Fiber Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣛⣉⣉⣉⣉⣉⣙⠛⠒⠒⠒⠒⠶⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠀⠀⠈⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠒⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢶⠶⠶⠶⠶⠶⠶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⡏⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣰⢈⡶⢠⡆⡰⠀⣠⠀⢠⣄⠀⠀⡀⠀⠀⡆⣾⠃⠀⠘⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⡀⠀⠀⠀⢸⡀⠀⠿⠀⠽⠸⠇⠂⠰⠃⠃⢸⠇⠳⡆⠀⠀⢰⢠⣿⠀⣤⣤⣍⣈⣁⣉⣘⣋⣛⣀⡀
⠀⠀⣠⡶⠋⠉⢙⣿⣄⠀⠀⠀⠀⠀⢸⡇⠐⠈⠀⠈⠁⠁⠘⠂⢸⡇⠀⢀⠀⡀⠀⣀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⡈⢸⡟⢰⢋⡁⢀⠀⢠⠀⠄⠀⡄⠈⣿
⠀⢀⡿⢷⣤⡞⠋⠀⢈⡗⠛⢶⡀⠀⠸⣇⠀⠀⠄⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠇⣿⡇⣾⢠⠰⠂⠴⡀⠆⡈⡧⡐⢀⡟
⣴⠁⠀⠀⠘⢷⣤⣤⣼⢧⣀⣼⠇⠀⠀⣿⠀⠀⠀⠈⠀⢀⣀⣀⣠⣧⣤⣄⣀⣀⡀⢀⣠⠖⢋⡵⠚⠉⠀⠀⠀⢸⢀⣿⢳⡇⡄⠂⠈⠄⠁⠦⠠⠆⠱⢸⠃
⠻⢷⣤⣤⡴⠿⢤⣴⡏⠀⠉⢸⡇⠀⢀⡩⠴⠖⢚⣋⠩⠥⠄⠀⠀⠀⠠⠤⠀⠈⡵⡋⣠⢞⠁⠀⠀⠀⠀⠀⠀⡟⣸⣿⢸⣇⣤⡌⠡⣾⠿⠗⠀⠘⠀⡟
⠀⣸⠋⣟⣷⣤⣄⣤⣿⢶⡾⢛⡇⡾⢁⡤⠒⠉⠁⣀⣠⣄⠀⠐⠒⢒⣀⠀⡀⢞⢀⡜⣡⡾⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⣸⠁⠿⢧⣴⣦⡄⢿⣿⣷⣠⡇
⠀⣿⠀⠋⠈⢻⡟⠛⣵⠿⠴⠋⠀⠻⢯⡀⠀⠀⠈⠉⠉⢉⣴⣦⠀⠙⠋⠚⢋⣀⣤⣶⣯⠀⠀⠀⠀⠀⠀⠀⡼⣠⡿⠃⣿⢰⣶⡚⠝⢩⠶⣦⡝⠛⣽
⠀⢸⡆⠀⠀⠘⠀⣰⠁⠀⠀⠀⠀⠀⢳⡍⠙⠒⠂⠴⠦⠤⠤⠤⠶⠖⢒⡋⠉⠀⢨⣿⠏⣤⣤⣤⣤⣤⣤⣤⠟⠁⠀⠀⣿⠈⠃⢀⣴⡀⠀⠉⠋⢀⡟
⠀⠀⢷⡀⣆⣀⡀⣿⠀⠀⠀⢠⡄⠀⠀⠻⣄⡳⣘⠥⠀⢠⡶⠰⠃⠀⠊⠁⠅⢐⣾⡟⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣿⣠⣿⠛⡋⢰⣿⣟⠆⣸⡇
⠀⠀⠘⢷⡋⣉⣹⠟⠀⠀⣠⠋⠙⢄⠀⠀⠘⢦⣙⡂⠀⠀⠞⠈⠁⠀⠉⠈⠐⢻⠟⠀⠀⢀⡴⠛⡉⠉⢉⠙⢶⣄⠀⠀⣿⠈⠛⣡⣿⡧⣠⣶⢀⡿
⢲⡤⣄⣀⠉⠉⠀⡐⠙⠄⣧⠀⠀⠀⣧⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠚⠁⠉⠉⠁⠀⢀⡼⢋⠀⠀⠀⠀⠀⠃⢠⡝⣦⡀⠸⣷⣤⣼⣉⣁⡛⠟⣸⠃
⠈⡇⠀⡈⢳⣤⡘⡆⠀⣰⠃⢠⡀⠀⡟⠀⠀⠠⣞⠋⢳⡀⠀⠀⢀⡞⠳⡄⠀⠀⠐⢿⡀⠈⠀⠀⠐⠀⠀⡄⠀⣿⡽⣇⠀⠀⠈⠉⠛⠛⠛⠛⠋⠀⠀⠐
⠀⠹⣆⠹⣆⠀⡸⠁⣠⢻⠄⢻⠀⠀⠹⣄⠀⠀⢸⣄⣲⠇⠀⠀⣏⠀⣤⡇⠀⠀⠀⠈⠛⣳⠶⢶⣦⣤⣤⣤⣤⣵⠞⠃⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠸⠀⣹⡀⢷⡀⢹⢸⠀⢿⡀⠀⢀⡽⠰⣦⣤⣬⣉⣀⣀⣀⣈⣙⣛⣁⣀⣀⠀⠀⠀⣿⣿⣶⣶⣷⣶⣾⡿⠀⠀⠀⠀⠀⢀⣠⣄⠀⣠⡀
⠀⠀⢆⠀⠈⠳⡄⢻⢰⡌⢳⠸⡇⠀⢸⡅⠀⢻⣤⣌⣈⡈⠁⠉⠉⠉⠋⠉⠋⢿⠀⠀⠀⢽⣽⣟⣙⣟⣙⣋⠃⠀⣀⡤⠶⠚⠻⠿⢭⠻⣭⠻⣦⡀⠄
⠀⠀⠈⠙⢶⠀⠳⡜⠶⡿⡜⢦⣷⢱⣀⡇⢀⡟⠴⠀⠀⠀⠶⠀⠀⠀⠀⠰⠆⠸⣆⠀⠀⠀⠀⠀⠉⢉⣡⡴⠞⠋⠁⠀⢠⡔⠀⠀⠀⠀⣈⡳⣌⠻⣆
⠀⠀⠀⠀⠸⣤⣤⣾⡄⠀⠙⣆⠈⢇⠉⠀⣸⠃⠀⠐⣢⢴⣐⢠⠀⡄⠠⢤⠀⠀⢹⣆⠀⢀⣤⣶⠟⠋⢁⡀⢠⡄⠐⠀⢀⠀⠀⠀⠀⢀⠿⡋⣼⣧⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠳⢌⢆⠀⣿⠀⡦⠀⠀⠀⠀⠠⡄⠉⠀⠁⠀⡄⠀⣿⠀⣼⡿⡿⠀⠆⠂⠁⠀⢀⢠⡀⠘⠀⠀⠀⡀⠀⠀⠠⢿⣿⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡆⢠⡄⠠⡄⢠⠀⡤⠀⠀⠀⣿⠀⢻⡜⠀⠀⠀⠀⠘⠆⠈⠀⢀⡠⠄⠂⠀⠀⠀⠀⣠⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣤⠀⠀⡀⠀⠀⡁⠈⠀⠁⠀⠚⢀⡟⠀⠈⣷⠰⡴⣦⢄⡠⠤⠒⠋⠁⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠠⣋⡉⠑⢄⡉⠀⣎⡉⢙⠆⡼⠁⠀⠀⠸⣇⣿⡿⠅⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢦⣀⠀⠀⠀⣤⠈⠐⠒⠒⠉⡾⠁⠀⠀⠀⠀⠹⣿⣷⣤⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣉⠛⠒⠠⠄⠀⠀⠀⠃⢸⡇⠀⢠⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠤⢤⣤⣤⣴⡾⠟⠀⠘⣄⠀⣄⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣯⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣄⠀⡄⣿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡎⣿⢿⡜⣧⢻⣧⣷
⠀⠀⠀⠀⠀⠀⠈⡆⣿⠘⣿⢿⣿⣿⢿⡎⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢶⣶⢶⣤⣤⣤⣤⣤⡤⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⢸⡗⣿⣤⣿⣾⣷⣿⢸⡇⣵⣾⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣶⠴⣦⣶⣾⢥⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣼⡧⢿⣿⣿⣡⢾⣿⣾⢿⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣮⣤⡶⠶⠶⠶⣶⠾⢷⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢿⡇⣿⠻⣷⡟⡿⣿⣧⣼⣧⢻⡇⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⢶⣾⣶⣤⣤⣤⣶⠷⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠸⡇⣿⢰⡟⣨⡇⣿⢹⡏⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠁⡿⢸⠣⣿⢸⠏⣸⠇⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⢠⣧⡆⢀⡾⡿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⡿⢸⣾⣹⣿⣼⣧⢳⣷⣷
⠀⠀⠀⠀⠀⠀⠀⣧⣷⠈⡟⣿⢿⢹⣇⡎⡟⣼⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⡿⠶⠶⠶⢶⣶⠿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢰⢿⣿⣾⣿⣿⢾⡎⢸⣇⣾⡏⠆⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⢿⠿⠶⠶⠶⠿⠷⣶⣶⡶⠶
⠀⠀⠀⠀⠀⠀⢸⡸⣿⣿⡟⡤⣿⣿⠿⢚⠇⣿⡀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠾⢿⡶⠿⠿⠛⢛⡿⠿⠿⠷⠶⠶⠶
⠀⠀⠀⠀⠀⠀⢸⣧⣧⢿⡿⢇⠟⡏⣶⡼⣧⣹⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠚⠿⠿⠿⠷⠶⠾⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣆⣼⢂⡿⢻⡅⣿⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡟⡿⠸⡞⠸⠁⡿⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠲⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢉⣉⠙⠻⢯⡉⠛⠛⠓⠶⠦⣤⣄⣈⠉⠛⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢰⡏⠙⡇⠀⣀⠹⣆⠀⠒⠶⠦⢤⣄⣉⠛⠲⢤⣀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⢸⢣⡖⢶⡈⠓⠚⠁⣼⠉⢳⢹⡄⠀⠀⠀⣀⠀⠉⠛⠲⢤⣉⠳⢦⡀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢺⠸⣦⣼⠃⠀⠀⠀⠙⠶⠛⢸⡇⢰⠀⠀⣿⠀⢀⡀⠀⠀⠉⠃⠀⠙⢦⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⣆⠀⣰⠞⢦⠀⡞⠛⣦⠀⣼⠁⢸⡆⠀⣿⠀⠈⠉⠙⠛⠲⠦⣤⣀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡻⣤⡾⠀⠳⠴⢏⣼⣃⣤⡼⠶⠶⠛⠒⠒⠒⠒⠲⠶⢤⣄⣉⠳⠄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠴⠃





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⡇⡄⢠⠠⣂⢠⠂⢀⣰⠢⠀⡯⠄⢠⢸⠒⣄⡔⠄⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠉⠈⠉⠈⠁⠈⠈⠁⠀⠀⠀⠈⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣠⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠞⠓⠘⠛⠃⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠰⠃⠀⠸⡄⣶⠘⡂⣀⣀⣄⢁⠤⠔⠒⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⡈⢀⠹⣿⡿⠋⠀⠀⠀⠀⠀⠈⠉⢀⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠊⢈⠣⠍⡀⢀⠀⠐⠁⡀⡤⠖⠊⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡌⠈⣔⣾⠊⠀⡀⠈⠀⠂⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠑⢄⡀⣤⡀⣀⠀⠂⠿⠋⡠⢊⢖⠔⡻⠃⠊⠋⡐⠶⠀⠰⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠊⢰⠰⠁⠀⢡⢃⠄⢊⡄⡀⢰⢾⠇⠂⠐⠀⠀⠆⡀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⠀⡽⠀⠀⡜⠃⠀⠀⠀⠀⠀⠐⡔⠂⠁⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠐⣤⢴⣽⣤⡀⠀⠀⠀⠀⠀⠀⠈⣂⢐⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠚⡀⠀⡄⣿⢶⣌⠉⠁⠃⠀⠀⠀⠀⠀⠀⢨⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠐⠀⠀⠉⠓⠦⠀⠀⢶⣤⠀⠀⠈⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠄⡀⠈⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡀⠀⠀⣠⠟⠁⢲⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣐⣚⢥⠁⠀⠃

⠀⠀⠀⠀⣀⠠⠒⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⡀⡀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠒⠲⠄⠒⠀⠐⠒⠦⠀⡀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡆
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠐
⠀⠑⡄
⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠒⠙⠂⠒⠒⠒⠢⠤⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠤⠀⠀⠀⠂⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡆⢀⠀⠀⠀⠁⡇⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢢⠞⠉⠠⠶⣴⢄⠀⡅⢀⠐⢠⡇⢠⡄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠚⠁⠈⢉⡰⠚⠀⠠⠅⠂⠀⠉⠐⠢⠆⢳⡟⠁⠂⠈⠘⠋⠠⢀⠔⠀⠀⠀⠀⠈⠰⠄
⠀⠀⠀⢀⣀⠀⣠⠤⠚⠓⠲⢤⣀⠀⠀⢨⠀⠢⠄⡀⠀⡇⠀⠀⠀⠄⢀⢠⡏⣀⡀⠀⠀⠀⠳⠉⠈⢂
⠀⠀⠀⠈⢭⡞⢁⠀⠀⠀⠀⡀⡘⢣⡀⡀⠀⠀⠀⠋⠺⡇⠐⢀⠀⠀⠀⠀⠀⠓⣧⡀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠈⠀⠴⠂⠀⠀⠉⢱⢇⢀⣧⠈⠒⠆⠀⡸⢳⣟⠀⠀⠀⠀⠀⠀⠃⠀⠸⣗⢦⡴⠆⠀⢂⠎
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⣿⣿⠏⣹⠠⣼⣀⡴⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣁⣉⠷⠔⠁
⠀⠀⠀⠀⠈⠠⡀⢀⠀⢺⠟⠋⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⠴
⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠠⠃⠔⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠁⠠⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠝⣯⢻⠝⣯⠿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡟⣲⢻⡞⣳⠛⣶⠻⡦⡤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣉⢏⣹⣏⡹⣉⣹⣏⠑⡞⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡿⠶⠷⠾⠷⠾⠶⠿⠴⣯⠯⡨⢏⡹⡇⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⢸⡇⢰⣲⡂⠀⠀⣖⣖⠀⣿⠛⠛⠚⠛⢻⣐⣲⣲⣒⣇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣈⣀⡀⣠⢀⡀⣀
⠀⠀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠚⠂⠀⠀⠛⠒⠀⣿⢸⣿⢽⠀⢼⡐⢲⢣⠃⡜⠉⠉⠉⠉⠀⠉⠀⢄⡀⠀⢤⡀⠀⠠⣀⠀⠢
⣀⡜⠆⠰⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣛⡋⢹⡋⣛⣳⠀⣯⠀⢀⠀⠀⢹⠬⠾⠷⠧⢼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠙⢦⣀⠈⠳⣄
⠈⠁⢀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠚⢻⡇⠀⣉⣏⣹⣉⣙⣹⠀⡿⠀⣖⣾⠄⣿⠀⣀⠀⠺⠠⠀⠀⠀⠀⠰⠒⢀⡀⠀⠀⠘⢦⠀⠙⣦⡀⠘
⠀⠈⠉⠉⠉⠉⠛⠦⣤⡴⠒⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠁⠀⡇⠀⠈⠀⠀⣿⠘⠙⠃⠀⢀⠀⡀⠉⠄⠀⠀⠀⠂⠀⠀⠀⠈⢳⡀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⢀⣀⠀⠀⠼⠷⠲⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠛⠚⠒⠒⠒⠚⠶⠶⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠴⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠈⢍⠜⢧⣄⣀⠘⣷⠀⢹
⠀⠀⠀⠀⢀⣠⡴⠁⠀⠀⠈⢳⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠦⠮⠦⠾⠄⠀⠉⠙⠣⢈
⠀⣀⠴⠚⠉⠀⡇⠀⢀⡀⠀⢸⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈
⠊⠁⠀⡞⠉⠙⠧⠀⠫⠠⠇⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣛⣲⡆⠠⠀⠄
⠀⠀⣀⣷⡀⠀⠀⢀⡤⠤⡄⠀⠀⠠⠾⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⢄
⠀⡼⠁⢀⣤⣤⡀⠈⠓⠚⠁⠀⡤⠴⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⢷⠀⠈⠢⠜⠃⠀⠀⢀⠤⢄⠈⠀⠁⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠑⢤⡀⠀⠀⡀⣀⠘⠤⠜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠸⡄⠀⠘⢍⠹⠃⠀⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⣀⣀⣀⣄⣀⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠤⠄⠂⠁
⠀⠀⢀⣄⠙⢦⠤⠤⣤⠶⠋⠀⢉⡉⠥⡤⠀⣠⡀⠀⠀⠀⡀⢀⠄⠀⠀⡽⠠⣦⠄
⠀⠛⠐⠗⠈⠁⠀⠀⠀⠀⡆⠀⠘⡃⠚⡟⠈⠀⠀⠏⠈⠠⠇⢨⠀⠙⠰⠇⠘⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⢬⡙⢞⠉⠙⠋⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⡈⢆⠳⣠⠔⠁⠀⠀⠀⠀⠀⠀⠠⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⢠⠟⠉⢢⠱⡀⢰⣄⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⠴⡻⡀⠀⠀⠣⡑⢄⢢⠙⠦⡔⠁⢠⠃
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⣄⡤⠊⠁⠀⢇⢣⠀⠀⠀⠙⢮⣓⡵⠼⠤⠔⢛⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⡈⡄⠀⠀⠀⠀⡨⢶⡀⣰⡊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢰⠀⣀⠠⠐⠒⠒⡇⣇⠀⠀⠀⣠⠋⠳⡝⡻⣼⢆⠀⠑⢄⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⠀⠀⡇⡏⠙⠢⡾⠢⣱⣼⡶⡷⠃⠀⠑⢄⠀⠑⣀⠀⠁⠑⠤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠢
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣀⠀⠀⣤⠧⠄⡲⠥⢤⣏⣴⡖⠁⠀⠀⠀⠀⠱⣄⠈⠢⡀⠀⠀⢀⡽⠖⠢⠤⣸⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡡⠋⠉⠀⠠⡊⠀⠀⢠⣃⠾⡀⠀⠀⠀⠀⠀⠀⡼⠒⠲⣮⢦⡞⠁⠀⠀⠀⠀⠀⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠱⡴⢩⠏⠀⣱⡀⠀⠀⠀⠀⠀⣇⠀⠀⠈⣣⠈⠒⠤⣀⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⠜⡾⠧⡀⠀⠸⡱⡀⠀⠀⠀⠀⠘⣄⣠⠊⠀⠀⠀⠀⠀⠻⠉⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣰⣧⠞⠀⠀⠈⠢⣄⣹⡧⠀⠀⠀⠀⠀⡎⢳⠀⠀⠀⠀⠀⠀⠀⠹⡓⠒⠪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⣈⠜⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣧⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⣴⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠁⠈⠉⢁⣘⣿⣿⣿⡟⠛⠀⠀⠀⠀⢀⡀⢀⣾⡿⠛⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⡇⢸⡀⠀⣠⣤⠘⠓⢸⣿⣶⣿⡶⠆⠁⡀
⠀⠀⠀⠀⡠⠐⠀⠈⠀⠀⠀⠀⢀⡀⠀⣀⡀⢀⣿⣿⣿⣿⣄⣀⣀⣿⣿⣿⣿⣧⣾⣄⠀⠙⣿⣷⣤⣤⣿⣿⣷⣶⣶⣗⣀⠄
⠀⠀⢰⣏⠀⣾⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⡟⣿⣥⣶⣿⣿⣿⣿⣿⣥⣤⣀⣿⣿⢰⠀⣿⣿⣯⣯⠰⢀⣴⣲⠖⡃
⠀⠸⠿⠯⣭⠄⠶⢷⡨⠉⠀⠀⠀⠀⠉⠉⠀⠉⢙⡛⠛⠛⠋⡛⡛⠋⠉⠙⠋⠉⠙⠋⠛⢛⠉⡉⠀⠀⠉⠉⠙⢃⠀⠁⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢀⣂⡀⢁⣀⠀⠠⣀⢀⢠⣠⠄⡁⣄⡀⡡⣀⣄⠤⠀⠀⠀⠀⢠⠀⢀⠀⠀⡈⡀⠄⠄⢀⠀⠀⠀⠠⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠘⡧⣥⠙⢠⠶⠁⢹⣇⡛⡒⣍⣳⡇⢠⣰⡗⢁⡺⣔⡧⠀⢰⠉⠋⢃⠛⣀⠇⢀⣴⢏⠉⡻⠰⢏⡔⢩⣬⣧⣘⡀⠁⢐⡄⢷⡃⢂⣀
⠀⠀⠠⣿⢸⢡⣞⠞⢠⡈⢻⡄⣰⣍⡏⢸⠊⢹⡇⠪⡽⢙⡿⣶⡆⠙⢰⢢⠟⠟⢠⠟⢙⣶⣋⡁⠜⣿⣌⡹⠄⢰⡴⣄⢠⢼⠁⢻⠿⠉⡉
⠀⠀⠐⣿⢸⡷⣻⠨⣄⣽⢿⡀⣷⣌⡷⠲⡗⢸⣧⣸⡵⠌⣃⢤⣹⡣⡆⣼⢖⠃⢰⡖⠁⠹⢘⢹⣇⣉⣮⠁⢈⡑⣚⣉⠙⣀⢠⡼⠳⢒
⠀⠀⠤⣯⣸⡇⣻⢰⡇⠀⣿⠂⠻⢮⣧⣈⣟⢼⡅⣽⣟⠈⡗⣘⢩⡅⠜⣩⣻⡇⢸⡇⠾⢧⣄⢨⣿⣡⢯⡿⠸⡷⣩⣯⡵⠈⠟⠶⠃⡤
