Blow air Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠠⠀⠘⠁⠐⢆
⠀⠀⠀⠀⢠⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠄
⠀⢠⠆⠀⠀⠀⠀⢀⡔⠋⠉⠈⠉⠙⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠁⠀⠀⢀⠉⠑⠆⠀⠉⠓
⠀⢸⠀⠀⢀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⡀⠀⠀⠀⠀⡴⠂⠁⠀⠀⠀⠀⠀⠒⠀⠐⢀⣄⠀⠀⠁
⣰⠋⠁⢠⠏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣳⠀⠀⠀⣸⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦
⡏⠀⠀⠈⠄⠀⠀⠀⡴⠚⠲⠀⢀⠀⠇⠀⠀⠛⡀⠀⠀⠀⠸⡄⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠹⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠦⠀⠦⣀
⠀⠑⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡀⠀⠀⠀⢀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠍⠹⠦⡔⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡊⠀⡀⠉⠁⠐⡄⠀⠀⢁⡂⡀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡷⢆⡀⢦⠀⠠⠃⠀⠀⠀⠁⢱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠠⡀⠉⠀⠐⠻⡤⠒⣠⣤⣄⣀⡁⢠⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠁⠀⠄⠀⢀⣍⠉⣾⠿⡏⠽⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⡄⠘⡊⠚⠑⠋⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⡴⠋⠀⠀⠀⠀⠂⠐⠢⡄⠀⠀⠸⢤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠨⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠂⠀⠀⠀⠀⠧⣀⡸⠤⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠈⠂⠀⢀⡀⠀⠀⠔⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠀⠀⢰⠃⠉⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢦⠀⠀⠘⠄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠈⢀⡼⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠸⠀⠀⣿⠀⠀⠀⠐⠉⣀⡤⠔⠋⠁⠀⠀⣀⣀⣀⣈⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠴⠶⣶⣶⣶⣚⣁⣀⣀⣤⣬⣥⣤⣤⣤⣤⣤⣄⡀⠀⠉⠩⠥⠤⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠭⠭⠭⠭⠤⠤⠤⢤⣤⣀⠀⠀⠀⣀⣀⠉⢻⡆⠀⡈⠑⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡔⠀⢹⡇⠀⢨⡅⠀⠀⣼⡟⠀⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⣧⠀⠓⠶⠚⠁⠀⠀⠙⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣤⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⣀⣽⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠋⠉⠉⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⢀⣀⡀⠀⠀⠈⢿⡆⠀⠀⠀⢠⡾⠛⠉⠉⠉⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢠⣾⠛⠛⠻⣷⠀⠀⠈⣷⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⡇⠀⣤⣶⠟⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣴⠾⢷⣦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢸⣏⠀⠈⠉⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣴⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣟⠛⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠛⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠟⠋



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠳⣄⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⠀⠒⠢⢄⠀⠉⠳⣄⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠞⠀⠀⠀⠈⠑⠢⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢽⡗⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠐⠬⣐⠒⠒⠶⠤⠤⠤⠤⠤⠴⠖⠒⠋⠉⠁⠀⣼⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⡀⠤
⠀⠀⠀⠀⠈⠑⠦⣈⠀⠒⠂⠀⠠⠤⠤⠤⠤⠤⠤⠴⢻⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⣄⠨⣉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠉⠉⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⢀⠉⠓⠤⣀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⣀⠨⠕⢦⣼⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠀⠀⠀⠀⢌⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠫⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⠜⢠⠇⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠘⡀⠀⢀⡟⢄⡀⠒⠈⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠝⠑⠀⠀⠈⠉⠉⠀⢠⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠘⣿⡀⠀⠀⢀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠈⠙⣿⣆⠀⠘⠳⠶⠶⠋⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⣸⣿⠀⠀⠀⠀⠀⣠⣿⡿⠁⢀⣠⣴⣶⣶⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⢀⣴⣾⠿⣋⣴⣾⡿⠟⠋⠉⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⣤⣾⡿⣛⣵⣾⠿⠋⠁⠀⠀⠀⣠⠖⠒⠂⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢋⣡⣴⡾⠟⣋⣵⡾⠟⠋⠀⠀⠀⠀⠀⠀⠰⣯⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢚⣉⣥⡶⠾⠛⣋⣡⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣤⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠚⢋⣉⣠⡤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠉⠉⠀⠒⠒⠚⠉⠉⠁


⣿⣯⢹⣟⠿⣿⢛⠟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣀⡀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡴⠞⠛⠉⠉⠙⠛⠋⠀⠀⠀⢀⣠⡴⠶⠖⠒⠒⠶⢄⠀⠀⠀⠀⠀⣿⣀⣀⣀⡀⠈⠙⠲⣄⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡞⠁⠀⠀⠀⠀⠀⠀⠠⠀⠠⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⣠⠋⠉⠉⠉⠉⠛⠶⣤⡈⠳⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠹⡄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⢀⣀⣠⠔⠋⠈⠒⠒⠀⠀⠒⠲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠂
⠀⠀⠀⠀⢸⠀⠘⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠣⠄⠀⠀⠀⠀⠀⢀⣀⠴⠋⠁
⠀⠀⠀⠀⠈⡄⠀⢻⡄⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣼⠁⠀⠀⠀⠀⠀⠘⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⣀⡀⠀⠀⣠⡟⠉⠈⡅⠈⠛⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠄⠀⠙⢦⡀⠀⠙⢳⠦⠤⠄⠀⠀⠀⠀⠀⠀⣼⠂⠁⠀⠀⠀⠉⠞⠁⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⣀⣇⣀⣀⡀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠾⡇⢉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠔⠚⠁⠘⠲⠤⣄⣀⣀⣰⠤⠚⠁⢰⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⢀⡠⠐⠛⡟⡷⡄⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣠⣀⢣⠃⠸⠋⠉⠐⠦⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠐⣎⢀⡙⠛⢲⠇⠤⡀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠁⠈⠃⣀⣌⠀⠀⠉⠀⠂⠰⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠒⠇⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠓⡞⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠊⢀⡴⠋⠉⠒⠤⠤⠤⠤⠐⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⡶⣶⣾⡿⠿⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣯⣿⣽⡿⢏⢔⡽⣿⢞⣫⣭⣭⣴⣿⣷⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣫⡭⠯⠙⣚⣫⣾⣿⣿⣿⣿⣯⡿⢋⣽⣿⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⢟⣵⣾⣿⣿⣯⠟⣩⡶⣿⣿⣿⣿⣿⣿⣍⡹⢛⣛⣛⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣭⣿⣿⣿⠿⣫⣿⣷⣦⡴⠶⠚⠉⠀⢁⡤⠀⢈⣿⡾⣋⣫⣟⣛⣛⡻⢽⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⣉⣽⣿⣶⢿⣿⣯⣾⣿⣿⡇⠀⢤⠤⠴⣻⠁⠀⠹⣿⣿⣯⣾⣿⡦⣄⠙⢷⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆⠙⠛⣿⠟⠂⠀⢀⡿⣧⣸⣿⣿⣧⠘⡇⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣿⠛⣩⣭⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⢿⣿⣿⣾⣿⣷⢻⡝⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⣉⣁⣤⣌⣩⣴⣿⣿⡀⠀⠀⠂⢢⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⣷⢻⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠉⠙⠻⣿⣿⣿⣿⣧⣄⣴⣿⠛⠧⣤⣀⡀⢀⣠⢾⢏⣿⣿⣿⣿⣟⡾⣹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⣈⣻⣿⣿⣿⣿⢟⡿⢟⣻⣿⣶⡎⣿⣫⠵⢫⡾⡹⢸⡟⣿⡿⣿⢫⣾
⠀⠀⠀⠀⢀⣠⣤⢤⣤⣄⠀⠀⠀⠴⠚⠉⠀⠠⠼⢿⣿⣿⣿⡿⣛⡭⣒⣭⠞⢻⣿⠇⣼⣧⢞⣽⡵⣣⣿⡇⣿⢿⡇⣿⠋
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣀⡾⢟⣫⡵⢋⣵⣾⣿⣿⣿⠿⣯⣚⡭⣞⠵⢋⣴⢏⡿⢡⣿⣾⣿⠇
⠀⢀⣠⠿⠶⠶⢦⡶⢇⣁⣸⣧⣤⣤⣤⡶⢖⣻⣵⣾⠟⣡⡾⣻⣯⣾⣿⣯⣶⣞⣯⠗⢋⣡⡶⠛⣡⡟⢡⢾⣿⠟⠋
⢠⡟⠁⠀⠰⣄⣠⣇⠀⠀⠉⠉⢉⣡⠴⣺⠕⣫⠟⣡⠞⠵⣾⣿⣿⣿⡿⢿⣻⡭⠖⣛⣩⣥⡶⠟⢉⣴⣿⣾⠇
⣾⠁⠀⠀⠀⣾⠄⠀⠱⣤⡴⠚⢉⡴⠋⣡⠞⢡⡾⠁⠀⣠⡿⠟⢋⣿⣷⡯⠷⠛⣿⠿⢋⣡⣴⣶⣾⠏⠉⠁
⣿⡇⠀⠀⠀⡼⠦⡀⢀⡿⢠⡾⠋⢀⡾⠃⣰⣿⣀⠀⠉⠁⢀⣠⣾⠟⠁⠀⠀⡴⣣⡾⢛⠿⠿⠟⠋
⣿⢱⡀⠀⠀⠀⢀⣴⡾⠷⠯⣀⣀⡾⠁⣰⢿⡉⠙⣷⡀⠐⠉⢸⡏⠀⠀⠀⢸⢱⠏
⢸⡾⠻⠶⣤⡾⠋⠉⠀⢱⠆⠈⡿⠁⣰⠃⠀⠀⠀⣿⡇⠀⠀⠟⠀⠀⠀⢀⣏⡟
⢸⡇⠀⠀⣿⠁⠀⠈⠚⠙⣷⠚⡗⠋⢳⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡎⢻⣷⣤⣀⣠⣤⠾⢷⣼⠀⠀⠀⠀⠒⠛⠉
⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⢀⡴⣋⣀⢀⠀⣿⠀⠀⡏
⠀⠀⠀⠀⢸⡏⠛⠛⠿⠻⡇⡞⠉⠹⣿⢷⡏
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⠀⠀⠛⠘⠃
⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢱⡌⠂⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⣿⣿⣿⠓⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣠⣾⠗⠋⡉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠉⠀⠀⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣒⣃⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠬⣭⣴⣒⣲⠤⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⡉⠉⠉⢳⠀⠀⠀⠀⠈⠀⠀⠀⠀⠡⡄⠀⠉⠉⠩⠭⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⠀⠀⠀⠐⠁⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠞⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠒⢒⡒⠒⠒⢦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡄⠢⠃⠈⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡵⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣰⡟⠀⢰⠛⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣬⡭⠤⠶⠶⠶⠶⠾⠛⠛⠛⠛⠛⠉⠀⠀⠈⠒⢀⣸⠇⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠖⠒⠒⠒⠛⠛⢛⣛⣋⣁⣀⣤⣤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⠒⠒⠒⠒⠢⠦⠦⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠒⠒⠒⠒⠒⠒⠶⠶⠶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠶⠦⠤⢤⣤⣤⣄⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉⠛⠛⠛⠳⢦⠀⠀⢄⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠉⠉⠉⠉⠙⠛⢷⡄⠀⢀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠃⠀⠀⠀⠀⡞⠁⠀⢈⡿⠀⠀⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⢦⡄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢛⣛⣛⣡⣔⡒⠲⠶⢶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣶⡛⠛⠓⠂⢉⠓⠘⢻⣶⣬⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣻⣿⣏⣩⣭⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣧⡙⢇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠟⢋⣭⣭⠍⠙⠋⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣰⣯⣴⣿⣿⣿⣿⠁⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠉⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⢠⣴⡶⠶⠦⣄⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠚⣉⡵⣶⡦⣅⠀⠀⠀⢠⡴⣶⡶⣄⠀⠀⢸⣿⡙⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣟⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠘⠛⠈⠁⠀⠀⠸⡇⢡⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡗⠛⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠁⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠰⠀⠀⠀⠀⢀⣤⣒⣀⣀⣀⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⡿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠐⠀⠀⠀⠘⢿⡤⢤⣄⣀⣀⣀⡀⠀⠀⣀⣠⣾⡿⠂
⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠁⠀⢘⣸⢨⣯⢩⣯⢉⡿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⢠⠇⢠⣴⢶⣶⣢⣶⣶⣤⣤⣤⣤⣀⣀⣰⣄⣴⢞⣶⣄⣴⣀⣀⣠⣤⣤⣤⣴⣶⣤⡲⢶⢶⣤⠀⢱⡀
⠀⢸⠀⢸⣏⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣹⣎⣿⠀⠈⡇
⠀⠸⡄⣸⣿⣾⣏⣿⡟⡟⣩⣭⢩⢻⣿⣿⣿⢩⡭⣿⡯⣭⢻⣿⣿⣿⢫⢩⣭⡙⡟⣿⢇⣿⣾⣿⡀⣸⠁
⠀⠀⠹⣿⣟⣿⣿⡚⠛⠛⠻⠟⠺⠿⠗⠟⠛⠿⠷⠿⠷⠿⠞⠟⠓⠿⠿⠞⠿⠻⠛⠛⣺⣿⣟⣿⡷⠃
⠀⠀⠀⠈⠻⣿⣿⣆⠀⢀⠀⠢⢄⡁⣒⣢⡤⠬⢛⣓⡬⠖⢚⣩⠥⠒⢊⡥⠖⠉⣠⢔⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⣵⣶⣚⣋⣉⣠⣴⣚⣉⣀⣴⣞⣉⣀⣠⣞⣉⣀⣀⣞⣁⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠘⠙⢖⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣗⠀⠀⠑⢹⣆⣠⣀⣤⣤⣤⣄⣀⣤⣾⠋⠀⠀⠀⡈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢻⡷⠆⠀⢸⣿⠿⢍⣹⡇⢘⣧⠀⠁⡉⡀⠀⠀⣿⠁⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⠁⠀⢠⣼⡿⣦⣾⣇⢷⣼⢩⣷⡀⠹⠃⡀⠀⣩⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣤⣾⣿⠿⣿⢸⡇⢹⡎⢡⡏⢸⡇⣀⠘⢶⡄⣼⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠙⣃⣠⣼⢤⡁⢾⡁⠈⡥⢘⣥⢭⡀⠈⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⠫⠻⠂⣧⢸⣷⠀⠁⡞⠙⠆⠌⠙⢳⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢙⣻⣿⣦⡀⠰⣾⡇⠀⣰⣧⢀⣲⣿⠂⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣟⠉⠘⠋⠙⠀⠘⢿⡭⠃⠀⠛⠛⠀⢙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢟⣢⠀⠘⠒⠶⠂⠃⠢⠄⠀⢀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣿⣦⣿⠶⣤⣤⠤⠶⠒⠊⠉⠁⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠉⠈⠀⠘⠀⠈⠀⠀⠀⠀⠠⠽⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⠿⣦⠂⠀⠀⠀⠀⠈⠙⢀⠴⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠙⠶⠦⠀⠀⠀⠀⠀⠀⠼⠟⠉⠀⠤⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠢⠴⣺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣂⠀⠆⠀⠀⠂⠀⢀⣢⣅⣺⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣻⣷⠀⠀⠀⠀⠰⢋⡙⠋⣉⣭⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⠞⠉⠻⣷⡇⠀⢸⡷⠋⠛⠋⠉⢛⠁⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡶⣶⡄⠰⣬⡁⠀⣨⣴⠆⣶⣄⠛⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠛⠛⠁⠀⠀⣸⠀⣿⠀⠀⠈⠉⠛⠁⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⣩⣠⣿⠀⠀⠀⠀⠀⣉⡛⠀⠀⣠⣾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣏⣶⡾⣻⣟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠈⢺⣿⡷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠙⠛⠛⠓⠒⠀⠀⠀⠤⠟⠛⠓⠂⠦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⣶⣾⣧⣤⣧⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠉⡟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢶⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣋⡉⠁⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⢿⡟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠁⠀⠀⠀⠀⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⡀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣤⣾⡟⠉⠀⠀⠀⠀⠀⠠⠎⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠙⢿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⡀⠀⠀⢄⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣆⡀⠈⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⡀⠲⣷⢦⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣼⣿⣿⣿⣿⣧⡀⠰⢣⡼⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠜⣿⣽⣿⢷⢽⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣗⢳⣿⣿⣿⣷⣎⡿⢿⡇⢈⣴⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣾⣏⣷⣿⣿⣿⣷⣴⣿⣌⢻⣿⣿⣿⣿⣿⣏⡸⣿⡏⣯⣭⣅⣼⣿⣿⣵⣾⣶⣤⣿⣇⣽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⣹⣿⠿⣭⣽⣿⣋⠛⠻⠋⣽⣿⣿⣿⡟⣿⣧⣙⣥⣤⣯⣭⣒⣉⣉⣛⠛⢛⣿⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠟⢉⣫⣾⣿⣿⣿⢿⣿⣆⠀⠉⣿⠻⠛⢁⣿⣿⣿⣯⣿⣿⠿⠛⠛⢛⣄⡈⠉⢛⣷⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣥⡦⠈⢿⣿⣿⣿⣿⠃⠀⠈⠀⠀⢨⣾⣟⣿⠿⡿⠧⠃⢀⣴⣶⣘⠟⠿⢶⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢿⣿⣿⣿⢿⡦⣉⡉⣿⣿⡆⠀⠀⠀⢠⣿⣿⣿⡇⢠⣤⣤⣰⢞⣫⣬⠁⣴⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣶⣭⡛⣿⣿⣿⡧⠀⠀⢰⣿⣿⣿⠋⡡⢀⢀⣠⠤⣾⢭⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠬⢛⣥⠽⠤⠛⠛⠛⢿⣛⠺⢟⣋⡽⠿⠚⠁⠀⠀⠠⠟⠳⣼⣿⣿⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠃⠀⠀⠉⠳⡾⠋⠀⠀⠀⠉⠀⠀⠀⠰⠾⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠓⠶⠄⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⠶⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⡠⠤⠖⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡻⠉⣧⠀⠠⣿⣿⣷⣏⠶⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠻⡄⠀⢿⣿⡿⣡⢰⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠼⠲⣿⠹⢦⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣸⠖⠀⣿⣧⢺⠁⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢰⡇⡠⡟⠀⠀⠀⣿⣿⣯⡖⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⠕⠁⠇⠀⠀⠀⣿⣿⡔⠱⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⡼⠀⠀⠀⠀⣿⣧⢷⣗⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣹⠃⠀⠀⠀⠀⣿⢏⡘⠁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⣿⠬⣼⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⢤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⣶⠂⠐⠂⣰⠶⠒⣺⣿⣿⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣰⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣠⣿⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠛⠋⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢰⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠀⠀⠀⢀⣿⣿⣾⢰⣿⠟⠉⠉⠙⣆⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠘⣿⣿⣿⠘⣿⢾⣯⣀⣠⡟⠀⠀⠀⣸⣿⣿⣷⣄⡀
⠀⠠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠃⠀⠀⣿⣿⣿⣷⣌⢳⣍⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⢨⣿⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢼⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣿⣿⡿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣠⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⣼⣿⡟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⠟⠋⠉⠉⢻⡆⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦
⠀⣼⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⣄⣀⣨⡇⠸⠃⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠁
⠀⢻⣿⣿⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⣿⣿⡄⠀⠀⠙⠻⠿⢛⠅⣠⣾⣿⣿⠃⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿
⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠹⣿⣷⣦⠀⠀⢠⣾⠋⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠈⠉⠉⠀⠀⢿⣿⣾⣿⣿⣿⠛⣿⠀⠀⢠⣄⣀⣾⣿⣿⣿⡋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠃⠈⠋⠀⠈⠉⠉⠉⠛⠛⠛⠃⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

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

⠀⢰⣶⣶⣾⣶⣶⣾⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⣾⣿⣿⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣷⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⢀⢰⡄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠹⢀⣿⣷⣴⠄⣷⣈⣼⣿⣠⣾⣤⣴⠖⣂⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⣼⣛⣻⣤⣼⣿⣿⣿⣿⣿⣿⣷⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟
⠀⠀⠀⠀⠀⠈⠉⠳⠀⣄⠀⠀⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠛⠉⠀⣠⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⢈⡙⠀⣠⢿⢤⣄⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⢿⣿⣠⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠩⠇⠀⠹⣀⢖⣼⢎⠉⣍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⠟⠛⣀⣻⣙⣻⣷⣶⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣨⣥⣾⣧⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⡛⠛⢀⣾⣿⣿⡟⠁⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡆⠀⣸⣿⣿⠏⠀⠸⠛⣻⣿⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠈⢿⣟⠁⠀⠀⠀⠀⠉⢉⣧⡄⣦⠀⠀⠀⢸⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡆⠀⠀⠀⣀⠀⢀⣀⣤⣤⣶⣶⣦⣤⡀⠀⠀⠀⠻⠏⢹⣤⣀⣉⣉⣶⡶⠆⠀⠦⣼⣟⣿⣿⠄⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⢀⣴⣟⣥⣾⣿⣿⣿⡿⢛⣉⣥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢠⠖⠛⠓⢻⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣴⣿⣿⣿⣿⣿⠿⠛⠫⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠚⠲⠀⣿⣟⠋⠀⠀⠸⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠿⣻⣿⣿⣿⣿⠿⢿⣦⣝⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⣠⣿⡿⠃⠀⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⢴⣿⣿⡿⠿⠟⠛⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⣿⣿⠇⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠏⣴⣷⣤⡤⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⣋⡼⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⡄⢀⡼⠁⠀⠀⠀⢀⣀⡀⡰⢡⣾⣵⣾⣿⣾⣿⣿⣿⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⡿⠁⣀⣤⣶⣿⡿⣋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠘⣿⣯⠓⠄⠈⠙⠛⠿⣌⠛⠿⣿⣍⠛⢿⣧⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠤
⠀⠀⢠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠈⠀⠀⠈⠛⠿⢿⣿⣿⡏⠻⠻⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⡋⠉⣻⠋⠀⠄⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠘⡃⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠘⠉⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⠰⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢱⣼⣿⣿⣿⣷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⠳⠾⠿⠿⠿⠷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣭⣭⣬⣭⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠦⣄⣀⣀⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⡹⠦⢄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠢⠈⠀⠀⠈⡇⠀⠘⠳⣄
⠀⠀⠀⠀⠀⢀⡤⠖⠒⠋⠀⠀⠀⠀⠀⠠⣤⡏⠑⠦⡀⠀⣀⡰⠚⣿⠀⠀⠀⠈⠓⠒⠲⣄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⢄⣤⣼⠏⠻⠖⠿⡏⠒⠲⢦⡀⠀⠀⠀⠈⠧⠠⢄⡀
⠀⠀⠀⢠⠔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⣠⠄⠀⠀⣹⣄⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢀⡯⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠰⠂⠒⠳⠘⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⡏⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠑⣆
⠀⠀⠳⣄⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠿⣿⠀⠄⡀⠬⣤⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠁⠉⠙⠢⢤⣬⡷⢤⣀⠀⠀⠀⣀⣠⠾⠧⠤⠤⣄⣀⠜⠻⢶⢿⣷⠦⢬⣍⡑⠤⠲⠤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠝⠲⠄⠀⠂⠄⢀⠐⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠠⢄⠀⠀⠀⠀⠀⠉⠑⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡉⠒⠄⡀⠀⠀⠀⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠀⠀⢠⣾⡟⠻⣿⣿⣿⣿⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⠛⠋⠀⠀⢀⣴⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠙⠿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⣀⣿⠿⠿⢿⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠃⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡆
⠀⢀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣾⣶⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠴⣿⡄
⠀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⠛⣿⣿⣯⡀⠀⣴⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢻⣿⣿⣿⡇⠀⣿⣿⣿⣿⡄⣹⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢨⣿⣿⡟⠁⢠⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠘⣿⠁⠃⠀⣸⣿⣿⣿⣿⣿⣟
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣤⠄
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⢻⣿⡄⠉⠛⠻⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⡀
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢕⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠑⠌⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠢⣀⣀⣀⠀⡀⢀⡤⠚⠁⡐⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠄⠀⠼⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠘⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⣀⣈⠀⠁⠀⠀⠀⢀⣀⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡟⣻⡏⢹⣿⣷⡄⠀⠀⣼⠛⣿⡛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⢇⢻⣿⠿⠛⢣⠗⠈⢁⣿⡿⢟⠏⡇
⠀⠀⠀⠀⡰⠚⠉⠉⠑⢹⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠐⠈⠍⠍⠩⠀⠀⠄⠈⠅⠭⢚⠇
⠀⠀⠀⡼⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠋
⠀⠀⠀⡇⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⣷⠁⠀⢠⠋⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡸
⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⢠⢈⡇
⠀⠀⠀⢧⠀⠀⣇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⠀⠀⠀⠀⢠⠃⡠⠁⣸
⠀⠀⠀⠸⡄⠀⠸⡄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⠀⣿⡞⠀⠀⡇
⠀⠀⠀⠀⠱⡄⠀⠙⢄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢹⡀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠙⢦⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡜⢱⡀⠀⠀⢸⢹⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠺⣆⠀⠀⠀⠀⠀⠀⢴⣁⣀⠮⢤⡀⣳⠀⠀⢸⡄⣇⠀⢣⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠤⢄⡀⠀⠉⢉⡷⠋⠉⢧⡀⠀⠈⢹⣄⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠂⠈⠉⠐⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⢀⣀⣀⠀⠠⠤⠐⠂⢉⡁⠄⠒⠈⠉⠁⠀⡉⠉⠒⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⣴⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⡔⠉⠈⢱⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠡⠱⣀⢱⠀⣀⠴⡪⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠔⠀⢈⠝⠉⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠔⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⣂⠭⠒⠉⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⡊⠅⣒⠥
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠉⠉⠡⠥⠄⠒⠈⢉⣀⠠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢤⠀⠈⢂⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢤⡀⠀⠀⠀⠀⢀⠔⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠤⠊⠀⠀⣘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠂⠤⡀⠀⠈⠢⣀⡀⡇⠀⣀⡠⠖⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢨⠀⠀⠀⣠⠎⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢞⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠶⠖⠥⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠚⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡴⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣉⠔⣁⠠⠔
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠤⠀⠒⠒⣂⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⡒⠭⠐⣉⡀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠨⠥⠌⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠅⠂⠈⠁⠀⠀⢀⡔⠂⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⡄⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠖⠒⣶⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⡀⠀⣈
⠀⠀⢰⡦⣄⡉⠀⠀⠂⠀⠀⠀⠛⠀⠀⠀⠀⠉⠀⠀⠀⠠⠏⠉⠂⠀⠀⠀⠀⠀⡼⢹⠃
⠈⡄⠀⠳⡀⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠰⠀⠀⠹⠄
⣀⡀⠁⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠉⠓⠲⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⢀⡀⠀⡿⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣤⡧⠤⠤⠤⠄
⠀⠀⠐⠒⠺⡶⠆⠀⠀⠀⠀⠀⠈⠳⠞⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣠⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡼⠁
⠀⠀⠀⠈⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠉⠓⠶⠤⣤⣤⣤⣤⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡄⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⡼
⠀⠀⠀⢸⠃⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡞⠁
⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠃
⠀⠀⠀⢸⡆⠀⢸⡇⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡟
⠀⠀⠀⠀⣇⠀⠘⠟⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⣀⣀⣀⣀⣀⣀⣀⣀⡤⠊⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣁⣀⣀⣀⣉⣉⡉⠉⢁⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣠⣤⡀
⠀⠀⠀⢾⡉⠉⠙⠛⠒⢺⣷⣤⠀⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⡴⠠⢴⣾⠋⠁⠀⠀⠀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠁⠢⡀
⠀⠀⠀⠀⠀⢄⣀⣀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠰⠁⠠⠂⢁⣀
⠀⠀⠀⠀⠀⠠⢔⠠⠄⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠨⠤⠐⠁⠔⠀⠀⠀⠀⠀⠂⢁⡠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠒⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⡠⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠈⠀⠂⠁⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠒⢄⠘⠀⠐⠒⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠁⠒⠀⠊⠄⠂⡀⠀⠀⠀⠀⠀⡰⠀⡇⠠⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠐⠒⠉⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⠠
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡀⠔⠀⠂⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠠
⠀⠀⠀⠢⠄⠀⠀⠀⠀⠄⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠀⣉⠒⠀⡀⣠⣴⡶⠾⢷⣼⠇⢤⠱⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⡠⢀⣴⣶⣽⠟⠋⠙⠶⠈⠻⣆⠀⠙⠩⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣠⣾⣿⢻⡿⠀⣀⣤⡾⠿⢿⣆⠀⠀⠀⠀⠀⠠⠞⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⠿⠛⠉⣳⣶⠾⠛⡉⠓⠐⠿⠂⢹⣧⣄⣀⡀⠀⠙⠛⠃⡌⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢁⣠⡄⣬⣿⣧⣾⣼⡿⣶⣤⢠⣤⠀⠀⠛⠛⠩⣾⣯⡉⡱⠢⣝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⣸⣿⡟⠈⠉⠛⠛⠓⣚⡿⠛⣂⡴⠶⡰⠄⠀⠀⢿⠁⠀⠋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣽⣿⠿⠁⣛⣠⣦⣴⣾⡁⠀⢉⠄⢂⡠⠀⠀⢀⡴⠿⣆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣽⣶⣶⡾⣿⣿⡟⠍⣀⠰⡠⢀⠀⡅⠰⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠟⢛⡟⢛⣿⣷⣶⣾⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠜⠃⠈⢀⡴⠿⣿⣯⡽⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⣴⣦⠘⠇⠉⠈⠛⠀⠀⠈⠁⣿⣿⣾⣧⣤⡤⢔⠢⣤⣖⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣿⡦⢾⣷⣲⣶⣤⡤⢶⡾⠟⠛⠟⠉⠛⠋⠁⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠷⣶⣶⣶⣬⠋⣃⣉⣉⣛⣀⣀⣠⣦⣤⠶⠶⠚⠃⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⡙⠀⠈⠉⢉⠍⠉⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⣤⣤⢀⣤⡀⡀⠀⡀⠀⠀⠀⣤⡶⠶⣿⠛⠛⠛⠛⠻⣄⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⡗⠂⠀⢀⣉⠀⠀⣀⣿⣷⡖⣻⣾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠙⠉⠈⠿⠹⠃⠀⠀⡘⠀⠀⠘⠂⠙⠶⠀⠙⠋⠿⣿⣿⣤⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠒⣦⡤⠐⠀⣜⠀⠇⠀⠀⠊⣰⠶⠄⠈⣞⢿⣇⣨⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡡⠿⠏⠴⢰⠒⢻⠀⢸⠇⢨⡔⠒⣛⣾⡟⠫⠁⠽⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣨⡇⡐⣴⠀⠀⠼⢩⣼⠋⠀⣷⣶⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⡴⢀⣴⠎⠋⠀⠰⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢉⣿⣏⠙⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠶⣄⡀⠀⠀⣼⠋⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⢷⠀⣴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡌⢳⠀⠀⠘⢦⣤⡴⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣒⣤⣀⡀⠀⠀⠀⠙⣩⢿⠆⠙⠶⠶⠃⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠁⢸⠀⠀⠀⠀⠀⢀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢭⣝⡒⠒⠛⠓⠁⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡟⠀⠀⠀⢀⣔⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⢗⡲⡤⠤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡱⠋⠀⠀⠠⠔⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠤⠀⠤⠤⠴⠞⠛⠉⠀⠀⠀⠀⠐⠈⠭⠭⢗⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠶⠶⠶⣒⡶⢤⣀⠀⢀⡬⣬⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢀⡙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⡈⢳⡘⢧⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠠⡀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠹⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣒⢄⣀⠀⠀⠀⣀⡤⠞⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠯⠭⠭⠥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⢳⠀⠀⢀⡤⠖⣛⣩⣍⣙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠨⡲⣄⢄⠀⠀⠀⠀⠀⢰⠟⠉⡝⣦
