Pirate ship Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⡟⠻⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠹⢿⣷⡀⣀⣀⣤⣤⣿⠶⠶⢠⡄⠀⢀⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣤⠀⠛⣿⣿⠋⠀⠀⠁⠀⠀⢀⣿⡃⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠂⣀⣉⡁⠀⠀⠀⠀⡀⠀⠈⢿⡖⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣀⣡⣍⣿⣾⣷⣺⣿⣿⣿⣛⣳⠈⠁⢡⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⡟⠙⠫⣯⠙⠙⠈⠻⠟⠀⠀⠋⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡏⠀⠀⢘⣿⡆⠀⠀⠀⠀⠀⡼⠻⠞⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣇⠀⠀⠀⡀⠁⠀⠀⢸⣧⢷⡆⠀⠀⣾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡄⢀⣁⡷⡶⠀⢚⠛⢻⣎⣿⡄⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠂
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠟⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟⠁⠉⠛⠳⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠿⠯⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⠀⠀⢠⣀⣤⡤⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢴⡖⠈⠃⠛
⠀⠀⠀⠀⠀⠀⣦⣠⣴⡾⠵⢿⡄⠀⡀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊
⠀⠀⠀⠀⠀⢀⠟⠉⠘⠒⣴⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣄⠀⠀⠀⣀⣀⣠⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡄⣀⠤⢺⣿⡌⠣
⠀⠀⠀⠀⠀⣻⢀⣀⠴⣶⣟⣛⣋⣛⣋⡴⠚⢻⣿⡇⠠⡄⢀⣈⡗⠂⠀⠀⠀⠀⠀⢹⡿⠻⣻⣿⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠠⣿⣟⣫⣄⣱⣼⠟⠛⠭⠃⠄⢦⢤⣦⣶⠶⠟⠛⠁⠀⠀⠀⣀⡀⠀⠀⠞⠴⢄⠀⠉⢢⡀⠀⡀⢰⣆⠀⠈⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⠃⣿⣸⣟⣿⣆⣀⣀⣤⣤⣴⡿⠋⢿⣧⠠⠀⠀⠈⢳⣤⣾⣷⠷⠿⠊⠉⠄⠀⣿⣿⣀⡠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⣺⣿⣠⠀⠉⠳⢼⣿⣻⡇⠀⠀⢀⣀⡀⡸⠛⣛⣻⡿⡶⠖⠛⠉⠉
⠀⠀⢲⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢼⣿⣷⣶⣶⣾⠿⡶⠦⠜⣿⣗⣞⣛⣛⣿⣿⣻⣿⣶⣾⣿⣿⣿⣿⣿⣿⠏⠂⠒⠃
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡀⠀⣠⣐⣷⡀
⠀⠀⣒⣛⣓⣚⢒⣾⣿⣿⢿⡟⢿⣛⣿⡽⢋⣉⣠⣴⣠⣉⣉⡩⣄⠩⠻⠿⠿⠷⠉⠉⠿⠉⠍⠋⢍⠛⠁⠀⠈⠉⠉⠉⠃⠘⠋⠙⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠉⢍⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣶⡖⠂⠘⡓⣤⣤⣀⡄⠀⢀⣤⣴⣶⣦⣧⣤⣬⣥⣌⣀⡀
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠀⠐⠓⠤⠬⣬⠈⣙⣉⣩⣽⣿⣿⣿⡿⢿⣧⣽⢿⡿⣿⣿⣿⣿⣿⣤⣿⣷⣼⣬⣿⣯⣹⣷⡶⠶⣤⠴⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠚⠛⠾⢕⣯⣼⣦⢮⣶⣤⣾⣿⠲⢿⢦⡉⣉⠩⡉⠈⠉⠉⠻⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠛⠉⠛⠓⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⣶⣼⠿⠛⠂⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢶⣼⡿⠟⠃⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⢰⣦⣽⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠆⠀⠀⠀⠀⠑⣦⣿⡈⠙⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠃⠠⣄⡀⠀⠀⠀⡤⠀⠀⠀⢸⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⡆⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⢀⣀⢀⢀⡀⠰⠒⢾⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡇⢄⠹⣷⣄⠀⠀⠀⠀⠀⠀⣘⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠏⠀⠠⣜⣾⣿⣿⣿⣷⠘⠻⢿⣿⣿⣿⡇⠈⢧⠹⣿⣷⡄⢀⣀⣠⣜⣻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⣿⣹⣿⣿⣿⣷⣦⡥⠄⠘⣧⠛⠛⢻⣨⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠰⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⠁⠘⣿⣿⣿⣿⡿⢶⣄⢸⣧⡠⠊⣹⣿⣿⡿⢃
⠀⠀⠀⠀⠀⠀⠘⡇⠀⣰⣦⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣆⣹⣿⠿⢿⡇⠀⣹⣿⣿⣤⣾⣿⣿⠟⢁⡟
⠀⠀⠀⠀⠀⠀⠀⢻⡀⣽⣟⣿⣵⣯⣿⣿⣯⣿⠏⣿⠏⠈⠻⠃⠛⣷⣾⣤⣼⣷⣿⣿⣿⡿⠛⠛⠉⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⢨⣿⣿⣿⣼⣿⣿⣿⣿⣷⣿⣶⣶⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⢟⣿⣿⣿⣿⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡭⠭⠯⠥⠥⠭⠤⠼⠾⠛⠛⠛⠛⠛⠛⠻⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠘⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠤⡤⠿⢿⡶⠶⠶⢴⣤⡤⠀⠀⠁⠈⠉⠉⠁⠉⠁⠉⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⣠⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣤⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⣹⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣀⣠⣴⣿⣿⣿⠿⣿⣠⣶⠿⠿⠀⣀⣀⡤⠤⠞⠛⠁⠀⠀⠀⢀⣀⣾⡶⢶⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣀⣛⣛⣾⣿⣿⣿⣿⡿⣿⠷⣿⣿⠃⡶⠶⠚⠋⠉⠀⠛⠒⠛⠃⠛⢷⣶⣾⣦⣤⣤⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⣴⣿⢠⡾⣲⠇⠉⠀⠀⠀⠀⠀⠀⣰⠞⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣾⡿⣹⣿⣯⡁⠀⠀⢀⣀⣠⣴⣾⣿⡿⠀⠀⠀⠀⠙⠛⠛⡀⠀⠀⠀⣀⣰⡇⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢙⣡⡴⠟⢻⡿⠻⠻⠟⣿⠛⢻⠿⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠋⠛⠋⠀⠀⠘⠀⢰⡧⠴⡿⠏
⠀⠀⠀⠀⠀⠀⠐⠦⠤⣄⣀⣀⣭⣭⣅⣬⣤⣤⢤⣤⣼⣾⣷⣶⣿⣿⣿⣿⣿⣿⣮⣤⣤⣤⣷⣤⣶⣤⣤⣤⣶⣷⣾⣾⣾⣷⣶⣷⣶⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⢿⢿⡿⠿⠿⠿⠿⠿⠿⡿⢿⣿⢿⣿⣿⣿⣟⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢀⣀⡐⠒⠶⠶⢔⢻⡾⠍⠉⠙⠉⠓⠚⢛⣛⣏⢹⣍⣄⣀⣐⣤⠒⣒⣠⡥⢄⠀⠠⠻⣿⣳⡿⡿⣿⢿⣿⠛⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣴⡤⠄⣀⡀⠀⢇⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢄⠈⠴⣿⠖⠂⠈⠉⠑⠊⣿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘⣿⡃⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠸⠟⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⢿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢋⡁⠀⠀⠀⠀⠀⢠⡴⠒⠀⠠⡿⠿⠿⠿⠿⣿⣿⡟⠉⠀⢀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡠⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⢢⡀⠀⠀⠀⣿⣿⣇⠘⠷⡼⠋⢻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⣁⣀⣀⣠⣀⣘⣶⡄⠀⠀⣿⡿⣟⡦⣸⡧⡶⢸⡿⠛⣋⣉⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡴⠾⢿⣷⢀⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣱⣶⣿⣿⣯⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠜⢧⡤⠐⠞⠋⠉⠉⠉⠁⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠙⡿⢻⣿⣿⡎⣻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⣧⠘⠛⠛⠛⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠀⠶⠀⣀⣀⡀⠀⣠⣄⣀⣀⣀⡀⠀⣸⣆⡀⠀⠙⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⣴⠖⠚⢙⠿⣿⡇⠀⣸⡟⣿⣿⣿⠿⠿⠿⠿⠧⣤⣀⣵⡀⠀⠀⠀⠘⠛⢿⣿⣿⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡿⠃⠀⠀⠐⠚⠛⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠁⠀⠀⠀⠀⠀⠀⠹⣿⣧⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠂⠀⠀⠀⣤⠄⠀⠀⠀⢹⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⣀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⣀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠻⣎⡄⠀⠀⢱
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠹⣆⠀⢀⣠⣶⣾⣋⠁⣸⡇⠀⣿⣿⣿⣿⣿⠟⠛⢛⣷⣦⣶⣿⣄⠀⢀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣣⡀⢀⡙⣿⣾⣿⣿⣿⡿⡿⠿⠧⠴⠿⠿⠿⠷⠶⠶⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⣻⠇
⠀⣀⡤⠶⠚⠛⠉⠉⠙⢦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣀⣀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠛⠂
⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⢀⠔⡡⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠛
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⢀⡀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⣿⣿⣷⣄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠂
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢠⣶⣟⢻⠟⠻⣿⣿⣇⣬⡃⠀⢿⣶⣤⡀⠀⠀⠀⢠⣤⣾⣏
⠀⠀⠀⠀⠈⢷⣄⢠⣇⣴⣿⡿⢫⢻⣶⡿⠛⠋⢹⣿⡰⢆⣬⠛⢿⣿⣶⣄⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣉⡶⠚⠻⣯⡀⠀⣤⣸⣿⡇⠀⠈⠙⠢⣬⡛⢿⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡠⢞⡷⠞⠁⠀⠀⠀⠈⢿⣀⣸⣿⣼⡇⠀⠀⠀⠀⠈⠙⠶⣌⡻⢿⣧⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠻⣴⡟⠀⠀⠀⠀⠀⠀⠀⠰⣆⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣝⢿⣿⢭⡟⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠸⣏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⡛⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣦⣀⡀⠀⠀⠀⠀⠘⠂⠉⠉⡆⠀⠀⠀⠀⠀⠀⢠⡀⢠⡀⠀⠸⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣿⣿⣼⣦⠀⠀⠀⠀⠀⢰⠱⡀⠀⠀⠀⠀⣾⣶⣿⣾⣧⣴⣦⡄⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠛⣻⣿⣿⣷⣄⠀⠀⡆⠸⣄⣿⠀⣠⣄⣰⣿⡿⣿⣿⡍⢻⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢫⣽⣿⣿⣟⣿⡄⣿⠀⣿⣿⣆⣿⣿⣿⣿⣿⣦⠙⠃⠈⠋⠈⠻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⢿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿⣿⣿⡇⢸⣿⣿⣿⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡇⢸⡿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⡤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⢫⠁⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⣀⠀⠁⠲⣄⣀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠐⡉⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠦⣐⢿⠋⠁⠀⠀⢸⡆⠀⠀⠀⠀⠀⠐⣿⡧⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⡀⠀⠀⠠⢷⠀⠀⠒⠚⠉⠉⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠙⠢⣀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢠⠤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠲⢜⣿⢀⣀⣤⡀⠀⠀⠸⡆⠀⠈⠲⣄⢻⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡠⠾
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣫⣿⣿⣷⡄⣄⣤⡗⡀⡤⠀⣎⠘⣇⠀⠀⠀⠀⠀⠀⢹⡀⠀⢐⡀⣠⣬⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣿⣿⣿⡓⢙⡧⠻⢛⠓⣖⠃⡄⠛⣤⠀⠀⠀⠀⣠⡬⣷⠾⠛⣋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⣤⣅⣉⣉⣀⣁⣰⣾⣂⣵⣾⣥⣤⣾⣥⣶⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡁⠀⠁⠀⠀⠐⠉⣿⣿⣿⢿⣿⣿⡿⣿⠿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⡦⣾⣿⣿⠿⠿⠿⠿⢟⣂⣐⣢⣶⢿⡿⣿⠿⠶⢾⠷⠞⠛⠛⠛⡙⣿⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠿⢯⢯⣬⣤⣢⠥⠽⢭⣑⢲⣚⠾⢿⡧⠽⢿⣯⠽⢭⡿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠶⠶⠮⠉⠁⠀⠀⠚⠉⠉⠑⠒⠊⠀⠙⠛⠒⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣈⣿⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠛⢛⢛⣿⡟⠛⠛⠛⠛⠉⠉⢙⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⢿⣿⣿⣷⣶⣶⣷⣶⣤⣾⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⡿⠿⠛⣛⠛⠛⠛⠛⠿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣿⠂⠀⠀⠀⢀⢠⡀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣀⣀⣀⣀⣿⣿⣿⢦⣤⡄⠤⣬⠤⠤⠤⠤⠴⢦⣤
⠀⠀⠀⠀⠀⠀⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠉⠉⢉⡉⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⡸⢿⣦⡀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠚⠋⠀⣽⡿
⠀⠀⠀⠀⠀⢠⢡⠘⣿⠘⢦⠀⠀⠀⢀⠞⠃⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢡⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢘⠀⠸⣧⠀⠙⠠⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⣆⡀⢀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣇
⠀⠀⠀⠀⢀⠀⢸⠀⠀⢹⣇⠀⣰⡟⢦⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣶⣍⣷⣶⣿⣿⣿⡄
⠀⠀⠀⢀⡄⠀⢨⠄⣴⣶⣿⣶⣿⠸⣦⡐⢤⡀⠀⠌⡍⢠⣿⣿⣿⠀⣿⡏⠉⠻⢿⣿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⣾⠓⠁⢨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⣈⡴⣹⣿⣿⣿⣶⣿⣧⣀⣀⡀⠈⠁⠮⣟⣿⣿⣿⣿⣷⡀⠀⣀
⠀⢰⣿⠃⠀⠀⣼⡿⢟⣿⡿⠻⣿⣿⣿⣿⣿⠿⣴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢹⣿⠙⠒⠒⠮⣭⣽⣻⣿⣿⣷⡀⢹⣦
⠀⣾⠇⣀⠀⠈⢹⠠⠞⠋⠀⠀⠘⣯⢻⠿⠟⣠⣿⣿⣿⡿⢛⢹⣿⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣷⣾⣿⣧⡀
⠀⠋⠀⠉⠀⠿⢻⣿⣾⣿⣿⣿⣷⢿⣿⣾⠿⠿⠿⠿⠿⡿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠁⠀⠀⢠⣿⣿⠀⠀⠀⢀⡔⠀⢢⠁⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠀⠀⡄⠁⠀⠀⠀⠓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⠦⠶⠦⠶
⣆⣠⣀⣀⣀⣀⣸⣁⣀⣴⣴⣶⣶⣶⣾⣿⣶⣴⣶⣴⣶⣶⣄⣀⣀⣠⣄⣀⣌⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢻⣿⡀⠀⠀⠙⠁
⢛⠉⠉⠉⠉⠉⢹⠛⡉⠉⠉⠉⠉⠈⠁⢹⣿⣏⠀⢠⣜⣍⠀⠀⠀⠀⠀⠀⠀⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡃
⠈⠄⠀⠀⠀⠀⠘⣧⣽⡴⣤⡄⠀⣠⡀⣠⣿⣿⣾⣃⣽⣏⣤⣤⣤⣤⣤⣤⣤⣮⣥⣤⣀⣀⣠⣴⣂⣤⣄⣼⣇⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠙⣿⣿⡿⠿⠛⢻⣿⠿⣿⣿⠟⠛⣿⣿⣿⡿⠿⣿⣿⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠋⠉⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⣿⠀⠀⠈⠀⠀⣿⢻⠏⡀⠀⠀⠉⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣶⡆⣿⣿⣀⠀⢀⡀⢀⣿⠸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⠀⠀⣰⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣧⣾⣿⣄⣦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠉⠙⠛⠉⠉⠙⠲⠤⣤⣀
⠀⠀⠀⠀⢠⣤⣤⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣶⣦⣤⣤⠄
