Muscle bee Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠞⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢧⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⣃⣀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡅
⠀⠀⠀⣀⢤⡖⢤⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠉⠀⠈⠉⠳⣄⢀⠤⠒⠂⠚⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠤⣄⡀
⠀⠀⡸⢥⠀⠘⣄⡵⠞⢶⠤⢤⣀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠶⣁⠞⠀⣈⠆
⠀⠀⡇⠀⠳⡴⠫⢀⠤⠈⠀⠀⠈⠉⠢⢄⡀⠀⢀⡇⠀⠀⠀⠰⣿⠂⠀⠀⡗⠾⠆⠀⠀⠀⠀⣸⣄⡀⠀⠀⢀⡠⠖⠋⠁⠀⡇⠠⢄⠼⢦⠔⠁⢸
⠀⢰⠙⠀⠀⠁⠀⣀⡠⠞⠀⠀⠀⠀⠀⢀⣽⠞⠋⠙⣄⠀⠀⠀⠀⠀⢀⡴⠣⣀⠀⠀⠀⢀⡴⠁⠈⠙⠳⣖⠉⠀⠀⠀⠀⠀⠙⠤⣀⠀⠈⠀⠀⠚⡄⠀⠠
⠀⢠⠗⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠈⠑⠢⠤⠤⠖⠋⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢡⡤⠀⠀⠀⠸⡄
⢠⡏⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡾⠁⠐⠲⠆⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣠⠀⠙⣦⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢳
⣸⠀⠀⠀⠀⠀⠀⢸⠄⠒⠛⢲⡞⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⣰⠃⠀⠀⠘⣗⠛⠛⠒⢴⡇⠀⠀⠀⠀⠀⠸⡆
⡇⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⡾⠀⠀⠀⠀⠈⣇⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡴⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣀⡀⠠⠴⠊⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠐⡈⢱⠦⠄⣀⣀⠀⠀⠈⣁⡀⠠⠄⢲⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠉⠳⢤⣀⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⠀⠀⠀⡏⠀⠀⠀⢸⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⢰⠏⠉⠓⠒⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⢀⡇⠀⠀⠀⢸⠀⠀⠀⢀⡷⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠒⠚⠻⡅
⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⣧⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠼⠦⠐⠒⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣼⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⣇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣸⠃⠀⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⢻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠉⠛⠛⠒⠒⠒⠲⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠖⠒⠚⠛⠛⠛⠉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⢖⢒⡒⠲⠶⣿⡇⠀⢀⡸⠿⠖⢚⣛⡻⢒⡶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠛⢋⠉⠀⡀⠄⣂⡽⠀⢯⣐⠀⠀⡀⠈⡙⠋⠁⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠑⠒⠂⠐⠒⠚⣉⣡⠴⠚⠉⠀⠀⠀⠈⠙⠲⢤⣉⡑⠒⠒⠒⠒⠉⣀⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠗⠀⠀⠀⠠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠀⣿⠿⡭⠋⣡⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠂⠀⠀⠉⢀⣠⣤⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣲⣶⣤⡤⠀⢀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠢⠀⠀⠂
⠀⠘⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠂⠁⠀⠀⠀⠀⠀⠈⠀⡀⠁⠀⠈⡐
⠀⠀⠐⠲⠯⠭⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠈⡏⣶⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⣿⣦⣄⣀⡒⠶⠦⢤⡄⠀⠀⠀⠀⠀⢀⣠⣀⣀⣒⠲⠦⠼⣥⣤⣀⣃⢻⣿⣿⣷⣦⡄⠀⢀⣀⣈
⠀⠀⠈⠙⠛⠿⢿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠰⠦⣭⣭⣍⣛⠓⠶⠦⢬⣈⠘⣿⣿⣿⣿⡇⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢛⣿⣿⣶⣶⣶⣶⣤⣤⣭⣀⣀⡀⠀⠀⣠⣿⣿⣿⣿⣷⡄⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⡿⠟⣛⣛⠻⣿⣿⠛⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡴⣢⠀⠀⠀⢀⡀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⡜⠋⠑⣦⣀⣞⠀⠀⠀⠀⠰⣿⣟⣟⣭⣳⣦⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣾⣯⣽⠿⠀⢀⣼⣿⣻⣿⡳⣄⡀⠀⠛⠛⠁⠀⢻⡜⣿⣯⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣯⠀⠀⠀⢀⡼⣿⣿⣯⣿⢿⣿⣯⣷⢻⢿⣷⡶⣶⣿⣮⣟⡶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣲⢾⣿⣧⣷⣷⣂⣿⣸⣿⣿⡿⢿⣷⣿⣾⡿⣿⣛⣿⢝⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣁⢀⠀⠀⠊⠁⠀⠁⠀⠀⠀⠀⠈⣿⣾⣿⣷⣿⠿⣿⣷⣿⣿⣟⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⡫⠿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣯⡿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣎⡉⣊⣿⣽⣿⣿⣾⣽⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡉⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣼⣿⣯⣽⣿⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠁
⠀⠀⠤⣴⣆⣠⣾⣿⣤⠀⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣽⢟⣿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⡤⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣯⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣧⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣯⡽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠛⠛⠛⠛⠛⠛⠲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣷⣴⣚⢿⣿⣿⣷⣶⣤⣴⢶⣖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣷⠝⣿⣿⣿⣿⣿⡎⣿⣷⣶⣶⣦⠤⣤⣀⣀⠀⠀⠀⣀⡼⠁
⠀⢀⣀⣀⡀⠀⠀⣀⣤⣶⣾⣷⣶⣍⢻⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣦⠈⠉⠙⠛⠛⠉
⠛⢋⣭⠭⠟⢳⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⢸⣿⣿⣿⣯⣿⡸⣿⣿⣿⣿⣿⣿⣷⡄
⠐⠋⠀⠀⣰⣿⣿⣿⠿⠋⠑⣿⣿⣿⣿⣿⣿⣿⣿⣟⣈⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢷⣮⣟⢿⣿⣿⣿⣿⣿⢘⣛⣛⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣧⠀⠀⠀⣀⣼⣿⣿⡿⣫⣿⣿⣿⣿⢼⠿⠟⣿⣿⣿⣿⣿⠟⣼⣿⣿⣿⣷⣮⡙⢿⣿
⠀⠀⠀⠀⠸⣿⣧⣴⣾⡿⠿⠛⠉⠀⠻⣿⠿⣫⡴⣾⣿⢹⣿⣿⣿⡟⣡⣮⣭⣝⡻⢿⣿⣿⣿⣦⡙
⠀⠀⠀⠀⠀⠙⢿⡏⠁⠀⠀⠀⠀⠀⠀⣡⣾⣿⣯⣻⢿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣮⣭⣛⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣿⣿⣿⣿⣿⣿⣮⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣌⡻⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣦⡹⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠐⠒⠶⠶⠿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⣶⡶⡶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣉⠉⠉⠙⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⣀⡠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣄⡀⠀⠈⠓⢄⠀⠀⠀⠀⢀⡀⠀⠀⢠⢀⡠⠀⠀⠀⠀⡠⠊⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⣀⡀⠀⠙⠲⣄⠀⠀⠱⡄⠀⢀⣀⡽⠿⠽⠺⠿⠤⣤⣀⡀⡜⠁⠀⣠⠞⠁⢀⣤⠶⠚⠛⠛⠲⣤⡀
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠛⢶⣄⠀⠈⠳⣄⣠⣽⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡞⠁⣠⡶⠋⠁⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣠⡾⠋⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠈⢿⣄⠀⠀⡐⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⢦⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡜⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢄⠀⠀⠳⣼⡟⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⣿⠀⢠⠎⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠑⢄⡀⣸⡇⠀⠀⣴⣿⡿⠛⠙⢲⡀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠛⢦⠀⠀⢹⡖⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠙⢿⡇⠀⢸⣿⣿⣧⡀⢀⣸⣷⠀⠀⠀⠀⠀⣿⣿⣿⣇⡀⢀⣼⡇⠀⣸⠇⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⢄⡀⠸⣧⠀⠘⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠃⢀⣿⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢭⡳⠿⣦⠀⠈⠚⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠚⠿⠟⠋⠁⢀⣾⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢽⣷⣄⠀⠀⠀⠀⠀⠀⠠⢤⣠⠤⠖⠀⠀⠀⠀⠀⢀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⣄⣀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣨⣿⢻⠟⠒⠒⠒⠒⠺⡟⣿⣥⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⣾⡟⣷⢃⡟⠀⠀⠀⠀⠀⠀⢷⡸⡿⣟⠳⣽⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡡⠞⠁⣿⢳⡏⣸⠋⠒⠤⠤⠤⠔⠚⢹⡇⢿⣿⠀⠈⠑⢦⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠹⣿⢁⣟⡑⠢⠤⠤⠤⠤⠔⠚⣇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢾⡷⢍⣓⠲⠦⠤⢶⣶⠟⣿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡼⠀⠀⠉⠙⠛⠛⠉⠀⠀⢹⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⢀⣠⣤⣶⣶⡾⠿⠛⠓
⢰⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⣴⣾⠿⠛⠋⠉
⣼⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻⣀⣀⡀⠀⠀⠀⢠⡟⠁⠀⢠⡾⠟⠉
⣿⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡴⠋⠉⠉⠉⠓⠒⠒⠿⠶⠶⣴⣏⡀⠀⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⠁⠀⠀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠈⠙⠚⠉⠉⣇
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣠⠟⠀⠀⣼⡟⠁⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⢰⣿⡗
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠘⡇⢰⠇⠀⠀⠀⠸⣇⠀⠀⠀⠠⣿⣿⣷⣄⠀⠀⠀⠀⠈⣿⣇⠈⠄
⠀⠈⢻⣄⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠙⠦⣄⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠙⢿⣶⣷⣶⣿⣿⣿⣿⣧⠀⢠⠀⠀⢻⣿⣷⣬
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠉⠳⣤⣠⠴⠟⠓⠛⠒⠾⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡇⠈⡇⠀⠀⠻⣿⡿
⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡿⠃⠀⠇⠀⠀⠀⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⡔⠀⠀⣠⣤⣄⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀
⠀⣶⠶⠦⠤⢤⣄⣀⣀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⢰⡏⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠘⣧⠀⠀⠀⠀⠀⠀⣈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣧⠴⠶⠚⠛⢋⣉⡉⠉⠳⣦⣤⣤⡶⡇
⠀⠀⠈⠳⣄⠀⣠⡾⠛⠉⠁⠀⠀⠀⢠⣤⡶⠾⣇⡀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠶⠶⠚⠛⠉⠉⠙⠳⢤⠆⠀⠀⢸⡇
⠀⠀⠀⠀⠈⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠋⠙⠓⠶⣤⣠⣾⣿⡷⠖⠋⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡟⠀⠀⣀⡤⣤⣀
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⣼⣤⠞⢡⡶⠋⠉⢀⣠⠼⠷⣄
⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠳⣦⠀⣹⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⡼⠋⠘⢧⣿⠀⠀⣴⠋⢰⠎⠀⠈⢷
⠀⠀⠀⠀⠈⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠒⠒⠒⠚⠳⠤⢤⡤⠶⠚⠛⠢⠀⠀⠘⣯⢀⡴⠛⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠈⠳⢤⡀⢀⡾⠉⠀⠀⠙⢲⣶⠶⣞⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⠀⠎⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢻⣿⣦⣄⣀⣀⠀⠀⣠⡟⢉⠀⠀⠀⣠⣶⠛⠁⠀⠘⢶⣄⣀⣀⣀⣀⣀⣠⠏⠉⠉⠀⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⠘⢿⣿⣿⣿⣭⣉⣉⣉⣉⣉⣩⣴⣿⣿⢿⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠻⣄⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢁⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠞⠳⠦⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣿⣿⣧⡀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠋⠁⢀⣾⣿⡏⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠁⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁⢠⣿⣿⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣾⣿⡿⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠉⠛⠻⠿⠿⣿⡿⠿⠟⠛⠁⠀⠀⣠⣿⣿⣿⠃⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠃⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣿⣿⣿⣿⣶⣤⣤⣀⣀⣀⣤⣤⣶⣿⣿⣿⡿⠟⠁⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣠⣴⡿⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣄⣀⣀⣀⣤⣤⣶⣿⣿⣯⣄⣀⣈⣻⣷⣄
⣶⣴⣶⣦⣶⣶⣴⣦⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣤⣴⣶⣤⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣟⠋⣉⣻⡋⣹⢛⢻⡝⢋⣿⠟⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠣⡀⠀⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠂⡑⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠈⠻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡛⠲⣄⣤⠀⠀⠀⠀⠘⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡑⠦⣠⠙⠆⠀⠀⠀⠀⠈⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡓⠚⠷⣆⡀⠀⠀⠀⠀⠀⢸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠤⠉⠀⠀⢀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠰⣄⠃⠀⠄⠀⢻⡆
⠀⢀⣠⣴⠶⠿⠿⠏⠭⠭⠙⠻⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⠁⠈⠂⠀⠀⢠⠈⣿
⣴⢟⡛⢗⡛⠿⣁⠂⠂⠀⠀⣀⣀⠀⠀⠉⠙⠲⢦⣄⣀⣀⠀⠀⠈⢷⣄⠀⢀⣲⣤⣮⣧⣼⣧⣤⣀
⠙⠶⢭⣽⣃⢷⡏⣭⣉⣐⠢⣄⣀⠐⠂⠀⠀⠀⠐⠬⡙⠿⣿⣷⣦⠀⣽⡷⠛⢣⠋⡀⠉⠩⡖⠩⢻⣿⡄⢀⣀⣤⣤⣀
