Jett Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠼⣦⠀⠀⠀⢀⡤⠒⠉⠁⠀⠠⣿⢽⠿⡏⠩⢛⠲⡝⠢⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⡶⠟⠁⠀⠀⠀⣠⣤⣌⣇⢀⡴⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠐⢁⡾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠻⠏⠀⠀⠀⠀⡠⠐⢤⣀⢈⣿⡿⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⣠⠔⢚⠗⠀⠀⢁⣴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⢦⣴⣿⡇⠀⠀⠈⠉⣿⠃⠀⠀⠀⠀⠀⠠⠀⢀⡤⠚⣡⡼⢋⣀⡠⠰⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⣾⡿⠐⠋⠉⠀⠀⢸⣟⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⠉⠒⠒⠲⠤⢤⣀⡈⣧⠀⢷⢰⣶⣦⣤⠀⢰⠏⠀⢀⠐⠶⣶⣄⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣿⡉⠸⡃⢀⣿⠀⠀⠘⢿⡟⠰⠻⢡⣿⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡶⠶⠶⣶⣤⣤⣄⣀⡀⠀⠘⣶⡊⠉⣿⣿⡄⠀⣻⢿⠀⠀⠈⠀⠀⠀⠠⡘⢿⣿⡇⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠷⢾⣿⣄⣇⠈⠙⢿⣇⠘⡄⠀⠀⠀⠀⢀⠠⣦⣬⠿⣧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣤⣀⡙⠂⢷⠀⡀⠀⢾⣿⡇⠀⣜⠁⡽
⠀⠀⠀⠀⠀⠀⠀⢶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠙⠳⢶⣤⡀⠀⠀⠀⠀⠀⠙⠻⣶⣦⣄⠀⠀⢨⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣤⣄⡁⠠⣀
⠀⠀⠀⠀⠀⠀⠰⣋⡈⠉⠛⠻⢿⣿⣿⣷⣦⣤⡄⠄⠄⠀⠀⡀⠀⣀⢈⣱⠄⣀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣿⣦⣨⡑⠄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣆⣀⣈⣉⣻⣿⣿⣿⣿⣷⣷⣶⣶⣶⠌⠘⠁⠀⠀⠈⠓⠠⣤⠀⠀⠀⠀⠞⣿⣿⣿⣿⣦⡜⢆
⠐⠟⠂⠀⠀⠀⠀⠀⡂⠀⠀⠈⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠆⠰⡶⣿⣿⣿⣿⣿⣿⣶⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⡀⠀⢀⣵⣿⡿⠿⠟⠛⠛⠛⠻⢿⡤⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠁⠈⠑⠋⠁⠀⠀⠀⣀⠀⠀⣠⣆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢠⡀⠀⣧⣴⣒⣴⡶⢊⣀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣹⠁⠀⢻⣦⣽⣿⣾⡃⢰⠃⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠛⠛⠛⠛⠻
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡏⠀⢀⡠⠟⣿⣏⢛⢃⣾⣦⡬⠐⠀⠂⠀⠐⠀⠂⣀⣀⢀⠂
⠀⠀⣿⣿⣿⣿⡁⢀⡀⠀⠀⠀⠀⣰⠋⠀⠀⣧⠶⠋⢀⡞⠙⢿⣿⣿⢛⡠⠐⢒⢒⣒⣲⣧⣭⢯⡭⣼⡇
⠀⠀⣿⣿⣿⣿⠇⠉⠁⠀⠀⠀⢰⠋⠀⠀⠀⠁⠀⢀⡞⣀⣠⠘⣿⣿⣿⣷⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⡄
⠀⠀⣿⣿⡿⠏⠀⠀⠀⠀⠈⢠⠏⠀⠀⠀⠀⠀⠀⢘⡿⠋⠀⠚⣿⣿⡟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆
⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣠⠖⢁⡀⠀⠀⣰⣿⡟⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠰⠇⠀⠑⠄
⠟⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣤⢞⣡⡾⢿⠁⠀⠰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠾⠀⠀⠀⠀⢠⠀⠀⣇⣠⡶⣿⡴⠛⢁⡠⠚⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠙⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠟⣡⡞⣩⣄⢰⣋⡁⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢄⠀⠀⢀⣤⡪
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⣦⡽⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠨⣷⣦⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣆⣴⡀⣶⡞⠁⣠⠾⣡⡟⠀⣴⠟
⠀⠘⣶⡆⣾⣶⣤⣶⣶⣿⣿⢻⠟⣻⠠⣄⣻⠐⣿⢣⡾⣭⡁
⠀⠱⣿⡇⣿⣧⡄⢸⡟⠀⢹⡞⣷⠸⣴⣿⣿⣾⡟⠁⠙⠃
⠀⢀⣿⡇⣿⣏⡁⢸⡏⠀⢸⡿⣫⣽⣿⣿⣿⣇⡀⠀⠀⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠈⠛⠃⠛⠛⠃⠘⠃⠀⢸⣿⡯⠖⠓⣳⣬⠍⠀⠀⠀⢁⡂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⣤⡄⣤⣤⡄⣤⣤⣴⠾⣫⣶⣶⣾⣧⣭⣿⣦⣎⠀⠀⠀⠀⠲⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟⠃
⠀⠀⣿⡇⣿⣏⠁⢹⣿⠠⢿⣿⣿⣿⣿⣿⣿⠟⢻⣧⣀⠀⠐⠀⠐⠊⠙⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⠀⠀⣿⡇⣿⡟⠃⢸⣏⠀⣸⣿⣿⢻⠀⡿⠁⠒⠉⠉⢹⣿⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠐⠿⠃⠻⠿⠃⠘⠟⠀⢻⣿⣿⡟⣶⠁⠀⠐⣠⣶⣿⣿⢀⠀⢸⡄⠀⢀
⠀⠀⢠⡄⣤⣤⣄⣤⣤⣄⣼⡟⣸⣿⣿⡠⠔⢊⣽⣿⣿⣿⡆⠁⠩⠟⠋
⠀⠀⢸⡇⣿⡏⠁⢹⣿⠉⢹⡏⠀⢹⣿⣷⣶⠾⠛⠛⢟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡛⠀⣠⡖⠋
⠀⠐⣾⡇⣿⡟⠃⢸⣿⠀⢸⡇⠀⢸⣿⡿⣥⣦⣶⣾⣿⣿⣷⠀⠀⠀⡠⠀⠀⠀⠀⣦⡴⠶⣿⡿⠜⠋
⠀⠰⠿⠇⠿⠿⠇⠰⠍⠀⠸⠇⠀⣼⡯⠤⠈⠃⣿⣿⣿⣿⣿⣿⡓⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣦⡄⠨⣿⣿⣿⣿⡿⢫⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣿⣿⠟⡻⣦⣄⣉⠣⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢟⣿⣿⡿⢷⣴⣢⣿⡭⣙⢻⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣾⣿⣏⠀⠀⢙⣿⣿⣴⡞⠜⡀⠈

⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⡀⠀⠀⠀⠀⠀⠸
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⡠⠾⠀⠀⠀⢸⠁⠀⡨⠀⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⠀⠀⠀⠀⠙
⠁⠀⠈⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢀
⡄⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⡐⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠉
⠀⠀⢆⠀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠛⢶⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠈⠁
⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋
⡀⠀⠀⠐⠈⢳⡀⠀⠀⢀⣴⣧⣴⣿⣦⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⢦⡀⠀⠀⠀⠙⣄⢠⠞⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠙⢠⠀⠀⠄⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠂
⠀⠀⠀⠈⠑⠀⡈⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⣿⣷⠄
⣄⠀⣶⠀⣤⠀⠑⠀⠁⣄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡀⠋⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠏⣼⡿⣀⠋⡀⠀⠀⠀⠊⣠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣾⡇⠀⠀⡏⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⣠
⣾⣿⣽⠏⠈⠃⣀⠀⠀⠀⠘⠡⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠀⠀⢡⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿
⣿⠟⡁⠀⠀⢠⡏⡹⠀⠀⠀⠀⠈⢄⢳⡀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢸⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿
⣿⣽⠂⠀⠀⢸⣇⣀⣀⠀⠀⡀⠀⠀⠂⢇⢀⣀⣤⠤⠒⠝⠁⠀⠀⠀⠀⢸⡶⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿
⣿⣿⣷⣶⣀⣾⡏⠉⠛⠓⢤⢶⣖⣠⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣳⣶⣿⣿⣿⣿⣾⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡉⠃⠠⢍⠂⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣃⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣏⣿⢙⡻⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣾⡃⠐⠉⢻⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣦⣼⣿⣿⣶⡗⠀⠀⠀⠀⠀⠀⣰⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣿⣿⣿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠇⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢤⣀⠀⠒⠶⢶⣶⣚⣿⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⠉⠐⢉⣤⡴⠶⠞⠉⠉⠁⠀⠀⠉⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠿⠋⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡞⠁⣠⡼⠛⠉⠉⠉⠉⠉⠙⠛⢦⣄⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠏⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢷⣄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣏⣴⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠃⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⣀⠀⠀⠀⠀⢘⣁⠀⠀⠤⠀⠀⠀⠠⠄⡶⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣧⠀⠀⠀⠈⢿⠛⠛⠒⠂⠠⢶⣶⡿⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣼⣧⡀⠀⠀⠀⠙⠶⠀⣀⠀⠒⠉⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣌⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠳⠟⡄⢻⡄⠀⠀⠀⠀⢴⠟⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠹⢿⡁⠀⠀⠀⠀⣰⣤⠀⠈⠑⢶⣄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⠀⠈⠙⣦⠀⠀⢠⡏⠈⢻⣆⡀⠀⢈⠿⠁⢸⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡇⠀⠀⠀⣤⣼⡇⠀⣼⠃⢀⠘⣿⡌⠳⣼⠃⠀⠀⢀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠉⠉⠀⠀⠙⣦⡀⣰⠆⠉⠹⢾⠁⠰⠋⠹⠏⠙⣤⡟⠑⠦⣴⣎⢂⣼⠟⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣷⠀⠀⠀⢀⠀⢨⣧⡰⡦⠀⠙⠛⣄⡀⠀⠀⠛⠶⡶⠟⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣐⠂⠀⠘⠛⠀⠈⢹⣇⠀⠀⢸⣏⠙⢳⣤⡀⠀⠀⠀⠀⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⠖⠲⢆⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠈⣣⠀⠀⠀⣼⠉⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡷⠀⠀⠀⢸⡇⠀⠀⢻⣄⠀⠀⠙⠳⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠒⠒⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⠀⠧⠀⠀⠀⢸⡇⠀⠀⠀⠻⣄⡀⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣧⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠙⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⡆⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠇⠀⣿⠤⢤⠀⠤⠤⢤⡄⠀⠴⢄⠀⣀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡦⠀⠀⣿⠤⠀⠀⠐⠤⢤⡇⠀⠀⠨⠀⠆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃⠀⠀⠀⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⢿⡽⠋⠀⠀⠀⠀⠀⠻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇
⠀⣶⣆⠐⠀⣿⠱⣦⠀⠀⠀⣠⠶⠶⣤⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇
⠀⡇⠈⠳⣄⠘⠂⠘⠳⣄⣀⣻⠀⠀⣽⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠲⡀⠀⠀⠉⠉⢛⠀⢀⠟⢷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿
⡘⢻⡛⠆⠀⠀⠈⠳⣀⠀⠀⠙⠉⠉⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿
⡇⠠⡿⠂⠀⣠⡄⠀⠈⠟⠀⠒⠐⠓⠒⠚⠛⠓⠚⢻⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠻⣄⠀⠀⠀⠉⣦⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠈⠳⣄⠀⠘⠏⠈⠿⠀⠢⠐⠀⠒⠀⠀⠀⠂⠰⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠉⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠳⠶⠦⠴⠶⠶⠶⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⠐⢂⣤⡼⠿⠞⣭⡔⠒⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⣀⣀⡐⠶⠿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⢶⣆⣉⣉⣉⣹⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢻⡟⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠁⠀⠀⣼⠀⣬⠱⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠘⠆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡇⠀⠘⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠸⣿⠀⠀⠀⢀⡄⠊⠀⠀⢸⡀⢻⣀⡎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠰⠈⠀⠀⠀⢻⣄⠻⣄⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢯⣹⡀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣷⡄⠀⠀⠀⠢⡀⠀⠀⠀⠹⣦⠉⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⢀⣠⡤⠞⠋⣁⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢼⠿⣦⣀⠀⠀⠀⢀⣀⣠⣤⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣿⢋⣥⣴⠾⢛⣩⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣸⣶⠴⠞⠛⠙⠛⠻⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣽⠟⣩⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⠉
⢠⡶⠶⢦⡀⠀⠀⢀⡄⣾⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⣡⣴⠞
⢿⣠⡴⠞⢧⡀⣶⢭⣀⠹⣇⠸⣧⣶⢀⠀⠀⠀⢀⣠⠤⠖⣏⢉⡝⠉⢸⡏⠉⢢⣄
⠀⢉⡵⠖⡛⢻⣿⣦⡽⠀⢙⣧⡈⠻⠿⠀⠀⠈⠀⠀⣠⠶⣼⣯⣽⣿⣿⣥⣤⠾⠿⠿⢳⡀
⠀⠈⠷⣦⠶⠞⠛⠲⠶⠚⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣬⣄⢀⣀⣀⣀⣀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⢹⣟⠉⠉⠉⠙⣯⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠞⢻⡷⠤⠴⣶⢿⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⢷⣶⣶⣿⠀⠋⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠇⠀⠀⠀⣾⠀⡄⠀⠀⠀⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡞⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠃⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⣏⣠⣤⣤⣤⡀⣿⠀⣿⠤⠤⡤⠤⠼⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⠛⢷⣶⣿⠀⠀⣿⠀⣿⠀⠀⣀⡴⠞⢿⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⠇⠀⠀⠪⢝⠻⣦⣿⠀⢻⣠⠞⠋⠀⠀⠈⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡟⠀⠀⠀⠀⠀⠁⠂⣿⠀⢸⠁⠈⠀⠀⠀⠀⢸⡆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠶⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⣭⣿⡄
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⠛⣲⣴⡦⣤⣄⠈⠉⠉⠉⠉⠙⠛⣻⡟⠛⢻⡿⠿⢿⡿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣿⢀⠈⣿⣿⣷⣮⠤⣴⣿⣷⣶⣿⡀⣠⣾⠋⠀⣾⠇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣀⠶⣶⣾⣍⣉⣡⡟⣛⣷⣿⣿⡟⠁⠀⠙⣿⣿⣿⣿⣿⡿⠁⢀⣾⣿⣰⠃⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢐⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠓⠉⠙⠾⣿⣿⣭⣿⣿⣿⣿⠿⠋⠀⠀⣀⣣⣼⣿⣿⡿⠟⢁⣴⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⡝⡆⠀⠀⠀⣠⣶⣾⣿⣧⣤⣤⣄⣀⣀⣿⣿⣿⠿⠟⠋⠃⠀⠀⠀⣛⣿⠟⣛⣩⣥⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⢿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡗⠐⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣐⡾⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣀⣸⣧⠘⢛
⠀⢀⠀⠀⢠⣿⣿⣿⣿⢇⢤⣿⠇⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠈⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠈⠻⣿⣿⣿⣿⣿⠏⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣟⣛⣻⣿⡇⠀⠀⠀⣾⣿⣿⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠩⠾⠿⠿⠿⠟⠋⠉⠉⠀⠀⠀⠀⠘⢾⣶⣶⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⣀⣠⢪⣿⡿⠳⣌⠉⠑⠒⠠⠤⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡠⠔⠂⠤⡀⠈⠀⠀⢹⠳⡶⣤⢄⣀⣢⠔⢋⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣠⠔⣀⣀⣀⣉⣿⣿⣶⣦⡄⢀⡄⠀⠀⢸⠀⠈⡇⠀⣨⡤⣮⣅⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⢒⡲⠶⢤⣄⡀⠀⠀⠀⠀⠈⢀⡨⠿⠓⠉⠁⠈⠛⠿⠭⢛⡛⢛⡤⠚⠁⠀⠀⠀⢠⡀⢩⡞⢡⠊⠀⣠⡪⣵
⠀⠀⠀⠀⠀⠀⡰⠡⡲⡫⡨⢊⡤⠖⠫⢗⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣀⡀⠀⠀⠀⠀⠀⣁⡾⠀⠁⠁⣰⣿⠀⢸
⠀⠀⠀⠀⠀⢠⡇⢻⠱⠰⢱⠋⣴⣾⡷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣌⠉⠀⠀⠀⠀⣠⣃⡇⠀⠈⠀⡿⢋⢀⡎
⠀⠀⣀⠔⢊⡽⠿⣼⡀⠆⢸⢀⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣄⠈⣷⡀⠀⡴⡊⡩⣻⠟⠻⣻⣤⡀⣠⣴⠿⠛⠉⠻⢤⣄⣀⡜⣥⠎
⢀⢞⣠⣞⠉⠒⠠⢀⡉⠙⣲⠋⠀⠀⢀⠤⢂⠡⢐⣨⣤⣞⠛⢷⣼⣧⡎⢎⠎⠜⡐⣡⢊⡤⠖⢿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠘⠫⣄⡀⠉⠓⠦⣄⡈⠁⢿⣀⠀⠔⢁⡨⠴⣚⡉⠀⠈⠙⠻⠶⣬⡿⡜⡌⡘⠘⣰⢃⣾⢸⢸⢸
⠀⠀⠀⠈⠑⠢⢄⡀⠉⠑⢮⠹⣛⡏⢸⠒⠤⣀⡉⠑⠲⠤⣀⡀⠀⡇⡇⡃⠀⠇⡇⢸⢏⠎⢄⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠓⠉⠒⠢⢄⡉⠓⠢⢤⣀⠈⠑⣳⠶⣶⠀⡀⡇⢐⡡⢊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⠀⠈⠑⠲⢤⣈⡽⠋⣀⠤⠛⠓⠧⢷⣬⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣠⣏⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠹⠉⠀⣄⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⣤⠴⠖⠯⢄⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡔⠋⠞⠀⠀⠀⢄⠀⠙⣄⡠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡙⠂⠀⠀⠀⠛⠂⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠀⠀⠀⠀⠀⠉⠒⣀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠒⢐⠒⢖⢇⠈⠣⣀⠀⠀⠀⢤⣀⠔⠁
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠈⢢⣀⣠⡑⢄⠈⠑⢄⠀⡄
⠀⠀⠀⠀⢠⠃⠈⢁⢂⡀⠀⠀⠀⠀⢃⠀⢱⠀⠀⣤⠋⢱⡎⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠆
⠀⠀⠀⠀⣨⠄⠀⠀⠀⠀⠐⠂⠀⠀⠈⡏⠉⠇⠀⠀⠀⢠⡽⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢁⡠⠖
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠀⠘⡀⠀⠀⠀⡇⠀⠇⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣠⣶⠁⣀⣄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠤⢤⡖⢮⠡⡀⠀⠀⠀⠀⣀⣤⡠⣤⡞⠋⢸⣷⣿⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⡇⠀⠈⠀⠉⠉⠐⠀⠅⢸⠀⢃⠀⠀⢀⡴⠭⡗⠊⠀⣧⡄⠸⠿⣻⠎
⠀⠀⠀⠀⠴⠥⠐⠛⠓⢎⠀⠀⣀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡀⢀⣸⢷⠀⠀⠀⠀⠰⣇⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡾⠋⢁⢼⠀⠀⢀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⡠⠖⢹⠀⠑⠂⡾
⠀⠀⠀⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠛⠒⠀⠄⣀⣀⣠⠞⠋⣀⣴⠋⠀⠀⠀⠳⠤⡄⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⣀⢀⣠⢀⣌⠀⠀⠀⠀⠀⠀⡙⣇⡤⠊⠁⡏⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⠀⠒⠒⠈⠓⢢⣄⠀⡀⡖⡇⣟⠒⠠⢤⡧⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢖⠟⢹⣠⡡⠗⠁⠀⠀⠉⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⠀⣀⣤⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠀⠀⣀⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠙⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡌⠈⠀⠀⠀⢀⣀⣤⣾⠛⠉
⠀⠀⠀⠀⢾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣨⣷⡄⢠⣤⣶⠟⠋⣡⡇
⠀⠀⠀⣀⣼⣧⠀⠀⠀⠀⠀⠀⣀⠤⣂⣥⣴⠾⠟⠋⠀⠿⠋⠉⡟⠛⠛⠋
⣠⣴⠟⠋⠀⠈⢷⡤⠤⠄⠂⣩⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⡶⠷⠶⠶⠾⢶
⠹⣄⠀⣀⣠⣴⠞⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠘⠿⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⡤⠤⠤⠤⠤⠼⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⢨⠀⠀⠀⢀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⡄⠀⠀⢸⡇⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣼⡇⠀⠀⢸⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠋⡇⠀⠀⢸⠻⣆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣸⠇⢨⠇⠀⠀⢸⡇⢹⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⡏⠀⢸⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⢷⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⢷⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⢠⡿⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠈⢷⡀⠀⢀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢙⡷⠟⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡆⠀⠀⠀⠀⠈⠳⣞⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠖⠛⠋⠉⣸⠁⠙⠿⣦⣀
⠀⠀⣾⣟⣧⡀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠒⠛⢦⣀⠀⠈⠉⠛⢷⡆
⠀⠀⣿⠈⠙⠿⣯⣻⣉⠉⠁⢠⡦⣄⠑⠀⣀⠀⠀⠈⠛⠶⣤⣀⡾⡇
⠀⠀⢿⠀⠀⠀⠈⠉⠙⠛⣿⡎⠙⣮⣳⠄⠀⠙⠒⣤⣤⣤⣶⣿⡾⠃
⠀⠀⠘⢳⣦⣄⣀⠀⠀⠀⣿⡏⠓⣬⣡⣤⣶⣿⠿⠟⢿⡉⠀⠘⣷
⠀⠀⠀⠀⢷⡀⠉⠙⣷⡶⠿⠿⠟⢻⡏⠉⢹⣧⠀⠀⠈⣷⣠⣤⡾⠷⢶⠖⢛⣷⣶⡄
⠀⠀⠀⠀⠈⣧⠀⠀⠹⣷⡀⠀⠀⠀⢻⡄⠀⣹⣷⣶⠟⠋⠉⠀⠀⠠⠁⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠹⣷⣄⠀⣀⣤⡿⠛⠉⢠⠃⠀⠀⠀⠀⠀⠃⠀⣼⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣈⡙⠛⠉⠁⠀⠀⠀⠈⠒⠒⠠⠤⠀⣸⠀⠀⡇⠈⣹⠋⣨⣙⣬⣿⣧
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣿⣄⣿⣾⠿⢟⣿
⠀⠀⣴⣿⣿⣿⣿⣯⡻⣿⣿⣿⣿⣿⣿⣞⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠙⢿⡲⣿⢻⡿⠋⠉⢳⠶⣄⡀⠀⣀⣠⣤⣤⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣽⣬⠋⠀⠀⡞⠁⡊⠑⠍⡻⣯⣿⣟⣿⣿⣿⣧⡀
⠀⢸⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠙⢦⠈⡹⠻⣿⣿⣻⣿⣿⣧
⠀⢸⣿⣿⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠳⢦⣄⡀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠈⢉⣀⠔⠉⠻⣷⣿⣿⣿⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠙⠳⠦⣤⣀⠈⠉⣒⣦⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠘⠧⢀⡉⠙⠛⢓⠲⢦⡄
⠀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⣀⣀⡴⠒⠂⠠⠄⢀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⢹⣿⣿⣿⣿⣯⢿⡯⣥⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠁⠈⠓⠦⣬⣭⣛⣒⣶⣤⣤⣄⣉⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠉⠁

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⣄⣐⠘⡄⠀⢀⣀⠀⠠⠄⣒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠏⠭⣵⣒⠛⠛⣂⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⡤⠔⠂⠹⠁⠀⠀⠈⣆⠌⠁⠀⠀⢱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡼⠟⠀⠀⠀⢁⠀⠠⡄⠀⣽⠐⠂⠀⣀⠎⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣉⣀⠀⠀⠀⠀⠈⠢⢄⡀⠔⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⠔⡆⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢺⠀⠑⠒⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⡆⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠁⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣷⣤⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡠⢦⡔⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢻⣿⣿⣿⡿⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠰⣤⣝⠒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢟⡉⠛⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣽⡀⠀⠪⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⡤⠊⢀⠝⢛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠤⠤⠤⠤⠤⠄⢠⢬⡤⠒⣁⠤⠊⠀⣀⠜⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠨⠉⣁⠒⠋⠉⠁⠒⠒⠒⠀⠀⠤⠬⠛⠳⠤⠦⠴⠶⠤⠴⠃⠺⠓⠉⠀⠀⠀⠰⠁⡠⠰⢄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⣠⡃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠈⠢⠷⠀⠮⠚
⠀⠀⠀⠀⠀⠀⠤⠁⢅⣀⡸⢁⣱⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡜
⠀⠀⠀⢀⠴⠛⠚⠢⠤⠤⠤⠧⠸⠂⠤⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠜
⠀⠀⠰⠁⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠘⠆⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠷⠰⣅⠧⠀⠀⠀⣌⣡⠄⣼⠀⠠⠀⠲⠀⠀⠈⠉⢳⠀⡖⠖⠲⠒⠂⠉⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠄⡁⠀⡆
⠀⠀⠓⠀⠀⠤⠤⠄⠀⠀⠀⣀⣘⠀⠤⠀⠚⠀⠀⠀⠀⡆⠀⢰⡇⠤⠅⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠴⠓⠉⠉⠉⠑⢦⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⠿⠻⣛⢷⣶⠾⠷⠶⠶⠶⣤⣤⣬
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠉⠉⢉⠁⠀⣀⣩⠸⡛⢻⠛⠛⠓⠒⠒⠒⢂⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢀⣧⡀⢠⣤⣤⠿⠿⠟⣻⣤⢸⣿⣀⣀⣀⣀⡀⠀⠀⠀⢡⡄⠀⠀⠀⢢⡀
⠀⠀⠀⡀⠀⠀⠇⡀⠀⠀⢀⠀⠀⣾⣿⣶⣿⣿⠿⠵⠶⠯⢿⣿⡿⣿⠹⢿⣿⣿⣷⡆⠀⢰⣾⣇⠀⠀⠀⢸⡇
⠀⠀⢸⠀⠀⠀⢰⣿⣦⡄⢪⣤⣤⣼⣿⡟⣽⠃⣀⣀⣀⡀⠀⣻⡇⠟⢀⣾⣿⣿⣿⡇⠔⠉⠉⠉⢰⡀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣿⣿⣿⠛⠛⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠠⠿⣿⣿⣿⣿⢵⠀⠀⠀⠀⣾⡇⠀⢿⡅
⠀⠀⡀⠀⠀⠀⢸⣿⣿⢸⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣴⣾⣭⣭⣥⣿⣿⠀⠀⠀⠰⣿⣿⣿⣿
⠀⠀⢷⣶⣶⣶⣾⣿⠙⠟⠛⠋⠛⠛⠛⡃⠀⠀⠀⠀⠀⣠⣾⣿⣟⠛⠛⠛⠻⠛⠛⢿⠀⠀⠀⢀⣿⡏⠉⠉
⠀⠀⠘⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠭⠍⠉⠛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⣿⣿⣿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣿⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣖⣶⣶⣿⣿⣿⡿⢿⣿⣿⡷⢶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⣿⡿⠋⢟⣽⣧⣶⣿⠿⡿⠇⢁⣾⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠢⣦⣤⣿⣿⡧⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠃⠀⠀⣀⠠⠊⠁⠻⣧⣿⢟⠙⡃⣤⣄⣾⣿⣿⣟⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⣐⡴⠥⠡⣤⣀⡀⠙⢯⡶⢶⣿⣿⣿⣿⣿⡟⠩⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⣴⠫⣹⣦⣤⣴⣶⣄⣩⠷⣄⠹⢿⣿⣿⣿⣿⣿⠫⢸⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢀⡾⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢷⣁⢙⠿⣾⣿⡉⣲⣿⣯⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣽⢿⠁⠀⢠⡾⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠹⣿⣦⣀⠀⢹⡟⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢻⡏⣿⢹⡉⣽⢹⣟⡦⣼⠿⢛⣄⢘⠞⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⣿⣿⠿⣣⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣿⣏⢨⣦⣿⣿⣿⣷⣿⣿⡟⣹⡏⠀⠁⡜⡿⠀⠀⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⣿⣿⠓⢠⡅⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣵⣿⣿⣿⣿⣿⡿⣿⢿⣿⣽⠟⠀⠠⢎⡞⠀⠀⠀⠈⢷⣝⠻⣿⣿⠿⡿⠟⠙⣦⣿⡿⣪⣝⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢻⣏⣠⣴⣷⢾⢶⢢⣄⠀⢀⠀⡙⣷⣷⣾⣶⣷⢼⡺⣟⡯⡮⠁⠁⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣵⣿⢿⡷⠿⠿⠿⢟⣹⣿⣯⣷⣿⢿⠿⡏⣧⣿⣿⣽⣿⡟⠀⢬⣻⣾⣿⣿⣿⣿⣿⣿⣎⢡⢀⡀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡿⢁⡤⠖⠒⠚⠛⢛⣿⣿⡿⢁⣿⣼⣿⣿⡿⡿⣿⣿⣯⣔⡙⢮⣿⣿⡿⣿⡿⠿⠟⠁⠀⡉⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠚⠁⠀⠀⠀⢀⣴⢿⣟⣧⡾⠟⣛⣽⣯⣢⣴⣶⣿⣷⣿⣿⣿⣷⣍⠛⠛⠂⠁⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⢀⡴⣫⣷⣿⣿⣷⠶⠿⠿⣿⡿⣿⣿⣿⣿⡿⠏⠛⠛⠉⠀⣀⣤⣴⡶⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⡿⠟⠛⠛⠻⣏⢻⣿⣿⡿⡻⠇⠀⠀⠀⣀⣤⣶⢿⣿⣿⣿⢟⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⡾⠛⢉⣣⡀⠀⠀⠀⠀⢿⣷⡙⠳⣄⡀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣋⣴⠷⣦⠁⠑⢄⡀⡀⠳⠄⠙⠻⣦⣬⢿⣿⣿⣏⣿⣿⣿⣿⣿⣿⡷⠬⠭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⡋⢑⣾⣿⣦⣀⣦⣿⣿⣄⠀⠀⠀⣰⠁⣾⣿⣿⣿⠻⣿⣿⣿⣿⣿⡍⢓⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠦⠞⣰⣿⣿⣿⣾⣿⣿⣿⣿⠿⠷⢄⢀⡌⣼⣿⣿⡿⣿⠙⣻⣿⣿⣿⣏⣠⢤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡗⠀⠀⣀⡤⣺⣿⣿⣿⣿⣿⣿⡿⠛⠁⣄⢀⣤⠟⣽⣿⡿⠋⢰⡏⢠⣿⣿⣿⣉⢕⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢋⠞⠀⠖⠛⢃⠜⣫⣿⣿⣿⡿⡿⠛⣀⣤⣼⠃⠀⠀⣾⠿⠋⠀⣠⣟⣴⣿⠿⠝⣫⣷⠞⠁
⠀⠀⠀⠀⠀⠀⢠⡎⠀⢀⠤⠀⠁⠀⠀⠉⠊⠁⠀⣠⣴⣿⠟⡇⠀⠀⠀⠀⠀⣀⣴⣿⣿⣟⣯⡶⠟⠋
⠀⠀⠀⠀⠀⢠⣿⠀⠐⣠⠂⠀⠀⠀⠀⠀⢀⢄⡾⠋⠐⠁⠈⠀⠀⠀⠀⠀⠸⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⢀⣾⠇⠔⢀⢀⣀⡴⠊⠠⢏⣀⡤⠂
⠀⠀⠀⢀⠃⠀⠀⢐⣡⣦⠞⠙⠉⠀⠀⠀⠀⠛⠋
⠀⠀⢀⣤⣴⠀⠰⠋⠁⢀⣀⡴⠊⠐⠋
⠀⠀⠀⠹⠃⣀⣤⣶⢾⠟⠁
⢀⠀⡰⠷⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⠀⢰⠛⠛⣻⣷⠀⠀⠀⠀⠀⠀⠀⢰⡋⢈⠿⢿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠂⢘⠀⠈⡉⠈⢅⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠆⠸⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣶⠈⡀⠀⢻⣶⣿⣻⠄⢀⠀⠀⠀⠀⠀⠘⠉⣛⣟⡀⢁⢦⡀⠐⣶
⠀⠀⠀⠀⠀⠀⢰⣶⣾⣡⢤⠞⠃⠀⠊⡀⠀⢾⣇⠀⠀⠀⠀⢀⡴⣻⠿⠟⠉⠙⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⣥⣴⣴⣄⣸⡇⢀⠹⠟⠀⢀⣄⢉⡴⠚⠁⠀⠀⣄⢸⡀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢏⣀⢛⠛⠋⣰⠏⠀⠀⠀⣠⣟⣯⡴⠶⢤⣶⡤⠋⠂⠙⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠋⠰⠖⠚⠇⠀⠀⢲⠀⡯⠿⠛⠕⡆⠀⠄⠀⠀⣠⣾⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠒⢿⠁⢆⠀⠀⡀⠀⠀⠀⠀⢀⡀⡄⠀⣹⣿⠏⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠈⠁⠹⣆⡀⠀⠀⠀⠀⡀⠁⠁⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⢀⠀⣿⣷⡄⠀⠹⡇⠀⠀⠀⠄⠀⠀⡄⠒⠇⢀⣈⠿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⡇⢠⠹⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢸⡆⠱⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⡇⠀⠀⠀⠀⡠⠀⢳⠈⣧⢀⠻⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠀⠀⠀⠈⣿⠀⠀⠐⢣⠀⠀⠀⢣⡘⣟⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⡻⡄⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠳⡌⠓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡶⠋⣁⣤⠶⢛⣡⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠿⢷⣄⣀⣀⣀⣀⣤⡴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣵⢶⣟⡩⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠼⠷⠚⠋⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⣠⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⢛⣩⠄
⠀⠀⠀⢿⣠⠶⢷⡀⣶⣅⠘⣧⠸⣇⢆⠀⠀⠀⢀⠠⠴⠞⢫⡛⠉⡶⠙⣥⡀
⠀⠀⠀⠀⣭⠶⠋⢹⣷⡬⠇⣈⠧⠙⠻⠃⠀⠀⠀⢠⠶⣾⡯⠿⠿⠧⠶⠿⠟⠻⡀
⠀⠀⠀⠀⠘⠳⠞⠋⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠻⠷⣤⣤⣤⡴⣶⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⢀⣿⣦⣀⣀⣸⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⣯⣭⣽⣧⠻⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠀⠀⠀⠀⠉⠀⠀⡆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠃⠀⠀⢸⠀⣧⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡞⠀⠀⠀⢸⠀⣿⠀⠀⠀⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠁⠀⠀⠀⢸⠀⣿⠀⠀⣀⣸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡿⢯⣽⣏⠉⢹⠀⣿⠈⠉⣩⠼⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡸⠀⠀⠋⢛⠧⣼⠀⣿⡴⠟⠁⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠃⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡟⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠘⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⢣⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠈⣄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣻⣆⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⠿⣷⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢟⣤⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣫⣶⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢋⣤⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢧⠹⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣫⣴⣿⣷⣞⣿⣿⣿⠿⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⠹⣥⣾⣿⣿⣿⣿⣿⡷⣟⡿⢷⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣋⣴⣷⣌⣻⢿⣿⣿⡟⣡⡴⠋⠦⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣡⣾⣿⣿⣿⣿⣿⣾⡿⢛⣿⠟⣫⣴⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣫⣴⣿⣿⣿⣿⣿⣿⠟⠉⠑⣝⢓⠼⣟⡔⢻⡛⠿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢟⣡⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⠋⠻⣿⡄⠀⠙⢦⡀
⠀⠀⠀⠀⣠⣴⣿⠿⣫⣶⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⣷⣶⣦⠀⠙⢦⡀
⠀⠀⣤⣾⡿⢟⣵⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣸⣿⡏⢹⣿⣎⢧⠀⠀⠙⢦⣀
⠀⠈⣿⣫⣴⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡗⡇⢠⡟⢿⡎⢧⠀⠀⠀⠙⢷⣄
⠀⠀⠉⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⡏⣽⡇⣀⡇⠈⢿⡜⣇⠀⠀⠀⠀⢹⣯⡻⣄
⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⠃⣿⡇⣦⡇⠀⠈⢿⡘⣆⠀⠀⠀⠀⠙⠿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⡏⠀⣿⡇⠋⡇⠀⠀⠈⢷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠁⠀⠙⠓⠃⠃⠀⠀⠀⠈⠗⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀

⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢘⢦⣄
⣿⣷⠀⠀⠀⢘⣋⠀⠀⠀⠀⠀⠘⣽⠀⠀⠀⠀⠿⠛
⠛⠀⠀⠀⠀⠀⣭⡄⠀⠀⠀⠀⢠⣥⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠷⠷⠀⠀⠀⢀⣾⣿⣿⣿⣇⡀⢶⠾⣿⣿⣷⣦⡄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠺⢿⠟⠣⢤⣤⢛⣬⣿⣍⣀⣀⣀⠀⠉⠉⠉⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠰⠶⢀⣤⣮⣤⠀⠀⠀⣀⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢲⣄
⠀⢀⣀⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣝⣦⠀⠀⠛⠿⢿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆
⠀⢀⣸⣿⣿⣿⣿⣷⣶⡶⣶⣿⣭⣭⣭⣭⣭⣭⣽⣿⢹⢷⡈⣋⣾⣾⡄⠀⠀⠀⠈⣩⣿⣿⡿⠤⢠⣤⣶⣶⣶⣶⣶⣿⣿⡿⠛⣿⡇
⠀⠀⠹⠯⠛⣿⣿⡟⠿⣿⣾⡿⠿⠿⠿⢿⣿⣿⣿⡿⢸⣏⣁⢋⡙⣿⡇⠀⠀⠀⠀⠸⠿⠿⠐⠀⠀⠿⠿⠿⠟⠛⠛⠛⠛⢁⣀⡙⠂
⠀⠈⠛⠛⠳⠿⠿⠷⠶⠾⠿⢿⣶⣶⣶⣶⣤⣬⣭⣥⣤⣷⣘⠸⣻⣿⣷⣦⣤⣀⣀⣀⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣁⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⡈⠑⠠⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠾⢇⠸⠷⠾⠗⡸⠷⠀⠀⠀⠀⠀⠀⢠⠁⢠⣀⣀⣀⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⡆⠀⠀⠠⠀⢰⠉⠱⠀⠀⠀⠀⠀⢠⠀⠸⠟⠛⠛⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⣴⠆⠐⣦⡈⠒⠊⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠈⠛⠓⠔⠐⠢⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⡐⢛⣛⣛⡛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠎⠉⠉⠾⠅
⠀⠀⠀⠀⠀⠀⠓⠛⠃⠂⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⢠⢤⡖⠁⠒⠉⢲⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⠿⠧⢠⠭⠿⣷⡄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠚⠿⠛⠛⠛⠘⠿⠛⠂⠀⠀⠀⠀⠀⠀⠛⠺⣤⣤⣤⣤⠓⠛⠀⠀⠀⠀⠀⠀⢀⠈⠉⠙⠛⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠿⠇⠨⠇⠆⠿⠇⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⣿⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣀⣑⣂⣒⣀⡿⠋⠁⠀⠀⠀⠀⠀⣠⣼⠟⠰⠆⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠙⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠒⠒⠒⠒⡶⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠀⠀⢀⡤⠤⣤⡀⠀⠹⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣠⠖⠋⠁⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠙⣆⠀⠹⣄⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡾⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠤⠞⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢻⠀⠀⠀⠀⣀⡤⠖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣇⠀⣧⡤⠖⣋⡥⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣁⠴⠚⣹⣚⡥⢶⡋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢀⡴⠁⠀⣀⣠⢼⣦⡀⠳⣄⣀⣠⡤⠤⠶⢲⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠉⣉⡥⠚⠉⠀⠉⠉⠙⡇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠠⣄⣰⣷⡟⠛⠉⠙⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠄⢹⣿⡷⠀⠀⢸⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠂⢸⣿⣇⠀⢠⣬⣿⣎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣯⠸⣿⠃⢀⣴⢾⣿⣷⣿
⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⡿⢻⠆⠀⠀⠻⣿⣷⡻⣿⣿
⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡿⠦
⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⢾⣿⣷⣄⡀
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣤⣀⣀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⠀⠀⠀⠀⠀⢸⣦⡀⠀⠈⠀⢀⡤⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⣿⡦⣤⠶⠻⠃⠀⠀⠀⠀⠀⠀⠀⠄⠘⣿⣿⣯⡶⡄
⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠰⠟⠀⠀⠋⠀⠈⠁⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠻⠿⣭⡛⢶
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠁⠀⠀⠀⠀⠰⣿⢿⡇⠀⠀⠀⠀⠀⢀⣽⠟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠀⠀⢹⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⠀⠀⠀⣠⣶⣿⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⣴⣽⢦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣻⣠⢀⣾⣿⣿⡟
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠌⡶⣚⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣽⡇⢸⣿⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢈⡷⠀⢹⣿⣻⡽⣿⣿⣿⣿⢿⣎⣷⣤⣤⣤⣆⣸⣿⡿⠀⠀⡝⠀⠀⢿⣷⣧⡀⠀⠀⠀⣀⠄
⠀⠀⠋⢠⡄⠀⠀⢀⠞⠱⢫⣶⡴⠋⠉⠉⣷⣌⠙⠏⣫⣀⡾⠵⠷⢋⣡⡾⣿⣿⠏⠠⠆⢡⡆⠀⠈⠛⠁⠀⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣪⣆⡘⢋⣤⣀⡀⢶⠿⢿⣿⣶⣤⣤⣤⣴⣶⣿⠏⠈⣿⠟⠀⠤⠒⠀⡁
⠀⠀⠀⠀⠀⠀⠈⠈⠙⠁⠛⠛⠛⠭⢼⣿⢢⠾⢻⣿⣿⣿⠟⠋⠀⠀⠀⠀⢠⣴⠀⠀⠈⠀⠙
⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢻⡿⣿⡟⢶⣄⣤⣶⣶⣾⢧⡟⠀⠀⠀⢰⡆⠀⠀⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⣶⣶⡾⠛⠛⠛⠁⠈⠋⠀⠒⠢⠤⣤⣽⣬⡛⠀⠈⠀⠀⠀⢹⣿⠁⣸⠀⠀⣘⢀⠀⠉⠉⠁
⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡿⠟⣓⣢⣤⣀⢀⡀⢺⡰⠃⠀⣠⠋⡚⠀⠀⠒⠲⠶⣿⣿⣷⣄⡀
⠀⠀⢠⣾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣶⣿⣈⠀⢹⣿⠿⣫⣴⢿⡻⢻⠇⠀⡼⣯⠲⣇⠲⢦⣄⣀⣀⣤⣿⣿⣿⡿⠁
⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡄⠻⠇⠈⠉⠁⢸⠇⢨⠁⠘⢠⡗⠀⠛⠆⠀⠀⠉⠉⠀⠘⢻⠟⠁
⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠙⠁⠀⠀⠀⠀⠀⠀⡶⠀⠀⣇⠾⣏⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢀⡀
⢀⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢿⣿⣿⣿⣛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣴⣴⣆⣻⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⣿⡇
⢸⡿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡀⠉⠹⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣹⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⣤⣿⢏⡅
⢸⣧⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠛⣿⣆⠀⠈⠻⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣆⡀⠀⢀⡤⠒⠿⣴⣠⡞⠛⠁
⠀⠘⢷⡀⣀⣴⣯⣴⣿⠁⠘⠂⠀⠙⢯⡡⠀⠀⠀⠙⠳⣦⢠⣤⣄⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠏⠻⣿⣷⣤⣏⠀⠀⠀⠀⠛⠐
⠀⠀⣰⡿⠟⣿⠟⠉⠀⠀⠀⠀⢤⣀⠀⠀⠀⣤⡤⠀⠀⠀⠈⠻⢏⠘⣻⣿⣿⣿⡿⠏⠉⠀⠀⠀⠀⠈⣻⣿⣿⡀⢀⠀⠀⣀

