄⠘⠂⠤⠄⠘⠿⠿⠿⠿⠷⢶⠖⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⠀⢀⣤⠤⠴⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡘⠛⠫⣉⠀⠀⠈⠉⠙⠛⠷⢶⣤⣄⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⠁⠀⠉⠙⠻⢦⣄⡑⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⢁⠤⡀⠀⠀⠀⠨⢿⠃⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠠⠴⠒⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⠛⠛⠛⠛⠶⢦⣜⡆⠀⠀⠀⠐⠋⠁⠠⠤⠀⠞⠐⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠶⢦⡀⠀⠀⠀⠀⢀⡠⠭⣿⠀⠀⣴⠿⠛⠛⠻⠿⠿⠷⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠽⡄⠀⠀⢰⣏⣤⡴⠶⠶⠶⠶⢧⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢷⣦⡀⠐⢤⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠐⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡛⢷⣾⣷⠤⠬⠷⠭⠿⢄⡀⢀⡏⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢟⣫⠭⠖⠒⠊⠉⠉⢙⡿⣿⣿⣧⣤⡤⣤⣤⣀⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⡊⠁⠀⠀⠀⠀⠀⠀⢀⣜⣾⠟⠉⠀⠀⠀⠀⠀⠉⠛⢿⣵⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠧⠠⡀⠀⠀⠀⠀⠀⠀⣠⠤⠺⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⢸⡆⠀⠀⠀⢠⡿⢧⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⢀⣀⣀⠎⠀⠀⠀⠀⡜⠀⢀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⣀⠀⠔⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠀⠀⠒⠂⠠⠌⠋⠙⠫⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢶⣶⣤⣴⣄⣄
⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠀⠀⠐⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣇⠀⠤⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣿⣿⣿⣿⡀⠀⠀⠿⡿⠇⢠⣤⣶⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⣿⣿⠯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠉⠙⠙⠿⠃⢀⣀⢤⣤⣤⣾⣿⠿⠁⠀⠀⠀⣠⣰⠋⢙⣿⣽⣿⠟⣀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠛⠉⣽⣿⣿⠁⠀⠀⠀⠀⣠⣌⣻⡟⠲⠆⠉⠈⠻⠟⢯⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣹⣿⡇⠀⠀⠀⠀⣉⣙⠋⠳⠀⠀⢠⠀⠀⠀⡉⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠀⠀⠀⠀⠙⠛⡠⠦⠀⠀⠈⢳⡄⢾⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣤⣄⢁⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡆⠀⠀⠀⠀⠀⠈⠐⣶⡄⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⣷⠀⠀⢀⡀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⣿⣿⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⢻⣿⣧⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⣿⣿⡆⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠶⡰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡟⡑⢿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢧⣼⠟⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣆⢹⠆⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡡⢚⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣿⣷⣶⣾⣿⣿⠀⠀⠉⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣛⣿⣽⣿⣿⣿⣿⣿⠿⠿⢿⣷⣾⣶⣿⣟⣿⣿⣿⣿⣿⣿⣳⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⣈⠛⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠘⠁⠸⣿⣿⣿⣿⣿⠋⠉⠉⢙⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣟⣤⣯⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⡿⠛⠿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣀⣴⣶⣿⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣻⣏⠉⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠈⢹⣿⣿⠿⣿⣿⣿⣿⡿⠁⣴⣿⣉⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣽⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣁⣸⣿⣿⣁⣨⣿⣿⣿⣿⣠⣤⣤⣴⣿⣿⣤⣿⣿⣿⡟⠀⣀⣉⣿⣿⡿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣧⣾⣿⣿⣿⣿⣿⣉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⣹⣿⡉⣿⣿⡾⠿⢿⣿⡟⠀⢻⣿⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣿⣿⣿⡿⢿⣿⡏⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⡿⠛⢿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⡿⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣾⣿⡟⠙⠛⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣷
⠀⣼⣿⣿⣿⣿⣏⢙⣥⣿⣤⣤⣾⠟⠻⣿⣶⣿⡿⠿⣿⣿⣿⣿⣿⡟⠙⢿⣿⣿⣋⣹⣿⣿⣿⣿⣾⣿⣿⣿⣴⣿⣿⣿⣶⣶⣾⣿⣿⣿⡟⠛⣿⣇⣠⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣟⠛⣿⣿⣿⣿⣟⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⡟⢛⣿⣟⣹⣿⣿⣿
⠀⢻⣿⠛⠉⢿⣿⢿⣿⣛⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡏
⠀⠈⠻⡆⠀⠀⠁⠀⠛⣿⢿⡇⠀⠀⡀⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⠟⠻⠿⠛⠿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠁
⠀⠀⠀⠈⠁⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣷⣄⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⣷⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣶⣷⣦⣀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠾⠛⠉⠉⠁⢹⣿⣿⣿⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠻⠃⠀⣠⣴⣾⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣹⣿⣿⣷⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⢠⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠙⢿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣽⣿⣿⣿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣴⣶⣶⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⡏⠀⠀⠀⠀⢀⠃⠀⢠⠟⠱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠡⠈⣀⢽⣴⣾⣿⣿⣷⣴⣾⣶⣝⠂⡉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢈⠳⣤⣹⣿⡿⠟⢻⣿⣿⣿⣯⣤⣼⡧⣼⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢣⣙⣿⣿⡀⠀⢸⣿⣿⣟⡻⠛⠋⠉⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠉⠀⠳⠛⣿⠁⠀⣼⣿⣿⢿⣅⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣰⣶⣤⣄⠀⠀⠀⠀⠀⠒⣨⠻⣧⣤⣿⣿⣎⣻⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢦⣄⣀⣰⣾⣿⣙⢯⣉⣡⣝⣏⢳⠀⡤⠂⠰⠋⠴⣿⣿⣿⣿⣿⣿⣷⡂⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣿⣟⣻⣯⣿⣿⣿⣿⣿⡿⠟⠁⢠⠀⠀⠲⠀⣦⠾⣟⣫⣭⣭⣭⣽⣿⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠉⠀⠚⠟⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠈⠠⠌⠀⠀⢀⣶⣿⣿⣛⣽⣿⣷⣤⢉⠙⠛⢻⣿⣿⡶⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣙⣿⣿⣿⢹⣷⡿⠂⠀⠀⠀⠀⢀⠀⢠⠙⠃⡿⣿⣿⣿⣿⣿⣿⣿⠖⠀⠉⢸⣿⣿⣿⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⠿⠏⠘⠋⠈⠀⠀⠀⢀⣰⣾⣿⣆⠈⡠⠲⠿⠿⣿⠈⣿⠟⢛⠀⠀⢀⣾⣿⣿⡶⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⢀⣴⢾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠏⢰⠏⠀⢀⣠⣼⣿⣿⠿⣯⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠀⠀⠀⢄⠻⡿⣿⣿⡯⠿⠏⠁⢸⣿⠃⠘⠁⠀⠺⢿⡿⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠈⡿⠃⠈⠀⠀⡿⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣰⡆⠀⠀⠀⠀⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣼⣧⠁⠀⠀⢀⠀⠀⠰⢿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⣼⢻⡏⠀⠀⠀⠆⠀⠀⠀⠀⠙⠿⣿⢿⡯⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣀⠀⣄⠀⣿⣾⡖⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⣸⡇⢴⣿⣴⣯⢸⣿⣿⠇⣾⣿⠁⠠⠀⠀⠀⠀⢠⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⣼⣿⡆⣿⢛⡍⣸⣿⡟⣲⣿⣷⡴⠁⠀⠀⠀⠀⠀⣹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣌⣿⣿⣿⢳⣿⣿⣿⡇⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⡄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⢶⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⡿⠋⠈⠹⠻⡇⠀⣿⡇⠹⣮⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠉⢁⣽⠿⠛⢿⣀⡷⠀⠀⣽⠉⠙⣿⠉⢿⣿⠻⡏⢀⣾⡅⠀⢀⣆⣴⠇⠀⠛⢃⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠉⠑⠀⠀⢺⠹⠇⠀⠀⠉⡦⠀⠘⡇⠈⡿⣄⠀⠄⠃⠀⢀⡞⠉⠉⠀⠀⠀⠸⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠺⠃⠀⠡⠀⠀⠠⠚⠁⠃⠀⠘⠀⠚⠁⠀⢸⠁⢿⠀⢩⡀⠰⢀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠇⠈⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⡶⠶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠿⣯⠀⠀⠀⠀⠀⢸⣧⠀⠀⠉⢹⣿⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⣹⡇⠀⠀⠀⣀⣀⡙⠻⠶⠿⠛⠁⢀⣼⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠻⢷⣤⣤⣴⠟⠁⠀⣠⡾⠛⠉⠻⣷⡀⠀⠀⠀⣼⡇⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢷⣤⣄⣤⣾⠟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠈⣿⣄⣤⣤⣴⣶⣶⠶⢶⡶⠿⠿⣿⡿⠶⠿⣷⡶⣶⣤⣤⣤⣈⡉⠉⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⣶⡿⣿⠛⠙⣏⠀⠸⡇⠀⠀⣇⠀⠀⢻⠀⠀⠀⣾⠀⠀⢠⡏⠉⠉⣿⠿⢶⣦⣄⡈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠁⠀⠙⢧⡀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⠀⣴⠟⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⡒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣾⣁⣀⠀⣼⠀⠀⢸⣿⠀⠀⠀⠀⠀⢻⣧⠀⠀⢦⡀⣀⣷⣤⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⣀⢸⣿⣾⣿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢸⠄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⡀⢸⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢀⢆⠎⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢀⡞⠸⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⡼⠀⠇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠐⠋⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⣤⠀⢀⢀⠀⠀⠀⢠⠀⠀⠀⡀⡀⠀⢰⠀⠀⢰⠀⢀⠄⡄⠀⠀⣀⣆⣀⠀⠠⠤⠄⠀⢸⢀⢻⡆
⠄⣯⠐⠐⠈⠀⠀⠻⠿⠿⠀⠀⢀⠀⠃⢸⢸⡖⢸⠀⠁⢀⡀⠀⠀⠻⠛⠛⠀⠀⢨⡈⠁⢸⠸⢿⡇
⠀⠗⠀⡰⠏⠀⢀⠀⠀⡀⣀⠀⠀⠳⢄⢸⠀⠐⢸⠀⡠⠞⠀⢀⣄⠀⣀⠀⣆⡀⠀⠛⠄⣸⠀⣻⡇
⠀⠋⡀⣄⠀⠐⠹⠁⣿⡆⠹⠋⠀⢠⠄⢈⠁⠁⢉⡀⢠⠄⠀⠙⠋⢸⢹⠀⠓⠁⠀⠤⠂⢌⠉⢹⡇
⠠⠩⠀⠀⣀⣴⡆⢠⣏⢧⠸⣶⣄⡀⠀⠀⠑⠀⡢⠀⠀⣀⣴⡾⠁⠾⠼⡆⠹⣷⣄⣀⠀⠀⠕⢸⡇
⠀⠀⡴⠿⢿⡟⠠⡇⠠⠸⡇⠸⡿⠿⠷⠄⠀⠀⠀⠰⠿⠿⣿⠁⢸⠀⠀⢻⠀⢹⡟⠛⠳⠄⠀⢸⡇
⠠⢄⠀⠀⠀⠁⢸⡇⣦⠠⡇⠀⠁⠀⠐⡔⠲⠀⠔⡔⠀⠀⠉⠀⢸⢰⡖⢸⠀⠀⠀⠀⢀⣄⠒⢸⡇
⢀⠊⠁⠀⠴⠀⠸⡇⠀⠀⡇⠀⠰⠆⠀⠀⠁⠀⠈⠈⠀⠰⠆⠀⢸⠀⠀⢸⠀⠀⠲⠆⠀⠀⠁⣸⡇