⠀⠀⠀⠈⠉⠛⠿⠿⣿⣭⣽⣒⣮⣛⣶⣤⠀⠀⡀⠀⠀⠀⠀⠩⡿⢷⣿⣠⣶⣾⣿⣿⣿⣷⣷⡂⠀⢼⣿⡿⠛⠉⠉⠙⠻⣦⡀⠀⠀⣀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣶⣤⣤⣤⣘⣶⢍⠛⠻⣿⣿⠿⠿⣿⣿⣿⣷⣿⣿⠇⢠⡶⠶⣄⠀⠙⣷⡠⢺⠗⠋⠉
⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⡇⠉⠁⣿⣿⣿⣿⢹⠋⠟⣹⣿⡏⠙⢿⣿⣿⣿⣿⣇⠀⢀⡀⠈⣉⣻⣯⢍⣿⡇⣿⡇⠀⠈⣧⢀⢸⣯⣴⢶⣄⣀
⠀⠀⢀⣴⣿⡿⠈⣿⣿⣿⣇⠀⠀⠸⣿⣿⣿⡇⠀⠀⢈⣿⣧⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠋⢱⣿⡄⣿⡅⠀⠀⣿⢨⣹⡟⠉⠀⠙⠢⢕⣂⣄
⠀⠀⣾⣿⣿⡕⠀⠘⣿⣿⣿⣆⠀⠀⠹⣿⣿⣿⣆⢠⣤⣽⣿⣆⠀⡹⣿⡏⠠⠏⠘⢿⠉⠸⣍⠀⠀⢴⣿⡇⢸⣷⣆⣴⠇⢀⣿⠀⠀⠀⠀⠀⠀⠈⠈
⠀⣸⣟⢻⣽⣷⠀⠀⠘⢿⣿⣿⣦⡠⣔⣜⣿⣿⣿⣶⡛⠷⠿⣿⡷⠿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠀⢀⣾⣿⣿⣀⢉⠛⠁⠀⣸⡟
⠀⣿⣿⡎⢿⣿⣷⡀⠀⠈⠻⣿⣿⣿⣾⣿⡿⢋⡵⢿⣿⣷⣦⣤⡴⣾⡏⠙⢻⣷⣼⣧⣶⡿⣿⣶⠿⠿⣿⢿⡛⠿⠿⠿⣤⣿⠇
⠀⣿⣿⣿⣄⠻⣿⣟⢦⡀⢠⣈⣻⣿⣿⠉⢠⡿⠀⣀⡀⠈⠁⠀⠀⢸⣧⠀⠈⣏⠛⠿⠿⢿⡿⡶⢶⣿⡇⠈⢷⠀⣀⠀⠙⠋
⠀⣿⢿⣿⣿⣦⡙⠿⣷⣿⣿⣿⡿⢿⣇⣠⠟⠁⠫⢝⣻⢷⣶⣄⡀⠀⣿⡆⣀⣿⠘⠷⣪⢴⡶⣶⡆⠸⣷⡀⠘⡇⠑⣪⠔⡊
⠀⢹⣄⣹⣿⣿⡿⠿⠿⢟⡿⢫⡴⠋⣽⠁⠀⠀⠭⢓⡪⢿⡿⠿⣛⡄⢸⣿⣾⠃⠀⠈⡀⠕⡋⢵⣺⣧⠹⣇⣀⣿⠀⢞⡭⣒⡡⢀⡀
⠀⠀⠉⠉⠉⠀⠀⣀⢀⣠⣴⠟⢀⡼⠁⠀⠀⠀⠀⢂⠩⠐⣂⠽⣿⠃⡾⠉⣿⠀⠀⠀⠂⠀⡪⠝⡫⠵⣦⡈⡇⢸⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠾⣯⣈⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡼⢱⢀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⠅⢧⣸⡇
⠀⠀⠀⠀⣴⣒⡿⠗⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠳⣄
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢳
⠀⢰⠒⢺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⢳
⠠⢺⠀⢸⠀⠀⠃⠀⠀⠀⠀⢀⠤⠶⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠴⠲⠤⡀⠀⠀⠀⢀⡇⠀⢸⠀⢸⠂
⣴⢸⠀⢸⠀⠀⢸⠉⠉⠉⠩⢽⠠⠄⠀⠫⠥⠤⠭⠭⠭⠀⠈⠉⠀⢄⠃⠀⠲⢰⠇⠤⠤⠠⠬⡇⠀⢸⠀⢸⢰
⢩⣸⠀⢸⡆⠀⢸⠀⠀⠀⠀⠀⣷⡤⠤⠊⠀⣀⡄⠀⠀⠀⢀⡀⠀⠀⠑⠒⣶⠃⢀⠀⠀⠀⠀⡇⠀⢸⠀⢸⡠
⠀⠉⡀⢘⡇⠀⢸⠀⠀⠀⠈⠉⢸⡀⠑⠠⣀⠉⠁⠐⢀⠀⠘⠉⢀⠔⠈⠀⡏⠉⠁⠀⠀⠀⠀⡇⠀⢸⡀⣸
⠀⠀⠉⠉⡅⠀⢸⠀⢠⠋⠀⠀⠘⡇⠀⣀⡄⠀⠠⠆⠀⠠⠀⠀⠠⣄⠀⢠⠇⠀⠀⠈⠢⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢹⠔⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠆⢈⠢⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⣆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢹⠀⠀⠀⠀⠠⠘⠀⠁⠠⠁⠤⠀⠀⠀⠀⠀⠀⢐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠤⢀⣀⡀⣀⡐⠀⠈⠀⠂⠀⣀⣀⣀⣀⣀⡀⡜⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠉⠈⠉⠀⠈⠉⠉⠉⠉⠉⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠐⠒⠒⠒⠒⠒⠀⣒⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢲⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⢤⠤⢾⠀⡸⠧⢤⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠈⣀⠠⠒⠁⠑⠠⢀⡀⠀⠀⡣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠁⢻⡀⠀⠀⠀⠀⠀⠀⣸⠇⠙⠷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣰⣾⣷⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⡄⢸⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠉⠁⠉⠙⣛⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢶⣛⣮⠙⠳⢮⡶⠛⢫⣼⣛⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣿⣿⡯⠴⠻⣷⣀⣼⣤⣠⣾⠟⠦⣿⣿⣿⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠽⠟⠀⠀⣠⣿⣾⣿⣿⣧⣧⡀⠀⠘⠿⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢿⣿⣿⣿⣿⠿⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⣠⣶⣿⣿⣿⣿⣦⣄⠠⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣱⣾⣿⣿⡻⠿⠿⣿⣿⣿⣶⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡏⡜⠙⢿⣿⣿⣿⣿⠟⠙⣇⣿⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣷⠟⠀⠀⠀⠉⢻⡟⠁⠀⠀⠈⢳⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠛⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⠿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣶⣶⣦⣤⣀⡀⠀⠀⠀⡴⠋⠉⢢⡀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⢤⣸⡀⠀⡀⠀⣇⠀⡠⠚⠉⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠒⡶⣶⣤⡴⣪⣥⠙⡾⠧⢄⣿⠜⠁⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣾⠞⢋⢟⣾⣯⣽⡿⣢⠤⣈⡟⠀⠀⢀⡖⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣴⣥⣾⣿⣿⠿⢡⣃⡰⡹⠁⠀⣰⠏⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠻⠟⠋⣁⡴⠟⣩⠾⢷⣤⣴⠗⠂⣡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣀⣠⣴⠾⠿⠶⠛⠁⠀⠀⠉⠓⢶⣚⣓⡛⠛⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢼⢽⣷⣌⡨⠦⣬⣦⡀⢈⢷⡈⣤⠀⠼⠽⣭⠭⢁⣠⠴⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣾⡿⠁⣠⣏⣽⠀⠀⡬⠿⣿⡇⠀⣠⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣯⣁⡉⠙⠻⢟⠁⠀⠘⣷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣌⠢⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠭⢿⡻⢿⣿⣿⣿⣷⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⣤⣀⡀⠈⠂⠹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⢀⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠻⢝⣚⣭⣭⣤⣽⣿⣿⠟⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠿⠋⠉⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⢻⣿⢻⣦⡀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣄⠉⠛⠟⣷⣶⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣿⣽⣷⣄⠀⠙⣛⠘⣵⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠈⢿⡻⣿⡿⠃⢻⣇⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠧⠄⠑⠁⠀⠀⠀⠻⣆⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣦⣆⠰⠀⠀⠀⠀⠁⢦⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣈⡧⠀⠀⠀⠀⠀⠀⢿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⣿⣿⡥⠀⠀⠀⢀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠳⣿⣻⣧⠉⠀⠀⣸⣷⣦⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⡇⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⢰⣿⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⣿⠷
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣶⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠙⠃⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⢰⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡟⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⡑⠢⢄⡀⠈⢿⣆⣀⡤⠤⣤⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠶⠶⠖⠀⠒⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⣤⡤⢁⣀⣨⠖⠉⠻⠇⠀⠀⠛⠁⠀⠉⢢⣀⡀⠀⠀⠀⠀⣀⡤⠔⡒⠩⠥⠴⠲⠦⠤⠤⠴⠒⠈⠉⠉⠉⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣠⡏⣤⣶⣤⣀⡤⠤⣄⣴⣿⣶⡄⣿⣿⣦⡤⠒⣊⣋⣉⣀⣀⣤⠤⢤⣤⢤⣤⡤⠤⠤⠤⠄⣀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢿⢿⣿⡿⠃⠀⠀⠈⠻⣿⡿⣻⣿⣿⣿⡏⠭⠭⢉⣉⡁⢈⠉⠉⠉⠒⠦⣤⢀⡀⠠⠄⢀⣀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡰⢌⢦⡀⠀⠀⠀⠐⠋⠀⢀⣼⣿⣿⣿⣿⢹⡗⠢⣄⠀⠀⠉⠉⠐⠒⠒⠂⠤⠤⠤⠤⠄⠤⠤⠵⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠈⠣⡉⠲⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⢇⣾⠗⠊⠙⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡿⢿⢯⠾⡫⠖⠉⡇⢸⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡼⠀⠀⠀⠀⠀⣀⠬⢛⠟⠻⣛⡝⣱⣵⡿⠊⠀⠀⣰⣷⠀⡇⢠⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠔⠉⠀⡠⢄⣒⡪⠭⠄⠂⠁⠀⢀⠎⣼⡟⠫⠕⡢⠤⣴⡿⠋⠆⠸⣾⡟⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠁⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠈⠒⠤⣴⣷⠆⡱⠀⢘⡀⠻⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⡌⠀⠀⠀⢀⠔⣩⠞⠁⠒⠚⠳⢄⠑⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠞⠁⠀⠀⠀⠀⠀⠈⢦⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠴⢒⠽⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠬⠆
⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠹⣆⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠑⢄⠀⠀⠸⠀⣼⠇⠀⡰⢶⣤⡀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⢘⣯⣤⣬⡛⢿⣿⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣑⣢⣄⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠻⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢛⠂⠀⣶⣦⢹⣿⣿⣿⣿⣿⣃⣴⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠋⣾⣿⣷⡀⠈⠻⣦⣍⡙⠛⠋⠁⠉⠠⠤⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠚⠉⢠⡄⠀⠹⣿⣿⣿⣦⡀⠈⠻⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠈⠻⣿⣿⣿⣶⠄⣶⡀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⡀⠈⠙⠿⣫⠆⠘⡷⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡈⠛⢿⣿⣿⠶⠀⣰⠏⠀⠀⡇⠱⣄⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⣰⠋⠀⠀⠀⠃⠀⠈⠓⠶⢤⣤⣤⡤⠶⠉⠛⠶⠶⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣛⣲⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣀⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣏⣥⣤⣄⡙⠓⠒⣉⣉⣛⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡽⣿⣿⣿⢟⣶⣿⣿⣿⣿⡿⡟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣻⣿⣸⣿⣾⣼⣿⣿⣯⢿⣿⣷⣹⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣸⣿⣿⣼⣿⣿⣿⣾⣿⣿⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣽⣿⣿⣿⢹⣿⣿⣿⣿⡿⠟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠻⠻⠿⣟⣾⢛⢛⣩⣽⣾⣿⣿⣿⣷⣒⡲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡟⠛⠻⢷⡿⣿⣿⣾⣿⣿⣿⣛⣾⣿⣻⣿⣶⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⣸⣑⣿⣿⢩⣾⢛⣿⣿⣟⠿⣿⣿⣟⣼⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣤⡀⠀⠀⠀⠀⢹⡟⣛⣟⣻⣿⣿⣾⣧⣿⣷⣶⣿⣿⡟⢿⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣸⣾⣿⣳⠀⠀⠀⢘⣿⡐⣿⣿⣿⣮⣽⣷⣿⣿⢟⣿⣿⣷⣤⣽⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡿⢿⣿⠏⠀⠀⢠⡎⢩⣿⣿⣿⢿⡿⠿⣯⣍⣳⠾⣿⣿⣿⢟⣥⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣖⣂⡒⢾⣄⢿⣵⣿⣿⣮⢹⣿⣿⢿⣿⢷⣭⣿⢳⣾⡿⣛⣭⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣿⣾⣿⣷⡿⣉⡻⢼⡿⣼⣿⣷⣾⣷⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣥⣤⣐⣮⣻⡿⣷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡽⣿⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⣍⣛⡻⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⠰⣾⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣮⡻⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣏⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣧⣽⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⢛⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⡏⠉⠉⡍⣿⠛⠓⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠞⠹⠿⠿⣿⣿⣿⣿⠿⠟⠁⠀⠀⣠⣿⣇⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⠤⠀⠄⠀⠀⠀⠀⠀⠾⠿⡿⠻⠿⠿⠿⠿⠿⠗⠀⠘⣓⣷⣶⠶⠶⠶⠶⠯⠿⡿⠦⢤⣿⣻⣬⣷⣄⣀
⠀⠀⠀⠀⢸⣿⡿⢿⡿⠻⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⠿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⢸⣿⣶⣾⣷⣶⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣷⣶⣿⣶⣞⣿⣤⣤⢄⣤⣤⣠⣄⡀⣀⣀⣀⠀⢀⣀⡀⡀⣠⣤⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⡍⣿⣿⣾⣿⣿⣇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⠶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⣡⢻⣿⣩⣭⣭⣭⣭⣭⣭⣿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣥⣿⡏⠀⠀⠈⠉⠉⢹⣿⣿⣗⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠈⠻⠟⠿⠿⠏⠄⠤⠤⠤⠤⠤⠤⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠞⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠳⢤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⣴⣦⠀⠀⠀⠀⠀⠀⢀⣤⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡫⠭⠤⢤⣀⠀⠀⠀⣘⣛⣻⠳
⠀⠀⢠⠖⠛⡍⢻⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠈⢳⡔⠋⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡱⢊⠗⠢⡀
⠀⠀⡏⠳⣀⣸⡞⠉⠉⡏⠉⠒⠤⡀⠀⠀⠀⢸⠁⠀⠀⠀⣤⡀⠀⠀⣇⣀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⣀⠤⠒⢚⠋⠉⠳⡏⢀⠖⢱
⠀⡠⠇⠀⠈⠁⠉⠉⣠⠆⠀⠀⠀⠈⠑⢦⡤⠞⡄⠀⠀⠀⠉⠁⠀⢠⡇⠉⠀⠀⠀⠀⡸⠛⠶⣄⣠⠔⠋⠀⠀⠀⠘⣄⠉⠈⠉⠋⠀⠸⡀
⠀⢰⠄⠀⠀⠀⢘⠂⠁⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠢⢄⣀⣀⣠⠔⠁⠀⠑⠂⠀⠒⠈⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⢤⡀
⢀⡏⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⣴⠋⠠⣤⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢀⠈⠳⡄⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢧
⢸⠁⠀⠀⠀⠀⠀⣧⠴⠶⠦⡼⠁⠀⠀⢠⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢠⠎⠀⠀⢹⡶⠖⠲⢤⣾⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠞⠁⠀⠀⣸⠁⠀⠀⠀⠸⡄⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⠋⠀⠀⠀⠀⢻⠀⠀⠀⠙⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢹⣄⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⡀⠤⠔⢁⡠⠞⠁⠀⠀⠀⠀⠀⠈⡇
⠑⢄⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢂⠉⡖⠤⠄⢀⡀⣀⠀⣠⠠⠤⠐⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⡰
⠀⠀⠈⠒⢤⣄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠱⡁⠀⠀⡰⠀⠀⠀⢹⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣠⡤⠒⠉
