Mischievous Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠉⠉⠉⣉⡵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠂⠀⠀⠀⣀⠠⠤⠤⠤⠦⠤⠤⠤⢤⣀⣀⠀⠀⠀⢀⣀⣀⣤⠬⠭⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠔⢩⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⢄⠀⠀⣷⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⡆⢀⡇⠀⠀⣀⣀⣀⣈⣙⡛⠀⠀⠒⠋⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠤⠬⣍⠓⠾⠀⠀⠀⢸⠁⠀⠛⠀⢸⠍⣹⠈⠉⠉⠉⢣⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠚⠲⢤⡀⣠⠞⠦⣅⣀⠀⢀⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠙⡇⠈⣹⢧⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⣀⠤⠤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⢱⢤⡀⠀⠀⠀⠀⠀⠁⠀⢤⣀⣀⣀⡼⠀⣸⠁
⠀⠀⠀⠀⠀⢀⡠⠖⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡴⠁⢀⠏⠓⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢏⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⣠⣋⣀⣘⣁⣰⡭⠭⠟⠒⠂⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⢦⣀⡤⡔⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣹⢃⣀⢿⡄⠀⠀⠀⠉⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡴⡊⠁⢀⡔⠐⠒⠒⠙⢷⠋⠀⠀⠀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠥⠞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣤⢤⡤⠴⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⣸⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⣾⣤⣤⣽⡤⢤⣷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠭⠟⠚⠉⠑⠺⠿⠾⠟⠃

⠀⠐⠂⠀⡄⢳⢄⠀⠀⠀⠀⠀⢼⠂⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⡴⠣⠁⣴⡇⠀⠀⠐⠀⠀⠠⣈⠞⠈⣆⠀⢸⢂⣼⣿⣿⣬⣀⣀
⠀⢀⠀⢰⡅⠘⡇⠈⠀⠀⠀⠀⣸⠀⢠⣾⡿⣿⡇⠀⠀⠀⠀⠀⡼⠃⠀⡰⢁⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⣸⠜⣽⠟⣿⠻⣿⣿
⠀⢸⣧⡘⠀⢀⠙⡀⠀⠀⠀⠀⡏⣰⣿⠟⠁⣿⣇⣤⣶⣾⣿⣿⡷⣤⡐⠀⣸⢒⠠⠀⠀⠀⡔⡇⠀⠀⢀⡄⠀⣿⣼⡿⢀⡏⠀⠑⠘
⠀⠸⣿⣧⠀⡀⠀⠸⡀⠀⠀⠀⣳⣿⠏⠀⣠⣟⣻⣿⣿⢫⣬⣄⢩⣷⡟⣆⡧⠈⠀⠀⢀⠎⠀⠀⠀⠀⢸⣷⣴⣿⣿⠃⢸⡏
⠀⠀⠸⣿⡄⠀⠀⠉⢿⡄⠠⠀⣿⡟⠀⠘⣿⠐⣿⣿⣿⡈⣙⡋⠠⣭⠠⣿⡱⠀⠀⢠⠏⠀⠀⠀⠀⠰⠇⢀⡻⣿⠁⠀⣾⠁⠀⢡
⠀⠀⠁⣿⣧⠀⠀⠀⣸⣷⠀⠀⣿⡇⠀⠀⠙⣦⣽⣿⣿⣿⣿⣦⣠⠄⠂⢺⠄⠀⢠⠏⠀⠀⠀⠀⠀⣽⣾⣥⠔⠋⠀⢰⣿⠙⢶⢾⡇
⠀⠀⠀⢹⡏⣷⡀⠀⢸⣿⡅⢠⣿⡇⠀⣐⣶⣿⠿⠿⢿⣿⡟⠻⣿⣿⣿⡿⠀⣠⠋⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⢀⣿⣿⡆⠀⢠
⠀⠀⠀⢸⡇⣿⡇⠀⢰⣿⠃⠸⣿⠀⣸⣿⣿⣿⣶⣶⣿⣿⣿⣦⣤⣤⣿⣀⣰⠡⠴⣶⣾⣾⣿⣿⣿⣿⠏⠰⠀⢠⣾⡿⣷⠈⠀⠀⣻
⠀⠀⠀⣸⣷⣿⠂⠀⢸⣿⠀⢀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣩⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⢏⠅⠀⠀⢀⣾⡿⠁⠈⠀⢀⣼⣿
⠀⠀⢀⡟⣿⣿⠀⢀⣾⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣼⣿⠿⢡⠟⠁⠀⠀⣾⣿⠃⠀⠀⢠⣿⡿⢹
⠀⠀⢸⠁⣿⡏⣠⣾⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠒⠶⣶⣾⡿⠛⢀⡏⠀⠀⠀⣼⣿⠏⠀⠀⢠⣿⡟⡰⠿
⠀⢀⡏⠀⣿⣷⣿⢧⠤⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢀⠀⡇⠀⠀⣴⣿⠏⠀⠀⣴⣿⠏⠀⠎
⠀⠸⠧⠄⣿⣿⡇⠘⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆⢀⠂⣴⣶⣦⠀⠀⠀⠈⣃⠀⢠⡇⠀⣰⣿⠏⠀⣠⣾⡿⠃⠀⠁
⠀⠈⠀⠀⣿⣿⣿⠀⠹⡄⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣿⣧⢤⣾⣿⣿⣿⣿⣄⠀⠀⠑⢄⢸⠀⣴⣿⠏⢀⣾⣿⠟⠁
⠀⠰⠤⢀⣿⣿⣿⡄⠀⢳⠀⠀⠀⢿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⢾⣿⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⡠⠚
⠀⢀⠀⢸⣿⣿⡿⠸⢄⠀⢇⠀⠀⠘⣿⣿⣿⣧⣠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢷⣆⡀⠀⢱⣿⡿⠃⠀⠀⠀⠀⠀⢀⢞⠱⡀
⠀⠀⢀⣿⣿⣿⡇⠀⠀⢵⢼⡀⡈⠀⠙⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣦⣤⣤⠻⠿⢶⣶⣿⠛⠻⢶⣖⡂⠀⢀⡿⡘⡄⢳
⠀⠀⣿⣿⣿⣿⢇⠀⠀⠀⢧⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠻⢿⣿⣷⡀⠀⠀⠙⢧⡀⠀⠈⠉⠋⣿⠀⣡⠘⢉
⠀⢸⣿⣿⣿⡇⠘⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣟⠿⠿⣿⣿⢿⣷⣦⠀⠈⢻⣿⣿⣶⣄⣀⡀⠙⣷⣄⠀⣸⡿⣍⠀⠀⠈
⠀⣿⣿⣿⣿⠁⠀⠈⢧⡀⠀⠀⠀⢀⠀⠀⠀⡠⠞⠤⣈⡁⠀⠀⣰⣿⡇⠀⠀⣿⣷⡆⠈⣧⠙⢿⣿⣿⣿⣶⣌⣿⣿⣿⠃⠈⠳⢤
⠀⣿⣿⣿⡿⠀⠀⠀⠈⢐⢄⠀⠀⠀⠘⢦⡞⠁⠀⠂⠈⠃⢠⣾⠟⣼⠈⠃⠀⣿⣿⣷⡀⣷⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⡝
⠀⢻⣿⣿⠀⠢⠀⠀⠀⠈⢢⠣⡀⢀⡴⠋⠀⠢⠀⠀⠄⣴⣿⡋⢸⠏⠒⠄⠀⠿⣿⣿⣧⢹⠀⠘⠁⠀⡈⣻⣿⣿⣿⣿⣿⣿⣷⣄⣱
⠀⣿⣿⣯⠀⢀⠀⡀⠀⠀⠀⢣⡸⠎⠀⠀⠀⠀⠀⢈⣼⠟⠁⣇⣾⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠂⣀⠞⠠⠃⠎⠳⣌⢉⠉⢻⠿⠿⢟
⠀⣾⣿⣿⠀⠈⠀⠛⠐⢂⠄⡟⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠠⢸⠇⠀⠀⠄⢀⡠⣾⣻⣿⣿⡁⠌⠡⠀⠀⠘⠀⠀⠟⣦⠹⣤⣀⢃⡘
⠀⢻⣿⣿⠀⠀⠀⢠⠜⠇⡄⠀⠀⠀⠀⠐⠀⢰⣿⠃⠀⠀⢀⡏⠀⢀⠤⠚⣡⠞⠁⣸⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠁⡷⡀⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠲⠤⠤⠖⠒⠒⠛⠛⠛⠛⠋⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⢸⣧⠴⠖⠛⠲⣄⠀⣄⣻⠉⠓⠲⠿⡍⠓⠲⠦⠿⠏⠛⠲⢦⣄⣀⠀⢦⣤⣀⠀⠑⠦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠢⢾⡀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⡞⠠⠌⠉⠓⠛⠀⠉⠓⠦⣌⣷
⠀⠀⠀⠈⠛⣟⣛⣒⣒⠒⠒⠠⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠶⢦⠀⠉⠁⠀⠀⠀⣀⣀⠀⢀⠴⠒⠓⣄⢠⠞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⡠⠔⣊⣁⣀⣀⡤⠤⣤⠤⣤⣤⡬⠍⠛⠛⢛⣿⡿⠶⠶⠶⠦⠈⠧⣄⣀⣀⣠⡄⠀⢠⡴⢋⡽⢩⠅⢰⠋⠀⠀⠀⡜⠈⣦⠄⠀⠀⣸
⠚⠋⠉⠉⠉⢻⡅⢠⠖⢾⠉⠹⡇⠀⣵⣔⣚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢻⡉⣵⢻⠀⢾⠀⠀⠀⡰⠃⠀⠈⢟⠛⣛⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠳⠜⠢⡀⢣⠀⡇⠐⢋⣁⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⣸⣿⠀⠀⠑⠒⠊⠁⢀⠀⠀⠨⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠉⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠳⣼⡄⠀⠀⠀⠐⢦⠉⠄⠀⢀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣇⢠⣤⣤⣀⣀⡀⣀⡰⢛⣶⡤⢾⣿⡆⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣙⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠛⠋⠉⠉⠹⣄⣈⣉⢙⡄⠀⠀⠉⠀⠁⠸⡅⢀⣼⠉⠹⣆⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⡟⠉⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣏⡃⠀⠀⠀⠀⠀⠀⠘⠲⢋⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢇⠀⠀⠀⠀⠀⠀⣎⠁⠀⠘⠛⠊⠁⠀⠀⠀⠀⠀⡀⠈⢧⣀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⡾⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⢀⣨⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⢶⢲⠊⡿⠀⠀⠀⠈⠓⠦⣄⣀⣀⡀⠀⣰⠁⣧⠔⣹⢿⣿⣤⡀⠀⣴⢮⣉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠁⠀⠀⠀⠀⠀⢰⠃⡏⠉⢩⠏⠁⠀⠉⠀⠀⠀⠙⠦⣙⢺⢁⣤⡨⢏⡁⣹⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣯⣄⡀⠀⠀⠀⠀⢿⠾⠗⠂⢺⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢰⡶⣄⣹⠁⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡜⠻⠏⡟⡿⡲⠶⡾⢶⣊⠻⠟⠻⠇⠉⢳⠀⣀⡠⠤⠒⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⢀⣃⡤⢖⡽⠿⣧⡀⠈⢑⣲⣭⣛⣓⠶⢲⡏⣟⡓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣴⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠥⣶⣛⡃⢂⣴⡧⠯⢬⢤⠖⠒⢺⠻⡉⠁⠀⠈⠉⡿⠿⣷⣶⣶⣤⣤⣍⣉⣉⠒⠒⠲⠦⠤⢴⠋⣠⣾⠟⣋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡸⠀⢸⠉⡏⠻⣳⠀⢸⠾⡆⠀⠀⡇⢧⠀⠀⠀⠀⢹⣤⣤⣤⣌⣉⣉⡛⠛⠻⠿⠿⢿⣶⣶⣶⠾⣋⡥⠊⢹⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠇⠀⢸⠰⡇⠀⠉⠀⠈⠉⠁⠀⠀⢳⠼⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢤⣄⣼⡧⠚⠁⠀⠀⢸⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⡦⠐⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠈⢻⣿⢿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⠀⠀⠀⡀⠐⡈⣋⠂⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣶⣄⠀⠀⠀⠀⢨⣴⣿⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠭⠿⡿⠟⠀⠀⠀⠀⠸⢿⡿⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣴⣶⠆⠀⠀⠀⠀⠀⠘⢧⣁⣙⣛⣉⣵⠋⠀⠀⠀⠀⠀⠐⣶⣦⣌⢲⣀
⠀⠀⠀⠀⢦⡴⣶⣶⡄⠈⠸⠛⠛⠁⠀⠀⡀⠀⠀⠀⠀⠈⢿⣻⢟⡿⠁⠀⠀⠀⠀⢢⠀⠀⠈⠛⠛⠃⠉⢠⣴⣶⢦⣴
⠀⠀⠀⠀⠈⠾⠿⡿⢷⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⡾⢿⠿⠷⠃
⠀⠀⠀⠀⠀⠈⡖⢄⢸⣧⣴⡶⠂⢀⠀⠀⠸⢷⣦⡀⠀⢀⣀⢀⡀⢀⡀⠀⢀⣴⡾⠏⡀⠀⡀⠐⢶⣦⣼⡇⣀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡬⣷⣟⡀⠀⠘⡦⡄⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠘⡏⢀⢴⡃⠀⠀⣙⣶⠥⣼⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣤⣌⣿⣁⡴⠊⠀⢻⠀⠀⢈⠌⠉⠻⡿⠋⠁⡀⠀⠀⡟⠀⠙⢦⣈⣿⣁⣤⡿⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠈⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠏⠀⠀⠀⠻⢿⣿⣿⡆⢰⣿⣿⣿⠟⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓


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

⠀⠀⢀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⢀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠣⢀⠀⠀⢀⡀⠀⠀⠀⣸⣤⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣶⣿⣶⣼⣿⣷⣧⣾⣼⣿⣷⢂⣤⡀⠀⠀⠀⠀⡠⠔⠋⠀⢠⠖⠋
⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢤⡈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⡤⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤⣞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠠⠤⠤⠴⠒⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠚⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣱⣿⣿⣿⠻⠿⠋⠉⠿⢻⣿⠃⣼⣿⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢈⣩⣽⡒⠀⠘⠋⠀⢘⣩⣤⣼⢿⣿⣿⠛⠁⠀⠀⠀⣀⠀⢈⡀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠈⠁⠛⠋⠀⠀⠀⠀⠉⠀⠛⠉⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡤⠄⠀⠀⠠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⣄⡑⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⣈⣤⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣴⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡛⢦⣀⣀⣀⣀⣠⣞⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠀⠉⠉⠉⠉⠉⠘⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠿⠽⠆⢠⡄⠘⠉⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠠⣤⠼⠧⡤⠄⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣴⠀⠀⠀⠀⣧⠖⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⠀⡴⠚⠛⠛⠋⠉⣿⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡼⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⣼⠈⢳⡀⠀⠀⠀⠀⠀⠀⣷⢰⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠐⣿⣀⡀⠈⠳⡄
⠀⠀⠀⠀⠐⠟⢃⣠⣴⣿⣴⣶⣿⠟⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠓⢿⣤⣶⡴⠿⠖⠲⠶⠶⠶⠖⠋⠁⠈⡿⠀⠀⡷
⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⢹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣌⡓⠿⢶⣤⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠙⠿⣿⣶⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠙⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⣿⣇
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⣼⣿⠟⠉⠁⠀⠀⠀⢻⣿
⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣠⡆⠸⣿⡇
⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⡙⠻⣶⣤⣀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣾⣿⡇⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷⡀⠈⡙⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠊⠁⠀⠀⠀⣀⣴⡿⠟⠁⠈⣿⡀⠘⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠉⠛⠿⣦⠹⣧⠀⠀⠀⠀⣀⣴⡾⢿⣉⡀⠀⢠⠀⠸⣧⠀⢹⣿⡄
⠀⠀⠸⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⡀⠀⣿⡇⠈⠀⢠⣴⣾⠟⠋⢤⣬⢿⣿⣷⡏⠀⣀⣹⣧⠀⢻⣿
⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⣿⡆⣿⣇⠀⠀⣾⣿⠀⠀⠀⠀⠉⠂⠋⠉⠓⠀⠈⢿⣧⣀⣾⡟
⠀⠀⠀⢸⣿⡆⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣷⡀⠀⠀⣙⣛⠁⢸⣿⠀⢀⣿⡏⠐⢦⡀⠀⢀⣰⣶⠦⠄⠀⠀⠀⠻⣿⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠹⣿⣄⠐⣿⠙⣧⢠⣿⠀⣸⣿⠇⠀⠈⢻⣶⡿⠻⣇⢠⣶⣄⠀⠀⠀⢻⣿⡀
⠀⣤⣶⠿⠟⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⠻⣷⣿⠀⣿⣿⠏⢰⣿⡟⠀⠀⠀⠸⠟⠁⠀⠈⣃⣈⡉⣷⣄⠀⢸⣿⡇
⢸⡿⢀⣤⣤⣤⣤⣤⣴⣿⣿⣧⣄⡀⠀⠀⠀⣿⡄⠘⠛⠀⣿⠇⣠⣿⡟⠀⠀⠀⠀⢀⣠⣴⣞⣛⣉⣍⣉⣻⣿⣷⣼⡿
⢸⢇⣾⠟⠛⠛⠉⠉⠉⢙⣻⣿⣿⣿⣷⣦⣄⠈⠿⣶⣶⡾⠋⡴⠟⣿⣷⠀⠀⠀⠀⠉⠉⠙⠓⠿⢬⣍⣋⣙⣹⢿⣧
⣼⡿⠃⠀⢀⣠⣴⣾⡿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣈⣉⣁⣼⣿
⠁⢀⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣤⣀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⡀⠀⠘⣦⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢿⣿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣻⣷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣆
⣿⣷⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠙⢲⡀⠀⣰⠞⠛⠛⠋⠙⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣔⣩⠤⣤⠀⠀⠀⠹⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠛⢉⡴⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⢀⢈⡗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠛⠁⠀⠀⠉⠓⠦⣄⣀⠐⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡉⠳⣄
⠀⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⠮⣷⣄
⠀⣠⠶⠛⢉⢉⠙⠲⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠠⡀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠙⠂
⢰⡇⠀⠀⠉⠘⢳⠀⠹⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⢹⡀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⢰⣤⣀⠀⠙⢷⡀
⠈⠛⠲⠤⠤⠤⢤⡄⠀⠀⠀⣀⣤⣴⠲⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⣀⣀⠘⣆⠙⠳⢦⣽⣏⠓⢦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣾⣿⣿⣇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⢀⣀⣉⣉⣉⠀⠸⡆⠀⠀⠀⠉⠀⠀⠈⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠀⠻⣇⣹⣾⠀⢹⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⡴⠚⠉⠉⠀⠈⠉⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢧⣈⡷⡁⢸⢦⡀⠀⠘⣆⡀⠀⢠⢀⣿⠘⠀⠀⠀⠀⢀⣠⠞⠀⠀⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡄⠀⠀⠙⢦⣼⠧⣄⣹⢦⡀⠀⠉⠓⢶⡛⠙⠲⠴⠶⠖⠛⠉⠀⠀⢀⣠⠼⠛⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠻⣦⡀⠀⠀⠻⣦⣼⠳⣦⣽⡷⣤⣜⡂⢃⠀⠤⠆⣀⠀⢠⠢⠀⢀⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠀⠐⣿⠻⢦⡐⠈⠀⠙⠓⢾⣟⠛⢮⣿⠛⠶⣦⣌⣀⣁⠀⠈⠈⠀⢈⠉⠉⠙⡏⠁
⠀⠀⠀⠀⣠⠞⠀⠀⢰⠃⠀⠀⠸⣆⠀⠙⠦⣀⠀⠀⠀⠈⠙⠺⢤⣉⣳⠟⠤⢤⣼⣉⣉⡟⠉⠈⠀⠀⠀⣿
⠀⠀⠀⡴⠃⠀⠠⠽⡟⠀⠀⠀⠀⠙⢆⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠉⠙⠒⠾⠧⢤⣯⡀⠀⠀⠀⠀⣰⠇
⠀⠀⡼⠁⠀⠀⠀⣸⠃⢀⣠⣤⣶⣚⡿⠃⠀⠀⠀⠀⡼⠻⡝⠛⠲⠦⢤⣤⣀⣀⣀⣀⣈⣀⣀⣠⡤⠞⠋
⠀⡼⠁⠀⠀⠀⠀⣯⠶⢋⣿⡿⠟⠛⢦⣀⣠⠖⣲⠴⠗⠒⠛⠒⠶⡞⣽⣶⠀⠀⠈⠈⠉⠉⠁
⡼⠁⠀⠀⠀⠀⠀⠛⠋⠉⠀⠙⠦⣄⡀⠉⠉⠈⡉⠓⠶⣄⠀⠀⠀⡗⢙⠿⠶⣦
⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠠⡈⠓⢦⡼⠀⠀⠀⠈⠋⠙⢺⠃
⠀⠈⢳⡈⠙⠓⠶⠦⣤⣄⣀⣀⣀⠀⣼⢈⠳⣄⣸⠦⡼⠃⠀⠀⠀⠀⡴⠒⠻⡆
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢉⣉⠉⣿⢸⡀⠀⠁⠀⠀⠀⠀⠀⢰⡋⠙⢶⡞⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⠀⠀⠀⠿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠐⣺⡛⠛⠻⣄
⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⢄⠀⠀⠀⠀⠀⠹⡟⠲⠶⠋
⠀⠀⠀⠀⣼⢷⣶⢤⣤⣀⣀⣀⢀⣀⣠⣏⣀⣙⣦⣤⣀⣀⢀⣀⡴⢃
⠀⠀⠀⠀⠙⠶⢍⣛⠳⠤⣬⣭⣏⣁⣀⣀⣭⢿⣩⣃⣈⣉⡿⠁⢀⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣸⣿⣧⠀⠀⠀⠀⣀⣈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡁
⠀⠀⠀⠀⢀⣾⠟⢉⡉⠙⢻⣿⠀⣤⣞⣋⠠⠤⠤⣤⣹⣿⣧⡀⠀⠀⠀⠀⣴⣿⠟⣉⣁⣉⣛⢶⣄⠀⠀⠀⢠⣾⠛⢉⣉⡛⣦
⠀⠀⠀⠀⣼⠃⠀⢀⣹⡆⠸⠟⢸⡏⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⣾⡿⠟⠛⠉⠉⠀⠀⠀⢛⡇⠀⠀⠀⠁⠀⣾⡟⠀⣼
⠀⠀⠀⠀⣿⡀⠀⢾⣯⣀⣠⡀⠈⢛⠲⠴⠤⠤⠤⠤⣼⣿⣿⠇⠀⠀⠀⣿⣦⣤⠀⢀⣀⣀⡠⠴⡟⠁⠀⢀⣀⣀⣀⣿⣷⢀⡿
⠀⠀⠀⠀⠸⠇⢀⣼⡿⠟⠋⢁⠀⠀⠘⠒⠂⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⣹⡿⠟⠛⠀⠩⠥⠄⠚⠀⠀⠀⠀⠉⠛⠿⣿⣧⡼⠃
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⣀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣾⠋⠀⠐⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠺⠶⠿⣦⣤⣠⣾⡿⠉⠛⠋⠀⠀⠀⠀⠀⣠⡶⠛⠉⠈⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⢹⡇⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣼⣷⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠼⠛⠉⠙⠛⠻⠷⢶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠋⠉⠀⢠⡿⠁⠀⠈⠙⠻⣦⠀⠀⢻
⠀⠀⠀⣠⣴⠞⠋⠉⠉⠉⣀⣤⡶⠾⠶⣦⡀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⡅⠀⢀⣀⣤⣄⣼⡇⠀⠘⣷⣤⣄⡀
⠀⢀⡾⠋⠀⠀⠀⠀⠀⢾⣏⣁⡀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦
⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⢹
⣿⠃⢀⣴⠇⠀⣠⣤⣄⠀⠀⢀⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⣤⠖⢻⡇⠀⠀⠰⣆⠀⠈⣿⡿
⣿⠀⢸⡏⠀⠀⡏⠀⠙⠷⠾⠛⠋⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⠸⠀⠀⠀⠀⣿⠀⣰⡟
⠸⣧⣼⡇⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⡆⠀⢺⡆⠀⠀⣠⡿⠞⠋
⠀⠀⠈⠙⠶⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣤⣿⡧⠀⠈⠻⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠄⠐⠒⠒⠶⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠉⢹⡿⠉⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠉⠙⠙⠛⠛⠛⠋⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⣀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠉⠙⠛⠿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣤⣤⣤⣶⣶⡶⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠿⣷⣄⠀⣀⣴⣾⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏⣽⣷⣤⣹⡍⣼⠋⢉⣉⠷⣆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠒⣀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠟⠀⣸⠃⣿⣺⣿⡆⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⡟⠁⠀⠀⣀⠀⠀⠀⠀⠈⠷⠶⠶⠾⢷⣦⣿⣏⣉⣀⡤⠟⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠀⠀⠀⠘⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⢘⣿⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠁⠀⠀⠀⣠⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢰⡿⢸⡇⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⡠⢿⡄⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⣼⠁⠈⢻⣦⡀⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⢸⡏⠈⢦⡔⠁⠈⠻⣦⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⢿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢠⡄⠈⢿⣤⠊⠑⢦⣄⡀⠈⢙⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⡀⢷⡀⠈⠛⠷⠶⠶⠶⠿⠛⠋⠁⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠻⣤⡙⠶⣤⣀⡀⠀⠀⠀⠀⢀⡴⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠠⣿⣧⣤⣌⣛⠶⣤⣀⠀⠀⠀⢀⣴⠋⠀⠀⡀⠀⡈⣧⣤⣤⣤⣶⠶⠿⠛⠃⢠⠇
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠉⠙⠛⠿⣯⣝⠓⣶⠞⠁⠀⢀⠀⢻⡀⡂⢹⡀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠘⠿⠶⣶⣶⠶⠶⠶⢶⣤⣀⠙⣿⠋⠀⠀⢀⣼⡇⠸⡇⢷⣼⠛⠛⠛⠛⠛⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣤⡶⠋⡘⢣⠀⣷⡞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣘⢉⡽⠀⢠⠃⠈⡷⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣠⡏⠀⣼⣁⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⢿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⠈⠉⠉⠉⣹⠋⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣷⡤⠹⣧⢀⡿⠀⠠⣾⣧⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⠶
⣤⣤⣀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣿⠶⠟⠋⠁⢀⣼
⢳⡀⠉⠛⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠃
⠀⠻⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⠾⠛⠻⠿⠶⣶⡶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢲⡀⠀⠠⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⠶⢺⠋⠀⠀⣤⡤⠄⠀⣀⠄⠆⣐⣠⢂⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡘⠀⠀⠀⢳⣦⠤⠄⢀⡠⠔⢫⠸⠒⠁⠀⠀⠈⣗⢈⡆
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⠾⠤⢤⢀⠞⠉⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠿⠶⠷⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠴⠖⠒⠒⠒⠒⠒⠶⠦⢼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠔⠚⠛⡿⠻⠿⠿⢿⣿⣿⣶⣦⣤⣤⣀⣰⠀⢀⣠⣤⣶⣿⠿⠟⠛⠉⠁⣧⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠡⠿⠟⠉⠉⠉⠉⣿⠀⠈⠟⠿⠿⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠢⢀⡀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⡤⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢱⡀⢨⡗⠒⠶⠤⣤⣤⣤⠤⡤⣤⠤⠐⠒⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣧⠔⣆⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣄⣀⡀⠀⠀⣷⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠂⠀⢸⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣷⣄⠀⠀⠉⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠉⢋⡄
⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠉⢿⡻⠏⠆⠀⠀⣠⣶⣿⠁⠀⠐⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⡀⠑⠀⠁⠀⢺⣿⣿⣿⣿⣶⠤⠍⠀⠀⠻⣿⣶⣦⡙⣶⠀⢀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⢀⠰⢴⣶⣶⣶⣾⡄⠀⠁⠀⠀⠙⠋⠀⠀⠉⠀⠠⢀⠀⠀⠀⠉⠉⠙⠀⠀⣿⣿⣿⣝⡋
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣽⣿⡿⠿⡇⠀⠀⣠⣀⠠⢰⠃⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢋⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠰⠻⠇⠀⠀⠤⣄⡂⣛⣯⣴⣿⠀⠂⠀⠀⠀⣾⠀⠀⠀⠀⠓⢶⣶⣾⣧⣰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠙⢿⣿⡻⠇⠀⠐⠀⣆⠘⢉⣼⠀⠀⠉⠻⣾⣧⣤⣿⡿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢰⠀⠘⠀⡿⠙⢤⣀⠘⢿⣏⣀⣼⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⠀⡀⣀⣠⣴⣾⡴⠀⠀⠀⠇⠀⠀⠉⠙⢋⣩⣿⠋⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⡷⠀⠘⠀⠀⠀⠀⠀⢀⡀⠋⠁⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡿⠀⠀⠀⠈⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣀⣀⡤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠛⠉⠁⠀⠀⠴⠾⠗⠒⠒⠒⢒⣲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⠶⠀⢀⣠⡤⢖⠲⡄⠀⠀⠀⠚⠉⠉⠉⢉⣭⣶⠶⠄
⠀⠀⠀⠀⠀⢀⣠⠶⠺⣍⠀⢀⣠⠟⠛⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⠵⠦
⠀⠀⢀⡤⣞⠭⠖⠒⠒⠦⣤⣋⠤⠤⠶⠲⣶⢺⠃⠀⠀⠀⠀⠀⣠⠖⠒⠲⡶⠿⠖
⢠⠖⠉⠀⢯⡉⠉⠉⣋⡽⣇⠀⠀⠀⠀⣀⡴⠋⠀⣴⢀⡤⠀⠘⠡⠾⢋⡾⠁⠀⠀⠀⠀⡀
⣏⠀⠀⢀⠤⠽⠯⣍⠁⠀⠈⠱⠒⠉⠉⠉⠉⡤⢾⡵⢋⣡⣾⣭⣽⠿⣥⣀⣀⠀⠀⣖⢼⢹⣞⡆
⠘⠦⣀⠘⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉⠀⢀⣀⡀⠀⡿⡶⠒⠦⠬⠭⣈⠉⢸⠁
⠀⠀⣆⣿⠿⣶⡶⣶⠒⠲⣖⠒⠛⠋⠉⢡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠓⠲⠤⣄⡀⠈⠙⠃
⠀⠀⠈⠑⠚⠄⡆⢘⠒⣚⠭⣦⠀⣠⠔⠁⠱⡄⠀⠀⠀⣠⠴⠦⢄⡀⢀⣀⣠⡔⠊⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠋⠥⠟⠀⠀⠀⠙⠛⠁⠀⢀⣀⡹⣄⡀⡼⠁⠀⠀⠀⠈⠙⠷⠀⣷⡾⠛⠛⠛⠭⣭⣕⣦⡀⠀⠀⠀⠀⠀⢀⡔⡄⠀⢠⢌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⣗⡢⢽⣷⣄⡀⢀⡾⣓⠦⠤⠖⠉⠉⠛⢿⢿⣼⠁⠀⢹⡧⠐⠆⠐⢀⢠⠾⠀⠖⠶⢬⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢥⣌⣓⣬⡣⣙⣼⡇⠀⠉⠒⠦⣄⣀⣠⠞⠮⠟⠤⠴⠚⠀⠀⠁⠀⠈⠈⠲⡆⢸⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣭⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⣀⡀⠀⠀⡀⢠⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠉⠉⠁⠀⠈⠉⠋⠉⠙⠛⠋⠉⠉⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⢀⢀⣤⣤⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢣⢠⡉⡈⡄⡀⡖⣶⠦⣴⢿⡶⢠⢷⣿⠁⠀⠀⠀⠀⠀⠀⡀⣾⢿⢸⡏⡟⢻⢸⣿⡿⣿⢷⣦⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⡀⠈⠋⠁⠁⡀⠐⠀⡀⢠⠠⣠⡘⡀⠀⡟⠀⠀⠀⠀⠀⠀⡄⠳⠼⠸⡃⠁⠣⢸⣟⣻⣿⣿⣆⣏⣿⡄⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⠃⣜⢠⢌⠠⠀⠀⡀⠁⢁⢨⣥⣁⡁⣼⠁⠀⠀⠀⠀⠀⢰⢳⡶⡆⣯⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠁⢡⢿⢸⠇⠇⠳⠹⢸⢿⠾⣾⠛⣟⣿⠀⠀⠀⠀⠀⠀⣼⣲⡄⢇⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⠁⠆⢈⠀⠈⠀⠀⠀⡀⠠⠠⣀⢠⠽⢽⡿⠀⠀⠠⠄⡀⠀⠍⠭⠭⠭⣙⠺⣛⡛⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣧⢠⢱⡇⡞⡀⣦⡼⢠⡁⡌⡀⠳⡈⠃⠀⢀⠨⠀⠀⠀⠀⠀⠀⠀⢛⣰⣏⠻⠏⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠑⢇⢣⠋⠁⠀⠤⠀⠰⠄⠀⠀⠀⠀⠀⠐⠀⠘⠛⠂⠀⠈⠉⢉⠉⠐⢠⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠘⡄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠙⠶⣶⣦⡀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⢀⣤⣀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢀⡀⠀⢀⣀⡀⢀⣀⠀⢼⣄⢳⣷⣄⠀⠀⠈⠻⣶⠶⠽⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣦⠀⣼⠁⢻⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠐⠖⠈⣧⡀⠈⠙⣋⣭⠹⢄⣀⣥⣾⠿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⡉⠙⠻⣦⣄⠀⣴⣫⣿⡿⠛⠀⠷⠶⠎⠰⠆
⠀⠀⠀⣦⣤⣦⡀⠀⡘⣃⡀⠀⠀⠰⠂⣞⣀⣴⣯⡶⠺⠷⠄⢳⡍⠷⠀⠒⠂⠞⠃⠀⠀⠀⠀⠀⠈⠻⠿⠦⣄⣀⣈⠖⠋⠙⠻⣧⠒⡰⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠸⣿⣿⣿⣷⠟⠉⠛⠢⠤⠀⢸⣿⢿⣟⣦⡀⠀⠀⠀⠀⠉⠻⠶⠄⠰⠶⠶⠶⠤⠤⠀⠀⠀⣀⠢⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢿⣿⣿⠏⣦⠀⠀⠀⠀⠀⣾⣿⡿⣧⠿⣿⡦⣤⣤⡀⠀⣤⣤⡤⠤⠤⠀⠀⣠⠀⠀⠀⢠⡈⢄⠀⠙⠛⠀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠐⠂⠰⡖⢲⡆
⠀⠀⠀⢠⠜⠈⠁⠀⠈⠓⣄⠀⠘⢼⣿⣿⡇⢻⡠⣿⠱⣿⣿⠓⢷⡟⠀⠀⡆⠐⠀⠀⠀⠀⠀⠓⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠁⠈⠐⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⢾⣄⠒⠘⠿⠿⠃⠀⠀⢸⡦⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⠻⠋⠁⠀⠈⠓⠢⠄⠀⡀⠀⠀⠈⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠆⠀⠖

⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠙⢦⡀⢠⠖⠛⠛⠉⠉⠉⠛⠓⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⣴⡭⠴⡦⠀⠀⠀⠳⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠈⢁⠔⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣤⣼⠉⠲⢤⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠓⠦⢬⣿⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⣌⡳⣄
⠀⠀⣠⠤⠶⠦⣄⠈⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠻⢦
⢠⠏⠀⠀⠂⢲⡄⢻⣄⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⢦⡙⡆⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⢀⡀⠀⠈⠳⣄
⠘⠦⣤⣀⣀⣀⡁⠀⠀⠀⠀⣀⣠⢤⠄⠀⠀⠀⢸⠀⠀⠀⠈⢳⡙⠀⠀⠀⠀⠀⠀⠀⢻⠓⠦⣄⣻⡗⠦⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣧⠀⢀⣴⣿⣿⡃⢘⣆⠀⠀⠀⢸⡆⠀⠀⠀⠀⢷⠀⠀⠀⢐⣚⣉⠁⠈⢧⠀⠀⠉⠙⠀⠈⠙⠮⣷
⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⢻⡉⣳⡏⠘⣦⠀⠀⠀⢧⠀⠀⠀⠀⢸⡧⠖⠋⠉⠉⠉⠙⣶⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣅⣹⢎⣈⡷⣄⠀⠘⣆⡀⠀⡄⣸⠃⠁⠀⠀⠀⣀⡴⠋⠀⢹⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢷⡄⠀⠈⠳⣤⠛⢦⣸⠶⣄⠀⠉⠓⢾⠉⠑⠒⠒⠚⠋⠁⠀⠀⣠⠴⠛⠉⠀⣹
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠙⢦⡀⠀⠀⠑⢴⡉⠲⣼⡓⠦⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡤⠞⠁
⠀⠀⠀⠀⢀⡴⠃⠀⡤⠀⠀⢇⠙⢦⡀⠀⠀⠈⠚⠧⣉⣳⠧⣉⣳⠖⠤⢤⡄⢠⣤⠤⠄⠀⠀⢷
⠀⠀⠀⣠⠏⠀⢠⣴⠃⠀⠀⠘⣆⠀⠈⠳⢤⣀⠀⠀⠈⠉⠒⠶⢯⣉⣙⡞⠲⡴⠃⠀⠀⠀⠀⣼
⠀⠀⡰⠃⠀⠀⢉⡏⠀⠀⣀⣀⣬⣧⠀⠀⠀⠈⢻⢶⠦⣤⣀⣀⠀⠀⠈⠉⠙⠷⠀⠀⠀⣀⡴⠃
⠀⡼⠁⠀⠀⠀⢸⣡⠴⣿⣿⡿⢯⡀⠀⣠⢤⣀⣯⡬⠧⣄⣠⠍⢭⡙⠒⠒⠒⠒⠒⠚⠋⠉
⣰⠁⠀⠀⠀⠀⠿⠷⠚⠉⠳⣄⡀⠙⠛⠃⠚⠧⢤⡀⠀⠀⢸⠞⢿⣦⣄
⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠠⡈⠲⢤⣽⠀⠀⠈⠓⠓⢶⡞
⠀⠘⢮⠉⠓⠲⠤⣤⣀⣀⣀⠀⠀⡏⡹⢤⣘⡦⡼⠇⠀⠀⠀⢠⠖⠚⣆
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢉⣉⢹⡇⣇⠀⠉⠀⠀⠀⠀⠀⣾⠉⠓⡶⠃
⠀⠀⠀⠀⠈⢳⡀⠀⠐⠀⠒⠒⠛⢧⠸⣄⠀⠀⠀⠀⠀⢈⣿⡉⠙⢧
⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⢀⡷⣌⠢⠀⠀⠀⠀⠀⢹⠙⠒⠋
⠀⠀⠀⠰⣟⠿⣗⡶⠶⠤⢤⣤⠤⠾⠦⣤⡽⡶⠤⢤⣤⠔⠋⣄
⠀⠀⠀⠀⠈⠓⠲⢭⣙⡒⠚⠶⢶⣒⣒⣺⣷⣧⣤⣴⡏⠀⣼⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠒⠉⢢⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣀⠈⡠⠚⠓⣀⢀⣠⣖⠂⢤⡄⣴⢀⠀⠀⡀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢰⢡⠁⠀⠀⢀⣼⣿⣿⣿⣿⠀⠀⠘⡶⢀⡄⡄⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢨⣅⠂⠸⡄⠿⢹⣿⠩⠽⠿⠄⣀⡤⠃⠈⢹⡆⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠘⢾⣿⡃⣄⠱⡀⠀⠣⠠⠖⠀⠀⠀⠀⠀⠀⡜⣠⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠈⢏⢷⣿⡄⠊⠀⠀⢠⠄⠀⠴⠂⢹⣷⡾⢊⠉⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠈⠲⣙⠻⣷⠾⠿⠁⠄⠀⢀⡴⠟⢉⠔⠁⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢈⠙⠓⠒⢣⣤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣶⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀⠀⠀⠀⠀⠀⢠⣾⢇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣄⠀⠀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣧⣵⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⣀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧⣾⣿⣿⣷⣶⣶⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠙⠿⢿⡿⠛⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣙⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡅⣼⣿⣿⣧
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣾⣿⣿⣿⣿⡄
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡍⢿⣿⡇
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠈⢸⣿⣿⡙⢿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣏⠙⢿⣾⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣄⣹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⡿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠙⣿⣿⣿⣿⣻⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⢻⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢸⣿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣾⣷⣦⢠⣶⣴⣿⣿⣷⣶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠋⠉⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣷⣶⣄⡀⠀⠀⠈⠻⣿⣿⡟⠁⣀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠃⠙⠲⣹⡏⢽⣶⠃⠀⣾⠀⠀⠀⠀⠀⠀⠠⠴⠾⢿⣿⡀
⠀⠀⠀⠀⠰⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⣍⠀⠠⠀⠀⠹⢦⣀⡀⠀⠀⠀⠀⠴⠆⡀⠉⠃
⠀⠀⠀⠀⢸⠁⠀⠠⠀⠀⠀⢀⡀⣠⣤⠘⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢸⠀⠀⠀⡀⡴⠊⠉⠀⠈⠋⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⢀⠀⠀⠀⠈⠡⣤⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⢶⣄⠀⢠⣁⣀⠀⡤⠀⢀⣴⠉⢠⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣷⡌⠉⠀⠈⠀⢠⣿⣉⣀⣈⠀⡆⠀⠀⠀⠀⠈⠀⢰⡶⡶⠀⠂⠀⠀⠙⢇⠀⠈⢿⡋
⠀⠀⠀⠀⢀⣿⣿⠋⠀⢀⠔⠀⠀⢘⡁⠈⠉⢹⣆⡁⠀⠀⠀⡀⢤⢀⡟⡀⠄⠀⠀⠀⠀⠀⢛⡗⠀⠈⢀⡀
⠀⠀⠀⠀⢾⣿⡗⠛⠊⠉⠀⠀⠀⠘⣧⠀⠀⣸⠏⣽⣷⣶⣿⣗⣨⡆⠻⡆⠀⠀⡾⠀⠀⠀⠘⠀⠀⠀⠰⣔⣶⣆
⠀⠀⠀⠀⣾⣿⡇⡀⠤⠚⠁⠀⠀⠀⠈⠀⣸⡿⠀⢿⠿⠟⠋⠉⠭⡝⠀⠻⣦⣖⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⢿⣾⡓
⠀⠀⠀⠀⣾⠟⠠⠄⠤⠒⠂⠄⠀⢱⠀⢈⣿⣖⢣⣨⣤⣀⣀⣠⣴⣦⠄⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡍⠃
⠀⢀⡀⢼⡿⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⣿⡳⠄⠙⠛⢿⣿⡿⠛⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣾⣏⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠠
⣤⠔⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢘⣿⢃⡀⠀⣠⣶⣿⣿⠶⢤⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢻⣿⡿⡄
⣿⣄⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⢾⡿⣷⠞⠛⠙⠋⠉⠀⠀⠈⠋⣶⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡈
⣿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠘⣠⠀⠊⡇⣠⣾⡿⣿⡄⠠⠤⠞⠲⠻⠋⣿⠋⠛⠀⠀⠀⠀⠀⠀⣀⢠⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣷⣾⢧⡴⢳⣀⣾⣼⣿⣿⡁⠋⠀⠀⠀⠀⠀⠀⠀⣻⣤⠀⠀⢀⣀⣤⣠⠀⢸⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣧⣀⡲⢆⡀⢠⣿⣾⣴⣄⡜⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣷⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠹⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠙⣭⡙⠿⣿⣧⣾⣜⣷⡀⣿⣿⣿⣿⣿⠿⠛⠛⡛⠉⠀⠙⢻⣿
⠀⠀⠀⠀⠉⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠁⠀⠀⠘⠁⠀⡬⣈⠛⡛⠻⢿⣿⣿⣿⡿⠶⡆⢰⠃⢰⡃⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⣸⠋⠓⠲⢬⣙⠛⡗⠀⠰⠃⠴⠛⠙⡦⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠘⢿⠟⠀⠙⣿⠸⣷⣄⠀⠀⡀⠀⠈⠀⠀⢂⠄⠀⠀⠀⠐⠀⢹⣿⠏⠀⡀⠤⠀⡀⠀⢡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣤⣤⣤⣴⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠉⠉⠋⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢖⠀⠐⢛⡓⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡼⠘⠁⠀⠀⠋⠀⢼⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡗⠓⠤⠤⠤⠄⢠⢀⠀⠀⡀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⢎⠀⠀⠀⠀⠀⠀⠈⢎⠻⠛⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⣻⠄⠀⠀⠀⠀⠀⠀⢈⠶⣀⣐⡉⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⣀⡠⠖⠊⠀⠀⠀⠀⣀⣀⣀⣀⣸⠖⠁⡇⠀⣬⣶⣏⡀
⠀⠀⠀⠀⠀⠀⢠⠿⢄⣉⡏⠀⠀⠀⠀⣠⠔⠋⠉⠁⠀⠈⠀⠈⠘⠓⠐⠛⠁⠀⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢱⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢄⣈⠑⡄
⠀⠀⠀⠀⠀⢸⠇⡈⠀⠀⢸⡤⢚⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠠⠘⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠄⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢱⡠⡄⠀⠀⠀⡀⣂⠤⠖⢹⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠳⠤⠔⠒⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⡈⢖⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⢳⠄⠀⠀⠠⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠂⠀⠀⠨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡇⡎⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠃⠀⠀⠀⠀⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴⢈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣶⣾⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣿⠉⠛⢿⣏⠀⠀⠘⡰⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⢪⠀⠒⢠⣼⡇⠀⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠁⠀⠀⠀⠀⠀⠃⡞⠀⣠⠀⠀⠀⠂⠀⠘⢿⣿⣿⠃⠃⠀⡄⠀⣩⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡆⠀⠀⢴⣿⠀⠀⠀⠸⠆⠷⠂⠿⠀⠀⠀⠀⣀⣀⢀⣿⣿⣇⡀⣆⠀⡀⢠⠀⠄⠐⠶⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠻⣿⡏⠈⠻⣿⠿⠷⠘⡷⣶⣶⠆⠸⠿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠃⠀⣤⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠫⣿⣷⣿⣅⠸⢆⣤⠀⠀⣴⣆⠀⠀⢀⣤⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢈⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢺⣿⣿⡟⠁⠀⠀⠀⠀⠸⣿⡇⠐⠀⠈⠉⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡾⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠺⣿⡏⠀⠀⠀⣀⣠⣄⣠⣏⣠⣀⠀⠀⠀⠀⢛⣇⣠⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢳⣿⠄⠀⠁⢼⣿⣿⠟⠉⢹⣿⣿⣿⣦⠀⠀⢸⣿⡿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⠄⠀⠒⠈⣤⡀⣤⣀⡀⠈⢩⣿⣯⡄⠀⣸⣿⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⡀⠀⢠⢿⣿⣿⣿⣧⣦⠴⠯⠉⠀⣀⣿⠋⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢽⢿⣿⡟⠾⠛⣿⣿⡿⠋⠀⠀⢀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠻⢿⡇⠀⢴⣤⣤⣼⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠃⢀⡎⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠛⢹⣿⣯⠇
⠀⠀⠀⠀⠀⠀⣾⣦⣄⠈⢿⣶⡄⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣦⣠⠀⠀⠀⠀⠀⣦⣴⣶⣿⣿⡄
⠀⠀⠀⣠⣶⡄⠈⠻⣿⣷⣾⡏⣀⠀⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢼⣿⣿⣿⣿⣯⣿⣆⡀⠀⠀⠀⠘⢿⣿⣿⠻⣿
⠀⠀⠀⠙⢿⣿⣷⣄⣀⣿⣿⣿⡏⠀⡄⠀⣌⠘⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠸⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠻⣿⣷⣿
⠀⢰⣧⡀⠀⠙⢿⣿⣿⠏⢸⡟⠀⠀⠁⠀⡟⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠢⠄⣽⡟⠀⣠
⠀⠸⣿⣿⣦⡀⣶⠟⣁⣾⠻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⢠⣴⣾⡿⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⢿⣿⣿⠏⠁⠀⠈⡷
⠀⠀⠀⠙⠻⠛⢳⣶⣿⠟⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⣜⠃⠈⢻⡿⢧⠀⠄⠀⠁⠀⠿⣿⢟⣻⣟⣻⣯⣾⣻⣿⡅⠘⠋⠁⢸⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠉⠁⢠⣄⠁⠘⠀⢸⠀⠀⠀⠀⣿⡟⠟⠛⠋⠉⠉⠉⠛⠓⢠⣯⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠀⠀⠀⠊⠃⠀⠾⣷⣄⠹⡅⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠘⡟⠆⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⣸⠁⠀⠀⢠⠀⠀⠀⡎⠀⠈⡆⢹⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣼⡆⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢸⡇⡀
⠀⠀⠀⠀⠀⣠⡆⠁⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⢸⠟
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿⣶⣶⣶⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽
⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣸⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢠⡿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣲⣶⣾⣿⣿⠿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⠁⡼⡄⠀⠀⠀⠶⣶⣶⣶⣿⣟⠉⢿⣿⣏⠁⡀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⡀
⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⡁⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⡿⠿⠟⠉⠙⢹⣷⣴⣄⣤⣄⡀
⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠟⠁⠀⠀⠉⢻⣿⣿⣶⣤⣶⣾⣦⣶⣶⣿⣿⣿⠛⠉
⣤⡿⠜⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣽⠟⣿⣶⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⠋⠘⣿⡏⠙⢻⣿⣿⣿⣄⣠⢾⣿⣦⣴⡿⣿⣿⣿⣿⡿⠛⠿⣿⡳⣿⣿⡇⠀⠀⠀⠀⠰⣼⠆
⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠛⢳⣿⣷⣦⣬⣿⢤⣞⢻⡿⣹⡿⡷⣴⣿⣶⠿⣸⣽⣿⣛⣿⣤⡰⣴⣾⣹⠾⣟⠅⠀⠀⠀⠀⠘⣿⣦
⣊⠀⠐⡇⠀⠀⠀⠀⠀⠀⠈⣀⣄⣤⣿⡿⣿⡯⢿⠛⠿⢾⣿⣾⣄⠠⣽⣻⣯⣿⡿⠟⢫⣮⣻⡿⠟⣽⠿⠶⣷⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⡟⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡿⠼⣿⠳⢦⣤⣀⣘⣻⣿⣿⣿⣿⡿⣿⣵⣰⣾⣿⠿⠩⣦⡴⢿⡾⡘⢻⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄⢗⡉⢱⣂⣯⣵⣶⣽⡿⠿⠟⣻⣿⣿⣭⣍⣛⠛⠿⣷⣿⣖⣛⣿⣉⣹⣓⡋⢹⡷⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣾⣿⡇⠀⣤⠀⠀⠀⠐⢿⣷⣿⠷⠟⢫⡭⡟⠁⢤⣶⠖⠀⠻⠿⣿⣿⣿⣯⣤⡀⠀⠉⠉⣿⣿⣻⣿⣿⣷⣾⣷⡦⠀⠀⠀⠀⠀⠀⠀⠈
⢰⣿⣿⢂⣾⣿⠀⠀⠀⠀⠘⠿⢿⣷⢶⣿⣿⣥⣤⣜⠋⣴⣶⣾⣿⣿⣿⣿⣯⣿⡿⠈⢻⠲⠛⠚⠊⢻⣼⣿⠿⠟⠂
⣼⣿⡿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣿⣯⡻⣧⣶⣿⣿⣿⣿⣿⠏⠉⠉⢀⣀⣀⣈⣁⣶⣶⣷⠟⢟⣡⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⣿⢿⡟⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣾⠯⢽⡿⢫⣷⣿⠟⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⣄⣿⡄
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠉⠀⠶⠀⠀⣿⠀⠛⣿⡿⣻⣿⠋⡘⣿⢿⠉⣿⣧⣶⣞⠋⣿⣿⣿⠀⠚⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠿⠆⠀⠠⠄⣿⠀⣠⣼⣿⣿⣿⣧⣇⢸⠈⠀⣿⣿⣿⣿⡟⠋⢸⣯⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣙⠴⠿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠓⠀⢸⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⢠⠁⠀⣾⣿⡇⠀⢸⣿⣿⣿⠹⣿⣿⡀⣻⣿⠟⠁⠈⠀⠀⢸⡇⠈⠉⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣯⣿
⣿⣿⣿⣿⣀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⠀⢸⣿⣿⣿⠀⢽⣿⣿⡿⣧⣀⣀⡤⠀⠀⣸⠇⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⣿⣿⠀⢺⡏⠋⠀⣿⣿⣿⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠈⠛⠛⠻⠀⠘⠇⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⢾⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠛⢧⡀⠀⠀⠀⠀⠸⣄⢂⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠻⣷⣤⣄⣈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡿⠷⣦⣄⡀⠀⠀⠘⢦⡈⢧⠀⠀⠀⠀⠀⠈⢿⣯⠙⠳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢰⠇⠀⠀⠉⠛⢶⣄⠀⠈⠳⣌⢣⡀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⣀⠀⠀⠉⠻⣦⡀⠙⢧⣻⡄⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⢶⡶⢶⡶⣦⣤⣀⣀⣰⣟⢸⠇⠀⣰⠟⠉⠉⠛⠲⢦⣌⡻⣦⡀⠹⣿⡄⠀⠀⠀⠀⢀⡽⢿⣿⣀⣀⣤⣤⣤⠶⢶⣶⠞
⠀⠀⠀⠀⠈⢳⣄⠙⠲⢤⣤⣍⡉⠹⣿⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⡉⠛⣿⣦⡈⢿⡄⠀⠀⢀⡞⢠⣿⠉⢉⣀⣤⠴⠒⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⡄⠘⢎⢿⡙⢷⣿⡆⠀⣿⠻⣿⡛⠶⣤⣀⣀⣼⣀⣿⣽⣿⣮⣷⡀⠀⢸⢃⣿⣿⠞⡿⢡⠏⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡈⠳⢄⣘⣷⠀⢸⡗⢦⣍⣭⣉⠉⡟⢃⠀⠐⢿⢿⣽⣿⣷⠀⠀⢸⢹⣇⡼⠁⣀⣀⡾⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣾⠋⠈⢳⣌⠻⣦⡀⠈⣿⣇⠈⣧⡀⠀⣾⣍⣀⠀⠀⠀⠀⠀⣠⣤⣿⣿⡆⠀⢸⢸⠋⢀⡼⠋⢉⡴⠉⢹⣿⣦⡀
⠀⠀⠀⣀⣴⣟⡉⠳⣄⠀⠙⠲⣄⠙⠛⢿⢹⡀⢻⣏⡂⠀⠉⠙⢷⡀⠀⠀⡾⠉⠁⢸⣿⡇⠀⢸⢸⠾⠋⢀⡴⠋⠀⣰⣯⣤⠿⠿⠦
⠀⠀⠀⠉⠙⠛⠛⠻⢿⣷⣄⠀⠈⠙⢦⣼⡈⡇⠸⣿⠇⣀⣄⠀⠘⢷⣶⡾⠃⡀⠀⢸⣿⠇⠀⠈⣿⢀⡴⠋⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⡀⠀⠙⣇⢹⠀⣿⡏⠉⠙⡉⢻⠒⠚⠒⠚⠉⠉⢹⣿⠀⠀⠀⣥⡟⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⣦⠀⢹⠘⡆⢹⣿⣦⠀⠀⠀⣀⣀⡀⠀⠀⢠⣿⣿⠀⠀⢠⣿⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⢳⡸⡇⢷⢸⣿⣿⣆⣠⡞⠉⠀⠉⠀⠀⣸⣿⣿⠀⠀⢸⡇⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣵⣇⠸⡆⢿⣿⣿⡏⠉⠁⢠⡀⠀⢙⣿⣿⡏⠀⠀⢸⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⠀⢷⠸⣿⣿⢷⣀⠀⠀⠀⢀⣼⣿⣟⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡆⢻⣿⡇⠉⠛⠷⠿⠛⢁⣿⢿⡇⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⢷⠸⣯⠙⢶⣤⣀⣀⣤⠞⠁⢰⠃⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣦⡙⢷⡀⠀⠉⠉⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⣦⣀⣴⣾⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠈⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣟⢻⣿⡆⣠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢻⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣷⣤⣴⣿⣿⡇⠘⠶⠖⠉⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢐⣀⣤⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⢀⣠⡤⢒⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡗⠒⡆
⠀⠀⢸⣿⣿⣿⣿⣛⣿⣿⣿⡁⢹⣿⣿⣿⣿⣶⣿⡄⢿⠿⣿⣋⡀⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⣿⣻⢿⣿⢿⣿⣧⣤⠇
⠀⠀⠈⢿⣿⣵⣿⣿⠟⠛⢛⣿⣿⣯⣿⡭⣥⣬⡿⣷⢸⡇⠉⠁⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⢀⣹⡿⡬⠟⠋⣻⣥⣌⠛⢻
⠀⠀⠀⠈⣿⡟⣿⣗⣩⣽⣿⣿⣿⣿⣿⣌⣽⣾⣿⣿⣽⣧⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣽⣽⣿⣷⡟⠀⣆⠀⠈⢻⡟⣿⡟
⠀⠀⠀⠀⠸⣷⣿⣿⣇⣀⣀⣽⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠻⢹⣽⣿⣿⣿⣿⣿⣏⣭⣿⡿⢛⣿⣿⣿⣿⡿⠞⢃⠣⠀⢹⣿⣿⣿⣷⣬⣙⣿⣿⣿⡿⣿⣿⡍⢚⣿⡏
⠀⠀⠀⠀⠀⠀⢱⣿⣿⢿⣭⣿⠟⠋⢛⣿⡾⣿⣿⣿⣿⣿⠋⠀⣴⣦⢣⣾⣿⣿⡟⠀⠀⠈⠳⣾⣿⣶⣧⣸⣷⣾⡿
⠀⠀⠀⠀⠈⠁⠘⢋⣤⢿⣿⣷⣶⣾⣍⢯⣾⣿⣿⣿⣿⠃⠀⢰⣿⣿⡇⢹⣿⣏⣛⠳⢦⣤⣄⠈⠻⣷⡙⢿⣟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢨⠴⣟⡏⠈⢻⣿⢿⣿⣶⣾⣿⣷⣄⣹⣿⡀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣤⢽⣿⠟⠛⢿⣿⣿⣿⠏⠀⠀⠐⡠⣿⠂⠀⠀⣾⣿⣮⡛⢿⣿⡻⣦⣙⣟⣇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⢉⣴⡿⠛⠲⠛⠖⠛⠛⠁⠀⠀⠀⠀⠀⣋⠀⠀⠀⠈⠿⣿⣿⡪⣿⣿⣿⣿⣿⣤⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠀⠀⠙⢯⡽⢶⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⢌⠀⠀⠀⠀⠀⠀⠹⣆⠙⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⢿⣿⣿⠿⣿⢄⠀⠀⠈
⠀⠀⠀⢀⣤⣾⡿⢿⠏⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠄⠨⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⢧⡉⠛⠭⣤⠆⢠⠀⠀⡄
⠀⠀⡾⣻⣿⡿⡛⢆⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡆⢁⠐⣠⣤⣔⢺⣿⣶⣤⠀⠀⠀⠀⠈⠻⣦⣄⡈⠀⠈⡆⠀⠃

