Physics Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣗⠀⢠⠆
⠀⠀⠀⠀⠀⣀⣴⣿⡟⠁⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡛⠻⢻⠛⠟⠛⢻⡟⠀⠀⠀⠀⠀⠀⠐⢄⢀⡡⠔⠒⠶⢍⡀⡤⠂
⠀⠀⠀⢀⡞⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⢰⠀⢸⡇⠀⠀⠀⠀⠀⠤⢤⢀⡞⠀⠀⠀⠀⠀⢳⡀⡤⠤
⠀⠀⠀⠸⡅⢿⣿⣿⠀⠀⠀⢀⣀⡤⠞⣡⠏⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⡁⢀⢸⠀⠀⣄⢸⡇⠀⠀⠀⠀⠀⣀⣀⠸⡇⠀⡤⠶⢤⠀⢸⠇⣀⣀
⠀⠀⠀⠀⠙⠦⣬⣉⣉⣉⣉⣉⡤⣴⠞⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡗⠟⠛⠛⠃⠀⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⣠⠹⣄⢃⠀⡸⣠⠏⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠶⣞⠻⣶⣄⡀⣀⣾⡇⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠉⠀⣰⢸⣾⣤⣷⡗⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡗⢲⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣏⠉⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⢀⣤⠴⠶⣾⠇⣠⣤⠙⠿⠶⠶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠃⢸⡇
⠀⠀⠀⠀⣠⢤⣰⡏⠁⠀⢹⡄⠀⠀⠀⢀⣹⣿⡧⠀⠀⠀⠀⢸⣇⠀⣠⡿⢁⡠⣤⡈⠐⣦⡀⢸⡿⠀⠀⠀⠀⢸⡇⢠⠄⢀⡀⣶⡷⠾⠤⠀⢸⡇
⠀⠀⠀⠀⠸⡌⣿⢷⣀⣤⣴⣷⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣴⠻⣯⣀⡇⢸⣤⣾⡿⠀⣄⣼⢿⡁⠀⠀⠀⠀⢸⡇⠘⠥⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⠙⣷⣦⣉⣉⣤⠶⣛⣁⢈⣷⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣾⣿⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⢿⡟⠛⠙⠛⡾⢯⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢁⡀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠈⠛⠉⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠯⠼⠂⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢀⡆⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⣤⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⢸⣷⡖⠀⡟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠼⠿⢴⣶⠿⠷⢤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠋⢹⠇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡿⢩⡍⠉⠉⠉⠉⠉⠉⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠖⢾⡷⠲⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠈⠀⢰⠇⢀⡇
⠀⠀⠀⠀⠀⢀⣷⣬⣥⣤⣤⣤⣤⣤⣴⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⡏⣶⢹⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡤⠖⠋⢀⡞
⠀⠀⠀⠀⠀⠀⣟⣻⣋⡉⣩⣽⣯⣭⣭⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠧⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⣀⣤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⡰
⢀⣴⣾⡟⠁⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠉⠙⠉⢻⡇⠀⠀⠀⠀⠈⠓⡤⠓⠉⠑⠣⡔⠋
⡏⣿⣿⡇⠀⠀⠀⢀⣠⢇⠧⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠆⢸⡇⠀⠀⠀⠈⠉⢰⡇⠀⣀⡀⠀⡗⠈
⠙⠬⣙⣓⠒⣒⣋⣩⡴⠉⠀⢸⡀⠀⠀⠀⠀⠀⣿⠷⠶⠾⠀⠺⢿⡇⠀⠀⠀⠈⠉⣀⢣⡸⠀⡸⡰⢃⡉
⠀⠀⠀⠙⠛⠛⠲⣟⢷⣤⣀⣼⡇⠀⠀⠀⢠⠎⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⠈⠀⠔⣷⣧⣿⠗⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠣⡤⠤⠤⠤⠤⠤⢤⠴⠃⠀⠀⠀⠀⠀⠀⠻⣿⡿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢳⠲⡄⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⢀⣀⠤⣏⠁⠀⠀⠸⣿⣇⠀⠀⠀⢀⡴⠒⢲⡟⠤⠬⠛⠒⠲⣦⡀⠀⠀⠀⡿⠀⠀⠀⠀⡀⡠⠈⢸
⢰⣳⣼⡀⠀⢘⡄⣀⣠⠤⠿⠟⠀⠀⠀⢈⣷⡄⢺⢡⠒⣶⡄⠑⢤⡾⠁⠀⠀⠀⣟⢰⠅⠢⠘⠛⠓⠁⢸
⠀⠳⠃⠛⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡤⠛⢿⣌⡻⢛⣡⡶⠋⢹⡀⠀⠀⠀⣿⣀⣁⣀⣀⣀⣴⣶⣼
⠀⠀⠀⠀⡾⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⣿⠛⠛⢳⢯⡹⠛⠀⠀⠀⠀⠀⠀⠀⠈⡇⡏
⠀⠀⠀⠸⠃⠿⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿

⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⡜⠉⠉⠛⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠪⠗⢀⡇⠀⠀⠀⠀⠀⠀⢸⠁⣹⠀⡄⠀⢸⠁⣹
⠀⠀⠀⢳⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠦⢴⣮⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣷⡏⢸⠉⢹
⠀⠀⡤⠤⠷⠤⠤⠴⠯⣤⣄⠀⠀⠀⠀⠀⠀⠀⣏⣉⣹⣏⣉⣹⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠃⢸⠀⡸
⠀⢀⣧⣿⣤⣤⣤⣤⣤⣯⣿⠀⠀⠀⠀⠀⠀⢠⣿⠁⠈⠁⢸⣫⢲⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⠤⠋⣠⠃
⠀⠈⣟⣟⣉⣹⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠈⢲⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢠
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡆⠀⢠⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⣦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠄⣀⢀⡀⠀⢠⠁⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⣣⠃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠁⠈⢀⠀⢀⠎⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⢤⡾⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠔⠁⡄⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠈⠢⠤⠤⠐⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣶⠶⣦⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⢰
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⠟⣿⠖⣦⢶⣄⣴⠆⣶⡶⢼⣆⣴⠶⠆⣶⡶⠄⠀⢠⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠀⠀⡿⠀⡿⠈⣿⠏⠐⢯⡿⢾⡟⢷⣤⡔⢯⡿⠂⢀⡇⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠆⠀⠈⠄
⠀⠀⠀⢀⠀⠀⠀⠠⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠎⠀⠀⣀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠠⡄⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠉⠠⠅⢘⠊⠛⠁⠀⠀⠀⠀⠂⠄⣀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣀⣀⣀⣀⣀⣀⣲⠘⡣
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠀⠀⣼⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⢤⣄⣀⣇⣀⡤⠔⠻
⠀⠀⠀⠈⠀⠐⠁⢠⠄⠀⠀⠊⠀⠈⠁⠀⠜⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣲⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⣀⣀⣀⣤⣧⣄⣀⣀⠀⠀⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⢛⠉⠀⠀⠀⡇⠀⠀⠉⠙⠲⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⡗⠲⠘⡲⠖⠒⠐⡓⠒⠒⠒⠒⠒⣾⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⢬⣉⠁⠀⠀⠘⠃⠀⠀⣀⣠⠴⠃⠈⠇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠉⠉

⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣔⠂⠘⠈⠓⠉⠐⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠁⣶⡖⢠⣆⣤⣆⠔⠒⠐⠃
⠀⠠⣎⠁⠀⡄⠀⠀⠀⠀⠐⠒⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣃⣋⣘⡘⠒⠃
⢠⠊⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠀⠀⠀⠀⠉⠉
⢸⠴⢃⠆⠈⠠⠀⠀⠀⠰⢄⠀⠀⠀⠀⢀⡔
⠀⠀⠃⡀⠀⠀⠀⠄⠀⡢⠔⠛⠒⠒⠒⡍
⠀⠀⠀⡇⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢰⠀⠀⢄⠀⡼
⠀⣀⠠⠄⠤⢄⠀⠀⠀⠀⠀⠀⠘⠽⠆⠀⠂
⠀⠀⢀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠨⠀⠍⠀⢀
⠀⡀⠀⠉⠁⣈⡺⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠉⠀⠀⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠈⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣾⣿⣿⡏⠛⠲⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠙⢷
⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠈⠛⠦⣄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣀⠀⠀⠀⠀⠀⠀⣀⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠳⣄⠀⠀⣠⠞⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢈⡵⢯⡁⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⣀⣤⣤⣤⢼⡧⢤⡴⠿⠤⠤⠿⢦⡤⢼⡧⣤⣤⣤⣀⣠⣞⡀⠀⠀⠀⢀⣀
⠀⠀⢀⣠⡤⠶⠒⠛⠉⠉⠉⠙⢧⡀⠀⠀⢸⡷⠋⠀⢀⣀⣄⡀⠀⠙⢾⡇⠀⠀⢀⡼⠋⠉⠉⠉⠛⢲⣿⣿⣷⡀
⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡴⢻⡇⠀⢰⣿⣿⣿⣿⡆⠀⠸⡟⢦⣴⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠳
⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠳⣼⠇⠀⠈⢿⣿⣿⡿⠁⠀⢸⣧⠞⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠓⠶⢤⣄⣀⡀⠀⠀⠀⢠⡞⠁⠀⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠈⢳⡄⠀⠀⠀⢀⣀⣠⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⡶⠛⠶⠶⠦⠤⢼⡧⣬⣷⣤⣤⣤⣤⣾⣥⢼⡧⠤⠴⠶⠶⠛⢶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⢈⣷⣾⡁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴⠋⠀⠀⠙⢦⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⠉⠀⠀⠀⠀⠀⠀⠉⡿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣰⣿⣿⣶⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢧⣄⣀⣤⠴⠛⠿⠿⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞

⠁⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠲⠮⠁⠲⠤⠔⠉⠀⠀⠀⠀⠀⡤⣨⣗⣻⡞⠡⠒⠀⠀⠀⠀⠀⠀⠈⠓⠯⠓⠁⠀⠀⠘
⠀⠀⠀⣀⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠛⠙⠒⠃⠀⠀⠀⠀⣀⣄⠀⠀⠀⣠⣄⠶⠂⠀⠀⠀⡀⠀⠀⠐⠀⠖
⣄⣴⣌⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⠀⠀⠂⠴⠟⠃⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⣖⠅⠀⠀⠀⠀⠊⠙
⠳⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⢉⣛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠚⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠐⢁⡀⠸⢭⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡅⠀⠉⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠱⡀⠀⢠⠀⠀⠀⠐⠃⠀⢀⡴
⠀⠄⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠉⢁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⠄⠀⠀⠀⢡⡀⠀⡀⠀⠛⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠶⠀⣠⠾⠋⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⢧⡤⠞⠁⠀⠀⣰⠿⣎
⣂⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠐⠃⠀⠈
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⢀⠉⠹⢦⡀⠀⠘⢦⡘⣎⠷⠀⠀⢀⣤⠽⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠛⣦⠀⠀⠹⠈⣤⢀⣄⣿⠁
⡀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠐⡷⣾⡄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠂⠀⠔⠂⠀⠐⠲⢤⡀⠀⠀⠳⡏⠙⡇⠀⢠⢞⡟⠈⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣀⣀⣄⠀⠀⠀⢠⠖⠿⣆⠀⠀⢳⠀⡿⠀⠸⠈⣷⣀⠀⠀⠀⠀⠀⠸⡍⢀
⠀⠀⠀⠁⠔⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⡤⠀⠀⠀⠀⠙⠘⠇⣿⠀⠀⠘⠶⠃⠀⠀⠀⠻⠽⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠈⠠⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠁⠒⠁⠀⡀⠀⠀⠀⠀⠀⢠⠄⠀⠁⠀⠀⢀⣀⣀⣤⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠈⠁⠀⠈⠉⠙⣦
⠒⣤⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⢀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠈⠳⠦⠶⠛⠓⠶⠒⠒⠤⠖⠤⠚
⢉⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠔⢉⠌⢁⠐⢄⢈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠲⡆⠀⠀⠀⠀⠀⠀⢇⠀⠘⢄⡈⠂⠊⣀⠜⠀⣹⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠈⠉⠉⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡚⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠛⠐
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⡠⠂⢰⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐⠉⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⢠⠀⡖⠒⡄⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠂⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠠⠄⠄⠀⠔⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⢽⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠁
⠀⠀⠀⠐⠀⠑⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠔⠀⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠘⠀⡃⠀⠀⠠⢴⡢⠀⣀⠠⠖⡤⡀⣠⠀⠀⠀⡄⢀⣠⣄⣄⠄⠀⢨⠀⠀⠀⠀⠀⢀⠂
⠀⢀⠀⢀⡂⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⢰⠎⠇⣀⡠⠇⠁⢖⠀⣰⠉⡇⠸⢀⡔⢆⡀⠀⢸⠀⠀⠈⠂⠀⠃⠀⠀⠠
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣀⠁⠀⣀⣀⡁⣈⣁⣀⣉⣀⣁⣀⣀⣀⡈⠀⣈⣁⢀⣜⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠉⠀⠈⠉⠀⠉⠁⠉⠀⠉⠉⠁⠀⠙⠙⠛⠛⠀⠀⠀⠀⠃⠈⠅⠀⠃⠀⠃⠊
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⡁⠀⠀⢠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠄⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠀⢀⠔⠀⠈⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠁⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀
⠀⠈⢒⠶⠚⠁⠀⠀⡴⣦⡷⠹⠂⠀⠀⣦⡆⡄⢐⣛⠀⠀⠀⡔⠩⢹⡟⠀⠀⠀⣀⣏⣀⣯⣈⣿⣀⡀⠀⡸⠁⠀⠐⠒⠒⠂⠀⢉⠀⠀⣟⣦⠀⠖
⠀⠀⢈⣏⠀⠀⠀⠀⢁⢀⣠⠀⠀⠀⠈⠈⠁⠀⠀⡆⠀⠀⠀⠷⠶⠗⠁⠘⠇⠀⠁⠉⠈⣏⠈⠈⠀⠀⠀⢡⠀⣉⣉⣉⣉⣉⣀⠈⢦⠀⢻⣿⣆
⠀⠀⠨⣿⠀⠀⢤⠶⣸⠄⠋⠁⠀⣀⠤⢖⣒⢦⠀⠑⢀⡀⠀⠀⠀⢀⣀⠤⠴⢲⡆⠀⠀⠉⠀⢀⣀⡀⠀⠈⠣⡀⠴⠄⠀⠀⠀⠰⡖⠱⡄⢻⣿⡆
⠀⠀⠠⣿⠂⠀⠈⢀⡤⣔⣒⠒⣎⢴⣿⣽⣿⡎⢆⠀⠈⠀⢠⡖⠺⠿⠿⢶⠆⡜⠸⡀⠀⢠⠤⣼⣡⣷⢒⡆⠀⢱⠸⠁⠐⠃⠀⠸⢡⠀⣱⠀⠉
⠀⠀⣨⠷⡄⠀⠐⡇⠪⢒⡪⡥⠘⡌⣋⢉⡤⢀⡈⢆⠀⠀⣎⣳⠀⢠⡄⠀⢠⣷⣷⣧⠀⠘⣧⣾⠻⢫⠟⢅⠀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠚⠀⠀⠛
⠀⠀⠙⠐⠃⠀⠀⠸⡌⢗⣍⠩⣅⠸⡙⠞⢛⣉⣁⣸⠄⢰⣋⣛⣇⠀⠀⢠⣤⡍⣭⡅⠀⠀⠣⠼⣏⡹⠒⠋⠀⠀⠀⠀⠀⣰⣲⣲⡀⢀⠤⠢⢄
⠀⠀⠀⣀⠤⠤⢄⡀⠱⡸⣿⢇⣩⣤⣷⡮⠛⠉⠉⠉⠀⠈⠉⢉⣉⣀⣀⣀⣀⣀⣚⣁⣀⣀⣀⠀⠀⠀⠀⢀⣀⠤⢤⡀⠈⣷⣧⢶⠚⢸⢼⣷⢸
⠀⢰⡿⠀⣀⣀⠀⠙⡄⠱⡶⠋⠁⠀⣄⣴⣄⡀⠀⣾⣿⣛⣿⣿⣿⣿⣿⣟⣻⡟⣿⢟⣻⣟⣻⣿⣧⠀⢀⠇⠛⢠⠠⡍⢣⠈⠁⡉⠀⢸⣹⣿⢸
⠀⠸⡀⠀⠯⡣⠃⠀⡇⠀⢀⠀⠀⢸⣿⡁⡱⣏⠀⣿⣿⣵⡿⣭⢸⣧⣾⣯⢻⡇⣿⡸⣿⣷⢽⣿⡟⠀⢣⠀⠺⣼⢰⡦⠙⡆⠀⡟⠀⢸⢺⣿⢸
⠀⠀⠑⠤⣀⣀⠤⠊⠀⠀⡈⠃⠀⠀⠓⠫⠎⠁⠀⠙⠿⠿⠿⣿⣿⣿⣿⣷⣿⣷⡿⠿⠿⠷⠿⠟⠃⡀⢧⡋⢆⣸⠋⡄⢺⠀⠀⡇⠀⢸⣸⣟⢸
⠀⠀⣤⠀⠀⣀⠀⡀⢴⠗⠙⠁⠀⣆⠀⡀⡆⢠⡦⣦⠄⠀⠀⠀⡀⡀⠀⢀⣤⣤⡀⠀⣠⠤⡄⠀⠀⣩⣄⢧⣀⡿⣄⠔⠉⠀⠀⠇⠀⢸⣸⣷⢸
⠀⠀⠀⠀⠐⡷⣂⠉⣠⡀⠀⠀⠀⣗⣚⡊⡇⡸⢟⠿⡄⠀⡇⠄⢹⡇⠀⠈⠪⠕⠁⠀⠳⠤⠆⠀⢀⡇⠸⣄⠀⡇⡇⠛⠀⢰⠲⡎⡇⢸⢼⣿⢸
⠀⠐⠇⠀⠀⠈⠁⠈⠸⡅⣤⠀⠀⠓⠀⠐⠃⡇⠀⠀⡇⠀⠋⠀⠾⡅⠀⠀⠯⢽⡀⠀⢸⠩⡇⠀⠀⠑⠚⠁⠋⠓⠋⠙⠀⢸⠀⢁⡇⠘⢽⣿⡝
⠀⠀⠀⠠⢟⠀⠰⠀⠀⠀⠀⢠⡔⡶⠀⢴⠀⠁⠀⠪⠅⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⡜⠀⡇⠀⢰⠓⠀⠀⢠⣰⢠⢰⠀⠀⠉⠉⠀⠀⢸⣿⠀⠤
⠀⠀⠀⠀⠻⠐⠀⣀⠀⠀⣴⠟⠙⠁⣤⠀⠀⣠⡀⠀⠀⢈⡠⢬⣀⠒⢀⠀⠘⢄⡈⣏⣀⠜⠀⠘⠗⠀⢀⣦⠈⠙⠈⠘⠀⠀⠐⠓⠀⠠⢀⠁⢄⠁
⠀⠈⠋⢀⣀⣀⠀⠉⠀⣀⣀⡀⠀⢀⠀⠀⠀⠳⠃⠠⢰⠃⠀⠈⠑⢱⠠⠄⠀⠀⣨⣍⣀⣀⣀⣀⠀⠀⠀⡷⡄⢀⣄⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢹⡍⣝⠀⠀⢸⠉⡏⡅⢩⠉⢧⠀⠀⠀⣀⡀⠈⢦⠘⣶⠃⡼⠀⠀⡿⠋⠾⠧⠴⡶⠉⠋⠀⠖⠀⣇⣇⣸⣸⣀⣏⣇⣜⣳⣠⣳⣠⣄⠀⣀
⠀⠀⠀⡏⠀⢢⡇⠀⢸⢰⣷⣷⣾⡆⢸⠷⠀⢰⢩⠱⠀⠀⣧⣿⡼⠀⠀⠀⡇⠀⠀⠄⠀⡇⠀⢠⠀⠀⠀⡏⡇⡍⢸⠈⠇⢇⡇⠘⠞⠈⠋⠉⢩⡉
⠀⠀⠀⣇⡀⣘⡇⢰⣺⠷⠶⠶⠶⠶⢟⣢⢰⡟⠒⠚⢳⠀⠻⡿⠃⠀⠀⢀⣧⣔⣈⣀⣩⣧⠽⠛⠀⠀⠀⠇⠑⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣁⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⡻⠿⢛⠿⢿⠿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢄⡀⠀⡘⠀⠀⠀⢡⢀⡴⣤⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢑⠧⡀⠀⢀⠨⡎⠛⠋⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⠇⠀⠀⢠⠀⣈⣖⣅⠀⣁⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠙⢌⠉⣹⠈⢁⢤⣀⠑⢏⠉⢉⠛⠐⣬⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢈⠶⣹⠀⣧⣤⣿⡇⢈⠵⡁⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⡤⢃⣀⣸⢢⡈⠛⢉⡀⣋⣀⡈⢦⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠘⠀⢈⠶⡉⠀⡁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢤⡤⡆⠁⠀⠀⠑⠧⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠙⠛⠁⢰⠀⠀⠐⣬⠇⠀⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠔
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢰⣛⣓⠀⢀⣤⣤⠀⢀⣠⣀⣄⠀⣠⡄⣨⡃⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣄⣀⠀⠀⢸⣭⣭⠀⠐⠒⠒⠀⢸⠀⡯⠸⠟⢧⡄⠀⠀⢀⣀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣻⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⡁⣃⣡⡀⢀⠀⠀⠀⢠⠰⠀⡆⢢⢀⠦⡄⠀⠀⠐⠁⡆⠀⠀⢸⠂⠪⠀⠀⠢⠇⠀⠀⠔⠀⠀⣀
⠀⠀⢃⡀⢂⠄⠀⠠⡠⠀⠀⠀⢸⠀⡇⠁⢸⠈⠀⠀⠀⠀⠈⠀⡇⠀⠀⢸⠁⢱⠀⠀⠀⠌⠀⠈⢄⠀⡨⠂
⠀⠀⠌⠘⠆⣒⠠⠋⠠⠀⠀⠀⢸⠀⠇⠀⢸⢀⠀⠀⠀⠀⢠⡀⠡⠄⠀⠞⢀⡘⠀⠀⠀⢨⠂⢠⠀⠀⠊
⠀⠀⠈⠀⠁⢀⠘⠉⠁⠀⠀⠀⣜⣀⡀⣀⣀⣀⡀⡢⠀⠀⠀⠑⠂⠤⡤⠰⠋⠀⠀⠀⠄⠇⠀⠀⠁⠠⠈