⠀⠀⠠⠿⢤⣀⣹⡈⣁⠃⣿⠐⠶⢓⡷⠼⠇⢠⡇⣺⣁⣘⣇⠐⢱⣍⡎⠉⢹⡃⢜⡃⠶⡫⠅⢼⣯⠘⢡⣿⣸⡟⣩⣟⡛⢘⡟⠸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠚⠓⠦⠤⢤⣧⣤⣄⣘⡇⠈⠁⢸⣟⠀⠉⠛⢘⡱⠈⠥⠀⢸⣟⠀⣀⣷⣸⡏⣰⠏⠋⢸⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠓⠶⠦⠤⢼⣅⡀⣀⠀⠰⡿⠀⠀⠁⢸⡏⠁⠑⠒⢺⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠓⠲⠰⠦⠬⠧⣀⡀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣤⠤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠉⣸⠃⢀⡏⠀⡿⠀⢸⠋⢹⡗⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣾⠃⢠⡏⠀⣼⠃⢰⡇⠀⣾⠀⣼⠀⠀⡟⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⠀⢷⣠⡟⠀⢰⠇⢀⡾⠀⢰⡇⢀⡟⠀⢸⠇⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠹⡆⠈⣿⠀⣰⠏⠀⡼⠁⢀⡟⠀⣼⠁⠀⡿⢀⡟⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⢻⡄⠈⢷⡏⢀⡼⠁⢀⡾⠁⣰⠇⠀⣸⡃⣼⠁⢰⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡀⢳⡀⠹⣄⠀⢻⡾⠁⢀⡾⠁⢰⠏⠀⣰⠏⠙⠳⠦⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢷⡀⠻⣄⠈⣷⠟⠀⢠⡞⠀⣰⠏⠀⣰⠏⠛⠦⣤⣀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⢳⣄⣹⠟⠁⢀⣴⠋⢀⡴⠃⠀⣴⠋⠳⢦⣄⡀⠉⠙⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡴⠟⠁⠀⣠⠞⠁⣠⠞⠁⢀⡾⠛⠦⣄⡀⠈⠙⠓⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⠞⠁⣠⠞⠁⢀⣴⠛⠦⣄⡀⠈⠙⠳⢦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣾⣁⣀⣠⡞⠛⠦⣤⣀⠉⣛⣶⡴⠛⠁
⠀⠀⣠⠶⠛⠛⠛⠓⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠈⠉⠛⠛⠛⢒⣾⠟⠋⠉
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣤⣤⣤⣤⡽⣞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠉⠛⠲⠶⠤⠤⠤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣀⠂
⠀⠀⠀⢰⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢴⠴⠀⠀⠐⠁⠀⠂⠀⠂
⠀⠀⠀⢀⠀⠀⠈⠐⠠⠀⠀⢄⠄⠀⠈⠀⠀⠀⠀⠈⠀⠁⢄⠁⠀⠀⠈
⠀⠀⠀⠈⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠂
⠀⠀⠀⠀⠐⠈⠢⣠⠎⠀⠀⣠⡴⠆⠀⠀⠀⠀⠶⠦⣄⠀⠀⠀⠀⢉⠄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠁⢾⣶⠀⠀⠀⠀⠀⣡⡄⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡀⠆⠐⠿⡧⠄⠤⠴⠀⠚⠛⠀⠠⢀⠀⠀⠈⢤⡤⠀⠀⠀⠀⠄⡀⠌
⠀⠀⠀⠀⠴⢟⡄⢀⠔⠀⠀⠀⠀⠀⠁⠀⢀⠐⠁⠀⠀⠀⠀⠀⣯⠀⠬⠙⠁⠰
⠀⠀⠀⠀⠀⠉⠁⠘⠂⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⢀⡀⠐⠃⢀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⢄⠈⠋⠋⠉⠀⡘⠒⠁⠈⠐⠐⠃⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢉⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⡔⠀⢸⣄⠁⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠡⡤⠊⠐⡩⢊⢓⠆⠀⠈⠉⠛⠋⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⡀⠀⡜⠀⠀⢍⠀⣀⠀⣠⡠⠺⠀⠈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠉⠳⡀⠀⠀⠀⠁⢸⠀⠠⢨⠀⢀⠜⠡⠀⠀⠈⠀⠀⠏⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢀⡠⠚⡅⠀⠀⠀⢠⣄⢢⠀⡘⣢⠦⠀⠤⣈⠆⠀⠀⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠤⢤⠟⠀⢿⣷⡀⠉⡀⠄⢀⡄⠀⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⡶⠁⠈⢻⠒⠨⠤⠮⠤⠄⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣒⣈⣠⡤⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡶⠚⠋⠙⠛⠃⠂⠀⠀⠉⠉⠀⠉⠁⠈⠀⠘⠛⠛⠛⠉⠛⠛⠃⠀⠀⠈⠉⠀⠀⠉⠀⠀⠐⠊⠛⠛⠋⠙⠛⠙⠂⠀⠀⠈⠉⠁⠉⠉⠁⠀⠀⠒⢦⡄
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠐⠠⠀⠀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠁⠂⠶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠈⠁⢀⠀⠘⠀⠠⢄⡀⠈⠉⠓⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠊⠁⢀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀⡀⠀⠉⠓⠢⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣠⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠚⠖⢠⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⡀⠠⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⢀⠀⢀⢀⠀⠀⠈⠄⣠⡀⢌⣠⣀⣌⣠⣡⠄⢄⡠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠁⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡎⠀⠀⠀⣴⡀⠀⠀⠀⢀⡴⠀⠀⣠⠎⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠋⠀⠀⠀⠀⠋⠀⠠⠀⠂⠀⠀⠀⠁⠀⠀⠄⠊⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠘
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁⠀⠀⠀⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠙⢦⡄⢀⣀⡀⠀⠀⠀⠀⡀⣀⠀⣀⣁⣈⣀⣀⣀⣀⣀⡀⣀⠀⢀⠀⠀⠀⢀⣀⢀⣀⣀⣀⢀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⢀⢀⣀⣀⢀⣀⣀⣀⢀⡴⠋
⠀⠀⢀⠀⠀⡀⠀⡀⠈⠉⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⠉⠁⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠀⠉⠁⠀⢀⠀⠈⠉⠉⢉
⣿⡟⣉⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⢄⣀⠀⠒⠀⢄
⣀⣀⣀⣀⡀⢀⡀⠤⠤⠤⠤⠄⠀⠀⠁⠒⠒⠒⠒⠒⠒⠐⠒⠒⠂⠙⠛⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠂⠈⠙⠉⠉
⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠁⠉⠁⠂⠀⠁⠀⠀⠢⠄
⠀⠀⠀⣴⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠤⠶⠚⠛⠃⠑⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⠀⠰⠶⣤⠠⠴⠚⠈⠉⠁⠀⠈⠁⠈⠉⠂⠀⠀⠀⠀⡈⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠤⠤⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠛⣛⡋⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠄⠤⠒⠒⠂⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠰⣾⣿⠟⠀⣿⣿⣿⡄⠀⠀⠤⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠠⠀⠛⠛⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠉⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣼⣯⣀⡀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠻⣯⡁⢰⡟⠻⣦⡀⢨⡿⠛⠷⣤⣠⡶⠟⠉⠙⠻⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣹⣇⣤⣬⣿⣾⡷⠶⠾⠿⠾⠷⠶⠶⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣴⠶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆⣀⣀⣤⣤⣤⡶⠶⠶⣶⠶⠶⣶⡶⠶⠶⢶⣶⢶⣤⣤⣤⣸⡇
⠀⠀⠀⠀⠀⢀⣠⡶⠟⠙⣿⠉⠉⠀⢿⡀⠀⠀⠀⣿⠀⠀⠘⣷⡀⠀⠀⢻⡆⠀⠀⢈⣿⠁
⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠘⣧⡀⠀⢸⡇⠀⠀⠀⢿⡄⠀⠀⠘⣷⡀⠀⠀⢿⡄⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣄⡘⣷⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⠀⠀⢈⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠈⠻⢿⣄⣀⠀⠈⣿⠀⠀⠀⠀⣹⣧⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⠶⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⠀⡴⠶⠶⠶⠶⠶⢦⡈⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⡇⠀⡴⠒⢦⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠂⠒⠒⠒⠀⠀⠀⢸⠀⡇⢀⣷⣤⣼⡀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠀⠉⠉⢉⠉⠉⠉⠁⢸⠀⣇⣯⣀⣀⣀⣹⣼⠇⡇
⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⠀⣤⣀⠀⠀⠀⠀⢀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠛⠀⠋⠁⠀⠐⠛⢻⡭⢥⣀⡬⢭⠟⢳
⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢯⠉⠉⠉⠉⠉⠉⠁⢸⡧⠤⡤⠤⢼⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⢶⠶⠾⠿⠶⢶⠶⠶⠶⠶⠆⠘⢇⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⣉⣉⣉⠉⠉⢱⡦⠤⠾⠦⠄⠀⠀⢀⢤⡾⣟⠛⣹⢶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠤⠤⢸⡟⠒⠤⣄⡀⢀⠜⠁⠀⠑⣌⠉⡡⠋⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠶⢦⡀⠀⢸⡧⣄⠀⠀⠉⠁⠀⢰⡆⠀⢸⠙⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠰⡃⣎⢙⡆⡷⢀⡸⠃⠀⠀⠲⢄⣀⡤⢺⡇⠀⠀⠁⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢮⣭⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣾⢸⡶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠒⣾⠟⢻⣿⠟⠛⢻⡿⠶⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⡟⠉⡿⠃⠀⣼⠁⠀⢸⡇⣀⣠⣾⡶⠶⠶⠾⠿⠿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⣴⠏⠀⢰⡇⢀⣰⣧⡴⠶⠛⠋⠉⠁⢀⣀⣀⣠⣤⣤⣤⣤⣬⣝⣛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣽⠃⠀⣼⠃⠀⣀⡼⠷⠛⠉⠀⢀⣀⡤⠴⠶⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⡟⠀⢠⣿⡴⠞⠉⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠶⠶⠶⠶⠶⢦⣤⣤⣙⣷⡄
⠀⠀⠀⠀⠀⢀⡴⢻⡟⠀⠀⡇⠀⢸⡏⠀⠀⢀⡴⠞⠉⠀⠀⠀⢀⣀⠤⠖⠚⠛⠋⠉⠁⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠙⠛⠿⣦
⠀⠀⠀⠀⢠⣾⠃⢸⡇⠀⠀⡇⠀⢸⡇⣠⠾⠋⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠙⠛⠶⢤⣄⡀⠙⣧
⠀⠀⠀⢠⢿⡿⠀⢸⡇⠀⠀⡇⠀⠸⣿⠁⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢾⣧
⠀⠀⠀⡿⢸⡇⠀⢸⡇⠀⠀⣷⠀⠀⣿⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣀⠀⢹⡆
⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⢻⡄⠀⠸⣧⡴⠋⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⢀⣠⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿
⠀⠀⣼⡇⢸⣇⠀⠀⢹⡆⠀⠘⣷⠀⠀⠹⣇⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠈⢻
⠀⠀⣿⡇⠀⣿⠀⠀⠈⣷⠀⠀⠸⣇⠀⠀⢻⡄⢠⡞⠁⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣬
⠀⠀⣿⣧⠀⢹⣇⠀⠀⠸⣆⠀⠀⠹⣆⠀⠀⢻⡟⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⢤⣤⣤⣀⣀⠀⠘
⠀⠀⣿⢸⡆⠀⢻⡄⠀⠀⠹⣆⠀⠀⢻⡄⠀⠀⢻⡄⢀⡞⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⣀⣤⣴⠞⠛⠋⠻⣯⠀⠀⠙⢦⣄⡈⠉⠙⠛⢾
⠀⠀⣿⠈⣧⠀⠈⢿⡀⠀⠀⠘⣧⠀⠈⢿⡄⠀⠀⢻⡏⠀⠀⠀⣠⡶⠋⠀⠀⠀⢀⣴⡞⠉⠈⠹⣆⠀⠀⠀⠈⢷⣄⠀⠀⠈⠙⠓⠶⠶⣾
⠀⠀⢹⡆⢹⡆⠀⠈⢳⡄⠀⠀⠈⢷⡀⠈⢷⡀⠀⠀⠹⣄⢀⣴⠋⠀⠀⠀⢀⡴⠋⠀⢻⡀⠀⠀⠙⢧⡀⠀⠀⠀⠈⠳⠶⢤⣤⣤⣤⣤⣼
⠀⠀⠈⣷⠀⢿⡄⠀⠀⠹⣆⠀⠀⠘⣷⡀⠈⢷⡄⠀⠀⠘⢿⠁⠀⠀⢀⣴⣏⠀⠀⠀⠀⢳⡄⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⢹⡆⠈⢿⡄⠀⠀⠘⢧⡀⠀⠘⣷⡀⠀⠻⣦⡀⠀⠈⢳⣄⣠⡞⠉⠹⣆⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠛⠶⠶⢦⣤⣤⣴⣶⣾⠃
⠀⠀⠀⠀⢿⡀⠀⢻⡄⠀⠀⠈⢳⡄⠀⠘⣷⡀⠀⠈⠻⣄⠀⠀⠙⢿⡀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠈⢷⡄⠀⠙⣦⠀⠀⠀⠹⣄⠀⠈⢷⡄⠀⠀⠈⠳⣄⠀⠀⠙⢦⣄⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠈⠳⣄⠀⠀⠙⢦⡀⠀⠙⢦⡀⠀⠀⠈⠳⣦⡀⠀⠉⠛⢦⣄⣀⠀⠀⠉⠙⠛⠓⠶⠶⣶⣤⣶⣤⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢷⣄⠀⠈⠳⣄⠀⠀⠙⢶⣄⡀⠀⠈⠛⢶⣄⠀⠀⠀⠉⠛⠶⢤⣄⣀⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⠙⢷⣄⠀⠈⠳⣄⠀⠀⠈⠙⠶⣤⡀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣿⣷⣄⡀⠈⠳⢦⣄⠀⠀⠈⠙⠷⣄⡀⠈⠙⠳⠶⠶⠶⣶⣶⣶⠞⠛⠁
⠀⠀⠀⠀⣀⣤⠞⠛⠿⠶⣦⣄⡀⠀⠀⠈⠉⠛⠛⠻⠶⣦⣌⣷⣦⣀⣀⠀⠈⠙⢶⣦⣤⣤⣴⡶⠛⠉
⠀⢀⣴⠞⠉⢀⣤⠶⠶⠶⣤⡈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⣿⠻⣿⠋⠁⠀⠀⠀⠀⠀⣀⣠⡤⠶⠶⠛⠛⠓⠶⣤⡀
⢠⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠙⠶⣄⡈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣼⠀⠀⠀⣀⡤⠖⠋⠉⣀⣠⣤⣴⣶⣦⣤⣄⠀⠹⣆
⣼⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠈⠙⠲⣤⣀⠀⠀⠀⢀⣠⡴⢟⣡⡾⠋⣠⡴⠛⠁⢀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠈⢷⡀⠹
⣿⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣉⣛⣛⣋⣉⣥⠶⠛⣡⡴⠛⠉⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢉⣀⣴⠞⠉⣀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣰
⢻⣄⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠁⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣴⠏
⠀⠙⠷⣄⡈⠙⠳⠦⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠖⠛⠋⢉⣀⣤⠶⠛⠁⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠞⢋⣡⡴⠞⠁
⠀⠀⠀⠈⠛⠷⠶⠶⢤⣬⣅⣀⣀⣀⣀⣀⣤⣤⡤⠶⠞⠋⠉⠁⠀⢀⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡴⠋⣁⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠄⠑⠀⠀⠀⠀⠀⢄⠐⡀⠀⠀⠄⠀⠠⡕⠀⠀⠂⠀⠀⠀⠀⠀⢀⠂⢤⣲⡄⠀⠀⠀⠊⠀⠀⡄
⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢧⣇⡬⠀⠀⠀⠀⣤⠄⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠂⣀⠐⠀⠀⠀⠀⠂⣀⠀⢄⠀⠀⠀⢀⣀⠀⠄
⠀⠀⠀⢀⣿⡃⠀⠀⢀⠀⢿⡈⡛⠀⠁⠀⢰⣿⣿⣿⡆⠄⢰⠀⠸⣭⠇⠀⠀⠀⠀⣾⣿⣶⠀⡂⠀⢰⣾⣿⣧⡀
⠀⠀⠀⠛⠛⠛⠀⠀⠀⠂⠈⠙⠛⠠⠁⠀⠀⡉⠉⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠈⠛⠀⠀⠀⠀⠌⠙⠛⢋⠔
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⣾⣿⣿⣿⣿⡄⢀⣾⣿⣿⣿⣷⡄⢀⣾⣻⣿⣟⣷⡄⢠⣾⣿⣿⣿⣷⡀⢠⣾⣛⣿⣻⣷⡀⢠⣿⣿⣿⡻⣷
⠀⠀⣿⣿⣿⣿⣿⠇⠈⣿⣿⣿⣿⣿⠇⠘⣿⣻⣿⣟⣿⠃⠘⣿⣿⣿⣿⣿⠃⠸⣿⣚⣿⣳⣿⠁⠸⣿⣛⣻⣻⣿
⠀⠀⠈⠙⠛⠛⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠁⠀⠀⠈⠛⠛⠋⠁
⠀⠀⣠⣶⠶⣶⣤⠀⠀⣠⣶⡶⣶⣄⠀⠀⣠⣶⣶⣶⣄⠀⠀⣠⣶⠶⣶⣄⠀⠀⣠⣶⠶⣶⣄⠀⠀⣤⡶⠶⣶⣄
⠀⠀⣿⡿⠄⠼⣿⡇⠠⣿⡀⠿⢦⣿⡇⢰⣿⠀⠄⠀⢹⡇⢸⣿⣧⠛⣼⣿⡆⢸⣿⠅⡁⠩⣿⠄⢸⡏⠁⠀⡉⣿
⠀⠀⠹⢶⣶⣶⠟⠁⠀⠹⢿⣶⡶⠟⠀⠀⠻⢶⣶⡶⠟⠀⠀⠻⢿⣾⡿⠟⠀⠀⠻⢷⣶⡿⠟⠀⠈⠻⣷⣶⡾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠑⠄⠀⠀⣀⠨⠓⠤⡀
⠀⠀⠀⠐⢄⠈⠢⡀⠀⠀⠰⡈⣖⡉⠔⢂⡠⠒⠉⠢⡀
⠀⠀⠀⠀⠀⠈⠉⠑⡀⠀⠀⢡⠘⢢⣔⣁⣀⣀⣠⠀⠱⡀
⠀⠀⠀⠀⢀⠀⢀⠀⠱⠀⠀⠀⠀⡀⢢⠠⠤⠤⠀⠒⠂⢳
⢐⡀⢢⢄⡂⠌⣐⠊⠩⢧⠀⠀⡄⡇⠈⢧⠤⠤⠄⠀⠀⠈
⢨⠀⠌⠄⠈⡱⢲⠬⣥⣒⣆⠀⠀⠹⠀⠘⡍⠉⠁⠀⠀⢠
⠈⡶⠀⠀⢀⢃⠎⡰⠃⠀⠸⠀⢆⢸⠀⠀⡏⠀⠀⠀⢐⡞
⠀⠐⡄⠀⠀⡜⢠⠁⠀⡀⠀⡆⠈⢆⠣⠀⡇⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠘⢄⠀⠀⢆⢂⠌⠀⠀⡄⠀⠈⠁⠀⡂⠀⡠⠓⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⢀⡇
