Samhain Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠰⠰⠆⠀⠀⠀⠘⠀⠀⡰⠀⠁⢠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣉⠀⠀⠀⠉⠠⠂⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⡠⠊⠔⠂⠉⠀⠀⠩⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⡌⠀⠉⢂
⠀⠇⠘⠀⠀⣤⣤⣤⠀⢃⠄⠀⠀⠀⠐⠀⠀⠀⠐⡘⠀⠀⠆
⠀⠰⠀⠃⠀⠈⠉⠁⠀⠈⡖⠀⠀⠀⠀⠀⠀⢀⡅⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠃⠈⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⢡⠁⠀⠐⠀⠀⠀⠐
⠀⠀⠈⡀⢹⠀⠀⠀⠀⠑⠲⣇⡀⠀⠀⠐⠁⠃⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠈⠀⠀⠀⠀⠠⠤⠬⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⢠⣾⢷⡔⢀⣤⡀⢼⡦⢠⣿⡆⠀⡀⠀⠶⠆⠀⢿⡿⠀⢰⣷⡂⣄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠀⠉⠈⠀⠘⠋⠁⠀⠆⠀⠉⠁⠀⠘⠂⠀⠠⣤⣆⣠⡄⠘⡿⣷⠉⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠿⢶⣿⣧⣶⡶⠄⣠⣾⣷⣿⣿⣶⠀⣘⠶⣿⣿⠟⢀⣄⠀⣘⡀⠘⠀⠀⠀⣳
⠀⠀⠀⡀⠀⠀⠀⢾⣷⠀⡀⠀⠆⠉⠁⠀⠻⠂⠹⣿⣛⢛⣻⡿⠐⠃⠀⠀⠁⠀⠈⣋⣘⣿⣿⣤⡖⠀⠀⢻⡄
⠀⢰⠀⣷⠊⠀⠀⢐⠂⠀⠀⠐⠀⠀⠀⠀⠀⣶⠆⣤⠬⣥⠠⡶⠰⡇⠀⢀⣴⠀⢀⡉⠛⣿⣿⠀⠀⡀⠀⢹⠁
⠀⠈⠢⣿⠀⠈⠀⣼⣥⠀⠉⠀⠀⠀⠿⢢⣤⢀⡀⣉⠀⣉⢀⣁⢠⡄⠺⠃⠉⠀⠯⣇⠀⠛⡟⠀⢸⡏⠀⣼
⠀⠀⠀⣿⠀⠈⠰⠁⠉⠀⢰⣶⣶⠀⠀⠈⠁⠘⠁⠛⠁⠛⠀⠋⣀⠁⢀⠀⠟⢃⣤⣿⣞⡛⢁⣠⡌⠀⢀⣿
⠀⠀⠀⢸⡄⠀⠘⠀⠀⠁⢿⣿⣿⠇⠀⣴⣶⡆⢀⣤⡄⠀⣴⠀⣿⠀⣸⣦⡀⢀⠀⣙⠛⠁⠈⣿⠓⢀⢤⣥
⠀⠀⠀⠈⠧⠠⠖⠀⠀⢦⠀⠀⡄⠀⢴⣿⣿⣷⠄⠉⠡⠼⠿⠦⠀⠀⢿⣿⠟⠠⠋⠣⠀⠉⠢⠄⠀⠘⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⡷⠄⠀⠂⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣾⡯⢤⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢻⡏⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢋⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⣧⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⢤⣿⣿⣿⣦⠄⠀⠀⠀⣀⣤⡴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡴⢠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣶⣀⠀⠉⠉⠁⣀⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠓⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⣤⡭⠑⠚
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣟⡿⢶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣡⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣱⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⢠⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣧⡀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⢐⡿⣿⡙⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⣠⣄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣼⢿⠏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣦⠈⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠿⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄⠀⢶⡦⠈⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠘⠗⠀⠀⣴
⠀⠀⠀⠀⠀⠀⢰⡀⠀⢈⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣁⠄⠀⠀⠀⠛⠂⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⠗⠊⠁⠀⠐⠀⠛⠿⢶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠉⠋⠁⠀⠀⠊⠰⢶⣦⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠨⣷⡀⠀⠀⠉⠿⣿⣿⣿⣿⣿⡇⢹⣿⡿⠻⡆⠁⠀⠄⢀⠀⠐⠒⠂⠀⠀⠀⠀⣿⡎⣦⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⢒⡘⠦⣰⠹⠿⠿⠟⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡄⠀⢸⣷⡙⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠫⢄⠀⠀⠀⠐⠄⠀⠤⠀⠀⠠⡀⡀⠀⠄⠀⠀⠀⠀⣐⡞⠉⢂⣼⢿⡶⢿⣧⣀⣀⠀⠀⢀
⠀⠤⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢢⠀⠀⠀⠠⠀⠘⡇⡔⠀⢀⠎⠁⠀⠀⠉⠙⠿⣏⢻⠾⣧⡙⠋⠙⢻⡿⠟⠁
⠀⠀⠀⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⠈⠙⡆⠐⠈
⠀⠀⠀⠀⠀⠀⡠⠓⠀⠀⠀⠀⢄⠀⡀⠀⣠⠚⢧⣀⠀⠀⠀⠀⢈⡀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⡃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡴⠖⠒⠓⠊⠁⠀⠀⠙⠒⠒⠖⠊⡁⠀⠉⠁⠀⠀⠀⢀⡀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠈⠀⠀⠀⠀⠀⠠⣀⠐⠀⠂⠠⠀⠄⡠⠀⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁

⣾⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣷⣶⣶⣗⡴⢋⣁⠀⠠⡆⢠⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⣿⣿⣻⣿⣿⡿⢿⣯⣿⣼⣟⣟⣩⣭⠜⠛⢋⠠⠂⣠⣷⣾⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠼⣲⡝⢽⠻⣭⡋⣡⠎⠀⢃⡤⠸⠤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢻⡟⣞⡿⢴⡉⢷⠻⠻⢟⣩⣯⠺⣞⣉⡀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢬⣿⣷⡿⣴⡉⢽⠈⣃⠀⢠⠹⠿⠀⢻⡗⠉⠔⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⢿⣿⣾⢿⠞⣸⣏⠃⠠⠛⢤⠶⣙⢿⢻⡌⠸⠿⠋⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣿⡸⣿⢀⣿⣿⣿⣿⡆⢋⡥⣢⠽⠛⠒⠓⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣷⣿⢶⣿⣿⣿⣿⣿⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣯⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢋⠡⢻⡎⣿⣿⣿⢟⣿⣿⣿⣿⣿
⣿⣿⣿⣻⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠃⠉⠉⠙⠛⠛
⣿⣿⣿⣿⣟⣷⣽⣟⢿⣿⣿⣿⣆⠀⠀⠀⠀⠛⠛⠃
⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣯⣿⣿⣷⣴⣤⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⡶⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣯⡗
⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣽⢿⣦⢦⡀⠀⡐⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⢀⠄⠀⠀⠘⠋⠀⠀⠀⢀⠀⠀⢠⠾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⣾⣿⣷⠀⠀⢸⣿⠀⠩⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠛⠋⠀⠀⣸⣿⣿⣿⡆⠐⠋⠁⣠⠀⣠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠳⢤⣿⣦⡀⠀⠙⢲⣿⣿⣿⣿⣿⠒⠃⠀⣘⣾⣿⠇⠀⠀⠀⠀⠈⢧⠐
⠀⠀⠀⠀⠀⢀⠃⠀⢰⠀⣀⣼⣿⣿⣷⡖⠀⢸⣿⣿⣿⣿⣿⠄⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠰⠀⢼⠀⡤⠂⠀⣩⢿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡃⢰⣿⣿⣿⣿⣏⠀⠀⠀⠀⠈⣆⣇⢀
⠀⠀⠀⣶⠀⣸⠎⠀⣤⣤⣤⣼⣿⣿⣿⣿⡆⣼⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣀⣀⣀⠀⠀⠀⠘⣿⡈
⠀⠀⠀⠘⢦⣿⣠⢠⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠢⣿⠗⠁
⠀⠀⠀⠀⠀⣿⡇⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢂⢠⣶⠀⣿
⠀⠀⠆⠰⡘⢹⡇⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠀⠁⢀⣿⠀⠃
⠀⠀⠀⠀⡇⢸⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⢸⡟⠀⠀⡀
⠀⠀⢷⠂⣧⠘⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⣘⣇⡀⣾⡇⠀⢸⠁
⠀⢀⣬⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⡀
⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿

⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠉⠀⠀⠀⠴⢿⣿⣦⣴⣤⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⣆⣀⡀⠀⠀⢸⣂⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠙⡇⣠⡀⣤⡠⣤⢠⢠
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠣⠂⠋⠃⡟⢹⠋⠘⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⠁⠀⠀⠀⠀⣀
⠀⠀⠤⡄⠀⠀⠀⢀⣠⣄⡁⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣤⣤⡿⠟
⠀⠀⠀⠈⡇⢠⠖⠛⠢⠀⢳⠀⠀⢹⡀⣀⠀⣀⣀⡀⢸⣄⡀⢠⣄⠀⠄⠀⠀⠈⠛⠋
⠀⠀⠀⣠⡹⡿⠀⠀⠈⠀⠉⠀⢦⡼⣱⢿⡆⠸⠉⠿⠘⠃⠇⠎⠻⠸⠠⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠋⠀⠘⣷⠂⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠤⠀⢸⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠀⢸⡇⠀⢀⣴⡄⣴⣆⣴⣶⣰⣦⣠⣦⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠚⠋⠉⢳⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⡇⠀⢼⣿⣷⣿⡷⣿⣿⣿⣿⣿⣿⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⢃⣠⠄
⠀⠀⠀⠀⠀⠁⠀⠘⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⣸⠇⠀⣼⠁
⠀⠀⠀⢀⣀⣀⡀⠀⣿⡇⣿⡇⣿⣿⣿⣿⢿⣿⢸⣿⠀⠀⠀⠀⠀⠐⠿⣿⠃⠀⠀⠀⠀⠀⣠⠚⠛⠓⢦⠀⣿⢱⠏⠀⣸⠇
⠀⠀⠀⢸⠻⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⣀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠈⠻⣿⣾⣴⠋⠉
⣰⠶⣤⡘⠳⣼⣦⠟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣦⣧⣾⣤⣿⣦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠶⣄⣸⣿⣿⡆
⢿⡀⢀⡉⠳⣿⡿⢦⡌⠀⠀⠀⠀⠀⢀⡠⠤⡤⠬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⡀⠀⠈⣿⣿⣿⡇
⠈⢡⣟⠙⢦⣹⣇⡀⢧⠀⠀⠀⠀⣰⠛⢦⠀⡇⡠⠚⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⣿⣿⣿⠇
⠀⠀⠙⠲⠀⣿⡏⢹⠄⠀⠀⠀⠀⡷⠐⠆⡽⣿⠤⠄⢾⠀⠀⢠⢤⠠⣶⡦⠀⠀⠀⢰⣋⣀⡀⠀⢀⣀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠁⠀⠀⠀⠀⠀⠘⢤⠊⠀⡀⠑⣤⠎⠀⠀⠀⢀⡇⣿⣧⠀⠀⠀⠈⠉⠈⢉⠉⠉⢉⣉⡿⠿⡟⠛⠛⠛⣆⡀
⠀⠀⠀⠀⠀⢸⡇⠀⣀⡄⣰⢠⡄⠀⣠⠉⠛⠃⠋⠁⠀⠀⠀⠀⠀⠻⠿⠧⠀⠀⠀⠀⠀⠀⠹⠖⢺⡟⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠿⠃⠀⠛⠛⠛⠛⠛⠚⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣠⣴⣾⡀⠘⠛⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣋⣁⣀⣀⣘⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⠀⠀⠀⢀⠰⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⣤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⢨⠯⠨⣿⡿⢿⠀⠀⠀⠐⠛⠀⠀⣴⣶⣿⣿⣿⣋⡁⠀⠙⠳⣾⣧⣶⠄⠀⠠⠞⠃
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠖⠂⠀⠈⠶⣖⣿⡽⠋⠀⠀⠀⣴⣶⣦⠀⠁⠁⠙⠿⣿⠟⠁
⠀⠀⠀⠀⣀⣸⠿⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣈⣷⣧⣴⣿⠿
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢤⠀⢀⣦⡀⠀⠀⢠⡄⣠⣿⣿⣿⣿⣿⣄⢠⡄⠀⢾⢀⣴⣷⠀⠀⠉⠋⠛⠻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣾⣿⣿⣄⠀⠀⠉⢹⣿⣿⣿⣿⣿⡏⠉⢀⣀⣤⣿⣿⡏
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣷⠃⠀⢸⣿⣿⣿⣿⣿⡇⠀⢉⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣈⡼⣿⣿⣿⣿⣿⡄⠀⣾⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠄⢠⠀⣀⠀⠀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣧⢀⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⢸⠀⢸⡖⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢼⡇
⠀⠀⠈⢦⣸⣃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠙⣟
⠀⠀⠀⠀⢸⣇⢠⣴⣇⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠘⠯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣴⣶⣄⡁⢸⡇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⡿⠨⠏⠀⣿⠁

⠀⠰⡄⠰⣿⣿⣧⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⢀⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠲⣄
⠀⠀⠙⢿⣿⣿⣿⡟⠀⠀⠀⢠⠄⠒⠀⠠⠈⠁⠀⠀⠀⠀⠀⣴⠾⢇⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⣹
⠀⠀⠀⣾⡿⠙⠟⣧⢀⡀⠠⢝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠘⡂⠀⠀⠀⠈⠳⢤⣀⣀⡴⠋
⠀⠀⠀⠘⣿⣶⣶⣶⡿⢟⠂⣴⠟⠀⠀⢠⣤⡤⠀⠀⠠⣴⡇⠠
⣷⣦⠀⠀⢙⣿⣿⣿⣿⣿⡀⠛⠣⢀⣤⡟⠉⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⣿⣿⣿⣦⣴⣿⣿⣿⣿⡏⠁⠰⠀⡀⠓⠀⡀⠀⡔⠀⠀⠀⠹⠷⠤⠂⠐⠐⠆⠀⠀⢸⣇⠀⠀⢠⡇⢠⡿⣄
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡷⠊⠁⠈⠡⢀⡞⢀⣠⡷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠸⢃⡿⠹⠏
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣼⣷⣿⠯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⣿⠏
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⠟⠒⠀⢀⢿⣤⡀⣀⠀⠀⠀⢀⣠⡶⠀⠀⠀⠀⠤
⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⠄⠉⠛⠲⠒⠟⠉⠁⠀⠀⠀⠀⠀⠀⠂
⠀⣾⣿⣿⣿⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣺⠛⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⣾⣿⡿⣿⡏⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⡀⠈⢃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⡙⣧⠘⡿⢫⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⠉
⣿⣿⣿⡇⣿⠀⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠐⠇⠨⠚⠓⠀⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡀
⠿⠋⢿⠀⡿⠀⣸⣿⣿⣿⣿⡦⠀⠀⢠⠀⠀⠀⠀⠀⢀⣴⣞⢉⠀⡀⠀⠊⠀⠀⠀⢰⠀⢷⠹⠀⠀⠄
⠀⠀⣼⠀⠁⢀⣿⣿⢿⡿⠋⠁⢠⡀⠀⠀⠀⡀⠤⠀⠁⢸⣿⣿⣶⠃⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⢢⠄
⠀⠈⠀⣀⡶⠛⠉⠉⠀⠀⠀⠀⠁⠀⠀⢀⣿⠷⠺⣍⠶⡼⡜⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹
⢀⠀⢸⠛⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠐⣹⠷⢤⡤⣀⠈⢏⢣⣿⠁⠠⢀⠀⠀⠀⠀⠀⢸⣄⠅
⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄⢠⠈⠀⣾⡏⢿⡤⠃⠀⠀⠀⠠⠀⠀⠀⣿⡀⠀⢀⠀⠂
⠀⠈⠻⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠆⠃⠘⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢼
⠀⠀⠀⠀⠀⢸⣆⠀⣀⡀⠀⠀⠀⠀⣠⣤⡘⢧⠋⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⣲⡌⢏⡸⣿⣿⣶⡆⠀⢼⣿⣿⣷⣄⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡇
⠀⠀⠈⠀⠉⠫⠏⠁⠘⠿⠿⠟⠀⡦⢿⣇⣻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢀⢀
⠀⠀⠠⠀⠐⠂⠀⠀⠀⠀⣀⠄⢀⠀⠀⠉⢳⡌⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠦⣤⣽⡦⠈
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣿⣛⠪⠥⠔⡓⣈⣄⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁
⠀⠀⠀⠀⠀⠘⠀⠚⢡⣤⣀⣽⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠾⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠂⠀⠀⠋⠀⠀⠀⢈⣀⣀⡀⠀⠀⠀⣢⣤⠀⢀⠿⠉⣿⡿⣷
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⣠⡶⠂⠀⠀⣾⣿⣿⣿⠀⠀⠀⣹⡿⠄⠀⠲⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠛⠛⠋⠀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠚⠁⠁⢠⠄⠀⣠⡀⠀⠀⠀⠀⠠⢄⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⣾⣷⡄⠀⠀⠘⠧⢴⣿⣿⣿⣿⣿⣿⡦⠼⠃⠀⠘⣦⣾⣿⡇⠀⠀⠀⠀⠀⠀⢹⡄⡴
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡀⠀⠀⢩⣿⣿⣿⣦⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣤⣤⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⡀⠀⣘⠀⠀⢀⠅⢠⣤⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠐⡄⢸⡆
⠀⠀⠀⠀⠸⠁⠰⡏⢀⡴⠃⠀⢠⡴⢻⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⢧⡀⠀⠀⠀⠀⠀⠌⣌⣧⠀⡀
⠀⠀⠀⠀⣧⠀⢀⣷⠋⠀⣠⣀⣀⣀⣸⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⡀⠀⠀⡀⠀⠀⠀⠀⢹⣿⢠
⠀⠀⠀⠀⠻⣄⢸⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⡀⣿⣇⠔
⠀⠀⠀⠀⠀⠈⢻⣧⠊⣰⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠈⢉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢡⠐⣯⠇⠀⣿⠃
⠀⠀⠰⠄⠸⡈⠙⣿⠀⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠀⢰⣿⠀⠘
⠀⠀⠀⠀⠀⡇⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⢠⣀⡀⣧⠀⢻⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡗⠀⠀⣿⡇⠀⠀⣴
⠀⠀⠈⢳⠀⢹⣀⣸⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣧⣤⣬⣿⣃⡀⢠⠇
⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠋⠉⠉⢉⣩⡽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢠⡶⠋⠁
⠰⣖⡒⠒⠲⠦⢤⣠⠏⠀⠀⠀⠀⠈⡇
⠀⠀⠙⠓⠦⣭⡶⢤⡁⠀⠀⠠⣀⠀⣇
⠀⠀⠀⠀⠀⢨⣿⡆⠭⣷⣦⣄⡀⠉⠙⠓⠒⠶⠦⣤
⠀⠀⠀⠀⠀⡸⠉⠀⢳⡷⠍⠹⡏⠙⠲⡖⠶⠒⠋⠁
⠀⠀⠀⠀⠸⢀⣤⡀⠀⠀⠀⠀⣧⠀⠀⠙⢦⡀⠀⢀⣀⣤
⠀⠀⠀⠀⠲⠞⢈⡷⠿⠶⢀⣴⠇⠀⢀⠀⠀⣙⠛⢿⣟⠃
⠀⠀⠀⠀⠀⣴⣋⣀⣠⣶⣿⠃⠀⣐⡞⢷⠶⠶⠶⠲⢮⣷⣄⡀
⠀⠀⠀⠀⠀⠈⠉⠉⢠⢿⣧⡶⠛⠉⠀⠈⢳⡀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠉⢳⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⡇⠀⠀⢠⡟⠲⢤⣀
⠀⠀⠀⠀⠀⣠⠴⠋⠉⢳⣠⠞⠀⠀⠀⡾⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⣴⣞⠣⠀⠀⣀⡾⠋⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠹⠶⠦⠤⠤⣶
⣠⣤⣠⣏⣍⣙⣦⣖⢧⡀⡀⠀⠐⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠛⠲⢶⣿⣶⣾⣭⠟⠀⠙⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⢀⣴⣋⣀⣀⣀⣦⣤⣤⠴⠤⠶⠶⠶⠶⠶⠶⣄⣀⡀
