Disagreement Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠤⢾⠉⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡆⠓⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣦⢀⡞⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠐⡀⠀⠀⣿⣿⣿⣿⡧⠀⠑⠠⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠶⣼⣿⣿⣷⡶⠛⠁
⠀⠀⠀⢱⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠻⡃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠚⣿⣿⠟⠓⠂
⠀⠀⠀⡴⠂⠀⢸⣿⣿⣿⣿⠀⠀⢀⠁⢑⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠄⢈⠀⢠⡀⠀⠀⠀⢿
⠀⠐⠉⠁⠀⠀⢸⣿⣿⣿⣯⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡈⠐⠂⠀⡀⣁⢀⡀⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠈⠵⠀⠀⠀⠀⢤⠀⠃⠘⣰
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠤⣴⢆⠻⠰⠵⠆⣙⠳⠖
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⡀⠁⠦⠀⠰⡄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⠀⠀⠂⠀⠒⠀⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢿⡀⣈⠀⢠⣤⣠⡷⡀⠶⠂
⠀⠀⠀⠀⠀⠴⠰⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⠈⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠏⢀⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢉⡀⠐⣀⡀⠀⠈⠁⠀⠠⠄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡤⠀⣀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠰⠎⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣾⣟⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠁⠠⣼⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠖⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⣀⠀⠲⢦⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡄⡠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠜⢁⣤⢄⢁⠘⠳⢷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡯⠠⠛⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⠁⠠⠆⢀⠨⡄⢙⠎⢁⠃⠐⢰⡶⣿⣥⣾⠀⠀⠀⠀⠀⠀⣀⣰⠷⡤⣾⣱⣗⠈⡱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢀⡾⠀⢀⠀⠀⠁⠀⢀⡖⢀⡔⠘⠀⢸⠿⠃⠀⠀⠀⢠⣶⣾⡿⣋⣡⣄⠉⢸⣿⣄⢠⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠘⣿⠙⣆⠄⠀⠀⠀⠀⣾⡃⢘⡅⠀⠀⠀⠁⠀⠀⢤⡾⠏⠛⢛⡋⠀⣰⣿⣤⣼⣿⣿⣃⣶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡜⠑⢄⡍⠉⠀⠀⠀⠚⣹⡇⣿⡗⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢈⠍⠘⣿⣿⣿⣿⣿⡿⢟⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⡁⠀⠀⢀⡠⠉⠦⠄⠀⠈⣛⣹⡇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠂⠉⠙⠀⡼⣮⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢈⣠⡼⣧⠀⠀⠘⠁⠀⠀⢀⣈⣥⡶⣿⣇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡁⢀⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢴⣿⠀⣿⠀⠀⠀⣀⣤⡀⠈⠀⠀⠉⣿⣿⠘⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠛⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⣾⡃⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⠀⢽⠉⣧⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢾⠀⣙⠸⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⣼⠀⢘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡄⠀⠀⠀⡼⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⣿⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢉⣥⠀⠀⢠⡍⠿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢎⠁⠀⠀⢀⣼⢿⡏⠀⠀⠀⠀⠀⢠⡇⠖⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠀⠀⠀⠸⣿⠀⠀⠈⠘⣿⠉⠃⠀⠀⠀⠀⠀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣅⣾⠀⠀⠀⡀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⢠⣾⠿⡆⠀⠀⠀⣿⡤⢦⣴⣴⣶⣤⣶⣤⣴⣶⣤⣷⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⢻⣿⣽⠁⠠⠀⠀⠀⠀⠀⢯⠆⠀⠀⡀⠀⠀⠠⠀⠁⡉⡄⠠⠆⢀⣀⣀⣀⠀⣀⠀⠈⣄⣠⣨⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣻⣿⣿⠀⡄⠀⠐⠁⠑⠀⢸⡁⠶⢀⡀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⢶⠴⠤⡀⠈⡆⠀⠙⢻⠟⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢪⡜⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⡼⠶⠄⠀⠠⢂⡄⠂⠀⠀⠈⠀⠰⠀⣿⡆⠀⠰⠀⠁⠀⠀⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢸⣿⣿⠄⠀⠀⠀⠀⠲⠘⡋⠈⠛⠳⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⢿⠐⠈⠀⠙⢊⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣾⡿⠎⠀⠀⠀⠀⠀⢂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠣⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠿⠇⠒⠃⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠒⠦⠾⠷⠖⠤⠤⠃⠀⠀⠀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⢁⣨⡰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⣐⠧⣤⠀⠀⠀⠀⠀⡆⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄
⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠈⠀⠀⠘⠁⠀⠀⠀⢡⠀⠈⡄⠀⠀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⠔⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠈⢄⠀⠀⠀⠈⠀⠀⠀⠁⡀⠀⠀⠀⠀⠟⣿⣿⣿⣄⠀⠀⠀⠀⠚⠀⠐⡂⡒⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠙⠢⡀⠢⡀⠀⠀⠈⠁⠂⠀⠂⠀⠀⠀⠀⠐⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢤⠤⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⡀⡒⠁⠀⠀⠀⠙⠹⣿⣿⣿⣧⣀⠠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠐⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⡀⠀⠠⠔⠀⠀⡀⠀⠀⢱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣀⠀⠐⣵⣤⣄⡀
⠀⠀⠀⢐⡀⠄⠀⠀⠠⠆⠀⠀⡄⠀⠀⠀⠀⣻⣿⠁⠀⠀⠀⣀⠉⢹⣷⣄⠀⠉⠀⠀⣤⣦⣄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄⡤⠚⡿⠢⣄⣿⣿⣿⣿⣦
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠂⢡⠀⠀⠀⠈⠻⡇⠈⠀⠀⠀⠈⠀⠀⢻⣶⣦⠀⡇⠀⣿⡄⣯⠢⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⢸⡟⣣⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠣⡁⠀⠀⠀⠀⠤⠆⠈⢻⠀⠀⠐⣄⠀⠀⠀⠀⠈⠡⠀⢀⠀⠘⣾⣿⡆⠘⢦⣿⣷⡹⠇⢡⡀⠀⣀⠀⢹⣿⣿⣿⣿⢿⡿⠧⠌⠁⠘⠟⢻⣭⣩⣍⣁⣴
⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠐⠀⡸⡂⠏⠀⠉⠁⠀⠀⠰⠆⠰⠛⠀⢦⠀⠙⡟⠻⠆⠀⠀⡛⠇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣤⡠⡀⠀⠀⠀⣄⣹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⡄⠀⠀⢲⣀⠦⠀⣆⠇⠐⠀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠘⠄⠣⠔⠉⠀⠠⠶⠶⠦⢤⠠⠔⠀⠀⠀⠀⠀⠀⠈⠙⡳⠶⠶⠶⠶⠿⠿⠉
⠀⠀⠀⠀⢠⡃⠂⠀⢸⣏⠀⠑⡜⡆⠀⠀⠁⠀⠐⡄⠀⠀⠀⢀⢌⠖⠂
⠀⠀⠀⠀⠀⡈⡇⠀⢸⣿⣷⡘⡆⠱⡀⠀⢀⢀⠀⠘⡆⠀⠀⠀⠋⣀⡤⠀⠀⠀⠄⠀⣀⡖
⠀⠄⠀⠀⠀⠙⡝⢦⡘⢿⣿⣷⠁⠀⢀⠀⠀⠊⠆⡠⠟⠘⢦⡤⠀⠘⠓⠈⠁⠚⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠳⠀⠈⠙⠛⢃⠀⠀⠈⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⢘⣄⠀⠀⠀⠇⡀⠀⠀⠀⠀⠔⠈⠀⠠⠤⠤⠤⠄⢀⡀⢠⡤⠞⠁
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠀⡋
⣀⣀⣀⣄⣠⣥⣤⣤⣤⣤⣤⣤⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢀⠡⠆
⠀⠀⠀⠀⠀⠀⠚⠉⣠⠤⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠬⠐⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢸⣿⠷⣼⣷⡄⠀⡐⡁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⢀⠀⠀⠀⡀⠘⠚⠈⡁⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠠⠀⠂⠁⠀⠀⠖⠃⠀⠚⠛⠻⠿⠿⣏⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠺⠅⠀⢠⣄⣀⣴⣦⡀⠀⠀⢉⡁⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢀
⠀⠀⠀⠀⣠⠄⠒⠀⠃⠸⠚⠀⠀⢀⠀⠀⢀⣾⣿⣧⠨⠛⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠑⢀⠀⠀⠚⠁⠀⠁⡴⣶⡏
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠸⡆⠀⢨⠀⠀⠨⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣧⡿⠇
⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⠠⠛⠀⢀⡳⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⡀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⡾⠸⠃⣼⣛⢴
⠀⠀⠀⠀⠀⣈⠷⠒⠉⠀⠀⠀⡀⠉⠁⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠐⠀⠐⠃⠀⠀⣸⠵⠊
⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠰⠀⠘⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠠⠒⠁⢀⣄⡀⠀⡀⠄⠠⡄⢀⣠⣵⡆
⠀⠀⠀⠀⠀⠈⠛⡓⢒⠲⠆⡃⠀⠀⠀⠀⠀⠀⠤⢄⣛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈⠉⠉⢠⠀⠀⠈⠀⠃⢀⡀⠀⡈⠱⡏⠙⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣴⠲⡀⠀⣍⡙⠓⠒⠒⠲⠴⠂⡠⠴⠛⠇⠀⠀⢀⡀⠀⡞⠀⠈⠇⠀⠀⢸⡵⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡸⠊⢙⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠁⠐⠁⠀⣀⠅⣐⣒⡭⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠚⠁⣜⠉⠃⠉⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠀⢀⠀⠀⠀⢀⠀⢀⢩⡼⢁⣄⣢⠽⣦⠀⠀⠉⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠲⠶⠓⠁⠀⠀⠀⠐⠛⠉⣾⡭⠊⠁⠈⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀⡀⠠⣠⣀⢀⡀⢠⡄⡀⢀⡤⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠃⠀⠁⠉⠉⠁⠈⠉⠉⠀⠉⠉⠉⠉⠉⠙⠋⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠁⠈⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠂⠐⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣧⣄⣠⡿⣿⣄⣨⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⠋⠁⠈⠛⢙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⢻⣿⠀⠀⠀⠀⠀⢸⣿⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⠀⠐⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠉⣉⠀⠀⣴⡆⢰⣦⠀⢈⣀⣈⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣿⡿⠿⠿⠏⠀⠀⠻⠿⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⣷⢰⡶⢀
⠀⠀⣠⣽⣥⡴⠖⠂⢴⣋⣙⣟⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣾⣖⣦⢠⣞⠹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣤⠉⠈⢠⣽⣧⡀
⠀⢰⢿⣿⣿⡀⠀⠀⠀⠉⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣟⠉⠀⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⣏⡧⢠⣏⣿⣿⣽⠆
⠀⠨⢿⣿⣿⡿⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣿⣿⡄⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠉⢹⣿⣧
⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⠗⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⣿⡇⠀⠀⠀⣾⠏⣿
⠀⠀⠸⡇⡿⠀⠀⠀⠀⠀⡏⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡏⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⡇⠀⠀⠀⡟⠀⢹
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⢀⡀⠻⠿⠿⠇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢃⣄⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡔⠀⠀⠀⠀⠀⡠⠠⡟
⠀⠀⢠⣾⣷⡶⠶⠒⠀⠀⣾⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣿⣿⡄⠀⠀⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣨⣥⠄⠀⣴⠻⣶⠈⠊
⠀⠠⢯⣿⣿⡇⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠁⠀⡴⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣧⠀⠀⠀⠈⢻⣿⡷⣄
⠀⠀⠀⢰⣿⣷⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢠⣾⣶⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠗⠀⠀⠀⣸⣿⡇⠈
⠀⠀⠀⣾⠇⢿⡆⠀⠀⢻⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠙⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠏⠀⠘⠃⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠟⠸
⠀⠀⠀⠀⠀⢀⡒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⡌⠋⠋⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠏⣶⠰⡄⠀⠀⠀⣠⡸⠧⠿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣠⣤⡄⣤⣄⣀⣀⣿⣃
⠀⠀⠀⢰⣾⡟⠛⠁⠀⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⣠⢶⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠉⠙⢿⡆⠽⠋⠀⠘⣿⣿
⠀⠀⠀⢨⣿⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢿⣿⡀⠀⠈⣿⣿⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⢿⣿⡿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣇⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⡿⣿
⠀⠀⠀⢸⠻⡇⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠈⣿⠀⢹⡏⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⡇⠀⠀⠀⠀⠀⠀⡾⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣀⠀⢰⠈⠉⠉⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⡿⣟⣥⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡾⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣽⠇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⢀⡀⠀⣠⣀⡄⠀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣾⢿⠿⢛⣡⠶⠞⣋⡵⣺⣵⡿⢟⡿⣛⣥⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠐⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠛⠉⣁⠀⠘⢉⠀⠖⠊⠥⠾⠛⠳⠾⠻⠞⠋⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠐⠚⠂⠴⠚⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠋⠉⠉⠙⠛⠓⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣋⣥⣶⣷⡄⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣵⣾⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠿⠿⢛⣋⣭⣤⣶⣾⣿⣷⠀⠀⠀⠀⠀⠀⠘⣆
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠛⠛⠿⠿⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢹
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠹⣿⠿⠿⠟⢛⣋⣉⣭⣤⣴⣦
⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠁
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣉⣤⣤⣶⣶⣾⣿⣿⣿⣿⡇
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⣤⣤⣤⣤⠤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⣠⠞⡆⠀⠀⠀⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣺⠋⠉⢤⡀⡇⠀⢀⡴⠚⠁⠈⠙⠉⠉⡉⠙⠓⢦⡀⠀⣼⠓⠤⢤⣄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⢹⠀⣿⠀⡇⠀⠀⠀⡇⠀⢼⠰⠂⠀⠘⠲⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢉⡇⠀⢸⠀⠀⠀⠙⠀⠛⠀⡃⠀⠀⠀⡇⠀⣹⡄⠀⠀⢰⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⡟⠉⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⠁⢀⡤⠞⠁⠀⠸⣄⣀⣠⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢾⣀⣀⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠀⠀⠀⠀⠀⣀⣼⠀⣧⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⣀⣀⠀⠀⠹⡄⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠓⠒⠊⠉⠱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠃⠀⠈⠳⣄⠀⡇⠀⠀⣠⠋⠀⡼⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡜⠉⠙⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠣⡀⠀⠀⠈⢻⣇⣠⠚⠁⠀⣰⠃⠀⠀⣀⣀⣀⠀⢠⡏⠀⡞⠀⠀⡰⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠢⣀⠴⠋⠈⠀⠀⣠⠞⠁⠀⠀⠀⠹⣄⠈⠣⣀⣳⡜⠳⢤⡴⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠉⠀⠀⣰⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡴⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠀⠨⠰⠡⢘⠃⠀⠘⠀⠈⠀⠈⠘⠀⠂⡀⠒⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠁⠀⠉⠊⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⣿⣷⡄⢳⣌⡁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠶⢆⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠁⠀⠀⠀⣀⠤⠀⠀⢄⣤⢀⢀⣿⣿⣿⣿⣿⡧⠀⠄⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠒⠄⠀⣀⠤⠒⠈⠁⠀⠒⠀⠀⠀⠀⠀⡾⡀⠀⢀⡀⠄⠒⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠛⠓⠀⠀⠋⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⢀⠀⡶⢷⠀⠻⠀⠀⢰⣦⣤⠾⣿⣦⡀⡀⠄⠠⠄⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⠆⠀⠀⠀⠀⡀⠀⢀⠤⠚⢀⡡
⠀⠀⠀⠀⢠⡤⠊⠁⠀⠃⠘⣇⠁⠀⠀⡀⠀⠀⣴⣿⣿⣷⠤⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠓⢄⢠⠔⢨⠁⠀⠁⣰⣮⣿⠃
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠁⠹⣇⠀⠀⡆⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣹⡜⣾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡀⠜⠃⠀⠘⡔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠙⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠀⠀⡀⣾⢀⡇⢸⡟⠁⣔
⠀⠀⠀⠀⠀⠀⢸⡴⠔⠊⠁⠀⠀⢀⠘⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠃⠀⠀⠨⣇⡵⠋
⠀⠀⠀⠀⠴⠊⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢐⠉⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⠭⠋⠁⢀⠀⠀⠀⠀⡀⢀⡁⠀⠀⣘⣥
⠀⠀⠀⠀⠉⠑⠲⢦⡤⢤⣄⣀⡘⠀⠀⠀⠀⠀⠀⠀⣀⠀⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠤⠤⠤⠰⠚⢁⠀⠀⠀⠋⣹⠀⠸⠀⠀⠠⢖⡼⢛⠟⠯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠖⠀⠀⠂⠀⠤⠄⣀⡀⠀⠀⠧⢽⣾⣋⣀⣀⡀⢀⡀⠄⢀⣤⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⢤⠀⠀⠀⢸⢂⠔⠃⠈⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠇⠙⠄⠀⢳⢂⢤⡀⠀⠀⠉⠋⠀⠊⠋⢀⠉⠀⠀⣀⡲⠀⢀⠇⠀⢀⠀⢀⣀⣀⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⢀⡤⠖⣠⣤⠲⡍⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢳⡴⠋⢱⠀⠀⠀⠀⠀⠀⠸⠀⢠⣀⣀⡀⠀⠀⠀⣾⠔⠉⠀⣾⡕⠊⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⡄⡀⢘⠀⣠⠀⣀⢀⣶⠂⡤⣍⡯⣛⣦⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠃⠐⠠⠟⠊⠀⠀⠀⠀⠐⠛⠉⢽⣫⠔⠋⠀⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠿⠀⠘⠛⠛⠛⠃⠈⠃⠛⠀⠛⠒⠛⠚⠛⠛⠻⠟⠛

