Too sweet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣪⡥⢡⣇⣿⢯⡳⢭⡹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡫⢄⠀⢩⡴⠟⠁⣹⠈⣧⠀⣷⠘⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡼⠿⠟⠲⣶⣛⣥⡾⠯⠴⡏⣠⡏⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡠⠔⠚⠛⠋⠉⣙⣃⡄⠤⠀⠊⢙⣣⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⣀⣠⣴⣶⣾⡿⠛⠛⣛⣛⡫⠟⣩⣿⢹⣿⣯⢻⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣤⠾⠋⠉⣀⠤⠙⠐⠊⣉⡬⠽⠆⠚⡹⠊⣼⣿⣿⠈⣧⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢣⡟⠁⣠⠔⠉⢀⣠⠴⠚⠉⢀⣀⡠⠤⠖⠒⢚⣱⠿⢋⣼⣟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠈⢾⡇⣴⠁⠀⢠⠏⠁⢀⡴⠊⠉⢀⣠⡤⠖⠚⣩⠅⠊⠉⠀⣼⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠛⢼⣆⡀⣯⠀⣰⡏⢀⣴⡞⠋⠀⢀⣴⣾⠧⢤⣴⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡉⠒⠢⠢⠀⣉⡛⡒⠿⠶⡿⠿⣶⣶⣿⣿⣩⣿⣷⣿⢿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠉⠙⡏⡟⢙⡛⢻⠛⡏⢹⡏⣧⡇⣾⢸⣧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢁⠁⠈⡅⢸⢨⠇⣸⢹⢛⣇⡯⡜⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡄⠄⠀⠀⠀⠀⠀⣇⢸⢈⠁⡎⣸⢨⣹⢹⣇⣿⣿⣿⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢢⣸⠸⢰⠀⡀⢀⠂⢹⢸⢸⢰⡇⣾⡶⡞⣿⣿⣿⣿⣿⣿⣿⣷⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠬⣿⣇⢿⣧⢱⣸⣰⣸⣿⣿⣸⣿⣯⣿⣧⣿⣿⣿⣿⣿⣿⣷⡛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢾⣷⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠟⠟⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣙⣿⣿⡇⢻⣿⣷⣿⣿⣿⠟⣽⣶⣮⠻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⣿⡇⠹⣿⣿⣿⣿⣇⠘⢿⣿⣿⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠛⢋⡀⠀⠈⢻⣿⣿⣿⣷⣜⠿⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⡺⣯⡻⣶⣦⣤⠙⠻⠿⣯⣻⣿⠶⣣⣴⣶⣾⢓⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠿⣮⣿⣾⣿⣿⣿⣦⡀⠀⢀⣠⣾⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠉⠛⣿⣿⣿⣿⡿⠀⣾⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢿⣿⣿⣿⣿⡿⠟⠀⢸⢿⣿⢿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠛⠈⢿⣿⣿⡄⠀⠈⠳⠙⠿⣉⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⣿⣿⡟⠂⠀⠀⣠⣾⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠙⠿⣿⣷⣶⣾⣿⡏⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠉⠩⢽⣏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⢀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠻⠻⣿⠟⠛⢿⣿⡿⠟⠻⢿⣿⣿⡿⠟⠻⣿⠻⣿⡿⢻⡿⢻⡟⠿⠛⣿⠟⠛⠻⡿⠻⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡖⢰⣿⠁⣴⣷⠄⣛⠀⣾⣷⠀⣿⣿⣧⣈⠓⢿⡆⠻⠁⠚⢁⣿⠀⢘⣻⣿⠀⣛⣻⣿⡇⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣸⣿⣧⣌⣉⣴⣿⣦⣈⣁⣴⣿⣿⣧⣉⣁⣼⣿⣄⣴⣄⣾⣿⣄⣉⣉⣿⣀⣉⣉⣿⣇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⡀⠀⠀⠀⠀⠀⠀⢤⠠⠐⡆⠀⠀⠤⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡴⠁⠐⠈⠰⡀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠁⣼⠁⢀⡀⠀⠀⠰⠷⣰⠞⠁⠀⠀⠀⠀⠀⠳⡀⠃⢀⠠⡄⠀⢅⡇⠀⢸⢀⠞⠀⠀⡠⠭⠔⢁⡤⠖⠲⠒⠲⡖⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⢀⢈⠒⠉⢰⠈⡄⠉⡠⠻⠀⠀⠀⠈⠀⠀⢀⡤⠤⡀⠀⠀⢱⠀⢸⠀⣿⡸⣸⠁⠀⠀⠈⠀⠀⠨⢀⠀⡰⠧⠄⠀⠶⠀⠀⠀⠖⠀⢠⡟⢦
⠀⠀⠀⠀⢦⢤⣈⠘⡓⠖⠚⠀⠁⠀⠁⢧⠔⣶⣀⡀⠀⠀⠘⠬⢂⠜⠀⠀⠀⠀⠈⣏⣻⠣⣬⠔⠒⠲⡖⠤⡀⠐⠞⢰⡁⠀⠦⠄⠀⢠⣶⣦⡄⠀⠀⣄⠀⢧
⠀⠀⠀⠀⠈⠢⡀⠉⠁⠒⢃⡠⠤⠤⣀⠀⠀⠀⠘⢁⡠⢲⠇⠀⠀⠀⠀⢀⠄⠂⠉⠉⠛⢴⡁⢠⡋⠎⡁⠀⠙⡄⠀⢸⠧⠀⠀⢀⡄⠈⠢⠜⠁⠀⠀⠉⠀⢸
⠀⠀⠀⠀⠀⠁⠘⣆⠀⢠⠋⠀⠀⠐⠈⠳⡄⠀⠐⠁⢠⣃⠀⠀⠀⠀⠀⡎⠀⠤⠎⠑⠊⠳⢳⠀⠯⠐⢳⣂⢀⡇⠀⠀⢧⢲⠀⡁⠀⠀⠸⠄⠀⡆⠀⠐⢂⡟
⠀⠀⣀⣀⠀⠀⡸⠇⡀⠞⠣⡔⠳⡤⠚⢆⡘⡄⢸⣀⠉⠯⣀⠀⠀⠀⡈⡋⠀⠀⣀⣀⡀⣠⢚⢋⢙⡒⣌⣩⠟⠀⠘⠃⠀⠓⢍⡒⠣⢀⣀⡀⢀⣀⠤⢂⠟
⠀⠀⠀⠀⠀⠤⠧⠄⠀⢠⠠⢠⡈⠲⠂⠠⠹⣼⡄⠈⠀⠀⠁⠀⠀⠐⡏⣵⣤⡊⠀⠀⡼⢉⠀⠀⠡⠑⡋⣇⠀⠀⢀⡀⠀⠀⠀⠈⠙⠒⠶⠷⠿⠴⠚⠁
⠀⠀⠀⠰⠰⠂⠀⠀⠀⠀⣆⠀⠀⠀⠰⠆⠑⠀⢹⡄⠈⡰⠀⠀⠀⡜⣱⠂⠉⠹⣗⠤⢷⡈⠀⠀⠀⠉⢀⠏⠀⣠⣭⡉⢳⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⡖⠁⡀⠀⠸⡁⠀⠀⡀⠀⠀⢀⡀⠹⡄⠁⠃⠀⠀⠟⢸⠀⢤⡀⠘⠵⠑⠺⠝⠓⠒⣶⡏⠠⢸⡃⢀⠽⠀⠀⠀⢀⡤⢤⣠⠏⠀⠀⠈⠉
⠀⠀⣠⡆⠀⠁⡠⠊⠠⢀⡀⢡⠀⠈⠃⠀⠀⠀⠀⢨⡽⠀⠀⠀⠀⠀⡌⠀⡀⠈⠑⠶⠤⢀⡔⠀⡴⡩⠀⠐⠘⢷⣄⠀⠀⠀⣠⠘⠲⢺⡇⠀⣀⠀⠀⣀
⠀⠠⠈⠀⠀⠸⠒⠒⢒⠆⠁⠀⢢⣤⣀⡤⢴⢛⣛⠁⠀⢀⠠⠀⠀⠀⡇⠀⠃⠄⢀⡀⠀⠀⢀⠾⠍⠀⠀⠀⠀⠂⢻⣦⠀⠔⢿⡀⢀⣼⠃⢸⢩⡾⢳⡌⢷
⠀⠀⠀⠀⠀⠀⡔⢠⠏⠀⠣⢠⢀⡀⠀⠀⠈⡏⠈⢆⠀⠘⠲⠉⠀⢀⠇⢀⠀⠀⠀⡠⠀⡠⠋⠀⠀⠀⠀⡾⠋⡠⠆⢻⣇⠀⢈⢙⢋⠁⠀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⢼⡄⣮⠴⠚⡿⠈⢁⠘⠀⠀⠀⠘⠦⠞⠀⠀⢠⡀⠀⢸⠀⠈⠙⢃⠀⣠⠞⢡⠞⣠⠞⢛⡷⠀⣾⠀⠀⢸⡿⠀⠀⣠⣄⡰⠞⠲⠀⢿⣼⣇⣠⢀
⠀⠀⢀⡄⠀⠀⡦⢄⡀⢰⣃⠖⠉⣧⠖⠀⡖⠀⠀⠀⣀⠀⠀⠂⠀⢸⠀⠀⠄⣠⠞⠁⠀⢸⠀⣯⠘⠉⠀⠀⠙⠶⣶⠟⠁⠀⠀⣠⣬⠀⣟⢹⡦⠟⣿⣯⣁⣾
⠀⠀⠀⠁⠀⠀⠀⠘⠁⠘⠃⠀⠀⠉⠀⠀⢁⡤⠤⣤⠠⠝⡄⠀⠀⠸⣀⡴⠊⠁⠀⠀⠠⠀⠀⢹⣇⠀⣰⣫⠇⠀⠀⢀⠹⣆⢸⣇⡾⠀⠻⣾⣥⠾⠁⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠰⣄⣀⡀⠀⠀⠀⠀⢀⠞⣅⠁⢘⠃⢷⠒⠛⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⣠⠴⠶⣿⠛⣿⠁⢰⣞⣦⢸⠇⣿⠘⠿⣤⣤⣾⣣⠔
⠀⠀⠀⠀⠠⠖⣁⣀⣯⡀⠀⠀⣀⠀⣼⡠⠱⠆⠘⢠⠏⠀⠀⠀⡀⡐⠳⣄⡀⠀⠀⠀⣾⠁⣠⢄⣿⠀⠻⣦⣼⣿⣃⣠⡾⢃⡾⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠁⠈⠀⠀⠀⠉⢧⣒⣹⠒⠒⠚⠁⠀⠀⠀⠀⠑⣦⢄⣯⠁⡀⠀⠀⠹⢦⣤⡾⠋⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⡟⢩⡐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣄
⠀⠀⠀⠀⠐⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡟⠁
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⢸⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⣷⣄
⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡀⢠⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⠃
⠀⠀⢸⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⠁
⠀⠀⢸⡟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣇
⠀⠀⠀⠻⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠙⢶⣄
⠀⠀⠀⣤⣾⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠟⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⠟
⠀⠀⡜⠁⢻⡀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣤⣤⣀⣴⣿⣿⣿⣶⣾⣿⣿⣧⣄⣤⣬⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡰⠁
⠀⠀⢃⠀⠀⠙⢳⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢿⡟⠛⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⢸⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣨⠿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣃⠀⠀⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠠⡄⠀⠀⠀⢾⣧⡀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⢀⡀⠀⠀⠀⠀⢹⡟⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿
⠀⠀⢀⠁⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠉⠀⠀⣠⠔⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⡀⡀⡀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡇
⠀⠀⠈⠃⠀⠀⠑⢢⡞⣡⠞⣡⠖⣠⠄⠀⠀⠀⠀⡰⠁⠁⠉⠁⠀⠈⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿
⠀⠀⠀⠆⠀⠀⠀⠀⠁⠋⠰⠏⣰⢣⡖⡟⣇⠀⠀⠀⠀⠀⢰⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣿
⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠠⡀⠀⠈⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠰⠻⠐⠃⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡇
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷
⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡄
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢰⣿⡆⢰⣧⡜⣿⡀⣰⡞⠂⣠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣧⠀⢹⢹⣿⠸⠉⠁⣿⡇⢈⣳⠀⣿⠁⡜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠈⠁⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⢀⡠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠲⣄⣎⠄⠉⠆⠀⠀⠀⠀⠀⠀⠈⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⢀⠀⠈⠐⠃⠀⠀⢀⠏⠉⠑⢶⠀⠀⠀⠀⢈⠇⡂⠀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡡⡈⠂⠀⠀⠀⠀⠈⢢⠀⠐⡸⠢⢀⣈⠤⠊⠀⠈⠙⠁⢠⠊⠉⠀⠀⠑⠊⠀⢠⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣀⡀⣸⡧⠔⠓⠓⠁⠀⢀⣴⣄⠀⠈⠢⡰⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠑⠂⠤⠄⠀⠀⣴⣿⣿⠏⠀⣀
⠀⠀⢀⠔⠉⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⢀⠀⠀⢀⣼⣿⠋⣩⠂⠀⠀⢀⣀⣀⡀⠰⠆⢰⣿⣿⠃⠀⢀⠠⡈⠐⡄
⠀⠀⠘⠤⠈⠀⠀⢰⡎⢿⣿⣆⠀⠀⠀⣿⠁⢀⣤⣀⠰⣁⢱⣀⢼⣿⡗⠊⠀⠀⢀⣾⣿⠋⠉⢹⠀⢀⣿⣿⠇⠀⠀⠘⢄⡀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠙⠐⢻⣿⣿⠀⠀⠈⠁⠈⠻⣇⠀⢀⡠⣾⣿⠧⠤⠐⠁⠀⢸⣿⡿
⠀⠀⠀⠀⣀⠔⠒⠀⠒⠢⡀⢻⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⢰⠀⠀⣀⠀⠁⠀⠀⣿⣿⠀⠀⠀⡄⠀⣼⣿⡇⠀⠀⢀⠔⠒⠢⢄
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢸⠀⣿⣿⡀⠀⠀⠸⣿⣇⠀⠀⡌⠀⠀⠉⠀⠀⠀⠀⠈⠛⠷⠔⠊⠀⠀⣿⣿⠀⠀⠀⡇⠀⠀⠀⠈⡆
⠀⠀⠀⠅⠀⠀⠀⠀⠢⠤⠊⠀⢸⣿⠃⠀⠀⠀⠙⠿⣄⠜⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠃⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡡⠃⠀⠀⠀⠀⠠⣒⠄⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠤⠒⠁

⠀⠀⠀⠀⠀⠀⡀⠤⠠⠤⢤⣀⠀⡚⠑⠂⠲⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡆⢱⢠⣀⠬⡁⠁⢀⢐⠄⠉⠱⣠⠞⠠⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠰⢄⡞⠫⡩⠉⠋⠍⢫⢳⡴⠈⣝
⢀⠜⠁⠈⠁⣑⠮⣆⠀⠀⢀⠉⡏⣷⣀⠈⢂⢩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠘⠀⠉⠲⣀⠀⡇⡆⢸⡀⢸⣸⢷⢠⠼⠂
⢏⠀⣠⢶⠐⠃⢄⣄⠂⢀⢀⠩⣤⠻⣌⣂⡌⠀⠀⠀⢀⠀⣠⠤⠤⢤⣄⣀⡀⠀⠳⢲⢠⡍⠚⠉⠢⠷⠂⠚⠓⠛⠁⠈⠁
⠀⠉⠀⠀⠑⠒⠊⠚⠮⠥⠶⠚⠁⠀⠀⠀⠀⠀⠀⠎⡁⠂⠨⠖⠟⠲⢆⠂⢭⡆⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠁⠀⠃⠀⠈⠋⠁⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⡇⠀⠀⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠑⣒⡠⡀
⠀⠀⠀⠀⡀⢲⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠁⠀⠂⠤⠤⠧⠤⠄⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠎⢑⠀⠀⡘⠤⠜⠈⡄
⠀⠀⠀⠀⣧⠈⠢⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠂⠊⠀⢄⠱⠴
⠀⠀⠀⠀⢣⠑⠤⣈⠑⠠⠤⠠⠤⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠉⠐⣂⠠⠤⢀⡀⠀⠀⠈⢃⡐⠀⠈⠂⠐⢊⡤
⠀⠀⠀⠀⠀⠑⣄⡀⠉⠀⠒⢒⠖⠉⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⢀⠀⠀⠊⠁⠀⠀⠀⠨⠙⠀⠀⠀⠉⠒⠐⠤⠚⢉⢤⠽⠂
⠀⠀⠀⠀⠀⢠⠃⠨⠋⠁⠉⠁⠀⠀⠀⠀⠀⣠⣈⣙⠒⣆⠀⠀⠀⠈⠀⠀⠒⠂⠠⠂⠁⠀⠀⠇⠀⠀⠀⠀⠀⠂⠈⠚⡀⢂
⠀⠀⠀⠀⢀⠎⢠⠁⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⢈⠜⠫⠦⠄⣀⣀⠴⣄⠀⢸⡄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢆
⠀⠀⠀⢀⡎⢠⠃⠀⠀⠀⠀⠀⠐⠣⢄⡂⢀⣀⣀⡀⠥⢀⣀⣀⣀⠐⠉⠂⠈⡆⠔⠉⠀⠀⠀⣀⡤⠤⠲⠦⣄⡀⠀⠀⠀⠑⠤
⠀⠀⠀⠸⠤⠎⠀⠀⠀⠀⠀⠀⠀⠢⢄⠘⡀⠀⢸⢀⢀⣘⠁⠀⠈⡇⢀⡸⠒⠃⠀⠀⠀⣠⠊⠁⠀⠀⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠈⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⠇⠀⢀⡔⢉⡉⢻⠎⠔⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠱⣀⠀⠀⠀⠀⠀⠀⢸⠀⠉⡏⠀⢫⣁⠜⢀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡒⢋⠉⠙⣏⣀⢈⠇⠀⠀⠀⠀⠀⠀⠳⣄⠚⢦⡙⠀⣠⡼⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠚⢉⠢⡀⠀⠈⢂⢃⢠⠿⠧⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁⠠⡈⢄
⠀⠀⠀⠀⠠⣀⠀⠀⠔⠈⡉⠢⢄⢄⠀⠀⢁⣁⣠⠤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠳⡀
⠀⠀⠀⢤⣀⠡⣈⣈⢉⡤⠈⠀⢈⣬⠦⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡆
⠀⠀⠀⡆⠀⡰⣬⠼⠁⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠁⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠉⠀⣠⡶⠋⠉⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⢀⣤⠶⠛⠁⢸⣿⠀⣰⡟⠀⠀⠀⠶⡦⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⢀⣤⠞⠋⠀⠀⠀⠀⢸⣿⢀⣿⡇⠀⠀⠀⡸⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⣸⣏⡾⠻⢷⣤⣤⠞⠁⢀⠀⣼⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣷⣀⠀⠀⠀⠀⠀⣠⡿⠛⠁⣀⡀⠀⢀⣀⡀⠛⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⢰⣿⠁⠀⠉⡛⠳⠶⠶⠶⠚⠉⠀⢠⠟⠉⢩⣾⠋⠉⡷⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢾⡏⠀⠀⢾⡈⠀⡠⠀⠀⠀⠘⠦⢾⠐⠀⣾⠃⡠⠞⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⣇⢰⠀⠈⣷⢰⠋⢰⡇⠀⣰⠀⠈⠑⠀⠻⡉⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠹⡄⠀⠉⠁⢸⣆⢀⡟⠚⠁⠀⠀⠀⠀⣀⣈⠉⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⡙⠋⢠⣦⠀⠀⠀⠀⢰⣿⣿⠀⠀⢻⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢠⣿⣿⠿⠿⣿⣆⠸⣿⡇⠀⠀⢀⣿⣿⡟⣠⣤⣤⣭⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢸⣿⣏⠀⠀⢸⣿⠀⢿⣿⠀⠀⢸⣿⣿⠃⣿⣿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠀⠀⠈⢿⣿⣶⣶⡿⠃⠀⠘⣿⡇⠀⣿⣿⡏⠀⣿⣿⣷⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣀⣀⣤⣤⣤⣭⣤⣤⣤⣤⡀⠸⣿⣴⣿⡿⠀⠀⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠙⠿⠟⠁⠀⠀⠻⠿⠿⠿⠿⠿⠂