⠀⠀⠀⠀⣼⠀⠉⠙⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡇⠀⠀⠀⢸⠀⠀⢀⡼⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠛⠋⠉⣧
⠀⠀⠀⠀⢻⡀⠀⠀⠀⣷⠲⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⣉⣐⣒⣚⣒⣈⣁⣀⣀⣀⣀⣤⣤⣤⠤⠤⠴⠶⣾⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠙⠦⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠤⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⡤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⣉⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠴⠲⠒⡒⠲⠤⢿⡇⠀⢀⡼⠷⠒⣒⣒⠖⢶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠈⠛⠉⢀⠀⠀⣀⡽⠀⢯⣐⠀⠀⡀⡈⠝⠁⠀⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠉⠐⠒⠒⢈⣉⡤⠖⠋⠁⠀⠀⠀⠈⠙⠲⢤⣍⣉⠁⠀⠉⢁⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⠫⠗⢿⣷⣤⡀
⠀⠀⠀⢀⣴⣶⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠈⠙⣿⡄
⠀⠀⢠⣿⣿⣿⡁⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡟⠀⠀⢀⠀⡀⠈⣿
⠀⢠⣿⣿⣿⣿⠿⣦⣀⠈⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢿⣿⠟⠀⣠⣾⣾⣕⣿⡄⢾
⠀⣿⣿⣿⣿⣿⡄⠈⢻⣷⣄⠹⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣻⢞⣥⣾⠟⢁⣴⣿⠗⠁⠀⠀⠘⠻⠿
⠀⣿⣿⣿⣿⣿⣇⠀⠀⢻⣿⣷⣌⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⡾⢣⡾⠡⠏⣴⣿⣿⠇
⠀⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⣟⢯⡳⠏⠘⢿⣿⣿⢆⡀⠀⠀⠀⠀⠀⠀⣀⣾⣵⠃⣰⡟⠁⣀⣴⢟⡷⣽⠃
⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⢿⣷⣽⣶⣤⡈⠻⣷⣿⣿⣦⡀⠀⣀⡤⢾⣱⡿⢫⣿⠟⣠⣾⢿⣵⢿⡿⠃
⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠻⣿⣯⣝⣷⣌⢻⣾⡛⢿⣆⢉⢄⠿⠗⣠⣿⣣⣾⢋⣿⣿⣹⡎
⢸⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣳⣝⣿⣦⣾⣒⣀⣴⣾⣿⡿⢫⣼⣻⡟⢹⣯⣿
⢸⣿⣿⢿⣿⣻⣽⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡼⣿⣝⠻⠿⠿⢋⣼⠏⣴⡿⢿⠋
⣸⣿⣿⣼⣿⣮⣿⠉⣽⣡⣶⣆⠀⠀⠀⠀⠀⣠⣼⣿⢏⣿⣿⣂⣤⣀⠸⣿⡆⣿⣋⡇
⣿⣿⣧⣿⣿⢻⡏⠀⠀⠉⠀⠉⠀⠀⣠⣶⣿⡿⣿⣿⣿⣿⣷⣿⣛⠻⢷⣿⣷⣟⣷⢻⡄
⣿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⣠⣾⡿⣯⣷⡿⢟⣿⣿⣵⠟⡛⣃⢾⣦⣻⣯⡻⣶⣝⢿⣢⡀
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣴⡿⣫⡾⠛⢁⣴⢿⣽⣿⣷⣾⣿⣿⣿⣼⣿⣿⣷⣌⠻⣿⣶⣿⣷⣄
⣿⣿⣿⣟⠃⠀⠀⠀⠀⣠⡾⣯⣽⠟⣣⡶⢟⣵⣿⢿⣲⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣦⠈⠻⢿⣿⣿⣆
⢿⣿⣿⡿⢀⣠⣤⣴⣿⠿⢛⣿⡿⣻⣍⣶⡿⢻⣾⣿⣿⣏⠙⣿⣿⣯⣿⣿⣿⣿⣿⣟⢷⣄⠀⠙⣿⣿⣧
⢸⣿⣿⣧⡿⠛⠉⢀⣴⣾⠿⢛⢻⣷⣿⡟⢻⣿⠟⠛⠛⠿⠤⠙⠻⣿⠋⠁⠻⣿⣿⣿⣿⣿⣷⣶⣍⠻⢿⣷⣄
⠀⢿⣿⣿⣤⣤⣾⡷⠛⢁⣀⣀⣉⣿⣯⣤⣶⣶⣶⣶⣶⣶⣶⣶⣲⣶⣦⣤⣄⣈⠻⢿⡿⣧⡝⠻⢿⣦⡀⠙⠿⣷⣀⣄⡀
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⡿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣷⣷⣮⣤⡄⣀⠘⢛⢄⠀⠈⠻⣿⣧
⠀⢀⣀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣄⣴⣿⣿⣇
⠀⠀⠙⠫⢝⣒⣬⡙⣛⣷⡟⢻⡿⠛⠛⣿⠿⣿⣟⠛⠛⠛⢻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⡄
⠀⠀⠀⠀⠀⠈⠙⠒⠬⠭⠍⠛⠩⠭⠭⠤⠤⠬⠿⠯⠭⠉⢛⣛⣐⣋⣠⠤⢌⣉⡉⢻⣿⠿⣿⣿⡿⣿⣿⠛⢋⣩⡥⣲⠿⠛⠁
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠩⠿⠖⢋⣋⠃⢉⠉⠥⠘⠛⠉
⠀⠀⣄⢀⣿⡀⣾⣶⠆⣤⣤⣄⣤⠀⣤⣄⡀⢀⣀⣀⡀⢀⣤⣄⣤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⣤⡄⢠⡄⣠⡄
⠀⠀⠈⢿⣿⣿⣾⣿⣰⣯⠙⣿⡜⣫⣻⡛⣷⢚⣿⣿⠿⠘⢿⣀⣿⠀⠀⠰⠶⣶⣾⠟⠁⠉⠀⢿⣿⠟⠀⢸⣷⣟⣥⣷⣶⡆
⠀⠀⠀⠀⠈⠙⠛⠿⣯⣾⣷⣿⣿⣯⣍⣻⣯⠈⢻⣿⠆⣤⡀⢶⣦⣀⢘⣿⣦⣸⣷⣼⣧⣼⣷⣾⣷⣿⣿⡿⢿⢛⡉⣭⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣟⠫⠿⢿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⠉⢛⣛⣛⣋⣽⣿⠿⠷⠒⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠋⠤⠤⠼⢿⣿⣄⠀⠁⠀⠀⠀⠉⠁⠁
⣿⣿⡿⠛⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠤⣀⠀⠀⠀⣠⣤⡴⠶⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠉⠢⢄⠁⠀⠀⠀⠀⣨⠟⡛⠉⠒⠾⣍⠁⠀⠀⠀⡩⠟⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⠀⠙⢦⡀⣰⠋⣁⡸⣈⢂⢀⢀⣜⡇⢀⡴⠊⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⢦⣀⡤⠿⠿⣄⣣⣙⣄⠉⠛⠋⢁⣷⣏⠀⠀⡜⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠠⠸⠷⢤⣀⢠⠶⠋⠀⠀⠐⠛⠉⠙⠛⠳⠶⠷⠋⠀⠈⢳⡎⠀⠀⢀⣠⡾⠷⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⣠⡏⠀⠀⠀⠀⠀⠠⢄⣀⣀⣀⣠⡀⠀⠀⢰⠈⢷⡤⠂⠉⠀⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⡀⠀⠀⢸⠃⠀⠀⠀⠶⣶⣿⣄⠀⠈⠉⢉⣉⣉⡙⠛⣷⠀⠀⣧⠔⠊⠁⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⡿⠀⠀⢀⣠⣴⣏⣙⣿⣷⡄⠉⡏⢸⣤⠇⣠⡟⠀⠀⠙⣦⡤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠰⠦⣸⡇⠀⠉⠙⣿⡁⠀⡷⣾⣶⣾⣿⣷⣤⠤⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠒⣟⠂⠀⠀⣰⠟⠛⠿⠿⠿⠿⣿⣿⣿⠓⡲⠲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠭⠟⠀⠀⢀⣿⣦⣤⣄⣠⣤⣤⣾⣧⢣⣀⣣⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⠉⠁⠀⠈⢻⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣤⣤⣴⣶⡿⠋⠀⠀⣯⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣛⡛⠋⠁⢀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠤⣀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⢀⠀⠀⠐⠁⠀⣀⠀⢋⡀⠀⠈⡀⠀⠀⠀⡀⠤⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠖⠛⢄⠀⠉⡀⠊⠁⠀⠀⠉⠒⢄⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀
⠀⠀⠎⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⠀⠀⢀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠠⡀⠸⡀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠄⠀⡂
⡓⠐⡂⠀⠀⠀⠀⣀⣀⡤⣴⢾⠁⠀⠀⠀⠈⠒⠌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣀⣀⣀⡈⠀⠀⢏
⠈⠉⠠⠄⠀⠉⠉⠉⢀⠎⠀⢸⠀⠀⠀⠀⠀⠡⠄⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢸⠒⠀⠤⠤⠤⠄⠀⠁⠀⠈⠀⠀⠤⠤⠤⠤⠤⠐⡇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡦⠄⢀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢭⣁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡏⠉⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⢤⠤⠼⡇⢀⠿⠤⡤⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⢄⡀⠔⠋⠙⠢⢄⡀⠁⠀⢈⡆
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⣁⣐⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⠡⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠆⣀⣀⣠⢞⠢⠀⠀⠈⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⡀⠀⡠⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠀⠰⠓⠚⡇⠀⠀⠀⠀⠀⠀⡔⠒⠳⠀⢠⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠰⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠉⠠⠠⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠒⠒⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠗⠒⠢
⠀⠀⣀⠄⠀⠀⢀⠀⠤⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡤⠄⠀⠀⠀⠀⠀⡀
⠀⠸⣤⢰⠆⠀⠀⠀⠀⠀⠀⠈⠒⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠊⠀⠀⠀⠀⠀⠀⠀⢶⢢⡼
⠀⠀⠀⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⠭⠶⡶⠒⠒⠒⠒⠒⠒⠒⠲⣶⠮⠭⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠓⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣧⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢀⣴⡏⠍⠀⠈⢹⣷⣄⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⡄⢸⣿⣷⠀⠀⠀⣸⣿⣿⠀⠀⠀⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣳⠀⠛⣿⣯⣾⣶⣿⣭⣍⡀⠀⠸⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠉⠋⠉⠁⠁⠉⠛⢿⣿⣿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⡄⠀⠀⠀⠀⠀⠀⠐⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⢀⣾⣏⣄⠄⠀⡀⠀⢀⢠⣤⣿⠷⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣶⣖⣶⢊⠿⠏⠀⢢⠆⠯⠿⣿⣷⣾⡷⣿⣾⣿⡿⣿⣫⠅⠄⠀⠈⢳⣶⣶⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠃⡤⣶⣶⠛⠉⠍⠁⠛⠋⠁⣳⢶⣖⢤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠘⢈⢿⠙⠛⡦⣤⠄⡤⠤⠴⠟⠛⠈⡍⠉⡀⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⠂⠀⠘⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠌⠀⠠⠀⢹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⡟⠙⠃⠀⠄⠀⠠⠄⠀⠐⠁⣉⠀⠰⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⠋⢰⡀⣀⠀⠀⠀⠀⠀⠀⢠⠰⣿⠀⠀⠙⢮⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠛⠙⠂⠀⠀⠄⠠⠤⠖⠂⡉⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⣀⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠉⠋⠋⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠞⠉

⠀⠀⠀⠀⠀⡂⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢉⡆
⠀⠀⠀⠀⠀⠈⠁⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣶⠄⠀⠀⠀⠀⣤⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢤⠤⣤⢤⠤⣄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠈⠁⢢⡄⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠀⡀⠀⠄⠜⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡞⠁⠀⠀⠈⠐⠠⢀⠀⣀⠀⠀⠴⠄⠀⡡⠄⠀⠀⢰⣀⠀⢀⠠⠐⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠀⠀⠀⢀⠔⠋⠈⠠⠀⡀⠠⠜⠑⠀⠀⠀⠁⠀⠙⢧⡀⠀⠀⠀⡇
⠀⢂⠀⠀⠀⣰⠃⠒⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠱⣀⡴⠚⡇
⠀⠀⠀⡀⢰⠁⠀⠀⢀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠀⢹⠀⠀⡇
⠀⠀⠀⣨⡇⠀⠀⠀⠀⠑⠄⢈⠐⠀⠠⠀⠀⢂⠁⠀⠈⠀⠀⠀⠀⡧⠈⡇
⢀⠔⠁⢸⠀⠀⠀⠀⠀⠐⠀⠀⠈⢁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⡇
⢸⠀⡴⢻⣀⣀⣀⠀⠀⠀⠈⠀⠄⠈⠠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠀⡇
⠈⡾⠅⢨⡆⢀⠉⠈⠙⠉⠉⠀⠈⠃⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⢠⡇
⠀⠛⠒⠒⢿⠀⠀⠀⠀⢈⠇⠁⠀⠀⠤⠤⠀⠀⠀⠒⠒⠒⠒⠒⡺⠀⠀⡇
⠀⠀⠀⠀⠀⠹⣤⣀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠜⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠈⠉⠀⠉⠉⠁⠉⣉⣀⠤⠖⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢀⣿⠀⢀⣧⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⠉⠀⠙⢀⠀⢀⠬⠂⠣⣀⠀⠈⠊⠀⠈⠱⡀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠦⣤⣤⣀⢀⣀⡀⡄⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣷⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⡭⢽⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⢱⣆⠀⠀⢿⣷⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣶⣿⡏⠁⠀⠀⡀⠂⠀⣤⠞⠉⢠⠏⠀⠀⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣬⣿⡵⠋⠠⠀⠀⣼⢶⢦⣀⢷⣿⣿⣿⣯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⢉⠆⠀⠀⠀⣼⣷⢸⣿⣾⠏⢻⡌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡃⠀⠀⡀⠀⣼⡿⠉⠀⣿⡆⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣶⣿⣿⣦⡶⠋⠀⠀⠀⠁⠀⠂⢀⣾⠉⠀⠀⠀⢸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠈⠻⢿⣿⡿⠁⠀⢀⠀⠀⠀⢀⣤⢻⣿⠀⠀⠀⠀⠈⢿⡆⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⢀⣿⠇⠀⣿⠀⠀⠀⠀⠀⢀⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⡷⢀⡄⡀⠀⠀⠀⢾⠀⠀⠀⢀⣀⣴⣿⠇⠀⢀⣿⡆⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢷⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢏⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢉⣼⣿⡏⠀⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣆⣿⣰⣶⣶⣵⣿⣿⣿⠿⠁⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠛⠛⠻⡿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣠⣤⣠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡴⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢧⣀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣯⣥⣤⣄⣀⠀⠀⠀⠀⠀⠀⣈⣉⣹⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠉⠉⠙⠻⣦⡀⢀⣴⠟⠋⠉⠉⠛⠻⣦⡀
⠀⠀⠀⣠⠴⠒⠒⠒⠒⠒⠤⢄⣠⣤⣤⣤⣀⡀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠦⢤⣄⡀
⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⢿⡿⢿⣿⣏⠀⠀⠀⠠⣬⡷⠀⠀⠀⣹⡇⠀⣼⠦⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⠀⠀⢙⣦⠾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⡿⠙⠷⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠙⢿⣦⣄⣀⣀⣀⣠⣴⡿⠃⠈⠻⢦⣤⣀⣠⣤⡾⠟⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⣠⡤⠀⠀⠀⠈⠉⡿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⢥⡀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠀⡆⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠰⡄⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠃⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢰⠏⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠓⢦⣄⣀⣀⣀⣀⣀⡀⠀⠤⠖⠋⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⠞⠁
⠀⠀⢠⡶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠳⠶⠶⠶⠾⠿⠿⠿⠿⠷⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠷⠶⠶⠶⠶⠶⠶⣄
⠀⠀⣸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿
⡴⠊⠁⠀⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠈⠑
⡗⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠀⠶⠔
⡝⠢⠄⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⡤⠰
