Facial expression Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢐⠂⠀⠀⡐⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣢⡠⠤⢤⣤⠄⡀⠀⠤⡤⢠⠀⢰⣒⣶⣵⢀⣶⣤⢦⡔⡒⡶⣴⡖⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⡁⠀⠀⠁⠈⠉⠀⠀⣀⠈⠁⢀⡈⢁⠁⢀⠉⡀⠉⠈⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠣⠾⠿⣠⢸⢡⠾⢾⡤⠀⢸⡧⡼⠎⠟⢫⠷⢹⡧⢙⡺⢟⡲⣼⢽⣩⠿⠳⠛⣸⠜⠃⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠁
⠀⠀⠀⠀⡆⠠⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠚⠃⢠⠀⠀⢀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢦⡀
⠀⠀⠀⠀⠱⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠖⠾⠄⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠦⣴⡄⠀⡂
⠀⠀⠀⡠⠖⡉⠉⠀⠀⠀⠈⠐⠲⢄⠀⠀⠀⠀⠀⢠⠊⠁⢀⣠⣤⠤⢀⢠⠀⠈⠳⡄⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠲⢄
⠀⢠⠞⡡⠐⠒⠖⠒⠨⠐⠀⡀⠀⠀⣷⠀⠀⠀⠀⠘⡀⠊⢁⠈⠀⠈⢁⡘⠳⡄⠰⠁⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠸⠀⠑⠦⡀⠀⠀⠀⣱
⠀⢼⢸⠀⠀⢀⠁⠀⢀⠀⠁⢌⠑⠂⠘⡇⠀⠀⠀⡠⡄⠀⠈⠀⠀⠀⠀⠁⠀⢀⢄⠀⠀⠀⢘⠂⠀⠠⠈⠚⠉⠀⠀⠉⠉⠙⠊⢲⠀⢀⠌
⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⡌⣦⡇⠀⠀⠸⡀⠆⠀⠀⠀⠀⠀⣀⠀⠀⢰⠈⠁⠀⠀⠀⠈⠱⢠⠇⡄⠀⠀⠀⢀⡄⠀⠀⢸⡐⠎
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠡⠁⣇⠀⠀⠀⠉⠣⠀⠀⡄⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠂⡺⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⣧⠂
⠈⠀⠘⠀⡆⡀⠀⠀⠀⠀⠐⢄⠱⡜⡀⢸⠆⠀⠀⠀⠸⡄⠀⢈⡀⣀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣤⣤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠉⠒⢤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠒⠒⠈⠉
⠀⡠⢂⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⢌⠢⡀⠀⠀⠀⠄⠀⡀⠀⣄⠀⢠⡀⢀⠀⠀⠙⡟⢎⠳⡀⠀⠀⠀⠀⡠⣞⡿⠿⠛⠋⠀⡀⢀⠈⢢
⢰⠁⠑⠀⠀⢀⡀⢀⡼⠃⠈⠦⡀⠀⢁⠁⠀⠀⠈⣄⣄⠘⠲⠼⠗⠒⠛⠒⠻⠋⠀⢰⡈⠆⢡⠀⠀⠀⠀⡈⠅⠠⠀⢀⡀⠀⢸⢸⠀⠀⡇
⣆⠀⠀⠀⠀⢠⡅⣀⠰⠒⠒⠀⠀⠀⠈⢸⠀⠀⠀⣎⠁⡀⠀⠀⠨⠅⠈⠀⠀⢡⠀⢸⡇⠀⢸⠀⠀⠀⣠⡅⢂⡀⠀⠀⠄⠀⡜⠈⣀⠘⡄
⠸⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠹⠃⠀⡄⠀⠀⠘⠃⠈⡃⠀⠀⠀⠄⠀⠀⢸⠈⡇⡁
⠀⢳⠈⠂⠀⠀⠈⣀⠀⠀⠠⡀⢐⠀⠀⠂⠀⠀⠀⠈⠂⠀⠸⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠸⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⠋
⣄⡰⠀⡏⢄⠀⠀⠀⠀⠀⠘⡡⡆⡀⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡄⠀⠃⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⢀⠜
⠸⠛⠊⠐⠁⠀⠠⠄⠠⠄⠊⠀⠙⠓⠒⠁⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠘⠦⠧⠬⠟⠁⠀⠀⠀⠀⠀⠈⠢⣄⠀⢀⠀⠔⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠀⢀
⠀⠀⠀⠀⠀⠑⠀⠀⢀⡲⠁⠀⡀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡓⠀⠄⡀⠀⠀⠀⠀⣀⣠⣆⠀⠀⠓⠁⠀⠀⠀⠀⡀⠀⠐⡁⠀⠀⠲⠑
⠀⠀⠀⡤⠀⠈⠁⠀⠀⠀⢀⠀⠂⠀⠀⠀⢢⠀⠀⠁⠀⠀⠀⠀⠄⢱⠀⠀⢀⠄⠻⠿⠿⠻⣿⣿⡷⢐⠄⠀⢠⠃⢠⠀⠀⠀⠀⠁⠀⠀⠇
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢄⠀⠀⠀⠀⠀⠨⠀⠀⠘⢄⠈⣿⣶⣶⣤⣶⠏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠁

⠀⠀⠀⠀⠀⢀⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠐⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠴⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⣀⠈
⠀⠀⠐⠄⠀⠐⠶⠂⠰⠖⠀⠀⡰⠀⠀⠐⡄⠀⠀⠶⠂⠰⠶⠀⠀⡰⠀⠀⠀⡄⠀⠰⠆⠀⠰⠆⠀⠠⠂⠀⠀⢠⠀⠀⣤⠀⠠⠤⠀⠀⠕
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⢢⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠐⢡⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠁⢋⠀⠀⢘
⠀⠀⠀⣀⠀⠰⡆⠀⠠⠆⠀⢀⠂⠀⠀⠀⢀⠀⠘⠃⠀⠘⠁⠀⢀⠁⠀⠀⠐⠀⠀⠀⣀⣄⡀⠀⠀⠘⠀⠀⢀⠃⠀⠘⠓⠀⠘⠒⢀⠀⠊
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⣿⡆⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⣀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⠍


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⣀⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⣰⡟⣋⡭⠿⠉⢿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⡂⠀⠒⠲⢤
⠀⣼⣿⡀⢏⢖⣻⠆⠐⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⡎⢱⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠈⢀⡽⠀⠀⠁⠆⠆
⢸⣿⣿⡇⠀⠀⠋⠠⠶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢶⠀⠀⠀⠀⠀⠀⠀⢰⡇⠉⠀⢈⠛⠘⠇⠀⠀⢀
⠘⠉⡉⠳⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡒⠀⠀⠀⠀⠀⠀⠠⢼⡇⠀⣰⢇⣤⣤⡄⠀⠀⠸
⠀⠀⢯⣁⡀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣞⣋⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠏⣾⠿⠿⢿⠀⠀⠄
⠀⠀⠀⠹⣧⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡘⠀⠴⠆
⠀⠀⠀⠀⢹⠉⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢶⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠈⠀⠀⠀⢠
⠀⠀⠀⠀⣸⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⢖⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠿⠓⠖⠛⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣲
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠤⣄
⠀⠀⠀⠀⣤⡴⠾⠟⠻⠿⣿⠷⠂⠀⠀⠀⠀⠀⠀⡶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠡⡿⡇⠰⢖⠮⢁
⠀⠀⠀⢸⡟⢔⠒⣛⠀⠳⠜⣷⠀⠀⠀⠀⠀⠀⠀⣆⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠒⢁⡃⣀⠈⠀⢸
⠀⠀⠀⠈⡗⠀⠐⢨⠈⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣦⢶⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣦⡀⢼⡟⠋
⠀⠀⠀⠐⢣⠀⠀⢠⣀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⣒⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⠼⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣆⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠤⠄
⠀⠀⠀⠀⠀⠀⠈⣶⣶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠈⠀⠁⠀⠀⣨⡀⢈⡀⠀⢈⣠⣄⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⠀⠠⣅⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⠀⡀⢀⡀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠤⠀⠀⠀⠀⠤⠂⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⢀
⠀⠐⠀⠐⠀⠀⠐⠀⠐⠀⠀⡰⠇⠰⢇⠀⠀⠀⠀⠀⠀⠀⢐⣀⡂⠀⠀⣠⣤⡀⠀⠀⠀⣀⡀⠀⠀⢠⣀⣀⡀
⠀⠈⠓⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠠⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠁⠀⠘⠥⠌⠁

⠀⠠⠀⠠⠀⠀⠠⠀⠠⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠀⠀⠐⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠐⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠊

⢀⠔⣈⣶⡤⣲⣄⢢⠀⠀⠀⠀⠀⢀⣀⠀⠠⠒⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⠒⢶⣄⠀⠀⠀⠀⠀⠀⢀⡴⢲⣤⠀⢶⣦⡀
⣞⠀⢿⠟⡱⡙⠟⢈⡇⠀⠀⠀⡜⣥⣿⠃⣇⢻⣿⡇⣷⠀⠀⠀⠀⠀⠀⡼⢁⣹⡟⠁⢰⣾⡏⣇⠀⠀⠀⠀⠀⣼⠰⣾⣿⠀⠿⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⢐⠢
⠻⣤⡤⠞⠀⠈⠓⠛⠁⠀⠀⠈⣧⣈⡡⠂⠘⢦⣀⣠⡟⠀⠀⠀⠀⠀⣸⣃⣤⠿⠥⠤⠶⠿⠶⠾⠤⠀⠀⠀⠀⢿⡄⠉⢀⡿⢦⣤⡿⠀⠀⠀⠀⠐⢡⣿⠀⣤⣷
⠀⠀⠀⡔⢚⢢⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⣠⠀⠀⢈⣁⣀⣤⣤⣤⣤⣤⡤⠤⠤⠄⣀⡀⠀⠀⠈⠻⡿⠟⢁⣀⢉⣠⠒⠒⠂⠀⠸⣬⠅⠡⡙⠋
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⢀⣀⠤⠔⠒⠂⠉⠉⠉⠉⠀⠀⠉⠙⠻⠛⣋⡩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠚⢉⣉⣉⡡⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉
⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠤⠖⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣄⠀⢰⣫⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢀⠀⡜⢁⣶⠀⠀⠀⣠⠄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣶⡄⠰⣶⡖⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣿⠛⡀⢸⡘⢃⡆⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣶⠙⢦
⠀⠳⢷⣤⠜⠀⠀⠀⣇⣘⡋⠀⠀⠀⠀⠀⠀⢾⡁⠻⠟⢃⡐⠟⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⠞⠀⢀⣙⠯⠤⠄⠀⠀⠀⠀⠀⢀⠘⠿⣿⠟⠀⣸⡇
⠀⢀⠴⠒⠒⢢⣤⣤⡬⠉⠀⠀⠀⠀⠀⠀⣄⡘⠿⠶⠶⠋⠛⠶⠶⠟⠁⢀⣠⡶⠂⠀⠀⠀⠀⠀⠉⣁⠔⠋⣁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣀⣠⣴⠟⠀⣴
⠀⡑⠶⠴⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⣒⠒⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢬⣍⣉⠀⣀⡾⠁
⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄
⠀⢀⡀⣠⠊⣯⢢⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢀⡭⣝⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⢃⣰⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⢀⠆
⢠⢣⣼⡏⠸⣿⠃⡇⠀⠀⠀⠀⠀⠀⢸⣡⣺⡟⢠⣽⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⡤⠘⠛⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⣾⢨⣷⣤⣤⢤⣶⢻⡄
⠸⣄⠉⡸⢧⣄⣴⠇⠀⠀⠀⠀⠀⠀⢸⠹⠟⠱⣄⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠴⠋⠙⣎⣋⡼
⠀⠙⠛⠁⠄⢩⡁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠈⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄⢀
⠀⠀⠀⣷⣾⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠗⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣲⣲⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣷⣾⣥⡀
⠀⠀⠀⢿⠙⡏⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠒⠒⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣙⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢝⢉⡋⡹⠃
⣀⣀⣀⣀⣈⣉⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣖⣁⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⢊⣼⡗⣽⡎⢂⠀⠀⢀⠄⣤⠀⢈⣿⣦⠀⠀⠀⠀⠀⣠⢶⣤⠊⣹⣦⠀⠀⠀⠀⢀⢎⣲⡆⣿⣦⡀
⠀⠀⢸⣄⣩⠌⠑⠤⠞⠀⠀⢨⠘⡋⠀⢌⡛⣸⠀⠀⠀⠀⢰⣁⣺⣇⣀⣿⣇⣇⠀⠀⠀⢸⡈⠋⣡⣌⣠⡇⠀⠀⠀⢀⡤⢀⣦
⠀⠀⠀⠀⢠⢐⢢⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⣀⠀⢀⣉⣥⣤⣤⡤⠤⠤⠤⠀⠀⠀⠈⠻⠾⢃⣈⣉⠔⠂⠀⠐⠬⠁⢘⣋⡌
⠀⠀⠀⠀⢺⣿⣿⠇⠀⠀⠀⠠⠤⠔⣂⣉⣭⡉⠉⠀⠀⠀⣍⣉⠭⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠁⠤⠄⠀⠀⠀⠀⣀⣤⠤⠍
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣄⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣀⡎⠼⡆⠀⢠⢀⡄⠀⠀⠀⠀⢠⠖⣹⡆⢐⣿⢲⡀⠀⠀⠀⠀⠀⠀⣇⠻⢋⠀⢯⡥⠀⠀⠀⠀⠀⠀⣠⣿⣷⠙⣆
⠀⠀⠀⠈⣙⢋⢀⡀⢘⠒⠃⠀⠀⠀⠀⠸⣦⣉⡴⢬⣁⣼⠃⠀⣀⠀⠀⠀⠀⠙⠖⣃⠄⠒⠉⠉⠀⠀⠀⠀⠰⣈⠛⠁⣠⠏⢠
⠀⠀⠀⡀⠤⠶⠞⠉⠉⠀⠀⠀⠀⠀⠈⠐⠒⢒⠀⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠊⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠬⣙⣋⢁⡰⠋
⠀⠀⠀⠈⠑⠉⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣠⠀⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣐⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢤⣴⠰⣿⢸⠀⠀⠀⠀⠀⢸⡥⡔⢉⣷⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⡅⠻⠋⡼⠀⠀⠀⠀⠀⠀⡎⣶⣄⣀⣤⣾
⠀⠀⠀⣏⣡⠷⣤⡾⠀⠀⠀⠀⠀⢸⠛⡣⢌⡋⣸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⠞⠁⠀⠀⠀⠀⠀⠀⠻⠖⠋⠱⢭⠜
⠀⠀⠀⠈⢁⣀⣰⣄⠀⠀⠀⠀⠀⠈⠉⠀⣀⣉⡁⠀⠀⠀⠀⠀⠀⠈⠛⣶⣄⣀⣀⡤⠀⠀⠀⠀⠀⠀⢰⣆⣄⡦
⠀⠀⠀⠀⢸⡟⡟⣿⠀⠀⠀⠀⠀⠀⠀⠻⠁⠤⠈⠹⠄⠀⠀⠀⠀⠀⠀⠈⡻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠟⠇
⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣿⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣭⣭⣿⣽