⢀⠀⡀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀
⠀⠀⡐⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠉⠁⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠀⠀⠉⠁
⢨⣀⠀⠀⣤⠤⢆⠄⠀⠀⠀⠀⠁⢠⡄⢀⡌⠀⠀⠈⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠀⠀⡀⠀⠀⠠⠀⢨
⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠈⠐⢋⡀⠐⠓⠀⠀⠀⠀⠈⣀⠀⢀⡡⢀⠴⣀⠀⠀⠀⠀⠀⢠⠲⠤⠄⠀⣀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠁⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠉⠀⠀⠈⠀⠀⠈⠀⠀⠀⡀⠀⢸⠀⠀⠀⠉
⠀⠀⠒⠒⠈⠁⠀⠀⠀⠀⠉⠀⠈⡱⠂⢀⠋⠀⢄⠘⢃⠀⠁⠒⣂⠠⠤⢄⡀⡀⠀⣀⡀⢄⠊⠀⡀⠴⢄
⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠘⡀⠀⠀⠡⢰⡁⠀⠀⠁⠀⠀⠀⢠⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠊⠀⠀⢀⡤⠋⠀⢀⠈⠢⠀⠈⠂⢈⠔⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠊⠀⠀⠀⠈⠐⠄⠂⠈⠉
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠥⢊⠔⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠀⡀⠀⠀⠀⠠⠐⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢌⠑⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⢀⠴⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣅⡀⣥⣄⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢰⡏⠘⣿⠀⣾⠁⢻⡧⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢻⣧⣼⠟⠘⣿⣤⡿⠃⠀⢿⡀⢠⡀⠀⠀⢰⡇⠰⣿⡒⠀⢾⣗⠒⠙⣻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢸⣧⡿⣷⣾⠀⢸⣏⣁⠀⣻⣉⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠁⠘⠃⠀⠻⠋⠉⠘⠿⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢸⡇⠀⠀⠀⠀⣠⠂⠀⢢⡀⣠⣄⠀⠀⠀⠀⢀⠂⣶⠀⣠⡄⠀⣠⡶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣸⡇⠀⢰⡇⠀⢱⣿⠀⠠⠏⠀⣿⠀⢰⡆⠀⠊⠀⠈⢰⡿⠀⣼⡏⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⣿⡇⠀⢸⡧⠀⣾⠇⠀⢰⠂⢸⡏⠀⣿⠃⠀⠀⠀⠀⣿⠇⢰⣿⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡸⠁⠀⣿⠁⠀⢼⡧⠀⢿⡀⢀⠎⠀⢸⡇⡰⣿⠀⢠⠂⠀⠀⣿⢀⠰⣿⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠙⠤⠈⢸⡇⠀⠈⠉⠁⠀⡤⡄⠁⠀⢿⣠⠆⠀⠀⠀⠈⠀⠀⠈⠑⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⣤⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⣿⡆⠀⢠⣮⣴⠂⠀⠱⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢀⡠⠋⠀⠀⠳⡄⠀⠴⠋⠈⣿⡄⢠⣿⠃⠀⠀⠰⡏⠁⠀⠀⠀⠀⣴⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢸⣧⢸⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣗⠀⠀⠀⡴⠁⠘⠇⠀⣿⠘⢧⣀⡀⠔⠁⢀⠤⠀⠀⠀⣿⠇⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⠃⠀⠀⠀⢣⡀⠀⠀⡰⠃⠀⠀⣀⡤⡀⠀⢉⣀⣀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢈⡤⡀⠀⣴⠏⢀⠏⣴⡟⠀⢈⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠐⠈⠈⣿⡆⠀⢠⡿⠁⢀⡀⠸⣇⡀⢰⣿⠀⠋⢸⣿⠀⠠⠊⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⠃⠀⣿⠁⠀⠀⠇⢸⡏⠀⢀⢾⡇⠀⠀⠀⠀⣿⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⡄⠀⠀⠀⣿⠀⣿⠀⣠⢿⡀⢀⠜⠀⠈⠳⠴⠋⠈⢷⡀⠀⠀⠀⢸⡄⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣿⠀⠀⠀⣿⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠐⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢀⡀⠤⠊⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣀⠀⠀⠠⣦⠀⠀⣠⣼⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣇⠀⠀⠛⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡸⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠿⠛⠉⠙⠋⣾⣿⣷⣾⠀⠀⢸⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⡀⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠿⣾⠇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠛⣉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠲⣶⡶⢦⡀⠀⠀⠀⠀⠋⠉⠀⠿⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⣾⣿⣿⣿⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠈⠙⠋⠁⠈⣹⣿⣿⣿⣷⣄⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠗⠢
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⣿⠀⠀⠀⠈⠉⡙⢻⣿⣿⣿⠟⠛⠛⠛⠛⢿⣿⣿⣿⣿⠿⠿⠀⠀⠀⠐
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠃⡀⠀⢹⡀⠀⣄⠀⠀⣁⣾⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣯⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣾⣿⣿⣿⣿⣿⣿⡿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡟⢻⣿⣿⣿⣿⠿⠛⣿⣟⣻⣿⡛⢿⣿⣿⣿⣿⠃⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡏⠙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠘⠢⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⠳⣄⠀⠁⠀⡜⠈⠀⠀⠖⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⢦⠀⣹⠀⠀⠀⣧⠀⢠⡞⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⢧⣄⣿⣿⠀⠠⡀⠈⢠⢠⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠻⣆⠀⢧⠀⠀⠸⠷⠎⠁⠀⠀⢀⡠⠄⠐
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢈⣄⠀⠀⣧⠀⠀⣀⡴⢫⠖⠀⠠⠂⠀⡄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢆⠟⠙⠒⠛⠒⠀⠀⠰⠃⢸⡠⠂⢀⡜
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠣⣄⣀⡀⠀⠋⢀⣠⡶⢚⣠⠴⠋⢀⠀⠈
⠀⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⠀⣯⠉⠉⠉⠑⠊⢉⣁⠔⢰
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠉⠙⠻⢯⣀⢀⠀⠈⠡⡴⠃
⠀⠀⢸⠁⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⠢
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠤⠄⠤
⠀⢸⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣼⠃⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⣿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠉⠉
⣀⣈⣳⣦⣤⣤⣤⣴⣔⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠠⣿⣠⡿⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢨⣷
⠀⠀⠰⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⣠⡿⣤⣅⣀⠀⠀⠀⠀⠀⠀⠀⡟⢹⣿⠟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⣠
⠀⠀⠈⠀⠈⠀⠈⠁⠂⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢼⠷⡀⢠⡋⠀⣀⠀⠀⠀⠸⣦⡈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⢠⠊⠀⠀⠀⠒⠀⠈⣧⣤⡄
⠀⠐⠒⢀⠄⠀⠀⠈⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠚⠒⠚⠿⣿⠟⠀⠀⠀⠀⠄⠉⢹⣿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⢋⠐⠄⠀⠄⠀⠀⠀⠻⣦⡶
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠀⠀⠂⠀⠀⠀⡘⠁⠀⠀⠀⠁⠀⠀⠀⠰⠀⠘⠧
⠀⠀⠀⠀⠢⢍⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠒⠒⠀⠴⣶⠁⠀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠃⠒⠀⠁⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⢃⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠓⢂⣠⣀⡀⠻⠆⠛⠻⠀⣵⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠲⢆⣀⣀⠚⠃⠀⠀⠀⠉⠉⠛⠛⢣⡞⡲⠞⣀
⠀⠀⠀⠀⣀⣻⣀⣤⠀⠀⠀⣀⣀⣀⡀⣠⡟⠉⠀⠀⠀⠀⠀⠀⠱⣀⠀⠍⠀⠀⠀⠀⠀⠀⣤⡼⠗⠛⠈⠉⠁⠈⠀⢢⠀⠂⠘⠀⠀⠀⠀⠷
⢀⣀⢀⡀⠿⠋⠀⠀⠀⠀⠀⠀⠉⠠⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠂
⠀⣈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠈⠀⢀⡄⠀⠀⠀⠘⠀⠀⠀⠄
⢞⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⢈⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⡤⠐⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠈⠁⠉⠁⠀⠀⠀⠀⣠⡶⣄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠺⣧⣴⢾⡇⠀⣆⠀⠀⠀⠀⢀⠀⠀⠀⢀⣿⠀⢀⡀
⠀⠀⠀⠐⠁⠀⠀⠀⠒⠀⠈⠀⠈⠀⠀⠀⠀⣠⣄⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡀⠀⠀⢹⡎⣀⡀⠹⣷⣶⣄⢶⡿⢤⢀⣠⣾⠇⠔⠋
⠀⠀⠀⠁⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⣸⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠻⠛⣾⣀⢉⡥⠟⣥⠴⡾⠋⠁⠀⠄⡀⠈⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⢐⣦⠾⠛⠀⠀⠀⠈⠀⡀⠀⠀⠈⡀⠀⠀⠀⠀⡀⠀⠉⠋⠉⠈⡓⠒⣥⡀⠁⢁⡀⠂⠀⠄
⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⠀⠈⠷⠲⡀⠀⠁⠀⠂
⠀⠀⠀⠀⠦⠀⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠐⠀⠀⠠⠀⠀⠀⠀⢈⠁⠀⠀⡁⠂⠄
⠀⠀⠀⠉⠲⠤⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠒⠁⢠⠀⠀⠂⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⢄⡄⠀⡀⠀⠀⢀⡠⠄⠒⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠈⠃⠄
⠀⠀⠀⠀⠀⠐⠄⠀⠣⠀⠀⠀⠀⠠⣤⡄⠀⠄⢠⣄⢿⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣠⠤⠀⠂⠒⠟⠀⠈⠁⠉⠺⠋⠀⠀⣀⠀⢿⣮⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⡉⢉⠀⠄⡝⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡀⢾⡳⢀⠀⠀⠀⠀⡀⡀
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⢀⡀⢈⠀⠀⡀⢚⢻⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠈⠉⠀⠀⠉⠉⠓⠈⢣⡀⣸⡊⠁⢈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠂⠐⠀⠀⠠⠀⠂⠄⠰⠋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⠋⠠⠐⠀⢀⠁⠀⡀
⠀⠀⠘⡀⠀⠀⠄⠀⡀⠄⢰⠁⠂⠐⠀⠂⠀⠁⠂⠐⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠴⠀⠀⠀⠒⣈
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡄⠈⠁⠀⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀⢈⠐⠀⠀⠀⠠⠀⡂⢀⡁⢀⠀⠠
⠀⠀⠀⠀⠀⠒⠀⠀⠐⠋⠁⢄⠉⠀⠀⠀⠁⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠑⢀⠖⠀⠆⢠⠐⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡟⠋⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢸⣿⣿⠀⠀⠀⢘⡍⢉⠀⠀⠀⠀⠀⠀⠶⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⡴⠁⠀⠀⢐⠀⠀⠀⠀⣴⡒⢢⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢳⠂⠀⠉⡠⠀⠀⠀⠀⠑⠬⠋⠀⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠿⠀⠀⠀⠐⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠴⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡴⣿⣧⡄⠀⠀⠀⡴⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠓⠙⠿⠁⠀⡠⠊⠀⠀⠐⠃⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢶⣉⣳⣄⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠂⠀⠁⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠔⠂⠐⠁⠠⢄⣼⢩⣿⡟⠹⣖⡾⠋⣿⣷⣀⣀⣠⣔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠃⠄⡠⠚⠳⡤⢿⣾⡷⠁⡜⠉⡇⠀⢻⡇⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡰⠁⠀⠀⠘⢿⡃⠀⠘⠁⠀⡥⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⣊⠉⠉⠓⠚⠣⠤⣄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡕⢠⡦⣄⠀⠘⣫⠈⣢⣄⡀⠀⠀⠢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⠊⣤⠞⠁⠈⠦⠟⠁⠉⠀⠀⠉⠙⠓⠛⠻⣽⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⡛⠛⠋⢉⠗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠾⠦⠯⣭⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣲⣴⠶⣶⣴⡂⣰⡇
⠀⠀⠀⠀⠀⠀⣠⠀⣾⣶⣄⡠⣺⣤⣴⣒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠷⢽⣿⣿⢖⣾⣿⢿⣿⣇⢰⡄
⠀⠀⠀⠀⠀⠀⢹⡀⣿⣿⣿⣿⣿⣾⣿⠟⠛⢲⣄⢸⣞⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡛⡉⠰⢣⡿⣵⣟⡿⣟⣦⣿⣿⣿⣀⣤
⠀⠀⠀⠀⠙⢿⣾⣷⣹⣿⣾⣿⠿⣯⣦⣛⣟⣾⣿⣮⣅⣷⢁⣘⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⡉⢈⡋⣀⣟⣡⣿⣿⡥⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⣄⣀⣈⣙⢿⣿⣿⣿⣯⣸⣿⣿⣿⣿⣿⣧⣿⣿⠞⠃⢬⢉⡀⠀⠀⠀⠀⡘⡿⢶⢈⢁⣍⣠⣿⣷⣿⣿⣿⣦⣬⣹⣿⣿⣿⣿⢷⠶⠆
⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣶⢠⣏⣿⣿⣿⣿⣿⣿⣼⣿⡌⠷⠠⡄⣌⡀⡀⠀⢘⠋⠆⡦⢈⢈⣇⠈⡻⣿⣿⣿⣿⣻⢛⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⢙⣿⣿⣿⣿⣟⢁⡰⣿⣿⣿⣿⣿⣿⣿⣧⣿⡖⣷⠴⡆⢤⣉⡀⠀⠈⣭⣖⢰⠆⢰⡇⢘⣣⣿⡗⣺⣿⣁⡦⣿⢽⠟⢿⠀⠁
⠀⠀⠀⣴⡿⣿⣿⣿⣷⡍⠈⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣷⡲⠤⣤⠀⠀⠈⠙⢿⣷⣿⣷⣯⣽⣿⣿⣿⠿⣿⣿⣣⣿⡿⠃
⠀⠀⠀⠀⣀⣿⡟⢉⡀⢦⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷⢶⠆⡀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠁⠐⡎⠃⠸⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠳⠘⣿⡙⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠿⣄⣙⣢⣙⣩⡭⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠯⠥⢤⣱⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣀⡀⣀⠀⡀⠀⣶⠀⠲⠀⠐⢁⡠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⢲⠀⢷⠀⢸⠀⠀⡆⠀⠀⢨⢸⡁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠒⠈⠚⠈⠊⠈⠊⠀⠉⠀⠈⠋⠑⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⢤⣖⣒⣛⣛⣛⣛⣛⣛⣛⣒⣶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠀⠀⢸⣤⣒⣒⠉⠛⠛⠛⠉⠑⣒⣒⣦⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠭⠍⠀⡇⠀⣠⠖⠉⠉⠲⢤⣤⡤⠖⠋⠁⠀⠀⡇⠀⠀⢸⡀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠥⠖⠀⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣒⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⣄⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠧⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⣠⡀⠀⢀⡠⢫⠋⣤⡀⠀⠀⢀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢧⠀⠙⠒⠾⢄⡙⠲⣽⣡⡷⠚⣁⠼⠖⠚⠁⢠⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠚⣦⣌⡶⠤⠤⠽⢶⣾⣿⣴⠾⠥⠤⢤⣖⣥⡚⠚⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⢋⡅⠀⠀⠃⠀⢀⡈⢉⡀⠀⠘⠀⠀⢨⡙⢶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡴⠀⠀⠈⠁⠈⠁⠀⠀⢦⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡄⠀⢠⠀⠀⢀⠀⢰⠄⠠⡄⠀⡀⠀⠀⡄⠀⢠⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠘⠂⠀⠀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠀⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡱⡐⡄⠀⠒⠀⠀⢀⡀⢀⡀⠀⠀⠒⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣜⢎⠢⡀⢠⡀⠀⠀⢀⡄⠀⢀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢧⡑⠬⡀⠀⠀⠀⠀⡀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡳⢄⠀⠀⠀⠈⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⠶⢶⣶⡶⠶⠾⠿⠿⢿⢿⡿⠿⠿⠿⣷⢶⠶⠶⠶⠶⢶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⣿⢸⡀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡏⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣧⣷⣀⣀⣀⣀⣸⣠⣇⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣿⣸⣀⣀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⣀⡀⣀⣀⣀⡤⡞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠳⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⡉⠉⠉⠉⣁⣠⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠹⣄⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠙⣦⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⢧⠈⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠈⢳⣽⡇
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⡤⠖⠛⠓⢦⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⢠⣿⡀⠀⢀⡞⠁⠀⠀⠀⣸⠃⠀⠀⠀⡟⠀⠀⠀⠹⣄⠀⠀⠀⣴⠖⠒⢦⡀⠀⠀⠀⣿⠷⠦⣄⠀⠀⢠⠶⠒⢦⡀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⡟⠙⠛⠉⠀⠀⠀⠀⠀⠻⣄⠀⠀⣠⠇⠀⠀⠀⠀⠙⠲⠤⠞⠃⠀⠀⠀⣇⠀⠀⢸⡏⠀⢀⡟⠀⠀⢸⠀⠀⠀⡇⢀⣼⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⣻⡇⠀⢸⣇⠀⣀⡼⠀⠀⠀⠉⠉⣿⠃
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣿⢿⣆⣴⣾⡆⠀⣀⠀⠀⣿⠇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠘⣿⠈⠛⠁⢸⣇⠀⣩⡀⠀⣿⠀⠀⣿⣴⣿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠹⠇⠀⠀⠸⣿⠀⢿⠇⠠⡿⠀⠀⣿⠉⠻⣦⡀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣿⣀⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣶⡀
⠀⢰⠆⠀⠀⠀⠀⠀⠹⣿⠀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⢰⠆⠀⠀⠀⠀⠁⠠⣶⡿⠁
⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟⠀⠀⠈⢃⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⠉⡼⠁⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟
⠀⠀⠈⠙⠺⡤⡴⠚⠁⠀⠀⠀⠀⠈⢳⣴⡿⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⢾⢿⡞⠀⠀⠀⠀⠈⠙⠺⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⡉⠁⣀⡀⢊⠀⠰⠉⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⢿⡟⠚⠖⠁⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠁⠘⠀⠀⡃
⠀⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆
⠀⠀⠈⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠞⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⢸⡆⠀⡇⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣾⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⣀⣀⣧⣤⣧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⣀⣠⣽⣤⠶⠷⠛⠛⠉⠉⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡖⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠹⣄⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢀⡼⠃⢀⣀⣤⡤⠴⠶⠚⠛⠉⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢠⡿⠷⢿⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠿⣦
⣿⠁⠀⠀⠀⠈⠉⠛⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠛⠋⠉⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢤⣀⣘⠆⣀⣀⣤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢸⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣴⡄⠰⡟⣷⠞⣹⠆⠸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢸⠀⠀⢰⠲⡆⠀⣾⣆⠈⣏⢿⢻⡘⡇⠀⣷⠀⢠⣏⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠙⢶⠇⠹⡆⢻⢸⣼⡇⣿⣀⣽⠀⣶⣝⢦⡀⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⠀⢀⡀⢹⣼⡎⣿⣧⣈⣠⣿⣧⡽⠘⠾⠁⢸
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⡟⣧⢘⣿⣦⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠚⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠠⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⣿⠀⠀⣒⠦⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⢠⣭⣟⡒⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠈⡇⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢹⡀⠀⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣧⡀⠈⠳⢦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞
⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⣸⣤⡶⠞⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠦⠐⠒⠂⠉⠉⠀⠀⠐⠢⠤⣄⡀
⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠒⢆
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠑⠂⡀
⠀⣴⠁⠀⠀⠀⠊⠀⡀⠀⠀⠠⠠⠔⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢰
⢀⠟⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀⠠⠐⠊⠁⠀⠉⠉⠓⠂⠤⢀⣀⣀
⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢀
⢈⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⠀⢀⠔⠁⠀⠀⠀⠀⡈⠈⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠙⣄
⠈⣄⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠎⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡔⠁⠀⠀⠀⠠⣁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠳⣌⠂⠄⡀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⠤⠤⠤⢀⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠙⢆⡀⠀⠉⣉⡠⠜⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⢀⡩⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠈⣀⡠⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢦⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⢀⠀⠀⠀⣇⠀⠂⠤⣀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣄⣀⠀⠉⠂⠄⡀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠟
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠈⠉⠉⠢⣀⣀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⡏⠉⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠙⠒⠤⠴⠒⠊⠉
⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠠⠤⠘⠀⠀⠀⠰⠀⠃⠀⠀⠀⠂⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡸⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠃⠀⠀⢠⠀⡂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠌
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⣈⡠⠤⠐⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⣀⡐⠀⠀⠊⠁⢀⡠⠤⠤⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠲⡄⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠄⡔⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⡖⠲⠒⠒⠂⠐⠂⠒⠀⠐⢺⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⠀⢰⠀⠃⠀⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⢰⠀⠃⠀⠀⠀⢰⠀⡄⠀⠀⡈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⠈⡀⢰⠀⠀⠀⠀⠀⢁⠀⠀⠀⡇⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⠀⠀⡇⢸⠀⠀⠀⠀⡇⠸⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠧⠤⠔⠃⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⢤⡈⢇
⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠖⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⠒⠒⢲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚

⠀⠀⠀⠀⠀⢠⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⢤⢄⡀
⠀⠀⠀⢀⣢⣄⣿⣤⣼⡃⠀⠀⠀⠀⢀⢀⢀⠀⠀⠀⠀⡐⠈⢸⣸⠿⣇⠀⠀⣐⣿⡿⠇⠀⠀⠀⠦⣷⢿⣿⣿⣧
⠀⠀⣴⡃⡜⢈⢹⣻⣿⡳⠶⠆⠘⠢⣾⣾⣼⠠⣠⠀⣰⠇⣆⡿⠗⢚⡿⠀⠀⠳⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣵⣿⣶⣄
⠀⢠⠋⡁⠱⣸⣭⣿⣿⣿⠀⠀⠠⣌⡉⠽⢿⣧⡇⢀⣿⣼⣟⣣⠍⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠈⠁⠀⣪⣿⣿⢾⡇
⠀⢸⣁⠠⢸⣽⣿⣿⣿⠟⠀⠀⠀⠈⠓⠊⠉⠉⠳⠘⠛⠛⠁⠀⠀⠀⠀⢀⢴⣴⣷⣀⡴⣄⡈⠀⠀⠀⠀⠀⢈⡵⣿⣿⠃
⠀⠈⢤⣭⣾⣿⣯⠏⠉⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣒⣀⡀⠀⠐⠿⢿⣿⣁⡤⣿⣿⣭⣶⣟⣅⠀⠈⠀⣿⡉
⠀⠀⠈⠙⠋⠁⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠹⣿⣷⢖⣫⠀⠀⠀⡟⠠⡸⣵⠿⣼⡿⣏⠛⣿⣷⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠠⣦⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠃⠛⡇⠈⠀⠀⠀⢗⣰⣿⣿⡠⠉⠷⠋⣻⣿⣿⠇⠀⠀⣄⠠⠶
⠀⠀⢀⣭⢿⡶⣿⣿⡷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⢿⡿⠿⠗⠈⠄⣼⣿⣿⡟⠀⠀⣾⣏⡗⢀
⠀⢰⡿⠐⠆⠹⠆⣿⣿⣦⠀⣤⢢⣙⠀⠀⠀⠀⠀⠀⣄⣀⣠⣧⠴⠀⠀⠀⠀⠀⠀⣴⣡⣄⣀⠀⠚⠿⠋⠀⠀⠀⠉⠉⠁⠉
⠀⢺⣦⢉⡘⣧⣿⣿⣿⣿⠀⢲⡈⢿⣷⡀⣂⠀⠐⢲⣿⣿⣿⠟⡶⢦⡀⢀⣲⢶⣶⠟⡐⠀⠜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⣿⡤⠹⣿⣿⣿⣿⠏⠀⠀⠁⠽⣿⣷⠃⠀⢰⠋⣁⠈⣉⡀⠐⠊⢻⠀⡅⠀⠖⠈⠛⠂⠀⡀⢣⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠻⣿⣿⡿⠟⠁⠀⠀⠀⢀⣀⣤⠽⠀⠀⠸⡄⠀⠄⠙⠇⢘⠀⢸⠀⠻⢦⣄⣁⡈⡁⣁⣀⣨⠀⢀⣤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠉⠃⠀⠀⠀⠀⠙⢦⡀⠂⠀⠀⢰⠃⠀⠀⠙⠤⠿⣟⠿⠟⠋⠀⠈⢿⣼⣯⡿⢋⣤⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⠛⠁⠙⠼⠿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡴⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣷⣿
⠀⢸⡿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢟⣭⣴⢿⣿⢿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠿⣻⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢽⣶⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣏⣽⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⣄⣀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⢀⣾⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠻⣷⡀⣼⡟⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣷⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡄⢻⣶⣿⠁⣿⣿⠀⠀⢿⣿⡇⠀⢸⣿⣇⠀⠀⠀⠀⢸⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠇⠈⠻⠃⠀⣿⣿⠀⠀⢼⣿⡇⠀⢸⣿⣿⣶⣶⣶⡆⢸⣿⣇⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣧⣧⣿⣿⣿⣿⣿⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠙⠉⢉⣭⣍⡙⠿⢶⣶⣶⣶⠶⠋⠉⠻⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠯⡭⠤⠤⣤⣀⡘⠻⣿⣇⠀⠈⠛⠛⠋⠠⣿⣆⢠⡀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢡⣿⠇⢠⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⣄⠀⣿⣀⠀⠀⣀⣠⣤⣤⣀⡀⠈⢆⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡄⣿⠀⠀⠉⠁⠈⠛⠛⠛⠙⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⢻⣻⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⠟⡀⠀⠀⠀⠀⠀⠀⡧⠞⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣾⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣤⣤⣤⣿⣧⣼⣿⣥⣤⣤⣤⣤⣤⣼⣿⣥⣤⣷⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣵⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣴⣶⣾⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣶⣤⣤⣤⣶⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢸⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⣿⣿⣿⣿⡿⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠋⠁⠀⠀⠈⢡⣶⣿⡎⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⣷⣴⣿⣿⡇⠀⠀⠀⠀⠉⠋⠑⠃⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⢳⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠯⠖⠀⠀⡀⣼⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⣿⣿⠉⠀⠀⠀⣴⣶⡄⠀⠉⠁⠀⠀⠀⠐⠇⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠉⠁⠀⠀⠀⠀⠨⡝⠁⠀⠀⠀⠀⠀⠀⢀⣴⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⢀⠀⣀⣀⡀⠀⠀⠀⠠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⡀⠀⠀⠚⠸⢿⣿⡿⠀⢐⣣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠶⠒⠛⠉⠁

⢿⣍⣉⣉⣉⣉⣉⣉⣹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⠋⢹⠉⢹⡟⢾⣝⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣇⣞⣠⠟⢉⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣛⣉⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⣾⡿⣭⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠜⠁⠈⢿⢦⣈⡽⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⡤⠉⠀⠀⠀⠈⠳⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⠎⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢿⣿⣿⣿⡿⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠛⠛⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠀⢸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠋⠉⠉⠁⠀⢹⣀⣸⠀⠈⠉⠉⠉⠓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠿⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠓⠐⠒⠲⠤⠖⠒⠲⢤⠴⠒⠒⠒⠊⠉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⠐⠲⠶⠶⠶⠶⠒⠒⠒⠒⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⣤⠒⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠉⢧⡀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⢶⣄⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠳⢦⣀⡰⠎⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠹⣆
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⡙⢇⠀⢠⠻⡆⠀⠀⢸⠋⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡉⡇⠀⣀⡤⢶⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠸⠃⠈⢷⡎⠀⠻⡀⠀⠘⠀⠇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡆⠰⠞⠁⣠⠏
⡇⠀⠀⠀⠀⠀⠀⠇⣠⢦⡀⢀⣤⡀⢣⡀⠀⡆⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠠⣞⠁
⡇⠀⠀⠀⠀⠀⢰⢀⡉⠀⠳⠎⠀⢳⠈⢧⠀⣇⢸⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⡇⢸⣧⠘⠦
⣿⠀⠀⠀⠀⢀⡇⡜⠁⠀⠀⠀⠀⠀⢧⡈⡇⣿⢈⠀⠀⣇⠀⠛⠛⠛⠂⠒⡄⠃⠸⠈⢧⠘⢣
⡇⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⠇⠈⠛⠀⠀⠘⠓⠒⠲⠖⠒⠋⠀⠙⠃⠀⠈⠳⠞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⠶⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠁
⡇
⢣
⢸
⢸⡆
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⠘⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟
⠀⠀⠻⣷⣶⣤⣤⢤⢤⣤⢤⣤⣤⢤⣤⣤⣤⣤⣤⡤⣤⡤⠤⣤⡤⠤⠤⢤⡤⠲⠲⠖⢶⠖⠒⣩⡟⣻⠟⠁
⠀⠀⠀⠈⢿⣀⣳⣀⠀⠳⠀⠈⠃⠀⠘⠃⠀⠉⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⣞⣡⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠂⡄⠰⢄⣴⣯⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢻⣿⣿⣿⣯⣿⣿⣿⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⠛⠙⠯⠿⠽⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁

⠀⠀⠀⣠⣤⣾⠃⠀⠀⢀⣰⠟⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠀⢀⣶⣖⣶⣆⢠⡀⠀⠀⠀⠀⢀⡀
⠀⠀⣼⣿⡿⠁⢀⣤⠶⠛⡁⠄⠒⠛⣿⡿⠃⣀⡤⡶⠦⣀⣺⠶⠚⢉⠁⠀⠀⣉⣯⣟⡻⠶⣄⡰⡄
⢀⣼⡿⠋⢀⡶⠋⠀⠀⠁⠁⠀⠀⠈⠙⢀⣢⣭⣿⡐⢛⣋⡄⢐⢎⡴⢒⡽⣚⡵⢋⡥⣒⠾⢛⣻⡄
⣾⡟⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣫⡵⢋⣴⢾⡫⠅⡰⠃⡾⣣⡟⢱⡷⡐⢉⡞⣡⣾⡿⡋⣧⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠏⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠣⣢⠽⣾⣁⡐⣚⠁⡸⢥⣏⢰⡏⣰⠣⢸⢰⠟⢋⠄⢠⡼⣿⣂
⠀⢀⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⢀⠟⠀⠐⣱⢞⡟⡱⣱⡇⢰⠃⠀⣡⠸⢰⢧⣤⢰⢀⠄⡀⠀⢀⡴⣿⢽⡰
⠀⣼⣿⠁⠀⣿⣆⡀⠀⠀⠀⡄⡛⣀⢂⣰⠷⠊⣴⣣⠟⣡⢇⡀⢰⢇⠀⣎⣟⡿⡿⠈⣴⠃⠂⢠⣆⣿⣷⠹
⢠⣿⠃⠀⢠⣿⣿⠛⣦⡀⠀⠀⢚⣦⠌⡋⠠⡠⠓⣫⣰⢫⠎⠜⣠⡎⢸⣹⠻⣱⠇⠀⣿⣷⠁⢊⢎⣯⣤⠐⠆
⣼⠃⠀⣠⣿⡿⢁⣾⡿⠋⢓⣶⠈⠀⢨⠁⢨⣰⢰⢗⢣⡍⡀⠀⠟⡄⠠⢧⠄⡃⡌⠀⢻⠋⣤⠆⠈⠛
⡏⠀⣴⣿⡟⢀⣾⠏⢀⣴⣿⠏⢠⡰⢸⡇⢎⡏⣼⢊⡴⢳⢡⡄⠾⢁⢖⠄⡐⠸⣵⡴⡼⢚⡄
⢇⣼⣿⣿⠠⠾⡋⠀⣼⡟⠁⠀⠀⠀⠸⢏⡜⠰⢃⣼⢡⢋⢫⠔⠰⠁⢀⡾⢿⢹⢏⠍⠀⠀⡀
⠀⣿⣿⣿⡄⠀⠀⣾⡏⡠⠀⠀⠀⠀⠀⠰⡆⠀⠘⠇⢣⡏⠀⠀⡰⢠⣼⡇⠾⠀⡌⠀⠀⠀⠂
⡀⠈⢻⣿⣷⣀⠀⡿⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢫⡀⠇⠁⣿⠰⡁⢸⠀⠀⠁⡀
⣷⡀⠀⠙⠿⣿⣷⣞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠋⠇⠘⢰⡜⢾⠁⠀⠀⠀⠀⠉
⣿⣿⣦⡀⠀⠀⠉⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠀⠃
⡘⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣌⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⢰⠷⡠
⠀⠀⠀⠀⠀⢀⠔⠀⠀⠁
⠀⠀⠀⠠⣴⣟⠀⠀⠀⠀⠀⠠⠘⠀⡠⠊⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⡰⠈⠀⠀⠉⠃⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⡀⠁⠀⢾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠅⣀⡃⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⢀⠀⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠆⠽⠀⠐⣩⢋⢡⡀⠀⠈⠸⡆⣰⣿⠖⠟⠉⠱⡄
⠀⠀⠀⠀⠀⣴⠖⣶⣾⣶⡄⣀⣹⡚⠋⠉⠉⠑⠀⠀⠀⢸⠍⠁⠀⠂⡀⣴⠻⠐⡱
⠀⠀⠀⠀⣾⠻⣅⣈⢿⣿⣷⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⡸⠀⡠⠀⠀⠐⣿⣢⠙⠏
⠀⠀⠀⠀⢹⣯⡐⠀⢥⣝⣿⣿⣿⣟⣛⡃⠀⠀⡜⠀⣴⡁⡀⠀⠀⠀⠀⠈⠛⠷⠒
⠀⠀⠀⢀⣀⣿⣷⣖⢾⣿⣮⣋⣹⣿⡇⠀⠙⠳⣜⠸⠇⠁⠈⠀⠀⠠⣶⣶⣶⣦⣤⣀
⠀⠀⢠⣿⠟⠙⣫⡭⣭⢹⢿⠟⢿⣷⣦⡀⠀⢀⠏⠀⠀⠀⢀⡆⣠⣾⣿⣿⠯⠊⠟⠩⠆⡀
⠀⠀⣿⠖⢀⠀⣁⡞⢿⠈⠀⣴⢿⣿⠿⢿⠀⡼⠀⣇⡴⢀⣿⣿⣿⣿⣿⣷⣤⡤⠤⢤⣀⣘⡀
⠀⠀⠹⣷⡟⣼⣿⣵⣼⣿⣿⣛⣿⠅⠁⠂⣀⠃⠀⠹⡾⣿⣮⣿⣿⣿⣤⠀⠈⠉⠓⠄⠉⠛⠁
⠀⠀⠀⠀⠀⠁⠉⢻⣟⣉⣉⣉⣉⡛⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⠿⢷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣾⣿⡿⢋⢸⣿⣿⣿⣶⡀⠀⠀⢀⣴⣷⣄⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠛⠟⠉⠀⠀⣠⣿⣿⣏⠺⣦⠀⠀⢸⢻⣿⡟⠳⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢛⣷⢊⣀⣠⣤⣶⠿⢟⡿⠋⣼⣷⣿⣧⣾⢛⠞⣿⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠀⢸⡏⡇⢀⡟⠀⠻⣿⣿⣷⣓⣻⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⡼⠁⠀⠀⠀⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠞⠀⠉⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡈
⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡷⠆⠀⠤⠀⠠⠂⠀⠀⠀⠀⡰⠒⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠘
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸⠀⠀⢀⠄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠃
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⢠⠑⠤⢀⠌
⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠁⠂⠀⠀⠄⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡟
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠆⡀⠀⠀⠙⠛⠉⠀⢀⡀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠙⠛⠉⠀⢀⣀⠀⠀⠀⠉⠙⠃
⠀⡄⠌⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣵⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡄⠌⠄⠀⠀⠁⠀⠀⠀⠀⠤⠤⠈⣽⠀⠀⠀⠰⡆
⠀⠘⠘⠀⠀⠀⠀⣀⡀⠀⠀⢸⡋⡁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⣀⡀⠀⠀⢸⡋⡁⠐⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢶⣀⡈⠁⠀⠀⠀⠋⠑⠛⠀⠀⢀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⡈⠁⠀⠀⠀⠋⠑⠛⠀⠀⢄⡀⠐⠂⡀
⠖⠀⠄⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⡄⠀⠀⠀⠄⠠⠉⠉⠁⠀⠀⠀⠀⠖⠀⠄⠀⠀⠀⠈⠑⠃⠀⠀⠀⢀⡄⠀⠀⠀⡀⠠⠉⠉⠁
⠁⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠄⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢀⡋⠀⠀⠀⠡⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠃⢀⠋⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢧⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢣⠈⠈⣀⠞
⠆⠀⠀⡌⢰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠳⢀⡄⠀⠀⡌⠰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠓
⠀⠀⠀⠹⣦⣄⡴⠏⠀⠀⡀⠀⠀⠀⣾⣆⠀⠀⠀⠐⠀⠀⠀⠁⠀⢸⠀⠀⠀⠹⣦⣄⡴⠏⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⢻⡿
⠀⠀⠀⠤⠀⠀⡀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⡠⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠤⡀⠀⠈⠙⠛⠉⠀⢀⣀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠙⠛⠉⢀⢀⣀⠀⠀⠀⠋⠙⠃
⠀⡄⠄⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣭⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠀⠀⠁⠀⠀⠀⠀⠠⠤⠈⣭⠀⠀⠀⠘⡆
⠀⠈⠘⠀⠀⠀⠀⡀⡀⠀⠀⢸⡋⡁⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⡀⡀⠀⠀⢸⡟⡁⠰⠆
⠀⠀⠀⠀⠀⢶⣀⡈⠁⠀⠀⠀⠉⠁⠛⠀⠀⢀⡀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⡈⠁⠀⠀⠀⠋⠑⠋⠀⠀⢄⡀⠐⠂⡀
⠖⠀⠄⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠉⠉
⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⠚⠃⢀⡋⣀⠠⠈⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢⠛⠃⢀⠋⢀⠀⠈⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢣⠈⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠧⠈⠀⣀⠞
⠆⠀⠀⡌⢰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⡳⢀⡄⠀⠀⡌⠰⠋⠷⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠳
⠀⠀⠀⠸⣦⣄⡴⠏⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠒⠀⠂⠀⠁⠀⢸⠀⠀⠀⠹⣤⣄⡴⠏⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠒⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⢻⡿
