Cake Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠋⠹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⡳⣞⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠖⢿⣤⣉⣉⣤⣿⠲⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⣀⣤⣀⠀⠀⣸⢧⡈⠓⠤⣏⠀⠀⣀⣤⣀⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠐⡇⠀⠈⠉⠉⠹⣆⠉⠲⢤⡏⠉⠉⠁⠀⢸⡂⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡖⠒⠋⠀⠀⠀⠀⠀⠀⢯⣑⣂⡼⠀⠀⠀⠀⠀⠀⠙⠒⢲⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠉⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠙⢦⣄⠀⠀⠈⠧⠤⠴⠒⠒⢦⠀⠀⠀⠀⡴⠒⠒⠦⠤⠼⠁⠀⠀⣠⡴⠋⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣇⠀⠀⠈⠙⠲⠦⣤⣀⣀⠀⠀⠈⠑⠶⠖⠋⠁⠀⠀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣸⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⡟⠓⠚⠙⣆⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⣰⠛⠓⠚⢻⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢿⡀⠀⠀⠘⣆⢀⡴⠛⢦⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⡴⠛⢦⡀⣰⠃⠀⠀⢀⡿⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠉⠁⠀⠘⢧⣀⣠⡞⠁⠀⢳⣄⣀⡼⠃⠀⠈⠉⠁⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⠀⠀⠀⠀⠀⠉⠳⠦⣄⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣰⠟⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠤⠤⢤⣤⠤⠤⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢦⣀⣀⣤⣄⠀⠀⠀⠉⠉⠛⠒⠶⠦⢤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠶⠒⠛⠉⠉⠀⠀⠀⣠⣤⣀⣀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠘⣧⠀⠀⣰⠞⠛⣦⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣴⠛⠳⣆⠀⠀⡼⠃⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠞⠃⠀⠀⠸⣆⠀⠀⠀⢠⡟⠀⠀⢻⡄⠀⠀⠀⣸⠇⠀⠀⠘⠳⠚⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠋⠀⠀⠀⠀⠙⠶⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄⠀⠀⢀⣾⠻⣧⡀⠀⠀⣠⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢈⣿⠀⠀⢸⣇⣀⣽⠇⠀⠠⣿⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣉⣛⣉⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣧⠀⠀⢸⣿⠉⣿⠀⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣯⣀⣠⣼⣿⠀⣿⣤⣄⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⢿⣇⣸⣿⠉⠉⠹⣿⣤⣿⠉⠉⠉⣿⣄⣸⡟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣌⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣡⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣯⠉⠛⠛⢻⣿⠛⠻⠿⠟⢻⣿⡟⠛⠛⢉⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⠷⣦⣴⠟⠻⣷⣤⣀⣤⡾⠛⠷⣦⣴⠾⠃⠈⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⢻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⡈⠉⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⣁⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡟⠛⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠟⠛⢻⣯⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠾⠛⣷⡀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⢠⣾⠻⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⠷⠶⠾⠋⠈⠻⣶⣤⣤⣤⡾⠟⠈⠛⠷⠶⠾⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⢸⣧⣼⠀⣼⣷⠀⣿⢹⡆⣿⢹⡎⢷⣴⠏⠀⠀⣿⣽⡇⣿⢸⡏⣷⠉⣿⠋⢸⣧⣼⡇⡟⢻⡆⣸⣷⡙⣧⡾⠁
⠀⠀⠸⠇⠹⠰⠟⠛⠇⠿⠋⠁⠿⠋⠁⠀⠿⠀⠀⠀⠿⠾⠇⠿⠸⠟⠿⠀⠿⠀⠸⠇⠹⠃⠷⠾⠳⠟⠛⠧⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⠤⠤⠤⡀⢠⠔⠒⠒⢤⡀⡠⠤⠠⢤⡀⠀⣀⣀⣀
⠀⠀⠀⠀⣀⡤⢄⡀⣰⣋⣀⡤⠽⠖⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠓⠾⠥⢄⣈⣱⡄⣀⠤⢄⡀
⠀⠀⠀⡞⠁⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⠀⠘⡆
⠀⠀⠀⣧⠒⠋⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠓⢦⡇
⠀⠀⠀⡇⠀⠀⠀⠀⣷⠊⠉⠉⠑⣄⣠⠔⠒⠦⡀⠀⣠⠤⢤⡀⢀⡤⠖⠒⢤⣀⡔⠉⠉⠉⢲⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⣀⣀⠴⣇⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⡏⠀⠀⠀⠈⡟⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡷⢄⣀⣀⠔⠁
⠀⠀⠀⠀⠐⡅⠀⠀⠈⠢⠤⠤⠖⠙⢆⡀⠀⣀⡼⢧⠀⠀⠀⣠⠿⣄⠀⠀⣠⠞⠑⠤⠤⠴⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠳⡄⠀⠀⠀⠉⠉⠁⠀⠀⠉⢒⠊⠁⠀⠀⠉⠉⠀⠀⠀⢀⡴⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡗⠒⠊⠁⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠙⠒⠒⡇
⠀⠀⠀⠀⠀⡟⠢⢄⣀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠖⠉⠀⠀⠀⠈⠑⠦⠤⠤⠴⠊⠀⠀⠀⠀⢀⣀⠤⠚⡇
⠀⠀⠀⠀⣠⡏⠒⠤⣈⡉⠓⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠋⣉⡠⠔⠊⣧⡀
⠀⠀⠀⢰⠁⡇⠀⠀⠀⠉⠉⠒⠒⠦⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠖⠒⠊⠉⠁⠀⠀⠀⡇⢹
⠀⠀⠀⠈⠢⣈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⢊⡡⠊
⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⣉⠙⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠚⢉⣉⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⣖⣒⣶⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⣖⣒⡶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⠭⢿⠉⠉⠉⢹⠯⠭⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠶⠶⠶⠿⠿⠟⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠦⡄⠀⠀⠀⠀⠀⠀⠙⠲⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⢀⡟⠀⠀⠀⠀⠘⠄⣀⣴⠇⠀⠀⠀⠀⠠⡊⠉⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢠⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣧⠀⠀⢃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢺⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡟⠀⠀⢣⣾⠟⠿⠶⣶⡀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠠⣒⣦⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠢⠀⠐⠂⠉⠀⠀⠀⠀⠀⢁⠤⠖⠛⠁⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠍⠻⠿⡁⡀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠄⠀⠐⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⢸⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠊⠀⡅
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠒⠂⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣼
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠰⡄⡐⠘⡄⠀⠸⣿
⠀⠀⠀⣇⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢸
⠀⠀⠀⢹⣶⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡤⠋
⠀⠀⠀⠈⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⣼
⠀⠀⠀⠀⡟⠔⠀⠀⠂⢦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⣰⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠐⠀⢀⣀⡀⠄⠂⠐⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢩⡟
⠀⠀⠀⠀⠸⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠤⠔⠂⠁⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡏⠛⠐
⠁⠀⠀⠀⠀⠀⠺⠗⣣⠘⢸⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠞⠉
⠀⠀⠀⠀⠰⠀⠀⠀⠈⠛⠢⢄⡁⠙⠛⠳⠶⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠈⠉⠒⠲⠤⣄⠀⠈⠈⠉⠉⠑⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⣧⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣅⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠭⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠷⠶⢤⣤⣤⣀
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⣀⡀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣧⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⣠⡞⠉⠉⠻⣦⡀⢀⣴⠞⠛⠛⢶⡀⠀⠀⠐⠀⣠⡶⠛⠋⠙⠳⣦⡀⠀⠀⠠⠀⣠⣤⢤⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⠶⠟⠉⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠙⠶⣤⠴⠞⠉⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠛⠉⠀⠀⠈⠻⢦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣤⠶⠛⠛⠛⠉⠉⠉⠉⠉⡉⡉⠛⠛⠛⠓⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣀
⠀⢸⡇⠠⠄⠀⠀⠀⠀⣀⣀⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉⢷⡀
⠀⠘⢷⣀⠀⢀⣀⣴⠟⠉⠉⠉⠻⢦⣄⣀⣤⡴⠖⠛⠲⢶⣄⡀⠀⠀⣀⣠⣤⠤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠖⢀⣀⣀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠙⠷⣤⣤⣤⣤⠶⠞⠋⠉⠀⠈⠙⠳⠶⣶⠞⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣴⠞⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⢦
⣼⣿⠀⠀⠀⠙⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁⠀⢸
⣿⠙⠷⣤⣀⡀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠳⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣠⣾
⠘⠷⣤⣀⠈⠙⠛⠶⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⢛⣩⠞
⠀⠀⠀⠙⠳⠶⣤⣄⣀⡀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⢤⣤⠤⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠋⢉⣉⣡⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠚⠛⠛⠛⠉⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄⢸⢚⡲⠄⡄⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠙⡟⠂⢠⣀⢂⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣤⣀⣀⣀⣀⣈⣀⣀⣭⡶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠉⢉⡉⠉⠉⠉⠁⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠁⠠⠤⠀⠀⠀⠤⠄⠐⣀⠀⠀⠂⠐⠀⠀⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠈⠁⠀⣀⠀⠀⠄⠀⠐⢄⢖⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣧⣴⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣁⣀⣀⣤⣤⠶⠞⠛⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣤⣤⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⡀⠄⢂⠀⠡⠂⠄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠍⠀⠀⠈⢵⡄⠈⠐⠢⢦⣶⣄⣀⣀⢀⡀⠀⠀⣀⢀⣀⠀⣤⠴⣖⡞⢀⠀⠀⠀⠀⠘⣡⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠂⠀⠱⠀⣀⣋⠀⢠⢙⠃⢠⠀⢼⠉⠷⢶⠀⡀⠠⢐⠃⠀⠊⠙⠂⢠⠀⠐⠤⠌⠀⢀⠀⣀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡷⠧⠉⠒⠄⠁⣄⡀⠂⠈⠀⠠⠀⠀⠂⠀⠙⠀⠁⠀⠈⠀⠀⠀⠁⠀⡀⠀⠀⠂⡙⠃⠢⠐⢉⡂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣇⡀⠀⠀⢀⠐⠒⠄⠈⠁⢀⠐⡀⢠⠀⠀⠀⠐⠀⠐⠀⠀⠀⢤⠦⠀⡄⠀⠀⠀⢐⣤⣶⣶⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠛⠻⣶⣦⣤⣤⣄⡂⠀⠀⠀⠀⡉⠀⠑⠂⠀⠀⠀⢈⠀⠀⠀⣀⣠⣤⣦⣴⣶⣿⡿⠛⠋⠁⡐
⡇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠑⠀⢬⡀⠀⠍⠉⠛⠻⠿⢿⠿⠿⢶⣶⣶⣶⣶⣶⣿⣿⡿⠿⡿⠟⠛⠛⠉⠉⢁⠀⢈⠀⠄⠊⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⣀⠀⠀⠀⠀⠉⠒⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣈⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠤⠾
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣤⡤⣆⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠒⠃⠀⠀⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⢠⡖⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢧⡤⢤⣀⣠⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⢀⣀⢀⡦⣄⡞⠲⠾⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠒⠈⠉⠉⢛⠋⠉⠷⢞⢹⡶⠎⠳⢶⠉⢲⠋⡙⠛⢋⠛⠁⠉⠠⠤⠤⠤⠤⠀⣀
⡇⠀⣀⠠⠀⠀⠒⠈⠉⠉⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⣀⠄⠂⠘⢀⠀⢀⡘⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⢠⢀
⡇⠁⠀⠀⡤⠀⠀⠀⠀⠤⠀⠀⠀⠤⡦⠀⠀⠀⠀⢠⠄⠰⡀⠀⠀⠩⠄⡆⠠⠣⠈⡃⠀⠐⠀⡠⠀⠀⣀⠐⡄⠀⠶⣀⠀⣀⢀⠀⢀⡀⢀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠀⠂⠀⠤⠬⢀⠤⠄⠀⠀⠐⠁⠀⠀⢀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠐⣨⠂⠀⠘⣈⠂⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠶⠀⠘⠠⠒⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑
⠏⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠙⢄⣠⠴⠚⢉⡀⠙⠒⢦⣀⡠⠞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⣄⡞⢁⣤⣶⣄⠀⠙⣦⣾⣿⣿⣷⣆⡜⠁⣠⣤⡀⠈⠳⣀⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢁⡰⠃⠈⣵⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⠞⠉⢀⡴⠈⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢋⡱⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠉⢀⠔⠋⢠⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠒⢋⡔⣈⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠏⡀⢠⠎⢸⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⠤⠖⡉⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⠲⢤⣁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠈⡞⠁⠤⠚⢡⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠁⠀⡇⢲⠀⠈⣷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⡿⡀⣇⠀⣀⡤⠊⠀⢀⠜⠋⡰⠉⠻⢿⡿⠿⣿⡿⠛⢿⣿⢿⡿⣿⠟⠋⠰⢈⠙⢧⠀⠊⢀⡜⠀⠀⡼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠘⢾⡦⣄⣀⢀⣰⠃⢀⡴⠃⡟⠀⠀⢻⠞⠁⢠⠏⠀⠙⢦⡾⠁⢀⡰⠃⢸⠀⠀⣇⣀⠀⣀⡠⠞⣡⠋⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⢬⣁⠹⣄⠀⢀⠔⠁⠀⢀⣾⠐⠚⠁⠠⠃⠀⠀⣧⡀⠈⠀⠰⠃⠀⣠⠏⠉⠉⢁⡠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠠⣄⡀⠀⠀⠈⠑⠺⠷⣶⣶⠖⠒⠋⠘⠦⣀⣀⣀⣀⡤⠞⠀⠉⠒⠲⠶⠖⠚⢁⣠⠤⠒⠉⠀⢀⡠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠶⠶⠶⠶⠶⠖⠒⠒⠚⠋⠉⠁⢀⡴⠒⠒⢀⣞⢀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠈⠙⠒⠒⠒⣞⠉⠉⠓⠦⡀⢀⣀⣠⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠖⠒⠂⠀⣮⣀⣀⡠⠞⠀⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠈⠙⠲⠤⠤⠖⠻⣄⡀⠀⠉⢓⣠⠖⠉⠉⠉⠃⣀⠜⠡⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠸⡧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⡋⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠶⠶⣤⣤⣴⣤⣤⣤⣼⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠻⣦
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣠⡶⠛⢷⣄⠀⠈⠁⢀⣤⡶⠶⣤⡀⠀⠀⠀⢀⠀⠀⢀⣤⡶⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠘⢷⣄⣀⣠⡾⠋⠀⠀⠀⠙⢷⣤⡴⠟⠁⠀⠀⠈⠻⣆⡀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠈⠻⣦⣄⣀⣈⣡⣴⠟⠋⠛⠻⣦⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⣴⠾⠟⠛⠛⠛⠉⠉⠉⠉⢉⣙⠛⠛⠛⠛⠷⠶⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣄⡀
⠀⢸⡏⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠻⣦
⠀⠈⣷⡀⠀⠀⠀⠀⣠⡶⠛⠋⠛⠻⣦⣀⠀⠀⢀⣠⣤⠶⢦⣤⣀⠀⠈⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠈⠻⣶⠶⠶⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠙⠻⠶⠶⠶⠾⠛⠉⠉⠉⠉⠉⠛⢷⣤⣀⠀⠀⢀⣀⣠⣴⠾⠛⠛⠛⠳⣦⣄⣀⣠⣼⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⣴⡟⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠛⣦
⣼⣿⠀⠀⠀⠈⠛⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠋⠀⠀⢸
⣿⠙⢷⣤⣀⡀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⣠⣾
⠙⢷⣄⡀⠉⠛⠳⠶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠞⣫⡾
⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠶⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡶⠶⠶⠾⠟⠛⠛⢉⣉⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣁⣀⣠⣤⣤⣤⠴⠶⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⢹⠁⠀⠀⠀⡏⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡠⢄⠀⠀⡠⠤⠀⢀⡀⢀⡀⠀⠀⢀⠀⣠⠲⡄⡠⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣠⠇⢸⠀⢰⠃⠀⠒⠋⡇⡏⠘⠀⠀⡏⠉⠁⠀⠧⠇⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠸⢤⠼⠀⠀⠀⢠⠃⢹⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠘⣤⠎⠀⠳⠴⠃⠀⠀⠀⠀⠀⠀⡏⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢀⣞⠁⢰⠉⡳⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⢞⠉⡦⠈⡳⡀
⠀⠀⠀⠀⠀⠸⡌⠂⠀⠀⠀⠀⠀⠀⠀⠉⠩⡙⠓⠒⠒⠒⠒⠒⠒⠒⢋⠍⠉⠀⠀⠀⠀⠀⠈⡠⠚⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡀⢢⠀⠀⠀⡀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢠⠇⡜⠁⢸⠀⠀⠀⠈⠋⠀⢱⠀⠀⠀⠀⢰⠙⢦⡀⠀⠀⢀⠀⠀⠀⣼⠵⠤⣄⠀⣴⡸⡀
⠀⠀⠀⠀⠀⠈⢲⠃⠀⠸⡄⢀⠆⠀⠀⠀⣼⠀⠀⢀⡀⢸⠀⠀⡇⠀⠰⠛⣆⠀⢀⡇⠀⠀⠘⠚⢹⠃⢣