⠀⠀⠀⡀⠔⠒⠒⠚⠓⠐⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠒⢒⡋⠉⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⣄
⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⢍⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⢾⣫⠿⠿⠛⠉⠀⠀⢀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣀⠔⡋⠉⠀⠀⠀⢀⠀⠁⠐⠢⡀
⠰⠀⡀⠰⣄⣈⣶⣦⣸⣦⣐⣦⠤⠀⠸⡌⢣⠘⡄⠀⠀⠀⠀⠀⠀⣑⠿⡀⠀⠀⠀⣰⡀⢻⢸⡆⠀⢃⠀⠀⠀⠀⠀⡴⢃⠤⢒⡒⠒⠒⣈⡐⠀⣄⠀⠀⠸⣆
⠀⡱⠴⠂⢴⠏⠹⠗⠄⠀⠀⠈⢆⠀⠀⣷⠈⠀⣧⠀⠀⠀⠀⠀⠘⠁⡀⠈⠀⠀⠁⡀⠈⠻⠘⠀⠀⡘⠀⠀⠀⠀⢸⠠⠁⠀⠀⣀⠀⠀⢑⠀⠐⢌⡑⠢⢄⢹⡄
⠀⢳⣀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠈⡤⢤⡟⠀⠀⡇⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⣖⠒⡆⠃⠀⠀⠀⠀⠸⡄⡇⡆⠀⢀⠀⠀⠀⠀⠀⠀⠇⢍⢢⣸⠇
⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠞⡠⠀⠀⢰⠁⠀⠀⠀⠀⠀⠁⠀⠀⡄⠁⠀⠀⠀⠀⠀⢸⡤⢃⠐⠀⠀⠀⢠⠞⠀⠰⠀⠁⡃⠀⠀⠀⠀⠀⠀⣏⠜⡰⢹
⠀⠀⠁⠀⠙⠂⠀⠀⣠⠀⠀⠀⢀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠅⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠸⠀⠐⡀⢢⠀⠀⠀⠀⠀⠀⠀⡈⠸⡀⡁⠀⢳
⠀⠀⠀⢀⠀⠠⠀⣼⠁⠀⠀⢰⠁⠀⢠⡀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠆⠀⠀⠀⠠⡄⠙⠄⠝⠈⠀⡞
⠀⠀⠀⠀⠑⠠⠤⠆⠀⠀⠀⠈⠷⢷⠤⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡠⢔⡂⠍⠉⠩⢍⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢔⠒⢊⣉⡭⠤⢤⣉⠓⣤⣎⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠱⠀⢀⣀
⠀⠀⢀⠔⠁⠈⠁⠀⠀⠀⠀⠀⠈⠑⠈⡳⣄⠀⠀⠀⠀⠀⡀⠳⣝⠒⣂⡭⢴⠖⠚⠒⠺⠳⠟⠩⠭⠥⠄⠀⠀⠀⢠⢊⠥⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⡈⢢
⠀⡰⢁⢂⠀⢀⠩⣐⠒⠒⠒⣂⣉⠀⠀⡘⢎⢧⡀⠀⠀⠀⢈⣤⠎⠉⠒⠋⠁⠀⠀⠈⠙⠓⠲⠄⠸⣷⠂⠀⠀⢠⠃⢣⠂⠠⠀⠀⠆⠈⠈⠄⠀⠲⠄⠀⠈⠆⠆
⢰⠁⢁⠇⠀⠀⠀⠈⠀⠀⠈⠀⠈⡆⠀⠘⣎⣇⢧⠀⠀⠀⠀⠋⠄⡀⠀⠀⠀⣠⢠⠀⠀⠀⠀⢰⠀⣂⡀⠀⠀⢼⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠
⠂⢠⠏⠀⠌⠀⠀⠁⠀⠀⠐⠀⠀⢡⠀⢂⠸⣽⢸⡆⠀⢀⠎⣹⠁⠁⠺⠆⢰⠁⠘⢧⠐⠗⠀⠀⠀⡍⠉⣦⠀⠸⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡄
⠄⢸⢠⢠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣇⢸⠀⣿⢀⡇⠀⠈⠦⣽⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣏⠤⠋⠀⠀⢣⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢠⠁
⠀⠘⣆⣇⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⠃⡿⢸⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠇⢰⢄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢸
⠀⠀⠈⠻⠷⣕⠦⣀⡉⠁⢀⡠⠔⠝⠋⡜⠀⠇⠁⠀⠀⠀⠀⠀⠈⠓⢤⣀⣁⠀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⢳⣮⢆⠎⡬⠁⢄⠀⠀⠀⠀⡠⠀⠱⣰⣌⣲⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣬⣵⣮⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣁⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⠁⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄⡄⠀⠀⠀⠀⠀⢥⠄⢠⡄⠀⠀⠀⠀⣄⠀⣠⠀⠀⠀⠀⣀⠀⣄⡀
⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠙⠁⠈⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠠⠄⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠐⠀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀

⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠒⠀⠺⠆⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⠀⠀⠀⠐⠷⠑⠾
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠞⠲⠃⠀⠀⠀⠀⠐⠻⠾⠂⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠
⠀⠀⠀⠘⠗⠘⠀⠀⠀⠀⠀⢀⣤⢐⣀⠀⠀⠀⠀⡄⠀⠀⢰⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⡆⣠⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⣤⠤⠆⠢
⠀⠀⡀⡤⠒⠀⠆⠂⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆
⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⠟⠁⠀⣠⣶⠾⠿⠿⠿⠃⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⣿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠿⣶⣤⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠠⣦⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠿⣶⡄⠀⠉⢀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠟⠀⠀⣿⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠙⠛⠻⣿⡏⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⣀
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠃⠉⠉⠛⠻⢷⣦⣄
⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣰⡿⠀⣴⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣶⣄⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣾⠟⠁⠸⣿⡀⠀⣽⡇⠀⠀⠀⠀⠀⠀⢿⣇⠀⢈⣿⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠋⠁⠀⠀⠀⠙⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠟⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣡⣤⣶⣶⣿⣿⣶⣶⣤⣌⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢠⣿⣿⣯⡈⠻⣿⣿⣿⣿⣿⣿⠟⢁⣼⣿⣿⡄⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣦⣨⣿⣿⣿⣿⣅⣴⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠟⢁⣴⣾⣿⣿⣷⣦⡈⠻⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⢿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⡿⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡈⢿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⡿⢁⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⡙⠛⠛⠿⠿⠛⠛⢋⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣼⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠟⢿⣯⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣀⣾⣿⣿⡿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣠⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠑⠊⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣤⣴⣿⠟⠋⠙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣥⡀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⠀⢀⣾⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⡘⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃

⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠈⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠉⠑⠢⡀
⠀⠀⡰⠁⢀⠤⣄⠀⠀⣠⢤⡀⠘⣆⠀⠀⠀⠀⠀⡰⠃⠀⣀⡀⠀⠀⢀⣀⠀⠘⢆⠀⠀⠀⠀⠀⣰⠃⢠⠀⢀⠀⠀⣄⠀⡀⠈⢆
⠀⢠⡇⠀⠈⠀⠈⠀⠀⠁⠀⠁⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠛⠃⠀⠀⠘⠛⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⢸⡄
⠀⠘⡇⠀⠀⢄⠀⠀⠀⠀⡤⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⢸⠃
⠀⠀⠱⡀⠀⠀⠙⠒⠒⠋⠀⠀⢠⠏⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠹⡄⠀⠊⠀⠀⠀⠀⠙⠀⢀⠎
⠀⠀⠀⠈⠢⢄⣀⣀⢀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⡀⢀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⣀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⢀⣤⠂⠀⠛⠀⠀⠀⠀⡀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⠒⠀⠀⢐⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠌
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢲⢢⡠⠤⢤⡠⡤⢄⠀⠠⢤⠤⡄⠀⢰⢒⣶⣲⢂⣴⣤⡴⣴⢲⢶⢆⣶⠖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠈⠉⠁⠀⠀⠀⠉⠁⠀⠉⠉⠉⠈⠉⠉⠉⠀⠁⠈⠉⠀⠁⠉⠁⠀⠀⠀⠀⢠⠀⠀⠠⠄⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⣾⣖⢉⣸⣃⣴⣦⣇⠀⠀⣾⣁⣶⣂⡯⢳⢯⣳⣏⢲⢭⣾⢭⣈⣿⡊⣱⡶⣼⠺⢭⠸⠃⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠠⡀⠀⠀⠀⠁⠈⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠁⠈⠀⠉⠀⠀⠀⠈⠁⠀⠀⠘⡄⠂⠀⠀⠁
⠀⠀⠀⠀⢰⠀⣀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠉⠀⠙⠛⠂⠠⣤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠊⠄⠀⠀⠈⠁⠀⠀⠄⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠘⠀⠀⠒⡆⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠉⠉⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠄⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡠⠤⠔⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠉⠀⠉⠀⠀⠂⠀⢀⠈⠐⠀⢀⠀⠈⠐⠀⣆⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡈⠀⠄⠀⠀⠑⠠⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠘⢆⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢐⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⠈⠳⣄⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢤⠤
⠀⠀⠀⠠⠁⠀⠀⠠⠂⠀⠀⡠⠁⠀⠙⠢⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⣀⣀⡀⠠⢴⣢⠄⠴⠀⠡⡀
⠀⢀⠔⠁⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠤⠀⠒⠀⠉⠀⠀⠒⠠⠀⠈
⠐⠡⣤⠐⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢃⠀⠀⠀⠀⡀
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠐⠀⠈
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠐⠲⡄⢀⢀⡤⢾⠓⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡇⡻⠛⠻⡆⠀⠀⠀⠀⠀⠨⢷⡄⠀⢠⡾⠉
⠀⠀⠀⠀⣄⠀⠀⠘⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⡄
⠀⠀⠀⣴⠂⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡔⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠈
⠀⠀⠀⠀⠈⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⡠⢄⠂⡄⢂⡤⢂⢀⡄⠀⠀⠀⠀⢐⠀⠀⡄⠀⠠⢀⡀⠀⠀⠀⠀⠀⠢⢀⣀⣄⣀⠄⠠⠠⠀⡀⠀⠀⠀⠀⠀⣀⠠⠠⠄⢠⡐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠁⠁⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠁⠀⠁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣤⠀⠠⢴⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⣤⣀
⠀⠀⠀⠀⠙⢦⡀⣤⣤⠞⠁⠀⠀⠀⠀⢀⡤⠖⠀⠀⠲⠶⢤⡀⠀⠀⠀⠀⢀⠀⠀⠙⠀⠰⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠨⠍⠓⠘⠖⠙⠆
⠀⠀⠀⠀⠀⡄⠉⠉⠁⠰⠆⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⡄⠀⠀⠀⢼⠀⠤⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠷⣄⡀⡴⠀⠳⢤⡤⠞⠀⠀⠠⠀⣤⠀⡞⠸⣆⠰⠆⠀⠁⠀⠀⠀⠈⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠉⠁⠐⠒⠃⠀⠀⠀⠀⠀⠐⠚⠋⠁⠐⠚⠛⠛⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠂⠀⠀⠀⠠⠚⠉⠍⠁⠂⠛⠁⠀⠋⠒

⠀⠀⠀⣰⠞⢩⣿⡛⢿⡷⣦
⠀⠀⢠⣏⣰⣿⣿⣿⣶⣿⡟⣷⣄
⠀⠀⢀⣟⣹⣯⡼⢻⣿⢻⣧⣿⢹
⠀⠀⢸⡟⢿⣟⣿⢿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠈⢷⡈⠻⢭⣽⣿⡿⠿⠿⠃⠀⠀⠀⣀⡴⠞⠋⠁⢀⣀⣤⣤⠀⣀⠀⣉⡙⠳⣤
⠀⠀⠀⠀⠉⠛⠛⠹⣇⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⡿⠿⠛⠉⠉⢠⢚⣼⡿⡑⠄⣤⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡾⠁⣀⣰⣿⠟⡁⢅⠂⡥⠌⣠⣴⠿⠋⢂⢃⣼⣿⣦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⢤⣾⢟⡠⡔⢁⣐⣬⣶⡿⠛⢁⠌⠒⣡⣾⡟⠀⡙⣿⣤⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣱⡿⠃⣪⣴⡾⢟⡛⢉⠑⠐⠎⣣⣬⡾⠋⠀⠻⣶⣔⠈⢿⣦⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⠟⣠⡿⢋⡁⢒⣠⣤⣶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠈⢻⣮⠘⢿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣌⢇⣾⢟⡔⣢⡷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢲⡀⢀⡀⠀⢻⣯⠸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⣼⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⢻⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⣟⣼⠋⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠘⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⠁⠀⠀⢿⡀⠀⣿⠀⠀⠀⣠⠄⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⣿⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣿⠀⠀⠀⠈⠛⠛⠁⠀⠠⣾⣁⠀⠀⠀⠀⡈⠛⠋⠀⠀⠀⠀⢿⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠈⠁⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⣾⣿⣭⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡇⠀⠀⢠⡾⠛⠙⠓⣦⠀⠀⠀⡗⠀⠀⠀⢠⣿⠐⡩⡻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡻⢷⠶⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠛⠛⠛⡛⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⣍⠙⠛⠁⠀⠈⠋⢋⣽⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⣤⡀⠀⠀⠀⡾⠀⠊⠀⠙⠷⣦⣀⣀⣠⣾⠛⠁⠈⣠⠈⣷⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠀⠀⢹⡆⠀⢀⡇⢨⣄⠀⠀⠀⣿⠉⠉⠈⣿⠀⠀⠀⣠⡀⢹⡄⠀⢰⡃⠀⠀⠛⠻⣯⣀
⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢶⡞⢛⢿⡄⢸⡇⢸⡏⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⡇⢸⡇⢠⣿⡛⢲⡶⢦⣄⡀⠙⢶⡀
⠀⠀⠀⠀⠻⠴⠞⠋⠁⠀⠀⢿⡀⠊⣿⡄⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⣇⣾⠣⠀⡾⠀⠀⠈⠙⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠜⣿⡇⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢺⡇⢸⣿⠫⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠃⠘⣷⣸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⣧⣾⢃⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠀⠽⠛⠿⠄⠀⠐⢿⠀⠀⠀⡿⠀⠀⠂⠿⠛⠋⠐⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠠⡄⠀⢠⡄⠀⠀⠀⠀⠀⠐⠂⠀⠚⠂⠀⠀⠀⠀⢈⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⣶⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄⠀⠀⠀⠀⠀⠀⠳⢤⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⣬⠓⠀⣰⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⡠⣬⠀⢉⡄
⠀⠀⠀⠀⠀⠖⠀⠲⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠈
⠀⠀⠀⢳⣠⣀⣄⣠⣄⡞⠀⠀⠀⠀⢸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣰⣶⡤⣤⣶⡀
⠀⠀⠀⠀⠙⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠶⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠄⠈⠁⠈⠀⠠
⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⣤⠂⠀⢠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠁⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠃⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠿⠿⠿⠿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠉⠉⠉⠉⠛⠿⣷⣄⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⠿⠿⠿⠛⠉



