Hbd Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⡄⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⣆⡟⠀⠀⠀⠀⢀⠎⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⠀⠀⠀⠀⠀⣏⡤⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⣧⠇⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⠀⠀⠀⠀⠀⢹⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⡯⢾⠀⠀⠀⠀⠀⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⣀⣴⣿⣛⣦⠤⣾⣒⣻⣦⣴⣶⣄⣸⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⠴⣿⣟⣻⡇⠙⠿⠿⠃⢾⣍⣛⣛⣹⠇⠈⢻⠻⣶⣿⣾⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⠇⠀⢸⣝⣛⣷⡀⠀⠀⠀⠀⣀⠉⠉⠀⠀⣠⢿⣷⣽⡇⠙⠛⠛⠉⢳⣾⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠁⠀⠀⣼⣯⣉⣩⢿⡗⠀⠀⠈⠛⠀⠀⠀⣾⠻⣦⣝⣻⣿⡄⠀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠛⠀⢀⣠⣄⠑⠢⠿⠖⠛⠀⠀⢀⣤⡶⣤⡀⠀⠈⠙⠲⠶⠶⠞⠁⣀⣀⠀⠀⣭⡍⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣷⣿⡃⠀⠀⢀⣀⠀⠀⣿⣼⣿⡿⡧⠀⠀⠀⠀⠀⠀⢀⡼⣷⣿⣿⡄⠈⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣴⣾⣿⣶⣬⠑⠒⠁⠀⢀⡘⠛⠀⠀⠻⢭⡭⠖⠁⠀⠀⢰⡶⠀⠀⠈⠳⠶⠿⠛⠁⢠⣶⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣾⠷⣿⣿⣿⣿⣿⡇⠀⢀⣠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣴⣆⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡿⢼⣷⣷⡚⢻⣿⣿⣿⣿⡀⠀⣾⣿⣿⣿⡇⠀⠀⣦⡀⢀⠀⠀⠀⣴⣦⡀⢠⣿⡀⠀⣸⣿⣿⣶⣿⣿⣿⡗⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢘⣶⣏⣸⣃⣹⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣧⠀⢸⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣿⣳⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⡽⢻⣏⣷⡀
⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⡟⠀⠀⠀⢹⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣦⣾⡟⢻⠟⢿⣿⣿⣿⡉⠉⢻⠁⠀⠀⣈⣿⣾⣿⣾⡿⢻⡇
⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠙⢦⡔⠙⣾⠛⢳⣿⣿⣿⣿⣿⠛⢿⣿⡀⠀⢈⣿⣿⣟⡉⣱⣦⣔⠉⣿⠋⠉⠹⣷⠾⢦⣀⣈⣿⣏⠀⠙⠷⣣⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠻⡛⢦⣔⡗⠉⠛⠿⣷⡄⠻⣿⣿⣿⠿⣶⣿⣿⣿⣶⣾⣿⣿⣇⡴⢩⠛⢦⣨⣿⠷⠤⠴⢋⣶⡇⠈⠁⠀⠁⣀⠤⢚⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⡙⠳⠶⣤⣀⡀⠀⢀⣈⣀⡀⠈⠛⠛⠋⠈⠉⠉⣭⠻⢷⣿⣶⡾⠟⠁⠀⠀⠀⢈⣋⣠⡤⠶⠚⣋⣥⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣉⡙⠛⢯⣁⣀⣹⣦⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⡴⠶⠶⠶⠛⠛⣋⣉⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠾⠷⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⢠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢻⣷⠀⢠⡿⠀⠄⠀⠒⠀⠒⠄⠐⠤⠀⠢⠀⠢⠄⠀⠢⠄⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣼⠧⠤⣾⠇⣡⡶⢳⡾⠐⣷⡖⢳⡄⢸⣶⠞⣷⠀⣾⠆⣰⡗⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡤⠃⣰⠏⠀⣸⣏⠠⣿⣠⣾⡃⣸⡏⣠⡿⠁⣿⢁⣰⠏⢰⣟⣴⣿⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠧⠈⡉⠀⠀⠈⠉⠀⠉⠁⠈⢁⡿⠉⠉⠀⣸⡏⠉⠡⠀⠈⢉⡼⠃⡼⠁
⠀⠀⠀⢠⡞⢁⣤⠤⢤⡄⢀⣤⠀⠘⠃⠀⠀⠀⠘⢁⣶⠀⠀⠋⠀⠀⢀⡶⢀⠋⠁⠞
⠀⠀⠀⢈⡇⢠⣿⢃⣰⠟⢤⡅⢠⣄⣠⡄⣴⣿⠄⣸⣇⣤⡌⠁⣠⢤⣼⠃⢀⡤⣤⡤⠠⣤⠀⣠⡄⢹⡄
⠀⠀⠀⣸⠁⣾⠛⢛⣧⢀⡿⠁⣸⡟⠁⢀⡿⠀⢠⡿⢋⡿⠁⣾⠃⣰⡏⢰⡟⢀⣾⠃⣸⠏⣴⡿⢀⡾
⠀⠀⠀⢻⡀⠛⠶⠚⠁⠘⠗⠀⠟⠀⢠⠘⠷⠚⠻⠁⠘⠗⠂⠻⠞⠛⠗⠘⠷⠛⠻⠂⠻⢚⣿⠃⣼⠁
⠀⠀⠀⠀⠉⠃⠀⠀⠈⠑⠀⠉⠀⠀⠀⠉⠂⠈⠒⠀⠁⠀⠉⠀⠈⠁⠀⠁⠀⠀⠀⠀⠰⠞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⢠⠐⠈⡁⠁⠉⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠
⠀⠀⠀⠀⠀⢫⠀⠀⠀⠠⠌⠀⠈⠀⠀⠀⠀⢂⠐⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⢏⠀⠀⠀⠀⡀⠀⠓⠳⠢⡀⠀⠀⠀⠐⠆⠀⠀⠀⠄⠁⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠄⠀⠀⠀⢀⠀⡐⠃⠉⠂⠀⠽⡃⠀⠀⠀⠀⠀⡀⠠⢆⠰⠀⠈⠀⠀⠐⠁⡘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣎⠦⣁⡇⠘⣠⣀⠀⡌⣅⠀⡀⠘⠀⠇⢀⠀⢠⠂⠘⠀⠀⠈⠀⡇⠈⠐⡎⠀⠀⠀⢊⠀⡀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⢀⣁⠉⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠁⠀⠀⠀⠉⣐⠴⠂⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠊⠁⠤⠨⠀⠀⠀⠀⠀⠂⠀⠀⠀⠅⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠊⠉⠁⣲⢄⡀
⠀⠀⠠⠀⠃⠑⠂⢠⡠⠚⠗⠂⢐⢲⠀⠀⠐⠒⠀⠐⠂⠐⠂⠀⠀⠀⠀⠀⡤⢠⠀⠤⣶⣶⠄⠀⠰⠁⢁⡨⠂⠀⡁⢡
⠀⢀⡀⠄⠀⠀⠚⠆⡐⠁⠀⠀⠈⣚⠄⠀⠀⠀⠀⢀⡆⠙⢲⠀⠀⠀⠀⡎⠀⠎⠐⢠⡀⠀⠀⠀⠀⠈⠀⠀⡤⠈⢱⡈
⠀⠀⠀⠠⣠⠌⠼⡀⠳⡄⠄⠀⡜⠁⠀⠀⠤⢠⡚⢁⠛⢆⡀⠑⠊⠑⠊⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠂⠃⠀⠀⠀⠀⠰⡊⠀⡰⠆⢚⡙⠁⠐⠀⠈⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢠⡁⠀⠀⠠
⠀⠀⠠⠄⠀⠀⢴⠄⠀⠐⠂⠀⢀⡐⠃⣀⠀⠄⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠉
⠀⢄⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠆⢀⠂⠈⡐⠀⠀⠀⠈⢑⡈⠁⠀⠀⠀⠐⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠉
⠀⠈⠀⠀⠌⠁⠀⠀⠀⡠⠀⠈⠁⠸⡀⠂⠀⠁⡀⠀⠀⡀⠒⢩⠞⢄⠄⠀⠀⠀⠈⠲⠀⠀⠀⠀⠠⠀⠁⠁⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⡀⠀⠀⠀⠀⠠⠀⠴⡆⠒⢧⠀⢌⡷⠢⠀⠀⠀⠀⠀⠂⠀⠀⣠⠀⢱⠠⠐⠂⠤⠐⡤⠊⠐⠂⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⣈⠢⢴⠀⡰⠄⠊⠲⠜⠆⠇⢀⠂⠀⠀⠐⡁⠀⠀⠀⢀⠀⢰⠃⠀⠀⠳⠀⡆⠄⠀⢀⠎⠀⠤⠘⠠⠖⠀⠀⡀⠀⠠⢀
⠀⠀⠀⠀⠴⢀⠀⠀⢀⠀⠀⢈⡀⠘⠀⠁⠚⠛⠂⠘⠚⠓⠒⠃⠐⠈⠀⠁⠀⠐⠀⠀⠀⠃⠀⠀⠀⠘⠤⠧⠀⠀⠈⠀⠤⢀⠀⠀⠒⠀⠂⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡤⠤⡄⠀⠀⠀⢀⠀⠴⢰⢐⠰⠀⣀⠀⢰⠀⠀⠀⡂⠀⡄⡆⠀⠀⠂⠰⠀⠂⠀⠀⠀⠀⠀⠀⠰⠐⣁⣠⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⡠⠈⠀⠀⠖⠠⡇⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠄⠠⠘⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠨⠂⠈⣐⠦⣅⠀⡀
⠀⠀⡀⠄⠳⠓⠛⠆⡠⡤⠠⠖⡳⠖⠀⠰⢰⠀⠀⠐⠲⠒⠀⠒⡖⠒⡒⠂⠀⠀⠀⠀⠀⠀⣤⠤⡤⠠⢤⣤⣤⣤⠀⠀⠰⠇⢇⠀⡹⠀⠀⠋⠈⡄
⠀⠀⠀⢀⡁⠀⠀⠥⢦⠈⡠⠊⠀⠀⢀⠃⣾⡄⠀⠂⠠⠀⠂⠀⣤⠀⠧⡤⠀⠀⠀⠀⢠⡴⠃⣸⠂⠤⡀⠉⠈⠀⠀⠄⠀⠘⠄⠉⠐⡂⠐⢣⠄⡋
⠀⠈⠉⠀⠁⠀⢀⠒⡆⠀⣇⠀⠀⠠⠂⡐⠖⠁⠐⠀⠀⠀⢠⠞⡻⣆⠀⠘⢤⡤⠦⣀⠼⠁⠠⠁⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠐⠺⠥⠀⠌⠋
⠀⠀⠀⠀⠑⠒⠁⠉⡱⠀⠙⠓⠈⣀⠐⠂⠀⠀⠀⠀⠀⡰⠤⠐⠃⢈⣳⣄⣀⠀⠀⡀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠁⠈⠀⠀⠀⠀⠀⠀⠫⡔⠤⡼⠿⠨⣪⠌⠀⠈⠁⠀⠀⠀⠀⠀⠞⣀⢀⡀

⠰⣦⡄⢠⡄⠠⠀⢀⣄⠠⣤⣤⡄⠀⡀⣲⠦⠶⠢⠐⠒⢲⣦⠀⢴⠦⣰⠖⠖⠒⠒⠒⢒⣶⡶⢶⣒⣶⡶
⠠⠌⠀⣀⣀⣑⡂⠸⠔⡊⡅⣄⠈⠚⠅⠈⠫⢔⠕⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠛⢋⢴
⠀⣮⠎⠀⠀⠀⠈⢳⡀⠀⠀⠈⢄⣀⣀⡠⡣⠀⠀⠀⢠⡀⡄⠠⠀⠀⡀⡀⣄⠀⠀⠀⠀⠀⠀⠀⢲⡥⢴
⠰⣿⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢈⣽⠀⣲⠀⠕⡻⠁⡠⠁⠀⠀⠀⠀⠀⠀⠀⣞⢼
⠰⡌⠀⠀⠀⠀⣠⠃⡸⡆⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠛⠲⠀⠇⠀⠃⠀⠀⠀⢰⠉⠒⡴⠒⡄⢸
⠀⠷⢄⣀⡠⠔⠁⡰⠁⡇⢀⠎⠀⠀⢢⡄⢱⣠⡀⠠⢴⠙⣫⡀⠒⢋⢭⢢⡀⠀⢀⡀⣘⣆⠀⠁⢰⠁⠸
⠀⠈⠐⡟⡠⠀⠔⠁⡸⡻⠁⠀⠀⠀⢈⠣⡊⣬⠀⡹⠋⡄⢠⠹⡇⠸⢀⢠⢋⠷⠀⣇⠀⠀⢱⠖⠛⣮⠂
⠀⢀⠂⠣⡄⠀⠀⠰⠛⠀⠀⠀⠀⠀⣎⠃⠄⠟⠀⠃⠀⠗⠺⠒⣃⣄⡫⠊⢘⣷⣦⣿⢻⢄⠙⠠⠤⠂⢻
⢸⣤⠆⠁⠀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢤⣤⡾⠃⠘⠃⠀⠀⠀⢸
⠈⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⠀⠀⡄⠀⠀⠀⢸
⢠⡌⢄⡴⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠆⠈⠠⠀⠀⠀⢸
⢸⣧⢸⠀⠀⠀⠙⢦⣤⣀⢀⣀⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢲⠁⢰⡇⡀⢠⠀⠈
⠐⢝⠙⣴⣊⡉⢑⣼⠀⢸⣋⡀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈⢙⠉⠀⡀⢀⠀⣀⢨⠄⠒
⠀⠈⠀⣸⣸⠉⠉⠙⢳⡟⠚⢻⡒⠬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠒⠀⠘⠒⠀⡦⢤⠐
⠀⠐⠀⢩⢹⠀⠀⠀⢸⡇⠀⠸⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣄⣀⣀⣀⣀⣀⡀⢀⣇⣐⢄
⠀⠘⠀⠀⢸⠀⠀⠀⢸⠇⠀⠅⠀⠣⠚⠢⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠀⡠
⠀⠀⢤⣴⢿⠀⠀⡆⢸⠀⠀⠀⡇⢀⢠⠲⠤⢄⣀⠀⢄⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⡜
⠀⢀⠮⠅⣙⣶⣤⣧⠧⠤⠜⠲⡇⣘⣊⣀⡀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢣
⢀⠀⠶⠦⢤⣤⡬⠥⢤⣤⠵⠒⢲⡠⢶⣶⣦⣤⣤⡭⠟⣺⡇⠆⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣀⣐⣈⣃⣀⣀⣃⣂⣊⣃⣃⣒⣙⣀⣀⣒⣹⣋⣂⣃⣈⣘⣂⣀⣀⣠⣀⣐⣒⣒⣉⣉⣉⣉⣐⣂⣀⣀⣊
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡶⠂⣴⠀⣷⣠⣤⣄⠰⣿⠞⠛⣷⡀⣾⡇⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠴⣶⢀⣿⡀⢸⡿⠀⣿⠁⠈⣿⠄⣿⠀⠀⣿⠁⢿⡆⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⣿⠘⣿⡤⢾⡇⠀⣿⠠⣤⠟⠀⣿⠌⠛⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⢿⠔⠀⠀⠘⠗⠀⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣿⠇⢀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⡿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠛⠃⣶⠂⠀⠰⣤⣠⣿⢀⣴⠆⠠⠤⠴⠟⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⠀⠀⠀⠀⠀⣿⠂⢿⡀⠀⠀⠀⣽⠇⣼⠇⠄⠀⠀⢀⣀⣼⡇⢀⡤⢀⡄⣰⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢠⡶⢶⣦⢠⣿⠀⠀⢠⡾⠃⢠⡿⠀⣿⠀⣀⠀⠀⣾⠁⢹⡇⣾⡇⣸⡇⣿⠃⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠀⠀⣿⡟⣿⠀⠀⣾⠁⠀⢸⡇⠀⢻⡆⢹⡆⠈⠿⠔⠙⣧⠙⠋⠹⠧⠛⠃⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠦⠟⠀⠉⠀⠀⠷⠚⠁⠘⠋⠀⠀⠙⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠓⠀⠀⠀⠀⠈⣁⡤⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠴⠀⠀⢁⠀⠈⠉⠀⠀⠰⠖⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢠⡴⠞⠞⠛⠀⠘⠙⠻⠛⠁⠠⠞⠗⠀⢀⣤⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⢀⡀⠀⠀⣠⡴⠀⠀⠶⠃⠀⠀⠁⠀⢀⣠⡄⠈⠙⠁⠀⠄
⠀⠀⠀⠀⠈⡤⠀⠊⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣸⠂⠀⠻⠂
⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⡄⣤⡴⠠⠶⠶⢢⣤⡠⢀⣀⠀⠀⣀⡔⠁⠀⠘
⠈⠀⠀⠠⠄⠀⡔⣀⣠⠄⠔⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠾⢦⡄⠚⠀⠀⠀⡤
⠀⠀⠂⣀⣄⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣢⣊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣷⣆⠀⠀⠀⣴⡟⠁⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⣿⠏⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣀⣼⡟⠀⢀⣠⣤⣀⠀⠀⣰⡿⢃⣤⡀⠀⣴⡿⣁⣄⡀⣠⣴⣶⠖⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣾⡟⠛⠛⢻⣿⠃⢠⣾⠟⢡⡿⠁⢠⣿⠗⠁⣾⡇⢰⣿⠟⢁⣿⠇⢩⣿⠏⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⡇⠀⣿⠃⡴⣿⡃⢀⣾⣿⣒⣾⠟⢠⣿⣷⣢⣾⠟⠀⣾⡏⡠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠀⢸⣇⠀⠀⠀⢿⣇⠀⠙⠋⠀⠛⠋⣽⡏⠈⠋⠁⠀⣾⡏⠙⠋⠁⠀⠀⠙⠛⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢰⡿⠀⠀⠀⠀⣴⠞⢛⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⣠⣶⠀⠀⠀⣤⡀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠈⠱⠿⡿⠂
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠶⠶⢤⠀⠐⠛⠛⠛⠛⠛⠲⠶⢶⣤⣤⣾⣟⣀⣾⡟⠁⠀⠀⢀⣿⡇⠀⣸⡿⠁⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⣿⠋⢡⣿⠃⠀⠀⣸⠇⠀⠀⠺⠟⢀⣤⡀⠀⠀⢀⣾⠏⢉⣿⠟⠛⠿⠿⠿⠟⠋⠀⢠⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⢣⣿⠷⡶⠶⢿⡋⠠⢶⣿⠃⠀⢸⣶⣧⡀⠀⣼⡟⠀⣾⢏⡴⢾⣿⠋⣠⡶⠟⢻⣿⠁⠀⠀⣴⡿⠋⣿⠇⠀⣼⡟⠁⣴⡿
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢠⡷⢀⣿⠃⢀⣴⠟⣥⡟⢁⣴⡿⠀⢸⣿⠟⢠⡿⠃⣼⠟⠀⢀⣾⠃⠀⣠⣾⡟⢀⣾⡏⠀⣸⡿⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣀⣠⣤⡶⠟⠁⠸⠿⠶⠋⠁⠸⣿⡶⢟⣿⠃⠀⢾⠃⠀⠺⠧⠖⠻⠷⠾⠋⠻⠷⠿⠋⠹⠷⠋⠺⠷⠊⠻⠷⢊⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠶⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣿⣏⣠⣤⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⣄⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣽⣿⠟⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣤⣤⣶⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠰

⠀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀⠀⢀⠀⡀⢀⢀⠀⢀⠀⡀
⣿⢿⠿⠿⢿⣟⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣿⣛⣿⣟⣛⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⡚⠉⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡐⠁⠀⠈⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡼⠁⣀⠤⠒⠞⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣸⢁⠎⠀⠀⠀⠈⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⡔⠀⠀⠀⠀⢠⠃⠀⠀⡇⡎⠀⠀⢠⠃⡸⠂⢸⢠⠇⢀⡇⡼⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠴⢤⣀⠀⡜⠀⠀⠀⠀⠠⠁⠀⠀⠀⡇⡇⠧⠀⠯⢰⠧⠀⠁⡾⠄⠈⠸⠇⢸⠇⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⢳⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⠄⠀⠴⠃⠀⢀⠞⠀⠀⡀⠀⢀⠇⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡴⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⠤⠤⡈⠑⣆⠀⠀⠀⠠⠤⠞⠁⠀⣠⠊⠀⢀⠎⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼⠁⠀⠰⠒⠛⠢⢄⡀⠀⠀⠰⡀⠃⠀⠀⡄⠀⢸⠀⠀⡠⠞⠭⠉⠆⣰⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠈⣀⣀⡬⠤⢠⣏⢀⡜⠱⠀⠀⡀⠀⢧⠤⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠛⠉⠀⠀⢀⡞⠀⡞⠈⠑⠒⠀⠤⠤⠤⠀⠀⠀⠀⠀⠘⢄⣀⣀⠔⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣀⡤⠶⠶⠶⢦⣄⢀⡾⠀⣠⠀⠐⠀⠀⢀⡞⠀⡜⠀⢤⠀⠀⠀⢀⡼⠀⠀⠀⢀⡄⠀⡄⠀⢀⡀⠀⠀⡰
⢀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢠⠞⠁⠀⠀⠀⠀⠀⢙⣿⠁⢠⠇⠀⠀⡸⠀⡼⠀⡸⠁⢀⡏⢀⠆⠀⣸⠁⢠⠂⠀⡼⠀⣰⠃⢠⡏⠀⢀⠜⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢠⠁⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠺⠄⠀⠸⠇⠀⠧⠀⠁⠀⠼⠀⠻⠄⠀⠧⠀⠷⠄⠈⠧⠰⠧⠀⡼
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⠥⠄⠒⡾⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢰⠃
⠀⠙⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⡼⠁⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⣠⠔⠉⠱⡄⠀⠀⠀⠀⣠⠞⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠊⠀⠀⠀⠀⢠⠋⠈⢳⡀⠀⠀⣴⠃⠀⠀⠀⣿⠀⠀⠀⡴⠃⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢷⡀⠀⢿⣀⠀⣀⠼⠃⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⢿⡤⠔
⣀⣀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣒⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢿⡟⠀⠀⠀⠀⣸⣦⡄⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢠⠞⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡄⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣷⡀⣾⠃⠀⠀⡧⠤⣽⠰⣒⡄⢰⠒⡄⡶⠲⡄⣆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⡏⠘⢿⠏⠀⠀⠀⠇⠀⠘⠘⠦⠇⢸⠴⠃⡷⠔⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⣋⣵⡀⠀⠀⢠⠦⣄⠀⠀⢀⣠⡀⣆⣀⠀⢀⣠⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣧⣈⣅⡴⠀⠀⢸⠓⣣⢸⠀⡇⢸⡁⣏⠨⡇⡇⢨⠀⡅⣺⠘⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣳⣆⠰⡿⢋⢉⡄⠳⣀⠈⠉⠁⠀⠀⠁⠀⠁⠁⠀⠀⠈⠉⠀⠉⠉⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠉⠟⠛⣤⣄⠋⠁⢨⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⠉⣿⣶⣶⢆⡜⢯⣿⡿⢭⣷⢙⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⢲⣏⠛⢱⣎⡛⣉⢘⡿⢤⣴⡉⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣷⣍⣠⣤⡞⠋⣿⢋⠹⣥⣌⣀⣹⡿⣤⡦⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠰⣯⡤⣵⣼⣟⣋⡁⢶⣤⣾⡇⠈⣿⡙⠻⠷⢠⢌⣩⡄
⠀⠀⠀⠀⠀⠀⢴⠿⣦⣤⠄⢩⡁⠘⣙⢣⣬⠓⡼⠁⣶⣍⢵⣦⣀⣀⣬⣻⠃
⠀⠀⠀⠀⠀⢰⡷⠀⠋⢼⣤⢬⡟⢻⡏⣸⣿⣶⣶⡿⢋⣱⡆⠙⡿⡝⠣⣿⡇
⠀⠀⠀⠀⠠⣯⠽⣷⣶⠶⠡⣄⠘⣦⣄⣝⢡⡐⠿⢀⡍⢹⡷⠖⢛⣃⠾⠀⣴⡆
⠀⠀⠀⢠⡄⠲⠄⠿⢃⡺⠀⡿⠛⣿⢠⣥⡞⢻⠟⢤⠠⣤⣠⣾⡉⢿⣶⡿⢿⡇
⠀⠀⠀⠼⠟⣶⢻⣀⣻⣇⣀⢸⣯⣠⡼⣿⣷⣤⣠⣿⠟⠘⠏⣨⠃⣈⠿⠠⢞⣩⡄
⠀⠀⣤⡳⠠⣍⣀⠹⠿⠚⣻⡿⣅⣤⢸⡷⡚⢫⣻⡏⠀⣿⣤⣼⢇⣉⢶⣤⣾⡇⠴
⠀⠀⣿⣏⡌⡉⣣⣜⠂⣴⣭⠁⣯⠉⣠⡄⣿⡄⠉⠲⠖⣙⣠⠟⢀⣉⡀⠛⠠⣷⣆⣄
⠀⣸⣯⠽⢟⢿⡿⣋⣻⣤⣵⡆⢿⠛⣿⠀⣿⢷⣶⣾⠟⣘⡉⢠⡦⠻⣷⢶⡞⣿⣠⣴⡆
⡶⣴⠠⡄⠸⠟⢬⡍⠛⣌⠟⠸⡶⠇⢤⠈⠻⣤⠸⠟⢤⠌⣿⠛⠷⠀⠿⠣⡬⠛⣧⠁⠴

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⡀⠀⢠⡄
⠀⠀⠎⠒⡂⠀⠀⠀⠀⠀⠒⠁⠀⠀⢀⠀⠀⢸⡷⣄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠬⢆
⠀⠀⠈⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢄⡊⠢⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠈⠏⠐⠀⠀⠀⠀⣆⢁⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠒⢀⣈⠵⠀⠀⠀⠐⠭⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠣⢄⣀⠌
⠀⠘⠃⠀⢀⡄⠀⠀⠀⠀⢠⠀⢄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⢸⠄⢠⠀⡴⠄⢸⡤⠘⡤⢄⠘⡆⠠⠀⢸⡇⡘⠀⢸⠀⣾⠉⠂⢸⡇⢰⠀⠀⢿⠈⠄⠀⠐⣆⢀⠂
⠀⢠⠄⠀⠿⠀⠀⣸⠃⠡⠈⠇⠀⡇⠠⠀⢸⠀⠀⠀⡇⠈⠀⠿⠀⣿⠀⠀⠘⠃⣼⠀⠂⢸⡠⢀⣞⡀⠘⡆⠀⠀⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠁⢀⢄⠀⠀⠁⠀⠁⠀⠀⠚⠀⠁⠀⠃⠀⢠
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢰⢲⠄⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠠⠄⠀⠀⠀⠘⠒⠀⠀⠈⡀
⠀⢠⠀⠀⠀⠀⠙⠐⠠⡅⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠠⠇⠂⡂⢰⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠃⢠⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢧⠭⠀⠒⠈⠈⡄⠀⠀⠀⡀⠈⠁⠨⠀⠀⠀⠀⡀
⡠⠐⢄⡀⢀⡔⠈⠠⠤⡀⠀⠠⡢⠀⠀⠀⠑⠚⠃⠠⠀⠀⠀⠀⠀⢰⡤⠶⠺⠙⠚⠋⠀⠀⠀⠈⠢⠔⠁⠀⣀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⢀⠤⠒⠀⡆⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠡⠀⠠⠁⠀⠀⢮⣀⣀⡠⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⡆⠀⡖⣠⠀⡰⢀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⡸⠃⠀⢺⠀⠀⢰⠁⡆⢠⣇⡼⠁⡇⠴⠇⢼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣀⡊⠀⠀⠀⠈⠓⠊⠀⠁⡡⢸⠉⠀⠀⣇⠞⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠐⠁⠀⡰⠀⠙⣆⠀⠀⠀⠀⣠⠀⣸⠁⠟⠠⠀⢀⡟⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠠⡀⣰⢁⠀⣠⠏⡔⡰⢠⠀⡏⢀⡏⢀⠀⡠⠀⣼⠁⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣏⡀⢀⠀⠹⣳⡇⡿⢰⡇⣸⠃⣼⢰⡃⠀⡇⢠⠇⣼⠁⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠱⠀⠀⡷⣇⠀⠁⠀⠀⠉⠀⠀⠀⠃⠈⠉⠸⡠⠛⠊⡇⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠲⠀⠀⠀⢠⠇⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣟⢠⠴⣆⠀⠀⡴⢢⡄⣠⣄⡤⡄⣰⣤⠶⡄⣠⡆⢠⠆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢿⡏⢀⡿⠀⡼⠧⢿⣷⢿⣿⡶⠋⣿⡿⠾⠋⢈⣛⣿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣧⢸⠁⠀⠀⠁⢸⠁⠀⠀⢿⠀⢠⡎⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠀⠀⠀⠘⠀⠈⠧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠁⠀⠀⠀⠀⠀⣤⡀⢠⣀⣼⠗⢤⣾⡇⠀⠀⠀⠀⠀⣴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠒⢻⡄⡴⠛⣦⠀⣶⠀⢷⣇⠀⠀⣾⠀⢸⡟⢁⣤⡀⠀⣠⢴⣧⠞⢠⡖⣶⢀⣴⢄⣴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢡⢒⡟⠉⢿⣠⠋⣾⠃⣸⢻⣠⠾⣷⠋⣼⢁⣼⣃⠞⢷⡴⠻⠞⠻⠎⠙⢛⣇⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⠀⠀⠀⠀⠀⠹⠔⠁⠀⠀⠀⠉⠀⠹⠞⠀⠉⠀⠀⠀⠀⢀⠴⠒⠋⣽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⣠⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠻⠟⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠏⣿⣿⠉⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣴⠀⠀⣾⣟⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣇⢠⡀⣤⣀⣴⣤⣠⣦⣠⣄⣰⠄⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢹⡟⣿⠛⡿⣛⠻⢟⠛⢟⠻⠏⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⢿⣷⣿⡏⡿⠟⠻⠿⢃⣿⠏⠰⢿⠿⠂⠀⠀⣿⣿⣿⣿⣿⣏⢈⡀⠉⡄⢸⣀⣠⣄⣠⡾⢀⣴⣯⡀⣨⣿⣿
⠀⠀⠀⣀⠀⠀⠉⠈⠉⠉⡁⠁⠀⠀⢀⠙⡅⠀⠀⠰⠆⠀⠀⠀⣿⣿⣿⢿⣿⣿⣾⣷⣶⣷⣿⣿⣿⣿⡿⣶⢻⣿⣿⣏⣿⣿⣿
⠀⠀⢸⡟⣳⢠⢇⢠⣌⡉⣟⠋⢷⡇⠈⡻⠀⣻⡃⢲⣞⠀⠀⠀⣿⣿⡇⢨⢸⣟⣸⡿⠷⠿⢨⣼⠈⢸⠷⣌⣿⠑⢾⣍⢨⣿⣿
⠀⠀⠘⡿⠷⡄⣿⢸⡴⠂⣿⣻⣾⢋⡇⠉⢦⣃⣧⠈⣇⠀⠀⠀⣿⣿⣧⣈⣈⣷⠐⡇⣈⣧⠰⠮⠱⡄⣾⣧⢹⠸⢸⣷⢸⣿⣿
⠀⠀⢨⣧⠴⠃⣿⣽⠹⢆⢹⡟⠰⢼⣦⡴⠻⣯⠻⡂⢯⠀⠀⠀⣿⣿⣗⠛⣋⣾⠁⠃⣇⡿⡄⣼⣏⣧⣟⢫⣔⢲⣤⣿⢜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠈⢀⣤⣔⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠛⠿⣿⡿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡀⠁⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠻⣿⣿⡿⠃⠀⠀⠀⣹⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠀⠐⠀⢹⣿⣿⡟⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣼⣿⣯⣿⡆⠀⠀⢠⣾⣯⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠠⠓⢂⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡴⠘⢢⠄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⣀⡞⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠄⣤⣰⢔⠒⡖⡆⠉⠲⠈⡰⡏⠰⡶⡒⢂⢤⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠤⠘⢞⡻⠚⢧⢍⣐⡓⢹⣢⣀⡐⣡⡨⢳⢓⠡⣰⠚⠠⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⢀⠛⠊⣤⡲⢰⡩⢀⣑⣅⠤⢭⡈⠓⠡⡦⣉⡁⠨⣡⠘⢗⡌⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠄⠀⠀⠀⠘⣟⠵⡠⠶⢑⠄⣾⠊⣙⠲⢾⠤⠪⢅⡸⢌⡺⣄⣢⡐⢆⠸⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠀⠀⠄⠀⠀⠙⠖⡇⢠⣒⠶⢸⣺⠊⣽⠀⠻⠜⠁⠰⢚⠆⡤⠜⡕⠋⡟⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢀⠈⠀⠀⠀⠀⠀⣇⠈⠓⠁⢸⠓⣖⣿⠀⠀⠀⠀⠀⠙⠮⠃⡟⠍⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⡀⠁⠀⠀⠘⣄⠀⠀⢸⠰⣢⡝⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠀⠻⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠠⠐⠁⠀⠀⠠⠀⠁⠀⠀⠱⡒⠂⠆⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠐⠈⠀⠀⠀⠀⡀⠄⠀⠱⠐⢒⣒⠒⠒⠒⢒⣒⣒⣒⣒⡒⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢀⠠⠀⠄⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠐⠁⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠼⠼⠓⠟⠰⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢤⡀⢻⡆⣠⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠸⠇⢹⣆⢀⡀⠈⣷⠈⠿⢤⣼⡷⠋⠀⠻⠄⢀⣿⡆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢀⣦⠉⠁⣿⡟⢸⣿⠒⠉⠀⢀⣠⣿⡇⢠⣶⢻⣷⢸⣿⢀⣿⡗
⠀⠀⠀⠀⣠⡖⢦⡄⢠⢺⡦⠀⣶⠘⣿⣀⠀⣿⡇⢸⣿⠐⣷⢠⣿⢣⣿⡇⣿⣧⠚⣿⠘⠿⢂⣿⠃
⠀⠀⠀⠀⠻⠆⢸⣿⠠⣞⡁⢠⣶⠀⢰⣿⠀⣿⠇⠾⣿⠘⣿⠈⠿⠃⠈⠁⠈⣡⡔⢠⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⠘⣿⡘⠀⠻⠂⢙⡁⠀⠀⠀⢻⣶⠀⢹⡷⠀⠀⢿⣄⠺⢃⡼⠃
⠀⠀⠀⠀⠀⠀⠙⠧⠀⣼⠟⢀⡤⣤⠀⠀⢰⡏⢠⡷⠀⡠⠊⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠐⠢⠴⠟⠀⠀⠈⠳⠤⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢄⢀⣀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠊


⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⡇⠀⠀⠀⠀⠀⠀⠀⢠⠴⠪⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂⠄⡢⠀⠀⠀⠀⠀⠀⠀⠹⡐⠛⠀⠀⠀⠀⠀⠀⠀⠚⠶⡺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⡠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢀⣀⡠⢒⠀⠀⠀⠀⡌⠐⠒⢂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣯⡠⡿⠃⠀⠀⠀⠀⠂⠁⠂⠀⠠⣴⡋⣏⠀⠀⠀⠀⠀⠀⠔⣨⣀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠪⣡⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠀⠀⠀⠀⢀⣀⡀⠀⢠⢄⡠⠄⡀⡠⣄⠠⢄⡀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣀⠀⡠⢄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡥⠆⣉⣈⠢⠀⠀⢸⠂⠀⠃⠀⠊⠀⠘⠀⠈⠹⠀⠀⠈⠃⠹⠀⠀⠑⣌⠀⠀⡿⠀⠀⠀⠀⢇⠀⠹⠀⡘⠀⠀⠀⠀⢠
⠀⠀⠀⠈⡄⠗⠎⠁⢱⠀⠀⢸⠀⠀⠰⠈⠄⠀⢠⡅⠢⢸⠉⠂⡠⠀⠰⡈⠕⡰⠉⡀⡠⡇⠀⢀⡀⠀⠀⠉⢢⠊⠀⠀⠀⠀⠠⠾⠶⠄⡀
⠀⠀⠀⠀⠈⠒⣄⠀⢠⠀⠀⠀⠀⠀⠀⠋⠈⠀⠁⠈⠀⠀⠈⠈⠀⠀⠀⠈⠉⢀⠤⡀⠀⠇⠀⢸⠈⠢⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠈⠑⡇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠂⡀⠀⡔⠑⠀⡀⠀⠀⢈⠠⠄⠀⡄⠀⠀⠑⠻⡀⡄⠒⠀⠀⠀⠠⡧⠌⠉⠉⠭⠱⠇
⠀⠀⠀⠀⢠⠀⣀⠠⠁⢀⡄⠂⠀⠀⡠⢄⣀⠤⢄⠓⠐⠁⠀⠀⠂⠐⠁⠀⠀⠀⠑⠂⠊⠀⠀⢰⠐⠄⣡⠶⠊⢀⠀⠀⠀⠄⣀⡀⣀⡀⠤
⠀⠀⠀⠀⠀⠁⠉⠃⠀⢁⢂⠃⠀⠘⣀⠘⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢙⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠮⠅⠛⠀⠀⠀⠀⠀⢀⠒⠚⠉⢢⠀⠀⠀⠴⢤⠀⠀⠺⢴⠤⠀⠀⠠⠆⠠⡀⠀⢀⠠⠀⠀⢼⠀⡐⠉⠁⢆⠀⢰⠀⠀⢸⡇
⠀⠀⠀⠀⢀⠤⡤⣄⠀⡴⠮⠔⠀⠀⢀⢺⣁⡂⠀⡘⢀⢀⠤⠀⠰⢠⡒⠄⠀⠀⡔⢠⢰⠀⠀⠁⣀⡎⠀⠀⢀⡈⢡⡀⠀⠀⢸⠠⢜⠀⠀⣸
⠀⠀⠀⠀⠀⠉⠁⠈⠢⡄⠀⠀⠀⠀⠈⠦⠅⠥⠐⠁⠐⠤⠕⠀⠐⠠⠊⠀⠐⠤⠥⠂⠀⠀⠀⠨⠟⠑⠤⠁⠄⠣⠔⠐⠀⠔⠉⠒⠁⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⡐⠉⠢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⡡⠃⠀⣀⣀⣒⣒⣒⣒⡒⠐⠀⠀⠐⠤⢈⢀⠔⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠑⠤⠂⠀⠀⠐⢌⡁⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⢠⣂⣀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢐⢒⡰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⣶⠓⠀⠀⠀⠀⠀⠰⡏⣸⠇⠀⠀⠀⠀⠀⠀⠀⠘⠬⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠉⠀⠀⡠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⠿⠄⠀⠀⠀⠀⠀⡠⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣊⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠇⠪⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⣉⣉⡛⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⠃⠀⠰⣄⡀⠀⠀⣚⣉⣤⣶⠿⠛⠛⠉⠛⠳⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⣷⠀⢿⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁⠀⣀⣀⢀⣴⡿⠆⠈⣷⢸⡇⢾⡆⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⠇⠀⠀⣤⣴⢶⣶⡆⣤⢶⣶⢴⣿⣭⣏⢸⣿⡃⠀⠀⣽⢸⠇⠘⣁⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠂⠈⠁⠀⠀⢀⣿⡿⣾⣯⢸⣿⣯⡅⠾⠦⠿⠋⠘⠋⠁⢀⡴⠃⠀⢀⣠⣤⣤⣤⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠉⡀⠀⠀⣼⡿⠴⠟⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⣩⡀⠀⠀⠀⣀⣀⢀⣤⡶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣴⡟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⣀⣀⡀⣸⣿⣷⣶⢠⣿⡽⠟⣩⡛⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⠇⠀⡔⠀⣠⣤⠀⣶⣶⢠⠿⢳⣾⠆⣾⣧⣿⢁⣿⡇⣿⡯⠘⠿⠟⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⠀⠀⢹⣿⡴⢻⣿⠏⠀⣼⣿⠰⠷⠿⠋⠘⢋⣁⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣆⠀⠸⠿⠁⠘⠋⠀⠀⠁⠀⠀⣀⣤⣶⠾⠛⠋⣉⣁⣤⣤⣀⡈⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡀⠀⣀⣀⣠⣴⠶⠟⠋⠁⠀⢠⡶⠟⠛⠉⠁⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢾⡅⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠠⠊⠀⠀⠀⠀⣠⡤⠄⠀⠈⠿⣦⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠒⠂⢻⣿⠀⠀⢤⣄⠀⠀⢸⣿⣀⠄⠀⠐⠀⠙⢿⣦⠤⠀⠒⢻⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡌⢸⣿⠀⠀⢸⣿⠀⠀⠠⠁⠀⡠⠈⢻⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⣸⡿⠀⠀⠘⠻⣄⠔⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢤⠟⠁⠀⠀⠀⠀⢀⣀⢀⣀⠀⠑⠒⠋⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⠀⠀⠀⣠⠀⠀⣀⢀⠀⢀⡀⠀⠀⢀⡄⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢿⡄⠀⠀⣿⠀⠸⣇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⢹⣆⠀⢸⡏⠙⠀⢸⡇⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⣷⡄⠀⣿⠀⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠃⠀⠛⠀⠀⠒⠋⠀⠘⠃⠀⠀⠘⠃⠀⠀⠑⠚