⣷⢘⣦⠀⡀⠀⣌⡓⠒⠘⣃⡀⠀⠀⣠⡖⢰⢻⠀⣷⡄⠀⠀⢠⣍⠀⠂⢉⣠⠀⠀⠀⣠⡇⢰⣿⡇
⢿⡀⠉⢀⠠⢤⠈⣿⡿⣾⡏⢰⠄⣀⠙⠁⣼⡼⡄⠉⠁⡀⠤⠄⠹⣿⢿⣿⠋⠴⠄⡀⠙⠁⡼⣿⡇
⠀⣷⠀⢀⢀⠀⠀⠈⢡⠉⠀⠀⡀⡁⠀⢸⠄⠀⢾⠀⠀⡀⡀⠀⠀⠉⡄⠁⠀⢀⠀⡁⠀⢸⠄⣿⡇
⠄⣿⠐⠁⠈⠀⠀⠻⠿⢿⠀⠀⠀⠀⠂⢸⢰⡤⢸⠀⠂⠂⠁⠀⠘⢿⠿⠿⠀⠀⠀⠐⠂⢸⢰⣼⡇
⠀⠗⠀⡰⠏⠀⢀⠀⠀⡀⣀⠀⠈⠳⡀⢸⠀⠀⢸⠀⢀⡼⠁⠀⡀⠀⠀⠀⡀⢀⠈⢦⡀⢸⠀⢽⡇
⠀⠋⡀⣄⠀⠐⠻⠁⣾⡆⠹⠛⠀⢀⠤⠘⠒⠒⠚⠀⢅⡀⠀⠺⠟⢰⢺⠀⠿⠒⠀⢀⠀⠚⠒⢺⡇
⠠⡉⠀⠀⣀⣴⡆⢠⣭⣧⠰⣦⣀⠀⠐⠈⠢⠀⠎⠁⠀⠀⣠⣴⠂⣸⣸⡄⢲⣤⡀⠈⠀⠉⠢⢸⡇
⠀⠀⡴⠿⢿⡟⢀⡇⠀⢸⡆⠹⡿⠿⢦⡀⠀⠀⠀⣠⡶⢿⣿⠃⢰⠀⠀⢷⠀⢻⣿⣷⣦⡀⠀⢸⡇
⠠⢄⠀⠀⠀⠁⢰⡇⣦⢀⡇⠀⠁⠀⠀⡠⠤⠀⡤⡠⠀⠀⠉⠀⢸⢠⣤⢸⠀⠘⠁⠀⠀⠠⡤⢸⡇
⢀⠊⠁⠀⡴⠀⠘⡇⠁⠀⡇⠀⠰⡄⠀⠉⠂⠀⠐⠙⠀⢠⡄⠀⢸⠈⠁⢸⠀⠀⢤⡀⠘⠑⠀⢸⡇
⣷⢘⣦⠀⠀⠀⣘⡃⠒⠘⣃⡀⠀⡀⢀⡆⢠⢾⠀⣦⡀⠀⠀⢀⡺⠂⠆⠺⠀⠀⠀⠀⢀⡄⢠⣾⡇
⢻⡀⠉⢀⠠⢤⠘⣿⣶⣾⡟⢠⡄⠀⠛⠃⣸⣸⡀⠙⠃⠀⣠⡈⢻⣶⣦⣼⠟⢀⡀⠀⠿⠃⣸⣿⡇
⠀⠃⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠁⠀⠐⠀⠀⠳⠀⠀⠁⠀⠀⠀⠛⠈⠋⠀⠁⠀⠃⠀⠠⠉⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⠀⣴⣯⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣂⡀⠁⠀⠀⠉⠛⠛⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠻⣿⡿⠓⣸⡇⣴⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣒⣲⠿⠭⠭⠭⣟⣛⣽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣭⣭⣭⡍⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣭⣶⣿⣟⣛⣿⣛⣿⣷⣄⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣫⣾⣿⣝⣃⣿⣿⣿⣿⣿⣿⣿⡿⢷⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣼⣿⣯⣽⣿⣿⣿⣿⡟⠷⢸⣿⣿⣧⡀⠹⣷⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣼⢱⣿⣿⣟⣂⣿⣿⣿⣿⣷⣿⠿⠿⠿⢿⣷⣼⣧⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣼⡜⢿⣿⣿⣿⣿⠿⠋⣡⣴⣶⣶⣶⣽⡪⣽⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣬⣭⣥⣖⣷⣾⡟⠉⠀⢨⢙⢿⣿⣾⣿⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠁⢸⣿⣿⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣭⣶⣶⢏⣉⡉⢦⡀⠀⢀⣠⣾⡿⠋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣯⣴⣿⣿⣿⣿⣿⣿⠿⠟⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣛⠉⠉⠉⠀⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⡴⡶⡛⣻⠉⠉⣿⢍⢟⢻⢶⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⡌⡜⡰⢸⡄⠀⣿⡌⣆⣣⠷⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠀⠀⠀⠙⠲⠭⣝⣛⣻⠁⠀⣸⠭⠽⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡟⠀⠀⠀⠀⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⡇⠀⢠⠇⢀⡤⠖⠊⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢶⣟⣹⡟⠉⣟⢒⣶⡄⢸⠄⠀⠀⠀⡞⠀⠀⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⡇⠀⣯⣭⠭⠴⠋⠀⠀⠀⡼⠁⠀⢰⠃⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⡏⠀⠀⠀⠀⠀⡝⠙⡶⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⢀⡏⠀⠀⡾⠀⠀⠀⠀⢀⡞⠀⣸⠁⠀⣀⡠⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢧⠀⠀⠀⠀⣸⠀⠀⢠⡇⠀⠀⠀⢠⠎⠀⣰⠃⢰⢋⣁⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣄⠀⠳⡀⠈⢳⡀⠀⠀⡏⠀⠀⢸⠀⠀⠀⢠⠏⠀⣰⠃⠀⠈⠮⣓⣮⠋⣱⠦⣄⠸⡄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠙⢦⠹⡄⠀⠹⣄⢸⡇⠀⠀⢸⠀⠀⠀⡟⠀⢠⠇⠀⠀⠀⠀⡸⠃⣰⠯⠭⠭⠞⠁
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⢀⣀⡠⠞⠀⠹⡄⠀⠘⢾⡇⠀⠀⢸⡀⠀⣸⠁⠀⢸⠀⠀⢀⡤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⢇⠀⠀⠀⠀⢳⡀⠀⠘⡇⠀⠀⠈⣇⠀⡟⠀⠀⣿⣠⠖⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠳⢤⣀⣀⡀⢧⠀⠀⢳⠀⠀⠀⢹⡄⡇⠀⠀⢸⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣹⠺⡆⠀⠈⢧⡀⠀⣰⠃⢧⠀⠀⢸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⢧⣀⣀⡞⠉⠉⠁⠀⠘⠦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢈⣡⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⡿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢤⡀⠒⠶⠾⢦⣔⢾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⣤⣤⣤⣈⠻⢶⣶⣦⣶⣶⣤⣄⣤⣤⣴⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣿⣿⣿⣿⣧⡄⠀⠉⠛⢻⣿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠶⢂⣤⣴⣶⢶⣾⣶⣶⣦⣾⠀⢠⡿⠛⣻⣟⣠⣿⠷⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡌⢿⡿⠟⠀⠃⠀⠈⣠⣾⣩⣹⡿⠑⡆⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⣶⠄⠀⠀⠀⠀⠀⠿⠿⠻⠛⠏⠁⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢦⠀⠸⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⡰⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣠⣶⣧⣄⠃⠀⠀⣸⣶⡀⢸⣄⠀⢀⣠⠎⠀⠊⠁
⠀⠀⠀⠀⠂⠀⢣⡀⣰⡈⠛⢹⣿⣯⠤⢦⣿⣿⣷⣼⣧⣖⣼⡃⠄
⠀⠀⠀⠀⠀⠀⠻⠋⠉⣿⣿⣻⣿⣿⣷⣄⣿⣿⢉⠙⡀⠟⠋
⠀⠀⠀⠀⠀⠀⠺⣟⠁⠈⢻⢛⣾⣿⠋⡙⣛⠋
⠀⠀⠀⠀⠀⠀⠈⠟⠓⠀⢠⠨⠅⠈⠁⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣸⣄⠀⢆⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⢀⡀⠀⣤⣤⡤⣤⣄⣀⣀⣉⡀⢀⣀⡀⣤⣴⣶⣎⣽⠿⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⡤⢴⡖⠀⠀⠁⡶⠂⣘⣿⡟⣫⣿⣿⣿⣿⣿⣼⣿⣾⣿⣿⣷⣌⣀⣀⡾⠛⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢤⣴⣶⣶⣶⣷⣾⣷⣴⡀⣼⡟⢿⣿⣃⠙⡼⡿⠻⣹⣷⠟⠻⠛⠟⣿⠿⢻⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣥⣠⡀⠀⠀⠀⠀⠀⠠⠄⢀⣴⣾⡟⢾⣿⠛⣿⡍⢻⣻⣟⣿⡋⠀⠉⣻⢇⠀⠀⠰⠁⠀⠀⠀⠐⢀⣿⣶⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡌⠂⠠⠄⠐⠆⢀⣤⣾⣿⣟⠛⡿⢿⣦⣈⣿⣦⡝⢿⠃⢁⡀⢠⣿⣿⡔⣠⣤⡄⣀⣤⢤⢴⣿⣿⣿⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣴⣿⣿⣿⠛⠯⠙⠝⠢⠴⣻⣿⣿⢿⣷⣿⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣟⡽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠉⠉⠻⠀⠀⡴⢦⡲⢤⣼⠿⣿⣿⣿⣿⣿⣷⡿⢿⡯⣿⣛⠛⠻⠿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠛⠛⠛⢩⡿⣿⣛⣷⢿⣧⣬⣿⠻⢶⣴⣷⣤⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⢺⣶⡿⡟⠻⣿⠿⢻⣿⡦⢀⠀⠀⠀⠀⠀⠀⠈⢲⣿⡿⣿⠿⣿⣿⣏⣿⣦⣄⣤⣈⣿⡂⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣤⣿⣿⣿⣏⣻⠿⢿⠇⣰⣧⡈⠳⢘⡁⠺⣆⠀⠀⢠⠀⠀⠀⠈⠉⣸⢷⣌⢨⣍⠻⣿⣤⣿⣛⠿⣿⠿⣣⣄⣰⣼⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠛⠳⣤⠀⣿⢹⣇⣧⣏⢀⡺⠋⠀⠀⠀⠀⠀⠀⠐⡿⠄⠸⣏⠰⣿⣷⡯⠿⢌⡛⢲⣶⣿⣭⡿⠟⣣⣈
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣯⡏⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠘⣄⠿⢆⣿⣿⣿⣷⣿⣴⣿⣫⠿⠛⠷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣯⣿⣿⣿⣿⣿⠟⠛⠛⢻⣿⣿⣶⣤⣀⠀⠉⠁⠀⠄⠐⡲⠾⠉⣳⡾⠁⢉⠻⢿⣿⣿⣿⡍⠀⠀⣠⠯⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠤⣮⣁⡁⠉⠉⠑⠀⠀⠀⠀⠀⠀⠋⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠁⠀⠉⠁⠀⠀⠀⢸⠏⠛⠛⢶⣶⣶⣶⣶⣦⣶⠆⠤⠶⠶⣦⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠻⠁⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠠⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⡀⠇⢀⠀⠀⠀⠀⠘⠉⠁⢐⣾⣄⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠂⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⡁⠈⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣭⣶⣿⣾⣶⣶⣶⣿⡀⠀⠀⠀⠀⠸⣄⠀⠃⠀⠀⠈⠒⠀⠠⠄⠀⠀⢘⣤⣌⣀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠛⠟⠛⣏⠙⠛⠃⠀⠀⠸⠿⠄⠈⠠⠀⠗⢀⡀⠰⠄⠀⠀⢀⡰⢻⠉⠉⠛⠙⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡴⠞⠀⠀⠀⢭⣉⡄⠀⢀⡀⡜⠉⢀⣀⣴⠦⠞⠀⡘⠀⣀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⢠⠐⠴⢇⣀⣬⣽⡇⠄⠈⠀⡉⠀⣿⣠⣴⣿⣿⣦⠈⠹⣷⡄⠀⠀⢠⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⢤⡀⣤⣄⢀⡀⢶⡁⠈⡀⢀⡈⣫⠙⢡⢂⡠⠒⢸⡇⠀⣿⣿⣿⣿⣿⣿⡄⠀⢹⣷⡀⠀⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢤⣤⣱⣿⠸⠏⠀⢀⣁⣦⡟⣧⠀⣷⣄⣿⠞⠁⢠⠶⠀⡞⠿⢿⣿⣟⣿⣿⡇⠀⣼⡿⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⠷⠆⠛⠍⣈⣳⣶⣈⣽⢟⣤⡖⠋⠀⢠⣴⣤⠘⣏⣹⡿⣿⣿⣥⠾⢋⡄⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⠛⠙⠛⠁⠀⠀⡜⠿⣿⠿⠿⠻⠿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠚⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣻⣽⣾⣽⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⡇
⣿⣺⣿⣿⣿⣟⣷⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⣿⣜⣿⣿⣿⣿⢟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠛⠛⠛⠛
⣿⣿⣷⣯⣯⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡰⠊
⠀⠀⢠⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⣶⣶⣦⣤⣶⣦⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣀⣀⠀⠀⡀
⠀⠀⠀⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣐
⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠉⠉⡭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸
⠀⠀⡶⢴⣤⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠋⠨⠕⠀⠘⠔⠬⠓⠳⠀⠀⢸
⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣸⣇⣠⣶⣿⣶⡄⣤⠀⠀⠀⢸
⠀⠀⣴⡶⣶⠠⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣤⢏⣹⠿⠿⠇
⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣾⠟⠛⠛⢿⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣀⣀⣀⣘⣋⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠡⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⢶⣶⣤⡀⠀⢹⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠹⣿⣿⡄⠀⠃⠀⣐⡺⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠠⢀⣴⣿⡿⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠠⠶⠶⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⣄⣠⡴⠏⠀⠀⠀⢺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠂⠙⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣘⡇⠀⠀⠀⠀⠀⠂⣀⢦⣴⣆⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠾⣦⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠛⣿⣏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⣀⠀⠀⢀⠀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠈⠀⣘⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠎
⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⣠⣀⣀⠀⠄⢄⣠⣤⣤⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠒⠒⠖⠐⠒⠒⠠⠄⠒⠚⠓⠋⠉⠉⠉⠉⠙⠳⢖⡀⠀⠀⠀⠀⠀⠀⠀⢩⡏⠙⠻⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⡀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠓⣀⠀⠉⢃⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠠⠄⠤⠤⠄⠤⠬⠑⠎⠉⠁⠘⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⠈⠁⠈⠀⣀⣀⠀⠀⠀⠀⠀⠦⠰⠢⠠⠀⠀⠀⠀⠀⠠⠈⠁⠢⠄⠀⡀
⠀⠠⠀⠈⠀⠘⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠄⠀⠀⠂⠀⠐⠤⠀⠄
