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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠴⠒⠒⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡏⠙⠉⠀⠀⠀⠀⣶⢲⠏⢹⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠈⠉⢉⣙⣿⠢⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠉⠓⠒⠒⡻⠛⠉⠉⠉⠉⠑⢮⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⠤⠤⣄⡀⠀⢱⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡞⢁⡴⡶⠄⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⣭⣯⠇⢠⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠈⠳⠤⠥⠴⠋⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠒⠒⠒⠒⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡾⢿⡿⢷⡄⠀⠀⠀⢋⣬⣭⣭⣼⣿⠀⠀⠀⢠⡾⢿⡿⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⣶⣶⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣛⣋⡅⠀⠀⠀⠀⠀⢬⣽⣫⣥⠀⠀⠀⠀⠀⢨⣛⣫⡅⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣍⢫⣶⣶⢄⣠⣤⡘⣦⡙⢋⣵⢃⣤⣄⡀⢶⣶⡍⢩⣶⣶⣤⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣷⢸⣿⡇⣿⣿⣿⣿⣬⡐⢃⣥⣿⣿⣿⣿⢸⣿⡇⢸⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢮⡛⣿⣿⢸⣿⢃⣿⣿⣿⣿⣿⣧⡌⣿⣿⣿⣿⣿⡌⣿⡇⢸⣿⣿⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⠳⡐⢿⣯⢸⣿⢸⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⡇⣿⡇⣸⠟⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢿⣿⢸⣿⣿⣿⣿⣿⡸⠇⣼⣿⣿⣿⣿⣇⣻⢟⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣙⡻⠿⣿⣿⣿⣿⣾⣿⣿⡿⠿⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠾⠿⠿⠶⠖⠚⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⡿⢿⠿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⢀⣴⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠒⠒⢲⡟⢀⡴⢋⣴⠞⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡷⢋⡴⠛⣡⠞⢋⡴⠆⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠋⣴⠋⠠⠞⠁⠐⠋⢠⠶⠃⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠛⠛⡿⠛⠛⣷⠀⠀⠀⣴⠞⠛⣛⠓⢦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣀⣀⣧⣤⣤⣿⠀⠀⣼⠃⠀⠋⠗⣫⠆⢻⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠉⠈⣯⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⠐⠋⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⠶⠶⠿⠒⠚⠛⠀⠀⡇⢀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠻⠃⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢀⣀⣤⣤⡤⠷⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠛⠒⠶⠶⠶⠶⠶⠾⠧⢤⣤⣿⣄⣀⣀
⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣧⣤⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠙⡶⠊⣀⡤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣶⣿⣷⡟⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠉⠘⢶⣟⣺⣿⢦⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⢉⣿⠞⣹⠮⠿⠟⠂⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⣷⠞⢉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠖⠻⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠌⣿⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠻⠿⣯⣟⢻⣿⣾⣿⣤⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠘⠛⠽⣿⢿⣽⣗⣖⣠⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣰⣧⣤⣿⣥⣤⣭⣭⣋⣉⣀⣀⣰⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣝⣛⣩⣛⣻⣟⣽⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣀⣤⣠⢀⠀⣠⠄⢠⢄⣄⡤⣠⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡀⣠⣀⡀⣀⢄⣄⣀⢠⠀⣈⠄⣀⣀⢠⢠⣠⣄⣀⡠⣠⠀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣶⣶⣷⣶⣾⣶⣶⡆⠀⠀⠀⠠⣤⢀⣹⣿⣅⠠⣤⠀⠀⠀⠀⠀⣦⣤⣤⣀⣯⠗
⠀⠀⠀⠀⢸⣿⣿⣏⡲⢽⣿⣿⡇⠀⠀⠀⣾⣿⣾⣏⡩⣿⣷⣿⣦⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⢸⣿⣿⣷⣭⣼⣿⣿⡇⠀⠀⠀⠛⠛⣿⣷⣵⣿⣿⠛⠋⠀⠀⠈⠙⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣤⣶⣳⣦⣤⣀⠀⠀⠀⠀⠀⣀⣈⡋⢀⣄⠀⠀⠀⠀⠀⠀⢀⣀⢠⣦⢘⡟
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⠸⣿⣸⡇⠘⠚⠁⠀⠀⠀⠀⠀⢘⣟⢰⣷⢸⣿
⠀⠀⠀⠀⠀⣿⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⢶⣌⡉⠤⣶⣦⣀⡀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿
⠀⠀⠀⠀⣴⣿⣶⣿⣾⣷⣾⣷⡄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠇⠀⠀⠀⠀⠾⠿⠾⠿⠾⠷⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣽⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠂
⠀⠀⠀⠀⢰⣶⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢤⡀⣠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⡇
⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣶⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⣿⣿⣿⣿⣽⣿⣷⣿⡿⠃
⠀⠀⠀⠀⡀⠒⡒⢀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠿⢿⣿⡿⠛⠻⠛⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠰⠖⠞⠦⠢⠔⠀⠶⠂⠷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠘⠃⠂⠃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣍⠠⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡾⡧⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡷⣥⡀
⠀⠙⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣽⣯⡧⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡷⠷⢚⠆⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠐⠀⠀⠀⣀⠀⠀⠀⣴⣖⣉⡁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⡿⠉⠀⠠⠤⠴⠦⠄⠀⠗⡆⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⢛⣻⣿⠒⠒⠒⠒⠶⠒⠂⠈⢷⡥⢶⠛
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡤⠄⠀⠀⢀⠀⠀⠀⡼⠓⢘⡆⠀⠀⠀⣶⣾⠄⠼⢾⡗⢀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠨⠇⠀⠀⠈⢉⠀⠀⠰⠛⠃⡠⠷⠀⠀⠀⣟⠻⠀⡈⢿⡅⠸⠁⣇
⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘⢶⡀⠀⠀⢀⡤⢾⠭⣧⢀⠀⣂⣀⡀⡰⠛⢠⣤⣼⣉⢛⣿⢻⠄⠀⢀⡇⠀⣧⠁⡞⣇⠐⡟⣿⠒
⠀⠀⠀⡇⠀⣼⣠⣸⣧⣷⣶⣶⣿⢶⠶⠿⠿⢷⣛⣛⣛⡙⠯⠭⢿⣷⢰⣾⠿⠻⣿⡿⣿⡟⣷⣾⣿⣴⣦⣻⣶⣯⣿⣅⣿⣿⣷⡆
⠀⠀⣠⣧⣸⡭⣟⣹⣿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠓⠦⠬⢭⣍⣀⣚⠛⠾⠿⢿⣷⣿⣿⣿⣿⠍⣿⣿⣿⣷⡽⠿⣷⠃
⠀⢠⠤⣗⣨⠶⣫⠽⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢶⣤⣤⣥⣉⡛⠺⠿⠿⣿⣿⣷⣶⣾
⠀⠀⢘⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⣷⣦⣤⣄⡉⠁
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡴⠓⠛⠛⠛⠛⠓⠛⠛⠛⠛⠓⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡴⢤⣦⣀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠻⠋
⠀⢠⡞⠁⠀⠀⠀⠀⣰⠛⢧⡀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠠⣶⢻⡁⣸⡀⢀⡀⢈⣿⣶⢄⡀⠀⠀⣤⣯⢿⡿⣷⣄⠀⠀⠀⡀⡀⣠⢄⢠⡄⣀⢠⠤
⣰⣟⣐⣒⣒⣂⣢⠞⢡⡶⡆⠙⣆⣂⣒⣒⣒⣚⣧⠀⠀⡎⢉⣉⣉⣉⣉⣉⡉⢉⣉⣉⠁⠀⠀⠚⡞⣓⡛⠚⠚⢳⠓⠀⠐⠋⠃⠳⠜⠘⠻⠛⠸⠭
⠀⠀⢽⠀⠀⠃⠛⠀⠈⠋⠋⠀⠘⠀⠃⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠈⢸⠀⡇⠜⠳⢻⠀⠀⠀⠀⡟⠿⠟⡤⢤⢸⠀⢰⠶⠶⢶⢲⡖⠆⠰⠶⡶⢶⢶
⠀⠀⣾⢸⡇⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⣿⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠇⠇⠐⢸⠀⠀⠀⠀⣧⢀⢠⡇⢸⣼⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣽⠈⠁⡂⠀⠀⣿⣿⣿⡇⢀⠀⡀⠉⢸⠀⠀⠀⠀⠉⠉⠈⠁⠉⠈⠁⠉⠉⠉⠉⠁⠀⡀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⣦⡆⠀⠀⠙⠉
⠀⠀⣿⠀⠀⠟⠛⢠⣿⣿⣿⣧⠘⠛⠇⠀⢸⠀⠀⠀⠀⠀⣠⠞⠙⠢⣀⠀⠀⠀⠀⣨⢗⣵⣮⡢⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣵⢝⡶⣄
⠀⠀⠻⠤⠴⠦⠤⠼⠿⠿⠿⠿⠤⠤⠶⠤⠼⠀⠀⢀⡠⠚⠁⠀⠀⠀⢈⡷⠀⠀⢾⠷⠻⠿⠿⠿⠿⣽⣿⣳⣄⡀⠀⠀⠀⢚⣉⣀⣛⣒⣛⣛⣚⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⢉⠢⣄⠀⠀⡠⠊⡇⠀⠀⢸⠀⢰⣻⠛⡇⠀⠛⠚⠛⠚⠛⡆⠀⠀⢸⣯⣩⣿⣿⣿⣯⣩⣿
⠀⠀⣀⣀⡀⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠰⠯⠤⠤⠧⠤⠤⢷⠊⡠⢘⡇⠀⠀⢸⠀⢠⠤⡤⡄⠀⠀⠼⠿⠹⠄⡇⠀⠀⢸⣿⣿⡏⠀⠈⣿⣿⣿
⠀⠀⣾⣿⣶⠿⣫⢬⡙⢿⣦⣀⠀⠀⠀⠀⠀⠃⠀⠄⠄⠄⠀⢸⣀⠜⣡⠇⠀⠀⢸⠀⢸⠠⡇⠇⠀⢤⠤⠤⠤⠤⠇⠀⠀⠘⠿⠿⠇⠀⠀⠿⠿⠟
⢀⣴⡿⢟⡥⠚⢁⣠⡈⠳⣍⠻⣷⣄⡀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣡⠞⠁⠀⠀⠀⠈⠂⠘⠘⠃⠓⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠿⣫⠖⠉⠀⠀⠈⠋⠀⠀⠀⠙⢮⡙⠿⠀⠀⠓⠚⠓⠓⠛⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣺⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢫⢝⠷⣄
⠀⣼⠀⢀⣀⣀⡀⠀⢀⡀⢀⡀⠀⡇⠀⠀⠀⣠⠖⠒⠒⠒⠒⠒⢒⡶⠢⡀⠀⠀⠠⣶⣿⠛⠛⢻⣿⡇⣍⣲⢤⣀⠀⠀⠀⠠⠞⠩⠾⠯⠿⠷⠌⠳⠄
⠀⢸⠀⢸⣿⣿⣗⠀⠿⠇⠿⠇⠀⡇⠀⠀⢺⠓⠒⠒⠒⠒⠒⠒⠿⠒⠒⠸⠆⠀⠀⣿⣿⡍⡍⣿⣿⡇⣯⡟⠛⣾⣗⠀⠀⠀⣿⣿⣿⢿⡿⠿⠿⣿
⠀⢸⠀⢸⣿⣿⡏⠀⠿⠇⠿⠇⠀⡇⠀⠀⢸⢰⣶⠠⣶⡆⣶⡆⠀⢰⣶⠀⠀⠀⠀⣿⣿⠀⠃⣿⣿⡇⣿⠅⠀⣿⣿⠀⠀⠀⣿⣯⢽⣿⡷⣯⣏⣿
⠀⢸⣀⣼⣿⣿⣯⣀⣀⣀⣀⣀⣠⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⠀⠀⠀⠀⠿⠿⠀⠀⠾⢿⠇⠷⠄⠀⠿⠿⠀⠀⠀⣿⣇⣸⣿⣿⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠉⠉⠀⠀⠉⠉⠙⠻⢿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠺⠿⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⠁⢸⡇⠙⣿⡷⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⢰⣿⣧⠀⢸⡇⠀⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⣧⣶⣿⠟⠀⠀⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉
⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⢿⣿⣷⣦⣄⣀
⣴⣿⠛⠿⢿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣶⣤⣀
⢻⣿⣤⣀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣷⣶⣦⣤⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡛⢿
⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣷⣶⣶⣶⣶⣶⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠈⢁⣾
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⡿⠿⠿⣿⣿⣿⠃⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠻⣿⣶⣤⣾⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡥⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⣩⡼⠷⡄⠀⠀⠀⠀⠀⠀⢠⢞⡭⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡞⢡⠘⣿⡆
⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣼⠛⣉⣥⠴⠾⡇⠀⠀⠀⠀⠀⠀⡏⡞⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠹⣌⠆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣋⠉⣤⠶⡟⠁⠀⠀⠀⠀⠀⠀⣧⢣⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡧⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣤⣿⣤⣧⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣯⣼⣤⣿⡇
⠀⠀⠀⠀⠀⠈⣳⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿

⠀⠀⠀⠀⠀⣀⢀⣤⢠⣄⢀⡀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣶⣤⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣞⠛⣶⡀
⠀⠀⠀⢀⣘⠋⢾⡏⠉⡽⠎⢛⣀⠀⠀⠀⠀⠀⠙⠦⣀⣬⣬⡃⣤⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡿⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡙⠿⠾⠏⡁
⠀⠀⠀⠾⠿⠸⡿⠀⠀⢛⡿⠸⠿⠆⠀⠀⠀⠀⠀⢠⣿⣿⡸⢞⣵⢟⣀⠀⠀⠀⠀⠀⣧⣿⣿⡇⠀⠘⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁
⠀⠀⠀⢶⣶⠰⣿⡶⢠⣾⡷⢰⣶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣧⣌⠡⠿⡋⠀⠀⠀⠀⠀⠹⣿⣿⣷⡗⣰⣿⣿⠇⠀⠀⠀⠀⠀⣠⠀⣾⣮⣩⣶⡆⢠⡀
⠀⠀⠀⠈⢩⣄⢶⣷⢻⣶⢆⣬⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢻⡂⢯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠇⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠻⢦⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⣿⣿⣷⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⢸⠟⢉⣉⠛⣿⠀⢠⢀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄
⠀⢀⡄⣿⣿⣿⣿⢰⣀⣠⣶⢾⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣴⣴⠀⠀⠀⠀⠀⣿⣿⡈⠀⢁⣿⣿⠀⠀⠀⠀⠀⡞⠀⢠⡀⢙⣶⣄⠀⠀⠸⡆
⠀⠘⠇⣿⣿⣿⣿⢸⣿⠶⠶⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣾⣷⣿⠀⢿⢿⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⠀⠀⢀⡇
⠀⠀⠀⣿⣿⣿⣿⠸⣿⠛⢻⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠿⠟⠀⢸⢸⠄⠀⠀⠀⠀⣿⣿⣈⣠⣀⣿⣿⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⡇⢀⡼⠁
⠀⠀⠀⠛⠋⠩⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣄⣀⣠⣎⡞⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⢫⣻⠇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⠰⠚⠋⠁⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⠀⣾⣭⣶⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣤⡤⢸⠩⣭⣭⡭⢻⠠⣤⡀
⠀⠀⠀⠉⠛⠗⣶⣶⣤⣀⡀⡶⠒⢦⠀⠀⠀⠀⠀⠀⢨⡿⢿⡅⠀⣴⡟⠉⢻⣆⠀⠀⠀⣶⡆⠀⢬⡥⠀⢠⣥⠀⠀⠀⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⠀⢠⣶⣶⣴⣬⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⣾⣧⠀⢸⣷⠀⠀⠀⠿⠟⣵⡎⠿⠿⠿⠿⠏⣶⠹⠿⠃
⠀⠀⠘⠿⣟⢙⡿⠿⠿⠿⠿⠿⢿⡟⢙⡿⠿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⢸⣿⣮⣴⣿⣿⣿⣿⣷⣭⣾⣿
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠉⠀⠀⠀⠃⠀⠀⠀⠙⠁⠀⠙⠋⠀⠈⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠉⠀⠈⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⣿⣿⣧⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠛⠛⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠋⠙⠁⢠⣿⣿⣿⡟⠀⠀⠀⣀⣤⣴⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⢿⣿⣿⣿⣄⣸⣿⣿⣿⡿⠋⠀⣠⣶⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⠛⠛⠛⠛⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⡇⢸⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⢸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠘⠀⠒⠀⠂⠐⠀⠀⠀⠣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⢄⠀⠀⠈⠀⠀⠐⠃⠁⠀⠀⢾⣿⢯⠉⠑⠲⠤⣀
⠀⠀⠀⠠⠄⢀⠀⠀⠀⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢠⢖⠻⠂⠀⠂⢐⠂⠀⠀⠀⠉⠐
⠀⠀⠤⠤⠀⠀⠠⠀⠈⠀⠁⡁⣀⠉⡀⡠⠄⠀⢉⢑⣒⠶⣶⠶⢶⡊⡁⠈⠁⡀⠀⡀⣉
⠀⠀⠀⡀⡄⡠⠀⠀⠀⠀⠀⠀⢠⢠⠀⡄⠒⠐⠀⠀⠈⠀⠨⠁⢸
⠀⠀⠉⠉⠉⠉⠉⠙⠉⠉⠙⠋⠉⠉⠁⠉⠉⠉⡉⡉⣉⠉
⠀⠀⠉⠉⠉⠈⠁⠉⠁⠉⠈⠉⠉⠉⠉⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




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

⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣄⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⣰⣿⣷⡀⣀⡿⣿⣟⡀⠀⠀⢀⡻⣿⡿⡀⢠⣶⣶⡄⢀⡻⣿⣿⡀
⠀⠀⠀⢰⣿⣾⣭⣷⣿⡆⠀⠀⣠⣼⣛⣻⣤⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣬⣻⣟⣥⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡟⠿⠿⠟⠛⠀⠀⠛⠿⠿⠿⢻⣿⣿⣿⣿⡿⠿⠿⠿⠟
⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣴⣦⡀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⣤⣿⣿⣤⣀⣤⣿⣿⣤⣀⣠⣿⣿⣤
⠀⠀⠀⠀⠘⢻⣿⡟⠁⠀⠀⠀⠀⣀⣶⣿⣆⣰⣿⣷⣀⠀⠀⠀⠿⠿⡿⣿⣿⣿⡿⡿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣧⢉⣽⣿⡍⣽⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠉⠀⠉⠉⠁⠉⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠙⠉⠁

⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⣠⣄⠀⠀⣠⠀⣄⢀⣀⡄⠀⠠⢀⡀
⣿⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡄⡇⣬⣷⢾⣉⢸⢿⡁⠠⣟⣯⡆⠀⠸⣾⢻⣾⢠⡏⣷⢸⢸⡇⣾⠽
⣿⠀⠀⠀⠀⡖⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁⠈⠉
⣿⠀⠀⠀⠀⠁⡀⠀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⢰⠋⠉⢉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠠⠀⢀
⣿⠀⠀⠀⠀⠀⠈⠉⠘⠁⠒⠀⠂⠁⠄⠀⠀⠀⠸⡄⢀⢀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡞⢿⣛⢸⡏⢙⣦⢼⢸⣉⡇⣿⢹⠀⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠸⠀⠀⠛⣶⠛⡆⠈⠀⠀⠀⠀⠀⠀⠈⠿⠄⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠷⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠠⠧⠍⠛⠁⠀⠀⠀⠀⠀⠀⠸⢀⡀⢀⠀⠀⠀⠀⠀⢘⣦⠀⠀⠀⠽⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠀⢠⠀⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⢀⠀⣀⠀⠀⠀⠀⠘⡇⠰⠄⠰⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡄
⣿⠀⠀⠀⠀⠀⠐⠀⣴⡆⠺⣇⡀⠀⠀⠀⠐⣛⡛⠂⠉⢻⠁⡀⠂⠀⠀⠀⠀⠀⠲⠂⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢁⠈
⣿⠀⠀⠀⠀⠀⠐⠒⠚⠚⠉⠁⠀⠀⠀⠀⠐⠊⠟⠉⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⣤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠀⠈
⣿⠀⠀⠀⠀⠀⣠⠤⢤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢰⣟⣀⢊⣿⡇⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠰⡂⠀⡁
⣿⠀⠀⠀⠀⣼⠅⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢉⣆⡰⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠛⠣⣜⠁⠀⠶⠀⣡⡀⡀
⣿⠀⠀⠀⠀⣯⡀⠀⠀⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠠⠒⢶⠁⠁⠈⠩⠁⢧⣹⢤⠴⣃⣀⣀⣀
⣿⠀⠀⠀⠀⠈⠸⠛⠹⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠁
⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠊⠐
⣿⠀⠀⠀⠀⠘⠠⠀⡀⢉⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠆⠁⠲⢀⣼⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠖⠢
⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡘⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠀⠀⢐⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠁⠀⠀⠀⠀⠺
⣿⠀⠀⠀⠁⠀⠀⠴⣀⠀⠁⠀⡀⠀⢠⣂⣄⣰⣦⠀⠀⠀⠀⢬⣀⠘⠀⠀⠁⠈⠀⠀⠀⢀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣽⣧⠀⠀⣀⣠⠌
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠈⠀⠈⢠⠑⠀⠘⣦⡄⠂⢠⠃⢄⣀⡀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠨⠉⠀⠀⠁⠠⠜⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠂⠐⠀⠸⠆⠤⠤⠤⠤⠎⠀⠈⠉⠀⠀⠀⠀⠸⠁⠀⢋⣃⠀⠝⠋⠁⠀⠀⠐⠀⣄⡴⠂⢀⣠⠐⠂⠀⠤⠄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠹⢴⡄⠀⠀⠀⠀⠀⠀⡿⡀⠀⠈⡵
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠀⠁⢁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠐⠂⠀⠀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⣠⡁⢈⡌⠀⢀⡠⠒⠄⠁⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠊⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡀⢘⠆⠀⠀⠀⠶⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⢴⠀⠀⠐⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠸⠁⠈⢨⠀⡅⡀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⠀⠀⠀⣄⣀⡀⠀⠀⠀⢀⡄
⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠨⠄⠥⠁⠧⠤⠤⠖⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠊⠉⠻⠄⠀⠀⠀⠈⠚⠉⠁
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⣧⡀⠀⢸⣷⡀⠀⣄⢸⡇⠀⠀⡀⠀⠀⣴⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢹⣷⣄⣨⣿⣷⣶⣿⣿⣷⣤⣾⣁⣠⣾⠇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠟⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⢸⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣏⠀⠀⢀⣀⣠⢤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠋⢠⣿⣿⣯⣻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣤⣠⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡻⢯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⣿⡻⠋⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠿⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠛⠉⠙⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡝⠘⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⣸⣿⠟⢻⡿⢿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⠎⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢠⡟⠁⠀⠚⠁⠀⣿⠟⡏⢀⡹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣠⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠈
⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠈⠁⠀⠀⠀⠀⠰⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⣽⠀⡰⡦⣤⣀⣀⠈⠀⠐⠤⠤⡥⣀⠀⣤⡄⠀⠀⠀⠀⠀⡯
⢀⣠⣄⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠉⠙⠗⠰⣶⣤⣀⣀⡀⠋⠉⠑⠐⠒⠀⠰⠯⡀
⠀⠀⠁⠀⠘⠃⠻⠶⢤⢤⣄⣄⡀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠙⣷⡤⠶⠤⠤⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡆⣁⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠘⠑⠢⠤⠤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣴
⠀⡤⣄⣀⣠⠀⣀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠈⠀⠉⠛⠇⣴⣆⡄⣀⣀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⠋⠈⠉⠑⠐⠗⠠⠷⠂
⠀⠀⠀⠈⠉⠀⠛⠲⠦⠀⡀⡀⠀⠸⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣷⣀⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⠀⠤⠦⣴⡦⡤⣀⠀⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⠃⠿⠦⠤⣄⣀⣀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠉⠘⠻⠗⠦⣄⡤⡠⠗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡏⠐⠒⠐⠶⠤⠆⣶⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠃⠤⠤⠤⡀⡄⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠚⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠐⠿⠷⠒⠶⠶⢖⠚⠃⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠒⠒⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠃⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠗⠦⢰⡄⢠⣀⡀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠙⠢⠖⢖⡲⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠢⢤⣀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⢠⣤⣿⣤⣤⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⣅⣿⣀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠉⣿⢙⣷⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠛⠛⣿⠛⠃⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠉⠀⢀⣠⣴⠟⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⠶⣦⣤⣀⠀⠀⠉⠛⠛⣿⠛⠛⠉⢀⣴⠟⠋⠉⠉⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⣿⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⠀⢠⣿⣤⡄⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠐⠻⣾⣷⡀⠀⣿⣴⠾⠿⢷⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠛⠿⣿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣶⣶⣶⣶⣶⣶⠿⣶⠶⣶⣶⣶⡶⢶⢶⣤⡀
⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡃
⠀⠀⠸⡿⠀⠀⢸⡏⠈⣿⠆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢰⣧⠀⠀⠈⠛⠛⠋⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⣤⣤⣤⣤⣾⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢹⣧⣠⡶⠾⠷⣦⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡾⠟⠉⠀⠀⠀⠈⣿⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠾⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠋⠉⢉⣉⣉⣉⣉⣉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⢴⡟⠛⠋⠉⠉⠉⠉⠙⠛⠛⣷⠄⠀⠈⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⣀⣀⣀⣤⠾⠋⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣀⣤⠶⢶⣤⣀⣀⣀⠀⢀⣀⣠⣴⠶⣤⣀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣠⡾⣿⡃⠀⠀⢀⣿⠉⠙⣿⠛⠹⣏⠀⠀⢸⣿⢷⣄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⣾⣋⢰⡟⠛⠳⠶⣾⣧⣤⣤⣿⣤⣤⣿⡶⠾⠛⢻⣄⣹⣷⡄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢠⣟⠉⠙⣿⠷⣤⣤⣀⣾⠁⠀⠈⣿⠀⠀⢹⣇⣠⣴⠾⣿⠉⠈⣻⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣿⠙⠳⣾⣏⣀⠀⠈⢩⡿⠛⠛⠻⣿⠛⠛⠛⣯⠁⠀⢀⣹⣶⠞⠋⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⢀⣿⠉⠙⠛⠷⢾⣧⣤⣤⣤⣿⣤⣤⣤⣿⡶⠞⠛⠉⣷⠀⢀⣼⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⠛⣾⣧⣤⣀⡀⠀⣿⠀⠀⠀⠈⣿⠉⠀⠀⢸⡇⢀⣀⣤⣿⡿⠋⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⣀⡿⠀⠈⠉⠙⢻⡿⠷⠶⠶⢶⣿⠶⠶⠶⠿⣿⠛⠉⠁⠀⣿⣠⣼⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⣽⠟⠶⣦⣤⣀⣸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣀⣠⣴⠶⠻⣯⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⣦⣄⣀⠈⠉⣿⠛⠛⠻⠶⠶⣿⠶⠾⠛⠛⢻⡏⠁⢀⣠⣴⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣌⠙⠻⢶⣿⣤⣀⣀⣀⣀⣿⣀⣀⣀⣀⣬⣿⠾⠛⣩⡿⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣼⠇⠀⠉⠉⠉⠙⣿⠉⠉⠉⠉⠀⢿⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⣿⣀⣠⣤⣴⠾⠛⠁⠀⢀⣀⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣰⡞⠋⠉⠛⢷⡀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡧⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠀⠀⠀⠀⠹⣦⣀⣀⣠⡾⠃⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠔⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢳⠤⠤⡼⠁⠀⠀⠀⠀⠐⠒⠁⠀⠢⠠⣀⠑⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⡤⡘⢄⠀⢠⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⡇⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⡤⣴⠖⢀⠴⠋⠀⠙⢆⠑⢬⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠠⠰⠉⣑⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢋⡠⠞⠁⠀⠀⠀⠀⠀⠙⠦⣑⠢⣴⠀⠀⠀⠀⡆⠐⣆⠱⠂⠀⣘⠤⠀⠀⢓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣉⠂⢄⡀⠈⠑⠄⠀⠀⠄⢄⡨⢀⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⣥⠴⠋⠁⠀⡀⣀⢀⣀⢀⠀⠀⠀⠀⣀⣀⡀⢀⣀⡀⡉⠒⠮⣅⡒⣬⠑⣴⣀⢈⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⣏⣉⠊⠄⠸⠀⠀⠀⠀⡽⢠⢠⠀⠀⡀⠃⠀⠀⠈⠀⠉⠀⠈⠞⠾⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣤⣸⣤⣆⠄⠀⠀⠀⢰⣥⣸⣠⣄⣶⠀⠀⠀⠠⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⢄⠀⣠⠀⢀⠄⠀⠀⠀⠀⠐⣤⠍⡤⡤⢤⡜⡀⠀⠘⠆⠀⡀⠘⢼⠀⠀⠀⠀⠔⡀⢀⠒⡄⣧⠒⡀⠀⢂
⠀⠀⠀⠀⠀⠈⣀⠀⢁⡀⠈⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⢸⠔⢀⡧⠙⣼⠀⠀⠀⣿⠀⠽⢸⠀⣿⡍⠀⡷⠇
⠀⠀⠀⠀⠀⠀⠋⠀⢈⠁⠀⠋⠀⠀⠀⡄⢸⠀⠀⠈⠁⠀⡇⠀⠸⠶⠬⠴⠶⠿⠀⠀⠀⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡿⡄⢸⠇⢀⢻⣠⣀⣀⡇⡗⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠯⠍⠀⣯⡅⠀⡿⡅⢈
⠀⠀⠀⠀⠰⠌⠗⠷⠰⠺⠶⠛⠗⠖⠴⠶⠳⠆⠖⠶⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠝⠦⠬⠦⠶⠦⠧⠶⠤⠖⠶⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠶⠒⠛⠉⠉⠉⠉⠛⠒⠶⢤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠋
⠀⠀⠀⠀⣰⠋⡠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠟⠀⢴⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠆
⠀⠀⠀⣰⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠉⢠⡀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂⡀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢠⡀⡀⠀⠀⡰⡛⠁⠀⠀⠀⠀⠀⠀⢘⡄⣀⣡
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢆⠀⢁⠜⡝⠁⠀⠀⠀⠀⠀⢀⠤⠚⠛⠛⠋⠀⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠐⠀⠀⠂⠁⠀⠀⠀⡀⠠⠀⠀⠒⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡇⡞⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣤⣴⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⣜⠣⢄⡀⣼⣸⠁⠀⠁⣀⡖⠋⠁⠀⠀⠀⡴⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠂⠈⠁⠀⠁
⠀⠀⠀⠀⠀⣍⠳⠄⢀⠀⠈⠉⠉⠉⢑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⡡⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸
⠀⠀⠀⠀⠈⠆⠉⠒⠢⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⡄⢪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⣀⣀⠀⢀⠔⠒⠀⠀⠀⠀⠀⢐⣉⣶⠆⠀⣀⢬⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡤⣚⡭⠟⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⡠⢚⡥⠋⠁⠀⠀⠀⠀⠐⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠛⣓⡶⠀⠀⠈⠑⣣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠐⢁⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⠞⠁⠀⠀⢀⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠂⠀⠰⠤⠀⠀⠠⠂⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠚⠢⠀⠈⠁⠀⢀⡴⠊⠁⠀⠀⡠⠚⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⡄⠀⠀⢯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠲⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⠋⠀⠀⠀⣀⠬⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡠⠔⠊⠁⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠔⠉⠀⢀⡴⠒⠁⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠐⠋⢹⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣰⣫⠖⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⣀⠀⢨⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡾⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢆⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠉⡄⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢲⡀⠀⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢠⡀⢀⡄⠀⣄
⠀⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⢠⣼⢱⣿⣿⡎⣧⡀⠀⠀⠀⠀⠀⠀⡆⠚⠉⠉⠀⠀⠀⠀⠀⢀⣮⣦⣴⣷⣤⣽⡄
⠀⠀⠀⣀⠀⠀⣃⠀⠀⣀⠀⠀⠀⠀⠀⢹⣆⢙⠋⣰⡏⠀⠀⠀⠀⠀⠀⠲⠶⢆⢀⠀⠀⠀⠀⠀⠀⡼⢤⠄⢠⡄⠀⣤⢹
⠀⠀⣀⣋⡀⣀⣛⡀⢀⣛⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣁⣻⣀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠷⠶⠿⠾⠄
⠀⠀⠉⠉⠁⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢾⡆⠰⣿⠂⢼⡷⠀⠀⠀⠀⠀⠸⡿⠀⠀⠐⠘⠃⠀⠀⠀⠀⠀⠀⢀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⣰⡀
⠀⠀⣴⣶⣶⣾⣶⣷⣶⣶⡆⠀⠀⠀⢰⣾⣾⣶⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡔⠁⠀⣀
⠀⠀⠘⣿⡟⢻⣿⡟⣿⣿⠃⠀⠀⠀⢘⣿⣿⠛⠀⠉⠈⠁⠀⠀⠀⠀⢠⣶⠄⣿⠁⣶⡄⠀⠀⠀⠀⠚⠂⠀⠶⠷⠀⠀⠛
⠀⠀⠀⠿⠇⢸⢿⡇⠿⠿⠀⠀⠀⠀⠈⢻⢿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠋⠙⢺⡆⠀⠀⠀⠀⠀⢀⡀⠀⢴⠀⣠⣧⡀
⠀⠀⣠⣍⠉⢉⣭⡉⠉⣥⡄⠀⠀⠀⠀⢀⠀⢙⡋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣃⣸⣃⣺⣿⠆⠀⠀⠀⠀⣼⣧⠸⣿⡇⢹⡿⠁
⠀⠀⠉⡅⠀⠈⡍⠀⠀⠩⠁⠀⠀⠀⠀⡄⢀⢠⡄⢀⢠⡄⠀⠀⠀⠀⠘⣿⡿⣿⡿⣿⡿⠀⠀⠀⠀⠀⢹⡏⢀⠿⠄⠘⠙⠂
⠀⠀⠲⠖⠀⠰⠶⠂⠀⠶⠆⠀⠀⠀⠐⠖⠀⠒⠖⠀⠒⠓⠀⠀⠀⠀⠀⠸⠇⣿⡇⠿⠇⠀⠀⠀⠀⠀⠭⠭⠸⠤⠄⠠⠠⠇

⠀⠀⠀⠀⣀⣤⠤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⡞⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣤⣾⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⣿⢸⡷⢰⣿⡆⢾⡇⣷⠀⠀⠀⢸⣿⣿⣗⣻⣿⣿⡇⠀⠀⠀⠀⠈⣿⣯⣭⣭⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⠿⣷
⠀⠀⠹⣶⣶⣦⣭⣴⣶⣶⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢩⡿⠿⢿⡍⠀⠀⠀⠀⠀⢘⣿⣿⣾⣷⣿⣾⡇
⠀⠀⠀⠉⠻⠿⠿⠿⠟⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠨⠿⠿⠿⠇⠀⠀⠀⠀⠀⠛⠛⠓⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⢿⠛⡟⣿⡟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⢀⠃⠀⠀⡠⠂⠀⢀⠅⠀⠀⢈⠀⠀⠠⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠀⢀⣑⡀⠀⠀⡃⡀⠀⠘⢄⣀⣀⠃⠀⡀⠬⢆⠀⢈⠤⠤
⠀⠀⠀⠀⢀⡴⠒⡇⠀⢱⡖⠆⠀⢰⠒⠊⠀⠈⢶⠠⠃⠀⠈⡆⠺⠀⠀⢈⡖⢂⠀⠀⡗⢦⡀⡠⢒⣐⠲⢄
⠀⠀⠀⠀⠸⡇⢤⡇⠀⢸⢁⡄⠀⢹⡀⡀⠀⢠⢸⡀⡀⠀⠀⣿⢀⠀⠀⠐⠇⢸⠀⠀⣇⣰⣿⡇⢢⣤⠅⢺
⠀⠀⠀⠀⠀⠈⠚⠃⠀⠸⠓⠇⠀⠸⠒⠻⠀⢀⢸⠚⡇⠀⠀⠽⠾⠀⠀⢸⠷⢿⠀⠀⠷⠋⠁⠈⠃⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠰⣏⣉⣉⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢨⡍⠚⠛⢇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⡇⠀⠀⢸⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⡇⠀⢀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⣥⡏⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡏⠀⠀⠄⠀⠀⠀⢠⠀⠀⠀⠄⡇
⠀⠀⠀⠀⠀⢀⠤⠆⠀⢀⡄⠆⠀⠀⣦⠤⠄⠀⠀⡤⡄⠀⠀⡤⠠⠀⠀⢸⠤⠤⠀⠁⡇⡀
⠀⠀⠀⠀⠐⠇⡄⡄⠀⢘⡇⠀⠀⠀⢿⠀⡇⠀⠀⡇⡀⠀⠄⣯⠀⠀⠀⢸⡃⢰⠀⡀⣷⠈⣷⠴⠾⣷⡆
⠀⠀⠀⠀⠈⠛⠤⢧⠰⢸⣯⣦⠀⡇⢹⣵⡇⢠⡂⣧⣧⢠⠇⣿⣴⡀⢰⢸⣧⣼⢀⡇⣟⣦⠟⠛⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠓⠐⠃⠀⠈⠢⠜⠀⠈⠒⠛⠁⠀⠑⠔⠃⠀⠀⠓⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠊⠐⠡⠊⠔⠁⠨⠂⠅⠨⠀⠕⠠⠂⠔⠠⠂⠅⠨⠊⠔⠡⠨⠂⠔⠡⠪⢊⠔⠅⠠⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣭⣵⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣸⣾⣭⣿⣟⣫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⣿⣉⣉⣉⣩⣭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣾⣧⡄
⠀⠀⠀⣠⣾⠿⢟⠛⡟⠿⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⢗⣒⣽⣆
⠀⠀⣴⡟⠁⠀⢸⢀⣇⡀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠄⠐⣵⡾⠿⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠹⢯⠥⢶⠒⠡⠤⡀
⠀⠀⣿⠀⠀⢠⢻⢿⡇⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣸⠀⢸⡀⠀⠀⠘⡄⠀⠉
⠀⠀⣿⠀⠀⠣⠚⠎⠑⠃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⣦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⡘⣾⣫⡏⢏⡆⠀⠀⢸⠀⠀⢀⡀
⠀⠀⠘⣷⣄⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠄⠀⢀⡿⠀⣠⡤⡜⠒⡈⠻⣦⠀⠀⠀⠀⠐⠂⠀⡆⠀⠈⠷⠙⠃⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⡀⠑⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠃⠃⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⡀⠑⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠒
⠀⠈⠀⠀⠄⠀⢀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠒⠒⠒⠈⠁⠀⠀⠐⠤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣛⣋⣀⡀⠀⠀⠀⠀⢀⣈⣛⣟⣛⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⢀⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣷⣶⣼⣿⢹⣿⣿⣿⣿⣿⡇⣿⣷⣾⣿⢹⣿⣿⣿⣿⣿⡇⣿⣷⣶⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⢸⣿⣿⣿⣿⣿⡇⠹⠿⠿⠟⢸⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣿⣿⢿⡇⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣾⢻⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣿⣿⣿⣿⠘⠿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⢸⣿⣿⣿⡇⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠸⢿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⡇⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠹⠿⠀⠀⠘⠿⠿⠁⠿⠿⠃⠀⠀⠀⠀⠘⠿⠿⠁⠿⠿⠃⠀⠸⠿⠛⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⢀⣼⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⢿⣿⣿⠟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠋⠉⠙⠋⠉⢻⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣶⣶⣶⣾⣶⣦⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⣿⠀⢸⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⣿⣿⣶⣶⣶⣶⣶⣿⣿⠀⠀⣿⣷⣶⣶⣶⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠚⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣏⠉⠉⠉⠉⢹⣋⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣏⢹⠉⠉⠉⠉⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢉⣉⠉⡏⠉⡇⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⡏⢉⣉⡉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢨⡂⡇⠀⡇⠀⠀⠀⠉⠉⠢⡀⠈⠙⢄⠀⠀⢠⢖⡶⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡆⡇⠀⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⢇⣰⠃⡾⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⡇⠀⠀⠀⣤⢄⡀⠀⡇⠀⠀⠈⠁⠀⣇⣀⠴⢚⡿⠖⠀⣾⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⡇⠀⠀⠀⢸⡆⠈⠲⠃⠀⠀⠀⠀⠀⠈⠁⠀⡞⠀⠀⠀⢻⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢺⠀⡇⢸⡇⡇⢠⠗⠒⠒⠒⢺⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⡗⠒⠒⠒⢻⡀⡇⢸⠀⡇⢸⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⠀⢧⡸⠃⣧⠏⠀⠀⢀⠔⠁⠀⠈⠳⡶⠋⠁⠀⠙⢶⠞⠁⠀⠉⢳⡀⠀⠀⢧⡇⠸⣤⠇⢸⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠸⠶⠤⠴⠤⠟⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠳⠤⠤⠤⠾⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡲⠛⠛⠛⢏⣩⠟⠛⠛⠫⣉⡽⠛⠛⠻⣖⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡼⠼⠧⣤⠤⣤⠬⠧⢤⡤⢤⠤⠿⠤⣤⢤⡤⠼⠾⣎⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣄⣤⣤⣈⣿⣥⣠⣤⣤⣽⣧⣤⣤⣤⣨⣯⣠⣤⣠⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢠⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⠀⠀⠐⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⣬⣽⣯⣭⣤⣤⣤
⠀⡤⠤⠤⣤⠼⠤⣤⠤⠤⣄⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠼⠦⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠉⠲⠂⠀⠶⠀⠀⠶⠀⠀⠶
⢠⣤⠀⢀⣤⠀⢀⣤⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢾⣷⠀⣾⡆⢰⣷⠀⣾⡆⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⣿⡇⢸⣿⡇⢸⣿
⣼⣿⡆⣴⣿⡆⢴⣿⡆⢰⣿⡦⠀⠀⠀⠀⠀⢰⣿⣿⡆⣿⣷⢸⣿⡦⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠃⠈⣿⠁⠈⣿


⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⢠
⠀⣶⡄⠀⠀⡇⢀⣤⡀⣼⣿⣄⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢰⡦⢀⡞⠠⣿⡦⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣶⣶⢰⣶⢰
⣤⣬⣤⡄⠀⡇⣠⣿⣅⢉⣍⠁⠀⠀⠀⠀⠀⣤⣤⣴⡄⠀⠿⠿⠆⠧⠀⣾⡆⠀⠀⠀⠀⠀⣴⣶⣶⡄⢸⠁⢻⡟⢼⡏⣾
⣿⣿⣿⠇⠲⡇⠉⣉⠉⣨⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⠇⠂⠐⠅⠀⠀⠈⣻⡋⠀⠀⠀⠀⠀⢻⣿⣿⠇⢼⠀⢨⡅⢨⡁⢩
⢸⣿⡏⠀⠀⡇⢨⣿⣅⢉⣉⠉⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⢰⡦⢀⡞⠠⣿⡧⠀⠀⠀⠀⠀⠈⣿⡿⠀⢸⠀⣶⣶⢰⣶⢰
⠀⠻⠃⠀⠀⡇⠙⠛⠋⣨⣿⡄⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠿⠿⠆⢧⠀⣾⡆⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠁⢻⡟⢼⡏⣾
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠁⠙


⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷
⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⡇⠀⢴⡦⠀⢸⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆
⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣇⢰⣶⣶⡆⣸⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⢴⡆⠈⢙⡟⠋⢰⣶
⣴⣶⣿⡇⠠⢼⡤⠄⣸⣷⣶⡦⠀⠀⠀⠀⠀⣾⣿⠎⡟⠘⣿⣿⠃⢹⠱⣿⣷⠀⠀⠀⠀⠀⠀⠰⢿⡿⠈⠉⠉⠁⠾⣿⠆
⠀⠀⣉⡁⠀⢸⠀⠀⣈⣉⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⡇⠀⢿⡟⠀⢸⠀⣴⡆⠀⠀⠀⠀⠀⠀⣚⠉⠉⣃⠀⠀⢘⠋⠉⢙
⠻⠿⣿⡇⠐⠚⠓⠂⢸⡿⠿⠗⠀⠀⠀⠀⠀⠾⠿⠆⠁⠀⠀⠀⠀⠈⠰⠿⠷⠀⠀⠀⠀⠀⢨⣿⡀⢠⣿⡄⢀⣿⣅⠀⣽
⣶⣶⠾⠿⠿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣶⣷⣶⣶⣿⣾⣿⣿⣷⣿⣿⣷⣿⣿⣿⣾⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⡠⡊⢕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠑⢄⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠔⠑⢄
⠀⠀⠀⠀⠀⣠⡪⠈⠀⠀⠁⢀⢀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢠⢠⠀⢸⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠼⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠄⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣽⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣴⣦⠀⠀⠀⠀⠀⠀⠀⠰⣖⣴⣿⣦⡀
⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠰⢻⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠃⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠿⠿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣤⣄⣠⣤⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠉⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠶⠒⠛⠻⣿⣿⠛⠛⠲⠖⠛⠛⣿⣿⠟⠛⠲⠶⠚⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣴⣿⣿⣶⣶⣶⣶⣤⣦⣿⣿⣦⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠸⠀⡸⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠃⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠋⠚⠑⠫⠾⠵⠣⠖⡴⣢⢶⣶⣤⣤⣤⣤⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠛⢻⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⢸⣿⡇⣿⢸⢰⡇⡇⣶⢰⢠⡆⡄⣤⢠⠀⢸⣴⡿⠒
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⢻⢸⡏⡇⣿⢸⢸⡇⡇⣿⢸⢸⡇⡇⣿⢸⠀⢸⣿⠗⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⡇⡇⣿⢸⢸⡇⡇⣹⢸⠈⡇⡇⣿⢸⠇⢸⢸⣎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⡇⣹⢸⢸⡇⡇⣿⢸⢸⡇⡇⣿⢸⡆⠸⢠⡧⠃⠬⠤⠄
⠀⠀⠀⠀⠒⠂⠀⠀⠂⠀⢀⠸⠂⠃⠁⠈⠀⠀⠁⠁⠉⠘⠸⠇⡇⣿⢸⠃⢀⠚⡇⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⢘⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣿⠙⠿⠯⠑⠒⠤⠤⠄⠀⣀⠀⠀⠀⠀⠐⢹⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⡟⢲⣶⣐⣋⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠽⠭⠄⠀⠀⠀⠀⠒⠄⣀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢄⣀⣀⣀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠛⡁⢈⣙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡘⠃⠘⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠇⠸⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣆⣰⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⣀⠀⠀⠀⠈⢹⡏⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣇⣠⣼⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠷⣦⣼⡟⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⢸⣿⣿⡟⠉⠉⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠐⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣉⣉⣉⣉⣉⣉⣉⡙⠛⠟⠿⠟⠿⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠐⣶⣶⣶⣆⠀⠀⢀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠟⠛⢋⣠⣶⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠠⡐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣇⠀⠀⠀⠀⣠⢇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠇⠀⢠⠖⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⡏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⢠⡀⣇⢸⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⢷⡈⠘⡄⠀⠀⡤⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⢗⡩⢛⣆⠀⠀⠀⢰⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢠⣛⠅⣊⠥⢚⠳⡀⠀⠸⠸⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣰⡫⢔⡊⠁⠈⠰⢂⡹⢆⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠥⣚⡅⠀⡿⣄⠈⠩⠔⣊⣧⠈⠀⠀⠤⠄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡾⢔⡫⠕⠀⢰⠃⠘⣆⠐⢋⡡⠚⣆
⣟⠷⣬⣍⡛⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣒⠭⡒⠉⣠⠃⠀⠀⠘⣆⠠⠖⣋⢽⠀⠀⠀⠒
⠳⢯⣖⡥⣋⠝⣲⢦⣤⣉⡙⠒⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢸⠫⢒⡉⠀⡴⠁⠀⠀⠀⠀⢸⡄⣋⡤⢾
⠀⠀⠀⠈⠛⠻⠶⣭⣞⠕⣉⠝⡲⢦⣄⣉⡑⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣚⠁⠀⡜⢀⠀⠀⠀⠀⠀⢸⠃⠡⢒⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣴⣫⢖⡡⢛⡵⣲⠦⣄⣉⠓⠲⠤⣀⡀⠀⠀⠀⠉⠁⠀⣯⠴⠂⢸⠀⠢⠀⠀⠀⠀⢠⠏⠠⢚⣭⠇
⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠷⢮⣥⢊⡥⣋⠝⣲⠦⣄⣉⠙⠲⠤⣄⣀⠀⢹⡫⠀⢺⠀⠩⠄⠀⠀⢠⠇⠰⢚⣥⡞⠀⠀⠀⠐⠂
⠀⠀⠀⠈⠱⠒⡭⠴⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢮⣴⣊⠴⢋⡽⣓⠦⢤⣉⠙⣲⠦⠤⢼⡄⠉⠄⠀⠀⢸⠀⠨⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠭⢒⡣⠴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢾⣥⡖⡡⣊⡿⢕⢴⢦⡀⡀⠀⠀⠀⠀⠈⠀⢰⠏⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢚⡭⠖⣂⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢧⡱⠣⢪⣾⢛⢔⢄⡀⠀⠑⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠥⠒⣊⡥⠄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢭⣉⣋⣘⣐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠴⢚⡥⠔⣀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠽⢒⡮⢗⡢⢔⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠥⠒⣫⠵⣚⠤⣂⡠⢆⡀

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡏⠉⠉⠉⠉⠉⠁⠀⠈⠀⠒⠤⢤⣄⠤⠄⣐⠒⠀⢨⡈⠍⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⡳⡄⠀⠈⠆⢠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡇⠀⠤⠴⠤⣀⠀⠀⠀⠀⠣⡤⠤⠤⣤⢄⠀⠈⠻⡀⢀⢞⠘
⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⠈⠢⡤⠨⠟⣛⣢⣄⠈⢣⠬⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⡛⠓⠦⠯⠹⠾⠷⠞⠀⢉⠕⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢍⣒⠨⢿⣍⣡⠶⠒⠀⢲⠂⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢐⡒⠂⡍⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢆⣀⣘⣄⡔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠍⡙⠃⠀⠀⠋⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣱⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠎⢀⡠⠤⢤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⡏⠀⢎⡀⢹⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠳⣄⣒⣁⡼⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠈⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠉⠒⠒⠒⠒⠒⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠁⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⢀⠄⠂⠈⠁⠈⠉⠀⠢⢄⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢠⠀⣆⣔⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⢈⠆⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠄⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠊⠄⠀⠀⠀⠐⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠂⠀⢸⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣽⠟⢻⣏⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢁⢠⢴⠊⠐⢀⣨⠉⠕⡢⢦⢬⡉⠁⠀⢁⠅⢤⣀⠀⢂⢤⣄
⠀⠀⠀⠀⠀⢀⠌⢠⠔⠚⣉⡀⠀⢠⡖⠗⢒⣦⣐⣂⣁⢄⣩⡆⣤⢷⠖⠢⡆⣐⣀⢌⠑⢿⣆
⠀⠀⠀⠀⠀⡈⠀⣎⠀⠞⣁⡅⢀⠏⠘⢠⠋⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⡇⢡⠀⢈⠽⠂⢈⠈
⠀⠀⠀⠀⠀⠠⡀⠈⠓⠤⠤⣉⣀⢤⣀⡆⡄⠉⠙⠀⠀⠀⠐⢡⡴⠾⢼⡧⣤⣭⢥⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠀⠼⠯⣈⡻⠓⠲⣒⠲⠶⠴⠲⢾⠷⠶⠊⡀⠤⠄⠰⠼⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠉⠠⠾⠅⠠⠀⠀⠢⠸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄⠂⠐⢀⠀⢠⠀⠀⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠈⠀⠌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠀⢁⣀⡔⢐⠨⡀⠐⠒⠑⠀⢸⠀⣀⡄⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠉⠚⠛⠉⠄⠂⠔⠨⢘⠳⡦⠄⡠⠐⠜⠡⢐⠠⠀⠁⠈⠪⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣄⠃⠀⠀⠀⠀⠀⠤⢈⠀⠀⠀⠐⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠒⠁⠀⠀⡄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠄⡈⠑⠀⣈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⣀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