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠰⣆⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⡀⠓⢢⡀⠀⣺⠹⣦⠉⠻⢿⠀⠀⡀⠀⠀⠀⠐⢈⠁⠆⣁⠁⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠲⠌⡉⡉⢹⠆⠙⠿⣦⣤⠀⠄⠀⣰⠀⠐⠀⠀⠀⠂⠽⠁⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⣀⠅⠚⠀⢷⠀⠁⠰⠆⠐⠉⠀⠀⠀⠀⠀⠀⠠⠀⠋⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢨⡄⠐⠂⠲⠀⠀⡐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡘⠀⢦⡴⠄⠒⠀⠐⠂⠀⠀⠀⢀⡈⣱⢄⠀⠀⠀⠀⠈⡍
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠴⡈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠆⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡉⠉⠈⠁⡀⡄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢾⠀⠀⠀⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡤⠤⣤⠄⢠⣤⣀⣀⡀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢨⡇⠀⣶⣤⣾⣿⢸⠙⣳⢾⣏⣹⡏⣹⠞⣇⠏
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⣽⡏⠀⠋⠟⠀⠘⠼⠍⠈⢯⠁⠐⠏
⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣼⠟⢳⣖⣴⠦⣶⠶⡔⠿⡿⠺⣿⣂⣀⣻⡿⣿⠿⢷⣤⣴⣦⠄⣠⣀⣠⣄⠁
⠀⠀⠀⠀⠀⠠⢀⣼⡷⢛⣅⣽⣴⣿⡴⠏⠀⡇⠀⢿⡿⠿⢿⣇⣿⣧⣼⢻⡇⢸⡇⢈⣧⣏⢉⡀
⠀⠀⠀⠀⠀⠀⠶⢸⡧⠽⣿⠙⠛⠃⠀⡀⠈⠇⠀⠘⠇⠀⠸⠃⢙⠛⠋⠚⠂⠀⠹⣼⡻⣄⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠛⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡈⠛⢾⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢤⢀⡀⠠⡀⠠⠄⠀⢀⠻⣦⡙⠳⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⣊⡄⠀⠽⢡⣄⣕⠈⠁⠀⠀⣿⡿⠓⠛⠻⠻⢷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⢩⠀⠈⠀⠀⠀⠉⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁

⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠠⠄⣀⣀⡠⠖⠁⢀⣀⠀⠈⢄⠀⠀⠀⠀⠀⠀⡰⠁⠀⠋⢁⣀⣀⠀⠀⠀⠳⡀⣉⣀⠀⠀⠀⢁⡔⠋⠀⠀⠀⠀⠙⢦⠀⣤⠈
⠉⠀⢀⣄⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠈⠩⠇⠀⠀⠈⠁⠀⣠⡄⠂⢄⠀⠀⣠⠞⠁⣀⠀⡞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⡇
⣈⡀⠈⠁⢀⣀⣀⠉⠁⠀⠀⠀⠀⠐⠀⢀⡤⠒⠛⠛⠒⢤⡉⠀⠀⠀⠈⣙⠅⠀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉
⠀⠈⠑⠈⠀⠀⠈⢳⡀⠀⠀⠀⣤⠀⢰⠋⠀⠀⠀⠀⠀⠀⠹⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⡀⠀⠀⠈⢧⠀⣠⡀⠐⠄⠀⠀⠀⠀⠀⢀⠜⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⢸⡅⢀⡤⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠑⠤⠐⠉⠯⠄⠑⠢⠤⠋⠀⠉⠁⠒⠀⠑⠢⠤⢤⡖⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠁⠀⠀⠀⠀⠐⠿⠀⠀⠤⠀⠀⠀⠀⠰⠆⠀⢀⠴⠒⠒⠒⢦⡀⠂⠀⠐⠚⠀⠀⣀⠘⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠰⠆⠀⠾⠆⠀⠀⠑⢄⡀⠀⠀⠀⠴⠋⢀⠀⢀⡀⢄⣀⠀⢀⠀⠀⣄⠀⢠⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘
⡙⠢⢤⡤⠔⠁⠀⢠⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠉⢙⣻⠄⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠃⠀⠀⠀⣠⡀
⠁⠀⠀⠘⠀⠀⣀⡄⠀⢀⠔⠉⠁⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠛⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢉⡁⠀⠎⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡄⠀⠛⠀⠃⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠛⠀⠀⠀⠀⠀⠰
⠀⠰⠗⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠜⢀⡀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⠃⡤⠀⠀⣀⣀⠀⠀⠀⢀⣄
⠀⠆⠀⣀⣀⣀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠀⠀⠀⠰⡖⠀⠀⠑⠶⡆⠀⠀⠈⣁⣀⣀⣀⠀⠀⠠⠀⠀⢠⡤⠄⠈⠠⣶⠀⠰⠋⠀⠀⠙⢦⠀⠘⠁
⡠⠖⠉⠀⠀⠀⠉⠲⡄⢀⠁⠀⠀⢀⣀⠀⠊⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁⠀⡰⠋⠁⠀⠀⠀⠙⢦⠀⠀⠘⠶⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠁⠂⠀⠉⢀⣀⣠⣀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⢠⣶⠀⠀⠀⠰⠷⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⢀⠔⠉⠀⠀⠀⠈⠙⣄⠀⢠⠄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⠀⠈⡀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠉⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣶⠆⠐⢄⠀⠀⠀⠀⡴⠃⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⡊⠀⠐⠂⠀⠀⠀⠿⠂⢀⣄⡀⠀⢀⠈⠢⢀⢀⡀⠁⠀⠄
⠑⣄⠀⠀⠀⠀⢠⠊⠀⠀⠈⠁⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⡸⠁⠀⣀⡀⠀⣶⠂⠀⠘⠃⠀⠉⠀⠀⠀⠀⢀⣄⠀⠀⠀⡏⠀⠙⢆⡀⠀⠀⠀⢈⠡⠀⠀⠘
⠀⠈⠓⢤⡤⠐⠁⢀⡈⠁⠀⠈⠀⡀⠀⠀⠀⠈⠑⠄⢀⠀⢀⠔⠁⠀⠀⠈⠁⠀⠀⠰⠆⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⢀⡠⠖⠁⠀⠀⠀⠈⠉⢑⡆⠀⠀⠀⠀⠠
⡀⠀⠀⠐⠂⠀⠀⠛⠁⠀⢀⠤⢤⠀⠀⣴⠀⠀⠀⠀⠀⢘⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈
⠃⠀⠀⠀⠂⡠⠒⠒⠢⠔⠁⠀⠈⡇⠀⠀⠀⠠⡄⠂⠀⠀⣉⢀⣀⠀⠀⣴⡄⠀⡠⠤⠤⢤⡈⢃⡀⢀⡀⠀⠊⠀⠀⠀⠑⡀⠀⠀⢀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⣄⠀⠀⢀⠀⠁⠀⠀⠀⠉⢢⠀⠀⠎⠀⠀⠀⠀⠈⠀⠀⠀⠈⢣⠀⠀⡀⠀⠀⡇⢀⠴⠋⠟⠒⠤⠞⠀⢀⡀⠀⠀⠠
⠄⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠛⠀⠀⣉⠁⠀⠂⠀⠀⠄⠀⠀⠛⠀⠀⢀⡀
⠀⡶⠂⠀⠀⢇⠀⢀⣀⣤⡀⠀⠀⠀⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢃⡀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠋⠁
⡀⢠⡄⠀⣀⠀⠀⠀⠀⠂⠈⠢⣄⣠⠇⠀⢠⠷⠀⠐⢄⠀⠀⠀⠀⠠⠊⠀⠀⠀⠙⢄⠀⠀⠀⠀⢀⠔⠁⢀⠀⠁⠀⠀⠀⠈⢢⠀⠄⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀
⠉⠲⡀⠈⠛⠀⠀⠀⠀⣶⡄⠆⢀⣀⡤⠤⣄⣀⠘⠃⠀⠉⠃⠀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠲⡤⠂⠁⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢠⠞⠉
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠛⠁⠀⠀⠰⠟⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠏
⠀⠀⢨⠀⠀⠻⠁⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⠀⠠⠆⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠘
⠀⠀⡜⠀⡀⠀⢠⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠞⠁⠀⠀⠀⠀⠈⠳⡀⠀⠈⠀⠀⠀⠀⠰⡖⠀⠀⠈⣛⠅⠀⠀⠀⣶⠀⠀⠘⠁⠀⠡⡀
⢀⠜⠀⠘⠛⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠊⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢠⡀⠀⠀⢀⠤⢤⡀⠀⣠⠄⠀⠀⢀⡀⠀⠀⣀⡤⣤⣁⠀⠉⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⢻⣿⡆⠀⠀⣰⡿⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⣸⡿⠁⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⢀⣴⠄⠐⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣥⣤⣴⣿⠁⢀⣴⡶⢶⡆⠀⣸⣟⠔⣲⡆⢠⣿⡣⢖⣶⠠⢾⣿⠋⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠉⠁⠀⣿⠇⢠⣿⠋⣰⡿⠀⢰⣿⣋⣠⣿⠃⣾⣟⣁⣼⡟⢀⣿⠃⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢼⡇⠀⠀⠸⣿⠀⠸⣧⠞⠹⡧⢴⣿⠻⠷⠟⠁⣼⡟⢷⠾⠋⠀⠸⢿⠔⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣖⠀⠀⠀⠀⠈⠓⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢰⣿⠁⠀⠀⠀⣠⣤⢤⣼⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠸⢿⣴⠾⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⡀⠀⠀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢂⣽⣿⠄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⠀⠠⠖⠒⠲⠶⠶⠦⣤⣤⣄⣠⣿⠋⣰⣿⠋⠀⠀⠀⣿⡆⠀⣼⡿⠀⠀⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡿⠋⣽⡟⠀⠀⢈⡇⠀⠀⢴⠇⢀⣀⠀⠀⠀⢹⣿⠛⣻⡿⠿⠷⠶⠶⠿⠛⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⣼⡿⣶⡶⠶⡏⠀⢴⣶⠆⠀⢻⣼⣇⠀⢀⣿⠃⢰⡟⣠⢶⣿⠟⣠⣶⠾⢻⡿⠁⠀⢀⣾⠟⢻⡗⠀⣰⡿⠁⣰⡿
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⣿⠀⣿⠃⢀⡴⢟⣽⠟⢀⣾⠇⠀⣿⡟⢁⣾⠃⣼⡟⠀⣠⣿⠁⢀⣴⣿⠃⣰⣿⠁⢰⣿⠁⣴⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⡶⠟⠁⠘⠿⠞⠋⠀⠸⠷⠾⢻⡟⠀⠸⠟⠀⠸⠷⠖⠛⠿⠟⠋⠻⠿⠟⠉⠻⠚⠙⠿⠚⠙⠷⠊⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣥⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⢶⣶⣶⣶⣶⣿⣷⣶⣶⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣰⣴⠂⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⣿⠶⠶⠶⠶⠶⠶⠶⣿⡶⠶⠶⠶⠶⠶⠶⣾⡷⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣠⣿⣄⡀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⣠⣼⣧⡀⠀⠀⠀⠀⣀⣼⣧⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⢀⣾⠉⣿⠉⢿⡄⠀⠀⣾⠋⣿⠉⢻⡆⠀⠀⣾⠋⢹⡏⢻⡆⠀⠀⣼⠏⢹⡏⢹⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⣀⣠⡾⠁⠀⠀⢻⣄⣀⣠⡾⠃⠀⠀⠻⣦⣈⣀⣼⠃⠀⠀⠹⣦⣈⣀⣼⠇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣧⡷⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⡞⣦⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣿⣟⣇⣗⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣉⠉⢉⠉⢉⠉⢉⠉⢉⡉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣤⣿⣦⣾⣦⣾⣧⣼⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣿⣀⣾⣄⡼⣄⣼⣆⣰⣇⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡷⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠛⠓⠒⠶⠶⠶⠶⠶⠶⠚⠛⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⣿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⠀⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠐⠒⠤⡀
⠀⠀⠀⢀⠐⠉⠀⠀⠉⠰⠁⠀⠀⠀⠀⠀⠀⠈⠳⡊⠉⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠉⡖⠁⠀⠀⠀⠀⠀⠀⠈⢖⠈⠉⠈⠁⠢⣄
⠀⠀⡐⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢂
⠀⢠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡱⢒⠤⠤⠐⢎⡁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢀⠕⠢⠤⠤⠐⣌⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠈⠁⠐⠖⠀⡩⠀⠈⠀⠉⠈⠩⠉⣀⣐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠈⠁⢰⠂⢈⠡⠀⠀⠀⠁⠈⠏⠡⣀⡒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢃⡠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⣇⠘⠉⠣⠀⠀⠀⠀⠀⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡱⠦⠀⠀⠀⠀⠀⡂⠘⠃⢀⠀⠀⠐⠋⠧⡀⠹⠍⠛⠀⠀⠠⡀⠚⠃⡄⠀⠀⠀⠀⢀⠴⢎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠗⠂⢀⠀⠀⠀⠀⢃⠠⢲⣤⠄⠰⠁⢰⠄⢸⠀⢴⠄⠰⠀⢤⡇⠄⠶⠀⠀⠀⠀⠀⡀⠐⠢⠂⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠈⠛⠶⣤⠖⡏⠀⠸⠀⠀⣷⣄⡀⠀⠈⠀⠀⣀⣠⡇⠀⡅⠀⢸⠓⢦⣤⠶⠛⠁⠀⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⠀⠈⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠁⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⣄⡀⠈⢻⡇⠀⠙⠉⠈⣿⠏⠀⠀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠎⠀⠀⢱⠀⠀⠈⠀⡖⠐⢮⠀⠀⠀⡸⠀⠹⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠀⠀⠀⢠⡆⠀⠀⢀⡇⠀⠘⡆⠀⢠⠇⠀⠀⠙⠒⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠃⠀⠀⠀⠀⠳⣄⣠⠼⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⢲⡶⠒⠲⣶⠒⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⢸⡇⠀⠀⣿⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⠿⠿⠿⠇⠀⠀⠿⢿⠿⠿⠿⠿⠦

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁
⠀⠀⠂⢠⡀⢀⠀⠀⠢⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠦⣤⡶⢠⣀⣀⠀⠀⠀⠀⢀⠀⠀⡀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽
⠀⠄⠐⠉⠳⢎⠀⢀⡆⠀⠀⠀⠀⠀⠠⠹⠌⠀⠔⠀⠀⠀⠀⠀⠈⠐⣗⣰⢩⢿⣉⡈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠆
⠀⢤⠀⠐⠣⠄⠃⠡⠀⠀⠀⠀⠀⠀⠀⡆⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠈⠀⢠⡤⠊⠀⠀⠀⠀⠀⠠⠀⠀⠤⠁⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠋
⠀⠀⢀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢆⣀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⢄⠀⠂⠠⠀⠀⠀⠀⠀⠀⠤⠷⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢖⡀⢏⠃⠠⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠠⠀⠀⠀⠀⠇⠠⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠐⠣⠖⠀⠀⠀⠆⠀⠀⠀⠀⡨⠁⠁⣀⡄⠈⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠠⢤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡉
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠂
⠀⠀⠀⠳⣄⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠛⠀⢰⠀⠀⠀⡄⠄⠀⢀⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠐
⠐⠎⣠⠔⠙⠀⠏⠁⠀⠀⠀⠀⠀⠀⠂⠐⠒⢀⠀⠂⡄⠀⠀⠀⠀⣧⠀⠀⢣⡼⠋⠢⠀⡀⠀⠀⠀⠀⠀⠀⠙⠣⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠲⠄⠘⠀⠂⢠⠀⠀⠀⠀⠀⠀⠃⠸⠀⠉⠀⠀⠀⠀⠀⢈⠑⠲⢡⣴⠶⠂⠀⡀
⠀⠀⠀⠃⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠈⡇
⠀⠀⢦⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡄⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠐⠀⠀⠐⠐
⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⡀⠤⣐⡖⣲⠤⠚⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡴⠉⢨⣧⣌⣀⡠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠿⠍⣤⡽⠃⠀⠀⠀⠀⠁⠀⠐⠰⠃⠙⠈⠁⠀⠀⠀⠀⠀⠈⠆⠀⠃⠛⠒⠠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣀⠜⠁⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⢸⡇⠐⠐⠂⠒⠐⠂⠒⠐⠂⠒⠐⠂⠒⠐⠂⠒⠐⠂⠒⠐⠂⠒⠐⠂⠂⠐⠀⠂⠐⠀⠂⠐⠀⠂⠐⠀⠂⠐⠀⢂⢸⡇
⠀⠀⠀⢸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇
⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⡇
⠀⠀⠀⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⡇
⠀⠀⠀⢸⡇⡁⠀⠀⠀⠀⡇⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠠⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢠⢸⡇
⠀⠀⠀⢸⡇⠄⠀⠀⠀⢸⠁⠘⠁⠊⡏⡼⢨⠴⢃⠆⠃⢘⠀⠀⠀⠀⡿⠀⡆⠀⠃⢰⢀⡇⠆⡀⠍⡐⢰⢠⠁⠂⠀⠀⠰⢸⡇
⠀⠀⠀⢸⡇⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⡏⠁⢸⠈⠀⠀⡎⠀⠀⠀⠀⠉⠀⠁⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢘⢸⡇
⠀⠀⠀⢸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇
⠀⣰⡆⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⢀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡇⢰⣆
⣰⣿⡇⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠁⠈⠋⠈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⣿
⠙⣻⣿⣿⡁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣿⣿⣟
⡇⠉⣩⣼⡿⠄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠐⢿⣧⣍⠉
⢁⠘⠿⢿⣵⣧⠐⡉⠙⠷⠾⡇⠀⠀⠀⠀⣠⡄⢀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡀⢠⣄⠀⠀⠀⠀⢸⠷⠼⠋⢉⠀⣼⣮⡿⠿⠧
⠂⠀⠀⢉⣈⣠⣼⠿⢦⡀⠀⢀⣀⢀⣾⡀⣿⣧⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣿⢀⣷⡀⣀⡀⠀⢀⡴⠾⣧⣄⣁⡉
⡴⢊⠉⠁⣣⣾⡇⢀⣠⣷⡆⠙⣿⢸⣿⡇⠙⠏⢛⣡⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣬⡛⠹⠋⢸⣿⡇⣿⠋⢰⣾⣄⠀⢸⣷⣜⠈⠙⡑
⣙⠻⣆⡴⡿⠌⡵⡫⢼⣿⡘⣆⠘⠀⠙⠁⡼⣋⣿⣟⡋⠉⠁⠀⠀⠀⣤⡈⢩⣿⣿⣿⣙⢧⠈⠋⠀⠁⡰⠣⣿⡯⢙⢮⠩⠿⣦⣠⠟
⣟⠁⠻⢭⣔⣹⡧⡈⢺⡇⠀⠈⢦⠀⣰⠖⣾⠟⠛⣿⠛⠳⡀⠀⠀⣤⡹⣿⣿⣿⠟⠻⠟⠃⠳⣄⠀⣴⣁⠀⢸⡗⢉⢼⣏⣢⡽⠟⠉
⣿⣄⣀⣼⠋⠼⣿⠾⠼⠗⠚⠶⣾⣦⣽⣿⣍⣀⡀⢽⣴⡲⢒⠤⡀⠸⣿⢘⣯⡥⠂⠀⢀⣨⣿⣯⣶⣷⠶⠛⠺⡧⠷⣿⠛⠙⣧⣀⣠
⠊⠻⣇⣾⣦⡶⢇⠀⣀⣸⣟⡉⠈⣷⣜⣿⣾⣿⣇⠸⡊⠻⣾⡉⣼⣆⣘⣭⡍⣄⣘⣿⡿⣮⣜⡕⡾⠁⢉⣻⣇⣀⠀⡸⢺⣴⣷⣸⠟
⣠⢞⣻⠇⣹⡷⠛⢻⣿⡿⣿⣿⣷⣾⣿⣯⡻⠟⠉⠀⠷⢶⡞⣻⠁⢀⣱⡬⢷⣴⣪⣭⣿⣷⢍⡻⡇⠀⢸⣿⢿⣿⡟⠛⢶⣇⠼⣟⡳
⣻⣿⢓⣕⣽⡇⠀⠙⣟⣿⣿⠑⢟⣹⣿⣏⣯⡴⠒⣦⣸⣾⠒⢻⣿⣿⣿⣦⠀⣸⣿⣷⠊⠀⠀⢿⡏⠳⠋⣿⣿⣻⠋⠀⢸⣦⣪⡘⣿
⠻⠿⡿⡛⣽⣿⣆⢻⡏⠋⠁⢀⡾⠋⢱⡿⠛⠁⢈⡝⡉⣘⣧⡀⢙⡿⣿⠷⠻⣤⠟⠁⠀⢠⡀⠀⠑⢲⡀⠈⠙⢸⡟⣰⣿⣯⢻⣿⠿
⡶⣶⢾⠿⠿⡿⢿⠿⢷⣾⣿⣿⣷⣶⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣿⣿⣶⣶⣿⣿⣶⣶⣾⢿⡿⣷⣾⡿⣿⡿⣿⡿⡷⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⡿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⣿⡿⠿⣿⣿⢸⡇⣿⣇⢿⣿⣸⣸⣿⢰⣿⣷⣿⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡶⠀⠀⠀⢿⣿⣿⣶⣿⣿⢸⡇⣿⢸⡼⣿⣥⣿⣿⢠⣿⣿⣮⣿⣿⣶⣿⡟⠀⠀⠀⢠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠁⠀⠀⠀⠈⠙⠛⠿⠿⠿⠾⠿⠿⠾⠷⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠘⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⠀⡠⠐⣶⠀⠀⣠⢶⡆⠀⠀⢲⢏⠀⠀⠀⢀⣾⡷⠀⠀⠀⣠⣾⠟⠁⣰⠁⠀⠀⢀⢶⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⣾⣏⠞⠁⣼⡟⠀⡼⢡⣿⠇⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣼⣿⠏⢀⡴⠁
⠀⠀⠀⠀⠀⠀⢠⠃⠀⣾⣿⠏⠀⣼⡿⢁⡾⢁⣾⡟⠀⢀⣠⡀⠀⠀⢤⣿⣿⠷⠖⠀⣼⣿⢏⠴⠋⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⣰⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⣸⣿⠇⡾⠁⣼⣿⠃⢰⣿⠃⡔⢀⣀⣾⣿⠁⠀⠀⢸⣿⢟⡵⣻⡿⢀⣴⡟⣵⠀⣾⡝⣀⣤⡀⠀⠏⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⣾⠁⢠⣿⡏⠀⣿⡏⢀⣳⠎⠁⣿⠇⠀⣀⢴⣿⣿⠞⣴⡿⠁⣾⡟⣰⠃⡠⠛⠚⣵⡿⠀⠀⠀⠀⢀⣹⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠀⠀⠻⡿⠃⠀⢸⣿⠁⠀⠙⢧⡾⠋⠀⠀⠙⠦⠚⠁⠸⣿⠏⣼⣿⠁⠀⢿⣷⡷⠋⠀⠀⠸⣿⣡⡶⠀⠀⢾⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢀⠞⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠖⠁⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⣿⡏⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠺⠓⠀⠀⢸⣿⣿⣿⣿⣦⣄⠀⠀⢰⣿⣿⣿⡀⢸⣿⣷⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡀⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⣿⣧⠈⣿⣿⣆⣀⣾⣿⠇⠀⠀⠀⠀⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠀⠀⠀⢸⣿⣧⠈⢀⣿⣿⡿⢠⣿⣟⢙⢿⣿⡀⠘⢿⣏⠈⣿⠏⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⠇⣸⣿⣿⠿⣿⣿⡇⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⣿⣿⠇⠀⠘⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠛⠛⠀⠀⠀⠛⠛⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢷⠀⠀⠀⠀⠀⠀⡟⠈⣷⠀⠀⠀⠀⠀⠀⢰⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡼⠀⠀⠀⠀⠀⠘⣇⣀⡿⠀⠀⠀⠀⠀⠀⢸⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⡀⠀⠀⠈⣿⠀⠀⠀⠀⣀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿⡄⠀⠀⣿⠀⠀⠀⣼⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠿⠶⠶⠾⠿⠿⠷⠶⠶⠿⠶⠶⠶⠿⠿⠿⠶⠶⠾⠷⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣴⠶⢦⡀⠀⣴⠶⣦⠀⠀⢠⡟⠉⠻⣆⠀⣼⠏⢿⡀⠀⡿⠙⣧⠀⢰⡿⣆⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠃⠀⢸⡇⢀⡿⠀⢸⡇⠀⣼⠁⠀⠀⠙⠶⠏⠀⢸⡇⢸⡇⠀⠘⠷⠾⠃⠙⢧⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⠞⠁⠀⠀⣷⢀⡿⠀⠀⠀⠀⠀⠀⠀⠈⠷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⣀⢀⣀⢀⣀⣀⢀⡀⢠⣀⢰⢀⣀⣄⣠⣀⢀⣠⢀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠓⠟⢹⠾⢻⠿⠋⡿⠁⠘⠞⠛⠛⠁⠳⠛⠘⠙⠾⠒⠟⢸⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⣀⢠⡀⢀⢀⠀⡀⡀⢀⡀⡄⢀⢠⡄⣀⡀⡤⣀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠘⠁⠀⠁⠁⠈⠁⠁⠀⠈⠀⠉⠀⠁⠉⠁⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠙⢦
⠠⠀⠀⠀⠀⠀⠌⠀⠀⢀⠀⠄⠀⠀⠐⠂⠄⠀⠀⠀⠉⠀⠄⠠⢘⡀⢀⠄⠀⠀⠦⠀⠀⠀⣀⣀⣸⣀⣀⠀⣸⡇
⠀⠀⠀⠁⠀⠀⠀⠈⢉⡉⠀⠀⠀⠀⠈⠀⠈⢄⠀⠶⢦⣘⣴⡔⢁⡠⠖⢒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠉⠀⠉⢈⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣀⢀⣉⠭⠤⠄⢿⡉⠉⠻⠭⠭⠭⠤⠤⠤⠄⠀⠀⠀⠒⣀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠠⠉⢐⢀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⡄⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠠⡆⠄⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠑⠶⠄⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠑⣆⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣱⠤⠨⠥⠃⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠆
⠀⠀⠀⠠⠒⠈⠂⠀⢤⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠈⠀⠄⠀⠠⢀⠀⠄⠀⠀⢴
⠀⠀⠀⠀⠒⠂⣧⡴⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠂⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠐⠀⠞⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠈⠈⡋⠈⠍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡂⣲⣤⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠤⡧⠠⡦⡄⠠⣤⠬⢤⣄⠀⠀⠀⠀⠀⠉⠤⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⣤⢠⣤⣤⡄⣤⡄⢤⣤⣤⠴⣤⣠⣤⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣼⡯⣿⣇⣿⡇⠜⢻⣿⠄⠟⠀⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠛⠃⠘⠛⠛⠃⠙⠛⠛⠁⠀⠘⠛⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢠⣶⠀⠀⠀⠀⠀⢀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡿⠀⠀⠻⠦⢤⣿⢻⠃⢻⡇⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠸⠁⣴⢷⡆⠀⣾⣯⠃⠀⡾⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⠀⣴⠀⢿⡜⠀⢀⣿⠏⠀⢰⠇⠀⡀⠀⢠⣾⢻⡟⠀⢠⣾⣿⡇⢀⣼⡟⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡇⣿⣸⡟⢀⣼⡟⠀⢨⣏⠀⢠⡿⠀⣼⡇⢠⡿⢣⣿⡃⠀⣿⣿⣿⢡⠎⣽⡷⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢰⡏⠸⣧⠎⢸⡇⡠⠋⠻⠄⠋⣴⡼⢻⠁⣸⠷⠋⠸⣇⠜⠛⠃⠿⠋⠀⠈⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣸⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠗⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠄⢠⣤⢤⣄⠀⢠⣤⠀⣠⡄⣤⡤⢤⣠⣤⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢰⡿⠀⠀⢹⣿⠟⠡⣿⡃⣸⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠙⠛⠛⠁⠀⠀⠘⠋⠀⠀⠛⠓⠋⠈⠛⠂⠛⠁





⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠄⠀⢠⠄⠀⠀⠀⡀⠀⠀⠀⠠⡂⠀⢠⠄⠀⠀⡀⠤⡄⠠⠄⡀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⢠⠀⣀⣀⢀⡀⣀⡀
⠀⡔⠃⢀⠌⡑⠄⠀⡈⠆⡀⢠⢲⠐⠃⠀⢘⣃⡄⠰⠈⠁⣾⠈⣰⠃⢄⠀⣿⢁⡽⣈⠇⢨⡇⣯⡋⣿⡇⡠⠂⠐⢄⡀⠤⢌⡀⠥⠄⣀⠄⠀⠠
⠀⢐⠀⠀⢈⢌⡄⠈⡥⠔⢀⠸⠐⠚⠀⠘⠌⢤⠠⢀⠬⡲⠍⠹⣤⡀⠤⠁⠻⠿⠿⣿⣿⡿⠿⠿⠿⠛⠀⠃⠰⢤⠸⢁⣰⡀⠑⣠⣀⠙⠀⣦⡄
⠀⠢⠸⢀⠂⠈⢥⠢⣒⢐⠈⠀⢘⠌⠊⠐⠤⠄⠊⠀⢨⡄⠃⢀⠘⠁⠀⡰⢄⢀⠀⠀⠛⠁⡘⡂⠀⢠⠄⠡⢈⣁⠠⠈⠒⢁⡌⠓⠉⡰⢀⣉⡁
⠀⠀⢀⠶⠀⡀⠤⢀⠀⠠⡢⠠⢊⢍⠢⠀⠀⢐⡄⠀⠉⡀⢀⠃⢀⠔⠈⠁⠁⠀⢰⠀⠀⢠⢜⠦⠀⢠⣜⠀⠀⠁⠀⠈⠒⠁⠈⠑⠋⠀⠈⠁
⠀⠔⠱⢆⠈⠁⠆⡐⡠⢒⣓⢔⠀⢡⠄⡇⡠⢚⠢⡀⠀⠀⠀⠀⣻⡈⠀⠤⠈⠀⢈⠓⢠⢸⠀⢸⠀⡧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢄⠃⠂⠄⠐⡰⠀⠅⣉⣈⢈⠘⠀⠊⠀⠰⠀⡰⠀⠰⠒⡂⠪⣈⠒⠒⢐⣀⠩⠄⠐⠁⢸⠀⢸⠀⢃⢈⡇⠠⠰⢆⠀⠸⠝⠀⠀⢉⠟⢄
⠀⠀⠈⠁⠀⠀⠀⢀⡂⠥⠥⠮⠀⠉⡝⠀⠂⠤⠄⠁⠀⠱⠁⠀⠀⠀⠀⣀⢸⡆⠄⠀⠀⠘⠂⠊⠀⣆⠠⠅⠀⠀⠁⠠⠀⠠⠀⠀⡖⡀⠈⠄
⢀⠄⠀⠠⡄⡀⠊⠀⣈⡢⠐⢙⠗⢖⠒⢒⠶⠒⠒⠒⠶⠒⠐⡲⠫⡀⠀⡄⠈⠁⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⢠⠀⠀⠸⠁⢁⠂⡈⠄
⢘⣠⣤⣤⢸⠙⡀⣾⣿⣿⠁⠀⠀⠈⢧⠎⡀⠐⢄⠌⠀⠙⠴⠃⠀⠀⠲⠀⠀⡂⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡐⠈⠑⠈⢱⠀⡇⣀⢀⠁⠀⠸
⠈⣿⣿⣿⠏⠀⠈⠚⠛⠫⣇⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢀⠠⠤⢐⠶⠒⠶⠤⢤⣀⠐⡀⠀⠀⠀⡸⠉⠀⠂⠱⣑⠀⡀⡄⠀⠀⠀⠀⢀
⠀⠈⢩⠉⠀⠀⠀⢀⠄⠀⢈⠷⡠⢀⣀⣸⢄⣀⠀⠀⠀⢠⡇⠀⠀⠀⠐⢠⠀⡄⢈⠀⢗⡈⠲⠒⠊⠀⠀⠀⠀⠀⠳⡀⠐⠁⠀⠄⠀⠘⠋
⠀⡀⢼⠤⡀⠀⠠⠃⠀⠁⡌⠈⠀⡁⠀⠈⠀⠀⢘⠀⠀⡖⠀⠰⠒⠐⠔⠠⠂⡇⢨⠀⠁⠂⠀⠀⡄⠀⠀⢀⠜⠀⠀⠑⠀⠀⠀⢄⠸⠄⠀⡠
⠀⠀⠈⠀⡀⡀⢨⠀⠀⠀⣨⠀⡎⠐⡠⡀⠀⠀⠔⢱⠀⠳⠀⢂⠀⢁⢀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠘⠋⠈⢀⠒⠐⢂⠀⠀⠀⠂⠐⢄⠈⢄⡀
⠠⠤⠀⢰⠻⡀⠐⡄⠀⠀⡖⢀⣃⠋⠀⠂⠐⠂⢈⢘⠀⠈⠖⡔⠀⠀⠉⠈⠁⠀⠉⣄⠖⠀⢀⠀⠀⣀⣀⣘⠁⠀⠉⣄⣀⣀⠂⠀⢀⣀⡀⠁⡀
⠀⠆⢀⠋⠀⠱⠀⠡⢀⢀⠃⢎⠣⢌⡀⠀⠀⣀⠠⢜⡁⠀⠀⠈⠀⠓⠐⠒⠒⣉⣀⠀⠀⠀⠟⠂⠘⢿⡟⠁⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠐
⠀⠀⡞⠀⠀⠀⢣⠀⠉⠀⣀⣠⣁⠀⡔⠀⠀⠒⠐⢉⡀⢤⡔⢀⠀⠀⢀⠐⠨⠂⢀⠄⣉⢀⣠⡀⠀⡊⡅⠐⠀⠀⠐⠀⢸⠢⠀⠔⠒⠉⠉⣉⠢
⠀⢸⡀⡀⢀⡀⡀⡇⠀⣸⠀⠈⠁⠈⠋⢁⠀⠀⣠⠊⠄⠐⢀⢀⡀⡀⠐⡮⠥⣐⡥⠊⠀⠈⠚⠡⠀⠀⠁⣀⠀⠀⢀⣠⠁⠀⢠⡀⠙⠁⡂⠡⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡄⠂⡂⡂⢨⠀⡰⠃⠀⠀⠂⣈⠜⠃⠀⠀⠡⠀⠀⢃⠀⢀⠄⠀⢖⠦⠀⣨⣀⣽⣼⣧⣠⡇⠀⠀⠁⠂⠀⠐⠀⠁
⠀⠀⠀⠘⠻⢃⡀⠀⠐⠦⠘⠐⡀⠂⠀⡈⠸⣥⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠑⠂⠬⠔⠁⠰⡷⢀⣀⣀⣈⣉⢙⣛⡋⠉⠁⢀⣀⡀⠀⣀⠠⢿⠂
⠀⣴⠋⢲⡶⣃⡱⢀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⠀⡸⠖⠉⠉⣣⡴⠒⠒⠀⠀⠀⠠⠀⠀⠁⢀⠇⠀⠞⠈⠀⠀⡅⡏⢉⠉⠈
⠀⡖⠝⠋⠉⣑⢡⠨⠁⢂⠀⡀⠀⠀⠰⡈⠈⠐⠀⢨⣑⠢⠤⢀⣀⠇⡀⠀⢠⠀⠀⠀⠘⠄⠀⠘⠋⠑⠇⠰⠐⠠⠃⠷⠛⠦⠀⠖⠀⠒
⠀⣀⠀⠨⠁⠀⠠⢨⠀⣠⣀⠀⠀⢀⠀⢓⢵⠃⠀⠀⢀⡀⠐⠀⠞⡰⠙⡄⠈⢱⠀⠀⡀⠀⢤⠀⢤⠀⡤⠀⠀⢀⢀⠀⢀⢀⢠⢀⡤⠀⢀
⠀⠈⠢⢴⠀⠤⠒⠊⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⢣⡈⠁⡜⠀⠘⡐⠓⠒⡺⠀⠀⡀⠀⢈⠃⢸⠐⠐⠂⠀⢨⠅⠀⣨⠌⠀⠂⠀⠀⡀
⠀⠔⠒⠂⠀⠀⠐⠂⠠⢠⠣⠀⠀⠀⠀⠀⠃⠇⠀⠣⠀⠀⠐⠀⠀⠀⢙⠀⠀⡄⠀⠀⠀⢀⠈⠀⠀⠀⡀⠈⡀⣉⢀⡀⢁⠀⠈⠁⣀⡀⢀⡀⣀
⠀⠀⠀⠸⠉⠇⠀⡠⢀⠀⠀⠁⠀⠀⡄⠀⢀⣀⢀⣀⣀⣀⣀⣀⢬⠼⢆⢢⡰⠀⠀⢨⠀⡀⠀⢀⡃⡇⠀⠘⠁⢘⠂⢰⠀⠈⠂⠠⠡⠄⠀⡂⠁
⠀⠓⠀⠀⠀⠀⠀⠋⠀⠈⠒⠀⠈⠉⠀⠀⠘⠀⠈⠒⠒⠒⠒⠒⠒⠐⠀⠀⠀⠀⠉⠀⠁⠘⠁⠙⠀⠃⠀⠃⠀⠈⠀⠈⠈⠈⠁⠀⠀⠈⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⡄⠀⠀⠀⠀⠀⣠⠞⢳⡀⠀⠀⠀⠀⠀⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡳⠀⠀⠀⠀⠰⡅⠀⢸⡇⠀⠀⠀⠀⢠⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢶⡶⠃⠀⠀⠀⠀⢀⣹⠶⠋⠀⠀⠀⠀⠀⠀⢳⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣟⠛⠽⢷⠀⠀⠀⠀⢸⡙⠛⠻⡇⠀⠀⠀⠀⢸⠿⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠒⢿⠀⠀⠀⠀⢸⠉⠙⠺⡇⠀⠀⠀⠀⢸⠓⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢺⠀⠀⠀⠀⢸⣄⠀⠨⡇⠀⠀⠀⠀⢸⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠙⠒⣿⠀⠀⠀⠀⢸⠈⠙⠲⡇⠀⠀⠀⠀⢸⠙⠦⢤⡇
⠀⠀⠀⠀⢠⡤⢤⣀⠀⠀⢀⣀⣸⣷⣄⣀⣸⣀⣤⣤⣀⢸⠦⣄⣠⠧⣤⣀⠀⠀⣸⣄⡀⠀⡇⢀⣀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⡏⠀⠀⠈⢳⡞⠉⠁⠀⠉⠳⡟⠉⠀⠀⠀⠈⢻⡖⠉⠀⠀⠀⠈⢷⠋⠁⠀⠙⢿⡿⠉⠀⠀⠉⣷⠋⠁⠈⢳
⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡁⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡏⠙⠒⠒⠓⠦⠤⣀⡠⠴⠲⢤⣀⣀⣀⣀⠤⠲⣄⠀⠀⠀⢀⡴⣆⡀⠀⢀⡼⠧⣄⣀⡠⠋⠑⠒⠒⠚⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡰⠋⠉⠙⠺⣇⡴⠋⠉⠁⠉⠉⢢⣀⡤⠖⠛⠛⠛⠓⢦⡀⣠⠖⠛⠉⠉⠙⠲⣄⡠⠖⠒⠒⠶⣄⣠⠖⠚⠛⠛⠲⣤⡷⠖⠒⠲
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠁
⠈⠳⡦⠤⣄⡠⢤⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡞⠦⣀⣀⡠⣴
⠀⢸⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠀⠈⠙⠒⠲⠖⠒⠉⠁⠈⠑⠦⠤⡤⠴⠚⠉⠙⠲⠦⠴⠚⠁⠈⠙⠓⠊⠉⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡜⠁⠀⠉⠓⢦⣠⠶⠚⠙⠋⠒⢦⣠⠴⠚⠉⠉⠉⠙⢢⡀⡴⠚⠉⠉⠉⠉⠳⣄⡴⠚⠛⠓⠲⡄⢠⠶⠛⠉⠛⠓⣦⡴⠚⠙⠛
⢇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠁
⠈⠙⠲⠦⠤⠤⠦⣄⡀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⠟⠳⢄⣠⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠲⢄⣠⣤⣶⣤⣤⠔⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⢻⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠹⡈⠉⠛⠒⣦⡄
⠀⠀⠀⠀⢸⠳⠦⢤⣀⡀⡞⠀⠀⠀⠀⠀⠀⢀⣨⠧⣤⣤⣠⡼⣀⡀⠀⠀⠀⠀⠀⠀⣧⠴⠚⠉⠁⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢳⣀⣀⣀⣤⣴⣾⠋⠀⣰⠃⠀⠈⢣⡀⠉⠹⣶⠦⠤⣤⣴⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⣤⣀⣀⠀⠀⠀⡏⠁⠀⢸⡼⠁⠀⢠⠧⢤⣠⠤⠖⢣⠀⠀⠘⢆⠀⠀⣿⣀⣠⠴⠒⢻⠁
⠀⠀⠀⠀⠀⡇⠀⠈⠉⠛⠲⡧⢤⣠⠞⠀⠀⢀⡏⠀⠀⡇⠀⠀⠈⢧⠀⠀⠈⢷⠚⣿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠯⢴⡆⢀⡾⠦⢤⣀⣇⣠⠤⠖⠛⢦⢠⡖⢤⡧⣻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠷⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢻⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠳⠤⣄⣀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠣⢤⣀⣸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣨⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⡇⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢀⣴⢃⣄⣀⣀⣀⣀⠀⣄⢲⣄⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣴⢀⣼⣷⠾⣟⣛⡛⠙⠛⠚⠛⠙⣛⣛⡿⢦⣿⣄⢰⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣼⡿⠿⠟⠶⠶⠶⠍⠙⠃⠀⠀⠀⠛⠉⠱⠶⠶⠞⠿⠷⣿⣄⢻⡆
⠀⠀⠀⠀⠀⢀⢀⣀⣤⣷⡿⣤⣭⠛⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢫⣍⡻⣬⣷⣄⡀⡀
⠀⠀⠀⢀⡄⠒⠘⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠃⠂⢠⡀
⠀⠀⠀⢉⣴⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣦⡉
⠀⠀⠀⠘⣛⣱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣎⣻⠃
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡚⡇⡆⡤⣄⡤⣤⢤⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠋⠻⠂⠹⠊⠱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢀⣿⠃⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢠⠠⢐⠦⣀⣆⣄⣠⠀⠀⠀⠀⠀⠀⣦⠈⣷⡀
⠀⠀⠐⢦⣿⣴⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠸⣦⣿⡴⠂
⠀⠀⠠⣤⣍⡻⣰⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣆⢟⣩⣤⠄
⠀⠀⠀⠀⠉⠙⢿⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢰⡿⠋⠉
⠀⠀⠀⠀⠈⠙⠾⣷⠃⢻⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⡟⠘⣾⠷⠋⠁
⠀⠀⠀⠀⠀⠻⢷⣬⣄⣞⢰⣦⣀⡞⠉⣻⠿⠦⡀⠀⠀⠀⠀⠀⠀⢀⠴⠿⡟⠉⢳⣀⣴⡆⣻⣠⣥⠶⠛
⠀⠀⠀⠀⠀⠀⢰⡦⠾⢻⣿⠻⣦⡍⣲⣆⠀⢀⢷⢠⠟⡝⢯⠻⡐⡦⡀⠀⣰⣖⢩⣵⠟⣿⡟⢷⢴⡆
⠀⠀⠀⠀⠀⠀⠈⢷⣵⡀⠙⣷⠜⠉⠀⠙⠁⠈⠻⣁⣔⣴⣦⣣⣀⠟⠁⠈⠋⠀⠉⠣⣾⢋⢠⣮⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢻⡁⠳⣄⢀⣥⠤⠂⠤⢟⠉⣉⣻⡿⡿⠤⠀⠤⣤⡀⢠⠖⢈⡟⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢀⣈⠀⠈⠀⠂⠀⡰⠛⡩⢏⠛⢆⠀⠀⠀⠁⢐⣁⡀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠒⠀⠀⠉⠉⠉⠀⠀⠉⠉⠋⠠⠀⠂⡀
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣿⣟⣿⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⡿⠉⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⢠⡟⢡⡶⢰⡇⠐⠷⠀⢀⡀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣷⣎⠁⣠⡴⡄⠀⣰⡇⢲⡿⠛⠀⢠⡟⢀⡾⠁⡼⢃⡶⠀⣠⣿⠁⣾⡥⣶⠄⢀⡴⣶⢀⣼⠟
⠀⠀⠀⠀⠀⢀⣾⠁⢠⡟⣼⣯⠴⢃⡔⢹⡇⡾⢁⡄⠀⣾⠁⣸⠃⡼⢡⡟⢁⡴⠁⡿⢸⡟⣰⠏⣰⡿⠖⢁⣾⣽⣤⡀
⠀⠀⠀⠀⠀⢸⣇⣠⠟⠀⠻⠤⠖⠉⠑⠚⠁⠳⠊⠀⠀⠙⠊⠙⠞⠁⠘⠖⠋⠙⠞⠃⠛⠀⠛⠊⠹⠶⠖⠛⢧⠞
⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠠⠀⢠⡄⠀⠀⠂⠀⠂⠀⠀⠄⠀⢁⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠃⠀⠱⠀⠀⠛⠀⡢⠀⢀⠐⠛⠀⠐⢆⢨⠃⠀⠀⠒⠀⠀⠈⠁⢄⡄⡐⠀⠠⠾⢦⡃
⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⡀⠀⠀⠐⠂⠀⠱⠀⠢⢃⣀⠀⠀⡀⠀⠀⡄⣨⠀⣀⠰⠀⠀⣀⠀⠀⠀⡀⠀⠀⡀⠔⢀⠀⠄⢀⠎⢀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠀⠀⠑⡀⠰⡏⠠⠀⠰⠊⠃⡈⣤⠀⠉⠀⠄⢠⠉⠀⠋⠀⠠⠁⠀⠗⠰⠉⡀⢴⠈⠃⠨⠦⢔⠁⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠴⠀⠀⠀⠢⠠⠆⠀⢀⠈⢃⠐⠃⠀⡁⠉⠀⠶⠗⠈⡀⠀⠔⠓⠀⠀⢤⡞⠑⠃⠀⢄⡤⠘⠀⠠⢀⡀⠐⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠴⠐⢀⠐⣎⢹⡀⠂⠀⠀⣴⠀⠂⠠⠀⠀⢹⣀⡨⠋⡀⢤⠌⡅⠂⠀⠡⣀⡀⠀⠈⠀⠀⠀⠛⣀⡖⢲⠀⢃
⠀⠀⠀⠀⠀⠀⡀⢄⠃⠘⠷⡆⠀⠀⠁⡁⠀⠀⠀⠐⡊⠀⣀⠀⠀⠀⠀⠄⢀⣄⠁⠂⠁⡀⢰⢈⡆⠈⠆⠎⠀⠞⢀⠠⠉⠈⠉⡉⠉⠀⠄⠁⡀
⠀⠀⠀⠀⠀⠘⢯⠀⠑⠂⠀⠅⠘⠁⢀⡈⠀⢦⡼⠋⠐⣀⣁⣴⠀⠒⠈⠁⢀⢠⡄⠒⠈⠉⢀⠡⠀⠀⠰⢐⠃⠈⡁⠀⠰⠥⠂⡡⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠐⠀⠀⡀⠀⢠⡄⢀⠐⡠⠣⠄⠸⠅⠘⢢⠀⢤⠀⠠⠄⠀⠃⠀⣗⠄⠈⠓⠠⠁⢀⠈⢁⠐⠚⠢⠊⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠣⡀⠀⠀⢀⣦⠘⠈⢁⢁⠀⠆⠠⠇⠀⠀⢢⡄⠀⡀⡈⠀⡤⠀⡰⠀⡀⠈⡀⠀⠄⢠⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⢌⢩⠃⠁⡀⠤⠀⢂⠰⠆⠙⠀⠞⢃⠀⢈⡛⠐⢦⡀⠄⠠⠀⣀⠀⠷⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣁⣈⠤⠤⠤⢶⡖⣢⡺⠄⠂⠖⡒⠶⠲⡗⢳⠠⠄⠤⢏⡀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠙⠑⠒⠺⡀⠀⠀⠀⣂⣄⣇⣨⣇⣠⠖⠳⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣇⠠⠐⣸⡅⠒⣨⣧⣬⣶⡾⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣧⠈⠀⠙⢹⠛⣿⡁⣉⡀⣤⡈⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⢁⠀⠀⠀⠆⢳⣟⣁⣉⣉⣉⣊⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡊⢀⠵⠀⠀⠀⠀⠨⠋⡏⠉⡏⠈⠝⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠠⠊⠐⠄⡇⠀⠀⠀⠀⡠⣃⠀⢱⠀⠕⢑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⡀⠀⠀⠀⠀⠐⠄⡪⠀⠐⢽⠀⠀⠘⠀⠰⢺⠀⢻⢂⠢⠔⠊⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠉⠉⠚⠚⠑⠛⣘⣀⣀⠧⠬⠚⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⣀⠀⠀⠀⠄⡀⢀⠀⠀⢀⠀⠀⢀⠄⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠊⠀⠁⠈⠐⠁⠀⠊⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠠⠲⡦⠀⠀⠂⠤⠠⣄⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠄⠀⠀⠀⠀⢦⣤⡶⠆
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠁⠀⡀⢢⡢⠀⠈⠀⢠⡀⠀⠾⠓⠆⠀⢰⠸⠁⠀⠀⡀⠀⠀⠀⢀⠰⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣅⠀⠀⠀⠀⢁⡼⠀⠁⣼⠁⠀⠀⠀⠄⢷⣄⠀⠀⣀⣴⡟⠀⠀⢠⠜⠁⠀⠀⠀⠀⠊⠀⠀⠀⡄⡘⠀⠀⠀⢀⠠⠘
⠀⢀⠠⠆⠀⠀⠀⢀⡷⡤⢠⣈⡀⠒⠖⠒⠛⠀⠀⢠⠇⢠⠀⠀⠆⠘⠢⡉⠙⠛⠛⠉⠀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣧⡇⠀⢀⢀
⠀⡁⠀⠀⠈⠄⢀⣤⣠⣴⣾⣿⡿⠒⠀⠀⠀⠀⠀⡾⠀⠈⠀⠀⠀⠀⠦⠀⠱⡖⡲⠲⣶⠂⡀⠀⠀⠀⠀⠀⡙⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠣⠄
⠀⠁⠤⠤⠞⠀⠸⣿⠿⠿⠛⢉⠑⠁⠀⠠⠆⢠⡄⢠⡄⠀⠀⠀⠀⠈⠀⠀⠀⠋⠀⢀⠉⠀⠲⡀⢠⡀⠀⠀⣓⡀⠀⠁⠀⠀⠀⠀⠀⠸⠀⠘⣧⠀⢻
⠀⢀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠊⠆⠀⠀⠀⠀⠦⡀⠀⢐⠎⠁⠀⡈⠀⠀⠀⠀⠠⠬⠐⠀⠀⢠⡀⣤⡀⠀⠀⠀⣄⡁
⠢⠈⠥⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⣄⣀⠀⠃⠀⠀⠀⡀⠀⠰⠞⠒⣆⠳⡄⠀⠀⠀⡠⠄⣤⡀⠀⠀⠀⡛⠀⠀⣞⠸⡃⠕⢸
⠀⠃⠀⡼⠻⣦⠀⠀⠀⠀⢂⠲⠙⠶⠠⠁⠉⠀⡀⠀⠠⠀⠀⢂⠀⠀⠀⠈⢧⠙⠆⢠⠊⠀⠀⣿⠇⠀⣤⣤⠀⠀⠀⢰⠀⠁⠈⢨⠀⢀⠄⠀⠀⠃
⠀⠀⠰⠛⡁⠘⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡘⠠⠄⠀⠀⠈⠓⠒⠃⠀⠀⢠⡿⠀⠀⣉⣉⡇⠀⠀⠐⠄⡣⣠⠞⠀⠘⠀⠠
⠀⠀⠀⠀⠤⠄⢧⠀⠀⠀⠀⠃⠀⠀⠀⠠⠀⠂⠀⠀⠀⢤⢨⠄⢲⢚⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣤⣤⠄⠀⠄⠀⠀⠀⠀⠄⢁⣲⡀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⠀⠀⠀⠀⠂⠀⠂⡀⠀⢸⢐⠀⡠⠴⡕⢾⡄⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋
⠀⠀⠐⠀⠀⠀⠀⠙⠓⠛⠀⠀⢂⠀⠀⠀⠀⠀⠘⠧⠀⠘⢨⠈⠳⠊⢀⣸⠃⢀⠆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐
⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⣟⠋⠁⠀⠂⠀⢀⡿⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠰⣶⣶⣴⣶⡦
⠀⢠⡄⠘⠃⠀⠀⠰⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣽⠀⢠⡃⠀⣠⡾⠁⠀⠀⠀⢀⡀⠁⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⡆⠀⢀⡀
⠀⠈⣗⣄⠀⠀⠀⠀⢫⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠀⠀⣠⣄⠓⠒⠋⠀⠀⠁⠚⠛⠁⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠘⠒⠒⠒⠚⠀⠟⢻⣿⣿⣦
⠀⠀⢋⡈⣳⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⡎⠙⠏⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⠤⠄⠀⠀⠋⠈⠉⡀
⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⢰⣲⠆⢠⠀⠐⢶⣯⠈⠛⠁⢀⣀⠄⠀⠤⠚⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢃⠀⠀⠀⠠⠂⠁⠢⠀⠀⠀⠠⠄⠐⠁
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈⠙⠀⠘⠉⡀⠈⠀⠀⠀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⣫⡇⠀⠃⠀⠀⣠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⢀⣀⡀⠀⣶⣖⠄⠐⠒⠂⠈⣁⠠⠼⠀⠀⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠂⠉⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠠⠂⡀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠀⣠⡀⠀⠀⢀⠄⠒⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠂⠄⠀⠐⠚⠙⠈⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⣀⡀⠀⠀⠈⢀
⠀⠉⢁⠰⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠠⠂⠀⣸⣷⠟⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣷⣿⠀⠀⠸⣭⣡⠴⣼⠁
⠀⠉⠀⠊⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠠⠤⣤⠄⣄⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⡏
⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀⠆⢰⠀⠀⢠⣋⡴⠃⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⣠⢲⢓⢀⡀⠀⠀⠴
⠈⢠⣴⣦⡀⠀⠀⠀⠀⠂⢀⡀⠒⠂⠉⢁⡀⠀⠀⠀⠈⠀⠀⢢⡄⠀⠀⠄⢀⠀⠘⡄⠀⠀⢠⠃⢰⠆⢀⠔⠲⠀⠀⠀⢹⢀⢂⠀⠃⠀⠀⠀⠀⠡
⠀⠘⣏⣼⠇⠀⠐⠠⠄⠂⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠘⢢⠀⠉⠀⠄⠁⠈⠀⠘⠇⠀⠀⠁⠀⣤⡶⢾⡿⠿⢿⠀⠀⠰⠒⠣⠼⠃⠠⠶⠤⣤⣤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠤⠤⠚⠁⢴⡒⢪⣹⡇⠀⠀⠀⠀⠀⠠⠜⠀⠲⢿⡟⠀⠀⠀⠘⡆⠀⠸⠀⠀⠀⠀⠀⠀⠀⢘⠖⠲⠘⠳⠚
⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠠⠀⠀⠀⠀⠘⠛⠉⠉⠀⠀⠀⢁⠀⠂⠐⠁⣠⠦⠀⠐⠀⠀⠀⠀⣧⠀⠀⠇⢀⣟⢟⠣⠀⠀⠀⠁⢀⠀⠐⡂
⠀⠈⠈⠁⠀⠠⠂⠚⠉⠀⠀⠀⠀⠀⠀⠬⠱⠤⢀⠀⠀⡠⣦⠀⠀⠘⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⢀⣨⡤⡄⠒⠀⢀⡀
⠀⠀⠀⠀⢀⣤⠤⠀⠐⢠⠀⠀⠀⡄⠀⠀⠏⠊⣹⢇⢼⠉⢉⡇⠀⠀⠉⠁⢠⣶⣶⠀⠀⠩⣤⡤⣧⠀⠀⠠⠀⠀⠈⠑⠠⠆⡰⠔⠃⠀⠻⣿⣾⣿
⠀⠀⠀⠀⠈⢿⡀⠀⡠⠋⠀⠀⠀⠀⠈⠀⠀⠀⠍⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⢄⣁⠤⠀⠀⠀⠙⠛⠁
⣀⣀⣀⣀⣀⣀⣁⣴⣁⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡷⠀⠰⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣶⡂⠀⠀⠀⠀⠀⢀⢹⡟⣹⠻⡿⡿⢻⡙⣿⢁⠀⠀⠀⠀⠀⠀⣲⡞
⠀⠘⡷⣄⢤⣄⠀⠁⣵⣷⣀⠀⠀⡄⢸⣾⡌⢿⢛⣙⣛⢻⠏⣼⣾⠀⡄⠀⢀⣰⣷⡅⠁⢀⣤⢄⡴⡟
⢀⣠⣬⡟⣤⣝⡇⠳⣦⠰⢧⣀⠀⢻⣼⠿⠞⠓⢹⡾⣿⠑⠻⢿⣿⣬⠃⢠⣠⠗⢠⡶⠃⣟⣥⡼⣯⣤⣀
⠈⠉⠀⣠⠞⠁⣸⡿⠻⣄⡐⠻⠁⡘⠁⡿⣣⡷⠂⣁⡃⠲⣧⡻⡇⠙⡀⠹⠓⣀⡼⠻⣿⡀⠙⢦⡄⠈⠉
⠀⠀⠀⠀⠀⠉⠁⠐⠋⠁⠀⠁⠳⠀⠀⠀⠋⠁⠀⠀⠀⠀⠉⠃⠀⠀⠰⠃⠁⠀⠉⠓⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠨⠂⠁⠅⠀⠀⠉⠘⠀⠐⠐⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⡀⠀⣤⣀⠠⠤⣀⠀⠠⡔⠀⠀⠀⣆⡀⠀⠀⠀⠀⣀⡆⠀⠀⡐⡤⠀⢀⡤⠤⢀⣠⡄⠀⡀
⢀⡀⠀⠙⢳⣄⣬⠷⣶⠋⠰⣾⡄⢱⡀⢿⡍⠿⢄⡭⣁⠼⠏⣽⠇⣰⠁⡼⡶⠈⢳⡶⢯⣄⣴⠛⠁⠀⣀
⠀⠉⡉⣷⠋⣡⠇⡶⠋⢰⣏⠈⠀⡾⢻⣿⣾⡴⣼⣷⣿⡴⠼⣞⣿⢻⡆⠈⢈⣷⠈⠳⡆⢧⡉⢳⡏⡉⠁
⠀⠰⠿⠋⠘⠁⠀⡀⡩⡿⠀⠀⠀⠃⢸⢻⠣⣾⡦⣥⡥⣾⡦⢹⢻⠀⠃⠀⠀⠸⡯⡁⡀⠀⠙⠈⠻⠶
⠀⠀⠀⠀⠀⠀⠠⠿⠇⠀⠀⠀⠀⠀⠀⣸⣧⣽⣶⣿⢷⣾⣥⣿⡄⠀⠀⠀⠀⠀⠀⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢿⡷⠄⠴⣿⢿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⣿⣿⡿⠏⠙⡋⠙⠛⠛⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⡿⠿⣿⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠚⣿⣿⠆⠀⣊⠀⠀⠀⠀⠀⠀⣺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣴⣶⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡉⠁⠀⠂⠀⠀⠀⠠⠁⢹⡇
⠀⠀⠀⠀⠀⠨⡿⠶⢚⠃⠈⠀⠀⠟⠀⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⠿⣿⠿⣿⣿⠿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠹⣯⣀⢠⡴⠀⠀⠀⣠⠄⢀⡘⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⠛⠀⠔⠀⠀⠁⠐⠉⢸⣧⠀⠀⠀⠀⠀⠀⠀⠙⡽⠉⠀⠀⠀⠀⠀⠀⠀⠈⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⣀⣰⡆⠀⠀⢠⡶⢀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⠂⠀⠀⠘⠃⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠋⠀⠀⠂⠀⠈⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡐⢾⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠀⠛⠉⠐⠀⠁⠠⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⢀⣾⣦⡀
⢀⠄⠀⠂⠀⠀⠀⠀⣾⠟⢿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣶⣿⣿⠀⠀⠀⠐⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣶⣿⣿⡿⠿⠿⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⡿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠛⠛⠛⠋⠉⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⣭⠇⠀⠀⠀⠀⠀⠀⠀⠘⢍⡹⠃⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠧
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣺⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢒⠢⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢠⠾⠥⠭⣭⣉⡆⠀⠀⠀⠀⠚⠶⠆⠀⠀⠀⠀⠓⢤⣀⣧⡇⠀⡀⠀⠀⠀⠀⣠⠖⠒⠒⠢⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢯⡉⢉⠐⠒⠒⠢⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⠁⠀⠀⣠⠤⠞⢧
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠜⠉⠁⢀⠤⠼⡆
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⡆⠀⠀⠀⠀⠀⢸⡏⠉⠉⢢⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠳⡤⠊⠉⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⡇⠀⠀⢸⠁⠀⣰⣷⡀⠀⠸⣷⣀⣠⠼⠁⣀⡀⠀⠀⢻⣆⠀⠀⡜⠀⠀⠀⠀⠈⠲⢤⣀⣠⠏
⠀⠀⠀⢼⠉⠑⠖⠁⠈⡇⠀⠀⠀⣿⠋⠉⠉⡿⠀⢠⡿⠀⣧⠀⠀⣿⠉⠀⠀⣿⠉⠉⢳⡀⠀⢿⣄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠸⣽⡆
⠀⠀⠀⠈⢦⡀⠀⠀⣠⠃⠀⠀⢰⣿⠀⠀⠀⡇⢀⣿⠋⠉⠹⡄⠀⢿⡄⠀⠀⣿⠀⠀⣠⠇⠀⢈⡿⠀⠀⠀⠀⢏⠓⣹⠂⠀⠀⠈⡆
⠀⠀⣀⠀⠀⠙⠢⠔⠁⠀⠀⠀⠀⠉⠀⠀⠈⠁⠸⠏⠀⠀⠀⢧⠀⠈⠁⠀⠀⣿⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠟⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣴⡏⠙⠳⡄⠀⠀⢸⡇⠀⢰⣾⠙⠳⡄⠈⠉⢹⡏⠁⢀⣀⠀⠀⡀⢸⡇⠀⠙⡄⠀⠀⠀⠀⢶⡄⠀⠀⢰⠀⠀⡸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠇⠀⠀⣾⠃⠀⢸⣟⣀⠴⠃⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠸⣷⠀⠀⢹⠀⠀⣀⠀⠘⣷⡀⢠⠇⠀⠀⡇
⠀⠀⠤⣀⡄⠀⢸⣿⠉⠉⠲⡄⠀⣿⠀⠀⢸⡟⠳⣄⠀⠀⠀⣼⡇⠀⣼⡷⠦⢼⠇⠀⣿⠀⢀⡞⠀⣼⠟⡄⠀⠘⣧⡞⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠇⠈⠉⠁⠀⢸⡇⠀⠈⠀⠀⠀⠈⠀⠀⣿⠀⠀⢸⠀⠀⠿⠖⠋⠀⣼⠯⠤⡇⠀⠀⢹⡇⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠻⠦⠖⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠻⠀⣠⡀⠀⠀⢴⠏⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣦
⠀⠀⠀⠀⠀⣠⡀⠀⣀⡁⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⡞⠳⠊⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠀⠀⠀⡇⠙⡖⠦⣄⡀⠋⠀⠀⠖⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣸⣄⣧⠤⠔⢲⠀⠀⠘⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀⡇⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⡭⠤⠔⠲⣚⣉⠽⡆⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠰⡶⠀⠀⠀⠀⢡⠀⢣⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠒⠉⠉⠀⠀⣀⣱⡀
⠀⠀⣀⠀⠀⠀⠀⠀⢸⡤⠊⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⡧⣤⢤⣖⣲⡭⠿⠽⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠴⠊⠀⠙⠦⣀⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢟⠏⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠁⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠘⢄⠀⠀⠀⠀⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠢⢄⠀⠣⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡰⠁⠀⠀⠀⠈⠳⡀⠀⠁⣲⢒⣀⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢱⠀⠈⠐⡖⣀⣘⡥⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠇
⠀⠀⢀⡠⠄⠄⡀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡴⠂⠉⠈⠑⢄
⠀⡰⠋⠀⠀⠀⠈⠱⡀⠰⡀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⢤⡊⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢃
⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⢙⡶⠀⢠⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠒⠖⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠓⠒⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⡠⠃
⠀⠈⠢⣄⣀⡀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣉⡀⢦⠙
⠀⠀⠀⠀⠘⠅⠒⠊⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠡⢤⡠⠤⠶⠴⠤⢤⣀⣈⣀⣀⡀⠉⠁⠀⠀⢀⡀⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠁⠀⠀⢰⠃⠀⠀⠀⠀⢀⠏⠀⠀⠀⢨⠃⠀⠀⠀⠨⡇⡏⢸⣠⡄
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⠀⠰⣦⢸⢹⡇⣼⠏⠇⢀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣹⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⡇⠀⠰⣦⣏⢿⢸⠃⡿⣸⣔⣽⣡
⠀⠀⠀⠀⠀⡴⠉⠉⠢⡄⠀⠀⠀⠉⣠⠴⢻⡅⠀⠀⢹⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⡟⠛⠒⡟⠻⠚⠛⠒⠣⢿⣿⣟⡁
⠀⠀⠀⠀⡼⠁⠀⣀⣀⣈⠶⠖⠒⣎⣀⡀⠨⡇⠀⠀⢺⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⡇⠤⠊⠀⡆⠀⠀⠀⠀⠘⠀⠈⢹⠉⠉⠉
⠀⠀⠀⢰⠁⠀⠀⠀⣀⡧⣤⡀⣀⢾⣀⡀⡀⣇⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠈⠁⠐⠀⠀⠀⠀⠀⢠⠀⣠⠃⡞⠉⣳
⡤⠤⠄⠛⠛⠛⢩⠟⠉⢰⠁⠀⠈⣆⠈⠻⡍⢳⠀⠀⠘⡆⠀⠀⠀⢸⠂⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⠀⡰⠁⠀⠀⡆⠀⠀⠀⢸⠀⠀⠘⣮⡆⠀⠀⢃⠀⠀⠀⢸⠀⠀⠀⠀⠸⠂⠠⠄⢴⠀⠀⠀⠀⠀⡃
⢡⣀⡀⠤⢰⠁⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⢸⣧⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⢏⠉⠉⠉⢩⠉
⠈⠀⠀⠀⠸⡀⣰⠄⠐⠀⠀⠀⠀⡟⠀⣧⠀⣸⢸⡄⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠨⣇⢀⣠⠶⠀⠀⠀⠀⠀⠀⠈⠓⠆⠒⣋⣠
⠸⡀⠀⢀⡀⠸⠯⣇⢴⠂⠀⠀⠀⠓⢤⠻⠴⠣⠄⣗⠀⠀⢸⡀⠀⠀⢹⠀⠀⠀⠀⢯⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⢸
⠀⠹⡁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⡆⠀⠀⣧⠀⠀⠈⡆⠀⠀⠀⠸⢰⡃⠀⠀⠀⠀⠀⢰⠀⢰⠁⠀⠀⠀⢸
⠀⠀⢱⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣⠀⠀⠹⡆⠀⠀⢱⡀⠀⠀⠀⢇⠑⠤⢇⠀⠀⠀⡄⠀⢸⣀⣀⣀⣀⡇
⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠢⠄⠀⠤⠓⠒⠒⠒⠒⠒⠚⠒⠒⠒⠓⠤⠤⠤⠓⠒⠒⠒⠚⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠻⣦⡀⠀⠈⣄⠹⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣀⠤⢤⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠶⠶⢬⣿⣦⡀⠈⢣⣱⠀⠀⠀⠀⠀⢀⠜⢀⠔⠉⠀⠀⢠⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠠⣄⣈⠉⠛⠶⢤⣨⠤⠶⡦⣄⠔⣃⣔⣁⠠⠤⠤⠴⢥⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠂⠠⠀⣀⠉⠉⠉⠉⠁⢸⣤⠀⢰⡃⠬⡁⠤⠤⠤⠤⠤⠤⢀⠈⢳⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠦⢬⡒⠖⠒⠂⠉⠉⠉⠩⠙⠛⠿⠷⠾⢇⡀⠈⣧⠐⠲⢤⣤⣀⠀⢠⠟⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⠈⣇⠠⠤⠤⠤⢄⣻
⠀⠀⠀⠀⠀⠀⠤⢺⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠉⠐⠒⠠⡔⠊⢹⠀⢸⠲⣤⣀⣀⣠⡟
⠀⠀⠀⠀⢠⢂⣀⣼⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⠏⠀⢸⠀⠀⠉⠉⠁⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⢸⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡾⢄⡀⡇⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣞⣠⢤⡀⢠⠇⠀⠙⡇⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢳⡼⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡇⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡄⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡇⠀⠀⡇⠀⠀⠘⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡔⠀⢠⠁⠀⠀⡇⠀⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠦⣄⡀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⢸⠀⠀⠀⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠤⠌⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡸⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⡇⠀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠃⠁
⣿⣟⢻⡟⠿⡿⠻⢟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠧⠒⠊⠉⠉⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⣀⣀⠤⠔⠒⠊⢩⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⣄⠀⠀⠀⠀⠀⡔⢄⠀⠀⠀⣼⡿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⡸⠀⠀⠀⠀⠀⢧⠈⡆⠀⠀⣿⡇⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢀⠔⠁⠀⢀⣠⣄⠀⠈⢳⣅⢀⠞⢻⡁⡠⠂⠀⠀⣠⣾⠛⠉⡆⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣀⠔⠁⠀⠀⣸⡿⠁⢸⠀⠀⠈⠹⣷⡄⠀⠀⠀⠀⢀⣾⡟⠁⠀⢠⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⣿⡁⠀⠀⠀⠀⣿⣇⠴⠃⢀⠄⠀⠆⠈⢿⡆⠀⠀⢠⣿⡏⠀⠀⡰⠃⠀⠀⠀⠘⣷⡐
⠀⠀⠀⠀⢀⡜⠁⣿⡇⢠⠒⢶⣆⠙⠷⠤⠔⠁⠀⢸⠀⠀⢸⣿⠀⢀⣿⡏⠀⡠⠊⠀⠀⢀⡀⠀⠀⢹⣧
⠀⠀⠀⠀⡞⠀⠀⠹⣷⢸⠀⡼⣿⡄⠀⠀⠀⠀⠀⢿⠀⠀⣸⡏⠀⣼⣿⠴⠊⠀⠀⠀⣴⡟⢉⠇⢀⠊⢿⣆
⠀⠀⠀⠀⠳⡀⠀⠀⠹⣿⢰⡇⣿⠁⠀⠀⠀⣰⡶⠈⠑⠚⠉⠀⢠⣿⠇⢀⠤⣤⠀⢰⣿⠕⠋⢠⠂⠀⠘⣿⣦⠖⠒⠲⣶⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠧⠟⠀⠀⠀⣴⣤⠀⠀⡖⠀⠀⣀⠸⣿⣠⠋⢸⣿⠀⠀⣿⣀⠴⠁⠀⢀⡾⢻⣿⡀⠀⠠⠋
⠀⠀⠀⠀⢿⡃⠀⣀⣴⣦⠀⠀⠖⣤⠤⢺⣿⠃⠀⢀⠽⣦⡞⠁⠀⢿⠇⠀⣿⡏⠀⡰⠁⠀⠀⠀⢰⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠁⠀⢠⣿⠋⣰⣷⠆⢸⠀⠘⠧⠠⠔⠁⣰⠙⣿⣆⠀⠀⠀⠀⠹⠧⠔⠁⠀⠀⠀⢰⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠔⢻⣟⢀⠎⠀⠀⠀⠀⠀⢀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠳⠤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠾⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠏⠉⠑⠀⠀⠉⠢⡀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣃⣀⣀⣀⣀⡀⠀⠀⢤⡈⢦⡀⠀⠀⡞⢀⡠⠂⠀⠀⠀⠀⢈⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⠀⠀⠉⠲⣤⣙⢦⣳⡒⡞⣠⠛⣁⣤⣤⣶⠶⠶⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡽⢶⣦⠤⣤⣤⣤⣿⣿⣿⣥⣍⣹⣷⣟⣋⡁⠀⣀⣀⣀⡠⢼⣃
⠀⠀⠀⠀⠀⠀⠀⡏⠓⡶⠋⠀⠀⢀⡤⠖⠋⡩⠋⠙⠛⠻⣟⠛⠻⠭⣉⡉⠀⠀⠀⠀⢢⠝⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠞⠁⠀⢀⣼⣿⣷⣶⠮⢄⣀⠀⠀⠀⠀⠑⠢⣀⣀⣤⣵⣤⡀⠻⠃⠇⠠
⠀⠀⠀⠀⢀⡴⠚⠕⠀⠀⢠⣾⣿⣿⣿⡿⠀⠀⠈⠛⠲⡦⠔⠒⠉⢹⠉⢉⠙⣿⡇⠀⠀⠸⣴
⠀⠀⠀⠀⠉⠉⢉⡿⢀⣴⠿⢿⣿⢻⣿⡇⠀⠀⠀⢈⣾⡇⠀⠀⠀⢸⠀⠈⠀⠘⡇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⣾⠵⡿⠃⠀⠀⡟⢈⢿⣿⠶⣤⣀⡀⣪⡇⠀⠀⣀⣸⡇⠀⠀⠀⣿⣄⠀⣶⡄⣣⡀
⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⠈⡞⡟⠀⠀⠉⠛⢿⡷⠖⠛⠉⠘⡇⠀⠀⢀⠈⠻⣦⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⡀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⣿⡀⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⢀⣴⣾⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⢄⡀⠀⠨⡇⠀⢀⣀⡤⠧⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠷⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⠀⠐⠂⠀⠔⠀⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠈⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠂⢄⡄⠀⠉⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠴⠀⠄⠈⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⣉⢨⢁⣀⡀⠀⠘⠄⠀⠀⠀⠌⠀⠀⡃⡆⠀⠠⠀⠐⠀⢒
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣀⠀⠀⠂⣂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢁⠈⢔⢤⠀⠐
⠀⠀⠀⠀⠀⢁⡹⠈⠉⠁⠠⣹⠀⠀⠉⠀⠈⡀⣄⡀⠀⠀⢀⡨⢃⠇⡈⠛⠛⠀⠀⠁⠐⠊⠀⠠⡀⠆
⠀⠀⠈⠀⠀⢒⠀⣄⠀⠀⠀⠚⠀⠀⠀⠠⢚⠧⠀⠣⠤⠦⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠅⠀⠁
⠀⠀⠀⠈⠀⠀⠡⠌⠄⠂⠁⠀⠀⠀⠄⠁⢠⡤⠟⠒⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⣐⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⡄⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢟⣋⣿⣿⣷⡀⠀⠀⠀⢣⠀⠄⡁⡀⡄⢄⡀⢐⠃⠀⠀⠀⠀⠙⠈⠉⠁⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡘⣿⣷⣶⢟⣽⣿⡾⠞⠃⠀⠀⢰⡆⡆⣶⠰⢂⢴⣞⣆⣮⢲⠀⠀⠀⠀⡀⢀⢀⡀⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢹⡄⣿⠙⣗⠷⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⠃⡇⢸⠀⡇⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⡄⢸⠆⠿⠘⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⣀⣨⠄⠀⠀⠀⠀⠁⠈⠀⠁⠁⠈⠀⠁⠀⠀⠈
⠀⠀⠀⢀⡤⠾⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣂⣀⣀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠉⠈⠀⠀⠐⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡨⢐⠠⠌⠀⠀⠀⠀⣤⠤⠭⣥⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⢡⠒⣈⠀⣀⢀⢀⢠⡀⣘⡛⣛⣛⣛⠂⠀⠀⠀⠀⠠⠄⠀⠂⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣥⢅⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠿⠸⠘⠺⠃⢇⠒⠲⠶⠆⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⡄⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠂⠀⢭⣭⠉⠋⠀⠀⠀⠀⠀⡴⡈⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠐⡗⠃⠂⠐⠀⠀⠀⠀⠀⠀⡀⢀⢠⣸⠇⡤⡃⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠠⠀⠐⠿⠭⠔⠤⠤⣴⣾⠀⠀⠀⠀⠖⢀⠀⠀⠀⠀⠀⠀⠀⠘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠁⠓⠒⠃⠋⠀⠀⠀⠀⠀⠀⠐⠐⠂⠁⠒⠀⠊
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡏⢰⠆⠠⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠋⠀⠀⠀⠀⢀⣴⣿⠟⢛⣉⡙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⢿⠘⠁⢀⡏⠀⡾⣰⣿⠟⠠⠞⠛⠉⠛⠆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⠆⠀⢀⣤⡀⠀⠀⠀⣼⠁⢰⠇⣿⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠏⢠⣿⠀⠀⢿⡅⢹⠀⣀⣰⡏⢀⡿⠀⠛⠀⢸⠇⠀⠀⠉⠀⠀⠀⠀⣼⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⢀⣾⠷⣄⠐⠈⠛⢾⡌⠀⣿⠋⣹⣣⣦⠀⣀⡿⢀⣤⡀⢀⠀⣠⠀⢠⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⡇⠘⠃⠀⢹⣆⡇⢀⠇⡿⢰⡟⠀⣿⢣⡟⡼⢩⡇⣯⢾⣡⣇⡴⣿⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣿⠁⣴⠗⠀⣸⠿⣇⡜⢸⢣⢸⣇⠀⠋⢸⡇⠷⠁⠉⠀⡈⠀⠉⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣀⣠⡟⠀⠉⠀⠘⠃⠀⠙⠂⡀⠘⠇⠀⠀⢀⣀⠀⡴⠚⢹⠇⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣉⠉⠀⣄⠘⣿⡄⠀⣴⡆⠀⣿⠀⠀⢀⢠⡏⠙⠸⣅⣴⠟⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⣿⠀⣿⡇⡄⢻⡇⢀⡏⣴⢰⡟⠀⣿⡄⣀⣀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⢸⣿⠀⢹⣷⡇⠘⣷⠚⡇⢿⠾⣅⢄⡼⣿⠉⠹⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢸⣿⠀⣾⠘⠆⡼⢹⠀⠀⠀⠀⣰⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢀⡏⠀⠀⠑⠋⢀⡀⠀⢠⡏⠀⣠⡟⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⠞⠁⢀⡆⣠⠀⠈⠀⠀⠈⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡏⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⠒⠛⢲⡴⠚⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢄⣸⠇⠀⠰⣶⡀⠀⣿⣤⣠⠴⠋⠉⠉⢳⢀⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢤⠏⠀⠀⠀⠀⠉⠲⣤⢄⠈⠉⠀⣿⡞⢢⠀⠀⠀⢀⣾⠇⣀⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⣾⣦⡄⢀⣀⣸⣴⣠⢀⠀⢠⣶⣶⣮⣯⡭⠄⣼⢟⠛⡽⢀⡀⣐⣼⣿⣿⢷⡶⣿⣿⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣟⡶⠚⠛⠿⣿⣟⡀⢸⣿⣧⣷⣻⣯⢹⡯⣢⢼⣷⠟⠋⠉⠀⠉⠙⢷⣷⣿⢚⠍⠀⠉⣆
⠀⠀⠀⠀⠀⠀⢀⣞⡁⢀⡠⣠⢾⡿⡿⡿⣮⣿⣹⣿⣏⣿⣿⣞⣵⣿⣿⣽⡆⠀⠀⢀⠀⠀⣟⣥⢥⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠸⣯⣻⢮⣍⡏⠈⠛⠺⠼⠼⠚⢛⣛⣿⣾⣿⣿⣿⣿⣟⣵⢁⣴⣧⣒⣠⣴⣷⢿⣽⣧⡚⠁
⠀⠀⢀⣠⠤⠖⣪⣿⣟⣛⣻⠀⡸⣟⡿⠛⠛⠉⠉⠉⢛⣿⣿⣿⣿⣿⣧⠉⠉⠉⠙⢿⣿⣛⣛⠋⠉⠉⢉⣉⣷⢶
⢰⠺⢥⣀⡀⠈⠁⠈⠻⢿⣿⣄⣙⣈⣇⢄⠀⠀⠀⠀⣾⠿⠛⣹⢿⣾⣾⢀⢄⠄⡄⡼⠳⠤⣄⣠⡴⠚⣫⡿⡡⣿
⣿⠀⠀⠀⠉⠙⠒⢤⠞⠉⠉⠉⠏⢛⢷⣾⣄⣇⣆⠀⣿⣿⣼⣛⠿⠾⠓⠿⠿⣾⠛⠏⠉⠀⠀⠙⣗⢕⠍⢜⢍⣿
⡧⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⣴⣯⣉⡙⠉⠙⠛⠛⠟⠉⠀⠀⣀⡤⣴⡞⡩⣧⠀⠀⢔⠅⠀⢻⡣⡪⡢⣲⡟
⢧⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠒⠦⢤⡠⡴⠚⡋⡯⡫⣫⢮⣺⣿⠀⠀⠣⠊⠄⢸⣗⣵⣟⡿
⠀⢸⠿⢷⣦⣤⣀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⢶⢎⢮⢎⢖⢕⢕⢕⣿⠀⠀⣠⡠⠆⣼⡫⠪⡶⡇
⠀⢸⠀⠀⠈⠉⠻⢷⠀⠐⣤⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣵⠕⠑⢁⡂⣰⣽⣿⣿⠠⠚⠁⠀⠀⣿⢕⣴⢾⡇
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⢶⢦⣤⣀⡀⠀⣿⣫⣪⣮⣾⢾⢟⢕⢽⡇⠀⠀⠀⠀⠀⣿⡎⡱⣹
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠙⢻⡿⣫⣝⣕⠕⡱⡣⡺⣻⡇⠀⠀⠨⡂⠀⢻⢟⡴⢿⣄⡀
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⢸⣯⡪⡺⣣⡪⡪⢊⢞⢽⡇⠀⠂⠙⠄⠀⢸⡾⣱⣿⣿⡁⢁⠀⠠
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣟⢔⢔⢝⢝⠕⡵⡱⣻⡇⠀⢈⢂⠄⠀⢸⣝⣯⣾⠒⠒⠀⠀⣀⡀⠁
⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡧⡣⡻⡣⡫⡪⢊⢞⣽⡇⠀⠀⠫⠊⠀⣸⡿⠟⢈⣉⠉⠋⠀⠐⢒⡒
⠀⠙⠛⠶⣤⣄⡀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢎⢊⢜⢕⡵⡕⣻⣿⡇⠀⢀⣠⣴⡿⢛⠋⠁⠉⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠈⠙⠻⣄⣦⡄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣟⡵⡱⡫⡪⢪⣾⣿⣿⣧⠾⠟⠋⠀⠀⠀⠀⠠⠤⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠶⣆⣷⣄⡀⠀⠀⠀⠀⣯⣮⢺⣞⣵⣟⡛⠓⠒⠂⠀⣙⠋⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠗⠶⣤⣄⣿⡵⡟⠉⠀⠤⠤⠀⠈⠉⠉⠁

