Hot day Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣬⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡙⠳⢶⣤⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⣤⡶⠞⢋⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡾⠋⣠⡾⠛⠉⠀⠀⠀⠀⠉⠛⢷⣄⠙⢷⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣄⡀
⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⢹⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⡏⠉⠛⢷⣦⣄⡀
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠙⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⣀⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⣤⣌⠉⠉⠉⠉⣡⣤⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡾⠃⠀⠀⠀⠀⠘⢷⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠒⠀⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠒⠂
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠉⠉⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠠⠄⠀⠒⠒⠒⠂⠐⠒⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠢⠤⠤⠤⠔⠊⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠄⡇⠑⢤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠔⠁⠑⠄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠁⠁⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠴⠚⣻⠃
⠀⣠⣤⣖⡛⠁⣠⠼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠾⠃
⠸⢋⠝⠉⢀⡜⠁⠀⠻⡀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠌⠚⠇⠀⠀⠐⠫
⣰⠋⡀⠀⠈⠓⣄⢸⡆⠙⣄⠀⢀⣠⢤⣀⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡖⣻⡍⣿⡇
⣯⠾⡇⢀⠀⡄⠙⢦⡀⠀⣌⠳⣼⡇⠀⠈⠙⠢⣄⠀⠀⠀⢀⡠⠴⠒⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⡇⣿⠇
⠙⠀⠙⢾⢸⢻⣰⠲⣝⢦⡛⠀⠀⠑⣄⠀⠀⠰⠿⠳⣤⣖⡡⠤⠤⠤⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⡇⣿
⠀⠀⠀⠘⠾⠀⠙⠀⠈⠳⡱⣄⠀⠀⠈⠳⣄⠀⠀⠀⠘⣮⣡⣤⠤⠴⠯⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⡇⣷⠁⣿
⠀⠀⠀⠀⠀⠀⠀⡠⠞⣩⢭⣝⠳⠀⢀⡄⠈⠙⠶⢤⣤⠾⡆⠐⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢠⣁⢿⣀⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠉⠉⣨⡄⠀⢸⡇⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⢨⡏⠑⠒⠢⠥⠤⠥⠤⠥⠖⠋⠀⠀⠀⠀⢸⠻⡤⢻⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⢸⠇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⢸⡷⠀⠀⠀⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣜⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⡖⠒⢛⡋⠉⠛⡏⠀⠀⠀⠉⠀⠀⠀⢀⡤⠤⠤⣄⢠⠀⣿
⠀⠀⠀⠀⠀⠉⠳⡤⣙⠲⢤⣇⠀⠀⠀⠀⠀⠀⣴⢋⡇⠀⠀⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣁⠀⠈⠙⠲⠤⣀⡀⠀⣷⠚⣳⠖⠳⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⢤⣀⠀⠀⠉⠑⠺⠭⣅⡀⠀⠀⠀⢸⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠈⢩⠷⡄⢸⡖⡸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠱⡦⠌⠓⠤⣀⣰⠃⠀⠙⣾⣦⢳⡑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⠀⣸⠀⠀⢀⣀⡵⠀⠀⠀⠀⠁⠈⠉⠉⠙⠟⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠺⡦⣄⣀⣉⣛⢹⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⡛⠫⢍⡇⢯⣽⠶⡒⠒⠒⠒⠒⠚⢛⠟⡟⠉⢉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡠⠚⢛⣃⠀⡴⠒⠶⠋⠀⠀⠀⠀⣠⢎⡜⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣏⠉⣝⡡⠞⠀⠀⠀⠀⠀⠤⠤⠾⣷⢾⡒⠂⣷
⠀⠀⠀⠀⠀⠀⠀⣴⠚⠙⣟⡯⠟⠒⠶⠤⠤⠤⠤⠤⠤⠴⠶⠳⠦⠤⣶⣾⣛⠒⢲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠶⠶⠶⠶⠶⠛⠛⣿⠆⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠘⠃⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠈⢁⣴⠞⠛⠛⠳⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢀⣀⣀⠀⣾⠁⠀⠀⠀⠀⠘⣧⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⢹⡇⠀⠀⠉⠁⠀⢻⣄⠀⠀⠀⠀⣰⡏⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⢠⣤⣤⣭⣥⣤⣤⠶⠶⠞⠛⠉⠉⠙⢦⣤⣤⣄⣿⠁⠀⠀⠀⠀⢀⡄⠙⠷⠶⠶⠞⠋⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠀⠀⢀⣽⢿⡇⠀⢀⣀⣤⣀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢘⣿⢿⡀⠀⠀⠀⠰⠋⠀⠀⠀⢰⡄⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⢸⡇⠀⠉⠉⣠⠉⠁⠀⠀⠈⠉⣤⠈⠉⠀⢸⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⢷⣼⡇⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠟⣠⣶⡄⢸⣿⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠏⣼⠃⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠴⠶⠶⠶⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠉⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠛⠉⠉⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠉⠉⠛⠛⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣶⠀⠀⠀⣾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣴⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠁⣀⡤⠤⠤⠤⣄⣀⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⢀⡴⠋⠁⠀⠀⠛⠓⠦⣌⠳⣄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠙⠓⠲⢤⡀⠀⢠⡏⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⡄⠀⠀⠀⠐⢦⡙⣆⠀⠳⡄⠘⡇⢳⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡾⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣴⠃⠙⣦⡀⠀⠀⠀⠹⠘⡆⠀⢀⡀⠹⣄⠳⣄⠀⠀⠀⠀⣀⡴⢋⡼⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡞⠙⠦⠴⠃⠈⠓⠶⠶⢤⡀⡷⠀⠈⣇⠀⠈⠓⠦⣄⣀⣀⣀⣥⠔⠋⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠳⡇⠀⠀⠈⠙⠒⠂⠀⣀⠀⠀⢀⡀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠀⠀⠀⠀⠁⠀⢠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠐⠶⠤⠀⠀⢀⡞⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⠤⡴⠋⠀⠀⠸⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢀⣀⣀⣠⡤⠤⠤⠶⠖⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣸⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⡀
⠀⠀⠀⠀⠝⠡⠀⠈⠓⠒⡀⠀⡆⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠛⠉⢀
⠀⠀⢀⡀⠀⠀⣿⠀⠀⣰⡆⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢀⠀⠀⠀⣴⠂⠀⠣⣳⡄
⠀⠀⠨⠄⠀⠀⠩⡀⠈⠈⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠇⢀⠀⠙⠃⠀⠀⠈⠄
⠀⠀⠀⠀⠀⢠⣀⣀⡀⠀⠀⠀⢶⡄⠉⠀⠀⠀⢀⡴⠞⠻⢶⡄⠀⡀⢀⠀⠀⠀⠁⠀⠀⠀⠒⠀⣚⡀
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠈⠀⠀⠈⠀⠀⢦⣠⠀⠀⠀⠀⠀⠀⠁⠀⠏⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠔
⠀⠀⢀⣀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⢄⠠⠤⢦⠀⠈⡟⢀
⠀⠀⠈⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁⠀⢣⠀⠙⠉
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠥⠴⠞⠊
⠀⠀⠀⣀⣀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⠤
⠀⠀⠀⠠⠀⠠⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠄⠱⠄

⠀⠀⣰⣴⣤⡀⡄⣄⣄⣠⣠⠀⠀⠀⠀⠀⢠⠀⣷⣧⣦⣴⣤⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⡀⡏⢛⠃⡻⣿⢹⣚⣉⡃⠀⠀⠀⠀⠀⠀⠀⡋⢛⢛⠟⠹⣛⢹⠁⠀⠀⠀⠀⠀⠀⢰⣴⣀⢑⣐⣀⣁⣤
⠸⠱⣞⠉⠷⠇⠃⠿⡸⠿⠿⢿⠃⠀⠀⠀⠀⢀⡇⣮⡿⢹⣦⠇⡏⡟⠀⠀⠀⠀⠀⡀⣎⡛⢛⡛⡟⠟⡋⡋⠐
⠀⠀⠟⠷⠞⠀⠀⠀⠽⠲⠖⠀⠀⠀⠀⠀⢀⠈⠀⡗⣦⣾⣬⡴⠀⢁⠀⠀⠀⠀⠠⠗⡾⢋⣤⢡⢸⣯⣿⠻⢿
⠀⠀⠀⠚⠛⠃⠦⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⡀⢀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⣀⣠⣤⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣇⠴⡤⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⢾⠸⡿⠟⢻⠁
⠀⠀⠰⣾⣾⣿⣿⣿⣽⣷⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠧⠉⠗⠂⠀⠀⠀⠀⠀⠀⠀⣤⢰⢦⡦⣄⣰⡇⡾⣰
⠀⠀⠀⣿⡙⠻⠛⢛⣛⣿⠀⠀⠀⠀⠀⠀⡾⣟⡞⡆⡟⣶⣗⡷⢷⠃⠀⠀⠀⠀⠀⠏⢸⠞⡇⡸⢹⢁⠃⠛
⠀⠀⠀⣟⣽⣟⣟⣫⣷⡟⠀⠀⠀⠀⢰⢀⢣⠶⣄⡾⡽⣬⡌⣆⣶⣤⣤⠀⠀⠀⠀⠀⠸⢹⢻⠹⠿⡿
⠀⠀⠀⠈⠛⠶⠶⠖⠛⠁⠀⠀⠀⠀⠋⠘⠉⠀⠉⠃⠁⠋⠉⠉⠈⠉⠋
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣆⠀⠀⢀
⠀⠀⠀⠀⠀⢤⠀⡀⠷⠀⠀⠾⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣰⡸⣿⠼⠸⣿⣴⣠⡾
⠀⠀⠀⣀⠀⠈⢁⣠⣴⣶⣶⣦⣤⡈⠡⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⣢⠟⣫⣴⣶⣶⣶⣶⣤⡙⠷⣣⣴⠆
⠀⠀⠀⠈⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⢵⡂⣀
⠀⠀⠒⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠋⠁⠀⠀⠀⠀⠀⠒⠛⠿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣛⠉
⠀⢀⣀⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣭⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣦⣤
⠀⠀⠀⠐⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⢶⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣟⠂
⠀⠀⠀⠊⠁⠀⡈⠙⠛⠿⠿⠟⠛⢁⡀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⡳⢦⣝⠻⠿⠿⠿⠿⢛⣡⣜⠯⠙⠓⠂
⠀⠀⠀⠀⠀⠘⠁⠈⢰⠆⠀⢰⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⠏⣿⠶⡖⣿⠙⠌⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⢀⠀⡴⣄⠰⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠄⣿⠀⢳⣀⡯⠀⠸⡇⠀⣴⠀⢠⠎⡄⢠⡏⠀⠀⠀⠀⠀⣠⢖⡇⠀⡰⢣⠄⢀⠀⣰
⠀⠀⡄⠀⠀⠀⠀⣷⠀⠀⢿⠀⢸⢿⡇⠀⠀⠈⠀⠧⠀⠘⠐⣇⢸⠃⠀⠀⠀⠀⠐⠅⢸⡇⠀⠃⠺⠄⠛⠐⡇
⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⢸⡀⠈⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠂⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣡⣤⣼⣿⣶⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣤⡶⠯⠷⠆⠈⠁
⠀⠀⠀⠀⠀⣠⡴⢞⣫⡥⠀⠀⣀⣀⡈⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⠏⣤⡤⠤⣵⠄
⠀⠀⠀⣠⢾⣿⠞⢉⡿⠗⠋⢉⣽⣋⣀⣀⣀⣀⠈⠲⣄⠀⠀⠀⠀⣎⣀⣬⣽⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠁⠀⢀⣴⠯⠭⣄⣀⠈⠳⡀⠀⢰⠀⢉⡻⠂⢀⣠⠴⠒⠉⠁⠀⠀⣀⡀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣍⡀⠙⡄⠘⣠⠎⣠⠖⠉⣀⣀⡐⠿⣖⠢⢄⣙⣍⠓⢦⣽⡷⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣌⡓⢌⣾⣣⠞⣁⣤⢭⣿⣬⠉⠉⠻⠆⠀⠈⠛⠀⠀⠈⠇⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢀⡄⢀⣠⡶⢂⣀⣿⣶⣇⣨⣟⢿⠛⢒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣾⣠⠋⣧⠎⠶⠚⠉⢰⠉⢹⢨⠀⢿⠳⣄⢸⡇⢰⡆⢈⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣴⣼⠈⠟⠀⠋⠀⠀⠀⠀⠈⣹⠅⣾⠓⠋⠀⠘⠟⢻⡞⣿⢸⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣿⠀⠀⠀⠀⠀⠈⠀⢽⡜⢷⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⣷⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠤⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣠⠴⠻⡄⠀⠀⣀⢀⡾⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡄⢹⡀⠀⠀⠀⣀⠙⣆⢰⣿⡎⣧⠊⠔⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢦⣣⠀⢷⠒⠚⠋⠁⠀⠈⢷⣇⠁⠹⠸⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⠀⣸⠦⠀⠀⠀⠀⢀⡤⠬⠿⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠁

⣏⠉⣹⣋⣛⣛⣛⣛⣏⣛⣛⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠠⠀⠀⠀⢸⡀⢀⡏⣰⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠒⣆⠀⠀⠈⠃⠀⠀⠀⠀⠃⠘⠈⠇⠀⢠⠎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠳⡌⠀⠀⠈⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠞⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢄⡙⠀⠈⠁⠀⠈⠀⠀⠠⠀⠄⠈⠙⠀⠀⠀⠀⡀⠀⠐⡀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⠂⠀⠀⠀⣤⠠⠗⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠁⠀⠀⢀⡀⠡⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⡄
⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠘⠁⠀⠐⠰⠺⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠁
⠀⠀⠀⠀⠀⡤⣤⣀⡀⠀⢀⠈⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠀⠤⠀⠂⢄⡀⠀⠀⠁⠀⠀⠀⠠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡘⠀⠦⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠗⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠐⡕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢧⡄⠀⠠⡉⠠⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠐⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠦⠀⡀⠀⠀⠈⡧⠔⠈⠀⠀⠀⠀⠂⠀⠀⡀⠃⡄⠀⡄⠀⠀⢢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⡴⠁⠀⠀⠀⣀⠔⠁⠀⢀⠔⡒⠀⠀⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⡠⣒⠕⠉⠑⠢⠄⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠶⠒⠚⠂⠤⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⢤⣧⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠟⣻⣿⣿⣟⢁⣤⡠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢿⣿⣟⢳⣿⡿⢿⣶⣟⣩⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡄⢚⣯⡀⣾⣿⡿⣿⣯⣶⣾⡷⣤⡤