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠑⠊⠀⠀⠀⠀⢹⠀⠀⣼⠐⠁⠀⠀⢧⣠⠇⠀⣿⠀⠈⡇⠀⠀⠀⠀⠸⣆⡼
⠀⠀⠀⡠⠤⡴⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠀⠀⠀⠀⢈⡗⢦⠤⣄
⢀⣠⠞⠧⠴⠃⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠘⠦⠼⠳⣄
⡏⢯⠀⠀⠀⠀⠀⠀⣙⣒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣒⣋⠀⠀⠀⠀⠀⠀⡽
⠘⢆⡓⢤⣀⠀⠀⠐⢅⡨⠂⠀⠈⠉⠉⠓⠒⢒⣶⠶⠶⠶⠶⠶⠶⣶⡒⠒⠚⠋⠉⠁⠀⠐⢅⡨⠂⠀⠀⣀⡤⢚⡱
⠀⠀⠙⠲⢬⣙⠓⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⢧⣠⠇⠀⠀⠀⠀⠸⣤⡼⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⣋⡥⠖⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⢤⣉⣉⡉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⢉⣉⣉⡤⠤⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⣀⢛⡦⢄⠀⣀⣀⣀⡳⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠚⠛⢇⣀⣭⠙⠤⡶⠆⠸⢉⣵⡐⠲⠗⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⠖⣂⠄⠀⢁⡀⠀⡀⢀⠀⡀⣤⠄⡤⠤⣢⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠤⢬⡕⣴⣆⡿⠁⢈⢰⠴⡇⣉⣣⣇⣰⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠶⡒⣄⡀⣫⠈⢉⢁⢸⣆⣀⣏⣉⣉⣀⣤⣭⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣁⠒⠤⠤⣌⣉⣈⣂⠉⡙⠉⠉⠈⠉⣉⣉⣠⣥⠴⠞⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣄⣉⡸⣂⠤⣠⡄⢤⣈⡩⣉⣉⣉⢉⣉⣠⢁⠄⢰⡆⠚⢠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⡌⢒⠛⠛⠲⠨⢤⣼⣤⣙⣁⣼⣀⣹⣌⣧⣭⣭⣲⠟⢟⢛⣹
⠀⠀⠀⠀⠀⠀⣀⣠⣸⠧⣄⠉⠃⠙⠓⠖⠶⢲⡤⡬⣀⣩⣭⣥⣤⣤⣽⡴⠶⠶⢛⢋⣩⣀
⠀⠠⡀⠒⠐⢅⠀⠀⠙⠲⢄⣁⠘⠙⠼⠶⠤⢤⣤⣨⣄⣄⡤⣤⣤⣤⣔⣦⢴⠶⠟⣋⣹⠏⠉⢓⠅⣀⠂⠄
⠀⠀⠀⠁⠐⠀⠠⠄⠀⠀⠀⠀⠉⠀⠂⠠⠄⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠶⠒⠒⠉⠁⠀⠀⠀⠁⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢰⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡞⠋⠁⠀⠈⢷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠈⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡖⢲⠀⠀⢀⡴⠓⠂⠀⣠⡄⠀⣠⡄⠀⠀⠀⣀⡀⢠⠏⢹⡆⣠⠶⡄⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣰⠃⢸⠀⠀⢸⠁⠀⠐⠚⠁⣿⢸⡇⠘⠀⠀⢰⠃⠉⠉⠀⠘⢧⡟⠀⣇⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠸⡄⠀⡼⠀⠀⠀⠀⠀⡏⠈⡇⠀⡆⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⡇⠀⠀⠉⠋⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⣇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⡎⠁⣹⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠀⢸⡉⢉⡇⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢻⠘⠆⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠙⣓⡒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⣒⡛⠋⠉⠀⠀⠀⠀⠀⠉⠉⢀⡴⠋⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⠆⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠋⢹⠀⠀⠀⠀⢤⡀⣀⠄⡉⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠋⠀⢸⠀⠀⢠⠀⠈⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⡏⠙⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠖⠶⢦⡄⠀⣴⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠸⣤⡟⠀⠀⢸⡆⠀⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⡀⠀⢠⠾⢯⡀⠀⠀⢸⠁⠀⠀⠀⠳⠴⠃⡗⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⣀⡞⠀⠀⠀⠀⠀⢿⠀⠀⠀⣴⠣⡰⠃⠀⠀⢸⡀⠀⡎⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠈⠷⠞⠁⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⠖⠋
⠀⠀⠀⠀⣠⠶⢶⠖⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⣿⠓⢶⠶⢦⡀
⠀⠀⣠⠾⣇⣀⡼⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⣄⣀⡿⢦⡀
⣴⢿⠋⠀⠀⠁⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠁⠀⠈⢻⠷
⢷⡈⢧⡀⠀⠀⠀⠀⠀⠀⡠⠖⠮⡉⠓⠲⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠒⠋⡩⠖⠦⡀⠀⠀⠀⠀⠀⠀⣠⠏⣰
⠈⠳⣄⠙⠦⣄⣀⠀⠀⠀⠳⢤⡴⠃⠀⠀⠀⠀⠀⠈⠉⠉⢉⡻⢟⠛⠒⠒⠒⠒⠒⠚⢛⠿⣛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠳⣤⠴⠃⠀⠀⢀⣀⡤⠞⢁⡴⠋
⠀⠀⠈⠛⠦⣄⡉⠛⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠇⠀⠀⠀⠀⠀⠀⢧⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠋⣁⡤⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⡉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⣉⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡎⠀⠇⠀⢀⠔⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡌⠙⠒⢼⠠⠐⠛⠠⡟⢀⣔⡿⠒⠒⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠡⡀⠀⣠⠏⠸⠁⢈⡒⢢⡀⠘⣄⠀⠁⠀⠀⠀⠀⠘⢏⠉⠁⢈⡸⠁⠑⣏⠈⣧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣣⣱⡯⠣⢴⡆⠙⡤⠸⠊⢙⣶⣿⣷⣶⣄⠀⠈⠀⠀⠸⠶⣈⣁⠴⡄⢴⠁⠈⢦⢠⡯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢤⣾⠑⢦⢸⠀⠋⢁⡔⠛⣾⡿⣺⣘⠽⢼⡽⣦⣄⢤⡧⢊⡥⢖⠛⠃⠀⠱⠀⠣⡀⣁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⡇⠀⠀⠀⠀⠀⡰⢻⡿⠒⣤⢞⣿⢿⢷⣷⣲⣏⢏⢱⣳⡿⠼⠒⠛⠲⠧⢄⣠⠁⠉⡤⠐⢇⠈⡦⢜⠉⠓⢤
⠀⠀⠀⠀⣠⠊⠁⠐⠀⠱⡀⠀⠀⠀⢠⡀⠀⠀⣼⣯⠮⣗⣏⠿⣻⣟⣿⣻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠓⡄⠸⠣⣀⡸⠈⠃⠀⠱⡀
⠀⠀⠀⡸⠁⠀⠀⠀⣀⡀⠈⠑⠒⠒⠟⠳⢄⣀⣿⣟⣿⣿⢎⣸⣛⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠛⠶⠚⠁⠀⠀⠀⣀⠀⠇
⠀⠀⠀⡇⠀⠀⠃⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⢿⣻⣮⣾⣿⠿⠃⠸⣏⡒⠄⢒⣒⣒⠦⣄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠪⠤⠷⠀⠀⠀⡄
⠀⠀⠀⡼⣄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡈⠉⠉⠀⠈⠀⠉⠢⣕⠢⣄⠀⠀⢠⡇⠀⠀⠤⠄⠀⠀⢀⠔⡇
⠀⠀⠀⡇⠈⠳⡌⠀⠄⠀⠠⠤⠀⠀⢀⣀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠲⠭⠿⠟⠁⠀⠀⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⡇⠀⠀⠈⢦⡀⣀⢀⡀⠀⠠⠌⢉⡀⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠶⠶⠄⠀⠀⠀⡰⠆⠠⠀⠀⠈⠁⣀⠤⠒⠒⠊⠁⣰⢻
⠀⠀⠀⠘⣄⣀⡀⠀⠀⠀⠀⠈⠒⠒⠒⠦⡀⣁⣉⣀⣁⢀⣀⠈⠀⠉⠀⠁⠒⠂⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣴⡿⢦⡀⣰⠿⠛⠛⠉⠨⡇
⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠀⠀⠄⠊⠁⠈⠁⠀⠉⠀⠀⢀⡾⠁⠠⠀⡇⢻⡀⠄⢀⠔⠁⣧⣀
⠀⠀⣠⢴⠈⠣⣉⡇⠀⣀⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⢒⢶⣄⣀⣀⣀⠀⠀⣾⠀⠀⠀⠀⡇⢸⣢⠐⠀⠀⠀⡧⠈⠓⣄
⢀⡔⠁⣸⠀⠀⠈⡇⠠⡟⢘⠥⡂⢙⢿⣷⣄⠀⠀⠀⠀⠀⣠⠖⠏⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡧⠀⢿⠀⠀⣀⠠⡷⡏⠀⠀⠀⢀⣠⠃⠀⠀⠈⢦
⡜⠀⠀⠈⣦⣄⠀⠑⠚⠀⠉⠀⠛⠐⠀⠨⢿⡆⠀⠀⠀⢸⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠃⣿⢠⠞⠛⠉⣻⠀⠀⠀⠀⠈
⡇⠀⠀⠀⡯⠘⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡎⠀⠀⠀⢀⣇⣸⠎⠀⠀⠀⣹
⢳⠀⠀⠀⠓⢦⡀⠙⢿⣦⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⡇⢀⣠⠞⠉⠉⠁⠀⠀⢀⡴⠉⠀⠀⠀⠀⡸
⠀⠳⡀⠀⠀⠑⠉⠢⣀⠉⠛⠿⠷⣦⡤⠒⠶⣾⠀⠀⢷⣤⣤⣤⣤⡀⠀⠀⣀⣤⣤⡤⠞⠋⠉⠓⣏⠀⡸⠁⠁⠀⠀⠀⠀⢀⠴⡪⠊⠀⠀⠀⢀⡔⠁
⠀⠀⠈⠢⣄⠀⠀⠀⠀⠉⡒⠤⣀⠀⠀⠀⠀⢸⡀⢀⡎⠀⠉⠉⠙⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⡀⠄⡪⠁⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠁⠀⠉⠔⠒⠠⢬⡅⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠆⠁⠀⠁⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⣀⡀⠀⠀⢸⠁⠀⡧⠋⠁⠈⠩⠟⠭⠬⠬⠛⠋⠀⠈⠙⠀⠀⠁⠀⠀⠀⢀⣀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⡏⠀⠀⢸⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠜⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣟⠋⠓⣻⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣉⣁⠀⢀⡿⡉⠫⢹⡀⠀⣈⣉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⣀⠇⠀⠉⠉⢳⣈⠒⡾⠉⠉⠀⠸⣀⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⠭⢤⡀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⢀⡤⠬⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢦⣀⠀⠀⢇⡠⠴⠦⡄⠀⠀⢠⠴⠦⢄⡼⠀⠀⣀⡴⠻⡆
⠀⠀⠀⠀⠀⠀⠀⡤⠊⢹⣆⣀⣈⠙⠲⠤⢤⣀⣀⣈⣒⣒⣁⣀⣀⡤⠤⠖⠋⣁⣀⣰⡏⠑⢤
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠸⡏⠁⠘⣆⠀⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⡠⣄⠀⣰⠃⠈⢹⠇⠀⠀⢳
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠳⣄⠀⠈⠓⠁⠈⢧⣀⣰⠋⠙⣆⣀⡼⠁⠈⠚⠁⠀⣠⠞⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⢸⠻⣄⠀⠀⠀⠈⠓⠦⢤⣀⡀⠀⠈⠀⠀⠀⠀⠁⠀⢀⣀⡤⠴⠚⠁⠀⠀⠀⣠⠟⡇
⠀⠀⠀⠀⠀⢸⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣠⡀⠀⠉⠓⠒⠦⠤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠴⠒⠚⠉⠀⢀⣄⣀⣠⡇
⠀⠀⠀⠀⠀⢸⠀⠉⠀⠘⣆⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣰⠃⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⢦⡠⠋⠈⢳⠀⠀⠀⣠⠖⠲⣄⠀⠀⠀⡞⠁⠙⢄⡴⠃⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠋⠀⠀⠙⠦⠤⠞⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠲⠖⠶⠒⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠒⠄⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢄⠠⠊⠀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠁⠠⠐⣂⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⡀⢀⢈⠉⣀⡠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠃⠉⠀⠀⢀⢠⢄⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠠⡐⢀⠁⢀⠥⠼⡁⢄⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⡗⡌⢄⡤⢾⢉⠉⠉⠑⡓⠀⠀⠘⡦⢀⣲⠃⠀⠀⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠄⠀⠀⠈⣉⣁⣈⣂⣀⣀⣒⡀⠀⠀⠀⠉⠠⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠠⠤⡀⠀⠀⠀⠠⡀⢀⠀⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠒⠤⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠐⠈⠁⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠐⠒⠂⠀⠤⠤⠀⠀⠀⠤⠤⠀⠒⠒⠒⠂⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡀⠀⠀⢠⣓⡄⠀⠀⢀⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⠇⠀⠀⣿⣩⡇⠀⠀⢿⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣭⡇⠀⠀⣟⢩⡇⠀⠀⣟⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⡇⠀⠀⣟⢥⡇⠀⠀⣟⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇⠀⠀⣿⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣴⡇⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠆⠒⠛⠛⠓⠀⠉⠉⠉⠉⠉⠁⠉⠛⠓⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡖⠉⢢⢀⡴⡄⡎⢸⠀⢸⠁⢸⠀⡇⢡⠀⡎⠀⢱⢠⠒⠀⢴⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠁⠀⢸⠈⡇⠓⠂⢸⠀⠸⠀⢼⠀⢃⠼⠀⡇⠀⣆⢈⠀⠓⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠁⠀⠀⢨⠀⣸⠃⠈⠒⠂⡇⠀⢸⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠤⠤⠆⠒⠒⠒⠚⠛⠉⠀⠉⠉⠈⠁⠒⠚⠒⠂⠤⠤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠒⢤⠀⢀⣀⢀⡊⢦⠀⡰⠋⢱⡀⡎⠱⠀⢠⠊⠑⡄⢀⢤⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⠇⠀⢰⠀⢂⠸⣠⠆⠘⠀⢃⠀⢘⠃⢣⢀⠀⢣⠀⠀⡇⢸⠐⠤⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⣄⡸⠀⠀⠀⡇⠀⠸⠀⠹⣀⡸⢸⠀⠈⡄⠀⠣⠞⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⡄⠀⠀⠀⠀⠀⠀⠀⠳⠤⠚⠀⠀⠀⠀⣈⠀⠀⡇⠀⠀⠀⠀⢠⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⢠⠒⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⡜⠒⡄⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠈⠲⠤⠍⣐⣒⠒⠐⠲⠆⠀⠀⠤⠤⠤⠄⠀⠀⠐⠶⠒⠒⣒⣒⡩⠤⠖⠁⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠬⠭⠍⣉⣉⣁⣀⣀⣀⣀⣈⣉⣉⠉⠭⠥⠤⠤⠐⠒⠂⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⡄⠀⠀⠀⠀⢠⠃⢆⢣⠀⠀⠀⢀⠇⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡆⢱⠀⠀⠀⠀⢯⢸⢸⢸⠀⠀⠀⢸⢠⠱⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⣄⡜⠀⠀⠀⠀⠈⣻⣿⢿⡄⠀⠀⠘⣼⣾⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⣯⢲⡀⠀⠀⠀⡞⠛⢩⣼⠃⠀⠀⡴⠻⠛⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⢸⡇⠀⠀⢸⣴⣿⡗⣿⠀⠀⠀⣷⣰⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢚⣿⡇⣀⡀⢈⣿⠦⣌⣏⠀⣀⠀⢸⣿⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⠶⣤⣶⣾⣿⣻⡏⠭⠛⣿⣿⣭⣿⡯⣑⠢⣿⣿⣭⣿⣿⠿⣋⣿⣷⣶⣶⡴⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⢾⣿⣷⣿⣛⣛⠋⠉⠉⠉⣸⡧⢐⣫⣧⠀⠀⢰⡗⠠⢍⣿⠀⠀⠀⣿⣈⠤⣿⠉⠙⠛⠛⠛⣷⣶⣽⣟⣛⣳⣆
⠀⠀⠀⠀⠀⢾⣷⢻⣽⣶⣿⣭⣭⣿⣶⡞⠲⣮⣩⣭⣭⣥⣤⡶⢮⣭⣛⣋⣭⣴⡶⢦⣭⡿⡿⢭⣴⡞⣛⣛⠺⣿⣷⣴⡟⣿⢻⡿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣯⣚⣿⡸⣿⣿⣿⣿⣾⣽⣶⢶⣾⣿⣿⣶⣮⣻⢯⣯⣭⣿⣶⣿⣯⣿⣷⣭⣿⡿⣿⠿⢿⣿⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡯⣽⣽⠿⣿⣿⣿⣮⣿⣯⣿⣾⣿⣯⣽⣿⣽⣿⣿⣟⣟⣫⣾⣯⣿⣷⣿⣷⣷⣿⡿⣿⣻⣿⣟⡾⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣏⡷⡏⣵⠟⣿⣿⣿⣞⣭⡭⢿⣟⣯⣿⣿⣟⣯⣭⢯⣟⣿⣯⣷⣿⣿⣷⡿⠞⢾⣿⢿⣿⢫⡷⣿⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⣸⠇⠀⢹⡟⣿⠉⣿⡝⣹⠟⠉⣿⣯⡿⠁⠹⣟⣿⣿⠿⣿⡿⢡⡟⠀⠀⠀⢻⣏⢃⡾⠀⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠀⠀⠘⣷⣻⡿⠃⣴⠏⠀⠀⠉⠉⠁⠀⠀⢿⣭⣿⣠⣿⠇⣼⠃⠀⠀⠀⠀⠙⠛⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣘⣻⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣹⠟⢁⣴⡟⠀⣀⣀⣀⣠⣤⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣭⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⣫⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠩⠻⣯⡛⠿⠗⢒⣺⣾⠶⢾⣖⠾⣿⣿⠿⢟⣫⡶⠞⠛⠻⢮⣟⣟⣩⡴⠞⠉⠉⣇⡀
⠀⠀⠀⠀⠀⣀⣀⣠⠶⣻⣄⣀⡀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠉⠓⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣠⡶⠿⣟⣷⣄⣀⡀
⢀⣠⠴⠞⠋⠉⠉⢿⡟⣽⢿⣯⣟⢛⣶⡶⢦⣤⣤⣄⣤⣤⣀⣤⣤⣤⣀⣀⡀⢀⣤⣤⣤⣤⣤⣤⣶⣶⣿⣻⣭⣽⣿⣿⣿⡿⠋⠉⠉⠛⠶⢤⣀
⣾⣧⣄⡀⠀⠀⠀⠈⠻⠾⢿⣽⣿⡿⡷⣿⣿⣷⣷⣿⢿⣿⣫⣴⣶⣮⣿⣿⣛⣿⣭⣴⣦⣽⣷⡾⠿⣶⣾⢿⣻⣿⣵⠟⠋⠀⠀⠀⠀⠀⠀⣀⣽
⠈⠛⠿⣿⣷⣶⣦⣤⣄⣀⣀⣀⠈⠛⠿⠿⠻⠷⡿⠿⣿⣯⣿⢾⣿⣿⠿⣯⣭⣽⡽⠿⣿⡿⠟⠛⠛⠛⠛⠛⠉⠀⢀⣀⣀⣀⣤⣤⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢢⠀⠀⠀⢠⠀⠉⣢⠀⠀⠀⠀⡠⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠃⠀⠀⠀⠀⠁⠛⠁⠀⠀⠀⠀⠲⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣤⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⢹⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣹⡇
⠀⠀⠀⠀⠀⣀⠄⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢐⣼⠟⠶⠶⢄
⠀⠀⢀⡤⠚⠁⠀⠀⠰⠷⠶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠉⠳⡄
⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠋⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠆⠀⠸⡇
⠀⠀⢸⣤⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡰⠃
⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀
⠀⠀⠀⣸⠄⠀⠁⠢⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣼⡅
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡀
⠂⠀⠀⠀⠨⣴⠤⠰⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠚
⠀⠀⠀⠠⠀⠀⠈⠒⠬⢉⡙⠓⠶⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠢⠀⠀⡀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠖⠒⡊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠢⡀⢀⡤⠶⠊⠉⠓⠦⣄⠀⢀⡠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⢠⠖⠉⣀⣤⡀⠈⠻⣄⣴⣿⣿⣶⡄⡰⠋⠁⣀⠀⠙⠢⡄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⡰⠈⠛⣦⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣦⡠⠛⠉⢀⠌⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠲⡄⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠠⠶⠋⢀⡴⠚⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣁⠴⢋⡈⢦⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣰⠋⠀⢠⠇⢤⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⣋⠴⢦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⡴⢤⣁⠰⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠓⣤⠖⠉⣀⠜⢀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢰⢠⡈⠙⢦⣠⢾
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡞⠀⠈⠁⣠⠏⠀⠈⣿⠟⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡙⠿⣏⠀⣠⠞⠀⡇⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠱⣼⣄⠐⠊⠁⠀⣠⠊⠀⢠⠃⡄⠈⠹⣏⣩⠟⠉⡤⠙⠿⡙⢯⠟⠁⠀⡼⢱⡆⠈⢦⠀⠀⠚⠀⢀⡴⢁⠾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⢕⡲⠤⠶⡇⠀⠚⠁⣠⠃⠀⠀⣿⢁⣠⠞⠀⡆⠀⠘⡏⠀⠰⠊⠀⡸⠀⠀⢸⠦⠤⠴⠒⢋⡴⠋⢸⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠲⢤⣙⢦⣀⠚⠁⢀⣀⡴⢧⠀⠀⠀⠈⠀⠀⢠⠟⠦⣀⡀⠈⠀⣀⡴⠋⠀⢀⣠⠔⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠒⠢⣄⠀⠀⠀⠈⠉⠛⠻⠯⠭⢤⣀⣈⣑⡲⠦⠤⠶⠚⠁⠀⠀⠀⣉⣉⣉⣤⠤⠶⠚⠉⣀⡀⠀⡴⠋⢸