⠀⠀⠀⠀⠀⢠⣤⣶⣿⠀⠀⠀⣿⠿⠛⡫⠇⠀⣐⣂⡢⢀⣄⠀⢀⠅⠉⠑⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠸⠅⠂⣁⠤⠒⠁⡴⣴⠰⡷⠌⠈⠀⠀⢀⣤⣤⠀⠀⠐⠂⠡⠉⣙⡂
⠀⠀⠀⠀⠀⡼⠛⠉⠹⠁⠀⠄⡄⠉⣶⡖⠆⠀⠭⠉⠒⠀⠀⠀⠀⣀⣩⣿⣙⣷⡤⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢂⡠⠀⣠⣔⢺⣇⠇⠀⢉⡡⡀⢀⣤⣤⡁⠀⢀⣂⠀⠈⠾⠉⢸⡿⠋⠀⠀⠀⠠⢴⠳⠆⢸⡟
⠀⠀⠀⢰⡎⣥⡎⠿⠘⢃⣾⠿⣇⡻⠍⠀⠀⣿⣿⣿⡃⢤⣿⠋⠀⣀⠀⠀⠈⠀⠀⠀⢀⢔⡲⢦⣤⣤⣞⡟
⣀⠀⠀⢸⣇⣣⣥⣶⣿⡻⣝⢲⠧⢱⡆⠘⡖⠓⠘⣮⣧⣺⣿⢰⣭⡚⢿⣿⣿⡿⠤⠼⠄⣿⣝⣆⣌⡃⡇
⡨⠣⣄⣶⡽⣿⠿⢂⣯⠝⢋⠐⠀⢸⣷⡐⣿⢳⣙⣷⣎⠑⠬⠁⡆⢿⣻⣉⣧⣝⠀⢺⣷⣿⣻⢯⠄⣻⠀⠀⣤
⣤⣵⠌⠁⠐⡻⠁⠛⠳⠞⢙⡀⠀⠀⣸⣇⣶⣜⣛⣻⢩⢆⢰⠀⡆⠪⡿⢭⡽⠜⡛⣿⡿⢿⣶⡎⠐⣳⠀⢉⣱
⣿⡉⠀⠀⣄⣴⣆⠗⠁⢀⣽⣁⢀⡧⠼⣸⣽⠉⠁⠈⠙⢛⣯⡄⠃⡱⣗⣒⣧⣨⡄⣠⣯⣽⣏⠐⠒⡯⠀⠈⢓
⣧⡻⠀⠀⠘⣮⠅⡈⠄⣼⡋⣉⠹⡟⢲⠁⡟⠒⣿⠲⡖⠹⢿⣷⣄⠠⣾⡏⡽⣶⡧⢻⣧⡞⢿⣶⣈⡿⠀⢀
⠏⠀⢠⣄⢼⣍⠉⣸⣟⡿⠃⣿⠁⠈⠻⣇⠇⣸⠃⢠⠝⢦⡈⣿⡇⠆⢿⡏⣉⡹⢣⠸⠁⠙⢦⡹⡄⣰⠀⡼⢷
⠀⣤⡾⡤⠾⣟⠚⡛⣿⣇⢰⡇⠐⠀⢀⠜⣱⠣⠆⠀⠀⠀⢳⡍⣿⡀⢢⣼⣯⡄⢾⡇⠀⠀⠀⠑⣷⢌⠀⠹
⠀⠉⠀⠀⣢⣼⡄⢿⡷⠟⠛⣷⠀⠀⢸⣷⢏⠀⠀⠀⠀⠀⠀⢣⣾⣿⣈⣷⣯⡷⢻⠇⠀⠀⠀⠀⠈⠺⣳⡀
⠀⠀⠀⠉⠉⠉⠙⣼⣷⠀⢸⡏⠀⠀⣸⡏⣞⠄⠀⠀⠈⣖⣠⣤⣿⣿⣿⣤⣞⣡⣿⣾⠟⠀⠀⠀⠀⠀⠑⢧⡀
⠀⠀⠀⠤⠴⠶⣤⣸⡗⠟⠿⠿⠶⢤⣿⣧⣿⣿⠢⣶⣶⣿⣿⣿⣼⣿⣱⣝⣿⣿⣷⣟⡛⠛⠒⠒⠒⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣀⡤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠛⢿⡇⠇⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡐⡍⢄⣀⠜⠀⠈⢄⠀⢀⢀⠀⠀⠀⠀⠙⢗⢆⠠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹⣥⠀⠀⡎⢣⠀⠀⠢⠈⡻⠄⠀⠀⠀⠀⠈⢎⢇⠀⠀⠉⠉⠉⠛⠒⣦
⠀⠀⠀⠀⠀⠀⡰⠇⣰⠀⠀⡇⣠⠞⢇⠘⠦⠤⢒⣡⡂⢳⡀⠀⠀⠀⠀⠈⢎⢆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣰⠃⠀⠀⢧⠀⠹⡅⠀⠈⠒⠤⠒⡁⠀⠑⠘⡉⠀⠀⠀⠀⠀⠈⢎⢆⠀⠀⠀⣴⠃
⠀⠀⠀⠀⢠⡇⡤⠐⠃⠈⠢⣀⠈⠢⡐⠒⠉⢁⡤⠖⡚⠓⠲⠤⢌⠀⠂⠀⠠⢌⡊⠂⠀⠘⠁
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⡋⠉⠲⣌⣆⡀⠈⡇⠀⠉⠀⠤⢖⠂⠈⠙⠲⠀⢀⡿⢀⠔
⠀⠀⠀⠀⢠⡇⡀⢀⠄⡸⠂⠭⣑⡒⠯⢄⣀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⣠⠴⣋⠔⠁⢀⣀⣄⡀
⠀⠀⠀⠀⣼⠁⢀⡞⠀⢧⠀⠀⢰⠀⣉⡐⠢⠭⣑⡒⠤⠼⠧⠤⠴⢖⣋⠥⠊⠀⠀⢠⠋⢀⠀⠹⡄
⠀⠀⠀⢰⡇⠀⡾⣀⠈⠙⠲⢤⣾⡀⠒⠭⣝⡒⠦⣌⠉⠐⠒⣂⠉⠉⢀⡠⠐⠁⢀⡇⣰⠋⠑
⠀⠀⠀⢸⠁⣸⠁⣸⢳⠶⢤⣄⡀⠉⠛⠲⢤⣌⣹⠃⠀⠀⠀⠈⢧⠀⠀⠀⣠⠄⠘⠀⢿⡀⡀
⠀⠀⠀⡿⠀⣏⠀⠿⠬⠥⠀⠈⠉⠛⠲⠦⣤⣀⠉⠙⠲⠦⠤⠤⠾⠶⠒⢋⣡⡶⠃⢀⠈⠉⢁⡴
⠀⠀⠀⠛⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⠴⠶⠶⠛⠋⠁⠀⠙⠺⠷⠖⠋