⠀⠀⡍⠉⠉⠉⠉⠉⠉⣕⠀⠀⠀⠀⠀⠂⠀⠀⠀⡦⠀⠀⢸⢉⠍⠉⠉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠴⡗
⠀⠀⡅⢐⠲⡪⡰⢔⠄⣴⠀⠀⠀⠀⠔⠁⠉⠠⠀⠀⠀⠀⠘⣦⠀⠀⡖⠀⢛⡙⠀⠀⠀⠀⢀⡘⣾⡧
⠀⠀⠂⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠃⡀⣀⠀⠀⠀⠀⠀⠈⡐⠂⠒⡂⠢⠀⠀⠀⠀⠀⠰⠘⢛⣀⣄
⠀⠀⠁⠀⠐⠂⠰⢀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠁⠀⠀⠀⣬⣬⣦⣧⣆⡇⠉⠀⠀⠀⠀⢠⡤⠜⣑⡑⡄

⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⢢⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠀⠀⠀⡦⠤⠤⠤⠠⠠⠠⠬
⠀⠀⠀⢀⠔⢂⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠘⠀⠀⠀⡇⡔⠃⠔⠁⢸⢹
⠀⠀⠀⠀⠀⠈⢁⠂⠀⠃⠀⠀⠈⡄⠀⡍⠀⠀⠀⠀⠀⠀⠀⡤⠀⠄⠀⠀⠀⢂⠀⠀⡧⣦⣦⣧⣦⡷⠶⠷⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠉⠉⠀⠀⠀⠀⠄⠀⡣⠀⠀⠈⠁⠀⠀⣇⠿⠿⠷⠶⠒⠂⢀⠇

⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠦⡼⡴⢆⡀⠀⠀⡆⠀⢀⡀
⠀⠀⠀⠠⠀⠀⡀⠑⠈⠀⠀⠀⠀⠂⢣⢀⠀⠀⠀⠀⠀⠀⢩⡆⠀⠀⢀⠀⢰⣭⠀⠀⡇⢀⠈⠐⡲
⠀⠀⠀⠐⠀⣸⣅⡠⢀⠀⠀⠀⠑⠅⠘⠀⠐⠲⡶⠂⠀⠀⢒⡏⠀⠂⠐⠀⢸⡖⠀⠀⡇⠁⠀⠀⠈⣀⠐
⠀⠀⠀⠀⠘⣗⡼⠀⠀⠀⠀⠀⠀⠉⠉⡀⠀⡠⢤⠀⠀⠀⠉⠳⡷⡶⢾⢾⡞⠉⠀⠀⡇⠀⠀⠀⠀⠐

⠀⠀⣠⠗⠛⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠓⠒⠂⠀⠀⠀⠀⠀⣠⠔⠒⠒⢤⡀
⠀⠀⠽⣄⣀⡼⣇⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠀⠀⠀⢰⠱⡿⢆⠼⡷⢱
⠀⠀⠀⡼⠳⡵⣇⢀⣽⠂⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⠶⠶⠂⠀⠀⠀⠀⠘⣄⢀⣾⣆⢀⡜
⠀⠀⠀⠳⠶⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⣀⠀⣀⠀⢾⣿⡷⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⡮⠁
⠀⠀⣾⠃⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⣤⣤⣿⡏⢠⡄⠀⠀⠀⠀⣰⣿⠋⢀⣴⡮⠁
⠀⠀⠡⢈⠛⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠇⠉⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣷⡆⠀⠀⠀⠀⠹⠟⠒⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⠀⠀⢀⣀⣠⣴⡄⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠖⠛⠻⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⣀⢬⣓⠮⠑
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⡁⢈⡀⠀⠀⠀⠀⠀⢾⣮⣟⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠃⠈⠃⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣶⣴⠀⠀⠀⠀⠀⠿⠁⢀⠈⠑⡀⠀⠀⠀⠀⠀⠠⠀⣌⡠⠆⠀⠀⠀⠀⠀⢠⡀⢠⣀⠆
⠀⠀⠀⠀⣠⡾⢋⣏⢀⠀⠀⠀⠀⠐⠀⠸⣯⠇⠀⡀⠀⠀⠀⠀⠰⠂⡹⣿⡿⡁⠦⠀⠀⠀⠀⢸⣿⣿⣿⣇⡄
⠀⠀⠠⠾⠯⠤⠴⠿⠀⠀⠀⠀⠀⠀⠁⠄⣀⡀⠔⠁⠀⠀⠀⠀⠀⠰⠀⢏⠁⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠷⠆

⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡠
⠀⢀⡼⠏⠀⠀⠀⠁⠀⢈⠈⠁⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⡀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠐⠈⠀⢀⠀⡀⠀⠰⠀⠀⠀⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠋⠀⠀⠂⠀⠒⠒⠤⠞⡡⠂⠀⠀⠀⠀⠀⡀⠀⢀⣠⡔⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠲⠴⠠⠀⠀⠈⠀⠊⠀⠀⠀⢠⡆⠀⠀⠀⠁⠈⠁
⠀⠰⡤⠀⣀⣀⣀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁⠈
⠀⠀⠑⢊⠉⢁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⣰⠃⠀⠀⣆⠀⡀⣰⢒⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈⠀⠀⠒⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠛⠀⣈⠐⠻⠄⠁⠀⢻⠃⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠄
⠀⠀⠆⠐⣤⡀⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠈⠀⠀⠦⠒⡠⠊⠀⠀⠀⠀⠄⠤⠀⣀⢀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠌⠠⡀⠀⠀⠲⠂⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢙
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⠀⠀⠠⣈⠋⠀⠀⠀⠀⠀⣀⡤⠀⠀⡀⠀⠀⠀⠀⠀⠌
⠀⠸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠀⠈⠀⠀⠸⢻⠂⠀⠀⠂⠀⠄⠀⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣧⣤⣝⣶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠛⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠀⣰⣁
⢀⡜⠁⢰⣿⣞⠈⢦⡀⠀⠀⠀⠀⢐⣶⡿⠁⠀⠀⠀⠀⠈⠉⠙⢿⣷⣼⣿⡆⠀⠀⠀⠀⠀⠀⡼⡿⢿⠃⠀⠆
⠮⢴⣎⣀⣉⣀⣱⣆⣳⠀⠀⠀⢦⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠦⠄
⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⠀⠀⣄⠀⠀⠀⡠⠋⢹⣿⡟⠑⢄⠀⠀⠀⠀⠀⢀
⠘⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠁⠀⠀⡇⠀⡱⠀⠀⣀⠀⠀⠀⢁⣤⣄⣽⣡⣤⣌⡀⠀⠀⠀⠀⠸⣷⠆
⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠈⠑⠁⠈⠲⠀⠀⠁⠉⠀⠀⠀⠺⡿⠋⠀⠙⠿⡟⠀⠀⠀⠀⢀⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠊⠁⠀⠀⠀⠀⠸⠿⣿⡿⠗
⠀⠀⠀⠀⡄
⠀⠀⣶⠀⡄⠀⠔⡀⠀⠀⠀⢀⣘⠁⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣷⠀⡇⠀⠰⠇⠀⠀⠀⠘⠛⢄⣀⠶⠀⡶⠀⡀⠀⠀⠀⠀⠰⠿⠿⢆⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣶⠦
⠀⠐⠛⠂⡃⠀⢰⡆⠀⠀⠀⢀⣤⠜⢻⠂⠘⠛⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣰⣄⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⢀⡘⠗⠀⠀⠁⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⡆
⠉⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢶⣤⣴⣧⣄⠀⠀⠀⠀⠀⠀⢀⠉⢁⡘⠙⣀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣆⠀⠀⠀⢀⢞⡟⠉⠉⢙⠲
⠀⢰⣏⣠⣟⣀⡀⠀⠀⠀⠀⠸⠉⠉⠉⠈⠉⠁⠘⡃⠀⠀⠀⠠⣾⣷⡇⠀⣸⣿⠀⠀⠀⣤⠞⠳⠄⠰⠚⠳⡇
⠀⠀⢉⣿⡋⢉⢹⠆⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠂⠂⢿⣿⣿⠿⠃⠀⠀⠀⠠⡀⡀⢸⡀⣠⠀⠄
⠀⠀⠀⠙⠛⢻⡟⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⠀⠀⠀⠀⠈⠛⠋⠉⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⡿⠿⢿⠛⠿⣿⣿⣷⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⢿⣃⡾⠀⡐⠌⠐⠋⠈⠀⠙⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⢛⡛⣉⣉⣁⣄⣀⣤⣀⣀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣬⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⠿⠻⠛⠛⠛⠃⠒⠟⠛⠹⠟⠟⠻⠿⠟⠻⠟⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠹⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⣤⣤⣤⣄⠀⠀⠀⠀⢀⢛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠝⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⣿⣿⣿⣿⣷⣾⣿⣿⣧⡄⠀⠀⡤⢈⢩⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣉⠛⠂⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⢎⣉⣉⡿⠿⠷⠾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣘⣛⣿⣷⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡠⡀⣀⢀⣤⢀⣄⡄⣠⠾⠆⠤⠐⠐⠒⠊⠁⠆⠀⠀⠀⠉⠛⡿⠉⠋⢭⣿⣿⣧⣍⢋⡉⠹⣿⣿⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⠟⠙⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠈⠙⠧⡄⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣾⣿⣿⣭⣤⣴⣤⣶⣶⣶⣶⣶⣾⣷⣶⣪⣴⣶⣶⣆⠴⣦⡶⣦⣤⣄⣀⣐⣠⣤⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠋⣨⣴⣿⣿⣿⣿⣿⣿⠿⡛⠛⠙⠉⠋⠉⠉⠉⠉⠛⠉⠀⠁⠀⠘⠻⠻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣡⢀⣂⣤⣠⣤⣴⣶⣶⣶⣤⣤⣴⣤⣀⣀⠀⠀⠀⠀⣹⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣹⢻⣿⣿⣷⣾⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠏⢛⣿⣛⡛⠻⠙⢟⢿⣿⣿⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣽⣀⡆⠠⠦⡌⠠⡝⢛⡚⢢⣮⣾⣿⣿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣥⣶⣷⣿⣿⣷⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⣠⣴⠾⠛⠛⠛⠛⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠛⠷⢶⣄⡀
⠀⠀⢀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠙⠷⣦⣤⣤⣠⣤⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢀⣞⢿⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢾⣽⣶⣶⢳⣶⣿⣿⡇⠀⡀⠀⠀⠀⠀⢀⣤⢾⠉⠁⣠⣄⣨⢽⡶⢄⠀⠀⠀⠀⠸⣟⣩⠇⠀⠀⠀⠀⠓⠈⢸⣀⣀⣀⣀⣀⣴⣭⠥⠚
⠀⠀⠀⢰⠀⠀⠀⣷⣭⠉⠉⠀⣘⣿⡇⠀⠉⠀⠀⠀⢠⢾⡵⡏⠀⠸⣿⣿⣿⢿⣧⡈⢳⠀⣄⡀⠀⠈⠁⠀⠀⠀⠀⣠⠶⠛⠋⠿⣌⠀⠀⡤⠀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⡏⠙⠀⠀⠀⢸⣚⡇⠀⠀⠀⠀⠀⣞⣋⣤⡽⠀⠀⠿⢯⣔⣓⣠⠟⠀⡇⠈⠹⠀⠀⠀⢀⣀⡀⣀⠹⡀⠂⠈⠀⢈⠑⢄⠑⠀⠉⠰⠇
⠀⠀⠀⠀⠀⠀⡼⠉⢳⠀⠀⠀⠚⠃⣇⠀⠀⠀⠀⠀⢏⠁⢀⣴⡶⣤⡀⠀⠈⠉⢁⣴⣶⠇⠀⠀⠀⠀⠀⠘⠉⣁⣹⣅⣘⢦⡀⠀⡀⠠⡈⢧
⠀⠀⠀⠀⠀⢸⣡⣶⣷⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⡜⢆⠻⣅⣴⣆⡇⠀⣠⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⣀⠭⠛⢷⣤⣤⡤⠎⠲⠄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠂⠈⠓⢬⣻⡛⣃⣀⣿⡿⠟⠁⠀⠀⠀⠀⣀⣠⢠⠄⣷⣄⡀⠉⠄⠀⠨⠀⢱
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⢖⣛⡲⡄⠀⠀⣀⣀⠉⠉⠉⠀⢀⠔⢤⠀⠀⠀⠀⠉⠁⣨⠷⠛⡉⠍⡻⣍⠙⠛⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠻⣿⣛⣛⣻⣿⣿⣿⡿⠿⠋⣾⢼⠦⢇⡇⠀⠘⢥⡾⠃⠀⠀⠀⠈⠲⠛⠀⢠⡚⣦⠀⠊⠁⠂⠁⠀⠀⠀⢈⡇⠀⠀⠀⠟⠻⠰⠆⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⢀⣀⡀⠃⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡔⠉⠀⠀⣆⣀⡇⠀⣏⣉⠁⣰⡄⣴⡀⠀⡆⠀⢯⡉⠀⠉⣿⠉⢸⣠⠞⠀⠘⠞⠁⠀⠒⠒⢿⣷⣾⠆
⠀⠀⠀⠀⢠⣀⡄⢠⡀⣤⢀⠀⡀⢀⢉⠉⠀⠸⣀⢀⡀⠐⠇⠀⠸⠀⠻⠤⠀⠃⠛⠁⠃⠀⠇⠀⠰⠛⠀⠀⠙⠀⠘⠀⠉⠀⠀⠂⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠸⠁⠇⠹⠀⠛⠘⠀⠉⣿⠻⠸⡇⣀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠐⠃⢠⡟⠛⡇⠀⠀⢸⠿⢷⠀⠃⠀⢸⠛⠛⡇⠙⠀⠀⠀⠀⠴⠀⣶⢻⡿⠿⠿⠟⠻⢷⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠁⠭⢉⠳⣴⡒⢲⠄⠀⠀⠀⢸⡇⠀⡃⠀⠀⢸⠀⢸⠀⠀⠀⢸⣤⣤⡇⢀⠀⠀⠀⠀⠀⠀⡟⠉⡇⠀⠀⠠⡣⢼
⠀⠀⠀⠀⠀⢊⠘⢀⡎⠀⠀⠀⠀⠀⠀⣡⣴⣸⡖⠋⠀⠀⠀⡇⢰⣟⣿⡇⠀⠀⢸⣀⣸⠀⠀⠀⢨⣿⣿⡇⢸⡀⠀⢠⣄⡄⠀⣧⣄⡗⠀⠀⢀⡧⢸
⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢀⠠⠄⣂⡥⢖⢛⠽⡇⠀⠀⠀⠛⡟⠛⣿⣿⡿⠛⠛⠛⣿⣿⠛⠛⠛⢻⣿⣿⡟⢻⠋⠀⠈⠿⠁⠀⣇⣤⣧⣤⣶⣾⣶⣾
⠀⠀⠀⠀⠀⣰⢯⢻⣄⡬⠴⠒⠉⠁⠀⢰⢎⡼⠁⠀⠀⠀⠀⡇⠀⣿⣷⣇⠀⠀⠀⣿⣿⠀⠀⣀⣨⣿⣿⣇⣼⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠉⠙⠦⣄⣀⣀⣀⣀⠴⠋⣠⣤⣤⡀⠀⠤⣿⠽⠿⠛⠛⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠀⠀⠙⠀⠀⣴⡄⠀⠀⠉⠛⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣉⣩⡀⠀⠀⣇⠻⣪⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣹⠃⠀⢀⣀⡋⠉⣟⡆⠀⠀⡄⢀⡤⠴⠧⣄⣀⣀
⠀⠀⠀⠀⢀⡐⠀⠀⣤⠤⠶⣿⣛⣛⡟⠉⣻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢹⠀⠀⠀⠈⢁⡴⠞⠋⠤⣹⢦⢸⠀⠀⠠⢰⠋⣀⠀⡈⠜⣿⠤
⠀⠀⠀⠀⠀⠀⣠⢴⠷⠾⢷⠖⢾⠉⠀⠄⢹⠀⠀⠀⣸⠀⡤⠖⢦⣀⣤⣤⣴⡁⠀⣹⠃⠀⠋⠊⡼⠛⢀⡴⠚⢹⠁⢊⡇⠀⠀⠤⢼⠀⣏⡉⣹⢆⣼⠧
⠀⠀⠀⠀⠀⠀⣧⠘⠒⠖⠋⠀⢸⠀⠄⠀⢸⠀⠀⠀⠘⠲⣇⠀⢠⠟⠉⠁⠀⠙⠋⠁⠀⠀⠀⡼⠁⡰⠃⠀⣀⡇⠠⣿⠀⠀⠀⠐⠘⣆⢹⢣⣽⣼⠟⠄
⠀⠀⠀⠀⠀⠀⡗⠐⠂⠀⠀⠀⢸⠀⠁⡂⢸⠀⠘⣤⡆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⢸⠃⠀⠀⢿⡇⢸⡟⡿⠁⠀⠀⠁⠀⢻⣇⣿⠉⠂
⠀⠀⠀⠀⢀⠀⡇⠀⠒⠀⢀⡀⢸⡀⠀⠤⠈⡇⠀⠙⠀⢀⣴⣾⠇⠀⣠⣶⡆⠀⣀⣀⠀⠀⢸⡇⠈⢳⠤⣀⣀⣹⣋⠉⠀⠀⠀⠀⠀⠀⣼⣼⣿
⠀⠀⠀⠀⠚⠁⡇⠀⠀⠀⠀⠀⠸⡧⠴⠖⠚⠁⠀⢀⣤⣞⠛⠛⠒⠲⣿⡿⠤⠿⠾⠼⠁⠀⠘⣧⠸⣯⡇⠀⠀⠠⣭⢉⡇⠀⠀⠀⡀⣀⣿⣿⣿⠀⠄⠄
⠀⠀⠀⠀⠀⠀⣣⡤⠖⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠁⠹⣿⣷⡀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠷⢶⣦⣤⣤⣤⣼⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠁

⠀⠀⠀⢀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡴⣔⡁⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⢃⠀⠀⠄⠀⢀⠠⠀⠀⠀⠀⠀⠀⠔⠖⠚⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⡀⠂⠃⠀⠀⠁⢢⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢐
⢀⠑⠚⠰⠦⠖⠁⠲⡤⢮⡆⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢀⠰⣠⠃
⠘⠈⠰⠐⠠⢀⠠⢶⠄⠄⠋⠀⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡉⠰⠂⠂⠈⠒⠀⡈
⠸⡷⠄⠄⠀⠀⢲⣦⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠀⡄⠊⠠⠔⠁
⠀⠘⢤⡠⠔⠂⠀⢤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡄⠒⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠖⠀⠀⠀⠂⡀⢀⠱⡄⠒⠄⠤⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠁⠀⡠⠏⠀⠒⠀⠈⡠⣜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠔⠀⠡
⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠁⣐⢠⠀⠠⣅⣉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⢀⠀⠀⠄⠀⠀⠀⠉⠉⠉⠀⠃⠀⠀⠀⠀⠉⠔⠆⠀⠘⠄⠀⠈⣁⠋⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠔⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⢋⠁⣤⣤⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡖⢋⡧⢸⠁⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⢈⡠⢋⠝⢦⡀
⠀⠀⠀⠀⠀⢰⠏⢂⡈⠀⢨⠁⡀⠘⠃⡈⣧⠀⠀⠀⠀⢰⠇⠀⠂⠠⣄⠉⡉⢉⣉⡹⡆⠀⠀⠀⠀⡼⢃⠄⠂⡀⠤⠤⠊⡠⠊⢹⡄
⠀⠀⠀⠀⠀⢾⠀⢅⡨⠀⢀⠤⡄⠻⠃⠀⢸⠀⠀⠀⠀⣿⠤⠤⠄⠒⠃⠘⠋⠘⠒⠂⣻⠀⠀⠀⠀⣟⣁⠄⢊⡀⠄⢂⡡⠔⢊⢵⡇
⠀⠀⠀⠀⠀⠸⡆⡀⡄⠀⠘⠶⠋⠀⠤⢀⡞⠀⠀⠀⠀⠸⡦⠴⠀⠚⠀⠫⠭⠥⠤⢠⡏⠀⠀⠀⠀⢳⡦⢊⠅⠂⠉⠐⠒⡈⠐⣹⠇
⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠁⠀⠤⢀⣠⠞⠀⠀⠀⠀⠀⠀⠙⢮⣭⠭⠥⠤⠶⢐⡶⠋⠀⠀⠀⠀⠀⠀⠻⢵⣨⠔⡂⠉⠍⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢞⡉⠒⠀⠤⠤⠬⠓⢦⠀⠰⠁⠄⡐⢉⣻⣟⣉⠁⠀⠂⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⡴⠋⣡⠀⠄⣒⡢⠉⠳⣄
⠀⠀⠀⠀⠀⢰⡇⠀⠉⠒⠤⢀⣀⣐⠍⣈⣧⠈⢄⠡⠑⢄⠑⢬⣉⣉⣉⣉⣉⣉⠉⠼⣇⠀⠀⠀⠀⡼⢱⠄⠀⢀⠂⠁⠑⠈⠛⠹⡆
⠀⠀⠀⠀⠀⢸⠉⣑⡒⠀⠤⠤⠤⠤⠤⠀⢾⠀⠀⠑⢄⠙⠟⠢⣄⠀⠲⠄⡀⠰⠐⠒⣿⣷⣄⠀⠀⡇⠀⠠⡠⠌⠀⠀⠂⠘⢃⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠓⡁⠄⠀⠀⠀⢀⡀⢰⡏⠀⠀⠀⠀⠹⡶⢌⡁⠸⠃⡀⠠⠉⡉⠀⠛⢀⠙⠱⡄⢱⡀⡑⠉⠂⠀⠰⣂⢠⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠐⠀⠀⠐⣢⡵⠋⠀⠀⠀⠀⠀⠀⠈⠶⣌⡳⠶⣤⣁⣀⠀⠛⠁⠀⠈⢀⠄⠀⠙⢦⣀⠐⠀⠈⢀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠊⠀⠈⠉⡝⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⡰⠛⠡⠡⠈⢈⡤⠸⢛⢦⠀⠀⠀⠀⠀⢀⡴⠯⠥⠬⠥⠤⠔⢋⠷⣄
⠀⠀⠀⠀⠀⢰⢣⠀⢤⠐⠁⠀⢀⡬⠛⠼⣷⠀⠀⠀⠀⢸⠃⡀⠰⡅⠴⢀⣀⢣⠱⠬⣇⠀⠀⠀⠀⡾⠬⣉⣑⠒⠀⠒⢊⣁⣠⢼⡆
⠀⠀⠀⠀⠀⢸⠀⠁⢰⠉⠉⠲⡌⠒⢦⠀⢸⠀⠀⠀⠀⢟⠢⡍⠀⡊⠉⠁⢎⠘⠲⢒⡿⠀⠀⠀⠀⡯⢑⣒⠂⢀⣩⡽⠴⢒⡪⠭⡟
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢡⠀⡰⠁⠀⠘⣠⠏⠀⠀⠀⠀⠸⣆⠃⠈⢀⠆⢀⠀⠉⠉⣴⠃⠀⠀⠀⠀⠹⣆⢀⠀⠉⠁⣒⡒⠉⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣘⡘⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠣⣤⣀⡠⣈⣳⡽⠞⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⣠⡬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠢⢤⣀⢀⣠⢤⣄⡠⢄⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠚⠉⠀⠀⠀⠘⠃⠀⠀⠈⠀⠈⠁⠀⠘⢇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠑⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠓⠲
⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠳⢄
⠀⠀⠀⠀⠀⠀⢀⢞⡀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠰⠂⠀⡀⠀⠚⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡀⠀⠀⠀⠀⠾⠟⢿⡄⠀⠀⠀⠀⠀⠀⣿⣷⣶⡔⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⢤⡀⠀⠂⡀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠠⢀⡴⠦⠦⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠋⠣⠴⠴⢲⡄⠀⠀⠀⠀⡰⠤⠴⠊⠉⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⠴⢰⡇⠀⠀⠀⢠⡧⠤⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣁⠀⠀⠐⠈⠷⠸⡤⠠⡸⠇⠠⠀⡀⠒⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣀⠤⣤⠶⣄⠀⡀⣠⠦⠶⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣘⠀⠀⠀⣿⠤⣀⣀⠴⠋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠁⣉⠺⠅⠈⣠⣀⡤⡤⢧⠄⠍⠑⠀⠀⢀⠛⠖⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠂⠀⠠⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⢀⡳
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠒⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠒⠒⠒⠃
⢀⣀⡀⡀⠀⡀⡀⠀⠀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣒⣚⣉⣉⣉⣍⣉⣉⣙⣒⣒⣶⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⢞⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣷⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠓⠶⠤⠭⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠤⠴⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⡶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⣴⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡀⢠⠀⠀⠠⠄⠠⠀⠄⠠⠄⠠⠀⣉⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡾⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠑⠠⢤⣤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠤⠄⠀⠀⠈⢧⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⠀⢯⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡜⢦
⠀⠀⠀⠀⠀⠀⣠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢧⡀
⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⡀
⠀⠀⠀⠀⡼⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡄
⠀⠀⢀⠞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⣄
⠀⢠⢎⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣆
⠠⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠃
⣦⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠
⢯⠳⢭⣓⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⣋⡵⢫
⠀⠑⠦⣄⣉⠙⠒⠲⠮⠭⢭⣥⣒⣒⣒⣒⣒⣢⣤⠤⠤⠤⠤⠤⠤⠤⠒⠒⣒⣒⣒⣒⣒⣋⣩⣭⡭⠤⠶⠒⠚⣉⣠⠴⠋
⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⠤⣤⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣀⣠⡤⠤⠤⠶⠒⠚⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠙⠂⠀⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠀⠁⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠘⠳⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠰⠀⠉⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⣠⠀⠈⠏⠀⠀⠀⠀⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠄⠀⠄⠒⢁⠀⠀⠀⠐⠊⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠐⠔⠀⠤⠋⠀⠀⠊⠙⡇⠀⠐⠰⣦⠸⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡠⡤⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢎⠀⠀⠀⠠⡄⣀⠀⠈⠃⠀⠜⠒⠃⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠱⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣈⣁⠙⠃⠀⠀⠀⡠⠀⠀⠀⢤⡆⠀⠀⠀⡀⠒⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘
⠀⠀⠀⠀⠀⠘⣦⠀⠐⠁⠈⢧⠀⠀⠀⢰⠇⠀⠀⠈⠁⢿⡄⠀⠀⠀⣀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⢸⡇⠀⢻⣀⡀⠀⠀⢀⡴⢚⠁⠀⠔⠋⠀⠈⠉⠻⣻⣄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠈⡑⠂⡊⠀⠢⠟⠫⠉⠁⠊⠐⠻⠁⠀⠀⠀⠀⠀⠁⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⢀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀⠆⠀⠀⠉⠛⠃⠠⣤⡐⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⣁⠀⠈⣀⠀⠠⣀⠈⠂⠀⠀⠀⢀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢸⡉⠙⢿⢥⡀⠀⢀⣀⣀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠀⠀⢙⠟⠀⠐⠠⠉⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠈⠉

⠀⠀⠘⠁⠀⢈⠂⠀⠀⠀⠀⠺⣿⣿⣿⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠿⠿⠇⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠣⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣷⡄
⠀⠀⠀⣀⣀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⣧⣦⣤⣤⡄⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⠀⠀⠈⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡟⣾⣿⡇⠀⠠⣶⣶⣶⣶⣶⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣠⣿⣿⣿⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠿⣿⣿⣧⣿⡿⠃⠀⠀⠙⠿⢿⠿⠟⠁⠀⠸⢿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⢤⣤⣤⠀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠈⠉⢱⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣀⣸⣄⣼⣀⣸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⢰⣤⣼⠀⢰⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢙⣉⣻⠋⢻⣉⣻⠋⡟⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⠿⢿⣿⣿⠿⢿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡂⢸⣿⣿⠀⢸⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣶⣾⣿⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⣶⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀
⠀⠐⠀⠀⠑⢄⠀⠀⠁⠈⠢⠀⠀⢠⢎⡵⠢⣝⠶⣫⠕⠮⡱⡄⠀⠀⠠⠈⠉⠐⢄⡠⠊⠉⠑⠄⠀⠀⣴⠋⠀⠈⠳⡴⠋⠁⠈⠹⡄
⠀⠢⡀⠀⠀⠀⠑⢄⠀⠀⡠⠀⠀⠘⣜⠦⠤⢊⠤⡙⠤⠴⣣⠇⠀⠀⠡⡀⠀⡠⠊⠡⡀⠀⣀⠆⠀⠀⢳⡀⠀⠀⣠⠶⣀⠀⠀⣠⠇
⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠈⠙⠋⠁
⠀⠠⠢⠤⠔⣄⠀⡐⠤⠰⠀⠀⠀⢠⡶⠛⠲⣄⠀⣠⠖⠛⢶⡄⠀⠀⣴⡶⠶⣶⣄⢠⣶⠶⢶⣄⠀⠀⢀⣴⠾⢶⣄⠀⣴⠾⠶⣦
⠀⠈⠀⠀⠀⠈⠑⠀⠀⠀⢌⠂⠀⢿⡀⠀⠀⣨⠿⣅⠀⠀⢀⡿⠀⠘⣿⡀⠀⣨⡻⣷⡀⠀⢨⣿⠀⠀⢺⣇⠀⢀⣝⢷⣄⠀⠀⣽⠇
⠀⠈⠒⠒⠁⠁⠈⠀⠒⠓⠃⠀⠀⠈⠛⠶⠚⠁⠀⠈⠓⠖⠛⠁⠀⠀⠙⠛⠿⠛⠁⠈⠛⠿⠛⠁⠀⠀⠀⠙⠛⠛⠁⠀⠙⠛⠛⠋
⠀⠀⣀⠤⢄⡀⠀⣀⠤⢄⡀⠀⠀⠀⢀⣠⣀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀
⠀⡎⡠⠒⢢⡙⢎⡰⠒⢦⠱⡀⠀⣼⠏⠉⠙⢿⣤⡾⠋⠉⠹⣧⠀⠀⣾⠋⠉⠻⣦⡐⠋⠉⠙⣦⠀⠀⠐⡇⠀⠁⢕⢦⡓⠀⠁⢨⡀
⠀⢣⡘⠦⠚⡹⢦⡑⠦⠊⡰⠁⠀⠹⣦⣀⣠⡾⠛⢷⣄⣀⣴⠏⠀⠀⠻⣄⣀⡤⠊⠻⣦⣀⣴⠏⠀⠀⠐⣅⠀⠀⡩⠓⢤⠀⠀⡮
⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⣠⡶⠶⣦⣄⢀⣴⠶⢶⣄⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠂⡠⠀⠀⠀⣠⡶⠶⣦⡀⣠⣴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠠
⠀⣿⡀⠀⢈⣿⣿⡁⠀⠀⣿⠆⠀⠠⠀⠀⠀⠰⡈⠇⠀⠀⠰⡀⠀⠀⣿⡀⠀⢈⣿⣿⡁⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠁
⠀⠘⠻⠶⠟⠁⠈⠛⠷⠿⠋⠀⠀⠀⠀⠂⠀⠊⠀⠐⠂⠐⠀⠀⠀⠀⠙⠻⠶⠛⠁⠈⠛⠶⠟⠁⠀⠀⠀⠀⠄⠄⠊⠀⠀⠤
⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠤⣄⠀⣠⠤⠤⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢀⣠⣄⡀
⠀⠊⠔⠀⠂⡑⢂⠔⠀⠀⠀⡀⠀⢰⢁⡖⠒⣄⠹⣤⠒⠲⡈⠇⠀⠀⣾⡿⠛⢻⣷⣾⠟⠛⢿⣧⠀⠀⣼⡟⠋⠛⣟⢶⡿⠋⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠢⡑⠀⠐⡠⠀⠀⠘⢄⡑⠚⣡⠷⣌⠓⠊⡠⠃⠀⠀⠻⣷⣤⣾⠟⢿⣷⣴⣿⠏⠀⠀⠻⣷⣤⣴⡿⠻⣷⣤⣴⣻⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⡀⠀⠀⠀⠀⠠⡆⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁
⠀⠀⠀⠀⠐⠁⣠⠀⠠⡌⠁⠀⠀⠀⠀⢀⣦⠈⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡂⠉⢂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠐⡐⢀⣀⡀
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠸⠅⠀⢨⡀⠀⢠⠀⠀⠀⠀⠈⠉⡅⠒⢦
⠀⠀⢀⡄⠴⠥⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⠀⠀⠘⠀⠀⠸⢧⡆
⠀⠀⠂⢵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⢸⡀⠀⠀⠀⣀⡴⠁⠀⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠉⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠨⠟⠀⠀⠈⠋⠀⠀⠀⠀⠴⠀⠀⠀⠀⠰⠀⢡⠀⠀⠀⠀⠀⠈⠁⠃⠙⠿⠶⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠰⢸⠀⠁⠀⠀⠰⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣤⣤⡤⠀⠀⠀⠀⢰⠁⣇⠀⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢧⠀⠀⠀⠀⠀⠁⠁⣆⡆⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠲⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣥⣉⠉⣀⢀⡀⣘⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⡿⡻⠿⣛⣛⣭⣾⣷⠘⣿⡟⣶⢻⡿⢹⡿⡃⣼⣿⣿⣷⣶⡹⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢩⣵⣾⣿⣿⣿⣿⣿⣿⣿⡼⡼⢧⣿⡜⢣⠸⣱⢧⣿⣿⣿⣿⣿⣿⣶⣶⡝⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⡇⣿⣾⣿⣷⣿⣼⡿⣼⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⠟⠛⠟⢸⣿⣿⣿⣿⣿⡇⠟⠛⠛⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢸⣿⣿⣿⣿⣿⠁⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⡁⠸⠿⠿⠿⠿⣿⠀⣠⣼⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢰⣯⣷⣶⣶⡖⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣾⣿⡟⣼⣿⣿⣿⣿⣿⡘⠻⣿⡶⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣫⣟⣽⣶⣯⢉⣾⢿⣿⣿⣿⣿⣿⣿⣔⢶⣾⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⡿⣛⣯⣥⣏⣁⣸⣭⣟⡛⠘⣣⣝⡈⠓⣙⣛⡿⢿⣿⣿⣷⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⢟⣫⣭⣭⣛⣫⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣝⣛⡿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠏⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠻⠇



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

⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠔⠉⠉⠙⣻⣶⡄⠀⠀⠀⠀⠀⠀⠘⣏⣿⣦⡀⠀⠀⠀⣠⢾⣭⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠠⢼⣿⣷⣄
⠀⠀⡖⠀⠀⠠⠀⠐⣽⣿⡀⠀⠀⠀⠀⠐⡀⡦⠀⣿⣷⠀⠀⢠⡁⠀⠀⠈⢲⣾⣿⡇⠀⠀⠀⠀⡀⢠⣀⣈⣽⣿⡆
⠀⠀⠁⠀⠃⠀⠉⠦⣹⣿⠀⠀⠀⠀⠀⠀⢁⣱⣮⣿⡿⠀⠀⠀⠻⣿⡆⢠⣿⣿⣿⡇⠀⠀⠀⠈⡩⢭⣹⣯⣽⣿⠃
⠀⠀⠀⠣⣀⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⠐⠈⣿⣿⠟⠁⠀⠀⠀⠀⠹⣧⣾⣿⡿⠋⠀⠀⠀⠀⠀⠂⣀⣩⣵⠿⠃
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠘⠛⣲⣦⡀⠀⠀⠀⠀⠰⠛⠹⣿⢶⣶⠻⣿⠀⠀⠀⠀⠘⠀⠘⢢⣅⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠠⣤⣤⡤⣤⡼⣷⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡄
⠀⠀⠀⠠⢶⣶⡴⢶⣶⣿⠁⠀⠀⠀⠀⠀⠈⡵⢟⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⣤⣬⡽⠋⠀⢀⠀⣀⠀⣀⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠈⢉⣻⡿⠏
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⣴⠋⠉⠉⠙⠿⣦⡄⣠⣴⠿⠋⠉⠉⠙⣦⠀⠀⠀⣿⣿⣿⠋⣴⣶⣶⣦⣄⠙⢿⡿⠋⣠⣴⣶⣶⣦⠙⣿⣿
⠀⠀⠰⣿⠀⠀⠀⠀⠀⢈⣽⣟⡁⠀⠀⠀⠀⠀⣿⠆⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⡷⠋⣠⢾⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠻⣄⣀⣀⣠⣶⠟⠋⠘⠻⣶⣄⣀⣀⣠⠟⠀⠀⠀⣿⣿⣿⣄⠻⠿⠿⠟⠋⣠⣾⣷⣄⠙⠻⠿⠿⠟⣠⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢈⣉⡱⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⡇⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⢃⠀⠀⠀⡠⠐⠉⠀⡀⠀⠀⢤⠀⠉⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢆⡔⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢀⠀⠠⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⢈⠀⠀⠘⡄⠁⠀⠒⠀⠀⠀⠀⠀⠆⠈⡂⡽⠒⢄⠐⠕⡠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡸⢃⡀⠅⠹⣄⡔⣉⣂⣄⣢⣠⣠⣄⣱⣤⣇⣁⣀⢑⢿⣋⣡⣴
⠀⠀⠀⠀⠀⠀⡐⠠⠊⠬⠾⠤⠲⢵⢬⣔⠷⢦⢹⡉⠑⢁⡬⠆⠄⠀⠀⠸⡬⠐⠒⢔⠀⠀⣊⡴⠇⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⢀⡴⢻⠜⠣⡪⡧⢯⡁⢕⣙⠒⠲⣶⠀⢐⣌⡪⣐⣗⡡⠪⠂⢕⢇⠁
⠀⠀⠀⠀⡰⠀⠀⠀⠈⠂⠊⡤⣨⡂⢕⣕⣄⡪⢐⣎⡗⠻⣕⠀⠀⣀⠀⡋⠀⠉⠖⡡⢈⢔⢕⡽⠸
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢍⠪⣺⣇⢠⡿⡊⣽⡿⣤⣷⡑⠉⢊⠤⡱⢂⢖⠭⡪⡱⢗⠵⡡⠁
⠀⠀⢠⠁⠀⠀⠀⠀⢅⠌⠀⠸⣋⣌⠞⡊⣕⠄⡪⢐⢡⣱⣹⡶⡋⢕⠅⡪⢚⢕⠈⣊⡔⢁⠔⠁
⠀⠀⠈⢔⠄⡀⠀⠀⠀⠀⠀⢨⠢⠐⠍⡪⠐⢅⢨⣲⡋⢇⡗⣬⣪⣕⡥⢪⠄⠵⠊⣉⠔⠁
⠀⠀⠀⠀⠈⠉⠀⠘⠁⠰⠶⠷⠾⠶⠿⠾⠿⠥⠉⠐⠊⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠠⡀⢀⠄⡀⢄⢠⠠⢀⠀⡀⡀⠄⠠⡀⢀⢀⠀⠀⠄⡀⠄⠀⡀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠐⠁⠊⠀⠑⠁⠊⠂⠅⠊⠐⠀⠐⠑⠨⠀⠅⠊⠔⡩⠂⠕⠨⢂⠕⠪⠐⠡⡪⠠⠃⠔⠔

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⢤⡀⠀⠀⠀⠄⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣽⣿⠂⠀⠀⠀⣰⡄⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⡤⠀⢀⢠⡜
⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠘⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠃⠓⠋⠁
⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⣀⢘⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠒⠲⠖⠊
⠀⠀⠼⣿⠷⠀⠀⠀⠀⠀⢨⡀⢰⠹⡄⠀⠀⠀⠃⢐⡄⠀⠀⢤⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢻⣖⠚⠖⠓⠉⠀⠀⠀⠸⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠈⠁⠄⠀⠀⢀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢽⡿⠄
⠀⠨⠄⠀⠀⠀⠀⠆⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⢤⠀⠁⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄⡐⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⢹⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⢀⣄⠀⠀⣿⡟⠡⣾⣿⣷
⠀⠀⠀⣼⣿⣅⠀⢀⠄⠀⠀⠀⠀⣀⡖⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠈⠙⠁⠀⠱⣦⣄⠀⡉⠁
⠀⠀⡜⠘⠁⠀⢰⠁⠐⣄⣀⡤⠚⡡⠀⠀⣤⡤⠀⠀⠀⠀⠀⡄⠀⢸⠉⠐⢢⣤⡀⠀⠀⠈⠀⠀⡀
⠀⠀⡂⠀⢂⠀⠈⠓⠒⠋⠉⠈⠁⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⢸⣠⠖⡋⢀⠙⢄⠀⠀⠀⠿⠃
⠀⠀⠠⠀⡂⠀⠀⠀⠀⠀⡔⠉⠈⡁⣐⡀⠀⠀⠀⠈⠀⠀⠀⣠⣀⠈⠣⡀⠀⠊⠀⡈⠢⣀⣀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⠀⡀⡀⠀⠀⠀⠰⠆⠀⠀⠛⠇⠀⠀⠙⢆⠈⠢⢃⣔⠁⠈⢢⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⡨⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠚⠹⡀⠙⣄⡠⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢶⣾⣄⠀⠀⢰⠉⠢⡀⢈⠦⠜⡆
⠀⠀⠀⠐⠦⣠⡠⠔⠁⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠑⠢⢬⣣

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⠄⠀⠈⠻⣿⣿⣿⣿⣿⣷⠿⠶⢀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡴⠚⠉⠀⠀⠀⠀⠉⠑⢦⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣈⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣤⠖⠋⠀⠙⡶⠤⠤⠤⠤⢶⠋⠀⠙⠲⣤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠖⠒⡇⠀⠀⠀⠀⢸⠒⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡇⠀⠀⠀⠀⢸⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⣠⠔⠒⡇⠀⠀⠀⠀⢸⠒⠲⣄⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣄⣈⡒⠦⠧⠤⠤⠤⠤⠼⠴⢒⣉⣠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡟⠉⠉⠉⠉⢻⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣇⠀⢀⠴⠒⠛⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣫⠄⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⢛⡀⠀⠀⠀⢀⠀⣑⠠⠵⠕⠃⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠁⠀⠀⠀⠀⠘⠘⢀⡀⠈⠁⠀⡄⠀⠈⠈⠀⠀⢸⠀⠀⠀⢠⢫
⠀⠀⠀⠀⠀⠀⠀⣰⣻⠀⡀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠿⠢⠀⠀⡀⠀⠀⠀⣀⢼⣆⠉⠁⢸⡑
⠀⠀⠀⠀⠀⠀⣰⠿⢹⠉⣃⡁⠅⢀⡀⠈⠠⢠⣤⠄⠈⠈⠀⡴⠇⠀⠰⠊⠀⢸⠀⠀⢒⡞
⠀⠀⠀⠀⠀⣰⡝⠃⠀⠀⠀⠁⠀⠀⠉⢣⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡴⠿⣶⣷⣴⣤⣤⡄⡆
⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠂⣖⠀⠐⠞⠁⠆⠀⠀⠀⠰⠀⠈⠃⠣⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠈⠁⠉⠀⠀⠘⠁⠉⠉⠉⢻⠆
⠀⠀⠠⠤⠜⠗⢠⣄⢠⣐⣛⠻⠅⠀⠀⢠⠀⠀⠀⠀⠠⠷⠆
⠀⠀⠀⠀⠈⠉⠛⠙⠻⠉⣹⣿⣯⢭⣿⠩⢥⡦⠄⢦⠽⠿⠶⠶⠮⠷⠖⠒⠺⠥⠟⠿⠷⠮⠥⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣠⣾⣷⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⢀⣀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣶⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⠿⣿⣿⣿⣿⡿⠋⠐⢦⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⡈⠻⢿⣿⣿⣿⣿⣿⠟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣄⣀⣤⡴⠖⣀⣤⡙⠳⠶⣤⣬⣉⣉⣉⣉⣉⣩⣤⣴⠶⠛⣩⣤⣀⠶⢬⣍⣉⣁⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠠⣶⣿⣿⣿⣿⡆⢰⠀⠀⠉⠉⢉⠉⠉⠉⠀⢠⡄⣾⣿⣿⣿⣿⡶⢨⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣙⡛⠿⠿⠋⣸⡇⠀⠀⠀⢸⡄⠀⡄⠀⢸⣇⠙⠛⠛⣛⣩⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠙⣇⠀⡄⠀⢸⡇⢠⡇⠀⢸⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣠⣿⣤⣿⣧⣼⣿⣄⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣛⣉⣉⣭⣭⣥⣤⣤⣤⣤⣤⣬⣭⣍⣉⣙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣶⠿⣿⣛⠛⣿⠙⣿⠉⢻⡿⢻⡿⠉⣿⢉⣻⡟⢻⣦⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢶⣤⣭⣹⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⣭⣥⣴⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⢸⡇⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⢸⡇⠀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⠋⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠙⢿⣿⣯⡆⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⢻⣿
⣿⡟⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣶⣢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿
⣿⣿⡀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠹⡿⣿⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⣸⣿
⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿
⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⣾⣄⣀⡀⠀⠀⢀⣀⣀⣀⣠⣿⣧⠀⠀⠀⠀⠠⠾⣿⠧
⠀⠀⠀⠘⢿⣿⣿⣏⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣤⡄⠀⠀⣺⣿⠆
⠀⠀⠀⠀⠘⠟⠉⠙⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠛⠻⣿⣿⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠋⠉⠁⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⣿⣦⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⣿⣶⣶⡤⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣾⣿⠃⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠛⠿⢿⣿⣿⣿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃
⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀
⠀⠹⢿⡏⠀⠀⠀⠨⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠶⣾⣿⣿⣿⡿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡀⣠⣾⣿⣷⠀⠀⠀⠀⠀⢸⣿⠿⣿⣇⠀⠀⠰⣶⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠾⠿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⠉⠁⠀⠀⠀⠀⠀⠀⠰⠿⠿⠛⠛⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⣦⣠⣤⣤⡄⠀⠀⠀⣤⣤⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣦⣶⣶⣶⠆
⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀
⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⢲⣶⣿⣤⡀
⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⡄⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⡀⠀⠸⠛⢿⠁
⠀⠀⠀⠀⡿⠋⠁⠀⠀⠀⠈⠀⢀⣴⡀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠘⠟⠃⠀⠀⠀⠸⡿⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣾⣿⣧⣀⡀⠀⠈⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣷⣶⡶⠀⠀⠼⣿⠇
⠀⠀⠀⠀⠰⣾⣿⡶⠄⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠛⠉⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡷⠆⠀⠀⠀⢀⣿⠟⠛⠻⠿⠀⠀⣠⣤⣿⣦⣀⡀
⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⢈⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠃

⠀⣀⡜⣆⣀⠀⠀⠀⠀⠀⢠⠧⠀⠀⠀⠀⠀⠀⠀⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⢠⡆
⠙⣤⠁⠡⡜⠁⠀⠀⠈⠳⠀⠀⣩⠋⠀⠀⠀⠉⢛⡽⡻⠋⠀⠀⠀⠀⢩⠰⠆⠅⠁⠀⠀⠀⠑⠀⠀⡈⠉⠀⠀⠀⠘⢻⡭⠶⠉⠁⠀⠀⠈⠻⡭⠀⠈
⠀⠷⠊⠑⠇⠀⠀⠀⠀⠸⠊⠁⠺⠀⠀⠀⠀⠀⠋⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⠀⠀⠀⠀⠀⠛⠁⠙⠂⠀⠀⠀⠀⠘⠃⠈⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠠⢢
⠀⢖⢣⢗⠆⠀⠀⠀⠀⠛⣟⠙⡴⠂⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠀⠘⣭⠁⠈⣭⠃⠀⠀⠀⠒⠆⠉⠲⠂⠀⠀⠀⠐⠖⠈⠐⠛⠂⠀⠀⠘⢭⠁⠀⢩
⠀⠘⠁⠑⠀⠀⠀⠀⠀⠸⠃⠛⠆⠀⠀⠀⠀⠀⠶⠀⠸⠀⠀⠀⠀⠀⠻⠖⠲⠟⠀⠀⠀⠀⠀⠟⠈⠳⠀⠀⠀⠀⠀⠴⠃⠸⠇⠀⠀⠀⠀⠠⠕⠒⠭

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⡄
⠐⢖⣚⣓⠖⠀⠀⠀⠀⢶⣾⣿⣶⠆⠀⠀⠀⢈⠉⠀⠈⡡⠀⠀⠀⠠⣄⠁⠀⠒⠂⠀⠀⠀⠠⣤⠻⡤⠀⠀⠀⠀⠰⣖⠋⢓⡶⠀⠀⠀⠰⢦⠈⠀⠒
⠀⠘⠛⠛⠀⠀⠀⠀⠀⠰⠿⠻⠧⠀⠀⠀⠀⠀⠡⠤⠄⠃⠀⠀⠀⠀⢀⠤⠀⢣⠀⠀⠀⠀⠀⠛⠛⢃⠀⠀⠀⠀⠀⠺⠖⠾⠆⠀⠀⠀⠀⢠⠔⠀⢷

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⣄⠊⠂⣤⠄⠀⠀⠠⡆⠀⠈⠀⡦⠀⠀⠀⢤⠐⠁⠓⣠⠀⠀⠀⢰⣟⠃⠘⢻⡆⠀⠀⠀⢤⡀⠃⢤⠄⠀⠀⠀⠤⠀⠀⣀⠠⠄⠀⠀⠠⣤⠘⠑⢤
⠀⣸⠴⢮⡅⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⣇⠤⣘⡅⠀⠀⠀⠘⢿⣤⣤⡿⠃⠀⠀⠀⠀⡤⠀⢦⠀⠀⠀⠀⠀⡁⢲⣿⡗⠀⠀⠀⠀⢨⡤⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⣀⡀⡀⠀⠀⠀⠀⣀⡠⠧⣀⡀⠀⠀⠀⣀⠴⠋⠢⢀⠀⠀⠀⢀⣀⣰⣀⠀⠀⠀⠀⢀⡠⠞⠀⠳⢆⠀⠀⠀⣀⠠⠇⠤⢤⠀⠀⠀⠀⠚⠁⠁⠙
⠀⠠⠛⢏⠀⠀⠀⠀⠀⠈⡧⢠⡍⠀⠀⠀⠀⠀⢆⣀⣰⠃⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⢹⠀⣀⠐⡏⠀⠀⠀⠈⠓⣠⠐⡆⠀⠀⠀⠘⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉

⣀⠤⠎⠣⠤⡀⠀⠀⠀⣀⣰⢧⣀⡀⠀⠀⠀⢀⣀⣾⣄⣀⠀⠀⠀⠀⠤⠎⠱⠤⠀⠀⠀⠀⣀⠀⡗⢀⡀⠀⠀⠀⣀⠠⠏⠡⢀⡀⠀⠀⠀⡀⢠⢇⣀
⠘⡆⠀⠀⡖⠁⠀⠀⠀⠙⣆⡀⣞⠁⠀⠀⠀⠈⣿⣥⣿⠋⠀⠀⠀⠈⡢⠀⠀⢔⠁⠀⠀⠀⠉⣃⠀⣍⠁⠀⠀⠀⠉⣷⢀⡀⡞⠁⠀⠀⠀⠙⢊⡺⡍
⠀⠓⠉⠑⠂⠀⠀⠀⠀⠘⠉⠈⠉⠀⠀⠀⠀⠀⠋⠁⠉⠀⠀⠀⠀⠀⠙⠋⠉⠊⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⠀⠛⠉⠙⠃⠀⠀⠀⠀⠘⠉⠀⠉

⠀⠀⡀⣀⠀⠀⠀⠀⠀⠐⠀⡀⠐⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡰⢳⣀
⠀⠨⠀⠈⡄⠀⠀⠀⢠⠀⣻⢿⠃⡠⠀⠀⠀⠑⠄⠀⠀⠌⠀⠀⠀⠈⢫⠀⢈⡝⠁⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠈⢣⠰⠷⢠

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⢀⠞⠛⠉⠉⠉⠉⠛⠓⠦⢄⢀⣀⣤⣴⣶⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠠⠾⠿⢿⣷⣄⢀⣠⣾⣿⠿⠛⠋⢻⣷⡄
⠀⠀⢸⡀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⢈⣿⣿⡿⠋⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣍⡁⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⣶⣿⡿⠻⣷⣦⣀⣀⣀⢠⣿⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⠿⠿⠟⠁⠀⠀⠈⠛⠿⠛⠉⠈⠁


