Piggy bank Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⣉⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢁⣤⠖⠋⠉⠀⠀⢸⠆⠀⠀⢀⣀⢠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⡤⠚⠁⠀⠀⠀⠀⠀⡰⠋⢀⡴⠚⡥⠞⠋⠉⠉⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢏⣠⠋⠀⠀⠀⠀⠀⢀⡤⠊⠀⢀⠎⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣾⠁⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⣸⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⢨⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⢉⣀⠤⠷⣜⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣄⡀⠀⠀⠀⠀⣰⠏⣉⣍⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⢀⡴⠚⠉⣀⣤⣶⣾⣷⣷⣦⠤⠔⠒⠀⠈⠉⠉⠉⠉⠉⠉⠛⢷⣤⡀⠀⢿⡀⢿⡿⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠑⠶⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⢶⡴⠟⠋⢀⣴⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠙⢶⡞⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣄⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⠒⠒⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠠⢤⡄⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⣤⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡏⠀⠀⠀⠀⠀⠀⢀⣤⣶⢿⣿⠿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢸⡿⠃⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣰⣿⠋⠀⣸⣿⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠻⣿⡿⠿⢻⣿⠛⣿⡟⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢠⡾⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⡏⣰⡄⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣇⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢸⠇⣿⡧⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣾⣿⡿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⣾⠀⢿⣷⢼⣿⢹⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢸⡆⠸⠛⣼⡏⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠈⠻⣦⡀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⣀⡤⠒⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⠲⠤⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⠶⠾⣯⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣁⣠⡾⣧⡀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠉⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⢀⡄⢈⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⡀⠀⢀⡄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁⠀⠋⠉⠀⠀⠠⠂⠁⢂⢀⣀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣐⠁⠀⠀⠀⠄⠂⠀⠈⠁⠀⠀⠀⡯⠚⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣀⠁⠀⠈⠁⠀⠐⠃⠀⠸⣄⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠉⠀⠠⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⡈⠳⣄⡀⠀⠀⡀⡀⢲⣄⡴⠀⠘⠁⠀⢰⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⢰⣭⣇⡀⢠⠧⢷⠀⢻⠉⠡⠥⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠳⠄⠀⠀⢄⠈⠇⠈⣧⣤⡶⠁⢠⡶⢀⠀⠠⡇⠀⣀⣀⠄
⠀⠀⠈⠀⠐⠣⠀⠀⠄⢰⠆⠰⠄⠠⠶⣄⣀⠀⠀⠁⠀⢀⡰⠌⠀⣀⣁⡂⠀⠀⠀⠈⠀⠒⠒⠃
⠀⠀⠀⠀⠄⠀⣀⣠⡄⠀⠠⡚⠀⠐⠋⠹⠤⠉⠀⠀⠬⠃⢀⡀⠤⠜⠛⢠⣤⣀⡠⠉⠀⠆⢀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠁⠐⠛⠧⡀⠀⠀⠤⠜⠳⠛⠋⠀⠄⠒⠒⠟⠁⢁⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠤⠖⠃⠀⠀⠀⠘⠂⠀⠤⠤⠶⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⡀⠄⠒⠂⠁⠈⠀⠀⠀⠘⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠔⠉⠀⠀⢀⠤⠖⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⢀⠤⡋⢑⠢⢄⠀⠀⠀⠀⠀⠡⡀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠃⣎⣰⢸⠛⠋⠀⠀⠀⠀⠀⠀⠃⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠙⠈⠁⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⠀⣑⠢⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⠴⣍⡇⡃⠀⠀⠀⠀⢀⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⣉⢀⡉⠜⠁⠀⠀⠀⢀⠼⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠄
⠀⠀⠀⢨⢤⣀⣀⣀⡀⠧⠤⠄⠀⠀⡈⢻⣶⠶⢦⠀⠀⠀⠠⣀⣀⣀⣀⣤⠤⠶⠛⡀⣀⣜⠘⠀⠐⠀⠩⡁⠀⠰
⠀⠀⠀⢐⠢⠤⠤⠤⡧⣀⣀⡀⠤⠤⡆⠒⠂⠀⠈⢠⡀⠀⣀⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣉⡠⠤⠔⠒⠒⠊
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠠⠄⠤⠤⠤⠧⠄⠀⠀⠠⡌⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠦⠴⠔⠒⠉⠓⠒⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠘⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠋⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠤⠨⣄⣠⣤⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠐⠛⠉⠉⠈⠉⠉⠀⠈⠲⡄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠇⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣴⢮⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⡀⠀⠐⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⣀⡀⢢⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⡼⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⢀⣀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠈⢦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢰⠋⠙⠋⣩⠑
⠀⠀⠀⠀⠀⠁⠀⠛⠛⡇⠀⣀⣀⣠⡤⢹⠶⣶⠀⠀⢀⡦⢤⡤⠤⣶⣿⠁⠀⣏⠀⠀⠀⢘⣠⡤⠤⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⡉⠀⠚⠴⠧⣀⣀⠼⠤⠀⠉⠉⠉⠛⠄⠚⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠳⠀⠀⣠⣤⣤⠖