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢀⣤⡁⠀⠀⠀⠀⡀⠀⠄⢠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠
⠈⠂⠀⠐⡀⠑⠀⠀⠀⠀⢲⠉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣰⣂⣈⡅⠀⠑⠀⠀⠀⡀⠀⢀⠀⠀⠀⠄⠀⢀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠤
⠀⠀⠀⠀⢀⡀⠀⠀⠀⡄⠐⢄⣀⠦⠂⠀⡄⠈⠉⠀⠀⠀⠀⠀⣾⠂⠄⡉⠀⠀⢁⠀⢀⠀⠀⠠⠄⢰⠀⠀⠀⠈⠀⣀⠂⢆⠀⠀⢀⠀⡄⠀⣀⠁⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠉⠄⠀⢀⠀⠀⠀⠀⠀⠀⠐⠦⣈⢐⢲⠀⠃⠀⠀⠀⠀⠄⠀⢀⠂⠀⠀⠰⠈⠀⠉⠄⠈⡄⠁⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠁⢀⡀⠀⠀⠀⠂⠀⠀⠀⠄⠤⡀⠘⠀⠂⠰⠀⠡⡀
⢀⠀⠀⠰⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠁⠈⠀⠠⠀⠀⠀⠄⠠⠀⠀⠐⠂⠀⠆⠀⠈⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠈⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠃⠀⠄⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠐⠀⠀⠀⠀⣅⡤⣴⠄⠂⠀⠀⠀⠤⠀⡀
⠀⠀⡀⠀⠐⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢦⠐⠇⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠂
⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠄⠠⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠘⣇⣸⡅⠔⣓⣆⠰⡦⢤⡀⣶⠤⣄⢠⡄⠀⡄⠀⠀⢹⢉⡱⠀⠦⢠⣤⢤⡜⢻⠛⠀⣧⢤⡄⢠⠤⣼⠰⣚⣦⠀⢤⠀⢠⠄
⠀⠀⠀⠀⠀⢠⡇⠸⣇⢮⣉⣿⠨⣇⣠⠇⣿⣀⡼⠀⢱⡼⠀⠀⠀⣨⡀⠀⢤⣧⢀⣇⠀⠀⢸⣄⢠⡇⠀⣇⢇⣀⣼⠸⣍⣹⡄⠈⢧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⣿⠀⠀⠘⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡁⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢈⢢⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⢁⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃
⠀⠀⠢⢀⡀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠨⡄⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠠⠀⠀⠠⠀⠄⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠱⠀⠀⠀⠊⠠⠀⠄⠀⡀⠀⠈⠀⠒⠀⠁⠀⠀⢀
⠀⠀⠀⠒⠂⠂⡀⠙⠒⠂⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠄⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐
⡀⠀⠘⠀⠀⠄⠀⠀⠀⠀⠁⡆⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠁⠐⠐⠀⠂⠊⠐⠀⠂⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⣄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢢⡄⠀⠂⠀⠠⠀⠠⠀⠀⠀⠂⡄⠘⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠒⠂⠀⠀⠀⠀⠌⠁⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡎⠀⠀⠀⢰⡓⡂⠀⠀⠀⠁⠀⠂⠀⠀⠄⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠠⡀⠘
⠄⡀⠀⠀⠀⠀⠀⠁⠁⠂⠀⠎⠀⠐⠀⠈⠀⡉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⡀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⡀⢠⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣷⣿⣿⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠀⢀⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠈⡇⠀⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⢸⡀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠉⢩⣭⣿⣿⡏⠉⠉⢩⣯⠉⢹⣿⣿⣿⡿⠋⣩⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠘⢿⣿⣿⡏⠀⠸⢿⣿⠀⠀⢛⣛⠉⠀⢸⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⡆⠀⢀⣉⠀⢠⣶⣄⣤⣄⢰⣿⣿⣧⢀⣈⡙⠋⠉⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠀⡀⣿⣿⣷⡸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠆⢼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⠀⣿⣿⣿⣿⣿⣧⣙⣛⣉⣀⣹⣿⣿⣿⣿⣿⠋⢠⣤⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⠀⠹⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣿⣿⣿⣿⣀⣀⣀⣀⣀⣼