⠀⠀⢀⠀⠀⠀⠠⣀⡠⠀⠀⠀⡰⠀⠀⠀⠲⠇⠀⠀⠾⠀⠀⠀⠂⠀⠀⠱⠔⠀⠀⠘⠂⠀⠀⢨⠄⠀⢤⠇⠀⢀⣨⠁
⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠈⠓⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠢⠌⠀⠂⢤⠀⡼⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⢀⡀⢠⠠⠀⠶⡒⠇
⠀⠀⠐⠁⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠀⠀⢢⣄⠀⠦⠀⠀⠀⡈⠀⠀⢁⠀⠀⠀⢠⠇
⠀⠀⠀⠈⠀⠀⠀⠑⠦⠤⠤⠞⠀⠀⠀⠀⠈⠃⠀⠠⠴⠀⠀⠀⠀⠀⠠⣸⢿⠀⠀⠀⠀⠀⠐⠄⠠⠜⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠠⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃
⠀⠀⠀⠈⠐⠂⠀⠀⡀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠄⠄⣤⠄⠄⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠂⠂⢸⡇⠒⡄⠆
⠀⠀⠀⠀⠀⡀⡀⠀⠁⠰⠀⠂⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠜⢹⠓⠚⠩⣤⡆⠀⠀⠀⡀⠀⣺⣁⠔⢀⠀⠄
⠀⠀⠀⠈⠀⠙⠃⠉⠀⠀⠀⠀⠄⠀⠠⡴⠃⠀⠀⠀⠀⠙⠊⠀⠀⠘⠂⠀⠘⠶⠚⠐⠚⠀⠀⠀⠉⢀⣤⣽⠇⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⢀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠢⢸
⠀⠀⠀⠀⠀⠀⠀⡰⠉⢀⠐⠁⠀⠀⢠⠀⠉⠒⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢁⠀⠀⠀⢐⣀⡤⠰⢤⠱⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⢀⠇⠀⠀⣈⡤⠴⠖⡛⢩⣴⣅⡴⡿⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠄⠁⠀⢠⠀⠊⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠸⠀⣠⠊⣁⡠⠖⠈⠀⠀⠍⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢀⡤⠴⡲⠊⢩⡶⠃
⠀⠀⠀⠀⠀⠀⠑⠛⠂⠐⠒⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⢀⡠⠔⡚⢋⠅⠀⠾⣷⠖⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⣤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣏⢒⢒⣊⣥⡀⠀⠈
⠀⠀⠂⠠⢄⠀⠀⠀⠀⠀⠀⠀⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⡃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡔⠤⠚⠙⠀⠀⠀⠀⠀⢀⡠⠒⠉⠠
⠈⠀⠀⠈⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⣀⠄⠒⠈⠀⠀⠈⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡤⠄⣀⣀
⠓⠦⠤⢤⡀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡔⠋⠀⠀⠐⠀⠀⠀⠀⠠⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢳⢤⣈⣤
⠀⠑⠴⡤⠈⠲⣄⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢴⣋⠑⡲⠦⣄⡀⠀⠁⠀⠀⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠙⠋
⠠⡙⠓⠒⠖⠒⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢝⠶⣾⠷⠀⠍⠙⡒⠦⢄⡀⢀⣃
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣌⣒⣒⣒⣹⠄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢤⡀
⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠤⢤⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠂⠉⡈⠈⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠐⠒⠀⠠⣄⡀
⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⠀⠐⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠀⠁⠀⠀⠀⠄⡈⠐⠤⡀
⠀⠀⠀⡼⠡⠀⠀⠀⠁⠀⣀⡤⡤⡖⢒⣶⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡀⠀⠀⠀⠀⠉⢐⣁⢤⠠⢤⡰
⠀⠀⢠⢃⠀⣠⡴⠖⠒⡉⠁⠰⣮⡶⢞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣤⠴⠖⠚⢩⣶⣧⡴⡾⠊
⠀⠀⠸⡀⣞⠓⠀⢚⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⡅⢠⠒⠁⠤⠲⠊⠀⠀⠙⠉
⠀⠀⠀⠈⠃⠉⠉⠉⠉⠑⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠐⠚⠒⠒⠒⠛⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⢄⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡁⠀⠀⢃⠠⠀⢀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠠⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⣄⠘⡔⠲⠤⣀⡀⠀⠄⠈⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠝⣻⠿⠃⠀⠢⣉⠙⠳⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣍⠍⠁⢿⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣶⣦⣀⣙⡛⠽⠿⢖⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠋⠟⠋⠿⣟⣷⣦⣤⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢾⡔⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣶⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠉⠉⠉⠁⠈⠉⠉⣹⣷⣍⡓⠠⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⠁⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣉⣀⣀⣀⡀⢀⡀⢀⢀⣀⣀⣤⣶⡾⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣗⣾⣧⢟⣾⣿⠛⣟⢹⣿⣿⣧⣿⣿⣿⡿⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠉⠉⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⣀⠠⣤⢶⡟⢿⣝⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⢀⡀⢸⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⠃⠀⠀⠈⠀⠀⠀⣈⠥⡓⠏⢟⣿⣿⣿⣿⣧⠀⢀⣤⣶⣶⣶⣦⣤⡄⠀⢳⣤⣾⠁⠀⠀⠳⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⢃⠀⠀⡠⠂⣰⡚⢧⡐⣬⣶⣿⣿⣿⣿⢿⠃⣴⣿⣿⣿⣿⣿⣿⣿⡷⠒⢻⣧⡈⣷⣿⣿⣦⢹⣷⡄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠀⣿⠅⠀⠀⠈⣙⣷⣿⣷⣟⣿⣿⣿⡏⠀⠘⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⢿⣿⣿⣿⡿⣸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⢀⡀⢀⢴⣿⣻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣯⢿⡯⠜⠀⠀⠀⢠⣿⣿⣿⠻⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣄⠰⣿⣷⣵⣿⣿⡿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠤⠌⠉⠙⠛⠋⠀⠀⠀⠀⣠⣿⣿⢿⠦⠇⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠙⠜⢿⡟⣡⣤⣤⣤⠶⣠⡶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣛⡘⠂⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣼⣶⣿⣾⣯⣿⣣⣿⣿⣿⣿⢫⣶⣦⡤⢤⣤⣄⠤⠀⠀⠀⢸⣿⣯⣿⠇⠁⠀⣀⡼⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣿⣹⣿⣽⣿⣿⣳⣶⣦⣤⣤⣤⣼⣷⣾⣭⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⡌⠙⠛⠛⠋⠠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣷⠀⠀⠀⠀⠀⢻⣿⣿⣍⡉⣿⢻⡿⢿⠿⡿⣿⢿⡖⠖⠉⠰⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⠻⡏⠹⠻⢷⣶⣧⣬⣀⣅⣘⣀⣁⣀⣤⣾⣿⣿⣿⣷⣄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢻⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠛⡻⠟⠿⠿⠿⠙⠛⢦⠙⣿⣿⣿⣿⡆
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠧⢦⠀⡤⠄⠀⠀⢀⣴⡿⣿⣿⣢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢶⣄⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣿⣶⣿⣿⡿⣿⣗⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⣿⣶⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠙⠿⣍⣉⠀⠉⠉⠉⠉⠁⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⠶⠒⠚⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⠻⠛
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣴⠉⢱⠮⠙⠛⠶⠲⢤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⡾⠷⠂⠀⠀⢰⣷⡤⢾⣫⡘⠟⢢⣽⣋⡻⣯⡴⢛⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠏⣦⠰⡴⣚⠿⢿⣧⡿⠻⠤⠟⢲⣋⠀⠀⣟⠀⣉⢿⡟⠀⠙⠁⠉⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢐⣠⠋⠀⢦⣵⠶⠛⠁⠀⠀⠀⠀⠘⢯⣷⣆⠈⠃⠽⣦⠙⢛⣧⣙⡀⣝⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢰⡥⠞⠒⠀⢴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣦⣤⣿⣈⣦⣽⡿⣇⠉⠀⣼⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣤⣹⡳⢄⡰⣿⠞⠁⠀⠀⠈⠙⠛⠀⠦⣄⠀⠀⠀⠀⠈⠛⠉⠉⠁⠈⠙⢿⣧⡆⣽⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⠋⡎⢉⣁⣸⡏⠀⠀⠀⠈⠁⠈⠁⠀⠂⠄⣉⣀⠂⡂⠀⡀⠀⠀⠀⠀⠀⢨⣻⠧⡟⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣤⠛⡾⢱⢻⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠠⢀⠀⠀⠀⢿⡋⡴⠡⠚⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢾⣃⣜⠀⠳⣿⠞⠁⠀⢀⣶⣶⣭⣶⣷⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠴⠻⣅⠂⠨⣯⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠙⣿⠁⠜⣇⠀⢀⠋⣈⡡⠤⣭⡙⠿⠛⢧⠀⠉⠀⣃⣤⣤⣗⠒⠆⠀⠀⣻⣾⡗⢦⣤⣼⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣽⣿⣏⡔⠃⠀⠙⠂⠉⠂⠠⣧⠿⠂⠁⢨⠀⠰⣾⠟⢉⣉⣈⠹⡷⠀⠀⣿⣿⠉⣖⠻⣮⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡸⣧⠙⠉⢋⠞⠀⠀⠛⠲⠄⣠⡤⠶⠀⠀⡜⠀⢸⡏⠀⠛⠈⢩⣿⣦⣄⣀⠿⠁⣆⣽⠏⣙⡟
⠀⠀⠀⠀⠀⠀⠀⠈⡿⣎⠀⠀⠸⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣠⠃⠀⣾⠀⠑⠀⠀⠌⠉⠉⠉⠀⢰⠀⠛⢠⣾⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠀⠀⢳⠀⠀⢠⠞⠁⠀⠀⠀⡆⢸⡇⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠐⣿⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠺⠀⢠⡎⠀⠀⢀⡰⠃⠉⠋⠻⣶⣾⣿⣷⢄⠀⠀⠀⠀⣀⠄⣼⠃⠀⣰⠿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⠀⣸⠃⠀⠀⠁⢀⣄⣀⣀⡰⠅⠈⠉⠀⠈⠆⠀⠀⠘⠁⠀⣿⠀⠀⠈⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⡏⠀⠀⠀⢠⠎⠉⢍⡉⠛⠚⠓⠦⣤⠀⠆⠀⣴⠃⠀⢸⡇⠀⢀⠚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡀⠀⠀⠀⠀⠀⠉⠀⠀⣤⠈⠉⢉⡉⠁⠸⠄⠀⡰⠋⠀⠀⣸⠃⢀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⢀⣼⠃⠀⠀⣰⠟⠀⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠉⠛⣄⡀⠀⣀⣀⣀⠀⣼⣷⣶⡿⠁⠀⠀⠀⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠠⠀⠀⠈⠉⠉⠉⠉⠙⠛⢻⠟⠉⠀⠀⠀⣠⠞⠀⠀⣾⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⣸⡃⢌⠒⢤⡉⡁⠀⠀⠀⠀⣀⣠⠾⢀⡴⠒⠈⢉⡕⢀⣠⣾⣿⣿⡿⢿⣆⣀
⠀⠀⠀⠀⠀⠀⣠⢖⣿⠋⠀⠴⠁⡇⠀⠙⠶⣄⠁⠁⠀⠀⠀⠉⢀⣴⡀⠀⢀⣠⣿⠶⠛⠉⠁⠀⠀⢀⣾⣻⣿⠿⠦⢤⠤⢄⡀
⠀⠀⠀⢀⡴⢋⣴⠟⠁⠀⠀⠀⡴⢇⠀⠀⠀⠀⡉⠒⠶⠶⠶⠞⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⠞⠛⠋⠀⠀⠀⠀⢹⣎⡭⠒⢦⣀⣀⡀
⠀⠀⣰⠋⢀⣾⠃⢠⣄⣀⣤⠊⠀⠈⣦⠀⠀⠀⠉⠓⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⡤⠚⠀⣀⠤⠚⣿⣷⣄⠐⣻⣭⡙⣷
⠀⣼⠃⢠⣿⣏⠀⠀⠀⠀⠈⠑⢤⣀⠈⠣⡀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠉⠀⠀⠀⡀⠔⠉⣠⣰⡵⠉⠀⣼⣿⢻⣿⠻⣿⣮⡙⣿
⡼⠁⢠⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠉⠢⠀⢰⡄⠀⠀⢀⣠⠴⠞⠉⠁⠀⠀⠀⢀⣠⠖⠈⠀⢀⠄⣰⠟⢀⡴⢡⣿⠏⣸⣿⠀⢻⡌⢿⣿
⢇⠀⢺⠉⣿⣿⡍⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⢀⡤⠐⠋⠀⠀⣠⠔⣡⠞⠁⡠⠊⣰⣿⣿⢠⣿⡟⠀⣿⡄⢸⣿
⢸⡆⢸⠀⢻⣿⣻⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣾⠡⠚⠁⢠⠞⢀⣴⣿⡿⠁⢸⣿⠁⢠⣿⠇⢸⣿
⠘⣇⢸⡆⠈⢿⣿⠹⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡔⠶⠛⠉⠀⠀⣀⠔⠁⣠⣿⣿⡿⠁⢠⣿⣏⠀⣾⣿⣥⠟⠁
⠀⢻⡄⢧⠀⠘⢿⣧⠀⠉⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⢀⠰⠒⡃⠀⣀⣼⣹⣿⡟⠀⢀⡿⠛⠃⣸⣿⠏⠁
⠀⠀⠹⣄⠳⡀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠚⠒⠛⠒⠒⠒⠂⠀⠈⠛⠃⠒⠉⢀⠔⢁⣱⢿⠞⠀⠀⣘⣁⣠⡴⠛⠁