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⢀⡀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠤⠊⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⣀⡀⠀⠀⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠐⠊⠁⣠⢖⠋⠉⠉⠉⠙⠲⣄⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⠡⠂⠀⠀⠀⠀⠀⠀⠈⢧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡸⠀⠀⠀⣇⢊⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡅⠀⠀⢀⣴⡹⣄⠂⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⣫⣾⢕⢝⠦⣄⡀⠀⢀⣀⡤⠊⢀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣚⢕⡵⡫⣪⣊⣬⣏⠍⡇⠀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣤⣀⠀⠀⠀⠈⢳⡩⣪⣾⣿⣿⣿⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢢⡀⠈⠙⢶⠊⣭⣂⣉⣻⣶⣷⣷⣟⣁⣒⡒⢍⠐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢸⠀⢱⠀⠀⠈⠃⠘⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠃⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⢀⡤⢤
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠘⠦⠀⢇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⣀⡙⠆
⠀⠀⠀⠀⢠⠏⢤⡀⠀⠀⠀⠀⢸⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⣏⠠⡙⢧
⠀⠀⠀⢠⠏⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣈⡹⠃⡼
⠀⠀⢀⡞⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠴⠚⠁
⠀⢀⡞⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠈⠳⠦⠤⢤⣤⣀⡀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣰⠋⠙⠛⠉⠉⢧⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡴⠞⠁⠀⠀⠀⢀⠠⠀⠂⠈⠻⠶⠶⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠉⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡇⠀⠀⢚⠝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡇⠀⠀⠸⣊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⣀⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⠈⢻⡷⢤⣤⠶⠚⠛⠉⠉⠉⠁⠀⠀⠀⠈⣉⣉⣛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⠽⠀⠀⠀⢹⡄⠙⢦⡀⠀⠀⣀⣤⠶⠖⣛⣛⣩⣭⣭⡭⠿⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠵⠚⠁⠀⠀⠀⢻⡀⠀⠉⠀⠘⠿⠒⠛⣛⣩⣭⣍⡀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠞⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠐⢁⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠘⣿⢷⡀⠀⠀⠀⠀⠀⡸⣰⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡼⢻⡄⠀⠀⠀⠀⠠⠏⣸⢻⣶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠃⠀⠀⠀⠀⠀⣴⢣⠏⣿⣀⡽⣷⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⣴⢻⡿⠶⠋⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡞⢡⣾⡇⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⡴⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣯⣉⡛⢳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢋⣤⢞⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⣤⡈⠙⠻⣴⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣵⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣹⡇⢰⣷⠘⣷⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡴⠛⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠛⠀⢹⣿⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⡴⠊⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⡤⣤⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡴⣣⢹⠷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⢢⡄⠀⠀⠀⠀⢀⣤⠖⢋⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠷⠞⠁⠀⠈⠹⡆⠀⠀⠀⠀⠀⠘⣩⠞⣡⣿⣀⣤⠤⠶⠛⡏⠀⣺⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣻⡵⣋⡿⠁⠀⠀⠀⠀⠀⠛⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠈⢉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢊⣭⣴⡶⠶⠾⠛⠛⠒⠂⠀⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠖⣚⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡽⢣⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠀⠀⠀⠀⠀⠀⠀⠐⣪⠍⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠂⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⠖⠉⠉⠙⠲⡄⠀⠀⠈⠁⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠰⡖⢰⡢⠀⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠘⢦⣀⣀⣀⣠⡯⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⣤⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣀⠀⠀⠀⠀⢠⡄⠠⠤⠒⣒⣉⣤⡶⡻⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠻⣿⠀⠀⠀⠀⢸⠟⠛⠋⣏⣉⣩⠽⢳⣣⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠦⢴⣧⣄⣀⣀⣠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡭⢭⣍⡛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⠋⠀⢴⡆⠀⠉⢣⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡇⠀⠀⢸⡇⠀⠀⠀⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⣇⠀⠀⣸⣧⡀⠀⢠⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⡳⢤⣈⣉⣠⠴⣋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡶⣤⡀⠀⠀⠈⠙⠲⠶⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⡆⠙⠶⠴⠖⠚⠛⠛⠛⠛⠒⠶⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠟⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠗⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⠀⣴⠛⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⣧⡈⣛⡿⠁
⠀⠀⠀⠀⠀⢠⣶⣻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⡉⠉
⠀⠀⠀⠀⠀⣾⣬⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹⠁
⠀⠀⠀⠀⠀⢻⣠⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⡏
⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣴⠶⠶⠶⡆⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⣀⣠⠿⠀⠀⠀⠷⣄⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠩⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡤⣍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠎⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠀⠀⠉⠉⠉⠉⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠐⠢⠀⠀⠀⠠⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⣚⣛⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣭⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠶⠶⡃⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣴⠀⠰⣾⢌⣽⠿⠁⣀⣤⣦⠀⢢⠀⠀⠀⠐⢄⡀⠀⠀⠀⣨⠞⢠⠀⠀⢀⠌⠀⠀⣾⠫⠥⠤⢆⣾⠀⢀⣀
