Compass Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⢀⣀⣤⢤⠴⠶⢶⠶⠶⢦⣤⣄⣀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣠⡶⠛⠉⠁⠀⠀⠀⠀⣶⣶⠂⠀⠀⠀⠘⠋⠓⢦⣄⡀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠙⢷⡄
⠀⠀⠀⠀⢀⣼⠋⠀⢀⣾⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠹⣦
⠀⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠀⠀⢀⠄⠊⡀⠀⠐⠒⢀⠀⣾⠐⡿⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⣧
⠀⠀⢀⡾⠁⠀⣴⠋⠀⠀⠀⠀⠀⢀⢀⡌⠃⠀⠀⠰⡈⢸⣿⠀⠀⠐⠀⠀⠀⢀⡠⢆⠀⣀⠀⠀⠀⠈⠹⣆⠀⠘⣧
⠀⠀⣼⠃⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⣦⠀⠀⣤⣾⣿⠀⠀⠠⠐⠄⡀⠈⠀⠀⠀⠁⣀⡤⠒⠀⠀⢿⡄⠀⢹⡆
⠀⢀⡿⠀⠀⣟⠀⡀⠀⠀⣸⠃⠄⠀⠀⠀⢉⠈⢷⣄⠸⣿⣿⠀⠀⠀⠠⠞⠀⢐⣠⣤⡶⠟⠉⠆⠀⠀⠀⠀⣷⠀⠀⣿
⠀⢸⡇⠀⢸⡇⣠⡀⠀⢀⡀⠀⠀⣀⠠⠂⠀⠀⠀⠛⣷⠼⢛⠐⢄⢀⣠⣴⣾⣿⣟⠁⣀⠀⠀⠠⠀⠀⠀⠀⢹⠀⠀⢹⡀
⠀⢸⡇⠀⢸⡗⣭⣿⡇⠀⠓⠦⢤⣤⣤⣤⣤⣤⣴⣾⣏⠰⣿⡷⠆⣿⡿⠛⠛⠛⠛⠛⠛⠓⠒⢀⠀⢸⡿⡿⢺⠀⠀⢸⡇
⠀⢸⡇⠀⠘⣇⠉⠀⠀⠀⡃⠀⠀⠀⣂⣼⣿⠿⠟⠋⠉⠀⠈⣠⡜⢷⣤⡀⠀⠀⠀⠄⠀⠀⠴⠈⠀⠀⠀⠀⣸⠀⠀⣼⠁
⠀⠈⣷⠀⠀⣯⠀⠀⠀⠀⠠⣠⣾⠿⠛⠏⠁⢠⠔⠀⠀⠀⠀⣿⣿⠂⠙⢧⠀⢀⠀⠀⠀⢀⠉⠃⠀⠀⠀⠀⡏⠀⠀⣿
⠀⠀⢻⡄⠀⠘⡖⠀⠀⠴⠚⠋⠀⠄⠀⠀⠀⠀⡘⠀⠘⠆⠀⣿⡏⠤⠒⠀⡁⠈⠀⠐⠆⠀⠆⠀⠀⠀⠀⣾⠁⠀⢸⠇
⠀⠀⠈⢷⡀⠀⠹⣤⠀⠀⠀⠀⠰⠀⠀⢤⡀⠀⢠⠀⣀⠀⠀⣿⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢠⡟
⠀⠀⠀⠈⢷⡀⠀⠙⣦⠀⠀⠀⠀⠈⠀⢴⠀⠐⠀⠀⢄⠀⠀⡏⠈⠉⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⡟
⠀⠀⠀⠀⠈⢻⣄⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠄⣴⠟
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠻⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⠏⠁⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠂⠀⠉⠓⠦⣤⣀⠀⢀⠀⠀⢛⠃⠀⢀⠀⢀⣰⡤⠴⠛⠁⠀⢀⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠈⠉⠙⠓⠒⠚⠓⠒⠛⠉⠉⠁⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⣴⣾⠿⠛⠉⠉⣿⡏⠉⠙⠛⠿⣶⣄⡀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠀⣀⠤⠒⢺⣿⢟⠒⠆⢤⡀⠈⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⡰⢎⣀⡴⠚⢻⣿⠸⠛⠲⢦⣈⡷⡄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⡜⢀⡼⠻⣶⣄⣿⣿⠀⡇⣠⣴⠟⣦⠘⢆⠀⠹⣷⡀
⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠀⣾⡇⠀⢸⢀⣾⣀⠀⠹⠿⣿⣿⣤⣿⣿⣧⣤⣸⣧⣘⡄⠀⣿⡇⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠸⣷⢿⣾⠁⠀⠒⠶⣿⡿⠿⣿⣶⣶⣶⣶⣶⣶⣾⡀⢀⣿⠛⠛⠛⠛⣛⣛⡿⠿⣿⡿⠒⠒⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⠀⢻⡉⢉⣽⣿⡏⢹⣿⣿⣿⣍⠉⢠⠏⢰⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠳⡈⢳⣾⠛⠁⠄⢸⣿⡇⠙⢻⣷⡏⢠⠆⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠙⠯⣈⠛⠶⢴⢸⣿⢧⠖⠋⣀⠝⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣷⣄⡀⠈⠑⠒⠼⣼⣿⠴⠒⠈⠀⢀⣴⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠈⠙⠿⣶⣦⣤⣄⣿⣧⣤⣤⣴⡾⠟⠋⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠯⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡇⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣾⡿⠟⠋⠉⣿⡇⠉⠉⠛⠿⣿⣦⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⡄⠀⠀⢠⣿⡇⠀⠀⠀⠰⠋⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⢰⣄⢸⡿⠇⢠⡾⠃⠀⠀⠀⢻⣿⡄
⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⢻⣿⣶⣶⣶⣾⣿⣧⣤⣤⣀⣀⣀⣀⠀⢀⡀
⠀⠈⠉⠀⠀⠀⠉⠉⠉⠉⠙⣿⣟⠛⠻⠿⠿⠿⡄⠀⠀⢀⣀⡀⠀⠀⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢴⠇⠀⣾⣿⠈⠛⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡴⠀⠀⠀⠀⣿⡏⠀⠀⠀⣳⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠻⣿⣶⣤⣀⡀⣿⣇⣠⣴⣾⡿⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠦⠴⢸⣧⠶⢠⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠗⠉⠁⠀⠀⠀⢸⣷⠀⠀⠀⠉⠻⠶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠞⣉⡲⠄⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢰⠎⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠫⠤⠤⠉⠀⠀⠀⡀⠀⠀⢸⣿⡇⠀⡐⠀⠀⠈⠤⠎⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠁⠀⠀⠀⠀⠈⠓⢤⡐⠀⠀⢸⣿⣷⠀⠀⣠⠋⠀⠀⠀⠀⠀⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣙⣇⠀⢸⣿⣿⣠⠞⠁⠈⠀⠀⠀⠀⠀⠘⣧
⢲⡀⣶⠀⠀⣀⣀⣠⣤⣿⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣦⣸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢰⡆
⠀⠟⠘⠇⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣻⠋⠋⠉⠁⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⡀⠄⠀⣴⠟⢻⣿⡇⠀⠈⣯⡉⠀⢀⠀⠀⠀⠀⠀⢠⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⠞⢁⠈⢸⣿⡇⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄⠞⡴⢀⠁⠀⠂⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠫⢦⣁⠙⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⣓⢔⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠵⣦⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⡀⣴⡾⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠸⡇⠰⠸⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⢿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣀⣴⣿⠿⠛⠉⠀⠀⠀⣿⣿⠀⠀⠀⠉⠙⠻⣿⣶⣄⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⣀⡤⠒⠚⢹⣿⡟⡋⠑⠒⠤⣀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣤⡊⠁⣠⡤⠶⣿⣿⡇⡟⠷⢦⣄⠈⠑⣦⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢀⠞⠀⣹⣿⣍⡀⠀⣿⣿⡇⢠⠀⠀⢈⣿⣿⠀⠑⡄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⡎⠀⣼⠃⠈⢿⣿⣶⣿⣿⡇⢘⣤⣾⡿⠃⠈⢷⡀⠸⡄⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⠀⠀⣸⣿⠀⠀⣸⣀⣸⡇⠤⠤⠤⠛⠻⣿⡿⠷⣾⣿⣿⣷⣶⣤⣼⣧⣀⣧⣀⡀⣿⣷⠀⠀⠀⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠈⣿⡾⢿⣿⠃⠀⠐⠒⠶⣿⣿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⡀⠀⣸⡟⠛⠛⠛⠛⠛⠛⠛⣛⣿⠿⣿⣿⠟⠒⠒⠀⢸⣿⣓⣛