⣿⠿⠿⢿⡿⡿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠟⠛⠛⠛⠛⠿⠶⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣩⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠟⢻⣿⠟⠟⠛⢿⡿⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⢿⡿⠛⠛⠻⣿⠟⠻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⠇⠀⠀⠀⢸⡇⢰⣿⠀⢸⣷⠀⣾⣧⠀⣾⡇⠀⣿⡆⢸⡇⠀⠀⠀⠸⣇⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⠏⠀⠀⠀⠀⢸⡇⠸⡿⠀⠸⡟⠀⠻⠏⠀⠿⠇⠀⠿⠇⢸⡇⠀⠀⠀⠀⠹⣇⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣴⠟⠀⠀⢀⣠⡶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⢦⣄⠀⠀⠀⢻⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⡏⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢻⣆⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⡟⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⢻⣆⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡟⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢿⡄⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣾⠁⢠⡿⠁⠀⠀⢀⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⣼⠇⠀⣼⠃⠀⠀⠀⣾⠃⠈⠙⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⣧⠀⠸⣇
⠀⠀⠀⠀⢠⡿⠀⢰⡟⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠾⢿⣇⠀⠀⠀⢿⡆⠀⣿⡀
⠀⠀⠀⠀⣾⡇⠀⣿⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⠀⢸⣧
⠀⠀⠀⢀⣿⠀⢰⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢿⡄⠀⣿
⠀⠀⠀⢸⡏⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢿⡆
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⡄⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠈⣿⠀⠸⣧⠀⠀⠀⠀⠘⣷⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣾⠃⠀⣿
⠀⠀⠀⠀⢿⡆⠀⣿⠀⠀⠀⠀⠀⢻⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⣰⡏⠀⠀⠀⠀⢠⣿⠀⢸⡟
⠀⠀⠀⠀⠈⣿⠀⠘⣷⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢰⡟⠀⠀⠀⠀⠀⡼⠁⠀⡿⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠹⣧⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡟⠀⠀⠀⠀⢀⣼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠙⢷⣄⠀⠀⠀⣠⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣄⠀⠀⠀⣠⡾⠃⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠷⣦⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣴⠾⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣦⣶⣶⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣤⣴⣶⣤⣤⣶⣦⣦⣴⣿⣦⣤⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣿⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣸⣷
⠀⠀⠀⠀⣿⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢈⣿
⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⣴⠾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢷⣄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠉⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣠⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⠤⠤⠤⢤⣤⣤⣤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠈⢳⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢿⡏⠉⠉⠁⠀⠀⠀⡀⢆⣸⣤⣼⣀⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⡽⠚⠚⢿⣷⣁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⠀⠀⠀⢸⣿⣇⣀⢀⣼⣿⠧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⣟⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⠸⠶⠾⠀⠀⠀⠀⠙⠻⣿⣿⣟⣋⣙⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠈⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⠿⠿⣿⢿⠿⠿⠶⠶⠶⠾⠶⠿⠿⠿⢿⣿⠟⠁⠀⠀⠀⢠⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠷⠶⣶⠿⠛⠛⠛⠛⠛⠛⠛⣿⠻⡟⠛⠛⠋⠀⠀⠀⠀⠀⢸⠯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⠟⣟⢷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣚⣼⠃⠀⠀⠀⠀⠀⠀⣸⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣋⣷⠀⠀⠀⠀⠀⠀⢠⡟⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣧⡀⠀⠀⠀⠀⣴⣟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣷⢦⣤⢶⣿⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠻⠃
⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡤⠤⠤⠤⠤⢤⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⡤⠞⠉⠉⠳⣄⡀⠈⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠈⠛⢿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠉⠳⣄⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠙⠢⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢹⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠈⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠉⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⣄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠈⠻⢿⣿⣿⣿
⣿⣿⠿⠋⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠉⠻⣿
⣿⣅⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠾⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣴
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⡞⠒⠒⠒⡖⠒⠒⠒⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣶⠒⠒⠒⣶⠒⠒⠒⣶⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⢀⣇⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⡏⠉⠉⠉⡏⠉⠉⠉⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠉⠉⠉⣿⠉⠉⠉⣿⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⣧⣀⣀⣠⣧⣠⣀⣤⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣤⣤⣤⣿⣤⣤⣤⡿⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⢤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⠚⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿

⠀⠀⢀⠀⣀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣿⣠⣽⣄⣿⡀⠀⠀⠀⢀⣽⣅⢸⡇⠀⠀⠀⠀⣠⣫⡀⣨⣧⡀⠀⠀⠀⠀⢀⠠⠚⠂⠀⠀⠀⠀⠀⡦⠀⣰⣦⠀⠀⠀⠀⢠⡄⣰⡄⢼⡆
⠀⠘⣿⠟⣿⠟⣿⠋⠀⠀⠀⠘⣿⡟⢸⡇⠀⠀⠀⠀⣻⣿⠓⢹⣿⠁⠀⠀⠀⠀⠚⠢⣀⠀⠀⠀⠀⠀⣼⣿⣤⣼⣯⣀⠀⠀⠀⢻⠇⢸⡃⠼⠄
⠀⠀⠛⠀⠿⠀⠛⠀⠀⠀⠀⠀⠿⠇⠸⠇⠀⠀⠀⠀⠙⠿⠀⠸⠟⠀⠀⠀⠀⠀⠀⠠⢽⠄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠈⠁

⠀⠀⣄⠀⣀⠀⢠⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡄⠀⠀⠀⠀⢀⠀⣶⡀⠀⠀⠀⠀⠀⢠⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⣤
⠀⠈⣿⠠⣾⡄⣹⡁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢾⡸⣿⠇⡷⠀⠀⠀⠀⢺⡟⣿⢱⡧⠀⠀⠀⠀⠾⠶⠿⠶⠷⠀⠀⠀⠀⠀⣴⡶⣦
⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⣆⠀⣲⠀⣰⡂⠀⠀⠀⢸⣷⣴⣾⡇⠀⠀⠀⠀⠚⠓⠉⡷⠧⠀⠀⠀⢰⣼⣤⣿⣤⣧⡆⠀⠀⢀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠁
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢺⠁⠄⠱⠂⠀⠀⠀⠀⣤⢀⣄⢠⡄⠀⠀⠀⢀⣼⣧⡀⠀⠈⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠃⢀⠘⠀⠀⠀⠀⠀⠀⢐⣻⡀
⠀⢀⣸⣷⣷⣿⡘⠀⠀⠀⢴⡷⣦⣭⣾⣷⠄⠀⠀⠘⣿⡿⠁⠃⠘⠀⠀⠀⠀⣤⠄⡿⢠⡄⠀⠀⠀⠘⠂⠠⠯⠀⠐⠂⠀⠀⢰⣶⣿⣿⣿⣶⡆
⠀⠀⠘⠓⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠴⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠘⠃⠛⠀⠛⠘⠃

⠀⢠⣦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⣤⠀⠄⠐⠀⠄⠀⠀⠀⠀⡀⣀⢀⢀
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⢠⣄⠀⣤⡀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⠀⠀⠀⠀⢰⣶⣆⣀⣀⣀⡀⠀⠀⢰⣷⣷⣿⣾⣾⡆
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣀⡀⢀⣀⠀⣀⡀⠀⠀⢐⣆⢀⣆⢀⣖⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠈⣿⠀⠀⠾⠆⠀⠀⠀⠈⠁⠃⠿⠘⠈⠁
⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠀⠀⠀⠀⠀⠉
⠀⠀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⣿⣴⣷⣴⣧⠀⠀⠀⠀⡺⠋⡭⠻⣆⠀⠀⠀⢀⣼⡉⣍⢹⣇⠀⠀⠀⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⢀⣛⡀
⠀⠸⠿⢿⣿⡿⠿⠀⠀⠀⠈⢳⣼⣿⣦⡏⠀⠀⠀⠈⣿⣶⣷⣾⡏⠀⠀⠀⠀⠸⠋⡇⢀⡄⠀⠀⠀⠀⠀⠀⣀⡘⠃⠀⠀⠀⢠⣷⣸⣿⢡⣾⡄
⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠸⠃⠿⠸⠃⠀⠀⠀⠀⠶⠆⠿⠸⠇⠀⠀⠀⠀⠤⠄⠀⠄⠄⠀⠀⠀⠈⠉⠁⠛⠀⠉⠁

⠀⠀⡠⠖⠒⠦⡄⠀⠀⠀⠀⣶⡶⠐⣶⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣆⣖⣰⡂⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⢸⢴⠠⣦⢰⠼⠀⠀⠀⠀⠿⠛⠒⠻⠇⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⢹⣟⣛⣻⡇⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠂
⠀⠘⢿⣷⣷⣿⠟⠀⠀⠀⠀⣢⣈⣌⣡⣔⠀⠀⠀⢀⣶⣀⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠈⠧⣼⣤⣷⢼⡇⠀⠀⢀⣷⣿⣿⣿⣾⡀
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⣤⣤⡄⠀⣿⣿⠀⣤⣤⡄⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠟⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠾⠿⠟⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⡀
⢀⣴⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⠀⢠⣿⠋⠉⠙⣿⡆
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣿⣦⣀⣠⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⠁⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠉⠻⣷⣤⣶⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⠀⠈⢻⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⣤⣤⣾⠏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⣄⣀⣀
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⡆
⠙⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣇⠀⢀⣼⡟
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣅⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢛⣛⣛⡿⠋⠀⢻⣆⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠈⠀⠈⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣾⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⠿⠿⠟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣡⡾⢗⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢗⡽⣋⣴⢟⡼⣛⣿⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠟⢃⡿⣹⣿⣿⣿⢾⣯⣵⣿⣟⣻⢟⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠠⠋⠀⠉⠉⠉⠰⡟⠩⣠⢟⣽⡫⢞⠽⣻⢟⡿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠎⠀⡰⠿⡾⢷⡾⢿⣿⣶⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠩⣶⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣶⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⣿⣯⢂⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣯⢿⣿⣷⣻⣿⡿⣣⠄⠀⠀⠀⠀⠀⠀⠠⣺⢟⣻⡿⢛⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠷⠾⠿⡷⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠔⠁⠊⠐⠽⣋⣴⣡⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡿⣀⣨⢅⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣠⣶⣿⣿⡾⣷⡴⢖⡠⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣷⠁⠀⠀⠀⠀⢐⣵⣾⣿⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠛⠛⠛⠛⠛⠉⠙⢻⡟⠁⠀⠀⠀⠀⠀⠀⢉⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣠⣾⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢉⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠂⠀⠀⠘⠒⠀⠤⡀
⠀⠀⠀⢀⣠⡤⠟⠓⠛⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⢀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⡞⠀⠀⠀⣀⠤⠴⣶⣻⡯⠥⠼⠥⠀⠀⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⢧⠀⣀⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠟⠁⠀⠀⠀⢀⣀⣤⡤⠤⠐⠒⠐⣒⡛⠩⠤⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⢰⣼⡿⠷⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠂⠄⡄⠀⠀⠲⠀⠀⠻
⠘⡷⠖⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⢸
⣾⠁⠀⠀⣀⣠⠤⠶⣖⣛⣫⣭⠥⠦⠔⠂⠀⠈⠑⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⡈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢻⡀⠀⠊⢉⡴⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⢗⣽⠋⠀⠀⢀⣀⣤⠤⠶⠶⣲⠾⠿⠯⢥⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠃⢸⠇⠀⠀⢰⢟⠉⠈⠙⠻⣼⠃⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠈⠢⠤⠴⠋⢸⡀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣼⠄⠒⠚⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⢀⠇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸
⠀⠙⢆⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠂⠒⠀⠚⠀⠀⠀⠀⠤⠧⠠⠴⠒⡶⠁
⠀⠀⠀⢻⠛⠉⠉⠛⠉⠉⠙⠯⠉⠙⠯⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣓⠀⡇
⠀⠀⠀⢸⡆⠀⠀⠤⠤⠤⠤⠤⠤⠀⠄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠠⠤⠤⠤⠤⢤⣤⠶⡶⠶⠀⡇
⠀⠀⠀⠺⡉⠀⠰⠶⠶⠶⠒⣒⣒⣛⡫⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠀⠠⠤⠤⠴⣶⣾⣿⠀⡆
⠀⠀⠀⢺⡅⠐⠶⠶⠶⠚⠛⣛⣚⣓⡭⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⣀⣀⣀⣀⣠⣵⣶⣶⢶⠀⡇
⠀⠀⠀⢾⡅⠀⠶⠶⠶⠒⠚⢛⣒⣛⠻⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⣤⣤⡶⣶⡃⣠⠇
⠀⠀⠀⠈⠳⣄⡀⠒⠒⠒⢚⣓⣛⣾⣥⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠉⠭⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠉⡭⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣛⣛⣿⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠠⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢛⣻⣛⣟⣻⣿⣉⣟⣛⣿⣛⣿⡛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣅⡀⠀⠀⠀⣀⣤⣀⠀⠀⠀⢀⣨⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠺⢿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡿⠗⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢰⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣏⠀⠀⠀⣹⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣈⣙⣟⣉⡀⠀⢀⣙⣛⣋⡀⠀⢀⣉⣛⣋⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠦⠤⠾⣿⣿⡿⠶⠤⠶⢿⣿⣿⠷⠤⠴⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣤⣤⣤⣼⣿⣦⣤⣤⣤⣴⣿⣧⣤⣤⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠂

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⢴⣧⡀⠀⠀⠀⢰⣧⡀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡈⠉⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠘⠋⠃⠀⠀⠀⠐⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠒⠀⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⢀⣀⡀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣀⠔⠊⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄⡀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡔⠈⠀⠀⢀⣤⣾⣿⡿⠿⠿⠿⠿⠿⢿⣿⣶⣄⡀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣤⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⡀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⣀⡔⠁
⠀⠀⠀⢀⠔⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠁⢲⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⣀⡀⠀⢀⡠⠔⠁⠀⠀⠀⠀⣼⣥⣥⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠁⢲⣶⣿⣿⣿⠉⠀⠀⠀⠀⠀⣀⠠⠚⠻⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⡟⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣀⣀⡀⠠⣼⣿⣿⣿⣿⠀⢀⣤⣤⡞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂⢩⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣿⣷⢸⣿⡇⠀⠀⠀⠀⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣏⣻⢸⣿⡇⠀⠀⠀⠀⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⡇⣿⣿⣿⣿⣿⢸⣿⣷⣶⣶⣶⣶⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠦⠕⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠻⣿⣿⣿⣦⡀⠀⠀⠀⣾⣿⡟⣡⣙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⢠⣤⡈⢻⣿⣿⣷⡀⠀⠀⣿⣿⡷⢦⠌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣀⣉⠙⢻⣿⣿⣿⡇⠀⠀⠙⢿⣿⣦⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠙⠿⠃⣠⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄⣀⠀⠉⠁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠶⠿⠿⠿⠶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣴⡾⠟⠉⢁⠤⢖⣒⣒⠒⠦⣄⣀⡀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠟⠛⠛⠻⣿⣾⣿⣅⣀⠀⡰⠁⠀⠀⢀⣽⠿⠿⠿⣿⣿⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣄⣀⢀⣼⠟⠁⠈⠉⠙⠽⡇⠀⢀⣴⡏⠀⠠⡶⢦⠺⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠈⢉⠛⢦⠀⠀⠀⠀⠀⢻⢇⡞⡻⠇⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⠷⣦⣄⠀⠀⠀⠺⣷⣤⣄⠀⠀⠀⠀⠉⢿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢱⠀⠈⠻⣏⡽⢷⣦⣼⣀⠈⠑⠶⣶⢶⠖⠋⠀⠈⠻⣕⢄⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡜⠀⠀⠀⣻⣿⠿⣿⣿⣿⣶⣮⣐⠾⣿⡗⠢⠄⣀⠻⠟⠒⠛⠐⠆⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠃⠀⢠⠎⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⣦⣝⡻⢶⣾⣷⣾⠀⠀⠀⠸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠠⠀⡞⠀⢻⣿⣦⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣶⣿⢻⣿⣯⣤⢄⠀⢣⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢿⣿⡿⣿⢿⣿⣷⣄⠀⢸⣿⣿⢿⣿⣿⣿⣿⠉⠻⢿⣿⣿⠘⣷⠈⣆⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡉⠻⣮⢣⢻⣿⣿⣦⣸⣿⣿⡀⠉⠻⣿⣿⣷⣄⠀⠉⠻⢧⣿⡀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠃⠀⢻⣇⣸⢏⣿⣿⣿⡉⠻⣿⣦⡀⠈⠙⢿⣿⣷⣦⡰⠎⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡀⠀⠀⠀⠀⠀⣸⣿⣿⣾⠟⣻⣿⣷⣄⣿⣧⣙⢦⡰⠆⢹⣿⣿⣿⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢇⠀⠀⠀⠀⣠⣿⣿⠿⡇⠀⠈⢽⣿⣿⣿⣿⣿⣷⣝⠶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠴⠚⠛⢷⣦⣄⣧⣀⡈⠒⠺⠿⣿⣿⣿⣿⣿⣷⣌⣀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠉⠁




⣶⡶⢶⢶⠶⠶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⡆⠄⠀⣤⢢⠰⡂⢰⢲⠂⢠⠖⡄⡖⠆⢸⠀⠀⠀⠀⡉⠔⠢⡔⠒⣤⡖⡄⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠁⠈⠁⠈⠊⠈⠊⠀⠈⠀⠈⠐⠁⠁⠀⠈⠁⠀⠀⠀⠁⠈⠊⠈⠊⠀⠁⠁⠑⠁

⠀⠀⠀⠀⠀⠀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣿⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠤⣤⣼⣷⠄⠀⠀⠀⠀⠀⢠⣶⢳⣦⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠿⠀⠿⠘⠟⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⡻⣫⣷⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣶⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠿⠼⠏⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣝⢫⣽⡵⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣽⠆⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠻⣦⡀⠀⠀⠀⠀⠀⠰⣿⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⢿⣽⠿⣅⣯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣼⣧⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠻⠟⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⢶⣦⡀
⠀⠀⠀⠀⣿⠛⠋⢸⡷⠙⠛⢿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣥⣿⣿⠁⠀⠀⠀⠀⠀⠀⠲⣦⡄⣴⡾⢿⣶⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣵⠂⠈⠛⠂
⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠟⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠲⠒⠛⠓⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⣴⣶⣄⠘⠁⠀⠀⠀⠀⠀⠀⣶⢀⢀⣿⣉⡇⠀⡀⡆⠀⠀⠀⠀⠀⠀⠰⠗⠮⣭⠧⠾⠄
⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⢶⢢⠀⠀⠀⠀⠀⠀⠤⢠⣿⡟⠿⣿⣷⠠⠤⠀⠀⠀⠀⠀⣿⣙⣹⡿⠿⠿⡿⡷⣷⠀⠀⠀⠀⠀⠀⠸⢷⣻⣿⣿⣾⡃
⠀⠀⠀⠀⣀⣤⣿⣧⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠤⢼⣿⠶⠷⠶⡶⠞⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿
⠀⠀⠀⠀⣧⣼⠿⡿⢿⢿⣠⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡼⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣭⢽⡀
⠀⠀⠀⠀⢀⣴⣧⡒⢿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⣄⣎⣀⣇⢰⣲⣄⠀⠀⠀⠀⠀⠀⠀⠈⢮⣶⡵⠋
⠀⠀⠀⢠⣾⣿⣿⡇⣠⣴⣇⡻⡀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠟⠛⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣾⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣶⡔⢂⢶⣶⡄
⠀⠀⠀⠘⠟⠝⠶⠭⢿⣿⣿⣯⠏⠀⠀⠀⠀⠀⠀⢸⣷⠏⠈⢿⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⢸⣿⣿⣿⡏⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣍⣾⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣶⡄⢸⣿⣿⣿⣿⣿⣿⡇⢠⣶⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠻⣿⣿⣿⣿⠟⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⣀⣤⣤⣬⣥⣤⣤⣀⠈⠻⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⡶⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢶⣶⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣆⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣸⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⢀⣠⣾⠋⠁⠀⢩⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠀⢀⣀⠉⠛⠻⠿⠿⠟⠛⠉⣁⡀⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⠟⠛⠻⣶⠀⠀⠀⠀⣶⠟⠛⠻⣦⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣄⣀⣀⣠⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡮⠓⠒⠐⠫⢕⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⢶⡶⠰⣶⣶⠆⠰⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣔⡗⠁⠀⠀⠀⠀⠀⠀⠑⢕⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠚⠒⠒⠒⠛⠚⠒⠒⠓⠚⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡅⠀⠙
⠀⠀⠠⠊⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠋⠙⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠷⢲⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠖⡎
⠀⠀⠐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠇⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⡇
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠠⢤⡤⠀⠀⠀⢤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣝⣢⣄⣀⣀⣀⣀⣀⣀⣤⣞⣵⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⢄⣀⠀⠀⠀⠀⣀⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠉⠉⠉⠉⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣁⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣉⣉⣹



⣿⠿⠿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⢦⣤⣄⣀⡀⠀⠘⡇⠈⡏⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣀⠉⠙⠛⠲⠮⢽⣿⣛⣺⡷⠶⣧⣤⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡷⢲⣦⣤⣄⣀⡈⠩⢿⡟⠲⠶⢭⣭⣛⡓⠶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠿⠞⠃⣼⣿⣭⡽⢻⡶⢲⡦⣤⣄⣀⡈⠉⠙⠓⠲⢬⡟⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⢀⡾⠁⠈⠉⠀⢸⡇⢸⡇⣸⣿⣏⢉⣿⣷⡶⢶⣼⣷⠟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢸⡇⣨⡟⢹⡏⠛⠋⣠⠿⣿⣶⠟⠉⠓⠶⠦⢤⣤⣤⣤⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣁⣀⡀⠀⠀⠀⠀⠀⣸⡟⢹⡇⢸⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠉⠁⣿⠫⢭⣉⡉⠛⠲⣦⣴⣏⣹⡇⢸⡇⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣠⡽⠟⠿⢧⣤⣈⡉⠙⣻⡷⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠙⣷⣈⣽⠟⣷⣦⣤⣄⣉⣽⣟⠿⢶⣤⣌⣉⠛⠻⠶⣤⣄⣀
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⢷⣤⠿⠛⣷⡾⣤⣄⣀⠈⠉⠙⠓⠶⢤⣌⣉⡽⣷⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣿⠀⢸⡏⢹⣿⡶⢦⣤⣤⣀⡀⣹⣀⣿⠖⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⢾⡟⠛⠋⠻⣿⣏⣽⠿⢯⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢻⡇⢸⡇⠀⢀⡾⠁⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣿⠀⢸⡇⢸⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⣿⠀⣸⡧⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⣿⠋⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⠄⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⠶⠛

⠀⠸

⠀⠀⢠⢤⣆⣴⣤⣦⣰⡄⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⢸⡟⣿⣿⣿⢹⡇⠀⠀⠀⠀⠀⢀⠀⣴⣶⣶⣌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣹⡏⠿⠀⢀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⢀⠸⣿⣿⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣷⡷⠞⠉⠀⠀⠀⠀⠀⠘⠚⠚⠛⠚⠛⠓⠈
⠀⠀⠀⢸⡿⣿⣿⣿⢾⣇⡀⠀⠀⠀⠀⠀⢠⠍⢛⠛⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠟⠉
⠀⠀⠀⠈⠁⠉⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⣀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡾⠻⣦⢸⣇⣸⠀⠀⠀⠀⠀⠀⣠⡾⠛⢹⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⠀⡀
⠀⠀⠀⠁⠀⠁⠀⣿⣿⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢨⣉⣶⢛⢡⠀⠀⠀⠀⠀⠠⠞⠁⠀⠈⠣⡔⢈⣠⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⠀⠀⠀⠀⢸⡿⣧⣀⢸⢀⣠⡿⣏⠀⠀⠀⠀⠀⢠⣤⣮⣽⣿⣿⣼⡄⠀⠀⠀⠀⠀⠸⠖⣶⢠⣄⣟⢹⣧⡄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠈⠻⠟⠛⠛⠛⠛⠿⠃⠀⠀⠀⠀⠀⠀⠚⠋⠸⡿⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠛⠂⠁⠿⠛⠉

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⠤⣤⣴⡤⠤⠤⣷⠀⠀⠀⠀⠀⠀⣠⣶⣶⡆⣀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡷⠶⢾⣿⡷⠶⠶⣿⠀⠀⠀⠀⠀⣄⡿⣿⣿⣾⣿⣿⡆⠀⠀⠀⠀⠀⣿⠿⠻⠛⠛⠻⠻⢿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡿⠒⠨⠿⠕⠒⠒⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣭⡻⠇⠀⠀⠀⠀⠀⣿⠀⠀⠰⠡⠄⠀⢸⡇
⠀⠀⠀⠈⠻⡿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⣟⠀⠀⠀⠀⠀⠈⠛⢫⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⣿⣠⣴⣶⣤⣶⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠈⠀⠁⠁