⠀⠀⠀⠀⠀⢸⠁⣻⣻⠒⢿⣁⢐⣓⣚⣿⠀⠀⠓⢄⠀⠀⠀⠀⠭⠙⠩⠥⠒⠁⠀⠔⠁⠀⠀⠀⢱⠍⠠⣭⡭⢿⣀⣻⡾
⠀⠀⠀⠀⠀⠘⠂⠾⠾⠭⣹⣇⠠⠭⢥⣿⣇⡀⢠⠈⠑⠀⠀⠤⢤⣤⡤⠤⢄⠀⠀⡀⢠⠀⢀⢤⠼⠅⠐⢻⣛⣽⡂⣚⣿⠖⢄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣋⢭⠀⠀⢈⢗⠶⠦⠤⠤⠼⠻⢦⣄⣀⣀⣤⡤⠁⠀⢻⣍⣀⣮⣭⠽⠗⠛⠉⠉⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣈⣁⣒⣛⣭⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣶⠿⢿⣦⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢿⣯⢠⡾⣻⡇⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡻⠶⠶⣛⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠦⣄⠀⠀⠀⢀⣀⣈⣉⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⠙⠚⠋⣉⣉⣁⣈⣉⣉⣉⠙⠓⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠙⠃⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣀⣠⠟⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢾⡧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠚⠣
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠙⠓⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣀⡴⠋⠉⠉⠙⠛⠛⠛⠉⠉⠉⠙⢦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠈⠙⠛⠓⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣶⣄⣤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣠⠶⣳⣟⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣼⣷⡾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣂⣀⣤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠘⢤⣤⡴⠤⣤⣴⡶⠿⣿⠿⠷⣶⣶⣴⣶⣾⣶⠮⠴⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠞⠀⠀⠀⠈⠙⠢⠾⠀⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⡠⠄⠉⠉⠁⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠱⠖⠋⠉⠀⠐⠛⠛⠚⠛⠿⠂⣴⢤⣌⡙⠒⠤⡄⠀⠀⣀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠑⠀⠀⠀⠀⠈⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠖⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣀⡀⠰⣶⠰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⣇⠘⠀⢃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠙⢿⡿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⣼⡷⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢻⡇
⠀⠀⠀⠀⠠⣄⡐⢾⣿⣄⣴⣌⣰⣿⣿⣿⣶⣦⡄⠀⠀⣠⠄⢀⣀⡀⠤⢴⣾⡿⠁⢸⡇
⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⢀⣾⣷⣦⣴⠼⠷⣤⠴⠞
⠀⠀⠀⠀⠀⠀⠘⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣤⣾⣿⣿⣿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠛⠻⠷⠾⠿⠖⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠿⠛⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠿⠿⠟⠛⠻⠿⣿⣷⣶⣶⣦⣼⡇⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠇⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠏⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠉⠁⠀⢀⣴⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠟⠉⠀⠻⠋
⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠉⠈⠻⣿⣿⣦⣀⣠⣴⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⣠⣾⣿⡿⠋⣠⠽⠻⢿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣦⣀⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠉⠉⠁
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡇⠀⠀⠀⣿⣿⣿⡆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢿⣿⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣀⣉⣀⡀⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟⠋⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⢀⣤⣤⣴⣶⣶⣶⣾⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠠⣿⣿⣿⡇⠀⢠⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡆⠀⠀⣷⣿⣿⡃⠀⢸⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⢸⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠀⠀⠸⣿⣿⣿⠇⠀⠀⣿⣿⣿⠇⠀⠀⣿⣿⣿⠇
⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣀⣀⣀⣀⡀
⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⢠⣶⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⢿⣿⣿
⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠘⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠈⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠻⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠙⠿⣿⣿⣦⣄⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣤⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣤⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣾⠟⠀⠀⠀⠀⠹⣷⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⢠⣿⣿⣿⣿⣧⡈⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⠐⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣠⣴⠾⠷⣦⣄⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠛⣉⣀⡙⠻⢿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠾⠛⠉⠀⠀⠀⠀⠉⠛⠷⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡈⣉⣤⣶⣿⣿⣿⣿⣷⣦⣌⡉⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⠟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣿⣶⢶⣦⣤⣾⣷⣶⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡶⣤⡀⠀⠀⢀⣴⣯⣤⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣾⣋⣰⣾⣿⢷⣴⣏⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⣿⠀⢠⡾⢻⣿⣿⣿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⣿⢐⣿⣿⡿⣿⣿⢿⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣸⡯⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣿⡗⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣾⣿⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢰⣿⣿⡧⠀⠀⠀⠀⢿⡄