⠀⠀⠀⠀⠀⠀⠀⢹⠘⢄⣀⠈⢆⢀⣠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⡞⠉⠀⠀⡼⠤⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠉⠱⢄⡀⠀⠈⠳⢀⡤⠖⠒⠦⢄⡀⠀⢀⣠⠤⢤⡀⠀⠀⡴⠋⠉⠁⢀⡼⠦⠤⠖⠋⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠙⠓⠋⠉⠑⠦⣄⣀⣀⡭⠖⣍⡀⠀⠀⣀⡤⠚⠑⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠴⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢱⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢠⡞⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⣫⣍⡀⠀⠀⠀⠀⠀⠀⣴⠞⣛⣥⡀⠀⠀⠀⠀⠀⠀⣀⣩⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠑⠚⢿⠀⠀⠀⠀⠀⠀⣯⠉⠉⠙⡏⠀⠀⠀⠀⠀⠀⣻⠳⠾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠒⠶⢾⠀⠀⠀⠀⠀⠀⡏⠉⠓⠲⡇⠀⠀⠀⠀⠀⠀⣿⠲⠦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣄⣀⣸⠀⠀⠀⠀⠀⠀⡿⠢⣄⣀⣯⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⣿⠀⠉⠒⢿
⠀⠀⠀⠀⠀⢀⡴⠢⣤⣀⠀⠀⠀⣀⣀⣀⣿⣦⣄⣀⣸⣄⣤⣤⣤⣄⠀⡿⢢⣤⣤⠷⠤⣤⣀⠀⠀⣀⣿⣄⡀⠀⣸⠀⣀⣀⣤⣀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⢳⣴⠛⠉⠉⠉⠉⠑⣦⡝⠋⠁⠀⠀⠀⠈⢻⣧⡞⠉⠀⠀⠀⠀⠉⢧⡞⠉⠀⠈⠙⢿⣿⡞⠉⠁⠀⠈⠙⣦⠞⠉⠉⠉⣷
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⣷⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠷⣄⣀⣀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠀⠉⠛⠲⠤⠴⠚⠋⠉⠓⠤⠤⢄⣤⠤⠞⠋⠙⢦⣀⣀⣀⣀⡤⠞⠻⢤⣀⣀⣠⠞⠉⠓⠦⠤⠴⠋⠀⠈⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣤⣄⡀⠀⡇⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡼⠋⠀⠀⠉⠛⢧⣤⠞⠁⠀⠀⠀⠀⠈⠻⣄⣠⠶⠛⠋⠉⠉⠉⠛⢶⡄⢀⡴⠚⠋⠁⠀⠈⠉⠻⣆⣠⠶⠊⠙⠋⠑⠶⣄⣠⠶⠚⠛⠛⠛⠛⢶⣜⡶⠞⠋⠙⠶
⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠟
⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢀⣼
⠀⠀⣟⠉⠉⠓⠚⠋⠙⠶⢤⣤⣤⣤⣤⡤⠖⠲⢦⣤⡀⠀⠀⠀⣀⣤⡴⠖⢶⣀⠀⠀⠀⠀⠀⣀⡤⠾⣄⡀⠀⠀⠀⣠⠞⠳⣤⣄⣀⣠⣴⠏⠀⠙⠒⠒⠚⠉⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠑⠒⠶⠖⠊⠉⠀⠀⠈⠉⠓⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡞⠛⠛⠳⢦⡀⠀⢀⣀⠤⠤⠦⠤⣤⡀⠀⢀⣀⠴⠶⠶⠖⠶⠦⣄⠀⠀⣀⡴⠆⠛⠛⠓⠶⢦⡀⠀⣠⡤⠤⠤⢄⡀⠀⠀⢀⡤⠶⠶⠶⠦⣄⠀⢀⡤⠦⠤⣮
⣯⠀⠀⠀⠀⠀⠙⣶⠋⠁⠀⠀⠀⠀⠀⠙⠷⠛⠀⠀⠀⠀⠀⠀⠀⠈⢣⡼⠋⠀⠀⠀⠀⠀⠀⠀⢹⡾⠁⠀⠀⠀⠀⠙⣦⡼⠋⠀⠀⠀⠀⠀⠘⣷⠋⠀⠀⠀⠈
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠲⠶⠤⠤⠶⢦⣄⡀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⢀⡜⠛⠶⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠉⠛⠓⠶⠶⠶⠞⠛⠁⠀⠈⠳⠆⣀⣠⣠⡴⠞⠋⠈⠛⠶⠦⠤⠶⠛⠁⠀⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄⠀⠀⢀⣾⠻⣧⡀⠀⠀⣠⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢈⣿⠀⠀⢸⣇⣀⣽⠇⠀⠠⣿⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣉⣛⣉⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣧⠀⠀⢸⣿⠉⣿⠀⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣯⣀⣠⣼⣿⠀⣿⣤⣄⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⢿⣇⣸⣿⠉⠉⠹⣿⣤⣿⠉⠉⠉⣿⣄⣸⡟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣌⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣡⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣯⠉⠛⠛⢻⣿⠛⠻⠿⠟⢻⣿⡟⠛⠛⢉⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⠷⣦⣴⠟⠻⣷⣤⣀⣤⡾⠛⠷⣦⣴⠾⠃⠈⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⢻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⡈⠉⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⣁⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡟⠛⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠟⠛⢻⣯⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠾⠛⣷⡀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⢠⣾⠻⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⠷⠶⠾⠋⠈⠻⣶⣤⣤⣤⡾⠟⠈⠛⠷⠶⠾⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⢸⣧⣼⠀⣼⣷⠀⣿⢹⡆⣿⢹⡎⢷⣴⠏⠀⠀⣿⣽⡇⣿⢸⡏⣷⠉⣿⠋⢸⣧⣼⡇⡟⢻⡆⣸⣷⡙⣧⡾⠁
⠀⠀⠸⠇⠹⠰⠟⠛⠇⠿⠋⠁⠿⠋⠁⠀⠿⠀⠀⠀⠿⠾⠇⠿⠸⠟⠿⠀⠿⠀⠸⠇⠹⠃⠷⠾⠳⠟⠛⠧⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⡄⠀⠀⠀⠀⠀⣠⠞⢳⡀⠀⠀⠀⠀⠀⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⡻⠀⠀⠀⠀⠸⡅⠀⢀⡇⠀⠀⠀⠀⢰⠋⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⡖⠃⠀⠀⠀⠀⢀⡹⠶⠋⠀⠀⠀⠀⠀⠀⢳⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠛⠽⢗⠀⠀⠀⠀⢸⡛⠛⠹⡇⠀⠀⠀⠀⢸⠿⠶⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠑⠒⢿⠀⠀⠀⠀⢸⠉⠙⠚⡇⠀⠀⠀⠀⢸⠓⠢⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢹⠀⠀⠀⠀⢸⣄⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠙⠒⢾⠀⠀⠀⠀⢸⠀⠙⠲⡇⠀⠀⠀⠀⢸⠑⠢⢤⡇
⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⢀⣀⣨⣷⣄⣀⣸⣀⣤⣤⣀⢸⠦⣄⣤⠧⣤⣀⠀⠀⣸⣄⡀⢀⡇⢀⣀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⡏⠀⠀⠈⠳⡞⠉⠀⠀⠉⢳⡝⠉⠀⠀⠀⠈⢿⡔⠁⠀⠀⠀⠈⢷⠋⠁⠈⠙⢿⡟⠉⠀⠀⠉⣷⠋⠁⠉⢳
⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡁⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡏⠙⠒⠒⠓⠦⠤⣀⣠⠤⠲⢤⣀⡀⣀⣀⠤⠲⣄⠀⠀⠀⢀⡴⢆⡀⠀⢀⠼⠣⣄⣀⡠⠃⠙⠒⠒⠚⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⠀⠀⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡴⠋⠉⠙⠺⣇⡴⠋⠉⠁⠉⠉⢢⣀⡴⠖⠛⠋⠓⠛⢦⠀⣠⠖⠋⠉⠉⠛⠲⣄⡤⠖⠒⠒⠶⣄⣠⠖⠚⠛⠛⠲⣄⣧⠖⠒⠲
⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠁
⠈⢳⡦⠤⣀⣠⠤⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡖⠦⣀⣀⡠⢴
⠀⢸⠄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠙⠒⠒⠒⠒⠉⠁⠈⠑⠦⢤⣤⠴⠚⠉⠙⠲⠦⠴⠞⠁⠈⠙⠛⠈⠋⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⡜⠁⠉⠈⠓⢦⣠⠶⠚⠙⠋⠒⢦⣠⠴⠚⠋⠉⠉⠙⢢⡀⡴⠚⠁⠁⠉⠉⠳⣄⡴⠊⠋⠑⠲⡄⢠⠖⠛⠛⠛⠓⣦⡴⠚⠉⠛
⢇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠁
⠈⠙⠲⠦⠤⠤⠦⣀⡀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⠟⠳⢄⣠⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣟⣛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢀⣼⣿⡿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋⣿⡇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣿⣤⣤⣼⣷⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⠿⠿⠿⢿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⣀⣀⣠⣤⡿⠛⠛⢿⣦⣴⡟⠉⠙⢻⣦⣄⣀⣀⠀⠀⠈⠉⠉⠛⠛⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠉⠉⢹⣇⡀⢀⣼⡏⢿⣆⠀⠀⣸⡿⠉⠉⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣤⣤⣽⣿⣿⣋⣀⣈⣻⣿⣿⣿⣥⣤⣶⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠛⠻⠿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⡿⠿⠟⠋⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣰⣿⣿⣷⡀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠁⠙⢿⣿⣿⣶⣶⣶⣶⣿⣿⠟⠛⣿⣿⣶⣶⣶⣿⣿⠏⠈⠻⣿⣿⣶⣶⣶⣶⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⢸⣿⣿
⠀⠀⠀⠀⣠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀
⠀⠀⣠⣾⣿⡿⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡿⢿⣿⣷⡄
⠀⠀⣿⣿⡃⠀⠀⠙⠿⣿⣿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⠿⠋⠀⠀⢹⣿⣿
⠀⠀⠻⣿⣷⣤⣀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⣀⣤⣾⣿⠏
⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠷⢄⡀⠀⠀⠀⠀⠰⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠤⠼⠶⡂⠈⢚⠺⡁⣨⣅⢈⡟⢛⡍⡦⣼⡃⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡶⢦⣤⢀⠈⠙⠛⠀⠈⠉⠀⠀⠈⠛⣉⠀⣈⣀⢀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣉⣀⠓⠄⢀⡔⣵⢦⢢⢒⡀⡞⠘⢷⢹⠈⡅⠸⣗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣈⣉⠙⠃⡿⠺⠺⠆⠰⣤⠢⢱⠶⠾⠟⠚⠣⢴⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⢄⣉⠊⠒⠢⠾⠠⢐⠴⢢⠧⢶⠞⢶⠶⠾⠾⠛⠛⢛⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠲⢦⣈⠉⠛⠒⠒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠶⠖⠶⢋⡉⣤⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠳⢆⠰⣿⣼⡉⠸⠓⢸⡖⢒⡶⠰⡠⢄⡶⣶⠢⠶⢌⣻⣀⣥⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠻⠆⣠⠀⣍⡙⡛⠺⠞⠷⠶⣂⡿⠥⠴⠷⠾⠷⡿⠻⠛⣉⣥⣤⢾
⠀⠀⡀⠀⠀⣀⠤⠴⢦⡟⠳⢆⣢⢉⣀⣁⠁⠛⠉⠓⢖⡲⠶⠲⠶⠶⠞⠷⠿⠛⠛⠋⣩⣠⣶⣤⣠⡀⣀
⠀⠘⠄⡉⠀⠃⠄⠀⠀⠉⠓⠠⢤⣉⠉⠚⠛⠒⠲⠶⠄⠶⠴⠴⠲⠶⠶⠶⠭⠖⠟⠛⣩⡤⠼⠃⠀⡈⠆⠄⠌⢰
⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠀⢀⡈⠍⠃⠐⠂⠲⡒⠶⠶⠲⠠⠤⠐⠒⠒⡒⠛⠉⠉⠀⠀⠐⢀⡁⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⡈⡀⡨⠂⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠱⡀⠀⠀⠀⠀⢀⠎⢠⠱⡀⠀⠀⠀⣸⢉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡆⢳⠀⠀⠀⠀⢸⢀⠀⡇⡇⠀⠀⠀⡇⡈⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣁⡼⠀⠀⠀⠀⠈⢳⣿⣼⡅⠀⠀⠀⢧⣧⣦⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣟⢤⡀⠀⠀⠀⡴⠿⢛⣁⡇⠀⠀⢀⡴⠿⠟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣤⣭⢸⡇⠀⠀⢠⣇⣾⡾⣽⡇⠀⠀⢸⡄⣴⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠽⣾⡇⠀⠀⠘⢿⣍⡑⢻⠀⠀⠀⠈⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⡶⢾⡆⢚⠿⣷⣾⣳⣶⣿⠢⢌⣹⣶⣖⣲⣶⣟⡯⢼⡷⣤⣤⣄⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣟⣟⣿⡿⠿⠿⠟⠛⠛⢻⣋⡡⠔⣏⠉⠉⠉⡿⣙⠒⢾⠉⠉⠙⢻⡇⠒⣹⡿⠷⠶⠿⢷⣶⣿⣝⣻⡶⠶⣦⣄
⠀⠀⠀⠀⠀⢰⡿⣭⣺⣫⣿⠷⠶⠶⣄⣠⣤⣄⡛⠶⢶⡾⠛⢀⣀⣐⡷⠤⠭⠿⢂⣀⣀⡘⣿⣿⣶⣓⣠⣤⡤⢤⣤⣾⣛⣻⣿⣿⣚⣿⠄
⠀⠀⠀⠀⠀⠘⢿⣧⣿⣾⢯⣿⡟⣿⣿⣷⣶⣌⡿⢿⣫⣍⣛⢿⡭⣍⣛⢷⣾⣟⣻⢿⣍⣽⣿⣽⣭⡝⢿⣶⣶⣶⣾⣿⣿⣿⣫⣿⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⠷⡾⢷⣽⣿⣽⣿⡻⣿⡿⣿⣽⣿⡿⢿⣿⡿⢷⢿⣿⣿⣻⢟⣿⢿⣯⢟⣿⣻⣛⣵⣿⣭⣿⣿⡻⣫⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣻⢿⣻⣭⣿⣿⣟⣿⠷⠿⣿⣟⡿⢾⣿⣿⡿⡿⣿⢷⣾⣶⢿⣻⣿⢿⡟⣟⣛⣻⢿⣯⣿⡾⣿⡷⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⠝⣰⠏⠘⣿⣻⣯⠿⡽⣏⣽⣵⠾⣾⢿⣳⡿⢷⣝⣿⣾⣟⣯⣿⡯⢻⡾⠋⠈⠻⣷⢽⡟⣸⠟⣷⣙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⣴⠏⠀⠀⢿⣔⣯⣠⡿⢃⡾⠁⠀⢿⣽⠟⠀⠀⣿⣼⣽⠋⢹⣾⠁⣾⠁⠀⠀⠀⠹⣧⣴⠏⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣀⠀⠀⠘⣷⣞⣋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⣯⢾⢷⣾⠏⢰⡏⠀⠀⠀⠐⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣷⣶⣮⣿⣯⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣯⣿⣁⣴⣿⣡⣤⣤⣤⣴⣶⣾⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡵⠶⠿⣯⡻⣛⣻⣿⡿⡿⢿⠿⠿⠿⠿⠿⣿⣿⣿⣾⣿⠿⠿⣾⣿⡿⢿⣿⣛⣫⠿⣋⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠊⠻⢦⣝⣛⣋⣩⣽⠿⠟⠻⣯⡛⠿⠿⠟⣋⣭⠾⠛⠉⠉⠛⠷⣯⣥⡴⠞⠉⠀⠀⣧⣄⡀
⠀⠀⠀⠀⣀⣀⣤⣤⣞⣻⣷⣤⣀⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣟⣛⣫⣻⣦⣤⣀⣀
⢀⡤⠖⠛⠉⠁⠀⠻⣿⢵⣟⣯⣾⣍⣻⡷⡖⣶⡶⢶⣤⣤⣤⣤⡶⠶⣦⣄⣤⣄⣀⣤⣶⢶⣤⣶⣶⣶⡶⠾⣯⣽⣦⣾⡿⣿⣿⡿⠋⠀⠈⠉⠛⠲⢤
⢿⣷⣤⣀⠀⠀⠀⠀⠙⠛⠻⣮⣿⣝⣿⠷⣽⡻⢿⣷⣿⢷⣾⣿⢶⣾⣷⣿⣿⣭⣻⣿⣶⣶⣶⡿⣶⣾⣿⣻⣾⣗⣵⣿⡷⠟⠉⠀⠀⠀⠀⠀⠀⣀⣠
⠈⠙⠻⢿⣿⣷⣶⣦⣤⣤⣀⣀⣀⠉⠛⠛⠛⠻⠿⠿⠿⣿⣷⡿⢷⣿⣷⠾⢿⣿⣼⡷⠿⠿⠿⠿⠋⠛⠛⠙⠛⠋⠁⠀⣀⣀⣀⣠⣤⣴⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢀⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⢀⡀⠠⠄⡔⠁⢴⠥⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⠔⠰⡉⠆⢀⠘⣄⠈⠀⠀⠀⠐⡉⢉⣠⠋⠐⣁⡢⢄
⠀⠀⠀⠀⠀⠀⢀⠄⢘⣺⢏⢰⠇⠕⠈⢄⣽⣿⢻⣦⢄⡀⢀⠋⢁⣢⡖⠂⡀⡐⠺⠳⡀
⠀⠀⠀⠀⢀⠠⠎⠀⠀⠉⠀⢽⡖⢄⣵⣶⣯⣾⣵⣞⣽⣿⠭⠵⠭⠄⣀⠂⢢⠠⠈⠂⡝⠂⢄
⠀⠀⢀⠖⠁⠀⠘⢄⡀⢀⣠⠀⢀⣿⣟⣾⢝⣿⣽⣿⠏⠀⠀⠀⠀⠀⠈⠣⠈⣠⠈⡠⠄⠈⠀⢣
⠀⠀⡌⠀⠄⠐⠂⠀⠀⠀⠀⠁⠚⢿⣿⣿⣾⡶⠋⡟⠄⠠⠤⢄⡀⠀⠀⠀⠻⡁⠉⢠⣀⡄⠐⠈
⠀⠀⡰⡀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠉⠉⠁⠀⢠⡌⠉⠈⠉⡉⠪⣔⠤⣀⣀⡇⠀⠠⠀⠀⠀⢘
⠀⠀⡇⠈⠠⡀⠀⠀⠂⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⢀⠉⠛⠋⠄⠀⠀⠀⠀⢀⡜
⠀⠀⠰⣀⡀⠈⠀⠈⠀⠀⠤⠁⡀⡀⣀⢈⠁⠀⠂⠂⠀⠈⠀⠀⠀⢁⣀⣀⣠⡶⣍⢠⠖⠒⠛⢻
⠀⠀⢠⢿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⠁⠀⣰⠃⠀⢸⢸⡀⠀⠀⢸⠤⡀
⢀⠔⢹⠀⠑⠇⣴⢛⠛⠻⣶⣄⠀⠀⠀⠀⣠⠔⠚⠋⠉⠉⠛⠒⠒⢶⠀⡇⠀⠀⢸⡜⠁⠀⠀⡸⠀⠈⢂
⡎⠀⠘⣤⠀⠑⠉⠉⠘⠀⠈⠿⡆⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠘⢸⢠⠒⠈⡇
⡇⠀⠘⣇⠑⢦⣄⠀⠀⠀⠀⠀⢁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⡠⠼⠜⠃⠀⢀⠇
⠘⢄⠀⠈⠑⠄⡙⠻⠶⣤⡤⠶⣼⠀⢡⣄⣀⣀⡀⠀⣀⣀⡠⠖⠒⠲⠁⡱⠈⠀⠀⠀⢀⠔⠁⠀⠀⢀⠎
⠀⠀⠑⢄⡀⠀⠈⠐⠢⠄⡀⠀⢸⠀⡎⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠉⠀⢀⡀⠄⠀⠁⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⢸⠀⢷⠀⠀⠤⠤⠀⠤⠤⠀⠠⠀⠒⠀⠈⠀⠀⣀⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⡆⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⠆⠀⢼⣻⠀⠀⣼⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⢺⣽⠀⠀⣞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠀⠀⢸⢾⠀⠀⡯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠀⠀⢸⣻⠀⠀⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠄⠚⠛⠃⠀⠉⠉⠁⠀⠛⠛⠂⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⢀⡀⡐⢆⢀⠖⢂⠰⢢⠀⠔⠢⠀⢄⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠐⠘⠘⠀⡀⠘⠀⠌⠘⠰⠈⡄⢰⢐⠐⢺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠀⣂⣠⠃⣀⠊⣘⢀⣂⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠔⠢⠀⡠⡀⡎⡆⢠⠊⢡⢰⠑⠀⡔⠁⠆⠠⠄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⠀⠐⠀⠆⠓⠃⠅⠰⠀⠌⠈⢠⠀⠃⠀⡆⡀⠐⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠀⠀⠁⠁⠀⠀⢇⠠⠀⠈⠊⢰⠀⠘⠀⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠉⢠⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠊⠀⠀⠀⡰⠃⡄⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠑⠠⠈⣁⠒⠨⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠨⠍⠒⣈⠡⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠉⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡆⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠀⠀⠀⠀⣰⡀⠀⠈⢳⠁⠀⠀⢀⢣⠀⠀⠀⢯⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⢸⣁⡇⠀⢰⠀⡇⠀⠀⣏⣸⠀⠀⠀⣈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⠀⠀⠀⢀⢿⠀⠀⢸⠀⣷⠀⠀⡼⢇⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣀⣀⣼⢸⡧⠤⢼⠀⣿⠤⠤⡇⢸⠤⢤⣀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⡇⡟⠉⠀⠀⢿⢸⡇⠀⠈⠳⠃⠀⢀⡇⢸⠀⠀⠀⣇⣸⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠑⠃⠀⠀⠀⠸⣸⠁⠀⠀⠀⠀⠀⠘⣇⡜⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⢸⠀⠘⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠶⠖⠒⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣾⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡆⡇
⠀⠀⠀⠀⠀⢹⡼⢸⢠⠞⠉⠙⠦⣄⣀⣀⣠⠤⠶⣄⠀⢀⠞⠉⠱⡄⢀⠖⠒⢄⡀⢠⠟⠚⠁⠀⢷⠇
⠀⠀⠀⠀⠀⠀⡇⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⡇⠈⡇⠀⠀⠉⠁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⡆⠀⠀⠀⠀⢀⣶⠀⠀⠀⠈⣿⠁⠀⠀⠀⡀⢳⠀⠀⠀⠀⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠀⠀⠀⠀⡟⠸⡆⠀⠀⡶⠀⡆⠀⠀⠀⡇⢸⠆⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⠀⠀⠀⠀⠀⢱⡞⠁⠀⠀⡇⠀⣧⠀⠀⠀⢳⠋⠀⠀⠀⠀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⠀⠀⢰⡎⣵⠀⠀⠀⡇⠀⣿⠀⠀⢸⡏⣴⠀⠀⠀⠀⡿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣀⣀⣠⠤⢼⠀⢹⠒⠒⠚⣇⠀⡿⠒⠒⢺⠀⢸⠶⠶⠤⠤⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢾⡇⣼⠉⠀⠀⠀⢸⠀⢸⠀⠀⠀⠘⠶⠃⠀⠀⢸⠀⢸⠀⠀⠀⠀⣧⢀⡏⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠷⠟⠀⠀⠀⠀⢸⡄⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠉⠓⠶⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠒⠛⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡆⣿
⠀⠀⠀⠀⠀⠀⠸⡇⣾⢸⠀⢠⠖⠚⠲⢄⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣰⠚⠒⢦⠀⠀⢠⠤⢤⡀⠀⠀⢀⡶⠦⠞⠁⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣠⠏⠀⠀⠀⠀⠉⠓⠒⠒⠚⠋⠉⠀⠙⢦⣀⡴⠃⠀⠀⠈⡇⠀⣯⠀⠀⠙⠦⠤⠞⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠴⠶⠶⠒⠒⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠒⢦⡀⣴⣋⣁⣀⣈⣱⣤⣮⣤⣤⣤⣠⣹⣤⣞⣁⣀⣈⣙⣦⣀⡴⠒⠓⠒⢤⡀
⠀⠀⠀⠀⢀⠴⠒⠒⠲⣄⣾⡤⠴⠒⠚⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠦⠤⣷⣠⠖⠒⠒⠦⡄
⠀⠀⠀⢰⡏⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠸⡆
⠀⠀⠀⠘⣧⠖⠛⠙⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠲⣴⠇
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣧⡴⠚⠉⠛⠢⡄⠀⣀⣤⣤⣄⡀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣠⣤⣤⡀⠀⢠⠖⠚⠉⠓⢦⣾⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠘⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢹⡞⠁⠀⠀⠀⠙⣦⡔⠋⠁⠈⠙⢦⣴⠋⠀⠀⠀⠈⢳⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠈⠓⡶⠶⠚⠉⢳⡀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢀⡟⠉⠒⠶⢶⠊⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠒⠶⠖⠋⠀⠹⢤⣀⣀⣀⡴⠛⢦⡀⠀⠀⢀⡴⠛⢦⣀⣀⣀⡴⠋⠀⠙⠲⠶⠒⠋⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠞⢧⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡞⠳⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡷⠤⠴⠖⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠙⠲⠦⠤⢾
⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠈⠳⢤⡀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⢀⣠⢾
⠀⠀⠀⠀⠀⠀⣷⣄⠉⠓⠦⢤⣄⣀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⣀⣠⡤⠴⠚⠉⣠⣼
⠀⠀⠀⠀⠀⡴⡇⠈⠙⠲⠤⣄⣀⠉⠉⠙⠒⠒⠲⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠴⠖⠒⠚⠋⠉⢉⣀⣠⠤⠖⠋⠁⢸⢦⡀
⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠤⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠴⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⣸⠀⣧
⠀⠀⠀⠀⠘⢦⡈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡉⠙⠓⠲⠦⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠶⠒⠋⠉⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣤⣤⣄⣀⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⣀⣤⣤⣴⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠮⠭⠭⣭⣭⣙⣛⣒⣒⣒⣚⣛⣉⣩⣭⠭⠭⠽⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⡇⠀⠀⠀⠀⢸⠶⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠤⠶⠚⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠻⣿⣯⣭⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣼⢿⡏⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣾⡇⡀⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⠟⠛⠋⠉⠉⠉⢉⣀⣰⡾⠻⢷⣄⣤⡾⠛⠻⣦⣀⡉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢠⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠛⣿⡀⠀⣸⡟⣿⡀⠀⠀⣽⠟⠛⠻⠷⣦⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⣤⣬⣿⣿⣏⣀⣈⣿⣷⣿⣯⣤⣴⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡿⣿⣷⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⢿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⠟⠛⠉⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⣠⣾⣿⣇⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣴⣾⣿⡿⠋⠙⢿⣿⣶⣤⣤⣴⣾⣿⠿⠻⣿⣷⣦⣴⣾⣿⠟⠙⢿⣿⣷⣦⣤⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⢸⣿⡇
⠀⠀⠀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀
⠀⠀⣾⣿⠿⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠛⢿⣿⣧
⠀⠸⣿⣿⡀⠀⠈⠙⠻⢿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⡿⠟⠋⠀⠀⢀⣿⣿⠂
⠀⠀⠙⢿⣿⣶⣤⣀⡀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠀⠀⢀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄⢸⢚⡲⠄⡄⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠙⡟⠂⢠⣀⢂⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣤⣀⣀⣀⣀⣈⣀⣀⣭⡶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠉⢉⡉⠉⠉⠉⠁⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠁⠠⠤⠀⠀⠀⠤⠄⠐⣀⠀⠀⠂⠐⠀⠀⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠈⠁⠀⣀⠀⠀⠄⠀⠐⢄⢖⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣧⣴⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣁⣀⣀⣤⣤⠶⠞⠛⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣤⣤⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⡀⠄⢂⠀⠡⠂⠄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠍⠀⠀⠈⢵⡄⠈⠐⠢⢦⣶⣄⣀⣀⢀⡀⠀⠀⣀⢀⣀⠀⣤⠴⣖⡞⢀⠀⠀⠀⠀⠘⣡⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠂⠀⠱⠀⣀⣋⠀⢠⢙⠃⢠⠀⢼⠉⠷⢶⠀⡀⠠⢐⠃⠀⠊⠙⠂⢠⠀⠐⠤⠌⠀⢀⠀⣀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡷⠧⠉⠒⠄⠁⣄⡀⠂⠈⠀⠠⠀⠀⠂⠀⠙⠀⠁⠀⠈⠀⠀⠀⠁⠀⡀⠀⠀⠂⡙⠃⠢⠐⢉⡂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣇⡀⠀⠀⢀⠐⠒⠄⠈⠁⢀⠐⡀⢠⠀⠀⠀⠐⠀⠐⠀⠀⠀⢤⠦⠀⡄⠀⠀⠀⢐⣤⣶⣶⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠛⠻⣶⣦⣤⣤⣄⡂⠀⠀⠀⠀⡉⠀⠑⠂⠀⠀⠀⢈⠀⠀⠀⣀⣠⣤⣦⣴⣶⣿⡿⠛⠋⠁⡐
⡇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠑⠀⢬⡀⠀⠍⠉⠛⠻⠿⢿⠿⠿⢶⣶⣶⣶⣶⣶⣿⣿⡿⠿⡿⠟⠛⠛⠉⠉⢁⠀⢈⠀⠄⠊⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⣀⠀⠀⠀⠀⠉⠒⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣈⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠤⠾
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣤⡤⣆⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠒⠃⠀⠀⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⢠⡖⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢧⡤⢤⣀⣠⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⢀⣀⢀⡦⣄⡞⠲⠾⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠒⠈⠉⠉⢛⠋⠉⠷⢞⢹⡶⠎⠳⢶⠉⢲⠋⡙⠛⢋⠛⠁⠉⠠⠤⠤⠤⠤⠀⣀
⡇⠀⣀⠠⠀⠀⠒⠈⠉⠉⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⣀⠄⠂⠘⢀⠀⢀⡘⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⢠⢀
⡇⠁⠀⠀⡤⠀⠀⠀⠀⠤⠀⠀⠀⠤⡦⠀⠀⠀⠀⢠⠄⠰⡀⠀⠀⠩⠄⡆⠠⠣⠈⡃⠀⠐⠀⡠⠀⠀⣀⠐⡄⠀⠶⣀⠀⣀⢀⠀⢀⡀⢀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠀⠂⠀⠤⠬⢀⠤⠄⠀⠀⠐⠁⠀⠀⢀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠐⣨⠂⠀⠘⣈⠂⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠶⠀⠘⠠⠒⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑
⠏⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠎⠳⠀⠀⡠⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡀⠀⡠⠟⠀⢀⡠⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠈⠍⠀⠀⣀⠥⠔⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡀⡈⠤⠔⠂⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣤⠤⣀⢄⡠⣀⠤⡄⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢁⠀⠁⠈⠀⠈⠀⠀⠀⣀⢰⢤⡀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠂⠀⡀⠀⠀⢀⠀⠎⠰⠁⠀⣸⣄⡎⠑⢄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠊⣣⡐⢄⢠⣠⣠⠖⠚⠒⠓⠠⡯⢀⣀⣈⠦⣄⠑⢤⣩⠍⠀⠉⠀⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠀⠈⠓⠓⠬⠿⢅⠸⣀⠀⠀⠀⣈⠃⠀⠐⠒⠒⠯⠝⠊⠓⠂⢀⣀⠠⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⡀⠀⠀⢄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⠒⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠔⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠙⠒⠂⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠐⠂⠈⠁⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠠⠄⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠤⠤⠤⠖⠒⠒⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⡀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⢰⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢦⣷⠀⠀⠀⠀⠀⣰⣷⣬⡇⠀⠀⠀⠀⢀⣼⣶⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣸⣿⠀⠀⠀⠀⢸⣿⢀⣼⡏⠀⠀⠀⠀⢾⣏⣀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣫⠿⣇⠀⠀⠀⠀⢀⣏⡹⠿⡄⠀⠀⠀⠀⢸⣛⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠖⡏⠀⠀⠀⠀⢸⣁⡤⢖⡇⠀⠀⠀⠀⢸⣁⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⢋⣿⠀⠀⠀⠀⢸⡧⠒⢉⡇⠀⠀⠀⠀⢸⠵⠚⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢚⣩⣧⠀⠀⠀⠀⢸⡶⢚⣩⡇⠀⠀⠀⠀⢸⠔⣋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⡤⡟⠀⠀⠀⠀⢸⠋⣉⡤⡇⠀⠀⠀⠀⢸⢋⣡⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡥⠒⣧⠀⠀⠀⠀⢸⣏⠥⠚⡇⠀⠀⠀⠀⢸⣩⠴⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⢚⡏⠀⠀⠀⠀⢸⡥⠴⢚⡇⠀⠀⠀⠀⢸⡤⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠷⠾⠥⠿⠤⠖⠒⠒⠚⠓⠚⠛⠓⠒⠒⠒⠒⠺⠶⠯⠽⢄⣀⣀
⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡤⠒⠢⣄⠀⠀⠀⣀⠀⠀⡞⠉⢧⠀⠀⢰⠋⠀⠉⢇⠀⡞⠁⠹⡄⠀⢀⡞⠉⠉⢳⡀⠀⣠⢤⡀⠀⢀⠀⠀⣷
⠀⠀⠀⠘⡇⠀⠀⡾⠀⠀⠀⢸⠀⠠⡏⠉⡇⠀⡇⠀⢸⠀⠀⡅⠀⠀⠀⣀⠀⢻⠀⠀⡇⠀⢸⠀⢀⠀⠀⡇⠀⡇⢰⡇⠀⢸⢳⣤⠏
⠀⠀⠀⠀⠳⣀⣠⠃⠂⠀⠀⢸⠀⠀⡇⠀⠣⠤⠃⠀⡞⠀⠀⢹⠀⠀⠀⡇⠀⠸⣤⠀⡇⠀⠸⡀⠀⠀⢸⠁⠀⡇⠀⠓⠲⠋⡼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⣹⠠⠀⠀⠀⠘⡇⠀⠀⠘⡆⠀⠰⡇⠀⠀⡇⢰⠇⠀⠀⢧⢀⠀⠘⢤⡤⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠙⡚⠁⠀⠀⠀⠀⠸⡅⠀⠀⢀⡇⠀⠒⠙⠒⠚⠁⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢀⣀⡤⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⣀⡀
⢀⡠⠖⠋⠉⠀⢀⣠⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣄⡀⠀⠈⠙⠲⢤
⡎⠀⠀⠀⠀⢼⠉⠀⠘⣆⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠉⣷
⠙⠦⣀⠀⠀⠈⠓⠦⢤⣀⣉⠓⠒⠲⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠒⠒⣋⣁⡤⠴⠚⠁⠀⠀⣀⡴
⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡈⠉⠉⠙⠒⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠋⠉⠉⢁⣀⣀⡤⠤⠖⠚⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣓⣒⣒⣶⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣖⣶⣒⣒⣒⣋⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡎⠀⠀⢰⠃
⠀⠀⠀⠀⠲⠬⠥⡤⣬⣭⣬⣭⡵⡇⠀⠀⠀⠀⠀⢀⠠⢼⠤⠀⢰⡧⠤⢤⡧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⣠⡒⠓⠶⠶⠤⠶⠶⠶⠾⠟⢓⣠⠀⠀⠀⠀⠀⠒⠶⠶⠦⠤⠤⠴⠶⠶⠖⠇
⠀⠀⢨⠉⠙⠛⠛⠳⠚⠻⠛⠛⠙⠋⢽⠀⠀⠀⠀⡏⠁⠚⢀⠆⠀⠀⠜⠀⠒⠀⡆
⠀⠀⠈⡄⠈⠀⠀⠀⠐⠚⠀⠃⠃⠀⣼⠀⠀⠀⠀⢷⠠⠤⠄⡀⢀⡠⢂⠤⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠙⠛⠒⠒⠶⠖⠶⠖⠓⠒⠋⠉⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉

⠀⠀⠀⠀⠀⢶⣈⣹⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⢸⠀⠀⠀⡆⠀⡄
⠀⠀⠀⠐⣛⣉⡥⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠔⠺⠖⠺⠛⠈⠿⠉⠏⠉⠏⠉⠟⠐⣢
⠀⠀⠀⠈⠁⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⣙⠒⠒⠒⠒⠒⠒⡒⠒⢒⣒⡋⡉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠠⠀⠀⠀⠀⠀⠀⢁⠌⠁⠖⠁⡐⠃⠈⠀⠛⠀⠉⢙⣐⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⠉⠉⠒⠚⠛⠒⠚⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢨⠀⠀⠀⠖⠄⢀⠒⡒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠶⠦⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀
⠀⠀⠀⣴⡪⡩⣍⣉⡆⠀⠀⠀⠀⣼⠉⠈⠒⠒⠶⠶⠶⠘⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠠⠟⡉⠟⠫⠍⠟⢓⡀
⠀⠀⠀⡇⠐⠊⠁⠂⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⢘⠛⠛⠛⠛⠛⠛⣛⣛⡀
⠀⠀⠀⡗⢒⠶⠒⠚⡇⠀⠀⠀⠀⣿⣿⠠⠤⠴⠂⠖⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⣉⣉⣉⣙⣛⣛⣋⡉
⠀⠀⠀⢠⣀⣀⣠⣤⠄⠀⠀⠀⠀⢸⣌⡊⠀⠀⠀⠀⠀⠠⣼⠀⠀⠀⠐⠂⠀⠖⡆⠀⠀⠀⢘⠿⣷⣭⣮⣾⣿⣿⡷
⠀⠀⠀⠨⠌⣩⡥⠠⠀⠀⠀⠀⠀⠸⢦⣄⣉⣙⣓⣒⣒⣂⡏⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⠚⠓⡀⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⣀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠾⠆⠀⢀⠟⡆⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢠⠎⠀⠱⠀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣃⠄⠀⡋⠀⢠⠀⡇⠀⡴⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠳⢄⡣⠜⡀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠇⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⡉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠙⠃⡅⠀⠀⡇⠀⠀⠀⠀⠤⠤⠀⠠⠀⠒⠒⠒⢂⣤⡔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠡⣌⠉⢛⠀⠀⢠⠤⣀⣄⡀⡀⠀⣀⡀⠀⠀⠀⠲⠂⡀⠱⠁⠀⠀⠀⠀⠀⡀⠀⣠⡀⠦⠤⣇⣀⠄⠐⠁⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⠉⠁⠒⠛⠓⠈⠁⠎⠤⠥⠤⠥⠤⠤⡤⠤⠤⠀⠠⠀⡀⠒⠒⢚⣂⠀⠀⠈⠉⠁⠀⠓⠀⠈⠀⠀⠀⠘⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣠⣤⣠⡤⣄⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠁⠈⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⠀⢤⣼⢶⣄⣠⢏⢧⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠳⠙⠡⠋⠐⠩⣻⡷⠄⠀⢰⠹⡇⠠⣤⡾⣫⢛⡽⣲⠖⡲⢋⠔⣡⢜⣿⡂⠀⣰⢟⢟⡽⡫⠣⠢⠪⠃⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡗⢈⣳⣒⠤⣰⣆⣀⡈⠛⣦⠠⠆⠈⠛⠹⠋⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠉⠘⡙⠁⢁⣀⡀⢠⡄⡀⠤⠰⠆⢺
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠎⠱⠦⠴⡤⢬⣸⣗⣈⣲⠠⠿⠔⣂⡐⠤⣶⠴⠂⣲⠠⡿⠤⠒⣂⡒⠚⠓⢊⣡⣤⣍⣉⣉⡴⢲⡳⠖⢻
⠀⠀⠀⠀⠀⠀⠀⣠⡗⣦⢆⠄⡀⢀⠀⠉⠀⠈⠁⠽⠉⠙⠛⠋⢽⠉⠛⠋⠛⠉⣤⠈⠓⠚⠋⣭⠉⠛⠛⠉⠀⠛⠀⢀⢀⢀⡼⣥⣤⢨⣀
⠀⠀⠀⠀⠀⠀⡼⠀⠛⢶⣥⣊⣔⡡⡢⢚⠔⡠⣠⢆⡤⡠⢀⠴⡤⢂⠄⡀⣀⢀⡀⡠⢠⢀⣄⣀⡰⣢⢂⡖⣴⡿⠿⠛⠛⠛⠚⠘⠉⠀⢻⠳⡀
⠀⠀⠀⠀⠀⢰⢥⡶⢤⣤⠀⡀⢀⠉⠙⠓⠾⠾⠵⠯⠞⢛⣷⣛⡒⠓⠮⠶⠵⣫⡾⠾⠓⠋⠉⠉⠉⠉⠉⠙⠋⠀⠀⢀⠀⠀⠀⣴⣲⣄⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠟⢧⡓⣷⡺⡫⣻⡀⠐⣧⣦⠀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣀⠀⠀⠀⢀⣤⣶⡲⣤⣴⢿⣧⣀⡠⠛⠛⠛⠗⡞⠁
⠀⠀⠀⠀⠀⣶⠠⠤⠀⠀⠈⠈⠪⠿⠴⠾⣻⣤⣿⣟⣽⠉⠀⣾⡻⣧⣤⣶⣛⣿⣦⣶⣣⣮⣧⡀⣤⣼⠟⠓⠙⠘⠋⠁⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣿⡀⢠⠄⣀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠈⠛⠳⠜⠛⠛⠛⠉⠈⢈⠁⠉⠀⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⢀⣲⡇
⠀⠀⠀⠀⠀⣿⣌⣙⣃⣬⣩⣝⠋⠋⣉⣉⠣⣐⣶⢀⣤⣀⣰⣄⢠⠄⠠⣀⣤⣀⣀⣄⡀⢠⡄⠀⠀⢠⡦⣀⠤⠤⠾⠷⠊⣈⣻⣉⣀⣤⡥⡇
⠀⠀⠀⠀⠀⣠⠈⠉⠉⠁⠳⠈⠙⠋⢩⡍⠱⠿⠿⠦⡶⠦⢤⣭⡤⠖⣓⠫⣭⣁⣠⠤⢬⣟⣋⣡⠬⠽⣯⣅⣤⠶⣶⠲⠛⠡⠮⠉⠉⠀⠁⡖
⠀⠀⠀⠐⣒⣻⡷⢹⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠸⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⣉⢀⠀⠄⠄⠂⠀⠀⣠⡴⠄⠀⣀
⠀⠀⣉⠤⠔⢂⡱⠦⢄⡉⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢄⡠⡠⢂⠄⡰⣠⢢⢂⠜⠜⠁⠀⠁⠀⠀⢀⣠⠄⢒⠽⠋⢀⡤⠊⠁⣀
⢀⡠⠄⠒⢉⡠⠔⠒⢉⡡⠝⠓⠒⣠⠤⠄⢀⣀⠀⠀⠁⠐⠁⠈⠊⠐⠋⠚⠱⠋⠊⠐⠁⠊⠈⠀⠀⢀⣀⡀⠤⣤⠖⠂⡩⠛⣁⠔⠁⡠⠖⠁⣀⠔⠋⣠⠖
⢠⠔⠂⠉⣀⡤⠖⠈⣁⡤⠒⠋⠁⣀⡤⠖⠉⠉⢀⣉⠍⠉⠀⡂⠴⣒⠴⠖⢒⡴⠖⠀⣉⠝⠋⢉⡴⠋⢁⠴⠋⢀⠔⢋⡤⠊⣀⠔⠋⢀⡴⠊⢁⡴⢊⡠⠖
⠀⠖⠊⢉⣀⠤⠖⠋⠁⢀⡤⠖⠋⠀⣀⡤⠖⠈⠁⣀⡤⢖⣩⠔⠊⢀⡤⠖⠁⣠⠔⠋⢀⣠⠞⢁⡠⠞⠁⡠⠞⣁⠔⠁⣠⠞⠁⡠⠚⠁⡠⠒⠁⠀⠋⠀⠈
⠀⠀⠊⠉⢀⣀⠤⠒⠊⢁⣀⠤⠐⠉⠁⢀⡠⠔⢋⣡⠖⠉⢀⡠⠖⠁⢀⡤⠊⠁⣠⠔⠉⢀⡴⠋⢀⠔⠊⡠⠞⢁⠴⠊⢠⠔⠉⠀⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠤⠒⠉⠀⢀⡠⠔⠊⠁⡠⠔⠋⠀⣠⠔⠉⢀⡠⠒⠁⢀⡴⠋⠀⣠⠞⠁⢠⠖⠁⠀⠊⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠱⢴⠢⣄⠀⠀⣀⡠⠤⠐⢲⢒⠩⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⢀⣰⠋⠀⠈⢲⠉⠀⠀⠀⢠⠃⠀⢂⡡⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡠⢬⡇⠀⠀⠀⠀⢇⠀⠀⠀⠙⠒⡻⠃⠐⢊⣁⣹
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠘⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠾⠥⢤⣌⣀⡠⠄⠣⡄
⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠈⠓⠶⠦⢦⣤⡤⠞⠀⠀⠀⠀⠀⢠⡔⠊⠁⠐⠂⠉⣳
⠀⠀⠀⠀⠀⠀⠀⡷⠮⢭⣄⣐⣒⡠⠤⠤⠤⠤⠤⠄⠀⠀⠀⣀⣀⣀⣉⣉⣛⣶⣤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣀⣉⠉⠉⠙⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡏
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠦⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠄⠠⠤⠤⠞





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⢀⠀⠀⢣⠆⠀⠀⢠⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠀⠀⠠⣋⡆⠀⡆⡆⠀⢠⣀⠀⠀⠀⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢠⢉⠀⠀⡇⢡⠀⢠⢃⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠤⠄⢺⢸⠒⠒⣇⡜⠒⢺⠸⠒⠒⢺⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠣⠇⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠒⠁⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣀⠤⠔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡆⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⢀⡀⠀⠀⣀⣀⠞⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣧⠏⠀⠀⠑⠒⠒⠒⠉⠙⢆⡴⠁⠀⠱⠀⡁⠈⠢⠴⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⣦⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠀⣏⢸⠀⠀⠀⣼⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠁⠀⠀⣹⣯⠀⠀⠀⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡇⠀⠀⣧⢾⠀⠀⠀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⣧⢾⠀⠀⠀⢿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⢾⢹⡗⠒⠒⢿⡼⠓⠒⠒⢿⢹⡗⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⢤⣄⣀⣈⣉⣡⣈⣀⣀⣈⣀⣀⣀⣈⣉⣡⣤⡭⠴⠞⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡆⠀⢀⡭⢯⡉⠉⠉⣭⡍⣩⣍⠉⠀⢠⢤⡄⢸⡿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⢀⡞⠀⠄⢧⣀⡼⠣⠙⡁⢹⡀⠀⡟⠐⠙⠚⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣈⣉⣀⣁⣐⣀⣉⣀⣐⣀⣀⣂⣛⣚⣁⣈⣀⣂⣐⣀⣟⣀⡀
⠀⠀⠀⠀⠀⢀⣞⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⣩⠽⠿⢭⡉⠉⢳
⠀⠀⠀⠀⠀⢸⠛⠲⠶⠛⡁⠐⠀⠈⠳⠤⡤⠴⠚⠁⡒⠀⠉⠓⠶⠶⠛⠁⠀⡁⠀⠉⡙⢹⡇