⠀⠀⠀⠤⠀⠀⣀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⢀⠀⢰⣶⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⢀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠱⠂⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠚⠳⠖⠋⢹⠂⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠠⣤⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠉⡆⠄⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣰⠁⠈⠿⣽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠀⠀⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⢀⢰⠋⠈⠉⠑⠠⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡔⠐⡄
⠀⠠⢠⡌⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡠⠀⠀⠀⠠⠈⠉⢉
⠀⠀⢈⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡤⠜⠚⠉⠓
⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠯⠙
⠀⠀⠀⠀⠈⠂⠤⠠⢀⣀⠀⣠⠆⠀⠀⠐⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢹⣀⠁⠀⠀⠄⠀⠀⠀⠀⠀⡠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⠈⠣⠄⡀⠀⠀⠀⠀⠀⡀⠋⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠈⠢⢀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠋⠀⢀⠀⠀⠤⡄⠀⠀⠄
⠀⠀⠀⢁⠤⠠⡀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠁⢀⠀⠁⠂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢐⠁⠀⠐⠀⠀⠸⡀⠈⠳⡄
⠀⢀⠀⠸⠤⠞⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡙⠳⡶⠶⡶⠶⠛⠀⠀⠀⠀⠀⠀⡋⠀⠐⠀⠀⠀⠀⠘⡆⠀⠈⢂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⡄⠀⠀⠀⠆⠀⠈⡀⠀⠀⢼⣷⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠿⠃⠿⠿⠿⠿⣆⠀⠀⠰⡀⠀⠁⠀⠀⣾⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠈⠀⢀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣧⡀⠀⠱⡀⠠⠂⢀⣭⣭⣏⣉⣉⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠀⣰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣦⡤⠊⠀⣴⣿⡉⠀⠙⢿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⢀⣠⠚⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡟⠠⠂⢸⡿⠃⢤⡄⠀⠀⠹⢿⣷⡄⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢰⣏⣉⠀⣩⣭⣭⣭⣭⡁⣩⣭⣭⣭⣭⣭⡸⠀⠀⡈⠀⠀⠠⠄⠀⠀⢰⡀⡘⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⡆⠀⣀⣀⣀⣀⣿⣿⡇⣿⣿⠿⠿⠿⠃⠛⣻⠿⠿⣿⣿⡇⠀⢀⠀⠀⠤⠀⠀⠀⠐⠀⢐⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠃⠉⠁⠀⠀⠀⠉⠉⠙⠓⠢⢄⡀⠀⠀⠀⠉⢬⠀⠘⢁⡿⠙⠁⢆⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠉⠳⡄⠀⠀⠀⠚⠀⠘⢇⣠⣤⣈⣦⣤⠀⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢻⠵⠶⠷⣖⣛⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⢃⠀⣀⣉⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⠄⠀⠀⠀⠀⠀⠑⡅⠐⠴⠀⠀⠉⠙⠝⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠛⢥⢀⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢠⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⣼⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⣼⡟⠀⠀⠀⠀⠰⣿⣿⡇⠀⢠⣶⣦⡀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣀
⢰⣿⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠻⠛⠁⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠛⠛⠛⠛⠛⠻⠿⢷⣶⣦⣄⣀
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀
⣿⡇⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣄
⢿⡇⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⢸⣿⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⢻⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠈⢿⣆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠙⢿⣆⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣦⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠉⠙⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣼⡟⠀⠀⠀⠀⢾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡀⠀⣾⡏⠀⠀⠀⠀⠈⠻⠿⠷⠀⠀⠀⢀⣶⡄⠀⠈⠙⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇

⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠻⠀⠀⠀⠀⠀⣀⡩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠁⠟⠀⠀⠀⠘⠠⢤⣤⢤⠜⠃⠀⠀⠙⠃⠀⢠⠴⣯⠈
⠀⠀⠀⠀⠀⠈⢆⠀⡬⠁⠀⠀⠀⢰⡦⠤⠀⠀⠄⣴⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⢻⡾⠿⣀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠘⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠐⠺⠉⠀⠀⠀⠀⠧⡀⣈
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠖⠀⠀⠀⠀⠀⣓⡀⠀⠀⠀⠀⠷⠆⢀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⢩⠙⢢
⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠊⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢏⡇
⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠶⠿⠶⠦⣴⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠈⢄⠀⠀⠀⢠⣴⠃
⠀⠀⠀⠀⠀⢀⣞⢑⡀⠀⠀⣈⣱⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⢻⠃⠀⠀⠈⠀⠄⢀⢠⠀⢸⠀⠀⠀⠀⠁⠲⡄⠊⠁
⠀⠀⠀⠀⠀⣨⡉⠲⠅⠠⠀⢨⣟⠀⠀⠆⠀⠀⠀⠐⣴⠀⡜⠀⠀⠀⠀⠈⠁⢰⡀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢹⣇⣀⣀⡀⠀⢸⣿⠀⠄⠘⠀⠀⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠰⠐⠤⠚⠀⠊
⠀⠀⠀⠀⠀⠀⠙⠦⠧⠤⠤⠼⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠚⠀⠀⠀⠀⠀⠲⠤⠴

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⢀⡖⣃⢀⣀⣠
⠘⠲⠤⡤⠀⣠⣠⡇⠀⠀⠀⠀⠀⠀⠀⠉⣧⠀⠀⠀⠀⠀⠀⢰⠉⠇⠀⢀⣀⠀⠀⢸⣈⣄⠀⢀⣄⢳⢺⠰⠾⡗⠛⠛⠋
⠀⠀⠀⠀⠈⠁⠂⠠⠌⠉⠀⠀⠀⠀⠴⠃⠿⢲⡤⠤⠤⣶⠀⠙⣃⡿⠀⠐⢸⡇⠀⠨⣃⣤⣉⠉⢻⡄⠀⡀⢠⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⡆⠚⠉⠀⠀⠀⠢⠐⠀⠀⠀⡉⠺⠀⢸⠧⠀⢠⠎⢣⠀⡆⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠗⠁⠀⠀⠠⢄⢠⢴⣇⠄⠀⠀⠀⠀⠀⠀⢀⠀⢠⡾⠀⠱⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠘⡇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠈⢁⣀⠆
⠀⠀⠀⠀⠀⠀⠈⡔⢁⠭⡉⢦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠆⡠⠒⠊⠀⠓⠺⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⡀
⠈⠠⢉⣉⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⢔⠀⠀⡃⠀⠀⠀⣀⣀⣀⠀⠀⠤⠀⠀⠠⠴⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠐⠀⠀⠁⠉⠁⡈⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠂⠀⠀⠂⠒⠒⡀⠀⠀⢀⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠚⢻⠛⠉⢹⠉⠉⠏⠉⢹⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣤⠼⠦⠶⠾⠖⠒⠛⣛⣋⣉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⠽⢦⡀⠀⠀⠀⠀⠀⣾⡏⠀⠙⡆⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⠚⠁⠀⠈⠳⣄⠀⠀⠀⠀⠻⣿⣦⠾⠁⠀⠀⢀⣈⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠛⠒⠶⠶⠿⣶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠤⠶⠶⠖⠒⢹⡇
⠀⠀⠀⢠⣟⣛⠓⠒⢶⣄⠀⠀⠀⠀⠀⠀⡷⠤⠤⠤⠤⠤⠖⠒⠒⠢⠤⣀⣀⣇⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡜⣷⠿⠿⠿⠿⠿⢿⣿⣶⣶⣶⣤⣄⠀⠀⠀⠀⢀⡇⠀⣄⣀⣀⠠⠄⣶⢸⣿⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣿⢿⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣯⢾⡇⠀⠀⠀⢸⡇⢸⡏⠉⢹⡜⠇⠙⠈⠁⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣧⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠙⠒⠒⠒⠊⠉⠉⠉⡏⣿⠀⠀⠀⠀⢸⡇⠀⠀⢠⠖⠳⣄⡰⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠉⢆⠀⡴⢺⡏⢦⣀⠜⠀⠀⠈⠀⣀⠀⢆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⢸⠀⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⡟⠉⠓⠶⢤⣄⣸⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣭⣒⣒⣒⣤⡤⢤⣔⣚⣥⠞⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠋⠀⠀⠀⠀⣀⣠⣤⠶⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⢀⣠⠾⠋⣼⠃⠀⣀⣴⠶⠛⠋⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠃⠀⠀⠀⣰⠟⠁⠀⣼⣏⣴⠟⠋⠁⠀⠀⠀⠀⣠⣾⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⡼⠃⠀⠀⢠⡿⠋⠁⠀⠀⠀⢀⣠⣴⠿⠋⠉⠉⠉⠛⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⠀⠀⢸⣇⢹⡄⠀⣸⠃⠀⠀⣴⠋⣠⠞⠋⢀⣴⠾⠛⢉⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣼⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠿⣾⣿⡘⣧⣰⠏⠀⢀⡾⠁⣴⠋⢀⣴⡿⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣫⣿⠻⣶⡀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣧⠸⣧⡀⠀⣾⠁⣼⠃⠀⣀⣤⠶⠒⠒⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠛⠁⠀⠙⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣶⠾⠛⠻⣧⠘⢷⡜⠃⣼⣏⣤⠾⠋⠁⣀⣀⣤⣶⠶⠿⠟⠛⠛⠋⠉⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⣀⣤⣤⣤⣶⠿⣧⣈⣻⡆⢹⠟⠁⠀⣴⠟⠋⠉⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠸⣿⡄⠀⠀⠀⣿⣦⠀⠀⢿⡆
⠀⠀⠀⠀⠀⣴⠟⠁⢀⠴⠛⠉⠁⢀⣀⣤⠔⣨⡿⠛⠁⠋⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘⠿⠀⠀⢸⣿
⠀⠀⠀⠀⣼⠃⠀⠀⣀⣤⣴⠶⢻⣿⠟⢁⡾⠋⠀⣰⠂⠀⠀⠀⠀⣼⠇⠀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⣰⠃⣠⣶⠟⢻⡿⠁⣰⡿⠁⣠⠏⠀⠀⣰⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠈⣿⡇
⠀⠀⣰⣧⡿⠋⠁⢀⣿⠁⢰⡿⠁⠀⠏⠀⢀⡼⠁⢿⡄⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⣶⡄⠀⠀⠘⠿⠀⠀⣿⡇
⢀⣴⠿⠋⠀⠀⠀⣼⡇⠀⣿⡇⠀⠀⠀⣠⡟⠀⠀⠘⢿⣦⡀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⣿⡇
⠚⠁⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⣰⡟⠀⠀⢠⣀⠈⠙⢿⣦⡀⠀⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⢠⡿⠀⠀⠀⠘⣿⡆⠀⠀⠙⢿⣦⢸⡇⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⠀⣾⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣷⢰⣦⠀⠸⣧⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠁⠀⢸⡇⠀⠐⣶⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⢀⡀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠁⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢸⣿⡀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⢻⣦⡀⠀⠀⠙⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣶⣦⣤⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠻⣷⠄⠀⠀⠘⢿⡧⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣶⣶⣴⣶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⢀⣴⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⡴⠆⠀⠐⠃⠀⠀⠐⠚⠂⡀⢀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⣀
⠀⠀⡘⠁⠀⣠⣄⠀⠈⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠘⡻⠁⠀⢰⠀⠠⠟⠂⠀⠀⢠⣄⠉⠁
⠀⠀⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠀⣀⠠⠁⠀⣾⠦⠐⠉⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡇⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠰⠦⠀⠀⢀⡤
⠀⠀⠻⠄⠀⢀⡀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠘⠁⠘⠃⠀⢀⣤⢤⡄⠀⣤⣤⣤⣤⣀⣰⣏⡼⢀⠀⠀⣠⢤⠀⠀⠀⠘⠺⢇⠀⠀⣉⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⣾⠇⢈⣋⠀⣨⢭⢩⣿⠭⡄⣮⣰⢒⢛⢮⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠸⠯⢠⣿⢸⣾⣧⣸⣾⣿⣞⠹⠧⠻⠅⢲⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣠⠿⣏⠟⠙⠳⢋⣈⣡⡿⢷⣋⣉⣱⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠘⠿⠿⠟⠋⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⢤⡀⠀⠀⠀⠀⠀⠀⣠⢇⣈⣁⠠⠤⣄⠀⠂⠀⣀
⠀⠀⠀⠚⢱⡄⠘⠁⠀⠀⣠⡀⠀⠘⠿⠀⠀⠀⠀⢀⠠⠞⠁⠠⡦⠄⠀⠀⣀⠀⠀⠀⠀⠀⠐⣿⠞⣉⠀⠀⠤⠀⠀⣀⠀⠁
⠀⠀⢸⣇⠀⠀⣄⠀⣤⡄⠁⢠⣤⡄⠠⢠⠆⠀⢠⡆⠐⠂⠀⠀⠀⠘⠀⠈⠉⠀⠙⠀⠀⠀⠀⣈⣤⡭⠓⠛⡛⠛⠁⠳⠄⣀⠁
⠀⠀⣾⣿⣿⣦⣦⣤⣠⣤⣄⣄⣀⣤⠤⠏⠀⠀⡰⠀⠈⠉⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⣄⠀⠉⠀⠀⣄⠀⠈
⠀⠀⠙⣻⣦⣤⠆⠈⠛⠉⠀⣀⣰⣿⣧⡾⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⡆⢀