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡟⠀⣽⣿⣿⡧⠀⠀⠀⠀⠘⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠙⠿⣿⣏⡛⠙⠋⠋⠙⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⢴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⣉⣿⡿⠟
⠀⠀⠀⠈⠻⢿⣦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢂⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡐⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠋⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⠈⠛⢷⣄⠰⢤⡀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠋⡵⠟⢁⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠙⠻⢦⡙⢷⣤⡀⠀⢀⠈⠋⠀⠀⣀⣰⣖⣡⣾⡿⢫⠔⠋⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠈⠙⠳⢬⣉⠓⠀⣠⣿⣦⠀⠀⠀⠉⠋⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠤⣍⡐⠂⢀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢉⡓⠢⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣀⠲⠤⢠⣾⣿⣿⣿⠿⠛⠃⣿⣿⣿⣷⣄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠬⢁⣴⣿⣿⡿⠛⣁⣴⣾⣶⣌⡉⠿⣿⣿⣦⡀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠠⠌⣡⣾⣿⠟⢉⣴⣾⠟⠋⠀⠈⠛⢿⣶⣌⠙⢿⣿⣆⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣀⣾⠟⢋⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣌⠛⢷⡀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠰⢋⣥⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣙⠂⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠙⠻⣿⣇
⠀⠀⠀⠀⠀⠐⠐⠢⠞⠔⠿⠫⠖⠖⡷⠾⠾⠖⡷⠺⠶⠕⠷⠶⢚⠶⡷⠲⠞⡵⠳⢞⠖⠣⢺⠖⠷⢺⠖⠷⠢⠾⠟⠶⠲⠂⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢇
⠀⢠⣤⣤⣤⣤⣤⣼⣿⡇⢹⣤⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⢿⡇⠀⢧⣤⣤⣤⣤⣤⡀⠀⢠⣄⣀⣀⣀⣀⡤⣿⣇⠘⢄⣀⣀⣀⣀⣤
⠀⠀⠈⠳⢶⣤⣄⣀⢻⠇⢀⣀⣤⣴⠾⠋⠀⠀⠀⠀⠙⢦⣄⣀⠀⠈⠃⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠈⠳⢦⣤⣀⣤⣽⣿⣤⣄⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠙⡟⠛⣡⣦⡙⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠁⡀⡀⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⢫⣿⣿⣯⠉⠛⠉
⠀⠀⠀⠀⠀⡸⢀⣼⣿⢿⣷⣄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⣬⡷⣧⣄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣼⣿⠿⣿⣆⠘⡀
⠀⠀⠀⠀⢠⣷⠟⠋⠀⠀⠈⠛⢶⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠛⠁⠀⠀⠙⠳⣼⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠈⠙⢷⣥
⣀⣀⣀⣀⣚⣁⣒⣒⣒⣒⣀⣀⣀⣙⣂⣀⣀⣀⣀⣀⣀⣒⣋⣒⣒⣒⣒⣒⣒⣒⣚⣓⣀⣀⣀⣀⣀⣀⣀⣚⣓⣒⣒⣒⣒⣒⣒⣒⣛⣂⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣶⡧
⠀⠀⠀⠈⠛⢶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠴⠿⣶⡀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠤⢠⣭⣥⡄⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣈⠉⠙⢿⣆⠀⣿⡿⠁⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⡗⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠛⠉⠁⢀⣾⣧⡙⣯⣙⠛⠻⠿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠈⠻⣟⣛⡁⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣏⡀⠀⠉⣿⣭⣤⣀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣯⣤⠄⠀⠈⢿⣦⣶⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⣽⣖⠄⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠟⠁⠀⠀⠈⠙⠶⣶⡄⠀⠀⠘⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⠛⢿⣯⣭⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢛⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢀⣾⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠙⣿⣠⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣿⢳⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠟⠛⠉⠁⠀⠀⢠⣾⣿⠀⠹⣿⣿⣿⣿⣷⣶⣤⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⣦⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠙⠷⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣴⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⡾⢹⡟⠆
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠁⠀⠀⣰⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⠟⠀⠀⢀⡴⠋⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠋⠀⠀⠁⠀⠀⠀⢠⣿⣷⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠘⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢋⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣋⣥⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢞⣫⣥⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣠⡴⠀⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣬⣭⡷⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢿⣧⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣠⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠛⢁⡀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⠀⠀⢠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢈⠛⠦⡄⠀⠀⣿⠀⠀⠀⠀⠀⠖⠋⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡿⠃⠙⣿⣆⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠤⠶⠋⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠴⠞⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣤⠖⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⡴⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢀⠴⠚⡁⠀⠀⠀⠀⠀⣀⣤⠀⡉⠻⢦⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⡆⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣴⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠉⣠⠀⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⢁⡤⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⣠⠀⣿