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⣾⠟⠛⢿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⢻⣦⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠋⠀⠀⠀⠈⢷⢳⠀⠀⢾⡇⠀⢀⣴⠟⠁⠀⣠⡶⠟⠹⠄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡟⠁⣀⣀⠀⠀⠀⠈⢧⢣⠀⠀⠀⣠⣟⡁⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⢻⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡾⠿⠛⠻⠿⣶⣄⠀⠈⢿⣧⡶⠿⣛⣛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡇
⠀⠀⠀⢀⣀⣠⣤⣴⣶⠾⡄⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⢠⠏⠾⠿⠟⣻⡇⠀⢠⣴⣤⣄⣤⣤⣤⣤⣴⣶⣷⣤⣀
⠀⠀⠀⡟⠿⠿⠿⠷⣶⣦⣽⣷⣶⣤⣤⣤⣤⣤⣤⣴⣿⣾⣷⣶⠗⣿⠋⠀⠺⠿⠿⠟⠻⢿⣿⣉⣭⣭⣭⣤⣬⣿⣿⣿⣿⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⢉⣽⠟⢉⣼⢛⡽⠋⠁⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠙⢿⣿⠛⠛⠉⠉⠉⠉⠉⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢠⡟⢁⡾⠁⠀⠀⠀⠀⣷⠈⢳⡄⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣷⣤⣤⣤⣄⣀⣀⣀⡀⢠⡾⠁⠀⢠⠏⠀⡼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠈⣧⢀⣀⣀⣀⣤⣾
⠀⠀⢀⡈⠉⣿⠛⠛⠛⠛⠿⣿⠏⠀⠀⠀⠘⠀⢰⣃⣀⣀⠀⠀⠀⠀⣿⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠸⠟⠿⠿⠻⡟⠛
⠀⠀⠘⠇⠀⣿⠀⠀⠀⠀⢰⢏⣠⡾⠟⠀⠀⢀⡏⠙⠛⠿⠿⣿⣶⣦⣿⣴⣾⣿⠿⢣⠀⠀⢠⣄⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡟⠉⠀⢰⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⠁⠀⠀⠘⡀⢀⡞⠙⢧⢀⡞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢧⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢃⡞⠀⠀⠘⡿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣦⣤⣄⣀⣿⢀⣀⣤⣴⣶⣿⡿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠄⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡠⢼⠀⠦⠀⠀⠐⣒⣀⣐⣀⠂⠂⠰⠂⠴⠀⠈⣷⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠼⢃⠀⠈⢓⠆⠀⢳⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⢦⠃⠀⡬⠗⠀⠀⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠯⡃⠀⢈⠆⠀⠘⠊⠉⠁⠀⢰⡆⠀⣤⠀⣀⠀⣤⢀⡄⢠⡖⠀⠈⠑⠒⠺⠀⠀⠀⣴⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⢻⡄⡟⠂⢸⢸⠇⠀⢻⡄⠀⠀⠀⠀⣠⣀⠤⡈⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣄⣠⣀⡀⠀⠀⠀⠀⠀⠠⠇⠈⠈⠃⠇⠀⠸⠘⠃⠀⠢⠃⠀⠀⠀⠀⠘⢮⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠃⠈⠀⠁⠙⢠⡇⠀⢀⡠⢄⠀⠀⠀⣀⡠⠤⠄⠂⠤⠤⣄⡀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⡤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠇⠔⠇⠿⠀⡇⠀⠣⠥⠎⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢸⠀⠘⠒⡄⠀⠸⡄⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠆⠶⠀⠀⡇⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠳⠒⠋⠀⠀⠀⢣⡸
⠀⠀⠀⠀⠀⠀⠀⠘⠗⠒⠖⠒⠖⠒⠒⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⡤⡄⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⣰⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣋⣥⠷⠺⠺⢧⣉⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣀⡤⠤⢠⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠃⢹⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⡄⣀⣀⢹⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⢸⡀⠘⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠁⠀⡸⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠸⡆⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦⠴⠤⢾⡁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡠⣄⠀⠙⠢⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡄⢠⡄⣤⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠣⢥⠎⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⢠⡦⣤⠀⠈⠓⢤⣀⠀⠀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⢤⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⢟⣛⣛⣛⣋⠗⠀⠀⠀⠀⢞⠳⢤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢀⠀⠀⠀⠀⣀⡰⠒⢦⠀⠀⣀⠤⠿⠧⢄⡀⠀⠀⠀⠀⠘⠒⠋⠀⠀⢀⡞⠀⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠘⠋⠳⡀⠀⠘⢧⡁⢀⠆⠀⣸⠁⡄⠀⣠⡤⠱⡀⠀⢀⣀⠀⠀⠀⢠⣀⡠⠏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⠖⠁⠀⠀⠀⠈⠉⠀⢀⠇⠀⠁⠀⡉⠍⠀⠇⠀⠛⠋⠀⠀⠀⢸⣀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⢀⣀⣈⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠉⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣿⣿⠁⢰⣿⡟⠀⠀⣀⣀⡀⠀⠀⢰⣿⡆⡠⢲⣦⠀⡀⠀⠀⢀⣙⠓⠀⣰⣿⡗⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⡉⠑⣾⣿⡏⢀⣿⣿⠃⣰⣾⡟⣹⣿⡇⠀⣾⣿⡏⠁⣾⣿⢠⣿⡆⡰⠋⣿⣧⢠⣿⡿⠁⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠃⢠⣿⡟⠙⣿⣿⠇⢸⣿⣟⡰⣿⣿⢁⣼⣿⡟⣀⣼⣿⠃⣼⣿⡟⣀⣴⣿⠇⢸⣿⣡⢊⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⢿⣿⡰⠉⠛⠋⠀⠘⠛⢨⣿⡿⠀⠙⠋⠁⢰⣿⡿⠁⠙⠛⣁⣤⡄⠉⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⢾⡏⠉⣷⡄⠸⣿⠃⠀⣀⠀⠀⣿⣿⠁⢀⡀⠐⣿⡇⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢾⣿⡏⠀⢠⣉⠑⠢⢄⣸⣿⡇⢠⣌⠀⠀⣿⣷⠘⣿⡇⠀⢸⣿⡄⠈⠙⠚⣉⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⡰⠋⣿⡆⠀⣉⡁⢀⣿⡿⠀⠀⢀⣿⣿⠃⣼⣿⣆⡴⠟⠁⠀⠈⠁⢀⣿⣿⠃⠀⠀⢸⣿⠁⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡰⣡⡾⠛⠀⢀⡻⠟⠈⣿⣇⠀⠀⣼⣿⠇⢠⣿⡿⢀⣀⠀⠀⠀⣀⣤⣼⣿⠏⠀⢀⣠⣤⣉⡃⢀⣾⣿⠇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠧⠀⢹⣦⡄⠀⣾⣿⠂⡰⢰⣿⡇⢰⣿⡟⠀⣾⣿⡣⢪⣿⡇⢠⣾⡿⢩⣿⡟⢀⣴⣿⠏⣼⣿⡇⣸⣿⠏⢠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠀⠀⢠⣿⡿⢰⣿⡏⡴⠁⣿⣿⢁⢼⣿⢁⠜⣿⣿⠁⣼⣿⠁⣿⣿⡥⢻⣿⣃⢼⣿⣯⠜⣿⣿⠠⣿⣟⡠⣱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⢀⣾⡿⠃⠀⠙⠋⠀⠀⠈⠛⠁⠈⠉⠁⠀⠈⠁⠀⢿⣇⠀⠈⣉⡀⠀⠉⠁⠀⠙⢁⣀⡈⠉⠀⠈⠉⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠻⢏⡀⠀⠎⢹⣷⡀⠀⠀⢠⣶⠟⠉⣲⡀⠀⠀⠀⢘⣿⡚⠁⠀⣿⡇⠀⠀⠀⣾⡟⠁⢸⣇⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⠾⠟⠀⠀⠀⢸⣿⠀⢶⡿⠁⠀⣠⠔⠋⠈⠛⠒⠚⠛⠁⠀⠀⠀⢿⣇⠘⠿⠃⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉







⣷⣾⣷⣶⣶⡷⢷⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⡶⢷⣶⠶⠶⠿⢶⣾⡶⠶⠶⡷⠶⠷⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⠤⠖⡒⠲⠤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡄⠀⣸⡇⠀⠀⠘⡆⠀⢀⠴⠊⠉⠁⢀⡴⡄⠀⢹⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠀⣿⠁⣀⡀⣰⠇⠀⠈⠀⠀⠀⠀⣿⡴⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠦⡄⢠⠃⢰⡿⠺⠵⠛⠻⣄⠀⢠⡖⢲⠀⢀⣿⠁⠀⢰⡏⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠻⢦⣿⠃⠀⣼⠇⠀⠀⠀⠀⣹⠦⣿⠔⢁⡴⠁⣿⡀⠀⠘⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠆⢰⠇⠀⢰⡿⢀⡤⠤⢄⣠⣿⠀⠙⢒⠁⠀⣀⢿⡗⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠶⠋⠀⠀⣾⠃⣯⣀⣀⣤⠟⠉⠉⠀⣿⠀⡞⠁⢸⡇⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠞⢛⣾⣏⠀⠀⣉⠉⠀⣠⡄⠀⢠⡟⠀⢧⣠⠾⠁⠀⣸⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢿⣶⠋⠁⣸⡤⢞⡁⠀⢠⣿⡇⠀⣸⠇⣀⠀⢀⡴⢦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡄⠀⠀⠉⠉⠉⢹⠀⢈⠁⠀⠈⣿⠀⠀⡿⠋⣸⠃⣾⠕⠃⡰⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢹⣆⠀⢲⡀⠀⡾⠀⣿⠀⢠⠎⣿⡀⢰⡇⠀⣿⠔⠻⠶⠊⠀⣀⣽⡧⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⢿⡄⢸⣧⣰⠃⠀⠿⠔⣁⡠⢿⣧⣄⡀⠀⠀⠀⠀⠀⢠⡎⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⠛⠃⠈⠁⠀⠀⢀⠞⠁⠀⢸⡇⠀⠉⠃⠀⠀⠀⠀⠈⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠘⢦⣀⣠⡾⠁⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣤⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⢤⣿⠤⠤⠤⠤⠄⠒⠒⠉⠁⡠⣤⠀⣸⡆⣼⠯⠤⡟⢡⡟⢀⡿⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⠀⣸⡇⠀⣰⠒⠲⣄⠀⠀⠀⠀⢡⡿⢀⣿⠀⢿⣤⠞⠀⠘⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣀⡤⠀⠀⠀⣿⠀⢸⡏⠉⢩⠃⠀⠀⠀⠀⠘⠷⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠤⠚⠷⠖⠁⠀⠀⣠⡴⠒⠛⠒⣺⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⠟⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠚⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⠶⠶⠶⠶⣤⡀⢀⣤⡴⠶⠶⠶⢦⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⠛⣇⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⢷⢸⡏⠀⠀⠀⠀⠀⠙⡆⠈⣧⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡏⠀⢻⡀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿⢰⡇⠀⠀⠀⠀⢀⣰⠇⠀⢿⡀⠀⠀⠀⠀⣾⢻⠄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣸⣧⠀⠀⠀⣾⣤⣤⣼⣷⠆⠀⣿⠶⠦⠶⠶⠞⠋⠀⢸⡷⠶⠴⠶⠶⠋⠁⠀⠀⠸⣇⠀⠀⠀⣼⠃⣼
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠋⠉⠉⢹⠁⠀⠀⣼⠃⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡼⠃⠀⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢰⠏⠀⠀⠀⠀⠘⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠻⠀⠀⠀⠀⠀⠀⢻⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⠴⠋


⠀⠀⠀⣠⢀⣤⣤⣄⠀⠀⡄⢀⣀⣤⡤⣤⡀⠀⢀⣀⣀⣤⣠⣤⣤⣤⢀⡄⠀⠀⠀⣠⠀⡶⠶⢤⣀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⠉⠀⠀⣸⠇⠀⡇⢻⠁⠀⠀⢀⡇⠀⠈⠉⠁⣿⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠠⡇⠀⠀⠈⠳⣆⠀⠀⢠⡏⣧⠀⠀⠘⡄⠀⠀⠀⢰⡿
⠀⠀⠀⣿⠤⠤⠾⠓⢤⠀⡇⢸⣀⣠⠴⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⢹⠀⠀⣞⣀⣸⡄⠀⠀⢧⠀⠀⢠⠏⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⡇⢸⠉⠳⣤⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠛⠙⠉⠉⣿⠀⡇⠀⠀⠀⢠⡟⠀⣸⠁⠀⠈⣧⠀⠀⠈⠳⡤⠏⠀⡇
⠀⠀⠀⣿⣀⣀⡤⠞⠃⠀⡇⢸⡀⠀⠀⠉⠳⢦⠀⠀⠀⢹⠀⠀⠀⠀⢺⠀⠀⠀⠀⣿⠀⡇⠀⢀⡴⠋⠀⢠⠇⠀⠀⠀⠸⡄⢇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠿⠀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢦⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣆⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⣠⠴⣤⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⠖⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣿⣿⣿⣿⣽⡇⠀⠀⠀⢸⢹⣿⣧⢠⠞⠁⣴⣾⡇⠀⠀⢠⣿⠿⣿⢷⡄⣼⡥⢞⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⡿⣷⣿⣼⣯⣛⠷⣿⠀⠀⠀⣿⣼⡟⢻⣿⣶⣿⣿⣿⡇⠀⠀⢸⣷⣾⣿⣿⢻⠇⠴⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠿⣦⣙⣿⣿⣝⢳⡏⠀⠀⠀⡿⢿⣿⣾⣿⣿⣷⢉⡟⠀⠀⠀⠘⣧⣾⣿⣿⣿⣦⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢷⣦⣉⠉⠛⠿⡿⠁⠀⠀⠀⢻⣿⣿⣷⡙⢿⢷⠞⠁⠀⠀⠀⠀⠹⣿⣿⠟⡫⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣬⣿⣷⣦⣰⠁⠀⠀⠀⠀⠸⣿⣿⣿⣇⣰⠋⠀⠀⠀⠀⠀⠀⠀⠻⣤⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀
⠀⠀⠀⠀⢸⡇⠀⢴⡄⠀⣿⠀⠣⠠⡄⠀⢼⡇⢸⡇⠀⠀⡆⠀⣴⠋⢰⡏⠀⣴⠋⢠⡆⢠⡄⠀⢠⡄
⠀⠀⠀⠀⠘⣇⠀⠘⡇⠀⣿⠀⠀⠀⡇⠀⢸⡇⢸⡇⠀⢰⡏⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡄⣸⠏⠀⠙⠂⠀⠀⠷⠀⠘⠇⠈⠷⠀⠸⣇⠀⠛⠂⠈⣧⠀⠉⠁⠈⠃⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⡇⣼⠃⠐⠂⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠁⠀⠀⠀⠀⢀⣴⣿⠟⠋⣁⣠⣄⡉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠁⠈⡇⠘⠀⢠⡏⠀⣸⢡⣿⡿⠁⠴⠋⠉⠉⠉⠓⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠀⠀⢀⣠⣄⠀⠀⠀⠀⣾⠁⢠⡏⣾⡟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢰⠃⢠⣿⠀⠀⢸⡏⠙⡇⠀⡀⣸⡇⠀⣾⠁⠙⠁⠀⣼⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡍⢀⣾⣧⡀⠠⠌⠻⢦⣇⠈⠉⣿⠓⢹⡏⣤⡀⠀⢠⡏⢀⣀⠀⠀⠀⢀⡀⠀⢸⠇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠾⠋⠈⢿⡄⣶⠀⢰⢉⡇⢸⡏⠀⣿⡏⣼⢃⡾⣻⢧⣟⣽⢃⡞⢀⢾⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⣿⡿⠀⣤⣄⠀⢸⡗⡟⢀⠇⣼⠁⢿⡇⠀⡿⢰⡟⢾⡠⠙⠎⠋⠙⠋⠳⠋⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⡟⠁⠀⣾⠃⠿⠎⠀⣧⠆⠸⣧⡀⠀⢸⣧⠀⠀⠀⠀⠀⠂⠀⣠⠶⣿⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⢷⣤⡾⠋⠀⠀⣤⡀⠀⠀⠀⠀⠀⣤⠀⠙⠀⠀⢠⡶⡆⢠⠟⠁⣸⠏⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣄⠀⠀⣶⠀⢻⣷⠠⢠⡟⡇⠀⣿⠀⠀⢠⡄⢺⣇⠃⠘⠦⠾⠋⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⠀⠀⣿⠀⢸⣿⢰⠈⢿⠁⢠⡏⣸⢣⣿⠃⠸⣿⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢰⣿⠀⢸⣿⠀⢸⡏⣺⠀⠘⣷⠻⠇⢹⠞⣧⠄⣡⢿⣿⠁⠙⠗⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⠸⣿⠀⢸⡇⠳⠀⡞⣿⠀⠀⠀⠀⢀⡾⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⣿⠀⣼⠁⠀⠀⠓⠋⠀⠀⠀⠀⣾⠁⠀⣰⠏⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢻⣤⠇⠀⢠⡄⡄⠀⠘⠁⠀⠀⢷⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⡇⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡀⠉⢳⡀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢏⠹⣷⡄⢳⡀⠀⠀⡼⠁⢀⡤⠞⠋⡷⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠙⢿⡌⢧⡀⣸⠁⢠⡞⣀⡤⠊⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠬⠷⢶⣤⣤⣼⣈⣉⣉⣀⡞⠋⠁⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠉⠉⠉⠉⠙⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⣀⣀⠀⠀⠀⠀⡤⠀⠀⠀⢀⡤⠤⢤⣀⣀⣀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠉⢣⡀⠀⠀⡀⠀⠀⢠⠏⠀⠀⠀⠙⢦⠀⠀⣽⠀⠀⠀⣀⡤⠴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⡏⢀⣴⠲⡀⠀⣧⠠⠤⡇⠀⠀⢸⠀⠀⡏⢹⣦⠈⣧⣀⡇⠀⣠⠞⠁⠀⢀⣀⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠙⢦⡀⠀⠀⢸⠀⣾⡏⠈⠁⢠⡇⠀⠀⡇⠀⠀⠈⢧⡀⠀⠈⠻⣧⢸⣸⠇⣰⠃⠀⣤⠞⠉⢩⠛⠄⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢠⢟⠙⠓⠶⣄⠀⠙⣄⠀⣼⢰⠟⠀⢀⣴⣋⡠⠴⠚⠳⡖⠲⠦⠾⢝⣶⣤⣀⡙⡆⠿⠶⠇⢀⣾⡡⠴⠊⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠉⠲⠤⢬⣷⡀⠘⠛⣉⣼⠴⠚⠋⠉⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠉⠉⠟⠐⠒⠒⢾⠤⣠⣤⣤⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣤⣀⣤⠤⡟⠊⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡸⠀⠀⠀⠈⠉⠉⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⠉⠁⠀⠀⠸⡀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⡆⠀⠀⢀⡀⣀⠤⠔⢺⡄⢀⡀⠀⠀⣰⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠹⠈⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠹⠀⠀⠀⠀⠗⠒⠠⠤⠄⢀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⡀⠤⠐⠀⠉⢣⠀⠀⠀⠈⣄⠀⠀⡀⠠⠄⠒⠈⢧⠤⠄⢀⣀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠘⡍⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡞⠒⠒⠒⠠⠤⠄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠤⠐⠒⠉⠁⠘⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣨⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣸⣤⡤⠖⠚⠛⠓⠶⠤⢴⣤⣀⣀⣀⣏⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣀⣠⡤⠼⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠾⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣿⣿⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢋⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣡⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢼⣿⠿⢛⣹⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢸⣥⣶⣿⣿⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠣⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠱⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣾⣿⣿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⣿⣧⣿⣿⣿⣿⣷⣶⢲⣶⣶⢲⣶⣿⣿⣿⣿⣧⣿⣿⣟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣹⣿⣿⣸⣿⣿⣸⣿⣿⢸⣿⣿⢸⣿⣿⣼⣿⣿⣹⣿⣿⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⡏⣿⣿⣿⣿⣿⣼⣿⣿⣸⣿⣿⣿⣿⡏⣿⣿⣇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣿⣷⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢿⣿⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣾⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡾⣿⣇⣿⣿⢹⣿⡿⣿⣿⣿⣿⢸⣿⣧⣿⡿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⢿⢽⣿⣽⣿⡟⣿⡏⣿⣿⣿⣿⢽⠿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠉⠁


⡿⠿⠿⠿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣻⣿⣛⣿⣟⣻⣿⣿⣿⣻⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⠀⠘⢿⠀⢃⣤⡀⠀⡖⠀⠀⠀⠄⢐⠠⢴⣦⠀⢱⣶⡇⡐⢸⡇⠀⡀⠜⠛⠀⠿⢀⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⢰⡆⠀⠆⢴⣀⠗⠆⠻⠧⠀⠒⠛⠀⠺⢰⡆⢦⡀⡄⡄⡿⢷⠏⢁⡌⣡⣄⣠⡤⣤⢚⠐⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⠠⠤⢤⣈⣡⣿⠁⠀⣰⠊⠾⣷⢰⣀⡌⠧⠀⢹⣷⣬⣷⠏⢰⡸⠇⠘⠿⠿⢀⡉⣹⣴⠟⣻⠆
⠀⠀⠀⠀⠀⠀⠠⡶⢁⠈⠢⡄⠀⠉⠉⠀⠀⠺⣿⢆⣰⢡⡟⢿⣤⡴⠂⠈⠛⢉⡀⠁⡞⠓⢲⠀⢰⣦⡀⣽⠟⣵⣌
⠀⠀⠀⠀⠀⠀⠀⢌⠀⢟⡎⢀⣴⡆⠀⣿⣶⠶⣿⠏⣽⡰⠇⡌⣿⣁⣤⢷⡄⣸⠋⢉⠻⣟⡸⠛⣿⣿⡿⠃⣜⠴⠋⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢶⢿⠧⣧⡌⢿⣷⣾⣿⡿⠀⠙⠛⠋⢠⡴⠃⢸⣿⣿⢰⣿⣿⠀⣸⡏⠛⠃⣸⣿⡟⢡⠞⢁⣦⣴⣌⡁
⠀⠀⠀⠀⠀⠀⠂⢀⣠⣷⡈⠣⢾⡌⠻⣿⣤⣤⠀⢀⣄⠀⣿⣰⠞⣿⣇⣿⢸⡇⣤⣻⡄⠸⠢⣿⡟⡴⠁⣠⡾⠟⡱⠋
⠀⠀⠀⠀⠀⠀⣀⡀⠙⣍⠻⢆⡈⠻⣆⠈⢻⣿⣤⣾⡟⠀⣼⢃⡇⢿⣼⣏⣁⡀⠈⠉⠁⢠⣴⣿⡟⠀⠈⢁⡴⠚⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠙⠢⢽⣦⣙⢷⣄⠹⣿⣿⣇⡼⠁⢸⡇⢸⣿⡇⣼⠻⡀⠰⣦⣄⣿⣏⣠⣤⠞⠋⠐⡀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠻⣿⠻⠿⠷⠿⠿⠿⢷⣶⣿⣷⣾⣿⣿⣿⡶⠿⠿⠿⠿⠟⠛⠛⠉⣽⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠐⠆⢀⠅⠀⠀⠹⣧⠠⣶⠦⡴⣶⠀⡴⠤⣄⠀⠀⠀⠀⠀⣾⠃⠀⢠⡄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠆⠀⠀⠀⠀⠀⢹⣧⠙⢦⡀⢸⡇⣇⠔⠋⢀⣾⠉⣆⠀⣿⣀⣴⠟⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠈⠛⠇⢀⡴⢦⢸⣿⠀⡇⢠⣿⠋⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⣠⢺⣧⠀⠀⢸⡇⢸⡎⢿⣦⣵⠏⣿⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢱⡇⠁⣸⡿⢦⡄⢸⡇⢸⠁⠈⣿⡿⠖⢻⠀⠎⢹⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⡇⣴⠏⠀⠈⣿⣼⡇⡟⠀⢠⡏⠙⣆⢸⡞⠀⣼⠀⢀⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡠⠀⣿⠏⠀⠀⠀⣿⢻⡟⠀⠀⣾⢀⠀⢸⡞⢿⠾⠃⢰⣋⡻⣦⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠁⢠⢿⠀⡠⢤⠀⣿⡾⣷⡀⣸⢻⣯⣀⣼⠇⠀⠰⠶⣄⠛⠁⢿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⠏⢸⣼⠁⠘⢰⠟⠀⠙⠛⠁⡀⠉⠉⠁⠀⣤⣼⢀⠹⣧⠀⣸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢠⡟⠀⣾⢻⣆⣠⠏⢠⢤⠀⣠⣄⠱⣄⣀⣄⠀⡇⠘⣇⠙⢻⡄⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣷⣴⠏⠀⠩⢤⡐⣦⠛⡐⣯⠈⣧⠘⣿⡻⣧⣿⡄⢻⡄⢸⣧⣿⢲⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⣱⠛⣦⠰⡈⣇⢸⣞⢻⡜⢧⣹⠀⡏⢳⡜⣯⢷⣼⡇⠀⣿⢻⣦⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⢦⡘⣧⢹⡟⢸⡇⡆⣿⠀⢹⣦⣠⠀⠻⣼⡄⢻⣷⣼⡿⠀⠁⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⠷⠇⠸⣾⡟⢼⡇⢹⣿⡄⢠⠈⢻⣧⠀⢻⣿⣸⡟⠙⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢹⣇⠈⣷⡀⢿⡇⠀⣠⡄⢻⡷⢼⡇⠻⠃⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠘⡌⢧⠹⣌⠈⢷⡄⠙⢧⣼⠇⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠁⡄⠒⠒⢄⣀⡀⠀⢂⠀⠀⠀⢀⣐⣀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠁⠀⠀⠋⠟⠒⠀⠈⠃⢀⡀⠀⠉⢹⠛⠥⣀⣀⣀⣀⣀⡤⠂
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢶⣤⡈⠍⠓⠚⠒⢀⡀⣉⢉⠁⠀⠄⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⣰⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⠲⣄⠀⠀⠀⠬⠟⢎⣀⠀⢀⠅
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠇⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡼⠀⠀⠀⢀⣌⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⣀
⠀⠀⠀⣁⠀⠀⡀⠈⠀⠀⢠⠃⢰⠁⡞⠀⡏⢀⠇⣼⠀⡝⢸⠁⡞⠀⠀⠀⠠⢀⡇⠁⢄⣀⡀⢀⡠⠂⠂
⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠁⢠⠃⠀⠀⠀⠰⠃⠀⠀⠀⠁⡸⢸⠐⠖⠘⠍⠉
⠀⠀⣼⢠⠄⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡠⠂⢠⠃⠙
⠀⠀⢉⡞⠀⠀⡴⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢐⠀⠀⡞⠀⡀⡀
⠀⠀⢸⠁⢤⡀⠀⠠⠀⠀⠀⣸⠀⡼⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠹⠯
⠀⣠⠃⠆⠀⡈⢠⠆⢰⠐⢢⠇⢰⠗⡴⢀⠆⣸⠁⡔⢠⠆⢰⠂⡖⠀⠀⢠⠀⠀⢠
⠈⠁⠉⢉⡉⠀⠈⠁⠈⠀⠈⠀⠈⠀⠁⠈⠁⠈⠀⠁⠈⠁⠈⢠⠃⠀⠀⠀⡄⠀⠀⠁⠈
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠥
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣒⢦⣤⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⢈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠓⢲⡖⠒⢦⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠘⣦⠤⢤⣀⠤⠤⠼⣄⡤⠦⠤⢄⡠⠦⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⡄⠀⢰⡇⢠⠀⢻⠀⠐⠃⢈⡇⠀⠒⠀⡷⣄⠉⢁⡤⠇
⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠿⣧⣀⣸⡀⡴⡆⠘⡆⠀⣖⠋⣇⠀⢰⢚⢁⣸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠠⣏⢩⡏⠩⠙⣏⠉⣹⠉⠏⢻⠉⡙⡿⢉⢹⡌⢋⡹⠂
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⣠⣏⣨⣇⣀⣸⠜⣆⣿⣰⣆⣼⣀⣡⢇⡴⣄⡿⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠰⠀⠠⠀⠀⠀⠀⠤⠘⠠⡔⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠐⠀⠀⠀⡃⠀⠐⢂⠐⣁⠂⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠐⠀⠢⠂⡀⠈⠀⣀⡀⠂⠀⠀⠰⢐⠮⣀⠁⠀⢀⠂⠈⠀⠀⠀⠀⠐⠒⠀⠂⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠴⠦⡤⠀⠪⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠂⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠲⡀⣂⠁⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⡂⢲⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣰⡴⠤⠀⠀⠀⠀⠀⠀⠈⠁⠲⠆⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠢⠁⠀⠀⠀⠀⠀⠀⠂⢰⢰⡀⠒⠀⠀⠢⠀⡄⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⡄⠠⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⠸⡀⠀⠀⠀⡀⢀⢥⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠄⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠂⠀⠀⠀⠈⠀⠀⢀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠆⠀⣀⠀⠀⠒⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠴⠟⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⢀⠉⠀⠰⠠⢂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠈⠀⠀⠢⠐⠤⠈⠋⠀⠤⣄⡠⠰⢎⣱⠀⠄⠀⠈⠀⠄⠀⠀⠢⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠀⠄⠀⢀⠀⠐⠀⠀⠀⠐⢧⡶⠒⢋⠛⠈⠀⠀⠀⠀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⠀⠀⠐⠀⠀⠀⠈⠛⠒⠋

⠀⠀⣤⣤⣀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⣀⣀⡠⠀⢤⣤⣤
⠀⠀⢹⣿⠋⠀⠹⣿⡏⠀⠹⣿⡯⠉⢽⣿⡷⠀⠲⣿⡶⠂⠰⣾⡶⠂⠰⣟⡶⠃⠈⢻⡿⠏⠈⢻⡿⠇⠀⠛⣿⠇⠀⠐⢿⠏
⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠻⠃⠀⠀⠸⠃⠀⠀⠸⠇⠀⠀⠈⠏⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⡴⠄
⠀⠀⠀⠀⢛⣼⡺⠀⠀⠀⠀⠀⣻⣿⣿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⣿⡟⠁⠀⠀⠀⠀⢠⣆⣀⠀⢿⣿⡘
⠀⠀⠀⠀⠀⠉⠁⠐⣶⣾⣇⠀⠈⠀⠉⠀⣾⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⡃⠈⠁⠈⠁⠀⠀⣀⠀⠘⢻⣿⣿
⠀⠀⠰⠖⠀⠀⠀⠰⠿⢿⡏⠁⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⣷⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⡞⡹⠇⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠘⠿⠾⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠁⠁
⠀⠀⠀⢀⡾⠿⠞⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣦⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠄⠉⠃
⠀⠀⠀⣿⣧⠀⠀⢈⣿⡇⠀⠀⣀⣀⡀⠀⠀⡀⠀⡀⠀⣀⡀⢘⢋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡀⢀⡀⣀
⠀⠀⠀⠻⣿⠀⠔⢺⡿⠃⠀⠨⢛⣧⣧⠀⠀⡟⠶⢻⠀⡧⠄⢸⢴⠃⣯⢼⠁⢱⠎⠀⠀⠈⠁⠁⠀⠀⢛⠋⣉⣡⣟⡃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⢀⠃⠃⠈⠀⡁⠁⠈⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠊⠀⠀⠀⠀⠙⠁⠘⠃⠑⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠈⣠⣤⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣂⠀⠈
⠀⠀⠀⠰⣾⣷⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠐⠽⠿⠯⠛⠄
⠀⠀⠀⢀⣝⣛⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡖⠀⠀⠠⣄⣀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠹⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠠⡾⠃⠀⠀⠀⡞⠛⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠤⠄⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠿⠿⠶⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣀⡀⢮⣷⡄⠀⠀⠀⠀⢀⡴⣆⣠⣄⣤⣀⡄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢮⠅⠘⠛⠀⠀⠀⠈⠀⠠⢤⣀⠀⢠⣀⣈⡀⠀⠁⠠⠄⢀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⠀⠁
⠀⠐⠀⠀⠄⠤⠀⠀⠀⢀⣠⡝⠁⡈⣿⠇⠀⠀⠀⠀⠀⠀⢠⣦⣴⣦⣴⣦⣤⠀⠀⠀⠀⢿⡏⠋⠁⠀⠀⠀⠀⠒⠒⠲⠆⠐
⠀⠀⠀⠀⡀⠀⠙⠿⠛⠛⣿⣇⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠒⠒⠒⠘⠋⠙⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠘⠶⣦⣤⡀⠈⢿⡆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⠀⠀⠀⠀⠀⠙⠻⣷⣌⢿⡿⠛⠛⠛⠛⠻⠿⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣿⣿⣷⣶⣤⣤⣤⣤⣤⣬⣿⣿⣿⠿⠿⣶⣦⣠⣤⠶⠟⠋⠉⠙⠻⢷⣦⣤⣀
⣤⣤⣶⣶⡿⠿⠟⠛⠋⠉⠁⠀⠈⠉⠉⠛⢿⣯⣍⢻⣿⡏⠀⠀⠀⠀⠉⢿⣆⣀⣠⣤⣤⣤⣤⣤⣴⡿⠛⠻⢷⣦⣄⡀
⣿⡟⠻⣶⣤⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠾⠟⠛⠉⠛⢿⣿⣄⠀⠀⠀⠀⢀⣿⣏⣉⣉⣉⡉⠉⠉⠛⠻⢿⣦⡀⠀⠉⣛⣷
⣿⡇⠀⠈⠙⢿⣷⣶⡶⠟⢻⣏⣉⡀⠀⠀⠀⢀⣀⣴⡿⢹⡿⠿⢶⣶⣶⣿⣿⠟⠛⠉⠙⠻⣷⣄⠀⠀⠀⢹⣿⡾⠟⠋⠹
⣿⡇⠀⠀⠀⠀⣿⠻⣷⣦⣿⡟⠛⠿⠿⠿⠿⠛⠛⢉⣴⠟⠀⢀⣾⠿⣄⢸⣿⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⣿⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⣀⣴⣿⠏⠀⠀⣾⣏⣀⣬⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⣿⣧⣀⣠⣤⣶⠿⠛⣹⡟⠀⠀⢠⣿⡿⠟⠛⢹⣿⣿⣦⡀⠀⠀⠀⢀⣠⣾⠟⠁
⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠿⠟⠛⠻⣿⣾⠿⣿⡇⠀⠀⠘⣿⡆⠀⠀⢸⣿⠈⠻⠿⣿⣷⣿⣿⠛⠁
⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⢻⣧⠀⠀⠀⠹⢷⡀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠈⢿⣧⣠⣤⣶⣾⠿⠆⢸⣿⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀⠀⠀⠀⣿⡇⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠄⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣿⡄⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⣷⡀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠻⣷⡀⠀⣿⡀⠀⠀⣿⡇⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠙⣷⡄⣿⠀⠀⠀⣿⡇⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⣿⡇⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣠⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢸⣿⠀⣀⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣿⠿⠛⠉
⣦⣶⣤⣶⣤⣤⣦⣶⣶⣶⣶⣤⣴⣶⣼⣿⣯⣴⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣦⣤⣶
⣿⣿⣇⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⣠⣐⠀⠀⠠⠀⢀⠀⠄⠀⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠆
⠀⠁⠀⠀⠊⡀⢀⠈⠀⠀⠱⡞⢁⠁⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢠⣞⣁⡉⠀⠘⠄⠀⠸⡀⠀⠀⠁⠀⢀⠀⠀⢀⡠⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠦
⠀⠠⠀⠀⠀⣀⠀⠀⠀⢀⠀⠠⣈⠤⡔⠀⢠⠀⠈⠀⡀⠀⠀⠈⣴⣣⠄⡈⠅⠀⠈⡀⢀⠀⠀⠀⠄⢰⠀⠀⡄⠈⠀⣀⠀⢂⠀⠀⢀⠀⣀⠀⢀⠀⠄
⠀⡀⠀⠀⠀⠈⠁⠓⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢌⡑⢸⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠈⠁⡋⠡⠀⠂⠁⠀⠀⠀⠄⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠐⢄⡀⠀⠀⣀⠒⠀⠀⠀⡄⠤⡀⠀⠀⠐⢀⠀⢣⡀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠐
⠀⡀⠁⠀⠦⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡆⠈⠁⠠⠤⠀⠀⠁⠠⠀⠀⠀⠂⠀⠔⠀⠀⠈⠁⠀⠂⠀⠀⡀⢀⠀⠀⠠⠂⠀⠂⠀⠀⡀
⠀⠁⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⠈⠀⠠⠀⠀⠐⠀⠄⠀⢀⡀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⡀⠀⣤⣤⡤⠤
⠀⠀⢄⠀⠀⠂⠠⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡄⠿⠀⠂⠐⠀⠀⠀⠃⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⡁⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢤⠀⠀⠀⠀⢠⡆⠀⣤⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣿⣤⡅⠐⣋⣳⡀⣶⠶⣦⢰⠶⢦⡀⢦⠀⡴⠀⠀⠩⣅⣼⠀⢲⠀⣶⠶⠆⢹⡏⠁⣿⠶⣄⢠⠶⢼⠳⣋⣳⡄⢶⡀⢰⠄
⠀⠀⠀⠀⠀⠀⡼⠀⢽⠸⢥⠼⠇⣷⢤⠞⢸⡤⠼⠃⠘⣶⠁⠀⠀⢠⣇⠀⠧⠾⠄⣷⠀⠀⠸⣧⠀⠷⠀⢿⠡⡤⢾⠘⢧⠼⠧⠀⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠸⠆⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋


⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠤⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡄⢈⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠠⠊⣄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠄⠀⡄
⠀⠀⠈⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠢⠐⠃⠀⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⡀⠀⠀⠀⠠⠑⢄⡀⠁⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠇⠀⢀⠂⠒⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⢀⠐⢀⠀⠜⠄⠂⠀⠤⠀⠀⠀⠉⠁⠀⠀⠀⠠
⠀⠀⠄⠈⠉⠈⢠⠈⠉⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡀⠀⣀⠀⠈⠁⠀⢀⣄⣉⣠⠠⠀⠀⠀⡀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈
⠠⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⡀⠀⠐⠀⠇⠀⠀⠀⠀⠰⠐⠀⠀⢃⡀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠂⠀⠀⠀⢠⡤⠦
⠀⠀⣄⠀⠀⠀⢀⠠⠐⠀⡀⠀⠈⡖⠀⠀⠀⢀⡀⢁⠂⠀⢄⡈⠠⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠁⠀⠀⠀⢀
⠀⣀⠀⠄⠀⠀⠀⣀⠀⢰⠀⠀⡞⠀⡀⠀⡈⢭⡬⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⠀⠀⠀⠀⠐⠀⡀⠀⠀⡀⢀
⠈⠧⠁⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠤⠤⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠰⠈⠂⠀⠀⠁