⠀⠀⠈⣛⡛⠛⠉⠙⠛⠛⣻⡏⠉⠁⣀⣀⠤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠘⠀⠈⠁⠠⠒⠂⠀⠀⣷⠉⢀⡀⠀⠀⠙⠀⢀⠀⠆
⠀⠀⢘⡹⠋⢹⣗⠛⠛⡿⢓⡖⠚⠛⠙⣋⡀⠀⠀⠀⠀⣀⠀⠠⠔⠒⠀⠂⠀⠀⢀⡀⠀⠀⠀⠸⡆⠈⠋⠀⠤⠀⠀⠀⠁
⠀⠀⢾⡿⣻⡗⠶⢦⡶⠶⠶⠶⠚⢋⣍⠡⢠⠀⠀⠀⠀⠀⣄⠀⠀⠀⣶⡆⠀⠀⠈⠁⣰⠂⠀⠀⠈⠲⠤⣀⡀⠰⠖⡠⠖
⠀⠀⠀⠛⠻⠷⠶⠍⢁⣀⣠⠤⠤⠤⠿⠚⠁⠀⠀⠀⠀⢄⠀⢤⡄⠀⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠆⠀⠠⡄⢊⢹⣿⣷⣶⢖⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠓⠛⠓⢚⣿⠻



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠄
⣶⣦⣴⣿⣶⣶⣦⣤⣤⣤⣤⣤⣶⣦⣦⣤⣤⣶⣤⣤⣼⣿⣧⣬⣽⣿⣦⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣛⣻⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⣉⣉⡉⠉⠉⠉⣉⣉⣉⡉
⢿⣿⣿⡿⠿⠿⠿⠿⣻⣛⣟⣿⣿⡿⠿⡿⣿⣿⠿⠟⠛⣿⣿⠿⣻⣿⡋⠀⠀⠁⠀⠀⠀⠠⠶⠶⠾⠿⠿⠶⠀⢻⣟⣻⣿⡿⣿⣿⢻⡿⣿⡿⢟
⣾⣤⣿⡾⢶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠿⠿⠁⢾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠉⢠⣶⣴⣶⣷⣴⣶⣶⣿
⣾⣿⣿⣷⣼⣿⣭⣯⡙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣦⣀⣀⡀⠀⢙⣿⣿⣿⣿⣽⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠐⠂⠀⠀⢰⣆⢈⣿⣿⣿⠿⠿⢿⣷⣶⣤⣝⠙⣿⡏⠉⣻⡿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣾⣟⢿⣿⣋⡁⠀⠤⠖⠒⠲⢦⣤⣤⣄⣀⣀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣦⣶⡀⢭⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⢹⡏⠉⠁⠀⢈⡽⡅⠀⠀⢿⣯⠉⠉⠉⢉⡟⡏⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⣿⠀⠀⠀⠀⡇⡇⠀⠀⢸⣿⠀⠀⠀⢸⡇⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣼⣿⣤⣤⣤⣴⣷⣧⣤⣤⣼⣷⣤⣤⣤⣼⣷⣧⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣿⢦⣄⣀⣀⣤⠴⡎⣿⠀⠀⠀⠀⠀⠀⠀⣼⢳⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣤⣤⣴⠶⠛⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠱⡄⠀⣷
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠙⣆⢹⡆
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⢮⣷
⠀⠀⠀⠀⢠⡏⠀⠀⣠⠖⠋⠙⣆⠀⠀⠀⡤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⢸⡧⢤⠴⠁⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠹⣄⣀⣠⠋⠉⠑⡄⠀⢀⡟⠉⢳⠀⠀⡞⠉⢱⡄⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠓⠤⢼⡇⠀⣏⠀⢀⡇⠀⠈⠓⢺⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠉⠉⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣴⣦⠀⣠⡄⠀⠀⠀⣶⠀⢠⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿⠘⠟⢹⡇⠐⣃⠀⣿⠀⢸⣤⡾⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⠃⠀⠀⠿⠀⠟⠀⠟⠀⢸⠏⠻⣆⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡴⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⠶⠟⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠒⠶⠶⠶⠶⠶⠾⠛⠋⠉
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢺⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣇⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠌⢀⠞⠉⢧⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⡎⠉⠑⢦⡀⠀⡿⠀⠘⠀⠈⠀⠀⣴⠛⠶⣄⡀⠀⠁⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⡇⠀⣠⠒⠉⠙⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⢨⣿⠿⠿⠶⢤⣤⣄⣀⣁⣀⡀⠀⠀⠀⠀⠈⢢⠀⣿⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢴⠉⣉⡵⠂⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠦⣀⣈⣥⣤⣀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⣦⣀⠀⠂⡏⠀⠀⠶⣉⠉⢳
⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠤⣤⠟⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⠝⣧⣼⡇⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡿⠀⠀⢠⡏⠀⠀⢀⣀⣀⣤⡷⠀⠀⠀⠀⠀⠰⡧⢤⣀⣹⠆⢹⡿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢰⡇⠀⠀⠀⢿⠛⠉⠉⠈⠋⣠⠇⠀⠀⠀⣶⠀⠀⠳⢤⠴⠋⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣼⡇⠀⠀⠀⠈⠳⢤⣤⡤⠞⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⠟⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣉⣧⠐⡄⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⢀⠀⢿⡀⠀⠀⠀⠀⢠⣾⣿⣷⣭⣿⠀⠀⠀⠀⠀⢀⡾⠃⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣅⣄⣀⣘⣧⠀⠀⠀⠀⣿⣘⣻⣿⣿⠏⠀⠀⠀⢀⡴⠋⠀⠀⢿⣐⠏⠉⡷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠉⠉⠛⠿⠀⠀⠀⣰⣟⣡⣅⣀⣠⡟⠉⠛⢺⠀⢠⠛⣦
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠖⢋⣩⠽⠚⠛⠻⣶⠒⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉⣷⠀⠚⡟⠲⣇⡠⠏
⠀⠀⣠⠖⠋⠉⠉⠁⠀⣴⠋⠀⠀⠀⠀⠀⠛⠶⢶⡞⠻⢦⣄⣀⣀⣀⣠⡴⠛⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠄⠐⠉
⠀⠀⣯⣠⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠉⠉⠉⠀⠀⣷⣼⣿⣄⣀⣀⣠⡴⣋⡴⠞⠁
⠀⠀⠀⢿⣠⡔⠀⣴⠤⠟⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠏⡔⠙⣇⠀⠀⠀⠀⢸⡿⠛⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣠⡟⣡⠊⢠⡴⢻⡆⠀⠀⣠⡼⢷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠏
⠀⠀⢀⣾⢋⡴⠁⣼⣿⢿⡅⢻⣄⣸⣿⡀⠈⢻⣦⣄⡀⠀⠀⠀⠑⠲⢶⣯
⠀⣰⠟⣡⠛⡄⢠⡞⠃⠀⢻⣄⣉⡽⠋⠙⠞⠋⠀⠈⠙⠳⠶⣤⡴⠾⣿⣻⠶⢦⣄⡀
⣾⣋⣞⣁⣠⠿⠋⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠆⣸⡇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⠾⣿⣾⡷⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢤⠤⠤⠍⣁⣀⣀⣰⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠶⠶⢦⣤⣯⣭⣉⣗⣚⣧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢈⣉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠈⠉⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠒⣛⣓⣦⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠞⠋⢉⡟⢉⣩⡽⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠒⣶⠟⠋⠉⢁⡴⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣰⠃⣀⣠⠴⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣉⣀⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⡤⠔⠒⠒⠒⠒⠁⠐⠚⠁⠀⠻⠹⠇⢤⠤⠤⠤⠖⠒⢂⡽⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀⠿⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⢀⡤⠶⠒⠉⠑⢶⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣦⡄
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠁⠀⠀⠀⠳⣀⠇⣄
⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⣠⠞⢁⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠘⣆⠀⢀⠀⠀⡀⠀⠀⢀
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣠⠶⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⠄⠀⢹⡄⠀⠘⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣠⠔⠒⠊⢹⠀⣠⠞⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⢰⣘⣇⠀⠈⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡰⠃⠀⠀⢠⠇⠊⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠘⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡼⠃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢇⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠁⠀⠀⠀⠀⢀⡜⠁⡀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠃⠀⠀⢸⣤⠁
⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢧⡄⠀⠁⠀⠀⢸⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⣼⡤⠆⠀⠇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠘⢆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠠⡿⠁⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⢷⠈
⠀⠀⠈⢣⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠁⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠻⣄⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣤⣤⣤⡤⠤⠤⠘⠇⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣦⡀⠀⡼⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡜⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠻⠧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⠻⣍⠉⠑⠶⣴⠲⠶⠖⠒⢫⣉⠁⠉⠑⢦⡀⠀⠀⠉⢲⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄⠀⠈⢳⣤⠤⠤⠒⠉⠳⣄⠀⠀⠀⠀⠀⠀⡟⠀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠙⠆⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⠚⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⡆⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⡆⠀⠀⠀⠀⠙⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠞⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣊⠉⠛⠲⣄⡴⠀⠸⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣧⣄⣳⡸⡇⢰⠀⠘⣦⠀⢀⡗⢠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀⠉⠙⠳⢼⡆⠀⣿⡴⢛⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢴⠆⠀⠀⠚⠋⠀⠀⠀⠀⠙⢦⡛⠉⠉⠛⠓⠒⠲⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣄⠀⠀⢀⡤⢀⠀⠀⠀⠹⣄⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡡⠀⢠⡷⠀⠀⠈⠉⠀⠀⠀⠐⠁⠀⠞⡰⠀⢹⡉⢉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡄⠀⠀⠀⠀⡈⣧⣈⠢⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠗⠀⠀⢞⠔⠀⠈⠕⠂⠀⠀⠈⠒⠁⢀⡴⡀⠇⣿⢹⡗⠒⠤⠌⠂⠀⠀⠀⠀⠀⠀⠀⢀⡏⣆⢀⡤
⠀⠀⠀⠀⠀⠀⢰⢧⠀⢀⣄⠀⠀⢀⠀⠀⠀⢠⡤⡀⠀⠀⣀⠀⠀⠊⠀⢠⡟⠂⠁⠀⠀⠀⠀⣠⠴⠖⠚⠛⠓⠶⣼⡇⢹⡏⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠋⠀⠈⠫⠆⢀⠀⠈⠉⠀⠀⠈⠥⠊⣠⠔⣠⠟⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⡀⠀⠀⠀⠈⢳⡄⠃⣸
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠴⢂⠀⠀⢘⡋⠔⣀⣀⠤⠴⣖⣚⣭⠶⠛⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠠⣤⡽⠗⠒⠀⠀⠀⠻⣶⣛⠲⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⣀⣀⣠⣬⠽⠶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣨⠶⠋⠁⠀⠀⢀⡴⠀⠀⠀⠈⢯⡙⠦⣄⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠠⣤⠞⢁⡤⠖⢊⣤⠞⠋⣀⢴⡂⠀⢘⢸⣧⣀⡂⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢈⡤⢚⣩⡴⢞⣉⣤⡖⠋⡇⠈⠃⠀⢊⣾⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠄⠀⠛⠛⡋⠓⢚⠏⠀⠘⠀⠀⠀⣀⣠⡴⠛⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣐⣁⣈⣁⣤⣤⡤⠶⣶⠾⠿⠭⠉⠀⡠⢢⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠭⣭⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠬⠞⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⣀⡴⠶⣤⠴⠶⠶⠶⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠐⠃⣠⠞⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠢⡀⠀⠀⠀⠀⠀⠀⣰⣲⡌⣧⠀⠀⠛⠛⠒⠋⠀⠀⠀⠀⢀⡴⠛⣦⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋⠀⣼⠃
⠀⠀⠀⠀⠀⢠⡟⠁⠈⢳⠀⠀⠀⠀⠀⠀⣀⣀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃
⠀⠀⠀⠀⠀⢘⡷⢦⡤⠞⠓⠀⠀⢀⣠⠎⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣞⡦⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠉⠈⠳⠤⠴⠚⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠠⢤⣤⣴⠖⠾⣏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⡤⠶⠋⠁⠀⠀⠀⠉⠳⠦⣄