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⣿⣿⠀⠀⠀⢠⣸⡇
⠀⠀⠀⣿⠀⠀⣀⢀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠾⣿⣿⣿⣿⣦⣄⠀⣾⣿⡇
⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⣿⣿⣿⣿⣇⠀⣀⣄⠀⠀⠀⠀⠀⠀⢸⣿⠃⣠⣿⣿⣿⣿⣿⣿⡇⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢿⠿⠉⠹⣿⣿⡿⠉⢹⡿⠿⣷⣦⣄⡀⠀⠀⢸⣿⢀⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⡇
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣽⣿⣿⣷⡄⢨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠛⠛⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⣿⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⢸⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⡀⠀⠀⡄
⠀⠀⠀⠀⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⡀⠀⠠⣄
⠀⠀⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣏⠉⣹⠻⣄⣀⠷⡄
⠀⠀⠀⠀⠀⠈⠢⢄⡀⠂⣀⠠⠃⠀⠀⢀⡄⢲⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⣀⠀⠄⠀⠳⢼⡀⠀⠣⡉⠁⠀⢀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⢐⠡⠋⠈⠀⠀⠀⠀⠁⠰⠀⣄⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⢄⢰⠀⠙⢦⡀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠑⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢌⣑⣒⣒⣈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡒⠂⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠉⠉⠐⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢸⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢧⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠃⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣄⠀⠀⠀⠀⣀⡠⣶⠶⠲⠒⠲⠖⠂⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⡗⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡄⣠⠀⢀⣤⠴⠟⣴⠖⠂⢤⠈⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⡦⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢤⡶⠞⠉⠀⠄⠀⢀⠘⠀⠈⠀⠀⠈⢿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⡀⠀⢠⠆⠀⠀⣼⡃⠀⠈⣿⠥⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢰⢁⡾⠉⠀⠀⠂⢸⡇⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⣀⠀⠁⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀⣀⣿⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⢀⣠⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠻⠛⠛⠛⣉⡉⠀⠀⠉⠉⠙⠛⠛⠛⠻
⠀⠙⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢈⣿⡟⠳⣿⣿⣿⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠉⠁⠁⠀⠈⠁⠐⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣀⣿⣿⡀⠈⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣽⣯⣭⣽⣿⣿⣿⠿⠛⢉⣽⣿⣿⣿⣿⡶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠋⠁⣠⣖⣭⣶⠿⠟⠋⠉⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠁⠀⣒⣛⠋⠍⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠗⠂
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠈⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠰⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠉⠉⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠂⠐⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣄⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣶⣤⣤⣄⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⢀⣾⡿⠁⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⢀⣾⡿⠁⠀⢰⣿⣿⣿⣿⠋⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⢀⣾⡿⠁⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣆⠈⢻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⠀⢺⣿⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠻⣿⣿⣿⠟⠀⠀⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠉⠻⣿⣦⣀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣤⣤⡀⠀⣀⡀⠀⠈⠻⢿⣦⣄⠀⠀⠙⢿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣀⡙⢿⣶⣄⠀⠀⠙⠻⣷⣦⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣦⣄⣀⣼⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⣿⠟⢿⣶⣴⡿⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⢀⣿⣇⣤⣴⣶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⢀⣙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣴⣦⣴⣦⣴⣴⣶⣾⠿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⢹⣿⠟⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⠄⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⣴⣿⢿⣦⠉⠹⣿⡆⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡍⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠰⣿⡌⣿⣿⣩⡀⢻⣿⠈⣥⡄⠀⢻⣿⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣧⠀⣤⣴⣦⣄⢸⣿⣟⡋⠀⠀⠀⢿⣷⢸⣿⣟⣡⣌⢿⣿⡿⠃⠀⠈⠉⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⡆⢿⣿⣽⣿⠈⣿⣿⠛⣀⢰⣷⣾⣿⠂⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⡀⢻⣷⠘⣿⡿⣿⣦⠹⣿⠿⠟⠃⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠛⠛⠛⠃⠘⣿⣇⠙⠋⠈⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⡿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢀⣴⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⡰⠋⣨⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠟⠳⡄
⠀⠀⠁⢲⠧⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⡀⠀⠀⠉⠲⣤⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠙⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠖⢀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠇⠀⠀⠀⠀⠀⢰⠀⠒⠀⠀⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢾⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⢸⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠉⡦⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢲⡴⠃⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⢸⠀⠀⠀⠀⠀⠀⢇⣠⠀⠀⠀⠀⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠐⠒⠒⠀⠀⠒⠒⠒⠒⠂⢸⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⣹⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣬⣑⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⣉⣉⣉⣉⣉⣉⣉⣉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢚⣉⣥⠤⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣍⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠘⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢉⣉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠐⣆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣠⠔⡻⠛⢳⣀⠀⣀⡀⠇⠀⠀⠀⠀⠀⠀⢠⢿⠁⠀⢸⠀⠀⠀⡆⢱
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⣆⠀⠀⠀⠳⣀⣠⠤⠴⠒⠊⠉⠹⣯⠀⢸⢁⢦⡀⠉⢹⡟⠉⠉⠓⠲⠤⣤⣀⣰⠃⡞⠀⠀⢸⠀⠀⠀⡇⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢈⢉⠀⠱⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⡇⠀⢇
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⣀⣠⠤⠴⠖⠚⠋⠉⠉⠋⠛⠙⠁⠉⠓⠒⠢⠤⢄⣀⣀⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡇⠀⠀⠀⡏⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠶⠶⠤⠤⠤⠤⠶⠦⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⡜⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡇⠀⠀⣰⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣄
⠀⠀⠀⠀⠀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⣞⠁
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣤⢠⡇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⢦⠀⠀⢷⠀⠀⠀⠀⠈⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⢧⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠷⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⢠⡀⠀⠘⣇⠀⠀⣧
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠉⣴⠒⣶⠒⣦⡛⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⡄⢠⡇⢠⡇⢠⡏⢹⡆⠀⣼⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠐⠗⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠇⣼⠀⣼⠀⣼⠀⣼⠉⣷⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠆⢿⠉⢳⢠⠇⠠⠇⢠⡇⢠⡇⢰⠇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⡆⠘⡿⠀⠀⠀⠀⠀⠘⠀⡾⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠋

⠉⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠘⠀⢀⣾⠟⠁⠀⠠⠀⠀⠀⠀⣿⡆⠀⠀⠀⢠⠏⠀⠀⣀⡤⠆⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⣠⠄⠀⠀⣠
⣤⡀⠀⠈⠳⠤⠀⠀⠈⠙⢦⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠞⠀⠀⣿⠃⠀⠠⠞⠁⠀⠴⠛⠁
⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠈⠂⠀⠀⠀⠺⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⠟⠋⠁⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⣤⠖⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀
⠐⠋⠀⠀⠀⠀⠰⣶⣦⠤⠄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⢀⣀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰⠶⠟⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠐⠂⠀⠀⠘⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠘⡇⠀⠀⠀⠔⠋⠁⠀⠀⢤⡄⠀⠀⠀⠀⢻⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠰⣆⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡟⠁⠀⠀⠀⡀⠀⠀⠀⢀⣤⠤⠀⠒⠀⠀⠀⠰⠿⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⢴⣦⠀⠈
⠗⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣤⡀⠀⠀⣶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⣿⡀⠀⣶⣶⣶⣶⠶⠆⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⢻⠀⠀⠐⠋⠁⠀⠀⠙⠷⡄
⠀⣼⠀⠀⠘⠇⠀⠈⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢰⣶⡆⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠹⠀⠀⢀⡤⠖
⠇⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠴⣦⣤⣄⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢠⣾⡿⠀⠠⣶⠤⠄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠠⠴
⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢶⣶⣤⣀⠀⠀⠀⠀⠀⠶⠛⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠻⡆
⠀⠀⠀⢴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠁⠀⠀⠀⠈
⢠⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⣠⣀⡀
⠉⠀⠀⠀⠀⠀⠀⠻⠦⡀⠀⠀⠀⠟⠀⠀⠐⠷⠶⠤⠄⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣄⠀⠀⠙⠛⠻⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠛⠋⠁⠀⢀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⣶⣦⣤⣀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣶⠶⠶
⠄⠀⢀⣤⣀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢸⠇⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡄⠈⠁⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠰⡄⠀⠀⠀⢀⡾
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠤⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡄⠀⠁⠀⠀⠀⠀⠶⣿⠿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠈⠀⠀⠀⠀⣀⡀⠀⠀⠀⣴⡄⠀⠀⠀⢀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⠀⠀⠀⠀⠀⠠⡶⠞⠁⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣴⡶⠿⠟⠛⠁
⣤⡄⠀⠀⠐⠚⠛⠁⠀⠀⠀⣿⠀⠀⠀⣴⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⣴⠶⢍⡇⢠⡎⠉⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡌⠠⡶⡤⣟⠄⠁⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⢘⡄⠀⠀⠀⠀⠀⠀⠆⠀⠡⣀⣙⠁⠎⢈⣒⠮⠐⢐⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠀⣼⠀⠀⠀⠀⠀⠀⠀⠰⡀⠤⠀⠂⠀⣁⣡⡤⠖⢛⣩⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠈⠈⡱⠂⠀⠀⠀⠀⠀⠰⠶⠒⠛⠉⠉⠰⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⢉⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠚⢉⣁⠤⠴⠒⠚⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠊⢉⣁⠤⠴⠖⠒⠉⠁⣀⣠⠤⠴⠒⠋⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡠⠤⠶⠒⠚⠉⠁⣀⡠⠤⠶⠒⠋⠉⢀⣀⡤⠤⠖⠚⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡤⠤⠖⠒⠊⠉⢀⣀⡤⠤⠖⠒⠉⠁⠀⣀⣠⠤⠴⠒⠉⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⠤⠴⠒⠒⠉⠁⠀⣀⡠⠤⠖⠒⠋⠉⠀⢀⣀⠤⡤⢖⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡤⠤⠖⠒⠊⠉⠀⢀⣀⣤⠴⠒⢊⣉⣥⠴⠊⠉⠀⣀⣀⠀⢹⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡠⠤⠴⠒⠚⠉⠁⢀⣠⡄⠀⠀⠁⢀⣀⣤⡴⠺⣿⣹⣇⠀⢻⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⢀⣠⡤⠔⠚⠉⠁⠀⣤⣦⡞⠻⣏⠙⣿⣫⡹⠿⠟⠀⠀⢻⡆⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠀⠀⣤⢰⣟⣁⢹⡄⢤⡄⠻⠆⠙⠉⠀⣀⣄⠶⠚⠉⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⠀⣀⣄⠰⣾⣃⠀⢹⣆⢻⣥⠆⠛⠋⠀⢀⡀⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣿⣽⣅⢻⣯⡼⠛⠛⠀⠀⣀⣤⡴⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠘⠷⠙⠃⠁⠀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠄⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣄⠴⠚⠈⠉⠀⡀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⠀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠆⣠⠆⠀⠀⠈⣠⣼⣿⣌⣀⠌⠀⢀⣒⢴⣿⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠃⡄⠠⣠⣴⢀⣸⣟⠀⣻⠟⠫⠖⠈⠁⢙⡿⢲⡅
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⢩⡠⠿⠇⠘⠛⠋⠀⠁⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡴⢀⡁⢸⡗⢥⣮⡀⠀⣤⠤⠒⠈⠁⠐⠄⠀⠀⠰⠃⠀⠀⠀⠒⡀⠀⠀⠀⠀⢴⠄
⠀⠀⠀⠀⠈⠡⣨⣷⣄⢜⢿⣿⣿⡦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠒⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢨⡛⢿⠎⣾⣿⣟⡧⡀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠶⢺⠗⡦⠀⣟⣿⠟⡇⠉⢀⡀⠀⠐⠉⠀⠀⡇⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠐⠒⠛⣷⠃⠀⣾⠛⠀⡇⠘⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⣸⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⢈⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⣻⡟⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⡆⠀⠈⠛⠁⠀⠉⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠋⠁⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡜⠁⢰⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣾⠉⠉⢹⡇⠀⠀⠈⠉⠉⠉⠉⣹⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢹⠆⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣀⣀⣸⠧⣄⣀⣀⣀⣀⣀⣈⡇⠀⠀⠀⠀⢯⣀⣀⣀⣀⡀⠀⠀⣸⢇⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡉⡌⠋⣇⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣆⣡⣻⠿⣄⡀⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡿⣿⣟⢇⢩⣃⣲⣄⡹⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⡀⠀⠸⣧⠳⡘⣿⣿⠸⠋⠀⣿⡟⣶⠌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠱⠀⠇⠜⢁⠄⠀⣿⣷⣜⢿⣿⡆⠀⢠⣿⢧⡏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⣠⣴⡲⣄⡤⠖⢻⣿⠏⣳⣭⢧⠸⢿⣔⡚⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠹⣼⡥⠶⠿⠞⠉⠀⠀⣾⡇⢠⠷⣄⠷⠶⠃⢘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⠀⠀⡀⠢⡈⠉⠂⠀⢸⣃⣀⡞⣨⣿⣿⣾⡟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⠀⠈⠀⠀⠀⣼⠋⢉⡴⠛⠁⠹⣯⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⡟⠀⠀⠀⠀⠈⢷⡀⠀⠟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⠀⢃⠘⠛⣊⡤⢶⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⣽⡇⠀⠀⠀⠀⠀⠀⠙⢦⣀⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠂⡞⣭⠝⠻⣿⡂⢀⠁⣄⠟⢦⣠⣤⣀⣀⣀⠀⠀⠈⣧⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣻⣄
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⠄⢷⡃⡴⣐⠘⣾⠪⢻⡛⢷⣀⡙⣆⠀⣨⠽⠷⣄⡀⢻⡀⢹⣇⣠⣴⣛⠲⢄⡀⠀⠀⠀⠉⠙⣿⣇
⠀⠈⠋⢕⢤⣠⣤⣤⡀⠀⠀⠀⢐⣱⡚⠛⢿⣟⣻⣭⣽⡿⣿⠞⢉⣵⠃⢰⣗⠦⣄⡀⠙⠺⡷⡾⠉⢀⣹⡿⠧⣦⠙⡆⠀⠀⠀⠀⠸⠿
⠀⠀⠀⢰⣿⡯⣿⣇⣳⢻⠉⠙⣻⣶⡶⠚⠋⠉⣹⢷⠒⠾⣷⣴⡿⠿⣳⣾⣏⣙⣓⣯⣷⣆⣀⡤⠞⠋⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠋⠉⠉⠁⠈⠉⠉⠉⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣰⠉⠙⠦⣐⣎⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡔⢀⠈⠢⡀⠀⠈⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠰⠿⠿⣏⠀⠀⠈⠻⢧⣤⠄⢊⠄⢱
⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⣤⣶⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢠⡄⠀⠀⠈⢣⡀⠀⠀⢸⣇⣉⠤⠀⠈⣇
⠀⠀⠀⠀⡼⠀⠀⢠⡔⠋⠉⠀⠀⠀⡗⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠤⠂⠀⠛⢧⡀⠀⠀⠀⠹⣤⠀⠀⢻⣖⠀⠀⠠⠺⡆
⠀⠀⠀⣰⠁⠀⢀⡟⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⡄⠀⢀⣀⠤⠔⠒⢲⠾⢖⠁⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠘⣷⡤⢄⠉⠁⠀⠀⠀⡇
⠀⠀⣰⠃⠀⠐⠞⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢶⣆⠀⠀⠀⠀⠈⢂⠀⠀⠀⠈⠀⠀⢱⠀⠀⠀⠀⣿
⠀⠰⡇⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠠⡀⠀⠀⠀⣀⠀⠀⠀⡀⠹⡂⠀⠀⠀⠀⠀⢳⡄⠀⠀⣈⠖⠁⠀⠀⠀⠀⣿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠉⠉⠉⡶⠀⠐⠶⠞⣶⣤⠀⠀⠀⠈⣇⠀⡆⠀⢈⡄⠀⠀⠀⠀⣿
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⣠⠎⠉⠸⣷⡀⠀⠀⠈⢢⡱⠔⠁⠁⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠎⠀⠀⠀⠀⠁⠙⠲⠴⠴⣟⠦⣀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⢦⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡈⠙⠒⠂⠀⠀⢸⡄
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣂⠴⠋⣹⣆
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠄⠀⢠⠃⢨⣷
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⡞⠁⠀⠀⠀⠀⠀⠊⠐⠀⠀⠐⠁⠀⠀⠙
⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⣀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣯⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣉⣉⠁⠐⠒⠈⠉⠉⠁⠀⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣒⣈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⠪⢭⣯⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠤⠤⠤⠬⣥⠤⠔⠒⡶⠚⣁⣴⡠⠴⢠⠀⠀⠀⠀⠀⠈⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠇⠀⠀⢀⡏⠀⠀⠀⠈⠉⠁⠀⠀⢀⠈⣆⠀⠀⠀⠀⠲⣌⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠸⢄⡀⣠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠀⠘⢆⠀⠀⠀⣀⡏⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠉⠑⠦⠔⠀⠀⠀⠀⠀⠀⠀⡤⠴⠉⢹⠳⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠒⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠊⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⢄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠤⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢍⣙⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠞⠉⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠰⡪⠗⡋⠐⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠿⡧⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠠⠼⠄⠐⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠧⢄⣀⣀⣀⡠⠔⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⣉⣤⣴⣀⣀⣀⣀⣀⣀⣀⣐⡂⣄⠈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠃⡰⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠡⡘⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠿⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⣰⠖⠂⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠂⠀⠀⠻⠿⠿⠋
⠀⠀⠀⠀⡇⠀⢀⣤⣴⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠠⣤⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡄⠀⠀⠀⠠⣤⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡤
⠀⠀⠀⠘⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣻⣿⣿⡟⠁
⠀⠀⠀⠀⠈⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠶⠶⠶⠾⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⡇⠋⠀⠀⠀⠀⠸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⣿⣿⣷⡆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡬⠂⠀⢀⣀⣴⣾⣶⣴⣄⣀⣤⣀⣀⢘⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⠵⠒⣲⣛⣿⣿⣿⣿⣿⣿⣿⣋⣿⢿⣿⣿⣿⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣯⡭⠶⠶⠿⠟⣻⣿⣿⣿⣿⠏⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⡿⢿⡛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠙⠀⠘⠹⡄⠀⠀⠀⠀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠟⠀⠘⢿⣿⣿⣿⣿⣿⠃⠀⠉⠀⠄⠀⠀⠀⠀⡀⠹⠦⣄⣠⣾⢛⠈⢫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠈⠀⣿⣿⣿⣿⣿⡿⢢⣄⡀⠁⠀⠀⠀⢀⣘⣻⣿⠑⠛⠛⠛⢶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢠⡀⠀⣿⣿⣿⣿⢿⣵⣂⠀⠙⢦⡀⠀⢾⣀⠉⠛⢿⣦⠀⠂⢠⠺⣿⣿⡋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⣿⠾⢿⣧⡟⠑⠛⠿⠿⣤⡀⢀⣹⡆⠈⠙⣿⣷⣛⠛⠀⠀⣼⢀⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣷⣾⠃⠀⠈⣿⣿⣿⣷⣤⠤⡿⠋⣠
⠀⠀⠀⠀⠀⠀⠀⣰⠛⢿⣿⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡞⢰⢿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠃⢠⣿⠇⠀⠀⠀⠀⠀⠸⡾⢤⢄⢀⡀⡇⠚⡿⠀⠀⠀⠀⣿⣿⣿⠿⢻⢿⠹⡟⠑⢿⠙⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢐⠁⠀⠀⠀⠀⡿⣿⣿⡄⠀⠀⠑⠄⡀⠘⡆
⠀⠀⠀⠀⠀⢠⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⠀⠀⠀⠀⠹⣆⣘⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⣼⣁⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢜⣠⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⢃⡘⢋⡉
⠀⠀⠀⢰⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠂⠀⠀⢀⢜⣯⣿⣿⣧⠀⡈⠐⠚⠃⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠋⣼⣿⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁

⣾⣷⠿⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢀⣴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣳⠀⠀⠀⣀⠤⠒⠊⠉⠔⠛⠉⠁⣸⠾⡷⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⣧⣀⠞⠁⠀⣀⣤⡴⣆⣤⡄⢠⡏⢀⢹⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡃⠈⠝⣄⡴⠋⢠⠏⠰⠫⠋⣷⠳⠀⢀⣼⡇⠀⠙⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢯⣉⠀⢺⣾⠀⠀⠁⠀⠁⠁⣰⠧⣾⠀⣠⠞⠁⠀⠀⠀⠙⠑⠢⣀
⠀⠀⠀⠀⠀⠀⢀⣴⡦⠞⠓⠒⢈⠟⠉⠸⡄⠀⠀⠀⠀⠀⠁⠀⠈⠙⠣⣄⡀⠀⠀⠀⠀⢀⠤⠤⢷⣄⠀⡀
⠀⠀⠀⠀⠀⣶⠎⠀⠀⠀⢀⣴⠯⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢑⣲⣶⠴⣒⡰⠂⠀⠀⠈⠑⢳
⠀⠀⠀⠀⠀⡏⣀⠀⠐⠲⡍⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣰⠞⡇⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⡄⢸⣿
⠀⠀⠀⠀⠀⢳⠈⠢⠀⠀⠀⠀⠸⡏⢻⢦⣌⣆⡀⢸⠀⣠⣷⡟⡳⣸⢣⠀⠀⠒⢫⠞⠉⠀⠀⠀⠀⠀⡇⢨⠏
⠀⠀⠀⠀⢀⡾⠣⡉⠓⠫⠉⠀⡄⠘⣌⡈⢻⠿⠳⣸⡴⠛⠛⠁⠰⠃⠀⣣⠀⠢⢤⣬⣵⣓⠀⠀⠀⢠⡷⠛⡆
⠀⠀⠀⠀⢸⠁⠀⠲⣩⡤⠖⠉⢧⡀⠈⡁⠀⠒⣀⣠⣤⣀⠁⠀⢰⢀⠔⠁⢀⣀⣰⣖⣹⠟⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡄⠀⠈⠉⣰⣆⣤⠄⠈⢢⣸⢀⣘⢷⣄⣰⠞⠡⠄⢸⡎⠠⢤⣶⣛⠧⠚⢭⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣾⡟⠀⠀⠰⣤⠖⠋⠁⠛⠿⣉⣿⡀⠒⢀⣠⣧⣄⣀⣁⣼⡷⠒⠉⠉⠀⠀⠀⠀⢻⠑⠢⠤⣴⠆⠀⢸
⠀⠀⣿⢸⠃⠀⠀⠀⢻⠀⠀⠀⢠⠔⠋⠙⣿⢹⣭⢦⣶⡔⣼⡇⣿⡇⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⡼⠁⠀⠀⡌
⠀⠀⢹⡀⠀⠀⠀⢀⡏⠀⠀⢀⠿⠀⡀⢀⢹⢾⣿⠀⠘⠀⢻⡟⣉⢳⡀⠑⢤⣀⣤⠎⢀⡟⠀⣠⠃⠀⠀⠘⣺
⠀⠀⠀⢳⠀⠀⣀⣸⠀⠀⠀⢸⡀⠀⠁⠋⠸⣷⣿⣶⢶⢶⣾⡷⣇⠈⠛⠢⡀⠀⡇⠀⡞⢠⠞⠁⠀⡀⠴⠊⠁
⠀⣠⠤⡟⠛⡟⠉⢹⠙⠲⢤⡀⠙⠲⠤⠤⠤⠷⣌⣉⣉⣣⡆⠞⠃⠀⠀⣀⣧⢀⠁⠀⣄⡏⠀⠈⣏⢀
⡞⡇⢰⠁⠐⣧⡴⣼⠒⢆⡠⢧⠀⠀⢰⣿⣿⣵⡊⡨⢿⠃⠀⡀⠀⢠⡮⣡⣾⣉⠙⣏⣉⣷⠀⠀⢸⣷⢷
⠃⣧⠹⠣⠱⠋⢣⡏⠀⠈⠷⠃⠀⠀⠀⠈⠉⠟⠋⠀⠀⠀⠘⠁⠀⠸⡁⠯⡇⢸⠼⢿⢠⠿⣴⣴⣬⣛⡜
⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠃⠀⠸⠃⠀⠈⠛⠛⠁

⠀⠀⣠⣤⣤⣤⣀⣸⣷⣦⣴⣾⣿⣦
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠘⠿⠿⢟⠋⡙⢛⡉⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⠙⠁⠼⠇⢸⣿⡿⠛⣿⣿⣿⠛⠉
⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠈⠇⡀⠀⢀⣀⣤⣼⣿⣍⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⣰⠟⠛⠉⠉⠁⣨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠟⠻⠿⢿⣿
⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⢹
⠀⠐⣠⡗⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⠀⠂⣹⡇⠀⠒⣀⠀⠄⠸
⠈⣼⠋⠀⢀⣴⡀⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣤⣹⡏⠛⢛⠇⢠⣿⡇⠀⠁⠿⠋⠀⡄
⠰⠛⠄⣠⠏⢹⡷⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠉⠛⠿⠛⣷⡐⠃⠀⠀⠉⠁⠀⠀⠀⠀⢰⠁
⡆⠀⠀⠁⢀⢼⣇⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢠⡟⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜
⠧⠠⠐⠊⠀⣸⠏⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⣿⠃⠀⣸⣿⠟⠀⢀⣀⣠⣀⠔⠁⢂⡀⠃
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⠀⢠⡿⠀⣴⣿⠁⡘⠂⠚
⠀⠀⠀⢀⣾⠁⡶⢠⣄⣀⣀⡠⣤⢶⢾⡆⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⡼⠁⠐⣿⡷⢠⣿
⠀⠀⠀⢸⡏⢰⡇⣾⢹⡇⢻⡇⠹⡎⣇⢻⡀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣾⠁⠀⣠⣤⡄⠠⢿⡀
⠀⠀⠀⢸⡄⣾⠀⣿⢸⣿⠘⣷⠀⣿⢹⢸⡇⠀⠀⠀⣸⡇⠀⠀⠀⢸⣿⠀⠀⣀⣀⡀⣷⢸⣷
⠀⠀⠀⢸⡇⣿⠀⣿⠈⣿⡄⢻⡀⣿⢸⢸⡇⠀⠀⠀⣿⠁⠀⠀⠀⢼⣿⠀⠈⢉⡉⣱⠋⢸⣿⡄
⠀⠀⠀⢸⡇⣿⠀⣿⡆⢻⡇⢸⡇⣿⠸⢸⡁⠀⠀⢸⡏⠀⠀⠀⠀⠈⣿⡀⠀⢸⡇⢿⠀⣸⣿⣿⡄
⠀⠀⠀⢸⠁⣿⠀⢸⡇⠘⡇⢸⡇⢸⡄⢞⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⢸⡆⢿⣿⣿⠇
⠀⠀⠀⠧⢠⣿⡠⠾⡁⢰⡇⢸⡇⢸⡇⢸⠅⠀⢸⡏⠀⠀⠀⠀⠀⢘⣿⠀⠀⢸⣿⠊⣧⠈⢿⡏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⣼⡿⢧⡿⠄⠈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠸⣅⣀⡀⠀⠀⠀⠀⠀⢹⣿⣷⡖⠒⠒⠠⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡄⠀⠀⠆⠀⠀⠀⠸⠳⡀⠉⠒⠢⠤⠄⣀⣈⣙⡃⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⣀⡹⢆⣀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠶⣌⣠⠐⠢
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠂⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠲⣎⣸⡒⢠
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣶⣶⣴⡆⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠡⠘⠬⠋⠤⣀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡿⢻⣿⣿⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢰⠈⠦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠚⣛⠠
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠋⠉⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡏⠛⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡄
⠀⢀⠀⢀⣉⠄⡠⠊⠀⠀⠀⠈⠀⢉⡝⠶⢄⡀⠀⠀⡜⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠸⣍⡵⢃⠀⠀⠀⠀⠐⠠⠁⠀⠀⡀⠈⠐⠶⠁⠀⠀⠀⠀⠀⠇⠀⠀⢀⣀⡤⠐⠸⠉⡁⠀⠢
⠀⠀⠐⠀⠤⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡀⠀⠣⠀⠀⠀⠀⠀⢛⠉⠱⠒⠒⠂⠒⠒⠊⠉

⣀⣠⣤⣤⣤⣴⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣵⣿⡿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣿⣿⣿⣿⣿⣿⣟⣻⠏⣉⠈⠉⠉⢉⠹⣷⡀
⣿⣿⣿⣿⢿⣿⡯⠒⡉⠀⠀⠐⠔⠁⠀⠀⢿⣦
⣿⣿⣿⣿⣿⣯⢴⡏⠈⢄⠀⠀⠀⡔⠀⠀⢸⣿⣷
⣿⣿⣟⡿⢋⣴⠏⡀⠀⠊⠀⠀⡜⠀⠀⠀⣾⠀⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⣿⣿⣿⣱⣿⢯⠎⠀⠀⠀⢀⡞⠀⡴⠀⠀⣿⠀⡈⢿⡄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣻⣿⣧⡏⠀⠀⠀⠀⡼⠀⣼⠀⠀⠸⠉⢰⠁⠘⡇⠀⠀⠀⡐⢁⠔
⣿⣿⣿⣿⡿⠀⠀⠀⠀⢰⠁⠈⡇⠀⠀⣿⡀⠟⠄⢸⡇⠀⠀⡠⠀⠁
⣿⣿⣿⣿⡇⠀⢀⠀⠀⡿⢰⠀⡇⠀⢀⣿⢠⣇⢀⣼⡇
⣿⣿⣿⣿⣇⠀⢸⠀⢠⠇⠂⢸⡇⠀⢸⣿⣼⠛⣦⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤
⣿⣿⣿⣿⣿⡀⣸⠀⢸⠐⠀⠀⠙⣡⣾⣿⡇⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣶⣠⣤⣀⠀⡀⠂
⣿⣿⣿⣿⣿⡇⢸⠀⡄⢀⡇⢰⣶⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣛⡛⠉⠉⠛⢿⡤
⣿⣿⣿⣿⣿⣧⣾⠀⡇⣼⢡⣿⣿⣿⣿⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⡿⠘⠛⠀⢼⣇⠃
⣿⣿⣿⣿⣿⣿⠯⠀⠇⣿⣿⣿⣿⣿⡏⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧⣅⠀⢠⣼⡿⠁
⣿⣿⣿⣿⣿⣿⣾⣤⣼⣿⣿⣿⣿⠟⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⢿⠥⠒⠀⠙⠢⡄
⠙⢻⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⢠⡀⠙⠲⠶⠖⠂⢁⠀⠀⢸
⢀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢷⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⢉⣿⡏⠉⠂⠀⠀⢀⢠⣯⡀⠀⢈
⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠶⠶⠖⠒⢁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣾⣿⣷⣶⣶⣿⡋⢀⠔⠉⠉⠀⢸
⣿⣿⣿⣿⣿⣿⣧⣶⣶⡦⠴⠖⠛⠛⠛⠿⢾⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣄⠈⠙⠳⣤⣀⡀⠀⣼
⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡷⠠⢴⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡄⠀⢉⣹⠛⠋
⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢼⠃⡷⢀⡃⠀⣦⠀⠀⣰⣿⣿⣿⣿⣟⣻⣿⣶⣾⡟
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠈⠇⠊⢉⠄⠋⣡⠖⠁⠀⢠⣿⠇⢸⠈⣩⠋⠹⡏⠉⣯⢿⡄
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠒⠐⠁⠀⡾⠁⠀⠀⠀⣾⣿⠀⡟⠀⣿⠀⠀⠁⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣀⣀⣀⣀⠀⣰⡇⠀⠀⠀⣰⡇⢻⢰⠇⡀⣿⠀⠀⠀⠀⠀⠀⢀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⠀⠀⠀⠀⠀⠀⠀⢣⡰⠋⠁⠀⠀⠈⣹⣿⠀⠀⠀⢠⣿⡁⡿⠈⠀⠄⣿⠀⢀⠀⠀⠀⠀⢈⡜
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣿⣿⣷⣷⣶⣶⣶⣿⣷⣾⣶⣶⣶⣶⣾⣿
⣿⣽⣭⣽⣽⣷⣽⣽⣯⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣶⣾⣿⣶⣷⣾⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣩⣴⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⡿⠿⠛⢋⣉⣡⣤⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡄⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⠉⠙⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢻⣿⠿⠿⠟⠛⣋⣉⣁⣤⣤⣴⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⡿⠿⠿⠛⠛⠋⣉⣉⣁⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⢀⣀⣠⠴⠞⠋⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⢛⠓⢶⣶⣶⣤⣤⣤⣤⣤⣤⣶⠶⠖⠛⠋⠉⡀
⣿⣿⠿⠻⣿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠂⠀⠀⠁⠀⠀⠀⠀⠀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠒⠋⠁
⠀⠀⠀⡀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠊⠊⠀⠀⠀⠘⠦⡀⠀⠀⢀⠔⡁⠁⠀⠀⡀⠀⠀⠠⢊⠔⠁
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡐⠀⠀⠀⠀⠀⠀⠀⠌⠢⠔⢁⠀⠀⠀⠀⠀⠁⡀⠐⠠
⠀⠀⠄⠄⠌⠃⠀⠀⠠⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⠀⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠄⠄⠊⠉⠁⠀⠀⠀⠀⠀⠀⣰⠲⠞⠙⠴⢲⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⣠⠤⠀⢰⠒⠒⠒⠒⠒⠒⠒⣲⡤
⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⡏⠁⠀⠀⠀⠀⣨⢉⡇⠀⡰⠁⠀⠀⠀⡠⡢⠙⡆⠀⠀⢸⠀⠀⣀⠀⢀⢔⠈⢻
⠀⠀⠀⢀⠠⣀⠄⡀⠀⠀⡀⠺⡁⠀⠄⢀⠀⡀⠁⢀⡹⠀⡇⠀⠁⡀⠊⠄⠀⠀⢀⠀⠀⢸⠀⠪⠀⠄⡁⠁⠀⢸
⠐⡈⠄⠀⠑⠠⠂⠀⠀⡰⠁⠠⣧⣄⠀⠀⠀⠀⢀⣬⠇⠀⠘⣄⠀⠀⠆⠀⠀⡠⠃⠀⠀⢸⠀⠀⠐⠄⠀⠀⠀⢸
⠀⠈⠢⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⠖⢢⡰⠒⠞⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄⡀⠀⠀⠀⠀⠠
⠀⠐⡁⠀⠀⠀⠀⠀⠢⠁⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠊⠀⠀⠀⠀⠀⢀⠼⡀⠀⠀⠁⠀⠀⠀⠀⠀⡐⠁⡄
⠀⢠⠀⡠⠀⢀⠀⡀⠁⠀⠀⢠⢀⠀⠑⠀⠀⡠⠂⠀⠀⠀⠘⠀⢠⡀⠀⢀⠐⠋⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈
⠀⠈⠀⠀⠀⡡⠂⠀⠀⠄⠀⠈⠀⠀⠀⠡⠈⡀⠁⠀⡰⠀⠸⠀⠀⠩⠒⡁⠁⠀⠀⠜⠀⠀⠀⠂⡀⠂⠁⠈
⠀⠐⢄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡀⠀⠈⠀⠀⠞⠁⠀⠀⠁⢀⠀⠉⠀⠀⡀⠞⠁⠀⠀⠀⠠⠈⠀⠀⠀⢐⠄
⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢀
⠈⠠⠀⠀⠀⠀⠀⠀⠖⢁⠀⠀⠁⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠂⠄⠀⠀⠀⠀⠠⠖⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢀⢀⡠⢀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡀⢄⠀⠀⡠⠀⠔⠁⠀⠀⢀⡀⠀⠀⠀⠠⠂⣠⠀⠀⠀⠀⠀⠄⠀⠀⠔⢁⠀⡄
⠰⠨⠑⠄⠑⠊⠠⠀⠀⠀⠀⠀⠀⠁⠀⡀⠊⡠⠂⠈⠀⠀⠀⠈⠁⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡇
⠠⠨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠀⠀⠈⠀⠀⠠⠑⠔⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⢀⡀⠀⠀⠀⠀⠀⡘⠁⡀⠀⠀⢀⠀⠀⠀⠀⢀⡀⣀⡀⠀⣀⠀⠀⢀⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣉⣉⣉⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠛⠻⠿⠿⠾⠿⠿⠷⠿⠿⠿⠾⠿⠿⠛⠛⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢠⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣿⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⢟⣭⣶⣶⣾⣭⡻⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢱⡿⣉⣿⣾⣿⠿⠛⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣇⠻⠟⠛⢉⣁⣤⣴⣾⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⡄⢶⣿⣿⣻⣿⣠⡿⣣⡿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿⣶⣽⣛⣛⣛⣭⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣻⣿⠉
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣿⢻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣇⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣁⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠒⠀⠰⠄⠠⡦⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠶⠀⠠⡆⠀⣤⠀⣠⡄⠀⣀⠀⠀⠁⠀⠘⠀⠀⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠓⠀⠰⠆⠠⣤⠀⣤⡄⢀⣄⠀⣈⡁⠀⡉⠀⠘⠃⠀⠗⠀⠰⠆⠀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠲⠀⠠⠆⠀⣿⠢⣠⣄⠀⣈⠀⢈⡉⠀⠛⣃⠘⠟⠀⠾⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⠀⠰⠂⠀⡤⠀⣤⡄⢀⣼⡆⠈⠳⣌⠛⠁⠛⠃⠸⡟⠈⢿⡗⢠⣶⠀⣤⡄⢀⣄⡠⣼⠃⠀⠉⠀⠘⠂
⠐⠀⠠⠄⠀⣤⠀⢠⡄⠀⣈⠀⢈⡙⣷⠀⠀⠈⠙⢴⣿⢆⠎⠀⠀⠘⡆⣠⣄⢀⣩⠥⠚⠁⣰⠛⠁⠐⠇⠀⠶⠀⠠⠄
⠤⠀⢠⠀⢀⣀⠀⣈⠁⠈⡋⠀⠛⠃⣿⡄⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠘⡬⠟⠉⠀⠀⠀⣰⠓⠿⠀⠶⠆⠠⣤⠀⢀
⡄⠀⣀⠀⢈⠉⠀⠛⠁⠘⣟⣐⣿⣆⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢰⡶⠀⣤⠀⢠⡀⠀⣀
⠀⠀⠉⠀⠙⠃⠐⠿⠀⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⣴⡄⢀⣄⠀⣈⠁⠀⠉⠀⠘
⠀⠘⠃⠀⠾⠂⢴⡦⢠⣶⡌⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣳⢈⡋⠀⠛⠃⠘⠃⠀⠶
⠀⠰⠆⠠⡦⠀⣴⡄⢠⣥⣠⠽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠻⠃⠐⠿⠀⢶⠆⠠⡄
⠀⣤⠀⣠⡄⢀⣬⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠐⠿⠆⢴⡦⠀⣤⠀⢀⡀
⢀⡀⠀⣸⠿⢭⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⠀⣤⡄⢀⣀⠀⠈⠁
⠀⠁⠀⠙⠁⠘⠟⠐⠿⠏⣿⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠙⢦⣉⠀⢈⠉⠀⠙
⠘⠃⠘⠟⠀⢾⠆⢰⣶⠀⣶⡄⣸⠇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣻⣿⠛⣛⡛⠒⠛⠂⠛⠃⠐⠗
⠺⠀⠰⠆⠀⣶⠀⣤⣄⢀⣭⣡⣿⡾⠿⡿⠛⠿⢿⣆⠀⠀⠀⠀⣼⣦⣀⠀⠀⠀⠘⡇⣙⡋⠈⠛⠃⠺⠟⠀⠿⠀⢰⠄
⠄⠀⣤⡀⢀⣄⠀⣈⠁⢈⡛⠁⠻⠋⠸⠿⠲⣿⡿⢿⡄⠀⠀⡼⣡⣯⢻⣷⣄⠀⠀⢳⠻⠋⠸⠿⠀⢾⠆⢠⡦⠀⣤
⠀⠀⡀⠀⢈⠁⠀⠛⠁⠻⠟⠰⠿⠆⢾⡶⢰⣶⡄⣼⣷⡀⣼⡁⣙⡋⠘⡛⠋⠳⣦⡘⣿⠆⢶⡦⢠⣦⠀⢠⡀⠀⡈
⠀⠀⠁⠀⠙⠃⠘⠟⠀⠾⠆⢰⣶⠀⣴⡄⣠⣭⢀⣩⡉⢿⡛⠁⠻⠋⠸⠿⠀⢿⠆⢹⣷⠀⣤⡄⢀⣀⠀⢈⠁⠀⠉
⠀⠈⠃⠀⠒⠀⠰⠆⠠⣶⠀⣠⡄⢀⣍⠁⣙⡋⠈⠛⠃⠻⠟⠸⠿⠇⢶⡶⢠⣶⠄⣤⡄⢀⣉⠀⢈⠁⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⣤⠀⢀⡄⠀⣈⠁⠈⠛⠀⠛⠃⠸⠿⠀⢿⠇⢴⣶⠀⣤⡄⢠⣄⠀⣈⠁⠈⠋⠀⠛⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠘⠃⠐⠿⠀⢴⡆⢠⣶⠀⣤⡄⢀⣉⠀⢈⠉⠀⠛⠁⠘⠃⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⢰⡦⠀⣤⡄⢠⣄⠀⣈⠁⠈⠉⠀⠙⠃⠐⠟