⠀⠀⠈⠉⠉⠉⠙⠛⠖⠓⣿⠁⠀⠀⠀⠀⠀⢀⣀⡤⠪⠤⠄⢀⡀⠘⣷⣬⣭⣽⡇⢸⡇⠀⠀⠤⠄⠀⠀⠀⠀⠂⠈⠉⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡝⠛⠉⠙⠒⢦⣄⣀⣠⣴⣪⣤⠤⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣀⣀⣀⣀⡀⢀⣀⠠⠄⠀⠈⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠙⠲⢤⡀⠀⠀⠀⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣾⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠖⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡛⠛⠿⣟⣶⡄⡄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣷⣦⣤⣭⣉⣉⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠋⠙⠛⢿⣍⣭⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣘⣼⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣻⡁⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⢰⣿⡇⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠠⠄⠒⠀⠀⠀⠀⣠⠰⠂⠀⠀⣼⠋⠀⠀⠀⣀⣿⠻⣆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣫⡟⠀⠈⢷⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣤⠞⠁⠀⣠⡴⠟⠁⠀⠀⠀⠀⠤⠾⠛⠷⠶⣤⣀⣹⣤⣾⠛⢶⣤⣤⠴⠶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣍⣿⠀⣀⣴⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠼⣧⣤⡿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⢰⣛⠓⠒⢒⠶⣶⠶⣤⣴⡏⠈⣿⣻⣿⣿⠿⠷⣴⡆⠀⠀⢀⢀⣠⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠉⣉⣉⣭⣿⡶⣤⠿⠀⠀⠀⠀⠀⠈⣹⡿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⢼⡇⠐⠂⠀⠋⣩⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠸⣟⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠶⣶⣤⣄⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠁⢿⠀⠉⠛⠛⠳⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠠⠂⠈⠐
⠀⠀⠀⡀
⠀⠀⠁⠉
⠀⠀⠀⡴⠛⠂
⠀⠀⠸⡇⡀
⠀⠀⠀⡟⠙⠁
⠀⠀⠀⠳⡀⢠⣤⣣⠤⢄
⠀⠀⢠⠞⠛⠉⠛⢡⡄⠀⢣
⠀⠀⢿⠀⠘⣃⣀⣬⣵⠇⡸
⠀⠀⣘⣦⡀⠉⣛⣉⣡⣜⣡⡄
⠀⡞⠁⠀⣈⠹⠋⠉⠛⠉⠳⣄
⢸⡃⢀⡰⠿⠀⠀⠰⣷⠀⠀⢹
⠀⢧⠈⠻⣾⣿⢧⣶⡶⠀⠀⣼⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⣠⠳⣴⣄⣤⣀⣠⣀⣤⠾⠉⠁⠉⠳⡄⣰⠀⠀⠰⠖⣤⣦⣸⣮⣥⣀
⠀⣬⡟⠁⠀⠃⠀⠈⠀⠇⢠⣄⠀⠀⠀⣹⡎⣠⠚⠉⠙⠻⠟⠉⠙⠃⠉⠛⣆
⠀⡟⠀⠀⠀⣾⣦⠀⠀⠀⠸⠿⢃⠔⠓⠻⣿⡿⣤⣤⣴⡆⠀⠀⢰⣧⠀⠀⠘⡆
⠀⣧⠀⠀⠀⣨⣥⣤⣰⣶⣾⣧⡏⠀⢠⣶⠈⠀⣀⠁⠈⢧⣤⣄⣴⣶⠂⠀⢸⠃
⠀⠙⢧⠀⠀⠈⠛⠻⠿⠙⠛⠋⣣⠈⢷⣵⣤⣠⣍⠁⠀⡸⠛⠉⠉⡁⣀⡴⢃⢠⠀⠀⠀⠐
⠀⠀⠈⠙⠶⣤⣄⣦⣤⡼⠷⠶⡻⠳⢤⣉⣉⣉⢁⣀⠴⠛⠚⠛⠋⠉⠁⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠙⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠛⠛⠻⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣶⣿⣿⣿⣿⣟⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⡟⠛⠛⣋⡙⠛⣿⠿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣟⡿⠛⠷⠀⠀⣈⠀⠀⣄⣸⣦⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⠙⠻⡁⠀⠀⠀⠰⠍⠁⠀⠭⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⠶⠒⠒⠀⠉⠀⠀⠀⠈⠡⡀⠈⠢⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢷⣤⣽⠃⠄⠀⠀⠠⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⠘⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠷⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠞⠉⠷⣼⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣹⣦⣀⣀⣤⠶⠋⠀⠀⠀⠀⠐⠀⢸⠀⠛⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠛⠋⠛⢿⠏⠁⠀⠀⠀⠀⠰⠷⠀⣀⣾⣆⠀⢀⠀⣰⣌⣻⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣹⣿⡟⢃⣤⠴⠛⠋⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⠀⠈⠀⠉⠁⡀⠄⠀⢶⣿⣷⣆⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢻⠇⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠘⠀⡄⢀⣰⣶⣿⡀⠀⠀⠀⣾⡟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣥⣴⣿⣶⣿⣿⣿⡿⢷⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢤⡞⠀⠀⠀⠴⠀⠀⠀⠀⠀⢻⣿⣿⣷⢻⣿⣿⣗⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡜⡇⠀⠀⠘⠀⠀⠀⠀⢰⠀⠀⠻⣿⣿⡎⢿⣿⣿⡀⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⢀⠃⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠛⡇⠀⢿⣿⣿⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢿⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣷⣿⠀⠀⠀⢰⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢹⣿⡟⠀⠀⠀⡇⠀⠀⢀⡀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⠀⡇⠀⠀⣸⠁⠄⠀⣾⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡧
⠀⠀⠀⠀⠠⠀⠀⢈⣿⠁⣿⣼⡇⠀⠀⡹⣷⡶⢠⣿⣇⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⡇⠀⢰⡇⢾⡇⣼⣿⡗⠀⢺⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠠⠀⠀⢤⡤⢸⡇⢠⣿⣿⡇⠀⢸⣧⣿⣿⣿⣿⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠈⢙⣥⠴⠧⣸⡇⢸⡏⣿⡇⠀⠘⢻⣿⣿⣿⣏⡀⣸⡟⠀⠂⠀⣰⣶⣶⣾⣇⠙⠀⠆⠀⡀⠈⠁
⠀⠀⠀⡤⠴⣾⠶⣿⡇⢸⡇⣿⣧⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠄⢒⠄⠀⠀⢀
⠀⠀⠀⣱⣶⣬⣷⣿⡇⢸⡇⢸⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡏⢀⣤⠀⠤⠀⠬⠐⠘⡁⠂⢖
⠀⠐⣶⣿⣿⣿⣿⣿⣿⣾⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣷⣽⢧⡉⠀⠀⠀⠀⠀⠁⠐⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠄⡀⠉⢉⡘⠻⠗⠂
⠀⠀⠀⠀⠙⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠿⠻⠶⠀⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢿⠇⣦⠹⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠄⠀⠉⠀⠛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⣿⣿⣄⡰⠆⠈⠀⣿⣿⣿⣿⣾⣿⡶⣄⣦⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣿⣿⣷⣶⣾⠀⣿⣿⣿⣿⣾⣿⠷⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⢹⣿⢹⣿⣿⠟⣴⣿⡿⠀⠉⠉⡷⡀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣞⣷⠋⡿⢃⠀⢹⣿⡄⠀⠠⠀⢠⣿⣦⡀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠻⠁⢈⡅⢸⠀⠀⢿⡇⠀⣠⣴⣿⣿⣿⣿⣄⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣸⡀⠈⢷⣶⣾⣣⡜⣽⣿⣿⣿⣿⣿⡿⡃⢤⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⢿⣿⣏⣼⣿⣿⣿⣿⣿⣿⠿⣿⣪⡍⠠⡤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⡤⠁⣼⣿⣽⣟⣿⡿⠿⣯⠀⠈⠡⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⠰⠇⠰⣿⣿⣿⣿⢟⡷⢸⠉⠇⠀⠀⣄⡁⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡿⣿⣿⣠⣿⡜⢣⠟⠀⠀⠀⠀⢹⣿⣎⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡐⠉⣾⣿⣴⡼⣿⠁⠈⠁⠈⡴⠃⠀⠀⠰⠈⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢧⡄⢿⣁⠟⣹⡟⠄⠀⠀⠀⠁⠐⠀⠀⠀⢀⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣤⣠⣼⣿⢀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣶⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⣿⣈⣽⡇⡸⣇⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣽⢿⠀⣇⠸⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿⣿⡏⢿⣘⡁⢘⠀⣼⠀⠀⠀⠀⠀⠀⠐⠁⡌⠁⠘⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⢾⣿⣯⣽⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠙⡄⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣼⣿⣿⡇⢈⡿⡟⠁⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⣦⣿⣿⢻⣿⣿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣌⡷⠀⢿⣿⣿⡏⠀⠀⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣿⡇⢸⣿⣿⢸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⠁⠀⠀⠘⠛⠁⠀⠀⠀⠙⠘⠧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿⠇⣸⣿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠛⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⣶⣦⣤⡧⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⣀⡀⠉⠉⢻⣷⡤⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠙⠟⠻⣠⣿⣿⣿⣏⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢰⣿⡿⠉⣯⣍⣩⣬⠀⠹⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣴⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣧⣤⣭⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣤⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⢹⣟⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢿⣷⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣏⠁⠀⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠷⠾⠉⠛⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢻⡆⣹⡧⢿⣿⠁⣿⡇⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠻⠁⣾⣿⠀⣿⠁⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠈⢰⣾⡾⠛⢿⣼⣿⣿⡿⠃⠜⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠘⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⢻⡟⠃⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⢶
⠀⠀⠀⠀⠀⠀⠠⠛⢿⡿⢿⡇⠀⠀⠀⣻⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⢀⡼
⠀⢀⣤⡶⠀⣤⣶⣦⣾⣷⡆⠀⠀⠀⠀⠉⣼⣧⣶⣤⢰⣦⡀⠀⠀⡜⠀⠀⠀⠀⠀⢫⡀⠀⢾⡀⠀⠀⢰⠏⠁⠀⠀⣀⠆⠀⠀⠈⠀⣴
⢰⣿⡟⣿⡛⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠋⡽⠹⠿⠀⠀⢡⠘⣶⣄⠀⢠⣾⡟⠀⠈⢻⣄⡄⠂⠀⠀⠀⠊⠁⠀⠀⠀⢰⠘⠿⠃
⠘⠉⠗⠛⠏⠀⠀⠈⠀⠀⠀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⠛⠀⠀⣩⠀⠀⠀⠈⠀⠀⣀⠀⣠⠤⠴⠃⠀⠀⠀⠀⢠⢤⡟
⠀⠀⠀⠀⠀⡀⠀⠐⣮⣤⣾⡙⠿⠭⢖⠀⠲⢄⠀⠀⠀⠤⠀⠐⠒⠛⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⣾⠷⠦⠇⠀⣀⠀⠀⠀⠀⠀⠌⠘⡇
⠀⠀⠀⢀⠎⠀⡴⠁⠀⠌⠁⠀⠀⠀⠀⠀⠆⠈⢳⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⠄⠈⡉⠓⠀⠀⠀⠀⠀⠄⡥
⠀⠀⠀⡞⠀⣸⠁⠻⣦⡀⠀⠀⠀⣀⣴⠇⠸⡆⠈⡇⠀⠀⢦⣴⣄⠀⣿⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⡇
⠀⠀⠀⣧⠀⡏⠀⠀⠽⠋⣰⣆⠈⠻⠟⠀⠀⠇⠀⡇⠀⠀⠀⠀⠈⠛⠋⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣤⠴⠂⠀⣀⠀⠀⠀⠺⠆⡤⠭⠂
⠀⠀⠀⠘⢶⣷⡀⠈⢦⣦⣀⣄⣠⣤⠞⠀⢀⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⢀⣀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠈⠙⠺⠧⢀⡋⠈⠃⢠⣤⠤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠈⠙⠈⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢠⣖⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣦⣼⣷⠀⠈⠛⢿⣿⣿⣿⣿⣿⣶⡿
⠀⢀⠂⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠁⠐⢦⡀⠀⠀⠀⠀⠀⣶⣾⣿⡿⣿⣿⣇⠀⠀⠀⣾⡿⠋⢼⣿⡇
⠀⣼⠀⠀⠀⡀⣽⣷⠀⣿⡇⢠⡆⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢈⢿⣯⠏⣿⣿⣔⣀⣱⣿⣇⣀⣠⣿⣴⣂⡀
⠀⢧⠀⠀⡌⠁⠻⠇⣦⠙⠃⠀⠡⠀⠀⠀⡤⢻⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠠⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡏⠀⣀
⠀⠈⠳⠬⢴⠆⣀⡄⣄⣐⠂⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡹⣾⡇⠀⠀⠀⢸⣿⣧⢸⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣾⣿⣿⣶⣿⠃⡀
⠀⠀⠀⠀⢠⠠⠍⠙⠭⠭⠠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣶⣾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⢻⠉⠙⠀⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣯⠀⣠⣄⡀⠀⠀⠀⣠⣽⡇⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣿⣿⠟⠿⣿⣿⢸⣿⡟⠀⣼⠃⠀⡇⢀⠤
⠀⠀⠀⠀⢀⣀⠀⢀⡜⠉⠉⠉⠀⢄⠀⠀⠀⠀⠠⠈⠉⠀⡈⠻⠛⠉⡁⠀⠀⠀⠀⠀⠀⠀⢻⣷⢻⣿⣓⣒⡂⡹⣿⣼⣿⣿⣷⣸⡗⠒⣡⡟
⠀⠀⠀⢰⢋⠈⠳⢊⣀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⣿⣿⡇⠂⠂⠀⠂⢿⣿⣿⣇⡘⠟⠷⡖⠁⠀⠍
⠀⣴⢤⡞⠉⢷⡼⠋⢩⣿⣤⣠⡤⢀⡀⡐⠙⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣿⣿⣧⣤⣶⣶⣒⠿⢿⣿⣿⣷⡀⠀⠈
⠈⠁⠈⠀⠀⠀⢀⣤⣿⣏⡀⠈⣥⣈⢤⠉⠹⠖⢛⣋⣥⡄⠹⠛⠛⠂⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠗⣠⣼⣿⣿⣇
⠀⠲⠴⣾⣿⣉⠉⠉⠉⣩⠉⠉⠉⠁⣠⣀⡠⠄⠠⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⣿⣿⣿⣿⣿⣿⣭⣐⣀⡀⠉⠉⠛⠻⣿⣿⣧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠋⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⣀⣾⣿⣷⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠁
⠀⠀⠀⢀⣠⣼⢟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠈⠑⠽⣷⣿⣯⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠬⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠳⠆⡟⢛⣱⡞⣿⣿⣿⣿⣿⣷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⡦⠾⣯⡾⠀⣿⣿⠛⠛⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢹⣿⣷⣶⣿⣷⠚⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢠⣴⣿⣻⣾⣿⣿⣿⣿⣏⣾⣽⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣣⣤⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠙⢿⣿⡿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠈⢻⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠀⣸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢰⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡇⠀⣄⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⢸⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢰⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣧⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠁⢰⣾⡂⠀⠀⠀⠀⠀⢀⠀⠀⠢⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣧⠀⠀⢠⣶⣶⠾⠋⠁
⠀⠀⠀⠀⠀⠰⣤⣷⣴⣿⣧⣶⠀⠀⠀⡀⠈⠀⡰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡋
⠀⠀⠀⠀⠀⢰⣎⣿⣿⣿⣿⣿⣆⠀⣀⣿⣾⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠣
⠀⠀⠀⠈⠀⠨⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣆⡀⣀
⠀⠀⠀⠄⠰⠋⠤⢿⣿⣻⣿⣛⣻⡿⠟⠛⠛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠿⠿⠃
⠀⢀⡄⠀⠀⠀⡲⢟⣷⣾⢿⡟⠿⠟⢿⣿⠿⢿⠿⠿⠿⠿⣿⣿⣏⡽⠛⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⠿⡿⠟⣿⣿⢿⣿⠻⠿⢿⣽⠿⣻⢶⡆⠘
⠀⠀⢀⠀⣤⠾⠿⠗⠋⠙⠛⠛⠤⠟⢛⣭⣥⣌⢁⡒⠲⠖⢻⡟⠃⠀⠀⠚⠍⠉⠙⠛⠉⠉⠈⠁⠒⠂⠘⠛⠛⠋⠹⠶⠘⠗⠒⠂⠀⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠂⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢸⡇⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣹⣧⠀⠀⢸⣇⣤⣶⠿⣟⡛⠛⠭⣀⠀⠉⠻⢍⠛⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⢛⣛⡿⠿⠿⣿⣿⣷⣤⣾⣿⠋⠀⠀⠀⠙⣆⠀⠀⠑⢦⠀⠈⢳⡀⠹⣿⡄
⠀⠀⠀⣠⣾⠟⠋⠀⢀⡠⠞⠉⣠⠔⢊⡽⠋⠉⣸⠛⠁⠈⢧⠀⠀⠀⠀⠘⡄⠀⠀⠀⠱⡄⠀⢱⡀⠹⣿
⠀⠀⣼⡿⠃⠀⠀⣠⠏⠀⣠⢞⣁⣤⣏⣀⠀⢀⡇⠀⠀⠀⠘⡆⢀⡶⠛⠛⠻⣆⠀⠀⠀⠘⡄⠀⢧⠀⣿
⠀⣼⡿⠁⠀⠀⣴⠃⠀⡴⢡⡟⠉⠀⠀⠉⣷⢸⠀⢀⣀⠀⠀⢹⣸⣇⣀⣀⣀⣽⡆⠀⠀⠀⠸⡄⠘⡆⢸
⢸⣿⠃⠀⠀⢰⠇⠀⢰⠃⠈⢿⣦⣤⠴⠶⠛⢸⢀⣾⠛⣦⠀⠈⡏⠉⠉⠉⠛⢻⠁⠀⠀⠀⠀⢳⠀⢳⢸
⣿⡟⠀⠀⠀⡿⠀⠀⡟⠀⠀⠀⣼⠀⠀⠀⠀⢸⣸⣇⣀⣼⡇⠀⢻⠀⠀⠀⠀⠈⡆⠀⠀⣀⣤⠸⡄⢸⣾
⣿⡇⠀⠀⠀⡇⠀⢀⡇⣤⣤⣤⣿⣄⣀⠀⠀⣸⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⣀⣷⣾⠿⠋⣿⠀⡇⠸⣿
⣿⣧⠀⠀⠀⡇⠀⢸⡇⢿⡇⠉⠉⢩⣿⠀⢸⠟⠿⠿⠶⠶⠶⠶⠿⢿⡆⠀⢀⣿⠉⠀⠀⢀⣿⠀⡇⢠⣿
⠸⣿⡄⠀⠀⡇⠀⠸⡇⠸⣧⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠋⠀⠀⣠⡾⠃⢀⡇⣸⣿
⠀⢻⣷⡀⠀⣧⠀⠀⣧⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠀⠀⢸⢡⣿⠇
⠀⠀⠻⣿⣄⢻⡀⠀⢹⡀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠋⡏⠀⠀⢀⣿⣿⠏
⠀⠀⠀⠈⠻⣿⣧⠀⠀⢷⠀⠀⠘⡏⠙⠛⢷⣶⣶⣶⣶⣶⠶⠾⠟⣿⠉⠀⠀⠀⢰⠇⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣌⣳⡀⠀⠹⡄⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡟⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⣝⣦⡀⠀⠀⢳⡀⠀⠀⠀⢠⡇⠀⣀⣴⣿⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣷⣶⣶⣶⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠉⠉⠉⠙⣿⠀⠀⠀⠀⢸⡋⠉⠉⠉⠉⠉⠙⠋⠙⣆
⠀⠀⠀⠀⠀⠀⣀⢠⣤⠄⠰⢿⣤⣤⣤⣤⣤⣤⣤⣤⣶⠄⠀⠤⠄⢼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⠷⠂⢤⣤⣀⡀
⠀⠀⢀⡠⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣄
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⠠⠤⠤⠤⢤⣤⠤⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦
⠀⠓⠒⠒⠒⠒⠂⣿⣯⡟⠉⠉⢉⣭⡝⠛⠉⠀⠀⠀⠀⠒⠀⠀⠈⠙⠒⣭⡽⠛⢉⣉⣉⣉⣤⣤⣶⣶⣶⠲⣤⠀⢠⣤⣤⣤⡼
⠀⠀⠀⠀⠀⠀⣰⡿⠉⢙⠛⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣃⣀⠋⠀⠀⠀⠀⠉⠉⠉⠀⢴⠄⠉⠙⣧⠈⢣⡞⣀
⠀⠀⠀⠀⠀⢠⡿⠃⠀⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣆⠀⣼⠃⡀
⠀⠀⠀⠀⢀⡌⢳⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⢸⠀⠇
⠀⠀⠀⠀⠸⠃⢼⠀⢸⡁⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⡆⢸⣇⢸⡇
⠀⠀⠀⠀⠀⣠⠀⠀⣸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠘⡇⠘⣿⠀⡇
⠀⠀⠀⠀⢹⣏⣠⠖⠛⣁⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣼⢇⡀⢻⣦⡀
⠀⠀⠀⠀⠈⢹⣟⠀⠀⣿⡀⢠⡏⠓⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⠉⡇⠀⢠⣿⠈⠛⣶⡏⠁
⠀⠀⠀⠀⠀⢸⣿⠀⢀⠈⢧⣸⡇⠀⠀⠈⠉⣶⠶⠦⢀⣀⣠⣴⣶⣄⠀⠴⠶⢾⠋⠀⠀⠀⠀⡇⢀⠸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢠⡆⣿⠀⠸⠃⣼⠉⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄⠀⠀⠀⠀⠳⢿⠀⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢻⠋⣰⠇⣰⠁⠀⠀⠀⠀⢀⣤⡾⠉⠒⠶⠆⢠⣤⣄⠀⠲⠖⠉⠉⣷⠒⠲⢦⡀⠀⠀⣦⠐⡆⠀⣼⡇