⠀⠀⠀⠀⠀⠙⡟⠛⠛⣋⣠⣤⣤⣤⡈⠉⠉⠉⠉⠙⣛⣻⣿⣿⣟⣙⣛⣛⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠹⢿⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣾⡿⠛⠉⠉⠉⠙⠛⠛⠿⠶⠾⠟⠛⠛⠛⠛⠛⠉⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠰⠤⡤⠤⢤⣀⣀⣀⣉⣃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠤⠀⠀⢘⠁⠀⠀⠙⠲⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠘⠛⠷⠾⡖⠒⠖⠂⠀⠀⠾⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠤⠐⠢⠴⡤⠄⠀⠐⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠛⠓⠲⠶⠶⠷⠤⢤⣤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⣀⣀⡔⠀⡄⠀⣀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣤⣄⠀⠀⠖⠚⠛⠙⠋⠉⠉⠀⠙⠀⣿⠁⠀⠉⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣀⢠⡐⣫⠦⠴⠿⠷⣤⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠐⠷⣾⣶⣼⣟⣉⡉⠀⠀⠀⢀⣧⣀⡀⣤⣤⣿⣦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢄⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠉⠉⠛⠻⠶⠾⢿⣯⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠦⣀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣟⣀⠀⠀⢀⣤⣤⣅⣈⣁⣰⣆⣙⣈⡓⠮⠤⠐⠀⠹⡆⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠴⠦⣤⣶⣀⣠⡄⠛⢋⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠥⠅⠀⠀⠀⠀⠀⠱⠄⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠐⢀⠀⠉⠉⠉⠓⠲⠶⠶⠤⣤⣤⣄⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠚⠒⠿⡀⠂⠤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠗⠁⠀⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⢃⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠲⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠀⠀⠀⢨⠀⠀⠀⡀⠀⣈⠇⠀⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠹⠀⣧⠀⠀⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢰⡏⠀⠀⠀⡟⠀⠀⠀⠀⢀⡜⠁⢀⡽⠆⢈⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡿⠊⡠⠂⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⠇⠀⠀⡸⠁⠀⠀⠀⠀⠙⠂⣀⠀⣤⠀⢸⠇⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⠀⢺⡄⢀⠀⠀⠀⠀⠊⠀⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠀⠼⠀⠀⠠⠁⠀⢀⡠⠄⢰⡷⢉⢹⣶⠀⠀⡌⢀⢰⠆⢀⣰⠶⠶⠛⠀⠀⠀⠀⠀⠇⠀⠀⠠⠄⠀⠔⠀⢀⠖⠉
⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣴⡎⣿⡀⠀⠈⡟⠸⡾⠁⢀⡜⣁⡴⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⣰⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡴⠐⠉⠀⢺⡇⢿⡇⠀⠀⣧⣐⣁⡴⠛⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠞⣡⢾⣀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⠘⠃⠀⢀⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⢰⠅⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⣸⡇⠀⠃⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡉⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣿⣄⡀⣈⣓⣲⣤⣀⡀⠀⣄⣉⣁⣬⡷⠁⣀⣷⣆⡄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣍⢉⣿⣿⣿⣿⣿⡛⣂⣿⣌⣩⣿⣴⠟⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢻⣻⣟⡻⣟⣿⠛⣻⣻⣟⠛⡟⣯⡉⣹⣽⣅⠀⠀⠀⡁⠈⠋⣠⣾⡏⣿⣾⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠐⠿⣿⠇⢿⡿⠯⠸⠿⠿⠀⠿⠿⠃⢀⣀⣤⣴⣶⠟⠫⣽⡇⡌⠙⢳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠒⠒⠒⠒⣖⢲⣶⢒⣖⣒⣒⣲⣒⣒⠖⠒⠋⠉⠉⠥⠀⠂⠠⡀⣼⣷⣆⣠⠍⠻⠧⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⠋⠉⠛⠛⠛⠛⡛⠛⠉⠈⠉⠁⡀⢠⡀⠀⠀⠀⡀⠀⠀⠀⠘⠻⠿⠛⢁⡀⠰⠦⠓⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠀⠠⠄⠀⠀⢀⡀⠒⣀⣀⠈⠁⠀⠀⠀⠈⠀⠑⠚⠛⠁⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⡾⠛⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢻⣿⣾⣮⡠⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡿⠟⢿⡀⠀⠀⠀⠈⣿⣧⡿⠿⠛⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠉⠁⠀⠠⠘⣧⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢟⠲⠶⠾⢿⣦⣄⣄⣀⡀⠐⡖⠶⠤⢴⣤⣷⣀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⠀⠀⠀⠉⠉⠛⠳⠷⠦⢤
⠀⠀⠀⢀⣠⣴⡾⣷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣸⣷⢤⡀⠀⠀⠀⡘⠀⢀⣀⣠⣄⣀⠀⠀⠀⠞⠀⠀⠐⠀⠶⢲⡀⣶⠆⠀⠀⠀⠐
⠀⢛⠶⢤⣄⣀⠀⠙⢧⡜⠂⠀⠀⣜⣁⠀⠀⠀⣽⢿⣏⣤⠀⠀⠀⠀⠿⠙⠛⠛⠚⠷⠿⠶⠤⢤⣤⣥⣤⣤⡀
⠀⠈⠓⠂⠼⠿⠓⠲⢾⣧⡀⠀⡀⠉⢏⠉⠛⠛⠛⠿⠿⢷⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣙⡳⢷⣤⣀⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠉⠉⠒⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣴⡶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣠⣬⡿⢏⠉⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿⣠⣼⣼⡾⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢺⡟⣿⣻⣯⣤⣴⢾⢿⡿⠋⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⠍⠺⢿⣿⣿⣿⣿⣥⢴⣶⣺⣿⣿⣿⣧⣀⠐⠚⠀⠀⡀⢀⣼⣇⣽⣿⣿⣯⣶⣶⠏⠀⠀⠀⠀⠤⠎⠀⠤⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣭⣷⣾⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⣀⣐⣃⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⣠⣡⡄⠀⠙⣿⡏⠉⠀⠀⠀⠀⠀⢠⡁⣀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢻⠋⠉⣉⡙⠉⠙⣿⡟⠉⠁⠀⠶⠓⠒⠶⣶⠭⠄⢀⣀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢆⢸⠁⠀⠐⠛⠡⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣿⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣦⡄⠀⠈⠀⢹⣇⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⠄⠐⢸⠈⢢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠀⠀⠀⠸⠠⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢃⠈⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡠⢾⣿⣿⣿⣶⠲⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣹⣽⣡⣮⣄⡼⠊
⠀⠀⠀⠀⠀⠀⣼⣀⣮⣭⣬⣽⣽⣆⣶⢤⣄⣀⠀⠀⠴⢄⡂⠒⠒⠀⠘⠤⠀⣸⢻⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⡿⣿⠿⢶⣶⣬⣍⠿⠄⠈⠈⠉⢉⡁⠈⣁⠀⠤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡳⠿⠷⠿⢷⣧⠰⠄⣤⠈⡉⢷⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣷⣶⣿⣿⣦⣀⣀⠀⠀⠀⣶⡄⠀⠀⢤⡄⠀⠀⠸⠂⠀⠉⠀⣴⠂
⠀⠀⠀⠀⠀⠀⣄⣨⣿⣯⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠚⡃⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠂⣹⢹⣟⣹⣿⣏⣻⣿⣿⡟⢻⣿⠿⠍⠡⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠋⠘⠿⠡⠀⠲⠿⠃⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢰⡎⠹⠶⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠛⠉⠉⠀⠀⠶⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⣀⣀⠀⠀⠀⣀⡠⣿⠆⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⣿⠍⠉⠀⣀⣪⡭⠐⠖⠛⠋⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠐⠋⠉⠹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣦⣤⣤⣀⠀⠀⠀⠀⣂⡀⢤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⡶⢶⣄⢘⡿⠋⢉⣁⣿⣷⣟⣳⣤⣀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⣻⠟⠁⣀⣾⣷⣤⣬⣿⡟⠉⠉⠉⠁⠀⠀⠘⠻⠋⠀⡃⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⢻⣿⠁⢀⣀⢤⠞⠚⠋⠉⠁⠀⠈⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⡴⠶⢺⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⡁⠀⠀⠺⢗⣺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣄⢀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⠟⡶⢞⠃⣠⡴⠃⠀⢸⣀⣀⣤⣤⣤⣤⣤⡀⢿⠿⠛⡿⠿⣿⣷⣦⣤⠀⠘⣿⣻⡇⠀⠀⢼⡇
⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⢸⣿⣯⣿⣴⣷⣟⣿⠁⠀⣠⡿⠟⠉⠙⣯⣿⣿⣿⡛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡇⠀⠀⢿⡧
⠀⠀⠀⠀⠀⠈⠳⢤⣄⠀⠘⣿⣿⣽⣿⡶⠿⠿⣤⣼⠥⠶⠖⠒⠛⠛⠛⠛⠛⣿⣿⣿⡿⠛⠛⠛⠉⠉⠛⠻⠿⣿⣿⣿⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡈⠻⣿⠏⠀⠀⢠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⣃⠀⠀⢸⠀⠀⣀⠀⣀⣄⠀⠀⠀⠀⠀⢸⣿⣇⣤⣀⡀⠀⠀⠀⠀⠆⢰⣿⣿⣿⣄⣤⣼⣿⣷⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡈⢿⣶⡿⠿⠿⠿⠿⢶⣶⣄⡀⢸⡿⠿⠿⢿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣷⣾⣄⡀⣤⣤⡀⢐⣿⡟⢿⣦⡁⠀⠀⠀⢘⣿⡟⢿⣶⣄⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣧⣀⡙⠻⣦⣀⠀⠀⣿⡇⠀⢻⣻⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣞⣋⣵⡿⣿⢟⡿⣿⣽⣟⣛⢿⡶⣶⣯⣶⣄⣿⣷⣤⣤⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠿⣿⡹⣿⠝⣳⣌⠀⢸⣿⣯⢿⡿⢶⣿⡤⣟⣉⣁⣛⣛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⣶⡤⣽⣿⣿⣿⣿⣧⣷⣘⣟⠇⣸⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣨⠐⠲⣜⣿⣾⠍⡓⠿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣿⣿⡿⣿⠛⢿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠦⢶⠓⠒⠉⠉⠉⠄⠙⠀⠂⠒⠅⠀⠊⠉⠳⠌⠛⠛⠻⠗⠐⠚⢉⣃⠤⠄⠉⡒⠳⡯⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢰⠟⠻⠦⠀⠠⠏⠀⢠⠀⠀⠀⠐⠂⠋⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢰⠀⠉⠀⠚⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⡠⠀⠀⢀⣠⠂⠀⢀⠄⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢒⣋⠀⠀⣻⠻⢷⣷⣶⠚⠠⣮⡀⠠⣄⣤⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⣯⣻⡿⠤⣀⠈⠻⠀⠀⠀⠀⠀⠠⠙⠻⢷⡚⠿⠉⠂⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⣾⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⡀⠈⢣⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣞⢻⣿⣿⣦⡾⣿⢁⡾⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠰⠄⠴⢄⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡛⣷⣯⣿⣿⣿⣿⣶⡞⠀⠀⠉⠈⣹⡇⣏⡁⠀⣀⣰⠋⢠⡄⢠⠄⠀⠀⠀⣸⠀⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⠛⢿⣿⣿⣿⢿⣴⣶⣶⣦⣿⣠⣿⣗⣾⣿⣿⣶⣾⣷⣿⢧⣄⢀⠀⣒⣛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣇⡒⠠⣬⡛⠁⠈⠛⠛⠻⠟⠉⠋⢉⣿⣿⣿⣹⣯⡉⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠩⠛⠋⠿⠊⠡⠍⠛⣿⠦⣤⣤⣀⣀⣀⣎⣀⣀⣀⣀⣀⣼⣨⣽⣾⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠊⠚⠋⠹⠏⠉⠉⢩⡉⠉⠉⠉⢁⣉⠼⠳⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠀⠀⠅⠀⠙⠁⠀⢄⠁


⣀⠀⣀⣀⣀⡀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⣀⣀⣷⠀⠀⠀⣀⡠⢤⠀⠠⠤⠤⠀⣸⣇⠀⠀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠤⣀⣀⣀⣀⡀⡄⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢴⠁⣤⣤⠄⠑⡤⠇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡔⠚⢢⣬⣩⢤⠚⢒⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠣⠃⠉⠀⠁⠑⠊⠀⢸⠀⠀⣘⣀⡀⠀⢀⣀⣀⣀⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⠤⠤⠤⡤⠤⠤⠀⠀⢀⡗⠂⠩⠤⠀⠀⢀⣇⣀⣀⣀⡈⠉⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠰⠒⠠⠤⠤⠀⣀⣀⣇⡀⠀⢀⢀⣀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢄⠀⢀⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⠒⠒⠒⠒⡦⠀⠀⠠⠤⠄⢀⣆⠀⠀⢺⣻⠿⢯⡉⡙
⠈⠑⡀⠑⢧⡀⠃⠉⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠀⠐⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⣤⣴⠟
⠀⠀⠘⡄⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡠⡪⠃⠀⠀⡘
⠀⠀⠀⠰⡀⠀⠀⠈⠉⠛⠛⠓⠒⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠛⠛⠓⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣒⣒⣒⣒⣂⣒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⣆⠀⠀⠀⣀⡤⠖⠲⣄⠀⠀⢀⡴⠒⠶⣄⠀⠀⢀⡴⠒⠶⣄⠀⠀⠀⣴⠖⠒⢦⣀⣀⣀⡰
⠀⠀⠀⠀⠀⠸⡀⡀⠀⠹⣇⠀⠀⣸⠂⠀⢸⡀⠀⠀⡽⠀⠀⢸⡀⠀⠀⣽⠀⠀⠘⣇⠀⠀⣨⠇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢣⠤⠀⠀⠉⠓⣚⣁⣀⡀⠀⠙⠒⠋⠁⠐⠒⠀⠙⠓⠋⠁⢀⣀⣀⣈⡙⠋⠍⠀⠤⡌
⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠤⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠤⠄⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡶⠻⠶⠖⠂⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠉⠉⠐⠲⠀⠀⠟⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⢸⢀⣀⣀⣀⠀⡟⢻⠙⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡶⠶⠶⢶⠻⠷⠾⠶⠾⠿⠧⣼⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⢣⠀⠀⠀⠀⠀⠴⡷⠶⠾⠿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⢷⡶⣎⢳⣎⡤⠤⣤⠤⢤⣄⣹⠀⠀⠀⠈⠻⣝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣀⣀⢳⠀⠀⠉⠻⣏⠛⠛⠛⠒⣒⣚⣿⣤⢤⣤⣤⣀⠘⠳⣄⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢉⣟⣀⣤⣤⣄⣘⡆⠀⠀⠀⠉⠻⡍⠉⠉⠉⠙⣿⣏⡒⠛⢦⠀⠀⠈⠓⢤⡀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢳⡉⠳⢬⣧⠀⢀⣤⢔⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠰⠆⣿⡰⣟⣻⣾⣓⣾⡳⠶⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢸⡇⢀⡼⢓⡾⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⠀⡏⠀⠀⣀⣠⣤⣾⡥⣶⣶⣶⣶⢶⢒⣞⡤⠖⠛⣻⣛⡛⠾⢷⣽⣧⡾⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠏⠀⢸⡧⢶⣿⣏⣀⡋⢸⡄⣾⣿⠁⠹⣻⣿⣥⣒⣮⢽⠴⣟⣸⣃⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣏⣍⢭⠩⠅⠆⢹⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣒⣂⣓⣘⣀⣿⣿⠿⠿⢟⣛⣯⣑⣪⣭⣷⣿⣿⣿⣿⣿⣿⣿⡷⠇⠢⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⠆⠤⣿⣿⣿⣿⣿⣿⣷⣶⣾⣟⣓⣿⣿⣿⣿⣿⠿⠛⠙⢋⡛⢉⡛⠋⠴⢖⠲⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡟⠒⢽⣄⣭⠍⠛⠻⣿⣿⡭⠉⠉⠛⠃⠄⠀⣉⠀⠀⢀⣈⣭⠍⢘⣿⠩⠝⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⣛⣛⣉⡉⠉⢁⢸⡷⠒⠀⠀⢀⠀⠀⠋⠉⠉⠠⣤⠤⠤⠀⠉⠉⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⣀⣀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠉⢁⣽⣿⣥⣄⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣽⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠈⠉⣿⡉⠈⢳⠀⣀⣀⣠⣤⡤⠀⠀⠀⢸⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣾⣴⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣼⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⢁⣀⢀⡀⣼⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣯⢿⣿⣿⠼⠫⠈⣡⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧
⢀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⣝⠿⣜⠒⣺⡿⢟⣵⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠑⢄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣟⣿⡿⠷⢌⣉⠱⠾⣿⣟⣿⣿⣿⡟⢄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣀⣀⣑⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣯⣭⣥⡄⢶⣿⣿⣿⣿⡶⣠⣤⣭⣽⣿⣧⠘⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢢⣿⣿⣿⠿⠟⠛⢻⡏⣉⣉⣩⣭⣿⡷⠞⠛⠛