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

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

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠤⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣤⢄⣀⠀⢀⠠⠦⠄⠀⠀⠀⠀⣀⣀⡩⠔⠶⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠈⠁⠀⠐⠠⠤⡄⠀⠐⠉⠉⠀⠀⠀⠄⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠧⢽⣻⣛⣓⣒⡦⡤⣤⣀⡀⠀⠀⠀⠀⣠⠔⠈⢠⣤⢤⡬⣔⠂⠀⢀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡾⠋⠁⢀⣀⣤⢋⠾⣫⡿⢿⣛⠦⢬⣭⠀⡞⣉⣤⣶⣿⡄⣇⣵⠷⠊⣀⠀⢁⠘⡏⢠⠖⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠦⢼⣷⠾⠥⣍⡀⡡⢉⠾⠗⠻⢋⡵⣻⢗⠌⠐⠃⢺⣿⣿⠿⠄⠉⠀⠀⡀⢁⡴⢶⣧⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⡀⣰⣧⣶⡄⠀⢀⡀⠉⠘⠛⠲⣖⣶⣿⣍⢀⠘⠲⠘⠙⠊⠀⢀⣶⠈⠠⢀⣾⣿⢾⢏⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠾⡇⠐⠀⡠⢢⣾⣟⣧⡧⣖⣿⣿⡉⠄⠀⠀⠀⠀⠙⠓⠛⠪⣴⡆⡴⣞⣵⢀⣿⣯⠀⡀⣽⠟⠀⣘⣥⠄
⠀⠀⠀⠀⠀⠀⠐⠁⠀⣞⡩⠀⠀⠊⠀⠀⠙⠋⠁⠰⠟⠉⢋⡠⠒⠂⠀⠀⢀⠉⠀⠀⠀⠭⠸⣽⢿⢃⡾⣿⠟⢀⡴⠉⣠⣠⠛⠂
⠀⠀⠀⠀⠀⠂⡸⠙⠆⠀⠀⢀⠀⠰⠀⢀⠀⠀⠀⠀⠸⠞⢉⠀⠀⠀⠀⢀⠔⢀⣁⡀⠨⠒⢼⣝⣹⣯⢀⣋⠴⠬⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠂⣀⡀⠀⠈⠁⠂⠀⠀⠀⠀⡌⠒⠂⢈⠁⠀⠀⠀⢀⡞⣱⣭⡻⢂⠀⠻⢌⡀⡱⠊⠀⠀⠁
⠀⠀⠀⠀⢤⣶⢄⡠⠀⠙⠛⠿⣶⣦⣤⡀⠠⠒⠒⡨⠀⠤⠀⠀⢒⡉⢈⡞⣁⣹⠇⡘⢨⡴⠒⠈
⠀⠀⠀⠀⠀⠿⢿⡃⠀⠀⠀⠂⠠⠀⡉⠀⠉⠚⠠⠀⢀⣀⣄⠠⠈⠀⡼⠀⣿⠟⢐⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⢄⣀⠀⠀⠀⣠⣤⣌⣀⠀⠀⡄⠀⣀⢀⣞⣢⠀⡇⣠⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣒⡚⠋⠉⠩⠶⠶⠒⢂⡁⠠⠬⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣿⣟⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⢿⣿⣿⣯⣿⣿⡟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⡆⠀⣦⠀⠀⠈⠙⠻⣿⣿⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⣿⠁⢀⣿⣷⣤⠀⠀⠀⣿⣿⡌⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⡟⣅⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡛⢷⣦⣄⣀⠀⠀⢀⣠⣾⡿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⠿⠋⠉⠉⠉⠉⠻⣿⣿⣽⣿⣿⣼⣿⣿⣿⣿⣿⣻⣷⣶⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⡿⠿⢣⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠟⢛⠟⢉⣝⠻⣿⣿⡏⠻⣿⣿⣽⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⠃⢉⣷⡾⣟⢕⣠⠜⣿⣷⣼⣿⣿⣟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⡿⣿⣿⣽⡄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠨⣿⣷⣯⢝⣫⠶⡋⣸⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣷⣄⣀⣀⣀⣠⣴⣿⣿⣿⣼⣿⣿⣧⡹⢵⣯⢉⣶⠋⣰⣿⣿⣿⣿⠋⣴⣿⣿⣿⣟⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣟⣶⣶⣿⣶⣿⣿⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣶⣿⡿⣿⣿⣿⣿⣿⣿⢿⡟⠋⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟⠽⢱⠃⢈⣾⣿⣿⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⠋⠈⠛⣿⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡰⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀⠀⠀⣿⠇⠀⠀⢀⣼⣿⣿⣾⣿⣿⣽⣿⡟⠘⠁⡴⢀⣾⣿⣟⣩⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⣇⠀⠀⡏⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠛⠋⠁⢀⣼⣿⠿⢻⣿⣿⣥⣶⣾⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⢿⣿⣿⣶⣿⣿⣽⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡄⠀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⢤⠤⠀⠈⡑⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣟⣡⣾⣿⣿⣋⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣶⣦⡀
⠀⠶⠶⠖⠒⠾⣷⣶⣶⣶⣶⣿⣦⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⣸⠏⠛⠁⠀⣾⣿⣿⣿⣿⡿⠩⢃
⠙⠳⠦⣤⣐⠒⠒⠿⢿⣿⣈⣛⣋⣙⡉⠦⡀⠠⠤⠴⠦⠀⠀⠀⣿⣿⣿⣿⠿⠫⠋⠁⠛⠫⣽⣿⣿⡟⠀⠀⠀⢀⣏⣠⡐⢦⡀⣿⣿⣿⡿⢤⣴⠄⣥
⠀⠀⠀⠀⢩⣿⣦⣦⣀⠀⠀⠈⡏⠉⠑⠂⢀⣠⣴⠶⠛⠉⠀⡰⣿⣽⣿⠇⠀⠀⢐⣗⣲⣤⡻⠋⡟⠁⠀⠀⠀⠸⣿⡟⠇⣀⢀⣿⣿⡿⠀⠐⡟⠿⠇⡀
⠀⠀⠀⠀⠘⢻⣏⡛⢿⣷⣦⣄⠃⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⣕⠏⡭⠁⠀⠀⠀⠀⣻⣽⠟⣡⡶⠇⠀⠀⠀⠀⠘⢹⡟⢈⣿⡇⠙⢯⡄⠀⠠⠀⢐⠀⠁
⠀⠀⠀⠀⠀⠀⣿⡍⠁⠙⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⢀⣀⣠⣇⣑⣂⣀⣀⠀⠀⠀⠁⠠⣀⠸⠃⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⡄⠸⡇⠀⠀⠀⢨⠆
⠀⠀⠀⢀⣤⡦⣿⣿⠄⠈⠻⣿⡿⠛⠁⠁⠀⠀⠀⠀⣨⣭⣿⡿⠟⢛⢉⡿⠀⠀⠀⠀⣤⣌⡁⠀⠀⠀⠀⠀⠠⣀⠟⡿⢸⣿⡘⣇⠀⢧⠀⣔⠀⠘
⠀⠀⠀⢨⠉⠃⠛⠛⠋⡈⢯⠉⡠⣔⣥⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣀⠀⠀⠀⢒⠟⠀⠀⠀⠀⠀⠘⠃⠉⠂⠁⠀⢻⠁⠻⣦⣬⣤⣬⣄⡈
⠀⠀⢀⣍⡀⠁⠀⠀⠀⣀⣠⣴⠾⠟⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡠⠙⣻⠦⣤⠿⠲⠶⢦⣤⣀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠀⠀⣠⣴⡾⠁⢠⡈
⠀⠀⠙⠻⠿⠷⠂⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠑⠂⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣤⠶⠶⣶⡶
⠀⠀⣠⣿⢃⢠⣶⣟⣧⠑⢂⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣄⢀
⢀⣾⣿⣟⣿⣿⣿⣿⡷⠂⠈⣁⣴⣃⠐⠟⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠉⠀⣀⣀⣤⣤⣄⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⡐⣶⣦⣤⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⠖⠉⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠶⣿⣷⠄⠀⣼⣿⠿⠃⠀⠀⠀⠀⠐⢦⡄⠀⠀⠀⠀⠀⠊⠁⢀⣈⠉⠉⠈⠁⠏
⣾⣿⢣⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠠⠛⢿⡇⣀⠀⠀⠀⢀⡼⠁⠀⣴⣿⢃⣀⣠⠀⠀⠀⠠⠤⠠⣿⣿⠀⠀⠀⠀⠰⠂⢀⣿⠋⠈⠀⠒⠢⢤⡞⠃
⡟⢈⢀⣿⣿⣛⣿⠋⠀⣀⣑⠢⡄⠀⣠⠼⣦⣿⠀⠀⢀⣾⠃⠌⣠⠟⣡⣿⠛⠁⣰⣦⣀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣀⣾⣦⢋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣼⣿⣿⡋⠹⠀⠀⠉⠛⠃⠀⢀⠻⠗⣿⣿⠀⠀⣾⢏⣰⡞⠏⡆⠸⣧⡆⠀⠙⠻⠟⠀⣴⢿⡿⠀⠀⠀⠀⣼⠃⠽⠋⢬⣿⣦⣤⡀⢠⣤⣖⡃⠈
⢀⡰⢿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠈⠀⡀⣿⣿⠆⠀⠹⠟⠻⣿⣶⣆⡇⠹⠀⠀⠀⠀⢀⣀⢀⣿⠁⠀⠀⠀⠠⠁⢀⣷⣄⠀⠈⠉⠙⠀⠉⠙⠉⡄⢰
⠰⠀⠀⢉⣿⣿⡇⠢⡀⠀⠀⠠⠶⠄⡠⠃⣿⠋⠀⠀⢠⠞⠛⣽⣿⣿⣷⣆⠀⠀⠀⢀⣌⠉⠘⠋⠀⠀⠀⠀⢀⠂⣾⡿⣿⡄⠀⠀⠀⠀⠀⠀⣰⠇⢺⠄
⠀⠀⠀⠘⠛⠍⣷⣶⣿⣦⣀⣀⣀⠜⠁⢸⠇⠀⠀⠀⣿⠆⡠⠜⠻⠋⢿⣿⣷⣄⣀⡘⠃⠁⠀⠀⠀⠀⠀⠀⠸⠀⣍⣴⣟⠙⠆⠀⢺⠷⢂⣼⣿
⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠁⠀⠀⠈⠀⠀⠀⠀⠠⠀⠠⠖⠀⠀⠀⠉⠈⢻⣿⠿⠡⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⣤⣴⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢴⠂⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⡕⡀⡀⡀⢀⠀⢀⡀⣤⣀⣀⣤⣤⣤⣤⣄⠊⠁⠀⢱⠀⠄⡀
⠀⠀⠀⠀⠘⠃⢃⠚⠀⠐⣱⣵⣶⣿⣿⣿⣿⣟⣾⣿⣿⣷⣄⣀⡀⠁⠀⠈⠘⠒⠄
⠀⠀⠀⠀⢠⠀⢹⡄⢁⡼⠸⠞⠛⠿⢉⣉⣁⠀⢀⠀⠀⠀⠀⠀⠈⠉⠛⠫⣉⠉⠓⠒⠤⣄⡀
⠀⠀⠀⠀⢙⠦⠌⠊⢣⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⣀⠀⠀⠀⠀⠀⠀⠂⢀⠉⠠⡢⣄⢀⣭⣷⡀
⠀⠀⠀⠀⠊⢢⡈⠀⠀⢹⡑⠂⢤⣾⡂⠀⠀⠀⢀⡶⣾⡗⠀⠐⠁⠤⣴⣒⠚⡇⣒⣿⡎⣮⣞⣧⠃
⠀⠀⠀⠀⢡⡀⣿⡢⣴⣼⣿⣦⡘⣿⣿⣌⢄⣀⠘⢿⡯⠄⢠⠞⠀⣸⣿⣿⣿⣿⣿⢹⡧⢂⠾⠇
⠀⠀⠀⠀⢁⠊⢹⡇⣿⠧⠸⣿⠦⣿⣡⣿⣧⡀⣙⣀⣢⠴⠉⡈⢤⡻⡛⠛⠻⢛⣽⣯⣿⣥⣀
⠀⠀⠀⢀⡐⠂⢠⣿⣿⢮⢆⠈⣲⣿⢿⣿⡿⠻⣾⣿⡖⢛⣛⣳⣒⠚⠾⠳⣶⣿⣿⣿⣧⢹⢿⣷
⠀⠀⠀⠀⢨⠀⠀⣿⣿⢀⣱⣶⣿⠁⠉⡍⣿⠀⠉⢈⡿⡊⠻⣿⡿⠬⠭⢍⣭⣭⣥⡿⢃⠏⢀⡟
⠀⠀⠀⠀⠘⠀⢄⡸⣿⣾⠿⢿⣿⣤⣼⣿⠃⠀⢀⡼⠿⠿⠟⠋⠀⠀⠀⠸⣿⣿⣟⣡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣠⣾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⠶⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⣌⣴⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣴⣶⣶⣌⠆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣇⣀⣶⣼⣭⣏⣻⡦⣄⠀⠀⠀⠀⠀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣻⠀⠀⠀⠀⡨⠐⠒⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡴⠃⠀⠀⠀⠀⠖⠀⠀⠐⠊⠦
⠀⠀⢠⠒⠒⠤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⠖⠀⢲⡀⠀⠀⠁⠶⠀⠸⡠⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠢⠤⣄⣬⣝⣸⣿⣿⠿⣿⣿⠿⠿⠓⠚⢿⣿⣿⣿⣿⣿⣿⡿⢻⣤⣄⡠⠜⠀⠀⠀⣠⠞⢄⡸⣑⠦⢀⠀⠀⣰⠁⠾⡒
⠀⠀⠀⡰⢊⣤⣤⡀⠈⠉⠙⠦⣌⠙⠥⣀⣀⣀⣀⣐⣿⠿⣿⡿⠟⠉⢉⣀⡈⠑⡄⠀⢠⠊⠀⠈⠈⢈⠈⠀⠠⡟⠉⠑⠉⠉
⠀⠀⢠⠃⢸⡏⠉⠉⠀⠀⠀⠀⠀⠉⠓⠣⠤⠤⠯⠭⠓⠊⠁⠀⠀⠔⠉⠙⣿⡆⣹⠀⡑⢠⠀⠀⠀⠀⢠⡴⠢⡕⠄
⠀⠀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢿⡄⠀⣷⣶⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠟⠀⡇⣰⢀⠎⣃⠀⠀⠀⠈⡻⠤⡐⢄⠀⡄
⠀⠀⡠⠷⠤⠤⠤⠤⠄⠄⠀⠒⠆⠀⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠤⣤⣀⣰⠂⠃⠛⡎⠀⠈⠲⣰⠏⠀⠀⡁⠀⠉
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠒⠒⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠉⢹⠒⠒⠒⠒⠒⠛⠛⡻⢹⠀⠀⠀⠀⠀⠀⢸⡦⠤⠤⠤⠤⠤⠤⢤⡖⠓⠒⠒⠛⣆⠀⠀⣶⠀⠀⢘⠒⠒⠒⣒⡂
⠀⠀⠀⠀⠘⡄⠀⠀⠠⠀⠀⢠⠇⠈⠉⠉⠉⠉⠉⠁⠉⡇⠀⠀⠀⠀⠀⠀⢸⠉⡿⣽⠄⢀⠙⠒⠁⠏⠀⠐⠏⠉⡿⠝⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠤⠊⠀⠉⠁⠀⠛⠒⠒⠚⠳⠶⠤⠟⠀⠑⠊⠁

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠒⠒⠀⢉⢍⡵⠋⠉⠉⠉⠑⠰⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⡄⠄⠐⠀⠀⢸⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠄⢂⠩⢇⠀⠁⠀⠀⢀⠠⠁⢁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠁⠀⠀⠀⡀⡁⠀⢀⠰⠐⠐⡀⡄⢆⢡⠸⡈⢆⠡⡰⡀⠣⣮⠧⡁⠠⢃⠂⠀⣤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠐⡛⣀⣀⣜⣔⡀⠠⠰⠀⢣⢡⣆⢂⢂⢢⠣⡡⡑⡘⢌⠢⡑⢌⢪⡣⣍⠪⣒⣌⡛⣾⡥⠶⠶⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠈⠙⠛⠛⠛⠉⣿⣏⠙⠳⠷⠿⠷⠾⠶⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠊⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢻⣾⣷⡿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⡾⢿⣿⡗⢔⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⠁⠂⢸⡇⠘⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢠⠇⠀⢀⢸⡇⠀⢁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⠀⠀⢸⢸⡇⠀⠈⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡼⠀⠀⠀⢠⢸⣧⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⠃⠀⠀⠀⢸⢸⣿⠀⠀⢲⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣤⡞⠀⠀⠀⠀⢸⢸⡯⡄⠀⢈⡄⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢋⡟⠃⠀⠀⠀⠀⣸⣸⣿⡇⠀⠀⠑⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⣤⣧⣼⣿⡇⠀⠀⠀⢣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠃⠀⠀⠀⣠⠞⠁⢸⢻⣿⣷⡄⠀⠀⠈⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡞⠀⠀⣠⡞⠁⠀⠀⠈⠚⣿⢼⠻⣄⠀⠀⠅⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸⠁⢀⡾⠋⠀⠀⠀⠀⠀⠀⢹⣿⠀⠙⣥⡄⢲⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⢻⣼⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢐⣭⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠙⡟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠈⠉⢨⣿⣇⠭⢂⡀⠀⢁⠐⠈⣁⠈⠀⡀⠈⠁⠹⠷⠈⢌⠰⠒⡠⣯⣷⡀⡀⠂⡀⠄⠀⢀⠂⢐⠀⠀⠀⠄⢀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠐⠀⠐⠈⠀⠂⠁⠀⠟⠋⠅⠂⠑⠊⠁⠈⠁⠁⠐⠈⠐⠈⠑⠈⠐⠁⠈⠁⠁