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠲⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⢳⡀
⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣧
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠰⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡌⠀⠀⣀⡀⡀⠀⠀⠀⠈⡇
⠀⠀⠈⠙⠒⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡇⢀⡜⠀⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡸⠓⠲⢤⣤⣴⠆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⠃⡼⠀⠀⡇⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣠⣌⡉⠓⠚⠹⣅⡀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠺⠶⣇⠀⠀⢹⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣄
⠀⠀⠀⠀⡌⠀⢀⡤⠖⠋⠀⠀⠈⠳⣄⢀⡷⣌⠉⠙⠒⠀⠀⠄⠀⣀⣀⠀⠀⠀⠈⠳⡄⠈⢧⡀⢰⣄⠀⠀⠀⠣⠤⠄⠉
⠀⠀⠀⠰⠀⠈⢃⠀⠀⠀⠀⠀⠀⢀⣨⡏⠀⠈⠳⣄⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠈⠳⢄⡀⠀⢀⡠⠔
⠀⠀⠀⡆⠀⠀⠀⠳⣤⣀⠀⠀⠀⠈⢹⡆⠀⠀⣾⣾⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠓⠒⠋⠉⣇⠀⠀⡠⠊⠀⠀⠀⠀⡄⢀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠉⠠⠀⢄⠈⡁
⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣄⣀⣐⣀⣀⣤⡤⠤⠶⠒⠋⠉⠀⢳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣀⣀⡀⣀⣀⣀⣀⣀⣀⡠⠤⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⣈⣒⡫⠭⠛⠳⠶⢖⣒⣚⣿⣋⣈⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠘⠉⠱
⠀⠀⢀⡤⠀⡴⠆⠠⣄⠀⠀⠀⠀⠀⠀⢠⡀⢪⡀⠀⡄⣄⠘⠀⠀⠀⢀⠤⡀⠀⠀⠀⡤⠀⠀⠀⠀⠹⢀⡴⣠⠆⠀⢀⡼
⠀⠀⠨⡇⠀⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⠫⠀⠉⠁⠘⡜⣆⠀⠀⠀⠘⠶⠎⠀⠀⣸⡿⠀⠀⠀⠀⣠⢊⠞⠁⢀⠀⣠⠇
⠀⠀⠀⠀⠀⠙⠀⠀⢉⠢⣀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠘⠈⠀⠀⢠⡄⢠⡄⠀⠰⠛⠀⠀⠀⣠⢞⠕⠁⠀⠀⠀⠉
⠀⢀⡴⠒⠒⠚⠉⠑⠀⠑⢬⡳⢄⠀⠈⠻⣕⣄⢀⣤⣤⣤⡀⠀⣤⣿⣿⣿⣇⣀⠀⠀⠀⠀⠘⠑⠁⠀⠀⠀⢰⢷⠋
⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢠⠪⠟⣿⢷⣄⠙⢄⢀⣎⢩⠟⢻⠛⣥⢄⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠆⣈⢊⣸⡿⣿⣠⡿⠻⠇⡸⠃⠰⢾⣿⣿⣿⣿⠿⢿⠻⠛⠻⣻⣧
⠀⠘⠦⠤⠄⠀⠀⠀⣴⣿⣶⣶⣿⡷⢶⡾⣴⣿⣷⣮⠡⣤⠏⠙⡄⢠⣤⣿⣷⡆⣸⣻⣿⡿⠃⠀⣴⣱⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢻⡋⠀⠀⢀⣷⣽⣶⠶⣖⣋⠀⠀⠀⠈⠢⢽⣿⣭⣾⣷⣿⣿⣮⣶⣶⡏⡆⡇⣿⣿⣄
⠀⡴⢡⡄⠀⠀⣿⣿⠿⢧⡀⠙⣤⣾⣿⡟⣺⡏⣠⠋⣠⠷⣀⣠⢲⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢰⡜⠲⡔⠋⠀⠀⢀⣿⣾⣿⣿⣿⡟⠁⠈⠣⠴⢏⢀⠀⠡⠾⡀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣰⢊⡱⡀⠒⠀⣰⢴⣾⣿⣿⡿⢿⣿⣿⣿⡄⢠⠄⠀⠀⠩⡀⢠⠰⡓⣵⠀⠀⣰⣿⣿⣿⡿⠋⣼⣿⣿⣿⡟⠀⢠⠄
⠈⠯⢴⣫⢆⠞⠁⠈⢿⣿⣿⣧⠀⢻⣿⣿⣿⡄⠀⠀⠠⠀⠈⠪⠗⠚⠁⢤⣠⣿⣿⣿⡿⠁⢰⣿⣿⣿⡿⠀⠀⠀⠀⣀
⠀⠀⠈⠈⠉⠀⠈⠁⠈⠿⠿⣿⣦⡄⠹⠟⠻⠿⠛⠀⠠⠀⠀⠠⠆⠀⠀⠀⠉⠉⠉⠛⠈⣙⠻⠿⠿⠿⠃⠀⢈⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣼⣷⡀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⢀⣾⣧⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣿⠈⠛⠟⠘⠟⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⠃⠻⠛⠁⣿⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣛⠉⠀⠀⢠⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⡄⠀⠀⠉⣛⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⡶⠀⠀⠙⠿⠛⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠿⠋⠀⠀⢶⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠀⠀⢀⣀⣀⣀⠀⠀⢀⣤⡶⠟⠛⠛⠻⢶⣤⡀⠀⠀⣀⣀⣀⡀⠀⠀⠾⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⠋⠉⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠉⠛⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠁⢸⣧⠀⣠⣴⣦⣄⣠⣴⣦⣄⠀⣼⠇⠈⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢹⣿⠋⠀⠀⣽⡏⠀⠀⠙⣿⡏⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣿⣧⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣠⡿⠃⠙⢿⣄⠀⠀⠀⠉⢛⡂⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠙⣷⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠡⠑⠄⠀⠈⠱⡀⠀⠀⠀⠀⣀⠎⠀⠑⢄⠀⠀⠀⠀⢀⡠⠀⡄⠀⠀⠀⠀⢀⣀⠀⡀⠀⣀⣀⡀⠀⠀⡠⠊⠀⠱⢀⠀⠀⠀⠀⢀⠌⠁⠀⠠⠊⠌
⠀⠀⠀⠠⢄⠀⠀⠀⠀⡈⠀⠀⡠⠊⠀⠀⠀⠀⣀⠅⠀⠀⠀⢈⠀⠀⠀⠀⠈⠀⠀⢘⠀⠀⠀⠀⠁⠀⠀⠀⠨⣀⠀⠀⠀⠀⠑⢄⠀⠀⢅⠀⠀⠀⠀⡠⠆
⠀⠀⠐⢄⠀⠐⢄⠠⠊⠀⠀⠀⠃⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⢀⠀⡀⠀⠀⢀⠘⠀⠀⠀⠑⠄⡠⠊⠀⡠⠂
⠀⠀⠀⠀⠑⠤⠂⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠑⠘⠀⠀⠀⠉⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠊⠈⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊



⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠉⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠄⠘⠋⠉⠑⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣳⣤⣄⠀⢀⣴⣎⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢻⡿⠇⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⠀⠀⠀⢠⠁⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠈⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠆⢤⣤⣤⠀⠶⠆⢸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡀⠀⢠⣥⠀⠀⠒⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠎⠁⠀⠀⠈⠑⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⠟⢣⡀⠂⢀⣤⠰⠂⠠⣀⠈⢢⡀⠀⠀⠀⢀⠘⣹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢪⡑⠦⡀⠀⠀⠉⠑⢦⡄⠀⠀⢠⠌⠀⢠⣿⠀⠀⢹⢠⡉⠀⠀⠀⠀⠈⠳⣀⠻⡀⢠⠶⠉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡰⠁⠀⠀⠈⠱⢮⡑⢂⠀⠀⠀⠈⠑⢦⡁⠀⢀⣾⣿⠀⠀⠘⣾⡁⠀⠀⠀⠀⠀⣠⣷⠀⢃⣤⠷⠊⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡄⠒⠢⡀⠀⠀⠈⠓⢬⡑⢠⡀⠀⠀⠘⠳⣬⡿⣿⡇⣠⠈⠀⠀⠀⠀⠀⢀⣄⠸⠄⠀⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠰⢡⡟⢀⡔⢀⠔⡀⠈⡆⢀⡀⠀⠀⠈⠑⢮⡔⢠⠀⠀⠀⠙⠷⣦⡈⠁⠀⠀⠀⣀⡘⠓⠒⠒⠈⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⡌⢠⠋⢠⠂⠀⠑⠂⠂⢠⡑⠀⠀⠁⠢⡀⠀⠀⠈⠑⢪⡕⣠⠀⠀⠀⠉⠓⢢⣄⢸⡟⠘⢷⡀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⢀⠘⡰⠃⡰⠂⡄⢻⠀⠴⠊⡑⠁⢴⠉⠒⠠⠂⠀⢀⣄⠀⠀⠈⠑⢮⡑⠂⠀⠀⠀⠈⠙⢦⡀⠀⠙⣦⡀⢀⠆
⠀⢀⠂⠉⢫⡴⠁⠀⠓⠒⠀⠈⠀⡀⠀⢀⢤⡄⢉⡄⠀⠀⡰⠂⣈⠳⠆⣄⠀⠀⠈⠑⢮⡕⠢⡀⠀⠘⠈⠑⢠⣈⠻⠎
⠀⢼⡀⠀⡜⠓⢤⡀⠀⠀⠀⠒⢰⡏⠀⠂⡄⠈⣁⡀⠀⡰⠁⣰⠆⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⢮⡑⠢⡀⠀⠀⠈⠑⠂⡀
⠧⡀⠉⠁⠀⠀⠀⠈⠐⢦⣄⠀⠀⠈⠒⢤⢀⡿⠁⢸⠆⢀⠲⠉⠀⠐⠀⠀⢀⠄⠂⠀⠰⠀⠀⠀⠈⠑⢮⡑⠢⠄⠀⠀⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠘⢤⡄⠐⠠⠃⡐⠁⠂⣄⠾⠁⡰⠈⢱⠀⣤⠂⠐⢄⠀⠀⠈⠑⢦⡉⠲⣄⠀⠀⠉⠁⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠘⠣⡀⠈⠒⠀⡘⠁⠞⠁⠀⠈⠈⠁⡤⢀⡀⢉⡶⢠⠀⠀⠈⠑⢦⡉⠲⣄⡀⠀⠀⠙⢢⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠙⠲⣤⡀⠀⠈⠑⠒⠀⡁⠘⠄⠐⢂⠀⢀⠜⠀⠎⢀⡛⢠⡄⠈⠒⡄⠀⠀⠈⠑⢦⡉⠂⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⠀⠑⠂⠀⠒⠁⢠⠂⠠⠁⢀⠎⠀⠣⡈⠑⢺⠂⠉⠒⢀⠀⠀⠈⠑⡌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠐⢦⠔⠀⡀⠀⡀⠒⢤⠀⡐⠀⡄⠁⠄⢸⠀⠀⢠⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⢤⡀⠀⠀⠀⠐⢅⠀⠑⠂⣀⠐⢠⠌⠀⠘⢠⠄⠀⡰⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠦⣀⠀⠀⠉⠒⠰⠏⠀⠃⠄⠂⡠⠁⠀⠞⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠈⠒⠀⠐⠊⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠢⣄⡀⠀⠀⠀⣴⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠲⣤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠤⠤⠤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣙⣶⣤⣤⣤⣤⣤⣤⣄⣉⢿⣷⠆
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⢀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍
⠀⠀⢰⢿⣿⣿⣿⣿⡛⣯⣽⣯⡿⡛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⣿⣿⣿⣿⠗⠈⡋⠉⠉⡳⠀⠉⣿⣿⢏⠓⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢻⣿⣿⠟⠋⠀⠀⠈⠉⠉⠀⠀⠀⣿⣿⠈⠀⣼⣿⣿⣿⣿⣿⣿⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⢠⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠨⠋⠟⠀⠁⠁⠀⠁⠀⠀⠈⠈⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⢻⣿⣿⣿⣧⡀⠀⠀⠠⠆⠀⠀⠀⠀⠦⠀⠠⠀⠒⠆⠇
⠀⠀⠀⠀⠀⠀⠐⠗⠢⠤⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠈⢁⢿⠃⠑⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡠⠠⠋⠀⠀⠀⢀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡤⠤⠊⠁⠀⠀⡟⢇⣀⣀⠀⠐⡠⠀⠁⠀⠀⢀⠄⠁⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠒⠐⠂⠀⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠶⣶⣒⠲⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠲⠶⠶⠶⠦⠤⣄⣀⣀⡀⠀⠤⣄⣀⡀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⠁⠀⠀⢀⣀⠀⠠⠄⠽⠏⠉⠉⠙⠛⠛⠛⠓⠀⠀⠀⠀⠜⠀⠀⠀⠈⠉⠑⠚⠂⠈⠉⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠰⡊⠀⠀⠀⠀⠀⠀⠛⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⡘⢧⠀⠀⠀⠀⣭⠷⠀⠀⠀⠀⣀⠤⠴⠖⠒⠒⠒⠒⠒⠤⢇⢀⣠⠴⠒⠚⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⠈⠳⣄⠘⡷⣌⢷⡀⠀⠀⡞⠁⠀⠴⠄⠀⠀⠀⠀⠀⢀⣸⠏⡄⠀⠀⠀⣠⣾⣁⣠⣤⡤⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈⣯⣷⠀⠀⠙⠒⠶⠶⠤⠤⠤⠄⠒⠚⠉⠙⠲⠶⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠲⢶⣖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣁⠤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠚⠉⠉⠁⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠒⠒⠚⢛⣻⠿⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣠⡤⠴⠚⠋⠁⠀⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⣯⡽⠛⠉⠀⢀⣠⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠤⢴⡖⠠⡤⢤⣤⣀⣀⣈⣊⠥⠤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠲⣀⠀⠇⠀⣰⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⠳⢤⣸⠀⡤⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⢢⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠲⢤⡀⢀⡍⢹⡇⠀⡉⠑⠒⠒⠒⠒⠂⢸⠓⠃⠀⠀⢀⡀⠀⢰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡀⠀⢙⠛⠀⠘⢦⠀⠘⣦⣄⡀⠀⠀⠀⢺⢤⣄⠚⢳⡏⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡁⠀⢸⡄⠀⠀⢸⠀⠀⠸⡄⠉⠛⠒⠤⢼⡿⠘⠦⡼⠁⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠈⡟⠀⠀⢳⠀⠀⠘⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢠⠋⠈⠉⠁⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⢰⠇⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⣧⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⣀⡀⣀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠠⣾⣿⣿⣿⣿⣿⠿⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣎⣡⣌⣠⠥⠀⠀⣼⡀⠀⢀⣀⣤⣴⠇⠀⠀⠙⢿⠿⠋⠁⠀⠀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠓⠒⠲⠶⠶⠶⠖⡾⠉⢻⠋⠉⠁⢠⡏⠀⠀⠀⠀⠀⠱⣄⠀⣠⡔⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣀⣇⣀⣾⣀⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠈⠛⡹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠧⣄⠀⠀⠀⠤⠬⠭⠕⣾⠉⢁⣀⣁⡬⠷⣠⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⠈⠉⢓⡶⠶⠒⠒⠋⠙⠣⠍⢡⡀⠀⢐⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠀⠐⠀⠀⠆⠂⠂⠀⠀⠂⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⠘⠀⠃⠁⠐⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣀⡀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⣀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡫⠂
⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠳⣤⠤⠤⠤⠤⠐⠒⠚⠉
⠀⠀⠈⠑⠶⠶⠤⠤⠶⠶⠤⣤⡎⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡳⠦⠤⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠤⠰⠀⠄⠀⠆⢀⠀⢄⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡀⠐⠁⡀⠀⠀⢀⡀⠄⣠⣤⡤⡴⠀⠀⠀⠐⠀⠀⠀⠀⢀⢄⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⡠⠞⠴⠋⠑⠈⠁⠀⠘⠧⠤⠤⠤⠄⠒⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⣤⣶⠷⠖⢲⣢⡀⠀⢰⠁⠀⠀⠀⠈⡂⠀⠀⠀⠠⡶⠤⠀⠀⠁⠀⠀⠀⠀⢀⡔⠁⠈⠢⠀⠀⠀⠀⡷⢺⣄⠀⠚⢺⡆
⠀⠀⠀⠀⢈⣻⠕⡶⡒⡦⡽⣓⠀⢸⠀⠀⠀⢀⡔⣓⡢⡀⠀⢀⣠⣤⣤⣄⢤⣤⡀⠀⠀⠘⢆⣂⣴⣶⠷⡀⠀⠀⡿⣀⣤⣤⣄⣸⣇
⠀⠀⠀⠀⢈⣭⠘⡙⣉⠀⢥⣭⠀⢸⢀⣴⣿⣏⢃⡥⢠⡇⠀⠈⠻⠉⡒⢈⣠⡠⢃⠀⠀⠀⠀⢹⠷⠦⠟⠁⠀⠀⡿⢈⣿⣿⣟⣻⣭⡇
⠀⠀⠀⠀⠈⠻⢗⡤⣤⣶⠗⠁⠀⢸⠀⠀⠀⠈⠛⡟⠋⠀⠀⡀⠘⣛⣃⠉⡥⠐⠊⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣷⣈⠉⠉⢉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⣀⣁⣈⡀⠀⠀⠀⣀⣈⣉⣉⣀⣁⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⢠⠔⣂⢄⠀⠀⠀⠀⠀⡏⢉⣀⣀⠀⣡⡀⠀⠀⠶⠒⣚⡒⠒⠒⠒⠲⠀⠀⠐⠷⠆⠉⠛⠒⡆⠀⠀⠀⠀⣾⠒⠙⡆
⠀⠀⠀⠀⠀⠘⠀⡀⡏⢀⢀⣀⠀⠀⡗⣾⡷⢺⣟⣌⠼⠀⠀⠀⠐⠛⢃⣭⡙⠁⠀⠀⠀⡀⣰⡞⢋⠛⣿⠃⠀⠀⠀⠀⣫⡦⣶⣅
⠀⠀⠀⠀⢀⠀⠾⢿⢛⣷⣛⠛⠇⠀⡟⢿⠿⠞⣋⡃⠀⠀⠀⠀⢯⣸⣟⣽⣭⡿⠀⠀⠀⡇⠚⠷⠤⡶⡣⡀⠀⠀⠀⡮⡽⢖⡠⡇⠇
⠀⠀⠀⠀⠥⠤⠨⠿⠟⠁⠬⠷⠄⠀⠇⠀⠀⠘⠧⠇⠀⠀⠀⠤⠯⠠⠯⠽⠯⠽⠤⠀⠀⠣⠠⠀⠸⠼⠏⠚⠀⠀⠀⠢⠤⠨⠱⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⡀⠀⢀⡀⣀⣀⣀⠀⣀⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⡏⢸⡷⠶⠀⡀⢸⡇⣿⡷⠶⢸⣏⠈⣉⠈⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠛⠘⠛⠛⠃⠙⠛⠁⠙⠛⠛⠀⠙⠛⠉⠀⠘⠁

