Love song Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⣿⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠃⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣼⣷⣶⣴⠖⠄⢻⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠛⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣝⡻⢯⣝⣿⠶⣤⣈⠘⠃⠀⢠⣼⠆⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣬⣙⠳⢮⣝⡓⠶⣯⣝⣓⠲⢮⢿⣀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠶⣍⣛⠳⢮⣟⡛⠲⢬⣍⣛⠻⠷⢬⣍⣛⠳⡶⢿⣤⣐⣛⠶⠶⣴⣾⣷⠄⠀⢤⡀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⢶⣭⣛⠶⣬⣝⣻⠶⣦⣭⣙⡳⠶⢤⣭⣙⣻⠷⢶⣭⣭⣟⣛⡷⠖⣠⡶⠆⢸⣏⢴⡀⡀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡶⣭⣛⠶⢮⣍⣛⡶⢾⣭⣝⣛⠶⢾⣭⣭⣛⡛⠷⠾⣿⣿⡛⢛⣍⣵⠟⣠⣿⠾⣋⣴⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣦⣍⣛⠶⢮⣍⣛⠳⢦⣬⣉⡛⠓⠶⢬⣍⣙⡛⠷⢶⣿⢟⣫⣭⣵⢾⣫⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣽⣿⣿⢿⣿⣻⢧⣭⣛⠶⣮⣍⣛⠻⠾⣯⣝⣛⠳⠶⢮⣭⣝⣻⣿⣿⣿⣿⣽⠿⢟⣻⣿⣱⡜⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠷⣦⣭⡛⠾⣍⡛⠶⢬⣍⣙⠓⠶⣬⣍⣛⡛⠶⠦⣭⣍⣿⣯⣽⢷⣾⣟⣻⣿⢿⣥⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡳⢦⣍⣛⠶⣬⣍⣛⠳⠶⣤⣍⣙⡛⠳⠶⣿⣿⣿⣶⣟⣛⣻⣭⢷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⣿⣿⣿⣿⣿⣿⣿⣻⣮⣝⡓⠮⣙⠛⠶⣬⣍⡛⠳⠶⢬⣭⣍⣛⣛⣻⣿⣿⣯⣭⣭⡽⢿⣿⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠆⢿⣿⣿⣭⣿⣯⣿⡿⠿⣍⡓⠾⣉⠓⠦⣬⣉⣙⠒⠶⠦⠼⣛⣻⣿⣿⣿⣿⠶⠶⠶⣋⡵⢖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣿⣿⣿⣿⣿⣟⡷⢦⣝⠳⢮⣝⡳⢦⣭⣭⣿⣿⣾⣿⠿⢿⣿⣿⣿⣿⣽⣿⣛⣥⢞⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣾⣿⣿⢿⣿⣷⣬⣙⠷⣬⣳⣦⣌⣙⠲⠟⠛⠛⠛⡛⢛⣛⣻⣿⣿⣿⣿⣷⣯⡭⠴⣻⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⢷⣿⣿⣿⣿⡻⢦⣿⣿⡿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠷⠾⠿⠽⢉⣿⠿⠟⡻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⣯⣾⠃⠀⠰⢌⣿⠀⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠀⠀⠚⠃⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠔⠊⠀⠀⠀⠀⠈⠲⡄⠀⡠⠔⠈⠉⠈⠉⠐⠢⣀
⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣸⠾⡄⠀⢤⡀⠀⠀⠀⠀⠈⢧
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⠈⡆⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠘⠀⠀⠀⠀⠐⠤⠤⢤⡄⣠⣄⢠⡴⠤⠠⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠒⠲⣄⡀⠀⠀⠀⢀⣀⡤⠴⠶⠤⢤⡀
⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢀⡴⢋⡤⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢀⠀⠈⢂⠀⠠⢀⣀⠠⠄⣄⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡁⠀⡖⣄⠀⠀⠘⠉⠩⠁⠀⣀⠔⠊⡀⢀⡄⡁⠄⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠒⠟⠉⠄⡐⠉⠑⠀⠁
⡆⠀⠀⠀⠀⠁⢉⠈⠑⠢⡀⠀⠀⡠⠚⠁⠁⠂⠉⢂⠃⣰⠄⠊⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠸⠀⠀⠀⡀
⠑⠒⠒⠒⠂⠐⠒⠀⠀⠀⠈⠢⠎⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⠋⠟⢡⣀⡙⠫⠓⢄⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠄⠀⣤⡈⢄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⣀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⠀⠀⡀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠉⢄⣠⣸⡄⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠜⣑⠯⡴⠄⠀⠀⠑⣄⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⣠⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣎⠐⢮⡓⠌⢢⠀⠀⠀⠈⠓⡤⠤⠤⡀⠍⡹⠃⠀⢀⡴⠊⠁⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⢰⠬⡤⣡⢤⢌⣒⠠⢎⠀⠀⠀⠈⠓⠲⣖⠊⠉⠀⠈⡱⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣎⣔⣄⠀⠈⢓⣴⣶⣶⣧⣸⠄⣆⣀⠀⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡇⠀⠀⠘⢿⣿⣿⣿⣧⢊⣈⡌⡤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠶⠒⠲⡍⠉⠁⠳⠊⢁⠌⠐⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣀⢀⣀⣠⠖⠁

⠠⠄⠂⠐⡦⠂⠀⠀⠀⠂⠀⡇⢰⠇⠤⠃⠤⠀⠆⠀⠐⠠⠆⠠⠀⡨⠀⠀⠀⠠⠁⠀⠦⠀⢀⠀⠄⠁⠀⠂⠠⠐⠃⠀⠀⠀⡀
⣀⣀⡆⢀⡄⢠⢖⡀⢀⣀⡀⣤⣤⡶⠤⠀⠀⠀⠀⠀⠰⣤⣄⠄⠀⠰⠤⠦⠠⠄⠀⠄⠤⠀⠀⡤⠶⠶⠀⠀⠄⠤⠀⠀⠀⠀⠠⠶
⠀⡀⡤⢀⡄⠀⠀⠀⠀⣄⣤⣾⣿⠿⠶⢶⣷⠀⠲⢾⣿⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⢀⡀⡀
⠀⠁⠀⠀⠉⡀⠀⢀⣼⣿⣿⠟⠀⣀⠀⠀⠙⠶⣶⣨⡟⣿⠠⠀⡄⣠⠀⠹⠀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠈⠁⠀⡀⠀⠀⠀⡀⠀⠈⠀⢀
⠀⠛⠀⠨⠂⠀⢹⣿⣿⣿⣿⣶⡿⣿⠋⠀⠀⠀⠈⢿⣶⣴⣾⣟⠿⠏⠀⠀⠉⠈⠁⠘⠛⠉⠉⠛⠋⠀⠻⠀⠈⠃⠈⠀⠀⠀⠀⠀⡁
⠂⠀⠙⠺⠲⢢⣿⣿⣿⣿⣿⢋⣀⡀⢰⣿⣷⡄⠀⠀⠻⠋⡬⠄⠁⠀⢀⠄⠀⠀⠘⠳⠎⠒⠀⠶⠴⠟⠛⠗⠰⠆⠒⠗⠂⠳⠿⠷⠷
⠀⠀⠀⣠⣴⣿⣿⣟⣰⠿⠛⠛⠛⠀⠸⢪⠗⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣂⡀⠀⢳⣠⠄⠠⣤⠄⠴⠆⠀⠤⠀⡄⠠⠄
⠀⢀⣠⣿⣧⣾⣵⣿⠛⠂⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠀⣿⢧⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⡈⣿⣿⣿⣿⣿⣿⣶⠂⣠⣤⡾⢟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⢈⡁⠉⠈⢈⣀⣁⠀⠀⠀⣀⡀⢁⠀⠀⡀
⠘⠓⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⣰⣿⣿⣷⣖⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣾⡀⠄⠀⠀⠃⠀⠘⠀⠀⠉⠈⠀⠀⠀⠀⠉⠁
⠿⠘⣻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠙⠍⠛⠉⢻⣷⠄⢰⣶⣷⣿⣿⣿⢷⡀⠀⠀⠃⠂⠀⠘⠀⠀⡀⠀⠐⠀⠀⠘⠂
⠂⢸⡿⠇⣰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠉⣹⣾⠄⠈⠃⠀⠀⠀⠀⠰⠀⠀⠁
⣀⠀⢠⣾⣿⣿⣿⠇⠀⠈⠙⢿⣿⣿⡟⠀⠀⠀⣴⢛⡎⠀⠀⠀⠀⠀⠻⠏⢤⠀⣠⢠⠠⣤⠀⢤⠀⠄⡀⢠⢤⠀⠀⠀⢀⠀⠴⠂
⠀⢀⣾⣿⠈⢻⡿⠀⠀⠀⠀⠀⠻⣿⣧⠀⢀⣀⣉⠻⠿⠒⠀⠀⠰⠁⢀⠀⢀⡀⠀⣀⣀⡀⢀⣀
⢀⣾⣿⣯⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⣿⣧⡈⠉⢯⠭⣥⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠈⠀⠀⠀⡀
⣿⣿⣿⣿⣷⡀⠀⠘⢿⣷⣾⢶⣦⢄⠘⢿⣷⣦⣀⣀⣀⡠⠀⠀⠀⠀⠀⡀⠈⠈⠉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⠻⢿⣧⠀⠀⠀⠻⡿⣆⠙⠀⠉⠪⢿⡿⢿⣿⡟⠀⣴⣾⣿⣿⡏⠉⠁⠀⠀⠄⠠
⣿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠌⠁⠀⠐⠦⣴⣮⠪⠽⠁⠀⠷⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⠀⢨⣿⡀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠀⠄
⣿⣿⡿⠿⠶⠾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⡀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⡄⠘⠁⠀⡴⠁⣹⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣇⠀⣶⠐⢳⣶⣿⠾⣀⠠⠋⢠⡄⠀⠀⠀⣠⠴⠚⠛⢻⠿⠋
⠀⠀⠀⠀⠀⠸⣷⣄⡉⠈⣧⣀⠀⠀⣉⣴⡏⣢⡇⣤⡿⠀⡐⢩⢏⠐⢄⣁⣢⣀⣀⣀⣴⣋⣁⣠⡤⠚⠁
⠀⠀⠀⣠⣄⠀⢳⣬⡇⠀⡇⠙⣆⣼⡿⠋⠉⠉⠈⠉⠁⠐⣤⣳⣴⡿⢿⠿⢿⡿⢿⠿⠿⠟⠋⠁
⠀⣠⢾⡟⠁⢂⡨⠄⠛⠀⡧⠀⣼⠋⠀⠈⠁⠀⠀⠰⠿⠟⢰⡿⣡⢶⢾⣷⡏⢃⢀⣧⠖⢦⡀
⠀⠸⡋⠈⢄⠰⠗⣃⢰⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⣇⣣⠞⣁⣿⠇⢸⠈⢻⣠⣾⠇
⢰⢦⡵⡀⢾⠆⠸⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣁⠤⠊⠉⣿⣏⠁⠂⠰⠶⢺⣋⡥⢤⣶⡶⠛⠉
⠀⠈⣾⡇⠀⠀⠀⠀⠀⠀⢰⡆⡆⣶⣶⣴⠂⠀⠀⣶⣶⡄⢠⣶⣦⡹⠿⠿⠒⠚⠉⠙⢀⣠⡿⠋
⣞⡙⠳⠀⠈⠀⠀⠀⠀⠀⢸⡇⣇⣿⢿⢿⡀⠀⠀⠙⠿⠿⢬⣿⣿⣇⣠⣁⣠⣁⣀⡋⠉⣁
⠈⢡⡺⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⣶⣆⣶⠂⠀⠀⠀⠰⠄⠐⠀⠄⢠⣆⢀⠂⢀⣿⠇⠀⠉
⠀⠘⠓⠺⢟⣧⣄⠀⠀⠀⠘⠃⠛⠙⠑⠋⠂⠀⠀⠀⣠⣤⠀⢈⣤⠛⡳⠿⡳⢸⣿
⠀⠀⠀⠰⣍⠀⣉⢤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠸⣃⣄⠹⠿⢧⠿⣿⠏⣀⠄⠀⣸⡿
⠀⠀⠀⠀⠈⢻⣄⢀⠔⣭⠇⣤⣤⡄⢀⣀⡀⠀⠈⢏⠐⣄⠁⠀⣤⣴⡁⢤
⠀⠀⠀⠀⠀⠀⠉⠁⠸⠻⢐⠏⠉⠀⣟⠛⠃⣀⠺⣄⣳⣜⢷⣮⣍⣤⡇
⠀⠀⠀⠀⠀⠀⠸⣄⠀⡇⢠⡁⡎⠀⡧⠀⣸⠀⠀⠘⢿⣙⣦⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⣾⠈⡷⣄⡧⢀⣿⣇⠸⠷⡆⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⡙⡗⠀⣀⠙⠷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⣿⣶⣄⣤⣤⣶⣾⣃⣀⡀
⠀⢀⢴⣾⣿⣿⣿⣯⠾⠋⠉⣹⣏⣁⣠⡝⢿⡉⠉⠉⠛⠳⠶⣤⣄⡀
⠀⣠⠏⡿⣒⣻⡿⠁⠀⠀⠀⢻⡍⠈⠙⠁⣼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠛⠛⠛⠷⣦⣄
⠨⠥⠞⣼⠟⣿⠃⠀⠀⠀⠀⠀⠽⠷⠶⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠈⠈⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣾⠁⣴⠋⠉⠳⡴⠚⠛⢦⡀⣶⡆⢹
⠀⠀⠀⣠⡾⠷⢦⠀⠀⡴⢀⣤⣤⣤⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢸⡇⠀⣇⠀⠀⠀⠀⠀⠀⢸⠇⣿⡇
⠀⠀⠀⣿⡀⠀⠀⠀⣼⢠⣿⠋⢀⣼⣝⡻⣦⠉⠙⠛⢻⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁⠀⠘⣇⠀⠹⣆⠀⠀⠀⠀⣰⠏⠀⠿⠃⢰
⠀⠀⠀⠘⢻⡿⠋⠀⠃⢸⣧⢴⣿⠘⣿⣷⣸⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠞⠃⠀⠈⠛⠶⠴⠛⠁⠀⠰⠗⢠⡿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣷⢸⣿⣷⡿⠋⠁⣿⡆⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠴⢦⣀⠀⠀⠀⠀⠀⣀⣠⡶⠛⠁
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠹⣧⣉⠟⠀⠀⣠⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠴⢦⣄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⠻⠶⠶⠾⠋⣠⡞⠀⢸⡇⠀⠀⠀⢀⣟⠁⠀⠀⠙⣦⠀⠀⢀⣀⣤⣤⠶⠶⡆
⠀⠀⠀⠀⠀⢻⣤⣀⣀⣀⣀⣀⣙⣒⣶⣶⣞⣉⣤⣤⠴⢿⡀⢰⣶⢿⡝⢧⣀⡤⠺⠇⠀⠀⠸⣍⣀⡤⠤⠖⢻
⠀⠀⠀⠀⢀⣿⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢠⠞⠉⢛⡷⠋⠹⣿⣿⣿⡍⠋⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⡇⠀⠀⣷⡀⠀⠀⢀⡶⠛⠛⣟⠛⣫⣶⣶⣄⠈⣢⡾⠻⠏⠀⠀⠀⠀⠀⠀⣠⣤⣼⡆⠀⢰⣿⣿⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣄⣀⣾⣁⣀⣠⣾⣴⣿⣿⣿⡿⠚⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠁⠀⠀⠈
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠉⠉⢹⡏⠉⠙⣿⡿⣟⠻⠋⢀⣀⣠⡾⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠐⠾⠆⠸⠂⠀⢸⣁⣹⡆
⠀⠀⠀⠈⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠳⣤⣀⡀⣀⣠⡾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠉⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⢩⣅⣈⠉⠉⢳⠄⠀⢀⡔⢩⣉⡉⠉⠉⣳