⠀⠀⠀⠀⠀⢸⣄⠀⢀⣠⣥⣄⡉⢀⣂⣠⣈⡀⠠⠀⠁⢀⣨⣤⣤⣀⡀⠁⠀⣀⡤⠖⠒⢾⡇
⠀⠀⠀⠀⠀⢸⣏⣉⣉⣤⢤⣌⣉⣉⣡⠤⣤⣉⠛⠒⣛⣉⡤⠴⠦⢬⣉⣉⣉⣩⡴⠖⠚⢺⠇
⠀⠀⠀⠀⠀⠀⣧⠉⢉⢀⡀⣤⢉⡉⢀⢠⡀⢩⠉⡉⠉⠁⡀⢰⠀⡀⣈⠉⣉⠀⣀⠀⡄⢸
⠀⠀⠀⠀⠠⠤⠿⠶⠾⠾⠷⠿⠴⠧⢼⣼⣧⣼⣤⣷⠼⠦⠷⠾⠶⠷⠿⠶⠿⠶⠿⠴⠷⠟⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠸⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠙⠲⢤⣄⣀⣀⣠⠤⠴⠒⠒⠒⠋⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣠⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠓⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠴⠚⠉⠀⠀⠉⠉⠉⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢦⡀⣠⠚⢦⡜⠳⣤⠞⢦⡴⠳⣤⠞⢦⡔⠳⣤⠚⢦⡴⠳⣤⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠙⡅⠀⠈⡄⠀⢰⠀⢸⠀⠀⡇⠀⠸⠀⢀⠃⠀⡘⠀⢠⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠰⡀⠀⢰⠀⠘⡀⢸⠀⠀⠁⠀⡇⠀⡜⠀⢠⠃⠀⡎⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢱⠀⠀⠇⠀⠇⠀⠀⠀⡀⢠⠁⢠⠃⠀⡌⠀⢰⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢃⠀⠸⡀⢸⠀⠀⠀⡇⡘⠀⡘⠀⢠⠁⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣆⣀⢃⠈⡄⠀⠀⡇⡇⢠⠃⢀⣎⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠓⠶⠶⠷⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⠀⣠⠤⠼⣤⠏⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠘⡄⠀⢈⠑⠢⠀⡠⡊⠀⠀⠀⠙⣴⣰⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠨⡙⣢⠵⠄⠀⠑⣀⠞⡄⠀⠀⠀⢀⠁⡘⣧⠤⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢼⠁⢈⡴⠆⠰⢇⠀⠐⠀⠘⢀⢤⡒⠂⢼⣿⡙⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠙⡿⠾⠷⠿⡄⠉⠈⢫⣌⣁⣨⣥⣾⡀⠀⠁⣼⠇⠉⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠫⣁⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠈⠓⠂⠉⠓⠚⠛⠓⠊⠀⠀⠀⠊⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⡀⠀⡼⠛⠉⠙⢳⠀⠀⢠⡶⠛⠉⠙⢏⠁⢠⠞⠉⠙⣦⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠿⣖⡲⡇⢰⣇⡀⠀⠀⡈⠀⠀⢾⡀⠀⠀⠀⣸⠀⢸⣤⣤⠶⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡇⢸⠟⠋⠉⠉⠙⠤⠤⠞⠙⠛⠋⠩⢽⠀⢸⠒⠚⠉⣻⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡁⠂⣴⢁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣀⣤⣴⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡀⠀⠀⠑⠶⠞⠋⠉⠉⠉⠀⠀⠒⠒⠂⠈⠉⢻⣻⡄⠘⢯⠟⠛⠉⠉⣾⡟⠻⢶⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠫⠻⢕⣿⣿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣧⣀⣸⣀⣠⡴⠟⣻⢿⣧⠀⠀⠛⢦
⠀⠀⠀⠀⠀⡞⠁⠀⢰⣿⣟⠊⡠⠤⣌⠉⠑⠒⠒⠲⠶⠦⠤⠤⠤⠤⠶⠶⠖⠛⠛⠛⠉⣉⡠⠴⠊⠕⢋⡟⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⣇⠀⠀⠈⠻⣙⡏⠀⠀⣸⡄⠀⠈⠓⠀⠀⠀⠀⠀⠤⠤⠤⠐⠒⠒⠊⢉⣽⠒⠲⣤⣠⠶⠋⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠘⢮⡢⡀⠀⠈⠙⠒⢶⠟⠛⠲⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡈⠉⠁⢰⣷⡀⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠺⢕⣠⢄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠁⠀⠀⠀⠙⠲⢛⣛⣭⣔⣮⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣲⣤⣄⣤⡤⠤⠤⠤⠤⠤⠀⢴⣀⣀⣀⣤⡬⠽⠟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⢹⠀⠀⠀⠀⠀⠀⡞⠁⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡞⠀⠀⠀⠀⠀⠀⠹⣄⣀⡼⠃⠀⠀⠀⠀⠀⠈⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡭⣭⣀⠀⠀⠀⠀⠀⢠⣶⣛⣩⣤⡀⠀⠀⠀⠀⠀⢠⡤⠷⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣍⡁⠛⢻⠀⠀⠀⠀⠀⠠⣧⣉⠉⢸⡇⠀⠀⠀⠀⠀⢸⡟⠛⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠙⠛⢻⠀⠀⠀⠀⠀⠀⡇⠀⠙⢻⡇⠀⠀⠀⠀⠀⢸⡋⠛⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠶⣤⣼⠁⠀⠀⠀⠀⠀⡏⠑⠦⣼⡇⠀⠀⠀⠀⠀⢸⠗⣤⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢸⡅⠀⠀⠀⠀⢠⣇⠀⠀⠐⡇⠀⠀⠀⠀⠀⢸⠁⠀⠉⢹
⠀⠀⠀⠀⠀⡴⠋⠙⠲⣄⠀⢀⡤⠤⠤⠾⣿⡲⢤⣼⠧⠒⠒⠒⠦⣤⡏⣱⠖⠚⠛⠒⠲⣄⠀⣠⠼⠷⢤⣄⣼⣄⡤⠴⠒⠦⣤⡀⣠⠤⠤⢤⡀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠉⣿⠏⠀⠀⠀⠀⠈⡿⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢸⡏⠙⠲⠶⠖⠲⢤⣄⡀⢀⣀⣤⠶⣄⡀⠀⠀⠀⢀⣠⠴⣦⡀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⢀⠾⣤⡀⠀⢀⡼⠉⠳⠶⠶⠖⢿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠑⠋⠉⠀⠀⠈⠑⠲⠶⠶⠞⠉⠀⠉⠓⠲⠚⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⡸⠋⠉⠁⠙⠻⣇⣰⠟⠉⠀⠀⠀⠉⠙⣆⣀⡴⠞⠛⠛⠉⠙⠛⢦⡀⢀⡴⠛⠋⠉⠈⠉⠛⢦⣀⡴⠖⠛⠛⠓⠶⣄⣀⡴⠚⠛⠛⠛⠳⢦⣸⡶⠞⠛⠲
⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠟
⠙⢦⣄⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣠
⠀⠀⡏⠉⠉⠐⠋⠉⠓⠶⢤⣤⣤⣤⡤⠖⠚⠢⣤⣀⣀⣀⣀⣀⣤⠶⠚⢶⣀⠀⠀⠀⠀⢀⣠⠾⢧⣀⡀⠀⢀⣠⡞⠛⢦⣤⣀⣠⡴⠃⠈⠙⠒⠒⠋⢹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠑⠓⠒⠒⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡇
⠀⠀⡇
⢀⣠⢧⡤⣄⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀
⣞⠁⠀⠀⠈⠙⢦⣠⠞⠋⠉⠉⠉⠙⠓⣦⡴⠚⠉⠉⠁⠀⠈⠉⠳⣄⣠⠞⠉⠀⠀⠀⠀⠉⠳⣤⡜⠋⠉⠉⠙⠳⣄⢠⡞⠋⠉⠉⠉⠙⢳⣤⠞⠉⠉⠉
⣏⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠳⠶⣤⣄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡴⢶⣄⠀⢀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⠶⠶⠶⠖⠉⠉⠛⠶⠦⣤⣤⣤⡴⠶⠋⠉⠻⣦⡀⠀⠀⢀⣤⠴⠋⠳⣤⣄⣀⣤⡴⠋⠉⠳⠶⠦⠶⠞⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣀⡀⡀⠀⠀⠀⠀⢀⡀⠁⠲⢾⡷⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠹⡛⠛⠳⣿⣿⣾⣟⠁⢀⡄⠘⢛⢀⣸⣿⣿⣷⡶⠄⠉⣨⣶⡅⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠲⠘⠀⣾⣿⣿⣶⣦⣤⣦⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣭⣥⣶⣶⣷⡿⠃⣘⠦⢀
⠀⠀⠀⠀⠀⠀⢀⡒⠉⠁⡤⠼⠆⠉⣿⠏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⠖⠀⠘⣣⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣷⣶⣾⣿⠏⠀⢶⡎⠙⢿⣿⣿⣿⣿⣿⣿⡿⠃⢠⠉⣿⣿⣿⣿⣿⣿⣿⣿⠏⠠⠶⠌⠛⣿⣦⣀⣒⣩⣡⣥⡞⣷
⠀⠀⠀⠀⠀⠀⠄⢩⡟⠿⢿⣿⣿⠟⠀⢀⣉⠲⡄⠘⢿⣿⣿⣿⣿⡿⠀⠈⠙⠁⠀⠨⣿⣿⣿⣿⣿⠿⠘⠀⠐⠢⠶⠘⢻⣿⣿⣿⠿⡋⠁⠻
⠀⠀⠀⠀⠀⠀⠈⣤⠒⢂⡤⠬⡛⢓⡒⠎⠤⠶⣥⣶⣿⣿⣿⣿⣿⠋⠁⠀⠀⠂⢉⡄⢨⣿⣿⣿⣿⣧⣤⣖⣠⠤⠞⡴⠊⠉⢡⠆⡀⣰⡔
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣶⣎⣉⣤⠀⠀⢀⣀⡠⠄⠀⠈⢉⠁⠀⠀⠁⠀⠀⣉⡑⠋⢉⡤⠀⠈⢉⣁⠀⠈⢁⡤⢀⡠⠊⣰⣫⣬⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢹⣏⠻⢿⣿⣿⣿⣿⣯⣶⣦⣐⣛⣩⣡⡄⠚⣁⠄⢒⣁⣠⠤⠚⠁⣀⡤⠞⣋⣀⣐⣚⣡⣤⣭⣴⣿⣿⣿⣿⣿⢁⡄
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣟⣿⡿⢀⡟⢻⣍⣛⣛⡛⡛⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⡏⢸⣇
⠀⠀⠀⠀⠀⢀⢢⣷⣸⠿⡿⢣⣾⣅⢹⣿⣿⣛⣻⣿⡿⠟⢸⠏⢻⣾⣿⣿⣿⣿⣿⣿⣿⠃⣼⡍⢿⡿⣿⣿⣿⣿⡏⢠⣶⣼⣿⣿⣷⣿⡿⢄
⡄⠀⠀⠀⢀⣆⣘⣿⣿⣾⣿⢿⣷⣿⣥⡿⣿⣟⣻⡿⠟⣠⣿⣷⡌⢿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣞⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⠻⠃⣼⡇
⢇⠀⠀⠀⠀⠹⣥⠂⣹⡟⠹⢿⣿⣿⣭⣿⣿⣽⣿⣧⣿⣿⢿⣿⣿⣦⣹⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣷⣽⣿⣿⣵⣿⣿⡿⣿⠟⠛⢉⣄⣦⡟
⠘⢧⡀⠀⠀⠑⢌⠻⣥⡄⣒⡦⠤⠈⢛⣉⡻⠟⠛⠛⡛⠿⠿⠿⡷⠾⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⡿⠿⠿⠛⠛⢋⠉⠃⡰⢁⡴⣴⣯⡾⢋⠄⠀⠀⠀⣠
⠀⠀⠉⠳⢤⣀⠀⠀⠀⠙⢳⠶⠾⣿⣭⣤⡴⠖⣉⣁⣤⡤⠖⣀⡤⠂⠀⣤⠤⠊⣉⣠⡤⠔⢈⣁⡤⠒⢀⡠⠔⢨⠤⣾⣶⠧⠚⠋⠃⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣀⡀⠈⠀⠀⠉⠙⠛⠛⠷⠾⠿⣯⣥⠶⢶⣿⣵⣶⣮⣭⡥⠶⠾⣿⣿⠶⠿⢿⠶⠟⠿⠛⠃⠈⠀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣠⣤⣤⠤⠶⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠰⠄⠈⠀⠀⠀⠀⠒⠂⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢢⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠤⠤⠀⠀⠀⠀⠂⠺
⠀⠀⠀⠀⠀⠀⠀⣈⠡⠀⠀⠐⠀⠀⢠⡀⢀⣈⠀⠠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⡖⠐⠀⠀⠀⠀⠀⡀⠀⣀⠜⠓⠂⡉⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢤⠀⠂⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠐⠚⠉⠀⠑⠒⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⡀⣠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠱⡤⠖⠀⢁⡀⠠⠴⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠀⠂⠀⠀⠄⠀⡀⢂⠀⣴⣡⠤⠁⢈⠉
⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠂⠛⠁⠀⠀⠠⠛⠠⠈⡗⠲⣶⠛⠶⠒⠒⣩⣤⡀⠤⣽⡂⢴⣶⢬⡶⠿⠿⠟⠛⠷⠎⣠⠄⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠄⡀⠀⡆⠀⠀⣤⠀⢸⠀⠐⠀⠀⠀⠤⠄⢙⡖⢃⡀⠀⠀⠀⢀⢀⠀⣠⠌⢹⡉⣤⢶⡴⠶⠶⠟⠛⠀⡀⠠⡄⠀⠲⠈⠠
⡀⠀⠀⠀⠀⠊⠀⣆⠀⠚⠣⠜⠁⠀⠀⢷⠐⠂⠂⠀⠄⢀⣄⣀⠈⣥⠀⠀⠀⢀⣀⡤⠀⣠⣄⣴⡈⢦⡴⠶⠾⠿⠂⣈⣙⡦⠔⢢⡴⠀⠀⢸
⡇⠀⠀⠀⠀⠰⣀⠉⠂⠄⠀⠀⠀⠀⠈⠀⣧⢒⣆⣀⡤⠮⢤⠄⡠⠈⠢⠄⣀⡀⢀⠀⣴⠤⢤⡤⣴⠮⣳⠿⠟⢃⣔⣉⣠⠤⡶⠀⠀⠀⢀⡞
⠘⣆⠀⠀⠀⠀⠐⢖⣂⠀⠀⠀⣉⠀⠔⠀⠼⠃⠽⠋⠀⠤⠤⣄⣀⡠⠘⠻⠶⠾⢦⠥⣤⣤⠤⣶⠶⣶⠿⠶⠞⠛⠛⠋⠉⠈⠀⢀⢈⣠⠞
⠀⠈⠓⢤⡀⠀⠀⠀⠈⠐⠮⠥⣤⣒⠒⠀⠠⠤⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣁⠀⠀⠀⡀⠀⡀⢀⣠⡀⠤⠟⠉⠀⠀⠀⠀⡀⠔⠉
⠀⠀⠀⠀⠉⠑⠢⢤⣀⡀⠀⠀⠀⠀⠈⠉⠒⠒⠲⠭⠤⣤⢄⣊⣠⣄⣀⣀⣀⡀⣚⣉⣤⣤⣄⣉⣤⠤⠤⣤⠴⠂⠴⠋⠁⠀⠀⠀⣀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⠤⣄⡀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢹⡇⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⠃⣾⠟⣁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠀⠉⠻⣄⠀⠈⠉⢉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⠀⠀⠴⠀⠀⠙⣦⡀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠤⠀⠘⢡⠄⠉⣁⠀⢸⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠀⠱⣆⠀⠘⠀⣥⣠⣟⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⠛⠛⠛⠉⠉⠉⠓⠶⠶⠛⠒⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⣠⣶⠛⢳⡄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣠⡴⢦⠀⠀⠀⠀⠀⠀⢠⡞⠁⠘⠷⠞⠁⢺⠀⠈⣧⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⣼⠃⠀⢸⡆⠀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⠟⣿⠀⠀⠀⣿⠶⣤⣤⠞⠃⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⣀⣴⡟⢳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠞⠋⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠉⣧⠈⣇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠉⢿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡇⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢹⠀⠀⢸⡎⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠶⠶⠖⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠘⠀⠀⠀⢀⠬⠀⠀⠀⠀⠐⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡵⢡⠀⠀⢸⠲⢤⠀⠀⢠⣦⡀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠇⠀⠀⠸⠁⠀⠀⢸⢀⣀⠀⠀⢀⠈⢀⠀⠀⠾⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠇⠀⠸⠆⣀⡇⠀⢘⠀⠩⡁⠀⠘⢠⣼⠀⠀⢠⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⢰⡄⠈⠇⠀⠘⠑⠶⠆⠀⢰⣀⢸⠀⠀⠈⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡈⠀⠀⠀⠙⡇⠀⢐⠆⣤⡅⠀⢀⠈⢹⡀⠀⠸⠰⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⡀⢰⡇⠈⢹⡔⢻⠆⣄⣥⠊⢹⠀⠀⣿⠖⣿⠑⠲⣯⣡⣸⢤⣼⣆⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠾⣷⣆⣴⣷⣿⡇⠐⢾⡿⢿⡄⠈⢿⠿⠿⠓⠶⡿⠿⢿⠦⣌⣿⣿⣿⢄⣈⣿⣶⣾⢟⣳⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣿⣋⢭⠤⡀⠛⢠⣼⠀⠈⡁⠳⢾⠀⠀⠦⠤⠀⠀⢸⡀⢈⡇⠀⢀⡀⢙⣃⠉⠙⡿⣄⠈⢳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢷⣿⣇⠉⠈⡄⢠⡀⠀⡉⢠⠐⠧⢦⣼⠀⠀⠀⠀⠀⠀⠐⣠⠙⣃⠀⠈⢫⠈⢻⡻⠶⢷⣤⣍⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠛⠲⢧⣄⣠⣵⣆⠙⡄⠲⣀⠸⣼⠳⡀⠀⣀⠀⢄⠰⡀⠂⣈⢶⣄⡛⢶⣬⣿⣦⡼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡀⠀⠀⠀⠀⠉⠛⠛⠓⠈⢷⣼⣿⣿⣦⢹⣦⣬⡷⣮⡽⠿⠆⠸⠿⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠗⠀⡏⢸⡀⢠⣶⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⣤⡀⠀⢀⣄⠀⠀⠀⠀⣾⣿⣧⠀⢀⣀⠀⠀⣷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠁⠀⠑⠜⠉⠀⡀⠀⣰⠋⠸⠀⠀⢸⠏⡩⡇⠀⢸⠏⠀⠀⠀⠀⣿⠟⠃⠀⢹⣯⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠋⠀⠀⠄⠀⢻⠖⠈⠁⣤⡜⢠⠀⢠⡄⠀⣷⠛⠀⢤⠞⠈⠃⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠲⠾⡇⠀⣯⠘⠀⠀⠀⢀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⠜⣾⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⡄⣀⠀⠀⠊⣁⣼⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⡿⠇⠀⡀⣤⣾⢿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡦⠀⡐⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠐⢁⡐⠀⠐⠀⠐⠈⠉⠋⠉⠉⠁⠂



⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀⠀⠀⠀⠀⠀⡠⠊⢢⠀⠀⠀⠀⠀⠀⢀⡔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣱⠀⠀⠀⠀⠸⡀⠀⢠⠁⠀⠀⠀⠀⠀⡁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣢⡐⠁⠀⠀⠀⠀⣠⣬⠯⣅⠀⠀⠀⠀⠀⢀⡙⢂⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡐⠚⢷⠀⠀⠀⠀⡷⠤⢈⣉⠀⠀⠀⠀⠀⢸⣉⠁⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⡇⠀⠀⠈⡄⠀⠀⠀⠀⢸⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⢸⠀⠀⠀⠀⡟⠲⠤⣠⡇⠀⠀⠀⠀⢸⠦⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣹⠠⠤⠤⠤⠷⠤⠤⠴⠧⠤⠤⠤⠤⠼⠄⣀⣉⣇⣀
⠀⠀⠀⣀⡠⠤⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠠⠄⣀
⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠢⣀⣀⣠⡀⠀⠀⠀⠀⢀⡴⠢⡀
⠣⣀⣠⠔⠁⠑⢄⡀⠀⢀⣠⠔⠁⠀⠀⠀⠈⠢⢤⠀⣀⢠⠤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠒⠉⠀⠀⠈⠑⢲⠒
⠀⠀⡇⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣧⠔⠊⠉⠉⠉⠑⠢⠤⣀⣀⣀⣀⣀⠤⠔⠊⠉⠉⠉⠉⠉⠉⠒⠤⢀⣀⠠⠤⠔⠊⠁⠀⠉⠒⠤⣀⣀⠠⠴
⠀⠀⡇⠀⢀⣀⡀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠢⢄⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⣀⡀⠀⠀⣀⢴
⠀⠀⡟⠉⠁⠀⠀⠀⠀⠉⠒⠒⠂⠀⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢀⠴⠠⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠒⠓⢄
⠀⠠⣀⢀⡎⠁⠈⢱⠒⠉⢒⠔⠒⠒⡤⠒⠒⡤⠒⠒⢢⠔⠒⢲⠔⠒⢢⠄⠒⠤⡔⠒⠲⡔⠈⠑⢺⠁⠀⢈⡆⠤⠊
⠀⠀⠀⠀⠙⠒⠒⠙⠢⠤⠜⠦⢀⠠⠧⢄⡠⠷⠄⡀⠜⠦⢀⠸⠦⣀⠸⢄⣀⡠⠧⣀⡠⠞⠤⠤⠊⠑⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣙⣋⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⡿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣫⣶⣦⡙⢷⡾⢋⣾⣷⣝⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣼⣿⣿⣾⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣛⣟⡻⣿⣿⣩⣿⡿⣛⠿⣿⣏⣹⣿⠟⣛⡛⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⢛⣿⣿⣜⡻⢟⣫⣾⡛⣿⣮⣛⣛⣵⣿⣟⢻⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡁⢑⣻⣿⣿⣿⣟⡈⢀⣻⣿⣿⣿⣓⠁⢙⣿⣿⣿⣿⣃⠉⣚⣿⣿⣿⣿⣓⠁⣙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣿⠿⣋⠻⣿⣶⣶⣿⠟⡙⢿⣷⣦⣿⡿⠟⡙⢿⣷⣾⣾⡿⢋⡻⢿⣷⣦⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣭⣭⣭⣴⣾⣿⣷⣦⣭⣭⣵⣾⣿⣶⣬⣭⣭⣴⣿⣿⣦⣭⣭⣥⣶⣿⣿⣶⣬⣭⣭⣥⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣟⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⡿⣻⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⢀⣿⠹⣇⠀⠀⢠⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⣧⣤⡟⠀⠀⢸⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⣬⣿⣤⠀⠀⣀⣹⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣻⣿⠀⠀⢠⣿⡴⣿⠀⠀⣿⣩⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⣡⣿⠀⠀⢸⣏⡴⣿⠀⠀⣿⡡⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢠⣿⠀⠀⢸⡏⡰⣿⠀⠀⣿⢁⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⢉⣿⠀⠀⣸⣟⣀⣿⣀⣀⣿⣁⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⣦⣤⣤⣤⣤⢦⣤⣀⣀⡀⠀⠂⠀⠀⣀⣀⣨⣤⡶⢦⣤⣤⠶⠶⠾⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠒⠉⠳⠤⠤⠔⠋⠻⣍⣛⣛⡿⠛⠻⢍⣉⣀⡤⢊⡁⠑⠢⠔⠊⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣤⣤⣬⣤⣤⣌⣀⣬⣥⣤⣤⣄⣐⣀⣀⣒⣀⣀⣀⣀⣀⣤⣥⣤⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣿⣤⣧⣀⣍⡀⣹⠉⢹⠁⢹⠀⢹⠉⢹⠉⢹⠉⢹⠉⢹⠉⡏⢹⢀⣇⣰⣼⡷⠾⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢠⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠳⠿⠶⠾⠶⠾⠶⠾⠶⠾⠶⠿⠾⠛⠛⣛⠉⠉⠀⠀⠀⠀⠛⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣠⣤⠀⢴⠄⣀⡀⠁⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⢀⡀⠉⠠⠀⢀⣿⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠻⣤⣤⡴⠟⠛⠛⢷⣤⣀⣀⣠⣴⠿⢷⣴⣦⡈⠈⠋⠀⣀⣴⠟⠛⠛⠶⠶⠶⠟⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠄⠀⠀⠀⠀⠀⠃⠀⠀⠉⠉⠁⠀⣀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠈⣁⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⣤⣤⣄⣀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢀⣀⣀⣠⣤⣤⢶⣶⠟⠛⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣷⣀⣠⡿⢿⡉⠛⠛⣿⣿⠲⠶⠶⣶⡶⠶⢶⣶⡶⠞⠛⢻⣿⠛⠉⣹⠿⣧⣀⣸⣟⣷⣾⠿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⠛⠛⠳⠾⠿⢶⣾⣯⣽⣦⣤⣾⣋⣷⣤⣾⣍⣿⣦⣴⣿⣽⠷⠾⠿⠟⠛⠛⠉⣉⣠⣤⣶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⢿⡟⠶⣶⣶⣤⣤⣤⣤⣤⣈⣉⣉⣉⣉⣉⣉⣀⣀⣠⣤⣤⣤⣤⣴⣶⠶⢻⡟⢿⣁⣿⠙⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡈⣷⡾⠃⢸⣇⣴⠏⠹⣇⢠⡿⠙⣯⢉⣽⠋⢻⡍⣼⠏⠹⣇⣼⠃⠘⣧⣾⠁⠈⣿⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢦⣤⣿⣃⣀⡀⣿⡟⠀⠀⢻⣾⠃⠀⠈⣿⡏⠀⢀⣻⣿⣀⣤⣼⣿⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀
⠀⠀⢀⠚⠁⠀⠀⡄⠀⢀⠀⠰⠀⠀⠀⠁⠀⠀⠘⠖⠀⠰⠂⠀⢀⡀⠀⠀⠀⡀⠘⠀⠀⠉
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⢀⡄⠀⠀⠠⢠⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠚⠁⠀⠀⠀⠀⠀⢆⠀⠀⢀⠄⠀⠀⠀⠂⠀⠀⢀⣠⣤⡀⠀⢀
⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⠐⠀⠒⢊⣁⠀⢀⡀⡀⠀⢀⠀⢀⡀⡀⡀⠀⠀⠀⠘⠉⠉⠁⣄⠛⠦⠄
⠀⠀⠀⠸⠦⠄⠀⠁⠀⠀⠀⠀⢠⢸⣯⡇⣼⣇⣧⡞⢸⡤⢸⣀⠗⣇⠄⣤⠄⠀⠀⠀⠄⡈⠀⠀⠓
⠀⠀⠀⠀⠀⠒⠂⠉⠀⠀⠀⠀⠘⠘⡇⡷⠇⡇⠏⢣⢸⠀⢸⠙⡄⢸⠀⠿⠄⠀⠀⠀⠓⠊⠃⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠤⣀⠀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠈
⠀⠀⠀⢀⣀⡈⠀⠀⠀⠄⠀⠀⠀⠈⢠⠂⠀⠀⠀⠄⠐⠀⠀⡀⠉⠐⠂⠤⠄⠀⠠⠄
⠀⠀⠐⠋⠐⠛⠉⠿⢶⠖⢒⡤⠤⣀⠀⠀⠴⠉⠃⠁⠠⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠂⠀⠠⠀⠈⠀⠀⠀⠤⠀⠤⠤⠄⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢆⣀⢀⠈⠀⠀⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠤⠂⠀⠀⠀⢭⡄⠀⠀⠀⠈⣻⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠆⠀⠀⠀⠀⡃⡄⠀⠀⠀⠀⠌⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢂⠀⠀⠀⠀⡃⠆⠀⠀⠀⠀⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠶⠖⠀⠃⠛⠀⠀⠀⠘⠙⠉⠉⠉⠛⠛⠉⠉⠙⠉⠁⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠘⠒⠲⢤⡄⠋⠀⠉⠂⠀⠠⠚⠀⠈⠀⠑⠂⠐⠊⠀⠉⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠤⠤⣀⡀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠠⢄⡀⠀⢀⡠⠖⠛⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡖⠒⠒⠢⢌⡉⠐⠂⠉⠉⣀⡤⠖⠒⠂⠤⠈⢉⣁⡤⠔⠒⢲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣇⣀⢀⣀⣀⣈⡁⣘⣛⣋⡉⢀⠀⢀⣀⣀⣀⣀⣁⣀⣀⣀⣠⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⣀⣀⡀⠀⣀⠀⣀⠀⠀⠀⢀⣤⣀⡀⢀⣠⣀⠀⣀⣀⠀⠀⢀⣠⣄⣀⢀⣠⣄⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⠋⠁⠀⠁⠈⠀⠁⠉⠙⠚⠛⠉⠀⠈⠀⠈⠀⠀⠈⠀⠈⠀⠈⠈⠁⠀⠉⠈⠁⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠈⠱⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⠊⠉⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⠁⠘⡆⠀⠀⠀⠀⠘⠶⠎⠁⠀⠀⠀⠀⢰⡉⠉⡂⠀⠀⠀⠀⠀⠓⠖⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣈⣑⠊⢁⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣙⣊⡁⠀⢀⣠⠤⢄⡀⠀⢀⣀⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣏⣁⣀⣈⣙⡉⢀⢀⣈⡓⠞⣉⣀⣀⣠⣬⣽⣭⣤⣤⣤⣬⣷⢭⠤⠤⠤⠈⠿⠃⠤⠤⠼⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⢀⡤⣀⣀⡴⣄⠀⠀⠈⢩⡭⠤⣄⡀⢀⣠⠖⢀⡀⢀⡰⠆⠖⠒⢦⣀⣠⢤⠴⠦⠤⠴⠓⠤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠉⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠤⠤⢄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡴⠆⠤⣀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣁⡤⠲⢄⡉⠲⠤⠔⠚⠉⣀⠤⣈⠓⢤⣀⡀⣀⣠⡴⠊⢁⣀⡈⠳⢤⡤⠞⠁⣠⠴⠢⣄⡑⠞⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠉⠓⠶⠴⠒⠋⠁⠀⠈⠑⠦⢌⣉⣉⣀⡠⠞⠋⠀⠉⠛⠒⠶⠚⠋⠁⠀⠀⠀⠉⢳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⡀⠀⢀⣀⣼⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⣁⡉⠉⠛⠋⣉⣀⠈⠙⠛⢋⣉⠉⠙⠳⠞⠋⠉⠉⠙⠳⠶⠋⢉⣭⡉⠛⠛⠉⣉⠉⠛⠋⣁⠤⡀⠈⠳⡄
⠀⠀⠀⠀⠀⢰⣖⣚⣒⣚⣓⣛⣒⣒⣚⣓⣚⣓⣒⣚⣓⣒⣛⣒⣒⣒⣛⣛⣛⣒⣒⣒⣒⣛⣒⣛⣒⣒⣛⣛⣛⣒⣚⣓⣒⣚⣓⣒⣛⣲⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⢍⡆⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠐⢣⡬⠷⣾⣔⡉⢣⠀⣧⠈⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡊⠁⠐⣄⢈⠿⠀⣠⠶⣏⠤⢤⡀⠈⣱⡇⠀⠀⡏⠘⠋⠹⠀⠀⠋⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠲⣤⠌⣿⠀⡜⠁⣸⠟⣖⠂⢋⡠⠋⣷⡶⠰⣧⡀⣰⣿⠂⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠖⢁⣤⣸⢮⣇⠀⠙⠦⣬⣾⠈⠁⠐⠃⢀⣤⣀⣮⠫⡁⢈⡉⢎⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣤⡄⠐⣧⡃⣴⠏⢸⡟⢳⣦⣤⡈⣢⣖⡶⠀⣠⡀⠆⠑⣿⠄⣿⡀⣷⡀⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡀⠹⠤⠄⠀⣱⣕⢐⡊⠈⠀⠈⡍⢩⣰⠏⢹⣷⡀⠈⠛⣺⢦⡄⢠⡟⡀⣌⢱⢰⠈⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠙⢬⡲⠶⣦⡈⠁⢀⣍⠀⠄⢂⠐⠛⣤⠈⠠⠴⠃⢠⣟⠢⠤⠤⠴⠛⠧⣬⠇⠋⣠⡾⠃⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⢤⣀⡈⠙⠓⠀⠠⠤⣤⣀⣈⣉⣉⣉⣐⣛⣛⣒⣚⣃⣉⣉⣲⣶⣾⠤⠶⠶⠛⠛⢁⣠⣴⢞⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢃⢉⠁⢀⠀⠢⡄⠑⢶⡶⠀⠠⣤⣀⣀⣀⣀⢀⠉⠉⣁⣀⣁⣈⣀⡀⠀⣤⣤⣄⡒⡂⠉⢙⣱⠿⠟⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣷⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠈⣁⡀⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠰⡄⢉⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠻⠷⠂⣄⡘⡲⠐⠊⠓⢠⡄⡠⠀⠚⠁⠄⠐⠀⢀⠈⠓⠀⣀⠀⠀⠀⠀⣀⢠⣤⣀⢰⡲⠿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡀⠀⠀⠀⠂⠉⠉⠁⠙⠦⢠⡶⠤⠒⠂⠛⠶⠃⠈⠁⠉⠒⢀⡈⠃⢐⠚⢤⡌⠁⠀⠀⠀⢀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡙⠤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⣀⣴⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⠿⣆⣁⠐⢂⣥⠴⠝⠡⠁⠠⠀⢠⠴⠛⠀⡀⠂⡠⣀⠠⠀⠀⠀⣀⠀⠁⣀⡀⡠⣞⣠⠀⠐⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⢀⢀⠀⡀⢠⠖⠒⢤⠤⠶⠌⠛⠁⠀⠀⠀⠀⠀⠹⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣡⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⢤⡾⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣁⢭⡷⢦⡤⣄⣀⣴⠄⠠⢤⡀⠀⠀⠀⠀⠀⠀⣠⣆⡀⡀⠀⠠⠄⢀⡀⣒⣠⣤⢷⣦⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⣀⠀⠈⠁⠀⠀⠒⠈⠷⠴⠖⠋⠀⠈⠊⠁⠀⠀⠀⠉⠀⠀⠉⠉⠁⣀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⢶⣦⣤⣤⣤⣀⣀⣀⣀⣢⣀⣀⣀⣀⣀⣭⠤⠴⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣿⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠟⠛⠛⠛⠉⢉⣉⣽⡿⠿⠿⣿⣭⣭⣭⡙⠛⠛⠛⠿⠷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⢀⣀⡀⠀⢰⡿⠟⠛⣿⠋⠀⠀⠀⠈⢻⡇⠙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⢰⡟⠛⠻⣶⣾⠇⠀⠘⣿⡀⠀⠀⠀⠀⣸⡟⠀⠿⠷⠿⠿⠿⠿⠷⢶⣦⣄⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⢀⡀⠙⢿⣦⣤⣤⣾⠟⠁⠀⠀⣴⡾⢶⣦⣄⠀⠀⠈⣻⡇⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⢰⣿⡋⠁⠀⠀⢸⡟⢻⣧⠀⠀⢩⣿⠶⣶⣄⠀⠀⢻⣆⠀⠈⠙⠻⠶⠾⠟⠁⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⣿⡋⠛⢷⣦⣄⣀⠀⠀⠙⠻⢷⣶⣤⣾⠇⠀⢿⣆⣀⣼⡇⠀⠈⠻⢷⣤⣼⡿⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⣻⡇
⠀⠀⠀⠀⠀⢹⣷⡀⠀⠈⠙⠛⠿⠷⣶⣦⣤⣤⣀⣀⣀⣀⡈⠙⠛⠋⠀⠀⠀⢀⣀⣈⣁⣀⣤⣤⣶⡶⠾⠟⠋⠉⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⣿⡋⠻⢷⣤⣀⡀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣴⠿⠋⣿⡇
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠈⠙⠻⠷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡾⠿⠛⠉⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⣿⡟⠻⣶⣤⣀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣠⣴⡾⠛⣿⡆
⠀⠀⠀⠀⠀⢻⣧⡀⠀⠉⠛⠻⠷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⡶⠿⠛⠋⠁⠀⣠⣿⠃
⠀⠀⠀⠀⠀⣾⠟⠿⣦⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⣀⣤⡾⠟⢿⡆
⠀⠀⠀⠀⠀⢻⣧⠀⠈⠙⠛⠿⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠾⠟⠛⠉⠀⢠⣿⠃
⠀⠀⠀⠀⠀⣾⠟⢿⣦⣄⡀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⣀⣤⣾⠟⢿⡆
⠀⠀⠀⢀⣤⣿⣇⠀⠈⠙⠻⠷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠿⠛⠉⠀⢀⣿⣧⣄
⠀⣠⡾⠟⠉⣼⡟⢷⣤⣀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⢀⣠⣴⠟⣿⡌⠙⠿⣦⡀
⣼⡟⠀⠀⠀⢿⣇⠀⠉⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠾⠟⠋⠁⢀⣿⠇⠀⠀⠘⣿
⢿⣆⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⢀⣾
⠈⠻⣷⣄⡀⠀⠀⠀⠉⠛⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠟⠋⠁⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠈⠉⠙⠛⠻⠿⠷⠶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶⣶⡶⠶⠿⠿⠛⠛⠉⠉⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⡶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢀⠞⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣧⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡤⠔⠢⢄⠀⠀⢀⠴⠒⠢⡼⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠁⠀⠀⠈⡄⠀⡎⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠒⠀⠚⠣⡀⠀⠀⠀⣀⠜⠒⠢⠧⡀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡩⣤⡤⠴⠶⠶⠬⠥⢶⣒⣫⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡮⠓⠉⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡏⠀⠀⠀⣎⣀⠜⠀⠀⠀⠀⡠⢢⠀⠀⠀⣀⠤⣄⠀⡟⠀⠀⢀⡴⠅
⠀⠀⠀⠀⠀⢧⠀⣀⠔⠉⠒⢄⠀⠀⣀⣀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠊⠀⠀⠀⠙⠒⠁⢠⣃⣠⣖⣋⡀⠠⡄
⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠑⠤⠤⠤⠞⢱⠀⠀⠀⢎⣑⡄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠒⡺⢻⠙⡝⠢⣀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢏⣑⠄⠈⠀⡤⠊⠀⡘⠀⠘⣆⡀⠙
⠀⠀⠀⢰⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠛⠀⠀⠀⠃⠠⠀⠀⠛⠒⡆
⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⡶⠒⢶⡶⣶⣶⣶⢶⣶⢶⡶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠐⠀⠀⢀⢠⠄⠂⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠁⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⣈⠉⢧⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠉⢀⠂⠈⠠⠍⠁⠀⢾⠀⠀⠀⠀⢀⠔⠚⠏⡒⢄⣀⡴⠶⢢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⣡⠀⠀⢠⡏⠀⠀⠄⠈⢤⠁⣀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⢎⠇⠀⠉⠃⠠⢀⠤⠀⣀⣀⠀⠁⠂⠈⠀⠀⠀⠀⠀⣀⠀⠃⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠐⠢⠄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠠⢤⡀⠀⠀⠁⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⡀⠂⠤⠤⣀⠀⠀⠈⠓⠀⠀⠊⠀⠡⠌⡀⠈⠀⠀⠂⠁⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠠⡀⠙⠂⢠⠈⠂⢤⠀⢤⣄⠀⠀⠀⠀⠑⠒⠸⠧⢄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠤⠀⠀⠲⠀⠂⠁⠀⠀⠁⠀⠢⠁⢀⡒⠒⢦⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠁⠀⠄⠂⠀⠀⠄⠀⢘⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠠⠀⢈⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⡱⢰⡄⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡇⢰⢳⠘⡇⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠘⣾⢸⣿⢸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⢸⣠⡿⣤⣿⢤⠜⡄⠤⠢⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠁⠀⠈⠟⠀⠉⠀⠛⠀⠋⠀⠀⠀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⡤⠤⠄⠀⠤⠀⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡤⠤⢤⠤⠢⠔⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢉⣀⡉⠉⠈⢁⣂⣀⣉⡠⠤⠬⠄⠒⠓⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣉⣉⣀⣀⣀⣀⣀⠤⣀⣀⡠⠤⠤⠚⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠒⠒⠉⠒⠉⠉⠑⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠦⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⢾⣉⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠠⠐⣞⡒⠒⠤⢬⣭⣍⣉⡉⠉⣭⡭⠤⠤⠔⣒⣋⠥⢞⡩⠗⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠓⠒⠂⠉⠉⠛⠓⠒⠒⠒⠒⠚⠛⠋⠉⠉⣉⣀⣠⠤⠤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡱⣞⠄⠀⠀⢠⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠐⠈⠀⠑⠢⠤⠤⠼⢄⣀⡠⠦⠄⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠲⣰⣶⢀⠄⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢸⠁⠀⠀⠇⢀⠤⠄⢠⠦⡀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡦⠀⣀⢀⣀⠀⡀⠄⠀⠰⡌⠈⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠋⠒⢣⡀⠀⠳⠃⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡐⠓⠂⣠⢤⡥⠀⠀⠀⠈⠉⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠍⠅⢊⡉⠤⢴⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⢿⣿⣷⣶⡒⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⡟⢣⢮⣿⡻⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠋⣷⠚⠿⠛⣾⠛⠛⠚⢻⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⣿⠀⠚⠂⢼⠀⠒⠀⢸⡷⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣿⡟⠳⣾⣿⣿⡗⢚⡿⠷⣿⣭⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣯⠼⣷⣏⣭⣻⣷⡾⢯⣩⡿⡾⣷⢶⣿⡅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢘⠉⣿⠓⣹⠛⢻⡟⠳⠞⢻⠛⠚⠟⠙⣿⠀⡅⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣧⣿⡴⣛⢣⡘⣇⡴⠓⣼⡆⡰⣋⠲⣿⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢶⣌⣿⣏⣀⢈⣿⣿⡉⢈⣻⣗⡏⣈⣹⣿⣥⣾⣛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣉⡻⣏⡹⠵⠮⣼⢻⡯⢯⣟⣻⡾⠶⢾⣤⡏⣥⢴⣟⣷⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣶⣿⡀⠀⠁⡟⠁⠻⢒⡛⠿⢖⠷⢹⠏⢉⠩⡑⠊⢠⠀⢂⣨⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣩⠀⡄⠐⠢⢀⡄⠠⠥⢨⡇⢡⠤⠄⣠⠀⢶⠀⣶⠀⠆⣿⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣼⣧⠀⠀⣇⠐⠓⢸⡇⠀⠾⢀⣧⠈⠛⠀⣿⡀⠙⠁⣿⡀⢀⣿⣿⣶⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡛⡿⠳⣿⣀⣘⣿⠃⠀⠸⣿⠀⠘⢋⣿⡙⠀⠐⢿⣏⠀⠈⣿⣤⣤⣿⣇⠺⣭⠟⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⡉⠁⣜⣻⡟⣻⠿⠷⣖⣿⣿⣤⣄⣿⣿⣿⣖⣤⣿⢷⠾⢻⡟⣿⠉⢯⣽⢃⣠⣴⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⣿⣷⣺⣿⣤⣙⣖⡃⣠⣿⣼⠋⠀⣼⣤⣿⣀⣈⣓⣾⣧⢤⣭⣿⣶⠟⢻⡿⠛⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠈⠏⠀⠘⠻⠛⠻⡿⠁⠻⢿⠿⠋⢿⠎⠻⢿⠟⠁⠙⠟⠋⢻⣏⡀⠀⠃⠐⢶⣠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⠁⠀⣤⡄⣼⠋⠀⠀⢃⡀⢰⠑⠀⠀⣘⡀⠀⠘⠻⢄⣤⣅⠀⠀⠈⠀⢺⡇⠀⢘⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠍⠂⢀⢸⡄⡛⡀⠀⡀⢹⡇⠎⡀⢀⡀⢹⡇⢀⠀⠀⠀⢸⡇⠀⠀⠀⡐⢸⡇⣂⣈⡯⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣸⣾⣾⡼⢣⣡⣱⣴⣁⣸⠧⡰⣵⣤⣁⠾⠃⣨⣶⣥⡀⠌⠀⣠⣿⣶⣦⠄⠙⢿⡟⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⢿⣅⣀⣹⡟⢿⡟⠀⠀⠹⡿⢻⡟⠀⠀⠻⡟⢿⡊⠀⠀⢈⣋⠹⡿⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠉⠙⠓⠒⠛⠓⠚⠓⠒⠒⠚⠛⠊⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠉⠱⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠀⠀⣦⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡁⠁⢀⡌⠈⡄⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡡⠀⠘⠀⠀⠰⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⡘⢤⠗⣁⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣠⡄⠄⠀⢻⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⢶⠡⡰⠄⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠠⡼⠀⠀⠀⠀⠉⠁⢈⠀⣀⡠⠄⣛⠡⠉⢄
⠀⠀⠀⠀⠀⠀⠰⡑⠀⠁⠒⠈⠐⠊⠑⠨⠦⠤⠤⠤⠤⢤⠤⠄⠤⠥⢀⠀⠀⢀⡂⠈⠐⠓⠀⠀⠉⠉⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⢡⣁⣀⣠⣤⣤⢄⣀⠀⠀⠀⢀⡀⠀⠀⢈⡀⡈⠀⠀⠀⣀⣠⣀⠀⠀⠀⣀⣀⠀⣠⠶⣄⣵⠻⣄⠇
⠀⠀⠀⠀⠀⠀⢸⠀⡘⠈⠑⠀⠁⠻⣕⠀⠀⠏⣇⣰⡶⡫⢋⠝⠱⢛⠋⠔⡡⡪⡿⣀⡰⠿⠫⠫⠓⠒⠒⠁⠁⡇
⠀⠀⠀⠀⠀⠀⢸⠥⠾⡫⢑⠾⠄⠠⢨⡖⢈⡀⣁⢈⠀⡀⢀⠀⣀⡀⢠⡀⣀⠀⣀⣤⣀⠄⠠⠠⠶⠔⣒⣙⣩⡇
⠀⠀⠀⠀⠀⠀⢸⡀⡀⠁⠈⠉⠻⠉⠒⠓⡖⠢⠭⠕⣒⠢⠽⠥⠖⡒⠭⠽⠔⣒⠦⠤⠔⠊⠯⠙⠒⠉⢸⢂⡀⡁
⠀⠀⠀⠀⠀⠔⠙⢮⣬⣐⡕⡡⡠⡠⠠⡠⢀⢀⢄⠄⡡⣀⢀⢀⢀⠁⡀⡀⡀⣁⡠⡄⡠⣠⣶⠶⠚⠒⠷⠛⠃⠙⡢⡀
⠀⠀⠀⠀⠠⣤⠦⣤⢀⣀⡈⠉⠛⠲⠾⠴⠕⠛⡿⢞⠚⠓⠷⢵⡟⠛⠛⠛⠩⠉⠙⠛⠙⠛⠀⢀⣀⠀⢠⣶⣶⣄⢻⡟
⠀⠀⠀⠀⢀⠋⠪⠹⠿⢝⣷⡄⢨⣶⠀⣦⣤⣤⠀⢀⣀⠀⠀⣀⣄⠀⠐⢂⡀⠀⠀⢲⢛⣙⣶⣿⣿⡧⠾⠉⠛⠛⣿⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⢼⣧⣠⣿⢽⡶⠟⠝⠻⠿⠛⠿⠿⠦⠿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢉⡹
⠀⠀⠀⠀⢸⡂⠽⢐⡒⠠⠷⠔⠒⢄⠠⡄⠀⠀⢁⡀⠀⡀⠀⣐⠀⠀⠀⠀⡀⠀⠀⣀⠀⣀⣀⣴⡠⠂⢼⠗⢈⣒⡟
⠀⠀⠀⣀⣨⠉⠉⠉⠸⠉⠒⠚⣍⠳⠿⠥⢥⠬⣬⣩⡤⢔⠻⣟⣉⣡⢭⣚⣃⣨⡤⣿⣉⣠⠤⡴⠴⠋⡍⠙⠉⠑⢠
⠀⠀⡀⠀⢸⡗⠧⢠⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠉⠀⠓⠀⠀⠀⠓⠀⠀⠀⢈⡁⠀⠄⠀⠀⢀⣠⡴⠂⢀
⠀⠀⠔⢂⠡⠜⢒⡤⢀⡈⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⣀⢀⠄⡠⣠⡢⣢⢂⠔⠔⠖⠐⠐⠉⠀⠀⢀⣠⡔⢊⠕⠁⡠⠖⢁⡠
⠐⣂⠨⠔⠂⣈⠤⠂⢂⠠⠉⠒⢀⡰⠶⠠⠤⠤⠀⠀⣁⠀⣀⠈⢀⡀⠁⢤⡤⠤⣠⠴⠂⡰⠛⡩⠔⡡⠔⣁⠔⠉⡠⠔⢁⠔⣈
⠉⠠⠄⠂⣉⠠⠔⠈⣀⡠⠔⠊⢁⡀⠄⠒⢉⡠⢔⡢⠕⢊⡡⠐⢉⡠⠒⢁⡠⠊⣀⠔⢁⡤⢊⠤⠊⡠⠊⣁⠔⢉⡠⠊⠡⠊⠑
⠀⠐⠊⠁⣀⠠⠐⠈⢁⠠⠐⠈⠀⡀⠔⢊⡡⠒⠁⡠⠒⠉⡠⠔⠁⡠⠒⢁⡠⠊⣠⠔⢁⠔⢁⠔⠉⡤⠊⠀⠀⠉