⠀⠀⠀⠀⣒⣒⣲⣒⣒⣒⣒⣒⡄⢀⠤⡠⠤⠤⠤⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠲⢒⠒⠀⠀⠀⠀⢀⣀⣔⣢⣀⣀⣀
⠀⠀⠀⠀⢡⣋⣉⣉⣉⣉⣉⣦⡁⢸⠊⠁⠂⣀⣀⠂⠑⡆⠀⢄⡀⢀⠀⠀⠀⢀⣠⠀⠀⢸⠸⠾⠌⢀⡀⠀⠀⠀⠸⡕⢾⣻⣏⠢⢸
⠀⠀⠀⠀⢸⠩⠛⠍⣩⣩⣉⢹⡀⢸⠀⢴⣾⢿⣛⣥⡀⡇⠀⢁⠄⢄⠀⠤⠤⠤⠈⠀⠀⢸⢩⢭⡿⣟⣫⠄⠀⠀⠀⡀⠢⠉⠫⢁⡍
⠀⠀⠀⠀⢨⠐⠒⠂⣈⣉⣉⢸⡃⠘⠈⠋⠉⠙⠻⣿⡗⠂⠀⠘⠌⠔⠀⠉⢡⠤⡄⠀⠀⢸⢸⣖⣿⢿⠋⠀⠀⠀⠀⢁⠐⠉⠊⢁
⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠉⠈⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⡀⠀⠀⣀⢀⣀⠀⢀⠀⢠⣶⡶⡄⠀⢀⢴⣶⡄⠀⢀⡴⠼⠟⡛⣧⡦⡀⠀⠀⡴⢑⡶⣴⡂⠒⣶⡀⢠⠀⠼⢄⠔⢢
⠀⠀⠀⠀⠂⡤⡉⣭⠍⠉⠠⠐⡇⠀⢿⠀⡈⣤⠁⠀⡸⠀⠀⢲⠁⣼⢀⠮⠻⠗⡔⠀⠘⢫⣣⡏⢩⡶⡴⡎⠀⠸⣀⠉⠩⠉⠩⢈⡍⠉
⠀⠀⠀⠀⠀⠤⠤⣠⣴⣾⣭⠕⡁⢸⠰⠌⠋⠀⡿⠣⠔⠃⠀⠜⢴⡯⠓⠫⠀⢀⡣⠀⠀⢸⣿⡠⠿⠄⡛⠄⠀⠀⠀⢄⣼⣶⣌⡁⣀
⠀⠀⠀⠀⠈⠉⠁⠛⣽⣰⠟⠉⠀⠸⡀⡀⠔⠈⠢⢄⢀⠄⠀⠈⠢⢺⣧⣤⡿⠕⠁⠀⠠⢿⠟⠀⠀⠈⢣⠖⠀⠀⠀⠘⢵⣲⠇⠈

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡴⡶⣤⡀
⠀⠀⠀⠀⠀⢼⣿⢝⠍⠨⠣⣕⢌⡺⢻⠟⠡⢣⡑⢕⢔⣵⠾⠋⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡉⢉⣉⢉⣡⢀⣠⡌⠙⢿⣕⢝⠷⣝⡿⣻⣇
⠀⠀⠀⠀⠀⢸⡷⣜⠳⣕⠥⡢⠓⢅⠂⡐⢅⢕⡉⢢⣾⠃⢰⠀⠀⢻⣷⢸⠘⢯⢰⣿⣧⣟⣥⣿⡟⣿⠶⢸⣟⡃⠀⠀⣧⡙⢪⠎⡑⡼⢿
⠀⠀⠀⠀⠀⠘⣷⡬⡫⣮⡻⣕⡡⠢⠀⡜⠢⡑⢐⢽⡏⠀⠘⠀⠀⠉⠁⠈⢈⢋⡈⠁⠁⠈⣏⢉⡉⣌⢩⠀⢉⣴⠃⠀⣿⡔⢡⢈⠚⡀⢻⡀
⠀⠀⠀⠀⠀⠀⣿⣾⣫⣮⣯⣾⠷⢤⣮⣦⢑⡌⠣⣼⡇⠀⣧⣄⡆⣿⠉⡏⢹⣼⡇⠀⠉⣿⠁⣿⡷⢧⣾⠀⣈⠁⢀⣾⠋⠎⡐⢁⠊⠐⢹⡇
⠀⠀⠀⠀⠀⠀⣿⠑⠝⣿⠋⠀⠀⠀⠀⠙⣿⠀⢁⠜⣧⣀⠉⠙⠁⠈⠀⠁⠈⢘⣀⣀⣀⣈⣀⣈⣀⣠⣤⣤⡤⢶⣟⠥⡠⢂⠐⠁⡁⠊⠚⣧
⠀⠀⠀⠀⠀⠀⣿⠻⣰⡇⠀⠀⠀⠘⠋⠃⢸⡦⠂⠀⢁⢉⡿⠃⣠⣶⢏⢋⠛⢯⡻⡍⡹⢟⢟⣿⠏⣴⢠⡄⠀⠀⠹⣖⠕⠠⡪⠂⡔⠁⠊⣿
⠀⠀⠀⠀⠀⠀⣯⡈⠺⡇⠀⠀⠀⢀⣀⣸⡟⠣⠢⣲⠞⢋⣠⠾⡫⡀⠁⢀⠀⠀⠙⢮⡪⠂⢊⣿⡆⠀⠀⠀⠀⠀⢀⣯⠐⠕⠠⠊⠐⠕⠁⢺⡄
⠀⠀⠀⠀⠀⢘⣷⠀⠈⢻⡀⠀⠘⠯⢽⣟⠕⢑⠘⢟⠚⢏⠁⡑⠄⠠⠀⡑⢅⢌⠈⢢⠳⢎⡢⣿⠗⠛⠶⠀⢀⣴⣟⠑⠀⠀⠀⠀⡀⠂⠐⢹⡇
⠀⠀⠀⠀⠀⢸⣗⢠⣼⠞⠻⣶⣤⣤⢴⡟⠀⢁⢂⢀⠪⠢⡑⠜⣤⢕⣌⣂⠢⠁⠁⣄⠠⡌⡱⠻⣦⣤⣴⣶⣯⠞⠻⣦⠐⠄⠠⠀⠀⠠⠂⠜⡇
⠀⠀⠀⠀⠀⢸⡇⣽⠃⠀⢤⡀⠉⠙⣷⡌⠠⠁⠁⠑⣁⣠⣬⡴⠟⢛⣽⠟⠀⠠⠀⠈⠳⣌⠈⡀⡐⣑⣿⠁⠀⠀⠀⠹⣮⠀⠀⢀⠀⠐⢈⢺⡇
⠀⠀⠀⠀⠀⢸⢵⡇⠀⠀⠀⠹⡄⠀⢸⡗⣰⣤⠶⠛⠛⠉⣇⠀⢀⣼⡕⠑⠁⢀⠎⠀⠐⢌⠸⠎⡈⣽⠃⣼⠃⠀⠀⠀⢹⡉⠂⠀⠁⠀⡑⢹⡇
⠀⠀⠀⠀⠀⣸⢺⡇⠀⠀⠀⠀⠳⠖⠚⠉⠁⠀⠀⠀⣀⣴⠟⠙⢛⠁⠈⠐⡁⢀⢀⠀⠈⡢⠁⢌⣸⡏⢰⠏⠀⠀⠀⠀⢸⡖⠁⠀⠀⡠⢊⣿
⠀⠀⠀⠀⠀⢻⣘⣁⣀⣀⣀⣀⣀⣀⣀⣠⣴⣖⣛⣋⣉⣀⣀⣀⣀⣄⣀⣀⣔⣁⣂⣡⣄⣦⣎⣰⣿⣁⣿⣤⣤⣤⣤⣤⣼⣧⣤⣤⣴⠴⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣴⠋⠀⣾⣙⣧⣴⠟⠀⠀⣤⣾⣤⣿⡤⠀⣿⠀⠀⢻⡄⠀⠀⠀⠀⠀⣼⠁⣼⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢈⣽⠋⠀⢈⣽⢿⡟⢳⠀⠠⣴⣧⣼⣧⠄⠀⠛⠀⠀⢸⠇⠀⠀⠀⠀⠰⠷⡦⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡘⠁⠀⠀⠛⠁⠈⠛⠋⠀⠀⠙⠀⠛⠀⠀⠀⠓⢀⣤⠟⠀⣼⠛⣿⠃⠀⢰⡿⠋⠀⢰⠟⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢦⣀⡟⠛⠛⠉⠁⠀⢰⣇⡘⢻⡶⠀⠀⠁⠀⢀⣟⣀⠛⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢹⣷⠏⠀⠀⠀⠀⠀⠀⠉⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢤⣄⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⠞⠛⠉⠛⠳⣄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠙⢷⡀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣦⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣽⣷⣤⣤⣤⣾⣯⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠙⢿⡉⠁⣠⣤⣤⣤⠤⠶⠶⠖⠐⠂⠘⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣆⠈⣧⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡄⣿⠀⠙⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠈⣇⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢷⡀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⢦⣄⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⢀⠈⡿⠶⣄⣠⠟⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠀⠀⠀⠀⢸⣾⠃⠀⠈⠙⠷⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠿⣫⣷⣶⣶⣯⣽⣛⣛⣟⣛⡋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⢨⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⢸⣿⣿⣿⣷⣶⣶⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣾⣿⣴⡎⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣱⣿⣿⣿⡇⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣧⣿⣿⣿⡿⣱⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠝⠛⠛⠛⠽⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀



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

⠀⠀⠀⢀⡶⢛⣻⡟⠶⣷⢦⡀
⠀⠀⠀⣿⢀⣾⣿⣿⣧⣼⣷⢿⣦⡀
⠀⠀⠀⣿⢻⣿⣩⡿⣿⡿⣿⣾⡏⣷
⠀⠀⢰⡟⢿⡿⢷⣾⣿⣥⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠈⣷⠘⢿⣛⣒⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⠉⠉⢉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠈⠳⠶⠼⠿⣿⠋⠉⠉⠀⠀⠀⢀⡴⠟⠉⠀⣀⣤⣶⡿⠿⠛⣁⣬⢓⣴⡗⣀⠈⡁⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⣴⠏⠀⣀⣶⣾⢟⠩⣀⠒⣢⠬⢐⣴⣿⢟⣐⠆⣼⣯⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠄⣶⡿⡋⢊⠀⠵⠔⣂⣥⣾⢟⡫⡀⠇⢆⣾⡟⡹⣿⣧⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⢪⣾⠏⡐⣎⣠⣴⣶⠿⢟⡋⠁⢈⠢⢈⣴⠟⢹⣦⡒⢊⢻⣧⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⢢⡿⢃⣵⡾⢟⣋⠁⠒⠔⣀⣁⣠⣴⠾⠛⠁⠀⠀⠙⢿⣮⠀⢻⣿⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠲⠋⣡⣿⢋⡰⣌⣥⡶⠿⠛⠛⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠹⣷⡠⢻⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠋⣰⡿⠅⣣⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⠀⠀⠹⣷⡘⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⢰⡿⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣱⠟⠀⠀⡦⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⡆⠀⠀⠘⣿⣿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢸⡏⠀⠀⠀⢿⣀⣠⡿⠀⠀⢀⣴⠆⠀⠀⠀⠹⣄⣀⣼⠃⠀⠀⠀⣿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⡇⠀⠀⠀⠀⠉⠉⢀⡀⠀⠿⢧⡀⠀⠀⠀⣀⡈⠉⠁⠀⠀⠀⠀⣿⣀⣴⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⢀⣀⡀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⣿⣿⣶⣌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠐⡇⠀⠀⢠⡾⠋⠉⠙⢳⡄⠀⠀⢸⠇⠀⠀⠀⣰⣟⠰⣀⠽⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡼⣟⡻⠿⠾⣆⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠀⠈⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠟⠛⠛⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⣛⠛⠛⠁⠀⠀⠉⠙⢋⡟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⡿⠀⠔⠁⠙⠷⣦⡀⢀⣀⣴⡾⠋⠁⠒⡀⢹⡆⠀⠀⠀⢀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⠆⠀⠉⢷⡀⠀⢸⡇⢘⣀⠀⠀⠀⣿⠉⠉⠉⣿⡇⠀⠀⠈⣕⠀⣷⠀⠀⣰⠏⠀⠀⠲⢾⣟⡀
⠀⠀⠀⠀⠀⣠⠞⠋⠁⣀⣤⣤⠶⠿⣧⠀⢸⡇⣸⡟⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠈⣿⡀⢻⠀⢠⣿⠷⢦⣤⣄⡀⠈⠛⢦⡀
⠀⠀⠀⠀⢸⣇⣤⠶⠛⠉⠀⠸⣆⠱⠹⣧⢸⡇⣿⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⢀⣾⢓⠀⣼⠁⠈⠙⠛⠶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢃⢻⣿⡇⢻⣷⠀⠀⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⣿⡇⢸⣾⢇⠂⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⡆⢻⣇⢸⣿⠀⠀⠀⣿⠁⠀⠀⣼⡇⠀⠀⠀⣿⢃⣾⠃⡌⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠈⢿⢾⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⢠⣸⣿⣿⠏⡳⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠉⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧⡀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⢀⣠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⣤⡤
⠀⠀⢀⡜⠁⡔⡲⣖⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠃⠀⠀⠀⠻⠀⣀⣠⡀⠠⣾⠿⠂
⠀⠀⢸⠀⢠⣿⢫⢺⡧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⡀⠀⠀⣞⣩⣅⢹⡄⠘⠁
⠀⠀⠘⢧⠀⢿⢺⡾⠛⢀⡇⠀⠀⠀⠀⡴⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⢿⠀⣾⣟
⠀⠀⠀⠀⠙⠲⠾⠖⢢⡜⠀⠀⠀⠀⠀⠘⠪⠔⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⣄⡀⠀⣠⠞⠛⠧⠟⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢦⡉⠛⠁⠀⠀⠀⠘⠲⣄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠒⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠛⡄⠀⠀⠀⠘⡟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢻⠀⠀⢠⣶⡧⡞⠁
⠀⠀⠀⠀⠀⠀⣠⠖⣲⠖⠊⠙⠒⠋⠉⢹⡓⢲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⠁⡼⠁⠀⣀⣀⠀⠀⠀⢠⠧⣿⣎⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠒⠒⠒⠛⡻⠇
⠀⠀⠀⠀⠀⡟⢰⢃⣠⠔⢉⣿⠀⠀⠀⢸⠀⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⡇⢸⠋⣠⡞⠉⠀⠀⠀⡠⠼⣶⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⢸⠉⠉⢹⠚⠉⢹⣇⣀⣤⠶⠋⠀⣠⢾⠦⢴⡋⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡤⠤⢾⠁
⠀⠀⣼⠉⠉⢹⢴⣄⣸⠴⠋⠁⢀⣠⠞⢡⠏⠀⢰⠋⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡆
⠀⠀⣿⠀⠀⢸⠛⠣⣵⡶⠲⢿⡟⠒⠚⠛⠛⠛⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⢸⠇
⡀⠀⣿⠀⠀⢸⣄⣀⣈⣇⠀⠈⣧⣀⣀⣀⣀⣀⣸⠀⠀⠀⡇⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸
⠻⠀⢿⠀⠀⢸⡀⠀⠀⠘⣆⠀⢳⠀⠀⠀⠀⠀⢸⡀⠀⠀⣟⠋⠛⠀⠾⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⡄⡟
⠀⠃⢼⠉⠉⢹⠉⠉⠉⠉⠙⣆⢸⡏⠉⠉⠉⠉⢹⡋⠉⠙⡏⡤⢹⣾⣷⠦⣄⡀⠀⠀⢀⣠⣏⣼⣿⣠⣻
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠸⡟⢳⡒⠒⠒⠛⠛⠛⠋⠉⠉⣉⣉⣉⣶⣶⣿⣧⠄⠀⠙⠛⠛⠿⠿⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣤⣤⣶⡶⠶⠿⠿⠿⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⢛⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢿⣦⣤⣠⣤⣶⠾⠟⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠙⠿⣦⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠈⠛⢿⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⣶⣿⠛⢿⣦⣀⠀⠀⠀⠈⠻⢷⣦⡀⠀⠀⠀⠉⠻⣷⣤⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣿⣄⣀⣨⣿⣷⣤⡀⠀⠀⠀⠙⢻⣷⣄⡀⠀⣠⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⢿⣯⣻⣶⣄⡀⢀⣼⡟⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⠆⣴⢦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡆⣤⢦⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠋⠰⣼⠇⠸⣠⣏⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠠⣼⡧⠸⢁⣾⢙⠀⠀⠀⡞⠁⠈⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠏⠉⠉⠒⠓⢦⠀⠀⠀⠀⠀⠁⠙⠚⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠙⠾⠿⠀⠀⠀⠳⣴⠂⡀⠀⠘⡆⠀⠀⠀⠀⠀⢘⠆⠀⠀⠀⣀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡡⠚⠧⡤⠞⠁⠀⠀⠀⠀⠀⠸⣆⣀⣐⣦⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠉⢁⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠁⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠔⣛⣉⠄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⡄⠏⠀⠀⢀⣴⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠒⠁⠈⠐⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠉⠀⠱⢤⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢠⠞⠛⠲⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠤⠤⠊⢘
⠀⠀⠀⠀⡴⠋⠀⠀⠒⣋⡭⠔⠀⠀⠀⠀⠻⣿⣿⣿⣯⣄⣀⠈⠁⣀⣀⣀⣠⣤⣤⣴⡶⠶⠶⠂⠀⠀⠉⢀⣠⣬⣿⣿⣿⡿⠃⠀⠀⠀⠈⠑⡆⢀⢨⣑
⠀⠀⠀⠘⣧⡀⠀⠀⠀⢉⡇⠀⠀⠀⣀⣀⣤⣬⣝⣛⣛⣫⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢉⣀⣀⣀⡀⠀⠀⠀⠣⠁
⠀⠀⠀⠀⠈⠉⠉⠓⠶⠚⠁⠀⣠⣾⣿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣛⡻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢱⣿⣿⣦⣝⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠐⠶⢦⣤⣄⣀⠀⠀⠀⠀⢀⣾⣿⢣⣾⣿⣿⢜⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣮⡻⣿⣿⣿⣷⣤⡀⢀⣠⣶⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣿⣿⢣⣿⣿⣿⢏⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⣝⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⡿⢣⣿⣿⣿⢏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠉⠻⢿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣵⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡡⠚⠉⠉⠉⠉⠒⢤⡀⢰⡕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠆⠀⢀⠖⠚⠓⠢⢵⠀⠐⠫⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠸⡀⢠⠎⠐⠃⠀⢉⠀⡇⠠⡄⢒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⡟⢳⠇⣤⠘⠀⠀⠀⠀⡇⢈⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠄⠈⣆⡼⢲⢄⣰⢣⣠⠒⢲⡦⡃⡏⢿⢸⢾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⣤⣧⠙⢘⠜⡻⢹⠉⢯⠙⢦⠈⢘⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠦⠀⠸⠀⡧⠊⠀⢸⢦⣘⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠤⢾⠁⠀⠀⡜⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⠀⠀⠀⣇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠫⣑⡒⠃⠀⠀⠀⢀⣼⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣒⠯⠤⢭⡭⠭⣗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⡞⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣇⠀⠀⡇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣯⣍⣭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣩⣤⣬⣍⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢁⣾⣿⣿⣿⣿⣧⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣾⣿⠟⣛⠻⣿⣿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⡀⢲⡄⠀⠀⢸⣿⢰⣿⠇⣾⣿⣿⡿⢃⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠐⣆⠈⣧⠀⣷⠀⠀⢸⣿⠸⣿⡀⣿⣿⣩⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⡟⠀⡿⠀⣿⠀⠀⠈⣿⡄⣿⣇⢻⣿⣿⡿⢩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⠟⠁⠀⠉⢀⡼⠃⠀⠁⠀⠀⠀⢹⣷⠘⣿⣆⠻⣿⣷⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣍⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄⠀⠀⠀⢿⣷⣾⣿⣷⣌⣛⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣴⣌⠻⣿⠟⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣦⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠙⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⣀⣀⣠⠞⢋⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣿⣿⣦⣄⠉⠛⠓⠲⠛⢛⠏⣋⡉⠉⠨⠍⣍⠱⣋⡭⢿⡏⠀⢀⣾⠿⠛⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠉⠛⢷⣄⡀⢯⣭⣅⡀⣠⣀⣘⣰⣮⣷⣂⣧⡴⠀⠂⣰⡏⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢻⡈⠻⣦⣟⣼⠿⣿⣿⣿⡿⢶⡾⠶⢋⣠⠏⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⣻⡶⠦⣠⢶⣩⣿⢯⣬⠶⠟⠗⠚⢿⠁⠀⠀⠀⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢀⣴⣿⣿⣿⣿⣿⣶⣤⣀⡐⠛⢉⣠⣴⣶⣿⣿⣿⣿⣿⣦⡄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⡙⣶⡋⢾⡿⣿⣿⣿⣿⣿⢿⠿⢻⣄⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⡇⠙⠿⣯⣭⣭⡵⠞⠃⠈⣵⣿⣷⣜⠻⢮⣿⣿⣿⠷⠋⠀⠈⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠠⠎⠸⣿⡇⠸⠆⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⢆⠀⢀⣤⠀⠀⠊⠀⠀⠂⠻⠀⠀⠀⠑⢤⡀⣀⡀⠀⠀⡼⢠⡄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣇⠀⠀⢀⣄⠈⠓⠢⣀⡀⠀⠀⣀⠤⠐⠁⡀⢀⡀⠀⣠⣴⣴⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⡗⣾⢛⡞⣷⢂⣬⣉⡖⢚⢩⣴⠘⠾⠏⠟⣉⣹⣏⣉⣹⣧⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡽⠿⢛⣛⣽⣯⢲⡆⠀⡾⠙⠟⠣⣂⡉⣶⢢⣶⡯⢹⠋⣿⣿⠿⠿⣏⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣶⣿⢸⣿⢸⠇⢸⡴⠿⣿⣿⣦⠁⡏⠀⣿⢸⠟⠁⡟⢸⡟⣿⡺⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣷⣿⣿⢸⣿⣸⠂⣼⡁⠿⣟⣃⣀⡄⢿⡆⣹⡆⠶⡐⣡⢼⣿⠿⠷⢸⣾⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣼⣟⣼⣿⣿⣾⣿⡟⢷⢟⢿⡱⠏⢻⡛⣿⡶⢲⡷⣿⣾⣗⣨⡿⢾⣻⣾⢻⣯⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⡿⣿⣿⡟⣼⢷⡏⠼⢇⡿⣿⠛⠿⠟⠸⣻⠀⣟⢣⡆⣹⣱⡆⣬⣽⣿⢙⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣿⣷⣿⣿⢃⣶⢸⡌⣴⣧⡄⣬⣽⣠⡄⣾⡹⠐⠻⠘⡉⢨⠹⡇⢼⣿⠻⣿⠟⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣿⣩⣿⣸⣿⣄⣧⡼⢠⣇⠛⠀⠉⢅⠟⠇⣸⣼⣘⡅⠂⠰⣧⣼⢿⣶⡟⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⢿⣻⣿⣧⠀⣴⣂⠀⢂⢲⠐⡂⠒⢀⣀⡍⠛⣉⡇⣠⢸⢋⣉⣿⡹⡞⣹⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⡎⣿⢿⡟⢀⡋⠹⠀⣼⠆⢆⣸⢸⣟⣡⣸⡇⣧⠹⠞⣾⠘⡛⣿⣇⢰⣧⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣙⣦⡁⡾⠇⢸⠀⡏⢸⡸⣿⠨⡿⠋⢹⡇⡏⣰⢂⣿⠸⠿⡏⣯⢈⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⣿⣿⡋⠁⠹⢟⣛⣷⢯⠷⢿⣶⠧⠾⠯⢷⡍⣵⡎⠥⡛⣿⢿⣿⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡧⠾⠖⢻⠖⡈⠁⣆⠈⠀⠰⠤⢉⠀⡹⡏⠸⠀⠀⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⣈⣉⡏⠀⡤⠚⠀⣷⣈⢉⠀⡷⠾⡄⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡧⠀⠀⠐⠀⠚⠈⠃⠈⠀⠁⠒⠊⠐⠃⠀⠁⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣧⣤⣤⣤⣔⢂⠀⠀⠈⠠⠀⠀⣀⣴⣂⡀⢀⣀⢀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⠘⢧⣬⠟⢏⣛⡿⠂⠑⢾⣟⡛⠿⡟⠛⠿⠏⣉⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠘⠟⠀⠀⢻⡇⠀⠀⠈⣿⠁⠀⢹⡿⠁⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠘⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠈⠙⠿⣶⣤⡴⠂⠀⠙⠻⣿⣄⠀⠀⠀⢈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡈⣿⣿⣁⣀⣀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠿⡿⠿⠿⠿⠿⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠤⣄⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⡍⠂⢄⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣄⠀⣀⡀⣿⢻⠤⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠁⠀⠉⠉⣿⡟⡰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠸⠀⠀⠀⣴⣯⠰⠁⢀⡆⠦⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣶⣗⣢⠾⢤⡞⣷⡇⠀⣌⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣩⠆⠀⠼⣿⡖⢛⣟⣹⡧⠛⢡⠄⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣁⠰⣧⠀⠈⠛⠿⣯⡭⠖⣉⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢩⣲⣟⠲⠀⢀⣤⢞⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠛⠛⠒⡟⠒⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⣩⠏⢹⠃⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⠇⡀⠀⠄⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠜⢠⠃⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣼⡇⠀⣼⠇⠀⠀⠀⠀⠀⢠⣹⣿⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣸⡿⠷⠾⡛⠀⠀⠀⠀⠀⢀⣼⠟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⠀⠀⠀⠱⣤⣤⣀⣀⣠⢞⣩⡀⠀⢹⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢤⠏⠁⠛⠉⠻⢿⠟⣇⠀⠘⣣⡌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡟⢻⡆⠀⠀⣰⠎⢠⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⣿⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠘⠶⠤⠋⠀⠘⠆⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⠿⢻⡀
⠀⠀⠀⠀⠀⠀⢀⣼⠃⡠⠀⠀⠀⠀⠀⠸⡀⠀⠈⣤⠀⠀⠀⠀⡼⢹⢠⡈⠁⠀⠠⡑⢄
⠀⠀⠀⠀⠀⢀⡚⠋⡄⠁⠀⠀⠀⠀⢠⡴⠉⢆⠀⠀⠀⠀⢀⣰⡣⠀⠏⠟⠒⠒⠶⣝⠪⢶⡄
⠀⠀⠀⠀⠀⠀⠻⠾⠃⠀⢀⠀⠀⣤⡋⣠⠀⠀⠛⢦⠤⣴⣻⠃⠀⠀⠳⠴⠂⠀⠀⠀⣲⣶⡏
⠀⠀⠀⠀⣀⠼⠁⠀⠀⠀⠾⠤⢈⣼⠃⢻⣆⠀⠀⠀⣠⡘⢟⣤⡀⠀⠀⠀⠀⠀⡠⠔⠿⠉⣷⡀
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⢀⣦⣴⡿⢻⠃⠙⠓⠛⣦⠄⠀⠠⢆⠐⠀⠀⠡⣘⣷
⠀⠀⠀⠀⠀⢀⣴⣿⣂⠀⠜⣹⡏⠀⠀⠀⠈⠀⡏⠀⠸⠀⠀⠈⣆⠀⠐⠁⠀⠈⠂⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠈⢳⣽⣭⠿⠿⢿⣄⣀⣀⣀⣠⣴⣇⡀⠐⠀⠀⠀⣼⠄⠀⠀⠀⠀⣀⣀⣤⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢛⡿⠛⠻⣿⠛⠟⠛⠶⢶⣿⣶⣶⣿⡿⢿⠛⠛⠥⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢘⠀⠀⠀⠈⠙⠒⠶⠗⠊⠀⠀⠙⠃⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⠦⠤⠤⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠒⠲⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣠⣿⣿⣿⣿⣦⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⠭⢿⡆
⠀⠀⠀⠀⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠸⠶⢾⡇
⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⠋⠙⠿⣿⠟⠙⠿⣿⡿⢻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠈⢻⣿⣿⣿⡇⠘⢿⣄⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⢰⣿⣿⣿⣿⣧⠀⠈⢻⣷⣦⡨⡓⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠸⡇
⠀⠀⠀⠀⠉⠛⠿⣿⡀⠀⠀⠙⢿⣿⣮⣳⣄⠀⠀⢰⠀⣰⣿⡿⠁⢀⣠⣞⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠀⠀⠀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢻⡓⠲⠽⣿⣿⣿⣗⠀⢸⣾⣿⣟⡠⠖⢩⠏⠸⡆⠀⠀⣰⡶⠚⢛⠉⠀⣦⠀⠀⣸⠀⣸⣇
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠙⢦⡀⠀⠉⣻⠿⣿⢸⡿⠛⡅⠀⢠⠏⢀⠀⢹⡄⢰⣿⠁⠀⠸⠀⠀⠃⠀⠘⠁⠀⣿⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠠⣄⠀⠀⠙⠦⣀⣀⡠⡟⠀⢿⡦⠤⠴⠃⢠⡟⡄⣸⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⢌⡛⠦⣄⡀⠀⠀⣰⠃⠀⠀⠙⠲⠤⠶⠋⢈⡴⠃⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠀⠀⠉⠛⠋⠁⢀⣤⠶⠶⣦⡀⠀⣠⠟⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠈⠻⠀⢹⡀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠲⠶⢶⣦⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣶⣷⣷⣄⠰⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠋⣱⣷⣿⣯⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⣴⣾⣿⡿⢹⠀⡇⣿⣿⡄
⠀⠀⠀⠀⢀⣠⣶⣿⣯⢏⣴⡌⢍⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠰⣿⣿⡟⠣⣾⡄⡁⠉⠻⡇
⠀⠀⠀⣠⣿⣿⢫⣿⣏⡞⡾⣿⢸⡇⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⡟⠁⠀⣿⣿⣷⣷⣾⠁
⠀⠀⢰⣿⣿⠃⣾⣿⣼⣿⣹⡟⣼⣇⠿⡿⠈⡀⠀⠀⠀⠤⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⢀⠔⠁⢿⣿⣿⡏
⠀⢀⣿⣿⡟⢸⣿⣿⣿⣧⣿⣱⣿⠎⠀⠀⠀⠉⡂⠀⠉⣹⠅⢴⡄⠀⠳⡀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠙⠉⠐⡄⣀
⠀⣸⣿⣿⡇⢸⢸⣿⣿⣿⣳⣿⠏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠶⠶⢄⡤⠀⠀⣐⣒⣂⠀⠀⠀⠀⢩⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣀⣤⣽⣿⡧
⠀⢿⢿⣇⢷⢹⠸⣿⣿⣿⣿⠋⠢⣄⡀⠀⠀⢈⡀⠀⠀⠀⢀⠖⠁⠀⠀⢈⢮⠛⠀⠀⠀⠀⠀⠰⡀⠀⠀⣊⡀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣷⣤⡀
⠀⢸⣸⢻⠸⣾⡆⢿⣿⣿⡏⠀⢀⡴⠛⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠚⠘⠁⡄⠀⠀⠀⠀⠀⠀⠓⠀⠈⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢻⡆⢧⣹⡇⠈⣿⣿⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣳⡾⠋⠉⠀⠈⠙⠓⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢸⠀⠀⠀⠀⠑⣠⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠈⡆⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⢸⠀⣄⣄⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⡿⣫⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠁⠀⠀⠈⡇⠙⠿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿
⣼⣷⡬⢻⣿⣶⣤⣀⣀⣀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡇⠀⠀⠉⠁⠈⠉⠉⠀⠀⣸⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣮⣿⣯⣿⣿⣿⣷⣽⣝⣿⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⡿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣻⣿⢭⣿⣶⠢⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⠋⡜⡼⣿⠟⠻⡿⣿⣿⣿⡯⣯⣿⡟⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿
⠋⠸⠋⠙⠁⠰⠊⠠⠛⠞⢻⠟⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠛⠛⠉⠿⠛⠙⠛⠁⠙⠋⠀⠈⠋⠉⠁⠙⠛⠃
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⡴⠋⠉⠛⠛⠒⢂⣠⣴⣶⣿⣿⣷⣶⣶⣶⣶⣷⣌⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡼⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠰⠷⢀⡞⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠉⠻⣯⣛⡛⢛⣋⣽⣿⣧⠹⣏⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠳⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠇⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠙⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⣵⣾⣿⣿⡿⢇⣤⡀⠀⠀⠀⠠⣄⠀⠀⠈⠳⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢛⣩⣶⣿⣿⠟⠀⠀⢀⠀⠈⠳⢦⡀⠀⢈⡷⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣫⣵⣿⣿⠆⠈⠓⣦⣀⣠⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠓⠒⠛⠁






