Draft Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠸⣁⡲⢄
⠀⠀⢀⠀⠀⠀⠀⠀⠚⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⢧⡀⠀⢀⢿⣿⡄⡣⡀
⠀⠀⠀⡀⠲⢄⠀⠀⠠⢹⣌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣻⡿⣿⣾⣿⣯⡻⣧⣷⢹⡠⡀
⠀⠀⠀⠹⢦⡀⠑⡔⣼⣇⢻⡌⢷⣦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⣉⣩⣍⣉⡚⠮⣍⠛⡿⣦⠂
⠀⠀⠀⠀⠀⣛⡚⣉⣛⡻⠗⠙⠈⢻⠻⣿⣦⣲⢄⡀⠀⣀⠤⣄⠀⠀⠀⠀⣴⣫⡼⠂⠳⠦⠭⢿⡾⢷⡇⡈⢻⣦⠚⡋⣦
⠀⠀⢀⢀⣠⠟⠓⠧⢤⣭⣓⣦⣄⠠⢎⠈⠻⣯⠻⣿⡫⣤⣿⡻⡀⣠⡾⠛⠉⠁⠀⠀⢠⡆⠀⠀⢬⡗⣿⣳⠀⢻⣿⣷⡙⡀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⣿⠞⡇⠀⠸⠀⠀⢀⣄⡁⣿⣿⣿⣆⠀⠉⠓⠈⠻⣿⢿⣷⣹⢻⡀⡇⠀⠀⠀⠀⠊⠀⣰⣿⡷⣷⢸⡈⣆⠀⢻⡿⡍⠻⡶⣦⣴⡉⢧⡾⠃
⠀⢾⡇⠸⠀⠀⡤⠀⢰⣿⣾⣷⣿⣸⢸⣯⡄⠀⠀⠀⠀⢸⠀⢻⢹⡜⣷⠃⠀⠀⠀⡄⠀⠀⠉⠩⠂⣏⣾⢿⠟⠗⠀⢣⢸⡆⢹⢘⡟⢧⠛⡄
⠰⣦⣧⠀⠀⠀⠀⠀⠒⠠⠉⠁⢿⡟⢸⠙⠈⠀⠀⠀⠀⣸⠀⣾⠻⡺⣿⠀⣈⣀⢰⡇⠀⠀⠀⠀⠀⣻⠇⢸⡄⠀⠀⠘⡆⡇⢸⠈⣿⣞⡇⢱
⠀⠙⣿⣀⣠⠀⠀⠀⠀⠀⠀⣴⡿⠃⢸⠈⢀⠀⠀⠀⠀⡇⢀⡟⠀⢸⡿⠒⢛⠛⠻⠦⣄⣠⠂⢀⠔⢹⠀⠈⡇⠀⠀⢻⡇⣇⢸⠤⡿⣿⣿⣎⡆
⠀⢸⠤⠤⣬⣅⠒⠤⣶⠀⣴⣽⢡⠀⢈⠀⣸⠀⠀⠀⣸⠁⣼⠇⠀⢸⣿⠙⣿⢹⡏⠑⣦⣉⠲⢧⣼⠃⠀⢰⢃⠀⠀⢸⡇⡟⢸⢀⢳⣷⡇⢻⣧
⠀⡼⢀⣰⣿⠀⠈⣶⠤⣙⠫⣷⡎⠀⣜⣾⢻⣆⠀⢠⠇⢠⡿⠀⠀⣿⡟⠀⠈⠀⠀⡼⠁⠀⠑⣾⠿⢄⣴⣿⡏⠃⠀⢸⢡⡇⡸⢸⡎⣼⣟⣊⢧⡀
⠀⣧⣿⡿⠃⠀⠀⠁⠀⣈⡯⣟⣠⣾⣿⣿⠀⠙⣀⡞⠀⣾⠃⠀⢠⣿⠇⠀⠀⠀⠀⠀⣠⠒⢛⣦⣴⡿⠋⠙⣿⠀⠀⡼⣸⡇⡇⡼⡸⠁⢹⣽⢦⣳⡄
⡼⠀⠉⠀⣠⡀⠀⢠⠶⢿⣴⣿⣿⠋⠈⢻⢀⡴⠋⣠⡾⠃⠀⠀⣾⠃⠀⠀⠀⣀⣀⣸⢙⣤⡞⣿⣿⡿⠀⠀⠻⠀⡰⣳⠏⠈⣷⣳⡷⡄⢸⠀⢻⣿⣿
⢧⠆⠆⢻⡎⢷⢠⣿⣤⡾⣿⣿⣿⣧⣠⠔⠉⣀⣼⠿⠀⠀⢀⡼⣿⡆⢀⠀⠠⣿⡷⡋⣿⣿⣷⡜⡻⣇⣀⢀⣴⠞⣱⠏⢀⣸⣿⣿⡴⡙⣾⠀⠀⢣⡠
⠈⢧⠀⠋⠀⠾⡾⡟⢸⠳⣽⣿⡟⠋⢁⣠⡾⠋⠀⢠⣏⢽⣯⠞⠁⠹⡜⠀⠹⠟⣵⡇⣿⡈⠙⠿⠦⠄⢉⣛⣠⣞⣃⡒⢛⣵⡿⢾⡏⢿⡄⠁
⠀⠘⠿⠦⠤⠴⠓⠃⠈⣷⣤⢉⣻⣙⠿⣿⣷⣶⣶⣿⠿⠟⠃⠀⠀⠀⠳⣴⣄⣈⣴⢵⣿⣿⡦⣄⡤⠀⡖⠚⢋⣉⣥⣶⣛⣽⣦⣤⣷⢫⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣛⠛⠛⣉⠴⠚⣉⠕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠳⡶⣇⠀⣿⠛⠛⢉⡰⠇⠁⢬⠴⠚⢁⠞⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠬⣹⡆⠻⣉⡽⠋⠀⣠⠶⠯⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣷⣼⠏⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣻⢖⣛⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡎⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡟⢳⠀⠀⠀⠀⠀⠀⠀⡟⣷⠀⠀⠀⠀⢰⠋⢳⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠷⠾⡀⠀⠀⠀⠀⠀⢰⠓⠻⡆⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣄⣠⠇⠀⠀⠀⠀⠀⢸⢀⡀⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢹⡏⣇⠀⠀⠀⠀⠀⢸⣌⣱⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣇⡇⢹⣸⡀⠀⠀⠀⠀⢸⢸⠁⡇⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⣸⢴⡦⣇⢧⠄⠀⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⣀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⢣⠇⠀⠀⠸⡸⡄⠀⠀⠀⢸⢸⠀⡇⠀⠀⠀⢸⠀⣹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡞⡼⠀⠀⠀⠀⢧⢣⠀⠀⠀⢼⢸⠀⡇⠀⠀⠀⢸⠉⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠗⠃⠀⠀⠀⠀⠘⢺⠀⠀⢰⣿⢸⠀⣷⠀⠀⠀⢸⠦⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⢾⣸⣠⠧⡄⠀⠀⠹⡤⡞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⣿⠀⠀⠘⢺⣹⣈⡞⠁⠀⠀⠀⣇⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠓⠒⠒⠒⠒⠓⠒⠛⠀⠀⠀⠈⠇⠛⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠂





⣏⠉⣹⣛⣛⣛⣛⣻⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄⣷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢀⠀⠀⡾⠀⣿⠀⠀⢹⠀⣀⣀⣤⣤⠤⠤⠴⠶⠒⠒⠒⠲⠶⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⢸⠃⢠⡟⠀⠀⢩⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠈⡇⠀⠀⠀⠀⠀⢠⡏⢀⣿⡗⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡀⢠⣼⠀⢸⡿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡄⠀⠸⡇⠀⠀⠀⠀⣾⠇⠀⣾⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿⠀⣼⡏⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠄⢸⣧⠟⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⢼⠍⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡅⠀⢸⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠇⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⡧⠤⠤⢤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣧
⠀⠀⠀⠀⠘⠀⠈⠙⠓⠲⠤⢀⣀⠀⣀⣀⣀⣤⠀⠺⠃⠀⠈⠉⠒⠚⠟⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢦⣀⣀⣀⣠⣄⣀⣀⣀⣀⣈⡈⠁⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠋⢉⠉⠉⠛⢿⣦⠀⠀⢈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡏⣠⣾⣿⣿⣿⣶⣄⢹⣇⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡁⣿⣿⣿⣿⣿⣿⣿⢌⡏⠀⠘⠀⠘⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢷⡹⣿⣿⣿⣿⣿⡟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠿⠿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠁⠀⠉⠉⠉⠁⠉⠉⠙⠛⠛⠈⠉⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠄⠀⠠⠄⠀⠠⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠧⠤⠤⠤⠤⢤⡤⠤⣤⣤⠤⠤⠤⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠀⠀⠀⠠⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⢠⡄⣀⠀⡀⣢⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠓⠞⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣄⠀⠀⠈⠉⠉⠹⠟⠁⠀⠠⣤⡠⡀⠈⠐⢤⣄
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠸⡏⠑⢄
⠀⠀⠀⢀⠄⡀⢀⢀⡴⠂⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡵⠀⠀⠀⠀⠀⠀⠀⡔⠀⣀⠀⠀⠀⢀⣄⠢
⠀⠀⢀⠊⠘⠀⠙⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⢶⠏⠀⠣
⠀⠀⠄⣰⠄⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⣀⣄⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⡄
⠀⢘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙⠻⠃⢠
⠀⠘⠀⠀⠀⠀⣴⣇⣀⣤⡄⠀⠀⠀⠀⠈⠻⢿⣋⠁⠀⠀⠀⠀⢀⠹⠤⠂⠁⠀⠀⠀⠀⠀⢀⠀⣠⣿⠁⠀⠀⠀⢠⡄⠈
⠀⠈⡀⣠⣄⠀⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣴⣿⣤⠁
⠀⠳⣣⡀⠻⣷⣄⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣄⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢟⣵⣿
⠀⠀⠻⢿⣷⡝⣿⣿⠦⣵⣢⠝⠀⠀⢠⣸⠃⡀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣶⣶⣿⠿⠁⣰⣿⣿⡏
⠀⠀⠐⡀⠉⠻⣮⣭⣄⠀⠩⣤⣤⣀⢌⣵⣾⠋⣀⣀⣀⣀⣀⡀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣤⣛⡿⠝⢉⣀⣴⣿⣿⠟
⠀⠀⠀⠘⢄⠀⠀⠙⠿⣦⣤⣬⣁⣀⣀⡈⠙⢿⣿⣿⣿⡿⠟⠛⠛⢛⣛⣣⣭⣍⢉⠈⢉⣀⣤⣶⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠉⠙⠛⠛⠿⣿⣦⣄⠀⠈⢁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⣀⠀⠀⠀⠉⠻⢳⠶⠛⢉⣉⣏⣽⣉⣿⣿⣻⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠠⠤⠤⠶⠬⠿⠿⠿⠿⠯⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠉⢭⠉⠙⠟⠛⠻⣿⣶⣆
⠀⠀⠀⠀⠀⣠⣿⣿⢿⣶⣴⣄⢤⡈⠂⠀⠀⠠⡈⠏⠻⣷⣄
⠀⠀⣀⣀⡾⠇⡜⢟⠾⢿⣿⣿⣷⣿⡄⡄⡆⣦⠉⡆⠀⠹⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠙⣿⣁⡾⡀⠀⣹⣷⣻⣿⣿⣿⣧⣿⡵⣏⠀⢀⢶⡆⡲⢽⣧⣀⣀⡴⠚⠛⠛⠛⠒⠛⠛⠻⢿⣿⣶⣤⣀
⠀⠀⢰⢡⣿⠒⢙⠛⢋⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣸⠇⠀⡀⠿⣏⡁⠀⠀⠛⠂⠀⠀⢀⣀⡀⠀⠈⠻⣿⣿⣦
⠀⢀⣼⣿⣿⣆⠈⠀⠋⡼⣟⣿⣻⡟⣿⣿⣿⣿⣧⣄⣿⣿⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠘⢿⣏⠻⡀⠀⠀⠸⣿⢿⣷⡄
⠀⢸⣿⣿⣿⣿⠀⠦⡄⠳⡿⢡⣿⣿⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⣿⣿⣽⣿⣤⡄
⠀⢰⠟⡴⠿⠟⣅⠠⢷⢻⣇⢸⣿⣿⣷⣿⣿⣿⡟⠉⠁⠀⣠⠇⢸⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⢀⠀⣿⣿⣿⣿⣷⣍⣓⠄
⠀⠀⠀⠀⠀⠀⠈⠒⠖⠚⢁⣾⣿⣏⣿⣟⣿⣿⣿⣄⡀⠘⠏⠀⢘⣿⡇⠀⢀⣀⣤⣾⣿⡿⢿⣿⠀⠀⡼⢰⣿⣿⣝⣿⣿⣯⣭⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⢩⣉⣉⣹⣿⣾⣿⣿⣿⠁⢀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠀⢾⠁⣿⣿⣟⣿⣶⡍⣷⣦⣈⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣟⠉⠉⠉⠀⠉⠙⠛⢿⣦⠁⠐⡿⠿⣿⣽⣿⣿⠟⠋⠁⠉⠉⣿⣿⡄⠀⢠⣿⣿⣿⣝⣿⢽⣿⣿⣿⡿⠿⠷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⣤⣤⡀⠀⠀⠀⠀⢻⡆⢸⡇⠀⣼⣿⣿⡃⠀⠀⠀⠀⠀⠈⢻⣿⡆⣾⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣿⣿⡌⣆⠀⠀⠀⢸⣿⠐⡇⣠⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠁⣿⣿⣿⣿⣿⣿⣟⡻⡿⢿⣿⣿⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠗⠋⠀⠀⠀⠀⢹⣷⣿⠟⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠋⠻⣿⣿⣿⣯⣽⡷⠤⠀⠁⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⡸⣄⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢳⡄⠀⠀⠉⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣾⣿⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⢣⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠉⢋⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⣤⠖⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠰⠆⠀⠀⠰⠶⠂⠀⠀⡀⠀⠀⢠⠖
⠀⠀⢀⠋⠀⢀⣀⡀⠀⠳⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⢟⠊⠀⠀⠔⠀⣴⠖⠀⠀⠀⢀⡀⠙
⠀⠀⠇⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠀⠀⠀⢀⠋⠀⣰⣆⡠⠔⠉⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⡆⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⡤⠶⠭⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⣄⠀⠀⢀⣀
⠀⠀⢿⡄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁
⠀⠀⠀⠐⠀⠐⠻⠀⠀⠀⠀⡶⠀⡾⠃⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⡾⢹⠀⠀⠀⠀⣀⠀⠀⠀⠁⢤⣤⡀⠀⠙⠃⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢋⣼⠇⠸⠟⠛⠛⠛⠷⣾⢟⡋⣘⢂⣤⡼⣧⠃⠀⠀⠀⠀⠈⠁⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⡏⢠⡶⢲⢰⡟⢹⣸⣿⣡⠏⣿⢸⡗⡡⣈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢁⣿⡧⣼⢻⣏⡞⠻⠏⢷⣄⡭⠥⢥⡈⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⡴⠋⠙⠓⢡⣤⣤⣴⡶⠟⠋⠙⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠴⠆⠀⠚⠛
⠀⠀⠀⢠⠄⠀⢀⡄⠀⠀⠉⠛⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢠⠄⠲⠒⠒⠈⠛⠢⠀⠀⠀⠀⠀⠀⣠⣏⣴⠶⠂⠈⠁⠒⠀⠁⠀⠶
⠀⠀⢀⡄⠘⠳⠀⠀⠀⠀⠴⠖⠀⠀⠉⠉⠀⠀⠀⠀⡴⢈⡃⠀⠐⠛⢂⠀⠀⣤⡄⢠⡄⠀⠀⠀⢹⡯⢠⣦⣀⣤⣧⣠⣤⡤⠀⠠⣄
⠀⠀⢸⣷⣄⡀⠶⠂⠺⠷⠀⠀⠾⠳⠂⠈⡜⠀⠀⠼⠁⢨⣅⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⡿⠋⠋⣩⡍⠁⠈⠙⠲⣶⠉
⠀⠀⢿⠿⠿⠿⠿⠿⣦⣿⠶⠶⠦⢶⣖⠛⣀⠀⠀⢎⠀⢀⡀⢀⠀⠀⠀⠈⠷⠐⠁⡄⠀⠀⠀⠀⠰⠦⠁⠴⠤⠀⠀⠀⠲⠶⠀⠈⠧
⠀⠀⠀⣽⣿⣶⣛⣠⣀⣀⣀⣠⣤⣿⠿⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣸⢀⡄⠀⠀⠀⢀⣄
⠀⠀⢀⣭⣍⡉⠁⢈⢉⡉⣩⠿⠀⢀⣀⣤⣴⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠙⠀⠀⢸⡇⠀⣤⡀⠀⠀⠉⠀⢠⠈⠇
⠀⠀⢈⣜⡋⠩⠿⠉⠉⠹⠏⠿⠉⢉⢉⣉⡥⡄⠀⠀⠀⠀⠄⠀⠰⠚⠀⠂⠈⠀⠁⠀⣤⠀⠀⠀⠀⢻⡀⠈⠁⠐⠂
⠀⠀⠺⡿⣩⣟⠻⢶⣶⠶⠓⠓⠚⠉⣩⡄⢐⡴⠀⠀⠀⠀⠀⡄⠀⠀⠀⠷⠖⠀⠀⠀⠁⢠⡆⠀⠀⠀⠙⠒⠠⢄⣀⡘⠃⠔⠒⠁
⠀⠀⠀⠙⠛⠛⠲⠮⠉⢤⡤⠤⠶⠤⠶⠚⠉⠀⠀⠀⠀⠀⠤⠀⠶⠆⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠂⠀⣠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⠇⠀⡈⢀⡀⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣇⠀⠀⢠⡏⠀⣸⠇⣸⠇⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣷⣤⡀⠀⠀⠀⠀⢻⠀⠈⠀⠁⢸⠀⠀⡟⠀⡟⠀⠀⠸⠁⠀⠀⠀⠀⠈⠁⠙⢲⣿⣷
⠀⠀⠀⠀⣤⣾⣿⣿⡄⠀⢻⣯⣏⠢⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠤⠶⠶⠿⠿⠿
⠀⠀⠀⠌⠁⢸⣿⣿⠃⠀⠀⠹⣿⣷⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣶⣤⠴⠶⠶⠶⠆
⠀⠀⠀⡀⢀⣿⣿⠟⠀⢀⣴⣶⣾⣿⣿⣠⠀⠀⣴⣀⢀⣀⣨⣿⣦⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠙⠿⠛⠁⠀⢀⢸⠛⢿⣿⣿⡝⢿⣧⠀⠻⢿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣀⡈⠈⡄⠈⣿⣿⣧⠈⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠠⠞⣿⣿⣿⠂⠘⠷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⠐⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢰⣴⣿⣿⡟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⣿⠿⢳⣶⣶⣶⣶⣶⣶⣾⣷⡀⠈⠁⠄
⠀⠀⠀⠀⠙⠛⠉⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡟⠀⠀⠀⠀⣰⠟⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⣼⠀⠀⠀⠀⡞⠋⠀⠐⠋⠁⠀⠁⠀⠀⢀⢰⡛⣛⡿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⡐⢠⡏⠐⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠻⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣇⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⢟⣷⠟⣼⢁⣿⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢿⢻⠁⠀⠀⠀⠀⠀⣠⣿⣭⣿⢋⠀⠀⢸⡇⠀⣠⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⣴⣣⡼⠟⣳⠋⠀⠀⢸⣇⣴⣿⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠟⠋⠀⢀⠄⠀⠀⠀⢸⡏⣿⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⢀⡀⣰⡿⠀⠀⠀⠀⠀⡿⡿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⣴⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠁⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣤⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠐⠀⠒⠒⠒⠂⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⢶⣿⣅⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡞⠋⣡⣿⠓⠛⣿⣻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢣⣿⡇⠀⢾⡏⢾⡙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠠⠦⣀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⠗⢺⣿⣅⠀⢸⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⠀⠀⢻⣷⡄⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡠⠀⠀⠀⣿⣧⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠓⠂⠀⠀⠙⠻⢿⣌⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡟⠀⠀⠀⠀⠀⠀⠀⣨⡇⣿⡧⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡃⣦⣾⡗⠀⠀⠀⠀⢻⣇⣿⠇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⢰⢿⣿⠿⣤⣿⡏⢦⡙⠀⢀⣴⠼⠻⣷⣆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣇⣩⣙⡂⠁⢀⡾⠉⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣷⣅⢨⣁⣀⠀⠀⡀⠀⠏⠀⠀⠀⠌⠀⢃⡀⠀⠀⢀⣠⣤⡤⠴⠶⢋⠛⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⣿⣏⠻⣿⠻⣿⣇⡻⢽⣶⣾⡾⣿⣿⣷⠶⢶⣾⡟⠉⠈⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠣⠹⡎⠿⣆⠀⠉⠁⠀⠈⠃⡆⠀⠀⣟⠀⠀⣿⠁⠀⠈⠀⢠⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢻⡀⠰⡄⠀⢸⡀⠀⠀⡇⠀⠘⡗⠀⠀⡅⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⡧⠀⢥⠀⠈⠧⠀⠀⠷⠀⠀⡇⠀⠀⠃⠀⠀⠀⠘⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠸⠀⢠⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣇⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠘⠀⠆⢠⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⢤⣴⣄⣀⣠⣠⣄⡆⢀⣶⣾⠦⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠛⠉⠉⠉⣙⡟⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⢀⣀⡤⠤⠒⠂⠘⠃⠉⠉⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣸⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣠⠏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠃⠀⠀⠀⠀⠀⢀⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⢠⠎⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠐⠒⠠⠄⠤⠀⠀⠠⠤⠤⠤⠤⠤⠄⠒⠉⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⣀⡤⠤⠤⠶⠶⠒⠛⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢸⢧⠀⠙⡄
⠀⠀⠀⠈⡆⢣⡇⡇
⠀⠀⠀⠀⢷⢸⠇⡇
⠀⠀⠀⠀⡼⠀⠀⡇
⠀⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⣾⠄⠇⠐⣸
⠀⠀⣸⡏⠀⢸⠀⢚⢇
⠀⠀⡏⠇⠀⠀⡆⠀⠘⣆
⠀⠀⡏⡀⡂⠀⠱⡀⢀⠘⢆
⠀⠀⡅⡇⢂⠀⠀⢣⠀⠡⢈⠢⡀
⠀⠀⠇⢣⠀⡡⠀⠀⠱⡀⠀⠣⠈⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠘⡌⢆⠰⢁⡂⠀⠙⢄⠀⠀⠁⢈⢹⢳⢶⢠⠤⠤⢤⣤⢤⡤⡤⠤⠤⠒⢒⠘⡯⠉⠍⢄⢢⠢⠰⡐⠄⠀⠀⢩⣷
⠀⠀⠀⠙⡌⢆⢦⢡⢀⠀⠀⠑⢄⡀⢈⠈⠘⠘⠈⠀⠁⠀⠀⡆⠃⠰⠈⡀⠃⠀⠀⠀⠀⠀⢀⡀⠁⣁⡠⠤⠖⢂⡄⡜
⠀⠀⠀⠀⠘⢌⠣⡀⡌⡌⡰⢀⠀⠈⠑⠻⠶⣤⡄⠤⠲⡄⠀⣀⣈⠀⠤⠤⣀⣀⡂⠭⠤⠐⢒⢉⢩⠆⢂⡠⣖⠝⠊
⠀⠀⠀⠀⠀⠀⠑⢝⠢⠁⠇⡄⡆⡌⢀⢀⢀⠀⡈⠉⠉⠐⠒⠒⠀⠉⡉⠁⡀⡀⠠⠰⠰⠀⢃⡡⣔⡮⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡃⠇⡇⠀⢸⢸⢠⢀⠁⡄⠀⠈⠈⠌⠰⢰⢀⠁⡇⠃⠤⣒⡺⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣈⣈⡈⠘⠀⠃⠀⠂⠂⠀⣀⣘⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣻⢦⣄⡀⠀⢀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⢄⣻⣿⣿⡟⣛⢧⡙⠛⠛⠿⢛⣟⠑⢦⣀
⠀⠀⠀⠀⠙⢿⣟⣋⣽⠁⣷⠁⠀⠈⠛⠻⢿⣿⡿⠳⣤⣼⡆
⠀⠀⠀⠀⠀⢶⡏⠟⣿⠀⣉⠰⢤⣠⡤⠀⠀⠻⠃⠀⢹⣿⣅⣀⡀
⠀⠀⠀⠀⠀⢈⡗⢀⣿⡄⢽⡇⠀⠈⢿⣦⡀⢀⠀⠀⢠⣄⢻⣿⠁
⠀⠀⠀⠀⠀⢾⣰⣿⣿⣿⡛⢉⢱⠀⢸⣿⣿⠈⡇⠀⣸⡿⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠈⣿⠄⠙⢿⣧⠘⠈⢉⢻⣿⣿⡇⠃⡴⠛⠁⡖⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣑⡂⡝⣿⣿⢁⠀⠀⣠⠞⠁⠀⠀⠼⠷⢦⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣿⡿⣥⣴⣿⣿⣾⡄⠀⠁⠀⢠⠀⢰⠋⠀⠀⠀⠈⢽⣿⠟⠻⠶⣤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣯⡻⣿⣿⡟⣿⣣⣿⠁⠀⠀⣠⡏⠀⡞⠀⠀⠀⠀⠀⠀⠉⠂⡄⠀⠈⢻⣶⣾⣿⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠻⣧⠸⡓⢿⣿⣾⣏⣿⣿⠟⠻⢄⠀⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢠⢻⣦⠀⠀⢻⣿⣿⣿⣆⠀⢘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢼⣯⡻⠟⡃⢀⣊⡀⠰⣀⣴⣿⡃⠀⠀⠀⠀⠀⠀⠀⣋⢸⡟⠀⠀⠀⢿⣿⣯⢻⣶⡄⣿⣿⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⣿⠟⢀⣠⣌⣉⣉⣠⣼⣿⡛⢿⣷⣤⣀⣀⣀⣠⣴⣾⣿⠈⡤⠀⠀⢀⢸⣿⣟⠈⢿⣿⣿⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣫⡾⠋⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢛⣽⣿⣾⣧⡀⠀⢸⢸⣿⣿⠂⠀⢈⠙⣷⡈⠂
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⡋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣷⣶⣾⣿⣿⣿⠿⠻⣿⣷⡀⠈⠸⣿⣿⣷⣬⣺⣿⣾⣷⣦⣤⡀⡀
⠀⠀⠀⠀⠀⣿⡿⠛⠿⠋⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢻⣆⡀⠈⠻⣷⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣍⡉⠁
⠀⠀⠀⠀⡠⠟⢷⣤⡀⢀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣄⡈⠙⣦⣄⠘⢿⣦⠈⢻⣿⣿⡿⣿⡿⣿⣿⣿⡿⠓⠂
⠀⠀⠀⠰⢷⣶⣤⣄⣸⠟⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⠘⡀⠈⣿⣇⠘⢹⣿⡓⠌⠳⠌⠙⠛⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⣇⠀⠈⢻⣆⠸⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠣⣤⡀⣿⡄⠳⢤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢠⣿⣷⡟⠁⠀⠀⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠛⠷⢿⣻⣿⡇⣠⣤⣰⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁⢼⣿⣏⣙⡭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡹⠀⢺⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠃⢠⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢲⠖⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡹⢁⡞⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢅⡴⠁⡼⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡻⡝⣬⢎⠄⣰⣇⣀⡈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⢋⠡⠃⣡⠞⠣⢈⡴⠁⢸⠃⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⢋⢳⠰⠀⢈⡠⡚⣄⠱⣠⢏⠲⠀⣼⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⢠⢞⡏⠈⣉⡒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣤⢴⠒⠋⠉⠹⠸⠸⡘⡘⠎⣀⠔⠋⠠⠙⢈⡞⠣⢄⠁⣰⢇⠰⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⢫⠉⠱⣦⣍⡓⠤⣀⠀⠈⠉⠁⠉⠉⠉⠉⠉⡏⡏⢩⠀⣤⢠⠸⢸⠈⠀⠀⠀⣀⣤⡰⡕⢃⠁⠀⠀⢀⡴⡋⠤⡑⠀⡰⠃⢀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠘⡜⡆⠀⣤⡝⡻⢶⣦⣍⣀⣂⠂⠖⠀⠘⠀⠇⠁⢘⣈⣀⣠⣤⢤⣤⣶⣶⠿⠛⠋⢅⢣⠸⡌⠀⣀⠔⠁⠀⠈⠀⢀⠞⠁⠀⠄⠁⡼⡟
⠀⠀⠀⣀⣀⣀⣀⣹⣌⡀⣡⠚⠕⠣⠆⠀⠀⠈⠛⠻⠿⠳⠒⢻⣿⣿⣯⠿⠞⠒⠊⠉⠁⠀⠀⠀⠀⠈⠀⣁⠤⠚⠁⠀⢀⠀⢀⠔⠁⠀⠀⠀⠀⢰⢳⠃
⠀⠀⣼⡵⠇⣘⡛⠛⠋⠀⠍⡝⢹⢛⠖⢲⢲⠖⡖⡖⢺⠋⠉⠉⠀⠀⠆⠃⠁⠀⠀⠀⠀⣀⣤⣶⠦⠒⣋⠡⡐⠀⠀⢈⣡⠞⠁⠀⠀⠀⠀⡀⢠⢧⠏
⠀⠀⠘⢆⢢⡀⠈⣕⡒⠤⢄⣀⠈⠘⠀⠢⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⢾⣟⠋⠉⠀⠐⠑⡌⠣⠈⣠⠔⠋⠀⠀⠀⠀⠀⠀⠁⢠⢯⠏
⠀⠀⠀⠀⠙⢽⣢⠈⠘⢰⣀⠀⣉⣁⣒⣲⣦⣤⠤⠤⠤⠄⠒⡒⠈⡉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⠀⡆⢠⢠⠀⠀⡰⣣⠋
⠀⠀⠀⠀⠠⣾⣏⡉⠉⠙⠍⡅⢸⢰⠀⡆⠀⠀⠐⢸⠸⠀⡇⢰⠀⠇⠁⠀⠀⠀⣀⣀⠤⢖⣂⠤⠒⠋⠀⠀⡀⠀⡇⢸⠀⡇⡘⡘⠐⣠⠞⠁
⠀⠀⠀⠀⠀⠻⡀⢦⠉⠒⠂⠤⣨⣀⣀⡀⠀⠀⠀⠈⠀⠃⠙⠀⠂⣀⣀⠀⡬⠭⠔⠒⠊⠉⠀⠀⠀⡆⢰⠀⡇⠀⠇⣸⢠⢁⣧⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢄⠑⢄⠀⠀⠓⢀⢈⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⡀⠤⠀⡆⠀⡀⠀⡇⡜⢀⠃⠰⢀⡧⢞⡽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣑⠦⣀⠉⠆⢸⠀⠁⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡇⡶⢠⠃⠀⠇⢰⠀⣃⠼⠐⣊⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠿⣶⣤⣤⣀⡀⣀⣀⡀⠀⠀⠀⠀⡸⢀⡇⣰⣀⣧⠼⠀⢈⣀⣠⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠋⠀⠤⠬⠍⠉⠙⠓⠒⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⢀⠶⣀⡤⠖⣾⣿⣉⣭⣭⣍⣳⣦⡀
⠀⠀⣀⣠⣤⣬⡿⠾⠛⣙⣛⡿⣿⡿⠡⠶⠶⠆⠉⠙⢿⣶⡀
⠀⠀⠀⠙⠻⠷⢶⣶⡾⠿⠟⣁⠉⠱⠄⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⣾⢫⡄⢰⡀⣀⡧⢸⣄⡀⠀⠀⠀⠀⠀⢀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠈⠀⡤⠿⠯⠁⠀⣿⣙⣷⣄⠀⠀⠀⠈⣷⢻⠃
⠀⠀⠀⠀⠀⠀⣽⠀⠞⠡⠞⡼⠀⣰⣷⡙⣿⣿⡆⠀⠀⢠⣿⠿⡗
⠀⠀⠀⠀⠀⢀⠏⠀⣤⣄⣙⡷⠞⣿⣿⡇⠘⣿⣧⠀⠀⣾⠏⡄⠹⣤⣶⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢮⣀⢳⡌⢃⡟⢀⣼⡟⣿⡇⠀⣿⣿⡀⠠⠏⣸⠃⠀⣼⠛⠛⠻⢿⣿⣦⡀⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠈⢯⣤⣤⡿⢠⣿⣿⡇⣿⣇⠀⣿⣿⠃⣠⠞⠁⡠⠀⢹⡄⠀⠀⠀⠈⢻⣿⡄⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⣿⣿⣇⣿⣿⡄⣿⠇⣾⣧⠎⣴⠃⠀⢾⡇⠀⠀⠀⠀⠀⢿⣿⣿⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏⣿⣾⣿⣷⠟⣧⣉⡀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⢿⣿⣷⣹⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⢇⣿⠀⣾⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣾⢦⠀⠀⣼⡿⠀⠀⠀⢀⣾⣧⣾⠇⣰⡟⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⢹⣟⣷⡿⠿⣿⣶⣿⢁⡄⠀⢸⠋⢀⣀⣤⣶⣟⣹⣿⠃⢠⣿⡇⢹⡜⢯⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢞⣽⡿⠃⠀⠀⣼⡿⢚⢀⣴⣿⣿⣿⡟⠛⣿⣿⠟⠁⢀⣾⣿⣇⢠⣙⢾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣾⠚⠀⠀⢀⣼⡟⢐⣭⣿⣿⣿⣿⠋⠀⢠⣿⡏⠀⡔⢻⣿⢿⣿⣮⡿⣿⣷⡉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⡇⠀⠀⠀⣿⣿⠇⡏⢻⣿⣿⣿⣿⣄⡀⠘⣿⡇⠀⢀⣾⣿⡈⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⡇⠀⢀⠀⢻⣿⢨⢃⠀⠉⠻⣿⣿⡍⣹⠀⣿⠃⢀⣾⣿⣿⣧⣾⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡳⠤⣾⣶⣼⣿⠈⣾⣶⡆⠀⠘⣿⣷⡟⣠⣿⠀⢸⣿⣿⣿⢿⣿⡿⣿⣮⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣹⣿⣿⠀⢈⠛⢻⠀⢰⣿⢹⠁⣿⠀⠀⣸⠁⠻⠟⠧⣙⢿⣜⣿⣿⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣆⣻⣷⡌⠀⠘⢷⢿⡏⠘⠛⣿⠂⣰⠃⠀⠀⠀⠀⠀⠉⠛⠉⠻⢿⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠉⣿⣿⣿⣆⢸⡆⢷⣤⣴⠟⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⢿⠿⣤⣿⣿⠃⠀⠠⠘⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣀⣠⣤⣠⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⠛⠛⣉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠍
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣷⣷⣶⣶⣾⣿⡿⢿⣿⣿⡿⠿⠿⠛⠋⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣛⠶⠶⠦⠤⠅⠀⠀⠀⠠⠄⠤⠤⠤⠄⠒⠒⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⠊⡁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠿⢿⠿⠿⠟⠛⠻⠶⠓⠚⠲⡶⠚⠿⠋⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣖⣄⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣻⡛⠽⠛⠿⢷⣤⣀⣀⣀⣀⣀⣠⠤⠖⠒⠚⠋⠉⠁⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣠⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠚⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢶⢶⣶⢿⡿⠉⠁⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠿⣿⣿⣿⣿⣉⣈⠉⠀⢀⠀⡈⣉⣉⢀⣀⣠⣴⡤⠶⢚⡉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡁⠪⠽⢛⡛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠄⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⣄⣀⣀⠀⠀⠀⠈⠀⠀⣀⣀⡤⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣿⡇⠀⠀⠀⠒⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢹⠁⠀⠀⠠⠀⠀⠀⠀⠄⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠐⠀⣀⠴⠒⣠⡀⣤⡄⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⣠⢾⣤⣤⣤⡝⣡⣿⡇⠀⢀⣷⠀⠀⠀⠀⠀⢀⠈⣷⡄⠀⠒
⠀⠀⠀⣴⢇⡞⠀⡀⠀⠀⠀⢴⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⠁⠀⠀⠀⠀⠀⠀⠈⠢⡘⢿⣶⣶⣄⠀⢀
⠀⠀⢸⣟⠛⠀⠸⡀⠀⠀⢀⠄⠋⠛⠛⠟⠻⠛⠛⣫⡿⠛⠛⢻⣿⣶⣀⢀⡀⠀⢠⡉⣽⠀⠙⠿⠟
⠀⠀⠸⣿⣷⡴⠀⠀⠘⠉⠁⠀⠀⢠⠄⠀⠀⠀⠊⠀⠀⠀⠐⠁⠁⢹⣿⣦⡈⠂⡀⢻⣿⡇⠀⠀⠁
⠀⠀⢀⢉⡻⣿⣷⣄⣀⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⢤⣽⣾⡋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢦⡈⠋⠹⡿⢿⣷⣶⣶⣦⣠⣄⣀⡠⠴⠶⠦⠶⠖⠚⠛⠉⠉⢐⣿⢻⣦⣿⡏⠙⣆
⠀⠀⠀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⢿⡿⠛⠀⠀⠈⠦⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠦⠤⠤⠤⢤⣤⣤⠤⠤⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠠⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣯⣿⡿⢷⢦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⣉⡉⣙⣻⡿⢶⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢤⣀⠀⠀⠀⠀⠀⣀⣴⡶⠚⠉⠀⠠⢋⣵⣾⡿⣿⣶⣦⣿⡇
⠀⠀⠀⠀⢠⣴⣶⠶⣦⠔⠋⠉⠀⡀⠀⠀⢹⣯⠙⠛⠿⠿⠛⢻⣇⠀⠀⠀⠀⢠⣿⣿⣿⣄⢿⢏⣹⡿⠃
⠀⠀⠀⢀⣿⣿⠁⢸⠁⠀⠀⠀⠀⢻⣧⠀⣼⣿⠀⠀⠀⠀⢀⡈⢿⣦⣀⣰⡇⣼⢳⣿⡏⠉⢻⣿⢹⣟
⠀⠀⠀⠸⣿⣿⠀⢸⡆⠰⣦⡀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣾⣿⣿⣿⣾⣿⣼⡿⠀⠀⠘⠿⠿⠋
⠀⠀⠀⠀⣿⣿⡇⠀⢻⡄⣿⣿⣦⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢠⣿⣿⡇⠀⠈⣇⣿⣿⡿⣡⣶⣶⣿⡟⠛⠻⠿⠿⠿⠿⣿⢸⣿⣿⣿⣯⣴⣤⣀
⠀⠀⠘⢻⣿⡟⠁⠀⣼⣿⣿⡿⠁⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣧⣿⠿⠁⠙⠛⠻⢿⣿⣿
⠀⠈⠿⠿⠟⠁⣠⠾⢿⣿⠋⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⢀⣀⠰⠋⣻⠋
⠀⠀⠀⠀⠀⡰⠃⠀⣾⡇⠀⠀⠀⠀⠀⠙⡌⠙⣷⣀⠀⡴⢚⣿⠏⠀⠀⠀⣼⢃⣠⣿⡏
⠀⠀⠀⠀⢰⡇⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⠀⠨⣿⡟⠁⢸⣿⡀⠀⠀⢰⣿⠿⠿⠉
⠀⠀⠀⠀⠘⣇⠀⣀⢼⣿⣆⠀⠀⠀⠀⠀⠸⣆⣀⣸⣦⡀⠈⢻⣷⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠠⢤⠤⢤⡤⠤⡤⠤⡤⠤⡤⠄⠀⠀⠀⠀⣰⣆
⠀⠀⠀⡀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⣤⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⣷⠀⠀⠀⠀⠀⢀⣼⣤⣼⣧⣤⣧⣤⣧⣤⣧⣄⠀⣟⠉⢹⡏⠉⢹
⠀⠀⠰⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⢰⣿⣿⡇⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣈⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣦⣿⣿⢿⣿⣿⣷⠀⠀⠀⣴⣾⣿⣥⣬⣿⣶⡆⠀⠀⢰⣷⡆⣷⣶⣾⣿⣿⣻⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠻⣿⡏⠀⢻⣿⠏⠀⠀⠀⣼⣿⣿⡿⢿⣿⣿⡆⠀⠀⢠⣿⣷⡧⣽⣿⣿⢸⣾⣿⠟⠁⠀⠀⠛⢿⡋⢹⡏⣿⠉⣿⠛⠛⠛⠛⠛⠛⠛⣟⠉⣿⠛⠛
⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁⠉⠉⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣻⠛⠀⣀⠀⠀⠀⠀⠀⢀⡁⣀⠀⠀⠀⠀⠀⠀⣸⣷⣄⡀⠀⠀⠀⣄⢀⣸⣇⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⡀
⠀⡤⠴⣂⡤⠀⠀⠀⠀⠘⠋⢈⣿⣿⠷⠾⠋⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿⠀⣉⡻⡆⠀⠀⠘⣏⢀⣸⣆⠀⠀⢠⣶⣾⣿⣿⣿⡏⠀⠉
⢀⣀⡤⠍⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠐⠟⠁⣻⣛⣻⡃⠙⠷⠀⠀⠀⠘⠧⠹⣧⠀⠀⠀⠀⠘⣏⢀⣸⣆⠀⠀⠻⠿⠋⠉⢻⣿⡄
⠀⡐⢦⣉⠓⠦⠀⠀⠀⠀⣸⣿⠛⠿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠸⣏⢀⣹⡆⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠉⠓⠌⠠⠀⠀⣤⣶⠾⠟⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠹⣏⣀⣹⡄⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠸⠿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠹⠅⠀⠁⠀⠀⠀⠀⠀⠀⠈⠿⠗

⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⣾⣿⣿⣿⡀⠀⠀⠀⠔⣻⣿⡛⠃⠀⠀⠀⢸⢀⣤⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣰⣿⣷
⠀⠀⣿⣿⣿⣿⡅⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⢸⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⢺⣿⣿⣿⣿⣴⣄⠀⠀⠀⢀⠞⠉⠉⠉⠉⠉⠙⣿⣿⣿⣶⣆
⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣶⣄⣸⣷⣶⣶⣶⠀⠀⠀⢠⣶⣿⣦⣟⣿⣿⡿⣿⣿⣿⡆⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠻⣿⣿⠏⠀⢠⣏⣹⣿⠿⠿⠿⢿⣿⣋⣿⢻⣿⣿⡟
⠀⠠⣿⣿⣿⣿⡧⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⢿⣿⠿⠋⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⠿⠀⠟⠁⠀⠀⠀⠿⠿⠉⠀⠀⠀⠈⠹⠿⠇⠰⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡀
⠀⠀⠀⢸⣿⡟⣻⡿⡿⡿⣿⠛⣿⣿⠀⠀⠀⠀⢸⠉⢩⠉⢩⠉⡍⡍⠉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⣧⣄
⠀⠀⠀⢸⣿⡇⢿⣟⣜⣇⣿⣉⣿⣿⠀⠀⠀⠀⠸⠤⠬⠤⠬⠤⠥⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣟⢿⣿⣿⣆
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢰⢒⠒⢒⠒⡒⢒⠒⠒⠒⡒⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⡷⠂⢸⣿⣼⢿⣼⣿⠘⢿⣿⠟
⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢸⣈⣘⣉⣉⣁⣘⣀⣃⣀⣀⣀⡴⠞⠁⠀⠀⠀⠠⠖⠛⠓⠒⠛⣿⣿⠿⠋⠈⠻⠿⠿⠿⠋⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠐⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⢻⣿⣿⣿⣿⡟⣼⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⣿⣆⢻⣿⣿⡟⣰⣿⣿⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⣿⣿⣎⢿⡿⣱⣿⣿⣿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣿⣿⣿⣿⡌⢡⣿⣿⣿⣿⠀⠀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠋⠀⠀⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠀⠀⠙⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⢦⡀
⠀⠀⠀⡠⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠈⢿⣷⢶⣴⠄
⠀⢠⢸⣁⣤⣠⣄⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⡿⠻⡆⢹⡇
⠈⢹⢻⣇⣿⣏⠹⢿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣛⣧⣀⢠⣼⡇
⠀⢠⣾⠏⢻⣿⠀⠈⠙⣳⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢨⣇⡁⠈⣉⣹⣷⠇
⠀⠈⠉⡴⣸⡟⠛⠛⠲⢿⡀⠘⠀⢱⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⠂⠠⣿⡿⠙⣿⢉⣧⠈
⠀⠀⢀⣭⣿⣼⣤⣽⡀⢠⡏⣐⣧⡾⣼⢷⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣠⡔⠈⢴⠛⡧⣽⠟⠂⠀⢹⣈⡝
⠀⠀⢘⡥⡌⠉⠁⠉⠹⡌⡏⠻⣏⡾⣿⠜⣇⠀⠀⠀⠀⠀⠀⠈⡆⠀⣀⣴⠺⠿⠗⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠙⠛⠀⠀⠀⠀⠘⣷⠄⢸⢷⣼⣧⣌⠳⡀⠀⠀⠀⢓⠞⠇⠛⠉
⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⠿⣸⡓⢠⣤⢛⣻⣟⣮⠢⠔⢶⣆⡾⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶
⠀⠉⠁⠀⠀⠀⠈⠉⠁⠉⠉⠙⠁⠒⠂⠈⠉⠉⠉⠁⠀⣠⣷⣷⡿⠿⠟⠛⠛⠉
⠄⠀⢠⣤⡀⠀⠀⢀⠤⠤⠤⣤⣤⣤⣤⣤⢤⠀⠀⣤⠬⠥⣬⡥⠀⢤⣤⣤⣤⣤⣤⡤⣤⠀⣠⣤⠄⢀⠤⡤⠀⠀⠠⠠⡄
⠀⠀⠰⢌⠄⣠⡀⠀⠀⠀⠰⣿⡿⠰⢿⡷⢴⣖⠀⠀⠀⠶⣿⡇⠀⢾⣏⡍⣿⣿⣿⣿⠅⠀⠿⠓⢀⣀⣶⠒⢢⣄⡖⠳⣆
⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⣿⡇⠀⠈⠀⠀⠹⠀⠀⠀⠀⣿⡟⠁⠀⠀⠁⠓⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠨⡝⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡤⣀⡖⠀⠀⠸⣿⣖⣨⣭⣍⣶⣦⡀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠄
⠀⠀⠀⢠⣴⣾⡒⠛⢸⢷⣷⡾⢃⣿⣿⣿⣿⣿⣿⣸⣤⣤⣤⣌⡀⠀⠀⠀⠀⠐⡎⣿⢸⣧
⠀⢀⣠⠉⢿⣴⣤⣯⣿⣿⣧⣿⣧⢺⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⢚⣾⣹⠀⣿⡀
⠀⢸⣹⠁⢐⣿⣿⠣⣾⣿⣿⣿⣿⣧⣿⣿⡛⢿⡆⠀⠀⠘⢯⡟⣿⣷⡆⠀⠀⣂⠀⣿⡆⢹⣇
⠀⠘⠛⡁⢻⣿⣿⣽⠡⠿⠿⠿⠟⠉⠈⠉⠁⢉⡡⠀⠀⠀⠈⠙⠛⠣⠉⠀⢀⡾⠀⠀⣔⠖⣧⣤⣤⣄⣀⡀

⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⡀⠀⠀⢀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢸⢀⠊⠠⢠⠀⠀⡄⠀⠀⠀⠠⠒⠈⠉⠀⡀⠀⠀⠬⢢⡀⠀⢰⠁⢸⠃⠀⠀⢠⠀⡎⢐⠈⡆
⠀⠀⠀⠀⡇⠠⡁⠀⠉⠀⠈⠀⢸⠰⠀⢀⡼⢀⢀⠂⢀⠂⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡌⠀⡎⠀⠀⠀⠀⠉⠉⣶⢶
⠀⠀⠀⠒⡇⠀⠀⠀⠠⠄⠀⠀⢸⠀⠑⠠⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠘⠀⠀⠒⡠⠀⠰⠀⠸⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⢂
⠀⠀⢩⠃⡇⠀⠒⠀⠀⠀⠐⠂⠈⠰⢌⠁⠀⠀⠀⡾⠀⠐⠀⣀⠀⠈⠑⠒⠠⠀⠂⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸
⠀⠐⠢⠈⠃⠀⡔⠀⠀⠀⠰⡑⠀⡇⠀⠤⠳⠀⠀⠇⠀⠤⠄⠆⠀⠰⠂⠀⠀⠂⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠤⠐⡂
⠀⡃⠂⠀⢰⠀⠀⠀⠒⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⠄⠠⠄⠀⣁⠀⠀⡈⠁⠀⠀⠸⠀⠀⠀⠀⠀⡇
⠀⠁⢀⣀⣸⣀⣀⣀⣀⣀⣆⣠⣀⣀⠠⠄⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⠀⠀⣁⠀⠀⠀⣈⠇
⠀⠀⠀⣠⠄⣀⡀⣀⣤⢥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠌⠔⠀⡈⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠄⠀⠉⠀⠀⠀⠉⡉⢉⡀
⠀⢰⠊⠡⠐⢁⡴⠋⢠⠊⠠⡆⠀⠀⠀⡔⠀⠀⠄⠀⠑⠄⣀⣠⢔⡡⢂⠀⠀⠀⠀⠠⠄⠀⠔⠊⠡⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆
⠀⠈⠈⠉⠀⠉⠁⠈⠑⠈⠋⠀⠀⠀⠀⡘⠑⠀⠤⣀⢠⡆⠐⠘⢁⣄⠀⡸⠀⠀⣀⠐⢀⠄⠊⠀⠀⠀⣀⠄⠐⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠉⠀⠀⢠⡈⠙⢚⠃⠀⢈⠄⠈⢀⢀⠀⠠⠔⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⢀⠀⡖⠠⠒⠀⢀⠁⠨⠴⣶⠀⠀⠀⠢⠀⡀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠠⠀⠀⡸⠂⠄⠀⠀⠀⠐⠁⠀⠀⠀⡈⠀⠀⠢⡄⠀⠐⠠⠄⠂⠑⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢈⠢⠠⠁⠊
⠀⠀⠀⠀⠈⠀⠊⠀⣴⡀⠀⠘⠉⠁⠂⢄⣠⠁⠀⠀⠀⠐⢀⡀⠠⠤⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠠⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠾⠟⢀⠉⠁⠀⠀⠐⠠⠄⠠⠤⠄⠀⠀⠀⠠⠅⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⢠⣀⡄⢈⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠠⡄⠀⠀⠀⠀⡼⠀⠀⠆⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠀⢰⠰⠀⠀⠀⠀⠀⠀⡱⣉⡱⠈⢈⠦⣁⠱⣈⡈⢎⠁
⠀⢀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠈⠀⠀⠀⡀⠀⠀⠠⠀⠄⠀⡀⠀⠀⢄⠀⠀⠀⠀⡨⠰⠎⠀⠀⠲⠀⠉⠀⠈⠀⠀⠁⠀⠠⠧⢄
⠀⠀⢸⠀⡆⣆⡎⠀⣿⠈⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠁⢀⣤⠀⠒⡀⠀⡇⠀⠀⠀⠀⢀⡄⠀⠀⢤⠀⠀⠌⠁⠊⠉⠁
⠀⠀⠈⠀⠘⠂⠊⠀⠈⠉⠀⠀⢀⠀⡔⢉⣉⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠢⠆⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠀⡀⢣⡘⢂⠇⠀⠸⠀⠃⠀⠀⠀⠀⡰⠀⠀⢀⡇⢀⡠⠒⢦⣨⠀⠀⠀⣀⣀⢀⠀⣠⠔⠦⡄
⠀⠀⣠⠂⠀⠀⠀⠐⣄⠀⠀⠀⠁⡐⠀⠀⠠⠐⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠔⠃⠓⠘⠃⠀⠀⠀⠀⠨⠧⠀⢏⠨⠄⠸⠤⠼
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⠀⣰⣇⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣆⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⠄⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣆⠻⣿⡿⠟⢃
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣿⣿⢇⡠⣶⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⢀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠉⠙⠛⠛⠛⠁⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⣾⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⢿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠀⠀⢸⣿⣿⣿⡇
⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠁⠀⠀⣼⣿⣿⣿⡆
⢠⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦⠀⣰⣿⣿⣿⣿⣿⠟⠀⠀⢠⣾⣿⣿⣿⣿⡇
⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡏⠀⠀⠀⠸⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏⠘⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠹⣿⡿⠋
⣈⣛⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⢋⣙⢛⣟⣻⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠐⢶⡐⢾⠛⢿⡿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⢠⣿⡇⠈⢇⣀⡁⠉⢿⡏⢸⣿⣿⣿⣿⣿⡷⠢⠤⢄⡀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠒⠒⠒⠦⢄⣀⣀⡀
⠀⠀⠀⠚⠑⠂⢰⠙⡖⣄⣁⠀⠀⡿⣿⡿⠋⠹⠋⠀⠀⢀⠄⠈⠉⠉⠉⠉⠉⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⢀⣀⡀⢠⡏⢀⡿⠀⠈⠳⣄⠘⠋⢁⠀⡠⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⢈⠉⠉⢸⠜⠁⠀⠀⠀⠀⠈⢣⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡆
⠀⠀⠘⠣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠳⣀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠤⠚⣁⡤⠶⠿⢦⡀⠀⠀⠀⢠⠃⣼⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⠀⠀⠀⡗⢲⠖⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠉⠲⡄⠀⠁⠸⣽⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣳⡀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⡆⠱⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣴⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣸⣿⡇⠈⢻⠿⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠻⣿⣿⣿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠛⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠿⠿⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠔⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⣦⣾⣦⡤⣤⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⣼⣾⡿⠻⣾⠐⢵⡀⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⣤⠟⠋⠉⣿⣧⣿⣿⠀⠈⣷⠄⣰
⠀⠀⠀⠀⠀⡄⠐⠔⠱⣎⠀⠐⠀⠀⢙⢻⠶⡦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡄⠀⣀⡾⠋⠁⠀⠀⠀⢿⡏⠈⠛⠿⣷⣶⣺⡅⠆
⠀⠀⠀⠀⠜⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⡄⠅⠀⠀⠀⠈⠙⠙⠓⠐⣿⢹⡤⣹⣦⣾⡷⠽⠱⠄⠀⠀⠀⠀⡆⣿⠀⠀⠀⠀⠀⠹⣿⣦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⡄⠀⠡⠀⠀⠀⠀⠀⠀⠸⣧⠀⣻⡿⠇⠉⠀⠀⠐⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢀⡇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠀⢐⡏
⠀⠀⠀⠀⠀⠀⠸⣾⢈⣌⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⣸⡇⡀⢸⣧⠀⠰⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠀⢀⡇⠀⣴⠿⠀⣠⣏⠀⢀⣔⣀⣿⣺⣿⣇⡁⢀⣿⣷⣿⢐⣼⣶⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣞⣤⣾⣃⣤⢺⣶⣿⣿⣿⣤⡀⢳⣿⣿⣿⣿⣿⣧⣵⣿⣿⣿⢸⡏⣿⡯⢀⣇⡽⠁
⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠿⣿⣟⣵⣿⣿⠚⠁⠈⠛⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣯⠴⠛⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣜⣿⠂⣰⠏⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢲⡀⢠⡇
⠀⠀⠀⠀⠀⠀⢺⢛⢧⡾⠃⠀⢻⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠸⠁⡞
⠀⠀⠀⠀⠀⢸⣿⣢⡜⠀⠀⠈⢻⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢸⡇
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠘⣷⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⡿⠁
⠀⠀⠀⠀⠀⠉⣾⡇⠀⠀⠀⠀⠀⠈⠃⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢻⢡⠸⡇⠠⠤⠤
⠀⠀⠀⠀⠂⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡆⠀⣧⣀⣀
⠀⠀⠀⠀⠂⣠⠉⣁⡄⠀⠀⠀⠀⠈⠂⢋⡀⠴⣆⠠⠤⠄⠀⠀⠀⠀⠀⠒⠒⠊⢤⡴⣇⡀⠛⢄⠀⠀⠀⠀⠉⠉⠈⠁⠐
⠀⠀⠀⠀⠀⠈⠾⠉⠹⠂⠉⠀⠀⠀⠀⡀⠉⠉⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠶⣈⣛⡦⠒⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣣⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠊⣉⣀⣠⠔⠀⠀⠀⠀⠀⠀⢺⠋⠉⣛⠋⡟⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠿⢿⣿⣏⣿
⠀⠀⠀⠀⠀⢀⡀⢠⡇⢸⡅⢻⣿⡯⠯⠄⠀⡀⠀⠀⠀⢸⠀⢠⣿⣰⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣃⣀⣀⡤⠔⢂⠀⠈⠀⠀⠀⠈⢹⣟⡛⠻⠟⠃⣀⡀⠀⡄⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠫⠈⡉⠃⠠⣤⣾⡿⣿⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠁⢿⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣛⡭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣇⠀⠀⡰⠀⠀⠀⠀⠀⠀⢸⠀⢰⢸⣇⡿⢾⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡉⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⡋⢸⡼⠙⣆⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣙⣁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡽⣻⠁⠁⠀⠀⠀⠀⠀⣶⠛⠚⢃⡌⠁⣀⡜⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⠿⠟⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠉⠓⣆⣸⣷⠀⣇⢰⡟⠁⡟⢦⣠⡶⠀⠀⠀⠀⠀⠈⠛⠂⠠⠀⠊⠉
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠷⠶⠛⠟⠹⢾⡇⢸⣏⣀⠇⢨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠱⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣙⣥⢯⣼⡆
⠀⠀⠀⠀⠀⠀⢠⣤⡄⡀⢀⢀⢀⣀⣀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢴⡊⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢸⡷⠀⢹⡏⢸⡅⢸⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⡀⠐⣀⠀⠀⠀⠀⠀⠔⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠋⠈⠘⠁⠈⠀⠔⠁⠀⠀⠀⠀⠈⢣⢰⠀⠹⠁⣤⠸⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢢⣀⠀⣫⠐⠀⣜⡤⠀⢠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠃⠱⠌⠃⠈⠡⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⢀⢾⣿⡿⣯⡳⠦⢴⣶⡿⣀
⠀⠀⠀⠳⣾⣯⡇⡯⠁⠉⠛⠲⣿⠗⢦⣷
⠀⠀⠀⠘⣧⢹⡇⢦⡑⠲⣦⡀⠈⠀⠘⠻⣦⠄
⠀⠀⠀⠰⣧⣾⣿⡞⡒⡀⢹⣷⣸⡄⢸⡇⠙⢄
⠀⠀⠀⠀⠙⠀⠙⣧⠁⡉⣿⣿⡏⠱⠋⣰⠀⠈⣧⡀
⠀⠀⠀⠀⠀⢀⡀⠈⣷⣆⣸⣿⣵⡀⠚⠁⡀⢠⠍⠉⠓⢲⣶⠦⣤⡀
⠀⠀⠀⠀⢰⣿⢸⣿⣿⣿⢿⣿⣿⡀⠀⣴⠁⡆⠀⠀⠀⠀⠙⢠⠀⠙⣶⣶⣿⠿⢷⣤
⠀⠀⠀⠀⠀⠙⣦⠉⠛⣿⣿⡿⠉⠝⢀⠀⣰⡇⠀⠀⠀⠀⠀⠌⡷⠀⠘⣿⣿⢧⡀⢿⡷⡄
⠀⠀⠀⠀⠀⠀⠹⣷⣴⡟⢁⣤⣀⣉⣴⣿⢻⣧⣄⣀⣀⣠⣾⡗⡣⠀⠀⢹⣿⠘⢿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⢀⡼⣫⠞⠁⠀⠀⠉⠛⠿⠿⣿⣛⣛⣛⣫⣿⣿⣷⡄⠸⢸⣿⣅⢀⣌⣧⡀
⠀⠀⠀⢠⣿⠿⠿⢳⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⡁⠛⢷⣄⠈⣿⣿⣿⣿⣿⣿⣛⠒
⠀⠀⠀⠜⠳⠤⣀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⡉⢶⡄⠻⣆⠙⣿⣿⢿⡿⣿⣿⡟⠂
⠀⠀⠈⠛⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢠⠀⠻⣆⢸⠿⠁⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠣⣀⢹⡆⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣅⣠⣿⣿⠁⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠬⠛⠋⣿⣞⣛⡻⠋
⣤⣴⣦⣦⣤⣶⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢰⣶⣶⣶⣶⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⢀⣴⠶⠆⠀⣴⣶⠀⢠⣶⠶⠆⣴⡶⠶⠰⢶⣶⠦⢶⣆⢠⡶⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠈⠻⢶⡆⢰⣿⣽⣇⢸⣿⠶⠆⣿⡷⠶⠀⢸⣿⠀⠀⢻⡿⠁⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠘⠿⠟⠃⠾⠃⠘⠿⠘⠟⠀⠀⠻⠿⠿⠀⠘⠟⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⡶⠄⣴⢠⣶⢦⡄⣴⠶⠤⢶⣶⠦⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⠀⠀⠀⠀⣿⠟⠂⣿⢸⡿⣿⡅⣛⣷⡆⢸⡇⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⡀⠀⠉⠀⠀⠉⠈⠁⠈⠉⠉⠉⠀⠈⠁⠀⠀⣠⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣀⠀⢀⣠⣴⣾⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣐⣄⡤⢠⣦⠰⠶⠤⣀
⠀⠀⠀⠀⠀⣾⣿⣿⡿⣝⠶⣤⣤⣴⣶⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣛⣙⣩⣮⡽⣿⡟⠭⠭⠝⠺⣷⡄
⠀⠀⠠⣦⡽⠞⣿⢱⡚⠉⠓⠶⣤⣴⣿⠦⣉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⢉⡉⠀⡀⡓⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⢨⡟⢿⡗⢘⠡⢄⣀⡀⠀⠙⠟⠀⢹⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⡇⠁⡬⠷⠞⠁⣷⢤⡀⠀⠀⠀⡈⣿⡀
⠀⠀⠀⢈⡷⢀⣷⡈⣷⠀⠉⢿⣦⠀⡀⠀⢠⣍⢻⣿⠗⠀⠀⠀⠀⠀⢠⠇⠊⢑⣣⠀⣰⣟⢾⣿⡀⠀⠀⣿⣿⠁
⠀⠀⠀⢸⣴⠿⣿⣿⠉⡘⣀⣸⣿⣧⢹⠀⣸⡟⠀⠙⢷⡄⠀⠀⠀⢠⠃⣄⠾⣶⠟⣺⣿⡏⢘⣿⡇⠀⣰⠟⢸⠁⢀⡀
⠀⠀⠀⠀⠛⠀⠈⢻⣆⢀⠀⢿⣿⣿⠈⠘⠁⢠⠇⠀⠀⠻⣦⣀⡀⠈⢏⣨⣴⢣⣶⡿⣿⡇⢸⣿⡆⠠⢋⡞⠀⣿⣿⣿⣿⡶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠆⢋⣿⣏⡆⠀⠚⠁⠀⠀⠀⠀⠀⠉⢻⠶⠶⢧⣤⣿⣿⡇⣿⡇⣼⡟⣡⢔⠉⡀⠀⡇⠀⠈⠙⣿⣦⠈⠻⡄
⠀⠀⠀⠀⢠⣾⡛⣶⡾⣷⣿⡿⣿⣻⡟⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠙⠂⣄⠸⣿⢏⣿⣿⣧⢿⡜⠛⢣⠞⠁⢸⡇⠀⠀⠀⠘⣿⣤⠀⣿⡀
⠀⠀⠀⠀⠈⠻⣇⠿⢿⣾⣿⣿⣿⡿⠷⢆⠀⠛⠀⣰⠇⠀⠀⠀⠀⠀⠀⣽⡇⣿⣿⣿⢻⣿⣦⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣆⠁⠠⣯⡿⠛⢁⣐⣠⠀⣢⣼⡟⠀⠀⠀⠀⠀⠀⠘⠹⢳⣿⢻⣿⣼⣿⣿⠿⠓⠣⠀⢀⣾⠇⠀⠀⢀⡼⣣⡟⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⢟⣠⠴⠦⢤⣤⣾⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣸⣯⢞⣿⣿⣿⡛⠇⡠⠂⢀⣟⣁⣀⣠⡴⢻⣿⠟⣰⡟⡼⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠟⡡⠛⠁⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣷⣷⣿⣿⠿⠃⠀⣲⣾⣿⠏⣙⠛⣻⡿⠛⠁⣰⣿⠁⠳⣹⡕
⠀⠀⠀⢠⣾⣿⣿⠿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣯⣿⣣⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣼⣟⢲⣿⠇⠐⢺⡿⣿⣧⣧⣽⡿⢶
⠀⠀⠀⣼⣏⣀⠀⠀⣽⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⡻⡧⣹⣿⣿⠀⣼⣿⣿⣿⣿⡏⢻⣿⡿⠀⣰⣿⡇⢻⣿⣿⣿⣧
⠀⠀⠸⠦⣬⣉⣱⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠁⠀⣸⣿⠀⠸⠿⡷⠙⣿⣷⢛⡿⠇⢸⣿⣿⣿⣿⣿⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡾⣬⡄⠈⠀⢷⣼⠟⣮⣼⠀⢀⠏⠙⠿⣟⢿⡹⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣧⡀⠀⢸⢼⣄⣠⡟⢰⠋⠀⠀⠀⠀⠙⠋⠻⣷⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣿⠋⠀⠜⢆⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠧⠬⠓⠉⣿⣿⣿⣿⡿⣧⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠒⠦⠤⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⣰⡗⠐⠀⣷⡄⠂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⣿⡀⠆⠆⢸⡇⠐⠀⢹⠀⠀⠀⠀⠀⢤⢤⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠘⣷⣶⠌⠁⣶⣿⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢀⡘⢷⣤⣶⡽⠁⢁⡴⢳⠀⠀⠀⣾⣶⡄⢉⠛⠛⠙⢩⠀⠀⠀⠀⠀⠀⢤⣄⠘⠿⣶⣷⠾⠋⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣈⡁⠀⠋⠁⠰⠀⣤⡀⡖⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄⣈⣤⡀⠀⠀⠀⣽⡿⠀⣼⣯⣯⣷⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣿⠁⢖⣿⣗⡢⢸⣼⡽⠀⠀⠀⠀⠀⠀⢀⡘⢿⣿⡿⠃⠉⣏⠗⠀⠀⠀⢀⣴⡀⠀⠀⠂⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⣀⣀⡈⠁⠀⠹⠇⠑⠰⠀⠀⠀⠀⣀⠀⠔⠳⣠⠒⠦⠐⠈⠀⠀⠀⠀⠀⠑⠻⠀⠀⠃⠀⠆⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣠⣉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠈⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠒⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢁⣈⢐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠰⡏⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠻⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠻⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠃⠈⠓⠀⠐⠒⠉⠀⠀⠀⠀⠚⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠀⠐⠆⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⢀⣤⣄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠟⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⢸⣇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣤⣄⡀⠀⠀⠀⠀⣀⣈⣉⣀⣀⡴⠋⠉⣷⠀⠀⠀⠀⡿⢩⠭⣍⠓⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠤⠤⠤⡀⢻⠋⠉⠉⣉⣉⠉⠉⠀⠀⠘⠂⠒⠉⠉⠉⠉⠙⠃⠸⣀⣠⠀⣏
⠀⠀⠀⠀⠀⢠⡞⠩⢬⠀⣁⠤⠔⠇⢸⠉⠉⠋⠀⠀⠱⠊⠉⠙⠲⠒⠛⠲⠞⠛⠒⠉⠉⠳⢢⣀⣠⠌⢉⡧
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠉⠙⠒⠚⠘⠦⠤⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠒⠒⠤⢌⣉⣑⡒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⢒⣒⣚⣉⠭⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠙⠒⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⢸⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠚⢁⣀⣠⣤⣤⣤⠖⢸⣿⣦⢿⣶⣶⣄⡀⣀⡀⢀⡀⢺⣆⢂⣼⠄
⠀⠀⠀⠀⣀⣀⠰⡀⠟⢶⣶⣾⣿⣿⣿⣿⣿⣧⡴⣿⠠⢲⣾⣷⣿⣿⣿⣿⣿⣿⣅⣼⣛⣯⣄⡀
⢠⣤⣶⠛⠩⣅⣂⠀⠀⠈⣟⠿⣿⣿⣿⣿⣿⣿⣷⠇⣾⠛⣯⣴⢇⢈⡟⣿⣿⣿⣿⣿⣿⣻⣇⣸⣧⠀⠙
⠘⠃⠀⠀⠀⠈⠻⡇⢢⢰⡸⣇⣿⣿⣿⣿⣿⣿⡏⡈⠻⣿⡿⠋⠀⠘⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠆⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠉⠓⠲⢤⣤⣈⣀⠄⢻⣿⣿⣿⣿⣿⣿⣿⣷⣽⣄⢿⣄⣀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡥⠄⠀⠀⠀⠀⠐⠛⠀⠃
⢠⣤⡀⠀⠀⠀⠙⠙⣇⠘⡀⠻⣿⣿⣿⣿⣿⣿⣙⢿⣿⣿⣏⣭⠏⠁⢹⣿⣿⣿⣿⠻⡿⣿⣿⣿⣿⡆
⠈⣷⣄⡀⣤⣤⣦⡀⠙⠀⣤⣤⡹⣿⣿⣿⣟⣿⣿⠶⠛⠻⠻⠏⠀⠀⠘⠻⣿⣿⣿⡆⠀⠙⢿⢟⠻⠛⡀
⢰⡿⣿⠷⣛⢧⣦⣴⣶⣿⣿⣿⣿⣜⠿⢿⣿⣿⠛⡂⡒⠆⢘⡇⠀⣠⡀⣀⡚⠛⠿⠿⡀⡀⢀⠹⢇⣼⣷⠃
⠘⡥⣴⣿⣿⡞⠉⠀⠀⠀⠀⣻⣿⣿⣶⣻⢇⣿⣤⠠⡉⠀⠀⠩⢬⣭⠍⢩⣭⠀⠓⢤⡇⠀⠀⠀⠀⠁
⢠⣿⣿⡿⣿⣷⣾⣶⣾⣧⣤⣿⣿⣿⣿⣿⣝⣹⢿⡔⠆⠀⠲⢷⡾⣷⠶⠿⠟
⢈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢻⣿⣿⣿⣤⣤⣄⣠⣄⣀⣠⣴⣶⡆⠠⠀⠀⠀⠀⠘⠃
⠈⠿⠟⠻⠟⠿⠿⠿⠿⠿⠛⠗⠀⠼⠏⠾⠟⠛⠛⠛⠉⠙⠁⠉⠉⠁⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠒⢛⡿⢲
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠈⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⢀
⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁
⢈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠸⢤⢀⣴⣶⠀⢠⣶⠀⢰⣶⣦⢠⣤⣤⢴⣶⣶⡆⣶⠀⣴⠀⠀⣶⣶⡆⣴⠀⣶⣶⡀⢠⣶⣦⣴⣶⣶⡆⠈
⢈⠀⠃⢸⣧⣈⠀⣼⢿⡄⢸⣇⣄⢸⣏⣉⠀⢹⡇⠀⢹⣿⡏⠀⠀⣿⣤⡀⣿⠀⣿⣹⡇⢿⣄⡀⠀⢹⡇⠀⠆
⠠⠤⠀⠀⠙⢿⡆⣿⣼⣇⢸⡟⠋⢸⡟⠛⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠋⠁⣿⠀⣿⢿⡅⠈⠛⣿⠀⢸⡇
⢠⠘⠀⠸⣶⡾⠹⠏⠉⠿⢸⠇⠀⢸⣷⣶⠀⠸⠇⠀⠀⠿⠀⠀⠀⢿⠀⠀⢿⠀⠿⠈⡷⠹⣶⠟⠀⠸⠇⠀⡁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠤⠀⠁
⠧⠿⠥⠭⠿⠷⠦⠤⠬⠭⠭⠩⠭⠁⠠⠤⠀⢀⣀⣀⣆⣀⣀⡀⠀⠀⠠⠩⠥⠤⠤⠭⠤⠤⠤⠤⠄⠠⠤⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹⠝⠉⠙⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⠀⠀⠀⣀⠘⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠃⠘⠛⠂⡴⠅⠀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠆⠀⠀⠀⠀⠀⠀⠃⡀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢄⠀⠀⠈⣥⠢⡀
⠀⠀⠀⢀⠀⠀⠈⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣮⣧⣀⣤⣿⡇⣤⣄
⠀⠀⠸⣀⠑⢄⢰⡜⣦⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣛⡛⠛⢽⡛⠿⣮⡀
⠀⠀⠀⠈⣗⣚⣛⡻⠎⠊⠻⡺⣧⣦⠄⠀⢀⠤⣄⠀⠀⠀⣖⡩⠃⠳⠭⠑⣿⢦⣁⠳⣦⢒⣢⡀
⠀⢀⣠⡴⠋⡙⠲⠬⠵⢦⣀⣆⠘⢿⡝⢯⣶⣾⡽⣠⡾⡋⠁⠀⠀⢰⠂⣀⡸⡟⡿⡀⠹⣿⣯⣅⡀⠀⠀⡴⢦
⠀⣼⢋⠋⠀⡁⢀⣴⣶⣼⣿⢿⡂⠀⠈⠀⠹⠙⣿⣿⣧⡇⠀⠀⠀⠀⠰⢿⡛⡇⣷⣾⣄⠹⡱⡌⢟⢿⣧⢹⠋
⢠⣽⡈⠀⠈⠁⠨⠉⠛⢹⡿⠘⠛⠀⠀⠀⢰⠂⣿⢿⣾⡇⡐⠀⡎⠀⠀⠀⠀⣿⠁⣇⠀⠀⢃⠇⠰⢸⣮⡆⢇
⠈⢻⣇⣠⠀⠀⠀⠀⢀⣾⠃⠀⠀⡀⠀⠀⠸⢀⡏⠀⣿⣓⣛⣛⡣⢄⡔⠀⠔⡹⠀⣿⠀⠐⣿⣴⠰⢼⢿⣿⣾
⠀⡓⠐⣶⠤⢐⠲⢤⣶⢁⠀⢠⣰⡇⠀⢀⠇⣸⠁⢠⣿⠋⠟⠛⢈⠷⠬⣻⣾⠁⢠⡇⡄⠀⡟⡿⠑⢌⣿⣷⢹⠆
⢠⣡⣾⡏⠀⠸⠉⠲⣽⠇⣠⣿⡇⠻⠀⡜⢠⡏⠀⢸⡿⠀⠀⠀⠈⢀⠤⣜⣩⣷⠿⣿⠀⠀⡇⡇⠀⡜⡜⢳⣢⡳⡀
⡸⠙⠋⣀⠀⠀⡤⢯⣼⣿⠟⠙⡧⡠⠋⣠⠞⠀⢀⡿⠁⠀⠀⡀⣰⢻⣠⢿⣿⡇⠀⠸⠀⡸⡿⠙⣷⣵⢄⢸⠙⣿⣿
⢣⠐⢘⡏⣧⣼⣧⡿⣿⣿⣷⡤⠊⣠⡾⠃⣀⣀⢾⢷⢠⡀⢨⣿⣋⣿⢿⣞⡻⢧⣀⡰⢊⡼⢁⣼⣿⢯⣎⡿⠀⠘⠤
⠀⣧⣈⠀⢻⡻⠸⣝⠺⢯⣡⣴⣾⣁⣀⣸⣖⡿⠁⠀⢳⢀⠉⠱⣏⣷⣄⠉⠛⠀⣈⣥⠭⢅⣩⢾⡛⢻⣹⣧
⠀⠀⠉⠉⠁⠀⠀⢿⣳⠾⠷⢊⡻⠛⡯⠋⠀⠀⠀⠀⠀⠉⠉⠈⠉⢿⡻⣭⣏⢸⡧⠶⠞⢻⠓⢛⣫⠟⣱⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣾⠟⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡮⣹⡌⢛⣻⠋⢀⡰⠮⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⣷⣞⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡷⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠉⠲⢌⡙⠻⢭⣒⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⡞⠛⠶⣄⡈⠉⠳⢦⣀