⠀⠀⠀⠈⢧⡙⢄⠙⣿⣷⢤⣀⠀⠀⠶⠦⠤⠴⠶⠂⠒⠒⠚⠛⠛⠃⠂⠐⠒⠀⠃⠀⢀⡤⠖⠁⣠⠞⣡⠋⣠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠙⢦⡱⣄⠉⠳⣌⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡠⠔⠊⢁⣠⣴⣞⡡⠜⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢮⣉⠀⠈⠀⠀⠈⠛⠲⠖⠒⠘⠛⠉⠉⢠⡤⠶⠒⠛⢉⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣭⡶⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣲⣶⣶⣿⣛⣻⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣸⣯⣹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠒⠉⠉⠉⠉⠒⠦⣄
⠀⠀⠀⠀⣀⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⡞⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢶⠉⠉⢩⠉⠓⣄⠀⠀⢣
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣠⠞⢣⠀⠠⡇⠀⢸⠀⠀⢸⡄⠀⢸
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⡇⠀⠀⠀⢀⡾⣁⠀⠈⡇⠀⡇⠀⡌⠀⢠⢾⠁⠀⡌
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇⠀⠀⢀⠎⠀⠈⠳⣄⢸⠀⡇⡸⢁⡴⠃⡎⢀⡼
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣧⢰⠀⠀⠀⡞⠀⠀⠀⠀⠈⠦⣳⡷⣷⣋⣠⡞⣠⠋
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⡹⠀⠀⣠⡇⣶⠸⠀⠀⠀⡗⠒⠒⠒⠒⠒⢒⡺⡷⢬⠳⣮⠞⠁⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⢰⠃⠀⠀⠀⠀⠀⣠⠶⡅⠀⡔⢁⡠⠞⡱⢡⠇⠀⢇⠀⠀⢇⠀⠀⠀⢀⠔⠉⢠⡧⣾⢷⡼⣆⠀⠀⠀⠀⢀⡴⡏⠀⠀⢣
⢀⡏⠀⠀⠀⠀⢀⣜⠁⠀⢣⣸⣖⣉⣠⡜⣡⠏⠀⠀⠘⢦⡀⠈⠓⠦⠤⠯⠖⣚⠭⠊⠀⠀⣇⢸⠀⠀⠀⠀⣾⢰⠁⠀⠀⠀⠑⢄⡀
⡸⠀⠀⠀⠀⣠⠋⠀⢉⣩⢕⣿⠓⢴⣫⠔⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠉⠀⢀⣠⠤⠤⡟⢰⠓⠲⡤⢼⣴⣾⠀⡴⠂⠒⢦⠀⠙⠦⡀
⡇⠀⠀⠀⢰⠗⠚⠉⢉⡴⣊⢾⠷⣴⠋⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⢀⡤⢴⠇⣸⠒⠠⢧⣀⣈⡟⠀⢇⠀⠀⡸⠀⠀⠀⠈⠲⡄
⢧⠀⠀⠀⠘⢤⣤⠴⠟⢊⠕⠋⠀⠈⢦⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠞⠁⢀⠎⢠⠃⠀⢠⣟⡭⡻⠁⠀⠀⠉⠉⠀⢰⠊⠉⠑⡄⠈⢦
⠈⢦⣀⢀⣀⣀⣀⠤⠚⠁⠀⠀⠀⠀⠈⣇⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⡼⠁⠀⢠⠎⣠⠏⠀⠀⡞⢉⠔⠁⠀⢀⠤⠢⡄⠀⠘⠦⣀⠴⠃⠀⠘⡆
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢳⠀⠀⠀⠀⠀⠀⣼⠀⢸⠁⢀⡴⢃⡴⠁⠀⠀⠀⣏⠁⠀⠀⠀⠘⠤⠴⠃⠀⠀⢀⠖⠓⡄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢇⠀⠀⠀⠀⠀⡇⠀⢈⡴⠋⣠⠎⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠢⢤⣀⠀⠀⠀⠈⠢⠜⣁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⢆⠀⠀⠀⠀⢧⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⢆⠀⠀⡠⠋⢀⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⢳⡜⠁⡴⠻⡀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⡎⠀⣾⡀⠀⠱⡄⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⡸⠀⠙⢦⠀⠙⣄⠀⠱⡄⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⣀⣀⣀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠷⡄⠀⠀⠳⡀⠈⢆⠀⠱⡀⠀⡴⠋⠀⢀⡠⠖⠋⠁⠀⠀⠀⠈⠳⣄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⢆⠀⠀⠹⡄⠘⡆⠀⢧⡞⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠉⣧⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠳⡀⠀⠙⡄⢳⠀⢸⠀⢀⡞⠀⠀⠀⠀⡠⠴⠲⠾⣉⣉⣉⣈⡇⠀⣿⠉⣉⠒⠭⡙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠹⡄⠀⠹⡞⠀⢸⢀⡎⠀⠀⠀⠀⢸⣖⠒⠲⢤⣤⠖⠒⢩⠋⠉⣇⣠⠤⠟⠒⠛⠢⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢱⠀⠀⠀⠀⠘⡆⢀⡇⠀⡇⡼⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠑⠒⠚⠒⠉⠁⠀⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⣇⠀⠀⠀⠀⢘⡞⠀⡼⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⡄⠀⠀⣠⠎⠀⣼⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠹⣀⠜⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣡⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣠⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⢂⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⣭⣭⣭⣭⣁⡉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡎⣿⣿⠿⡉⣵⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢺⣿⡞⠓⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠟⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡄⠀⠉⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣵⣤⠀⠀⠀⠛⠃⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣿⣿⡷⠒⠀⠀⠀⠀⠀⠤⠄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣶⣶⣶⣶⣴⣦⣤⣶⠤⢶⠶⠶⠞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢶⠄⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡤⣿⣿⣿⣿⢿⣿⡗⢾⢷⡸⠿⠛⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣧⣿⣿⡟⣁⠸⢿⣷⣶⠈⠛⣠⣶⣿⠇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣷⣦⣉⠻⣿⣿⠿⣋⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣿⣿⣿⣿⣿⣿⣿⠷⢈⣅⠚⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⣿⣿⣿⣿⣤⠀⢠⣾⣿⣿⣿⣦⡍⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣱⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠛⣿
⠀⠀⠀⠀⠀⠀⢀⠀⡀⣠⣄⣤⣠⣤⣤⣤⣼⣧⢸⣽⣿⠿⣿⣿⣿⣿⣶⡶⠦⠤⠐⢿⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠱⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣷⣶⣶⣶⣶⣆⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣭⣉⣉⣉⣉⣉⣁⣀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠛⠳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠭⣒⠒⣿⣽⣾⣷⠿⢿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢌⣉⣉⠉⠀⠴⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⢒⣒⣒⣒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢋⠉⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠈⢙⡄⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⣎⣉⣩⠗⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠴⠚⠛⠒⢦⡀⠀⠉⠀⠀⣀⡤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣄⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡎⠀⠀⠀⠀⠀⣠⠇⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠈⢦⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠑⠦⠶⠒⠒⠋⠁⠀⠀⠀⣏⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠈⠒⠒⠉⠉⠉⠀⠀⠀⠀⣠⣄⣀⠈⠲⠶⠶⠖⠚⠁⠀⠀⠀⡠⠤⣌⢧
⠀⠀⠀⠀⡾⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢸⡾⡆
⠀⠀⠀⠀⡇⠀⠀⠸⡅⠀⠀⠀⠀⠀⢀⣠⡤⠚⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⠀⠀⠀⢀⡴⠲⢤⡀⠀⠙⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠂⠀⠰⠒⠊⢉⣠⠴⢖⡋⠉⠙⢯⠀⢹⠀⣯⢸⠀⡜⢀⠎⠀⡠⠋⣩⠛⢒⣦⣈⠳⣄⣀⡷⠀⠀⠀⠀⡧
⠀⠀⠀⠀⢹⡀⠀⡤⠴⢤⠀⠀⣠⡖⠋⠒⠤⣀⡉⠲⢄⡈⢣⠸⡆⢸⢸⢰⢣⠏⣠⢞⡴⢋⡤⠚⣉⡠⠜⠳⣄⠀⠀⠀⡠⢄⢠⠇
⠀⠀⠀⠀⠀⠙⢆⠃⠀⠜⠁⡼⠡⠬⢍⣛⣒⠦⣍⣙⠒⠽⣦⣱⡧⠴⠶⠾⠧⢼⣟⣵⠾⠵⢒⣋⡥⡤⠖⣒⣙⣇⠀⡎⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢷⠒⠒⠲⠒⠒⠛⢛⣛⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣛⣛⣋⠭⠭⠥⠤⢠⡟⢀⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣻⣮⣉⣉⠩⠭⠶⢚⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣦⣤⠶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠓⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠄⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢴⠶⣚⠛⠂⢤⣶⠖⣿⡵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠀⣀⡤⠄⠒⠂⠀⠐⠁⠀⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠑⢦⣄⢀⡀
⠀⠀⠀⠀⠃⠀⢀⣰⠆⢐⡢⠠⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠉⠁⠀⠁
⠀⠀⠀⢀⡠⠔⠋⠱⠚⣉⣀⣞⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠉⠀⠀⢀⡀⠀⢉⢀⢈⠄⡀⣀⠀⢔⠀⣴⠃⢀⣀⣀⣀⣐⣦⡀⢠⣤⣄⣀⣈⠣⡤⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠛⠁⢠⣇⣀⣶⣿⣷⣿⣠⣾⣦⡟⠰⣿⣿⣿⣿⣿⣿⣿⣏⠙⢿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣿⠟⠁⠈⢿⣿⣿⠃⠀⣿⣿⣿⣿⡿⠟⠋⢈⣾⣿⣭⣿⢻⡁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠻⣭⡀⠀⠈⠛⠛⠉⠀⠀⠀⣾⣿⣿⠀⠀⠀⢳⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣶⢰⠶⣤⣶⣾⡤⠀⠀⠉⣁⣀⠀⠲⠄⠉⠉⠉⢰⣇⣀⡍⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠟⠀⠀⣼⠿⣿⣿⢿⣶⣿⣿⠄⠙⠒⠂⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣀⣶⣤⡄⢠⣿⣿⣿⡟⣾⣿⡆⢻⣿⡞⣿⣿⠄⠀⠀⣾⡇⠨⠨⠠⠤⢹⡿⠛⠿⠆⠐⠦⠤⠤⡄⠶⠒⠈⠁⠁
⠀⠀⠀⠀⠹⡏⠀⠈⠉⠁⠀⠈⢻⣿⡜⣿⣷⡄⢹⣿⡈⠻⡀⡀⠀⠁⠐⠀⠀⢀⠀⣾⢁⣿⣤⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⣷⣿⣼⠷⠀⢢⡁⠀⠀⠀⠀⠀⠀⠀⢧⠸⣿⣏⣼⣤⣴⠂⣀⣤⣶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⣿⡻⣿⣿⣿⡿⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣼⣶⡿⡟⠛⠛⠁⠾⠟⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢰⡶⣾⣿⣷⣦⣍⠁⠉⠉⢋⡿⢶⣦⢄⣤⣤⣤⡾⣷⣾⠾⠿⠻⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣰⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠂⠀⠀⠀⠐⢲⣛⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣠⢄⣀⣤⣾⣿⣎⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⢤⡎⠁⠸⣿⣿⡿⣿⣿⣆⠀⠀⠀⠀⠀⣰⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⢧⣿⣿⣿⣆⠀⠀⠀⠀⢳⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣶⣶⣾⣿⣿⣿⣇⡾⣼⢟⣿⣿⣿⣆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡿⠞⠅⣠⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢏⡵⠀⠀⠀⠀⣸⣿⣿⡿⠟⠋⠀⢰⣇⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡏⠈⠋⠂⠀⠀⢀⣾⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢀⡀⡰⠃⢀⣾⢇⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡟⠁⠉⠀⠀⢀⣾⠏⣾⣿⣿⣿⣿⣿⣿⣿⣟⡿⣬
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢾⣧⣾⡆⠀⣴⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⡘⠂⠋⡀⣸⢿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢸⣷⣎⢠⣵⡿⣋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣻⣛⣋⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⢠⠞⠛⣄
⠀⠀⠀⠀⢀⡴⣆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢹⡄
⠀⠀⠀⠀⠈⢧⡜⠀⠀⠀⠀⠀⠘⣿⣿⠿⠟⠛⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣋⣽⣿⣧⠴⣶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣰⣿⣆⣀⣠⣤⣁⣤⣠⣶⣄⣀⣄⣉⣈⠁⢀⣀⠐⠂⠈⠛⣦⣄
⠀⠀⠀⠀⣰⣏⡴⡛⠛⠉⠀⠈⠪⣻⠁⠙⡭⡹⢹⣹⢹⡙⡏⣉⢋⢋⠟⡯⢴⣦⣀⡉⡤⠈⢣⡀