⠀⠀⣶⡄⡆⣀⠀⣠⡆⠀⢀⡀⢀⡀⣀⡄⢠⣤⣠⢴⡀⠀⣄⢀⠀⢀⡔⢰⣀⢀⡀⢀⢀⡀⢀⣠⠀⡀⢀⠀⢰⢦⠀⡀⢀⡀⠀⣀⢀⣀
⠀⠀⣿⠷⠿⡿⠾⡴⠿⠀⠸⡽⠿⠷⢿⠇⠀⠟⠟⠸⠿⠯⠧⢿⠇⠈⢧⠸⠿⢿⠿⠇⢺⡿⠧⠸⠸⡿⠿⠀⢸⡼⠺⡿⢸⠶⠿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠆⠈⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⡞⣷⠀⢮⡉⣹⢿⡆⠀⠀⠀⠀⠀⠀⣮⣄⠘⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠰⠁⠀⠀⠀⠠⠔⡁⢠⠀⢠⡄⣀⣾⡅⠀⠀⠀⠀⠀⠀⠿⢿⣦⡄⠻⠛⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡢⠴⠛⠉⠚⠂⠂⠀⠀⠀⠀⢀⣄⣀⣼⣦⣐⠀⠀⢀⣈⣿⠈⡀⣤⠛⢻⠗⠃⢠⡄⠀⠀⠀⠀⠀⣠⣿⣦⠤⣽⡆⠁⠛
⠀⠀⠀⠀⠀⢀⠔⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠺⠋⣩⠖⠋⠀⠀⢀⠈⢿⣻⣎⢋⣀⣀⠀⣆⠀⢨⣿⠀⠀⠠⠄⣨⠵⠋⠁⠀⢠⡁⠀⢿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⣤⠳⣎⡁⠀⠀⢰⠖⠉⠀⠈⣿⢻⡟⠻⣿⠶⢏⡴⠂⠀⠀⢐⣷⣞⣡⣄⣀⣀⡾⠈⠳⣀⠐
⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⣀⣤⡀⢈⣷⡅⠀⠀⢀⠀⡀⠙⠳⣶⣈⡇⠈⠀⠀⢹⣆⠺⠆⣄⠤⢾⠀⠀⢠⠠⣏⡿⠟⢡⡈⠉⠉⠀⢀⣰⣿⡷
⠀⠀⠀⠀⠀⢠⠀⡄⠀⢸⣿⣮⠽⢧⣹⡄⠀⠀⠀⢷⠷⢦⣤⣼⣿⣇⠀⠀⠀⠀⣹⠆⠀⠠⢶⡂⢤⠀⠈⠀⠁⠀⠀⠁⠀⢀⠀⣐⣿⡟⠀⠑
⠀⠀⠀⢠⢴⡇⣄⠃⣀⠀⠙⠛⠂⠈⠉⠀⠀⠀⢂⠛⠀⠀⡸⣧⢿⠁⠄⠀⠀⠀⡩⠤⠀⠀⠀⠃⠈⣆⡀⠀⠀⠀⠀⠘⠈⠀⠀⠈⢿⡀
⠀⠀⠀⣚⡿⠟⣿⣤⣥⠤⣶⣾⣿⡀⠀⠀⠰⣦⡛⢸⢄⡠⠀⣿⠸⣇⡀⠀⠀⠀⣷⡆⡀⢸⣿⡆⠀⠘⣗⠀⠀⠀⠄⠀⢠⠐⠯⠀⡈⠿⡄
⠀⠀⠀⠉⠀⠐⠏⠙⠋⠈⠙⠟⠋⠃⠀⠀⠀⠻⠟⠊⠘⠇⠀⠀⠉⠉⠳⠀⠀⠘⠟⠃⠻⠷⠾⠷⠆⠀⠻⠀⠀⠐⠷⠿⠋⠀⠀⠌⠐⠗⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠀⣀⡀⠀⠠⠤⠌⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣴⠂⡌⢉⣁⣠⡶⠿⠛⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣴⠷⠋⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣠⠟⠁⣠⠶⠚⠉⠉⠉⠛⠷⢶⣄⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣼⠃⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣦⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢰⣁⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻⠉⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢠⣤⠀⠀⠀⠸⣅⣀⠐⠀⠀⠊⠁⠒⢲⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠿⡄⠀⠀⠀⠻⡟⠛⠁⠀⠲⠶⣾⠟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣽⣦⣀⠀⠀⠈⠋⠰⠦⠀⠈⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⣌⣻⡿⢳⡀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⠉⢴⣯⣧⠀⠀⠀⠀⠚⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡌⠉⣿⠀⠀⠀⠀⡿⢷⣄⠀⠈⠑⢦⡴⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⣹⠆⠀⢠⡟⠀⢻⡷⢦⣀⡸⠁⢀⠀⡟⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠎⠙⠳⣄⠀⠀⡾⠛⢷⣴⠟⢀⡶⣤⡿⢀⡘⢇⣀⡄⢀⡘⢀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠳⡾⠀⠀⠀⠙⣆⠈⠀⠀⠀⠈⢿⡠⣄⠈⠻⢶⡟⠁⠀⠛⢶⣄⣠⠴⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⢳⡆⠈⠱⣿⠀⠀⠀⣿⠳⣤⣀⠀⠉⠉⠀⠀⡀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢤⠄⢤⠄⠀⠀⢙⠀⠀⠀⢻⡇⠈⠙⣷⠀⠀⢀⡰⠛⢦⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⢻⡀⠀⠈⠳⣦⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⢆⡀⠻⠀⠀⠀⠘⠃⠀⠀⠈⢳⣄⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠳⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⠻⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠙⠳⡴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⢸⡷⠰⠆⠶⠂⠲⠦⠀⠾⠦⠄⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠃⠀⠘⠇⠀⠀⠀⠀⠰⠦⠄⠀⠠⠶⠀⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣼⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⡏⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣠⣤⡴⠛⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⢶⠆⠀⠀⠀⢸
⠀⢀⣄⣤⣄⣸⡧⠛⠀⠀⠀⠀⣀⣀⣹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠘⠉⢦⡀⠈⡅⠻⣦⡀⠀⢟⠁⠙⢋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠙⢦⡀⠀⠈⠛⠶⠞⠀⢠⣷⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⣧⢴⡆⠀⠀⠙⠤⠀⠀⠀⠂⠀⠞⠀⠉⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡆
⠀⡇⢘⡧⠀⢀⡀⠀⢁⠖⠒⠒⠒⠐⠒⠐⠂⠈⠻⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇
⠘⢧⠈⠀⠀⠉⢹⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠳⣄⠀⠀⠞⠉⠿⠆⠰⠶⠶⠶⠶⠶⠶⠶⢾⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠳⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠳⣄⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⣤⣀⣀⣀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠉⠀⢻⠙⠛⠒⠖⠒⠛⠛⠁⠙⠛⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣿⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⢀⣀⣀⣰⣶⡤⠴⠛⠉⠘⠒⠭⣤⣖⣀⣀⡠⠤⠄⠠⠀⠤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠚⠛⠖⠒⠚⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⡛⢛⡛⠛⣛⣛⣛⣛⣄⣛⠛⣿⡛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣤⣭⣭⣭⣭⣿⣿⣿⣏⣉⣉⣿⣏⣉⣉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠰⠀⠀⠀⢀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⣩⣴⣦⠠⠖⠞⠈⠀⢠⣤⣾⠋⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣴⣿⣿⣿⣿⣄⠤⢀⣄⣶⠿⠦⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣼⣿⡿⢟⣿⣿⣿⣿⣦⠊⡏⠀⠀⠀⠀⣰⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣰⣿⡿⡏⠟⠈⠉⠉⠙⠛⢽⠷⣄⠀⢀⠀⢐⡾⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢰⣿⡿⣱⠋⠀⠀⠀⠀⠀⠀⠈⢻⠙⢿⣦⣄⢛⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⢰⡆⡄⢰⠏⣠⡼⣿⣂⣹⡀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⣼⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⠈⠀⠀⠀⠀⠀⠎⣸⣿⣯⣤⣿⣿⢇⡀⠀⠀⠀⠀⠀⣸⠆⢀⣾⣛⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠖⠂⣀⣼⠿⠳⠱⠿⣿⣿⣷⣿⠂⠶⠆⠾⠛⢀⡤⣮⣼⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⢀⣀⣀⡀⠀⡠⠖⠋⠁⡀⠄⠈⠋⠃⣸⣿⠉⠃⠀⠀⠀⠈⠸⢗⡷⡿⠛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠋⠉⢁⠴⠊⣀⠴⠃⡁⢀⣤⠀⣠⠞⠛⠬⡁⠀⢄⣠⣤⣾⣿⣳⡴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⠈⠀⠒⠋⠀⠈⢀⠜⠁⣀⣀⣠⣬⣿⣾⠿⢿⠝⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⢀⣠⣴⣾⡻⡇⠠⠴⣋⣾⣾⣿⣿⠿⠛⠉⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⣣⣾⣟⡻⣿⣿⣿⣄⠙⢦⣙⢿⡟⠉⠀⠀⠀⠀⠀⣻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠞⢫⣵⣾⣿⢯⠻⢿⣿⣿⣦⣽⠁⠈⢸⡀⠀⠀⠀⢀⣲⡃
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⠐⣠⣾⣿⣿⣿⣄⡀⢀⣙⣿⢿⠇⠀⢐⠸⣦⠀⠀⠀⣴⡟⠠⠁
⠀⠀⠀⠀⠀⠠⡿⠃⡠⠔⣺⣿⣿⣿⣿⡿⠋⢉⡿⠟⢫⠎⠀⠀⠈⣸⠃⠀⠀⡲⠏⠐⠁
⠀⠀⠀⠀⠀⡰⠁⢀⠠⠊⠉⠛⠿⠋⢉⣠⣾⠿⠀⠀⠀⠐⠀⠀⠐⢁⠠⠞⠉
⠀⠀⠀⠀⢼⠇⢐⡅⠀⠀⠀⠀⠀⣴⠏⠙⠁
⠀⠀⠀⡀⠀⠀⠞⢀⡤⠤⠒⠋⠐⣥⠖⠁
⠀⠀⣀⡤⠀⠴⠋⠉⣀⠄⠴⠂
⠀⠀⠹⢃⣠⡴⠾⠋
⠀⠀⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣴⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡛⠃⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠇⣠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣇⠀⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠛⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣴⣿⣾⣿⣿⣿⣿⣿⣤⡶⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣁
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠉⠹⢄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣹⡷⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡉⡙⠛⠶⡧⣀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣿⣿⠿⠟⠿⠋⠉⠉⡉⠒⠦⣜⡃⠌⣹⠖⠈⠉⠒⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡾⠛⢶⣶⣶⣥⣀⠠⢙⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣳⡻⠶⢤⣌⡙⠿⢷⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣮⣏⡳⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄

⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⣿⣇⣀⡀
⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⡿⢩⣙⣛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠇⠀⠀⠉⠉⠛⠒⠶⠭⡝⣿⣇⣀⡀
⠀⠀⠀⠀⣿⠇⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠛⠛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⢸⣿⣀⡀⢀⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣂
⠀⠀⠀⠈⠉⠛⠛⠿⠟⠈⠉⠙⠛⠿⠷⣶⣦⣤⣼⣿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⣸⡟⢻⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣆⠀⠉⠉⠛⠻⠿⢶⣶⣤⣤⣿⣷⣾⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⡏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣿⡇⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣿⡇⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠿⠿⣶⣿⣷⡶⠿⠟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠉⠛⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠐⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⢀⣴⠶⠶⠶⢤⣄⠉⠓⢦⣄⠀⠀⠀⡤⠐⠊⠁⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣀⡟⠁⠀⠀⠀⠀⠈⠙⠲⢄⡈⠙⠲⣤⣆⠀⣀⡠⣄⡤⣮⠛⠁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⡤⠶⣿⡄⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠉⠻⢮⣥⣬⡶⠾⠛⠛⠉⠉⠈⢿⡀
⠀⠀⠀⠀⡼⣛⣦⠀⠀⠻⣄⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⡴⢿⡉⠉⠙⣧⠀⠀⠈⠳⣄⠙⢿⣦⣄⣀⠀⠀⠀⠀⣀⣠⠶⠋⠁⠀⠀⠀⠀⠀⢀⡠⠴⠦⣤⣴⠿⠋⠙⠳⣶⣤⣀
⠀⢠⡇⡤⠹⡄⠀⠹⣷⡀⠀⠀⠀⠙⠲⢌⣙⠻⠿⠿⠿⠟⠋⠁⠀⢀⡤⠖⠒⠒⠶⠶⠾⠷⢤⡀⠈⠳⣄⠀⠀⠀⠈⠢⡙⠳⣦⡀
⠀⠀⣧⠀⠀⢻⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠈⠉⠒⠠⠀⠀⠀⠀⠐⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠈⢣⠀⠀⠀⠀⠱⣜⠝⢿⣦
⠀⠀⢹⣦⣀⠸⡇⠀⠀⣿⠉⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠉⠉⠉⠛⠳⢦⡀⠀⠀⠀⠀⠀⠈⠳⡀⠡⠀⠀⠀⠀⢿⢀⣸⣿⣧
⠀⠀⠀⢳⡍⢰⡇⢀⣠⠏⠀⠀⠀⠈⠉⠻⢦⡀⠀⢀⣀⣥⣤⣄⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⠿⡿⣾⢿
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠟⣛⣿⠿⠿⢿⣷⣄⠀⠀⠀⠀⣀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠓⠲⠦⣄⡀⠀⠙⢷⣄⠀⢾⠟⠉⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣴⡾⠟⠲⣄⠙⢦⡀⠀⠙⣷⡌⢆⠀⣴⣷⣿⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⡎⡜⠀⠀⠀⠀⢳⡄⢳⡀⠀⠸⡟⢦⣍⠙⠛⣁⣀⣤⡤⠴⠖⠚⠋⠁⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢳⢱⡀⠀⠀⠀⠀⣧⠀⣧⠀⠀⡇⠀⠹⣷⣿⣧⣀⠀⠀⠀⣀⣀⣤⢶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠳⣽⣦⣤⣤⣤⠇⠀⡿⢀⡼⠁⠀⠀⠀⠙⠻⢭⣽⣿⣭⡭⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣛⠛⠛⠉⢀⣼⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣀⠘⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡎⠀⢱⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠃⠀⠀⠈⠉⠁⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠇⠀⠀⠀⠀⢀⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⠔⠄⠀⢀⠔⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⣒⢶⠌⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠌⠁⠀⠀⠁⠀⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡢⠀⠐⠐⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠢⠐⠀⠉⠀⠂⢄⠀⠠⠀⠀⠀⢀⠄⠀⠈⠁⠃⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠟⠁⠀⠈⠚⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡠⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠥⠒⠥⠀⠀⠀⠁⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⢀⠁⠄⡀⠔⢹
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠐⠁⠀⠀⠒⠦⠛⡊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡈⡈⡀⠊⢀⠄⠀⢰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠊


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

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

⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣧⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⢛⣿⣿⣿⣿⡄⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣾⣿⣄⡀⠀⠀⠆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀⠀⠀⡜
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢹⡶⠁
⠀⠀⠀⠀⠀⠙⠛⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣇⠀⡀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠹⡇⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠦⣠⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠚⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣤⣾⣿⣿⣶⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣯⠉⢻⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⢋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣿⣿⣿⣧⣾⣿⣿⣿⣿⡿⣣⣿⠿⣛⡽⠟⢋⣡⠞⠁
⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⠞⠉⠀⢠⡏
⠀⣀⣠⣴⣿⣋⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⠟⠉⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠐⣦⣄
⠀⠀⠀⠠⠄⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡀⠠⠀⠀⠂⠀⠀⣿⣿⠦⠤⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⣀⣤⣀⣀⣠⣤⠤⠚⠁⠀⣾⡾⣿⠿⠿⠿⢆⣄
⢀⣴⣾⣿⣿⣿⣷⣶⣖⣛⣿⣟⣋⠟⠉⠀⠀⠀⢠⡀⢠⠀⠉⠀⡀⡇⢹
⢸⣿⣿⣿⣿⣿⡿⠛⢰⣎⣩⡟⠙⠦⠀⠀⠒⠒⠀⠀⣗⢠⣴⣶⠶⠀⠚⢲⣴⠒⢒⡤⡀
⢸⠿⠻⣿⣿⣿⣇⠱⠀⢘⣯⣁⣐⠀⠀⠀⠀⠀⠤⡴⠾⡒⠤⢤⡼⢦⣄⣤⠐⢫⢟⣒⠻
⠈⣇⢠⣿⣿⣿⣿⣿⢷⣌⠳⣎⠁⠠⡀⠤⡤⠶⠶⠧⡴⡆⠅⠚⠋⠁⠀⠀⠀⠈⠛⢿⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⡄
⠀⠘⢿⣿⣿⣿⣿⣿⣧⣽⣿⡚⠀⠀⠈⠀⢰⣶⣾⡂⠀⠀⠀⠀⠀⠵⠐⠤⢀⠀⠀⠀⠑⢇⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢽⣧⠀⠀⠀⠀⠘⠿⠿⡧⢶⣶⣶⣶⣄⡀⠀⡰⠋⠈⠀⠀⠀⠀⠙⢷⡒⢪⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⣀⡀⠔⢺⣾⢷⣿⣿⣿⣿⣞⣛⣳⣶⣶⠀⢰⡆⠀⠶⡷⠑⣄⣿⣿⣿⡟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⢺⣿⣿⣿⣿⣿⣿⣿⠾⣿⣿⣿⣿⡾⠧⣈⠀⠀⠀⠈⢌⠟⠋⠉⠀⠈⠂⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢿⣿⣿⣿⣿⠟⠙⠍⠉⠉⠡⢄⡀⠤⢷⢱⠄⢠⡎⠀⠀⠀⠀⣀⡉⢵⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣞⣽⣿⣿⣿⡀⠀⣤⢰⠀⠀⠀⠀⣀⣚⣨⣏⠩⠥⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠈⠑⠤⠼⠂⠈⠉⠁
⠠⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⢢⡞⣰⣿⣿⣿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⣠⣶⣄⠀⠀⠀⠈⠀⠀⢠⣴⣾⣿⣿⣾⡇⣿⣿⣿⣿⡀⢈⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⢄⣴⣺⣽⣿⣿⣿⣿⢹⣷⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⣴⣳⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣧⡙⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣶⣬⣍⠉
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠓⠒
⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⣿⠇⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣦⣝⠻⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⡀⢀⣠⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⡇⠻⣿⠿⠟⠻⠿⠛⠁⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸
⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⡏
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢠⣿⡿⠁⠉⠛⠋⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠘⣿⣷⡙⣿⣄⠀⠀⠀⠀⠙⠏⠁⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣼⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣧⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠙⢿⡀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣀⣀⣀⣈⣻⣧⣀⣀⣠⣿⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠠⠀⠰⠆⠰⠆⠰⠦⠀⠀⠈⠀⠀⠀⠀⠀⠈⡀⡀⠁⠈⠀⠀⠅⠀⠁⠀⠀⠰⠆⠀⠆⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢭⣀⡀⢀⢬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣾⣟⠀⠈⠁⠉⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣻⣧⡢⠀⠀⠀⠀⠀⠀⣿⣿⡇⠁⢠⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣟⡛⢿⡦⡀⢀⣀⠤⠊⢻⣿⣷⣤⠀⡾⢿⣦⣌⢀⢄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣩⡿⢷⣌⡛⢷⡼⠒⠛⠮⣻⣿⣿⡼⠁⢸⣿⣧⣄⣄⠞⠣⡘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣘⡤⢖⣉⣀⣤⣤⡴⠛⠁⠀⠀⠀⠀⠻⣿⢟⣵⣷⣾⣿⣿⣿⣿⣷⣄⠀⢐⣽⡙⠲⢤⣀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⢸⠀⣀⣀⣤⢶⣉⠉⠁⠛⣼⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣿⣿⣇⣀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣄⢚⣿⠏⠀⠀⠀⠹⡷⣄⣾⣿⡿⢿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡇
⠀⠀⠀⠀⠀⠈⠳⣤⣐⠒⢶⣿⡿⠿⣿⠀⠀⠀⠀⢰⠙⢮⡫⡉⠉⠢⣹⠿⢷⠲⢧⡀⠉⠉⠙⠛⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢸⡄⠀⠀⠂⠁⠀⢸⢿⣮⠢⠄⠀⠀⠀⢳⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⠀⠀⠀⠈⢈⠏⢷⠀⡄⡐⢲⣌⣇⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⢰⠇⣠⠀⠈⠀⣸⡇⡇⢣⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⣿⢰⡇⠀⢰⣆⣿⣷⡀⠀⡙⠛⢛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠻⠿⠴⠶⠿⠿⣿⣿⣿⣿⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣰⣶⣶⡆⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣴⣤⣤⡄⠀⢀⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠠⣀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠆⠀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠻⣿⣿⣿⣿⡀⣀⣿⡷⢃⣠⡴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣌⣢⣷⣾⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣆⠚⣀⣼⣿⣿⣿⣿⡿⠯⠒⠛⣵⣾⡆⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢀⣼⣿⠋⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⠀⢀⣿⡿⠛⠁⠀⠀⠉⠘⢿⣿⣿⣿⣤⣤⣶⣿⡆⠀⢀⣾⡟⢁⣼⣇⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⡿⣿⣷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢟⣿⣿⣿⠟⢀⣴⡿⢋⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣏⣹⣿⣿⣷⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣿⣿⠟⢁⣴⡿⢉⣴⣿⣿⣿⣿⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣡⣶⠟⣡⣾⣿⣿⣷⣿⣿⣯⠙⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠸⣿⣿⣿⣿⣿⣷⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣧⣦⣼⡟⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠛⢻⣿⣿⣿⡿⠋⢀⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡄⠀⢀⣿⡿⠋⠁⠀⢾⣿⠿⠟⠃⢀⣀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⣫⡞⠀⠀⠀⠀⣀⣠⣴⡾⢛⣩⣶⣿⣿⣿⢛⣿⣿⣿⡿⠛⠛⢿⣿⣿⠟⠛⠉⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣰⢽⣿⣶⣶⣶⣿⠟⢋⣥⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠒⠒⠒⣿⡏⠀⠀⠀⠀⠀⣈⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢥⣸⠿⠛⣉⣴⣾⣿⣿⣿⣿⣿⣏⡼⠛⣿⣯⣿⣿⣿⣷⣦⣤⣾⣿⡆⠀⠀⠀⢀⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠋⣿⣿⣿⢿⣿⣿⣿⣾⣿⣑⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠮⠀⠀⠀⠄⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠁⠈⠙⣾⣿⣿⣿⢿⡿⠿⠟⠛⠛⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠃⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠸⠟⠇⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⢦⡀⠀⢀⡴⠋⡁
⠀⠀⠀⣀⡤⠴⠶⣦⠤⣄⡀⠀⣠⠊⠾⣆⡀⠀⠀⠀⠀⠠⣼⣷⡀⢹⣿⡏⠀⣼⡧⠄⠀⠀⠀⠀⠀⣠⠴⠒⠲⡖⠲⠦⣄⡀
⢀⣴⣾⣿⣶⣦⣦⣿⣦⣦⣽⣾⣧⣴⣚⠛⢿⣆⠀⠀⠀⠀⠀⣈⡻⣯⡀⣹⢞⣁⠀⠀⠀⠀⢀⣤⣾⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷
⠸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣿⡇⡿⣿⠀⠀⠀⠀⠺⠟⣀⣬⠻⣦⡀⠻⠗⠀⠀⠀⠸⣿⣿⣿⡻⣿⡿⠿⠿⣿⣿⣿⠱⠋⠋
⠀⠀⠘⠿⠞⠁⠀⠀⠀⠀⠀⠘⠾⠞⠁⠐⠇⠹⠀⠀⠀⠀⠀⠀⠧⠏⠀⠈⠫⠀⠀⠀⠀⠀⠀⠈⠉⠳⠿⠃⠀⠀⠈⠿⠛⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣭⣭⣷⠾⡿⡿⢾⣭⣭⣝⡇⠀⠀⢰⣧⣰⠀⢠⣀⡄
⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡿⠆⠀⠀⠀⢷⢿⡏⣁⠛⠛⠛⢋⢩⡿⣼⠀⠀⠀⠀⢻⡞⠀⠈⣿⠁⠘⣶⠂⢠⣀⡀
⠀⠀⣾⠇⠀⣠⣴⣤⣤⣤⣤⣤⣤⣦⣶⣶⣶⠄⠀⠀⠀⠘⣏⣷⣉⣤⣤⣤⣉⣼⣳⠃⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠈⡏⠀⢦
⠀⣴⣿⠀⢰⡟⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⡶⠄⠀⠀⠀⠀⠘⢮⢿⣿⣿⣿⡿⣳⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⡇⠀⢸
⠀⠻⠿⠀⣿⡇⢀⣿⠀⣤⣤⣤⣤⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠈⠳⣝⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⣼⣇⠀⢠⢿⡄⢀⠿⡀⢀⢧⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠉⠀⠈⠋⠀⠙
⠀⠀⠀⠀⢀⣀⡀
⠀⢀⡴⠟⠋⠉⠙⠳⣦⡀⠀⠀⠀⢀⣸⣿⡇⠒⠒⢺⣿⣇⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⢀⣿⣁⣀⣤⣤⣤⣀⣸⣷⠀⠀⢰⣿⣶⠶⣶⣶⣶⣶⠶⣶⣿⡆⠀⠀⣼⣿⣿⣋⣿⣙⣿⣿⣇⠀⠀⠀⠀⢀⣤⣭⣿⣯⣤⡀
⠘⣿⠉⢻⣏⣋⣻⡋⢉⣿⠀⠀⢸⣿⠠⣷⠄⣿⣿⠠⠤⠄⣿⡇⠀⠀⣿⣿⣏⣹⠿⣏⣽⣿⡿⠀⠀⣧⣾⣿⣿⠏⠶⠹⣿⣿⣿⣆⣾
⠀⠙⣷⣿⠉⠉⢙⣿⡿⠃⠀⠀⢸⣿⣦⣭⣴⣿⣿⣦⣤⣴⣿⡇⠀⠀⠘⢿⣿⣿⣶⣯⣾⡿⠁⠀⠀⠏⠻⢿⣯⣤⣤⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃⠛
⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣼⣿⣭⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⡆⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⡀⠀⢠⣄⠀⠀⠀⠀⢠⡶⠶⠶⠶⢶⡄⣴⣶⣶
⠀⠀⠀⡘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⣷⠀⠀⠀⠀⠀⠀⢼⠁⢰⣆⠈⣧⠀⠈⣿⠀⠀⠀⠀⢸⡗⣶⠶⣶⢸⡇⣿⣿⣿
⠀⠀⠀⠱⡄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣷⡄⠀⠀⠀⠀⠀⠘⢧⣄⣠⡼⠃⠀⠀⢿⠀⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⢹
⠀⢀⣠⣴⣮⡛⣫⣶⣤⣀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠸⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠚⠛⠓⢾⡄⠀⠀⠀⠀⣿⣟⠿⣻⡏⠀⠀⢸⡇
⣼⣿⣿⣿⣿⡗⣿⠿⠿⣿⣿⡄⠀⠀⠀⢼⡟⠀⠀⠀⢻⡧⠀⠀⠀⠀⠀⠈⢹⡯⡅⠀⠀⠀⢸⠇⠀⠀⠀⠀⢿⣯⣾⣽⡇⠀⠀⠀⡧
⠿⠿⠿⠿⠿⠗⠿⠷⠶⠿⠿⠇⠀⠀⠀⠲⣤⡤⠶⠚⠛⠲⠦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⠶⠋⠀⠀⠀⠀⠀⠈⠉⢿⡉⠀⠀⠀⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠰⣦⣾⣿⣤⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠴⠛⠋⠛⠇⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠒⠒⠚⠛⠛⠛⠛⠛⠒⠲⠶⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢋⡥⣤⡉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⡿⠀⠈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⣇⢷⣤⡴⢃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⠶⠤⠶⠛⠁⠀⠀⠀⡴⠖⠒⠒⠻⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣧⣂⣀⣂⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⢰⠫⣟⣛⡳⠶⠶⣶⣤⣤⣤⣭⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠈⠣⣕⡻⢿⣿⣗⣚⣛⣳⢒⡿⠀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠈⠉⠉⡛⠒⠛⢻⡋⠀⠰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡷⠀⠀⠀⢀⣴⣟⠋⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⣿⣿⣿⣷⠆⠀⢸⡶⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⡿⣿⡿⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢷⣄⡀⠘⠋⠀⠀⢀⣴⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡤⠖⠋⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⢦⠤⠶⠒⠐⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣀⠩⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣤⡿⢀⣆
⠀⠀⠀⠀⠀⠀⢰⣿⡀⠸⠆⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠉⠉⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠃⢒⠃⢄⣠⠃⠈⠀⠀⠈⠁⢀⠀⠀⢐⠻⡳⡄
⠀⠀⠠⡄⠀⠀⠀⠉⠻⠿⣮⠼⠘⠁⠀⢀⡄⢀⠀⠀⠀⠀⠀⠈⡇⣃⢱⠀⠰⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡐⠊⠉⢨⢰⣍⣷⢄⢈⣇⠀⠳⡹⣿⣸⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣥⡴⠷⣶⣾⠿⠷⢧⣸⠟⢻⡿⠛⠿⠿⠒⠁⢀⣾⠐⠂⠀⠈⠘⡋⠀⢀⡿⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠉⠉⠀⠀⠀⣀⣄⠀⠀⠙⠓⡗⠁⠀⠀⠀⣀⣴⣾⣯⡀⢀⣠⡴⢞⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠻⠀⠀⣦⣼⠍⠀⠀⠀⢾⢿⣧⠘⠛⣻⡭⠒⠋⠝
⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠙⠳⢶⣄⠀⠀⠀⠀⢉⣼⣄⢦⡀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡟⡈⠁⠀⠀⢀⡴⢚⢧⠀⠀⢖⡎⢙⣓⠂⠴⠂⠀⢠⡾⠁⠀⠁
⠀⠀⠀⠀⠀⣾⠧⠄⢀⣠⣴⡾⠁⠀⢺⠆⠀⢠⡟⢸⣿⠀⢻⠀⠀⠸⣷⣀⠁
⠀⠀⠀⠀⠀⢹⣧⠾⠝⠟⠉⠀⠀⣠⡟⠀⠀⢾⠃⢸⣿⠀⠘⡇⠀⠀⠌⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⢀⣀⡾⠛⠀⠀⢠⠟⠀⠸⡿⢄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣠⡤⠊⠀⠃⠀⠑⢬⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⡀⠀⠀⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣿⡓⠒⠛⢋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢋⠩⠛⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠈⣵⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⢀⣰⣿⣿⣿⣿⣿⡿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⠀⢰⣿⠟⠙⠿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠈⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠊⢿⣦⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡄⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠈⠟⠉⠙⣦⡀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢳⣄⠀⠀⠀⢰⣾⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠇⠀⠀⠀⢀⠀⠈⠀⠀⠀⠹⣷⣄⣀⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡾⠃⠀⠀⢀⠔⠻⣦⠀⠀⠀⠀⠀⠈⣛⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠓⢶⣎⠀⠀⠀⠙⢷⣤⣤⣆⡤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠳⣄⠀⠀⠐⠉⠛⠈⠀⣠⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠙⡦⢀⠐⠚⠛⠀⠀⠀⣰⠇⢰⣿⡿⠀⣿⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋⠀⠀⠀⠀⠀⠀⠀⢀⠋⢱⠆⠸⡟⢻⠁⢿⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠁⠀⠀⠀⢸⠀⠀⢃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣶⣄⣼⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡴⢴⠾⠿⣦⣿⣷⣤⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢦⣴⡠⠤⠒⠉⠁⠀⠀⠀⠹⡽⠃⠉⠉⠁⠉⠐⠠⣿⡎⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣎⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠙⣿⡀⠀⠀⠀⠀⢀⣾⠏⠀⢀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠙⢿⣄⠀⠀⠀⠘⣷⡈⠚⠀⡀⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠈⠹⣧⠀⠀⠀⣿⠛⠂⠙⠃⡔⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠟⠛⠁⠀⠠⠙⣷⡄⢰⡟⢳⠖⡀⠐⢻⠃⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⡷⠶⢿⣿⠃⡙⠈⣿⡟⠁⠠⢈⡁⢀⠂⢀⠁⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠚⢛⣤⣾⡱⠓⠀⠉⠀⠉⠀⢀⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣴⠀⠙⡿⠀⠀⠀⠀⠁⠀⠀⠀⣧⢄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠞⡀⢸⠉⠟⠐⠀⢰⣿⣆⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣄⣠⣿⡈⢠⠀⠀⠀⣸⣿⣿⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠈⠃⢀⠏⠋⠙⢿⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡗⡀⢀⠠⡀⡤⠄⠀⡹⣧⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⠐⠉⠀⠀⠉⠃⠀⠀⠉⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⢰⣧⠀⠐⣞⡆⢴⣷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⡆⠀⠀⠠⣼⡟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⡠⠠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿⢹⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠄⠀⢻⣄⠀⣿⠀⣿⠰⣸⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡿⣿⡀⣿⠸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⡾⠁⠸⣧⣿⠀⢻⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢀⣿⠁⠀⠀⠈⠁⠀⠈⣿⡀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠓⠒⣥⢸⡏⠀⠀⠀⠀⠀⠀⢠⣿⠻⠗⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⠶⠟⠋