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠃⡼⠀⠀⣡⠖⠋⢁⣠⣤⡈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡘⡌⠢⣱⢄⡀⠀⠀⠘⠷⡴⠃⡼
⠀⠀⠀⠀⠀⠈⠒⠤⣀⣀⡀⠀⠀⠀⣀⣀⡤⠞⣁⡴⠋⠀⠀⠀⣟⠀⠈⠃⣼⠀⠀⠀⠀⠀⠀⠀⠸⣇⡸⣣⠇⠀⠈⠛⠻⠷⢶⣦⡤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠬⢉⣉⣉⣉⣉⠤⠴⠚⠁⠀⠀⠀⠀⠀⠙⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠞⠃⡟⠀⣠⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠛⢳⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠸⣇⣀⡶⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠞⣰⠃⠀⣠⠔⠒⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⡥⠚⠀⠀⠀⢈⣉⣠⢾⠋
⠀⠀⠀⠀⠀⠉⠒⠢⠤⣀⣀⣀⣀⣀⡤⠴⠊⡡⠔⣩⠴⠚⠋⠉⠙⢦⡀⠙⡆⠀⠀⠀⠀⠀⠀⢐⣒⣒⣶⣺⣒⣚⣉⣀⣀⣀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣐⣢⡥⠖⠉⠀⠀⢠⡞⠛⣷⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⣆⣛⣋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢻⠄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⡴⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠉⠙⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⣠⣄⣠⣤⠀⠀⠈⢻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣤⣿⠉⢻⡟⠉⣿⣥⡄⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣾⣷⡾⠛⠻⣶⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⠟⢻⡟⠻⣷⣿⡋⠀⠀⠀⣸⣿⣿⣷
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣶⣶⣦⣤⣀⠀⠀⠀⠀⠉⣿⣤⣾⣿⣦⣿⠏⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠟⠋⠉⠉⠛⠻⢿⣿⣿⣶⣤⣄⡀⠈⠁⠈⠁⠈⠁⠀⠀⢀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣤⣴⣾⣿⠿⠛⠉⠀⢀⣀⣠⣤⣤⣀⡀⠈⠛⠿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢀⣤⣾⠿⠟⠛⠛⠻⣿⣿⣶⣄⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢠⣴⣤⣶⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠀⠀⢠⣄⣄⣤⡀⠙⣿⣿⣧⠀⠠⣀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⢐⣾⣿⡈⣿⢁⣼⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣬⣯⡼⢷⡀⠸⣿⣿⠄⠀⠈⠛⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⣿⣏⣉⣿⣿⣿⣉⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢛⣟⢳⡾⠁⢰⣿⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣦⣄⠀⠐⢿⣿⢁⣿⡈⢹⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⠋⠛⠛⠁⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⠈⠻⠛⠿⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣀⣀⣤⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⠞⠉⠚⠶⠛⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣠⡤⠾⠁⡀⠀⢀⠀⠠⣦⡤⢆⠈⠁⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠋
⠀⠀⠀⢀⡽⠀⢀⠰⠟⠘⠋⡀⠀⡄⠀⠀⠛⢦⡄⠈⠉⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⡏⠀⠀⣠⠄⠀⣳⠀⠉⠙⠁⠐⣧⣀⠈⠳⡦⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠰⠃⠀⢰⠟⠀⠘⠉⢠⣴⣶⣶⡄⠀⠀⡀⠈⡧⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⢠⡀⠀⢻⡀⠀⢷⠀⠸⣿⣿⡿⢇⠀⠹⡁⠀⣠⠂⠸⠇
⠀⠀⠀⠟⠀⠀⣀⠀⠴⠦⣄⣿⣿⣷⢀⠶⢄⣀⠐⣯⠀⣀
⠀⠀⠀⠠⣄⡀⠙⠁⠀⡀⢸⢻⣿⡟⢾⣄⣸⡁⠀⠀⣸⣇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠸⣇⣀⡀⠀⠙⠛⢺⡿⠦⠀⠋⠁⠀⢠⡏⠉⠉⢠⡤⠼⠋⢋⡉⠑⠒⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣦⠶⣦⢸⣧⣄⡀⢠⠞⠋⠋⠀⠀⡀⠚⢀⣠⠀⢉⠘⠃⠰⢦⡀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡏⠀⠈⠁⠀⠀⠀⠀⢀⠇⠠⠏⢠⠘⢋⣀⡙⢃⠀⢳⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⠄⠶⠀⢮⢀⣿⣿⠇⠈⡃⢘⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢦⣼⣿⣿⡥⡄⠎⠀⡋⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣀⡿⠋⠸⢤⠞⠐⢏⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠑⠤⠴⢤⡴⠒⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⣸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢏⣾⠇⣿⢸⡄⢏⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠏⣼⢻⡀⣿⡌⢻⡢⢈⣉⡙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣼⠇⠀⣿⢸⢣⠙⡿⣄⡙⠶⣤⡤⠤⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢸⣿⢀⠈⢻⡆⢆⢣⡘⢌⡛⢶⣤⣄⣈⣐⠂⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⡇⢸⣿⠈⡀⠎⢿⣌⢢⠙⢦⡉⠢⢌⡙⠿⣛⠻⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⢷⠘⣿⣧⠑⠈⢆⠙⢷⣷⣄⡉⠳⢦⣌⡑⠢⢌⡒⢝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⣿⢠⠘⣧⠹⣿⣷⣄⣀⡑⢄⠙⠭⣙⠳⢶⣬⣙⠳⣄⠈⠓⢌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡄⢂⠘⡑⢌⠿⠿⣿⣿⣦⣄⡀⢀⠀⠐⠬⣻⢷⣌⠻⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠌⢿⣿⣄⠣⡈⢦⡙⠷⣮⡙⠻⣿⣿⣷⣮⣕⠦⣀⠙⣿⣷⡘⠄⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⡌⢻⢿⣷⣌⠢⡙⣆⠈⠻⣦⡀⠙⣿⣷⡝⢗⢌⠱⡌⢿⣧⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣷⡝⣿⣷⡈⢎⢳⡄⠘⡟⣦⠈⣿⣿⡌⣆⢆⣨⡾⠋⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣈⣆⢻⡄⠸⡘⣧⣼⣿⣷⠾⠟⠋⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢰⣿⣏⣾⣿⣿⠟⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢋⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢿⣶⣌⠉⡝⠳⢄⠀⠀⠈⠻⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠛⠷⠦⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⠶⠿⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠁⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂⠘⢀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁⠀⠀⠈⠈⠁
⠀⠀⠀⠠⠡⠁⠂⢂⠘⡈⢁⠈⣀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⣀⠀⠀⠀⠀⠠⠤⠀⠦⣀⠀⠀⠀⠀⠢⠀⠈⢢
⠀⠀⠀⠀⡀⠀⠌⠌⢠⢁⡈⢀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠁⠉⠐⠀⠄⢠⠈⠀⠀⠀⠀⠀⠀⠠⣀⠈⠉⠛⠲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠽⣍⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⢋⡾⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⣀⠔⠚⠉⠉⣀⣀⣤⣤⣀⠀⠀⠀⠀⠠⠐⠒⠒⡒⠒⣲⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⡞⠁⣮⢻⡀⠀⠀⡞⠀⠀⠀⣠⠴⢲⠄
⠀⠀⠀⠣⣄⣀⣰⣚⣩⠤⠤⠖⣃⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⢛⠿⣗⡒⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠚⢁⡼⠁⠀⠀⠳⢤⣠⡮⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠴⣿⠗⠚⢉⣠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⢤⣤⣤⡤⠤⠶⠚⠋⠀⠀⠀⠀⠀⠀⢺⡀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡾⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⠀⠀⠀⠀⠈⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡇⢀⣠⠞⠁⠀⠀⠀⠀⢠⠋⠉⠉⠳⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⣤⡶⠝⠁
⠀⠀⠀⢰⣛⡆⠀⢀⣠⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⣿⡴⠋⠄⠀⠀⠀⠀⠠⠴⠛⠍
⠀⠀⠀⠐⠉⠓⠊⠉⠀⠀⠀⠀⠀⠀⣠⠖⠲⢤⡀⠀⠀⠉⣉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠚⠋⢙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣷⣠⡴⠞⠁⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⣀⣁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡄⠀⠀⢉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠶⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠿⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠊⠁⠀⠀⠀⠀⠀⢀⡴⠛⣉⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣼⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⡿⠀⢾⡳⠀⠀⢯⡗⠀⣻⣼⣁⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠒⠶⠶⠞⣛⣷⣶⣚⣛⣒⣒⣋⠉⠉⠁⠈⠉⠀⠉⠛⠒⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢔⣪⠵⠚⠁
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠭⠭⠭⣒⣒⣚⣉⣉⣤⣤⡤⠤⠤⠤⠭⠭⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⡼⢃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁
⠀⠀⠀⡠⠤⠶⠿⠯⠭⣍⣓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠄⢀⡠⠴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠲⣍⡓⠤⣄⠀⠀⠀⠀⠐⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢭⣔⣒⣒⣶⡾⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⠀⠀⠀⣴⠋⢹⡆⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⢸⡑⠚⢀⣠⢔⣩⠔⠒⠉⢉⡭⣝⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣧⠔⠃⠀⠀⠀⣧⡤⠋⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠭⠭⠽⠒⠉⠀⠀⠀⠀⠘⠧⠭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠲⠤⠤⠖⠚⠉⠙⠓⠒⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣪⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠂⠙⠳⡆⢴⣶⡞⡿⣿⡿⢣⣶⠞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠅⠀⣄⡆⣠⣿⡄⠱⢿⡧⠀⣄⠀⠀⢾⠋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣊⣮⠁⡝⣆⠃⣦⠿⡷⢎⠩⡆⠤⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠐⠙⠑⣂⠛⠈⢿⡙⠀⢃⣊⠉⠛⠈⠁⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠖⠚⣽⢵⠀⡾⡇⢰⣯⣾⠟⠐⢶⣄⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢢⣤⠀⠈⠀⠀⠀⠈⠀⠀⣤⣴⣁⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣺⡋⠀⠀⠈⠓⠉⠉⢻⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠈⠁⠉⡉⠙⢀⡀⣿⣿⣿⣟⣿⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠉⢡⣿⣿⣞⣟⡱⣽⡿⢷⣎⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠖⠠⢶⣶⠿⠟⠣⠭⣵⣤⣷⣻⣷⣼⡀⣦⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⣴⠴⡈⢡⡶⣗⢲⣿⣟⣇⠘⣷⡌⠛⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠁⠀⠀⠀⠐⣻⣯⣮⣶⡷⠶⠍⣦⣙⣷⠘⣧⠙⢋⡀⢛⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠈⠁⢀⣉⣥⡴⠶⢶⣌⠙⣿⠇⢹⡆⠘⠿⠝⣡⣄⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⡀⠀⠀⠐⣿⣀⣤⣤⣦⣄⠙⠿⣼⡆⠻⡁⢼⣷⢠⣮⠉⣚⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣦⡀⠀⠩⠟⣀⣨⣉⣙⣛⣟⣛⠁⠰⣋⣀⠀⣬⡍⢀⣿⠎⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣉⣠⡀⠀⢉⡉⠉⠋⢛⣛⠟⠛⠃⣉⣭⣀⡘⠳⢦⣬⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⠙⠻⢿⡞⠙⠛⠛⠟⢛⡻⠀⠰⢶⠾⣆⡉⠙⣷⡄⢭⡁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⣸⠃⢠⣴⡶⠾⠛⠁⠀⠀⠘⠓⠛⣉⣤⡄⣙⠲⠚⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠛⠋⠁⠀⠀⠀⠀⢴⡿⠿⣭⣄⠨⠍⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⡽⠀⠘⠆⠀⠂⢁⡘⠒⠄⣁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⢠⣿⣧⣴⣦⣤⣤⣽⣶⣭⡿⣟⢿⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠓⠿⣄⣀⣀⠀⠀⠈⠉⠻⣯⡍⠉⢭⡉⢻⡄⢳⣸⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠠⣼⡿⠃⠠⢾⣃⣿⣵⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠾⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣒⠒⠒⠒⠲⠿⠯⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠉⠒⠠⢀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠚⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⢲⣶⣤⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠀⢰⣦⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠋⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠈⠀⠠⢰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠸⠀⠀⠈⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⡇⠊⠀⠀⣁⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣾⣿⣿⣞⢿⣿⣆⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣟⣻⣛⣫⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠎⠀⠀⠘⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢻⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢿⣿⡆⢹⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⠤⠖⠊⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠀⠀⠀⠀⠉⠙⠳⣤⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠐⠛⠳⢦⡀
⠀⠀⠀⠀⠀⡄⠀⢰⠋⠀⢠⠖⠉⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠈⣆⠀⢻⡀⠀⡆⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠓⠀⠻⠦⠈⠢⡀⠀⠀⠀⠀⠀⠀⠉⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⢀⡴⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠤⠤⠶⣞⠉⠀⠀⠠⠔⠊⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣠⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣀⣀⣤⠴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠶⢤⣄⣀⣀⣀⣠⡤⢴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡾⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢠⡞⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⢀⡏⠀⠀⠀⠀⡼⠁⠀⠀⠀⣰⠋⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⡞⠀⠀⠀⠀⡼⠁⠀⠀⢀⡼⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢀⠞⠀⠀⠀⢀⠞⠁⠀⢀⡴⠋⣠⠞⠁⠀⠀⠘⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣯⣀⠀⠀⡴⠋⠀⢀⡠⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⡻⠿⢦⣤⣴⣋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢀⠴⠊⠀⡠⠒⢁⡽⠋⠙⠛⠛⠷⠶⠶⠶⠶⠶⠾⠛⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⢀⠤⠊⠁⠀⠐⣁⠤⠊⠁⠀⠀⠀⠈⠐⠒⠀⠀⠀⠀⠀⠤⠤⠔⠊⠁
⠀⢀⣀⣠⡤⠴⠒⠉⠁⠀⠠⠐⠈⠀⠀⢀⠠⠐⠊
⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠒⠈⠁
⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣿⡟⠿⣛⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⡩⠟⣻⣿⡆
⠀⠀⠀⠀⠈⢧⡀⠀⠈⠑⠀⠈⠻⡑⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠙⠋⠀⠀⠉⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠹⣦⠄⠤⠄⠀⠀⠈⠛⢿⠳⣦⡀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⣴⠞⡹⠟⠁⠀⠀⡀⠤⠆⠠⣺⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣗⠄⠀⠈⢃⠠⢀⠀⢳⣌⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠟⡡⡾⠁⠀⠄⣂⢉⠀⠀⠔⣺⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠈⠁⠀⢀⠉⠓⢦⣄⠲⡆⠉⠻⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠛⠀⢠⠗⡀⡴⠞⠉⣁⠀⠀⠈⢐⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡁⠀⠀⣀⣀⣤⣤⣧⡉⢶⡀⠀⢰⢿⣆⠀⠀⠀⡀⠀⠀⠀⣴⡿⠇⠀⢀⡶⢋⣴⣥⣤⣮⣤⠤⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠐⠢⠤⣀⡀⠀⠉⠙⠻⣄⠘⠈⢿⡆⠀⣢⠉⣆⠀⠸⡿⠀⠃⣰⠞⠉⠁⠀⢀⣀⡤⠔⠂⠈⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠀⠘⠷⣄⠸⢸⣤⠀⠀⢰⣴⠇⠇⣰⠞⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢤⣬⣿⣷⡽⠀⠀⠈⣵⠿⡭⢤⡖⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠒⠒⢀⣰⡶⠶⠿⣶⣦⣶⣭⡶⢖⠾⣿⠀⠀⠐⣟⠯⣛⠷⠬⠷⢶⣶⡿⠶⠶⢶⣄⠐⠒⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣒⡽⣯⠅⠀⢰⣖⡴⢋⣆⡏⠀⠀⠀⣹⣴⡈⢧⣶⣆⠀⠀⢭⡿⢭⣙⠻⢕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠵⢋⡴⢋⣡⠎⣥⠔⠂⠹⡿⣁⣾⢹⢷⡀⠀⢰⠯⠋⣷⢌⣻⡏⢀⠒⢦⡭⠦⣌⠑⢄⠙⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣯⣀⣉⣀⡞⢁⡼⢃⣼⡻⢡⣿⠀⣎⡇⠀⣼⣼⠀⣿⡎⢻⣷⡄⢳⡄⠙⣦⣤⣉⣲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⡽⠀⠚⠡⠞⠋⠀⣼⠿⣾⣿⢧⠀⡿⣿⡗⠿⣧⠀⠈⠛⠦⠉⠀⠘⡄⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠏⡠⢹⠇⣾⠀⣷⠹⡎⢆⠘⠑⢄⠀⠀⠀⠀⠀⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣤⡀⣀⣤⣤⠞⠀⠀⣰⣡⠋⠀⠘⣦⡏⠀⠹⣌⣆⠀⠀⠙⢦⡴⣄⣀⣰⣼⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣭⠞⠁⠀⢀⠀⡟⠁⠀⠀⠀⠸⠀⠀⠀⠈⢻⠀⠀⠀⠀⠙⢦⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣰⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⣦⣤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⡠⠄⡀⠁⡰⠇⠀⣠⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠑⠛⠲⠤⢤⡤⠄⡀⠁⣰⠇⠀⣠⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠛⠛⠲⠦
⠂⠀⠐⠂⠁⠀⠞⡁⠀⣚⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠙⠙⠛⠳⢄⠂⠀⠐⠀⠁⠀⠞⡁⠀⣛⠃⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⠉⠛⠳
⠀⢤⡀⠀⢀⠰⠋⡀⢀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢁⡉⠐⢄⠀⢤⡀⠀⢀⠴⠋⡀⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⢉⣁⠐
⣔⠶⢠⠂⠀⢀⠅⠞⢻⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⢠⠎⣄⠶⡄⠚⠉⣀⠄⢾⢻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣘⡠
⠀⠑⠊⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡖⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡖⡒⡄
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⠓⠀⠘⡆⢀⠈⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⠓⠀⠘⠀⢀⠈⠑⠃
⠀⠀⠀⠀⠀⠈⠁⢀⣉⣉⣓⠒⠲⠤⠄⠀⢀⠾⠁⣠⠴⠒⣒⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣉⣉⣛⠒⠲⠤⠤⠂⢀⠞⠁⣠⠴⠂⣒⢢⢠
⠀⠀⠀⠀⠀⠀⠞⠉⣉⣉⠉⠙⠢⠐⠒⠀⠁⠀⠐⣁⠀⠘⣜⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⣉⣉⠉⠙⠆⠀⠒⠀⠁⠀⠐⣡⠀⠨⣝⢀⡰
⠀⠀⠀⠀⠀⠀⠠⠚⠁⣠⡄⠑⡄⠀⠦⠀⠠⡰⠊⡡⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⣤⡄⠑⡄⠀⠦⠄⢠⡐⢊⡡⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠁⠔⠁⠌⠓⣄⣀⡀⠠⠊⠏⣫⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠁⠔⠁⢌⠓⣄⣀⣀⠠⠊⠏⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣤⠂⠠⡀⠀⠸⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⡀⠀⠺⠤⢀
⠙⠁⠀⠃⢀⡄⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣄⣀⡀⠀⠙⠁⠀⠇⢀⡄⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⣀⡀
⠉⠁⠀⡀⠞⠃⣠⠞⠉⣩⡙⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣉⣉⠉⠁⠀⣀⠞⠃⣠⠞⠉⣩⡙⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣍
⠀⠀⠀⠀⠀⢀⡥⠂⠀⠲⠄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠴⠠⢈⠀⠡⠀⠀⠀⠀⢀⡥⠂⠀⠲⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣠⠴⠠⢌
⢀⡙⠃⠀⠠⠋⠄⣢⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠢⠀⡡⢀⡙⠋⠀⠠⠋⠄⣢⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⢲
⠓⣍⡳⡦⠀⠐⠁⠈⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠘⠁⠓⣍⡳⡤⠀⠐⠁⠉⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢢⠀⠀⠣⢂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠂⢠⠀⠀⠃⢆⡧
⠀⠀⠀⠀⠀⠤⠖⠒⠢⠤⣤⣀⣀⡉⠀⠀⠁⣰⠄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠒⠲⠤⢤⣀⡀⣈⠀⠠⠀⣰⠄⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠒⠶⠦⢤⠀⡉⠀⡀⠔⠉⢀⠞⠁⢠⢶⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠶⠦⣤⠀⡈⠁⠠⠔⠉⢀⠞⠁⢠⢶⠙⢐
⠀⠀⠀⠀⠀⠈⠁⣠⠖⠒⠒⢀⠀⠀⣀⠀⠀⢀⡠⠖⠀⠈⠓⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠖⠒⠒⢀⠀⠀⡀⠀⠀⠀⡠⠖⠀⠀⠒⠐⠊
⠀⠀⠀⠀⠀⠀⠈⠀⢠⡏⠇⢀⠇⠠⣍⠀⠁⠎⠐⠨⡵⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡟⠇⢀⠇⡠⣍⠈⠁⠂⠐⡨⡵⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠑⠄⠹⠆⠐⠈⠀⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠁⠄⠹⠂⠀⠉⠀⠁⠚⠁

⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠
⢷⠄⠈⠆⠀⠀⠉⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠄⠈⡆⠀⠈⠉⠴⠂

⠀⠀⡞⠉⠉⠑⠲⢤⣀
⠀⢸⠇⡴⠋⠉⠒⠲⢬⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢄
⠀⢸⡴⡁⠀⠀⠀⠀⠀⠈⣳⣽⡦⠴⠶⡖⠒⠒⠒⠒⠒⠦⣤⣀⡀⠀⢀⠴⢚⣉⠤⠤⠤⣀⠈⡇
⠀⠘⡇⡇⠀⠀⢀⡤⠖⠋⠛⢤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠉⠙⢳⣾⡉⠀⠀⠀⠀⢸⠆⡇
⠀⠀⡇⡇⣠⠾⡁⠀⠀⠀⠀⠀⠈⠓⢤⡇⠀⠀⠀⠀⠀⢰⣇⡠⠔⠊⠁⠀⠈⠳⣄⠀⠀⢸⢳
⠀⠀⢧⠟⠁⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠷⣄⡼⡸
⠀⣰⠛⠒⠦⠶⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠘⢧⡇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠦⠼⣇
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡗⠉⠉⠉⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢸⡀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠓⠒⠒⣾
⠀⢳⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣰⠃
⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠈⠓⠤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃
⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⠉⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⢠⠏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡰⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡞
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠘⠾⣅⣀⣼⣁⣇⡴⠀⠀⠀⠀⠀⢠⠧⢄⡤⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣄⣜⣀⣀⡴⠋⠀⠀⠙⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠏⠀⠀⠀⠀⠈⠠⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠘⠉⠉⠙⡆⠀⠀⠀⠀⠀⠀⡿⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡅⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠓⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠶⠆⠀⢀⡔⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠈⠳⣤⣀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⢸⣿⠀⣰⠋⠀⠀⠀⡀⢀⡴⠁⠀⢻⣄⢢
⠀⠀⠀⠀⠀⠀⢹⣄⣦⣗⣋⡤⠈⣿⠀⡟⠓⣒⣀⣘⣷⠫⠿⣶⡷⠀⠈⢡⣀⠴⠗⠓
⠀⠀⠀⠀⠀⢀⡌⢻⢿⠋⡈⢀⣠⣋⡼⢷⣟⡚⠻⢿⢿⡀⣠⣿⣷⣾⣿⣿⣄⣀⣒⡂
⠀⠀⠀⠀⠀⢘⡷⠟⠃⠀⢷⣽⡿⠯⡤⣤⣤⡭⠑⣲⠦⠈⣹⣿⣿⣿⣿⣿⣿⡌⠉
⣶⣦⡀⠀⠀⠸⢧⣀⣴⣦⣘⠻⢦⡀⢀⣴⣦⣠⡀⠈⠾⡿⣿⣾⣿⣿⣿⣿⣿⣇⠠⠄
⣿⠿⠛⠛⠛⠉⠓⠬⣅⡘⢦⡈⠘⣿⡾⢻⠾⠫⠄⠀⠀⠘⢻⣿⣿⣿⠻⢿⣿⣿⡟⠃
⠁⠀⠀⠀⠀⠄⣀⣠⣶⣿⣧⢱⣶⠯⠾⣿⣶⣤⡀⠀⠀⢀⣾⢹⣿⣧⣾⢃⣿⣿
⡴⠾⠶⡶⠖⠒⣲⣿⣿⣝⣯⢰⡄⣠⣿⣿⠛⠻⣿⣶⡀⢼⣿⣇⢻⣿⣿⣿⣿⣿⣆
⠀⣄⣼⣾⣿⡽⠻⠟⡉⡉⠙⢾⡵⣿⣧⢿⡗⢀⡻⠛⠐⢿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣷⢤⣀⣀⡀
⠼⡿⠛⠉⠀⠀⡀⠀⢳⡇⠀⠈⢷⡈⢿⣦⣾⣿⣷⣷⣾⣿⣾⣿⠿⣿⣿⣾⣯⣾⣏⠀⢸⣿⣿⡿⠆
⠈⡇⠀⠀⠀⢀⣳⣆⠉⣹⠳⡀⠸⣿⣾⣿⡟⢻⣿⣿⣿⣿⡿⠋⠴⢿⣿⣿⣿⣿⡘⣷⢸⡿⠋⠀⠀⣤⡀⠀⠀⠀⠀⢀⣀
⠀⡄⠀⠀⠀⠽⣣⣿⡿⠿⣿⣷⣾⡿⠛⣩⡷⠾⠛⠹⠿⣿⠀⣠⢂⣼⣿⣿⣿⣿⣿⡼⣾⣵⡆⠀⠀⢸⣿⣦⣀⠀⠀⣸⣷
⠀⡃⠀⠀⠀⠘⠁⢠⣷⠀⠀⣽⣿⣣⣾⡏⠀⠀⠀⣤⣄⣷⣐⣵⣿⣻⣿⣿⣿⣿⣿⡷⣿⣿⡇⠀⠀⣠⣿⣿⣿⣷⣶⣿⠏
⠀⡇⠀⠀⠀⢀⣀⠘⠿⣷⣶⣿⣿⣿⣿⣥⡀⠀⢀⡿⠻⣿⠟⢿⠇⣴⣦⡩⣿⣿⣿⣿⣿⣿⢷⣶⣾⣟⣩⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠿⠋⠙⠉⠉⣁⣀⣠⣤⡴⣾⣿
⠿⣿⣿⣦⣄⡀⠀⠀⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠛⠋⣀⣴⣾⣿⣿⠿⠛⠉⠉⠀⠀⣿⣿
⠻⢷⣶⣦⣭⣟⡷⢶⣤⣭⣭⣍⣉⣉⣉⣈⣙⣲⣤⣄⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⠿⠋⠁⣠⣴⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣤⣈⣉⠛⠛⠿⠿⢾⣯⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣄⣤⡶⣿⣿⣿⣿⣿⣿⡏⢀⣀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠉⠛⠛⠿⠿⠷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣛⣛⣿⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⠃⠈⢉⠉⠳⢄⠀⠀⠀⠸⣿
⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⢋⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣉⣵⣾⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢠⡟⠁⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣀⣤⣤⣶⣶⡟⣿⣿⣿⡿⠿⠿⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣇
⠿⠛⠛⠋⠉⢁⣁⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿
⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⢛⡻⠍⠉⢀⣴⣿⣿⣿⡿⠿⠛⢛⣩⣽⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿
⡛⠛⠋⠉⠉⠉⣉⡀⠄⠒⠊⠁⠀⠀⢀⣴⣿⣿⣿⡿⠋⢀⣠⣶⣿⣿⣿⡿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣆⢿
⠙⠻⠿⠿⠿⠋⠁⠈⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⣋⣤⣾⣿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡸
⠀⣀⣀⣀⣐⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⢟⣵⡿⠟⢛⡽⠋⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠉⢀⡠⢖⣩⣴⣾⣿⡿⠿⢛⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢰⠀⣿⣿⢸⣿
⠿⠿⠉⠛⠛⠛⠿⠿⠛⠛⠉⠀⠀⢀⣤⣬⣷⣾⠿⠟⠛⠉⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣛⣽⣿⣿⡿⢧⠀⠀⠀⢸⠀⠸⣿⣼⣿
⣀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⠿⠟⠛⠉⠁⠲⠤⢴⣮⣶⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣤⣾⣿⡿⠛⠁⢀⣤⣧⠀⠀⢸⡇⢆⠹⣯⣿
⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠿⠛⣋⣁⠀⠀⠀⣀⣤⣾⣿⡿⠛⠁⢀⣠⣶⣿⣿⣿⡄⠀⢸⡇⢸⣆⠹⣿
⠀⠀⠀⠀⠤⣀⣀⣀⣀⣀⣀⣀⣠⣴⣶⣿⣿⠿⢿⣻⡽⠗⠞⠛⠉⠉⢀⣠⣴⣾⣿⡿⠟⢉⣠⣴⣾⣿⠿⣿⠟⠉⢹⠃⠀⢸⡇⢸⣿⡄⢻
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠉⠀⠀⢠⡿⠛⠁⠀⠀⣠⣴⣺⣽⣿⡿⠛⣩⣴⡾⠿⠛⠋⠉⠀⡴⠁⠀⢀⠏⠀⢠⡾⣡⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⣠⣴⡿⠿⠿⠛⢋⡥⢖⣋⣉⣀⣀⠀⠀⠀⢀⡾⠁⠀⢀⣊⣠⠴⣫⣴⣿⣿⡿⣣⣾
⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠩⢭⣉⣛⣛⣓⣚⣛⣋⣭⣵⣾⣿⣿⣿⣿⣿⣿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣶⣾⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢋⡥⠖⢒⡪⠖⠀⠨⣿⠩⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢋⠜⡩⠔⠚⣉⣴⠟⠰⠋⠉⠐⢄⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⡜⢀⢐⢋⠋⢁⣀⣀⠀⠀⠀⠈⢳⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡟⠀⡃⠀⣼⠟⠉⠉⠙⠻⠿⠇⣤⣼⡇⢡⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣧⠞⢁⡼⠁⣤⡾⢿⣿⡷⢄⠀⣴⣾⣇⢹⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠾⣡⠲⢿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⢶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣾⣿⣦⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡍⢿⣝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠠⢀⠒⢠⣿⣆⠁⢸⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠲⢴⡀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠋⠉⠉⠉⠛⠒⠦⣄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠛⠶⡏⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠙⣆⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠘⡆⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⡸⠳⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣠⡄⠀⠀⠀⠀⠀⠑⢄⠀⠀⣠⠏⠔
⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢸⡇⠉⠁
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠳⢤⠤⠞⣇⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠒⠦⣄⡀
⠀⠀⠀⠀⠀⢀⠛⠂⠤⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠴⠃⠤⢄⠀⠀⠁⠒⢤⡀⠙⢦⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢀⡀⠀⠈⢢⣀⣹⣆⣀⣀⠈⡄⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢠⣀⠞⠀⠀⠀⠀⡠⠟⢺⣭⣵⣤⣼⣷⣤⣤⣩⡐⠒⠠⣈⠑⢤⡀
⠀⠀⠀⠀⠀⡀⠳⣄⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡠⢊⣤⠞⠋⠉⢣⠀⠀⠘⣧⠈⠙⠛⠶⣤⡀⠑⢄⠙⢆
⠀⠀⠀⠀⠀⠰⡀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠜⣰⠟⠁⠀⠀⠀⠀⢃⠀⠀⢸⡆⠀⡆⠀⠈⠻⣦⡀⠳⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠉⠑⠒⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠨⣸⠃⠀⠀⠀⠀⠀⠀⠈⡀⠀⢸⡇⠀⠇⠀⠀⠀⠈⢷⡄⠱⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡟⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡇⢸⠀⠀⠀⠀⠀⠈⢿⡄⢱⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⡿⠀⡞⠀⠀⠀⠀⠀⠀⠘⣷⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⢻⡄⠀⠀⠀⠀⠀⡠⠁⣠⡾⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⢿⡄⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠻⣦⣀⡀⣀⣁⣠⡾⠋⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣉⣙⣛⣋⣉⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⢁⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢧⠀⠀⠀⣿⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡃
⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⣿⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣷⠟⠤⠂
⠀⠀⠀⠀⠀⠀⠙⢦⣴⣌⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⡇⣄⣾⠋⣠⠖
⠀⠀⠀⠀⠀⠈⠛⠳⣿⣿⣗⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣳⣿⡄⢰⢡⣿⠃⠰⠃
⠀⠀⠀⠀⠀⢄⡲⣄⣿⣿⣿⣮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣾⣿⣧⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣿⣿⣿⣿⡷⠵⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠠⣸⣿⣿⣿⣿⠓⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣽⣬⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣿⣿⣿⣿⡀⠸⡄⠀⠀⠀⠀⠀⠀⣿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠐⢤⡽⣿⣿⣿⣿⣷⡆⣿⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣾⣿⣿⣿⢿⣽⣰⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣇⣾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣟⣟⣹⣻⣮⣛⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣭⣭⣽⣯⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⢻⡄⠀⠀⠀⢀⣠⠴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣾⠉⢿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⠀⡧⠀⢀⡴⠋⢁⡶⠛⠂⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠶⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⣸⠇⣠⠞⠀⠀⢸⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠤⣤⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢃⡴⠃⠀⠀⠀⠈⠛⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢉⣓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⢚⣩⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⣉⣭⡛⣦⠀⣼⢫⠍⣳⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⢁⣼⠀⠙⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠈⠉⢳⡄⣿⠀⠀⡙⣧
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⡆⠀⣿⠘⠶⠞⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠙⠓⠖⠋⠁⢠⡟⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⢦⣀⠀⠀⠀⣴⠟⢉⣤⣍⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⢀⣠⡴⠏⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡶⠳⣄⠙⣦⠀⠐⣯⠀⣿⢀⡈⣷⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⣙⠛⠒⠛⠛⠉⠁⣀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠳⣄⣁⣠⠏⠀⣿⠀⠀⠹⢦⣈⣉⣴⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠉⠀⢀⡼⠃⠀⢀⣀⣀⣉⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠒⠚⠉⠀⠀⢀⣠⡤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣠⠤⣄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠖⠚⠉⣠⡤⠤⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡶⠓⠈⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⢤⣄⠀⠸⣆⠈⠳⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠧⣤⡴⠏⣰⠇