⠀⠀⠀⠀⠀⢹⠀⠀⠀⢠⣿⠀⠀⠀⠀⠐⠛⢁⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠚⠋⣁⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠛⢁⢀⣿
⠀⠁⠀⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠐⠖⠋⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠰⠚⠁⣾⡇⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠖⠋⣸⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠛⠛⠟⠛⠛⠿⠛⠛⠛⠀⠀⠀⠀⠙⠛⠟⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠛⠛⠛⠛⠟⠻⠟⠛⠛
⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣦⡄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠛⠉⠁

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⣽⣿⡟⠁⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⣽⣿⡟⠁⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠚⢿⠷⠀⠀⠀⢀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠷⠀⠀⠀⢀⣠⣆⡀
⠀⠀⢰⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⠀⠀⠀⠀⢶⣿⡶⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠀⠀⠀⠀⢶⣿⡶
⠀⠀⠛⠙⠏⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠛⠛⠏⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉
⣤⣀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢹⠿⠏⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⢠⣤⣀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢹⠿⠏⠀⠀⠀⠀⣀⣴⡀
⠻⠏⠀⠀⠀⠀⠀⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠋⠀⠀⠀⠼⠻⠏⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠋
⠀⠀⢀⠀⠀⠀⠠⠿⡿⠷⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠿⡿⠷⠀⠀⠀⠀⣠
⠀⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠂⠀⠀⠀⢤⣾⣆⠀⠀⠀⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠂⠀⠀⠀⢤⣾⣆
⠀⠘⠋⠉⠁⠀⣀⣴⣆⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠘⠋⠉⠁⠀⣀⣴⣆⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⠋
⠀⢀⠀⠀⠀⠀⠸⠛⠟⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠛⠟⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⢀
⢶⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠇⠀⠀⠀⠘⣿⡷⠂⠀⠀⠐⢶⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠧⠀⠀⠀⠘⣿⡷⠂
⠛⠙⠃⠀⠀⠀⠀⣤⣿⣞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⣤⣿⣞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⢀⣀⡄⠀⠀⠀⠙⠉⠀⠀⠀⢸⣾⣷⠆⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠙⠉⠀⠀⠀⢐⣾⣷⠂⠀⠀⠀⣀⣤⣀
⠀⠀⣼⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⣼⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⠿⠏
⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⢠⣴⣧⡄⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⢠⣴⣧⡀⠀⠀⠀⠀⣤
⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠿⢿⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⢶⣿⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⢶⣿⡤⠀⠀⠀⠀⠀⢀
⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⣶⣿⡦⠀⠀⠀⢸⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⣶⣿⡤
⠈⠁⠀⠀⠀⠀⠀⣼⣷⣇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣼⣷⣇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋
⠀⣀⣴⣄⡀⠀⠈⠉⠋⠉⠀⠀⠀⢤⣾⣤⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⣴⣄⡀⠀⠈⠉⠋⠉⠀⠀⠀⢤⣾⣤⠀⠀⠀⠀⠀⣠⡀
⠀⢹⡿⠿⠅⠀⠀⢀⡀⠀⠀⠀⠀⠸⠛⠛⠀⠀⠀⠀⣽⣿⡷⠀⠀⠀⢹⡿⠿⠅⠀⠀⢀⡀⠀⠀⠀⠀⠺⠛⠛⠀⠀⠀⠀⢽⣿⡷
⠀⠀⠀⠀⠀⠀⢲⣿⣷⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣷⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣴⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⡗⠀⠀⠀⢠⣤⣄⡀⠀⠀⢀⣀⣴⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⡗⠀⠀⠀⢠⣤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠱⠄⠈⢁⣀⣵⡙⠳⡿⠄⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⢀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢕⣤⠀⠀⣠⠊
⢀⣴⣕⠄⠛⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠉⠀⠛⠀⢀⡀⠀⠀⠀⡔⢠⠀⠀⠀⠘⠁⠀⠜⠁⠸⣆
⣝⠂⠙⢀⣴⠄⠓⠀⠀⠀⣠⠰⠲⠲⠆⠁⠀⠀⠴⠀⠒⠠⠀⠀⠈⠀⠀⠀⠀⠀⠁⠤⡄⠀⠀⠀⠀⠠⠀⠀⠈
⡙⠂⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡚⠶⠄⠀⠀⢀⠥⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠘⠉⠁⢢⠀⠀⠀⠈⠀⠀⢸⠒⠒⢀⡀⠀⠀⠀⠁⠀⠔⠀⠀⠀⡠⠚⠁
⠀⠀⢠⠀⠤⠰⠇⠀⠀⠀⢰⠃⠚⠁⠠⡄⠀⠀⢁⠇⠰⠀⠀⠀⢰⠤⠐⣀⠸⠂⠒⢀⠰⡄⠀⢀⡀⠀⠀⠀⠠
⡰⠈⠙⠂⠀⠈⡱⠔⠶⠤⠌⢠⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠸⡀⠀⠉⣠
⠳⠦⠀⠀⠀⠀⣁⠀⡤⠬⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠄⠀⠠⠅⠀⠠⠒
⣦⠀⠀⠀⢰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⠂⠀⠀⠉⠉
⠀⠈⡁⠀⠘⠠⠂⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⡤⠤⠄
⣤⡦⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡶⠖⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣼⡷⠖⣋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⢴⡶⠟⣋⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠢⣌⠙⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣵⣷⣮⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣤
⠀⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠓
⠀⠀⠀⠀⠀⢹⢿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢑⣶⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣤⣔⠁
⠀⠀⠀⠉⠉⠉⠉⢹⣿⣿⣿⡋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠿⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡟⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣷⣄⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⣀⣠⣿⣆⡀⠀⠀⠀⠀⢀⡰⠣⡠⠀⠀⠀⠀⣀⠀⠂⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣀⣀
⠀⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⢠⠂⠣⠀⠀⠀⠀⠈⣃⠤⣄⠁⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢈⡟⢯
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⢘⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠇⠀⠛⢀⡉⠀⠀⠀⠂⠀⢤⠄⠈⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⢳⡄⣚⠃
⠀⠀⠀⠈⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠐⠂⠉⠃
⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠂⠺⠠⠀⠃⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⢠⢠⡾⢤⠤⠀⠀⠠⣶⣿⣶⣀⣠⠔⠁⠀⠀⠄⠈⠃⡔
⠀⠀⣠⡀⠡⠂⣴⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⣙⠶⣻⡀⠀⠀⠀⠟⠙⠳⠀⠀⠀⠀⠀⠀⡤⠐⠨⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠂⠀⠀⠀⠁⠀⡀⣀⠘⡀⠀⠀⠀⠀⠀⢀⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣷⣤⡄
⠀⠐⣾⣿⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠁⠀⠀⠀⠉⣹⠿⣶⡀⠀⠀⠀⠰⠶⠴⠂⠀⠀⠀⠀⠀⢙⣿⣿⣇
⠀⠀⠋⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣶⣼⠧⠖⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠳⠄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⢀⡀⠁⠠⠀⠀⠀⡀⡀⠸⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠤⠀⠀⠀⠠⡖⠂⠡⡶
⠀⠠⠂⠄⠀⠤⠄⠀⠀⠀⠀⠈⢓⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠇⠊⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠾⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠟⠛⠉⠉⠀⠀⣤⣤⣤⣬⡄
⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠐⡿⠛⢻⣧⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣀⣀⣀⡀⠀⠀⢸⡇⠀⢸⡿⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⢀⣠⡾⠁⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡟⠛⢻⡇⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠐⠶⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣤⡤⢤⣤⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠶⣿⠀⠀⠀⢸⠇⠀⣸⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡀⢀⣿⠀⠀⠀⠀⣧⣀⣠⣧⠀⠀⠀⢸⣧⣀⣸⡇⠀⠀⢸⣇⣀⣀⡇⠀⠀⠀⠀⣿⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠈⠛⠛⠛⣿⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⢻⡿⠛⠛⠛⠛⠛⠛⠛⣿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⡿⡏⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇
⡀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠂⣀⣤⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⠀⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠼⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠈⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢲⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⢺⣿⡂
⡀⠀⠀⢒⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣠⡀⠀⠀⢙⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⣠⡀⠀⠀⠠⣾⡤⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠰⠿⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠸⠿⡏
⠙⠃⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠉⠁⠀⠀⠀⡄
⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢺⠿⡍⠀⠀⣄
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⡀⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠄⠀⠀⣀⣤⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠄⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣤⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠹⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠛⠧⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠛⠃⠀⠀⢲⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢺⣿⡍
⡄⠀⠀⠐⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢐⠿⡏
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⡄
⣰⡀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠰⠛⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣿⡖⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠼⠻⡏
⠙⠃⠀⠀⠀⠁⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠁⠁⠀⠀⢀⣄
⣀⣤⣀⣀⣀⣀⣀⣀⣀⣘⣿⣏⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣹⣿⣅⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠐⠈⠉⠉⠑⠢⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠙⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠂⠀⠀⠤⠄⠀⠀⡄⠀⠀⠀⠑⠖⠖⢀⣀⠄⠀⠀⢀⣄⣀⠄⠀⠀⠈⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠒⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⢀⡉⢣⠀⠀⠀⠀⠐⠦⠤⠄⢀⣀⡀⠀⠢⠤⠔⠊⠒⠃⠀⣠⢀⠄⠀⠀⢠⠀⠀⠀⠈⢢
⠀⠀⠀⠰⠊⠁⠀⢀⠀⠀⣈⣧⣀⠉⡉⠀⠙⠓⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡜⣠⠀⠀⠀⠀⠁⢳
⠀⢀⡤⠂⠀⠐⠊⠹⣄⡀⢀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⡀⠀⢀⡜⠀⠀⡀⠈⠹⠄⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠀⡴⠀⠀⠈⠀⢤
⠀⠊⠀⠀⠒⣆⠀⠀⣀⡍⠀⠀⠀⠑⢪⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠀⠠⣄⡀⠐⠆⠀⠒⠁⠒⠒⠋⠀⢀⠦⠔⠄⠈⠇
⡀⠀⠀⠀⢤⡈⠁⠀⠰⡁⠀⠸⣄⠀⡀⠑⠂⢄⠀⠰⣄⡀⠄⠀⠀⣀⠀⠀⠣⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⣰⠤⠂⠈⠋⠁⢀⠄⠀⠀⡀
⢇⠱⠄⠀⠀⣉⠁⣠⠀⠉⠀⠀⢀⠉⡀⠀⠀⠀⠉⠁⠀⢀⠀⣉⡉⢀⣀⡠⢄⣈⠉⣉⣁⡀⠠⣄⣠⡄⠀⠀⢤⡀⠀⠀⠈⠉⠁⠠⠖⢊⡁
⠈⠢⠀⡄⣄⠈⠀⠻⠀⠀⠙⠤⠈⠉⣁⡜⠐⠋⠈⡐⠊⠁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⣀⣜⠃⠀⠀⠀⠠⠴⠒⠊⠀⠄⠈⡄⠊
⠀⠀⠀⠈⠢⡁⠀⠤⡀⠄⠐⠂⠀⠀⠀⠈⠋⠉⠉⠀⣿⠀⠀⠀⢰⠀⡇⠰⠀⠀⠀⠀⡼⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⣀⠀⠚