⠀⠀⠀⢰⣦⠤⠤⠤⠤⣤⣄⡠⠂⠰⢷⡴
⠀⠀⣀⣸⡿⠀⠀⠀⣠⡼⠟⠂⠀⠀⠻⠿⠻⠿⠷⠐⢒⡒⠒⠒⠒⠲⠦⢤⣀⡀
⠀⢸⣽⣿⠀⢻⣭⣿⢟⣛⡛⣉⠉⣉⣹⣿⣿⣦⡶⠿⣶⣌⠑⠊⠀⣈⡑⠦⣈⠉⠛⠿⣦⡀
⠀⠸⣻⣿⣿⡾⠿⣿⣞⣟⣣⣻⣿⣝⣻⣿⣿⣏⣾⣿⣷⢻⡇⠈⣿⣭⣤⣀⡠⢍⡲⠤⡀⠙⠷⣤⣀⡀
⠀⠀⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠸⣏⢿⣿⣟⣼⣷⡏⢹⣿⣿⣭⣛⣿⣿⣷⣦⣤⣦⣼⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠿⠟⠋⠀⠈⠛⠛⠛⠿⠿⠿⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠙⠛⠳⠶⠶⠶⠶⠶⠛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡄⠀⠀⣸⡿⠀⠀⠀⠀⢠⣾⠟⠛⠻⣦⡀⠀⠀⠀⠀⢿⡇⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠘⢿⡄⠀⣿⠇⠀⠀⠀⣴⡿⢇⣀⣀⣀⡹⣷⡄⠀⠀⠀⢸⣇⢀⣾⡟⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣤⣼⣿⣼⣿⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢸⣿⣼⣿⣀⣀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⣿⡇⠀⠀⣸⡿⢁⣤⠴⠶⠒⠲⠶⣤⡘⣿⡄⠀⠘⣿⡿⠀⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⠃⠀⢠⣿⣵⠟⠁⠀⠀⠀⠀⠀⠈⢻⣼⣷⡀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⣿⠀⠀⣼⡟⢿⣤⣤⡴⠶⠶⠶⣤⣄⣤⠟⢿⡇⠀⣿⡇⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⠀⠀⢸⡏⠀⠀⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣿⠂⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡿⠁⠀⣾⡇⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣀⣀⣀⣿⡇⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⣿⣧⣄⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣌⣉⣉⣩⣿⣷⡿⢿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠟⣿⣿⣧⣈⣉⣉⣉⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢉⣡⣾⠟⠁⠀⢸⡇⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⢰⣿⠀⠈⠻⣿⣬⡉⠉⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣇⣴⡿⠋⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠇⢿⡇⠀⠀⠀⢸⡏⠀⠀⢠⣯⣿⢿⣦⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢰⣿⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠊⠹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠒⠒⠒⠐⠊⠉⢉⢹⣿⣉⠓⠂⢼⣿⠀⢸⣿⠤⠤⢴⣿⣀⣀⣀⣀⣀⣀⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠒⠉⠉⠉⠉⠀⠀⠀⠀⢿⡏⠉⠒⢺⣿⠀⢸⣿⠀⠶⣾⡿⠴⠶⠦⠄⠠⠤⢤⣬⡉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠿⢿⡷⣶⣼⣿⠀⣾⣧⣤⣴⣿⣷⣶⣶⣦⣤⣴⣶⣶⣶⣦⣿⠇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣀⣀⣤⣿⢻⣧⣀⠘⠳⠿⠶⠿⠿⣿⣽⣥⠟⠀⠀⣼⡿⣿⡀⠀⠀⠀⣸⡿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡙⢻⡛⣿⠁⠀⠙⣿⡟⢻⡿⢿⠶⣾⠿⢶⠿⣶⢿⣿⠟⠀⣿⡿⡿⢿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠃⠀⢸⣿⣤⣾⣤⣼⣤⣤⣤⣼⣤⣽⣬⣿⡄⠰⣿⣶⣷⣾⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡶⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣠⠄⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠃⠀⢸⠀⠿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠃⠀⡀⠀⠀⠆⠈⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡆⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠿⣶⠀⠁⠀⠀⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⡀⠘⠀⢀⣴⡿
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⢸⡟⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡀⢀⣾⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣯⣶⠛⢿⣽⣿⡤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⣿⣿⣿⣿⣿⠛⢛⣛⠿⣿⣿⣿⣿⠈⠀⣋⡉⢸⡷⢺⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠟⢫⣿⣯⣭⣭⣴⣿⠿⣷⡄⠹⣿⣿⣆⠐⠉⠙⢺⠁⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣼⣿⣿⣿⣿⣿⣯⢻⣀⡤⠋⠀⣺⣿⣿⣿⣦⣤⡞⡆⢹⣿⡏⣻⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣮⣴⣾⣿⣿⣟⣛⣻⣿⣿⣆⢻⠀⠀⢉⣽⣿⣿⠿⡿⠟⠳⠟⣸⣿⡇⢾⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⢋⡭⠿⣿⢿⣿⠀⣦⣤⣤⣭⣛⣁⡸⠛⠫⠠⣶⡿⠝⣄⣸⣟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢠⣾⣟⠰⣾⣧⣿⣷⠘⣄⣀⣠⣾⣿⣷⣦⣶⣿⣟⢳⡄⢸⠇⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⠘⣦⣬⣦⣤⣭⣽⣿⣇⣸⣧⣿⣦⣤⣤⣤⣶⣿⣮⣧⢻⣼⠠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⢠⣝⢿⣿⣿⣿⣿⣛⣿⣿⣿⣿⡿⣹⠇⠦⢰⣠⡾⣿⣸⡟⢐⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⠄⢸⠙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣿⣛⠛⢣⣿⡇⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣿⠄⢸⣿⣉⣉⣉⣉⣿⣿⣍⣉⣉⣉⣉⣉⣉⣉⣉⣠⣴⣿⠿⡇⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡁⣸⣿⣿⣝⣋⡈⠁⣠⣽⠿⢿⣫⣿⣿⣯⣭⣩⣭⣾⣿⣴⡇⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣗⠂⢼⡇⣬⣭⣭⣭⣭⣭⣽⣭⣭⡭⠭⠭⠭⠵⢶⣆⢿⣿⣿⡇⠨⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠚⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣘⣿⣼⣿⢛⣇⠰⣾⣟⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣖⣓⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣑⡦⠴⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⢽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁

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

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

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

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