⠀⠀⠀⠀⠀⠈⠙⢷⡾⠃⠀⠀⠀⣠⠞⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠙⢦⠀⠙⣆⣸⣦⡯
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⢲⠟⠁⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⠈⢳⣀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⡰⠎⠁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠘⣻⠦⢤⣀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡠⡤⠚⠿⣉
⢀⡴⠊⠁⠀⠀⠉⠑⠦⣄⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠉⠁⠀⠀⠀⠙⢶⡄
⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⡀⠈
⠀⢹⣅⣶⠂⣠⡀⢀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⡀⢠⣄⢹⡏
⠀⠀⠀⠿⡰⠿⣀⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣷⣀⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠈⠳⠶⠤⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢹⡧⢤⡤⠤⡶⢾⡟⠚⠒⠒⢿⡷⠶⠶⠶⠶⣾⡇⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⠤⠤⡤⢼⡇⠀⠀⠀⢾⣦⣤⣤⣤⠤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠒⠒⠚⠓⢺⣇⠀⠀⠀⣼⡏⠛⠛⠛⠓⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠶⠄⠘⠋⢹⠀⠀⠀⠁⠈⠛⠀⠐⠒⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠀⠀⠀⠀⠀⣠⡄⠀⢠⡇⠀⠀⠀⣀⠀⠀⣤⣄⠀⠀⠀⠀⠘⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⣀⣀⡀⠐⠒⠒⠈⠉⠑⠒⠚⠁⠀⠀⠀⠈⠁⠚⠁⠉⠑⠒⠒⢀⣀⣤⣤⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠆

⠀⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⠄
⠀⠉⠛⠓⠀⠀⠄⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠠⣀⣏⣠⣤⡇⠀⠀⠈
⠀⠀⠀⠀⠀⣴⡖⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣍⡓⠴⠒⠲⡼⣥⡈⣷⡄⠀⢠
⠀⠀⠀⠀⣶⣿⡇⣿⣛⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⣊⠀⠀⠙⠿⣷⣴⠟
⠀⠀⠀⠘⣿⣿⣇⣽⣿⡏⣿⣿⣿⡇⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣚⣉⣩⡿⠟⠷⣶⣿⣏
⠀⠀⢦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠝⠉⠉⠉⠉⠀⠀⠤⣄⣻⣿⣆
⠀⠉⢺⠃⣹⣯⣽⣿⠿⣿⣯⣿⣿⢱⠞⠠⠄⠀⠀⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠂⠀⠀⠀⠀⠀⢿⣿
⠀⣀⣼⣯⣼⣿⣿⣿⣶⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⣾⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣦⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡏⣰⢂⡤⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⢠⣀⣈⣁⣤⢀⢀⣽
⣿⣿⡛⠿⠽⠶⢿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢿⣿⣷⣶⣛⠛⠖⠲⠂⣠⣶⣶⣾⣤⡀⣀⣴⡀⢈⣙⣿⠉⣥⡀⣠⣄⢀⣠⣴⣤⣭⣭⣤⣴⣶⣾⣿⡛⣿⣿⠿⣷⣾
⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣧⣼⣿⣻⣷⡾⢿⣿⣿⣿⣿⣷⣶⣤⣟⠳⡞⣿⣭⣿⣀⣏⣁⣹⣿⣿⣤⣼⣦⣿⣿⣿⣭⣿⣯⡽⣿⣿⣿⣿⣿⣍⠻⠯⠿⠿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣉⠋⣉⣉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠻⣝⠛⠒⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠚⠉⠉⠀⣠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣄⣼⠠⡀⢠⠏⠀⠀⠀⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⡤⠶⣻⠫⡀⠀⠀⠈⠓⠢⠤⠤⠤⠤⠤⠤⢖⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⡿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠉⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣼⡄⡏⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠉⠀⠀⣀⡤⣾⠀⠀⠀⠀⠀⠂⢤⣀⡀⠀⠀⣠⠖⠁⠀⠸
⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⠀⠘⢷⣦⠶⠋⠁⠀⠀⠠⠔⠒⠋⠉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡽⡿⠁
⠀⠈⠉⠛⠲⡦⣤⣀⠀⠀⣠⠀⠀⠹⣇⣸⣧⠀⠀⣰⡟⠉⠳⣤⠤⠦⠤⠶⠒⠂⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⡤⠒⠁⢸⠃
⠀⠀⠀⠀⠀⠈⢢⡈⠙⠛⠻⣶⢶⣶⣿⣿⣿⣿⣾⠟⠀⠀⣀⣌⣷⣤⠤⢤⠤⠴⠃⠀⢠⡀⠀⠀⣉⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⢷⣞⣛⡯⠋⠁⣠⠈⠶⠤⢤⠀⢳⣄⡤⠽⠖⠉⠀⢀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡴⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⡁⡀⠀⠉⣍⠙⢻⢓⣶⣾⣾⠾⠯⠥⠦⠤⣼⣄⣀⣀⣘⣆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡠⠞⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠉⢷⣸⠀⠀⡼⠀⢈⠈⢦⠏⠉⡄⢀⡁⠠⣴⡁⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠲⠿⠦⠤⣤⡀⣀⡙⠢⣄
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⢸⠇⠀⢀⣿⣆⠁⠀⠀⠀⣷⠌⡀⠀⠙⡉⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⡠⠒⠁⠀⠈⠃⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⠠⠚⣿⠶⢄⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠰⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾
⠀⠀⠀⠀⠀⣀⣾⣣⠤⠶⠏⠑⠠⠄⡏⠀⠀⠁⠀⠀⠀⠀⠀⢻⡾⠋⠀⠀⠀⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠟⠳⢤⣀⠀⠀⠀⠀⢠
⠀⠀⠐⠚⠛⠉⠉⠛⠛⠛⠓⠤⡀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠸⡄⠀⠀⠀⠤⠒⠒⠒⠲⢦⣀⣤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢾⡏⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣠⠏⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⡇⢠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⢿⡀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠠⠤⣧⡌⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⢰⠀⡛⣒⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢴⠖⠈⠈⠁⣿⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⣃⣀⣀⣀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠠⠄⣿⡀⠀⠀⠀⠀⠀⠀⢸⣀⠀⡀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⢸⡇⠀⠀⠀⠀⠀⠀⢠⠠⠄⠄⠌⠀⢹⡆
⠀⠀⠀⠀⠀⠈⠁⠁⠁⠈⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⠀⠀⠀⠀⠄⠰⢀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⡆⠂⠢⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⣦⠀⠸⠁⠀⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⣉⣩⡤⠿⠀⠀⠀⠀⠀⠀⢠⡶⡤⣤⠤⣤⠤⣴⣧⠀⠀⠀⠀⠀⠀⠈⠠⡀⣨⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣻⡿⠟⠛⠛⠛⠻⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠛⠛⠶⠶⢾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠉⠀⠀⠀⠀⢀⣀⣤⣤⣶⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡦⠔⠀⠀⠀⠀⠀⠀⣠⡀⠙⢿⣿⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠟⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣆⠀⠀⠈⠀⠀⠈⢻⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⣿⣿⡇⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⡀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣿
⣿⣿⡇⢠⡄⢀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣥⣄⢀⣶⣿⡿⣿⣿⣿⢷⡄⠀⣠⣤⣼⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⣿⣿⣷⡄⠀⠠⠁⠀⠀⠀⣴⣟⠛⣿⣿⢛⣙⣿⣿⡿⣿⣤⣼⡟⣧⣼⣿⣾⣿⣏⣙⣿⣟⣹⣷⡄⠀⠰⠆⠀⠀⠘⣿
⣿⣿⣿⣿⣄⠀⠀⠀⠉⠀⣿⡙⠿⠿⠿⠛⢋⣡⣿⣷⡈⡙⠻⠛⠟⠋⣰⣿⣏⡛⠻⠿⠿⢛⣽⡇⠀⠀⠃⠀⠀⣠⣿
⣿⣿⣿⣿⣿⣷⣦⣤⣤⣥⣽⣿⣷⣶⣶⣾⣿⣿⣯⣿⣿⣷⣶⣴⣶⣿⣿⣼⣿⣿⣷⣶⣾⣿⣿⣶⣶⣤⣴⣶⣿⣿⣿
⣿⣿⣯⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⠉⠉⢉⣩⡽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠀⠀⠀⢀⣠⠞⠉
⠀⠀⢀⣤⣤⣤⣀⣀⡀⠀⠀⣴⠋⠀⠀⠀⠐⠚⡇
⠀⠀⠈⠙⠶⣤⡀⠺⢍⣙⠺⠧⠀⠀⢀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠉⠛⢾⣏⠳⢦⣄⡀⠈⠓⠦⣄⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠶⠮⣝⡶⣤⣀⡀⠉⠉⠙⠛⠒⢒⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠹⣿⠢⠉⢹⡩⠙⠛⣞⠒⠓⠚⠉
⠀⠀⠀⠀⠀⠀⢀⡎⠀⣀⢀⠀⠀⠀⠀⠈⡇⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⣥⠞⠙⣇⣴⣄⡀⠀⣠⠇⠀⠀⠀⢀⠙⠲⢶⣞⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠉⠀⢀⣠⣾⠏⠀⢀⢰⣆⢀⠉⣁⣂⣈⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠞⢩⣿⠋⠠⣀⣴⠟⠘⣯⠉⠉⠀⠈⠉⠳⢝⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣯⣴⡋⠁⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠈⠈⢷⡀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⢸⡏⠀⠀⠀⣼⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⡟⢷⡄⢤⠿⠀⠀⠀⢠⠏⠀⠀⢈⠉⠳⠦⣄
⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⠀⢈⡿⠏⠀⣀⠀⢀⡾⠀⠀⠀⠀⠀⠄⠀⠙⣦⣀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢸⠿⠦⣝⡆⢀⣶⠛⠉⠀⢀⡀⢁⡾⠁⠐⠀⠅⠀⠀⠀⠀⠀⢉⠉⠉⠋⠉⣹⠇
⠀⠀⡴⠶⠶⣏⡶⠶⢬⣿⣭⠙⣦⠱⡀⠀⣨⠿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⢠⠏
⠀⠀⠉⠛⠓⣟⣷⣲⢲⡶⢋⡀⠀⢹⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⡁⣠⠞⠀⢈⠠⣶⣏⣀⣀⣀⣐⡷⢦⣤⠴⠶⠶⠛⠳⠛⠛⠒⠒⠲⣤⣤⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠶⠓⠛⡟⠀⠀⠀⠀⠂⠐⠀⣀⣐⣤⣾⣭⣤⣦⣄⡀⠈⣷⣤⣤⣤⣽⡆⢸⣿⠀⠀⠉⠤⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⠊⠻⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣏⠸⠗⠛⠛⠲⠶⣄⡀⠀⠠⣤⡖⢂⣀⣀⣠⡶⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠠⢀⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠒⠒⠶⢾⣞⡒⠙⠋⠀⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡀⠀⠀⣼⠻⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠀⣀⣠⡞⠕⢒⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠉⠉⠈⣬⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡜⣖⠤⣄⡀⠉⠁⠠⢄⠀⠀⠀⠀⠀⣀⡤⠶⠋⠁⠀⢦⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⠀⢸⠦⡀⠁⠉⠂⠠⠄⢵⣤⠤⠐⠊⡁⠀⠀⡀⡄⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡺⣱⢶⡜⠒⠤⢤⣀⠀⠀⠀⢉⡙⠫⣙⠒⠄⠙⠀⠀⠂⠿⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣣⣾⠃⢰⡴⠀⠀⡀⠀⠰⠄⠀⠁⠸⡗⢴⣾⠀⢀⣤⣀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣤⣾⡉⠙⠡⡀⠁⡀⠁⠊⠀⠀⠀⢠⢰⡄⠑⠌⡏⠀⠀⠁⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠈⡶⠤⣀⠈⠂⢄⡐⠤⡀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠥⠟⢀⡰⠼
⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⣿⣿⣿⡡⣪⠤⠤⠄⠹⠦⠦⠌⡓⢄⣄⠀⠀⠘⠎⠁⠀⢠⡠⠀⣴⠉
⢀⠀⠀⠀⠀⡀⠀⢑⢂⡀⣻⣻⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⢦⣄⣠⣀⡤⠓⠒⠛
⠀⠀⠀⠀⠈⠉⠈⠈⠙⠀⠓⠓⠉⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢢⡉⠀⣤⣀⣠⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡛⢿⡇⣭⣤⣄⡉⠭⣭⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠈⠗⢦⡀⠀⠉⠓⠦⢤⣩⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣄⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⢠⡀⠘⢆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡓⠲⣄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠁⠀⠈⠑⣄⠀⠀⠀⢤⠀⠀⠀⢀⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢘⠴⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣀⣀
⠀⠄⠀⠀⠀⢀⣀⣰⣖⣀⣁⡀⠉⠉⠉⠉⠑⠒⠀⠀⠀⠀⠤⠤⣄⡀⠀⣀⣀⢀⡀
⠸⣄⠀⠀⠀⠈⠽⠟⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠀⠉⠃⠀⠀⠀⠀⣶
⠀⡉⠓⠢⣄⣀⣀⣤⣄⣠⣄⣤⣤⣄⣀⣀⠀⢀⡀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⢀⠁⠀⠀⠘⠀⠉⠄⣍⡹⢿⠻⠿⠟⠋⠍⠍⠉⠙⢩⣿⠋⠉⠙⠋⠛⠛⠟⠟⠛⠛⠒⠚⠋⠀⢹⣿⣿⡇
⠀⣀⠀⠀⢸⣦⣀⣼⡛⣧⢨⡀⠠⢀⠂⠀⠀⢰⣰⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢦⣀⠄⢸⣿⡿⠁
⠀⠈⠓⢄⣸⣿⡿⡏⣯⣿⣬⣼⢠⡤⠦⣶⣿⠻⢿⡾⢻⢼⠿⠷⢶⡤⠄⣄⣤⡤⢹⢿⠋⡿⠁⢸⡟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠶⢺⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⢆⢰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠴⢒⡾⢧⠀⣀⣘⣷⣿⡿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⢿⡥⠼⣿⣿⣿⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠞⠁⠀⣼⣷⠶⠦⠤⠤⠤⠤⠶⠶⣺⠿⣆⠀⠀⠈⠳⡄⠉⠉⡇⢹⢄
⠀⠀⠀⠓⠲⢤⣀⡤⠊⠁⠀⠀⣀⡴⠋⠁⠀⣠⠜⠁⢈⣷⡤⣤⣤⣄⣀⣤⣴⡋⠀⠘⢷⡀⠀⠀⠘⠧⣄⠁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⡏⠉⠛⢶⡶⣏⣁⠀⣀⡴⠞⠁⠀⣠⠞⠙⣧⠀⠀⠀⢀⡾⠉⠑⢤⡀⠀⠙⠦⣄⡀⢀⣈⣳⣤⠤⠤⠶⢻⡖⠒⠚⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠉⢻⡟⠦⣤⣴⣞⠁⠀⠀⠘⣧⠀⢠⡟⠁⠀⠀⠀⣙⣦⡤⠶⣾⠏⠉⠁⣼⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠇⠀⢸⡏⠉⠛⠶⢦⣄⣼⠷⠛⣦⡤⠖⠛⠉⣹⠋⠀⣰⠏⠀⠀⡼⠃⠀⠀⠀⡜
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠇⠀⠀⣼⠀⢀⣾⣁⣤⣤⠴⠶⢺⣧⣄⣴⠿⣄⠀⠀⢰⡏⠀⣰⠏⠀⠀⡼⠁⠀⠀⢀⡜
⠀⠀⠀⠀⣠⡏⣀⣀⣠⣿⣤⠴⠖⠻⣟⠉⠙⢷⡀⠀⢀⡴⠟⠁⢸⣿⠀⠈⠛⣦⡟⠀⣸⡏⠀⠀⣼⠃⠀⠀⢀⡾
⠀⠉⠉⠉⠉⠛⢍⡀⠀⠈⠳⣄⠀⠀⠈⢷⡀⠀⣷⡶⠯⢤⣤⣀⢸⣿⣠⠴⠚⠉⠙⢦⣿⠀⠀⢠⡇⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢧⡀⠀⢈⣷⣾⣭⣄⣀⠀⠀⠈⠻⣿⠁⢀⣠⠶⠛⠉⠙⢷⣄⡸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢿⣠⣞⣁⣀⡀⠀⠉⠙⠳⠦⣄⣿⡶⠋⠀⠀⢀⣀⡠⠴⠛⢷⣄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⡾⠋⠀⠉⠉⠉⠙⠓⠲⢤⣀⠈⣿⠀⣠⡴⠚⠉⠀⠀⠀⠀⣀⣨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⠦⠤⣤⣀⡀⠀⠀⠀⠀⠈⠙⢻⠟⠁⠀⠀⢀⣠⠤⠒⠉⠁⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⡀⠀⢸⡄⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢼⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⢀⠄⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠀⠀⠀⠘⣄⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠆⠀⠀⠀⡜⠁⠀⠑⢄⣀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠴⠉⠒⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠂⠀⠀⠒⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠃⠀⠀⠀⡤⠤⠐⠁⠀⠀⠀⠀⠀⠁⠀⠠⠤⠤⠤⠀⠐⠂⠈⢁⣀⡀⠀⠀⠀⠀⠈⠄⠀⡀⢀⠁
⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⢺⣌⣴⣿⡇⠀⠀⠀⠀⠀⢸⣇⣠⣿⡇⠀⠀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠏⠲⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠔⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⣀⢈⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⢠⠊⠀⠀⢃⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠠⠄⠀⠀⠠⠌⡱⠃⠀⠀⠀⠈⠆⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠀⢃⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⡀⠉⠁⠉⠉⠁⠒⡄⠀⠀⠀⠀⠀⠀⡴⡁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⣀⠀⡴⠂⠀⠀⠀⠀⠀⠓⢤⠐⠂⡄⠀⠀⠀⠀⠀⠀⡤⠐⠤⡀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠤⠒⠒⠁
⠀⠀⠀⢠⠊⠁⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⡠⠂⢁⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠸⠈⠂
⠀⠀⠀⠈⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⢀⡄⣠⢌⠀⠀⠀⠏⠉⠀⠀⠀⢀⡀⠀⠀⠀⠁⠲⠄⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠈⡇⠰⠀⠑⡆⠐⢄⡀⣀⡤⠂⠁⠈⠢⡀⠀⠀⡘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠉⠁⠀⠀⠀⠀⢀⠜⠳⡁⠀⠀⢡⠀⠀⠀⠀⡰⠄⢀⡀⠀⠉⠐⠚⠉⡞⠀⠀⠀⠀⠀⠀⠀⢠⠀⢄⠀⠀⡇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠀⠀⠀⠀⠁⠀⠀⠈⠲⢀⠀⠉⠀⠒⠀⠀⠀⠀⢃⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠁⢀⠁
⠀⠀⠀⠀⠂⣀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠸⠄⠀⠀⠀⠀⠒⠂⠁⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⡘
⠀⠀⠀⠀⠀⢮⠉⢹⣀⠔⢘⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠌⠀⢀⠊⠀⡀
⠀⠐⠀⠀⢀⠀⡀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢘⠀⢀⠔⠀⠀⢀⡴
⠀⠀⠀⠀⠈⠂⣱⡖⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠈⠀⠶⣀⠒⠉⠀⠉⠐⠠⡎
⠀⢈⠀⠀⠀⠀⢹⣅⠈⠀⢀⣀⠀⣀⠀⠄⠀⠒⠀⠀⠀⠈⠁⠀⠐⠂⠠⠤⠤⠠⠄⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠠⠀⠀⡠⠐⠁⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⢀⠘⠁⠢⠀⢠⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠤⢀⡀⠀⠀⠀⠈⠄
⠀⠐⠀⠀⣁⠔⠒⠢⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⢀⠄⠂⠴⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠈⠢⠄⠠⠄⠂
⠀⢰⠀⠀⠁⠀⠀⠀⠜⠡⠈⣉⠀⠀⣀⢀⣀⠀⠤⠤⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⣐⠤⠄⠀⠀⣀⡀⠀⠀⠀⠠⠤⠴⠦⠤⠤⠀⠀⠤⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠦
⠀⠀⠰⠂⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠒⠖⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⣠⠀⠐⢦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠
⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠁⠉⠀⠈⡂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠔⢖⠛⠁⣴⣧⡀