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠙⠻⢿⣿⡆⠀⢿⡏⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⡼⢟⠹⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⡇⢹⣶⣕⣄⠀⠉⢻⡄⠈⠳⡀⠀⠀⠀⠀⠀⢠⠃⠀⠈⡟⠻⢿⣿⣇⣸⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠘⢯⡉⠉⠛⠻⢿⣿⣌⣿⣿⣿⣷⣴⣌⣳⣀⣀⠀⠢⢀⠀⠀⠀⡼⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠙⠢⠄⡀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡄⠀⢀⣀⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢂⣠⣤⣤⠶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠤⠖⠋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣦⡀⠀⢹⣷⣶⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⣸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⠛⠋⠙⣿⣿⣿⡉⠉⠉⢹⣟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣽⣿⣿⣿⣶⣶⣾⣿⣀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣛⣉⣠⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⣿⣷⣾⣧⣤⣄⣀⡙⣆⢀⣯⠈⠉⠉⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⡉⠛⠟⠹⢿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣧⣤⣀⡈⡇⠈⠁⠂⠀⠀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣯⣽⣟⣿⣿⣿⣿⣶⣶⣶⣦⣤⣭⣿⣿⡿⢿⣿⣿⣿⠏⢐⣆⠀⠀⠀⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⡿⣿⢾⣿⣯⡿⠁⢸⡇⢘⠛⣋⠥⠴⡛⠁⢀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠿⠟⣿⣿⡏⡀⢠⡟⡹⠏⡀⣀⣰⣧⣷⣶⣾⣿⣾⣿⣶⣏⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⢀⣇⡀⠀⢰⣦⣴⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠿⠶⠶⠻⠟⠛⢛⣛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠙⠏⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⣀⡤
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠄⠀⣀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠺⢡⣆⡤⣼⠁⠏⡷⣦⣴⡐⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠄⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⣬⣷⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⢀⠇⠀⢯⡜⠛⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠈⠢⡀⠀⣼⠀⠀⠀⢻⣦⠀⠀⠀⡶⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠿⢦⠄⠀⠈⢢⡏⠀⠀⠀⡾⠉⠑⢄⢰⡇⠈⢹⠇⠰⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠢⡀⢰⣏⠑⠲⢾⣦⡀⠀⠀⢈⣄⣠⢞⡑⠃⠀⠀⠻⡁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠑⢄⡀⠈⠱⠾⣶⣾⣯⣙⡛⠟⢊⡀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠉⠢⡀⢸⠷⡉⠿⣿⣿⡏⠘⠭⣀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠐⣤⢰⢾⡇⠂⠀⠀⠀⠈⠻⣬⢙⡀⠘⠻⡇⠀⠀⣼⠓⠀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠿⢛⢿⣼⢙⣷⠀⠀⠀⠀⣾⣿⠸⡇⠀⠀⠈⠢⣤⣿⠀⠀⠀⢀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⠒⠂⠀⣸⡇⢹⣿⠆⣤⣀⣰⣿⠫⠀⠖⠀⡀⠀⠀⠈⠱⠄⠀⠀⠀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⠟⠃⠀⠘⠾⢻⣾⣿⣷⠿⣮⣤⣼⠓⣠⠂⠀⠀⠈⠓⢄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⣈⣷⣿⡁⢽⣯⣭⢭⣅⣈⣹⠛⠤⢟⣿⣅⣤⡀⣀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⠻⠷⣾⣿⣿⡏⠙⠶⢬⣻⡙⢿⡷⠠⠼⣑⠹⠷⢿⡆⡜⣏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⡇⠀⠀⠀⢉⢻⣿⢠⠀⠀⠉⠙⠺⢧⣀⠀⣿⠁⠦⣈⢿⠿⣏⠻
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣧⢆⣼⡍⡄⠀⠀⠀⣤⣿⡆⣦⠆⠀⠀⠀⠀⠈⠙⠻⣤⣄⠈⢷⡤⣀⡆
⠀⠀⠀⠀⠀⠀⠀⡜⣿⠓⣾⣿⡿⡧⢠⠀⣰⢩⣼⣿⣇⠠⠄⠀⠀⠀⠀⠀⠀⢸⠉⠳⣶⣷⠀⠁⠢⢄⡀⠀⠀⢀⡴⠗
⠀⠀⠀⠀⠀⠀⣜⢐⡻⣾⣟⣿⡷⢾⣿⣾⣏⣼⣿⣿⠄⠰⡆⢐⠂⠀⠀⢀⢀⣄⣦⢘⢿⣿⣷⣴⡰⡆⠈⠳⠶⣏
⠀⠀⠀⠀⠀⠂⠀⠀⠴⣿⣿⣿⣿⣾⣿⠇⢩⣿⣿⣿⣷⣶⣷⢸⣷⣥⣇⠚⠀⢠⣇⣾⢈⣿⣿⡿⢳⣹⡄⣰⡄⠀⠙⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡒⣬⠎⡙⣢⣻⣟⣸⣿⣟⢿⣿⡿⠏⢻⠘⠏⠹⡜⣗⣀⣟⣿⣧⣾⣛⣛⣐⣼⠄⣷⣏⡆⢰⠁⠀⡸
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⣁⣀⣘⡟⠋⢛⣃⠭⣶⣾⡿⠟⣛⣫⣭⣭⠿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⠂⣽⣴⣾⠃
⠀⠀⠀⠀⠀⣾⠁⣶⡿⠇⠙⠛⣿⣿⣿⣿⣥⣵⣿⣷⣚⠭⡖⠾⠛⡋⠈⠬⠄⠠⢄⣴⣾⣿⡿⢸⣿⣆⠀⠀⠀⢈⡞⠁
⠀⠀⠀⠀⠐⣿⣘⣒⠃⠭⣅⡒⢂⠽⠿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⢒⣒⣂⣀⣒⣾⣿⡿⣡⣏⢻⠟⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢻⡁⠺⢡⣒⣲⠦⠤⡭⢨⡿⣩⣿⡛⣷⣽⠈⠠⠂⠀⠈⠡⠤⣤⣬⣽⣿⣟⣾⠿⡝⠁
⠀⠀⠀⠀⠀⣿⣿⣁⣲⡴⠬⣭⡥⠛⣐⣋⡉⣫⣿⣿⡷⠄⠐⣃⠠⠠⢖⣒⣶⣶⣿⣿⡿⣣⠋
⠀⠀⠀⠀⠰⠿⢟⣃⣛⣿⣿⣒⣛⣂⣰⣾⣶⣶⠿⢛⣉⣛⣛⣷⣶⣾⣥⠞⢿⣿⣿⣏⣱⣷⢄⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠈⠉⠙⠛⠛⠂⠀⠙⠛⠿⠟⠟⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⢰⣤⡆⢰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡏⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣀⣾⠃⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠏⠙⢿⠦⠄⠀⠀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠧⣉⡒⠂⢦⣿⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠙⠻⠷⠤⣤⣿⣀⢿⠻⠶⣤⣈⣀⢠⡄⠀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⡏⠀⠀⠀⠀⠙⣟⠓⠲⢬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⢤⠂⠀⠀⢀⣠⠆⠀⠀⢀⢾⣿⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠘⣅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠢⠎⠀⠖⣘⡿⠀⢀⣤⣾⣿⣋⠀⠀⢀⠀⡀⠀⠀⣿⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡟⠛⠻⠿⢿⣿⣾⣾⣶⣾⣧⣳⣿⣿⣟⠛⠻⠿⢿⣿⣀⠉⢰⣦⢿⡄⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢈⡁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣧⣄⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠉⠁⠒⢾⡀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡿⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃⠀⠀⠒⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⡴⡴⢁⢆⣤⣠⣎⣠⣤⢄⣴⣿⡿⠁⠀⣴⠀⣠⠂⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⡀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠊⢾⢶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣄⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⠃⢀⣹⠾⠷⣤⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣿⣾⣾⣶⣿⣙⣿⣿⡿⢛⡿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠈⠐⢶⣼⣿⣿⣠⣟⣁⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⣤⣬⣟⠙⠛⠿⢿⣿⡟⠈⠉⢠⣯⣬⣛⣺⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠓⠲⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⡞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠀⠀⠀⢁⣠⣴⣾⣯⣤⡝⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠟⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀⣼⠇⠀⢀⣀⣶⣾⣿⣿⣿⣿⣿⣿⡇⣼⠉⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⡶⠶⠄⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢾⣛⣩⣥⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣋⣤⣿⣿⡄⠀⠀⠙⠟⢎⢻⡛⠛⠛⢻⡇⠸⠿⠄⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠘⠶⣤⣀⣘⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣏⣛⡛⠿⢿⡅⠀⠀⠀⠀⠀⠴⠳⣌⠢⣶⣿⣷⣄⡠⣄⣀⣀⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠒⠶⠒⣛⣉⣭⣭⣭⣭⣍⣉⠛⢿⣦⣍⠳⠀⠀⣢⣒⣷⣶⣼⣿⣶⣭⣿⣿⣿⣿⣿⣿⣛⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠀⢐⣂⣭⣽⣿⣿⣿⣿⣿⣿⣿⣤⠈⠛⢿⣦⡀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣛⣋⣉⣤⣤⣿⣿⣿⣿⣿⣿⣟⠃⢰⣦⠙⠧⣄⡈⠑⠾⣷⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠉⠁⠛⢿⣿⣿⣏⡞⠻⠀⣬⣿⣶⣄⣄⡉⠛⠛⠶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠭⢽⢿⣷⣤⣄⠐⠌⠙⠻⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠂⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡎⢓⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⡆⠀⠀⠈⢽⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⠽⠶⡖⠀⠒⢸⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠚⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠥⠟⠙⠿⠓⡯⢭⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠸⠆⠔⠴⠦⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣠⣶⢶⠶⢶⣶⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠆⠘⣡⢤⣬⣀⣸⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠔⠒⡒⡒⠁⠚⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠴⠃⣀⢀⢀⣀⣀⣀⡇⠀⢀⡠⠤⠒⡪⠉⢅⣀⣀⣭⠭⠴⠶⠖⡶⠖⠒⡶⠦⢼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠸⠛⡆⠚⠻⠀⢊⡩⠷⠪⢑⣠⡭⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠐⠁⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢠⡖⣺⡟⠒⢲⣶⣯⣴⠾⠋⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⣀⡚⠉⠉⢉⣭⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡠⠀⠠⢧⣀⡇⠠⠲⢼⠟⠃⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⣤⡗⢒⡟⠛⠉⠀⣿⠀⠀⢀⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⡇⠁⠺⠋⢩⡅⠀⠀⠀⠈⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡀⢸⣁⣀⣀⣀⠀⡀⢠⢤⢤⡍⣤⠤⡄⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠈⠉⠹⠙⠷⣶⠶⣶⣤⣤⣤⣭⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠉⠉⠉⠛⠋⠹⠿⠿⠟⠋⠙⠩⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠶⠶⠊⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⣀⡀⠀⣠⠀⣄⠀⣠⣾⠃⡀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢿⠀⠀⠀⠀⠀⡉⠀⠀⣿⣦⢿⣿⣿⣵⣴⣿⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⣀⡤⠖⠃⠀⠀⠀⠘⡇⠀⠀⠀⢰⣿⡤⠔⡭⢗⣿⣿⣿⣿⣿⣟⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⣁⣀⣀⣀⣀⣀⣠⣤⣷⠴⠶⠛⠋⠁⣠⡠⠞⢻⣿⣿⣿⣿⣿⣿⢿⣶⣿⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣈⢳⣬⣬⣭⣍⣉⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⢰⡟⢀⣄⣠⠀⠀⠙⠛⢋⠛⠈⠉⣛⣧⣤⣾⣷⣶⠀⠠
⠀⠀⠀⠀⠀⢀⡀⠉⢘⣉⣹⣛⡛⢛⣻⣿⣿⠿⠿⣭⣽⣶⣶⣶⡶⢶⣶⣬⣬⣽⣶⢿⣿⣷⣶⣿⣶⣿⣯⣰⣿⣶⣌⢻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠼⠿⠿⠿⢿⣿⠿⠟⣛⣛⣉⣱⣶⣷⣾⣿⡿⠟⢁⣀⣀⣲⢿⣿⣉⠀⠀⣘⣛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣦⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠾⠿⠿⠿⠛⠛⠛⠛⠛⣛⣉⣀⣀⣈⣛⡿⠿⠇⠀⠀⠀⢀⣿⣿⣷⣶⡄⠈⠉⠹⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣷⣤⣶⣦⣝⣻⣟⡿⠿⣿⠛⠋⠑⠂⠀⠀⠀⠈⠛⠉⠉⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠛⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠉⡟⢉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡒⠒⠚⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠄⠀⠀⡏⠀⠉⠉⢉⣭⣭⡉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⡐⢠⠁⠀⣷⠀⠀⠀⠸⠿⠿⠇⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣶
⠀⠀⠀⠠⠀⡘⢤⣤⣷⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⠁⠀⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⠀⣾⣿⣿⡷⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⣀⣀⣈⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠋⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⢾⣿⣿⣤⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⠤⣿⡌⠁⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣻⣀⢺⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⣈⣷⠀⠉⠉⠉⠉⠛⠛⠛⠓⠲⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠈⠫⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⣤⣴⣾⣷⣿⣿⣿⣭⣷⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠯⠥⢬⣯⣿⣽⣿⣛⣿⣀⣀⣀⠀⠀⠀⠄⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⢳⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣬⠤⠤⠀⠠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣀⣀⣤⣤⠤⠤⠔⠤⢄⣾⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⣔⣉⣾⣤⢼⣿⡦⠶⠒⠒⣾⡟⠛⠋⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢀⣿⣧⣤⣤⣾⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣶⣿⣿⣿⣿⣿⣿⡧⣷⡙⠢⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡍⢻⣿⣿⡿⠿⠿⢿⣷⠙⠿⠃⣄⠹⡿⣫⣽⣧⣤⢤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⣁⣭⣵⣶⣿⣿⣿⣿⣿⠙⢦⡼⣿⣷⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⣿⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⠉⠛⢿⣷⣤⢙⣻⠿⣿⣿⠇⠀⣾⣿⠇⣾⡅⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⣿⣿⣿⣿⣯⣮⡁⠘⠛⢟⣄⠀⠀⠙⠻⢶⣾⣿⠿⢶⣿⣧⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣜⣣⡄⢉⠈⠉⠿⠛⠛⠻⢦⡀⠈⠙⠚⢃⣀⡀⠀⢈⡀⠀⠀⠹⣿⣷⣬⡙⠓⠂⠀⠀⣠⣄⣀⡀
⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡁⡀⠐⢦⡉⠉⠻⣷⣤⣈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣶⣦⣤⣀⣭⣍⣁⣀⠀⠐⢶⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠛⠉⠁⠀⠉⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠰⠆⣶⡦⠀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣽⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣿⣿⡀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⡍⠓⠚⠗
⠀⠀⠀⠀⠀⠀⠀⣄⣀⣈⠙⠏⠸⣿⠀⠃
⠀⠀⠀⠀⠀⠠⠀⣈⣯⠉⠉⠉⠋⠻⠚⠒⠓⠢⠤⠤
