Taxonomy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠹⠁⠀⠀⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢀⢶⣂⡀⠀⠈⡀⠠⠾⠀⢸
⠀⠀⠀⠀⠀⡄⠀⠙⢂⠀⠠⠟⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠂⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠃⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸
⠀⢴⣾⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⡖⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠺⠿⣾⢾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠇⠀⠀⠀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⡁⠀⠀⠀⠁⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡀⠰⣟⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣿⣷⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡝⣿⡝⢿⣿⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣟⡐⢆⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⢺
⠀⠀⠀⠠⣤⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢟⡼⠄⠁⠀⠀⠀⢀⡀⠀⢠⣄⣠⠄⠁⠀⠀⠀⢸
⠀⠀⠀⠈⢩⠰⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠁⠀⠀⠀⠁⠈⠘⢧⠀⠚⠛⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⢀⡼⣷⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠞⠣⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠀⠀⠀⠀⠀⠀⡀⢠⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⢸
⠀⠌⠀⠀⠀⡀⠀⠀⡀⠓⣆⣿⠃⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣿⠀⠀⢸
⠀⠀⠀⠀⠴⢀⠠⢢⣾⣷⡿⠃⠀⠀⠀⡄⠁⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⡿⠃⠀⠀⢸
⠀⠀⠀⠀⣹⢾⣿⣾⣿⠟⠁⠀⠀⠀⠰⠃⠀⠀⠀⠭⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣸⣿⣿⡇⠀⠀⠀⠀⠀⢀⣐⣾⠆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⣹⠛⣷⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢂⣉⢵⣿⣿⡷⠀⠀⠀⠀⠐⠾⠿⠋⠀⠀⠀⠀⣠⠀⢸
⠀⠐⠀⠀⠻⡀⣿⣷⡀⠀⠀⠀⠀⠀⣏⡀⠀⠁⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣓⢪⣿⠂⢸
⠀⠀⠀⠀⣶⠀⢿⣿⠇⠀⠀⠀⠀⠀⣷⠭⢀⡛⢹⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠤⣭⣛⣒⣲⣾⡟⠀⢸
⠀⠀⠀⠀⢈⣤⠾⠿⣆⠀⠀⠀⠀⠀⠘⢿⣯⣷⣾⣿⣿⡿⠃⠀⠀⠀⡀⠀⠸⣿⣳⣯⣿⣽⡟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣸⡿⠿⢿⠿⠋⠀⠀⠀⢸
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂⠐⠂⠀⠀⠀⠀⠄⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠋⠁⢀⡀⣀⣀⣀⣀⣀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢠⡦⣤⢠⡖⠀⣴⠂⠀⢀⡴⢤⡠⡶⠂⠀⣤⠀⢠⣤⡖⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠸⠛⠧⠸⠥⠀⠿⠄⠀⠘⠧⠼⠓⠏⠁⠀⠿⠄⠸⠺⠋⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣠⢠⣤⡄⣰⢠⡄⣤⢠
⠀⠀⠀⢴⠰⡂⢀⠆⡶⣀⠐⠀⢦⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⠁⠉⠀⠉⠀⠉⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀⣀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⠋⢁⠀⠀⠠⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆⠤⠀⠀⠿⠀⠸⠀⠀⠈⠉⠀⠋⠁⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⢀⡀⠀⠈⠉⠉⠀⢠⠄⠀⠓⢠⠐⠒⠒⠀⢸⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠩⣀⠀⠈⣀⠀⡀⠀⢰⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⡅⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⡀⠀⠠⡅⠀⠀⢀⠀⠀⠀⣸⠃
⠀⠈⠁⠀⠀⠀⠁⢈⣠⠇⠀⠀⠀⡐⠀⠒⣤⠲⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠳⡀⠀⢷⠀⠀⡜⠀⠀⣰⠃⠀⣀⣀⡄⢀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⣠⡆⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠸⠆⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠦⡈⢧⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠺⠟⠀⢈⠀⢀⠀⠛⠀⠀⠀⡆⠀⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⡄⠘⠃⠀⠠⠋⠀⠀⠀⠀⠀⠔⠀⠖
⠀⠀⠀⠀⠀⠀⢴⡄⠀⠒⢄⠀⠸⠀⠈⡄⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣦⣀⡀⠀⠀⠘⠂⢘⠀⠘⠀⠘⠀⠀⠤⠠⠄⠀⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠈⡄⠀⡇⢀⠇⠀⠀⡰⠁⠀⠁⠈⠛⠉⠀⠀⠁⠉⠉⠈⠀⠀⠲⢯⡱⣤⡟⠁⠀⠀⠀⠀⠀⢰⠀⠤⠄⠠⢀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⣽⠁⢡⠀⠀⡇⠀⠁⡜⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⠛⠀⢠⠀⠀⠀⠀⠀⡇⢀⡀⣀⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⠀⠈⢧⠀⢣⠀⣰⡡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢦⡀⠀⠀⠀⡇⠀⠁⠁⢀⠀⠀⡔
⠠⠂⠠⠆⠠⠀⠆⠀⠀⠂⠑⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣀⣀⢀⡀⡈⡀⠈⠉⠈⠀⠁⠁⠈⠉⠉⠢⡀⠀⡇⠀⠀⠀⢈⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠙⠀⠓⠛⠙⠋⠒⠃⠃⠀⠀⠀⠀⢐⢄⣢⠀⠀⠀⠘⢦⢡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠀⠀⠀⠀⠀⠀⢿⠀⠀⠞⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡶⡠⢤⣠⠀⠀⢠⠀⢀⡄⣤⠃⠀⠀⠀⠀⠀⡀⠈⢉⣁⡀⠀⣀⠀⡀⠀⠀⢸⢀⠎⠀⣀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠞⠀⠀⢀⠄⡄⠀⠤⢄⠀⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⢄⠀⠀⠀⠀⢸⠇⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢠⠃⠀⠀⢀⢁⡴⠃⠁⠄⠤⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠻⣆⠀⠀⠘⠛⠀⡀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣈⣦⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠂⠐⠂⠀⠂⠀⠀⠀⠀⢰⡄⠀⢀⡤⠊⠀⠀⠀⠘⠃⠊⠀⠈
⠠⠤⠄⠠⠤⠤⠀⠤⠤⠀⠀⠀⠠⠀⠀⠤⠄⠀⠉⠒⢤⣷⡀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠦⣤⡤⠴⠀⢳⡀⣼⠁⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠹⠇⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡔⠀⢄⡀⠀⠀⠀⠀⠈⠉⠓⠦⢄⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠚⠁⠀⠀⠀⠂⠠⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⠀⢳⡀⠀⠀⠀⠀⠀⡰⠀⣰⠋⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢺⠃⣇⡿⢺⢢⡾⡤⣋⠈⡏⢻⠐⡿⡇⢢⢿⠓⠒⠠⠤⣀⠀⠀⠀⠀⣣⠀⠀⡇⠀⠀⠀⠀⠀⣇⡰⠁⠀⣠⠴⠒⠀⢰⣟⡙⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠈⡆⠀⠀⠀⠀⠀⢀⡴⠋⣠⠴⠋⢀⣄⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠚⠁⠀⠀⡤⠰⡆⠠⡶
⠀⠀⠀⠀⠀⠀⣰⡀⣴⢤⡴⢂⣤⡄⣤⢠⡦⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠈⠀⠀⠉⠉⠈⠁⠀⠀⠀⠈⠁⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⠶⡆⣂⡆⡇⢖⣲⠄⠓⢰⣆
⠀⠀⠀⠀⠀⠸⠧⠼⠷⠮⠄⠹⠸⠟⠸⠟⠇⠼⠧


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠚⠂⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠄⠈⠉⠁⠀⠀⠀⠀⠠⠄⠌⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠐⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⠄⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠤⡀⠠⡀⠠
⠀⠀⠀⠀⡉⢁⠈⠉⠁⠀⠀⠀⠀⢠⡤⣤⣤⠤⠀⠀⠈⠑⠒⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠋⠀⠀⠀⢀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠒⠀⠀⢐⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⡀⣠⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠈⠁⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠢⠤⠴⠤⠄⠀⠀⠀⠦⠤⠤⠖⢡⡀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠚⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⢸⠀⠀⠐⠒⢒⠀⡀⠈⠠⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠤⠤⠈⠤⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠀⣄⠀⠀⠈
⠀⠀⠀⠀⡰⠁⠀⠱⠄⠄⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⣀⣀⣀⣠⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⡀⠀⠀⠀⡿
⠀⠀⠀⢸⠀⠀⢒⠒⠂⠂⠀⠀⠂⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠁⡄⠀⠠⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡀⠈⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠠⠄⠀⣿⠀⠀⠀⠀⠀⠘⡀⢀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠉⠀⠠⠄⢹⠀⠀⠀⠀⠀⠀⠑⢈⡄⢠⠀⠀⠀⠀⠿
⠀⠀⠀⢰⠀⠀⠐⠀⠄⠀⠀⢄⢡⣀⣀⣄⣀⠀⠀⠀⠄⠀⠰⡁⢠⣀⣀⢀⡀⠀⠀⠄⢀⠘⠀⠀⠤⠤⠤⠀⠀⠈⠁⠀⠐⠂⠈⠀⡐
⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⢸⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠁⠀⠀⠀⠀⠀⠀⣰⠀⠀⡆⠀⠀⣿
⠀⠀⠀⠘⡀⠈⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⢀⠸⡀⠀⠐⠀⠀⠀⢀⠆⠀⣼⠀⠀⠀⢄⠀⣿
⠀⠀⠀⢀⡃⠀⡀⢰⠀⢠⡄⠈⣆⠀⣀⠀⠀⠀⠀⠀⠘⡀⢐⡀⠀⠀⠀⠀⡄⢠⡆⢸⠀⡃⠀⠀⡀⠀⠀⡘⠀⢠⡏⠀⢀⠔⠀⣰⠁
⠀⠀⠀⠈⢹⡆⠁⠈⠣⠈⠳⣄⠘⣦⡈⢸⠄⠀⠀⠁⠀⡷⠈⢰⠀⠤⢴⠌⠀⠘⠂⠛⠀⡟⠂⠈⠇⠀⠘⡇⠀⣿⡀⠀⢙⠀⠈⣿
⠀⠀⠀⠀⠈⣅⠀⠣⡀⠀⠠⣌⠳⣌⢳⣘⢃⠀⠀⠀⠀⣄⠀⢹⠂⠀⠀⢸⢠⠟⠀⡧⢼⠀⠀⠀⠁⠀⠀⡿⣄⡿⠀⣀⡦⠀⠀⡿
⠀⠀⠀⠀⠀⠘⢦⡀⠈⢦⡀⠈⠱⢌⠢⣿⠀⠀⠀⠀⠀⢉⡀⠀⢢⡀⠳⢸⠈⠀⢸⠓⡄⠀⠀⠀⠈⠀⠀⠀⠸⣧⡴⠛⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⡄⢷⣄⠈⠙⢄⠀⠀⢱⡗⠀⠀⠀⢠⣴⡄⠰⣀⠀⢹⡌⠻⠙⡇⠀⣰⠁⠀⠀⠀⠀⠁⡀⠀⠀⣿⠁⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⣹⡄⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⢆⡈⣧⠠⣘⣠⠖⠁⠀⠀⠀⠀⠀⠀⠶⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠶⣿⠀⠀⠀⠀⠀⠀⠀⠁⢀⡉⣿⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣿⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉⠁⢸⡧⠀⠀⠀⠈⠁⠉⠉⠈⢉⣿⠏⠈⠁⠀⠈⠁⠀⠄⠀⠉⠘⠉⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣄⠀⠀⠀⠀⠐⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⣀⠀⠀⠸⣿⡄⢀⣀⣀⠀⣐⣢⣶⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣦⣀⣿⣿⠿⠛⠛⠛⠛⠛⠉
⠀⠀⢘⡃⢀⡂⠀⠐⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣷⠈⠓⠐⠂⠀⠀⠀⠐⠂⠐⠀⠒⠂⠐⠀⠀⠖⠒⠐⠒
⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣹⣿⣿⣷⣄⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⠀⢠⣤⣄⠀⠀⣄⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⢀⠀⢿⠁⠈⠙⠁⠀⠀⠛⣀⠀⠿⠋⠂⢰⡆⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⣠⡆⠈⠑⠁⠀⠰⡷⠀⠀⠀⠈⠻⠟
⠀⠀⠀⢰⠀⠘⠀⢠⣿⣷⠀⠀⣼⣿⠀⠀⡄⠀⢸⡇⠀⠛⠀⠀⠀⠀⡄⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢲⡆⢸⣿⣿⠀⠀⣿⣿⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠉⠓⠀⠀⠻⠿⠂
⠀⠀⠀⠀⠀⠉⠁⠈⣿⡏⠀⣴⣿⡿⠀⠸⠏⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠺⠋⠁⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢤⡄⡤⠀⠠⣀⣧⣼⠀⠀⠠⣤⡄⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠋⠀⠀⠀⠈⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⠀⠀⠀⠀⣀⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⣀⢀⡀⠀⢰⣾⣷⣿⡷⡗⠀⢀⣤⡄⠀⢸⡰⠀⠀⠀⠀⠀⠹⢿⡷⠧⠀⠀⣴⡗⢶⣤⠀⠈⠁⠀⠉
⠀⠀⠀⠀⢿⣿⠇⠀⢘⢟⣿⣯⡾⠃⠀⠀⠈⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠃⠙⠁⠀⠈⠉⠁⠈⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠰⣷⠄⠀⠀⣤⡄⠀⠀⠀⠀⠉⠃⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⡀⠀⠿⠃⠀⠸⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠲⣤⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠁⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⢀⡀⠀⠰⡇⢡⣷⠀⠐⠁⠀⠀⠀⠛⠃⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠠⢀⡙⠀⢰⠛⡟⡏⠀⢤⡤⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠘⠀⠀⠈⠘⠓⠁⠀⠘⢻⠏⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠄⠀⠀⢠⡀⠀⣶⠆⠰⣦⡄⠀⣀⡀⠀⢠
⠀⠀⠀⠠⢤⡃⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠀⠀⠘⠻⠃⠈⠀⠀⠀⠁⠀⣿⠀⠀⣴⠀⠘⣩⠃⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠘⠠⢤⠄⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⡶⠴⠀⠀⠈⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣤⡄⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠓⠃⠀⢀⣀⠀⡀⠀⢰⣶⣶⠀⠀⠀⠀⠐⢿⠟
⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠙⠀⠀⣄⣄⠀⣀⣠⣤⡀
⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢾⡂⠀⠠⠄⢤⠀⠀⠀⠀⠀⠀⠽⠇⠀⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠂⠀⠀⠀⠠⡄⠀⠀⠀⢲⣷⡄⠀⠀⠀⠀⠏⠛
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠈⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠈⠻⠄⢹⣋⠀⣷⡀⠀⣤⣤⣄
⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⣤⡀⠀⠀⠀⢆⡤⠀⠀⣤⣀⠄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣤⠀⠀⠆⠀⠀⠿
⠀⠀⠀⢴⣾⣧⡀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠋⠀⠀⠀⠂⠀⠀⠀⠓⠢⠶⠆
⠀⠀⠀⠀⠙⠻⡆⠀⠐⠀⠀⠀⣄⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣤⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠦⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠄⠀⠠⠶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣜⠀⣠⣄⣈⠁
⠀⠴⠶⢶⣶⣶⣤⣤⣤⣤⣤⣍⣮⡧⠀⠀⠀⠀⠀⠠⠀⠙⠀⢤⡀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠿⣸⣿⡷⢢⡀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⡀⠀⢀⣤⠠⣤⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣟⠻⣦⣁⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⢠⣽⠀⠶⠂⠀⠐⠀⠤⠤⠀⠈⢧⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠓⠐⠾⣧⣶⣤⣤⣿⠇⠀⠘⠛⠶⠶⠶⠶⠶⠶⠾⣿⣷⣶⣄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⠗⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⢀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⢤⣤⣴⣧⠀⠘⠓⠀⠀⠈⠉⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠉⠀⠁⠀⠀⢀⡀⠈⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⡄⢻⡟⢠⣿⣯⡿⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡾⣿⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⡎⢷⣾⠿⠋⠁⠀⠀⠀⠀⠛⠛⠒⢦⠷⠶⠦⢶⡂⣤⠀⡀⠀⠀⠀⢠⡤⢿⣶⣿⣿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⢶⡄⢠⣧⣄⠀⢿⡾⠁⢼⣿⡷⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠶⠚⠉⠉⠐⠛⠁⠐⠺⠛⢷⡈
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢀⣠⠤⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠀⢀⠀⠀⠀⢠⣤⠾⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣄⠁⠖⠂⠀⠀⠀⠀⠘⠂⠤⢄⣈⡀⠀⠀⠀⠀⠸⣌⠑⠂⣄⢀⡀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠈
⠀⢶⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⡀⠙⡆⠀⠀⠈⠁⠀⠈⠃⠈⢦⡀
⠀⠶⣿⣿⣷⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃⠀⠀⠀⠀⠀⡄⠀⠀⠡⡀
⠀⠐⣿⣿⣿⣿⣿⣿⣾⣿⣷⣦⣤⡄⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣹⣿⣿⣄
⠀⢈⣿⣿⣿⣿⣿⣿⣦⣿⣿⣾⣿⣿⣿⣟⠁
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⢌⡛⢻⡶⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠁⠀⠉⠛⠿⠿⣿⣿⣿⣦⣍⠉⠲⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠘⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦⡀⠀⠀⠀⠀⠀⠀⠨⠈⠀⠐⣦⣤⣠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⠿⣶⠻⡿⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠁⠀⠁⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣹⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣏⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⡼⣯⣄⡀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣏⣣⡀⠐⠉⠀⠰⠄⠀⣮⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢁⢣⡀⠀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠊⠃⠀⠀⠀⢀⡀⠀⠠⠂⠒⠀⠐⢉⡿⠁⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠰⠠⠾⠃⠀⠀⠲⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢠⣄⠀⠀⠀⠀⠀⣠⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⣶⡇⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⣿⣿⣁