⠀⠀⠀⣴⢏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣙⣲⣤⣀⠈⠛⠦⣄⡀
⠀⠀⠀⣿⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡈⠙⠶⣄⡀⠙⠳⢤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣉⣳⢤⣀⡈⠙⠲⢤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠆⠀⠈⠉⠙⠓⠲⠬⣽⣶⣤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠛⢻⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⢀⡴⠛⠁⠀⢀⣀⡤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣠⣞⣩⡤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠈⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡬⠜⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣸⣁⣠⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⢀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠘⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣂⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⢀⠤⠜⠓⠐⠀⠀⠀⠰⡇⣦⡔⡉⠤⠠⢾⠀⠀⠀⣀⠃⠀⠀⠈⠈⣡⠃⠀⠈⠀⢀⠤⠚⠃⠒⢀⠀⠀⠸⡑⣤⢐⠩⠄⠠⡎⠀⠀⠀⡐⠁⠀⠀⠉⢈⡔⠁
⢿⡃⠀⠀⠀⣠⣾⣄⠀⢀⠜⠁⢀⠤⠄⠀⣨⢀⡠⢴⡉⠡⠖⠊⠉⠈⠉⠉⡹⠶⣾⠁⠀⠀⠀⣠⣿⡀⠀⣠⠋⠁⢠⠤⠀⢀⡌⢀⠠⢞⠉⠰⠒⠊⠉⠉⠉⢉⡵
⠀⣱⠖⠦⠐⠋⠈⠉⠙⡿⠓⠒⠛⠓⠢⣤⠼⣏⠀⠀⠀⠀⠀⠴⠶⠦⠐⠾⠀⠀⠀⡶⠖⠆⠘⠉⠉⠉⢙⡟⠒⠒⠛⠒⢦⡤⢿⡅⠀⠀⠀⠀⡠⠶⠶⠆⠰⠮
⠨⢂⠀⣀⡴⠤⢤⣤⠜⠁⢀⣠⠄⣄⣀⣀⣠⠌⠉⠟⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠹⣀⠀⣀⡴⠤⢤⡤⠞⠀⣀⣠⠤⣄⣀⣀⣠⠈⠉⠟⠉⠀⠀⠀⣀⡀
⠛⠋⢉⣼⣄⠀⣀⣠⠀⠉⠈⢂⠀⠀⠀⠀⠴⠞⠀⢀⣀⡤⠔⠋⠉⠉⠀⠀⣨⠚⠛⠋⣁⣼⡀⠀⣀⣠⠂⠁⠙⢄⠀⠀⠀⠀⠴⠇⠀⢀⣀⡤⠒⠉⠈⠉⠂⠀⡥
⢴⡊⠀⠀⠈⠉⠁⠘⣦⠤⠐⠈⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢀⡠⠐⠒⠚⠳⠦⣴⠊⠀⠀⠉⠉⠀⢹⣄⠄⠒⠋⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣀⠤⠒⠒⠛⠲
⣀⣧⣀⡀⠀⢀⠠⢘⣻⠶⠟⠋⠀⠀⡠⠴⠦⣾⡤⠾⠿⠛⠞⠉⢿⠀⠀⠀⢀⣀⣈⣧⣀⡀⠀⢀⠤⣚⡷⠶⠚⠁⠀⢀⡠⠴⢴⣦⡴⠾⠟⠳⠞⠙⢗⠀⠀⠀⣀
⠀⠀⠀⠈⡹⡡⠞⠛⢦⡤⠖⠙⠒⠋⠀⠀⢀⡏⠀⠀⢀⡴⡀⠀⠀⠈⣻⠋⠁⠀⠀⠀⠀⠈⡟⡴⠚⠛⢤⡤⠊⠙⠒⠉⠀⠀⢰⠃⠀⠀⢀⠜⡀⠀⠀⠙⡿⠉
⢄⠀⢀⡴⠗⠁⠀⠀⣠⠗⠊⠀⢀⣀⠀⣠⠞⠀⢀⣄⢀⡈⡤⠖⢉⠚⠧⠤⣠⠄⠄⠀⢀⠼⠊⠀⠀⠀⣸⠗⠁⠀⣀⣀⢀⣠⠋⠀⢀⡆⢀⠀⡤⠚⢁⠛⠧⢄⣠
⠔⠛⡋⠀⠀⣀⡤⠚⠁⣀⠔⠈⠀⡘⠉⠛⠦⠒⠋⠉⠠⠀⣸⣤⣾⡀⠀⠀⠀⡠⠚⠛⡃⠀⠀⣀⠄⠛⢁⣠⠀⠁⢀⠘⠉⠓⠄⠚⠉⠉⡤⠀⣧⣤⣆⡀
⣠⣒⠁⠐⠈⢿⠆⠉⠀⠉⢧⠀⡸⠁⢀⠔⠀⠀⣀⡀⠤⡦⠋⠁⠀⠈⡣⠐⠔⢁⣴⣒⠥⠐⠉⡿⠊⠁⠈⠙⣆⠀⡞⠀⣠⠄⠀⠀⣀⡠⢤⠜⠉⠁⠀⠉⡦⠂⠊
⠀⠀⠀⢀⡤⠻⣀⣀⣀⣤⣊⠴⠁⣲⣁⠄⠚⠉⣡⣴⠊⠀⠀⠐⠥⠺⠤⣤⠔⠉⠀⠀⠀⢀⠔⢧⡀⣀⣠⡴⣣⠾⢐⣞⣁⠔⠚⠉⣠⣴⠃⠀⠀⢀⠕⠺⠤⣤⠞
⠀⠀⡠⠊⣀⡔⢹⠁⠀⠀⣠⠖⠋⠉⣡⠤⠒⢈⡡⠋⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⡐⠁⣀⡌⢹⠁⠀⠀⡤⠚⠉⠉⣠⠔⠂⢉⡵⠋⠀⠀⠀⠀⣀⣤⡾⠃
⢠⠞⠀⡈⠘⠁⣼⡀⠀⣠⡧⠀⠲⢔⣡⢶⣯⠥⠤⠤⢤⣀⠠⠊⠀⣠⢧⡠⢤⡄⡠⠎⠀⡁⠙⢀⣾⠀⢀⣼⡤⠒⠦⢎⡠⣶⣯⠤⠤⠤⢤⣀⠔⠈⢀⡼⢇⡠⢤
⣴⣀⣘⣀⡴⣚⣀⣉⣉⣙⣦⣐⣚⣋⣠⣮⣌⣿⣀⣀⣀⣠⢂⣐⣉⣀⣤⣠⣞⣁⣄⣀⣚⢠⡖⣋⣀⣉⣉⣻⣤⣒⣛⣉⣴⣧⣬⣟⣀⣀⣀⣸⣂⣚⣉⣀⣄⣰⣊
⣿⢿⠿⠿⢿⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣛⣿⣟⣿⣿⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⡠⢤⣶⡖⠚⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢦⣄⣀⣧⡿⣿⣍⣀⣾⣿⡿⠟⠛⠛⠛⠻⠾⣿⣷⣄
⠀⠀⠠⢤⣶⣶⣶⡾⠛⠛⠉⣴⣥⣤⡟⢿⣿⠃⠘⠛⠋⠉⠋⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠉⠛⡷⠶⢺⣷⣾⠿⠿⠋⣁⠈⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢋⡄⠀⣆⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⠁⢀⢻⣶⡾⠛⠀⢸⣿⠷⣦⣄⠀⠀⠀⠀⠀⢸⣆⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⢀⡾⠉⣠⢁⠁⠀⢸⡟⢶⣽⣿⣦⠀⠀⠀⠀⢸⣿⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠏⠀⠞⣁⡞⠀⣠⣿⣿⡌⢻⣿⣿⡆⠀⠀⠀⣸⣿⠇⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⣴⣨⣹⡷⠞⢻⣿⣿⡇⠀⢿⣿⡇⠀⠀⢰⣿⠃⣄⠈⢳⣶⣿⣯⣿⣿⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢰⡉⠐⢧⣌⠻⠽⠟⠀⣠⣿⢿⣿⡇⠀⢸⣿⣿⠀⠀⣴⠋⣸⠟⠀⠀⣿⠿⠿⠿⣿⣿⣿⣄⡀⠉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢷⠛⠘⡋⣰⡟⢀⣼⣿⣿⢸⣿⡇⠀⣼⣿⣿⡇⠀⢁⡼⠋⢀⠀⠀⣿⠀⠀⠀⠀⠉⠻⣿⣿⣆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠝⠃⣼⣿⣿⣿⢸⣿⣿⠀⢸⣿⡟⢀⣴⢟⡴⠀⡾⠀⠀⣿⠄⠀⠀⠀⠀⠀⠈⢿⣿⣆⣦⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⡄⣿⣿⡇⢸⣿⠀⣿⣿⠟⢠⡾⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢸⣷⣿⣿⣧⣿⢹⣦⣁⣀⠀⡞⠁⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢀⣿⠃⣿⠁⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⢀⣾⢃⣼⡟⠀⣼⡟⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⣌⣿⣿⣿⣿⡟⣿⣿⠟⠓⠀⠀⣼⣿⠋⠀⠀⠀⢀⣴⡿⣿⣿⡟⠁⣼⣿⠁⣿⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣮⣿⣿⠻⠿⠿⣿⣿⡇⣠⠆⠀⢸⣟⣀⣀⣠⣴⣾⣿⣏⣰⣿⠟⠀⢠⣿⣿⠀⠻⣆⠻⣎⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⡼⢋⣽⡿⠏⠀⠀⠀⣸⣿⡷⢫⢀⣴⣿⣿⣿⣿⣿⠛⢻⣿⣿⠟⠋⠀⢠⣾⣿⣿⡀⢄⡈⠳⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢡⣣⠞⠁⠀⠀⠀⣰⡿⠋⠔⣣⣿⣿⣿⣿⣿⠟⠁⠀⣸⣿⡟⠀⢠⡄⣿⣿⢻⣿⣷⣜⣾⣷⣿⣍⠛⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⠃⠀⠀⠀⢠⣾⣿⡇⢠⠺⣿⣿⣿⣿⣿⣟⠀⠀⠀⣿⣿⡇⠀⠘⠀⣿⣿⠀⢿⣿⣿⣷⣽⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢹⣿⠀⠀⠀⠀⢸⣿⣿⠃⡾⠀⠻⣿⣿⣿⣿⣿⢦⣤⡀⠸⣿⡧⠀⠀⣼⣿⣿⡆⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣸⣟⠀⠀⢠⡄⠈⣿⡟⠠⡇⡀⠀⠀⠙⢿⣿⣿⡆⣸⡇⢀⣿⠃⠀⣾⣿⣿⣿⡇⣾⣿⢿⣎⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣷⣤⣾⣷⣆⣹⣿⠈⣷⣷⣴⠀⠀⠀⢻⣿⣷⡿⢁⣼⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣹⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠈⣿⣻⣿⡄⠘⠿⠏⣷⠀⠀⣼⣿⢹⠃⣸⡏⠀⠀⢢⡟⠙⢿⢿⣧⡙⢿⣧⠘⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣀⣼⡗⠹⣿⣆⠀⠆⠀⢼⣦⣰⣿⡏⢸⣆⣿⣧⠀⢀⡾⠁⠀⠀⠁⠀⠙⠲⠿⣿⠺⢿⣿⣦⣝⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⣦⣼⣿⣷⣄⡀⠀⢩⡟⢻⡀⠀⠉⣸⡏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣿⣿⣧⢀⡷⠈⣷⣤⣴⡿⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠿⡿⠻⣿⣿⣁⣴⣿⣿⠏⠀⠀⣠⠜⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢉⣿⡟⠀⠀⠀⠀⢀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣾⣷⣿⡾⣷⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣉⣀⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣹⣩⣿⣱⣌⣭⣏⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣤⣼⣿⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠄⣐⠒⠒⠀⠀⢠⡄⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢐⠦⢛⡀⢓⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣘⣁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⢠⢾⡉⠓⡄⢀
⢇⠀⠀⡠⠄⠀⠀⡀⡄⢚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠱⠆⠀⠀⠀⠀⡖⠄⡁⠀⠀⠀⠀⠸⠤⠤⠴⠁⠀⢁
⢸⣷⡞⠀⠀⠂⠑⠁⠀⠀⠀⣈⣉⣀⠖⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡎⠀⠄
⠀⠀⢅⠀⠀⠀⠈⠈⠛⠻⣿⡩⠍⠀⠀⠀⠀⢀⡰⢆⡄⠀⠠⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠒⠒⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⢧⡀
⠀⠀⠀⠀⠀⠀⠃⠤⠖⠒⢦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢹⠀⠀⢠⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣸⠀⠇⠀⠀⠀⠀⠀⠂⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢸⠀⠀⣀⣶⡀⠀⠀⠀⡇
⠀⠨⠀⣠⣤⣤⡁⠀⢀⣤⣤⡀⡀⠀⠀⢀⡴⠀⠀⠙⠃⠀⠈⢀⣄⠀⡀⠀⠀⡀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠉⠿⠉⠀⠀⠀⣇
⠀⠰⠄⠺⣘⡪⠷⠴⢯⣒⡾⠳⠂⠀⠀⠘⠄⠀⠰⣾⡧⠈⠠⠃⣺⡧⠂⠀⠠⠀⠀⠀⠀⠒⠀⡢⣬⡐⡄⠀⣴⣩⡑⡤⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠀⠀⠀⠻⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠓
⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣇⣀
⣄⣀⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⠀⡁⠀⢄⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⡖⠀⠐⠄⣀
⠈⣸⠷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⢌⠁⢈⡇⠀⢀⡎⠉⢱⠄⠀⠆⠀⠀⠀⣠⢐⣛⣛⠁⠀⠖⠁⣀⣈⡀⠈
⠐⠈⠓⠶⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠋⠉⠀⠀⠀⠉⠒⠉⢠⣶⠀⠀⠀⣖⣋⡿⢽⡌⣷⣀⣠⣿⢹⡬⢽⣄
⠀⠀⠀⠀⠀⠀⠑⠠⠀⠉⠉⠈⠉⠗⠂⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⠖⠚⢿⡎⠇⠀⠀⠀⠀⠙⠳⠚⠁⠀⠀⠈⠓⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠐⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠁⢀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⠈⡲⢰⡿⠷⠀⣾⡇⢰⡿⠿⢰⡶⠶⠿⣿⠿⠸⣇⣼⠃⠀⣿⠿⠃⣷⢰⡿⣷⢠⡿⠿⠸⣿⡿⢂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⣷⣤⢠⡏⣷⢸⣷⡶⢸⣷⣦⠀⣿⠀⠀⢻⡟⠀⠀⣿⣶⠄⣿⢸⣧⡿⠘⢷⣦⠀⢸⡇⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣈⣿⣸⠿⢿⡘⡇⠀⢸⣇⣀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⢸⡏⢷⢠⣄⣿⠀⢸⡇⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠉⠁⠉⠀⠈⠁⠁⠀⠈⠛⠋⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠉⠈⠁⠈⠁⠉⠁⠀⠈⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣄⣀⣶⣄⣀⣁⣐⡒⠂⠀⠀⢀⣀⠀⢀⣀⣠⡀⣀⠀⠀⠀⠐⠒⣀⣀⣀⣂⣀⣀⣀⡀⢠⣤⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡟⠛⠋⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠵⢆⠀⣴⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣟⣝⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠶⠀⠂⠂⠂⠀⠐⢄⠐⠐⠐⠀⠀⠀⠀⠤⠂⠀⠶⠖⠈⠂⠀⠚⠠⡤⡄⠀⠀⢀
⠀⠀⠀⠄⠀⠄⠀⠀⠀⠘⠄⠀⠜⡀⠀⠠⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢄
⠀⠀⠀⠀⠰⠀⠇⢄⡰⠀⠐⠀⠆⠹⠠⣀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠰⢆⣆⠘⡀⡀⡼⢁⢀⠄⢢⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠂⠐⠐⠀⠀⠰⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠈⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠶⠠⠅⠀⠀⠀⠀⠠⠂⠀⠀⠀⠻⡏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠷
⠀⠀⠀⠀⢅⠀⡊⠐⠒⠀⢠⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⢡⡆⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⢀⠀⢺⡄⠀⠀⠨⢟⡃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢶⠸⢇⠀⠀⠀⠀⢈⣿⣤⠉⠀⠀⠈⠡⠼⢇⠀⠀⢌⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘
⠀⠀⠠⣀⣦⠀⠄⢀⣤⠀⠚⠻⠤⠆⠆⠀⠐⠀⠈⠁⠀⠈⠀⠸⣿⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⣠⠆⠀⠀⠐⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⠀⠀⠘⠺⠉⠐⠒⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⢈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠀⢀⣴⡄⣠⣤⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠃⠀⢸⠀⠀⠀⣀⡄⠀⢀⡀⡄⢀⣀⣀⣀⣀⠀⠀⠈⠉⠉⠀⠘⠻⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⢸⠀⠀⠍⡏⢱⣠⣴⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⠄⣠⣄
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⡀⠸⢻⠀⠀⠀⡇⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠀⣠⣶⠷⣾⠿⠻⠿⠄⠘⠛⠛⢻⣿⡿⠿⠿⠿⠿⡿⠿⠶⠼⠷⢤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠛⢰⡏⠁⠐⣶⣦⣄⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⣠⣤⣤⣌⣛⣛⣀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡧⠴⠶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⣿⢀⣄⠀⢸⡇⠐⠒⣿⣿⡧⠿⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡄⣤⣿⡇⣿⠘⠛⠀⠘⣷⠶⡾⠋⠀⠀⠀⠠⠄⠀⠁⠂⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣿⣧⣻⣶⣶⣶⣶⣶⡞⣰⣿⣿⣷⡀⠀⢰⡟⠛⣿⠛⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣻⡟⣶
⠀⠀⠀⠀⠰⢾⣿⣿⣯⠻⣿⣿⣿⠋⢻⡯⠭⣭⣿⣿⣿⣰⡿⣋⣉⣻⣷⠀⠈⠁⠛⠛⠛⠉⠀⠸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠸⡇⣿⡃⣿⣿⣿⣿⣄⣸⣇⣀⣏⠀⢼⣿⣿⣿⠿⠟⣿⡟⣷⠒⠒⠒⠒⠒⠒⠛⠒⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠻⠷
⠀⠀⠀⠀⠀⠤⠿⠿⣷⣿⣿⣿⣿⣿⣿⣭⣍⣛⣻⣿⣿⣿⣿⣷⣆⠤⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⣿⣿⣿⣿⣿⣿⠿⣡⣦⡇⣾⣤⣤⣤⣤⣤⣤⣤⠄⠀⠐⠀⠒⠒⣶⣶⣶⣶⣶⣶⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠻⢻⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓
⣦⠤⠦⠤⠤⠤⢤⡤⢤⡀⠖⠒⠶⠤⠦⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠒
⠀⠶⠶⠶⠦⠠⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠦⠤⠤⠄⠀⠀⡀⠠⠄⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⢀⠀⣤⣤⣄⣀⣀⣀⣀⣀⡀
⣦⣿⡿⠛⠿⠯⠿⠿⣿⣿⣿⡟⠛⠒⠒⠛⠙⠛⠉⠉⠉⠉⠉⠛⠛⠉⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠍⠩⠛⠛⠓⠀⠛⠉⠛⠛⠛⠛⠛
⣿⣿⣿⡿⠶⠶⠶⢶⡶⣖⣀⣠⣀⣀⠀⠀⠀⠀⠤⠄⠀⣠⣤⣀⣀⡀⢀⡀⠀⠄⠀⠠⡄⠀⠠⠄⠀⠤⠤⠄⠀⢀⡀⠤⠠⠤⠤⣤⠄⢀⣀⡀⣀⣀⣀⣀⣀
⢠⣤⣤⣄⣀⣀⣴⠔⠀⠀⠀⠤⠤⣤⣤⣤⣠⣤⣤⣄⣠⣄⣠⣄⣀⣀⡀⣀⣀⠀⣀⣀⡚⠛⡛⠚⠛⠒⠒⠂⠲⠤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠬⠷
⠀⠀⠀⣀⣈⣱⣶⠶⠶⠶⠶⠶⠆⠀⠐⠶⠄⠀⠠⠩⣭⣴⣶⡦⢤⣭⣯⣍⣭⣉⣉⣉⣙⣛⣛⣛⠛⠓⠒⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠘
⠀⠀⠶⠿⠿⠟⠉⠉⢉⡙⠙⠋⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⢀⠀⠀⠈⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣿⡟⠒⠶⠦⢤⡄⠀⠈⣄⡁⠀⠀⠀⠀⠀⠄⠀⠀⣤⠀⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠐⠂⠈
⠐⠒⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⠄⠄⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠠⠤⠒⠀⠠⠤⠤⠤⠤⢤⠀⠒⠒⠒⠀⠠⠤⠄⢠⣄⠀⠀⠲⠴⠀⠠
⠤⠄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣐⣒⣶⣶⣦⣤⣄⣹⡟⠛⠙⠀⣐⣒⣲⣶⣶⣶⣶⣦⡴⠀⣀⣠⣄⣘⣓⠀⠠⠬⢤⣭⠙
⠀⣀⣀⣀⣀⣀⣀⣲⣤⣤⣤⣤⣤⣤⣤⡶⢷⣾⣿⣻⡿⢛⠛⠛⠶⠶⠶⠶⢤⣤⣄⣀⡛⠛⠛⠛⠿⠃⠠⢤⣤⣤⣤⣍⣉⣛⣉⠉⠙⠚⠛⠻⠗⠒⠒⠂
⣭⣭⣭⣍⣉⣙⣛⡛⠿⠿⠶⠶⣶⠿⣿⣿⣶⣾⣯⣭⣭⣭⠽⠽⠿⠐⠒⠶⠶⠶⠾⢯⣭⠉⠉⠙⠿⠶⠶⠶⠶⠶⠾⠍⠉⠉⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⢀
⠆⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⣄⣤⣤⠤⠶⠶⠷⠿⣿⣿⣟⣛⣛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢛⢛⣛⣛⣛⣛⣛⣛⡛⣛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⢻⣿⣿⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣛⣛⣛⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣠⣄⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢵⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⢿⣿⣿⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣀⣄⣀⣀⣀⣀⣴⢲⣭⣭⣭⠁⠄⠀⡌⢩⣭⡅⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣭⣭⣭⣀⣃⣐⣁⣨⣭⡅⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢻⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠰⠦⢀⡀
⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣤⣬⣥⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⢉⣳⣿⣷⢦⣄⡤⠤⠤⢤⣀
⠀⠀⠎⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠀⣼⡟⠉⠹⡆⠀⠉⠛⠻⢅⣈⠙⠢⢄⡀
⠀⠀⣦⡏⠀⠀⢀⡟⡿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢠⠀⠈⠐⠢⢤⣀⠀⠈⠉⠒⠶⣤⣷⣦
⠀⠈⠛⠃⠀⠀⢸⡌⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢹⣷⡆⠀⠀⠘⡌⠉⠒⡦⠤⣀⠈⠙⠂⠤⡀
⠀⠀⠀⠀⠀⠀⡿⡇⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⢸⢿⡇⠀⠀⠀⣧⠀⠀⢸⠀⠈⡇⠑⡖⢤⣬
⠀⠀⠀⠀⠀⢀⣦⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣄⠘⡇⠤⠄⣀⣈⣀⡀⠈⡄⠀⠠⠀⢸⠀⢙⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠙⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⢸⠀⢻⠀⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠈⢤⣿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⣀⠀⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠠⠆⣠⡀⠀⠠⠾⣿⡇
⠀⠀⠀⠀⠀⢸⣤⣀⣠⣤⣬⣤⣤⣤⠄⣤⡤⠠⠶⢶⡶⠶⣀⣀⣠⣤⣴⣶⣿⡀⠀⠇⡇⢂⣀⣀⡤⠤⠤⠤⠶⠶⠒⠚⠛⠻⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣔⣏⣿
⠀⠀⠀⠀⠀⢠⣘⣀⣀⠀⠨⣯⣄⠀⠀⠀⠀⠀⣰⣶⣶⠆⠀⢀⣀⣀⣀⣠⡤⢄⠀⠀⣿⣴⣿⣿⡆⠀⠀⢀⡀⠤⠀⠀⣿⡏⠉⠅
⠀⠀⠀⠀⠀⢈⣿⣿⣯⡄⠀⢺⣷⡂⠀⢤⡄⠀⣤⣽⠓⠀⢠⣿⣿⣿⣿⣿⣇⡨⠀⣀⢿⣿⣿⣿⣇⠈⠀⠀⠀⣀⣠⣴⣿⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡅⣀⢒⣶⣲⡶⠶⠶⠶⣶⠶⠶⠂⣐⣒⣒⣒⠒⠒⠒⢂⠀⠁⢸⣿⣿⣿⠏⠀⣤⣤⣼⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠍⠉⣉⣽⣋⣀⣀⣀⠀⠈⠁⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⡏⠀⢀⣵⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣟⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠿⠘⠉⠽⠿⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣶⣦⠒⠐⣷⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠖⠈⡔⠀⠀⢸⣄⠀⠀⢀⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠀⠀⣿⣀⣀⣴⣿⣿⢀⡴⡻⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣰⣿⡿⠟⠛⢉⣿⠏⠈⣼⣱⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢴⡿⠃⠀⠀⠀⣾⣯⣴⢾⣻⣵⣿⣿⣷⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡾⠿⣿⣿⡿⠿⠋⢉⣿⠷⣄⡀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⠤⣄⣀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣾⣿⣷⣶⣾⣿⡶⠟⠁⠀⠉⠙⠛⠛⠛⠛⢋⡅⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣶⣿⣯⣭⣍⡙⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⣽⣿⠛⠉⣡⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⡉⠉⠻⣿⠇⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠃⠁⠀⠀⠙⠃⠀⢠⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣇⣤⣾⣿⡞⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⡄⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣷⠀⠀⠀⠀⠀⠀⣼⠇⣾⣿⣿⣿⣥⡴⢋⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⣄⣶⣮⣭⣉⣿⣆⠀⠀⣿⢱⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡗⠀⠀⠀⠀⣠⡾⣏⣾⡟⣼⣿⣿⣧⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⣿⡿⢿⣿⣷⣶⣾⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣋⣭⣿⣦⣀⠀⠀⠀⠀⢹⣿⢠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣟⠀⠀⠀⠉⠉⠉⠉⠛⠙⢿⣯⣯⣿⣿⠻⠿⠿⠿⠿⠟⠛⠛⠙⣿⣿⣿⣿⣿⣿⠷⣦⣄⠀⠈⠛⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠻⡿⣆⠈⠻⣦⣄⠀⠠⣝⡻⢿⣷⣶⣦⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣸⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠘⣮⡓⣄⠀⠉⠛⢦⡈⠿⡎⣿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣚⣉⣴⡾⠿⠋⠀⠀⠀⠀⠙⢶⠀⢻⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣻⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢁⣾⠉⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢏⣀⣻
⠀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣤⣤⠶⠶⠒⠒⠒⠀⢀⣤⣤⣼⣇⣄⣉⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠒⠒⠲⠶⠾⣧⣤⣤⣏⣀⠀⠀⢀⣀⣀⡀
⢰⡏⢀⣬⣹⣏⢉⣩⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣉⠉⣹⣫⣥⡀⢹
⠘⣷⡈⠳⢿⡟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⡿⠟⢀⡾
⠀⠈⠙⢷⣬⣷⠀⠀⢀⡔⠁⣉⣭⠿⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠿⣭⣉⠐⢦⡀⠀⠀⣾⣵⠾⠋
⠀⠀⠀⠀⠈⠻⣷⣠⣿⡴⠛⠉⠀⠀⠀⠀⠈⠙⢶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡶⠋⠁⠀⠀⠀⠀⠉⠳⣦⣿⣄⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿

⠀⠀⠀⠀⠀⢀⣶⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣆
⠀⠀⠀⠀⠀⢸⣟⣿⣦⣤⢤⣤⣤⣤⣤⣥⣼⣗⣭⣭⣭⣭⣥⣬⣥⣭⣭⣭⣽⣤⣭⣤⣤⣤⣤⣤⣴⣾⣽⣿
⠀⠀⠀⠀⠀⠀⠙⣯⣿⣷⣾⣿⣷⣶⣿⡿⠿⠿⠿⠟⠛⠛⠛⠙⠛⠛⠻⠿⠷⠿⣿⣿⣾⣿⣿⣿⣾⣯⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣤⡚⠛⠷⠦⠤⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠷⠖⠞⠛⣤⡀
⠀⠀⠀⠀⠀⠐⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣼⣷⣤⣴⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠂


⣿⣿⡿⠿⢿⡟⢻⠿⢿⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣷⣾⣶⣾⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⠀⣀⠀⢀⣔⡁⣸⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢀⡠⠠⢄⡀
⠀⠀⠀⠀⢠⠉⠈⠀⠀⠀⢀⢍⠱⡀⠈⠉⢳⠀⠀⠀⡋⠀⢀⠤⠬⠤⠤⡀⠈⡇⠀⠀⠀⠀⢠⠀⠀⠀⠈⠑⠢⢈⠆
⠀⠀⠀⣀⠬⢉⡂⠆⠀⢠⠤⣭⠚⠀⡠⠤⠤⠃⠀⠀⣉⠉⢹⠀⣤⠞⢸⠉⠉⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠰⠽⠑⢢
⠀⢘⠉⠁⠀⠀⠘⠦⠔⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡉⠉⠑⠒⠒⠂⠘⠁⠉⡇⠀⠀⠀⡀⢀⡠⠓⠒⠤⣄⣀⠤⠀⠴⣀⣀⡀
⠀⠀⠁⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣠⠤⠀⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠎⡠⠤⠈⠄
⠀⠀⠀⢀⠁⠀⠀⠀⠀⢸⠈⠊⡊⢂⠀⠀⠀⠀⠀⠀⢎⠉⠁⢺⠧⠀⠀⢀⡨⠂⠀⠀⠀⠀⠀⠀⡐⠘⠄⠠⠀⠣⠤⠁⡇
⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⡄⠁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠑⠤⣌⢸⢅⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⡠⢀⠀⡇
⠀⠀⠀⠈⠉⡟⠛⠟⠛⠛⢲⠒⠲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠂⢲⠤⢎⠀⢀⠟
⠀⠀⠀⢂⣀⡈⠋⠀⠀⡀⠂⠛⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡟⠃⡆⠀⡟⠋⠖⠒⡄
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉


⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⢰⠁⠆⠆⡔⠒⠀⠒⡆⣴⠉⢪⢂⠀⠀⠀⠀⠀⠀⠀⡐⡟⠁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠿⠓⠢⡰⠒⠒⡕⡄
⠀⠀⢠⠀⠑⠁⠈⠊⠀⠑⠁⠈⠂⠊⢸⠀⠀⠀⠀⠀⠀⣔⡀⢇⣀⣘⠀⠈⡆⠀⠀⠀⠀⠀⠀⡌⠀⡘⠂⠐⠊⠐⠂⣒⠁⡀
⠀⠀⠈⠂⢼⣀⡷⠀⠀⠀⠀⢎⣀⡗⠊⠀⠀⠀⠀⠀⠀⣇⣈⠖⠀⠐⣖⣒⠇⠀⠀⠀⠀⠀⠀⠈⢏⣀⡧⠤⠤⠤⢎⡀⡙
⠀⠀⠈⠭⠉⠍⠉⠭⠭⠭⠉⠩⠭⠭⠉⠀⠀⠀⠀⠈⠩⠭⠉⠉⠭⠭⠍⠉⠭⠍⠀⠀⠀⠀⠈⠩⠭⠍⠉⠩⠭⠉⠉⠭⠭⠍
⠀⠀⠉⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠶⡚⠻⣿⠷⣎⢻⣿⣿⡿⣿⢲⣄
⠀⠀⠀⠀⣴⠟⡵⠋⠀⠀⠈⠣⠙⣇⣿⢻⡇⢜⢧⢻⣹⣷⣄
⠀⠀⠀⠾⠟⠈⣡⡤⠤⣦⡲⣶⣤⣈⠻⣮⣻⣾⣿⣿⣿⡿⢿⡶⣄⠀⠀⠀⠀⢀⣠⣤⡤⠶⣾⣿⠿⢿⣶⣦⣄
⠀⠚⠻⣶⠒⢴⣠⡀⠉⠻⣶⣦⣭⣝⡳⣜⠿⡍⠻⡝⢿⣝⣷⣝⢿⣷⣤⠴⠾⢿⣍⢠⣶⣶⡖⠺⢽⣷⣬⡻⣿⣷⣄
⠀⠛⢻⡿⡏⢡⣼⣇⠀⠀⢹⡟⠫⣻⣿⡜⣆⠀⠀⢣⠈⣿⣿⣿⣼⣿⡇⣠⡈⠛⣯⣲⣯⡛⢿⣦⣱⡌⠉⡙⠜⣿⣿⣧
⠀⠀⢸⣿⣿⡋⣉⡈⢂⡤⣾⣏⠀⢹⡿⣿⣿⡀⠀⢺⣰⢟⠟⢹⡿⠋⢰⢣⣿⣆⠘⣿⣿⣷⠘⣿⣿⡇⠀⢹⡄⢹⣿⣿⡀
⠀⠀⠀⠙⢯⣳⡌⠻⡦⠶⣼⣿⠀⣸⠇⠹⣿⣇⠀⠀⠙⠀⡀⠘⠇⠀⣿⡸⣿⣿⠀⣿⡏⢨⢸⣿⣿⡇⡆⢸⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⢮⡒⢾⣟⣻⡄⡟⠀⢷⢈⣿⡄⠀⡄⣼⠁⠀⠀⢀⣿⡇⣿⣿⣿⡿⢁⣿⣿⣿⣿⢡⡇⣾⣇⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢻⠟⠀⠈⠳⠿⠿⠃⣿⠀⢸⡏⢉⠙⠂⠹⣿⠀⣀⢀⣼⣿⣧⠃⣿⡟⢡⣾⣿⣿⣿⣥⣿⣱⣿⡿⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⡆⠘⠓⠀⠀⣴⣿⠣⠿⣿⣿⣿⠜⣯⣶⣿⣿⣿⠟⢿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢨⣃⣀⣄⠀⠀⢹⣿⣰⡀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⣿⣿⣿⡟⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡿⣿⣿⠛⠛⠿⢿⡏⠀⢹⣿⡿⠉⠘⢿⣿⣿⣷⣄⠀⠈⣿⣿⣇⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⢻⠇⠀⠀⠀⠈⢿⡀⠈⣿⡇⠀⠀⠀⢹⣿⣿⡿⠀⣇⣿⣿⣿⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣁⡞⠀⠀⠀⠀⠀⠈⢷⠀⢿⡇⠀⠀⠀⢸⣿⣿⠁⣼⣿⣹⣿⡿⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉⡟⠀⠀⠀⠀⠀⠀⢸⠐⠛⣧⠀⠀⢀⡿⢻⡇⠀⢯⡾⢸⣿⡇⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣾⠇⠀⠀⠀⠀⠀⠀⢸⡀⣼⡇⠀⣠⠏⠀⠘⣧⠀⠸⢀⣾⠟⢳⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⣿⣿⣿⣶⡿⠀⠠⣯⠤⣂⣸⡇⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠉⠉⠉⠉⠀⠀⠀⠻⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢂⠀⢻⡇⠀⠀⠀⠀⠀⠀⢠⣿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⣦⣾⡇⠀⠀⠀⠀⠀⠀⣿⡇⣀⢲⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠞⠛⠛⠉⣩⡵⠚⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠞⠛⠋⠉⠁⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠿⠶⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⣀⣴⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⢀⣀⣠⣤⠤⠴⠒⠒⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⣀⣤⠾⠋⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⢛⣉⣀⣤⠤⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⢸⡏⢙⣷⡄⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣶⣿⣿⠓⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠉⠈⠻⣦⣀⣠⣴⠾⠛⠛⠒⠲⢦⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠉⠙⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣶⣶⣶⠶⠶⠶⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣤⣀⠀⢸⡇⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⠿⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣤⣤⣴⣾⣿⣟⣋⣭⡭⠄⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⣷⣶⣤⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⣍⠉⡁⢍⠹⣓⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠨⡓⠀⣷⢸⡆⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢝⢸⡇⢸⣆⣿⠀⠀⠀⠀⠀⠀⣀⡀⠉⠀⠀⠀⠀⢀⣀⣀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠸⣧⠈⢿⣿⡇⠀⠀⠀⠐⠛⠿⠛⠀⠀⠀⠀⣤⣦⣀⣾⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠻⣧⡈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⢩⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡏⢿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣝⠚⢿⠀⠀⠀⠀⠀⠻⣭⣟⡃⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣛⠓⣤⠑⣦⡀⠀⠀⠀⠀⢉⣴⣿⠟⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠀⠉⠁⠀⠙⢦⣘⠯⣗⡶⠶⡶⠿⣿⠏⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠀⠀⠐⢤⡀⠀⠀⠀⠉⠀⢩⡤⠎⠀⠈⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠈⠙⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⣄⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢂⠀⠀⠁⠂⢂⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠐⢤⡏⠈⠡⠂⠠⡂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢠⡄⠁⠀⠀⠀⠀⠈⠀⣹⠀⠀⡀⠘⠀⠀⠀⠀⠀⣀⠀⢈⡘
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⡀⠤⠀⠀⠀⢐⠀⠀⡄⠀⢠⡤⠂⠉⠁⠀⠀⠉⠀⠔⠄⠀⠀⡢⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠂⠐⡜⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣵⡴⡄⠈⠀⠈⠀⢣⠀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠀⡈⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠘⢰⠓⠄⠊⠀⠍⠀⡀⠀⠒⠀⠆⡗⢢⠔⡠⠂⠠⢈⡒⠡
⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠃⡀⢄⡀⠒⠀⠀⢠⡟⡜⡀⠀⢠⠄⣡⠫⠐⠍⠀⠴⢆⠖⠁⠀⠠⣈⡬⠀⡛⠠⠂⢁⠰⠀⡜⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⡢⠀⠠⠠⠀⢢⣄⣀⣠⠋⢉⠀⠉⡒⢡⠚⣋⠤⣄⠈⢤⠲⠧⠤⠤⣤⣈⣼⢡⠨⢥⠈⡁⠡⠤⠂⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⡋⠙⠂⠀⠁⡤⣄⣀⣈⣸⠆⠀⠉⢈⣉⣀⢀⠀⠐⡀⠀⣢⠂⢀⢹⣴⠃⣾⠷⠧⣔⡁⠁⠐⡀
⠀⠀⠀⠐⠐⠀⡀⠀⠀⠈⠀⠙⠢⡄⢀⣾⡁⠶⠛⡦⠀⢀⡀⠴⠊⠈⠙⠲⠖⠁⠛⠊⢈⡇⠀⠀⡀⠀⠍⣀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠀⠁⠀⠁⠀⠁⠈⣑⡄⢠⣖⣟⣁⣀⠨⣁⡴⠖⡀⢉⡐⠂⠁⢀⠠⢀⠀⠀⠐⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠂⠄⠀⠈⠘⠀⠂⢐⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠄⠉⢁⠄⠴⠿⠍⠒⠛⠀⢀⠀⡐⠀⠂⠀⠀⠉⢉⡉⢀⢀
⠀⠀⠘⢀⠀⡨⠘⣯⠁⢂⠀⠀⡠⣞⠀⠁⠀⠀⢐⠃⣼⣫⡝⢀⡀⠀⠔⠀⠑⡄⢁⠀⠆
⠀⠀⠀⠘⠐⠻⢿⡁⠀⠈⠉⢉⠐⠀⠀⠀⠀⢓⠡⠊⠁⠀⠀⠀⠀⠀⠀⠈⢀⠁⠨⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⣇⠀⠀⠀⠃⠛⠀⠄⠐⠈⠁⠀⠰
⠀⠀⠠⠀⠀⠀⢀⢈⡷⡂⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠠⣀⠼⢋⢛⢂⠀⡀⠀⠁⠀⠤⠔⠸⠄⢐
⠀⠀⠀⠀⠚⢿⠨⠐⠂⡐⠂⠀⠄⠀⠀⠀⠀⣀⠒⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣒⡒⠖⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⡌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠀⠀⠀⠒⠒⠠⠤⣀⠀⠀⠀⠀⠸⠀⠀⠙⢄⠀⢠⣚⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⣲⣦⣤⣤⣤⣀⣀⠀⠀⠀⠈⠑⠤⡀⠸⠀⠀⠀⠈⡆⠉⣠⡴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⣻⠿⠛⠛⢿⣿⡶⢦⣄⣀⠈⠑⣄⠀⠀⠀⢡⠀⠀⠈⠓⠊
⠀⠀⠀⠀⠀⠀⣄⠀⢸⡆⢀⡆⠀⠀⠀⢸⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⢠⢝⡛⠛⢢⣌⢆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠙⠂⠈⠁⠘⠃⠀⠀⠀⢸⠀⠀⠞⢹⠄⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⢿⣿⣆⢠⠁
⠀⡤⠒⠒⠒⠀⠀⠠⠤⢤⣤⣤⣤⣤⣄⣬⡀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠁⠱⠀⠀⠀⢸⣿⡟⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠻⠙⢿⣿⣿⣟⠲⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠁⢸⣿⠀⠰⠒⣹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠊
⠈⠢⠤⠤⠤⠄⢀⣀⣠⣿⣄⠀⠀⠀⠀⣾⡯⠊⠉⠋⠥⣀⣀⣀⣀⣀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢢⠀⠀⠀⢡⡤⠴⡖⠛⣿⣧⣀⣀⣠⣤⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⠀⠀⢳⡈⠸⢠⣼⠿⣛⣛⢍⠩⡍⠙⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⡀⠀⠀⠀⠙⠛⠉⣷⢋⠰⠀⠀⣀⠈⠂⣼⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⡄⠀⠀⠀⠀⠀⢸⡎⡇⣀⣈⢙⣡⡾⢋⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣶⢶⠒⡛⣷⣭⣵⠾⣛⡉⠊⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⠙⠉⠁⠀⠀⠀⠁⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇

⠀⠀⠀⠀⠀⠀⠉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⣱⠄
⠀⠀⠀⢀⠀⠀⠀⠀⡈⠀⠀⢀⣀⡀⠀⠀⠀⢄⡙⢢⠄⠀⠀⠀⢦⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣤⠈
⠀⠀⠀⠀⠀⠠⠀⠒⠈⠉⠓⠚⢺⠂⠀⠀⠀⠀⠈⠂⢐⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠅⠁⠀⠂⠁⠀⠀⠀⠀⠀⠁⠀⢀⣡⠊⢁⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⢯⣑⢂⡤⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠁⣄⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⡬⠀⠄
⠀⠀⠀⡐⢠⠛⠓⠒⠒⠒⡎⠁⠀⠀⠀⠀⠀⠀⠘⠤⢃⡩⠢⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡘⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⡮⡀⠠⠤⠐
⠀⠀⠀⠀⢾⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡒⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣥⠔⡂⠀⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠁⡰⠈⠐⡈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢅⣀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠊⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠦⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⠀⠀⣴⠎⠡⠄⣉⡳⣄⠀⠀⠚⠶⠀⠀⠀⠀⠀⢨⠀⠀⣈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢁⠀⢀⡁⢢⡀
⠀⠀⠀⠀⠀⢤⠜⠃⠙⣤⠄⠀⠀⠀⠀⢸⢡⡀⠀⠀⠙⠁⢸⢀⠄⡀⠂⡀⠀⠀⠀⠀⠈⢠⠛⢧⠘⠁⢛⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡜⠓⠘⣆⡅
⠀⠀⠀⠀⠠⠴⠆⠀⣀⠴⡄⠀⠀⠀⠀⠘⣄⡀⠈⠀⠘⠃⡜⣧⣿⠈⢀⠈⠀⠀⠀⠀⠀⠸⡀⠚⢣⢠⣼⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠀⢀⡸⣄
⠀⠀⠀⠀⠀⠀⠘⠁⠹⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠿⠓⠓⢂⣈⣁⡠⡜⠀⠂⠀⠀⠀⠠⠀⠙⠢⠻⠔⠉⠻⢤⡀⠀⠀⠀⠀⠀⠈⠠⠀⠸⠋⠹⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⡴⠛⣄⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠉⠈⢠⠀⠀⠀⠀⢠⡾⠣⣼⠟⠄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡄⠀⠀⢀⡄⠀⡣⠀⠀⠀⠀⠀⠀⠸⠓⢦⣄⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠲⡿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠄⣀⣀⠠⠄⠐⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠠⣈⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣀⠀⢀⣀⠀⢀⣀⣿⣿⡀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⣀⣤⣶⣴⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⠋⢀⣀⣤⣤⣴⣤⠦⢄⣤⠤⢀⣀⣾⡇⠀⣠⡆
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠻⡿⣛⣩⣅⠲⣿⣿⣿⣿⣿⡟⣁⣯⡴⢒⡿⣻⣿⣷⡿⢿
⠀⢠⣿⣿⣛⣻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⡾⣿⣿⣛⢻⣀⡙⡹⣿⣿⡥⣰⣯⠏⣻⠿⡿⢛⣋⡍⠐⢦
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣭⣿⣿⣿⣿⡿⣿⣿⣸⣷⣸⣿⢟⢷⣽⣇⠙⠜⢻⢿⣿⣿⣿⠷⠤⣿⢏⠛⠁⠀⠀⢀⠀⣀⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⣼⣿⣟⠀⠀⠈⣿⣷⣿⣯⣢⣄⣯⣽⣀⣀⠀⠀⠈⣁⢈⣧
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⣿⣿⣿⣿⣇⠀⢀⡸⠿⣏⣿⣿⣧⣿⣿⣿⠋⠀⠀⠀⢣⣠⠿
⠀⠈⣼⣿⣿⣿⣿⣿⣿⢘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡛⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⠿⣾⣿⣆⡀
⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣿⣿⢿⡿⠀⠀⠀⠈⢿⣿⣿⢻⣆⢬⣻⣟⣥⣶⠆⠠⠄
⠀⣿⣿⡌⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠍⢱⣟⣻⡁⡀⠀⠀⠀⠀⠸⠟⠛⣿⣽⡆⢻⣿⣷
⠀⣿⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡟⣳⣮⣠⠟⣡⣹⠀⡀⢠⣶⠂⠀⠀⠀⠙⢻⣋⢴⠋
⠀⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⢿⣿⡀⠙⠁⢷⠀⠳⠀⠀⠀⠀⠀⠘⠻⠌⠆⠀⢠⣄
⠀⣿⣿⢏⠙⠊⠲⡈⠍⣿⣿⣿⣿⣿⣿⣯⡹⣿⣿⣿⣿⣿⣣⠚⢛⣻⡾⠗⢿⣷⣶⣶⣦⣾⣷⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠈⠉
⠀⣿⠏⠎⠀⠲⣴⣴⣀⡈⠨⠛⣿⣿⣿⡿⡓⢻⢿⡛⣯⢿⣧⣼⣿⣿⣿⣶⣶⢇⣽⣿⣿⣿⠇⠄⠃
⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⣶⣴⣾⣪⣭⣥⣿⡜⣿⣷⣶⣿⣿⣿⣿⣿⣝⣿⣿⣿⣯⣿⡉⠡⣶⠀⠀⡄
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠰⠄
⠀⠀⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⡷⣿⣿⣿⣿⣿⣯⣙⣿⣿⣿⣿⣿⡿⠀⠀⠀⠐⠄⠰
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠌⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠿⢷⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⡄⣀⣀⡀⡠⠤⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠄⠈⢉⠇⡃⠈⠉⠁⠀⠒⢒⢒⡗⠢⠤⠤⣤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠤⠾⠓⠒⠚⠽⠒⠛⠒⠓⡟⠫⡀⠉⠀⠙⢷⠈⢹⡌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⠀⠉⠉⠤⣠⡽⠟⠂⠀⣀⣀⣀⠀⠠⢤⠀⢢⣦⡶⣿⣶⠀⠶⢾⡟⢻⠇⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢠⢐⠐⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⡕⢺⣿⠀⢠⣠⡀⠴⠟⠛⠹⢻⠂⠀⢸⠇⢺⡠⠊
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠠⠄⣋⠀⠀⠀⠠⣀⣤⡽⠇⣈⡃⠈⠁⠀⠀⠀⠠⠀⢰⢾⠀⢀⣿⡴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣉⣱⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣟⠕⠁
⠀⠀⠀⠀⠀⠀⠸⢶⡼⠂⠀⠀⠀⠀⠀⠀⡆⡉⡦⠀⢀⢀⠀⢀⣦⣾⡿⠿⠛⠟⠃
⠀⠀⠀⠀⠀⠀⡠⠁⡒⠂⢀⣀⠠⠤⠄⠀⠈⠻⠉⠀⢨⣴⣿⣿⣿⠿⡗⠒⠂
⠀⠀⠀⠀⠀⠀⠑⢲⣧⡤⣟⣛⣿⣿⣄⠀⠠⢙⣀⣠⣯⣿⠿⠋⠁⠁
⠀⠀⠀⠀⠀⠀⢀⡩⠗⠋⠉⠉⠉⠒⠚⠒⠻⠿⢟⣭⣛⠁
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠷⣄⠀⢀⡾⠉⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⣄⠘⣧⣾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠟⠀⠴⢀⡴⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡏⠻⣦⠸⡟⢰⠟⣷⠀⠀⣿⠀⠀⠀⢀⡾⠋⠀⢀⡴⠋⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⡀⠘⣧⡇⣿⠀⣿⠀⣰⠇⢀⣠⡾⠋⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⢻⡄⠸⣿⡏⢠⣏⣴⣧⣶⣿⠵⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠙⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣈⣳⣶⣿⣒⠻⠷⣾⣿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣭⠽⠟⠋⠉⠉⠙⣦⠘⣆⠻⡷⡄⠀⠀⠐⠃⠈⠀⢀⣀⣠⣤⠶⠶⠛⠁
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠐⠀⠂⣀⣀⡤⠴⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⠀⢷⢸⣦⣤⠴⠶⠞⠛⠋⠉⠁
⠀⠀⠀⣾⠀⠀⠀⠀⣠⠴⠒⠛⠻⣍⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣡⡿⠀⢸⠀⢻⡀
⠀⠀⠀⣿⠀⠀⠀⣸⠁⠀⠀⢀⣠⣬⣧⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢀⣠⡴⠛⡿⠁⢀⡏⠀⣸⠃
⠀⠀⠀⢸⡆⠀⠀⢸⡀⠀⠀⢸⠃⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⣰⠇⣠⢿⡇⢠⡏
⠀⠀⠀⠈⣷⠀⠀⠀⠳⣤⡤⠟⠛⠛⢹⡇⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⢠⣟⡾⠁⣸⠇⣿
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⣞⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠁⠀⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡇⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡇⠀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣇⠀⠀⢹⣇⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣦⠀⠀⠻⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠈⠳⣄⠀⠹⣆⢈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠞⠛⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣿⣯⣁⡄⠀⠀⣾⡛⠰⣴⢲⡆⢲⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢻⣿⣿⠉⠋⠀⠀⠀⠸⠤⠰⠛⠘⠇⠸⠀⠠⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⡔⣦⡀⠀⠀⠶⠿⠃⢹⡷⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⢴⢆⢀⡀
⠀⠀⠀⠀⠀⠀⣽⠟⢛⣇⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠘⠛⠀⠀⠠⠀⠀⠀⠵⠋⣍⠋⢳⡄⠈⠉
⠀⠀⠀⠀⠀⢺⣶⢚⠉⠀⠀⢀⡀⠀⠀⠐⠂⠀⢀⣤⣤⠀⣀⣀⠀⢀⣀⡀⠀⠀⡀⠘⢆⠀⠻⣆
⠀⠀⠀⠀⢤⣼⣿⣶⠀⢤⠀⠈⠀⠛⠀⠀⠀⠀⢸⠀⢰⡆⡿⢈⡇⣿⠉⠁⠀⠀⠠⠄⠀⠳⣄⠘⢧⡀⠀⠂
⠀⠀⠀⠀⠀⣻⣿⣷⣴⣹⣇⠀⠀⠀⠀⠀⠙⠃⢸⡟⠀⡇⣟⢻⡀⣿⣁⡀⠀⠀⣀⠀⠀⠀⠙⠤⠜⠁⠀⣤
⠀⠀⠀⠀⣶⡟⠛⠛⠋⠋⠉⡄⠄⠀⠀⢀⣴⢦⠼⠇⣀⠗⠛⠀⠷⠉⠉⢁⡀⠀⠙⠁
⠀⠀⠀⠀⠹⠛⠂⠠⠀⠐⠉⠁⠀⠀⠀⠘⠷⣄⡀⢰⣿⠀⢸⠛⢡⡤⠍⣿⣏⠀⢠⡀⠀⠀⠀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⠀⢈⣷⣾⣿⡇⣿⠆⢹⡤⠀⣿⠘⣷⡟⠁⠀⠀⢇⡗⠘⠛⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠋⠀⠛⠿⠀⢿⣤⡤⠉⣠⠏⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⢀⠀⠤⠤⡤⠤⠀⠀⠀⠘⠟⠀⠀⠀⠠⠀⢺⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⣉⠁⢹⠇⠀⠀⠀⠀⠀⢰⡷⣤⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠑⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣀⣤⡈⠙⠋⢩⣽⠅⡁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⢻⣿⢲⡂⣈⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢹⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⡇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⢹⡏⠀⠀⠀⣸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠒⠚⠓⠒⠚⠓⠒⠶⣾⠋⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⠤⠤⠤⠤⠤⠤⠤⠤⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣿⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡏⠉⠉⣿⠀⢀⡞⠋⠉⠉⠙⢷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠶⢦⡀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠻⢦⣤⣤⡴⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⣨⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣉⣙⣉⣀⣀⣀⣸⡇⠀⠀⣿⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣇⣀⣀⣿⠀⠀⠈⡏⠉⠉⣉⣉⣉⡉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⢼⡋⠀⠈⣻⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡇⠀⠈⠛⠓⠛⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⠀⢄⠀⡀⢰⣀⠀⢄⢀⢄⠀⢀⡠⡀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠰⠼⠘⠤⠘⠳⠰⠼⢸⢪⡘⠘⠇⠃⡗⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⠀⠘⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⡔⠀⠀⢠⠄⢈⠐⠀⢀⠀⠘⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣇⣾⡇⠀⣠⠏⠀⠀⠀⠀⠀⠐⠊⠁⠔⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣄⡀⢀⠀⠀⠀⠀⠀⡀⢠⣿⡿⠃⢸⣏⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊
⠀⠀⠀⠀⠀⠼⢳⣞⠀⣤⣀⣠⣾⣷⢋⣵⣦⡴⠚⠋⠀⠀⠀⠀⠀⠀⠉⢱⣄⠀⠀⠀⠀⠀⠘⠁⠀⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⣿⠟⠻⢣⣿⡟⠉⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠖
⠀⢠⣤⡄⠀⣼⣿⣿⡏⠙⣿⡙⣷⣿⣏⣤⣾⣿⠀⣀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠉⠓⢺⣯⠭⠛⠂⠿⠇⠸⠯⠿⠟⠛⠋⠿⢛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠒⠒⠉⠤⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢀⣙⣷⡑⢿⡀⣆⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠈⠙⠟⢮⡁⣺⣿⣿⠀⠀⠄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⣆⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠃⠈⢻⣿⣿⣾⢠⡿⠂⠀⠀⠀⠀⠀⠀⠘⡀⠐⡗⡀⠀⠀⠀⠀⠀⠀⡀⣜⢷⣿⣿⡁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣎⣧⠀⠠⣿⣿⡇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠐⣼⡷⡄⣠⣀⢀⣿⣿⠗⣾⠟⣁⡀
⠀⠀⠀⣤⡤⣠⣀⣠⣶⣿⢿⣿⣷⣄⣿⣿⡄⢻⣿⣷⣤⣐⡤⣄⠀⠀⢠⣤⡄⢻⣿⣿⣍⢫⡿⠟⢱⡷⣸⣯⣞⣁⣠⠀⢀⠤
⠀⠀⠀⠙⠺⠭⠍⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⢙⣛⡋⠁⠉⠀⠀⠀⠉⠓⠋⠉⠉⠛⠛⠛⠛⠛⠛⠋⠭⠭⠛⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⣷⣾⣿⣿⢦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠉⠉⠙⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣤⣄⢀⣀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣟⠉⠀⠀⠀⠀⠀⡀⣠⣾⣿⣷⡄⢹⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡧⠾⢿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠙⠛⠿⢏⠁⠀⠀⠀⣄⣼⣰⣿⣿⣿⡿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣥⣾⣿⣿⣤⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⠿⡀⠀⢹⡇⣺⢄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠰⣿⢰⡗⢀⡄⠀⠀⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⢀⣘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⢊⣾⣷⣿⣷⠦⣤⣦⣤⣟⣿⣷⣦⣤⡤⠚⢳⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⢠⡴⠆⠉⠉⠛⠻⠿⠿⣿⡿⠿⠿⠿⢿⣿⣿⠁⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠁⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⠻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣯⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢸⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣷⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣷⠗⠂

⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⢿⣦⣾⣧⣄⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢻⣽⣽⢯⣿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠺⣰⠚⢻⣿⢿⣿⣮⣦⣆⣟⢷⣄
⠀⠀⠀⠀⠐⢡⠃⣶⣿⣷⣿⣿⣿⣏⢿⣿⣌⠻⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⣇⣶⣇⡽⠛⠁⠙⣿⣿⣿⣮⡟⣿⡄⠂⠘⠉⣶⣤⣤⣴⣶⣯⣿⣭⣙⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠙⠻⠛⠀⠀⠀⠀⠘⣿⣿⣿⡟⣼⣿⢈⣦⠄⠜⣃⠡⢴⣾⣿⣝⣻⣯⣳⣹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠟⣷⣾⣷⡄⡀⡁⠰⣹⣯⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣸⣿⣿⣿⣗⣻⣿⣖⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⣿⣽⣽⣷⣿⣿⣿⣿⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⣽⣿⠿⠿⠿⠿⠿⠛⠈⠀⠹⣿⣿⣿⣟⣻⣿⢯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡿⣿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡝⣿⣿⣧⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠹⣿⣿⠃⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠀⣿⣿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢄⠀⢠⠙⠀⣦⠀⠈⡀⠀⠠⠁⠠⠄⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡼⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⢀⠰⠾⠂⠀⠀⠀⠀⠀⠀⠂⠀⠁⠉⠁⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣼⡿⠁⠈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⣾⡿⠁⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠟⠋⠉⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⢐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢠⡚⠟⠶⢧⡀⠀⠀⠰⠂⠉⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠃⠀⠀⠀⢸⣧⣀⣺⣤⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⢔⠀⠀⡔⡁⠀⠀⠀⠀⠀⠘⠃⠹⣿⡿⣮⣤⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶⢠⠠⢄⡖⠀⠀⢀⢢⣴⣾⢿⣷⣤⡷⡠⠃⠀⠀⠀⠀⠁⢀⢀⠋⠀⠀⠀⠉⠛⠳⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠤⠌⡕⠀⠀⠀⢉⣿⣿⡁⣾⡟⠂⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣛⣷⠙⠀⠀⠀⠀⠀⢈⢿⣿⡃⣹⣦⡤
⠀⠀⠀⠀⠀⠀⣀⡈⣧⡀⠀⠀⠀⢠⣽⠃⡀⠀⠀⠀⠀⠀⠀⠈⠈⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡀⠢⢔⣶⣤⣒⢠⣦⣴⣄⡄⠀⢀⠀⠆⠈⢄⠀⠠⣽⣦⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠁⣤⠀⣿⡟⢿⣦⣼⣿⣿⣿⡿⢿⣞⣿⣷⣶⣷⣴⣶⣿⣶⣾⣟⡟⢧⣴⣶⡶⠞⠟
⠀⠀⠀⠀⢀⣴⡌⢠⡿⠀⢹⡿⠛⠋⠉⠉⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡥⠀⢀
⠀⠀⠀⠐⣿⣿⣧⢸⠁⠰⣼⠁⢀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠁⢻⡏
⠀⠀⢀⢾⡟⣣⡇⡀⠀⠀⠁⠀⠘⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠸⢱⣿⡇⢰⡀⡾⠀⠀⠸⣷⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠅⣦⡆
⠀⠀⠠⡟⣹⣿⣧⡌⠁⠃⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠉⠁
⠀⠀⠀⠁⠘⠏⢿⡁⠀⠀⠀⣀⣀⡀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⣾⠀⡠⠐⢻⣿⣿⣟⠀⠀⠞⢿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠀⢨⢠⠴⠋⠻⣿⣿⣷⣶⣶⣤⣀
⠀⠀⠀⢸⣿⣿⣿⣴⣧⣥⣐⣿⣿⣿⣆⣦⣤⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣴⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠈⠋⠛⠛⠿⠿⠿⠛⠛⠛⠋⠋⠙⠋⠋⠉⠉⠋⠉⠋⠉⠙⠛⠛⠛⠛⠙⠛⠛⠛⠋⠉⠁

⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠰⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠓
⡇⢴⢿⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⠿⠆
⢳⡂⣸⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⢠⡐⣰
⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢠⠦⡀
⠀⠰⡠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡰
⣤⣤⣵⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣴⣶⣶⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣿⣧⣴⡷⠀⠀⠀⠀⠀⢀⣴⣾⣶⣄⠀⠀⠀⠀⠀⠰⣷⣤⣿⣧⣶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠋⠀⠀⠀⠀⠀⣠⣿⠟⠁⠙⢿⣧⡀⠀⠀⠀⠀⠈⠻⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⢸⣿⡀⣿⣧⡀⠀⠀⠀⣼⣿⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠈⢿⣧⠀⠀⠀⢻⣷⣌⠛⠁⠀⢀⣴⣿⠃⠀⠀⢰⣿⠏⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⢸⣿⡇⠀⠀⠀⠙⠻⠿⣿⠿⠿⠋⠁⠀⠀⠀⣿⣏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⡿⠋

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


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⢀⡄
⠀⠀⣿⡞⠁⣶⠁⣷⣴⡃⢘⡆⣿⠀⠀⠀⠺⢶⡀⠀⡇⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⢳⡄⠀⠀⠘⠳⡆⢴⠐⡦⠀⠀⢬⣧⢸⡍⡧⠴⣿⠰⢾⡇
⠀⠀⠛⠃⠀⠙⠒⠋⠈⠑⠊⠁⠙⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠁⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⡿⢹⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠺⠛⠉⢹⣆⣽⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⢃⣦⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⣞⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⢰⣿⣿⣧⣿⡃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣤⣀⠀⠀⠀⢹⣿⣿⠷⣀⣤⣀⡴⠧⣄⡀⠀⠀⠀⠠⠆⠀⣿
⠀⠀⣿⢸⣿⣿⣿⣿⣏⣀⡀⠀⠀⠀⣿⣿⣙⣿⡛⢿⣿⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⣦⠚⠟⣉⠻⠅⢸⣿⣿⣿⣢⣤⣴⣶⣦⣄⡀⠀⠀⣿
⠀⠀⣿⢸⣿⣿⣿⣿⣯⣈⠙⠛⠶⠶⢾⣿⣿⠿⣿⣊⠛⠶⣀⣀⣠⣶⣿⣿⣿⣿⠟⠋⢁⡒⣠⠀⣨⣤⠴⣷⣨⣿⣿⣿⣿⣿⣿⣿⣿⣯⡥⢞⠀⣿
⠀⠀⣿⢸⣿⣻⣿⣿⣿⣿⣧⣀⣀⢠⣾⣿⣏⣴⡭⠭⠟⢒⣵⣾⣿⣿⣿⡟⠋⠀⠀⠐⠊⣉⠙⢷⠖⠒⠂⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⣹⡅⣿
⠀⠀⣿⠘⣿⡿⢿⣾⣿⣿⣿⡿⠗⠚⠉⠉⠁⠀⠀⣀⣶⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠇⣿
⠀⠀⣻⠀⠀⠀⣶⡿⣿⣭⠄⠀⠀⠀⠀⠀⡀⣤⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣹
⠀⠀⣿⠀⠈⠀⢉⡟⠩⠄⠖⡀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⢀⠀⠀⠀⠀⠀⢠⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⣿⠀⠀⠀⠈⢛⡓⠀⠀⠀⣠⣾⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣿⣿⣿⡿⣿⣿⣿⡗⣿
⠀⠀⣿⠀⠀⠘⠛⠻⠉⢀⣤⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣭⣾⣿⣧⣿
⠀⠀⣿⠀⠀⠀⠀⠙⠓⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⢉⠋⠉⠉⠃⣛⣛⣿⣃⣿
⠀⠀⣿⠀⠀⠀⠀⠲⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⡶⢛⣧⡤⠀⣩⣦⡅⢿
⠀⠀⣿⡇⠀⠀⠀⠰⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠒⠛⢷⡄⠀⠈⡹⣇⠘
⠀⠀⢹⠁⠀⠀⠸⠷⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠥⡆⢀
⠀⠀⢸⡇⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⣀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠠⣺⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢹⡿⣆⢀⣽⣤⡄⠀⢀⣤⡄
⠀⠀⢸⡇⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠶⠶⠞⠉⠳⡏⣩⡄⢘⣋⡠⣾⠟
⠀⠀⢸⡇⠀⣀⣿⣿⣿⡆⠀⢀⠀⣼⡛⠂⠀⠀⠀⣸⣿⣤⡟⠉⣤⠦⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢸⡧⣿⣿⣿⣿⣿⡇⠀⠀⠠⠏⠉⠀⠀⠀⠀⢱⡶⢻⠄⣾⣿⣧⡤⠞⠋⠀⠀⠀⠀⠀⠤⠤
⠀⠀⢸⡇⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡅⠆⠉⠈⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢐⡒⠚⠒⠒⠒
⠀⠀⢸⡏⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⡏⣀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢸⡇⠰⠴⠶⠶⣛⣛⣷⡀⠀⠀⠀⠀⡌⢀⣶⠄⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⢸⡇⣴⣶⣾⣿⣿⠟⠋⢉⣤⠀⠀⠀⢻⣿⣿⣠⣆⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠻⠟⠛⠁⢀⣠⣴⣿⣷⣶⣀⣠⣀⣿⣿⣿⡟⣻⠏⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠐⠛⠋⠉⠀⣀⣨⣿⡿⠛⣿⡿⣿⠿⣻⣥⣶⣿⠿⢿⣤⣤⣠⣤⣄⣀⣀⣀⠀⣀⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⣀⣀⣴⣶⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠛⠿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣟⣁⠐⠀⣐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢸⣇⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⡉⠁⠀⠀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣛⠛⠛⠛⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⡁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣈⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡄⠠⠠⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣀⠆⣀⠀⠀⠀⠀⠀⢀⡀⠄⠦⢠⣀
⠃⠐⠀⠀⢆⠀⡀⠀⠀⠀⠀⠬⢅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⡶⡀⠠⠁⠀⠀⠀⠀⠈⠩⠔⡎⡁⢀
⢦⢂⠀⠀⢤⢠⠅⠀⠀⠀⠀⠀⡨⠅⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⡠⠀⠀⠀⠀⠀⠠⢀⠀⡇⡐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⢐⠒⣖⠒⣆⠒⠀⠀⠀⠀⠀⠀⠴⠄⠀⠰⢄⠀⠀⠀⠀⠀⠠⢨⡄⡠⣠⠇⠀⠀⠀⠀⠀⢀⡀⠀⠶⠀⣀
⠸⠿⡾⠿⡞⠿⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠰⡀⠀⢸⠀⠀⠀⠀⠀⠀⠪⠀⠀⣶⠀⢀
⠸⠖⠹⠶⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠉⠉⠖⠀⠀⠀⠀⠀⠀⠢⠀⣀⠀⠔


⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠈⡵⢆⠲⡜⠀⠀⠀⠀⠀⠀⠀⠀⠶⡖⠂⠀⡇⠀⠀⠀⠀⠀⠋⢘⠉⠁⠁⠀⠀⠀⠀⠀⠈⠽⠂⠀⡉⢭
⠀⠄⠩⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⢁⣂⠀⠀⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠀⠁
⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠈


⠀⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢐⢐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠓⠎⠓⠀⠀⠀⠀⠀⠀⢠⢴⣞⣿⣷⡦
⠀⡁⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠢⠦⣼⡢⠄⠀⠀⠀⠀⢸⣿⣿⣿⡟⣶
⠀⠄⠰⠷⠼⠜⠀⠀⠀⠀⠀⢠⠸⠹⠤⠤⢂⠀⠀⠀⠀⠀⠠⢀⠀⠤⢈⠀⠄⠀⠀⠀⠀⠈⠛⠿⣻⠿⠛
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣷⣄
⠀⠀⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⡏⠠⣶⣄⣻⣿⡏⢀⠹⣿⣿⠀⣤⣤⣼⡇⢠⣤⣤⣾⣦⡄⢠⣤⣶⡄⠻⠟⢠⣾⣿⣿⣿⣿⣦⡀⠛⣷⣄
⠀⢀⣴⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⡿⣦⣤⡈⢻⡿⠀⠿⠀⢹⣿⠀⣤⣤⣿⡇⢠⣤⣤⣿⣿⡗⢸⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⢷⣄
⣴⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣉⣠⣾⣅⣾⣿⣿⣄⣿⣠⣿⣿⣿⣧⣈⣉⣉⣹⣿⣧⣸⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢹⣷
⠻⣿⣆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⣾⠟
⠀⠈⠻⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⠿⣿⠿⠿⠿⢿⣿⣿⡿⠿⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠈⠻⣿⣄⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⣶⢶⣿⠀⣿⠀⣶⣶⠆⢹⣇⠰⠶⣦⣽⣷⣶⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣷⣤⡈⠻⣿⣿⣿⣿⣿⣿⡇⢰⣶⣾⣿⠀⣿⠀⣶⣄⠘⣿⡟⢷⣶⡄⢹⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⡈⠻⣿⣿⣿⣿⣧⣼⣿⣿⣿⣤⣿⣦⣿⣿⣦⣼⣷⣦⣤⣴⣾⣿⣿⣴⣿⣿⣿⣿⣿⡿⠋⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠻⣿⣿⣿⣿⣿⡿⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠻⣿⡿⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠁⠈⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⠐⠒⠒⠂⠃⠀⢸⠀⣐⡤⠠⡠⠀⠀⡶⠶⠶⠶⠄⠀⠀⠀⠀⢀⡦⠀⢀⠌⠠⠀⢠⠀⢀⣀⣀⣀⡀⠀⠴⠘⠀⠀⠀⠀⢀
⠀⣀⣤⣶⡾⡿⡇⠀⢸⠀⠀⠛⠢⠴⠄⠌⠶⠶⠶⠴⠶⠀⠀⠀⡴⠁⠀⠀⠈⠂⠀⠐⠘⠀⢀⠄⠀⠀⠀⠀⡀⠘⠀⠀⣀⣥
⠀⣿⣘⣻⡷⠿⠇⠀⠈⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠈⠀⢀⣴⣿⠀⠂⠳⠏⢿⣶⡾⠀⠁⢰⡖⡆⠠⠥
⠀⠛⠉⠁⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡆⢰⠀⡉⠈⠁⠆⠀⠀⠀⠀⠀⢠⡾⣏⡼⠛⠀⠀⠂⠰⣶⣀⠀⠈⠀⢸⠿⠏⠁⠤
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣄⠀⡇⠌⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡿⠋⣀⠀⠀⠠⡀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠐⠀⢸
⠀⢾⣒⣶⣖⣲⡇⠀⠀⣼⣻⣿⣿⣽⣷⠅⠀⠐⢀⡤⠦⠻⠀⠀⠀⠀⠀⠀⠀⡌⠉⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠐⠀⠤⠤⠐⠃⠀⢀⠙⠺⠿⠿⠛⠃⡆⠀⠠⠈⠀⠀⠀⠀⠀⣄⠐⠢⡀⠀⠀⡊⠐⠀⠐⠀⠀⠠⡚⢆⠀⠀⡐⠀⠀⠀⠀⢸
⠀⠀⣀⡀⡀⠔⠀⠏⠀⠁⠀⠀⠀⠐⠈⠀⠈⠘⠀⠀⠀⠈⠀⠀⠀⠈⠁⠚⠂⢀⠠⠐⠈⠘⠀⠦⠀⢀⡃⡄⠀⠃⠂⣀⠀⠀⠘
⠀⢀⠀⠀⠤⠔⠘⠂⢤⡄⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠇⠈⠐⠂⢀⢠⠀⠀⠀⠓⠉⠀⠯⢅⠀⣅⡑
⠀⠸⠉⠀⠂⠄⠲⠖⠃⡀⠀⠀⠀⣿⠉⢉⡅⠀⠀⠀⠀⠀⢸⠉⠻⣿⠀⠀⠀⠀⠁⠀⠘⠛⡉⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠂⠀⠀⠘⡿⠦⠀⠁⠈⠉⠀⠈⠁⢄⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢏⠗⠒⠀⠗⠂⠠⠀⠤⠄⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠘⠷⠦⠶⠷⠶⠶⠶⠶⠷⠶⠶⠶⠖⠀⠀⠀⡂⠀⠀⠂⠈⡳⠏⠀⠐⢄⠀⠀⠠⠀⠄⢀
⠀⠀⠀⠀⠀⢰⣟⡼⠇⠀⠀⠀⡄⠤⠤⠀⠠⠤⣀⣤⣤⠤⠤⠤⡄⡀⡀⠀⠀⠈⠁⠘⠂⠁⠀⠀⠀⠀⠀⠁⣀⣒⠀⣀⠀⢀⣀
⠀⠠⠀⠀⠀⢸⡟⡁⠄⠃⠀⠀⠇⢀⡤⡶⡿⢟⠿⣫⡏⡍⡆⠀⠇⠀⠀⠈⠀⠄⣤⡀⡀⢠⠀⣰⢰⢰⠀⣰⡟⠁
⠀⢀⡀⢀⠉⢉⡁⢀⣀⣀⡀⠀⡀⢸⡸⣣⣦⣬⣴⣶⡿⠞⠃⠀⡇⠆⠤⠀⠀⠂⠈⠀⠀⠉⠀⠈⠈⠁⠰⠿⠁⠀⠀⠀⠀⠠⠤
⠀⡆⡉⠈⠁⠀⠀⡤⠀⢠⡟⠀⣂⣀⣉⣉⣉⣉⡉⡀⣿⣭⣿⡀⡇⢠⠐⢠⣢⣠⠄⢂⢄⠀⠀⠀⠀⠀⠀⠀⢠⠄⡄⠐⡤⢄
⠀⣤⠤⠤⠤⠤⠤⠭⠭⠭⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠰⡈⢿⣳⠜⣲⣕⡟⠽⠇⠀⠀⡀⡈⡀⠀⠈⠒⠢⠠⠃⠈
⠀⠀⠀⠀⠁⠀⠈⡀⠀⠀⠅⠀⢸⣸⡰⢆⣍⠆⢸⣑⡁⠀⡋⢱⠀⠑⢄⠀⠀⠀⠀⡠⠊⡤⠂⠀⠀⠀⠀⠙⢀⠶⠄⠀⠰⠦⡀
⠀⡀⠀⠀⠀⣖⡖⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠤⢤⡄⠀⡀⠁⠤⠤⠈⡀⠀⠁⠀⠈⠁⠀⠄⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠇⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⢸⠀⠀⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⡇⢀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⢀⣰⡾⠛⢳⡆⠂
⠀⠀⠀⠠⠤⡀⠄⡠⠂⠀⠀⠀⠀⠀⠐⣿⣧⣭⣿⣴⣾⣿⠀⠀⠀⠀⡆⠀⢂⣁⠀⠃⠀⠀⠄⠀⡀⠀⠀⠀⢸⠈⠙⠛⠋⠁⡁
⠀⢀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠐⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⢀⠂⠀⡇⠀⡉⡉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⢠⠅
⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠃⠀⠑⠀⠀⠀⠐⠁⠀⠉⠀⠒⠒⠀⠀⠀⠀⠀⠐⠀⠂

⠀⠀⠀⢠⡀⢀⣀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠤⠀⣀⠄⠂⡆⠀⠀⠀⠀⠔⠛⠉⠛⠢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢐⠌⠉⢈⠁⠔⡀⠀⠀⠀⠀⠀⠙⠒⢠⠔⠁⡠⡊⠀⠀⠀⠀⠐⠨⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠩⢉⠩⠋⠍⢂
⠀⠀⠈⠀⠀⠀⠀⠈⠑⠂⣤⠀⠀⠰⢒⡠⠀⡠⠊⠡⠡⠀⠀⠀⠀⠀⠑⣴⣀⣤⠆⠀⠀⠀⠀⣿⠄⠠⠁⠠⠀⠀⠀⡁
⠀⠀⠓⢅⠎⠦⠤⠴⡰⡱⠉⠀⠀⠀⠨⠴⡈⡀⠀⢧⢘⠁⠀⠀⠀⠀⠀⢸⠛⡃⠀⠀⠀⠀⠀⣨⣤⡤⡤⠀⢤⢤⢤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠈⠑⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⠀⠀⠀⢀⠠⠤⠤⠤⠤⠄⡀⠀⠀⠀⠀⠀⢀⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡋
⠀⠀⠴⠒⠞⢋⠉⢡⠷⠠⡀⠀⠀⠃⠄⠀⢀⠰⠀⡆⠀⠀⠀⠀⢀⠠⠟⠿⠿⠤⣒⠀⡰⠀⠀⢀⢀⡌⠖⡀⢁⠈⡐⢀
⠀⠀⢇⠈⢐⠛⠋⠠⡀⠀⡘⠀⠀⡧⠥⠤⠥⠔⠀⡧⠤⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠛⠤⡘⠊⠓⠉⠚⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⠵⠀⠈⠀⠑⠓⠖⠀⠀⠀⠈⠁⠈⠀⠉⠈⠁⠈⠀⠀⠀⠀⠐⠁⠉⠉⠉⠉

⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡅⠀⢸⠀⠀⠛⢯⠇⠀⠀⠀⠀⠀⣿⠰⠉⠂⣆⠀⠀⠀⠀⢠⣄⢀⣤⡀⢸⢀⠀⠀⠀⠀⣏⠀⠀⠁⠀⠾⠇⠀⡁
⠀⠀⠁⠒⣚⡚⢂⣀⠨⠤⠀⠀⠀⢀⡉⣀⡆⠀⠀⡇⢑⡀⠀⠀⡀⡩⠨⡉⠉⢉⠀⢀⢿⠀⠀⠀⠀⠀⢂⠤⠠⣈⠀⣰
⠀⠀⠀⠃⣒⠐⠐⠂⠃⣛⠀⠀⠀⠈⠁⠙⢦⢀⠤⠋⠉⠀⠀⠀⠈⠁⠁⠉⠛⠉⠈⠐⠀⠀⠀⠀⠀⠀⢈⣓⣄⢠⣘⡡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆
⠀⠀⡄⠀⠀⠀⠀⠀⠒⠒⠒⠂⢄⠀⠀⠀⠀⢠⠤⠤⠤⠾⠭⠽⠦⣖⠲⢤⡀⠀⠀⠀⠀⠛⡉⠉⢉⠛⠀⢶⢲
⠀⠀⣏⠉⣭⠉⣍⠉⣭⡋⢩⡋⡍⡀⠀⠀⠀⠮⠥⠕⠥⠴⠪⠨⠎⠉⠱⡉⣣⠀⠀⠀⠀⢀⣂⣒⣐⣐⣀⣘⣘⣰⠄
⠀⠀⣃⡀⣀⡄⡀⠤⠤⠤⠤⢄⣠⢈⡀⠀⠀⢅⠉⡩⣅⠉⠉⠉⢀⢀⢀⣠⢄⡀⠀⠀⢸⡏⢠⣾⡄⢀⡀⢀⡀⢨⣳⡀
⠀⠀⠀⠁⠑⠑⠁⠀⠀⠀⠀⠈⠛⠚⠁⠀⠀⠈⠈⠓⠛⠁⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠈⠈⠛⠋⠙⠛⠙⠋⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠻⣿⡿⣛⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣴⣶⣶⣦⣤⣤⣦⣶⣶⣦⣶⣦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣩⣥⣭⣙⢿⣿⣿⣿⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⢅⣾⣿⣿⣿⣿⡆⢩⣭⣭⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⡜⣿⣿⣿⣿⣿⡟⠸⠿⠿⡿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣷⣜⠿⠿⠿⢟⣵⣶⣶⣶⡇⠀⠸⠿⢛⣛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣷⣦⣝⡻⠿⠿⠿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣶⡤⠤⣄⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠃⠂⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠑⠒⢤⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⣈⡓⢤⣔⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠀⠁⠀⣉⠝⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠉⢁⡀⢫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣠⣞⠇⠲⡄⠀⠀⣀⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⢀⠤⡤⠀⠀⢈⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠂⠀⡟⡄
⠀⠀⠀⠀⣀⠤⣖⠀⠀⠀⠀⠘⡄⠚⣄⠈⠲⣶⣶⡶⠋⡔⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣇⡎⠀⡰⠁⠘⢆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠛⠒⠉⠉⠉⢹⠒⠚⢚⣁⣀⣀⣀⣀⣀⣀⣀⡌⠀⡜⡜⠀⣰⠁⠀⠀⠈⠣⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠜⠉⠙⢄⡀⠀⠀⠀⠀⠀⠀⡜⠀⡰⡜⠀⡰⢇⠀⠀⠀⢀⡠⢿⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⢆⠀⠀⠀⠘⠣⡀⠀⠀⠀⡨⡶⡒⠒⠒⠒⠒⠓⠒⠛⠓⠚⢧⠏⠈⡉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⡦⠤⣀⣀⣨⠶⠤⠊⠀⡇⠙⢆⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⢰⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠷⢄⠀⠀⠙⣾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠘⠛⠛⠓⢲⠂⠐⠢⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢄⣀⠀⠑⠢⠤⠋⠑⠤⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠒⠓⠢⠤⠤⠤⠤⠤⠞⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠂⠊⠀⢀⡀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠴⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⡁⠀⣀⣤⡀⠀⠐⠀⠀⠀⠀⣠⣄⡀⠐⠂⠀⠀⠀⠛⠀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⢧⣀⡿⠀⠀⠀⠀⠀⠘⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⠛⠛⠋⠛⠋⠛⠋⠛⠉⠙⠛⠓
⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⣶⡦⠘⠀⠀⠀⢠⣾⣦⣶⡦⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠆⠀⠰⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠰⣛⡖⠐⠒⠂⠐⠐⠂⢞⣳⠰⣛⡆