⠀⠀⠀⠀⠀⡀⡀⠀⢈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠀⢨⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣾⠿⠛⣿⣿⣿⣷⣾⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢉⣥⠉⠉⠉⠉⠉⠛⠚⠛⠚⠻⠶⠶⠶⠤⢤⣥⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣰⣃⠠⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡀⣤⡴⠀⠄⠠⠄⣼⣇⣀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣴⡭⠴⠶⢺⣿⠒⠚⠛⣿⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣴⣿⣶⣶⣾⣿⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣿⣿⣿⣿⣿⣽⣿⢿⣦⡑⠢⣠⣴⠶⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠿⠘⣛⣻⣭⣭⣽⣿⡦⣍⢡⣶⣼⣵⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⠀⢘⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⡟⠻⣿⣯⣉⢻⢿⣿⣿⡟⠀⣼⣿⠇⣿⢀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣴⡃⠘⠻⢏⡉⠀⠈⠻⢧⣶⣿⠿⢾⣿⡎⢷⣄
⠀⠀⠀⠀⠀⢠⣼⣿⠄⣉⠈⠹⠟⠛⠛⠲⠄⠈⠙⠘⣀⠄⠀⠀⣁⠀⠀⠻⣿⣦⣍⠓⠂⢀⣀⣤⣄⣀⡀
⠀⠀⠀⢠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡐⠦⠀⠙⠿⣦⣌⡉⠉⠉⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣶⣶⡶⠦⠤⠀⠙⠻⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣿⣿⡯⣹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠿⢷⠀⠀⠀⢹⣯⡿⠿⠟⠻⡆
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠠⠸⡆⠀⠀⠈⠁⠀⠀⠀⢸⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣥⣄⣿⣄⣀⡀⠀⢠⣄⣀⣀⣻⠀⢢⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠚⠂⠀⠈⠉⠉⠓⠓⠶⠤⠄
⠀⠀⠠⠶⠞⢷
⠀⠀⠀⠀⠀⢸⣆⡀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠄⠀⠀⢠⢤⢠⡄⠀⠀⠰
⠀⣒⠦⣄⣀⠙⣧⠃⠀⢠⣏⣀⣀⣹⣿⣧⠀⠀⠀⠻⠛⠛⠚⠻⠷⠤⠤⠴⢤⣄
⠀⠈⠁⠈⠋⠙⣿⡦⣄⡀⠐⡀⠉⠉⠙⠛⠓⠶⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣷⠚⠛⠙⢇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣒⣃
⠀⠀⠀⠀⠀⠀⡀⠈⣇⢀⣄⡙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣤⣤⡄⢀⣠⣤⡴⠶⠚⠓⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠠⠳⢶⣿⣿⣿⣧⣴⣴⣶⣶⣆⣀⠀⠊⠀⢀⣸⣉⣿⣿⣿⣯⠟⠁⠀⠠⡤⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠿⠿⠿⠿⠟⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢿⣿⣿⡏⠉⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣤⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣿⡛⣿⠛⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣼⣿⣿⣿⣿⣧⣀⣄⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢠⡄⣋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣚⣀⣀⣀⣸⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⡟⢻⠋⢻⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣦⣤⣤⣤⣀⣀⣀⣀⣈⣉⣉⣉⡉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠛⠛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢖⣚⣫⣭⣭⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⢸⡟⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣀⣤⡤⣿⢸⡇⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠽⠖⠛⠛⢻⣿⣃⣀⣠⣤⡴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣤⡤⣾⣟⣛⣭⣥⡤⠴⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠻⠶⠶⠦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣤⣼⡇⠀⣿⣥⢤⢴⡶⣶⣶⣲⣻⣟⣟⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣷⣶⣶⣦⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠖⢚⠛⣛⢉⡍⠅⠰⢸⡇⠁⣿⠛⠚⣛⣛⣉⣉⣭⣭⣥⣤⣤⡤⠤⠶⠶⠶⠶⠦⢤⣤⣤⣭⣭⣭⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣸⣘⣁⣁⣨⣤⡥⠤⠶⢒⣚⣿⢃⠁⣿⠛⢛⣉⣉⣉⣥⣤⣤⡤⣤⢤⡤⢤⢶⡶⡦⢤⡤⡤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⢸⣯⣭⣭⣥⣶⣶⣶⡶⠿⠿⠟⣛⣿⠀⠄⣿⡯⢹⢹⡇⢸⠄⣿⢸⣇⣿⣸⣇⣸⣸⣇⣇⣸⣇⣇⣸⣸⣼⣿⡿⠷⠶⠲⢶⣄
⠀⠀⣀⣀⡴⠾⠧⠤⠒⠒⠒⠒⠒⢚⡉⢉⠉⢉⠀⡤⠀⣿⠷⠾⠞⠓⠛⠋⠉⠉⢉⣉⣀⣤⣤⡤⠶⠾⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣸⢿⣷
⠀⠰⣏⣉⣁⣐⣂⣚⣀⣃⣘⣀⣁⣈⣡⣬⣤⠤⠴⠶⠒⠻⢦⡤⠤⠶⠶⠚⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠒⠛⠛⢉⣿⡿⠋
⠀⠀⠈⣹⠉⡉⢉⠉⠉⡉⢉⢀⡀⡀⢀⢀⡀⡄⠠⠀⠄⠐⠂⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠴⠶⠒⠛⠋⠉⣉⣀⣀⣴⣶⣤⠤⠶⠖⢺⡿⠁
⠀⠀⠀⠙⣶⣶⣦⡤⠤⢥⠬⣤⠤⢥⠤⠤⠤⠤⠤⠶⠶⠖⠒⣒⣛⣛⣉⣉⣭⣤⡤⠤⠴⠶⠒⠚⠛⠋⠉⠉⢡⡈⢿⡿⠃⠀⠀⠀⣾⠁
⢠⣤⣶⣄⣿⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⠵⠶⠖⠀⢰⡇
⠀⣀⡀⠈⠻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠉⣉⡭⢽⣾⢯⡭⠟⠛⢷⣶⡶⠦⠤⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠶⠦⠶⠛⠒
⠚⠁⠀⠀⠈⠳⢤⣀⣀⡞⠉⠛⠿⣓⠶⠖⠃⢈⡽⠿⠯⣭⣭⡟⠛⠻⢿⣛⠛⠓⣶⠲⢶⡒⠒⠲⠒⡶⢶⣖⠒⠚⢛⣿⠿⠭⣼⡏⠛⢶⣄⣀⣀
⠀⠀⢀⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⠈⠙⠒⠖⠋⠀⠀⠀⠀⠈⠉⠓⠒⠲⠾⠿⠟⠃⡀⠀⠉⠳⠶⠤⠇⠀⠈⠛⢦⣜⣋⣹⠆⠈⣷⠤⣤⣈⠉⠁
⠒⠒⠛⠁⠈⠉⠙⠓⠒⠒⠦⠤⠤⠞⠉⠛⠲⢤⣄⣀⡀⢀⣠⠟⠙⠛⠶⢤⣄⣀⣀⠏⠛⠲⠤⣄⣀⠀⠀⣰⠗⠦⢬⣉⡁⠀⣠⡿⠀⠀⠉⠳⣄
⠀⠀⠀⢀⣠⡴⠞⠋⠉⠙⠒⠢⢤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⣀⡤⠖⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⣀⠴⠛⠯⠥⢤⣀⡀⠀⠀⠈
⢤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⢤⣤⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠾⠷⠖⠒⠒⠒⢢⡀
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢯⢧⠀⠀⠀⠀⠀⡼⣁⡤⠴⠖⣶⡞⠛⠛⠓⠒⠒⢦⣾⠀⠀⠀⠀⠀⠘⡿⢿⣷
⠀⠀⠈⣧⠀⠀⠀⠸⡌⢧⠀⠀⠀⣴⠿⠵⠖⠒⠒⠋⠛⠛⠉⠉⠉⠉⠉⠈⢳⡀⠀⠀⠀⠀⡇⠛⠻⢷⣶⣤⣄⡀
⠀⠀⠀⢸⢳⡀⠀⠀⣇⠈⢧⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠚⠙⠛⠉⠉⠉⢳⡀
⠀⠀⠀⢸⠀⠳⡀⠀⢹⠀⠈⢧⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢸⠀⠀⠹⡄⢸⠀⠀⠈⢧⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⡾⠀⠀⠀⠹⣼⠀⠀⠀⠈⢧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⢧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⠤⢤⣠⠤⣄⣀⣀⠀⡼
⠀⠀⣸⣀⣄⣀⡀⠀⠀⠀⡀⠀⠀⠀⠈⢣⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⢸⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣠⣤⠄
⠀⠀⠀⠀⣤⢤⣿⣏⣓⠲⠏⠙⣶⠦⣄⣀⢳⠷⢤⠀⡴⠤⠤⠤⢤⣤⣀⣀⠀⡟⠀⠀⢀⣸⢸⣀⣤⡤⠤⠤⢤⡶⠞⠛⠋⠁
⠀⠀⣴⠶⠟⠀⠀⠀⠈⠉⠉⠧⠿⠧⣄⣈⣙⣃⣸⠀⡇⠀⠀⠀⠀⠀⠀⣸⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠉⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣇⣀⣀⡀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⢸⣦⣄⣹⠉⠻⡇⠀⠀⣸⣶⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣧⣄⡀⠀⢀⣿⣿⣿⣷⣤⡀⠀⡰⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⣸⣿⣿⣿⣿⣿⣿⡆⢳⣦⣤⡃⠙⢿⢠⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠟⠛⡛⠿⡿⢜⡛⠻⠿⢿⣿⣿⣿⣷⢨⣿⣿⣿⣶⣦⡀⢷⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⡗⠀⠁⠈⣿⣶⣤⣿⠂⠉⠻⣿⠚⠛⢿⡿⣿⣿⣷⣜⣿⡄⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄⡹⣷⣦⣼⡇⠈⠹⣿⣿⣿⣿⡄⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⡙⣿⣿⣿⣿⣆⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠎⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠿⠿⠿⠇⠀⠀⣸⣿⡿⠿⢿⠛⠛⢿⣿⣿⣿⣿⡿⠿⠿⣿⠁⠀⠈⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣧⣴⣤⣭⣤⣄⣀⣸⣀⣀⣈⣛⣉⣉⣠⣦⣤⣤⣥⣤⣤⣤⠴⠒⣽⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠴⠿⢛⡛⠻⠿⠟⠛⠛⠿⠛⢛⡛⠻⠿⠿⠛⡛⠿⠿⠿⠛⣛⠋⠀⠀⠒⠒⠒⠒⠒⠒⢒⡀
⠀⠀⠀⠒⠀⠀⠀⠁⠀⣀⠤⠤⠀⠀⠀⠀⠒⠋⠃⡀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠐⠂⠀⢀⡤⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⡄⣰⠏⠷⢦⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠛⠉⠁⠀⠀⠖⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⣀⣀⠀⠀⠀⣀⣠⡷⠔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢸⣿⠉⠉⠀⣐⣫⠅⠰⠞⠛⠋⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠘⠋⠉⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣦⣤⣄⡀⠀⠀⠀⢀⣀⡠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⣶⡶⢦⣄⣸⠟⠋⢉⣭⣿⣧⣝⣦⣀⡀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⠀⣿⠟⠁⣀⣿⣧⣤⣽⣿⠟⠉⠉⠉⠀⠈⠀⠟⠿⠋⢐⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢻⡿⠁⢀⣀⣤⠟⠚⠋⠉⠁⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡴⠶⣿⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡁⠀⠀⠻⢗⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⢀⣀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⠛⡶⠒⠁⣠⡼⠁⠀⣆⡀⣀⣀⣤⣄⣠⣄⡈⣿⠿⠛⠿⢿⣿⣶⣤⡀⠀⢸⣟⣿⠀⠀⠀⣿
⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⢸⣿⣧⡮⢠⣴⣞⣿⠀⠀⣰⡿⠟⠉⠻⣭⣿⣿⣿⠻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⣿⡀⠀⠰⣿⡄
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠘⣿⣿⣵⣿⣷⠿⠷⣀⣾⠥⠴⠖⠶⠛⠛⠛⠛⠻⣿⣿⣿⠟⠻⠛⠛⠉⠛⠛⠿⣿⣿⣿⡇⠀⠰⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⠻⣿⠏⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⡂⠀⠀⣾⠀⠀⡀⢐⣀⡀⠀⠀⠀⠀⠀⣼⣿⣇⣄⣀⠀⠀⠀⠀⠰⠂⢸⣿⣿⣿⣀⣤⣿⣿⣦⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡈⢷⣶⡿⠿⠿⠿⠷⢶⣦⣄⠀⢻⡿⠿⠿⣿⣷⣦⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣶⣿⣀⠀⣠⣄⡀⢸⣿⠻⣷⣌⠀⠀⠀⠈⢸⣿⠻⣷⣄⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣄⣀⡙⢷⣤⡀⠀⢸⣿⠀⠈⣝⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⠿⣋⣵⡿⣿⣟⢷⣿⣭⣟⡻⢿⣶⣬⣭⣦⢸⣿⣦⣤⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠿⣿⢻⣯⣛⣧⣉⠀⢻⣿⡝⣿⡾⣶⣿⣭⣃⣛⡘⠿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣿⣦⣬⣽⣿⣿⣿⣿⣮⣧⣸⡿⢀⣻⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣉⠐⢦⣸⣿⣿⠉⠳⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠨⠤⣼⠒⠀⠉⠉⠉⠄⠃⠐⠀⠺⠀⠑⠋⠙⠦⠙⠛⠿⠿⠕⠂⠛⢉⢀⡥⠀⠉⠖⢾⡉⠫⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⠿⠻⠿⠛⠟⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⡇⠀⡿⠛⢷⣏⠀⠾⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢂⠀⠀⠀⠀⣼⠀⠀⠙⣯⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠄⠀⠙⢦⠀⢠⡇⠀⠀⠀⢨⣿⣄⠀⠀⠀⡟⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣰⠿⠞⢀⠀⠀⠀⠁⢼⠃⠀⠀⠀⡼⠁⠀⠳⠄⢸⡏⠀⠀⡟⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀⠰⣯⠉⠐⠶⢌⡦⡀⠀⠀⠀⠑⣄⣠⠞⠑⠂⠀⠀⠀⠻⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⠀⠠⡀⠀⠀⠉⠄⣤⣴⣞⣋⠁⠾⠖⠄⠂⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠈⠳⢄⠀⠀⣼⡍⠻⣿⣿⣷⡆⠠⢍⡂⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢠⣿⢠⠀⠀⠀⠀⠑⢾⡇⢸⠖⠉⢻⣿⡷⠀⠀⠈⣷⢄⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠄⢎⣰⢂⡧⣀⡀⠀⠀⠀⠀⢠⣹⡇⢲⠆⠀⠈⠻⣄⠀⢰⡿⠀⠈⠂⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠊⠠⢡⣿⢇⣙⠃⠀⠀⠀⠀⣼⣿⢇⠙⠀⠀⠀⠀⠀⠳⢾⡟⠀⠀⠀⠈⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⢏⡉⠁⠀⣸⡟⠀⣏⠑⠂⡦⠀⣰⣿⢏⠂⠀⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣴⠛⠂⠀⠀⠐⠾⢹⣮⡿⡓⠦⠟⡤⣂⣜⠃⡀⠀⠀⠀⠀⠀⠓⢄⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠃⠀⡻⢾⣏⠀⣽⣯⣭⣤⣈⣀⣊⡴⠿⣀⡗⠼⣋⢀⠄⠀⠀⠀⠀⠑⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⢦⣤⣻⢷⣿⠙⠻⣦⣉⠻⠛⢻⡷⣶⠠⣀⡃⠌⢿⢠⣟⡄⠀⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢠⡆⠀⠀⠉⠛⠻⣿⣷⠀⠀⠈⠛⠻⢦⣝⡃⠀⠀⢸⡗⠀⡀⠸⣧⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⣸⠧⠀⠀⠀⠀⠂⢈⣿⠀⡀⢀⠀⠀⠀⠈⠛⠳⠦⣼⡁⠀⠀⠳⣍⠃⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢰⣿⢦⡆⠀⠀⠀⢈⡏⣽⡇⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣦⣀⠙⣇⠲⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠺⡸⡆⣱⡿⣿⡏⠿⣠⠂⢂⡞⢰⣿⣷⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⣿⣿⡁⠀⠈⠓⠠⣀⠀⠀⢀⡠⠛⠁
⠀⠀⠀⠀⠀⠀⢀⠊⠁⢙⢮⡿⣽⣿⡹⢻⣼⣹⠿⣡⣿⣿⣿⠀⠀⢺⠀⠈⠀⠀⠀⢀⠀⠀⣤⡆⢈⠿⣿⢼⡆⠄⢸⡄⠀⠙⠒⠿⡁
⠀⠀⠀⠀⠀⠰⠆⠤⠤⠤⢾⣿⣻⣿⣟⢾⣧⡿⠂⣱⣿⣿⣿⢰⡆⢸⡆⣸⣶⢀⡄⠘⠁⠀⢼⡇⣾⠀⣿⣾⡿⠇⢸⢁⡀⢠⡄⠀⠈⠓⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣱⠛⠣⠄⢟⡛⣂⢲⣟⠿⣿⣿⡼⠿⠺⡇⢺⡇⠸⡀⠷⠀⠠⢻⣷⣃⣼⠟⠛⠃⣀⣊⠈⣥⡟⡅⠀⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⢶⣴⣶⣾⣧⡔⣶⡟⠋⠉⠉⠿⣽⣳⢾⠇⢛⣀⣤⣶⣤⣶⠶⠶⠚⠛⢻⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠘⠁⢂⠐⣃⢀⣴⠇
⠀⠀⠀⠀⠀⠀⣸⠏⢚⣹⠻⠻⢿⡿⣿⢶⣶⣾⡞⠀⢂⣷⢚⣩⠿⢟⣋⡥⣴⡞⠛⠛⠛⠉⠉⠛⢫⣹⣟⣻⡟⣹⣷⡀⠀⠀⠀⠈⠌⣿⠋
⠀⠀⠀⠀⠀⢠⡏⢀⣘⠭⠉⣀⠀⠀⠿⠿⣿⣛⣿⣚⣏⣻⣵⣲⡌⠛⠈⠁⠀⠀⠐⣈⣁⠐⠈⠒⣲⣿⣿⡿⢡⢿⡿⣷⠀⠀⠀⡀⠞⠁
⠀⠀⠀⠀⠀⠘⣧⠀⢥⣀⠀⠲⠤⠌⠁⣚⠋⢡⣺⣷⢿⣿⠛⠟⠋⡀⠀⠀⠀⠀⡉⠤⠤⠤⢤⣬⣭⣽⡟⣡⡿⣄⡿⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣄⠒⠠⢤⠉⡑⠒⠠⠔⠀⠬⢡⣄⣻⠈⠷⣤⡀⠀⠐⠀⠀⠀⠀⠒⠶⠦⠤⣽⣿⢯⣾⠏⡳⠈
⠀⠀⠀⠀⠀⢀⣟⣿⡉⣑⡀⠒⢬⣭⡤⠉⢁⣋⣀⠀⣭⣾⢿⡶⠄⠐⢋⡀⠀⠄⠐⢐⣲⣖⣶⡿⣳⡿⢃⠜
⠀⠀⠀⠀⠀⠸⠶⠟⠓⠹⠿⣿⣒⠲⠖⠒⠈⠬⣅⣀⣴⠶⠟⠛⠚⠷⢟⣢⣴⣮⣭⡥⢶⣾⣿⣱⡏⢱⣮⡄⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠙⠛⠋⠁⠉⠉⠁⠀⠈⠉⠛⠉⠙⠛⠻⠿⠿⠤⠀⠘⠿⢯⣿⣯⣽⠕⠛⠛⠛⠛
⠀⠈
⠀⢸
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⢲⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡶⠦⠤⣤⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠢⠈⠹⠉⠛⠝⢛⠒⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣀⢸⣶⡤⡖⡀⣀⡀⡠⠚
⠀⠀⠀⠀⠠⢠⣤⣤⣄⣴⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⡐⠃⠀⠀⠰⠿⣾⠿⣷⣶⣷⡇⡿⠆
⠀⠀⠀⠀⢸⣽⣿⣟⣿⣭⣽⣹⣓⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣦⣄⣀⣀⣤⣴⣻⠇⠀⠀⠀⠐⣶⣿⣷⣿⣿⣿⣿⠷⠄
⠀⠀⠀⠀⠀⠈⠻⣿⣿⢻⣿⠿⡏⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⡿⠿⠿⠿⣿⠟⠁⠀⠀⠀⠀⠀⠹⢿⣿⢿⡟⣫⡝
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣇⠀⢀⣆⡀⠀⠀⠀⠀⠉⣴⣤⡄⣸⣇⣄⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣥⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣾⣷⣦⣤⠄⣀⣴⡿⠿⠇⠛⠃⠛⠂⠀⠀⠀⠀⠐⠉⠁⠀⠀⢀⣤⣾⣿⡧⠀⠀⠀⠠⣄
⠀⠀⣠⠄⠠⠄⠠⠄⠀⣹⢿⣿⣿⣿⣽⣿⣧⣾⣭⢤⣴⣀⡠⡤⣤⣄⣤⣤⣡⣈⠡⠄⠀⣠⣤⣶⣿⣿⣿⠟⠀⠀⠀⠸⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠾⣃⡴⢟⣯⣽⡿⠿⠿⠿⠟⠛⠛⢛⣛⣙⣩⣷⣶⣶⣶⡿⠟⢓⣛⣛⣋⣭⣭⣉⣁
⠀⠀⠀⠀⠀⠀⠈⠁⢁⣼⣶⡿⠟⠉⠀⠤⠶⠶⠻⠛⢛⣩⣽⣿⣿⣿⣿⣿⣿⣶⣿⠟⣋⣭⣷⣾⣿⣿⣿⣦⡀⠐⠂
⠀⠀⠀⠠⠄⠤⠀⠀⠾⠯⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠉⠿⠛⠟⠛⠉⠉⠉⠉⠙⠻⠵⡾⠿⠛⠙⠿⠿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣄⣭⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣇⣅⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡉⠀⠀⠀⠁⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢼⣇⣀⡇⣤⣄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡆⠀⠉⠙⡟⠲⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⠡⠀⢰⡇⢀⣰⣿⡁⠀⠀⠀⢹⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠉⠙⠛⠿⢿⣿⣾⣿⣿⠉⠛⠛⠤⢸⣞⡆⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠈⡏⠉⠀⢳⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⡀⢀⣀⢀⣀⣠⣴⠏⢀⢀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣴⣫⣐⣑⣺⣿⣿⣿⣿⣿⣧⣤⣴⣬⡄⠀⢀⢀⣼⢁⡼⠛⢂⡤⠊