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

⠉⠀⠀⠀⠀⠀⠀⢠⣦⠀⠈⠁⢀⣾⠟⠁⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠠⠏⠀⢀⣠⡴⠂⠀⠀⣠⣤⣤⣤⠄⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⣀⠄⠀⠀⣠
⣦⣄⠀⠘⠓⠄⠀⠀⠉⠳⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢀⡀⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠎⠀⢠⡿⠁⠀⠺⠃⠀⠰⠛⠁
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠀⠀⠀⠙⠻⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠸⠁⠀⠀⠀⢀⣀⣀
⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠘⣿⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⠇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⢴⠏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣀⡤
⠈⠀⠀⠀⠀⠀⠙⠛⠛⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄⢠⡶⠖⠂⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠠⠦⠤⠤⠀⠀⠐⡇⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠁⠀⠀⢹⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠙⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠀⠸⠃⠀⣀⡀⠀⠂⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠈⠁
⡦⠀⠀⣀⡀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⢀⡀⠀⠀⣠
⠀⠀⠀⢻⣧⠀⢠⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣿⠇⠀⠀⢻⠀⠀⠠⠖⠀⠀⠀⢶⣄⠀⠀⠈
⠀⡶⠀⠀⠹⡄⠈⠉⠉⠉⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣴⣦⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠏⠀⠀⣠⡤⠂⠀⠉
⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⣤⣤⡄⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⢠⣾⡿⠀⠠⣤⠤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠴
⠀⠀⠀⠀⠘⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠾⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠐⠿⠶⢤⡀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣴⣶⠆⠀⠙⢧
⠀⠀⠀⠾⠟⠉⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠰⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⡴⠂⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣸⠇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠉⠀⠀⠈⠉⠙⠓⠂⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠙⠀⠀⢠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠿⠟⠉⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⠛⠛⠛⠶⠦⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁
⠁⠀⠘⠻⠶⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠿⣦⡀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠀⠀⠀⣄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋
⠀⠀⠀⢠⡾⠃⠀⠀⠀⠘⠋⠀⠀⠀⢀⡀⠀⠃⠀⠀⠀⢠⣴⣦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⠠
⠀⠀⠀⠈⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣴⡦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⠀⠀⠉⠀⠀⣴⠶⠂⠀⠀⠀⠀⢀⡴⠃⠀⢀⣠⣴⡾⠿⠟⠃
⣤⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⡃⠀⠀⣴⡿⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢀⣠⣶⣾⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠈⠻⣿⣿⡀⠀⠀⠀⣿⣿⡟⠉⠀⠀⠙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠚⠂⢸⣿⠃⠀⠀⠀⣿⣿⠀⠓⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣤⣄⣤⣿⡿⠀⠀⠀⠀⠸⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠟⠋⠀⣤⣴⣶⣶⣶⣤⠀⠈⠛⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢛⣟⢿⣿⣿⢛⣙⣿⡟⣿⣿⢻⢿⡿⣻⣻⡿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣪⣕⠿⣸⣿⣌⠻⣿⡇⣿⠆⡖⢰⣷⣽⡽⡵⣛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣛⣛⣥⣛⣿⣛⣩⣾⣟⣿⣷⣧⣿⣿⣟⣾⣯⣛⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣚⡛⠙⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⢆⠊⣉⠷⠶⣟⢊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡈⠉⠛⠶⢄⠈⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⠠⠂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠒⠦⢤⣀⠀⠀⠀⢠⠖⠁⠐⠦⣠⠖⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⠄⠀⠀⠀⠀⠈⠙⠲⢴⢻⠀⠀⠀⠀⠈⠀⠀⠰⠊⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠴⢾⣞⡁⢙⠤⠤⠤⣀⣀⠀⢀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⢔⢥⠀⠀⣡⣠⣌⠚⢤⢀⡂⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⡘⢢⠀⠀⠀⠘⢆
⠀⠀⢴⡞⠋⠀⠀⣌⠈⠙⣦⠀⠷⡕⢄⠀⠇⠀⠀⢀⡤⠈⠓⠄⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠙⢦⠀⠀⠈⠢⡀
⠀⠀⠀⣿⢶⣄⠀⠈⠁⠂⠁⠙⢄⠈⢢⣑⣄⡠⢚⡡⠃⢀⡤⠞⢋⡜⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠑⢄⠀⠀⠈⢆
⠀⠀⠀⢸⠀⠈⠳⢤⡀⠠⢠⠀⠈⠢⣀⠭⠐⠉⣁⡴⠞⠛⠋⠉⢉⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡱⠜⢆
⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡈⠃⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⣠⠖⠁⠀⠀⠈⠙⠂⠤⠄⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡦⡀⠑⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣦⣤⠞⢻⠁⠀⠀⢈⣦⡴⠊⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣈⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀⡏⠀⠀⠀⢀⢀⡠⣤⠶⠭⠤⡤⣤⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠭⠹⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠈⠘⡅⠀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⢰⠻⡀⠀⠀⠀⠸⡀⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣴⡞⢻⠛⡟⠉⡟⢙⡟⣦
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⠀⠹⡄⠀⠀⠀⢣⣿⠀⠀⠀⠀⣿⠀⠈⣧⠀⠀⠀⠀⠀⢿⣿⡿⡟⠻⡞⢳⡞⣿⡏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⣤⠞⠀⠀⠹⡄⠀⠀⠈⣇⠀⠀⠀⠀⠻⠀⠀⣿⠀⠀⠀⠀⠀⠸⡟⠓⠷⠶⠷⠚⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣿⠛⠉⠀⣀⣀⣸⠀⠀⠀⠀⠹⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠹⣄⣀⠤⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⣋⣭⠉⢡⡦⣤⠈⢭⣭⡙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⠉⠀⢿⡄⢻⡶⠂⢸⡇⣤⠀⣸⡇⢀⣿⡛⠆⢙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠰⣏⣥⠀⣬⣿⢘⣛⣯⣬⣿⣯⣤⣿⣀⡾⣯⠉⣴⠟⣿⠌⠹⣷⣄
⠀⠀⠀⠀⠀⠀⣴⡟⣱⢶⣆⣙⢷⢞⣴⡾⠛⠉⠀⢀⠀⠀⠀⠉⠉⠛⢶⣮⡓⠾⠃⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣼⡟⠀⠙⠻⣯⢙⣴⠟⠁⠀⢠⣤⠀⣾⢿⠀⠾⠇⠀⠀⠀⠈⠻⣦⣀⣀⣀⣠⣤⣿⣧⣴⣶⠆
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⡿⠃⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⢛⣛⣉⡉⣩⣥⣤⣄
⠀⢀⣀⣠⣿⣧⣤⣴⣶⠶⠿⠿⠟⠛⠛⠋⢋⣉⣉⣉⠀⣠⣶⣶⣄⠰⢶⣾⠿⠃⣿⡟⠛⠁⢸⣿⠉⣿⡇
⠀⠈⠉⣉⣭⣤⣄⠀⣴⣶⡶⠆⠀⠀⣿⡇⢸⣿⡋⣉⠀⣿⡇⠘⠛⠀⠈⣿⡇⠀⢿⣿⠿⠃⢸⣿⡀⣻⡇
⠀⠀⠀⢹⣿⢙⣿⠇⢻⣷⣤⡆⠀⠀⢹⣯⠈⣿⡟⠛⠀⢻⣧⠀⣶⡆⠀⣿⣇⠀⢸⣿⣶⣶⠀⠿⠿⠟⠃
⠀⠀⠀⢸⣿⠛⣿⡆⢸⣿⣀⣀⠰⣶⣸⣿⠀⣿⣷⠶⠆⠘⠻⠿⠛⢀⣀⣈⣉⣀⣤⣤⣤⣤⣶⠶⢶⣾⠟⠛⠃
⠀⠀⠀⠈⠿⠇⠻⠿⠀⠛⣛⣛⣁⣉⣩⣥⣤⣤⣶⡶⠶⠶⠟⠻⠛⠛⠛⠋⠉⢉⣿⠏⠀⠀⠀⠀⣸⡏
⠀⠀⢀⣶⡶⣶⣾⠿⠟⠛⠛⠛⢿⣍⠉⠉⠀⠀⢠⣤⠀⣦⣖⠀⠾⠂⠀⠀⣠⡾⢣⣤⣄⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣠⡙⢷⣤⣀⠀⠈⠁⠈⠙⠋⠀⢀⣀⣴⡾⢋⡤⠻⣦⣙⡷⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⢀⣴⢯⣽⢆⣬⣝⠛⠷⠶⠶⠶⠶⠿⠛⢫⣥⡖⠻⣶⣃⡀⣉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣅⡿⢁⣾⠷⠌⠀⣸⠇⣾⣛⠀⣿⠙⠂⠈⣷⡀⢘⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣭⣓⡐⣧⡟⠀⣿⣭⠀⠻⣶⣟⣀⣬⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠶⠶⠾⠿⠛⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠄⢠⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⠀⡀⠀⣀⡤⢧⡤⠆⢠⠐⢠⣄⠀⠀⠀⠀⠀⠀⠘⠿⢦⡤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢴⡖⠋⠁⠄⢀⠀⠃⠈⢀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠄⢠⠘⠀⡀⢺⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡖⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⠶⠤⢶⣦⣤⣤⣶⣶
⠻⢿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⣿⣿⣻⣿⣿⢟⠿⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠤⠤⠄⠀⠈⠉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⢸⣿⡋⠓⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣻⣿⣿⡿⠛⢉⢽⣿⣿⣿⡶⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠋⢁⣤⡾⢷⠟⠛⠉
⠀⠀⠀⠀⠀⠘⠛⣛⣿⣯⣥⡀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠐⠒⠒⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⠇⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⣠⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⡀⢨⣿⢿⡿⠛⠀⣀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠀⠀⠀⠀⣯⣷⣀⣀⠀⠈⢻⣿⣿⠟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣤⣀⣠⡼⠿⣿⣶⣦⣬⣿⣯⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠉⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⠛⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣤⠀⣀⣤⣾⡿⠋⠀⢨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠹⠟⣿⠟⠁⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠿⠟⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠰⠚⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠒⠒⠶⠤⢤⣄⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢯⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠐⠺⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠡⠖⣀⡴⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣠⠴⢏⣉⠓⠒⠢⠤⠄⣈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡤⠖⠋⠀⠀⠀⠈⠳⡀⠀⠠⡀⠀⠈⠉⠉⠛⢒⡶⠂⠀⠀⠀⠸⠧⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢧⠀⠀⠀⠀⠀⢠⣀⠌⠀⠀⠈⠳⡀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠹⡓⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⢰⣶⣼⠂⠀⢸⠃⢀⡤⠤⢤⣄⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠒⠮⡙⠒⠒⠂⠘⣄⠀⠀⠈⣡⠏⠀⠀⢉⣉⠉⠐⢦⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⠤⠤⠄⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⢀⡀⠻⠽⠆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠚⠉⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠠⢄⠀⠀⠀⠀⢀⣠⢤⡎⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⣀⣤⠤⠴⠚⠉⠀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢾⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⢤⡤⣀⣤⣤⠤⠤⠤⡃⢸⠤⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠀⣬⠀⠀⠀⠀⠈⠉⠋⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡗⠛⠂⠀⠀⠀⠀⠐⠋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⣧⣄⣠⠼⣇⣀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣯⠈⣾⡀⣰⠛⡏⢫⢹⡙⠓⣒⣒⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠋⠩⠭⠭⠭⠭⠷⠖⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣲⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⢦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⣴⣿⣿⣿⣿⡏⠙⠿⡇⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣅⣀⣰⣿⣿⣿⡟⠛⠁⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⣠⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠲⠋⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡉⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⠎⠉⠉⠉⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⡰⠂⠀⠀⣠⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⣼⣿⡿⠋⠉⣻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⡟⠀⠀⣰⣿⡿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣆⠰⠛⠋
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣤⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣴⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣌⡉⠛⢿⣿⣦⡙⢿⣿⣿⣿⣿⠟⣡⣾⣿⠿⠛⣉⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁⠀⠈⠻⣿⣮⢻⣿⣿⢋⣾⡿⠋⠀⠀⠘⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⢀⣶⣌⢻⣯⣿⣿⣿⢋⣴⣆⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⢣⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⢉⣉⡉⠉⠉⠉⣉⠉⠉⢉⣭⡿⣯⣍⠉⠉⣉⡉⠉⠉⠉⢉⡍⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣰⣙⢦⡵⢤⡴⣽⣶⢎⣶⠋⣀⣀⣀⡈⢳⣾⢢⢾⡵⣄⠜⡣⢞⣳⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣱⣭⢾⡵⣯⠞⣜⠶⢻⣷⠈⢸⡁⢹⠌⢘⣿⡳⢾⡕⣮⢚⡵⢮⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⣎⠃⡜⣹⠟⣜⣷⢞⢎⠶⡳⣻⣦⣀⠉⠁⣠⣾⡳⡹⠻⣝⢦⣻⡝⢦⠻⡄⢋⣷⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣆⣞⡰⠛⣌⡳⢾⢎⡛⡳⣼⠾⣟⣛⡟⢫⠾⢵⡙⡳⢌⠶⠯⣜⢫⡻⣄⡾⣿⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠇⢻⣯⠳⣉⢞⡷⠫⠻⣉⡽⣌⠓⡱⣍⣳⡹⢳⣞⣜⣻⣯⠓⣿⢎⡻⣡⣿⠀⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⣷⣿⠻⣿⣡⢯⠜⡱⡱⣏⣌⢎⡛⡴⣮⡑⡴⣣⢎⣎⣴⡷⡙⡼⢫⣱⡟⢿⣷⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣀⣿⣧⡧⢛⡱⡱⣭⣎⢮⣙⡜⣧⡝⡞⣱⢪⢮⣼⡳⣝⡾⣣⣿⣇⣸⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢸⣿⣿⣿⡿⢹⣷⣋⠴⣝⠦⢫⠻⣭⢜⡴⠛⢬⡌⡳⠣⢾⡙⡬⢳⡿⢸⣿⣿⣿⣿⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣷⣠⣿⣷⣎⣜⠶⢳⡳⣤⢋⠴⣾⢪⢾⡵⠳⢎⣞⣴⣿⣷⢠⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢿⣿⣿⣿⣼⠇⢹⣿⣈⠓⡷⡱⢮⢫⡖⢉⢣⢾⡷⡱⢋⣺⣿⠀⣿⣸⣿⣿⣿⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣧⣾⡿⣿⡙⡗⡽⢶⢣⣝⡓⡷⢺⠝⡝⣱⡿⣿⣶⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢺⣿⣿⠃⢻⣿⡝⣜⢳⡵⣌⡙⡷⡻⢎⣼⣿⠃⢻⣿⣿⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢾⣿⣿⡷⣿⡟⣿⣌⣛⣵⣭⣛⣜⡙⣾⡟⣿⣗⣿⣿⣿⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡗⣿⣀⣽⣧⠁⢀⣀⣁⠈⢡⣿⡀⣹⡯⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⠛⠃⣸⣿⣿⡄⠛⠛⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⡄⠀⠀⠿⣿⣿⠇⠀⠀⣴⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⢛⣯⠀⠀⠸⠿⠿⠀⠀⢸⡟⢻⡟⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠦⣿⠠⣿⢸⣿⠰⠿⠷⠶⠶⠾⠷⢸⡇⢸⡇⣿⡷⠾⠷⠶⠶⠿⠷⣾⡿⠿⠿⠾⠷⠿⠷⠿⠿⠿⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠘⠁⠀⠀⠀⠀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢰⣿⣿⣆⡀⠀⣀⣤⣤⣼⣿⡇⠒⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣀⡄⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣷⣾⣿⣶⣶⣦⡀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣹⣿⡇⠀⠘⠘⢻⣿⣿⣿⣿⣿⣿⣿⣇⠈⠙⡟⠋⠛⠿⠿⠿⣿⣤⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣄⠀⠀⠿⠋⠈⣿⣿⣿⣿⣿⣿⡄⠀⢰⣿⠷⢰⣶⡄⡿⢸⣿⡿⣿⠰⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣄⣸⠟⠋⠉⢉⣙⣷⣀⣤⣤⣤⡈⠻⠛⠛⠿⠿⢷⠈⣶⣶⠶⣌⣙⣡⣦⣬⣿⣁⣼⣷⣮⠆
⠀⠀⠀⠀⠀⠀⠐⣿⡿⠟⢛⠻⣿⡷⣾⡿⠀⠛⠛⠛⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⣻⣿⡟⠿⣿⣿⣿⠋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣶⣿⡿⡗⢠⣴⣟⣳⣶⠊⠉⠉⠉⠁⠀⣀⠀⠀⠀⠀⠀⢠⣀⡀⢀⣀⣙⠛⠥⠀⢠⣉⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢋⣿⣦⢂⣧⣬⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⢉⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⡇⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⡾⣱⣿⣿⠿⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠉⠻⣿⡏⠙⢁⣀⠀⢸⣿⡉⠘⠂⠀⠒
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣏⡁⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡤⠀⣿⣷⣿⣿⣿⣿⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠹⠟⠋⠙⣿⣿⣿⠃⠀⠀⢀⣸⣿⣇⠀⠀⢠⣾⣿⣿⣦⠀⠀⠈⠁⠀⠀⠀⠉⡟⠀⠉⠉⠉⣿⣷⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠀⠻⡇⠀⣰⣿⣿⣿⠿⠛⢡⣀⣿⠏⠁⠀⠈⢻⣿⣤⣴⣶⣶⣶⣤⣇⢉⣀⡈⢃⡏⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠁⣠⣿⣿⣿⡿⠠⠄⠲⠤⠭⠤⠀⠀⠤⠘⣿⡏⠛⠛⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⣿⣿⡇⠀⠀⠴⠶⠶⣶⣾⣶⣤⣶⣿⡇⠀⠀⢻⣿⣿⠿⠁⠉⠉⠧⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠘⠛⠛⠛⠛⠻⠟⠛⣴⣿⣿⣦⣄⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣾⣟⠋⠉⠉⠰⣶⠂⠉⠉⣉⣿⣿⣤⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠹⣿⣉⣉⣉⣹⡿⠁⠀⠀⠀⠀⠘⢿⣍⣉⣉⣩⡿⠃⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣉⣀⣀⣉⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠉⠙⠛⠷⢿⠆⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣄⣀⣀⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠐⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⠔⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠥⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⢧⣀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠧⠤⢴⠄⠀⠈⠳⡄⠀⠀⢧⠀⠀⢀⣋⣀⣀⣤⣄⣼⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧⢀⡤⣍⡦⡀⠀⠳⣞⠉⠀⠀⠀⠀⠀⡀⣹⢀⣀⣰⣃⣀⣀⣠⡤⠤⠶⠖⠚⠋⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡟⡾⠀⠈⠳⣿⢆⠀⠈⠓⠒⠦⠤⠴⠚⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠴⠚⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠙⠉⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠤⠀⢀⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣠⠜⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠉⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⡴⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡓⠠⠧⢤⣀⣀⣀⣀⣀⡀⠀⠊⢀⢀⣀⣀⣀⣸⣓⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢬⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣼⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣇⣀⡉⠉⠐⠒⠠⠤⢤⣀⡀⠀⠀⡜⠉⢧⣿⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⣥⣄⣀⡀⠀⣀⡉⠉⠉⠉⠉⠓⠐⠒⠒⠘⠳⠶⠤⠤⢸⣇⣣⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣿⡿⢸⣿⠀⣿⡿⠙⠇⠀⢰⣶⠆⢰⣶⠠⣤⠀⣠⠄⣀⢀⡀⣀⠈⠉⠉⠉⠐⠒⠒⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⡏⣿⣦⢀⣿⡗⠟⠀⠀⢸⣿⠀⣾⣿⣰⠁⣼⣿⠀⠹⠺⢹⣿⠋⠇⣿⡿⠸⡆⢲⣶⢠⣄⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠛⠃⠻⠟⠸⠿⠇⣼⠺⡟⣿⡿⢀⣿⡇⢈⡄⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⡇⡎⠀⣿⡿⢸⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠘⠓⠒⠁⠀⠿⠿⠀⢸⣿⣇⣴⢀⣿⡇⣾⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠉⠁⢛⢲⠒⠒⡶⢤⠤⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡌⢧⡼⢁⡼⠃⠀⣀⣀⣀⣀⡍⢸⣍⣩⡭⠭⢶⡶⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⡉⠉⠉⠁⠀⡼⠛⠃⠉⠉⠉⠁⡏⢰⣷⡌⢱⡄⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣀⡀⠀⣀⠔⠁⠀⠀⠀⠀⠀⡼⢁⣏⠙⢇⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡟⠉⠉⠁⠀⠀⠀⠀⠒⠛⠛⣛⢋⣉⣑⣂⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⠽⠿⠿⠷⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠓⠛⠛⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⠒⠒⠒⠒⠒⣛⣛⠉⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠶⠭⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠱⢶⡀⢻⠔⣠
⠀⠀⠀⠀⠀⠀⢸⠻⣮⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠸⣞⡵⢋
⢀⣀⣀⣀⣀⣤⣏⡓⠀⠈⣹⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡴⢃
⣿⠉⠉⠁⣴⣾⠳⠁⠀⡰⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢋⡄
⢸⠀⠀⠀⢶⣧⠉⠀⣴⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢋⡄⠀⠀⠀⠀⠀⣠⢊⣝⠒⣷⠓⢲⡄
⠸⡇⠀⢈⣿⡂⢀⡾⠷⠆⠀⠀⣀⣀⣀⣀⣤⠤⠤⠴⠀⠀⣀⣀⣀⣀⣠⡤⠀⠀⠀⣧⢋⡤⠀⠀⠀⢀⡜⡵⢻⡇⢀⡼⣻⢿⡟
⠀⡇⠀⢒⡧⢀⣞⡛⠃⢈⣭⠭⠭⠤⠴⠶⠒⠒⠚⠋⣉⣉⣉⣡⡤⠤⠤⠄⠀⠀⠀⢷⢎⡴⠀⠀⣠⣾⠞⣠⠛⠙⠯⣞⣱⣿⠋
⠀⡇⠀⢽⣇⡾⠄⠀⠀⠠⠤⠤⠶⠒⠒⠒⠋⣉⣉⣉⣩⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⢸⠎⡴⢃⡴⢻⣣⠞⢁⡰⢌⣓⡞⠉
⠀⣧⠀⣻⣿⠂⠀⠀⠀⠀⠶⠶⠒⠒⠛⠉⠉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣰⣏⣠⠿⠁⢀⠲⣝⣲⠟
⠀⢸⠀⢾⡏⠁⠀⠀⠀⠀⠤⠴⠶⠒⠚⠋⠉⠉⠉⠀⢀⣀⣀⣀⣀⣤⠤⠄⠀⠀⠀⠀⢿⣔⡴⠉⠀⣜⠦⣙⠞⠁
⠀⢸⠀⠨⡇⠀⠀⠀⠀⠀⠤⠤⠴⠒⠒⠒⠛⣋⣉⣉⣩⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⢠⡓⠬⡽⠋
⠀⢸⡆⠨⣷⠀⠀⠀⠀⠀⠒⠒⠒⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡠⣌⠲⣭⠟⠁
⠀⠘⡇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢠⡲⢽⣪⠟⡁
⠀⠀⡇⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠋⢨⣒⡬⢿⣧⠎⡀
⠀⠀⢧⠀⢩⡇⠀⠀⠀⠀⠀⠀⢀⡞⢁⡶⢀⠆⣠⢀⡀⠈⡉⠉⠉⢁⡀⣠⣯⣦⠵⡋⢁⡈⣧⢎⡄
⠀⠀⢸⠀⠐⣧⠀⠀⠀⠀⠀⢠⡞⠀⡼⣡⢏⡴⢧⢾⣠⠞⠥⠔⠲⠞⠉⠉⠑⠋⣚⣥⣠⠖⣵⠾⢀⡄⣠⢀⡀
⠀⠀⠘⡇⠈⢿⠀⠀⠀⠀⠀⣙⣤⣞⣉⣁⣤⣤⠤⠤⠤⠶⠶⠒⠒⠚⠛⠋⠉⠉⠉⠉⠀⠈⢻⣵⢋⡞⢡⠞⣡⠋⡰⢂⡄⡀
⠀⠀⠀⣧⠀⢽⡄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠏⡴⢁⡴⢁⠞⣡⢋⠜⣱⢋⡔⣠⢂⡀
⠀⠀⠀⢸⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⣁⠀⠀⠀⠀⠁⠋⠘⠁⠚⠰⠣⠞⠼
⠀⠀⠀⠘⡇⠐⣿⠘⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⢯⡜⢁
⠀⠀⠀⠀⣧⠀⣿⡄⠋⠉⠁⠁⣀⣀⣀⣀⣀⣀⡤⠤⠤⢶⣒⣒⣚⣿⠭⠭⠿⠷⠒⠚⣧⠞⣰⠋⡴⠃
⠀⠀⠀⠀⢹⠀⢨⡿⠯⠭⠭⠽⠿⠛⠛⠓⠒⠛⠉⢉⣉⣉⣉⣠⣤⢤⡤⢤⡴⣶⠖⣲⢋⡴⣡
⠀⠀⠀⠀⠸⣦⣤⣤⠤⡴⢶⡶⣶⠒⡶⢛⡟⢉⡝⣩⢋⡼⢁⡞⢡⠞⢠⠏⠰⠃⠐⠁⠘⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⢻⣿⠟⠻⣿⡿⢻⣿⣿⡟⣿⠟⣿⣿⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⡀⢻⣿⣈⠃⠈⠰⢿⣿⣥⢀⡤⢠⣼⣿⢸⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⣴⣿⣿⣋⣴⣄⢃⣸⣿⣆⣴⣄⣴⣾⣿⣸⣿⣿⠃⠀⣀⣀⠀⠀⠀⢸⠏⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠟⠻⠛⠛⠟⠻⠛⠿⢿⣿⠟⠟⠛⠋⠁⠀⢰⣿⣏⡀⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠀⠀⠉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⣤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣛⣉⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⢀⣀⡀⢠⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡈⣿⠀⠀⠻⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡘⣿⣿⣿⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⡙⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⠙⠻⣿⠟⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⡇⠋⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⢹⣿⣿⣿⠀⣿⣿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣬⠛⢛⡏⠀⠀⠀⠀⠀⠀⣈⠀⢸⣿⣿
⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠋⠀⠀⠀⠀⠀⠀⠠⢈⢽⢸⣳⣜
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⣰⣶⣯⠾⠋⠛⠛⠻⠿
⣿⣿⣷⡖⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⢰⣇
⣿⣿⣿⣏⠀⢑⣶⣿⣓⡶⠂⠉⠀⠀⠐⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⠀⠀⠐⠔⣀⠀⠀⠀⠹⣿⠇⠀⠀⡤⠀⣼⣿
⣿⣿⣿⣿⠀⡀⢸⡆⡘⡆⠐⠀⠀⠀⢨⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠄⠀⠘⠻⠇⠀⠀⢀⣨⣿⣧⡀⠀⠀⠉⠀⢀⠀⠀⣸⣿⣿
⣿⠿⠿⣋⠀⠀⠰⡹⠃⠙⠦⡀⢀⣠⠀⠀⠈⠀⠀⠐⠂⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣀⣠⠄⣘⠙⠻⣿⣷⠀⠙⣀⠀⣴⠀⢠⣿⣿⣿
⠁⠀⠹⢻⡄⠀⠐⠃⠀⠀⣠⣾⠿⠋⠀⠀⠀⠈⠀⠄⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠋⠘⠛⠛⠁⣿⣿⣿⣶⣝⡋⠀⢠⣷⣟⠛⠀⠘⠛⠿⠟
⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢏⠉⠻⠋⠀⣀⢀⣠⠍⠀⡈⠲⡴⡄
⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⠀⠑⠀⠀⠾⣷⣯⠀⠀⠆⠹⣆⠀⠘⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢁⠠⠐⠆⠙⠢⠀⠁⠀⠀⠐⠀⡽⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣷⣾⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢶⣀⡀⣀⠿⠸⡟⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣿⣶⣤⡀⠀⢠⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇⣾⣿⣿⣿⣷⣤⣤⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠟⢋⣀⣽⣿⣿⣿⣿⣿⣿⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣚⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⡇⠻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡹⡿⠋⡁⣤⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⡟⠀⠀⠀⢴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⠀⠀⠀⠀⠀⣼⡇⢾⣧⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⡀⣾⣿⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣠⣿⣿⢧⠄⠀⢀⣤⣤⣤⣸⡇⠈⠉⢺⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠉⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⠀⣤⠞⣿⢹⣿⣿⣿⣧⢰⡟⢸⡿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⠆⡿⠀⣀⣸⣿⣿⣿⠿⣿⡷⣯⡀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣆⠿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⡀⣾⣿⣿⣿⣿⡏⢁⣾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣭⠉⡉⠁⠙⠛⠛⠏⠻⢻⠟⡟⢛⡻⠿⠿⠿⡿⠿⠿⠟⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠒⠛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠄⠉⠉⠉⠋⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⣿⣶⣄⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣿⣘⣿⣛⣧⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣾⠂⠀⠀⠉⠙⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠄⠀⠀⠀⠀⠈⠀⠁
⠀⠀⣠⣴⣾⣇⠀⠀⠀⡀⢰⠀⠙⠇⢀⠐⡆⠰⡆⠀⠀⠄⠀⠀⠐
⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡀⣠⣀⣠⡤
⠀⠀⠀⢀⣄⠀⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣟⢙⣿⣧⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠬⠫⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠘⠉⢿⠛⠏⠃⠀⠀⠀⠆⠀⠀⠸⠋⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣽⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣶⣾⣿⣛⣷⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⠿⠟⠛⠛⠉⠉⠉⠉⣉⠙⠛⠿⠻⠿⣿⣿⣷⣶⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣶⠿⠛⢉⠀⠀⠀⢀⠘⣧⡀⠀⠰⢦⣿⠠⠄⠀⣀⣴⠀⠀⠈⠉⠛⠿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠋⠁⠀⢀⣤⡷⣆⠀⠀⠸⠛⠃⠀⠀⠚⠛⠆⠀⠀⠞⢛⠋⠀⢰⣤⡄⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⣳⣴⡆⠀⠀⠇⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⢶⣤⣤⣄⣀⡀⠀⠚⠻⠇⠁⠀⣄⣠⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣶⣾⠏⠁⠀⡀⠈⠉⡟⠋⠀⣠⣴⠾⠛⠉⠉⠑⠲⠶⠦⠤⠄⣀⣀⠀⠈⠉⠛⢶⣦⣄⠀⠀⠴⠿⠿⠄⠀⠀⠙⣿⣥⠆
⠀⠀⠀⠀⣼⣿⡿⠁⠀⣻⣾⣇⠀⠀⢀⣴⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠲⠶⠭⢿⣦⡀⠐⠀⠀⢰⣀⡤⠈⠻⣿⢃⡀
⠀⠀⠀⣼⡿⠋⠀⢀⠈⠉⠉⠀⣠⣶⠏⠀⠀⠉⠉⠐⠂⠀⠀⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠀⠀⠀⠀⠹⣇⠀⠉⢹⠟⠦⠀⠀⠹⣿⡅
⠀⢀⣼⡿⠉⠐⢶⠿⠦⠄⠀⣰⢟⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣰⣀⡀⠹⣶⣦
⠀⣼⠟⠃⠀⠀⠉⠙⠂⢠⡼⠃⠀⡦⠈⠉⠀⠒⠐⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢉⡿⠿⡄⠀⠸⠿⣀
⠀⢻⡶⠀⡀⢴⡀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠢⠤⠤⢄⣈⡁⠐⠘⣦⠈⠀⠀⣠⠀⠀⢻⣿
⠀⢠⠁⠀⠸⠿⡄⠀⠀⣿⡖⠲⣆⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠀⠤⠤⢄⣀⡀⠀⠀⠀⠠⠤⠄⣉⣁⢻⡆⠀⢻⣿⡗⠂⠈⣿⡇
⠀⣾⠀⠀⡀⠀⠀⠀⢀⣿⣀⡠⠿⠃⢨⡟⠉⠙⠗⠀⠀⠀⡖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠀⠀⠉⣷⠀⠉⠀⠉⠀⠀⢹⣗
⢀⡛⠀⣤⣷⡶⠂⠀⢸⡏⠰⣧⠀⠀⣼⠃⠐⠶⠀⠀⠀⢘⠃⢰⡟⠀⠉⠏⣠⠖⠙⡷⢆⡦⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠐⢦⣾⣄⡀⢨⣿
⢸⡇⠀⠈⠏⠙⠂⠀⠉⠁⠀⠛⠗⠠⣿⣀⣀⣤⠰⣷⣀⣾⠇⣾⠃⠒⠾⢠⣉⠀⠀⠀⠀⠀⢬⠃⠈⠃⣻⠓⠒⢶⠀⣤⠤⣄⠀⠀⠀⠛⠩⠁⠀⢸⣿
⢾⡇⠀⠠⡀⢀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠰⠿⢄⣠⣴⠘⣿⣤⣠⡖⠀⠀⢠⠀⠀⢠⡿⠠⢤⡄⢠⡛⠀⢸⣧⠀⠀⣀⣠⠀⠀⢸⣿
⣸⣿⠀⠴⣿⡿⡄⠀⢿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠼⢬⠀⠀⣼⡇⠀⢀⡀⣾⡇⠀⣾⡿⠀⠠⠿⡟⠢⠀⠺⠋
⢨⣽⡇⠀⠘⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠑⠿⠧⠶⠛⠁⢀⠀⠀⠀⠀⢀⡂
⠈⠉⣿⠀⠀⢠⣴⡆⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣜⠀⠀⣸
⠀⠀⠸⣇⠐⠛⢿⠋⠀⠀⠡⣤⠄⣀⣀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠐⠒⠠⠄⡤⡀⣀⡀⠀⠀⠀⠀⠨⣭⡉⠉⠒⠒⠂⣶⠃⠀⠀⠀⠛⠉⠁⣠⠛
⠀⠀⠀⣿⣆⠀⠀⢤⣠⡆⠀⠻⣦⡄⠤⢀⣉⠀⠐⠀⠀⠀⠄⠀⠉⠁⠒⠒⠳⠤⠤⢄⣉⡁⠒⠂⠀⠀⠀⠀⠀⢀⡾⠃⢀⣸⣶⡖⠀⠀⣰⣿⠃
⠀⠀⠀⢻⣿⣇⠀⠤⠶⡝⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠐⢠⡶⠉⢀⠀⠀⠸⠏⠛⠀⣸⡟⠃
⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⢠⣤⣧⣀⠈⠻⢦⣄⡀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠄⢠⣀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠠⣌⢳⡖⠀⠀⣠⣞⠏⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠠⠟⠻⠀⠀⢠⡄⠈⠛⠷⢦⣤⣀⣉⣀⠀⠀⠀⠀⠀⣀⣈⣉⣴⠶⠟⠉⢁⣀⣶⠀⠀⠚⠉⠃⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢶⣀⠀⠀⠘⣶⣾⡋⠀⠠⣄⣀⠉⠉⠉⠋⠛⠛⠋⠋⠉⠈⠁⠀⢀⡄⠀⠀⣽⢼⡏⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⣀⠀⠀⠀⠀⠲⣿⣯⠀⢀⣹⣄⡃⠀⠀⣷⣾⣄⠀⠀⠸⠃⠂⠀⠀⠀⣀⣤⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡈⠁⠀⠀⠈⠹⠏⠀⠀⠈⠉⠋⠁⠀⠀⠀⢁⣀⣤⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠤⣤⣤⣤⡤⣤⡤⠴⠶⠶⠚⠋⠉⠁