⠀⠀⠀⠀⠑⠢⢏⣈⠀⠀⠀⡆⠀⠀⠀⡠⠔⠋⠀⠀⠀⠙⢄⠑⠄⡀⠀⢀⡠⠊⠁⠀⠀⠀⠈⢀⠠⠔⢂⡡⠄⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠤⠒⠁⠀⠀⡀⠤⣒⡉⠄⠒⠉⠀⢀⣀⣀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠀⠀⠒⠒⠒⠒⠂⠚⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠭⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⣉⣍⡉⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢸⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠘⢿⣿⠏⢨⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣦⣤⣶⣿⡟⣡⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠱⣿⣏⠙⢁⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠹⢿⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠛⢿⣿⣿⠟⠛⣉⡙⠻⣿⣿⡿⠛⣋⡉⠛⢿⣿⣿⠟⢋⣉⠛⠻⣿⣿⡿⠛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣴⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣦⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⡿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣶⣶⣾⣿⣿⣷⣶⣶⣿⣿⣿⣶⣶⣾⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⣿⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⣿⣿⡿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠙⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢀⣠⠴⠞⠓⠚⠓⠲⡄⠀⠀⣠⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡤⠞⢋⣁⣀⣀⡀⠀⠀⢀⡏⠀⢸⠷⣄⣰⢯⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣧⠀⠀⢀⣴⠇⠀⠀⠀⠀⢀⡤⠞⠋⣉⣤⣶⠋⠁⢤⣄⠀⣹⣀⡴⠋⠀⣰⠋⣠⠟⠁⠀⠈⣆
⠀⠀⠀⢲⠒⣶⣦⣌⡣⣑⣻⡏⠀⣰⣿⠧⢰⠀⠀⠀⢠⡏⠀⠀⢸⡇⠌⠋⠀⠀⢈⡁⠀⢻⠁⠀⠀⡴⠁⣰⣯⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⢧⡀⠙⢿⣿⡄⢹⡇⢰⣿⣿⠀⡌⠀⠀⠀⠘⢧⣀⣀⣼⠛⠀⠀⠀⠀⠁⠬⣿⠋⠀⢀⡜⠁⣼⠃⢹⣄⣀⠀⠀⠀⣘⣧
⠀⠀⢤⣄⣀⠛⠶⠤⠬⠷⣾⠃⢸⣟⣩⠔⡁⠀⠀⠀⠀⢀⣭⣽⠏⠀⠀⢳⣤⣀⣀⣈⡏⠀⢠⠞⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠸⡌⢹⣿⣷⣦⠀⠀⣟⠟⢩⣾⡿⢫⠃⠀⠀⠀⣰⠏⠀⣿⡀⠀⠀⢸⣿⠟⠉⣏⠉⣻⠋⢀⣾⣄⡀
⠀⠀⠀⢳⡈⠩⣿⣿⡇⢸⣹⢠⣿⢟⣵⣧⡶⠾⢿⠛⣿⣀⣀⣈⠛⡶⠶⡿⢣⣤⡤⣽⡰⠃⢠⡞⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠉⠓⠒⠚⢳⣇⡟⠉⢈⣴⠟⠁⠀⠀⢸⡎⠛⢻⡉⠙⣧⢻⡸⣱⠋⣸⣛⠛⠳⢤⡟⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⢠⡟⠁⠀⠀⠀⠀⠘⡇⠀⠘⡇⠀⠈⠊⠃⠃⣴⢯⣉⠙⠂⠀⠹⣆⠀⠀⢀⣀⣀⣬⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⡿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⣴⠿⢤⣈⠁⠀⠀⠀⢻⣤⣞⠋⠉⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⠀⢠⢀⡿⠀⠀⣿⣶⡶⠶⢶⡿⠶⣤⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⢤⣤⣤⣿⡆
⠀⠀⠀⣀⣤⠞⠛⣻⠿⠿⠾⠟⠓⢦⡀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⣠⡿⠁⢀⡾⢿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡀
⣴⠞⠋⣽⢶⣤⣾⣥⣄⢀⣤⡀⠀⠀⢻⣄⡀⣿⡟⠀⠀⠀⠀⠀⣰⠟⠀⢀⡾⠃⠀⠉⠙⠛⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠻⣤⣤⣷⠀⠘⣿⡀⠈⣏⠈⢹⡟⠻⣼⠏⢻⣿⠁⠀⠀⠀⢀⣼⠏⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⢻⡆⠀⠙⡇⠀⢻⠀⠸⡇⠀⠹⠀⣼⡟⣀⣤⣤⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠘⠳⢦⣤⣿⣄⣼⣄⣀⣧⣀⣠⣾⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣿⣶⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠂⡀⣀⠀⠀⠂⡠⡖⠂⠀⠢⡀⡆⡄⡄⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡋⠀⠀⠰⣻⡟⡇⠀⠀⠸⣴⢾⠀⠀⠀⠽⣿⣧⠀⠀⠈⡿⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣴⠀⠀⠀⡬⠊⢠⠁⠀⠀⠀⠀⠸⠃⠀⠀⠇⢳⣿⡄⠀⠀⠐⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣥⠄⢨⠅⠀⠀⢠⣥⡤⢬⠀⠀⠀⢼⠤⡤⡄⠀⠀⢤⢼⡿⣥⠀⠀⠀⣤⢿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠉⠀⠀⠀⣀⣈⣀⣁⡀⠀⠀⠈⠀⠀⠀⠀⠂⠈⠐⠓⠁⠀⠐⠂⠈⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠄⠀⠀⠀⢰⠉⠉⠉⠹⡇⠀⠀⠀⠈⡌⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢤⠇⠀⠀⠀⠀⠀⠘⠤⠤⠤⣴⠇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢸⠀⠁⠉⢙⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠲⠲⠒⠚⠇⣀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⡀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢉⠀⠀⢀⢈⣁⡀⣀⣀⣀⡀⢀⣀⣈⢀⡈⣁⣉⣀⡀⣈⣁⣉⣉⣀⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠀⠒⠒⠂⠀⠁⠐⠂⠒⠀⠘⠃⠀⠀⠀⠀⠀⠛⠀⠀⠚⠘⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠤⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠀⠀⡄⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠂⠀⠂⠂⠂⠀⠀⠂⠐⠀⠐⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠁⡉⡁⢁⡀⣉⣀⢈⣈⢈⣉⢈⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢄⡀⣀⠀⡄⠀⠀⠀⣀⣀⣀⢠⠀⠩⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢀⡠⠀⠒⠀⠀⠀⢀⠘⠟⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⠁
⠀⢠⠃⡀⠀⠀⣀⠀⢀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⠀⠀⠀⠀⠀⠄⡀⠀⠠⡀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠔⠀⠒⡺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠒⠒⠀⠊
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⣷⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠋⠀⠀⠀⠈⠙⢿⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠂⠀⠀⠠⠀⣈⣻⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣀⣠⣤⣶⠂⠀⠀⠀⢀⢀⡀⣀⡀⢀⠀⡀⠀⣀⣀⠀⡀⢀⢀⡀⢀⠀⠉⠀⠀⠀⡀⠐⠁⠀⡠⠂
⠀⠀⡔⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⡼⢺⠠⣟⠢⠿⠇⢧⡤⢸⠇⢸⠓⠾⠀⢹⠇⠀⠀⠀⠀⠁⠀⠀⠀⠊
⠀⠀⠁⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣿⣿⣷⣧⣿⣿⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠁⢠⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠄⠀⠀⠀⠛⠛⠛
⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⠛⠛⠛⠁⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⢻⣦⣄⣀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠘⠃
⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⣿⣿⠿⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡀⠀⠀⠀⠀⠀⢸⣩⠏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠁⠈⣉⠑⢢⣿⡘⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢌⣻⡇⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠣⠀⠠⢻⣿⢃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⢻⡄⠀⠀⠀⢀⢀⡜⠘⢯⢗⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠣⠀⠀⠀⠘⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠏⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡘
⠀⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡔⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⣀⣠⠤⠄⠨⠈⠨⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⢠⠈⠀⠉⢻
⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠂⠀⠐⠻⠿⣟⣛⣦⣝⠓⢯⣙⣛⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣐⡚⣻⣍⠻⣿⣷⣦⣤⡤⠾⠛⢉⣉⣻⣄⠻⣤⡼⢷⡈⠹⣽⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⢸⣿⠋⢹⣿⣆⠈⢷⣤⠼⠿⠡⠴⠒⠞⠉⠙⠛⠛⢷⡈⢳⣶⠷⣄⠘⣥⠴⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⣾⡇⠸⡟⠀⠿⠛⠛⣧⡀⣧⣤⣤⣶⠶⠖⠲⢶⣶⣾⠊⢛⣿⣄⠱⣦⢾⡄⠘⠋⠲⢽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄⠿⠿⡇⠀⠇⠀⣿⣿⠟⠉⣇⠈⢿⣿⣿⣷⣦⣠⣤⣟⣁⣴⣿⣿⠟⣤⠈⣏⣉⡄⠘⠷⣦⣌⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣀⣠⣴⣿⡀⡇⠀⡿⠷⠖⠛⠻⣆⠈⣿⡏⣉⣉⣉⣉⢉⣉⣉⣉⣛⣚⣻⣄⠘⣟⣻⡄⠐⡶⠧⣝⣾⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣷⣿⣯⠴⠛⠛⢿⡇⠠⣧⡶⠲⣾⠛⢻⡄⠈⣏⢙⡟⠉⢁⣾⣿⠟⠉⣽⣿⡍⢿⡄⠙⣿⣧⠀⢹⡆⣦⢬⣽⡀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢹⠋⣀⣤⢶⣾⠛⣇⠀⣟⡛⢶⡋⢰⠻⢿⡄⠘⣿⠁⢀⣿⣿⣋⢠⡾⠿⠿⠿⠲⢿⡄⢹⡿⡄⠈⣟⠛⠓⠛⣷⠀⢷⡀
⠀⠀⠀⠀⠀⠀⡀⢈⣽⠀⠘⣿⣿⣆⠜⢁⣼⣧⠐⡎⣗⠀⠙⢿⣷⣤⠷⠀⠹⣏⠈⡁⣈⣁⣀⣤⣤⣤⠄⡶⣾⣇⠘⡿⢷⠀⢹⡈⠀⠁⠉⡇⢸⡷⡄
⠀⠀⠀⠀⠀⠈⡁⢸⠿⢧⠀⠸⣿⣯⣠⣾⣿⣿⠄⡆⢹⣧⠴⠛⠁⣀⣴⣧⠀⠸⣏⠙⣉⣉⣤⣤⠆⠰⠋⠉⠀⢨⡁⢐⣸⡆⠘⢷⣶⠒⣿⣷⠘⡿⣧
⠀⠀⠀⠀⠀⢰⡆⠈⡷⢾⡆⠀⢷⢤⣤⣀⣀⣹⡆⢸⠀⣇⡠⠔⠛⠉⢉⣀⣅⠀⢻⡹⠋⠉⠀⣀⣀⣤⢦⠴⠶⢟⣧⠈⣿⣷⠀⢇⣻⣤⠿⡇⢠⣷⢿
⠀⠀⠀⠀⠀⢸⣷⠀⠹⣧⡻⡀⠈⢿⣧⠀⡸⠟⢣⢈⡇⠈⣦⣴⣓⠙⠫⠦⢼⡆⠀⣷⡶⠻⠿⠿⣿⣷⡓⣶⣤⣼⣿⡀⢴⣾⠀⢘⠋⣀⣴⣡⠾⢋⡼⡇
⠀⠀⠀⠀⠀⠀⠻⣆⠀⢿⠙⠳⣀⠀⢿⡋⢁⣤⣾⡀⢇⠀⢻⣴⠟⠻⢶⣄⡀⠱⡀⠈⢷⣦⣀⣀⠀⢈⣉⣭⠿⠟⢛⡇⠈⣶⠇⢸⡿⠗⠋⣡⣶⡏⢘⡇
⠀⠀⠀⠀⠈⢦⡀⠙⣇⠈⢿⣄⣸⣇⠈⢳⣟⠛⠛⣷⠘⣇⠘⣯⠤⣴⡿⢿⣛⡷⣿⠀⠘⡦⠀⠙⠛⢿⣁⣤⣤⣶⠾⠷⠈⢻⡆⠀⣦⡴⠋⡝⢹⠇⣼⠁
⠀⠀⠀⠀⠀⠘⣷⣄⠘⢧⠈⢿⣋⡼⠷⡀⠩⢷⣄⣹⣦⠸⡆⠘⣆⡀⠐⠈⠉⠉⢙⡇⠀⠹⣶⣄⠀⠀⠉⠻⢥⣀⣠⣼⠀⢺⡇⠀⣿⠃⣰⣷⡌⢠⠇
⠀⠀⠀⠀⠀⡀⠹⣟⢳⣤⡑⣆⠙⢷⣄⣽⣆⠀⠻⣿⣿⡄⠱⡀⠹⣿⣷⣶⣶⣤⡤⣷⡄⠀⠻⠉⠑⣤⣄⡀⠀⠉⠙⠿⣄⢸⣇⠀⠃⠐⠛⣹⢁⠞
⠀⠀⠀⠀⠀⠑⢦⣘⢏⣴⠿⠾⣿⣄⠈⠯⣍⣹⣄⡈⠣⠌⡄⠻⡀⠙⣧⢀⣀⣀⣀⣠⣧⡀⠀⢳⣯⣤⣬⣍⣻⡦⢤⠀⢿⠀⣿⠀⣶⣀⣤⡷⠋⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⢴⣿⣿⣷⣦⡈⠫⢟⠳⣄⠀⠹⣆⠙⡄⠈⢏⡈⠈⠋⠁⡈⢳⠀⠀⢻⡟⠋⠉⠁⠀⠀⣨⣿⠀⣟⣀⣿⠟⠋⢀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠉⠉⠉⠋⠛⠓⠘⠟⠻⠓⠢⠬⠳⡷⠤⢄⣿⣿⡿⠟⢛⣋⢣⣀⣀⣧⣤⣤⣶⠾⡿⠣⠼⠛⠛⠁⠀⣠⢴⡟⣯⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣝⠛⣍⠓⠲⣦⠶⢤⢤⣤⣁⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⡀⣀⣀⣀⣠⠴⣶⠉⠏⣰⣯⣀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⢾⣷⣄⠈⠻⣷⣶⣶⣷⠟⠿⢻⣧⣄⡘⠻⣿⣏⠉⠛⣿⣟⠙⠛⠛⣿⡏⠉⢻⡋⣹⣽⣿⡿⠟⠀⣿⢀⡿⠟⠿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣷⣤⡀⠙⢿⣿⠶⠖⠛⠛⠋⠉⢛⣷⡌⢳⣤⣀⠈⠻⢶⣿⡶⠛⣄⠀⠳⣷⠛⠉⢀⡍⠀⡂⣸⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠤⣤⡝⣢⣀⡈⠳⠶⣴⣶⣶⣟⠉⠙⣷⡀⠻⣮⣝⠶⣤⣀⡉⠙⠛⠧⠤⣽⣶⣾⣿⣇⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢦⣄⡀⠉⠉⠛⠳⠶⠮⢿⡄⠙⡏⠉⠳⠦⠉⣉⡒⠦⠤⠤⠤⠤⠴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣯⣤⣬⣾⣷⣤⡤⠤⠤⠤⣽⣆⠈⠳⣤⡶⣶⣍⡉⠛⢿⠷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠭⡤⠶⠖⠛⠛⠛⢳⣤⠈⢻⢿⣿⠿⠚⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡏⠉⠃⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢧⣤⠆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠳⢤⣀⣀⣠⠴⠋⠀⠀⢠⠤⢤⡀⠀⠀⠀⢀⡴⠖⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡼⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢻⣀⣸⠇⠀⠀⠀⣾⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠉⠀⠀⠀⠀⠀⣄⣇⣠⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⣀⠘⠦⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⢠⠞⢉⣀⣀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣠⠟⠀⠀⣰⠛⠛⡆⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⠀⠈⠁⠀⢸⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠘⠲⠞⠃⠀⠀⠀⠹⣾⡜⠁⠀⠹⣄⠘⠦⠶⠂⢀⡾⠀⠀⠘⢾⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠓⠶⠶⠖⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠠⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⢀⣠⣀⡀⠀⠀⣀⣄⡀⠀⠀⢀⣠⣀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠙⠓⠚⠁⠀⠈⠓⠚⠋⠀⠀⠙⠒⠋⠁⠀⠉⠓⠚⠉⠀⠈⠙⠒⠋⠁⠀⠙⠓⠚⠁⠀⠈⠓⠒⠋⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⠀⢀⡴⢦⡀⠀⠀⣤⠦⡄⠀⠀⢠⠴⢤⠀⠀⠀⡴⢦⡀⠀⠀⣠⠶⣄⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠘⠳⠞⠁⠀⠀⠳⠶⠃⠀⠀⠘⠶⠞⠀⠀⠀⠳⠖⠃⠀⠀⠙⠶⠋⠀⠀⠈⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⣤⡀⠀⠀⠀⣀⣤⡀⠀⠀⢀⣠⣄⠀⠀⠀⣀⣤⡀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟⠀⠀⠻⣤⠟⠀⠀⠀⠿⣤⠇⠀⠀⠸⣤⡼⠀⠀⠀⢧⣤⠇⠀⠀⠸⣤⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄


⡖⢲⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡔⠒
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⡆⠐⠄⡆⠐⡂⢀⢦⠀⣆⡠⢰⠀⠀⡂⠀⠀⢄⠀⠀⡆⠀⡂⡠⡠⠀⠢⢰⡠⠂⣖
⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣇⣀⣂⣇⣀⣀⣜⣈⣀⣇⣘⣸⣀⣀⣅⣀⣀⣀⣙⣀⣇⣀⣉⣡⣑⣄⣜⣸⣉⣂⣧⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢀⣴⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⣰⠏⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⡿⠯⣤⣤⡤⠭⠭⠽⣿⣾⢏⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢟⡵⣫⠾⠭⢭⣙⡒⠮⢍⡛⢿⣍⣾⠋⠼⣿⣷⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢣⢎⡼⠋⠉⠓⠲⠤⣍⡑⠢⢌⡓⢮⡙⢮⣵⡿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣡⢃⣾⣉⠽⠶⣶⣤⣄⡀⠉⠓⢤⡙⠢⣌⠲⣝⢯⣝⣚⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⢣⠇⣾⡭⠖⣚⣩⠥⠴⠒⣛⣲⣤⡀⠉⠢⣌⠳⣌⢣⡹⣮⢕⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⡞⣸⡧⠖⢋⣩⠴⠒⣋⣭⠭⢶⣾⣿⣢⡄⠈⢢⡈⢦⡙⣜⣷⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⠇⡿⠤⣞⣉⠤⠒⢉⣡⠴⣲⡿⢫⠤⠖⠚⢢⡀⠙⣄⢳⡘⣾⣷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⢰⣏⠒⠦⣍⣙⡚⠻⣦⣾⡿⠡⠤⠶⠒⠚⣉⡹⣄⠘⢆⢱⡘⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⢸⣟⢷⡦⢤⣀⣩⡿⢻⡿⠁⠒⠛⠛⠉⠉⣁⣴⠋⣆⠘⡆⢧⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡳⣝⣶⣾⢏⣩⡿⠁⠓⠒⠒⠚⣉⣉⣴⢃⡜⢸⡆⢹⠸⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣬⣾⡟⢭⣳⡟⣠⣉⣉⣉⣉⣉⣥⡔⣡⠎⡰⢃⣿⢈⣿⠏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡑⢮⣳⡟⣐⠮⢝⣓⠶⢤⡴⣫⠞⡡⢎⣴⣿⣿⠟⣁⡀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⣷⡟⣐⡮⢭⣓⣺⢟⡫⣞⣵⣮⣶⣿⡿⠛⠁⠚⠙⠛⠛⠛⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠛⠛⠿⠷⠿⠿⠿⠛⠛⠉⠁



⠀⠄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⡛⢯⣿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣩⣛⢿⣿⣿⣿⡷⠜⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⣯⣾⣷⣆⣬⠟⠁⠀⠀⠈⠛⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡺⢣⣮⣴⣄⣻⠟⠁⠠⣭⠝⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⣯⣝⣙⣻⡿⠋⠀⠀⠀⠀⠀⠶⢯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣷⣮⣦⣜⣩⡿⠃⠀⠀⢀⣀⡀⠠⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⢛⡟⠋⠀⠹⣄⠀⠘⠲⠝⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⣿⠀⠘⣇⠀⠀⠀⠙⡷⡤⠴⠾⠛⠯⡀
⠀⠀⠀⠀⠀⠀⡴⠋⢶⣄⠀⠀⠘⣆⠀⠈⠓⠶⠤⣤⠞⠀⢰⠂⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡼⠁⢈⢣⣻⣆⡀⠀⠈⠛⠶⠶⠶⠚⡟⠲⠚⠁⢄⡆⠀⡀⢹⡄
⠀⠀⠀⠀⢰⠃⠀⠸⣿⠇⠻⢷⢠⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⣰⣿⠀⣧
⠀⠀⠀⠀⢸⣶⣶⣄⠘⢆⠀⣾⠟⠀⠀⠀⠀⠀⢠⡇⣇⠀⠀⠀⠉⣿⠇⠀⢸⡄
⠀⠀⠀⠀⠀⢹⡇⠈⠙⠲⢤⣸⡇⠀⠀⠀⠀⠀⠸⣤⡼⠀⠀⠀⠀⢿⡤⠖⢿⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣤⡀⠐⠆⠀⣶⠀⠀⡼⠀⠀⢸⢿⣿⠀⠀⠀⠀⣠⢔⡎⢙⣲⡀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⣷⣴⢄⣾⣉⣻⠙⣶⣾⠿⣛⣛⢿⡿⠶⠶⠾⠿⠿⠶⠶⠶⠮⠗⠋⢀⡾⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⣿⠂⢸⠉⢑⢈⢇⢻⠋⢠⠋⣨⣤⡉⢶⡀⠀⢀⠚⠛⠛⠛⠉⠉⠓⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⡗⢺⣤⡼⢢⠔⡸⠀⣼⠞⢣⠀⠈⠲⣛⡿⣿⣀⣀⣤⣤⣤⣤⡄⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠉⠓⠺⣟⠧⢼⣾⡱⣏⣽⣷⡦⣿⣿⣿⡿⣦⠀⠀⠀⠀⠈⠉⢻⣷⣍⡵⣿⡀⠄⣤⡤⣄⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣝⠿⣶⡬⣤⣍⣷⠶⠾⠿⢳⠦⣤⣤⣤⣤⣀⣹⣼⣻⠀⡓⣄⠀⠀⠀⠀⠈⠉⠁⠐⠒⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣤⣄⣀⣀⣀⡎⣿⠗⠸⡓⠓⣿⡇⠀⠀⠠⠔⡠⠈⠱⢆⠀⠀⣽⣿⡿⢀⠁⠋⢦
⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠟⠛⠛⠉⠉⠉⠉⢹⢹⣏⠀⠀⢸⣾⣾⡟⠛⠛⠓⠲⠮⠶⠴⠾⣧⣀⣈⣿⠓⠿⠴⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠲⢀⠀⠃⠄⠻⣼⣿⠀⢠⡏⢉⣯⠗⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠟⠛⠓⠶⠤⣄⡀⠀⠀⠀⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠈⠉⠛⣉⠩⠤⠀⠤⠒⠒⠉⠉⠀⠀⠀⣀⣀⡠⠤⢤⣒⣒⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⣀⣀⠠⠤⠤⣐⣒⣪⡭⠥⠖⠒⠚⠉⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⣒⣢⡭⠥⠶⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡠⠀⢤⣀
⠀⠀⠠⠖⠐⢶⡄⠈⡂⠀⠀⠇⠀⠠⠀⠀⢠⣅⣈⣀⣹⠐⠀⢤⠊⠰⢸⡆⠄⠤⢴⡆⠀⠈⠄⢠⡉⣱⣶⣧⡀⣦
⠀⠀⢀⣄⠀⠤⠀⠔⠁⠀⢸⠀⠆⠂⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠁⠤⢰⠀⢀⠀⠀⠄⠄⠈⠀⠘⢁⣨⠋⢩⡴⠃
⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⢁⡀⠀⠀⠀⠤⠭⠭⠭⠥⠀⠀⠢⠴⠀⠀⠀⠀⢀⢀⡴⠦⠐⠀⠾⠀⢈⣩⠀⣄⡀
⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⢀⠈⢉⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠁⠀⠀⠄⠠⠠⠟⣛⣼⡟⢁
⠀⠀⠀⠀⠑⠁⠒⠀⠈⠀⠠⠀⠃⠄⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⢤⢄⠀⠀⠀⠀⠀⠀⠀⡀⠘⢿⣿⣦⠄
⠀⠀⡀⠀⣀⡀⠀⠢⠀⠀⠘⠒⠒⠃⠀⡀⠀⠀⠀⠐⠀⠀⠀⢀⠄⠀⠐⣃⠏⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠉⠀⢄
⠀⠀⠁⢀⠉⠁⠄⠀⠀⡠⢖⣱⣞⡭⠙⠀⠑⠀⢀⣀⠠⠀⣀⠀⠀⠀⠮⢄⠐⣄⠀⠀⠀⠂⠀⠀⠈⠐⡄⠀⠀⡰
⠀⠀⠀⠀⠀⠈⠂⠀⠨⠑⠒⠀⠐⠀⠀⡍⠁⠀⠈⢹⣿⣿⡏⠀⠠⡀⠀⠀⠁⠀⠀⡠⡀⡀⠀⠸⢀⢠⣄⠉⠉
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢄⣀⣀⢠⢀⠀⠁⠴⠀⠀⠉⠉⠀⠐⠄⣀⣤⣴⠇⠀⠀⠑⠁⢀⠀⠀⠀⠉⠊⠀⢀
⠀⠠⢄⡠⠀⠀⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⢁⡀⠈⠀⠈⠋⠀⠀⠀⢉⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⣀⡅⠀⠀⠀⢀⢡⡾⠀⠀⠂⣀⠀⠀⠀⠠⡰⢆⡀⢐⡌⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠁⠍⠀⠀⠁
⠀⠀⠀⢀⠂⠀⠀⠠⠂⠆⠈⠀⠱⣀⠀⠀⠀⢐⣲⠂⠈⠉⠀⠈⠀⠀⠀⠀⡀⠄⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤
⠀⠀⠐⠀⠀⠂⠀⠀⠁⠀⠀⠀⠠⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠄⠤⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠃⠀⡎⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⢴⣶⣶⢶⢶⢶⢤⣄⣰⠃⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡎⢏⣧⢃⢷⡟⣱⣟⣕⣵⣩⣾⣿⣦⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⢿⣿⢸⣿⡾⠟⣋⣭⡥⠷⣒⣶⣾⣫⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⢿⢯⡻⡼⠞⣡⠦⣕⣾⠷⣛⢟⠿⣫⣵⣾⡱⠾⢋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣶⠏⣌⣾⢏⣴⠞⡡⡪⡪⢗⢝⠕⣡⣾⣿⣿⡷⢾⣛⣲⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢿⡄⡧⢡⡾⢡⢊⢎⢎⡴⡡⣢⣾⣿⡿⢛⣵⢾⡻⡇⠅⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢿⢧⢡⢻⢡⢃⢆⢏⡞⣴⣿⡟⣱⢟⣵⣿⣵⢸⡄⣿⠀⡄⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣼⡏⡆⡎⡘⣬⢎⣾⡿⣱⣿⢣⠯⢙⡻⠿⠟⢶⢷⡞⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣾⡇⡇⡇⣿⡏⣾⢧⢣⣿⡇⡾⠭⠅⣐⡛⠉⠀⠠⣭⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⣷⢸⢻⢹⠸⡡⣧⣷⡫⠄⠠⠄⠒⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⣼⡜⡕⠟⡿⣿⣯⡫⣛⠝⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣷⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠐⢂⣀⡀
⠀⠀⠀⠀⠀⠠⣄⠉⠁⡗⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀
⠀⠀⠀⠀⠀⠘⠁⢸⡲⣤⠀⠀⠀⢉⣀⡁⠀⠀⠀⠂⢀⣴⢿⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠀⠀⠀⠀⠯⠤⠐⠆
⣋⣠⣄⣶⣀⣀⣀⢸⡇⠰⠄⠀⢀⣌⣡⢁⡀⠀⠐⡀⠙⠁⠈⠉⠀⠻⠆⠀⠀⠀⠀⠀⠀⢠⡄⠐⠀⠀⠈
⠞⠋⠉⠉⠉⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠉⣀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⡤⠀⣤⣤⣤⣬⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠿⠿⠷⣶⣶⣦⣤⣀⣠⡤⠶⠇⠀⠀⠀⠀⢰⠄⠉⠉⠉⢦⡀
⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀⡤⠴⠟⠉⠀⠀⠀⠄⠀⢿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⣀⣀⠀⠄⣐⠀⠀⣤⠠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⢀⠀⠠⠂⢘⠈⠉⠰⢶⠿⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠀⡆⠤⠀⠀⠋⡉⠤⠄⡒⠒⠛⡀⠈⠈⠉
⠀⠀⠒⠈⢀⡁⠤⠐⠀⠉⠗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠤⢒⠫⠴⠓⠂⢁⣀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣿⣄
⠄⠀⠀⠠⠀⠔⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣴⣿⣿⣧⡀
⠤⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣺⣿⣿⣷⣿⣾⣤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⠴⠒⠊⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡆⠀⠀⢀⠔⠁⠀⣠⠔⠁⠀⠀⠀⢀⣠⣤⠶⡖⡖⡲⠲⢖⢴⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⡇⠀⣰⢃⡠⠔⠊⠀⠀⢀⣠⡴⢾⢋⠁⠀⠀⡇⣧⣗⠬⢖⡮⢃⠭⢛⠭⢛⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⡰⠋⠁⠀⠀⠀⢀⡴⡏⢸⠇⣾⡄⡀⣇⡶⡫⢋⡵⢊⡥⠊⡡⠂⣡⢖⠥⡊⠝⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢠⡾⠁⢀⠔⡆⠀⣰⠛⡀⠇⢸⢸⡏⢇⢧⢟⠞⡱⠋⡐⠋⡠⠊⡠⢊⠵⣛⠩⠔⣪⠔⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠱⣾⢁⡴⠁⡰⠁⡰⡏⠆⠱⡸⡈⡎⡿⣼⢧⢃⠎⣠⠊⢀⠞⣠⢊⣴⣛⡭⠅⢒⡩⠔⢊⡤⢚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⡀⠀⣏⡞⠀⡴⠁⣰⠇⢿⡜⢆⢱⠳⡸⣽⠏⡏⠸⢠⠃⢠⠋⡰⢡⠾⣅⣐⠲⠬⠅⠒⣊⣥⣐⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢃⠀⣿⡠⠞⠀⠀⡏⡄⡿⠹⣌⢢⡳⡙⣼⢰⠀⠇⡌⢠⠃⡼⢡⢿⣥⢤⡶⣄⠲⢎⡋⠹⢖⣒⣒⣚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⡇⠀⠀⠀⠘⣇⢡⢷⢇⢯⡷⡝⢮⣷⢸⢸⠀⡇⡎⡰⢡⡟⢆⢫⡑⢭⣛⠓⠶⣈⣑⣶⣶⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡇⡇⠀⠀⠀⠀⡏⢂⢮⢎⢾⠃⡜⡷⢼⣾⡘⡀⡅⡇⠇⡿⡜⢆⠣⡙⢄⠙⢍⡒⢦⠠⣭⠭⣽⠿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⣼⠀⠀⠀⠀⢹⣌⢪⣮⢮⡟⡸⢡⠎⡽⣯⡻⢟⠷⢦⣟⢮⣢⡳⣜⠦⡑⠦⣙⢢⣵⣾⣛⣉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⠀⠀⠀⠀⢏⠣⡹⣷⣷⡇⡎⡜⢰⠁⡜⣳⢽⣢⢍⣻⠭⣙⡛⠿⠿⠷⣞⣛⣹⣤⣤⡴⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣆⠘⢮⡫⣿⢀⠃⠇⡘⢰⠁⡾⣿⣓⠾⢭⣒⡭⢕⣒⣒⣒⣒⠢⠶⠶⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠘⢕⠦⡙⠪⢿⡐⠀⡇⠸⢸⣍⣥⡐⠪⠥⠖⡋⠵⣒⠶⣲⠶⣲⡿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢌⣳⣌⡓⠢⠽⣇⡇⢸⢸⡓⠒⠒⠢⠍⠡⣒⠩⢔⡪⢔⣩⡯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠂⠻⠶⠾⠷⢒⡂⠀⣬⡥⠴⠨⠥⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⠀⠀⠀⣠⣶⣞⣲⣶⣄⠀⠀⣠⣴⣤⡀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⢠⣷⣄⠀⠀⠀⠀⠀⣴⣶⡀
⠀⠀⠀⠀⠀⢠⠤⣼⣿⡶⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣟⣻⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠈⠻⠿⠿⠃
⠀⠀⠀⠀⠀⢨⣿⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣴⣿⠀⠀⠀⣠⣶⣄⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⣠⠖⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⣾⣿⣿⣿⠀⠀⠸⣿⣅⣠⣿⠇⠀⠀⡇⠀⠆⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠙⠢⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣦
⠀⠀⠀⠀⣀⣴⣶⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⢲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣀⡀⠈⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⣶⣿⣿⣿⣿⣿⣷⣶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡞⢿⣿⣿⣿⣿⣿⡟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠉⠙⠛⠋⠉⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠿⠿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣤⣤⣴⣿⣿⣛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣀⣀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣰⣿⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠱⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⡀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⠀⣠⡆⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠮⢉⠛⢀⠁⠰⠇⠈⢠⡇⠀⣏⡉⠳⠢⣤⡀⠀⠀⠀⡴⠋⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠎⠁⠉⠐⠂⣤⠙⠋⢁⡄⠂⠉⣰⠀⣸⡡⠐⠊⠁⠠⠙⠓⣤⠞⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⡀⠒⠠⣈⠕⠂⢀⠤⠂⢁⡠⠐⢁⡜⠀⣛⡀⠄⠂⠉⡀⢤⡞⠁⣰⠏⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠄⢀⠀⠲⠖⠀⡠⠀⠀⠀⠔⠈⡠⠐⢈⡇⢠⡏⠀⡀⠔⠋⡰⠋⢠⡞⠈⠂⡝⢦
⠀⠀⠀⠀⠀⢀⡞⠁⠠⠀⣀⠛⠀⠀⠀⠀⡀⠈⢀⡠⠃⠀⡠⢘⠃⢸⠷⢆⠉⢲⠎⠀⣴⠏⠁⡀⠀⠇⠘⢣
⠀⠀⠀⠀⢀⡞⠡⠆⢀⠀⠁⣠⠊⠀⠠⠂⠀⠀⠀⠀⠄⠀⠀⣬⠀⡴⢄⢀⠞⠁⡠⠞⡅⠠⠀⠇⢀⠀⠈⠀⢷⡀
⠀⠀⠀⠀⡼⠀⠀⠠⠀⠀⠔⠁⢀⠚⠀⠀⠀⢀⠔⠁⠀⡠⠎⠉⠠⣇⡠⠋⢀⠜⠁⠄⠸⠀⢠⠀⠸⠀⠀⠀⠈⣧
⠀⠀⠀⢸⡇⠂⠠⠁⢀⠊⠀⡠⠁⡠⠊⠀⠐⠀⠀⡠⠚⠀⠈⠲⣄⠈⠡⡴⠋⠠⡀⠈⠇⠀⡈⢀⡀⢰⠀⠸⠀⣿
⠀⠀⠀⣸⢡⡔⠀⠠⠀⢀⠊⢀⠜⠀⡠⠂⠀⣠⠊⠑⢄⠀⠀⠀⠈⠳⢄⠈⠂⡀⠁⠦⠀⠀⠁⠀⠀⡀⠀⡀⠀⠀⡇
⠀⠀⠀⢿⡾⠀⠠⠁⡀⠂⠠⠈⢀⠄⠁⠀⠐⡙⠂⡀⠀⣴⠀⠀⠢⡀⠈⠃⠀⠰⣀⠀⠀⠉⠀⠃⠀⠃⠀⠀⠄⢸⡇
⠀⠀⠀⢸⠇⠀⠀⠠⠁⠠⠁⢀⠆⠀⣠⠃⠀⠁⠀⣬⠞⢀⡼⢆⠀⠈⢀⠀⠀⣠⠋⠀⡈⠀⠘⠀⠘⠀⠈⢠⠀⣸⡇
⠀⠀⠀⠸⣧⠁⢀⠃⡀⠁⠰⠂⠀⠀⢩⠀⠠⢀⡜⠁⣠⣿⠀⣜⠣⠀⠀⢂⠀⡏⠀⢠⠁⠀⠃⠐⠂⡘⠀⡀⠀⣿
⠀⠀⠀⠀⢹⢀⠀⠠⠀⠠⠁⢀⡜⠀⠐⠀⡼⠊⢀⠜⢁⡟⠀⣿⠀⠁⠀⢀⠗⠀⠀⠀⢀⠉⢀⠃⢀⠁⣀⠁⢰⡏
⠀⠀⠀⠀⠈⢿⡀⠃⠀⠂⢀⠘⡅⠀⣠⠞⠁⡴⠟⠀⢸⠇⢠⠈⢀⠀⠰⡎⠀⢠⠀⠀⠌⠀⡈⠠⠄⢠⠙⣆⡿
⠀⠀⠀⠀⠀⠈⢿⣀⠁⢀⡆⠀⣇⡴⠃⣠⠞⠀⠀⠀⣿⠀⠸⡁⠋⢦⡐⠀⣀⠃⠀⠘⠀⠠⢀⠂⠀⠾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡘⢇⣰⠋⢀⡜⠉⠀⠀⠀⠀⡿⠀⡾⠁⢀⠊⠀⡐⠀⢀⠆⠀⡌⠀⠌⢠⣌⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠟⠁⡴⠋⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠂⢀⠔⠀⠀⠂⠀⠐⢀⠈⢠⣡⠞⠉⠉⠉⠉⠙⠉⠁⠒⠲⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣠⠞⢧⣄⡀⠀⠀⠀⠀⢰⠀⢠⡔⠀⢠⠂⠀⡐⠀⠠⠀⣀⢂⠴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠢⢤⣀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⣠⠊⢀⡜⠁⠀⠀⠈⠁⠒⠦⢤⣄⡸⠀⣺⢀⡄⠁⣀⣈⠀⣴⠆⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⢣
⠀⠀⠀⠀⢀⠞⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠋⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣸
⠀⠀⢀⡴⠋⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠞⠋⣀⡴⠃
⠀⣠⠋⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠶⠀⠛⠉⢀⣀⠠⠖⠛⠁
⢰⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠀⣀⣠⠀⠶⠒⠉⠉
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠙⠓⠠⠶⠦⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠲⠶⠶⠶⠤⠤⠤⠤⠄⠀⠀⠀⣀⣀⣄⠀⠀⠙⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⣄⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⠤⠚⠁⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⣀⣀⣀⡀⠠⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠰⣇⢀⣹⣠⣤⡿⣧⢸⡏⣗⢀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠘⢧⠼⢃⣩⣥⣴⣶⣶⣶⣶⣤⣭⣁⠺⢽⢍⣴⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢊⣴⣾⣿⠿⠛⠛⢉⣉⣉⠙⠛⠻⢿⣿⣶⣌⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⢀⣴⣿⠟⠉⣀⣀⣠⣴⠏⠉⠉⢷⣤⣀⣀⡈⠛⢿⣷⣄⠐⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⣿⡿⠃⣰⠟⠉⠙⡿⠛⠣⣀⡤⠛⠻⡟⠉⠙⢷⡀⠻⣿⣧⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⣿⡿⠁⠀⣿⣄⡀⣠⣇⠀⣀⠿⢇⡀⣀⣦⡀⣀⣼⡇⠀⠹⣿⣇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⠇⢀⣼⢿⡏⠀⠀⣿⢿⡃⠀⠀⣹⢿⡇⠀⠈⣿⢿⣄⠀⢿⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⣾⠁⠀⠈⣶⣶⠀⠀⢐⣶⡖⠀⠀⢰⣶⡎⠀⠀⢹⡆⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⡄⠘⢷⣤⡎⠀⠀⣶⣴⡂⠀⠀⢢⣴⡆⠀⠀⣦⣴⠞⠀⣼⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣿⣧⠀⠈⣿⠟⠂⠲⡟⠋⠓⣤⡔⠋⠛⡷⠂⠚⢿⡏⠀⢠⣿⡟⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠹⣿⣧⡀⢻⣄⠀⢀⣦⣀⡠⠛⠣⣀⣠⣇⠀⢀⣼⠃⣠⣿⡿⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢿⣷⣄⡉⠛⠛⠻⢿⡅⠀⠀⣽⠿⠛⠛⠋⣁⣴⣿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢿⣿⣶⣤⣀⡀⠙⠛⠛⠁⣀⣠⣴⣾⣿⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠿⢢⡬⣙⠻⠿⢿⣿⣿⣿⣿⠿⠿⠛⣩⠤⠘⣞⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠉⣿⠵⠶⢲⡶⢰⡆⢲⡝⣧⠹⠯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠂⠘⠓⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⠿⠒