⠀⠀⠀⠈⠀⠀⠐⠊⠁⣀⠄⠒⠉⡠⠒⠁⡠⠔⠈⡠⠔⠊⢀⠔⠋⢀⠔⠁⠠⠊⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⠀⠀⠀⠀⠀⠀⣠⠎⠱⡄⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠈⡆⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡋⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠜⠁⠀⠀⠀⠀⢀⣑⣒⡊⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠯⠥⢶⡄⠀⠀⠀⠀⣿⡉⠒⠚⡇⠀⠀⠀⠀⠀⢸⠳⠮⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠢⢼⡇⠀⠀⠀⠀⣿⠈⠉⠓⡇⠀⠀⠀⠀⠀⢸⠙⠒⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠤⢤⡇⠀⠀⠀⠀⣿⠈⠙⠒⡇⠀⠀⠀⠀⠀⢸⠉⠲⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠧⠤⠔⠒⠓⠒⠒⠒⠒⠋⠀⠀⠀⠉⠈⠀⠐⠒⠒⠛⠒⠒⠂⠾⠤⠤⣀⣀
⠀⠀⢀⣠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠤⣄
⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢰⠁
⡇⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⢦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠦⡀
⢧⣀⢀⡠⠎⠈⢧⡀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠑⠤⣀⣀⢀⣀⣀⡠⠔⠋⠀⠀⠀⠀⠁⠀⠈⠒⠤⠤⠤⠔⠉⠀⠀⠈⠓⠤⠤
⠀⠀⢹⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⣀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠶⠶⠦⠤⢄⡀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⠤⣀⠀⠀⠀⠀⢸
⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⢹
⠀⠀⢸⣀⡠⠤⠖⠒⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠀⠀⠈⠑⠦⢄⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠢⢄⣀⣠⠤⢚
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⢀⡸⠆⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠒⠚⠢
⠀⠸⡀⠀⣰⠋⠁⠉⠦⠒⠒⠒⡤⠖⠒⢤⠤⠒⠢⡤⠔⠒⢤⡠⠔⠒⣤⠴⠒⢤⠤⠤⢤⡠⠔⠢⢤⠒⠒⠒⣎⠁⠀⠈⣆⣀⠴
⠀⠀⠈⠉⠉⠢⠤⠔⠳⣄⣀⡠⢇⣀⢀⡸⣄⠀⣀⣧⣀⢀⣠⢆⡀⢀⣧⣀⠀⣧⡀⠀⢀⣇⠀⢀⡸⢇⣀⣠⠞⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⢲⡀⠠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⢹⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢢⠠⠀⠀⠀⠀⠀⢈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠒⢠⠄⠀⠀⠈⠉⠀⠀⣀⠀⠀⠀⠀⠈⢀⡵⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠤⠀⢀⠐⣤⠛⠄⠀⠀⠀⠀⣄⠀⠨⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⣤⢤⠀⢀⠀⠀⠐⠌⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡂⠚⠋⡤⠉⡉⠭⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠐⠶⠂⠐⡊⠈⡤⠙⠶⠾⠛⠇⣙⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⢀⢀⣀⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠀⠀⠀⠤⠈⠀⠀⠙⣱⢰⢀⠀⠺⢧⠀⠀⠀⠀⠀⢀⡶⠋⠀⠑⠆⠌⠳⣤⣤⠗⠋⡉⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣠⡁⠈⣆⠀⠀⠀⢠⢾⠀⠀⠀⠤⠀⠈⢤⡄⠂⣀⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢛⡀⠀⠑⠒⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠵⠃⢀⠀⠀⠀⠀⠦⠀⠀⠠⠿⠀⢬⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠢⠥⠠⡄⠀⠀⣉⠁⠖⠈⠉⠙⠛⠶⠂⠄⡠⢀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠒⠒⠠⠄⢀⣈⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠀⠊⠁⠉⠓⠲⠀⠀⣀⠀⠀⠂⠈⠤⡀⠐⠀⠁⠑⠂⠶⢦⣀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⢡⡈⠰⠤⣄⢠⣄⠀⠀⠀⠀⠈⠓⠒⠀⠀⠐⠋⠀⠈⣀⠉⠀⠄⠉⠀⠀⠠⠈⢃⠰⠢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠁⠁⠀⢀⠈⠀⠘⢯⡄⠀⡉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⢤⣀⣀⢤⡄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢁⣈⣀⡀⠈⠗⠄⠀⠠⠘⠔⠂⠈⣘⠳⠈⠉⢽⡲⠖⠠⢀⣀⣀⠀⠀⠀⠈⠉⠉⠛⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠢⠤⠄⠀⠉⠦⠀⠓⠈⠄⠀⠀⠀⠁⠈⠐⠲⠀⠀⢍⠛⠛⠛⣦⣀⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠠⠄⢀⡀⠀⠐⠂⠈⢂⠀⠦⠀⠈⠀⡀⠀⠠⠘⠍⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠠⢤⣁⣀⡀⠀⠀⠘⠀⠤⣐⠒⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠲⠤⣬⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠐⠁⠸⡀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠒⠒⠓⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠖⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠤⠒⠒⠂⠤⣄⠀⠀⠀⠀⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠒⠊⠉⠒⢤⡀⠀⠀⠀⠀⢁⠔⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠲⢄⠀⡠⠖⠲⠀⠀⣀⠝⠒⢦⡔⢊⠕⠒⠢⡀⢀⠤⠒⠲⣄⢀⡸⠓⠲⡖⢈⡥⠒⠪⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⠀⣀⠀⠀⠀⠁⠀⠀⠀⠙⠁⠀⠀⠀⠙⠁⠀⠀⠀⠈⠃⠀⠀⠀⠘⠋⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡅⠀⠀⢠⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡶⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠀⢤⠀⠀⠀⡇⠈⠂⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡌⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⠠⠠⠭⠭⠍⠁⠀⠀⠀⠀⠀⠀⠀⠉⠭⠭⠭⠥⠤⠀⢀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⠬⠍⠁⢀⣀⣀⣐⣂⣐⣀⣀⣀⣀⣀⣀⣀⡀⠉⠭⠤⠤⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢴⠟⠻⠿⣿⣶⣤⢸⡞⣇⠀⠀⠀⠉⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠒⠴⣄⣀⢙⣿⣿⣻⠏⠀⢀⡀⠄⠈⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠔⢁⣠⣶⣿⣿⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣆⣠⠀⢀⣀⠀⠀⠀⠀⠀⣤⣩⣰⣾⣿⣿⣿⠿⢋⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠻⢷⣿⣿⣷⡄⢀⣀⣴⣿⣿⣿⣿⣿⡏⣠⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣻⣤⣤⣄⣉⠉⠃⠸⠿⢓⣿⡿⠟⢋⣤⣾⣿⣿⡿⠛⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⢟⡟⡹⢿⣿⣇⣰⣦⣤⣤⣷⣿⣿⣿⣿⠿⢃⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⠙⠚⠚⢿⣯⣻⣿⢿⣿⣿⡿⠟⣡⣵⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣮⠽⣽⣷⠶⣤⣤⣀⠈⠙⠛⣛⣉⣠⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣛⣽⡗⣭⣿⣶⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢭⣇⣻⣟⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂
⠀⠀⠀⢀⠀⠀⢀⠂⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡳⣄
⠀⠀⠀⡡⠀⡉⠳⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣥⣭⡆
⠀⠀⠀⣥⠀⠀⠂⡍⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣢⠒⠉⢀⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡘⠉⠈⠁⢠
⠀⠀⡏⠈⠓⠒⠛⠟⠓⠂⠒⠋⠈⢻⠀⠀⠀⠀⠀⣨⣶⡇⢧⡼⠈⢼⡌⣿⠀⠀⠀⠀⠀⠀⠀⢻⠃⠴⠒⠀⠈⠀⢸⡉
⠀⠀⣥⣄⡀⣠⢀⡀⡀⣠⣀⢠⡴⠊⠀⠀⠀⠀⠐⣉⠉⠈⠁⠀⠀⣀⣁⢉⡁⠀⠀⠀⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠁⠀⠈⠉⠉⠈⠁⠈⠀⠀⠀⢀⠀⠀⠀⠀⠴⠛⠤⠄⠀⠀⠀⠩⠽⡿⠷⠀⠀⠀⠀⠄⠋⠉⠀⠀⠀⠀⠀⠀⠙⠂⠉
⠀⠀⠁⠶⠐⠦⠀⠀⠀⠀⠖⠀⢴⠊⠀⠀⠀⠠⣄⡀⣀⠀⠀⢀⠀⠀⡈⢁⣠⠄⠀⠀⠀⢠⣀⣀⡠⠄⣀⠀⠀⠀⠀⣤⠄⠞
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢩⠉⠉⠉⠉⠈⠉⠈⠈⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠣⣤⢀⠀⠀⠀⢀⡴⠇⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢣⡀
⠀⠀⠀⠀⣨⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⡆⠀⠀⠀⣴⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⣀⣿⡀
⠀⠀⠀⠀⣡⣀⠄⠉⠈⠑⢂⠀⠀⠀⠀⠀⠀⠀⠚⢧⣸⣅⠀⠳⠾⢯⣏⡀⠈⠂⠀⠀⠀⠀⠀⠈⠁⠁⠐⡛⠋⠉⠀⠈⠆
⠀⠀⠀⠔⠪⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⠦⢤⣀⣠⣄⣤⡠⠴⢧⡀
⠀⠀⠴⢀⠉⠄⠤⠀⠀⠀⠀⠀⠈⣡⠀⠀⠀⡠⠀⠀⡇⡄⠀⠐⠀⠁⠰⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠘⠄⠀⠴
⠀⠀⢀⡀⠀⣠⠒⠀⠀⢀⠀⢀⠀⣉⠀⠀⠀⠰⠾⠶⢤⡄⢀⣀⣀⡀⣠⣤⣤⡶⠀⠀⠀⢠⣀⠀⣆⠀⠀⠀⠂⠀⠀⠸
⠀⠀⠹⠉⠁⠀⠉⠀⠈⠀⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠃⠀⠀⠀⠈⠉⠈⠉⠁⠀⠊⠑⠀⠀⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠥⢴⡆⠠⠤⠤⢀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⣀⢀⡀⣀⣀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢠⠈⠀⠠⠀⠀⠀⢀⠀⠀⠀⠠⠀⡤⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠉⠈⠀⢈⠀⠀⠀⠀⣽⣉⡛⠓⠛⠓⠒⠚⠛⠛⢀⢛⣫
⠀⠀⣖⠒⠂⠄⣀⣠⢀⠤⠤⠄⠐⢉⠅⠀⠀⠀⠰⣤⣤⠄⠀⠀⠀⠐⠲⡖⠣⠀⠀⠀⠀⣭⣭⡉⠛⣛⣛⣛⡛⢚⣛⣩⣭⡏
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠉⠉⠀⡀⢠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⠐⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠔⠚⠉
⣿⣿⡷⠾⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣿⣷
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠻⣆⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢀⡞⠹⡆⠀⠀⢻⡀⢘⡇⠀⠀⠀⣸⠋⢳⡄⠀⠀⢀⡾⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⡄⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⣭⣍⠀⠀⠀⠀⠹⣄⣠⡟⠀⠀⠘⢧⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡴⠃⠀⠀⠀⠀⢙⣛⡅⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⣤⣵⡄⠀⠀⠀⢰⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢲⡆⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⡇⠀⠀⠀⠀⢸⡇⡗⢀⣀⣀⣿⢸⣤⣤⣤⣤⡄⣿⢸⡇⢀⣀⣀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠶⠒⠛⠋⢸⡇⣿⠀⠀⠀⢿⡼⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠈⣿⣸⠟⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣧⣿⠀⠀⠀⠀⠸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⠀⠀⠀⠁⠀⣠⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⢀⣀⣤⡴⠖⠋⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠙⠓⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣆⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡴⠋⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠙⢧⡀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢳⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠈⠳⢦⣤⡤⠞⠃⠀⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢦⣄⡀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠛⠲⠦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠶⠚⠋⠉⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠖⠚⠋⠉⠁

⠀⠀⠀⠀⢀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⠀⣀⣀⠀⠀⠀⣤⢀⠀⠀⢀
⠀⠀⠀⠀⢸⠁⢸⡇⢀⣿⡀⠀⡞⠙⢷⢠⡴⠶⣄⢳⡀⡼⠁⠀⠀⠀⢸⠉⢙⡆⠀⡇⢰⡏⠙⣦⠈⢹⡏⠁⣿⠀⣼⠀⣿⠉⢷⠀⣸⢿⠈⢷⣀⡞
⠀⠀⠀⠀⢸⠛⢻⠀⣼⣹⡇⢰⡷⠖⠋⢸⣧⡴⠟⠀⣿⠁⠀⠀⠀⠀⣼⠺⢯⡀⠀⡇⢸⣧⣴⠋⠀⢸⡇⠀⣿⠋⣿⠀⣿⢀⡾⢠⡿⢾⡆⢨⡟
⠀⠀⠀⠀⠸⠀⠘⢠⡏⠈⠷⢸⠇⠀⠀⢸⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠿⠦⠞⠁⠀⡇⠸⠇⠙⠷⠄⠘⠃⠀⠿⠀⠉⠀⠛⠉⠀⠚⠀⠈⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⡿⠛⠛⠉⠉⢠⣶⣿⣷⣦⠀⢠⣄⠙⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⠉⡉⠙⠿⠇⢸⣿⣟⠀⣿⣿⣿⣿⣿⡇⢘⣿⠀⠙⠛⠛⠛⠛⠻⠿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣇⠘⢿⣷⣦⣴⣿⣿⣿⡄⠙⢿⣿⣿⠟⠁⣼⣿⣶⡶⠞⠻⢿⣿⣶⣶⣤⡈⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⣀⣹⣿⣿⡟⠋⠙⢿⣶⣤⣤⠤⠴⠾⣿⣿⣿⠀⢰⣶⣤⡈⠙⠻⠟⠃⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣀⠙⠻⢿⣿⣿⣿⣧⡈⠛⠿⢿⣿⠇⢸⣧⠀⣿⣿⡏⢠⣶⣤⡈⠻⣿⡧⠈⣿⣿⣿⣷⣶⣶⣾⣿⠿⠛⢁⡀
⠀⠀⠀⠀⠀⢿⣿⣦⣄⣈⠙⠛⠿⠿⣷⣶⣤⣤⣴⣿⣿⣦⡈⠉⣀⣼⣿⣿⣿⣦⣤⣤⣴⣿⣿⡿⠿⠟⠛⢉⣀⣤⣾⣿⠇
⠀⠀⠀⠀⠀⣀⠙⠿⣿⣿⣿⣷⣶⣤⣤⣄⣈⣉⣉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢉⣉⣉⣀⣤⣤⣴⣶⣿⣿⣿⡿⠟⢁⡀
⠀⠀⠀⠀⠀⢿⣷⣦⣄⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⣁⣤⣶⣿⠇
⠀⠀⠀⠀⠀⡀⠙⢿⣿⣿⣿⣶⣦⣤⣄⣈⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣉⣉⣀⣤⣤⣶⣾⣿⣿⣿⠟⠁⡀
⠀⠀⠀⠀⠀⢿⣷⣤⣈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢉⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣶⣤⣄⣈⣉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠋⢉⣉⣀⣤⣴⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⣿⣶⣄⡉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⣁⣤⣾⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣈⣉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⢉⣉⣠⣤⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣦⣄⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢁⣤⣾⡇
⠀⠀⠀⣀⠀⠘⠿⣿⣿⣷⣦⣤⣄⣉⡉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⣉⣁⣤⣤⣶⣿⣿⡿⠟⠀⢀⡀
⢀⣴⣾⣿⠀⣶⣤⣈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠛⢉⣠⣴⡆⢸⣿⣷⣄
⣾⣿⣿⣿⣆⠘⢿⣿⣿⣶⣤⣄⣉⡉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠉⣉⣁⣤⣴⣾⣿⣿⠟⢀⣾⣿⣿⣿
⠙⢿⣿⣿⣿⣷⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⣁⣴⣿⣿⣿⣿⠟
⠀⠀⠙⠻⢿⣿⣿⣿⣷⣦⣤⣈⡉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⣁⣠⣤⣶⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠀⠀⡀⠀⢀⠀⠀⢀⢀⠀⢀⣶⠆⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣆⠀⠰⢡⢣⢂⠎⠀⡘⢸⡄⠀⢸⢸⠀⣼⠏⢀⠀⡄⡄⠀⣏⡏⡀⠀⣀
⠀⠀⠀⠀⠀⠠⡀⢀⡀⠀⠀⠀⠀⠘⡜⠀⠀⠀⠏⠎⠀⢀⠃⢻⠀⠀⠘⠀⠰⠏⣰⠃⠀⣧⡇⠸⡻⡰⢁⢞⡁
⠀⠀⠀⠀⠀⠀⣈⠢⡙⢦⠀⠀⢀⠀⠘⠀⣿⡀⠀⠀⠀⢸⠀⠋⠀⠀⢀⣴⠀⠀⠁⠀⠀⠃⠀⢀⡁⠰⡣⠋⠀⠀⠀⢀⡤⣚⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢌⠳⠅⢀⣿⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣮⡟⠀⠀⠀⡄⠀⠀⠀⣞⡇⠈⠁⠀⠀⢀⠐⠉⢈⣁⡠⠤⠄
⠀⠀⠀⣀⣐⠢⠤⣀⡀⠙⠂⢘⣼⠀⠀⠀⡞⣷⠀⠀⠠⢳⡀⠀⠀⢰⢻⡆⠀⠀⢰⢹⡀⠀⢠⣾⡆⠀⠀⠀⢰⣻⠀⠈⠐⠒⠒⠘⠃
⠀⠀⠀⠀⠈⠉⠓⠒⠪⠄⠀⣴⣷⡆⠀⠀⡇⣿⠀⠀⠀⢻⡇⠀⠀⢸⢀⡇⠀⠀⠈⣾⡀⠀⢸⠀⡇⠀⠀⢠⣾⡇⠀⠐⠒⠲⠶⠭⠭⢥⣄⡀
⠀⠀⠀⠀⣒⡒⠫⠭⠙⡁⠀⢻⠀⡇⠀⠀⡇⣿⠀⠀⠀⡓⣿⠀⠀⢸⢸⡇⠀⠀⢸⠋⣿⠀⢸⠀⡇⠀⠀⡸⢰⡇⠀⠲⠭⣒⡢⠄
⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⢸⠀⣷⠀⠀⡇⢿⣀⣠⠬⠄⣿⠤⠤⢾⢘⡧⠤⠤⢼⡆⣿⣀⣸⠄⡇⠀⠀⡇⣸⠀⠀⠀⠀⠀⠉⠓⠲⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⡀⢿⠉⠹⣳⣞⡦⠀⠀⡇⣿⠀⠠⡞⣼⢷⡄⠀⠘⡇⣿⠀⡸⠀⣏⡒⢲⠁⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⣇⣸⣄⠀⠈⠉⠀⠀⠀⡇⢸⠀⠀⠉⠉⠉⠀⠀⠀⡇⢹⠀⠑⠿⠷⠃⣸⢰⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠞⠗⠋⠀⠀⠀⠀⠀⣖⠣⢾⣦⠀⠀⠀⠀⠀⠀⢮⣳⣞⡷⠀⠀⠀⠸⢜⣺⠽⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣃⡤⠞⡗⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⣀⣤⣀⡀⠀⠀⢠⠖⢦⣀⡟⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠑⠛⠉⠉⠑⢄⣀⡤⠞⠋⠉⠑⢄⣀⣠⡾⠁⠉⠁⠉⠛⠉⠀⠀⠉⠳⢤⡼⠀⠀⠀⠀⣻⣢
⠀⠀⠀⠀⠀⠀⠀⠘⣟⠣⣀⠀⠀⠀⠀⢀⣀⢶⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⣠⠀⠀⢀⣀⣤⣾⢿⠁⡠⠤⠤⢤⣀⣀
⠀⠀⠀⣠⠴⠒⠋⠉⣸⠉⠻⢝⣓⡒⢖⣼⠂⠈⣽⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⡁⠀⢯⡽⠿⠻⠟⠉⠁⢸⡧⡄⠀⠀⠀⠀⠈⠙⢦
⠀⠀⣞⣁⣀⠀⠀⠀⢝⣄⡀⠀⠀⠈⠉⠉⠚⠑⠻⠦⠯⣽⣒⣒⣒⠶⡶⣖⢒⡒⣏⡭⢽⠴⠲⠉⠉⠁⠀⠀⠀⠀⢀⡼⡣⠇⠀⠀⠀⠀⣀⣀⡼
⡠⠊⠉⠉⠉⠀⠀⠀⠀⠙⠼⣶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠛⠋⠉⠛⠁⠀⠀⠀⠀⠀⢀⣀⠤⣶⣮⠽⠋⠁⠀⠀⠀⠀⠈⠉⢙⡇
⠙⠦⠤⠤⡤⠶⠃⠀⠀⠀⠀⠈⠉⠓⠽⠿⣗⢦⣤⣀⣤⠤⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⣤⣴⣒⣺⣭⠽⠛⠛⠁⠀⠀⠀⠀⠀⢠⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠳⣄⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠙⠓⠚⠛⠓⠋⠙⠛⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢯⣁⣀⣀⠀⣀⣠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢽⣷⠤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠳⠦⣤⣤⣤⣀⣤⣤⡾⠗⠀⠀⠀⠀⣠⣾⠃⠀⠀⢀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣦⣄⠀⠻⡧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠁⠈⠁⠀⠀⠀⣀⣈⣬⡉⡛⠿⣷⣄⠀⠀⢸⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⣉⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⡇
⠀⠀⠀⠀⢀⠤⠐⠈⠁⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⣸⠇
⠀⠀⠀⠀⠦⠤⠴⠦⠤⠤⠤⠤⠤⠤⣤⣠⠀⠤⢀⢠⣠⠠⣄⣀⣤⣤⣤⣀⢠⣤⣤⣀⣤⣄⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⢈⢻⣿⡟
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⢀⡤⠄⠀⠀⠀⠀⠉⠉⣉⠉⠉⣉⠉⠉⠛⢻⠁
⠀⠀⠀⠀⠙⠊⠁⣤⠀⡔⠠⣠⠎⠐⡀⠀⢰⠀⠂⠚⠁⠘⢆⠼⠉⠂⠀⠀⡰⠓⠆⠼⠁⠙⠤⠞⠁⠀⢡⠀⠀⡼⠣⠼⠁⠒⠞⠁⢇⠀⡰⢚
⠀⠀⠀⠀⡦⠀⠀⠰⠾⠀⠀⠀⢀⣀⣈⠐⢁⡀⣀⣀⣀⢀⡀⠀⠀⢸⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠈⢉⣡⣿
⠀⠀⠀⠀⡃⠀⣀⣀⣀⣀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠷⠾⠿⠿⠿⠿⡿⡷⣦⣔⣶⣶⣤⣿⣽⢯⣽⣿⣥⣶⣲⠶⠶⠦⣤⣄⡒⣆⡀⠀⣀⣀⢀⡀⠀⠀⠚⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠛⠛⠙⠀⠀⠀⢘⢻⡇
⠀⠀⠀⠀⠀⠁⠐⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠠⢴⣤⣠⣀⣀⣀⣀⣀⣤⣤⣄⣄⣀⣠⣀⣀⣠⣄⣠⣀⣀⣀⣀⡀⣀⣀⠈⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⡔⠱⢶⠁⠀⠀⢀⡄⠀⠀⠀⢀⠔⠈⠉⠉⠉⠁⠢⡀⠠⠊⠀⠀⠀⠀⠉⠑⡄
⠀⣀⣠⣔⣗⣒⠶⠛⢙⠍⢣⣳⠆⠀⠀⠘⠦⢴⣀⣀⡀⣀⠠⢳⠸⡒⠤⠤⠴⠶⠟⢒⠇⠀⠀⠀⠀⠀⣀⡄
⣜⡹⠭⠽⠒⣏⣀⣀⣀⣂⠞⢳⠃⠀⠀⠀⠑⠒⠲⠦⠤⠄⢀⣀⣀⣈⣑⣒⠒⠒⠉⠁⠀⡠⠐⠈⠉⠘⠻⠛⠘⠒⠒⠐⢄
⠙⢷⡒⠙⡖⠚⡲⠚⢑⡖⢩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠀⡀⠀⠀⣁⠀⠀⠀⢪⠀⢀⡀⠄⠀⡴⠒⢂⠀⡀⠀⡀
⠀⠀⠉⠐⠤⠔⠒⠤⠚⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⣀⣐⡒⠉⢉⡼⠀⠀⠀⠘⣧⣤⡴⣶⢿⣾⠶⣶⢷⢿⠞⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⡿⠾⠿⢷⡷⡶⠺⣼⠗⣖⣶
⠀⠀⠀⣠⣶⡤⣾⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠓⠒⠒⠚⠛⠛
⢀⡾⣵⣶⠿⢶⣦⢤⣀⣤⣦⡀⠀⠀⠀⢀⡤⣒⡒⠒⠿⠃⠺⠷⠂⠾⡶⠤⣴⢄⡀
⠈⠾⣧⣔⣿⣶⠖⠢⣩⣶⣬⡇⢀⡄⣤⡼⢿⡷⣿⣯⣖⣤⣄⠀⠀⠄⠀⠀⡈⠉⢀⣝⠢
⢀⠘⠉⠩⠉⠩⡉⠑⠓⠛⠚⡇⠈⠳⡛⠋⢠⣆⠀⠒⡄⠐⠀⠀⠒⠁⠀⠀⢁⡀⠈⠛⢁⡇