⠀⣾⢻⡖⠐⠂⠺⠃⢰⡶⠒⠀⠁⠀⠈⡁⠒⢶⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⣄⣤⠀⠀⠀⢈⠀⠹⣟⣭⣻⢉⠀⠁⠀⠀⠀⠛⡆
⣤⣿⣸⣧⣤⣤⣼⡆⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣛⣔⣔⣄⣄⣢⣠⣻⣅⣸⠀⢰⣶⣶⣦⣤⣿⣘⣿⣄⣴⣶⣶⡀⠀⣿⣇
⣀⣀⣀⣀⣀⣀⠀⠀⣶⡆⡀⣦⠀⠛⠀⢰⢀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠿⠿⠿⠏⠿⢻⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⢸⣿⠀⢶⣶⣶⣒⣒⣒⣒⣒⣶⣶⣶⡦⠀⣿⣿⣿⣿⣛⣻⣿⣿⣿⡇⠀⡹⠿⠏
⢩⣭⢩⢡⡈⣌⠀⠀⣿⣟⣛⣻⠶⠶⠶⣞⣛⣻⣿⠀⢸⡇⠉⠉⠉⣿⣿⣿⣶⣶⣶⡇⠀⢸⣿⣿⣇⠛⢳⣿⣿⣿⡁⠀⢠⡁⡄
⣸⣿⣸⣰⣄⣦⡀⠀⣿⣧⣥⣽⣴⣤⣦⣯⢤⣼⢿⠀⠸⣷⢶⡶⣶⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⢸⢿⣿⣿⠇⠀⣰⣄⣆
⣀⣀⠤⠔⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⠦⠦⠥⠼⠦⠦⠦⠆
⡀⣠⡄⣠⡄⣠⣄⢀⣙⠒⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚
⡇⡇⡁⡏⡁⠈⢹⢸⣿⣷⠀⠀⠀⢠⣿⠟⠛⣿⣷⣿⣯⡭⣿⠀⠀⢸⡷⠾⣿⡶⣿⢷⡾⢿⢾⣷⢿⡿⡿⢿⠿⣿⡿⢿⡅⠀⣿
⠇⣧⠅⣈⡅⢮⡵⢸⢿⠿⠀⠀⠀⠰⠿⠀⠀⠿⢧⠿⡯⣤⠿⠀⠀⢸⡟⠛⢺⡏⠋⢸⡗⠡⢸⣿⣿⡇⠁⠈⠁⣸⡀⢸⠂⠀⠿
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣤⣄
⣴⣖⣫⣭⣝⣳⣦⣤⣀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣤⣤⣼⣿⣭⣭⣭⣭⣿⣧⣤⣤⡄⠀⠀⠀⢀
⣿⡇⢹⠁⣽⠉⣿⠉⠉⠀⠀⢹⣿⡿⠿⢿⣿⠟⠿⣿⠿⠿⣿⠿⠻⣿⠀⠀⠀⠙⠀⠀⠃⠄⣶⢿⠿⣶⠀⠁⠆⠰⠁⠀⠀⠀⠉
⣿⡇⢸⡀⣸⠀⣿⣀⠀⠀⠀⢸⣿⣇⢀⣸⣿⣤⣴⣿⣤⣴⣿⣤⣴⣿⠀⠀⠀⢀⣀⣐⡀⠂⣿⡞⣷⣿⠐⢀⣂⣐⡀
⣿⡏⢹⠉⣿⢉⣿⠹⠉⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠸⠿⠿⠷⠶⠿⠷⠾⠿⠶⠿⠿⠿⠇⠀⠀⠀⠈
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⣠⣤⠀⠀⠀⡤⣤⠀⠀⢠⢤⡄⠀⠀⣤⡄⠀⠀⠀⢠⣤⠤⡤⠤⠶⢤⢤⣤⣤⡄⠀⠀⠀⠿
⣸⠆⠀⢼⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⡿⠛⢿⣿⠛⣿⡟⠛⢿⡆⠀⡀
⢹⠅⠀⢽⡟⠉⠉⠋⠋⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⢸⣿⠀⣿⡀⠀⢸⡇⠀⠛⠙⠙
⢸⠅⠅⢽⡇⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢸⠅⠅⢽⡇⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⡄⠀⠠⢤⣀⣀⣆⣚⣃⣆⣠⣠⡤⠄⠀⢸⣿⣿⠿⣿⡿⣿⣿⣿
⡾⢷⣶⠿⣧⡀⣄⢀⢠⠀⠀⢸⣿⢰⣶⣶⣿⠀⠀⢸⡿⠛⠻⣿⠛⢻⣿⠛⠛⣿⠀⠀⢸⣿⣿⣿⢉⢹⣿⣿⣿⠀⠀⠀⣆⢠⡀
⠷⠾⠿⠶⠿⠿⠿⠿⠿⠀⠀⠸⠿⠾⠿⠿⠿⠀⠀⢸⡇⠀⠀⣿⠀⢸⣇⠀⠀⣿⡀⠀⢸⣟⣛⣿⣤⣼⣟⣻⣿⠀⠀⠠⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡿⣯⣭⣭⡿⣿⣿⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣴⣦⣤⣤⣤⣴⡦
⠉⣉⣉⡉⠉⢙⣿⡏⠀⢈⢁⡄⡇⠀⢸⡇⡉⣠⠀⠠⣶⣶⣶⣶⣶⡖⣖⡒⣶⣶⠄⢰⠶⠶⡿⠷⢶⣶⡶⠿⢷⠶⢶⡄⠀⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⣴⠞⢁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⠿⣿⣶⡄⠀⠀⠀⢀⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣀⣄⣲⣻⠎⢿⣶⣿⣿⣿⣿⣷⣦⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣧⣾⣿⣿⣿⢉⢀⠘⣿⣿⣿⣿⣿⣿⣿⠁⣷⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣩⣶⡿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠚⠉⠹⣫⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⣿⣿⡎⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢞⡻⣯⣿⣿⣿⣿⣿⣻⣿⣿⡟⢁⡈⡏⠀⠀⣿⠙⠿⣿⡿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠸⣷⢯⠈⢳⣻⣿⡿⢙⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠃⠀⠀⠀⠀⠰⡾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠰⠚⠛⠛⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣄⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠟⣠
⠀⠰⠀⠀⠀⠀⢀⣴⠛⡖⠁⠀⠀⠀⢠⣿⣿⣿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠑⠠⣀⡄⠀⠂
⠀⠀⠀⠀⠀⠛⠻⡯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⣾⣿⣿⠏⠉⣁⣿⣿⡗⢷⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⣀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⣯⣿⡋⣸⣿⠄⠒⣻⣿⣍⡻⢮⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡵⠋⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠿⢿⣧⠋⣙⣳⣾⣿⣿⣿⣷⠛⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣶⣦⣾⣿⣾⣿⣿⣿⠿⠿⣿⣿⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠋⠁⠉⠋⠈⠀⠀⠀⣿⣿⡿⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣭⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠶⠖⠒⠂⠠⠶⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠧⢤⡤⡤⢰⠆⢤⡄⠈⢡⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⢀⠐⢧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣩⠃⠙⠃
⠀⢀⢴⣒⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⣀⣀⣀⣀⠀⠀⡾⠀⡀
⠀⠉⠑⠀⠀⢰⠀⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠒⠀⠀⠀⠀⠀⢀⣀⡄⠉⠁⠀⠀⠀⡀⠀⠀⠁
⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠉⠁⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⣴⣆⠀⠐⠒⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣜⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡼⢶⡶⢈⠤⣠⡶⠋⠀⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠛⠋⠁⠀⠘⠠⠃⠀⠇⠹⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡺⠁⠀⠀⠀⠀⠀⠐⢲⡖⠉⠀⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠦⠄⠀⠀⠒⠒⡐⠊⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢠⠀⠀⠀⠀⡉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢬⣬⣄⡀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣤⣇⣀⣈⠈⠀⠀⠀⢀⡨⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠈⠉⠉⠉⠉⠉⠉⠉⠈⠙⠃⠀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠶⡄⠀⠀⠀⠀⠀⠀⢰⠶⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⡀⠤⣶⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠤⢀⠀⠀⠀⠀⠀⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠡⡀⠀⠀⠀⠎⠀⠀⠈⠃⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠸⠁⠀⠀⠱⡀⠀⠀⢀⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡀⠀⢡⠀⠈⢦⠀⡌⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⡀⢱⠀⡴⠁⢀⠃⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠳⡀⠀⢳⠀⠀⡀⠀⠈⠢⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠴⠀⠀⢀⡀⠀⡾⠀⠀⠼⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣧⠀⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⢀⠀⠀⡀⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡄⠀⠀⣼⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡎⠀⣀⠀⠉⢀⣠⣄⠲⠲⠆⣤⣀⡀⠀⠀⠀⠉⠉⠀⠀⠀⢀⡀⣤⠰⠖⡖⣠⣄⡀⠙⠀⣠⠁⢳⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡼⠋⠀⠀⡰⢇⣠⡇⠀⠀⢈⠥⣠⠩⢭⠗⣶⣬⢁⣀⡀⠀⢀⠀⠀⢀⣀⡈⣥⣶⠛⡭⠭⣤⡬⡁⠀⠀⢸⣄⣸⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢀⡧⠀⠀⠀⠃⣸⠏⠧⠀⠀⠀⠂⠀⠙⠺⢮⣭⠿⠘⡛⢿⠋⠁⠈⠘⡿⠛⠃⠿⣭⡴⠗⠋⠀⠐⠀⠀⠀⠜⢸⣇⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢀⣼⠀⢀⣀⠑⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠈⠆⠀⠀⠨⠁⠀⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠈⠖⠆⠀⡇
⠀⠀⡰⢉⠚⠣⣿⢸⡀⣸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡞⠢⠞⠁
⠀⠀⡇⡄⠀⠀⡴⡺⢙⠃⠀⠆⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡋
⠀⠀⠱⡑⠔⢊⡴⠃⠎⡀⠀⠀⠸⠀⠀⠀⠀⠀⢀⣀⢤⠤⢤⠖⠛⠀⠀⠀⠀⠀⠀⠓⠲⡤⠤⠤⣀⡀⠀⠀⠐⠊⠀⠆⠀⠀⢀⠱⡀
⠀⠀⠀⠉⠉⠉⠀⢸⠠⠅⠀⡐⠀⠀⠀⠀⣀⠤⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠤⣀⠀⠀⠀⠀⢂⠀⠨⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢒⣄⠹⠆⢀⡀⢂⠀⠀⢸⡀⠰⣳⡦⣄⠀⡀⠀⠀⠀⠀⠀⠀⣠⢴⣞⠇⣠⠇⠀⠀⡀⢀⠀⠴⠏⣀⡒⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢩⣀⠀⢂⠉⢶⣁⠜⣦⣽⣦⡤⠤⣤⠿⣷⡀⠀⠀⢠⣾⠾⣤⠤⢤⣴⣯⣴⠧⣄⡿⠉⡐⠀⣀⡍⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡐⠉⢱⠤⠍⣀⡤⠀⣀⣀⠐⠶⠂⢀⡀⠀⠙⣦⣴⠋⠀⣀⡀⠐⠶⠂⣀⣀⠀⢤⣀⠨⠤⡎⠉⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠤⡄⠀⠤⠀⠀⡉⠛⢶⣶⣆⡉⠄⣉⣀⣀⣉⠠⢉⣰⣶⡶⠛⢉⠁⠀⠤⠀⢠⠤⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠁⠂⠄⠀⠈⠑⠦⣄⣉⠙⠛⠛⠓⠚⠛⠛⠉⢉⣠⠴⠊⠁⠀⠠⠐⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢭⣈⠀⠀⠀⠈⢙⡷⠶⠶⠶⠶⢶⡋⠁⠀⠀⠀⣉⡥⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣀⣀⡤⠔⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⡞⠉⠈⠻⣿⡌⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⢿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣦⡤⣄⡀⢀⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⢿⣦⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢘⣿⡻⣦⣿⣿⡿⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣰⣿⣿⠏⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣿⣿⣷⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣻⣿⣝⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⣀⠀⡀⢀⠀⠀⣷⢘⢰⡇⣀⠀⣀⡀⠀⢸⡇⣀⠘⡃⢀⡀⢀⣀⢘⠂⣀⣀⠀⣀⣀⠀⠀⣷⣀⠀⢀⣀⠀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⢻⡿⠱⣯⣽⠞⣧⣽⠀⠀⣿⢸⢸⡿⣯⠸⣿⡿⠀⢸⡿⣏⢸⡇⣻⡧⣸⣷⢸⡇⡏⣿⠸⣯⣽⠀⠀⣿⣽⠇⣿⣽⠇⢻⡾⠁⣿⡧
⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠐⠻⠟⠀⠀⠀⠁⠀⠀⠁⠀⠺⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⣾⣻⡇⣾⣻⡆⣿⢻⡆⠁⣿⠃⠀⢻⣼⠃⣾⣻⡆⣿⣸⡇⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠻⡄⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠉⠈⠁⠀⠉⠀⠀⢼⠋⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⢀⡴⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢻⡀⠀⠈⠀⠂⢄⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠈⢧⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⡖⠒⠒⣶⠒⠂⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣸⠇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⡆⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠈⠦⠀⠀⡿⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢻⠀⠀⠀⠉⠉⠉⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢽⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠟⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠚⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠘⠁⠷⠊⠁⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⢠⠞⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡳⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆

⠀⠀⣠⣤⠀⣤⣄⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠾⠛⠛⠀⠈⢻⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⡀⠀⠀⠀⠀⠐⠰⠛⢷⡆⢀⡄
⠀⠀⠀⠀⠙⢦⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠳⠀⠑⠀⠀⠀⠂⠀⠀⠀⠄⣰⣾⢆⡀⠀⣀
⠀⠀⠀⢤⣀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣀⣁⣰⣃⠛⡸⢿⠛⠲⢤⡀⠎⣀⠈⣉⣘⠁⣧⠌⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠉⠻⣾⣟⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⣀⣤⡘⢩⣭⣝⠀⠉⠲⣽⣿⣷⢦⣤⢿⣧⣿⣿⣀⢸⣇⣄⠀⣲⣶⣾⠁⠤
⠀⠀⠀⠛⢦⣄⢸⡿⠛⠀⠀⠀⠀⠀⠉⠙⠻⠿⢷⣾⣿⣶⣿⡶⢂⣀⣽⣯⡘⠛⠦⣬⣙⢢⣤⣽⣿⣧⣤⣤⠼⣿⣯⡹⠛⠁⢉⠘⣿⡻⠋⠉⠐⠀⠑
⠀⠀⠀⠠⣄⡈⢻⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠁⠀⠈⠙⢿⣿⠛⠓⢶⣿⡋⣁⣽⣿⠯⠹⣿⡟⠀⠀⠻⠁⡄⠀⠀⠐⠘⠁⠀⠀⠀⠚⠁
⠀⠀⠀⣄⡀⠙⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⡀⠀⢀⠄⠀⠸⢶⣶⡿⠋⢋⣻⣿⣄⢠⡏⢳⡄⢠⣄⡁⡀⠀⠠⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠉⠓⢾⣿⠓⠀⠀⠀⠀⠀⣀⡀⢀⣴⣿⣿⣿⡿⢿⠿⡿⠇⢀⡄⠀⠛⣶⣤⣾⣿⣿⣯⡙⠻⡞⠃⢀⡀⠈⠑⠁⠺⠃⠈⠆⠘⡇
⠀⠀⢀⠀⢤⣄⣈⣿⠻⠀⠀⠀⠀⢹⣿⣟⣾⣿⣿⡿⣿⣿⢿⣿⢢⣼⠏⠀⠀⠼⢉⣍⣻⡏⠀⠻⡄⠛⣧⡄⠸⠀⠰⢇⠀⠀⠀⢀⣦⠊⠀⢀
⠀⠀⠀⡀⣄⠈⢹⣿⠃⠀⠀⠀⠀⠠⢶⣿⣿⣿⣿⣿⣿⣿⣾⡿⢿⣿⣷⠂⠀⣀⣼⣿⣿⠛⠳⠀⠀⠀⣸⣶⡄⢠⣐⣤⡑⠀⠀⠈⡝⠚⠛⠃
⠀⠀⠀⠘⣾⡿⣾⣿⣉⠁⠀⠀⠀⣀⠀⠈⠛⠓⠻⠟⠛⠿⢫⣄⣠⠻⠿⠭⠉⣉⣉⣉⣹⠷⣄⠀⡂⠀⠙⠻⣦⣀⠀⢘⡁⣀⣿⠺⠇⠀⠈⣤⠄⡀
⠀⢀⠀⢀⡈⠻⣾⣿⣄⣠⣄⣀⣠⣿⣶⠟⠒⠶⠿⠶⠆⠀⠀⠉⢉⣃⣴⣒⣿⣿⣿⣿⣷⣶⡜⠛⢿⣄⢐⣳⠛⢉⣻⡾⣛⠛⠃⠀⠀⠀⠀⡤⠊⠤
⠀⣿⣦⡄⠉⠳⣾⣿⣿⠛⢷⣾⠋⠙⢉⣠⣤⣶⣤⣤⣤⣤⡶⠞⣨⣙⡂⠒⠒⠈⠽⣧⣤⡀⠉⠉⠛⣿⣋⠻⣆⢿⣈⢻⣿⡿⠿⠛⡄⠀⠀⠅⣀⡰
⠀⠼⠿⣿⣿⠀⣸⣿⣿⣷⣿⠛⣤⠚⠋⠉⠁⢀⣤⠍⠉⠀⠀⢤⡀⡀⠙⠶⠶⠞⠙⠛⠛⠻⢶⣶⣤⡈⢩⣿⣦⣿⣟⢁⣾⣷⣆⡈⠙⣀⣀⢀⣦
⠀⣠⣴⣾⣿⣷⣿⣿⣿⣿⡷⠋⠀⢀⣠⣤⡴⠏⢠⠖⠀⠀⠀⠀⠀⡉⣤⡀⣀⠀⠉⠛⠢⠄⠀⠀⠘⢿⣿⣧⡀⠉⠙⠓⠹⣿⡿⠿⢿⡉⠋⢫⠁⠀⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⠟⠁⢀⠰⠂⠀⣶⣶⠾⠛⠉⠉⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢻⣿⣄⣠⣶⢷⣀⠉⠳⣶⣿⠙⠃⠈⣦⡀
⠀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣀⠀⣰⠏⠀⢀⡌⠀⠀⣸⣿⠇⢀⣴⣿⣿⣿⣿⣿⠶⠤⢷⠀⠀⠀⠀⠀⠀⠀⣿⡯⠙⣿⠞⠛⠙⣿⠋⠁⣀⡆⠀⠸⡇⠈
⠀⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⡟⠀⢠⡿⠁⣠⢐⣿⣿⣾⣿⣿⣿⣋⣉⣁⡈⢳⣄⠀⠀⠀⣢⡀⠐⡆⠀⢹⡟⠈⠁⣠⠀⠀⠈⠙⠻⠏⠹⡄⠀⠽⠋
⠀⣿⣿⣿⣿⣟⠁⠙⢿⣿⣿⣿⡇⠀⣮⠀⠙⢷⣼⣿⣿⣿⣿⣿⣿⣿⠉⠛⣿⣿⣿⣷⠔⠞⠈⠁⠠⠀⠀⣼⡇⠀⠀⢉⣤⣠⣄⠀⠀⢀⣀
⠀⠿⠿⠛⠛⠛⠉⢉⡟⠻⠿⣿⣀⡀⠝⠀⠀⢻⠁⣹⡿⣿⣿⣻⣿⢻⣄⡈⢩⣿⣿⣿⠀⡀⠀⣴⠀⠀⠀⣿⡧⠄⠀⣿⣻⣿⣿⠆⠀⣿⡌⠃⢠⣄⡀
⠀⣴⣦⡤⠀⢜⣶⣿⠟⠻⣦⣤⣝⣿⠀⠀⠀⠈⢻⡌⠢⠈⢿⣿⣿⡿⢿⣻⣶⣿⣿⠟⣾⠇⢀⠏⠀⠀⣰⣿⠇⠀⣿⣷⣿⣿⣿⡇⠀⢸⣟⠀⣼⣹⣿
⠀⣿⡟⢰⡄⠋⠛⡀⠀⣹⣿⡷⣽⣯⣄⣀⠀⣀⡀⠉⢶⠀⠀⠽⢿⣿⣿⣿⠿⠟⠏⣼⣡⣴⠁⠀⠀⣴⣿⡿⠀⣼⡟⠉⢿⣿⣿⣧⣠⣾⣿⣿⠟⠛⣯
⠀⣿⣿⣤⣧⡴⠊⣱⣾⠛⣿⡟⡙⠉⠉⠿⡀⠀⠙⠻⣿⣗⠸⢶⣶⠶⠻⠶⠀⠶⠛⠁⡹⠁⠀⢀⡴⠋⢻⣷⡼⠋⠁⢀⣾⣿⣿⣿⣿⣿⣿⣇⣠⣴⡿
⠀⣿⣷⠎⠁⠄⠀⢿⡇⠐⡮⠐⣧⡀⠀⠀⠐⠄⣀⠀⠀⠉⠂⠄⠈⠉⠙⠋⠉⢁⣈⣡⣤⣤⡴⠊⠁⠀⢘⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿
⠀⢛⠁⠀⠠⡄⠀⠀⠃⠠⠀⢠⣿⡃⠀⠠⠶⠾⢿⣿⣶⣦⣤⠔⠛⠶⠶⠶⠒⠋⠉⠻⣿⡏⠀⠀⠀⠀⢸⣏⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠛⠿⡆
⠀⣿⠦⣤⣤⣴⣶⠆⠀⠀⣠⣾⣿⣿⣄⡀⠀⠀⠀⣿⣿⣭⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠀⠀⠀⣠⣿⣿⠿⠿⠿⢿⡟⠉⠛⠛⠻⠿⡇
⠀⣈⠂⠈⢿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠙⠟⠛⠛⠛⠛⠳⠶⠶⡶⠶⠶⣶⣶⣶⣾⡶⠾⠋⠁⠀⠀⣴⣾⣷⣤⣤⣤⣀⣀⡀⢠⣦⡄
⠀⣿⣷⣦⣽⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⡿⠋⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠁
⠀⣷⣙⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⣀⣀⣀⣀⣀⣠⣤⣀⣸⣿⡁⠀⢀⣤⣿⣿⣿⣿⣿⠟⠛⠋⠛⠿⣿⣿⣿⣦⡀
⠀⣿⣿⣿⣧⣼⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⡈⠛⠻⢿⣷⣦⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣧⡀⠀⠀⠉⠛⢿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⣨⣿⣿⣿⣏⠙⢿⣿⣿⡿⠛⠛⠉⠻⠷⢦⣤⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣶⣧⣴⣿⠟⠁⠀⠀⠀⣠⣾⠧⠀⣀⡈⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠀⠀⢰⣿⣿⣿⠿⢿⠟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⢿⣿⠂⠀⢠⡴⠛⠉⠀⠀⠀⠈⠙⢏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢖⢀⣿⡿⠋⠀⣂⡶⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⢠⡾⠋⠀⠀⣀⡤⠄⠀⠰⠶⣯
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣼⣾⠏⠀⣠⣴⣿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⠞⣡⡄⠀⢻⣥⣴⣿⣿⣷⣤⡈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢖⡢⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠲⡕⠀⠙⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠖⠲⢄⠀⠀⠀⢀⠜⠳⢄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⢇⠰⡄⠈⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⠜⠒⢄⠃⠀⠰⣡⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣐⣇⠀⣀⡸⠀⢠⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⡉⠛⠉⠀⠀⠈⢭⣭⡉⢦
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⡶⠤⠤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⡄⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠺⣼⠓⣄⠀⠀⢀⣗⠖⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⣄⠀⠀⠈⢫⣔⠱⠖⠈⢀⡨⠄⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⠴⡇⠀⠈⠃⢑⡄⠊⢁⠀⢀⡠⠒⠁⢀⡀⢠⡟⠛⠃⠇
⠀⠀⠀⠀⠀⠀⡠⢖⠒⠀⠐⠇⠀⠉⠉⢀⠁⠘⠁⠀⠀⠙⠉⠀⠀⠀⢸⠀⢩⠄⠀⠼⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⠑⠒⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠉⠀⠈⠲⢄⣨⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠢⢤⡄⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⡜⠀⠀⠀⠀⠀⠀⡘⠀⢤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⣅⣇⠀⠀⠀⠀⠀⠀⡧⠼⠢⢔⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⣤⣀⣸⡉⠀⠀⠀⠀⠀⣸⣀⠼⢟⡉⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠛⠂⠬⠭⠴⠉⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⢀⠀⢀⡀⢀
⠀⠀⡀⠠⠀⠐⠂⠄⠂⠀⠀⠀⠀⠂⠀⠀⠀⠉⠙⠉⠩⠉⠀⠀⢀⣀⣀⣀⠀⠈⠀⠃⠀⠀⠀⢀⠉⠁
⠀⠊⠂⠈⠐⠀⠆⠀⠐⠈⠀⠐⠀⠀⠀⠠⡀⠃⠆⠀⠶⠐⠀⠀⠈⠀⠀⠀⠀⠀⢤⠀⠀⠀⡀
⠀⠐⠃⡀⠀⡤⠀⣀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⢀⠀⠘⡀⠒⠀⠂⠀⡀⠐⠄⠛⠁⠀⢀⠀⠠
⠀⠀⠀⠁⠀⠀⠄⠁⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠤⠧⠄⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⣅⠀⣈⠐⠀⠀⠀⢠⠄⠀⡄⠀⠠
⠀⠀⠀⠠⠖⠀⠳⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⡠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠿⠀⠻⠀⠀⠀⣶⠂⣴⡂⠐⠈
⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣩⠀⠀⠐⡂⠀⢐⠈⠀⠀⠄
⠀⢀⠀⣀⠁⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠁⡉⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣉⢈⠇⠀⠀⠄⣶⣴⡦
⠀⠀⠀⡀⠀⠈⠻⣦⠞⠃⠀⠀⠐⠀⢀⠀⠀⢀⣄⠀⠀⠀⠁⠀⠀⠴⠃⠀⠀⠀⠠⠩⠀⠡⠄⡀⢠⣤⡀
⠀⠀⠀⠀⠂⢠⡆⣀⢰⣏⠀⢶⠆⠀⠀⠀⢠⡀⠁⢠⡀⠀⠀⠀⡄⡂⢀⡀⠙⠋⠙⠀⠶⢀⣄⣉⣄
⠀⠀⠀⠂⠀⠛⢇⠛⠞⠛⠀⢠⠀⠄⠀⠀⢉⢠⠀⠄⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠁⣄⣀⠈⠀⠀⠀⠉⠀⠀⢆⢌⢠⠀⠀⢀⢀⠀⠀⢠⠀⠀⠄⠀⠀⠄⠀⠀⠀⢀⠂⠀⠀⠄
⠀⠀⠈⠛⠋⠛⢢⣀⠀⠈⠀⠈⣿⡄⠀⠀⠐⡀⠀⡀⠀⠀⠀⠒⡤⡀⠁⠠⠁⠔⠀⠀⠀⢀⠀⠅⠈
⠀⠦⠶⣤⠆⠀⠉⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠰⢤⠀⢀⡼⠁⠀⠀⡀⠀⠄⡀⠀⠙⠀⡁⠘⠁
⠀⠀⢼⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠂⠀⠀⢠⡄⠀⣤⠀⠒⠒⠋⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠠⠈⠃⠠⠉⠀⠀⠁⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠂⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢔⣾⠭⠐⣢⠿⠋⠁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣸⠧⣚⣍⣲⢴⣿⡏⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢊⣠⠖⣋⣵⡾⠋⠽⣃⣥⡟⠁⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⢞⣥⠖⠋⢀⣴⠋⠀⣀⣤⣾⡿⠋⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⠞⢉⣿⣿⢟⣡⠴⢚⠡⢓⣤⣾⡿⣻⠟⠁⠀⠀⠀⢀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡤⢊⠝⢁⣤⣿⢿⣿⣿⢖⣪⠵⠚⡻⢁⣼⠟⠁⠀⠀⠀⢀⣴⢿⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⣿⢐⡥⠚⠏⠁⣰⣿⣿⣟⡯⢅⡶⣀⣾⡿⠃⠀⠀⠀⢀⣴⣿⡿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⣷⣏⠀⣀⠴⠚⢋⣿⣯⣟⣭⣴⣷⡟⠉⠀⠀⠀⢀⣴⣿⣿⣿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠃⠹⣧⣾⢃⣠⢶⠞⣿⣿⣏⣩⡶⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣾⣏⣀⣤⣤⣤⣤⣴⣶⣾⣷⡾⠟⠉
⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⣿⣿⣦⣀⣀⣸⣿⣯⣧⣾⣾⣿⣿⣿⣿⣧⣤⡶⠾⠛⠿⠛⣿⠿⣿⣿⣿⣿⣯⣿⣿⣞⣯⡿⠞⠋⠁
⠀⠀⠀⠀⠤⣬⣽⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣴⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡻⢭⣄⣙⣿⣿⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⡶⢋⣡⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣯⣿⣿⣿⢿⣃⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⡡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢉⣹⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣧⡘⣿⣛⣷⣬⣽⣿⣿⣗⡺⠯⢵⣆⡑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠿⣿⣻⢿⣾⣿⣿⣿⣿⡿⢭⣒⠨⠙⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠻⢯⣿⣿⣿⡒⠍⡑⠫⢙⠳⣾⣷⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠈⠙⠿⣍⡒⢬⡙⠠⣠⣈⠋⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠯⣅⡢⣍⡛⠷⢶⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠯⠽⠷



⣿⠿⠿⠿⡿⡟⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣛⣻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⢀⣠⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣃⡟⠉⠉⠉⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠷⣽⣿⣿⣄⠘⠛⠀⣀⠀⠙⡄⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣶⣿⣧⣶⣶⣶⣾⡅⠀⠚⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢇⠠⡀⠱⣾⣿⣿⣿⣿⣿⣿⠋⠛⠛⠻⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⣯⠀⠤⣼⠷⢿⣿⣿⣿⣿⣿⣿⣄⠀⡀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣿⣿⣦⣄⣀⣴⠛⠛⢿⣿⣿⣿⣿⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣶⣿⢂⣀⣴⢢⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣏⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠈⢻⣛⣿⠇⠘⠻⣿⣿⣿⣿⡿⠋⣠⠟⠛⠻⢿⣿⡆⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⢸⣿⡧⢄⠀⠀⠀⢈⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⣀⣼⣿⣿⣶⣵⣌⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⣿⣿⡿⠛⠉⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠈⠉⠉⠀⠈⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠀⠄⢀⠀⠀⠀⠀⢀⡠⠀⢀⣸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⢰⡇⢇⡮⣅⠒⠭⡒⠶⢚⠭⣂⠵⣚⡽⣿⡏⣿⣿⣿⣆⠀⡀⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣾⣶⣍⢶⣼⡇⢀⣤⣦⣌⠂⠀⡝⡱⠊⠀⢈⣠⣦⣄⡘⣿⣿⣿⠟⢉⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠻⡿⢙⣿⣿⣧⣿⣿⢿⣻⣷⣤⣧⣦⣴⣾⣿⣛⣿⣿⣿⣿⣿⣇⣼⣿⣿⡟⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢜⢿⣿⠟⠻⡛⣶⣿⣿⣿⠃⠸⣿⣟⡃⠿⣿⣿⣿⢿⣿⡿⣿⣿⡿⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠙⠿⠛⣡⠿⠓⠈⢻⢻⡍⠛⠻⠟⣫⣾⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣩⡿⡷⡎⣡⠳⣦⡀⠀⣰⣾⣿⡆⢹⣿⣿⣿⣿⡙⢛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢇⠀⠈⠻⠿⡿⠿⣿⣿⣸⣸⣿⣿⡇⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣿⡓⠚⠲⠶⠒⣲⢶⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⡅⣈⠻⠿⠿⣥⢊⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠁⢀⣴⡀⠘⣬⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⡿⢻⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠈⠊⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢷⣦⣄⠀⠀⠀⠀⢀⣿⣿⢿⠻⢿⣷⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠆⠀⠀⠘⣿⠀⠀⠀⠀⠀⣀⡈⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠛⠛⠀⠁⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⡆⠿⠶⠒⠻⠿⠋⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣾⣧⣠⡄⢀⣠⣤⣀⢀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⢀⠀⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⣁⠙⠊⢉⣽⣾⣿⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢰⣿⣿⣿⣿⡟⠛⠻⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣠⣼⣿⣿⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢲⢄⠀⠀⠀⠀⠀⡠⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡮⠃⠀⠀⢰⣧⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡌⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡴⠁⣰⡟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣷⡄⢀⣼⣿⣦⡄
⠀⠀⣀⠀⠀⠀⢀⣄⣠⣀⠀⠙⠾⣿⣧⡏⠀⠀⠀⠀⠀⠀⠰⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣸⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⠭⢭⣭⣝⣝⣙⣳⣤⡀⠀⠉⠀⠠⣤⣀⣀⠀⠀⠀⠛⠉⠉⠈⠁⠉⠉⠉⠉⠉⠉⠉⠹⠟⠙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡙⠿⣦⠀⢀⣄⣿⡟⠛⠉⠁⢀⣄⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⣶⣮⡙⠟⠋⠉⠉⠻⠂⢴⣾⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠟⠛⠉