⠱⢄⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣠
⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣦⣌⣉⡉⠉⠋⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣡⣴⠿⠛⠋⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⢲⣶⠶⠶⠶⠲⣶⡟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠲⠖⠞⡓⠒⠶⢾⣿⡀⠀⠀⣠⡿⠷⠒⠛⠛⡻⠖⡶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠑⠿⡚⠉⠉⠀⢀⠠⢘⣿⠀⢾⡃⠤⣀⠀⠈⠉⢑⠿⠋⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠢⠤⢀⣀⣀⠠⠬⠖⠒⣉⡤⠖⠋⠁⠀⠈⠙⠳⢦⣍⡐⠒⠧⠤⠤⠤⠤⠤⠒⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣀⣤⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣤⣤⣤⣴⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠴⠋⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠈⢱⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⣯⠀⠀⠀⠀⠀⠀⢸⡦⠖⠒⠛⠒⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠞⠀⠉⠉⠑⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠽⠯⣅⡀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣴⣿⡄⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠙⢦⠀⠀⢠⡟⠉⠙⠒⠦⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡤⠒⠒⠒⠦⣼⠀⠀⠀⠀⣀⡀⠀⠀⠘⣧⣄⣸⣆⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠛⠃⠀⠀⢀⡇⠀⠉⠉⢳⣤⣀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠻⠊⢣⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠚⢯⠁⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⢠⠎⠈⠓⠒⠒⠒⠋⠡⢄⡀⠀⠀⠀⠸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠶⠤⠤⠤⢤⣀⣀⣠⡞⠁⠀⠑⠒⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣸⢳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠠⣄⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⣀⣾⠁⢸⠀⡇
⠀⠀⠀⣠⠖⠒⢆⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣄⡀⠀⠙⠦⣄⣀⣀⣀⠤⠴⠒⠋⠁⢀⡠⠞⠁⠈⢳⡎⠀⡇
⠀⠀⠀⢯⠀⠀⡎⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⣀⣀⣠⢤⡤⠒⠚⠉⠹⡄⠀⠀⡸⠁⠀⡇
⠀⠀⠀⠈⢦⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⠀⠀⠀⠀⠹⣄⠜⠁⠀⠀⡇
⡠⠒⠈⠉⠫⡄⠀⠈⢣⣀⡀⠀⠀⠀⠀⠀⣀⣤⠿⠛⠧⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⣇⠀⠀⣀⠴⠋⣀⡤⠖⠛⡇
⢧⡀⢀⣀⡴⣇⠀⠀⠈⠃⠈⣀⣤⣴⠶⠟⠋⠁⠀⠀⢀⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠦⢤⡤⠤⣿⣶⠾⠷⠚⠉⠀⠀⠀⠀⡇
⢰⡋⠉⠀⠀⡼⢣⡀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⡴⠋⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠑⠶⠖⠋⠀⢠⣧⠞⠲⠊⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⢧⠀⢀⣀⣠⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉⠀⡇
⠀⠀⠀⠑⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠈⣷⡀⠀⠀⢀⣀⣠⡤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠻⣞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠉⠀⠀⠀⠀⢀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢾⣏⣉⣁⣀⢀⣀⣀⣀⣤⣤⠴⠶⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⡀⠘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⠘⢦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓⢤⡀⠀⠀⢣⡀⢻⡆⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠉⠓⢄⠈⢷⠈⣷⠀⠀⠀⣧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢢⠈⢧⢹⣇⠀⢠⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣆⡑⢬⡳⣿⣦⠘⢇⣤⣤⣶⣦⣄⡟⢀⡤⠤⡄⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠝⣶⣭⣿⣿⣿⣧⡈⠉⠻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⢠⣶⣶⣶⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠟⠛⠛⠛⠛⠁⠙⠛⣿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣸⣷⣶⣶⣶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⠿⣿⡿⠋⠙⠉⢿⣿⣿⣿⣿⣿⣴⣻⣿⣿⣿⡠⣦⣌⡀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⠟⠋⠁⠀⣰⣿⡃⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡻⣧⢾⣝⠻⢷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣼⢟⠟⠁⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⠛⠋⠈⠳⣮⠛⠦⣬⡉⠛⠒⠠⣄⡙⠛⠏⠩⠭⠙⢲⡀
⠀⠀⡀⣀⡴⠞⠉⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢻⣿⢿⣄⠀⡀⠀⠠⠀⠀⢀⡉⠉⠀⠂⠀⠉⠉⡁⠉⠐⠀⣸⠟
⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⢀⣸⣿⠈⢿⣷⣷⠀⠀⠀⠐⠠⠙⢶⣤⣄⣀⠀⠀⢀⢑⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢄⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠤⣀⠙⢿⣿⣿⣿⠿⠋⠀⣾⡏⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠛⠛⠉⠀⠀⣠⣾⡿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣿⡟⠁⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⢸⠶⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣧⠀⠀⠀⠀⠀⠀⢀⡿⠋⠉⠀⠈⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠋⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠚⠳⢦⣀⠀⠀⠀⠀⢾⡀⣀⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣾⣿⡧⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡼⠉⠉⠉⠛⠲⢤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡤⠔⠒⠒⠒⠦⣼⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⣧⣤⣀⣷⡀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⡏⠀⠉⠉⠛⢦⣄⣀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⢿⠞⢧⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠠⢾⡏⠉⠙⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡜⠙⠶⢤⣤⣤⡤⠖⢋⡀⠀⠀⠀⠀⠀⢷⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⠟⠙⠦⣄⣀⣀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠀⠉⠉⠉⠉⠛⠲⢦⡾⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣠⠏⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠐⢦⡀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢀⡼⣇⠀⡿⠀⡇
⠀⠀⠀⢠⡶⠛⠓⡆⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢦⣀⠀⠀⠙⠢⢤⣤⣤⣄⠠⠴⠒⠋⠁⠀⣀⣤⠞⠉⠀⠘⣷⠇⠀⡇
⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⣀⣀⣀⣀⣠⣤⡴⠖⠋⠁⠹⣆⠀⠀⢀⡞⠀⠀⡇
⠀⠀⠀⠀⠳⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠙⣆⣠⠞⠀⠀⠀⡇
⢀⠤⠒⠒⠺⣅⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡾⢿⣉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢹⡄⠀⠀⠀⣀⠜⠁⢀⣀⡤⠶⡇
⡇⠀⠀⠀⠀⣸⠂⠀⠀⠈⡟⠙⠄⠀⢀⣀⣤⣶⠿⠋⠁⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢽⣄⡀⠀⠀⠈⣷⣠⣤⣚⡥⠖⠚⠉⠁⠀⠀⡇
⢘⡷⠒⠒⠋⠙⣦⡀⠀⠀⠀⠀⡻⠿⠛⠋⠁⠀⠀⠀⠀⠀⣰⠋⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⢉⣉⣭⠽⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⢧⣀⣀⣠⠴⠃⢸⡄⣠⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⡏
⠀⠀⠉⢏⣀⢀⣀⡼⠟⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢳⣄⣤⡤⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⢀⣀⣿⣄⠀⠀⣀⣀⣤⡤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠙⢿⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣶⣞⠛⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⡇⠀⠀⠀⠀⠀⣤⣾⠟⠛⣿⣾⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⠓⠲⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢼⢻⣲⡄⠀⠀⠀⢸⡇⢹⠀⠼⣿⣿⣩⠽⣿⠇⠀⠀⡠⠂⠐⠊⠀⠴⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⢼⡯⣬⠇⠀⠀⠀⠀⣇⢸⣤⣾⡏⠉⠠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡀⠊⣿⠦
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢿⣶⠀⠀⠀⠀⠀⠸⡘⢇⣸⢿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⣮⢀⡆⠋⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣟⠉⠀⠀⠀⠀⠀⠀⢻⡿⢿⣭⣇⡃⠀⠀⠀⠀⠀⢀⠀⠠⠂⣷⣾⡇⢀⠃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⣿⢻⡇⠀⠀⠀⠀⠀⠈⢀⣼⣆⠸⣿⠄⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣶⡇⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⢿⡔⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡜⢿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣦⣻⠃
⠀⠀⠀⠀⠀⠀⡔⣴⡔⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⣇⡫⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⣆⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠉⠉⠛⠶⣤⣀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡼⠁⠀⣈⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣸⠴⢒⣾⣷⢴⡏⠁⠀⠘⠛
⠀⠀⠀⠀⠀⠀⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣠⠋⠀⡜⠁⢀⠈⢳⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠙⢄⡈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠇⠀⠀⢄⠻⠿⠃⡜⣀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠲⠄⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠑⠒⠉⠐⠭⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⣿⣷⣍⠉⠉⠻⣿⣯⠱⣄⠀⠐⢦⣀⣀⣀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠈⢿⣿⣦⠀⠀⠙⣿⣧⠈⢳⡦⠤⠭⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄⠀⠀⢻⣿⣇⠀⠀⢹⣿⠀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠈⣿⣿⠀⠀⢸⣿⢟⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⢿⣿⣿⣷⠀⠀⣿⣿⣀⣤⠞⢁⣼⠇⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠛⠛⢩⡿⠁⣾⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠟⢠⣾⠃

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⡔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⣢
⠀⠀⠀⠀⠀⠀⠳⠤⠚⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠿⠇⠀⠀⠀⠀⠀⠾⢶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠉⠑⢆⡠⠒⠀⠙⠢⡀
⣰⠚⠉⠉⠉⠁⠒⠄⡀⠀⠀⠀⠇⠀⠀⢀⣄⠀⠈⣆⡀⠀⠀⠀⢱⠀⠀⠀⠀⢀⡠⠤⠒⠚⠛⠒⢤
⡇⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⠞⢇⠀⠀⠈⠁⠀⡰⢄⠁⠀⠀⢀⠎⠛⠦⣤⠒⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡠⢤⡀
⠰⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠑⠂⠤⠔⠊⠁⠀⠁⠀⠈⠁⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠆⠀⡁
⠀⠱⡀⠀⠀⠀⢀⡞⡠⠴⠂⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡀⠀⡣⠒⠀⠠
⠀⠀⠈⢆⠀⣠⡞⠀⠀⠘⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡰⠃⠀⠸⡄⠀⠀⣠⠊⠀⠀⠀⠀⢼⠀⠀⢱⠤⠄⠘
⠀⠀⣠⡴⠛⡿⠀⠀⠀⠀⠘⡦⢀⠀⠑⠤⣀⠀⠀⠀⢀⠀⠴⠊⣀⠤⠊⠀⠀⠀⠀⢻⢲⠶⠥⣤⣤⣤⣤⣤⠀⠀⠀⡙⠢⠠⡴
⠀⣾⣁⠴⢲⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠔⠑⠓⠠⠐⠃
⠀⡜⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠑⢄
⢸⠀⢈⠷⢸⣄⣀⣀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⢀⣁⠠⠜⠫⠇⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠈⡆
⢀⣗⠞⠀⠸⡇⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⢸⠁⠀⣀⠜⠁
⢇⠘⡀⠀⠀⢳⡤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡞⠉
⠀⠑⠠⠤⠤⠚⢧⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⢲⡖⠒⠒⣶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢶⠲⣒⡒⠲⠾⣇⠀⣀⠿⠒⢒⣒⢒⡶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⢙⣀⠀⢀⣠⠞⠈⠣⢄⡀⠀⣄⣉⠀⢀⡀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣈⣩⡤⠴⠚⠉⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣤⡤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⠆⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣿⠀⢀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⢀⣻⠁⠀⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠤⢤⣤⠶⠋⠁⠀⠀⠒⣩⣄⠀⠀⠀⠈⠪⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠀⠀⠀⠀⢿⣖⠁⠀⠀⠀⠈⣲⡿⠀⠀⠠⠐⡄⢻⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣡⣦⣉⢀⠐⠤⣀⠈⠻⢿⣶⠂⠈⣽⣿⡀⠀⠠⠀⡝⠄⠹⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣷⣤⣈⣷⡀⠈⠛⢻⣿⢩⣙⣿⡄⢠⡀⡀⠀⠢⡄⠀⣴⢒⣈⠥⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠿⣿⣿⣿⠿⠾⠿⣄⣠⣼⣿⠀⣽⣿⡀⠀⢻⣧⠀⠀⠹⢄⡇⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⡄⠀⠀⠀⢻⡟⠀⢠⣿⣿⠀⠐⠀⠈⢷⡄⠐⡌⣧⠀⠠⠏⣰⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡄⠀⠐⣼⣷⣰⣾⠏⠤⠀⠁⠀⠀⠀⠙⢷⣦⡘⠄⣀⠖⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠙⠂⣶⣿⡟⢿⠃⠀⠒⠀⠁⠀⠀⠀⠀⠀⠈⠙⠆⠐⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⡿⠟⠁⠀⢡⣀⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠀⠀⢤⣻⡇⠀⠁⠀⠠⠖⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣻⡿⠃⠀⠀⠀⠤⢊⣥⠠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⣀⠘⠟⠃⠀⠀⠀⠀⠖⠋⣶⣦⡄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠠⣿⣿⠀⠀⠀⠐⠀⠀⠐⠋⠀⠀⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡖⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣿⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠀⠉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠈⠐⠐⣠⠈⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠒⠰⠤⠒⠉⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠡⠆⠠⠌⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⠕⠁⠀⠀⠀⠀⣀⣔⣤⣂⣠⢄⡤⢒⣈⡩⠤⣔⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⣿⠿⣶⣄⠀⣀⣔⣮⣷⠿⢶⣀⡤⠝⠫⠤⠤⠜⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣴⣿⢿⡿⣿⣿⣿⣿⣿⡿⣟⡿⠿⣿⣭⣶⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠨⣼⣿⣿⣿⣿⡵⣫⣵⣶⣿⣿⣿⣿⣿⣿⣿⢻⣷⣶⢀