⣶⣶⡶⣶⣶⣶⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⣀⣤⣤⣭⡙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⣤⣤⣤⣄⡀⠀⠀⣠⣤⣤⣬⣉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠀⠈⠋⠀⠈⢻⣿⡟
⣴⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣾⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⣀⣴⡿⠋
⢻⣿⣿⣿⣿⣿⠏⠀⠈⠉⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠟⠉
⠀⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠘⠿⣿⣿⣿⣷⣤⣀⣴⡿⠋
⠀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣠⣋⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣈⣣⣀
⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣶⣄
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢰⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢳
⡼⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠋⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣑⣒⣲⣤⣤⣤⣤⣤⣤⣤⣶⣒⣒⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⢿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣰⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠢⠀⣱⣿⣅⣈⣿⣿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⡟⠑⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢼⡿⠁⠀⠀⠀⠈⢉⡴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⡄⠀⠀⠀⠀⠀⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡆⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣤⠀⢠⣶⣴⣿⡾⠧⠀⠀⠀⠀⠀⢀⣭⣶⣿⣷⢖⡴⣧⡄⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⡞⣻⠷⣾⡻⠃⡹⠁⠀⠀⠀⠀⢀⣠⡿⢟⣽⢞⣩⠞⣿⡯⢚⣠⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣻⣿⣿⣓⢾⣷⡾⠀⠀⠀⠀⣠⡴⢟⣿⢖⣫⣿⣿⣷⣾⢿⡷⣫⣿⣷⣮⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠾⣿⣿⣿⣷⣭⡷⠏⠀⠀⠀⠀⢤⣶⣊⣿⣿⣯⠔⣫⣿⣿⣵⣫⢾⣿⢟⣽⢿⣃⣤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⡯⣟⣿⣿⡟⠁⠀⠀⠀⢠⣼⣛⣯⡭⢟⣩⡴⢛⣥⡿⣿⣿⣷⡛⣷⣿⣵⣟⣯⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣽⣿⠋⠀⠀⠀⠀⠀⠸⠿⠥⠤⢖⣫⠴⣛⣭⢞⣻⢟⡛⢻⣿⣄⢛⡿⢟⣵⢟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣯⣭⣿⣿⣤⣶⣿⣦⣤⣴⣿⡶⠖⣚⡻⣞⣻⣽⢾⣛⢫⡗⠀⠀⢛⣿⠺⣀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⠽⣿⡯⠷⢛⣻⢾⣿⠃⠀⠠⠊⠱⢟⣯⣿⣿⣿⣿⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣷⣾⣿⣿⡏⣽⣿⣿⡾⣿⣿⣗⣚⣊⣭⣾⣿⡽⣾⣿⣿⣤⢄⠔⠀⠀⠛⣷⣿⣿⣿⣾⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣽⣿⣿⣾⡗⠃⣿⣿⣿⣻⣽⣟⣯⣽⣷⣻⡏⠀⠀⠀⠁⡀⠀⠐⣿⣯⠾⣟⡿⣟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡏⠁⠈⣿⣏⠁⢘⠙⢿⣿⣷⣿⣧⣀⣤⡄⠀⠀⠐⠻⣿⡻⣶⡈⢳⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣭⣿⣰⣀⣿⣿⣅⣨⣤⣿⣽⡶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠙⡿⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⢴⡶⠖⠒⠒⠒⣶⠶⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠒⠶⠋⠀⠀⠀⠀⠀⠙⣆⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⠖⠚⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠲⢤⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣶⠀⠀⠹⡄⠀⠀⠀⠀⠀⣠⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠉⠓⢦⣄⢀⡼⠁⠀⠀⢰⠚⠉⠀⠀⠀⠹⣄⢀⣤⠖⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠈⠓⠒⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠛⠛⠋⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣄⣀⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠋⠉⠙⢿⣿⠟⠉⠉⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⢀⣤⣤⣤⣤⡀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⠙⠛⣿⣿⠟⠋⠙⠻⣿⣆⠈⠻⣿⣦⣀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣻⣿⠀⠀⠈⠛⠿⣷⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⠿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠿⠛⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣶⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣸⣿
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⡿⠛⠋⠉⠙⠻⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉⠉⠉⢿⣷⣤⣄⣀⣤⣤⣾⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠙⠛⠛⠛⠛⠛⢻⣿⠟⠋⠉⠉⠙⠻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣘⣿⣷⣤⣤⣤⣤⣶⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠢⡀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⢹⣦⡀⠀⠀⠛⠀⠀⢻⣙⠳⡀⢸⣷⣄⠀⠀⠀⢰⠘⣆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⣷⣀⠠⠀⠀⠀⠐⢻⠛⢧⢸⣦⡙⣇⠀⠀⠘⢧⣹⠻⢦⠀⣷⡀
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠈⡳⣿⣿⣇⠠⠀⠒⠀⣉⡈⢯⠉⢹⡟⠻⣟⢀⣄⡄⠀⡇⣂⣈⣁⡟⢷⡄⠤⢀⡄⡀
⠙⠛⠿⢿⣋⠉⠉⠉⠂⡴⠋⠀⣿⣿⣧⡠⠔⠙⢋⡀⢤⣴⡆⠐⡧⠄⣻⠻⠟⣟⠀⡁⣸⡄⠐⡇⠈⢿⠀⢈⣡⡇
⠀⠀⠀⠀⡀⠉⠙⢲⢾⣄⣠⠊⣿⣿⠋⢹⣶⣿⢿⣷⣶⡉⢁⣤⡏⠀⠂⠀⡂⣗⢀⢼⡿⠃⢀⡇⢀⠱⣶⣍⡩⠊⠳⣄
⠀⠀⠀⠈⠛⠀⠀⡎⢸⣿⣿⡆⣻⣿⣄⢿⣯⡹⠧⡀⠙⡇⠈⣛⣫⠀⠀⢀⣤⡯⡀⠤⢪⣶⣶⡟⠿⠷⠞⠉⠀⠀⠀⢙⠆
⠀⠀⠀⠀⠀⢰⣾⣦⣤⣟⠋⢰⠁⠙⡿⣦⣽⣓⣂⣨⠿⣅⠀⠛⠋⠀⠀⠈⠁⠀⠀⠠⠊⠉⠉⠀⠀⠀⠀⠂⣀⣻⡷⣋⣀⠸
⠀⠀⠀⠀⠀⣼⣿⡿⠀⡏⠉⢻⠲⢴⣇⣀⡏⠉⠁⣀⣄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢡⣼⢛⣃⡤⣴
⠀⠀⠀⠀⠀⣿⣿⡇⠀⡇⠀⣸⣀⣘⠇⠈⣿⣷⡄⠙⠛⠒⠁⠀⠀⠀⣶⡄⡆⣴⣦⣶⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢩⣏⢱⠔⠋
⠀⠀⠀⠀⢰⣿⣿⡁⠀⢣⣞⣉⡏⠻⣧⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡂⠃
⠀⠀⠀⠀⣸⣿⡿⢣⠀⠈⠏⠛⠳⡀⠘⣆⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣷⣝⣏⣿⣘⡇⠀⠀⠀⠀⠀⣀⠀⠀⣴⣠⡀⣸⡶
⠀⠀⠀⠀⠛⠛⠳⠾⢦⠤⢼⣄⣀⣱⡀⠈⢆⠀⠑⣄⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⢳⣿⠀⠀⠀⠀⠀⠀⠋⠘⢷⣿⡙⠓⠻⠃
⠀⠀⠀⠀⠀⠾⠂⠀⠀⢣⣤⡨⢆⠀⢹⣿⣿⠱⡄⣿⢧⣿⠂⠀⠀⠀⠉⠀⠁⠁⠉⠉⠀⠁⠀⠀⠀⠀⣰⡦⠘⢦⡈⠍⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠐⠻⣷⡤⠛⡷⢉⡱⡌⠛⢻⣤⣬⣦⡀⠙⢌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠘⢅⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠫⡀⣈⠶⡁⠈⢻⡃⠈⣶⡟⠛⢫⡓⠀⠀⠀⣴⠆⠀⣴⢆⣶⢀⢿⡓⠀⠀⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⣿⣠⠞⠣⣀⠜⠺⣿⠿⠦⣿⠼⠿⣀⠈⠢⡀⠀⠡⠀⠀⠐⣈⢧⠔⠁⠳⣄⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⡾⠋⣀⡔⠊⠉⠢⣄⠈⡱⣴⣶⣷⢄⠀⠑⢄⣨⣶⢄⢱⡠⠂⠘⢆⠱⣤⣿⠘⠃
⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠐⣿⡌⢿⣆⢀⡠⠚⠙⢄⡈⠙⠫⡀⠙⠢⡀⢹⠣⡀⡩⢳⣀⣴⡿⠋⠈⢇
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠘⣿⣬⢻⣧⠀⠀⠀⠀⣉⡦⡖⣿⣿⣷⠈⠲⣆⡈⣀⣼⠟⠉⠀⠀⠦⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠻⣷⣠⠴⠊⢁⣤⣈⠑⢍⣩⠗⠦⣸⣾⠟⠁⠀⠀⠛⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠒⠉⠀⠀⠶⣦⠈⠉⢠⡾⠉⠢⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⢿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣶⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠄⠀⠀⠀⠶⠶⢤⡀
⠀⠀⠀⠀⠴⣶⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣤⣗⣿⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⣿⣿⠂⠀⠀⠁⣿⡽⠃⠀⠀⠀⠀⠀⣄⣦⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠶⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠒⢦⣤⠀⠀⢠⣤⠴⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠎⠁
⠀⠀⠀⢀⣤⣤⡀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⢀⠀⢙⣳⡄⠀⠀⠂⠈⣟⣢⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖
⠀⠀⠀⠸⣯⣦⡇⠀⠘⣟⣢⡇⠀⠀⠀⠀⠀⠘⢶⣩⡽⠃⠀⠀⠲⣾⣭⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠘⣿⣒⠂⠁⠀⠀⠹⣷⣿⣾
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⣀⣠⠆⠀⠀⠀⠀⠠⣄⢀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⢉⣷⣆⠀⠀⠀⢀⣴⣮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣙⡋⠀⠀⠀⢙⣓⣋⠁⠀⠀⠀⠀⠀⢠⣶⠶⡄⠀⠀⣶⡶⣆
⠀⠀⠀⠘⠯⠍⠀⠀⠀⠈⠭⠽⠁⠀⠀⠀⠀⠀⠀⠣⠿⠃⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⣼⣽⠇⠀⠀⠀⠹⣭⡿⠀⠀⠀⠀⠀⠀⠨⣀⣏⡿⠀⠀⣀⣻⣹
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠠⠴⠶⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰


⠀⠀⠀⠠⠖⠲⠆⠀⠀⠶⠒⠲⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⢀⢤⣄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣠⡴⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠓
⠀⠀⠰⣧⣴⢿⡇⠀⢸⣤⣾⣻⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⠇⠀⠈⣏⡿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠙⠻⠚⠁⠀⠸⠿⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠋⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠳⠊⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⢒⢢⠒⠢⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠐⠀⠁⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⠐⠤⢾⠠⠃⠩⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠬⠚⠬⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠿⠿⠿⠿⠿⠿⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣤⣤⡤⠔⡊⣀⣀⣀⣀⣀⡠⣄⣀⣀⣀⣀⣈⠒⠤⣤⣤⡤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠟⠛⠛⠛⠤⠤⠤⠤⣤⠓⢣⠤⠤⠤⠤⠜⠛⠛⠛⠧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣈⣀⣈⣒⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⣰⣿⡟⠁⢀⣴⣾⣿⣷⣦⣠⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣤⣤⣶⣿⣿⣶⣄⠀⠙⣿⣷⡀
⠀⢠⣿⡟⠀⢀⣿⡿⠋⠀⠉⠻⣿⠟⠁⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠈⠙⢿⡿⠋⠀⠈⠹⣿⣇⠀⠘⣿⣧
⠀⣿⣿⠁⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢹⣿⡇
⢸⣿⡇⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⡿⠋⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠈⠻⣿⣷⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⣿⣿⡀⠀⣰⣿⣷⡀⠀⣸⣿⡇⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣴⢶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣻⣿⠿⢻⣟⣬⣡⣾⣽⣧⣿⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣊⣥⡶⣶⢼⠿⣿⣿⣿⣿⣿⡋⣷⣾⣟⣿⡇⢿⣾⣿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢉⠔⠛⠉⠋⠈⠀⠰⠃⠨⠉⠾⣿⣿⣿⣿⢿⡿⡿⠧⠘⢿⣿⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣿⡛⠿⣿⣜⠳⠾⠶⠚⠺⠿⠿⣾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠘⠆⣠⠀⠈⠦⣺⣼⣶⡀⡁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⢎⣽⣷⡔⣀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠟⠇⠀⠁⠀⠀⠀⠀⠈⠙⠏⠳⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⣼⡟⣿⣿⣿⣿⡎⣴⣄⣤⣶⣾⣿⣿⣿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⢃⣿⣿⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⢿⡿⢻⣿⣿⣿⢯⣿⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢁⡏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣤⡀⠀⣄⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⠏⠁⠀⠀⠉⠻⣿⣦⡙⣧⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣼⣷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⢀⡴⣶⠲⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠒⠶⡦⣄
⢸⠉⠛⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡜⠛⠸
⢸⠀⠀⡇⠀⠀⠰⡆⠀⠀⢠⠦⡀⢢⢀⡖⣔⡂⠠⣦⣄⠀⢏⡀⠰⠒⡆⢰⢆⢰⢀⠞⣑⠠⣏⠀⠀⠀⡇
⢸⠀⠀⡇⠀⠀⠀⣇⡠⠀⠘⠴⠃⠘⠞⠀⠧⠄⠀⠛⠁⠀⠤⠝⠘⠦⠃⠎⠈⠋⠘⠤⠟⠀⠬⠇⠀⠀⡇
⢸⠀⢀⡇⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⡇⠀⢰
⢸⠀⢸⡇⠀⠀⠀⠀⢸⣿⡿⠟⠻⢿⣿⣿⣿⣿⣇⠀⠀⠀⢠⣿⣿⣿⣿⡿⠏⠹⢿⣿⡇⠀⠀⠀⠀⠀⡇⠀⢸
⣿⠀⢸⡇⠀⠀⠀⠀⢸⣿⣇⡀⢀⣸⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⢀⣹⣿⣿⠀⠀⠀⠀⠀⡇⠀⢸
⣿⠀⠸⡇⠀⠀⠀⠀⠸⠿⠿⠿⢿⣿⣿⡿⠿⠿⠥⠤⠤⠤⠤⠽⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⢀⠇⠀⢸
⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⢸⠀⠀⠳⠶⠦⠤⠶⠲⠲⠒⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠶⠾⠀⠀⢸
⣸⠀⠀⠀⠀⠀⠀⠀⡔⠀⠉⠉⠉⠉⠉⠉⠙⠓⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡄⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠰⡪⠑⠄⢠⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢱⢀⠔⣉⡲⡀⠀⣿
⢹⡴⡱⠐⠬⠥⠃⡆⠀⠀⢿⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠸⠷⠀⠀⠀⠏⢦⣓⡵⠳⢷⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠀⠤⠋⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡞⠉⠉⣷⠀⠀⠈⠻⣦⣀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢈⣤⡴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠑⠾⠀⠀⢀⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣠⣤⣶⠾⠿⠛⠛⠛⠛⠛⠛⠿⠷⣶⣦⣤⣀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣰⠏⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⡾⠋⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣄⠀⡀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠠⢤⠞⠉⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠙⣧⡠⠔⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣶⣦⣤⣤⣤⣶⣶⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⡆⠀⣠⣤⡀⢁⡔⠠⠔⣊⣥⣶⣥⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⢸⡏⠈⡧⠀⠡⠴⠛⠋⠁⡤⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⡇⣸⢇⣠⣀⣾⢰⢆⣐⠒⠀⠈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠤⢾⡟⠁⢾⡾⠁⠻⠋⣿⠛⠴⠆⣠⣏⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠛⣻⠟⠛⠛⠶⣾⡇⠀⢿⣠⠇⢠⡿⡽⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣬⣭⡭⠥⠤⣤⡀⢀⣍⡁⣀⠈⣉⠀⣈⡉⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢉⡏⣿⢞⡟⣸⣩⣿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠲⠤⣤⡟⠋⢿⠾⠁⡟⢸⣇⡿⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢩⣶⠶⠶⠶⠶⢶⣬⣥⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠻⣶⣤⡾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⣀⣠⣤⣤⣤⡶⠶⠶⠶⠟⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠉⠉⢻⡆⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⠄⠈⣿⣷⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣸⡇⠀⣿⠀⠀⣾⠛⠛⠛⠋⠉⠉⠉⠀⠀⣹⡇⠀⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⡾⠛⠙⠛⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠳⣦⣿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠞⠋⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠾⠛⢋⣉⣥⣤⠶⠾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠟⠛⢉⣉⣤⣤⡶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⠞⠛⠛⠉⠁⠀⠀⠀⠺⠛⠛⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠄⠆⠐⠣⡈⡨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡴⠶⠖⠚⣄⣁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠛⢰⠈⠉⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⣷⡏⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢾⠀⢐⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡇⢠⡞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡊⠤⡠⠭⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣚⡥⣤⡽⠿⠗
⣶⣾⡿⠶⠾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣿⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⢰⣶⣶⠀⠀⠀⢀⣾⣿⣶⣾⣿⡆⢠⠀⢤⣄⡄⠀⢸⣿⣿⣿⣿⣿⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⣶⣶⣶
⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃⠘⠔⠆⠋⠣⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠸⠉⠐⡆⠀⠀⢀⣍⣬⡀⠀⠀⠀⠠⢀⠀⡀
⠀⠀⠀⢰⣦⢶⡆⠀⠀⠀⠀⢰⣶⡶⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⡖⠈⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⢀⢻⣿⡟
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⢰⠊⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⣤⣀⣤⣄⠀⠀⠀⢀⡀⢀⣀
⠀⢠⣾⣷⢶⣿⣦⠀⠀⣀⣀⣀⣀⠀⠰⣶⡏⠻⣿⣄⡀⠀⠀⢿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿
⠀⠈⢻⣿⣸⡿⠃⠀⠸⣿⠉⠉⣿⠇⠀⠘⠀⠂⢸⠛⠁⠀⠀⠀⠁⠠⣤⡄⠀⠈⠻⠿⠋⠀⠀⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠷⡾⠋⠀⠀⠀⠃⢣⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⢀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣤
⠀⠀⠀⢿⣿⣿⡇⣀⣀⣄⠀⣀⣀⣀⢰⣿⣶⣶⠀⠀⠀⠐⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⣤⣄⠀⠀⢀⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠙⠋⠸⣿⣿⡿⠘⣿⣻⣿⠋⠻⠟⠋⠀⠀⢠⠤⠚⠻⠿⠋⠀⠀⠀⠀⠀⠿⣿⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈⠉⠁⠀⠀⠀⢀⡀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣴⣦⣴⣦⠀⠀⢀⣀⠀⢀⣀⠀⠀⢀⣬⣭⣉⣩⣥⡀⠀⣴⣿⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟
⠀⠀⠀⠈⠻⠟⠁⠀⠀⢿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠈⠻⣿⣿⣿⠿⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿
⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⡎⠉⠉⡦⠀⠀⠀⠀⠀⡏⠁⢿⣿⣿⣿⠇⡇⠀⡇
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠛⠁⠀⠈⠒⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠐⠒⠒⠂⠒⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣼⣅⡀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⢀⣨⣧⣄⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⢸⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠠⠤⠤⠄⠤⠤⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠿⠿⠿⠿⠶⠒⠒⠚⠓⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠳⢂⠄⠀⠤⠀⠠⠤⠤⠖⠒⠖⠲⠶⠒⠒⠒⠒⠒⠒⠐⠒⠀⢀⠉⢹⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠰⣾⠛⠛⡄⠠⠀⠀⠀⠀⠀⠐⠂⠀⢰⠟⠻⡍⠀⠀⠈⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡃⠀⠀⠘⢯⣢⡴⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⣧⣶⡁⠄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠉⠉⣩⣽⡿⠿⠿⠿⠷⠾⠶⠖⠻⠛⠛⢻⣿⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣻⠏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠹⣿⡧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢘⠛⠀⠀⢸⡏⢠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢹⣥⡀⠀⢺⣦⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣧⣀⣘⣏⡻⣏⠁⠀⠶⠶⠶⠈⠀⠀⠶⠿⢏⣉⡂⠸⣶⣁⣰⣎⣠⣾⣿⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠹⡛⡛⠃⠋⠛⠛⠛⠛⠛⠛⠛⠛⡉⣙⡛⢉⠉⠉⠉⠉⠏⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠋⡁⣡⠏⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⡀⠀⢠⡶⠆⠀⠁⣟⢬⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠔⠂⡠⢄⡀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣠⣐⡤⠖⠓⢊⣁⡞⢀⡴⢸⠆⠉⣀⠴⠚⠉⢀⣁⢤⢲⡶⠆⡻⠁⣢⢯⣀⣤⣬⠼⡇⢹⣦⡾⣥⠞⠁
⠀⠀⠀⠀⠀⠀⢀⢔⡠⠖⠛⢉⡇⠀⡴⠁⠀⠀⠀⡸⠟⠆⡜⠁⠈⠑⠊⠁⢀⢢⠖⠉⠁⢸⡏⠡⠴⠋⠴⠃⠊⠟⠁⣠⠼⠋⠡⡴⠋⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠎⢀⡀⠠⠞⠀⠀⠀⠂⠘⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢀⣀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢎⠀⢰⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒





⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⣶⡶⢶⠶⣶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡗⠐⠋⣠⠀⣀⣤⣤⣤⣌⠙⠻⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⣠⣶⡿⢡⣾⣿⣿⣿⠿⢿⣿⣆⠈⠻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠀⣼⣿⡿⢀⣿⡿⢋⣥⡶⠶⣶⠆⠙⠇⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⣿⣿⣷⢸⣿⠁⣾⡟⠐⠺⢯⣳⡄⢠⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⣿⣷⡈⢿⣿⣆⢻⠀⣿⡇⢹⣿⡈⣳⠇⠘⣿⣿⣿⣿⡟⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡙⢿⣿⣆⠙⢿⣦⠰⣿⣿⣦⣬⡍⠁⣤⡀⢹⣿⣿⠋⡀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣌⠻⣿⡇⠰⣭⣀⣉⣉⣉⣡⣴⣾⣿⣷⠀⠻⢃⣾⡇⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠘⢃⣀⡈⠻⠿⣿⣿⣿⣿⣿⣿⠟⠀⣠⣿⡿⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢰⣿⣿⣿⣿⣶⣦⣭⣍⣉⣉⣡⣴⣾⣿⡿⠁⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣀⣤⣤⣤⣬⣭⡛⠻⠿⠿⠿⠿⠿⠋⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠛⠿⣦⡐⢶⣶⣶⣶⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⠿⠟⣋⣤⡈⠀⢠⣬⣭⣝⣋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⢉⣴⣦⣿⣿⡇⣿⣷⣦⡙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⢁⣶⣼⣿⣿⣿⣿⡇⢿⣿⣿⣿⣦⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠏⢠⣻⣿⣿⣿⣿⣿⡿⠁⠈⠻⣿⣿⣿⣧⡘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⣠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⠛⠿⢿⣬⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢁⣴⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢋⡴⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠊⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠛⢃⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣏⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣽⢛⣿⠀⠀⠠⠆⠀⠀⢀⠀⠈⡀⢀⠀⠀⠘⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠋⣥⠀⣄⡖⠃⠀⠀⠁⠀⠀⠀⣾⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⢀⣀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢾⠖⢠⡆⠀⢰⡾⠁⠀⣼⠀⣠⡷⢠⢤⣤⣤⡤⠀⣤⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⢦⡘⢿⠀⣸⠙⢤⣿⣷⣮⣿⢋⣼⣿⣷⣧⣾⣿⡟⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠻⠟⠿⢿⣿⣿⣌⢷⣴⣤⣾⣿⡿⠟⠁⢉⣿⣿⡟⠉⠁⠀⠀⣀
⠀⠀⠀⠀⠄⣀⣀⣀⣈⣰⣤⣤⣤⣤⣼⣿⣿⣿⡆⢿⣽⠃⠀⠀⠀⣠⠊⠁⠈⠛⠋⠳⣶⠊⠀⠛⢲⡞
⠀⠀⠀⠀⠀⣈⠉⠉⢉⣽⠟⠛⠙⠏⠉⣿⠟⡻⣦⡰⠿⠋⣹⠾⠛⢻⣶⣤⣤⣄⣀⠀⠈⣷⠀⠀⠀⡻⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠔⠁⠀⠐⠇⠀⠀⠉⠀⣴⡶⠾⢉⣀⡌⣩⣽⡿⠛⢿⣦⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠀⠀⠀⠀⠀⣀⣶⠀⢸⠛⢁⡴⢛⣼⣽⣶⣿⢦⡴⠀⠀⢻⢿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠂⢠⡄⠒⢦⡄⣻⡞⠉⢳⡀⣿⣄⣼⢷⢸⣭⣿⡏⢻⡏⠐⠓⢻⠟⠀⢻⢿⡆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠺⠛⢛⣶⣷⣿⠂⠀⠈⢿⣿⢻⡏⢾⣼⣽⡿⢅⣏⠛⠁⠀⠃⠀⠀⣷⠸⣿⡄⠀⣀⡞⡠
⠀⠀⠀⠀⠀⢘⣦⣀⣞⠀⢀⣤⣤⡼⠇⠈⠉⣆⠀⠀⢸⡣⡘⢷⠒⠭⠾⠓⢀⣁⠀⠀⣸⠄⠀⡐⠁⠀⠻⣿⢶⣿⣿⠁
⠀⠀⠀⠀⠀⠁⢺⣿⣿⠀⣿⣾⠋⠀⠀⢠⠞⠛⠁⠀⢸⡇⠈⢦⣧⣀⣀⡀⣿⡟⠁⠁⠀⠀⢰⠇⠀⠀⣼⡿⣿⣿⣷⠶⢶⣤⠀⡾
⠀⠀⠀⠐⢉⠛⠉⣡⣬⢷⣿⠉⠁⢂⡀⠈⠛⠲⣤⠀⠻⢷⣄⠀⣶⠈⠙⠛⠳⣤⡀⠀⠀⢠⣿⠀⣴⣼⣿⡽⠋⠀⠀⠀⠀⠘⡟⠁
⠀⢀⡀⠂⠀⠠⢞⡿⢳⠀⠉⠳⡄⠀⠀⠀⠀⠀⠈⢧⡀⠀⠹⣶⣬⠿⠿⠿⠛⠉⠛⠷⣤⣼⡟⢀⣿⣿⣿⡷⠄⠰⣴⣶⠶⠚⠁⠀⠀⠀⠐
⠀⠈⠀⠀⠀⠰⢿⠃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡤⠠⡤⠄
⠀⠀⠀⠀⠀⠀⣆⠀⣆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠷⠶⣶⣦⣴⣶⣾⣶⣶⠾⣿⡿⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣠⡘⣧⡤
⠀⠀⠀⠀⠀⠀⣿⡟⠋⣀⣀⡀⢠⣤⣤⣀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⠉⠛⠁⠙⠿⣶⣶⣌⠀⠀⣀⢀⣠⡴⠋⠀⠀⠀⢀⡀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠠⠜⡛⠄⠂⣉⣤⣬⣹⣿⣶⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠉⠉⠉⠙⠛⠳⢶⣶⠮⣉⠡⢶⣶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠴⢾⣶⢆⣿⣿⠯⢺⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠙⢆⠈⠃⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠉⠉⣽⡶⣛⣻⣿⣿⣷⠠⠤⢴⡀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠁⠀⢁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⠐⠢⣀⠀⠠⣤⡘⣦⣀⣿⣿⣿⣿⣿⠙⣿⡄⠀⠀⠙⠳⣄⠀⠀⣀⡄⢁⡀⠀⠀⣀⡤⢘⡩⢉⠛⠀⡀⠀⠀⠀⠀⠀⠣⡈⣧⡀
⠀⠀⠀⠀⠀⢀⣠⠒⠺⠋⠀⡴⠎⠉⡉⣹⣿⣿⣿⠸⣧⣿⣿⣦⠀⠀⠀⠀⠈⣴⠟⠁⣼⣿⡷⣾⣿⠞⠛⣯⠙⣩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣠⠋⡌⢀⣤⠀⠖⠀⠀⢲⣿⣿⣿⣿⠿⠖⠻⢿⡿⠀⢀⣁⣡⠤⢶⣧⣄⡀⢱⣿⣿⣯⠛⢧⡉⠉⠀⠉⠁⡀⡴⢆⣀⣴⣦⣤⣤⣀
⠀⠀⠀⣾⠃⠚⠁⠉⠀⠀⠀⠀⠀⣿⣿⣿⠟⢻⡦⠀⠀⠈⠳⣦⣉⣽⠄⢀⡼⢿⣿⣁⠞⢻⡇⠀⢳⡄⠙⢦⡀⣠⣾⣶⣴⠟⠋⢉⡁⠉⠀⠀⢱
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⡏⠠⠿⠃⣀⠀⠀⠀⠉⠀⠀⠀⢸⡇⠈⢻⣿⠀⠞⣿⣄⠈⠀⠀⠈⠻⡈⠙⠻⠷⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⠁⠀⢰⡏⠢⢄⣀⡀⠀⠀⠀⢮⠙⠄⠀⠉⠀⠀⠈⠏⢻⡆⠀⠀⠀⢹⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠰⣤⣄⣙⠋⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠲⢰⣼⠇⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡿⣉⣩⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠌⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⢰⠃⡞⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠟⠈⡱⠋⠀⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠘⠰⠁⠀⠀⢸⡇⠁⢀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣤⠔⠊⠉⠉⠉⠙⢭⡉⠑⢦⡇⠀⢀⠀⠀⠀⠀⢠⣃⠴⠓⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⡠⠚⠉⢀⡀⠴⠥⢄⣀⣀⡀⠀⠀⠀⠙⡄⠀⠹⣄⠘⠶⠤⠤⠰⠋⠤⣖⠉
⠀⠀⠀⠀⠀⠰⡎⠉⠀⠀⠀⠀⠀⠀⠉⢢⠞⠁⠀⢠⠋⠀⠔⠀⠉⡀⠀⡀⠈⠤⡀⠀⠘⡄⠀⠱⣉⡶⠒⠒⠒⠲⢤⡀⠃
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠐⠢⢤⣀⣹⡄⠀⠀⠈⠀⣠⢆⢀⢪⠄⠀⢈⡆⠀⠘⠆⠀⡇⠀⠀⠏⠉⠒⢤⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⣠⠔⠋⠀⢡⠀⠀⠀⠰⡏⢆⠙⠤⠤⠖⢈⣃⠄⢈⠀⣠⠃⠀⠀⢰⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⢑⡞⠁⠀⠀⠀⠀⡆⠀⠀⢀⠆⠈⠢⡀⢀⠀⠈⠁⠀⢸⡜⠁⠀⠀⠀⡎⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⡄⢠⠖⠉⠀⠀⠙⣄⠀⠸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡃⠀⣀⢠⡌⠀⠀⠀⠀⠀⢰⣆⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠦⢤⣄⣀⣀⣱⠁⠀⠀⠀⠀⠀⢈⠳⣬⡄⠀⠀⠀⠡⠀⠀⠀⢀⡠⠖⢋⠔⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠧⠒⠢⡀⠀⠀⠰⠛⠓⠲⣄⠀⠀⠈⠁⠓⠂⠉⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢀⣼⣡⣔⠈⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⣀⣀⣃⠀⠀⠀⠀⠀⠀⠈⠻⢔⡂⠤⠤⢀⣲⡞⠉⠀⠀⠀⠀⠀⣀⡤⡶⠋⠀⠀⠈⠙⢦⣠⠞
⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠁⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⢈⠟⠈⠒⠒⠒⠛⠉⠁⣴⠛⠒⠄⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⡀⢀⡤⠔⠒⢺⣷⠤⢄⣀⣀⣀⣀⣀⠠⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠋⠀⠀⡴⠃⠈⡄⠀⠀⠋⠀⠀⢸⠓⠦⢤⣀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠜⠁⠀⠀⣿⢄⡀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣰⠇⠀⠉⠓⠶⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠛⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠉⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢷⣦⣀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣁⣉⣉⡠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡈⠳⣦⣴⣦⣄⠀⠀⠀⠉⠛⠛⠋⠉⢁⣴⣤⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠝⢿⣿⣿⣿⣦⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠙⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⢀⠂⠀⠒⠁⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣀⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠁⠫⠀⠸⠀⣦⠀⢀⡥⠀⠈⠀⠀⠀⠀⡀⠀⠀⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡖⠀⠘⡔⠄⠸⣄⣁⣸⣧⡑⣤⠀⠀⠂⠀⠀⠀⠒⢤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣹⢷⠀⠀⠈⠚⢦⡹⡝⠉⠛⠳⢆⠳⠶⢴⣄⣠⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣮⣣⠈⢿⠦⣄⠀⣀⠉⠳⣶⡒⠂⠀⠐⠀⠰⠶⠟⠁⣠⡀⠀⢠⡆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣯⢿⡇⠀⠸⣇⣦⠙⠉⠧⣄⣀⠉⠓⠤⢉⡀⠀⢲⣾⡟⠚⠀⠀⠋⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⡏⠨⢻⡄⠀⠘⢷⣀⣴⣠⣀⡀⠉⠃⠀⠰⣻⡿⢿⢛⠁⠀⠀⠀⠀⠀⡴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⢧⡄⠀⠸⠿⡶⣤⣤⣉⠣⢴⡋⠉⠉⠛⠚⠈⠈⠉⠉⠁⠀⣀⣀⠦⣀⢸⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠹⣷⣄⣀⡀⢇⢿⣿⣷⣶⣮⢳⠀⢠⢈⡐⢲⣶⣶⣶⣮⠾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣶⣆⠸⠏⠉⠙⢿⣰⣿⡿⢿⢿⠀⡆⠈⣼⡰⠿⠛⠋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡵⣟⣿⣿⡻⣄⠀⢸⣿⣿⠡⡿⢿⣆⢱⡞⠡⠔⠊⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⠇⡈⣀⢸⣿⣧⠖⠋⠉⢁⡴⣿⢦⣤⢾⠯⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣭⣿⣇⠹⠀⢀⠀⠀⣿⢟⢻⣿⡿⠃
⠀⠀⠚⠛⠁⠁⠉⠛⠚⠓⠉⢁⣾⣿⢿⡿⣿⣿⣿⡲⣦⣾⠛⣿⣿⣷⡿⠛⠁
⠀⠀⠀⠀⠀⢤⣶⠶⠴⠖⠚⠋⠉⢀⡾⠀⡜⠁⠈⠉⠉⠛⠻⡟⣏⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠌⠀⠀⠀⠀⠀⠀⠀⠙⣼⣜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⢿⡟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⢁⣀⡀⠀⠀⠀⠀⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣼⣿⣿⣿⣿⡍⣡⣶⣿⣿⣿⡆⠀⠀⠔⠁⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⠇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣿⣿⡏⢉⣡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢦⣬⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣏⠀⢹⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡿⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡼⢻⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⡀⢷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⢸⡇⠠⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡶⢄⠀⠘⢷⣿⡾⠁⢁⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢆⣑⣀⣘⣻⠔⠒⢮⠋⠕⢒⣒⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⢈⡶⠋⠁⢀⣈⢙⣷⡀⠘⠓⣌⠿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⢣⢺⠓⣩⠤⣬⠓⡝⠆⠀⢸⠈⠙⢶⡀
⠀⠀⠀⠀⠀⣶⡫⠶⠋⠺⢷⡄⢰⣿⢀⡇⢰⣄⡞⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⣇⠀⠀⣸⣤⡭⠝⠃⡜⢠⠚⢉⠇⠀⣠⠞
⠀⠀⠀⠀⠀⠀⣰⠗⠮⢤⣀⣀⣤⣞⣁⠟⢰⣿⣠⠏⣰⠂⣠⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣽⣋⣉⣁⣠⠴⠞⠁⢀⡴⢣⠞⣿⣷⡀
⠀⠀⠀⠀⣰⣯⣬⣣⡀⠀⠀⠀⢠⠟⠀⠀⣀⣀⣾⢁⡏⢸⣿⣿⡇
⠀⠀⠀⠈⠛⠛⠿⠿⠾⠗⠒⣈⢓⣒⡦⠭⠤⠤⠤⠸⠀⣾⣿⣿⢇⠤⢔⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣆⠻⢿⣿⣶⡿⠿⠶⢤⣘⡿⠋⠠⣾⣿⡿⠁
⠀⠀⣀⣠⡤⠤⠤⠤⣄⡘⠸⢟⣿⣤⣴⣟⣁⣀⠀⠀⠀⠀⠉⠛⢻⠟⠋
⠀⠀⠻⣿⣷⣷⣶⣦⡀⡈⢠⡿⠁⠀⣠⡤⢀⡉⠛⢦⣶⠦⣄⠀⠈⠲⣄⠀⠀⠀⣠⠴⣶⡖
⠀⠀⠀⠈⠻⢿⣿⣿⣿⡷⣺⣇⡤⡶⠃⣀⣴⠾⡟⢿⣻⣆⠈⠳⣦⣀⡈⣧⢠⠞⣡⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⡩⠚⣹⠋⢰⡇⣞⣽⣿⣿⣾⣆⣷⢻⡇⠀⠸⡇⠙⣧⠀⢰⣿⣿⠟⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣿⣆⢸⣇⢻⣯⢞⡡⠏⣿⠃⣸⠀⠀⢠⡇⠀⢸⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣁⣉⣜⡵⠆
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠘⠻⣇⠙⠫⣭⡴⠞⣡⠞⢹⡗⠚⢩⠇⠀⣸⠇⢒⣊⣭⣷⡶⣰⢶⠊⢉⢉⡙⠓⠲⡞⣁⡀
⠀⠀⠀⠀⢀⣀⠀⣿⠃⠀⠻⣄⠀⠙⠲⠖⢋⣡⡾⠋⢀⣼⠃⣠⠏⠀⣰⣿⣰⣿⣿⣿⠟⣰⣏⣻⠟⡉⠉⡻⢟⡛⠹⣉⠉⣀⣩⣾⣿⠇
⠀⠀⠐⢾⣿⣶⣶⣜⢧⡀⠀⠈⣻⣖⠒⠊⠉⣁⣠⣴⣛⡵⠞⠁⠀⣾⣿⠃⠛⠛⠻⣤⡘⠿⢿⡉⣾⣷⣖⣋⠥⢚⣃⠿⠸⣿⡟⠛⠁
⠀⠀⠀⠀⠈⠛⠻⣿⠗⠑⠄⠀⢈⡩⠿⠿⢿⣿⣉⠉⠁⠀⠀⢀⡴⢿⣵⣷⣗⣦⢄⣀⠙⢦⡀⠉⢡⣍⣀⠐⢄⢠⣥⣤⣤⣀⣉⡓⢦⡀
⠀⠀⠀⠀⠀⣠⠎⣡⣴⣶⣿⠀⣍⢧⠀⠀⠀⢹⡿⠛⠛⢛⠋⣵⣿⠈⢿⣿⣿⣿⣿⣿⣶⣄⢻⣆⠘⢿⣿⣿⣎⣧⠻⣿⣿⣿⣿⣿⣶⣿⣄
⠀⠀⠀⠀⢰⣧⣾⣿⣿⠿⠋⡄⣿⣿⠛⠶⠶⠾⠀⠀⠀⣆⣼⣿⠟⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⡄⠀⠉⠛⠻⠿⠆⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⢳⡿⠃⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠛⠿⠿⣿⣿⣿⡟⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡈⠛⢿⣷⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⠉⠙⠻⠆⠀⠈⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⢷⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⡀⠈⠳⣦⡀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣆⠀⠻⣦⡀⠙⣿⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣦⠀⢻⣿⣄⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠉⠉⠉⠙⠓⠒⠤⢀⠀⠀⠀⠀⠀⣀⣰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢡⠀⠀⢀⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢿⠀⠀⢸⣿⠃⠈⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣤⢊⡥⠖⠒⠀⠸⠀⠀⠀⢹⠀⠐⠒⠦⣝⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢰⣍⠀⠀⠠⠔⠒⠋⣡⡎⠀⠀⣀⣠⠄⢳⠂⠀⠀⠈⠀⢠⣀⡀⠀⠁⠘⣿⢿⢿⣿⣦⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⣏⠙⠲⠄⠀⠐⡋⠆⠁⠀⠀⠀⠀⠀⡴⠋⠀⠐⢧⡀⠀⠀⠀⠀⠀⠀⠁⡿⠄⣿⣿⣿⣷⣤⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⡎⢣⣀⠀⠀⣀⣑⠄⠀⠀⠀⠀⠀⢲⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⡿⣁⠈⠳⣄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⠀⠈⠑⣶⡴⠛⠀⠐⠲⣤⣀⡀⣸⠀⠀⢀⣀⡜⢣⡀⠀⢀⣀⣤⠶⠒⠘⠿⣿⡿⠛⠓⠚⠓⢶⠾⠟⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣀⣀⣠⠞⠁⠀⠀⢀⡾⢥⣾⣿⣷⣦⣴⠋⠈⢣⡈⠙⣟⠉⠙⣮⠀⢀⡤⠞⠁⠀⠀⠀⣠⣴⠭⠽⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⢠⠀⣸⣿⠀⠀⠀⠀⠀⠀⠞⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠙⠲⣌⡳⢀⠈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⣸⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⢸⠋⠑⢺⠦⠤⠤⠤⠴⠒⠠⠤⠤⠤⠄⠒⠛⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢠⣧⣾⣿⣿⡧⢤⣤⣀⣤⡤⠶⠊⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⠤⠖⠒⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠁⠉⠉⠒⠂⠄⠀⣀⡀⠠⠠⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠒⠄⢀⣀⣀⣠⠤⠤⠔⠲⡄⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠀⣠⠊⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⡔⠁⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠉⠁⠒⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⠎⢀⠀⠀⠀⠀⡀⠒⣊⠉⠉⠉⠲⣄⠈⢳⠀⠀⠀⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⠀⡊⠀⠀⠐⠈⠔⠀⢀⡉⠂⢄⠀⠀⡇⠈⡆⠀⠀⠀⠀⢷⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠂⢹⠀⠂⠀⡀⡄⠀⡈⠉⠓⠀⠁⠠⠇⢀⠇⠀⠀⠀⠀⢸⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠑⢀⣰⠛⡁⠘⠀⠰⠮⢾⡠⠀⠀⡼⠀⠀⠀⠀⠀⡜⠀⠀⠀⣸⢀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠀⠀⠀⠀⠀⢤⡀⠀⠀⡇⣰⢷⠀⠈⠑⠀⠀⠀⠀⠀⠐⠋⠀⠀⣰⠁⠀⠀⠀⠀⡜⠁⠀⠀⣰⠁⠀⢡
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠟⠀⠱⠀⠀⠀⠁⠀⠀⠀⠀⡠⢂⠔⣹⡀⠀⠀⡠⠊⠀⠀⢀⠴⡁⠀⠀⢸
⠀⠀⠀⠀⢀⠀⠙⢄⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⡷⣤⡀⠀⠈⠂⠄⠀⣀⣀⡀⠾⠐⣁⡴⠋⠀⠠⠔⡀⣀⠄⠊⠁⠀⠘⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⠈⢦⡀⠀⠀⠀⢇⠀⠀⠐⣼⣬⡁⠐⠒⠒⠒⠀⠀⠀⠚⢉⣡⠄⠂⣁⡼⠋⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⡈
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠑⠀⢀⡙⢦⡀⠀⠈⠳⢄⣀⠈⠳⠀⠉⠉⠉⠀⠀⠈⠁⠠⠋⠤⠲⠋⠁⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠘⠄⠙⢦⡀⠀⠀⠈⠉⠙⠂⠒⢎⡉⠉⠀⠀⠁⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠁
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⢆⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠃⠀⢀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠈⠉⠉⠉⠁⠀⠀⠀⢀⣀⡤⠔⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠘⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠻⠁⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠄⠤⠠⠔⠉⠀⡔⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⠀⠈⠉⠉⠝⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⡠⠤⠐⠉⠂⠄⠀⠀⠤⠔⠒⠂⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔⠒⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣠⠀⠀⠉⢛⡀⠀⠚⠶⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠰⣂⠫⣀⠀⠸⠢⠁⡄⠀⠀⠀⠀⡀⠀⠙⠋⠀⠀⣶⢯⣤⡤⠤⡀⠈⣤
⠀⠀⠀⠀⠀⠀⠰⢇⢪⡏⠀⣀⡄⠾⠛⡯⢱⠀⡸⢀⠀⠀⠀⠀⠙⣿⣦⣄⡾⠀⢀⢴⡎⠁⠳⣔⠏⢠⠸
⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⣦⡌⣆⠳⠔⣡⡎⠀⠁⢀⠻⠀⠀⠀⢀⠈⢡⣾⠇⢰⣿⡀⠓⢀⢀⣼⠃⠸⠇⣠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠦⠙⢃⠤⠼⠭⠐⠋⠀⣠⣎⠄⠀⠀⢀⡴⠞⠋⢝⢌⡐⠻⠳⠒⣻⠗⠁⢀⣔⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠈⠉⠑⠠⡄⠹⣦⡀⡠⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⠢⠛⠉⠁⠐⠀⠀⠨⠁
⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⣈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠀⡦⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢐⣠⡍⠁⠚⠉⠁
⠀⠀⠀⠚⠋⠁⠘⠛⢂⡠⠌⠙⠛⠆⠀⠀⠀⠃⠯⠊⠛⠛⠛⠚⠛⡟⠛⠁⠀⠀⣰⡟⠁⠀⠉⣉⣻⣾⠀⡀⠤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⣶⣯⣤⡖⠶⣤⡀⠀⠀⠀⠀⢠⢖⣅⢔⣀⡄⣀⡀⠀⠀⠀⠀⡍⠀⣠⣞⡑⡟⠓⣒⡦⣌⠃⠹⣾⣄
⠀⠀⠀⠀⣠⠊⠂⣡⣞⠉⠑⠮⣛⠀⣦⠀⠀⠀⠀⠘⠣⣜⡁⠋⠃⠋⠀⠀⠀⠀⢿⢰⠈⢋⡞⠁⠀⠨⠃⣹⣿⠀⡞⡳⡀
⠀⠀⠀⢸⣿⠀⣴⡆⠙⠀⡀⢠⡿⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠸⡇⠈⠻⣶⣰⣦⡶⠼⠟⢊⡴⡱⠁
⠀⠀⠀⠈⣏⠂⠘⣿⠤⠶⣻⠏⠀⠀⠀⠀⢀⠄⡀⠘⠣⠄⠀⠄⠀⠀⢀⣀⠀⠀⠙⠦⢰⣦⢀⠀⠉⠓⠶⠶⢖⠞⠰⠁
⠀⠀⠀⠈⠛⠕⡒⢄⠶⠚⠁⠀⠀⠀⠀⠐⢀⡘⠋⣀⣠⠂⣄⠀⠁⠻⠋⠀⠀⠀⠀⠀⠙⠉⢀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⢄⠈⠆⢸⢹⠛⡯⣀⠀⠻⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣿⣷⡤⠀⢀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⣤⠴⣧⣠⠀⠀⠼⠌⡒⢪⠝⣠⠀⣇⠜⠃⡀⠀⠀⠀⠀⣠⡴⠶⠙⣿⢀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠹⣂⠞⢃⡴⠁⢰⣿⠁⠀⠀⠀⠐⠉⠀⠀⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠁⠐⠚⢃⠀⠐⠒⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⣠⡤⠞⢻⠃⠀⠀⠀⠀⠀⠀⠀⢰⣏⠳⣍⡽⣿⠀⠀⠀⠀⠀⠀⠀⣠⠚⡔⠿⠷⣨⠓⣦⠀⠀⠀⠀⠀⣰⣇⣸⣮⣁⣩⣿⠷⡶
⠀⠀⠀⠀⣜⡜⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⢳⠘⡄⠀⠀⠀⠀⠀⠀⠷⣤⡙⠶⣩⠕⠚⠁⠀⠀⠀⠀⠀⢹⠀⠙⠺⢤⣺⠏⠀⡇
⠀⠀⠀⠀⠘⠦⣄⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣘⡼⠁⠀⠀⠀⠀⠀⠀⠀⠳⣕⠀⠁⣀⡼⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠙⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⠤⠒⢐⣒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⠒⢲⡀⠀⠀⠀⠀⠀⣠⢴⣯⣭⣍⠓⠲⡄⠀⠀⠀⠀⠀⠀⢀⡴⠦⣖⣓⢦
⠀⠀⢸⣅⠀⢺⠟⠛⣷⡈⢧⠀⠀⠀⠀⠀⠀⣜⣥⡞⣩⠿⡇⣼⢩⡇⠀⠀⢀⡞⠱⣏⣶⠖⡟⢯⠑⣷⣄⠀⠀⠀⢀⡞⠉⡤⣞⠯⠏⣬⡓⣆
⠀⠀⢸⡈⢧⡈⢓⣖⣋⠗⢀⣧⠀⠀⠀⢠⠊⢺⣤⣵⣤⣤⠿⡻⢸⠁⠀⠀⠘⢦⠘⣍⠫⣝⠿⣫⣢⠏⣸⠆⠀⠀⠈⢧⢧⠹⠈⠥⣄⢋⣇⣸
⠀⠀⠀⠑⢤⠙⠀⠀⣉⡽⠋⠀⠀⠀⠀⠈⠙⢦⣐⠤⡤⠶⣚⡵⠋⠀⠀⠀⠀⠈⠹⣌⠑⠶⠚⢉⣇⠞⠁⠀⠀⠀⠀⠀⢯⠓⠚⢛⡋⢈⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣠⠤⢤⣔⠒⠲⡄⠀⠀⠀⠀⠀⠀⣠⠤⣴⣒⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⠔⠒⣦⣄
⠀⠀⠀⣰⢃⡴⣯⠬⣝⢳⡗⢄⠀⠀⠀⠀⡴⠃⡼⣿⣻⠝⡎⠹⡄⠀⠀⠀⢀⡴⠋⣹⣷⡾⢿⣺⣟⡙⡆⠀⠀⠀⢀⡞⡱⣚⣾⡫⡉⣿⠳⣄
⠀⠀⣾⠀⡿⣌⣧⡼⢎⣿⠇⢸⠄⠀⠀⢰⡇⠀⡀⠈⠁⠴⠻⡄⣻⠀⠀⠀⠀⠳⣸⡟⡏⠓⠒⣺⢿⠇⢹⠀⠀⠀⡼⡷⠉⡧⣤⣨⠟⣮⢳⣈⡇
⠀⠀⠹⣄⣻⠮⠿⣛⣫⠾⡶⠋⠀⠀⠀⠸⡟⠲⣌⠓⠒⢂⡠⠃⣸⠃⠀⠀⠀⠸⡍⠓⠬⠲⡍⠒⠋⢛⡏⠀⠀⠀⠙⢿⢦⡙⢶⣶⠚⢁⡾⣿
⠀⠀⠀⠈⠉⠳⣄⣀⣤⠞⠁⠀⠀⠀⠀⠀⠙⠶⠬⠿⠿⣉⡤⠖⠃⠀⠀⠀⠀⠀⠙⠦⠤⠖⠳⠤⠞⠉⠀⠀⠀⠀⠀⠘⣦⣭⣭⠽⠯⠭⣥⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢋⡟⢻⢽⣶⡄⠀⠀⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⢰⠒⠊⡇⣰⡟⢧⡘⡴⠃⢹⣄⠀⠁⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⢯⣵⣼⠀⠀⣷⢳⠇⠀⠙⢷⡶⠟⠘⡇⠀⠀⠀⠃⠀⡀⠀⣠⠴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠋⠛⣏⠓⢦⣿⣖⠒⢒⣶⣿⣶⡖⢻⡅⠀⠀⣀⠀⠀⢿⡞⢹⡄⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⣿⣠⣾⢯⣉⢱⡄⣏⠻⡍⠉⠉⠉⠉⢁⡜⢀⡠⠔⢹⡀⠀⣿⡇⣼⠃⣸⣷⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⣾⠀⣿⡇⢧⣾⠟⡟⢿⡟⠀⢹⠿⣯⣍⠉⠉⡰⠋⠀⠀⠈⡽⠀⣿⣿⣿⣾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⢀⡟⠈⢣⣽⣧⣀⣹⡾⠷⢻⠃⣠⢯⠀⠈⣯⢻⣴⡇⢀⣼⠃⣸⢧⣠⠟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⡀⠀⠈⠳⣤⣀⣀⣀⣠⣿⠞⠁⢸⡇⢀⣿⣼⡇⢳⣾⡷⣾⣵⣯⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠠⡤⣤⣐⠒⢤⡀⢸⠀⠙⠶⢤⣀⣈⣻⡿⠟⠋⠁⠀⢀⡼⠀⣸⣭⣭⣿⠁⠒⠻⣥⣀⠙⠛⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠑⠦⣬⣭⡱⢾⣘⣦⣀⠀⠀⠊⠉⠛⠛⠛⠒⢲⣶⣯⣥⣤⡟⠉⣧⠹⣤⠀⠢⣈⢿⢮⠀⢄
⠀⠀⠀⠀⠀⡠⠊⡩⠁⠀⣼⡟⠻⣶⣌⠙⠦⣄⠀⠀⠀⠀⣠⠞⢿⣿⣯⠀⠀⠐⢬⡳⡈⠻⠷⢬⡙⣮⢣⡀⠑⢄⢲⡄⠀⣠
⠀⠀⠀⠀⠀⢀⠎⢀⡼⠆⢿⣽⡀⢠⣽⠀⣦⣬⣿⣶⣶⣾⡿⢳⡀⠙⠿⠷⣤⣦⣀⣱⣇⠀⠀⠀⠈⢺⡇⠉⠒⠒⠋⠁⠉⠁
⠀⠀⠀⠠⠴⠧⠶⠟⠁⠀⠈⢧⡇⣸⢇⣴⡿⠋⣸⠁⠀⠙⡇⠘⢇⠀⠀⠀⠈⠀⠀⠉⠻⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⣡⣿⠋⠀⠀⢹⣦⠀⠀⢻⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡿⠁⠀⠀⠀⠀⠉⠻⢦⣸⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⣿⣋⣌⣿⣿⣿⣿⣿⣿⣷⣾⣷⣮⣽⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣭⣿⢿⣿⣿⡘⣿⡄
⠀⠀⠀⠀⣾⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠛⠉⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣷⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣋⣠⣼⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠿⠿⣿⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣷⠉⣰⡿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣻⣿⣿⠀⣿⣟⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣤⢻⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡔⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣤⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⢽⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⣿⣧⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⣿⣿⣿⣿⡌⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠻⣿⣷⢻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠹⢿⣿⣿⣷⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⢸⣏⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣷⣤⣭⣿⣿⣿⣜⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣸⣿⡏⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣶⡀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢀⣼⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡌⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠃⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡆⠀⠀⣿⡏⠈⠀⢸⣿⡏⠛⣾⠦⠤⢤⣴⠤⠔⠋⠀⣿⠀⠁⣿⡟⠻⠛⣿⣿⣿⣿⣿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⠸⢿⣇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⣿⡄⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡅⠀⣾⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄⣿⣄⣾⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡃⣄⠹⠿⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠇⣰⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡿⢻⡟⣿⢸⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⣿⣿⣿⣿⣸⣿⣿⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⠿⢿⣿⣼⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠈⠉⠋⠉⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⡀⠀⡀⠈⠃⠀⠀⠀⠀⠈⠙⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄⠀⠀⠀⠀⠀⣰⠟⢀⣿⣯⣾⣽⡧⠀⠀⠀⢀⠈⠀⠀⠀⢋⠀⠁⠀⠀⠀⠀⠀⠀⣤⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⢿⣯⢋⡼⠃⠀⠀⠀⠛⠃⠀⠴⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢠⡄⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠂⠐⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠨⡆⠀⠀⠀⠀⠀⠛⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⡀⠀⠀⠀⠀⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠀⠀⠑⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠠⡀⠀⠀⠃⠀⠀⠢⠀⠀⠦⠦⠄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠀⠀⠈⠁⠀⠊⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⣀⣲⡄⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠔⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠟⠀⠀⢀⠀⢀⠀⠀⢀⠀⢠⠄⠀⠀⡀⠐⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⠉⠈⠉⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠒⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⡿⣦⡀⣄⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡀⡀⠠⣄⡛⣴⢏⠀⠙⠛⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⡶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⡀⠰⢄⠐⢶⣄⢶⣄⠠⣀⢛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⠭⠀⣘⣇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⠻⠾⠿⠧⠴⢤⣰⣤⡀⠀⠀⠀⠀⠘⠷⠀⢰⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠹⢄⠀⠀⠀⠉⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠼⠃⠀⡀⠖⠀⡀⠀⣄⠀⠈⣷⣄⠀⠐⢀⠀⠐⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠆⠸⠢⣧⣱⠿⣤⣏⣀⠟⠀⠀⣾⡿⠛⣉⠉⠑⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠄⢐⣿⡿⠶⢤⡉⢠⡆⠀⡰⣽⡛⣹⣤⣴⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠄⢀⣬⠹⣦⠨⣿⠎⢀⠄⢀⡿⠃⢿⣩⠭⠭⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡼⢄⣲⠝⠁⣀⠄⠁⣴⡟⠃⡟⣤⠄⡀⠄⠂⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢀⣀⡤⠞⣁⣠⡴⠛⠙⠿⣦⢾⠆⢻⡀⠐⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠓⠚⠋⣿⣇⢈⠄⠀⠀⠙⠿⣧⣀⠠⠄⠓⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠒⠒⠒⠤⠤⣄⣀
⠀⠀⠀⠀⠀⣤⢀⠴⠊⠉⠀⢀⠠⠔⠒⠉⠉⠉⠉⠉⠀⠈⠙⢦⡀⣀⠀⢀
⠀⠀⣔⠻⠁⠘⠁⣀⠠⠀⠀⠀⠀⢀⠠⠤⠒⠒⠋⠉⠀⣠⡤⠖⣋⢵⡴⠃
⠀⠀⢈⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⢉⡵⠞⣇⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⣲⠶⠓⠒⠒⠲⠤⣄⠀⠀⢀⣠⣤⣄⡀
⠀⢠⠏⠐⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⢀⣀⣠⠴⠞⠉⠀⠰⣹⡀⠀⠀⣠⠔⡶⢊⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⢦⣋⣤⠂⠈⠙⣆
⠀⡞⢠⠁⢠⠃⠀⠀⠀⢠⠀⠀⠒⠒⠚⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⠃⢠⡞⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⡝⣆⢉⡀⢸
⢠⡇⢸⠀⠈⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢹⢀⣿⢠⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⢀⡽⣼⡀⠀⣸
⠈⡇⠘⠆⡴⢁⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣸⢸⠺⢀⠧⣄⣀⣤⣀⡀⠈⠉⠒⠤⣀⠀⠉⠒⠦⣤⡴⢞⠅⣧⡤⠖⠁
⠀⣿⠀⢰⢡⡏⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⢀⠇⠈⢇⣸⠠⠀⢀⡄⠀⠉⠙⢦⡀⠀⠈⠳⡀⠀⠀⠀⠐⠋⡴⡇