⠈⡥⠀⠀⣀⣁⣀⣐⡀⠀⠁⠁⢠⠀⠈⠓⢌⣉⠀⠺⠗⠀⢠⣶⡀⠠⣷⡄⠚⠞⣀⠤⠊⡇⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠲⢞⠢⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠚⠉⠉⠀⣀⢰⡇⠀⢀⡠⢔⣒⠶⠃⢹⢷⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠓⠒⠊⠇⣐⠒⢄⡀⡠⡄⢀⢀⡔⢒⣰⠃⠋⡇⠀⢮⣮⠕⠒⠒⠊⣁⠞⢀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠈⠁⠀⠈⠈⠀⠀⠈⠘⠁⠀⠁⠀⠖⠃⠀⠀⢋⢳⠲⡚⡟⢳⠚⣟⡟
⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠲⠶⠶⠶⠖⠒⠛⠉⠀⠀⠀⠀⠀⠈⣾⡇⡇⡇⢸⢸⡿
⠀⣠⡶⠉⠫⠝⠉⢈⠁⢢⡄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠒⠒⠐⠒⠒⠀⠈⢉⠖⡢⣄⠀⠀⠀⠀⠘⢧⣀⣡⣸⣬⠇
⢰⢡⠶⠀⠠⡢⢀⣍⠀⣙⣹⣤⣀⣀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣌⣤⠈⡆
⠀⠳⣦⢤⣀⡀⢀⠕⣋⠁⠠⠆⠀⠈⠑⢆⠀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⡇⢰
⠀⠀⠀⠉⠉⢹⠁⠈⢚⡁⠀⠀⣀⣞⠖⠛⠀⢿⡀⡤⣠⢄⣀⠤⠤⠤⠤⠄⠀⡇⡟⠿⡇⢸⠀⣀⣀⡀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠘⣄⠛⠈⣥⠀⢨⡞⠁⠀⠀⠀⢸⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢰⢀⢇⢶⣍⣎⣉⣰⣛⣓⠆⠀⠈⣱
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠅⣀⡨⠽⠆⠀⠀⠀⠀⠓⠤⠄⢀⡀⣀⣀⣀⣀⣀⣀⣀⣣⡩⠊⠈⢄⠈⠉⠉⠉⠉⠉⠉⠉⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡀⠀⠀⡀⠀⢌⡀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⢘⢧⠀⠃⡐⠢⡀⣋⡤⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠀⠨⠌⢿⡉⣿⠓⠦⢯⣠⠶⠛⠀⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠉⢙⠀⠀⠀⡀⡀⠒⠢⢿⡚⣓⣬⣤⣑⣜⠁⠈⣠⡄⠒⠀⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠁⠀⠀⠀⢀⠸⡄⣨⠖⢉⣉⠁⡱⠚⠋⢉⡄⢩⠈⠙⣆⠛⠤⡈⡑⠀⢂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠱⠁⠀⠀⠀⠀⠀⠀⣀⠽⠫⠀⠐⠒⢲⣷⠪⢆⡀⠈⠊⠄⠂⠘⠷⣄⠑⡠⠔⠉⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡰⡄⠀⠀⠀⠀⠀⣼⠇⠑⠀⠈⢁⢀⣀⣼⣷⡒⢤⠀⣠⠀⠖⣠⣐⡾⡁⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡨⡷⢄⡀⠀⠀⠈⠺⠛⠛⠋⠉⠉⠀⠉⠀⠈⠙⠓⠚⠛⠉⠉⠁⠈⠀⠀⠀⡀⠀⠰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣙⠂⠫⡉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣁⣀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡼⡀⢢⢘⡇⢠⡖⣄⠀⢀⡠⠤⢄⣀⡀⠀⠀⠀⡩⠭⢁⣀⣠⡀⠀⢠⠎⢀⠞⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠡⠑⠈⢃⠓⠞⠀⠈⠛⠁⠊⠉⠢⠉⠹⣄⠀⡜⠠⡰⠋⠙⣦⠀⠀⠁⠀⡼⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢠⠀⠌⢧⠀⠀⠀⠀⠀⠀⡐⠀⢦⠀⠀⠉⠀⡐⠀⢀⠀⣨⢷⡄⠀⣼⠃⠀⠘⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠘⠀⠁⠀⠑⢠⠀⠀⠈⠀⡇⠁⠀⢢⠀⢀⠄⢀⠐⢸⠰⠇⠀⢹⣰⡇⢐⠰⠆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡀⣣⡠⠄⠀⣿⠀⠀⠀⠀⡟⠀⠀⠠⢣⡇⠀⠐⠀⡘⡀⠰⡀⠀⡿⢁⠒⣀⠠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡑⠄⠉⠒⠚⠹⣂⡈⢀⠂⢳⠀⠂⠀⡼⣇⠀⠐⠀⠀⠱⠀⣀⡴⠁⠈⠋⡀⠄⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀⠠⠢⠀⢀⠉⠉⠉⠛⠢⠐⠀⠀⠈⠛⠀⠉⠀⠈⠈⠁⠀⢤⢀⣁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀⣀⠂⠘⠔⠀⠄⢠⠀⠀⠀⠂⠠⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⡀⠀⠸⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⠐⠒⠉⠉⠉⠉⠙⠋⡛⠻⢿⣷⣄⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠈⠈⠻⣷⣠⠀⠘⠓⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠘⠒⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣤⣤⣤⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠉⢱⡄
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠁⠀⠀⢀⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣷⡀⢸⡇
⠀⠀⠀⠀⡔⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠊⠛⠂⠛⠚⠛⠚⠚⠛⠛⠛⠚⠐⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠠⠶⠶⠶⠶⠶⢶⣦⢤⣭⣿⠃
⠀⠀⠀⠀⢠⣀⠄⠀⠀⠀⣀⠀⢀⠄⡀⠀⠀⡰⢢⠀⣠⠖⢀⠀⣠⢤⠀⠀⠀⢀⣀⠀⢀⡞⢢⡀⢠⡞⠉⢢⠀⠀⢀⣄⠀⣴⠂⡀⣴⢂⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠸⢀⡞⠀⠐⠋⠀⠰⡀⣠⠇⠀⠈⠁⠀⠈⠒⠁⠀⢇⠀⢀⡎⠀⠑⠊⠁⠀⠙⠉⠀⠀⠀⢂⠀⡼⠉⠙⠁⠀⠉⠁⠈⠦⠰⢋⡍
⠀⠀⠀⠀⢙⠉⠀⡈⠙⠐⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠸⠦⠾⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣾⣇
⠀⠀⠀⠀⢠⢠⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣤⣄⣀⣀⡀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣀⡀⣐⡂⠀⠀⠀⠀⠀⠰⠶⠗⠛⣉⡉⠉⢠⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠙⠉⠙⠛⠛⠛⠛⠛⠛⠓⠛⠻⠿⠿⠿⠯⠿⠿⢿⢶⣿⣿⣽⣿⣏⣙⣛⣉⣙⣇⣈⣀⣉⣉⣉⣋⠉⢙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠖⠶⠶⠶⠤⠤⠴⠤⠦⠴⠶⠦⠶⠤⠤⠦⠤⠤⠶⠤⢦⣤⣤⣄⣠⣠⣤⣄⢀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⣆⠀⢈⣩⣉⠉⠉⢉⣩⣉⠀⢀⡞⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣇⠀⠀⠈⠛⠉⠀⠈⠳⠶⠋⠀⠈⠙⠋⠀⠀⢀⡟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣉⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠶⠚⣋⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠁⠙⣆⠀⠀⢀⣀⣀⠀⠉⠉⠁⢀⣀⣀⠀⠀⢀⡞⠁⠈⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡇⠀⠀⠀⠘⠳⠖⠋⠁⠈⠳⣄⣀⡴⠋⠀⠈⠓⠶⠚⠀⠀⠀⠀⡏⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠚⠉⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠦⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⡉⠙⠓⠲⠦⠤⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠶⠖⠛⠉⣁⣠⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⢹⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣴⠁⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠳⢤⣠⠴⠋⠁⠈⠳⣄⠀⠀⢀⡴⠋⠁⠉⠳⢤⣠⠴⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡠⡠⢀⢀⢀⠀⡀⡀⢀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠠⠂⠈⠀⠀⠀⡀⢈⠀⠀⠁⡀⠊⠄⠀⢡⣁⠂⠊⠊⠐⡁⢀⠀⢂⠀⢀⡀⠢⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⡈⠐⠈⠀⠀⠀⠆⠱⡀⠀⠀⠀⡈⠱⡀⠀⠀⠀⠜⢣⠀⠀⠀⡌⠱⡀⠀⢘⠈⣆⠀⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠄⢀⠇⠀⠀⠀⠄⠇⠅⠀⠀⠸⡀⠈⠇⠀⠀⠇⠀⠑⠀⠀⢐⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⡡⢂⠀⠁⡴⠂⠐⢠⠀⠀⠀⠀⠓⠀⠀⡀⠀⠌⠚⠢⠀⢀⠐⠛⢠⠁⠐⠉⠹⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠀⢸⠀⢰⣤⠡⡊⠀⠀⡲⢽⠠⡪⠐⠴⠀⠠⠀⠈⡄⢠⢸⡆⣳⢀⡀⡄⢀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠧⠻⠈⠂⠀⠦⢰⢸⠀⠐⠄⠨⠠⠀⠀⠕⡇⠜⡇⠂⠐⠸⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⢨⢤⠀⠀⡇⠸⡇⠀⠘⠨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⡆⠀⢸⡀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠈⢈⠁⠀⣀⢈⢉⠉⡁⡈⡩⠈⠉⡉⢀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⡀⢒⣈⡀⠀⠀⠠⠂⠔⣁⣊⣊⣊⣀⠁⠐⠔⠡⠢⠀⣂⣊⣈⡐⠐⠀⠌⠄⠄⢂⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠈⠉⠂⠈⠓⠒⡰⠊⠉⠁⠑⠀⠀⡉⠉⠓⠒⠒⠒⠓⠁⠀⠲⠋⠉⠑⠂⡮⠉⠉⢉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠶⠷⠶⠶⡖⣖⡶⠿⢶⣶⣶⢶⣶⢶⣶⡖⣴⡶⣖⢢⠯⠯⠽⠿⠭⡟⠿⠽⠽⠿⠏⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠈⠀⢠⡄⠈⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢂⠀⠀⡀⠄⢐⠆⠂⣀⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣥⠤⠠⢤⢤⣄⢠⣭⠤⡤⡬⠥⠤⡤⢠⣤⣐⣂⢀⢤⢤⢤⢽⠉⣉⡉⢹⣭⡉⠙⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡉⠉⠉⢩⠉⠉⠉⠩⠈⠉⠉⠉⠩⠁⠉⠩⠉⠉⠙⠛⣉⠉⢙⣩⣉⣉⣛⣝⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠒⠀⠐⠀⠈⠉⠀⠀⠀⠀⡉⠁⡈⠉⠁⢉⠀⡁⢈⢉⠉⢀⢉⡩⠄⠄⡉⡈⣉⢉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠧⣀⡹⠦⣄⠠⢑⡴⠦⠞⠉⠳⢌⣠⠷⢢⡐⠐⡡⠖⣄⠁⢠⠞⠙⠳⣬⢐⠝⡇⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⡘⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⢧⣋⠞⠁⠀⢀⠀⣀⣀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⡀⠀⣀⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠑⠀⠑⠁⠊⠓⣐⣓⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⡆⢠⢒⣔⢢⠀⠜⣩⡍⠍⢉⠤⡤⢌⠀⠩⢭⠍⡍⢩⢭⡬⢍⠉⡕⡬⢢⠀⠔⣦⠒⠰⣞
⠀⠀⠀⠀⠀⢀⡀⣀⣘⢋⡀⠈⠊⠩⠃⠀⠈⠟⠿⠃⠀⢪⡱⠞⠀⠐⠒⣴⠃⠈⢟⣲⠏⠀⠁⠉⠞⠀⠩⠙⠂⠀⠹⠀⡀⣠⡀⡀⡠⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠐⠉⠛⠉⠛⠉⠛⠙⠛⠰⠒⠒⠒⠲⠒⠶⠒⠶⠲⠶⠰⠶⠴⠖⠶⠖⠶⠒⠲⠂⠜⠫⠋⠑⠉⠊⠈⠁⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣶⠄⠀⢰⡶⠀⠀⣶⠆⠀⠀⠀⠉⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣴⣾⠟⠋⠀⠀⠸⠗⠀⠀⠿⠃⠀⠸⠟⠀⠀⠻⠇⠀⠐⠿⠀⠀⠺⠇⠀⠈⠻⣿⣦⣄
⠀⣀⣴⣶⣿⠿⠛⠉⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⢀⣀⠀⠀⣀⡀⠀⢀⣀⠀⠀⣀⡀⠀⠀⣀⠀⠙⢿⣷⣤⡀
⣼⣿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠁⠀⠘⠋⠀⠀⠛⠁⠀⠘⠋⠀⠀⠙⠃⠀⠈⠛⠀⠀⠙⠃⠀⠈⠛⠀⠀⠀⠈⠛⢿⣷⣤⡀
⣿⣧⠀⠀⠀⠀⢠⣤⠀⢀⣤⡀⠀⢠⣄⠀⠀⣤⡄⠀⢠⣤⠀⠀⣤⡄⠀⢠⣤⠀⠀⣠⡄⠀⢀⣤⠀⠀⣤⡄⠀⠀⠈⢻⣷
⠹⣿⣦⡀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣠⣿
⠀⠈⠻⢿⣶⣤⣄⣀⠀⠠⣶⠄⠀⢰⡦⠀⠀⣶⠄⠀⢰⡶⠀⠀⣶⠆⠀⠰⣶⠀⠀⢶⡆⠀⠠⣶⠀⠀⠀⠀⠀⣠⣶⡿⠏
⠀⠀⠀⠀⠉⠙⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣦⣤⣤⣤⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠟⠿⠿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠛⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠸⣆⠀⠀⠀⢀⠧⠠⠤⠤⠤⠤⢄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⢉⣭⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠒⠳⡄⠀⠀⠀⡰⠊⠉⠉⠂⠀⠀⠀⠀⠀⢠⠚⠉⢳⡀⠀⠀⢠⠏⠀⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⠜⠁⠀⠀⢻⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠳⣄⣠⠏⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣄⠀⠀⠀⠳⣤⠏⠀⣀⠀⠀⠀⢠⠀⠀⢀⣼⣄⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠷⡤⠋⠈⢢⠀⠀⠀⠀⠀⡴⠁⡱⡀⠀⠀⠉⠀⢩⠏⠈⢆⠀⠀⠀⠀⢀⠘⠀⠘⣤⠗⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢄⣀⣀⢳⣦⣴⠼⠉⠉⠛⠛⠻⢤⣠⣥⠟⠚⠒⠒⠛⠣⣦⣼⠼⠛⠛⠋⠉⠙⠲⠦⡶⢃⣀⣀⡤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⡉⠛⠛⠛⠛⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠋⢹⡆⢠⡖⠛⢢⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⠀⠀⠀⠀⠀⣠⠖⢦⠀⠀⠀⠀⡔⠉⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡏⠀⠘⢧⠼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢸⠃⠀⢸⡇⠀⠀⢠⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠤⣀⡤⠏⠀⠀⠀⠙⠦⣤⠞⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢌⠒⠁⠀⠀⠀⢀⠀⠀⠀⠘⡄⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⢻⡀
⠀⠀⠀⠀⠀⠀⢀⣰⠙⠀⠈⣆⡀⠀⠀⢠⠎⢳⡀⠀⠀⠀⠉⡰⠳⠀⠀⠀⠀⠀⣠⠛⢆⠀⠀⠀⠀⠀⠊⠱⡄⠀⠀⣀⡜⠁⠀⢸⣄⡀
⠀⠀⠀⠀⠀⠰⢧⣌⣛⠚⠛⠉⠉⠙⠛⣏⡂⣀⡹⠶⠶⠶⢞⡀⢀⢸⡶⠶⠶⢶⣉⠀⢈⡷⠶⠶⠶⣧⣀⣀⡽⠛⠉⠉⠙⠓⢚⣋⣠⠼
⠀⠀⠀⠀⠀⠰⡆⠀⠉⠉⠛⠛⠒⠶⠶⠾⠿⠯⠤⣤⣤⣤⣬⣽⣿⣭⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⠤⠭⠷⠶⠶⠖⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⢰⡏⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⣠⠶⠶⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠘⡇⠘⢄⠀⠀⠀⠀⢠⠏⠀⢳⡀⠀⠀⠀⠀⠀⣠⠖⠒⢤⡀⠀⠀⠀⠀⠀⢰⠃⢸⡄⠀⣸⠃⠀⠀⢹⡀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⠢⠤⠴⠃⠀⠀⠀⣷⠀⠀⠀⠀⢰⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⡎⠀⠀⠓⠞⠁⠀⠀⠀⠀⢳⠀⠀⠀⢠⠃⣺
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣠⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣟
⠀⠀⢀⠔⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠢⡀
⠀⠀⠹⢄⡀⠀⠈⠓⠦⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠶⠚⠁⠀⣀⡴⠃
⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⣤⣀⣈⣉⡉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣉⣉⣁⣀⣤⠤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⣉⣙⣋⣩⣉⣹⣙⣩⣟⣫⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠐⠀⠀⢀⢠⠄⠂⠀⠀⠀⠐⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣉⠐⡠⠉⠣⣀⣠⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠈⠠⠐⠈⠀⡤⡀⡀⢾⠀⠀⠀⠀⢠⠔⠋⠭⡑⢄⣀⠶⠒⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠂⠤⢀⢧⡀⠀⣠⠏⠀⠀⠀⠈⠠⠀⣤⠈⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠎⡃⢀⠉⠁⠀⢀⠂⠤⢤⡀⠀⠈⠉⠉⠀⠀⠀⠀⠩⠀⠀⠁⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠂⠂⠀⢀⠀⠀⠉⢀⠀⠀⢀⠠⢤⡀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠡⠄⠁⠢⠴⣄⠀⠀⣈⠉⠀⠀⠈⠀⠠⠄⣀⢀⠀⠀⠂⠁⠑⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠠⠄⠉⠁⠰⠄⠑⠦⠐⢠⣤⠄⠀⠀⠀⠉⠛⠚⠳⠤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠢⠄⠀⠑⠀⠂⠀⠀⠀⠀⠀⠒⠁⠠⠐⠛⠢⢄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠀⠂⠃⠀⠀⡄⢀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⠀⠤⢍⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣇⠀⠀⠀⠀⠀⠀⠀⢠⡷⣄⠀⠀⠀⠀⠀⠀⠀⣸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⠘⡆⠀⠀⠀⠀⠀⢠⡟⠀⠘⡆⠀⠀⠀⠀⠀⣰⠃⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⠸⠃⠀⠀⠀⠀⠀⠈⠳⣽⠴⠃⠀⠀⠀⠀⠀⠘⠆⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠳⡄⠀⠀⠀⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⢠⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡄⡗⠀⠀⠀⠀⠀⠀⡇⣠⣿⠀⠀⠀⠀⠀⠀⣿⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⢻⠀⠀⠀⠀⠀⢰⡇⣠⡟⠀⠀⠀⠀⠀⢀⡟⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢸⡀⠀⠀⠀⠀⢸⡇⣁⡿⠀⠀⠀⠀⠀⢀⡇⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠶⠾⠶⠶⠶⠶⠶⠾⠷⠿⠿⠶⠶⢶⣶⣶⣼⣧⣭⣿⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡁⣄⠀⣰⡟⠁⠀⠉⠻⣦⡐⠦⠤⠤⠖⣡⣴⠟⠉⠉⠉⠻⣦⣄⠐⠤⣤⣤⡤⠔⢀⣴⠟⠉⠉⠻⣦⡀⢠⢆⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠷⣶⡶⠟⠀⠀⠀⠀⠀⠈⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠾⠛⠁⠀⠀⠀⠀⠈⠻⠷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⠈⣿
⠀⠀⠀⠀⠀⠀⣠⣴⠿⢷⣤⣤⣤⣤⣤⣤⡶⠾⠛⠛⠛⠛⠛⠛⠶⣶⣤⣤⣤⣤⣴⠶⠾⠛⠛⠻⠶⢦⣄⣀⢀⣀⣠⣤⣤⣴⣿⣀
⠀⠀⠀⠀⠀⠰⣿⠀⣀⣀⡀⠀⢉⣉⣁⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣉⣉⡀⠀⣀⣀⣀⣀⠀⠀⢀⣉⣛⠛⠋⠉⣀⣀⣀⢈⣿
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣭⡶⠾⠛⠉⠉⠙⠻⠶⣦⣭⣥⡶⠶⠟⠛⠋⠉⠉⠉⠛⠷⠶⣮⣭⡶⠶⠟⠛⠉⠙⠛⠷⠶⢾⣿⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠐⠶⠶⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠶⠶⠶⠖⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠖⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠓⠲⢤⡈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢼⢱⡀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⣀⠀⠈⢣⠀⠀⣹⠦⢿
⠀⠀⠀⠀⠀⠀⠀⡼⠚⠀⠀⠀⢀⡀⠸⣁⣀⡀⠀⠀⠀⠀⠀⠙⠢⠤⠴⠦⣄⢰⠃⠀⠀⣀⣈⣉⡿⠥⠖⠋⠁⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⢯⣈⡉⠁⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⢻⠈⡏⠉⠉⠉⠀⠀⠀⠀⣀⣠⢴⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡞⣿⠇⠀⠀⡠⠤⠭⢷⣶⣶⣤⣤⣤⣤⣤⣤⡤⠤⢤⠤⠬⠿⠥⠤⠶⢖⣒⣒⣋⡭⠵⠚⠉⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠸⡗⣇⠀⠀⠀⠓⠤⣄⡀⠀⠀⠉⢉⡽⠯⡍⠉⠉⠉⠉⣹⠿⢯⠉⠉⠉⠉⠉⠀⠀⢀⣀⡤⠴⠋⢸⡇
⠀⠀⠀⠀⠀⠀⣰⢳⡼⠃⠀⠀⠀⣠⠖⠛⠒⠒⠒⣾⠦⠤⠭⠤⠤⠤⠤⠥⠤⠬⣥⠤⠶⠖⠒⠚⠋⠉⠀⠀⠀⣠⠞⣧
⠀⠀⠀⠀⠀⠀⢻⠸⣀⠀⠀⢀⣀⢹⣄⣀⣀⠀⠘⠛⠁⠀⠀⠰⣤⡴⠆⠀⠀⠘⠛⠁⠀⠀⠀⢀⣀⣠⣤⡶⠞⠁⢀⡏
⠀⠀⠀⠀⠀⠀⣼⢳⣌⣳⠀⢳⠬⠭⣤⣀⣈⣉⣉⣉⣛⣛⣛⣒⣒⣒⣒⣒⣒⣒⣛⣿⠿⠽⠟⠛⠛⠉⠁⠀⣀⣴⠏⢧
⠀⠀⠀⠀⠀⠀⢧⠀⠙⢿⡀⣾⠤⢤⣀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⣖⣺⠭⠟⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠉⠉⠙⠓⠒⠒⠻⠿⠯⠭⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠓⠚⠛⠉⠉⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢒⣠⣦⠖⢊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⣿⢻⡮⢵⡟⢻⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣸⠲⣽⡇⢸⣸⣇⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸⠔⠉⠈⠲⡭⠷⢣⣹⡀⣶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⢿⢷⡶⣤⣤⣤⣼⣤⣶⢘⢛⡦⡡⡥⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⡇⡇⣿⢸⢸⢸⢸⡇⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⢸⢸⡇⡧⠃⡇⣿⢸⢸⢸⡘⠣⡇⡇⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡉⢸⡜⠃⠀⠀⠀⠛⠼⠸⠊⠁⠀⠐⠧⠋⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡜⡙⢂⡽⠷⣼⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣿⣿⢾⢹⣹⣿⣿⣿⡏⡏⡏⡏⣿⣿⣹⢸⢸⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⡇⣿⣿⡸⠈⠀⠘⢸⣿⡇⡇⡇⣧⠟⠀⠙⣼⢸⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠓⠛⠉⠀⠀⠀⠀⠀⠙⠃⠇⠛⠁⠀⠀⠀⠈⠚⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠻⣷⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⠿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