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⢠⣿⣿⡀⠀⠀⣿⠿⠿
⠀⠸⢫⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠈⠉⠁
⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠈⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠛⠀⠀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠂⡃⠀⠈⠀⠂⠀⠀⠀⠀⠂⠀⠀⠘⡀⠆⠀⠀⠀⠀⠀⠈⠀⠈⢂⠀⡍⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⡔⡂⡠⠀⠀⠠⠀⠀⠀⠄
⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠧⠈⠘⠀⠀⠠

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠿⠟⠛⢛⣷⣶⣶⣶⣽⣸⣿⣶⣤⣶⣿⣿⣟⣛⡛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⣀⡴⠿⠛⠛⠻⢯⣾⠏⠉⠉⠹⣿⣿⣿⠛⠩⠙⠢⢤⡀⡀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡠⠟⠋⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⢿⣯⠃⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⢾⠏⢀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢿⡦⢀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⢂⣴⣿⣦⡀⠀⠀⢀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣀⢀⣶⣄⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡛⢛⢷⢾⣿⣄⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣭⣿⡿⣿⣧⣄⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⡇⠃⠈⣸⣿⣿⡿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⣿⡿⠏⠩⢹⣿⣧⡀⢹⡗⠄⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⠓⢠⡔⣿⣿⠁⠀⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⢠⣾⣿⡿⠑⠂⣿⣦⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⠂⠛⠛⠛⠓⠛⠁⠈⢻⣿⣟⢣⠀⠀⠀⠀⠀⠀⠄⠾⠿⢿⡻⠝⠷⠶⠋⠿⠿⠦⠴⢿⣇⡇⠀⠀⠀⠀⠀⠀⢣
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⢥⠺⠀⠀⣠⣦⣄⣀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣫⡇
⣿⡃⡆⠀⠀⠀⠀⠀⠀⠨⢸⣇⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣬⠀⢀⣼⣿⣿⣿⣿⡅⠀⠀⢸⣿⣷⡀⠀⠀⡆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢠
⣿⡁⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⢁⣾⣿⡿⡟⢹⣿⣇⠀⠀⠘⣿⡇⡇⠀⠀⠁⠀⠀⠀⠀⣿⣿⣷
⣿⡔⡄⠀⠀⠀⠀⠀⠘⢢⣈⣿⢿⣰⠀⠀⠀⠀⠀⠀⢻⣿⣿⣳⣿⣿⣿⣿⣀⣠⣿⣿⣧⡀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⡛
⣿⡇⢀⡀⠀⠀⠀⠀⢠⣼⣿⢿⡿⣿⣂⡄⠀⠀⠀⠀⣿⣿⡿⣏⣍⠉⠉⠟⠉⠉⠉⠉⠉⠁⣼⣿⣿⣇⡘⠀⢀⠀⢠⣦⣿⣟⠻⣧⠀⠀⢀
⣿⣷⢸⣧⠀⠀⠀⠀⠀⣼⡟⠗⠣⠼⠿⠳⣾⣧⣤⡄⣣⣿⣷⣩⡻⠃⢀⠀⠀⠀⠀⠀⠀⠂⣽⣿⣿⣁⠃⣤⣿⣷⣾⣿⣿⣦⣿⣧
⣿⣿⣆⢸⡆⠀⠀⠀⠀⠙⢣⣀⣤⣧⣤⠾⡟⢿⣿⣿⣛⣿⣿⠛⣷⣶⣿⣴⣷⣄⣴⣦⣠⣾⣽⣿⣿⣿⣿⣿⣿⠟⠨⣿⣿⣿⣟⡇⠀⠀⠀⠀⠀⢰
⣿⣿⣿⡎⣷⠀⠀⠀⠀⠀⠛⠺⢻⣿⣿⣷⠋⠀⢸⣿⣿⡝⠛⣻⣿⣿⣯⣿⣿⣿⣿⡿⡿⣿⣿⣟⠇⠠⢻⣿⣿⣤⡤⣿⣿⣿⣿⢰⠂⠀⠀⠀⢀⣿
⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⠀⠀⠻⣟⣻⣿⣷⣴⢿⣿⣿⡀⢀⣽⣿⣏⠁⠈⣿⣿⡏⠀⢁⣿⣿⣿⣆⣠⣿⣿⢿⣿⠛⣿⣿⡵⣁⠛⠁⠀⠀⢠⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⡉⠀⢹⡽⠻⣯⢿⡣⣒⣙⣿⢻⣄⣼⢿⣿⡟⢴⢫⣛⣟⣄⢻⠿⡿⠉⠙⠃⣼⣿⠟⣿⠉⡠⠀⠀⣠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⠈⠀⠹⢾⡉⢿⠟⠛⢾⠏⢻⡽⠛⢻⣽⡾⠛⠂⠀⠀⠀⠀⣴⣿⣟⡡⢏⢠⠃⠀⣵⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⢿⣶⡄⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠂⠀⢠⣾⣿⣿⢿⣽⣿⣿⣱⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠠⣄⡀⠙⠿⣧⠈⠀⢘⢺⣿⡄⡤⡀⠀⠀⠀⡀⣼⣿⣿⣿⣄⡠⣎⣴⣿⡿⢟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣿⣿⣿⣷⣿⣿⣽⣿⣶⣀⣠⣽⣿⣿⣥⣾⣿⣾⣿⣿⣋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁⣠⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡟⢿⣿⠋⠀⠴⠟⢻⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣾⢇⣸⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣽⡏⣠⡇⠀⠀⠀⠀⠀⠀⣿⣿⣯⣭⣽⣤⣷⣷⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠜⠩⣇⡃⣿⠀⠀⠀⢀⣠⣴⣾⣿⡿⢿⣻⣿⠟⠋⣹⣿⢟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣸⢋⢶⢰⣿⣾⣾⠀⢀⣴⣿⠿⣛⣫⣵⢾⣿⠏⠉⠀⠀⢹⢿⡞
⠀⠀⠀⠀⠀⠀⣿⣹⣆⠋⢹⣿⣿⠀⠀⠉⠉⠉⢹⢸⣧⠀⡑⢤⡀⠒⢂⣾⡷
⠀⠀⠀⠀⠀⠀⣿⢉⢑⣦⣜⡻⡿⣇⠀⠀⠀⢀⣼⣾⠊⣴⠈⠳⣽⢷⣿⠟⠇⣆
⠀⠀⠀⠀⠀⠀⢿⣌⣇⣿⣿⣿⢿⣿⡄⢀⣴⢟⡝⠁⠀⢩⢣⢱⣻⣿⣝⡀⣼⠜⠄
⠀⠀⠀⠀⠀⠀⠸⣧⢟⣅⣐⠚⢁⢨⣿⣿⣿⣿⣿⣶⣶⣾⢸⢸⡟⠿⠾⠟⣳⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠹⣾⢧⣤⠚⣡⣾⣿⢟⣿⣿⣿⣿⣿⣿⢸⢸⣟⣛⣛⣟⣛⣛⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡷⣞⠫⣿⡟⠼⢿⣿⡻⢿⣿⣿⡾⡌⣏⢷⣿⡟⢝⣿⡭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⣱⠈⢇⡄⠬⣿⣿⣷⢍⡻⠿⣿⣮⣦⣍⣛⣋⣩⢚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣝⣻⠬⢷⠀⠙⣦⡿⣶⣿⣧⡉⢽⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠿⢬⣭⣽⣧⡶⠷⠾⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⢻⣿⣷⣔⣦⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣏⣿⣾⣶⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⢷⡾⠷⣤⣄⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⡟⢻⢅⡨⣑⢤⣈⣲⣉⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢻⡄⠀⠑⢮⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⣟⣩⢉⣺⣼⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⣀⣠⣴⣿⣿⠟⠋⠀⠀⣦⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⡟⠋⠛⣿⠻⣿⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣛⣛⠳⠆⠈⠙⠉⠀⠖⢛⣳⣌⡙⠿⣟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢿⢭⡴⠦⠿⢦⠑⠀⠸⣠⠿⠴⠦⣯⢿⢦⠈⢣⡓⣵⣌⠝⠋⡏⣋⢬⡯⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡟⡁⢺⡾⡙⠒⠚⠁⠀⣀⠀⠈⠑⡒⠚⠁⢸⡇⡇⠀⣝⢎⢻⢑⡦⣥⡿⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣷⣾⠿⣿⠏⡜⣿⠈⣦⠙⢷⠀⠀⠀⠠⠄⠤⠀⠀⠚⢃⣇⠎⡄⡇⠀⠻⡄⢣⡿⠟⠉⣀⠻⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡟⠛⣉⠽⡆⣴⢃⠎⣼⣮⠜⠁⡇⠸⠇⠀⢈⣤⣶⣦⡁⠀⠀⠀⡝⡘⡸⠀⣰⠃⢱⢸⡇⢠⡀⢸⣷⢹⣧⡦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣉⡛⠴⣧⣵⡏⢸⣴⠟⠁⢀⠜⢁⣼⠀⠀⠺⣿⣿⣿⡿⠂⠀⠀⣿⠟⢀⣴⠋⢠⢇⡾⠀⢠⣿⢸⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠷⣜⣿⢥⣸⠃⢀⡔⢁⡔⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⡴⣹⠃⡰⣣⣾⠄⢀⣿⣟⣙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡾⣟⣀⡇⠀⡞⢠⣿⡀⠱⣸⠈⠲⢤⣄⣀⣀⣤⠴⡟⡏⡜⢠⠃⡜⡴⣿⣿⣾⣥⣍⡙⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⢀⣤⡾⣿⣷⡀⢇⠸⡣⡱⣄⢹⠁⠀⠀⠀⠉⠀⠀⠀⡃⡅⡇⢸⠀⣹⠀⡹⣏⣉⣉⣭⡻⠸⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠁⠻⣯⣻⠺⣷⣝⢿⣬⡞⣸⡟⠀⣴⢂⡴⠖⠀⢧⠱⣵⡀⠣⡙⢆⠣⠽⠏⣻⣭⡷⣶⣛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠑⡿⢾⣿⣧⠟⣠⡟⢳⣳⣿⡛⠉⢻⣿⠛⠉⠀⠀⠀⠀⢑⣬⣻⣦⣙⡆⢳⣆⣊⣼⠻⣦⠀⢸⡇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣑⣷⣌⣦⣤⠖⣱⢣⣾⣿⠁⠉⠢⣄⠉⠁⠀⠀⠀⣀⠔⠉⢸⢹⢱⠈⡿⢸⠀⡇⢹⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣴⡞⢡⢻⣦⣤⠃⣾⢱⢰⠀⠀⠀⠈⠑⢄⣾⡤⠊⠁⢀⠔⣡⣿⡷⣞⣵⠧⠊⢠⡾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣻⡘⠻⡅⢇⠈⠢⣮⣧⣀⠀⣀⡝⢡⣶⣿⣧⣀⣮⡾⣫⢖⣟⠭⣤⣶⠞⡹⠁⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣶⡎⠓⠛⠶⠤⡌⢻⣧⠉⣇⠹⣿⣿⡿⠁⠹⣿⡀⢸⡁⠐⠋⢁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⡾⠃⠀⠁⠙⣟⠉⠁⠀⠀⠈⠓⠂⠹⡆⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⡆⢠⣿⡏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⡇⢸⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢠⠇⠀⠀⠀⠀⠀⣀⣤⣄⠀⢸⣿⡇⠀⡇⢸⣿⡇⠀⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⠀⡞⣧⠀⠀⣼⣿⠋⠈⡇⢸⣿⡇⢸⠁⢸⣿⡇⠠⠀⠀⢠⣸⣏⣙⣦⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣸⠀⣿⠀⢸⣿⣿⢀⠼⠁⢸⣿⣷⠃⠀⢸⣿⣷⠀⠀⠀⣿⡇⠉⠛⣿⡏
⠀⠀⠀⠀⠀⠀⢀⠴⢿⣿⣧⡏⠀⣿⡀⠈⣿⣿⡀⠀⠀⢀⣿⣿⡄⠀⠀⢿⣿⡄⠀⢠⣿⣇⠀⢀⣿⠃
⠀⠀⠀⠀⠰⠚⠁⠀⠸⣿⣿⣧⠀⢿⣇⣠⠋⠛⠿⠶⠶⠋⠈⢿⣿⣦⡴⠋⠻⢿⠷⠋⠘⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠶⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⣰⣿⣿⣿⣿⣯⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⡿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣷⣾⣿⣿⡄
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠭⠥⣋⣉⣑⠦⠤⠝⠛⢿⣿
⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⢀⣦⡀⠀⠀⠀⠀⣴⣄⠡⡀⢳⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡏⠀⡆⠚⠛⠃⠀⣲⣄⠈⠉⠉⠀⢳⠈⡇⠀⣀⡀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡿⠶⣇⢘⣿⣿⡇⠘⡇⢀⡀⠀⠙⠛⠛⢃⡀⢠⡄⢰⢀⡇⢘⡿⣿⡖⣖⢐⣻⣿⠇
⠀⠀⢀⣴⣿⣿⣿⣻⣿⡏⠀⠀⠈⡟⢿⣿⣧⡀⢣⠘⣿⣶⠿⣷⣶⡿⢿⠿⠃⠀⡼⢠⠃⠠⠂⣵⣠⠞⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⡆⠀⠑⢸⣿⣿⣷⣤⡀⣠⡈⢠⠈⠉⠀⠀⢀⣠⣾⠁⢞⠃⠀⣤⣿⣥
⠀⠀⣿⣿⣍⢈⣻⣿⣇⣀⠂⢀⣈⣸⣿⡿⢿⢿⡟⠛⢻⣿⣿⣄⠰⢳⣾⣿⣿⠋⠴⡆⠙⠐⠛⠛⣻⡄
⠀⠀⠘⢿⣿⡎⣿⠶⣿⢾⡿⢿⠶⡿⠿⡗⢻⣿⣿⢰⣿⣿⣿⣿⡺⣁⠽⠿⣿⣶⣰⢁⠘⣿⣿⡏⣿⠇
⠀⠀⠀⠀⢹⢁⡿⢳⣿⢶⡷⢾⡶⣿⢾⡟⢻⣿⡏⢸⣿⣿⣿⣿⣿⣿⣷⣶⣉⣹⠿⢏⣙⣯⢿⠝⠁
⠀⠀⠀⠀⣿⠈⠀⡜⢹⣶⡷⢾⠶⣿⢺⡿⢻⠙⠅⣸⣿⣿⣿⣿⣿⣿⣿⣧⣽⠟⠛⣿⡿⡷⠃
⠀⠀⠀⠀⠹⣄⡜⢰⠃⠘⣷⢾⡷⠿⣃⡄⢸⣄⢀⣿⣿⣿⣿⡋⢻⣿⣿⣿⠶⣛⠷⣭⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠮⡛⢹⠟⣡⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠉⠤⣿⣿⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣴⢺⠉⠻⣦⣢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⢴⠏⣠⠿⣿⣿⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣠⣴⠚⣷⢛⡿⢘⡇⠀⠘⢻⡛⣟⣙⣿⡿⣿⣿⣿⣿⣿⡟⡉⣾⠊⠑⣤⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢰⣻⡟⣻⡛⣿⣹⠁⢈⡇⣠⠶⢿⡛⣿⣉⣿⣴⣿⣿⣿⣿⣿⠟⢿⠀⡷⡊⢸⣿⣿⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣏⣽⣯⣿⣉⠳⣼⣯⡥⢶⣾⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⠴⢹⡧⣀⠼⣿⣿⣯⣀⢺⣿⣿⣿⡿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠉⠰⠛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⡸⡇⠉⡢⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⢸⠀⠀⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⣧⠛⢄⣿⠊⠀⠈⢯⠉⠉⠉⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢰⢿⡀⠈⡟⣦⠊⢢⡸⣇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⢀⢧⡀⢡⡿⠀⠸⣧⡊⢁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡄⢸⠀⠀⡿⠿⠿⠟⠛⠁⠀⣰⢲⣅⣸⢿⠃⠀⠀⢣⡙⠢⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡆⠀⠁⠀⠀⠀⠀⠰⣟⡑⠯⣯⣟⡟⠀⠀⠀⢻⣶⣶⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣤⡁⠇⢰⣀⡄⠀⠀⠀⠀⠀⠰⠖⠉⠉⠁⠀⠀⠀⠘⠿⠾⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢘⣿⣦⣯⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣈⣍⠀⠀⠀⢘⣁⡀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣥⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣌
⠀⠀⠀⠰⠋⠈⢁⣀⣀⠈⠉⠳⠀⠀⠀⠀⠀⢀⠀⠉⠙⠃⣀⠀⠛⠉⣀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠠⠾⠏⢀⠀⠙⠿⠆
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣀⠀⠀⠴⠄⠀⠉⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣈⣁⠀⣠⠞⠁
⠀⠀⠀⢸⣿⠇⠙⠛⠛⠁⢹⣿⡄⠀⠀⠀⠀⠀⠻⣷⣾⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠘⢶⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣨⡅⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⣈⣉⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⢢⡀⣐⠀⠀⢀⢀⠖
⠀⠀⠀⢀⠀⠀⣿⠃⠘⠟⠀⠀⡀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⡀⠀⠼⠿⠃⠈⠻⠏⠀⠠⡀⠀⠀⠀⢠⡀⠙⠉⠀⠀⠈⢉⡄
⠀⠀⠀⠸⣧⡀⡀⢈⠀⠀⣰⣶⠇⠀⠀⠀⠀⠀⠈⠛⠛⠁⠤⠈⠛⠁⠀⠀⠀⠀⠀⠰⣀⠀⠀⠁⠈⠀⠀⣀⡴⠁⠀⠀⠀⠀⣿⣶⣄⣰⣶⣶⣿⠃
⠀⠀⠀⠀⠙⠿⠻⢿⡼⠾⠛⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠙⠋⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⢂⣀⠀⠀⠀⣁⣌⠑⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⣿⡋⠁⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂⠀⠲⠾⠋⠀⠀⠀⠀⠀⠀⡓⠀⠀⢀⣤⠀⠀⠀⡒⠀⠀⠀⠀⠀⠉⠈⠃⠀⠘⠁
⠀⠀⠀⠀⠈⡉⣀⢀⣀⣉⠉⠀⠀⠀⠀⠀⠀⠈⢳⣤⣄⣀⣁⣠⣤⡶⠁⠀⠀⠀⠀⠀⠛⣷⣤⣤⣀⡤⣤⡿⠃⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⡄
⠀⠀⠀⠐⠶⠿⠛⠛⠛⠻⠻⠧⠄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⠛⠨⠿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠲
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢚⡓⠀⠀⠀⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣠⡆⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣷⡄
⠀⠀⠀⠈⠛⠛⠒⢠⡄⠴⠛⠛⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⡀⢨⡿⠻⠄⠀⠀⠀⠀⠈⠲⢦⣤⠄⠐⢶⠶⠋⡀⠀⠀⠀⠀⣠⣄⡉⠉⡤⠀⠛⠁⠀⣠
⠀⠀⠀⠠⣶⣶⣶⣶⣦⣀⣴⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣦⡀⣀⢘⣃⢀⣄⢾⠇⠀⠀⠀⠀⠁⠘⣿⣤⣴⣦⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠘⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢸⣿⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⠿⠟⠛⠻⠿⠿⠻⠿⠿⠻⠿⠛⠻⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⢸⣿⣿⡿⢿⣿⣿⠿⠿⡿⠛⠻⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⣿⡇⠸⠟⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⣰⣿⠋⢰⡆⢸⣿⠃⠀⠀⠀⠀⠀⠹⣿⠇⠀⠉⠀⠙⠻⣿⣷⣹⣿⡆⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢠⣾⣿⠁⠀⠘⠇⠘⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡆⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿⣿⣆⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣼⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⢻⣿⣇⠀⠃⣸⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⣀⣿⡷⠘⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿⣿⣧⠀⠁⢹⣿⡄
⠀⠀⠀⠀⠀⢠⣿⠇⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠹⣿⣧⠀⠀⠹⣿⣄
⠀⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠸⣿⡀
⠀⠀⠀⢀⣿⡇⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⢸⣿⡀
⠀⠀⢀⣾⡏⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄⠙⢿⣿⣧⠀⠀⢹⣿
⠀⠀⣾⡟⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠹⣷⠀⠈⢹⣷⡀
⠀⣾⡏⠁⣤⡟⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⡀⠀⠹⣷
⣿⣿⣤⣴⣿⠃⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠈⣿⣷⣤⣤⢿
⣿⡇⠀⣹⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣀⣀⣿⣶⠟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣾⣛⠏⠁⢈
⢻⣿⠀⠻⣿⠀⠀⢺⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⢶⣶⣿⣿⣿⡟⠉⠐⢿⡗⠀⣶⣶⣤⠜⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢹⣿⠀⠀⣼
⠈⣿⡄⠀⣿⡇⠀⢸⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠛⠿⣶⣴⣿⡿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿
⠀⣿⡇⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⣾⣿⠀⢸⣿
⠀⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢻⡟⠀⢸⣿
⠀⣿⡇⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢸⣿
⠀⣿⡇⠀⢸⣇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣿⠀⠀⢸⣿
⠀⢸⣇⠀⠈⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⣾⡇
⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⠀⢰⣿⡇
⠀⠀⣿⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠇
⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⣿⡇⠀⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸⣿⠀⢸⣿
⠀⠀⣿⡇⠀⢻⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⣸⣿
⠀⠀⢸⣇⠀⠸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⣦⠀⠀⣿⡏
⠀⠀⠈⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠃⠀⢸⣿
⠀⠀⠀⢻⣇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⠀⣼⡿
⠀⠀⠀⠈⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⠀⢀⣿⠃
⠀⠀⠀⠀⣿⡄⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⡟⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⢸⣿
⠀⠀⠀⠀⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠁⠀⣾⠏
⠀⠀⠀⠀⠀⣿⡀⠸⣿⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿
⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡟⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⣸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⢰⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠀⢸⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢸⣇⠀⠀⠀⠀⠈⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢀⡿⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠈⠁⣸⣷⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣷⣻⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⢹⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠰⣶⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡇⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠘⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣿⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢘⣿⡀⢀⣾⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣠⣴⣦⣤⣾⡇⢀⣿⡟⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡦⠼⣿⣿⠿⠟⠛⠛⠛⠓⠾⠿⠶⠶⠶⠿⠿⠷⠞⠛⠛⠻⠿⠿⠷⣾⣿⣷⠞⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⣠⠖⠛⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠉⠻⠤⣤⠤⢤⡤⠼
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠲⠤⠶⠋⠀⠀⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠿⠤⠼⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠲⠤⢤⣀⣀⣀⣤⠤⠶⠒⠛⠛⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣴⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⢶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿
⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠘⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⠿⢟⣋⣥⡀⠀⠐⠢⠄⡉⠛⠿⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⢛⡩⡴⠞⠛⠉⠀⠀⠀⣀⣀⠀⠀⠉⠂⠠⡈⠉⠛⠿⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⣋⡥⠖⠚⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣶⣤⠀⠀⠀⠀⠀⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⠛⣉⣥⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⢀⠜⠁⠀⠀⠀⠀⡜⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⡀⠄⠚⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⡸⠀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⡰⠋⠀⠀⠀⢀⣴⡿⢿⣶⣤⣀⠀⠀⣠⣶⡿⠛⠀⠀⢀⠔⠁⠀⠀⣀⠀⠀⠀⡰⠁⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠁⣼⡅⠀⠀⠀⠰⢿⣋⠀⠀⠈⠙⠻⢿⣾⡿⠋⠀⠀⢠⠜⠁⠀⠀⠀⠀⠛⠁⠀⢠⠃⣸⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡟⢁⣼⠟⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⢴⣧⣀⠀⠀⠀⠀⠀⠀⠀⢠⢃⢢⣿⡿⢻⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⣿⠏⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀⠀⠀⠀⢀⠂⠎⣼⣿⠁⣸⣿⢡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢠⠞⠓⠦⣀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⡸⢠⣿⡏⠀⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢀⡔⠁⠀⠀⠀⠈⠉⠲⢤⡀⣰⠃⠀⠀⠀⠀⡇⢸⣿⠃⠀⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⣴⠏⠀⠀⠀⠀⠄⠀⠀⣠⣾⡿⠋⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⣠⠃⣸⢃⣾⡟⠀⢠⣼⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢀⡾⠃⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⡶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡠⠊⡠⢊⣴⣿⣿⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢠⡞⠁⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠔⣋⠔⣡⣾⠟⣻⢹⣿⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⠋⣰⠟⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⡵⢊⣵⡿⠟⠁⠀⣾⣿⣿⣆⣾⠇
⠀⠀⠀⠀⢠⣼⡿⠁⣴⠋⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⡥⢚⣡⣾⠿⠋⠆⠀⢀⣼⣿⡿⣿⣿⠃
⠀⠀⠀⢀⣾⡟⢀⡾⠁⠀⠂⠀⠀⣠⣿⠏⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡥⠞⣡⣴⡿⠛⠁⠀⠀⠀⢠⠸⣟⣿⣾⣿⠇
⠀⠀⢠⣿⠏⢠⡞⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢚⡥⠞⣡⣴⡿⠟⠁⠀⡄⠀⠀⠀⠀⣰⣶⣿⣿⠟⠁
⠀⣰⣿⠋⣰⠏⠀⠀⠀⢠⣾⠟⠁⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⡥⠞⣉⣴⡾⠟⠉⠀⠀⠀⠀⢠⣶⠀⣠⣿⣿⠿⠋
⣰⡿⠁⠰⠁⠀⠀⠀⢰⡟⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⣁⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣈⣵⣿⠿⠋
⣿⣅⠘⢄⡁⠀⠀⠀⠀⠈⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⢀⡤⠾⣍⣴⡾⠟⢉⠄⡄⢠⠆⠀⠀⠀⢀⣴⣾⠟⠋
⠸⣿⣿⣦⣝⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢁⣤⡾⠟⠁⠀⢠⢎⠞⠀⠁⠀⢀⣤⣾⠟⠋
⠀⢿⣿⡙⠿⣿⣦⣈⠒⢤⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⣰⠣⠏⠀⢀⣴⣾⠟⠋⠃
⠀⠘⣿⣿⡀⠀⠉⠛⢿⣦⣌⡐⠳⢠⠴⣻⣿⠿⠋⠀⠀⠀⠀⠀⠰⠃⢀⣤⣾⠿⠛⠋
⠀⠀⠈⠻⣷⣤⡀⠀⠀⠈⠙⠻⠷⣶⣾⢛⠁⠄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠈⠹⢿⣷⣦⣀⠀⠰⠂⠀⢸⡇⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⠀⠀⣇⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣼⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⢀⢠

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⢮⡓⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢆⢀
⠀⠀⠀⠀⠱⣄⡀⠆⡤⠄⠀⠀⠀⣀⡄⢉⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠑⠌⢃⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢠⡓⠈⠀⢛⣡⣼⠎⠃⠰⡆⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⡇⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⠄⢸⠿⠋⣤⠀⠱⠀⠀⠀⠀⠀⠀⣠⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⣤⠟⠀⠀⠀⢘⣿⢀⠐⠀⠀⠀⠀⠀⠀⠟⠛⠟⠻⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⣡⡞⢀⠀⠀⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠴⠁⠢⠸⠄⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⡇⠀⣇⠀⡇⠰⡆⠀⣧⠀⠁⢰⠀⠂⠀⠀⠀⠀⢸⡇⠀⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠿⠀⠀⣀⡴⠒⠀⢼⡃⠀⣿⠀⣇⠀⠁⠀⣿⠀⠀⠈⠀⡟⢀⠄⣀⠀⢼⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡀⠀⠉⠀⠖⠀⠈⣧⡀⢸⡄⢹⡤⣄⠴⢻⠈⢧⡀⣼⠀⣤⣀⠉⠒⢸⡇⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠰⠊⣧⠐⠰⣦⡔⢷⡄⠘⡇⢸⠁⠀⠀⠃⠀⠘⠀⠈⣧⢿⣄⠀⢉⠀⠙⣿⠀⣿⡀⠀⢠
⠀⠀⠀⢤⡀⣠⡆⠀⠀⠀⠀⠀⢸⡀⢀⠀⠈⢆⠀⠈⠇⠀⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣷⠘⠳⣤⣇⠴⢻⡇
⠀⠀⢠⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⡇⠋⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠈⢷⡀
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢞⡿⢿⡿⣿⢶⣦⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣅⡈⠀⢀⠀⡈⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡠⠖⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢦⠹⠿⠃⠀⠘⠻⠿⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡉⠑⣿⠛⠓
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣞⢻⢿⣿⢿⡿⣣⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠛⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣾⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠄⠀⠀⠈⠉⠁⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠒⠿⢏⣵⣟⢉⠍⠿⡍⢯⠉⡷⠂⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⡴⠀⠐⢰⡀⠀⠀⠀⠀⠉⠉⠉⠒⠚⠃⠚⠚⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⠶⠦⠤⠤⠤⠼⠿⠇

⣠⣤⣀⣀⣀⣸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣎⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠊⠈⠙⠻⠿⠿⠿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠉⠙⠃⠉⠁⠙⠋⠉⠛⠉⠉⠈⠙⠂⠉⠉⠈⢧⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣈⠉⠀⠈⠃⠢⠀⡀⣸⠀⢀⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣴⣷⠄⢸⣿⠀⠀⠀⠀⢀⣿⠆⠀⠀⠀⠃⠀⡀⢈⠸⢤⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣶⣷⣿⡿⡋⣹⠟⠳⠀⠀⠀⢻⣿⣦⡼⢿⡄⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⢿⠿⠿⡏
⠀⠀⠀⠀⣶⣿⣿⣿⠀⠀⣿⠃⠀⠀⠀⠀⠾⠃⠀⢹⣶⣥⣶⣿⣿⣿⣿⡤
⠀⠀⠀⠀⣿⣿⣿⣿⣀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠼⠷⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⡻⣿⣿⣿⡗⠶⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡙⢿⠟⠀⠀⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣁⣀⣀⣼⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣼⣧⣄⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤
⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⢿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉
⣿⣿⣿⣌⣀⣶⣶⣿⣿⣷⣄
⢻⣯⣻⣿⣿⣿⡻⣿⡿⣿⣿⣗⠢⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠙⢿⣿⣯⡟⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢁⣴⣀⢶⡙⠿⣿⡾⣿⠿⠷⠓⠲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⣀⣈⣈⣉⣭⣴⣿⣶⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣏⣭⣵⣤⣍⣉⣹⡛⠛⢛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠟⢻⡍⠍⠉⠉⠛⣻⣿⣿⡿⢛⡉⠉⠉⠀⠀⠒
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡧⠀⠀⠀⢡⣿⣟⣥⣿⣷⢶⣶⠛⠛⠿⠛⠉⡓⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣿⣿⡟⢯⡁⢄⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠈⠓
⠀⠀⠀⣀⣼⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⡿⢿⡏⢳⣤⡙⠆⠁⢄⠀⠀⠀⠀⠀⠀⠾⡅
⠀⠀⠀⠘⠩⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠇⠀⠀⠀⠘⣷⡌⠳⢦⡁⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠊⠵⢾⣀⡀⠀⠀⠀⠀⠀⢠⢳⣐⠲⠀⠀⠀⠐⡀⠀⠀⠙⣄⢀⠀⢈⡀⠢⠤⣄⣀⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⡀⠀⢘⣮⡳⣾⣆⢂⠀⠡⠀⠀⠀⠈⡙⢷⣦⡉⠑⢦⣄⣀⣈⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⡆⠀⠙⣿⣾⣿⣿⣴⣀⣷⣄⠀⠀⠈⠻⣿⣿⣦⣀⠀⠉⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢿⡠⢀⠈⠿⢿⣿⣇⠈⠙⣿⣷⣤⣾⣾⣶⣽⣻⠿⣿⣷⣦⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠈⠟⠗⠰⠎⠻⣿⣷⡄⠀⢻⣿⣿⣿⠹⣿⣿⢿⡿⣶⡾⠋⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⠀⢸⣿⣿⣿⠆⠹⣹⠷⠒⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣷⣸⣿⠏⢨⣞⠉⠡⠄⢠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠫⡈⠁⠀⠉⠀⢁⡀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⣀⢀⠠⣭⡄⠥⠄⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠐⢦⣴⣞⣷⣾⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡤⠄
⠀⢀⣴⣦⣤⣤⣼⣿⣿⣿⡿⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⣾⣿⡿⠟⠻⠁⠀⠙⠉⠀⠀⠀⢀⡀⠀⠀⠛⠳⠶⠀⠀⠀⣸⣧⠤⣿⣧⡀⠴⢢⡀⢰⡆⢶⠄⣴⠰⣦⠀⢀⡀⢠⡤⠀⠀⢀⡄⢠⣿⣿⣿⣿⡀
⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡏⠃⣤⢹⡇⢸⡇⢺⡆⣿⠀⣿⠀⠈⣧⡼⢡⣤⣤⣤⢶⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡟⠀⠾⠆⠀⠙⠘⠁⢿⠁⠛⠁⣿⠐⠋⠀⠀⣸⠁⠈⢿⣿⣿⣧⣝⠻⠿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠀⣹⡇⠀⠀⠀⠘⠀⢀⠠⠀⠀⠀⠀⠀⠝⠁⠀⠀⠛⠀⠒⠥⠴⠃⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣿⡇⠀⠀⠀⠀⠘⣿⢠⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡏⠉⣹⡿⣦⠰⢒⡆⠀⣿⠈⡏⢠⣶⢿⣿⣶⣶⡀⢺⠃⣸⠇⢸⡇⣴⠛⣦⠠⡾⢻⡆⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢴⡀⣷⠘⣿⢸⣿⢸⣿⣶⣿⣾⣿⠇⣿⠂⣹⡀⢸⡇⣿⠀⠈⢰⡇⠀⠁⢸⡇⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠠⠟⠳⠄⠈⠉⠁⢀⡏⢸⣟⠀⠙⠛⠛⠛⠁⠀⠀⠈⠉⠉⠉⠁⠈⠓⠁⠀⠙⠊⠀⠚⠇⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⡀⠶⣦⣤⢀⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣀⠀⠀⠀⠤⠛⢿⠓⢆⡈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢸⡁⠑⠄⠀⠀⠀⠀⢀⡀⠈⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢸⠀⠀⠤⠀⠙⠒⢂⠤⡀⢶⡊⠉⣱⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢀⡠⠤⠼⡄⡄⣓⡘⡆⣀⡳⠤⣂⢀⣈⠉⡁⠀⠀⠀⠀⡸
⠀⠀⠀⢠⠖⠀⠈⡉⠛⠚⠁⣀⡀⠀⠙⣮⢸⡑⣣⠤⠐⠚⠤⣌⡠⠀⠁⠀⠀⠀⡰⠃⠂⠄⡀
⠀⠀⢠⠃⡀⠀⠯⠬⢂⡤⡘⠓⠚⡤⠒⣶⣿⡏⠀⠀⠀⠀⠀⠘⣄⣀⣄⡨⠔⠊⠀⢀⡄⠀⠀⠢⡀
⠀⠀⢸⠀⠁⢰⠖⣆⡤⠭⠤⣄⡜⠀⢠⠄⣬⠁⢠⠀⠄⠤⠀⠀⡏⠉⡀⢀⡀⠀⡄⠠⠃⠀⠀⠀⢣
⠀⠀⠈⢧⡂⢀⠧⢀⠀⠀⠀⣀⡇⠀⢈⠭⢤⠤⠭⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠙⠾⡆⢈⠩⠛⠉⠉⠉⠉⠙⠲⡊⠑⠐⠚⠑⠒⠈⣼⠀⠀⠈⠠⠈⠁⠀⡀⠀⣼⢀⡼⠓⠒⠲⢤⡀
⠀⣠⠔⠊⠉⠁⠉⠬⠆⠀⠀⡀⠀⠀⠀⠀⠘⣄⣀⣀⣀⠤⠞⠉⠉⠑⢦⡁⠐⠂⠁⣀⡼⠛⠋⠀⠀⠀⠀⠀⠙
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢃⠀⠀⠀⠀⣀⠠⠒⠂⠠⢹⠒⠚⠉⠁⠀⠀⠔⠊⠉⡂
⡇⠀⠀⠀⠀⢀⢀⠀⢠⠀⠤⠂⠀⠀⠀⠀⠀⡮⢉⠩⠉⢁⡑⠒⠋⣤⠀⢠⠂⠄⠀⠀⠐⠀⠈⠉⠉
⢇⠀⠀⠀⠀⠀⠈⡉⠀⠀⠄⠀⠀⠀⠀⠀⡰⠁⢀⢀⠀⠃⢞⢀⣀⡏⠀⡾⠀⠀⠀⢀⠀⠈⠠⠀⠀⠀⠀⠀⢀
⠘⢄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⣀⠠⠊⠱⢄⣀⠀⠄⢀⠡⠴⠋⣠⠞⢧⡀⠀⠁⠀⠠⠂⢀⡀⠁⣀⣀⠴⠋
⣀⣀⣑⣢⣤⣄⣠⣤⣒⣉⣉⣉⣁⣀⣀⣀⣀⣀⣉⣑⣒⣒⣒⣒⣉⣀⣀⣀⣉⣓⣶⣦⣴⣒⣉⣉⣉⣉⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡠⠚⠉⠉⠲⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠃⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⢶⡀⠀⠀⠀⣵⢸⡆⠀⠀⠀⠀⣼⢀⠇⠀⠀⠀⢀⡖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⣧⡠⠤⣒⣛⡊⠩⠭⠭⠭⠭⠭⢙⣓⣒⢤⢄⣼⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠴⣋⠵⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠪⢝⡢⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡰⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣦⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏⠀⠀⠀⠀⠀⠀⠀⢠⣠⣆⡄⠀⠀⠀⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠃⢘⡂⠀⢰⠛⠟⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠊⠀⠀⠀⠓⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢿⣀⡼⠀⠀⠀⠀⢸⡀⠀⢠⠇⠀⠀⠀⠀⢸⣀⡼⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⢄⡀
⠀⠀⠀⠀⠀⠀⡜⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣠⣒⠾⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⠀⠀⢸⡀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⡜⠀⠐⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄
⠀⠀⠀⠀⠀⡼⠀⠀⢠⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣷⠀⠀⠸⡀
⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⢰
⠀⠀⠀⠀⠘⡄⠀⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢇⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠒⠒⠒⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⠈⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠘⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶
⠀⠀⠀⠀⣄⣼⣦⡼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⣴⣶⣤⡾⢿⣿⠇⠀⠀⠀⢦⡀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏
⠸⠛⠋⠉⠀⠀⠀⢤⣀⠔⠀⠀⠉⠉⠀⠀⣿⠉⣿⠆⡉⡇⣿⢸⡈⡇⣯⠐⣆⡜⣀⣀⣂⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡿⠀⠿⠀⠓⠃⣿⠘⢸⡇⠋⠀⡸⠀⠻⣿⣿⣝⣿⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣤⢠⠀⠀⠀⠁⠀⠀⠀⠈⠊⠁⠀⠀⠀⠉⠛⠿⠿⠿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⡏⢹⡷⡄⢲⡄⣧⢹⢠⡶⣶⢶⡄⣶⢰⡆⡗⢰⢲⣠⡖⡆⢺⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣇⠸⠾⠇⡟⢸⠘⢿⣿⠿⠃⠛⠼⠧⠗⠻⠄⠘⠧⠄⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢻⣿⣦⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠄⠀⠀⣸⣿⠛⡉⢩⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠂⠠⠤⠮⠂⠀⢀⣀⡾⣫⣿⠀⠁⣈⣰
⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⢀⠀⠀⠀⣀⠄⠀⠀⢰⣻⡿⣻⡿⡿⠁⢂⣬
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⣘⠤⠒⠉⠀⢀⣀⣠⠤⠝⢋⡩⠂⠀⠀⢿⠃⠀⠉⠑⠢⢄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⠔⠒⠋⠉⣀⣠⣴⣶⡿⠁⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣧⠀⠈⠐⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠉⠉⠉⠙⡿⠷⡀⠐⠦⣈⠐⠠⡀⠀⠀⠘⡄
⠀⢀⣿⣿⣿⣿⡿⠁⠀⠈⠻⣿⣿⣿⣿⣿⠟⠉⠀⣼⣶⣤⣤⠞⠀⠀⠙⠦⣀⡈⠙⠒⠬⣐⠄⠀⢸⣠⡴⠖
⠀⢸⡿⠿⡿⠟⠁⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀⡈⠉⠐⢠⡀⡠⠜⠚⡅
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠤⡤⠙⠙⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⠍⠀⠀⠀⠀⢀⡀⠀⠀⢠⠔⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢉⡻⠿⠒⠒⠚⢋⡝⠁⠀⢀⠔⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⣣⡤⠖⠋⠁⠀⠀⠀⠀⢠⠎⠀⣀⠔⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⣀⣀⣠⡴⠖⠚⣋⣠⣬⠷⠿⠀⠀⢄⠀⠀⠀⠠⢾⠿⠃⡠⠊⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡭⠩⠞⡥⠖⠋⣸⣿⠟⠁⠀⠀⠀⠀⠀⠉⠢⠀⢰⠅⡡⠈⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠴⣚⠯⢀⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢴⠗⢀⠜⠁⠒⠒⠒⠒⠉⠉