⠀⠀⠀⠀⠀⠀⢀⣴⠋⣁⣤⣤⣄⠈⠳⣄⠹⡆⠀⣰⠟⣛⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⠀⠈⣇⠀⠹⡆⢹⡀⣟⢸⡇⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⡀⠑⠀⠀⣠⠏⠀⠀⢻⠀⡇⢻⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠛⠁⠀⠀⠀⢸⠁⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⠳⣄⢰⢏⡴⠲⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⠇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢛⡙⢳⡄⠀⣿⠠⠶⢦⠀⣿⢸⡌⢃⣰⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠌⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⣹⠀⣿⠀⠻⠶⠶⠛⢠⠏⠀⠉⠋⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⣠⠏⠀⠀⢀⣠⠔⠃⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠘⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠠⠴⠐⠐⠚⠁⠉⠉⠉⠁⠉⠉⠁⠀⠐⠒⠀⠀⠤⠤⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⠠
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠁⠈⠐⠀⠒⠒⠐⠂⠒⠒⠋⠉⢉⠅⠤
⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢼⣿⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣺⠑⠂⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣷⣖⣢⢾⣿⠕⠿⣀⣤⢶⣷⠀⣦⢠⢰⣧⣿⠃⠓⠠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸⡟⠻⠉⠑⣐⡙⢆⠈⣼⣇⣬⣹⣿⣿⡆⣾⣉⣽⠟⣒⣶⡾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣷⡗⢖⠀⢙⡅⠈⠒⠀⢠⣷⣜⡛⣿⠄⠐⣁⣷⣄⠙⢻⣿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠐⠀⢀⣁⡘⡆⢸⣸⣿⣿⣱⡇⣆⠀⣿⢋⣰⣮⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣤⠖⣶⠒⣬⣁⡃⠈⠿⢿⡏⢻⡁⡿⠐⠁⠄⠉⢊⡀⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠋⠉⢵⣦⣽⣧⣁⣨⡇⠀⠀⢼⢰⡏⣠⡖⠿⢉⡿⢻⠿⢷⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣈⡀⠀⠀⠈⢛⣻⣿⣄⣿⡄⠀⠠⢿⣷⣿⣛⣶⣯⣾⡿⠗⣉⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠉⠁⠀⠀⠈⢑⣿⣟⣿⡇⠀⠀⣿⣿⣿⣿⠿⣿⣷⡿⠲⢞⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠴⠲⠀⠀⠀⠈⠉⠀⠉⠡⢼⣶⡽⠿⠿⣿⣷⣈⠋⠷⢿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣂⠀⠀⠀⠀⠀⠀⣘⠋⠀⠀⠀⠚⠋⢙⣿⣾⣿⣿⣓⣾⣟⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠩⣤⡀⠀⠀⠀⠉⠙⠛⠂⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣽⣿⣿⡯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠿⢧⣄⣀⡀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣬⣿⣿⣿⣿⣯⣽⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣾⣿⢫⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠖⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠀⠨⡏⣰⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⡻⠋⣼⣿⣿⣟⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠁⠀⢼⣿⡿⣿⣿⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢤⣿⣿⣬⣿⣿⡟⠛⣟⣿⣷⣿⣿⠟⠏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠛⠋⠛⢹⡴⠾⠿⠛⠋⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠛⣿⡒⣿⣷⡴⣘⡋⡐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⣴⣶⠊⠛⠑⡀⠀⠈⠩⠁⠑⡾⠛⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠞⠛⠀⠀⢁⣽⠖⠠⢄⠀⠺⠃⠀⠀⠀⠰⠦⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢶⡟⠋⠀⠉⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢥⡈⠀⠀⠀⠘⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣒⡀⣀⠤⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣑⡙⠻⠉⠀⣐⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠞⠁⠌⠀⠀⠉⠉⠉⠉⠉⠤⡠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠤⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⣿⠛⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⡦⠀⠀⠠⠀⠄⠀⢠⡞⠃⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣄⠁⢀⡠⠨⠀⠀⢰⣄⠠⠆⠐⠢⣹⡇⢋⣠⠄
⠀⠀⠀⠀⠁⠴⣤⣦⠄⠀⠀⢿⣯⣀⠸⡀⠸⠀⠀⠛⠏⠰⡀⢀⢀⣬⠴⠛⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠀⠛⠶⢤⣤⣤⣞⢷⣶⠂⠀⠋⠙⡷⢒⠄⠃⠄
⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢰⢄⡄⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢄⠀⠀⠟⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠈⠀⠂⢁⠖⠠⠠⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡳⠀⣴⠶⢿⡗⠀⠀⠀⠀⡀⠠⡶⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⢀⡀⠄⠸⠃⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡅⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠖⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠈⠋⠒⢿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⢀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠼⠖⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠞

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⡩⠝⠛⣩⠽⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⡴⠯⠶⠒⢋⣡⡴⠖⠒⠲⠦⣌⢷⠤⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⣠⣶⠿⠋⠀⣀⣤⡶⠛⢁⡴⠖⢒⣒⣒⣨⣿⠢⣄⡉⠑⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⣠⡾⠋⣀⡴⠞⠋⠉⠀⣀⡴⣯⠖⠋⣡⠞⠉⠀⠀⠉⠲⣍⡓⢄⣻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⣼⡟⢀⡴⠋⠀⢀⡤⢖⣿⡷⠟⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠙⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⢸⠀⡾⠁⢀⡴⢋⡴⠁⡞⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡘⡜⡟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⣾⡼⠃⠀⡞⢀⡞⠀⣸⣧⣶⠿⠛⠛⠛⠿⢿⣿⣶⣶⣶⡆⠀⠀⠀⣿⢣⢱⠰⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⣿⠁⠀⢀⣧⠞⢀⢔⣿⠏⠁⠀⠀⢀⣀⡀⠀⠈⠙⠛⠋⠐⠿⣿⠷⣿⡏⡎⡇⡇⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⡿⢀⡴⡿⠁⠀⣠⠟⠁⠀⢀⣤⣾⣿⣿⣿⢿⣶⣄⠀⠀⢠⣄⣤⡶⣿⡇⣧⢳⣱⠀⣿⣿⣿⣿⣿
⣿⣿⣿⠏⢠⣧⡿⠊⠀⣀⣴⡏⠀⠀⢺⣿⣟⠁⠸⡿⠿⠶⠿⠉⠓⠂⠘⠏⠿⠷⠿⣏⡿⠘⣟⠀⣿⣿⣿⣿⣿
⣿⣿⡟⣠⢿⠏⣠⠴⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢷⠀⡿⣴⣿⣿⣿⣿⣿
⣿⣿⡾⠁⣾⣼⣿⡀⠠⡈⢷⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⡧⣷⢸⢿⣿⣿⣿⣿
⣿⣿⢁⣼⣿⣿⣿⣿⣆⠣⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⣾⠁⢸⡇⠹⣾⣧⠻⣿⣿⣿
⣿⣯⣼⣿⣿⣿⣿⣿⣿⣆⣀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣂⡐⢢⣾⣿⠀⢸⡇⠀⣿⣟⢷⡹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠥⣈⠀⠉⢁⣾⣿⣿⣧⠀⡇⠀⡿⢸⠈⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣼⣿⣿⣿⣇⢻⠀⡇⢸⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⢀⣰⠿⠛⠉⠁⠉⠉⠓⢧⣺⠀⡼⣹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⢣⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡞⠀⠀⠘⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⡞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠊⠀⠀⠀⢰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠈
⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠑
⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢄
⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠀⢴⠇⠀⡸
⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢰⣿⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠁
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠙⠲⠤⠄⠒⠂⠀⣀⣀⣀⣀⣀⣠⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡌⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠔⠀⣠⣴⠶⢶⣤⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⡾⠋⠀⠀⠀⠈⢻⡶⠛⠉⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡟⢀⠊⠁⠈⠢⠀⠀⣿⠐⠒⠒⠲⢤⡀⠀⠀⠈⠙⢷⣄⠐⠄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢦⣤⡀⠀⠀⠻⣦⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠹⣧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⣿⠋⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣿
⠐⢤⣒⠒⠶⠦⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠖⠛⠋⠉⠀⠀⣸⡇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⡠⠔⠊
⠀⠀⠉⠓⢦⣀⠀⠠⠤⣈⣉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠓⢦⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣀⣠⡟⠀⢀⣀⣒⣒⣛⣋⣁⣀
⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠉⠉⠓⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠻⣆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠉⠐⠦⢤⣀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠿⠶⣤⣀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡐⢄⡀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀⠀⠀⠈⠙⠲⣤⡀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠙⢢⣀⠤⠴⠖⠶⠽⢷⣾⡄⠀⠀⣠⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⠋⠁⠀⠀⣀⡀⣤⣀⡈⠙⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠓⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⣠⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡇⡌⠳⣌⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣿⠀⠸⡄⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⢀⣀⠤⠋⠀⣼⠃⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠈⡁⠀⠀⠀⠀⣸⠷⣄⠈⠓⠒⠒⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢖⡂⣠⠀⠀⠑⠤⣤⣤⠶⠋⠀⠈⠳⠦⢤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⢷⣤⡀⣀⠀⠀⠀⢀⣀⡀⠀⣀⣤⡞⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠻⣷⡖⠙⡷⠿⠀⠚⡟⢼⣿⣿⣿⡅⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠇⠉⠃⠘⠈⢀⠐⠀⣴⠌⠋⠉⠛⡇⢀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣋⣁⡈⠄⢀⡀⡟⡇⢸⣇⡀⣔⡀⠀⢰⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⣉⡄⠀⡜⢷⣼⡧⠆⠹⠇⡙⠁⠀⢀⠁⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣋⡹⣽⣧⣜⡧⡸⣿⣧⡶⣴⣾⣥⣤⣚⣧⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢡⡿⣿⣿⣷⠗⣿⠿⢿⣿⣿⢷⣤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⠷⢺⣿⣇⡁⣰⣟⠀⠀⠚⠃⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠰⣶⣿⣷⡎⡀⢀⠄⠀⠠⡄⠀⠀⠀⠀⠀⢀⠔⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⢻⡿⣿⣿⣷⣭⠘⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⠰⠃⠒⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠡⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡼⠿⠟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠗⣀⡤⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠶⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠈⡉⠀⠀⠐⠒⠀⠀⠀⢀⠀⠈⢀⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⣤⠶⠒⠀⠀⢹⢀⣴⡆⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣽⡄⠀⠀⠀⠸⣿⣿⠀⣀⣤⣴⡾⠀⠀⠀⠀⢀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠄⠀⠀⣿⣿⠂⠈⠈⣺⡧⢠⡄⠰⠀⠸⠋⣉⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⣴⣯⡇⠀⢀⠀⠂⠘⢿⠀⠀⠀⢿⡅⠈⠁⠀⢀⡤⠐⠀⡀⠀⠀⠠⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⣣⠘⡇⠀⠉⠀⠀⠀⠸⠆⠀⠀⠀⡿⢻⠺⣱⢿⡿⠁⢰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠊⣸⣧⣾⣯⣗⡁⢸⣤⡾⡅⠀⠒⠢⣼⡅⠀⡸⠋⠮⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠋⠮⠡⠿⠛⢟⡣⠀⠀⠘⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣉⣁⠂⢀⣠⡤⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⡧⢸⠁⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠓⠚⢁⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢁⡎⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣊⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠃⡘⠠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠁⠀⠉⠉⢹⡆⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢠⠄⠀⠀⠀⠀⠀⠀⢼⠁⠀⠲⠤⠀⠀⠀⢸⠷⠛⠛⠛⣦⠀⠈⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠙⡆⠀⠀⢀⡖⠻⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠸⠶⣄
