Snotty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣰⠞⠉⠁⠀⠉⠛⠞⠉⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣠⠶⠾⠷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⣠⣀⠀⢀⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠛⠲⠶⠖⠛⠁⠉⠛⠋⠙⠲⡶⠛⠉
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⡼⠷⠀⠀⠀⠀⢠⡞⣁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢿
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣾⠐⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠋⠀⠸⡆
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠸⣆⠙⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠐⢿⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⠛⢛⣓⣲⠦⢤⣤⡤⠴⠖⠛⠉⢉⡗
⠀⠀⠀⢙⣦⠄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡇⢀⣄⣈⡉⠉⠙⠳⠦⣤⣀⣀⣀⣠⠟⠁
⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣦⡀⠉⠙⠳⢦⣀⠀⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⢤⣤⣄⣀⣀⡀
⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣒⡚⠛⠋⠀⢻⡞⢻⡷⢦⣀⠀⠉⠳⢦⣄⣀⡤⠋
⠀⠀⠘⢧⣀⠀⢀⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡆⠀⠸⡇⠈⣿⡀⠉⠳⣄⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠉⠉⠉⠙⠦⣤⣤⠴⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢷⡀⠀⠀⠀⠀⠈⢳⣄⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣼⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠛⠶⣤⣀⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⠉⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢷⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⠘⡇⢠⡟⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⡞⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣤⠞⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡏⠀⠈⠀⠀⠀⠀⠀⢠⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡋⢺⣓⣤⠑⠂⠀⠀⠀⠀⠀⣿⡤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠏⠀⢸⠓⢮⣤⠀⠀⠀⠀⠲⠀⢼⣷⠦⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡾⠀⠀⣏⠀⠀⠀⡀⢄⡄⣀⠀⣀⣻⣿⣷⠠⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠰⢴⢏⣄⡀⢋⢁⡧⠴⠋⣠⡝⢮⠻⡛⠃⢠⡈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣡⣇⢠⠟⠊⢰⣿⡝⡍⢶⠲⠛⠛⢈⡷⠄⠱⢀⣿⡖⢀⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡯⠀⡄⠄⠀⣜⢡⠀⠃⠃⠦⠀⠀⠩⣻⡭⠂⠈⢦⣇⠡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⣠⠛⠀⠀⢱⡋⡔⠀⠀⠀⠀⠀⠀⠀⣷⠄⠀⠈⡏⣇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠜⢠⣿⠰⡄⠀⠸⣥⠰⠀⠀⠘⠀⠀⠀⢡⡏⠐⠂⠀⡇⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⣠⢻⣷⢦⢡⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⢸⠁⠀⡁⠀⠑⢶⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠃⡨⢯⢘⡀⠁⡐⠀⠙⠦⠀⠀⠀⠀⠀⣃⡤⢀⠀⢰⡁⣀⣀⡠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠈⠉⠃⢀⠀⠀⢨⠀⡄⠀⠀⡼⠅⠰⠘⢠⠄⡄⡗⡝⡄⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢄⡤⠀⡀⠊⢶⣰⠀⠠⡆⠷⠀⠀⢠⣊⡞⢴⠟⡿⠁⠀⠙⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢪⣙⠼⠀⠰⡍⣶⠘⠡⡄⠀⣀⢸⣿⣉⢼⡟⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠈⠢⠁⠴⡠⠀⠉⠀⠀⠁⠀⠀⠈⠉⠉⠃⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠇⠈⠁⠀⠀⠀⠀⠀⢀⣤⣴⣶⠶⠶⢠⢸⡇⠀⢀⣀⣀⡀⠀⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢉⣵⡶⠟⠛⢻⡘⡿⠿⠿⠛⠻⠿⢿⣶⡄⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣾⠁⠀⠀⣴⢸⡇⡇⣲⠶⠶⣦⡀⠀⣻⡟⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣿⠀⠀⠀⢻⣌⠃⢠⡇⠀⠀⠈⣷⠀⢹⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⢀⣴⠏⠀⠀⢀⣼⣿⠀⠸⣷⣀⠀⠀⠘⠀⠀⠙⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⣠⡴⠛⠁⠀⣀⣴⡿⠟⠁⠀⠀⠈⠉⠛⠛⠻⠷⢶⣤⡀⠈⠙⠶⣄⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢠⠞⠁⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠸⠆⠀⠙⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⠀⠀⣠⠞⢁⡄⠀⢠⣤⠶⠾⠟⠻⣦⠀⠀⠀⠐⠛⠛⠓⠦⠝⣷⡀⣄⡀⠀⠀⠀⠈⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⠏⢠⠎⠀⠀⢀⡤⠶⣒⣲⠄⠘⢆⣠⣤⣤⣝⡛⢻⡶⠄⠘⢿⣮⣙⠳⠶⣶⣤⡄⠈⠃⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠶⠒⠛⣋⣀⣤⡴⠟⢁⠀⠀⠀⠈⠒⠤⣄⡀⠀⣭⣛⠿⠶⣶⣶⠶⠄⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⢀⠀⢸⡆⣿⡄⠀⠀⢀⣴⠟⠋⠉⡡⣶⠞⠋⠁⠀⣾⠱⡄⣶⣆⠙⣦⣿⠈⠛⠶⠶⠂⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡍⠁⠀⠈⢇⠀⠓⠸⣿⡄⢀⡾⢋⡾⣦⡞⢁⠗⡀⠀⠀⡴⢿⡇⢻⡇⢹⡞⡿⣧⣠⡴⠂⢀⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠳⣤⡀⠘⢿⡟⠀⡼⢡⡟⠀⡾⡰⡇⠀⢰⣷⠸⣧⠈⣷⠸⣧⡇⠘⣯⡀⠺⠿⠛⢁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⣤⣌⡻⣦⡀⠀⠀⢀⣴⠟⡾⣿⢃⡞⠀⡼⢡⡇⡏⠀⠸⡟⣧⢹⡀⢻⡆⢿⡇⠀⠈⢷⡄⠰⠟⠛⠋⡉⠛⠛⠿⣶⣄
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠉⠻⠦⠀⣤⡶⠛⠁⡼⢁⡏⣸⠃⣼⢡⡟⣼⠃⠀⠀⢻⡼⡄⠙⠀⠀⠘⣷⠀⠀⢀⡿⠀⠊⢀⢀⠈⢢⠀⠀⠀⠙⣷⡄
⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⠢⠀⠙⣧⠀⢸⡇⣼⢠⡏⢰⠃⡾⢰⡏⠀⠀⠀⢸⡇⠉⠀⠀⠀⠀⣹⡆⢀⡞⠁⠀⠀⠀⠐⢁⡼⠀⠀⠀⠀⠈⢿⡆
⢠⣿⠃⠀⠠⠰⠲⢤⠀⠀⠸⠀⠀⠀⠐⠁⠀⢹⡆⢸⡇⠀⠘⠀⠈⠘⠃⢸⣇⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⣤⠟⠀⠀⠀⠀⠀⠉⠁⢀⠠⠄⡀⠀⠀⠘⣿
⣼⡇⠀⠀⠀⠈⠑⠀⡁⠀⠀⠀⠀⠀⠀⠀⡀⠸⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢃⠔⠀⠀⠑⢆⠀⠀⢰⠁⡠⠀⡄⢡⠀⠀⣿
⣿⡇⠀⠀⡀⠢⠤⠊⠀⡠⠒⠀⠐⠄⠀⢰⢡⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇⠘⢠⠊⠡⠀⢠⠀⠀⠈⠀⠣⢈⡠⠎⠀⠀⢸
⢿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡠⠀⠈⢦⣀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠸⣷⠀⠀⠀⠀⢸⡇⠀⠀⠓⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣇⠀⠀⠀⣀⠠⠄⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⠞⢉⣀⠁⢢⠀⠀⠀⡠⠒⠀⣿
⠀⢻⡄⠀⡞⠀⡀⠀⠈⡆⠀⠀⡔⠂⡉⠐⡄⢀⡟⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⠀⠀⢻⡆⢀⢄⠀⠘⡀⢄⠀⢀⡼⠀⠀⢠⠁⡄⢠⡿
⠀⠀⠻⡄⠡⡀⠑⠀⠔⠁⠀⠸⡀⠨⢐⠠⠃⡼⠁⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣀⣤⣴⣦⢼⡇⠀⠀⠀⠀⢿⡈⠀⡇⠀⠐⠀⠈⠁⠀⠀⠀⠀⠂⠀⣼⠃
⠀⠀⠀⠙⣄⠈⠀⠀⠀⠀⠀⠀⠑⠤⠠⠀⡼⠁⠀⠀⠀⠀⠀⣼⠀⠷⣦⣄⣤⣾⠟⠋⠀⠀⠸⣇⠀⠀⠀⠀⠘⡇⠈⠀⠀⠀⠔⠊⡠⢌⠢⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢄⢀⡴⠀⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠤⠄⠐⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⠂⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⢀⢀
⠀⠀⠀⠠⠊⠁⣀⡀⡀⢄⣀⠀⡀⠀⠈⢀⠀⠀⠀⠀⢀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⡁⠐⠁⠀⠀⠀⠀⠠⠀⠈⡄⠀⠀⠇⠀⠀⡐⠁⡠⠁⠘⠠⠤⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠎⠀⢀⠀⡠⠀⣀⣤⠄⡀⠀⠀⠀⠁
⠀⠀⣠⠁⠀⠐⠂⠀⠀⠀⠐⠀⠀⢡⠀⠀⠰⠀⠀⠇⡀⠀⠀⡁⠀⠀⠈⠐⡂⢄⠀⠀⠀⠀⠰⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⡀⠀⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢰⠀⠸⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⢈⡀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠃⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠘⠀⠀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠆
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢸⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠆⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊
⠀⠀⠀⠀⠈⠁⣒⢶⡀⠤⠆⠲⠒⠣⣀⠀⠀⠀⠀⠆⡀⢉⣉⠖⠤⠤⠴⢖⡩⠉⠑⠢⡀⠀⠀⠃⠀⠀⠑⣄⡀⠀⠀⣀⢤⡤⠒⠠⢄⡀
⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠁⠀⠀⠈⢆⠀⠀⠀⠀⠈⡜⠡⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⢠⠐⠊⠉⠁⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠔⠊⠐⠚⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡌⠀⠀⢀⠀⡀⠀⡀⢀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠑⠢⡀⠐⠒⠠⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⠈⠈⠀⠀⠀⠀⢀⡀⠉⠀⠀⠀⠀⠃⠀⠀⢀⠎⠄⢀⠄⠂⠁⠐⠤⡀⠀⠀⠘⠗⠀⠀⠐
⠀⠀⡌⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠉⠁⠀⠀⠀⠉⠉⠀⢰⠀⠀⠀⢠⠀⠀⣰⢃⠖⠑⠂⠀⠀⠒⠀⠀⡄⠀⠀⠈⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢰⠀⠈⠁⠀⠀⠀⠉⠀⠀⠐⠐⡂⠀⠘⠀⠀⢙⠁⠀⠀⠄⠀⠀⠄⠀⠀⠰⠀⡀⠀⡇⠀⡠
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡌⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠇⠀⠁
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠜⠐⠒
⠀⠀⠀⠑⠦⢄⣀⡀⠀⠀⢀⡠⢔⠊⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⡠⠒⠛⠈⠁⠋⠀⠔⠁⠈⠐⢄⠀⠀⠀⠀⠀⠀⣠⡔⠛⠀⠀⠀⠀⠐⠶⢄⡀⠀⠀⠀⠀⠀⠀⡩⠒⠶⠤⠔⠈⠍⠀⠢⡀
⠀⠀⠀⠎⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢠⠊⠀⠐⠂⠀⠀⠀⠐⠊⠘⡀⠑⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣿⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣄⠀⠙⠁⢀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⠟⠁⢀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⣿⣿⣷⣴⣿⣷⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢛⣩⣻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠋⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⢦⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠐⠠⠀⠰⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⣀⠄⠀⠛⠂⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁⣀⠈⠁⠀⠀⠀⠀⠉⠙⠈⠲⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠢⠀⠀⠀⠀⠊⠀⠀⠀⠀⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⠀⡀⠀⠀⠀⠀⠀⢐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡀⠑⢇⠀⣄⡀⠀⠀⢀⡟⠁⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠊⡇⠀⠀⠀⠀⠰⠁⠀⠉⢂⠾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠈⠢⠀⠀⠀⠐⠄⠀⠐⠁⡀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠰⠮⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠠⡀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠀⠄⠐⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢆⠀⠀⠀⠀⠀⠀⠀⠸⠾⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠘⠈⠀⠀⠀⢀⡀⠆⠀⠀⠣⡀⠀⠀⠀⠀⢠⠃⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡶⠈⠀⢨⢆⠀⠀⠀⠈⠀⠠⠤⠖⠁⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⠀⠀⢠⠃⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣰⠃⠀⠀⢸
⠀⠀⠀⠀⠔⢒⠲⢄⠑⢄⡠⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠔⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⢀⣀⠀⢀⠀⠀⠀⣙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣀⣔⢆⡸⣟⣷⣾⢷⣶⣊⡙⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⣖⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣶⢮⡪⣤⠀⠀⠀⠀⠀⠀⢈⣙⣟⣛⠋⠋⠈⢀⣁⡚⠐⢮⣉⡛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣿⣿⡏⢻⡿⣿⢿
⠀⠀⠀⠀⠀⠀⣿⡉⣻⢍⠉⠺⡦⡀⠀⠀⠀⠒⠚⣡⢮⠀⣠⠎⠿⡗⠀⠊⠉⠑⠿⣓⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠽⠏⠁⠀⢾⣇⠉
⠀⠀⠀⠀⠀⢺⣾⠀⠀⠀⡇⢠⢽⢱⠀⠀⠀⠀⠶⠈⠛⠉⣽⡿⢷⡿⣿⣿⠒⡒⠱⠒⡬⢱⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣬⣾⣀⠀⢀⡄⠊
⠀⠀⠀⠀⠴⣿⣿⣧⠐⢁⠄⠁⡸⢰⠀⠀⠀⠀⠀⠀⠀⠀⢷⠶⣎⣳⠶⠟⢀⡔⢹⡏⠘⣿⠀⠀⠀⠀⢠⠀⠱⣖⠃⡀⢞⠛⣿⣷⣧⣉⣹⣿⣦⡀
⠀⠀⠀⠀⠈⣩⠉⣋⡗⣶⡒⠊⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⡁⣙⣧⠄⡊⡀⠚⠁⠀⡇⠀⠀⠀⠀⢀⡖⠀⠈⢾⢳⡄⠩⣾⣿⡿⠿⣿⣿⡿⠿
⠀⠠⢄⣶⣿⣿⣿⣿⣿⣿⠿⣦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣥⡇⠠⣢⢴⡇⠀⠀⠀⠀⠀⠀⠉⣇⠤⠤⠃⠉⠀⢘⣥⣤⣴⣶⠂⠈
⠀⠀⢸⣬⣉⡛⠡⡅⣀⠤⠶⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⣿⣿⣷⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⡇
⠀⠐⠊⠡⡗⠉⢩⠋⡇⠀⣶⠌⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠋⡇⠀⢸⠀⢹⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣧⣀⣨⣇⣼⣲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠰⠷⠽⣛⠻⠯⣛⣕⣛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣧⣤⣶⣿⣏⣙⡟⠙⡆⠀⠀⠀⠒⣀⣲⣶⣶⣴⣿⣿⣿⣿⡿⡛⠫⠝⢋⣀⡡
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠙⠛⠿⠟⠛⠀⠀⠈⠀⠉⠛⠸⠿⣿⣻⠿⢿⣮⠑⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠀⣠⠀⠙⠀⡀⠀⠄⠀⠘⠀⡀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⡍⠢⡉⠀⠁⢀⠈⠢⠘⠀⡀⠀⠁⢃⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢄⠂⠐⢦⡐⢄⡰⢸⢢⡀⠆⠃⠤⣀⠡⡀⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠢⡉⠀⠒⡈⠢⠀⣌⢀⠱⡠⢮⠞⠉⠳⡘⠈⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠒⠀⠁⣈⣲⡄⢰⡄⢹⡄⠝⢦⡹⠁⠀⠉⢠⠃⠀⠂⡠⠂⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⢁⣻⡈⠉⠻⡌⠋⡹⠀⠈⠇⠈⢀⣴⠁⠁⠠⠦⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⡝⠓⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⠘⡶⠨⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⠀⢰⡕⠀⠀⠀⣠⡂⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠉⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⠀⠀⠀⠀⠐⠂⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⠢⠀⠤⠄⢌⡤⡁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠉⠂⠄⠀⠈⡰⠑⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠨⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢊⠄⠀⠈⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠄⡰⢑⠀⢀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠤⠀⢸⠁⡠⢀⠀⠘
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⢜⡉⠀⠈⠄⠀⣀⠀⠀⠤⠷⠃⣴⣄⣠⠀⠀⠀⣾⡵⣶⢗⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠒⢔⠀⡠⣿⠶⠶⠢⠂⡄⠾⠀⠀⠀⠀⠒⠘⠿⢓⡽⡂⢸⠁
⠀⠀⠀⠀⠀⢄⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠵⠚⠁⠀⠳⣀⠓⠄⠀⠀⠀⠀⠐⠘⡗⢊⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⠁⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⠀⠀⠀⢩⡄⡠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠄⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⡀⠀⠀⠀⠀⠈⡇⠒⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⠤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠂⠂⠀⠀⡔⠁⢂⠄⠔⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢂⠀⡠⠀⡀⠀⡠⠈⠀⠐⠂⠀⠐⢊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠈⠔⠁⠠⠀⠠⢊⠔⠀⠔⢁⠔⠠⠁⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠐⠁⠠⠀⠄⠀⠊⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠋⠟⢻⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠛⣻⣿⡛⠁⢠⣤⣤⣉⡛⠿⣷⣤⣄⠈⠉⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⡵⠚⠉⠀⠀⠞⠋⠁⠐⢛⡛⠛⢿⣯⣉⠁⠠⠿⣿⣦⡀⠐⠶⣮⡙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢃⡼⠁⢀⡄⠤⠶⠶⠚⠴⣾⣿⣯⣁⠲⠿⠿⠛⠃⣢⡀⠻⣿⣿⣆⠀⠈⢿⠂⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠈⠛⢢⣿⣾⣶⣶⠶⠖⠀⠀⠠⠾⠟⠛⠃⠀⠀⠈⢻⣷⠦⠀⠙⣿⣷⣦⡀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠒⢦⣤⢈⣫⣿⣿⠟⣇⠀⠀⠀⠈⠉⠛⠳⢦⣀⠀⠀⠀⠀⣿⣦⣀⠚⠻⠿⠟⠃⠀⠀⠀⠀⠠⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⠁⠾⠷⠦⢍⠒⠾⠃⣠⠀⠀⠀⣀⣀⣀⣀⣙⣦⠀⠀⠀⣿⣿⣿⣧⡀⠘⣷⠞⠁⠀⠀⠀⠀⢷⠸⡆
⠀⠀⠀⠀⠀⢀⡼⣷⣋⣤⠴⠒⣁⣤⣤⣤⡗⢠⣌⣙⠛⠶⢤⣿⡞⠛⠿⠀⢨⣭⡤⢀⣴⠟⢸⣿⡏⠃⠀⢹⠀⠀⢀⡀⠈⢶⣼⡇⠃
⠀⠀⠀⠀⢀⡾⠁⠈⠉⢀⡴⠛⠉⠀⠒⠈⠁⢤⡌⠙⠿⣤⣀⡈⠉⣀⣶⡾⢛⣿⠁⣾⠃⠀⢸⢻⠁⠀⠀⣾⠀⠀⠀⢷⣀⠈⣿⠓
⠀⠀⠀⠀⢸⠻⣶⣶⡶⠛⣠⣴⣾⣿⣿⣿⡆⠈⣁⣀⣀⡀⠉⠛⣿⣿⠁⢀⣸⣿⡄⣿⡀⠀⠀⠸⠀⠀⠀⢸⢰⠀⠀⠈⢿⣷⣽⡆
⠀⠀⠀⠀⠸⣦⣨⣿⣦⣴⣶⣿⣿⣿⣿⣯⣤⣤⣬⣍⣀⣡⡴⠂⠸⣿⠀⢸⣿⣿⡇⢹⣧⠀⠀⠀⣷⢠⠀⠀⠸⣇⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⡖⢒⣋⣭⣴⣎⠉⢻⣿⣿⣿⡿⣿⡏⢥⡤⠀⠀⡿⠀⠘⠁⣿⡇⢾⡏⢀⠀⣀⡏⢸⠀⠀⠀⢿⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢫⠉⠻⣿⣆⠘⣿⡇⢻⡇⣿⣷⡄⠁⠀⣰⡇⠀⠀⠀⡟⢀⡾⠁⣼⠀⢹⠇⣾⠀⠀⢰⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⠉⢿⠹⣧⣽⣿⠘⡇⢸⣿⣿⡀⠀⢹⠇⠀⠀⠀⢀⡾⠁⣴⠏⠀⠀⢰⡏⠀⠀⣼⠇⣿⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⣿⡝⠿⠀⠁⣼⣿⣿⡇⠀⠀⠀⠀⠀⣠⠞⢀⡼⠋⠀⣠⢀⣿⠁⠀⠚⡟⠀⣿⠇⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡙⣛⣿⣤⣤⣾⣿⣿⣿⢃⡜⠁⠀⠀⠀⠀⠀⡜⠁⠀⣠⣿⡾⠃⠀⠀⠀⠀⣼⣿⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠉⠀⠀⠘⠻⢫⣾⣀⡾⢸⣠⣾⡇⠀⣀⣴⣾⣿⠟⠁⠀⠀⠀⠀⣼⣿⠟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⣀⡀⠀⣀⣴⠏⠈⡿⠁⠸⡿⢿⣷⣾⣿⣿⠟⠁⡀⠀⠀⡀⢀⡾⠋⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠛⢿⡟⠀⠀⠀⠀⠀⠁⠈⠻⣿⠟⠁⣠⣾⣴⡶⠞⠁⡜⠁⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡴⠂⠀⠀⠀⣠⠀⠀⠉⠀⡼⠋⠀⠁⠀⢠⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣁⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣽⣿⡞⣠⡾⠀⣠⣶⣿⣾⠟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠿⣿⣿⣿⣥⣾⢻⣿⡿⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⡿⠁⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠀⠀⠀⠀⣦⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⣠⣴⠶⠛⠛⠒
⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠰⢄⠀⠀⠙⣀⠀⠀⢠⡉⣆⠀⣦⡀⠀⠄⠀⠀⢤⠀⢧⠀⡹⢀⡤⠀⠀⠀⠀⣨⠗⠈⣰⠦⠦⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣴⡦⢀⣁⡈⠐⠂⣀⣠⡄⠀⢰⡋⠹⣾⠁⢀⡖⠀⢰⡄⢾⠀⠘⢶⠷⠲⠂⢠⠀⠀⠀⣵⡶⢾⣁⡶⠾⠋⠁⡀⣍⡻⢦
⠀⠀⠀⠀⠀⠞⢉⡁⠀⡄⠠⠝⢳⡿⠿⠃⠙⠀⡉⠉⠀⠙⣷⡀⠰⠀⣸⡇⠀⠀⢀⣌⠀⠀⠀⠃⠀⠀⢸⠉⣿⢠⠉⢶⣤⡀⢴⢞⠻
⠀⠀⠀⠀⠀⢠⠻⠗⠖⢡⠀⠀⡄⠀⢰⠈⢳⠀⣯⠀⠘⠛⠿⢻⡀⠙⣿⡏⠉⠈⣿⠛⠋⠘⠂⠘⠻⡗⢁⠄⡹⠀⠓⢂⠀⠀⠢⠀⣦
⠀⠀⠀⠀⠀⠀⡼⢦⠐⠃⣸⡟⢰⣶⠰⢀⣬⡶⢣⡀⠀⠂⠀⡀⠻⣄⢻⡉⢲⣾⡗⠒⠂⠀⣀⠀⣬⢀⡇⠈⠣⠀⠠⡜⠒⣦⠈⠓⠀⠀⠀⠀⣰
⠀⠀⠀⢰⡄⠀⠁⢀⡧⠀⡰⠀⠰⣆⡀⠟⠉⣁⢹⡂⢠⠄⠀⣤⠀⠈⢫⠙⠛⢦⣬⠀⠀⠀⢉⢸⣰⡟⠁⢠⣴⠿⣀⠀⣤⠛⠀⠐⣀⢠⣤⣾⡟
⠀⠀⠀⢸⣿⡾⢃⣼⣁⣎⠀⠀⢠⣄⣤⣤⠶⠻⠈⠓⣼⠃⠀⢈⣤⣤⡞⣀⠀⠐⣇⠀⡄⠀⢈⢳⣟⠿⠶⠀⠙⠘⢋⣠⡌⠳⠀⠀⢯⠘⠟⠉⣤
⠀⠀⠀⠀⠈⠀⠞⠾⠷⣿⡁⠀⠀⣙⢻⣦⣀⣀⢢⣤⣵⣾⣷⠋⠀⣀⡿⡉⠛⠒⠛⣿⡙⢾⡁⠰⣋⢀⣄⡀⣾⠿⠟⠿⠛⣄⣠⣤⠈⣦⠰⣾⠏
⠀⠀⠀⠀⠠⣾⠀⠀⠼⠛⢁⣦⠖⠻⠷⣶⣭⣉⣿⠿⠁⣿⡇⠀⣸⣯⢵⣷⣤⣦⣼⣿⣗⠀⢻⣿⣿⡟⠛⠋⣉⣴⣦⣀⡀⠊⠉⠁⠀⠱⡈⠁
⠀⠀⠀⠀⠀⠘⠇⢀⣴⠞⠻⠿⠿⢷⣶⣶⣌⣿⡿⠀⣴⣿⣇⠀⠈⢀⣴⠃⣈⠀⠈⢙⣿⡆⠀⢿⡈⢠⣴⠿⠟⠿⠛⠟⠛⠳⣄⠀⣤⠀⡟
⠀⠀⠀⠀⠀⡇⣠⣎⣠⣶⣿⣶⣶⣦⣬⡉⣻⣿⣷⠀⢸⣿⣿⡀⣀⣈⣽⣆⠈⠀⠀⢸⣿⡇⠀⣿⡵⢋⢀⣤⣶⡶⠻⢶⠶⢷⣿⣿⡏⣀⢸⣤⡾⠃
⠀⠀⠀⢠⡀⣱⣿⠟⠉⣡⣤⣾⣤⣈⡻⢿⣿⣇⣿⡀⠸⢻⣿⣧⢈⣿⣑⣩⣿⠀⠺⣿⣿⠃⢠⡿⣷⣟⠋⠉⢀⣠⣤⣤⣄⠀⠁⠻⢿⡛⠂⣿⢇
⠀⠀⠀⠈⣿⠟⢡⣼⣿⣿⣿⡟⢁⡉⣿⢿⡟⠋⠸⣧⠀⢻⣿⡿⢻⣇⣉⣧⣽⣦⠀⢿⣧⣠⡾⢇⣿⢿⣴⠿⡿⠛⣿⢿⣿⣷⣴⠀⠈⣷⠀⣧⡈
⠀⠀⠀⢸⡟⢠⣿⣿⣿⣿⣿⠀⣀⣼⣏⡸⣿⣆⠐⠛⠆⠹⠋⠁⠈⢻⡿⢡⡿⠄⠂⠀⠱⠆⠡⣯⣽⣿⣟⣘⣃⣀⣽⣿⣿⣿⣿⣗⠀⢻⡆⢩⣀
⠀⠀⠀⢸⡇⢸⣿⣏⣽⣿⣿⠀⠉⠉⣽⣷⣿⡇⡀⠠⠀⠀⣀⣴⣴⣾⣁⡼⠁⠤⠐⠶⠖⣰⡖⢿⣿⣿⠏⣉⡿⠋⣾⣿⣿⣿⣿⡯⠀⣼⡇⢤⠃⡆
⠀⠀⠀⢸⡇⠈⢻⣿⣿⣿⣿⣤⡆⠀⠸⣿⠿⢁⣙⣶⣾⡷⣿⣿⣿⡿⠛⠁⢀⣡⣤⣤⣤⣤⣤⣘⠻⣿⡀⢙⠶⠟⢹⣿⣽⣿⣿⠁⣠⣿⠁⢈⠁⡅
⠀⠀⠀⠸⣷⡄⢻⣿⣿⣿⣿⣿⠁⠀⢀⠏⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⣿⣿⣯⣽⣿⣿⣿⠂⠘⡙⣿⢳⣀⣿⣿⣿⣽⣏⣤⣼⡟⠀⢸⣄⠁
⠀⠀⠀⢠⣿⣶⠊⠻⠛⣿⣿⣷⠀⠀⡘⣧⠈⢻⣿⣋⣵⣜⣿⠿⠋⢰⣾⣇⠹⠿⣿⣿⣿⣿⣿⠟⠠⠀⣿⢷⡆⢡⣿⣿⣿⣿⣿⡿⢻⣿⠷⢋⣈
⠀⠀⠀⠀⡿⢿⣴⣦⣀⠀⢀⠙⣳⣄⣳⠉⠀⣤⠈⠉⣁⠈⠐⢶⠀⣿⣿⣿⠀⠂⠠⠌⠉⣉⠀⠀⠀⢀⡿⠦⢁⣾⡿⠋⠁⠀⠀⣡⠞⢻⣀⣿⡿⣄
⠀⠀⠀⢸⣿⠈⣿⣿⣿⣷⣄⠀⠀⠛⣳⡑⠦⣤⣸⣆⠙⠟⢀⢈⣾⣿⣿⣿⡇⠀⠙⠂⠋⢀⣀⣀⡴⡿⣏⣴⠟⠳⠀⠰⡄⣠⣿⠯⣙⣿⢻⡻⣧
⠀⠀⠀⠀⡹⢧⠟⣧⣿⣿⣿⣷⣀⠀⠉⢵⡄⠌⡜⢻⡅⢠⠉⠈⣿⣿⠟⠿⠇⠀⠀⡀⡆⣿⢷⠎⣴⢩⡾⠻⢦⠀⠀⣶⡾⠛⡋⠀⠈⠉⡿⣁⠈
⠀⠀⠀⠀⢁⠘⠟⠟⢷⡾⢿⣿⣭⠁⠠⠈⣧⠀⠰⠎⢿⣤⣤⡄⣄⣡⢰⠀⡀⠠⣘⣷⣾⣇⠛⢀⣶⡿⠧⡄⠀⢘⣶⠏⢷⠟⠀⠀⣤⠀⣆⣉
⠀⠀⠀⠀⠘⠁⠀⠐⢿⣷⣌⣿⣿⡀⣤⠀⣿⠀⠈⣷⠈⡿⠿⢹⡿⢿⣿⣷⣿⡶⣿⢟⣿⢩⠀⠆⣾⡧⠴⠂⠀⣸⠇⣸⡇⠀⠀⠀⣿⡶⠊
⠀⠀⠀⠀⣴⣷⡀⠀⢀⣤⣿⣿⣿⣿⣼⣇⣸⡇⢠⣟⢰⠹⣿⡾⣤⡿⢿⣿⡏⣡⢫⣿⠃⢘⣤⣦⣼⣗⠐⠀⣶⣿⣴⣿⡆⠀⡀⠸⣿⠃⣴⣄
⠀⠀⠀⠀⣾⠇⢻⠿⠋⢹⣬⠉⠙⢿⣿⣟⠛⡀⢸⣿⡈⠀⠉⠻⠿⣧⣤⣈⢿⠟⠺⠇⠘⠘⣇⠋⢋⣙⣛⣶⣶⡿⠋⡀⣹⢾⡍⢰⡭⡶⠀⠉⠛
⠀⠀⠀⠀⠁⠀⠘⠂⢀⣯⠀⢇⢠⡌⠿⠟⢫⠁⣾⡟⠈⢳⠀⠂⣤⣀⠀⢀⣈⠁⠀⠀⣠⢊⣹⡶⠏⠙⢛⣿⣀⠀⣠⠀⠻⠟⠉⡛⣤
⠀⠀⠀⠀⠀⠀⣹⠗⣼⣿⠀⠈⣾⣆⠰⠟⠗⡓⢿⣿⠙⠊⠓⠤⠤⠤⠴⠄⠉⠠⠄⠛⢰⣿⠿⠗⢒⣴⣯⣀⡼⠛⠁⠐⠀⠆⠀⠈⠂
⠀⠀⠀⠀⠀⠀⣌⠶⠇⢡⡄⠴⢿⣭⠃⢤⠀⠅⠀⢉⠑⠀⢀⡄⠀⠀⢀⡀⠀⡀⠀⢀⠋⠝⣢⠚⣤⣿⣴⠿⣄⡙⢂⡀⠀⣠⣎⣤⣤
⠀⠀⠀⠀⠀⠀⢈⣦⣱⡋⡄⠃⠘⣿⡄⠀⠑⢀⡃⡈⠈⠁⠆⡅⠐⢀⣾⢧⡀⣁⠀⠶⠀⠋⣠⡞⣽⠟⠁⠀⠉⠀⠦⢩⣤⣤⡔⡍⢀
⠀⠀⠀⠀⠀⠀⠙⢻⣌⢡⣄⠀⡀⠈⠓⠀⣀⣈⣡⣤⣤⣶⣦⣀⣀⣈⣿⢻⣁⢀⣀⣴⣶⣶⣥⣤⣤⣤⡤⠀⠀⠀⢀⣘⣿⡿⠛
⠀⠀⠀⠀⠄⠀⢘⡃⢰⣼⣿⡆⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⠆⢉⣯⣿⣿⣿⣿⣟⣿⣿⣤⠄⠀⠀⠀⣿⣿⠛⠿⡷⠖
⠀⠀⠀⠀⡀⠀⠀⠁⡈⠉⠻⣷⣄⠄⠘⡤⠘⠋⠩⢻⣿⣽⣿⣿⣿⣿⢿⠶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠳⠚⠉⢡⣾⣿⢋⠶⠐⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⢂⠈⣙⡃⠐⠀⠘⢃⠀⠙⣛⣉⠛⠿⠛⣿⠿⠀⣿⡟⠋⢉⡀⠁⠙⢚⠑⠂⣀⢐⡋⡉⢀⠁⣠⡃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⢀⠘⠬⠁⠘⠀⠁⠈⠀⠀⠈⠀⠈⢤⡘⣿⣿⣿⣿⠒⠃⠀⠉⠀⠀⠈⠁⢠⣄⣀⠠⠀⢩⡄⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠰⡤⠎⣩⠙⠁⠀⠀⠀⣤⠀⡀⠀⠲⠈⠛⠛⠘⢋⠀⢤⠰⠆⠠⠐⠋⠀⢀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠈⢀⡀⠀⠀⢤⡄⠠⠌⠱⠆⠐⠐⠂⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣻⣟⣿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠋⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣯⣷⣶⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠿⠛⠿⢿⣷⣤⠀⠀⠙⢶⣤⡀