⠀⠀⠐⠛⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡤⠄⢋⠀⠀⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⢸⡠⠀⡢⠀⠀⠀⠑⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠚⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⡄⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠂⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠀⠀⣠⢂⠊⠀⠀⠀⠈⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠉⠢⡥⠴⠒⠋⠁⢀⠀⠀⠀⡰⡁⠁⠀⠀⠀⠀⠀⠀⠁⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⣈⠀⢀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⢀⡀⠼⡀⠀⡘⡜⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠀⠀⠈⠁⠠⣁⠀⠀⠀⠀⠑⠢⣀⡖⣂⣩⠥⠐⠈⠀⠀⢧⠃⠀⠀⠀⠤⢾⠶⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣄⠀⠀⠀⠀⠀⠀⠑⠦⡁⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢈⠒⢈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈
⠀⢀⣠⠤⠖⠊⠉⠙⠒⠆⠠⢤⣄⡙⠢⢄⡀⠀⠀⠀⠀⠀⠠⠑⠆⡙⠢⣄
⠀⠀⠞⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⡉⠐⠠⠀⠀⠀⠀⠀⠀⠈⠡⠈⠑⡄⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠌⢐⡀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠑⠒⢤⣁⠂⠤⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠢⡀⣀⠀⠀⣉⡐⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠁⠀⣀⣉⠀⠘⠀⡘⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠈⠫⣀⠂⠈⠉⠑⠲⠆⠀⠀⠀⠅⢐⠠⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣇⠀⠀⠀⠀⠀⠈⡍⡁
⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⠀⢤⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⡀⠀⠀⡘⠓⠤⢄⣄⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢌⣐⠀⠄⣈⣀⡀⠤⣒⠽⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠤⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠒⠶⠆⠤⠀⣀⣴⠿⠏⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⠀⠀⠀⠀⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⡀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠋⠋⠁⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣷⣦⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⡆⠺⣷⣦⡂⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡖⠀⠀⠀⠈⠋⣿⣦⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⣿⣷⣬⣀⡀⠀⠶⢶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣶⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⡼⠟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢠⣦⣤⣢⣤⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣰⣿⠛⠉⠁⠀⢸⠃
⠀⠀⣠⡶⠖⠐⠒⠒⠒⠒⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⢿⣷⣶⣤⣾
⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠠⣤⣶⣦⣄
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⡰⠿⡆
⠈⢿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣟⣲⣦⣶⣞⠛⠁⠀⠀⣰⠏
⠀⠀⠀⠈⣦⠀⠀⢷⣄⠀⠀⠀⠀⢻⣿⠟⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠘⢷⣤⣶⡿⢦⣀⠀⠀⢸⣿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⢿⣷⣤⣾⡟⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⢹⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣤⣩⣿⣂⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣖⠀⢆⣤⡶⣦⣀⠀⠀⠀⠀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⠇⠀⠀⠉⠛⠶⠾⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠠⠰⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂

⠀⢠⡤⠤⠤⢤⣤⠠⠄⠠⠤⠤⠤⠤⠠⠤⠤⠄⠤⠄⣤⡤⡤⡤⠤⠤⠄⠤⠤⠤⡤⡤⢤⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⢠⣤⢤⠄⠠⣤
⠀⢸⠃⠀⣰⠞⠉⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠄⠁⠁⠂⡀⠀⡀⠀⠐⠐⠀⠀⠑⢄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢧⠀⠀⢿
⠀⢸⠀⠀⢣⠀⢀⠏⠙⠌⠐⠦⠤⣤⣤⠴⠶⠛⠉⠀⠀⢐⡀⠀⠀⠉⠒⠐⠁⠁⠀⠐⣄⠀⠀⠁⠪⠵⣤⣤⣄⣤⡤⠔⠺⠉⠀⠀⢨⠀⠀⢸
⠀⢸⡇⠀⠸⣧⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀⠠⢚⡋⠈⠠⠀⠀⢀⠀⠀⠀⠀⠰⡌⠻⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢄⣾⠀⠀⢸
⠀⢸⣇⡴⠃⠆⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠔⢁⡤⡠⣇⡒⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠇⣶⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠈⡄⠣⣀⣿
⠀⢸⣉⡱⠞⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡾⠁⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠀⠈⢻⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣏⣹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠖⠉⣰⠏⠀⠀⣰⢿⠀⠀⡀⢀⡀⡄⠈⠀⠀⠱⠀⠀⠀⠻⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⣠⡾⠁⠀⠀⢠⢟⠇⠀⢀⣀⣸⣇⡇⠀⠀⠀⠀⢣⠀⠀⠀⠙⢧⡀⠑⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⢀⣴⠋⠀⠀⠀⣰⠏⣾⡴⠚⠌⠁⠀⠀⠘⡄⠀⠀⠀⠀⢇⠀⠀⠀⠈⢻⣆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⣰⢏⡜⠁⠀⠀⣀⡤⠄⠒⠒⠚⠆⠠⡄⠀⠈⣧⠀⠀⠀⠀⠙⣿⣄⠀⠈⠙⢦⣄⠀⠀⠀⢸
⠀⢸⢀⣠⣴⠟⠉⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⣴⢃⠎⠐⠀⡠⢚⠡⠀⠀⠀⠀⠀⠐⠆⡸⠀⠀⠘⣆⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠉⠻⣦⣄⣸
⠀⢸⠟⠋⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢀⡼⠃⢾⠀⠀⢊⡜⠁⠂⠀⠐⠶⠞⠂⢰⡾⠁⠀⠀⠀⠘⣆⢀⠀⠀⠀⠀⠙⢯⡳⣄⠀⠀⠀⠉⢻
⠀⢸⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡴⢀⡾⠁⠀⠘⢄⢸⣌⠈⠿⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⣆⢧⡀⠀⠀⠀⠀⠹⣌⢳⣄⠀⠀⢸
⠀⢸⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⣠⡟⠁⠀⠀⠀⢀⣿⠹⣄⠁⠀⠀⠐⡆⢄⡾⢿⣧⡆⠀⠀⠀⠀⠀⠘⠈⢷⡀⠀⠀⠀⠀⠈⠢⡙⠳⣄⢸
⠀⢸⣄⣴⡟⠁⠀⠀⠀⢀⡼⠃⣰⠟⠀⠀⠀⠀⠀⣸⡇⠀⠘⡇⠀⢄⠀⠀⠁⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠀⡈⣻
⠀⢸⡍⠀⠙⠁⠒⢆⠴⠿⣄⣾⣏⡀⠀⠀⠀⠀⢠⡿⠀⠀⣠⠋⠠⡀⠈⠙⠋⠐⣖⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣆⡀⠤⠄⠒⠚⠉⠁⢹
⠀⢸⠉⠐⠂⠠⠽⠥⣀⡀⢀⣤⠀⠈⠉⢏⠉⠀⠛⠓⣲⠞⠁⡄⠀⠇⠁⠰⡀⣠⠊⣡⣔⠄⠒⠖⠖⠒⡊⠉⠉⠙⣤⠁⠀⡂⢀⡈⠿⠤⠒⢺
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠛⠀⣈⠹⠴⠁⠀⠀⠘⡀⢨⠀⠠⡏⢡⠀⠀⡷⡤⠤⠽⠤⠄⠗⠐⠒⠂⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⠊⠉⠁⠀⠐⠀⠠⠋⠀⠀⠸⣾⠄⠶⠛⢳⠆⠤⢳⠈⠂⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠙⢄⠀⢠⠌⠀⡄⠀⠀⠀⠀⠘⣆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡀⠈⠑⣠⠆⠰⠁⠐⠀⠀⢀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠃⠀⠀⠈⡖⠀⠀⠀⡄⠀⠀⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⢿⡠⠀⠀⠀⢱⠀⠀⠸⠀⠀⢰⠄⠀⠀⠀⡇⠀⠀⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⠈⠀⠀⠀⠀⡇⠀⢰⠃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢆⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⡘⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢀⠟⣇⠀⠀⠀⠀⠀⠀⠈⢢⢸⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡁⣸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠠⣠⠟⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⢀⡷⢸⡇⠀⠀⣼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡆⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⡇⢸⠃⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠹⡄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⡇⢸⠂⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢳⡼⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⠀⠀⠀⠀⣗⠈⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣆⠀⠀⠀⠘⡀⢹⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⠸⣆⠀⠀⠀⠀⣠⠎⠀⠀⢀⠄⠉⢃⠀⠀⠀⢡⡞⠄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⢷⡒⠊⠉⠀⠈⠑⢦⡌⠀⠀⠈⡄⠠⠞⠁⢻⡡⡇⠀⠀⠀⡀⢳⡀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠈⠇⣻⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⠠⠄⢷⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡂⠀⠀⠀⠈⢿⡀⢚⡀⠰⠄⠈⡇⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢉⣿⣿⠋⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣾⣻⣿⡀⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣈⣉⠉⣻⡟⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠒⠚⣼⣯⣭⣭⣤⡛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣼⣿⣿⣿⣿⣿⣿⠀⠀⢈⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣶⢀⣼⡟⠉⠙⢿⠟⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢾⣟⣠⣴⣶⣶⠾⡷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠿⠛⢶⣿⣷⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡓⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⣾⡅⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠈⠀⠀⠀⢻⣿⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢮⡈⠉⠂⢀⠀⠀⠀⠀⡀⠀⠐⠀⠁⢠⠖⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⢘⠶⣦⣀⡅⠀⠀⠊⣘⠳⣦⢰⢃⡅⣠⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠈⠀⠀⢀⠁⣢⠞⠁⠀⠀⠀⡀⢻⠂⡾⢠⡾⠛⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢡⠊⠁⠀⠀⠀⠀⠀⠀⠀⣷⣷⣬⣤⣤⣠⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⢳⠈⠀⠀⠀⠀⡈⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠟⠛⠛⠛⢙⣛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠈⠂⠄⠀⠀⣀⣀⣧⠁⠀⠀⣜⣀⣀⢫⡆⣠⣤⣉⣀⠐⠈⠁⠀⢀⣔⣩⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠠⠀⠀⠀⠀⠀⠀⣤⣾⣮⣛⠝⢿⣿⡾⡀⠀⢆⢻⣴⡿⣩⠟⣄⠘⠁⠀⠙⢄⠀⠀⣈⣿⣤⣸⠿⠟⠃
⠀⠀⠀⠀⠀⠀⢀⠖⠊⠁⠀⠀⢀⠔⣢⠐⠉⣡⡤⢝⣿⣷⣾⣿⣤⢀⣠⣤⠄⠿⣟⣳⡖⠚⠛⠛⢐⠶⣬⡃⠀⠀⠈⣽⡿⢠⡀
⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⢁⠨⠀⣴⡋⠁⠴⢿⠍⡱⡢⠊⣿⡟⠉⠀⠀⠀⠀⢙⢧⡀⡐⠀⠀⠁⠀⢑⣆⡀⠀⠀⢉⠣⣹⠆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠔⢀⡴⠋⠀⢞⠀⡠⠁⠀⢪⠀⠈⠀⣼⡟⢀⠆⠀⠀⠀⠂⠀⠀⡹⡌⠀⠀⠁⠠⡀⠙⢳⡀⡀⠀⢱⠈⣧⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡌⢠⠏⠀⠀⡠⠂⠐⢀⣀⡀⠁⠀⠀⢸⠏⠄⠎⠔⠀⠀⠀⠿⠀⠀⠠⣹⠐⠀⠀⠀⢀⣀⣀⠹⣞⡆⠀⢧⠙⠌⡄
⠀⠀⠀⠀⠀⠀⠀⡌⢀⠡⠀⣰⠀⠁⢠⠁⣥⣿⣿⣿⣶⣤⣮⣅⠀⠚⠀⠀⠀⠀⠀⠃⠀⠀⣣⣤⣶⣷⣾⣿⣿⡇⠀⢿⠘⡄⠘⡀⢰⠇
⠀⠀⠀⠀⠀⠀⢠⣹⡏⠁⠀⠁⡄⠀⣆⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠰⠀⣦⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⢷⠀⠃⠀⢰
⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢰⠁⢠⡜⡆⠘⢟⠻⣿⣿⣿⣿⣿⡿⢃⠀⠀⠀⡀⠀⠀⠈⢌⣻⣿⣿⣿⣿⣿⠟⠀⠀⠀⡇⠦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠇⠰⡇⠀⠸⠆⢈⢇⡇⠀⠀⠁⠺⠯⣭⠿⠏⠁⡄⠀⢀⣾⣿⣦⠀⠀⠀⠛⠂⢼⡿⡏⠁⠀⠀⠀⠀⡇⡇⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⢰⢀⠀⠁⠀⢠⠇⢸⣞⡆⠀⠀⢀⡀⠀⠈⠀⠀⠀⠁⠀⡎⠿⣿⡿⢧⠀⠀⠀⠀⠀⠁⠀⠀⢀⡄⠂⠘⡏⠇⠀⠀⠘⡈
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠆⢰⠛⡿⡀⠀⠀⢿⣿⣶⣾⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢠⠀⢠⣴⣾⣿⠇⠀⢠⠱⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢠⠀⠀⠀⠀⠀⠄⢧⠇⠀⠀⢼⣿⣿⣿⣿⡀⠀⢀⣶⣶⣶⣶⣶⣶⣿⡀⠀⣠⣼⣿⣿⡟⠀⠀⠸⠇⢠⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢣⠀⠀⠀⠀⢳⠘⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⡠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡣⡀⠠⠈⠈⣧⠀⠀⠀⠀⠐⠌⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣿⣿⣿⣿⡿⠂⠀⢰⠀⠀⣠⢎⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⣓⣘⣷⡀⠀⠀⠀⠀⠘⢶⣍⣛⠻⠿⡏⠉⠋⡅⢹⢺⣿⣟⡋⠑⠀⠀⠀⢠⣏⣀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠿⠿⢿⣦⠀⠀⠄⠀⡘⣷⡈⠉⠉⠁⠀⠀⠁⠀⠉⠁⣿⠀⢡⠔⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣿⣧⣹⣮⣿⣦⣀⠀⠀⢰⠐⢀⣀⡾⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠙⠓⠛⠛⠋⠁

⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠟⠛⠛⠛⠛⠿⠿⢶⣦⣄⡀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣦⣄⣀⣤⣤⡶⠖⠋
⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠹⣿⣿⣿⣿⣿⣿⢿⡛⠛⠳⣦
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⣸⣿⣿⠿⡙⡝⣎⣁⣲⣶⠛⠃
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣛⣿⡇⣧⡇⣵⣿⠘⠻⠿⢦⡀
⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣻⣯⡭⠽⣛⣋⣴⡿⠃⢠⣤⣠⡾⠃
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣮⣽⣿⣿⣾⣷⣄⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢻
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠞⠋⣁⣤⠶⠛
⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿⠛⢷⣦⣄⣄⣀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣤⠶⠛⣿⣷⣶⠛⠉
⢸⣇⠀⠀⠀⠀⠀⠀⣶⣟⣻⠉⠀⢀⣀⣤⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣠⡶⠟⢋⣡⣶⣻⣯⡵⠟
⠀⣿⡄⠀⠀⠀⠀⠀⢻⣯⣵⢎⡾⠛⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⡟⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣡⣴⣾⣿⣿⡏⠉
⠀⠘⢿⡄⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠉⠀⣠⡿⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⢀⣴⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣤⣀⣀⡀⠀⢀⣀⣠⣤⣶⠿⠛⠁⠀⢀⣠⡼⢿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⣀⣤⠶⢋⣡⣶⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣉⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡾⠛⢿⣟⣡⣴⠞⠋⠁⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⢋⡭⢴⣿⠀⠀⣸⡿⠋⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢀⣤⡾⠛⢁⣠⠴⠚⣩⠔⡫⢟⣿⠟⠋⠁⢀⣴⡾⠿⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⣠⡾⠛⢁⡤⠖⠉⣀⡴⢊⡵⢋⣴⠟⠁⢤⣤⣄⣋⣁⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢸⣿⣠⠖⠉⢀⡠⠞⣡⠔⢋⣴⡿⠁⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⢿⡿⠋
⠀⢿⣇⢀⡴⠋⢀⠞⠁⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⠈⠁⠀⠉⠀⠈
⠀⠈⠻⠿⠶⠾⣿⡶⠾⠟⠁

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠄⡀⠀⢤
⠀⠀⢏⢀⠀⠀⠀⠀⠀⠒⠁⠀⠀⣀⠀⠀⡼⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠠⠘⠆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⠙⣆⠀⠀⠀⠠⡠⡄⠀⠀⠀⠘⠉⠂⠀⠀⠀⢇⡠⠀⠈⠉
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠅⠤⠒⠁⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠂⣤⠁
⠀⠈⠀⠀⣴⠀⠀⢀⡀⢰⡆⢢⠀⠀⢠⠀⠀⠀⡤⠠⡀⢰⡄⣀⢀⠀⢤⠄⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠙⡀
⠀⠀⠀⠀⡟⠀⠃⣼⢡⠀⡇⠁⡆⠀⠈⣧⠀⠀⢿⠐⠄⢸⠀⣿⠀⠀⢸⠀⡏⢀⠸⡆⠀⢠⠄⢳⡐⠀⠀⠄
⠀⠈⠀⠀⠁⠀⠘⠃⠀⠀⠁⢠⡇⠈⠀⠇⠀⠀⠸⠤⠂⠈⠀⠟⠀⡀⠈⠀⠃⠀⠀⠃⢠⠟⠂⠀⡇⠀⠰
⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠑⠆⡄⡈⠀⠐⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⡄⠀⠁
⠀⠈⠀⠀⠀⠆⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢗⡂⠨⠈⠐⢱⠀⠀⠀⠎⠑⠊⢁
⡀⠀⢀⠀⢀⠖⠁⠤⡀⠀⠀⡤⠀⠀⠈⠂⠐⠄⠄⠀⠀⠀⠀⢠⣤⠴⠦⠒⠛⠁⠀⠀⠈⠢⠠⠁⠀⣀⠀⠊

⠀⠀⠀⠀⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡟⠃⠘⠀⠀⠀⠀⢀⣾⠟⠠⢡⣴⢦⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢡⠀⢄⠈⠁⢀⡀⠀⠟⠀⢁⠀⠉⠸⠀⠀⠀⣼
⠀⠀⣸⡿⠾⠴⠤⠤⠀⣽⣿⠀⣤⡧⡂⢰⡿⢁⠆⠀⠀⣴⠛⢳⠀⢀⠔⠀⠀⠀⠀⠀⠁⠘⠄⠀⠀⠀⠀⠀⠂⢄⠀⠛⠀⡶⠀⠾⢀⣼⠇
⠀⠀⠁⠀⠀⠀⢀⣀⡀⣤⡌⠁⢯⣱⡜⠀⠃⠰⡇⠀⠀⠈⠛⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡄⠀⠀⠀⠀⠑⠠⢄⣀⡴⡻⠋
⠀⠠⠄⠒⠐⠈⠀⢠⢾⣿⡇⠀⠀⠉⣈⣀⣠⠀⠙⣦⠀⠀⠁⠀⠁⠀⡀⠀⡁⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠔⠒⠂
⠀⠀⠀⠀⠀⣀⠚⣱⢫⠇⡃⠐⢲⣟⣿⣿⡝⠓⠀⢸⡇⠀⠂⠀⠀⠡⠠⠀⠀⠀⠄⠂⠀⠐⠀⡂⠠⣆⠀⠉⠉⠑⠆⠀⠀⣠⣴⠎⠁⠀⢀⡄
⠀⠀⠀⠠⠎⠀⡼⠁⡮⠀⡂⠀⠉⠀⣉⠁⣀⣀⠀⠾⠃⠀⠄⠀⠀⠀⠐⠀⠀⠀⡀⠈⠀⠀⠀⠑⢈⣊⠦⠐⠀⠈⠀⣴⣭⠀⠀⠀⣀⠀⠘
⠀⡠⠏⠁⠀⡜⠀⠰⠁⠀⠂⠀⠀⠀⠴⢿⣿⣿⣷⠀⠀⠐⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡳⠀⠶⢸⡏⢷⣤⣀
⠀⠀⠀⠀⠉⠀⠀⢀⡤⠖⠉⣠⡄⠀⡀⠀⢉⠉⠀⠁⠀⠐⠀⠂⠀⠐⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠂⠀⠑⠀⠁⠐⠀⠐⡀⠐⠀⢘⡀⠨⡹⢿