⠀⠀⠀⠀⠀⠉⠒⠂⠐⠤⡍⠉⢀⣐⡊⡐⠒⠀⠀⠖⡿⠀⠀⠀⢸⠀⢇⡇⠀⡈⠠⢸⠃⠒⠓⠒⠂⠀⠀⠉⡈⣁⣀⡠⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠑⠄⠀⢺⡀⢈⠸⠈⡇⠸⠀⢃⠇⡆⡿⠖⠤⠒⠐⠊⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡆⡀⡇⡄⠠⣸⢰⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⡇⠁⠀⡇⡏⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡼⢇⡠⠴⢧⣇⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠌⠉⠁⠀⠀⠀⠀⠀⠁⠘⠋⠳⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⢠⣴⠦⣴⠠⢤⣤⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠲⠄⠠⣿⡇⣿⠀⢸⠀⢸⢣⠔⠢⠤⠆⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⡿⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⡇⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢳⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠴⢦⠀⠀⢈⠤⢤⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠣⠀⠀⠀⠀⠋⠑⡀⠀⠀⠉⠀⠀⠁⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠀⠀⠔⢢⡤⡈⠂⢀⡀⢩⠤⠒⢤⢤⡀⠙⠀⠐⠳⡄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠀⠠⠄⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠐⠆⠀⠀⠑⠲⠀⠈⠓⠀⠈⠈⠋⠂
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠛⢿⣿⣿⣦⣤⣾⣿⣿⠟⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠀⠀⠀⠙⢿⣿⣿⡿⠟⠁⠀⢰⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⢠⡄⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡀⣀⣄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣤⣠⣶⠋⠀⠀⢀⣴⡤⣤⠘⠿⢿⡧⢘⣟⡃⠀⣀⠀⠀⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠈⣛⣃⣈⢁⠀⠀⠀⢘⣻⣿⣾⠟⠀⠀⠛⠴
⠀⠀⠀⠀⠀⠀⠀⠾⠀⢈⠉⠉⠙⠃⢴⣶⣦⠀⠀⠿⣿⡿⢸⣿⣷⠀⠉⠉⠉⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣾⢻⣿⠀⠀⠀⠀⠀⢸⡏⣿⡀⠀⠀⠀⠀⠀⠀⢹⡿⣿
⠀⠀⠀⠀⠀⠀⢠⢿⡇⠀⠀⠀⠀⢠⡏⠘⡟⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠀⠀⣾⠁⢿⡀
⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠙⠀⠈⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠙⡏⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣷⡆⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⢸⠇⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⣿⢡⡟⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⣾⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡏⣸⣧⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢸⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⠀⡿⢸⠇⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡌⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠸⡾⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢤⣤⡤⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡽⢮⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠊⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⠻⢿⣿⣶⣶⣤⣤⣀⡠⠁⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠉⠛⠿⣿⣿⣿⣷⡀⠀⢰⣿⣿⡿⠦⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⠻⢿⣷⡀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⣒⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣠⣬⢕⡩⣤⣶⣶⣶⣶⣾⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠁⢸⣧⠀⠉⠻⢿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⠀⣿⣿⣧⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⣀⠤⠘⢿⣿⣿⣧⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢀⢤⠞⠀⠀⠀⣴⡿⠋⢀⠠⠐⠈⠀⠀⠀⠀⠻⣿⣿⣆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣷⡀⢀⠄⢊⣴⠏⠀⠀⢀⣾⠫⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⣤⣤⣤⣤⣤⣤⣤⣅⠈⣿⡏⢀⣴⣿⠃⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠇
⠀⠀⠀⠈⠐⠨⢍⠙⠛⠻⠷⣼⡴⠟⠛⠃⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⢈⣶⣶⠟⣹⡍⠻⣶⣶⣦⣬⣅⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⠠⣿⣧⠀⡇
⠀⠀⠀⠀⠀⢠⢟⠡⠊⠁⠀⠈⢿⡆
⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⢶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢟⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢍⠉⠃⠀⠀⠀⢀⠄⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣺⣯⣶⠄⢀⠤⠄⠂⠀⣀⠀⠀⠔⠃⠀⠐⠤⡀⠐⠄⠀⢀⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡝⠟⠉⡀⠎⠉⣶⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠀⠀⢀⡈⢹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣫⡞⠀⢼⠀⢀⡀⣷⡀⠀⢄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡇⠊⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣾⣦⣼⣶⣤⣽⣿⣿⣶⣾⣷⣤⣿⣿⣦⣄⣀⣀⣤⣄⡀⢀⣀⣁⡀⠀⠀⠁⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣝⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⡿⠿⠶⣶⣦⣴⢀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠳⣭⣿⣥⣝⣿⣷⣶⣿⣿⣿⣿⣁⡀⠀⣀⡀⢠⣿⣿⣿⣾⣿⡿⠿⣿⣫⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⢸⠿⢿⣿⣷⣿⣏⠉⢻⡟⣿⣇⠉⠋⠀⠀⡟⠃⣼⠟⠉⣹⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣬⡉⢿⡙⢿⣆⠀⠁⣿⣿⠀⠀⠀⢠⡇⢰⠏⢀⣾⣿⠟⣡⣾⣿⠁⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠲⣝⠆⠙⢧⠀⣿⣿⠀⠀⠀⠈⡇⠀⣠⡿⢋⡴⡚⠱⠂⠀⠀⠀⠀⠘⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣭⣙⡛⠂⠁⠀⠀⢰⣿⣿⠀⠀⠀⠀⢱⡐⠋⠀⠑⠊⠁⠔⢛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣭⣽⣛⣛⠟⠃⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⢧⠀⠀⠀⠴⠒⢛⣛⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣾⣿⣿⣿⡄⠔⠒⢂⣀⠜⠆⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠁⡀⠀⠀⠈⠺⣿⠟⠛⠻⣿⣦⣴⣈⡀⠀⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣖⣁⣀⣀⣀⣈⣙⢿⠶⣶⣀⣀⣠⡤⠈⠛⠉⠙⠛⢿⣿⣁⣙⣒⣲⣷⣶⣶⣀⣾⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣟⣹⢛⣟⣻⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢻⣷⡄
⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣷⣌⡻⣆⠀⣀⠠⠄⣀
⠀⠀⢄⠀⠀⡠⠄⢸⣿⣷⣏⢷⣽⣿⣮⡇⠀⠀⠀⠈⣷⠞⠉⠑⠢⣄⠀⢀⡀