⠀⠀⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣼⢿⣤⡏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⣷⣿⡅⣀⠠⣤⡄⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢸⣿⢿⣿⡇⠀⠀⠀⠀⠀⠤⠄⠤⠀⠀⠀⠀⠐
⠀⠀⢛⡋⢴⡿⠷⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣹⣟⣪⣿⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢐⡏
⠀⠀⠀⠳⠞⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠰⣏⣼⣿⡗⠂⠀⠀⠀⠠⠒⠁⠀⠀⢀⡞⠁
⠀⠀⠐⠠⣾⣶⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣱⣮⠇⣞⣿⣿⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡴⠂⠙⠛⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⡦⣿⣿⠾⡝⣿⣛⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢵⣳⢚⣿⠾⠘⠿⣿⣷⡀⡀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⢯⣿⡃⠰⢮⡈⠋⠘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣿⠙⢷⡂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠞⠉⠉⠁
⠀⠀⠀⠀⠀⢠⣿⡟⢀⡿⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢉⡾⠏⠁⠀⠀⠀⠀⠀⡀⠠⢶⣟⠀⠀⣸⣷⣶⠛⠁⡇⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⢰⡆⡸⠁⣠⣤⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠈⠉⠀⠈⠀⠀⠁⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢀⣰⣆⣷⠉⠐⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣼⠇⣌⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄
⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡄⣰⠀⣤⣀⢲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢴⣦⡄
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⠈⠳⣦⢍⡰⣽⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⣿⡇
⠀⠊⠀⠀⠀⠀⠀⠒⠠⠀⠈⠈⡲⢄⡆⢸⣵⣮⠁⠀⠀⠀⠀⠀⠰⡈⠀⠀⠩⣶⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢴⠂⣀⣘⠛⢢⣟⣸⣝⣿⠏⠀⠀⠀⠀⠀⠀⢖⠀⠀⠀⠄⠀⣉⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠂⠈⠛⢋⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣌⣿⣻⣿⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⡴⠃⢀⠠⠀⠀⣀⠵⢶⣻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠤⣀⠀⠄⢠⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠘⣷⣻⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⣿
⠀⠀⠀⠀⠀⢠⡤⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠑⠀⢀⡀⠀⠒⢍⣻⣏⣿⣿⣧⠀⠀⠀⠀⠀⠀⠠⣄⣼⣾⣿⠟
⠀⠀⠀⠀⠀⠈⢠⣿⠉⣽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⠄⠈⡀⠦⢴⢻⣽⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠠⣀⣂⢆⣾⡧
⠀⠀⠐⠂⠀⠀⠻⢿⠀⢺⣟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢈⡀⠀⡁⡀⠀⢻⣻⣦⣿⣾⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣀⡘⠂⠴⠧⠏⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⡜⠁⢼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡎⣡⠄⡖⠛⣹⡿⢶⡟⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣤⣏⣹⣛⣛⢻⣻⣿⠏
⠀⠀⠀⠀⠀⠠⠾⣇⠀⢸⡏⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⢦⡜⠷⣀⣽⣿⣿⣿⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⡶⢶⢏⣹⣏⣻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡸⠷⠾⠛⠙⡮⡇⣀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠇⠺⠿⠿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠘⣤⡀⠀⢼⣿⣷⣟⣶⣿⣮⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢣⣼⣿⠉⠉⠻⠛⠉

⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⠂⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠰⠁
⠀⠀⠀⠀⣰⠿⣷⡄⠀⠀⠀⠀⠀⠀⠈⣗⠫⣟⣿⠀⠀⠀⠀⠀⠀⠀⢺⣥⣼⣿⡆⠀⠀⠀⠀⠀⠀⠸⠿⢿⣽
⠀⠀⠀⠀⣻⢶⣿⡁⠀⠀⠀⠀⠀⠈⠀⡼⠛⢿⣥⠀⠇⠀⠀⠀⠀⠀⡤⠟⢻⣧⡄⠀⠀⠀⠀⠀⠀⣨⢿⣿⣯⡀
⠀⠀⠀⠸⣧⣀⣼⡷⠀⠀⠀⠀⠀⠀⠄⣷⣶⣾⣿⠄⠀⠀⠀⠀⠀⠙⣷⣠⣼⣿⡟⠁⠀⠀⠀⠀⠒⣿⣔⣺⣿⡗
⠀⠀⢠⠆⠉⠙⠛⢳⣤⠀⠀⠀⠀⠀⡾⠏⠙⡛⠟⣷⠁⠀⠀⠀⠠⡾⠉⠉⠛⣽⣿⣦⠀⠀⠀⢠⡞⠁⠉⠙⠉⣹⣦
⠀⠀⣼⠀⠀⢀⠀⢨⣿⡄⠀⠀⠁⣸⠀⢀⣶⣿⣠⣿⣄⠀⠀⠀⢈⣷⣀⣀⣶⣿⣿⣷⡀⠀⠀⢸⡇⠀⡀⢰⡀⣸⣿
⠀⠀⢿⣿⣦⣿⣾⣿⣿⠇⠀⠀⠀⠹⡷⠿⠿⠿⣿⣿⠉⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⢻⡿⡶⠷⠿⣿⣿⣿⠛⠄
⠀⠀⠈⣿⣄⣭⣽⣿⠏⠀⠀⠀⠀⣈⢷⣄⣠⣤⣾⡿⠀⠀⠀⠀⠀⠹⣯⣤⣴⣿⡿⠁⠀⠀⠀⠀⢇⣀⣀⣀⣴⣿⡟
⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀⠍⠉⠡⠀⠀⠀⠀⠀⠀⠻⠛⠿⣿⠿⠻⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡔⠀⠀⠀⠀⠀⠀⠀⢰⢤⡦⣔
⠀⠀⠀⠀⣿⣽⣿⠆⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢯⣽⡿⠀⠀⠀⠀⠀⠀⠀⠸⣟⣹⡿
⠀⠀⠀⢠⣞⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⢢⡊⢹⣷⡄⠀⠀⠀⠀⠀⠀⢠⡎⠈⣷⡀⠀⠀⠀⠀⠀⠀⣰⠋⠙⣧
⠀⠀⢀⡀⠿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣀⣺⣿⣿⣿⡃⢀⠀⠀⠀⠀⢀⡈⠙⠿⠟⣁⠀⠀⠀⠀⠀⢈⠻⠷⣾⣿⣁
⠀⠀⣸⣇⣠⣾⣿⣿⣧⠀⠀⠀⠀⠄⢸⠁⠀⠀⢹⣿⡅⡀⠀⠀⠀⢸⠁⠀⢠⡀⣿⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣧⣠⣿⣿⣿⣇⠁⠀⠀⠀⢾⣄⣠⣼⣧⣿⡷⠀⠀⠀⢸⣆⣀⡆⣤⣤⣿⣿
⠀⠀⢸⣽⣤⣶⣿⣿⡿⠀⠀⠀⠀⠊⢹⣍⠉⢙⣿⣿⠏⠂⠀⠀⠀⢸⡄⠀⠀⣠⣿⠇⠀⠀⠀⠈⣟⠛⠛⠛⢋⣿⡏
⠀⠀⠈⠿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⣈⠛⠿⠿⠿⠋⡀⠀⠀⠀⠀⠐⠿⣷⣿⣿⠟⠀⠀⠀⠀⠀⢹⢷⣶⣾⣿⢿⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⡖⢉⡏⡸⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠦⣰⣿⣤⡐⠀⠲⠀⢀⡀
⠀⠀⠀⣰⣿⡤⠛⢻⡉⠋⣼⡇⠀⣹⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡟⠿⡆⢸⣷⣿⡟⠙⢂⣀⠀⢠
⠀⢠⣿⡏⢻⠀⠀⢈⣷⣿⣽⣷⣾⡟⠇⠀⠀⣠⣶⠀⠀⠀⠀⠀⣰⠶⠏⠀⢣⣴⣇⢀⡙⠉⠿⡿⠛⠁⢰⣿⡷⠶⠄
⢀⣉⡉⢰⠦⣻⣤⠟⣹⢿⡇⠀⠟⠻⠐⠶⠉⠁⠀⡄⠀⠀⠀⣐⠁⠀⢸⠃⢸⢶⣧⣬⠳⠀⠀⠀⠀⣤⣄⡃⠀⠀⢀
⣘⣁⣹⣟⣦⣿⣿⣃⡟⠻⠃⣲⡆⢠⡄⣤⠠⢀⣾⣅⡠⠀⠀⠻⡤⠖⠁⠀⠈⣿⡿⢿⡓⠶⣤⣼⠋⠹⣌⢙⣇⠠⢤
⢯⡉⢀⡝⠉⠎⠙⠿⠣⣄⠈⣉⣡⣴⡿⢶⣴⠛⠛⣻⣗⠀⠀⢠⠓⠂⠰⡛⠈⡛⠛⢺⣏⣿⣿⣯⣴⠷⠞⠦⣍⣠⡾
⢠⣿⠉⠙⢢⠄⠀⢅⡀⠓⠿⢿⠷⣾⣀⠀⢹⠂⠙⣿⠟⠀⠀⠲⡂⣠⣴⢉⣧⡁⢰⡍⢯⠀⠀⣿⣥⣄⣀⠴⠁
⠀⢸⣦⣄⣠⡆⡴⠂⠈⢿⣿⣶⣶⣀⣏⠙⠃⠀⠂⠈⠀⠀⠀⠐⠇⣼⣿⡛⢿⡷⣄⠻⢈⢀⠀⢛⠀⠛⠉⠲⡀
⠀⠈⢉⣿⡏⣀⡘⣰⡆⠰⣄⠻⠿⠉⠉⠉⠁⠘⠂⠀⠀⠀⠀⠀⠈⢻⣿⢣⣾⡗⠛⣷⡉⡆⠄⠠⠄⠀⠀⢀⡟
⠀⠀⠀⠙⠏⣈⢹⡟⠁⠒⣡⣶⠻⣍⣴⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣯⣀⣠⡿⢄⣀⡀⠀⠈⠿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠺⠁⠠⠆⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⡄⢶⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⠄⠠
⠀⠀⠀⠀⢤⠀⠞⣸⡗⠒⢖⣘⠟⠋⢡⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠦⣤⠾⠁⢸⡇⠀⠀⠀⠀⠰⢄⡀
⠀⠀⠰⢤⣼⡗⠈⠁⠉⢰⡏⠙⢷⠃⡘⠲⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠂⠐⠄⠈⠠⠌⠁⠀⠀⠠⠔⢱⠈
⠀⢘⣁⣰⠉⢹⠃⠀⣤⡀⢻⡖⠋⢀⣠⣤⣄⠀⡠⢿⡄⠀⠀⠀⠂⠠⠈⢁⡶⠶⡄⠀⠀⠐⠀⠀⠀⠀⠂⡰⠓⢦⣠
⠰⠋⠙⣧⠰⡅⠀⠲⠀⢙⠞⣣⠤⠘⢧⣀⣬⠂⠧⢤⡇⠀⠀⠀⡀⠀⡀⢟⢀⣰⡧⠂⠁⠀⠀⠀⠀⠁⠀⠹⣦⡜⠹
⠑⠶⣾⠻⠄⠃⠀⣰⠋⣻⠒⡉⠶⠂⠀⠉⠁⠄⢴⣾⠷⠀⠀⠀⠀⢄⡀⠀⠀⣼⠁⠀⣼⠁⠄⠂⠈⠀⠱⡤⣔⠀⢾
⠀⣾⣁⣰⠿⡴⢇⡙⠛⠓⢠⠄⠐⠡⢂⠬⠁⡐⣢⣼⣦⠀⠀⠘⠀⠋⡀⣞⣇⢙⠛⠛⢡⠒⠂⠀⠀⡤⠈⠁⠀⡉⢸
⠈⢠⠉⡏⢰⡆⠉⠱⡈⠐⣋⣄⠈⠐⠀⠛⠡⠶⣽⠀⠀⠀⠀⠀⠠⠄⠤⠼⣍⠉⣿⠀⡀⠛⢠⡅⡙⢢⣤⡄⠆⠠⠋
⠀⠀⠞⠁⠁⢈⣀⡀⠁⠐⠇⢰⡟⢹⠆⠲⢤⡴⠁⠁⠀⠀⠀⠀⠠⢈⣦⡤⠌⠀⣯⠀⣸⢻⡄⣈⠀⡆⡀⠈⢐⡀⠁
⠀⠀⠀⠀⠀⣟⠀⠀⣷⢀⠀⢀⡗⣿⡄⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠙⢠⡿⣤⠰⠁⠀⠉⠛⠇⣈⠉⢣⣴⣄⠚⠁
⠀⠀⠀⠀⠀⠈⠛⠲⠏⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠔⢠⡌⠳⡄⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠀⢐⡂⠀⠀⠀⠛⢠⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣜⠁⣸⠃⢄⠚⠂⣀⣀
⠀⠀⢠⡌⠿⢃⣴⡄⢀⣤⣬⠉⠙⠊⠁⢢⠤⠀⣀⠀⠀⠀⠀⠀⢀⡀⡏⢀⠤⣄⡈⠉⢀⠀⣠⢀⡞⠁⣟⠳⣄
⠀⡜⢯⠀⣿⢹⠋⠀⠻⣽⢋⣥⠸⠃⠀⠈⠤⣷⢧⢀⠀⠀⠀⠀⠛⢀⠠⠂⠴⢌⢀⣀⣯⠘⣿⣿⣿⣶⣟⣶⡾⠃
⣸⠀⣈⡆⢹⣬⠀⠄⢀⣨⠟⢠⠀⣀⠀⣠⣦⣹⣟⠯⡄⠀⠀⠈⠣⣇⠄⠘⣄⠈⣿⣌⣿⠻⢛⠼⢿⢸⠛⠃⡶⠋⠳
⡜⣿⡟⣾⡽⠶⢶⣿⡍⢤⠒⡃⢼⣅⡻⠋⡘⠛⢏⣀⣅⠀⠀⠀⠀⢡⡀⣀⡞⠻⣋⠉⠋⣶⣈⣶⡍⠠⠖⠈⠷⠶⢞
⡟⣿⣿⡿⣇⡸⠀⣁⣺⣧⢼⡗⣡⠶⣌⢣⠘⣧⡘⢿⡿⠀⢀⣧⡀⢸⡿⠉⠛⢢⡌⠄⠋⢁⡹⠀⠸⢇⠈⣤⠙⣦⡎
⠀⠙⢿⡷⣯⡿⠀⠈⣿⢿⡏⢸⣷⠞⡟⠀⡺⠶⣧⡀⠀⠀⠈⣙⠿⢿⡤⢠⡁⠀⠀⢀⣉⠉⠁⠄⣳⡀⠀⠀⠱⠎⠞
⠀⠀⠷⣿⣿⣧⣀⠶⢿⡛⠁⢠⡎⡹⢶⡾⣇⢀⣳⡍⠀⠀⠀⠙⡀⣧⡴⠞⠀⠈⠂⢤⠍⡀⠱⣾⢿⣅⢨⠐⠒⢈
⠀⠀⠐⠶⢈⡹⠛⠶⢾⣃⠶⣸⠇⠁⠰⠴⠛⠯⠀⠀⠀⠀⠀⠀⠈⢀⣽⡶⢃⣴⣶⡄⢈⣷⠀⠈⣟⠋⠠⠀⠉
⠀⠀⠀⠀⠀⠒⠦⠀⠀⠁⡶⣠⣤⡼⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢲⠛⢻⣄⡛⠁⠁⠀⠈⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣄⣶⣰⣧⣴⢄⡀⣼⡆⠀⠀⣴⣆⠀⠀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⡀⠀⠀⠐⠶⢢⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⡿⠇⣸⡇⠀⢀⣿⣿⠀⠈⠉⠁⠈⣳⣤⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠂⠀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⣄⠀⠀⠀⠀⠀⠀⠈⠈⠋⠉⠋⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⢠⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠛⠃⠀⠂⡀⠀⠀⠀⠀⠀⠀⡄⠤⣄⣀⣧⠸⡆⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣆⠀⠀⣿⡇⠀⢸⣿⣿⠀⠈⠙⠃⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣿⠥⠀⠀⠀⠀⠀⠁⠀⠀⠉⠸⡀⢁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠿⠇⠀⠘⠛⠛⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣄⣀⣈⡱⠾⠯⠀⠀⣀⡀⠀⣀⢀⡀⢀⡀⠐⣧⠈⡄⠀⠀⠀⠀⣠⠠⠉⢉⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠿⠛⠉⠀⠀⠀⠀⠶⠋⠀⢀⡄⣸⠘⣀⡀⠀⠈⠄⠀⠀⠀⠀⠀⠠⣶⣶⡦⠀⠀⠀⠈⠉⠉⠉⠉⠈⠈⠁⠁⣠⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡐⠤⠾⠧⠄⠐⠾⣒⡙⠒⠒⣩⡥⠤⠃⢈⠀⠀⠀⠀⠀⠀⠾⠿⠗⢀⠀⢠⣤⡀⢠⣶⣶⡄⢰⣶⠀⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠒⠂⠲⠼⡇⢘⠉⠁⠀⡤⣠⠄⠀⠀⠀⠀⠀⠀⣂⠐⠒⣀⠀⠸⣿⠄⢚⣿⣿⡃⠸⡿⠀⠘⠻⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⠐⠈⠐⠦⠦⠀⠀⢓⣢⠖⠃⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠛⠃⠀⠀⠀⢹⣿⣿⡗⠀⢀⣤⣄⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠀⠀⠀⢀⣼⠛⠁⠑⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡂⠀⠀⢠⣾⣷⣆⠀⠀⠈⢀⠀⠈⠻⠋⠀⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡆⠀⠀⠺⠿⠿⠿⠃⠄⠀⠹⠃⠀⠀⠀⠀⠙⠋⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣠⣧⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢶⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠒⣶⠤⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠿⠿⠛⠁⠀⠀⢚⠀⠀⠀⠀⠀⡌⢹⠀⣤⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠈⠥⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡄⠀⠳⡙⠁⣀⠘⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠠⣤⣄⣴⣦⣤⣤⣴⠀⠀⠈⠿⠿⠃⠀⠀⠈⠒⠛⠂⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠰⡄⠠⡆⠄⢀⣠⣄⣀⠑⠀⠀⢀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⣴⡄⠀⠤⡄⠀⣀⣀⣄⣀⡀⠀⣸⡃⠀⠀⠐⠀⡀⠀⠸
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⢈⠀⠄⠉⠐⢀⠨⣄⣀⠀⠀⠀⠀⠀⢀⣠⣶⣦⠀⠀⠉⠁⠀⠀⠀⠀⠛⠃⠀⠀⠸⠀⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠈⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠈⠛⡉⠀⠠⣶⣶⣶⣶⣾⣶⣶⠶⠦⠀⠄⢀⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠁⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⢢⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠈⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠴⠖⠒⠒⠒⠒⠒⠒⡟⠛⢛⣺⡒⠒⠒⠒⠒⠶⠶⠄⠀⠀⠀⠒⠒⠒⠒⠔⠒⠒⠲⣶⣶⣶⣶⡶⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠲⠤⠤⠶⠥⠤⠾⠿⠷⠴⠶⠶⠦⠤⠒⠒⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠴⠿⠿⠿⠿⠧⠤⠤

⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⠀⠀⠀⠀⠄⠀⠀⣤⣤⡄⠀⠀⢄⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣠⡀
⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣀⠀⢀⣀
⠀⠛⠛⠀⠘⠛⠃⠀⠀⠃⠀⠘⠓⠃⠀⠀⠈⠀⠉⠛⠀⠈⠉⠉⠀⠉⠉⠀⠈⠉⠀⠀⠈⠁⠀⠀⠃⠀⠀⠉⠉
⠠⠴⠦⠀⠴⠀⠀⠀⠤⠀⠀⠠⠤⠄⠠⠐⠃⠀⠀⠀⠀⠀⠆⠀⠀⠰⠶⠀⠐⠂⠀⠐⠒⠀⠀⠀⠒⠀⠀⠐⠂
⠀⡠⠤⠀⢀⣀⠀⠀⣀⡀⠀⢀⡤⠀⠀⣠⡄⠀⢀⡀⠀⠀⣤⠀⠀⣤⡤⠄⠀⣤⠀⠀⣤⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⡀⠀⢀⣀⣀⠀⢀⡀⠀⠀⡀⠀⠀⣀⡀⠀⢀⡀⠀⢀⣀⡀
⠀⠉⠉⠀⠘⠛⠃⠀⠉⠉⠁⠘⠛⠁⠈⠙⠇⠐⠀⠀⠀⠉⠛⠋⠀⠉⠉⠀⠈⠉⠁⠀⠀⠘⠀⠂⠈⠂⠈⠋⠂
⠀⢠⣤⠀⠰⠷⠆⠀⠶⠶⠀⢰⣶⡆⠨⣔⠀⠀⠖⠶⠀⠐⢃⠁⠀⠴⠧⠀⢰⠿⠇⠀⠒⠒⠀⠰⠖⠂⠈⠛⠒
⠀⣀⡀⠀⠠⣤⡤⢀⣀⣀⠀⡤⣄⡀⠀⣤⣄⠀⣠⣤⡄⢠⣤⣀⠀⣠⣤⡤⠀⠤⣤⣤⡤⣀⠀⠀⠄⠀⠠⠤
⠈⠀⠁⠀⠀⣀⠀⠀⠉⠉⠀⠁⡈⠁⠀⢀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣀
⠀⠛⠉⠀⠘⠉⠃⠀⠾⠓⠀⠲⠙⠂⠐⠎⠃⠀⠐⠃⠀⠐⠓⠂⠀⠾⠀⠀⠀⠙⠁⠀⠈⠙⠃⠘⠙⠀⠰⠊
⠀⢀⡂⠀⠀⠰⠀⠀⢘⣶⠂⣀⣦⠀⠀⣴⣇⠀⢀⠢⠀⠀⠹⠆⠀⣰⣦⡀⠀⠶⠄⠀⢳⡂⠀⠄⠲⠆⠀⠽⠂
⠀⠀⡄⠀⢀⣤⡀⠀⢠⡀⠀⢀⣠⡄⠀⢀⣀⣤⣤⣄⠀⢠⣦⣄⠀⢀⣄⣠⠄⢀⠴⢦⠔⠀⣰⡤⡄⠀⢤⡦⠄
⠀⠁⣈⣤⣬⠙⡀⠀⠀⠉⠀⠉⠉⠁⠀⢉⡉⠉⠉⢉⠀⠈⡉⠉⠀⠉⣉⠉⢀⣈⡁⠀⠀⡀⠀⠀⠀⡀⠀⢀
⠀⠸⠿⠿⠯⠴⠟⠀⠻⠯⠀⠻⠟⠂⠀⠸⠞⠀⠛⠛⠃⠘⠛⠁⠀⠻⠛⠂⠈⠉⠁⠀⠐⠛⠀⠐⠉⠁⠀⠉⠁
⠀⠰⠒⠀⠐⠶⠖⠀⠶⣶⠀⠸⠖⠆⠰⠾⠷⠀⠶⠶⠂⠰⠲⠃⠀⠖⠗⠀⠰⠒⠂⠀⠚⠦⠀⠄⠇⠀⠀⠀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⣦⣆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⠢⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⢙⣧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠰⣅⣩⣥⡴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠙⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⣴⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡟⣀⢀⣄⣀⣀⠉⠛⠁⠀⣤⠀⠀⠀⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠟⠋⠙⠿⣦⠀⠀⠀⢼⣦⠀⣸⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⢴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢎⡄⠀⠀⠀⠀⠘⢿⡆⠀⠀⡹⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣄⣤⡿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠈⠁⢀⣠⡴⠟⠀⠀⠓⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⠉⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⡷⠆⠚⢍⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡤⣾⣷⡦⣶⠶⢿⡾⠿⢿⣦⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣾⠟⠋⠀⠃⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⢦⡀⣀⡄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠷⣤⡴⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢸⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡴⠛⠁⢀⣠⡴⠶⠒⠲⠶⣤⣀⠀⠙⠳⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⢿⡉⠀⠀⣰⠟⣡⣄⣠⣿⣧⣀⣠⡙⢷⡄⠀⠈⣹⠋⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⡟⠙⠓⣸⠃⠀⣻⣿⡿⠛⠻⣿⣿⡇⠀⢻⡐⠛⠙⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢹⡄⠀⣿⣿⣶⣤⣴⣾⣿⡇⠀⣼⠁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠻⣆⡙⠋⠙⣿⡟⠉⠛⣁⡾⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡮⠙⠶⠦⢤⠤⠶⠞⠩⣆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢻⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⢀⣀⣘⣀⣀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠶⠋⠁⢀⣤⣾⣿⣿⠛⢻⣿⣿⣦⣄⠀⠉⠳⢦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢤⣄⢠⣾⣿⡉⠉⠁⠀⠀⠉⠉⢹⣿⣦⢀⣤⢼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⣾⣿⠟⠁⢀⣴⣿⣷⡆⠀⠙⢿⣿⡆⠀⠀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢿⣿⣤⡀⠈⢿⣿⣿⠏⠀⣠⣼⣿⠇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⣿⣿⡁⢀⡀⠀⠀⣀⠀⣹⣿⡟⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⠻⢿⣿⣿⣤⣼⣿⣿⠿⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠋⠀⠉⠉⠙⠋⠉⠁⠈⢳⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠶⠶⣶⠿⠿⣶⠴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⢄⠀⠀⠀⠀⠀⢠