⠀⠀⠀⠀⠀⢀⣴⣶⣦⣀⠀⠀⢤⡶⢠⣿⠒⣃⠙⡟⣿⣿⣿⣿⣿⣿⢿⢾⡿⠄
⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠆⠀⢰⣧⣼⡟⣉⣹⣤⣿⣿⣿⣿⣏⠻⠋⢠⠈⠁⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣾⣿⣿⠟⠁⠽⠝⢛⢤⠿⣿⣿⣿⣿⣻⣟⣶⣿⣗⣶⠂
⠀⠀⣀⣄⠀⠀⠊⠸⢻⣿⣿⣿⡟⠁⣴⡋⠟⡶⢯⣤⣼⣿⣿⣿⣷⣋⣍⣵⣄⣀⣉⡀
⢀⣿⣿⣿⣷⣤⣄⠀⠘⣿⣿⣿⣷⠈⢥⣿⣷⣶⣿⣿⣿⣿⣿⡿⠟⡿⢿⣏⠉⢉⡀
⠀⠻⢿⣿⣿⣇⣹⣶⡟⣼⣿⣿⣿⡷⢓⣾⢮⣋⣩⣿⣿⣿⣿⣧⣼⣤⣼⠄⣾⣿⣿⣷⣴⣷⣶⣦⣄⡀
⠀⠀⠀⢸⣿⣿⣿⣿⢹⣿⣿⣿⣿⠘⢀⣿⣿⣟⣿⣿⠟⢻⠟⢻⣿⣿⣥⡘⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠈⣿⣿⣿⡅⠈⣿⣿⣿⣿⣿⣉⣷⣿⣽⣿⣿⣤⣼⣤⣼⣿⣿⣝⡢⠄⢀⣬⣤⣼⣿⡟⠛⢿⣿⣿⣿⠦
⠀⠀⠀⢠⣿⣿⣿⠃⢈⣿⣿⣿⣿⣿⣿⣃⣽⣿⣿⣿⣿⣿⣿⣿⡟⠙⡟⠁⠀⢅⣤⣤⣾⠟⠀⠀⣿⣿⣿⣿⠂
⠀⠀⠀⠘⢿⣿⣿⠀⢸⣿⣿⣿⣿⢿⡴⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣧⣀⢰⣤⣌⣠⣶⡀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⣾⣿⣿⠀⢸⣿⢻⣿⢁⢴⣷⣿⠿⣻⠟⢻⣿⣿⠿⣿⣿⣿⣿⠟⠋⡩⠛⠫⢸⡡⠀⣿⣿⢿⣿⡀
⠀⠀⠀⠸⣿⢽⡟⠀⣸⡇⢈⣿⠾⠋⠉⠀⢸⡇⠀⣜⡛⠘⣤⠉⠙⢫⠟⠀⠀⡇⠀⠀⠀⠄⠀⣸⣿⣸⣿⣷
⠀⠀⠀⠀⠿⣾⣧⡀⢹⣗⠈⢻⣇⠀⠀⠀⢸⡇⠁⡎⠁⠀⠿⠀⠀⠈⠀⠀⠀⠃⠀⠐⠀⠀⢀⣼⡿⢈⣽⣿
⢠⣀⣀⣀⣀⣈⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣬⣭⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡐⢦⡀⠀⣸⡇⢀⣤⣤⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠿⢿⣿⣿⣄⣿⣷⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣾⣿⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⢻⣿⣿⣿⣿⣿⣍⠉⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢠⣾⠃⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⣾⣿⠟⠿⠁⢻⣿⡄⠀⠀⠉
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣿⣷⣿⣷⣿⣟⠻⢿⣷⡄⠀⠀⠀⠀⣸⠏⠀⢀⣼⣿⡗⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠁⠀⠀⢀⣠⣴⡿⢿⣿⣿⣟⠙⠛⢷⡄⠈⠁⠀⠀⠀⠀⠟⠀⢠⡾⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡿⠉⠀⢠⡿⣿⣿⣿⣆⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈
⠀⠀⠀⠀⠀⠻⠀⠀⠀⣿⡇⠀⢻⡝⢿⡄⠀⠸⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠻⣎⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣿⣿⠀⣿⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣾⣿⣿⣾⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠻⠿⠿⠟⠛⠛⠃
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣤⣦⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣄⠀⠹⣿⣿⣇⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣐⣺⣿⣿⢀⣿⡿⢂⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣅⣘⠿⣿⣷⣼⣿⣿⣾⣏⣴⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣫⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⣠⣶⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡿⣿⣿⣗⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣯⣾⡿⠛⣵⣿⡟⠀⣿⣿⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣀⡀⢀⢱⡇⣰⣴⣶⠦⠀⠀⢠⠟⢁⣾⡿⠛⣡⣾⣿⡿⠀⠀⣿⣿⠈⠘⢻⣿⣿⡄
⠀⠀⠠⢄⣛⣿⣿⣿⣆⣿⣿⣿⠁⠀⠀⢀⠀⠀⠸⠉⠠⣾⣿⣿⡿⠁⠀⠠⣿⣿⠀⠀⠀⣿⡿
⠀⠀⠀⠀⢈⣹⣾⣿⣿⣿⣿⡗⠀⣠⣿⣿⣿⣷⠄⠀⠀⠉⠉⣿⠃⠀⠀⠚⣿⣿⠀⠀⠀⠉
⠀⣠⣤⣴⣾⣿⣻⣿⣿⣿⣿⢇⣼⡿⠟⠉⠉⠉⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠁
⠀⠈⠉⣉⣭⣭⣿⣿⣿⣿⣿⣾⣿⣷⣾⣿⣿⡿⠇⠀⠀⢰⡟
⠀⠀⠺⠟⢋⣻⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣷⣧⡀⠀⢀⣿⠁
⠀⠀⢀⣴⠿⠻⢫⣿⣿⣿⣿⠈⣿⠿⣿⣦⡉⠻⠻⡀⣼⡇
⠀⢀⡾⠁⢀⣴⡿⣹⣿⢻⣿⣇⠘⣇⠘⣿⡏⠀⠀⢰⡿
⠀⠈⠀⠀⢸⡿⠱⣿⡿⠈⣿⣿⠄⠘⣆⠉⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⢹⡄⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣿⣾⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣴⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣼⣿⣿⣿⣿⣿⣿⣿⣯⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⠀⠀⠀⣨⠃⠀⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠃⠀⠀⢀⣴⣶⠀⠀⠀⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣰⣶⡮⣿⠀⠀⠰⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠁⣹⡆⠀⠀⠺⠾⠽⣽⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡇⠀⡴⠟⠁⢠⡄⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⡙⠃⠀⢀⣀⣀⠀⠀⠀⣾⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢾⠗⠀⠀⠘⢿⣄⣀⡤⠖⣾⡇⡴⢉⣿⡀⣰⣿⠟⣿⢀⣴⡿⢛⡷⢠⣿⠀⣰⣿⠋⢹⡆⠀⢰⣿⠃⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢻⣷⡀⢠⣿⡿⢁⣼⡿⢠⣿⠧⠞⠁⣸⡟⠀⠉⢀⣿⡟⢠⣿⡇⠀⣼⣿⢀⣿⡟⠀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡴⠋⠀⠀⢸⣿⠇⣼⣿⣡⣾⡿⠔⠋⢿⣀⣠⠔⠻⣇⣀⡤⠺⣿⣡⠞⣿⣠⠞⢿⡧⢾⣿⠁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⣠⡿⠋⢰⣿⠏⠉⠀⠀⠀⣀⢤⣁⠀⠀⠀⠀⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠀⠀⢾⡟⠀⠀⠀⠀⠈⠳⣴⣿⣤⣴⠶⠶⠟⠛⠛⠛⠛⠿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣤⣤⣀⣿⡏⠀⣠⣤⢄⠀⠀⢀⣄⠀⠀⣤⠀⠀⠀⠀⣠⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⢤⠀⠀⠀⢰⣿⠏⠀⣿⣿⢁⣾⡟⠁⢸⣧⢀⣾⡿⠀⢠⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣰⣿⡇⢸⣿⠁⢠⣿⡟⣸⣿⠁⢠⣿⡏⠀⠀⠀⠀⡠⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠤⢄⠀⠀⠀⠙⠧⠼⠁⠿⠴⠛⠿⠔⠁⠿⠖⠙⠿⠔⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⢠⣿⠇⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠘⠃⠀⠀⢿⣇⣠⡿⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠈⠐⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⡜⣀⣀⠠⠤⠤⠤⢉⣀⠀⠀⠠⠒⠒⠒⠒⠁⠈
⠀⠀⣀⡀⠀⠀⠀⠠⢀⠀⡇⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠈⢦⠀⠀⠀⠀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠉⠒⠤⠤⠂
⠀⠀⠀⠀⠸⠄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢀⣀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠰⣅
⠀⠈⠀⠑⢄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠱⣄
⠀⠀⠀⠀⢀⠕⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠓⢒⠃
⠀⠀⠠⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⠠⠐⠁
⠀⢠⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢄⠁
⠀⠒⠈⠀⠁⠂⠤⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠘⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠛⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⣓⠶⠒⠒⠒⠒⡛⢍⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠔⠂⠉⠀⠀⠡⡀⠀⡎⠀⠀⠑⠢⠄⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠁⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠉⠉⠁⠀⠉⠉⠉⢙⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣷⠀⣿⠛⠛⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⢿⡄⠀⢸⡇⠀⢰⡟⠀⠀⢀⣴⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠿⠟⠛⠛⠻⠿⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠠⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠀⣿⠀⠀⠀⠀⣿⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠈⠉⠋⠀⠀⠀⣰⡟⠀⠀⣀⣠⣿⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢻⣷⢶⣤⣤⡶⣾⡛⠉⠻⣦⡀⠀⠀⠀⣼⠏⠀⢠⡾⠋⠉⠉⠉⠛⢷⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠾⠃⠀⢸⡇⠀⠘⠷⠀⠀⠈⠻⠆⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⠻⣦⣤⣀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠉⠉⠉⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠛⠁

⠀⠀⠀⢠⣼⣦⣠⣾⣥⡄⠀⠀⠀⠀⢀⣭⣿⣦⣴⣶⣂⠀⠀⣈⣿⣧⣾⣯⡄⠀⢤⣶⣿⡆⣠⣴⣤⡀
⠀⠀⠀⢀⣼⣿⠛⣿⡿⠗⠂⢀⣦⣄⣾⣾⠉⡟⠉⣿⢹⠀⠈⢩⡿⢻⣿⠏⠋⠀⠿⣿⠁⡆⢿⡍⠻
⠀⠀⠀⠀⢰⠃⠀⠈⠛⠀⠀⠛⣟⠙⡏⠃⣼⠁⠀⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⡇⠀⠛
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠐⣦⣼⣀⡤⠀⠀⡼⣿⣷⣴⣿⣿⠂⠀⢐⣶⣿⣷⣴⣿⣏⠀⠀⢀⣠⣤⣾⣿⣯
⠀⠀⢀⣰⣦⡿⠿⢍⣿⡿⢿⣗⠀⣾⢹⡏⢩⠉⠙⠃⠀⠈⢹⡟⣹⠻⣿⠿⠁⠀⠽⢿⠛⢋⠙⠣
⠀⠀⠠⠟⣿⠙⢦⠏⠀⡟⠀⠈⠀⠈⠀⠁⠘⡆⠀⠀⠀⠀⠈⢁⡇⠀⠉⠀⠀⠀⠀⠈⠀⠘⡄
⠀⠀⠀⠀⠈⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠢⡄⠀⠘⡅⠉⠑⢦⡀⠀⠀⠀⢰⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠈⣦⢠⠃⠀⠀⠀⢱⠀⠀⣠⠋⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠊⠃⠒⠢⣄⠀⡞⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⠃⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠈⠻⡄⠀⠀⢀⣀⣀⣤⡧⠀⠀⣤⣠⣎⡀⠀⠀⠀⠀⠀⠀⣇⡠⠔⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡖⠙⠉⠉⠉⠙⠲⢤⡇⠀⠀⠀⠀⢀⡽⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀⠀⡼⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠙⢄⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡴⠚⠉⠉⠉⠉⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⣠⠔⠒⠂⠙⢒⠶⣐⡻⠭⠝⠛⠙⠋⠩⠿⢶⣢⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣖⡶⠿⠋⠙⠋⠙⠋⠽⢜⣣⢄⡀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⢀⠞⠀⠀⢀⣀⡰⡩⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠤⠤⠤⣤⢞⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡝⢦⣒⣫⢄⣀⡀⡄
⠀⠀⠀⠀⠘⠒⠒⢴⡃⢸⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢻⡠⠤⢤⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡅⠘⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣀⡼⠧⣄⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣬⠀⠀⠀⠇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠴⠀⣠⠞
⠀⠀⠀⠀⢠⠎⠁⠀⠀⠘⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠃⠀⠀⠀⠸⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠓⠒⢏⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠘⢮⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡵⠃⠀⠀⠀⠀⠀⠙⢞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠃⠀⠀⠀⠉⠑⠒⠒⡆
⠀⠀⠀⠘⠒⠂⠁⠒⠢⣄⡀⠀⣻⠪⢗⣤⠤⠄⡠⢤⣤⣒⠾⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠷⣲⣤⠤⡤⠀⠤⣔⡪⠕⣻⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠙⡆⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⠧⠤⣄⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⣠⠔⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⣄⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠳⠊⠁⢀⠜⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠙⡶⠤⠤⠤⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢇⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣄⣀⣴⠋⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠉⠀⠀⠈⢳⢤⣀⠀⠀⢀⣈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡠⠚⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⠈⠉⠙⠛⠀⠲⠂⠚⠋⠉⠉⢇⠀⠀⠀⠀⠀⢀⡎⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡞⠀⢸⠀⠀⠀⠀⠀⢀⠜⢆⠀⠀⠀⠀⡇⠈⠢⣀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢦⠀⠀⠀⢠⠋⠀⠈⠓⢤⣀⣀⣣⠀⠀⠀⠉⠑⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⣶⣾⣿⣾⣶⣿⢶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⠾⣶⠷⡶⢶⢶⣶⣿⠷⠶⠾⠷⡾⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⢸⣿⡆⠀⠀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⣦⠀⠻⠛⠇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣤⣀⠈⢉⣤⣶⣿⣿⣿⣶⣤⡉⠀⣠⣴⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠟⠀⡉⠻⢿⣿⣿⣿⡿⠟⢋⡀⠛⠻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠀⢠⣤⣤⠀⠺⣿⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡀⠠⠄⠀⠀⣠⡿⠋⠀⠀⠘⣿⡇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⣀⠄⠊⠀⠀⠀⠀⠹⣤⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠟⠀⠀⠀⠀⠈
⠀⠀⠠⡀⠀⠀⠀⠱⣄⡎⠀⠀⠀⠀⠀⠸⡀⠀⣿⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⢳⣠⠁⠀⠀⠈⠙⠶⣤⣠⠀⠀⠀⠙⣦⡏⠀⠀⠀⢠⠎⠀⢀⡠⠂
⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠹⣦⠀⠀⣠⡟⠀⠀⢀⣠⠞⠉⠉
⠀⡀⠀⠀⠀⣀⡤⠞⠋⠉⠉⠉⠙⢿⣧⣼⣯⣴⠶⠿⠻⠷⢤⣀⠀⠀⠀⠀⣀⠠⠄
⠀⠈⠉⠈⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⣏⠛⠛⠉⠉⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠋⣿⠉⠛⠶⣤⣤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠸⠀⠀⠀⠀⠙⢆⠐⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⣿⣯⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⣍⡁
⠀⠀⠀⠀⠀⠉⠉⣛⣿⣿⣿⣿⣿⣿⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⣾⠿⢛⡽⣿⣿⣿⣿⠻⣿⣦
⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⡻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣴⠋⣸⣿⠏⢿⣿⡄⠀⠙⠇