⠀⠀⠀⢀⣤⠴⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⣾⠀⢀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢤⡇⢦⣸⣠⠂⣧⠄⠀⠀⠀⠀⠀⠀⣴⣼⣿⣷⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⡄
⠀⠀⠀⠹⣄⠀⢀⣰⠃⠀⠀⠀⠀⠀⠀⠁⠴⣟⢾⢞⡷⠄⡁⠀⠀⠀⠀⠀⠀⣿⡏⠘⠛⢿⡧⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⡆⠤
⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠽⡊⠽⢺⠺⠍⡾⠅⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⡈⢂⠊⡘⣂⢋⡀
⠀⠀⠀⠰⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠑⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠒⠿⠗⠒⠿⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⡇⢰⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠱⠗⡠⠊
⠀⠀⠀⠀⠀⠀⣰⠳⢦⣨⠇⠀⠀⠀⢳⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⣀⠀⠀⠀⠘⡆⠀⠀⣇⢀⡞⠉⠉⠙⠓⠒⠶⢦⣀⡴⠊⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢦⠀⠀⢹⡀⠀⢹⡈⢳⠀⣠⢤⣤⣄⣀⡸⠊⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⡀⠈⡆⠀⣀⠷⠚⠺⣇⡏⢠⡟⠾⣄⡴⠋⢀⣴⡚⢧⡀
⠀⠀⠀⠀⠀⢸⡀⠀⡎⠀⠀⠈⢂⠹⡋⠁⢀⠴⠒⠋⠀⠸⣧⡾⢋⣀⣜⡧⠤⠽⠔⠛⠲⡄
⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠈⡄⢳⠀⡎⡤⠀⠦⠤⣀⠈⢱⠉⠀⢀⣀⣀⣀⣀⣀⠀⢻⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⡆⠈⢆⠀⠀⠀⡀⠀⢿⢰⠁⠀⠀⠀⢀⠀⢸⠀⡎⠁⠀⠀⠀⠀⠈⡆⢸⠋⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡔⠒⠻⡀⠈⠳⠤⠴⠃⠀⠈⠇⠀⠀⠀⠀⢸⠀⡽⠀⢰⠀⠀⠀⠀⠀⠀⢣⠈⡆⠀⢀⣄⠀⢠⠘⡄
⠀⠀⠀⠀⠀⢳⡄⠸⠥⣀⣀⣀⣀⣀⣀⠀⠈⣆⠀⠀⠀⡇⢰⢦⡆⢸⠀⠀⠀⠀⠀⠀⠘⡄⣧⠜⡅⠈⡇⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⠤⢀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠁⠀⡆⠀⢧
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢢⠀⠀⣇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡟⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠉

⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢋⣿⣿⣏⢿⡁⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠔⠶⠆⠀⠀⠀⠀⠀⠀⢀⣻⣃⢀⣛⡀⣘⣃⡀
⠀⠀⠀⠀⡀⠀⠈⣭⠁⠀⢀⡀⠀⠀⠀⠀⠀⠸⢧⠘⢿⣿⠟⢸⠿⠀⠀⠀⠀⠀⠀⠸⣈⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠿⠛⠻⠛⠛⢧
⠀⠀⠀⣠⡄⠀⠀⣤⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣺⣧⣄⣇⣴⣿⡂⠀⠀⠀⠀⠀⠀⠀⠉⠉⡱⢂⡔⢄⠀⠀⠀⠀⠀⢀⣇⣿⡃⢀⣿⡀⢘⣟⣸⡀
⠀⠀⢠⣬⣥⡄⣤⣭⣤⢀⣬⣥⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣬⣧⡬⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⢠⣶⡄⢀⣶⡆⢀⣴⡄⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡤
⠀⠀⠀⣤⣽⣅⣤⣍⣤⣠⣝⣥⡀⠀⠀⠀⠀⣠⣬⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⡘⠛⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣯⣿⣿⣯⡇⠀⠀⠀⠀⣿⣿⣿⣽⠇⠰⠆⢰⠆⠀⠀⠀⠀⠀⠀⣀⠈⣿⡏⠀⣀⠀⠀⠀⠀⠀⠀⣲⡀⠠⣀⣿⣀⠀⢀⣖
⠀⠀⠀⢸⣿⡇⢸⣿⡏⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⣯⡇⠘⠿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠉⠛⠃⠀⠀⠁
⠀⠀⠀⠸⠟⠇⠸⠿⡇⠘⠟⠇⠀⠀⠀⠀⠀⠈⡿⢿⠀⠀⠀⠆⢰⠆⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠠⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠾⠻⢦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠰⡆
⠀⠀⠀⠠⠀⠀⠀⠶⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠶⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⢠⣦⠀⣼⣧⡄⠀⠀⠀⠀⠀⣸⣇⠀⣼⣧⡀⢿⣿⠇
⠀⠀⠰⠿⠆⠀⠰⠿⠆⠀⠰⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣴⣵⣦⣿⣿⠁⠀⠀⠀⠀⠐⢿⡿⠂⢹⡟⠀⣸⣿
⠀⠀⠀⠐⠀⠀⠀⠒⠀⠐⠀⠂⠀⠀⠀⠀⠐⠰⠎⠠⠀⡶⠀⠄⠰⠂⠄⠀⠀⠀⠀⠘⣿⡟⢻⣿⢻⣿⡟⠀⠀⠀⠀⠀⠀⣸⣇⠀⠚⠛⠄
⠀⠀⠘⠛⠃⠀⠘⠛⠃⠀⠘⠛⠃⠀⠀⠀⠈⠛⠛⠀⠙⠛⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠿⠇⠸⠿⠘⠿⠃⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⡀
⠀⠀⠀⣠⡾⠋⠉⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣻⣁⣙⣃⣸⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⢠⡏⣤⡄⢀⣤⣄⢀⣤⡹⡆⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⡿⠿⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣤⡀
⠀⠀⠸⣇⡻⣃⠘⢿⠟⣈⢟⣡⡏⠀⠀⠀⠀⣿⣽⢯⣿⢼⣷⣿⣾⡇⠀⠀⠀⠀⠀⠀⡿⣿⣷⣿⡿⡿⠀⠀⠀⠀⠀⠀⢠⣄⣻⣟⣿⣻⣟⣠⡄
⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣞⣓⣛⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⣿⣿⢿⣿⣇⡀
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠛⠛⠛⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠛⠛⠛⠛⠛⠛⠀⠀⠀⣉⣉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠛⠃⠘⠛⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠁⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇





⣾⠉⣷⢸⣹⠆⡞⢋⡁⣸⢳⡀⡿⣆⡇⣿⠘⡿⠃⣸⣷⠘⣿⠃⣿⣴⠋⢳⢸⢷⣸⠀⠰⣏⡁⢻⡟⢸⣿⡇⣿⠀⡇⣾⠋⠉⢻⠋⣾⠀⣿⢸⣽⡆⣿
⣽⣶⣯⣼⣬⣧⣽⣶⣧⣯⣽⣧⣧⣼⣧⣿⣾⣷⣦⣯⣽⣧⣿⣤⣿⣼⣶⣯⣼⣤⣿⣤⣴⣾⣧⣼⣧⣼⣬⣧⣽⣾⣧⣽⣶⣦⣼⣤⣼⣶⣯⣼⣬⣧⣿
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⢀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⢀⣼⣿⣦⠀⠀⠀⠀⠀⢿⣇⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣋⡀
⠀⠋⣿⣿⣯⠑⠀⠀⠀⠀⠀⠈⢹⣿⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢧⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⢹⢿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢤⣤⣤⠀⠀⠀⢀⣀⣀⢼⣿⡆⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠈⠳⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠸⠸⠀⣶⡆⠀⠀⠀⠀⠀⠸⠏⠇⢸⣿⡟⠀⠀⠀⢸⣿⡿⠸⠿⠀⠀⠀⣈⣶⡆⠀⠀⠀⣶⣶⣶⠞⠈⣿⡟⠀⠀⠘⢆⠀⠀⠀⠀⠀⠛⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠋⠉⠀⠀⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠇⠛
⠀⠀⢀⣀
⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⣿⡟⢧⣀⣀⡀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣮⠁⠀⠀⠀⠀⠀⠀⢸⣷⣆
⣀⣠⣿⣧⠋⠀⠈⠀⠀⠀⠀⢸⣿⡀⠑⢢⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⠓⠀⠀⠠⠾⠶⠀⠀⠀⠀⣼⣿⠉⠙⣷⠀⠀⠀⠀⠀⠀⣼⣟⠈⢦
⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⠒⠒⠒⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⡷⡆⠀⠀⠀⢠⣿⣿⠀⠀⣣⣀⣤⣄
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢸⠖⠒⢲⡄⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠈⠀⡇⠀⠀⠀⠈⣿⡟⠀⠀⢻⣻⠿⠷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢩⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢾⠷⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡃⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⣤⣠⠄⣺⣛
⠀⠀⣸⣿⢃⣴⣆⣰⣶⡄⠀⠀⠀⠀⢠⡏⣿⡏⢳⣀⠀⠀⠀⠀⠀⠀⣯⡴⣷⣿⣄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡓⢬⡄⠀⠀⠀⠀⠀⠰⠿⠿⣿⣸⣿⣿
⠀⢠⣿⣿⡀⢸⣿⣿⡏⠀⠀⠀⠀⠰⠶⢶⡞⠳⢺⡿⠳⠂⠀⠀⠀⠀⠀⢸⣿⣿⠙⠆⠀⠀⠀⠀⠀⢡⣿⣿⣿⡇⢸⣿⠀⠀⠀⡖⠒⠒⠒⢲⠀⢩⣿⣿
⠀⠸⣿⣿⣷⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢙⡶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣷⠀⠀⠀⠀⠀⠈⠙⡿⣿⠁⠀⠋⠀⠀⠀⡇⠀⠀⠀⢸⠀⠸⡿⡿
⠀⠀⠿⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠃⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣶⣶⡶⠶⠶⠶⠷⠾⠷⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⢹⡇⠀⢸⡟⣿⡆⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⢸⡇⠀⢸⡇⣿⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣼⡇⠀⠸⠷⠿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⢻⡿⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⣠⡿⣱⡟⠁⢀⣼⢿⣶⣾⠛⣿⡄⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣯⣙⠻⣶⣟⣼⠟⠻⣶⡾⣣⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠉⠛⢿⣿⣿⠻⢶⣴⡟⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣿⣾⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣹⣿⣿⣿⡛⠛⣛⣛⡛⠛⠛⠛⣷⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⡇⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⡇⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠿⠷⡿⠃⠀⠛⠛⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣦⣀⠀⠀⢀⣽⠟⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠈⣻⣟⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣼⢿⡿⢶⡿⠃⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⢸⣏⣹⣧⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠶⠶⠶⠿⠿⠿⠿⠶⠶⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⣠⡾⠋⠀⠀⠘⢷⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⢀⣤⡾⠛⠁
⣠⣤⣤⣶⠶⠟⠛⠉⢀⣠⣴⠾⠛⠁
⠉⠁⣀⣀⣠⣤⡶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣄⡀⠀⢀⡀⠀⢻⠁⠀⠀⠀⣼⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⡇⠰⣏⡽⠀⢸⠀⠀⣠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣇⣀⣼⣀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠀⣸⠗⠋⠁⣀⣀⣀⡀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⢦⣀⣀⣠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⢩⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠀⠙⠋⢀⣰⣋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠶⠶⠶⠶⠶⣾⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣿⣷⡄⠀⠀⢸⡟⠛⠛⠛⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⠀⠀⠀⢸⣿⣿⣿⡁⣹⣿⣿⣿⡇⠀⠀⢸⠧⠤⠤⠤⠤⠤⣞⣹⠄
⠀⠀⠀⠀⠀⠀⠀⣴⠓⡶⠋⠉⣀⣀⠀⠘⣆⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡆⠀⢀⣞⣀⣀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠧⠼⠛⠛⠛⢦⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⠞⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⣀⣀⠀⢀⣀⣠⠴⡞⠁⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠉⠉⣽⠀⠀⡗⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⠀⢠⠞⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⠋⠀⠀⢀⡾⢦⠈⠛⠁⠀⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠉⠹⠤⠏



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢢⠀⠀⢄⣔⣤⢰⠂⠴⢦⢂⢠⣒⡄⡴⠆⢸⠄⠀⠀⠀⠈⡠⠒⢤⠒⢄⣴⢢⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠈⠀⠉⠀⠡⠀⠐⠁⠀⠈⠨⠈⠢⠀⠁⠀⠈⠃⠀⠀⠀⠀⠈⠤⠉⠠⠀⠉⠈⠐⠌

⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⣶
⠀⠀⠀⠀⠀⠀⣴⣾⣯⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣨⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⡦⢻⣿⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣤⣜⣿⣷⢘⣿⣟⣤⡀
⠀⠀⠀⠀⠀⠀⠛⣿⡟⡃⣹⣇⠀⠀⠀⠀⠀⠀⠀⡴⣶⣿⠈⡏⢹⣷⡶⡄⠀⠀⠀⠀⠀⠙⠻⠟⠻⣶⡋⢁⡾⠗⠀⠀⠀⠀⠀⠛⣿⢽⣿⣿⣿⢹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣟⠆⣻⡿⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⠀⡇⢸⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠛⠙⠋⠋⠛⠙⠋⠙⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⢿⡆⣀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠐⠒⠶⣆⣀⣠⣤⣴⣳⠄
⠀⠀⠀⠀⢀⣟⠁⢿⣿⣿⣿⠇⢹⣇⠀⠀⠀⠀⠀⠀⠛⠛⣠⣤⡘⠛⠃⠀⠀⠀⠀⠀⢀⣀⣤⡟⠛⠈⢠⣾⡏⠀⠀⠀⠀⠀⠀⣸⣿⣻⣀⣄⣸⣻⣿
⠀⠀⠀⠀⠯⠈⠇⠘⠿⡿⠟⠀⠇⠨⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⣷⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣥⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋⢙⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⢶⣶⡶⣶⣶⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⡦⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢠⡄
⠀⠀⠀⠀⠀⣀⣤⣤⣄⢠⡄⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣼⣯⢿⢣⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣛⡿⠃⠀⣴⢶⠀⠀⠀⠀⠀⠀⣰⣿⡿⣻⣵⣶⠒⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠾⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣧⣽⣿⢿⣹⡿⣘⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠾⠀⠰⠗⠀⠺⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⣿⢟⣿⡟⢯⣾⠏
⠀⠀⠀⠀⠀⠷⠿⠿⠟⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠸⠿⠗⠾⠿⠒⠿⠿⠀⠀⠀⠀⠀⠀⠸⠶⠾⠶⠗⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⠸⠚⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⢺⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢬⣿⠂⠀⠀⠀⠀⠀⠀⠀⣰⢖⣧⣾⢻⣦⡓⢶⡀⠀⠀⠀⠀⠀⠀⠸⣿⣍⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣬⣟⠛⣩⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⠿⠨⣟⣵⣿⡇⠀⠀⠀⠀⠀⣤⣀⠈⠉⢁⣠⣤⣄⡀⠀⠀⠀⠀⠀⠉⠀⢿⣿⣨⣿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣆⢦⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⢿⣯⣾⣿⣮⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣜⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠖⠀⣿⣿⡇⠐⠄
⠀⠀⠀⠀⠀⠀⠙⠋⠙⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣠⢴⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡨⡀⢀⢁⠀⣠
⠀⠀⠀⠀⠀⣴⣯⣿⣟⣻⣷⣯⣷⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠁⣡⠞⣋⣑⠷⡁⠈
⠀⠀⠀⠀⠀⢸⣿⣿⠈⠅⢹⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢧⢾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣌⠻⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠙⠯⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠊⠙⠉⠑⠁⠾


⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⡇⠀⠀⠀⡔⠲⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢠⣤⣤⣤⣤⣤⣤⡄⢸
⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠄⠀⢸⠀⠀⠀⠀⡇⠀⠈⠉⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠷⠤⠤⠤⣤⣤⡦⠤⠤⠼⠀⠀⠀⠀⠷⠤⠤⢤⣤⣤⡦⠤⠤⠾⠀⠀⠀⠀⠷⠤⠤⢤⣤⣴⡤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⣶⣶⣶⢲⡆
⠀⠀⠀⠀⢰⣶⣶⣶⣆⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⣾⣗⣶⣶⣶⣶⣺⣇⡀⠀⠀⠀⠀⢰⣶⣶⣶⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠉⠁⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣤⣥⣴⣦⣬⣤⣾⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇


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

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

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⡤⠀⠤⣤⣤⣤⡤⡤⠀⠀⠀⠀⢀⣠⣤⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠉⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⢀⡴⢋⠉⢉⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠤⣀⣀⡜⢁⠂⠉⠋⠀⢸⠀⢸⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡰⠋⡆⠀⢩⠀⠸⠄⠀⡇⠀⢸⠀⠘⣠⠴⠒⢒⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡜⠁⠀⡖⠚⠒⡆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢻⠀⠀⠀⠧⠤⢄⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠘⡄⠀⠀⡄⠀⠀⡇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠦⣀⠉⠹⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠦⣄⡀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⣧
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠉⠉⠉⠉⠛⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠘⠇⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⡟⠁⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⣧⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠃⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣿⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⢄⣼⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣤⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠘⢷⣌⠳⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣫⣿⢷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠚⠳⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠘
⠉⠉⠛⠒⢲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡏⠉⠒⠒⢦⠤⠤⠀⠀⢀⣀⠀⠀⠀⠀⢸⡏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠸⣧⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠃⣀⣤⡶⠶⠖⠒
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠶⠶⠚⠋⠋⠀⠀⠀⠀⠀⠀⣰⣧⡾⠟⠁
⠉⠉⠉⠒⢺⠖⠒⠢⠤⢴⡧⠤⠀⠀⣠⡀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠀⠀⠠⡄⣿⠄⠀⠀⠀⠀⠀⢸⡇
⠛⠛⠛⠛⢛⠓⠛⠓⠒⢚⠓⠒⠒⠒⠛⠂⠒⠀⢸⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠼⠀⠀⠀⠀⣀⣤⣄⣠⡀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⢸⠀⠀⣀⣀⣸⣀⣀⡀⠠⣾⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠚⠛⠉⠉⢛⠉⠉⠁⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠶⠶⠞
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⣀⡀⣼⠇⢀⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⢀⣺⣀⣠⣤⡤⢶⠶⠖⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠹⡷⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⣿⡆
⠒⠛⠛⠉⣽⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠤⢿⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠁⡀⢸⡇
⠀⠀⠀⠀⣺⠀⠀⠀⠀⢸⡀⣀⣀⡠⡶⠒⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓
⠀⠀⠀⢀⣽⣠⠤⠴⠒⢻⠋⠁⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣀
⠶⠒⠋⠉⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠤⢤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⢀⣠⣤⡶⠾⠃⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⣴⠶⠟⠋⠉⠀⠀⠀⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠠⢀⠰⠃⠈⠜⠁⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿
⠃⠄⡰⠊⠄⣀⠃⠀⠐⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢡⠈⠁⢀⠆⢉⠀⡘⠃⠀⠰⢠⡼⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠍⢡⠗⠂⠐⠎⠑⠒⡘⢓⡒⢲⠛⣒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠛⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠦⡀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠍⠀⠀⠀⠴⡀
⡇⠀⢀⣇⡔⢲⠀⠤⢤⣤⡤⠀⡏⠀⡇⢠⣤⣤⣤⡄⢀⣆⣰⣒⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈
⡇⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠃⠀⠀⠀⠀⠀⢀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⣽
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠒⠻⡶⠺⢧⣄
⡇⠀⣀⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢠⢀⠠⣴⢇⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢱⠀⢸⠁⠙⠶⣄
⡇⠀⠈⠋⠐⠁⠀⠈⠉⠉⠁⠀⡇⠀⡇⠀⠉⠉⠀⠁⠈⠙⠰⠋⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠴⣄⠀⠀⠀⢸⠀⠈⠀⠀⠀⠙⢦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⣠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠀⠀⠀⠀⠀⠀⡆⠻⠿⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠈⡆⠀⠀⠀⠀⠀⠀⡇⠐⠂⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⡇⠀⣠⣧⣬⡙⠀⠀⠠⠤⠄⠀⣿⣿⡇⠠⠤⠤⠤⠀⢠⣷⢤⡍⠀⠀⡇⠀⢁⡀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢳⠀⠀⠀⠀⠀⠈⠳⣤⠒⠒⠂⠶⡖⢦⣤⠀⠀⠀⠀⠀⠀⠠⠎
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠳⠄⠁⠀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠒⠀⠈⠉⠉⢇
⡇⠀⠀⢠⡀⢤⠀⣀⣀⣀⣀⠀⣿⣿⡇⢀⣀⣀⣀⡀⠀⠤⡄⠠⡄⠀⡇⠀⠀⠀⠀⠈⠷⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠾⣇
⡇⠀⠐⠋⠙⠀⠀⠈⠉⠉⠁⠀⣿⣿⡇⠀⠉⠉⠀⠁⠀⠛⠙⠂⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢻⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡀
⡇⠀⠀⢤⣉⠇⠀⠀⠠⠀⠄⠀⣿⣿⡇⠀⠤⠤⠰⠀⠀⢠⣍⠇⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇
⡇⠀⠀⠀⠠⡄⠀⢀⣀⣀⡀⠀⣿⣿⡇⠀⣀⡀⠀⠀⠀⠀⠠⣤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰
⡇⠀⠀⠈⢤⠄⠀⠈⠉⠉⠁⠀⣿⣿⡇⠀⠉⠉⠀⠁⠀⠀⢦⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⡇⠀⠀⠀⣴⡇⠀⠦⠴⠶⠶⠀⣿⣿⡇⠰⠶⠶⠶⠆⠀⠀⠀⣼⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠀⠀⠀⢶⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⢀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡿⠀⠀⠀⠻⡀
⡇⠀⠀⢰⠀⣰⠀⣀⣀⣀⡀⠀⣿⣿⡇⢀⣀⣀⢀⡀⠀⠀⢰⠀⣶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠉
⡇⠈⠁⠘⠀⠉⠀⠉⠉⠉⠉⠀⣿⣿⡇⠈⠉⠉⠉⠁⠈⠀⠘⠀⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢡
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠟⣿⣿⣿⣶⣳⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⡇⠀⠀⠀⠀⠀⠀⠀⢀⢸⡟⣶⣿⣿⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⡗⠀⠀⠀⠀⠀⠀⠀⠀⠛⠼⣻⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢧
⣿⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⣀⣀⠀⣀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠤⠐⠺
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣠⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⢉⡉⣢
⠀⠠⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⣞⡀⣀⢀⢀⠈⠬⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀⡀
⣶⡶⠶⣶⠶⢶⡶⡶⣶⢶⢶⢶⣶⢶⣶⠶⣦⠀⠀⠀⠀⠁⠀⠀⠀⡀⠈⡛⠛⠀⠁⠀⠀⠀⠀⠀⢀⠄⡠⠊⠁⠙⢦⡠⡀
⣿⡷⢶⣿⣴⣼⡧⢦⡿⠷⣾⣷⣿⠶⣿⠶⣾⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠥⠤⠆⠐⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠙⢢
⣿⣇⣘⣿⢈⢸⣃⣸⣇⣀⣿⠀⢸⢀⣿⣉⣿⠀⠀⠀⠀⠀⠀⡀⠀⢈⢤⣶⠛⢲⠤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⣐⠀⠁
⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠉⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⡠⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣰⣀⣊⣐⣿⢀⠀⢀⣐
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃
⠀⠀⢀⢤⡄⠀⢠⡤⡤⢤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠎⢄⢗⠰⡢⢀⡊⡀⢀⠀⡄⡐⠝⡇⠀⠀⠀⠠⢤⣠⣮⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢰⠁
⡔⢎⣄⣡⡑⠡⡾⠋⠙⢆⠢⣥⣉⣤⣡⣹⣦⠀⠀⠐⠉⢀⠀⠀⠈⢦⣀⣀⣖⣌⣱⣄⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸
⠑⠂⠐⡀⠒⣲⠁⠻⠓⠀⡉⡌⠉⢩⡉⠉⠁⠀⠀⠀⢠⣀⣀⣀⣀⠀⣿⣽⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸
⠀⠀⣸⣗⠀⡷⠀⠀⠀⠀⢠⠧⠀⡧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠒⠐⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠠⠤⠀⠤⠤⠴⠒⠒⢣⠤⠤⠀⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡑⡓⢖⡶⢶⢞⠠⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⡈⠀⣞⣛⣛⣫⢂⢃⣘⣧
⠀⢀⡞⡡⡀⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⢭⣭⣶⣶⣶⣶⣶⣗⣒⠆⠀⠀⠀⠠⠤⠼⠤⠬⠽⠭⠥⠯⠦⠼⠴⠼⠧
⠀⣎⠔⠀⠈⡀⢀⢀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⣼⣏⣉⣿⣿⣿⣉⣸⣿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠂
⠀⡆⠠⠵⠤⠀⠀⢀⡀⠀⢰⣖⣀⣰⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡁⠁⠀⠀⠀⠈⠈⠁⠁⠈
⠀⠁⠄⠄⠆⠀⠀⠀⠈⠀⢸⠁⢰⠁⠀⠀⠀⠀⣿⣧⣬⣿⣿⣿⣮⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⡆⡇⠂⠀⠀⣤⣴⡆⠤⢼⢀⠘⠀⠀⠀⠀⠀⣿⣭⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢐⢐⠀⣶⣿⠠⠀⠀⡶⣿⢀⢐
⠀⠇⠃⠒⠂⠀⢹⣷⡇⢀⢸⠀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⠘⠀⠀⣿⣿⡔⠀⢁⣿⣿⠈⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢩⡍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⣧⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⠿⡿⢿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠀⠀⠀⣀⣴⠞⠋⠙⠳⣦⡀⠀⠀⠀⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣆⣿⣸⣷⣦⡀⠈⠀⠀⠀⠀⠀⠀⠁⢀⣴⣿⣇⣿⣰⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠻⣿⣿⠏⠀⠀⠀⢀⣠⡶⠻⢶⣄⡀⠀⠀⠀⠹⣿⣿⠟⠀⠀⠀⢀⣠⡶⠟⢶⣄⡀⠀⠀⠀⠹⣿⣿⠟
⢀⣤⣦⢰⡆⣶⣤⡀⠐⠛⠁⠀⠀⠀⠉⠛⠂⢀⣤⣦⢰⡆⣴⣤⡀⠀⠛⠉⠀⠀⠀⠈⠛⠂⢀⣤⣶⢰⡆⣴⣤
⣿⣿⣿⣾⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠁⠀⠀⠈⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣤⡀⣰⣄⣠⣄⢀⡦⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢻⠛⠃⠙⠋⠙⢻⡇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠈⣧⠀⠀⠀⠀⣾⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢿⠀⠀⠀⠀⡟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡆⠀⠀⢠⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛⣛⣛⣛⣛⣛⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣭⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠶⠶⠶⠶⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣛⣛⣻⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