⠀⠀⠀⢀⣤⡾⢛⣩⡶⠛⠛⠉⣩⡽⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⠀⠀⣿⢿⣷⣄
⢀⣤⠶⠛⣡⡴⠋⠉⠀⣀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⣿⢿⣢⡀
⠟⣁⣴⡞⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣄⠀⠀⠀⠘⢿⣧⣿⡛⣄
⣾⡟⠁⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⣻⣿⣿⣿⣿⣷
⠏⠀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠉⠉⠉⠙⠳⣦⣆⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⡰⣷⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⠀⣌⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⣿⣿⣿⣿
⣼⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣛⠁⣀⣠⡤⠀⠁⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢙⣷⣿⣿⣇⣠⣴⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣿⣿⣿⣿
⣿⡿⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠛⠘⠲⠿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣠⢀⠔⣄⣠⠀⠀⢈⣿⣿⣿⣿⡟
⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢼⣮⣅⣾⣿⡟⠉⢻⡆⠀⣠⠄⢀⡼⣱⠏⠎⠀⠀⠀⣀⠈⣼⣿⣿⣿⠇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣿⠉⣣⣔⣧⣨⣧⣼⣿⣴⣿⡿⡃⠂⠀⠀⠀⠈⠛⠆⣿⣿⣿⡟⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡰⠃⣴⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⠁⡀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣶⣿⡿⠁⢀⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⣯⠞⠁⠀⡀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⡿⠋⡀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢂⠀⠀⢀⡼⢃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡿⠛⡟⠀⠈⢙⣦⡙⣿⠃⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⢠⡔⣴⡟⢱⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⠃⠀⠀⠈⠻⣿⣿⠀⠀⠀⣿⣿⠀⠉⠛⠻⣿⣿⣷⣶⣶⠏⠘⣾⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⡍⠨⠃⠀⠀⠀⠈⠛⠀⠀⠀⣙⠘⢄⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠾⠀⠀⣸⣿⠛⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣶⣾⡿⢃⣰⣿⣿⣿⡟⢻⣷⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⣿⠏⣹⠏⠀⢨⣿⣾⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠐⢀⠀⣴⣦⠄⠀⠀⢀⣈⣤⣾⣿⣿⣿⣿⣿⡿⣿⣴⠊⠁⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣷⣀⣠⣴⣿⣿⣿⢿⣻⣿⣿⡿⠋⢐⡁⠀⠶⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣴⣿⣿⠟⣿⣦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣟⣏⣥⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠐
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⢟⣩⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣩⣾
⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠟⠀⠀⠀⠀⣀⣤⢸⣿⣿
⣿⣷⣤⡀⣀⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣶⣿⣿⡿⠛⠁⢀⣤⣶⣾⣿⠿⠃⠻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠈⠓⠎⠻⠁⠔⡥⠉⣁
⠀⠀⠀⢀⠔⣉⢀⣀⠄⠀⡀⣀⠤⣀⡀⠀⠀⠚⠫
⠀⠀⠰⠟⠉⣤⡊⠤⢒⠁⠀⠀⠀⠀⠠⡄⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⢢⡀⠈⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠲⡆⢠⠤⡀⠀⠙⣄⢼⠀⠀⠀⣀⣀⠀⠀⡠⠉⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡇⠀⠘⠄⠀⠈⢎⢧⢀⠊⠤⠈⠣⠘⡀⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠸⡄⠀⢸⠀⠀⠈⠿⠋⠘⠁⢀⡄⠀⡇⠀⠙⠃⢐
⠀⠀⠀⠀⠀⢀⡠⠐⠀⠉⠢⠤⣴⠋⠓⠓⠀⠀⠀⢀⡀⠘⢇⠉⠀⠀⠈⠑⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⢤
⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠈⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡌⠀⠀⠀⠀⠚⢢
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠁⠘⠁⡀⣀⣶⣿⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠀⠄⠀⠀⢤⠀⠀⠀⠐⠂⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢠⡡⠖⠂⠈⠁⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣁⢎⠀⠀⠀⠀⠀⠀⠦⣀⠤⢧⡨
⢇⠀⠀⠀⠀⢀⣀⠤⠔⠂⠁⠀⠀⠈⢁⠐⠈⠉⠉⠈⠉⠉⠛⠇⠀⡆⠀⠀⡠⢄⡀⠀⣐⠩⠃⠀⠈⠢⠕⢀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⣀⠠⠐⠚⢦⡀⠘⠀⢰⠀⢠⠉⣳⠉⠛⢄⠈⢄⠀⠀⡴⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⢰⢦⡸⠷⠤⡇⠀⠒⠈⠁⢀⢧⠀⣀⣁⡦⠑⠢⢄⣠⠀⠃⠒⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠑⢄⣤⠆⠀⢀⣠⢊⣠⡘⢮⡁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠉⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡨⠷⠦⠤⡤⠔⢉⡸⠋⠁⡟⠈⣳⠀⠀⠈⠢⣀⣀⣠⠴⠒⠒⠂⡏⠀⠀⠀⠀⠀⢹⡀⠀⢀⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠘⢊⠝⠃⠌⠀⢸⠺⡇⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡎⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠘⠃⠐⡄⠀⠐⡄⠀⢐⠂⠒⢸⡀⢹⣍⠀⠀⠀⠀⢀⡀⠤⠐⠒⡖⠀⠀⠀⣀⣤⡞⠥⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠠⠰⡅⢀⡼⠛⠛⠛⠀⠀⡌⢣⡆⢹⡯⡑⠶⠈⠀⠀⠀⠀⣸⠀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡄⢀⡀⢀⢒⠟⠁⠀⠀⠀⠀⠀⢈⠷⡆⠐⠀⠓⠈⠆⠡⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢧⠀⠀⠀⠀⠀⠀⠘⠓⠢⢌⠁⠀⠀⠀⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⠀⠀⠀⣀⣀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⢉⣭⠡⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠤⠤⠐⠲⠈⠙⠛⠛⠉⠃⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⣹⡿⠂⠀⠰⠿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⣫⣿⡿⠟⠉⢸⣿⣥⣶⣦⡄⠀⠀⠀⣸⣿⣿⣿⣧⠲⡌⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⢸⣿⣿⡟⠃⡀⠀⠀⢀⣀⠘⠋⠀⠀⠀⡽⢿⡉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⡰⢋⣤⠴⠛⠛⠉⠀⠀⠀⠀⠀⣄⣸⡇⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠙⢧⣻⠿⠓⣒⣈⣉⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡔⣡⠞⠏⠀⠀⠀⠈⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣠⡇⠀⣀⡴⠚⢩⣼⣿⣧⡀⠀⠀⣀⠀⠀⠀⠻⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠓⠛⠀⢀⣙⣒⣎⡁⠈⠟⠫⡴⢋⡞⢉⡿⠁⠓⠀⠀⣿⣄⠀⠀⠀⠙⢲⠇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⣈⡷⢀⡴⣾⠟⣽⣿⡿⠿⠀⣤⡀⠀⣟⣠⠞⠁⠀⠀⠀⢀⣿⠇⠀⠀⠀⣠⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢰⡿⢚⡡⢞⣥⢠⣞⣠⣴⢂⡉⢀⡿⠀⠉⠁⠳⣄⠀⠀⣠⠞⢯⣀⡀⠀⣰⠃⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣙⡶⠋⢐⣎⡼⠟⠛⠻⣯⠉⠁⢾⣁⣀⣀⠤⢤⣈⠙⠛⠁⠀⣠⠟⠉⣿⠃⡰⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠾⢫⡄⢱⢋⡞⠑⢒⡤⣄⡈⠑⠒⠲⢦⡀⠀⠀⢀⠜⠀⠀⠀⠀⠷⠒⢶⡇⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠷⣾⢸⡇⡸⢏⣸⣿⣥⡤⠼⠿⠀⠀⠀⠘⣷⢠⠞⠋⢀⡶⣶⠄⠀⠀⢀⣼⣀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡶⠫⣄⣿⣅⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⡓⣚⠽⠛⠁⠀⠀⠀⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡤⠈⢿⣉⣙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠶⠶⠶⣤⣤⣄⣀⣸⡃⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣭⢀⣀⣤⣤⣤⣤⣀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠉⠉⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⠶⠛⠉⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣸⣇⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣏⣹⣍⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠛⢁⣤⣶⣤⠀⠀⡹⣯⠀⠀⠀⠠⠽⠯⠁⠀⣀⣀⡈⠛⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣦⠐⢻⣿⣿⠃⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠛⢀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢀⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠁⠀⣠⠴⠐⠄⠀⠀⠐⠒⠐⠀⠀⣿⣿⡆⠀⢙⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⣶⣷⡄⠀⠀⠀⣾⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⡟⠃⠀⠀⠀⠸⠉⠃⠀⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣠⣦⣱⠄⢁⣀⠈⠀⠀⠂⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣷⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⢢⣤⠆⢠⡾⣿⣿⠎⢀⣠⣶⣖⠤⣤⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢼⣿⣦⠀⠈⣶⡥⠏⠀⡸⡟⠉⢁⡴⣿⣿⣿⣿⣶⣪⣬⢇⢰⣨⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠀⣸⣿⣿⣆⠀⠀⠀⢠⠀⠃⠀⢀⣠⣤⣻⣿⡿⠛⠻⠿⣿⣿⠈⣷⣂
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠰⣬⣷⣾⣟⠟⠁⠸⠧⣄⠘⣤⣴⣾⣯⣾⣛⣉⣿⣟⡄⠀⠀⠀⠻⡆⢹⣿⡤⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⡤⣦⡑⣜⡿⢯⠄⠀⠀⠶⣶⡉⠃⢿⣰⣿⣿⣿⡿⢻⣿⣿⣧⠀⠀⠀⠀⠀⣼⣾⣿⣧⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⠿⠅⢈⡡⠂⠀⠀⠀⠘⠛⢷⡇⠊⢿⣿⠿⠻⡄⠀⢻⡿⠃⠳⠀⠀⠀⠀⢱⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠒⠉⠀⠀⠀⠀⠠⠀⠀⢰⡇⠀⠈⠁⠀⢀⣵⣶⣷⣿⠓⢀⡄⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄⣸⣾⣄⡠⣀⣤⣾⣿⣿⣿⣿⣿⣷⡿⠿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢈⡛⠛⣋⢿⣿⣿⣿⠟⣩⣿⣿⣭⣯⠃⠀⠀⡀⠀⠀⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⢸⡥⠶⣾⣟⠞⢁⣿⡜⣿⣿⡏⢀⣿⣿⣿⢻⡇⡀⠀⠀⠀⠀⢤⣀⡿⣿⢷⣤⣄⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠹⣿⠀⠸⠟⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠙⠿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡀⠀⠀⠀⠀⢀⠄⠆⡉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢽⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⠁⠀⠀⠀⠀⠀⠿⠓⠄⠀⠀⠀⢠⣄⠀⠀⠈⠇⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠸⠆⢀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡴⢿⡿⠀⠀⠀⠀⢠⡄⠀⠀⢀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⣿⡇⢸⣿

⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠴⠤⠐⠀⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⡴⡋⠁⠀⠐⠋⠉⠀⠀⠀⠀⠀⣉⠁⢦⠀⠀⠀⠀⡀⠒⠊⣀⡩⢀⣁⡶⠊⠉⠉⠉⠳⠀⣍⠃⠬⣵⢦⡀
⠀⠀⢀⠞⠸⠁⠀⠀⠀⠀⠀⠀⠀⠄⣀⣦⡴⠖⣿⠀⠀⠀⢸⠀⢀⣀⠦⡄⠣⡿⠧⣀⠀⠘⣦⠀⠀⠀⢳⡀⠀⠀⠹⣆
⠀⠀⡖⢈⠁⡀⠀⠀⢀⣀⠴⠓⣻⡯⢟⠝⠀⠀⢻⠀⠀⠀⡸⠁⢀⣰⠚⠀⡘⠀⠀⠈⢷⣦⣘⢀⡀⠀⠘⢧⠀⠀⠀⡎⣇
⠀⣘⡔⠁⠀⠁⠀⡠⠁⣠⢶⣰⣿⠓⠀⠀⢹⣷⢀⠀⠀⠀⠀⠠⢴⡷⠈⠀⠱⣿⠍⠀⠀⠸⣟⣦⣽⣆⠈⠊⡀⠀⠀⠆⢸
⢠⣢⠀⠀⠀⠄⠠⠂⢾⣬⢮⣿⣿⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢫⠸⣭⡋⣦⠀⢷⡀⠀⠇⢸
⢀⢋⠐⣤⠀⠀⡞⠃⠀⢹⣿⡏⡿⠀⠀⡀⠀⢀⢣⠤⠀⠀⠀⠠⠒⠀⠑⢲⣔⡀⠀⠀⠀⠀⠀⠰⣧⡾⣿⠀⠘⢱⠀⢀⡾
⠸⠇⢀⠈⠉⠺⡇⢀⣴⢉⣿⣿⣄⡀⠀⠘⢰⠿⠟⠁⠀⠀⠀⠀⠀⠠⠀⠈⠘⢶⣌⠂⠀⠀⠀⠀⣿⡟⣿⡻⠀⠀⣸⡶⡇
⠀⢁⠘⠀⠷⣄⡀⠂⡹⡦⣽⡿⣿⣷⣷⣶⣉⡂⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠀⠀⠀⠀⣴⣿⡟⣿⣷⢀⣴⣷⣷⡇
⠀⠠⠤⠂⠒⠓⢩⠴⠾⠁⠀⢹⠿⠛⠛⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⢘⣟⣤⣿⣿⣇⣾⣥⢄⣇⣞⣇⡾⠉
⠀⠁⠀⠀⠀⠈⠈⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⠀⠒⠀⠀⢀⠀⠀⠳⠿⠛⠉⠉⠙⢿⠛⣩⢿⡿⣿⣉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠻⣦⡴⣶⠂⢲⢦⣶⠾⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢲⡋⡆⠀⠀⠀⠀⢀⡀⠀⠀⣤⢰⣄⡘⠯⡧⠸⠿⢻⡎⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠂⡶⢶⢀⣏⣹⠀⡀⠿⢠⢰⡷⣦⠋⣿⣁⣷⡀⠀⣼⣿⡇⢸⣦⣾⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⢸⣿⣿⣨⣹⡄⣈⣣⣩⣬⣴⣲⢺⡡⠌⡄⣿⣿⡇⢸⢻⣿⠀⣿⣸⠃
⠀⠀⠀⠀⠂⠀⢀⣤⡀⣸⢟⣿⢸⡷⣿⢸⢿⡇⢢⣁⣋⠇⠹⢮⣌⣥⣬⠁⡏⠙⡇⣸⡇⣿⠀⡿⢻⠀⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⢸⣿⣿⡊⣿⣼⡸⣧⢸⣋⢨⢼⡤⣥⠌⠉⠀⢰⣿⠘⣧⣿⡄⡿⠀⡏⢻⣇⢿⡇
⠀⠀⠀⠀⠀⠀⣿⢈⠀⣿⢸⣿⣿⡓⣿⣿⢇⣿⢨⡹⠜⠘⣳⡨⠵⡀⠀⢸⣯⠘⡿⣿⡏⣼⢸⡇⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⡘⣿⣿⡧⣿⣿⣼⡇⠀⠀⠀⠟⠛⠉⠁⠁⠀⢸⣯⠸⣿⣿⡇⣾⣸⡇⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⡏⣸⣶⣿⡇⣿⣿⣯⣼⣿⡟⣿⢠⣶⣾⠀⠀⠀⢾⣿⡆⢰⣇⠸⣿⣿⡳⢻⣿⡇⢾⣿⠊⡇
⠀⠀⠀⠀⠀⢰⡁⣿⣿⢿⣿⣿⣿⣿⣿⠿⢸⣷⣟⣿⡟⠀⠀⠀⢸⡏⠹⡌⣿⣆⠋⣿⡽⣠⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠙⣿⣎⠻⣿⣿⣿⣡⣿⣿⣿⢴⣿⠁⠠⠀⠀⢸⡄⠰⣿⣿⣿⣿⡇⢃⡬⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠸⡿⢿⣿⣿⣿⠌⣿⣿⣿⣿⢿⠻⣿⡿⠀⠃⣀⠀⠘⣿⡀⠿⣿⡙⡏⢀⣽⣿⣿⢿⣿⡅⢺⡇⢰⣦
⠀⠀⠀⠀⠀⠈⢿⠘⣿⣿⡛⣰⢿⣿⡿⡿⣸⣿⡿⠁⠀⠰⣬⠀⠀⠘⣮⣄⡹⢻⢸⠀⣾⣿⣿⡯⡏⢰⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢷⣌⣿⣷⣿⡻⢏⣑⣓⣿⠟⠁⠀⠠⠆⠀⠰⠀⠀⠘⢷⣒⣲⠜⡿⠛⣛⣉⣅⣶⣿⠃
⠀⠀⠀⠀⠀⠘⠂⠀⠉⠛⠛⠛⠛⢛⠋⠉⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠈⠙⠳⠾⠿⠿⠿⠿⠿⠛⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠇⠀⢀⠔⠁⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠤⠂⠁⢀⠞⠀⠈⠉⢉⣥⠏⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⢀⠀⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢄⠂⢄⡀⠐⠒⠒⠒⢆⡁⠀⠀⠀⠀⠀⠈⠒⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠀⠀⢲⢀⠀⠀⠀⡀⠈⠐⠠⠀⠀⠀⣥⠑⠒⠒⠦⣤⢤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠈⠐⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢃⠀⠀⠈⣷⠀⠈⠀⣤⣤⣴⡾⠏⠀⠀⠀⠀⠀⠙⠻⠶⠖⠂⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢢⠀⠀⢉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠨⠴⢠⣕⢄⡀⠀⠀⠀⠀⠘⠳⣦⡄⠀⠀⠀⠀⠀⣴⣿⣁⠀⠀⠀⣄⠸⠀⠀⠊⡀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠙⠄⢈⠐⠀⠀⠀⠠⠖⠛⠁⠀⠀⣀⣀⡀⠀⠈⠉⠀⠀⠀⡉⠲⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⢹⠆⠈⠂⣄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠈⠒⢀⡀⠀⠀⠀⠀⠀⢀⠔⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠉⠣⡀⠀⠀⡔⠀⠋⢀⡎⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠒⠦⠀⠀⠀⠀⢀⠤⠖⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠐⠁⡀⠤⠄⢤⣄⠀⠀⠀⠈⢀⠀⣘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⣤⠤⠤⠀⠐⠈⠇⠀⠀⠀⠀⠀⡈⢐⡌⠀⠀⢀⣾⢿⡗⠀⠁⠀⣸⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠤⣀⣀⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠒⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡲⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢎⣴⣾⠟⠿⢦⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⠋⠀⠢⠀⣈⠚⠁⠈⡙⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⡂⠖⠲⠁⠀⢿⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠠⠀⠠⢀⠀⠂⠀⣼⠃⠹⠁⢲⡏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡰⠒⠒⠀⠄⡃⠀⢢⠐⣅⠀⣧⡸⣅⡀⣠⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠍⢀⠄⠀⠀⡴⢹⠋⠀⢈⠉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡕⠀⠀⠀⠱⠂⠁⠈⠀⠄⠤⢠⠋⠶⡌⠀⢀⢴⠆⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡈⡀⠰⠀⠀⠀⠨⠀⡐⠆⠰⢀⠀⡺⠀⠀⠀⠄⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠀⠀⢀⠀⠀⠀⠀⠀⣷⣤⣽⡀⠀⠀⢀⠀⠀⢨⣤⡤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⡃⢦⡸⠄⠀⣂⣀⠀⢉⠈⡻⡇⠀⠀⠀⠀⠀⠐⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢁⠀⠀⠁⢧⠀⠀⠀⠀⢸⡴⣿⢵⡖⣒⡲⠒⢪⣇⠼⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠂⠀⠔⣆⠀⠀⢀⠈⠳⠀⠀⠀⠈⠀⠀⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣂⠀⠀⠈⣄⠀⠀⠀⠀⠐⡆⠀⠀⣄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⣍⠂⣈⢆⠈⢀⠀⠀⢂⠀⢀⡉⠆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠁⠑⠀⠐⢤⡎⠀⠸⡇⡘⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⢱⠇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠆⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡥⠀⠹⠇⠨⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡤⠴⠆⠰⠤⢤⡤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣠⠸⠆⠀⣿⣿⠃⠀⣿⢿⡇⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⠆⢰⣞⢦⠀⠊⠀⠈⢹⣿⠁⠀⢿⠺⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡻⣿⣿⣿⣇⣦⡀⠀⠰⣼⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣇⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣏⣿⣽⣏⡀⠀⠸⣿⣿⡇⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣽⣿⣼⣼⣾⣧⣅⢠⣼⣿⣿⡇⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠾⠿⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢉⣽⣾⣿⢿⣿⣿⣿⣛⠉⠀⠀⢀⠀⠀⠀⢠⡄⠀⠰⣆⠐⣿⡄⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢹⣿⣿⣦⣀⢩⣿⣿⣿⠀⠀⢠⣿⣦⣾⣿⣿⣿⠐⢾⣿⣤⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠲⣼⣿⣿⣿⣿⣿⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⠿⣿⣿⣿⣿⣧⡀⣼⣿⣹⣿⣿⣿⣏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠿⠈⠭⣭⢭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢠⡿⡯⠈⠉⠉⣠⡟⢘⣿⡿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⣾⡏⠀⢀⡈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢻⡇⠀⠈⣉⣁⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠂⠖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠀⠘⠀⠉⠀⠉⠉⠛⠓⠒⠲⠦⣄⣠⠞⠀⠀⢀⣀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠁⠀⠉⠓⢦⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⠎⠁⠈⠳⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡤⠤⣄⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⢰⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣏⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⢀⡠⣿⡗⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⢚⠟⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠎⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠉⠰⣄⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⡴⠃⠀⡀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣁⡀⢰⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠘⠛⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠧⠤⣄⣀⣀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢰⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⢳⡀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣀⡀⠀⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢠⣴⡶⣦⡄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⣿⣿⣿⣷⣶⣶⣶⢂⣼⣿⠟⣿⣿⡇⢸⣿⣿⣯⣭⣍⠁⠛⣿⣿⡟⠛⠁⣼⣿⢿⣿⣤⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠿⠿⠿⠁⠀⠀⠀⠠⠿⠿⠛⠛⠿⠿⠇⠶⣾⢿⣿⡿⠟⠀⠰⠿⠿⠀⠀⠀⠙⠛⠛⠙⠛⠃
⠀⠀⣸⣿⣿⣿⣿⣤⣤⣤⣤⣤⡄⣾⣿⠇⢰⣿⡿⢠⣿⣿⠿⢿⣿⡷⢀⣾⣿⢂⣴⣾⡿⠿⣿⣷⡄⣸⣿⡗⢠⣿⣿⢂⣶⣿⣿⠿⡿⠟
⠀⢠⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⢸⣿⣿⣠⣿⣿⠃⣾⣿⠿⣿⣿⣍⠁⣼⣿⡏⢸⣿⣿⣀⣴⣿⡿⢠⣿⣿⣁⣼⣿⠏⣨⣛⣛⣻⣿⣿⠆
⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠀⠈⠉⠉⠀⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉










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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣤⢄⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⣠⡞⠣⡈⢢
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣷⣄⠬⠌⠘⠘⠳⠀⠀⠀⠒⠢⡤⠄⢣⠴⣎⠀⢴⠀⠘⠺⡄
⠀⠀⠀⠀⣿⣿⠟⣋⡀⠆⠈⢉⠳⡀⠰⠀⡀⠠⠐⠒⠒⣒⡂⠀⠘⡴⢷⡇⠀⠢⢀⣶⠃
⠀⠀⠀⠀⢙⠋⣆⠬⠌⠊⢘⣸⡷⠈⡂⠉⢡⠀⠀⠀⠀⠰⠄⠀⡠⠘⠤⣨⡡⠔⡋⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⣿⣷⣵⡶⠥⢺⡟⠁⠀⠇⠀⠀⡀⠀⠠⠄⢀⠄⡪⠊⡠⠊⠀⠑⠒⠁⠀⠀⠢⡀⠈⢂
⠀⠀⠀⠀⢀⠅⠋⠁⠑⢠⠿⣷⠀⠀⠀⠀⢀⠇⠀⠀⠔⠁⠀⠀⢰⠁⠀⡰⠄⠀⠀⠀⠀⢀⡇⠀⡸
⠀⠀⠀⠀⡎⠀⠠⠔⠂⢫⠀⢸⠀⢐⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠒⠁⢺⠀⠀⢠⣎⣁⢈⠕⠁
⠀⠀⠀⠀⢃⠀⠀⠀⣀⠼⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠒⠀⠈⠁⠐⠂⠀⠀⠀⠀⢀⠎⠉⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠤⠄⠤⢄⡀⠁⠢⡀⠀⡠⠔⠒⢧⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣤⠀⣀⠀⠙⠳⢄⢰⠋⠀⠀⠀⠘⢀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠈⠀⠀⠈⠑⢧⡤⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢴⡏⠀⠀⠐⠢⡄⠀⠀⠙⠠⢄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣀⣀⢀⠀⠀⠀⠀⠘⠾⠃⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⡬⠥⠈⠀⠀⠀⠀⠀⢠⡈⣡⠇