⠀⠀⠀⠀⡟⠁⠀⠀⠄⢀⠂⠄⡐⠈⢶⣄⠌⣱⡾⣏⣧⣕⣼⣏⣡⣪⢊⡴⠓⠉⡹⠶⣗⣄⠀⢱⡀
⠀⠀⠀⠀⢷⡀⡀⠀⠀⠠⠬⣒⣬⣑⣦⣜⡷⠿⠷⢿⣿⣿⣿⣿⣥⣿⣥⢤⣴⣮⣴⣵⣮⠿⣌⣋⠡
⠀⠀⠀⠀⠀⠻⣤⠒⣒⠺⠷⢶⣬⣽⣻⣿⡇⠀⠀⢰⡟⢿⣿⣶⣦⣤⣄⣀⡤⠍⠉⢉⡬⠟⢻⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⣥⠶⢒⣛⣻⣿⠀⠀⠀⢸⠀⢸⣧⣙⠡⡉⠙⡻⠽⢍⠛⡻⠶⠄⣼⡛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⠀⠀⠀⠀⡆⢻⣿⣿⡀⠀⠀⠈⠂⡀⣑⣌⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⡇⣼⠻⢽⣧⠴⠖⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠠⣶⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠂⣤⡀⡻⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⣵⣿⣾⣿⣿⣷⣮⣬⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⡀⣤⣀⣼⠟⠋⠁⠸⠸⡟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣗⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⣀⣻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣭⣭⠀⣡⣤⣭⠽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⠂⠀⠀⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⠘⠃⠀⠀⢸⠆⡀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠾⣓⡲⠤⠤⠤⠤⠤⣶⣋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠓⠶⠦⠤⠤⠤⠤⠤⠶⢶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⠂⡅⠀⠀⠀⠀⠁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⢑⠀⡒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⠠⠀⠀⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣶⡏⢛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⢀⠀⠀⢀⣀⡠⠤⠤⢈⣉⠉⠉⠉⠙⠋⢹⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⡇⠀⠀⡴⢉⣀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⡇⠀⠰⡇⣿⣿⣷⠀⢰⣾⣶⡄⢳⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠳⣍⠛⠁⠀⠘⠿⠿⢃⡜⠀⠀⠀⢸⣿⣻⣿⣿⣿⣽⡇⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠠⡀⠈⡇⢰⡶⣴⡆⢰⠋⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣿⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⣿⡇⢰⠁⡈⠒⠙⠋⠙⣋⡙⠚⠀⠊⢸⡀⠀⢸⣿⣷⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣇⠀⢄⡀⢀⡉⠅⠒⠊⠭⢛⡀⠀⠙⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⢿⣷⣦⣵⣉⡀⠀⠀⠀⠀⢀⡈⠣⣈⠁⠀⢸⣿⣿⣿⣿⣿⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡉⠻⠿⣿⣿⣿⣿⣶⣶⣾⣾⣶⣶⣶⣶⣿⣿⣿⠿⠛⢁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣌⣉⣉⠉⠛⠛⠛⠛⠛⠛⢉⣉⣁⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⡤⠄⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠁⠀⠀⠀⠀⢰⠀⠀⠠⣠⡶⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠉⠉⠀⠀⠠⠤⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠉⠉⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠇⢨⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢎⣁⡀⢢⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⣂⡤⠀⠀⠢⠈⠇⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠃⠀⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⡔⡀⠀⡠⡀⠀⠀⠀⠽⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠁⠀⠀⠉⡀⠘⠁⠱⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⡨⠀⠀⠠⠀⢀⡀⠀⠀⠄⠂⠀⠀⠀⢸⠀⢐⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠒⠒⠊⠉⠀⠀⠀⠂⣀⣀⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠂⢠
⠀⠀⠀⠀⠤⠦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠢⠔⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⣳⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠈⠉⠉⠉⠉⠀⠀⠙⠒⠲⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⢠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⠿⣿⣦⡀
⠀⠀⠀⢀⡴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣾⣿⣿⣿⣦⡀
⠀⠀⢀⣾⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢘⠿⣿⣿⣿⣷⡀
⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣿⣶⣿⣿⣿⣿⣷⡀
⢰⣿⣿⣀⣄⡀⣤⠄⠀⢀⣀⣀⣠⣤⣤⢤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣼⣽⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⡯⠯⠷⡻⢿⢋⡍⣩⡟⠕⠋⠵⠛⢵⡿⣿⣿⠃⠀⢸⣿⣿⣟⣒⢯⠊⡉⡒⢬⢒⠨⡛⢿⣽⣿⡻⣿⣿⣿⣿
⠻⢿⣿⣧⣤⣨⣄⣅⣍⣨⣫⣠⣬⣨⣮⣴⣯⣼⣿⡇⠀⠀⠀⣿⣿⣃⣲⣁⣋⣪⣪⣢⣳⣕⣌⣳⡕⣌⣻⣮⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣉⢩⡙⡡⠛⣏⡉⠛⡛⠳⢚⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣷⡦⢞⠷⣾⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⡶⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠎⠀⠀⠀⠀⠁⢣⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⣈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⣄⠄⣰⣤⣴⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⢿⣿⣾⣿⣿⣿⣿⠻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣁⣙⣟⣟⣙⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⡖⠛⠛⠛⠛⠛⠛⠛⡶⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠈⠉⠳⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠦⠚⠉⠀⠀⠀⢀⠖⠋⠉⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠉⠀⠀⠀⠀⣀⠠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⠀⠘⣧
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠱⡄⠈⢧
⠀⠀⢠⡟⢀⡠⠔⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠘⣧
⠀⠀⡾⢀⠎⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⢀⣠⠜⠁⠀⠀⠹⡆
⠀⣸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⢿
⠀⡟⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢸⡇⠀⠀⠈⠲⢄⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠻⣄⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⢈⣳⣤⣀⣀⣠⣤⣵⣦⣤⣤⡤⠤⠤⠤⠤⠤⠤⠾⢥⣤⣤⣤⣤⣾⣁⣀⣀⣀⠀⠀⣠⠞⠀⠀⠀⣠⠄⠀⠀⠀⢀⣼⠃
⠀⠀⠈⠻⢶⣤⣤⣤⡤⠴⠾⠟⠛⠛⠛⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠉⠉⠉⠛⠛⠛⠶⠶⠿⢥⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠾⠿⠦⢤⣤⣤⣄⣠⣤⣤⠴⠞⠛⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⢚⣤⠓⠂⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢨⠯⠅⠀⢺⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣠⣴⣤⣦⣄⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⡏⠉⡏⠉⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⢾⡀⠀⠁⠀⠃⠀⢸⡇⡏⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠹⡀⠜⡌⢷⣧⣤⣤⣤⣶⠏⡴⡘⢰⠋⢙⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠃⠀⠹⣌⠈⢆⢻⣿⣿⣿⠏⡜⠁⣴⠋⠀⠈⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠴⢤⣄⠹⣦⠈⢦⢻⣿⢏⠞⢀⡾⠃⠀⡄⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⣓⠀⠀⠉⠓⣼⣷⡴⠶⠛⠺⠤⣿⡁⢀⡾⠁⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢹⣴⢶⡶⠞⠛⠉⠁⠀⠀⠀⢀⣀⣀⣩⣿⣧⣤⣄⣀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡟⠀⢸⡁⠀⠀⣠⠔⢀⣠⠞⢹⡏⠹⠁⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⡀⠀⢸⠁⠀⠘⣧⠶⠋⣠⡴⢿⣧⣀⠘⣷⠀⠃⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⢤⣤⣀⣹⣷⣾⡋⠀⠀⣿⠟⠷⣿⣇⣀⣀⣧⣀⣤⣤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠒⠉⠁⠀⠑⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣼⣿⣄⣀⡀⠀⠀⠀⠰⢜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠀⠉⠙⠛⠿⢿⣿⢷⣶⣾⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠁⠹⡿⠯⠀⠀⠀⢈⡉⡤⠖⠈⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠨⣦⡀⠀⣠⣮⣄⣀⢣⠀⠀⣴⠚⠻⢿⣶⡦⠤⢤⡀⠀⠀⢷
⠀⠀⠀⠀⠀⣤⣤⣤⣶⣶⣶⣶⣶⠶⠯⢷⣄⠉⠃⠀⢹⢛⡠⠤⠌⡑⠦⣀⡽⠃⠀⣴⣿⣷⢤⣸⣆
⠀⠀⠀⠀⠀⠓⠀⠭⠟⢻⠁⢈⣗⠒⠚⠋⠁⢀⠀⠀⣼⠋⠀⠀⠀⠘⡄⣠⣇⠀⢠⠋⠤⠀⣀⣬⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⡻⠀⠀⠠⡀⢸⠀⢠⣅⣀⣀⡀⠀⢀⡇⡹⠉⠑⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡆⣧⠀⠀⠀⣵⢸⠀⠀⣏⣿⣉⢹⠿⠟⢡⠁⠀⡘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡇⠸⣿⣛⣺⠃⢈⡀⠀⣧⠁⢘⠋⠅⡠⠃⠀⡰⠁⣼
⠀⠀⠀⠀⠈⣦⠸⡵⣎⠭⢧⣿⠀⢻⡁⠙⣴⣾⡇⢰⠇⠀⢸⣾⠏⠁⢀⣴⠁⠠⡋⠠
⠀⠀⠀⠀⡀⠈⢀⢼⣿⡀⠀⣿⣀⡈⣧⢠⠹⣿⣧⣿⡄⣲⣺⣤⣶⣶⣟⠁⢠⡄
⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣴⡞⠓⠢⡀
⠀⠀⠀⠀⣰⣿⣿⣿⠻⢿⢟⡻⠟⢛⢻⣏⢉⢻⣄⣀⣈⣙⠿⠽⣿⠿⠿⣿⣏⣾⣿⣷⡆⠉
⠀⠀⠀⠀⠊⠉⠁⠀⠀⠉⠉⠁⠈⠈⠉⠉⠉⠓⠛⠛⠀⠚⠛⠛⠻⠛⠛⠛⠛⠛⠋⠑⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠉⡉⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⠀⠀⠀⠀⠀⠊⠀⠀⠠⡉⢒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣁⢄⠀⠐⠠⣇⣙⢘⣭⡁⠀⠀⠀⠄⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⠄⠁⢀⣶⣿⡇⠙⢻⣿⣿⡿⠶⠄⠀⡠⠼⠥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⣀⣀⣀⡁⠀⠂⠓⠀⠘⡟⠙⢑⢀⡰⠊⠀⠀⠁⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢤⢒⠈⠉⠓⠠⡄⣐⠀⠁⠁⠀⠀⢰⡅⠀⠀⡴⢃⣀⣤⣴⣶⣶⣶⣶⠶⣬⡣
⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠠⠀⠀⠔⠀⢀⠙⡦⣤⠃⠀⣸⡿⠄⢾⣋⣉⣈⡋⣧⠀⠹⠻⢝⠒⠛⠁
⠀⠀⠀⠠⢪⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠉⠀⢸⠀⠀⢇⣉⣼⠤⠤⣾⠁⣠⠎⢀⡟⠀⠤⣧
⠀⠀⠀⠋⡁⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⡶⠒⠀⡠⠃⠀⡞⢁⣀⣠⣬⣯⠟⠃⣤⠌⡿⢠⠔⠋