⠀⠀⠀⠑⠄⣈⠀⣸⣿⡟⣟⣸⢿⠇⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⠋⠁⠉⠑⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣶⡿⠄⠛⠠⠤⠄⣀⠼⠄⣄⡀⠀⠀⡏⠀⠀⠀⠀⣸⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠙⠛⡟⠀⠀⠀⠀⠀⠸⠁⠀⠀⠠⠁⠀⢸⠁⠀⠀⠀⢠⡏⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⠀⠲⡀⠀⠀⠀⠀⠰⠧⣀⣴⢇⡜⠀⡟⠀⠀⠀⢀⡞⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠲⣤⣀⣀⣀⣤⣴⠋⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⣴⣷⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⡿⡟⠹⡿⢶⢶⣿⣷⣴⢶⣯⣀⣀⣠⣾⣽⣏⢻⣾⣿⡿⠋⣡⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠞⠁⠃⠀⠀⠈⡇⠛⢿⡀⠀⠉⢍⣍⠙⢿⣯⣻⣫⣾⡿⠀⣮⡈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⢢⠈⠻⣿⣿⣿⡇⢸⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⡄⠀⠀⠀⢡⡀⠙⠹⣿⠀⠘⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⡀⠀⢀⣀⣼⣣⠀⠈⠀⠀⣀⠀⣟⠦⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠙⠛⠛⠛⠋⠥⢤⣤⣤⣤⡼⠋⠦⠖⠚⠙⠓⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠠⢄⣀⠀⠀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣓⠄⠰⡱⠙⠦⠠⢀⣩⠤⢀⠝⡦
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⠁⠀⠀⠉⠛⠒⠀⠒⠛⠋⠉⠁
⠀⠀⠀⣀⠤⣆⡒⠺⡺⢻⡅⡀⠲⣦⠴⠀⠀⠁⢐⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠰⢤⣀⡀⠀⢀
⠀⠠⠎⠘⠒⠛⠒⠬⠠⣈⠽⠛⠚⠁⠀⠀⠀⠀⠀⠈⢡⠄⠀⠀⠀⠀⢀⠖⠀⠀⠠⠂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠠⠔⢠⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠿⣥⠭⡅⠂⠀⢀⢄⠞⠁⠀⠀⢀⠀⡂⠤⣀⠠⡀⢀⣼⠇⠒⠒⠀⣲⣤⣤⢀⣀⣠⡷⢶⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠲⡀⠀⢀⠄⠠⠂⠁⠀⠀⠀⠐⠃⢈⡠⠤⢍⡿⢀⠾⢇⡀⠀⠀⠈⢚⣈⣡⠁⡠⢄⡀⠹⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⣩⠔⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠀⠀⠢⠀⠀⠀⠉⠀⠀⡀⠞⠩⠝⡟⡳⠁⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠰⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢆⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠸⠀⠀⠀⠀⠀⠊⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠢⢧⣀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠥⠠⠄⠀⠀⠀⠐⠒⠦⠒⠂⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠕⣒⠀⠀⢰⠽⠒⠰⡦⠤⣔⢃⢷⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠒⠒⠒⢀⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⠚⠉⠁⠊⠀⠑⠫⣢⡰⠂
⠀⠀⠀⠀⠀⠀⣀⠀⡀⣀⣀⡠⣄⠀⠺⠄⢲⣆⣀⠨⣟⠐⠯⠕⠶⠤⠤⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢥⣀⠐⣀⠀⠀⠈⠑
⠀⠀⠀⠠⡊⠁⠀⠏⠃⠁⢠⣄⡜⢢⠀⠈⠀⠀⠀⠈⠁⠛⠬⠤⠤⣥⡤⠤⠍⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠉⠀⠉⠐
⠀⠀⠀⢠⡁⠂⢅⠱⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡖⠚⠒⠲⠄⡀⠄
⠀⠀⠀⠈⢏⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠯⠷⡷⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠉⠊⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆⢁⣠⠬⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠋⠉⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⣠⣶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣛⣟⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠉⠛⠋⠙⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⣿⡶⠿⠿⠿⠿⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢼⡇⠀⢸⣿⠀⣶⠀⠀⠀⠀⢠⣶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢸⡇⠀⢸⣿⢀⣿⠀⠀⠀⠀⢀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠋⠀⠀⠀⠀⣿⡟⠛⠿⠛⠻⢿⡟⠀⠀⣿⡆⠀⠀⠀⠀⢹⡇⠀⠀⠻⠿⣿⠀⠀⠀⠀⠘⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠰⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⠀⠀⠀⢸⣧⠀⠀⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸⣏⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⢀⣤⣤⣼⣧⣤⣤⣤⣤⣼⣿⣤⣤⣿⣧⣤⣤⣤⣤⣼⣿⣤⣤⣿⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣿⣤⣴⣿⣤⣴⣿⣤⣤⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣇
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⣠⣶⣿⣿⣿⡿⢿⣿⣿⣿⣶⣄⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⣀⣴⣿⣿⣿⡿⠛⠁⠀⠀⠈⠛⢿⣿⣿⣿⣦⣀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⢠⡄⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⣀⠀⠀⠀⠀⡀⢀⠐⢀⠁⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⡖⠀⠀⠉⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣧⠀⢀⠀⠀⠐⠦⠤⠄⠂
⠀⠈⠢⠀⠀⠀⡀⠀⠀⠘⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⣽⣿⠋⠀⣤⡀⠂⠀⠃
⠀⠀⢀⠀⠀⠀⢀⠀⣠⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢲⣻⡖⠀⠀⠀⠀⠀⠀⠀⡰⣆⠀⠉⠀⠀⠉⠀⠙⠂⠀⠉
⠀⠀⠀⣠⢆⠀⠀⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠰⣶⠶⠃⠘⠖⣶⠆⠀⢀
⠀⠈⢫⠁⠈⡭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠄⠀⠈⡷⢀⣀⢸⠁⠀⠀⠙⠀⣴⡀
⠀⠀⠸⠚⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⡀⠀⠀⠀⠛⠉⠈⠛⠇⠀⠀⠛⣿⣿⣿⠟⠁
⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⡵⢀⣀⡀⠀⠀⢀⣤⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠟⠉⠻⠄⠀⠀⣄
⠀⠀⠀⠄⠀⢠⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠉⠁⠀⣀⠀⠸⠁⠠⣀⣀⠀⠀⠀⠀⠰⠶⣆⣼⣂⣠⠄
⠀⠀⢤⣤⠴⢊⡴⠛⣦⣦⡠⠄⠀⠀⠀⠀⠀⢆⠀⠀⠀⣿⣿⣿⣿⣯⠀⠀⢀⢀⣶⡀⠀⠈⠓⠀⠀⢀⠀⠀⠀⣠⢀⠀⠀⢰⠞⢹⡉⠁
⠀⠀⠀⠡⣶⠟⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃⠘⠋⠀⠀⠈⠛⠀⠀⠈⢻⣿⣿⠁⠀⠴⠚⠀⠀⠇⠀⠀⠚⠙⠀⠀⠉⠀⠀⠉
⠀⠘⠃⠀⡥⡾⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠉⢀⡀⠀⠉⠁⠀⣀⡾⢇⡀⠀⠐⢞⣱⠖⠀⠀⠅⠇
⠀⠀⠀⠐⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠐⣾⡖⠀⣠⡤⠎⠥⢤⡄⠀⠸⠍⠀⠈⠩⠟⠀⠉⠉⠁⠶⠻⠦⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⢳⢀⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⠀⢶⡒⠒⣃⣤⣈⠓⢒⡶⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠋⠁⠈⠃⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⣤
⠀⠙⢿⣿⣿⣿⡿⠛⢁⠀⠒⠀⡀⠙⡆⠹⠿⠏⢰⠋⢀⣀⣼⣀⡀⠘⢿⡿⠻⢓⡴⠀⠀⠀⢠⣼⣷⡤⠀⠀⠀⠀⠀⠀⠠⣔⠚⠀⠑⣲⠆
⠀⠀⢰⡿⠿⢿⣧⠀⠀⣀⠀⡈⠀⢰⣇⡤⠒⠦⣈⡇⠀⢹⠿⣏⠁⠀⠀⡾⠛⢿⡄⠰⠀⠀⠀⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⢨⣃⡄⣘⡁