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣤⢀⣎⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⢾⡋⠈⡆⠀⠈⠓⣠⡄⠀⠀⠀⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⢸⡄⠀⣀⣀⣀⣂⣀⣀⠇⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠳⡄⠈⣶⠚⠳⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⡴⠟⠀⠀⣯⠀⠀⠈⠳⣄
⠀⣠⠖⠒⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢹⠇⡀⠀⠀⠈⢧
⢰⡍⠀⠸⢀⡴⣈⣭⣤⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣤⣤⣄⡼⢚⡇⠀⠀⠠⠀⢷
⣞⠀⠀⠀⡿⠀⢯⣀⡤⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡏⢀⡼⠀⠀⠀⢀⠞⠀⡇⠀⠀⡇⠈⣧
⢽⠀⠀⢸⠃⠀⠀⠉⢰⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡷⠋⠀⠀⢀⡔⠁⠀⠀⡇⠀⠀⢁⠀⠘⡆
⠘⣧⡀⡞⠀⠀⠀⠀⠈⠀⢠⠞⢁⡀⢄⠀⠀⢀⡴⠋⠀⢸⠀⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢱
⠀⠀⢉⠇⠀⠀⠀⠀⢠⣄⡜⠢⣤⢁⡜⠉⠉⠁⠀⠀⠀⣸⠀⠀⠀⢀⡀⠀⠀⠀⣸⠃⠀⠀⢸⠀⠀⠘⡄
⡄⠐⠋⠀⠀⢰⠀⡀⠀⡄⠀⠀⠘⠴⠀⠀⠀⠀⠀⢀⡴⠛⣧⡴⠚⠫⠉⠀⠀⠀⣿⠀⠀⠀⠘⡄⠀⠀⣇
⠀⠑⢤⡀⠀⡘⠀⡇⠀⡇⠀⠀⠀⠀⠈⠶⠶⠶⠞⠉⠀⠇⢸⠇⠈⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣧⠀⠀⡿
⠀⠀⠀⢳⡀⣇⣀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢻⡀⢀⠃
⠀⠀⠀⠀⠘⠃⠈⠇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠦⠿⠄⠀⠀⠀⠀⠀⠀⠀⠠⡙⠦⡄⣄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡲⠤⠬⠀⠀⠀⠐⠠⠬⠻⣆⡀⣀⣴⠋⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣉⡽⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡿⠋⠒⠒⠒⢹⠋⠀⠀⢀⣄⣩⡄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣠⠏⠀⠀⠀⠀⢀⠇⠀⠀⡔⠉⢊⠁⠀⠀⠧⣀⠐⢂⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣠⡞⠁⠀⠀⠀⠀⣠⠏⠀⠀⠸⠀⠀⠀⣇⠀⠀⠀⠈⢷⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢻⡄⠀⠀⠀⢰⠃⠀⠀⢰⠃⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠈⢣⠀⠀⠀⠸⣄⠀⠀⢘⡇⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡸⠀⠀⠀⠀⠈⢧⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⢸⡇⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠓⠲⠞⠃⢸⠀⠀⠀⠀⠀⡀⣽⠀⠀⡿⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢰⠁⠙⠓⠛⠛⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠊⠁⠈⠀⠀⠀⠀⠀⠀⣷⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣠⠿⠒⠉⠁⠀⠀⠀⠀⠀⠐⠂⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⠤⢀⣀⣀⣀⣀⣀⣀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⣠⣤⣤⡴⠶⠶⠞⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡛⠲⢤⣀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡒⠶⠶⠦⠤⠤⣤⢽⣄⡀⠉⠳⠏⠰⢴⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠛⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⢶⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡤⠚⠉⠓⢦⡴⠚⠉⠁⠀⠀⠀⠀⠘⣷⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠤⢤⡀⠀⠀⠀⢀⠴⠒⠲⠄⠀⠀⣟⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⣏⠀⣞⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⢉⣕⣬⠿⠀⠀⠀⠀⢶⡋⠀⠈⠀⠀⠀⢰⠃⡼⣫⠍⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡰⠚⡇
⠀⠀⠀⢠⣠⠶⠋⠁⠀⠰⡀⠀⠀⠀⡟⠒⣄⢲⣤⣾⠤⠤⣄⠀⠀⠀⠀⠀⠸⠶⢡⣌⡠⠟⠐⠒⠒⠤⠤⠤⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀⡃⠛⠀⣰⠗⢚⡆
⠀⠀⠀⢽⠃⠀⠀⠀⠀⠉⢉⠒⠒⠲⢹⡆⠉⠀⠀⠠⠤⡴⠋⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠐⠢⠀⠀⣀⠀⠀⠉⠉⠉⢉⡍⡇⠀⠈⡇⠀⠞⢁⣠⠞
⠀⠀⠀⠈⠳⣄⡀⠰⠤⠈⠉⣀⠀⠀⡈⣇⠀⠀⠐⢒⡿⠤⠤⣤⣤⡤⣴⡞⠋⠀⠀⠈⠉⠛⠲⠤⢄⣀⡀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠰⢯⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣥⣀⣀⣳⣈⠲⣉⡿⢁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠛⠃⢸⠷⠤⠖⠓⠢⢄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡽⠉⠙⠉⠁⣿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣼⠁⠀⠀⠀⠀⠁⣆⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠒⠠⠊⡟⠀⢠⠃⠀⠀⠀⢰⠒⠀⠈⢱⣀⠇
⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠉⡇⡎⠀⠀⠀⠀⠈⠲⣀⡴⠋⡌
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠚⠉⠀⣧⣤⣤⣀⣀⣀⣀⣠⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠶⡤⠤⣤⡼⣼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠑⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣇⠀⣠⠴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠙⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⣤⢤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣃⢹⡎⢣⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⡤⠽⢿⣌⢇⠈⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⣾⠁⠀⠀⠀⠈⢾⡰⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⢤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣆⠉⠛⠒⠲⠒⠚⢛⣉⣡⠶⠋
⣀⣀⡀⢀⣀⢀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣈⣉⣉⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠛⠶⢤⣤⣄⣀⣀⣠⣴⣶⢟⣻⡿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣻⢷⣶⣤⣀⣀⣀⣠⣤⠴⠞⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠉⠹⣅⡜⠁⣠⠞⢁⡜⢁⠞⢓⠛⢆⠙⣄⠙⢦⡀⠙⣄⡽⠉⠁⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢄⠀⠀⠀⠀⠀⢰⣟⠀⡰⠁⠀⢸⣅⠸⣄⠈⢀⡼⢀⣽⠀⠀⠱⡀⠘⣷⠀⠀⠀⠀⠀⢀⢜⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢻⡳⣄⠀⡀⠀⣀⢏⣴⠇⣠⠾⢁⡾⢠⠟⠀⠘⢇⠸⣄⢹⢦⡀⢷⡈⢇⠀⠀⡀⢀⡴⣻⢏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠩⠠⢦⣼⣿⠿⣷⣤⠶⢛⣭⢏⡼⢁⡾⢋⡼⠃⠀⡄⡄⠀⠻⣌⠻⣆⠻⣌⢯⣝⠳⢦⣴⡿⢿⣿⣤⠆⠨⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⣴⠟⢓⣿⡴⢲⡶⢪⢿⠃⣾⠁⢸⡇⡾⠀⢣⠀⠛⠃⢠⠇⠸⡆⣿⠀⢹⡆⢻⢯⠲⣶⠶⣼⣗⠛⢷⣄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⢡⠊⣠⣿⢠⠏⠀⣾⢸⠀⣇⠀⡀⢻⡇⠀⢸⡄⣠⡀⣼⠀⠀⣿⠃⠀⢀⡇⢸⢸⡆⠈⢧⢸⣧⡈⢦⢹⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⣏⢹⡼⡀⠀⡿⢸⣠⠻⣄⠈⢆⢷⡀⠀⠿⢋⠻⠇⠀⣰⢇⠎⢀⡼⢣⣸⠸⡇⠀⡸⣼⢉⣷⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣸⣇⠘⣇⠳⡞⠁⢸⡿⣄⢈⣓⣦⡳⣝⢦⡀⠀⠈⣠⢞⡵⣫⣖⣋⣀⡼⣿⠀⠙⡶⢃⡿⢀⣿⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⠀⢹⡿⢦⣽⣦⣸⠧⣼⠉⢩⠿⢷⡘⣆⢳⡄⣴⢃⡞⣰⠿⢯⠉⢹⡤⢿⣠⣾⣡⠞⣿⠁⢸⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⣾⠀⣼⠃⠱⣮⣀⠈⠓⠭⣀⡼⣠⢆⡧⣿⠀⣷⡇⢸⡷⣇⢦⡸⣄⡨⠕⠋⢀⣨⡷⠁⢻⡄⢸⡄⠙⢶⣄
⠀⠀⠀⠀⠀⠀⢰⢿⡁⠹⣠⠏⣸⠃⣠⠺⣏⢿⣏⠓⢶⣶⣶⡷⣿⣘⣁⣼⠏⢿⣄⣛⣸⡷⣷⣶⣶⠖⢋⣿⢏⡿⢢⡀⢻⠈⢧⡸⠁⣹⢷
⠀⠀⠀⠀⠀⠀⠸⣄⠉⠳⣽⣆⠘⠷⣏⠹⠿⣦⣈⣻⣒⣾⠷⣶⡿⠋⢩⠏⢀⠈⢯⠉⠻⣷⡶⢿⣗⣚⣋⣠⡾⠿⢉⡷⠟⢀⣾⡵⠋⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣟⠈⡷⢄⡈⢣⠀⠀⠈⠉⣛⠋⠉⠁⠀⠀⢈⠳⣤⡴⠋⠀⠀⠀⠉⠉⢛⡋⠉⠀⠀⢠⠋⢀⠴⡏⢘⡷⠖⠉
⠀⠀⠀⠀⠀⠀⢠⠶⠶⠀⢸⡇⡗⠀⠱⡄⠈⢧⡤⠎⠉⠀⣠⢎⡳⣏⡧⢣⢹⢡⠳⣏⡷⣻⢦⡀⠈⠉⢦⣤⠇⠀⡴⠁⠐⡇⣿⠀⠰⠶⣦
⠀⠀⠀⠀⠀⠀⣿⡙⠒⠒⢋⡼⣁⣀⠀⢱⠈⠒⠀⣂⡀⠀⣿⣅⡀⠙⠒⠂⠀⠈⠒⠚⢁⣀⣿⡇⠀⣀⡂⠐⠊⢰⠁⢀⣀⡹⣌⠓⠒⠚⣹⡇
⠀⠀⠀⠀⠀⠀⢿⡛⠛⠛⣫⠞⠁⠈⣷⠘⣤⡆⠘⠃⠈⠳⣄⡀⠙⠳⣼⣄⣤⣄⣼⡴⠛⠁⣀⡴⠋⠀⠋⠀⣦⡞⢰⡏⠀⠙⢮⡛⠛⠛⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⢀⡟⣰⡿⣝⠢⣄⡄⠀⠘⣿⣟⣶⡾⢯⣥⣭⠿⣶⣞⣿⡟⠀⢀⣄⡤⢚⡽⣷⠘⣇⠀⠀⠀⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡵⢫⡇⠈⣦⡈⡿⣆⣤⣬⣿⣿⣧⣼⣿⣿⣤⣿⣿⣯⣤⣄⡾⡏⣠⡎⠀⣯⠳⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠉⢀⣾⠔⠋⠀⠙⢿⣼⡇⢹⡈⢻⡿⣿⣿⣿⡿⣿⠋⣸⠁⣿⣼⠟⠁⠈⠑⢼⣆⠈⠙⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣹⡿⣌⠻⣿⣾⣧⣿⣧⣿⣾⡿⢋⡼⣿⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⡮⢝⣿⣿⡟⣛⢛⣿⣿⣿⠭⣮⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡀⠀⠤⠤⠤⠄⠀⡀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣄⣀⣀⣀⣀⣬⠔⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠎⠉⠉⠱⢤⡄⠶⠀⠀⠰⠦⣄⠀⢠⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠹⠖⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⡀⣀⡀⣤⠒⢤⠀⢀⣀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠐⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠀⠀⠀⠀⠀⠀⢠⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠄⠒⡤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢀⣀⠀⠀⠀⠀⠈⢶⡉⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠓⠀⠀⢸⡆⣰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢹⠀⠀⢀⠊⠀⠀⢓⡄⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠛⠃⠈⠀⠀⠀⠀⢃⠀⢸⠀⠀⠛⡀⠀⠰⠋⠙⠢⠤⠀⠀⠤⠤⠀⠒⠒⠛⠛⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠘⢆⠸⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠰⠿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⢄⠠
⠀⠠⠄⠚⠁⠀⠀⠀⠀⠀⠁⠷⠀⠀⠀⠀⠇⠀⠀⢀⡳⠆⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢡⠔⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢣⡀⠀⠀⠰⠀⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠰⠇⠀⠀⠀⢸⠀⠀⠀⠀⣧⣀⠤⢿
⠀⠀⠀⠀⢀⡴⠁⢀⠎⠁⠀⠀⠀⠙⢄⡴⠃⠀⠆⠀⠀⠀⠀⡴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠦⡀⠀⠀⠀⡞⠢⠤⠔⠃⠀⠀⠀⡸⠁⠀⢀⡌
⠀⠀⠀⠀⠸⣇⡠⠛⠀⠀⠀⠀⠀⠀⠸⠆⢤⣄⣸⣄⠀⠀⠀⢧⡀⠀⢀⣀⠀⠀⠀⠀⠑⢄⣹⡀⠀⠀⠀⠱⡄⢀⡜⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⡿⠶⠄⢀⡀⢹⠀⠀⠀⠀⠀⠉⠑⢦⡀⠀⢱⠄⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣠⠞⣠⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⢣⣀⡀⠀⠀⠀⠀⠀⠘⠶⠶⠀⠀⠀⣼⡀⢰⠀⠀⠀⣀⠰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠋⠘⠋⠙⢧⠀⠐⢲⠀⠀⠀⠀⠀⠀⠹⣶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠉⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠶⢤⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡄⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠙⢤⣤⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠤⠄⠒⠂⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠸⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠉⠀⠑⠒⠒⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠒⠠⠤⠤⠤⠤⡤⠋⠀⠀⠀⠀⠀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠠⠤⠤⠀⠀⠤⠼⡄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⡶⠠⠤⠄⠘⠃⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⣰⣆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠋⠀⠈⠛⠁⠀⠈⣦⠀⠀⠀⣀⠔⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠙⠙⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠦⠀⠀⠋⢹⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠲⢶⣤⠐⢒⣄⠤⠄⠂⠀⠀⠀⠀⠂⠲⠶⠦⠤⠤⠄⠠⠀⠀⠀⠀⣀⣀⡀⠄⠀⠀⢀⡠⢄⣀
⠀⢀⡴⠦⠤⠤⡌⠱⠀⢥⣄⠀⠀⠐⠛⢉⡤⠅⡀⢠⡄⠄⡀⢀⠀⣀⠄⠛⠋⠉⣀⠀⣠⠘⢋⡀⠀⠀⢀⠀⠉
⠀⡎⢁⣤⡼⠔⠒⠆⠀⠀⠘⠓⠒⠐⠀⠀⠀⠂⠀⣿⠁⠀⠈⢩⠉⠙⢡⠈⠈⠉⠻⣷⡇⠉⠿⠇⠠⠀⠠⠀⠄
⠀⠹⠋⠀⠀⠀⠀⠀⠐⠿⣿⠋⠂⠀⣀⣀⠀⣴⣾⢿⡀⠀⠀⠀⢠⢦⡰⠠⣄⠀⠀⢘⣿⡄⢀⡆⠀⢀⠂
⢠⠃⢀⠞⣿⣦⠀⠀⠀⠀⠀⠀⠀⠰⢉⠀⣼⠟⠁⢀⠀⡀⠀⠠⣴⢺⡆⣄⠙⢹⠖⣼⠛⠉⢀⡭⠀⠀⠀⢐⡀
⠘⡄⠈⠀⠉⠉⣤⣀⠀⠀⡴⠞⠂⠁⠀⠈⠀⠀⠀⠀⠀⣳⣀⠀⠞⠋⠙⡆⠿⣤⡶⠶⠶⠛⠁⠀⠀⠀⠙
⠀⠙⠦⠄⠀⢀⠼⣯⣉⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⣧⡁⠈⠀⠀⠀⠈⠁
⠀⠀⡂⣴⠞⠀⠀⠈⠳⠄⣀⠉⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠙⠦⣄⡀⠁⠠⡀
⠀⠀⢙⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⠀⠀⢦
⠀⠀⠉⠀⠀⣠⠒⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⣀⢀⢀⡉
⠀⠀⢀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⢀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠊⠀⠀⢹⣀⠀⡀⠀⠨⠁⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠁⠀⠀⠀⢀⡀⢳⠀⠻⡀
⠀⠀⠀⠀⠀⠀⢙⣫⣥⢤⣤⠆⠀⠀⠀⠂⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠆⠀⠀⢠⠀⠀⠹⡠
⠀⠀⠀⠀⣠⠾⠛⢡⠈⠂⢀⣤⠐⠈⠀⠓⠊⠀⠀⠠⠀⠀⡀⠀⢆⠀⢸⡗⠆⠀⠀⢀⡀⠈⢉⠂⠤⡄⠱⡀⠐
⠀⠀⠀⢰⣿⠀⠀⢈⣇⠀⠀⠀⢰⠀⠀⠀⠁⠈⠀⠓⠀⠀⢳⠀⠀⠀⠀⢷⡤⣀⡀⣠⣀⠀⠀⠀⠀⠀⢶⠁
⠀⠀⢀⣾⣿⡆⠀⠘⣿⠓⠀⠆⢀⠄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⣦⡀⠀⣼⠃⠈⢠⣭⠣⠖⢀⡄⠀⠀⢀⣦
⠀⣀⣴⣿⡿⣧⠀⠀⢸⡿⣿⠾⢻⡀⠀⠀⠀⢀⣀⡰⠖⠀⠀⠀⠈⢁⡾⣿⣦⡁⢀⡿⠃⠀⢀⡇⠀⠀⢠⢿
⢈⠛⠃⠈⠃⠉⢦⡀⠀⠀⠈⠋⠼⡕⠛⢸⠿⠋⠁⠀⠀⠀⠀⠀⢰⣿⣧⣿⣯⣷⣾⣿⠃⡀⢾⠀⠀⠀⡐⢸
⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠲⠷⣄⠀⠀⣀⡀⠀⠀⠀⣀⣀⣸⡟⣡⠟⠻⣿⣿⣏⣷⡴⠎⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠠⠉⠀⠠⠂⠈⠈⢳⣶⣦⣤⣄⣁⣀⣀⡀⠀⠀⢀⣴⠁⡀⢠⡀
⠓⠀⠀⠈⠉⢰⡄⢠⡇⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠱⣦⣿⠌⣿⣿⣿⣏⢻⣿⣿⣱⡶⢿⠛⠁⠄⠀⣸⣿
⠸⠂⠀⠀⠀⢸⡇⣼⠁⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⠜⡁⠀⠊⣻⣷⣿⣿⠟⣇⠙⣿⣿⡿⠒⠶⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⢸⣷⠏⠸⠃⠀⠀⠐⡤⢀⠀⣴⣿⠋⣰⠄⠀⣱⣿⣿⣿⣿⣎⣿⡆⠟⢉⠁⠀⠀⠀⠀⠀⡰⣿⡛
⠒⠀⠀⠀⠀⢸⡿⢰⢾⠀⠀⢀⣦⡶⣞⣭⡟⠁⠘⢡⡴⣿⣿⠟⠻⢍⢹⠟⠉⣿⣆⢸⣇⣀⣀⣠⡴⣷⣇⣹⡇
⠁⡄⠀⠀⠀⣾⢀⣎⣈⣤⣶⠊⠙⠟⠉⠁⣀⣤⣾⡿⠋⠉⠀⠀⠻⣄⢤⡉⠀⠀⢻⣄⡿⢿⢾⣲⡺⠋⠿⢻⣿
⣆⠄⠐⠒⢠⣏⣾⡟⣿⣿⣤⣶⣴⣶⣿⣿⣶⣾⠿⠛⠂⠀⣀⣀⣂⣠⣶⣦⣤⣤⣌⣹⣿⣖⠒⠈⠳⠤⣴⡿⠁
⠋⠀⠐⠃⣼⡏⠉⣍⣹⣿⣿⣿⣿⡿⠟⠋⠡⠠⠈⠁⣨⣿⣿⣿⣿⣿⡿⠿⠛⠻⠍⠫⠭⢿⣿⡶⢦⣜⠋⠁⢀
⣼⣿⠞⢚⣯⣥⣶⣿⠟⠋⠉⠉⢇⣂⣀⣤⣵⣶⣶⣿⠿⠛⠛⢋⣁⠐⠊⢉⡳⠶⢶⠶⠤⢤⣭⣭⠉⠛⠄⠢⡇
⣿⠏⠀⢿⣋⣉⣀⢀⣤⡶⠾⠿⠛⠛⣻⡛⣫⠛⣷⣾⣶⣶⡿⡤⠋⠸⠎⠻⠟⢟⣛⠻⡛⠦⡏⠁⢳⣤⣼⠂⣩
⣏⡄⡀⠈⠓⠚⠛⢻⣯⠀⠀⢠⣀⠞⠉⠓⠉⠉⠁⠉⠀⠀⠠⣤⡞⣧⠸⡗⠈⠙⠚⠛⠟⢲⡿⠒⡏⢹⡃⢀⣿
⡟⠄⠁⠀⣤⣼⠌⠟⠛⠳⠶⣾⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣧⣼⡲⠾⢿⠿⣿⣶⣦⣠⠃⠈⢠⣼⠟
⣧⣤⣿⣶⣯⣭⣴⣦⣤⣤⣤⣤⣤⣤⣦⣤⣤⣶⣶⣶⣶⣤⣤⣤⣤⣤⣭⣭⣭⣬⣧⣭⣽⣴⣶⣿⣧⣿⣿⣧⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠻⣦⣀⣤⣤⠶⠶⠟⠛⠛⠛⠛⠛⠛⠳⠶⣦⣤⣠⣴⠟⠁⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠰⢶⣤⣀⡀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣀⣠⣴⠶
⠀⠀⠀⠀⠉⠛⣿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⠛⠋⠁
⠈⠛⠛⠷⠶⢶⣿⣄⣀⣉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⢋⣁⣠⣿⣤⠶⠶⠞⠛⠋
⣀⣀⣀⣀⣀⣀⣇⢈⣉⣉⡛⠛⠶⢶⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⡷⠶⠞⠛⣋⣉⣉⣿⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⢻⡏⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⣹⡟⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠟⠋⣿⠃⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠛⠛⠛⠻⢷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠟⠛⠛⠛⠻⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣤⣄⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⣠⡏⠀⠀⢰⡟⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢿⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠸⣿⠷⠶⠟⠋⠀⠀⣠⡿⠁⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠰⣿⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠐⠿⠶⠶⠟⠋⠁⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣦⣄⡀⠀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⡀⠀⢀⣀⣴⠏⠀⠀⠀⠀⠈⢷⣄⣀⠀⢀⢀⣴⡟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠎⠉⠉⠱⢤⡤⠶⠆⠀⠲⠦⣄⠀⢠⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠹⠖⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢠⣀⢀⡀⡤⠒⢤⠀⢀⣀⡀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠀⠄⠐⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⢠⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠄⠒⡤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢀⣀⠀⠀⠀⠀⠈⢶⡉⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠈⠀⠀⠐⠀⠀⢸⡆⣰⠄⠀⠀⠀⣀⡀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢹⠀⠀⣠⠂⠀⠀⢓⡄⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠐⠒⠃⠈⠀⠀⠀⠀⢃⠀⢸⠀⠀⠻⡀⠀⠰⠋⠙⠢⠤⠀⠠⠤⠤⠀⠘⠒⠛⠛⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠘⢆⠘⠀⠀⠀⠈⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠰⠿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⢄⠠
⠀⠠⠄⠚⠁⠀⠀⠀⠀⠀⠁⠷⠀⠀⠀⠀⠇⠀⠀⢀⡳⠆⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢡⠄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢣⡀⠀⠀⡠⠀⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠈⠙⠂⠀⠰⠇⠀⠀⠀⢸⠀⠀⠀⠀⣧⣀⠤⢿
⠀⠀⠀⠀⢀⡴⠁⢀⠎⠁⠀⠀⠀⠙⢄⡴⠃⠀⠆⠀⠀⠀⠀⡴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠘⠢⡀⠀⠀⠀⡟⠢⠤⠔⠁⠀⠀⠀⡸⠁⠀⢀⡌
⠀⠀⠀⠀⠸⣇⡠⠛⠀⠀⠀⠀⠀⠀⠘⠆⢤⣄⣸⣄⠀⠀⠀⢧⡀⠀⢀⣀⠀⠀⠀⠀⠑⠄⢹⡀⠀⠀⠀⠱⡄⢀⡞⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠁⠾⠒⠄⢀⡀⢻⠀⠀⠀⠀⠈⠉⠑⢦⡀⠀⢱⠄⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣠⠞⣠⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠽⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⢣⣀⡀⠀⠀⠀⠀⠀⠘⠶⠶⠀⠀⢀⣼⡀⢰⠀⠀⠀⣀⠠⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠋⡘⠋⠙⢧⠀⠒⢲⠀⠀⠀⠀⠀⠀⠹⣶⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠉⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠶⢤⡀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡄⠀⠀⠀⡠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠙⢤⣤⠀⠀⢀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠤⠤⠄⠒⠂⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⡆⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠸⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡌⠁⠀⠑⠒⠒⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠒⠠⠤⠤⠤⠤⡤⠋⠀⠀⠀⠀⠀⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠠⠤⠤⠀⠀⠤⠼⡄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⡖⠠⠤⠀⠘⠃⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⢀⣸⣆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⠋⠀⠈⠛⠁⠀⠈⣦⠀⠀⠀⣠⠖⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠙⠛⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⠦⠀⠀⠋⢹⡀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⠛⠛⠋⠉⠉⠛⠛⠛⠓⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠶⠄⣀⠀⠸⠀⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠹⡇⠀⠀⠀⣠⡴⠯⠚⠉⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⢤
⠀⠀⠀⣀⡀⡀⢀⣀⣰⠲⢾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣷⣰⠶⢤⠴⠒⠲⢤⣀
⢀⣴⣭⣍⡑⠀⠉⠀⢡⠘⣾⠃⠀⠀⠀⢠⣀⣀⡼⢤⣤⠚⠀⠀⠀⠘⣿⠀⠈⠁⢀⠞⠉⠉⠙⡀
⠎⡠⠤⠤⣙⠀⠀⡄⠀⢁⣿⠀⠀⠀⠀⠀⡤⠈⠀⠀⠁⠐⠄⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠄⠠⢵
⠀⡆⠀⠀⠀⣁⣇⡇⠀⠀⠻⣤⣀⡀⠀⢀⡑⠚⠤⠤⠜⢁⡠⠀⢂⣿⠀⠀⠀⢀⣌⠐⠂⠉⠙⡞
⠀⢿⠀⠀⠀⡏⠁⠉⠀⠀⠀⠘⣇⠠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡞⡿⠀⢠⠔⠉⠈⣧⠀⠀⢠⡇
⠀⢸⡄⠀⠀⣇⠀⠀⠀⠰⡀⠀⠹⣆⠫⠀⡀⠀⠀⣀⠨⠁⠀⠀⣹⠃⠀⣼⠀⠀⢀⡏⠀⠀⢸⠃
⠀⢸⠇⠀⠀⠈⠦⣀⠀⠀⠹⣄⠀⠘⢧⡉⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⢀⡇⢀⡤⠚⠀⠀⠀⢻⡀
⢠⠇⠀⠀⠀⠀⢀⠈⢣⠀⠀⠙⣦⠀⠀⠙⠚⢷⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁⢀⠀⢄⠀⠀⠀⠀⢱
⠇⠂⠀⠀⢉⡀⠙⣤⠂⠀⠀⠀⠈⢷⣄⠀⠀⠈⠀⠀⠀⠈⠂⠀⢀⡾⠃⠀⠈⡧⠋⠀⠀⠀⠀⠀⡵
⠈⠲⠦⠑⠦⠜⠒⠋⠀⠀⠀⠀⠀⠀⣹⠷⢤⣤⣀⠀⠀⣤⡤⠴⣏⠁⠀⠀⠀⠈⠒⠅⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⢀⡤⠴⠀⠀⢠⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠁⠀⠀⠀⠀⠁⢄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⠀⠀⠀⠈⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠀⠀⠀⠀⠀⡀⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣽⣵⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣷⣶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡟⣩⣭⣭⡍⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⢋⣥⣍⠻⢃⣿⣿⣿⣇⠸⠟⣡⣬⡻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⣿⣿⣿⣿⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡷⢨⣿⣿⣿⠟⣋⣭⣍⡙⢿⣿⣿⣿⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⣶⣶⣾⣿⣿⠇⣾⣿⣿⣿⣿⣦⢹⣿⣿⣷⣶⣶⢸⣿⣿⣿⡟⣉⣉⢻⣿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⡿⢸⢋⣴⣮⣥⣿⣿⣬⣥⣦⡙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣦⠘⣿⣿⣿⣬⣙⠛⠛⣋⣴⣿⣿⣿⠡⣶⣶⠟⠃⣹⣿⢟⣩⣭⡛⣿⣯⠐⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠌⣿⠸⣿⣿⣇⢸⣿⣿⡗⢸⣿⣿⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣝⠿⢟⣥⡍⣿⣿⣿⡟⣩⣌⠻⠟⣡⣾⣿⣷⠆⣹⣿⣦⣭⣭⣴⣿⣯⡰⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣇⡛⠛⠛⢃⣾⣿⣿⣿⣿⣿⣿⣿⣦⡙⢋⡉⣿⣿⢋⣙⢛⣵⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣥⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠉⠉⠙⠛⠛⢿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⠿⠟⠛⠋⠙⠻⣿⣿⣿⣿⣿⡿⠟⣛⣩⡭⣿⣿⣿⣿⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣤⣤⣄⣀⠀⠀⠀⠀⢹⣿⣿⡟⢼⣿⣿⣿⣧⣴⣾⣿⣿⣶⠀⠀⢹⣿⣿⡿⠋⢀⣭⣿⣾⣿⡆⣿⡟⠁⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣷⣶⡦⠙⡿⠿⠿⢿⠿⠛⠁⠀⠀⠞⠟⠋⠀⠀⠿⠿⠿⢿⡿⢣⣿⡇⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠈⠻⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⣀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠛⢳⣆⣀⣀⣴⣶⣶⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠋⣠⣶⣿⣿⣦⢻⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠚⠛⠃⠸⠿⣿⣿⣿⣿⣿⣷⡇⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⠃⢰⣿⣿⣿⣿⣯⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⡿⠀⠀⠀⠀⠀⢀⣠⡤⠤⣀⣀⠉⠉⠙⠻⣿⣿⡇⢠⡿⢋⣥⣶⡝⣿⣿
⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣴⣶⣦⡀⠀⣔⢠⡄⢀⠜⠿⠿⠋⠀⠀⠀⠀⢀⠀⠋⠑⠾⠿⠿⠟⠋⠀⠀⠀⠀⠉⠁⠉⣠⣿⣿⣿⣷⣿⣿
⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⠿⠟⠛⠀⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⣶⣿⣷⡆⣰⣶⣶⡈⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⠀⠹⢿⣿⣿⠟⠁⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣧⡛⣿⣿⣿⣿⣵⣿⡿⠿⠿⠶⢾⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⣠⣼⣿⠇⠀⠀⠀⠀⣀⣀⣄⢸⣾⣄⣿⣧⣷⣴⡇⡀⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⢋⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣿⣏⣷⣾⠀⢠⣷⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠈⠙⢿⣿⣿⡿⢟⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⣉⡙⢿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⣩⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⣾⣿⢟⣼⣇⠛⢁⣰⣶⣄⠀⠹⣿⠻⠿⠹⠟⢿⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⢡⣿⣿⣿⡿⠿⠛⠛⠿
⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⠿⠄⠀⠀⠘⣡⣄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠻⠿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣆⠘⣷⠄⠀⠀⠀⠉⠀⠹⣶⣾⣶⣶⡀⠹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣶⣦⣦⣴⣾⢟⣤⡤⠘⢿⣿⣿⣿⡏⠀⣿⣦⠀⠀⠀⢀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣄⣀⣀⣀⣾⣿⣿⣶⣿⣄⣈⠛⠿⠿⠿⣿⣾⣿⣿⠆⢸⣿⣿⣿⣿⡄⠙⢿⣿⣶⣾⣿⣷⣶⣦⣄⣀⣀⣤⣴⣶⡄⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⣀⣀⣀⡠⠄⠒⠒⠛⠓⠢⠤⠄⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠳⠶⣶⣶⠊⢀⣠⣶⠆⢀⣠⣀⡱⡄⠀⡀⡀⠀⡀⣀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡵⡄⠀⢀⡄⠀⠚⠁⠀⡄⠉⢸⡅⢿⠗⠘⠶⠇⠟⠲⣣⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡸⠆⠀⣀⡼⠀⠸⣇⠱⣌⠁⠀⣉⣉⠤⠴⠖⣙⢆⣩⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⠁⠀⠀⣨⠗⠀⢸⠃⠠⠃⠀⠜⡽⠉⠁⠀⠀⠉⢿⣿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⡇⠀⠠⠟⢻⠃⠀⢰⣅⠀⠈⠆⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠰⢲⠀⠈⠳⣄⠀⠀⣠⠏⣰⣿⣷⣶⡖⣶⣶⣶⡾⣤⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡎⢠⡄⠀⠀⠀⠀⢿⣄⠈⣆⡄⠀⠘⠄⠀⣵⢒⢻⣿⢻⢻⣷⣿⡿⠁⣸⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣤⡼⠧⠈⠃⠀⣠⠀⠀⠀⠈⠃⠹⠅⠀⠀⡄⢠⡏⠈⣾⣿⠇⢸⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠴⠶⢶⡄⠀⢀⣿⣀⡀⠀⢀⠀⢀⠀⠀⣼⡁⠸⣧⣦⢹⣿⠸⢿⡉⣿⠇
⠀⠀⠀⠀⠀⠀⣠⠖⠠⠥⠤⠤⠖⠒⠳⡄⠻⢦⣌⠙⠻⣿⠀⠸⡆⠸⣠⡆⠀⢻⡀⠻⡇⢼⣽⣷⣿⣿⣥⣆
⠀⠀⠀⠀⠀⠸⣡⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⡄⢹⡇⠀⢿⣄⠁⠸⣄⡄⠁⢀⠉⢸⣷⣤⠤⠤⠤⡔⡴⠒⠐⠒⠲⠖⠀⠶⠚⣉⣑⣢⡴⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣰⠇⣼⠃⢠⡇⢠⣤⠀⠀⢹⣆⡀⠀⠀⡀⠈⡇⠀⣾⠋⣀⣀⡙⠃⠀⠘⠂⠀⠻⠷⠤⠤⠔⠁⠀⠀⠀⠀⡄⢄⠀⠀⠬⣄⠀⠀⠈⢳⣤⣀⡬
⠀⣠⠶⠦⣾⣿⢀⣿⠀⣾⡀⠸⣿⡆⠀⢿⠉⠉⠙⠲⣣⠀⠃⠀⠁⡴⠋⠉⠉⠁⢀⣤⠖⢀⠼⠀⠀⣀⠀⢴⠀⢀⣷⢠⡗⢸⣷⠷⠶⠙⠿⢶⣄⢠⠦⣹⠟
⡀⡁⣠⠞⠛⢃⣼⣿⢰⣿⡇⠀⡏⡄⠀⠈⡇⠀⠀⠀⠸⣄⣀⣀⣀⣁⠀⣀⡤⠶⠿⠏⠀⠀⠒⠒⠚⠉⠁⠈⢶⣾⣷⢂⢄⠸⠟⣤⡀⢠⡖⢤⠿⠿⣦⠟
⠷⣷⢃⡀⡔⠉⠁⢰⠟⠋⠀⠀⠈⢮⠚⣨⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣸⣥⠾⠛⠿⠿⣇⠎⠀⠘⠉
⠀⠸⠺⢶⠇⣄⠀⢸⣀⠀⠀⣠⣼⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⣶⠿⣶⣾⣏⢿⣻⣛⡵⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠇⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠔⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⠜⠇⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢨⡀⠀⣨⠇
⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠑⢄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠐⣀⠀⠘⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠉⣗⠒⢇⠀⠀⠀⢸⡤⠼⠘⣿⠳⡂⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠘⣆⠈⡄⠀⠀⠈⠙⢄⢠⣿⠀⢣⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠘⡗⢶⠀⠀⠰⠀⠈⢻⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠸⡀⢆⠀⠀⠱⣄⣿⣿⡆⠀⠈⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠈⠳⢄⡀⠈⠙⠿⡇⠀⡰⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⢸⠉⢱⣶⣤⣈⡁⠀⠁⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⠁⠀⠀⠀⡞⠀⢀⠟⠋⣌⠀⡀⠀⠀⣷⣾⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⡜⠀⢀⡎⠀⠀⠈⠀⠣⣠⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠁⠀⠀⢰⣁⣀⡞⠀⠀⠀⠀⢠⠈⢸⠿⠿⠿⠾⡇⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⡀⠀⠀⠰⠛⠯⡆⠀⠀⠀⠀⠀⢔⠡⡔⡚⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣉⡙⠻⠄⢀⡉⠉⠀⠐⠂⠚⠀⠀⠀⣀⠱⠳⢈⠢⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣦⣤⣴⣶⠶⠾⠿⠶⠶⣶⣦⣤⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠁⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠘⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⢀⣀⡀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⢀⣾⠃⠀⠀⠰⣿⣿⣿⣿⣷⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠹⣿⠋⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠘⣧⠀⠹⣦⡀⠀⠸⣿⠀⠀⣠⡾⠁⢀⡿⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠛⢿⣷⣄⡘⢧⣄⠀⠙⠓⠾⠿⠶⠛⠉⠀⣠⡾⢁⣠⣾⣿⡉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⣠⣿⠋⠙⠿⢿⣿⣷⣦⣤⣀⣀⣀⣠⣴⣾⣿⡾⠟⠋⠁⠹⣿⣄⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣴⣾⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣷⣦⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⣉⣤⣴⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣤⣀⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠉⠀⠀⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⠿⠿⣿⣿⡿⠿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣶⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣶⣶⠿⠛⠉


⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠾⠛⠛⠛⠛⠙⠛⠛⠛⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⣸⣷⣤⣠⣶⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⢿⣾⠟⠉⠉⠛⢿⣤⣄⠀⠀⠀⣀⣀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡀⢠⣼⣿⠀⠀⠀⠀⢸⣿⢉⣤⣀⣠⣿⣿⣤⣠⣶⡄⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢘⣿⡶⠿⣧⣌⣍⢻⣦⣤⣠⣴⡿⢿⠎⣽⡿⠛⠉⠉⠙⢿⣿⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠿⣿⠃⠀⠀⠈⣿⡅⠘⠋⠉⠹⠿⠀⢰⣶⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣦⣀⣀⣠⡿⠛⠃⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⢀⣴⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⡿⠛⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠻⢿⣿⠿⠛⠿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠚⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣄⣀⣿⡄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⡿⠟⠛⠩⠛⣿⡘⡋⡈⠛⢻⣏⣉⡘⢹⠘⡿⠬⠛⣻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠄⣤⣄⣠⠤⠄⠘⡟⠋⠋⠛⠛⡏⠉⠉⠛⢻⣃⡀⢀⡿⠤⡞⣻⣿⡟⠢⠉⠛⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠙⠀⡄⣈⣧⠀⠀⠀⠀⣀⠤⠄⠄⠐⠇⠀⠀⠀⠀⠀⡉⠛⠁⠀⠀⢻⣤⣀⠀⠀⠁⢀⣼⠟⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡁⠀⠀⠀⠀⠈⠁⡀⠔⠒⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⡆⠀⠀⠈⠉⠒⣵⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠⠞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠒⠀⠀⠀⠀⢠⠎⠀⠀⠀⠠⠊⠀⠀⠀⠀⠤⠒⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⡿⠌⠊⠙⡧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠖⠓⢤⣉⠉⠀⠲⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⡅⣿⡿⠀⠀⢸⠁⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠉⠙⠛⠂⠀⡀⡀⠂⠀⠀⠀⠀⠀⠀⠻⠋⠁
⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣿⣧⣀⡀⣸⠀⠀⠀⠠⢀⡀⠸⠓⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⡯⠉⡉⠉⢙⠛⠒⠢⠤⠄⢀⡉⠁⠐⠠⠄⣀⠉⠓⠀⢀⠀⡠⠒⠊⠀⠀⠐⠁⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠄⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠢⠍⠀⠀⠀⠀⢀⠄⠐⠉⠀⠐⠒⠂⠀⠒⠒⠒⠂⠀⠂⠈⠈⠉⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⣉⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠒⠒⠦⠤⣄⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠛⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⣁⠒⠒⠚⠉⢀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⣿⣿⣿⣿⣿⣯⣿⣿⢬⣦⣀⣠⣩⣦⠀⠀⠀⣀⣀⡤⠔⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠃⠀⠒⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⢋⠉⠈⠘⣷⠛⠉⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠀⢀
⣿⣿⣿⣿⣿⣿⣽⣟⣹⣿⣽⢷⣀⣶⡿⠟⠀⢀⡠⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠘⠲⠶⠲⠶
⣿⣿⣷⣴⣿⣿⣿⣿⡧⡿⢿⣧⠉⠁⢀⣴⠚⠉⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐
⢿⣿⣿⣿⣿⣿⣿⣿⣳⡿⣛⠟⢰⣿⠟⠋⠁⠢⠀⣠⡶⠉⠀⠀⠀⠀⡘⠀⢣⠧⠄⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠐⡀
⣬⣿⣿⣿⣿⣿⣿⣿⣯⡿⣅⣶⢟⠛⠷⣢⡀⠀⠀⠉⠠⣀⠀⠀⢀⣼⣀⣀⡆⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⡤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢫⠖⠀⣠⡿⠇⢄⠀⠀⠀⡙⠲⣾⠟⠀⠀⠀⠉⠈⠃⠀⠀⠀⠀⡏⠉⠉⠀⠀⠀⠀⠀⠤⠒⣏⠀⠀⠈⣆⣀⣠⡴⠞
⣿⣿⣿⣿⣿⣿⣿⣿⢛⣔⣿⣵⡳⣬⣾⠟⠁⠀⠀⠉⠲⠞⠀⣰⡏⠀⠀⠀⠀⠀⠀⢤⢄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⢧⣤⡾⢿⣿⠡⠄⡠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠉⠐⣿⣤⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⢉⡶⠒⠒⠒⠈⠉⣄⣀⣶⡴⠶⡋⣼⠀⠀⣠⣷⣎⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣸⣿⣿⣖⣛⠿⡶⢰⡤⣾⣤⣤⣄⣠⡷⠀⠀⠀⠀⠀⢿⣄⣀⢤⣍⣷⣼⣦⡶⢟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣿⣮⣶⣷⣠⣦⣬⣭⠀⠀⠀⠀⢀⠀⢼⣿⣽⣾⣿⣿⣿⣿⣿⣈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠙⠦⠀⠀⠘⠃⠐⠀⠒⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠳⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢟⡿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢔⣆⠀⢀⠐⢦⠄⣀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⠠⠈⢁⠀⢧⠀⠙⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠤⣤⠒⣉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠐⠒⠀⠑⢄⠀⠀⠀⢂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢢⡀⢀⡀⠀⠀⠀⢃⣼⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⠤⠤⠀⡀⣑⠪⢠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠐⡄⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠑⢄⡑⢄⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⣎⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠵⢄⠱⡤⠂⠀⠁⠉⠟⠂⠀⠤⢀⠀⠀⠀⢀⠀⡈⣢⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢀⠀⠁⠈⠀⠀⠁⠀⠜⠁⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢻⡾⣦⣾⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣶⣶⣶⣶⣶⣿⣷⣤⠀⠀⠀⠀⠀⠀⢎⠠⣄⠒⢶⠶⣶⣦⣄⡀⢀⠀⣠⣴⠆⠀⢳⣾⢧⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⢀⠀⠀⠀⠈⠀⠀⠸⠀⠑⠉⠢⣹⠈⠁⢸⣿⡁⣀⡤⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠏⠉⠻⠋⠉⠁⠀⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠒⠉⠀⠀⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠎⡥⠀⠀⠠⠀⣀⠛⣀⠀⠒⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣑⣀⡐⣠⣬⣄⣀⣀⣀⣀⣈⣵⣶⣦⣤⣤⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠉⠙⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⢂⣴⣿⣿⣿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣦⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠇⣾⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠛⠀⠿⠿⠿⠟⠛⠋⠁⣸⣿⣿⣿⣆⠉⠛⠿⠿⠟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠔⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⡿⣿⡿⠀⡅⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣀⠘⠇⠀⠀⡀⢉⢀⡀⡀⡀⢤⢀⣤⣄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠈⠿⠿⠃⢀⢰⢶⢀⢀⣅⣙⣈⣉⡏⢾⣿⡟⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇⣦⣾⣿⡆⠘⠹⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠇⣿⣿⣿⣇⢺⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⣿⣿⣿⣿⡞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣧⠛⢿⣿⣿⠿⡇⠻⠿⠁⠋⠉⠘⠹⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣏⢠⣠⠁⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠧⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠈⠛⠉⠁⠀⠀⠀⠂⠬⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠓⠚⠙⠋⠐⠢⠤⢀⣀⡀
⢀⣀⣤⣶⡶⠞⠛⠉⠁⠀⠀⠀⠠⢀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⣙⣒⣲⣶⣶⣦
⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛
⣿⣿⣋⢛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⢲⣤⣬⣭⣍⡉⠉⠉⠉⠉⣉⣭⣭⣤⣤⠒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⣇⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⠔⢤⠲⡚⠉⠹⢻⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠘⢧⡀⠀⣤⣀⠀⠀⢸⡆⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠉⠙⡖⢢⠔⢄⣀⡀
⡔⠃⠀⠈⠆⠈⡧⠀⠞⠂⠸⡇⠀⣠⡶⠋⠁⠀⠀⠀⠀⢀⣀⠀⠀⠙⠶⠏⠉⠛⠲⠾⢃⣀⠀⠀⠈⠙⠓⠶⠦⣤⠤⠀⡿⠀⠚⠆⠀⡋⠀⠎
⠧⠤⠄⠤⠈⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣉⠻⠂⠀⠀⠀⠀⠀⠀⠺⢋⣁⣀⡀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠈⠠⠀⢤
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⡇⠀⠀⠀⠀⠘⠋⠙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠙⠀⠀⠀⠀⠀⣿⣀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⢹⠀⠉⠓⠲⠤⠤⢤⢀⠀⡞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡆⢀⡀⠤⠤⠴⠒⠛⠁⢸⠂
⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠈⠑⡙⠦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠞⡑⠁⠀⠀⠀⠀⠀⠀⠴⡏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣧⠀⠀⠀⠀⠀⠀⡤⠰⢤⣤⡤⠤⠤⣤⣤⠴⠢⡄⠀⠀⠀⠀⠀⢠⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢄⠘⢧⠀⠀⠀⠀⠀⠹⣠⠎⠀⠀⠀⠀⠀⠈⢦⡸⠁⠀⠀⠀⠀⢠⠟⢀⠜⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣆⣀⣀⣀⡀⢀⣙⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣞⡁⠀⣀⣀⣀⣀⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⠝⠋⠀⠀⠀⡾⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠋⡀⠀⠀⠈⠛⢅⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠈⠉⠀⢤⠒⠘⢺⠃⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠚⠛⠉⠁⠀⠘⣦⣗⠐⠋⢢⠄⠈⠁⠘⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢂⠟⠂⢠⠜⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⠀⠘⠄⠀⠚⠑⢢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠰⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠉⠙⠋⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠢⠇⠀⠀⠂⠀⠀⠇⠀⠀⠦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁⣠⡀⣀⣀⣀⡄⠀⠀⣅⣀⣀⣀⣠⣀⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⡽⠀⠘⣅⣀⣀⣀⣀⣀⣀⡁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⣀⠀⠀⣠⡦⢰⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶
⠰⠐⠇⠙⢙⣿⣻⢏⠈⡄⠀⢠⠀⡀⣀⠈⠈⠀⠀⠀⠀⠀⢀⣀⠀⠐⠆⠀⡀⠀⣠⠀⢀⠀⠃⠀⠀⠹⠿
⢠⣴⣿⣶⣿⡏⠀⢧⠶⣧⠀⢘⡟⢱⣯⠅⣿⡳⠀⠀⠀⠀⠈⠙⠁⢀⡦⠞⠛⠉⠉⠙⠻⢤⣴⣚⣶⡄
⠘⠉⣉⠙⠻⠷⠒⠆⠀⠸⠀⠈⠁⠘⢳⡘⠮⠇⠀⢀⠀⠀⠀⠀⠐⡞⢀⠀⠀⠀⢀⠃⡀⢀⢈⡿⢁⠙⠆
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⣇⠀⠤⣦⣷⣨⣤⣷⣧⣾⡵⡏
⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠘⢦⣐⣚⣭⠭⡙⠛⣫⠭⠕⠃
⠀⠀⠈⠕⠂⠶⠠⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠤⡀⠀⠀⠀⠸⠀⠁⠀⠡⠀⠀⠀⢰
⠐⠽⠆⠀⢂⠁⠀⠀⠀⠈⠨⢉⠀⠻⠀⠀⠠⠁⠈⠀⣀⣤⡤⠀⣄⡠⠇
⠀⠀⠁⣰⠜⡿⢬⡀⠀⠀⠀⣁⠈⠃⠀⠀⠁⢰⠾⠛⠋⠉⢀⡶⠳⠶⠾⠶⠄
⠀⠰⣀⠻⣗⡀⠀⠀⠑⠀⣠⣄⢀⠀⠀⠀⠁⠀⡀⠠⢰⠺⠁⠉⠂⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣤⣄⢆
⠀⠀⠁⠠⠄⠑⠀⠤⠒⢚⠙⢋⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣵⡿⣿⡆⡿⣿⣿⣯
⢠⣶⣶⣶⣶⡖⠊⠁⠛⠃⢓⠃⠈⠐⠆⠀⠀⠀⢀⡤⣤⡂⠀⠀⠀⠀⠀⣶⠀⠀⠈⣉⠛⣍⠳⠷⣯⡽⢻
⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣶⣴⠤⡶⠀⣿⢾⣦⠡⣿⢰⡧⡄⠀⠈⣶⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡆⠀⠀⠀⠀⡠⡈⣄⣙⣋⠤⡛⢛⣀⠒⠂⣒⠘⠛⢐⣻⣝⣳⠇⢀⣴⣿⣿
⠸⠒⠀⠀⢀⣠⠝⠻⣿⣿⣿⠕⢾⣶⣶⣶⣷⣎⡺⠊⠙⠲⠮⢭⣤⣉⠉⠉⣉⣉⣁⣭⣭⡵⢾⣿⢿⣿⡿
⠀⠀⠀⠐⠐⠀⢲⡀⠀⠀⢀⡖⠀⠀⠀⠀⠈⠈⠈⠈⠀⠀⠈⠉⣩⣉⣧⣮⣭⣿⡅⡉⢩⠉⠉⠉⠉⠈⠘
⠀⢠⢿⣤⣏⣠⡀⢁⣄⢀⣀⡀⡀⣀⡴⣦⣸⢳⡟⠟⠦⣤⢠⣠⠄⢻⣿⣿⣿⣿⣧⣿⣇⣤⡖⢀⡀
⢠⠟⠈⠿⠻⠼⠃⠿⠿⠿⠼⠛⠯⢋⣛⡏⡟⠚⠀⠀⠀⠀⠀⢀⣠⣿⣿⢿⡛⣛⢿⣋⣯⣹⣙⣫⡀
⢀⠢⠤⠂⣀⣄⣀⣄⣀⢤⣴⣶⣽⡿⠔⠓⠀⠀⠀⠀⠐⠀⠀⠚⠛⠛⣿⣶⣿⣿⢿⡿⢿⡿⢯⠙⠃
⠾⠿⠧⠀⠠⡥⠿⠇⠀⠀⠉⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠽⠿⠿⠿⠿⠼⠇⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⢲⡴⠋⠉⠓⠋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢻⡀⠀⢘⡞⠁⠀⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠓⠀⢸⡇⠀⠀⠀⠀⠈⢳⡀⣠⢦⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘⣇⠀⣤⠀⠀⠀⢸⣷⣧⡆⣷⣄⠀⠀⢨⣧⡀
⠀⠀⠀⠤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡾⠀⠉⠀⠀⠀⠈⠛⠡⠴⠃⠈⢧⣠⠏⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣨⠴⠒⠛⠛⠛⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣨⠟⡶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣧⣀⠀⠀⢀⡴⣣⠞⠁⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠚⠷⠶⠤⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣏⠉⡿⠉⠉⠉⠀⠀⠀⠙⢦⣀⣀⣤⠰⡄⣶⢼⢧⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⣷⣿⡾⢮⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠋⠀⠈⠛⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠛⠛⠓⠒⠒⠲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣤⡤⠖⠚⠛⠛⠽⠿⠛⠈⠉⠉⠻⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠻⢦⣶⣶⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣸⠟⠉⠀⣄⣤⣶⣶⣶⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⡈⠉⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⠋⠀⠀⢀⡄⢿⣿⣿⠿⠿⣫⡾⡇⠀⠈⢀⣤⡾⣿⣾⣿⣿⣷⡄⠀⠀⠈⠣⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠀⠀⠀⠀⠈⠟⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠿⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠻⠟⣳⢶⣦
⠀⠀⠀⠀⠀⠀⢀⡾⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣛⣭⡾⠇⠀⠀⠀⠀⠀⠐⣾⢹⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣇⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢰⢻⣧⠀⠀⠠⠀⠀⣴⡀⠀⠁⠀⠀⠀⠞⣿⣿⣿⣶⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⡘⣿⡷
⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡾⣿⠂⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⠾⠃
⠀⠀⠀⠀⠀⠘⢿⡦⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⠀⠀⠐⣻⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠀⠀⢿⣿⣿⣿⣄⡀⠈⠛⠋⣭⣙⡫⣿⠋⠀⠀⠀⠀⢐⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠰⠀⠙⠻⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⢠⢶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣴⣂⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣈⠻⣦⠀⠁⠀⠀⠀⠀⠀⠸⠿⠯⠍⠀⠁⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⢶⣶⣤⣤⣟⣻⣧⣠⣼⣏⣡⣀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠿⣯⡽⠿⠁⠀⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⡏⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢋⣵⣦⠀⠀⡇⠀⠀⣠⠍⡓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠡⠞⣁⣘⣿⠥⠶⣺⢷⠲⠤⣬⣉⣙⠒⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣋⡤⠤⠖⠒⠋⠉⠀⠀⠀⣼⠃⠈⢧⠀⠀⠀⠉⠙⠓⠲⠦⢤⣉⣓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢠⡶⡄⠀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢳⡀⠀⠀⢶⡖⠋⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⠿⠀⣧⡞⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⠀⢠⡶⠒⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢧⠀⠀⠀⠀⠈⠀⠀⢀⠞⠀⠀⣀⣀⠀⠀⠀⣀⡀⠀⠹⡄⠀⠀⠐⠿⡏⠀⠀⣸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⠀⠘⡆⠀⠀⠀⠀⠀⢠⠎⠀⠀⠚⠉⣹⠆⠀⡾⠉⠹⡆⠀⠙⣄⠀⠀⠀⠀⠀⢀⠇⠀⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣿⢀⣧⢻⠀⠀⠀⠀⢠⠏⠀⠀⠀⣠⡞⣁⠀⠀⣧⠀⢀⡇⠀⠀⠘⣆⠀⠀⠀⠀⡼⠀⠀⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣿⢧⢉⡘⣇⠀⠀⣠⠋⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠙⠋⠀⠀⠀⠀⠘⣆⠀⠀⢰⠇⢰⣳⣹⠞⣿
⠀⢰⣾⣟⣿⠒⢦⣄⢀⡤⠖⠛⢦⣿⠀⣞⠁⢹⡄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡞⠀⠈⠻⢺⡴⠛⠲⢤⡀⣠⡴⠒⣿⣻⣷⡆
⠀⠀⠳⣞⠛⠲⣼⡌⢿⠀⠀⠀⠈⣿⠀⠁⠀⢀⣷⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣁⠀⠀⠀⢸⠀⠀⠀⠀⡿⢡⣧⠖⠛⣳⠞
⠀⠀⠀⠈⠻⢯⣉⠭⢿⣷⣄⠰⣆⣿⣀⠴⠚⠉⠀⠙⢄⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⣠⠞⠁⠈⠓⠦⣄⣸⣰⠆⣠⣾⠿⢭⣉⡽⠟⠁
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣭⣿⢫⣿⣧⣤⠤⠖⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠻⠦⣤⣴⣿⡝⣿⣭⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⡴⢿⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⡿⢦⣄⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣽⠋⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⣤⠤⠤⢤⡤⣤⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠙⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⣳⡀⠀⢀⣠⠴⠶⡾⢿⣏⢿⣿⣳⣦⣌⣻⡇⠀⠀⠀⣿⣡⣴⣞⣿⡿⣹⡿⢿⡶⠦⣤⣀⠀⠀⣾⠁⠀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠙⢶⣋⣀⣀⣤⣼⣿⣽⢳⣶⠋⠀⠀⠙⣧⠀⡄⢰⡿⠁⠀⠈⢷⡞⣯⣹⣯⣤⣀⣀⣈⣷⠞⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠇⠈⠻⣄⡀⠀⠲⣼⢿⡳⢯⡴⠆⠀⢀⡼⠃⠸⣇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣀⡀⠀⠀⣰⢿⡀⠀⢠⡴⠛⠛⣛⣯⡻⠿⣺⣟⠛⠛⠯⣄⠀⢀⡟⢦⠀⠀⢀⣀⣀⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⢳⠀⣼⠁⠀⠉⠛⣯⠀⠀⠀⢉⣼⢻⡟⢷⡈⠁⠀⠀⣸⠛⠋⠀⠈⢳⠀⣼⠁⠈⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠈⠷⣤⡴⢻⠇⠀⠁⠘⡟⠦⣤⡴⠋⠀⠀⠀⠀⠘⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡟⣉⣻⠟⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⣸⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⣿⠇⠸⡄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⣿⡟⣴⡦⢥⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⡿⡿⠻⠃⠈⡄⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⣿⣿⠛⠙⠃⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣱⣿⡿⠛⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⣟⠛⣙⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣵⣟⡞⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣽⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⡸⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠰⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣱⢦⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⣾⣣⠏⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⣴⣧⣴⣦⣾⣇⣀⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢀⣾⢣⠏⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡔⣡⣾⠿⠛⠉⠈⠁⠀⠉⠙⠓⠶⣤⣧
⠀⠀⠀⠀⠀⣾⢯⡗⢴⠟⠀⠀⠀⠀⠀⠀⠀⢀⡜⢱⣨⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⣾⣯⡟⠴⠛⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⢢⡀
⠀⠀⠀⣼⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡐⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠇⢿⣄
⠀⠀⣼⡟⣼⠃⡀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⠀⠻⢳
⠀⣼⣿⣥⣅⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡇⢿⠀⡱⡹⣦
⠀⣿⡇⣭⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣦⣾⣦⣷⣤⣦⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣡⢧⡎⠀⠰⡀
⠀⠸⡇⢤⠰⢟⣄⠀⠀⠀⠀⢈⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠇⢠⡀⠀⠀⠀⢀⣰⡿⢋⡾⠀⠀⠀⠹⣄⢳⡄
⡀⠀⢻⣌⢳⣟⣷⣗⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣠⣿⠶⠿⣋⠴⠋⠀⠀⠀⠀⠀⠘⣦⠻
⠉⠉⠋⠙⠛⠿⣿⣿⣿⣛⣿⣟⣰⣎⣹⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠐⢫⣜⣧⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⣿⣿⣿⣿⣛⣿⣿⣿⣿⡿⠿⠿⠯⢍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⢿⣿⣿⡻⠛⢿⢏⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢸⡄⠹⣧⣉⣙⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢷⠀⢻⡀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠈⢆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢠⡤⠠⠄⢠⣤⠀⠄⢠⠤⠀⣬⠇⢠⡄⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠀⣀⠀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⡱⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⡠⠒⠀⠀⠀⠀⠀⢀⠀⢀⡠⠗⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⡰⢡⠆⠀⣀⠤⠤⠊⠑⢪⡶⠷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⠀⠀⠀⠀⠀⠀⠈⠳⣇⣏⣴⠋⠐⠄⠀⠀⠀⠋⠴⠒⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠁⠀⠀⢀⣀⠠⠤⠐⠒⠚⠦⣈⠉⠁⢆⠇⠀⠀⠀⠀⠊⠉⠀⠀⡠⠊⠑⡂⠤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⡠⠚⠉⠀⠀⠀⢀⣀⣀⡀⠈⠑⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡎⠃⠀⠀⡛⠢⣄⡈⠁⠁⠒⠂⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠉⠢⡀⠑⢄⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢀⡇⠀⠀⠉⠒⠤⣄⣀⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠳⡀⢀⣠⡴⠚⠁⢡⠀⠀⡌⠹⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠉⢣⡀⠀⠀⠘⠂⠔⢳⡀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠱⡀⠀⠰⠄⠀⠈⢿⣄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠱⡀⡄⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣱⠃⠀⠀⠀⠀⠀⢋⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⠀⠀⠈⡄⠙⠻⣦⣀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠙⠳⠾⠁
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣷⣶⣶⣶⡾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠾⠿⠿⠿⠿⠿⠿⠶⢦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣀⣀⣠⣤⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠑⠦⣄⡰⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⡯⠉⠉⠉⠙⠒⠚⣒⣀⣈⡁⠉⠹⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⢀⡤⠒⢤⣀⡠⠔⠊⠉⠁⠀⠨⣧⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⣄⡀⠀⠀⢀⡤⠖⠦⠀⠀⣽⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠤⡀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⡯⢀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠁⢸⡴⠛⠁⠀⣀⠘⠟⠀⠀⠀⠀⠀⣇⠞⠥⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡔⢉⡇⣀
⠀⠀⢠⡶⠋⠀⠀⠠⠧⣄⣀⣸⡅⠱⠜⠛⠛⣋⡳⠀⠀⠀⠀⠀⢉⡴⠒⠋⠀⢈⡉⠉⠙⠒⠒⠶⠦⠤⠶⡖⠒⢲⠀⢀⠞⢉⡼
⠀⠀⠈⢧⡀⠀⣀⡀⠖⠀⠀⢀⢡⠀⠀⠠⢥⣞⡀⣀⣀⣀⣠⠴⠋⠑⠒⠢⢤⣀⡀⠀⠐⠃⠀⠀⠀⠀⠋⡇⠀⢸⡄⠀⣴⠋
⠀⠀⠀⠀⠈⠑⠲⠤⢤⣍⣀⠘⣆⠢⣒⡖⠉⠀⠈⠉⠉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⣄⡀⠐⠖⢠⡧⠤⠼⠢⢄⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢩⠏⠙⠋⠁⠇⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⡏⠀⠀⠀⢰⠸⢤⠀⡰⠁
⠀⠀⠀⠀⠀⣜⠉⠉⠾⡄⡸⠀⠀⠀⠸⣉⠀⣠⢇⠃
⠀⠀⠀⠀⠀⠪⠤⡠⠄⠃⣇⣀⠀⠀⠀⠈⢚⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣙⣍⣭⣭⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡆⠈⠑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⠀⣴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢇⠈⠍⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣦⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡎⣧⠱⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⠔⠚⠷⡝⡀⡇⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠁⠀⠀⠀⠈⠗⠁⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣚⠳⠤⣄⣀⣠⠤⠴⢚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠖⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡄⠁⠀⠀⠀⠀⠢⡅⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⡌⠀⠀⠀⢐⠋⠀⠐⢰⠐⢦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠂⠀⠴⠋⠑⠒⠂⠈⠑⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣄⢠⠈⠂⠀⢀⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⢈⢁⣼⡇⠈⡀⠈⢀⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠐⠁⢀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠉⠪⢙⡿⠀⠀⡀⠀⠀⠁⠀⠀⠈⡢⠀⠈
⠀⠀⠀⠜⠀⠐⠋⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠑
⠀⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⢀⢂⡀⡉⠲⡢⠤⡤⠄⠀⢀⠜⢀⡀⡘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⢀⠀⡤
⡈⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠸⡀⠙⠄⢠⠋⠳⡅⠀⠔⠁⡀⠄
⠁⠀⠀⠀⠀⡰⠂⠰⠀⠀⠀⠀⠀⠀⠁⠢⢕⣄⠴⠋⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠒⠄⠤⠠⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠠⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢏⣁⢀⡀⣀⡀⠀⠀⠀⠂⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠁⢀⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣹⣦⣀⣴⠀⠀⠈⠀⠀⢰⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢙⡛⠛⠴⣶⣶⣵⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠁

⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⣿
⠙⢿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⣤⣶⠷⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠾⣶⣤⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣾⣶⡀⠀⠀⠀⠀⠀⣯⡁⢤⣴⣿⣿⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣟⣿⣦⡤⢀⣽⠀⠀⠀⠀⠀⢀⣶⣷⡀
⠀⠀⠸⣿⣿⣿⣷⣦⣠⣀⡀⣻⣿⣶⣮⣿⣿⡇⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢸⣿⣿⣴⣶⣿⣟⠀⣀⣠⣠⣶⣿⣿⣿⠇
⠀⣀⠀⣀⣀⣀⣀⣀⣀⡀⣠⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⣜⢿⣿⣿⣿⣿⣿⡄⢀⣀⣀⣀⣀⣀⣀⠀⣀
⠻⠷⠾⠋⠉⠉⠁⠉⠀⠀⣾⣿⣿⣿⣹⣾⢿⡿⠷⠀⠀⢈⢿⣶⣶⣶⣶⡿⡁⠀⠀⢾⣿⡟⣿⣏⣿⣿⣿⣷⠀⠀⠉⠉⠉⠉⠛⠷⠿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⢷⣌⣿⣿⣡⡾⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⡀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣼⣿⣿⣿⣿⣿⣿⣧⣤⣤⣭⣭⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣝⡁⣈⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⠛⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣩⣯⣸⣿⣿⣇⣽⣅⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠙⠛⠻⢿⣯⣿
⣿⡟⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿⣿⠿⠿⣿⣿⡇⠸⠋⣹⣿⣿⣿⣿⣏⠙⠇⢸⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢻
⠋⠣⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣤⣴⣾⡿⣿⠦⠄⠀⠀⢿⣿⢏⡹⣿⡿⠀⠀⠠⠴⣿⢿⣷⣦⣤⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠜
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡋⠀⠈⢠⢡⣃⣤⣸⢳⢶⡶⡍⣇⣦⣘⡄⡄⠁⠀⠙⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢠⡤⣤⣤⡦⣤⣿⣟⣿⡝⡴⡄⣅⢼⡜⡏⣿⣙⣮⢾⡷⣱⣏⣿⢹⢣⡷⣨⢲⢦⢻⣯⣻⣿⣦⢤⣤⣤⠤⡄
⡤⠀⠀⠀⠀⠀⣀⠤⠈⠙⠙⠿⣿⡟⣟⣗⣹⣿⣸⣿⣷⣿⣿⠿⣇⣼⠿⣿⣿⢾⣿⣇⣟⣏⣾⣻⢻⣿⠿⠋⠋⠁⠠⣀
⣠⣤⡤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠈⠓⢿⡿⡟⡿⣿⢻⢙⠏⠁⠀⠀⠀⠀⠈⠹⡋⡟⣿⠿⢻⢿⡿⠚⠁⠀⠀⠀⠀⠀⠈⠳⢦⣤⠤⣤
⠋⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠁⣀⡄⠈⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣄⠀⢠⡆⠀⢸⡀⠀⠀⠀⣿⣿⠀⠀⡀⢀⡇⠀⢰⠄⠀⡀⠀⠀⡂
⢀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠀⢸⡀⠀⠈⠀⠀⣣⠀⠀⠀⠸⠀⠀⠉⠉⠀⠀⠇⠀⠀⠀⣜⠀⠀⠋⠀⢀⡇
⢢⠀⡀⠀⠈⠀⠰⠀⠸⠀⠀⠀⠃⠀⠀⠀⠀⢩⠀⠀⡄⠘⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⡍⠀⠀⠀⠀⠈⠀⠀⠀⠇⠀⠆⠀⢁⠀⢠
⢸⠀⠑⠀⠀⠀⠀⠀⠀⠁⠀⠀⡃⠀⢼⠀⠀⠈⡀⢀⠁⠀⠃⠀⠀⠀⠀⠘⠀⠈⠀⠀⠁⠀⠀⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠛⠛⠛⠛⠷⠶⠶⠶⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⠀⢀⣤⡄⠀⣠⠞⠙⢦⡀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣀⣼⣯⠴⠋⣸⡤⠊⠁⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⣀⠖⠋⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡶⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠀⠀⠀⠀⠀⣿⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⠀⠀⠀⠀⠰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⢠⡟⢀⡿⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⣿⠀⠀⠀⣀⣀⠀⢀⣠⡶⠛⠛⠛⠛⠳⢦⣄⠀⢠⣄⠀⠀⠀⢸⡷⠾⣇⠀⠀⠀⢀⣴⠛⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢻⡀⠀⢀⣠⣴⠞⣻⢿⡟⣻⡆⣾⡛⢿⠷⣍⠳⣦⣀⠀⠀⠀⣼⠁⠀⠈⣧⠀⠀⠛⠉⠉⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢸⣧⠶⠛⠉⢠⡞⣱⢏⡴⢋⡿⣯⠙⣮⠳⣜⢷⡀⠙⠳⣄⣠⡟⠀⠀⢠⡟⠀⠀⠘⠻⣟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡀⠀⠀⣰⢏⡼⣡⠟⣠⡾⠁⠘⣧⣌⢳⡝⢦⡻⣄⠀⠈⢻⡷⠶⠞⠋⠀⠀⠀⠀⠀⢹⡆⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣼⠃⠉⠴⢫⡞⣡⠏⠀⠀⢳⡌⢷⡙⠆⠁⠹⣆⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠉⠰⣏⠀⠀⠀⠀⣽⠀⠉⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⢿⡛⢛⡿⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣄⡀⠀⢈⡷⢾⡁⠀⣀⣤⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡼⠃⠈⣷⡿⠛⡻⠿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣧⣭⣤⣤⣤⣤⣬⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣭⣉⣉⣉⣽⣏⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠛⠛⠛⠻⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠻⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢡⠀⡀⠀⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢦⣀⠀⢣⣍⡙⠙⡄⠀⠲⠀⢀⠄⣀⢤⠀⠀⠀⠀⠀⠁⠀⠳⢄⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⢈⡇⠛⠢⣈⠖⠒⢧⠒⠀⢆⡠⠊⠀⢠⠀⠀⠀⠈⠋⢉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠛⠐⣀⡀⠀⠛⠀⠉⡐⠢⣀⠀⠰⢇⠀⢼⣿⠇⠁⢺⠈⡱⣞⢱⡆⢀⠚⠁⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⡀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠐⠌⢂⣀⠈⣂⢷⣜⡧⡤⣼⡤⢕⣦⣶⢭⣛⡄⠋⠭⠁⣰⠆⢁⠀⠀⠄⠐⣆
⠀⠀⠀⠀⠀⠀⢀⡀⠁⠠⠀⠄⠈⠀⠂⠄⠀⠄⠀⠀⢌⠁⡚⢈⠻⣯⣭⠝⢣⠬⢈⣧⣁⠀⣈⣶⡧⢿⠊⠑⠈⡋⠤⠉⠓⢑⠄⡀⠈⠊⡀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠬⠄⠢⢀⠀⢁⠠⠤⠧⠤⠤⢄⢡⠂⠀⠀⠄⠙⣧⠀⠴⠴⠲⢮⣭⣻⣿⢖⣳⠶⢧⠀⣀⠚⠓⠚⠢⠉⠀⠀⠻⣄⡁⢀⠈⠆⢻
⠀⠀⠀⠀⠀⠲⠠⠄⠄⠀⠀⠄⢈⠀⠀⠀⢀⠀⠀⠀⠈⠳⣤⠀⡀⠠⠜⠛⡆⠄⢨⣜⣍⠛⠋⣹⣀⠀⠤⠔⠈⠂⠀⠀⠀⠹⣆⠐⠀⠈⠀⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠾⠀⠀⡀⠉⠀⠀⠈⠳⡄⠐⠈⠣⣄⠀⠀⠀⠒⡮⢴⡡⠄⠰⠛⠛⠂⣹⠇⠀⠀⢺⡀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⢀⠁⠈⢦⡈⢸⡇
⠀⠀⠀⠀⣸⠀⠂⣁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡙⠀⠀⠒⠀⠡⠄⠀⢈⠿⠁⠈⢻⡇⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⡈⠀⠀⢄⣀⡠⡄⢺⠁
⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠉⠐⡟⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠤⠀⠀⠀⡰⢀⢀⠆⣟⠦⣤⡤⠬⣝⠐⣄⠀⠀⠀⠀⠀⡔⠣⢆⠐⠄⠠⠀⠀⠀⢩⣼
⠀⠀⠀⢸⠠⠂⠀⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠰⠤⠤⠘⣄⢀⠀⠀⢦⠠⡦⠌⠒⠂⡌⠉⠒⢈⡀⠃⠠⡍⠲⠀⠑⠧⠌⠀⠀⡀⠰⣄⣀⡀⢨⡟
⠀⠀⠀⢸⠀⠆⠀⠉⠉⠉⠉⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢦⠰⢦⡆⣄⠀⠀⠴⡇⠁⠐⠆⡀⠚⢄⡆⠀⢈⠀⠀⠐⠀⠀⠀⢀⣾⠁
⠀⠀⠀⢸⡁⡆⠀⠀⠀⠀⠀⡈⢣⡀⠰⢦⣀⠀⠀⢸⡳⣄⡀⠐⢧⡢⠄⠀⠀⠙⢂⠅⠋⢶⠘⢷⣹⠙⠷⣥⠡⠄⣐⡦⢤⠠⠃⠔⠚⢢⡤⣰⠩⡍⢆⢀
⠀⠀⠀⠈⡇⡔⠤⠄⠀⠤⢆⠐⠈⡈⢦⠉⠀⠉⠆⠀⠀⠈⠂⠀⠀⠱⡆⠂⠀⠘⠀⠀⡐⠀⠘⠷⠸⣄⢀⢘⣋⣋⣭⡴⢂⣙⣆⠀⠀⢚⣰⣨⣷⡞⣛⢡⣹⡄
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢈⡀⡦⢬⠤⢕⣤⡀⣡⣹⠂⡲⣄⠀⠀⠀⠈⠙⠲⢤⡀⢂⡈⢀⣄⣀⣏⠙⠉⠋⠉⢁⡤⣼⣧⡘⣿⠳⣤⣨⣿⣿⠟⢿⡿⠞⠷
⠀⠀⠀⠀⠀⢻⣆⡀⣀⡀⠰⢄⠍⣃⠀⠀⠉⠻⠈⠙⠺⠀⡄⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣁⣺⡉⠁⣹⢈⣃⡐⣉⣛⣁⢰⣋⣉⣙⣉⣁⣁⢀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢢⠀⠙⠉⡩⠉⠒⣄⠺⣄⠀⢤⠀⠈⠑⠒⠐⠆⢠⡀⣀⣀⡴⠿⢶⠚⢿⠀⠉⠉⠡⠙⢩⣭⠏⠂⢯⡓⠈⠩⠉⣿⣉
⠀⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⠈⠁⠲⣄⠒⠒⠴⠚⡄⠉⠑⣇⠲⢀⠁⠸⠄⠒⣀⢠⠀⠀⠀⠐⠀⠁⠀⣈⣂⣀⡤⢬⡦⣬⠙⢂⠈⠣⣢⣸⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢪⡀⠀⠀⠀⢀⠈⠓⠒⠶⣥⠱⣤⣤⠷⠈⠉⠩⠁⠈⢠⣀⠄⣀⠁⠀⢀⡠⠴⠒⠉⠁⠀⠀⣌⢆⠈⠻⢅⣇⡀⢌⡀
⠀⠀⠀⢀⣠⣤⣄⠀⠃⠙⢫⡷⢦⣀⠁⠀⠀⠀⠀⠘⠀⠓⠸⡇⠰⢤⣄⣠⣬⡁⠀⣀⣤⠴⠛⠉⠀⠀⠀⠀⣄⠀⠀⠈⠀⠠⠀⠘⡌⠇⢀⡈⠁
⠀⠀⠐⠓⡿⡕⠚⢷⠀⢀⡴⢉⠓⠠⢛⠿⡒⠶⢅⣀⣀⣀⣀⣀⣀⣀⣡⡤⠤⠗⠊⠉⠀⠀⢀⣠⠄⢀⡀⠀⠐⠀⠁⠀⠀⠀⠀⢀⠇⢀⠈⠉
⠀⠀⠀⠀⠈⠻⢶⣦⣷⣭⣀⣠⠤⠚⢅⠒⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢽⣃⠒⠺⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⢀⡄⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣯⠉⠉⢠⡄⢠⡄⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣄⠀⠀⢠⡄⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣼⡷⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣹⡄⣀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢮⣿⣿⣿⣻⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠋⢿⡛⢿⣽⣿⣭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡘⢸⣾⣷⢷⣦⣙⠛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢳⣤⠀⣟⣿⢣⣿⢱⣟⡏⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣦⡟⣧⣽⣧⡿⣿⣿⣿⡾⣺⡿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢸⡟⣰⡟⣹⣿⡡⢋⣞⡿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣾⣾⣯⣴⣿⣾⣿⣿⣽⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠖⣏⣿⣷⣿⣽⣿⣿⣟⣉⣉⣙⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⣀⢸⣿⡻⣿⣿⠷⠾⠿⠻⣽⣿⣟⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣽⣉⠛⣿⣿⣦⡙⡏⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡘⣿⣦⣽⣿⣷⣿⡿⣿⣿⣿⣿⡆⣌⣿⣿⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⢿⣿⣿⣿⡿⣟⣻⣦⣝⠻⣿⢿⣿⣾⣿⣟⢛⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢶⣿⣿⣿⣝⣿⣿⣿⡾⣿⣎⠻⢷⣿⢸⣿⣿⡿⣿⣷⣷⣰⡖⣶⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡀⣿⡿⣿⣿⣿⣿⣿⡿⡏⠻⣿⢷⣻⣼⣿⣿⣶⣿⠃⣿⡟⠛⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⣿⣯⣽⣧⣘⣷⣿⠿⠟⠋⢁⣻⡼⡆⢹⡘⣿⣿⣿⣟⣒⣿⣧⣸⡇⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⢠⣼⡘⢿⡇⣧⢀⡿⠁⠀⣿⢿⣭⣭⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⢰⣿⣿⣿⣿⣇⣿⠈⠀⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⣄⠀⣴⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠀⠻⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⠙⠋⠀⠀⠐⡆⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⡏⠹⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡎⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠰⠁⢰⣦⢴⡦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠐⠀⡈⢡⣾⣷⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡄⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠃⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⣀⠀⠛⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡛⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⡿⣿⢿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⢿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⠈⢀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣄⠀⠀⣬⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⢈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣤⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⢨⣿⣿⣿⡀