⠀⠀⠀⠀⠀⠟⠁⠀⣿⣿⠹⣿⣏⠻⣄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡴⠟⠁⠀⢘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⠛⠦⠘⢦⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⣿⣴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⡇⠀⣼⠀⠀⣠⠋⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠈⠢⡀⠑⡄⢀⠹⠀⡀⡇⢃⢧⠂⡔⠁⢀⡴⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠈⡀⠈⠀⠁⠁⠀⡇⠈⢀⢎⢔⠔⢉⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢄⣀⠅⠈⠀⠀⢀⠠⠀⠀⠄⠑⠀⠠⢒⠡⠀⡀⠤⠒⠋
⠀⠀⠀⠀⠀⠙⠛⠒⠒⠦⠤⠤⠀⢁⡀⠀⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢤⣶⠤⠒⠖⠋⠑⠒⠚⠉⠓⠒
⢠⡤⠤⠤⠴⠒⠋⠉⠐⠒⠤⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡄⠤⠤⠤⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠒⠃⠈⠉⠀⠉⠀⠀⣀⡠⠔⠃⠴⠉⠀⠀⣀⠠⠔⠒⠀⠈
⠀⠀⡀⢀⡀⢀⠀⢉⠐⣂⡀⢀⣁⣀⣀⣐⣻⣯⡍⠉⠉⢉⣩⠋⠀⠀⡀⠉⠀⠈⡳⠨⡖⠢⡀⠄⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⢞⡤⠿⠼⠞⠲⢥⣀⣀⣀⡬⢤⣄⣈⣩⣄⣯⣄
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠶⠪⠉⠑⣚⢋⠩⠍⠀⠒⠒⠒⠒⠒⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⡠⢀⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣧⡅⠢⡀⠐⣿⣦⡀⠀⠠⢊⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⣧⠬⠢⡙⣿⡰⠠⢁⣾⠛⢁⢄⣤⡶⠦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠿⡷⣶⣤⣐⠺⣗⡑⣘⣧⡇⣽⣭⣯⣴⠟⠛⢉⣁⠠⣄⢤⡨⣤⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣈⠍⣉⠁⡛⠻⠿⣵⠢⡘⣯⣅⠟⡵⢋⣰⣦⡙⠧⠙⠳⠙⠃⠙⠈⠑⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⢟⡾⣻⠇⡾⢃⡾⣱⡾⡄⠨⣙⢺⣼⣿⠏⠔⢃⣨⣍⣭⣭⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⢠⣂⣥⡦⣶⡶⠖⠐⣾⣡⠛⡏⢙⡭⠥⠬⣍⢩⡉⠛⠫⢛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢛⠝⠃⢡⣠⢞⣩⠿⣀⣸⢈⡿⠫⡐⢦⣦⣼⣳⡗⢰⡄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡄⠿⢈⣧⣾⢃⡈⠀⡇⡁⢨⡌⢦⡄⠩⠙⠋⠇⠘⢱⣷⡲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣴⣆⠻⠀⠘⠇⡇⢬⡏⠀⠁⠁⠐⣯⡈⡼⡂⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⠉⠀⠀⠀⠀⠡⡸⠍⢀⡄⣀⠠⠞⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠋⠈
⠀⠀⠀⠀⠈⠒⠐⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠶⠖⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈
⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⣦⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⣶⡀⢸⡇⠀⠈⠁⠀⠰⠿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣾⣿⣶⣤⡀⠀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡇⠀⠾⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠻⠿⠛⠋⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣠⡄⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⡇⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢀⣿⣿⡇⠸⣧⡀⠀⢀⡀⢀⣀⣤⣤⣬⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣾⣿⣿⣷⣆⠙⣿⡀⠀⠙⠛⠛⠛⠁⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠲⢶⣶⣶⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⠿⣿⣿⠿⠋⣰⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣦⡀⢀⣴⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠻⠿⠿⣷⣬⣿⣾⣷⣶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠶⣤⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⣻⣿⣿⣿⣿⠏⣿⣿⣿⣿⡿⣿⣷⣌⠛⢿⣦⣀⣀⡀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⣾⢩⡾⢋⣿⣉⣸⣿⠇⢰⣿⣿⣿⣿⣧⡈⠙⠻⣆⠀⠙⢿⣍⡙⠻⣶⣤⣼⣯⣶⠶⠺⠲⢤
⠀⠀⠀⠀⢲⣴⣿⣿⣿⣟⣁⣤⣼⣧⣄⣸⣿⡟⠉⣿⣿⣷⠀⠀⣠⠶⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⡛⠓⠦⡀
⠀⣠⠖⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢿⡇⠀⠀⠙⣿⡆⠐⠋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣹⣿⣽⣷⣤⡀
⠀⠀⣴⡾⣻⣿⣿⣿⣿⣿⣿⠿⣿⣦⡈⠀⠘⠃⠀⠠⠀⠈⠃⠀⠀⣠⣴⠿⣫⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣆
⠀⣼⠏⡼⢻⣿⣿⣿⣿⣿⣿⣷⣄⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠟⠛⣿⣿⣿⣿⡇⠸⣿⠈⠻⠀⠉
⠀⠇⠀⠀⣸⡏⣼⡟⢸⣿⡇⠙⢿⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⣿⡇⠀⠸
⠀⠀⠀⠀⣿⠁⠹⠀⠀⢿⡇⠀⠀⠉⠣⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠨⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢀⣷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⠿⠛⠾⠿⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣀⠀⠐⠋⣀⣀⣀⣀⣀⣠⣤⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣾⣿⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⢀⣈⣋⣀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⠟⠛⠛⠛⢿⣦⡉
⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⢀⣀⣀⢠⣿⠃⠀⠀⠀⠀⠀⢻⣷⢀⣀⣀⡀
⠀⠀⠸⠿⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⢸⣿⠀⠀⠀⠈⠉⠉⠘⣿⡆⠀⠀⠀⠀⠀⣸⡿⠈⠙⠉⠁
⠀⠀⢀⣀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣙⢿⣦⣤⣤⣤⣾⠟⣁
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⠀⢸⣿⠀⠀⠀⠀⠀⢰⡿⠋⠀⠈⢉⣍⠉⠁⠈⠻⣷
⠀⠀⢸⣿⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⢸⣿
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⠀⣼⣿⡀
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⣾⡟⠹⣷⡄
⠀⠀⢸⣿⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⢛⣿⡏⠀⠀⠹⣿⡄
⠀⠀⢸⣿⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣼⣿⡀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⢿⣶⣶⡾⠟⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⡟⠀⠀⠀⢀⣾⡏
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠘⠿⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠿⠟⠀⢸⣿⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢠⣿⠏⠀⠀⠀⢠⣿⠏
⠀⠀⢸⣿⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠸⠿⢿⣷⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⣻⡧⠀⠀⠀⠀⣹⣷⠀⠀⠀⠀⢹⣷
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠋
⢀⣤⣼⣿⣦⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠰⠇⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠐⣬⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⣀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣉⠙⠣⡀⠀⠀⢂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣦⣀⣈⠀⢬⠉⠠⣠⡀
⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣏⠏⠉⢳⡄⠀⠀⢸⡝⠆⠈⠂⠒⠆
⠈⠛⠡⢀⣀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⣀⢿⡆⠈⢳⣲⣾⡧⠀⠂⠀⣻⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠀⠐⠂⠐⠉⠉⠁⠀⠀⠀⠀⡈⠚⠛⠻⠇⠀⠉⢿⡏⠀⠀⠀⢰⡋⠹⠀⠀⠀⠀⠀⠀⡲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣼⣦⣤⡄⠀⠈⠁⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠈
⣷⡆⠀⢀⣠⡔⠀⠀⠀⠀⠀⠈⢦⡂⠰⣟⠻⣿⡿⠻⣿⠐⣶⣾⠀⠀⣇⠀⠘⠁⠀⠀⠀⣦⢠
⢤⣀⣠⡼⠇⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢁⡮⠁⠀⠈⠙⢿⡏⢿⣿⣷⠀⠀⢠⠀⠀⠀⣿⠀⠀⠈⠉
⠀⠉⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣿⢁⡿⢁⣀⣀⠀⠀⠀⠛⠆⠙⡿⠂⠀⠀⣧⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣍⠀⠀⣸⠛⢻⠿⣿⣿⣿⠿⣿⣿⣆⠀⠀⣄⡀⢠⠀⠸⡄⠀⠀⣻⠀⡠
⠴⢦⣤⡴⢾⢿⡟⠻⠮⠏⠁⠀⠀⣿⡀⠀⣻⠶⡄⠉⠚⣽⠇⠘⠻⣤⡏⢱⣿⠃⣤⢹⡄⣴⡿⠉⢀⡀
⠀⠉⠀⠀⠈⠀⡀⠀⠀⠀⠀⢀⣴⣯⣴⣶⣏⡀⢱⡀⠀⠙⡷⢦⡀⠘⢷⢸⣿⣧⢸⣆⣷⣿⠟⠂⠀⢳⡀
⠀⠀⠀⠤⣀⡀⢳⠀⠀⠀⠀⢸⡷⣾⣿⡿⣿⣷⣾⣷⡀⠀⢄⠀⠑⠠⣈⣼⡏⣸⡇⣸⣿⠟⠀⠀⣤⣈⠁
⠀⠀⠀⠀⣈⡳⢃⠀⠀⠀⠀⣿⢁⣸⣿⣿⣦⣼⢻⣿⣧⠀⣻⣆⠀⢀⢸⡏⡇⣿⣿⣿⠋⠀⠀⠀⣠⠟
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣶⡠⢴⣿⣿⣏⡾⢋⡟⣿⣿⣿⣿⣿⣷⣿⣾⣷⣧⣿⣿⠉⠀⠈⠁
⠀⠀⠄⠀⠘⠶⠀⠈⠁⠈⠻⢻⣷⣾⣽⡿⣷⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣴⣿⣿⠿⠟⠉⣉⣉⣁⣤⣽⣷⣶⣿⣿⣿⣿⣿⣏⠃⠘⣂⣀⣀⣀⣀⣀⣤⣤
⠀⠀⠀⢀⣀⣰⡟⠋⠕⣛⣯⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿
⣴⣶⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡽⠯⣾⣯⣤⣶⣾⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣴⣷⣶⣶⣦⣬⣵⣶⣤⣬⣤⣵⣤⣭⣙⣯⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣛⠿⠿⠿⠿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣶⣦⣴⣶⣶⣶⣶⣯⣿⣿⣿⣿⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⡆⢀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⡀
⠀⠀⠀⠈⡇⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⡏⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣇⢠⡇
⠀⠀⠀⠀⢻⡄⢹⡄⢻⡄⠀⠀⠀⠀⠀⠹⣆⠹⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⢸⠃⣼⠃⡷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⣷
⠀⠀⠀⠀⠈⣇⠀⣧⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⢠⡏
⠀⠀⠀⠀⠀⡏⠀⠟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢳⠀⠇



⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⡞⠀⡏⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢳⡠⡏⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼
⠀⠀⠀⠀⢷⡀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠈⢷⠈⢷⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⢯⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡄⠻
⠀⠀⠀⠀⠀⢳⠈⢷⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧⠘⠇⠀⠀⠀⠀⠀⠀⠹⢀⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⣆
⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡼⠁⣼⠀⢻⣄⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⢰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⡀
⠀⠀⠀⢧⡀⠘⠷⣄⠈⠙⠆⠀⠀⠀⠀⢻⡄⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠰⡜⣆⠳⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡷
⠀⠀⠀⠈⠻⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢳⣌⢷⡙⣆⠇⠀⠀⠀⠀⠀⠀⢸⡀⠙⣆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣾⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⠀⠉⣙⣻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣴⣧⣾⣷⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡹⢿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡩⠿⣿⣿⣄⠀⠀⣠⣾⣿⣟⠻⠿⠟⢛⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠑⣞⣿⣿⣦⣴⣿⡿⠯⠝⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣻⣷⡶⣶⣶⣾⣿⣟⢿⡷⡶⣖⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⡿⣿⢿⣟⡿⣿⣿⢒⣿⣿⣟⣽⣿⣿⣿⣿⣯⣿⡷⣿⡉⠛⠟⠚⠓⠁⠙⠈⠎⢿⠛⡏⢻⢶⣄
⠀⠀⠀⣀⣴⣿⢿⣿⡿⢫⢞⠞⡡⠋⠊⠐⠋⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠙⡷
⠀⢀⣼⣿⣿⢫⡾⠋⠐⠁⠈⠈⠀⠀⠀⢀⣾⣿⣿⣿⡿⢱⠏⢻⣿⠻⠟⢻⣿⣿⣿⣿⣆⡀
⢰⣿⡟⣹⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠙⠀⠀⠀⣸⣿⠀⠀⠠⠘⢸⠘⢿⣿⣿⣦
⠨⡟⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠘⣿⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡝⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠁⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢻⡿⣄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⠈⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣩⣵⣶⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠋⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣊⣥⣶⡿⠿⡿⠟⣛⣉⣭⣤⣴⣶⣶⣶⠾⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠉⠀⣰⣿⣿⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⠀⠴⠞⠛⠉⠁⠈
⠀⠀⠀⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣶⣦⣀⢀⠀⠀⡀⠀⢀⠀⠀⣠⣾⣿⡏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⢿⣿⣿⡟⢁⡤⣄⣦⣁⢨⣷⢠⣾⣿⣿⠅⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉⠀⠛⠛⠋⢠⣿⣵⣶⣯⢽⣷⣿⣾⣻⣿⠟⠀⠀⠀⢘⠂⢈⠃⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣿⠟⣻⢇⣠⣂⠀⠀⠈⠀⠐⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢠⡴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣟⣿⠟⣡⣾⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣈⣀⣉⣉⣻⣿⣿⣿⣿⣾⣿⢟⣀⡀⡠⣄⣤⡤⠄⢀⣀⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢿⣿⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡿⡯⠊⣿⢶⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠃⠁⣀⡾⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⡨⢿⣯⡙⢿⣿⣯⢿⣿⣆⠀⠢⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣠⣿⠉⠀⠀⢀⣼⡿⢈⣼⣿⢿⡷⠙⢰⣿⡇⢿⣻⣿⣷⣾⣧⣀⠻⣿⣿⡟⣿⣅
⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⢀⣾⠟⠀⣾⠏⠟⠏⠁⠀⢸⣿⠇⠀⠻⡟⣿⣿⣿⢯⡀⠉⠹⠻⠹⢿⣦
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣸⠃⠀⠈⣿⠊⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠊⣷⣯⣟⠆⠀⠀⠀⠀⠸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⡽⠀⠀⠀⠀⠀⠉⢹⣾⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢠⠄
⠀⠀⠀⠀⠀⢦⠀⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠀⢻⡀
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠻⣦⠙⢷
⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡟⣸⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡞⠁⡼⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠜⠁⠜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠡⡎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡇⠀⢆
⠀⠀⠀⠀⠀⢸⡄⢠⡏⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⢋⡵⠲⣄⠀⠀⠀⠀⠀⠀⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣼
⠀⠀⠀⠀⠀⠘⣇⠸⡇⢰⡏⠀⠀⠀⠀⠀⠀⣠⠶⣄⠈⢳⠈⢯⠒⠃⠀⠀⠀⠀⠀⠀⢰⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠧⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠳⠘⠁⠀⠀⠀⠀⠀⠀⠙⠾⠈⠇⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄
⠀⠀⠀⠀⠀⡏⠀⣸⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡇⢀⡄
⠀⠀⠀⠀⠀⢻⡄⠁⠀⣸⠀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠏⣠⠞
⠀⠀⠀⠀⠀⠀⢹⠄⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⡇⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢀⣀⡤⠖⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢿⣛⣿⢩⣶⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⣫⣷⣾⣿⠿⣋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⣾⣿⣿⣯⣹⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⣫⣾⣿⣿⣿⣿⡋⠙⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠉⣛⠿⢷⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣾⢟⣾⣷⣾⠟⢿⣾⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢷⡄⠈⠉⠉⠛⠛⠋⠉
⠀⠀⠀⠀⢀⣾⢫⣯⣿⣿⣿⣶⣿⡿⠷⠐⡈⠀⠙⠻⣿⣿⣌⡻⠿⣿⣿⣯⣉⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠆
⠀⠀⠀⠀⣾⢯⣷⣾⣿⣯⣼⣷⣿⣷⠀⡁⠁⠀⠀⠀⠙⢜⣿⣄⡀⠀⠀⠉⢉⠉⣘⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣄⡀
⠀⠀⠀⣸⡟⡾⢿⣷⣿⣿⣿⣧⣤⢿⡆⠀⠀⠀⠀⠀⠀⣾⠱⠸⢏⠄⠀⠀⠀⠀⠀⠀⠀⣀⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠉
⠀⠀⠀⣿⢻⣿⡾⠻⣿⣿⣿⣿⣿⡾⡁⠀⠀⠀⠄⠀⢰⣏⠁⠉⠀⠀⠀⠀⠀⣀⠀⠀⣠⣿⣷⣟⡿⣿⣿⣿⣿⣿⣎⠻⣿⡆
⠀⠀⢠⣿⣘⠉⣻⣿⣿⣋⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠚⠟⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣆⠹⣿
⠀⠀⢸⣿⢾⣿⣿⣻⣿⣿⣿⠦⣿⣆⠀⠀⠀⠈⠂⠀⡀⢤⣿⣿⣿⣿⣶⠂⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⣿⣿⡆⡏
⠀⠀⠀⣿⢸⣿⡿⢿⣿⡿⣧⣀⣼⣿⣷⡀⠀⠀⠀⠈⠿⠎⣿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣇⢈⢿⣿⠁
⠀⠀⠀⢿⡟⣿⣦⣼⢿⣿⡿⣼⣿⣿⡿⢿⠄⠀⠀⢴⠀⠀⠉⠁⠈⠉⠙⠛⠛⠶⢤⡀⠀⢿⡍⠛⠻⢿⣏⠻⣿⣿⢸⡎⢿
⠀⠀⠀⠘⣿⣻⣶⣿⣴⣿⡶⣿⣽⣟⡿⢛⡄⢀⣤⣿⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠙⠄⠘⠃⠀⠀⠀⠈⠀⠘⣿⡆⣿⢸
⠀⠀⠀⠀⠹⣷⢻⣻⣧⣼⣿⣿⣤⣿⡑⢿⡷⠟⠉⠀⠀⣠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⢰⡟
⠀⠀⠀⠀⠀⠹⣧⣻⣴⣿⣿⣿⣇⣨⠃⢀⡄⠀⠀⠀⢴⣿⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⣥⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣝⢏⣩⡟⢿⠏⠀⣼⡇⠀⣰⡦⠀⠀⠀⣾⣿⡦⠀⠀⠀⣴⣶⡆⠀⠀⠀⣿⡀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣝⢿⡏⠀⠀⠈⠁⠀⠉⠀⣶⣶⡆⠈⠉⠁⣶⣾⣆⠈⠉⠀⣶⣶⡦⠈⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠠⣤⡀⠙⠛⢁⣤⣦⡀⠹⠛⠁⣤⣦⡀⠙⠛⢁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⡘⠇⢀⣀⠀⠻⠿⠋⠘⠄⠀⠿⣟⣣⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠷⣾⣶⣶⣶⣶⣶⠶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⣱⢸⠀⠀⠀⠀⣱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣋⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⡀⡠⢚⠈⠢⠂⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠂⠬⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⣂⡵⣶⠤⠤⠐⡋⠡⠰⡆
⠀⠀⠀⠀⢀⣀⡂⡀⠀⠁⡀⠸⢀⣠⠂⢠⠂⢈⢄⠀⠤⠀⠀⠐⠀⠀⠀⠀⠖⠸⠎⢉⠙⠒⣦⡄⡒⠚⠁⠁
⠀⠀⠀⠀⠐⠀⠀⠁⣶⠞⣛⣿⣻⠿⣶⡉⠀⠀⠄⡀⣀⡰⠃⠒⠀⠤⣤⠈⠍⠑⠀⢈⡅⠉⣠⢀⠀⠀⣀⠴⠦⡀⢀
⠀⠀⠀⠠⡄⠀⠙⣻⣫⣻⣿⣿⣿⣷⠚⣿⠊⠁⠘⣿⠛⠑⠦⡀⠀⠀⠸⡉⢠⠅⣁⠼⠆⡄⢨⣏⠉⠒⠦⢥⣀⠢⢠⡁⠂
⠀⠀⠀⠀⠀⢀⠈⠙⣿⣣⡩⡿⢫⢟⢒⡾⠀⠀⠠⣎⠀⠀⠀⠹⢻⢹⠦⠻⡖⠺⣅⠀⢀⠀⠎⠈⠈⣙⡒⠆⠤⢁⡒⠊⢷⠉⠁
⠀⠀⠀⠀⠠⣀⡀⠌⡀⢉⠟⠓⠫⡙⠛⠁⣀⢀⡬⢡⡗⠂⠠⠌⠀⠒⠀⡀⠣⣀⢄⡑⠀⠀⠀⠀⠀⠘⣇⠃⠕⠒⠒⡿⠋⢀⣄
⠀⠀⢀⡀⠀⠀⠑⠠⢀⠀⣀⠀⠴⠁⡶⠇⠘⡣⢀⢘⠁⠀⠀⠈⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣉⠣⢀⣔⡈⠀⡀
⠀⠀⠀⢉⣁⣄⠀⠀⠋⠑⣀⠓⡿⠛⡓⣄⠀⠧⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡴⠉⢰⠈⢤⣄⢈⡄
⠀⠠⠤⢼⠪⡠⢍⡲⡵⣄⡏⠃⢻⠡⠘⡄⠣⠀⣆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢏⠀⠁⠀⡐⠀⠀⡈⠀⢓⢂⡀
⠀⠀⠀⠄⠳⡈⢆⠙⠠⡙⡓⠀⢢⣂⢳⠈⢆⠁⡟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣱⣶⠟⠿⢦⣔⡃⢀⢀⠎⢀
⠀⠀⠀⠈⠂⣄⠑⡉⢢⠈⠺⡓⠶⠷⢆⠱⡀⠣⡀⠀⢀⠀⠀⠀⠜⠛⠌⠐⠀⠀⠀⠂⣀⢀⣸⣿⢩⣷⣶⣶⣮⡛⣟⣁⠀⢀⠓
⠀⠀⢀⠉⡖⢨⣛⠘⠃⠑⠦⡈⠢⡀⢀⠀⢔⣄⠳⠤⡠⣄⠀⠀⠀⠀⢀⣀⢠⣠⣀⣄⠤⢠⣞⣧⣿⣿⣿⣿⣿⣎⡟⠀⠀⢀
⠀⠀⠀⠀⢤⣰⠨⢙⠂⠀⠀⠀⠀⠈⢢⠷⡘⠂⠑⣆⠀⣨⢠⣂⢀⠦⠊⠀⠀⠈⠈⠛⠊⠉⣟⣽⣿⣿⣿⣿⣿⡿⢿⣷⣾⡓⣟
⠀⠀⠀⠀⠀⠈⠉⠲⠾⠄⠔⠂⣀⢠⠚⠂⠀⢹⠦⡛⢿⠙⡌⣿⣰⠀⠀⠀⠀⠀⡅⠀⣹⣉⡁⠉⠉⣹⡉⢹⡍⢿⡀⠹⣤⣽⠧
⠀⠀⠀⠀⠀⣠⠄⣤⠥⠍⠄⠁⠉⠣⠐⠢⣶⢃⠠⡁⠤⠏⣥⠟⠁⠀⠀⠀⠀⠀⠀⠀⡛⠓⠡⡠⠎⠸⠀⢈⠃⠀⢀⣹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠒⢐⣣⡀⠤⢠⢅⡜⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠹⡼⠉⡑⠋⣠⠙⠘⡇
⠀⠀⠀⠀⠀⢖⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡌⢁⡴⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⢀⢩⠣⠀⡆⡅⣾⠀⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠨⡁⠸⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠐⠐⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣀⣀⣀⣀⡀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣸⡇⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠘⠷⣦⣀⣀⠀⣀⣠⣴⠞⠛⠛⠛⠻⢦⣤⣀⣠⣤⠾⠋⠁⠀⠈⠙⠿⣦⣀⠀⠀⠀⢀⣠⣴⠾⠃
⠀⠀⠀⠀⢠⣄⡀⠉⠙⠛⠋⠉⢀⣠⣤⣴⣤⣤⣀⠈⠉⠉⠀⣠⡴⠞⠛⠻⢶⣄⡈⠙⠛⠛⠛⠋⠉⢀⣠⡄
⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⡶⠾⠛⠁⠀⣀⡀⠈⠙⠛⠶⠾⠛⠉⢀⣠⣤⣤⣀⠉⠻⢶⣤⣤⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠘⠳⣦⣄⣀⣀⣀⣠⣴⠞⠛⠛⠛⠻⢶⣤⣄⣠⣤⠾⠋⠁⠀⠈⠙⠳⣦⣀⠀⠀⠀⢀⣠⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣌⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣨⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣥⣶⣾⣿⣷⣄⡀⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡄⢉⣹⠟⢉⣿⣿⣿⡛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠷⡋⠀⣠⣿⠋⢻⣿⡟⢿⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠱⢼⣿⣿⠷⠋⣩⠍⣫⣻⡿⠁⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠿⣿⢿⣿⡿⠉⣃⣳⠛⠙⠛⢦⣤⣤⡖⢋⡽⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⠰⠛⠁⢐⣿⣿⣗⡛⠒⠚⣛⡛⡑⣿⣿⣶⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠖⠐⢛⠋⠁⠹⡄⢰⣴⣾⣿⣿⣿⣿⠚⢳⣿⢿⣷⣼⣿⣛⠛⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣾⠃⣠⣴⣿⣿⣿⣿⣿⣿⣻⣿⣿⡠⠞⠻⠤⢜⣈⠛⠿⠻⣛⣏⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡿⠞⣽⣻⣽⠿⠋⣭⡿⢿⣥⡿⠿⠿⣶⣶⣦⡀⣠⠟⢀⠀⢉⣬⣤⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⠀⠠⣆⠚⠉⢁⣀⣰⣿⠦⣴⣷⣒⠦⠤⠬⢿⣖⣉⠈⠻⠿⣷⣾⣋⣾⣀⠀⢀⣀
⠀⠀⠀⠀⢀⢠⣀⡀⠀⠀⢾⡄⣄⣦⣄⢀⣘⢦⣠⠿⠋⢉⣠⢶⡄⠙⠻⣶⣤⡀⠀⠙⣻⣷⣶⣤⣀⢠⣀⣴⣴⠾⠾⠏
⠀⠀⠀⠀⣞⣸⣿⢻⠏⠀⠈⠙⠋⢿⣯⣴⡿⠛⠉⢤⣴⢿⣏⣹⣿⡆⠀⠙⢿⣥⣤⠐⣁⣨⠏⠻⣿⡿⠻⠃⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⣹⠿⣣⠏⢀⣤⣤⣄⠀⣠⣿⠟⠁⣠⣤⣾⣿⣿⣇⣠⣿⣧⡄⠀⠘⠻⢿⣷⡶⢦⣦⣀⡀⠙⠒⠤⣀⣀⣠⣶⣷⣸⣷
⠀⠀⠀⠀⠇⢀⡟⣾⡌⣿⣿⣿⠟⡉⠀⣠⣾⣎⢿⣯⣯⣤⣄⣠⣿⣿⣧⣶⣧⢰⣦⣼⡛⠳⣄⠸⠿⠻⢷⣷⣬⣍⣹⣿⡿⢻⡯⢤⠄
⠀⠀⠀⢸⡆⢸⠃⢸⣿⡸⣿⣿⡇⠳⣾⠛⣉⣙⣾⣿⣿⣿⡿⠛⠉⠉⠉⣻⡇⣼⣿⣿⠃⣴⢋⣀⣀⠀⠀⢹⢫⠽⢻⡏⠀⣸⠿⠓
⠀⠀⠀⢈⣷⠈⢿⣉⣝⠃⣠⣤⣤⣿⣹⣾⣿⠟⠛⢻⣿⣿⠀⣠⣴⣶⣄⡟⠁⠛⠻⣿⣾⣷⠿⠋⠛⣷⡀⣾⢸⣧⠟⡡⢢⠇
⠀⠀⠜⣯⣿⡄⠈⢿⣿⣿⣿⡇⠀⠸⣿⣿⡟⠀⠀⠈⣧⠿⣴⡏⠁⠀⣽⠉⠉⠉⠻⣿⣿⣿⡀⢀⠀⣿⣵⣛⣩⣵⣾⠷⠛⠒⠤
⠀⠀⠀⠛⠟⠉⣶⣏⣿⠟⠁⠀⡀⣺⣿⣿⣧⣅⢠⠀⣿⠰⣿⣧⡠⠄⣿⢀⡀⠀⠳⣿⣧⠈⣅⣈⡠⠉⣿⣥⡶⠟⡛⠙⣿⡞⠶⠘
⠀⠀⠀⠀⢰⢶⡉⢻⡛⠿⣿⣋⣤⠋⢉⣿⠁⠈⠐⠚⠋⠀⣏⢳⣤⣤⡾⠧⡠⡤⠲⣿⣷⣷⣦⣤⣠⠞⣁⠩⠓⢠⠧⠄
⠀⠀⠀⠠⣏⡈⣃⣀⣹⣄⣀⠈⣍⡚⠛⠿⣶⣤⣀⣠⣤⣼⣿⣿⣿⣿⣆⣠⠋⢱⣤⣾⣿⣿⡿⠟⠛⠉⠀⠀⠀⣼⣤⡀⠈⡀
⠀⠀⠀⢰⣫⡔⠁⣿⣿⡿⠟⠅⣿⢹⢳⠀⠀⠀⠙⠛⠛⠻⠿⠛⢛⠛⠛⠋⠁⠉⠉⠁⠁⠀⠀⠀⠀⠀⣀⠀⢰⡿⠟⠁⣀⠇
⠀⠀⠀⣾⣴⣇⠀⢣⠃⠃⠀⠀⠿⢸⣸⢠⠖⣤⠀⡄⢀⢀⡀⢸⠘⠁⣿⣀⢠⠀⣄⡤⢰⠂⣶⣶⣦⣿⠹⠂⡾⠃⢲⠋
⠀⠀⠀⣿⠁⢸⣦⡞⠘⠲⠦⣄⣀⣀⠈⠈⠀⠓⠀⠇⠸⢼⠃⠺⠀⠀⠷⠼⠸⠀⠷⠘⠛⠁⠋⠉⠉⣀⡠⠸⡆⢠⠏
⠀⠀⠀⢻⣄⣾⠋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀⢀⣈⣀⣀⢀⡀⠀⣀⠀⢠⣤⢤⡤⠔⠂⠉⠉⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠁⠀⢸⠃⠀⠀⡜⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⠀⠀⠀⠀⢀⣾⠃⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⠀⠐⣄⠀⠀⠀⠀⣼⡇⠀⠀⣠⠟⣿⡀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠸⣷⣦⣤⡾⠋⠙⠒⠛⠁⠀⠈⠛⢲⡞⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⡀⢀⣿⠀⠀⢀⣠⣤⠴⠦⣤⣄⡀⠀⠈⠳⣤⣤⣤⣤⣤⡶⠞⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⠀⠀⣻⠉⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⢠⡟⠀⠀⠠⠚⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⣧⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⢀⡿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⡀⠀⠉⠙⠲⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⣾⣁⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢀⣈⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢀⣰⡏⠁⠈⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠠⠶⠒⠀⠀⠀⠰⠯⣄⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣤⠶⠋⠁⠀⠀⢰⠟⠉⠃⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡿⠀⢀⣀⡀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣾⠷⠛⠉⠉⠙⣇⣀⡀⠀⢀⡤⠶⢤⡴⠚⠲⢦⣘⣇⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠉⠉⢻⣤⠏⠀⠀⠀⠁⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡏⠀⠀⢠⠀⠀⠀⠘⣦⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠸⠁⠀⠀⣿⠀⠀⠀⠀⠈⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣠⣾⣟⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣏⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠁⠈⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣶⠖⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠰⣤⣤⣴⣦⣤⣤⣴⣦⣦⣶⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈⠒⠒⠢⢄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⣸⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠻⣏⠉⢳⡄⠀⠀⢿⡆⠀⠀⠐⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠃⡤⠤⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⣠⣻⠀⢘⣵⣿⠧⠘⠆⢸⣧⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿
⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣠⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⣠⠈⢉⠙⠀⠀⠸⡏⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⣤⢿⣴⣿⣿⡞⣀⣠⠀⠀⡀⠀⡿⠀⠀⠀⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿
⠛⠛⠛⠛⠋⠋⠉⠉⠙⠛⡋⠀⣠⠞⠃⠀⠀⠀⠀⠀⠉⢀⣾⠆⣙⠃⠈⠓⢼⣿⢶⣶⡇⠀⠀⠀⠀⠠⣿⠁⠀⠒⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣼⢇⣀⣀⠀⠀⠹⠆⠹⡿⠀⠀⢱⠀⠀⠙⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠿⠿⠿⠿⠿⣿⣿⠿⣿⡟⠀⠀⠀⠀⢀⣀⣠⣤⣆⠀⢀⡟⠛⡟⢿⢿⣿⢻⣿⣧⣀⠀⣆⡀⡀⠘⣆⠀⠈⣇⡀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠛⠛⠛⠛⠛⠛⠋⠉⢹⡷⢲⠶⠖⠛⠻⠟⠋⠉⠁⠈⢈⡧⠀⢸⠛⡆⠈⢫⣟⠀⠙⢶⡇⢹⣗⢰⡘⣆⣾⡏⠀⣄⠀⠀⠉⠙⠛⠀⠘⠛⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠰⡀⠀⠀⠀⢠⣿⣾⣿⣿⣄⣸⡄⠀⠉⠙⢆⠈⢃⣸⠿⣾⢳⣿⡿⠁⠀⠈⠆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠿⠿⠿⠿⠿⠟⠛⠛⠻⠟⠀⠀⠈⠐⣦⠃⠀⠀⠀⣼⠋⣿⣿⣍⢹⣿⣷⡄⠘⣆⠀⠀⢶⢿⢰⣷⣼⡟⠁⠀⠈⣳⠀⠘⠛⠛⠛⠛⠿⠿⠿⠿⠿
⣀⣀⠀⠀⢀⡀⣀⡀⢀⣀⡀⠀⠀⡉⠁⠁⠀⠀⣠⠁⣸⣿⡿⣹⠟⡾⣿⣷⣾⣿⣦⣶⣿⣿⣽⣿⡟⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣤⣄⡀⢀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠰⠄⠀⠀⠀⠻⢷⣮⣿⣿⣷⣮⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠀⠀⠀⣠⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣀⣿⡿⣿⣿⣿⣿⡿⠿⠿⠿⣿⠛⣛⣻⣿⣿⣿⡿⠷⡀⠀⠀⠀⢀⣤⣠⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢠⡾⢟⣻⡿⢟⣋⣩⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣤⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡏⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠋
⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣤⣭⣯⣥⣤⣍⣭⣭⣭⣭⣩⣭⣙⣻⣏⣭⣷⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣮⣵⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⣯⣭⣿⣿⣏⣭⣭⣭⣭⣩⣍⣉⣽⣿⣯⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⣿⣿⣿⣯⣭⢽⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣭⣿⡿⣓⣀⡀⠀⢀⡴⢋⣀⣀⣀⠀⠀⠀⢳⣀⠃⣠⡴⠒