⠀⠀⠀⠀⠀⣠⠶⠋⠀⢀⠀⢉⣥⠚⠇⠀⠈⠈⠁⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠘⠀⣃⠐⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠈⢅⠐⢛⠴
⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠀⠈⢃⡖⠀⢰⠂⠀⠠⠾⠇⠀⠀⠀⢀⠖⠁⠀⠄⠀⠀⠀⢀⡹⠇⠀⠠⠄⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⢠
⠀⢠⣾⡥⢀⠀⠀⠀⠀⠘⠂⠀⠈⠷⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⣁⡀⠀⠋⠀⢀⠎⠀⠀⠠⡦⠀⠀⠷⠀⠀⠀⡐⠙⠃⠀⠟⠀⠳⠄⢮⡀⡇
⠀⠁⢋⡴⣣⢂⡔⢀⠀⠀⠀⠀⠀⡠⠛⠀⣴⣀⣀⠀⠀⡠⠾⠻⠿⠿⠿⢇⠀⠸⡀⠀⠀⠀⣠⣠⣤⣴⠄⠀⠀⠁⠰⠀⠠⠄⠀⠀⠀⢘⡖⢻
⠀⠀⠈⠘⠁⠚⠀⠎⡰⠀⠀⠀⠈⠀⠆⠀⣯⠙⠁⠠⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⡀⠀⢀⣄⡉⠉⠐⠂⠠⠊⢛⣓⢶⡤⠀⡰⠦⠀⠈⣿⢷
⠀⠠⠀⠀⢄⡀⠑⠀⡁⠀⠀⡀⠀⢀⡴⠋⠀⠀⢀⣴⣲⣄⠀⢰⠸⠂⠀⡀⠀⠀⠀⠀⣩⣤⡤⡀⠀⠀⣀⠀⠈⠁⡏⠈⠀⡈⠣⠄⠀⠀⠘⡞⠃
⠀⣿⡆⠀⢀⣇⠤⠤⠴⣊⣡⡄⠀⠁⢀⡔⠀⠀⠁⢸⣿⣿⠀⠈⠲⣤⡾⠃⢀⣀⡤⠼⠒⡈⠞⠃⠀⠘⢧⣻⠂⠀⠂⢄⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠓⢒⣉⡁⠐⠀⠙⠉⠀⣤⠀⢀⠞⠶⠆⠀⠀⢸⣿⣿⠀⠀⠀⢀⡴⢚⡭⠅⣀⡴⠊⠁⡡⢴⡖⠤⡄⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⠙⠉⠁⠀⠀⠐⠇⠀⠀⠀⠒⠀⠄⠀⠐⠀⠈⠿⠇⠀⠀⠀⠘⠧⠤⠴⠚⠉⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈⠉⠉⠈⠁⠁⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢁⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⣠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣖⠀⠀⠀⠂⣄⢠⠂⠒⠀⠤⠠⡄⠀⠀⠀⢤⠀⠀⠀⠀⢃⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢁⠄⠈⠐⠀⠀⠀⠀⠀⠨⠘⠀⠀⠀⢀⢀⡇⠀⠀⠀⠸⠀⠀⠐⠀⡜⠂⠚⠣⠀⠠⠤⠤⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠒⠺⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢸⠉⠉⢙⣛⣛⡛⠛⠉⠀⣀⣀⠀⠐⣀⣐⡒⢲⠀⠀⢀⠀⠀⠀⢠⠤⠤⠀⠀⠀⠀⠀⠀⢘⠀⢀⠆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢸⠀⡔⠁⠀⠀⠉⠢⡀⢺⠀⠀⢱⢸⠀⠀⢹⢸⠀⠀⡀⠂⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠢⠀⠰⠋⠀⠀⠀⠀⣀⡀
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠇⠈⣲⢖⠁⠈⡱⢖⡁⠨⠀⠀⠀⠀⠀⠀⢸⠀⣸⡀⠀⠀⠠⣀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠦
⠀⠀⢸⠘⡄⠀⠀⠀⠀⠀⡆⢸⠀⠀⡱⢸⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣅⠀⠀⣀⣀⣀⣀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠅⠄⢀⠔
⠀⠀⢸⠀⠈⠢⢄⣀⠤⠊⠀⠈⠓⠚⠁⠀⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⡄⠀⠀⠀⢸⠉⡇⠀⠀⠀⠠⡊⠁⠀⠀⠀⠀⠉⢢⠀⠀⢄⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⡼⢀⡇⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠈⣠⡶⢤⠤⣤⡐⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⢙⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠎⢏⠙⠧⡌⠇⣟⣆⠀⠀⢀⣤⣠⣀⠀⠀⠀⠀⠀⠀⠁⠁⠙⢰⠈⠁⡆
⠀⠀⠀⡏⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠐⠈⠀⠉⠰⠀⠀⠘⣆⣗⢀⢤⠇⣴⠂⣿⠀⡜⠁⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠡⣀⡀⠀⡠⠀⠀⠀⠙⢤⡠⢤⡴⣑⡫⠃⠀⢣⡀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠇⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⢄⠈⠉⠉⠁⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⢠⠀⠀⠀⡇⠀⡆⠈⠉⠰⠀⠀⠀⠀⠀⠄⠀⢰⠁⠀⠀⠀⢑⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠤⡄⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠈⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠐⠐⡂⠀⢸⠀⠀⠀⠀⢈⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠃⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⡄⠀⠀⠃⠀⠀⠁⠀⠀⠣⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠸⣀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⡄⠀⠀⠀⠀⠀⣴⠫⠭⠂⠀⠀⠀⣈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠟⠙⠀⠀⠀⢠⡎⠓⠾⠀⠀⠀⠀⠈⠉⠹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡥⠀⠀⠀⠀⠀⠀⠀⣴⡓⢮⠆⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠟⠁⠀⠀⠀⠀⠀⠀⠿⠒⠎⠀⠀⠀⠐⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠎⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡁⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢲⣾⡤⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⣶⡆⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠐⠠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢂⣀⠀⠀⠀⠤⡂⠈⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠈⠰⠂⠐⠀⠀⠀⠀⡀⠀⢧⢡⡤⠈⠁⠀⠀⠀⠀⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠡⡀⠀⠂⠔⢄⠀⠀⠸⡌⣦⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡶⢖⣴⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢀⣼⣦⡀⠀⠀⠙⠆⠷⠊⠀⠀⠀⠤⠠⡀⠀⠀⠀⠀⣠⡖⠂⣾⠇⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣠⣿⣿⣿⣿⠦⡀⠛⠋⠠⡀⠀⠀⠀⠄⠠⠂⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⢀⣾⠏⠀⢠⣾⣿⣿⡿⠋⠁⠀⠀⣱⣤⡀⠐⡄
⠀⠀⠀⠀⠀⣼⠏⢀⣴⣿⣿⠟⠁⠀⣀⣤⣶⣿⣿⠿⠛⠋
⠀⠀⠀⠀⣼⠏⣠⣿⡿⠋⢁⣠⣴⣾⠿⠟⠋⠉
⠀⠀⠀⣰⢃⣾⠟⣉⣴⠾⠟⠋⠉
⠀⠀⢠⣿⡿⠗⠋⠉
⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢛⠃⠂⢀⣔⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠤⠊⠀⠀⠀⣀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠠⠤⠄⠀⠀⠀⠀⠉⠐⠀⠀⠐⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡉⠐⠒⠬⢧⡠⠒⢈⠀⠀⠀⡉⠐⠠⡀
⠀⠀⠀⠀⣠⢴⣂⣀⡀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠠⡀
⠀⠀⡴⠊⠐⡡⠤⠒⠊⡟⠀⠀⣀⣂⣀⡀⠀⠀⣀⠤⠒⠊⠀⠀⡘⠁⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢂⠡⠁
⢠⠊⠀⠀⠈⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⢀⠜⠀⠀⢀⡥⠖⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣉⣁⠀⠀⠀⠀⣀⣤⣴⠭⢶⡟⢀⡤⠾⠿⠶⢶⣶⡄⠀⠀⠀⡄⡰⠊
⠘⠢⣀⠀⠀⠀⠀⠈⠂⡤⣀⡀⣉⣉⣉⣉⠉⢀⣀⣤⣴⣾⠗⠉⣠⡤⠠⠀⠀⠙⠇⣰⡠⠞⠉
⠀⠀⠈⠑⠲⢤⣀⣀⠐⠀⠡⠍⠁⠀⠰⢐⣚⣋⣭⠿⠛⢁⣴⣾⣿⣿⣦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠚⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠻⠿⠛⠋⠁⠀⣾⠇⣿⠀⠀⠙⠻⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠚⣆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢛⣷⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠃⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠷⣤⣆⣠⡀⢰⡃⣠⣀⣤⣤⡶⠞⡏⢀⣴⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⢀⣀⠀⠈⠁⠙⠉⢹⠉⠙⡩⠃⢀⣼⣾⡶⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣉⣉⣻⣶⣶⣷⣶⣾⣦⣺⣷⣞⣉⣉⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣋⢸⠁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣱⣷⣎⡓⢒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⣾⣇⣏⣿⣾⣷⢿⣶⣶⢶⣶⣶⣶⣶⡶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⣩⡴⠞⢛⡿⠟⠋⠉⠉⣹⠟⠛⠛⠻⣿⡷⢿⣶⡖⠛⠛⠷⢤⣈⠉⠑⠦⣌⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡤⠚⠁⢠⡖⠉⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢹⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⣄⠀⠈⠳⡄⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⣠⠞⠀⠀⡴⠋⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠘⢷⡀⠀⠙⣆⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⣰⠋⠀⠀⡼⠁⠀⠀⢀⡴⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⣤⡌⠂⠀⠀⠀⠈⢷⡀⠀⠸⡆⠀⢻⣆
⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⠁⠀⠀⣸⠁⠀⠀⣠⠟⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣾⣇⠹⣄⠀⠀⠀⠀⠈⢷⠀⠀⢻⡀⠈⣿⡄
⠀⠀⠀⠀⠀⣾⡏⠀⢀⠃⠀⠀⢀⠇⠀⠀⣴⢋⣼⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣿⣿⣿⣷⡜⢦⠀⠀⠀⠀⠘⡇⠀⠘⡇⠀⢹⡇
⠀⠀⠀⠀⢠⣿⠁⠀⡜⠀⠀⠀⢸⠀⠀⠚⠓⠿⠿⣿⡿⠿⠿⣷⡄⠀⢀⣤⠀⢀⣴⣿⣿⣿⡿⠿⣿⡷⠞⡁⠀⠀⠀⠀⢿⠀⠀⣿⠀⢸⣧
⠀⠀⠀⠀⢸⣿⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡴⢋⣼⣇⠀⠈⡍⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠘⣿⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡞⣰⣿⣿⣿⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠀⠀⣿⠀⣾⡏
⠀⠀⠀⠀⠀⣿⣇⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⠀⢀⡏⢠⣿⠇
⠀⠀⠀⠀⠀⠸⣿⡀⢹⡀⠀⠀⢹⡀⢠⣄⡀⠀⠀⠇⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⢀⡟⠀⣸⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠹⣷⡀⢧⠀⠀⠘⣇⠈⢿⣿⣿⣶⡄⢀⣄⣀⡀⠀⣀⡀⠀⣀⣀⣈⠀⣤⣴⠶⠀⠻⠿⠋⠀⠀⠀⠀⢸⠃⣰⡏⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡈⢧⡀⠀⠹⣆⠀⠙⠃⠸⣷⣾⣿⠛⣿⣠⣿⣧⣸⣿⠿⣿⣶⣿⡏⠐⡿⠋⠀⡞⠀⠀⠀⣰⡟⣰⢟⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣳⣄⠀⠻⣆⠀⠀⠀⠀⣮⠉⠀⠛⠛⠻⠿⠿⠟⠀⠙⠛⠋⠁⠀⠀⠀⡼⠀⠀⠀⣴⣿⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣝⣧⣀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠁⢀⣤⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠷⣶⣦⣼⣷⣦⣀⣀⣀⣀⣴⣿⣅⣠⣤⣴⢾⣿⣿⣿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠒⢒⣚⣛⣉⣉⣭⣭⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠁⡼⣏⠙⠉⠋⠉⠀⠘⠃⠹⠳⣄⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡻⠁⡾⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⣝⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡴⣁⡼⢁⠾⠀⠀⠀⣤⠀⣀⣠⣤⣄⡟⠀⠀⠀⠀⠀⢙⢷⡎⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡼⢡⡿⢁⡎⠀⠀⠀⠀⠘⣧⠋⠉⢾⣏⣧⠀⠀⠠⠀⠀⠈⠃⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⣾⣷⡿⢁⠞⠀⠀⢸⡗⠤⠤⠛⠀⠀⠀⠈⠀⠶⢶⣆⠀⠀⠀⠀⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣧⠸⡚⠂⠀⠀⣧⠐⠀⠁⠀⠀⠀⠀⠀⠀⡀⠿⠤⠀⠀⢀⢸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡿⣿⣧⠱⠀⠀⠀⢹⡆⠀⠀⠀⡀⠀⠀⠀⡥⡶⠒⠺⠀⠀⠈⠘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣧⠙⢻⣻⣇⠁⠀⢦⠀⢻⡴⠚⠟⣿⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⠈⢇⣿⡄⠀⠀⠀⠀⠁⠀⠀⠸⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⡄⠈⢎⢿⡄⠀⠀⠀⡂⠀⠀⠀⢻⡀⠀⣿⠰⡀⠀⠀⠀⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡌⠈⠈⣷⡀⠀⠀⠀⠀⠀⠀⢸⠇⠴⣛⡡⠕⠂⠀⠀⠘⡀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⢿⡄⠀⣸⣧⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠃⠘⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢻⡄⠘⡿⣇⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢿⡄⠰⡹⡆⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢳⣄⠑⢻⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢿⡄⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢿⡀⠘⣷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢷⠀⢸⣧⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠩⢧⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⣿⡈⢿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⣷⠀⢻⡘⡄⠀⠀⠀⠀⡀⠀⠀⠄⢤⠭⠞⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣮⣷⡀⣸⣃⠰⠐⠃⣀⣀⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣧⣠⡴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⠀⠀⢀⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣻⣿⣿⣿⣿⣿⣿⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⣿⣿⣿⣷⣶⣤⣤⣄⡀
⠈⠒⠒⠒⠤⠤⠄⣀⣀⣀⣀⣀⣠⠶⣿⠟⣁⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣦⣀⠻⢿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⠿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢯⣀⣀⣰⣿⣿⣶⣶⣶⣶⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⢻⡏⠉⠙⠛⠛⠛⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠙⠀⠀⠀⠀⣽⣦⠀⠀⠀⠀⠀⠉⠛⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⠋

⠀⠀⠋⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠂⠑⡄
⠀⡀⠀⠀⠀⠀⠁⠈⠉⠐⠒⠂⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠤⠤⠤⠔⠒⠂⠀⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡏
⠀⠀⠀⢃⠈⠑⠦⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⡄⠴⢺⠁⠀⢰
⠀⠀⠀⢸⢀⡀⢸⠀⠀⠀⠀⠉⠐⠒⠠⠄⣀⠀⠐⢀⡀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠰⠖⠂⠁⠀⠀⠀⠈⣄⡀⢸
⠀⠀⠀⣼⠖⠱⣢⠀⠀⠀⠀⠀⡀⠀⡀⠂⡈⠀⠀⠀⠙⣀⠀⠘⡄⠀⠀⠀⠀⡠⠁⠀⠁⠀⠀⠀⠀⠀⠹⡕⢾⡀
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⣀⢈⣀⠀⠀⠀⡁⠀⠀⠀⠀⡀⢣⡀⠈⠢⡠⠔⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠑⠄⠀⣁⠴⠃⣉⡀⢉⡈⠋⢦⠘
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠈⣀⠃⠔⠉⠀⠀⠀⠀⠑⢌⢱⡠⠂⠉⠈⠉⠒⣄⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠗⡈⠀⠀⠀⢀⡀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡰⠠⡁⠀⣀⣀⣀⣀⣀⣤⠀⠐⠀⠀⠀⠀⠀⠀⡰⠀⠀⠘⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢠⡘⠋⠁⠈⠉⠁⡰⠇⠀⠀⠀⠀⣠⠖⠁⠀⠀⣠⠼⣳⢄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⡊⠀⠠⠆⢤⡒⣛⠓⠂⢲⣄⠀⠀⠀⠑⠂⠀⢃
⠀⠀⠈⠉⠐⠢⣀⠘⠀⠀⠀⠐⠗⢀⠀⠀⠀⣀⠀⠋⠂⠀⠀⠈⠒⠢⢄⠲⠐⠲⠡⠞⠁⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡆⢀⠖⠲⠒⠶⠦⠱⢶⡆⠡⠀⢀⠘⠠⠄⠠⠐⠤⠀⠷⠃⠀⠀⢤⠤⠔⢉⠀⣀⠄⠒⣀⢘
⠀⠀⢀⠀⠀⠀⠀⣇⢼⠀⡠⠀⠀⡀⠀⠘⡰⠆⠀⠀⠆⠀⠀⠰⠄⠀⠄⠀⠉⠛⢳⠊⢲⠚⠖⡳⡴⠶⢦⡉⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡶⡄⠺⠀⠁⠐⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠉⢣⠈⠹⡆⢠⣇⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠈⠀⠀⠀⡃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠀⠤⢺⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠐⢊⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠈⠄⠀⠙⠂⠤⡀⠀⢀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠀⠸⡀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠒⠦⣄⡀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠈⠐⠠⢀⣉⡐⠂⠤⠤⠤⠤⠐⠊⠁⣀⠤⠊⠁⠀⠀⠀⠉⠓⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠇⠀⠀⠀⠀⠩⠉⠉⠋⠛⠃⠐⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⡀⠤⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⣶⠀⠀⠐⢁⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠴⠙⢦⡀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠊⠀⠀⠀⠉⠒⠒⠊⠀⠀⠀⠀⠀⠈⠢

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀
⣿⣛⣛⣛⣛⣋⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣽⣿⣯⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠉⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⢀⣤⠞⠋⠉⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣄⠀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⣀⡴⠟⠉⠀⠀⢀⠀⠀⠀⢀⡾⠛⠛⢶⣦⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠋⠀⠀⠀⢀⣴⠋⠛⠳⠶⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠃⠀⠀⠀⠀⠺⢧⣤⣀⠀⠀⠀⠀⠀⠛⠳⢦⣤⠀⠀⠀⢀⡿⠀⠀⠀⣼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣠⣤⣤⣀⣀⡾⠃⠀⠀⠀⣸⠃⠀⠀⢰⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣴⠋⠀⠀⠉⠉⠁⠀⠀⠀⣠⡟⠀⠀⠀⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⣾⡇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣠⡾⠋⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣠⠞⠉⠀⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠠⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⡀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⣠⡾⠋⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣽⠀⠉⠙⠛⠶⢦⣄⣀⡀⠀⠀⠀⠈⠉⠛⠾⠋⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣄⣀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣄⣀⣿⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠶⢒⡚⣛⣩⢉⢉⡍⢱⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣿⢱⣾⣶⣿⡌⣿⡃⢸⡇⠘⣿⠇⢸⠃⠈⠉⢉⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁⠀⢿⠸⠏⠘⠉⠃⣀⣉⣀⣤⣭⣥⣤⣽⠤⠤⠤⠬⣽⣦⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠉⡽⢁⣠⠾⠶⠖⠚⠛⠉⠉⠀⠀⢀⡀⠀⠀⢠⣄⣀⠀⢀⣀⣀⣤⠴⡞⠁⠀⢔⠃
⠀⠀⠀⠀⠀⠀⡇⡇⢸⠁⠸⠟⠋⠀⣀⣀⣀⣀⣀⣀⣤⡔⢺⠹⡇⠀⠀⢸⡿⠉⣧⡟⠻⣿⠉⠳⠇⢀⡆⣿⡆⢀⢀
⠀⠀⣠⠤⠐⠒⠷⠷⠾⠤⢄⡀⠀⠀⣿⡏⢹⣉⡇⣿⣿⣇⠈⠗⠛⠀⠀⠈⠀⡦⠈⣠⡀⠀⢀⣤⡀⢸⣇⢿⠇⠻⠟