⠀⠀⠀⠀⠀⣠⠞⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠳⣄
⠀⠀⠀⠀⠀⡇⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸
⠀⠀⠰⣄⠀⡇⡇⠀⢀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣄⠀⢀⣤⣄⠀⠀⢸⢸⠀⣠⠖
⠀⠀⢀⣈⡁⡇⡇⠀⠈⠳⡀⡴⠃⠀⢸⠀⠀⠀⠀⠀⣏⠀⠈⠁⠀⢸⢸⠀⣁⣀
⠀⠀⠈⢉⡀⡇⡇⠀⠀⠀⢹⠀⠀⠀⢸⠉⠉⠉⠀⠀⠈⠉⠳⡄⠀⢸⢸⠀⡉⠉
⠀⠀⠰⠋⠀⡇⡇⠀⠀⠀⢸⠀⠀⠀⠸⠤⠤⠤⠤⠀⢤⣀⡴⠃⠀⢸⢸⠀⠙⠦
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⢤⠀⠀⠀⠀⠀⣸⠼⣤⡞⠛⢦⡀
⠀⠀⠀⠀⠀⣇⠳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⡀⠀⠑⢄⣰⠚⠺⣇⠀⠈⠳⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠈⠳⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⢦⡀⠀⠙⢆⠀⠈⠳⣄⠀⠈⠓⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠙⢆⡀⠈⠓⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡭⢭⣙⣦⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠢⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⡄⠀⠀⢀⣀⣉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠘⢦⠀⠀⡇⠀⡴⠋⠁⠈⠙⢦⠀⠙⠲⢤⣀⡀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠈⢳⡀⡇⠐⡇⠀⠀⠀⠀⣸⠀⠀⠀⢸⡇⠉⠙⢳⡞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠙⠇⠀⠙⠦⣤⣤⠴⠃⠀⠀⠀⢸⡇⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠑⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁

⡤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠
⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠂⡆⢰⠀⡔⡀⡆⡔⢰⠀⢠⠀⠀⢆⡀⢰⠀⣆⡆⡔⠐⡆⣦⠂⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡔⠢⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣿⢫⣝⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⡉⢳⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣟⢶⡯⣻⡿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢌⡩⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣷⢾⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⣤⣤⠤⠤⠤⡤⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣻⣛⡟⢿⣛⣛⣛⣟⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⡠⠤⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⡶⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⢆⣀⣀⠀⠀⠀⠀⠀⠙⠦⣀⡰⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣹⡿⢿⣿⣿⣿⣿⣿⣮⣝⢿⣏⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢾⣡⠜⢲⣄⠀⠀⠀⡀⠢⣌⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧⠾⣣⡍⠻⣿⣿⣿⢿⣝⠷⢼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠓⢶⣁⡴⢳⡔⢦⣈⡶⠊⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣮⡉⠾⣫⣬⣿⣝⠷⢉⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣶⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⢤⣄⡀
⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⠿⢿⡿⠿⠟⠃⠀⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⣿⣧⠀⡩⠝⠓⠋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢀⣀⠀⢠⣄⣀⡀
⠀⠀⠀⠀⣿⣿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣾⣄⣠⣷⣾⣿⣗
⠀⠀⠀⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣁⣤⢾⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢉⣌⢁⠚⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣿⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢈⠉⠉⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡇⢠⠀⠉⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠻⠻⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⠀⣷⡄⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠿⠃⠀⠐⠿⢿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠛⠁⠈⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠋⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀⢀⣠⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⣿⠀⣴⣿⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣽⣿⢿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠻⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⣥⣶⠎⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠿⡏⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣿⠂⣀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠉⢡⠀⠀⠁⢀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣻⠀⠀⡀⣘⣠⣤⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣰⣷⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡋⠁
⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢻⣿⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧
⠀⠀⠀⣿⣿⡙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⡆⣾⣿⡇⣸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⢹⣿⡟
⠀⠀⠀⣿⣿⣡⡾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠉⠀⡁⣿⣿⣿⢻⡟⣼⣿⣿⣦⡀⠀⢀⣠⣾⣿⣿⢻⠋⡀⠸⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣰⣯⣼⣿⣿⣷⡄⠀⠀⣠⣾⣿⠟⠋⢸⡀⣠⣿⣿⣿⣿⣾⡿⢻⢏⣻⣿⣷⣶⣿⣿⡟⠋⡞⢸⢀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠙⣿⣿⣏⣼⠇⣹⣿⣿⣦⣾⣿⣿⠋⠀⢀⣸⣴⣿⣿⠟⠙⢿⣿⣧⣿⡿⣿⠙⣿⣿⢻⠏⣠⠀⡇⠆⣸⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡾⢻⠇⢻⣿⣿⢛⡏⢀⡆⣸⣿⣿⠟⠁⠀⠀⠀⠻⣿⣿⣤⣿⢰⣿⣿⠀⡀⡇⣰⣷⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⠀⢸⣿⣿⠀⡇⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣼⣿⡟⣀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣸⣿⣿⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⣠⡶⠀⠀⠀⠀⠀⠀⢶⣄⠀⣤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣤⣴⣤⣄⡀⠀⠀⣴⠟⢹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡏⠻⣦⠀⠀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⢠⣾⣿⣿⣿⡟⠙⣿⠀⠀⠀⠀⣀⣤⢀⣠⡦⠀⠀⠀⠀⢰⣄⡀⣤⣀⠀⠀⠀⠀⠿⠹⠏⣿⣿⣿⣿⡄
⠀⠀⠀⣾⣿⣿⠿⠋⠀⡦⠉⠀⠀⡴⠞⠋⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠿⠉⠳⢦⠀⠀⠀⠷⠀⠘⢻⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⠀⠀⠀⠀⠀⢠⣿⠟⠁
⠀⠀⠀⢿⣿⣿⣿⣷⠤⠀⠀⠀⠀⠀⠀⠼⠷⣶⣶⠆⠀⠀⠀⠀⠰⣶⣶⠾⠧⠀⠀⠀⠀⠀⠀⠚⣿⣿⣄
⠀⠀⠀⠈⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣴⣿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠉⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⢈⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠥⠀⠀⠠⠍⠀⠈⠿⠛⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠄⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠀⠀⠀⠀⠀⣀⡴⠀⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠋⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠻⠿⠁⠀⠭⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⠍⠀⠈⠡⠀⠀⠹⠟⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠤⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⢀⣠⠆⠘⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠢⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠴⠛⠛⠋⠁⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠁⠀⠀⠀⠀⠈⠁⠀⠀⠈⠱⡄
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⢠⠎⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣼⠁⠀⢀⣤⡄⣠⣠⢀⣴⢶⡄⡆⡴⠀⠉⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⡙⠁⠲⡆⣄⣀⣶⣆⠀⠠⣾⡆
⠀⠀⠀⢿⠀⠀⠹⠴⠋⠘⠎⠹⡵⠿⠈⠉⠀⠀⠀⠀⠑⡄⠀⠀⠨⡷⠀⠀⠠⣏⡗⠀⠆⡉⠈⠟⠉⠀⠀⠘⡏
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⡰⠃⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠈⠉⣿⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢉⠇⠀⢀⡰⡫⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠊⠁⠀⠀⠀⠀⠀⠀⠠⠶⠭⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢲⡀⠀⠀⠀⠀⠀⢀⡞⣰⠃⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠀⢷⣀⣀⣀⣀⣀⡼⢡⡇⠀⠀⠀⠀⠀⠺⡁⠀⠙⢶⠛⠛⠛⠛⠛⣲⠞⠁⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢨⣧⠘⣧⣤⣤⣤⣼⢃⡟⡇⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠙⢟⠛⣻⠞⠁⢀⡴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠈⢧⠘⣆⠀⢠⠇⣾⡀⡇⠀⠀⠀⠀⠀⢠⡏⢠⡟⠲⣄⠀⠙⠁⠀⣰⠛⢧⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠈⢧⠘⣦⡞⡸⢹⡇⣧⠀⠀⠀⠀⠀⢸⠀⣼⠀⢀⠞⠀⠀⠀⠺⣅⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠈⢷⡈⢱⠃⢸⡇⣿⠀⠀⠀⠀⠀⠸⡆⢸⡶⠋⠀⢀⡤⣄⠀⠙⢦⣰⠇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠳⠏⠀⢸⠇⣿⠀⠀⠀⠀⠀⢀⡿⠋⠀⢀⣴⡋⠀⠈⢧⡀⠀⠻⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣉⣉⣉⣉⣉⣉⣛⣛⡛⠀⣿⠀⠀⠀⠀⠀⠙⢦⣀⡴⣏⡈⠉⠉⠉⠉⢙⣦⡀⠈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠛⠛⠛⠛⠉⠈⠻⠞⠁