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⠉⠛⣦⣼⣟⡋⠉⠁⠀⠀⢀⣀⣈⣷⣾⣋⣀⣀
⣿⣿⣿⣿⣿⡿⢿⣿⣇⣠⡴⣶⣿⡟⢹⣆⢻⡄⠀⠠⠚⠉⢀⡼⠻⡅⠀⠈⠙⢇
⣿⣿⣿⣿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣷⣾⣿⣶⣿⡀⠀⠀⣴⠟⠁⠀⢿⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣤⣤⣤⣼⣤⣀⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠙⢉⠏⠈⠉⠋⠁⠁
⣿⣿⣿⣿⣿⣿⣿⠟⠀⠉⠁⢀⠎⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡰⠃
⣿⣿⣿⣿⣿⠿⢅⠀⣠⡾⠁⠀⠀⠀⣀⠴⠊
⣿⣿⣿⣿⣿⣶⣶⡿⠋⢠⣤⣶⣾⠇
⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠊
⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣷⣶⣾⣷⣿⣷⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⢆⠈⡄⢠⢠⡄⡌⢀⠆⡰⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠈⢆⠘⡄⢣⠸⡄⣧⢸⢸⡇⡇⣼⢠⠇⡜⢠⠃⡰⠁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⠑⣄⢳⣘⣮⠷⠛⠋⠋⠉⠉⠙⠙⠛⠾⣵⣃⡞⣠⠊⡠⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡑⢦⡑⣮⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣵⢎⡤⢊⡠⠔
⠀⠀⠀⠀⠀⠀⠀⣈⠒⠤⣙⣲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⣋⠤⠒⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡒⢮⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡵⢒⡉
⠀⠀⠀⠀⠀⠐⠒⠢⠤⣭⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠤⠔⠒⠂
⠀⠀⠀⠀⠀⠉⠉⣐⣒⣒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣒⣒⣊⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠬⠭⠭⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⠭⠥⠀⣀
⠀⠀⠀⠀⠀⠤⠀⠒⢒⣛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⡒⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠒⢈⣉⠥⢞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⠬⣉⡁⠂
⠀⠀⠀⠀⠀⠀⠈⢁⠤⠒⣩⠼⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠧⣍⠓⠤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⡠⠊⡱⢛⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡓⢌⠑⢄⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⡰⢋⡼⢹⢟⡶⣦⣄⣄⣀⣀⣠⣠⣴⢶⡻⡏⢧⡙⢄⠑⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⠎⢠⠃⡜⢰⠃⡟⢸⢸⡇⡇⢻⠘⡆⢣⠘⡄⠱⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠎⠀⠁⡈⠈⠃⢁⠈⠆⠱⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠘⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⠁⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡬⠿⢿⣿⣶⣦⡀⠛⣸⡇⢀⠖⣁⠀⢈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠙⠿⣷⠀⣼⣣⠏⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣭⣭⣶⣦⣬⣵⣟⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣡⣤⣜⣿⣿⣾⣶⡭⢙⣷⣶⡶⠦⣝⣣⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠿⢿⣿⣿⣯⣾⣟⡟⢩⣿⣿⣦⡘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠸⣿⠋⣿⢛⣿⣿⢸⠈⠉⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢻⠋⡰⠈⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣤⣏⠀⠀⠀⠀⢀⠀⣀⡀⡀
⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣤⣤⡉⣭⣭⣽⡿⠿⠩⠈⣿⠤⢬⣕⣛⠚⢻⠉⠉⠩⠁
⠀⠀⢠⣤⡴⠦⣴⣿⣿⣿⣽⠻⠷⠛⠉⠁⠀⠀⠀⢀⣹⣧⠀⠉⠙⠻⢛⣛⣛⠓⢲⣖⣀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠠⢶⡶⢶⣶⣦⣭⣙⢂⣀⡀⡀⠀⠀⠀⠀⠈⠉⠳⣄⠐⠀⠀⠙⠻⢿⣷⠍⠉⡉⠈⠉⠉⠛⠛
⠀⠀⠀⠀⣤⣤⣶⣿⣶⣬⣭⣾⣶⣿⣿⣿⣧⣈⠓⠒⠓⠈⠀⠈⠳⠄⠀⠀⠀⠴⠾⠒⠿⠿⠶⠆⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠤⠒⠶⠶⣾⣯⣤⣀⡈⠛⠋⠀⠲⠾⠿⠿⠿⠻⠒⠒⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣀⣀⡀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡾⠉⠀⠙⣧⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠆⠀⢀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣇⣀⣀⣀⣸⠀⢸⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡷⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠇⠀⣿⣿⣿⣿⣿⠀⠸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠆⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣼⣿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠘⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣄⡀⠉⠉⠉⠉⠉⣀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠈⠃⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣀⢀⣄⠀⠀⡀⠀⠰⠒⣶⠀⠰⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠒⢀⡀⠈⠉⠉⠉⠛⣮⣀⠠⠀⠚⠓⣢⡦⠀⠀⠀⠀⠓⠄⠐⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡰⠀⠀⠀⠀⠀⢉⣽⣧⠄⠀⠈⠁⠀⣷⠀⠀⠀⠀⠀⠀⠈⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣤⣼⣐⣥⣷⣿⠿⠀⠀⠀⠁⣶⠐⢻⡀⢀⡀⡀⢀⠀⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠉⢭⠉⠉⠠⡀⢙⡄⠉⠉⠳⣨⡿⠋⠉⠙⠻⣅⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠂⠀⠀⠀⠀⠉⠀⠙⢪⡄⡀⢸⡄⠀⠀⠀⠀⢹⡧⠔⠂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢀⠀⢀⠀⢀⠀⡄⡆⡄⢀⠀⠀⠀⠁⠈⠿⣦⣀⠀⢀⣼⠓⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⣀⠠⣁⣌⣷⣾⣿⣶⣯⡔⢀⠠⠘⠀⣄⠁⠹⠻⣿⡁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢽⣿⡟⠁⠀⠀⠀⠹⣾⠗⠀⠀⠀⡉⠂⠠⠀⠘⣯⡀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠁⠉⢛⣻⡆⠀⠀⠀⠀⢀⣿⣿⣭⡁⠀⠠⠿⠀⠀⠀⠁⢴⠀⠸⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠁⠉⠔⡿⣶⣤⣠⣤⣾⡛⠦⠀⢀⠐⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢀⠘⠁⡿⠛⡟⠋⠣⠙⠂⡀⠂⠩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⡇⠘⡀⢠⡀⢨⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⢻⡠⢆⡇⡀⠘⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣈⠛⠻⢿⣦⡀⠀⠀⣿⡇⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣠⣤⣤⣤⣄⣀⣉⠉⠛⠻⢿⣶⣄⠙⢿⣆⣠⣿⣶⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⣶⣤⣉⢻⣷⣾⣿⠿⠛⠉⠉⠙⠛⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣹⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣤⡀
⠀⠀⢀⣤⡶⠾⣿⣿⣶⣄⠀⠀⢀⣠⣶⡿⠟⢛⣽⣿⠋⠉⠉⢹⣿⡿⠟⠉⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠻⣷⣶⣾⠿⠛⠉⠉⢀⣴⠿⠋⠁⠀⣴⣿⠋⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⣀⣀⣠⣾⣟⠁⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢼⣿⠇⠀⠀⣜⢳⣾⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣭⣭⣽⠿⠗⠀⠀⠀⠀⣠⣤⡙⣿⡄⠀⠀⠀⠈⠛⠀⠀⣠⣌⢿⣿⡻⣿⣿⣦⡀⠀⠀⠀⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣰⡟⢉⣿⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⢿⣿⣧⠙⢷⣌⠻⣿⣿⣦⡀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣿⡏⠀⢀⡀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠈⢿⣿⣧⡀⠻⣧⡙⠿⢻⣿⡄⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⢀⣠⣴⣶⣾⣿⠟⠀⣿⣿⠁⠘⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣤⡈⢻⣿⡷⠀⠈⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⠁
⠀⢀⣾⡟⢁⣠⣿⡿⠃⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠲⠿⢷⣦⠈⠉⢻⣷⠀⢻⣇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠈⠿⣿⠿⠟⠋⠀⠀⠀⢸⣿⡇⢀⣀⡀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⠀⠈⣿⡄⠀⠀⠀⢸⣿⣧⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠹⣿⡷⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⠀⠀⢹⣇⠀⠀⠀⢸⣿⣿⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⣿⣧⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠰⣿⣿⣦⠀⠀⠀⠀⠀⠙⣭⣤⣀⣈⣿⡄⠀⠀⢘⣿⣿⡿⠛⠁
⠀⠀⠀⣀⣤⢶⣿⣿⠃⠀⠀⢹⣿⡄⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⠀⠀⠀⣿⣿
⠀⠀⣼⣿⣡⣾⡿⠃⠀⠀⠀⠀⢿⣷⡀⣰⣿⠃⠀⣀⣤⣶⡿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⣿⣧⠀⠀⠸⣿⡄
⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣴⡿⠟⠋⠁⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣾⣿⣷⡀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⣀⠀⠀⠻⠶⣦⣤⣀⡀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠟⠉⠻⣷⡀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠈⠙⠿⠷⠀⠙⠃⠀⠀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠹⣷⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣄⣀⣀⣀⣀⣤⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⢰⠂
⠀⠀⠀⠈⠳⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⢄⠀⢀⡎⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠐⠉⢠⠃
⠀⠀⠀⠀⠀⠈⢢⡈⠑⠒⠤⣄⡀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡵⠮⠤⠤⠤⠤⣤⣀⠣⠠⠐⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡀⠤⢄⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠉⠒⠒⢚⡠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⢀⠇⠀⠀⡀⠤⠒⠈⠁⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠓⠉⠁⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠖⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢴⣁
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠉⠒⠒⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⡠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠑⠤⡀
⠀⠀⢀⠤⠚⠁⠀⠀⣀⣀⣀⠀⠤⣄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣀⡀⠀⠀⠀⠀⠀⠑⠢⣀
⠤⠮⠅⠒⠒⠉⠉⠁⠀⠀⠀⠀⡰⠃⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠈⢇⠀⠈⠉⠁⠒⠂⠤⠤⣄⣑⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡀⠤⠬⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⢡⢄⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠒⠉⠀⠀⠀⢠⠃⠀⠈⠉⠓⣒⠢⠤⠤⠔⠒⣊⠉⠀⠀⠀⢸⠀⠀⠉⠒⠤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡠⠊⠁⢱⠀⠀⢀⠎⠀⠱⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣠⠊⠀⠀⠀⠈⡆⢀⠎⠀⠀⠀⠘⢆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠞⠁⠀⠀⠀⠀⠀⢣⠎⠀⠀⠀⠀⠀⠈⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⣠⣶⡦⠀⠀⠠⠄⠠⠤⠠⠀⠤⠠⠄⠀⠄⠆⠀⠀⠄⠀⢠⣦⡠⢀⣤⣀⣀⡀⡀
⠀⠻⣿⡆⠀⠀⠠⠠⠀⠀⠀⠠⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠻⠐⠟⠱⠿⠯⠿⠿⠶⢶⣤⣆⣀
⠀⠀⠈⠀⠀⠈⢰⢲⣠⢦⣀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠆⠈⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⣆⣰⣀⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢛⠈⠁⠁⠁⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠟⠻⠘⠃⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡤⠀⠀⠄⠈⠀⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡶⠩⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠒⠞⠀⠀⠀⠠⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠋⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠗⠒⣀⠀⠀⠀⠀⢀⣾⣄⠄⠀⣀⠀⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢤⡤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠈⠀⠈⠉⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢶⣄⠀⠀⠄⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣥⣤⣤⣤⣋⣶⢀⠀⠀⢠⢴⣟⠿⠿⢿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⡽⠟⠋⠉⡁⠺⣀⣀⣠⣂⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⠯⠿⠋⠈⢁⡴⠋⠀⠀⣀⠘⣷⠈⣷⡄⠻⢟⠟⣤⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠈⡐⡘⠲⣿⣷⠀⠈⠰⢼⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠑⣿⠀⢹⣿⡆⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⠘⣿⡇
⠛⠛⠛⠛⠛⠛⠛⠿⠟⡏⠀⣠⣿⠿⠋⡙⠂⠿⣿⣤⣽⠉⣿⡇⠀⠙⠛⠻⠛⠻⠿⠿⠿⢿⣿⡟⠻⠛⠻⠿⠿⠯⠟⠛⠛⠛⠚⠛⠛⠻⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉⠉⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠋⡀⠥⠠⠄⠀⢀⣀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠒⠶⠦⣠⣄⡋⠁⠁
⠁⠉⠉⠉⠐⠛⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠤⠤⠤⢶⡤⢄⡈⠛⠓⠀⠀⠐⠒⠲⠶⠆⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢴⡆⠀⠀⡀
⠀⠀⠀⠀⢄
⠀⠀⠈⠁⢈⡤⣬⠤⠤⠴⠀⠦

⡠⠒⠲⡴⠒⢢⡔⠒⢦⠖⠒⣤⠒⠢⡴⠒⢦⡴⠒⢦⠖⠲⡤⠒⠲⡴⠒⢦⠔⠒⢦⠖⠲⡤⠒⠢⡴⠒⢦⠔⠒⢦⠖⠲⡤⠒⠲
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⣴⠂⣤⣢⣠⣶⠐⡶⢠⠀⡆⢸⠃⢣⡖⢀⣔⡆⣤⣦⢐⠆⣠⣲⢠⣦
⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠈⠈⠈⠁⠘⠿⠃⠈⠁⠁⠁⠀⠉⠁⠈⠉⠉⠀⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣹⠆⠀⠀⠀⣀⣀⡀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠲⠠⠀⠰⠂⠀⠀⠀⣠
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠈
⡜⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠐⠒⠀⠀⠀⠙
⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠰
⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡜⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣲⠦⡟⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣱⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⢧⡀⠀⢀⣀⣀⣀⣀⢀⡀⢤⣤⣤⣤⡤⢄⣠⣤⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣎⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡼⠃⠀⠨⠤⠄⠠⠄⠤⠤⠄⠁⠀⠖⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡱⠄⠀⠀⠀⠀⠀⢠⣀⣀⠀⣀⠀⠀⠐⠂⢀⢀⣀⣀⢠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤
⢿⡀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠈⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⡾⠀⢀⠀⠀⠀⠀⢠⠀⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠠⠀⠀⠀⠀⠁⢀⠀⠉
⡼⢃⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠻⠤
⢟⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣼⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠿⡿⡁⠀⠀⠒⠀⠒⠀⠀⠀⠀⠈⠛
⢳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢶⣂⠀⢀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡤
⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⡍⠉⠉⠉⠋⠙⠛⠛⣛⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡾⠁⠐⠂⠀⠀⠀⠐⠢⠀⢀⠀⠀⠀⢀⣇⣀⠀⢀⣀⣀⡀⠐⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣼⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⡄⠀⠀⠄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⢳⠄⠀⠖⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⢷⡀⠀⡀⠈⠀⠈⠀⠀⠀⡀⠈⠀⠁⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡾⠁⢀⡧⠀⠀⡀⠠⢀⡀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢄⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡉⠀⢀⡀
⠘⠒⠚⠓⠒⠊⠓⠒⠊⠓⠂⠀⠰⠒⠛⠒⠒⠄⠄⠚⠓⠖⠄⠐⠛⠓⠒⠊⠓⠒⠊⠒⠒⠛⠲⠚⠓⠒⠚⠒⠒⠚⠳⠒⠂⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠈⠉⠓⢦⡀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⡽⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢯⣀⠀⠀⠀⠀⠀⠀⣀⡤⠤
⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁
⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠙⠲⢤⣀⣀⡤⠔⠋⠀⠀⠀⠀⠙⠲⢤⣀⣀⡤⠖⠋⠀⠀⠀⠀⠙⠢⢤⣀⣀⡤⠖⠋
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠉⠓⢦⣀⣀⣀⣀⡤⠚⠉⠁⠈⠉⠓⢦⣀⣀⣀⣀⡴⠚⠉⠁⠈⠉⠓⢤⣀⣀⣀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢀⠄⢸⡀⡀⢀⢀⠄⡀⠀⢀⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⣀⢄⢠⢰⣉⠀⡇⢁⢘⠀⣙⢈⣢⢸⠁⢸⢸⢸⠒⠁⠀⢸⠃⠀⡇⡈⢁⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢸⠀⠘⠌⠠⠀⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠊⠈⠤⠌⠘⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡤⠤⡤⠜⠙⠉⠓⢲⣀⠀⠀⠀⠀⠰⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⢀⠔⠃⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠀⠋⠙⠊⠳⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⠔⠋⠁⠀⠀⠀⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⡈⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢐⠉⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠠⢌⣒⡲⠄⢺⠁⠀⠙⠀⠀⠀⣠⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣄⣀⠦⠬⠭⠹⣇⠀⠀⠀⢀⡠⠟⠁⠘⠤⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣸⢸⠀⡝⣿⢿⠀⠀⠀⠸⢤⣀⡶⠞⠀⠀⠀⠀⠀⠈⠉⠓⠋⠒⠩⡃⠀⠀⠀⢦⣀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⡏⡼⢠⠃⠉⠀⠀⠀⠀⠀⢸⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢅⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢰⠃⡾⠀⠀⠀⠀⠀⠀⠤⡄⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡆⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠜⠀⠀⢴⠆⠀⠀⠀⠀⢁⠡⡀⠀⠀⠀⠠⣞⡀⠀⠀⠀⠀⣤⠀⠀⠈⢆⣀⠀⡀⢀⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⢷⢤⡀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⢀⠴⠩⢋⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⣦⠤⠤⠤⢤⢴⣖⠉⠁⠴⡄⠸⣗⡲⠤⠄⢤⡤⠤⠤⠤⢶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠡⠀⢻⡇⠀⠀⡾⠞⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠧⠴⠟⠁⠀⠀⠟⠃⠰⢷⠄⢀⡀⠀⢤⠄⢢⢄⡘⠚⠷⡀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠦⠀⣀⡀⠈⠙⠀⢸⠀⠀⠉⠉⠂⠼⠟⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⣠⠄⠀⠉⠀⡒⠦⢼⣀⢀⠀⡀⢀⠀⡀⠠⣐⠢⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠩⠧⠤⠾⠦⡤⠴⠾⠳⠈⠀⠑⠚⠂⠈⠁⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠀⠈⡀⠀⠀⠈⢂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡄⠌⠸⡄⡤⠂⠀⠀⠐⠤⢠⠃⠈⠢⡄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⠀⢠⣀⠧⡀⠀⠀⢠⠲⠴⠀⠀⡠⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣆⠀⠀⠀⣠⢠⣼⣾⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⣄⣀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡆⢀⣾⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⣿⣿⣿⣶⣤⣆⠀⠀⢿⣿⣿⣿⣷⣾⣿⣟⣯⣿⣇⣰⣶⣴⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣿⣿⣿⣷⣦⣼⣿⣽⣿⣿⣿⣿⣿⣡⣾⣷⣿⣯⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠈⠙⠿⢿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣴⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠿⣟⣿⠟⢿⠿⢿⣶⣾⣿⣝⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⠻⢿⣯⢻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⣀⠐⠋⣽⡿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠁⠈⠛⡙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡏⠀⢈⣵⣿⡿⢿⡟⣿⣿⣿⣿⢿⢿⣿⡟⢿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣦⡀⠀⠀⠀⣰⣿⣿⠋⢀⣴⣿⣿⡿⡝⣿⠀⣿⣿⣿⣿⣎⠟⠹⠋⡌⡿⢻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⢀⣼⣿⣿⠀⢠⣿⣿⢻⡝⠣⠉⠀⠀⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣋⣰⣿⣿⣹⣧⣵⣶⣶⣶⣤⣿⣏⢸⣟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢣
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢿⡿⣿⣿⣷⣶⣀⡀⢨⣿⣿⣿⣿⡟⢉⣿⣿⣿⣿⣿⠟⠋⠉⠈⠉⣿⣿⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠁⠀⢲⣿⣿⣿⣾⣿⣿⣿⣿⣵⣾⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⣿⣟⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⣿⡯⠀⠀⠁⠃
⠀⠀⠀⠀⠀⣴⣿⢿⠛⠉⠈⠁⠀⠛⠛⢛⣩⣿⣿⣿⣿⣿⣿⣿⣏⠉⠁⠀⠀⠀⠀⠉⠛⠀⢠⣿⡇
⠀⠀⠀⠀⠰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢹⣿⡟⣿⡿⣿⡿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣿⣿⠀⣿⡇⣿⡇⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢹⡇⠀⢻⣇⢹⣧⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠁⠀⢸⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣖⣶⣿⣭⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣶⣿⣿⠫⡦⢤⡀⠀⠀⢩⡙⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⢰⣿⣿⢾⠀⣶⡇⢸⣇⡔⡀⢸⠃⡄⣤⣨⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠈⢿⣿⣾⣬⢸⡗⠉⢸⣃⣇⣼⣠⡿⢸⢿⡇⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣿⠿⠛⠒⠛⠿⠄⣠⠤⣤⠤⠴⠆⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⡁⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢸⡁⣶⣿⢰⣴⢠⢤⢀⡴⡀⠀⠀⢺⡀⠃⡀⢹⡶⠶⠆⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⣿⣿⣿⠻⣦⠈⠙⢿⡆⠀⠀⠉⡁⣿⢸⠀⢸⡉⢸⡋⠀⠀⣴⢫⠔⣞⠁⠀⠀⣀⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣿⣿⠀⠹⣧⠀⠀⠁⢠⡏⡄⢢⢉⣠⠀⣀⠉⣠⡈⢀⢀⡘⠎⣁⠞⠀⢠⡞⢉⣠⣍⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠙⢿⣄⠀⢻⡆⠀⠀⢸⡇⡇⢸⢿⠁⣸⠟⢹⡇⡗⣸⡏⡇⠀⠈⢳⣄⡘⢧⣸⡄⢸⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠀⠘⣿⠀⠀⠀⠙⠒⠋⢈⡷⢬⡷⣌⠋⠙⠉⠀⠉⠀⠀⠀⠀⠙⠀⠉⠁⣸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡄⠀⣿⡇⠈⢷⢀⡀⠀⡟⢰⢸⣇⡸⢦⣤⢀⢤⢀⡤⠀⡤⢤⠄⡤⡀⢠⠏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⣿⡇⠐⠛⠋⠉⠀⠙⢈⢸⡏⣳⣾⠉⢹⠚⢸⡟⠀⣰⠋⣸⡗⣱⢏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡄⠀⠻⣿⣷⠛⠃⠀⠀⠀⢀⣤⣶⠾⠿⣷⣭⡈⠀⠈⠉⠀⠉⣈⣉⣉⣠⡟⣡⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⣤⡈⠻⢷⣄⡀⠀⢠⣿⠋⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⣸⠏⠉⠈⠉⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠬⣙⠒⠘⣿⣄⣀⡀⢀⣀⣠⡶⢦⣄⠈⠙⠻⠶⣦⣀⣀⣀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣴⣦⣤⠁⠐⢚⣿⠟⢻⡿⠛⠛⠛⠲⣬⡛⢦⣀⠀⠈⠉⠉⠉⠳⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠐⠚⠉⠀⠀⣻⣄⣀⠀⣀⣀⠀⠙⠢⣍⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡏⢸⢀⢀⠄⠀⠀⠀⠆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⡁⢸⠸⣈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠁⠀⠠⢀⡤⠀⣀⠀⠀⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⣿⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⡱⢿⠀⠀⣸⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣴⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠿⠟⣿⣿⣿⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣡⢠⠀⠀⠀⠈⠛⢿⣷⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⡆⢀⡂⠸⠅⠀⠀⠈⠀⣼⣏⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⣦⠸⡇⢹⡀⠓⠉⠀⠀⠀⠀⠀⠂⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠓⠀⠀⠀⣼⠀⢠⠀⠀⡀⢀⣠⣾⣿⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⣀⡀⠉⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⢊⣁⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠕⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⢡⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡁⠀⠀⠀⠌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣀⣀⣂⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡿⠿⠿⣿⣿⣶⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠴⠚⠋⠁⠀⠀⡀⠀⠀⠉⠉⢁⢠⢀⡘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡉⠉⠉⠀⠀⢀⣠⣄⠠⠰⡃⣅⠀⠀⠀⠘⠘⠀⣥⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠟⢠⡦⣄⢤⠄⣤⠰⡞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠌⣿⣧⣿⢠⣯⣼⡆⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠛⠀⣏⣼⠇⢸⣇⠾⠀⠛⠉


⣿⠿⠿⠿⠿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣻⣿⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⣦⡀⠀⠀⢰⣽⡇⠀⠀⠀⠀⠀⣤⣾⡿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⣀⡸⢿⣿⣷⣄⠘⣿⣧⠀⠀⢠⣠⣾⡿⢃⣤⣾⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⢉⠛⠉⣉⣩⣿⣿⣿⣷⣿⣝⢷⣿⣏⢀⣼⣿⣿⣟⣡⠿⢻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢛⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⠉⣀⣀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢯⣉⠹⣿⣄⣀⠙⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠈⠁⠀⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡍⠻⠿⣿⣶⣄⠀⠉⢻⣿⡙⠂⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣳⣿⠆⠀⠀⠙⠓⣠⡀⠀⠈⠙⠀⠸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠻⠁⠁⣼⣿⣿⣿⢿⣇⠉⢻⣟⣻⢦⠀⠀⠀⠀⠸⡿⣶⡆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⣿⣿⠂⣯⢻⡅⠀⢸⣯⢹⣷⠀⠀⠀⠀⠀⠈⢏⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⣇⣾⠁⠀⠀⢿⡗⣿⠀⠀⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠁⠀⠀⢻⡇⠀⠀⠀⠸⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣧⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠿⠃⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠀⣀⣠⣴
⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡏⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⣄⡀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡾⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠘⠂⠀⠀⠀⠃⠀⠀⢀⠞⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⠶⠶⢤⣄⡀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠀⠀⡞⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆⠀⠠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⢳⡀⠀⠀⠘⢦⣀⣀⣠⠟⠀⠀⢀⡼⠁⠀⠐⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⡀⠀⠙⠲⢤⣀⠀⠈⠉⠀⢀⣀⡴⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠞⠀⠀⡠⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⢀⠀⠀⠈⢷⠀⠀⠀⠀⠈⠙⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡟⠀⠀⠀⢻⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⣿⠀⠐⠋⠀⠠⠀⠀⠀⠀⠀⠀⠂⣠⡤⢶⡥⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠔⠘⣫⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢂⡀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠀⠀⠀⠈⠊⠉⠄⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⢀⡠⣤⡶⠂⠀⠈⠢⣄⡀⠀⠀⠀⠐⠂⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠤⠿⠷⠂⠉⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠤⠀⠀⠀⢸⣿⣿⡀
⣀⣀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣇⣀
⣿⣿⣿⡿⠋⣩⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⢩⣿⣿⡿⡟⠋⠉⠋⠉⡉⠋⢉⠉⠙⠋⠋⠉⠉⠙⠋⠉⠉⢩⠉⠛⣿⣅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠛⢉⣙⣛⡒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢠⡀⢀⣴⣿⠟⠁⣀⣴⣿⣿⡿⠟⢛⣉⣡⣤⠤⠤⠀⠈⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣷⣿⡿⠁⢠⣾⣿⣿⠟⣉⣴⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢰⡀⢹⡟⠀⣰⣿⣿⠟⣡⣾⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡆⠘⣧⠀⠀⣰⣿⣿⠋⣰⣿⡿⠿⠿⠿⠿⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢸⣿⠀⣿⠃⢰⣿⣿⠇⠀⠉⣀⣀⣤⣤⣤⣤⣀⣀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡞⣿⠀⠀⠀⡔⠀⠀⠀⢠⡀
⠀⠀⠀⣼⡟⢠⣿⠀⣼⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣷⣦⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣾⣿⠸⣰⣆⢸⣇⠀⠀⢠⣿⣷
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⡖⢀⣴⣿⣿⣿⠀⣿⣿⣦⣿⡀⣶⠘⡿⠁
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⡆⣽⣿⣿⣿⣷⣿⡇⠀⠀⡄
⠀⠀⠀⢿⣿⠈⣿⡀⢸⣿⣿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⣿⣿⣿⣼⣴⣿
⠀⠀⠀⢸⣿⡄⢻⣇⠀⢿⣿⡇⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢼⡔⢋⡡⠖⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⠘⣿⣿⣿⣿⣿⣿⡿⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣧⠘⣿⡌⠘⣿⣿⡄⠹⣿⣆⢤⣤⡤⠶⠶⠚⣛⣩⡥⠎⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⠉⠿⣿⠿⠋⠀⠀⠀⣸⣿⣿⡿⠋
⠀⠀⠀⠀⠹⣿⣆⠘⣿⣆⠘⢿⣿⣄⠙⣿⣧⡐⠒⠚⠛⠋⠉⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣭⣤⣤
⠀⠀⠀⠀⠀⢻⣿⣆⠘⢿⣦⡈⠻⣿⣧⡈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⠤⠤⠤⠤⠀⠈⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣧⡈⠻⣿⣦⡈⠻⣿⣦⣈⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣛⠿⢷⣭⣛⡦⠄⠀⠀⠙⠻⢦⣤⡀⠀⠀⣀⣴⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣌⠙⢿⣦⣈⠛⠿⣷⣦⣄⡉⠛⠻⠷⣶⣤⣤⣄⣛⣛⣶⣾⣯⣭⣛⣛⣛⡤⠤⠐⢀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣌⡙⠛⠦⣌⣙⠛⠿⢷⣶⣤⣤⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⢁⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⡿⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢦⠀⠀⠀⡴⠖⠲⣄⠀⠀⠀⣠⠤⢤⡀⠀⠀⢀⡤⠶⢤⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⠶⠶⠾⠿⠿⢿⡿⠿⠿⠿⠿⠟⢿⡟⠿⠿⠿⠟⠿⣿⠿⠶⠾⠷⠶⢶⣿⣶⣦⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠶⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠳⠦⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⡧⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⢤⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡔⠒⠒⢲⢠⠒⠒⠒⠂⡔⠒⠒⠒⠀⡖⠒⠒⢢⡔⠒⠒⠒⡴⠒⠒⠒⡆⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⡀⢣⣤⣤⣼⠘⣤⣤⣤⡄⢣⣤⣤⣤⠀⣧⣤⣤⡜⢣⣤⣤⣤⠳⣤⣤⣤⠇⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠁⠀⠀⢹⡇⠀⠀⠨⢸⠀⠀⠀⠁⡉⠀⠀⠈⠀⡇⠀⠀⢸⡇⠀⠀⠈⡏⠀⠀⠀⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠥⠼⠛⢅⡩⠭⠤⢬⢈⠥⠬⠽⠅⡨⠭⠥⢬⠀⡭⠭⠤⢍⡨⠭⠤⢬⣨⠤⠤⠾⡁⠀⢸⣧⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠰⣀⣀⣀⡸⣇⠀⠀⢸⠠⡀⠀⠀⡀⠇⠀⠀⢀⠀⡇⠀⠀⣸⢇⠀⠀⢸⢟⠀⠀⢀⠇⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠉⠉⠙⣶⠉⠉⠉⡇⡎⠉⠉⢹⣴⠚⠛⠛⣧⣶⠛⠛⢻⣦⡞⠛⠛⣧⡞⠛⠛⣿⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣷⡤⢤⠔⣛⣤⣤⡤⢃⣣⣤⣤⢼⣃⣀⣀⡀⢉⣰⣀⣶⠾⠻⠴⠦⠴⠟⠧⠤⠤⠏⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣰⡟⠁⠀⣇⣀⣀⢀⢳⠀⠀⠀⡟⡇⠀⠀⢀⣿⠀⠀⠁⣉⡉⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⡿
⠀⠀⠀⠀⠺⠿⠤⣤⣤⣄⣀⣀⣀⣈⠉⠉⠉⠁⠉⠉⠁⠈⠈⠉⠙⠛⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⣾⣟⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠢⠀⠠⠤⣄⡀⠀⠀⢀⣀⠀⠀⠀⣄⣀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⢅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡂
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠒⠢⢄⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠰
⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣨⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢶⠀⠀⠀⠸⡇⠀⠀⡾⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚
⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⢀⠀⠘⢦⠀⠀⠈⢧⠀⠀⠀⡇⡀⢸⠃⠀⠀⡴⠃⠀⠀⣀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠑⢄⠀⠳⡀⢢⡈⠆⢰⢰⠃⠃⠏⡜⢀⠞⠀⠀⡠⠞⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⡑⢄⠈⠂⠑⠈⠈⢸⡀⠘⠈⡠⢃⠜⡡⠊⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠈⡑⠌⡂⠀⠀⡁⠀⣀⡈⣉⡀⠘⠄⠁⢀⠠⠚⣁⠄⠀⣀⣠⠴⠒
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⠀⠈⠉⡓⠨⠁⠀⠐⠊⡔⠉⠀⠀⠀⠈⠐⡄⠂⠁⢀⠉⠀⠒⠉⠁⣀⡀⢀⣀⢀⡤⠤⣤⣄⣠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠉⠉⠉⠁⠒⠂⠀⠄⠥⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠒⠋⡭⠛⠚⠉⠉⠉
⠠⠶⠒⠒⠒⠚⠉⠁⠀⠀⠀⠉⠑⠢⣀⣀⠀⠀⠀⠠⠸⡀⠀⠀⠀⠀⠀⣀⡼⠄⠒⠖⠒⢂⡶⠊⠁⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢄⠒⠋⠈⠉⠉⠉⠉⠀⠀⠀⣀⣀⠤⠚⣠⡔⠉⠀⠀⠀⣀⡠⠔⠒⠂⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠠⠤⠠⠀⠄⠑⠔⠀⠀⠎⠿⢯⣉⠀⠐⠒⠒⠉⣱⠋⠀⠠⠀⠉⠂⠀⠀⠫⡀⢘⠤⢄⡠⢄⡀
⠀⠀⠒⠂⠘⠓⠘⠂⠀⠒⠀⠉⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣰⠚⢁⣆⢠⣀⣊⡢⠀⠀⠐⠊⠀⠊⠅⠀⠒⠶⠎⠡⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣁⣤⠼⠵⠞⠉⠉⠉⣁⠀⠉⢓⣒⣒⠲⠚⠙⠛⠒⠶⠒⠚⠒⠓⠷⠄
⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⠤⣤⠖⠒⠛⠛⠉⠒⣐⣀⠫⠅⠠⠸⠂⠁⠉⠉⠁⠀⣉⣉⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀

⠀⠀⠀⠀⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⡶⣪⣭⣭⣭⡲⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⡇⢸⡇⠀⠀⠀⠀⠀⠀⠄⢸⠃⠀⠀⠇⡠⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⣿
⠐⢾⣸⣿⣿⣿⣿⣿⣸⡖⠀⠀⠀⠀⠀⠀⣿⠀⡇⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣠⣿
⠀⠿⡸⣿⣿⣿⣿⡿⣸⠷⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⢸⡇⠀⠀⠀⠀⠀⡠⠐⣎⠀⠉⠀⢰⡖⢤⠀⠀⠀⠀⢀⠠⠟⠁⠀⠈⠻⠄⡀
⠀⠀⠟⢮⣽⣿⣭⠞⠇⠀⠀⠀⠀⠀⠀⠀⡏⢸⣁⡱⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣴⣿⡇⠀⡇⠀⠀⠀⡇⣠⣷⣤⣀⣤⣶⣆⢨
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠒⠚⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠀⠃⠀⠀⠀⠃⠘⠛⠛⠉⠛⠛⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⡀⡐⡞⣎⢤⡲⣤⡀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠈⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⣀⣞⣗⡀
⢀⣾⣹⡇⣿⢹⣏⠿⠜⠿⠀⠀⠀⠀⡞⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀
⠘⠛⠉⠁⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠀⣠⠀⣠⠀⣠⠀⡇⠀⠀⠀⠀⠀⢨⢠⠴⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⡇
⠀⠀⠒⠒⠖⠒⠳⠒⠒⠒⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣀⠨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡤⢴⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣬⣤⡅⣤⣬⡄⠀⠀⠀⠀⠀⢀⣠⣤⣭⣬⣭⣥⣤⡀⠀⠀⠀⠀⠀⢰⣤⣥⣦⣬⡆⣼⠇
⢸⣿⣿⡟⠙⠋⢻⣿⠉⠉⢣⠀⠀⠀⠀⠀⠸⣿⣿⣸⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⢿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣇⣀⠼⢿⡇
⢸⣿⣿⣧⣤⣤⣼⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⢻⣿⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢢⣼⡇
⠾⢿⣛⡿⠿⠿⠿⠿⠿⣟⣻⠻⠇⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠉⢃⣀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡠⣤⠤⢀⣀⣀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⣠⣖⡽⣾⣿⡿⣟⠔⠋⠀⠀⠀⢠⣿⣿⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣤⣀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⠿⠿
⠀⡜⠇⠀⠉⠉⠉⠛⠋⠁⠀⠀⠀⠀⢠⣿⣿⢿⡇⡿⣻⣿⣧⠀⠀⠀⠀⠀⠺⣿⣿⠆⡙⢛⣋⠀⠀⠀⠀⠀⠀⢠⣟⣻⡄⠈⠙⠻
⠉⠫⣤⡀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⣾⣿⡿⠃⠀⠀⠀⠀⠀⢰⣬⣉⠶⡏⠻⠏⠀⠀⠀⠀⠀⠀⣤⣘⢻⣀⣤
⠀⠀⠁⠀⠋⠉⠉⠉⠩⠛⢄⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠚⠒⠓⠒⠂⠀⠀⠀⠀⠀⠐⠒⠒⠚⠒⠒⠒

⠀⠀⠀⠀⠀⠀⣀⡀⣠⠴⢦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⡤⠒⠒⠒⢦⣞⠙⣆⣀⡤⡄
⠀⠀⠀⠀⢰⡏⠉⠙⣷⠀⠀⠳⡄⢰⠋⠛⢳⣸⡁⠀⣤⠈⢳⠀⠘⠃⢀⡟⡆⠈⣁⡤⠟
⠀⠀⠀⠀⠀⢧⠀⠀⠉⠀⠀⠀⣹⣼⠀⠲⠀⢻⡇⠀⢀⡤⠾⡆⠀⣾⠉⠀⣇⣰⠃
⠀⠀⠀⠀⠀⠸⡆⠀⢠⣶⡀⠀⠸⣏⣀⡞⢆⣈⠧⢤⡾⠤⢤⡉⢩⠷⠶⢤⡼⠛⢆⣀⡤⢦⡤⠖⠶⢤⡀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⢳⡀⢀⡟⠣⢴⣶⣧⡀⢠⡞⠛⠉⣹⡇⠀⣦⠘⣿⠀⡀⠈⣟⢦⠈⢁⣠⢾⣅⡀⠀⢾⡁⠀⠀⢰⡏⠀⠘⡇
⢀⣀⣀⡴⠒⢲⡈⢉⡽⠒⠲⢼⡇⠀⣷⠀⠙⡇⢸⣏⡇⠀⠛⢀⡇⠀⡤⡄⠘⣾⡇⣸⠀⢺⡀⠉⠀⢀⡇⡴⠒⢾⡇⠀⠀⣧⠴⢦⡀
⣯⠉⠈⢿⠄⠀⠙⣾⠀⢰⣶⠈⡇⠀⠙⠛⣏⣁⣀⣹⠷⠶⠒⢋⣙⠚⠁⠙⠚⣁⡉⠁⠀⠀⠉⠓⠚⠉⠘⣇⠀⠀⢁⢀⣀⣃⢀⣠⣇⣀⣀
⠸⡆⠀⠀⢀⠀⠀⢿⣄⠀⠉⢠⠿⠤⠖⠚⠋⠉⠀⠀⠀⠀⠀⡏⠈⠳⣤⣤⠞⠉⢹⡆⠀⠀⠀⡟⠛⠉⠋⠉⠉⠅⠀⠀⠀⠈⠁⠈⠉⠉⢹
⠀⢹⡄⠀⢿⣇⣀⣈⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠾⠧⠤⠤⠃⠒⠦⠀⠐⠷⠶⢶⠀⣷⠀⠀⢀⠄⠀⢣⠀⠀⠀⠐⠀⠀⠀⠀⢸
⠀⠀⠳⡤⠞⠈⣉⠁⡴⢲⡾⢹⣤⠖⡆⠀⠀⠀⠀⢸⠆⠄⠤⠄⠀⠠⠀⡀⢠⠀⠀⠒⠒⣺⡄⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡏⠙⣧⠈⡇⣸⠃⣰⠷⡄⠀⠀⠀⢸⡏⠉⠉⠉⠉⡉⠀⠇⠈⡭⠉⠉⠀⢸⠃⢻⠂⠀⠀⠁⠀⠠⠀⠀⠀⠘⠁⠀⡠⠊⢸
⠀⠀⠀⠀⠀⢺⣟⣦⣈⣦⡇⠯⢼⠥⠾⠷⠶⠒⢻⠀⡇⠀⠀⠀⠀⠟⠀⠀⠂⣾⠂⠀⠀⢸⠀⢸⡄⠂⠀⡠⠀⠈⠃⠀⠀⢀⠀⠊⠀⠀⢸
⠀⠀⠀⣶⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇⡠⠂⠀⠀⠀⠁⠀⠀⢨⠀⡀⠀⠀⢸
⠀⠀⠀⢹⡄⠠⠤⠤⠤⠆⠀⠀⠀⠒⠒⠒⠂⠂⣻⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⡏⢀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡗⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣷⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠶⠒⠒⠒⠒⠒⠒⠛⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠛⠛⠚⠓⠚⠒⠒⠒⠒⠓⠒⠛⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠠⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣃⠈⠲⡄⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠁⠀⠀⠀⠀⢀⠀⠈⠻⢿⣿⡅⠀⠈
⠄⠀⠀⠀⠀⠀⠈⢱⠶⠲⠜⠛⠃⠠⣀⠀⠀⠀⠀⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠆⠀⠀⠈
⠀⠀⠐⠂⠀⢠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠛⠛⠉⣀⣠⣤⣄⣄⣈⠉⠙⠛⠿⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⣆⣴⠾⣶⠜⠛⠙⠛⠛⢿⣿⣷⣶⣶⣤⣄⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠋⠙⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣧⣼⠛⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣷⣀⣠⣬⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⣾⣠⣿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠻⠟⠛⠛⠋⠉⠉⠉⠙⠛⠱⠄⠈⠉⠀⠸⠿⣧⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡀⢀⣀⣀⣾⠇⢀⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣾⣿⣷⠶⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⣿⣿⣿⠁⣤⠘⢻⣿⣿⠟⣷
⠀⠀⠀⠀⠀⠀⢠⡄⢀⣿⣿⣇⣤⡌⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠀⢻⣿⣦⠈
⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿⠏⠘⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠻⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠠⠄⠀⠀⠀⣀⡀⠀⣀⢀⣀⠀⠀⠤⠤⠤⠤⠶⠆⠴⠶⠄⠀⢻⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠶⠶⠀⠀⠉⠛⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⢸⣿⠀⢠⣤⣶⠖⠀⢷⣄⠀⠠⠆⠀⣴⠆⠠⠤⠐⠀⠀⠀⠠⢘⣿⣦⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⡇⠸⣿⠟⠁⠈⠁⠀⣀⡘⢿⣆⢀⣤⠈⠋⣀⣤⣴⣶⣶⣶⠄⠀⣿⣿⣿⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢃⡼⠿⣧⣾⣿⣿⣿⠿⠿⣿⣿⣷⡄⠻⣿⣽⣿⣷⣿⣿⣿