⠀⠀⠀⠀⠀⠀⢸⡟⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠛⠃⣼⣿⠟⢰⣷⡀
⠠⠄⠒⡨⠑⠓⢾⡇⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠐⡂⠀⠛⠃⢠⣿⣿⡿⣦
⠀⢠⠎⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣻⣿⣿⣿⣿⣿⣿⣾⣿⡶⢀⡀⠈⠙⠟⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠋⣁⣩⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⣀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⣼⣿⠟⠛⠋⠙⠛⢻⣿⣿⣌⠛⠛⢻⣿⣿⣿⣿⣿⣿⣶⣿⡿⢛⣩⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⠀⠀⠀⠀⠀⣾⣿⠛⢿⣆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢻⣆⠀⠀⠀⣠⣿⠏⠀⠘⠛⠷⠄⣽⣿⡈⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠈⢿⡄⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠈⠙⠿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋
⣿⣿⠛⢿⢻⠿⡿⢻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠥⢀⠀⠀⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⣀⠦⠤⢄
⠀⡠⠠⢅⣈⡉⠀⠒⠘⠂⢴⠀⠀⠀⠀⡀⠀⠀⠁⠀⣀⠀⠙⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠀⠈⢉⣁⠠⠤⠒⠚⠈⢱
⠀⠧⠄⣀⡀⠀⠉⠉⠀⠒⠚⠠⠀⠀⠀⢀⠴⠊⠀⠀⠉⢀⠔⠀⡀⠀⠈⠑⢦⡈⠀⠀⠠⠂⠋⠉⠁⢀⣀⣠⡄⠒⠂⠉
⠀⠀⠀⠀⢘⠙⠋⠑⡆⠐⠠⠠⠀⠀⡰⠋⢰⠂⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⡀⠙⡄⠀⠈⠂⠐⠀⠉⡇⠀⠀⠸⡄
⢀⡀⠀⠀⠣⠀⠆⢀⡧⣤⣀⣀⣀⡰⠁⠀⠀⠆⠈⢄⠀⠀⠀⠀⠀⢀⠃⡐⠀⠀⠘⣄⣀⣤⣤⡤⠶⠧⣌⣃⢈⡇⠤⠔⠒
⢇⠀⠀⠈⠉⠐⠛⠃⠠⠀⢈⣉⠩⠇⠀⠀⠀⠈⠠⠄⣁⠀⠒⡀⠀⡔⠊⠀⠀⠀⠀⢳⠁⠀⢀⠀⠐⠀⠈⠁⢀⣀⡠⠤⠔
⠀⠸⣉⣉⠐⠐⠒⠠⠠⡤⡠⠁⢸⠀⠀⠀⠀⠀⠘⠄⢰⠀⠀⡇⠀⣡⡄⠀⠀⠀⠀⢸⠈⠰⡘⠤⠠⠒⠀⠉⡁⠤⢤⠖⠈
⠀⠀⠀⠁⠠⠭⠀⠂⠐⠀⢇⠀⠸⡒⠒⠀⠤⠄⠤⠤⠈⠀⠀⠃⠬⠤⠤⠤⠤⠐⠒⡚⠀⢀⠇⠀⠀⠁⠉⠃⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢧⠤⠄⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⢍⣉⠀⠀⠀⠀⠀⠀⣉⣉⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⠆⠠⢇⢀⠼⠀⠴⣦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⠀⢑⡠⠔⠊⠈⠓⠤⣐⡀⠀⢀⡱

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⣰⠁⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⠓⢴⠁⠀⠀⠋⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⢱⠀⠀⠀⠈⡃⣠⠶⣀⡜⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠸⠇⠀⠀⠀⠒⠧⣄⢸⠀⠀⠀⠀⠀⠀⠐⠁⢀⡴⠁⠀⡰⠁⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠴⢆⣀⣔⢦⠀⠀⠈⢦⡀⠀⠀⠀⠀⠌⢀⡴⠋⠀⠀⠀⠠⠄⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⡅⡀⠈⡀⠨⠈⠀⠀⠀⠈⡇⠀⠀⢀⢊⡼⠁⠐⠒⠀⣈⡡⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠑⠣⠥⠜⠀⠀⢰⠆⢠⠙⡄⠠⠂⡾⠐⠈⠉⠉⡩⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠒⠂⠀⠀⢀⠔⡩⣠⡃⠀⠈⠀⠊⠀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣤⣴⣶⣿⣿⠟⣿⣿⣯⠙⢿⣿⣿⣷⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣿⣿⣿⠟⠁⠄⢸⣿⣿⡆⠀⠹⣿⣿⣿⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⠉⠀⠀⠀⢸⠀⣿⣿⣿⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⢀⣠⣿⠀⣿⣿⣿⣄⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⠊⣀⡱⠀⠀⡠⠘⢿⡤⣟⠛⡓⢌⠑⠠⣖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⡄⠙⡄⠀⠀⠑⠲⠀⠀⣘⡠⠼⣦⠿⠭⡊⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠰⣉⣀⡠⠗⢹⡇⠀⢃⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⠒⠒⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⣀⣄⠀⠀⠀⠀⠀⢀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠯⠤⣀⠀⠀⢀⣛⣻⠇
⠀⠀⠀⠀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠱⡎⠁⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠸⠁⠀⠀⠀⠀⠈⠑⠢⡀⢀⣼⠀⠀⠰⠷⠀⠀⡗⠆⠀⠀⠀⡤⣄⡀⠀⡠⠔⠉⠁⠀⠀⠈⠱⡄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠢⣀⣀⣀⣠⠞⠑⠠⠀⠠⠐⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡴⠁⡠⠄⠀⠀⡴⠈⠁⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠱⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠠⠀⠀⠘⡆⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠈⠢⡼⠁⠀⠀⠀⠠⣄⠀⠑⢄⠀⠀⠀⠀⠀⠀⡠⠎⢀⠔⠁⠀⠀⠀⢹⡀⡠⠊
⠀⠀⠀⠀⠀⢀⡠⢴⡇⠀⠀⠀⠀⠀⢡⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⡰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓⢄
⠀⠀⠀⠀⡇⠀⠀⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠁⢂⣉⠖⢆⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⡆
⠀⠀⠀⠀⠳⣄⣠⡼⢧⠬⢍⠉⠉⠉⠓⠐⠒⠒⠒⠚⠛⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠉⡵⠳⠖⢤⠴⡁
⡶⠶⠶⠶⠶⠃⠀⠁⠘⡀⠈⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠀⢀⠁⢠⠀⠈⠶⠶⠶⣦
⡇⠀⠀⠀⠀⠓⠊⠳⠶⠋⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠙⠦⠞⠑⠊⠀⠀⠀⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⣛⠒⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣀⡁⠀⠐⠚⣂⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⢘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⡶⡄⠀⢀⠖⠂⢸⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡈⠉⠁⢀⡴⠁⠘⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣈⠀⠸⠑⠒⠒⠁⠀⠀⠀⠠⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⣠⠄⠀⠲⣄⠀⠀⠀⠀⢀⣀⠀⠀⢃⣠⣤⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠁⠀⠈⠑⠄⢀⠎⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⡜⠀⠀⠐⠒⠤⠀⠀⠀⠙⠦⣤⣤⠀⠀⠀⠀⠀⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⢦⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠢⡀⠰⢻⢡⣄⣀⠀⢀⣀⣀⣀⣀⡤⠖⠋⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠣⡅⠀⠀⠈⠉⠀⠁⠀⠀⡠⠀⠀⠀⠘⢦⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠂⠤⢄⣐⠀⠁⠒⠒⠀⠀⠀⠉⣠⠔⢁⠀⣀⠤⠓⠁⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢄⡀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠈⠀⠔⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣉⠓⠢⢄⠀⠀⠁⠀⠐⠒⠒⢤⡀⠊⠁⠀⠀⣀⠀⠀⠀⡠⠀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⡒⠄⠀⠀⢄⠀⠠⠀⠀⢱⠀⠀⠀⠊⠁⠀⣠⡬⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠢⡄⢣⣸⠊⠀⠒⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⣀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣠⣤⣤⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣀⠀⢀⠀⠀⠀⠀⢠⣶⢾⣏⠁⠉⠁⢨⣿⠷⣶⡀⠀⠀⠀⠀⠀⢀⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⡀⠀⠀⣿⣿⣦⣽⡀⠀⠀⣾⣷⣴⣿⡇⠀⠀⢀⣤⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡈⢿⣿⣿⣿⣷⡀⣀⣿⣿⣿⣿⢃⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⠃⠀⠀⠀⠀⠀⢬⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣾⣿⣯⠀⠀⠀⠀⠀⠀⡀⣢⣿⣿⣿⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠜⠈⢉⣽⠟⠛⠻⠿⢿⣿⣠⣆⣀⡀⢠⣥⣿⡿⠿⠿⠿⢿⣟⠶⠟⢧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⣴⠟⠁⠂⣠⣶⣾⣿⠟⠉⠋⠉⠹⠈⠃⢹⣾⣶⣤⡀⠀⣻⣦⠀⠈⠹⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢀⣴⠟⠉⠀⠀⢸⣿⠟⣿⣥⣤⢀⣀⡀⣤⣴⣶⣾⣿⣿⣿⣿⡆⠀⠙⢧⡀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⠀⠀⢼⣿⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⡇⢹⣷⠀⠀⢈⠻⣦⡀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⠀⠀⠀⠀⢁⣿⣿⢸⣆⣀⣀⣈⢀⡀⠉⢉⣀⣠⣴⣾⡇⠸⣿⣧⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⠀⠀⠀⢀⣾⣿⠁⢸⠹⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠀⠙⣿⣦⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠈⣷⣆⣤⣄⣠⣄⢀⣤⣴⣶⣶⣾⡇⠀⣄⠘⠻⠷⣄⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⣴⠾⠋⡀⠀⠀⠀⠀⠸⣍⣛⠛⠛⠿⠿⠟⢛⣛⣩⣿⠁⠀⢸⢆⠀⠀⠙⢦⣤⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣾⣿⣿⣿⡿⣿⠇⠀⠀⠀⠁⠀⠀⠀⠀⠛⠋⠀⠀⢀⣡
⠀⠀⠀⠀⠀⠀⠸⠷⠤⠤⠴⠞⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣠⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠛⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢶⣄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣤⠶⠚⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⣷⡀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⠋⠁⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠸⠿⠋⠙⢷⡄⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠷⠿⠶⣤⣀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠙⢳⣶⣶⠚⠛⣿⣿⣶⣾⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⠤⣄⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿⣿⣿⡇⠈⣿⣿⣿⠉⠙⣿⣯⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⢰⣿⣿⣿⠁⠀⣿⣿⣿⠀⢀⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣠⣾⣿⠟⠀⢠⣿⣿⣿⠏⠀⣸⣿⣿⡟⠀⢸⣿⣿⠇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⢧⣀⣴⣿⣿⣿⠏⠀⢠⣿⣿⣿⠁⢠⣿⣿⡟⠀⣸⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠃⠀⣠⣿⣿⣿⠃⢀⣾⣿⡿⠁⣰⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣾⣿⣿⡿⠁⢠⣾⣿⣿⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠤⠤⠤⠔⠒⠒⠲⠾⠌⠉⠉⠙⠶⠒⠒⠂⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⣀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠄⣀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢦⣀⣠⠤⡶⠛⢄⡀⠀⣀⠄⢠⣀⡀⣀⡠⠿⣦⢄⣀⡠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢀⢉⠿⠛⠂⠀⠛⢿⡁⡀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⡦⠄⠀⠀⠀⢴⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⠤⠀⠀⠤⡤⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣬⣄⣠⣀⣠⣤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣿⣿⣿⣿⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡇⢻⠃⢻⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢻⠀⠈⠀⢸⠀⠘⠀⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⢠⠎⢣⡀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠀⡸⠀⠸⡇⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⡧⠀⠀⣇⢀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠰⣇⠀⢀⡿⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⢸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣀⣀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠉⠉⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⢦⣤⡀
⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣤⡀⠀⠀⠀⠀⣠⠟⠁⠈⠀⠀⠀⠀⠁⠈⠻⣆⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⣿⠟⠉⠉⠉⠉⠻⣿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠾⠟⠛⠉⠁⣴⣯⣤⣤⣤⣤⣤⣤⣽⣦⠀⠉⠛⠻⠷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣠⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⠿⠛⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠉⠛⠿⣌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣼⣷⡄⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⣀⡴⠛⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠛⢦⣀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠛⠛⠛⠛⠛⠟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠚⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢧⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡭⠭⠦⠤⣄⡀⠀⠀⠀⢀⣛⣋⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠙⢦⡰⠊⠉⠀⠀⠉⠳⣄
⢀⡴⠚⠋⠉⠉⠙⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠶⣤⡀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⣨⡇⠀⠀⠀⠀⠾⠗⠀⠀⢀⡗⠾⠃⠀⠀⠀⠀⢀⣧⣄⡀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠛⠁⠹⣄⠀⠀⠀⠀⠀⠀⢀⡜⠣⣀⠀⠀⠀⠀⣠⠞⠀⠈⠙⠷⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠖⠋⠀⠀⠈⠉⠑⠊⠉⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⡤⠄⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠘⢄⠀⠀⠀⠀⠀⣼⠃⠀⠊⠉⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠈⠳⡀⠀⠀⣸⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠞⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠘⢦⣠⡏⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠒⠀⠀⠈⡙⠒⠦⠤⠤⠤⠤⠒⠒⣉⡠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⠁