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠓⠋⢁⠆⠀⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠾⢤⡀⠂⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠘⣆⣰⢠⠀⠀⠀⠀⠀⠀⡰⠋⢀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⢀⣴⣾⠶⠿⠓⠒⠦⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠉⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⢷⡄⠀⠀⠀⣸
⠀⠀⠀⠤⠤⠒⠚⠉⣀⠤⠒⠀⠀⠀⠈⠉⠑⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠤⠐⠁⠀⠓⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⢀⡖⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠐⠋⠒⠒⠶⠦⠤⠶⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣀⠈⠀⣠⡟
⠀⠀⠀⠀⠀⡠⠔⠊⠁⢀⣀⠠⢤⣀⠀⠀⠀⠀⠒⠢⢄⣠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠺⡝⢠⣄⠈⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡙⢦⠀⠀⠀⠀⠀⠈⠑⠄⠀⠐⠶⠖⠂⠀⠀⠀⠀⠀⢹⡆⢸⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠀⢹⠄⠃⠀⠀⠀⢸⡋⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣘⠒⠊⠀⠀⠀⠀⠀⠀⠻⣈⡠⢆⠀⠀⣶⡄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠹⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣆⣤⣤⣀⣠⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣝⣻⣿⣿⣿⣿⡟⣫⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠤⠝⠀⠀⠉⠉⠘⠟⣹⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣧⣄⠔⡀⣀⠀⠀⢀⣀⣠⡤⠚⣡⣤⣶⠀⠀⠀⠀⠀⠈⠁⠣⣀⣀⡴⠈
⠀⠀⠀⠀⠀⠐⠛⠿⣿⣿⡿⠿⢃⣀⣴⠟⢋⣵⡾⠉⣹⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢀⠤⠴⠶⣖⣦⣤⣍⣙⣙⣩⣿⠃⣴⢛⠻⢷⣤⠄⠂⠀⠀⠀⢀⣶⣦⣀⣀⣤
⠀⠐⠁⠤⠒⠉⠉⠀⠀⠈⠉⣨⡼⠃⣸⣇⠈⡃⠘⣡⣀⡖⠀⠀⠀⠺⠹⣿⢿⣭
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣁⣡⣤⣶⣿⣿⡆⣴⡄⢀⠭⢷⣤⠀⠀⠀⠀⠀⢀⡽⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣻⢟⣿⣿⣿⣿⣷⣤⣤⣿⣖⠸⡿⠂⠀⠀⠀⣠⣼⣄⣾⣿⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠴⠃⣸⣿⣿⣿⠋⠀⠙⣿⣿⣯⡄⣀⠀⠀⢀⠀⠿⠉⣽⣿⣿⡇⣀⣀⣀
⠀⠀⠀⠘⠛⠛⣉⡄⠄⣊⡴⠋⠛⠁⠋⠀⠀⠈⣻⣿⣿⣾⣵⢣⡆⠾⠁⣠⣶⣿⣿⡦⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠈⠉⣉⡤⠴⠶⠤⢀⠀⠀⠀⠀⠈⢿⣿⣧⣿⠖⣿⣿⣶⢹⡟⠏⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⢿⣿⣿⣿⢻⡀⠁⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⣾⢿⣿⣿⣿⣷⣶⣦⣠⡄⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠃⠀⠈⠛⢿⣿⣿⣿⣿⣷⡆⢠⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠳⣄⠀⢤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⣸⠟⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⡀⣀⣀⡀⠀⣀⣀⠀⠀⡀⠀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⣀
⣿⣟⡋⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⣒⣢⣦⣤⣤⣠⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠴⢋⣡⣴⣾⡿⠟⣻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣁⠔⣡⣴⣿⠟⠋⢁⣴⣿⠇⣀⡶⣂⣁⣀⡀⠀⠀⠀⢀⡀⢸⣿⣿⣆⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⡪⢊⣼⣿⠟⠁⢀⣼⠟⠀⣿⣿⣿⡧⡟⣝⣷⣽⡅⠀⠀⡤⢀⣾⣿⣿⣊⡇⠀⣼⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠁⡾⣽⠏⠀⠀⣾⡏⠀⠀⢻⠈⠻⡂⢻⣿⡿⠉⠻⣧⠔⣡⣾⣿⡗⣍⠍⠁⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡈⣸⢟⡟⠀⠀⢰⣿⡇⠀⠀⠀⢃⡀⣄⡘⠿⠷⢆⣨⣶⣿⣿⡿⠟⠉⢀⣤⣾⣿⣿⣽⣿⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡄⣿⣡⡇⠀⠀⠈⣿⣇⠀⠀⠀⠀⠙⢼⣿⣷⣶⣤⣄⣄⣄⣠⣤⣴⣾⣿⣿⣿⣿⢷⣿⡟⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⡅⢿⡿⢷⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⡥⠟⢋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣷⢸⣏⠐⠂⠀⠀⠀⢻⣿⣿⠀⠀⡀⢄⣠⣄⠀⠙⠻⢿⣿⣿⣿⠟⠿⠓⠉⣁⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣿⣏⢿⣇⠀⣄⠀⠀⠈⢟⢻⣷⣄⣱⣾⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣶⣶⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣽⡜⣿⣗⡀⢠⡀⠀⠈⠃⠙⠻⣿⣏⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡼⣿⣷⡄⣿⣦⡀⠀⠘⣦⣈⠻⢷⣯⣎⣙⡛⡿⠻⣛⣋⣭⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣤⡛⢿⣷⣤⣈⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣶⣯⣽⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠤⣄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⡸⠀⢸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠠⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁⢀⠆⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡠⠐⠘⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⠔⠁⠀⠆⠇⠀⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⣊⠥⠖⠉⠀⠀⠀⡠⠖⣀⠥⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠬⠉⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⡞⡰⢉⡤⠒⠒⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠆⢀⡠⢊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⣀⠀⠀⠀⠠⡇⠁⡎⡠⠊⠑⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢎⠂⡀⠤⠠⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢁⡤⠤⣀⠀⠀⠨⡢⡀⠀⡇⢰⠀⢣⡘⠒⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⡃⣡⠮⠒⠉⠉⠑⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠁⠀⠀⠀⠙⢆⠀⠀⢱⠀⡇⢸⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣴⠞⠁⢀⠄⠒⠒⠄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠸⢄⡠⠇⠀⠀⣸⠀⠀⣾⢰⡁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁⠀⠀⡏⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⢀⡴⠁⠀⣴⠣⡳⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣀⠴⢚⢑⠞⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠥⠒⣁⠄⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠬⠥⠤⠤⠐⠉⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⡁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠉⠀⠀⠀⠀⢀⡠⠞⠀⠀⣀⡠⠤⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⠀⠉⠁⣀⠠⠔⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⠀⠀⠑⠦⠤⠊⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠉⠑⡄⠀⠀⠀⠀⡰⣰⠀⠀⠀⠀⠀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡔⡡⠁⠀⠀⠀⢀⠜⡠⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡜⠀⡠⡪⠊⠀⠀⠀⣀⠔⡡⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣉⠒⠠⠀⠀⣀⠀⠀⠀⢀⣀⡤⣖⠵⠊⢀⡨⠊⠀⠀⠀⠠⠊⠕⡪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠀⠀⠉⠉⠓⠂⠉⠀⠒⠈⠀⠀⠀⠀⣀⣉⠤⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⠀⢀⣼⣿⣹⡄⠄⢠⣄⠀⠀⡀⣠⣸⡶⠅⠀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⣿⣿⠛⠏⢹⢯⠀⢺⣿⢘⢻⣴⣿⣁⡾⢻⣧⠤⣤⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣷⡶⠦⠀⢛⠉⠃⠎⢉⣯⣽⠿⣿⡍⢙⣻⣅⠛⢻⣧⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠉⠊⠄⠈⣀⢳⢰⡾⣿⣿⢳⣇⡄⢸⣟⣉⣮⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⠖⣶⢲⣬⣘⠀⠻⢻⡟⣻⠸⠇⢚⢀⠉⣘⡁⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⠋⠉⠱⢦⣽⣦⡤⢽⡀⠀⠸⢰⡏⣴⡚⢏⣽⢛⡟⢻⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣄⣀⠀⠀⢭⢿⣷⣼⣇⠀⠀⣻⣿⣿⣽⣿⣾⣿⣙⢠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣐⠀⠀⠀⣬⠏⠻⢯⢀⣀⣽⣿⣿⣯⡙⣿⣟⣈⣩⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣥⠀⠀⠀⠀⠀⠀⠉⡿⠋⠉⠛⣿⡷⢦⣬⣭⣿⣟⣻⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀⠀⠀⠀⠀⠰⠶⠄⠀⠀⠀⣠⣾⣿⣿⣿⣷⣿⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣿⣶⣤⣀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣹⣿⣿⣿⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⣿⣿⣿⣿⢿⣓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠂⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠡⠋⠀⣿⣿⣿⣿⢹⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⢀⣠⣿⡟⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠹⠿⠻⢻⣥⣼⣿⠿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠓⢦⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⡽⢧⣀⣀⣀⡽⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠻⣄⠀⠉⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢍⣹⠆⠀⢀⣠⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⣀⡙⠧⠼⠗⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠈⠳⠤⠟⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⢤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣤⠴⠤⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⣖⣒⠒⠲⠤⢬⣳⡌⠳⣄⡀⠀⠀⣠⠏
⠀⠀⠈⠓⠒⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠲⠀⠀⠈⠁⠀⠈⠳⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒⠒⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⡉⣄
⠀⠀⠀⢶⣉⣩⠇⠀⠀⠈⠓⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢸⡆⠀⠿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⡼⠢⣄⠀⠀⠀⣠⠛⠲⢼⡀⠀⠀⢀⣀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉⠀⠀⠈⠉⠛⠋⠁⠀⠀⢸⠇⠀⠀⢸⠈⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠓⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠋⠉⠉⠉⠉⠉⠁