⠀⠀⠀⢀⠀⠀⠀⠄⢧⠀⡄⠀⠀⡀⠀⠀⠀⠀⢰⠰⠀⠀⠀⠀⠀⢆⠀⡄⠀⠀⠀⣀⢀⡀⠀⠄⢠⠀⢀⠄⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⢘⡆⢑⠀⠀⠀⠀⠀⠀⠀⠸⠸⡆⡆⠀⡄⡄⠐⠀⠃⠀⠀⢰⢃⡞⠀⠂⠀⠈⡠⠀⣠⠊⠁⢀⡴
⠀⠀⠰⡀⠀⠀⠀⠁⠀⢣⡈⡆⠀⠀⣰⠀⠐⠀⠀⡇⡇⣷⢸⡇⠃⠀⠀⣠⠀⢀⠏⡼⠀⠀⠀⢀⡜⢡⠞⠁⢀⡴⠀⠀⠄
⠀⠀⠀⠙⢄⢀⠡⡀⠰⡜⣧⠸⠀⡀⢿⡀⠀⡆⠀⢇⣷⠈⢸⡇⢰⠀⢰⡏⠀⡜⣰⠁⠀⠀⠀⠊⠐⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠳⡙⠄⠈⠘⣧⢁⢡⢸⡇⢀⠃⢸⢸⢸⣀⣼⡇⣸⠀⣾⠁⣸⢱⡇⠀⣠⠈⠀⠀⠀⠀⠀⣀⣠⠊⠀⠀⡠
⢀⠀⠀⠀⠀⠰⡄⠙⢔⣄⠀⠘⢧⠀⡜⣷⠀⠸⢸⡎⣘⡇⢹⣷⡇⣇⡇⢆⢃⣿⠁⣤⠏⡰⠁⢀⠄⢀⠼⠋⠁⠀⠀⠊⡄
⠀⠃⠀⢠⠀⠀⠙⣆⠸⣾⡣⣜⠘⢄⠃⣿⣧⠀⣿⡇⣏⡇⢸⣿⡇⢹⡸⠈⡾⢃⣾⢃⡼⠁⣠⠞⠐⣡⡾⠃⠀⠀⠀⠀⡀
⠸⠆⠀⠀⠣⠀⠀⠘⢆⠀⢦⠸⡷⣌⠀⢻⣿⡆⠸⡇⣿⠃⠘⡇⠁⣿⣷⡼⢡⠟⣡⡿⢁⣴⣿⣶⣿⠋⠀⠀⣠⠆⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⠎⡆⢽⣮⡀⠘⣿⣷⠀⠣⢿⡄⠘⡟⠀⣿⣿⡅⡫⣺⡿⡡⣿⡿⢻⡿⡡⢊⡴⢞⠑⢉⡥⡪⠞
⠀⠀⠀⠠⡀⢠⡄⠀⠀⠸⣿⡌⠈⣿⡇⠀⠘⣿⢀⠀⠸⡇⠀⠃⠀⣿⡟⠁⡼⠛⢃⣾⠟⣴⣿⢋⣀⣬⢞⠏⠐⠋⠀⢀⡀
⠀⠀⠀⠀⠈⢷⣌⠀⡄⠀⢻⣿⡄⠘⣷⠀⢀⠸⢸⡀⠀⠃⠀⠀⠀⠋⠀⠘⠁⡠⢈⠁⣺⡟⣱⣿⠟⣡⠞⢉⣤⠴⠚⣋⠄
⢈⡓⠂⠀⠀⠀⠙⠷⣌⡀⠀⢻⣷⡀⠘⠄⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⢃⡼⢋⣼⡿⣫⣾⣫⠔⣋⣀⠴⠊⣁⠤
⠀⠉⠉⠓⠂⠄⢀⣳⣬⡻⣆⠀⠙⣗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⢋⡼⢛⣩⣶⡟⠉⠀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠉⠉⠛⠛⣛⣂⠈⠀⠈⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⣟⣋⣀
⢰⡶⠶⠶⠶⠶⠶⠤⠤⠜⠛⢁⣀⣠⠶⢋⣡⣶⠆⢀⡀⢀⡿⢂⠀⣤⡄⣀⠒⢦⡘⠚⢛⣋⣉⣉⣿⣿⣿⠿⣿⣿⣿⣿⣿
⣾⣷⡶⠶⠿⠿⢀⡀⣐⣛⣛⣓⣠⣴⣾⣿⡽⢂⣴⡟⣰⣞⣥⣣⣿⡹⢥⡌⠷⣔⢌⡻⣷⣍⠻⣯⣙⠊⠉⠉⠉⠛⣛⣻⣿
⣿⠧⣤⣄⣠⣤⣤⣤⡈⠉⠉⢉⣉⡀⡛⢉⣴⣿⡋⣼⣭⠉⠉⠛⣿⣿⣌⢻⣦⠀⠁⠉⠊⠉⠀⠈⠉⠁⠐⠻⠶⣦⣄⣙⣿
⣿⣶⣦⣬⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣭⣴⣶⣤⣤⣤⣬⣭⣭⣬⣭⣥⣶⣾⣿⣦⣾⠿⠿⠷⠶⠶⠶⠶⠶⠾⠿⠟
⣿⣿⢟⠿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣶⣾⣿⣿⣿⣿⣷⣶⣶⡀
⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⣯⢴⣶⣶⡆⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⢇⣿⣿⢩⣭⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣜⠿⠛⢸⣿⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡞⣿⣿⢹⣿⣿⣿⡟⠁⠀⢸⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣿⣿⣿⡇⣿⣿⢸⣿⣿⡿⠀⠀⠀⢸⣿⢸⣿⢸⡷⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢸⣿⣿⣇⣿⣿⣸⣿⣿⡇⠀⠀⣤⢸⣿⣾⣿⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣭⣭⣭⣭⣭⡍⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣯⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢰⣾⣿⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣀⣩⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣤⠤⠤⠶⠶⠿⠿⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠚⠁⠀⡴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⡀
⠀⠀⣾⠏⠁
⠀⢰⡟⠀⢠⣶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠃
⠀⢸⣇⣠⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃
⠀⣼⣿⣿⣹⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣰⣆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣴⣿⡇⠀⠀⠀⠀⠀⠰⢻⡿
⠀⣿⠈⢿⣿⡿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠀⢿⠀⠀⣿⡇⠈⠙⠻⠿⠿⠿⠋⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⣀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠋
⢸⣿⣷⣄⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠉⠛⠛⠟⠛
⣀⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣡⣤⣤⣤⣤⠀⣀⣠⣤⣤⡀
⠸⣿⣿⣿⣿⠾⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣶⣶⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣒⣂⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡩⠤⠒⠂⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⣂⡩⠥⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣩⠤⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⢂⣈⡁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⡈⠭⠄⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⠤⠐⢒⣈⣉⠉⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠐⣊⣉⡀⠠⠀⠀⠀⠀⠀⠀⠈⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣈⠩⠤⠄⠐⠂⠀⠀⠀⠀⠀⠀⡐⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠐⠒⣒⣈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣈⠩⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠌⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⡠⠤⠀⠒⠂⠀⠀⠀⠖⠒⠋⠋⠙⠋⠖⣲⠤⣀⠐⠠⠡⢡⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠒⣒⣂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠭⠷⢄⠉⠂⠎⡜⡀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⣀⡈⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡩⠭⢵⣄⠈⠑⡐⡴
⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⠚⢷⡠⠘⠴⡡⡀
⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢀⢒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠉⠭⠭⢍⡒⢷⢆⡄⠱⣕
⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⡊⢉⡓⠲⢌⡑⢯⠌⡀⠈⢾
⠀⡴⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠱⡜⢗⢬⡑⢤⡙⣼⡜⡠⠀⠀⢋
⡼⠀⠀⠀⠀⢀⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⡌⢆⢣⠘⡆⡱⡑⡀⠱⣸⠞⠕⡄
⡇⡄⡀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⡰⢠⢀⠇⡃⢸⢈⠆⡇⢁⢱⢱⠀⣿⠊⡜⡤
⠻⣤⢡⢢⠂⠄⢠⢀⠄⡐⢠⠂⡤⢠⠂⠄⡄⡀⠀⣀⢤⡤⡤⠄⠤⢤⢤⣀⠀⠁⠊⢰⢁⠆⡸⢀⡇⡜⢸⠸⣠⢃⠌⡰⠁
⠀⠈⠛⠶⢼⣠⣇⣌⣰⣁⣆⣠⣀⣇⣸⣸⣰⣀⡈⢁⣈⡀⠁⠈⠐⠃⠞⠴⢫⠖⣤⡀⠈⠘⠀⡘⢠⠃⡌⢸⢏⠌⡴⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠹⡙⠩⢉⠏⠛⡛⠙⢓⢦⡀⠀⢀⣝⡇⢀⠔⢠⠃⣰⠕⣡⢊⢞⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠁⠈⠘⠘⠀⠃⠊⡜⢩⠓⡦⢤⢤⡧⡶⢻⢋⢋⠔⠡⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠐⠀⠁⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⡶⢿⣿⣿⠿⢿⣟⣛⣛⡁⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⣵⣾⠟⣱⡿⡩⢖⣛⣛⣛⣻⣇⣤⢄⡸⣦⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡵⠿⠿⣗⣠⣿⢠⢞⣭⣽⣷⣶⣶⣿⠿⢟⢡⣿⡇⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⣛⣒⣚⣻⣿⣿⣿⣧⣵⣦⣄⣛⡛⠿⢿⣿⢿⣛⡍⣼⡿⠀⣟⣛⣛⠇
⠀⠀⠀⠀⠀⣛⣛⣿⣶⣿⣿⣿⣛⣉⣙⣻⣿⡟⣩⡛⠿⠿⠿⢛⣠⢹⣿⡟⣱⡿⢡⢸⡌⣿⣿⡇⣆
⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣻⣿⣿⣿⣣⣿⢏⣼⠏⣠⡟⣸⠇⣿⣿⢡⣿⡆
⠀⠀⠀⠀⢼⠂⡶⢙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⢋⣾⣿⠟⠡⠚⠵⣾⣿⢣⡿⢸⣿⠇⣾⣿⡇
⠀⠀⠀⠀⠈⠀⠃⠈⣿⡮⢻⣿⣿⣿⣿⣿⣿⣿⡇⣸⡿⠿⢛⡅⠠⢞⡻⡿⣎⠋⣾⢃⣿⡟⣼⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡏⣿⣿⠻⠏⢿⣿⣿⣇⣿⣷⣾⠏⣴⣾⣿⠿⡇⢈⣄⣿⢸⣿⢸⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⣿⣿⣿⣿⣿⣿⠿⠛⣿⣟⠇⣼⣿⣿⣿⠇⢀⣾⣷⠻⣌⠇⠋⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠘⢿⡿⢏⣼⣿⠘⠉⠉⢵⣒⡶⣾⣿⣿⠀⣿⡟⣼⡿⣰⡿⢟⠋⣁⠬⣠⣾⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⡟⠀⠀⠀⠀⠈⢻⣶⡝⣿⣇⢻⡇⣯⠇⢉⡴⢣⡞⣡⣾⡿⢋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡷⠈⢿⣎⢿⢹⣧⡾⣱⡏⠼⣛⣩⡶⣣⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⡌⢻⣎⢇⢿⢳⣿⣿⠿⣛⣭⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⢧⡈⠛⠵⡾⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣍⡓⠀⠉⢻⣶⣍⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣼⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣾⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⣽⣶⣾⣿⣿⣿⠁⢀⣰⣿⣦⣄⣀⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠙⠋⠉⡰⠛⠏⠀⣼⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠉⠉⠛⠹⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣤⡀⠀⠀⣀⠂⡀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠁⠀⠀⠈⢻⣶⣾⣿⣷⡇⠀⠀⠀⠀⠂⢀⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠔⠠⡤⢂⣒⣼⣿⣭⡀⠀⠀⠀⠀⣀⣴⣿⣿⣟⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⢠⣤⣭⣉⠛⠻⢿⠀⣰⠟⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⠿⢿⣯⣿⣿⡃⠀⢠⡜⠁⢀⣼⣿⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⢿⣻⣇⠀⠀⠈⡍⠁⠀⡬⠄⡰⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠙⢿⣆⠀⠀⠀⢳⡾⠀⠀⢠⣿⣿⣿⡇⢱⣆⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑⠄⠀⠀⠀⠀⠁⢀⠀⢰⣇⣀⠀⣼⣿⣿⢿⣷⠀⢿⣦⢸⣾⠿⣿⡯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠀⠀⡈⠀⠀⣀⣠⡆⢸⡿⣿⣿⡟⢰⣿⣿⣏⣸⣷⠀⠘⣷⣾⣧⠾⢟⢀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠃⠀⠘⢦⣸⣿⡿⣿⢘⣗⣿⠃⣇⣾⡿⣹⣯⣿⣿⡀⠠⠛⢻⠃⠀⠀⠀⠀⠘⠳⢀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⡁⠀⢼⣿⣿⣿⣿⣿⣿⣿⣻⣿⠂⣾⡟⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⢠⣿
⢠⡔⠀⠀⠀⡀⠀⢠⠃⣿⣿⣿⣀⣽⣿⣿⣿⢿⣮⣿⣿⣷⣴⡏⣠⣿⣿⣿⣿⣿⢿⣿⣄⠀⠀⠀⣰⠃⡄⣰⣇⡴⣡⢣⡟
⢸⡅⠂⡀⠼⡀⢠⠇⠀⢻⣿⣝⢿⣿⣿⣿⡝⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠉⢻⡦⠄⢠⡟⢠⢃⣿⠟⣰⢃⡾⢱
⠈⠀⠀⠈⠀⠁⠉⠀⠀⠘⠈⠿⠆⢹⣿⠿⣿⠀⠸⠷⠿⢿⡿⣿⣿⣿⣿⣏⡾⠀⠀⠀⠘⠀⢠⣿⠁⣾⡾⢁⠾⠃⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣹⠿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⣠⢤⡄⠀⠀⠀⢠⣤⡀⠠⣤⡀⣤⡄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⣤
⠀⠀⠀⠀⠀⣀⣤⠊⠀⠀⠆⠙⠀⠀⣀⣸⡇⢲⡀⠛⠿⣦⡘⢷⣼⣷⠿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠐⢀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⢠⠊⡉⠁⠀⠀⢀⢀⣤⣤⣶⢻⢸⡇⢈⢃⠰⣦⡌⠙⠣⣍⠀⠀⠀⠈⠷⣶⡖⠀⠀⠀⠀⠀⠀⠀⢤⣤⣶⣦⣶⣾⣷⣶⣤⣦⡀⠐
⠀⠀⢠⢺⣿⣿⣠⡶⣶⣏⡿⠇⠛⠘⣛⠋⠁⢘⠋⠉⠠⠀⠀⠀⠀⠈⠁⠐⠒⠲⠻⣿⣄⠀⠀⠀⣧⣤⠠⢤⣹⡟⢻⢿⣿⣿⣿⠿⣿⣁⣨
⠀⠀⡰⠀⣿⠿⠟⠛⠁⠀⠀⠑⠂⠁⣀⢀⣤⣌⡀⠠⠀⣀⠖⠀⠠⠄⠒⠠⢂⣁⠾⣭⣉⠁⠂⣠⣆⢉⣁⢀⣠⣤⣀⣀⣌⡶⣥⠆⡮⢹⣾
⠀⠀⠱⢄⡀⠀⣈⠁⢀⣈⣠⡀⣀⢿⠁⠀⠉⠛⠳⣄⠀⠈⠀⠤⢰⡶⠓⠆⢷⡝⠷⠈⠁⠀⠀⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⠉⠉⢋⠋⠋⣹⡟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⡆⠀⠀⠐⠀⠀⠀⢠⣿⣿⣷⣨⣭⡻⣛⡛⢡⣾⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠻⢿⣿⠆⠀⠀⠀⠀⠀⠘⠿⣟⣶⣼⣯⠁⠉⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠹⣀⠠⠀⠳⣄⡀⠀⠀⠉⠀⢸⡿⠁
⠀⠀⣢⡤⠒⢀⠇⠁⠀⠉⡛⠒⡒⠂⠀⠀⡀⠀⠀⠀⢹⣿⠿⢃⡀⠀⠀⠀⠸⢷⣤⠆⠛⠉⠀⠂⣴⣶⠾⠀⠀⠀⠀⠹⣷⠀⠀⠀⢀
⠀⠸⣿⠀⣀⣁⣐⣀⠀⠀⠉⠈⢁⡰⠃⢀⡇⠀⠀⢸⣇⢱⡾⠛⠿⢤⣀⠴⠖⠒⠺⢄⠀⠀⠀⠀⠐⠆⢠⢴⡞⠀⠀⠀⠀⠀⠀⠒
⠀⠱⢧⣬⣭⡝⠛⠉⠚⢛⣾⣿⡾⠋⠀⢺⣿⠀⣀⣼⡟⠉⢠⣤⣶⣾⠁⠀⣾⣿⡆⠀⣹⣛⣦⠀⠀⣀⠁⠀⢻⣷⡆⢘⠟⠁
⠀⠀⠈⠛⠻⠿⠿⣿⢿⣿⡏⠙⠇⣴⡆⣼⣿⠀⣿⣿⣷⡼⣦⣽⣏⠟⣸⣶⣽⣋⣴⣦⣿⣏⣠⡟⠛⠁⠀⠀⠀⠀⠀⠜⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⢰⣶⣦⣿⡇⣿⣿⣆⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⠁⠈⠿⣷⣆⠀⠀⠀⠀⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⣿⣷⣿⣿⠃⠀⠀⠁⠀⠙⠻⣿⣯⣿⣿⡯⢹⣼⣿⣤⣇⠀⠀⠀⠈⠻⣧⣀⠀⠄⢀⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠸⠿⢿⢿⡀⠔⠀⠀⠠⠀⠀⢀⣽⣿⣿⣷⣦⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠘⣢⢉⠻⡤⠒⠁⣀⣶⣾⣅⡠⠀⠀⠠⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⣶⡖⠋⣉⣠⣤⣤⣀⣀⣓⡻⠛⠱⠂⠺⠟⠍⠀⠀⠁⠀⠠⠐⠀⠐⡀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠈⠋⠁⠉⠓⢆
⠀⠀⠀⠀⢀⣿⡧⠤⠞⠉⠀⠉⠋⠁⠀⠈⠛⣿⠆⠀⠀⠀⠀⠀⠀⡠⢀⡖⢋⠉⠽⠩⠇⠌⡆⠀⡀⢾⠁⠬⢄⣄⢦⣦⣀⡀⠈⡉⢻
⠀⠀⠀⢠⡞⣁⣤⣤⡤⣰⢈⣀⣠⣤⣤⡀⢀⡀⠑⣤⡄⢠⣷⠀⠀⣇⣸⣇⣢⣄⣩⣈⣧⣿⣏⡀⠃⠀⢿⣿⠛⠹⣧⣼⣿⣿⣿⣷⡞
⠀⢀⠀⢼⣿⣿⡁⣸⡿⣿⣸⡿⣯⣤⣼⢷⣾⣿⠟⣼⡇⢸⣿⠀⠀⠈⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⠀⠰⡸⠿⢆⠀⠈⡩⣄⡥⠛⢋
⠀⠈⠀⠈⠳⣝⡛⠋⠑⢤⠀⠀⠀⠀⠀⠈⠁⠋⠛⣹⠃⣸⣿⠀⠀⠄⠸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣀⡀⣠⠌⠹⠾⠷⠉
⠀⠀⠀⠀⠀⠀⢋⠙⢧⣀⠉⣀⣤⡉⠈⢀⣀⣰⠈⠇⢠⣿⠇⠀⠀⣠⣒⡋⣉⣉⡉⠉⠉⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣤⣉⠉⠉⠉⠉⠟⠛⢿⠟⠈⠾⠟⠋⠀⠀⠀⠀⡀⠄⢡⣶⢿⡦⠄⢀⣠⣦⢰⣀⣄⣬⡰⣰⠀⠀⡤⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠇⠀⠀⠀⠀⠀⠀⢠⡈⡀⠀⠀⢸⡀⠈⣾⣿⢻⣷⢠⣿⣯⣿⡈⣤⣾⣿⡧⢟⣹⣷⡆⢀⣤⡀⢀⣤⡄
⠀⠠⠀⠤⠏⡦⡄⢀⠀⠀⣒⠂⢀⡤⢤⠀⠀⢨⣠⡤⠀⣼⣷⣶⣾⣾⣇⠀⠼⠿⠾⠟⠀⣠⡉⢙⣣⠉⢉⡻⣷⠿⢿⣿⣴⣿⡿⣴⣿⣿⣧
⠀⠀⠀⠨⠀⠉⠐⠇⡥⠍⠀⠀⠁⠀⠀⠁⠉⠐⣘⠓⠂⠀⠀⠀⠉⣉⣉⠓⠓⠓⠔⠢⠚⠿⠿⠿⠿⠿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠴⠩⣟⣶⢀⣐⢀⣤⡀⣤⣤⣦⠀⢡⠀⠀⠀⠀⠀⠀⠉⢲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⢍⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠃⠘⣯⣿⢾⣽⢻⣬⡇⡿⢿⢸⡇⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠄⠀⠀⠀⠀⡀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠳⡤⣍⣁⠉⠑⠉⠈⠛⠒⡈⢘⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢁⡀⠈⠁⠀⠀⡀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠂⠗⠚⠛⠒⠒⠚⠛⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⢠⣤⣄⡀⢠⢠⡄⢠⠀⠐⠒⢲⡒⠒⠒⠀⡀⠀⢦⣶⣾⣿⡟⠀⠀⠀⠀⠀⠰⣤⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠢⠀⠀⠸⠀⡀
⠀⢸⣿⠿⣳⠻⠆⠸⣜⣃⣄⣀⣤⣄⣧⣴⢋⢋⠳⠾⣿⣿⡿⠁⠂⠀⠀⠀⠀⠀⠐⢀⣠⡤⢆⠀⠀⢀⣴⠀⠀⠀⠀⠀⣄⣠⡕
⠀⢸⣿⣶⡄⢬⡷⣿⣼⣿⣿⣿⣟⣻⣿⣶⣾⣷⠇⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⢢⡈⢡⣀⡠⠀⣀⠏
⠀⠘⠛⠊⠛⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠋⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⢠⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠘⠴⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣸⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢹⣀⠀⠶⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⣠⡴⠞⠣⢦⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣴⠏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠂⠁⣠⣼⡿⠧⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⢀⣇⠀⠀⢀⠀⠀⠒⠀⠈⠀⣶⣿⠑⠀⣠⣾⠛⠃⠀⢀⢀⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠆⠀⠈⠋⡜⠛⡁⠀⣰⠆⠀⠀⢰⣟⠃⢀⠼⣿⡋⢀⣤⢆⣽⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠛⠀⣬⠁⡼⡍⠀⠀⠀⣼⠇⢀⡌⠀⢹⣟⢩⣷⠟⠂⢈⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠴⠃⢀⡄⣿⠂⢱⠀⡀⠀⠀⠻⠤⠞⠀⠀⠀⠉⠈⠁⠀⠠⠀⠣⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡷⢟⠀⣤⡿⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡐⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⡇⢸⠀⢀⡷⠉⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠘⣇⡰⢦⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡧⠀⠀⠃⢸⠀⢸⢱⠀⡀⠀⣀⡄⠀⣼⠄⠀⠀⠀⠀⠀⠀⢀⠈⣷⡰⣧⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⣀⠀⠛⠀⣦⠀⢿⡏⠀⢁⣀⣈⣱⣬⣯⠶⣄⠀⠀⠀⠀⢀⣾⠀⣾⠱⣿⣾⠋⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠰⠀⢸⣿⣇⣀⣤⣿⠃⠘⠷⠶⣌⠁⠉⠞⠛⡛⠛⢧⣰⣔⡄⠴⠿⢿⡀⢰⣶⠎⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠸⣿⠛⣿⣿⡇⠀⢀⣀⠀⠙⡻⠋⠀⡿⠃⠀⠀⠩⣭⡰⣦⢤⣶⡅⠛⠹⠄⠀⠀⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣤⣿⣿⣿⡿⠁⠀⠘⣉⡷⣄⠉⢢⠈⠁⢀⡀⠀⠀⢹⣷⢸⣆⢿⡆⣼⡖⡆⠀⠀⠀⢰⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠻⣿⣿⣇⣀⠀⠚⠉⠀⠙⢷⣄⠀⠀⠊⣠⡀⠀⠀⢿⡄⠻⣦⠋⣿⠀⠁⠀⢲⣘⢿⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣥⣄⣤⢄⣤⢄⠀⠙⣧⠀⠟⣋⢀⣤⠀⠨⣯⣾⢏⠘⣁⠄⡀⢠⣯⠻⡶⣝⡀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢿⣿⣽⣦⣤⣤⠶⢾⣷⡄⠘⣇⠀⢩⠛⠅⠀⢀⠴⡿⢈⡶⢁⣴⠁⠘⢿⣦⠀⣌⢉⠳⡄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⢤⠘⣿⣏⠉⠉⠀⠀⠀⢘⠇⠀⠛⢀⡃⠀⠀⠀⡏⠀⢳⣯⣶⣏⠀⠀⠀⢸⠗⢴⠉⢌⠀⡈⠂⡈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢮⣇⠘⠀⡼⣿⡛⠶⠦⠒⠚⠋⠀⢀⣾⡘⢱⠀⠀⢠⡸⣦⣹⣟⣿⡇⠀⡆⡱⠈⠰⣴⢷⠄⢴⣧⣦⣜⠷⠀⠁⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡆⢸⡧⠠⠎⠀⠉⣿⡇⠊⠉⠂⠀⠀⠘⡀⠁⠀⠀⣠⣿⣽⠛⢹⣿⢋⠁⢸⡁⢿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠃⠞⠓⠂⠀⠔⣠⡄
⠀⠀⠀⠀⠀⠀⢀⡾⣛⠁⣀⣯⡄⠀⠀⢰⣇⣾⠙⡄⠀⣀⣀⡀⢀⣼⠇⢀⡀⠈⠻⣾⡇⠐⠋⣿⠈⣀⣾⠇⠌⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⡴⠋⠻⠏⣹⡿⠄⠀⣼⣹⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀⣾⣷⠂⠀⢶⢻⡵⣦⣾⠀⣿⣴⠃⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠸⣠⡟⠠⠐⢟⣧⠀⠀⡇⠜⠃⠀⢀⡉⣩⣿⡿⠋⠐⢉⡝⣧⠀⠀⠈⣿⣆⣿⣿⠄⠛⠈⠀⣠⠀⠃
⠀⠀⠀⠀⠀⢈⣶⡿⣴⢁⣶⣸⡿⣄⣤⢠⠂⠀⠀⣰⣴⣏⡁⠀⡀⠀⠉⢀⢛⠀⠀⠀⢻⣇⠀⣍⡀⠄⠀⢺⠋
⠀⠀⠀⠀⠦⣿⢿⣷⡟⢸⡇⡹⢃⡟⢡⠎⢠⡎⣰⣿⠏⡟⠀⣼⠁⠀⠀⢈⠈⣷⠀⠀⠀⠛⠀⠙⠁⠀⠀⠃
⠀⠀⠀⡞⠸⠋⠚⠁⣷⣿⠿⢁⡜⣲⣷⢠⡿⢿⣿⢹⣼⠇⢰⣿⠀⠀⡀⠀⠀⠟⠀⠀⠀⡆⠘⡿⠂
⠀⠀⢀⠇⠀⠀⠀⢠⡿⢻⣿⡿⢿⣿⣿⢿⢃⣾⠋⢈⠃⣄⠘⠿⢸⣾⡇⠀⠀⢸⠀⠀⠀⡁
⠀⠀⠸⠀⠀⠀⠀⠘⠈⣿⡟⠁⡸⠙⣷⡏⣸⡏⢠⢸⠀⢿⠀⢰⡄⡿⣿⠀⠀⣿⠀⠀⠀⣿⡎⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡄⢻⣇⣼⠁⠀⣸⠇⣿⡇⠈⡧⠀⢀⡤⠈⢃⣷⣸⡄⡆⢸⠀⠀⠀⡿⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⡀⠀⢿⢠⠿⣷⠀⢰⠀⠘⣧⢠⡇⣿⣿⠀⠀⡾⠀⠀⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠚⣼⠃⠃⠀⠀⡻⡄⢿⠆⢸⠀⠀⢠⣾⠁⢹⡏⣠⢰⠀⠀⠀⡸⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠘⠃⠁⠀⠀⠘⠀⠠⠟⠁⠀⠛⠀⡈⠄⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⡏⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠓⠈⢰⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢟⣩⣴⣶⣾⣿⣿⣿⣶⣶⣬⣙⠿⣿⡿⢀⣛⣛⣛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢋⣴⣿⣿⡿⠿⢛⣛⣛⣛⡻⠿⣿⣿⡷⢈⠰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣰⣿⣿⡟⣫⣶⣿⣿⣿⣿⣿⣿⣷⡮⠉⣴⣿⣷⡜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣰⣿⣿⠏⣼⣿⣿⠟⣫⣭⣭⣭⡛⢋⣴⣿⡌⢿⣿⣿⡘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡟⢸⣿⣿⢏⣾⣿⣿⡿⢋⣴⣎⢿⣿⣿⠘⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡇⢻⣿⣿⡸⣿⣿⣿⣶⣿⣿⡿⣸⣿⣿⠃⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⢿⣿⣿⡘⣿⣿⣷⡙⠿⣿⣿⡿⠟⣱⣿⣿⡟⣸⣿⣿⠇⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡌⢿⣿⣷⡜⢿⣿⣿⣷⣶⣶⣶⣿⣿⣿⠟⣰⣿⣿⠏⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣌⠻⣿⣿⣷⣍⣛⠻⠿⠿⠿⢛⣋⣵⣾⣿⡿⢋⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⣷⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣭⣙⣛⣛⣛⣩⣭⣴⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢡⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠩⠀⠴⠔⢢⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢀⡀⢎⣈⡑⢚⠉⠇⠀⠀⠁⡂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⣈⠧⠉⢸⠗⠆⠘⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠙⣡⠀⣌⠙⢀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⢰⠄⠠⠰⠊⣴⢴⠀⠁⠠⠐⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠙⠒⠤⡀⡎⢠⠈⡇⣠⠒⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠐⠀⠀⠐⠄⣦⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠐⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠁⠄⢀⡀⢰⠆⠀⠀⠀⠀⠀⠠⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⠀⠀⡀⢶⠜⠋⠀⡀⠀⠀⠀⣄⠀⠀⢠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢎⠃⠀⠀⢀⠃⠀⠀⠀⠀⢻⡀⠀⠀⠈⠓⠀⠻⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠠⡟⠲⠶⡟⠶⢹⠀⠀⠀⠀⠈⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⢴⡆⠀⠀⣧⢈⢸⠀⠠⠀⠀⠀⠙⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠘⡺⠈⢆⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⣹⠆⡮⠀⣰⢙⣻⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢀⡠⡠⢞⠃⡐⠁⡐⠃⣸⡿⢀⠸⡉⢁⣄⠠⡀
⠀⠀⠀⠀⠀⠀⣀⠸⠁⠠⠀⠒⡂⡀⡀⠀⢁⣤⡇⠐⣆⠀⣇⢿⠇⠀⠀⠀⢧⡐⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠉⠄⢈⠲⢊⣠⠈⠉⠇⠘⠉⠀⢐⢸⣟⡀⠈⠉⠀⣁⡰⠋⠀⡑⣀⠆⠀⡀⣠⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠳⢠⠀⡤⠈⢤⣑⠂⢈⠠⠈⡿⠈⡛⠉⠉⠒⠀⠀⠈⡀⠐⠠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠀⠁⠂⡄⠐⣁⠀⢠⡶⡈⠉⠋⢋⠊⠀⠑⢾⠅⢀⠀⢀⢠⠘⠒⢐⡤⠀⠉⠱⠂