⠀⣿⠰⠷⣿⡘⢛⣧⣚⣂⣿⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡅⠧⠴⠇⠀⠤⠒⠔⠀⠲⠁⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠈⠛⠛⠉⠛⠛⠛⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠀⠀⠀⠀⠀⢂⡠⠐⠁⠀⠰⠃⠄⢸⠤⣄⠀⠘⡀
⠀⠀⢀⡀⠀⠀⠀⠀⡄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠡⣀⠬⠈⠁
⠀⡠⢸⠱⡀⠀⠀⠆⠜⠂⠀⠀⢄⠀⠀⠀⠀⠀⢆⣤⣲⢄⡤⡄⢠⠔⢶⡀⣐⠀⡌⠀⠀⠀⣶⡆⠀⠀⠀⢄⠀⢀⠀⠀⠀⡀
⠀⡆⢸⠀⢸⠈⢠⢸⠀⠀⠀⠈⠈⠇⠐⠂⠀⠀⠀⠯⠤⠞⢇⡠⠾⣄⡬⠃⠉⠊⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠈
⠀⠀⠺⠀⠀⠀⠄⠀⠢⡀⠀⠂⠐⠀⢁⠀⠀⡀⠀⠀⡄⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡄⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠉⠅⠐⠒⣤⠉⠀⠀⠀⢸⡆⠈⠉⠀⠀⠴⠤⠇⢰⠀⢄⡙⡄⠀⢄⠀⠈⠂⠰⠖⠊⠁⠀⣀⣇⣀⣀
⠀⢠⠀⠀⠀⠀⢀⡀⢀⠀⡀⠀⠀⢠⠒⠢⣸⡇⠉⠀⠀⡀⠀⠀⠀⣇⠀⠀⠇⠹⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀
⠎⡈⢃⠬⠀⠐⡴⠄⠀⠀⠁⠂⠀⠁⠀⠀⠈⠉⠐⠂⠀⠉⠒⠖⠀⠘⢶⠤⠔⠁⠀⠀⢰⠋⠀⠀⠀⠀⠨⢀⠀⠂⠀⢀⣬⠀⠠
⠀⠀⢁⡠⠤⠄⠀⠤⣄⡀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠋
⠀⣔⡻⣿⠄⡤⠐⣫⣽⣿⣦⡀⠀⠀⠀⠀⠉⢬⡃⢠⡗⠜⡄⠀⠀⠀⠂⠀⠀⠀⠀⢂⡀⠀⠀⠀⢀⠆
⢰⠹⣻⣄⠀⠀⠾⢛⡛⢻⣿⡇⠀⠀⠐⠁⠀⠉⠐⠀⠹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠇⢸⠒⠁
⢸⣿⢿⠅⠀⠀⣴⣿⣿⣮⡝⡇⠀⠀⠤⡀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⣿⣷
⠀⢧⣤⣴⣄⠀⠈⠉⢹⣿⠱⠃⠀⠀⠀⠃⠀⢂⠀⠠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠈⠋⠉⠋
⠀⠈⠙⠿⢿⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⢒⡓⠂⠀⠀⠀⢠⠨⣄⠡
⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠭⠌⠉⠁⠁⠠⠂⠀⠀⣀⠴⠒⠀⠀⠘⡇⠀⠀⢠⠟⣛⢻⠋⢻⣷⠀⠀⠀⠈⠁⠔
⠀⠠⡀⠄⠐⠠⠎⠈⡆⠄⠘⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠠⡞⠁⠄⠀⠀⠀⣀⠇⠀⠀⠘⢷⣤⣾⢶⣶⡿⣦⡄⠀⠂⠀⢆
⠀⠀⠀⠒⠀⣀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠞⠁⢀⡿⠑⠀⠀⠀⢀⣸⣧⣙⣹⣡⣜⡿⠀⠀⠀⠈
⠀⠀⠀⠀⣊⣀⠤⠴⢚⠓⠀⠀⠀⠀⠂⠀⠤⠀⠐⠁⡀⠀⠀⢐⠁⣀⡴⠋⠀⠌⠀⠀⢠⣿⢉⢸⡏⡏⢬⣿⡇
⠀⠰⠂⠈⠁⠀⡠⠔⠀⠀⠀⠀⠈⠁⠁⠇⠨⢨⠠⡀⡃⠀⠀⠸⠚⠁⠒⠀⠀⠀⠀⠀⠀⠛⠻⠟⠛⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⢀⠐⠂⠒⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠟⣆⠀⠬⡁⠄⢉⡟⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠁⢈⡃⠠⠆⠀⠀⠀⠀⠀⠀⣡⠀⠀⡀⡙⠇⡐⠀⢨⠄⠈⠵⠊⠀⠀⠘⠛⠀⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣿⠀⠀⠀⠀⠀⣸⣿⣄⠀⠆⠀⠀⠈⠈⠉⢰⠆⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢄⠀⠀⠈⠀⠀⠀⠀⠈⠈⠻⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣿⡀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣁⡆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⡀⢸⠿⡷⢦⣻⢲⢲⡖⣾⡗⡦⡏⡷⡟⡟⢹⣶⣠⡇⠈⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠄⠀⠀⠁⠀⡀⠀⠉⠉⠈⠈⠀⠉⠀⠀⠁⢉⠁⠁⠀⠈⠀⢉⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢒⠇⠀⠙⠛⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⣠⠀⢤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡖⠀⠀⠀⠂⢀⡀⣙⠕⠀⠀⠘⠦⠤⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⡟⠚⠃⠀⡆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣉⠑⡋⠐⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣄⣀⠀⡀⠀⠈⠛⣷⣶⡄⣰⠇⠀⠰⣄⠀⠀⠀⠀⠀⠁⢀⠨⢁⠊⠁⣄⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣼⡟⣻⡟⠀⠀⠀⠘⠛⠛⠥⠂⠀⠀⠹⣷⡆⠀⠀⠀⠀⠀⠀⠋⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠉⠀⠀⠀⠐⠀⠉⠐⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠰⠇⠀⠈⠁⠀⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠰⣤⠤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣖⠼⠒⠼⣧⣄⢛⣤⣄⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣴⣿⡿⠞⠙⠳⣾⣯⠿⢿⡁⢀⡏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠻⡏⠀⠀⠀⡠⢨⣁⢀⣄⡇⡫⢾⣟⠅⠃⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣄⣤⣾⣴⣿⣿⣿⣿⣿⣷⠟⣡⣤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠄⠀⠚⠾⢭⣭⣿⣦⣶⡶⠶⠟⠛⠋⣡⢞⣁⠀⠀⠀⠀⢀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⢶⠶⠶⠞⠛⣹⡿⣳⡙⠲⠦⠤⠄⠷⡟⠁⠀⠀⢼⠿⠋⠞⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠰⡏⠀⠉⠉⠳⣶⣄⡀⠀⣀⡶⠖⠒⠀⡀⣀⣀⣀⣠⠴⠖⠂
⠀⠀⠀⠀⠀⣠⣤⣥⣶⠶⠞⢓⠒⠒⠚⠛⠒⠢⢄⣷⡀⠀⠀⠀⢹⣿⣷⠞⠁⢀⣀⠴⠛⠋⠁⠀⠋⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠛⠋⠀⠀⠀⠀⢀⡀⠀⣈⣻⣦⣤⣤⡿⠿⠑⠲⠾⢗⡀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⠿⣻⠏⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣸⠵⠛⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠖⠋⠁⠀⠀⠀⠀⠀⠀⡜⣀⣠⣤⣶⣴⡾⢥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠉⠉⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠈⠈⠀⠀⠀⢀⠠⠐⠈⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠀⠠⠂⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠁⠄⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⡀⠀⠀⣀⠀⠀⠀⣄⡤⠀⢀⣾⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠠⠀⠀⢀⡇⠠⠴⣧⡔⠂⠘⣅⣆⠿⠻⣛⡛⠛⣇⣄⢲⠶⠳⠢⣆⡀⠀⡇⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⠴⠋⠀⢩⡾⠁⠀⠀⢠⠟⠉⠙⢿⣷⡿⠾⣿⣟⣿⣄⠀⠀⠀⠑⢦⣳⣄⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡆⠀⢀⠔⠉⠀⠀⠀⡰⠃⠀⠀⠀⡸⢛⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠈⠛⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⢀⡼⠁⠀⠀⠀⡠⢁⠒⠳⣄⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣶⡄⠀⠀⠀⡴⠋⠀⠀⠀⠀⡰⠁⠇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠉⠻⣷⣴⡁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠉⠁⢀⡠⠞⠀⠀⠀⠀⣠⠎⠀⠀⣈⠦⣁⠀⡀⠀⠑⣄⠀⠀⡀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠐⠾⠲⠀⠀⣰⠃⠀⠀⠀⡃⠀⠈⠳⡀⠀⠀⠈⠑⠾⣁⡠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⡤⠴⢟⣅⠃⠀⠠⠞⠛⠒⠤⣀⣨⡳⣴⠶⠴⠀⠀⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠠⠀⢀⡀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⢠⡀⠀⠉⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣗⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠻⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠁⠂⠀⠀⢶⠃⠀⡇⠀⢀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠐⢀⢐⠀⠀⠀⣴⡦⠀⠀⠸⠂⠀⠇⠀⠈⠀⠀⠀⢈⣁⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡸⢬⠂⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⣨⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⠇⢀⠀⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⡆⠀⠀⠀⠀⡀⠀⠀⠄⠘⠁⠀⠀⠀⠛⢭⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣀⠀⡄
⠀⠘⠇⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢀⠀⢄⣀⣀⣀⣤⣀⣀⠑⠀⠀⠀⠀⢁⠀⠀⠀⠀⢠⠄⠈⠐⠂
⡀⠀⠃⠀⠰⠀⢨⡀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠿⡿⠛⠛⠉⠙⢻⣷⠀⠀⠀⠁
⡧⠬⣁⠀⠀⠀⢐⠁⠠⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⠳⠂⠀⠀⣠⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⢿⠀⠈⠀⠂⠀⢸⡀⢀⠀⠀⠀⠀⠀⠘⣧⣴⡔⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠈⢀⠀⠀⠤⠂⠃
⣀⡧⢄⠈⠀⠀⠸⡆⢀⠤⠀⠀⠀⠀⠀⠈⣿⠃⡠⣶⠀⠀⠙⠀⢀⠠⠀⠠⣰⠧⠀⠀⠀⠠⡀⠒⡠⠄⠁
⠁⢠⣄⠃⠘⡀⢸⡦⣾⠀⠀⠀⠀⠀⠐⢢⠈⠆⣐⡅⢨⠄⡀⢠⡋⡄⢉⠀⣯⠆⠀⠀⠀⠀⠉⠲⣀⠀⠘
⠀⠋⠀⣧⠀⣁⣺⠀⡿⣧⠒⠀⠀⢀⢰⡆⣰⠀⠀⠀⠺⠄⠟⠒⠀⢦⠘⢜⠿⠀⠘⢄⡠⠀⠀⠀⠀⠀⡄
⡠⠀⠐⠬⠺⠅⢹⠠⠑⢫⡷⡄⢀⠈⡼⠁⠘⠦⡀⡈⢈⠀⠀⠈⠁⣮⣷⣄⡀⠀⢀⡨⢬⡤⠨⠀⠑
⠀⣦⠦⠤⣴⡟⢻⠖⠁⠸⡇⠰⠀⠶⣶⠂⡀⠪⠜⠏⢺⠉⠁⡰⠀⢐⣿⣄⠆⠀⣬⡀⣼⠎⠉⡀⠀⠐⢆
⣁⢰⣶⡄⠹⠧⣿⣧⠈⠐⡄⣄⣉⣿⣅⣐⠉⠁⠀⠀⣯⢄⡎⣸⡻⢸⠈⡎⠼⠠⠉⠀⠁⠈⠱⠎⢤⢶⣂
⡨⢶⢿⣾⠟⢷⠜⠃⠀⣀⢁⣟⣿⣿⣿⣻⠋⢄⡀⠩⡷⡿⠅⢿⢾⣾⠶⢀⠥⠛⠂⡄⣸⡶⣎⣠⣿⣄⠈
⠁⢼⣇⣤⡈⣉⡴⣷⣶⢭⣜⣏⠉⣴⡞⣿⣀⣴⠂⣴⣟⡆⢁⡘⠲⡗⠀⠘⡀⠶⡆⠺⠽⢣⣻⣽⢷⡻⠇

⠀⠀⢀⣀⣤⣦⣜⣶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠲⠇⡩⠛⠓⠁⠣⠘⠕⢊⡡⣨⠜⡟⣶⠃⠄⡈⢄⡔⠈⡀⠀⠁⢀⡰⠀⠉⠀⡜⣸⠂⠀⠀⠀⠀⠂⡄
⠀⠘⣯⣿⢽⣟⣿⠃⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠌⠂⠑⠙⠊⠔⠡⢊⠄⡁⠠⢑⢸⠇⠀⠈⣀⡭⡀⠈⢀⣼⣷⡞⠁⠀⠠⢰⢁⡟⠀⠐⠀⠀⢊⠀⠀⠐⠃
⠀⠀⢋⡡⢞⡾⠁⠀⠀⠀⠀⠀⠔⠉⠈⢀⠀⠤⠰⠂⠐⠀⠂⠒⠀⠄⡈⠊⣤⣴⣴⣌⣠⠁⠎⠀⠀⣡⣾⣿⣿⡟⠀⢀⢠⡇⡘⠀⠀⠀⠀⠀⠄⠒⣂
⠀⠀⢟⣅⣮⡆⠀⢀⡠⠂⠁⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢼⡏⣿⡧⣻⡎⠉⠆⢂⣼⣿⣿⣿⠟⠀⢀⢀⠊⠐⠀⠀⡀⠤⠔⣢⠫⠐⠀⠂⠄
⠀⢀⣄⣿⢿⠀⡐⠂⠀⡠⠘⠀⠀⠀⠀⢀⣠⣴⣶⣞⣛⢿⣿⢿⣶⣶⣤⣈⠻⠿⠿⠟⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⢀⠎⡴⠀⠀⢨⣴⡮⣜⣖⣫⠜⡄⣇
⠀⢘⣯⠷⣿⠂⠀⡠⠊⠀⠀⠀⣀⣴⣾⣿⣿⠿⠋⠁⠉⠀⠉⠙⣶⣛⣿⣿⣷⡀⠀⠀⠠⣼⣿⣿⣿⡟⠀⠀⠀⠀⢊⠜⡡⢪⡴⣟⡇⣯⢵⡸⠆⡫⠩⠈
⠀⢘⣾⡾⠁⠀⠌⠀⠀⠀⣠⣾⣿⢟⣻⠟⠋⠀⡀⢨⣔⣧⡄⢔⠘⣿⣿⣴⣿⣿⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣴⣻⣃⡇⢻⠜⢃⠐⠂⠀⠆
⠀⠀⡟⠀⠀⠂⠀⠀⣠⣾⣿⠟⡝⡡⠱⠀⣀⢨⡾⣿⣿⣿⣻⠐⡆⢸⣿⣭⣿⣿⡇⢸⣿⣿⣿⡟⠀⠀⠀⠁⠄⠀⢈⣾⡋⢓⠃⠜⠠⠀⠀⠀⢀⣀⣤⢤⠄
⠀⠀⠀⠀⠂⠀⢀⣾⣿⣿⣧⢞⣤⢄⡆⡐⣠⣛⢏⠀⢆⠋⢿⠀⡇⢸⣿⣿⣿⣿⠁⣿⣼⣿⣿⡁⠀⠠⠀⠀⠠⢀⣿⣛⣉⠃⡘⠀⠁⢀⡴⠛⠉
⠀⠀⠀⠌⠀⣰⣿⣿⣫⢻⢿⠻⠍⠿⠀⢤⣯⣦⣷⣥⣂⣳⣻⣿⠇⣼⣿⢟⡧⠜⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⡠⠚⠋⠉⠉⢀⠀⠀⠐⠋⢀⠀⠤⠀⣠⣤⡀
⠀⠀⠘⠀⣰⡿⣟⣃⠄⠀⠀⠐⠀⡌⢸⠋⠣⠊⢿⣿⣄⣞⣧⠂⣰⣿⢣⡔⠄⠀⠠⠹⣾⣿⣿⣿⣿⣶⡦⡄⠖⠘⠙⡉⠉⠉⠩⠛⠫⠱⡄⡀⠲⢿⡏
⠀⠀⠃⢰⣿⡷⡷⣧⠀⠀⠀⠈⡂⢱⠹⡀⠀⢀⣼⣶⠟⠃⢁⣼⣟⣿⡿⠃⢀⠀⠀⠑⣝⠿⣿⣿⠟⠋⠁⡀⡄⢠⢢⠂⠄⠆⠦⠤⣀⠀⠈⠂⠑⠆⡉⠠
⠀⠀⢠⣿⠯⡜⠉⠍⢐⣁⣀⠀⣑⠇⡈⠀⠀⣾⣿⣏⣠⣶⢿⠷⠷⣋⣀⠀⠰⠀⡆⢤⠀⢠⠈⠃⣀⠔⠈⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠉⠂⢄⠀⠀⠘
⠀⠀⣸⡟⠀⢄⠀⠀⣾⣿⠗⡨⡨⠁⠅⠀⢸⣿⣿⣿⠽⢒⣍⣿⣼⣿⣿⣷⣦⡀⠁⠈⠠⠈⢀⠆⠁⢀⣰⣾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣄⠀⠙⢄⡀⠈⠄
⠀⠀⣿⢮⢰⠆⡑⢾⠁⠘⢁⠠⠆⠧⡁⠀⣿⣿⡿⢁⣜⣷⠟⠉⠀⠈⢿⣿⣿⣿⡄⠀⢀⠔⠁⠀⡴⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠙⢷⡄⠈⠳⡀⡁
⠀⠀⣟⠀⢘⠃⢌⡄⠀⣴⢶⢾⣷⡠⠀⢸⣻⣿⠁⣾⣽⠃⠀⠀⣦⣄⣀⠀⠉⠙⡎⡖⠚⠀⢠⠞⠤⠤⠞⢳⡄⠀⠀⠀⠀⠀⠀⢯⠀⠉⠳⣌⢻⡆⠀⠙⡅
⠀⠀⠘⣤⣶⣶⠄⠑⡀⢸⢹⢹⠞⣯⠀⢸⣿⣿⠀⣱⣿⡀⠀⠀⠘⣿⣿⠇⠀⠀⢇⢈⠀⠀⡎⢠⠖⠲⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠁
⠀⠀⡌⠜⠿⣻⣔⡤⠢⡐⠀⠠⡂⢸⡄⠘⣿⣿⡄⠹⢞⣷⣄⠀⢰⣿⡿⠆⠠⠀⣠⠖⠀⢰⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⢺⠂⠀⠁
⠀⠀⣦⠘⠢⡏⠽⢧⠀⢀⠀⡀⢧⣇⣻⡆⠘⢿⣿⣦⣈⠁⠋⠿⠋⠉⡀⣠⠴⠋⡡⠀⠀⠸⣠⣆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠠⣏⣆⡁⠀⡌⠌⣴⣖⣦⡘⠀⢙⡯⣿⠀⠀⠈⠛⡡⢍⢁⣀⣐⣉⠍⠥⠚⡡⠐⢀⠀⠀⢻⣿⣄⠀⡏⠀⠙⢆⢠⡀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠆⠀⠀⡇
⠀⠀⢿⡿⠈⠑⢄⡐⢿⢟⣞⠛⢆⠀⢻⣾⣷⡀⠀⠀⠀⠀⢄⠀⠀⢀⣀⡄⠂⠁⣡⣤⣖⠀⠈⠻⣿⣆⠙⠢⠤⢄⡀⠈⠓⢦⢣⠀⠀⠀⠈⣲⠏⠀⢀⡼⡇
⠀⠐⢸⢲⣷⣶⣦⡁⠈⠸⢹⠀⡀⠃⠌⠹⢿⣿⣷⣤⣀⠀⠚⠈⠋⠘⠀⣁⣤⣾⣿⡿⡀⢀⠂⠀⠙⠿⣷⣄⠀⠀⠳⡄⢸⡜⢰⡀⢀⣠⠞⠁⠀⡄⠪⠯⡗
⠀⠀⠈⠨⠨⢩⢿⡿⣔⠐⠀⢀⠁⠣⣖⣀⠀⠙⠟⢿⣿⣿⣿⣷⣷⣿⣿⣿⡿⡿⢋⠀⢀⠀⠠⠁⢀⠀⠈⠙⠳⠤⠄⢌⢼⠡⠤⠖⠉⠀⡀⠔⢉⡒⣝⣝⡃
⠀⠀⣄⠰⣄⢆⡆⠘⠻⠧⠈⠈⢢⡀⢀⠁⠠⣖⡄⡠⠄⠍⠩⢙⢙⢋⡉⠡⡔⠈⠈⠀⠀⠀⠐⢆⣠⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠄⠖⠈⢀⠠⢎⢪⡵⢣⠁
⠀⠀⠠⡠⠉⢻⣿⣿⣆⢰⠀⠢⠄⡵⡯⣅⡲⡀⠁⠀⠐⠀⢁⠉⠊⠁⢀⠀⣠⠄⠖⢲⠂⢀⡄⢬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢎⠝⣪⠟⡨⠃
⠀⠀⢁⢋⠊⠐⠎⢿⣷⣿⣦⣤⡁⠀⡁⠊⣔⢎⡉⠺⢄⡤⣠⢄⡂⠱⠵⣯⣇⣤⣧⡦⣆⣴⣿⣿⣿⣯⡟⠁⠀⠀⠀⠀⠀⠀⢀⡠⠿⣚⡱⠞⣡⠞
⠀⠀⠀⡀⠈⠐⠲⣦⣿⣮⣿⣿⣿⣲⣶⡀⢠⠋⢄⠐⡀⠖⢈⡁⠂⠀⣘⠻⡿⢿⣟⡫⠿⠛⠻⣿⠿⠉⠄⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⣩⡴⠞⠁
⠀⠀⠐⠊⠵⠒⠦⠄⡛⠿⠿⢿⢟⠹⡛⠛⠻⠶⢆⣻⢷⡨⠀⠹⡗⡶⠿⠻⠏⠹⠈⠁⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡾⠾⠔⠃⠉⠁⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣾⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣷⣾⣶⣷⣶⣷⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢒⣀⣩⣿⡟⠻⡓⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠈⠄⠒⠛⢋⠿⡿⢿⡟⠀⢇⠠⠐⢛⡈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⢏⡀⠀⠀⠀⠀⠑⠖⠁⣀⣵⠀⠀⠀⠀⠀⠀⣠⡠⣛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⡼⠀⠀⠀⠀⠿⢖⡠⠒⠃⠈⠀⠀⠀⠀⠀⠀⢈⡟⠉⡛⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠁⠀⡊⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠿⢿⠦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡄⢀⠊⠉⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠠⠏⡣⠀⠋⡉⠛⣆⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠉⠉⠣⢄⠀⠀⠀⠀⠀⠀⠈⠢⠤⠄⢤⠀⠀⡰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⢠⡴⠁⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠃⠀⡀⢏⢂⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡰⠀⠀⠘⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠀⡀⠙⠦⣀⠀⠀⠀⠀⠣⠠⠞⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⠐⠃⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡄⢈⣢⣄⣀⣀⠀⢤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⢀⡄⠠⠆⡲⣦⡄⠙⠢⢴⣄⣀⠄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠠⣀⣠⡦⠄⠉⢠⣴⢆⠰⠄⢀⡀
⠀⠴⠒⡾⠁⠀⠙⠁⠀⠄⠀⠈⠂⠀⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠐⠁⠀⠠⠀⠈⠋⠀⠈⢷⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⣀⡄
⠀⠀⠸⡆⢸⡆⠀⠀⢠⡇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈
⠀⠀⠀⢷⠸⡇⠑⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠰⠀⠄
⠀⠀⠀⠘⠀⠇⢀⡇⠈⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⢲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠳⣀⠃
⠀⠀⠀⠀⡰⠀⠀⢠⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⡇⠈⠠⡆
⠀⠀⠀⡀⢃⠀⢰⡸⡄⢻⠇⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠲⡿⢀⣀⠄
⠀⠀⠀⠇⡞⢰⠀⠷⣧⠋⡆⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⣳⡀
⠀⠀⠀⢸⠃⠸⠀⠀⣿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⡿⠘⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣧⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣤⡰⠦⠀⣙⡛⠀⠀⠀⠀⠋⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣾⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣦⣃⡀
⠂⡐⠀⠈⠀⠀⠈⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⡛⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⡈
⠀⠀⠀⠀⠀⢤⢠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⡞⠅
⠀⠀⠀⠀⠀⠈⢀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠇⢼
⠀⠀⠀⠀⠀⠢⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⣾⣉⣈⣆
⠀⠀⠀⠀⠀⢀⡀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠈⠁⠀⢡⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠆⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠐⠢⣬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢾⣶⣤⢏⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣬⣯⣤⣿⣬
⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡶⠎⢻⡄⠀⠀⠀⠀⠀⠀⢠⣾⠛⠙⠋⠿⠟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡙⠚⢻⣿⡄⢀⣠⣞⠟⠛⠛⠟⠀⢀⣀⣤⣤⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣧⣿⠙⠈⠀⢀⣰⣶⣾⣿⣿⢟⣿⣿⠃⢰⡖⣲⡤⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣿⠇⠀⡞⢡⣿⣿⡼⢂⣼⣿⣻⠿⢶⣼⣿⣿⣷⡿⢉⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⡿⣿⠿⣿⣿⣿⡿⢋⣀⣉⠈⣿⣿⣿⣿⣿⣿⣏⠀⢄⠀⠘⠛⣿⣷⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢤⣤⣶⣿⣿⣿⣷⣿⣿⣿⣷⣤⣽⣿⣿⣯⣟⣹⡄⠀⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠛⢛⡿⠋⠉⠻⢿⣿⡿⠿⣿⣾⣿⣿⡟⠁⠀⡁⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠚⢃⣁⣄⡀⠀⢀⣀⠀⠶⠶⢾⣿⣶⣿⣷⣄⠀⡁⠈⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠈⠚⠹⡶⢿⠿⠿⣶⣄⢀⣿⣿⣿⣿⣟⡁⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠙⠂⠀⠀⠀⠠⠀⢠⣀⣈⣿⣿⡏⠸⠟⢿⡉⠑⠶⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⢤⣀⡀⠃⠀⠀⣠⣿⠿⣿⣿⣿⠄⠀⠀⠈⠙⠀⠀⠀⠀⣠⠴⢚⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠄⠉⠓⠄⢸⣿⣯⢰⡿⣿⣿⡄⠀⠀⣀⠀⢀⣤⡴⠟⢁⣼⠋⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⢠⡖⠁⢸⣿⣿⡟⣇⣿⣿⣷⡶⠿⠏⠀⣼⣿⠓⠀⣨⢁⡄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡅⢠⣾⣿⣷⣾⣯⣿⣿⣿⣤⡀⠀⠀⣿⣿⠂⢠⣇⡾⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢿⡦⢄⡸⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡄⠀⣿⣯⠕⣸⣏⠒⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣍⠁⢸⣿⣵⣦⣪⣿⣷⣿⣿⣿⣿⠋⣿⠈⠓⠘⣿⣆⠹⣿⡄⣿⣯⣿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣽⣿⣿⣿⣿⣶⣾⣿⠛⠛⠛⠁⠀⠋⠀⡤⠀⣬⣿⡄⠘⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⣩⣿⣿⡷⢹⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⣴⣖⠒⠶⣦⣀⠈⢛⣿⣿⣿⣷⣈⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣄⠉⠻⣶⣤⡀⢹⡟⢿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⡌⠻⣦⡀⠀⠈⣿⡄⠀⠀⠙⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣂⠘⣻⣆⠀⠀⢷⠀⠀⠀⠀⢿⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣄⡸⣿⣷⠀⠘⡄⠀⠀⠀⢸⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣀⢻⣷⣄⣷⡀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣾⣿⣿⣿⣦⡀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠙⠻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢯⠁⠁⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠐⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠏⣾⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⠊⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣴⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇
⣿⣿⣿⣿⣿⣿⣿⠀⣴⠇⢾⡿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⠐⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠈⣿⣿⣿⣿⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠻⢿⣿⣿⣿⣿⣿⣿⣷⣀⠘⣿⣿⣿⣯⣿⣿⣯⡇⠀⠀⢿⣷⡿⠃⢠⣿⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣧⡈⠻⢿⣿⠍⢛⣾⣝⠀⠀⠘⠻⠦⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⢤⡀⠈⢣⡄⠀⠀⠀⠀⢠⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡇
⣠⣠⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⢱⣦⡄⠀⠀⢸⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⡇
⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⠣⠀⠈⠛⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠐⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡈⣀⢡⣤⣤⣄⡀⢸⣷⣽⣿⢿⣯⡟⣿⡾⣝⣿⣟⣻⣷⣾⣿⡇
⢠⣴⣄⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⡟⢿⡏⣸⣿⣿⣟⠿⣿⢿⣿⣿⡏⣸⣟⣿⡿⢙⣿⡇
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠿⡌⠇⠸⠈⡼⣰⣷⡾⣿⣇⣿⠉⣄⠈⠉⢯⣿⠋⣽⣿⡇
⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠿⣌⣿⣿⣷⣯⣀⣠⣰⡟⡄⠻⣾⣿⣩⣷⠗⢼⡢⢬⡚⣏⣆⣸⣿⡇
⠿⣻⣛⣿⣿⣿⣿⣿⣽⣿⢿⣿⡻⣢⡟⢻⡿⣿⣿⡿⢡⠘⣟⠎⠳⣾⢫⡇⢺⠙⣿⣴⣆⣼⣿⡟⢂⣿⡇
⠠⣷⠿⠻⣿⡿⢿⣿⣿⣿⢋⣴⣭⣿⠀⢈⡇⢸⣿⡇⢸⡇⢈⠳⢄⣩⠚⠁⠊⢁⠘⠃⠀⣹⣿⣿⢻⣿⡇
⠀⡁⣀⠋⠙⢐⡼⠟⣫⣴⣿⣿⣿⡯⠀⣽⠇⠘⣿⡇⠸⣿⠈⠓⠒⠈⠉⠙⢦⣁⠀⠐⠀⢀⠀⠈⠘⣿⡇
⣠⣼⠟⢀⠖⠁⠐⠀⠤⣠⡌⣉⠉⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠂⠀⠔⢀⠀⣿⡇
⠋⠀⡰⠁⠈⣀⠁⠘⠉⠻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠀⢀⡈⠸⣿⡇
⠠⣰⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⣡⠀⠀⠂⢠⣿⡇
⠀⣻⢀⣀⣀⠀⠀⣰⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄⠀⢻⠀⠀⠀⠈⣿⡇
⢂⣹⣼⡿⣠⣠⣦⣼⣶⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡇⡃⠀⠀⠀⠀⣿⡇
⠀⠙⣿⣿⣦⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢷⠀⠀⠀⠀⠀⣿⡇
⠀⣴⣿⡟⠛⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣇⠀⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠀⠋⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠎⠉⠉⠁⢈⣉⠡⣬⣩⠲⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣶⣾⣿⠳⠓⠉⠀⠀⠀⣠⣾⡿⠇⢫⣯⡀⠘⢛⣻⣟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⢋⣶⣽⡯⠊⠀⡄⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⣮⠠⠖⠋⠁⠉⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⡎⣰⢿⣿⠏⠑⠒⢶⢧⣀⠀⣠⣿⡿⠁⠀⠀⣠⠞⢹⡆⠀⠀⠀⠀⣼⠉⠻⣼⣇⡀
⠀⠀⠀⠀⠀⠐⣿⣟⡛⠀⠄⠀⡠⠎⠀⡸⢋⣿⣿⠁⠀⣠⠞⠁⠔⠘⠁⠀⠀⠀⢠⡏⠀⠀⠬⢿⣿⡀
⠀⠀⠀⠀⠀⢸⡏⣟⠀⠠⠼⠊⠀⢠⣟⣀⣼⣿⣅⣠⣼⣁⣠⣎⣀⣾⢀⣀⣀⠀⣎⡠⠊⠀⠀⠸⠁⢷⡄
⠀⠀⠀⠀⢠⣿⠖⣦⣶⣶⡿⢿⣷⠛⠛⢻⣿⣿⡿⠋⢉⠕⠛⠋⠉⠈⠙⠛⢛⡿⠛⠿⠿⣶⣦⣦⡄⠘⣧
⠀⠀⢀⡴⣾⣿⠾⠋⠁⠀⣠⠋⠀⠢⠀⢸⣿⣿⢀⣴⠂⠲⣶⡄⠁⠐⢀⣴⡞⠀⠀⠀⠀⠀⡜⠉⠻⣴⣾⡇⠠
⠀⠀⣿⣷⣿⣿⣇⠀⠀⡼⠁⠀⠀⠀⢐⣿⣿⡯⠊⢠⠀⠀⣸⣿⢠⣀⣈⠿⡀⠀⠀⠀⢀⡼⠀⠀⠀⣿⢻⠗⠒⢂
⠀⠀⠘⣞⠃⠛⢿⣦⣾⣀⡀⠀⠀⢀⣞⣷⣿⣧⠀⠀⠉⠚⠋⠁⢨⠉⠁⠀⠀⠄⡀⠀⡘⢁⣀⣴⣾⠟⠁⠀⣠⠊
⠀⠀⠀⢿⠿⣶⣤⡈⢉⡛⠛⠷⠶⢈⢘⡛⢛⣧⠀⢀⡀⠠⢴⣧⠀⢀⣀⠀⣤⡐⠩⠭⠁⠉⢉⣩⣤⣴⣶⣿⡇
⠀⠀⠀⠈⣦⣹⣿⣿⡤⠚⠋⢀⡖⠀⢲⣶⣤⡄⠀⠀⢀⢤⠤⣤⡄⣤⣴⠂⠀⡲⠂⠀⠊⣀⠴⣶⣽⣿⠉⠋
⠀⠀⠀⠀⠸⡇⠹⣶⣷⡀⠀⣸⠁⠀⣼⣿⣿⡇⢀⡠⠖⠁⠈⠉⠉⠋⣿⣤⡎⠀⠀⣠⠞⢣⣴⣿⣿⠋⡄
⠀⠀⠀⠀⠀⠘⠄⠙⢳⣅⣤⡇⠀⠀⢿⣿⣿⡛⠹⠁⢠⡶⠃⠀⠀⢰⣿⠁⠉⠐⠲⠅⠠⢿⣿⡟⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠢⡈⠻⣿⣿⣶⡖⢻⣿⣿⡇⠁⣠⡟⠁⢀⠄⠚⣼⡇⠀⠀⣀⣼⣶⣷⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣌⠛⢿⣿⣟⣿⣛⣃⣰⡏⠁⠀⠀⠀⠀⣿⣧⣶⣾⣿⡿⣿⣯⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣯⡉⠛⠻⠾⣷⣤⣤⣬⣬⠁⣠⠿⠛⠛⠉⢁⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠶⠶⢞⣿⣷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢼⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡰⣰⠃⠀⠳⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣔⠤⠂⠐
⠀⠀⠀⠀⠀⠚⠁⠖⠀⠀⠀⠀⠙⠄⠈⢨⠆⡤⠶⠄⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠈⠉⠀⠀⠠⠄⠀⢠⠀⡀
⠀⠀⠠⣀⠀⠀⠀⢀⣿⡆⠀⠀⣰⡆⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⢉⠀⠀⠀⠀⣴⡐⠀⠀⠳⡽⣦
⠀⠀⠀⠈⡁⠀⠀⠀⠋⠀⠀⠂⠛⠁⠀⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠁⠁⠀⠁⢹⣧⠀⠀⠀⠻⠇⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠻⠇⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠉⠀⠃⠠⠄⠀⠀⠀⠀⢀⣈⠀⣠
⠀⠀⠀⠀⠀⠀⠉⠲⠠⠤⡴⠀⢀⠀⠈⠐⠻⠂⠀⠀⠀⠀⠀⠀⠰⠛⠉⠈⠉⠻⣦⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠐⠤⠦
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠘⢦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠂⠀⠀⠄⢠⠀⢁⡠⣄⡀⠀⠈⠾
⠀⠀⠀⠘⠚⠁⠀⠸⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡈⠅⡆⠅⢆⢱⠀⠀⠺⠤⠖
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠁⢀⠨⠐⡀⡵
⠀⠀⠀⠀⠀⠀⠄⠣⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠘⠰⠖⠘⠒⠉
⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢠⠀⢀⣠⡄
⠀⠀⠀⠀⠉⠉⡄⡀⠈⢀⠀⠱⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⡄⠀⡀⠐⢀
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠛⠾⠶⠶⠤⣤⣤⣤⡴⣶⠖⠛⠛⠛⠛⠋⠉⠉⠻⠛⠛⠛⠛⠳⠶⠤⣤⣀
⠀⠀⠀⡠⡞⠁⠀⠀⠀⡄⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣉⡂⢤⣀
⠀⢠⡞⠁⠀⠀⢀⣤⡿⠛⠉⠙⠛⠛⠛⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣿⠷⣻⡛⠾⣦⡈⠙⢦⣄
⠀⣺⡀⠰⣄⣦⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⢳⣦⣤⣀⡀⠀⠀⠀⢀⣠⠴⠲⠋⠉⠙⡏⢣⡞⣡⢀⡤⣯⣹⡄⠀⠙⢳⣄
⢽⣭⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣔⢉⣳⠖⠉⠁⠀⠀⠀⠀⠀⠈⢦⣙⣶⣫⢞⣽⡿⠁⠀⠀⠀⠈⠳⣤
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣝⣽⣼⣿⣿⣻⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢈⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣧⣯⣿⣿⣿⣥⣠⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠻⠶⢶⠶⠶⠾⠟⠛⠉⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⢀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠋⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⣤⠘⠂⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠋⡉⠛⠛⠛⠒⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠄⠀⠀⠀⠀⢀⣽
⠀⠀⠀⠀⢀⠖⣹⡿⠷⠮⡢⣄⡀⠠⠴⠋⠁⠀⠀⢈⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠰⠦⣼⡿
⠀⠀⠀⠀⠸⡌⣿⠀⠲⢦⣽⠀⣷⠈⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠘⠿⡴⢶⣯⡭⠾⢿⣴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⣌⠁⠀⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠘⢷⣄⠀⠀⠂⣈⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣆⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠷⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠠⠤⠄⠄⠠⠶⠒⠈⠉⠛⠛⠷⠶⢶⣦⣴⣤⣶⣶⣷⣾⣦⣠⡄⠀⠂⠀⣀⣴⣦⡀⠀⠀⠀⠀⠉⠵⢦⡀
⠀⠀⠀⠀⠀⡠⠔⠂⠁⡈⡄⡆⣇⣠⣼⣰⣄⣇⣤⣀⢘⢰⠀⡀⡀⡄⠀⠀⡀⠈⠉⠉⠉⠉⠙⠛⠛⠛⢿⠿⡿⠿⠿⢶⣤⣆⣸⠀⠆⠁
⠀⠀⢠⠀⠁⣀⣴⣼⠷⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⠷⠾⠷⠯⠿⠶⠶⠷⠾⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠃
⢀⡆⠁⢴⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣀⡀
⡞⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠠⣉⢆
⣗⠀⠀⢆⠀⠀⠀⠀⠀⠀⣠⢎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣽
⠹⣧⡀⠈⠙⠲⠠⠀⠒⢫⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢃⡾⠃
⠀⠈⠳⢦⣄⣐⣀⢀⢀⣰⣾⣴⢆⢷⡀⢀⠀⠀⢠⠀⠂⣀⣩⣤⠾⠋
⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣛⣛⣾⣿⣷⣶⣷⣾⣾⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣅⣠⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⡿⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢈⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣦⠀⠀⠀⠀⣠⣾⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢿⣿⣿⣿⣇⠀⠀⢠⣿⣿⣿⠟⢸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠉⠛⠛⠋⠀⠀⠈⠛⠋⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠛⠛⣿⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣏⣻⣀⠲⢦⣵⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠛⠋⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣏⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠤⠤⠴⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣬⢤⡐⢦⠲⠦⡴⢤⢤⣄⣀⣀⡉⠉⠒⠲⠤⣄⣀
⠀⠀⠀⢀⢀⣀⡠⠴⢚⠉⣽⠋⢹⣿⣭⡙⣛⠒⠻⠭⢭⣇⣲⣴⣄⣀⡀⠉⠓⠢⠤⣀⣀
⠀⣀⡴⢋⡾⠀⠀⠐⠀⢈⠄⠓⠚⠏⠛⠛⠁⠀⠀⠀⣲⣤⣌⡉⠙⠛⠤⣍⡂⠤⠀⠀⠈⠉⠓⠒⠤⣄⡀
⠺⡆⠀⢸⢧⣦⣀⣠⣴⣥⣤⣤⣤⣤⣦⣤⢠⠀⠁⠒⠻⠿⣿⣾⡗⠢⣄⠀⠈⠙⠢⣌⠀⠀⠀⢀⠀⠀⠈⠁⠢⢄⡀
⠀⠙⣰⡏⠜⢿⣿⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣾⣦⣠⣄⣍⣛⣦⣧⣉⡲⣄⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⢻⣻⠆⠘⠣⠱⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠑⢆⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠰
⠀⠀⠀⠳⣧⠀⠠⠎⡠⠙⠛⠛⠿⠿⡿⢿⣇⠹⡌⢩⣿⣿⣿⣿⣿⣿⣿⡁⠀⣀⠀⠈⠳⣄⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠉⠳⣦⢀⣠⣦⣄⣀⡀⠀⠀⠀⠉⠂⠙⠎⠉⠙⢛⠛⡁⠉⡩⠀⠠⠀⠀⢠⣾⣿⣧⡀⠀⠂⠀⠀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣦⣶⣄⠤⠤⠀⡀⢄⢀⡈⠆⠁⠀⠁⠠⠀⡈⡄⡛⢀⣼⣿⣿⣷⣤⡀⠈⠙⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠦⠀⠐⢷⣾⣾⣷⣼⣷⣿⡧⠿⠭⠾⠓⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⡇⢻⠀⠀⣾⡆⠀⠀⠀⢰⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⢐⡇⢸⠀⠀⡗⣇⠀⢠⣄⢸⡀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣄⠀⠀⡿⢿⡆
⠀⢸⠁⢸⡄⢸⡇⢹⠀⣸⢻⡀⣇⠀⠀⠀⢀⠄⠬⠁⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⡼⠀⠀⠀⠈⣷⡀⠀⡇⢸⡇
⠀⣸⠀⠈⡇⣸⠁⠸⡇⣿⠸⡇⠘⣆⠀⠀⣈⣄⠀⡔⠛⡀⠀⠀⠀⠀⠈⠀⠀⡅⡰⠁⠀⠀⠀⣀⣿⢻⡶⠇⢸⠃
⠁⡿⠀⠐⣧⡿⠀⠀⢿⡟⠀⣇⠀⠈⠳⣈⣣⣤⠀⠲⣀⣙⠄⠀⠀⠀⠀⠀⣠⠞⠀⣠⡧⠀⠀⠘⣿⢀⣿⠀⢸
⣐⣇⣀⣀⣿⣧⣤⣶⣟⣷⡿⢿⣀⠀⠀⠈⣛⠦⣄⣀⣀⠀⠀⣀⣀⡤⠴⠋⠁⠀⠀⣹⣇⡿⢻⣿⣿⠛⣿⠀⢺⡄⠸
⣿⠾⠟⠛⠛⠻⠿⠿⢭⣄⣐⣀⣭⠷⠚⠋⠉⠛⢦⣀⠉⠉⠉⠉⢠⡶⠀⠀⣴⣶⣶⣿⣏⣿⣸⣧⡿⠶⠿⠿⢛⢟⢻
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠤⣤⣤⣭⣳⣦⣤⣤⣽⣿⠤⠶⠿⠗⢚⠻⠩⢁⡀⢠⣤⣤⣴⣦⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠢⠄⢠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡠⠄⠒⠃⠀⠀⠀⡤⠠⠄⠀⠀⠄⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢀⠍⠊⣁⠤⠀⠚⠠⠤⠄⠐⠁⢀⠠⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠥⠂⠉⠀⠀⢀⣀⡀⠤⠐⠂⠈⣁⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⠋⠀⠙⣭⠀⠀⠐⠂⠈⠁⢀⣀⠠⠤⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠱⠟⠰⠖⠊⠀⡼⣁⠄⠋⠉⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠈
⠀⠀⠀⠈⡉⡲⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠
⠀⠀⠀⠀⢯⣮⢂⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣾⠃⠀⠀⢀⠤⠆⡴⡋⠁⠌⡈
⠀⠀⠀⠀⠀⠳⡁⠰⡉⠀⠐⢎⡂⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⢀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠐⠇⠈⠠⠀⡫⢊⠎
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠨⠠⢀⡑⢦⡈⠰⠄⠐⡈⠉⡏⠣⣀⠀⠑⠀⠀⠁⠒⠂⠚⠉⠀⠀⠀⠀⣄⠆⢀⡠⢊⢔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠢⡀⠠⢂⠜⠆⠄⡀⠀⠐⡼⠀⠐⠢⣵⡠⢄⡀⡀⠀⠀⠀⠀⠀⠀⢀⢞⣈⠖⣊⢔⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠒⠓⡀⠀⠀⠘⢇⠀⠈⠂⢴⠂⢝⠳⣌⠲⣎⠓⠀⠀⠀⠀⠀⠠⠠⠔⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣠⡀⢀⠲⡂⠀⠀⠈⢧⡀⠧⠄⣤⡞⠁⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠸⠄⠀⠀⠀⠙⠃⠐⠰⠌⠶⠆⠌⣂⡀⠀⠀⠀⠄⣀⡀⠀⡲⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠂⠠⣄⠀⠀⠀⠀⠐⠂⢰⠆⣀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠒⢁⠴⡄⠉⢢⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂⠀⠀⠀⠀⠑⠕⠀⠀⠀⠀⢬⠆⠀⠙⢀⢈⡁⠀⠀⠀⠀⠀⠀⢤⣀⠉⠠⡤⠣⣤⠀⢰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⢈⠑⠢⠀⠀⠀⠕⣀⡸⠑⠠⠈⠒⢆⠀⠀⠀⠒⢅⣸⡄⠀⠗⠪⠀⠘⢣⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠈⠡⠒⠂⠉⠁⠈⠀⡨⠀⠀⠀⢤⢡⡈⠂⠄⠀⠀⠡⠀⠀⠀⠈⢀⣀⢆⡀⠀⠦⠔⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠉⠀⢺⡆⠒⠒⠀⢲⡈⠀⠀⠀⠀⠀⠐⠴⢄⠀⠀⣠⠞⠃⠀⠀⠀⠀⡁⢄⠉⠐⠂⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠧⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠒⠢⠤⢄⣀⡂⠀⢠⠁⢠⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠓⠧⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⡄⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⣠⠔⠒⠲⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⢦⡠⠊⠀⠈⠀⡀
⠀⠀⠀⠁⢀⠀⠀⠐⡁⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠⠖⠛⠓⢤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠈⠉⠛⠓⠒⠒⠊⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠒⠤⠄⠀⠤⠄⠚⠁⠀⠀⠈⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠐⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠙⢆⡀⠀⠀⠀⠤⠞⠁⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣠⠞⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡇⠀⠀⠀⠀⠀⢀⠄⢀⡤⠊⠁⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢌⠛⠶⣤⣀⣀⠀⠀⠀⢨⠃⠀⠑⣤⣀⠤⠞⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠀⠈⠉⠙⠛⠛⠛⠓⠒⠈⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷
⠀⠀⠙⢷⣶⣾⡃⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠁⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⣿⠃
⠀⠀⠄⠀⠈⠛⠿⣦⡀⠀⠀⠀⠻⣧⡀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠛⠀⠟⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡅⢰⠋
⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣓⠮⣭⣻⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣄⠒⣠⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣯⣽⡷⠿⠤⣤⡏⢙⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢦⡀⠀⠁
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢄
⢀⣠⣾⣷⣿⣾⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⣑⡄
⣿⣿⡿⣛⣫⣟⣛⠻⡿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⣠⣶⣤⣾⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⡾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡝⠻⣏
⠀⠉⠑⠃⠀⠀⢸⠒⠉⠁⠀⠀⠀⠀⢰⠟⠿⣿⡿⢠⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⢎⢢⣽⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⠈⠛
⠀⠀⠀⠈⢦⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠲⡿⢿⣿⡟⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡟
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⢬⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡇


⢠⠀⡀⢘⠀⠀⠈⠂⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣷⣶⣶⣶⣾⣶⣄
⠘⠛⣿⣿⣷⠀⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠛⠉⠁
⠀⠀⣿⣿⠻⠰⢿⣿⣿⠿⠀⠀⠀⠀⠀⢺⣿⣶⣷⣾⣷⣾⡧⣾⡦⠄⠀⠀⠀⢀⣄⣴⣿⣿⣿⣿⣯⣿⣦⣤⣤⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠
⠀⠀⣿⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠉⣹⣿⠉⠘⠃⠈⠁⠈⠀⠀⠀⠀⢀⣾⣿⣿⣿⣏⣉⣉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⡘⠀⣶⠟⡿⠁⡇
⠀⠀⠛⠻⠀⠀⠀⠟⠇⠀⠀⠀⠀⠀⠀⠐⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⠆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡍⠩⠭⠭⠍⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢡⡃⡆
⠀⣠⣤⣬⣭⣭⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠨⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣄
⠀⣿⣿⣿⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⡖⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠰⠾⠿⢿⣿⣿⣿⡷⠶⠶⠆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡯⣝⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠋⠁
⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠛⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⡠⠀⠀⠀⠀⠀⡀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠂⠀⠀⠀⠐⠀⠄⠂⠐⡤⠔⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠤⠂⠃⠀⠤⢀⣠⡄⢽⡨⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⢰⠤⠂⠀⠀⠀⠀⠀⢠⡠⠂⠀⡀⠀⢀⡼⠀⢀⠀⠔⡠⠔⡲⣺⠛⢛⣌⣀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⠁⠀⠀⠀⠀⠀⠀⠀⠌⢀⡜⠁⠄⠐⢸⠀⠀⠉⠧⠐⠊⡻⠃⣀⠤⠝⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⠃⠀⠀⢠⠤⠿⠂⠅⠀⠊⠀⠀⢈⣡⡀⠢⢀
⠀⠀⠀⠀⠀⣡⠇⢚⠲⣼⡄⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡀⣤⠠⠇⠐⣒⡂⣹⠉⠁⠒⠒⡤⠤⠀⠁⠀⠀⢀⡤⠑⠀⠀⠈
⠀⠀⠀⠀⠉⣹⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠘⡠⠤⠚⠓⠘⠁⠤⣠⠂⠁⠀⠀⠇⠀⠀⠄⠀⠐⢄⠀⠉⠠⡁⠉⠠⠢⣀
⠀⠀⠀⠀⠣⢹⣇⣿⣻⠏⠁⠀⠀⠀⢀⡨⢴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⢞⡠⠐⡀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠲⢄⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡢⠥⢎⣁⣀⡀⠀⠐⢂⣰⠂⢄⣢⡀⠀⠀⠙⡄⣈⠑⠀⢄⠫⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠠⠚⣣⡼⠊⠁⠀⠀⠀⠀⠈⠣⡉⡭⠉⠉⠙⠒⠌⣡⠊⠉⠉⠉
⠀⠀⠀⠠⢂⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡭⢀⡀⠂⢄⠝⠀⠀⠀⠀⠐
⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⠿⠗⠂
⢨⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠠⢄⣀⢀
⠀⠀⠀⠀⠀⠀⢠⡖⠦⡀⠀⠀⠀⠀⠀⢰⠏⢀⣴⢱⡿⢛⡎⣑⢒⣳⣶⣾⣦⣴⢯⡠⡄
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⢶⡀⠀⠀⠀⠰⢬⢣⡄⠈⠹⢮⠑⣮⣿⣿⣿⣿⣿⣿⣦⡷⣿⠄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣾⣇⠀⠀⠀⢠⣡⢻⠙⢉⢳⣼⣀⠸⠉⠉⣿⣿⡟⠉⠻⢧⢿⣆⡴⣠⡄
⠀⠀⠀⠀⠀⠀⠠⣳⣶⣛⣿⠃⠀⠀⣀⡏⠙⠳⣄⣹⡆⠧⠬⣉⠙⠲⠣⢽⣿⣵⣶⡛⠛⣿⠃⣸⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠶⡄⠠⠀⠃⠈⣟⠻⠉⠛⣧⣰⠂⠈⠓⠀⠀⠀⠛⡻⠭⣤⣤⣿⣝⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣓⡛⣆⠉⢽⣿⣿⣧⣼⣏⣛⡉⡉⠈⠙⡷⣾⣻⣯⣽⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⡀⢱⠀⠋⢿⡡⢩⡗⠂⠤⠄⠤⠋⢹⣿⣏⠛⢙⣿⣿⣿⢯⣠⠾⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣿⣿⣦⣶⡵⣿⡾⡫⠊⠐⠀⡤⠰⠒⠺⠛⠪⡟⡛⢿⣮⣍⣙⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⢌⣻⣵⣿⡭⠴⠂⢀⣶⠆⠀⠀⡆⠀⠀⢠⠤⠶⠯⢁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣷⡖⣦⣈⣿⠁⢀⠆⢰⣿⡔⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣆⡝⠀⠘⢁⣠⣴⡎⣩⠂⠀⠀⢢⠀⡀⠀⠀⢠⡞⣹⣋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢁⣀⣀⢹⣄⡇⣾⠿⠿⠛⢻⣀⣠⠘⣦⡰⠀⡠⣰⣿⣟⣫⠏⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠚⠻⠿⣗⠷⢦⣧⣀⢀⠀⠀⠘⢧⡅⠀⠀⠀⣿⣏⣇⠀⣼⡇⢰⠟⠋⢰⡍⢀⣰⣢⣤⣴⣾⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢶⣿⢷⣌⣄⠀⠀⠣⡀⠀⠀⡆⢿⡟⠀⣹⠠⠞⣶⣯⣾⣾⣫⠭⠥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡭⠷⠿⢿⣾⣴⣤⣈⡐⢤⡇⠘⡇⠔⠁⣠⣞⠋⣙⣻⣉⣴⣟⢂⣄⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⡤⠟⠊⠉⠉⠉⠉⠙⠻⢷⣛⣿⣛⠻⠛⠷⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⢿⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢺⡇⠀⠀⢀⣠⣤⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣦⣤⣀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣤⣀⣀⣠⣴⣶⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠉⠉⠛⠋⠁⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠛⢷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠈⠛⠻⣶⣶⣤⣤⣤⣶⡾⠛⠁⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⣠⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠉⠙⠿⢶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣠⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⣴⡟⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣼⡏⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢠⣾⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢠⣾⠛⠁⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⢀⣠⠿⣧⡀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⢀⣾⠋⠀⠈⢻⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⢿⣤⣠⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠟⣿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣿⠇⠀⢨⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⢻⡟⠀⠀⠀⠀⠹⡟⠛⠛⠛⠛⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢿⣄⠀⠀⢀⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⢰⣧⢀⣤⢰⣟⣄⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡃⣼⣿⡧⠿⢃⡿⠟⣡⣶⡎⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣻⣿⣷⣶⣿⠍⠛⠿⠋⠻⣿⢣⡄⠈⠃⣾⣿⣿⠇⡚⢛⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠳⡙⠛⠿⢋⠈⢻⣿⣿⣷⣦⣿⣿⢰⣆⠹⠛⣡⡖⢀⣿⣿⣥⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣷⣀⣾⡇⢸⣷⡄⠙⠿⣿⣿⡿⡟⣹⣟⠀⣾⣿⠗⣿⣿⠿⡣⠾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⢷⣶⣷⣤⠿⠃⢈⣻⣟⢸⣧⢀⣠⡆⢻⡅⠉⢀⣿⡯⣠⡆⠀⢨⣴⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢾⣧⡝⠄⣙⠿⠟⢷⡛⠿⣿⣿⠟⢧⣸⣿⠇⢸⣿⣴⣿⣥⣴⣻⡁⠀⣼⣻⣟⣋⣀⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⡟⣒⡛⢦⣄⣀⣿⣆⣀⠻⣿⣾⣿⡔⣦⣼⣿⢋⣀⣠⣾⣭⣷⠿⢋⣷⡶⠀⠸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠿⢋⣾⣿⡿⠉⣽⡿⠉⠙⠛⢿⣷⣦⣿⣧⣼⣿⣿⡿⠛⠉⠀⠘⠻⠗⠻⣿⣿⣆⠀⠙⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠛⢛⣿⣿⣿⠿⢷⣦⣤⣀⣀⣤⣤⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢹⣶⣷⣾⣿⣿⣿⢻⣥⠴⠆⠀⠀⠉⠛⢿⣿⣿⣿⣿⢦⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣤⣷⣾⣿⣿⣿⣿⣿⡿⠇⣠⣤⣤⡀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠚⠉⠉⠛⠿⠿⠶⠆⠀⠀⠈⠻⣆⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢲⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠈⠓⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣾⢡⣿⣿⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⠀⠀⠀⠟⠛⣿⣿⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠙⠉⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⠁⠙⠻⢿⣟⣉⡈⠉⠛⠛⠙⢛⣋⣹⣿⠟⠛⠵⠂⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⣿⡏⢻⡏⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠁⠀⠐⠇
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠀⡴⠺⠓⠆
⠀⠠⡇⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠺⣀⡁
⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢱⣶⣶⡾⢿⠷⢶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠐⠀⠐
⣆⠠⠤⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠋⢀⣄⣀⠀⠈⠛⠁⠀⠀⠐
⣯⠙⠲⡄⡈⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⢀⡄⡀⢀⠀⠀⠉⠀⠀⠀⠸⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔
⠙⡤⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠘⢿⡿⡇⠀⢀⠀⢀⢠⠂⠀⠀⠀⠘⠀⣠⠀⠀⠀⠂⠀⠈⡚⠀⠀⡄
⠄⠗⠓⡄⢀⡄⠈⡗⢠⡌⠂⠀⠀⠀⠀⠀⠀⢿⠁⢂⠏⢠⠀⠈⠀⣠⠄⢃⠄⠳⠟⠂⠀⠀⠀⠑⣌⠱⠚⠂⢀
⠀⣰⠲⡄⠀⡅⢀⡟⢾⣇⠀⠀⠀⠀⠀⠈⠒⠠⠃⠤⠆⢰⠂⣤⡰⠇⢸⠀⡄⣸⡞⠀⡀⠀⠀⠀⠀⠙⠖⠀⠈
⠀⠁⠀⢱⢀⣤⢨⡇⢸⢻⣎⠁⠀⠀⡄⣼⠆⡷⠀⠀⢌⢙⠂⠃⢍⡁⠻⣤⠳⠝⠃⠀⠑⣤⢀⠀⢀⠀⡀⠁⠆
⠔⠄⠈⠒⠋⢗⣠⡇⢂⠉⣿⠳⠀⠄⠀⡏⠀⠈⠳⣔⡀⢈⣀⡀⠀⠀⢾⣾⡧⢤⠀⠐⣤⠒⡖⣆⡓⠀⠈
⠀⢾⠾⠔⣶⣟⢙⡟⠋⡐⠿⠀⠇⢐⢟⡟⢠⣄⡒⠊⠭⣹⠀⠀⡴⢀⠈⣽⣦⡔⡀⢠⣤⢰⡿⣄⡁⡀⠀⠘⢣
⣥⢰⣿⡦⡰⠿⢾⣷⠄⠀⢣⠐⣌⣩⣿⣤⣜⣀⠀⠀⣀⡽⢒⣟⢠⣽⠈⡄⠤⢼⠃⠆⠀⠀⠀⠀⢳⠏⣔⣿⣠
⡤⢺⢿⣷⠟⠻⡖⠃⠀⢀⡈⢸⣫⣿⣿⣿⡿⠉⢄⡀⠹⡳⡷⠆⠻⣿⣿⠗⠂⡬⠜⢓⠠⢀⣯⣾⣁⣴⣣⣄⠈
⠁⢼⣗⣤⣄⢀⣥⣺⣶⡦⣦⣻⣉⢡⣶⠚⣗⣀⡴⠀⣤⣟⣆⡁⡘⠻⡞⠂⠀⢃⠰⢺⠐⠿⡝⣜⣧⢿⢷⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⢐⡥⠈⠂⠒⠐⠑⠂⠒⠂⠐⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠄⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠊⠀⠀⠀⠀⢂⡁⠤⠀⠀⢀⠀⠈⣀⠦⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣠⢄⡀⠀⠉⠈⠁⠀⠀⠀⠀⠐⠆⠲⠋⠀⠀⠀⠀⠄⠀⡀⠅⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠀⠈⠈⠳⣕⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠘⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠳⡄⠀⠠⡢⡢⡁⠀⢁⠽⠟⠀⠀⢀⠀⠂⣠⠀⠀⠀⠄⠪⡄⠀⠀⠈⠁⡀⡄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠸⣗⠄⢀⡌⠔⠊⠁⠀⠀⠀⠀⠀⠜⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠂⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠘⣆⠀⡀⢀⡠⠖⠊⠁⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠖⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢶⠟⠓⠊⠁⠀⠀⠀⡀⠀⠂⠀⠀⢤⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⢐⣈⠵⠼⠛⢕⣉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠸⠈⠰⠀⠀⠂⠀⠀⠀⠀⠀⠦⠄⠀⠀⠄⠀⢘⣀⢔⠄⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠳⠾⠉⠀⠀⠀⠀⣈⣩⡶⢦⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⢤⠄⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠄⠢⡎⠀⠀⠈⠀⠤⠠⠴⠈⠀⠀⠈⠀⠀⢀⣠⣎⢊⢌⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⠒⠩⠢⠁⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⢸⠀⢡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠃⠐⡐⠂⠒⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣟⠀⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠂⢰⠻⠾⠀⠀⠲⣠
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣸⠀⠀⠀⠀⠀⢉
⠀⠀⠀⢀⣀⠀⣄⣦⠀⠔⠛⠐⠀⠠⠀⢠⣠⡌⣤⢀⠀⠀⣀⠄⠠⠀⠄⠠⠀⠄⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠸⠇⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⠖⠉⢠⠴⠚⢩⡍⠓⠦⡄⠉⠲⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡟⠶⢴⠃⢺⣿⠿⠿⣿⡗⠘⡦⠶⢻⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢿⠸⢿⣇⠀⠀⣸⡿⠇⡿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣆⠺⠿⢿⡿⠿⠗⣰⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣨⠓⠦⠬⠥⠴⠚⣅⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⡧⠤⠤⠤⠤⠤⠤⢼⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⣀⣘⣃⣀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠋⠀⣴⡿⠿⠏⠹⠿⢿⣦⠀⠙⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡏⠙⣸⡿⠇⢀⣴⣦⡀⠸⢿⣇⠋⢹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢻⣷⡄⠘⠿⠿⠃⢠⣾⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠻⣧⣶⣄⣠⣦⣼⠟⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡔⠈⠙⠛⠛⠋⠁⢣⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠚⠛⠛⠛⠚⠓⠻⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠈⢸⡇⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣧⠔⠉⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣶⣶⣶⡆⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⡄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⢸⠇⠀⠀⠀⠀⠠⠞⠉⠻⠟⠛⠒⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢼⣿⣿⡟⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿
⠀⠀⠀⠀⠠⢅⠀⣴⠀⠀⢀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡁
⠀⠀⠀⢉⠀⠠⠂⡿⡅⢀⡸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣦⠀⠀⠀⢀⠀⠂⠀⢸⠁⡀
⠀⠀⠐⣩⡀⠁⡖⣿⡃⢜⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⠲⡆⣿⢹⣇⣴⣹⠆⠀⠀⠀⢂⠀⠈⢸⠆⠀⠀⡀
⠀⠀⠐⠢⠸⢣⣗⣷⣢⠟⠚⠉⠀⠀⠀⠀⢸⣅⠓⢀⡔⡆⣼⠉⣇⢠⣸⡦⠻⠴⠃⠛⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⣼⣧⠎⠁
⠀⠀⠀⠀⠀⠀⢁⣿⡄⠈⡈⠀⠀⠀⠀⠀⣠⠍⣿⠘⣯⠄⠛⠒⠙⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠰⢠⣵⢿⣻⣟⣷⡶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡠⠞⠙⠿⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⣶⣶⠤⡄⠀⠀⠀⠀⠀⠀⣐⢄⢳⡤⡕⡀⣀⠀⠀⠀⠀⡄⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⡂⣌⡁⢺⡳⠆⢡⣂⠀⠀⠀⢸⣧⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠙⢿⣿⣿⣷⣦⡄⠀⠴⡄⡤⢞⢹⡬⢖⡾⣴⠲⠄⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡟⠁⠀⠈⣗⣦⣝⢸⡱⣾⣿⡾⠖⠀⠘⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡻⢺⡟⠉⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⠈

⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣁⡠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠴⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣛⡶⠤⣴⣷⢆⠶⠀⠄⠀⠀⠀⡀
⠪⣔⡀⠀⠀⢠⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⡀⠀⠀⠉⠰⠦⡠
⠀⠈⠃⣤⡄⠀⠄⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡰⠼⢎
⠀⠀⠀⠀⠿⣤⠃⠀⠘⣄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠁⠀⠀⠣⠷⠶⢿⡆⠀⠀⠀⠀⠀⠔⠦⠁⡀⡀⠀⠢
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠹⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⢀⣤⣤⢀⣤⡰⡄⠀⠀⠀⢠⠠⠟⢃⡅⡀
⠀⠀⠀⠀⠀⠀⡟⠐⡆⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠃⠐⠆⠀⠀⣀⠈⠉⠉⡛⠣⠈⠂⠀⠂⠁⡀⠾⡛⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⡀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⠀⠀⠀⠘⠀⡀⠋⠀⠄⠀⠀⠄⠀⠑⠠⢈⠋⠈⡡