⠀⠀⠀⣠⠖⠛⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠠⠄⠀⣀⡈⠉⠀⠀⠉⡙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⣀⡀
⠀⢠⠞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠲⡄
⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⡠⠄⠀⠤⡤⠚⠳⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣆
⠈⣇⠀⠀⠀⠀⢸⡦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠺⢤⣀⣀⣀⣠⠤⠔⠋⠰⠧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣼⠀⠀⠀⠀⠀⣿
⠀⠙⢦⡀⣀⣀⣸⠧⣤⠤⣄⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠒⠒⠛⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⡤⠿⢤⠶⢤⣤⣴⣃
⠀⠀⢠⠟⠁⢸⠃⠀⡇⠀⠈⢣⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠀⠀⢸⠀⠀⢇⠀⠈⢦
⠀⠀⢾⠀⠀⢸⠀⠀⢹⡀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡀⠀⢀⡎⠀⠀⣼⠀⠀⢸
⠀⠀⠈⠳⠖⠉⢧⣤⡞⠙⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠋⠻⣦⡴⠋⠙⠖⠋
⣿⣿⠿⠿⡿⡻⠿⡿⠟⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⣿⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣿

⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠉⡆
⠀⠀⠀⠀⠀⠑⠊⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣠⣤⠀⠀⠀⠀⢠⣄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠓⠒⠢⡀⡠⠌⠹⢇
⡠⠒⠀⠀⠒⠠⢀⠀⠀⠀⠃⠀⠀⣀⠀⠘⣀⠀⠀⠀⢁⠀⠀⠀⠀⡀⠤⠒⠒⠲⣄
⡇⠀⠀⠀⠀⠀⠀⢈⡶⠚⢃⠀⠀⠉⢀⡴⢈⠀⠀⡀⠊⠓⢦⡔⠁⠀⠀⠀⠀⠀⠘⠄⠀⠀⠤⢤
⠐⡀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢉⠐⠂⠉⠀⠀⠈⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⠁
⠀⠐⢄⠀⠀⡼⠁⠘⡁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡄⠈⢧⠀⠀⠀⡐⠁⠀⠀⢀⠂⠀⣇⠀⣀
⠀⠀⣀⡶⣾⠁⠀⠀⠑⣄⠀⠙⢄⡀⠀⠀⠀⠀⢀⠔⠀⡠⠊⠀⠀⠘⡦⣤⣌⣀⣀⣀⣀⡉⠀⠀⠯⣀⣠
⠀⣞⡡⠴⡇⠀⠀⠀⠀⠸⡄⠀⠀⠈⠁⠀⠀⡤⠒⠀⠁⠀⠀⠀⠀⠀⢻⣀⠀⠀⠉⠉⠁⠀⠔⠒⠦⠤⠂
⢠⠻⣆⢠⡇⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡀⠇⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦
⠸⡀⡘⠸⡧⠤⠤⢀⣀⣀⣀⣀⣁⣂⣀⣀⣀⣒⣉⣃⣀⣀⣀⡤⠤⠤⣼⠀⠀⢈⠆
⡔⠏⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⠔⠉
⠐⠈⣀⣀⠸⣍⠉⠉⠓⠒⠒⠒⠒⠂⠀⠐⠒⠒⠒⠒⠒⠒⠚⠛⠀⡐
⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠈⠉⠉⠉⠁⠀⣀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡏⠉⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⡇⠀⢸⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠒⠊⠀⢈⡱⠐⣏⠀⠀⢘⠒⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠳⢌⣀⣈⡬⠵⠊⠁⠀⠀⠀⠉⠒⠦⢭⣀⣡⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢼⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣤⡔⠿⠟⣠⣤⣄⠀⠀⠀⠀⢀⣤⡴⣾⠀⡷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆
⠀⠀⠀⠀⢸⣿⣿⣷⡇⣰⣿⣿⣿⠆⠀⠀⠀⣿⢿⣷⣾⠀⣷⣿⡟⣿⠀⠀⠀⠀⠀⢠⣶⣆⡻⢛⣴⣶⡄⠀⠀⠀⠀⣠⣶⡺⡅⡟⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⠛⠃⠛⠻⣿⣿⠀⠀⠀⠀⡿⣿⣯⡿⠀⣿⣿⡿⢿⠀⠀⠀⠀⠀⣻⡁⠀⢠⠁⠀⢸⡇⠀⠀⠀⠀⣿⣿⠿⠇⠿⢿⢿⡇
⠀⠀⠀⠀⢸⡟⢻⢤⡄⣤⣀⡏⣿⡇⠀⠀⢠⣄⠹⡿⡃⠀⢻⢿⠃⣸⠀⠀⠀⠀⠀⣿⡇⣶⣮⣷⣶⢸⡇⠀⠀⠀⠀⠛⢸⣶⠄⣴⣎⠘⠃
⠀⠀⠀⠀⣾⠇⢸⣿⣧⣾⣿⡇⢸⡇⠀⠀⢸⣿⢀⡷⠀⠀⠀⢾⡀⣿⡆⠀⠀⠀⠀⣿⢀⣿⣗⣿⣯⠘⣷⠀⠀⠀⢸⡿⢠⠟⠀⠸⣧⢰⣷
⠀⠀⠀⠀⣛⢰⣿⣿⣿⣿⣿⣿⠈⡇⠀⠀⠸⠇⢨⣦⣄⢀⣤⣮⡁⠹⠃⠀⠀⠀⢰⡏⣼⣿⣿⣿⣿⣦⢻⠀⠀⠀⢸⠇⣽⣦⣀⣤⣾⡄⣿
⠀⠀⠀⠀⠟⠘⣿⣿⡛⢻⣿⢿⠈⠇⠀⠀⢸⠆⢸⣿⣿⣿⣿⣿⡇⠴⡆⠀⠀⠀⢨⠀⣿⣿⡟⢿⣿⣿⢨⡄⠀⠀⣘⢀⣿⣿⣿⣿⣿⡇⢘⡀
⠀⠀⠀⠀⠀⠀⡌⣿⡇⢹⣿⣰⠀⠀⠀⠀⠀⠀⢉⣿⣿⠈⣿⣯⡅⠈⠀⠀⠀⠀⠈⠀⣷⣿⡇⣾⣿⡇⠀⠁⠀⠀⠋⠘⣸⣿⡇⣿⣯⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣿⣿⠃⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⢿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⣹⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⢻⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⢸⣦⡇⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⢀⡰⠈⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⠜⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠳⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⣠⣤⡄⠐⠒⠂⠀⠀⠤⢄⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢹⣟⣟⣃⡀⠀⠀⠀⠐⠿⠿⠎⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⠖⠉⠀⠀⠀⠉⠳⣀⡤⠒⠉⠉⠑⢦⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠂
⢀⡴⠚⠛⠀⠒⠒⠢⢄⡀⠀⠀⠀⢀⡿⡎⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠶⠦⣄⡜⠀⠀⢧⣀⣀⠠
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⢸⣧⡇⠀⠀⠀⠻⠟⠀⠀⢸⠓⠛⠀⠀⠀⠀⢸⢤⣇⡀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⣀⠀⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠙⢄⡀⠀⠀⠀⢀⡠⠋⠢⢀⡀⠀⣀⠔⠁⠀⠈⠙⢾⣅⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⡀⠠⢀⣆⠀⠀⡈
⠈⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⢉⡓⠒⠚⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⣠⡾⠋⢀⡏⠀⠀⠀⠈⠁
⠀⠈⢆⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢩⠉⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⢠⣀⣠⡶⠟⠉⠀⢀⠞
⠀⠀⠀⠣⡀⠀⠀⣀⠌⠀⠀⠀⠀⠈⡆⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣰⠀⠀⠀⠀⠈⢿⡉⠀⠀⠀⢠⠎
⠀⠀⠀⠀⣈⣶⣾⡇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢁⡴⠁⠀⠀⠀⠀⠀⠘⣇⠀⢀⠔⠁
⠀⢀⣴⠞⠋⢀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠗⢄⡀⠈⠙⠲⠦⠠⠤⠄⠖⠒⠊⢁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠉
⠀⢹⣧⠴⠊⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⡏⠒⠢⣤⠄⠀⢠⡤⠄⠒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⢸⡧⢤⣀
⠀⡴⢻⣆⠀⢰⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⡇⠀⠈⠳⣄
⢰⡇⠀⢹⣦⢼⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢅⣀⠀⠀⡇⢀⣀⠜⠋⠃⠀⠀⠀⠀⠀⢠⠁⢰⠀⢀⡇⠀⠀⠀⠘⡆
⠀⢧⣤⡼⠀⠘⡏⠉⠀⠘⠒⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⣤⣬⣭⣭⡤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠊⠀⠘⠛⢻⠇⠀⠀⠀⣰⠇
⡔⠻⡌⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠀⠀⣼⣤⡤⠤⠚⠁
⠱⡀⢡⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⡀⣀⠀⠀⠀⣰⡏
⠀⠈⠒⠤⠴⠖⠚⠹⠆⠠⠖⢐⠖⠀⠐⠲⡆⢢⠀⠀⠠⠂⠀⠀⡀⠢⠀⠀⡞⠈⡏⢁⢉⠁⡀⢁⣀⠁⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠆⠂⠐⠛⠛⠂⠐⠰⠀⠀⠒⠀⠄⠀⠂⡐⠀⠀⡗⠒⠓⠚⠒⠒⠂⠚⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠲⠶⣦⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⡶⠶⠖⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢠⠀⠀⢠⡀⠄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠦⠄⣀⡀⢀⠀⡀⡀⠠⠐⠲⠶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠑⠓⠚⠉⠀⠀⠀⣈⡇⠸⣅⠀⠀⠀⠉⠁⠚⠈⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⠒⠀⠤⠤⠀⠒⣀⣠⠴⠚⠁⠀⠀⠈⠙⠲⢤⣀⠐⠒⠂⠀⠒⠈⢀⡷

⠀⠀⠀⠀⠀⡈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢉⡆
⠀⠀⠀⠀⠀⠈⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡶⠆⠀⠀⠀⠀⣦⣔⠁
⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡼⠒⠒⠲⣄⣠⠖⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡞⠁⠀⠀⠈⢠⣾⢷⢶⣾⠀⠀⡤⠀⢸⡇⠤⠀⠀⢸⣄⠀⢀⠠⠒⠉⠉⠉⠙⢆
⢃⠀⠀⠀⠀⠈⢁⠔⠉⠘⢦⣀⣀⣠⠞⠳⢄⣀⡤⠞⠀⠙⢧⡀⠀⠀⠀⠀⠀⠘
⠀⠂⠀⠀⠀⣰⠃⠐⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠱⡄⠀⠀⠀⠠⠁
⠀⠀⠁⣀⣴⠃⠀⠀⠀⡀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠃⢀⠄⠀⠀⢱⠀⠀⠐⠁
⠀⡤⢚⣉⡇⠀⠀⠀⠀⠈⠒⠌⣐⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣧⣄⡀⠀⠀⠀⢀⠔⠒⠐⣢
⢀⢿⡁⢸⠁⠀⠀⠀⠀⠀⢂⡀⠀⢠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠙⠻⡓⠶⠦⠞⠀⠀⢋⠵⡁⡆
⢸⠀⡳⢸⣄⣀⣀⠀⠀⠀⠀⠁⠒⠈⠐⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⡄⠀⠀⢱⠀⠀⢙⢂⠴⠏⠗⠊
⡤⡟⠁⠘⡆⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡇⠀⠄⠃
⠁⠐⠀⡀⢹⡐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⡾
⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠈⠉⠉⠉⠉⢁⣀⣀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣀⣿⠀⢀⣧⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⡴⠉⠀⠙⣈⠀⢀⡼⠃⠣⣀⠀⠀⢊⠀⠈⠱⡀

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣤⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠐⠒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⢀⠙⠿⠋⠀⠀⠀⠀⠀⣤⣦⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠡⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠐⠠⠄⠄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣬⣙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⠃⠀⠘⣿⣿⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⠿⣷⣦⡙⢿⣿⣿⣿⣇
⠀⢸⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⣿⣿⡄⢿⣿⣿⣿
⠀⢸⡿⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⡆
⠀⢸⡇⠀⠈⠊⠉⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡆⢹⣿⣿⣷
⢂⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⣿⡿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠘⠋⠁⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⠀⠀⡜⢳⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡁⠀⢠⠶⣐⠇⣸⠸⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠓⢾⡄⢻⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠹⡌⢠⠖⠊⠁⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⣀⣀⡀⠀⢼⠄⢀⣀⠀⡀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⢹⡖⠒⠛⠉⠉⠁⠙⠁⠀⡾⡄⠀⠉⠀⠉⠉⠙⢺⡏⠙⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⢀⣤⠾⠤⣷⡄⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣶⡏⠀⠀⠀⠀⠹⡿⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⢿⡀⠀⠀⠀⠀⢀⣿⠶⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠈⠳⢤⣀⣀⣤⠞⠁⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⣤⣤⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⣤⣦
⢸⡇⠈⠛⣶⣤⣀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣀⣤⣶⠛⠁⢸
⠈⣷⡀⢸⡀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⢀⠇⢀⣾
⠀⣿⡉⢰⣋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣙⡆⢉⣿
⠀⠈⣿⡁⠸⣅⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣨⠇⢈⣿⠁
⠀⠀⠈⠛⠛⠛⠋⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⠀⢀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢦⣀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠤⣀⠀⠀⠀⠀⠀⣀⡴⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢽⠛⠷⠦⣴⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⣦⡴⠶⠛⡯⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣬⣷⣤⣤⣤⣤⣭⣭⣽⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣯⣭⣭⣤⣤⣤⣤⣾⣥⠴⠞⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠠⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⠐⠀⠆⠀⠄⠂⠐⠀⠆

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⣎⠀⢉⣓⠐⠲⠤⣀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⣤⠔⠒⣋⠉⠉⢱
⠀⠀⠀⠘⠄⠀⠀⠓⠢⡀⠀⠉⠒⠤⣀⠀⠈⢦⠀⢳⣠⣀⣄⡞⠀⡴⠁⠀⣠⠔⠊⠁⣠⡞⠚⠃⠄⢠⠃
⠀⠀⠀⠀⠈⠀⣉⠄⠀⠌⠳⣔⠱⣄⠀⠑⠦⡄⠑⣤⡟⠉⣻⣤⠞⣀⠴⠋⠀⠀⡤⢊⠥⠈⠉⠉⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⠈⢉⠲⡒⠠⣄⠀⢐⣮⠷⠤⢿⣧⠊⢀⠆⠴⢒⢋⡁⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠠⢠⡈⠒⠤⣤⡞⣴⠟⠛⠳⣜⢦⡤⠂⠀⣀⠠⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣬⠽⣿⣷⣟⠀⠀⠀⢹⢸⡿⢧⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣶⣦⠻⣶⣤⣴⠟⣼⣴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣴⡎⣥⣶⣶⣶⣤⡝⣦⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢀⣾⡔⢷⣤⣭⣭⣭⣥⣴⣯⣿⣆⠀⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⣇⢣⣭⣉⣉⣉⣉⣵⡏⣿⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠈⣟⠈⢨⣭⣉⣫⣭⡟⠁⣻⠁⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠙⠻⠟⠋⢀⡴⠋

⠀⠀⠀⠀⢀⣾⣷⣀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⢀⣀⣀⣼⣿⣿⠿⠛⢿⣿⣿⣿⡟⠉⠙⠻⢿⣿⣿⣿⣿⠿⠛⠉⠉⢻⣿⣿⣿⠟⠛⢿⣿⣿⣆⣀⣠⡀
⢹⣿⣿⣿⣿⣏⣀⡀⠀⠻⢿⣿⡿⠆⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠰⢿⣿⠿⠋⠀⢀⣀⣿⣿⣿⣿⣿⠃
⠀⠙⢿⣿⣿⣿⡿⠁⠀⠀⡠⠖⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠢⡀⠀⠀⠘⢿⣿⣿⣿⡟⠁
⠀⠀⢸⣿⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿⣿
⢠⣶⣾⣿⣷⠀⠀⠀⢀⡀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠀⠀⢰⣿⣿⣶⣦
⠀⠻⣿⣿⡟⠀⠀⠀⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣞⠀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢹⡇⠀⠀⢸⣿⣿⣿⣿⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣉⠉⠀⠀⢨⣿⡏
⠀⢀⣠⣾⣿⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠋⠀⠀⢸⣿⣧⣀
⠀⠘⣿⣿⣿⠀⠀⠀⠈⠘⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠀⠁⠀⠀⢸⣿⣿⡿
⠀⠀⢈⢉⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢹⠁
⠀⠀⢸⣾⣦⣿⡶⠤⢄⣀⣀⣀⣀⣀⣤⣤⣤⣽⣿⣿⣧⣤⣤⣀⣀⣀⣀⣀⣠⡤⠾⣿⣧⣼⣼
⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠈⠉⠁⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠉⠁⠀⠀⠀⠀⠙⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⠟⠤⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⡄⠀⠀⣴⣿⣿⣿⣿⣯⠅⠘⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⣀⣼⣿⣿⣿⣿⣿⣿⣶⣦⣼⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⣿⠭⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢂⣤⡜⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣭⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⠀⠀⢱⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⣠

⣿⣭⣭⣍⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⠿
⣿⣿⣿⣿⣿⣿⠷⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣶⣶⣶
⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿
⣯⣭⣭⣭⣭⣭⣍⣉⣉⣉⣉⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠛⠛⠛⠛⠉⠙⠉⠙⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐
⣟⡛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣒⣲⣶⣾
⣿⣿⣿⣛⣛⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣻⣿⣿⣿
⣛⣛⣉⣉⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠿⠿⣿
⣿⣿⣿⣿⣿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶
⣿⣿⣿⣛⣛⣛⣛⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣚⣛⣿⣿⣿⣿
⣷⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛
⣿⣟⣋⣉⣉⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣩⣽⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩
⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶
⣿⣿⣿⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿
⣿⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⠿⠿⠿
⣿⣯⣭⣭⣭⣭⣭⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣭
⣓⣒⣒⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠾⠿
⣿⣿⣿⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠣⣄⣤⣤⣠⠞⠀⠀⢀⣾⠏
⠀⠀⠀⠒⠂⠐⠒⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢠⣶⣿⣿⣿⣿⣶⡄⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠂⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑⠂⢀⡀⠀⠀⠈⢿⣦⡼⠟⠛⠙⠋⠛⠻⢧⣴⡿⠁⠀⠀⢀⡀⠔⠊⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⢠⣄⣀⣯⠀⠀⠀⠀⠀⠀⠀⠀⣽⣀⣠⠄⠂⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣽⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢠⣾⡿⠝⠛⠛⠛⢛⣭⣭⡛⠛⠛⠛⠫⢿⣷⡄⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⣾⠉⠀⠀⠉⠙⠛⠻⠏⠹⠟⠛⠋⠉⠀⠀⠉⣷⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⣸⡿⠀⠀⢆⣀⣀⡀⢀⣀⣄⡀⢀⣀⣀⠰⠀⠀⢿⣇⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢿⠇⠀⠀⡀⠈⠉⠉⠉⠁⠈⠉⠉⠉⠁⢀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⡆⠀⠀⠹⣶⣶⣤⣤⣴⣶⣦⣤⣴⣶⠏⠀⠀⢰⣷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣴⡇⠈⣷⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⣟⠻⠉⢋⣽⣿⣿⣿⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡿⠇⠈⣧⢈⣧⠰⠶⣿⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣷⣤⣽⣿⣾⣏⣠⣸⣿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⣤⣾⣿⣦⣽⣦⣠⣿⢳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣿⡿⣿⣿⣸⣿⣿⣿⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⢿⣿⠈⡿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⣿⢻⣿⡥⠐⡧⠠⣿⣿⢻⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏⢻⣿⣦⠟⠞⢧⣼⣿⠛⡇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠇⢸⡿⡗⠂⡕⠠⡾⣿⠀⢇⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣸⠇⢸⣿⡇⠀⠀⠀⣿⣿⠀⢷⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠀⢸⣧⡗⠒⠗⠒⢧⣿⠀⠸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡟⠀⢼⣿⡀⠀⠀⠀⠹⣿⡄⠘⣿⣩
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣾⡟⢰⣀⣀⡀⠀⣽⡆⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣼⣿⣿⣦⣤⣾⣿⣿⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⣀⡈⠉⠀⠀⢀⡿⢿⣶⡿⠿⣷⣾⠿⣇⠀⠀⠈⡁⣀⡀⠀⠀⠀⢀⡘⢙⠀⠀⠀⣿⡏⠀⣻⣿⠁⠈⣿⡇⠀⠀⠘⡊⣀
⠀⠀⠀⠀⠀⠀⠙⠙⠋⠀⠀⢸⣇⣀⢻⣧⢠⣿⢁⢀⣿⠀⠀⠈⠃⠈⠁⠀⠀⠀⢻⣑⠛⠀⠀⢠⡿⣷⣶⣿⢿⣷⣾⡿⣴⠀⠀⠈⢓⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡄⣿⢸⡇⡞⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⢸⣿⣿⣷⣿⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠷⡏⠈⡿⢿⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡏⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣄⠇⠀⣇⡜⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⢿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣻⠀⠀⢹⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠾⠃⠀⠙⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠋⠀⠀⠈⠃⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠙⠀⠀⠈⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣿⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⠀⠀⢠⣾⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⠀⠀⢸⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⠀⠀⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠇⠀⠀⠈⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠟⠀⠀⠐⠆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡇⠀⠀⠀⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢃⠑⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⢠⣾⣿⣿⣿⠾⡄⠰
⠀⠀⠀⠘⢄⣼⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⣿⣿⠟⠉⠁⠀⣷⠒⠓⡄
⠀⠀⠀⣠⣶⣿⠿⠢⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡄⠀⠀⠹⣿⣄⡀⠀⢀⡍⠀⠀⢰
⠀⠀⠀⣿⣿⣇⠀⠀⠀⠱⡄⠉⠒⣄⠀⠀⠀⠀⠀⠀⡆⠰⡀⣀⣀⠈⢿⣿⠏⠀⣉⠏⠒⣼
⠀⠀⠀⠈⠻⣷⣄⣀⡀⠤⠦⠔⠒⠈⠧⣀⠀⠀⠀⠀⠰⡰⠉⠀⠀⠀⢀⠎⠉⠉⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠠⠃⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠣⠤⠄⠒⢩⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠄⠐⢲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠱⢮⠀⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⢹⠤⡤⠔⠓⣄⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠑⠄⣀⠄⠊⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡠⢦⠀⠀⠀⢰⠀⠀⢠⠃⠀⠀⢀⠔⠲⢄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⡜⠀⠀⡠⠊⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠑⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠊⢉⠂⠀⠀⡘⠀⢠⠞⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡰⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠩⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠲


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣰⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠳⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⡼⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡾⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠿⢿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠙⡆⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠴⠒⠲⠦⣄⠀⠀⠀⢀⣀⣀⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡑⠒⠒⠉⠧⣀⡴
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠹⣤⠊⠁⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⡤⠖⠋
⢠⡞⠋⠉⠉⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣄⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠛⠛⠳⣾⡟⠉
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⠀⣠⣼⡄⠀⠀⠀⠘⠿⠃⠀⠀⣼⠙⠛⠀⠀⠀⠀⠀⡾⢦⣄⡀⠀⣠⡴⠚⠁⠀⠀⠀⠀⠀⣠⡾⠋⢹⡄
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠻⣄⠀⠀⠀⠀⠀⢀⡴⠋⠢⣀⠀⠀⢀⡠⠞⠀⠀⠈⠛⢿⣅⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢸⡇
⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⢀⣴⠟⠋⠀⠀⠀⠀⣸⠁
⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠈⢳⡾⠛⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠈⢣⡀⠀⠀⠀⢀⡾⠁⠀⠴⡁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠑⢄⣀⣤⡾⠁⠀⠀⠀⢳⢄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⡜⠁
⠀⠀⠀⢀⣤⠾⠛⣽⠃⠀⠀⠀⠀⠈⡄⠉⢲⠤⣀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡴⠋
⠀⢠⣶⠟⠁⣀⣤⡟⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢸⠋⠈⠙⢻⠉⠛⢻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠘⣿⡶⠋⠁⢸⡇⠀⠀⠀⠀⠀⠀⠈⢦⠢⠤⠤⠼⠤⠤⠤⠼⢤⡄⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠦⣄⡀
⠀⣰⠛⣷⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢦
⠀⡇⠀⠘⣷⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢀⡀⠂⠐⠊⠁⣢⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⣧
⠀⢧⣀⣰⡏⠀⢺⡟⠓⠶⠶⠶⠤⢤⣤⣤⣤⣄⣀⣀⣉⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠴⠶⠶⠶⠖⣿⠀⠀⠀⠀⢀⡿
⣠⢼⣩⠋⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣠⡤⠊
⢧⠀⢹⠀⠀⠀⠀⢿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⠋⠉
⠈⠳⣌⡃⠀⢀⠀⣨⣧⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⢡⡏
⠀⠀⠀⠉⠛⠉⠉⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣠⠶⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡏⠉⠉⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⣿⡇⠀⠀⠀⣿⣇⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⢽⣌⠡⠴⠀⠈⠉⠛⢧⡀⣠⠞⠋⠉⠀⠒⠴⢬⣴⠍⠉⠛⠶⣄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢅⣀⠀⠀⠀⢉⣢⡀⠄⢀⣈⡴⠞⠀⠘⠶⣤⣀⠂⠠⢤⣀⣀⡀⣀⣀⡠⠜⢧
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠀⠉⠉⠉⣁⣡⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣄⣁⡀⠀⢀⣀⣠⡾⠁

⣿⣿⣿⣿⣿⣿⣿⣷⡝⣆⠹⣿⣿⣿⣿⣿⣿⠀⣷⠀⢻⡆⢸⣿⢸⣿⣿⣿⣿⣿⣿⣼⠁⠏⢠⣿⡟⣾⣿⣿⣿⠟⢀⣾⣿⡿⢋⣾⣿
⣿⣿⣿⣿⠋⠻⣿⣿⣿⣌⠣⠘⣿⣿⢿⣿⡿⡆⠘⡆⠘⡇⠀⣿⢸⣿⣿⣿⣿⣿⠇⠇⠈⠀⣾⠏⣼⣿⣿⣿⠋⣰⣿⣿⢟⣵⣿⣿⠿
⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣦⠀⠈⢿⣯⢻⣷⠑⠀⢁⠀⠣⠀⢿⠸⣿⣿⣿⣿⡟⠈⠀⠀⣸⠏⡼⣻⣿⡿⢁⣾⣿⠟⣡⣿⣿⠟⣡⣾
⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣿⣷⡀⠀⠻⣆⢻⣇⠀⠀⠀⠀⠀⢸⠀⣿⢻⣿⣿⠁⠀⠀⢠⠏⠘⣱⡟⠁⣠⣿⠟⣡⣾⡿⠋⣡⣾⠿⣫
⣿⣿⣿⣷⣝⠻⣿⣿⣦⡀⠙⢿⣿⣄⠀⠙⣆⢻⡆⠀⠀⠀⠀⠘⠀⣿⢸⣿⡏⠀⠀⠀⠈⠀⣰⠋⢀⡼⠋⢀⣼⠿⢋⣴⡿⢛⡵⢚⣥
⣿⣿⣿⣿⣿⣷⣮⡙⢿⣿⣦⡀⠙⢿⣦⡀⠘⢆⢻⡀⠀⠀⠀⠀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠐⠁⡠⠋⢀⣴⠟⢁⣴⠟⠩⢒⡡⢞⣫⠴
⢽⣛⠿⣿⣿⣿⣿⣿⣦⡈⠻⢷⣄⠀⠙⢷⣄⠀⠀⠳⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⡠⠊⢁⠴⠋⠀⠀⠐⠩⠐⠉⣀⡤
⣤⣈⠉⠀⠉⠛⠿⣿⢷⣽⡢⠀⠙⠳⣄⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢒⣩⣵⣾
⣉⠙⠛⠶⢤⣀⡀⠀⠀⠈⠉⠓⠄⠀⠈⠑⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠁
⣻⠿⠷⣶⣤⣄⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⣤⣶⡾⣿
⣉⣉⡁⠀⠀⠈⠉⠙⠓⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠄⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣚⣋⣩⣭⣴⣶⠿⠿
⣿⣭⣭⣭⣍⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⢻⡽⣿⢂⢿⣂⢸⡼⠃⠀⠀⠀⠀⠀⠀⠐⣀⣉⣉⣩⣥⣤⣴⣶⣶⣾⣿
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠖⠒⠒⠀⠀⠀⠀⠀⠀⠹⠟⠘⠀⠈⢁⣈⡅⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠲⠿⠿⣿
⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⣿⣾⢻⣂⢙⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⣓⣶⣶⣿
⡿⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⣀⡀⠀⠀⢈⣉
⣤⣴⣶⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠐⠒⠤⣄⣉⠙⠛⠷⢶⣶
⠛⠉⢀⣀⣤⡴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⢦⡀⠀⠐⢤⣀⠀⠀⠀⠈⠙⠓⠶⣤⣄
⣶⡿⠟⣋⠥⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⢷⣌⡓⣮⣛⢶⣦⣤⣀⠀⠀⣀⠉
⡵⠖⠋⣀⠄⢀⡤⣒⠅⢀⠴⠊⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⣄⠀⠻⣿⣮⣝⢿⣿⣿⣿⣷⣦⣄⣉
⣠⠴⣊⣵⣞⡽⢊⣡⠞⢁⣴⡄⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⠀⠀⠀⠈⢧⠰⡄⠀⠻⣷⣄⠈⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿
⣵⣿⣿⠞⣡⣴⠟⣡⣴⡿⣫⣾⡿⠃⠀⠀⠀⢠⠀⠀⠀⢸⡇⠀⢸⠀⡆⠀⡀⠀⠀⠘⣧⠹⣆⠀⠙⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿
⡿⣋⣴⣿⠟⣡⣾⣿⢟⣵⣿⡟⠁⣠⠆⠀⣰⠇⠀⠀⢀⣿⠀⡄⣼⡀⡇⠀⣤⠀⠀⠀⢹⣧⡹⣦⠀⠈⢿⣿⣦⡐⢌⠻⣿⣿⣿⣿⣿
⣾⡿⢋⣵⣾⣿⢟⣵⣿⣿⠏⠀⣰⣯⠎⢰⡏⠀⠀⡆⣸⡟⢠⡀⣿⡇⣧⠀⢹⡆⠀⠀⢀⢿⣷⣹⣷⡀⠀⠻⣿⣿⣦⡳⣝⢿⣿⣿⣿
⣫⣴⣿⣿⣿⣿⣿⣿⡿⠃⢀⣼⣿⡟⣰⡿⠀⣸⡿⢀⣿⡇⢸⣿⣿⡇⣿⠀⢸⣷⠀⠄⠘⣾⣿⣷⣿⣿⣄⢢⡙⣿⣿⣿⣮⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⣾⣿⡿⣱⣿⠇⣰⣿⠇⣸⣿⠁⣿⣿⣿⡇⣿⠀⠘⣿⣇⢸⡀⣿⣿⣿⣿⣿⣿⣦⠹⣌⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠶⠶⠿⣿⠻⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢠⡀⠀⠀⠀⠀⣀⠀⠀⡀⠀⡀⢰⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣼⣷⣶⣴⣶⡞⠛⠀⢀⡇⢸⡇⢘⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡤⠞⠋⣻⣿⠟⠉⠙⠻⠷⣾⣿⣧⣾⣥⣼⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇
⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠖⠛⠉⠀⣿⣿⠃
⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⣀⣀⣀⣀
⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠿⠟⠛⠋⠉⠉⠛⠻⢶⣤⡀
⣿⣿⣿⣿⣦⣀⠀⠀⠀⠉⠉⠁⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣤⣤⣶⡶⠶⠶⠶⢶⣤⣄⣀
⠹⣿⣿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣀
⠀⠈⠻⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⠿⢶⣤⡀
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⠷⠶⠒⠀⠀⠀⠙⠷⣦⣤⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⠋⠁⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⢀⣀⣠⣶⣄⡀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣄⠀⠀⠀⠀⠀⠙⠿⣿⣦⣼⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⣠⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⢠⣛⣦⣴⣶⣶⣦⣔⣿⡄⠀⢀⣤⠞⠋
⠀⠀⣠⠔⠒⠪⣉⣉⡉⣑⣒⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠹⣝⣦⠈⠿⣿⡿⠿⣿⣿⣿⠿⠁⣴⣫⠏⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⣒⡊⣉⣉⣉⠕⠒⠢⣄
⠀⠸⣇⠀⠀⠀⠀⢀⣔⣀⣩⡭⠤⠻⢤⡸⣍⠒⠢⠤⣄⣀⠀⠈⠻⢷⣾⠃⠀⠀⠀⣼⣿⣷⡾⠟⠁⠀⣀⣠⡤⠔⠒⣩⢇⡤⠟⠤⣭⣍⣁⣲⡄⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠉⢍⡉⠉⠀⠘⠢⠄⣀⣀⠨⡕⠒⠒⠉⠒⠢⠤⢦⣿⣿⣷⣿⡗⠀⠀⠀⣐⣒⣾⣇⣿⣾⣿⣿⠧⠦⠔⠒⠋⠒⠒⢺⡅⣀⣀⡠⠔⠂⠀⠉⢉⡩⠉
⠀⠀⠀⠀⠀⠉⠒⣒⠚⠉⠀⣀⡠⠔⠊⣳⠒⠒⠒⠒⣒⣒⣿⣿⣿⣿⣿⡄⠀⠀⢐⣺⣿⡯⣿⣿⣿⣿⣓⣒⠒⠒⠒⠒⣞⠙⠢⢄⣀⠀⠉⠑⣒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠭⢥⠤⠒⠘⢓⡒⠒⠋⡩⠝⠩⠴⣚⡿⠿⢿⣿⣗⣒⣒⣾⣿⣽⠿⢿⣓⠦⠍⠫⢍⡙⠒⢒⡚⠃⠒⠤⠬⠭⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⢍⠤⢔⣋⠤⢒⣚⣭⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⡶⣶⣶⣭⣓⠒⠤⢙⡢⠤⡉⠗⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⠉⣩⡾⣻⣿⣿⡋⠉⢹⣿⣟⢿⣟⠳⣭⠉⠙⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠛⠛⠋⠁⠀⠀⣾⣵⡿⣿⠿⢿⣿⣿⣿⣿⣿⣷⢻⢿⣮⣷⠀⠀⠈⠙⠛⠛⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠋⢰⡇⠀⠀⠀⠀⠨⢿⣿⣿⡜⡆⠙⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣺⡟⠀⢸⣷⣶⣤⣤⣤⣬⣭⣿⣿⡇⡇⠀⢻⣗⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⢸⣎⠉⠙⠛⠛⠛⢿⣿⣿⠇⡇⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠘⣟⠻⠿⣷⣶⣾⣿⣿⣿⢳⠃⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⢻⣶⣤⣤⣄⣠⣿⣿⣏⡞⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣭⣭⣽⣿⣯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣿⠿⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⣨⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⡀⢠⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠁⢠⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢉⠂⠠⢬⠀⠀⢠⡾⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠀⠀⠀⠘⠣⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢠⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠄⢠⠀⠂⢳⣄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠐⣿⣷⣤⣿⣏⣤⣤⣌⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⢿⣿⣿⣿⣿⣿⣿⠈⠀⠈⠳⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠈⠉⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠴⠢⣄⡀⠈⠀⠀⠀⢀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⣿⣿⣿⣷⣶⣀⠢⠄⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠂⠸⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⠔⠉⠀⢸⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⣀⠸⠎⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⣶⠏⠒⠀⠀⠀⠀⣄⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡡⠔⠒⠀⢲⠲⡀⠀⠀⠀⠸⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⠀⢈⠀⠱⡦⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠲⡀⠀⠀⡍⠙⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠤⠴⠒⠈⠁⡧⠤⠊⠀⠀⠀⠀⠑⠤⠤⠁⠀⢡⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⡀⠈⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠇⠀⠀⠑⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠘⡀⠀⠀⠀⠀⣠⠎⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⠀⠀⠀⠃⠀⢀⡠⠊⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠓⢄⠀⢰⠤⠄⢀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠱⡀⠀⠀⠸⠖⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢈⡅⡆⠀⠀⠈⠁⠐⠒⠂⠀⡴⠋⠀⠀⠀⠱⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⠢⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠈⠛⠓⠀⠀⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠒⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠎⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠳⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡌
⠀⠀⠀⠀⠀⡰⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⡸
⠀⠀⠀⢀⡰⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠘⠛⠹⡀
⠀⠀⣴⢉⣧⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡀⠀⠀⠀⠘⢆
⡴⣏⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⢋⡀⢂⣀⠀⠑⢦
⠑⠪⡓⢄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠳⠶⠋
⠀⠀⠈⠲⣷⣄⠈⢦
⣀⣀⣀⣀⣀⣉⣳⣦⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⣴⠟⠋⠙⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠻⣟⠙⠛⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⠷⣶⣤⣀⣸⠃⠀⠀⢀⠀⠈⡇⠀⡴⠛⠉⠉⠛⢦
⠀⠀⠀⠀⠀⠀⣴⡟⠉⢀⡤⠶⠶⠦⣤⡀⠉⠻⢿⡀⢰⣦⣼⣷⠀⡇⡼⢀⠀⢠⡄⠀⠈⡇
⠀⠀⠀⠀⠀⣸⡏⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢄⣛⣛⣣⠞⠁⢧⢸⣷⣿⡿⠀⣰⠇
⠀⠀⠀⠀⠀⣿⡇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢳⡄⠀⠀⠈⠳⠭⣉⣤⣴⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⣀⠛⠶⠤⠤⠀⣀⣴⠟⠁⠀⠀⠀⢀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⠙⠛⠿⢿⣷⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣤⣤⣤⣴⣿⣦⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠉⠁⠀⢨⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⡞⠁⠀⢀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃⠀⠀⢸⡇⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⣦⡀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢻⡇⠀⠀⠈⢿⣶⣶⠶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢶⣾⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠀⠀⣀⡾⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⠛⢷⣦⣤⣀⣡⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁



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

⠀⠀⢀⡴⠛⠉⠉⠉⠉⠉⠙⢦⣀
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⡏⠀⣠⣄⢠⣤⣤⡀⣠⣄⠀⣤⣤⡀⢸⡇
⡇⠀⣯⣌⠀⢸⡇⢸⡇⢸⡇⣿⣸⡇⢸⡇
⡇⠀⣤⣽⠃⢸⡇⠘⣧⣼⠇⣿⠉⠀⢸⡇
⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⡂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠
⠀⠀⠀⠈⠳⠶⠶⡶⡶⠶⠶⠋⠀⠀⠀⠈⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡶⠀⠀⠀⢀⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠊⠉⠐⡤⠉⠀⠀⠑⡄⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⢰⡉⡛⠀⠙⡎⠀⠀⠀⠁⠂⠄⣀⡤⠆⠀⠀⠒⣅⠀⠐⠂⠀⠤⣀⠀⠂⠀⢀⡀⢀⣬⡄
⠀⠀⠀⠀⠀⢳⠄⠴⡆⣀⡷⣄⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠊⠀⠈⠑⣄⢠⠏⠉⡏⠀⢹⠤
⠀⠀⠀⠀⠀⠈⠓⡎⠛⠁⠀⠈⠙⠲⢤⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠈⢦⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠢⠀⡎⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⠔⠁⢠⠀⣞⠉⠙⢇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢼⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠈⠑⡄⠈⠑⡄⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⢀⠊⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡋⠁⠒⠒⠒⠒⠂⠀⠀⠐⠂⠀⠀⠂⠒⠒⠂⠈⠉⣹⠉⣉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠤⠤⠀⠤⠤⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢍⣉⡉⠁⠀⠀⠀⠀⠀⢈⣉⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢇⠀⢸⡧⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⠀⠉⢄⡠⠴⠃⠓⢤⡀⠨⠁⠀⠈⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠙⠲⡀
⠀⠀⠀⠀⠀⡀⠤⠐⠒⠒⠒⠀⠄⡁⠀⠀⠈⠠⡀
⠀⠀⠀⠀⢤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⣍⣀⣀⣀⠐⢤⠐⠀⠁⠀⠙⣶⣴⢷⣶⣦⣤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⡷⣶⣖⣦⣀⣈⡱⣴⣄⣤⣿⣿⢿⣿⣏⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣧⣶⣿⣿⣿⣿⣟⢟⠛⢿⣼⣿⣿⣤⣿⠏⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⣿⣿⣄⠘⣿⣿⣿⣿⢻⣿⣦⣻⣿⣿⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⠻⣿⣿⣟⣿⡟⢙⣿⠇⠉⠹⡧⠃⢳⡈⠛⠷⠖
⠀⠀⠀⠀⠀⠀⠀⣰⠈⢿⣿⣧⣶⣬⣷⣿⠋⠀⣸⠿⠀⠀⠀⠀⠀⠀⠻⣆⢀⣄⠀⠀⠀⠀⠂⠁⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣈⡝⣫⣿⡟⠁⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠊⠀⠀⠘⠋⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⡿⢻⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢺⣶⠌⢡⠀⠀⠀⢀⡤⠐⠀⠁⠀⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠰⣧⣗⡡⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠊⠀⠀⠀⠀⣻⣿⡑⢂⡁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢽⣯⡍⠀⠀⠀⡀⢀⣼⣿⣟⠛⠛⠓⠦⠀⠀⡀⠀⠀⠀⣀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⣀⣄⣀⣠⣶⣿⣿⢧⡈⠉⠀⠀⠂⠀⠀⠀⠈⠑⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠔⣻⢿⣾⡟⣿⡏⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠉⡞⠏⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣎⣷⠀⠀⠀⠀⠘⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⣭⣿⡟⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠤⠴⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⢿⣿⣍⠁⠀⠀⠀⠀⠀⠐⠒⠲⠶⠿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣿⠖⠂⠀⠀⠀⠀⠀⠀⠚⣓⣐⣪⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠆⠀⠀⠀⠀⠀⠠⠤⠶⠶⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣆⠀⠀⠀⠀⠀⠀⢐⣺⣿⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⢙⣻⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠴⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣿⣿⠒⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣿⣿⣿⡽⢂⣀⠀⠀⠀⠀⠘⣫⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠿⠟⠛⠉⠀⠀⠀⠐⡚⢿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣙⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠉⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⣀⣀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠻⣄⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⢤⣠⠞⠀⠀⠀⠀⠀⠀⡠⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⡾⠿⠇⠀⠀⠀⠀⠀⣶⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠠⠎⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⠐⠠⢄⠀⠀⢀⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠎⣠
⠀⣀⡠⠤⠄⣀⡀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠱⡞⠁⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢳⣦⠖⠈⠁
⡜⠁⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⣨⠀⠀⠀⢰⡦⠀⠀⠰⠦⠀⠀⠀⠀⣧⡀⠀⠀⢀⡠⠒⠈⠁⠀⠀⣸⠟⣄
⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠚⠉⢢⡀⠀⠀⠀⢀⡼⠣⠀⠀⠀⣀⠜⠀⠉⠳⢾⡁⠀⠀⠀⢀⣠⠞⠁⠀⠹
⠈⡄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⡙⠒⠛⠉⠀⠀⠀⠈⠉⣀⠀⠀⠀⠀⠈⠻⣄⢀⡴⠟⠁⠀⠀⢀⠃
⠀⠈⢆⠀⠀⠀⢀⡾⠁⢀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠣⣀⣠⡾⠁⠀⠀⠦⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠠⠋
⠀⠀⣠⡴⠛⣻⠃⠀⠀⠀⠀⠈⠰⠀⠀⡑⣤⣀⣀⣀⣀⣀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠔⠁
⠀⢾⣁⠤⠒⡟⠀⠀⠀⠀⠀⢀⣄⠀⠀⡅⠀⠀⡅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀
⠀⡼⢷⡀⢰⡇⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠑⢆
⢰⠁⠈⣷⢸⣇⣀⠀⠀⠀⠀⠀⠀⠑⠆⢠⠄⠀⠊⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⣧
⢈⡷⡴⠃⠈⣏⠉⠉⠉⠛⠛⠒⠒⠒⠒⠂⠀⠀⠀⠠⠄⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⢹⠃⠀⠀⡠⠋
⠏⠙⡁⠀⠀⢻⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿
⠈⠢⢁⣀⡀⠨⢧⠁⠉⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠋⠉⠉⠉⣹⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣍⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⣤⣀⣀⣀⣀⣀⣤⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⡇⠀⠀⣿⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠛⠫⡭⠍⠉⠉⠛⢧⠀⡴⠋⠉⠁⠛⢩⠝⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠤⠄⠤⠤⠖⢂⡥⠖⠋⠀⠉⠓⠤⣀⠒⠂⠤⠤⠄⠐⢹⡄