⠀⠀⠀⠀⠀⠀⠐⠦⢭⣽⡛⠻⢷⡿⠁⠫⣯⣟⣟⣿⣷⣶⣶⣿⣿⣿⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠀⠀⣀⣶⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠀⠀⠀⢀
⠀⠀⠀⠄⠀⠀⢀⣤⠤⣃⣠⣴⣶⣿⣿⣿⢿⣼⠉⠿⣿⣿⣿⣿⡿⢩⡖⠂⠀⠀⢀⠎
⠀⠀⠀⠀⢦⣀⣹⣿⣿⡿⢿⣿⣿⣿⣟⡷⢿⡏⠀⠀⠈⡿⣀⢤⣿⣈⣅⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠈⢉⠉⠈⠈⠭⣵⣿⣿⣧⣍⠻⢮⣁⠐⢾⣿⣿⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣯⣭⣿⣿⣿⣿⡇⢰⡝⠦⣄⣺⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠻⢿⣦⣁⠘⠻⢷⡦⠬⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⣀⣀⣀⠀⠀⠀⢰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⡿⠿⣿⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⠈⠙⢃⢻⣿⣿⣧⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢠⣸⣇⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⣾⣀⣼⣦⣘⣿⣟⣁⣀⣀⣀⣼⣷⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠁⢶⡟⠉⣿⡟⠈⠉⠻⠛⢻⣿⠿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⡇⠀⡏⣿⠀⠐⠀⠀⠸⠙⡇⠀⠈⠙⠆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣴⣤⠀⠁⢼⣷⣀⠁⡟⠄⠀⠀⠀⡂⣅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡗⣦⠄⢱⠹⠀⡏⠀⠀⣰⡖⠇⣸⢘⣋⣴⡗⢤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⢤⡀⢺⡧⠀⡇⢀⠈⠁⣴⣶⣿⣿⣿⣿⣷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⣿⣿⣦⣧⢤⣀⣀⡼⢿⣷⣾⣿⠿⠿⠿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠻⠶⢶⣛⣂⣀⠀⠂⠐⠀⠁⣈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡎⠘⠀⠊⢩⣍⠁⢠⣤⠀⠐⠆⠈⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠰⣆⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⡿⠿⣤⣴⡘⣁⡆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠁⠐⢋⣏⣟⣽⣿⣽⡿⣿⣿⣿⠛⣁⠂⠈⠃⠀⠄⠀⠂⠑⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠛⠋⠁⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠉⠀⠀⣠⠆⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⠘⠛⠈⠉⣏⣁⣀⣆⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠄⣤⣤⣤⣀⡞⠋⠉⢹⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⢟⣨⠟⠉⠉⠉⢉⣧⣤⣤⣸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⣿⠎⣰⣆⣀⣀⣀⠈⠏⠉⠁⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡃⠀⠀⢘⠋⢠⢤⡬⠿⠿⠿⠿⠗⢦⣾⣄⣹⣭⣄⡤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄⠀⠀⡮⡀⢰⣇⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠬⣿⢷⣤⣿⡟⠋⡁⣿⠿⣶⡤⡄⡶⠶⢢⠄⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡋⠛⠿⢋⣽⠻⣂⠹⢤⡸⣇⣤⣼⣶⣤⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣷⣶⣿⡿⣿⣷⣶⣦⣽⣥⡤⣬⣭⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⢄⡁⠠⠜⢁⣀⣀⠉⠛⠫⠍⠁⢰⣌⣉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠉⠉⠈⠒⠀⠒⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠀⠀⠞⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⢩⠄⢹⠀⠀⠀⠈⠋⠉⠉⠉⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠴⣧⣤⠀⢠⠾⢧⡄⠀⠀⡆⢀⣀⡠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣄⣸⣇⣘⣲⣟⣁⡐⡾⠧⠤⠼⡧⣤⣤⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠶⢿⡍⠉⠉⠉⠉⠉⠉⣻⡆⠀⠀⠀⠀⠈⠝⢷⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣅⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣰⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣎⣠⣤⢶⣶⣶⠚⠛⢿⠇⠀⠀⣀⣀⣠⣤⣴⣿⠀⣰⡏⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠲⣿⣿⠙⠛⠺⠿⠶⣶⣶⣮⣶⣾⣿⣿⠛⠀⠀⠉⠃⢀⡿⠀⠀⠰⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⠾⠷⣾⣷⣶⠀⠀⠀⠀⠈⠛⢿⡍⠋⠛⠛⠛⠻⠿⢿⣟⡽⠀⠀⢠⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡼⣿⠆⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢨⣿⣇⠀⠠⠃⠀⠀⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⢃⡿⠀⠀⠀⠀⠀⠀⢸⣹⡇⠀⠀⠀⠀⢀⣾⡿⢙⡔⠁⠀⠀⣠⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⡴⠟⡙⠀⢀⣠⡾⠁⢀⣀⣤⣤⢴⣾⣿⣿⠁⠀⠀⠀⢀⣾⣟⣡⣊⣤⠄⠚⠛⠛⠛⢣⠀⠀⢀⣦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡷⢄⡰⠓⢸⣿⢿⣶⣿⣿⠉⣿⣿⡄⠀⢀⠃⣀⣤⣴⢶⣿⠛⠋⠁⠀⠀⢠⡀⠒⠂⠤⣬⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣼⣿⣇⣲⣿⡇⣸⣿⣿⣇⠀⣿⢻⣇⠀⠊⣽⣿⠈⣿⣸⣿⣬⠤⣾⣷⣶⣿⣷⢶⡴⢞⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⢹⣿⡟⣽⣭⢻⣿⣿⣤⣿⣱⣿⣤⣤⣛⣛⣌⣩⠅⣦⣴⡆⢻⣿⣿⣿⣾⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡶⠬⢅⣙⠙⠓⠷⠹⠏⠿⠹⠯⠯⠿⠋⠛⠀⣛⣀⠥⣿⣷⣜⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡤⠀⠬⠉⠉⢒⣒⠒⠒⠒⠚⢚⣉⣉⠩⢠⣔⣾⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣠⣤⣤⣽⣛⡿⣾⣯⣶⣶⡶⢶⣶⣖⣒⣒⣲⡶⠿⢿⣿⣿⣿⣿⢿⣿⣧⣤⣤⠤⠶⠒⠶⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠙⠛⠯⢭⣛⡛⠿⠷⠶⢶⠶⣛⣛⡳⢾⠯⠿⠶⣿⣛⡛⠷⠾⢵⣶⡟⢛⠷⠤⠈⠉⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠠⠄⠉⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢣⡖⠐⡂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠘⠆⠉⠀⠀⠀⠁⠀⠘⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠙⢉⠀⠀⠀⠀⠀⠀⠀⠂⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢘⢗⠄⠄⠐⠊⠃⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣙⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⠶⠀⠂⠀⢀⡴⠁⡀⠈⠃⠀⢀⠀⠀⠀⠀⠃⠈
⢄⠀⠀⠀⠀⠀⠀⣶⠘⠁⠀⠄⠘⢀⠂⠉⠃⠀⠀⠀⠀⠡⠀⢀⠼⠜⢂⠀⠠⠀⡀⠀⠀⠀⠢⠁⣄⠀⠁
⠀⠁⠀⠀⠀⠀⢰⠹⠄⠀⠀⠀⠀⠆⠀⠀⠀⢠⠠⠀⢘⡄⢠⠂⠂⠀⢻⡁⡈⡀⠉
⠀⠀⠀⠀⠀⠀⢸⠾⠂⠀⠀⠀⠀⠘⡀⠋⠀⠀⠀⠀⠄⠀⠘⠀⠄⠀⠠⠈⠈⠠⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠠⢀⠂⠈⢘⢀⠁⠰⠀⠀⠀⠀⠩⣏⡁⢀⡄⠖⠔⠃⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠇⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⡤⠈⠀⡀⠋⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠚⠁⠄
⠀⠀⠀⠀⠀⠈⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠈⠂⡄⠀⠄⢣⠀⠀⠅⣀⢀⡅⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠠⠀⠀⠂⡈⠨⠀⡀⠀⠀⠁⠀⠀⠀⠁⣠⢃⢖⠁
⠀⠀⠀⠂⡅⠀⠁⠀⠙⣂⠚⠀⠐⠠⠄⠂⢈⡀⠂⠀⠀⠀⠀⠀⢠⠈⠀⣀⣼⢿⣟⡈⢂⠀⢀⠠
⠀⢠⣆⣼⣷⢠⡄⠀⡀⣰⡁⠃⠠⠄⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼⣿⣿⡽⠋⠀⢸⡄⠉
⠀⢀⠀⡀⠀⢘⡀⠂⠰⠄⢄⠂⣀⠀⠀⠀⠨⠩⠀⠀⠀⢀⡰⢂⡍⢀⡹⠋⡡⠖⠀⠀⡹⡇
⠀⠸⠀⠂⠀⠸⠁⢨⠁⣐⠀⡁⠘⠃⠄⢡⠘⠀⣐⡆⣀⠌⠱⠖⠊⣡⠄⠀⠀⠀⠠⠊⢸⠁
⠀⠈⠉⡽⠙⢓⠀⠀⠀⡍⠀⢈⠈⠁⠋⠀⠀⠉⡉⠁⠀⡂⠤⠐⠘⠀⠀⠀⠀⠠⢂⢀⠋
⠀⠀⡀⡅⡷⠐⢄⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠆⠤⠶⠔⠋
⠀⠀⠄⠋⢹⣒⡂⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡨⠉
⠀⠀⠀⠒⠘⣷⢍⡭⣤⣐⣂⠒⣂⣀⣀⣤⡤⢀⡂⠚⠉
⠀⠀⠀⠐⠂⢉⣙⠻⢿⣧⠿⠛⡁⠋⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⣴⠰⣾⣾⡇⠆⢀⠤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠈⠇⠠⡻⣺⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠀⠐⣠
⠀⠀⠀⠀⠀⠈⠑⠿⠷⡇⠀⢰⠀⠀⠀⠀⠀⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⢴⣤⣼⠄⢸⠖⢿⣶⣠⣤⠆
⠀⠀⠀⠀⢸⣦⣀⡀⠐⠀⠀⠀⠠⣀⣠⡄⠈⠙⣿⣿⣶⡄⠀⠀⠀⠀⠀⣼⣿⣿⣶⣭⣟⣂⣉⣸⣬⣿⣿⠏
⠀⠀⠀⠀⢾⣿⣿⣿⢷⣤⣄⠀⠘⣻⡿⣷⣠⣾⣿⣟⣻⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⢿⣛⣿⠻⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣵⢿⣿⠛⣶⢶⡀⣿⣼⣿⣿⠛⢻⣷⣖⠠⣄⡀⠀⣿⢠⠏⣾⠼⢻⣭⣶⣾⣿⠿⠇
⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣷⡾⠿⣭⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣾⣢⣴⣜⣿⡆⢿⣷⣿⣿⣏⣠⣦
⠀⠀⠀⠀⠛⠻⢿⣿⣿⠿⣷⣿⣄⣸⣿⣿⣿⡿⠛⠈⢉⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣶⡶⠄
⠀⠀⠀⠀⢲⣷⣶⡿⣅⠀⠰⣶⣯⣿⣿⡫⣭⣶⣿⣿⣿⣿⣯⣿⣟⣙⣛⡻⣟⣩⣭⣭⠭⠀⠀⠲⠿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠜⠻⠿⠿⢟⣛⠛⠛⠻⠿⠿⠿⠿⣿⠿⣿⠉⣉⡉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠂⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⣤⠶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠂⠀⠢⠄⠄⠀⢰⡆⢰⣄⣠⣾⢟⢀⣄⡀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠒⠉⣷⠀⠀⠀⠀⠀⣤⠀⠀⢹⣷⢞⣿⣿⣿⣿⣿⣿⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⣐⣿⠷⣒⠽⢚⣽⣿⣿⣿⣿⣿⣯⣾⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣯⣥⣀⣀⣀⣀⣀⣤⣤⡴⠷⠶⠛⠋⠉⠀⢀⣤⠞⠉⣽⡿⣿⣿⣿⣿⣿⢿⣾⣿⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣻⣤⣤⣤⣭⣭⣬⣭⡉⠁⠀⠀⠀⠀⠀⠀⠀⢐⡾⢁⣤⣀⡄⠀⠈⠙⠛⡉⠛⠁⠈⣻⣧⣤⣾⣿⣷⡆
⠀⠀⠈⠀⠀⠈⠀⠈⠓⣋⣉⣛⣛⠛⠛⣿⣿⣿⡿⠿⢿⣭⣷⣶⣶⣶⡶⣶⣦⣆⣬⣽⣶⣾⣿⣿⣾⣼⣧⣾⣿⣇⣶⣿⣷⣌⢻⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠠⠿⣿⠷⢿⣿⣿⡿⠿⠛⢛⣛⣩⣤⣷⣶⣿⣿⡿⠛⢁⣀⣀⠶⣿⣯⡉⠁⠀⣈⣛⡛⠿⣿⢿⣿⣿⣿⣿⣿⣿⣦⠉⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣶⣶⠾⠿⠛⠛⠻⠓⠿⠿⠛⢉⠀⡀⣙⡻⢿⣿⠆⠀⠀⠀⠀⢘⣿⣿⣤⣤⠀⠉⠉⢹⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣠⣤⣤⣐⠿⡿⢿⣿⣿⡿⠿⠓⠂⠀⠀⠐⠀⠘⠿⠛⠛⠛⠋⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⠛⠿⠿⠻⠛⠃⠀⠈


⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠃⠒⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣁⣀⢹⡄⠉⠀⠉⠒⢖⠢⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⣿⠇⠀⠀⠀⠀⠈⠆⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⢲⡒⠛⠻⠿⠯⠿⠯⣁⣲⡶⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠧⠴⢿⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣀⣼⠄⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⣦⠀⠀⠤⣌⠽⠯⠶⠶⢶⣶⣶⣷⠒⡒⠂⡿⣆⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡆⠀⠀⢸⣧⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠟⣿⡄⠀⡀⠀⠢
⠀⠀⠀⠀⠀⠂⢀⡀⠀⠁⠀⠀⣷⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠹⣿⠀⠨⣤⠶⠋
⠀⠀⠀⠀⠀⢸⡆⡇⠀⠲⡄⢤⣿⠀⣀⣼⣿⡿⡇⠀⠀⢀⣀⣠⣤⣀⣀⣱⡀⣿⡷⢾⣁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣧⢊⣩⣟⡗⢀⣷⢞⠋⣿⣃⣿⣠⣿⣷⣾⣿⣇⣲⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣯⣿⣿⠿⣿⣿⣿⣇⣿⣠⣼⣿⣿⣿⣿⣿⣿⣾⣿⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠉
⠀⠀⠀⠀⠀⠀⠀⢨⣭⠴⠶⣶⣂⣤⡭⠭⠭⠤⢤⣬⣭⣉⠁⣤⣤⣀⣁⣤⣭⣤⣵⣦⠶⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡒⠉⠉⠁⣐⠶⠶⠿⠃⢈⣁⣤⣭⣽⣟⠋⠉⠉⣭⣽⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠉⠁⠀⠠⠤⠉⠉⠋⠉⠀⠀⠀⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡤⠪⠃⠀⡠⢄⡠⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠉⠰⠅⢀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠒⠂⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⣹⣅⠀⠀⠀⠀⠀⠑⡆⢀⠀⠀⡠⠂⠠⠃⢤⠀⣠⡪⠔⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣎⠁⠘⡀⢂⣠⠶⡰⣎⡔⡡⠝⠻⡅⣖⢠⣈⠴⠚⠁⠀⠀⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⡶⠞⠒⠋⠀⠀⢱⢹⣸⠆⠀⢀⡧⠚⠉⠀⠀⠀⠀⠀⢰⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠔⠙⡆⣸⠶⠊⠁⠀⠀⢊⡇⡄⠀⠀⠀⡼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡂⠤⣀⠔⢿⡇⠀⠰⠋⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢂⠊⢀⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣀⢌⠁⠚⠀⠈⡇⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠤⣰⠃⠀⢸⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢙⡄⠘⠀⣧⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠽⠙⠀⠀⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠫⢀⣰⡏⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡏⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠱⡌⠙⠲⣄⡏⠀⡜⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈⡆⠈⠑⢄⡀⡹⠀⠀⠈⠳⣰⠟⡌⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠈⠙⠒⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠸⢋⠏
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⡤⠖⠰⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⢳⡢⢄⡀⠀⢰⡧⠐⡾⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠲⣌⠑⣿⠀⡀⠃⠁⠀⠀⠀⠀⣠⡀⠤⠄⠀⠀⢴⡋⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠈⠗⢯⡀⣹⠷⣄⠀⡀⠀⠀⣿⢀⣀⡔⠈⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠄⠘⠓⢄⡁⢰⡟⠃⠸⠥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⠳⣄⣴⡏⢤⠄⣠⡋⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠈⠑⢾⡅⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠈⣿⡟⢦⣄⡼⠁⠀⠳⢤⣀⡴⠋⠀⡀⠀⠀⠀⠀⠀⠙⠲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣿⡇⠀⠈⠿⢦⡀⠀⣾⣿⣛⠒⠶⢆⡀⠀⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣾⡇⠀⢰⠀⠀⠹⡶⣿⡇⠉⠳⣤⡀⠈⠙⢦⣀⠀⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢠⣤⣿⠀⠂⠸⠀⠀⠀⠃⣿⠁⠀⠀⠇⠉⠲⣄⢠⡿⠳⣅⠀⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⢤⣄⣀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠙⠳⣄⠈⠳⣍⠀⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠳⣤⡉⠙⣶⢏⠀⠀⡀⠀⠀⠀⢀⣾⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣙⣆⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢙⡶⠃⠃⠀⢠⠀⠀⠀⠀⡼⠁⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠏⠻⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⣏⣁⡑⠀⢠⠎⠀⠀⡀⢾⣷⡶⢦⣤⣴⣁⠀⠀⢀⠂⠄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡿⢮⣍⡙⠛⠲⣤⣔⠈⡼⣇⠙⠳⣄⡀⠙⡙⢶⣼⡃⠀⠀⣠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠸⣇⠀⠈⠙⢶⣤⣄⠈⣻⣿⣿⠀⠀⠘⠉⠳⣄⡞⢸⡟⠦⣀⡄⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠰⡄⠉⣻⣿⣿⣇⠀⠀⠀⠀⠀⠈⠙⠳⣇⡀⠈⠣⡀⢀⠇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀⠀⠀⣿⠀⠀⠀⠀⠛⣠⣿⣿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣄⡀⢾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠠⡁⠀⢠⡏⠀⠀⠀⠀⢠⢷⢧⠏⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⣟⠶⣧⡼⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡑⠀⠀⢸⠇⠀⠀⠀⢠⣏⣏⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠛⠿⣧⠀⠀⠀⠀⠀⠀⣨⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣋⣁⡀⠀⠂⢀⣿⠈⠀⠀⢠⠟⡞⡰⠀⢀⡇⡀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢈⣸⡇⠀⠀⠀⠀⠑⠢⣄⠀⣠⡾⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠉⠙⠓⣾⠃⠀⠀⣠⡯⡽⣡⠃⠀⣼⠣⠁⠀⠀⠀⠀⢀⠃⠀⠀⡄⠘⣿⢁⠀⠀⠀⠀⠀⠀⠀⠛⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⣰⠏⠐⠈⢡⣟⣸⢳⠇⡀⣼⢏⠓⠀⠰⠃⣀⣀⣌⣀⣀⣰⣁⣻⠏⠏⠀⡰⠀⠀⢠⠀⠀⠀⠀⠀⠉⠑⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣼⣋⣀⣀⣠⣿⣬⣧⣽⣴⡜⢃⣡⡤⠶⣟⠋⠉⠍⣻⣿⣁⠈⣽⢿⣭⠿⣿⣉⣛⣲⡶⢤⣄⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠿⣿⣛⠭⣉⣀⣼⣰⠇⡴⢠⣿⠞⠉⢉⣉⣻⣷⣶⡞⣛⣯⣽⡟⢛⣛⣛⣛⡛⠳⠿⣽⠟⣷⡀⠈⠑⣤⢇⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠲⠽⢟⣻⡾⠿⢿⢷⡶⣿⣿⣟⣻⠿⢽⡛⣯⣁⠀⠈⠉⠞⠛⣩⡭⢭⣭⣭⣙⡳⣩⣿⠿⣿⠀⠀⡸⠈⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⡒⠠⠍⣉⣉⢩⣍⢩⡍⢡⡧⠰⠗⠚⠉⣡⡤⠶⠀⢒⡺⢭⣁⣀⣀⣀⣉⡙⣱⣿⠍⢲⠊⢀⣴⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣆⢈⠑⠒⠲⠶⢤⠀⣀⣀⣴⣂⡒⠛⠋⣉⡀⡄⠀⠀⠨⢟⡊⡁⠤⣤⣍⣽⣷⡿⢦⡼⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡤⠴⠓⠛⠿⣄⣈⠉⣶⣶⣦⣤⡬⣔⣲⣴⣚⡛⢁⣀⣀⣤⣖⠒⠓⢚⣛⣛⡶⠿⠿⣿⣿⣧⠾⠖⢒⣒⣂⡀⠄