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣷⣶⣾⢿⣿⣿⣷⣿⣿⣯⠓⠄⠈⠣⠄
⣿⣿⣿⣛⠀⠋⠍⠙⢿⣿⣿⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣠⡝⠿⠟⠙⣿⡟⠀⠀⠀⢠⣤
⣿⣿⣿⣯⠀⠀⠘⠃⠀⠉⠛⢷⣎⠉⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠹⠹⢿⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⣰⠆⠻⠁⠀⢀⡼⠛⣻⡆
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠙⠻⠿⢿⣶⣤⣀⡠⢄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠘⠇⢰⣾⠋⠀⠃⠀⠀⠿⠇⠈⠿⠁
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⠀⠀⠀⠙⠿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⠀⠀⣠⡀
⣿⣿⣿⣿⣿⣇⢠⣶⠟⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠙⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠀⠀⢠⣿⣿⡃⠀⡀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣮⣻⣿⠟⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣿⣿⠉⢻⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⣤⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢸⣿⣿⣷⠾⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣍⠻⣷⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⡟⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣷⣶⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⢸⣿⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣧⠀⠹⣿⣿⠿⠶⠤⣤⡀⠀⠀⠀⢠⡀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⢿⣿⣿⣿⡿⣿⣿⣷⣿⣿⣿⣿⠀⠀⠘⢟⢷⣄⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠉⠙⠛⢷⡄⠉⠀⠀⢸⡿⣿⣷⣦⣄⠈⢆⠙⠳⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠈⢿⣿⣿⣷⡄⠣⡀⠀⠈⠑⠉⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣾⣵⣿⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⣿⣿⣿⣿⣿⣿⠛⢿⡄⠀⠀⠀⠀⠀⠀⠂⠀⣠⣾⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⢻⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠇⣴⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣾⣿⣿⣟⣿⡇⣸⣿⣿⢃⣼⠋⠀⠀⠀⠀⠀⣸⣸⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣄⣴⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⠷⠟⠓⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⡿⢿⣿⢃⣠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⡡⠄⠀⠀⠀⠀⠀⠀⢤⣶⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠿⠖⠀⠀⠀⠀⠒⠀⠀⠹⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠚⣫⣿⣿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠞⠀⠀⠁⠂⠁⠀⠀⠺⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⡇⠀⠀⠀⠒⡀⠀⠈⡀⠀⠈⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⠩⢽⣿⣿⣦⡁⠀⠀⠀⠔⣒⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣰⠀⠈⠀⠀⠙⠛⠱⣴⡤⠀⢒⣻⣟⡳⣼⠁⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡠⣄⣀⣲⣺⠟⣉⡓⡥⢄⢨⠱⢜⣷⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⣿⢯⡛⠙⢟⣩⡄⢠⠉⢋⠢⢄⣠⣦⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡍⠀⠀⠀⠀⠀⠈⠿⣷⣴⡤⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣴⣶⡾⠃⢀⡔⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣷⣤⣶⡛⠀⠀⠀⠀⠀⠀⠀⣹⣿⣶⡄
⠀⠔⢀⢰⡖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣷⡿⢿⣿⣿⣷⣶⣶⣶⣶⡿⣿⣟⣷⣿⣿⢷⣤
⠀⢱⠈⣠⠌⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⡿⣿⢿⣿⣷⣿⣿⡿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⢽⡆
⢀⡠⠛⠃⢴⠄⢄⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣞⣿⣽⣾⣯⣿⣟⣿⢛⣿⣾⣿⣿⣻⣿⣽⣾⣿⠄
⠀⠈⠀⠤⣀⠀⠀⠰⠈⠆⠀⠀⠀⠀⣴⣿⣞⣿⣟⣿⣿⣿⣿⣷⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠢⡀⠀⢰⣿⣮⣿⣽⣹⣿⣿⣷⣯⣻⣿⢿⣟⣾⡷⣿⣿⡻⣿⣻⣿⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠲⣿⣽⣿⡿⣿⣾⣿⣿⣾⣿⣿⡿⢿⠹⣿⣷⣿⣟⣷⡿⣿⣛⣿⢛⣭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣼⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠿⣛⣻⣛⣿⣾⣟⡻⣯⣻⣷⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣤⡴⠿⠿⠯⠿⣿⣿⣿⣿⣎⣿⣿⢿⡿⣿⣿⣿⣿⣿⡿⣯⣽⣿⣻⠟⣯⣿
⠀⠀⢀⡀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠚⠿⣿⣾⣿⣿⣿⡿⣿⣿⣯⣿⣿⣿⣿⣽⣻⣾⣟⣿⡟⣻
⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢐⠢⠤⠲⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠁⠛⣿⣩⣿⣿⡿⠿⠟⠛⠉
⠧⡀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⡀⢁⠀⠀⠀⠠⠀⠒⠠⠤⢀⣀⣀⡀⠀⠀⣀⠀⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⣀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⢎⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢔⣠⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡞⠡⢩⠏⣉⠢⢄⡀⠏⠱⠲⠄⣀⡴⡱⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢩⠁⡜⠠⢁⠣⢀⠂⠍⡢⠈⢳⠀⠂⣠⢋⠾⣄⠀⠀⠀⠀⢀⡠⠔⡪⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠆⠐⢀⢃⠏⠒⠄⠑⠢⡈⠢⣙⠀⡖⡵⠉⡆⠤⡑⣄⢔⡢⠵⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡄⠀⠸⠀⡅⡄⡸⢈⠢⢕⠈⡢⡈⠢⣈⣯⡞⢰⠀⣅⠁⠢⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⡀⢀⠀⢠⠑⡈⢅⠢⡑⢌⠪⠢⣸⠁⠀⠘⠀⠉⢍⠢⣄⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡆⡇⠸⠀⠸⡄⠘⡄⠢⠈⠢⡑⡀⡎⠀⠆⡄⢀⢇⠈⢓⢄⠑⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢻⠀⠃⠇⠀⠀⢠⠈⢄⠈⢢⠑⢄⠐⣼⠁⡸⠀⠀⡘⢀⢂⠈⣃⠑⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⢣⡀⢠⠘⠀⢠⠑⢄⠑⣄⠑⢄⣰⠂⠠⠁⡄⢠⠣⠈⡀⠄⢙⠢⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⠄⢳⡈⡀⠀⢸⡑⢄⠐⢄⠓⣤⠃⠠⠁⡜⢠⠟⡀⠂⢠⢰⢀⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢜⡷⡈⠦⠑⣼⡐⡀⠅⢠⡉⢢⡕⠀⡔⢁⠌⡠⠋⠰⢁⢀⡠⠚⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡊⠕⠂⠁⠈⢊⠢⣱⢄⠛⢥⠘⢠⡨⠋⡠⠊⡠⠃⠜⠁⡠⠤⠚⡉⢐⣠⠇
⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⠥⠒⠉⠀⠀⠀⠀⠀⢀⠝⣡⠊⡑⠢⡙⠣⣤⠎⠠⠞⠐⡓⠈⠁⠀⠂⢁⡂⠔⠁
⠀⠀⠀⣀⠴⣊⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⣣⠎⠀⠈⠂⠕⣤⣑⠢⢍⠛⠥⠄⣒⣀⢠⣴⡶⢋
⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠛⠂⠉⠉⠀⠀⠈⠀⠀⠀⠀⠤⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠀⠀⠀⠐⠒⠀⠀⠀⣀⠀⡀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠠⢀⡀⠀⠀⠀⠀⠀⢀⠨⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⡈⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠅⠂⠁

⠀⢰⣫⡄⢷⠃⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣖⢦⠀⠀⠀⠀⠀⠀⢠⣄⡶⢶⣠⡄
⢰⡦⠭⣤⠤⠴⣤⠤⢶⡄⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣵⣿⠂⠀⠀⠀⠀⣸⢷⣭⣽⣭⣭⡾⢇⡀
⢸⡇⠀⠈⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⢸⣿⢘⣠⣤⣤⣄⣘⢸⣿⡀⠀⠀⠀⠀⠀⠀⣸⡿⠮⠥⠞⠋⠀⠀⠀⠀⠀⣻⠿⣶⣾⣶⣶⠾⣟⡁
⢸⣿⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣈⠉⠛⠛⠋⢩⢸⣿⠃⠀⠀⠀⠀⣠⡾⠋⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠙⢿⢿⣶⣶⡿⠿⠋
⠚⠛⠛⠛⠃⠘⠛⠛⠛⠃⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⠃
⠀⣾⣿⢀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡀
⠀⠀⢼⣿⣯⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠞⢷⡟⡔⠉⠉⢢⠀⠀⠀⠀⠠⠞⠛⣻⣦⣜⠿⠃
⠀⣀⣀⣙⣿⣟⣀⣹⣇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⠄⠈⠛⠣⣀⣀⠜⠀⠀⠀⠠⢄⣀⣾⡭⠽⠞⠚⠒⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⡆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⡶⠶⡆⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⣥⣬⡲⡀
⢰⣷⣶⣶⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡧⠍⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡇⣴⠇⡇⠀⠀⠀⠀⠀⠀⢠⠃⡈⠿⠿⢃⠄⣄
⢻⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡤⣤⣤⡇⠀⠀⠀⠀⠀⠀⣺⣄⣈⡢⣔⣋⣠⣼
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉

⠀⠀⠰⢶⠝⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄
⠀⣴⣾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢀⢄⣤⡰⡶⣀⣄⢤⠀⠀⠀⠀⠀⠀⢿⣿⡷⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭
⠀⣯⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣧⠸⠿⣣⣦⡻⠿⢀⡇⠀⠀⠀⠀⠀⠠⢤⠔⠒⠔⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⢻⣜⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣶⣦⣤⣶⣾⡿⠁⠀⠀⠀⠀⠸⣷⣶⣶⣿⣾⣷⣾⣾⠂⠀⠀⠀⠀⠀⠀⢹⣖⣖⡿
⠀⠀⠙⠻⠟⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⢀⣾⣿⣿⢟⣠⣤⣄⠀⠀⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢼⣿⣿⡿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⢀⣾⣿⣿⣿⣿⡎⡻⠯⣞⠁⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⢀⣾⡿⢿⣿⡿⠟⣾⣿⣿⣿⣧⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⣾⣿⣿⣿⣷⣌⢿⣿⣿⣿⣇⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣿⣿⣿⣿⣿⢿⡞⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⡤⢀⣯⣅⢿⣿⣿⣿⡿⢏⡾⣹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠀⠀⢾⣿⣿⣷⣝⣛⣷⣚⣫⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⢷⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣮⣭⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡔⢲⢶⠂⠤⢀⡀⡠⢊⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠥⠀⠐⠀⡎⡌⠀⡄⢠⠈⠓⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣒⠀⠈⡝⢰⢰⠁⣸⠁⡞⠀⠀⢿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠐⡀⢐⡲⢠⢃⠂⡴⡏⡼⠁⠀⡀⡸⠀⣇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠣⡰⢈⡔⡡⢃⠊⡼⢁⡅⡀⠀⠰⢀⡋⠀⢸⠀⠀⠀⢀⠴⠚⠉⠉⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⡵⢊⠔⡡⠃⡜⠀⡜⢡⠁⠁⠁⠘⠀⠀⠸⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⠔⢁⠜⢁⠊⢀⠞⠀⠂⠈⡄⡬⠭⠉⣩⢃⡴⠋⠀⠀⠀⣠⠄⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⢶⠁⠔⢁⠔⠁⠆⠀⠀⡔⡘⠩⢄⡰⠋⠉⠀⠀⠀⠀⠀⠙⢦⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⠃⠀⠙⠶⣅⣀⡜⢀⠄⠜⠂⡠⠄⠋
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡠⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠯⠽⠯⠩⢭⣭⣽⣭⡿⠿⠿⠿⠭⠭⠽⠽⠯⠭⠭⢿⣭⣿⣻⣿⣻⣓⣲⡖⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⠀⡀⢀⡀⠀⣀⠀⣀⢀⠀⠀⠀⠀⠀⠈⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⣅⣿⣼⣧⡟⢹⣶⣯⡷⠀⢸⠀⠀⠀⠀⠀⠀⢸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡀⠀⠀⠀⠀⠀⠈⢀⠤⠿⠯⠽⠼⠧⠽⠟⠉⠃⠀⡤⠸⠀⠀⠀⠀⠀⠀⣼⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡇⠀⠀⠀⠀⠀⡤⢾⣤⠲⡤⡤⢠⣤⡔⣦⡖⢲⣶⠷⣴⡆⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⢀⣆⡁⣀⣏⡃⣸⣹⡃⣻⣇⢸⣽⣆⢳⣳⡀⠀⠀⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣇⠀⠀⠀⠀⠀⠉⡟⠛⠛⠛⣻⠛⠟⢛⠉⠉⠈⡹⠉⠉⠉⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⡇⠀⠠⠚⠁⠞⠀⢸⠀⠀⢰⠁⠀⢠⡇⠀⠀⠀⠀⢸⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⠀⠀⠀⠀⠀⡐⠀⠀⠒⠂⠑⠒⠒⠒⠂⠀⠀⠂⠀⢈⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⡾⠯⣭⣽⣿⣯⣿⣿⣿⣽⣿⣿⣉⣝⣭⣿⣿⣿⣏⣭⣿⣿⣿⣏⣽⣭⣇
⠀⠀⠀⠀⠀⠀⢀⠤⢚⡡⠒⡒⢦⡐⢒⣄⡒⢠⣐⠒⠒⠤⠒⠀⡤⠰⠒⡶⠶⡲⡖⢒⢦⠒⡐⣴⣒⠓⠲⣕⠦⡀
⠀⠀⠀⢀⡤⠚⠁⠀⠉⠀⠈⠁⠈⠉⠁⠉⠉⠁⠉⠀⡹⣒⣒⣒⣒⣒⣒⢏⠉⠉⠉⠉⠉⠀⠉⠀⠉⠀⠀⠈⠁⠈⠓⢄⡀
⠀⠀⣞⡗⠒⠒⢒⣖⢶⢖⢶⠶⢲⠶⡶⣲⢶⠶⠶⡲⡷⡶⢲⢞⢗⠞⣺⠟⢶⠲⠶⢲⡶⡲⡶⡖⢲⡶⣶⢶⠶⣲⢶⢖⢿⣦
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠓⠒⠓⠓⠛⠚⠓⠛⠓⠓⠛⠛⠛⠛⠓⠛⠚⠛⠚⠛⠛⠓⠛⠛⠛⠚⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠁⠂⢈⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⣄⠬⠒⡁⠁⠠⠀⣉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡅⠄⠘⠀⠄⠁⠖⡽⠰⢄⣚⠓⢉⢩⣀⠀⡁⠀⠂⠀⠈⠀⠂⢸
⠀⠀⢀⠴⡘⡈⠁⠆⠦⠀⠁⠉⠐⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⠆⡁⢀⠢⣀⡑⠀⠀⠢⠀⠰⠧⠀⠂⢈⠁⡄
⠀⠀⢏⠣⣻⢖⢄⡤⠒⣰⠀⠀⠀⠀⠡⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠠⠀⠀⠈⠄⠑⡀⠱⡀⠀⠀⡄⠄
⠀⢸⠜⣲⠡⢡⠆⣍⠠⠒⠇⠀⠀⠀⠀⠐⠄⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡀⠀⠀⠐⠀⠰⠀⠡⠀⠂⠠⠄⡘
⠀⠀⠣⣃⢠⣁⠬⡀⢉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠌⣀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂⠃
⠀⠀⠀⠈⠙⠐⠚⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠌⠀⠄⠀⠐⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠌⠀⠒⠀⠀⠉⠀⢐⠀⢂⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠤⠤⢤⠀⡈⠀⠀⢀⠁⠀⠄⠀⢀⡡⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢅⡀⡀⠈⠀⠀⠀⠄⠀⠀⠂⠀⠐⠀⠀⠠⠀⠑⠒⠂⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠑⠂⠒⡄⠀⠀⠈⡁⠀⠁⠀⠀⠁⠀⠀⡐⠐⠐⠀⢂⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⢀⠟⠒⠲⠂⠪⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡦⢀⣴⣶⣶⣶⣶⣶⣦⣄⡀⢀⣤⠀⢰⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠠⠈⠛⠛⠻⠿⠿⠿⠿⠿⣧⠘⠕⢂⣶⣆⠀⢀⠔⠒
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⢀⣤⡄⠀⢠⠂⠀⠀⠦⠀⠀⡀⠀⠀⠀⠙⠠⡆⣸⣤⣤⡄⢏
⠀⠀⠀⠀⠈⣿⢿⡀⠰⠄⠀⠭⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠥⠘⠉⠓⠀⠂⠀⠰⠆⠰⣌⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⡁
⡠⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠌⢃⠁⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿
⠁⡤⠀⠀⠀⠁⠘⠇⠀⠃⠀⠒⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣴⠀⠀⠐⠀⠀⢃⠀⠀⠀⠀⠀⠂⠠⣄⡀⠠⠀⠲⣶⡶⠆
⡀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢠⡀⠉⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠄⠀⠀⠂⠀⠀⠀⠀⢀⡀⢀⣴⢶⣦⡀
⠁⠀⠀⠀⠀⠀⠈⠐⠀⣀⠀⠈⠀⢠⠆⠀⠐⠀⠀⡀⠁⠀⠀⠀⠠⠀⠲⠀⣀⠀⠀⠀⠀⠠⡴⠀⠀⠶⠀⠀⠻⠃⣔⠥⣚⡞⢁⣴⠂
⠀⡀⠀⠀⣂⣥⡤⣤⣐⣒⡂⠀⢀⢀⡀⢰⠄⢳⠞⢳⣤⠄⠐⠀⠀⠀⠀⠀⠀⠐⠠⠀⣤⠀⢀⣶⣶⡄⢀⣴⠀⠀⢹⠚⠞⠀⣾⠫⡂
⠀⠀⠃⠁⠸⢿⡷⠀⢾⣿⠗⠦⠈⠤⠡⠄⠐⠛⠳⡏⠀⠀⠘⠀⠀⣀⠀⢀⡀⠀⢀⠀⠟⠀⠈⠿⠿⠃⠀⠈⠃⠀⠈⠀⠸⠀⠁⠀⠁
⠀⠀⠈⠀⠀⠀⠄⠐⠈⠉⠀⢰⣷⠀⠀⣾⣧⣶⣴⣦⣶⣶⣦⣰⡐⣿⡆⢸⣷⠀⣼⣷⠀⠲⣦⠀⠀⢰⡶⠂⢠⠦⠀⠀⠀⠀⠀⢰⣿
⢀⠀⠀⡠⠀⢰⡶⣗⡢⣀⠀⢈⣿⣄⣤⣿⡏⣩⡾⣦⠁⢭⣬⣍⢀⣿⡗⢿⣿⠃⣿⣿⣠⣄⢠⣤⠀⣴⣦⠀⠓⠁⠀⠀⠀⠀⠀⣈⠉
⠛⠀⠀⠀⠀⠀⠺⠟⢃⠄⠀⢸⣿⠉⠉⣿⠀⢿⣞⣫⢀⣤⣼⣿⠘⣿⡇⢿⡇⠀⣿⡏⠈⣿⡏⣿⣧⣽⣿⠀⠀⠈⠠⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡀⠿⠀⠀⠉⡁⠘⠷⣼⣿⠀⠿⠀⠘⠿⠗⠻⠃⢠⠙⠁⠈⠉⢻⣿⠀⠀⢀⠀⠀⠀⠈⠁⢘
⠀⣰⣦⠀⠐⠠⠀⠀⠀⠀⠀⠘⠀⠀⡁⠀⠀⠐⠀⠉⣠⠐⢂⡄⠒⠆⠢⡀⠠⠤⣠⠀⠆⠀⠖⡀⢰⣼⡿⠀⠀⠀⢰⡄⠀⠀⠀⠠
⠀⣿⣿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⠁⠀⠀⠀⢠⠒⠈⠀⠀⠍⠀⠀⠈⠁⠠⠀⠠⡄⠀⠀⠉⠀⠀⠀⡀⠈⣀⠀⠘⠃
⠀⣿⣿⡇⠀⠀⠉⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣄⠒⠡⠀⠄⠀⠤⠀⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⢾⣶⠆⠀⠄
⠀⣿⣻⡇⠀⠠⠠⠀⠐⡄⠀⠀⢀⡀⠀⠀⠀⠈⠉⠀⠅⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠀⠀⢀⠀⠰
⠀⢿⣿⡷⠄⠀⠀⣀⠀⠀⠄⠽⡲⢤⠀⠀⠀⠀⠐⠶⠈⠠⠀⠀⠀⠀⠀⢁⠈⠀⡀⠀⠀⠎⠂⣀⠀⠀⢀⣊⠔⠁⠀⠀⠀⠀⠠
⣀⠈⠋⠀⠀⠀⠀⠉⠀⠀⠈⠙⢁⠄⢑⡂⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠐⠈⠀⠀⠀⠀⢀⡄⠀⢠⠂⠀⠁⠀⠤⠀⠁⢠⣴
⠛⠃⠐⢄⠀⠘⠀⠠⢐⣆⡀⠂⠀⠀⠁⠀⠀⠰⠠⡬⣿⠁⣼⠁⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⠖⠁⠀⠐⠁⠀⠐⠀⣀⡀⠀⠀⠙
⠈⠀⡀⠈⠀⠀⠀⠀⠈⢻⣄⠀⠘⠓⠐⠆⠀⠀⠀⠑⣺⣦⡸⠇⠘⠀⠀⠀⠀⠀⣀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠉⢆
⠆⠠⠀⠠⠀⠀⠀⠀⠀⡈⠻⣦⠈⠀⠀⠀⠀⠳⠀⢠⠀⠀⢀⣀⢀⢻⣄⠴⣶⠻⠃⠀⣠⠈⠀⠈⠁⠀⡀⢀⡀⠐⠈⠀⠀⠀⠈
⠠⠀⠁⠀⠀⠀⠀⣦⣦⠀⠀⠉⠾⠇⠀⠀⠀⠀⠐⠁⠀⠀⠈⠈⠛⠀⠰⡀⣤⣤⣤⡀⠉⠀⠀⠐⠷⠀⠐⠎⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠖⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⡽⠂⠀⠀⠀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠁⠀⠀⠀⢠⠏⢠⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠀⠀⠀⡰⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⢉⣙⠒⢤⡀⢰⡇⡇⠀⠀⡴⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡴⠉⠈⠉⠲⣝⣼⣱⠃⢀⡞⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠃⣀⣴⡒⠉⠹⡉⠻⡉⢻⡺⣅
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⢏⠞⠓⠦⣍⣉⡉⠙⣄⠙⢦⡙⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢠⠋⡴⠋⠉⠙⠢⡹⡄⠀⠀⠀⢀⡠⠊⣀⠴⢣⣏⠙⠒⠦⠤⢤⣭⣭⠭⠷⣄⠙⠢⣌⠛⣆
⠀⠀⠀⠀⠀⠀⢫⠀⡇⠀⠀⠀⠀⡇⡇⣀⡤⠖⣋⡤⠚⠁⠀⣼⣨⠟⠒⡤⢤⣄⣀⣀⣤⠤⡼⠓⢤⡀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠓⠬⣑⣒⣒⢚⣋⣩⠥⠖⠋⠁⠀⠀⠀⠀⢻⣀⠤⠊⢀⡠⠊⣠⠎⢀⣾⣛⣫⠭⠜⠓⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣠⠔⢋⡠⠞⢁⡴⣟⡒⠤⠴⠒⠒⢋⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠒⢉⣠⢶⡉⠢⣄⡉⠉⠓⠒⣺⠏
⠀⠀⠀⠀⠀⠀⢀⡴⢋⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠙⠯⣼⣄⠑⢤⣀⣉⣑⠶⠚⠁
⠀⠀⠀⠀⠀⢠⠋⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠁⣀⣈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡇⡰⠁⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠳⡀⠙⣄⠀⠀⠀⠀⠀⠀⣇⢱⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⡑⠦⠤⠤⠤⠤⠜⣦⡙⠯⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠊⠉⠀⠉⠓⠦⢤⣈⣉⣉⡉⢉⣉⣉⣀⡠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⣀⠀⡀⠀⠀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⣀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣸⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠷⠶⣿⣿⣿⣵⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣶⣍⡑⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⡿⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⠋⠇⢸⣿⠌⠻⣝⣝⠉⠒⡄⣀⡀⢀⣠⣤⣬⣵⠀⠀⠀⠀⠀⣻⣻⠃⠁
⠀⠰⠖⠂⠀⠀⠒⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣋⡙⠿⢠⣿⣯⣿⠶⠆⣀⣴⣼⣴⣄⣀⠠⠤⠤⠶⣶⣶⡆
⠀⠀⠶⠶⠶⠶⢦⣠⣤⣄⡍⠿⠿⢝⣻⣟⣻⣿⣿⢸⣿⣃⢸⣿⣿⣿⣏⣽⣦⣶⣼⣿⡏⠉⠉⢹⣿⣄⣀⣤⣤⣿⣿⣛⣭⠁⠐⠒⠂⠐⠒⠒
⠀⢠⣤⣤⣤⣤⣤⣤⣬⣤⣖⣲⣿⣿⣿⠿⠿⠟⠛⠘⠛⠋⣻⣟⣛⠛⠋⠙⠛⠛⠘⠛⠃⠀⠀⠚⠿⠿⠿⠛⢿⣭⣷⣴⣾⣴⣶⣶⣤⣤⣠⣤⡀
⠀⠸⠿⠿⠟⠛⠛⠉⢉⡩⠁⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⢙⣿⣏⣤⣤⣰⡤⣴⣤⠴⣶⣤⣤⠦⢀⣀⣀⡀⣀⣛⣻⠿⠿⠿⡿⠿⠟⠻⠿⠿⠇
⠀⠀⣀⣿⣇⣤⣜⣅⣀⣦⣤⣏⠙⣿⣟⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣟⣱⡇⠢⣦⣿⣿⠃⣠⠈⢻⣿⣤⠈⠛⢛⣦⣤⣤⠂⠀⠀⠀⡀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣶⣀⡀⠀⠀⡀⠀⠀⠈⠙⠿⣿⣿⣿⣷⢀⡟⡿⠿⣢⣿⡀⢸⣿⠿⣆⠀⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⡀
⠀⠠⣶⣾⣿⣽⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡷⠗⠖⠺⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣇⣹⢇⣿⣿⠗⣾⣶⣾⣿⡶⣶⣾⣿⣾⣶⣶⣶⣶⣿⣿⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣨⣯⣗⣀⣐⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣻⣻⣛⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⡿⡍⠉⠉⢹⠉⠉⠉⡏⠉⠉⡿⠉⠉⢉⡟⠉⠉⠉⡹⠋⠉⠉⠉⠉⡩⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠹⡄⠀⠘⡇⠀⠀⡇⠀⢠⡇⠀⢀⡞⠀⠀⢀⠞⠁⠀⠀⢀⡴⢊⡂⠤⠀⠀⠀⣀⠤⠀⠀⠋
⡇⠀⢹⡄⠀⢻⠀⢀⡇⠀⣸⠀⠀⡾⠀⠀⣰⠋⠀⠀⣀⠔⠉⠀⢀⡀⠠⠀⠀⠀⠛⠁⠀⠀⠀⠀⣀⠠
⡷⣄⠀⢳⡀⠸⡇⢸⡇⢀⡟⠀⡼⠁⢀⡞⠁⠀⣠⠞⠁⠀⣄⢐⣵⡄⠀⠀⠀⠀⠀⣀⡤⠔⠂⠁
⡇⠘⢧⡀⢷⡀⣷⢸⡇⣸⠃⣼⠣⠴⠏⢀⡴⠞⠁⠀⠀⠀⢀⠀⠘⠃⣀⡤⠴⠚⠉
⡟⢷⣌⠻⣌⢷⠹⠞⠇⠿⣼⣧⡴⠂⣿⣿⣷⣶⣶⣶⣤⣤⣽⡵⠶⠌⡏⠀⣀⣀
⡟⢶⣍⡳⠉⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣇⣝⣙⡿⠻⢿⣿⢻⠃⣰⣧⠤⢶⣽⣿⣷⡄
⣟⡳⠮⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⢸⣿⠿⣿⣹⡙⠿⢿⠸⣦⡟⡀⠈⣿⣿⣿⣿⣻⠉⠉⠉⠀⠀⠀⠀⠀⡀
⣯⣭⡅⠀⠀⠀⠀⠀⠀⠒⣆⠀⢠⣤⠾⠏⢼⡿⡛⣾⣿⣿⣿⣿⡟⢇⣠⣿⡟⠀⣿⢋⣀⣀⠀⠀⠀⢀⠞⠁
⡷⠶⣖⡀⠀⠀⠀⠀⠠⡷⢾⠤⠄⠸⠦⠀⠸⣆⠢⣸⢿⣿⡏⠉⢁⣆⠙⢀⡋⣰⠏⠀⠀⡩⠋⠀⠀⠀⠰⠈
⡷⠞⠫⠶⠀⠀⠀⠀⠤⡭⠠⣤⠤⠤⢤⣤⣘⣹⣼⣾⣬⣟⣀⣀⣠⣩⣇⣸⡁⢛⣂⣐⣀⣀⣀⣠⣤⣤⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠐
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢾⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⡆⠚
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⣀⠠⠤⡀⠀⠀⡠⠒⠒⠈
⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠰⠁⠀⠀⠘⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠸⠀⠀⠀⠀⠁⠀⢀⠊⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠐⠃⠉⠍⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⠀⠀⠀⡠⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠁⠀⢀⠌⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠓⠤⢔⠡⢀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⢂⣀⠀⠉⠈⣁⣴⣀⣀⣀⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠣⠀⢤⠠⠤⠀⠠⠀⠀⠀⠠⠠⠸⠦⠤⠤⠤⠤⠥⠤⠦⠤⠬⠥⠤⠤⢤⠥⡵⠬⠽⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⠀⠀⠚⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠐⠀⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠐⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠜⠁⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⡆⡆⠂⠖⠆⠀⠀⠂⠠⢠⠄⢴⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠇⠀⠀⠠⢠⣌⡆⣧⣄⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⠀⠁⠞⡅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡆⠀⠀⡂⠀⠀⠀⠀⢧⡀⢰⣴⠀⠀⠈⠰⠠⡀⠀⠀⠀⠀⡀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢤⡠⢤⡴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣾⣟⠷⣶⣤⡤⠽⠛⣛⣳⡝⢯⡽⢆⠘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡏⣿⠃⣾⣷⡌⢷⠒⠋⣒⣛⣋⣈⣉⣙⣦⠱⣟⢣⠈⣶⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⣿⡇⢹⠀⣧⣶⠟⡌⢿⣿⣿⣍⣈⣹⠟⣠⣾⡷⡘⠿⠧⠀⣬⣝⠿⣆
⠀⠀⠀⠀⠀⠀⡀⡼⢋⣠⣴⣷⠠⠀⡿⠗⠚⢻⡈⢿⣋⣉⣉⢉⣉⣉⣛⣒⣳⡘⣟⣧⠐⡿⠽⣾⡄
⠀⠀⠀⠀⠀⢸⡇⢸⡿⠒⢉⣩⣷⠐⣷⣏⡿⢩⣧⠘⢯⡟⢀⣾⡿⢃⣼⣿⣌⣧⠘⣿⡄⢹⣷⣖⣿⡀⣆
⠀⠀⠀⠀⠀⢘⣃⠀⣶⣯⡩⢋⣼⠠⣟⡌⠳⣿⣍⣧⠘⣧⠚⠛⣂⣉⣁⣨⢡⣬⡇⢳⢧⠀⡇⠀⠚⣇⢸⣆
⠀⠀⠀⠀⠁⢸⡿⡀⢸⡿⠤⢾⣿⡆⠎⣷⠖⢋⣠⣼⠆⠸⣏⣉⣭⠤⠖⠐⠉⠀⣩⠀⣮⡆⠻⣶⠲⣿⠸⢿⡄
⠀⠀⠀⠀⣷⠀⣯⢷⠀⢫⣶⠀⣾⣇⢰⠰⣆⡩⠤⣴⠒⡄⠹⣁⣠⣤⣶⣞⣟⡛⢻⡆⢙⣇⠸⡼⠛⡋⣸⠟⣇
⠀⠀⠀⠀⠹⣆⢸⡟⠦⡈⢿⠞⣀⣼⠸⡀⢻⡽⠶⢤⣈⠹⠀⢻⣤⣀⡉⠙⣚⡿⠿⢷⠈⣭⠀⡷⠞⢋⣵⠎⡷
⠀⠀⠀⠐⣄⠙⣆⠹⣦⡿⡈⠻⣍⠙⣧⢳⡈⣷⠲⠟⢿⣟⣧⠈⣶⡈⠙⠻⣤⡶⠶⠛⡀⣹⠀⣶⡞⣹⠹⢡⠇
⠀⠀⠀⠀⠹⡷⣌⢣⡙⢾⡉⣦⠘⢷⣾⡆⢃⠸⣦⣤⣀⣀⣘⡄⠈⠿⠦⣀⠀⠙⠒⢾⡇⢹⡄⠟⢰⢿⠇⡜
⠀⠀⠀⠘⢦⣙⣏⡷⢿⣦⡙⢏⣓⣆⠙⠍⡄⢆⠙⣍⣉⣉⣁⣽⡄⠐⣶⣭⣝⣲⡤⠀⣹⢸⡇⣰⣠⣼⠊⢀
⠀⠀⠀⠀⠀⠉⠉⠓⠺⠿⣿⣦⣍⢻⣳⣄⠘⢆⣃⠈⢧⣥⣁⣠⣹⡀⠘⡟⠁⢀⣀⣤⢿⣠⡧⠿⠛⣡⡴⠇
⠀⠀⠀⠀⠈⢛⠶⣆⠤⣄⣠⣀⣀⠀⠁⠉⠉⠉⠉⠁⠚⠙⠋⠓⠒⠛⠒⠛⠋⠋⠉⠁⢉⣠⡄⠶⣫⡏⢙
⠀⠀⠀⠀⠀⠀⢩⡿⣶⣌⠳⢶⣾⣿⠿⢻⣯⣄⣛⡷⡂⠓⠶⣗⠛⢻⣷⠘⢿⢙⣿⡿⢟⠈⡇⡾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣙⢛⠳⢤⠙⠻⣟⣉⣉⣁⡬⢿⡌⢷⡦⣌⠛⠿⢋⣦⡈⢻⣉⣤⡾⢠⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠶⣌⡉⠛⠻⠦⢬⣿⡌⢻⠓⠿⠓⢂⠠⢌⢉⣹⣟⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢯⣬⣿⣶⠦⠠⢠⣽⣆⠙⢦⣶⣮⣉⡻⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠉⠉⠛⠂⠙⠛⠋⠁

⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⢠⣤⣛⢿⣾⠗⠀⠐⢿⡷⠀⡠⣽⠀⠀⣶⣦⣾⠆⠀⠀⠀⠈⠻⠃⠀⠀⠀⠰⠀⠘⠁⠀⠀⠀⢎⠀⠛⠛⠃
⠀⠀⠙⠛⠉⣿⣿⡇⠀⢀⣴⡺⠝⠁⠀⣠⣼⢇⣵⣿⣿⡆⠀⠀⢠⣶⣶⣦⡀⠀⠀⠖⡀⡀⠀⠘⠀⠣⠀⣠⡄
⠀⢀⣀⡀⠀⠈⠉⠑⠀⠀⠀⠰⡄⠀⠀⠀⠠⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⡿⠃⠀⠀⠈⠀⠓⠀⣠⣵⣦⠸⠿⠋
⠀⠘⢿⠇⠀⠀⠀⠰⡄⣄⢤⠢⠘⠂⠀⠀⠀⠈⠉⠻⠿⠟⠀⠀⠀⠀⠈⠀⢠⠞⡉⢧⢦⠀⠀⠘⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢤⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⣾⣃⣀⣄⠀⠀⠀⠀⠀⡀
⠀⠀⣵⣾⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣰⣇⠀⣤⠀⠀⠀⠀⣏⠤⢹⠂⠀⠀⠀⠻⢿
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⡄⢰⣶⠀⠀⠀⠀⣠⣤⡄⣿⡈⣿⣏⡀⣿⣤⠀⡀⣀⠈⠛⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠈⠛⢿⣿⠟⠁⠀⠀⠀⢸⣷⣼⣿⡄⣾⠿⠟⣿⠿⠇⠛⠋⠙⠋⠐⠛⠛⠘⠗⣿⠁⠠⣤⣤⣤⣶⡾⠟⠃
⠀⢀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⠇⠙⠃⢉⡉⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⣴⣴⡟
⠀⣼⢅⣤⢴⠆⠿⠃⠀⠀⢠⣶⠀⢀⡀⢠⡟⠉⣀⣄⠀⠿⣅⢹⣿⠁⣴⣠⡆⢨⡅⢀⣤⡄⠀⠀⠀⠀⠐⠄⠂
⠀⠘⠀⠉⠀⠀⢰⣶⡦⠀⢸⣿⠀⢨⡅⣻⡟⠸⣿⣛⠀⠒⠋⠘⠟⠃⠉⢹⡇⣿⠃⢿⣯⠁⠀⠀⠐⠈⠀⠀⣀
⠀⠀⠠⣄⠀⠀⠀⠉⠁⠀⢸⣿⡶⢸⠇⠛⠁⠀⠀⠀⠀⢀⡠⠄⠀⠸⠷⠟⠀⠛⠛
⠀⢀⣼⣿⠆⠔⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠻⠏⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣿⣿⣿⠀⠈⠢⠀⡥⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⢀⣤⣀⡠⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣜⣷⣦
⠀⠀⠉⠁⠀⠀⠀⠀⢀⡄⠀⢶⣀⣸⡷⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⣀⠠⢄⠀⠀⠀⢈⣿⣹⢯⣆⡀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠆⠀⠀⠀⠀⡳⠿⣤⣀⠀⠈⠉⠉⠁
⠀⠀⠀⠐⡀⠀⡈⠀⠀⣀⣨⠿⠿⠛⠋⢉⡦⣀⡀⠀⠀⠤⠞⠙⠀⠀⠀⢺⣧⠈⠀⠀⠀⠋⣜⢡⠄
⠀⠀⠀⡀⠀⠀⠀⣠⣾⣿⡿⠀⣀⣀⣤⡅⠀⢉⢳⡿⣤⡄⠀⠀⠀⣴⣿⣿⣶⣶⣦⠠⠆⠀⠀⠈⠀⠀⠠⠂
⠀⠐⠿⢿⠀⠀⣴⣿⠟⠁⠀⠸⣿⣿⣿⡯⠀⠈⠉⠀⠈⣠⣤⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡁⠀⠀⢰⣷⡆⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢠⠋⠀⠀⡠⠖⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⢷⡖⠉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣼⠇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⢠⣶⣶⡉⠀⠉⠉⠉⣠⣤⣇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠉⠀⠀⠈⠉⠁⠰⣏⠀⢈⡧⢀⣀⣤⡰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴⡤⠊⠙⡛⠛⠁⠀⢈⠛⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢆⡀⢀⣦⣄⡀⠀⠀⣠⣌⠁⠀⢀⠁⠀⠀⠀⠿⢇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠈⣷⠒⠂⡻⠟⠀⣰⣾⣄⠀⢀⣀⣀⣀⡿⠛⠻⣆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣾⣿⣷⠄⠒⠂⠉⣉⠈⠳⢦⡾⠋⠀⠔⠀⠀⠀⠙⠛⠳⣄⣀⣀⡀⠈⠷⣤⡴⠋⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠘⠿⠣⣀⢀⣤⣄⠊⠀⢀⣀⠀⠀⠀⠀⣾⠋⠉⢻⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡷⠂⠙⠛⠀⠀⠀⢀⡻⣦⣤⣾⡃⠰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠓⠛⠁⠢⢄⠀⣀⢀⡴⠋⠀⠀⡇⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢾⣿⠀⠀⠀⠑⢄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀
⢀⣨⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⣶
⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣽⣿⣥⡀⠀⠀⡀⣸⢿
⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣆⠸⡟⢹⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣿⣿⣿⣿⣿⢾⡷⢾⣸⣤⣾⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣸⣿⠀⢹⡇⢀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⣾⣧⠋⠉⠉⠛
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣤⡤⠼⠿⠿⠿⠃
⠉⢻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⣿⣿⣻⣿⣿⣿⠋
⠀⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣼⣿⣇⣿⣿⣿⣿⡀
⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢁⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⢀⡤⣖⠒⠒⠒⠂⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⢀⡔⠫⢍⣢⣙⠦⣹⢦⠀⠑⣝⢦⡀⠀⠀⣀⡠⠤⠐⢤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠟⡙⣿⢉⣭⣟⡻⠲⢤⡀
⠀⠀⠀⢀⠜⠒⠈⣙⢿⣛⢶⣞⢯⣳⣳⣽⣿⡷⠀⠛⡂⠈⠓⢦⢢⣳⣢⢯⢳⣄⠀⠀⠀⠀⠀⣀⡔⠛⢹⢻⢸⢣⢧⢹⢸⣿⠋⠻⣦⣙⣦⡀
⠀⠀⠀⠘⠕⢒⣃⣲⣤⣬⣲⣼⣿⣿⠿⣿⣥⣌⣐⣊⠺⢴⣕⣆⢫⢷⣹⢸⠀⢼⢷⡄⠀⠀⣰⢟⡀⢄⣿⡇⡎⢟⠎⡮⣿⣖⡭⣔⡦⣝⣿⣷⡀
⠀⠀⠀⠂⢈⠴⢻⠿⢻⠋⢩⣹⣯⠀⠰⣤⠪⢙⡿⣯⣽⣦⣹⣾⣮⣸⣿⣿⢜⡾⣭⢿⠀⠀⣿⣯⣿⣿⣿⡟⢣⡞⢠⠃⣿⡲⠬⣕⡫⢺⢽⣿⡇
⠀⠀⠐⣦⢋⡄⣼⡆⣏⢮⡸⡈⢮⡫⣦⣖⣭⣲⣽⣿⣿⣷⣿⣿⣿⡿⠿⠿⠥⠖⠽⢻⡆⠀⣿⢿⡟⣹⢽⡇⢿⠞⢸⣷⢻⣄⣶⡿⣿⢶⣽⣿⣿
⠀⠀⠀⠹⣩⢽⣻⡧⣌⠗⢽⣝⡲⢍⡢⣿⢾⣾⣿⣿⣿⢿⢋⠙⢍⡛⠓⠲⣶⢻⣴⣿⠀⠀⢻⣧⢟⡿⣿⣿⣾⡇⠘⢻⢸⣿⢋⠔⢡⢮⣿⣿⠇
⠀⠀⠀⠀⠙⠿⣅⢷⡉⢷⡾⣦⣭⣷⣿⠶⠻⣿⢿⢋⢿⢸⡯⣿⣿⠲⢌⢳⣗⣿⣾⠃⠀⠀⠈⢿⣿⣹⣵⣿⡘⣧⠄⣿⡾⣿⣿⡖⣩⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠘⠾⣾⣻⢸⣿⣾⣭⡛⣿⣿⣿⣿⠋⠀⠀⠀⠀⡈⠳⣿⣶⣿⣷⣟⢼⣷⣧⢛⠉⢞⡛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣸⠿⠿⠿⢿⣿⠽⠋⠀⠀⠀⠠⠂⠉⠀⠀⠀⠙⠻⠿⣿⣼⣿⣷⡾⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠳⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠐⠒⠂⠀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠯⢋⣡⡾⢟⠍⡨⢍⣷⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⣒⢖⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⢊⢽⡿⠃⣠⢞⣥⠞⠑⣩⣞⣡⣽⡀⡌⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣵⣲⠿⢿⣿⣿⠏⣨⣾⠟⣦⡀
⠀⠀⠀⠀⠀⢠⠎⠁⣸⠏⢠⠞⣕⠴⠃⣠⢟⣳⠦⣍⡻⢧⡹⠘⡌⢧⡀⠀⠀⠀⠀⠀⢠⣾⣿⡭⠴⣂⣤⣭⠷⠾⠷⡿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣥⣤⣶⠃⣠⢋⢞⡕⠀⡼⠭⣴⣦⣉⠓⠯⣲⢝⢦⣁⢸⣧⠀⠀⠀⠀⠀⣞⣯⡹⣞⡯⣗⣚⣿⣭⣭⠷⡿⣭⣹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⣙⣿⢃⣰⠃⡂⡜⢀⣾⡷⣶⣤⣄⢯⠷⡶⣾⣕⢥⡝⢦⣿⡆⠀⠀⠀⢰⣿⣷⣝⡶⣝⡿⣿⣽⣱⣶⣷⡾⠾⣿⣷⣽⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣧⡏⣼⡏⡜⣸⠁⣼⡿⠞⢋⠔⣩⡞⢯⣉⢻⢭⡳⢧⣌⣿⡇⠀⠀⠀⢸⣿⢿⣿⣿⣾⣽⡲⠯⠟⠳⠿⣯⢭⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡇⣿⢸⢣⡇⢠⣟⠭⢂⡏⣾⢋⡞⣡⣿⢳⣝⢽⡻⣝⢿⡇⠀⠀⠀⠘⣟⢷⣌⡉⣛⡛⢣⣤⡦⠤⠨⠽⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⡇⡏⣯⢿⢳⢸⣧⣞⣽⢸⣵⣿⡼⢿⢇⢸⣿⢳⣽⣾⡟⠀⠀⠀⠀⠀⠙⣿⣷⣧⣥⣍⣛⣛⣛⣋⣉⣛⣫⣽⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠹⡇⡇⠀⢸⢸⣸⣷⡿⣻⡞⢟⡝⣄⡿⡸⢸⡎⣼⣻⡟⠁⠀⠀⠀⠀⠀⠀⠙⢯⣿⣷⣿⣿⣿⣟⣶⣿⣾⣿⣿⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢧⣿⣼⣸⡿⣿⣾⣯⣳⡹⣿⡿⢡⢳⠏⣴⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣿⣿⣿⣿⠎⣻⣌⠳⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⡩⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⢉⠴⠆⠖⠂⡀⠈⣁⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⣀⣈⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢃⡄⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣀⠄
⠀⠀⠀⠓⢜⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⡏
⠀⠀⠀⣀⡐⣶⣷⡂⠀⠀⣴⠾⢤⣀⣀⠈⠀⠒⠀⢀⣠⠄⠊⢸⣿⠋⡹⢣⡀
⠀⠀⠀⠈⠁⠀⠠⢤⡖⢀⣻⢲⡶⢻⣿⣟⣦⣤⣞⣉⠤⠀⠘⣼⠃⣠⠇⣸⡏⢉⣳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠉⢲⣼⣾⣥⣬⣇⣈⣚⣉⢣⠀⠀⠉⢡⢇⣼⠃⠀⢻⣇⢈⡟⠻⡗⠛
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡼⣻⡟⡏⡟⠙⠟⡿⢻⠿⡿⣿⣷⣾⣿⣾⣿⣼⣾⣿⣾⣷⣤⣿⠰⠦
⠀⠀⠀⠴⣶⣶⣦⣬⣯⣀⣐⣧⣶⣿⣿⣷⣷⣮⣤⣄⠧⣇⡅⡥⡁⡻⡿⠟⢟⣟⣷⠛⢿⣶⠄⠤
⠀⠀⠀⠀⠀⠀⠨⣭⣍⣀⣀⣀⣈⡉⠉⠙⠛⠻⠷⠿⢾⣿⣛⣽⢓⠛⢧⠀⢘⠁⠈⡄⠈⠉⠁
⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣶⣶⣶⣶⣴⠦⠶⠶⠦⠤⠤⠤⢤⡭⣭⣝⣛⣳⡆⢀⣀⠀⠄
⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣸⣽⡵⠮⠶⠷⠷⠶⠦⣄⠀⠀⠀⠀⢀⡴⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣝⢮⡻⣷⣒⣒⡩⠭⠭⠭⠭⠭⣒⣛⠦⣠⠖⠉⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⡙⢮⡓⢝⠢⣝⠶⣖⣒⣒⣒⣒⣒⢒⣤⠾⢵⡀⠀⢀⣠⠴⠿⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠽⣛⣦⣙⠦⣙⠢⣝⣦⡿⣻⣻⣏⣷⢏⢫⣙⣯⣷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢴⡻⣯⡌⡸⡳⢌⣱⠞⣩⢞⡵⢃⢽⡜⡜⡆⠇⢷⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢣⠱⡜⢧⣇⡷⢞⡡⢊⡵⢋⠔⢁⣴⣳⢱⢹⢸⣸⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣳⡼⢞⣩⠖⣩⠖⣡⠖⣡⣴⣿⡾⣿⡏⡞⡖⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠵⢚⣉⣴⣿⠵⣋⡥⠾⠯⠿⣒⣫⢭⣿⡇⠇⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⣹⢝⢛⣩⠵⠿⠭⡭⢭⠭⠭⢖⣚⢭⣿⠱⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠊⠁⠀⠉⠋⢛⣟⡛⠿⠛⠿⡛⠿⡵⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⢀⣠⠤⠖⠒⠒⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⢤⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⣀⢢⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠿⠈⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠆⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⢠⠄⠒⠒⢄⡀⠀⣀⠔⠉⠉⠁
⠀⠀⠀⠴⠀⠐⠀⢄⡀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠌⠢⠀⠀⠀⠀⠀⠀⠀⡔⠲⠁⠋⣝⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢢⡀⢀⡠⠊⠀⠀⡠⠃⠁⠀⠀⢠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠋⠢⠤⢔⠣⢀⡀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠋⠦⡀⠀⠀⠑⠒⠋⢩⣵⠄⠀⠀⠀⢀⡀⢀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠸
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠰⡗⠒⠒⠛⠓⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⢒
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠉⠉⢉⣍⣉⣉⢉⡉⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠏⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠶⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⢐⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⢀⣀⣘⣓⢒⡒⣒⣒⣒⡒⡒⢒⣒⠄⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⠀⠨⠉⠁⠀⠀⠈⡉⠉⢨⠇⢩⡇⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⢰⠀⠀⠀⠀⠀⡇⢰⡦⠦⢸⣦⡤⠀⠤⠠⡎⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠈⠀⠀⠀⠀⠀⢀⡼⠀⠀⠰⠃⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠘⠀⠀⠀⠀⢸⡀⠠⢀⡄⣌⠀⠀⠀⠃⣤⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠸⠂⠀⠀⠀⠀⠈⠞⠑⠸⠶⠧⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠸⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣽⣧⣧⣧⣧⣧⣿⣼⣾⣼⣒⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡶⠓⠀⠀⠀⠒⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠃⠀⢸⣿⠄⠀⠀⠀⠀⠉⠉⣪⣿⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣄⣀⠀⠀⡤⠀⣠⡄⠀⠀⣸⠁⠀⢀⣤⢴⣖⡿⠋⢹⣿⣽⣻⣿⣆⠀⠀⠀⠀⠀⣾⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⣷⣜⡿⣿⡿⠂⠀⢾⣽⠃⠀⠀⣿⠶⡿⢟⠀⣁⠲⣅⠀⣰⣿⣿⣿⠟⣋⣴⠀⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡔⠀⠈⢲⢿⣝⣿⠃⠀⠀⠊⣿⠁⠀⠀⢹⠀⠀⡀⠃⠬⡱⣿⣿⣿⠿⣋⣴⠞⣉⡴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢱⠄⠙⠒⣶⡆⣰⡷⣠⢾⣿⡟⠀⠀⠀⠾⢿⡀⣆⠀⡘⡀⠘⠁⢀⣶⣷⠞⣋⣵⠿⣋⣵⣾⣿⣷⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡈⠀⠀⠀⣻⣴⣟⣺⣷⣌⣻⣧⠀⢀⠂⠀⠄⡀⢿⡀⠰⢹⣼⣷⡟⣫⣴⠿⣫⣴⣾⣿⣿⣿⣿⣿⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢋⣀⣘⣉⣥⣶⣿⣿⠿⠛⠛⢿⣄⣾⣌⠂⠂⣧⣷⣥⠀⢷⣻⣷⣿⣯⣷⡿⣿⣷⣿⣿⣿⣿⣿⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡙⣿⣿⣿⣿⣿⠛⠀⢀⠀⠀⠠⠂⠙⢿⣿⣷⣾⣿⣿⣇⠘⢿⣿⣿⣿⣯⣷⣫⣿⣿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣯⣿⣷⣿⠁⠣⠀⢡⣀⣠⣶⣄⣀⡠⠟⣫⡴⢛⣽⣿⣿⣾⠿⠛⢻⡿⠛⣛⠛⢻⣿⣿⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠠⣄⡲⠄⢉⣛⣠⠿⣋⡴⢚⣩⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⡏⣘⣿⣿⣯⡹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⢷⡤⢋⣥⠞⣩⣶⠛⢿⣿⣿⣿⢑⠰⠀⢂⣤⡄⢀⡀⡼⢿⣿⣟⣟⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠾⢋⠴⡯⠞⠃⠀⠀⠈⠙⠛⠋⠁⠐⠁⢇⠀⠈⠘⢡⢏⣟⣻⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣶⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠾⠄⣼⣿⣿⢏⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⢿⣛⣿⡶⢟⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠛⠛⠋⠉⠁⠙⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡞⠠⡀⠀⠀⠀⡠⠂⡀
⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠸⣹⣮⠗⡲⠄⠀⡁⢀⠀⠐
⠀⠀⠀⠀⠀⠘⡀⠈⡁⠠⡀⢀⡤⠛⠻⣌⠠⠆⠀⢧⠀⢀⠂⠀⠄⢀⠀⣀⣤⠀⣴⠀⠀⠀⠀⠠⠒⠄
⠀⠀⠀⠀⣀⣀⠡⢤⣑⡄⠀⣾⠀⠀⠀⣼⠀⠂⠀⢰⠄⢨⣀⡴⠤⠚⡋⢰⣿⠀⡿⢄⣀⣀⠚⠁⠀⡀⠔⡆
⠀⠀⣀⠘⣱⠠⢻⡲⠬⢙⡆⣿⠀⠀⣰⣧⣶⢶⡄⠈⡉⠀⠠⠀⢀⠁⡰⢫⡿⠺⠓⠞⢻⡇⠑⢹⡔⢠⠆⡇
⠀⠀⡃⢈⠤⣵⡚⢻⣷⣆⢹⡇⣀⣼⡿⢱⢟⣾⢅⡀⠀⣀⣠⡾⢋⣿⣵⠟⣷⣄⠀⢀⣸⣧⢸⣨⣷⡦⠼⢄
⠀⠀⡰⡘⠄⣞⢯⡓⠉⢻⢸⣿⢩⢿⡇⣹⣿⡟⣷⠶⡾⡛⣿⠁⡾⢸⡏⢼⢯⣭⣻⣯⣾⡵⣽⣟⡅⣒⢊⠅⠤⠃
⠀⠀⠈⠉⠀⢁⠑⢅⠀⢨⣿⣿⣶⣿⣧⠸⣿⠃⣏⣘⣤⣷⣵⣤⣷⣸⣧⣏⣌⠘⣩⡟⠁⠀⠀⠈⢳⠀⢀⢀
⠀⠀⢰⢦⣾⠧⣙⢦⣱⡏⣴⣾⣿⣿⣿⠦⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣤⡛⢧⡀⠀⠀⢠⡿⢀⣴⣾⡿⠏⡄
⠀⠀⠀⢇⢿⣵⣾⣟⣿⣇⠟⠱⣺⠟⡱⣳⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⡦⠖⣻⠋⠉⠉⢻⡈⠉
⠀⠀⠀⠈⣎⢯⡻⣮⢷⠹⣧⣙⠋⣴⣿⣿⣿⠿⣇⡿⢋⣽⣿⣿⡿⣿⣿⣿⣿⡿⣿⣧⣶⡝⣶⠃⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢦⡉⠑⢿⡄⣸⡏⣸⣿⠛⣿⠏⣰⣿⣷⣝⣛⣛⡛⢀⣩⣥⣼⡏⣼⣿⣿⣿⡏⣿⢀⣤⠞⠋
⠀⢀⠤⠄⡠⠄⠵⠦⣄⣹⣿⢀⣿⡿⢹⡟⣰⡿⠋⣹⢋⣿⣿⡁⣼⡟⡟⢹⠀⣹⠟⢷⣨⡾⠟⠋⠀⢀⢀⣀⡤
⠀⠈⠢⢄⠀⠾⢶⣤⣠⣽⣿⣌⣯⣶⣬⠁⣶⢳⣬⣿⣦⡌⢻⣇⣫⣄⡇⢘⡀⢏⠰⡿⣹⠇⠀⠀⠀⢒⣧⠇
⠀⠀⠀⠀⠀⠠⠄⠙⣦⡾⠷⠉⠉⣿⣿⢸⣿⣸⠅⠙⠿⠟⢛⣹⣿⣿⡇⢸⣟⣛⣗⣞⣁⣀⡠⠤⠞⠭⠀⠂⡀
⠀⠀⠀⠀⠀⠀⡀⠺⣏⠃⠀⢀⣴⠟⣿⣘⣷⡿⡆⠀⠀⠹⣿⣿⣿⣿⢇⡾⠉⠣⡉⠣⣤⠌⠑⠆⠤⡀⠀⠀⠀⠂
⠀⠀⠀⣀⠐⠀⠀⣴⠛⠦⠠⢾⡇⠀⠈⠉⡹⣹⣳⡀⠀⠀⢉⣿⣿⡿⠟⡇⡀⣜⢟⡎⢈⢓⡦⠘⠆⠀⠉⠈⠉⠉
⠀⠀⠰⠁⠀⡀⠠⠀⣲⠄⣥⡿⡟⠒⠒⠊⠛⣷⠷⢟⡦⣀⣬⢞⠇⠀⣐⠹⣿⣮⢽⠧⠜⠀⡙⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠋⢸⠐⢀⠀⠀⠀⠀⠓⠈⠁⠀⠈⡓⠻⢤⠎⠩⣀⠰⢹⡄⠕⢫⠘⠛⢄⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠀⠀⠀⠈⠙⠒⠋⠙⠁⠀⠀⠀⠉⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⢒⠒⠛⢉⣍⠉⣩⡉⠙⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⠀⠉⠀⠀⢸⠈⠀⢧⠀⠀⡇⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⡀⠀⠀⠘⠃⠀⠘⠀⠀⣠⡿⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠙⠒⠬⠤⠤⠤⠤⠤⠤⠤⠴⠖⠚⠋⠁⣤⡶⠀⠉⠉⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⡤⠴⠞⠋⠁⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢣⣼⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⡀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣟⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣥⣤⡤⠤⠤⠴⠦⠐⣒⣒⣒⣢⣬⠩⠉⠛⠉⢿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⠦⠀⡿⠤⠤⣤⠶⠶⠾⠿⠭⢽⠟⠒⠒⠒⠒⡾⠋⠉⠉⠉⠉⠉⠙⢦⡀⠀⠒⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣟⡰⠊⢀⡴⠞⠛⠳⣄⠈⢷⡀⠈⠀⡸⠁⣠⠖⠃⣀⠢⣄⠀⠀⠱⣄⠀⢂⡴⢻⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⢀⡴⠋⠒⠀⢀⡀⠘⡀⠀⠙⠒⠚⠁⠀⠁⠠⠄⠀⠀⠈⢢⠀⠀⠈⠉⠉⢀⠼⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇⡼⠁⠀⠀⠀⠈⠀⠀⠡⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⣀⢳⡀⠀⠀⡰⢁⠀⡇⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣡⣇⣀⣀⠁⠀⠀⠀⠀⠀⠙⢦⣤⣎⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠼⠶⣆⠀⠐⠒⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⢸⡤⣬⣉⣉⣩⡭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠶⠶⠒⠒⠒⠚⠛⠋⠉⠉⢩⡍⠉⠉⢹⡀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢰⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠠⢴⣿⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⣤⣄⣀⣀⡇⠀⠀⠀⠀⠀⠀⠈⠶⠦⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡄⡇⡉⡝⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⣠⢾⢶⠛⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠞⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣋⡟⠚⠎⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠤⣤⣀⣀⣀⣀⣀⡀⠀⠀⢀⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣭⣝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣿⣿⣿⣯⣽⢿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣯⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣄⣀⣿⣿⣿⣿⣿⢿⣿⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣻⣿⣾⣿⣾⣿⣷⣿⠞⠽⣿⣿⣿⣯⣷⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣳⣿⣫⠞⠋⠁⠀⠀⠈⠙⢯⣻⣿⣿⢻⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⡿⢷⣷⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⣿⣻⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣟⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢻⡝⣷⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣷⣽⣾⣿⣯⣾⣿⡀⡒⢄⠀⠀⠀⠀⠔⡒⠒⠦⡀⢻⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⢿⣿⣿⣷⡈⠈⠀⠀⠀⠀⠈⢀⣀⠀⠀⢸⡯⣿⡿⠿⠾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀⠹⡿⠛⠀⠀⠀⠀⠀⡀⢿⠟⠁⣠⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦⡤⠄⡀⠀⠀⠀⠀⢀⠀⢤⣾⣧⣤⠖⠂⠀⢀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⣿⣿⣿⡍⣿⣳⠙⠻⠿⠦⠊⠀⠤⠀⠄⠀⠀⠤⠼⠿⠉⣤⣤⣶⣿⣅⡀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣿⣿⣷⣿⣿⣷⣿⡿⣇⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢼⡁⢠⠒⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣟⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣯⣷⣤⣷⣤⠆⠀⠀⣠⠴⠊⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣽⣿⣿⣿⣼⣿⣿⣄⠀⠀⠀⠉⠀⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣟⢻⣯⢿⣿⡟⠀⢀⡾⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣾⣿⣷⣿⣿⣫⣿⣿⣿⣿⣿⠲⢄⣀⠀⠀⠀⢀⣠⠞⢁⡷⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⢠⡟⠀⠀⠀⡰⠁⡖⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⢿⣿⣿⣿⣻⣷⡯⢿⠀⠀⠈⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣟⡿⢡⡟⠀⠀⠀⡐⠀⠒⠇⠀⠺⠂⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣿⣏⡿⢷⣧⣾⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢻⣷⣷⣿⣷⣻⡟⠋⠀⠞⠀⠀⠀⠰⢁⢠⠀⠀⢤⠀⠀⣈⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠉⠋⠛⠛⠚⠋⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠃⢀⠀⢀⡦⠀⠀⢸⠁⡰