⠀⠀⠀⠀⠸⣦⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⢤⠞⠀⠀⠀⠀⠀⢀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⢠⡏⠀⠀⠈⠉⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣸⠀⠀⠀⠀⡴⠒⢢⡀⠀⠀⠐⠢⠀⠀⢺⡄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢠⡏⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⢠⡞⠁⡀⠀⠀⠘⠳⠶⠖⠃⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠘⢳⡆⠀⠀⠀⠈⠳⠤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠋⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⢴⡞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⡀⢠⡀⠀⢸⠁⠀⠀⠀⢹⠔⠒⠊⠉⠀⠀⢀⡴⠛⠲⠶⠶⠚⠓⠒⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢧⡀⠀⠀⠀⠉⠁⠈⠙⠛⢦⣤⣤⠶⠋⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⣤⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠴⠶⠶⠶⠶⠶⠦⠤⠤⠶⠋⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⣤⡼⠇⠀⠀⠀⠀⣠⣤⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⡟⣋⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⣙⣃⠈⠀⠅⠈⠀⠀⢀⠈⢠⠀⠸⢧⣀⡀⠠⣿⣭⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠮⠵⠛⠀⠀⡀⠈⠒⠲⠖⠊⠁⠀⠀⣁⡧
⠀⣿⠁⠀⠀⠄⠀⠀⠀⠈⢀⠀⠀⠀⠀⠁⠉⠻⣿⣧⣀⠀⠀⠐⠀⠠⠰⡄⠀⠀⠉⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠀⠀⠀⠀⠀⠀⠀⠲⠖⠁
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠤⠀⠄⠎⠁⠉⠉⠁⠀⠐⡾⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⡄⣠⡤
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣇⣑⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⣾⠁⠀⠀⠀⠀⠀⢠⡼⠏
⠀⠀⠃⠔⠀⠒⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠒⠀⠀⠀⠀⠀⠀⠑⠘⠃⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⢀⡺⠇⠀⠀⢀⡰⠖⠊⠉
⠀⠀⠆⠀⢐⣔⡼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠠
⣡⣆⡀⠀⠀⢻⣡⠶⠞⠛⠁⠉⠉⠙⠃⢀⠀⠀⠀⢤⠀⠀⠀⠐⠀⠀⢠⣤⠀⠀⢀⡘⠳⢻⣷⡶⠑⠮⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠿⡷⠄⠀⠺⢣⣽⣦⠴⠤⢤⡀⣤⠊⠀⠘⠀⠀⠀⢀⣀⠀⣀⠐⠀⠀⠌⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀
⠋⠠⣄⣤⣈⡿⡟⠃⣿⡷⠿⢿⣄⣁⡀⠈⠓⢠⠤⠿⠛⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠂⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣴⠂⠀⠈⠉⠐
⡄⣀⠝⣯⡯⠰⢇⠀⡿⠀⠀⠀⢢⢽⣧⠀⠀⠈⠀⠀⠀⡀⠀⠀⢉⡼⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠆⠢⠀⠀⠀⣮⠾⠠⠄⠀⠀⢀
⠛⠋⢁⢿⣶⠶⡌⠑⢇⠀⠀⠚⡙⣸⡿⠀⠀⢀⣀⣀⡄⠠⢠⡶⠁⠀⠀⢰⢀⠀⠀⠀⠀⢀⡀⣁⡠⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠈⠙⠂⠀⠀⠀⠀⠙
⠀⣠⠟⠣⠀⠊⠀⠀⠈⠆⢀⢉⣴⠟⠁⠀⠁⠁⣀⢼⡁⢤⡯⠀⢀⡠⢤⣏⣠⣠⣷⣦⣄⠀⠀⠉⠀⢀⠂⠌⠀⠁⠲⠄⠀⠀⢀⠀⣀⠀⠀⠀⣄⡀
⡰⠃⠀⠀⠀⠀⢠⣀⣀⠈⠓⠊⠁⠀⠀⠀⡀⠀⠈⢻⠗⠾⣶⣠⣯⣤⣿⠏⠁⠘⠲⡜⢻⣧⠀⠀⠀⡀⠀⠈⠡⣄⠀⠀⢲⠁⢀⡀⠈⠆⠀⠘⠻⠃
⠁⠀⠀⢀⣤⣶⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠗⢪⣿⠃⠙⢻⡄⠀⠀⠼⢂⠘⣿⠀⠀⠀⠗⠀⠀⠀⠘⢦⡄⠁⠠⠎⠀⠀⠠⠀⠀⠀⠀⡠
⠀⠀⠀⠟⠈⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣭⡠⠅⣈⣉⡟⢳⡦⣤⡻⠦⢀⠀⠈⠈⡿⠀⠀⠀⠸⢦⡀⠀⠀⠈⣿⣭⡇⠀⠀⠀⡈⠀⠀⠀⠀⠠
⣢⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣉⡙⠆⠰⢾⣽⣹⡆⣁⡥⡄⠀⠀⠉⠓⠚⠁⠀⢀⢀⠄⠐⠛⠒⠛⠒⠿⣿⣷⣤⣤⠄⠀⠀⠀⠲⢶⣖
⡉⠁⡀⠀⠀⠀⠀⠀⠀⠈⠠⠇⠀⠐⠦⢠⣤⠔⠈⠀⠀⠀⣢⠘⢟⠿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⡺⠟⠀⠸⡟⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⡡⠊⠀⠌⠀⠀⠀⠡⠀⠀⠀⠀⠿⠀⠑⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⢦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣻⠀⠀⢀⠀⠀⠀⢄⡲⠿⣦⣄⠀⢸⠀⠀⠀⠻⡁⠀⠀⠀⠀⢀⠼⠇⠀⠀⠀⠁⠀⠈⠙⠃⠈⠁⠄⠀⠀⠀⢀⣤⠞
⠀⠀⣠⡀⠀⠀⠀⠀⠁⠀⠀⠀⢿⡀⠀⠠⠀⠀⠀⠀⠈⠑⠺⠟⡅⠈⠂⠀⠀⠀⠙⢄⡀⠀⠀⢼⡝⠁⠉⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠈⠉⠁
⡢⠞⠁⠈⡄⠀⠐⠠⢉⠀⠀⠀⢺⣧⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠠⠄⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠁⠀⠀⢨⠇⠄⢠⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣈⠄⠀⠀⠀⠁⠀⠀⠀⠀⠲⠀⠀⠀⡀⠀⢀⣶⢆⠀⠀⠀⡀⠁⣀⠂⠀⠤⠴⠖⣴⠆⠀⣴
⣠⠔⢢⣿⡶⠶⡛⡻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⢠⠁⡀⠈⠀⠀⢠⣜⠁⠀⠆⠀⠀⠀⠉⠁⠀⠈⠀⠐⢢⣼⡄⣾⠃
⢷⢶⣿⡏⠀⠀⠀⡝⠨⣿⡆⠀⠀⢀⠠⠀⠄⢠⠀⠀⠀⠀⢺⣿⠐⠀⡆⠀⠀⠰⠃⠀⠀⠀⢀⡴⠋⠈⠀⠀⠄⠀⠀⣀⠀⠀⠀⠈⠄⠀⠀⠝⣉⣟⣻
⣧⢀⠈⢷⡀⠀⠈⠉⠀⣹⡇⠀⠀⡀⠀⠙⡄⠀⠈⠢⠀⠀⠠⠀⠀⠀⠡⡀⠠⠀⠀⠐⠀⢄⣼⠁⠀⠀⠀⠄⠀⠀⠐⠿⠇⠀⠀⠀⢠⣂⠤⠀⠙⠡⢾
⣀⠛⠸⠒⠙⠑⠢⣀⢀⡼⠁⠀⠰⠁⠀⠀⠈⡷⠀⠀⡗⠀⡑⡀⢦⠀⠀⠑⠀⠀⠀⠀⠀⣾⡇⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣷⠂⠐⢠⣦
⠧⠠⠞⠳⠀⠀⠀⠀⠀⠀⠀⠠⠰⠆⠀⠀⠠⠱⠆⠠⠂⠴⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠈⠆⠀⠰⠀⠀⠀⠀⠀⠀⠈⠔⠺⠁⠚⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠤⠼⠤⢠⣄⣠⣰⡀⡇⣰⣶⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠛⠁⠉⠢⡀⠀⠀⠀⠀⠀⠀⠐⠔⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⠋⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠈⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠳⣄⣀⠠⠤⠖⠒⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣠⡠⠤⠔⠒⠈⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠀⠀⠈⠉⠉⢉⡍⢹⠀⢀⡀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⣀⡀⢀⠀⢀⠀⢸⠀⢸⡇⣼⠁⠀⠇⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠸⠁⠃⢸⠀⣿⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠈⠀⠛⠀⢸⠋⢷⡄⢀⡀⠀⣠⡠⠴⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠄⠒⠒⠒⠒⠒⠒⠒⠀⠁⠀⠀⠬⠭⢼⠗⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠚⠋⠙⠚⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣩⣵⠄⠀⣀⣀⠤⠼⢦⣈⢧⣚⣻⣤⡷⠧⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣧⡔⠊⠉⠁⠀⠀⠀⠘⣮⠏⠃⣟⠉⠀⠀⠀⠀⠀⠈⠙⢢⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⡀⢣⣀⣀⣀⠤⢤⡄⠀⠿⠃⠯⢿⠀⠀⢠⢤⠤⠤⠤⠤⣼⣇⡤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠿⣯⠉⡇⢀⢤⠔⢁⠇⠀⠀⠀⣀⡸⡆⠀⠈⣎⠳⣄⠀⢀⣷⠈⣅⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣿⠀⡇⡎⠀⠀⡾⠀⠀⠀⠀⠳⠋⡇⠀⠀⠘⡄⠘⡆⡏⠁⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⠀⡇⡇⠀⢠⠇⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⢹⡀⢧⢳⠀⠀⠸⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡏⠀⠧⠃⠀⢸⠀⠀⠀⠀⠀⠓⠃⣃⣀⠀⠀⠀⣧⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⡏⣳⢒⡶⠶⠿⢏⠉⠉⡟⠛⠯⢿⡁⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢠⣧⠀⠀⡇⡇⢹⡀⠀⠀⢺⠀⢰⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠛⠀⠀⢧⠃⢠⠃⠀⠀⣞⣀⣸⠀⠀⠀⢹⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠳⠛⠉⠉⠉⠀⠉⠉⠩⣇⣀⣀⠔⠉⠉⠈⠢⠤⣀⣀⣹⠀⠀⠀⠀⠀⠉⠉⠉⠉⣙⡦
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠦⣀⠀⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢐⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠀⠀⠈⠉⠉⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣀⠀⠄⠐⠀⠀⠀⡀⠀⠀⢀⠛⠁⢂⠀⠀⠀⠤⠜⠉⠀⠀⠀⠈⠀⠀⠂⠀⠉⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠞
⡔⠈⠁⢱⠀⠀⢲⠂⠀⠀⠁⡄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣌⡀⠀⠀⣄⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠐⠄⣀⣀⣠⢴⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⢨⠁⠈⠉⠃⠀⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠘⠁⠐⠀⠀⠈⠛⠁⠠⠖⠀⢀⠀⠀⠠⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠃⠈⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠢⡀⠀⢄⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⠈⠁
⠀⠀⠀⡤⠯⠤⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡓⠀⠂⠀⠂⠀⠀⠆⠀⠀⠃⠐⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠰⠀⠀⢀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣦⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢄⡀⠀⠀⠀⢸⢻⣴⢈⠈⢻⡄⠀⠐⡇⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠤⠤⠄⣴⠇⢠⡠⠀⠹⠀⠀⠀⢸⣿⠆⠗⢃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢑⣄⠀⠀⠀⠀⠀⣀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠺⣶⣺⡛⠤⡼⠀⠀⠀⠠⣿⡇⣤⣄⢀⠐⢺⠈⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⠤⢀⣈⠈⠁⢹⠋
⠀⠀⠀⠀⠀⠀⢸⡟⡛⠋⢻⠆⠀⠈⠀⠛⣹⢿⠛⠉⠀⠀⠀⠀⠀⣿⡏⣦⠈⣤⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣆⠘⢩⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣠⣖⣧⣤⣀⡄⠺⡏⠀⠀⠀⠀⠀⠀⠀⣽⡉⣄⡿⠀⢀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠁⢂⠎
⠀⠀⠀⠀⠀⠀⠀⢻⣯⡭⠤⠈⢠⡈⠉⢰⢢⠃⠀⠀⠀⠀⠀⠀⠀⣿⢷⡴⠟⠩⢾⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠈⣧⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡿⣤⣀⠀⠶⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⠀⠀⠀⢖⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⢀⣀⣀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⢀⡌⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠢⣠⡀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣌⣄⣴⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠒⠂⣀⠶⠶⠔⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡆⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⠄⠉⠉⢀⠀⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⡄⠐⠐⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡀⢀⡀⠀⣄⠈⠇⣐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⢿⡆⠀⠀⠰⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣀⣶⠒⣳⣦⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢾⣷⣄⣤⣼⢘⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⣩⠋⠁⠀⢨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢟⡉⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢰⡧⠳⠢⠔⡀⢀⡸⠰⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡛⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⠜⠁⠀⠀⠙⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣀⣄⠄⠈⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠁⠘⠈⠺⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⢀⣴⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠇⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠒⠒⠒⠛⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠉⠚⠂⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣀⣠⡴⠶⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⣴⠛⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣀⣠⡴⠾⠟⠛⠛⠛⠛⠳⠶⢿⣋⠉⠛⠛⠛⠳⠶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣼⠋⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⡾⠃⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣿⠁⠀⢠⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⡴⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣀⣀
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠉⠻⠶⣦⣀
⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣤⣶⣶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠋⢹⡟⠋⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣆⠀⠀⠀⣾⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣍⠻⢦⣴⡟⠀⠀⠀⢀⣼⠟⣷⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠹⢦⣀⣿⠀⠀⠀⠀⣾⣿⠀⠘⢷⣄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡏⠀⠀⠀⢰⡏⠹⣆⡀⠀⠙⠛⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠐⢷⡀⠉⠳⠶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣀⣠⣤⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⢉⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⣤⣤⣤⣶⣶⠶⠾⠛⠛⠋⠉⠀⠠⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⡀⡰⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠌⠁⣲⣾⣿⡏⠥⠉⠙⣦⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠴⠖⠊⠁⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡞⣥⡶⠀⢙⠿⢯⠱⠄⠀⠀⠉⠶⣭⣔⠒⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠐⠒⡀⠀⠀⠼⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⢰⡄⢫⠔⠟⠀⠀⠀⠀⢁⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠐⢃⣀⠀⠀⠀⠴⠃⠄⠈⢷⡐⠀⡀⠀⠀⠀⠀⠀⠸⠑⡃⠘⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣠⢀⡄⠀⠰⠀⠉⢠⢃⡀⠐⡀⠀⠀⠀⠀⠘⠓⠇⠜⠁⠀⠀⡀⠀⠀⢠⠀⡀⢀