⠀⠀⠀⠀⠀⠈⠛⠛⠒⠂⠀⠀⠈⠽⠟⠛⠛⠛⠃⠀⣀⣀⣽⠿⠛⠋⠉⠈⠉⣻⣿⠿⠿⢭⠉⠁⠀⣉⣠⡤⠤⢤⡌⣉⡉⠀⠀⠠⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⣿⠀⠑⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⢒⠳⠤⠚⠁⢸⡇⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣽⣿⣿⣿⣿⣿⣿⣟⣛⣟⢛⣟⣿⣿⣾⣟⣓⣛⣛⣒⣒⣒⣛⣛⣒⡒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣴⣾⣛⣛⣛⣛⣛⣿⣛⣛⣛⣛⣡⣾⣿⣿⣟⣛⣋⡉⠉⣿⠋⠙⠛⠻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⠭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⣍⣽⣟⣋⣉⣉⣉⣉⣉⣉⣁⡀⢿⣀⣀⣀⠀⠀⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣶⣶⣶⣔⢶⣶⣶⣶⣾⣿⣿⣿⣯⠀⠉⢭⣭⣭⣭⣭⣍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠙⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣠⠤⠛⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⡏⠀⠀⠀⢀⣤⢞⣫⡵⢣⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⠀⣠⡶⣫⣴⣿⠟⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣳⡃⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⠇⣿⣿⡟⠀⣠⢞⣥⣾⣿⣿⠏⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⠟⣠⢞⣵⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣭⡍⠀⠀⠼⢿⣿⣿⣿⣿⣿⣿⣿⢃⡿⣫⠞⢁⣋⣭⠤⠴⠶⠀⠈⠹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣓⣿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⠿⣓⣫⣽⣶⠿⠟⣽⠃⢠⠀⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣞⣇⣴⡿⠟⠛⠋⢹⣿⠁⣀⠭⣒⣭⣶⣿⠿⣛⣭⣶⣾⢻⡏⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣧⢿⠋⠀⠀⠀⠀⠀⣸⢟⣪⣵⣾⣿⣿⣭⣶⣿⣿⣯⣽⡇⡾⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣽⠭⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⠃⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⣭⣤⣶⡖⢒⣒⠒⣒⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣽⣿⣴⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⠋⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡱⠃⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠞⠁⢀⣾⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠑⠁⠠⠾⠟⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⣉⣀⣀⣀⣀⡀⠀⠀⠘⠛⠛⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠈⠉⠁⠐⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⣻⣷⠖⠚⠻⣶⠤⣄⣀⢸⠛⠛⠛⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠁⠀⠐⠒⠀⠀⡸⠀⠉⠉⠒⠛⠤⠤⢙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠴⠶⠶⠐⠶⠶⠒⠒⠚⢀⣇⣄⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⣠⡴⠛⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⢤⡬⠁⣼⣿⠿⢿⣿⣾⣷⣧⣶⣄⡀⢁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⣲⠞⢿⣿⣆⡀⠉⢻⣷⣆⠀⢡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠃⠀⠈⠙⣧⠈⠉⠉⠓⢥⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⠔⠀⠀⠀⣿⠀⠀⠀⠀⠈⡀⢀⣼⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢿⣷⣶⣀⡄⠀⠸⠋⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⠋⠀⠀⣠⠄⢀⣠⣠⡶⢰⣿⣿⣿⡿⣷⣼⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸
⣿⡑⠶⠶⠲⣄⠀⠀⠀⠀⠀⠂⠀⣀⣴⡿⠴⢶⣿⣾⣿⣿⣿⣿⣿⠁⠂⠀⠀⠈⠉⠐⠻⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠴⠒⠢⢄⣀⠀⡰⠉⠀⢹
⢸⣯⢖⡂⠀⠨⡇⣀⠤⢤⣤⠴⠲⣽⣷⢦⣖⠛⠺⣿⣿⣿⣿⣿⡧⠼⠀⠈⠀⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀⣿
⠀⢻⣅⠀⠀⠀⠉⠉⠉⠀⠈⠆⠀⠀⠛⢾⣭⢷⣴⠿⠿⣿⡿⠟⢿⡷⢾⣷⣴⣤⡄⢀⠇⡇⢀⣄⣷⢶⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣿⣷⣴⣟⣀⣀⣸⣷⣾⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣅⣨⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁
⠀⠀⠀⠈⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣯⣽⣿⢿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣿⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣖⠀⣤⣶⣄⣀⣁⣾⣶⣀⣿⣿⣿⣿⣿⡿⣿⠿⠛⠛⠿⠿⣿⣿⣿⣿⡆⡸⠶⣤⣤⣴⡿⠿⢦⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣮⣭⣭⣿⣷⣆⢤⣟⠿⠿⡿⢻⣷⣶⠦⠴⠶⣾⣿⣿⣿⣿⣯⣽⣿⣟⣉⣽⣿⣽⣷⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢿⣿⣿⣿⣿⣶⣦⡴⢶⣶⣾⠟⠻⢿⣯⣝⣿⣶⣶⣾⣭⣤⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠲⠿⠿⠿⠶⠾⠿⠛⠛⠛⠛⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⢿⣽
⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⣆⠀⢸⡄⠀⠈⢾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣆⠰⣿⡄⠀⢸⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠛⠚⠿⠛⠽⡄⠸⡿⠿⠿⢿⣿
⠀⠀⠀⠀⠐⢒⣶⣤⣤⣄⣀⡀⠀⠀⠀⢰⣴⣶⠰⡔⠊⠁
⠀⠀⠀⠀⠀⢹⣿⣾⣽⣿⣾⣷⣶⣶⣶⣿⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⡄⠀⢀⣞⣋⣻⣉⣉⣹⣅⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠃⠀⢀⡤⠀⠰⠀⠀⠠⠆⠀⠀⢸⢶⣀⡀
⠀⠀⠀⠀⣦⠀⣿⡄⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣿⡀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⢀⣿⣧⢸⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠸⢿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⠋⠉⡏⠉⠛⠃⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠘⠀⢰⠀⠂⠀⢀⣀⡀
⠀⠀⠀⢠⣶⣷⣦⣤⣍⣙⣈⣿⠉⠉⠉⣻⣤⣴⣷⣶⠶⠊⠀⠀⠀⠀⠀⢠⠐⠉⠈⠁⠿⠀⢋⠰⠈⠀⠀⠲⠃⠀⠈⠀⡀⠘⠉⠁
⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡤⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣰⠀⠀⠀⠀⠀⢀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠊⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⢠⣔⡀⠀⠀⠀⠀⢸⣎⣇⡀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠢⠼⡇⠈⠀⠑⠒⠦⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⠤⣤⠄⠀⠐⢿⠀⠀⢠⡤⠀⠐⠀⠀⢀⣀⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠤⠄⠐⠒⠀⠩⠭⡞⠹⠆⢤⠛⢧⠀⢤⠈⠀⠀⠈⠀⠀⠠⠖⢊⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣷⠀⠠⠴⠾⠶⢚⠋⡁⠀⡻⡌⠙⠛⢷⣶⡂
⠀⠀⠀⠀⡀⠀⠠⡴⠂⣐⣷⣿⢿⣿⣿⣶⣄⣉⣁⣀⣴⣿⣧⠀⣠⣾⣏⠁⠀⠀⣀
⠀⠀⠀⠫⢾⣿⣿⢿⣿⠿⣋⣥⣶⣤⣷⣾⣿⣟⣋⣹⠿⣻⣷⣦⠻⣧⣿⣿⣟⢻⣿⣷⣤⣬⣭
⠀⠀⠀⠀⠀⠈⠡⠜⠛⠻⠎⠏⠉⠉⠙⠿⣆⣉⠀⠀⢸⣿⣿⣿⠄⠻⢿⣿⣿⣿⣦⣌⡉⡩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣆⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠆⢲⠤⣀⠀⠀⠀⠀⣠⠔⠊⠀⣀⣧⡼⢒⠗⡲⡲⣂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⡄⠊⡛⢻⠁⣿⡀⠀⡙⢛⡻⠋⠐⠀⣴⡿⢫⠏⡀⢁⢨⠞⠙⠩⠛⠃⠀⠀⠰⠀⠙⠃⡋⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠉⠉⣁⡜⢡⡴⠂⣉⢬⢉⣥⡜⠁⠀⠀⢀⠄⠈⢰⣻⡇⣾⠀⡃⢘⠈⠄⠀⠀⠀⠀⠀⠀⣶⠒⠒⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠚⣡⠔⠡⣶⣥⠓⣈⡟⣿⣱⣀⡠⠊⠀⠀⠀⠸⡎⢷⡺⢦⠷⠤⢦⠈⢢⡀⠀⠀⠀⠀⡏⠀⢈⣅⣅⢀⣀⡇
⠀⠀⠀⠀⠀⣀⠔⠈⠁⠀⠀⣀⣠⡴⣾⠿⣿⠟⣡⠟⢽⣿⣷⣦⣀⠘⢾⣡⣴⣶⡶⢚⡗⠛⢛⢓⡶⣶⣤⣤⣀⠀⠀⠀⢠⠹⠿⠀⠀⠀⠀⠀⠀⣠⣶⡟
⠀⠀⣀⠴⠊⠀⠀⡈⢀⣤⠞⠻⠋⢩⠗⢻⠇⢸⢡⣶⠓⠋⠋⢉⣭⢿⣿⢹⡿⠋⡋⣸⠏⠀⣰⣿⠋⣈⣭⣴⡿⣿⣦⠠⠃⣠⣼⡂⠂⠀⢈⡽⢟⣩⠞
⠀⢳⡙⡤⠀⠀⣀⣠⡟⡿⠀⡤⠀⡍⠀⡀⠀⠎⠉⠉⡀⢀⣴⣿⠇⣼⠁⡈⠛⡞⢡⢿⢣⣼⠏⢲⠖⢹⡄⠀⠀⠎⠠⠁⠔⢙⠻⠇⢀⠲⣡⠴⢂⡎
⠀⠀⢻⢄⢤⠀⢹⣿⣃⡷⠊⠁⠀⡇⠀⠁⠠⠀⠀⠀⢐⣾⠁⣿⠠⡇⠀⡇⠀⠀⠀⠈⠀⡅⠀⠎⠀⠈⠑⣀⠎⡀⠁⠊⠀⠼⡀⠔⡤⠐⠁⠀⡜
⠀⠀⠀⠌⠢⠺⣸⡿⡏⢻⡀⢰⠀⠰⠀⠀⠀⠀⠀⠀⢸⣽⡆⣟⠀⡇⠀⡇⠀⠀⠀⡀⠀⡇⢈⠀⠀⠀⠀⠜⣔⡀⣡⠔⠁⣽⠲⣞⣃⢀⡤⣔⣀
⠀⠀⠀⠈⡄⠱⡘⢧⢻⡄⢳⡄⠢⠀⢂⢀⠀⠆⠀⢁⠸⡾⡅⣧⢔⡇⠀⠀⠀⠃⠀⠁⠀⡇⠀⡄⢠⠀⢀⡴⣞⣣⡕⢻⢿⢻⡀⠱⢲⡀⡿⠧⣽
⠀⠀⠀⠀⠐⡀⠈⢎⠳⣝⢦⣽⠖⣻⡟⡏⠑⣶⣄⡀⠀⣷⢿⢻⡍⢿⡀⠀⣀⣨⣄⣈⡀⢳⠀⠐⡄⠺⣭⠴⠉⠀⢠⡯⣿⢸⣧⣀⣭⣿⣀⣀⣸⣄⡄
⠀⠀⠀⠀⠀⣡⣤⡄⡧⡌⣓⣏⣸⡿⠁⡇⠀⢒⣻⠉⠺⣾⡘⣇⢻⣮⠟⣿⣿⣺⣿⠈⣿⣶⣇⣀⢷⣄⠀⠢⡀⠀⣼⠯⠏⠼⠿⠿⢿⡿⡟⢻⣿⠟⠁
⠀⠀⠀⠀⠸⡓⠪⢯⢱⣛⣖⠺⢭⡯⢯⠿⣞⢛⣛⣒⣢⠈⢳⣹⣿⠋⢰⣁⣷⣺⣿⠀⢿⢿⡇⢹⣿⣿⣷⣄⠀⢴⠷⡂⠠⡶⠧⡐⢛⣻⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠐⠊⣤⣍⣉⣉⣁⠛⠀⠓⠚⠳⠭⡆⠀⠳⢬⣿⣜⣳⣿⢸⣿⠀⢸⣻⣃⣀⣘⣁⣹⣭⡿⣦⣶⢭⠤⠭⠤⠷⠾⡿⡁
⠀⠀⠀⠀⠀⠀⠀⢀⢔⣜⣹⣿⣿⣽⣿⣿⣿⣷⡆⢢⣆⡴⠒⢲⠆⢸⠛⠾⠯⡞⠿⠶⣾⠾⠞⢠⣤⡦⢠⣤⡄⢤⢴⠀⠶⣶⣴⣶⣾⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠈⠈⣿⣿⣾⣿⣿⣿⡶⣧⢌⠉⠁⠁⠈⠁⠈⠀⠉⠀⠁⠉⠀⠉⠀⠉⠀⠀⠅⠀⠀⠉⠤⠬⡁⠠⢝⣟⢻⠟⡓
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠸⠵⡯⣿⣯⣿⣭⣿⣯⡍⠉⠁⠀⠘⠛⠃⠀⠘⠛⠛⠀⠘⠛⠃⠀⠚⠛⠃⠀⠛⠛⢃⣈⢋⣛⣈⣛⢻⣿⣶⡤⣶⠶⡗⢶⠶⠶⣶
⢸⡿⠛⢟⠝⡹⣫⢛⠝⠛⡖⡲⠛⣿⣯⣿⣭⣭⣭⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣒⣒⣒⣒⡒⡒⠒⠒⠒⠒⢒⣒⣂⣈⣩⣭⣽⣯⣹⠏⠀⠀⠀⠊⠃⠀⡆
⢰⡄⠀⠀⠀⠁⠁⠀⠈⠀⡞⠃⠃⣅⡂⠈⠙⠻⠿⠛⠉⢛⡙⠛⢿⣿⣿⠛⠛⠛⠛⣛⡿⠛⡻⡿⣿⡟⠛⣟⡛⣛⠛⠛⠛⣛⠛⠋⠀⢀⡠⠊⠀⢘⠀⠀⢡
⢐⣒⣒⠀⢀⠐⠒⠒⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠠⠤⠍⠲⠄⣈⣀⠀⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠉⠁⢀⠀⠬⠤⠔⠒⠊⠁⠀⠀⠋
⠀⠘⠛⠓⠊⠉⠙⠛⠓⠒⠒⠒⠈⠉⠁⠀⠘⠓⠒⠒⠀⠀⠀⠠⠤⠤⠤⠭⠥⠤⠤⠤⠤⠤⠭⠭⠭⠭⠭⠤⠤⠀⠠⠦⠄⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣶⣷⣶⠶⣷⠿⣷⢷⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⣧⣀⣤⣀⣼⣠⣀⣁⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⡶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⣿⡆⠀⠀⠀⢘⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣸⣀⣀⣿⣧⣿⣧⣽⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⡁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣂⣀⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠛⠛⡟⣿⠛⠻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢥⣼⣼⣥⣭⣧⣽⣭⣼⣧⣆⠀⠀⣰⣶⣶⣾⣷⣾⣷⣾⣿⣿⣷⣾⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣩⣯⣨⣯⣨⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈


⣦⣴⣤⣦⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣦⣤⣤⣶⣦
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣸⣀⠀⠀⠀⠀⠀⠠⠦⢤⣿⣀⣀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢀⠀⢹⡇⠀⠰⠤⢤⣿⣧⠀⠀⠀⠀⠀⠐⠲⠾⢿⣷⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠘⠓⢺⡗⠐⠂⠀⡜⢹⠀⠀⠒⠀⠀⡂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠸⣥⠆⠀⠴⠀⠸⢠⠀⢀⣠⣤⣦⣤⣤⣤⣼⣾⣶⣶⣶⣾⣿⡟⠁
⠀⠀⠰⢶⣶⣿⣿⣿⣶⣾⣟⣻⠻⢟⣿⣿⣿⣭⣿⣿⣿⣭⡯⣿⢽⡿⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠸⠛⣭⠍⠉⡉⠙⡁⢰⠶⠧⢶⣾⠿⣿⣿⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠀⠒⠂⠀⠀⠀⠠⠀⠀⠐⠒⠈⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⡛⠋⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠅⠒⠁⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠀⢴⠒⠚⠓⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⢸⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡈⢸⡀
⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢻⠘⠀⠀⠈⠁⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⢸⡇
⠀⣠⠶⠴⣧⡸⣒⣒⠒⠒⠒⠚⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣀⠀⠀⠙⢹⣯⡠⠂⠀⢀⡴⠋⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡟⠀⠀⠀⢸⡇
⠀⡏⡂⠀⠀⠁⠄⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠞⣫⠏⠀⠀⣠⠔⡿⢁⡀⠀⢸⠁⠀⣹⠀⠀⠀⣀⣠⠤⠚⢻⠂⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⡇⡇⠀⠀⡁⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠠⠊⣴⣣⢴⡾⢳⠃⣀⣽⡈⠀⠀⠈⠞⣽⣻⠶⠒⠉⠁⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⡄⠀⠀⡇
⠀⣷⠁⣀⣀⢀⡇⠠⠶⠃⠀⠀⡀⠀⠀⠀⠐⢦⣀⠙⠒⠲⣤⣤⢡⠞⢉⡴⠋⠀⠘⠛⢙⣾⡅⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣇⠀⠀⠀⢀⡇
⠀⠈⠉⠙⣏⣸⠛⠉⠉⠙⠛⠓⣌⠀⠀⠀⠀⠀⠈⠑⠦⣛⠧⢄⡀⣰⠎⠀⠀⠀⢰⠋⠁⢸⠃⠀⠀⢠⠞⠁⢸⢠⠀⠀⠀⢀⠠⠀⢸⡄⠀⡀⠀⣿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠃⠀⠈⠱⣦⣈⢱⣶⠶⢤⣤⣸⣤⣤⣼⣇⣠⣀⣸⣄⣠⣾⣛⣛⣛⣿⡛⣛⡛⢟⣛⠙⡟⠀⢻⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⢽⡞⠃⠀⠀⠛⠉⠓⠀⠇⠰⠦⠔⠆⠰⠯⠝⠆⠐⠩⠷⠍⠈⠄⣯⣥⠇⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣲⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⡀⠀⠉⣠⣼⡃⠀⠀⠀⡈⡇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⠀⠀⠀⠀⠀⠀⠀⠻⣧⣐⣀⣱⣂⣈⣀⣀⣀⣀⣀⣤⣈⣁⣀⣠⣭⣴⠞⣏⡡⡌⣧⠀⠀⠀⠀⢻⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⠀⢠⣈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠚⠛⠙⡄⠀⠸⠀⠀⢀⠀⠤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠇⠐⠊⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣡⣤⣶⠖⠛⣿⠉⣹⣁⡭⠍⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣉⣧⣼⣶⡶⠿⡟⣻⢧⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠐⠉⠀⠀⠀⠁⢿⠀⠰⡶⠿⠿⣿⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣀⣤⣴⡶⠖⠛⢙⠀⠀⠀⠀⠰⡚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⣩⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠋⠉⠛⢻⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⠀⠀⣸⠀⠀⢰⣼⠠⠤⢤⣤⣄⣀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢠⣼⣿⢀⡄⠀⠀⣹⣿⣅⠘⠛⠛⠷⠤⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⣿⣿⠈⠁⣶⣶⡜⠿⠿⣷⣄⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⣢⠉⣿⡀⡀⠀⣿⣿⣀⠀⠘⣿⣿⣄⠀⢀⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠁⠘⣣⣹⣷⣀⢀⣿⣿⣿⣾⣉⣭⣿⣿⣵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣤⠤⣭⣽⣭⡭⠖⠂⠀⠔⠛⠋⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⣴⡂⠁⠀⠀⠸⠀⠿⡿⠀⣔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⣻⣛⡋⣩⡝⠻⠿⠷⢀⣂⠈⣁⣀⣄⣠⣌⣩⣭⡌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠀⠚⠫⣿⠛⠝⣹⢿⡟⠫⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣦⠀⠀⠀⠀⣀⡀⠠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠒⠐⠒⠉⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⣀⠠⠄⠒⠒⠒⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⠵⠶⠶⠏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠐⠒⢺⠀⠀⠀⠀⠀⢸⠒⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡈⠢⠄⡀⢀⠈⠒⠠⠤⠤⠔⠊⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⢀⠠⠠⣀⡒⠒⠂⠀⠀⣠⣤⡜⠻⠶⣿⣶⣶⣦⣤⠈⠈⠉⢭⣭⣭⣽⣿⣿⣿⡟⠛⢃