⠀⠸⣇⠀⠈⠣⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⢇⡞⠀⠀⠀⢿⢀⠄⠀⠁⠀⠀⠀⠀⠓⢸⡍⢣⠁⠀⠀⢀⠀⡰⢣⡇
⠀⠀⠹⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣀⠴⣫⠋⠀⠀⠀⠀⢸⡘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠶⠎⠀⠀⠀⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⢬⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣡⠞⢁⡀⠀⠀⠀⠀⠀⢳⡤⠤⠞⠁⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠤⣄⡀⠀⣠⣤⠤⠶⠚⠉⠀⠀⠙⠃⠀⠀⠀⢀⣄⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠘⡎⠀⢒⣈⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢽⡀⠾⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠀⠈⠉⠓⡆⠀⡠⠒⠲⠿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⠀⠉⠢⡀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⠀⠀⠀⠀⡠⠂⠑⠒⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢄⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠈⡄⢇⡀⠀⠀⠀⠈⠛⠿⠃⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠂⠀⠀⠀⠀⠀⢠⠀⠈⠑⠒⠤⢤⠖⢦⣤⠤⠖⠋⠹⠀⠀⠀⠀⠀⢸⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢏⣢⠀⢀⡠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠈⠉⡩⡅⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡤⡆⠀⠀⣠⣴⢺⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠊⠀⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣈⠙⠒⠛⠂⠛⠋⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠙⢓⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣊⣑⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠭⠃⡔⢚⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⢋⣿⣿⣿⣿⣿⣿⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠐⢰⢶⣏⣰⣷⣿⢿⡇⠀⠀⠀⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣦⣘⣻⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣟⣿⣿⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣯⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⢿⣿⣿⣿⣿⡻⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡿⢿⢿⣯⣿⣇⣛⣶⡏⠂⠈⢻⣿⣿⣿⣿⣿⣿⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣃⣴⣿⡦⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣮⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢉⣡⡤⠉⠉⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠛⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠞⠋⠀⠀⠀⠈⠛⠢⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢻⣷⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⡘⠀⠁⠐⠇⡀⠀⢹⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡁⠈⠈⠀⠃⠀⠠⠇⡾⠃
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠞⡀⠀⢃⠀⠀⠠⠊⣼⠁⡇
⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠃⠀⠀⠀⠀⠀⢸⠂⠀⡞⠂⠤⠈⠘⠀⠀⠀⠀⡠⢞⢿⢠⠇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢈⣀⣠⠴⠏⠂⢠⡾⠋
⠀⠀⠀⠀⠸⣷⣆⣀⣀⣀⣀⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⣛⣿⣿⣿⣷⣤⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠈⠪⣒⣦⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠫⠟⢻⣉⣁⡿⢶⡲⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢛⣿⢹⠀⠈⡄⠀⠈⠛⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠈⠉⠀⠀⠀⠷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠃
⠀⠀⠀⠀⠀⠠⠠⠀⠤⠶⠚⠉⠀⠀⠀⠀⠀⣀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡛⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡭⣿⡄⠀⠀⠀⣦⣿⣢⣴⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠛⢻⣦⣶⣮⡹⠿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣢⠹⡟⢆⠻⠧⣴⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠙⠛⢲⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢢⡄⣇⣠⣴⣷⣴⣅⣏⠀⣛⣥⠴⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣟⣳⠦⡻⡿⠟⣦⣻⣿⣿⣿⡽⢟⣛⡋⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢼⣇⠘⢳⣰⣽⣿⣿⡟⠋⣕⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣷⢒⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣶⣄⡩⡟⠻⡿⢇⡸⣝⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠚⠿⣷⣷⣧⣰⣎⠳⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠠⢺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠃⠀⢀⣵⣿⣿⣆⠀⠐⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠐⡲⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣿⣿⣴⡇⠀⡀⠀⢣⠀⣀⣶⡿
⠀⠀⠀⠀⠀⠀⢀⠏⡠⢀⣀⣼⠀⣀⠀⠀⠀⠀⠁⠲⣄⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠘⠰⣿⣿⣿⣿⡴⢒⡇⠀⢸⣿⣿⡿⠳
⠀⠀⠀⠀⠀⢀⠂⠰⣤⢼⣯⣿⣞⣿⣏⡲⣦⣤⡶⡀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⢄⡁⠹⣿⣿⣿⣿⠛⣁⠀⢸⣿⠋⡀
⠀⠀⠀⠀⠀⠃⢠⠰⣯⡠⠬⢥⣽⢏⠿⣯⣿⠿⡇⢀⠀⣿⡄⠀⠀⠀⠀⢻⣿⣶⣤⣼⣷⣤⣀⣠⣄⣤⣽⣿⣿⣤⣧⣴⡿⠉⠙⠁
⠀⠀⠀⠀⠀⡀⠀⠈⠙⢷⠒⣚⣩⣅⡶⢿⣿⡇⢀⠇⢰⣿⡇⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠱⣀⠀⠙⠻⠁⠙⢧⠞⠛⠂⠉⠁⣀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠈⠉⠉⠉⠛⠿⠿⣿⢸⠁⢀⣦⣄⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠄⢀⣀⣤⣶⣾⠟⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⠀⠀⠈⠉⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⣶⣿⣿⣿⣿⣿⣿⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠰⠤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢆⠀⠀⡀⠀⠀⢀⣮⣴⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡀⢸⠀⢀⣿⣱⠄⢼⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣧⡾⠀⠸⣿⣿⠂⣾⣿⣷⠟⡿⠁⠀⠀⠀⠀⣴⠀⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠒⢤⠂⠿⠻⠜⠛⠋⠑⠈⡁⢄⠀⠀⢰⠀⣫⣭⠍⠁
⠀⠀⠀⠀⠀⠸⡶⢞⡓⣦⠋⠀⠀⢀⡟⣰⢀⣤⣤⠈⣤⣄⠈⠂⠑⡄⢻⣭⡄
⠀⠀⠀⠀⠀⠀⢳⣀⠘⡏⠀⢠⣾⠿⢁⡿⠘⣟⢦⣴⣦⠈⣷⠀⣾⣿⣤⠜⠂
⠀⠀⠀⠀⠀⠀⠈⢻⣷⣡⣠⣿⠃⢠⢸⣦⠘⢿⣾⠇⢻⠆⠛⣰⢿⠚⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡀⢠⣘⡿⠋⣤⣤⣾⣿⣥⠀⠀⠙⣿⠏⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⢿⣾⣿⣷⣿⣿⠛⠉⠋⣥⡶⢒⣻⣯⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢾⠉⠀⠉⠉⠉⠀⠀⢠⣟⢣⣶⡮⠁⣠⠟⠁⡿⠃⠤⠖⠂
⠀⠀⠀⠀⠀⠀⢀⣀⣈⢄⣀⡀⣸⡇⣤⢀⡀⢠⣴⣆⠈⢿⠘⠋⠅⠰⣿⣤⡤⣱⣦⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠉⠡⡀⠀⠅⠙⢿⣇⣿⣸⣏⣹⣛⣇⢙⣶⣷⠖⠀⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠻⣬⡟⣫⠄⣿⡿⢿⣿⣉⣿⣿⣄⣸⠿⠋⠀⢠⡴⠚⢛⣻⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣿⣿⣿⣷⠻⣶⣾⣿⢂⣸⣿⣿⣿⣷⡄⢰⣯⣾⣿⣿⣿⣸⠇
⠀⠀⠀⠀⠀⣤⣶⣶⣿⣯⣙⣿⣿⢿⣿⣧⠎⠹⠟⠻⢄⢋⠉⠈⢿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⡛⣿⠿⠛⡁⢶⡶⠆⠘⠟⢻⣇⠈⣿⣿⣿⣿⣿⣟⡛⢶⡄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡛⠃⢠⡿⠁⣿⣴⣛⢿⡀⠀⠉⠀⠉⠛⢿⣿⣿⣿⣿⣦⣕⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠋⣷⠀⣾⠃⠀⡈⢻⣿⣿⣷⢼⣿⣆⠀⠀⡤⠙⠿⣿⣿⣿⠏⣡⣆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀⠀⢿⣆⣹⠀⠀⢳⠞⠋⣁⣤⠾⠛⠋⣠⡄⠀⢰⣀⣴⣴⡟⢴⣯⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢙⠿⢿⡇⠀⠈⣿⣿⣧⡀⢠⣦⠞⠉⠀⠀⣠⣾⣿⡇⠀⢾⠿⢿⣌⡛⠦⠄⠉⠁⢀⠴⠚⠛⠛⠓⠃
⠀⠀⠀⢀⣀⠘⣷⡜⣷⡄⣿⡄⠀⠻⣿⣿⣿⣾⣿⣶⣶⣾⣿⣿⡿⠃⠀⣠⣾⣿⣿⣿⣿⣶⣶⣿⣠⣧⠆⠀⠀⣠⠎
⠀⠀⠀⠉⠛⣿⣿⢿⠿⠿⠟⢻⣶⣄⣈⣽⣿⡿⣿⡿⠿⠛⠉⠉⠁⠀⣀⣈⢻⣿⠿⢿⣿⣿⠟⠉⠁⠈⠉⠓⠄
⠀⠀⠀⢀⣾⣻⢟⡻⠏⠀⠀⠸⣿⣿⡿⠟⣉⢷⣄⠀⠀⠀⢀⣤⡄⢰⣿⠿⠷⠤⣤⣤⡀⠁⠐⠀⠀⠈⠂⠀⠀⢂
⠀⠀⠀⠈⠉⠙⠛⠓⠛⢀⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⡿⠛⠛⣩⡟⠁⢠⠴⠚⠛⠟⢃⣠⣄⠀⠘⠒⠀⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⡤⠠⢴⡏⣰⢿⣯⠹⣍⢳⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⡾⣿⣿⣿⣿⡟⣩⣿⣿⣏⣹⣅⠀⠙⣿⣿⡇⠀⠀⠺⣿⣿⡟⢿⣃⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠛⠋⠉⡉⠠⢻⣿⣿⣿⡏⢨⣿⡷⢾⣿⣿⠂⣀⡠⡘⣷⣤⣴⣦⣄⡻⢟⣋⠁⣂⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣢⡷⢛⡽⣇⠸⣿⣷⠾⢉⠉⣿⣬⡿⢷⠈⠙⠛⠛⠛⠁⢉⣹⣾⡉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⢸⡇⠘⢧⣿⠏⠀⠀⠸⣿⣿⠷⣞⢷⡒⠀⠀⠀⠐⠏⠹⠋⠀⠒⢶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣡⣿⣧⠀⣸⣗⣄⣹⣆⠙⣎⠀⠀⠀⠐⢄⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⢹⣟⠧⠎⠙⢧⠈⠳⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⢣⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠸⡁⢿⡿⠀⣈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠘⠃⠀⡇⠀⡁⠀⠀⠀⠀⢸⡏⠴⡟⠃⠀⠃⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣷⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣇⠀⢗⠀⠀⠀⣠⠏⠀⠀⠁⣠⡖⠀⠀⠀⣀
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⢀⣘⡷⡤⢦⡀⠀⠀⢿⡃⣤⣿⣛⣭
⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⡿⠛⠉⠙⢿⡷⠼⣶⠀⢰⣿⠛⠀⠉⠛