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⡀⠀⣤⡀⠃
⠀⠀⠀⠁⢰⡞⢳⡆⠀⠓⠓⠛⠛⠋⣴⠋⢳⡄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢉⣥⡀
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠴⠋⠉⠙⠦⠋

⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠓⠒⠛⠚⠚⠛⠛⠓⠀⠀⠀⠀⠐⠛⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠈⠀⠂
⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⢀⠅

⠀⠀⢰⣿⡍⠑⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠴⠶⠶⠶⠒⠒⠒⣲⣦
⠀⠀⢸⢈⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⡄
⠀⠀⢸⠈⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢨⠇
⠀⠀⠀⢧⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠞
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣀⣼⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣄⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⢧⣤⣠⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠦⠦⠴⠦⠶⠶⠦⠶⠴⠶⠦⠶⠶⠶⠶⠦⠴⠤⢤⣶⠇

⠀⠀⠀⠀⣤⠶⠖⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⣠⠿⠀⠀⠀⠉⠋⠃⠀⠀⠀⠀⠀⢘⡏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣄
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡴⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡆
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣰⠏⢹⠀⠀⠠⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⢸⠆⠀⠀⣅⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠈
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡄⠀⢸⠀⠀⠀⣏⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣣⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠘⡇⠀⢈⠆⠀⠀⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⠁⣩⠀⠀⣷⠀⠘⠇⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣰⠏⢻⠀⣿⡀⠀⡟⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⡼⢿⠀⣿⠀⣼⠃⠀⡿⠀⢀⡅⠀⠀⢿⠀⠀⠰⠀⠀⡀⠀⠀⠀⠀⢀⡼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⢿⡅⢸⡀⣸⠀⢹⠄⠀⡇⠀⠈⡧⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠒⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⡇⢸⡁⢻⠀⢰⡄⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠸⡇⢸⠇⣼⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠜⣯⣧⠀⠀⣄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠰⡇⢸⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⡏⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠈⠀⠀⠀⢀⣠⠤⣤⠄⣀⡀⠀⠀⠀⠀⢰⡃⡆⠀⠀⢸⡇⠀⠀⣿
⠀⣿⠀⠀⡤⠒⠛⠛⠓⠢⢄⣀⠀⠀⡃⠀⠀⠀⣀⣀⣀⣠⡄⣀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⠀⠀⣠⣼⡏⠀⠀⠀⠲⡌⠑⢄⠀⠀⠀⠘⡇⡇⠀⠀⢸⡇⠀⠀⣿
⠀⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠁⣴⠊⠉⠁⠉⠈⠋⠑⠦⢿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⣴⢻⡉⠁⠀⠀⠀⠀⠀⠹⣆⢡⡀⠀⠀⡇⡇⠀⠀⢸⡇⠀⠀⣿
⠀⣿⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿⠛⠃⠀⠀⠀⠀⠀⠈⠁⣀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠇⠀⠀⠀⠀⠀⠀⠔⠛⣼⠁⠀⢰⣿⣇⠀⠀⢸⡇⠀⠀⣿
⠀⣿⢃⠀⠀⠀⠀⠀⠀⠀⠠⠀⡋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⠀⠀⠀⠰⣁⡠⢶⠀⠄⠸⠀⠀⢸⠁⠀⠀⣿
⠀⣿⠘⢦⡀⢀⠀⠐⠂⠀⠀⠂⢡⠀⢳⡀⠀⠀⠲⠀⠰⢄⡤⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠆⠐⠋⠁⠀⠀⠀⡖⠉⢀⢠⠖⢶⢸⠀⠀⢸⠃⠀⠀⣿
⠀⢹⠀⠀⣙⠾⡃⠀⠀⠀⠀⠀⡏⠀⢸⠗⠀⠀⠠⠂⠰⠂⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡄⡠⠈⠁⠀⠀⠀⣤⠀⠀⠚⠁⡰⠋⠀⠀⣸⠘⠀⠀⢸⠀⠀⠀⣇
⠀⢸⡤⠊⠀⠀⠀⣐⣶⢆⡘⠉⠀⠀⠘⠓⠒⠶⣦⢀⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠁⠀⠀⠀⠀⢀⡿⠀⢀⡤⠚⠁⠀⠀⠀⢸⢘⠀⠀⣿⠀⠀⠀⣿
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣄⣀⣴⣿⢾⡀⠁⠰⣷⡤⠤⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠁⠀⠀⣺⠗⠈⠀⠁⠀⠀⠀⢠⡛⢉⡼⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢹⠀⠀⠀⢸⠟⢺⡏⢠⡆⢹⠂⠀⢻⡇⠀⠀⠀⢀⠀⣴⠏⠀⠀⢀⡴⠂⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡼⢆⡀⢧⡘⠂⠀⢸⠀⠀⠀⠀⠀⡼⠁⠀⠀⡴⠋⠀⠀⠀⠀⣶⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⣿⣅⠀⠀⢈⣋⠉⣛⡏⠀⠀⠀⠀⠰⠀⠀⢀⡾⠆⠀⠀⣀⠀⢸⡇⠴⠂⠉⠀⠀⣀⢱⡄⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠙⠏⠡⠟⠆⢶⣀⡶⢩⠀⠀⠐⠋⠉⠀⠀⠈⠉⠁⠈⣿⣿⣄⠀⠀⠀⡖⠀⢠⡜⡷⠦⠀⠀⣠⠞⠡⠀⠀⠀⠀⠀⠸⠃⠀⢡⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠀⣤⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣀⠀⡰⠃⠀⠜⠀⡁⠀⠀⠀⠀⠀⠀⠃⠃⢀⠄⠻⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⡇
⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⣽⠃⠀⣸⠀⠀⣠⣬⠀⢻⡤⠂⠀⠀⠀⣿⠶⠊⠁⠀⠀⠆⠀⠀⠀⢇⠀⠀⠀⠀⠀⢠⣠⠀⠀⢀
⠀⢀⡀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠉⠀⡄⠀⠀⢉⣾⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⣷⡀⡀⢠⡾⢤⠤⠲⣦⡤⣴⢺⠀⠀⢸
⠀⠀⠀⠉⠉⠀⠀⢠⠀⠀⠀⡇⠀⠀⠹⣦⣙⠦⣄⣰⠖⠒⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠠⠀⢀⠘⡷⠤⠤⠀⠀⠺⠭⠉⣿⠀⠀⠀⠀⣀⠴⠆⢀⡏⢨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⡇⠀⠀⠀⠀⠉⢹⠎⠛⢾⡎⠛⢶⠄⣀⡀⠀⠀⠀⢠⡿⠏⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠌⠛⠉⠀⠠⠤⡷⢰⡄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢠⠊⠀⢀⣾⠃⠀⠈⠀⠀⠈⠉⠐⠰⡿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢸⣄⠀⢾
⠀⢰⡿⠀⠀⠀⠀⠏⠀⠀⠀⠃⠀⠀⠀⢠⠏⠀⢀⡜⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⡆⢠⠀⠀⠀⢰⠀⠀⠀⡞⠀⠀⣸⠁⢠⠀⠀⡇⠀⠀⠀⠀⠰⡁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠸⠇⠀⠀⠀⡷⢸⡄⠀⠀⠈⠲⢤⣼⠀⠀⢰⠀⠀⢸⡀⠀⢰⣄⠀⠀⠀⠀⡙⠳⠆⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⡤⢄⠀⠈⠙⢢⡘⠒⠒⠒⠦⣤⡇⠀⠀⠈⠀⠀⢸⡇⣀⣀⠉⠉⢀⣲⡖⢁⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡾
⠀⠀⢣⡀⠙⠚⠒⠁⠀⠀⠀⠀⠀⠻⣀⡀⠰⠚⣄⠀⠀⠉⠈⠁⠀⠉⠉⠉⢄⠀⠈⠁⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠉⠳⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣤⣤⣵⣤⣤⣤⣠⣤⣀⣀⣀⣀⣱⣄⠀⠀⠙⣤⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⢠⠶⠶⠂⠰⠰⠖⠀⠰⠟⠲⠶⠄⠸⠀⠶⠀⠆⠀⠷⠲⠀⠲⠠⠒⠶⢲⠷⠶⠂⠸⠐⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠛⠛⠛⠛⠛⠻⠿⠿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⡴⢤⢰⠖⣶⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⢀⣀⣀⣤⣤⡤⠀⠶⠀⠀⢶⡟⠻⠈⣿⠀⢹⡇⢹⡇⢸⣦⣈⠁⠀⣿
⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⣴⡀⠀⢲⡏⠙⠏⢿⡁⠙⠘⠀⣷⠈⠈⢿⣄⠃⠀⠀⢸⡧⣦⠀⣿⡀⢸⡷⢿⡀⢀⠙⠻⣧⠀⢹⡇
⠀⠀⠀⠀⠀⢸⣧⣈⠁⠀⠟⣧⠀⠸⣧⢼⠀⢸⡗⠧⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⠸⣇⠀⠀⣸⣇⢀⣿⡈⠿⠼⠷⠴⠋⠀⠚⠛
⠀⠀⠀⠀⠀⢀⠙⠻⣧⢰⠒⢻⣇⠀⣿⠀⠀⢸⣇⣀⡶⠀⠽⠆⠀⠀⠟⠂⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠘⠧⠤⠏⠺⠂⠐⠛⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣦⣤⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣨⣭⡶⡦⢤⣤⡴⠶⠛⠉⢉⣁⣭⣭⡽⠶⠦⣤⣤⠴⠶⠛⠉⢉⣠⣤⣬⡭⠷⠦⠤⣤⣤⠴⠞⠛⠉⣁⣠⣤⣭⡭⠶⠦⠤⠤⠴⠞⠛⠉
⠀⠀⠀⠀⢀⣠⣤⣴⠦⠴⠶⠶⠶⠿⠛⠋⠁⢀⣠⣤⡴⠶⠶⠶⠶⠶⠿⠛⠋⠁⢀⣠⣤⡶⠶⠶⠶⠤⠶⠾⠟⠛⠉⠀⣀⣤⣤⡶⠶⠶⠶⠶⠾⠛
⠀⠀⠀⠀⢉⣉⣠⡤⠴⠖⠒⠒⠶⢤⣔⣒⠛⠛⢉⣁⣤⠶⠒⠒⠒⠶⢤⣐⣚⠛⠛⢉⣁⣤⠴⠒⠛⠓⠲⠦⣤⣒⠚⠛⠋⣉⣠⡤⠖⠒⠛⠒⠶⠄
⠀⠀⠀⠀⢉⣉⣀⠒⠛⠛⠻⢶⣦⣤⣈⣉⣙⣋⣉⡁⠘⠛⠛⠿⢶⣦⣤⣈⣉⣉⣉⣉⣁⡀⠉⠛⠻⠷⣶⣤⣀⣉⣙⣛⣉⣉⣀⠈⠙⠛⠿⢶⣦⣄
⠀⠀⠀⠀⣈⡉⠛⠻⠷⣦⣤⣤⣄⣉⣩⣭⣍⣉⡉⠛⠻⠷⣶⣤⣤⣄⣈⣭⣭⣭⣍⡉⠙⠻⠷⣶⣤⣤⣤⣈⣉⣭⣭⣭⣉⡉⠛⠻⠷⢶⣤⣤⣀⡀
⠀⠀⠀⠀⣈⠙⠓⠶⠤⠤⠴⠞⠫⢥⣤⣤⣄⣉⠙⠓⠶⠦⠤⠶⠒⠩⠥⢶⣤⣤⣈⡉⠛⠶⠦⠤⠤⠶⠚⠭⠤⣦⣤⣄⣉⠙⠛⠶⠶⠶⠖⠚⠩⠤
⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⢶⣦⣤⣀⡀⠉⠛⠛⠓⠲⠒⠒⠲⠶⣶⣤⣄⡀⠉⠙⠛⠳⠶⠶⠒⠒⠶⢶⣦⣤⣀⡀⠉⠙⠛⠛⠒⠒⠒⠲⠶⣶⣤
⠀⠀⠀⠀⠴⠒⠚⠛⠛⠲⢦⣤⣀⣉⣙⣛⣻⠽⠒⠛⣛⣛⠓⠶⣤⣀⣈⠉⢛⣛⡯⠕⠒⠛⣛⡛⠲⢦⣤⣀⣉⣉⣉⡩⠽⠒⢛⣛⣛⠳⠶⣤⣀⡁
⠀⠀⠀⠀⣴⠾⠛⠋⠉⣉⣀⣀⣍⣉⣉⣡⣤⣴⠾⠛⠋⠉⣉⣁⣠⣬⣍⣉⣉⣤⣤⡶⠟⠛⠉⢉⣉⣀⣤⣬⣉⣩⣤⣤⡶⠿⠛⠉⠉⣉⣁⣠⣬⡁
⠀⠀⠀⠀⢀⣤⣴⡶⠿⠛⠉⢉⣀⣤⣤⣤⣤⣀⣤⣴⠶⠿⠛⠉⢁⣀⣤⣤⠤⣤⣤⣠⣤⣴⠾⠟⠛⠉⢁⣠⣤⣤⢤⣤⣤⡤⠶⠾⠿⠛⠉⣁⣠⡤
⠀⠀⠀⠀⠠⠄⠒⠶⠶⠶⠞⠛⣉⣤⣴⠶⠶⠬⠍⠓⠲⠶⠶⠞⠛⣉⣤⣴⠶⠶⠆⠤⠐⠲⠶⠶⠶⠛⢋⣡⣤⡶⠶⠶⠬⠉⠓⠲⠶⠶⠛⢋⣩⣤
⠀⠀⠀⠀⠶⠶⠖⠒⠒⠒⠚⠛⠋⠁⣀⣠⣴⡶⠶⠚⠒⠓⠚⠛⠛⠋⠁⣀⣠⣴⡶⠾⠛⠛⠛⠒⠒⠛⠛⠉⢀⣀⣤⣶⡶⠟⠛⠛⠛⠛⠛⠛⠉⡁
⠀⠀⠀⠀⣠⣤⠶⠖⠛⢛⣉⣙⠛⠻⢉⣉⣁⣀⣤⡴⠾⠛⣋⣉⡙⠛⠫⢉⣉⣁⣀⣤⡴⠞⠛⣉⣉⣉⡛⠛⠋⣉⣉⣀⣠⣤⠶⠟⢛⣉⣉⣙⠚⠉
⠀⠀⠀⠀⠠⢤⣶⣊⣉⣉⡉⠉⠛⠷⢶⣤⣍⣩⣤⣴⣄⣉⣉⠉⠙⠛⠷⢶⣤⣭⣭⣤⣴⣬⣭⣁⡈⠉⠙⠻⠶⣤⣬⣉⣥⣤⣴⣬⣅⣀⡉⠉⠛⠗
⠀⠀⠀⠀⢤⣤⣀⡀⠉⠙⠻⠷⢶⣤⡤⠤⢤⣤⣤⣀⡈⠉⠛⠻⠷⠶⠦⠤⠴⠶⢶⣤⣄⣀⠉⠙⠛⠿⠶⡶⠤⠤⠴⠶⣦⣤⣀⣈⠉⠙⠛⠷⠶⠦
⠀⠀⠀⠀⢤⣤⣉⠛⠻⠶⠖⢂⠠⠴⠶⠶⠶⣤⣌⣉⠛⠛⠶⠒⡂⡄⠴⠖⠲⠶⢦⣤⣉⡙⠛⠶⠶⢒⣀⠤⠶⠒⠶⠶⣤⣌⣉⡛⠛⠛⢚⣂⠤⠆
⠀⠀⠀⠀⣀⣈⠉⡛⢛⡛⠛⠛⠛⠿⠶⣦⣤⣀⣈⣉⣛⣛⣛⡉⠙⠛⠛⠿⣶⣦⣄⣀⡉⠛⣛⣛⣋⠉⠙⠛⠻⠷⣶⣤⣄⣈⣉⣙⣛⣉⡉⠙⠛⠓
⠀⠀⠀⠀⠉⣉⣉⣉⣉⡙⠛⠷⢶⣤⣄⣀⠈⠉⣉⣉⣉⣉⣉⠛⠻⢶⣦⣤⣀⣈⠉⡙⠛⠋⠉⠉⠙⠛⠳⠶⣦⣤⣠⠌⢉⣉⣭⣭⣉⣙⠛⠿⢶⣦
⠀⠀⠀⠀⠛⠉⣉⣠⣬⣭⡕⡲⠦⢤⡤⠴⠖⠛⠉⣁⣠⣤⣬⠽⠶⠦⢤⣤⡤⠶⠞⠋⠀⣀⣤⣤⣤⠴⡶⠦⠤⠤⠶⠞⠛⢉⣠⣤⣤⣭⣿⡶⠦⡤
⠀⠀⠀⠀⠶⠿⠛⠉⠁⣀⣤⣤⡶⠶⠶⠶⠶⠶⠟⠛⠉⠁⣀⣠⣴⡶⠶⠶⠶⠴⠶⠾⠓⠃⠀⢀⣀⣤⣶⠶⠶⠶⠶⠶⠞⠛⠛⠉⢀⣀⣤⣶⠶⠖
⠀⠀⠀⠀⠰⢤⣞⡛⠛⠛⣉⣠⣤⠶⠞⠛⠒⠶⢤⣚⣛⠛⠋⣉⣠⣤⠶⠚⠛⠛⡲⢦⣔⣛⠛⠛⢋⣁⣠⡴⠖⠛⠛⠓⠶⢤⣝⣛⣛⣉⣁⣤⡴⠖
⠀⠀⠀⠀⠙⠓⣈⣉⣙⣋⣉⣁⣠⣴⡶⠿⠛⠛⠁⣈⣉⣉⣉⣉⣁⣤⣴⡶⠿⠛⠉⠉⣀⣉⣉⣙⣋⣉⣀⣤⣶⠾⠟⠛⠉⢀⣀⣉⣉⣉⣉⣀⣤⡦
⠀⠀⠀⠀⠾⠟⠛⢉⣉⣉⣭⣍⣉⡠⢤⣤⣴⡾⠟⠛⢉⣉⣩⣭⣍⣉⡠⢤⣤⣴⠾⠟⠋⢉⣉⣤⣭⣭⣉⣁⠤⣤⣤⣶⠾⠛⠋⢉⣁⣤⣭⣅⣀⡀
⠀⠀⠀⠀⠴⠖⠛⠫⠭⢤⣤⣤⣈⡉⠓⠶⠤⠤⠶⠚⠫⠥⢤⣤⣤⣉⠙⠓⠲⠤⠤⠶⠚⠫⠥⠴⣦⣤⣄⡉⠛⠲⠤⠤⠴⠶⠛⠭⠥⢤⣤⣤⣈⡉
⠀⠀⠀⠀⠰⠶⠶⣶⣤⣄⡀⠈⠙⠛⠷⠖⠶⠶⠶⠶⣦⣤⣀⡀⠉⠙⠛⠓⠒⠒⠒⠒⠶⢶⣤⣄⣀⠈⠙⠛⠻⠶⠒⠒⠒⠲⠶⣦⣤⣄⣀⠉⠙⠛
⠀⠀⠀⠀⠲⠶⣤⣀⣈⣙⣛⣻⠷⠞⠛⠛⠛⠲⠶⣤⣀⣉⣉⣛⣻⠷⠛⠛⠋⠛⠛⠶⣤⣄⣈⣉⣛⣛⣻⠶⠛⠛⠛⠛⠳⠶⣤⣄⣈⣉⣉⣩⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠠⠀⠒⠒⠀⠀⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠓⢡⣨⠴⠆⠀⠀⠀⠀⠀⠀⠠⠧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠍⢙⠀⢄⡀
⠀⣦⣠⠈⢀⢀⠤⠂⠑⠂⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢩⠅⠀⠀⠀⠉⠲⠀⠀⠀⠀⠓⠒⠉⠀⢰⠈
⠀⠉⢃⠀⠀⠀⠀⠤⢤⣖⡒⠂⠁⠀⠀⠠⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠈⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠠⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡏⠉⢂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡇⠀⢈⣤⠀⠀⠀⡆
⠀⠀⠀⢠⡴⣬⠀⣠⠤⣄⠀⠀⠀⣠⠆⠁⠉⢀⠀⢀⢤⢀⠀⠀⠄⠀⠀⠜⠀⠀⡇⠀⠉⠛⠁⠀⠀⡇
⠀⠀⠃⠙⠿⠚⠓⠳⠽⠛⠃⠀⠀⠁⠀⠘⠿⠆⠁⠁⠿⠟⠀⠐⠀⠀⠀⠫⠠⠲⣦⠒⠀⠾⣮⠷⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠧
⠀⠄⣤⠤⠤⠤⠤⠤⠤⠤⠬⠴⠀⠀⠀⠀⠠⠐⣄⠀⠀⡤⠤⡀⠀⠀⠀⠀⠀⠀⣴⠄⠀⣀⠇⠀⠅⢶⠂
⠀⠌⠯⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠐⠣⠔⢁⣄⠀⠀⢠⢞⡖⣛⢦⡀⢡⡴⣚⢲⡀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠐⠿⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⠶⢾⡝⠆⠀⠈⠉⠓⠿⠊⠉⠉⠙⠽⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