⠀⠀⠀⡐⢡⠊⠁⣀⣀⡠⠤⠤⠤⠤⢴⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⠉⠉⠉⠉⠉⢹⡇⠀⢸⠒⠒⠒⠒⠲⠶⠶⠶⠦⠶⢶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠘⡂⠝⣃⠤⠠⠀⣉⣀⣉⣀⣭⣀⣀⡀⠀⠀⠀⢸⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⠒⠒⠒⠒⠒⠒⠒⠀⠠⠤⠤⠤⠄⣀⡀⠈⠉⠙⠛⢶⡦⡀
⠀⠀⠀⠀⠰⣀⠀⠈⠁⠐⠂⠂⠠⠭⠍⢉⣙⡛⠛⠛⠛⠛⠗⠠⠤⠤⠿⣶⠶⠶⠶⠶⠾⠇⠀⠘⠋⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠙⠓⢢⣀⠰⣳⡇
⠀⠀⠀⠀⠀⠹⣿⣷⣦⣤⣄⣀⡴⠶⢦⡀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠐⠒⠢⡑⠄⠀⠀⢀⢤⣠⣠⣇⣉⣉⣉⣉⣉⣉⡉⠉⠉⠭⠭⠭⠍⢛⣛⣋⠩⠕⠊⣰
⠀⠀⠀⠀⠀⠀⠩⣙⡛⠿⠿⣿⠃⠀⠀⣳⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣀⣉⣉⣉⣤⣤⣴⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠹⡛⠛⠷⠾⣷⣤⣶⣟⣛⣛⣛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣟⣛⣛⣭⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣴⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠿⠛⠛⠛⠉⣩⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠄⣀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠦⠄⠠⠠⢠⣼⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⡧⠖⠾⠿⠿⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣴⡿⣬⣽⠟⠓⠂⠁⠁⠁⠉⠙⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢳⡶⠂⠀⢀⣠⡤⢬⠶⠶⠧⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠘⠋⠀⢀⣺⠏⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⣇⣠⢀⡤⣤⣿⡿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠐⠊⠉⠁⠠⠄⢀⠻⣼⡗⢻⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠐⣂⡠⠾⠳⠋⠋⠙⠏⠉⡉⣻⡟⢿⣿⣄
⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠊⠀⠁⠀⠀⠀⡀⠀⠀⡄⢂⢹⣷⣼⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢡⢀⠀⣷⢸⢸⣿⣿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣂⠘⣆⣇⢴⠺⠘⠆⢛⣷⠳⠿⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⡘⠆⠋⠈⠀⠀⢠⣠⠸⣿⡄⣎⣧⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣌⡆⡀⢀⡆⢘⣸⣿⣷⣈⣷⣿⣿⣿⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠀⠌⣻⠀⣑⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠸⢠⣆⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⢹⣨⣼⡿⠛⠛⠉⠉⠀⡀⣀⢽⣿⣍⠛⠛⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣩⣌⣀⡠⣤⠶⡮⣻⣩⣷⣿⣿⣿⡟⠁⠀⠀⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣛⠒⠂⠀⡤⠤⠴⠶⠛⠿⠈⠀⣿⣿⣿⣷⡀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡍⢹⢃⣖⡛⢀⠀⢄⠀⢴⢀⠩⣿⣿⣿⣿⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠷⠎⠁⠀⠉⠉⠉⠈⣀⣘⠀⠉⠀⠰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠉⠛⠛⠓⠉⠀⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣭⣥⣭⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣶⣾⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⢀⣀⡀⠀⠀⢀⡀⠤⠒⠂⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⢛⣉⠤⠒⠊⠁⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⢈⠉⠬⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠓⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠓⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠋
⠀⠀⠐⠊⠀⠀⠀⢀⣤⠾⢿⣿⣯⢒⣤⡆⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠃
⠀⠀⠀⠀⠀⣀⣴⠿⠃⠙⠁⢠⣤⣾⡿⠀⠀⠀⠀⢀⣤⣶⣿⡿⠍⠉
⠀⠀⠀⣠⣾⡿⠛⠒⠀⠀⠈⣹⣿⣿⣅⣤⣤⣶⣾⡿⠿⠛⠛⠚⠂
⢀⣴⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠓
⣿⣟⠋⠁⠀⠀⠠
⠟⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⢚⣿⠫⠝⠃⠙⠁
⠀⠀⠀⠀⠃⢂⠀⠀⠀⠀⠒⠄⠉⠀⠀⠘⠐⠁⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣉⡖⠉⠉
⠀⠀⠀⠀⠀⠀⠸⡀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠆⠀⠀⠀⠀⠀⠀⣴⣯⣷⣒⣶⡦⢠⢾⣁⡵⠋
⠀⠀⠀⠀⠀⠀⠤⠷⠼⢠⡀⠀⠀⠀⠀⠀⠀⠠⠒⠣⠮⣤⠀⠀⠀⠀⠀⠀⢳⣿⣥⣤⠛⡟⡏⡹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⢀⡐⠾⢧⢸⡇⠀⠀⠀⠀⠀⠀⢀⠰⠞⠆⣿⠀⠀⠀⢀⣀⣀⣸⣻⣻⠟⢦⣡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⢸⡇⠈⠀⢀⠀⠀⣶⡆⠀⠀⠀⢹⠁⠁⠀⠀⢠⣿⣿⣿⣿⠀⠈⠉⢹⣋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⡉⠉⠉⢹⡏⠓⠢⠛⡇⣰⣛⢳⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆⠀⠈⢩⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠹⣿⣦⣄⣘⣃⣠⠸⠀⢣⠇⣬⢭⣟⡢⣄⣘⣀⣄⣀⠀⠘⠶⢦⣤⡸⠀⠀⠀⢸⣿⢗⡄
⠀⠀⠀⢀⣹⣿⣿⣿⣥⣄⡀⣭⠾⠚⢻⡿⢮⢍⠀⠐⠳⠦⠬⣉⣛⠒⠶⠬⣿⠿⢏⡉⠉⣳⣿⡇
⠀⠀⣖⠿⡿⢿⡭⣟⢶⣾⣏⠀⠀⠀⠘⢷⣾⣯⣆⠀⠀⠀⠀⠀⠈⠉⠀⠀⢨⣴⡀⠈⠛⠠⢤⣴⣿⡀
⠀⢼⣋⣷⡇⣎⣷⣽⡿⣷⣿⣀⠀⠀⠀⠸⡗⣿⠁⠀⠁⠀⠂⠠⠀⠀⡐⠀⠀⣍⡀⢰⣶⡄⠈⣿⠀⠀⣴⡶⠤⣄⡀
⠀⠘⣿⣧⢸⡿⡇⠛⠓⠯⢽⣛⡿⣷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣤⣄⣈⡉⠓⠃⠤⠿⠢⠤⡿⠋⠀⠀⠉⢢⡀
⠀⠀⠉⠹⣿⠀⠀⠀⠙⠛⠢⠤⢏⡙⠛⠿⢿⣿⣿⣶⣦⣤⣄⣀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣴⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⢓⡦⠴⣈⠉⢛⡻⠯⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⠄⠈⡉⠛⠲⠮⣍⣙⠛⠛⠣⣈⣿⡿⠟⠿⠭⠥⠤⢃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⠓⠶⢼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠿⠾⠟⠻⠶⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠋⠁⠁⠀⠀⠀⠀⠀⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⡇⠘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⣠⣤⣤⣤⣤⣤⣇⡂⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡟⠛⣛⠼⣍⣙⣙⠟⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣧⣠⣬⣦⢤⣬⡿⢤⠤⢤⠤⢤⠤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠚⠋⠉⠋⠉⡏⠀⢨⠀⠃⠀⠁⠀⠘⠂⠘⠀⠈⡀⠀⠀⠷⠀⡇⠹⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠘⠄⠀⠀⣀⣤⣤⣧⣤⣧⡶⣷⡶⠶⢷⠶⠷⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠇⠀⣅⣤⡴⠟⠿⣟⠉⠹⣏⠀⢿⣧⡳⠺⣿⠀⠈⡇⠘⣟⣧⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⠻⢯⡙⣷⡄⠀⠈⠆⠀⠈⡄⠀⢪⡆⠀⠸⣆⠀⠰⠀⠘⢿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢀⡀⠀⠀⠈⡻⣆⠈⢂⠀⠈⠄⠀⠐⡀⠐⢿⡂⠀⢿⠀⠀⠀⢾⣞⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣇⠀⠈⠀⠀⢰⠄⢣⠻⣧⠀⠆⠀⠈⡄⠀⢡⠀⠸⡇⠀⢸⡆⠀⣤⠈⢻⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠰⠄⠀⠆⠀⠈⡆⠈⣆⢻⡇⠈⠆⠀⢰⠀⠀⡆⡁⣿⠀⢸⠇⠀⣽⡀⣽⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠈⡧⠀⠘⣄⠀⢸⡆⠙⣬⣿⠀⠘⡀⠀⣇⣀⣇⣹⣟⣠⣾⠀⢨⣿⢠⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠳⡀⠀⢢⡐⠀⢷⣀⡙⣿⣀⣠⣷⣾⡿⣿⡿⣿⣿⣿⣿⡶⢾⣟⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢃⠀⠈⠳⡀⣀⣿⣽⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢚⣦⣴⠶⢻⠋⠹⢠⡿⢈⡇⣇⣠⣰⣼⣾⣿⣿⣿⣿⣿⡟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣹⣿⣩⣧⣠⣦⣤⣦⣼⣿⠾⠟⠛⢋⡽⢧⣭⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣩⣉⡀⢀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠠⠤⠄⠲⠒⢿⣟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠀⠛⠛⢛⣿⣿⣿⣿⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣥⣌⢩⡁⢠⠀⡀⠀⡀⡤⢤⠤⡤⡤⠶⠶⠀⠓⠃⡻⣿⣿⣿⣿⣇⣀⠉⠢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⠉⠓⠛⠲⠤⣄⣀⣀⢠⣡⡠⡥⣀⣀⣠⣦⠦⠦⣦⣴⠲⢲⢿⣿⣿⣻⣿⣿⣿⡇⡀⠁⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⢀⡀⠀⠀⠉⠛⠳⠷⠶⠿⠛⠛⠁⠀⠀⣀⠠⠬⠍⣛⠉⠛⠳⠿⠿⠿⢛⠔⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠂⢄⡀⠀⠀⠀⠠⠄⠒⠀⠀⠀⠀⠀⠈⠉⠙⠓⠊⠁⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠒⠂⠐⠒⠊⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠖⠶
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⢤⠿⣷⡤⠄⠈⠀⠀⠀⢠⠛⡧⠀⡀⢸⠀⠢
⠀⠀⠀⠀⠀⠘⢷⣤⣉⡻⠶⡄⠀⠠⠶⢴⠠⠀⠀⣸⡄
⠀⠀⠚⠛⠿⢏⡿⣿⣿⣿⣿⣷⣤⣶⣶⣲⣦⣦⣴⣿⣷⣆⠤⢤⣄⢀⣀⣢⣤⡀
⠀⠀⠀⠈⠀⠀⠠⠿⠿⠿⠾⠿⠿⠿⢿⣿⠿⢯⣙⣻⣿⠯⠿⠿⠆⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠈⠀⠀⠀⠂⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⡠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢴⡞⠉⠁⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠈⠒⠠⣄⠀⠀⠀⠀⣼⣤⡄⠀⠀⠀⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠎⠙⠦⣀⢠⣿⣿⠒⠄⠀⠉⣯⠀⠀⠀⠀⠀⠀⢀⣶⠄⣀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣇⠂⠀⠁⠀⢻⡄⠀⣀⣠⡤⠤⠖⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⣿⠳⣄⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠮⠀⠂⠁
⠀⠀⠀⠀⠀⡀⢠⣦⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⢦⡀⢸⡆⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢉⠝⠁
⠀⠀⠀⠀⠀⠈⢒⣿⣧⣶⣿⣇⣀⠀⠀⢻⠀⠀⠀⢙⡆⢷⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡀⠀⢀⠔⡉
⠀⠀⠀⠀⠀⠀⢿⢿⣾⣿⣿⣿⣧⢳⣦⣿⣥⣴⡂⣸⡇⠸⡇⠀⠀⠀⠂⠀⠀⠸⣆⣀⣤⣤⠔⣳⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⠄⠜⢋⣱⡆⣸⣏⡱⣁⠱⡆⣀⡀⠀⣀⠴⢊⣽⣋⣭⣥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣶⣶⣖⠀⠲⠂⠀⠿⢿⣴⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢀⣉⣸⣿⣿⠻⢻⡻⢿⣻⣿⣛⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠲⠶⠿⣿⣿⡟⡛⠻⢛⢋⣭⣬⡥⣾⠟⢻⡻⢿⣛⣛⣿⣛⣏⡉⠉⠩⢍⡿⠃
⠀⠀⠀⠀⠠⠤⣤⡾⠿⠷⠿⠶⣶⣽⠴⠖⠓⣆⣙⣿⣛⠻⡿⠶⠛⡿⠥⠿⠯⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠿⠏⠈⠁⠀⠐⠐⠋⠈⠑⢒⠚⠁⠈⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⡀⠀⠄⠀⠀⠘⠀⢇⠀⠀⠏⢛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣆⢨⣇⠀⠀⡇⠸⡄⠀⠀⠸⡆⠙⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⣇⠉⠉⠈⠉⠀⠐⢢⣀⡇⠀⠀⠀⠹⡄⠘⣆⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠸⣿⠅⠐⠶⠂⠀⠷⠀⠈⢧⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣆⠀⠘⢧⣀⣀⣘⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡿⠀⠀⠀⠀⠔⢰⣞⣿⠀⢄⠀⠀⠀⠀⢻⡤⢠⠴⠼⢿⣷⣄⣈⡹⠒
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠒⢹⡧⠖⢲⡏⣿⠈⠉⠙⠏⠘⣊⣥⢤⡴⠂⢸⡣⠂⣀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡟⠶⠶⡾⡷⡶⠮⣀⣷⣀⣤⣼⣤⣥⠿⠾⠿⣷⣶⣿⣿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⣧⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡉⣠⣴⣶⣶⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠸⣿⣿⣿⣿⣿
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠔⣾⣿⡷⠶⠤⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠊⠉⢀⡾⣿⣻⣿⡆⠀⠀⠀⢉⠉⡀⣘⣛⣶⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣖⣩⣥⣤⣤⣤⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣤⣄⡀⠀⣀⡀⡆⠀⢉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⢸⣿⠋⠀⠀⠀⢣⢀⡼⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⣠⠟⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⡿⣿⡆⠀⡀⠁⠘⣁⠀⣀⢻⠟⡿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⠎⠀⠀⠀⢡⠀⢌⡟⠀⠐⠃⠘⣿⠙⠿⣿⣿⣦⢀⣰⡿⠉⢿⣿⣿⣿⣿⣿⡧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣧⠀⠀⠀⠈⣗⢸⠁⢹⡇⠀⣰⣿⠀⠀⠈⣿⣄⣿⠛⠳⢀⡌⢈⠻⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠉⠀⠙⢻⠏⠀⡀⠈⠀⠀⠀⠀⢻⡜⠀⠀⠠⣤⣿⢏⣤⣄⣀⣿⣿⣽⣤⣶⣶⣶⣦⣤⣬⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢆⠀⠀⢀⡴⠁⠀⢠⠀⠀⠀⠀⠀⠀⢨⣷⣤⡀⠀⠀⡀⢀⠟⢿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠿⣠⡹⣿⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠈⠀⢠⡞⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠈⠀⠋⠀⠀⠀⢻⣿⣶⠟⠁⣿⠛⢧⠸⡀⠂⢀⡀⢀⠉⢡⣿⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢆⣴⠏⠀⠀⠀⠀⠸⠀⠀⠀⡜⠀⡴⠀⠀⣁⣀⣴⣠⣾⣿⡧⡀⢀⣂⠤⠜⡒⢷⠄⣼⢿⣿⠁⠘⣿⢣⠃
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣿⣷⣶⣄⡀⣴⣦⣭⠀⠀⠀⠁⢰⠃⢰⣾⣿⣿⠟⢩⣿⣿⠀⡈⠋⠀⠀⠀⢧⠘⡀⢱⢀⣈⠙⢣⣼⠆
⠀⠀⠀⠀⢀⣾⣿⣿⡏⣿⣿⣿⡛⠿⢿⣿⣿⣿⣦⣶⣦⣶⣇⣤⣼⣿⡗⢁⠔⠁⣿⣿⡂⠀⠀⠀⠀⠀⠸⡀⢣⡈⣾⡿⠀⢀⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣀⣤⣄⢀⣗⢩⠝⢛⢿⢿⣯⣾⣿⣿⡐⠊⠀⢀⣿⣿⠅⠀⠀⠀⠀⣠⣤⣿⣿⣷⣾⣿⣦⣾⣅
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⠿⢿⣿⣾⣷⣬⣽⣆⣿⣿⣿⡿⣶⣶⣯⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣦
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣍⣀⣷⡈⣍⣽⣛⣿⣿⣿⣿⣥⣼⣧⠿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣤⣩⣽⣿⣷⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣯⣿⡟⢻⣿⠿⣿⣶⣿⣧⣽⣧⣸⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⡆
⡴⣶⣶⣶⣶⣿⣿⡿⠟⠛⠛⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⡿⡷⠟⠓⠛⢋⡀⣉⡉⠉⣛⣹⣟⣿⣿⡿⠛⠛⣻⠟⠶⠶⣶⣶⣦⣤⡀
⠉⢉⡉⣉⠉⠀⠐⠛⢗⣛⣉⣽⣿⣿⣿⣛⡿⣾⣥⣤⣍⣿⣿⣿⣿⣿⣽⠿⢽⣤⣴⣶⣿⣿⣶⣶⣿⣿⣿⣿⡿⠟⠿⠿⠚⠻⠶⠿⠯⣤⡽⢿⠿⣿⣥⠄