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡀⢰⠣⢭⣤⠀⠀⠀⠀⠀⠀⠉⠘⣏⠀⠉⠛⠶⣦⣈⡙⣿⣟⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠪⢥⡀⠀⠀⠛⠐⣄⠺⢿⣦⣄⡀⠀⠀⠈⠑⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠁⢔⠈⠃⠀⡏⢡⡄⣹⢿⡿⣿⣶⠿⠿⠛⠛⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠖⡇⣼⠧⠔⠉⣠⠏⠀⠣⣄⡀⡠⠞⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠴⠃⠈⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢴⣄⠋⠀⠀⠀⠀⠀⠁⢀⣿⣡⣤⣤⡀⠀⡄⠠⠔⠊⠉⢀⠆⢈⣏⡄⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⠀⠀⠀⢺⣿⠋⠁⠀⠀⠉⠢⡀⣆⡀⠀⠀⠈⣀⣼⠃⣰⣿⣯⠇
⠄⠠⣄⡀⠀⠀⠀⠸⣇⠀⠀⠀⣀⣼⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢘⠀⡅⠂⠐⢠⡿⢹⣿⡟⠛⢁⣠
⣷⢾⡿⣿⡄⠀⠀⠀⢹⣄⣠⣟⣉⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⢀⣦⠐⣿⢀⡴⣡⣿⠀⢿⠙⡄⢿⣿
⠦⠤⠶⠛⠀⠀⠀⠀⢀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡽⠷⠾⣶⣤⠀⢀⡤⠒⠀⠱⢤⠀⠀⣁⣹⡼⠟⠸⣻⣿⡿⢀⠸⠀⢸⠈⠙
⠀⠀⠀⠀⢠⠆⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣈⣶⣾⣾⣗⡢⣍⠱⠆⢤⣔⠠⣶⠀⠠⣀⣠⠴⠃⠀⠀⠙⠋⠀⠀⠑⢶⡟
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠉⠛⠂⠀⠉⠫⠙⠋⠛⡻⠟⢇⣤⠮⣹⣆⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⡔⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢴⣾⡾⡿⣧⡉⣎⢦⣹
⠀⠀⠀⠀⠀⠀⡀⠰⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡃⠀⣴⠞⠀⠀⠀⠀⠀⠀⡾⢲⣿⠃⠘⣿⣗⠏⠈⠉
⠀⠀⠀⠚⠓⠉⢁⡙⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⠃⠀⠀⠀⠀⠀⠀⠀⢆⢘⡏⠀⠠⠈⠻
⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠄⠀⢠⠀⡤⣶⣶⣮⡴⠂⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠆
⠀⠀⠀⠀⣀⡀⠀⠈⠳⡄⠁⠀⡀⣤⠀⢳⣿⠟⡷⡝⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⢠⡾⠏⠀⠀⠂⠈⠀⢧⢀⠸⣯⡄⢚⣹⠗⠋⠘⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠨⢹
⠀⠀⠚⠀⠈⠀⠟⠁⠀⠀⠀⠈⡄⠀⠿⣩⡕⠚⠆⠀⡴⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⢗⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠀⠀⠀⠀⠀⠀⠿⣶⣤⠋⢁⡴⠃⠀⠀⠀⣠⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣤⡀⠀⠀⠀⠀⢈⣷⠞⠉⠀⢆⠀⢠⠉⣀⣴⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠈⠹⠦⠆⠁⠁⠙⠛⠛⣻⣤⣤⣱⠾⣿⣋⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⢓⡄⠐⢤⣂⣼⡎⢳⣄⣀
⠀⠀⣠⣮⠿⠶⠿⠿⠦⢀⠂⣠⣀⠦⣄⡀⠀⠀⠀⢜⣭⣶⣿⠋⣸⣿⡟⠿⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⢀⠙⢷⡂⢉⡇⢐⣬⣾⣿⣆
⠀⠘⠛⣥⣖⣿⣿⣵⣶⣼⢀⣨⣛⣷⡉⠱⡄⠀⠀⠙⠋⠉⡇⠰⠁⢸⢰⣶⣶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠛⡄⠀⠀⠈⢳⣶⣷⠘⣷⣊⣀⢀⣿⣫
⠀⠀⠘⠛⠛⠋⠉⠋⢋⣠⣽⣷⣮⠁⣎⢲⣷⠀⠀⠀⠀⠀⢣⣀⣠⠃⢸⠟⢿⣷⣈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢠⠃⠀⠀⠀⠀⢬⡘⢧⡈⠋⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡛⣿⡎⠉⢿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠳⣄⡙⠻⣿⡇⠀⠀⠀⠀⢀⠆⠀⠀⠹⡾⠀⠀⢀⣠⢴⣊⣦⡙⠻⠿⠶⠿⣿⣏

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄⠢⡀
⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⡴⠴⠀⠂⡀⠀⣼⠀⠀⠀⠀⠈⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠚⠀⠐⢂⠀⢀⠀⠀⠀⠄⠰⠀⠀⠑⠀⡠⠀⠀⠀⠀⠠⠀⢀⠀⠖⠀⠎⠀⡠⠀⠀⣄
⠀⠐⢐⠀⠀⡀⢀⠀⠀⠄⠀⡀⠀⠈⠀⠤⠀⠁⠀⠄⠐⠀⠁⠐⠇⠊⠀⢅⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠂⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠑⢀⠀⠘⠃⡄⠀⠀⠄⠢⠀⠀⠀⠀⢀⡄⠀⠀⣀⣤⢀⠀⢀⠘
⠀⠠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⢀⡀⠐⠀⠀⡁⢀⣀⡄⠤⠀⠀⠀⠀⠁⠀⠀⠀⢀⢀
⠀⠁⠀⠀⠀⠄⠄⢀⠀⡎⠀⠠⠀⠀⠗⠤⠤⠊⠉⠀⠈⠑⠁⠐⠿⡧⠀⠀⠀⠉⢈⣁⣀⠀⠀⡠⠊⠀⠀⠀⠰⠀⠲⡇⠐
⠀⠀⠤⡀⠀⠀⡀⠀⠀⢀⠀⠀⠁⠀⠄⠀⢀⠐⠀⠔⠀⠀⠄⠀⠀⠀⢠⢰⠀⣴⣿⣿⣟⣷⡀⢀⡀⠀⠈⠀⠀⡀⠀⠐
⠀⠁⠈⡤⠀⠀⠮⠀⠃⠀⠀⠀⡄⢠⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠀⠻⣿⣿⣿⡿⠁⠠⢅⠀⠀⠀⠄⠀⡀
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠓⠀⠀⠄⡘⠀⠀⠀⢀⠡⠄⠐⢁
⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠰⠀⣀⠔⠂⠤⠤⠊⠉⠀⠀⠄
⠀⡃⠄⠀⠀⢀⠄⠊⠌⢀⡟⢀⠟⠛⠉⢹⡇⠀⠀⠀⠄⠁⠠⠀⠂⠀⠀⠈⠐⠂⠁⠀⠀⣐⡄⠀⠐⠂⠀⠂⠀⠘⢆⠉⣀
⠀⡌⠃⠀⠀⠀⠀⠀⠱⠈⢷⣤⣤⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠊⠀⠎
⠈⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠦⠀⡀⠀⠀⠀⠄⠠⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⠃⠂⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠂⠠⠈⠀⢠⢠⠀⣈⣀⣀
⠀⠀⠀⡀⠈⠀⠀⠀⠒⠆⢀⠀⠙⠀⠂⠀⠀⠀⠁⠁⠀⠀⠀⠀⠔⢀⢀⠀⢈⣠⣶⠄⠀⠀⠀⠀⠀⠀⠈⠁⣼⡟⠛⢻⣷
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣻⣏⣤⢸⣠⡦⢰⡖⢠⠰⡄⠻⣿⣶⣿⠟
⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⡆⠀⠀⠐⠀⢀⠀⠀⣀⠈⠀⡐⠀⠀⠀⠀⠄⢸⠿⢯⡜⢣⡿⢠⠟⠀⠄⠐⠀⠀⠋
⠀⣀⠁⠤⠤⡀⠀⠐⠆⠀⠀⠀⡇⠸⣾⠇⠀⠀⠙⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⠀⠀⡀⠀⠐⠀⠈⠀⠀⠀⠃
⠀⠔⠀⠀⠀⠣⠈⠀⣀⠀⠀⠀⢃⠀⠈⠀⠀⢀⢽⠀⡆⡀⠀⠀⠐⣿⠗⠀⠀⠀⠀⠃⠀⠀⠠⠊⠀⠀⠃⠈⠀⠀⠀⠀⠂
⠀⢀⡀⣀⠀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠰⠃⢀⠉⠈⠁⢉⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⢀⠀⠀⠀⠀⠠
⠀⠈⣡⢉⡌⠃⠀⠈⠀⠠⠄⠀⠔⠐⠀⠀⠀⠀⠀⣦⠄⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠁⢀⣀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠘⣟⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠫⠟⠀⠀⢴⠾⠓⠀⠠⠄⢸⡇⠀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠛⠃⡕⣧⠀⡇⠀⠀⠘⢿⣿⣫⣭⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⢻⣟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠂⡀⠈⠛⠋⣠⣼⣯⠻⠏⠊⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠏⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⣐⠈⠙⠛⠛⠀⠀⢀⣴⣶⣶⣤⣀⣂⠀⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢈⣀⡤⠀⠲⠿⣟⠉⢻⣿⣿⣿⡀⠘⢿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣋⣝⠀⢴⣷⣾⣿⡷⠀⣿⣿⣿⣧⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠂⠈⠋⢛⣫⣤⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠰⠂⠀⠾⠛⠛⠛⠉⠉⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⠀⠀⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣄⡀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⢹⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⣄⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢰⠸⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣀⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⠁⢀⣱⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣄⢀⣴⣾⣿⣿⣶⣄
⠀⣠⣶⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡏⣶
⠀⠀⠀⠀⢠⣤⣤⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⠞⠷
⠀⠀⠀⠀⠀⠐⢻⣿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠂⠉⠈⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠁⠀⢀⠀⠤⠤⣴⣚⣃⡒⠀⠀⠀⠀⠀⡀⣤⣶⡾⣿⣟⣙⠳⣀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠃⠀⡺⠟⠒⠒⠒⠛⢿⣉⠀⠢⡄⠀⠐⢋⡜⠩⠀⠃⢰⠈⠆⠑⣌⢳⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢧⡀⣀⡢⢂⠎⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⠃⠀⢀⡞⠠⠀⠀⡀⢸⠀⠠⠀⠈⡆⢷⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⡎⠀⠀⠀⠀⠀⠀⠀⡰⠄⣄⠀⠱⠀⠀⠃⠘⠀⠀⠁⠀⠀⠀⠀⠀⠂⢸⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡁⠀⠀⠀⣠⣾⡃⠜⠁⠀⠈⢷⠀⢃⠰⡇⠀⠤⠄⠀⠀⠠⠄⠀⠀⠀⡞⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⣀⢀⠀⠀⠇⢸⠐⠷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠔⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣤⣀⠒⠛⠃⠀⠀⠉⠀⢀⠜⢀⠆⠀⠀⠈⠐⢠⣤⣀⣤⡄⠖⠁⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡸⠟⠁⠀⠀⠀⠀⢀⠠⠴⡿⠦⢄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⡖⠀⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠆⠙⠲⣄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⠉⡄⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⣤⠀⠱⠀⠀⠀⡈⢢⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⡇⠀⠱⡀⣸⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡘⠁⠀⢰⠀⠀⠀⠁⠀⠀⡹⢻⠀⠀⠀⠀⠀⠀⡔⠋⠙⢦⠘⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠁⣠⠶⠚⠤⠀⠀⠰⠴⠊⠀⢸⠀⠀⠀⠀⠀⠀⠱⠤⡤⠚⠀⠀⡅⠈⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⡠⠃⠀⠉⡄⣀⠴⠁⠀⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣹⡄⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⠏⠀⠀⢀⠔⠁⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠁⠀⢨⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⣏⣀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇⠀⢸⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡦⠀⠀⠃⠀⢸⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠉⠉⢹⠋⠉⠁⠀⠀⠀⢸⠀⠀⢸⠀⠇⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⢠⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⠀⠀⠀⠀⠀⣼⠤⡀⠀⠀⠀⢀⠁⠀⠀⡇⠀⠸⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠀⠀⡇⠀⢅⠀⠀⠀⠘⢲⡖⠶⠃⠀⠐⠖⢰⡎⡁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠐⠁⣇⠀⣦⡀⠀⠀⠸⠱⣄⣀⣥⣤⣄⣉⣈⣉⣠⣤⣤⣤⣀⠁⠀⠀⠲⠄⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⡀