⠀⠀⠀⣠⡴⠾⠿⠿⠿⠶⣦⣄⠀⠀⢀⣠⣤⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡄⢀⣴⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⢳⣶⡿⠟⠉⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⢿⣾⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⢧⣀⠀⠀⠀⠀⢀⣠⣾⡿⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣠⣾⢿⣦⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠉⠛⠛⠻⠟⠛⠛⠉⠀⠀⠀⠙⠻⠶⢶⣶⡶⠾⠛⠁⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠁⠈⠻⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣠⣤⣶⣶⣶⣦⣤⣀⠀⠀⠀⢀⣤⣶⡿⠿⢿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣤⣀⠀⢀⣤⣾⣿⣿⣿⣦⣄
⠀⠀⣼⠋⠀⠀⠀⠀⠉⠛⠿⣷⣦⣴⡿⠋⠁⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠛⠿⣿⣿⣿⣿⣿⡟⠉⠀⠈⠙⣿⣷⡀
⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣽⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⢈⣽⣿⣿⣿⣷⣦⣄⡀⠀⢸⣿⡇
⠀⠀⠀⠙⠿⣶⣤⣤⣤⣴⣾⠟⠃⠀⠈⠙⠻⢿⣷⣶⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣿⣿⠟⠉⠙⠿⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣷⣦⡀⣀⣤⣶⣶⣶⣦⣤⡀
⠀⠀⣠⣿⠻⠿⣿⣶⣶⣦⣤⣄⣀⣴⠟⠁⠀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠻⣿⣄⠀⠀⠀⠉⠉⠙⠛⢻⡿⢿⣷⣶⣶⣤⣤⣀⣀⢘⣿⡇⠀⠀⠀⠀⠈⢿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⡟⠁⠀⣠⣾⣿⣿⡷
⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⣤⡴⠟⠁⠀⠀⠀⠉⠉⠛⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠈⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠋⣩⡶⠔⢋⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⠿⠛⠛⠻⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣠⡴⠞⢉⡤⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣤⣤⣤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⣀⠼⠉⣁⠴⣚⠽⢊⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⣾⡥⠖⠛⠟⣋⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⣖⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⣀⣾⢻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠁⠀⠀⠀⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠆⠙⢳⣿⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠍⢲⣀⢀⠏⣛⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠑⠀⠐⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠿⢿⣷⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠤⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⣀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⡶⡀⢀⠆⠀⠀⠀⠀⠈⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠐⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⢀⠀⠀⠀⡢⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⡂⠀⠀⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢁⡇⠀⠀⠀⠂⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣁⣈⣐⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢠⣊⠴⠋

⠀⠀⢀⠀⢀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⢀⠄⠀⡀⠈⠀⠠⠀⠁⢁⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷⡖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣭⠀⠀⠀⠀⠀⠢⣤⣥⡤⠂
⠀⠉⠠⠀⠀⠀⠀⠀⠁⠀⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⣻⣿⣾⠟⠁⢠⡀⠀⠂⠐⠆
⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⣤⢿⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠚⠀⠠⠟⠋⠛⢇⠀⠙⠛⠀⠠
⠀⠀⠈⠀⣄⠀⠀⠁⠀⣣⢬⡇⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠞⠙⠇⠀⠀⠀⠀⠀⣀⣀⣠⠃⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠠⣤⠞⠘⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠉⠲⡔⠀⠀⣠⠊⠀⠀⠴⠆⢠⡀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⢨⣃⣤⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠈⠀⠀⢸⣥⠖⠦⣸⡀⠀⠀⣤⣤⣾⣧⣤⡤
⠀⠀⢈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣰⢇⠀⠀⠐⠐⠀⠐⠀⠀⠀⠈⠀⠀⠀⠈⠡⠀⠀⠀⣹⣿⣿⡏
⠀⠀⠈⠀⠄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣤⣾⣿⣏⣤⣤⣄⡄⠀⠀⢶⣶⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠋⠁⠈⠓⠀⠀⠈⣇
⠀⠀⠀⠐⠀⢀⡾⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠠⣄⠀⠋⠀⠐⠦⡤⠀⠀⠀⠀⠈⠛⠻⣦⣯⣤⠔⠂
⠀⠀⠲⢶⠶⢋⡴⠟⣻⣶⣦⠶⠂⠀⠀⠀⠀⠀⠱⡄⠀⠀⢸⣿⡿⠿⣿⣷⠀⠀⢀⣀⣼⣧⣀⡀⠈⠁⠀⠀⢠⡀⠀⠀⢤⣦⡤⠀⢀⡿⠁⢷⡀
⠀⠀⡀⠈⢶⡿⢁⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠛⠁⠀⠀⠈⠙⠂⠀⠀⢹⣿⢿⡏⠀⠀⠖⠉⠀⠐⠇⠀⠀⠘⠉⠃⠀⠈⠀⠀⠀⠉
⠀⠘⠋⠀⢨⢶⠿⠿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠂⢀⣀⡰⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⢧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⡆⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠂
⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠠⣤⠤⠶⠞⢀⠈⠳⠦⠤⣤⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠸⠛⠉⠛⠇⠠⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠀⢀⡄
⠀⠲⣶⣶⣿⣿⣷⣶⠖⠀⠀⠄⠀⠈⠳⡄⠘⣿⣿⠂⡴⠊⠀⠀⣤⠀⠀⠰⣶⣶⡿⠧⢀⡄⠀⠀⠀⣀⣴⣆⣀⠀⠀⠁⠀⠀⠀⠀⣀⡤⠞⠙⠤⢀⡀
⠀⠀⠈⣿⣿⣿⣿⠁⠀⠂⠀⠀⠐⠂⠀⡇⠀⣁⣉⠀⢧⠀⠐⣾⣿⡶⠂⠀⠈⣹⣮⣶⡏⠀⡀⠀⠀⢙⣿⣿⡁⠀⠀⠀⠀⠀⠐⠀⠈⢲⠀⠀⢰⠋
⠀⠀⠸⠛⠁⠈⠛⠇⠀⠀⠂⠀⠃⠀⠸⠗⠉⠀⠈⠙⠻⠀⠀⠋⠉⠓⠀⠀⠀⠉⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠟⠊⠙⠺⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢡⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠐⢖⠚⡞⡒⠒⠒⠒⠒⠒⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠃⠀⠀⠀⠀⡂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣸⣀⣀⣀⣙⣇⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⡤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡏⡇⠀⠉⣉⡉⢹⡇⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⢡⠀⠀⠀⠆⢸⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⣤⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠘⡄⠈⣁⡁⢸⡇⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⣇⠀⠀⡃⢸⡇⠀⠀⠀⠀⠀⢿⢆⠀⠘⠝⣀⠺⢠⣿⠇⠀⠀⠀⠀⠀⠀⣿⣇⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣣⣤⣼⣧⣬⣭⣴⣪⠇⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣯⣶⠵⠃⠀⠀⠀⠀⠀⢀⣴⣻⣧⣤⣼⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠛⠛⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⠒⠒⠒⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠒⠒⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠠⠬⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠚⢻⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠐⠒⢷⣻⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠉⠩⣇⣿⡄⠀⠀⠀⠀⠀⠀⣿⡏⠉⢹⢿⡉⢩⣍⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣺⡾⣷⠀⠀⠀⠀⠀⠀⠸⡿⡄⠈⠉⢀⣬⠁⣤⠀⠀⠀⠀⠀⢠⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠹⣝⢄⣀⣀⣉⣀⣉⣀⣀⣀⣀⡴⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠖
⠀⠀⠀⠙⠻⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢀⣠⣾⣷⣄⡀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⣠⣴⣿⣿⠟⠻⣿⣿⣦⣄⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⢀⣴⣾⣿⠿⠋⠀⠀⠀⠀⠙⠿⣿⣷⣦⡀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢏⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄⡸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠁
⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⡀⠀⠂⠀⣠⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠽⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠿
⠀⠀⠀⢰⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠈⠁⠀⠀⠀⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⢺⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿
⡀⠀⠀⠀⠠⠀⠀⠀⢹⡿⣿⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⢻⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣶⠀⠀⠀⢴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⢴⡦⠀⠀⠠⣼⣷⠄
⠀⠀⠀⠤⣽⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⢀⠀⠀⠀⠰⣾⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠃
⠃⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠟⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⢀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠠⣦⠄⠀⠀⢸⣿⡷⠄⠀⠀⠀⢀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⣻⣿⡶
⠿⡗⠀⠀⠀⠟⠻⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠀⠸⢿⠂⠀⠀⠨⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦
⠀⠿⠆⠀⠀⠀⠀⠂⠀⠀⠛⠙⠂⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠠⣼⣧⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡀
⡦⠀⠀⠀⠀⠈⠛⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⢲⣿⠄⠀⠀⠀⠀⠘⠃⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⣶
⠁⠀⠀⢀⣠⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠻⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄⠀⠀⠀⠀⠀⠀⠚⠻⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⢀⣄⠀⠀⠀⢀⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⢠⣄⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⢰⣀⡀⠀⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⡿⠿⠄
⡦⠀⠀⠈⠝⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⣴⣄
⣴⡄⠀⠀⠐⣾⣷⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠚⠛⠏⠀⠀⠀⢠⣴⡀⠀⠀⠰⣿⣗⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⠀⠛⠻⠇
⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀
⣀⣶⣄⣀⣀⣀⣀⣀⣀⣈⣿⣿⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣈⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣈⣿⣯⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠙⡧⠀⠈⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⣰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⡞⠹⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠶⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠀⣐⠰⣟⠀⠀⠀⢰⠉⢻⠇⢠⠀⢀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⠋⠙⣿⠀⠀⠀⠀⠐⠪⠃⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠠⡀⠳⡴⠛⠀⢀⡈⢁⣀⡪⢊⣠⠾⠛⠟⣻⣿⡷⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣖⣫⣿⣶⣮⣝⢿⢜⣋⣠⢦⣿⠁⠀⠀⠀⢱⣿⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡆⢿⡘⣿⣧⠻⢆⡀⣀⣠⣬⠁⠃⠂⠀⠀⠀⢀⣴⢿⣿⣦⠀⠀⠀⠀⣀⠁⠈⡇
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣇⣛⣽⣿⣿⣿⣿⣿⣿⣏⡚⠋⡁⡘⠉⠁⠁⢀⡀⠀⠀⣼⠀⠀⠘⡿⢃⠀⡄⠀⠘⢆⠀⡄⠰⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⣷⡔⠜⠆⣤⠝⠳⠟⠀⠀⠈⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣀⢄⣀⣠⣤⣴⣄⣤⡤⡄⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠙⢿⣿⣿⣿⣿⣿⣿⣅⠹⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠰⠆⠀⢠⠉⢸⠟⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡄⠀⠛⠁⠀⣀⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣒⡔⣄⣶⣶⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⣿⣿⣿⣆⠹⠛⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣛⣉⣩⣭⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣄⠀⠀⠀⢀⡠⠤⠗⠀⠁
⠀⠀⠀⠐⠠⡒⠠⢤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⡤⠀⠀⠓⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀⠄
⠀⠀⢐⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⡠⣆
⣄⠀⠀⠀⠀⠀⠂⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣿⣿
⡏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣅⠒⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣟⣷⠟⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡘⠙⠻⠿⣿⣿⡿⡿⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⢯⡇⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠑⡀⠀⠀⠀⠀⠘⠛⠷⠀⠀⠀⠐⠉⢃⠀⠀⣶⡴⠄⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠠⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠝⢫⣤⣴⣄⠂⠀⠘⠗⢾⡱⠀⠀⢀⣀⣄⣶⠀⠀⠀⠀⢹⠘
⠀⠀⠀⠁⠀⠀⠀⡀⢤⠘⠀⠐⠢⡄⢀⠀⠀⠀⢀⡀⠀⠀⢻⠿⠿⣠⡀⠧⠀⣸⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠀⠀⠁⠲⠀⠠⠀⠀⠀⠀⠠⠿⢇⣤⠀⠈⠀⣠⢤⣤⡤⣄⣀⠀⠀⠀⠀⠈⠻
⡀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠋⠁⢀⡀⢀⣀⣀⢀⣀⠀⡉⢹⡆
⠬⠫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡠⠆⠦⠄⠀⢂⡀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉
⠉⠀⠁⠀⠀⠀⠀⡤⠀⢤⠀⠀⠀⠀⠀⢠⠀⠸⡀⠣⠜⠀⡕⠙⡎⠀⠀⠀⠀⠀⠠⠐⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠑⠀⣈⠁⠀⢁⠠⠊⠀⠀⠀⠀⠀⠀⢣⠀⡄⡠⠃⠀⠀⠀⠀⠀⠀⢰⢢
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣷⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠠⢺⠔⠊⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠤⡀⡔⢠⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠰⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⡟⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢠⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠌⠠⡈⠋⣁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠄⠀⠀⠀⠈⠱⠂⢠⡄⡀⠀⠀⠀⠀⠐
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠷⣦⡀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⣿⡏⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⠄
⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⠇⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⣀⠀⠀⠀⠾⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠀⠀⠀⠀⣿⠟⢻⣿⢀⣀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⣉⡀⠀⣀⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡟⢹⣧⢸⡇⠀⣿⡇⢾⣍⣉⠀⢹⡇⣾⠋⠉⠀⢸⣏⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⣿⠀⠨⣿⠀⢸⣿⠘⢷⣼⣿⡇⢠⣍⣽⡷⢸⡇⠻⣦⣠⣴⢠⣍⣹⡷⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠄
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠁⠐⢿⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⣼⡇⠀⠉⠁⠀⠈⠁⠀⠈⠉⠀⠀⠉⢁⠦⠀⠀⠀⣀⣤⣶⠷⠛⠁⠀⠀⡤⠀⠀⣄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⢤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠘⠻⠛⠁⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⣴⡾⠗⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠂⠀⠈⠀⢠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠉⠔⠀⠠⠀⠀⠀⠀⠈⣈
⠀⠀⠀⠀⢀⠀⢀⣶⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠈⠃⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠁⠶⢶⡔⢂⡀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⠛⣀⠆⠑⢺⣄⣠⣄⣶⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠇⠀⢠
⠀⠀⠀⣠⣆⠀⠀⠀⠀⢠⣴⡂⠁⠀⠀⠉⠉⠁⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢓⡀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠠⠀⠛⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣭⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢰⠖⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠙⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⡤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⠀⣀⣀⡀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠐⠂⠣⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣷⣿⡿⠫⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣟⠉⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢿⣿⣿⣿⣏⣿⣏⣿⣿⣿⣿⣿⣿⢯⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⡣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠄⠀⠊⠀⠈⠁⠀⠀⠀⠙⠓⠂⠥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠂⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃⠀⢢⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⠆⠀⠀⠀⠀⢀⣰⠄⠀⠀⢠⠀⠂⠾⠀⠣⣢⠔⠃⢀⡄⠀⠀⠙⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡋⠀⠸⠀⠀⠲⠄⠀⠘⠛⢀⡀⠀⠈⠁⠠⠀⠔⢀⡁⠀⠀⠀⠀⠡⢚⣋⣄⡷⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢱⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⠈⠁⠉⣦⣙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⠤⣹⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⣠⣶⣤⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⡀⠀⣶⡄⢹⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢮⣿⡇⠀⠈⠱⠶⠖⠂⡀⠤⠐⠉⠀⠀⡀⣠⠄⢀⡀⠀⠀⠐⣀⣤⠄⡀⠈⠩⣏⣴⣿⣷⣼⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢐⣿⠡⠀⠀⢀⠀⢠⡀⣀⡃⣈⠀⠀⠈⠄⠘⠛⠛⠉⠲⡀⢱⠀⡈⠉⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢙⣧⡀⠀⢀⡊⣉⣧⠀⢈⡉⢀⣭⣄⣀⣀⢀⠶⠞⢛⣡⣿⣤⣈⠃⠀⣾⣿⠟⣛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⢿⣇⢀⣈⡳⢔⠯⣭⡽⣿⠿⢿⠿⣿⡿⣿⠟⣻⢿⣿⣶⡄⠼⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣹⣷⠎⠹⠿⡷⣾⠷⠒⠶⢰⣀⠸⣿⣿⡟⠿⢷⡶⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⣿⡏⡠⣤⢴⣴⡇⣀⣈⢊⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣤⣄⣈⡙⠃⠈⡙⠿⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⢠⠏⢾⡇⢰⠒⠃⣠⡩⠍⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠟⢸⣀⣴⡇⠘⡆⣾⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡁⣾⣿⡇⠛⢠⡤⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠛⠛⠀⠟⠂⠀⠀⠈⠁⠘⢿⣧⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠘⣁⣀⡀⠀⠀⠈⠉
⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣤⣭⣿⣇⣲⣈⠷⠶⠤⣴⣧⣼⣿⣯⠟⢠⡶⠤⠼⢭⣿⣾⣧⣤⠀⠀⠶⠾⠧⢶⣦⣤⡶⣤⠤⡄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠿⠿⠿⠌⠀⠂⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉
⣴⣶⣶⣤⣶⣤⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣦⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠉⡆⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠵⡒⠁⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢦⣀⡼⠀⠀⠸⡯⠀⠁⠀⠀⢀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠓⠀⠀⠠⠄⠀⠈⠛⢇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠁⠀⠀⢸⠀⠀⠘⡏⡇⠀⢀⣤⡈⠀⠑⠀⠀⠀⡀⠀⠀⠀⢱⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠘⠻⢤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⢀⠊⠈⠁⠀⠀⠃⠀⠀⠙⠿⠿⠿⠋⠀⠛⠉
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡖⠀⠀⡀⠀⠹⡆⠀⠀⡇⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢣⠀⠀⠅⠀⣠⠇⠀⠀⡇⠀⠀⠀⠀⠀⣸⠘⣦⠀⠀⠀⠀⡄⠀⠈⢩⠉⠉⡇⠀⠀⢠⡮⡀⠀⠀⢘
⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠈⠟⠛⠟⠛⠏⠀⠀⠀⡇⠀⠀⠀⠀⠉⢻⠀⠉⠻⠉⠉⠋⠁⠀⠀⢸⠀⣗⡇⠀⡖⠉⠀⠀⢀
⠀⠀⣠⣤⣿⣀⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣷⣧⣤⣷⣤⣤⣤⣼⣤⣬⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⠦⠀⢀⠀⠀⠀⠀⠀⣤⠾⣿⠋⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⢀⣛⣩⡥⠄⠐⠒⠐⠆⡹⠀⠀⠀⠀⢸⣿⣤⡏⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣛⠛⠛⠛⠃⠠
⠀⠀⠀⠀⠀⣾⣸⣿⣿⡇⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠘⣿⣬⣇⠠⠄⠀⠁⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠑⢀⠀⠀⠀⠀⠀⠀⠘⣋⣮⣅⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠿⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣖⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠐⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠⣿⣯⣥⡴⠶⠒⠛⢻⡆⠀⠀⠀⠀⢰⣿⣿⡇⢀⣀⡤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⠀⣀⡾⠀⠀⠀⠀⠀⠀⣿⣟⠛⢃⠀⠀⠀⠈⡇⠀⠀⠀⠀⣠⡾⣿⣿⠇⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠘⢿⣿⣧⣉⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠘⢿⣴⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠿⣿⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣻⠟⢉⣠⠀⠀⠀⠀⠀⢠⠛⠺⡏⠀⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣁⢴⣏⢹⡏⠀⠀⠀⠀⢠⣾⣿⣟⡐⢄
⠀⠀⠀⠀⠀⠀⢠⣿⣭⣿⡏⠒⠉⣀⠀⠀⠀⠀⠀⠀⢴⡞⠀⠀⡀⠠⡦⣄⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⣼⠋⣡⣾⠏⠀⠀⠀⠀⠀⢸⣿⣿⠉⣄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣭⡟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⢠⠏⡀⠉⠹⣷⡄⠀⠀⠀⠀⠀⠀⢀⣏⡻⣟⡯⡴⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠛⠉⠉⠉⠛⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠛⠋⠉⠉⠙⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⡀
⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡁⢀⠀⣁⣠⢀⡅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠀⠤⠤⠤⠤⠜⡀⡀
⠀⠌⠀⡄⢠⠆⢀⡀⢠⠀⠀⢹⠁⠀⠉⠈⠀⠀⠀⠀⠈⡇⠀⠆⠰⠀⠠⠤⠄⠂⠁⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁
⠀⠉⠀⠁⠈⡏⢈⠀⠠⡀⠀⣸⠀⣀⠀⠀⡀⠀⠀⠤⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢀⡀⠠⣈⠃⠀⠀⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⢀⠀⠀⠠⠀⠂⣀⠐
⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⡀⠀⢁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠒⠂⡑⡄⡅⠀⠀⠀⠐⠠⡤⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⣁⡒⠀⠠⠀⠒⠈⣀⠀⠐⠀⠀⠀⠀⠈⠀⠐⠄⠀⠀⠠⠄⠀⠆⡄⢀⠀⡆⠀⠑⠁⠀⠀⠀⢎⠀⠀⠀⣀⠘⠀⢀⡂⠶⡅
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠐⡱⠦⠤⠔⠊⠀⠀⠀⣇⢀⠙⣠⠀⢀⠠⠦⠀⢰⠀⠭⠃⢠⠠⡄⠀⠁⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⠁⠀⠀⠐⠀⠀⠀⠀⠘⢦⡀⠈⠁⠉⠁⢂⡠⠬⠀⠀⠉⠀⠋⠀⠠⡀⠀⠀⠐⠈⠐⠉⠐
⠀⠂⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠰⠀⠀⠀⠀⠀⠀⠉⡁⠒⠒⠊⠉⠀⢠⠀⠀⢰⠀⠑⠀⠠⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠔⠀⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⡊⠐⠀⠄⠀⠘⠀⠑⠁⠦⠀⠀⢀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠂⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⣄⡀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠈⠉⠉⠁⠁⠀⠀⠀⠈⠁⠀⠀⠀⡀⢸⣿⠀⠀⢸⠟⠉⢻⡇⣿⡇⢹⡇⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠍⠀⠁
⠀⠀⠀⠀⢨⠈⡅⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠃⢸⣿⠀⠀⢸⣆⣀⣸⡇⢸⣧⣾⠇⠀⠀⢀⣀⣀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠆⠺⠀⠆⡀⠠⡄⠀⠀⠀⠀⠱⡀⢀⠔⠀⢸⣿⠀⠀⢸⡇⠀⢸⡇⢸⡏⣹⡇⠀⠀⢸⠉⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠘⠃⠀⠘⠁⠘⠛⠛⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠂
⢀⠀⠐⠉⠀⠀⠀⡆⠀⠂⠀⠀⠀⢠⢀⠀⠀⣀⠀⠀⢸⡛⡟⣻⠻⢿⠛⢟⣿⢻⡇⠀⠀⠀⡄⠀⠆⠐⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠈
⠀⠉⠀⠀⠒⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣷⣷⢾⡾⣾⡷⣷⡿⡾⡇⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠑⠀⠀⢸⠀⠠⠀⠀⠀⣀⠀⠀⠠⠀⠀⠐⢀⠀⠌⠐⢆⡃⠂⠀⠡⠠⢀⡐⠀⠀⠀⠀⠀⠠⡀⠠⠂
⠀⢀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⢀⣀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠵⠀⠀⠀⠀⠊⢔⠀⠀⠀⠀⠀⠙⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠀⠀⠁⠀⠂⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠂
⠀⠀⢰⡆⢰⡆⠀⠀⢠⡾⢷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠐⠁⠊⠁⡐⠀⠀⠀⠉
⠀⠀⢸⣧⣼⡇⡶⣦⠸⣧⢸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠈⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⢸⡇⢀⡿⠀⣿⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⡄⣡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠈⠛⠘⠁⠙⠛⠀⠙⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠞⠒⠤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⡴⠛⠁⠀⣆⣇⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠃⢀⠴⠊⢁⡤⠛⠁⠀⠀⠀⠀⠈⠁⠀⢴⢿⠆⠀⠀⣠⠀⡀⠀⠱⡀⠀⠀⠀⠀⢀⠜⠀⠀⠠⣳
⠀⠀⠀⠐⠈⠀⢿⣅⣠⠖⠉⠀⠀⣤⡄⠀⢠⠆⠀⡇⠀⠈⠉⠉⠀⠀⠁⠀⠇⠀⠀⠈⠒⣤⣴⠶⠁⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⠈⠀⠀⢺⣀⣞⣀⣀⠀⠀⡄⢀⡀⠀⢠⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀
⠀⠀⠀⠀⠀⠘⠁⢰⣖⠤⣀⠀⢀⢀⠀⠀⠀⡟⢉⣁⣹⠃⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠈⡕⠂⠉⠰⠦⠀⣀⢠⡯⠚
⠀⠀⠀⠀⠀⠀⢀⡮⠉⣻⠚⢁⠀⠂⠂⠀⠀⠙⠋⡼⢉⣹⠂⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⡎
⠀⠀⠀⢀⡤⠒⠊⠁⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⡹⠋⠉⢱⠄⠀⠀⠑⢄⠀⠀⠀⣀⠇⠀⠀⠀⢠⠀⣽⢒⡲
⠀⠀⠀⣿⣿⣶⣦⣄⠘⡆⠀⢰⡠⡇⡀⢰⠆⠀⠀⠀⢧⠄⠀⠉⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⣀⡀⠀⠸⡟⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣷⠇⠀⠘⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠘⣄⡜⠀⠊⠀⢀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢀⠤⣶⣶⣶⣐⡦⠀⠟⠳⠄⠇⠀⠀⠀⢹⡀⠀⢀⣴⠷⣴⠁
⠀⠀⠀⢀⠠⡞⢶⠀⠀⠀⠀⢾⣁⣀⣀⡤⠆⠀⠀⠈⢺⢻⠟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣴⠛⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠚⠋⠿⣅⠀⠀⠀⠀⠀⠀⠀⡞⣷⡏⢹⡄⠀⠀⠀⠀⠀⣤⠀⠀⠘⠂⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠚⢧⠀⠀⠈⢛⣦⣄⣼⠆⠀⡀⠀⠀⠀⠀⢹⢻⣷⠘⣇⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣉⣁⣹⠀⠀⠘⠉⠈⠀⠀⠠⠈⡼⣿⡄⢿⠀⠋⠉⠹⡇⢱⠆⠀⠀⠀⠀⠀⠀⠘⠬⠜
⠀⠀⠀⢠⣄⠀⠀⢠⠖⠊⠛⢿⡉⠀⠀⠀⡤⠄⢤⠀⡀⠀⢣⢿⣷⣸⡇⠀⠀⠀⡸⢧⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠈⣁⠀⠀⠙⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠯⠚⠀⠀⠀⠘⠃⠀⢱⡄⠀⠰⣮⣭⣭⣵⠂
⠀⠀⠀⠘⠖⢃⣘⡛⠀⠀⠈⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠇⡇⣀⡤⠒⠁⣠⣀⠀⢸⠀⠉⡇
⠀⠀⠀⠀⢐⠛⠉⠉⢹⠛⠉⡇⠀⠀⠀⠀⠀⠀⣀⡀⠤⠶⠶⠤⠤⣀⣀⣀⣀⣩⣤⡄⠀⠑⠋⣠⠜⠀⠀⠡⡀
⠀⠀⠀⠀⠈⡄⠀⠀⢸⠇⠀⢰⠀⠀⠛⠋⠉⠉⠙⠓⠤⠤⠤⠤⠜⠋⠉⠁⠀⠀⠀⠠⣶⠀⠀⣷⣶⣿⣿⣿⡟⢄
⠀⠀⠀⠌⠀⠉⠉⠹⠥⠤⠔⠚⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⡐⠠⠥⣖⣀⠨⠊⣁⣀⡈⠀⡀⣰⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⢂⠰⠈⢣⠤⠆⠀⠈⠃⠉⠀⠉⠀⠀⠙⢀⠙⠉⣸⣧
⠀⠀⠀⠀⠀⠀⡀⠸⣏⠾⠁⡛⠀⠀⠯⣤⠰⣄⡀⢠⠀⠄⠀⢀⣀⣠⣼⣁⢠⠹⠇⠐⣚⣣
⠀⠀⠀⠀⢠⡜⣧⡤⢿⣀⣢⡿⠋⠦⠸⣦⣀⣌⣿⣯⣼⣿⣿⣿⣽⣶⡌⢉⣷⢠⣆⠐⣦⠏⡐⢦⡄
⠀⠀⠀⠀⢸⢁⣭⡗⠀⠘⠿⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣽⣎⢄⠑⣖⣣
⠀⠀⠀⠀⠋⣀⣈⢳⣤⣞⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠘⠸⢿⣿⣿⣿⣿⣟⣿⣿⣿⣷⣿⣯⣵⣯⠃
⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⡿⢿⡿⢿⣿⠿⢿⡀⠀⠀⠀⠀⢡⡿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⡀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣎⣴⣴⣶⣶⣶⣶⣶⣶⣦⣄⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⠛⠿⠘⠛⣿⠳⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣦⠀⣦⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣿⢿⣿⣿⣷⢻⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣶⠅⠾⢧⣘⣻⣆⣿⣂⣷⣲⣔⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣺⡟⢛⢋⡋⠻⠟⡷⡼⠮⢤⢤⠷⡋⣉⠿⣿⣿⣷⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠈⠉⠈⠀⠨⠜⠀⠉⠈⠀⠠⠀⠠⠀⠠⠀⠠⠀⠈⠉⠙⠟⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠋⠛⢿⣯⠁⢸⣾⣷⡶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⠀⠀⠀⠀⢄⣀⠹⢻⣶⡿⣿⣿⣃⣿⣿⣿⣯⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⣤⣤⣀⡘⠛⠛⢛⣽⣤⢾⡿⢿⣿⣿⣿⣿⣿⡻⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣦⡤⢄⠀⠀⣉⡿⠓⠡⣄⣄⠂⣮⠉⣻⣿⠗⣺⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡠⣾⣿⡷⠚⡀⠸⢯⢗⣧⠀⠿⠿⠋⠀⠀⠈⠪⠀⠛⡻⠀⠾⢿⣿⡿⠯⣷⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠳⣹⡟⣛⢈⠦⢄⠄⣠⣞⢿⠀⠀⠀⡀⠀⠀⡀⠠⠄⠄⠀⠀⠈⠉⠉⠑⢊⣿⣧⣄⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⣿⣯⣽⢼⣿⣾⣿⢼⣷⣻⣾⡓⣬⠖⠶⢀⡠⣤⠆⡴⠠⠀⠁⠀⠐⠏⠁⠈⢾⡊⣿⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠟⣸⠟⠋⢱⣿⣿⣿⡿⢿⣿⣿⢿⠋⠉⠂⠀⢺⡿⠃⠍⠁⠀⠀⠀⣠⠐⠉⠀⠀⠙⠿⢿⣿⣿⣻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠽⠢⣰⣿⣿⣿⣧⣝⣿⣿⣷⠇⢀⢄⠀⠀⠀⠰⠀⠀⠠⠀⠀⠆⠀⠀⢀⠀⠰⣿⠁⠐⠩⠹⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⡀⠈⣉⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣶⣄⢀⣤⡾⠂⠀⢠⡄⠒⠂⠀⠀⠻⠀⠁⢰⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣼⢲⣽⣶⡿⠸⣿⣿⣟⣿⣯⣼⣇⣠⢤⣆⠉⣻⣷⣿⡟⢡⠋⢴⠇⠀⠀⠀⠀⠐⠂⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⠈⣿⡐⠦⣍⢶⣭⣿⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣆⠂⣈⣶⡄⠀⠀⠀⢠⣲⠄⠀⠠⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⠀⠇⡿⡸⣯⣶⣿⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⡶⡂⠀⡀⢽⣿⣧⣀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠉⣹⡿⣟⣻⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣿⡝⣬⡛⠉⢡⠀⠀⠀⠈⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⠉⠀⠀⠬⢃⡺⠿⡿⣿⡏⠰⠂⢀⠀⠀⠈⢿⣿⠏⠁⡴⠀⡤⠐⣀⠀⠀⠘⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠤⠄⠸⠽⢇⠶⡿⣧⠀⠀⠀⠀⢠⣀⣿⣟⡀⣀⠀⠀⠀⡀⠀⣰⣼⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⠀⠀⠁⠙⡳⠶⠤⡄⢴⣿⣿⣿⣿⣦⡿⣦⣴⣴⣿⣿⣧⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢭⣿⡄⠦⢿⣷⣾⣂⣰⡾⣿⣿⡍⣿⣟⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠪⢯⣍⡹⡛⣶⣿⣛⣵⣶⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠁⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡠⠠⠤⠄⠄⡀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⡀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠉⠂⠐⠁⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠸
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣄⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠘⠢⡀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⡠⠔⠉⠑⠤⡀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢀⣀⣀⣀⡀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⢀⣀⣀⣀⠀⠤⠐⠊




⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⠉⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣦⣄⣠⣴⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣤⣤⣤⣤⡄⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⣾⣿⣷⣍⢿⣿⣿⡜⣿⣿⡿⣼⣿⣿⢟⣵⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣮⣝⢿⣿⣎⣿⣿⣧⣿⣿⡇⣿⣿⣟⣼⣿⢟⣭⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢟⣻⣭⣭⣟⣛⠿⠿⠿⠿⠿⠷⠿⠿⢟⣛⣯⣭⣽⣛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣴⣿⣿⣿⣿⣿⢛⣵⣾⣿⣿⢸⣿⣿⣶⣍⢻⣿⣿⣿⣿⣿⡜⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠇⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡟⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⡆⣶⣶⡶⣴⣶⡆⣶⣶⡆⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣟⣿⣿⡿⣛⣵⣿⣿⢣⣿⣿⣷⢻⣿⣿⣝⡻⣿⣿⣟⣯⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⣵⣶⣿⣿⣷⣶⢏⣩⣭⣭⢩⣭⣭⣉⠵⣶⣿⣿⣿⣶⣭⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣿⣿⣿⣿⣿⡟⣴⣿⣿⣿⣿⢸⣿⣿⣿⣷⡌⣿⣿⣿⣿⣿⡇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣹⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠃⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡆⠐⣄⠀⠂⠀⠀⠀⣆⡆⠀⢐⣰⠀⣠⠀⠠⠀⠔⠀⡂⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠊⠘⠁⠀⠈⠀⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠐⠂⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠠⡄⠀⠀⠀⠀⢸⠀⢀⠀⠀⠁⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢤⢈⣀⣀⣀⡀⣐⠠⠜⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⢸⢥⠄⠠⠄⠄⠀⢬⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠸⠾⣿⣼⣇⠼⠦⠴⠀⠀⠀⠀⠀⠤⢀⡃
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢉⢀⠉⠁⠀⠀⡇⠀⠀⠀⠀⢹⠩⣙⢛⠉⢉⡙⣀⣀⡤⠀⠠⠆⠉⡦⢈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠁⡉⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢾⡁⠀⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠈⠀⠀⠀⠀⢀⠊⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢄⠀⠀⠘⠆⠀⠀⠁⠀⠀⠈⠉⠉⡏⠀⠀⣠⠤⠚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⠀⢈⡗⠠⡀⢀⡀⠀⠈⠀⠀⠀⠄⢫⡧⠴⠊⠀⠀⠀⣀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⠋⠀⠀⠈⠺⠧⢀⠼⡀⠀⠊⠀⢹⠁⠀⠀⠐⠀⠈⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⣀⡱⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⣀⠀⠄⠀⠒⠈⢠