⠀⢸⡇⣀⠀⣀⡀⢀⡀⢉⣄⢹⡆⠀⠙⠁⠘⠉⠁⠈⠀⠀⣄⢀⣖⣰⡆⠀⡾⣷⠀⣸⢧⠀⠈⣟⣦⠈⠿⠀⢠⣄⣀⠰⠖
⠀⠘⣷⠟⣿⣯⡇⣟⣿⡟⣿⢸⠇⠀⠀⠀⣶⣾⣧⡄⠀⠀⠸⠉⣎⠠⠇⢸⣭⡿⠀⣿⠒⣇⢰⣯⣽⠧⠀⠦⡽⣿⡏
⠀⢀⣽⣇⢠⠌⠛⣃⠁⢠⣘⠿⢤⡀⡖⠒⣻⣿⣉⣉⡦⠀⡖⣆⣰⢗⠀⠀⡠⢀⢀⣙⠛⠋⠀⠀⠀⠀⢀⡀⡙⠁⢠⣀⡀
⠀⣿⠐⠈⢉⣉⢉⣭⠉⢉⡁⢀⡀⣷⢹⠉⢹⣿⠀⢠⠇⢰⣿⣿⣿⣿⡆⢠⢃⣻⠤⣱⡦⠃⢸⠋⠙⣶⠋⠀⢸⠀⠈⠛
⠀⢹⡟⢳⣾⢸⣾⢻⣇⡏⢷⣼⠹⡾⠘⡄⢸⣿⡀⣸⠀⠈⣧⠀⢱⢾⠀⠀⠀⢀⡤⠤⣄⠀⠘⢧⣀⠀⠀⢠⡟⢸⣷⡶
⠀⠸⣇⣬⣃⢌⠻⣀⣙⣂⣬⡿⢔⡇⠀⢩⠈⣅⢀⣀⡀⠀⠻⠼⠿⠞⠀⠀⣀⣼⠁⠀⠘⣇⡤⣄⠈⠙⢶⠏⠀⠈⢉⡀
⠀⢀⠛⠭⢭⣋⣛⡭⠧⠽⠒⠛⠉⠀⠀⠙⣴⣿⣿⣤⠋⢠⣤⣀⣀⠀⣀⣾⡁⢿⠀⠀⢠⡏⢀⣸⡇⠀⠫⡀⢀⡞⠁⠀⠈
⢤⠟⢶⠀⣀⣠⣤⣴⣶⢀⣀⠈⣯⡗⠸⡇⢸⡇⢸⡁⢨⡿⣿⣿⠋⣼⠁⠀⢹⣞⣆⣠⢾⡟⠉⠈⢷⡀⠀⣽⢸
⠸⠗⠚⠂⠷⠋⢉⠁⠁⣸⣾⠱⠯⠷⠀⣇⠘⡇⣿⠀⣼⢹⣿⣸⠀⣧⠀⠀⣸⡟⠀⢹⣾⠃⠀⠀⣼⠁⠴⠃⠈⣷⠤⠴⠞
⢸⠶⣆⠀⢠⠒⢻⠷⠀⢑⣋⣠⠢⡂⠀⠉⠉⠉⠉⠉⢁⣈⠉⠁⠀⠈⠳⡞⢹⡇⠀⣸⠏⢳⡤⠞⣁⣀⡀⠀⢸⠁⢀⣷⣶
⢉⠛⠀⠐⢵⠞⠉⢔⡸⣷⠼⠛⠐⢣⣾⡀⠀⠀⣠⡞⢧⣼⣢⡀⠀⠀⠀⠀⢨⠻⡴⠋⠀⠠⢠⠋⠀⠀⠈⢦⠘⡄⠉⠯⠋
⠉⠁⣴⠦⠤⠤⠤⢄⡉⠀⠀⢀⣖⣓⣒⣚⣦⠈⡖⣬⣥⡥⣦⡞⠀⡅⠀⠈⠠⡀⠀⠐⠠⠁⢸⠀⠀⠀⠀⡼⠀⣸⢀⣶⡆
⠀⠀⣿⠘⢁⣠⣴⠟⣿⢤⠀⣼⠃⠬⠔⠢⣚⡆⠣⣿⣷⣿⡿⢀⢀⡿⣀⠀⠀⠙⣦⠀⠁⠀⠀⣷⡤⠴⠚⠁⠐⢃⡀⠉
⠀⠀⠻⢾⡋⢁⣠⢞⡭⠋⠀⠸⣽⢿⣏⣿⣹⠃⠀⢀⡀⠀⢠⠃⠀⠀⠀⠘⢢⠀⣿⠀⠀⠀⡎⠀⣠⣀⠀⠛⢀⠘⠿⠃
⠀⠀⠀⢼⣛⡭⠖⠋⢀⣸⣾⡀⠈⠉⠉⠉⠉⠐⣧⣿⣷⠆⠈⢳⢶⣦⣴⢲⡏⠀⠟⠀⠀⠀⢱⠈⠛⠙⢀⡖⢹⡤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⠊⠀⠀⠈⣿⣿⠿⠿⠃⠀⠀⠀⠀⣠⠋⠀⠀⠀⠘⠉⠺
⣿⣿⢛⠿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⣶⣴⣿⣧⣔⣿⣷⣋⠁
⠀⠀⠰⠾⠿⠿⠽⠿⠛⠻⠿⠦⠹⢷⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⣒⡲⠿⣿⣿⣷⣶⣦⡰⡤⠀⣠⠤⠄⢰⣆⢀⣸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡯
⠀⠀⢰⣿⣿⡄⠘⢿⣿⣿⣿⡃⠀⠈⠙⢳⣾⠷⠛⠓⠾⠟⣹⡁⠀⠀⠀⠀⠀⠀⡀⢠⣀⠰⣀⣤⣍⠿⠿⠟⠂
⠀⠀⠀⢈⣉⣶⣶⣶⣯⣽⣭⠁⠸⣿⣷⡿⠃⠀⠀⣠⠀⣴⡀⠀⡀⢈⠘⢀⣄⢀⣇⢸⣿⠗⡈⢁⠈
⠀⠀⢀⣉⣉⣉⣭⣥⠤⠤⢤⣴⣾⣿⣿⣶⣶⣾⣿⣧⣾⣿⣿⣾⣿⣾⣾⣿⣿⢿⣿⢨⠿⣽⡿⣿⣷⡄
⠀⠀⠨⣅⠒⠀⡒⡒⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣠⠙⢿⣿⣿⣿⢿⡋⠀⠀⡾⠀⣠⣤⡰⢾⣶⣎⣁⣀
⠀⠀⢀⣹⣾⠝⠋⠉⠉⢛⣋⣛⣿⣿⣿⠛⢿⣿⣿⣿⣏⣾⡆⣿⣿⣿⣿⡛⣠⡴⣇⢘⣿⣾⣿⣿⣿⣿⣿⠿
⠀⠀⠈⠫⣟⣛⠀⠀⠀⣾⣿⣿⣿⣿⣾⣦⠀⢿⣿⣿⣿⢻⣗⣿⣿⣶⣿⣿⣿⣿⡇⢨⡿⠿⠻⣿⣿⠉
⠀⠀⠀⠐⠉⠡⣄⣄⣼⣿⣿⣿⣿⣿⣿⣷⣄⠘⣿⣿⡿⠀⠉⢉⢛⣿⡿⠟⠋⠍⡙⠈⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠡⡼⣿⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⡓⢹⣿⣷⡞⠉⠁⠀⠁⠀⠀⠀⠀⠱⠀⠆⠰⠀⠽⠇⠂⠀⠀⠡⠆⠄⠆
⠀⠀⠀⠀⠀⠐⠋⣅⣈⡨⠁⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⣀⠀⠀⠀⠀⣏⠧⡄⡆⡤⡄⢠⢀⣄⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠠⢭⡋⠉⣿⣿⣿⠟⢹⣿⠋⢀⠀⠁⠀⠀⠀⠀⠃⠘⠃⠃⠃⠐⠛⠚⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢶⡆⠤⠁⠭⠙⣿⣿⣷⣤⣿⠐⠋⠁⠂⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⢟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣿⣶⣯⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣇⣾⣿⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣽⣲⢶⣤⣤⣤⣤⣤⣤⣶⣾⣿⣯⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠶⣞⣻⣭⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⠷⠄⠛⠂⠙⠛⠠⠾⢿⡝⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢿⣿⣆⠀⠐⣤⠄⠀⠀⢀⡼⡀⣿⣯⣿⣿⣿⣿⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢿⣿⣦⠀⢶⡶⠀⢀⣴⣿⢃⣿⣿⢸⣿⣿⣿⣿⡽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠁⣼⣿⣿⣷⣄⣀⣴⡿⠟⣡⣾⣿⣏⣿⡿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣀⣀⣠⡾⣿⣿⣿⣿⠟⢻⣡⣴⣾⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⣿⣾⣿⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⢟⣫⣾⣿⣿⡿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⠞⠋⠉⠀⣴⣿⡿⠟⣫⣿⣿⣽⣾⣿⣿⣿⠿⠋⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣼⡿⢋⣴⣿⣿⣿⡿⠿⠟⠛⠉⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠋⠁

⠀⠀⠀⠀⣠⢤⡖⠤⠤⠤⠒⠁⠒⠤⠠⠤⠂⠑⠤⠀⠔⠂⠒⠤⣀⡀⠤⠖⢲⠤⣀⣀⠤⠒⠐⠢⠀⠤⠊⠐⠤⠀⠤⠒⠊⠐⠤⠤⠤⠒⠢⣄
⠀⠀⠀⢰⠦⢄⣑⡀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠈⢧⡀⢰⠃⢀⠀⢀⡀⢀⣀⣀⡀⢀⣀⣀⠀⣀⣀⠀⣀⠀⠀⡀⢠⠄⢀⣈⡆
⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠉⠐⢄⠀⠀⠀⠀⠙⠁⠀⢿⣤⣸⡇⣼⣋⣹⡇⣿⣍⣿⢾⣏⢙⣷⠹⣧⡾⠁⠠⠀⠀⠉⡃
⠀⠀⢠⠊⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⣿⠁⢸⡇⢻⡋⢹⡇⣿⠉⠁⢸⡟⠋⠁⠀⣿⡃⠀⠀⠂⢀⡤⠖⡄
⠀⠀⠘⢄⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠙⠀⠀⠁⠙⠁⠀⠁⠙⠀⠀⠈⠋⠀⠀⠀⠈⠁⠀⠀⠙⠉⠀⡠⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⣀⡆⣾⣲⣴⠆⠀⡆⠀⣴⠶⣴⣆⣤⢠⠶⣟⡁⣾⡛⢰⣦⡀⣤⠀⡇
⠀⠀⠀⠎⠀⠀⠀⠀⡄⠀⠰⠿⠇⠀⠀⠤⠤⠀⠀⠀⠿⠷⠀⠀⠀⠀⢸⠀⠷⡏⠹⠿⢦⡤⠷⠤⠻⠶⠃⠹⠋⠟⠀⢯⡤⠿⠤⠸⠇⠙⠿⠀⠱⡀
⠀⠀⠸⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⢡⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸⠁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄⠀⠀⠀⠀⠀⡎
⠀⠀⠀⡸⠀⡘⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣀⠀⡃⠀⠀⠀⠈⡂⠘⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠰⠆⠀⠀⠀⠀⢇
⠀⠀⢘⠁⠀⠑⠔⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⣠⠁⠠⣄⣀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⢳⡀⠀⠀⠀⠈⡃
⠀⠀⠈⢃⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠚⠉⠉⠈⠉⠁⠐⠂⠈⠉⠁⠉⠙⠒⠦⠄⡀⠀⠀⠈⠑⠠⣦⠀⠀⠀⠘⠁
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢤⡀⢀⣀⣀⣀⣀⠀⠠⠤⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣
⠀⠀⠌⠀⠀⠀⢀⠤⠐⠊⠉⢹⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠁
⠀⠀⠘⡄⠀⠀⠈⠲⡄⠀⠀⢺⠀⠀⠀⠸⠀⠀⠀⡠⢊⣴⣶⣶⣍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⣥⣶⣮⣕⢄⠀⠀⠐⠆⠀⢠⠃
⠀⠀⢀⠇⠀⠀⠀⠀⠘⢦⠀⢌⠀⠀⠀⠇⠀⠀⠸⢰⣿⣿⣿⣿⣿⣧⢹⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⢠⢱⣿⣿⣿⣿⣿⡎⡀⠀⠀⢰⠀⠸⡀
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢰⠀⠀⠀⠘⡜⢿⣿⣿⣿⣿⠟⡜⠀⠀⠀⠀⣼⣿⣿⣦⠀⠀⠀⠀⠾⠸⣿⣿⣿⣿⣿⠇⠉⠀⠀⠀⡀⢤⡥
⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠢⢝⣛⡛⠭⠊⠀⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠀⠀⠑⠬⣛⣛⡛⠥⠊⠀⠀⠀⠀⡇⡸⠁
⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡄⠀⣴⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡀
⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⡄⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣀⣸⣿⣿⣿⣿⣇⣀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡁
⠀⠀⠀⡉⢉⣉⠝⠂⠀⢰⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⡶⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢚
⠀⠀⢸⠗⠉⠀⠀⠀⠀⠀⠈⢄⠀⠀⣼⣿⣿⣿⣧⠀⠀⡠⠂⢘⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡃
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡜
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠇
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⣀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠘⡄
⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢠⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⣄⠀⠀⠀⡇
⠀⠀⢀⠜⠀⢲⠒⢠⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠉⠀⠀⠀⡠⠏⠈⢣⡀⠀⠣⡀
⠀⠀⠸⡀⠀⠀⠳⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⢑⡄⢀⠇
⠀⠀⠀⢸⠀⠀⠀⠹⡀⠁⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠀⠀⢠⡏⠠⠒⠐⠉⠀⠈⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠃⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠳⠄⠀⠤⠀⠤⣀⢀⡠⠤⠀⠤⣀⢠⠴⠛⢦⡀⣀⠔⠒⠀⠐⠂⠸⠇⠀⠒⠢⣨⣆⡴⠀⠿⣄⣀⠤⠀⠤⢄⡀⣀⠤⠠⠤⠀⠠⠎
⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠶⠄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣀⣴⠇⠀⠀⠀⢀⠀⣤
⠋⠐⠀⠀⠀⠀⠀⣸⣏⠀⠠⡐⠀⣴⢠⡆⠀⠀⠀⠀⠀⠀⡰⠶⠶⣿⡉⠀⠂⣀⠀⣠⡀⠟⠈⠀⠀⠀⠀⣼⠿⠛⣃
⣀⡄⠀⠀⠀⠀⠐⠉⣉⣀⡀⠀⠀⡆⠘⠀⠀⠀⠃⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠁⡙⠀⠀⠀⠀⠀⠀⢰⣧⣴⠿⠛
⢸⡇⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⣤⣿⠿⢦⣤⣤⣀⠀⠀⠀⠀⠐⠆⣴⡇⠀⠀⠀⢠⣅⣴⠞⠃⠀⠀⠀⢿⣭⡏
⠀⠀⠀⠀⠀⠀⣾⣻⠋⠁⠀⠀⢾⠾⣾⡿⠟⠛⠛⢦⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣿⡉⠀⠀⠀⠀⠀⠈⠁⡁
⠀⠀⠀⠀⢰⣶⣍⣍⠀⠀⠀⠀⢠⣤⣿⡇⠀⠀⠀⢸⠀⠀⢠⣀⣌⡁⠀⠀⠀⠀⠀⢨⠱⣶⣿⣿⣿⣿⣷⣦⣿⡀
⠐⠃⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠘⠛⠋⠃⠀⠀⠀⠘⠒⠚⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠊⠀⠈⠙⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣟⣿⣿⣿⣿⣟⠻⠿⢽⣿⣿⣿⣿⠿⠿⢿⣉⠙⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⠿⣯⣽⣿⣶⣶⠶⠗⠾⠛⠒⠤⠀⣶⣶⣶⣤⣤⡄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣜⠻⢿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⡀⠗⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⡡⣉⡔⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠛⡵⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠝⠿⠁⠜⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣽⣿⣧⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢋⣴⣶⣷⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣬⣽⣭⡍⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣻⣿⣧⣴⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣏⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣿⣿⡿⣟⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣍⡩⠘⣩⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡷⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣋⡥⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⣻⣿⣷⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⣐⣂⣀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠿⠃⡀⠙⠁⠀⠀⠠⠈⠀⠻⠟⢋⠈⢿⡷⠀⠈⠁⢀⣾⣿⠇⠀⡀
⣿⣿⣿⣷⠀⠀⣀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⠀⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠘⠋⠀⠀⠀⢨⠾⠀⠀⠀⣾
⠿⠿⠟⠁⠀⠀⠹⠀⠈⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⣄⠄⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⠠⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠂⠀⠀⠀⠀⠀⣤⣤⠀⢀⣀⢸⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠃⠀⠀⠀⠀⣄⠀⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠰⠆⠀⠙⣿⣷⣿⣿⣿⠀⠀⢤⣤⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢰⡆⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠄⠀⢂⠤⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⣿⡀⠀⠈⣹⣿⣦⡀⠀⠀⠀⣄
⣿⠀⠀⠀⡀⠀⠀⠀⠛⠛⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠃⠀⠀⠀⠉
⣿⣷⡀⠘⠛⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣴⣾⠟⠓⠀⠀⠀⠄⠀⠜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⢀
⠄⠙⠛⠀⠀⣠⣿⠀⠈⠋⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢤⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣿⡄⠀⠀⠛
⠐⠀⠀⠰⣿⠟⠻⣧⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣷⣄
⢈⠀⠀⠀⠀⠐⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⠟⠀⠀⠉⠋
⠒⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠋
⠀⠀⠀⠐⠸⠂⠀⢰⣶⠀⠀⠀⠀⢰⠗⠀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠈⡐
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⣶⣦⣀⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⣦⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀
⣒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢿⣿⡇⠀⠀⠀⠰⠀⢠⠀⠀⠘⠀⠀⠀⠀⠀⠐⠈⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠋
⠒⠀⠅⠀⠀⠀⠀⢤⠦⠀⠀⠀⠀⠰⡿⠇⠀⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣴⡶⠄⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⠀⠀⠰⠶⠀⠀⠁⠀⠁⠀⠀⠀⣼⣿⡇⠀⠶⣦⣤⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠉⠙⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠘⠋⠛⠀⠀⠠⠀⠁⠈⠀⠀⠀⡄⠀⢠⡄
⠀⠉⠀⠁⠀⠀⠀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⡀⠀⠀⢀⣤⣤⣧⣤⣄⠀⠀⠠⣤⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠘⠃⠀⠀⠀⠐⠀⠠⠄⠀⠸⠀⠀⢀⣀
⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣷⠀⠀⣀⣰⣀⡀⠀⠀⠀⢀⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡄⠀⡶⠀⠀⠈⠉⠙⠋⠁⠀⠀⣀⣼⣿⡟⠀⣾⣿⣿⣿⣿⠀⠀⡄⠀⠀⣰⡾⢿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠛⠛⠋⠀⣰⣿⡄⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣾⣇⣠⣶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣶⠀⠀⠀⣾⠟⠙⢻⣧⠀⠀⠀⣴⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠠⡿⡧⠀⠀⢹⣿⠿⠿
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⣿⣷⣤⡀⠀⠀⡈⢠⠀⢀⣤⣾⣿⣧⠀⠀⣀⡀⠉⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⣤⡤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣶⣶⣦⣶⣤⣤⣤⣴⣦⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⢉⡀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠈⠒⠿⠿⠿⠿⠻⠛⢻⠂⡉⠒⠿⠒⠶⠿⠷⣿⢶⠶⡲⢶⠟⠿⠿⣿⠟⡿⡟⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢥⡀
⠀⠀⢀⣰⣾⣿⣿⣿⣴⣿⣦⡏⠀⠄⠀⠀⣠⡐⢛⣯⡁⠈⠔⠀⠀⠀⠀⣌⢠⣿⡟⣏⣿⣭⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⢸⣿⣷⣿⡽⣿⠸⠷⠇⠃⠀⠀⠀⠀⢾⠆⠀⠀⠁⠙⠓⠓⠚⠛⠋⠉⠛⣿⠻⣟⢻⡟⠿⢿⠿⠃⢠⡌⢺⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣷⣦⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡆⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠀⢀⠙⠁⠀⢸⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⣀⠀⠀⢀⡀⠄⠠⠚⢿⣿⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢏⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠸⢿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡿⠀⠀⠸⡷⢄⡀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢉⠢⣈⠺⣿⣿⣿⣿⣿⣿
⠀⢸⣾⣿⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣤⠀⢴⣤⡷⢀⠻⣴⣦⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠼⠉⠃⠑⠹⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⠃⠇⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠉⢡⣬⣳⣄⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⢸⢼⣿⠿⠀⠀⣾⣿⢿⡧⢙⡿⠇⠀⠀⠠⠈⠀⠀⠀⠈⠀⠀⠤⠤⠤⢤⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡾⢿⣿⣿⣿⢻⡦
⠀⢸⠘⠉⢀⣀⣀⣼⣽⡿⠁⢰⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣺⣶⣧⣼⣬⣿⣇⣼⣇⡀
⠀⢨⣷⣶⣿⣿⣿⣿⣿⣇⡀⠶⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⡏⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⡏⢿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⣧⠞⢉⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠐⣿⣿⣿⣿⣇⣸⠇⠀⠀⠀⠀⠀⣠⣠⣴⣦⣄⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⣀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢹⠀⡙⠆⡸⡿⠻⠿⠿⣿⣿⣿⣿⡇
⠀⢰⣿⣿⣿⡿⠁⣿⡇⠀⢀⠀⠀⠀⠻⣿⣿⣿⣿⣧⠀⠐⡗⣰⣿⣗⢺⣿⣿⣹⣷⠀⠀⠀⠸⠷⠀⠀⠀⠀⢀⣿⣟⢡⡁⢸⣷⡀⠈⡻⢿⣿⡇
⠀⢸⣹⣿⣿⣅⣸⣟⠀⣄⣠⠀⠀⠀⠀⢹⣿⣿⣟⢠⣴⡦⢲⣿⣿⣿⠀⠁⢸⣿⣿⡆⠀⠀⠀⠃⠀⢀⢀⡀⢸⣏⣿⣈⠛⣸⣿⣿⣦⣷⡄⠙⡷
⠀⢘⠋⣹⣿⣾⣿⣿⣦⣼⣿⠀⢠⣶⣿⣿⣿⣿⡏⠉⢿⠇⢪⣻⣿⡏⠀⠀⠀⣛⡥⠀⠲⣤⡄⠀⠀⠨⡇⠁⡈⣿⣿⣿⣶⣿⣿⣿⣿⣿⠟⢁⡇
⠀⢸⣯⣸⣟⠿⡿⢏⣴⣿⡤⢌⠀⣸⣏⣻⣷⣿⣷⣈⢛⠛⠋⠉⠉⠢⣌⣀⣿⣿⣷⠶⠷⠿⠷⠓⢰⡞⠐⠈⠽⠿⣿⣿⣿⣿⣿⣿⡻⠏⡠⢏⠁
⠀⣸⣿⣿⡿⢻⣳⣿⣷⣶⡆⠉⠐⢿⣻⣿⣿⡿⣿⣿⡦⠀⠇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠀⠛⠁⠀⢼⣿⣂⣤⡾⠆⠉⢻⣿⣿⣻⣿⣼⢛⣁⣨⡀
⠀⢸⣿⣿⢔⠂⣾⣽⣿⣿⣇⠉⣁⣿⣿⣿⣿⠞⣿⠆⢐⢀⠀⠉⠀⠁⠀⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⢀⠉⠟⣿⣿⠙⠤⡤⣲
⠀⠀⣿⡇⢰⠟⣿⣿⣿⣿⣿⣄⢹⣿⣿⣿⣿⣴⣿⣦⣾⠆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⣺⡆⢀⢻⣿⡄⣙⣴⣿⠂
⠀⠀⣿⠁⣿⠘⣿⣿⣿⣿⣿⣄⡐⢿⣿⣫⣭⣽⣿⣼⣿⡫⠂⠀⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⢀⣴⠟⢁⠂⢰⣿⣧⠈⢿⣿⣿⡿⠿⠋
⠀⢸⠇⠀⡹⢂⣿⣿⣿⣿⣿⣿⡇⢰⣿⣯⡇⣼⣿⡙⣧⠀⠀⠀⣿⡇⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⡶⢇⠀⠠⣎⣿⣸⣿⣿⡇⠈⣿⣿⡿⠚
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣝⢿⣿⣧⣷⣴⠀⠀⣿⠀⠀⠀⠀⠁⠀⠤⠀⢸⡅⢸⡏⣧⠀⡆⡿⢸⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠰⠄⠀⡈⠛⣻⣟⣿⣿⣿⠏⠙⠏⢋⣠⣯⣿⡇⣿⣀⣀⣀⣏⣀⣐⣄⣠⣰⣀⣀⣄⣿⣧⣽⣾⣿⣴⣿⣷⣼⣿⣿⣿⣾⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢀⠀⢀⡀⡄⣠⠀⢀⡀⠀⠀⢀⠀⢠⡄⠀⠠⢠⢀⡄⣠⠀⠀⡀⡀⠠⢀⢠⡀⠀⠠⢀⠀⠀⢀⡀⢄⠀⢀⡀⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⡀⢈⠀⠁⡀⢀⣀⢀⠀⠀⡀⠀⣀⢀⢀⠈⠀⢀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣈⣀⣀⣉⣁⣁⣈⣈⣈⣈⣀⣁⣈⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