⠀⠀⠀⠀⠈⠚⠳⠄⣄⠐⠀⡴⠉⠁⠚⠀⠀⠈⠠⡉⠉⡄⢀⡳⢈⡢⠴⠀⢀⠙⠆⠐⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠈⠓⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⡟⠻⢿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣦⡀⠀⠀⠀⠀⠀⠰⠶⡀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⡆⢻⣿⣦⡀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⣿⡀⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⣘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⢁⣉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣠⡂⡈⠻⠋⣼⣿⠃⠀⠀⠀⠀⠀⣀⣼⣟⣻⠇
⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⢸⣿⣮⣛⡁⠉⣠⣾⣿⣿⣿⣦⣤⣴⣶⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⣿⠏⠉⠛⠻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠘⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷
⠀⠀⠀⠀⠀⣀⡀⢀⣀⣤⣴⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⣀⠀⠲⣿⠄
⠀⠀⠀⠀⣿⣿⢻⣿⡿⠿⠛⠋⠁⠀⠀⠈⣿⣄⣀⣀⣤⣾⣿⣿⣿⣶⣶⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⠙⢿⣿⣿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠿⢿⣿⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠙⠏
⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠲

⠀⠀⠀⠀⢀⠠⠀⠒⠒⠂⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠈⠐⠂⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠤⠆⠐⠒⠂⠀⠠⢄
⠀⠀⠀⠔⢁⠠⠤⠄⠲⠤⠐⢆⡀⠀⠠⠀⠀⠀⢀⠜⢁⢤⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⢸⠀⠁⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠇⠀⢠⠁⠔⠁⡀⠁⠐⠦⡀⠀⠀⠀⠀⠈⠢⡀⠀⡈⠀⠀⠠⠖⠒⠀⠓⠢⢄⠀⠀⠈⡄
⠀⢠⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡘⠀⠀⠀⠐⠀⠀⠀⠀⠋⠢⠀⠀⠀⠀⡁⠀⡇⠐⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡅
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠇⠀⡅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡇⠀⢣⡆⠀⠀⠀⠀⠀⠈⠀⠀⢒⠀⠠⠁
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠎⠀⠀⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠁⡠⠃
⠀⠀⠀⠉⠒⠤⢀⣀⣀⣀⢀⠤⣔⠉⠀⠀⠀⢣⠀⠑⠰⢤⣄⣀⣰⡤⢤⠒⠦⡀⠀⠀⠨⠀⠀⢣⡀⠀⠀⠀⠀⡀⡀⢄⡊
⠀⠀⠀⠀⢠⠖⠉⠃⠁⠀⠊⠀⠀⠑⡄⠀⠀⠀⠉⢒⠞⡁⠈⠉⠀⠁⠁⠀⠀⠘⡖⠂⠀⠀⠀⢀⠎⠁⠒⠒⠃⠋⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⠠⢀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠁⠠⠄⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⠔⠡⠐⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⢠⠁⠀⢀⡀⢀⠀⡀⢀⡀⠀⠀⠑⡄⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠈⠐⢄⠀⠐⠠⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⡕⢁⣀⠀⠀⠀⣀⡀⠈⡆⠀⠀⠘⠀⠀⠀⡜⠕⡠⠔⠊⠀⠑⢄⠀⠀⠀⠳⠀⠀⠘
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⡈⠁⠀⠰⠀⠀⠀⡆⠀⢰⣏⠊⠀⠁⠀⠀⠁⠀⠱⡀⠀⠀⠇
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠄⢠⠀⠀⢸⡁⠀⠀⠂⠀⠐⠂⠀⠀⠓⢠⠀⡀⠀⡐
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡇⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡇⢰
⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⠌⠀⠉
⠀⠀⠀⠈⠉⣰⡖⢤⣠⡔⢂⠟⠠⣀⠀⠀⠀⠀⠀⠉⠂⢄⣤⠄⠄⠀⠲⢁⡀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⠀⢠⠤⠴⣀⠁
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢠⠒⠩⣀⣀⣀⣀⠀⠔⠉⠑⡄⠀⠀⠀⠀⢠⠊⠀⠉⠉⠉⠈⠁⠀⠑⢄
⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠻⠆⢸⡿⠛⣷⡆⠘⢻⡿⠃⠛⢻⣿⠛⠃⠛⣿⡟⠀⣴⡾⠛⠻⠀⢀⣾⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢺⡿⢾⣿⠁⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⣿⡄⠀⠀⠀⣼⣯⣸⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠟⠃⠘⠃⠀⠙⠗⠐⠟⠻⠂⠀⠘⠛⠀⠀⠺⠟⠗⠀⠈⠛⠷⠛⠐⠛⠁⠀⠙⠇⠘⠛⠿⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⠶⠆⢰⡦⠀⢰⣶⠀⠲⣶⠶⠀⣶⣶⡀⢰⡆⠀⣶⠆⣠⡶⠀⠲⣶⡶⠀⣶⣶⡄⢰⣦⠀⣠⡶⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡿⠿⢿⣿⠀⠀⣿⠀⠀⣿⡝⢷⣾⡇⠀⣿⣿⣯⡀⠀⠀⣿⡇⠀⣿⡟⢿⣼⣷⢸⣿⠀⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠸⠗⠀⠸⠿⠀⠶⠿⠶⠀⠿⠃⠈⠿⠇⠀⠿⠂⠙⠿⠄⠶⠿⠶⠀⠿⠇⠈⠻⠟⠀⠙⠷⠶⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠈⠉⠛⠶⠟⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠾⠃⠀⠀⠀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡾⠁⠀⠀⠀⠀⠀⠀⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⣄⠀⣴⠶⠶⠶⠶⣄⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠌⠻⠶⣤⣤⣤⠈⠿⣤⣤⣤⣴⠟⠀⢷⣤⣤⣤⣴⠟⢠⣤⣤⠾⠋⠘⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣄⣀⣴⠶⠶⠶⠀⠀⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⢿⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⣧⠘⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠻⢦⣤⣀⣀⠀⠉⢉⣹⡇⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠈⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣄⣀⣠⡄⠀⠀⣀⣠⣤⡶⠞⠿⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠛⠋⠉⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⢦
⠀⠀⣠⡾⢋⣡⣤⡶⠶⢦⣤⣉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀
⢀⣾⢋⣴⠟⠁⠀⢀⣀⠀⠀⠙⠳⣄⠻⣆⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⣿⣄
⣼⠃⣾⠃⠀⣰⠞⠛⠟⠛⢶⡄⠀⢹⣆⢹⡆⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⠀⠀⠀⠀⠀⣤⡾⠋⢙⣷⠛⠁
⣿⢰⣏⠀⢰⣿⠄⠀⠀⠀⢼⣷⠀⠀⣿⢸⡇⠀⣿⣉⣉⣉⣉⣉⣹⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⢠⡶⣦⣀⡀⠻⣶⡞⠁⠀⠀⢀⣀⡴⢦⡀
⢿⡀⣿⡀⠀⠻⣦⣀⣄⣠⡼⠃⠀⢰⡏⣸⠇⢀⣈⣉⣉⣉⣉⣉⣁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠘⠳⠟⢹⣧⣤⠈⠀⠀⢀⣤⣼⡏⠛⠟⠁
⠈⢷⡈⢷⣄⠀⠈⠙⠛⠉⠀⢀⣴⠟⣰⡟⠀⢈⣉⣉⣉⡉⠉⣉⣉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣿⠀⣴⢶⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⠶⣄
⠀⠈⠻⣦⣉⠛⠶⢦⣤⡴⠶⢛⣡⡾⠋⠀⠀⢈⣭⣭⣭⣥⣤⡍⠉⠁⠀⣿⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⣿⠀⠙⠟⠋⢉⣉⣉⠀⠀⠀⠀⣉⣉⠉⠙⠛⠋
⠀⠀⠀⠀⠉⠛⠳⣶⢶⡶⠟⠋⠁⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⠀⠀⠀⠀⠀⠰⣿⣀⣀⠀⠀⠀⣿⠀⢠⡶⣦⣼⡏⠉⠀⠀⠀⠈⠉⢹⣧⡴⢶⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣼⣧⡀⠀⠀⠠⡦⠤⢤⣤⣤⣤⣤⡤⣤⠤⢴⠄⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⣿⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢀
⠀⠀⠀⠀⠀⠀⣿⣤⠀⢸⡇⠀⠰⠀⠶⠀⠆⠀⠀⠀⠀⠀⠀⠶⠶⠆⠀⣿⠀⠀⠀⠀⠀⠀⣠⠄⣿⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠻⣦⡾⠛⠛⣻⠟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠰⣦⡀⠀⢠⡾⠋⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠻⣶⠏⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠓⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣤⣤⣴⣤⣤⣤⣾⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⢀⣠⣤⡶⠞⠛⠋⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⢤⣹⠀⠀⣠⣴⣾⣿⣿⣷⣦⣤⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠻⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡠⠴⠶⠶⠞⠀⢀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣆⡃⣿⣿⣿⣿⣿⣿⣿⣿⡟⢒⣛⡂⠀⡀⠀⡀⣄⠒⠁⠄⠀⣸⡵⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠁⣽⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠁⠡⠶⠚⠀⠀⢠⢃⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣼⠏⠈⠛⠛⠛⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡻⢊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⣴⣏⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⢀⠷⡄⠀⠀⠀⠀⠀⢠⣇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠠⣫⣿⠀⠀⠀⠀⣠⠚⣅⡆⠀⠀⠀⢀⣀⠽⡀⠀⠀⠀⠀⣾⠟⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⠀⠀⠀⠰⠃⠀⠀⠙⢦⣤⡴⠟⠁⠀⠃⠀⡄⠀⢠⢿⣒⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣴⣈⣹⣶⠠⢀⠀⠀⢀⠀⠀⠀⠇⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢻⣿⣿⣧⢀⣴⣤⠴⠞⠇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣀⠀⠲⠄⡀⠄⠀⠀⠤⠔⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡟⡇⠈⠙⡶⡀⠀⠀⠀⠉⠉⠀⠀⠀⣰⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡺⠛⠁⠀⡇⠁⠀⠀⢷⠟⠳⠶⠦⠤⠤⠶⣶⠋⠁⡟⣿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠐⠉⠀⠀⠀⠀⢧⠀⠀⠀⠈⠜⡄⠀⠀⠀⠀⢠⠃⠀⢠⠃⠛⠃⠀⠙⠲⢤⡀
⠀⠀⠀⢠⡶⠾⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢠⠟⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢤⣀
⠀⣰⠏⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠢⠄⠀⣀⡀⠀⣀⣀⡤⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠦⣄
⢠⡏⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⢦
⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡇⠀⠀⠀⠀⠈
⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣛⣋⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠐⠫⠘⠓⢣⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠤⠄⠀⠀⠠⢄⠀⠀⠀⠀⢘⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⢀⣀⡀⠤⠀⠀⠤⡀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⠁⠀⠀⠤⠀⠀⠀⡠⠀⠀⠀⠀⡠⠃⠠⡨⠐⠉⠄⠐⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⡆⠀⠈⠀⠀⠀⡠⣂⠙⢦⡐⠊⠐⠀⠁
⠀⠀⠀⠀⠀⠀⢰⡄⠜⠀⠉⠀⠀⢆⠆⠈⠉⠑⠄⠢⠊⠱⡔⠤⠠⠙⠒⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⡂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢈⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠒⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢄⢇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠱⣁⣈⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⡀⣀⡀⣀⢀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠸⠺⠨⠇⠸⠀⠧⠻⠤⠼⠳⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢠⣄⡄⣤⢠⣤⣤⣤⠀⡆⣤⣠⢠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠃⠛⠈⠈⠉⠉⠃⠃⠉⠙⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠁⠀⠺⠂⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣰⠂⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠕⠤⠄⢪⠭⠒⢯⠽⢠⣔⠡⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⢖⢄⣓⣚⡍⠁⠉⢹⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠈⠑⠢⠤⢄⡠⠤⠚⣀⣀⣀⣀⣰⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠗⣉⡚⢕⡄⢀⠀⠀⣀⡀⠀⠀⢠⠚⠁⠀⠀⣰⣢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢾⣀⡸⢨⡲⠰⠥⠽⠀⡇⠀⠀⡇⠀⠀⠦⣜⠞⢀⡤⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣦⣭⣴⠿⠃⢸⣯⠝⠀⡇⠀⢠⠃⠀⠐⡓⣒⠀⢐⡒⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡆⠐⠚⠛⠛⠻⠀⡇⠀⢈⡇⠀⠦⠭⠥⢄⠤⡭⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡇⠒⠒⠒⠒⠒⠚⠉⠶⠁⠈⠒⢲⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣁⡄⠒⢀⣀⣰⣞⣿⣿⡦⣄⡀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⠛⢁⡤⣞⣭⣷⣿⣿⣿⣿⣿⡿⣿⣶⣄⣤⣹⡤⢤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠴⠀⠐⣫⣾⣿⡿⠿⠛⠋⠹⠟⡟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣪⠎⢰⢆⣴⣿⢿⠛⠓⠈⠀⠀⠀⣀⠀⢀⠛⢙⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢋⡴⣠⣿⣿⣿⣿⠟⢿⣿⡇⠀⠠⠿⣿⡿⠿⢦⣽⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⠟⠋⠙⠋⣀⣀⠀⠀⠈⠙⠻⢟⡛⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣼⣿⣿⣿⣿⣿⣿⣿⣧⠃⠀⣠⠼⣿⣿⣿⠋⠀⠠⡀⢦⣿⡞⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣠⣾⣿⢿⠻⠿⣶⣄⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣶⣾⣶⣷⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⢛⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⡘⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢃⠀⣾⣿⣿⣿⣿⣿⡿⢷⣿⢺⡷⡾⢟⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣷⣈⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣼⣿⣿⣿⣿⣿⣿⣿⡆⠸⠋⠛⠛⠙⠋⢹⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠸⣿⡟⠙⢛⣿⣿⠹⣿⣿⣷⣶⣶⣤⡓⢠⣤⢿⣿⣿⡿⠀⣿⣿⣿⣿⡿⣿⠿⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠁⠛⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠛⠿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⠆⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢤⣾⣿⡀⢼⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣂⣺⠏⠁⠙⠛⢿⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣰⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⡿⣿⣿⣿⣿⣭⣼⣷⣿⣿⣷⣶⠛⠻⠎⢴⣶⣄⠁⢈⢹⣘⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⣭⣭⣶⣴⢾⣿⣿⣿⣿⣿⣿⣯⡶⢘⢿⣿⣿⣦⣳⣌⠿⣷
⠀⠀⠀⠀⠀⣀⢤⣾⣀⡾⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⡳⣙⣿⣿⣿⣿⣷⢹⠷⠢
⠀⠀⠀⠀⠀⣘⣾⣿⣿⣵⢟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠨⣶⣄
⠀⠀⠀⠀⢰⡏⣿⣿⠛⢁⠸⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢧⡌⢣
⠀⠀⠀⠀⣾⢱⣿⠃⢸⣥⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢦⠹⡌⡀
⠀⠀⠀⣄⣿⣼⠃⠀⢸⠏⣠⣽⣿⠟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢦⢷⣷⡄
⠀⠀⢰⢸⣿⣿⡄⡄⠋⢸⣿⣿⡟⠸⣽⣿⣿⣿⣿⣿⠟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣿⣿⣏⢃⢿⢻⡄
⠀⠀⣆⣿⣿⡿⢧⠁⠀⣸⣿⣿⠁⠀⣌⠉⣻⣿⣿⢋⠼⢣⣾⣿⡿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣮⡀⢌⡧

⣈⠙⠷⠋⠖⠲⢦⣤⣖⣇⣠⣿⡛⠐⠞⠚⠀⠀⠀⠀⠀⠀⠀⢆⣀⣱⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠨⠀⠂⠀⠀⠀⠘
⣱⡖⡶⠒⡤⠟⢻⣇⠀⠈⢻⠍⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⡠⠄⠀⠎⠀⠀⠁⠀⠤⠀⢀⣀⠦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠑⠂
⠃⣰⠶⡀⢀⣀⠀⡤⠄⠀⠀⠀⠀⡀⠀⠈⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⡀⠀⠀⠀⡠⠄⠁⡒⠀⣭⠎⠈⠁⠀⠁⠉
⠀⠀⠀⠄⡀⠁⠨⣻⣤⠴⠆⠁⠐⡿⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠠⠄⠀⠀⢲
⠂⣉⢘⣧⠠⠚⠀⠉⠀⡼⠻⠬⠴⠁⠀⠀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⢀⣶⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠈⠧
⠷⠚⢷⣈⡙⠁⠄⠜⢒⣹⣦⡀⠀⠀⢀⡁⠀⠀⠈⠂⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣧⣀⢹⡛⠛⠛⠋⠁⠀⠀⢀⡀⡄
⠐⣀⡉⢛⣧⣄⣴⣆⣉⠉⠁⠀⠀⠀⠋⠒⡆⠀⠀⠀⠀⠐⠉⠀⣀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠄
⣲⣬⡵⣿⠋⠰⣯⠀⠌⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠈⠑⠆⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢯⣾⣤⣄⡩⢟⣉⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆⠀⠠⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠞⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⣐⡆
⡼⠿⠿⠯⠟⣒⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠀⠀⠀⢀⡀⡄⡀⠀⠀⠀⠀⠑⣾⣧⠀⡣⠀⠊⢀⠀⠀⠀⠀⣤⠖⠚⠓⠀⠐
⠲⠖⢶⣢⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣋⠀⠀⠠⠀⠲⠀⢸⣏⠉⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⢠
⢶⣟⣌⡓⠁⣹⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠼⡁⠀⠀⠀⠀⠀⠀⠈⠱⠤⠄⡤⣠⡄⠱⣦⠤⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠘
⣈⡈⢻⣷⣬⡟⠟⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠦⠀⠀⡀⠀⢀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈
⠋⣄⣀⠇⠠⠥⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠈⠀⠎⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠎⣵
⠊⠻⡟⠚⠀⠂⠐⠀⠀⠄⠀⠀⠀⠀⠀⡀⠖⠀⠚⠃⡸⣄⣀⠀⣀⡀⠀⠴⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠈⢀⣾⠴⠟⠶⠋
⡆⠈⡳⠥⠄⠀⠀⠄⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠁⠑⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢂⣀⠇⢠⠒⠈⠋⣠⠸⠀⠀⠂⡐⠂
⣥⣆⠖⠀⡁⠀⠀⠀⠑⠂⣠⡶⠁⠀⠃⢈⠳⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠰⣦⡤⠀⠀⠀⠀⠀⠀⣀⠐⡀⠖⠓⢀⡔⡃⠈⠑⠂⠀⠀⠄
⢠⠾⣎⡩⠃⠀⠀⠀⡐⢌⠁⣁⢀⠀⠠⠤⢔⠃⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⣀⠘⡀⠀⠀⠀⠀⠀⠀⢈⠁⢠⠃⠀⠀⠈⠀⠁⠀⠀⠁⠀⡀⠀⠄
⡴⠎⠫⠀⠀⢄⡀⠀⠀⣈⠐⠉⠁⠀⠀⠰⠎⠥⠜⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣄⠠⣽⣁⢀⠄⠃⡰⠂⣎⠀⠈⠠⢀⡄⠀⠀⠀⠀⠀⠀⠃⡀⠠⠂
⢁⣶⢶⣶⣶⣞⣳⣄⣻⠶⠹⡷⠶⣶⣦⣀⡆⠀⠬⢉⠛⣉⣀⡔⠤⠤⠀⡀⠤⠀⠚⠂⣉⠋⠁⣴⠗⠊⠉⠀⠀⠋⠍⠀⠀⠀⠌⠁⠀⠁
⡾⣷⣾⢭⡝⢉⡃⠞⠙⡟⠀⢉⠐⡶⣧⢞⣉⣈⡐⠆⢀⣋⣉⣬⣤⣤⣠⣶⠒⠓⠛⠲⠡⣤⠾⢟⢀⣃⡀⠀⠂⠀⠠
⣀⠀⠀⢀⣧⠼⡟⠻⠾⠧⣤⡄⢘⡷⣵⢞⣯⢿⠓⣾⠉⠁⣹⣄⣻⣷⣿⢟⣷⣶⠦⣄⠐⠙⢢⣾⣧⣀⠀⠆⠀⠀⠠⢂⠒⠦⠄⢠
⠀⠹⠀⠈⢡⡐⠉⠳⣦⡀⠩⠄⠄⠤⠚⠋⣑⡷⢾⣿⡀⣠⣿⣿⣿⣿⣿⣿⣂⠿⠁⣄⢰⡆⠙⠛⠏⠉⠀⠀⠀⠈⢀⡌⠥⠀⢠⠆
⣤⢠⠴⠚⣈⠃⠁⠀⠈⠀⠀⠀⠂⠀⢀⡠⠰⠓⣦⣭⣿⡿⣿⣿⣿⣿⣯⣛⣿⣷⡀⠙⠁⡀⠄⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠇
⣅⠆⠁⠋⠁⠼⡖⢀⠀⠀⠀⠀⠰⠂⠀⠁⠀⠁⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡇⠈⠁⠀⠄⠀⡀⠈⠀⠈⠀⠀⢀⠀⠀⠀⠀⣀⠴⠄⠆
⣦⡀⢐⠚⠒⠈⠁⠀⠋⠀⡲⢁⠤⣀⠀⡀⠀⠁⠀⢺⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡾⠓⠂⠀⠰⠉⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⢸⣿⣶⣴⣶⠩⢻⣿

⠀⠀⠀⠀⠀⠀⠠⢔⠒⠒⠒⠛⠛⠂⠀⠀⠀⠒⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡑⡆⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠔⢣⣆⡔⠁⠀⠀⠸⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠣⡎⠀⣴⣁⡠⠤⡄⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠑⠒⠦⠴⡀⠀⣅⠀⠀⠀⣸⠀⢰⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠉⠉⠁⠀⠈⠄⠀⠀⡏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠀⠀⠀⠤⠐⠂⡙⣀⠔⠦⡀⠀⣀⣸⠀⠀⠁⡠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡫⠿⠥⠦⣶⣶⠀⡇⠀⠀⠀⠐⢀⣊⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠂⠉⠱⠾⢿⡇⠧⠤⠞⢄⡀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠒⠀⢰⣦⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⣄⣸⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠙⢄⡈⢙⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠠⢖⡪⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣀⡐⠠⡑⠬⡑⠊⠀⠐⢄⡠⡒⡄⠀⠀⠀⠀⠀⠀⠁⠆⢣⢄⣀⠅⠴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢇⠠⡈⠇⠦⠀⠀⠁⠒⢠⣄⣁⡇⠘⡖⠂⠀⣖⡰⠂⠠⢌⠳⡀⡽⠤⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⢈⣟⢀⠭⠭⠤⠭⠅⢿⠁⠓⠘⠓⠦⠇⠀⢀⠇⠈⠀⠀⠠⠁⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⠀⠈⠂⠀⠸⡀⢆⠀⠀⠀⠙⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⡔⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠃⠀⡰⡙⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⡤⢤⣷⢎⠎⡵⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⠀⠀⠖⠁⠀⡇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⢤⠤⠼⠞⣄⠀⠀⠀⠀⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠠⠤⠐⠚⠻⠤⠬⠇⠀⠀⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡶⣶⣶⣶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⣦⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣴⠾⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠴⣾⣿⣿⣿⠏⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡩⢓⡥⠀⠀⠰⣶⣿⡿⢫⣾⣿⣿⣿⣿⣟⣉⣿⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠛⠛⠹⣿⣿⠿⣛⣿⣿⣿⡿⠟⢹⡷⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣹⡿⠂⠉⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⣿⡿⠟⠉⠀⠀⣾⣰⣡⣾⣿⣿⠶⠆⠀⠀⠀⠀⠀⣾⠿⣿⣿⠿⡜⠎⠉⢢⡀⢹⣼⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⢸⡇⣿⠈⠐⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⣠⠔⠂⠀⢠⣴⣶⠾⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣧⠀⢩⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⡼⢁⣴⡾⠟⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣸⠏⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠈⠳⣄⠳⡿⠿⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠀⡨⠛⣠⣴⠒⠋⠉⠉⠉⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡞⠁⠀⡱⠀⠀⠀⠀⠀⢀⣠⠤⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⢀⣴⠃⠀⠀⢀⡴⠞⠋⢀⡦⣾⣿⣿⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣧⡴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢰⠶⠀⠀⠀⠙⠦⠤⠖⠋⡀⠙⢠⠶⣋⢀⣀⠏⣡⠞⠛⣸⠇⠘⢷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⠀⠀⠀⢀⣠⢶⡿⢢⢞⣵⣾⡁⠀⠀⠀⠈⠛⠀⡼⠃⠀⣴⠏⠀⠀⠀⠀⠀⠀⡾⣼⡇⠀⠀⠀⠀⠀⠀⣼⠃⢀