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢇⠀⠀⠀⠀⠀⠀⠀⠲⠋⠙⠟⠲⠃⠿⠃⠢⠊⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⣀⢀⡀⢀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣼⣿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢤⣀⠀⠀⣿⡟⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠈⠑⢞⡶⢦⣄⠉⠁⠀⠘⠁⠀⠀⣠⠀⠐⣿⡇⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⠷⠄⠀⠀⠀⠀⠀⠟⠀⠀⠹⡇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠹⠆⠀⠠⡀⢀⡇⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣶⠶⠖⠀⠀⢀⡤⠀⠀⠀⠀⠈⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠁⠄⠀⠙⢿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠒⠛⠛⠻⠷⠶⠤⢀⠀⠂⠀⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠁⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠃⢘⣿⠋⣼⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡁⣾⡿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⢿⣿⣯⠀⠀⠈⢻⣿⣯⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⢛⠉⠀⢺⣿⣿⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠸⣿⣿⡄⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⠿⣿⠿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣾⣷⢦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢽⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⢸⣇⠈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⡇⢻⣿⡇⠸⣿⡆⠻⢦⡍⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡿⠋⠩⣻⣟⣛⠟⠛⠛⠛⠃⠙⠉⠀⠈⢿⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⡿⠋⢠⣑⣾⣿⣿⣿⣿⣿⡿⠆⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⠁⢀⣴⠛⣏⣹⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⣿⣿⡇⢰⣿⣧⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⡇
⠉⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠶⠖⠦⠀⠶⠶⠶⠶⠶⢦⠤⠤⠤⠤⢤⠤⠿⣿⣿⣿⡧⢸⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⢘⠛⣿⢛⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡛⣿⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⣦⣽⣧⡴⠀⠴⣿⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣶⠃⣀⣤⣄⡀⠈⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣟⡁⠀⢲⣶⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣅⡀⣛⣻⣛⡃⣠⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⣀⣹⣯⣀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⢷⠹⣿⡿⢡⡟⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⢠⡟⣿⣿⣿⢹⡆⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⡇⣿⢿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⢸⣷⣿⣷⣾⢃⣿⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⢸⠇⣿⡿⣿⡘⡇⠀⠀⠀⠀⠀⠀⡇⣾⠟⢻⡇⣿⢸⣿⣿⡇⢻⠀⠀⠀⠀⠀⠀⢸⢰⡿⠉⣿⣿⣿⣿⣿⣹⣾⡏⠀⣿⡠⡇
⠀⠀⠀⠀⠀⠀⢸⠤⡿⠀⢹⡧⣇⠀⠀⠀⠀⠀⠰⡃⣿⠀⠘⡗⣻⢼⡏⠙⣷⢸⠀⠀⠀⠀⠀⠀⠺⢹⡇⠀⢹⡯⣿⡿⣿⡏⣿⡇⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢀⡇⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡷⠋⢸⠁⠀⣿⠨⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠆⣿⠀⢸⡇⢾⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢸⡁⡿⠀⠸⣇⡼⠃⠀⠸⢤
⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠉⠂⣸⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠀⠀⠣⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡆⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠆⠸⡄⢻⣿⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢷⠍⣉⣉⣛⣻⣷⣤⣻⡇⠀⢀⠴⠖⠛⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠙⠻⣇⠀⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣤⣤⣌⣳⣦⣆⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣷⣾⡿⠟⠻⡟⠁⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠱⠀⠀⣠⡶⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣀⠀⠀⣇⣰⣯⣔⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢮⡳⣦⣼⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⣧⠀⠀⣀⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣟⠟⡉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⡾⠚⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠟⠉⣡⡴⢛⡽⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠁⡆⢰⢏⠜⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⣮⡳⣄⠀⠀⣿⣤⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠏⠹⣆⠀⠀⠀⠀⣾⡇⠀⠀⠀⣷⣄⠀⠀⠀⠀⢸⡇⢭⣿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢂⠀⠀⣠⢿⢿⠀⠀⢠⡿⠏⠳⢦⡀⠀⢸⣠⣾⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠀⠀⠀⠘⡆⠀⢩⠈⠘⡇⠀⠸⣣⠀⠂⠌⣷⠀⠈⣿⣿⣷⡿⠟⠛⠉⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢠⣴⠃⠀⠈⠣⠀⣜⠀⠀⠉⠛⢶⣷⣣⡴⠚⠉⠀⠉⠉⠒⠤⠤⠤⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⠏⠀⠀⠀⠀⠀⣿⠀⠀⢀⡠⠼⢯⣥⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠲⢾⣷⠤⠤⠤⠤⠤⠤⣾⠖⠊⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣦⡀⠀⠀⠀⢰⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡹⣧⠀⠀⣀⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣧⡹⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣉⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⡄⠀⣄⠀⠀⠀⠀⠀⢸⣍⠋⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⡴⠋⢊⠔⡡
⠀⠀⠀⠘⡀⠣⢀⣀⣀⠀⠀⠀⠀⠀⠈⠟⠀⣔⠉⣦⠀⠀⠀⠀⠀⠘⡓⠠⢀⡀⢠⠓⡄⠀⠀⣰⣍⡊⠐⡡⢀⠔
⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⢼⡣⠀⠀⠈⡶⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠉⠉⠁⠀⢸⡌⠈⠉⢀⡴⠃
⠀⠀⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⡀⠀⠀⠀⠀⠀⠀⠑⠚⠊
⠀⠀⠀⠰⣊⠾⣸⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⢤⢀⠔⡭⠁⠀⠀⠀⠀⠙⡗⠯⠳⡀
⠀⠀⠀⠀⠙⡆⠙⢆⠀⠉⠀⠀⠀⠀⠀⣔⢎⡹⢥⣮⡀⠀⠀⠀⠀⠀⠈⢢⡰⣉⢎
⠀⠀⠀⠀⠀⠈⢆⠀⠑⢄⠀⠀⠀⠀⠀⢠⠧⣪⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⢀⠄⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⡴⠢⣄⡀⠀⠀⠀⠀⠀⡃⢊⠥⣰⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡼⡓⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠽⠲⠶⠯⣅⠀⠀⠀⠀⠀⠐⠤⠭⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⢷⡦⠀⠀⠀⠀⠀⣇⣰⠶⠲⣆⢨⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠴⡦⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⣁⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠠⢀⣀⠔⣁⣠⠔⣀⠀⠀⣘⣻⣕⣤⣤⣄⡘⠄⠀⣃⣀⠀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠈⢦⡘⠂⡉⠻⢧⠀⠈⠳⡈⠈⠷⡀⠙⠓⠢⠄⠀⠀⠈⠂⣞⡁⠀⠀⠀⠨⠓⠨⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠒⠂⠁⠐⠀⠓⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢲⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣥⣀
⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⠂⠂⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠀⡂⠄⠲⢀⡄⠈⠆⣰⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⡀⠐⣄⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠒⠙⠚⠛⠛⠛⠉⠙⠛⠁⠀⢸⢿⠷⠀⢀⠘⠋⠀⠀⢋⠴⢙⠖⠀⠀⠈⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠂⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣞⣏⣧⣤⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣾⣾⣼⠏⡀⠀⢀⣠⠴⣩⠖⣄⠀⠓⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⢻⣷⠆⠀⠠⠊⢋⡀⠀⣴⢯⡤⠄⠀⠀⠠⠐⠯⡇⠺⡀
⠀⠀⠀⠈⠆⠀⡤⠀⠀⠠⡀⢰⡄⠀⣿⡶⠊⠠⠠⠊⣤⢞⡵⠟⠃⠄⣠⠀⠀⠀⠀⠁⡀⢀⠒⠤
⠀⠀⠀⠀⠀⠀⠱⠀⡇⠐⠀⠙⣷⣦⣿⣷⣒⡒⢬⠵⠚⠁⠀⠀⠀⠀⠂⠀⠠⡀⠒⠀⡏
⠀⠀⠀⠀⠀⠀⠐⠈⠈⠕⠂⠤⠤⠉⠁⠂⠀⠁⠀⠀⠂⠂⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠉⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠔⠀⠀⢀⣀⠎⠀⠀⠀⠂⠀⠀⠸
⠀⠀⠀⠀⠀⢀⠀⢀⠠⢚⡀⠀⡀⣀⣀⣀⣀⠀⠄⠤⠤⠔⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⢒⠈⡀⠀⣀⠀⠉⣁⡀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⢀⣴⠿⡿⠀⠀⠀⠀⠀⡀⢀⠀⡞
⠀⠀⠀⠈⠉⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠿⠗⡰⠁⠀⠀⠀⠀⠀⠀⢹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⣛⡩⠍⢊⡠⠊⠀⠀⠀⠀⠀⠁⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⢛⡡⠐⢊⡁⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠛⢁⡤⠂⠁⣠⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⡇⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⠾⢩⢀⠔⠁⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠃
⠀⠀⠀⠀⠀⢀⠔⣡⠞⢁⣠⠚⠁⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣜
⠀⠀⠀⠀⢠⢫⡾⡡⠒⠋⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡠⠎⡴⠁
⠀⠀⠐⢢⣏⡉⡇⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡂⠊⢀⡵⠒⠒⠞⢉⡅⣰⠉
⠀⠀⠀⢷⣿⣿⠵⠋⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⠠⠌⠀⢈⠅⠀⡀⢂⠬⠄⠂⠔⢀⣨⡕⠅
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⠄⢀⡀⠀⠀⣀⠀⠐⠋⠀⠀⠀⢐⡴⠀⠂⠀⠓⢂⡥⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠈⢀⡀⠀⠠⠒⠀⢈⡄⡀⠐⠠⠄⠅⣈⠈⠔⢀⠲⠊
⠀⠀⠀⠀⡄⠠⣌⣠⠤⠄⠈⠀⠀⠀⠐⠑⠂⠀⠈⠀⠉⠈⠐⠢⣀⠤⠞⠁
⠀⠀⠀⠀⠌⠉⠠⠤⠉⠁⠒⠛⠂⠶⠤⠤⠤⡥⠶⠄⠐⠒⠛⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠈⠉⠀⠈⠀⠉⠀⠐⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠚⠁⡂⠀⡀⠀⢈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢁⠰⠟⠁⠀⠀⠀⠀⠌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠞⡰⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠠⢂⡀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⠀⠂⠀⠀⠀⠀⠀⢦⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠂⠀⠀⠈⠁⠏⠀⡀⠀⠀⠀⠀⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⠢⠀⠀⠐⠀⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⢠⠏⠀⠀⠰⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠱⡄⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠒⠀⢹⡄
⠀⠀⠀⣰⠏⠀⠀⠀⠂⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠔⡀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠹⣆⠀⠀⠹⣆⠀⠐⠀⠀⠀⠀⠀⠀⣇
⠀⠀⣰⢃⠀⠀⠀⠀⠁⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠊⠐⠡⠊⠀⠀⠀⠐⢦⡀⠀⠀⠘⢆⠀⠀⠘⢶⠁⠄⠘⠆⢠⡀⠀⢀⡀⠩⠀⢻
⠀⢰⠏⠀⠀⢄⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢦⣀⠀⠀⠀⢀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠈⢧⠈⣀⠠⠘⣆⠀⠀⢥⠀⠀⢸
⠀⡾⠈⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠸⡄⠀⢼⡄⠀⢸
⢠⠁⡈⠀⠀⢤⠙⡁⠄⠀⠀⠀⠀⠂⠀⠀⠀⠈⠓⢦⣄⠀⠀⠔⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢳⠀⢘⡇⠀⢸
⢸⠀⠀⠀⠐⠀⠐⡂⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢳⡀⠤⠀⠀⠀⠈⢧⠀⠀⠈⣇⠀⣿⠀⢸
⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠈⢳⠀⠀⠀⠀⠀⠈⠇⠀⠀⢹⠄⢸⡀⢸
⢸⠀⡄⠀⠘⠿⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⡇⢸⡇⢸
⢸⠀⠸⣈⠗⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠉⠉⠉⠉⠉⠓⠢⣄⡀⠀⠀⠇⠸⠇⠘
⠸⡇⠀⠈⠀⠀⢀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠰⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡀
⠀⣧⠀⢠⡇⠀⠀⡆⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢠⡇
⠀⢹⡀⠘⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠀⠀⠐⠈⣁⠄⢸⠃⠀⠀⠀⠀⠀⣀⡀⠀⠂⡀⠀⠀⠀⠀⠐⠀⠘⣦⢸⠇
⠀⠘⣇⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠛⠉⠉⠀⠀⢀⡴⠖⠛⢉⡴⢀⣼⠀⠀⠀⠀⠀⠊⠁⠀⠱⡋⠁⠠⠖⠀⠀⠀⠠⠀⢹⡞
⠀⠀⢻⡀⠀⡄⠀⠀⠀⠀⠠⢳⢄⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⢀⢉⠀⠈⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⠶⠀⠀⠁⠀⠀⡀⢸⠇
⠀⠀⠀⢷⣴⡁⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠏⣠⠂⣠⡿⢦⣄⠀⠀⠀⠀⢀⣴⣇⣀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠈⢣⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⡟⠘⠁⠀⠋⠀⢸⡏⢟⣷⠚⠚⠛⠷⢄⠉⠙⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠳⡀⢼⡀⠀⠀⠈⠀⠀⠀⡉⡛⠛⠁⠀⠀⠀⠸⠃⠀⠀⠐⠀⠀⠈⠡⠸⠞⠃⠘⠄⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢮⠙⠒⠀⠀⠀⡀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⢦⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⡀⡀⠄⠀⠈⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⢤⡀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠒⠠⠄⠀⠀⠀⠀⠀⠤⠐⠂⠠⠀⠊⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡉⠘⣦⣀⠀⠀⠀⠀⠀⠐⠂⠴⠂⠀⠀⠀⣤⠴⠆⠀⠈⠀⢀⠀⢠⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢐⣶⣾⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠾⠯⠭⠤⠤⠽⠿⠅⠀⠒⠚⠛⠛⠉⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⡀⠀⠀⠀⠀⠴⢿⡿⠦
⠀⠼⢿⡿⢷⠀⠀⠀⠀⠀⠀⠀⢨⣍⣿⣩⠄⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣭⠇⢀⣀⣀⠀⠹⣾⣷⠏
⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠋⠀⠰⣤⣧⡴⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣋⠀⡛⠿⠛⠂⣤⣾⣧⣤
⠀⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⢀⣹⣿⣁⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⢿⠹⣿⠟⢰⢻⣿⣿⡿
⢸⢻⣿⣿⡟⣇⠀⠀⠀⠀⠀⢰⡇⣿⣿⣷⢹⡿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⡏⣼⣿⣿⣸⣷⣿⣷⣿⢸⡿⢿⡇
⠿⢸⡿⢿⡇⠿⠀⠀⠀⠀⠀⢸⣁⣿⠙⣿⢸⢃⣿⣷⡇⢰⠀⠀⠀⠀⠀⠀⡇⣿⠁⢻⣿⣿⣿⣿⣿⣿⠃⢸⡧
⡞⢸⠇⠘⡇⢺⠀⠀⠀⠀⠀⠘⠉⡇⠀⢻⢸⢿⡏⠈⣿⢸⠀⠀⠀⠀⠀⠈⠃⣿⠀⢸⡇⣿⠛⢿⡉⣿⠀⠀⡇
⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⢸⡇⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⣿⠀⢸⠃⢿⠀⠀⡇
⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠸⣄⢸⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠘⢦⡇⠀⢸⡖⠋⠀⠀⠙
⠀⠁⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⡀
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣿⣦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄⠀⡠⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠰⠀⠀⢀⠀⠁⠀⠐⠃⠸⠀⠙⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⢀⣾⣿⡦⠄⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⡞⠀⠁⢀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⣤⠂⠀⠀⡴⠻⢷⡟⠀⣠⠀⠀⠀⠚⠉
⠀⠀⠀⠀⢀⡀⢀⠀⢀⢼⣿⠛⠁⠀⠀⢸⣦⣤⠈⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠋⢢⣴⠀⢀⡏⠁⠀⡐⠛⠹⠀⠀⠀⠀⣄⣴⣄⡀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡞⠀⠀⡼⠀⠀⡌⠀⠀⣤⣾⣿⡏⠉⢳⡄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⠀⢰⣿⠋⠉⠀⡀⠀⠋⠀⠀⠐⠱⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⡀⠰⠆⠀⢁⡠⣀⡀⠁⠀⠀⠀⠆⠘⠋⠀⠀⠀⡀⢀⣾⡄⠀⣳⡄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡁⠚⠃⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠙⠋⢠⣤⠾⡟⠂