⠀⢀⡀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡀⣠⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⢠⣿⡿⠦⠀⠀⠈⠀⠀⠀⠀⠸⠃⠀⠀⠀⢀⠀⠀⠀⢠⣿⡿⠦⠀⠀⠈⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⢤⣦⠄⠀⠀⠀⠀⢘⣶⣿⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⣦⠄⠀⠀⠀⠀⢘⣶⣿⡄
⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠈⠉⠏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⠀⠀⠈⠉⠋
⠀⠈⠛⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣾⣤⡀⠀⠀⠀⠉
⠀⠀⣀⣷⣶⠁⠀⠀⠀⠚⠛⠇⠀⠀⡀⠀⠀⠀⠴⠀⠀⠀⣠⣷⣶⠁⠀⠀⠀⠚⠛⠇⠀⠀⡀⠀⠀⠀⠴
⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⢀⣀⠀⠀⠐⢾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⢀⣀⠀⠀⠐⢾⣿⡟
⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠁⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠁
⠀⢀⡀⣀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡀⣠⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⢠⣿⡿⠦⠀⠀⠈⠀⠀⠀⠀⠸⠃⠀⠀⠀⢀⠀⠀⠀⢠⣿⡿⠦⠀⠀⠈⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⢤⣦⠄⠀⠀⠀⠀⢘⣶⣿⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⢤⣦⠄⠀⠀⠀⠀⢘⣶⣿⡄
⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⠀⠀⠈⠉⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⠀⠀⠈⠉⠋
⠀⠈⠛⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣾⣤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣾⣤⡀⠀⠀⠀⠉
⠀⠀⣀⣷⣶⠁⠀⠀⠀⠚⠛⠇⠀⠀⡀⠀⠀⠀⠴⠀⠀⠀⣀⣷⣶⠁⠀⠀⠀⠚⠛⠇⠀⠀⡀⠀⠀⠀⠴
⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⣀⠀⠀⠐⢾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⣀⠀⠀⠐⢾⣿⡟
⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠁⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⡠⠖⠚⠓⠢⡀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠂⠎⠀⠂⠀⠴⠀⠄⠀⠀⠀⢀⠀⠈⠣⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠞⠁⠀⠀⠈⠈⠀⠠⠂⠀⠀⠠⠀⠈⠈⠀⠀⠀⠐⠀⠀⢀⡀⠱⠐⠒⠦⡀
⠀⠀⠀⠀⠀⠀⢰⠋⠔⠠⠒⠀⠀⠊⠀⠒⢀⠀⡠⡄⠀⠀⠀⠀⢄⠀⠀⠀⠈⠐⠀⠀⡀⠀⠀⠀⠀⠱⠠⡀
⠀⠀⠀⠀⠀⡴⠁⠀⢠⠀⠠⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢀⠀⠄⡧⡀
⠀⠀⠀⠀⡠⠓⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠬⠀⠀⡠⠀⠀⠀⣄⠀⠈⠀⢀⠠⠀⠀⠂⠀⠈⠐⠀⠀⢸
⠀⠀⠀⠀⣇⠁⠀⠀⠢⠀⠀⠀⢠⠀⠀⠀⠀⡀⠈⠀⠄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠃⠀⠠⡀⠀⠃⠀⠰⠁⠊
⠀⠀⠀⠀⢦⡀⠐⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠁⢰⣦⠤⠶⢤⣠⣤⣀⣀⡄⠈⠀⠀⠀⠀⠀⡈⠀⠀⠄⠰⢀⠇
⠀⠀⠀⠀⠀⠙⠥⡌⠀⠀⠀⠀⠀⠀⠀⠂⠃⠐⢣⣹⠀⠀⠀⠉⠀⠉⣟⡼⠃⠀⠀⠀⠀⠐⠁⡠⠀⢀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣈⣀⡀⠀⠀⠀⠐⠠⠀⠈⢹⠀⠀⠀⢀⠀⠀⡏⠀⠀⠄⠠⠀⠀⠀⠀⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠔⢄⣀⣀⠀⢠⠀⠀⠀⢸⠀⠃⣇⡀⠀⠀⡀⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡅⠀⠀⠀⠀⠀⠈⣷⠀⠅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢋⠙⠊⠛⠊⠑⠊⠛⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⢀⠀⠀⢀⠀⠂⢀⡰⡁⠀⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠚⠦⠤⠶⠤⡤⠦⡤⠔⠲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣇⣀⡀⠀⣀⡀⠀⠀⢀⠔⠚⠳⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠴⣚⡩⠎⠩⠙⣎⡹⠤⢍⡱⠚⠹⢢⡵⠲⡓⢦⠩⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡉⢱⠛⠡⠱⠫⠙⢦⠏⠀⠒⠠⠋⢉⠲⠞⠛⠢⠀⠔⠢⣼⠢⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠒⠒⠒⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣄⡀
⠀⠀⠀⠀⡄⠁⢀⠄⢄⠀⠉⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠉
⠀⠀⠀⡗⡁⠀⢈⠀⢈⠀⠀⡁⢈⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⡌
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠐⡁⢸⢀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⣇⣂⠿⡀
⠀⠀⠀⡇⠐⠄⢠⠀⠀⠀⠀⡇⢰⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣈⠛⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠂⠘⠈
⠀⠀⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠄⠘⠚⠀⠀⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠄⠈⠁⠠⠭⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⡀⢀⣤⠀⣠
⠀⠀⠀⠀⢰⠋⠙⠦⠤⠤⠤⠤⠜⠋⢱⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⠖⠶⣧⠀⢀⡇⠠⡟⣋⡛
⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡎⠑⠒⠢⠤⠄⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⢐⡂⢸⠁⢙⡀⢠⢣⣛⣸⠆
⠀⠀⠀⠀⢸⢀⣒⣒⣒⣒⣒⣒⣒⣒⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡁⢸⠀⠐⢅⢨⠀⡆⠒⡀
⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠒⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⢘⢨⡅⢺⠀⠸⡤⠰⠀⠊⠯⠇
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠄⢺⠀⠀⡇⢸⠀⢸⢻⡼
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣿⠀⢘⡇⢘⠀⢸⣶⣾⡆
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠉⠉⠿⠆⠾⠇⠠⠀⠀⠯⠤⠇

⡿⢿⠿⠿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣿⣿⣿⣻⣿⣛⣿

⠀⠀⢼⡿⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠃
⢀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠻⠂⠀⠀⠀⠀⣲⣿⡥⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠐⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠁⠀⠀⠀⣄⣠⠀⠀⠐⠀⠠⣴⣄⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⢻⠟⠂⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠠⣶
⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢿⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟
⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡧⠀⠀⠀⠀⠀⣺⣷⡥⠀⠀⠀⠀⢀⣠⡄
⣶⡇⠀⠀⠀⠀⠀⠀⠄⢠⣷⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠄⠀⠀⠘⠻⠋
⠛⠉⠀⠀⠀⠀⠀⠃⠀⠉⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⣽⣯⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠋
⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣄⠄⠀⠀⠀⠀⠀⠀⢀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠜⡿⠏⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⢀⠀⠐⠂⠀⠁⠀⢀⣄⠀⠀⠀⠀⠀⠚⠟⠃⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⢠⣀⠄

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⣏⣹⣿⣿⣿⣙⡆⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⢴⢂⡀⢦⣀⠀⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠘⠌⠋⠞⠉⠀⡇⠀⠀⠀⠀⠀⢸⡿⡉⡇⢉⠵⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢸⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣲⢶⣶⡖⠒⠒⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡼⠋⢸⡇⠙⣦⠀⠀⠀⠀⠀⠀⠀⠈⠹⠯⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠐⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡆⠀⠀⠀⠀⠀⢠⣾⣿⣏⣉⠻⣿⣆⠀⠀⠀⠀⠀⠀⣤⡿⣿⣯⣿⣷⣼⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⢫⣼⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢋⡽⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣹⣿⣿⡟⠀⠀⠀⠀⠀⠘⢿⣿⣽⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡽⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⠊⠻⣄⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣾⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦
⠀⠀⠀⠀⠀⡏⠁⣩⠐⡒⢽⠉⢹⠇⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⢿⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣶
⠀⠀⠀⠀⠀⡨⠪⣅⢿⡿⢀⡵⢮⡀⠀⠀⠀⠀⠀⢸⣡⣄⣀⣀⣀⣠⡆⠀⠀⠀⠀⠀⢸⣟⣻⣿⣿⣿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠧⣤⢬⠐⠒⡽⠀⡼⠇⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠻⣿⣭⣿⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⠟⣤⣤⣠⣤⠀⠀⠀⠀⠀⠀⢀⣀⡧⠈⠠⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣉⣉⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠉⣿⡄⠀⠀⠀⠀⠀⢀⣈⣽⡄⠘⣿⣿⣿⣿⠗⠀⠀⠀⠀⠰⣿⣿⠐⣾⡷⢨⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠮⠯⠯⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⣶⣀⣀⣶⣿⡇⠀⠀⠀⠀⠀⢨⣿⣿⣿⣾⠎⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠙⡳⢠⠐⠟⠙
⠀⠀⠀⠀⠀⠛⠛⠛⠛⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠠
⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠽⡗⠀⠀⠐⠀⠀⢀⣀⠀⠀⠀⠲⠆⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⣀⠀⠀⢰⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⣷⠇⠀⠠⠀⠀⡀⠀⢠⠀⠀⠀⠀⣤⣀
⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁
⠄⠀⠀⠰⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠀⣠⡄⠀⠀⢀⡀⠀⠀⠸⠗⠀⠀⠀⠂⠀⠀⠀⠐
⠀⣤⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠃⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠸⠇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐
⢰⠄⠀⠀⠂⠀⠀⢀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠀⠠⠄⠀⠁⠀⠀⠀⠀⠐⠷⠀⠀⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⢠⡄
⢠⠄⠀⠁⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢷⡧⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⢰⣴⡄
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠉⠀⠀⠀⡄⠀⠀⠀⠶⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠈
⢀⠀⠀⠈⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⣠⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⣰⣦⠄⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠆⠀⠀⠛⠀⠀⠀⠰⠀⠀⠀⠀⠆
⠀⠘⠿⠅⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠂⠀⠀⠛⠀⠀⠀⠀⠀⠐
⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠙⠃⠀⠀⡀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠂⠀⠄⠀⠀⠰⠆
⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⡀⠀⠰⠆⠀⠀⢀⡀
⠙⠃⠀⠠⠀⠀⠁⠀⠀⠀⠛⠃⠀⠀⠠⡄⠀⠀⠀⣠⡤⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐

⣹⣿⡟⠉⠀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠿⠛⠁⠀⠀⠀⠀⣹⣿⡟⠉⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠿⠛⠁
⠉⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠠⣤⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼⢿⣿⠁⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠠⣤⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼⢿⣿⠁⠀⠀⠀⢠⣴⣦⡄
⠀⠀⠴⠿⣿⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠴⠿⣿⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠛⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠂⠀⠀⠀⠀⣤⣼⣧⣤⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠂⠀⠀⠀⠀⣤⣼⣧⣄⠀⠀⠀⠀⢠⣤
⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⠿⠁⠀⠀⠀⠀⠀⡿⣿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⠿⠁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⢠⣴⣦⡀
⣴⣦⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠰⠟⠻⠁⠀⠀⠀⠀⣤⣼⣇⡀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠰⠟⠻⠁⠀⠀⠀⠀⣤⣼⣇⡀
⠛⠏⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⠋⠁⠀⠀⠀⠚⠛⠏⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⠋⠁
⠀⠀⠀⡀⠀⠀⠀⠰⠿⣿⠷⠆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⠷⠆⠀⠀⠀⠀⣤
⠀⢤⣾⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠂⠀⠀⠀⢠⣤⣶⠀⠀⠀⠀⢤⣾⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠂⠀⠀⠀⢠⣤⣶
⠀⠀⠿⠛⠓⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠰⠿⣿⠃⠀⠀⠀⠀⠿⠛⠓⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠘⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠁⠀⠀⠀⢀⣄
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⢿⣿⡃⠀⠀⠀⠠⣤⣦⣀⠀⠀⠀⢀⣀⣴⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⡃⠀⠀⠀⠀⣤⣦⣀
⣹⣿⡟⠉⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠿⠛⠁⠀⠀⠀⠀⣽⣿⡟⠋⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠿⠛⠁
⠉⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠠⣤⣴⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼⢿⣿⠁⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠠⣤⣴⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼⢿⣿⠁⠀⠀⠀⢠⣴⣦⡄
⠀⠀⠴⠿⣿⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠴⠿⣿⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠂⠀⠀⠀⠀⣠⣴⣷⣄⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠂⠀⠀⠀⠀⣠⣴⣧⣀⠀⠀⠀⠀⢠⣠
⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠿⠁⠀⠀⠀⠀⠀⡿⣿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠿⠁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⢠⣴⣦⡀
⣶⣦⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠰⠟⠻⠁⠀⠀⠀⠀⣤⣼⣇⡀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠰⠟⠻⠁⠀⠀⠀⠀⣤⣼⣇⡀
⠛⠏⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⠛⠁⠀⠀⠀⠚⠛⠏⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠿⡿⠷⠆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠿⡿⠷⠆⠀⠀⠀⠀⣤
⠀⢤⣾⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠂⠀⠀⠀⢠⣤⣶⠀⠀⠀⠀⢤⣾⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠂⠀⠀⠀⢠⣤⣶
⠀⠀⠿⠛⠓⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠋⠁⠉⠀⠀⠀⠀⠰⠿⣿⠃⠀⠀⠀⠀⠿⠛⠓⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠐⠿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠁⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢀⣄
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⡃⠀⠀⠀⠠⣤⣦⣀⠀⠀⠀⢀⣀⣴⣆⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⡃⠀⠀⠀⠠⣤⣦⣀

⠀⠀⠀⠀⠀⠎⠆⠢⡀⠀⠀⠀⠀⠀⢀⡄⣄⠀⡄⢠⠀⡠⢀⠀⠀⠀⢈⠍⢩⠁⠀⠀⢸⠍⠨⠁⠀⠀⠀⣱⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢰⠁⢨⠡⠒⠢⠍⠀⢑⠀⠀⠀⠈⡏⢸⠐⠃⢸⡂⠂⠙⠁⠀⠀⢀⠂⢸⠀⠀⠀⢸⠖⠐⡀⠀⠀⠀⢈⠆⠀⠲⠀⠀⠀⢎⡱
⠀⠀⠀⠶⣈⠸⠃⡨⢠⠱⡃⡀⠀⠀⠀⡇⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠸⢀⠀⠀⠸⡃⠀⠀⠀⠀⠀⢆⢇⠀⠀⡕⠀⡰⡁
⠀⠀⠘⡠⢘⡁⠤⠄⣏⣀⡑⠁⠀⠀⠀⡇⠸⠠⠀⠘⠁⠄⠠⠀⠀⠀⠘⡣⡀⠑⠕⠨⠊⢀⢬⠂⠀⠀⠀⢈⠆⠐⠺⠀⠀⠈⡸
⠀⠀⠀⠁⠀⠑⢀⠔⠀⠀⠀⠀⠀⠀⢐⡃⣀⡀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠈⠊⠤⠂⠠⠤⠓⠁⠀⠀⠀⠪⠹⠀⠀⠀⠑⠐⠏⠕

⠀⠀⠔⠔⠂⠂⠀⠀⠀⠂⠢⠢⠀⠀⠀⠀⢀⠠⡄⠀⠀⠀⠠⡀⠀⠀⠀⡖⡒⠒⠒⠂⡂⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⣤⣶
⠀⠀⡇⠀⡤⣀⢤⠠⣀⢤⠀⠀⠀⠀⠀⡐⠕⣡⠂⠔⣄⠢⠈⠀⠀⠀⠀⣇⣉⣀⣀⣀⡇⢠⠤⡤⠀⠀⠀⠀⠀⠀⣼⣤⣏
⠀⠀⡇⠀⠓⠉⠒⠒⠉⠚⠀⠀⠀⠀⠀⢀⢈⠄⢎⡱⠰⡁⠀⠀⠀⠀⠀⠜⣃⣠⣀⡃⠄⠀⠓⠁⠀⠀⠀⠀⡀⢾⣤⠓⠋⠂
⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⡀⠢⠙⠔⠢⢋⢔⠌⠀⠀⠀⢀⡸⣀⢀⠀⠄⡈⢇⡀⠀⠀⠀⠀⠀⠁⠅⣢⠴⣦⠂
⠀⠀⠀⠀⠠⠘⠁⠘⠃⠄⠀⠀⠀⠀⠈⠂⠈⠈⠁⠀⠂⠁⠀⠀⠀⠀⠸⠬⠧⠧⠧⠧⠧⠀⠀⠀⠀⠀⠀⠀⠦⠤⠭⠬⠬⠵

⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⠀⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⡐⠄⠀⠀⠀⣆⣀⣀⣀⣀⣀⣀⢂⣀⣰
⠀⠀⠠⡀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠂⢸⠀⠀⠀⠀⠓⠀⠀⠀⢆⡂⣂⠶⠀⠀⠀⡂⠀⡤⣀⡰⠂⢰⠒⡄⢸
⠀⠀⠀⠈⠪⡀⠸⠢⣀⢀⢠⡄⠀⠀⠀⠦⠀⠀⡴⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⡂⡉⢠⢈⠀⡀⣜⣉⣢⢸
⠀⠀⠀⠀⠀⠈⠢⡠⠂⠀⠀⠀⠀⠀⠀⠘⡐⠂⠇⣀⣀⣀⡸⠀⠀⠀⠀⠐⠩⠠⡃⠀⠀⠀⣬⠆⠀⠀⡟⣹⣽⣿⡿⠯⠏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠘⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠈⠔⠀⠀⠀⠀⠀⠀⠀⠯⠘⠚⠛⠚⠚⠋⠉⠀⠸

⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣠⣆⣰⣄⣦⠀⠀⠀⠀⠆⠀⠀⠤⡀
⠀⠀⠀⠂⠠⠂⠌⠁⠒⡀⠀⠀⠀⠀⠀⡠⠵⡀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣯⠐⠀⠀⠀⠀⠂⣽⠂⠀⠀⡃⠀⠀⢋⢣⢄
⠀⠀⠀⠀⠀⠁⢲⡆⠀⠀⠀⠀⠀⠀⠈⡰⡀⢸⢘⣂⣀⣀⣊⡀⠀⠀⢈⣷⠄⠀⡁⢈⠀⠀⣿⣁⠀⠀⡃⠐⠃⢰⠀⠱⠀⠠⡲
⠀⠀⠀⠐⠈⢢⠾⠷⡔⠁⠂⠀⠀⠀⠀⢩⣀⣈⠀⠀⠈⠛⠁⠀⠀⠀⠠⣟⠁⠀⠀⢀⠀⡄⣻⠄⠀⠀⡆⠀⠀⠰⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠘⢇⠼⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄⠀⢄⢒⣢⠀⠀⠀⠀⠈⢻⠛⡟⢻⠋⡟⠁⠀⠀⠀⢇⣀⣀⣈⣀⣀⣈⣀⣀⡀

⠀⠀⠀⠀⡞⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⢻
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠛⠿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⡴⠋⠀⢰⡆⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⣠⣾⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⢆
⠀⢸⣁⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣀⣠⣤⣄⡀⢈⣧
⠀⢸⣿⣿⡛⢳⣶⣤⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡀⠈⠗⠀⢠⣬⡁⠈⢧
⠀⠈⢻⣿⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠙⠋⠀⠀⠀⠹⠿⠃⠀⢈
⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⡄
⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⢿⣿⠇⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣀⡀⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢒⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠻⡇⠄
⠀⠀⠈⢩⣶⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡾⠇⠅
⠀⠀⣴⣾⣿⣷⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠭⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠿⡿⢣⠆
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⡏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣮⡀⣸⡖⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⢈⣆
⠈⠢⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢦⢴⡷⠛⠳⣷⢤⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢈⣧⠤⢹⡇⠤
⠀⠀⠈⣇⢰⠁⠹⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠫⣀⣨⢻⡄⠚⠂⡼⢧⣀⡸⠗⠀⠀⠀⠀⠀⠀⠠⠤⡷⠤⢵⡁⠀⣠⡧
⠀⠀⠀⢡⢸⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⣏⠿⣏⡏⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⡵⠒⠛⠁
⠀⠀⠀⠸⠇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠊⠱⡀⡰⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺

⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⠋⠹⢿⠟⢁⡤⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣠⣴⣿⣃⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠙⠋⠉⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠙⠉⢉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠠⣴⣶⣶⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⡁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂⠾⠛⢛⡃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡒⠒⠂⠀⠉⠉⠉⠀⠈⠣⢠⢿⣠⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠞⡛⠋⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣆⠻⣧⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡤⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢂⡈⢿⡦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣷⣶⣬⣄⣱⡀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⡂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡏⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣃⠈⡙⠛⠿⢿⣿⡿⢿⣿⣿⣷⣶⣲⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⡿⠿⠿⠿⠿⣿⡅⠹⣾⠟⠛⠻⢿⣾⣿⠊⢷⡀⣹⣿⣮⣾⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣾⡿⠳⣮⣄⠀⠀⣬⡳⣠⡏⠀⠀⠀⠀⠉⠉⠘⠋⢃⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⠛⠉⠀⣠⣾⠏⠉⢃⠞⠀⠁⣁⠗⣈⢀⣿⡛⠻⣦⣀⣤⠀⠤⣶⡴⢎⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠐⢿⣻⣿⣷⣟⣃⣴⡿⣿⣾⣵⢟⣙⣧⣴⣾⣿⡋⢾⣿⣯⣴⣿⣷⣿⣿⣿⣿⠿⠿⠟⠛⠁⠀⠀⠘⠉⠐⠤⠠⠝
⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⣀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠉⠛⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡊⠉⠀⠀⠀⢖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠦⠴⠶⠿⠅
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠂⠂⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⡀
⠀⠀⠀⠀⠀⢀⠚⠀⠀⡉⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠉⣂
⠀⠀⠀⠀⠀⢘⠀⠀⠙⠇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠁⢨
⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠈⣀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⣀⣤⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢
⠀⠀⠀⠀⠀⠀⡔⠊⠁⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠃⠈⠁⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠭⠀⠉⠲⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡤⠀⣀⣀⢸
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡂⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠁⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡠⢀⡂⢀⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠐⠽⠞⠀⠀⠀⠀⠱⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠸⢨⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣇⡀⠀⡀⠄⠀⢨⡉⠆
⠀⠀⠀⠀⠀⠣⡳⡀⠀⠀⣰⣭⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠚⠉⠒⠈⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠢⢄⠘⠚
⠀⠀⠀⠀⠀⢀⠃⠀⢀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡋⠀⠀⠀⠀⡸⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠒⢄⣠⠒⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠘⠢⠀⠄⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠓⠎⣀⣀⡱⠚⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡰⠊⠁⠀⠈⠓⢄⡄
⠀⠀⠀⠀⠠⣲⣇⡀⠀⠀⠀⣀⣸⣖⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣰⣀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠱⡉⠈⠉⠉⢁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠑⠦⢤⢜⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠒⠺⣿⠛⠛⠛⠛⠛⠛⠛⣿⠗⠲⠶⠶⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠞⠛⠉⠉⠻⢦⡀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣤⣀⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢻⡄⠀⠀⠀⢰⡏⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⣤⣭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠈⠉⠳⢦⣄⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⣸⠇⠀⠀⠀⢠⡟⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠃⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⠶⠚⠛⠛⠛⠓⠶⠦⣤⣄⣀⣀⣀⠀⣀⡀⣀⠀⡀⢀⣀⣠⣤⠴⠶⠚⠛⠛⠛⠓⠶⣤⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠉⠉⠉⣿⠉⠉⠉⠉⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⣀⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⣤⣤⣨⣷⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣿⣀⣤⣤⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠈⠉⠉⣿⠛⠛⠛⠛⣶⠚⠛⠓⣶⠛⠛⠛⠛⣿⠉⠉⠁⠀⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡇⠀⠀⠀⢻⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠶⠶⠶⣤⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⢀⣾⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠻⢦⣄⡀⠀⠀⠀⣤⠶⠶⠶⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠞⠋⠉⠉⠀⠀⠀⠉⠛⠓⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠛⠛⠉⠁⠀⠀⠉⠉⠙⠳⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠧⣤⣤⣤⣄⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣤⣤⣤⠿⠶⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⡏⢙⠀⠀⠀⡆⠀⠸⡆⠀⢰⡄⡇⠀⠉⠁⠀⡂⢸⠀⢸⡆⣾⠀⠀⠀⠀⠉⠁⠀⡇⢸⠀⠀⣄⠃⠀⢏⠙⠀⢸⠀⢸⠉⠁⠀⢌⠙
⠀⠀⠀⠀⠀⡇⣼⠀⢀⠀⡃⠀⡠⢇⠀⢈⠰⡇⠀⠀⠀⠀⡇⢸⠀⠀⠟⣿⠀⠀⠀⠀⡇⠁⠀⡏⢸⠀⠀⢹⠀⠀⡈⢣⠀⢸⠀⢸⠀⡀⠀⡀⢱
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁


⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠧⠤⠟⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡁⢸⠁⠉⣦⢸⠈⢹⠁⣐⢋⡀⢸⣉⡇⢸⠀⠀⣏⡁⠀⠐⢯⡁⠈⡏⠁⣿⣨⠇⡜⠁⠙⡂⣣⡜⠀⣯⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣤⣸⣤⣴⣃⣸⣀⣸⣀⣏⣈⣣⣸⣤⣇⣸⣤⣀⣧⣤⣀⣠⣤⣝⣀⣇⣀⣏⣀⣇⣑⣤⣼⣁⣇⣘⣄⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡴⠚⠛⠙⠛⠶⡄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡴⠋⠁⢀⡀⠑⠉⣀⠀⠀⠠⠠⡤⡈⠆⠀⠀⠈⠙⢳⡄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣴⠟⠉⠀⠀⠘⠁⠐⠉⠁⠀⠔⠉⠂⠔⠒⠤⢀⠀⣀⣀⠀⠤⡀⠀⠀⠙⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠐⠉⠋⠉⠀⠀⡠⠀⠀⠀⠀⠀⢀⣀⡀⠊⢰⠁⠀⠀⠀⠈⠐⠢⠀⠀⢀⡀⠀⠀⠸⠄⠐⠒⠶⣄⡀
⠀⠀⠀⠀⢀⡴⠚⡉⠉⢁⡀⠒⠲⠀⣀⠖⠀⢀⡔⠂⢀⡀⠀⢀⡀⠀⠀⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠒⢄⣀⠀⠀⠙⠊⠂⠀⠀⠀⠠⡄⠈⢳
⠀⠀⠀⢀⡏⡤⠚⠁⠀⠃⠈⠀⠀⢈⡁⠀⠀⠁⠀⢰⠁⢀⠄⢎⠀⠀⠀⠀⠀⠀⠀⠓⡆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠤⡀⠀⠀⠀⠀⠘⡀⠸⠁⠓⣦
⠀⠀⡴⠊⠀⠀⠀⡠⠀⠀⣠⠴⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⢐⠀⠀⠀⠀⠀⠀⡀⢸⠂
⠀⠀⡇⠀⢠⠀⠀⠳⠄⠀⠁⠀⠀⡺⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⠀⡀⠃⠈⠈⢳
⣠⠂⡈⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠣⠀⠀⠀⡐⠂⠀⠀⢀⠔⠐⠉⠀⠀⠚⢢⢀⡀⠀⠀⠈⠀⢄⠘⡦⢀⠀⠈⠙⠀⡀⠀⠀⢸⠀⠠⠀⢠
⣇⢰⠃⠈⢁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠀⠀⠞⠁⢠⡀⠀⠀⠀⠀⠠⡄⠦⠀⠀⡀⠁⠀⠀⠀⠀⠈⠇⠀⠘⢁⠀⠀⠀⡸⠀⠈⠁⢠⠖⠡⠞
⣘⠢⠁⠀⠨⠀⢀⠈⠀⠀⠀⠀⢤⠛⠀⠀⠠⡁⠀⢀⡮⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢤⡀⠀⠀⠀⠀⠀⠀⠉⡢⠀⠀⠀⢠⠀⠈⡀⠀⣠
⠸⣆⡀⢀⠑⠂⠜⠀⠈⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⢠⣿⡷⠶⠻⡿⢶⣤⣴⢾⣦⣀⣀⣤⡆⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠚⠀⠸⠁⣠⠇
⠀⠈⠙⣍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠂⠀⠲⢏⠈⣿⠀⠀⠀⠀⠀⠏⠀⠈⠉⣿⠇⣹⡶⠆⠀⠀⠀⠀⠀⠠⠤⠊⠀⠀⡀⠀⠀⠀⠰⠚⠁
⠀⠀⠀⠈⠉⢋⠀⠀⠰⣀⡀⢤⣀⡀⠀⠀⠀⢀⠀⠀⠀⢀⡈⡇⣿⢠⠀⠀⡇⡀⢠⠀⠀⠀⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢹⢸⠀⠀⡇⡇⢸⠀⠀⡆⢹⠀⠀⠀⠘⠤⠐⠢⠤⠀⠀⠀⠀⢄⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢤⣀⣀⣀⣠⡆⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠃⠸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠳⠦⠤⠤⠊⠒⢺⢸⠀⠀⠀⡆⢠⠀⠀⡇⢼⠒⠤⣤⣀⣠⠴⠈⠒⠲⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠇⠸⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⡇⠶⢸⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠊⠐⠂⢸⡅⠀⠀⠀⡀⠀⠐⢀⠀⠀⢻⡇⠠⢤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠒⠀⢦⡿⠿⠶⣤⠴⠷⣄⠦⠾⢤⠶⢾⡧⠀⠐⠁⢲⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⠒⠳⠀⠠⠂⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠠⡄⠀⠀⡪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠄⠞⠀⠀⢀⠆⡁⠀⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣌⣁⣴⣄⠉⠀⢀⣄⠁⠉⠀⣀⠀⠉⢁⣤⣉⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⠛⠛⠛⠛⢙⠛⠋⠉⠹⠛⠋⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⣀⣰⠞⠛⡛⢦⠶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠶⢯⡽⠛⠺⢿⣛⠶⢶⡞⠛⢷⠦⣤⣠⣮⣬⣇⠴⢎⣉⣤⣯⣉⠉⠳⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⠰⠋⣁⡥⣴⠚⠆⠀⠂⠰⠹⣤⠴⠛⠒⠳⣤⠎⠁⠀⠒⠈⢳⠞⠱⠂⠤⠍⢳⠘⠓⢦⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠃⢉⠶⠻⡏⢀⡁⠴⣋⣙⡛⣦⢰⠋⡠⠢⡔⢀⡼⠛⠉⠛⢲⡴⢛⣛⠲⣄⠀⣠⠴⢧⣄⣸⠇⣜⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠛⠀⠋⠂⠈⠀⠈⠊⠀⠈⠘⠘⢈⣀⠀⠀⠀⢸⠀⠒⢶⠁⠸⠀⠈⠀⠁⠈⠆⠁⠀⠀⠘⡇⢀⡿⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡴⣚⣛⡒⢦⡀⢀⡴⢒⣛⡒⢦⡀⠀⢀⣴⡾⠿⢶⣤⡀⢀⣴⡶⠿⣶⣤⠀⠀⢀⠔⣊⣉⣉⡒⢤⣀⠤⢒⣉⢩⡑⢦⠀⠀⣠⡴⠶⢤⡀⠀⠀⣠⢀⣤⠶⠶⣄
⢸⠸⡅⠀⢈⠗⣩⣍⠾⣁⠀⣨⢆⡇⠀⢺⣏⠀⠀⠙⣿⣿⣿⣝⠁⠀⠀⣽⡇⠀⢸⢸⡁⠀⠀⣩⠶⣹⠶⣉⠀⠀⢸⠀⠇⠀⡏⠀⠀⠈⢃⡴⣟⡵⢋⡀⠀⠀⢸
⠀⠑⠢⠤⠔⠋⠁⠈⠛⠾⠭⠵⠊⠀⠀⠀⠙⠷⢶⡿⠯⠛⠛⠿⢿⡶⠿⠋⠀⠀⠈⠢⢭⣀⡩⠴⠊⠉⠓⠬⢁⣀⠥⠈⠀⠀⠙⠶⠴⠶⠋⠞⠉⠀⠀⠳⠦⠴⠞


⠀⢀⠤⠀⠀⠀⠀⣠⣖⡭⠿⢥⡀⠀⠀⠀⠐⠂⠀⠀⢀⣤⢆⣠⣴⣶⣦⣄⠀⠀⠀⢀⢤⣤⣤⣀⠀⠀⠀⣠⣤⣤⢤⡀⠀⠀⠀⠔⠲⢒⠠⡀⠀⢀⠄⡒⠖⡠⡀
⠀⣇⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⡟⠀⠀⢰⣀⢀⣀⡴⢋⣷⡿⣫⡴⠚⠉⠘⠁⠀⢀⡃⡋⡎⠀⠈⢑⢶⡋⠀⠈⡧⢱⢱⠀⠠⠁⠌⠀⠀⠈⠊⢿⠅⠋⠀⠀⠈⡎
⠀⠈⠻⣿⠯⠞⠁⠀⠀⠀⠀⠊⠀⠀⠀⠈⠳⢿⣷⠾⠟⡥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠑⠽⠾⠶⠚⠁⠀⠙⠲⠾⠴⠵⠋⠀⠀⠑⠬⣂⣐⠪⠂⠀⠐⠕⣂⣔⠪⠊


⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⡀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣀⠄⢀⡀⠀⠀⣠⣤⡶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠀⠀⠉⠛⣦⡶⠟⠉⠉⠉⣻⠀⠀⠐⠋⠀⠀⠈⠙⣷⣾⡟⠁⠀⠀⠉⠀⠀⠀⡞⠀⠀⠀⢀⣴⣾⡿⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⢦⣤⣤⣶⠿⠋⠻⣦⣤⣤⣰⠟⠁⠀⠀⠐⢦⣤⣴⠾⠋⠉⠻⣷⣦⣤⡶⠃⠀⠀⠘⢦⣠⣰⣷⠟⠉⠑⠢⢤⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉

⠀⣀⣤⣄⡀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⢠⠒⠤⢂⢄⠀⠀⠀⠀⣀⣀
⢰⡿⠋⠙⠿⠄⣠⣾⠟⠋⠉⢻⣷⠀⠀⣰⣿⠟⠛⠿⣷⣤⣀⠦⠑⠒⠉⢂⠀⠀⢀⢣⠀⠉⠁⠠⣡⠖⡠⠒⠉⠀⠠⠪⠂⠀⣰⡿⠛⠛⠻⣶⣄⣤⣾⠿⠿⢿⢆
⠸⣿⣤⣠⣴⡾⠟⠁⢺⣦⣤⣼⡿⠀⠀⢹⠡⠀⠀⡀⢌⠻⢿⣦⣄⣀⣠⣿⠆⠀⢈⠈⠀⣀⡀⠔⠠⠶⠁⠢⣀⣀⡐⠈⠀⠀⣝⣆⠀⢀⣠⣴⣿⣏⡀⠀⠀⢈⣷
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠑⠊⠉⠂⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠑⠠⠖⠌⠀⠀⠀⠀⠈⠐⠒⠈⠀⠀⠀⠈⠻⠿⠿⠛⠉⠀⠙⠻⠶⡶⠿⠋

⠀⣠⣤⣤⣄⡀⠀⠀⣠⠤⠄⠤⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⣠⠤⣐⣤⠤⣀⠀⠀⠤⣒⣀⠤⣀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⣀⣤⣤⣄⡀
⡞⠁⠀⠀⠉⢻⣶⣏⠁⠀⠀⠀⠈⡀⠀⣰⡟⠉⠉⠙⠻⣦⣜⠛⠁⠀⠉⢿⡆⠀⡜⡕⠉⠀⠈⠙⠪⣳⠖⠉⠀⠀⠙⡎⠆⠀⣾⠋⠁⠈⠙⢻⣶⡾⠋⠉⠀⠙⣷
⠀⠤⢀⠤⠖⠋⠈⠛⢷⣤⣤⣤⠞⠀⠀⠘⢷⣤⣤⣤⡾⠊⠛⢷⣦⣤⣴⡿⠃⠀⠱⡱⢄⣀⣀⠤⠚⠯⡳⢤⣀⢀⠴⣣⠃⠀⠹⣷⣤⣤⣶⠟⠉⠻⢷⣤⣤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠈⠙⠒⠒⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉

⠀⣠⠤⠤⢤⣀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⢀⣤⣤⣤⣀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⡀⠤⠤⣀⠀⠀⠀⡀⠤⠤⣄⠀⠀⠀⠀⢀⡀⡀⡀⠀⠀⠀⣀⢠⡀⣀
⡞⣡⠚⠉⠓⢮⣡⢞⣩⠖⠋⠙⢦⠹⡄⠀⣿⠁⠀⠈⠙⢻⣶⡟⠋⠀⠀⠈⣷⠀⠀⣼⠊⠉⠁⠊⣕⣤⡬⠒⠀⠀⠸⡆⠀⠀⠤⠃⠀⠀⠈⢀⡰⠶⠙⠀⠀⠉⣖
⢳⡘⠦⠤⠴⣊⡵⠋⣙⠲⠤⠤⢎⡴⠁⠀⠻⢦⣤⣴⡾⠛⠉⠻⠷⣦⣤⡼⠋⠀⠀⢻⡀⠀⠀⣠⠜⠋⠝⠲⠤⠤⢞⠃⠀⠀⠘⢢⢀⡄⢴⠘⠃⠐⢤⡀⡀⡰⠏
⣤⣬⣿⣾⣿⣥⣤⣤⣬⣽⣷⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣯⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣼⣧⣀⠀⠀⠀⠀⠀⠀⣀⣣⡀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀
⠀⠀⠈⠉⢻⡟⠉⠁⠀⠀⠀⠀⠀⢑⡠⠫⡀⠀⠀⠀⠀⠑⠆⣀⣀⠚⠁⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠁⢱⢿⡏⠁
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⠃⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠐⠋⠀⠑⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣦⠀⢠⠆⠩⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⣤⡀⢛⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠢⣤⡀
⠀⠀⠀⠀⠐⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠒⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄
⠀⠀⠁⠘⢃⠂⡈⠀⠀⠀⠀⠄⠀⠀⠑⡒⠀⠀⠀⠀⠀⠤⢠⠟⢲⡶⠀⠀⠀⠲⣿⣿⣿⣤⡤⠞⠁⠀⠀⠐⢀⠈⡠⠂
⠀⠀⢠⡄⠈⠚⢠⠶⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠀⠀⠀⠀⣙⠶⢭⡆⠀⠀⠀⠸⠋⠉⠃⠀⠀⠀⠀⠀⠀⢠⠄⠁⠰
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⠀⠐⠀⡀⠀⡀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⡄
⠀⠠⢴⣿⡶⠂⠀⠀⠀⠀⠀⠀⠉⠈⠀⢉⠐⠁⠀⠀⠀⠒⠿⣭⢭⡉⠀⠀⠀⠠⢤⣥⡤⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡍
⠀⠀⠞⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠛⠀⠈⠑⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠟⠁⠉⠓
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⠀⠀⠀⣀⡀⣿⢄⣠⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡟⣿⡉⠁⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀
⠀⠀⠀⢠⠀⠤⠀⠀⠐⠀⠀⠀⢄⡄⠀⡷⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠒⠀⠀⠀⠀⢺⠋⠀⠰⠛
⠀⠠⠁⡐⠀⠰⠖⠀⠀⠀⠀⠀⠀⣑⡤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡀⠀⠀⠀⠐⠿⠈⠈⠳⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠋⠀⠈⠻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣀⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡀⢠⣶⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⠉⢹⠳⠦⣄⡀⠀⠀⣀⣤⠶⢻⠋⠁⠈⠉⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⣩⠷⢯⣁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢰⣇⣠⠖⠋⠀⠀⠀⠈⠓⢦⣀⣷⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⣤⡴⠋⢸⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⢸⠈⠑⢤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄⢸⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⢸⢀⡴⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠋⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠘⡇⠙⠲⣄⡀⠀⠀⣀⡴⠚⠁⡾⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣹⠶⢾⣉⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣀⣀⣼⡶⠚⠉⠀⠀⠀⠈⠙⠲⣾⣄⣀⡀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠺⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣠⠴⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