⠀⠀⠀⠀⠀⠛⣠⡄⢀⣤⠛⢃⣤⠖⣋⠴⢋⡴⣣⢎⡾⠹⠃⠀⣐⣦⡀⠀⢸⡇⢠⡞⠁⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⡼⠃⢀⡞
⠀⠀⠀⠀⠀⢀⡈⠀⣾⣯⠖⠋⣠⠞⣡⠀⢀⡴⠃⢀⣠⠤⠒⠉⠀⢨⡷⠀⠈⠉⠁⠙⣦⡀⠀⠀⠀⠀⢀⡞⣿⠀⠀⠀⠀⠀⢀⡾⠁⢠⠞⣸
⠀⠀⠀⠸⢣⣬⣡⢖⡋⢀⡴⠟⢁⢺⠙⣦⡾⣲⣏⣃⠬⠴⠶⠂⢠⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠶⠋⠀⠈⠙⡶⠶⢤⣀⡞⠀⢠⠎⣰⠃
⠀⠀⣀⣴⠞⠋⠠⣀⡵⢋⠀⢀⣸⣆⡽⠋⠉⠉⠉⠹⣇⠀⠀⠀⠺⣧⣀⣠⡤⠤⠶⠖⠲⣤⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠟⠀⢠⠏⣰⠃
⠀⠀⠉⠀⢀⣠⠞⠋⡆⠀⢳⠟⢡⠞⠙⠒⠦⣤⣀⠀⠈⠙⠓⠒⠲⢦⣄⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢿⠤⠀⠤⣾⠀⠀⠈⢰⠇
⠀⠀⢀⣴⡟⢁⣤⠀⡇⠀⢹⣶⣋⣐⣒⡂⠉⠁⠈⣹⣦⡀⠀⠀⠀⠀⢹⣷⠀⠀⣠⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏
⣠⡶⠋⠛⠋⢛⡵⠀⣇⢠⠞⠑⠤⠴⠶⠶⠶⠚⠋⠉⠋⠀⠀⠀⠀⠀⠀⣿⡆⣼⠋⠀⠀⣀⡼⢋⡽⠃⠀⠀⠀⠀⠰⠾⣤⡤⣤⡘⠁
⠉⠀⠀⢀⡴⠋⣴⠄⣽⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡈⠓⠚⣉⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣿
⠀⠀⠀⠈⠀⠀⠀⠚⢁⠈⠙⣿⡶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠾⣧⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⢸⣷⠦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠉⠉⠛⠛⠿⣷⣶⣶⣶⣤⣷⣤⣿⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⠄⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠈⠉⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣶⠒⠾⣁⣤⡾⠟⠋⠁⠀⠀⠀⠈⠉⠙⠳⢦⡿⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣛⣋⣀⡀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣛⣻⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠋⠠⠆⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⡬⣷⣃⠀⠅⣈⠀⠀⠀⠀⠀⠸⣦⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡬⠀⣿⡅⣾⠈⠉⠀⠀⠀⠀⡀⠀⢿⡏⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡞⠀⢀⣪⠉⠳⣶⠂⠀⠀⠀⠀⠳⠀⡺⣏⠃⠢⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⠃⠠⢸⣇⡀⠁⠀⣀⠦⡠⢀⡀⠀⡀⣙⣿⣯⠂⢶⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⠣⠀⡦⠺⡍⣄⡂⠎⢅⡼⢤⠞⠉⣸⠝⡌⠫⡛⠑⠄⣄⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣁⣿⠀⣰⠳⠃⢰⣟⡿⡛⢱⡠⠾⠷⠍⢙⡆⣄⠳⡂⢻⡶⠆⠠⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⠈⣀⢁⠀⠀⡏⣋⠁⠇⠊⢠⡀⠀⠐⢋⢧⣒⠄⠐⢇⡹⠐⡉⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠯⠀⡹⡄⠀⠐⣣⠛⡠⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀⠁⠈⢽⣳⡀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⣸⢡⠀⠀⠀⡷⠆⣃⠀⠀⠀⡀⠀⠀⠠⢂⡷⠀⡄⠀⢸⠈⠃⢈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠁⡸⣿⡂⢩⠀⠀⠸⡷⠜⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⠀⢸⡀⠄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢰⡸⠁⣯⢟⡆⢧⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⡞⠀⠄⠀⠀⣠⠉⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢁⠤⠻⠔⣵⣀⠀⠀⠀⠈⢃⡂⢐⠀⠀⠀⢰⡁⠖⢠⠀⠀⣧⠤⢤⡤⢰⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡠⢀⠨⠀⠙⡌⠐⠀⠀⢀⠄⢸⠀⠀⢠⠷⠆⠺⠈⠠⠆⣠⠸⠡⡂⡄⡄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢦⣣⠀⢁⠐⠸⢦⣰⠀⠀⡇⢸⠀⠀⠀⣦⢊⡞⢴⡝⠺⠉⠀⠀⠙⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠈⢂⡙⢢⡇⠁⢸⡙⣤⠼⠃⣤⠀⠀⠀⣼⢿⠑⢘⡿⠃⠀⠄⠀⡀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡝⠡⣜⠁⡤⠑⠉⠓⠀⠀⠒⠉⠀⠈⠉⠛⠒⠳⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⠈⠁

⠀⠀⠀⡐⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢰
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⢠⡄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡇⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠁⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣇⢀⣁⣀⣀⣈⣉⡀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⢠⠴⠂⢙
⠀⠀⠀⡇⣤⠀⠠⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡂⠚⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⢸
⠀⠀⠀⡅⢸⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠒⠀⠀⠂⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀⣀⣀⢁⡀⢀⡉⡀⠀⠀⠀⠸
⠀⠀⠀⡟⠓⠒⠒⠚⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠁⠀⠈⠉⢈⠉⠉⠉⠁⠀⠀⠀⢨
⠀⠀⠀⡇⠀⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⡅⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⢀⢨
⠀⠀⠀⡆⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡙⣈⣙⣏⣀⡋⣉⣿⢙⣸⡟⢻⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠠⠐⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⣞⣉⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢉⣠⣭⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⣿⣿⠹⣿⣿⣿⡙⣿⣿⣿⣿⣿⣷⠀⠀⠉⢦⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠈⠛⠇⠈⠛⠿⣷⠀⠙⠻⣿⣿⡿⠀⠀⠀⠈⠓⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡇⠀⠀⠈⣓⣦⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⡇⠀⠀⠀⠁⠀⣀⡤⠤⠤⢤⣀⠈⠙⠀⠀⠀⢸⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⢿⠀⢀⣠⠴⡾⣻⢛⡆⣼⡳⡿⡗⢦⡀⠀⠀⣼⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗⠉⢠⢞⡝⡵⢋⡞⢧⠱⣝⢎⢧⡉⠳⣴⣿⡟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢳⣴⠋⢫⢞⡖⡹⠀⠸⡟⢮⢮⠳⠳⣤⣾⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢳⡀⠀⠋⢸⣥⣤⣀⣹⠌⠃⠀⢠⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠹⡟⣻⠃⠀⠀⣰⠏⠹⣿⣿⣿⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠓⢦⣠⣧⣷⡤⠖⠋⠉⠢⡀⠙⠉⠛⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣿⢾⣿⡇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⣼⠻⣿⣟⣿⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣰⠃⠀⠙⠛⠀⢧⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⡴⢦⡀⠈⢧⡀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣠⣼⣁⣀⣀⣀⣙⣋⣀⣀⣀⣳⣄⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⡘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡽⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠴⠙⣶⡆⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢷⡦⡤⣱⣀⡸⣧⠴⠟⣿⠇⠀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⣀⡉⠈⠹⠉⠀⣈⠂⢈⡁⢃⡄⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠐⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠓⠙⠁⡀⠀⠓⠐⠇⢀⠸⣇⢠⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⠀⠻⡎⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⢸⣧⠀⠀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣈⠀⡄⠐⣐⠂⣽⢃⡎⠀⠀⡌⡀⠀⠀⢀⠀⠀⠀⠠⣄⡀⠀⠠⠄⠀⠀⠀⠀⠀⠁
⢠⢺⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠙⣶⠀⠘⢉⠉⠁⢸⠀⠀⢸⠇⡇⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀⠀⠀⠀⠀⠀⠂
⢸⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣨⣿⣷⡄⡏⠛⠤⠴⠦⠖⠹⣧⣴⢿⡄⣷⡁⡀⢀⡀⢀
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡷⠟⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⢹⡿⠡⠛⠛⠿⠋⠀⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠈⢁⠀⠀⠈⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⡈⠀⢠⠀⢠⡟
⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠁⠁⠂⠉
⠀⠀⠰⠀⠀⠀⠃⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠶⠶⠖⠛⠋⠩⢿⠮⠛⠛⢛⣻⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠉⠛⡳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠠⡀⠐⣶⣶⣯⣶⡦⣉⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡄⢀⣀⠀⠀⠀⠐⢯⣭⡿⠟⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠻⣿⣿⣿⣧⠉⢿⢮⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢈⣴⡿⠋⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⢿⠘⠘⠏⢈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⠃⢨⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⡀⢀⡴⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠇⠀⡇⣿⡎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠰⣿⣿⡆⠀⠀⠀⣸⣛⣢⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠄⣿⣿⣿⣟⣿⠗⠀⠀⠀⡀⢸⢿⣶⣦⡰⣶⡀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣾⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠀⠁⠀⠀⠀⠀⢹⡇⣿⣿⣿⣿⣼⢁⠀⢀⠀⠀⠀⠀⢀⠠⢿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠗⡀⠀⠀⠀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠁⠡⠀⠄⠀⠀⠀⠀⠁⠹⠝⢿⣿⣏⣄⠀⠠⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⢐⣿⣿⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⣿⣻⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡤⠀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠃⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣛⣛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⣀⠀⠀⠀⢠⠐⠀⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⡄⠀⠙⠻⢿⣿⣿⡧⠀⠀⠀⠀⠏⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣵⣯⣿⠀⠐⢀⠀⢈⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠟⢠⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠿⠻⣯⣽⡟⠁⠀⠈⣠⣮⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠄⠘⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⡀⠀⠀⠀⠀⣲⣾⣆⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⢀⠀⠉⠿⣿⣿⣿⣿⣽⡧⠁⠀⠀⠀⠀⠆⢠⣶⣤⣞⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣀⠂⠠⠀⠉⡉⠉⣥⣤⣄⣤⣄⣀⣸⣾⡵⠟⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⠄⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠂⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠄⠂⠙⠃⠀⣔⡰⠰⡀⠀⠀⠀⠀⠠⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠧⣄⣫⠅⢰⡐⡀⠀⣠⣄⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢀⠄⡐⢂⣦⢠⡌⢠⡆⠁⠀⣀⠀⠀⠠⠂⡔⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠀⠰⡄⡰⡄⠉⢬⡙⡟⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡁⠀⡘⠓⡆⢀⠄⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⠋⡌⠛⡄⠄⠋⠀⢀⡀⠀⢀⠄⠘⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠆⠀⢄⠃⠐⠂⠀⡁⡝⠛⢻⡄⠀⠀⢉⣀⠐⣚⠃⠀⠀⠠⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢁⠀⠀⠈⠋⠙⠲⡀⢿⠀⢤⠀⣶⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠂⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠉⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠋⠀⠠⠀⠀⠀⠀⠀⡐⠁⢏⠀⠀⠀⠀⢀⣨⠠⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⣾⡷⠈⠇⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠂⠀⢀⠐⡄⡛⢅⠢⠀⠄⣂⠀⠀⠀⠀⠲⣄⠀⠀⠠⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⡈⠖⠀⠀⠀⢈⠁⠀⠀⠀⠈⠂⠈⣇⠀⠀⠀⠀⠘⠁⠀⠘⠇⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣄⢀⡀⣤⣄⡀⣧⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇⠀⠟⠀⠙⢈⡟⠘⡇⢩⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⣇⢺⣇⠀⠀⠀⣿⡌⡀⢸⠀⠀⠆⠀⠀⠀⠈⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠱⠆⠉⢳⡀⠀⠈⣿⡇⠸⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⡄⢀⡛⠀⢀⡟⡷⣸⠀⠀⠀⢀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣸⣼⠁⡹⠁⠀⡼⠀⣧⣿⠀⠀⣤⣼⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⣠⠤⢀⠖⣡⠁⢸⠁⠀⠜⠙⣤⢿⠛⡀⡄⠣⡀⢠⣤⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⢂⠐⠴⠖⡌⠀⠄⠀⠜⠁⣠⣇⠀⠘⡄⠀⢾⡘⣟⢧⠀⠀⠀⠀⢷⡀⠂⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⠠⣄⡰⠆⠀⢑⠒⡆⠀⠗⠞⠉⠙⠀⣼⠿⠄⠀⠓⠿⠄⠓⠂⢰⠭⠂⠻⠄⠀⡀⠀⠀⠀⠀⡀
⠀⠁⠀⠀⠀⢀⡄⠀⠀⠁⠀⠒⠉⡴⠚⡇⠈⠀⠙⠄⠡⣄⠈⠀⡇⠸⢫⣮⠀⢤⣤⠴⠶⢟⡁⠀⠀⠲⡄⠛⢃⡀⢠⠐⠒⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠘⢡⠇⢐⠦⠀⠐⣦⣌⣀⣀⡤⢲⠀⢿⠀⣀⡀⠀⠀⠀⠙⡌⠈⠀⠠⠄⢈⠰⣉⡀⠐⠒
⠀⢠⠀⠀⠀⠀⠀⠦⠀⠀⠐⢠⠀⡘⢁⠀⠈⣐⡾⠁⠀⠀⠁⠊⡐⠀⠀⠈⢦⠩⠀⢀⡀⠀⠀⡴⠠⠘⠂⡰⣆⠀⡀⠉⠐⠛
⠀⠼⠀⠀⠀⠀⠢⠼⣥⠔⣀⠂⠄⠀⢉⡔⠙⠋⡰⠀⠀⠀⠈⠂⡉⢈⠙⣰⠀⠐⢅⠘⢃⠈⣥⠀⣀⠐⣶⠆⠁⠒⠂⠀⠠⠣⡴
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠐⠃⠤⠄⠀⠐⠦⠀⠀⠦⠀⠈⠀⠊⠡⢀⡸⣀⣐⠟⠈⠂⠀⠘⠂⠠⠀⠀⠀⠈⠁
⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⡟⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⢘⡇⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⢀⣀⡠⣼⣧⣄⣀⡀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣉⣀⡤⠤⠖⠚⠋⠉⠀⣠⠋⠘⣆⠀⠉⠙⠓⠲⠤⢤⣀⣉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠓⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢧⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⡤⠖⠛⢯⡉⠙⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⡹⠛⠲⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣚⢁⣀⡤⠴⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠳⠦⢤⣀⡈⣑⡾
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣀⡤⠴⣖⠛⠉⠙⠣⢄⡈⠳⣄⠀⠀⠀⠀⣠⠞⢁⣠⠜⠋⠉⠛⣲⠦⢤⣀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⣁⣀⣠⣬⣷⡀⠀⠀⠀⠉⠓⠮⣧⡀⢀⣼⠵⠚⠉⠀⠀⠀⢀⣾⣥⣄⣀⣈⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣁⣀⣀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣀⣀⣈⢳⡄
⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠀⠘⡆⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢰⠇⠀⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣦⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠙⠂⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠳⣄⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣓⣒⣲⡤⣤⡘⣆⠨⡄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢉⣩⣤⡤⠬⠽⣷⣯⣿⣄⡵⠁⢀⣀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣠⠞⠋⢁⣠⣤⣶⣶⡶⢾⣿⣿⠿⣿⣿⣿⣿⣭⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁⢠⣶⠿⠛⠉⠀⠀⢀⡾⠉⢳⣄⡀⠈⢻⡆⠉⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣴⡿⠁⠀⠀⠀⠀⠀⠘⣇⠀⢺⣼⠟⠦⣼⡏⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡟⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣦⣤⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡵⠟⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣤⣤⣾⡿⠟⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣶⡆⠀⠀⣠⣾⠋⠁⠸⡍⠙⢶⡀⠀⠰⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠁⠀⠀⣼⠋⠘⡄⠀⠀⢧⠀⠀⢻⣄⠀⠀⠈⢹⣿⠛⠛⠛⠛⠿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⠏⠀⠀⢣⠀⠀⢸⣀⣤⠾⠛⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⠀⡇⠀⠈⣤⠴⠛⠉⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡇⠀⣠⡴⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠸⣷⠞⢉⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣄⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠶⠟⣿⡿⠿⠿⠿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⡷⠶⣦⣤⣀⡀
⠀⣀⣠⣀⣠⣴⡿⠛⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⠈⢳⣄⠀⡠⠞⠉⢋⠉⠲⣄⠀⣠⠟⠹⣿⠀⠀⠈⠙⠻⣶⣄⡀
⣼⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⢸⠀⠀⢈⡿⠋⠀⠀⢻⣇⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣄
⠸⢿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢸⠀⣠⠟⠁⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠂
⠀⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢛⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣿⡿⠿⠿⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⠼⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡤⠴⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⢀⣆⣀⣤⣆⠙⣦⠀⠀⠀⠀⠀⠀⠀⣰⠋⢡⣤⣄⣀⣅⣀⣴⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣭⣥⣤⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣶⣶⣦⣤⣄⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣰⡗⠀⠀⢀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣤⡀
⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠠⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷
⠀⠶⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⣿⠇⠀⠀⠀⠀⢸⡇⠠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢰⡆⠀⠀⠀⠀⠀⠀⠈⠛⠃
⢀⣾⡟⠁⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇
⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢸⣿
⠀⢀⣿⠃⠀⠀⠀⢶⡄⠈⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⠀⠀⢀⣴⠀⣠⡿⠃⣀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⣿
⠀⣸⡿⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣽⣿⠟⠁⣠⡟⠁⣴⡟⢠⣾⠋⠈⠻⣿⣏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠉⠁⡀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠙⠁⠈⠋⠀⠛⠁⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⣀⠈⠻⠃
⠀⠀⠘⣿⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⣿
⠀⠀⢠⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⢠⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣯⡀
⠀⢸⣿⠋⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⠋⣿⣤⣾⡇⢸⣧⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠹⣿⡆
⠀⢺⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⣷⡄⠀⢾⠟⣿⣀⣿⠀⣿⠁⠀⣿⠻⠆⠀⣠⣿⠿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⢀⣿⡇
⠀⠈⢿⣇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠙⢿⣦⡀⠀⠈⠙⠁⠸⣿⠀⢀⣿⠀⢀⣼⡟⠃⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣼⡿
⠀⠀⠈⠻⣷⣄⣘⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠛⠷⢟⣁⣴⠿⠋⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣴⣿⠟
⠀⠀⠀⠀⠈⠙⠿⣿⡆⠀⠀⠀⠀⠀⣠⣾⠏⣠⣴⡿⠿⠿⢷⣦⣍⠛⠿⣷⣶⣾⠿⠟⢛⣡⣶⡿⠿⠿⣷⣦⡄⢻⣧⡀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣸⡿⠃⢰⣿⠋⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠈⠙⣿⡄⠙⣿⡄⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⢻⣷⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⣠⣿⣷⣦⣿⠇⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣴⡿⠋⠀⠙⠿⣷⣤⣄⡀⢀⣠⣿⣇⣀⣀⣀⣀⣼⣿⣄⡀⣀⣤⣴⣾⠟⠋⠀⠻⢿⣦⣴⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⠋⠙⠛⠻⠛⠛⠋⠹⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣦⣤⣤⣤⣴⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠈⠉⠉⠉⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⣀⣀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣄⣀⡀⣀⣠⣾⠟