⠀⠀⠀⠀⠀⠂⡄⠀⠐⠁⠀⠀⠀⠀⠚⠀⠀⠀⠇⢸⡃⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣩⠶⠶⠗⢶⢤⣩⣄⠀⠤⠀⠥
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⢁⠀⠐⠋⠀⣀⠄⠒⡀⢳⠝⠂⠀⠀⠀⠓⠤⣆⡤⠄⣉⠔⠋⠁⠃⠀⠰⠀⢀⠀⠐⡈⢠⡀
⠀⠀⠀⠀⠀⢀⠋⠀⠀⡀⠉⠠⠔⠀⠀⡁⠀⠐⠁⠀⠀⠀⠀⠺⣠⠆⡔⢈⠙⡚⠁⠀⠇⠀⠨⠆⠀⢀⠘⠀⠀⠇⠐⡈⠂
⠀⠀⠀⠀⠀⠘⣀⠀⠊⠀⠀⠠⠄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⡌⠠⢘⢃⢀⣰⡃⡆⠀⡀⠀⡂⠀⠀⠠⠀⠀⡐⠀⠀⠁⣴
⠀⠀⠀⠀⠈⠀⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⡈⠤⣠⠇⠀⠀⠃⠀⠁⠀⢠⠀⢀⠄⠀⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡄⠁⡄⠐⠀⠀⠸⠄⠀⠃⠀⡈⠀⠈⠀⠐⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠊⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠃⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢀⠀⠇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢔⣹⣃⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠋
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⢡⣍⣤⡀⢀⣀⠀⠠⠭⠀⠐⠀⠀⠙⠪⠢⠀⠀⠁⠐⠤⢤⣀⣀
⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠉⠉⠀⠉⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢒⠱⣧⣧⠭⣀⠀⠀⠀⠀⠉⠛⠿⣭⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣢⣀⡅⠀⠀⠀⢶⠂⠁⢠⠂⢡⠀⠀⠠⡍⠀⠀⠀⠀⠀⠀⡀⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⡷⣿⡆⢅⣄⠀⢶⢂⠀⡤⠀⠀⢰⠐⠂⣤⠀⠀⠀⠀⠀⠀⡅⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠁⠀⢰⠂⢹⣄⠝⠖⢲⡌⡁⢠⠉⠰⠆⠀⢧⡌⠀⢀⣠⣤⣦⣴⡂⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣤⢊⢿⠀⠈⣉⢰⠔⠈⡄⠑⢄⠢⠐⠆⠜⠈⠀⠘⠏⠌⠁⠝⢿⡝⡶⠡⠞⡿⢿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡉⣌⠙⠀⠰⢀⠈⠀⠘⠀⠁⠀⠀⠁⠠⠼⠂⠀⠐⠃⠃⠚⠀⠀⠺⣡⣷⠆⢠⡁⠸⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⡀⠱⠀⠐⠂⠨⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⣩⠈⡄⢡⠟⣅⢿⢈⡁⡂⠙⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡘⠆⠙⠀⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠄⣁⠶⠀⠁⠀⢥⡀⣌⠁⢣⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠎⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠖⠒⠒⠒⠲⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⣀⠀⠀⢀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⣿⣿⣷⣦⡀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⠿⣿⡇⢰⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠟⠋⠁⠀⠀⢀⣤⡀⠀⠘⡇⢸⠁⠒⠀⠤⠤⠤⠤⠤⠀⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡦⠀⠈⠁⠀⠀⠀⠀⠀⣼⣿⣟⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⣿⣆⠀⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢀⣤⣄⡈⠛⠿⠻⠿⣿⡿⠃⠀⡆⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⠀⢀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⣠⣤⣾⣿⣿⣿⣿⡇⠀⠀⠀⠸⢿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠙⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⢀⠜⠀⠀⢸⣰⢀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠢⠤⠤⠼⠿⠿⠟⠛⠒⠁⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠿⠿⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⣰⣾⣷⣦⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⢠⣾⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠻⣿⣿⡿⠀⠀⢀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣿⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣸⣿⠋⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠘⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠈⠻⠿⠟⠁⠀⠀⣤⠀⠀⠛⠿⠟⠁⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⣶⡆⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢀⣤⣦⣤⠀⠀⠀⢀⡀⠀⠀⢀⣠⣀⠀⠘⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⢸⣿⣿⣿⠇⠀⠀⠙⠁⠀⠀⣿⣿⣿⣷⠀⢸⣿⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠁⠀⠀⣻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣶⣶⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣃⣀⣚⣃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠯⠯⠿⠽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠟⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠠⠤⠤⠤⣤⣤⣤⣦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡉⠀⣠⣶⣿⠇⠀⠀⠀⠭⠉⡁⠀⡐⢀⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⣿⠏⠓⡉⢉⠀⡀⠀⠀⠄⠁⠀⠹⠿⠿⠷⣁⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣷⣰⡟⣿⠤⢤⣀⣝⠲⣀⠀⢀⣠⢄⣂⣀⣰⠾⣚⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡈⠒⠚⠋⢾⠃⣤⣀⠈⠋⣉⣉⣿⣗⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠋⠉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣔⣨⣻⠿⠿⠻⢒⡁⠈⠓⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠶⣿⠿⠠⢆⣐⠠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠊⠀⠜⠛⣿⡷⡛⠂⠐⠿⣿⣷⡤⡀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠸⣻⣆⠠⣤⣴⣤⡀⠘⠿⠿⠃⠠⠘⣿⣧
⠀⠀⠀⠠⡖⠁⠀⣠⠀⣀⣤⣌⠀⠀⣤⣤⣄⠀⢈⠹⣟⣷⡀⡌⣇⠀⠀⠀⠀⠀⠀⠸⣿⣇⣆⡤⢙⡛⢦⡁⠀⠀⠀⠀⠀⢀⠁⠰⡀
⠀⠀⠀⢸⣷⡀⢀⠠⠾⠿⠿⠏⠰⠀⠙⠻⠿⠗⠈⠉⠨⠝⡁⣸⢿⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣄⠹⣿⠆⠹⣆⠰⡈⢷⡄⠹⢿⢀⣧
⠀⠀⠀⠀⢿⣼⡧⣀⠁⠠⡄⢀⡀⣴⣶⣿⡷⠠⠤⣔⣲⣯⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⣄⢀⠃⠧⡘⠛⠣⡀⢰⡿⠋
⠀⠀⠀⠀⠀⠈⠛⠻⠶⢦⣤⣵⣦⣤⣦⣤⡴⠶⠿⠿⠛⠋⠀⢀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢼⣶⣿⣲⣾⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⣠⣤⡄⠻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠶⢶⠤⠖⡾⠖⠚⠋⠛⠒⠶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠉⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠉⣑⠦⠤⢠⡀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⡴⠋⠀⠀⠠⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠐⠀⠱⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣀⠀⠙⢦
⠀⠀⡼⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠤⡠⠇⠀⠀⠀⠀⢀⠄⠢⢄⠀⠈⠀⠀⠈⠳⣄
⠀⣸⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠸⡀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠠⡀⠀⢀⠇⠀⠀⠀⠀⠀⣸
⢠⡏⠀⠀⠀⠀⠐⠁⢱⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠔⠠⡄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢻
⣏⠑⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢸⡀
⢻⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠊⠒⢄⠀⠀⠀⠤⠀⠐⠀⠀⢱
⣼⠐⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢧⠀⠁⠀⠀⢀⠀⠜⠀⠄⠀⠀⠀⠀⠀⠁⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣸
⣸⠀⢄⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⡄⠁⠀⢱⠀⠀⠁⠀⠀⠀⠀⣀⣀⡀⠀⠂⠀⠀⠀⢀⡠⢤⡼⠁
⢻⡀⠈⡆⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠣⣀⣀⠜⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡀⠀⣇
⠀⢳⡀⡃⠀⠀⠀⠀⠑⠒⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠓⠄⠀⠂⠀⠀⠀⠀⠀⠀⣹⠏
⠀⠈⣇⠐⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠹⣄⠀⠈⢂⡀⠉⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠊⠀⠀⢀⣰⡿⠋
⠀⠀⠀⠙⢷⣄⣀⡈⠂⠠⠤⣀⡀⠀⠀⠠⠒⠀⠐⠄⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠀⣉⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⠀⠈⠙⣇⡀⠀⠠⠇⠀⢒⣉⣥⡤⠤⠴⠶⠖⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡏⠚⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⡻⠿⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⢿⣿⡟⠁⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⢀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠂⠑⠛⠘⠀⠃⠀⠃⠂⠃⠋⠞⠁⠐⠃⠀⠋⠘⠁⠐⠑⠃⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣤⠖⠀⢖⠂⢰⠀⡀⠐⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣤⣤⣠⣄⣬⣄⣠⣸⡀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣯⣤⣤⣬⣤⣶⣴⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣶⣿⣿⣿⣯⡿⠛⠛⠛⠉⠉⠻⠯⠯⠿⠯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣶⡞⠛⢿⡛⠛⠙⠉⣿⡏⢰⡖⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇⣰⠼⠁⠀⠀⠀⣿⡇⢙⡋⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢲⠿⠗⠻⢳⣶⣶⣶⡶⠿⠷⣿⣷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠹⢿⣦⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⢀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣾⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⣿⣿⠁⢀⣾⣿⡟⠉⠁⠐⠲⠿⢿⣛⡺⠟⠿⠿⠟⠛⠟⠛⠿⠍⠛⠉⠉⠀⠀⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⢸⣿⡟⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠀⣴⣿⣿⢃⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⡇⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⠿⠄⢴⣿⡿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢠⡀
⢀⡴⢂⡀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⢇
⢿⡆⠘⠀⠀⣴⢿⣷⣾⡟⠀⠀⠀⠀⠀⠀⢣
⢀⣷⠀⣠⣾⠏⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠸⣛⣛⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠉⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠰⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣀⠀⡠⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠒⠒⢤⣠⠤⡤⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⢀⠔⡠⢂⠆⢠⠀⡀⠀⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡠⢀⠄⡠⢂⠔⡡⢚⠔⢁⠔⡡⢂⠔⢁⠜⢠⠊⡔⠁⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠠⠂⠔⡡⠊⢐⠑⡠⢀⢐⠕⢈⠔⡪⠂⡥⢊⠔⡡⢊⠔⡡⠊⠔⢁⠔⡡⢊⠔⢁⠔⢁⠔⡡⢊⠔⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣀⣀⣁⣈⠀⠁⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠊⠀⠁⠊⠊⠚⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⡤⠀⢀⣠⠀⠀⡀⠀⠀⠉⢉⠏⠉⢩⡭⠧⢉⠜⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣔⠃⠀⠉⣀⠀⠘⠁⠀⠈⠁⢀⡸⢃⡀⠀⡰⠃⠀⠀⠚⡠⠀⠁⠀⠀⠀⠓⠁⠀⠀⠀⢠⠞⠰⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠞⠀⠸⠁⠀⠀⢠⠞⢁⠜⠁⠀⠀⠀⠚⠒⠒⠈⠉⠭⠭⠄⡠⠋⠀⠀⢀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣴⠃⠀⢀⡄⠀⣠⠀⠀⣀⠖⠀⠀⠰⠇⠐⠒⠒⠒⠲⠒⠀⠀⠀⡨⢁⠔⠉⠀⡼⠁⠀⠀⣠⠄⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠀⠟⠀⠸⠂⡀⠀⠉⠀⠀⠀⠀⠀⡴⠀⢀⠔⠁⠀⠀⠠⢢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡨⠀⠀⢀⡦⠀⠈⠃⠀⢠⠆⠀⢠⠄⠀⠀⡠⠊⠀⠀⠀⡀⠜⠁⡀⠀⠀⠀⣰⠂⠀⠀⠰⠃⢠⡆
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠈⠁⠶⠀⠈⠀⡾⠀⣠⠀⠀⠀⠀⠋⠀⣠⠊⠀⠀⠀⡠⠊⠀⠀⠞⠁⡴⠀⠀⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠓⠀⠀⠀⠈⠁⠀⠁⠀⠐⠟⠀⢀⠔⠁⠀⠀⡠⠊⠀⠀⣀⠀⠀⠀⠀⠀⣠⠀⠉⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⣤⠀⡴⠁⠀⢀⠄⠉⠀⠀⠀⠜⠁⠀⠀⠀⠀⠈⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⡌⠐⠊⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡘⠀⠀⢀⠤⢛⡩⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠀⠀⣢⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠒⠉⠀⠀⠠⠞⠀⠀⠀⠀⠀⡴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀⠀⠄⡐⠠⠀⣒⠠⠄⡒⠠⢄⠢⠐⡂⠄⠒⡈⠄⢀⠨⠐⢈⡡⠄⢂⠁⠄⢐⠠⠅⠒⠨⠀⠶⠒⠶⠂⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠁