Bengal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡖⢲⣤⣤⣀⣀⡀⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⡶⣀⣤⢤⣴⣤⣄⣠⣾⣫⣽⡧⠤⣾⡉⠙⠛⠛⠛⢷⣤⣌⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣢⡞⣡⣞⣽⣿⣋⡴⡛⣿⣉⣛⣻⠟⠉⣀⣶⣾⣿⣿⡂⠀⣠⠞⠁⠀⡉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣼⣿⣿⣿⣿⣿⣿⣫⣾⣿⠘⠿⠋⣁⣴⣯⠿⣩⣯⠶⢮⡭⠺⡀⡤⣶⣾⢁⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢄⡆⣠⡴⠖⠚⣿⠋⠻⠶⠦⠤⣀⠀⢾⣿⠈⢏
⠀⠀⠀⠀⠀⠀⠀⡈⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⣿⠆⢐⡞⣻⣿⣿⣟⣟⡂⠀⠀⣹⡁⣀⠙⠛⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⣷⠇⡇⣼⣿⣿⣿⣿⣿⣿⣿⡟⣹⣿⣯⣄⠉⣨⣿⣿⣻⡅⢨⣹⣴⠤⡟⢃⠘⣷
⠀⠀⠀⠀⠀⠀⢸⣿⢠⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣬⢀⣻⠉⡟⠛⡧⢀⣥⣶⣾⣿⣼⡆⠘⣧
⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡿⣿⣿⡽⣿⣿⡇⠀⠠⢁⣿⡩⠿⢋⡤⠈⢿⠀⣿⠀⠁⠈
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣶⣿⡏⠁⠀⠈⣿⣿⣑⠚⠋⠠⠶⣿⠀⣿⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⡄⠻⣯⣹⣿⣿⠅⢠⡄⠀⠿⡏⠉⠳⣄⢠⠴⣫⣾⡏⠀⠰⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡹⣿⡹⣿⡇⢻⣿⣿⣿⠀⠉⠛⣿⣿⡤⠿⠷⢀⡤⡦⠞⠁⠈⣠⡞⠛⠹⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⡇⣿⡇⢸⣟⣿⣫⣭⣄⠈⠛⠛⠛⠻⠟⠁⠀⠐⣠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⢹⣧⠸⣿⣇⣭⣿⢿⣷⡜⣃⠐⠚⠛⠲⠒⢲⡿⣿⣽⡋⠡⠃⠀⠀⢀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣹⣿⣿⡿⣿⣧⢸⣧⣿⡇⡋⢿⣶⣿⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣜⢿⣿⣿⡇⠻⣿⣦⣿⣼⣿⣥⠀⠙⣿⣿⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣬⣍⣿⣿⣿⡇⠀⠙⢿⣿⣿⣿⢿⣧⣄⡠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⠋
⠀⠀⠀⠀⠀⠀⠀⢲⣤⣬⣿⣿⣿⠁⠀⠀⠈⢻⣿⣿⢁⡛⣿⡉⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡟⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣌⠉⠉⢙⢻⡏⠀⠀⠀⠀⠈⣿⡿⠛⠋⠀⣿⣿⡦⠀⠀⠘⠈⠀⠶⢾⡿⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⣾⡅⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠛⠿⠃⠀⠀⠀⠀⢠⣷⣾⠧⠒⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠈⠹⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠻⠗⠒⢀⡤⠜⠁
⠀⠀⠀⠀⠀⠀⢰⣷⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠾⠿⣿⡷⣾⣏⠷⠀⠀⠀⠀⠀⠈⠆⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠿
⣼⣿⣦⣀⣠⣶⣦⡄⠻⠁⣇⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠻⣿⣿⣿⡿⠟⠀⠈⢩⣬⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠀⠀⠀⣰⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀
⠀⠈⠉⢿⣿⣿⣶⢀⠈⠀⠙⠹⣷⣾⠀⠀⠀⠀⠀⠀⢠⡄⠃⠀⠀⠀⠘⢀⡡⡤⢀⣤⣤⠔
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠚⠓⠔⠛⠛⠂⠀⠀⣉⡰⠁⠁⠈⠀⠀⠀⠀⠎⠿⠀⠁⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⣸⠡⠶⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠔⠺⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠈⠳⡜⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠈⢧⡀⠀⠈⠀⠙⢦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⣣⡴⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢇⣆⢻⣦⡀⠀⠀⠀⢻⣄⠁⣀⣀⡤⠤⠤⠀⣀⠈⡀⡀⣰⠋⣼⣿⣁⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⠸⣦⡙⢿⣦⣤⣄⠈⢿⠛⠉⢀⡴⠀⠀⣦⡀⠈⠉⠉⠋⣼⣿⣿⡷⠆⢀⢀⣿
⠀⠀⠀⠀⠀⠀⠸⡄⢹⣿⣿⣏⠻⠈⠀⠀⡀⠀⠘⣴⠀⢰⡈⠃⠀⠀⠀⠈⠩⠿⠿⢃⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣮⡛⠓⠀⠀⠀⠀⠀⣀⠀⠀⣿⣄⣿⠃⠀⣤⡀⣀⡀⠀⠀⠀⠈⠈⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⠀⠀⠀⢀⣉⠳⣿⠀⠀⣿⣿⣿⠀⠀⠸⣋⣩⣉⠁⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠀⡴⠛⠻⣿⢆⠀⢸⢹⣿⠿⡄⠀⡴⠛⣿⡍⠻⠂⣤⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⣿⡀⢧⡀⠿⠟⢸⡂⠀⢸⣿⠀⡁⢀⣇⠈⠛⣃⠀⢀⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⡟⠦⢉⡀⠀⢸⡇⡆⠈⠀⠀⡄⢸⡃⠈⠀⠡⠔⠋⣼⣇⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠒⠛⢦⡀⠀⣴⠟⠃⣧⡀⠀⣀⣁⠀⠉⠂⣀⠀⠴⠋⠉⠉⢣⠃⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⣀⣀⠀⡁⠀⠰⣇⡀⠀⣤⡽⠀⠤⠄⢌⠠⠀⠀⢈⣴⠋⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠒⠈⠉⢀⠀⠈⢷⣲⣤⠄⠰⠀⠈⠃⠖⠋⠀⠀⠦⢠⣄⣀⡰⠖⠫⣀⠈⠑⠀⣿⡄
⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⠤⠂⣹⠀⠉⠑⠀⢀⠤⠒⠛⠒⠶⠤⠤⠔⠈⠀⢀⠀⢄⠀⠁⢠⠀⡟⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠊⢸⠀⠀⢠⠀⠈⠑⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠱⡀⠑⣴⣸⡟⠀⢸⡹⠝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡴⠋⠋⠀⠀⠀⠃⡆⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⠚⠹⡀⡀⣀⠀⠀⠀⠀⠀⠠⠎⠈⠀⠀⠀⠀⢠⡇⡄⠀⠀⢸⡇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠃⠀⠀⣾⠃⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠋⠀⢀⡀⣼⠏⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠑⢤⡀⢀⢷⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⢠⣸⡿⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠙⠿⠆⠙⠛⠿⣦⠷⠦⠄⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⣀⡤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠤⢤⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⢤⣤⣤⠤⡤⢤⡤⢄⡀⢀⠇⢄⢢⠴⡌⠈⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠋⢇⠎⡰⠁⢴⢃⠊⢀⡋⠀⠙⠀⠈⠀⢃⡬⢤⠀⢀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡏⠟⣽⢠⠟⢰⢁⢆⡏⢨⢰⢋⢀⢰⠀⠀⠀⠀⢠⠢⡅⣆⠀⠯⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⢦⡄⣿⠘⡀⠈⢠⠀⠃⢎⢋⠘⡌⠀⡄⠠⠀⣾⠀⠀⠀⣙⣢⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡈⠁⠈⡇⡇⠧⡁⢧⠲⠀⠂⢸⠀⡾⣳⢠⠁⢀⠃⠸⡄⠀⠄⣼⠏⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠀⠀⠀⢈⢻⢦⠳⡈⢦⠀⠀⠎⡄⢇⠏⠘⡄⡀⢀⠀⠉⠘⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⢀⣐⡓⠷⢕⡌⠠⢒⠄⠘⣎⢦⡂⠃⠡⠀⠂⠀⢀⠀⠔⢁⡀
⠀⠀⠀⠀⠀⠀⡰⠓⠀⠀⠀⠀⠀⢮⢨⡙⢓⠆⣈⣉⡍⠀⠀⠈⠉⠃⠀⠀⠈⠛⠉⠈⠀⡠⠋⠁
⠀⠀⠀⠀⢀⡄⠋⠀⠀⠀⠀⠀⠠⠬⠑⠨⠁⠀⠰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣁
⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡯⠀⣈⣻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣞⣁⠀⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠉⢉⣩⣿⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣿⣍⡉⠉⠻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠛⠛⢋⣠⣶⡿⠟⠛⠉⢁⣤⣶⣿⣿⣿⠿⠿⣿⣿⣷⣶⣤⣀⠉⠙⠻⢷⣶⣄⠙⠛⠛⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣴⣿⠿⠟⢀⠀⣠⠞⠋⣉⣥⣀⣀⣌⣃⢘⣣⣀⣀⣬⣉⠙⠻⣆⠀⡀⠻⠿⣿⣦⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⡅⠀⠀⣾⠀⠃⠠⠞⠛⠛⠛⠿⠿⠛⠻⠿⠿⠛⠛⠛⠳⠄⠘⠀⣿⡀⠀⢨⣝⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⢸⡏⠀⠀⠀⣠⡶⢀⡇⠀⠀⠀⠀⠀⠀⢸⡀⢶⣤⠀⠀⠀⢹⡇⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⣿⠁⠀⢀⡀⠈⠃⣴⠿⠻⠋⠀⣼⡥⠊⣠⢂⡘⣄⠑⢮⣧⠀⠙⠟⠿⣦⠘⠁⢀⡀⠀⢈⣿⣎⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣰⣿⣟⡀⠀⣁⡠⠤⣀⠀⣿⣴⣟⣡⠞⠳⣌⣻⣦⣿⠀⣀⠤⢄⣈⡀⠀⣻⣿⣆⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣾⣿⣿⢁⡈⡻⣿⣿⣷⣄⢹⠙⠉⠀⠀⠀⠀⠈⠋⡟⣠⣾⣿⣿⢟⢁⡈⣿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⡀⠀⣿⡇⣇⠸⣆⠈⠺⢭⣙⠿⠀⢠⠀⠀⠀⠀⠀⢰⠀⠀⠿⣋⡭⠕⠃⣰⠇⣸⢸⣿⠃⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⣀⡀⢿⠃⢻⡄⠻⣷⣄⠀⠈⠃⢀⣎⠀⠀⠀⠀⠀⠀⢧⠀⠐⠁⠀⣠⣾⠟⢠⡟⠘⡿⢀⣀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣻⣧⠀⠸⣿⡄⡀⠈⠻⠆⠈⠉⢁⡀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠸⣇⡆⢀⡈⠉⠁⠴⠟⠁⢀⢠⣿⠇⠀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢳⡌⣿⡜⢄⣄⠀⠀⣰⡏⠀⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢹⣦⠀⠀⣠⡠⢢⣿⢡⡞⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠇⡿⢻⣎⠙⠛⡀⢿⠁⠀⢺⣀⣈⣉⡒⠒⠒⢒⣉⣁⣸⡇⠀⠈⡿⢀⠛⠋⢰⡟⢿⠸⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠘⣦⣀⠀⠀⣦⢹⡀⣾⡁⠘⠆⠀⢸⡿⠿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠰⠃⢈⣷⢀⡏⣤⠀⠀⢀⣴⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡆⠛⠀⠇⠻⣝⡓⠒⠀⠀⠀⢀⢿⣿⣿⣿⣿⡿⡀⠀⠀⠀⠒⢚⣫⡟⠸⠀⠛⢰⣤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡶⡄⠀⠀⢯⣙⠛⠛⢉⡀⠀⠁⠉⣻⣟⠉⠊⠀⢀⡉⠛⠛⣋⣹⠀⠀⢠⢶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠈⢻⣭⣉⣭⣤⣤⣴⣾⡿⢿⣷⣦⣤⣤⣭⣉⣭⡟⠁⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⣦⠈⠉⢿⣿⠟⠉⠘⠋⠁⠈⠙⠁⠉⠻⣿⡿⠋⠁⣰⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⣻⢇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠇⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣴⣿⠋⣧⣶⣀⠀⠀⣀⢶⣼⠙⣿⣦⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠈⠃⠹⣾⣷⠏⠘⠁⠀⡾⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢠⠔⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣄
⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠹⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠖⣩⡶⠛⢳⣼
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣘⣮⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣴⡟⢡⡾⠋⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣽⡙⠧⢈⣿⣍⡶⡶⣿⢿⣿⣿⣿⠿⣿⠿⣿⡶⢞⣽⢏⣴⡯⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⡀⠘⣇⠠⣼⠏⠻⡏⡇⣿⡄⢿⣿⠟⢀⣿⡄⣿⠛⠋⣋⡈⢁⠀⢀⡠⠊⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⡷⢤⣽⠀⠉⠀⠀⡁⡁⢸⣿⢠⡅⣶⣿⠏⡾⠏⠀⠈⠛⠿⣯⣄⠘⡵⠞⣛⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠐⠛⠂⡠⠒⣀⣄⠲⣼⡁⠘⡇⣾⣧⣽⠁⢠⣷⠋⣀⣄⣀⡀⠘⠿⠛⣰⡟⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢠⠎⠀⣾⣭⢭⣧⡙⠁⠀⠁⢻⠟⠀⠀⠘⢃⣠⣭⣭⣭⣑⠀⠀⠀⠙⠣⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⢏⡤⡄⢸⡄⢸⣿⠿⡆⠘⠀⠀⠀⠀⠂⢠⣿⠻⡿⠓⣹⢉⣤⣀⠀⢀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠟⢀⣹⣦⣙⠲⠤⠶⡇⢀⣤⣶⣄⠀⠀⣾⠟⠶⠶⢚⣡⣞⣀⣹⣷⡈⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠿⠯⣥⡈⠁⠀⣰⠅⢸⣿⣿⣿⣧⠀⢻⣀⠀⠉⠉⣁⡤⠶⢮⡛⡧⣹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠳⠶⠖⠂⠐⠋⠀⠀⢫⡉⠉⣉⡻⠀⠀⠈⢉⠂⠀⠴⠶⠶⣼⠀⣹⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⠀⠀⡀⢀⣐⣃⣀⡁⠀⢦⡤⠊⠀⠀⢀⢀⠰⣧⣀⠀⠀⠴⠒⠚⠉⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠚⡟⣧⣉⡳⠔⡚⢿⣟⢉⣁⠤⢀⣠⠷⣄⡀⠀⠀⠒⢶⣤⣿⠟⡳⣄⣤⡼⠿⠃⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡷⣿⣍⠔⠈⠀⡠⠟⠛⠖⠚⠋⠁⢤⡀⠉⠓⠲⠶⠛⠋⠳⡄⠈⠢⡉⠢⣀⣰⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡡⢏⠓⢦⣴⣊⠀⠀⠀⠀⠐⠶⠦⣤⣄⡴⠖⠒⠂⠀⠀⠀⠈⢂⣀⣼⢶⠋⠑⣴⠈⢿
⠀⠀⠀⠀⠀⠀⢀⣿⡀⠸⡄⠃⠈⠉⠙⠓⠲⠶⢶⣶⠖⠒⠒⠒⠂⢐⣦⠤⠵⠶⠶⠛⠫⠀⢀⣷⡾⠋⠉⢸⡀
⠀⠀⠀⠀⠀⠀⣸⠉⠡⠈⢻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣋⠑⠃⣴⣿
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢉⣛⠿⢿⣯⣤⣤⣄⣠⠀⣄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢁⣴⢀⡇⢹⡀
⠀⠀⠀⠀⠀⠀⠿⣿⣇⠀⠠⣀⠻⠿⠶⠶⠶⠶⠛⠁⠠⣽⣿⣿⣿⡶⣤⣤⣤⣶⡶⠛⠉⠀⠴⠛⠁⠞⣰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠘⠳⣦⣴⢦⣄⡀⠀⠀⠀⠈⠙⠋⠙⠛⠷⠶⠶⠒⠒⠊⠀⠀⠀⠀⠀⣰⡟⠁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠉⣶⡈⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠀⣄⣠⡾⠋⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢶⣿⠻⢦⣄⠀⠀⠀⠐⠶⣶⣶⣶⣶⠿⠛⣉⡠⠴⠚⠛⢉⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀⣤⠴⠾⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠃⢶⠀⢞⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠹⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⠀⢀⡄⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣉⣉⠀⣀⣀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣧⠀⠀⠀⣠⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠈⠰⠆⠚⠁⠀⠀⠀⠀⠀⠧⠄⠀⠢⠀⠚⠉⠐⠀⡤⢙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠈⡀⠀⠀⠈⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⢀⠀⢰⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠸⠈⣉⠀⢰⠀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢐⡀⠁⠐⠒⠂⠀⢈⡠⠂⠐⠙
⠀⠀⠀⠀⠀⠀⣄⡒⢠⡈⠉⢀⣀⡀⢁⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠡⡆⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢀⠉⡑⢠⠉⠤⣉⡐⢀⡀
⠀⠀⠀⠀⠀⠸⠃⠀⠈⣫⡠⠾⡅⡵⠦⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠐⠠⠠⠤⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⡤⠤⢀⠾⠀⠀⡩⣈⡄
⠀⠀⠀⠀⠀⣁⠲⢠⠖⠈⠗⠀⠁⠀⢰⣠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠈⠀⢈⡐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠠⢀⠀⠉⠀⠀⢠
⠀⣤⠶⠒⣒⡿⠁⠈⠈⡄⠀⠀⢰⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠑⠈⠀⢤⠤⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠀⠀⠏⠘⠓⠠⡀⢀⣨⠄
⣴⣿⠘⠋⠉⠐⠀⠀⠀⠡⠀⠀⠈⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢶⠉⠒⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⣰⣧⠀⢠⣄⠀⢨⣧⡀⢠⣁
⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⠒⠂⢈⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠛⠒⠘⠛⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠃⠐⠤⢉⠀⢀⢘⠘⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠾⣱⣙⣀⠐⡀⠉⠀⠘⠈⠀⠆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠁⠐⠂⠀⠠

⠀⠀⠀⠀⢀⣴⡶⠶⠄
⠀⠀⠀⠀⡼⡟⠁
⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠱⢿⣦
⠀⠀⠀⠀⠀⠸⡉⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠈⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⠋⣻⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⣽⢦⠀⢀⣀⣀⣤⡤⡤⢄⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⡓⡴⠓⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣯⡋⣩⢷⢰⣆⢻⣿⠚⣽⠋⣛⡻⣻⡖⠶⢶⡶⣶⡾⢿⡺⢓⠈⢒⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⢣⣳⠈⠫⠠⣰⣄⠙⠂⣾⣹⡋⠚⡁⣻⡿⡇⢘⢁⡛⣜⢦⠳⡘⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠀⡄⣠⡝⢂⣀⠚⣇⢉⣦⠉⣓⠁⠃⣿⣗⡇⠟⠀⡗⠘⢦⡡⠳⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠀⢙⡳⣝⡫⠿⠌⠋⢀⠁⣄⣽⣿⡟⠈⡄⠃⠟⣰⣼⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠾⣷⠀⢠⣀⣙⡾⠣⠄⠈⠀⠀⠈⠅⠟⣿⣧⣠⠶⠦⢶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡲⠟⠀⢶⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⠚⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠈⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⢯⠁⢤⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⢀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡀⠀⠀⡌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠗⠀⠑⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠂⢠⠆⠅⠍⠀⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⡮⠛⡌⡒⢾⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⢠⡀⠈⢢⡬
⠀⣆⠀⢀⠠⠄⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠰⠇⠿⠐⠁⠁
⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠬⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣋⣐⢒⠩⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣊⢦⠯⣕⡀⠄⠀⠀⠠⢆⡀⡀
⠀⠀⢰⠄⠀⢀⢐⠰⢮⣥⡦⢈⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⢀⣩⠠⢷⢦⣇⢊⠺⢿⡞⢋⡜⠁
⠐⠶⡋⠒⣾⠏⠁⠑⢾⠌⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠉⢧⣀⡤⠈⠓⠉
⠀⠀⠈⠓⠙⠀⢲⢦⡏⠀⠀⠀⣄⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠋⠼
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢨⠌⣥⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢡⡿⠋⠝⢤⣡⡤⣍⡛⢛⠢⢤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⢸⡶⣷⡶⡙⡥⣃⢺
⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠐⠧⠻⠈⠁⠀⠀⠐⢢⢄⠀⠠⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢠⡴⢂⣦⠐⠺⠻⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⣀⠐⠀⠀⠀⠀⠀⠀⠀⡺⢾⠆⢠⣆⠦⣄
⠀⠚⠲⡐⠚⢿⡄⣀⠴⡗⣀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠠⣧⢄⠀⣰⠿⠖⡱⠛⠂
⠀⠀⠀⠋⠀⠀⠀⠅⠿⣝⠺⠐⡋⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠶⠆⢔⡵⢳⡮⡅⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠉⠬⢕⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠠⢘⢰⠒⣩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢳⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢎⡬⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢩⠝⣍⠁
⣚⡗⢒⠦⣴⠊⠁⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠸⣠⡿⠟⠗⣤⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡾⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡧⣽⣿⠻⡇⢸⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣀⢾⡀⡁⣿⡜⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠂⡠⠈⡠⠃⢿⡇⣸⢟⡾⢻⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣼⣣⠀⠀⠀⠈⠞⣡⢊⠵⣿⠏⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⠉⠳⣄⠀⡀⠀⢄⡴⠛⣡⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⢟⣿⣇⠀⡈⠠⡀⠐⢃⢠⣾⣿⣿⣿⣿⣧
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⣿⡿⠃⣿⠛⠋⢼⢋⠇⣿⣿⣿⣦⡙⣆⠁⢠⣯⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠟⠁⣤⣤⣄⣠⣄⣤⣤⣤⣤⣶⢲⣾⣿⣿⡿⠛⠇⣹⣼⣽⣦⣿⣦⣴⣦⣔⣦⣄⠀⣼⣿⣿⣟⢿⡍⠹⣆⡿⣧⡇⣟⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠈⠃⠁⠛⠛⠀⠈⠈⠀⠁⠘⠉⠋⠀⠀⠂⠺⠿⠛⠛⠛⠋⠛⠛⠛⠛⠋⠈⣿⣿⣿⠛⠇⠑⠂⠾⠇⠠⣧⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠾⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠄⠎⡊⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠿⠟⠃⠀⠀⠀⠀⠈⠙⠛⢻⠋
⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⡄⢈⠀⡥⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠀⠀⠀⠀⠀⠀⠉⠒⠓⠁
⠀⠀⠀⠀⠀⢣⠀⠄⠀⡀⠠⠀⠘⣀⠂⠁⠰⠀⠈⠃
⠀⠀⠀⠀⢀⠨⣐⢄⠀⠈⢦⠀⠀⠈⠠⡼⢧⡀⠈
⠀⠀⠘⠀⠀⠀⠈⠳⢅⡠⠀⠀⡀⣀⠔⠙⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢈⡐⠉⠀⡈⠉⠉⠂⡤⠀⠀⠠⡑⡄⢆⠆
⠀⠀⠷⠄⠀⠀⠀⠑⡀⠠⡀⠈⠠⠀⠄⠁⠔⠂⠐⠉⠆⠐⠰⣧⠘⢎⡇⢸⡇⢠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠁⠄⠈⠂⢄⠀⠀⠀⠁⢠⠃⠀⡐⢀⣀⠀⢀⠀⠀⡠⢀⡤⠀⠨⠛⠀⠃⠀⠋⢠⠀⠁⠘⠀⠂⠀⡄⠀⣤⠀⡀⠀⡀⢀⠀⠀⠈⠿
⠀⠀⠀⡆⠀⠀⠀⠀⠐⡤⠴⠤⠠⠌⠰⠋⠀⠀⠨⠀⠑⠐⠛⠂⠊⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠐⠀⠃⠂⠀⠛⠓⠀⠘⠂
⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠


⠀⠀⠀⠀⠀⠀⠘⠒⠔⠒⠀⠀⠀⠀⠀⠐⠦⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⢔⠖⠂⣲⣶⢶⡶⣶⣶⣤⣤⣤⣤⣀⣤⣤⠶⣢⣶⣿⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⠟⢔⢝⡵⣀⠾⡫⠫⢟⣴⣿⣯⣴⠍⣩⣾⠿⣫⣵⠟⣋⢤⣾⠕⣂⣉⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⠋⠈⢆⢎⡞⣽⢋⡞⡵⡱⢋⢊⣾⢿⢏⣾⢹⣡⠊⢐⠵⠟⣁⢎⡁⢈⠀⣶⣻⣗⣂⣤⢀⡐⢉⠙
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣾⡇⡀⡸⠘⠀⠈⣴⡞⡼⡐⢀⡆⡜⡜⡜⢨⡅⠜⠃⠀⠁⠴⠛⠉⣾⣷⠔⢀⡤⡧⢾⠾⠩⡢⡉⢤⠈
⠀⠀⠀⠀⠀⠀⠀⣼⢸⠘⡇⠃⡇⢰⠈⢸⡟⢸⣿⠁⣼⡇⠀⠇⠁⣾⣆⠀⢀⡆⢀⠀⠀⢀⠉⡠⢚⡇⡀⡙⢻⣟⡑⠀⢻⢨⡀
⠀⠀⠀⠀⠀⠀⣀⠋⢹⠀⡇⠀⡇⢸⠀⢸⡇⢸⣧⡶⣻⢱⠆⠀⣰⡟⣽⠀⠸⠀⡆⠀⠀⠀⡾⢡⡬⣧⣻⠃⢿⠀⠿⠀⢋⢶⣹⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠶⢸⢆⠀⢧⠈⢸⠀⢸⡗⣾⢹⠀⡏⢸⠀⠀⣿⡇⠻⠆⠀⠀⡇⠀⠀⢠⡇⠸⠸⣄⠦⣷⠄⠀⠀⣗⠞⠙⡛
⠀⠀⠀⠀⠀⠀⠷⢶⡞⡟⣆⢲⠀⢇⠀⠸⡇⣿⢺⢠⡿⣾⡇⢂⢸⠡⠀⠀⡇⠀⠱⡀⠀⠘⠷⡞⣑⣊⡃⠉⠀⠀⠀⠈⠀⠚⠣
⠀⠀⠀⠀⠀⠀⣶⣾⡇⠸⣜⢎⢣⠘⡀⠀⠉⠸⣌⢸⡅⡟⡀⣸⠈⣇⠀⠀⠹⡄⠣⡱⡌⠀⠀⠈⡓⠈⢚⣤⠠⡀⢀⠄⡔
⠀⠀⠀⠀⠀⠀⣀⣸⡃⠀⢯⠻⣷⣳⡐⢄⡙⠆⢘⠏⣷⠀⣧⢹⣤⠈⠀⠀⠀⠄⠀⠐⢄⠠⠄⢘⢷⡏⠘⠫⠅⠼⡀⠈⢰⣟⠢⡀
⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⢁⢻⡈⢧⡱⣦⣈⠃⢈⡰⣜⣷⢘⣇⠻⡇⠀⠀⠀⠈⠢⡀⠀⠑⠆⠠⠊⠤⠂⠀⠀⠀⠀⠀⠘⣿⡀⡀
⠀⠀⠀⠀⠀⠐⠛⠿⠀⠀⠴⠀⣾⣎⣻⡢⣫⠃⠀⠀⠈⠐⠾⠿⡗⠰⢦⢠⠇⠀⣀⣑⡢⣄⠀⠋⣀⡤⠀⠀⠀⠀⠀⠀⠀⠈⢧⠐
⠀⠀⠀⠀⠀⢈⣹⡷⠀⢀⢶⣤⡘⢦⣭⣙⡛⠃⠀⠀⠀⠀⠀⠀⠙⠿⠷⣀⡄⠈⠁⠈⠳⠀⠴⠋⠁
⠀⠀⠀⠀⠀⠤⢤⠀⠐⢃⢨⠳⠝⠶⣶⣮⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⣦⠄⢀⠘⢌⠻⠛⠋⠱⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣼⢷⠁⢠
⢀⣀⡀⠠⣬⡍⠀⢸⡯⠑⠒⠀⢈⠙⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠆⣰⢋⡞⠁⠰⢂⠀⠄⠀⠄⠊⠀⠀⢀⣶⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡅⠀⡆⠀⡄⡜⠀⠄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣡⣄⠘⠒⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠐⡀⢸⠃⠀⠁⠀⡜⠀⠁⠀⠀⠃⠈⠀⠀⠀⠀⠀⡨⢩⢭⡉⠀⢄⠻⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠘⠀⡆⠀⠜⡀⢀⡇⡅⠀⠀⠀⠀⠀⠀⣶⠄⡘⠰⠈⠤⢘⣆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢃⠀⡀⢇⠘⡀⠀⠑⢠⢰⢀⢸⠀⡇⠀⠀⠀⠀⠀⠰⠓⠳⠀⠀⠼⠛⢛⠜⡎⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢶⠀⠀⠀⠀⠈⣄⠁⠀⠀⢸⠀⣸⡸⠀⣇⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠠⣰⢉⣟⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡆⠰⠂⡀⢦⠘⡆⢣⡓⠘⢆⢹⡀⠀⣿⢀⠀⠀⠀⠀⢀⠀⠐⢴⠆⠀⢠⡖⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠀⠰⠒⠀⠷⡑⢀⠩⡐⢦⣢⡿⢦⣿⣘⣧⡀⠀⠀⠈⠀⠐⣌⠐⠐⠛⠐⠋⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠀⠀⢞⣳⣄⠀⠙⢷⣝⠢⣿⣢⣼⡛⠙⣿⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠀⠈⠀⠙⢙⡲⣤⣉⣿⣿⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠀⠀⠀⢠⡾⣶⣦⣉⣉⣹⣿⣭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠂
⠀⠀⣀⠀⠀⠀⢴⣷⣦⠀⠀⠀⠀⠀⠈⣤⠄⠉⠛⠻⣤⣿⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣿⣟⡀⢴⣷⡈⠛⠁⠀⠀⠀⠀⠀⡄⠉⣑⣢⠄⣼⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠘⠇⠿⠇⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠤⢴⣿⣧⠤⠀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠨⠉⢿⡂⠄⠂⢉⣝⣿⣿⢟⣿⣻⣿⣿⣯⡉⠑⠢⢖⡿⠉⠄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣶⠾⠍⠀⠶⠚⣫⡴⠾⢻⣿⡟⠷⢶⣝⠓⠖⠀⠙⠶⣶⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠏⠈⢡⡶⠒⣩⠵⣞⣋⣿⠀⢿⣝⣓⡮⣍⠓⢶⣄⠁⠹⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⣱⡏⠀⣡⡛⢁⠾⡇⠘⢸⠷⡈⢙⣌⠀⠹⣎⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠰⢿⢠⣼⡍⠀⡟⠐⠁⠀⠈⠃⢻⠀⢨⣧⡄⡿⠇⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣾⡆⣿⣤⣤⣤⡈⢿⡆⠀⠀⠀⢰⡿⢁⠤⣤⡤⣿⢰⣷⢻⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢃⣼⣏⣿⠠⡈⠓⣍⠛⡷⢸⣿⠀⠀⠀⢿⡇⢼⠃⣩⠊⣁⠆⣿⣸⣧⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⣿⡟⢺⠀⢙⣋⠀⠀⠷⣸⠃⠀⠀⠀⠈⣇⢺⠀⠀⢙⡋⠀⡟⢻⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢺⣷⣧⣵⢖⡐⠀⡌⠀⠀⠀⠀⠀⠠⠀⢢⡒⣦⣼⣶⡟⣾⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣾⣟⣢⣯⡞⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠳⣾⣔⣻⣿⣾⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⣿⣿⣭⣽⠻⡅⣀⠀⠲⣄⠀⠽⢀⣠⠞⢀⣀⢨⠟⣭⣭⣿⣿⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⣿⠿⣿⢡⣤⣉⣀⠉⠛⠤⠛⠉⣀⣩⣤⣌⣿⠿⣿⡿⣝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠑⠓⠀⠀⠁⠻⣄⡐⠀⣠⣶⣄⡀⣒⣤⠟⠈⠀⠀⠒⠋⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠳⣦⣄⠿⠛⠋⠉⠛⠉⠙⠛⠿⣠⣴⠟⠀⠀⠀⢰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣇⠀⠀⠀⠛⣿⣷⣤⣄⡀⠀⢀⣠⣴⣾⡿⠛⠀⠀⠀⣰⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠁⠀⠀⠀⠈⠻⢿⡿⣿⣿⣿⢿⡿⠟⠀⠀⠀⠀⠈⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠙⢒⡒⠒⠶⠶⠒⠒⠶⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⣙⠛⠲⠀⠀⢀⡀⠀⠀⠀⠈⠉⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⡿⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠓⠤⢠⡀⠠⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡇⠀⢸⡇⠀⠸⠃⠀⡀⠀⢰⡇⠀⠘⠀⠀⠀⠉⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢸⡄⠀⢸⡇⠀⠀⠀⢼⡇⠀⠈⠉⠛⢷⠁⠀⠠⠖⠢⠄⠰⣆⣄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠀⠈⢷⡀⠘⣧⠀⠀⠀⢻⡆⢸⠀⢰⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣇⠹⡇⠀⠘⣧⠀⠹⠀⠀⠀⠈⠧⠀⠁⠈⠀⠀⡄⠃⣯⠉⣹⠶⠾⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢿⡀⠁⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⢹⡄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣦⡀⠙⠯⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠘⢿⣆⠘⢷⣄⠀⠀⠰⣤⣀⣀⠀⠀⠀⢸⡿⢁⣾⡏⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠀⠙⠷⢦⠀⠀⠉⠉⠀⠀⠀⢸⡇⠸⠉⠿⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠹⣇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀⡄⠀⣶⡆⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠙⢦⡀⠀⠙⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⢸⡏⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡜⢷⡀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠞⠋⠀⣠⡶⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣷⠀⠻⣦⣀⣀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠇⠹⣧⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠆⠘⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢷⣼⣶⣷⠞⠁⠀⠘⠁⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡙⠋⠁⣠⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⣄⠀⠀⠹⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣤⠶⠖⠛⠂⠀⠀⢠⣿⣿⣿⡇
⣇⠀⡀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡈⠐⠂⠀⠹⣿⣿⣧⠀⡔⠀⠀⠀⠀⠙⠾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⡇
⠸⡄⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠙⣿⣿⣾⡇⠀⢠⠀⠀⠀⠀⠘⣿⣿⣇⣀⣀⣀⠀⠀⠀⠀⣸⣧⣶⣿⣿
⠀⢹⡀⣀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣀⣀⠈⢻⣿⣷⣄⡏⠀⠀⠀⠀⠀⠘⢿⣿⠉⠉⠀⠀⠀⠀⢠⣿⣿⣿⢿⡏
⠀⠀⢣⡈⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠈⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠐⠲⣾⣿⡄⠀⠀⠀⠀⢀⡟⢉⣩⣴⡿
⠀⠀⠈⢷⡀⣤⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⠄⠀⠀⠙⠿⣷⢀⡀⠀⠀⠀⠈⢻⣧⠀⠀⠀⢀⣾⠿⠟⠛⡙⠁
⠀⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⢿⡄⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠳⡄⠴⠛⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠙⢧⡘⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠘⣧⢀⣾⣿⣫⣍⣴
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⡟⠲⠦⣤⣤⣤⠶⣿⡟⠁⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡘⠀⠀⠰⡏⠀⠀⠀⠹⠷⠀⠀⣀⠴⠋⠁⡟⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣁⣀⣀⣀⣀⡤⠖⠋⠁⠀⢀⡞⠁⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆⠀⠀⡄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣤⣠⣿⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⣧⠀⢠⣆⠀⠀⢠⣆⠀⢠⡿⠋
⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠓⠛⠛⣦⣴⠟⠛⠛⠋⡀⠀⢀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⡒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠑⡄⣶⣽⠷⡲⣴⣶⣶⣶⣶⣏⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⣂⠀⢀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾⠷⠈⣵⣿⣿⠇⠌⠹⢏⡠⣿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠏⠀⢀⣤⣄⡀⠀⢉⣭⣶⡇⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠡⠀⠆⢸⡏⢡⢀⡀⠀⢀⠀⢉⠓⢲⣄⠀⣠⣴⡟⠿⠉⠉⠁⠰⠇⠸⣿⣿⣿⢂⣈⠉⣿⠁⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠎⢿⣿⡄⠀⠁⣠⢾⣒⢤⣬⣦⠤⣬⢸⣿⣷⠸⡿⣾⣃⡄⠀⠛⠁⠀⠁⠈⠙⢹⣿⣿⠀⣡⡄⠀⠀⠀⠙⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⣀⣀⡂⠉⠉⣡⣀⣀⡤⠞⠅⣿⡏⢸⣦⠃⠿⢻⣼⡇⢠⣶⣶⣦⡘⠃⠘⢿⡟⠀⢻⣿⢀⣾⢇⡶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠁⠈⠁⠴⢨⣁⠀⠄⣠⢊⣴⠋⠰⣋⣼⣰⠀⡸⡿⠀⠘⣿⣿⣿⠃⠀⠀⢀⣀⠀⠈⠛⣿⠯⢊⡴⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠞⠀⢀⣤⣾⡥⢖⡻⢒⣉⠤⠾⠛⠉⠉⢸⣧⠓⠀⠰⠆⠈⠛⠁⣀⠀⠀⣾⡿⠀⠀⠀⣥⠔⠋⠤⠞⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠿⢀⡯⢐⠱⠃⢀⣚⠛⢃⡀⠀⠙⢄⠀⢠⣤⡀⢰⣿⣿⣷⠀⠋⠀⠀⠀⠀⢿⣛⢿⡄⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠳⠿⢋⠔⠉⣀⡔⠀⠒⠲⠮⢛⣀⡀⢿⡇⠀⠙⠗⠀⠙⠛⠁⠀⠀⠀⠀⠀⢖⣼⢮⡳⠷⡞
⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣥⡤⣥⠬⣗⠒⠱⠌⠳⡈⠛⣦⡀⠀⠀⠀⠉⠁⠈⢎⡳⡀⢠⣄⠀⣠⣾⣆⣠⡊⡔⠀⠈⢀⡡⠷⠞⠁
⠀⠀⠀⠀⣤⡄⢀⠀⠀⠀⠄⠠⠄⢴⡿⠀⠀⣦⠀⢹⣄⠈⠉⠉⠛⢶⣤⠀⠀⣼⡿⠧⠐⠉⠉⠉⠀⠀⠈⠛⠳⠻⠛⠁
⠀⠀⠀⠀⠛⠛⠻⠷⠶⠦⠤⠤⠤⠦⠤⠵⠒⠚⠚⠒⠛⠛⠒⠒⠒⠒⠛⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡄⡤⠤⠤⡄⠀⡄⣀⠤⠀⠀⣄⠀⡄⠀⠀⠀⢀⡤⠀⠀⣤⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠟⣗⣉⠁⠍⢢⡿⣑⡸⡧⠾⠎⢣⡇⠀⠀⠀⣏⣀⡤⠴⠦⠳⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⡀⢀⡤⡠⠤⠴⠶⣿⠖⠲⠤⠤⡀⠀⡀⠀⢀⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⠻⡇⠉⠀⠀⠤⠄⠬⠿⢍⡁⠤⠀⠀⠀⣉⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⠀⠀⠐⠒⢋⣩⡭⢷⣿⡿⢭⣝⠓⠒⠀⠀⠐⣣⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠿⠁⢰⡞⠑⢋⣭⣴⠆⣷⠤⣦⣬⡑⠈⣷⠀⠈⣿⢀⡕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⢃⣤⠀⠀⠠⠖⢢⠶⠛⠶⣄⠲⠄⠀⢀⣄⠐⣄⢻⡄⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢡⡟⡄⡌⣛⠰⠆⠀⠀⣇⠀⠀⠀⡸⠀⠀⠴⢈⡛⡁⢶⢦⡙⡆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠳⡏⣽⠛⢇⣂⡄⠁⠀⠀⠀⠀⣀⡐⡺⠻⡇⠟⢸⣼⡇⢀⠰⡄⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣿⠀⠀⡅⠻⡀⠁⠙⠟⠀⠀⠀⠀⠀⠿⠁⠈⣠⠇⠀⠀⢹⣷⢸⠀⠐⡈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢽⣧⣶⡿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣷⣦⣾⠛⠘⠀⠀⢡⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠹⣿⣄⠲⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢀⣼⣿⠋⣠⠎⠀⠀⢸⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠠⠈⠉⠙⠻⢍⡀⠀⠶⣄⠀⢠⡴⠀⢂⣩⠿⠋⠋⠐⠈⠀⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡀⠀⠘⢷⣆⡀⠀⠀⣲⠁⠀⢀⣰⡞⢠⣤⠔⠀⠀⣠⡀⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠳⣄⠀⠈⢻⡶⠖⠛⠛⠛⠒⠶⠏⠀⠋⠁⠀⠀⢠⡿⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⢧⠀⠀⠳⠤⢀⢀⡠⠤⠀⢀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⣧⠀⠀⠀⠳⣄⠀⣠⡶⠋⠀⠀⡀⢀⡾⠃⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⣆⠀⠈⠳⣄⠀⠀⠈⠀⠁⠀⢀⡤⢊⡴⠋⠀⠀⠀⠀⢀⣞⣠⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄⠀⠀⠈⢣⠀⠀⠙⢦⡀⠀⠀⣴⢞⣩⠴⠋⠀⠀⠀⢀⡆⢀⠊⠛⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⢳⡄⠀⠀⠙⣦⠀⢠⠋⠀⠀⠀⠀⠀⣠⠟⠀⠄⢀⣩⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠢⠀⠀⠀⠳⣄⠀⠀⠸⠄⠀⠀⠀⠀⣀⣴⠞⠃⠀⡸⠀⠈⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡈⠁⠀⠀⠀⠀⠀⣶⠟⠋⠁⠀⠀⠈⠀⠀⠾⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣔⠠⣄⠀⠀⠁⠀⣀⡠⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠾⠿⣰⠏⠁⠀⠀⠀⠀⢀⡘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠄⡀⠰⡤⠖⢻⡵⠂⠀⠀⠀⠀⠀⠘⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⡄⢀⠅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠐⠠⠀⠀⠄⠀⠀⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠐⠿⠒⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣄⣸⡇⠀⣠⠀⠀⣤⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠚⠛⠛⠋⠑⠒⠋