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣷⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠀⠿⠿⠿⠿⠿⠈⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣤⠄⠀⠀⠀⠙⠉⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣉⣙⠻⢿⠀⠀⠀⠐⠒⣶⠖⢄⠀⠀⠆⢀⠤⣶⡖⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⠀⠀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠸⠆⢁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠌⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠂⠈⠐⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡀⠂⠀⢂⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠈⠀⠀⠀⠘⠋⠀⠀⠙⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡀⠀⡀⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠐⠀⠁⠘⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠈⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠂⠐⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣶⢦⡀
⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣼⣿⢳⣦
⠀⠀⠀⠀⠀⣿⣹⣯⣞⣿⡟⣿⣸⢻⠆
⠀⠀⠀⠀⠀⣏⢻⣯⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢦⣝⣳⣾⠟⠛⠛⠁⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⡴⠋⢀⣠⣴⠶⠿⠶⢶⣤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣴⠟⠉⢀⣀⣤⣄⣀⠈⠻⣦⡈⠑⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⣀⡴⠃⠀⠈⠁⣠⡾⠛⠉⠉⠉⠉⠻⣶⡘⠃⢀⣤⣤⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⡉⠉⠀⠀⠀⠀⣴⠟⠀⠀⣠⣶⣶⣶⣤⡈⠛⠀⠈⠉⠈⠙⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢶⣤⣤⣤⡾⠋⠀⠀⣰⠿⢦⣍⠳⡌⠻⠄⠶⠶⠶⣄⠀⢹⡇⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⣀⠈⠉⠀⠀⠀⢀⣼⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠘⣧⠈⢿⡀⣧⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠃⢨⡟⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⢸⣆⡈⠳⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠘⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠛⣿⠀⠀⠈⣇⠀⣸⠁⠀⢀⡴⠀⠀⠀⠸⡀⢀⡾⠀⠀⠀⣿⠛⢷⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣤⣿⠀⠀⠀⠈⠉⢁⣀⠀⠛⠶⠀⠀⠀⣀⡉⠉⠀⠀⠀⠀⣿⣤⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢿⡀⠀⠀⠀⣰⠋⠁⠀⢀⣀⣀⡀⠀⠈⠙⢷⠀⠀⠀⢀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠀⠘⠀⠀⠰⠏⠁⠀⠙⠦⠀⠀⠘⠀⠀⢀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⣶⡶⢶⣶⡖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠾⠏⠉⠉⠉⠽⠒⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⡞⠁⢀⠹⣧⡀⠀⠀⠀⣠⡾⠁⠆⠙⣧⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣽⠷⠂⠀⠙⣆⠀⢸⡇⣠⡥⠠⠌⠛⠛⠾⠛⠋⠐⡄⢠⡁⢹⠀⠀⣾⠁⠀⠐⠿⣯⡀
⠀⠀⠀⠀⢠⠞⠉⣀⣤⠶⣾⠛⢻⣆⢸⡇⣿⠁⢣⡄⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⡆⣰⡟⠛⡶⠶⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠘⣆⠀⢻⣼⡇⣿⡄⡆⡁⠀⠀⠀⠀⠀⠀⠈⢸⡇⢸⣷⡟⠀⣸⠃⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡇⢿⡇⢁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⡐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠌⢿⢼⡇⢈⠀⣀⠀⣂⡒⠂⠀⢂⣸⣷⡟⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠒⠠⠄⠀⠀⣀⣠⡀⠴⠊⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠘⠁⠐⠀⠀⠀⠀⢀⠀⠀⠉⠀⠘⠁⢦⠀⡄⠀⠀⠀⠄⠀⢰⡲⠀⠁⠀⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠃⠀⠀⠠⠄⠀⣀⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠣
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠱⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⣀⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⢀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠐⠀⠂⠀⠀⠀⢒⠀⠈⠀⠁⠀⡆⠀⢀⡀⠀⠀⠀⠀⠀⠄⠄⠀⠀⡐⠐⠁⠀⠀⠋⠀⢠⠋⠉⡑⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡅⠁⣀⠁⠀⠀⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠘⠂⠆⠀⠐⠠⠀⠀⠆⠀⠀⢘⠀⠀⠀⠤⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢁⠀⠀⠀⠀⠀⠀⠇⠤⠀⠀⣢⠀⠀⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠈⡓⠀⢦⠀⠀⠀⠈⠉⢀⠀⢀⠈⠁⢀⠄
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⢰⠀⠁⣸⢏⠀⠀⠀⠀⠀⠰⠉⠿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⠤⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠢⣀⡛⣄⠀⣀⠸⡓⠀⠀⡠⠔⠒⠃⠀⠀⠀⠀⢠⡤⠒⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠾⠱⣿⡄⠀⠧⢤⣈⠀⠀⠀⠀⠐⠺⣜⣠⠤⠖⠛⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⡤⠔⠊⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣒⠿⠛⠉⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠠⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⡙⢲⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡙⠁⠘⡆⠈⢶⢤⣀⠀⠰⡶⢄⠀⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢦⠀⢸⡀⠀⠇⠀⠘⣆⠈⠉⠐⠛⠈⢺⠥⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠃⠀⠈⠀⠀⠘⠆⠀⠀⠀⣰⡃⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢾⣿⡀⢿⠿⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣖⡀⠛⢁⡀⠳⡄⠰⠒⣻⠁⠀⠀⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣇⣠⣅⠀⢠⠇⠀⠀⠀⠀⠈⠒⡛⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠦⣀⠀⠀⠈⢑⣿⠉⠬⠤⠞⠢⠄⠀⠀⠀⠀⠘⢅⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣴⣾⣿⣷⣦⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠛⠋⠛⠛⠛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⣶⣶⣶⣶⣤⡀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣼⣿⣿⣿⣆⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠉⢻⣄⣹⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⣿⡀⠀⠀⠀⠀⠀⢹⣿⣿⠛⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠪⡀⠀⠀⠀⢸⣿⣿⣿⡿⠛⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⠀⠀⠀⠈⠳⣰⢌⡀⡿⠿⠋⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠀⢀⠀⠀⠀⠈⠲⡀⡈⠔⠁⠀⠠⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠸⠴⠀⡖⠒⠃⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡐⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢎⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠈⠒⠀⠠⠰⣦⠐⠀⠁⠀⠀⠀⠑⠄⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⢻⣧⡘⠃⠀⠀⢄⠀⠈⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣉⠭⢿⡿⡀⠀⠀⠻⣷⣄⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣆⣀⣀⡀⠀⠄⠀⠀⠀⠙⠻⢷⣤⡀⠀⠠⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣠⠀⡀⢀⡠⣄⡀⠌⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡴⣋⡀⢈⢀⣤⡈⢳⡐⠀⠀⠀⠀⠀⠀⠀⡤⠖⢂⠀⠀⢀⡀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠿⠟⢹⠸⠿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠥⠵⠎⠥⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣸⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⢣⡄⠷⠀⡸⠁
⠀⠀⠀⠠⢞⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠚⠉⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠒⠋⠙⠒⠊⢁
⠀⠀⠀⠐⣡⠖⠒⢄⠀⠀⣀⣀⣀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣤⠤⠶⣲⠚⠂
⠀⠀⠀⠀⡇⠀⠀⠈⣷⠋⠀⠀⠈⢣⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣤⠤⠀⣠⣤⣀
⠀⠀⠀⠀⢧⢠⣶⡄⡇⠀⠠⣿⡆⠈⡇⠀⠈⠀⠀⠠⠖⣂⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⡀⣀⣤⡆⠀⠘⠿⡿⢋⡔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠔⠚⢆⠭⠀⠧⠠⠤⠤⠤⣤⡃⠀⠀⠀⠀⠀⡼⠉⣧⠀⡴⢳⠀⠀⠀⠀⠀⡌⣤⢹⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢃⣴⡿⠐⣧⣾⠀⠀⠀⠀⡀⠉⠧⠞⠑⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢡⠔⠊⠉⠲⡀
⠉⠐⣦⣀⣀⠈⠁⠁⠀⠀⢀⣀⡀⡄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⣉⣀⠀⠀⠀⣸⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢿⣆⣈⣸⡷⠀⠀⡇
⠀⠀⠹⡷⠶⠭⢭⣭⣉⣉⣩⠞⠁⠁⠀⠀⠀⠀⠀⣜⠉⢉⣿⣿⠏⠉⠁⠀⠀⠀⢿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⡃⠀⠈⣆⣠⠜
⠀⠀⠀⠈⠒⠤⣄⣠⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⣁⠔⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⣦⠀⠀⡔⠊⠉⠑⡄⠀⠀⠀⠀⠈⣉⠒⠒⣊⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⢂⠤⠬⣉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠻⡆⢀⢰⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢏⠀⠀⠈⢧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠈⢊⠉⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠁⣀⠤⠤⢄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠣⣀⣀⡜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠒⠭⠒⠉⠉⠉⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣀⣤⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢱⠦⡝⣲⠶⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣛⠛⣉⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⣦⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⣃⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⠩⠿⣁⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣎⠒⠶⢤⠘⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡨⢠⠈⢀⡰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠁⠀⠈⠑⠢⢤⣠⡏⠀⠀⠁⠃⠀⠨⠓⢄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢴⡾⡋⠢⠤⠄⠀⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⢀⣠⠖⠋⠁⣀⠐⢱⠀⠀⠴⠆⠑⠂⢄⡀⠀⠈⡌⠀⠀⠀⠀⠀⠀⠜⢀⡀⠀⠑⡀
⠀⠈⢻⣄⠀⠘⠄⠸⠃⠈⠢⡀⡀⢀⠤⡪⠀⣈⡴⢻⠁⠀⠀⠀⠀⠀⠘⠀⠀⠑⠀⠀⠈⠠⡀
⠀⠀⢸⠈⠓⢤⡀⠀⡀⠈⢆⣀⠍⠓⣨⠴⠚⠓⠂⠣⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⡠⠣⡀
⠀⠀⢸⠀⠀⠀⠙⠢⡉⠀⢀⣠⠖⠋⠁⠀⢀⠴⠋⠀⠀⠉⠓⠂⢀⡴⡀⠀⠀⠀⠀⠀⠀⢑⡢⡈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⢹⠚⠉⡇⠀⠀⢺⠚⠁⠀⠀⠀⢀⣠⠔⠚⢁⣀⡜⠀⠀⠀⠀⠀⠀⠐⠠⠬⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⢖⠈⣿⠀⠉⠉⣏⢸⡇⠀⠀⠀⠀⠀⠈⠒⠤⠥⠃
⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡜⢆⠀⠀⠈⡄⣿⠀⠀⠀⡿⠈⡇⠀⠀⠀⢠⣶⣻⣛⡏⣻⢙⣓⡄
⠀⠀⠈⠳⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠈⢆⠀⠀⠸⡇⠀⠀⠀⡇⠀⣷⠀⠀⠀⠈⣿⣟⣏⣹⣉⣯⣿
⠀⠀⠀⠀⠙⢦⠀⠀⢸⠤⠒⠋⠉⢹⠀⠀⠈⢧⠀⠀⢣⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⢸⠖⠒⠋⠉⠁⠀⠀⠀⠈⢧⣀⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠻⢿⣿⣿⠿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⣠⣤⣤⣄⢠⣿⣿⣄⢛⣸⣿⣿⡘⢃⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣹⣿⡏⣼⣏⣹⣿⣿⢸⣿⣿⡿⢛⣩⣭⣍⣙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣙⣻⣇⠻⣯⣭⣭⣿⡌⣿⣿⣰⣿⣿⣿⣿⣿⣧⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠈⠙⠋⠉⠀⠈⣩⣤⢹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣾⠟⠁⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣡⣾⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⣿⣟⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⣘⡂⢀⣛⠀⠀⠀⠐⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣀⡔⠦⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠈⣤⢄
⠀⠀⠀⠀⠀⠘⠃⠁⠘⠈⠀⠀⠀⠀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠉⢠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⡘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠐⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠈⠀⠈⠀⠁⠉⠀⠀⠀⠀⠈⠀⠀⠉⠀⠉⠁⠀⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉
⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡆⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠰⠄
⠀⠀⠀⠠⠐⣝⠁⠖⡁⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠠⠊⠉⠁⠀⠀⠀⠀⠐⠈⠉⠱⠀⠔⠉⠉
⠀⠀⠂⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⡆⠀⡔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡶⢦⠀⠀⢰⠿⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣀⠴⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⢋⡄⠀⠈⢉⠁
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⡀
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠠⠠⠂⡘⢠⠰⠦⠔⠀⠀⠀⠀⠐⠀⡆⠀⠀⢢⠄⡂⠀⠀⠀⠀⠀⢀⠔⠀⢀⢆⠤⠂⠀⠀⠀⠀⠀⠆⠀⠂⣐⣀⡤⢐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⣤⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⢿⢟⣿⣿⡏⠻⣿⣿⣿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠿⡿⠁⢠⣿⣿⡿⢹⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⢿⢶⣷⡆⠧⠀⢿⣿⣟⠁⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣻⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠞⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣥⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢀⡤⢤⣤⣀⠀⠀⠀⠀⠀⢀⣤⢤⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠋⠀⣀⣀⡉⠁⠀⠀⠀⠔⢉⣀⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⢏⡄⢿⡀⠀⠀⠼⢻⡯⠉⠀⠀⠀⠀⠀⠙⠫⠙⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣟⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡄⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠗⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡇⠀⣉⡿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⠀⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣾⣿⣿⣶⣷⣾⣿⣿⠿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⣤⣤⣬⣿⠀⠀⣼⠟⠉⠁⡀⠀⠈⢲⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⢾⣿⣷⣾⠄⠀⠀⠀⠀⠀⢸⣿⠟⠟⠻⣿⡇⠀⢰⡟⠈⠂⠀⠈⠃⠀⠀⠐⠃⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⡿⠲⣦⡀⠀⠀⠸⣿⡀⠀⠀⣿⡇⠰⠉⠀⠀⠠⢄⠀⢢⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⠀⠀⢸⠁⠈⣆⠀⠻⣷⣤⣤⡿⢇⠁⡢⠀⠀⠀⠀⠀⠈⠀⠴⢁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠟⠙⠷⠀⠉⠉⠀⢰⡶⠁⠀⠀⠀⠀⠀⠀⣠⡖⢋
⠀⠀⠀⠀⠀⣠⡶⠞⠋⠀⠀⠀⢰⡏⠽⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡤⣤⣤⢶⠟⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⠀⠈⠀⠀⠠
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢳⣷⣿⣿⣿⣻⣿⣿⣿⠿⣿⣄⣀⣀⠀⠀⠀⠀⣠⠃⠀⠉⠗⠀⠀⠀⠀⠀⣠⠊⠉⠙⢿⣦
⠀⢀⣀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠘⣿⡄⢠⡏⠈⠉⠉⠑⠛⠀⡴⠃⠀⠀⠀⠠⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠙⣧
⢠⡿⠁⠀⠀⠀⣠⣴⠶⠶⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠉⢷⡞⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⢠⣤
⠈⢿⡀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢧⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠙
⠀⠈⢷⡀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡙⠙⡃⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⢻⣆⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠹⣆⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡿⠁⠀⠙⢇⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⣧⠀⡞⢻⡏⠛⠛
⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣠⠃⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⢷⡀⠀⠀⢸⠇⡇⠀⠀⠀⠀⠀⠀⠘⣷⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⢿⡄⢀⡟⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣀⣠⣶⡆⠀⢳⡞⠀⠶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⠒⠂⠉⠉⠉⢻⡇⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡆⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⠀⠀⠀⠀⠀⣠⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣄
⠀⠀⠀⣠⢿⢤⣄⣀⡤⠽⢶⠚⠒⠀⠀⠀⠀⠀⠀⣰⣻⠦⠤⠤⠟⠛⠻⡋⠀⠀⠀⠀⠀⠀⠀⠀⡼⡳⢤⣍⣿⠿⠛⢻⡛⠂
⠀⠀⢰⡿⠋⠖⠂⠐⠲⠄⠈⢧⠀⠀⠀⠀⠀⠀⢰⣿⠁⠘⠉⠊⠙⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⡟⢁⠔⡀⠀⢲⡄⠀⢳
⢠⠶⡿⠁⠀⠰⡆⠸⠆⠀⠀⢸⡤⣄⠀⠀⢠⠒⡟⠃⠀⠀⠿⠸⠇⠀⠀⠈⡶⢤⠀⠀⠀⢠⠴⡟⠁⠠⡊⠉⠸⠉⠁⠀⠈⡧⢄
⢸⣀⡇⠀⠀⠰⣍⡤⠆⠀⠀⢈⣇⡿⠀⠀⠸⠤⡇⠀⠀⠀⢦⣠⠆⠀⠀⠀⣇⡸⠀⠀⠀⢾⣐⡇⠀⢀⡰⢯⣤⠶⡀⠀⠀⡇⢸
⠀⠈⣧⠀⠀⢠⠤⠤⠤⠀⠀⢸⠃⠀⠀⠀⠀⠀⢳⠀⠐⢧⣀⢀⣠⠔⠃⢠⠃⠀⠀⠀⠀⠀⠈⣷⠀⠉⣿⣦⣤⣶⡟⠂⢰⠋⠁
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠽⠏⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣿⡿⢟⠁⢀⠎
⠀⠀⠀⠘⢦⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣈⣉⣁⡴⠋
⠀⠀⠀⠀⠀⠈⢉⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠍⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠂
⠀⠀⠀⠀⠀⣠⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣶⡤⣄⡀
⠀⠀⠀⣠⢯⢶⣿⣿⠷⢙⣦⣤⡀⠀⠀⠀⠀⠀⠀⢈⣿⠯⣿⣿⣿⣿⣳⣤⠤⠄⠀⠀⠀⠀⠀⠀⢀⢎⣿⠿⢿⢷⣮⣿⣆⣄⡀
⠀⠀⢰⡿⠫⠶⣯⡉⣹⠿⠄⢣⠀⠀⠀⠀⠀⠀⢀⣿⢃⡴⣄⢩⡉⣩⡅⠹⡄⠀⠀⠀⠀⠀⠀⢀⣷⠎⣀⣉⠛⠋⣉⡉⠹⡆
⢀⡤⣿⠁⠠⣤⣿⢹⡇⡠⠀⠈⣦⡀⠀⠀⠀⣠⣼⠇⢠⡇⢾⠟⢿⢉⡆⠀⣧⢄⠀⠀⠀⠀⠴⣼⡏⢠⡍⢿⠟⠿⢩⡅⠀⣿⠤
⢸⡁⡏⠀⢀⡴⢯⣨⠷⣄⠀⠀⡇⡿⠀⠀⢸⡁⢹⠀⠀⢿⣖⠁⣀⣾⠁⠀⣿⢨⡇⠀⠀⠸⣄⢘⠀⠀⢉⠮⣀⣤⣋⣀⠀⢹⢀
⠀⠉⣷⠠⣿⣶⢻⢻⣹⡟⠇⢸⠗⠁⠀⠀⠈⠓⢻⠀⡴⢁⣨⡽⠟⣹⠷⠀⡿⠞⠁⠀⠀⠀⠈⠸⠀⢰⠟⣒⡿⠟⠋⠙⠃⡿⠉
⠀⠀⠘⡆⠀⠙⢿⡛⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⢷⠓⢯⣅⣠⣼⠇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠘⢦⡀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠛⠋⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠉⠛⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡒⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣈⣉⡉⣀⡀⣀
⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣄⡀
⠀⠀⠀⢀⠾⣽⣿⣿⣿⣿⠷⡤⠄⠀⠀⠀⠀⠀⠀⠀⡠⢮⠴⣮⣿⡿⠿⣶⠦⠄⠀⠀⠀⠀⠀⠀⢀⢴⢤⣞⣚⣖⣊⡽⣦⡤⠄
⠀⠀⢀⣿⡞⢁⡼⠉⢯⡀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢰⣷⠃⠖⠒⠔⠒⠲⠈⣧⠀⠀⠀⠀⠀⠀⠀⣾⡎⠠⠬⠉⠩⠅⠀⠈⣧
⠀⣤⣼⡟⢀⣫⠔⡉⠢⢗⡀⠀⣷⢤⡀⠀⠀⢠⢤⡷⠇⠀⠤⠄⠀⠤⠄⠀⢸⡤⣄⠀⠀⢀⡔⢾⣿⠀⠀⢿⡀⠿⠂⠀⠀⢸⡖
⠸⣇⣿⠁⠈⠐⢯⡤⠽⠂⠁⠀⢸⢀⡇⠀⠀⣷⢠⠀⠀⠀⠰⣁⣨⠆⠀⠀⠈⡇⣸⠀⠀⠸⣇⣸⠁⠀⠀⠻⠤⠞⠀⠀⠀⢠⣇
⠀⠈⢹⡀⢀⣤⠴⠒⠶⢤⡄⠀⡸⠉⠀⠀⠀⠈⠙⡇⠀⠀⠀⣠⣴⣶⡆⠀⢰⠟⠁⠀⠀⠀⠀⢸⡄⠀⠀⠠⣤⡤⠄⠀⠀⣸⠁
⠀⠀⠀⢧⠈⠁⠀⠀⠀⠀⠁⢠⠃⠀⠀⠀⠀⠀⠀⠸⡀⠀⠸⣿⡿⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠓⠂⠀⠀⢠⠏
⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⣛⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣿⣿⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢹⣿⣿⣷⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣼⣿⣿⣿⡀⠀⢿⠋⠛⣿⣿⣿⣿⣟⠋⢹⡇⠀⢸⣿⣿⣿⡂⣀⣀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⢉⣰⣿⣿⣿⣿⣷⣄⠀⢀⣼⣿⠛⠛⢻⣿⣄⠀⢀⣰⣿⣿⣿⣿⣿⡄⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠾⠿⠟⣻⣿⠿⠿⡿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠿⢿⡿⢿⣿⡛⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠰⠁⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠛⠉⠉⠉⠙⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⣿⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⣼⣿⣿⣇⠀⠀⠀⣿⣿⣿⣠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⢀⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⢹⣿⣿⣿⣿⠛⢿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠾⢷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠉⠻⣷⣄⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⣀⣀⣀⣀⣀⠀⠀⠀⢈⣿⡿⠛⠋⠉⠙⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠁⠀⢠⣿⠏⢀⣴⡄⠀⢠⣦⡀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠛⠛⠁⠀⠈⠛⠛⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣴⡿⠿⢿⣦⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⡉⠁⠀⠈⢉⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⣀⣀⣠⣴⣾⠟⠉⠙⠻⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠛⠛⠛⠋⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣶⣾⣟⠁⠀⠀⠀⠀⠀⠀⢘⣿⣶⣦⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠈⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣶⣶⣾⡿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣶⡶⠶⠶⢀⠀⠀⠈⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢲⡄⠀⠀⣰⡟⠀⠐⠷⠙⠃⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠋⣉⣠⠤⠖⠂⠈⣇⠀⠀⣿⠀⠀⢴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠤⠖⣚⣋⣥⠀⢹⡄⠀⣿⠀⠀⠘⣆⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⢩⢴⠖⡀⠈⣧⠀⢻⣇⠀⠀⠈⠳⣄⠙⠻⣦⣄⠀⠀⠀⢰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢏⣛⣠⡃⠀⢹⡄⠈⠻⣦⡀⠀⠀⠈⠑⠦⡈⠙⠃⠀⣰⡿⠁⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣀⣀⣭⢼⡿⠛⠋⠁⠀⠀⣹⡿⣶⣤⣀⣀⣀⣀⣠⣴⡾⠋⠀⢰⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠈⣇⠀⠀⠀⠀⣰⢿⡇⠀⠉⠛⠛⢿⣿⢿⡍⠀⠀⠀⣾⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢀⣴⠋⣼⠇⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡾⠃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣰⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢿⠷⠶⣶⠶⣦⣤⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣻⡄⢸⡇⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⠋⠀⠀⠀⠀⡄⠀⠀⣊⡤⠀⠀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠋⠀⠀⠀⠀⡄⠀⠀⡰⠁⠀⠀⣈⣳⠀⠀⢠⠏⠀⠀⠀⠙⡆
⠀⠀⢾⣤⣀⣠⣴⡆⣷⡄⣸⡷⣶⡄⢶⣦⣷⠀⠀⣠⢰⣅⢀⢀⣤⢸⣄⠀⢠⣼⢀⣠⠀⢀⣀⣯⡄⢀⣧⣤⡶⢦⡅⣌⣇⠀⣼⣤⣤⣦⣦⣶⣧⡀
⠀⠀⢸⠙⢧⠌⠋⢠⠟⠀⢸⡅⠉⣻⣊⠁⢸⠀⠈⢿⠈⠂⢀⡄⠙⢸⠟⠀⠈⢿⠛⠙⡀⡛⠛⣿⠃⢣⡇⠉⠡⠠⠍⢁⣯⠃⢹⡟⢻⣇⡇⠉⡿⠁
⠀⠀⠘⢆⠈⠊⢁⢾⠀⠀⠀⢃⠸⢿⡿⠀⡆⠀⠀⠀⢂⠀⠈⠀⠁⡜⠀⠀⠀⠈⢆⠀⠻⠁⣠⠇⠀⠀⠹⡈⠉⠐⢀⣸⠁⠀⠈⠃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⢹⠉⠁⠘⠀⠀⠀⠀⢰⠄⢶⠈⠀⠀⠀⠀⢀⡏⠐⠂⢹⡀⠀⠀⠀⠀⠀⠋⠐⠈⠁⠀⠀⠀⠀⣸⠐⠳⠋⡇⠀⠀⠀⢸⠓⠒⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄⠀⠀⠐⠀⠀⠀⠀⣠⠐⠉⠈⠐⢂⠀⠀⠀⡐⠀⠉⢆⠀⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠔⠈⠉⠑⠲⣄
⠀⠀⠠⠁⣤⠄⠀⠠⣜⡆⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⣧⠖⠂⠀⠒⢾⡇⠀⣠⣧⣆⢰⣾⣄⠀⣠⣧⣄⡀⢀⣠⣼⡄⠀⠀⣼⠆⠒⡲⠄⠀⢈⡇
⠀⠀⡞⡯⣶⡆⠀⣶⣾⠓⡄⣞⡷⢶⡶⠶⣶⣶⣿⠆⢸⡿⠇⢃⠀⠿⠽⣱⠀⢹⡏⢩⣬⢸⣿⠀⣽⠛⠿⢻⢛⠿⢛⣽⠀⠀⡟⣃⠘⠛⠃⠀⣾⣷
⠀⠀⠹⡇⠀⠒⠒⠃⢠⡏⠀⠙⣇⠀⢰⡥⠀⢀⠏⠀⠘⡇⢠⣸⣾⣴⠆⡏⠀⠈⢇⠠⠤⢸⠁⠀⠸⡏⠢⢑⣒⠕⢹⡇⠀⠀⡏⠋⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠑⡄⢀⡀⠤⡞⠀⠀⠀⠈⣶⠈⣛⠠⣏⠀⠀⠀⢠⠄⠁⠄⠁⠾⠁⠀⠀⠸⡄⣤⠌⠀⠀⠀⠙⣄⠀⠁⣠⠎⠀⠀⠀⠧⣄⠠⠀⠒⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⣐⡀⠀⠀⠀⠠⠚⠀⣲⠀⠀⠀⠀⡈⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠁⠀⠀⠀⠈⣦
⠀⠀⣴⠤⡶⠂⡨⣤⣧⠀⢀⡀⠀⠱⣄⠿⠃⠀⠀⢸⡿⢶⡤⣠⠶⢻⡇⠀⢰⡿⣦⡤⢠⣾⣿⠀⠀⠀⣄⠀⠴⠏⠸⡿⠀⠀⣢⣭⡀⣨⣭⠀⣾⡦
⠀⠀⠻⡆⠀⠤⠄⠀⡏⠀⢻⡇⠀⣶⣿⣷⣆⠀⠀⠀⢱⣠⡦⠴⣄⣸⠁⠀⠀⢳⡤⠤⠴⢶⡏⠀⠀⠀⠻⡀⠀⢀⣶⣅⠀⠀⠞⣻⡁⢈⡋⠀⡞⠁
⠀⠀⠀⠡⠀⠐⠂⡐⠁⠀⠀⠱⣨⠛⠉⣀⣹⠂⠀⠀⠘⠏⣀⣀⠹⠏⠀⠀⠀⠀⠄⠠⡄⡹⠀⠀⠀⠀⠀⠁⢀⠿⠿⠿⠀⠀⠐⠹⢿⠟⠁⠜⡄
⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠱⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡆⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠛⠉⠁⢀⠤⠒⠒⠢⣄
⠀⠀⠰⠀⠀⠀⠀⠀⠐⡀⠀⡀⠀⣔⡀⠀⣙⣇⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠁⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⢠⡇⠀⠀⠀⠀⠈⢧
⠀⠀⣴⠀⢠⣤⡄⢀⣦⠇⠀⣇⠀⠉⠋⢠⣟⠉⠀⢰⡇⢀⣉⡁⢀⢀⡰⠀⢋⡓⢠⣠⡄⠀⢀⡦⠀⣷⡆⢶⣒⠀⢀⣤⠇⠀⣼⡇⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠙⠃⠀⠙⠁⠈⠉⠀⠀⠻⣆⠀⢠⡟⠿⣆⠀⠈⠣⠈⠑⢁⡈⠋⡁⠀⠌⠨⠀⠉⠀⠀⣸⠟⠀⠈⠁⠈⠁⠀⡸⠟⠀⠀⢹⠇⠠⢾⡷⢦⣾⡟
⠀⠀⠀⡘⠠⢀⠀⠀⡀⠀⠀⠀⠈⡗⠢⠀⢰⠂⠀⠀⠈⠂⠐⠿⠃⠀⠀⠀⠀⠈⠃⠙⠀⠼⠃⠀⠀⠈⠁⠀⡀⠔⠀⠠⠀⠀⠀⠣⡀⢠⣴⣶⠠
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠴⡿⠛⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠀⠀⠀⠀⠀⣰⠃⠀⠠⠙⢿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣤⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠂

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

⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠑⠒⠢⠄⡀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⢠⠏⣠⣄⣀⠀⣲⡶⠆⠀⠀⠀⠈⢷⠚⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄
⠀⠀⠘⠶⣤⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⣾⠋⠁⠹⣧⠞⠉⠙⣆⠀⠀⠀⠀⠈⣷⠒⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢦⣤⣠⣏⣀⣀⣀⣽⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⢀⣤⠶⠶⠶⠒⠒⠲⠶⠶⠤⢤⣤⣼⣀⡘⢿⢿⠃⠉⣿⣧⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⢀⣾⠇
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣯⡛⠓⠦⣄⣠⠏⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⡴⠟⠃
⢸⣧⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠙⣿⢿⣶⣿⣦⣤⣀⣄⣀⣴⣇⣀⣀⣠⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀
⠀⢰⠃⠀⠀⠉⠙⠛⠛⠛⡟⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣏⠀⠀⢀⡀⠀⠀⢀⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠻⣄⡀⠀⠉⠛⠛⠋⠁⠀⠙⠻⠶⠶⠶⠶⠶⢦⣤⣤⣤⣀⣀⠀⢀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠈⠙⠛⠛⠛⢳⣄⠀⠀⠀⠀⣀⡶⠶⠶⠦⣤⣤⣄⣀⣀⣉⣀⣋⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⣧⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰⠇⠀⠀⢀⡴⠒⠀⠉⠑⠒⠦⢄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⢧⠀⠀⠀⠀⢰⡇⠀⠀⢠⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡇⠈⢧⠀⠀⠀⢸⠃⠀⠀⣸⠃⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠇⠀⠈⢧⡀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⣠⠴⠦⢷⣤⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⡜⠁⠀⠀⠀⣸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡿⠁⠀⠀⣀⣀⣿⠀⠀⠀⡟⠀⠀⠀⣴⠖⠛⠒⠶⠦⣤⣄⣀⡈⢳⡀⠀⠀⢷⠀⠀⠾⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠖⠚⠃⠀⠀⠘⠷⢤⣴⠛⠉⠉⠉⠀⠀⠘⠛⠛⠛⠻⡇⢀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠀⠀⢸⡆⣴⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⣿⢸⠀⠀⢰⡇⠀⠳⡄⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣼⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⢠⡟⠀⠀⠀⠀⢸⠀⠀⠀⣿⢿⣄⣀⣸⣇⣀⣠⡟⠛⢓⣶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣀⣀⣀⣀⡿⠛⠛⠛⠛⠛⠒⢾⡇⠀⠀⠀⠀⣸⣀⣀⣠⡟⠀⠈⠉⠁⠀⠈⣁⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⡙⠛⠷⠾⠛⠋⠉⠉⠉⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠛⠲⠶⠶⠤⠤⠴⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠠⠔⠊⠀⠀⠀⠁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀
⠀⠀⠀⠠⠁⠀⢀⢄⠀⡚⡑⡐⢀⡆⠀⠀⠀⠀⡀⠔⠀⠈⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠉⠑⢦⡀⠀⠀⠀⠀⣠⠈⢁⡀⣀⣀⣁⣉⣓⣄
⠀⠀⠀⢸⠀⡴⡥⠛⠚⠙⠚⢷⣡⣿⠀⠀⠀⡐⠁⡀⣔⠖⠀⠁⢢⡁⡀⢣⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⢷⠖⠉⢉⣉⣉⡉⠉⣉⠙⠅
⠀⠀⠀⢈⠦⡹⠁⠀⠀⠀⠀⠀⣿⡎⠀⠀⢠⠀⡸⡰⠃⣀⠀⠀⠀⠱⡄⠀⢁⠀⠀⢔⠂⠠⣒⡄⠀⢠⣤⡤⠁⣇⠀⠀⢠⢛⠧⢙⣛⡛⠋⣓⣻⠝⠾⡀
⠀⠀⠀⢀⠘⠀⠘⠋⠀⠘⠒⠀⠹⡅⠀⠀⠸⢰⠗⠀⠈⠛⠁⠈⠛⠁⢷⠁⢠⠀⠀⠯⡀⠀⠀⠀⠀⠀⠉⠀⠀⢻⡁⠀⠸⣃⠀⠈⠋⠁⠀⠙⠉⠀⠰⡜
⠀⠀⠀⠀⠈⢃⠀⢀⠀⣀⠀⠀⠂⠁⠀⠀⠀⢣⡙⣆⠀⢀⣒⡀⠈⢀⣼⡄⠈⠀⠀⠀⢂⠀⠀⡐⠆⢀⡀⠀⣨⠁⠀⠀⠀⢸⡇⠀⠀⠠⢄⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠈⣦⣀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠑⠸⡦⣀⠀⠈⣀⢼⡃⠀⠸⠀⠀⠀⠘⠳⡀⠀⠀⠀⢁⡴⡉⠀⠀⠀⠀⠀⣿⣄⠐⠉⠀⠉⠀⣼⡶
⠀⠀⠀⠀⡠⠰⢧⣉⠉⢉⡡⡦⢄⡀⠀⠀⠀⠀⠀⣄⠇⠙⠹⠋⠅⢈⠚⠀⠂⠀⠀⢀⡴⢆⡉⠓⠒⠚⠹⠐⡳⡀⠀⠀⠀⠀⡠⢮⠑⠦⢤⣴⠊⣽⢄
⠀⠀⠀⠜⠀⠀⠀⠉⠉⠉⣉⠀⠀⠈⢦⠀⠠⢶⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⢢⠈⠠⡀⠀⠀⠉⠱⠆⠉⠉⠀⠉⠲⡤⠒⠉⠁⠢⡉⠰⢄⡠⣆⢅⠜⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠒⠒⠄⢄⠁⠐⠂⠈⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠈⠉⠑⢄⡀⠀⠀⠀⢀⣀⣀⣀⠼⠀⠀⢀⠀⠐⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⠎⠀⠠⢀⠔⠈⠐⡔⡀⠈⠢⠀⠀⠈⠀⠀⠀⠀⡀⠀⢕⡈⡂⣀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠆⠀⠀⠠⠄⠐⠀⠀⢸⡀⠀⠀⢠⠁⠀⣰⢀⠂⠀⠀⠀⢸⠀⡆⠐⡀⠀⡨⠂⢀⠂⠜⠀⠀⠀⠈⢞⠫⡪⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡿⠘⠟⠃⠁⠘⠟⠃⡤⠸⠀⠀⢸⠀⡄⢃⠛⠛⠀⠀⠻⠛⡇⣪⡀⠀⠀⡥⠀⡄⡈⠰⡶⠀⠀⣶⡾⡎⢂⢀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢂⣀⣠⠤⣀⢀⠴⠋⠁⠀⠀⢈⠀⣴⡇⠀⠀⢀⠀⠀⠀⢘⡆⠃⢰⠀⡁⠀⠃⡇⠀⠀⠀⠀⠀⠀⠑⣄⠑⠤⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣄⠀⠉⠉⠚⠁⡌⠀⠀⠀⠀⠘⠄⢸⢻⡄⠐⠊⠁⠢⣠⣿⢃⠀⡸⠀⡟⠀⢼⢷⡀⠠⠔⠂⠄⠀⣰⡏⣂⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠶⣤⣤⠷⡏⡇⠀⠀⠀⠀⢀⠇⠸⠟⡟⠦⢤⡤⠾⢫⢯⡇⢠⠃⠀⠈⠑⠤⡙⡝⢦⣄⣀⣠⣾⢸⠄⠥⠭⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠣⢄⣀⠀⠀⠉⠀⠀⢉⡢⠤⣀⠀⢈⠆⠲⢆⡁⠀⠁⠀⠀⠈⢓⠭⢄⡁⠀⠀⢀⠴⣶⠁⠀⠈⠢⠂⠉⠈⠐⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠠⠉⠉⠋⠉⠉⠁⠐⠂⠀⠑⠀⠐⠀⠀⠉⠑⠂⠤⠐⠂⠥⠀⠀⠙⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘

⠀⠀⢀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠔⠀⠀⣢⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠔⠑⠀⠙⡄⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠑⠐⡁⢀⠀⠀⠀⠀⠎⠀⡀⠀⠂⠀⠀⠀⠀⠀⠔⢢⡐⠐⡄
⠀⠠⡀⠆⣰⢀⠇⠀⠀⠀⠀⠀⠀⠁⠙⠉⠀⠀⠀⠀⠀⢤⠉⠈⡁⠃⠀⠀⠀⠀⠢⢤⡃⠩⠀⠀⠀⠀⠀⠀⢖⡛⠋⡰⠃
⠀⢄⠈⠁⠔⣃⡀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⢈⡩⠀⠀⠀⠀⠀⠀⣠⣄⠠⠒⠀⠀⠀⠀⠀⠀⠀⠐⠂⢁⠠
⠀⠀⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠅⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠔⠠⠁⠀⠀⠀⠀⠀⠘⠁⠀⠃⠀⠀⠀⠀⠀⠜⢀⡚⠆⠀⠀⠀⠀⠀⠐⠄⠀⠠
⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠐⢤⠈⢄⠔⡀⠀⠀⠀⠀⠀⠐⠦⠰⠀⠀⠀⠀⠀⠀⠘⠋⠛⠁⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⢀⠱⣃⡹⢁⠀⠀⠀⠀⠀⠈⣅⣔⣡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣴⠖⠁⠀⠀⠀⠀⠀⠠⡆⢱⡤
⠀⠀⢭⣽⣿⠋⠀⠀⠀⠀⠀⠀⠋⠉⠟⠀⠀⠀⠀⠀⠀⢣⡠⠔⠂⠀⠀⠀⠀⠀⠈⠋⠹⠆⠀⠀⠀⠀⠀⠀⠄⠀⠬
⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠃⡀⠲⠄⠀⠀⠀⠀⠀⠊⠰⠀⠘⠆
⠀⠠⠀⠁⢐⣠⠄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠘⠀⠃⠀⠀⠀⠀⠀⣀⢇⡀⠀⠀⠀⠀⠀⠀⠀⣂⠰⡀
⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⢰⠟⢁⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠁

⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣄⠀⢀⠤⠄⠶⠒⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡁⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣶⣦⣤⣀⡀⢳⠒⠉⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣿⣿⣿⣿⠿⠟⠋⠹⡿⠛⠋⢸⣿⡿⠋⣸⣿⣿⣿⡟⠉⠀⠀⠈⠙⠻⢿⣿⣿⢑⠢⠤⠤⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⣿⣿⡍⠁⠀⢤⣤⠴⠞⠁⠀⠰⠟⠋⠀⠐⠛⠋⠉⠀⠙⠳⠦⣤⠄⠀⠀⢸⣿⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡿⠃⠀⠀⠀⠰⢶⣲⡶⠆⠀⠀⠀⢰⠂⠀⠀⠀⠰⢶⣶⠶⠀⠀⠀⠀⠘⢿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠭⠥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣤⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠈⠓⠢⠔⠊⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⠀⠀⠀⠀⣀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡏⠈⠉⠉⠀⣭⣤⣤⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠈⣿⣿⣿⣷⣄⠀⠀⣠⣿⣿⣿⡟⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣅⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣍⣉⣉⣉⣽⡟⠁⠀⠀⠀⠀⠀⠻⣯⣉⣉⣉⣩⣿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠿⠿⠶⢶⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⣀⣄⣀⡀⠀⠀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠤⣤⣄⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣧⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⠿⠅⠀⠀⠀⠀⠀⠿⠿⠿⠟⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⡞⡅⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡟⠀⡋⢸⡗⠀⡶⢰⣆⣿⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠰⢡⡟⠀⣠⡄⢠⢤⡄⢰⡇⣶⣸⡇⠰⢧⣾⠱⠞⠛⠈⠁⢻⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠋⣳⠀⡟⠡⢿⠜⣧⠞⠃⠉⠸⡷⠃⠀⢻⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⣴⠏⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠏⠓⠬⣿⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⢐⠂⠀⠀⠤⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠁⡀⠈⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⣔⣀⡀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠴⢈⠢⠤⠤⠄⠊
⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⡿⠓⢦⡀
⠀⠀⠀⠀⢸⡆⠀⠸⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠳⣄
⠀⠀⠀⠀⣀⣈⠻⢤⡴⠞⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⠀⠀⠈⠣⣄
⠀⣠⠖⠋⠉⠁⠀⠲⣍⣤⣞⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠘⢦⡀⠀⠀⠀⠈⠙⢦⣄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠶⣶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠈⡹⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡔⠁⢀⠀⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣥⡥⢄⣹

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⢩⠟⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⠉⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠈⢲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠐⢄⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠈⢣⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⡆⡄⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠒⠶⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⢢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⡷⣄
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢈⣥⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⠢⡀⠄⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡠⠖⠁⠀⠈⠢⣄
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠳
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⠒⠢⡀⢀⠞
⡇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠉⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠄
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⡆
⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠴⠊⠀⠀⢀⠀⠀⠀⠐⡸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠤⠔⠂⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣤⣀⣤⠤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⣀⣀⡠⠤⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠔⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠉⠉⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣐⠠⢄⣀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠉⢢⠀⠀⠀⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢍⡉⠒⠒⠒⠀⠀⠁⢀⡠⠜
⣶⣶⡶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⡦⢄⣀⣀⡠⠔⠒⠀⠀⠂⠀⠀⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠑⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣠⣔⡒⢄⠀⠀⠀⠀⠀⡀⣤⡤⠤⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡼⣿⣿⣿⠆⠃⠀⠀⠀⢰⣿⣿⣿⡆⠘⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢴⠀⠱⣙⠛⠋⡠⠃⠀⠀⠀⠨⡙⠿⠟⠁⡠⠂⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⠤⠄⢀⣀⣀⠀⠀⠤⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡤⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠗⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠚⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⣄⠤⠀⢤
⠀⠀⠀⠀⢶⡌⠓⠖⠋⣴⠆⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠤⠇⠀⠀⠁⠀⠀⡰⠀⠀⢀⠖⠢⡠⠤⡀
⠀⠀⠀⠀⢀⡟⠀⠀⠸⣇⠀⠀⢦⡙⠶⠤⠞⣡⠆⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠘⢣⠀⢠⠼⠄⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠋⡴⠛⠙⢶⡌⠃⠀⠀⣽⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠹⡀⣀⠴⡀⡏⠈⠁⠀⠑⠤⡄⡀⠀⢀⣀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡵⠚⠛⢦⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠀⠀⠀⠰⠏⠙⠤⠎

⠀⠀⠀⡠⠊⠁⡉⠉⠓⣄⠀⠀⢰⠶⠶⢶⣶⣤⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⡉⠉⠓⡄⠀⠀⢰⡶⠶⢶⣶⣤
⠀⠀⢰⢀⣠⠔⠉⠢⣄⠘⡄⠀⢀⡏⢠⠄⢠⠀⢣⡀⠀⠀⠀⠀⠸⢀⣠⠞⠉⠢⣀⠘⡄⠀⢠⡏⢠⠀⢠⠀⢧
⠀⠀⢰⣺⡀⠎⠀⠓⢠⣳⡇⠀⠑⣇⠈⠐⠈⠀⡼⠃⠀⠀⠀⠀⢰⢻⡀⠏⠀⠓⢰⣳⠇⠀⠑⡇⠈⠀⠈⠀⡾⠃
⠀⠀⠈⢢⡑⣤⣤⡴⢋⠜⠀⠀⠀⠈⢢⣤⣤⡞⠁⠀⠀⠀⠀⠀⠈⢦⠑⣤⣠⡴⢋⠜⠀⠀⠀⠈⢢⣤⣥⠚
⠀⠀⠀⠀⣹⠇⠆⠸⡇⠀⠀⠀⠀⢠⠋⠜⠢⠱⡀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠆⠹⡅⠀⠀⠀⠀⢠⠋⠜⠀⠳⡀
⠀⠀⠀⢠⠋⠀⠀⠀⠹⡀⠀⠀⠀⡎⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⢱⠀⠀⠀⠀⡏⠀⠀⠀⠀⢣
⠀⠀⠀⡎⠀⠀⠂⠀⠀⣇⠀⠀⢰⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡎⡄⠀⠂⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠃⡇⠀⠀⠀⠀⢹⠀⠀⢸⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠇⠀⠀⠀⠀⠸⠀⠀⠸⠰⠀⠀⠀⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉









⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿
⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠈⠑⡄⠀⠀⢀⠖⠁⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⠀⠀⠇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣯⠤⠐⠒⠛⠒⠒⠚⠒⠠⠤⣱⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⡄⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⣠⠞⠀⠀⠙⢦⣄⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣠⠊⢠⠔⡏⢡⡌⠄⠀⠀⠀⠆⣦⢨⠓⠤⠱⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡤⠚⠁⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠉⠉⠉⠂⠀⠀⠈⠢⢄⣀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠤⢤⡤⡒⠉⠁⠐⠦⣀⣀⡠⠔⠀⠈⠑⡲⢤⠤⠰⠤⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⣀⡠⠼⢤⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠒⠒⠷⢤⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣙⠤⠒⠒⠒⠤⣄⠚⠉⠁⠀⠉⠑⠋⣡⠤⠭⠴⠤⢤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⣀⢔⠀⠀⢀⣄⠀⣀⠜⠀⠀⠘⢄⡀⠀⣀⠀⠀⠀⣄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠒⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢒⠶⠖⠒⠈⠉⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠠⠤⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣤⡀⠀⢿⡇⢀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠿⠇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⠀⣧⠀⠀⠓⠈⠀⢀⠠⠀⠀⠀⠈⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠈⢦⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠉⠓⠦⠤⠤⠤⠤⠔⠊⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⢏⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠘⡆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠰⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠒⠂⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⡐⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⢄⡀⠀⠀⠀⡄⡠⠊⠐⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠰⠋⠀⠠⠃⠠⠄⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢢⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠱⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡰⠀⠀⠀⠀⠀⠀⠱⢱
⠀⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⠀⡌⡀⢀⠂⠀⢀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠈⠀⢀⠀⠀⡄⠀⠀⠀⠀⢇⠀⢀
⠀⠀⠘⠾⢃⠁⡈⠀⡈⠀⠐⠍⠑⣄⠓⠈⠓⠔⠃⠐⠈⠀⠂⠀⠐⠂⠐⠂⠀⠀⠭⠀⠀⠅⠚⠀⠀⠒⠘⣆⡠⠤⠤⣀⠂
⣆⠀⡄⠪⠄⣀⠀⠨⠃⠤⠀⠀⢄⠀⠀⠀⡀⢀⠀⠀⠀⠁⢀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠁⠄⢄
⠈⠓⠁⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡊⠀⠀⠀⠄⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠈⠂⡴⣞⡁⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠒⠈⠓⠀⠀⠂⠠⠄⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣶⡖⡐⠀⠀⠠⢀⣤⣤⢄⡀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠁⠀⠀⠄⢀⠀⠙⠉⠉⠀⢀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠒⠂⠤⢠⣤⣤⢀⡀⠑⢀⠜⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠞⠛⠛⣛⣒⣒⣛⠙⠂⠀⠁⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡙⠈⠀⠀⠀⡒⠨⠉⠑⠒⠒⠐⠒⠲⢶⢘⠀⡥⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢄⠂⠀⢠⡇⠀⠀⠀⢀⠐⣁⡀⣼⠝⠠⠊⠂⢤⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠉⣄⡀⢀⠒⡤⠐⠃⠡⠎⣉⣉⠉⡁⣀⠤⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠉⠐⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡌⣑⢀⠠⠠⠊⠉⠑⠆⠤⣈⣀⠀⠀⠀⠀⣨⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡒⠒⠓⠓⠚⣐⣉⣁⣥⡭⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡊⠛⠛⠚⠒⠓⠒⠒⠒⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⣴⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣀⢀⣈⣁⣙⢉⡉⠉⠉⠁⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠛⡙⠋⠉⠈⠙⠠⣍⣭⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡒⠌⠀⠀⠀⠀⠀⠀⠘⠇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡁⠈⠃⠀⠀⠀⠀⠀⠐⠂⠙⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⠳⠀⠀⠀⠀⠀⠘⠲⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢈⠀⠀⠀⠀⠀⢨⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣊⣰⣀⡠⠀⠀⠀⠀⠘⠀⠙⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⡤⠤⠼⠖⠋⠀⠂⠐⠸⣀⣀⣀⣀⢈⣓⡦⡀⢄⠤⡠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⢀⣶⡿⢀⣀
⠀⠀⠀⠀⠀⢠⣄⠀⣾⡿⣠⣦⣿⡿⣵⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣤⣄⠸⣿⣆⣿⡇⣿⠁⣿⣙⣇⣠⣤⣤⣤⣶⣾⣿⣿⣛⣉⡀
⢀⣤⣤⣙⢿⣧⡹⣿⣼⣷⣾⡿⠟⠛⢛⣿⡿⠛⠉⠉⠛⢿⣿⡉⠛⠛⠛⠿⠷⣶⣦⣤⣀⡀
⠈⠛⠿⢿⣿⡙⣷⣾⡿⠋⠀⠀⠀⠀⣾⡏⠀⣠⣤⣀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣶⣦⣄⡀
⠀⢠⣶⣶⣯⣻⣿⠋⠀⠀⠀⠀⠀⠀⣿⡇⠀⣯⣽⣿⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⠉⠀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄
⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⣋⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣠⣴⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣼⣿⠏⠉⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⡀⠀⠀⢴⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠈⠻⣆⠀⠀⠈⠉⠉⠛⠛⢻⣿⡿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠟⠋
⠸⣿⣦⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠙⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇
⠀⠀⠀⠀⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⠶⠿⣿⡟⠛⣿⡆
⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠘⣿⠉⠉⠉⢉⡿⠉⠉⠉⠀⠀⠹⣆⠀⠀⢀⡿⠀⠀⠘⣿⡄
⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠹⣆⢀⣾⠃⠀⠀⠀⢻⣷
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠁⠀⠀⠀⠀⠈⣿⡇
⠀⠀⣿⡿⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⢠⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣾⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣿⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠾⢿⣿⠿⠿
⢀⣀⣤⠀⠀⠈⣿⡅⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣤⣴⣦⡄⠀⢸⣿
⠸⣿⣿⣇⣀⣠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠙⢿⣿⣶⣶⠾⠋
⠀⠀⠉⠉⠉⠉⠁⠀⣠⣿⣧⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣀⠀⠀⠀⠀⠀⠈⠹⣷⠀⠀⠀⣴⠟⢉⡉⠁⠀⠀⠀⡍⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠿⣦⣼⠛⠛⠷⠶⣶⣦⣤⡾⠟⠀⠀⠸⣷⣤⣿⠛⠿⢶⣶⣴⣧⣴⠾⠏
⣿⣿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠒⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠢⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⡄⡀⢸⠠⠊⡃⠰⠢⠀⣤⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠉⠀⠓⠊⢸⡠⢁⡃⡐⠰⠦⠃⡃⠂⡌⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠓⠂⠀⠂⢠⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠉⠐⠀⠠⠤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠋⠁⢀⣀⡈⠛⠏⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⣈⡳⡄⠨⣴⠖⠀⠀⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡌⠁⠇⠀⠀⠀⠀⡠⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢦⡀⠚⠃⢀⡼⢁⡼⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡋⠀⠀⠀⠉⠙⠫⡥⠴⠋⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣠⠽⠓⠒⠧⣤⠤⠤⠤⣄⠀⢀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠴⠗⠛⠉⠉⠙⢓⣦⣀⠀⠈⣳⠞⠉⠙⢦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣤⠞⠋⠁⠉⠉⠉⠁⠀⠀⢀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠒⠋⠁⡿⠓⠦⣄⣀⣀⣀⣀⣀⣴⠋⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢸⠃⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠉⠁⠘⡆⠀⠈⡇⠙⠒⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⣾⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠓⠲⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⣤⣤⣾⣄⡀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠐⢻⠏⠉⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠀⠀⠀⠀⠀⢀⣾⠐⠁⠀⠌⠻⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⠄⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣷⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠶⠚⠛⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠁⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣠⡴⠞⠛⠉⠈⢳⣄⠀⠈⠉⠉⠉⠉⠙⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⠷⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⢀⡴⠟⢁⣠⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠾⣻⣧⠶⠋⠁⠀⣰⠟⢀⣤⠾⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢰⣿⡵⠞⠉⠀⠀⠀⢀⣼⣧⠶⠋⢁⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⢸⡿⣄⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⢋⣥⠶⠛⠉⠀⠀⠀⠀⠰⠶⠦⣤⣬⣽⡷⢻⡗⠶⢾⣧⣙⣷⡀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢰⡆⢶⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣼⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⡀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⢀⣤⠶⠚⠋⠁⠀⠀⠉⠙⠛⠶⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠶⠶⠶⠶⠶⠶⠦⠤⣤⢶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠋⠁⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⣠⣽⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⣴⣦⠀⠀⠀⠀⢰⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣾⡿⠛⢻⣿⡇⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⣿⣿⣷⣿⣶
⠻⣿⣶⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⣀⣀⡀⠀⠀⠀⠹⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣽
⠀⠀⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠿⠛
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠻⠛⠉⠙⢿⣷⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⣶⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠔⠚⠛⠛⠛⠲⢦⣄⠀⠀⠀⠀⠴⠖⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⣠⣤⣤⣤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠈⢮⣥⠄⠘⠀⠀⠀⠀⠀⠐⠃⠵⣭⡥⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣤⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣘⣻⣿⣛⣛⣛⣻⣟⣟⣻⣛⣛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⢄⠀⠀⠀⠀⠀⠀⡠⠔⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⠀⠀⠈⠐⠀⠐⠂⠁⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠈⠉⠉⠀⠉⠉⠁⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣶⣶⣶⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠞⠛⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠛⠓⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢺⣿⣿⣿⣶⠒⣶⠒⣶⣿⣿⣿⡗⠀⣸⠀⠀⠀⠀⠀⠀⠙⠿⠏⢹⣿⣿⣿⠛⠻⠻⠟⠛⣿⣿⣿⣿⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡖⠀⠙⠛⠛⣋⣴⠉⣶⣙⠛⠛⠋⠀⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠁⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠆⠁⢠⣬⣁⣀⠀⠈⣄⠁⢀⣀⣈⣥⡔⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⢀⢀⡀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠄⠀⠈⠉⠒⣠⠖⠉⠁⠀⢠⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⡀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢶⡂⠀⠀⠀⠀⠀⠀⠀⢐⡶⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⡟⠂⠳⣄⠀⣠⠞⠀⢽⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠾⠜⠀⠀⠀⠣⠷⢸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠲⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣓⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠯⣍⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣤⣤⣀⣀⠀⠀⠀⢀⡀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⣀⠀⠀⠰⠖⠒⠒⠒⠒⠚⠓⠀⠀⠀⡠⠉⠈⠉⠁⠒⠚⠏⠁⠒⠲
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⢄⡈⢳⡀⠀⣄⢰⡓⠀⠀⢀⡤⠒⠛⠉⠉⠉⠉⠁⠚⣤⠴⠚⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠦⢿⠳⡽⡄⠀⠻⣄⣀⣉⡀⢀⣀⣀⠠⠴⣇⣂⣀⣠⠴⠾⠖⠒⠛⠋⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢋⣩⣭⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⠉⠁⣈⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠒⠛⠛⣻⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠖⠒⠋⠁⠀⠀⠉⠒⠤⠀⡀⠀⠀⠀⠀⠀⠘⢉⡵⠚⢁⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢉⠀⠸⠉⢙⣐⠒⠒⠒⠒⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠑⠰⣄⣸⠉⣁⡦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠢⣄⠀⡉⢳⡇⢈⡑⠒⠀⠄⠐⠒⡖⠂⠀⠀⣀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡀⠈⡙⠁⠘⢦⠀⢳⡦⣄⡀⠀⠐⣷⢦⠘⢻⠁⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣄⠀⢳⠀⠀⢸⠀⠀⢳⡀⠉⠙⠒⠛⠀⠓⣏⣀⠤⠼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⡇⠀⠀⣇⠀⠈⠀⠀⠀⢣⡀⠀⠀⠀⠀⡈⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠀⠀⣈⠄⠀⠀⠀⠀⠀⢳⠀⠀⣠⣾⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢜⣩⣌⡽⠍⠀⣴⠀⠀⣀⣠⣴⠃⠀⠘⢿⠿⠛⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠚⠒⠒⠒⠒⢺⠉⡛⠉⠁⢠⡇⠀⠀⠀⠀⠑⣄⣀⡴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠤⢶⡴⣷⣦⠀⣾⠀⠀⠀⠀⠀⠀⠈⣡⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡑⠦⢤⣀⣀⣀⡭⠽⡄⠴⠒⠒⠉⣲⠄⠀⠀⠀⠀⠀⠙⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠁⠀⠀⠀⠀⠉⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⡄⢠⡀⢠⡄⣤⣤⡄⢠⣤⡄⣠⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣧⣼⣇⣾⠁⠈⡇⠀⢸⡷⠄⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠙⠃⠙⠃⠀⠈⠃⠀⠘⠁⠀⠛⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠎⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⢿⣦⡀⠀⣀⣀⣰⣿⣧⢀⣀⡀⠀⣠⣾⡿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠠⣤⡍⠛⢿⡇⠀⠘⣿⡇⠀⢠⣿⠟⠋⣥⡤⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡷⠶⠂⠀⠀⠉⠀⠀⠻⠁⠀⠈⠁⠀⠀⠲⠶⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⡀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⣀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠉⠀⠀⢏⣨⠹⠇⠀⠀⠀⠈⠟⣡⢸⠀⠀⠈⠉⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⣤⣤⣤⠀⠀⠈⠉⠛⠀⠀⠒⠂⠀⢘⠋⠉⠀⠀⢤⣤⣤⣤⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡠⠴⠒⠒⠦⣄⣀⠇⠀⠀⠀⠀⠀⠀⠇⣀⠤⠖⠒⠲⠤⣀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠖⢰⠋⠀⠀⠀⠀⠀⠀⠙⣷⣶⣦⣤⣤⣶⣶⡏⢀⠀⠀⠀⠀⠀⠈⢣⠰⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠀⠀⠀⢠⢀⠀⠀⠌⠛⠿⢿⠿⠟⠋⠀⠀⠀⢡⠀⠀⠀⠀⢸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢣⠀⠀⠀⠀⠣⣀⠀⠀⠀⢀⣼⣄⠀⠁⠀⢐⡠⠃⠀⠀⠀⣠⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢵⣄⡀⠀⠀⠀⣿⣽⡿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⣠⡶⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡇⠉⠓⠒⠒⠒⠒⠛⠁⣇⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣮⣕⡒⠒⠒⣒⣭⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠀⠉⠻⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢄⡀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠐⠒⠉⠀⠀⠀⠀⠀⠘⠢⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠃⠀⠀⠀⠀⢰⠞⠀⠦⣄⡛⢦⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⣠⠞⢹⡶⠛⠛⠢⠈⠃⠀⢩⡀⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠤⠊⠀⢀⠴⠁⠀⠀⠀⠀⢠⡃⠠⢀⠄⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠉⢉⠀⢀⠀⡀⡀⡀⢠⠀⡀⣀⠀⠀⠀⣀⡴⢟⠝⠣⠶⡆⢬⡁⢀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡠⠊⡴⢁⣘⣬⡵⣅⠥⣤⢠⡄⠀⠰⢾⠁⠀⢠⠖⠒⠀⡇⠀⠙⢮⡁⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢎⡠⢋⢠⠈⠀⠀⠀⠀⢀⠞⡸⠂⠀⠀⠀⢀⠀⠈⠀⠀⢀⠃⠀⠀⠀⢣⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠸⡀⢿⠀⠀⢊⠔⡡⢊⣧⠀⠀⠀⠀⢸⠁⠋⠀⣀⡠⣤⢤⢤⣄⡀⠀⡞⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢙⠄
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠘⢇⠀⠈⠞⡴⢡⠈⢧⠀⠀⠀⠀⠀⢰⢊⢇⢃⢢⠁⠖⡸⢁⡞⠀⠀⠀⢀⠴⠁⠀⡀⠆⠁⣠⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠐⣡⠎⡀⠳⣄⠀⠰⠀⠘⣦⣸⡼⠞⠛⠋⣠⠏⠀⠀⢀⠔⠉⠀⡄⡇⠁⢀⠜⠁⠐⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢀⡴⠁⣀⡠⠒⠁⠀⢀⠀⠃⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⡄⠀⠀⠀⣀⠄⠀⢠⡤⢄⣲⠾⠓⣂⠄⠀⠀⡄⡇⠇⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⣨⠥⠂⡉⠰⠒⣡⠴⢒⣉⡥⠖⠀⠰⠁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢀⡭⠐⢊⡡⠔⢊⣁⠤⠒⣉⠥⠒⣉⠅⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣀⠠⠐⢊⠥⠔⢊⡡⠤⠒⢉⡠⠔⢊⡡⠔⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣈⡥⠖⣊⢡⠴⠊⣁⣠⠴⢊⣡⠴⢊⣡⠔⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠰⠊⡡⠔⠊⢉⡠⠔⠋⣉⠤⠚⡉⠔⠊⠁⡀⠄⣃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠔⢉⡀⠔⢂⡡⠄⠒⣉⠄⠊⢁⡀⠄⣒⠥⠐⣈⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡤⠊⣁⡤⠊⣡⠴⠒⣉⡤⠔⣊⣡⠔⣉⡤⠒⣉⠤⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠔⢋⣡⠔⠋⣠⠀⠊⣁⠔⠊⣁⠴⠊⡁⠔⢋⡥⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠔⢉⡠⠖⢉⡠⢴⠻⠀⠐⢋⡠⠖⣉⡤⠒⣁⡤⢂⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⢉⡤⠚⡡⢐⡇⠀⡆⠀⠈⡤⠈⠁⠔⠛⠁⠐⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠖⣉⠔⢋⠔⡾⠀⠀⡇⠀⠀⠖⢋⠰⠞⠁⠰⠞⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⡥⢊⡡⠚⣇⡀⠀⠃⠀⠀⠚⣡⠖⢛⡥⠖⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢁⠔⢁⡴⢊⡠⠛⣳⡇⠀⠀⣉⠤⠚⣁⠤⠒⡉⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢁⡴⣋⠔⣁⠔⢊⠤⡇⠀⢀⡥⠒⢋⡤⠖⣉⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠖⡡⢚⡡⢚⡡⢞⡇⠀⠠⠖⢋⡥⠖⣋⠔⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠔⣋⠔⣋⠔⢋⡇⠀⠐⣊⡡⠔⢊⠤⡚⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠈⠁⠀⣁⠔⠁⢁⠀⢀⡥⠒⣊⠥⣊⠥⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣠⡞⠁⠀⣤⠶⡟⠶⢒⣠⡞⠀⠀⠐⠋⠶⠋⠴⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠒⠂⠠⢬⡟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠉⠐⠒⣂⣀⣀⣀⠠⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠶⠖⠒⠒⠠⠭⠉⠒⢒⣂⠈⠉⠭⠄⢐⣒⠂⠈⠭⠭⠀⢒⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠋⠀⠀⠐⠈⠙⠉