⠀⠀⠀⢸⣻⣏⣾⣎⣿⣷⣻⡟⠀⢀⣴⣖⣒⠲⢾⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣠⡴⠶⢾⣿⣷⡀⣿⣷⣷⣿⣷⡿⡇⠀⢀
⠀⠀⠀⢸⣿⡿⣿⣿⣞⣷⣿⣧⣶⠟⢙⠻⣿⣿⢶⣮⣷⣙⠂⢀⡀⡄⠀⠠⢚⣤⣶⣶⣶⣾⣿⡿⣟⠻⠿⣿⣿⡿⣿⣿⡇⠸⢸
⠀⠀⠀⢸⣽⡽⣽⣟⣎⣾⠏⠁⠀⠂⠈⣾⣥⢶⠃⠛⣻⣿⣷⡤⠔⠁⠢⢶⡿⢛⣿⠿⠻⠶⢤⣹⡧⠀⠀⠛⣿⣿⣿⡿⡅⢀⢈
⠀⠀⠀⠸⣟⣞⣿⣎⣾⡏⠀⠀⠀⠀⠈⢹⣿⣀⢬⣬⣌⠹⢿⡛⠀⠀⠀⢺⡥⠹⣯⣤⣄⣤⣸⠏⠀⠀⠀⠀⢻⣿⣿⡿⡅⠈⢸
⠀⠀⠀⢸⣟⣞⢎⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣵⡂⠴⣇⠀⠀⠀⡿⠀⢀⣪⡽⠟⠛⠁⠀⠀⠀⠀⠀⣼⣿⣿⡿⡇⠀⠈
⠀⠀⠀⢸⣿⢎⣾⠟⠋⢿⡀⠐⠲⢾⣷⣶⣶⣤⣄⣀⠀⠙⠶⠖⠓⠓⠒⠖⠐⠊⠀⠀⣀⣠⣤⣤⣷⡤⠀⢠⡏⠙⣿⣷⡇
⠀⠀⠀⢸⣛⣎⡿⢸⣆⠈⢷⡀⠀⠀⠀⢰⠟⣻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⣿⣆⠈⠀⠀⣠⠟⣰⡿⢾⣿⡇
⠀⠀⠀⢸⣿⢼⣇⠀⡟⠱⣦⣿⠂⠀⠀⠘⠿⢶⠾⣿⣿⡿⠿⣿⣶⣄⣴⢿⢿⣟⣭⣿⣷⠿⠃⠀⠀⠙⣧⣾⠙⡇⢸⣟⡇
⠀⠀⠀⢸⣿⣾⣿⡰⡇⠀⣿⡃⠀⠀⠀⢀⠀⠈⠓⠛⠛⠁⠀⠀⠘⠝⠁⡄⠀⠈⢗⡓⠋⠀⠀⠀⠀⠀⢹⠟⠀⡇⣿⣿⡇
⠀⠀⠀⢸⢿⣿⡿⣿⡟⡦⠼⡇⠀⠀⠀⠈⠁⣄⠀⠀⡀⢀⣠⣤⣤⣤⣤⣤⣀⠀⢀⠀⢀⠄⠈⠀⠀⠀⡹⠄⠘⣸⣿⣟⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡌⠠⢻⡄⠓⠀⠀⠀⠀⠀⠀⠸⠯⢿⣭⡁⢁⣩⣭⠿⢷⠀⠀⠀⢀⠀⠠⠄⣰⠃⠀⣴⣿⣿⣿⡇
⠀⠀⠀⢸⣿⢯⢻⣿⡟⣿⣄⣼⡏⠢⡀⠠⠃⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⣂⠀⠈⠀⠃⠀⠘⠀⢂⠄⢫⣄⣾⣿⣿⣿⢛⡇⢀⢠
⠀⠀⠀⢸⢯⢯⢯⣿⣿⣿⣟⣽⣿⡀⠈⢃⠀⠀⢰⠀⠀⡀⢠⡙⢛⠉⠉⣀⠀⠀⠀⢸⠀⠀⠖⠀⢀⣿⣿⣿⣿⣏⣷⣻⡇⠘⠈
⠀⠀⠀⢸⣯⣎⢮⡿⣿⡿⣿⣟⣿⣷⣤⡀⠀⠀⠈⠳⡀⠉⠛⠙⠛⠻⠛⠉⠂⠀⡴⠃⢀⠎⢀⣤⣾⣿⣿⢿⣿⣜⣿⣞⡇
⠀⠀⠀⢸⣿⢫⢺⡽⣽⣽⡜⣿⢾⣿⣿⣿⣷⣦⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⢊⣽⣿⣿⡽⣽⣿⢯⣿⣻⣻⡇⠀⢀
⠀⠀⠀⢸⢣⢣⢻⡿⡿⡞⣾⣏⣞⣾⣻⣿⡌⠻⣿⢳⣦⣶⣶⠤⠀⠀⠠⡴⢖⣤⣾⠏⢠⣿⣿⣟⡜⣳⣿⣏⢳⣷⢣⢳⡇⢸⢸
⠀⢰⠀⢸⣤⣧⣿⣿⣿⠾⠟⣿⠋⢉⣭⡟⢹⡀⠘⢷⠙⢿⣿⣿⣷⡾⣿⢿⠟⢉⡟⢰⠋⡟⢻⣍⡉⠻⡿⠻⠿⣿⣿⣧⡇⠘⠈
⠀⠸⠀⢘⡾⣿⣣⡏⠀⠴⠒⠁⢀⣼⣿⡇⠘⣇⠀⠈⢣⠀⠳⠿⠔⠂⠷⠃⢀⠎⠱⠁⢀⡇⠸⣿⣧⡀⠈⠒⠦⠀⠙⣮⢧⠠
⠀⠀⠀⢸⢾⣇⡟⠻⣄⠀⠀⠀⣼⡟⣿⠁⠀⣿⠀⠀⡀⠡⠀⠀⠀⠠⠀⠀⠋⢀⠀⠀⣿⠀⠀⣿⠛⣷⠀⠀⠀⢀⡼⠛⡎⣿
⠀⠀⠀⢸⣻⣇⠀⠀⠘⣦⠀⠀⢿⣠⣿⡉⠀⢹⠳⠀⡆⠀⠑⠄⠀⠀⠰⠀⠀⢸⠀⠘⡟⠀⠀⣿⣆⡿⠃⠀⢠⠞⠀⠀⢇⣿
⠀⠀⠀⢸⣽⣿⠀⠀⠀⠘⣄⠀⠀⠙⢧⡲⢦⢸⡄⠀⣇⠀⠀⠄⠀⠀⠁⠀⠀⡰⠀⢰⡇⣰⣖⡼⠋⠀⠀⢠⠇⠀⠀⠀⡟⣿
⠀⠀⠀⢸⣿⣸⠀⠀⠀⠀⠈⣧⠀⠀⠀⠙⢿⣎⣷⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢠⣿⣱⠿⠋⠀⠀⠀⣰⠏⠀⠀⠀⠀⣁⣿
⠀⢠⠀⢨⠹⣏⡄⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠉⠻⢿⣄⠀⠀⠀⠀⠀⠀⡀⢀⣴⡿⠛⠁⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢠⢸⡏
⠀⠈⠀⠸⠀⢻⡜⢄⠀⠀⠠⠀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠓⠶⢧⣄⣠⡾⠖⠛⠁⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠂⠀⢀⢂⡿⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢦⡀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠻⢿⠟
⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠈⠙⠿⣿⣿⢿⠄⠀⠀⠀⢀⣠⣾⠀⠁⢀
⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠈⠌⠃⠌⠀⢀⣴⣿⣿⣿⣄⡀⠈⢷⠀⠀⢀⣀⣀⠐⢢⣄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⣀⡀⢄⠀⠀⠀⠀⠘⢿⣿⡫⠞⠻⣟⠁⠀⠀⠀⠈⠻⢿⣿⣷⠜⠃⠀⣠⣴⣶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠀⢰⣿⣿⣿⣮⠇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣿⡭⠝⠟⠁⠀⠀⣠⣤⡀⠀⡟⠀⠀⠀⠀⠰⢃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⣠⡆
⠀⠀⠀⠀⠀⠀⠀⢷⠂⠜⠻⠃⠀⠀⠶⣤⣂⢙⣻⣴⣾⡇⠐⠀⠀⠀⠀⣾⠀⠀⠀⠀⠘⠳⣶⡶⢿⣆⡼⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠙⠿⣟⣟⡼⠁⠀⠃⠰⣀⠐⠉⣸⠀⠀⠈⠙⢷⣿⣤⣼⣯⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠛⠀⣾⠙⢄⡀⠀⢻⡿⠉⣠⣿⠟⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣧⢠⠀⠀⠀⡇⠀⠀⠙⠓⠒⠛⣹⡿⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣶⣾⣿⣿⣿⣿⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢿⣦⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠛⠻⢿⡆



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⣤⡾⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢠⣦⣶⠘⣧⣿⢣⣿⣱⡷⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡘⣼⣭⠾⣗⡿⠿⠲⢤⣿⡾⠋⣉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡄⠀⠀⢀⣈⠛⠦⣧⣿⣽⠾⡁⠀⠀⠀⣀⡝⢷⣏⣥⣐⠀⠀⠀⠀⠀⢻⡆⡾⢁⣠⡄
⠀⠀⠀⠀⣄⡀⠈⢻⣄⣾⠁⠀⠀⣾⠭⢳⡜⣽⠟⣡⡀⠀⠀⠀⠈⣡⠴⢄⢻⣿⠽⠂⠀⠀⠠⣄⣸⣿⣷⠋⠁
⠀⠀⠀⠀⠈⠙⠲⣬⣿⠛⣧⠀⠀⠴⠖⡘⣼⠏⠰⣿⣿⠀⠀⠀⢸⢡⣶⠈⠈⣿⠀⠀⠀⠀⠀⠀⣿⣤⠟⠙⠛⠒
⠀⠀⠀⠀⠐⠺⠟⠉⣿⠾⢿⣦⡀⠀⠀⣿⡟⠀⠠⣈⣡⠂⠰⠆⠈⠢⠵⠃⠀⣻⣧⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠳⡙⢦⡀⢻⡗⡀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⢠⣿⡟⠀⢀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣦⣿⣦⡹⡜⢦⡀⢰⣟⣫⣬⣼⡟⠀⣰⡿⠋⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠹⣮⣦⣝⣷⠮⣍⡉⢀⣴⠟⠋⣠⠾⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠉⠀⠙⣯⠛⠚⡿⠋⢷⣴⡿⠁⢀⣼⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⡄⠀⠈⠳⣾⠃⠀⠀⠉⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠉⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠻⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠯⣤⣀⡀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⢀⣼⠻⣿⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣶⣾⠃⠀⣿⣧⣤⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠉⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⣄⣀⣀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡴⠚⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠴⠖⠋⢹⣿⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠳⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠂

⠀⠀⠀⢀⣀⠀⢀⣀⡀⢀⣀⡀⡀⠀⠀⠀⢀⡀⠀⣀⣀⠀⣀⣀⠀⢀⠀⢀⠀⣀⡀⠀⡀⢀⠀⣀⣀⠀⠀⠀⣀⠀⢀⠀⡀⢀⠀⡀⠀⡀⣀⢀⠀⡀
⠀⠀⠀⢿⣭⡿⢸⣭⡅⣿⣭⠈⡇⠀⠀⢰⡏⢹⡆⣿⣭⢸⣯⡅⠀⠘⣦⠏⣸⠉⢹⢸⡇⢸⡇⣯⣽⠇⠀⠻⣭⡁⣿⣾⠃⢸⢸⡿⣄⡇⣿⢸⠀⡇
⠀⠀⠀⠛⠀⠀⠘⠖⠂⠛⠲⠒⠓⠒⠀⠈⠳⠞⠁⠛⠀⠘⠃⠀⠀⠀⠙⠀⠘⠳⠚⠈⠳⠞⠁⠓⠘⠃⠀⠛⠶⠋⠛⠈⠓⠘⠘⠃⠙⠃⠒⠐⠀⠇

⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠘⢶⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠘⣿⠀⠉⠉⠛⠛⠷⣦⣄⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣤⣤⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⣀⣀⣀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣤⣾⣯⣤⣤⣤⡀
⠀⠀⢿⣏⠛⠛⠻⠷⠿⠇⠀⠀⠀⣴⣶⣿⡿⠿⠛⠁⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠈⠻⢷⣤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠸⠛⣿⣧⡴⠖⠀⠀⠀⠀⠘⢷⣤⣶⡟⠉⠀⠀⠉⠛⠛⣿⠛⠁⠀⠀⠀⠀⠀⠐⠟⣿⡶⠟⠁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⣰⣟⢀⢀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣤⣤⣤⣶⠾⠿⠂
⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠻⠷⢶⣶⣤⠄⠀⠀⠙⠳⠶⡴⠟⠀⠀⠀⠀⠀⠈⠹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⠤⢴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣀⠤⠒⠉⠀⢠⡿⠚⠉⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡜⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢹⡞⠁⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⣄⡀⠀⠀⢇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠀⠀⢀⡠⠖⠊⠉⢁⣤⣦⡀⠀⠀⠉⠳⡄⢸⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⠴⠊⣁⠀⠀⠀⠀⠸⣿⡿⠃⠀⠐⣿⠆⠹⡈⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠎⠀⠀⢀⡔⠁⠀⠐⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠱⣄⠀⠸⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠇⡞⠀⠀⢠⠏⠀⠀⢀⣤⡀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠾⠗⠘⢧⠀⠷⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠁⠀⢀⣿⠃⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢴⣷⠆⠀⠀⠀⠀⠀⠀⡈⢇⠀⡜⠀⠀⠀⠀⠀⠀⡠⠂⢠⠞⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢸⠀⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⣾⡿⠊⠂⡇⠀⠀⠀⢀⡆⡼⠁⢠⠟⢉⡴⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠘⡀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠭⢠⡀⠈⣿⠁⠀⠀⠀⠈⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣼⡀⠀⠀⠀⠀⠱⡟⠆⠀⠀⠀⠀⢀⡜⠁⢠⠟⢆⠹⣄⠀⠀⠀⠀⠀⢠⣤⡀⠀⢡⢠⡃⠈⢾⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⢠⣄⡀⠀⠀⠱⡈⠓⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⢀⠞⠀⠀⣜⠀⠈⢣⠘⣆⠀⠀⠀⠀⠻⠟⠁⢠⣄⢇⠙⣤⢂⠀⠀⠀⠀⡸⣁
⠀⠀⠀⢠⠇⣀⠈⠁⠂⠤⣴⣄⠀⠀⠀⠀⠀⠀⠰⡀⠀⢠⠋⠀⠀⠜⡏⠀⠀⠀⢃⠘⣆⠀⠀⠀⠀⠀⠀⠀⠁⠘⡟⣁⣸⠀⠀⠀⠀⡿⠃
⠀⠀⠀⣾⢰⠁⠛⢦⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⣆⠀⠋⠀⠀⠠⠊⠀⠀⠀⠀⠈⡄⢹⡄⠀⠀⠀⠛⠀⠀⣠⠖⠋⠀⠀⠑⣄⠀⠀⡌
⠀⠀⠀⡇⢨⠀⠀⠀⠹⢆⠄⠀⠀⠉⠳⢤⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡇⠀⠄⠀⢀⡴⠋⠀⠀⣾⡇⠀⣾⡞⡆⠀⠱
⠀⠀⠀⢿⡾⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⡄⡸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠃⡇⠀⠀⣰⠋⢠⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀
⠀⠀⠀⠸⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢐⠁⠀⡆⠀⠀⠀⢳⠃⡆⢰⢷⠆⠀⠀⢀⣤⡄⠀⠀⠾⠀⠀⢳⠀⠀⢣
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠰⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠇⠀⠀⢠⠏⠀⠁⢸⠀⣴⡄⠀⠈⠛⠃⠀⠀⢀⠀⠀⠈⣇⠀⠈⡇
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⠃⠀⠀⣤⠸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢻⠀⠀⢿
⠀⠀⠀⠀⠀⢶⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠣⠀⣠⠄⠀⠘⢤⡄⢠⣿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠀⠀⠀⢤⣵⣄⠀⠀⠀⠀⠈⠢⡀⢀⠔⠁⠀⠀⠀⠀⢀⡤⠖⠒⠒⠀⠀⠀⠀⠀⣇⢸⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢷⡈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⡴⠁⠀⢠⠀⣠⡧⢮⠀⡰⠛⠲⢧⡔⠒⢦⡀⠔⢢⡀⠀⡔⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⢠⠎⢩⠖⠲⣤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠴⢃⣼⠋⠀⠈⠃⡇⠀⠀⢸⠀⠀⢸⠁⠀⠀⠳⠎⠀⠀⠤⠊
⠀⠀⠀⠀⠀⢸⡀⢻⠀⢸⡀⠈⠑⣄⡀⢀⠎⠁⠆⠀⢀⡀⠄⠀⠀⠀⢀⣀⡔⠉⢸⠀⠀⢀⣆⣵⡀⠀⢸⠀⠀⢸⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠈⠑⠊⠓⠀⠀⠀⠈⠙⣄⣀⣀⡤⠞⠖⠒⠒⠒⠉⠙⠊⠑⠒⠉⠑⠒⠁⠀⠀⠙⠒⠉⠑⠒⠉⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠘⠁⠚⠀⠀⠋⠁⠂⠃⠀⠈⠐⠄⠀⠀⠀⠃⠘⠀⠈⠀⠘⠀⠀⠀⠀⠀⠇⠊⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠉⠈⠉⠉⠙⣦⣄⠀⢀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣶⣤⣀⣰⣾⡋⠻⠆⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⡄⠀⠀⡣⢠⡀⣠⣐⢻⠏⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣑⣂⣉⣼⣃⠀⠰⠄⠀⢠⣄⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣯⣛⠀⠉⠫⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⡛⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠛⠛⣿⣿⣿⣿⣿⣯⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠿⠻⠟⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠇⠈⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠉⠁⠀⠀⢀⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋⠀⢀⣾⣿⣿⣿⣿⡟
⠀⠀⢀⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣝⣌⠀⣼⠿⠋⠛⠋⠉
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⠀⣸⠋⣟⣦⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠐⢻⠁⢿⠀⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣯⠀⠉⠛⠟⢦⣩⣅⠈⠠⣴⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠃⠴⢾⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⣀⡀⠀⣀⡤⠿⠁⠙⢿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡏⢁⡖⠀⠂⠈⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠈⠙⠿⢿⣿⣿⣶⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄⡀⣀⠀⠉⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠻⠿⠟⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠂⠙⠿⣿⣿⢿⣿⣿⣿⣿⡿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩⣭⣭⠭⢭⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⢉⣩⣭⣭⣽⣿⣿⣯⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⠽⠒⠒⠉⠉⠉⠉⠉⠙⠻⢗⣦⣄⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠈⠉⠙⠻⢽⣦⣀
⠀⠀⠀⠀⣠⢞⠵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡼⣛⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠀⠀⠉⠪⣷⢄
⠀⠀⢀⣴⡕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠞⠉⠳⣵⡄⠀⠀⠀⠀⠀⠀⡀⠃⠃⠀⠀⠀⠀⠀⠈⠳⣵⡄
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⢠⣿⠃⠀⠀⠀⠘⢞⢆⠀⠀⠀⠀⠀⠀⡁⠁⡀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⣾⠏⠀⠀⠀⠀⠀⠀⣞⠈⠉⠀⡀⠀⠀⠈⠀⠀⣾⠇⠀⠀⠀⠀⠀⠈⣞⡆⠀⠀⠀⠀⠀⠀⠀⡐⠤⠄⠀⠀⠀⠀⠀⠀⢿⣆
⢰⣿⠀⠀⠀⠀⠀⠀⡈⠀⠁⢧⠱⡁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⢉⡀⠆⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠄⠀⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣖⣖⣒⣶⣶⣶⣦⡼⣾⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣲⡍⠀⠀⠀⣿⠂⠀⠀⠉⡽⣢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⢳⣣⠰⠶⠶⣿⠶⠶⠆⣰⡽⠉⠓⢝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠃
⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢳⣣⠀⠀⣿⠀⠀⡴⡳⠁⠀⠀⠀⠳⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⠃
⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠑⣕⣄⠿⣦⢮⠞⠁⠀⠀⠀⠀⠀⠈⢪⢢⡀⠀⠀⠀⠀⣠⢞⡵⠁
⠀⠀⠀⠀⠀⠙⠻⣦⢄⣠⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠵⠛⠣⣷⡤⣀⡀⠀⠀⠀⠀⠀⠀⠳⣳⡀⣀⠤⣪⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠾⠿⠿⠽⠭⠭⠭⠿⠛⠋⠉⠀⠀⠀⠀⠀⠉⠙⠻⠿⠴⠶⠶⠶⠶⠾⠷⣯⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢈⠨⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⡀⠀⠊⠅⠇⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡸⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢞⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⢬⡬⠐⠂⠁⠁⠀⠐⠀⠨⠻⣿⣿⣉⣤⣄⢠⣤⣀⡀⣠⡀⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⡀⣀⣀⣤⣤
⣶⣴⣴⣾⣷⣿⣷⣭⣁⣤⣄⣀⣥⣹⣦⣿⣤⣦⣠⣤⣴⣶⣤⣄⣄⣶⡜⢿⣿⣿⣿⣏⣞⣾⣣⣿⣿⣶⣿⣶⣿⣿⣿⣷⣷⣿⣿⣿⣾⣯⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣯⣼⠿⣿⣿⡿⢟⣋⣛⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣙⣛⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠐⠂⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠗⢤⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠸⣄⠀⠀⠀⠀⠈⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⡘⢦⢄⠀⢘⢰⠠⡉⢲⡙⠀⡏⠀⠀⠈⠁⠀⠀⠀⠀⠈⣆⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠚⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠀⢆
⠀⠈⠓⠶⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠂
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡐⠂⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠐⠐⠀⠂
⠀⠀⠀⠀⢀⣤⣴⠤⠀⢀⣤⠀⣄⢀⠤⢀⡠⠀⠠⡤⣠⢤⠀⢀⠤⣠⠤⡀⢠⣠⢤⣀⢤⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣷⢺⠇⠀⠸⠿⠾⠻⠿⠤⠼⠣⠀⠀⠇⠳⠼⠃⠨⠵⠣⠤⠏⠹⠳⠴⠻⠤⠿⣧
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⣟⠑⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠐⢦⠘⢦⠙⠖⠨⢧⡹⣄⡱⠌⠁⠀⣠⠈⠳⡄⠠⠀⠀⠈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡲⠸⠍⠁⠉⠂⠘⠀⠀⠀⠀⠀⠉⠁⣰⠉⠓⡯⠤⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣟⣛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠿⡿⠿⠿⠿⠷⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⠀⠀⢀⡤⣄⢠⡄⠀⢠⡄⢀⣤⡄⣤⣠⠀⢀⣴⡄⠀⢠⣠⣤⡄⣤⢠⢠⡄⢤⡤⢠⡤⠄
⣿⣿⣿⣿⣿⣿⠀⠀⠀⡷⣯⢸⡇⠀⣿⣧⢸⡇⡅⣿⣧⠀⢠⣿⣥⠀⢸⣿⣿⠀⣿⢿⢸⡇⢸⡇⢸⡗
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠋⠘⠛⠀⠃⠙⠈⠛⠁⠛⠘⠀⠈⠛⠉⠀⠀⠃⠛⠀⠋⠘⠘⠃⠘⠃⠘⠛⠃
⣿⣿⣿⣿⣿⣿⠗⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒
⣿⣿⣿⣿⣿⣿⠀⣿⡟⠋⠉⢁⣈⠀⠈⠉⠻⠀⠀⠀⠀⠀⠀⢃⡄⠀⠇⠀⠀⠀⠀⠀⠀⠤⠤⡀⠀⠠
⣿⣿⣿⣿⣿⣿⠀⡿⠀⠀⠛⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⢐⠀⠊⠙⠐⣉⠀⠀⠀⠀⢠⣔⢂⠀⣥⣀⡀
⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢮⢤⡄⢾⣴⠸⣶⠀⠀⠀⠀⠈⠐⢶⢸⣿⠇
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢉⣀⠂⠀⠀⠀⠀⠀⢻⣶⣶⣶⡾⢷⡆⠀⠀⠀⠀⠀⠀⡼⣷⣄
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠛⢢⡀⠀⠀⠀⠀⠂⣳⣿⠀⢸⠀⠿⢿⡄⠀⠀⠀⣠⠴⠳⠿⣧⡄
⣿⣿⣿⣿⣿⣿⠀⣀⠀⠀⠈⠀⠒⠀⠏⠉⠀⠀⠀⠰⠾⢿⠈⣷⣿⡄⠀⠀⣤⠀⠀⠀⢻⡀⠀⠀⠈⠙⠛⢦⣄
⣿⣿⣿⣿⣿⣿⠀⣾⣰⢀⡇⢠⠀⠀⣀⣀⡀⠀⠀⠀⣾⠿⣄⣚⠓⠯⠿⣧⣟⠀⠀⠒⠛⠉⠀⠀⠀⠀⠈⠘⠋
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⢀⠉⠀⢠⣤⣄⡀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠞⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⡄⠀⡥⠠⣼⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢺⠁⠤⣅⣀⣇⡆⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⢇⠀⡰⠇⠀⠀⡔⡀⠀⠐⠀⠆⠀⠀⠀⠀⠀⢀⡏⠀⣷⡞⢈⣿⡷⣴⡠⠄
⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⢁⣴⣀⠀⡠⣰⣧⠄⡶⠆⠀⠀⠀⠀⠀⠀⠀⡷⢶⡀⠀⡼⣿⣦⡙⠃
⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣾⡛⠛⠈⢙⡡⠘⠃⠃⠀⠀⠀⠀⠀⠀⠀⠠⢠⣾⣦⣶⣿⣯⡉⠹⣶⣤⡄
⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣳⠦⠭⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠛⣳⡶⣀⣼⡿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⡿⠀⠁⠠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⢈⡇⢠⠁⠀⠻⠜
⣿⣿⣿⣿⣿⣿⠀⠀⠀⣧⣴⡆⠀⠀⠀⠁⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣸⠀⠈⠛
⣿⣿⣿⣿⣿⣿⠀⢠⡾⠿⢿⣷⣤⠀⠠⠀⠰⢞⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣁⡀
⣿⣿⣿⣿⣿⣿⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠉⠉⠉⠁⠉⠁⠀⠉⠁⠉⢩⣮⢉⡉⡉⡉⠉⣉⠉⢹⣯⢉⢉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣄⣁⣀⣁⣀⣀⣀⣀⣁⣀⣈⣛⣘⣈⣃⣃⣀⣛⣀⣘⣁⣘⣀⣂⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⣤⠤⠄⢈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢰⣇⡴⠒⠒⠒⠦⢤⣀⣠⠤⠖⠋⠀⢀⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠁⠀⠀⠀⠀⠠⡶⢮⡓⢤⣠⠴⠚⠉⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⣠⡴⠀⢀⣤⠄⠁⠀⠙⢦⡀⠀⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠸⠿⠶⠂⠀⠘⣿⠿⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⣆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⢧⡀⠀⠀⢤⠀⠀⠀⠀⢀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠶⣄⣸⠀⠀⠀⠀⡇⠀⢀⡼⠋⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡞⠳⣄⠈⠙⠓⠶⠤⠤⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢰⡇⢃⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠧⠤⠟⠁⠈⠓⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡾⠋⠙⠓⠶⠦⣤⣤⣄⣀⣀⣀⣤⣤⣤⠴⠶⠖⠛⠻⣅⠀⠀⠀⣰⠋⠈⢉
⠀⠀⠀⣀⡤⠖⠋⠉⠁⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠲⠶⣏⡀⠀⠹
⠀⠐⢯⡁⠀⢠⡴⠚⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⢸
⠀⠀⠀⠙⢦⡀⠙⢦⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣀⣀⡀⠀⢸
⠀⠀⢀⣀⣀⣹⣆⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⢀⡏
⠀⣀⣿⠉⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣟⠛⠋
⣨⢧⡙⣦⠴⢖⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠙⠦⣱⠶⠚⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠶⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠶⠾⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣠⠖⢻⣿⢧⣄⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢁⡔⣿⢸⣇⢊⢳⡀⡑⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⣤⠶⣚⣥⠞⠹⡆⠉⠺⣆⠻⣕⠈⠙⠦⠭⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠟⢉⣴⡾⠋⠀⠀⠀⠹⡄⠀⠹⣄⠙⠦⣄⡠⠔⠊⠀⢉⡷⡄
⠀⠀⠀⠀⠀⠀⢰⡄⢠⣿⣿⣇⡒⠀⠀⠠⢒⣻⣄⣀⠈⠓⠷⣖⣮⣀⣐⣴⣿⣆⢠
⠀⠀⠀⠀⠀⠀⢿⡃⣿⣿⣿⠛⠛⢠⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⣍⢏⢻⣿⡿⣦⣬⣳⡄
⠀⠀⠀⠀⠀⠀⠈⢧⣿⡟⣽⡀⠀⠆⠀⠀⠀⠀⠀⠀⢿⢿⣿⡿⣻⠻⡄⣿⣻⡾⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢋⣴⣿⣷⠀⣤⣤⣄⡀⠀⠀⠀⡘⣠⠟⢡⠏⢀⡇⣿⡏⢷⢹⣶
⠀⠀⠀⠀⠀⠀⢀⢠⠞⣿⢿⣿⣧⡉⠓⠈⠀⢀⣠⣮⠞⢁⡴⢃⠞⣹⠃⣿⣇⡜⣸⠟
⠀⠀⠀⠀⠀⠀⠀⣮⠀⣿⢼⢻⣿⣷⣤⣤⣶⣿⠟⣡⢞⡡⢒⡥⠤⢛⡄⣿⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢀⣿⡄⢻⡈⢳⣿⣿⣿⠏⡟⣡⣞⡵⢋⡠⢋⣾⡷⠋⠻⡞⣿⣷⣀
⠀⠀⠀⠀⠀⠀⡞⡏⣸⡄⢳⡀⢀⡿⢋⣼⢱⣟⡎⣰⠋⢠⠏⢾⠁⢀⠞⠉⣿⢻⢮⡳⢄
⠀⠀⠀⠀⠀⢸⠼⢠⡿⣿⠀⢣⢈⣴⡿⣿⣿⡏⢣⡇⢀⡿⠀⡟⠀⠐⠀⠀⠘⡎⢣⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⢸⠀⠘⠀⣿⣠⡼⠟⢽⢃⣿⣿⣿⣸⡇⢸⡇⢀⡇⠀⠀⠀⠀⠀⠸⡄⠹⣦⣉⠢⡙⢦
⠀⠀⠀⠀⠀⠘⡄⣠⠷⠛⠁⡀⢊⢎⣼⣿⡇⣿⣿⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢻⣆⠈⢮⡓⠽⣆⢳
⠀⠀⠀⠀⠀⢀⡿⠗⣒⣚⣭⣴⣣⣾⢟⣿⢹⢹⣿⡇⠘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⡷⣄⡙⠲⣬⣸
⠀⠀⠀⠀⣴⣡⣴⡿⢟⣟⣛⢻⡿⢋⣾⡏⣎⣿⢻⡇⢸⠁⣸⠀⠀⠀⠀⠀⠀⡄⠀⢹⣿⡦⡙⢻⠉⠃
⠀⠀⠀⡴⢻⢋⣾⣿⡫⡍⣿⡟⣾⡿⢋⣠⣾⠏⣾⠁⢸⢠⡟⠀⠀⠀⠀⠀⣰⢃⣠⡾⠹⢧⡿⡫⠊
⠀⠀⠈⢧⡘⢌⣿⣧⢇⢷⡈⢱⣿⣿⣿⣵⠋⣼⡏⠀⣾⢛⡣⠤⠤⠤⠒⠉⠉⠘⣉⡠⣴⠿⠊
⠀⠀⠀⠀⠈⠳⣮⣟⠻⠦⢭⣽⡤⠿⠛⡡⣪⠎⢀⣼⣻⡴⠿⠷⠒⠋⢛⣿⡟⢿⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠯⣵⣢⣤⣄⣀⣤⣚⡥⡖⠻⡇⢏⠀⢐⡀⡠⠬⠍⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠟⠛⠁⠀⠉⠠⠼⠷⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢀⡾⢻⠟⢷⡀⠀⠀⡾⢻⠟⢷⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡇⠹⣄⣼⡇⠀⢀⡇⠸⣄⣼⡇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠻⢤⣤⠟⠁⠀⠀⠻⢤⣤⠞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠐⠿⠿⢭⣍⡛⢦⣄⢸⡆⠀⣀⣀⣀⣤⣤⣴⡶⠖⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⠙⠳⣝⣦⠙⠋⠉⠉⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⢤⣤⡤⠴⠶⠚⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⠛⠛⠛⠛⠛⠻⣟⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⡠⠀⠀⠀⡀⠀⠈⠉
⢀⠀⢀⡀⣔⣽⣦⡀⠀⣠⡀⣆⣀⠀⡂⢀⣀⠀⠀⣀⣀⠀⣀⣀⢀⣀⢀⣀⠀⢀⣀⠀⣀⣀⠀⠀⡂⢀⣀
⠈⠀⠒⠺⢿⠏⠉⠁⠀⢸⡀⡏⢸⡇⡇⣙⢪⠀⠀⣏⢸⡇⣗⢚⢸⠃⢘⢲⡅⣇⢈⡇⡇⢸⠀⠀⡇⢘⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠏⠉⠀⠈⠁⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡞⠙⡆⣼⠉⣷⢰⠀⡀⣀⣀
⠀⠀⢀⢤⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⡀⠀⠀⠀⠀⢸⡇⠀⣇⢀⡇⢿⣀⡿⢸⠿⡀⣿⢸
⠀⠀⡆⠀⠈⠢⡀⠀⣀⣯⠑⠒⠤⡀⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁
⠀⠀⡇⠀⠀⠀⠈⠛⠤⣀⡀⢀⠀⠀⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⢠⣖⣦⢐⣒⡆⣶⠂
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⣀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠈⠳⠆⠘⠶⠃⠛
⠀⠀⠈⢆⠀⠀⡰⠋⣡⣄⠀⠀⢈⣶⣎⠱⠀⠠⢔⡁
⠀⠀⣀⠄⢸⢰⠁⠀⣿⡿⠀⠀⠘⣿⡿⠀⠇⠀⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠙⠆⢈⠨⠦⠀⠘⠋⠀⠀⠀⠀⠀⠈⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡏⢺⣀⡷⢾⣀⡇⣿
⠀⠀⠀⠓⠂⠑⠤⢄⣀⠀⠒⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠚⡤⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⡒⠀⠀⠀⠀⡄⠀⠱⡀⠀⠀⢸⢭⣟⣦⠀⠀⠀⡗⢲⢠⡖⢲⣰⡄⢰⠰⣖⠂⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠘⡟⠛⠟⠀⠀⠀⠃⠘⠈⠳⠞⠘⠳⠞⠐⠶⠃⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣇⣀⣀⣘⣀⣀⣠⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢠⣿⢁⣠⠴⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣷⣤⠿⢻⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢋⣿⣷⡶⠞⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠒⣶⣉⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⣥⠀⠀⠈⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠿⣇⣠⡇⢈⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠁⠀⠀⠀⢀⠠⢤⣤⡿⣟⣾⣿⢿⡴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣻⣾⣷⡦⠀⠀⣠⠋⠰⠟⠛⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣶⣿⣿⡿⢷⣦⡷⠟⠃⠀⢲⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢀⣀⣤⣾⣽⡿⠛⣩⣿⡿⠿⢣⡶⠀⢼⡟⠻⣿⡄
⠀⠀⠀⠀⠀⢀⠀⠚⠁⣠⡴⣋⣭⣽⢁⣿⠳⠖⣿⠇⠐⢶⠾⣝⠳⠤⣶⠞⠛⢻⢷⣄
⠀⠀⠀⠀⠀⠸⣷⡴⣾⣭⠾⣋⣭⣜⢿⣋⢸⣋⣿⡶⣶⣤⣿⣧⣄⣤⣬⣆⡀⠈⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠹⠷⡴⢶⣿⣿⣿⣁⣾⣿⣿⣿⣿⣶⣿⣄⢻⡷⢿⣿⣿⣋⡴⢊⣉⣿⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⣻⣿⣿⣿⣿⣿⣿⣽⣿⣧⢿⣿⣿⠛⢿⣿⣿⣿⡿⠀⠿⢷⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣿⠏⣿⣿⣹⣿⣧⠋⣻⣿⣿⡟⣀⠀⠀⠉⡟⠋⠁⠀⠀⠩⣥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⠉⠀⠈⠙⢳⣿⣿⣿⣟⡁⣀⠀⠀⣧⣶⣾⣿⣾⣿⣟⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣽⡟⣛⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢠⡀⣢⣤⡄⠀⣴⣶⣿⣿⣿⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⡿⠁⠀⠀⠿⠿⠿⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠦⠀⢤⣟⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣤⠈⣉⠉⢣⡀⠀⠀⠀⠀⢀⡄⠰⣿⣻⠿⠿⠛⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣖⣻⣀⠘⠀⠀⠀⠀⠀⠀⠀⠆⠠⠾⠃⢹⡀⢘⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⢻⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣧⣾⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠋⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠧

⠀⠀⢠⡏⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢀
⠀⠀⠘⣄⣀⣞⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣠⠃⠀⢀⡇⠀⠀⠀⠀⠀⢀⣀⠀⠈⢦⠀⣾⠀⠀⠀⢀⡤⠖⠒
⠀⠀⡴⣫⠤⣄⠀⠉⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠒⠒⣲⠖⠊⠉⠉⠉⠉⠑⠦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠾⠼⠧⠤⣤⣤⣞⣠⡤⠤⠄
⠀⡜⠁⣅⣠⡼⢠⠖⢾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰⢳⠢⢰⡄⠀⠀⠀⡴⢒⢝⢷⠒⠦
⠀⢻⡀⠀⠁⡀⠻⠧⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡔⣐⢲⡀⠀⡤⠦⡄⠀⢹⠀⠀⠀⠀⠀⠈⠫⣲⣚⡁⠙⠒⠁⠑⠂⣊⠼
⠀⠀⠱⡀⠋⠉⠡⣞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣈⠣⠭⠞⠁⠀⠧⠥⠋⢀⡼⠀⠀⠀⠀⠀⠀⠳⣄⠀⡿⠀⠀⢻⠉⣉⡴
⢀⡤⢾⠃⠀⠀⠀⠀⠙⣍⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡖⠀⠐⠲⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⡟⠁
⠞⠀⢸⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢤⠀⠀⣠⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠷⠚⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⣴⠃
⠀⠀⠘⢤⣀⡀⢀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡖⠒⠶⢲⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⢩⡇
⠀⠀⠀⠸⠃⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄
⠀⠀⠀⠲⢄⠀⠀⠀⣱⣄⣿⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠂⠒⢦⣀⡷⠒⣋⠁⢀⡤⣬⠙⡄⠀⠀⠀⠀⠀⠐⠓⢦⣈⣷⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠲⡤⠖⠛⠉⠉⠉⠉⠓⢯⡉⠁
⠈⠉⠙⢲⠋⢰⠻⠆⣳⠘⠮⠞⢀⠏⠀⠀⠀⠀⠀⢠⢢⣟⠳⡄⢀⣤⡊⡕⠀⠀⠀⠀⠀⠀⢰⣣⣞⢳⠌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠘⡇⠈⠓⠛⠱⠶⠛⢰⡏⠀⠀⠀⠀⠀⠀⣏⠸⠥⠞⠁⠘⢭⡴⠃⣧⠀⠀⠀⠀⠀⠸⣌⠉⠉⣶⣻⢆⡿⠛⢦⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡶⠙⢆⠀⠀⠀⠀⠀⠙⠲⡄⠀⠒⠂⠀⡀⠀⠹⡄⠀⠀⠀⠀⢀⠏⢹⡖⢾⠗⠋⠀⠀⡘⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠃⠀⠻⣄⠀⠀⣰⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣇⠐⢦⡀⢰⠃⠀⠀⣻⠀⠀⠀⠀⠀⡄⢨⡅⢸⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⡏
⠀⠀⠀⣯⠀⠀⠀⠈⠙⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠓⠈⠃⠀⠃⠀⣼⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠸⣄⡀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡦⠤⠶⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣠⣤⠚⠁
⠀⠀⠀⠀⠈⠹⡟⠋⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠈⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡤⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⣸⡀⢀⡤⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠢⡤⠤
⠀⠀⠀⠀⣰⣧⣤⣾⣅⡤⠶⠆⠀⠀⠀⠀⠀⠀⠐⠉⠉⢓⡶⠺⣛⢯⣩⠽⢍⠉⠲⢤⠤⠦⡀⠀⠀⠀⠀⢋⣡⡧⡋⡉⣆⢀⠔⡒⡜⠦⡄
⠀⠀⢀⡴⠁⠀⠀⠀⠀⠉⠳⡗⠦⠀⠀⠀⠀⠀⢀⡤⢲⠏⠀⠓⠚⠃⠘⠢⠚⢀⣀⠈⣇⠀⠀⠀⠀⠀⠀⠚⢸⡇⠻⢥⠏⠘⠦⡡⠎⡆
⠀⠀⡞⠀⠀⣀⣀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠈⠉⠉⠉⠉⠉⢉⣉⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⣀⣀⠀⠀⠀⣧⡠
⠀⣴⠃⠀⣎⠱⢎⡧⠀⢾⡭⡾⣧⠀⠀⠀⠀⠀⠀⢀⡾⡄⠀⠀⠀⠀⠀⣾⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣧⠀⠀⠉⠉⠀⠀⠀⢻
⠸⠙⣇⠀⠈⠉⢩⣠⣀⣈⠉⢡⠆⢳⠀⠀⠀⠀⠀⠚⠀⠙⣄⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⡀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠈⠳⡶⠦⠤⠤⠤⠖⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠤⠤⠤⣤⠴⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢸⣦⡀⠀⠀⠀⠀⠀⠀⠸⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠙⢲⣤⣀⡀⠀⠀⠀⠛⠉⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣌⡀⠀⢀⣠⣤⣄⠂⣿⢈⡟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣤⠈⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⢹⡖⠋⠀⠀⠈⣟⠁⠸⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⣀⣀⣀⣀⣀⣀⠛⠒⠚⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡿⠄⠀⠀⠀⠀⡿⠀⠀⢹⣄⠀⠀⠀⠀⡼⠁
⢠⡞⠉⠀⠀⠉⠉⠉⠉⠉⠙⠉⠉⠙⠛⠒⠚⠛⠛⠛⠓⠒⠒⠒⠒⠶⠶⠶⠶⠶⠼⠷⠦⠤⠖⣷⠀⠀⠀⠀⣼⠃⠀⢰⠇⠀⠀⠀⠀⢰⠇
⠸⣆⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⠖⠛⠁⠀⠀⢻⡀⠀⠀⠀⠀⠸⡄⠀⠀⣀⡀
⠀⠈⠙⠛⠚⠶⠶⣶⠶⠶⠲⠖⠲⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠋⠉⠉⠉⣹⠋⠁⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⣻⢦⣄⠀⠀⠀⠉⠉⠉⣹⠁
⠀⢠⣞⣳⡦⠐⠊⣳⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠚⠁⠀⠀⠀⠀⠀⢀⣠⠴⠃
⢀⣏⣿⣇⣻⠀⣏⢹⣲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⠋
⠋⠀⠀⠉⠁⠀⠈⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢠⠎⠈⠙⠓⠶⠾⠷⢤⣤⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⣦⠀⢀⣎⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⢶⣻⠀⡸⣿⡷⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣴⡞⡇⠀⠀⠉⣠⠗⢫⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⡤⠒⠒⡾⢹⣿⣿⠏⠉⢶⣷⣴⡶⠛⣷⣶⣬⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡘⢇⠹⡄⠀⠀⡼⢁⡞⠛⣇⢂⣉⣡⠾⠗⣸⠄⢁⣤⢀⣼⣿⠁⣴⣦⡘⣿⣷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠚⠷⠙⠒⠚⠁⡾⠀⠀⠉⠉⠰⣿⠧⣶⣷⡆⣠⣍⠘⠿⠟⠀⡙⢛⣡⣮⣙⡓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⠲⠾⠟⠂⢀⣀⣀⡈⠳⠤⢦⡀⢀⡴⠏⡀⠙⢻⡄⠛⢛⣤⣄⠀⣼⡋⢿⣧⠞⠉⠉⠳⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠛⠉⠉⠀⠀⠙⠳⠶⠊⠀⣾⠆⠺⠿⢀⡈⠋⠀⢿⣿⣿⡇⠩⣯⡄⠀⠀⠀⠀⠀⢻⡀⡇⠀⠀⢀⡴⢋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢀⣶⣦⠈⣴⣶⣶⡌⢉⡁⠀⠀⣿⠃⠀⠀⢀⣀⠀⢈⡆⠸⠴⠚⣥⣞⡩⠴⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡛⠃⠀⠛⠛⠀⠻⢿⠿⢃⠘⠛⠀⣴⡏⠀⠀⠀⢾⣈⣉⣉⣀⣀⣀⠀⠴⣦⣶⡖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠸⠿⠀⣠⣶⣶⡄⠙⠃⣶⣿⣿⢻⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠙⠲⢤⣍⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣦⣤⣄⣿⣿⣿⡟⢀⣤⡛⠿⢋⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢧⠀⠀⠈⠉⠉⠛⠛⢿⣶⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣤⣤⣸⡟⣆⠀⠀⠀⠀⠀⠀⠸⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣀⣀⣀⣀⣀⣈⣓⣊⣥⣜⣷⡄⠀⠀⠀⠀⢠⣿⣜⣒⠢⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⢟⡒⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⣹⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢨⡱⢤⡤⠤⠤⠴⣫⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣴⡶⡄⠀⠀⢠⠴⠄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⡁⢸⣇⢀⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠁⠈⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⣀⡠⠀⣴⠀⠀⠀⢸⡴⠒⠃⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⢠⠖⠛⠋⠉⠉⢹⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⢤⡀⠀⠀⠀⠀⠀⠈⣇⠀⡿⠀⠀⢠⡇⠀⠀⠀⠀⠀⣀⡾⠳⢤⡀
⠀⠀⠀⠀⢀⣄⠀⠀⣠⠞⢉⣀⣤⠞⠀⠳⡄⠀⠀⠀⠀⠀⣼⣰⠃⠀⢀⣾⣀⡀⠀⠀⢀⣈⡽⠦⣄⠀⠙⡆
⠀⠀⠀⠀⠈⢡⣶⣾⣿⣿⡿⠟⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠈⠙⠛⠓⢚⣉⣉⣭⣭⣭⣭⣁⣀⣤⠞⠀⢀⡇
⠀⠀⠀⠀⠀⠙⢛⡽⣌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⢀⣀⣀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠸⣇⢯⣳⣄⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠞⠋⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣺⣍⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⠋⠈⠉⣱⠂⠙⠒⣂⠼⠍⠈⠒⣒⣋⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠾⠤⠀⠶⠠⠤⠄⠐⣜⠂⠀⠓⠈⠁⡏⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠐⠀⢀⡠⠤⠠⣤⢸⣠⠊⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣱⠖⣍⠀⠀⢀⠟⠸⡟⠀⡀⠀⠀⠀⠀⡰⠰
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠒⢚⣯⡀⣀⡠⠔⠉⠀⠀⡟⣫⠌⡇⡀⠘⠛⠧⠮⢤⡄
⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⡴⠆⠉⠁⣀⠀⣠⡀⠀⠀⠵⣺⢫⡽⡀⡇⠀⠀⣜⢀⡤⠠⢄⡀⣀⣤⣀
⠀⠀⠀⠀⠈⠓⠤⠤⣀⣄⡤⠤⢖⠊⠁⠀⠀⠀⢞⠀⢀⣰⡇⠀⠀⠀⠈⡿⢁⠗⠇⠀⡸⢣⠀⠈⠁⢄⠈⢶⠧⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠉⠀⢠⠇⠀⠀⠀⣰⠃⡘⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀⠀⠈⠀⠀⠀⢰⠃⠀⠇⠀⠀⠀⠀⠀⢸⠀⢀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⢠⠃⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⢸⣇⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣱⠓⣶⠦⠇⠀⠀⠀⠀⠀⠀⣼⢒⣾⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⡯⣠⣴⡆⠀⠀⠀⠀⢠⠋⢀⡿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⣌⣷⠤⣄⣀⠀⣠⠋⠀⠞⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢻⡀⢸⡏⢉⠇⠀⠀⠀⠀⠀⠀⡸⠲⡉⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢲⣤⡄⠛⢿⣇⡸⠓⣀⠀⠀⠒⠂⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⡇⠀⠀⠀⠁⠀⠙⢬⣶⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠓⣄⡀⠀⠀⠀⠀⠀⠙⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣈⠟⢦⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠐⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠⡀⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠓⢐⠉⢁⠐⠁⠂⢄⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⣛⢂⢴⡁⠀⡈⠁⠉⡀⠀⣱⢄⢚⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠉⢀⠀⠀⠀⠀⠀⠀⠠⢒⠄⠊⠀⠄⡬⠹⠉⠀⠀⠀⠀⠹⠉⡄⠄⠉⠂⢔⠢⡀⠀⠀⠀⠀⠀⢀⠈⢡
⠀⠀⠀⠀⠀⠘⠀⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠆⠐⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡔⠀⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠈⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠤⣤⣀⢠⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣠⢀⣀⠀⠀⠀⠤⠀⠀⠈⠉⠢⣱
⠀⠀⠀⠀⠀⠀⡆⠀⠀⣆⡄⢸⠀⢘⡯⠀⠀⠁⢀⣀⠂⠀⢠⡀⠀⠀⣠⠀⠀⢈⣁⡀⠁⠀⠌⠉⠀⠀⠀⣄⡆⠠⣐⣄⣀⡼
⠀⠀⠀⠀⠠⡤⠅⠀⠈⠟⠃⠀⠈⠋⠉⠚⠉⢭⣘⣯⡉⠺⢿⠇⠀⠀⢿⠿⠎⣹⡟⣨⠅⠚⠊⠁⠀⠀⠀⠛⠏⠀⠀⠭⠥
⠀⠀⠀⠀⠀⣤⠹⡴⡻⢋⠝⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡸⢅⠻⡳⡼⢡⡄
⠀⠀⠀⠠⢜⡙⣺⡔⡘⠳⡶⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡄⠀⠀⠀⠀⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⠋⡐⣸⡚⡙⠤
⠀⠀⠀⢐⣺⠃⠞⠹⠣⢀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⠀⠻⠙⠆⢺⣒
⠀⠀⠀⠩⡶⢀⡞⢶⠆⢸⠀⠀⠀⢀⠤⠚⢹⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⣸⠙⠢⢆⠀⠀⠀⢸⠀⢶⠞⣆⠰⠮⠁
⠀⠀⠀⠐⢋⢃⡠⠀⠖⠈⡆⠀⢠⠏⠀⠀⠀⠱⢄⠙⠻⡄⠀⠀⠀⠀⠀⠀⡼⠛⢁⠴⠃⠀⠀⠈⢧⠀⠀⡎⠐⠆⠠⣀⢉⠃
⠀⠀⠀⠀⠗⢏⠴⠁⠂⠀⡇⠀⠋⠙⢦⣄⡀⠀⠀⠑⠒⢧⡀⠀⠀⠀⠀⣠⠗⠒⠁⠀⠀⣀⣤⠖⠁⠀⠀⡇⠀⠂⠱⣌⠗⠇
⠀⠀⠀⠀⠘⢋⡤⠂⠙⢃⢣⠀⠀⠀⠠⡙⢿⣝⡳⣳⠒⠖⠷⢤⢤⢠⠴⠗⠖⢲⡓⣛⣽⠟⡡⠀⠀⠀⢠⠁⠛⠑⠤⣈⠛
⠀⠀⠀⠀⠀⠋⡄⠚⠇⡤⡂⠘⡀⠀⠀⢙⠊⡝⠛⠽⢹⣨⣀⡄⡂⡂⣄⣪⣸⠭⠐⠋⣡⠞⠁⠀⠀⡘⠠⡢⡄⠛⠦⡘⠇
⠀⠀⠀⠀⠀⠀⠀⡠⢊⡸⡰⢇⠑⠀⠀⠈⡈⠈⠰⡂⠀⣀⣈⡉⠉⠉⣉⣀⡀⠀⠒⠉⠀⠀⠀⠀⠐⢁⠷⢸⣌⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣰⠳⡀⠀⠀⠀⠀⠁⠉⠀⠀⠘⠉⠁⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢳⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⠀⠀⠈⠐⠤⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡠⠔⠈⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣍⠀⠠⠁⠀⠀⠡⠀⢈⡥⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣐⣒⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠖⣦⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⡾⠛⡆
⠀⣸⠆⠉⠰⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠂⠀⢸⣇⡀
⣾⠁⢠⡾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⡆⠈⢷
⢿⣄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣼⠳⣤⡀⠀⢀⡼⠻⣦⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⣠⡇⠀⣸
⠀⠙⣧⠀⠙⢷⠀⠀⠀⠀⠀⠀⢠⠏⠙⢷⣄⡼⠁⠀⠈⠻⣤⠞⠁⠀⠘⣧⢠⡟⠈⢷⡀⠀⠀⠀⡼⠋⢠⡾⠁
⠀⢠⠟⠀⢀⣼⠁⠀⣰⠿⢦⣠⠏⠀⠀⠀⠙⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠹⠋⠀⠀⠈⣧⠀⠀⠸⣇⠀⠸⣇
⢠⠏⠀⢰⠏⠀⠀⣰⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠘⢷⡄⠈⢻
⢸⡄⠀⢸⡄⢀⣀⣏⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⠙⡇⠀⠀⠀⣿⠀⢨
⠀⢷⣄⠀⠻⣼⡉⠉⠀⠀⠀⢀⡔⢉⡟⠉⠙⣳⠀⠀⠀⠀⠀⠀⡴⠋⡏⠉⠉⣷⡀⠀⠀⠸⢿⡆⢀⣴⠃⠀⣼
⠀⠀⠙⠳⣦⡌⠁⠀⠀⠀⠀⢸⠀⠀⠳⠶⠖⢩⡇⠀⠀⠀⠀⠀⡇⠀⠉⠓⠛⢹⠇⠀⠀⠀⠸⠷⠛⠁⣠⡾⠃
⠀⠀⠀⠀⠿⣅⡀⠀⠀⠘⣆⠘⣄⡀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠶⣋⠆⠀⠀⠀⢠⣴⠶⠛⠁
⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠂⠀⠉⠉⣉⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⢸⣀⡞⠓⠚⠛⠻⣄⡞⠀⠀⣱⠀⠀⠀⠀⠀⠀⢠⣶⡛⠉
⠀⠀⠀⠀⠈⠻⢦⣤⡀⠀⠀⠀⠀⠘⢧⡀⠀⣠⠤⠤⣤⠴⠶⢤⡀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠸⢿⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠶⢶⡇⠀⠀⣴⣤⠀⠀⠀⢀⣄⠀⠀⠀⢰⣤⠀⠀⢻⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠇⢿⡀⣠⠞⠋⠉⢷⡀⣠⡞⠘⣧⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⠈⢷⠏⠀⠀⠀⠸⠟⠉⠀⠀⢸⡄⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠶⢾⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣤⣤⣤⣤⡾

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣶⣶⡋⠭⠙⢢⠴⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣯⣿⣿⣿⣿⣿⡄⣔⢈⡁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣏⡠⣀⡀⠠⠀⠰
⠀⠀⠀⠀⠀⣾⣶⠖⠋⠀⢠⣶⣶⠀⡀⢠⣄⡸⠿⢿⣿⣿⣿⠏⠁⠀⠁⠀⠀⣀⠡
⠀⠀⠀⠀⠀⠀⠔⠲⣶⣶⣾⣯⡹⣷⣠⣶⣭⣴⣆⠈⣩⣽⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⣂⣿⣿⣽⣿⣿⣗⣶⣟⡶⣥⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢨⣍⡡⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣴⠟⠁⠀⣼⣷
⠀⠀⠀⠀⠀⢺⣧⣼⣿⣼⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⠀⠀⠸⡿⣷⣾⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⠿⠻⠿⠻⢛⠁⠀⢀⡄⢀⣾⣄⣀⣀⡐⠀⡼⠿⣿⣿⣿⣯⣶⣤⣬⣻⣿⣿⠿⠟⣉⣠⣤⡿⠿⣻⣿⢟⣥
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⢠⣤⠀⢈⣽⢷⡔⠾⣖⠟⠛⣷⢫⣿⣿⣤⣿⣦⣷⣼⣯⣍⠛⣛⠋⠉⢉⡐⠚⠛⠿⣿⣷⣿⠿⣫⣴⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣥⣤⣿⣿⣿⣿⣿⣹⣏⠁⠆⢘⣗⢚⠭⡜⢿⣿⣿⣿⣮⣽⣿⣿⣿⣷⣬⣿⣶⣶⣿⣿⣿⣿⡿⠟⢁⣠⣾⣿⢿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣷⣿⣷⣭⠳⢶⡿⢛⡿⣿⣽⣞⡻⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣛⣫⣴⣾⣿⣿⣿⢷⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⢟⠿⣟⣿⢿⣿⣶⣭⣟⣻⣿⣿⣿⣿⣛⣽⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⠐⠒⠲⠦⢤⣄⡀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠸⡛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢀⣠⡤⠶⠒⡶⢞⣡⡴⠒⠋⠉⠉⠉⠉⠉⠉⠓⠢⣄⡀⠀⠈⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡤⠞⠋⠁⠀⢠⣾⠵⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢙⡶⠛⠀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⡴⣿⠁⣴⠶⠄⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢷⡄⠀⠀⡞⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⠋⢠⡏⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠀⠀⠀⠀⢀⣼⣤⡀⠀⠀⠀⠀⢀⣤⣤⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣀⡀⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢯⠀⠀⠙⣆⠀⠀⠀⣾⠀⠀⡷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠞⣉⡤⣤⡀⠀⠀⢸⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠙⠛⠙⢶⡀⢸⠀⠀⠀⠀⠈⠳⣄⠀⡸⠋⠙⢶⡝⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠙⢦⠀⢸⡀⡴⠋⠁⠈⠙⣦⢀⡤⠤⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⡇⠈⣆⠀⠀⣀⣠⣤⣸⡂⣇⠀⢀⡏⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⣷⡄⠀⠘⣦⣬⣖⢧⣤⣤⣤⣤⠀⢺⡀⠀⠘⠃⠀⠠⠶⠚⠛⠛⠛⠛⢿⡷⠀⠘⢦⡞⢠⠃⠀⠈⠁⠉⡓⠋⠀⠀⢀⡞
⠀⢰⡆⢀⡴⠚⠛⢾⠃⠀⢠⠉⠀⠀⡇⠀⣸⠟⠋⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠸⠁⠈⢳⠖⢛⡆⠀⠇⠀⢠⡶⠋
⡆⠘⡇⢾⠀⠀⣀⣼⡀⠀⢸⡀⢠⠀⠀⢰⡇⠀⠀⢠⠔⠋⠉⠉⢉⡱⠶⠢⢄⣀⣀⣀⣀⡴⠋⠀⠈⠁⠀⠀⢀⡤⠴⠋⢀⡞⠁⠀⠀⠀⣼
⠱⡄⢻⡄⠓⠚⠁⠀⠙⣶⠚⠀⠘⢦⠀⠈⣇⠀⠀⢸⡀⠀⠀⠰⠋⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠏⠀⠀⠀⠀⠀⣿
⠀⠙⠆⠙⠆⠀⠀⠀⠀⠈⠛⠻⡏⠀⠀⠀⠘⠧⣀⠀⠙⠲⠤⠀⢀⠀⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠀⠒⠂⣤⠄⠠⠤⠤⠀⠤⢄⠠⠀⠀⡰⠀⠂⠈⠙⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣿⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡠⠤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣤⡴⣚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠉⠓⠤⠤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⢳⣶⠖⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢤⣄⣀⠀⠀⠀⡸⢳⡀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡏⠈⠛⠲⣤⡇⢸⣇⣤⡶⠖⠚⢹⣟⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠙⠻⣄⠀⠀⡟⠁⠈⠛⠚⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⢤⣤⡬⠷⠦⣤⣸⣇⣸⡛⡄⠀⢀⣀⣶⠖⠉⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠈⠓⣆⠀⠀⠀⠈⣽⡟⠀⠀⢏⡹⠷⢖⣻⠶⠿⠿⠿⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠳⢤⣀⣀⠀⠀⠀⠀⣸⣃⣀⡤⢖⡽⠃⠀⠀⣸⣇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢬⣉⣉⣉⣉⣉⣉⡩⠵⠞⠋⠀⠀⠀⠀⠈⢧⡙⠳⢤⣀⣀⣀⠀⣀⣤⠞⠁⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣈⣉⣉⣁⣀⠤⠚⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣿⣿⡶⠿⠛⡿⠟⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢈⣡⣴⣿⠛⠁⠀⠀⠀⠓⠿⢿⡿⠿⠿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣇⡀⠀⣤⡤⠤⠄⠀⠈⡿⠓⣤⣠⣀⣔⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⡄⠀⠉⣶⡶⠦⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠒⠁⠈⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠆⣹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣰⣾⣶⣆⣀⣴⣿⣧⠐⠞⠃⠀⠀⠀⠀⠀⠀⠀⢠⣥⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⡆⠀⠻⣷⣽⣿⣿⣿⣿⣿⡶⣗⣛⠆⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⡿⣿⣿⣿⡁⠀⠀⠀⠀⢰⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡇⡇⢠⢀⣤⣟⣉⠙⠻⣿⣿⣿⡈⠁⠀⠀⠀⣀⣠⠄⠈⠀⣿⡿⠛⢁⣿⠻⢻⡣⡀⠀⠀⡀⢸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠰⣄⠸⣿⡇⣧⣾⣾⣿⣿⣿⣿⡆⣺⣿⣿⣿⣿⣾⣿⣿⠟⠁⠀⠀⠀⢿⣄⣤⣾⡇⢀⣼⣇⠱⡀⠀⣿⣾⢸⣿⠇⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠿⡇⠹⣧⢸⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⡿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⠇⢰⢧⡇⣼⠇⣸⠿
⠀⠀⠀⠀⠀⠀⠀⢠⢹⡷⣿⣌⣿⠏⠈⣹⣿⣿⣾⣷⣶⣤⣴⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⠞⢛⡿⠿⠿⣿⣿⣿⣆⢸⣾⢡⣿⣻⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡞⢿⣏⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠃⠀⠀⣤⣴⣾⣯⡶⠚⠁⠀⠀⠴⠾⠿⢿⣿⣾⣟⡼⢻⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢿⣾⡟⣹⣿⠛⠛⠿⣿⢿⡿⠙⣿⣿⣄⣀⣤⢾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣳⡿⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⣿⣿⣿⣶⣿⣶⣶⡿⠁⠀⠈⢿⣿⣍⠴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢋⣼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍⡻⣿⣿⣿⡿⡿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣨⣿⣟⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⡿⠟⢱⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡆⠻⢿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣐⢦⣀⣤⡟⠷⣄⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠤⠄⠀⠀⣠⡾⢻⣤⣀⡴⢂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢵⣿⡿⢗⣈⡳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣋⡺⢿⣿⡮⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣿⣿⣿⣷⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣾⣿⣿⣷⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠉⠟⣻⣿⣇⢀⣉⣏⡁⢰⣿⡟⠛⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣾⡟⠻⣷⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⣀⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣈⣁⣀⡀⠀⠀⠀⠀⢻⡆⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠤⠒⠉⠉⠉⠀⠀⢀⣀⣀⠀⠰⣦⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠑⢄⠀⠤⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⠔⡞⠋⠀⢀⠴⠒⠀⠀⠀⠀⠐⠌⠣⡀⠀⠑⣄⠀⠀⠀⠀⣠⠖⠉⠓⢄⠀⣀⠤⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡞⠀⢀⠀⡎⠀⠀⠀⠂⠀⠀⢀⠀⢀⠱⡀⢉⠇⠀⠀⢀⠔⢁⠖⢋⠑⡌⣯⠔⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢴⠋⠀⡇⠀⡤⠊⠉⠑⠶⠧⠀⠀⠰⠑⣬⠀⠀⢠⠎⡰⠉⠀⠀⡞⠙⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢄⠰⠘⠀⠀⠀⠀⠰⡀⠀⠀⢠⠀⠸⣇⡴⢁⠜⣀⡠⠜⠉⠳⢤⡠⠤⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢉⣙⠒⠂⠤⢀⣱⣵⠁⠀⠤⠤⠛⠉⠐⠒⠁⡀⠀⠈⣠⠇⠐⠁⠀⠀⠀⠀⠀⠗⡤⠆⢀⠤⠼⡄⠀⠄
⠀⠀⠀⠀⠀⡤⠔⠒⢺⣀⠀⡏⠀⡏⠉⠒⠠⢬⣉⡀⠀⠀⡄⠀⠀⠀⠀⠀⠈⣦⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢀⠆⠀⠀⢡⠀⠘⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠲⡷⠤⠦⠤⠼⠤⠭⢽⣶⡁⠀⠀⠀⠀⠀⢀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠰⡇⠟⠁⢀⠠⢄⠀⡆⠀⠃
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⣀⣠⣴⠃⠀⠀⠀⠀⠀⠀⢾⣿⣷⣤⣤⣤⣤⣶⠿⠟⠛⠿⣄⠀⠀⠀⠀⠀⠀⢰⢠⡴⠊⣁⣀⠀⡷⠃⢀⡌
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⣜⣈⢢⠀⠀⠀⠀⠀⠀⠘⢿⡿⠛⠉⠉⢉⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⡎⠀⠀⠊⠁⠀⡠⠁⡠⠋
⠀⠀⠀⠀⠀⠀⢱⠀⢲⠶⢤⡀⠈⢳⠱⡄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣀⣀⡀⢀⣠⣞⣁⡀⣤⠤⢤⠎⢠⠏⠢⣤⠤⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠈⡄⠘⡄⠀⠱⠤⠞⠣⡈⠢⡀⠀⠀⠀⠘⣦⣄⣠⠔⠉⠀⠀⠈⠉⠓⠠⠥⢄⡠⠖⠁⣠⢋⠔⠊⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠱⠀⢱⠀⡴⠋⣉⢙⣛⠀⠈⠒⠖⣛⡉⣩⠖⠃⣠⠂⠀⠀⠀⢀⡠⠝⠓⠶⢤⠴⠚⠷⠁⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠳⡀⠹⠊⠁⠀⡙⠢⠤⣀⣈⡁⠴⠊⠀⠉⠁⠐⡏⠁⢱⡤⢤⠴⠁⠀⠀⠀⠹⡀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠬⣒⠀⠠⠚⠢⠤⣀⡀⠀⠀⡴⠒⠤⢄⠠⡔⢻⣀⡠⠟⠀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠀⠞⠁⠑⣄⠀⢀⡠⠖⠀⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⠀⠀⠉⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣀⠀⠀⠀⡜⠀⠀⠀⢸⡏⡇⠱⠤⣄⡌⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠦⡼⠀⠀⠀⠀⢸⢗⡏⠚⠀⢨⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠐⠁⠀⠀⠀⠀⣼⣱⠃⠀⢀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡀⠀⣦⣿⡽⠀⠀⡎⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡠⢴⠚⡅⠀⠀⠀⠀⡠⠊⢹⡔⣿⣿⡇⠀⡜⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⠒⠖⠉⠉⠁⠀⠀⠘⡆⠘⡄⢀⡠⠊⠀⠀⠀⢻⢊⡿⢠⠎⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣨⣤⢾⠉⠀⠀⠀⠀⠀⠀⠛⢷⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣵⣤⣀⣀⣀⣀⣠⣤⣚⣬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢯⡛⠯⣯⣍⣛⣛⣿⣿⠷⡶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⡍⠉⠀⠀⠀⠭⠭⠭⠝⠛⠛⢛⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠄⠀⠤⠤⠤⠀⠒⠒⠀⠀⠉⠉

⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⢀⡾⡄⠀⢀⠐⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠈⠩⠉
⠀⡃⠉⠔⠀⠀⠣⠀⠀⠀⠀⠠⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⣻⠀⠀⠀⠀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠀⠰⢦⣾⠃
⠀⣧⣄⠈⢃⠁⡀⠀⠀⢂⠀⢀⠀⡆⠀⠀⠀⠀⠀⢖⡆⠀⠀⠀⠀⠀⠘⠋⠐⠀⠴⠵⠂⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠞⠉⠓
⠀⠂⠀⠀⠀⠲⠐⠀⠘⠀⠢⠐⠒⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠙⠆
⢀⣀⣀⠀⠀⠀⠱⣿⣆⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⣠⣄
⠀⠻⠉⠑⠶⠀⠀⠘⠿⠇⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⢑⠂⢄
⠀⠀⠐⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⡠⢋⠄⢠⠸⡀⢨⠀⠀⢱⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⡉⠩⣄⣐⠀⠰⠀⠀⢰⠀⢱⠀⡧⠀⠈
⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠊⠀⠀⠈⣙⣷⣄⠀⣀⠀⡄⠈⢰⡇⠀⠀⠀⠁
⠀⠀⠀⢇⡀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠈⢆⠀⠀⣀⣠⣿⣿⣿⠀⠳⡀⣴⣴⠟
⠀⠀⠀⠋⡴⠐⠀⢉⢻⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠳⢅⡛⠿⠿⠟⢁⣀⠠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⣠⣤⣄
⠀⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢩⣶⣦
⠀⠀⠀⠀⢀⠀⠤⠐⠀⠀⠈⣇⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠠⠀⠈⢷⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠐⠀⢁⣁⡼⠄⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠰⠒⠋⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠵⠒⠊⠉⠀⠀⠀⠀⠀⠐⠉⠀⠀⠄⠀⠀⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠟⠿⣿⢹⠃⣿⠃⢸⠸⠈⠀⠁
⣿⣿⣿⣿⣎⠻⣿⣟⢿⣯⠻⡿⡿⣟⢿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⡇⣨⣥⣤⡄⢹⠈⡇⡿⠄⣘⠀⠀⡄⠀⢠⡂
⣿⣿⢿⡌⢛⡢⠈⢋⠀⠈⣣⡑⠘⠘⠎⢻⣿⣿⣿⣿⣿⢿⡈⣿⣿⡏⠋⢿⣫⣽⣷⡇⢰⡇⢰⡆⠘⢀⠔⢋⠅⢸⠃⡄⢰
⣿⣿⣷⡤⠿⠛⣿⡾⠿⣿⣾⣷⣶⣄⣀⣀⠁⠹⠈⢹⣿⡌⢧⠘⣷⠁⠀⠼⣿⠙⢉⠀⣠⢗⠤⠀⡼⠁⠠⡉⢢⢸⡀⠧⣇
⣿⣿⣎⢻⣄⠱⡌⢧⠔⡖⢰⡉⣽⠿⣿⣿⣿⣿⣷⠦⡉⠑⠀⠄⢹⡂⠀⠀⠸⣦⢥⡆⠀⠀⠈⠓⢴⣀⠃⣁⠨⠞⠰⠖⡸
⣿⣿⡻⠑⠭⠒⠌⣺⠀⢁⡘⢧⠴⢲⣾⡟⠻⣦⣥⡜⠀⠐⢄⡈⠀⢧⠀⠀⠀⢸⣿⠁⠀⡀⠀⠀⢀⠱⡁⡏⠰⡶⠀⡞⠁
⣮⡽⠷⣼⠀⣿⠄⢹⣄⠐⠃⡸⠯⡤⠛⠳⣤⡏⣫⢹⣷⣄⠀⠀⢣⠈⠂⠀⠀⢹⣿⣧⣾⣟⢿⣟⢁⡀⠠⡉⠢⠴⠊⠱
⡈⠻⣆⠙⢷⣄⡆⢨⠹⣿⣮⡁⠀⣴⣤⣦⣉⣠⣿⣾⣻⡿⡅⠀⠀⢀⠀⠀⠀⠈⣿⡿⠉⠈⣿⣿⣋⣀⠀⠁⠰
⠻⣦⡈⢢⣴⡙⢷⡅⠑⠌⠻⣯⠀⣿⣿⣿⡿⣿⡟⢹⡇⠛⠀⠀⠀⠀⢡⠀⠀⠀⠸⠁⠀⣻⣿⡿⠋⠠⣀⣠⠆⠀⠀⠀⢠
⠱⢮⡙⢆⠈⠻⢄⠙⠂⠀⠀⠈⠓⠘⣿⣿⡇⠸⣿⡿⠀⢀⣀⢂⠀⠀⢬⠀⠀⠀⠸⢀⣰⣿⣿⠿⠂⠀⠀⠙⣆⠀⠀⢀⠬
⠀⠀⠉⠂⠀⠀⠀⠁⠀⠀⠀⠀⢀⡈⠨⠩⠶⢧⣼⣿⡾⣯⠙⠁⠀⠀⠈⢇⠀⠀⢣⢎⠌⣿⡷⠉⠀⠀⠈⠀⠛⠻⡿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⡁⠤⡀⠀⢀⣀⣽⣿⣯⠿⠉⠁⠀⠀⠀⠈⠘⢟⣪⣼⡇⠻⠋⠁⠀⠀⠀⡀⠀⢐⠀⢺⣞
⠿⣤⣶⣶⣀⣀⣤⣾⣿⣿⣷⡀⣶⠠⢀⣤⡤⢦⣿⣿⠇⠀⠀⠀⠀⠀⡰⣱⣷⡸⣿⣿⠇⠀⠀⠀⣠⣴⡎⢰⡀⠞⠀⡏⡿
⠀⠀⠉⠛⢿⣯⠉⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠐⢡⠻⡿⠇⠛⠩⠀⠀⣠⣶⠘⠋⡇⢸⡇
⠀⠒⢤⣀⣀⠬⠑⠢⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠉⠀⠀⣑⠄⠠⠀⣠⣶⣗⣿⠟⠀⠀⡇⠸
⡠⠬⡐⢽⠡⣶⠄⢠⠀⠀⠔⠄⠀⢂⡀⠢⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠹⡆⠀⠀⠹⣿⠟⠀⠀⠀⠀⠃
⢙⡄⢃⠈⣴⣾⣶⠜⢂⢠⡰⠀⠄⡀⠉⠘⠀⠀⠈⠳⣤⣄⣤⣴⡀⠀⠀⠀⢨⡀⠀⠀⠹⣆
⠀⠰⡀⡂⢿⣷⡿⠁⢘⡦⠤⠾⠮⣄⠀⠠⠀⠀⠀⠑⠌⠛⢿⣿⣟⠀⠀⠀⠺⢦⣤⣤⡄⠉⠄
⣈⠀⠁⠄⠈⠻⣿⣄⡐⠀⠀⠀⠠⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠙⠿⣆⠀⠀⠀⠀⠻⣿⢹⣾⡀
⠀⠉⠁⠀⢠⡴⠿⠋⠀⠀⣠⠆⠂⡁⠛⠛⠷⢾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠇
⠀⠀⠀⢐⠋⠀⢀⣐⣰⢟⠽⠔⠲⣖⢪⣶⠦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠢⠀⢀⡀
⠀⠀⣠⣦⡂⣼⣦⣁⡀⠏⠀⠐⠒⠁⣡⠐⠀⠀⠈⠑⠂⠀⠀⠀⣀⠀⠀⠀⠁⠀⠐⠀⠂⠈⠉
⡆⣴⣿⡿⠋⠀⠀⠙⠿⣿⡿⣿⣿⣷⢧⡀⠀⠀⠀⠀⠀⠀⠠⠴⣀⡑
⠋⠹⣋⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⡟⠁⡀⠀⠙⠲⣄⡀⠆⠐⠀⠉⠉⠻⣶⡆⣴⣦⠄
⠤⠀⠁⠀⠀⠠⢤⣤⣤⣤⣴⣦⣤⣇⣺⣇⣀⣄⡐⣾⣶⣦⣴⣾⣗⠠⣴⣦⣅⠙⠛⠂⠀⠐⠄⠀⢀⠤⠤⢤⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⠀⠀⠀⢀⣤⠀⠀⠀⠘⠃⠀⠁⠀⠤⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡧⠄⢀⡀⣾⠁⢀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠘⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠁⠈⠁⣾⣿⣅⣖⠀⠂⡀⠄⠀⠀⢀⣈⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⣀⠀⠀⠰⠽⣇⠀⠀⠀⠀⠉⠉⠀⠞⠋⣁⡀⠀⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⣦⣤⣼⣷⠀⠀⠀⠄⠑⠀⢰⣾⡏⠠⣄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⠀⠂⠌⠀⠈⠁⠀⠀⠀⠀⠀⠁⠈⠙⠁⣴⡅
⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣤⣟⣤⠼⠁⡄⠀⠰⣀⠀⠀⣴⡶⣇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣉⠁⠀⠀⡄⣀⢀⣙⠀⠀⣛⡷⠉⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⡖⠀⢀⣀⡀⢀⠴⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⠻⠿⢿⠋⢷⠻⣁⡟⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⢀⣤⠄⠈⠁⠈⠁⠀⠤⠼⠻⣿⠿⢱⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⠓⠀⠀⠠⠀⢌⣡⣦⡄⣤⣸⠇⠀⠀⠀⠀⠒⢀⣠⣞⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠟⠀⠂⢀⢀⣀⣨⡧⠉⣐⣘⡞⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠴⠊⠷⠋⣙⠋⠀⠀⠀⠀⠀⠈⠁⠙⠿⣟⠛⠁⠀⠹⣗⣶⣄⠀⠁⠠⣶⠋⠀⠀⠀⠀⣤⡖⠺⣿⣉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣤⠀⠋⠿⠿⡀⣄⠀⣾⣤⡀⠀⠒⠃⠀⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⠀⣠⣅⡬⠒⠉⠀⠀⠛⠠⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣹⠋⠀⠰⢤⡄⢠⣀⣀⣉⣼⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡴⠀⠐⠦⢁⡀⠈⠀⠀⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠒⠒⠀⠀⠀⠀⠚⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠙⠀⠀⠀⠀⠀⠤⣴⣭⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⠀⢀⠀⠙⠁
⠀⠀⠀⢀⡀⢀⣀⣈⢉⠁⣀⡀⠀⠀⠀⠀⠀⠀⠛⢉⣤⠀⠐⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠀⠦
⠀⠀⠀⠀⠀⠈⠁⠟⣀⣴⣾⣿⣶⣄⠀⠀⠀⠲⡖⢻⣯⢀⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠜⢿⣷⡄⠀⠀⠿⡌⠄⠈⢳⡄⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠂⠀⠀⡀⠀⠀⠀⢠⣄⠈⢧⠀⠙⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡍⠉⢀⡟⠀⠈⠳⡄⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢦⡤⠀⣸⡷⣴⣾⠁⠀⠀⠀⠙⢠⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠉⢁⠔⠈⠃⢹⡏⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣹⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢠⠏⠀⠀⠀⢸⡟⣤⠀⢸⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⣠⠚⠀⠀⠀⢠⡿⠀⠹⠀⠙⢀⡧⠀⡴⠁
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠟⠀⠀⠀⣰⠿⢃⠀⠀⠀⠀⠸⠇⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠒⠀⠀⠀⠀⠏⠀⠈⠀⠀⠀⡀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢀⠀⠀⠐⣮⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠤⠀⠀⠀⠘⢧⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠑⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠀⠀⢰⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⡀⠀⠀⠀⠀⢠⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣴⠃
⠀⠀⠀⠀⣠⠴⠒⠒⠛⠛⠒⢺⡆⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣾⣷⠀⠀⠀⠀⣾⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢰⡗⠒⠛⠛⠓⠒⠦⣄
⠀⠀⡠⠊⣡⠾⠛⠛⠙⠛⠛⢿⡃⠀⡍⠙⠦⣀⠀⠀⠀⠀⠀⠀⢳⡙⠃⠀⠀⠀⠀⠘⢋⡞⠀⠀⠀⠀⠀⠀⣀⠔⠋⢹⠀⢘⡿⠛⠛⠛⠛⠛⠶⣄⡙⢆
⠀⢰⢁⡞⠁⠀⠀⠀⠀⠀⠀⢸⠄⠀⡿⣦⡀⠈⠳⢄⡀⠀⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⡠⠞⠁⢀⣴⢿⠀⠨⡇⠀⠀⠀⠀⠀⠀⠈⢳⠈⡆
⠀⠘⡌⣇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠳⡌⠻⢷⣤⡀⠉⠑⢒⣩⢞⠃⠀⠀⠀⠀⠀⠀⠸⡷⣍⡒⠊⠉⢀⣠⡾⠟⢡⠞⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣸⢡⠃
⠀⠀⠱⠸⡄⠀⠀⠀⠀⠀⠀⣀⣙⣶⣄⠈⠑⠒⠬⠟⠛⠛⠋⣁⡞⢠⡀⠀⠀⠀⠀⠀⡄⢳⣈⠙⠛⠛⠻⠥⠒⠊⢁⣠⣶⣋⣀⠀⠀⠀⠀⠀⠀⢠⠇⠎
⠀⠀⠀⠁⡃⠀⠀⠀⡠⠖⣋⣉⣉⣉⠙⠻⢷⣼⠤⣦⠼⠶⢛⠟⠀⢼⡇⠀⢀⡄⠀⢸⡧⠀⠻⡛⠶⠧⣴⠤⣦⡶⠟⠋⣉⣉⣉⣉⠲⢄⠀⠀⠀⢸⠈
⠀⠀⠀⢀⠇⠀⢀⠞⣠⠞⠉⠀⠈⠉⠛⢶⣀⠈⠑⠒⠒⠚⠁⠠⣬⠿⠇⠀⢸⡇⠀⠸⠿⣄⠄⠈⠓⠒⠒⠊⠁⣠⡶⠛⠉⠁⠀⠉⠳⣄⠳⡀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⣾⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠧⠼⠦⠞⠛⠁⡜⠀⠀⣼⣧⠀⠀⢣⠈⠙⠳⠴⠧⠼⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⣳
⠀⠀⠀⠀⠀⠀⠸⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⠀⡸⠃⠘⢦⡀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠑⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⢄⡼⠞⠁⠀⠀⠈⠳⢦⡀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣇⠀⠀⠀⠀⢀⣠⠖⠋⠁⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠈⠙⠲⣄⡀⠀⠀⠀⠀⣴⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⡴⠋⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⠙⢦⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⠁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠈⢧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⠢⢄⣀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡁⠀⠀⠀⣠⣀⣳⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣥⠀⢤⣊⣀⢀⣽⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡀⠀⢸⣋⠈⣄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣄⣲⠩⣷⣞⣿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠀⠀⠀⠈⠓⡦⣉⠳⣌⣯⠼⠋⣉⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⣠⡴⡾⠛⣛⢼⣿⣷⣄⡀⠁⠁⠉⠀⣀⣠⣿⣌⣀⡀
⠀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠩⢿⣤⠤⠂⠁⠀⠿⣿⡋⠁⠀⠀⠀⠛⠹⣿⣏⠀⠉⠙⠛⠳⢦⣤⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣡⣤⠞⠉⠀⠀⠀⠀⠀⠀⣿⡇⠐⠂⣄⠀⠀⠠⠉⠃⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠟⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⣿⠀⠀⠈⠀⠀⠀⡤⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠂⡀
⠀⠀⠀⠀⠀⠀⢄⡁⠀⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⢸⠉⠇⠀⠀⡀⠀⠀⠰⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⢠⡄⡡
⠀⠀⠀⠀⠀⠀⠙⠿⠧⢯⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⡄⠀⠀⡀⠀⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠛⡧
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠘⣷⠀⠀⡝⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣧⡀⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⣧⠀⢳⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢶⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣀⡴⠀⠀⠀⠚⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠂⣸⠟⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡎⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣍⠉⠐⠒⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠉⠓⠦⣄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠈⠑⢦⣫⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣀⣀⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⠃⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠈⠉⠉⠓⠂⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠝⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⡞⠁⠀⠀⠉⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣩⠴⢋⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠒⠋⢣⠀⠀⠀⣦⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠊⠀⠀⣠⣤⠀⣆⡄⠀⢻⡀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢼⣧⣴⡎⠁⠘⡄⠀⠧⠀⠀⠀⠀⣈⢦⠀⠀⠀⠀⠀⠠⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣀⣀⠀⠀⠀⠀⠀⠉⠒⠐⠀⠀⠀⠁⠀⠀⠀⢸⠁⠉⠀⠀⠘⡄⠀⠀⠀⠠⡮⣤⠀⠀⠀⠀⠀⠀⣶⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠂⠀⠀⡀⠀⠀⠀⣀⠑⠂⠀⠸⠂⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱⡖⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠈⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠣⣈⠗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠳⢮⣱⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠒⠦⠤⠤⠄⠀⠔⢲⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡌⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⠃⠀⠀⠀⠀⡀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠉⠖⠲⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡧⠤⣄⣀⣀⣀⣀⡇⣼⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢠⠃⠀⠈⠓⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠋⠁⠀⠀⠈⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢂⢾⣶⡶⢲⣦⣤⡄⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠻⠷⠷⠿⢿⠶⡦⠴⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡢⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠛⠁⠀⠀⠨⠿⡶⣄⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣀⣀⣀⣀⡠⠐⢀⠀⠈⢢⠀⡠⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢊⣉⡉⢥⡤⠀⠀⢀⡤⠀⣉⡑⢢⡇⠑⠀⡀⠀⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢾⣿⣿⣿⣶⡦⠰⣶⣾⣿⣿⣿⠂⣱⠀⠘⠴⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢒⡌⠻⣿⣿⡿⣁⣠⢻⣿⣿⠿⠋⣀⡘⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠆⠨⠁⠊⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠤⡳⢤⣀⣀⠀⠁⠀⢀⣀⡴⠋⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠍⠀⠀⠀⠀⢈⣶⠞⠻⢏⣉⡹⠟⠒⢦⢀⠄⠀⠄⢀⠀⠀⠀⡀⠤⠀⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⡀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⣸⠈⠀⠀⠀⢄⠁⠀⢀⡈⠲⣤⣀⡀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠢⢀⢾⡄⠀⠀⢠⠂⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠿⠳⠈⠉⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠘⢀⣀⣀⣶⡀⣀⡴⢄⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠄⠀⠀⠀⠘⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⢁⠐⠉⠀⠀⠈⠉⠡⡀⠘⣾⠁⠀⠀⠀⠀⠀⠀⣇⡼⡟⠐⠈⠀⠭⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⡞⠀⠀⠀⠀⠀⠉⠀⢻⢿⣅⡀⠀⢀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠴⢤⣁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡏⠀⠀⢀⣿⠀⠀⠀⢸⡄⠉⠀⠘⠉⠀⠀⠀⠂⠈
⠀⠀⠀⢠⠊⠀⠀⠀⡈⢳⠀⠀⠀⠀⠀⠀⡠⢺⠁⠀⠀⢸⠸⡀⠀⠀⠀⣇⠠⠀⠀⠂⠈⠈⠐⠄
⠀⠀⠀⠰⡀⠀⠀⠀⠀⢹⠀⠀⠂⠒⠒⠊⠁⠛⡂⠀⠠⠞⠀⣇⣀⠀⠠⡼⠀⠀⠀⠂⠐⠀⠀⠆⡀
⠀⠀⠀⠀⠈⠃⠤⠤⠶⡹⠉⢉⠃⠁⠀⠀⠀⣠⣖⡄⢠⠀⠀⠀⠀⠀⢀⢴⡂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡈⠀⡠⣀⠐⠊⠀⠀⠀⣠⠴⠃⠎⠀⢀⠀⠀⠀⠀⠃⠉⢛⠙⠷⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⠂⠐⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡷⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁

⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⡀⠀⠀⠀⠀⠈⠙⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢰⣿
⣿⣿⣿⣿⡄⠀⠀⠰⢶⡶⠶⠶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠉⠉⢿⡉⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⡿⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⡇⠀⢀⣤⣾⣿⣿⣿
⣿⣏⠉⠙⠛⠛⠀⢰⡇⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣿
⣿⣿⣷⣤⣀⠀⠀⠈⢷⠀⠀⠀⠘⢿⡿⠟⠀⢠⣤⡄⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠃⠀⠀⢀⣴⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠖⣺⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⡤⠚⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠴⠛⠒⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠦⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⠀⠀⠀⠀⣿⠃⢰
⠀⠀⠀⠀⠀⡄⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠈
⠀⠀⠀⠀⠾⣁⡷⠒⡌⢑⠄⢀⠀⠀⠀⠀⣸⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠐⠒⣥⡄⣼⡟⢺⠟⣡⠂⠀⢠⠛⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠇⢘⡛⠉⢡⣀⡨⠀⠀⠀⠸⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣧⠀⡐⢠⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡇⢀⠔⠁⠀⠀⠃⠀⣸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠀⠀⢹⡄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠸⡷⡔⠟⠳⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⡇⠀⡈
⠀⢀⢀⢖⡀⠠⡁⠰⣶⣦⣤⣤⣤⡴⣾⠀⠀⠠⠠⠁⡀⢀⣴⣧⡌⠀⠀⠊⠑⠀⠋⠠⠀⠀⠀⠀⡰⡄⠀⠀⠀⠆⠀⠀⠀⣸⡇⠀⠃
⠀⣿⣷⣦⠹⡄⢙⠶⣍⡳⣀⠀⠀⠀⠘⣄⣤⠖⡠⢊⡙⢛⠀⢀⣙⣀⡈⠀⠀⣎⠀⠘⠆⢀⠖⠀⠐⣄⡠⠀⠀⠣⡀⢠⣴⡿⣧⢦⡇
⠀⣿⣿⡟⠀⠃⢸⠀⠈⠑⢤⣳⠀⠀⠀⢿⣧⠂⠀⠀⢁⢡⣸⣿⣿⣿⣿⡆⠀⢻⠀⠀⠀⢆⣠⡀⠀⢸⣧⡄⠀⠀⡆⣸⣿⣇⠀⠀⠰⡀
⠀⢹⣿⡟⠛⢶⡟⠀⠀⠀⠘⠉⠀⠀⠀⢸⡜⣧⠀⠀⡸⠀⠸⣿⣿⣿⣿⠃⠀⣿⠀⠀⡀⠘⠆⠓⠀⠘⠁⣄⡀⠐⠁⠓⢿⣿⣯⡀⠰
⠀⠘⣏⠁⠧⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⣷⣶⡾⣦⡄⣠⠄⠉⠛⠶⠤⡏⠘⠦⠀⠱⣔⠙⠂⠠⢼⡇⠀⠁⠀⡇⢐⠀⠀⠈⠙⠻
⠀⢀⠛⠲⢾⡜⢣⠀⠀⠀⠀⠀⠀⠀⠀⣽⠯⣉⣽⡷⣿⣇⣧⠀⠀⠀⢠⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠐⠀⠀⠀⡹
⠀⠈⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠛⢿⣫⠭⣿⣿⠀⠀⠀⠀⠖⢚⠀⠀⠀⠀⠿⠄⠀⠀⠀⠐⠐⠀⠀⠈⡗⠂⠂
⠀⢀⠆⠀⠀⠀⢸⠛⢳⣶⡤⢴⣶⠒⠋⢱⡗⠀⣿⣿⣿⣿⣿⣤⣀⣀⣀⣀⢨⠀⠀⠀⣰⠈⠁⢰⢂⠂⣠⡦⣤⠴⠏⢰⡄⠀⠐⠒⠞
⠀⠘⠓⣶⠶⢶⠁⠀⢰⣴⠀⠀⢐⠀⠀⢾⣿⣶⣿⣿⣿⠇⠈⡀⢀⡀⠀⠹⠞⠓⠶⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣼⣄⢸⡀⠶⢛⣿⡀⣷⣿⣀⡀⠀⢹⣩⣋⡏⠀⠀⠀⠁⠈⠃⠀⠸⠢⠦⠤⠖⠃⣀⣄⣀⢀⣄⠀⠠⣤⢄⢠⣄⡀⡀
⠀⠀⠀⠛⠛⠉⠛⠁⠘⠻⢿⣿⡟⠻⠿⠂⢸⡿⣇⡷⢤⡀⠀⡆⠀⡆⠀⢲⣤⡖⠐⣲⡄⠡⠽⠿⠶⠦⠥⠶⠙⠋⠉⠋⠁⠁⠈⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠭⣉⡻⠿⠀⠙⠒⠻⠔⠋⠁
⠀⢀⠔⠁
⠔⠁
⣾⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⣹⣶⡀⠀⠀⠀⠈⠀⠀⠤⢄⣀⣀⣀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⠀⠀⠸⡿⠿⠟⠳⠟
⠀⠀⢠⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢰⠗⣶⣶⣾⠟⠉⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⣲⡿⠻⠆⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⡉⣥⠤⠤⠴⠠⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⡿⣿⣿⢻⡟⠻⢿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⠿⠟⠷⠿⠏⠼⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⠖⣆
⠀⠀⠸⣿⣿⣿⣿⣷⠀⠄⠀⠀⢀⡴⠛⠀⠐⠂⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠛⣋⠁⠆
⠀⠀⢠⠿⠿⣿⣿⣇⠀⠈⠠⠴⠋⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆
⠀⠀⢸⣄⠠⠘⣻⡟⠂⣠⠇⠀⠀⢠⣿⣿⣿⠶⠄⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢸⣿⣦⣀⣿⡟⠞⠁⠀⠀⠀⠀⠉⣿⠃⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⢀⣠⣤⡀⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣄⡐⠄⠈⢻⣄⣸⣿⣿⣿⣿⣿⣴⠀⢀⣀⠿⢿⣿⡟⠿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠃
⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⠆⠀⠳⣌⣿⣿⣿⣿⣿⣿⣏⣷⡉⠛⠀⠀⢩⠀⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠹⠇
⠀⠀⢿⣿⣿⣿⣿⣿⠀⣠⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠸⠁⠀⣈⡆⠀⠀⠀⠂⡠⠀⠀⠀⠀⢀⡤
⠀⠀⢸⡟⠛⠛⣿⣿⡿⠙⣿⣆⣠⡄⠀⢬⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣀⡄⣼⠋⣠⡖⠀⠀⢰⡇⠀⢠⡤⠴⣿⠀⠀⠀⣦⣀
⠀⠀⠠⢭⣴⣶⣿⣿⣤⣦⠉⢹⣿⣿⣷⡀⠀⠀⠿⣿⣿⠿⣿⣿⣿⣿⣿⣇⢓⠀⢿⣶⠶⠶⠌⠳⡄⠀⠳⠚⣿⠃⠀⠀⣿⠙⠻⢦⣄⡠⠀⣠
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⣌⠛⠛⠿⠿⣿⣷⣿⠁⠀⢹⣿⣿⣿⣿⣿⠌⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠸⠋⢀
⠀⠀⠀⠀⢸⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣶⣦⣴⡇⢸⠀⠀⠀⣿⣿⣿⣿⣿⡆⣰⡟⢀⣀⣀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠏⢹⣿⢿⣿⣿⣿⠀⠘⠹⣧⠀⠀⠀⢠⡏⠀⠀⠙⠷⡄
⠀⠀⠀⠀⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠀⠀⠀⠀⣬⡁⠸⡇⢹⡏⠀⠀⠀⢹⡂⠀⢠⣞⡁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠒⠆⠀⠺
⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣴⣧⡇⠀⠀⠀⣽⡆⠀⣶⣼⡃⢀⠀⢀⣼⠃⢀⠿⠋⠁⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣠⣿⣿⣀⢸⣄⠋⣿⠀⢈⠠⠤⠀⣀⡀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠙⠿⣿⠴⠤⣤⣉⠝⠛⢻⣿⣿⣿⣿⣿⡇⠀⠀⣀⣿⡿⠋⠈⢛⣯⠉⠉⠉⣻⣤⡦⠀⠀⠀⠀⠉⠱⠷⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠛⠀⠈⠛⠲⠻⡯⠻⣿⣿⣿⣶⣾⠟⠁⢶⣤⣠⡾⠋⠁⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠠⡢⠀⢀⠄⠠⡖⠐⢦⠤⢄⠀⠀⠀⢀⡐⡤⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⡠⠒⠆⠀⠘⢄⠀⢀⠄⠀⡰⡷⢎⣽⡎⢹⡯⠖⣢⡀
⠀⠀⠀⠀⠖⢑⠔⠈⠉⢉⡉⠢⣠⠦⡀⠀⠦⣁⠌⠀⠀⠀⠀⡖⣄⡘⠄⠿⠀⠏⣀⢀⣀⠙⠀⠀⠑⠀⠁⠰⢷⣶⣤⠚⡏⢩⠁⡩⠛⡍
⠀⠀⠀⡜⠉⠃⠀⢼⢹⣿⣽⡆⠐⣀⣄⠀⠀⠁⠀⡄⠐⠀⠀⢸⠼⢃⠁⡆⠀⡘⡧⠸⣧⢈⠀⠀⡁⠀⢀⣷⡶⢁⣀⢣⣰⣆⡬⠤⢐⡦
⠀⠀⠀⡘⠙⡀⠐⢨⠚⠫⠎⠅⢀⣅⡠⠂⢀⠀⠀⠐⠒⠁⠀⠀⠘⠄⢀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⠈⣄⡥⠀⢒⣸⡇⠀⠱⡿⠋
⠀⠀⠀⠘⢰⠑⣄⠀⠑⠒⠀⡠⣎⣀⠅⠀⡄⠫⡳⠀⠀⠀⠀⠀⠁⠀⢁⢄⠀⠁⠀⢀⡀⠁⠀⠀⠁⠀⡀⠈⠉⠙⠛⠂⠙⡄⠀⠃⠀⠀⣲⠆
⠀⠀⠀⠀⢈⡒⠦⡈⣫⡁⢩⠤⠜⠀⠀⠀⠉⠁⠀⢀⠀⠀⢄⡥⠑⠀⠀⠀⠀⠀⠀⠡⡠⠁⢠⠄⣀⡔⠉⣆⡀⠤⠀⠀⠀⡄⠀⢸⠀⠎⡱
⠀⠀⠀⠀⢘⣻⡧⠤⢼⠉⣡⠪⢉⡜⠀⣀⠀⠀⠀⠡⠺⣇⠘⠳⡀⠀⠀⠀⠙⢶⠀⠀⠀⠀⢈⡖⢁⣄⢠⣤⡑⣼⠀⠀⢠⠀⠀⡸
⠀⠀⠀⡾⠽⢇⣀⠀⢀⡖⠁⠂⠁⢀⡎⡀⢑⡀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⡂⠀⠢⣁⠀⠳⠬⠑⠤⠃⢀⡈⠆⠀⠑⠂⠱⣲
⠀⠀⠀⠀⠀⠘⠭⠋⡸⠀⠀⠀⠀⡜⡗⣱⢫⠱⠀⠀⠀⠀⢀⠀⠡⠀⠀⠀⡀⢀⣀⡁⠄⠀⢀⠃⢁⠈⠥⢀⡠⠫⠀⠀⠀⠀⡂⢒⣸⠀⢢⡔
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠀⠀⡁⠑⠆⢚⡀⢃⠀⠀⢠⣏⢱⡖⠃⠀⠈⠐⢹⣧⠿⡀⠁⠈⠈⠉⠣⣀⠋⠉⠀⡠⠀⠀⢀⠀⠀⢰⢉⡢⠄
⠀⠀⠀⠒⠩⠀⠀⠁⠀⠀⠀⠀⠀⠑⢼⡟⠀⡿⠋⠀⢀⠔⡩⠀⠉⠄⠀⠐⠄⠁⠀⠀⠄⠀⢀⡀⠀⠀⠄⠀⠂⢰⠀⠀⠀⠀⠆⠐⠀⠠
⠀⠀⠀⠀⠀⠀⡀⢤⡊⠭⠑⣄⠀⠀⢀⠃⠀⡇⠀⠀⢶⡌⠀⠁⠊⠀⢀⢀⡄⣠⢀⠀⠀⠈⠂⠄⠀⠨⠀⠀⠀⠀⠆⠀⠀⠐
⠀⠀⠀⠀⡠⠺⣘⣰⣄⡙⣤⢉⠣⠀⠘⠄⠠⠃⠀⢀⠀⠀⠀⠀⠀⠠⠋⡤⣤⠴⣌⠓⣀⣀⠀⠀⠠⠀⠓⠀⠄⡀⣪⡉⠉⠣⣀⠀⠀⠚
⠀⠀⠀⠀⢻⣁⣲⡾⢪⡶⠭⢦⢺⠀⠀⠀⠀⠀⡰⠁⡆⠀⠀⠀⠀⢛⠀⠐⡿⠙⡿⢸⠈⡻⠀⠀⠆⠀⠀⠀⠠⣆⠿⠀⠀⠀⡱⠀⠀⠀⠀⠸
⠀⠀⠠⡢⠀⠠⠋⢀⠚⠛⠛⠓⠂⠀⠀⠒⡀⢨⠔⣶⣸⠀⡐⠀⠀⠸⡄⡀⢐⠀⡂⣨⠋⠁⠀⠀⠈⠂⠄⠀⠈⠈⢢⠤⠤⠔⠀⠁⠐⠠⠐⠁
⠀⠀⠀⠀⢀⠃⠀⡆⠀⠀⠀⠔⠤⢐⠀⠀⢈⠃⠜⠛⠛⠋⠀⠀⠀⠐⡁⠀⠀⡁⠈⠙⠀⠀⠔⡀⠀⢈⠀⠀⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠘⠄⠔⠁⠀⠀⠀⠈⠐⠁⠀⠀⠦⠐⠂⠙⠢⢬⠀⠀⠀⠀⠓⠉⠉⠉⠉⠃⠀⠘⠉⠙⠀⠀⠑⠀⠀⠚⡀⠀⠀⢀⠓⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⢢⣔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⠤⢤⠀⣨⠉⠀⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⡌⢀⣀⣀⠀⡠⠋⠀⢠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⣀⣀⣀⡧⠚⠓⠒⠛⡄⠀⣼⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡹⠿⢍⣁⡡⠿⠜⠁⠀⠹⢋⡭⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣇⣇⣼⠀⠀⢀⡤⠤⠄⠒⠒⠊⠉⠀⠠⢰⠆⠈⠀⢨⠏⣴⠀⢠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠉⡲⠤⠃⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⣠⡋⢲⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⠁⠀⠀⠀⠀⠓⠦⣄⣀⣀⠤⠔⠉⠀⠀⣠⡔⠧⣨⠏⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢣⣀⡠⢤⠤⣄⣀⡤⠧⠀⠀⠀⠀⠴⠚⢅⣉⠶⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠁⠀⠈⢧⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠀⠀⠀⢀⣠⡃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢖⠙⣄⣀⢸⡴⠊⠑⢦⡀⠀⠀⠀⠀⢀⣀⠤⠖⠂⠉⠉⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠢⠤⠚⡞⠀⠀⠀⠈⠦⠤⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⣠⡞⣁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡠⠀⠴⠋⠀⠀⠀⠀⠀⣀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠩⣛⣅⠄⡰⡖⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣡⣦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢫⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠈⠙⣍⣩⣯⣕⡻⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⠉⠛⠷⠦⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠠⢄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⢊⢀⡾⡇⠨⢂⡪⠈⠡⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠘⣳⡀⡗⠻⠛⠛⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⠀⠢⠅⡀⣰⣏⡀⡂⢡⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⢀⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⣇⣠⠉⡼⠁⡷⠧⠒⢼⡀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢄⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠗⢻⠋⠀⠀⠀⠀⠠⢁⡒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠼⡀⣂⢂⠀⠉⠀⠀⢰⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠋⠁⠀⠀⠙⠀⠠⡄⠀⠀⠀⠀⠀⠀⢧⠈⠉⡀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠓⠢⠈⠆
⠀⠀⠀⠀⢀⠤⠒⠉⠀⠆⣀⠀⣤⠶⠒⠀⠀⠀⢀⣄⡀⡀⠄⠀⣹⡟⠁⠄⠀⠒⠈⢣⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⡈⢄⠀⢠⡀⠋⠁⠀⠀⠀⠀⠈⠀⠀⠀⠠⠒⠚⠁⣅⠀⠀⢀⠆⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⠀⠂⡤⢰⠶⣦⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠈⠦⡄⠀⠘⠁⠄⠀⠀⠂⠤⠀⠀⠁⠠⠰⣂⡾⠃⢄⠀⠀⠂⠐⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠄⠀⠠⠀⠌⠒⣛⠷⣦⡀⠀⣡⡤⠖⡹⠇⠀⠐⠀⡆⠁⠀⠀⠀⠥⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠶⠶⠷⣶⣶⣅
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡙⠷⠼⡋⢩⠐⠒⠊⠀⠀⠀⠖⠋⠂⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠢⢀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡠⡈⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠀⠀⠐⢬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠐⠠⠄⠊⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡾⠕⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠆⠄⠰⡇
⠀⠀⠀⠀⠀⣠⣶⢫⠀⢠⣠⣤⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠷⣿⣿⣿⣅⣀⡀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⠘
⠀⢀⣀⠀⢾⠛⡾⢧⣠⠌⠺⢏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⠃⠀⠓⢀⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⠀⢻⣦⠀⠄
⠀⠈⢙⣿⣟⣥⠇⢒⡵⠋⠁⣱⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠆⠋⠛⠷⢈⠻⠄⡀
⠀⠀⠀⠙⠻⢾⣖⣭⣹⣧⡿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡐⠐⠂⡀⠚⠁⠀⠀⠀⠡⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠙⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣄⡀⠀⠀⠈⠀⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠑⠤⠀⠀⠉⠑⠤⠤⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣾⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠃⠀⠀⠀⠀⠀⠞⢹⡋⠳⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠃⠌⠡⠘⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡱⠀⠀⠀⠀⠀⠀⢎⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⣠⣾⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣷⣄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⢀⠾⣿⡟⠓⠒⠏⢹⠃⠀⠀⠀⠀⠀⠀⠘⡋⠙⠒⠚⢿⣿⠷⡀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⣸⡷⡆⠀⠀⠋⠁⠀⠂⣄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⠀⠀⠈⠙⠀⠈⢰⢾⣇
⠀⠀⠀⠀⠀⠀⠰⣟⠥⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠘⢿⡿⠃⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣷⣖⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣛⣛⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣴⣴⣦⣴⣦⣴⣶⣦⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢀⣤⢄⡀⠀⢠⣧⡀⠔⠜⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡶⢿⣅⣤⣤⣑⢾⢣⡂⢀⢀⣤⠂⠁⠀⠀⢀⣦⡀⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠐⠒⠿⡚⢚⢿⣵⢬⠿⢟⣤⡬⢘⡿⣵⣧⠎⠀⣠⡤⠶⠞⢁⡈⠛⢦⡀⠘⠄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⢶⢶⣶⣦⠈⠲⣵⣽⣿⢠⣾⣷⣼⣿⣟⡿⠃⢠⣿⣿⣷⣶⣤⣄⡈⠲⣤⣉⠲⡄⠀⢄⠀⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠃⠸⡇⣼⣷⣄⠀⣽⣿⡯⠍⣠⣹⣿⢟⡁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡆⠀⠀⢳⡆⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⢰⠇⠀⣼⣿⣦⡈⢻⣿⠟⠛⠿⠿⣿⢄⣼⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣇⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⡇⠘⠛⣿⣿⣿⣶⠇⠀⢀⣀⡀⠀⠙⢅⠁⢛⠛⠿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣿⢱⡈⣿⣿⡿⣿⣿⣿⠏⠀⣨⣿⣿⣷⡀⠀⠀⠠⣽⡆⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠛
⠀⠀⠀⠐⠀⣀⣬⠽⠶⠛⢉⣿⣟⣿⣿⣿⣮⣿⣿⡟⠀⣴⣿⡏⢃⢻⣿⣦⣀⠀⣻⣷⡀⣤⣤⣤⣌⣻⣿⣿⣿⣿⣿⠿⠟⠋⠔
⠀⠀⠀⠀⠋⠉⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⠟⢠⢾⣿⣯⣽⣽⣭⣿⣿⣿⣿⣿⣯⡙⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡟⣨⣧⣌⣡⣿⡿⠏⣀⣏⡘⠛⣿⠻⢣⠙⣿⣿⣿⣿⣿⣿⣯⣦⣨⣿⢿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠀⠈⠳⣅⡁⣀⣀⡀⢫⣿⣿⣶⢹⣧⠀⠀⢸⠿⣿⣿⣿⣹⣾⡻⣿⠛⠄⠹⣿⢿⠙⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠀⠀⠉⠃⠛⠿⣹⠿⠃⡌⠛⣷⡀⢈⣢⡙⣮⣿⣿⣿⣿⣽⡆⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠋⠀⠀⠇⠀⢸⣿⠛⠙⠿⠿⠿⣿⣿⡿⡟⢇⣠⣶⣶⣶⣖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢻⣄⠐⢤⠀⠀⠉⠛⢷⠀⠈⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣷⡀⢣⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠈⢿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣤⣤⣤⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⠿⠟⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣯⠶⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠓⠖⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠙⠙⠁⠛⡉⢙⠘⠛⠘⠋⠃⠋⠛⠙⠋⠙⠛⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠚⠛⠛⠛⠙⠀⠟⠛⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⠂⠀⠤⠀⠀⢡⠈⠒⢄⡠⢢⣶⡆⠈⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠁⠀⠠⠂⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠂⠀⠀⡤⢰⣾⡆⠀⢠⣀⣄⢂⠀⠀⠠⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣂⠀⣅⠸⣿⡇⣀⢀⠛⠛⣄⠇⠀⠐⠄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡄⢀⠘⠚⠉⠄⠄⠀⠈⠁⠶⠛⠀⠖⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢈⠖⡀⠀⠀⠀⠀⠒⢄⡀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠰⠥⠀⠀⠀⠀⠀⢀⡳⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣶⣉⣀⡬⠘⠷⣁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠖⣴⠟⠉⠈⠉⠁⠓⢾⡟⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⡠⣤⣄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⣑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠉⢐⠓⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⣿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠓⠒⠒⠢⡀⠀⠀⠀⠀⢀⣀⠤⢄⡀⠀⠀⠀⠀⢹⠀⠻⠄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣰⠋⢁⠀⠀⠘⢦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⢠⡖⠀⠀⠀⢸⠀⠀⢰⠇⢰⡏⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢇⠀⠘⠁⠀⠀⠀⡟⠀⠀⢹⡀⠀⠀⠀⠀⣠⠴⠃⠀⣠⠟⠛⠓⢦
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣇⣠⠆⠈⢷⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠘⠓⠒⢶⠖⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠉⠉⠛⠶⢄⣀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠈⠙⡦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠖⠛⠁⠀⠀⠀⠀⣟⡠⠒⠓⠒⠒⠲⠀⢤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠢⡄
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣇⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⣀⡀⠀⢀⠞⠻⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣤⣐⠿⠟⠋⡉⢹⡷⠀⣤⣾⣆⣉⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢴⣷⡘⠿⢻⡿⣀⢠⡦⣼⣉⡀⠀⣽⢾⡙⠋⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠈⠁⠘⠻⠃⠲⠼⢦⢻⣿⠃⢸⠏⣸⡇⠀⣠⠘⠻⣿⡘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠛⠳⣷⣀⣼⣷⡄⠁⢰⣄⠘⠿⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠹⡋⢸⣿⢾⡇⣀⠛⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠅⠀⢤⣉⡼⠟⠛⠻⡆
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⣴⡾⣿⣅⣴⠟⠀⣱⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣴⣯⢀⣤⡝⠛⢦⢂⣘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡀⠀⠀⠀⠀⠠⠋⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠶⠒⠀⣀⠆⠈⠉⠙⠿⠿⢿⣦⠀⠸⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⣈⣁⣐⣶⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣦⡌⠻⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡉⠉⢀⠀⠀⣉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠒⠻⣿⡃⣀⣨⡿⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠉⠁⢶⡔⠀⠀⠀⠐⢀⣬⣅⡙⠉⢰⣿⠁⣶⣿⣷⠈⣿⣆⠀⠀⣾⣇⠀⠀⠀⣿⠋⢿⣿⠁
⠀⠀⠀⠀⠀⠀⢤⣴⣾⠆⠀⠀⠀⠀⠈⢻⡆⣀⣠⣶⣿⡿⠛⠃⠘⠛⠁⠀⠛⠃⢈⡤⠸⣿⡄⠀⢨⣽⣿⠀⠀⠉⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣾⡀⠀⠀⣤⣄⣄⠘⣿⢻⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⡅⠀⢀⠿⣏⠁⠀⣶⣶⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣙⣿⡄⠀⠀⠘⢛⡀⣿⡌⣯⣤⡆⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣀⠚⠁⣰⣶⣿⠟⠈⠀⠻⣿⠞⠿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⣀⣀⣾⣇⣿⡇⡿⠛⠁⠤⠀⠀⠀⠀⠀⠀⠀⠙⣄⠹⠟⠀⠀⠀⠀⣤⣶⠂⠀⠀⠁⠸
⠀⠀⢀⣀⡀⠀⢀⣠⣿⣿⡿⠈⠻⢿⣿⣿⣿⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⣈⣷⣶⡿⠃⠀⠉⣀⣤⣾⣿⣿⡇
⣴⣿⡟⠁⠀⠘⢯⣙⣿⣿⠃⠀⠀⠀⠙⠛⠁⠀⢹⡟⠀⠀⠀⠀⣠⣤⣚⢻⣿⠟⠓⠀⠀⠤⣿⣿⣿⣯⣭⠀⢀
⣿⣿⣧⣤⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⣠⣄⠀⠀⠘⠃⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⢀⣴⣶⣶⣶⣮⣽⡿⠛⠀⠈⡀
⣿⣿⡛⢥⡀⠔⠿⢿⣿⣿⣧⠀⠙⢿⣟⠛⢋⣣⡀⠀⠀⠀⠀⢀⠿⠋⠀⠀⠀⠀⠀⢹⣿⣿⠿⡟⠛⠃⣴⣿⡿⠄
⣿⣿⡇⢀⡹⠆⣴⣾⣿⣿⡟⠈⢿⣿⣿⠀⣸⣿⣧⣴⣤⣤⡎⠁⠀⠀⠀⠀⠀⢀⠀⣀⣿⣿⣶⠁⠀⣾⣿⣿⡇⠐⡆
⣿⣿⡇⠘⡟⢴⣿⣿⣿⠏⠀⢦⡘⠿⣿⢱⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⣤⡇
⣿⣿⣷⣤⠀⣿⣿⠿⠿⢃⠀⠀⣸⣦⡉⢠⣿⣿⣿⣿⣿⣘⠗⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠏⣴⡇⠀⠛⠿⢿⣿⢿⡇
⣿⣿⣿⣋⣿⣿⠏⣠⣴⣿⠃⠋⠀⣿⣿⣾⣯⠛⢿⣿⣿⡟⣠⡀⢠⡄⢹⣿⣿⣿⣿⣿⠏⠀⣿⣷⢀⣤⣴⣶⠠⣬⠋
⣿⣿⢁⣿⣿⣏⣰⣿⠿⠃⠀⠀⢀⣿⠟⣡⣿⣿⣷⣝⠛⡇⠻⠇⢸⣿⣼⣿⣿⣿⣿⡏⠀⣾⣿⣷⠛⠛⠛⠁⣰⠟
⣿⠀⣸⣿⣿⣇⠹⠋⠀⣠⣀⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⡆⢠⣾⠛⢻⣽⠿⠿⢱⣾⡛⠃⠈⠀⢠⡄⠸⣿⡀
⠁⢠⣿⣿⣿⠟⠀⠀⣸⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⠿⣄⡈⠸⡇⢾⣿⣿⡇⣴⣿⣿⡇⠀⠀⢀⣿⠇⢠⣿⡇
⠀⠘⣿⣿⡿⢰⣦⣼⡿⠿⠟⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣦⡽⣿⠆⢰⣶⣿⣏⣻⣿⣿⣿⣿⣶⣤⣿⠏⢠⡾⣿⠃⣠
⠀⠀⠘⢿⣿⣌⠛⠟⣣⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠀⢀⠛⣿⣿⣷⡘⣿⣿⣿⣿⠟⠁⠀⣿⡆⠻⢣⣿⠃⣴⣶⣶
⠀⠀⠀⠀⠻⣿⣷⣿⣿⡿⠛⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡟⢶⣾⣿⣾⣿⣿⣿⡇⣿⣿⣿⣿⡄⠀⢠⣿⣟⠀⣾⣇⠀⣿⣿⣿
⠀⡞⠀⠀⠀⠘⢿⣿⣇⠀⠀⠀⠀⠀⠀⠘⣿⡻⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⠛⣿⠘⢻⣿⣿⡇⠀⣾⣿⡟⠀⠊⠈⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣾⣟⣻⣿⣿⣿⣿⣿⠏⢰⣿⡇⢸⣿⣿⡄⠀⣿⣿⣶⡆⠀⣤⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣭⡟⢀⡿⠏⠁⣼⣿⠟⠀⢸⣿⢿⣿⣿⣶⠋⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⢀⣼⠁⣠⢸⣿⣿⣇⣰⣿⣧⠈⣿⣿⠟⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣿⣿⡟⠙⢻⣿⣿⣿⣧⣈⣉⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣿⣿⣿⠛⡆⠘⠿⠿⠿⠿⠿⠻⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣤⣴⢿⣉⣁⣀⣤⣴⣶⠞⠃⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣽⣿⣿⣿⣿⣿⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣬⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣾⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡀⠀⠱⣦⣬⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡈⠉⢠⠀⣬⣹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⡿⠛⠀⣽⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠉⣀⠐⠞⠋⠀⣾⠂⢹⡿⢿⣤⣄
⠀⠀⠀⠀⠀⠘⠃⣀⣀⣤⠃⣶⣇⠰⣿⣧⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠠⠀⢸⣿⡻⢱⣿⣿⠀⣿⣿⡟⣿⠛⡄
⠀⠀⠀⠀⠀⠀⠦⠀⢿⡋⣿⣿⢿⣷⣾⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟⠙⠁⠙⡜⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣭⣿⣿⣿⣿⠿⣿⢷⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣿⣿⣿⣿⣟⣭⡉⠁⠀⠀⢀⣿⣿⢻⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠜⣟⣽⣿⡿⠋⠉⠁⠀⠉⠙⠿⠿⣿⣿⣿⣿⠟⡿⠛⠛⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣕⡢⢤⣀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠈⠀⠀⠀⠀⣿⣿⣿⠇⠀⣀⠤⢔⣮⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣶⣄⠙⢦⣿⣿⡿⣛⣭⣁⠂⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠐⣈⣍⡛⢿⣿⣿⡴⢋⣠⣾⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠣⠙⡿⢿⣿⣷⡄⢻⣿⡵⡿⠿⢿⣦⡀⠀⠀⠑⠚⠓⠊⠀⠀⢀⣶⠿⠿⢿⣮⣿⡟⢠⣾⣿⡟⠿⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢻⣿⣷⣸⣿⠟⢠⣴⢶⣿⣷⣄⠀⠀⡄⢠⠀⢀⣠⣿⣷⡶⣤⡄⠛⣿⣇⣾⣿⡛⠁⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⣿⡿⢻⣿⣿⢗⣾⣿⢛⣉⠻⣿⢿⣿⠿⠿⣿⢿⣿⢛⣉⠻⣿⣷⠻⣿⣿⠟⢿⣿⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣬⣥⣾⠋⠀⠠⠿⣷⣜⣛⣠⡿⣿⡿⠀⠡⢿⣿⣷⣜⣛⣡⣿⠿⠄⠀⠹⣷⣬⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣧⣀⠀⠀⠀⠙⢛⣩⡎⣁⡁⠀⠀⢈⣌⣱⣍⠛⠋⠀⠀⢀⣠⣼⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⡿⣫⣶⣾⣞⠟⠋⠉⠛⠻⣶⣶⠟⠛⠉⠉⠻⣷⣷⣶⣟⠻⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣙⣿⢿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⡶⣿⣿⡿⣿⣋⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡜⢯⡻⣿⡤⣨⡁⣉⢈⡅⣨⣴⣮⠟⣽⢣⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣷⣤⡙⠓⠺⠼⠼⠗⠓⢋⣠⣾⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠛⠷⠶⠶⠶⠶⠾⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⣄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢰⣿⣿⣆⡀⠀⣀⣤⣤⣼⣿⡇⠒⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣀⡄⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣷⣾⣿⣶⣶⣦⡀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣹⣿⡇⠀⠘⠘⢻⣿⣿⣿⣿⣿⣿⣿⣇⠈⠙⡟⠋⠛⠿⠿⠿⣿⣤⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣄⠀⠀⠿⠋⠈⣿⣿⣿⣿⣿⣿⡄⠀⢰⣿⠷⢰⣶⡄⡿⢸⣿⡿⣿⠰⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣄⣸⠟⠋⠉⢉⣙⣷⣀⣤⣤⣤⡈⠻⠛⠛⠿⠿⢷⠈⣶⣶⠶⣌⣙⣡⣦⣬⣿⣁⣼⣷⣮⠆
⠀⠀⠀⠀⠀⠀⠐⣿⡿⠟⢛⠻⣿⡷⣾⡿⠀⠛⠛⠛⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⣻⣿⡟⠿⣿⣿⣿⠋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣶⣿⡿⡗⢠⣴⣟⣳⣶⠊⠉⠉⠉⠁⠀⣀⠀⠀⠀⠀⠀⢠⣀⡀⢀⣀⣙⠛⠥⠀⢠⣉⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢋⣿⣦⢂⣧⣬⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⢉⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⡇⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⡾⣱⣿⣿⠿⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠉⠻⣿⡏⠙⢁⣀⠀⢸⣿⡉⠘⠂⠀⠒
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣏⡁⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡤⠀⣿⣷⣿⣿⣿⣿⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠹⠟⠋⠙⣿⣿⣿⠃⠀⠀⢀⣸⣿⣇⠀⠀⢠⣾⣿⣿⣦⠀⠀⠈⠁⠀⠀⠀⠉⡟⠀⠉⠉⠉⣿⣷⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠀⠻⡇⠀⣰⣿⣿⣿⠿⠛⢡⣀⣿⠏⠁⠀⠈⢻⣿⣤⣴⣶⣶⣶⣤⣇⢉⣀⡈⢃⡏⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠁⣠⣿⣿⣿⡿⠠⠄⠲⠤⠭⠤⠀⠀⠤⠘⣿⡏⠛⠛⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⣿⣿⡇⠀⠀⠴⠶⠶⣶⣾⣶⣤⣶⣿⡇⠀⠀⢻⣿⣿⠿⠁⠉⠉⠧⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠘⠛⠛⠛⠛⠻⠟⠛⣴⣿⣿⣦⣄⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣾⣟⠋⠉⠉⠰⣶⠂⠉⠉⣉⣿⣿⣤⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⡓⣾⠏⢀⡰⢤⣀⡄⢉⠉⠉⠀⠀⣾⢘⣘⣶⠖⠛⠿⠿⣿⣯⣾⣶⣶⠟⠁⠀⠀⢀⣈⡿⠆⠀⣠⢦⣴⣿⣴⣿⣿⠟⠀⠀⠀⠀⠀⣀
⠉⠁⣮⡮⣍⣶⣿⡵⢯⣳⣿⠀⠀⠛⠛⣋⣀⣴⢦⡀⠀⢉⣀⣉⠉⠁⠀⠀⠀⠀⣨⡛⢂⡠⡎⠁⢀⣼⡎⠻⠁⠀⠀⠀⡀⣄⣾⠏⠟
⣠⣞⡻⣽⣿⣋⣞⣨⣝⠟⠁⠀⠀⢀⢬⡽⣷⣿⣿⣷⢶⣟⣤⣷⣷⠀⠀⠀⠀⠀⠉⣷⣯⣶⣷⣶⣿⣿⣿⡆⢀⠁⠀⠳⠙⠁⠁
⠯⣲⢾⡿⡽⢽⣺⣿⠊⠀⠀⠀⣰⣶⡻⣽⣿⣿⣿⣟⡷⣿⣿⠛⠉⣆⢀⠀⠀⠀⠀⣿⠿⠯⠿⠿⢿⠿⢿⣿⠷⡄⠠⢠⠀⠀⣀⣀⣤
⣽⣿⢋⣿⢯⣌⡿⠁⠀⠀⠀⢄⢾⣓⣶⣽⢫⣿⢽⣿⡝⠁⡀⠀⠀⠀⠈⠀⢠⣄⠻⠀⣴⣶⡦⢰⠆⠀⢼⣧⣶⣷⣼⣿⣧⠤⡞⠛⢿
⣞⣷⣛⣾⣿⣯⠀⠀⠀⠀⣼⢯⣹⢿⣿⣹⣟⣦⣼⠿⠀⠀⠉⡉⠂⠀⠀⣀⡬⠉⣵⣻⣿⡿⠳⠠⠒⠀⠀⢋⠻⣿⡿⢿⣷⣿⣿⣀⣀
⡾⣗⣺⠗⠀⠁⠀⠀⣐⣜⢿⣥⣿⠛⠓⠈⠙⠋⢩⠠⠄⣰⢬⠱⠂⠀⠉⠁⠀⣀⡜⠙⣿⣿⣧⠀⢰⣶⣿⣿⣟⣿⠀⢺⣁⠥⣀⠈⠈
⣽⣿⠋⠀⠀⠀⣠⡰⣟⣺⣿⣟⠛⠂⠀⠀⢀⡆⠤⠶⠚⠁⠀⠀⢀⣠⠴⢿⢋⣭⠄⠀⡟⡟⠟⢀⣘⠛⣿⠏⢉⣀⢠⡇⠻⠟⠋⠀⢤
⠟⠉⠀⡄⢀⣤⣿⢭⣼⣽⡋⢙⣂⠀⠀⠐⠀⠀⢀⣀⣤⡶⠞⣿⡯⣶⡶⣩⣵⡤⣤⣤⣧⣷⡦⣾⡿⠇⠈⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄
⠀⠀⠘⢁⣲⣓⡭⣛⣾⡉⠋⠻⠟⠂⠀⠀⠀⣿⣛⣭⡷⢾⣿⣿⣿⣾⣿⢿⡟⠓⡓⠓⠋⠉⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣤⠮⢻⢾⡷⣿⡈⠉⠀⠀⠀⢠⣄⡶⣤⣽⡛⠹⣿⡿⠟⠛⠫⠄⠀⠉⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣟⠯⣭⣿⡾⢷⠄⣀⣠⣤⠀⠀⠌⡟⢷⢸⠿⠟⠨⠍⠑⠈⠀⠁⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡟⠛⣉⣩⣤
⠷⣍⡿⡵⢻⠇⣤⣾⣿⣿⡿⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢉⣡⣬⣦⣶⣿⣿⣿⣿⣿
⣽⡫⢋⠴⣥⣆⣿⡿⠟⠋⠶⠂⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣉⣤⣴⣶⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿
⣿⣿⢃⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣅⣤⣶⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿
⢻⠈⠘⠋⠛⠂⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣁⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣹⠋⠶⠞⣋⣉
⢹⠀⠀⠀⣥⡀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⣀⣴⣾⣿⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⠟⢛⣩⠋⠲⢿⣋⣡⣤⣶⠾⠿⠛⠋
⣼⡅⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢋⣡⣴⢶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⠟⠳⠶⠋⣀⣥⣴⡐⠿⠛⠋⠉
⠁⠁⢀⣤⣄⠉⣻⣿⣿⣿⡿⠋⣁⣤⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠿⢳⠫⣵⡞⠑⣿⣩⣴⡶⠟⠛⠉⠁
⡀⡀⢹⣿⣿⠈⠱⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣡⠤⢚⣁⣠⣤⣶⠷⠟⠋⠁
⠀⠁⣸⣿⣿⣄⡀⣿⠻⣿⢿⣿⣿⣿⣷⣟⣿⡿⢟⣩⡤⠄⠀⣈⢠⣾⠿⠛⠉
⠶⠘⣿⣯⠕⠁⠄⠈⠀⡀⠒⠤⢿⣿⡿⠟⢛⠓⠺⠃⣠⣶⠿⠛⠁
⡀⣰⣿⣿⡀⠀⠀⡶⢚⡿⠀⡈⠀⠀⠵⠚⢋⣡⣾⠷⠋⠁
⣍⢻⣿⣿⢷⣤⣤⣤⣬⡄⠙⠀⠀⡀⢀⣄⡿⠋⠁
⣿⡿⢻⡿⣶⡿⣄⣀⣸⡇⠀⠃⠀⢔⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⣠⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣬⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠻⠟⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣢⠀⠀⠀⠀⠀⠀⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣤⠻⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠘⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠋⣽⢇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⢴⣶⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢀⣅⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠒⠿⠋⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⣿⣻⣷⣿⣟⡁⣿⣟⢸⣿⠀⠀⢠⣿⠻⣷⣼⣿⡋⣿⣟⠁⠀⢻⣿⡟⣾⡟⣿⣾⣿⢸⣿⢸⣿⣿⡇⠀⢾⣿⡋⣿⣷⡿⠱⣿⢸⣿⣾
⣿⠋⠁⣿⣯⡅⣿⣯⢸⣿⣤⠀⠘⣿⣴⡿⢻⣿⠁⣿⡏⠀⠀⠀⣿⡇⢿⣧⣿⠟⣿⣾⡿⢸⣿⣿⡄⠀⣤⣽⡿⣿⡿⣿⣘⣿⢸⣿⣿
⠉⠀⠀⠈⠉⠀⠉⠉⠀⠉⠁⠀⠀⠀⠉⠀⠈⠁⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠉⠀⠈⠁⠈⠁⠀⠀⠉⠀⠈⠁⠉⠀⠉⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠉⢷
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠙⢦⣤⠴⠶⠒⠒⠒⠒⠲⢿⡆⠀⠙⢷⡀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠙⠛⠻⠷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠃⠀⠀⣀⣀⠀⠀⠈⠛⠿⣷⣶⣦⣤⣄⣀⠀⠀⠀⠀⣀⣠⣯⣤⢤⡄
⠀⠀⠀⠀⢰⠦⣤⣬⣷⡄⠀⠀⠾⠿⠟⠛⠉⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠗⠀⠀⠈⠉⠁⠀⣠⡾⠃
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠋⣽⣠⠖⠃⠀⠰⣟⠉
⠀⠀⠀⠀⠀⠈⠳⣄⠐⠋⠹⣤⠞⠋⠀⠀⠀⠰⣆⣀⣴⠞⠉⠙⠓⠶⣶⠞⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣋⣉⣉⣉⡙⣛⠛⠓⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⣊⣵⠶⠶⠖⠒⠂⠈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠦⠤⠤⠤⢬⣍⣉⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣟⠛⠭⠍⣐⡒⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠒⠒⠀⠤⠤⠤⣤⢻⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⡐⠒⢰⠏⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠙⠛⠛⠓⠒⠒⠲⠶⠶⠦⠤⠤⠤⣤⣤⣤⣀⣀⣀⣉⠉⣹⠏⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣷⢀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣄⠀⠀⠀⢀⣤⣽⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠛⠋⠻⡅⠀⡴⠁⠀⣀⣤⡙⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⠀⠀⠀⡴⢦⡀⠀⢠⠖⡳⠀⠀⠀⠀⠀⠀⢺⠀⢰⣿⣯⡷⠀⣇⠀⢸⣿⣿⣭⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⡴⠒⠦⣷⡀⠱⡄⢸⡀⣅⠀⠀⠀⠀⠀⠀⠈⣷⢤⠽⠛⠁⠀⠈⠢⣤⣉⣽⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⣨⠷⠤⢤⠉⠀⢈⡏⠀⠈⡇⠀⠀⠀⢀⣀⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⡇
⠙⠒⢒⡶⠂⡀⠀⠣⠀⣠⣝⣒⣚⣋⣩⡤⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⡇⠸⢸⢸⡇
⠀⠀⢯⠴⠚⠉⠓⠒⠛⠁⠀⠈⠉⠉⠀⠀⠀⣿⠀⠀⢧⡀⠀⠀⠀⠀⠀⣨⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣄⠀⠀⡇⠀⢸⢸⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢶⠲⠴⠒⠋⢁⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⢤⣇⠸⠸⢘⡧⠏⢈⡇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢧⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣈⠙⠛⠏⠀⠘⣯⠔⠚⢉⣁⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡞⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡇⠀⠀⠀⡸⠀⠈⣉⣁⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢹⣦⣄⣀⣠⡀⢰⣌⡉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣼⢻⡇⠀⠀⢣⣨⠆⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣸⠋⠉⠉⠉⠙⠛⠛⠻⣟⠒⣶⠲⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳⡀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠙⢄⡀⠀⠀⠀⢰⠇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⠓⠶⠴⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣍⣀⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⠉⠁⠀⠀⠘⢦⡀⠀⠀⠀⢀⠀⠀⠀⠁⢀⣠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⣄⡀⢀⣠⠆⠀⢨⣿⠀⠀⠀⠢⠤⣠⠒⣐⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣽⣿⣇⣠⣫⡾⠿⠖⠁⠀⠀⠈⣀⣴⣿⣯⣶⣶⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⢁⠸⣿⡍⢛⣿⣿⠰⣶⡄⠀⢠⣾⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠂⢀⠼⡧⣀⢰⣿⣿⣏⣴⣈⣵⣾⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠛⢳⣆⣶⣷⡾⣇⠴⣈⣿⣿⣿⣿⣿⣿⣿⠫⣾⡿⠁⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣿⢡⣄⣻⣿⣿⣿⣿⣿⣋⡽⣿⣿⢿⣷⠆
⠀⠀⠀⠀⠀⠈⠉⠻⣧⣸⣿⣿⣿⣿⣿⣿⣿⣥⣦⡙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠘⢫⣽⡿⢿⣿⡿⠾⠿⣿⡿⠗
⠀⠀⠀⠀⠀⢀⣴⡿⢿⣦⣽⢥⣾⣿⣇⡀⢀⣿⢿⡌⠎⡀
⠀⠀⠀⢀⡔⠋⠁⢀⣩⡿⠿⠽⣾⣞⣿⣤⣴⣿⣠⠍⣹⣁⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣺⠏⠙⠛⠛⠛⠛⠿⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢯⣫⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠹⢄⠀⠂⠀⢠⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠿⠿⠿⠿⠿⠦⠀⠠⠷⠦
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡆⠀⠀⠀⠀⠈⠀⣠⠟⢿⣾⣷⣦⡄⠀⢀⣴⡆
⠀⠀⢠⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠻⠀⠘⣿⠋⠉⡿⠀⣿⠻⠯⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢺⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣼⡟⠻⢿⠻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣶⠏⠀⠀⢿⡄⠀⠀⠀⠀⠁
⠀⠀⠈⠹⣿⡿⠟⠁⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢰⣶⠀⢀⡁⠴⠀⡄⠀⠀⠀⠀⣴⣦⡀
⠀⣀⢄⠂⠈⣁⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢰⣏⠉⠀⠀⠀⠀⠀⠀⠙⠙⠃⣘⡃⠀⠀⠀⠙⠂⠀⠀⠛⢛⠛⠓⠒⠛
⠀⢹⡄⠀⠈⢡⡶⠃⠀⠉⠙⣮⣤⡀⣠⠀⠀⠀⠠⠊⠀⠀⠠⠀⠀⠀⠀⠀⠈⠙⠋
⣀⡀⡇⠀⠀⡍⠀⠀⠀⢸⣧⣴⣶⣶⣾⣿⣶⡆⣤⡀⠀⠀⢀⠀⠀⠀⢀⣀⡀
⣿⠷⠀⠀⠰⠁⠀⠀⣀⡈⢉⡀⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⢀⠀⠀⠀⡄⠂⠐⠀⠀⠀⠙⠛⠂⠀⠀⠔⠈⠀⠈⠉⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠉⠭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠍⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠄
⠀⠀⠀⠅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣴⣾⠟