⠀⠐⠚⠛⠛⠛⠛⠛⢩⣽⣯⣙⣶⣦⣴⣾⣿⣿⣭⣴⣶⣾⣯⣤⣭⣿⠿⣓⣆⣈⣙⣿⣿⣻⣿⣯⣙⣋⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢠⣤⡤⣤⣤⣀⣈⣉⣛⡿⢿⣿⣿⣯⣯⣿⣿⣾⣿⣿⣷⣾⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⡿⠿⠿⠶⠾⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⢿⡛⠋⠉⠁⠀⠀⠘⢟⣋⠉⠉⡛⠛⠻⠿⢶⣤⣤⣥⣼⣿⣿⠛⠋⠁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣶⣶⣶⣶⣿⡿⠿⠿⠿⠿⠛⠛⠿⢯⣭⣽⡿⠟⠋⠁⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡾⠆⠤⣶⣶⣶⣶⠶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶
⡠⠤⣄⠀⠀⢀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢤⣤⡶⣿⢿⣿
⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣬⣤⣤⣤⣤⣤⣤⣬⣿
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠇⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⢋⣡⣤⣤⣤⣾⡿⠋⠉⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣾⣿⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣤⡄⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠀⠀⣰⣿⡟
⠿⢿⣿⣷⣶⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⣼⣿⣿⢿⣿⣷⣶⣶⣶⣾⣿⡿
⠀⠀⠀⠉⠉⠉⠉⠁⠀⢻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠋⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⢾⣿⣷⣦⣤⣤⣤⣶⣿⣿⣿⣷⣦⣤⣄⠀⠀⠰⣿⣿⣶⣦⣤⣤⣴⣶⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠉⢁⡀⠀⠀⠀⠉⠀⠀⠻⠁⠀⠐⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠥⠄⠐⠚⠉⠁⠀⣠⠀⠀⠀⢐⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠐⡖⠀⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁⠀⠀⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠆⠀⠠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡀⠄⠠⠤⠤⠀⢨⠿⠀⠀⠀⠀⠐⢒⡀
⠀⠀⠰⣮⣀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡧⠀⠛⠂⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠉⠑⠢⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⠛⢃⠂⠒⠤⠈⠠⠛⠀⠤⠒⠂⠧⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⢠⣀⣇⢠⣥⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄⡤⠀⠀⠀⠀⠀⠀⠠⠶⠦⠀⠀⠠⠀⠀⠀⣤⣴⠀⢱⠄⠀⣰⣿⡤⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⡈⠉⠋⠡⠀⠠⠀⢀⠘⡟⢓⠓⡊⠃⠰⠛⠛⢻⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣀⣀⣀⠈⠀⠁⠀⠁⡈⣀⠈⢀⣈⡀⠀⠀⠀⠄⠰⠶⠀⠚⠋⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠌⠛⠛⠉⢳⠀⠀⠀⡇⠀⠀⠀⠀⣀⣠⢤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠤⠓⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠊⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠀⠀⠀⢀⣀⣂⠀⠀⣴⣶⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣠⣤⣶⣶⠟⠛⢻⡏⠉⢩⣄⣠⡤⠄⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⢛⢀⣸⣤⣬⣼⡿⡿⢿⢿⡁⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠶⠚⠛⡫⠟⠛⠋⠁⠰⠁⣼⡄⡑⡀⢀⣠⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠃⠀⠘⠟⠛⣛⢛⣿⠾⠷⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⣤⣶⣶⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠘⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣛⡃⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠲⣿⠿⠿⢿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠀⠀⡀⠀⠘⠀⠀⠉⠉⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠩⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⡀⠀⠀⠀⢀⡇⠀⠀⠀⣄⡞⢀⣤⣄⣀⣀⣀⡐⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢀⣾⡇⠀⡀⠀⠙⠓⢲⣶⡤⠬⠝⠻⣿⣿⣆⠈⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣧⠃⠀⡇⠀⢈⣯⣿⡇⢰⣇⢀⠀⠀⠺⣿⣿⣦⡀⠀⠀⠀⠉⠉⣡⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⢻⣇⠀⠃⠀⠌⣿⣿⣇⠀⠀⠘⣿⣿⣇⠛⠛⠻⠿⢦⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⢺⡈⠸⣿⡄⢀⠀⠀⣿⣿⣿⢠⠀⠀⢹⣿⣿⣆⠀⠀⡠⣨⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢑⠂⠸⠷⡀⢻⣷⡘⡀⠀⣿⣿⣿⣿⣖⠒⠒⢻⣿⣿⠗⣡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡄⠛⠛⠿⠷⣦⠶⠿⢟⣋⡉⠙⠀⢈⣿⡿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⣷⣖⣚⣛⣶⣿⠿⠟⠋⠁⡀⠄⠐⠉⣀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦⣴⡞⠁⠀⠀⠀⠀⠹⠀⠾⢿⡧⠀⠔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠿⠿⠿⠛⠋⣙⠿⢿⣿⣿⣂⠀⢤⠈⠛⠀⠁⡀⠀⠀⠐⠉⣉⣉⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠱⠚⠛⠛⠁⠻⠷⠆⠀⢠⣤⢶⣶⣶⠖⡒⢲⣿⣿⣿⠿⣿⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠟⠁⠈⠀⠛⠛⠑⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠖⢉⠀⠃⠃⠀⠀⠀⠓⣦⣥⣠⡀⠀⣴⠉⠀⠀⠈⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⡰⠒⠄⠄⠀⠀⣼⣿⡿⠿⢶⣷⣀⠀⠀⠈⠈⢁
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠉⠐⠂⠀⠀⠛⡉⠁⠀⠙⣆⠨⡿⣦⣄⠄⠀⣠⡌⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⡀⠤⠠⠀⢠⢺⠆⠦⠙⢶⣴⡄⡈⠹⣧⣄⡙⢳⣤⡟⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠦⣄⡇⣽⡿⣿⣮⣿⢦⡀⢈⣞⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣢⣤⡄⠐⣼⠇⠑⣄⢿⣯⠈⠛⡿⡤⣯⠵⢹⠟⣿⡼⠿⠄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢰⣦⣶⣿⣿⣿⣿⣿⣷⣿⣿⣾⣊⣍⣨⣿⠀⠾⣏⠋⣿⣴⣿⣏⢹⡷⠠⣄⣦⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡇⢀⣼⣷⣾⣿⣿⣿⠏⠀⣄⠠⠯⣿⣵⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠋⠙⠋⢁⣤⣬⣧⣆⣸⣿⠿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠋⠉⠁⠀⠀⠀⡀⣴⣶⣿⣿⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠒⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢐⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠀⠀⠀⠀⠀⠄⠓⠀⢤⣀⠀⠀⠘⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⢆⢣⡄⠀⠀⣼⠤⠀⣂⠠⢄⡟⡍⠙⠁⠓⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⢰⠁⠀⠀⠀⠈⢸⠀⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠈⠉⠀⠒⠒⠒⠓⠒⠂⠠⢄⠀⠀⠇⠀⠀⠀⠀⠀⢬⠠⠦⣀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠀⠀⠀⠉⠉⠉⠈⢁⠀⡟⠢⡑⢾⡀⠀⠀⠀⠀⠀⣿⠠⢀⢈⠦⡈⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠶⠀⠀⠒⠒⠒⠒⠒⠒⠂⠐⠢⢼⡎⡲⢍⠦⡀⠀⠀⠀⡏⠀⠒⢼⠀⠀⠀⠈⠃
⠀⢀⠠⠤⠤⠄⢀⣀⣀⡁⠀⠀⠀⠀⠀⠀⢰⠤⠤⠄⠤⠤⠤⠤⠠⠤⠀⠤⠤⠤⠤⠄⡌⠑⣄⠺⢳⣬⡆⠀⠀⡇⠀⠀⠸
⠀⠈⠐⠢⣄⠀⠀⠀⠀⠉⠉⠁⠒⠒⠒⠠⠤⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⢰⠙⢄⠈⠻⣦⡸⠀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠦⡄⠒⠒⠒⠒⠒⠒⠠⠤⠤⠄⣀⣈⠉⠘⠛⠒⠦⢄⣀⠀⠂⠀⡆⡄⠀⡊⠷⡀⠈⠳⣄⠀⠨⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠠⠤⣀⡉⠙⠓⠤⢇⣘⡈⠁⠌⠢⣀⠀⠑⢬⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠉⠑⠒⠤⢄⣉⠑⠳⠦⣈⡓⢄⠀⠑⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠈⠉⠒⠒⠤⠤⢄⣉⠐⠢⠤⣉⠓⠵⣤⡀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡠⠀⠒⠈⠉⠉⢀⣀⣀⠀⠀⠉⠉⠉⢀⠀⠀⠀⠀⠠⠍⣁⠒⠂⠬⢕⣂⠭⢲⣄⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠬⠍⣐⠐⠂⠭⢴⡈⠙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⣭⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣆⠸⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣠⣤⣤⢤⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠣⢨⣀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣠⡤⠭⠚⠛⣩⠏⠀⠀⠀⠉⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠒⠬⣇⣀⣀⣐⣢⣤⠴⠶⠾⠒⠂⠉⣩⢋⠖⡵⠋⠁⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠈⠀⠈⠑⠄⠁⠀⠂⠍⡩⠝⠉⠀⠀⠀⢀⣠⣤⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠫⡉⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠄⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣾⣿⡿⠶⠄⢰⡆
⢠⣴⡶⠛⠿⠶⢤⣀⡤⠶⣤⣿⣿⡟⠿⠟⠛⠋⠀⠀⠀⢸⡇
⠟⠻⠀⠀⠀⠀⠀⠀⠁⢰⠟⠛⠉⠁⠀⠀⠀⠀⠀⡀⠔⢺⡇
⠀⠀⠀⢀⣰⡿⠛⠉⠉⠉⠀⠀⠀⠴⢶⣶⠶⠖⠈⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⢸⠃⢿⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣀⠀⠀⠀⠀⣼⠁⢸⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠻⡏⢿⠹⣍⠻⡟⠉⣿⣶⣿⠷⠶⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⢿⣗⠲⠶⠶⠦⠤⣿⣍⣉⣉⣁⡀⠀⠀⣡⡄⣿⠁⢸⠀⠀⠀⠀⠀⢀⣈⠁
⠀⠀⠀⠀⠀⢻⣯⣭⠀⠀⠀⠀⠀⠀⠀⠘⠭⠭⠭⠭⠙⠛⠒⠚⠂⠘⠛⠂⠐⠿⠿⠧⡄
⠀⠀⠀⠀⠀⠀⢻⣯⣭⣅⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⠀⠀⠀⠠⣄
⠀⠀⠀⡠⠀⠀⠘⠻⣭⠉⠉⠹⠂⠘⠁⠀⠀⠖⠒⠀⠀⠴⠒⠀⠀⠀⠀⠐⠲⡍⠉⠉⢁⣉⣙⣂⣶⣾⡂
⠀⠀⠀⠳⠶⠖⠒⠢⣌⠓⠚⠋⠉⠓⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢿⣿⣷⠶⣶⣮⣿⢶⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠍⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣧⣤⣴⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢰⣿⠿⠟⠛⢻⠏⠉⠉⠉⠙⠛⠛⠇⠀⠀⠀⠀⠀⠀⠓⢒⡦
⠀⠀⠀⢀⠀⠀⠀⠈⣿⣆⠀⠀⠀⢠⣭⣤⣤⣤⣶⣾⣷⣶⣶⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠦⠤⢄⡀
⠀⠀⠀⢸⣦⠀⠀⠀⢻⣿⣆⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣷⣶⣶⣶⣿⣦
⠀⠀⠀⢸⣿⣧⠀⠀⢸⣿⣿⣆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢸⣿⣿⣧⠀⠘⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣧⠀⣿⣿⣿⣿⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⡿⠿⠿⢿⣿⣿⡟
⠀⠀⠠⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣷⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⢠⣀⣼⡈⠉⠛⠃⠉⢻⠻⠿⣿⣆⠛⠛⣿⠟⠛⠻⠿⠿⠿⢿⣿⣿⡇⠀⠀⠀⢸⣿⣀⣀⣀⣠⣤⣤⠴⠒⠋⠉
⠀⠀⢠⣤⣼⣿⣿⣿⣿⣿⣿⣤⣼⣀⣀⣀⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡄⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⢹⣿⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⡷⣶⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⡀⠀⣀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⡀⠀⡴⣠⠀⠀⢀⢀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣞⡋⠉⣙⣿⣹⣗⡃⠫⠄⢨⣧⠨⠤⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠑⠢⡀⠐⠀⠀⠀⠀⡀⣸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣴⣦⣶⣴⣿⡇⠀⠀⢀⣀⣀⣸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠚⠒⢲⡟⣺⣗⡃⠁⠩⠭⠿⠼⣧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣷⠀⠀⠀⠀⠀⠀⠀⠐⠄⠙⠛⠓⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⣻⡆⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣿⣶⣶⣆⣴⠆
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠛⠲⠷⣷⡦⡄⣤⣾⠇⣶⣿⡏⠀⠀⠀⠀⢀⣰⣀⡿⣽⠿⠃⠀⠀⠀⠉⠉⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⠻⠿⠿⣿⠀⡀⣤⠆⠉⢀⣽⣇⣾⣦⣤⣤⣤⡤⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣷⣴⣄⣠⣿⣉⣠⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⢀⣀⡀⣀
⠀⠀⠀⠰⢶⣦⣴⣶⠒⣶⡖⣴⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⢿⡿⢰⣿⣶⣯⣭⣬⣾⣿⢲⣤⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⠛⡋⢉⡉⢉⣉⣭⡿⠉⠙⠟⠋⠉⣰⣾⣿⠃⠘⣙⣳⣾⣿⡏⣍⣙⣛⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠲⠚⠋⠁⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠋⠀⠉⠀⠉⠉⠁⠀⠀⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠹⠏⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢠⣏⣇⠀⠀⠀⠀⡤⠀⡀
⠀⠀⠀⠀⠄⢷⠛⢙⢫⠑⣚⠆⠀⠀⠀⠐⠀⠀⠞⠀⢹⠐⠢⣀⠀⡖⠀⠈⠋
⠀⠀⠀⠀⠀⠘⢻⣶⣤⣃⠘⡸⠷⣆⠀⠀⢀⣰⣇⡆⡆⡆⠈⠀⠀⣥⡀
⠠⠤⣤⣤⠶⠶⢲⢿⣷⣽⣿⡷⣴⣿⣄⢀⣠⣤⣤⣄⣇⣹⡀⢀⣾⣿⣧⡤⠒⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣀⢉⣁⢞⢸⣿⣿⣝⣛⣻⢿⢿⡾⣥⣧⣿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣭⣉⣐⡒⣂⣔⣤⣦⣤⡿⠦
⠀⠀⠀⠀⠀⠀⠈⠊⠿⠿⠿⠿⠿⠟⠛⠻⠟⠿⢿⣏⡽⠿⢦⣍⣙⣫⡿⠷⠧⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠉⠀⠀⠀⠒⠦⢀⡐⠛⠋⠓⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠳⣦⣴⠟⣇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠠⠤⠀⢢⣲⠀⣿⡠⠤⠒⠆⠂⠀⢀⣀⣀⠁⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢠⣤⠖⠘⠁⠀⠀⣀⡤⠞⠋⠉⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢻⠂⠀⢀⣄⣤⣿⣿⡀⠀⠀⠀⠸⡗⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⡤⣶⣾⣸⣿⡿⠿⠃⠀⢀⣀⡠⣧⠈⢢⠄⠊⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡿⠟⠉⣁⠼⣷⣶⣶⠾⠿⠛⠁⣿⡂⠀⠠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠊⠁⠀⠐⠹⢻⡇⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠈⢿⠁⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡏⡆⠀⠀⠀⠀⣤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡇⠁⠀⠀⠀⠀⣸⣿⢀⠀⢀⣠⣼⣷⢳⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⡇⢀⣆⣼⡠⣆⣿⣿⣽⣿⠿⠟⠛⣿⠸⣦⣤⣄⣀⣄⣀⠀⢤⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣟⣿⣿⣷⣿⣿⣿⣏⠀⠀⠀⠀⢾⡄⠛⠛⣛⣛⠛⢿⣿⠺⣿⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠬⣷⡿⢿⣿⢻⣿⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠳⠈⡇⣶⣿⣿⣿⣿⣿⣿⣿⣿⣄⣘⠆⠐⣿⣧⣾⣿⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⢹⣶⣽⡇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⣯⣣⣼⣽⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣌⣠⠾⠿⢻⣁⣠⣴⣤⣶⡒⠆⠀⠀⠀⠀⠀⠀⣀⣀⣚⢿⡟
⠀⠀⠀⠀⠀⣾⡟⣀⣤⣟⡉⠩⠉⠀⠀⠀⠀⠬⢸⠍⠩⣹⣿⡿⠟⠻⢀⣤⣾⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⡀⠭⡉⢹⣿⡇
⠀⠀⠀⠀⠀⢻⣾⣿⡾⠟⡀⠀⠀⢀⣤⣤⣴⣶⣶⣖⣳⣠⣿⣧⣾⣿⣿⣿⠿⠿⣿⣿⣿⣿⡎⠀⠀⠀⠈⠀⠀⠉⠉⠁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣤⣀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣴⣶⣤⣬⠋⠉⢻⣿⡴⣄⠀⢀⠀⠀⠰⣆⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⠛⠀⢀⣈⡻⠿⠿⡿⠿⠿⠿⠿⠿⠋⠀⠘⢁⣠⣾⣿⣴⣿⣦⣿⣿⢸⣿⣦⣷⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢷⣶⣼⣶⣿⣶⠆⠠⠶⠶⠾⠿⢿⣻⣿⣿⣿⣿⣿⢟⣼⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣶⣤⣤⣤⣬⣽⣿⠫⠘⣻⣿⠟⠁⠚⣻⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠾⣿⡿⣻⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⣀⣥⣾⣷⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⠿⠿⠿⣿⣿⣷⣶⡾⠿⠿⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