⠀⠀⠀⠐⠠⠐⠂⠉⠙⢛⡋⠀⠀⠈⠫⠐⠀⠀⠀⠀⢀⢂⡈⢛⣏⠜⠁⣠⠞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢠⡔⠰⣠⠋⠀⡔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢃⠀⠀⠈⠀⢠⣸⠀⠀⢿⣀⠔⠁⢀⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⢈⢣⠀⠀⠀⠀⠹⢷⡤⠘⠻⠀⣠⠏⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⢢⣷⡤⢄⠀⠠⠤⡍⠀⣶⡿⠻⠶⡉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⡏⠐⠂⢤⣁⣀⣠⠗⠻⠠⣤⠒⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠅⣀⣠⣤⠤⠛⠃⠒⠒⠒⠛⠊⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡂⠈⠀⠀⠀⠀⠀⠉⠁⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠂⠀⠀⠀⡶⢶⡆⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠘⠏⢸⠃⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⣿⡀⠀⢰⠆⠀⠀⣬⣄⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠻⠿⠿⢷⣶⣶⣶⣿⣶⣿⡿⠿⠿⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡶⣶⣶⣶⣤⣤⣤⣶⣾⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠤⠛⠛⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢈⣉⣉⡡⠀⠲⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠘⠋⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣮⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⡿⢋⣼⣿⠁⠤⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠙⣿⣷⡔⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣟⠁⢮⡍⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣿⣿⣮⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣿⣿⣶⣦⣥⣄⣐⣒⣒⠒⠂⢀⣀⣒⣒⣀⣀⣀⣁⣤⣼⣿⣿⠛⢻⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⢸⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⠿⠛⡡⢮⡙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⡡⠒⠂⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠒⠒⠒⠬⠭⣍⡻⣿⣿⠸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⡟⢰⠃⠀⣀⡀⠀⢀⣤⣤⡀⣷⢠⠞⠳⢀⡴⣤⣠⡀⢀⣹⡾⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⣿⠰⣿⠉⠻⡆⣼⢁⡈⡇⣿⠈⠳⡄⣸⣇⢸⣿⡇⢸⣩⡆⣿⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⣿⠀⢿⠀⢠⡏⢻⡌⢩⡇⣿⢠⣀⡽⢿⡇⣸⢻⣇⢸⠀⢰⣿⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⣿⠀⢸⡷⠋⠀⠈⠛⠛⠁⠉⠀⠉⣀⠈⠛⠋⠾⢸⣇⠀⣼⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⡇⣿⠀⢸⣇⠀⠀⠀⠀⡠⠂⠈⠉⠈⠉⠻⣄⠀⠀⠘⠋⠸⠛⣿⡇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⡜⢱⠃⠀⠘⠉⠀⠀⠀⣰⡅⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠙⣾⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡌⣧⠀⠀⠀⡤⣄⠀⠛⡇⣴⣷⠀⠰⣿⠇⡿⠀⠀⢀⠀⠀⣼⢡⡎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢸⣧⢸⡀⠀⢰⠁⢻⣄⠀⣆⡀⠁⢴⡦⠀⠀⠀⡀⠀⣸⠀⠀⢻⣿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⢸⡇⠀⠀⠙⠣⢌⣑⠮⣉⠣⢤⡤⢆⣠⣈⠀⠈⠁⠀⢼⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⣿⠀⣧⠀⠀⠀⠀⠀⠀⢉⠲⠯⣐⠺⠷⠍⢀⣀⠀⠀⠀⡸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣇⠹⣄⠀⠀⠰⣌⣀⢀⡰⠞⠉⠉⠓⢦⣄⣠⡿⠀⣠⢃⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⡯⣿⣿⣷⣌⡙⠢⢄⣀⡉⠋⠁⠀⠀⠀⠠⣤⣌⠭⠤⢘⣡⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣇⢻⣿⣿⣿⣿⣷⣶⣶⣬⣍⡓⢦⡴⠊⣩⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣍⣉⡛⠛⠿⠿⠿⠷⠶⠶⠶⠶⢾⣟⣛⣋⣩⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⢿⣾⠿⡇⠀⢠⣄⡀⢠⢀⣴⣤⡶⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⠛⠉⠁⠘⡅⠀⠁⠁⠈⠙⠻⠀⡎⠉⢘⠋⣤⠀⢸⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⠀⡄⠁⠀⠀⡘⠈⠀⠘⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⡄⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⡀⢿⣇⢠⣷⣇⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠸⠳⠿⣿⡷⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⣹⡄⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠤⠂⠶⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠐⠀⠀⠸⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠏⠁⡀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠰⢧⢰⡄
⠀⠀⠀⠀⠀⠀⢰⣆⡆⠸⠃⢠⠀⠀⠿⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠘⣾⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣅⠠⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠧⠄⠀⠀⠀⠀⠀⡆⡆⠀⣿⣶⡄
⠀⠀⠀⠀⠀⢰⣾⣿⠈⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠐⠆⠀⠀⣀⣤⣾⣿⣿⡿⠟⠛⢃⡀⣲⠀⢳⠰⢀⣿⣿⣷
⠀⠀⠀⠀⠀⣼⡟⡟⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⣶⣦⠶⠆⠀⠀⡀⣀⣀⣀⣤⣤⡶⣶⣾⣿⣟⡻⡯⣖⡯⠛⠀⠀⠀⠀⡠⠁⠀⠸⠆⠈⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣷⡆⠀⡃⠀⠀⠀⠀⠀⠀⢀⣴⠃⠛⢀⣠⠶⣿⣿⣭⡙⣄⣸⣼⣷⣿⣿⣿⡥⠚⣋⠅⠀⠀⢀⠀⠀⠄⠀⠀⠐⠀⣄⠀⢿⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣤⣖⠛⣉⣩⣽⣿⣿⣿⣿⣿⣿⣿⠿⠛⠰⣶⣿⣿⣿⡿⣶⣾⠶⠁⠀⠀⠀⠀⠀⠸⢸⣼⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠛⠋⠉⠭⠖⣫⡿⢛⣁⣤⣀⣀⣰⣾⣿⣿⣿⠿⠟⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⡆⠛⠀⠀⠀⠸⢿⣿⣿⣿⣾⣦⠄⠀⢂⢀⣴⣾⣷⣿⣿⣿⠛⠹⢏⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠒⠒⡟⠋⠉⠉⠉⠉⢻⣡⠀⠀⠀⠀⠀⢀⠆⠀⡄⠀⠀⠀⠀⠀⡠⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣀⣀⠀⠞⠄⠀⠀⠀⠀⠀⠀⠆⠀⠈⠄⠀⠀⡀⠀⠘⡟⠀⠀⠀⠀⠀⣈⠣⠠⠇⠀⠀⠈⠀⡀⠀⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡉⠳⣄⠈⠀⢠⡀⠠⡀⠀⠀⠀⠐⠒⠂⠀⡀⡜⠀⡷⠀⠀⠄⣰⣿⠁⠀⠀⠀⠀⠀⠀⢄⣡⠞⣷⡏⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣶⠆⠐⢲⡀⠉⠀⠹⣰⣃⠀⠀⠀⠀⠀⠘⡇⡄⣿⡆⠘⢸⣿⣿⣧⠄⠀⠀⠀⠀⠀⠈⠁⠀⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⢠⣛⡁⢰⣤⠀⣿⣿⣖⣀⣤⣤⠀⠚⡁⠀⠘⣿⠇⠀⠹⣿⠿⣷⣶⣦⣀⣀⣀⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⣹⣴⣦⣿⡞⢿⣿⣿⠟⢻⣿⣝⣴⠁⠀⢀⡿⠂⠠⡅⠠⣰⡟⠻⣧⠋⠉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢾⣿⣽⣷⣯⠛⠿⣾⡁⡀⢤⣿⣦⣄⢈⣤⠖⡁⠀⢠⣿⠀⢲⣿⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢇⢃⡀⢠⣿⣿⣿⣿⣿⡏⣿⡄⢱⣾⣿⡿⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⣍⡻⢭⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⢶⡾⡿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣿⣾⢯⣿⡟⣦⣤⣤⣤⣴⣶⣾⣟⢿⠛⠋⠛⠋⠈⠈⠁⠁⠛⠏⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠊⢹⣿⠿⣫⣤⠉⠉⠙⠛⠙⠉⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠍⠁⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⣤⣾⠁⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠩⣽⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠽⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⢚⡟⣟⣟⣻⢿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⡿⢿⡷⢶⣶⣶⡶⣶⡿⠛⣋⠉⠀⠀⠀⠐⠉⠉⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣄⠀⠀⠂⠠⢏⡎⡈⠀⠁⠀⠁⠀⠀⠀⢀⣀⣠⠤⠴⠖⠒⠛⠹⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⠻⣦⣤⣄⣀⣁⣁⣀⣠⣤⣤⣶⣶⣿⣿⣥⡀⠀⠀⠀⣀⣴⣶⣶⣬⡻⣿
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣿⣿⣿⡀⣀⠀⠻⢿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⢹⡏⣴⣷⣿⣿⣿⡟⢟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣧⠀⠀⠹⢿⣿⣿⣿⠿⠟⠁⠀⠈⣰⣿⣿⡼⢿⣿⡿⠈⢢
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⣿⣶⣴⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣻⣇⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⣾⣄⠀⣾⣷⣿⣿⡿⠋⠀⣀⣠⣄⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠛⠋⠈⠁⣶⠦⣤⣤⡏
⠀⠀⠀⠀⠀⢠⡟⣿⣿⣷⣾⣿⣿⣿⣶⣶⣾⣿⣿⣿⡟⣷⣶⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⢸⡇⢻⣿⣿⡇⠀⠉⠉⠉⢹⢫⣿⣿⣿⠟⠯⠛⠛⠛⢻⣿⣿⣿⡶⢲⠀⠋⠘⠀⠉⠙
⠀⠀⠀⠀⠀⠀⢿⡀⠛⣿⣿⣶⣤⣄⣀⣠⣷⣾⠟⠋⠀⣠⣶⣿⣿⣶⣦⣝⠹⠛⣾⡐⠂⠘⠊⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠛⠙⠛⠛⠋⠃⠁⢀⣠⡴⠟⠑⠇⠱⠄⠠⠘⣈⣳⣦⡀⢻⣶⡞⠚⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠶⠚⠛⣏⠁⠀⠀⠀⠀⠓⠀⠈⠀⠒⠀⠞⠻⣆⠘⣿⣄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⠹⣷⣄⡸⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣠⣤⠤⠴⠶⠖⠒⠒⠒⠲⠦⠤⠤⠷⠮⠛⠿⠶⣶⣶⣶⡶⠛⠁

⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄⠠⣤⣴⣄
⠀⠀⠀⠉⢛⣂⡸⢻⣟⣷⣢⣞⣩⣟⢦⡀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⢀⣴⣤⣀
⠀⠀⠀⠀⠈⠉⠤⠌⢙⠫⠟⣿⠛⢹⣞⣷⣤⣄⠀⠀⠻⠟⠀⢀⣤⠖⡝⣟⢻⣁⣀⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠸⠁⠦⣈⠉⣩⠉⠁⢀⡔⢺⣿⣻⣯⣿⣬⢟⣽⡿⣿⣿⠙⢇⡠⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠘⢷⡀⠃⢚⡀⠸⡀⠆⠸⡁⠧⠐⠦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣤⡦⠀⠀⠀⠘⠀⠃⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠈⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠈⠮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢘⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣀⣀⡡⠤⣀⠤⠄⠀⢀⣀⢀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠩⢀⣀⣱⣤⣴⣤⣦⣤⣬⠄⠙⠛⢷⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⣐⣒⠢⢽⣽⣿⡏⢸⣿⣿⣿⠿⣛⣛⣀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢈⣁⠤⠒⣡⢞⣿⠀⢸⡏⢿⡯⣑⠲⠀⠤⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠤⠤⣞⣁⡞⢹⠀⢸⡈⢀⡈⠢⠅⠒⢀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⢠⠿⠛⠀⠀⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠉⢦⣼⡄⢸⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⣡⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠀⠀⠈⣿⠀⠘⠀⠀⠀⠀⡜⢻⠂⠀⠀⠀⠸⣌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⣹⡄⠀⠀⠀⠀⣸⣭⢠⣬⣤⣤⣤⣽⡾⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢟⢻⠠⠃⠀⠀⠀⠀⠳⠤⢦⣟⡿⣿⣿⠂⣄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡶⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⡇⠀⡸⠀⠀⠀⠀⢀⠜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡚⡯⠀⠀⡇⠀⠀⠀⡠⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠃⣀⡔⢀⣠⠴⠊⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣁⣀⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⣿⠿⣿⣦⣀⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣶⣾⣷⣨⣿⣿⣿⣛⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⠛⠛⠛⠉⠉⠉⠉⠛⠻⢿⣿⣧⡀⠙⢷⡄⠘⣿⣿
⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⡀⠀⠈⢿⣿⣆⠀⢿⣤⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢰⣿⡿⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢻⣿⣇⢘⣿⣿⠋⠻⢶⣶⣶⠶⠿⠛⠋⠉⠉⢉⣽⠏
⠀⠀⢀⣿⣿⠁⣰⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣾⣿⠿⠶⣤⣤⣀⣀⣀⣤⣴⠶⠿⠿⠿⠋
⠀⠀⢸⣿⡇⣴⣿⠟⠛⠛⠟⠋⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠘⣿⣿⣿⠁⠀⠀⢀⡄⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣄⠀⢀⣾⡿⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⢋⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣷⣄⣀⣀⣀⣴⣿⣥⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣤⣤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣿⣿⣍⢌⠁⣾⡙⢳⢁⡐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢋⠛⣾⠛⠃⢟⠃⣠⠈⢷⣠⠀⠀⠀⠄⡀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⣸⢏⠐⠀⡼⠉⢀⣀⣌⠷⢸⣧⠈⠉⣀⣀⣀⣤⣩⣤⣰⡶
⠀⠀⠀⠀⠀⢀⣼⠁⠀⠂⠓⢀⣌⣁⣤⣶⡾⠿⢿⣿⣿⣻⣿⣿⠩⠛⠋⠁
⠀⠀⠀⠀⣀⣾⣤⣶⣶⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠟⣹⠋⢳⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⡴⠁⠀⡉⠀⠙⡀⢠⡀⠠⡀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠂⢀
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢰⣤⣠⡈⣋⠀⠀⠉⠃⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁
⠀⠀⠀⠀⠠⠈⠁⣀⣠⡶⠷⠍⠙⠈⢀⠀⣤⠀⠈⠀⠀⢈⡀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠒⠀⠀⠀⠀⠘⠿⢙⠣⣄⠙⠂⠀⣄
⠀⠀⠀⠀⠀⠰⡛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠆⠀⠀⠈⠐⢫⠿⠙⣦⣄⣤⡶⠀⠈⠙⠻⢿⣤⡀⠀⠲⣄⠀⠀⠀⠾⠃⡀⠁⠚⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠛⡀⣐⠀⠀⣌⢧⡄⣠⣄⠛⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⡀⠀⠌⠁⠙⠳⣤⡘⢿⡄⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⣤⣾⠿⠶⢻⡿⠶⠷⠹⡟⡿⠛⠘⠃⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣾⣥⠄
⠀⠀⠀⠀⠈⠉⠀⠀⢀⡾⠗⠒⠀⣿⣺⣿⡼⠦⠸⡎⢻⠟⠂⠖⠀⠀⠐⠀⠀⠀⠀⠐⢠⣄⡀⠀⠀⠀⠁⠀⢀⠙⠳⢦⣄⡀⡀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡾⠛⠿⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠁⠁⠀⠈⠀⠀⠀⡸⠳⠇⠀⠀⠙⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢠⠀⠉⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⣀⠤⣤⡀⠀⠀⠀⣀⣀⠀⠀⢀⡤⠠⣄⠀⠀⠀⢀⠠⡄⠀⣀⡀⠀⢠⡤⣤⡀⠀⠀⠀⠀⣤⡀
⠀⢀⣼⣇⣀⣀⣿⡄⠀⡜⠈⣿⣿⣧⠈⠗⢺⡏⠀⠀⢀⠈⢉⠁⢴⣥⣷⠠⣿⣀⣸⠏⠀⣤⡀⠘⠿⠇
⠀⣼⣥⣽⣿⡿⠛⢋⢼⣻⣷⣭⠋⠁⠀⠑⠚⣡⠔⠈⠙⢷⣄⠀⠀⢸⠀⢤⡈⠉⠧⠀⢀⣁⡿⣆
⠀⠁⠟⡟⢹⢧⠀⠈⠛⠾⠃⢹⢒⣄⢀⢄⠘⢿⣤⣄⣀⡸⠿⢶⡀⠀⠒⠾⠋⠀⡋⢀⣲⡄⠠⠿⠆⢠⣄
⠀⠀⠸⡇⠉⡇⠀⠀⠈⠀⠀⢘⢿⠋⠛⢺⠆⠀⠘⠋⠟⠋⠁⠈⠁⠀⠀⢸⠀⠐⠃⠈⠉⠀⠀⠀⢴⣛⣿
⠀⠀⠀⡇⠀⡇⠀⠀⠀⢳⣦⡜⡌⠀⠀⢽⠀⠀⢄⡾⢀⠀⠀⢤⣄⠀⠰⡜⠀⠀⠀⠀⠀⠀⢰⡆⠈⠙⠋
⠀⠀⡀⠷⣶⠇⡄⣠⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠞⠙⠩⣷⣦⣼⠶⢓⣤⣔⠒⠒⢦⡀⠀⠀⢀⡆⢀⢨⡄
⠀⡞⢿⡀⠀⠀⠈⢉⡀⠀⠀⡀⠠⡰⣤⠉⠉⠀⢀⣀⠀⠀⢰⠀⠀⠻⣟⠛⠻⢶⣶⣿⣄⠚⠊⢀⣈⣧
⢀⡇⢸⣧⠀⠐⡲⠈⠁⠀⡸⡧⡑⣽⠻⡎⠀⢠⣿⣻⣷⡀⠘⡄⠀⠀⠀⠙⣺⠛⠛⠉⠁⣴⣶⡄⠉
⠘⠷⣾⠟⠃⠀⠀⠀⣤⠀⠈⢺⣿⢿⠨⠆⠀⢾⣾⣽⡿⠋⣤⣊⠉⣷⣦⠠⣽⠀⠀⠈⠻⢿⡶⠷
⠀⠀⠾⠀⠀⠀⠀⠀⠀⣴⠀⢴⡄⠘⡄⠀⠀⠀⠀⠈⡇⠀⠛⠻⣻⠟⠋⠀⣿⠀⢶⡷⠀⠈
⠀⠀⡄⠀⠀⠀⠈⠉⠀⠻⠿⠗⠃⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⠀⣆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠦⠤⢀⣀⠀⠀⠀⠈⠠⠄⠀⠀⠀⢤⡀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⣤⣴⣤⣦⡀
⠁⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠂⠀⠀⠀⠠⠅⠀⠀⠏⠁⠂⢤⠀⢀⣀⣤⠀⠈⠀⠒⡄⡀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠄⠀⠘⠛⣉⡀⠅⠂⡐⡤⢰⠐⣦⣤⠂⠤⣲⣶⣷⣿⣿⡏⠿⠉⠓⠶⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⢀⣀⣀⠀⠁⢠⠄⠀⠈⡉⠽⠁⠀⠉⡳⠂⣈⢆⡤⣫⠛⣝⣿⣿⡟⠃⠁⠀⠀⠀⠀⠀⡀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠜⡂⣀⠘⢩⠄⠓⣄⠫⠀⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠈⠉⠀⠀⢐⣄⠈⠀⡊⠀⢒⠄⠀⢳⠑⡀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⢳⣀⠀⠀⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠄⠐⠂⠀⠉⠀⡠⠃⠀⠀⠀⠁⣸⢠⠸⡒⠀⠀⢽⣿⣿⡀⠀⠠⡄⠀⠀⠀⢿⠀⠀⠀⠰⠀⠄⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠄⠂⠁⢈⡁⠀⠄⡵⢊⡵⢢⠈⠅⠀⠰⡙⣿⣿⡇⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢁⡠⠧⠀⠔⡫⠞⣡⢥⠘⠒⠀⠘⠻⣿⠇⠀⠀⠀⠘⠂⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⡠⠊⡡⠄⢀⠈⠁⠚⠁⡣⣄⠙⠢⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⢉⠀⠀⠈⠔⡇⢠⣨⡈⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⣤⠀⠉⠙⠃⠠⠦⠤⡶⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣿⣷⣠⣷⣴⠶⠀⠀⠤⠤⢤⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠶⠶⠶⠶⠿⠿⠿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⡼⠛⠀⠀⠀⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢆⡇⠈⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⣴⣾⣁⡉⣠⣤⠎⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⡆⡀⠘⠻⢿⣦⣀⣄⡀⠀⠀⢠⡦⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⢠⡟⢰⠀⠀⠀⠀⢈⢩⡅⢀⡔⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⣸⢰⢹⠇⠀⠀⠀⢸⠘⢠⠏⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣾⣷⣿⡟⠁⠀⠀⠀⠠⠃⠀⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠡⠶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⣆⣠⡄⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣤⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠁⣿⣿⣿⣍⡁⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⣀⠲⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⠃⠙⠃⠈⠉⠁⠀⢉⣌⠉⠹⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⣁⣀⣀⣠⡴⠿⢟⠻⣶⣶⡶⠾⣿⣦⢴⣿⣶⣿⣿⣷⣶⣦⣶⣴⣶⣶⣆⣀⣠⣤⡄⠀⠁⠀⠀⢤⠠⠿⢦⡀
⠀⠀⠀⠀⠀⠀⢰⣯⣠⣾⣋⠋⠉⠭⢀⡀⠠⣙⢮⡻⡀⠢⠙⢮⢆⡹⠈⣏⢹⠈⡇⢸⡁⢠⡸⢁⠏⢹⠟⢻⠶⢾⣷⣄⣀⡀⢠⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢠⡿⠟⠁⠀⠀⠈⠁⠲⠄⣀⠀⠀⣸⠙⢦⣀⠀⠈⠸⣡⡇⣿⠘⣦⠙⢦⢁⣼⢣⠎⡰⢋⡴⢋⣴⡋⠝⠛⢿⣤⣼⣇⡈⠀⠈⢣⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠒⠂⢄⠉⠲⢤⣁⠘⠿⢷⡜⠀⣿⣧⢹⣧⣽⣿⣶⣿⣷⣵⡮⢴⣯⡤⠞⠉⡠⠂⡴⠊⠈⠛⣶⣿⠀⢠⣤⣧
⠀⠀⠀⠀⠀⠈⢿⡈⢭⡍⠁⠒⠒⠪⠭⠴⢊⣥⠖⣨⣑⢦⣌⣿⡾⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⣶⢾⣥⣴⢶⣥⣶⣿⣦⣭⡶⣭⣴⠻⣦⡈⣁⡈⠆
⠀⠀⠀⠀⠀⠀⠈⠻⣖⣂⣤⠤⢤⣴⣶⣦⣭⣄⣛⡻⠿⣷⣮⡿⠀⠀⠀⠀⢨⣿⢿⣿⣿⣭⣤⠄⠀⠀⠀⠉⢐⣀⡚⠓⠚⠉⠀⢘⣷⠾⢟⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⡖⢒⣠⠤⠍⠉⢛⣛⣛⣻⣿⣿⡏⠀⠀⠀⠀⢸⠀⠀⣿⡟⠿⣻⠿⢛⡿⣿⣿⣶⠤⣤⢤⣄⣘⣛⠋⠤⠄⣻⣿⣃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⢤⣾⠿⠛⣉⣩⣭⡽⣿⡇⠀⠀⠀⠀⢸⠀⠀⣿⣷⣦⡇⠰⡠⢈⠣⢍⠳⣛⠦⡈⠩⣙⠟⠛⠖⣠⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣽⣮⣿⣿⠇⠀⠀⠀⠀⠸⣄⠘⣿⣿⣿⣟⠢⡀⠀⠀⢀⠑⢌⠡⡈⠳⡌⠳⣌⣶⣿⣿⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡿⢯⡶⠐⢀⣁⣀⣴⣤⣷⣼⠶⠾⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢰⣿⠀⡏⠙⠳⠿⠿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡠⠀⢀⣴⣤⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⡜⠛⠃⠈⢻⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢀⢀⣾⠁⣸⣷⣄⡤⠹⣍⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⣰⣏⣾⣷⣾⣿⣿⣿⣿⡿⣮⣵⣌⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⠀⢀⠀⠀⣼⣿⣿⠿⠋⠁⡷⢸⣸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡷⠶⠶⠒⠚⢧⣦⣾⠛⠁⠀⠀⠀⠈⠓⠸⡏⠉⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠘⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⡆⣼⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⢧⡈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢠⡞⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡦⣀⠀⠀⠐⡿⣋⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⣭⣤⣤⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣢⣶⣿⠿⢋⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡾⣿⠹⠟⠉⠀⠀⠨⠁⢄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠂⠋⠌⠁⢀⠀⠀⠀⠂⢘⠀⠈⠈⡀⠀⡀⠂⠀⣀⠠⠄⠂⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠄⠐⠀⠠⠐⠰⠄⠀⠂⠰⠀⠤⠀⢤⣀⣪⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⣀⡔⣀⣀⣀⡀⠀⠀⠀⠀⠑⠒⠬⡶⠞⠛⠛⠉⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠃
⠀⠉⡀⢰⣁⠕⠋⠀⠀⠈⣆⠀⠀⠀⠀⠄⠀⠘⠓⢤⠀⠤⢤⣤⢀⡀⣀⢀⡰⡄⡁⠀⠀⠀⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠘⢟⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠁⠉⠉⠂⠒⠤⣠⣀⣀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣾⡟⠃⠲⠦⢷⡆⠀⠀⠀⣧⠀⣺⠀⠀⠀⠀⠀⢷⣤⣤⠤⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠟⠋⠀⠀⠀⠀⣰⡇⠀⠀⣴⣿⣼⣿⣦⡀⠀⠀⠀⠀⠀⡀⣂⠒⠄
⠀⠀⠀⠀⠀⠀⠁⠃⠋⠀⠀⠀⠀⠀⠀⣒⡉⣀⡀⡸⠛⠙⠙⠛⠛⠃⠙⠢⠤⠤⠜⠻⠿⠖⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠻⠿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣴⣦⣴⣾⣷⣷⣶⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠿⣿⣿⣿⢿⡿⡿⡿⣟⠿⠻⠻⠛⠛⠛⠟⠻⠛⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠙⢛⠿⣟⣿⣿⣿⣿⣿⡿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠶⠾⠟⠻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠛⡻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣴⣇⠀⢀⣀⣤⣤⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢿⣿⣿⣿⣿⣾⣿⡿⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠟⠛⠛⢯⣍⡉⠉⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⡹⠶⣤⣀⡀⠀⠈⠙⠛⠶⠶⠶⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣠⣋⣀⣀⣀⠉⠛⠳⠶⠶⣤⣤⣤⡤⠶⠶⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠾⠛⠛⠛⠛⠛⠻⠿⣶⣦⣤⣤⣤⣀⣀⡀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣳⠆⠀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣯⣉⠙⠛⠛⠋⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠸⠀⠈⠧⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠖⠲⠶⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠛⠉⠳⢄⡀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠪⣍⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠈⠓⠦⢄⣀⡙⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠉⠛⠳⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⡶⠞⠋⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠒⠲⠤⣤⣄⣀⣀⣀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣠⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠶⣤⣄⡀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠉⠙⠛⠶⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⣌⠉⠉⠉⠉⠉⠉⠀⠀⢿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⣼⠟⣿⣦⣄⣀⣀⣀⣀⣀⣤⣤⠶⠛⣿⣟⠳⢦⣤⣄⣀⣠⣴⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠛⠶⣤⣄⣀⣀⣀⣀⣤⣼⠟⠀⢹⣷⡈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣹⣿⠀⠀⠀⠈⠉⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠉⠉⠉⢉⣼⡟⠀⠀⠀⠻⣿⣿⣒⣒⣲⣖⣒⣒⣿⣿⣿⠏⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠶⣶⠶⠿⠟⠋⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠉⠛⠶⠶⣤⡴⠶⠟⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⣛⡛⠛⣻⠿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠋⠉⠛⠻⠷⢶⣄⡀
⠀⠀⠀⠀⢀⣠⡞⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣶⣦⣶⣤⡄
⢀⡾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣧⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⣿⡆⠘⠈⣿⣿⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⢿⡷⠀⠀⢉⣉⣐⣿⣿⠂⠀⠀⠀⠀⡴⢶⣾⣿⣿⠏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠛⠂⠀⠀⢄⣠⠾⠟⣻⣿⣟⠋⠁⠀⠀⣠⡺⣿⠦⠌⠋⠻⠷⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠙⣿⣷⠀⠀⠀⠐⣙⡛⠻⠄⠀⣀⠀⠐⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢾⣿⡇⠀⠀⠉⠙⠡⠶⠂⠀⠙⠷⠸⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣶⣤⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⡇⠀⠀⠀⠀⠠⢄⣄⠀⠉⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣹⣿⣧⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠙⠛⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢽⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⣧⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨⣿⣿⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣼⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣺⢙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠟⢽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣃⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣾⢛⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡼⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣿⣿⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⣄⠀⠀⠀⠰⣿⣿⣥⣿⣿⣿⣿⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠉⠉⡏⠉⢹⡇⠀⣠⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣯⣿⣿⠀⢀⣴⡾⣷⣾⡇⠀⢸⣥⣾⣿⣯⣨⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⢹⠋⠉⠁⣴⣿⣿⣦⣾⣿⣿⡄⣾⣿⣿⣿⣿⡿⢿⣿⡿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⢲⣄⠀⢁⠀⡆⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣿⢿⣿⣧⠈⡄⠇⠸⣿⣿⣿⣿⣮⣿⣿⣿⣿⢳⠅⠀⡙⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢛⠛⠋⢐⣀⡀⡆⠀⠀⠀⢸⠀⢈⡇⠀⠀⢸⠀⢀⠇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⢣⠈⢣⠀⠀⠀⡀⠀⢰⠀⢀⣀⠞⠀⡜⠀⢀⣾⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣬⡆⠀⡆⠀⠀⠇⠀⣨⣾⣻⣿⣆⢰⠁⢰⣿⣽⣷⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⡿⢷⠀⠠⣀⣸⠀⢸⡿⣿⢯⣿⣿⠀⠀⢀⠏⠉⠉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⡄⢱⠈⡇⣼⣿⣿⡆⢸⢃⠃⢰⡠⠋⠀⢀⡎⡐⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⣧⡱⠈⢣⢣⠀⡃⢸⠀⣼⠎⢠⡏⠀⢀⢀⠘⢆⡀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢷⠛⡅⢣⠈⢻⡀⡇⢸⢀⡟⠀⡞⠀⠔⠱⢋⠄⢀⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⡘⢄⠳⡀⠙⠇⠸⠊⠀⡼⠅⠐⠀⠈⢀⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣿⠄⠀⠀⣉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⠀⠀⠰⣿⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠂⢀⣿⣷⡄⠀⠀⢀⣤⣶⣤⡀⠀⠀⢠⣾⣿⡀⠐⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠁⠐⠀⣿⣿⣿⣿⣷⠀⠂⠘⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠟⢸⡟⢗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢘⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂


⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⠀⢀⠀⠴⠤
⠀⠀⠀⠀⠀⠀⢀⣬⡁⣨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡤⠀⠀⠀⠀⠀⠈⠃⠈⠀⠆⢀
⠀⠀⠀⠀⠀⠀⢨⣅⣉⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⢀⡝⡪⠇⠱⣦⣰⡇⠀⠀⠀⠀⠀⠀⠀⡗⢿⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢀
⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘
⠀⠀⠀⠀⠀⠀⢠⣀⡘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣶⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⢦⠀⠀⠀⠀⠀⠀⠀⠠⠉⠻⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⢉⡙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠉⡓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠒⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠂⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⡠⣄⣸⣣⡀⠀⣀⡴⣀⣠⢆⣀⣠⠁⣄⢤⣠⢦⣀⡤⣀⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠞⠛⠉⠉⠉⡉⠉⠙⡗⠀⠈⠈⠁⠄⠀⢠⠋⠁⠀⣍⣉⡩⠭⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠒⠒⠒⢲⠃⣀⠀⣤⠀⣀⠈⠸⡉⠉⠉⠀⢀⠠⠐⠉⠁⠉⠕⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠉⠀⠉⠁⢹⠒⠁⠀⠀⢀⡁⠈⠐⠀⠀⠀⠒⠰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⡀⠀⠰⢋⣩⣏⣉⠻⢻⠖⠲⢶⡤⢤⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⡷⠀⢨⠋⠀⠀⠈⠙⠲⣄⠀⠸⡇⠅⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⠃⢠⢎⢆⠀⣀⣤⠔⠲⢺⡄⠀⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡔⢠⠃⠀⡆⢠⢴⣿⠁⠈⣧⢲⡞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠎⣰⠃⠀⠀⠸⢍⣎⣸⡄⠀⢸⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡜⠀⠀⠀⠀⠀⠆⠹⡄⠀⠀⠀⠀⠀⡾⠀⠀⠀⣟⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣁⣀⣀⣀⣀⣀⣀⣴⠃⠀⠀⠀⠀⠀⠧⠴⠦⠤⠷⠤⢄⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠸⡟⠃⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠆⠘⠀⠀⠀⡀⠀⠀⠀⠠⣷
⠀⠀⠀⠀⠀⢀⠈⢁⡓⢄⠐⠊⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠄⣠⠁⠀⣠⣴⢿⣄⢻⣿⡷⠃
⠀⠀⠀⠀⡀⠉⠐⡀⢘⠣⠛⢀⠓⢼⣤⢄⣀⠀⠀⠀⠀⠀⠀⢘⠡⢀⡀⠘⣷⣾⡿⠋⣠⣿⠿⠋
⠀⠀⠀⡈⢡⠀⡆⣀⠨⠀⠀⠈⠓⠘⣷⢀⠈⠁⠀⠀⠀⠀⠀⡬⢙⣶⣿⣦⣿⠏⠀⠀⢀⡜⠀⠁⠾⠶⠀⠐
⠀⠀⠀⠈⠠⡀⠆⠘⠠⠉⡆⠘⠛⠂⣦⠿⡄⠢⣤⡀⠀⠀⢈⠢⢉⣽⣿⣿⡟⠀⠀⠀⠈⠀⠀⠀⢀⠂⡀
⠀⠀⡅⡸⠀⢠⠘⠃⠀⡰⣅⢢⣠⣴⠈⠳⠈⣄⠀⠀⠀⠀⢰⣟⣙⣹⡿⢟⠇⠀⠀⠈⢲⠈⠢⣀⣼⠃⠀⠁
⠀⠀⠀⡆⢰⠀⠀⡀⢀⠰⠈⡀⣁⢉⠒⢿⣷⠀⢁⠀⠀⢠⣟⠙⠿⢿⣿⢻⠀⠀⠀⠀⠘⡄⠀⠫⡉⠳⢤⣤⠛
⠀⡀⠄⠇⠀⠀⠈⠀⣉⠂⠁⠘⠲⡌⢢⣿⢾⡿⠠⠙⠂⢸⣿⣆⣠⡟⣻⣿⠀⠀⠀⠀⠀⠈⠄⠀⠹⣆⠀⠚⠷⠊
⠀⠀⠀⡀⠃⠘⡀⠈⡀⠐⠄⣤⣠⢡⠨⣝⣉⠻⢷⣬⣁⠈⠙⣫⣼⡥⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀
⠀⠋⡁⠀⠐⠄⠣⠄⠘⢀⠠⣤⡈⢟⠰⡥⡌⢳⡆⠉⠛⠛⠟⡿⠿⠾⠿⣿⣅⠓⠂⠐⠂⠀⠉⠲⣄⠀⠐⠉⠉⠒⣀
⠀⢠⢃⢰⡀⠘⠆⠀⢃⡀⠀⡽⣌⢻⡻⢦⣌⡀⠻⣼⣛⠒⠶⠴⣶⢤⣤⣤⣽⡧⠈⠓⠄⠀⠁⢚⡀⠀⠀⠁⠀⠀⠈⠂
⠀⠠⠆⠊⠁⠀⠘⠀⡄⠃⣤⢰⣨⠀⡻⠛⠻⢧⣄⠀⠙⢫⣄⠐⠂⡠⠄⠤⢷⡄⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠱⠄⠀⣠⡀⠀⠀⢀
⠀⠀⢠⠶⢀⠠⡀⠙⢠⡀⡏⠺⠋⠀⡴⢼⣇⠈⠑⢯⡒⢤⡈⠳⠦⠀⢰⣦⡶⠀⠀⠱⠀⠉⠀⢀⡴⡆⠀⠐⠃⠀⢀⠀⠁⠤⠀⣠⠜⠀⠠
⠀⠀⠈⣰⠌⠃⠀⠀⠈⠧⢿⠀⡠⣙⠾⡄⠿⣗⣤⡀⠙⠣⣤⡁⣬⣷⣬⣿⠁⠘⢄⠀⠀⠀⠂⠀⠘⠇⠀⠀⢠⠀⡀⠐⠀⠀⣠⠊⠀⡄
⠀⠀⠀⡅⢀⠀⣠⢰⡄⠀⡈⠛⣌⡳⠇⢈⠃⠘⢿⣟⣷⢢⣄⡙⠻⢟⣿⠱⠀⠀⠀⢡⠀⠀⠀⡀⠴⢦⡀⢀⡄⣨⣡⣄⣠⠎⠀⠀⡔
⠀⠀⠀⣇⢸⡆⡿⡀⢰⣄⠙⣷⣿⢳⠄⠘⠱⣤⡀⠉⠻⢾⣿⣿⢷⡦⢍⡂⡀⠀⠀⠀⠑⠆⠀⠱⢄⡀⠙⣿⢻⠟⠀⠀⠀⣠⠖⠊⠁
⠀⠀⢀⡉⠀⡇⠀⢧⠈⠄⣻⡌⣍⠘⠑⢄⠀⠀⠹⣶⣄⡀⠈⠻⢆⠙⠊⠋⠙⠦⠀⠀⠀⠈⠲⣔⠀⠀⠀⠈⠻⡄⠀⠀⠠⡁⢢⡄
⠀⠀⠈⢣⠄⠱⠃⣌⢇⣸⣷⡍⡈⠓⢄⠀⠀⠀⡀⠀⠘⢿⣷⡖⠤⢁⢀⠀⠀⠀⠀⠀⣠⡾⡡⠈⣻⢖⣶⣄⡠⠈⠂⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣦⣉⢲⣸⣀⠏⠁⢆⠀⠑⢤⡀⠈⠑⢄⡀⠙⢿⡄⠀⠉⢲⡄⠠⢄⠐⠋⠀⢄⣎⣡⣾⣿⣿⣿⣄⠲
⠀⠀⠀⠀⠀⠀⠛⡘⠛⠃⠉⠀⠣⡀⠑⣄⠀⠙⢶⣄⠀⠻⣷⣄⡙⢶⣄⡄⣀⡀⣠⣤⣀⣦⠳⣿⣿⣿⣿⢿⣌⠻⢷⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠢⠑⡀⠱⡄⠈⠳⣄⠀⠙⣧⡀⠹⣿⣿⣦⡙⢷⣭⡴⠶⠟⠋⢿⠀⠈⠛⢿⡛⢧⡙⢦⡀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⠀⠀⠀⠈⢄⠀⠈⢧⡀⠘⣿⣄⠹⣿⣿⣷⣌⣦⠀⠀⠀⠀⠈⣇⠀⠀⠀⠁⠈⢷⠀⠈⠀⠈⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⡀⢂⠀⢢⡀⠈⢿⣆⠘⣿⢦⡿⠃⠈⢿⣿⡀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠸⣆⠀⣷⡀⠘⣿⢇⠸⣼⣿⡆⠀⠀⢿⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷⣿⡄⢹⣷⠀⣿⠈⡆⣿⣿⢷⡀⠀⡸⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⢉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣧⣸⣿⡆⣋⠐⣰⣿⣿⣦⡇⠘⠀⠀⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⣿⡿⢿⣿⡟⠾⣿⣿⡿⣱⡇⠀⠑⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣠⢟⡿⠁⣹⣿⣟⢸⠏⢿⡆⠻⠀⠀⠀⠀⠉⠢⡈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣹⢁⡞⡱⣾⣿⠟⢻⠈⠀⠩⠛⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢿⣸⡼⡃⠉⠋⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢙⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠒⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠜⠁⣀⣈⡀⠀⠠⠤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠇⠟⠉⠐⢲⣾⡟⢻⣿⡞⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢀⠀⠀⣀⡤⠤⠐⠒⠒⠒⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡟⠕⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠘⠦⣄⣀
⠀⠀⠀⠀⢰⡄⠀⠀⠀⣀⡀⠀⣄⠀⡄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠿⠟⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠶⢟⣁⢤⠖⢀⣿⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢀⡠⠖⠛⠀⠀⠋⠁⣤⣤⡾⡅⠀⠀⠀⠀⠀⠐⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⣠⣖⠋⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⡡⢤⡀⢈⡄⠸⡄⠃⣾⠃⠀⣀⣠⣤⣄⣀⣳⣆⠀⢠⣤⣴⣀⣀⡈⠳⡔⢾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠊⠀⢠⡆⠀⣠⣿⣿⣷⣿⢄⣶⣿⣤⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣾⠍⠻⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⡟⠁⠈⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⡿⠏⠙⣠⣾⣟⠻⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣽⡿⠋⠀⠀⠀⠈⢿⣿⠟⠀⠀⠻⢿⡟⠻⠝⠋⠈⠀⠀⣱⣿⣿⡟⠛⠋⠘⣆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣿⣉⣀⡀⢀⠀⠀⠀⠀⠑⣿⠆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠸⣿⡿⢿⠇⣀⡀⠀⣸⠀⠐⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⢿⣿⡇⢼⠿⢶⣾⣶⣛⣤⠀⠀⠀⣀⣤⣤⣀⠘⠓⠀⠀⠀⠀⠀⢯⣥⣠⡦⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⡽⠇⠀⠀⣸⣿⣿⣿⣿⢿⣷⣾⣿⣿⠀⠫⠃⠒⠀⠀⠀⠀⠀⠈⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⡀⠀⣻⣿⣿⣟⡛⣼⣿⣇⠘⣿⣿⡜⣿⣿⣏⠙⠀⠀⢠⣶⠀⣈⢈⢀⣄⡈⣿⣯⠳⣿⢶⠀⢠⣄⣀⣀⣀⡀⡤⠖⠒⠉⠉⠂
⠀⠀⠀⠀⠀⢿⡟⠛⠛⠛⠲⠼⠛⠛⠻⣿⣧⠹⣿⣿⣄⠙⣟⣿⡜⢿⡏⠈⡀⠄⠜⠋⣀⠉⠀⢀⠀⠈⣿⠁⡴⢀⠀⠀⠀⠀⠉⠉⡉
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣻⣿⣿⡄⣹⢻⣃⡀⢑⠤⠄⠀⠀⠀⠀⠀⠘⠐⠒⢲⡇⢰⣿⣏⠀⡀⠀⠀⠀⠘⠛⠚⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣾⠟⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⣿⣿⣼⣗⣶⡿⢁⣤⣶⣿⣿⢕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⣿⣏⠹⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⡿⡝⠙⠙⠋⠀⠻⠟⠛⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣾⣿⣷⣦⣌⣉⠀⠙⠘⠻⣻⡟⣶⣤⡄⣀⣤⣀⣤⣾⣿⣶⡶⢾⣿⠋⠟⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⡙⠛⠛⠋⠋⠀⠀⠀⠀⠉⠁⠘⠿⠅⠉⠟⠻⠿⠁⠋⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