⠀⠐⢄
⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠘⣦⡤⠤⡄⠀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠲⠀⡀
⠀⠀⠀⠀⣠⠔⠍⢀⠀⢐⠀⡀⠀⡔⢁⠜⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⠆⠷⣀⠃⠀⠠⡀
⠀⠀⢀⢼⠁⡀⢡⠈⢧⡐⠠⠤⡜⠴⠆⠛⠐⢺⡓⠲⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢉⠀⠘⠲⠄⠀⠈⠁⠀⠠⠄⠠⠀⡀
⠀⢀⠮⠀⠳⢔⢀⡓⢊⠐⡈⠁⢛⣦⠀⠰⣂⠛⢃⠀⡀⠉⢆⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠤⠄⠘⠈⠀⠅⡢⡄⠀⡀⠀⠠⢨⠀⠐⠀⠀⠀⠂⠰
⠐⣾⣀⠉⠂⣄⢋⠐⢠⠀⠀⠐⠎⣉⡖⠗⣋⣨⡘⠙⢦⣿⣾⠄⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡀⠘⢀⢼⣄⡳⡀⠀⠈⠁⡀⡇
⠀⢁⠁⠊⠡⠃⠆⡄⠀⠀⠀⣀⠖⡰⠒⠁⠀⠲⡤⣭⣾⢂⡞⠳⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠐⡔⠀⡀⡋⠻⠊⠂⡌⢀⡇⠲
⢸⠈⠘⠒⢲⡈⠘⢾⣶⡔⡦⡠⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠂⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠰⠠⠀⠈⡇⠂⢐⡀⢈⡣⠂
⠸⠛⠒⡚⣙⠲⠄⠁⠀⣵⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠆⠘⠈⠰⢚⠬⠓⠉
⠀⠣⠥⠅⠒⣇⠠⢀⠀⢞⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣔⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡈⠁⠂⠈
⠀⠀⠀⠀⠀⠀⡰⣵⣮⣸⢽⡄⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠻⡿⣤⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁
⠀⠀⠀⠀⠀⣇⠘⢢⠞⣉⠌⡁⡰⠤⠢⣀⠀⠀⠀⢀⠵⢄⠀⠈⠁⠀⠀⠀⠈⠉⢸⣽⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢹⣦⡠⠀⠑⣛⢔⠂⡀⣬⣿⠁⠂⡜⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠈⠛⢳⣶⢶⣶⠶⠟⠛⠀⠀⠀⠀⠐⠄⣠⣾⣿⡆⢀⣀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢉⠀⠒⢤⣴⠄⠀⠀⠀⠀⠈⠙⠋⠡⠀⠀⠉⡛⠁⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠢⠠⠄⠊⠀⠀⠀⠀⠀⠀⠈⠉⠂⢇⣾⣿⠖⠁⠄⡀⠀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡟⠉⠉⢻⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣿⣿⣄⠔⠀⢠⣿⣿⣿⣿⣿⣿⣶⣶⡂
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⡏⠙⡏⠉⣿⡏⠙⡏⠉⣿⡏⠙⡏⠉⣿⢫⡍⡭⢉⣉⡩⣉⠭⣉⠙⣿⠉⢹⠋⢹⣿⠉⢹⠉⢹⣿⠉⢹⠉⢹⣿
⠀⠀⠀⣿⡇⠀⡇⠀⣿⡇⠀⡇⠀⣿⡇⠀⡇⠀⣿⣘⣋⣒⣈⣙⣳⣋⣒⣙⣀⣿⠀⢸⠀⢸⣿⠀⢸⠀⢸⣿⠀⢸⠀⢸⣿
⠀⠀⠀⣿⣷⣶⣷⣶⣿⣷⣾⣷⣶⣿⣷⣾⣷⣶⣿⣿⠿⠛⠋⠁⠈⠙⠛⠿⣿⣿⣶⣾⣶⣾⣿⣶⣾⣶⣾⣿⣶⣾⣶⣾⣿
⠀⠀⠀⣿⡏⠉⡏⠉⣿⡏⠉⡏⠉⣿⡏⠉⡏⠉⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⠉⢹⠉⢹⣿⠉⢹⠉⢹⣿⠉⢹⠉⢹⣿
⠀⠀⠀⣿⡇⠀⡇⠀⣿⡇⠀⡇⠀⣿⡇⠀⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⠀⢸⣿⠀⢸⠀⢸⣿⠀⢸⠀⢸⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧⣤⣤⣤⣤⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⣰⣆⡀
⠀⠀⠀⡀⠠⠾⣫⣽⡐
⠀⠀⠘⠉⠁⠉⠀⢀⢃
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⡀⠀⢀⢀⠀⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⣸⣄⡀⠒⠂⠀⠀⡠⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⣇⠀⢠⠄⡸⠋⠀⠀⠈⢠⠎⡀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⡄⠀⠀⢸⣿⠀⠀⠘⢤⠐⠁⠀⠀⠀⠛⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠸⣿⣄⣄⡞⣆⡀⠀⠀⠀⠀⠀⠘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⠀⠘⢁⣘⠘⠿⢿⣿⣴⡷⢂⠀⣠⡄⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠢⢁⠴⠋⠙⢧⠀⠀⠘⡿⢿⡽⡆⠀⠀⠀⢷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡰⢦⡀⠀⠀⢿⡇⠀⢴⡀⠀⠘⠛⣁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠀⠀⠀⠀⠀⠀⢙⣄⡀⠤⠁⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢳⠀⠀⠀⢀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⣴⡀⠠⠆⠡⢀⣀⣾⣶⡾⠃⢰⣶⣶⣦⣤⣤⡄⢸
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢴⣿⡿⠋⠁⣠⣤⣀⣽⣿⣿⣿⣿⣿⡟⠀⢰⣄⠀⠀⠻⣶⣴⣶⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⣤⡴⠚⠛⠛⠻⠾⢮⡤⢶⡜⣋⡴⠿⠝⠋⠄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠉⠁

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⢤⠀⣀⠀⠀⠿⠿⣿⡁
⠄⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣦⣿⣁⠴⠂⠀⠙⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠚⢿⣿⣿⣿
⠀⣀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣤⣀⣀⣀⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠙⡿⣿
⢾⣿⣿⣦⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⣤⣤⣀⠈⠁
⠀⢁⠀⠉⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣿⡿⠿
⣤⡏⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠋⠀⠰⢤⣘
⠛⠀⠀⢀⣰⡏⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠶⣿⣿⣿⣿⠏⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠙
⣀⣠⣴⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠈⠻⣿⣿⠀⠀⠈⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⡀
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⣀⣀⠀⠉⠂⠀⣴⣿⣿⣄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⣿⣿⡿⠀⠀⠀⠉⢋⠉⠙⠁⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠋⠀⠠⣄⠰⣗⣠⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⢻⣿⣿⠟⠋⠁⣠⣾⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⠀⠀⠀⢀⣀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣙⠉
⠋⠟⠁⠀⠀⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣀⣿⣷⣦
⠂⢀⣶⣶⣶⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠈⠋⠉⠻⠋
⠀⠀⠛⠻⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⣆⠠
⠖⠉⠀⠀⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠐⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠸⣿
⠀⠀⠀⢀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢹⣿⣿⣿⡿⠆⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀
⣤⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢳⣦⡀⠀⢸⠿⡟⠉⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢶⠿⢻⡿⠃⢈⠉⢀⣀⣀⣘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣆⠠⣀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣲⣾⡗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠙⠀⠀⠀⠀⢈⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⣽⣿⡆⠀⠆⣠⣀⡄⡧⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⣤⣄⡠⣡⡴⢲⣅⠀⡴⣸⡇⠀⡠⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⢃⣾⣿⣿⣶⣴⣦⣥⡟⠀⣏⠰
⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣻⣇⣿⣄⣿⣿⠚⣧⣿⠃⠟⠁⠀⠀⠀⠠⠀⠀⠀⢺⠿⠟⣼⣿⣿⣿⡟⠁⠐⠛⠧⣸⡞⢤⡀⣀
⠀⠀⠀⠀⢐⠂⠀⠀⣷⣿⣿⣭⠽⢿⡛⠉⠾⣿⠋⠀⠀⡀⡄⢀⠐⠀⢠⣀⡀⠂⠀⢰⣿⢿⣿⣿⡇⠀⠀⢾⠰⣿⣷⡤⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠐⡇⠋⠁⣿⠿⢻⣿⣿⢲⡌⢓⣦⣀⡆⠘⣀⠀⠀⠀⠉⠀⡄⢠⣾⣿⣼⠟⠆⠁⠀⠀⠈⠓⠈⠉⡁⢀
⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⣀⡀⣀⣀⣑⣀⣰⢻⣧⡀⢹⣷⠘⠀⠄⠀⠠⠈⢴⣆⠙⢿⣿⡿⠴⠶⠖⠖⠛⠛⠓⠛⠉⠉⠈⠁
⠀⠀⠀⠀⢀⣀⣀⣈⣩⠏⡉⠉⢀⣈⠁⣠⣿⣀⠷⣶⣾⣿⡇⡀⠠⣮⡀⢀⣿⣿⣆⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡹⢢⡉⠙⣾⣻⠋⠁⡀⠀⠉⣿⡽⠀⠀⢡⣖⣀⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠆⢠⡖⠈⢣⠀⡌⠁⢠⣾⣯⣤⢶⣆⠉⣯⣉⠻⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⢀⡀
⠀⠀⠀⣀⠤⠴⠖⠐⠋⠉⠀⠀⠈⠀⣤⣦⠼⠷⠦⠜⠩⠌⠀⢸⠿⢦⣿⣿⣿⢿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣰⠋⠃⠀⢠⣄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠦⣤⠄⠋⠁⠀⡠⠴⠒⢷⣼⣄⣈⡖⠲⣶⣿⣿⣿⣿⣿⢟⣛⢻⣄⣀⡀⠀⣄⠜⠧⠘⣆⠀⠸⢩⣶
⠀⠀⠀⠀⢀⠀⠀⡠⢀⠠⠂⠁⠀⠘⠃⣉⣷⣾⣿⣿⠿⠿⣿⠅⣴⠿⢿⣿⠿⠛⠉⠈⠙⢟⠛⠛⢀⡾⠁⠀⠀⢁⡘⢲⣶⠘⣇⣀⡘
⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⢀⡀⣠⠤⢤⡿⠋⠀⠀⠀⢠⡄⠀⣿⠞⠉⠀⠀⢀⡄⠀⠀⠀⠀⢱⣿⠀⠐⠂⠀⣿⠰⢌⡃⢠⠉⡽⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣿⡄⠀⢀⣠⠀⢁⡒⠦⣄⢀⣤⣀⠶⠀⠀⠀⢈⣴⡿⠃⣘⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⢠⣀⡜⣁⣿⣷⠴⠋⢠⣀⣤⣤⣜⣉⡼⢟⣩⢶⠽⠛⠺⢿⣷⣿⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢨⡁⠛⠀⠈⠁⠐⠟⠀⢷⠙⡏⠠⣶⣤⣧⣿⠅⢀⠙⠋⢁⣨⣿⡶⢤⣠⣽⣾⣿⣷⢰⡽⠟⠁
⠀⠀⠀⠀⠀⠈⢲⢾⢆⠀⣠⡀⠐⠀⠀⠄⣀⢀⣀⠆⠀⣤⣾⣿⣿⣿⣦⣀⡴⢚⠛⢁⣶⣤⣾⠖⠉⠀⣀⣻⣿⣼⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠙⠀⠘⠀⢈⡵⠛⣶⣦⢀⣉⡉⠉⣉⡠⠾⠿⡿⣿⠟⢻⣿⣿⠔⢃⣰⣿⣿⣿⣿⠶⢿⣿⣿⣿⣿⠟⠉⡇⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠁⠀⢈⡉⠛⢉⠞⠙⣻⣷⡇⠀⠀⠀⠻⠦⣌⣿⣿⣻⠟⠋⢤⣾⠿⠛⠉⠀⠠⠶⡆⠢
⠀⠀⠀⠀⠀⠀⠘⠛⠒⠂⠀⠀⣶⠀⡀⠉⣧⠴⠧⠀⣠⡞⠋⠀⠘⠀⠀⠀⠠⢽⣿⣿⡷⣄⠀⡸⠀⠀⠀⠰⠎⠁⢠⡠⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⠘⠤⠆⠘⣿⣇⠛⢶⣀⣲⣿⣤⢀⣸⣿⡏⢠⣾⣿⡇⠀⠀⠀⠀⠃⠀⢈⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡿⠟⠁⠼⣦⠝⠛⠢⠘⡿⠿⡤⠛⠊⣻⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠓⠂⠐⠠⢀⣀⣀⠈⠀⠀⠀⡄⠠⠛⠇⠰⠶⠖⢻⠟⠢⠀⠀⠀⠁⠀⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⠀⠀⠀⠀⠀⠀⠀⣈⢃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠈⠙⠛⠒

⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆
⠀⠀⠀⠀⡼⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠠⠠⠄⠄⠠⠤⢤⠤
⠀⠀⠀⢠⠃⠀⠀⠀⠓⡤⠒⠒⠒⠛⠛⠛⡆⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠃⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆
⠀⠀⡎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡄⠀⣤⡄
⠀⠠⣇⣀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠈⠋⠛⠒⠒⠚⠦⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠓⠲⠼⠄⣀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⡄⢠⠀⣠⣠⢀⣄⡠⡀⡠⣀⡠⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⠄⣠⣠⠀⣄⡠⡀⡠⣀⣀⡀
⠀⠀⠀⠄⠈⢸⡷⠀⠸⣿⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠒⠒⠓⠚⠒⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠌⠱⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠈⢄⠰⠀⠒⢄⢀⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠠⠏⠅⠀⠀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢁⠀⢀⠆⣵⡄⠀⠀⢰⣏⢢⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⡤⠆⠈⠀⠻⠇⣀⠀⠸⡿⠀⠀⠐⢓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠂⠀⠀⠀⠠⡖⠤⡔⠀⠀⠀⢀⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠢⠄⠄⠙⣒⣥⣶⡚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣔⣶⡿⠿⣅⡈⠓⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠀⠀⢀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⢠⠁⢰⡇⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢀⡤⠤⣤⡀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⡃⠂⡐⠃⠠⣤⡀⠹⣄⠀⠀⠀⠀⠸⡄⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣀⣿⡇⠁⣀⣀⣼⣿⠀⢻⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⣿⢃⡄⢿⣿⣿⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠈⠉⢁⠆⢳⡈⠻⠿⠋⠀⣸⡇⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠿⠛⠉⠀⠈⢷⣄⡀⣀⣴⡟⠀⠀⠀⣀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣤⣄⣀⣀⠀⠙⠛⠛⠋⠀⠀⠀⣀⣿⣦⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠟⠛⠋⠉⠉⠙⠛⠛⠿⠿⠶⠶⠚⠛⠋⠉⠀⠙⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣠⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⡇⠀⠘⠛⠉⠁⠀⠀⢀⣀⣀⣀⣠⣤⣤⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⢀⣴⠟⠋⣁⠤⣼⡇⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣿⠁⣀⣀⣤⣤⣿⡇⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠉⢀⣀⣠⠆⠀⠀⠀⢻⡄
⠀⠀⠘⠿⠛⠋⠉⠉⠁⣿⡇⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣾⠿⠛⠁⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣠⡶⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠉⠉⠉⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⡾⠛⠉⣀⢼⣿⠀⠀⠀⠀⣾⡿⠛⠛⠿⢶⣤⣄⣀⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⣴⡾⠋⠁⠀⠀⠁⣠⣼⣿⡀⠀⠀⠀⢻⣷⣀⠀⠀⠀⠈⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣾⣯⣀⣀⣤⣤⡶⠟⠋⠀⢿⣇⠀⠀⠀⠀⠙⢿⣷⣦⣤⣀⣀⣀⣤⣤⠶⠟⠛⠋⠁⣀⠀⠀⠀⠀⣼⡇
⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁⠀⠀⣀⣤⠶⠛⠁⠀⢀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠳⣄⡀⠙⢦⡀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢣⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡈⠙⢦⣄⠉⠳⢬⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⣈⠙⠲⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⢶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣧⠀⠀⠀⠸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠈⠙⠛⠛⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣤⣤⣤⣴⠶⠛⠁
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⠙⠲⣶⠚⠉⠉⠙⠓⠲⢤⣀⠀⢀⠴⠋⡼⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠘⣦⠈⠀⠀⠀⠀⠀⠀⠀⠘⢟⠁⠀⡼⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣆⠀⠀⠘⠃⠀⢀⡀⠀⠀⠀⠀⠀⠈⣧⠞⠁⠀⠀⠀⠀⠀⣸⠁⠀⢀⠄⡠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢹⣶⣤⠀⢀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁⠊⢀⣾
⠀⠀⠀⠀⢀⣴⣾⣿⣿⡟⠛⠻⣿⡿⠿⣿⣿⣿⣿⣿⡀⠀⢠⡀⠀⠙⠛⠉⠉⢿⡝⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⡀⢺⠇
⠀⠀⠀⠰⣿⡟⠁⠉⠛⠁⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣇⠀⠸⡏⢻⣦⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⡿
⠀⣤⣤⣤⣿⣷⡤⠴⠶⠶⠶⠒⠒⠒⠀⠒⠒⠒⠶⠿⣿⣷⣄⠙⠻⣽⡿⣷⡶⠤⠾⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠇
⠀⠘⣿⠋⠉⠉⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⠀⠀⠙⢿⣷⣆⠀⠉⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡾
⠀⠀⠹⣧⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⠛⣿⣿⣿⣷⣦⣀⠈⠻⣿⣿⣄⡀⣧⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠹⣧⡀⢀⣴⣿⣿⣿⣿⣿⡀⣤⣠⣾⣿⣿⣿⣿⣏⠛⠷⣾⣿⣧⠈⠙⢦⡀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣀⠜⣿⣿⣿⣿⣇⠀⠈⠙⢿⡇⠀⠀⠻⡄⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⡠⠊⢀⡬⠻⣿⣿⣿⣿⣆⠀⠀⠀⣿⡄⠀⠀⢻⡆⠀⠀⡼⠀⠀⠀⢸⣛⠉⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠗⠁⣠⠞⠉⢰⣿⣿⣿⣧⠠⣤⣿⣿⣀⣴⣾⡁⠀⢰⠇⠀⣤⠖⠿⣿⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣤⣾⠅⠀⢀⣴⣿⣿⣿⣿⣷⠈⢹⣿⠋⢻⣿⣇⠀⠟⠀⣸⣅⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆⣀⣉⣿⣿⣿⣿⠿⠿⠛⠛⡿⠟⠿⢿⠖⠒⢲⣶⠛⠙⣆⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠷⣾⢿⣿⣿⡿⠃⠀⢀⡞⠀⠀⣼⠙⣧⠀⠸⣆⣠⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⣿⣧⣤⣽⣿⣟⣡⣀⣠⡞⣀⣀⣰⡿⠟⣛⣷⣾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⣿⣿⣋⣿⠿⠿⢛⣿⡿⠿⠿⣛⣿⣿⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠛⣁⣤⣾⣿⠏⣀⠴⣺⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠷⠚⣩⣿⡿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠈⠉⠁