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠸⠀⢀⠀⠀⠀⠀⠀⠀⡌⠀⠇⢰⣶⣶⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⡆⠛⠀⣀⠁⠀⠹⠉⢰⠁
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠘⡄⠀⠀⠀⠀⡰⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠇⠀⠈⠇⠀⢰⠚⢁⠇
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠈⠢⣀⣀⠴⠁⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠰⡏⠀⡄⠚⠀⡘
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⢀⢇⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡥⠀⢠⠁
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣤⣘⣎⣤⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⠛⠉


⣿⠛⠛⣿⢿⢿⢿⠿⠷⣿⡿⠿⠿⠿⢿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⣿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠖⠊⢹⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣶⣶⣤⡀⠀⠸⢀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⣿⣿⣶⣶⣮⣄⣲⣲⣷⣴⣶⣶⣾⣿⣿⣿⣗⣦⣸⣿⣿⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⣿⣿⣿⣿⣿⣿⢷⣟⣿⣟⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⡇⣿⢿⣿⠛⠛⠛⠷⠿⠿⠿⠿⠛⠛⠻⣿⣿⡿⣿⣿⣿⣿⢻⢇⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠏⡀⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⡿⣧⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡄⢤⣤⡤⢸⢣⣭⠀⠀⠉⠐⠒⠐⠐⠐⠚⠂⠀⣿⣽⣿⣛⣷⣿⣿⣿⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣍⣉⣍⣉⣉⣉⣁⣀⣀⣠⣁⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠭⠯⠽⠽⠿⣿⢿⣿⣿⣿⣿⣿⣻⢿⡿⠿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⣾⣧⣤⣄
⠀⠀⠀⢾⣿⣿⣿⠯⡹⢹⡦⠄⠃
⠀⠀⠀⠀⠀⠀⢹⣄⠁⡀
⠀⠀⠀⠀⡀⠄⣟⠫⣥⡏⢲⢀⠀⠞⡓⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣠⡀
⠀⠀⠀⡞⠀⡀⠉⠑⢿⠋⡉⠀⠑⡞⠀⠀⠁⠀⠀⠠⠄⠀⠀⠔⠡⠦⠙⢿⣿⣿⣷⡄
⠀⠀⠀⣧⠀⠹⣦⢶⣌⣢⢵⣂⠘⠛⣰⣄⣠⡴⠀⠀⠀⠀⠀⠠⣦⢰⡆⢰⣀⣶⣿⣯⡀
⠀⠀⠀⢹⣷⣫⡽⣟⠛⠋⠻⠂⠰⢾⣿⢿⠟⠃⠀⠀⠀⠀⠀⠒⠒⠒⠂⠚⠛⠛⠛⠛⠛⠲
⠀⠀⠀⠀⣿⣿⣤⡈⠑⠤⣤⠶⠤⣤⢀⣸⠖
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣤⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣾⢿⣻⠄⠀⡠⠀⢸
⠀⠀⠀⠀⢸⣿⡯⠀⠀⢰⡇⠀⡜
⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⠃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣃⡛⡛⢛⣛⢃⡘⣛⠛⠛⢛⡘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⢠⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⠈⣿⣿⣿⣿⣷⣄⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⢋⣴⣷⣌⠻⠿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢛⣡⣴⣾⣿⡿⢸⣿⣿⣶⣦⣙⠻⣿⣿⡌⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⣰⣿⠟⣋⣥⣶⣶⢸⣿⣿⣿⣿⣿⣷⡘⢿⣷⢸⣿⣿⣿⣿⡿⠟⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣸⣿⣿⣾⣿⡛⠛⠿⢸⣿⣿⣿⣿⣿⣿⣿⡈⢿⢸⣿⣿⡟⣡⣶⣿⣿⣿⢡⣷⣦⡙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⠿⣿⣿⣦⠸⣿⡟⢰⣿⣿⣿⠟⡁⢾⣿⣿⣷⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⡆⣨⣭⣥⣾⣿⢸⣿⣦⣭⣭⡀⣦⣽⣿⣿⢠⣿⡇⠿⢛⣋⣴⣾⣿⣶⣌⡛⠻⠇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣿⣿⡟⣻⡟⣿⢸⡟⣿⣿⣿⣿⣿⣿⣿⢃⣼⣿⡇⢾⣿⠻⠿⣿⣿⡿⠿⢻⣿⠂⣿⠟⡁⡈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢻⡏⢰⣿⡁⠿⠘⠇⣸⣿⣿⣿⣿⠿⠃⣾⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⢡⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣿⣿⣿⣷⢰⣿⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⣷⡜⢿⣿⣯⣭⣿⣿⢏⣴⣿⣿⢃⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠻⠿⠸⠿⢻⣿⣿⡿⠃⢸⡇⣿⡿⢛⡙⢿⣿⣶⣌⣛⢛⣋⣴⣾⣿⣟⣡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⢰⣿⣿⡿⠋⠀⠀⢸⡇⢉⣴⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⣍⣛⢈⣋⣥⣶⣧⠀⠀⢈⣴⡿⢋⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢃⣴⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠄⠀⠁⢀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⢳⠀⣿⣿⡦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡀⢺⡗⠈⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⣴⣷⣶⣄⠀⠀⠀⢀⠀⠀⠀⠈⠛⠁⣴⠏⡀⠈⠋⠀⠀⢀⠙⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣛⢷⣤⡄⡰⢀⣯⣽⡈⠀⠀⠀⡤⠀⣿⣿⣿⣿⠆⠀⠀⠈⠁⢀⠀⢢⡄⠐⠃⠸⠝⠶⡗⢰⢶⡏⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠖⢻⡿⠟⠃⢀⠓⣿⣿⣿⣿⠀⠊⠀⣀⠀⠙⠛⢻⡁⡄⠀⠠⠃⢆⣉⠓⠀⠀⣠⢸⡇⢴⠀⣠⠀⢨⣀⡲⣼⠀⠀⢳
⠀⠀⠀⠀⡴⢂⣶⣿⠀⠐⣶⠿⣤⠌⠛⠋⣁⠘⢧⡀⠁⠄⠲⠀⠀⠷⢀⣠⡄⠀⠀⠈⠁⣀⡀⠀⠛⠠⣾⣽⣷⣾⣷⣍⠁⣠⣦⡀⢘⡣
⠀⠀⠀⠸⠇⣸⣿⠟⡀⢿⣿⣿⣷⠀⠀⠸⠏⠀⢈⠉⠀⣴⣤⢀⣀⡉⠉⠁⠀⡀⠀⠀⠀⠈⠻⠲⠂⣼⣿⣿⣿⣿⣿⣿⡄⠿⠿⠃⠀⠘
⠀⠀⠀⠀⠀⣿⠏⠘⠇⠸⢿⠟⠁⠰⡄⠄⢀⢤⠘⣇⠀⣴⣷⣿⣿⣿⡄⠀⡍⠀⠀⠀⠒⠀⠀⠀⢠⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⠄⢄
⠀⠀⠀⢴⠀⣥⣄⠰⣧⣀⠀⠀⠀⣎⡷⠘⠘⢂⣀⣀⣄⠹⣿⣿⣿⡿⠃⠀⠀⠀⠐⣆⠀⠀⠀⠀⢈⣄⡬⠻⣿⣿⣿⠟⠁⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠸⡆⢸⣿⣦⣝⣿⡾⠷⢀⣠⡦⢦⣰⣌⠉⢉⠠⠀⠈⠉⢤⣀⠸⠾⠺⠂⠐⢦⣀⡀⠀⠀⠺⣿⡿⠀⢀⠁⢰⠞⠉⠈⠂⠀⠀⢀⡾⠂
⠀⠀⠀⠀⠀⠀⣤⠀⣩⣿⣷⡆⢸⠁⠆⢈⣧⠏⠀⢐⡄⠀⣶⣿⡇⠀⢰⣦⣴⣾⡆⢠⡍⠁⣰⣦⡀⠀⢁⣰⣿⡇⠀⢰⣾⣿⣾⣿⡆⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠙⠿⣿⠃⠈⠛⠒⠛⠛⢲⡞⣛⡹⣧⠈⠿⠁⣠⣴⣶⣽⣏⡀⡌⢢⠘⠧⠿⠃⠀⣿⣿⣿⠇⠀⡈⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡈⠀⢶⣿⣿⠁⠀⠘⢧⣌⣡⠏⣸⢠⣾⣿⣿⣿⣿⣿⠁⢅⠀⠑⠀⣰⣶⡄⠈⠉⡁⠠⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⢦⣀⠉⠉⠐⠀⠀⠀⠀⡀⠰⠃⠈⠻⣿⣿⣿⠿⠋⢀⣨⣴⣦⠸⢿⠿⠃⠚⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠼⣯⡦⠶⠀⠀⠀⠀⣧⠸⠀⠀⠀⠀⣠⣶⣾⠀⠘⢾⣿⡟⠠⣶⣿⡞⢠⣶⣧⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢰⡎⠓⠀⠀⢶⣶⡧⠀⢈⣙⠋⠀⠀⠀⡠⠀⠤⠈⠉⠃⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄⠀⢀⠀⠘⠛⠀⠀⡀⡀⠑⠠⠓⠀⡇⠀⠀⠘⠃⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠲⢻⣾⣶⣄⠀⠰⠄⠃⢀⣤⣤⣄⡈⠓⢀⣰⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡆⠀⠠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠟⠁⢀⠀⠻⣿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠛⠇⠀⠀⠛⠃⠀⠀⠀⠉⠁⠀⠀⠰⣶⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⢾⡿⠱⢁⠀⢀⡘⠹⠟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅⣠⣶⡐⠎⠀⠁⠂⠀⠁⢈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢆⠼⠋⠀⠀⠄⠒⢀⣈⣬⣤⣛⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⠆⠂⠀⣀⣰⡖⠂⠜⠙⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⣠⣴⡏⠉⠈⠉⠀⢀⠀⠀⣐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣠⣄⡀⠀⠀⠀⠀⠈⢻⣾⠇⠉⢆⠀⠀⠀⡤⡐⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⣠⣶⣿⣶⠶⣟⡿⢿⡷⣤⠀⠀⠀⠀⠈⠙⠶⠦⣤⣤⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⣾⠝⣯⣿⠿⠁⠭⠟⠘⠧⣉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⣧⣢⠞⠁⠀⠀⠒⢈⠥⣴⣾⡍⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠈⣷⠽⠋⢀⣠⡴⣿⡁⠊⠈⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢹⣄⣶⡿⡅⠀⠀⠁⠠⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠙⠻⢄⣈⡐⢶⢶⢋⡲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠁⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢷⠀⠛⢢⡴⢅⡀⠀⠀⠀⠀⣜⠉⠀⢣⠀⠀⠀⠀⣎⠀⠀⠂⠉⠉⣢⡎⠉⠒⠒⠀⢰⠋
⠀⠀⠀⠀⠴⠊⠉⡑⠂⠀⠀⠀⠀⠈⠄⠤⢈⡀⠀⠀⠀⢿⠀⠀⠀⣧⠀⠀⠀⡎⠉⠉⠁⠀⡝⠓⠃⠄⠐⠒⢠⡞
⠀⠀⠀⠔⠉⠣⡀⢀⠤⠀⢀⠀⠀⠀⠠⣀⡤⠉⠀⠀⠀⠻⠢⠘⠓⠁⠀⢀⠔⡇⠀⠀⠀⠀⢱⠦⠤⠤⠤⠤⣾⠃
⠀⠀⠈⠀⠀⢀⡔⠁⠀⠀⢸⠀⠀⠀⠀⠜⠱⡀⢈⠀⠀⠀⠀⠀⠀⠀⠒⠁⠠⢇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣯⠇
⠀⠀⠘⡀⠀⡜⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠤⠄⢼⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠈⠂⢆⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⡀⠀⠐⡛⠂⠠⠤⠤⠐⣳⠈⠑⢆
⠀⠀⠀⠀⢀⡀⠙⠒⠀⠀⠐⠋⠓⠒⠊⣁⠀⠀⡠⠄⠠⢄⡀⠉⢶⠀⠀⡔⠉⣀⢀⡈⢢⠀⠈⠉⠁⠀⠀⠉⠁⠔⠒⠁
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡐⠡⠀⡎⠀⡆⢰⠄⢱⠀⠈⣦⠀⢄⠐⠇⠸⠁⡰⠀⠀⠄⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠐⠁⠁⠀⠣⡀⠃⠈⢁⡜⠀⠀⠀⠣⠀⠐⠠⠤⠒⠁⠀⠀⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠆⠈⢆⠀⠀⠀⡰⢀⠆⠀⠀⠀⠈⢀⣀⠁⢀⣀⣀⡔⢀⣀⣀⣤⣤⣤⣤⣄⣀⡃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡄⠈⢆⠀⡀⠁⠎⠀⠀⠀⢀⠞⠁⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢦⠁⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⢠⠤⢄⠀⠀⠀⠀⠀⡔⠉⠉⠑⡄⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠼⢶⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠣⣀⠰⠀⠀⠀⠀⠀⢇⡀⠀⣠⠇⠀⠀⠀⢸⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠈⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢘⣇⠠⠄⠒⠋⠉⠒⠢⢄⠀⠉⢉⠕⠚⠒⡄⠀⢸⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠤⠒⠃⣌⠁⠀⠰⠀⠀⠀⠀⠀⠀⠉⠉⠯⢭⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⢄⣀⣠⠇⠀⢸⣀⣸
⠀⠀⠀⠀⠐⡡⠊⠁⠠⢸⠀⠰⢀⠨⠑⢦⡀⠀⠀⢄⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⢸
⠀⠀⠀⡌⣰⠁⠀⢀⠀⡜⠀⠅⢪⠀⢀⠀⡇⠀⢀⣠⢴⢙⡞⡧⢤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⡅⢧⣀⠔⣡⠞⠀⠀⠈⠢⠬⠤⠚⠀⡰⠁⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠑⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠤⢄⣀⣀⣠⣤⣴⣤⣂⣀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠈⢻⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⢻⣿⣿⣿⡀
⠀⣠⣿⣿⣿⠟⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⢻⣿⣿⣿⣄
⣼⣿⣿⣿⣏⣀⣀⣀⣀⣼⣿⣿⣿⣿⣀⣀⣀⣀⣼⣿⣿⣿⣿⣇⣀⣀⣀⣠⣿⣿⣿⣿⣧⣤⣤⣤⣤⣽⣿⣿⣿
⠹⢿⣿⣿⣿⡿⣿⡿⠟⠙⠿⣿⠿⠋⠙⢿⣿⠿⠋⠻⢿⡿⠟⠙⠿⣿⡿⠟⠙⠿⣿⠿⠋⠻⢿⡿⢿⣿⣿⠿⠿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠛⠛⠛⠲⡖⠒⠒⠒⠲⣶⠶⠶⠶⢤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠀⢀⡏⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠒⠒⢻⡖⠒⠒⢺⡗⠒⠒⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⠀⠀⣸⠁⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠒⠚⠛⠚⠒⠛⠓⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠻⣟⠛⠛⣻⠟⠻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⢀⡼⠀⠀⢧⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠶⠶⠶⠶⠶⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠳⠒⠒⠒⠒⠒⣚⣋⡩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠲⠦⠤⠤⣤⣤⡤⠬⠿⠿⠻⣄⠀⠀⠀⢠⣴⣒⣋⣓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⣀⣠⣤⠀⠋⠉⠀⠀⠀⠩⣭⣽⠀⠀⠀⢸⣿⣀⣀⣀⡠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣿⠈⠉⢘⡦⠤⠤⡤⡤⡴⠶⠷⢾⡇⠀⣰⢃⣀⣀⣀⣀⣀⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⣉⠉⡉⠁⠀⠀⠁⠃⡁⢀⠀⠀⡇⠀⡿⠟⠀⠀⠀⠀⠀⠸⠇⡇
⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠤⠤⠤⣀⠀⢸⠀⢹⠼⢹⢹⢷⣤⡿⣝⣾⠇⢸⣇⠀⡇⠀⡇⢰⡶⣶⡆⣆⠰⣆⠀⣿
⠀⠀⠀⠀⡾⢴⣥⠄⠀⠀⣠⣴⣢⠼⠀⢸⠀⠈⠀⡈⢫⣌⢩⣥⢩⡄⢉⡀⠈⠀⡇⠀⣧⣈⣁⣉⣉⣉⣡⣥⡤⣽
⠀⠀⠀⠀⣿⣶⡖⠉⠉⠉⠀⢠⣤⣠⠀⢸⠤⠤⠤⠽⠿⠿⠯⠿⠯⠭⠭⠤⠤⢴⠇⠀⢻⠒⠒⢒⠒⠒⠒⠒⠚⢻
⠀⠀⠀⠀⡇⠀⡉⣉⣉⣉⣉⡉⠉⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⢺⠀⠀⢸⢀⢔⡵⡫⠀⠚⣉⣓⣿
⠀⠀⠀⠀⢣⠀⠓⠛⠛⠋⠉⠁⠀⢸⠀⠸⡄⠀⡆⣦⡄⠀⠀⠀⠉⠩⠭⠭⠭⢽⠀⠀⢸⠀⠈⠘⠁⠀⢈⡥⠤⢿
⠀⠀⠀⠀⠈⠑⠒⠒⠒⠀⠀⠒⠒⠋⠀⠀⠉⠓⠓⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⢀⡀⢀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⣾⢹⢸⠀⣿⠀⡼⡄⡏⣙⢸⠁⡷⣼⠀⢻⠀⡏⣷⣿⡽⢸⡆⣿⠻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠙⠜⠘⠆⠻⠀⠋⠃⠳⠞⠸⠄⠃⠛⠀⠚⠀⠣⠟⠛⠘⠘⠙⠙⠲⠞