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⣶⣤
⠀⠀⠀⠀⣰⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣼⣶⣾⡖⣼⣿
⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢠⠀⡂⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣶⡾⠋⠉⠛⠛⠟⠁⡈⠱⠄
⠀⠀⠀⠀⢸⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠁⠦⠀⡀⠀⢻⣷⠀⠀⢸⣶⡏⢤⠁⠀⣴⡞⠁⠀⠀⠀⠀⠀⢾⣿⣆⣀⢀⣀⠀⠀⣼⡇⣰
⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠆⣸⣿⠀⠀⣼⠀⡿⢸⡇⠀⣿⡇⠘⠊⠀⠀⠀⠀⠀⠿⠛⠉⠛⢻⣿⣤⣿⢁⠈
⠀⠀⠀⠀⠀⠈⠑⣾⡇⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣿⡿⠀⢸⡇⠀⠇⢺⠃⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣷⡌
⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣇⠀⠘⢧⡀⠀⡹⢄⠀⢘⡻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢹⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣠⠌⠁⢸⠇⣸⠁⠈⢳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠛⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⣾⡇⠀⢠⣿⠀⠀⡼⢀⣿⠀⠀⢰⣿⡏⠀⢠⡀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢸⣿⡇⠀⢘⠋⠀⠐⠀⠀⠉⠀⠀⣿⣿⠁⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠸⣿⣇⠀⣾⠤⠧⠘⠀⠀⣾⠆⢠⣿⣟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣠⡄⣤⣶⠀⣺⣿⣿⣿⣿⣿⣿⣏⠀⢿⡆⠂⠀⣦⣾⣿⡟⣸⣿⣿⣿⣿⣿⣶⣤⡄⠠⣤⡀⠀⣤
⠀⠀⠀⠀⠀⢠⡿⠋⣹⣟⣿⡁⠀⠹⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⠀⠀⢬⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⡙⣿⣦⣼⡇
⠀⠀⠀⠀⠐⠋⠀⠀⣿⣿⠋⢳⣴⣄⣙⣻⠿⢿⣿⣿⢿⡇⠘⠁⢀⡀⠸⠛⠃⢸⣿⣿⣿⣿⣿⣿⠿⣋⡀⢀⡿⠛⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⢻⣿⣿⣷⣕⣦⣠⣽⣛⣧⡀⠠⠀⣠⡀⠀⠐⣿⠃⢻⣍⣉⢭⠶⢾⣿⡷⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⠏⣹⣥⣤⣄⡀⢠⡄⢈⠀⠀⣴⣿⠿⠉⠶⢿⣿⣿⡀⣤⣄⣀⣶
⠀⠀⠀⠀⠀⣤⠀⣠⡄⠸⣶⣿⣿⠿⣿⡏⣽⡿⠋⠁⠀⣘⢋⠹⡍⢷⣄⣀⣨⠀⠀⠈⣿⣬⡀⠸⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣶⣿⣿⡀⠹⣿⣯⣀⡀⢠⣿⣤⣄⡀⠀⠉⣤⣀⣰⣳⣿⣿⡟⠀⠀⠀⣬⣽⣿⡶⣾⠿⠗⣀⣨⣿⠟
⠀⠀⠀⠀⠀⣿⣿⡏⠈⢷⣆⠈⠛⢿⣿⣦⣾⣥⡄⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⠀⢀⣬⣭⣅⠐⢢⣶⣾⠿⠛⠁⠀⢀⣶
⠀⠀⠀⠀⠀⣻⣿⠀⠀⠘⣿⣷⣾⣏⡈⠙⠛⢛⣿⠒⠦⠀⣤⣝⣻⣿⡉⢀⣀⣶⣶⣾⣿⡇⢠⣶⠿⠋⢀⣀⡀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣟⠀⠀⠈⣿⣿⣿⣿⣿⣶⣿⣯⣽⠛⠻⠿⠿⠿⠋⠉⠻⡶⠿⣿⣿⠟⠁⣰⣿⣿⣶⣾⡿⠋⠛⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣿⣿⡏⢻⡟⠉⠉⠁⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⣠⡌⠙⢿⠿⠿⢿⡿⣿⣿⣿⣿⣶⣶⣴⣶⣤⣤⣴⣾⣿⣿⣿⡟⢸⣧⣶⣠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢉⡴⠀⢹⣿⣦⡈⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠇⣾⡟⢿⣿⣽⡿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⣿⣿⣿⣦⠀⠀⠀⠠⠦⠛⠛⠿⠿⠿⠿⢻⣿⣿⣿⠉⠹⠘⠇⠙⠚⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢹⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣴⠆⠘⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣴⣶⣶⣬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠘⠄⢀⠀⢠⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡬⣁⡴⣢⣕⢒⡾⣍⢾⠟⠲⠤⣩⡾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣶⢁⢼⡏⢙⣄⣬⣷⡄⡀⣿⣃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢧⡘⠃⠀⠀⠁⠈⢙⣳⠍⡆⣿⡌
⠀⠀⠀⠀⠀⢠⠊⠉⢀⣴⣿⣿⠿⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⡋⠀⠀⠀⠀⠀⡄⢧⡼⢣⡟⠠⠇
⠀⠀⠀⠀⠀⢧⡀⠀⣾⣿⡟⠀⠀⠀⣀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣦⠲⣆⠀⡀⠊⠀⢠⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠳⣄⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣦⠀⠓⠤⠀⠰⢏⠀⠀⠠⠪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡀⢻⡦⡀⠀⠀⠀⠘⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⢸⡇⡆⢀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣶⡊⣿⡁⡟⢿⣟⠳⡜⣦⠙⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡏⢿⣇⢻⣇⠘⣧⠁⠈⢿⣧⣷⡘⠦⡀⠀⡗⢌⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠿⡿⠀⠀⠀⠀⠀⢀⠠⣾⠃⣿⡯⣷⠈⢿⡆⢻⡆⠘⢀⡇⢤⡻⣎⠹⣆⠘⡄⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣅⣀⠀⠠⠀⠀⠀⠀⡀⢾⣿⡘⣿⡀⢹⣇⠸⣧⠀⣿⡈⢇⢠⡿⢀⠈⡇⠘⣆⠹⡄⠃⢠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠻⣿⡗⠁⠀⠀⢀⣼⡇⠈⣿⡇⢻⡏⠘⣿⡆⠹⣇⠹⡇⠀⠀⠃⢷⣾⠿⠿⠶⢄⠙⡄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣤⡀⢀⠊⠀⠀⠀⣰⣇⠹⢿⡆⠘⢧⡈⠃⣶⡹⣷⡀⠻⡆⠉⠀⠀⠀⢸⠁⠀⠀⠀⢄⠁⢸⡆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠻⣿⠃⠀⠀⠀⣼⠻⢿⣇⠀⠛⢀⡀⠀⠀⢸⣷⠘⢷⡀⠀⠀⠀⠀⠀⡇⠤⠾⠷⠦⠈⠳⣬⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⢀⠃⠀⠀⢀⢾⣿⣄⠀⠙⠃⠀⠀⡀⠉⠑⠚⢿⣧⡀⠀⠀⠀⠀⠀⠀⣟⠒⣀⣀⡀⠀⢠⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⡟⠀⠀⠀⣾⣆⡈⠉⠛⠦⠀⠀⠀⠉⠛⢶⣤⣍⣿⠷⠄⠀⠀⠌⠀⠀⣸⣏⠉⠉⢁⡀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⡇⠀⠀⢸⡌⠙⠻⠶⠤⠀⠀⠀⠴⠶⠶⣤⣍⠻⢿⡧⠀⡠⠂⠀⠀⣰⠟⢿⠟⠛⠉⠶⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢇⠀⠀⣞⠻⣷⣦⣄⡀⠀⠀⠀⠲⠶⣶⣌⡙⠿⣿⠇⠊⠀⠀⠀⠔⠁⠀⠘⣷⣶⣶⠆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⣶⣶⣿⡀⠀⣿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⡀⠙⢿⣶⡞⠀⠀⢀⠄⠁⠀⠀⠀⠀⣯⣴⠶⠄⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡅⢀⣤⣷⣄⣸⣧⣍⡙⠓⠀⠀⠀⠈⠙⠛⢿⣷⣦⣝⣠⣴⠞⠁⠀⠀⠀⠀⠀⠀⢟⣀⣚⡛⠉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⠋⢠⣿⡿⠛⠿⠶⠀⠀⠀⠀⠘⣓⡶⠾⢿⡿⠿⢿⡿⠿⣿⣿⣿⣿⣶⡄⡾⠛⠛⠉⠀⠸⢿⣿⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢄⡈⠿⣧⣠⣤⡤⠀⠀⠀⠀⢾⣿⠁⢰⠟⠀⠄⠠⣤⣐⡖⢤⡈⠻⠟⠃⠘⢄⡀⠀⠀⠀⠄⡹⣖⢢⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⠤⠤⠐⠒⠒⠚⠛⠒⠚⠲⠶⠦⢤⣀⣀⠽⠀⠋⠀⠀⠀⠀⠀⠙⠶⠦⠤⠤⠼⠜⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡠⠤
⠶⢆⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠑⢒⣀⠀⡀
⠠⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠒⠉⠂⠒⠒⠀⢒⡒⠰⠦⠠⢤⣀⠀⠀⡀⡦⠫⠁⠈⠄⠈⠀⠉
⠀⠘⡷⢤⣀⣀⣀⣤⠀⡀⠀⠠⠅⠄⠀⡤⠂⢰⡐⠈⣀⢆⠘⠒⠀⢡⢘⢠⣆⠩⡅⢧⠓⠑⢄⢐⡮⡄
⠀⠀⠀⠙⠙⠋⠃⠈⠠⠐⡆⠐⠈⠀⠓⢰⡄⠆⠠⡄⠈⣀⢀⡖⡂⡈⠂⠤⢁⠀⢰⣏⣒⡞⠀⠈⠫⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢋⡠⡍⠀⣄⠰⠠⠄⡰⠆⠀⠁⣤⠀⢴⠂⢺⡑⠼⠸⠇⠀⡠⢤⡰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢦⡐⠆⠘⠆⠀⠀⠁⠣⠜⢁⠰⢀⠟⠀⠄⡀⠞⡀⠰⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⠳⣆⡿⠆⠁⠈⠀⠈⠉⠁⠀⠈⠃⠘⢸⣃⣀⡀⢸⡤⢐⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠳⢤⣳⢯⠥⠏⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠛⠁⠀⠉⣭⡽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡐⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡤⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣿⠟⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣤⣿
⠀⠀⠀⠀⠀⢀⡀⢠⣤⡤⠀⢰⣶⠉⠉⡩⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠰⣾⣏⠀⢸⣿⠧⠠⠼⣿⠷⠮⢡⣖⣽⡿⢋⣵⠟⣩⡶⠛⠉⣉⣤⣤⣤⣬⣶⣤⡀
⠀⣴⡇⠀⡹⠛⠉⠀⠀⠀⠀⣰⣡⢒⣈⢻⣿⡏⢠⡿⢫⣾⡟⠀⣴⡿⠛⠉⠁⢀⣠⣤⣬⣍⣢⡀
⢀⠻⣿⣿⠀⠀⠀⠀⠀⠀⣨⢙⡷⢋⣤⣿⣻⡇⣿⠀⣿⡟⢀⣾⡟⠀⣀⣤⣾⠟⠉⣀⣤⣤⣤⣙⣦⡀
⠰⡀⠀⣸⡄⠀⠀⠀⠀⣀⣿⣾⣡⡋⠉⢭⣿⣀⡟⣸⢿⠃⢸⣿⠀⣞⡵⠋⠕⣣⣾⡛⠋⠀⠀⠙⢷⣷⣄
⠀⠻⡿⠿⠋⢸⣶⡆⠹⣿⣿⣿⣿⣶⣿⢿⣿⣿⡇⣿⢸⣤⢸⣿⡖⠋⡙⠳⣿⣿⣭⣤⣶⣶⣶⣶⣤⠉⠙⢷
⠀⠀⠀⠀⠒⠚⠛⠁⠘⠉⠉⡏⠙⠀⠐⠀⠀⠘⣷⣹⣎⣿⡟⡽⣩⠟⠶⢟⡽⣻⠟⠉⠁⠀⠀⠈⠙⢷⣤⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⣤⡧⣌⠧⡼⠋⠛⠿⣿⣿⡇⢃⡴⢱⣏⣞⣡⠶⠟⠛⠻⢿⣷⡄⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣸⣷⢿⣟⢉⡤⠶⠶⠶⣤⠀⠿⠿⣶⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡏⣿⡩⠙⠒⢯⣟⠶⠼⣟⠳⣆⠘⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣿⠇⡉⠐⠁⢃⡚⠮⢭⡓⠎⠧⢽⡶⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⣩⣤⢩⢊⣌⠁⠁⠰⢆⠘⣠⠰⣬⠷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⣿⣴⠹⠇⠀⠁⢠⡆⠲⢠⠀⣷⣞⣉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣄⠃⢻⡗⣅⣡⡔⠒⣹⣆⣽⠆⡆⢛⡋⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡐⣸⡐⠛⠲⡽⢻⠾⣛⣩⠾⠃⣼⣶⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠷⣦⣴⠴⠡⠀⠤⢤⣠⣾⣯⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠛⠲⠾⣰⣾⠉⣿⣿⠏⣅⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⠉⠒⠐⡾⠛⢻⡞⡁⠫⠒⢦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⣴⣆⢠⡁⣀⠘⣼⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢠⠇⠀⡁⢀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⣌⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣇⢀⣰⢀⠻⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⣠⡶⠛⠛⠷⣤⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⡇⠘⣇⢰⡏⠀⠀⠀⠀⢸⡇⢸⡇⢰⡏⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⢠⡟⠁⠀⠀⠙⣷⠀⠀⠀⣿⠀⢿⡈⢷⡀⠀⠀⢀⣼⠃⣾⠀⣾⠁⠀⠀⣾⠋⠀⠀⠈⢻⡄
⠀⠀⠀⠸⣧⠀⠀⠀⢀⡿⠀⠀⠀⢹⡆⢸⣇⣀⣙⣛⣛⣋⣁⣰⡏⢠⡟⠀⠀⠀⢿⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠻⠶⠶⠛⠁⠀⠀⠀⠈⣷⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⠇⠀⠀⠀⠈⠛⠶⠶⠟⠁
⠀⢀⣀⣤⣴⠆⠀⠀⠶⣤⣤⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣤⠶⠀⠀⠰⣦⣤⣀⡀
⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⣿⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⢈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⡄⠀⠀⠀⠀⠀⠀⢠⡄
⢻⡆⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⣼⠇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠸⣧⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠻⣾⡇⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠙⢷⣿⠀⠀⠀⠀⠀⠀⢸⣷⠟
⠀⠀⢸⡇⠀⠀⢠⡀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⢠⡶⢶⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⡄⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡇⠀⠀⣸⠇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⡗⠀⠀⠀⠀⢻⡄⠀⠀⣿⣼⠁⠀⠀⣿⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⢸⡿⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢸⡆⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠸⠇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⡇
⠀⠀⠀⢸⣇⣀⣸⣇⣀⣼⠇⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠸⣧⣀⣸⣇⣀⣸⡇
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠠⠀⠀⠀⠀⠄⠀⢠⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠸⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠙⠱⠀⠀⠡⠗⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢇⡀⠒⡦⠂⣐⠊⠁⢀⠀⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠒⠓⢸⡟⡀⠀⠨⠁⡄⠀⠠⠀⡀⠁⢐⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠄⠘⠹⠿⠟⠉⠄⢀⠡⠃⠠⠎⠀⠀⠬⡄⠀⠀⠈⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⠂⠈⠥⠂⠀⡉⡂⠒⠄⡓⠖⠊⡀⡈⡖⢀⠀⠈⡠⠘⠁⡬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠐⠀⠀⠀⠀⠀⠄⠤⠨⠀⡌⡱⠐⡣⡂⠆⠁⠁⠀⢠⢐⠤⠔⡁⢵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢦⡀⠂⣤⡀⠠⠞⠃⠩⠉⠌⣑⡌⡀⡁⣿⠐⢁⠀⢠⡈⠰⡦⢃⡊⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⠛⠳⣾⠟⠃⢀⠠⡉⣀⢩⢊⠩⣞⣍⠋⠸⠀⢶⡰⢆⡵⣇⢼⠺⣀⢩
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠐⢠⠈⠧⠀⠂⢅⠰⣀⡧⠞⠋⠉⠙⠛⠋⠶⢀⡀⠀⠻⢯⠯⡁⣻⡱⣤⠓
⠀⠀⠀⠀⠀⠀⢀⢁⠂⡁⠙⠀⠰⠀⡀⣣⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠉⠀⠐⠻⠷⠶⣐⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡀⣺⢇⣀⡠⣤⡴⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⢀⠰⠞⠋⠁⠈⠛⠋⠉⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⡶⠶⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡗⢻⣟⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠄⠀⢀⣀⢀⠘⠀⠀⠀⠀⠀⢠⣿⣿⠍⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣿⠟⠃⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣥⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⣾⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣷⠀⠀⢀⣀⠐⢦⣤⣼⣿⣿⣿⣿⠷⡤⢾⣇⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⣶⣿⣶⣿⠿⠈⠀⠀⠀⠀⣰⣬⣭⣴⣶⡞⠁⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢉⡀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡏⣀⡀⠀⠀⠀⠀⣤⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡗⠙⠁⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠙⠀⠀⠀⢰⣿⣿⠿⠀⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠄⠀⣰⣦⢀⣴⣿⠟⠁⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⠼⠋⠀⠀⠀⠀⠀⠀⠀⡠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠻⣦⣶⣦⣴⣶⣶⣾⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⣋⡉⠉⠉⣾⠏⠀⠀⠀⠀⣸⣦⣶⣾⣿⡋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠃⠈⠉⢳⢀⣼⠋⠀⠰⠀⠀⣠⣿⣿⣿⣿⣿⡿⠓⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⠀⢀⡴⠋⣸⣇⣠⠾⠅⢀⣴⣿⣿⣿⠻⠿⠏⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠐⠒⠀⠀⠀⣠⣼⣿⣿⣿⣿⣦⣭⣤⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠛⠛⢻⣶⣿⣿⣿⣿⠟⠹⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡁⢸⡿⠿⠟⠋⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠉⠙⠛⠛⠛⠋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠋⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠘⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣻⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢽⣿⣿⣿⣽⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡪⠽⣿⣿⣿⣯⡔⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⠻⡿⣿⣽⠾⣋⣴⠆⡀⠀⠀⠀⢀⣤⣿⢡⡄⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢻⣷⣾⣯⣵⡾⣋⣠⡾⢀⣼⡣⢁⣾⠃⠀⠀⠀⣈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡿⣿⣿⣷⣿⣿⣯⣶⣿⣿⣥⡿⣯⣴⣶⣾⠷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⠿⠋⠙⢻⣿⣿⡯⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠙⠋⠿⠿⣟⣋⠁⠀⠀⠀⠈⢱⣬⣁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣶⣶⡾⢿⡏⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣻⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢬⣿⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣭⣽⡯⠅⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠃

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠻⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⣀⣴⣾⣶⣦⣀⠀⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡞⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⢱⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢿⣿⣿⣿⣿⣿⣿⡿⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣷⣶⡿⡶⣒⡀⠶⣔⣂⣐⡂⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢛⣭⣦⣶⣿⢫⣾⣿⣿⣝⣯⠛⢿⣶⣌⠱⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢰⡿⢻⠟⣩⡆⣾⡿⣍⢻⢿⣿⣷⠀⣿⣯⡄⢾⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⣴⣿⢻⡿⣾⡿⣽⣿⣷⢺⣿⣿⣿⣿⡇⣿⡗⣧⢸⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢀⢁⣿⠟⣡⡿⠀⡏⢠⡽⠟⠚⠛⠛⠯⣿⣿⣿⡟⢠⡏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⡄⣿⠁⢠⡇⠈⣁⠐⠊⠉⣀⠬⣯⣿⡿⢁⡾⣁⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣾⢿⠃⣿⣶⣞⠀⠀⣠⠖⣀⣠⣶⣿⣦⣝⢛⡋⣤⣷⣿⡿⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢈⣷⢻⣿⣶⣤⣈⣥⣴⣿⠟⢻⣿⣿⠟⠟⠁⣹⣿⣿⢁⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣼⣿⣻⢎⣽⠟⣉⣤⣤⣴⣿⡤⠀⢀⣼⣿⣿⣇⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿⣩⡿⠿⣋⣴⣿⢿⣿⣿⠏⢀⣴⡿⣿⣿⡟⣼⣿⢯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠉⣿⣷⣌⢉⠀⢁⣼⣿⣥⣶⣟⣫⡾⢏⣍⣱⣿⢟⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⠿⠷⠾⠋⡀⠙⣿⣿⣿⡿⠟⣽⠆⢿⣿⠿⠣⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣴⣴⣶⣾⣿⣿⠃⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⣿⣫⣋⡵⡞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⢀⣰⣿⣷⣦⠀⠘⠛⠳⣶⣾⠇⠀⣲⣶⣶⣶⣶⣷⣶⣷⡀⠀⢳⡴⠚⠃⠀⣾⣷⣶⣶⣦⠰⣿⣿⣿⣿⣿⣿⣿
⣠⣶⣿⣿⣿⣿⣿⣷⣶⠖⠳⢬⢻⠀⣰⣿⢿⢿⢿⣿⣟⣿⣿⡇⠀⠈⠴⣿⣿⡿⢿⣻⣷⠙⣿⡀⢽⣿⣿⣿⣿⣿⣿
⢿⣿⣿⡟⣿⣷⣇⠸⣧⠘⠀⢨⣧⡀⣿⣷⣿⣿⡿⠁⠈⠻⢿⡇⠀⠀⠁⢸⣿⠇⣼⣿⠄⠀⠹⡇⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣺⣻⣿⠿⣻⣷⣌⡛⠾⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠁⣼⣷⡾⠛⢁⣾⣿⠟⣀⠀⠀⣗⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢻⣿⣿⣽⣿⣿⣿⣷⣌⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⣿⣻⣤⣾⠟⠀⢰⠃⢠⣿⣿⣿⣿⣿⣿
⠀⠸⠃⠀⣻⡿⣿⣟⣿⣾⣭⣿⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡏⠉⢁⣄⠀⣄⣄⣀⢸⣿⣿⣿⣿⣿⣿
⢀⠀⣤⣬⣭⣽⣿⣿⣶⣿⣿⣿⣿⣿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⠻⢁⡼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⠿⣿⡏⠉⠷⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠘⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠁⠀⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠹⣷⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢴⣤⣤⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⠇⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣨⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠛⣻⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄⠈⠛⢿⣶⣤⠀⠉⢀⡤⠖⠁⡰⠾⢋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣭⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⡀⠀⠀⠙⢿⣦⣤⠟⠀⠀⣀⣀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⣈⣭⣉⣁⠀⠀⠀⠉⠀⠠⢤⣩⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠂⠀⢀⣤⣶⣿⣦⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡶⢶⣤⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣧⡀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⢷⣄⠀⣀⣀⡀⠉⠛⠛⠛⢋⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠿⠾⠶⠶⠾⠂⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⣦⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠹⣇⠀⠀⣠⣤⣤⣤⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢷⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠫⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⣀⣠⣄⠀⠀⠀⠸⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣴⠶⠛⠋⠁⠹⣦⠀⠀⠀⠹⠷⠾⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣷⠶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⠶⠿⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⣺⣿⣗⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣴⠟⢫⣶⣿⣿⣷⣮⣢
⠀⠀⠀⠀⠀⠀⢠⠣⣿⣿⣿⣿⣿⣦⣝⠻⣦⡀⢀⡤⠴⣶⡎⠉⢀⣿⣿⢸⡇⠀⠀⣼⣿⣿⡒⣤⣾⠟⢁⣜⠿⣿⣿⣿⣿⣿⡇⡆
⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣿⡇⠀⣿⣿⠀⣼⡇⣼⣿⣇⠀⣿⠟⠋⠉⣰⣿⢋⣾⣾⠁⣛⣭⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢃⠘⣿⣿⣿⣿⣿⣿⣿⡟⢂⣿⣄⢀⣿⡟⠀⣿⠇⣿⢺⣗⠀⣿⣆⣠⣴⣿⢧⣞⣿⠇⠀⠥⣶⣾⣿⣿⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠈⢆⢼⣿⣿⣿⣿⣛⡻⠗⢸⠛⣿⢺⣿⠇⠀⣿⡀⠙⢹⣿⠀⢹⣯⠿⢿⣿⣼⣻⣷⣀⠀⠉⠽⢿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⣿⣿⣿⣿⣿⡂⢰⣶⡟⠋⣺⡿⠀⢠⣿⣇⢰⣾⡇⠀⢸⣿⢧⣀⡀⡁⠛⢿⣿⣾⣿⢿⡿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⣿⣛⠙⠂⣘⠛⠀⢰⣿⡇⠀⢸⣿⣿⣿⡟⠀⠀⢘⣿⠈⡉⠙⠳⣤⠈⠩⣽⣥⠈⢈⣿⣿⡿⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⡆⢀⣝⠃⠀⣠⣿⣿⡇⢠⣼⣿⣿⣿⢁⡀⢰⣿⡿⠋⢷⣄⠀⠉⢞⠀⠩⠿⡆⢘⣿⣿⣳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡓⠓⣀⣴⠶⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⣿⣷⣧⠈⠻⣷⡀⢫⡦⠀⢘⣧⣭⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠋⠁⣾⣿⣿⣿⡇⢨⣿⣿⣿⡿⣿⡇⢠⣿⣿⣿⣦⣀⡿⢀⣼⠇⠀⠀⣾⣿⣿⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣸⣷⣾⣿⢿⣿⡇⠀⣿⠈⢻⣇⣿⠄⠀⣿⣿⣿⣿⣿⣶⣿⣿⣥⠀⢾⣿⣿⣿⣿⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣾⣿⣿⣷⣿⡄⢸⡆⢸⡧⣿⠀⠘⣟⣭⣤⣶⣬⣍⣙⣿⣿⣶⡿⠋⠻⠿⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣿⣿⠸⣿⡘⠿⠿⣿⣷⠀⣿⣾⣷⣿⠀⣼⣿⡿⣿⣷⢿⣿⡯⣿⣿⡟⠁⢀⣀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⢻⣷⣝⣿⣿⣿⣿⣿⢸⡏⢼⡏⣿⢠⣿⣿⣿⣦⣶⣾⡿⢃⣿⡟⢷⣿⣿⡿⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡺⣿⣯⠉⠑⣠⣙⣽⣿⠧⠘⠠⣿⣾⣿⣿⠉⠉⣩⣥⣾⣿⠟⣠⣾⣿⠟⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⢐⣿⣿⠟⢻⡟⠀⠀⣄⠘⣿⠿⣿⣟⠋⠉⣩⣯⣄⣰⣿⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⣛⠶⠀⠘⣷⡷⣶⡾⣿⡇⠀⣨⡽⠿⠿⡿⠟⢉⠩⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠉⠀⠀⠘⢷⣾⣯⡟⠀⠀⢒⣛⠋⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡐⠲⠿⠿⢿⡷⠶⠈⣡⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠀⠀⢀⠀⠈⡀⢄⠐
⠀⠀⠀⠀⠁⢀⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠪⣱⣶⠛⠁⠀⠀⠀⠀⠀⠀⠄⠀⠈⠐⠀⠀⠈⢆⠈⠂⠘⠀⠇⠁⢐⠆⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⡀⠀⠢⢁⠀⠀⠀⠀⠈⠈⢦⣆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣦⡘⢸⡀⡌⠀⢀⠀⡌⠀⠀⠈⠈⢶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠰⡄⠠⣤⡝⡻⠃⠀⢏⣀⠀⡀⠇⡀⠀⢴⠀⢈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡷⠀⢄⣽⣰⠋⢩⠅⠰⠁⠊⢹⣷⣠⢆⢀⣤⠜⠀⠈⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠩⠿⢋⡡⣶⣢⠀⠈⠻⣆⣀⣐⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠁⠞⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠁⠀⠤⠦⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣀⠀⠀⠀⣶⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⣛⣠⣿⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⡀⠀⠀⣀⢀⣴⣶⣿⡿⠄⠀⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⠿⠋⠁⠀⠀⠀⠲⣆⠁⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢰⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠉⢹⡏⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⡀⠀⠀⢠⣿⠃⠀⠀⡀⠀⠀⢀⡠⠎⠀⠀⠑⠄⠀⠁⠀⠀⠀⠀⠘⠁⠀⢸⡇⠠⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⠄⠼⢤⠦⠤⠤⣤⠄⢠⡀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⣿⠀⠀⣀⠠⠄⠉⢀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠷⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣵⣿⡶⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣷⡀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠘⣷⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡈⢻⣦⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣦⣀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⣾⠁⠀⠀⣀⣤⣶⣦⡴⠄⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣷⣶⣦⣤⣼⣿⡀⠀⠀⢸⡟⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠛⠻⣶⣦⣼⣷⡀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣦⣄⣠⡛⠳⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠓⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⢸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⠊⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡅⠀⠀⠀⢠⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠀⠀⢀⣠⣤⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢸⠇⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣼⡁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡹⠃⠀⢠⣿⣿⣿⣯⡙⠀⠙⢿⣿⣿⣿⣿⠃⠠⠀⠀⠀⠀⢀⡦⠰⣿⠆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣥⡴⠋⠁⠀⢀⣿⣿⣿⣿⣿⡇⠀⠐⠀⠉⠻⣿⠛⠓⠀⠀⠂⠀⣤⠎⠁⠂⣽⣶⡆
⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠈⠉⠉⠒⠉⠁⠀⠀⠀⠀⠈⢹⠇
⠀⠀⠀⠀⠈⢻⣦⡀⠠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠌⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢰⠗⢀⠀⢠⡄⠀⠀⠀⠀⠄⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠈⣧⣔⣀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⠈⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⢀⣴⣾⣿⣏⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡃⠈⠿⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡽⠛⢛⣿⡷⣿⣿⡟
⠀⠠⠤⣴⣤⣄⡀⠀⠀⠀⠀⣾⡟⠀⠇⠀⢰⡄⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣎⣹⣷⣿⠋
⠀⠀⠀⠀⠉⠻⣿⣦⠀⠀⣸⣿⠁⠐⠀⠀⠸⡇⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠆⢀⣻⡅⣼⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣠⣿⡇⠀⠀⠀⠀⠀⠻⣄⢀⣄⠙⠀⠋⠉⠁⠉⠈⠉⠑⠆⠉⠙⠂⠉⠉⠈⠉⠁⢠⣤⡄⠘⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⣀⣴⡆⠀⠀⠀⠀⠛⠈⠿⠓⣶⢴⡆⢠⡀⠀⠀⣀⠀⠀⢘⡀⠠⡧⠴⠆⠁⡸⠏⠃⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠙⠀⡤⢠⠀⠈⠁⡀⠉⠉⠉⠈⢉⠁⠀⠀⢠⡄⠞⠃⠀⠀⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠀⠂⠀⠃⠘⠀⠘⠃⠀⠀⠀⠀⠀⠠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠠⠈⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠰⠂⣶⡄⢴⠆⢠⡆⢰⡄⢰⡆⠲⠀⢰⡆⢲⣦⢰⡆⢰⣦⠰⠶⠀⠐⠆⢠⡦⢴⡆⠲⠀⠰⡆⢰⡦⠰⡆⢴⠀⡆⠀⢶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠇⠸⠀⠈⠀⠟⠀⠽⠂⠙⠀⠇⠈⠟⠀⠀⠸⠂⠀⠟⠀⠿⠀⠿⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀⢀⣄⠀⣠⣀⣀⡀⢀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠻⣏⠙⠛⣿⠋⠈⣿⠛⠛⣿⠙⡯⢉⣿⡿⢻⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣶⣿⣶⣷⣿⣷⣶⣾⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣏⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⣽⣿⣿⡿⣹⣽⣾⣿⣿⣿⣟⡿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣶⣿⠇
⠀⠀⠠⠤⡀⠀⣀⢀⣀⣤⣴⡶⡶⡾⣠⣰⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠙⢻⢿⣿⣿⠁⣤⡄⠀⡉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡇
⠀⠀⠀⠀⡀⣀⡟⡿⠹⠻⣸⣓⠁⠀⠀⠀⠈⠃⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣼⣄
⠀⠀⠀⢀⣃⣴⣆⠀⠀⣸⠿⣏⣠⡌⢙⣷⣄⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿
⠀⠀⠀⠸⣧⣌⠛⣀⡀⠉⠉⠛⠷⠶⢶⣷⣿⠏⣠⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣧⡀⠙⣇⠀⠀⠀⠀⠉⠉⢙⣁⣼⣿⣿⣾⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⡰⣒⣟⣻⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⠀⠀⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠲⢩⣷⣿⣛⢻⣷⣿⣿⣿⣿⡏⠀⠀⠠⠶⢤⣄⣠⣾⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⠿⢥⣖⡲⢧⣽⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠓⠲⠆⠉⢙⣦⣿⡋⠁⣀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠰⠟⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐⠋⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⢁⠤⠒⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠯⢟⡛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣠⣾⠟⠁⢠⠞⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠲⢌⡙⠳⢦⣤⣤⣴⠶⠾⠛⠛⠛⠋⠉⠉⠛⠋⠀⠀⣰⠃⠒⣍⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⢂⠠⢀⠙⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⣶⡎⠴⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠰⠶⣿⣀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢⣼⡲⡀⠀⠀⠀⠀⠀⢈⠻⡿⢇⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢜⠛⠁⡱⠀⠀⠀⠀⠀⠐⠤⠒⢉⣀⣠⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠅⠀⠀⠀⠂⠁⢀⡈⠆⠀⠀⠈⠠⠤⠀⣀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢀⡀⠤⠀⠀⠒⢀⠐⠆⠀⠀⠀⠈⡾⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠢⡀⣀⠠⠻⠂⣲⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠒⠁⢀⠄⠀⠀⠀⠀⠀⠀⢰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⣠⡶⠺⢶⣄⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⣸⡟⠃⠀⠀⢻⡆⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⣿⠃⠀⠀⠀⠀⣷⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⣯⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⣟⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏
⠀⢹⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠰⣿
⠀⠀⢧⡀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠇
⠀⠀⠈⠳⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢀⡤⠶⠚⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠘⣦⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠘⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣦⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣹⠀⢘⡆⠀⡄⢀⡽⠈⢿⣄⣠⠆⠀⢀⡀⠀⢠⡆⣰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠧⠴⠾⠤⠾⠓⠋⠁⠀⠀⠙⠳⠦⠤⠾⠤⠶⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢹⠄
⠀⠀⠀⠾⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣇⠀⠀⣇⠀⠀⢸⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠀⠀⡄⠙⢦⠀⠀⢀⢠⣀⣠⡤⠤⠴⠶⠿⠦⠤⢤⣄⣀⠄⡀⠀⠀⡴⠋⢠⠀⠀⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢤⣻⣶⣦⣁⠽⢚⡉⠡⠶⠚⠉⠉⣙⣋⠙⠙⠓⠶⠌⢉⡓⠯⣉⣴⣶⣟⡤⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠲⣴⣶⣿⡿⢋⡥⠚⢁⣠⠴⠛⠭⠿⠛⠋⠙⠛⠿⠭⠙⠶⣄⠈⠓⢬⡙⢿⣿⣶⣦⠖⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⣶⡟⣫⡼⣡⠤⠀⠨⠴⠚⠀⠠⠶⠟⢛⡛⠛⠶⠄⠀⠓⠦⠅⠠⠤⡌⢧⣝⢻⣶⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢋⣾⡟⡰⣡⡶⠃⢀⡴⠿⠋⠥⠶⢛⣩⣉⡓⠲⠬⠙⠿⢦⡀⠘⢶⣌⢦⢻⣷⡍⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⠋⢜⠁⡹⢃⣀⡉⢠⡞⢠⠆⣰⠎⠤⠤⠰⣆⠰⡄⢱⡄⢉⣀⡘⢏⠈⡣⠛⠷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠃⣰⠇⠘⢁⣛⣩⢭⣄⡁⠉⡄⠁⡈⠀⠀⠃⠈⢰⡉⢈⣠⡭⣍⣛⡈⠃⠘⣆⠘⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⣿⡇⢰⡏⠰⡇⢠⠸⣇⠰⠿⢻⡄⠁⠈⠁⠀⠀⠀⠀⠈⢠⡟⠿⠆⣸⠇⡄⢸⠆⢹⡆⢸⣿⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣾⣿⠀⣿⢀⡆⣄⠀⠳⠭⠛⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠛⠭⠞⠀⣠⢰⡀⣿⠀⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⣿⢰⣿⠈⢧⡈⠻⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⢁⡼⠁⣿⡆⣿⣿⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢻⣿⠘⣿⣇⠈⢷⣦⠛⡫⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢝⠛⣴⡾⠁⣸⣿⠃⣿⡟⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⡇⠸⣿⣧⠀⠙⠂⣴⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⣦⡐⠋⠀⣼⣿⠇⢸⣿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⡌⣷⠰⣤⣀⠿⣷⣌⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣠⣾⠿⣀⣤⠆⣿⢡⣔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢿⡌⠁⠹⣿⣶⣤⣙⠇⠀⠀⠀⢰⣀⣄⡀⠀⠀⢀⣠⣀⡆⠀⠀⠀⠸⣋⣤⣶⣿⠏⠈⢡⡿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠃⠳⣦⣤⡙⢛⡿⢆⠀⠀⠀⠀⠉⠛⢿⣷⣾⡿⠛⠋⠀⠀⠀⠀⡰⢿⣛⠋⣤⣴⠞⠘⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣆⡙⠛⠿⡿⡟⠠⠄⠁⠀⠀⠠⠀⠀⢻⡟⠁⠀⠄⠀⠀⠈⠠⠄⢻⢿⠿⠛⣋⣰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠶⢤⢼⣤⠆⠠⠠⠄⢠⡀⡤⠾⠷⠤⢀⡄⠠⠄⠄⠰⣤⡧⡤⠦⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⢀⡀⠀⠀⠀⠀⢀⡀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⣿⢟⡿⢛⣛⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡆⠟⣼⣇⣛⣛⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣴⣿⣯⣭⣭⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡬⢛⣭⢭⣝⢿⣿⣿⣿⠿⣋⣭⣽⡭⢁⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢛⣩⣶⣿⣿⣦⢻⣧⢻⣿⠏⣾⠿⣛⣭⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣛⣭⣦⣙⢿⣼⡿⣸⣿⡸⣿⣿⠿⣛⣥⣮⡙⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣴⣿⣿⣿⣿⣿⣷⣬⡁⠿⠿⠇⣩⣶⣿⣿⣿⣿⣿⣷⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣽⣿⣿⣧⣶⣿⠟⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣛⣻⣿⣿⠇⠀⠀⠀⠀⠈⢿⣿⣛⣛⣛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⢾⣿⣶⣾⣿⢇⡄⣀⣀⠀⣀⡀⣄⢿⣿⣶⣾⣿⢆⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣌⡳⢭⣍⣭⠵⢚⣴⣿⣿⣷⣌⢳⣜⠲⢭⣭⣭⠵⣋⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠿⠿⠿⠷⠶⠶⠾⠿⠿⠏⣿⣿⣿⡆⠿⠿⠶⠶⠶⠿⠿⠿⠏⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣾⣾⣶⡾⣛⡛⣶⣾⣶⡆⣿⣿⣿⡧⢰⣾⡞⣛⡻⣶⣿⣾⡆⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣧⣛⣃⣿⣿⣿⡇⣿⣿⣿⡇⢸⣿⣆⣛⣣⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣬⣭⣭⣭⣭⣭⣭⣭⣭⡅⣿⣿⣿⡟⢨⣭⣭⣭⣭⣭⣭⣭⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⢠⣤⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⢠⠀⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢠⡄
⠀⠈⠀⠀⠈⠉⠀⠄⠀⠉⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁



⣀⣀⣀⣀⣁⣤⣀⣀⣀⣤⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠉⠢⡀⠙⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⠀⣀⡟⠀⠈⠁⠀⠀⠀⢀⣀⣠⡀⣀⣤⣤⣶⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⡡⠄⠖⠀⠀⠀⠀⣴⡿⢋⠟⠙⠋⠁⣀⡤⠶⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⡀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠊⠀⢀⣄⢸⡏⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠣⣷⠛⣭⣿⣶⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⡆⠻⢞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢦⡀⢇⢶⣿⣿⢇⡇⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠈⠒⠤⠖⠋⠀⠀⠀⡀⠀⠀⠀⢈⣰⣿⣿⣿⡤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠉⠀⠄⠤⠄⠐⠒⢀⠠⠀⠀⠀⢌⡻⢿⣫⠞⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⣀⠀⠐⠀⠲⠀⠄⠀⠀⠀⠉⠉⠁⠀⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠙⠢⠤⠠⣤⡶⠟⠓⢄⠀⠠⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠤⠄⠀⢀⡴⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠋⠀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⡇⠀⠀⠠⢾⠀⠀⠀⠀⠀⠀⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⠶⡴⠞⠃⠀⠀⠀⣀⣠⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠠⠤⠄⣀⣀⢀⣿⠋⠀⠀⠀⣠⠄⠀⠀⠀⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⢉⠀⠀⠙⠢⣀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠋⠀⢀⢀⡀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠿⠤⠴⠤⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⠇⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢿⣿⣿⡻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⣀⣴⡟⣿⣽⣷⣶⣿⣿⣿⣮⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣽⣿⣽⣿⣿⢦⣄⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⣴⣿⢮⡷⣯⠾⣿⡿⡟⣿⡫⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣀⠀⠀⠀⠀⠀⠀⠀⣿⣩⣟⢾⣗⣿⣿⣻⢿⠟⣦⠀⠀⠈⢿⣷
⠀⠀⠀⣾⣻⢗⣾⣹⡝⣿⡿⠿⣋⣿⣽⣿⣀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⣠⡙⠛⣿⣷⣄⠀⠀⠀⣠⣾⣿⠿⣿⣿⣵⢿⢿⣿⡿⣿⣧⠀⠀⣸⣿
⠀⠀⢸⣟⣻⢸⣿⣽⣿⣷⡿⣿⣿⣼⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣟⣿⣿⣿⣿⣷⣿⣾⢯⣼⣧⠀⢠⣿⡿⠿⠿⣿⣿⣿⣿⣿⣓⣿⣷⢾⣦⣶⣿⡟
⠀⠀⣸⣿⡿⣷⣯⡻⣽⣗⣿⣟⣳⣿⡇⠀⠀⠉⢻⣿⡇⠀⠀⠀⣙⣿⣿⣿⣣⣺⣟⣾⣿⣺⣿⠀⢸⣿⠇⠀⢠⣿⣿⣿⣿⣿⢿⣾⣬⣳⣾⡿⠛
⠀⢀⣿⣿⣶⣿⣁⣿⣮⣿⣾⣿⠗⢿⣿⣶⡄⠀⠀⠀⠀⠘⠻⠻⣿⣿⣿⣽⣇⣾⣿⡾⠗⠻⣾⡆⠀⠉⠀⠀⣼⣿⡏⢙⣿⣿⣾⣿⣿⣾⠁
⠀⠈⠛⠻⣿⠿⠿⠿⠿⠿⢿⡿⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⡟⠁⠀⠀⢠⣿⡇⠀⠀⠀⠚⣿⠟⠁⠘⠛⠛⠛⢿⡟⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣓⠂⠄⠀⠀⠀⣀⣀⡴⣘⣑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⡷⠐⢸⢻⡷⡙⡋⣴⣿⣿⣗⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠱⢓⠓⣘⠶⠝⢳⣈⠻⣿⠏⣾⣦⣄⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡴⠂⠄⢏⠥⢘⠈⢁⠋⠃⢷⡌⢿⡳⣤⣤⠒⠿⣌⡠⡈⠉⠀⠉⠳⡌⠻⣴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡏⠀⣠⠟⠫⡉⠘⠼⣧⡀⡜⡇⢈⠿⣏⠛⠳⠀⠈⠣⠙⣆⠘⣆⢱⡀⢠⠙⡀⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠈⠀⠨⡥⢐⠥⠂⣿⢱⠇⠈⢰⠈⣇⢰⡄⠀⠈⣷⠨⡀⠸⡄⠙⠀⣇⢣⢹⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠞⡶⠂⢖⣄⢠⣌⣠⡜⣡⠏⡰⠀⢸⢆⠋⠀⡇⢠⡀⠙⠆⢳⡀⢻⡀⡄⢸⢸⠘⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢝⣃⣉⣽⣏⠳⡍⡴⠋⣼⣣⢀⠌⡘⠀⡄⠀⢸⡇⠀⠐⠀⡇⢸⠁⠇⠀⣤⣴⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡑⢸⡧⡰⠟⣁⣠⠂⢀⡷⠁⣧⡿⢠⡆⡟⠀⠁⢀⠆⠰⢹⣲⣯⠶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠷⡀⠀⡴⠉⢀⣠⠍⠀⠀⠹⡇⣴⠀⠁⠀⣠⣭⠄⣠⢻⡟⣿⡶⠋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠓⣦⠒⠀⡀⢦⡄⠀⠀⠀⠙⣷⣥⣦⣤⣾⡟⣡⡤⣂⣿⠀⠹⣷⡾⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠿⣿⣦⣝⣾⣠⡄⠀⠀⠀⠛⠋⠉⠈⣿⣿⡆⠈⠹⡷⡀⠀⠘⢷⣷⠋⢰⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠓⠚⠋⣿⠿⠀⠀⠀⢸⠀⠀⠀⠀⣸⣏⠉⣡⢼⠇⠀⠀⠀⠀⠈⠓⠿⢧⣞⣤⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⠀⣿⠻⠶⠆⠀⡆⠀⠀⠀⢠⣯⡤⠀⠴⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠿⠶⣀⣿⣷⠀⠀⠸⠀⢀⡀⠐⠋⠁⢀⠀⣾⠗⠈
⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⣯⣤⣰⣀⡄⠀⡈⠁⠀⣏⡀⢾⣤⣶⣀⡀⣀⣼⣻⠶⠛⠉⠁
⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⢛⣷⣜⡓⢆⣚⡙⠉⠀⠀⠀⠈⠉⠉⠁⠐⠒⠀⠀⠀⠉
⠈⠀⠄⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠙⢷⣦⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⡿⠋⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠷⠶⣶⣶⣶⣾⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣧⣤⣤⣬⣤⣤⣤⣤⣄⣈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡄⠀⠀⢸⣿⡇⠁⠉⢉⣩⣭⣭⣉⠉⠉⢹⣿⡏⠉⠉⣩⣯⣭⣭⣍⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣾⠟⠁⠀⠀⣾⣿⡇⠀⣴⡿⠋⠉⠉⠻⣷⡀⣸⣿⠇⢀⣾⠟⠁⠀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣦⡀⠀⠀⣿⣿⠅⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⢸⣯⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠉⠻⠆⠀⣿⣿⠁⠀⠻⣷⣤⣀⣠⣴⡿⠁⣿⣿⠀⠀⠻⢷⣦⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣿⣿⣦⣤⣤⣬⣿⣿⣿⣯⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣌⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⢻⣧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⣀⣴⡿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⡄⠀⢠⠖⣲⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀⠀⠈⠳⠇⠀⠸⠞⠁⠀⠀⠀⠀⣠⠖⠋⠁⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠳⠦⠶⠒⠒⠒⠒⠒⠒⠳⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠦⢤⣉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡠⠴⠖⠢⣄⠀⠀⠀⠀⠀⠀⢰⠃⠀⡖⠓⡄⠈⠑⢦⡱⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣰⠋⠀⣠⠚⡳⠀⢳⠀⠀⠀⠀⠀⠘⣆⠀⠉⠛⠁⠀⠀⠀⢳⠘⡆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢰⠃⠀⠀⠈⠋⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣸⠀⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⡤⠤⠞⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠳⢄⣀⣀⡤⠞⠁⠀⢀⡖⢦⠀⠀⠀⠀⡼⠳⡄⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡆⠀⠀⠀⡇⠀⢳⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⡇⠀⠀⠀⡇⠀⠘⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠲⠶⠶⢶⠃⠀⠀⣧⠀⠀⠀⠄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠲⣟⣉⣀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣄⣀⣀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣞⣠⠶⣄⢀⡤⠤⠤⣤⣤⣀⣠⣬⣭⡁⠀⢀⠀⣀⣠⠴⠚⠉⠁⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠃

⢀⡴⠲⣤⡀
⢸⣷⣄⢼⣿⡦⣀
⠀⠉⠉⢳⡀⠀⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠙⢦⡛⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⣤⣤⣤⣤⣾⠁⢸
⠀⠀⠀⠀⠀⠀⠻⣄⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡴⢉⣠⡀⡤⠄⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠉⢙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡂⢴⣄⠀⢠⣶⠆⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠠⢾⢙⠲⣤⠴⠶⢶⣾⣟⠛⡿⠛⡛⣷⠒⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⡤⠾⣍⢿⠑⣒⠀⢤⢄⠠⣔⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠘⠀⠁⠘⣶⢀⡈⠃⡤⠸⡃⣩⡽⠇⢀⡈⠷⠆⠸⡏⢉⢻⡟⣻⡙⡿⠹⡇⣶⠈⠈⡛⠶⣴⠋⠳⣥⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⣰⠇⠺⠈⠃⣀⢶⢸⡏⢻⣵⡄⣿⡝⠰⠇⠛⣰⠟⡴⡀⣿⡧⠳⡀⡇⠈⢳⠘⢮⡙⢦⡈⢉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡤⠖⠂⣙⠀⣉⢈⡀⠁⠛⠁⢠⡆⣴⠛⠻⠆⣙⢻⢥⠁⢹⡆⣀⠣⣄⡄⣄⠓⣀⠙⢦⣬⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡶⠤⠄⠉⡄⠿⠄⠿⣀⠀⢰⢇⣤⠡⣦⠄⣴⠃⢨⣬⠀⣾⠁⣛⠀⣸⢤⣈⡰⢀⠐⢄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣤⡀⢿⡆⠷⢀⡄⢹⣇⢤⡜⢋⣤⣤⠀⣴⣷⠸⠟⢁⡏⢠⡟⢠⠤⢠⡬⣧⡀⠹⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡈⠙⡻⠆⢶⣄⢸⡇⢨⡍⠈⠃⠘⠋⢋⠀⣛⠉⣀⠀⡀⣰⠇⡁⠴⢂⠉⣁⡈⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⢫⣌⠻⣦⡀⢻⣿⢟⣙⣶⠤⠤⠤⠤⠼⠯⠿⠬⠭⠤⠧⠤⢾⠑⢋⣉⣁⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠈⣻⣆⢨⡿⣏⢵⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠒⣤⣭⣿⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣙⣀⡿⢛⣿⠇⠛⠦⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢛⣴⣿⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⡏⠀⢸⡛⠛⠒⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⢠⣿⠀⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠙⢦⡀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢂⣽⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⢧⡀⠀⠀⠀⠙⢦⡀⠙⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⢺⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠏⠀⠀⠀⠀⠀⠙⠛⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠘⣿⡯⠀⠐⠀⠀⢀⡔
⠀⠀⠀⠀⢀⠚⡅⠀⠠⠀⠀⠛⠁⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀
⠀⠀⠀⠀⠐⢀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠙⠲⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⡆⡤⠀⠀⡀⣀⠐⠁⠀⠀⠀⠀⢀⣰⣂⣠⣤⣶⣾⣦⣄⣀⠀⠀⠀⠀⠁⠙⠔⠒⢼⣿⣿⣿⣿⣿⣷⣦⣴⣾⡏
⠀⠀⠀⠂⠀⠛⠚⢤⡽⠞⠁⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣷⣿⠏⢿⣿⣿⣧
⠀⠀⠀⠀⢀⡤⠒⢁⢗⣦⣠⣶⣟⣋⣉⣀⣹⢟⠋⣽⣿⣿⣿⡆⠙⢿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⡆⠀⢛⣿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⣿⣹⣟⣿⡿⠛⢻⣿⣿⣟⠀⢿⣿⣿⡟⣿⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠉⠻⢿⣿⡯⠉⠑⠄⠀⠞⣹⠃
⠀⢤⠀⠀⢀⣰⠾⣻⣿⣿⣿⠿⠇⠀⠘⢟⣿⣿⣶⠀⣌⣿⡅⠉⠀⠀⠀⠉⢿⣿⣿⣷⣶⣤⣄⡀⠀⠈⠑⠆⣀⣄⣒⣼⠧⢄⣀
⠀⠀⠀⠀⠀⠀⣈⣿⡿⠏⠙⠀⡄⠀⢀⠀⠛⠻⣿⠃⠀⢹⣿⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣄⣀⣀⡀⢀⠉⠉⠙⢲⡯
⠀⠀⠀⠀⠀⠛⠻⣿⠓⡀⢀⣀⠀⣤⠀⠀⣠⡜⠁⠀⣤⣺⣿⣷⡄⠀⠀⠀⠀⠟⠉⢩⣿⣙⠻⣿⣿⣯⣿⡍⠉⠙⠓⠺⠶⠖
⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠘⠷⣶⡛⠰⠚⠋⠀⠀⣼⣿⣿⣿⣿⢰⡃⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠉⠁⠀⠀⠀⣄⣼⣿⣿⣿⣿⡟⠊⠀⢀⣀⠀⠤⠘⠁⠀⠀⠀⢠⣴⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠄⠛⠒⠒⠛⠀⠰⠞⠋⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠋⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣤⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠔⠛⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⣶⣄⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠲⣴⣿⣆⣀⣾⠃⠀⠀⠀⠀⠀⢸⡿⠀⠀⣠⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠰⠶⠛⠛⠛⠉⠙⠛⠻⠛⠛⠛⠓⠶⠀⣾⡇⠠⣾⠋⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣿⣧⣴⠇⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⢰⡿⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡾⠛⠁⠀⠀⠈⠉⠉⠉⠉⠙⠋⠉⠉⢳⠀⠀⠀⢠⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣠⡿⠀⠀⣠⡿⠁⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠿⣏⠀⢀⣴⠋⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⡏⠁⠀⠈⢠⣾⣇⡀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣝⠻⣏⡻⣧⡀⠀⢰⠏⠀⠈⠁⢀⣴⣿⠏⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⣬⡙⠾⠛⠿⣏⠀⠀⢀⣶⣿⣯⡁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⠀⠻⣷⡈⠙⢦⣄⡀⠙⠷⠶⠿⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⢹⣷⠀⠀⠉⠙⠛⠳⠚⠒⠛⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡄⠀⠀⠀⠀⠀⣠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⣀⣀⠀⠀⠐⠻⠞⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢤⣴⣿⣿⡿⠿⠞⠛⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣮⣝⣛⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢻⡿⠃⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡼⣧⢱⡟⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⡄⠀⢿⣷⠁⠟⣼⣿⠀⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⠀⠀⠸⣩⠞⡳⢮⠻⠀⠀⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠆⠀⣿⣿⠀⠀⢀⡻⣌⣇⡿⡀⠀⠀⣿⣿⠀⠠⢾⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣶⢰⣾⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⠀⠀⢰⣶⣶⢰⣶⡦⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⢰⠉⠓⠚⠉⠛⠚⠙⡇⢠⣂⢢⠀⠀⠀⢀⡃⠀⠀⢀⡎⠢⠀⠤⣀⡀⠐⡆
⠀⠀⠀⠀⢹⠀⠀⠁⢠⡄⠐⠀⠇⢾⠁⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⡰⢠⣤⡄⣤⠀⢱
⠀⠀⠀⠀⢸⠀⠀⠀⠠⠂⠀⠀⡄⢸⠈⢲⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠋⠄⠛⠋⠀⢸⡴⣄
⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠣⣈⠀⠈⠶⠁⠀⠀⠀⢰⠲⣤⡤⢠⣀⣀⣀⠄⢀⠞⠀⢈⠉⠐⠂⠤
⠀⠀⠀⠀⠀⢹⠀⠀⠁⠀⠀⠀⢀⠏⠓⠀⠂⠆⠤⠊⢀⠈⠀⣀⠀⠀⣀⠀⠂⢣⣿⠀⠀⠉⢠⡀⢠⠀⢰
⠀⠀⠀⢀⡀⢸⠀⠀⠀⠀⠀⠀⡌⠀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠁⠶⠆⠉⠀⢸⠇⢿⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⢸⠀⢀⠦⡀⠀⠀⠰⠛⡀⠀⠀⠀⠲⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⠏⠀⠀⠀⠀⠀⣴⠊
⠀⠀⠀⢘⡀⣼⠀⠀⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡀⠻⡤⠃⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢨⠔⡟⠀⠈⠃⠠⠈⠁⠈⡃⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠎⠑⠂⠉⠀⠐⠒⠂⢢⠀⠀⢸⡇
⠀⠀⠀⠈⢹⡇⠀⠀⠀⠂⠐⠀⡦⠒⠒⠀⠀⠀⠀⠙⢣⠄⠀⡀⣸⠀⣀⡀⢠⣤⢀⢆⠀⣠⣆⢀⣸⠁
⠀⠀⠀⠀⢸⣇⡀⠀⠤⠠⢄⢀⠀⠀⢀⠀⠀⠀⡀⠀⡘⠀⠀⠀⠀⠀⠀⠁⠀⠀⡖⠀⠀⠀⠀⠈⠘⠀⠁
⠀⠀⠀⠰⢼⠁⢄⣠⠀⣠⣤⢸⠀⠀⠈⠀⠐⠂⠀⢰⡇⠀⠐⣐⡂⠀⠀⢱⠀⠀⡇⠀⠀⠘⠂⠀⠀⡼⠐⠈
⠀⠀⠀⠀⢸⡀⠘⠛⠤⠛⠋⢸⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠉⠁⠀⠀⡞⠀⠀⠈⡆⠀⠀⠀⠀⠈⢁⡀
⠀⠀⢨⡀⠀⢳⠀⠀⠈⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⢰⢃⡀⠀⠀⢀⡀⠈⢣⠀⠀⣸⠁⠀⢀⡀⠀⠘⣶⠃
⠀⠀⢸⣴⠀⡌⠀⠀⠀⠀⠀⠀⠱⣴⠦⠤⠐⠂⠀⠉⠉⠑⠀⠀⡞⠓⠒⠚⠃⠠⠇⠠⢶⠋⠑⠂⠄⢸⣀
⠀⠀⠀⠻⢶⡇⠀⠀⠀⣀⠀⠀⠀⡟⠀⠀⠀⠀⣀⠀⠀⠀⢂⡰⠀⣴⡀⢶⣤⠀⣤⡄⠈⡄⠿⣀⣠⡈
⠀⠀⠀⠀⠸⠛⠋⠉⠉⠑⢇⠀⣤⡇⠀⠀⣠⡈⠋⠀⢀⣄⠨⠧⠤⣈⢛⠀⠘⣳⡈⠀⠀⡇⢀⡀⠈⢰⠇
⠀⠀⠀⢠⠁⠀⢀⡄⠠⠀⠸⣼⠋⢀⠀⠀⠙⡇⠀⠀⡄⠀⣀⠀⠀⢀⠘⡄⠀⡇⠉⠉⠉⠉⠁⢸⢠⠏
⠀⢠⡤⠺⠀⠀⠀⠠⠀⠀⠀⣼⠀⠀⠘⠀⠀⢸⠘⡎⢆⠀⠀⠘⠃⠀⢀⠇⠰⠀⠐⠂⣀⠀⠀⠸⠋⠄
⠀⠈⡆⠐⠁⡦⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠸⣦⠘⠈⢢⠄⠀⠀⢠⡌⠀⡀⠀⠀⠀⠁⠀⠀⠀⡄⢼⢀
⠀⠀⣶⠀⠀⠀⢠⠁⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣿⡐⣠⠋⠀⠀⠀⠈⠳⡄⡄⠀⠀⠀⠀⠀⠀⠀⡇⣼
⠀⠀⠙⠆⠀⠀⡼⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⢱⡇⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠀⣧⠿⡇
⠀⠀⠀⠐⡀⠸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣸⢽⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣘⠜
⠀⠀⢀⣤⣾⣦⣽⣦⣀⣀⣀⣷⣄⡀⠀⣀⣠⣴⣥⡸⣄⠀⠀⠀⠀⠀⣀⣼⡓⢄⣀⣀⡀⣀⣀⣴⣥⡀
⠀⠀⠈⠛⠛⠿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣿⣶⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣧⠀⠀⢐⠂⠤⣿⡟⠀⠁
⠀⠀⠀⢈⣵⠄⠀⠀⡀⣀⠴⠂⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣧⠀⡾⠇⠈⠙⣧⡀⠘⠀⣀⣴⡄
⠀⠀⠀⠟⠡⠦⢠⢀⣹⡁⠀⠀⢀⣀⣀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣷⣄⠈⠻⣿⣿⣁⠉⣀⡀
⠀⠀⢘⣴⣿⣿⡿⡋⠀⠀⠁⠀⠸⣿⣿⣿⣷⣆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡛⠿⣀⣤⡠⢸⣿⣿⣿⣷
⠀⠀⣾⠟⣿⠿⢹⣿⡑⠀⠀⠀⠐⠛⢻⣿⠈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣉⡉⢶⣤⡍⠁⠀⢈⣇
⠀⠀⠸⣿⣧⢃⣾⣿⣧⠘⠀⠀⠀⠀⣸⠿⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠾⠟⠋
⠀⢀⡟⣾⣿⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠄⠀⣸⡟⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⣷⡄
⠀⢀⡀⠚⣿⣽⡿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⡟⠀⠿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠿⠋⠁⣀
⠀⠸⣄⡴⡿⣿⡇⡴⠁⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂
⠀⠀⠀⠀⡖⠍⣹⢡⡀⣀⠁⡀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣧⣤⡁⢀⡁⣈⢀⡁⠀⠋⠀⢘⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀
⠀⠀⠀⠀⠋⠀⣅⣬⢽⠋⡋⣏⢁⠐⣊⢈⣽⢷⡍⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡰⠃
⠀⠀⠀⠀⠛⠋⢀⢉⢘⡏⠁⠀⠀⠀⠀⠀⠈⡄⠻⣃⠀⣯⡏⠙⠟⠿⡿⢿⣿⡿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⢰⠀⠋⠀⠈⠁⠀⠀⠀⣀⠀⢘⣳⡼⠃⠿⢷⡤⢺⣿⣿⠇⣠⡄⣿⡇⢸⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠈⠁⠈
⠀⠀⠀⠀⠸⡉⠁⠀⠀⠀⠀⠀⠀⢉⡾⠁⠙⠂⠀⢰⣆⢀⣼⣿⣿⠀⠘⡿⠞⢠⣿⠿⠿⠿⠏⠙⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢢
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠂⠀⠴⠊⠁⠀⠀⢀⣤⣿⣿⣏⠻⡟⣣⣴⢎⣶⠴⣋⣥⣶⣶⡷⣦⣤⣾⣿⣿⣿⣫⠞⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢐⡀⣻⡻⠛⠛⠁⠀⠀⣧⡿⠁⠀⢚⣡⣾⣿⣿⣯⣍⣭⢿⣾⣿⣿⣿⣥⡿⣷⡜⡿⠻⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠋⠀⣙⠁⢰⣶⣶⣾⠷⠋⣀⣶⡞⣿⣙⣶⣶⣬⣭⣍⣳⣿⣿⣿⠟⣿⠟⠋⣉⡾⢿⣶⣦⢀⠿⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢙⡀⠘⣿⢸⣧⢶⡿⠿⣏⣿⡿⠛⢿⣿⣿⣿⣿⣿⡿⣿⣧⣠⣿⣷⡘⠉⠀⠛⠿⠟⢡⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⣿⠋⠁⣀⡿⠈⢵⠏⠀⠀⠈⢡⡀⠀⠀⠈⠙⣴⣿⣟⣀⠸⣿⠁⡠⠉⠻⣶⣤⣀⣀⣠⣿⣧⣤⣽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⣿⣿⣿⣿⣿⡿⠃⠀⠀⠻⡿⠀⣶⣿⣗⠀⠐⢠⣿⣿⣟⣛⣾⣣⣾⣁⡄⠀⠈⣿⣿⣿⣿⢿⣻⠿⠟⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠐⠋⠉⠀⣼⡟⠀⠀⠀⠀⠤⡎⠳⣿⣿⡟⠀⠀⠘⠿⠏⠋⠉⠛⠉⢿⣿⣿⡇⠀⣸⣿⣿⣄⢈⣍⡀⠀⣸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡘⠃⠀⠀⠀⣠⠀⡞⢈⠹⣿⡄⠀⠀⠀⠘⠂⠀⠲⢄⠀⠀⠈⠀⠀⠺⣿⣿⣿⣿⣷⣮⣟⣶⣮⣭⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣷⣶⣄⠀⠸⠟⠀⣀⠘⠁⣿⡇⠀⠀⠀⣀⣤⣧⠀⢈⡦⡀⠀⠰⢶⡀⢻⣿⣿⣿⣿⣿⠿⠃⠹⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡄⠙⣿⡌⠀⠀⠀⠒⣂⣽⢰⣿⡇⠀⠀⠖⠽⠞⢠⡀⠈⠃⠈⠀⠀⠀⠀⠈⣿⠟⠋⢏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠈⣧⠀⠀⠰⠟⠉⠀⠀⠿⠃⠀⠀⠀⢶⡠⠘⠃⢀⣴⠆⠀⣴⠀⠀⣼⠿⢧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⣄⠰⠾⠿⣷⣶⣆⠀⠀⠀⠀⠑⠂⠀⠈⠅⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⢲⠶⠿⠟⣛⠂⢀⣀⠀⠀⠴⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣿⣶⡌⠃⢠⡄⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣇⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⡔⣶⣴⣶⠒⠆⠀⠒⠔⠀⠈⠀⠀⠎⠈⢡⠤⠠⠀⠄⠀⡸⠁⠊⢀⠀⡀⡀⢀⠤⠀⢚⡶⢄
⠀⠀⠙⣠⢿⡟⢻⡃⠍⢍⡁⢦⣀⣰⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡀⠀⠀⠶⠀⠈⠀⠀⠀⠁⣎⣦⠞⠁⢻⣷
⠀⠀⠀⠾⠼⠥⠞⣀⡾⣆⠳⡌⣿⡼⠀⠀⠀⠀⠇⠈⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⢳⡄⠀⠀⠀⠀⣟⣿⣆⡸⢻⣿⡆
⠀⠀⡆⣷⡆⢰⣿⣿⣁⢈⣳⣿⣿⣷⣤⡰⠀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣸⠖⢺⣏⡇
⠀⠀⢻⣿⠁⠀⠈⡻⢿⡼⢿⣥⣿⣿⡍⡀⠀⡄⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣆⣹⠀⠀⠀⠀⠀⣿⢙⣿⢡⣴⣾⠁
⠀⠀⠀⠩⣶⣿⣲⣻⡿⠁⠀⠋⢿⣽⣷⠃⠀⠀⠀⠐⣄⣷⠀⠀⠀⠀⢀⠈⢳⣦⡙⠀⠀⠀⢀⠲⣿⣻⡏⠀⣽⡝
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⠆⠀⠀⠈⠂⠳⣤⣄⠂⠈⢷⣶⣝⠀⠀⠀⠀⢀⣱⣿⣿⣷⠶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣴⠀⠀⠀⢐⣻⣿⣿⣧⡀⢰⣮⠉⠐⢄⠀⠀⠖⢿⣿⣿⢿⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸⣟⣻⣿⣛⠀⠉⠀⠀⠈⢀⠀⢀⡐⣿⠻⣷⣿⢈⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢙⣯⣷⠸⠃⠀⠀⠀⠀⠀⠈⠠⣩⣤⡿⢇⠁⢽⡿⣀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣸⡆⠈⣼⣬⠙⠀⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠇⣹⡇⠀⠙⠃⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢾⡟⡁⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠜⣶⡿⠀⠀⠀⠀⠈⠓⢼⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡾⠧⠇⢈⣽⠀⠀⠀⠀⠀⠀⠠⠂⠀⢰⣿⡇⠀⠈⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠛⠂⠰⠒⠛⠋⠀⠀⠀⠀⠀⠀⠓⠒⠉⣉⣭⣬⣿⣿⣿⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣟⣿⣭⡭⢯⣭⣿⣻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣯⣓⣿⣿⣿⡿⢷⣾⣿⣿⣿⣿⣿⣯⣽⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣶⣿⣿⡷⣟⣵⣿⣾⣯⣾⣿⢿⡿⣳⢿⣿⣿⡽⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⢄⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⢠⣿⢟⣿⡿⢿⡾⣿⠟⠷⠋⣽⢿⣿⡷⠻⠟⣿⣻⣾⣿⣶⢻⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⣸⠀⠀⠀⣷⠀⠀⠾⠏⢈⣁⡀⠋⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⣾⣯⣿⣷⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⣠⡼⣧⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⢰⢠⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⣹⡟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠁⠀⢹⡀⠀⠀⢠⡟⠂⠀⣸⠁⠀⠀⠸⡆⠠⢤⣤⣤⢤⠀⠀⠀⢀⣀⣀⣠⣀⠈⢻⡟⣛⡻⣮⢿⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢀⣼⣇⠀⠀⣾⠀⠀⢀⡏⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠸⠾⣏⠁⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠀⢻⡀⢀⡟⠦⠀⣸⠇⠀⠀⠀⠀⣧⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢧⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢠⣄⠈⠳⣶⠶⠂⠀⣠⠤⡀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⣤⠔⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇⢲⠌⡑⠒⠶⠶⠒⠋⠁⠀⣹⡄⠀⠀⣰⠏
⠀⠀⠀⢀⠼⣇⠀⠀⠀⠠⡄⠀⠀⢀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⣌⠢⢄⠀⠈⠀⠈⠀⠈⠀⡼⠁⢀⡼⠋
⠀⠀⠀⣮⠀⠙⣧⠀⢰⡏⠋⣻⠟⠧⣴⠒⠒⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠡⠴⠶⠒⠂⠉⣀⣴⡿
⠀⠀⠀⠘⢦⡀⠘⢧⠸⡗⠾⠁⠀⠀⡈⢷⣤⣤⡀⠈⡇⠀⠀⠀⣀⣀⣀⣤⠟⠙⠷⣦⣤⣤⣤⠖⠋⠁⠘⡇
⠀⠀⠀⠀⠈⣷⡀⠈⠳⣝⣶⣤⡀⠋⠀⠀⠀⠀⠀⠈⣧⡶⠿⣏⣿⣿⣿⣿⣿⣿⣶⣦⣬⣤⣤⣤⣤⣶⣶⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠹⣟⠢⠤⠞⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡷⣄⠀⠳⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣿⣷⡄⠀⠀⠀⠈⠙⠛⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣤⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⣿⣿⡟⢡⣣⢽⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢛⣟⣻⢛⣿⣋⡛⢻⡟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣻⣿⣛⣿⣟⣛⣛⣻⣛⣛⣛⣻

⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡠⠊⢱⠤⠒⠒⠈⠙⢆⠀⠀⠀⣰⣿⡘⣯⣵⣶⣶⣶⡄⠀⠀⢀⢾⡁⠀⢀⡤⠤⣌⢢⡀⠀⢠⡸⣿⣿⠿⣛⣿⡛⢦
⠀⢠⣇⠀⠀⠣⣀⡠⠤⣄⣈⡆⠀⠐⢿⣿⣷⣝⠿⣟⣛⢛⡿⠀⠀⡎⠀⠉⠉⠁⠀⠀⠀⢣⠇⠀⣿⣿⣶⣶⣿⣿⣿⣿⡎⡇
⠀⠘⡏⠣⣄⢀⡎⠀⠀⠇⢀⠇⠀⠸⣮⡻⢿⡿⣸⣿⣿⣸⣿⠀⠀⢇⠀⣀⣀⡀⠀⠀⠀⡜⡆⠀⣿⣿⠿⠿⣿⣿⣿⣿⢇⡇
⠀⠀⠙⢄⣨⠏⠀⠀⣼⡠⠊⠀⠀⠀⠙⢿⢗⣵⣿⣿⢇⡿⠃⠀⠀⠈⠻⣀⠀⠈⠙⠒⣫⠜⠀⠀⠈⢵⣿⣿⣶⣭⣯⣥⠞
⠀⠀⠀⠀⠉⠑⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⣠⡶⠓⠀⠺⣶⣤⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠻⣦
⠀⠀⠈⠁⢀⣤⣤⡆⠀⠉⠀⠀⠀⢀⣾⣿⡟⠛⠛⣿⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⣠⡦⠙⢷⡄⠀⠀⢀⣾⣿⣿⠏⢙⣷⣌
⠀⢰⣦⠀⠸⢿⡿⠇⠀⣠⣶⡆⠀⠈⠻⣿⣅⡀⢀⣽⣿⠟⠙⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣾⣿⡿⠁⣰⣿⣿⠏
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢦⣀⠙⠁⠀⠀⠀⠀⠀⠀⢈⠻⣿⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠐⢾⣿⡶⠁⠀⠀⠀⠀⠀⠀⠙⠏⠁⠈⠹⠟⠉⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠘⢷⣦⡉⠻⠛⠁

⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣦⡀⠀⠀⠀⠀⢀⡤⠂⠈⠀⠒⢀⡀⠀⠀⠀⢀⢰⣶⣿⣷⣶⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⢁⠇⠀⠀⡴⠚⠉⠑⡀⠀⣰⡇⣾⣿⡿⢫⣴⣾⣷⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⣧⡞⠀⠀⢸⠁⠀⠀⠀⡁⠀⠛⣼⣿⣿⠅⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⠟⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⢠⠃⠀⠸⣿⣿⣿⡅⢿⣿⣿⡿⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠈⠂⠄⠀⠉⠶⠒⠁⠀⠀⠀⠈⠻⠿⢿⡦⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣀⡠⠄⢀⡀⠀⠀⠀⢀⣿⣧⢀⣀⣀⣀
⠀⠀⠈⢿⢆⠀⠀⠀⠀⡀⠀⠀⠀⠙⣤⡀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠈⢀⠞⠁⠦⡀⠀⠈⢢⠀⠀⣾⣟⣵⣿⣹⢿⣿⣷⣄
⠀⠀⠀⠀⢂⠁⠒⠈⠁⣔⠀⠉⢐⡄⠘⣿⣶⣶⣿⡟⠽⣷⣶⣶⠦⠀⠀⢸⠀⠘⠂⠉⠀⠀⠀⡇⠀⢿⢹⣿⣧⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠢⠀⢀⣀⡁⠤⠀⠀⠀⠀⠘⠿⠿⠿⠷⠽⠟⠛⠛⠉⠀⠘⢌⣄⠀⠀⠀⠀⢀⡔⠀⠀⠘⠯⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁
⣟⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣛⣛⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡰⠀⠀⠀⠘⡝⣿⣿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡃⠀⠀⠀⠀⠴⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠫⠁⠽⢧⠀⠀⠀⢀⣂⣿⣅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢸⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠭⢽⣛⣷⣾⣿⣿⡘⣧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⣥⠠⠤⠤⣤⡄⣴⣾⠩⢿⣻⣛⡛⠡⠌⠳⢶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠂⠀⠙⠛⠛⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣀⣀⣀⠀⣠⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠒⠶⣼⣇⣼⣳
⠀⠀⠀⠀⠀⠀⠀⢸⣀⠘⠛⠀⢀⠘⠛⠀⢸⠖⠒⠋⠉⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣀⡀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⠀⠠⠽⡤⠀⣬⠾⠀⢀⠔⠀⠀⠀⠈⡟⢄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣄⡀⠀⠉⠀⠐⠚⠓⠀⢘⡇
⠀⠀⠀⠀⣀⠤⠒⠒⢫⠞⢉⠒⣒⠒⠉⡏⠀⢀⣸⣤⠖⢄⣀⠀⣸⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡼⠷⣗⠠⣼⣿⣀⠀⠀⡐⣺⢧⡀
⠀⠀⢠⡚⠁⠀⠀⠀⠈⠲⠶⠋⠲⠒⠒⠛⢶⠻⠤⠷⣀⣈⡡⠞⠁⢀⡠⠽⠆⠀⠀⠀⠀⠀⠉⠀⠀⠈⠑⠲⠤⣥⣤⠤⠞⠳⡄⠘⠆
⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⠤⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄⡀⣀⡠⠤⠒⠒⠊⠉⠁⠀⠀⡠⠒⡄⠀⠀⠀⡠⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⠹⠘⠤⠤⠜⠵⠻⢸
⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⢠⡔⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣼⣿⠳⣄⣀⣀⣀⣀⣰⢏⡿⡇⠀⠀⠀⠀⢠⠊⠀⠈⠁⠀⠀⠉⠁⠀⠈⠦⡀⠀⠀⢰⡰⠀⠀⠀⠀⣰⣫⠳⠴⠶⠤⠞⡥⣧
⠀⠀⢀⡿⠘⠀⠉⠉⠉⠉⠉⠉⠛⠀⠷⣄⠀⠀⠀⠘⢄⡀⠀⠸⡞⡖⠁⠀⠀⢀⠴⠃⠤⢴⠟⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠄⠀⢸⡆
⠀⠀⣿⠁⠀⠛⠛⠃⢀⣀⠈⠉⠋⠀⢸⡏⠀⠀⠀⠀⠀⠉⠒⢶⣬⣤⠴⠒⠊⠁⠀⠀⠀⢰⡀⠀⠀⠀⠀⣹⣶⣢⠀⠰⠤⠆⢤⣸⠟⠦
⠀⠀⣿⠀⠀⠴⣿⠃⢹⡟⠰⣿⠂⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⣀⣀⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠐⠋⠉⠓⠒⠒⠒⠋⠙
⠀⠀⠈⠙⠿⠾⣧⣠⠴⠿⣴⢾⠷⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡅⢸⠀⠀⠀⠀⠠⠼⣠
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢠⢶⡀
⠀⠀⠀⠀⠀⡼⡀⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣇⣀⣀⣀⣀⡞⣹⠀⠀⠀⠀⠀⠀⣠⠟⠞⠓⠚⠓⠲⠟⠶⢷
⠀⠀⠀⠀⠀⣱⡇⠄⠀⠤⠳⢣⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠀⠀⢹⡀⠀⠀⠀⠀⠠⡇⠀⢤⣤⠀⠀⢰⡶⠄⠈⡇⣴⣦
⠀⠀⠀⠀⢠⠀⣀⡀⠀⢀⣀⠀⠀⠠⡺⠟⠀⠀⠀⠀⠀⠀⣾⠀⠶⠀⠀⠀⠀⠆⠘⡇⠀⠀⠀⠀⢀⡵⣞⡤⠀⢠⣤⡄⠀⢠⡿⡇⠘⣿⠇
⠀⠀⠀⠀⣸⠀⠉⠁⠀⠀⠉⣀⣇⠎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⢀⡀⠀⡐⠈⣇⠀⠀⠀⠀⠀⠀⠋⠙⢳⣶⣶⣶⣶⡏⣩⢤⣾⠟
⠀⠀⠀⠀⣺⠠⠀⠀⠒⠂⠠⣒⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⡀⠀⢀⠀⡇⢀⠀⠰⢼⣆⠀⠀⠀⠀⠀⠀⡤⡾⠀⠉⠉⠘⡿⡏⠉
⠀⠀⠀⠀⠁⢀⣀⣀⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢦⣀⠀⠉⠛⠋⠀⠘⣲⣏⠀⠀⠀⠀⠀⠀⢧⡇⢰⠂⢰⡄⣇⡽
⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠉⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠾⠷⡞⠉⠉⠹⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠉⠉⠙⠳⢦⣄⠀⠀⠀⣀⣤⣤⣤⣄⡀⣤⣤⣤⣤⣄⡀⠀⠀⢀⣤⡶⠛⠉⠉⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⠾⠿⠛⠋⠛⠛⢋⡉⠙⠛⠛⠛⠿⣷⣄⠀⠀⣶⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣦⠀⠀⣀⡤⠟⠐⢊⣠⣤⣤⣤⣤⣤⣶⠏⠻⣶⣦⣤⣄⡀⠉⠈⠃⠈⠻⣄⠀⠀⣤⣴⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣿⣷⡾⠋⠀⣀⡤⠀⠠⠚⠛⠉⠉⠉⢁⡄⣀⠀⠀⠈⠉⠛⠧⠈⠳⣤⡄⠈⠻⣾⣿⡿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢉⡰⠁⢰⣿⠁⠀⠘⢷⣦⣤⣀⣴⡿⠀⢸⣷⣄⣠⣤⡴⠒⠁⠀⢹⣿⠀⠘⢮⡉⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠞⠁⠐⠛⢡⣶⣶⣶⡤⠀⢹⠟⠉⣠⡞⢠⣌⠙⢿⠏⠠⣴⣶⣶⣦⠙⠓⠦⣄⠛⢅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢠⠆⢠⡀⠀⠟⠉⠈⣀⣈⠳⠀⢠⡞⠛⠁⢸⡛⣿⠀⠐⠋⣁⣀⠉⠙⠳⠀⢀⠀⠀⠈⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣰⢏⡀⠘⣷⣄⠀⠶⣏⠸⠿⢗⡀⢸⡄⠀⠀⠀⠀⢸⡇⢰⠻⠿⢉⡷⠦⢀⣴⡿⢠⡀⢹⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⡿⢸⣧⠀⢻⣿⣷⡦⠈⠉⢻⡿⢣⠈⠇⠀⠀⠀⠀⠘⠃⡹⣿⠋⠉⠀⣶⣿⣿⠃⠀⣷⠀⢿⡆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⣼⡇⢸⣿⡄⠀⠉⠻⣷⡀⠀⠊⠰⣯⡄⠀⠀⠀⠀⠀⠀⠀⢿⠈⠃⠀⣰⡿⠋⠁⠀⢠⣿⠀⢸⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣿⣷⠈⣿⣿⣦⡀⠀⠈⠙⠂⠀⢰⠋⣴⠀⠀⠀⠀⠀⠰⡄⠈⠀⠀⠚⠉⠀⠀⠀⢀⣾⡏⠀⣿⣿⡆⠹⣦
⠀⠀⠀⠀⠀⠀⢰⡏⠀⢾⣿⡇⠀⠘⣿⣿⣿⣿⡿⠟⠛⠂⠀⢠⡆⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⠲⠶⣾⣿⣿⣿⡟⠀⠀⢹⣿⡇⠀⢻⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⣷⣄⠤⠜⠻⣿⣿⠁⢀⠀⢀⡀⠘⢿⣷⣶⣤⣤⣶⣾⠟⢀⡀⠀⠀⠀⢸⣿⡿⠋⠀⠀⣠⣿⣿⠃⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⣿⣿⡟⠀⠀⠀⠀⠉⠳⣾⠀⠀⠤⠉⠀⠀⠉⢻⠟⠉⠠⠖⠨⠄⠀⠠⡧⢚⡉⠀⠀⠀⠀⢻⣿⣿⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠈⢻⣇⠀⠀⡀⠀⠀⠀⢸⣧⡀⠀⠀⠀⡀⠀⣸⣄⠀⠀⠒⠒⠠⢄⣼⡇⠀⠀⢰⡇⠠⡀⣼⡟⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠈⠙⣦⠀⠘⣷⣾⠊⠁⠙⠿⣶⣤⡯⠔⠚⠉⠉⠓⠮⣥⣤⣶⠟⠋⠀⠀⣀⣼⠏⢁⡼⠋⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠁⠈⣧⡀⠀⢄⠀⠤⠖⠈⠀⠀⠀⠈⠒⠀⠐⢋⡄⠀⠴⠛⠁⠉⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⣄⠀⠈⠉⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⣠⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⢿⣶⣶⣦⣴⣶⢾⡟⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠿⠏⠀⠈

⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⠛
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠲⣾⡿⠛⣤⡀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣅⡀⠚⢹⣷⡄⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣿⣿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡉⢹⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠿⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠙⣿⣿⡆⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⣿⠁⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠐⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⣿⣄⣠⣤⣄⠀⣤⣄⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠈⢻⣿⠉⠿⠁⠙⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢿⣄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣴⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣼⡇⠀⢸⣾⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⢻⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠀⠀⠀⣸⢿⣤⠛⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⣿⡀⠿⠀⢠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡀⠀⣀⢸⣿⠀⣶⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⢹⣿⠟⠿⢿⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠀⠀⠀⠉⣥⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⣤⣶⣾⠛⠁⠀⠀⠀⠀⠈⠛⠛⠓⢲⣤⠀⠀⠈⠉⠛⠿⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣤⣀⣠⠀⠘⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⣼⠟⠁⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠀⣤⡀⠀⠀⠀⠀⢀⣠⣤⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣤⠋⠀⠀⠀⠈⢰⡄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠈⠛⢀⣀⣶⠇⠀⠛⠛⠁⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣤⣆⠀⠚⠀⠀⠀⠀⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⣶⣤⣿⣿⣿⣶⣦⣤⣶⣤⣄⢀⣤⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⣿⡟⠋⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⢹⡧⠀⣿⣿⠆⠘⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣤⠀⢾⠏⣀⠀⢸⣿⠋⢿⣿⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣿⣄⠀⣀⡨⠆⠸⣿⠄⠘⠁⠀⠀⠀⠀⠀⢀⣿⣦⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢈⣿⣏⠻⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣧⣄⣹⣿⣷⠀⠃⠰⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡟⠻⣿⠁⠀⠀⢣⣤⣿⠀⢀⣤⣤⣶⣾⣿⡏⠙⢿⣷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠻⣦⡀⠀⢀⣽⣿⠀⠈⠻⣿⠉⢸⣿⠇⣶⠀⠿⠀⠀⡤⠐⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⠈⠁⠀⢼⣿⣧⣄⠀⣿⠁⠀⢸⣿⠀⠉⠀⠀⠀⢀⡀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡅⢰⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⣿⠀⠀⢿⣿⠀⠀⠀⠀⠀⣾⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⣦⠀⠀⠀⠀⢠⣿⣿⣿⠀⠘⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢸⣿⠀⠀⠐⠀⠀⣼⠛⠋⠀⠀⣀⠀⠀⠀⠤⠀⠠⠀⠀⣀⣿⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣿⡀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠻⣷⣄⠀⠀⢰⡄⠀⠀⠙⠋⠀⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠐⠂⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣶⡆⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠠⠀⣀⠀⠀⠀⠈⠉⠛⢿⣿⣿⡿⢿⣷⡄⢠⠀⠀⢰⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠻⠄⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠉⣿⢇⣀⡈⠀⠀⠙⠁⠀⠹⣿⡿⠋⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣤⣾⣧⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣧⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⠙⢛⣛⣀⣄⡀⠀⠀⠀⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⣆⠐⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠺⠿⠿⢿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠈⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣤⣤⣤⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⣠⡀⠰⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣋⠁⠤⠆⠒⠮⣙⠈⡉⠡⠶⠀⠌⢳⣤⠖⠋⠠⠁⣀⠄⠊⢚⠀⠤⠀⠁⠄
⠀⠀⠀⠀⠀⠀⣄⠊⡤⢉⠡⠆⡀⠨⢉⣍⣠⡜⠟⢀⡐⠟⠰⡄⡀⠙⠶⢢⡜⠙⡄⠀⡠⠀⠤⠀⣤
⠀⠀⠀⠀⠀⠀⡛⢸⠐⣋⡰⠠⠩⠙⢰⡉⠈⠙⠐⠿⠁⣠⡶⣠⣴⢄⠐⠠⠻⣦⠙⠶⣤⢘⠁
⠀⠀⠀⠀⠰⠶⡟⠔⠀⠀⡑⣧⡰⠎⠀⡀⠳⠶⢾⠄⡠⠉⢁⠈⠏⢉⠉⡥⠒⠈⡰⠦⠌⠒
⠀⠀⠀⠀⣑⡀⣀⢀⢁⠐⡈⠖⠂⠀⠘⠀⠀⠀⠀⠠⠔⠘⡇⢰⠠⠎⠠⡶⠘⠃
⠀⠀⠀⠀⠙⠿⢿⣤⡌⣨⡄⢹⣤⠘⣠⠑⢶⡄⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠀⠀⠈⠃⠈⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⣷⢤⣀⣄⣀⣀⣄⡤⠞⡕⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠳⠄⠉⠀⠈⠀⡄⠘⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠴⢲⠶⠀⠀⢀⡴⡶⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣦⠀⣀⢠⡀⠀⠻⠃⠀⢀⢠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠋⠀⠀⠀⠈⠉⡫⠌⠤⠤⠞⢢⠤⠓⠯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡀⡼⠁
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⠀⠘⣆⠀⠀⠀⠀⢸⡄⠀⡰⠃⢈
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠘⣆⠀⠀⠀⠈⡷⠋⠁⠀⣸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠀⠹⠀⠀⠀⣟
⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢘⣿⡇⠀⠀⠀⣷⠀⠀⢸⡀
⠀⠀⠀⢿⡄⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠤⠒⢋⣡⡴⠖⢲⠀⠘⣦⡀⠀⠓⣄
⠀⠀⠀⠈⢿⣆⡀⠀⠈⠉⠛⠛⠉⠉⠉⠛⠉⠉⠙⠉⠉⠉⠁⠀⣠⢿⠀⠀⡀⢭⢢⠽⠚
⠀⠀⠀⠀⠀⠈⠛⠷⠶⢤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠖⠋⠁⠈⠓⠂⠓⠚⠋

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣾⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⣭⣿⣾⡿⡜⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡯⣿⡿⠾⠙⢻⣗⣿⣮⣉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⢟⣶⡶⢞⣿⣿⣧⣋⣾⣟⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⢛⣷⣿⣿⣿⢙⠿⣏⣿⣞⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣾⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣽⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣾⡟⣻⣯⣽⣿⣿⣿⣶⣟⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⢿⣿⣿⡟⢿⡿⣿⣿⣿⣿⡟⣿⠇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣤⣶⡾⢿⣿⣿⢘⡻⢿⣷⣿⣷⣿⣾⣿⣿⣮⡀⠈⠋⣩⢌⣍⣹⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣶⠈⣷⣿⣿⡟⢻⣿⣿⣿⣿⣿⡗⢸⣷⣶⣾⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣛⣿⣿⣼⡟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣭⠘⢋⠉⢣⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣆⣼⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡴⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣼⣷⣿⣿⣿⣿⣿⣧⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠿⢿⣿⣿⣿⣿⡟⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⣿⣾⣾⣿⣿⣷⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠖⠒⠒⠦⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠛⠋⠙⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠾⠇⠀⠀⠠⠶⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣦⣤⣄⣀⣀⣀⠀⢀⣀⣀⣀⣤⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⣞⠐⠊⠀⠀⠀⠀⠀⠀⠑⠂⣳⡄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣾⣁⡛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢛⣈⢷⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⣃⣘⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣈⣧⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⢠⣟⢘⣛⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣛⣃⢿⡆⣄
⠀⠀⠀⠀⠀⡀⣰⡏⡿⢟⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡻⠿⢹⣆
⠀⠀⠀⠀⢰⡇⣿⣾⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣷⣿⢸⡆
⠀⠀⠀⡄⣿⣇⣏⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣝⣼⣿⢠
⠀⠀⢸⣿⢿⣿⢟⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡻⣿⡿⣿⡇
⠀⢠⢸⣿⣼⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣧⣿⡇⡄
⠀⣿⡟⣿⡿⣫⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣝⢿⣿⢻⣿
⢀⢹⣿⣼⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣧⣿⡏
⢸⣯⢿⣿⡿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⢿⣿⡿⣽
⠘⣿⣎⣿⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣿⣱⣿
⢷⡹⣿⣿⣿⣏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣽⣿⣿⣿⢏
⠸⣿⣎⢻⡟⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⡟⣳⣿
⠀⠹⣿⣦⣷⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⣿⣼⣰⣿⠏
⠈⢷⣮⡿⣿⣿⠇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠸⣿⣿⠿⣣⡾
⠀⠈⢿⣷⣌⢻⡄⣿⣿⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⢠⡟⣡⣾⡿⠁
⠀⠀⢀⡙⢿⣷⣿⣿⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣾⡿⠋⡀
⠀⠀⠈⠻⣶⣝⡻⠿⣿⠘⣿⣷⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣾⣿⠇⣽⠿⢟⣫⣶⡟⠁
⠀⠀⠀⠀⠙⢿⣿⣦⣜⢷⣽⣿⡇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢰⣿⣯⡾⣣⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣉⠻⠿⣷⣿⣿⣇⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⣸⣿⣿⣾⡿⠟⢉
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣭⣉⠙⢷⣬⡻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢟⣥⡾⠋⣉⣥⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣬⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣭⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⠻⢷⣶⣤⣄⣀⣀⣠⣤⣶⡾⠟⠋⠉⠛⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿⣟⣛⣛⣻⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡔⠉⡍⠁⠀⠑⡄⠀⠀⠀⡠⠚⠡⠊⠀⠉⢢⡀⠀⠀⠀⡴⠋⠀⢀⠀⡙⢆⠀⠀⠀⡠⠊⠁⣀⣀⠈⠳⡀
⠀⢸⠲⠀⠀⠐⠐⠐⢽⠀⠀⠀⡇⠈⠦⠀⠀⠐⠀⡇⠀⠀⢸⠀⠀⠀⠈⠠⠄⢸⠀⠀⠀⡃⢰⠋⣶⡆⢱⠀⠁
⠀⠘⣄⠅⠀⠣⡀⣀⡜⠀⠀⠀⠳⣁⡆⠀⠀⠀⣰⠃⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡈⠢⠽⠥⠊⣠⠃
⠀⠀⠈⠑⠒⠒⠛⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠉⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⡠⢚⡽⠋⠉⠲⡄⠀⠀⠀⢀⣴⠉⠉⠉⠑⣄⠀⠀⠀⠀⣠⠖⣩⡙⢉⡲⡀⠀⠀⠀⣀⡠⠤⠤⠤⢤⣀
⠀⢰⢱⠃⠀⠀⠀⠀⣸⠀⠀⠀⡏⢸⠀⠀⠀⠀⡇⡇⠀⠀⢸⠡⠈⢄⠈⠀⠐⢻⠀⣠⢚⠁⠀⠀⠀⠀⠀⢸⡓
⠀⠸⡈⢆⡀⠀⣠⠞⡸⠀⠀⠀⢧⠈⠲⠤⠤⠞⢠⠃⠀⠀⠸⡔⢢⡀⠡⠀⠤⡻⠀⠘⢿⣍⠉⠀⠀⠉⣉⡽⠟
⠀⠀⠙⠢⠬⠭⠕⠊⠀⠀⠀⠀⠀⠑⠦⠤⠤⠒⠁⠀⠀⠀⠀⠈⠳⠤⠦⠥⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⢀⠤⡶⢲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⢒⢢⠛⡿⠾⡆⠀⠀⠀⠀⣠⠴⡶⠖⡤⣀
⠀⢠⠃⠐⠀⠀⠀⠀⢳⠀⠀⢠⡖⠉⠁⠀⠀⠉⠑⢦⠀⠀⠀⢸⣿⢸⢰⢻⣸⠁⠀⠀⠀⡜⢡⠛⢊⡅⠂⡏⡆
⠀⠸⡀⠀⠀⠀⠀⠀⡿⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢻⣾⣾⡾⠃⠀⠀⠀⠀⣗⠰⠐⡆⠐⣩⠒⡇
⠀⠀⠑⠤⣄⣀⡤⠔⠁⠀⠀⠀⠈⠉⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⡿⠿⡇⠀⠀⠀⠀⠀⠘⠲⣥⣔⣷⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠤⠄⠒⡢⢄⡀⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⠀⠈⢣⣈⣯⡰⠋
⠀⢠⠊⠀⠀⠀⠀⠈⢳⠀⠀⠐⢭⣁⣠⠀⣟⣭⠆⢱⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⡠⠚⠛⠙⠣⣄
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣲⣷⣖⣧⣜⣋⡇⢸⠀⠀⠀⠀⢀⣎⣀⣧⠀⠀⠀⠀⠀⡜⡐⢡⡀⠀⠈⡘⡄
⠀⠈⠣⣄⡀⢀⣀⠴⠃⠀⠀⢸⣞⣻⢽⣿⣁⠤⣤⠎⠀⠀⠀⠀⢸⠦⠢⢾⠀⠀⠀⠀⠀⢡⡀⠀⠀⢀⠠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠳⢶⣄⡀⣀⣠⣤⣤⠶⣶⣶⡶⣶⣤⣤⣄⡀⢀⣤⡶⠛⠻⣦
⠀⣀⣤⡶⠶⠶⣤⣄⠀⠀⠀⣿⡄⣤⠀⠉⠛⠛⣿⡟⠀⢀⣿⣿⢰⣿⡇⢀⣿⣿⠟⠁⢀⠔⠂⢻⡄
⢰⠟⠁⠀⣄⠀⡴⢌⠻⣦⠀⢸⡇⣇⡅⠀⠀⠀⠻⣿⡄⢸⣿⡇⣾⣿⣠⣾⣿⠏⠀⠀⠳⣄⡠⢸⡇
⣿⠰⡀⡀⢢⠇⠱⠴⠀⢹⣧⢸⣧⣬⣤⣤⣤⣤⠀⠈⠁⢸⡿⠀⠿⠁⠈⠁⠀⠀⣀⣤⣤⣬⣅⣸⣧
⢹⡆⠙⠁⡠⠴⠦⡀⠀⠀⣿⣼⡿⠟⠋⢉⣉⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⡛⠛⠻⢿⣿⡄
⠈⣷⠀⢸⠁⠀⠀⠈⡄⠀⢸⣿⠁⠀⠜⢹⣿⣿⣿⠋⢹⣦⠀⠀⠀⠀⣴⡟⠉⣿⣿⣿⣏⠳⡄⠀⠹⣇
⠀⢹⡇⠈⡄⠀⠀⠀⣷⠀⢸⡇⠀⠀⢣⡘⢿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣻⣿⣿⣿⣿⡿⢃⡴⠁⠀⠀⣿⣀⡀
⠀⠘⣿⠀⠘⢄⣀⡠⠟⠛⢿⡷⠶⣦⠀⠈⠑⠊⠩⠉⠑⠋⢀⣀⣀⡀⠙⠚⠩⠭⠑⠊⠉⠀⢀⣛⠛⣿⠛⠋
⠀⠀⢿⡄⠀⠀⠀⠀⠴⠶⢿⡷⠶⠆⠀⠀⣠⣤⠀⠀⠀⠀⠉⢹⠋⠁⠀⠀⠀⠀⢠⣦⡀⢠⣌⡉⢻⡿⠷⠶
⠀⠀⢸⡇⠀⢀⣀⡀⢀⣠⣼⣷⡾⠗⣀⣼⣿⡏⢀⣄⠀⠐⣤⢾⠧⡴⠂⠀⣰⣆⠀⢿⣿⣦⣍⣙⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠘⣿⣿⣿⠟⠁⠈⠉⠀⠈⠻⣾⣿⣿⠟⢀⣾⣿⠀⠀⠈⠂⠐⠁⠀⠀⢻⣿⣆⠈⢻⣿⣿⡿⠁⠈⠉⠀⠀⠀⠀⢀⣤⠾⠛⠉⣿⣿⡇⠀⠀⠈⠙⣷⣦⡀
⠀⠀⠀⣿⠉⠀⣀⣠⣤⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠷⣶⣶⣦⣤⣤⣤⣤⣶⠾⠿⠿⠛⠛⠋⣹⣇⠀⠀⠀⠀⠀⠀⣴⣟⠁⢠⣄⠀⠸⠿⠁⣠⣿⡆⢠⣿⡿⢿⣄
⠀⠀⠀⢹⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⠀⠀⠀⠀⢀⣾⢿⣿⣆⠹⣿⣿⡶⠶⠶⠿⣿⣀⢘⣩⣤⠀⢻
⠀⠀⠀⠈⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⢀⡾⢃⣤⣤⣀⣰⡿⠁⠀⠀⠀⠀⠀⢹⡿⠛⠉⠀⠘
⠀⠀⠀⠀⠈⢻⣄⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣇⠀⢀⣾⣇⡀⠙⢿⣿⡟⠀⠀⠀⠀⠀⣀⣴⣾⣃⠐⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣷⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣾⣿⣿⣿⣿⣿⡀⣼⣿⣿⣿⠄⠀⡿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢻⣷⡟⠀⣤⣶⣶⣾⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠙⠿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢴⣶⣶⣾⣿⣿⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⢻⡿⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢠⣶⣶⣶⣿⡇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠛⣿⠄⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣤⣤⣴⣿⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠟⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠉⠛⠛⣿⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⡇⠀⠀⢰⠀⠀⠘⣷⣦⣤⣤⣤⣤⣼⡇⠀⠀⢠⠀⠀⠰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢀⣇⠀⠀⢀⣇⢀⣸⠏⠀⠀⠀⠀⠀⠘⣷⣀⠀⣸⠀⠀⣠⣀⣴⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣽⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⣤⠂⢠⡀⡴⠀⣰⠀⡆⠀⢠⠆⣴⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⠅⣇⠀⡾⠷⣷⢂⡏⢰⣧⠀⢺⣤⡏⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢿⣙⢦⣀⣀⣀⣠⢆⡟⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⢸⡝⠁⠹⡿⠋⠉⠘⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡋⠉
⠀⠀⠀⢸⣐⠿⢄⡁⣼⠗⢤⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠿
⠀⠀⠀⢸⢾⣥⠀⠁⠀⠛⠛⣠⣦⣄⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣾⡟
⠀⠀⠀⠀⢳⡶⠄⡤⢜⣶⣾⠹⢿⠇⣹⢻⡄⣉⡁⠘⣟⠋⠀⠉⠺⠿⠿⠿⢿⠿⠿⢿⣿⡆⢒⠀⠠⡄⢲⣿⠂⠛⠁
⠀⠀⠀⠀⠈⢝⢂⣙⣛⡵⢣⣠⠟⠤⡟⢸⠁⢹⣿⠀⣰⣿⣿⡆⣴⣶⣄⢠⣌⠉⠀⠀⣀⡀⠛⠿⡿⣩⣊
⠀⠀⠀⠀⠀⠈⠁⠉⠉⢑⠁⠄⣴⠄⠀⣼⠀⢸⡱⠆⠿⠟⢋⣀⠈⠉⠁⢿⡿⣤⣦⠸⠿⠇⣀⠀⢈⢳⣿
⠀⠀⠀⠀⠀⠀⠆⠀⢈⣉⣐⠂⣀⢀⡜⠁⠂⢸⡇⣴⣶⡄⣿⣿⠃⣾⢆⣴⠸⠿⠋⣀⠀⠿⠵⠚⢹⣬⣿
⠀⠀⠀⠀⠀⠀⠐⡦⢀⣈⣹⣧⣙⡋⠀⠸⡃⢸⣧⠻⠟⢣⢨⣧⣤⣱⠋⢉⣀⡀⣸⣿⠇⣠⣴⡄⣨⢛⡭
⠀⠀⠀⠀⠀⠀⠀⢀⠲⢿⣍⠹⣯⣅⢙⣾⣀⡾⠃⠘⠷⠶⠘⠋⠉⠈⠑⠛⣁⣠⠌⢁⡀⠠⠤⠈⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢷⣎⡈⠻⢾⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢁⣀⠀⠀⣠⠀⢀⠀⣰⡆⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠁⢴⡇⠐⠿⠀⠁⠀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠛⠆⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⢀⡤⠀⠀⠀⠈⠄⢀⡇
⠀⠀⠀⠀⠀⣤⢊⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⠚⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠮⠶⠶⠖⠋⠀⠀⠀⠀⠐⠂⠐⠒⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣄⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣿⣿⡿⠿⠛⠏⠉⠉⠙⠉⠛⡏⠉⠙⠉⠛⠓⠲⢶⣦⣌⣉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣽⣾⠿⠿⠉⠁⠈⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠁⠍⠻⠿⣶⣞⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⣠⡾⠋⢁⡠⢴⠾⠿⠿⠋⠋⠉⠉⠁⠘⠀⠈⠉⠉⠙⠓⠒⠆⠤⣀⡀⠀⠀⠀⢄⡙⢷⣄⡈⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⣴⠟⢫⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠈⠲⢟⢿⣿⣟⠙⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⢟⣿⠏⢈⣴⠟⢁⠀⠀⠀⠘⢦⣀⠀⠀⠀⣤⡀⠤⠤⠠⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢦⡀⠀⠈⢷⣟⣿⣷⡙⣿⣿⣿
⢸⣿⣿⣿⡿⢁⣾⠋⣠⣿⡏⣰⠇⠀⠀⠀⠀⠘⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠳⡄⠀⠀⠙⣷⠹⢿⣿⣿⣿
⢸⣿⣿⣿⢧⣾⠇⣰⣿⠉⣰⡿⠀⠀⣠⡾⠁⠀⠀⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣧⠀⢻⣿⣿
⢸⣿⣿⠟⣾⠏⣰⣿⠃⣴⡟⠂⠀⣰⠟⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢂⠘⣷⡏⢿⣿
⣾⣿⣿⣸⡟⢰⣿⠇⣸⣏⡀⢀⣾⠏⠀⠀⣠⠎⠀⢼⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⣰⣾⣿⣿⡿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⡀⠸⡄⣿⣿⣿⣿
⣿⣿⢷⣿⣅⣿⡏⣼⡟⠀⠀⣼⡿⠀⠀⣰⠏⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⢠⣇⠘⣿⣿⣿
⣿⣿⣼⣿⣼⡟⢰⣿⠃⠀⣸⡟⠠⠀⣰⡟⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠄⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠸⡇⠈⣿⠂⣿⢿⡿
⣿⣿⣿⣿⣿⠁⣾⡿⠀⠀⣿⠃⠀⢠⡿⠀⠀⡶⠀⢲⣶⣿⣿⣿⣿⣿⠟⠉⡍⠻⣿⣿⣿⣿⡿⢿⡖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠇⣄⣿⣿⣷
⣿⡇⣿⣿⡟⢸⣿⠃⠀⢰⡿⠀⠀⣸⠇⠀⢸⠃⠀⠀⣩⣿⣿⣿⣿⣇⠀⠀⠗⠀⣽⣿⣿⡏⠶⣸⠇⠀⡄⢦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⣿⣿⣿
⣿⡇⣿⣿⠁⣾⣿⠁⠀⢸⡇⠀⠀⢘⣠⠤⠤⢀⡀⠀⢥⣾⣿⣿⣿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣾⡿⠄⢠⣷⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡹⢼⣿⣏
⣿⣿⣿⡇⢀⣿⡯⠀⠀⠹⠁⠀⠀⣀⣀⠀⠀⠀⠙⠁⠈⢉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣯⣽⣿⡧⠀⠀⢼⠗⠀⠀⠒⠚⠃⠀⠀⠀⡜⠀⠀⠀⠃⣿⣿⣿
⢿⣿⣿⣧⢸⣿⠀⠀⠀⣀⣴⡾⠛⠛⢿⣷⣌⠀⠲⠀⠀⠐⠂⠐⠛⣿⣿⣿⡋⣿⣿⣿⣿⠟⠁⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣀⠀⣿⢹⣿
⢸⣿⣿⣿⢸⣏⠀⠀⣼⣿⠏⠀⠀⠀⠀⢻⣿⣆⠀⠀⡠⠂⠀⠀⠀⣿⣿⠃⠀⠀⠐⠀⠀⠀⠀⠓⢄⠀⠀⠐⠦⠀⠀⠀⠀⠀⣾⠀⠀⢀⡏⠀⣿⢸⣿
⣼⣿⣿⣿⢸⣿⠀⣼⣿⡟⣤⠀⠀⠀⢀⡄⠻⣿⣄⠀⠀⠠⠄⠀⠀⣿⣿⡃⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⡞⠀⢠⡟⢸⣿
⣿⣿⣿⣿⢼⣿⠀⣿⡏⠀⢻⡆⠀⠰⣿⣧⣤⣿⡟⠀⠀⠀⠀⠈⠒⢿⣿⠇⠄⠀⠛⠋⢁⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢠⡿⠀⢀⠜⠀⠀⢸⡇⢸⡯
⢻⣿⣿⣿⠜⣿⣄⣿⠃⠀⠸⣷⡀⠀⠘⠿⠿⠋⠁⠀⠰⣦⠀⠀⠀⢾⣿⡆⠀⠀⠀⠈⠉⢀⣤⠞⠁⠀⠀⣠⠃⠀⠀⢀⣼⠃⢀⡎⠀⠀⠀⡾⠀⣿⡇
⢸⣿⣿⣿⠀⠸⣿⡿⡀⠀⠀⠹⣷⠀⠀⠀⠀⢤⡀⠀⠀⠈⠓⠄⡀⢹⣿⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠐⠁⠀⠀⢀⡼⠃⠀⡸⠀⠀⠀⣼⠃⢀⣿⠧
⢸⣿⣿⣿⠀⣾⣿⡃⣿⡄⠀⠀⠹⣷⡀⠀⠀⠀⠛⢶⣄⠀⠀⠀⢠⣾⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⡼⠃⠀⠠⠁⠀⠀⢰⡇⠀⢸⢿⡆
⢸⣿⣿⣿⠁⢿⣿⡁⠘⣷⡀⠀⠀⠙⢷⣀⠀⠀⠀⠈⠹⢷⣦⣀⢸⣿⣿⣿⠀⠀⢀⣀⣀⠶⠖⠉⢀⣀⡀⠀⢠⠎⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣾⢹⡿
⣼⣿⣿⣿⡆⠸⣿⡇⠀⠹⣷⡀⠀⠀⣸⣿⣷⣄⠀⠀⠀⠀⠈⠙⢹⣿⣿⣿⠂⠒⠉⠉⠀⢀⣠⣴⣿⣿⠁⠔⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⠇⣿⣇
⣿⣻⣿⣿⣧⠀⠘⣿⡆⠀⠙⣷⡀⠀⢿⣿⣿⣿⣷⣄⠀⠈⠑⢠⣼⣿⣿⣿⡄⠀⠀⣠⣴⣿⣿⣿⣿⡟⢀⡀⠀⠀⠀⢀⠀⠀⠀⣸⠁⠀⢠⡟⢀⣿⡇
⣿⣟⣿⡿⣿⡦⠀⢉⣿⣦⡄⠈⢷⡄⠈⢿⣿⣿⣿⣿⣷⣄⠀⣸⣿⣿⣿⣿⣇⣠⣾⣿⣿⣿⣿⡿⠋⠠⠊⠀⠀⢀⠐⠁⢀⠀⣴⠃⠀⢠⣾⠁⣸⣿⣷
⣿⣿⣿⣿⡿⣿⣧⡀⠈⣻⣷⣦⠀⠈⠠⠀⠙⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠊⠀⠀⢀⣴⠋⠀⠀⣠⠟⠁⠐⢃⡾⠃⠰⣿⣿⣿
⣿⣿⣿⣿⣷⣜⣿⣷⣶⣉⣛⠻⢿⣷⣤⣧⡀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⡴⠋⠀⠀⣴⡿⠁⢀⣴⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠷⣦⡈⠿⠿⢿⣦⣤⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⣴⠟⠀⠀⢀⣾⠟⠁⢀⣼⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠛⢷⣤⡀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⢀⣠⠶⠛⠁⢀⣠⠾⠟⠁⠀⠀⢠⡾⠃⠀⣼⡿⢻⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣆⠀⠀⣤⣴⠏⠀⣠⣾⠛⣿⣿⣿⣿⣿⡯
⣿⣿⣿⠟⠿⠟⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⣻⣿⠿⢿⡿⠿⣿⠿⠛⠛⠛⠛⠛⣽⣾⠟⠁⢀⣴⡟⣁⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣰⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⡴⡶⢶⢶⡾⠿⠛⠀⢀⣀⣴⣿⢟⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣶⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠒⠂⠐⠘⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠋⣀⡤⠒⣀⢀⣀⢉⣉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠞⢁⡴⣄⣉⢡⠥⢤⡅⠠⠴⣄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢀⡴⢋⠀⠈⣿⡶⠆⠀⢤⣔⡶⢨⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⢁⡟⠰⠟⠉⠀⠒⣛⣂⢛⡲⠶⢙⠛⠶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⡾⠇⠀⡄⠀⠉⠩⠭⠅⡀⠬⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠁⠀⡸⢠⡄⡖⢀⠐⠃⠋⡛⢂⠰⡆⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠇⠼⡓⢦⣄⣀⠁⠀⢀⣤⡴⠒⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣇⠀⡆⠁⢀⢠⡈⢀⠈⣿⠛⠛⢿⣟⠀⢋⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠸⠆⣸⣧⠘⠦⠀⢠⠀⡏⠀⠀⠘⣧⠈⣩⡔⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢴⣿⠻⣶⣄⣰⠍⠀⠃⡄⢀⡀⠁⠀⣤⣴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣼⣿⢦⡈⠛⠻⣄⡀⡀⢸⣏⠀⢀⣴⠛⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢻⣿⣄⠹⠄⠀⠈⠹⣤⣀⣤⡼⣋⣴⠾⠥⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡇⣹⣆⠀⠀⠀⠀⠀⠉⠁⣰⠏⠁⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣾⡇⠀⠙⠳⢄⠈⠶⣦⣄⡘⠁⠀⠈⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣤⣀⠀⠱⠄⠀⠀⠀⢀⣠⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠒⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⢀⠀⠀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣸⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⣴⠿⠀⠁⠀⢠⡄



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

⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⣖⠱⣭⠎⢤⠀⠀⠀⠀⠀⠀⠘⢦⣙⣋⡼⠁⠀⠀⠀⠀⢀⢠⡴⣐⠿⣠⢦⡄⡀⠀⠀⠀⠀⣀⣀⡀⢀⡠⢄⢄
⠀⠸⣿⠃⣿⠰⣿⡇⠀⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠈⢻⡞⢹⣿⠉⣿⡎⠀⠀⠀⠀⡀⠛⠛⢋⡙⠋⣟⢕
⠀⠠⠿⠜⠉⢣⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠿⠻⠃⠻⠇⠀⠿⠗
⠀⠴⠤⠶⠤⠷⠤⠦⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠈⠁

⠀⠀⢠⣴⠶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⢆⡄⠀⠀⠀⠀⠀⠀⢠⣤⠀⣴⡄⢠⡄⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⣼⣿⢈⠁⣿⣥⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣬⣥⣤⣽⣤⣬⣥⡄⠀⠀⠀⠀⣤⣽⣥⣼⣧⣀
⠀⠀⣿⠛⢿⡟⠋⣿⠀⠀⠀⠀⠀⠀⠀⡼⠿⡀⠀⠀⠀⠀⠀⠘⢿⣿⢻⣿⡿⣿⣿⠃⠀⠀⠀⢀⣁⡿⣧⣼⢧⣄
⠀⠀⠘⢷⣾⣷⡞⠃⠀⠀⠀⠀⠀⣏⣉⣇⣀⣏⣀⡆⠀⠀⠀⠀⢸⢿⢸⣿⡇⢿⡿⠀⠀⠀⠀⠤⠭⠤⠬⠭⠤⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢾⣿⡟⠟⣿⣿⠂⠀⠀⠀⠀⣀⢀⣤⠀⡿⠫⠿⠇⠀⠀⠀⠀⡀⠘⢀⣁⠀⠃⡀
⠀⠀⢹⣿⣾⣟⠁⠀⠀⠀⠀⠀⠈⢲⣥⠞⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣹⡇⠀⢠⠀⠀⠀⠀⣤⣶⢦⡄⣤⢶⣶⡤
⠀⠀⣿⣏⢙⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣧⠀⠀⡄⠀⠀⠀⠀⠉⠙⠛⠁⠉⠛⠉⠁
⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠂

⠀⠀⢠⣤⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⡀
⠀⠀⠈⣇⠶⠶⣺⠃⠀⠀⠀⠀⠀⠈⠧⣜⣫⠼⠁⠀⠀⠀⠀⠀⢰⡆⢠⣯⡄⢹⡏⠀⠀⠀⠀⠀⣠⣤⣤⡀⢴⣶
⠀⠀⠀⠻⢷⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⣿⠁⡸⠇⠀⠀⠀⠀⠀⠁⠾⠻⠟⠯⠁
⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠜⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠈
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈⠀⠉⠁

⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⣼⠠⠲⣌⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡵⠒⢾⡇
⠀⠀⠀⠀⠀⢫⢀⠀⠈⠳⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠈⢧⡀⠀⠀⠀⠙⢦⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⢞⣡⡞⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢣⣆⢻⣦⡀⠀⠀⠈⢷⣄⠀⣀⣤⠤⠤⠄⠀⢀⣘⢐⣀⡼⢱⣿⣿⣤⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⠹⣦⡙⢿⣦⣤⣄⠘⠿⠻⠁⢠⠆⠀⠐⣆⠀⠈⠉⠘⢁⣾⣿⣿⠖⠂⡀⣰⡟
⠀⠀⠀⠀⠀⠀⠸⣆⢼⣿⣿⣏⠛⠀⠀⠀⠀⠀⢋⣆⠀⣶⠈⠀⠀⠀⠀⠈⠸⠛⠟⣠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣮⡛⠉⠀⠀⠀⠀⠀⣤⠀⠘⣿⣴⡿⠁⠀⣦⢀⣀⣀⠀⠀⠀⠀⠈⣼⠟
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠷⠀⠀⠀⣀⣉⡻⠇⠀⢠⣿⣿⡇⠀⠀⢛⣩⣤⣉⠁⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⠀⡞⠛⢻⡿⢦⠀⡏⢸⣿⢻⠀⠀⠞⢙⣿⠉⢋⢠⣤⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡾⣿⡄⢧⡘⠿⠃⣸⠀⡀⣼⣿⠸⠀⢸⡅⠈⠋⡀⠀⡸⢿⣷⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⡏⠢⠍⠀⠀⢸⣆⡇⠀⠀⠀⡄⢺⡀⠀⠀⠐⠊⣡⣾⣀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠚⠙⢦⡀⢀⡞⠛⢘⣧⡄⠀⢠⣧⠀⠉⠢⣄⠀⠚⠁⠀⢉⡞⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠳⣤⡀⠀⠀⡀⠀⠘⢧⠀⠠⣤⠟⠀⠄⠀⠀⠀⠈⠁⣭⢏⡁⠈⢿⡐
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣿⢳⣖⡂⢛⠀⠀⠑⡋⠁⠀⠈⠄⠦⡨⠠⠞⠛⠠⢀⠀⠁⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠠⠂⠀⡠⠔⠉⢼⠈⠀⠀⠂⢄⠒⠉⠉⠙⠓⠒⠒⠂⠀⠐⢌⠐⢄⠀⠁⣦⣸⠹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⢦⡀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠡⠀⢱⣶⡿⠃⠀⢳⠑⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠳⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡤⠟⠈⠀⠀⠀⡘⠀⠇⠈⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣄⣄⣆⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⣾⢳⠀⠀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠑⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡟⠈⠀⠀⢸⡟⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠏⠀⠀⢠⣰⡟⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⢦⣄⣸⠳⣼⣄⡀⢠⡀⠀⠀⠀⠀⠴⠋⠀⠀⠀⣦⡿⠋⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠉⠛⠂⠙⠙⠻⠾⠙⠒⠂⠀⠀⠀⠀⠀⣠⠾⠋⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠊⢁⡤⠴⠚⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣲⣯⣭⣭⣭⣍⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃
⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠸⠿⠿⣿⣼⠛⠿⠃⢸⣿⠟⠦⠾⠿⠃⠀⠀⢀⣿⣿⣧⠀⠀⠀⢸⣿⣿⡀⠀⣸⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣤⣤⣤⣤⠀⠀⠀⣾⡿⠘⣿⣆⠀⠀⢸⣿⢻⣧⢀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡆⠀⣡⣿⠀⠀⠀⢸⣿⠉⠉⠉⠉⠀⠀⣼⣿⣷⣤⣿⣿⡄⠀⢸⣿⠘⣿⣼⡿⢸⣿⣿⣏⠘⠃
⠀⠀⠀⠀⠸⡿⠀⣿⣿⢀⡤⠂⢸⣿⣶⣶⡶⣶⡆⣰⣿⢏⢱⣤⠬⢻⣿⡀⢹⣿⠀⢿⣭⡇⢸⣿⡿⠇
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠞⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣦⠀⠀⢠⣤⠄⠀⠀⣠⣤⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣠⠈⣇⡀⢀⣤⣤
⠀⠀⠀⠸⣿⡇⠀⡾⠁⢀⡀⢀⣿⡏⢀⣾⡿⠉⠁⠉⠋⠹⡄⢀⣿⡏⣩⣭⠙⣿⡇⠀⣿⣶⢉⣴⣿⠟⠁
⠀⠀⠀⠀⢿⣷⢸⣶⠀⢿⣧⢸⣿⠁⢸⣿⡇⠀⠀⠑⢄⣾⡇⠀⣿⣷⣾⣿⣿⠟⠃⠀⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⡏⠀⠸⣿⣿⡏⠀⠈⢿⣷⣀⣀⣀⣼⣿⠃⠀⣿⡇⠀⠙⢿⣷⡀⠀⣿⣿⠁⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠛⠛⡡⠄⠀⠛⠛⠁⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠛⠃⠀⠀⠈⠛⠛⠀⠛⠛⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠐⠉⠴