⠀⠀⠀⣤⣤⣤⡄⠰⡿⣿⡆⢸⣿⣶⣤⡾⢶⡤⣾⠿⠀⢸⣿⣿⣿⣄⣀⠀⣴⣿⣿⣿⡷⠀⠘⣿⣿⣿⣿⣿⣿⣦⠄
⠀⠀⠈⠛⠉⠀⠀⠀⠀⠈⢀⡀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⢀⡈⢩⠍⠉⠀⠀⠀⣿⡟⠻⠿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣶⣾⣿⣶⠀⠀⣀⠀⢸⣟⠉⢙⠋⢻⣀⣠⣄⣿⡇⠀⢿⠳⣶⣄⠈⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⢃⣸⠟⢹⣉⣹⣿⣿⣿⣇⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⠀⠈⢿⡿⠀⠈⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⣾⡟⢰⣾⢉⣉⣉⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠋⠰⡿⠟⠀⣼⣿⠿⢿⣿⣿⡟⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠃⢰⣀⣀⣀⣠⡆⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⢠⣶⠄⠀⠠⣀⣰⠟⣿⡄⠘⠛⠛⠁⠀⣰⣧⠀⠀⢤⠞⢲⡄⠀⠀⠈⠉⠁⠙⣷⡞⠛⠟⠉⠉⠃
⠀⠀⠀⠀⠀⣼⢃⣰⣿⢃⡈⠻⠊⢻⣷⣶⣿⣿⡖⣿⣿⣿⣧⣀⣀⠀⣺⣿⣤⣤⣴⣶⣶⣿⣿⣿⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣴⡇⢸⡟⠀⠿⠁⠀⠀⠀⣥⣻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⢀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠙
⠀⠀⠀⠀⣿⠀⠀⢀⡾⠀⠰⠇⣰⡄⣿⣧⣿⣿⣟⣿⣿⣿⣿⠿⠿⠿⠿⠟⠻⠟⣻⣿⣿⣤⣿⡟⠀⠀⠀⣼⠀⠀⠀⠀⢀⣿⡇⠀⢸⡇
⠀⠀⠀⠀⢉⣹⠃⠘⣷⣌⠀⠈⣻⣿⡿⠛⠻⡿⠟⠛⠻⠟⠛⠿⠛⠛⠻⠓⠀⠻⠋⠛⠃⠀⠀⠀⠀⠠⠖⠀⠀⠀⢀⣰⡿⠋⠀⠀⢸⡇⢠⣾
⠀⠀⠀⠀⠘⢿⣷⣶⣞⣛⡷⣾⣿⠋⠀⠀⠀⠀⠀⣴⣶⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢠⣤⡿⠇⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⢰⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠤⠄⠊⠉⢁⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⡄⠈⠛⠷⠂⠲⠂⠀⣀⢀⣀⡀⠀⠀⠀⠀⠤⠤⠤⠖⠂⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⡇⣨⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢀⡀⣿⡇⢠⡴⠶⠂⠲⠖⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠀⠈⠉⠉⠁⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⡗⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣤⣤⣤⣄⠴⠶⠿⠿⠃⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣿⣿⡄⠀⠀⠚⠛⠛⠓⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⠀⠀⠀⠚⠃⠀⢸⣿⠟⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡾⠿⠿⠿⣧⡄⠀⣤⠀⢠⣶⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡙⠻⣶⣌⣷⣶⣿⣶⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠻⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣶⣽⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⠛⠟⠋⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣴⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⡿⡿⠿⢸⣿⣽⣿⣿⠇⣻⣿⡛⠹⢾⣿⣿⣿⣶⣶⣤⣴⣤
⠀⠀⠀⠀⠀⠠⣄⠻⣷⣼⡇⣠⡄⠀⠀⠀⣠⣶⣿⠿⠟⠋⢁⣾⣿⣿⣧⡀⠀⢾⣿⢻⣿⣿⣿⣿⣇⢀⣀⣀
⠀⠀⠀⣰⣿⡿⣿⣷⣿⣿⣿⡿⢀⣀⡀⣸⠿⠋⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⡿⢿⣿⣬⢻⣿⣿⣿⡿⣟⣻⣿⠟
⠀⠀⠘⠛⠛⡛⣿⣍⠻⣿⣿⣿⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⢻⡿⡷⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣿⣯⣭⣭⣿⡟⠛⠛⣯⠙⠲⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⠿⠛⢿⣿⣯⡿⢻⣿⣿⣯⣝⣛⣿
⠀⠀⠀⣸⡟⠉⠁⠿⠿⠁⠹⣶⣝⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⢞⣵⣿⣿⣇⢨⣿⣭⡉⠛⢿⣿⡄
⠀⠀⠀⠉⠀⠀⠀⠀⢀⣶⠀⠀⠙⣷⣄⠀⢹⣇⣽⣿⣰⡏⠀⠀⠀⣿⠀⠀⢰⣿⠈⣿⣿⡇⢿⣾⣿⣿⢿⣷⣶⣯⣃
⠀⠀⢸⣷⣶⣦⣴⢟⣿⣷⣄⠀⣀⣈⡙⢿⣄⢿⣿⣿⡿⣢⣤⣤⠀⣿⠀⠀⠸⣏⠀⠛⢹⡇⠈⠘⣿⣿⠀⢹⣇⠉⠉
⠀⢐⣶⣾⣿⣿⣿⣾⣿⣿⣿⡶⠦⣬⣙⢿⣿⣾⡿⢿⣿⣿⣙⣛⠇⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠘⠿⠀⠀⠃
⠀⣿⣛⣉⡙⠛⣿⠁⠴⣿⣷⣦⣴⣾⣿⣿⠿⣿⣧⠸⣏⣉⡉⠛⠓⣿⠀⠀⠀⠀⠀⠀⣿
⠀⣼⠿⠛⣿⣿⢻⣧⡶⢾⡇⠹⠇⠉⠁⢠⣾⣿⠛⢻⣿⡙⠻⢿⠆⣿⠀⠀⠀⠀⠀⢀⣿
⠈⠁⠀⠀⣿⣿⢾⠙⢧⢸⣷⡀⠀⠀⠞⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠁⢸⡄⠈⢹⡈⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠂⠘⠛⠛⢻⠷⠶⠶⠶⠶⣶⣶⣾⣿⣶⣦⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄
⣀⣀⣠⣀⣀⣀⣀⣀⣀⣉⣉⣻⣿⣿⣉⣉⣉⣉⣈⣉⣉⣙⣻⣿⣿⣿⣿⣋⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡍⢽⡟⠛⠛⠛⠛⠛⢛⠻⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠦⡄⣠⡾⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢠⣿⢁⡼⣤⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣾⡿⠊⣴⠄⢠⣿⠁⠀⣴⡿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⡿⢀⣾⡏⠀⣰⣿⠧⠊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⣄⣀⣿⣇⡀⠘⠷⢺⣿⣇⠔⠙⠿⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡯⠿⠋⡉⠛⠿⣦⣼⡏⠉⡆⣀⠀⢠⠀⡶⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⢑⡛⢓⠀⣿⡇⡆⠯⢿⠀⠿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⢐⠂⡜⣺⠃⣾⢠⠻⠷⢁⡤⠄⢠⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣿⠜⠸⡅⣯⠸⠇⡯⠀⠀⢾⠰⡧⢾⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⡆⠸⡽⠶⢴⠴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡿⢠⣿⡎⡾⠡⣞⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⡠⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠋⣽⢻⢋⢛⡭⣿⣿⣿⢩⡙⡟⢋⡏⢽⣿⣿⡟⠉⡛⠻⠻⢋⡽⣿⣿⣿⠋⢿⢻⣿⣿⡟⠩⡏⠝⡛⢽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣭⣼⣦⣼⣮⣥⣿⣶⣿⣬⣵⣥⣮⣧⣿⣿⣾⣧⣾⣧⣷⣼⣬⣥⣿⣾⣯⣴⣼⣼⣷⣾⣧⣬⣧⣿⣯⣤⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⢠⡾⣿⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠹⣟⠀⠙⣧⣠⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢰⡏⠀⠀⢻⡄⠀⠘⣿⠀⢻⣤⣤⣤⣤⣤⣴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡍⠉⠛⢾⣇⣠⡴⠾⠓⠛⠲⠿⣤⡼⠿⢤⡀⠀⠀⠈⣿⢀
⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣈⣷⡀⠀⣠⡟⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍⠉⠛⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣤⣀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⢾⡇⠀⠀⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠉⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠚⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠞⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠛⠛⣿⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⣠⣤⠿⣳⣦⣄⣀⣀⣀⣤⣾⠀⠀⢸⡇⠀⠀⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠶⢻⡟⠁⠀⣰⡟⠀⢨⡏⠉⠉⢻⣾⠀⠀⢸⡇⠀⣀⣉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡇⠀⢠⡟⣷⠀⠀⣷⠀⠀⢸⣿⠀⠀⢸⡇⠀⠉⣉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣾⠁⢸⡇⣰⠟⢷⣄⣿⢹⠀⠀⢸⡇⠀⠀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠈⣿⠏⠀⠀⠈⢉⣿⠀⠀⠸⣇⠀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣅⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣦⣄⣀⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣯⠉⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠉⣿⡿⠋⠀⢸⣿⠀⠙⣿⣿⣇⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠸⣿⠃⠀⠀⢸⣿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣁⣀⣠⣾⣿⣿⣷⣦⣸⣿⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⠿⣿⣿⡙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⣿⠏⠀⢹⣧⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠿⠀⠀⠀⣿⡆⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⡃⠀⠐⠂⠀⢸
⠀⠀⠀⡃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡄⢄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣀⠠⢤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⠀⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠈⠐⠀⠀⠀⠀⠑⠠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣄⣤⣤⣤⣤⣀⣤⣄⣀⣀⣄⣀⣀⣀⣠⣄⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⢀⣀⢀⡀⣀⡀⣀⢀⣀⢀⡀⢀⣀⣀⠀⡀⢀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠰
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠎⢀⠀⠀⠀⢰⠀⢈⠐⢳⠂
⠀⠀⠀⠀⠐⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡠⠎⠀⠀⠀⠀⠀⠀⠓⠊⠀⢢
⠀⠀⠀⠀⠀⠀⢄⢨⣇⡓⠀⣀⠀⢀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⡄⡀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠐⠀⠀⢏⠈⡄⡂⡹⢦⢩⣷⢀⠀⠐⡆⠘⡀⢢⣄⢀⣆⣰⡀⣤⣀⠠⣃⠀⢄⡀⡀⠀⠈⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠉⠀⢸⠉⠀⠀⠰⢷⠀⠁⠣⠁⡁⠇⣼⠟⢉⠁⠀⡜⠇⠙⣷⢾⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠠⠼⣧⢸⡯⣄⡆⠈⣿⡂⠠⣀⡄⡇⠂⣡⠀⢸⣂⠀⡆⠀⡀⠠⢀⠉⠃⣽
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣃⡏⣹⣨⣍⢬⣣⠀⢠⡇⠀⠐⠃⡇⢪⢽⡰⠨⠦⠀⢧⠃⠹⠤⢘⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⢛⣙⡋⡧⠽⠚⡀⠘⡧⠀⠸⠙⢳⣠⢸⣠⠀⡆⠤⢸⣤⢰⡀⠌⣎⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡟⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⡧⢀⠀⠀⡀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⢹⡇⣾⠹⣀⣿⠋⣹⣶⡏⠅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠂⠀⠀⠛⠘⠃⠻⠦⠀⣿⠘⠛⠘⠓⠂⠀⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⢀⣄⡀⢠⠀⢀⢰⣇⠀⠀⠀⠀⡟⠀⣾⢱⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⡟⠣⢼⡀⣾⠈⠛⢶⡆⠀⠀⢀⣠⣿⠄⣠⣄⡀⢠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡿⠈⠙⠁⠈⠁⠘⠂⠀⠸⣇⠀⠀⢿⠀⣿⠀⡏⢀⡇⣼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠇⠀⠀⠠⠄⠀⠀⠀⣠⢤⠉⠁⠀⠈⠉⠈⠁⠉⠈⠉⠈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⠂⣠⠀⠂⠐⠀⢰⡇⠀⠀⠀⠀⣰⢢⢰⠢⠀⠀⠀⠀⠠⠏⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⢳⣆⡀⠀⠀⢸⡇⢰⡆⢰⡆⣿⡀⢸⡀⠀⠀⢠⢶⠄⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⠧⠀⠘⣿⠀⠀⠘⡏⠸⡇⢾⡇⣿⠀⢸⠁⠀⠀⢻⣄⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢛⡤⡄⠰⠃⠀⠀⠈⠉⠙⠂⠈⠃⠀⠀⠀⠀⠀⢸⠇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠒⢄⡠⠔⠒⢤⠀⠀⣾⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⣿⠊⣸⡆⢸⠰⣇⠞⡷⠀⢀⡾⠁⡇⢸⣠⢻⠆⡖⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢻⠉⠘⠃⢹⡇⠛⠀⠧⠀⠈⠷⠀⠷⠘⠃⠺⠜⠇⠘⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠲⢤⠀⣠⢄⠀⠾⠀⡞⢑⠀⠀⢠⠒⠢⠀⠀⢠⡆⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠸⠀⠀⢸⡇⠀⣶⠀⠈⠀⠀⠀⠀⢸⡇⠠⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢠⡞⠃⣿⠃⣰⠛⢸⡷⢶⡝⢳⣶⢀⡾⢓⡆⢸⡇⢰⡆⠀⣾⣶⢀⡆⡰⣶⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠸⣇⠀⢿⡀⠻⡆⠘⠇⠨⠃⠀⣿⠸⣇⢸⡇⢸⡇⠘⣧⡸⢿⣈⠉⠟⠀⣿⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠈



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⡄⠀⠀⠀⡜⢸⠀⠀⠀⠀⠀⢀⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢢⡀⠀⠀⠀⠀⢹⠀⠣⡀⠀⠤⣧⠀⠳⡀⢀⡄⢠⠊⢀⡇⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠙⢦⣀⡀⠀⠤⢷⡄⢈⣳⣧⣽⣤⣤⣿⣏⣈⣿⠘⠀⡆⠀⠀⣠⠖⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠌⠙⢳⣴⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢦⣟⢨⡴⠃⠀⡞
⠀⠀⠀⠉⠠⡈⠓⠦⣄⡀⠀⠀⠀⢄⠿⢦⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⠀⣰⠃⠀⠀⢀⣀⣠⠄⠀⢠⠄
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠁⠈⠉⠛⣶⣿⣿⣿⣭⣉⣛⣻⣿⠁⠀⢠⠶⠶⠞⠛⠓⠀⠀⠀⠘⢿⡋⡽⣁⠔⠋⠀⡠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢓⣲⠦⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣰⣾⣶⣤⡀⠀⠀⠀⠈⢿⠉⠁⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠷⢤⣿⣿⣿⣿⣿⣿⣭⣭⣿⣧⠀⠀⠀⠪⠥⢭⣌⠻⠿⠀⠀⠀⠀⠸⣥⣤⢞⠁
⠠⠴⢒⠒⠒⠒⠒⠊⠉⠀⠀⠒⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠮⠐⠒⠉⠉⠒⠂⠄
⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠩⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⣀⠄⠒⠚⠛⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣛⠏⠁
⠀⠀⠀⠀⠀⢀⣀⡠⢚⡠⢊⡠⢖⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠉⠒⠒⠒⠦⣄
⠀⠀⢀⡴⠋⢡⣤⣤⡥⠖⠋⠀⠘⢰⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣠⣤⣶⣿⠋⠲⡶⠖⠀⠤⣄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠈⠙⠧⡁⠀⠀⠀⠀⠉⠓⠤⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⣠⠃⡧⠴⡏⠙⣿⢿⣿⣿⣿⣿⣿⣿⡿⠟⢿⢍⣸⠓⠦⢄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠤⠒⠊⠁⠀⠀⢠⠁⢠⠟⡀⠺⡁⠀⢻⢦⡸⢧⡀⠈⠳⡄⠀⠀⠀⠱⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠎⠀⠀⠀⢡⠀⢸⠀⠀⠀⠈⣄⠀⡳⠀⠀⠀⠀⠙⢄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠊⠀⠀⠀⠀⠀⡏⢀⠏⠀⠀⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠁⠀⠀⠀⠀⠀⠀⠑⠅
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