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠰⠀⢻⠂⠀⢸⣷⢠⡄⠀⢀⠈⢀⠀⠀⠀⠀⠀⠁⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢰⠈⠙⠂⡀⢋⠀⠈⣇⢲⡿⠷⠋⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⢠⠉⢾⡀⠀⣏⡼⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠂⠀⣷⡄⡟⢀⣶⣧⠀⠀⠀⠐⠂⣤⣁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⣄⡄⣧⠉⡄⠸⣿⠋⡀⢀⣃⠀⠀⠛⠛⠋⠀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢳⣿⡅⢹⡀⠁⠀⠉⠀⠷⠏⣿⠏⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢻⠁⢸⡁⠀⠀⣀⣼⣶⣶⠠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠸⡇⣸⣵⣦⣾⠿⢋⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠛⠓⠀⠀⣧⡟⣟⣉⣥⣶⣼⢿⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⢹⠇⣿⢀⢙⣿⣿⣿⣷⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠈⠛⢦⣀⠀⠀⣿⡠⣿⠏⠿⠿⣿⣹⡟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠻⣦⠀⣿⣷⡇⠀⠀⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⢀⣧⣤⡄⠒⠒⠂⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠿⠟⠁

⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠅
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠐⠃⠀⠀⠀⢀⠂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⢄⠀⠀⠆
⠀⢸⠀⠀⢀⢀⡀⠄⠀⠀⠀⠀⠐⠂⠐⠄⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠰⣿⡷
⠀⢸⠀⠀⣠⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⠀⠀⠀⠈⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⢀⠀⠀⠀⠀⠀⠀⢂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⢄⠀⠐⠈⠀⠀⠀⠀⠀⠀⢂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠤⠄⠀⠀⠀⠀⠁⣀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠤⠀⢠⢄⡈⠉⢅⡢⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢤⠀⡠⠀⠐⠓⠁⡈⠠⠀⠡⠀⠀⠀⠀⠁⠂⠀⠀⠤⠤⠄⠔⠐⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠌⠠⠊⠉⠈⠣⣝⠀⢀⡀⠀⠀⠻⢢⢅
⠀⢸⠀⠀⠀⠀⠀⠀⠢⠂⢀⢄⠀⡸⠀⠀⠀⠁⠀⠀⠀⢇⢀⡱
⠀⢸⠀⠀⠀⠀⠀⢀⢠⠀⠈⠈⠢⠙⢆⠀⠀⠦⡀⠀⠀⠀⠈⠂⠀⢀⠠⢄⠐⡄
⠀⢸⠀⠀⡠⠐⠈⠀⠸⠀⠀⠀⠀⠀⠢⢉⡒⠐⠊⢁⠡⠐⠓⠈⡁⠀⠀⠀⡱⠁⠀⢀
⠀⢸⠀⠄⠱⠀⠀⢀⡀⠧⠰⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡁⠠⠐⢈⠔⠁⠀⠀⠀⠈⠀⢄
⠀⢸⠀⠄⠀⠱⠂⠙⠀⣠⠁⠀⠀⢀⣀⠌⠠⠀⠀⠠⠂⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⢸⠀⠀⠀⠀⠀⢂⡐⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠓⠤⡀
⠀⢸⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠄⠒⠈⠀⠀⢀⠠⢤⠊⠁
⠀⢸⠀⠀⠀⣠⠁⠀⠀⠀⡠⢊⠌⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⠠⠄⠂⣁⣤⢐⠊
⠀⢸⠀⠀⠠⡀⠈⡢⠔⠈⡠⢀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠤⠂⣉⠠⠔⠀⠃⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠜⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠢⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⠜⠀⠀⠀⢀⡁⠀⠀⡠⠐⠀⠀⡤⠤⢄⠀⠀⠀⢀⠤⠠⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⡆⠀⠀⠀⠀⠉⠑⢄⡌⠀⠀⠀⠀⠈⠒⠈⣐⠢⢄⠛⠐⠒⠒⠤⡀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡄⠀⡗⠂⠉⠀⠈⣄⠀⠀⠓⠤⣦⣀⡙⡄⢀⡴⠃⠀⠀⡘⠑⠉⡾⠀⠀⠀⠀⠀⠈⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⢣⠀⠀⠀⠀⠘⢆⠀⠀⠀⣀⣄⠈⣏⠡⠄⠀⠀⡰⠁⠀⢠⠁⢠⠃⢰⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠣⣀⣔⠙⢧⡿⣇⡠⠌⠦⠺⡅⡰⠀⡌⢀⡶⠀⡼⠋⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠛⠲⠼⢤⡆⠀⢸⠀⣀⡠⠤⠐⠊⢁⡨⠏⠲⠊⢹⠉⠀⠀⠀⠀⠈⠑⣴⠀⠀⡱⠞⢤⡄⣀⣁
⠀⠀⡠⠒⢲⠄⢀⠤⣀⠀⠠⠒⠀⠙⣆⣋⢉⡀⠤⣴⡂⠉⠀⢠⠒⠒⠀⠉⠉⠉⡖⠤⠤⢄⣀⠥⢄⠜⠀⠀⠀⠀⠀⣀⠤⢤⠄⠒⢤
⠀⠀⠀⠉⠁⠨⢄⡠⠤⢃⡴⠢⡤⢔⣺⡋⢠⡊⠉⠀⣑⠀⠀⠀⠉⠒⡄⠀⠀⡔⠁⠀⠀⠀⠀⣤⡤⠷⠖⢣⡀⠀⠘⡄⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⡀⠀⠀⣏⣹⠁⠀⠙⠤⠜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠤⢰⣞⣅⠀⠀⠑⠒⠢⠤⠤⠜
⠀⠀⠀⢀⣠⠖⠒⠌⠗⢨⠤⣤⠞⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠦⡠⠔⠉⠐⠒⠊⠁⢰⠁⠀⠈⢢
⠀⠀⠀⢹⠀⢀⠏⠾⠸⢿⠖⠊⠀⠀⠀⢀⠖⠉⠀⠀⡉⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢃
⠀⠀⠀⠑⠤⠃⠀⠀⠀⠌⠀⠀⢀⠤⠤⠆⠀⡤⠜⠂⠁⠚⠢⡂⠀⠀⠀⠀⠀⡤⠚⠂⠉⠒⠢⡄⠀⠔⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠃⠀⠀⠀⡰⠁⣠⣾⣿⣷⡄⠘⡄⠀⠀⠀⡜⠀⣴⣾⣿⣶⡄⠙⡄⠓⠄⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⢄⠈⠢⣀⠀⠀⣃⠀⢿⣿⣿⣿⡿⠀⡇⠀⡀⠀⡄⠀⣿⣿⣿⣿⡇⠀⡧⢀⠀⠀⠱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠱⡄⠀⠀⠀⠘⢆⡈⠙⠛⠋⢀⠔⠀⠋⠀⠁⠘⢄⠈⠛⠛⠋⣀⠜⠀⠀⠉⠂⠔⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠑⠦⣅⣀⡠⢺⣂⠀⡈⠑⢆⡈⠁⠀⠀⠀⠀⠀⠀⡜⠉⠁⠂⠈⠁⣀⣀⡫⠒⠤⠤⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⣀⡀⠀⠀⠀⡐⠊⣅⠀⠀⠈⠑⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠶⣂⠀⠀⡠⠄⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠙⢦⣰⡬⠿⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠿⠑⣆⠎⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠄⡄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠈⡄⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠒⠉⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠧⠤⠀⠀⠀⠤⠴⠓⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣦⣴⠟⠉⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢤⣀⡀⠀⢆⠀⠀⠠⠼⠛⠉⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠈⡩⠋⠙⣦⠞⠛⠲⠆⣤⣤⣴⣖⣋⣁⡀
⢀⣥⣤⣤⣶⣾⠀⠀⢀⠏⠀⣠⡤⠶⠻⣿⣿⣿⣿⡿⠟⠓⣀⣀⡀
⠐⢟⠋⠹⡏⣿⠀⠀⣮⢠⣾⡋⡠⡠⠑⠀⠀⠀⠀⣠⠴⠊⠀⠀⠠
⠀⠀⠑⣆⠁⡁⠀⠀⣧⣿⡿⣳⡞⠁⠀⠀⠀⢀⡞⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠸⣾⣿⣆⠀⡟⣻⣹⣟⡀⠀⠀⣀⡴⠋⢀⠄⠀⣸⡏⢀
⠀⠀⠀⠜⠻⢿⡟⠛⢻⣧⠟⠛⠛⠛⠛⠋⠀⠀⠀⢀⡀⢹⡇⠀⡓
⠀⠰⠀⠀⢀⡞⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⣰⣮⠀⠀⢣⠀⡇⠀⠀⢄
⠀⠀⣠⣾⠛⠻⠦⢀⡀⠀⠙⢷⡀⠀⠀⠀⠀⠀⣿⡃⠀⠀⢀⡀⢱⠀⠀⢸⣷⡀
⠀⠘⢿⠙⠛⠓⠂⢀⡠⠤⠤⠀⠙⢦⡀⡀⠀⠀⣾⣧⠀⢀⣼⠃⢨⣧⡀⣸⣿⣷⡀
⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⢧⡤⣄⣿⣿⡿⠊⠀⠀⢀⣿⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢹⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⠀⠀⠀⠀⣀⠈⠻⣿⢤⡀⠀⠀⠄⢧⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⠋⠁⠀⠀⠀⠀⠚⣿⡄⣀⢠⡝⢧⣂⣄⠉⠛⢹⠁⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⣀⡐⠀⠠⠄⠀⢨⣿⣿⣇⠈⠀⠉⠳⢆⣤⣼⠀⠀⠀⠀⠈⠃⠀⠰⣴⡔⠀⠀⠁⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠴⠂⠙⠷⠀⠈⠀⠉⠙⠳⢾⣿⠋⣍⢹⣿⣶⣀⠀⠀⠈⠙⢿⣶⡀⠀⠀⠀⣠⡞⠳⣾⣧⠀⠀⠀⠸⣿⡏⠹
⠀⠀⠀⠀⠀⠀⠀⠤⠖⠚⠿⣶⣶⣤⣦⣾⣛⣿⣿⣾⠋⠈⠉⠙⠳⢦⣄⡀⠙⢿⢦⣀⠈⠉⠀⠀⠙⠋⠅⡀⠀⠀⢀⠹⣷
⠀⠀⠀⠀⠀⠀⠲⠗⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⢀⡀⢈⠛⢦⣄⠳⡸⡆⠀⠀⡄⠀⠀⢀⣷⣀⠀⠈⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠋⠉⠉⠉⠛⠿⣇⡻⣿⣄⠀⢸⣲⡴⣾⣿⣾⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⣻⠿⠿⠿⠿⠿⢿⣏⣽⡿⠀⠀⠀⠀⠀⠀⠈⠱⡝⠻⣆⡬⡛⣦⣼⣿⠛⣧⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣶⡾⠿⠶⢤⣀⠀⠀⠀⠀⠀⠀⣀⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢻⡄⠘⣽⡛⠿⣀⢩⣽⠞⠙
⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠈⠑⠐⠂⢀⣐⣂⣉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠈⡆⣷⠀⠈⢻⣷⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⡿⡿⣿⣿⣿⡵⡶⠉⠁⠀⠀⢀⠄⠛⠋⢶⣦⣔⢸⡄⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠟⠉⠀⢠⣤⣤⡤⣴⡁⠀⠀⠀⠀⠈⢏⢸⡇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠋⠁⠀⠀⢠⣿⣿⣿⣽⠟⢳⡄⠀⠀⠀⠀⠘⣼⠁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡏⠀⠀⡟⠀⠀⠀⠀⢳⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠛⠓⠶⠶⠶⠂⢺⣿⣿⣿⣿⠟⠀⠀⡰⠁⠀⠀⠀⢰⠿⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⣯⡤⠖⠋⠀⠀⠀⠀⠀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢠⡶⠚⠛⠛⠳⢦⡀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠓⢶⡄⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡟⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠛⠋⠀⠀⠀⠀⠙⠛⠲⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠶⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠉⢹⠟⢹⡿⠿⣤⢤⣤⣤⣤⣤⠽⢿⡏⠻⡏⠙⢿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣰⡟⠀⣿⠁⢸⡇⠀⣼⣇⠀⢸⡇⠘⣿⠀⢿⣆⠈⢷⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀⠀⢀⣀⣤⠞⠋⣠⢾⡿⠁⣸⢿⠀⢸⡇⠀⡿⢿⠀⢸⡇⠀⡿⣇⠈⢿⠷⣄⠙⠳⣤⣀⡀⠀⢀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠸⣏⠀⠀⠉⠉⠉⠉⠉⣀⣴⠞⢁⡾⠁⢠⡏⢸⡆⢸⡇⢰⡇⢸⡆⢸⡇⢸⡇⢻⡄⠈⢷⡈⠳⣤⣀⠉⠉⠉⠉⠉⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠖⠚⠛⠉⠀⣠⠟⠀⢠⡟⢠⡟⠀⣸⠇⢸⡇⢸⡇⠸⣇⠀⢳⡄⢻⡄⠈⠻⣄⠀⠉⠛⠓⠶⠶⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⢀⣴⠏⠀⣿⠀⣰⢿⡆⢸⡇⢸⡇⢸⡿⣆⠀⣷⠀⠹⣦⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣀⡴⠟⠁⠀⠀⡿⠀⣿⢸⠇⠸⡇⢸⠁⢸⡇⣿⠀⣿⠀⠀⠈⠻⢦⣀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠛⠉⠀⠀⠀⣀⡼⠁⢠⡇⣾⠀⠀⡇⢸⠀⠀⣧⢹⡄⠘⢧⣀⠀⠀⠀⠉⠛⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⣠⡟⢠⡏⠀⢰⡇⢸⡄⠀⢻⡀⢳⣄⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠞⠋⠀⢸⣇⣠⡞⠀⠈⢳⣄⣸⡇⠀⠙⠳⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢂⠀⠀⠀⢠⣤⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠔⠀⢰⣾⡿⠂⠀⠀⠀⠙⢛⣁⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⣀⣯⡗⢀⣈⣽⣿⣷⣀⣀⡀⣹⣿⣿⣍⣿⣿⡠⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣛⣀⣤⣘⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⡟⠈⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣿⡏⣡⣿⣿⣿⣿⣿⠩⠿⣛⣫⣅⠉⠛⠿⠟⠟⠙⣤⠄⠻
⠀⠀⠀⠀⠀⠀⣿⣿⢡⣾⣿⣿⣿⣿⣿⣧⣾⡿⠟⣉
⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿⣿⠏⢉⣷⡖⠈
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢇⡮⢿⣿⣿⡴⠀⠷⠤
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣥⣍⠺⣿⣤⣦
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⢉⣿⣿⣿⣿⡿⢀⢾⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣮⣿⣿⣿⣼⣿⡤⢻⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣾⣿⣾⡽⠦⠂⠈

⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠗⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣇
⠈⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡲⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⡊⢡
⢀⡼⣃⣁⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⡙⠆⣂⠞⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⡲⢬⣾
⢸⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⡇⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⣁⣀⣀⣤⡄⢠⡤⢤⣤⣀⣾⣿⣿⣿⡇⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢸⣷⣶⣿⣿⣿⣿⣿⣿⡇⢸⣷⣶⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣆⣀⣀⣀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⣀⣀⣠⣴⣮⣤⣾⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⢥⣦⡀⠀⢀⣼⠇⠀⠈⠻⠿⠭⢭⣭⣯⣿⣿⣿⣿⡿⢿⣿⣿⡤⠤⠄
⠀⠀⢠⣤⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠟⠋⠀⢀⣄⠀⣀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠘⠻⠄⠀⢀⣀
⠀⠀⠙⠇⠀⠀⠸⠋⠀⠀⡀⣠⠤⠰⠒⠒⠂⠒⠒⠢⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⣀⡿⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣤⣤⣀⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠺⠇⡇⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠉⠙⠿⠿⠿⠛⠉⠉⠉⠁
⠀⠀⠈⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡗⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠈⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⠀⠀⠀⠒⠀⣿⡆
⠀⠀⠸⢷⣾⣾⠿⠃⠀⠀⠀⠀⢨⠶⠟⠂⠀⠐⢿⠿⡁⠀⠀⠀⠀⡏⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣀⡤⠄⠀⢈⣁⣀⣈⣁⣀⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡦⠀⢰⡆
⠀⠀⠉⠓⠂⠀⠀⠠⠤⠤⠖⠒⠒⠛⠋⠁⠀⠀⠀⠚⠀⠀⠴⢮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⠶⠄⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠘⠶⠖⠚⠛⠋⠁⠀⠀⠠⠃⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⣠⡴⢶⡀⠀⠀⢸⡇⠙⠿⢶⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠉⠉⠉⠉⣻⣿⠿⠗⠒⢲⡤⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉
⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠜⠁⣠⠤⡔⠀⢨⠀⡤⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠁⠂⠀⠻⣦⠊⠁⢸⣶⣶⠿⢦⣃⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⠀⠀⠈⡄⠀⠊⠉⠀⠀⣿⣧⠴⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣷⢤⣀⡟⠀⠀⠀⠀⠐⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢛⡿⠻⣾⣥⡤⠄⣀⠀⠴⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⡂⠀⠢⣠⠀⠀⢹⢀⣠⠄⠀⡄⠀⠀⢠⠋⠀⠀⠀⠉⠛⠶⠦⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣠⣴⣴⡿⢶⣤⡶⠃⠙⠀⡰⠀⠀⡄⠁⠀⣀⡀⠠⠒⡄
⠀⠀⠀⠀⠐⠈⠀⠘⠂⠉⠑⠾⠈⠻⡇⠀⠿⠁⠀⣴⠞⠁⠀⡾⠀⠀⡔⢸⡀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⣃⢄⡇⠀⣀⣠⠞⠉⠀⠀⢺⠁⠀⢨⡇⣼⡁⠁⢀⡇
⠀⠀⠀⢀⡤⠶⠒⠛⠢⠄⠀⣄⣨⡽⠗⢺⡗⠒⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣧⣴⡇⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠤⠤⠢⠖⠁⠀⠀⠈⢻⡄⠀⠀⠀⡈⠀⡠⠈⢀⡿⠁⠀⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢠⠁⠀⠀⢠⠟⠐⢆⣣⣤⣅⡐⠃⠀⢀⡁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣼⠠⠀⢀⡎⠀⠀⠼⢮⢙⣟⠟⢋⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠱⣜⡀⠼⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠉⠐⠦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠱⣼⠉⠫⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣿⠀⠀⢠⡿⠉⠓⠒⠤⠤⢤⣀⢀⣀⠀⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠉⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠋⠙⠛⠛⠻⠿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⠚⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣤⡿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⡿⠈⠀⠀⠀⠀⠀⢀⣦⠀⠀⢀⣿⡇
⣿⣿⡄⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣴⣿⣧⣀⡀⢰⣿⡇
⣿⣿⢿⣆⠀⠀⢠⣿⣿⣿⣟⣿⣿⠇⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⠛⠁⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣉⣾⣿⡇
⣿⡏⣶⣿⡆⠀⣸⣿⣿⣿⣿⣿⠏⠀⠀⣼⣿⡆⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠹⣿⠁⠀⠀⠀⢀⣤⣾⡟⠈⣿⣿⣿⣿⣿⣿⣿⡇
⣿⡇⢻⣿⡇⠀⠘⢿⣿⣿⣿⠃⠀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⢀⣴⣶⣿⣿⡿⠀⠀⢸⣿⣿⡿⢿⣿⣿⡇
⣿⣷⠀⠙⠁⠀⠀⠀⠉⣻⣡⣤⣴⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⢠⣿⡏⠁⠀⠈⣿⡇
⣿⣿⣿⡄⠀⠀⣠⠀⢀⣽⣿⠿⣿⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣶⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⠀⠸⠏⠀⠈⢟⠏⠀⡾⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⡇⣀⣤⣀⡀⠰⡄⣠⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⠀⠀⠉⠉⠀⢡⣏⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⠶⡄⠀⠀⠀⠀⣸⣿⣯⡀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠒⠀⢠⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠈⠀⠀⠀⣴⣿⣿⣿⣿⣤⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣤⣤⠖⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⡿⢫⠀⠀⢸⣿⡏⢤⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡇
⣿⣿⠿⠿⠛⠉⠐⠁⠀⠀⠈⣿⠀⠈⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠀⡀⠀⠀⣿⡇
⠋⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡽⠀⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⢸⠃⠀⠀⠀⠀⣿⡇
⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠚⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⠐⠒⠀⣩⢉⠉⠫⠝⢁⢐⠒⠢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⢘⠶⠂⠀⠀⣁⠀⢖⡰⠉⠘⠀⡠⠂⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢰⠁⠀⢀⣀⣠⠄⢰⡤⠰⠁⠴⢠⠀⢀⢀⠠⠀⢀⣈⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠃⠀⠀⠀⠀⠀⠈⠒⠊⣁⠀⢠⢤⡔⠀⠀⠀⠀⠘⠤⠥⠐⠒⠈⠀⠐⠦⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣮⡀⢀⡀⠀⠀⠀⠀⠀⠐⢀⠠⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠟⢰⠁⠈⠑⠒⠤⡀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⠀⠀⠀⠀⣔⣁⣵⡀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠘⡓⠢⢤⣀⣘⠐⢮⠀⠸⣖⣤⢀⡀⢰⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⠀⢄⣐⣠⠴⠟⠛⠓⠒⠈⠉⢉⠅⢰⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠐⢄⠀⠈⠉⠛⠂⠤⣀⡈⠛⢖⠀⡂⣠⢀⠐⣂⠔⡨⠔⡡⠔⡠⠂⠔⠡⢊⠔⡡⢐⠵⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡣⡀⠀⠀⠀⠀⠀⠈⠑⠢⣿⡰⢙⠟⣈⢔⡨⢐⡡⡒⠕⡈⢐⠡⢊⡴⢚⠗⠁⠀⠀⢃⠀⠀⠀⠀⠀⠀⡐⠁⡘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⠐⠡⢐⠅⡊⢔⠡⢤⢁⠾⢃⠗⠁⠀⠀⢠⠃⠀⠑⠄⠀⠀⢀⠊⡀⠀⠁⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠒⠀⠁⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠔⡩⢒⠡⡢⢃⠡⡸⠋⢀⠀⠀⠀⠞⠀⠀⡗⣐⢀⠔⠁⡄⠁⠀⢰⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣂⡀⠀⠠⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡔⠛⠩⡒⢅⡞⢁⠂⠀⢱⠀⠀⠘⠤⡤⣴⡯⠁⠀⡁⡇⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢒⠒⠀⠀⠡⠤⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⠊⣲⠋⠀⠈⢢⣴⠋⠀⠀⠀⣀⠔⠁⠰⠀⠀⠀⠁⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⠒⣂⣀⡉⠉⠤⠕⢄⠠⠀⠀⠂⠀⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠘⠀⢠⠀⠂⣤⠖⠋⢀⢌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡐⠒⠀⠤⠀⢈⣐⡻⢦⡀⠀⠄⡀⠀⢸⠇⠀⠀⠄⠂⢀⡠⠂⠡⢀⠁⠂⠀⡄⠀⣨⠴⠋⠁⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡤⠞⠋⠉⠉⠉⠙⠛⠓⠤⠦⠤⢠⣮⡈⣠⡴⠾⠓⠚⠉⠙⠲⡔⢘⣠⠶⠛⠉⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⡆⠀⡁⡏⠉⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡪⠜⠀⠀⢀⡀⠐⠈⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠖⠤⠤⠤⠄⠀⠂⠉⠀⠀⠀⠀⠀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠃⢁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠌⠀⡀⡀⠀⡀⠀⠀⢀⠀⠄⠃⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠀⠒⠒⠀⠀⠤⠤⠠⠤⠤⠀⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⢀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠠⠐⠈⠄⢂⠁⠐⡈⠀⡂⠅⠂⠁⠐⠀⠐⠈⠐⠠⢀⠀⠊⠔⠠⠂⠅⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠒⠒⠂⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣶⡥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣷⣤⣤⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠛⠹⠿⡻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣶⣢⠕⠀⠈⠑⠢⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡁⠂⠄⠀⠀⠀⠀⠀⠀⠰⠁⠛⣻⣷⣦⣖⡦⢂⡄⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⣫⣭⣷⣞⡵⠣⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣖⢃⣻⣦⡀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⠁⠀⠀⢢⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠾⣺⣿⣿⣿⣿⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢹⣦⣶⣦⣀⡀⢀
⠀⢀⡂⣨⣿⣄⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠺⣛⣭⣿⣿⣿⣷⣷⡀⠀⠀⠀⠀⠀⠀⠈⠘⢀⣿⣿⣿⣯⣴⣏⡧
⢠⠈⢻⠟⠉⣹⠀⣠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡩⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠈⢻⡿⣿⣏⠊⠃
⠰⡇⠀⣀⠀⠈⠀⠁⢈⠎⢀⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠸⣿⣯⣴⠀⡀⠄⢠⣴⣂⠀⠁⠀⠀⠀⠀⠈⠐⢱⣿⡿⡫⠀⠀⠀⢀⡜
⠀⠁⠀⠀⢢⣤⠀⠀⣀⣺⣭⣭⣭⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢐⣻⣽⣾⣿⣷⣶⣵⢛⣷⠞⠀⠀⠀⠀⠀⠀⠀⠚⠻⠛⠀⠀⠀⣤⡞⠁
⢄⡀⡀⠀⡀⠈⣀⠀⠀⠙⠟⠏⠛⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠉⣩⠽⣿⣿⣯⣿⣟⡻⣿⣲⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣾⠋
⠈⠳⣏⣼⠓⣦⠽⠁⠀⠀⠀⢀⣴⠏⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡻⣿⣿⣿⣿⣯⢟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉
⠀⠀⠰⣶⡦⡔⠒⡢⣤⣶⠾⠋⠡⢴⣽⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣭⣿⣿⡿⣗⣿⠇⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠁
⠀⠀⠀⠈⠳⢬⣾⣾⣷⠿⠯⠛⢿⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡤⠿⠟⠛⠉⠀⠀⠀⠀⣀⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣦⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣴⣾⠶⠶⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠒⠒⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠔⠒⠈⠁⠀⠀⠀⠀⠀⣀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⢀⣠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⣀⣀⡀⠀⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣿⣟⠋⠛⣿⢷⣦⡀⠀⠀⠀⠀⢀⣴⠾⠻⣿⣿⣿⣿⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣾⣷⣤⣶⣿⣿⣿⡏⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠹⢿⣻⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣟⡿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⡴⠲⡆⠀⠀⠀⠙⢦⡀⠉⠙⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠋⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⣇⢀⡇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠰⠆⠰⠄⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠸⡄⢀⡏⢸⡇⠀⠀⠀⠀⠀⠈⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⡘⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠉⠉⠀⠀⢰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⡿⠤⠤⠖⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡶⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⠶⣶⡶⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠏⠀⠀⢠⡇⠀⠀⠀⣀⡤⢤⣤⠴⠋⠀⠐⣄⡠⠀⠀⠓⠦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠐⠀⢸⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠈⠂⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠹⡄⣸⠄⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢻⡇⠀⢰⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠃⠀⣾⡟⠀⡷⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣾⣿⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡳⣄⡀⣻⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠇⠀⠀⠀⠿⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⢻⣀⣄⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠉⠀⠀⠀⠀⠼⠛⠓⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢀⣸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢈⣿⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣷⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠸⣿⣿⣿⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢐⡬⠃⠰⠀⠀⠀⠀⠀⠀⠀⠰⠆⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠳⣿⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣾⣀⣿⡀⠀⠀⠀⠀⠀⣠⠆⠀⡼⢃⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣤⣤⣽⣯⡻⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⢠⣾⣿⣶⣿⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠛⠛⠿⠿⠿⣿⢿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠰⠿⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣤⣴⠀⠀⠀⠀⣀⠀⣈⡨⠿⣿⣿⠿⠿⠌⠛⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠹⣿⣿⣿⣿⡦⠶⠿⠿⢿⠛⠙⢷⣄⠈⠛⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢴⣆⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⣶⣶⠟⠀⠀⠀⠀⠀⢀⡾⠛⠋⠁⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣷⣴⣶⣄⡀⠀⣠⣶⣾⣿⣋⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⢀⠐⣤⣻⣗⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⢸⡗⣯⣩⣿⡿⠀⠀⠀⠀⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠟⢹⡿⢻⡟⠀⠉⠻⣿⡿⠻⣿⡿⠛⣿⠏⠀⠀⠀⠀⠀⠀⠀⠛⠧⠾⠋⠁⠀⠀⢻⣿⣿⡄⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⢁⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣿⣿⡶⠋⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠⠶⠟⠻⠿⠿⠷⠀⠀⠿⣿⣦⡀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠆⢠⡟⠀⣿⣿⡇⣠⣾⠃
⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⡀⢉⡀⢀⣀⣀⡀⠺⢦⣠⣴⣿⡏⢀⣾⠏⠀⣰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠉⢙⣿⠟⠛⠁⠀⠀⠀⠈⠛⠻⣇⢸⡏⠀⢰⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣤⣄⡀⠀⠀⠀⠀⠀⠈⠀⢠⣾⠏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠐⠁⠀⣸⣿⣿⣿⡷⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠚⢶⡾⠿⢿⡿⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠞⠛⠃⠀⠀⠀⠀⠀⣠⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⢋⣉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⠿⠷⡄⠀⠀⣀⣴⢿⣶⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠙⢿⣶⡀⠀⠀⣼⡿⠿⠛⣿⣿⣿⣿⡄⠀⠀⣰⣿⠃⣼⠏⠀⠀⠀⠀⠀⠀⠀⣤⠂⣲⣈⡶⢆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣤⣘⣋⠁⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢹⣂⣽⡋⠂⠌⠀⠀⠀⠀⠀⠀⠀⣶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⣿⣿⣿⡿⣿⣿⠻⢿⣿⡍⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠚⠿⠛⢋⣡⣾⡿⠋⠋⠀⡟⢹⣇⢀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⠛⠁⠀⠀⠀⠀⢻⡈⣿⡈⠉⠉⣡⣾⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⣷⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⣿⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣧⠀⠀⠀⠀⠈⠙⢛⣻⣶⣦⡄⣀⠀⠀⣀⣀⣠⠞⠁⠀⣰⡿⠃⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣆⠀⠀⠀⠰⠾⢿⣿⣿⠟⠰⠿⠿⠟⠛⣩⣵⣦⣀⠊⠉⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⢀⣠⣴⣾⡿⠟⠁⠀⠀⠀⠀⠀⢰⣿⣿⡟⢹⡆⠀⠀⠀⠀⠀⣸⠏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⠿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠁⠈⠀⠀⠀⠀⠀⢠⠏⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⢷⡄⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣰⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⠇⠀⠀⢀⣠⠶⠛⣉⣉⣭⣽⣿⣿⠀⠀⣜⠀⠀⠀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡟
⠀⠀⢀⣿⣿⣿⣿⣿⣿⢿⣻⣵⡿⠋⠁⠀⢀⡴⠋⢁⣴⣾⣿⣿⣿⣿⡿⠋⢀⡴⠃⠀⠀⠀⣿⣁⣿⣿⣿⣧⣵⣿⣿⣏⡶⠀⠀⠀⠀⠀⢠⣶⣿⣽⠏
⢀⣴⣋⣻⣟⣻⣭⣷⠾⣟⣯⣷⡿⠀⠀⠀⡟⠀⢀⣿⣿⣿⠿⠿⠟⢋⣠⡴⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⣴⣿⣿⣿⡿⠁
⠘⠛⠛⠛⠛⣭⣷⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠻⠿⠷⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠛⠀⢀⣠⣤⣤⣄
⠀⠀⠀⡀⠛⠿⠶⠿⠷⠶⠛⠙⢆⠀⠀⠀⠀⠀⣠⡾⠂⠀⠀⠘⢃⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠒
⠀⠀⢰⣀⣀⡀⠀⣀⠀⠀⠀⢀⣀⡇⠀⠀⠀⢰⣿⣧⠀⣤⣤⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠄⠻⣿⣿⣿⣿⠇⠀⢠⡀
⠀⠀⢸⡉⠙⢿⠟⢿⣟⠛⢛⣛⣫⡇⠀⠀⠀⢸⣿⣿⣿⣶⣿⠛⠛⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠆
⠀⠀⠈⢿⠿⢿⣿⣿⣿⣿⡿⠷⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⣨⣿⠃
⠀⠀⠀⠈⠑⠤⣴⣶⣶⣶⠶⠚⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡇⣤⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣀⣴⣻⠏⠀⠀⠀⠀⠀⠀⠛⠿⣷⣶⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠤⠶⢟⣽⣿⠏⠀⠀⠀⠀⠀⠘⠿⣿⢟⣥⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢀⠤
⠀⠀⠀⠀⣾⠿⣸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⡿⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣿⣿⠋⠀⠙⠻⡿⠃⠀⠀⢠⣎⣤⣀⢀⡴⠋
⠀⠀⢀⣾⢟⣥⡜⢿⣿⣿⠁⠀⠀⠀⠀⠀⢀⠀⠙⣿⣿⣿⠋⠰⣿⡇⠀⠀⠀⠈⢷⡄⠻⣦⡉⠳⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⡻⣿⡄⠀⠀⢀⡀⠀⠁
⠀⠀⠉⣠⡿⢟⣵⠷⠿⠃⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠙⠷⠀⠀⠙⠃⠀⠀⠀⠀⠀⠙⢶⣌⡙⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣇⡤⠒⠁
⠀⠀⠐⠉⠠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠁

⠀⠀⢶⡀⠀⣤⡀⠀⠀⢠⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣆
⠀⠀⠀⠻⣦⣽⣥⡀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠘⣿⣤⢠⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⣀⣠⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠼⠟⠛⠉
⠀⠀⠀⠀⢸⣿⡿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣥⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠉⣿⠋
⠀⠀⠀⠀⢸⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠛⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠻⢿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⡿⠿⣿⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢻⠛⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠠⠁⠀⠀⢹⡀⠀⡼⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠳⢤⠀⠀⠀⠸⠄⠐⠐⠀⠀⠃⠰⠃⢯⠀⠀⡰⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠃⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠆⠈⠖⠀⠀⠐⠀⠀⠀⠀⡀⠐⠲⠀⠀⠀⠀⢀⠀⠀⡀⠘⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⠀⠀⠠⢠⡆⠈⠘⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠀⠀⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⢒⠒⠒⠀⠀⠀⠀⠀⠃⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠢⣀⡤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆
⠀⠀⠀⠀⠀⠘⠛⠓⠒⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⡄⠀⠋⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠉⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢷⡂⠀⢐⠄⠐⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠘⠂⠀⠀⠀⠀⢈⠗⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠴⠀⠰⠀⠀⠀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠊⠀⠀⠀⢀⠠⠊⠀⠀⣠⢔⣅⠁⠀⠀⣠⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⡤⡪⠖⠁⠈⠑⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠒⠚⠉⠒⠠⠤⠐⠋⠉