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⡀⠹⣏⠉⠛⢻⣿⣿⡟⢻⣿⣿⣿⣶⡆⠉⠉⢹⡏⠀⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢧⠀⠹⣆⣰⣿⠿⠛⡆⢸⣿⣿⠛⢿⣿⣆⣠⡿⠀⡰⠁⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠈⠣⣀⡽⢿⣅⡀⠀⠃⢸⣿⣿⠀⠀⣨⡿⢿⣀⡼⠁⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠙⠦⣀⠈⠙⠳⢾⢸⣿⡿⠾⠛⠉⣀⠴⠋⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣦⡀⠀⠀⠉⠒⠢⠼⣼⣿⡧⠴⠒⠊⠁⠀⢀⣤⣾⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠛⢿⣷⣦⣄⣀⡀⠀⣿⣿⠀⠀⣀⣠⣤⣾⡿⠟⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠤⠤⢸⡤⠤⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⠔⣈⠠⠄⠒⠀⠁⢸⡇⢀⠐⠂⠌⢁⠂⠜⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣠⠄⢀⠈⠠⠀⢘⣠⣌⡄⢸⣧⣠⣄⡊⠀⠄⠉⠀⢀⠄⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠀⢄⠀⡡⣴⠞⠓⠈⠁⠀⢸⣿⠈⠉⠛⠹⢶⢔⠁⣠⠈⠔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠂⡀⠐⠄⣠⡪⠊⠀⠀⡀⠐⠀⠀⢸⣿⡀⠂⠄⡀⠀⠙⢝⢄⠠⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⠐⠂⣰⠝⠀⠀⠐⢦⡀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠴⠀⠀⠹⣣⠐⠂⠡⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⢠⠁⠒⣰⡝⠀⢀⠂⠀⠀⠙⢷⣄⠀⢸⣿⡗⢀⣴⠋⠀⠀⠀⠀⢸⢇⠐⠀⠆⢀⠉
⡃⠀⢀⡀⠀⠀⠀⠀⠀⡇⣄⢐⣀⣽⣁⣀⣤⣤⣴⣶⣾⣦⣝⠆⢸⣟⠴⠋⠁⠀⠀⠀⠀⠀⠀⠿⠀⠂⢰⠸⠀⠀⠀⠀⠀⢠⣀⣀⣠
⠖⣋⣩⠄⠀⠀⠀⠉⠉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡄⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢶⢶⠖⠒⠒⠀⠀⢀⠀⠄
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠇⢁⠒⠂⢭⡄⠀⠀⠀⠀⠐⣠⡶⢋⣾⡇⠈⢦⡉⠫⠉⠉⠉⡉⠀⢀⣳⠀⠒⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⡈⡄⠂⠘⣶⡀⠀⠠⠀⡴⠋⠀⢸⣿⡇⠀⠈⠙⠶⣄⠀⠀⠀⠀⣴⠃⠐⠀⠁⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⡐⠉⠘⢕⣄⠀⠈⠠⠀⠀⠨⣿⡇⠀⠀⠀⠀⠈⠂⠀⢀⣾⠂⠈⠂⠌⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠈⠁⣨⠳⢷⢄⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠀⡠⣴⠟⢁⠈⠁⠀⠐⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠂⢉⠚⠶⢤⣀⣼⡇⠀⣀⣤⠶⠻⡊⠐⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢈⠀⠀⠀⢌⡀⠐⠀⠀⢸⡇⠀⢠⠀⠐⠀⡠⠄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠆⠉⠒⠤⢀⡁⢐⣚⡇⢐⡂⢈⡁⠤⠐⠈⠰⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⢀⡀⢤⣶⣂⡔⠒⢦⣤⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠶⠒⠒⠓⠒⢐⣀⣂⣂⣒⣒⠢⠤⠤⢉⣛⠱⢤⡑⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠠⠔⠊⣍⡤⠤⠐⠒⠉⠉⣰⠀⠀⠀⠀⠀⠈⠉⠙⠒⠤⣈⠙⠦⣤⡀⢀
⠀⠀⠀⠀⠀⠀⡠⢀⠕⠊⢀⡠⠒⠋⠁⠀⠀⡀⢀⠀⢠⡏⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⠙⠢⣈⢻⣦⠉⡀
⠀⠀⠀⠀⡠⢊⠔⠁⡠⠒⠁⠀⠀⡀⠀⠀⠈⠀⠀⢀⣾⡇⠀⣀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⢢⣿⠁⠈⡋
⠀⠀⢀⠊⡰⠁⡤⠊⠐⠀⠀⠐⠛⠋⠠⢤⡀⠀⢀⠞⡛⡇⠚⠁⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣷
⠀⠀⠉⡸⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠰⢈
⠀⢀⣶⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠈⠀⠀⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐
⠀⢸⢿⡜⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂⠀⠀⠀⠐⠀⠠⠌⣠⠇⠀⠄
⠀⠈⢝⡇⠰⠀⠀⠀⠀⠀⠀⠀⢀⢄⠘⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣰⡟⠀⠀⠈
⠀⠀⠐⡟⢄⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠊⡸⠁
⠀⠀⢀⠀⠀⣛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠴⠁⠄⠀⠁
⠀⠀⠈⢳⣄⠀⠀⠁⠻⠗⠄⠀⠀⢀⣀⣀⣀⣁⣀⡀⠤⡤⠔⠒⠈⠁⠀⠀⠄⠀⢀⠠⠐⠁⠀⡀⠊
⠀⠀⠀⠀⠙⢭⡂⢤⢀⠀⠠⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠈⠀⠀⢀⢹⠏
⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠁⠒⠀⠠⠠⠤⠀⠠⠀⠠⠀⠠⠀⠀⠂⠈⠉⠀⢀⡶⠐⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠒⠰⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠒⢲⠈⠉⢹⡍⠉⠁⡖⠂⠠⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡊⠀⠀⠃⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⠀⠈⢱⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢀⡏⣿⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣁⡴⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠁⠈⠻⣷⣄⡀⠀⠀⣀⠥⠐⠂⠉⢹⠇⣿⡉⠉⠐⠂⠭⣀⠀⠀⢀⣤⣾⠟⠀⠈⠁⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠄⠀⠀⠀⠀⠙⢮⠻⣶⣊⠀⠀⠺⠇⠀⣸⠀⣿⡇⠀⠸⠇⠀⠀⣱⠶⣫⡾⠁⠀⠀⠀⠀⠰⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⡠⠳⣌⠻⣷⣤⡀⠀⠀⣿⠀⣿⣧⠀⠀⢀⡴⠚⣡⣾⠛⢄⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⢤⡄⡜⠁⠀⠈⢧⡈⠻⣿⣦⣄⡇⠀⣿⣿⣠⠖⠋⣠⣾⡟⠁⠀⠈⢆⢠⡤⠀⠀⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠀⡸⠀⠸⠇⠀⠀⠙⣄⠈⠻⣿⠇⠀⣿⣿⡅⣠⣾⡿⠋⠀⠀⠺⠃⠈⡆⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⢄⠤⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣆⣀⣠⣤⣤⣤⣤⣾⣷⣶⣿⣷⣄⣿⡿⠛⠛⠛⠷⠤⠤⠤⣤⣄⣀⣰⣀⡀⠀⠀⠀⠀⠀⠁⠀⡄⢠
⠀⠀⠀⢀⡖⣿⠀⠀⠀⠛⠒⠺⠿⠿⠿⢿⣭⣝⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠿⠓⠒⠾⠀⡼⢉⠁
⠀⠀⠀⠈⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠉⠉⠉⠉⣹⣶⣶⢾⣿⣿⠈⢻⣟⠛⠻⣏⠉⠉⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⢀⡘⡀⠸⠇⠀⢀⣴⣿⠟⠁⣸⣿⣿⠀⢸⣿⣷⣄⠘⢦⡀⠀⠺⠆⢠⢃⡀⠀⠀⠀⠀⠀⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡠⠀⠀⠀⠀⠉⠁⠱⡄⠀⣠⣿⠟⢁⡴⠊⠁⣿⣿⠀⣸⠈⠛⢿⣷⣄⠱⣄⠀⣠⠃⠈⠉⠀⠀⠀⠀⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢈⣾⠟⣡⠞⠉⠀⠀⠀⢿⣿⠀⡟⠀⠀⠀⠉⠻⣷⣌⢳⡁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⠁⠀⠀⠀⠀⣰⣟⡵⠛⠣⣀⠀⠻⠃⠀⢸⣿⠀⡇⠀⠘⠇⠀⣠⠜⠛⢷⣝⣆⠀⠀⠀⠀⠘⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠄⢠⡾⠟⠁⠀⠀⠀⠀⣍⡂⠂⠤⢼⣿⢸⠥⠄⠖⢀⣭⠀⠀⠀⠀⠈⠻⣷⡀⠠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠋⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⢙⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡆⠀⠀⢀⠀⠀⢸⡇⠀⠀⡀⠀⠀⢲⣀⡠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⠠⠄⢼⠅⠀⠀⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣦⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⢀⣀⣀⣰⣿⣀⣄⣀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⣀⣠⡶⠟⢻⠉⠁⠀⡙⣿⡁⠀⠉⢹⠛⠵⢦⣤⣀⡤⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⣠⡴⠛⢏⠁⠀⠀⠘⠀⣀⣀⠃⣿⣇⣀⠀⠈⠀⠀⠀⢈⠟⠷⡤⢀⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡝⠁⡀⠀⠈⣂⠤⠒⠊⠉⠀⠘⠀⣿⣻⠀⠈⠉⠒⠠⠄⡊⠀⠀⢀⠛⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠁⡤⠜⢉⠔⠉⠀⠀⠀⠀⠀⠀⡸⠀⣿⣿⡄⠀⠀⠀⠀⠀⠈⠐⢄⠙⢷⠄⠑⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢎⠀⠀⢀⠔⠉⠳⣦⡀⠀⡠⣔⡲⠘⡇⠀⣿⣿⡿⠴⣒⡤⡀⠀⣠⣴⠋⠑⢄⠀⠀⢈⢶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠁⢀⠎⠀⠀⠀⢉⡻⣿⣶⡉⠀⢸⠀⠀⣿⣿⣿⠀⠀⣁⠞⣫⣟⠁⠀⠀⠈⢂⠀⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠀⠀⠀⡌⠀⠀⠀⢀⣊⠟⢊⠻⣿⣦⣌⠀⠀⣿⣿⣿⡤⠚⣡⣾⠟⢟⣢⠀⠀⠀⠈⣄⠀⠀⠘⣣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⢘⠀⠀⠀⠀⢾⠃⠀⠀⢡⡈⠻⣿⣆⠀⣿⣿⠏⣠⣾⣿⠃⠀⠀⢣⠆⠀⠀⠀⠸⠀⠈⠉⢩⡄
⢤⡄⣀⠀⡄⠀⠀⠀⠀⡇⠀⢀⣀⣃⣠⣤⣤⣾⣷⣶⣾⣿⣿⣿⣶⣬⡿⠛⠉⠛⣾⠟⠋⠁⠉⠉⠐⠚⠛⠦⠠⠤⣀⣇⣀⠀⢈⡗⠀⠀⠀⣤⠤
⠈⣷⠙⣦⠃⠀⠴⠶⠶⣿⣿⡿⠿⠿⠷⠿⠿⠟⠻⠟⠛⠿⠛⠿⠛⢻⠀⢸⣿⠆⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡷⠶⠶⠆⣹⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡏⠑⠒⠂⠰⣤⠤⠀⢀⣀⣀⣤⣾⢷⣤⣤⢴⣟⠻⢿⣿⣿⡿⠿⢿⣿⠛⠛⠋⠉⡍⠉⠀⢸⡇⠀⠀⠀⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣄⠀⠤⠱⠀⠀⠀⠀⢛⣆⠀⠀⣼⣿⠟⢁⣾⣿⡇⠈⢿⣷⣄⠙⢄⠀⢀⣚⠃⠀⠀⠀⢰⠡⠄⢀⣞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣵⡄⠀⠀⢇⠀⠀⠀⠀⢡⣷⣾⠟⣡⠔⢻⣿⣿⡇⠀⢀⠛⢿⣷⣌⢲⣧⠉⠀⠀⠀⢀⠎⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⡠⠌⢆⠀⠀⠀⣠⣟⣥⡾⣁⡀⢸⣿⣿⡇⠀⠸⠀⣀⡽⣻⣷⣅⡀⠀⠀⢀⠎⠤⣀⡰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢯⡄⠀⢀⠳⣀⡼⠟⠁⠀⠀⠉⠺⢿⣿⣿⡇⠀⣿⠿⠚⠁⠀⠀⠙⠻⣄⡰⠊⠀⠀⣸⠟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠃⢆⡈⠳⢀⡀⠀⠀⠀⠀⠀⢻⣿⡇⢠⠃⠀⠀⠀⠀⠀⣀⠤⠊⢤⡗⣀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢷⣄⠀⠀⠀⡈⠑⠠⠄⢀⡀⢸⣽⡇⢰⠀⣀⠀⠤⠐⠉⡀⠀⠀⢀⣴⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠐⠃⣠⣎⡀⠀⠀⠀⠄⠀⠈⣿⡇⡌⠀⠀⠄⠀⠀⠀⡈⣦⡔⠃⠁⠀⠀⠙⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠐⠋⠉⠉⠓⠦⣾⣤⣀⣀⣹⣧⣁⣀⣠⣼⡦⠶⠋⠋⠉⠓⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠉⢹⣿⠉⠈⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢄⣴⠿⠛⠋⡙⠉⢹⣿⡿⣷⣆
⠀⠀⠀⠀⠀⠀⣯⠸⡇⠀⠀⣾⣷⣶⠞⠛⠓⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢧⣌⣒⠄⠙⠿⣿⣄⡀⣀⣸⣿⠇⢀⣀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠶⢿⣿⣿⣿⠿⠛⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣡⠖⣱⠺⣾⣤⣤⣴⡤⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⣠⣾⣿⣶⢿⡀⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢰⢱⣿⡿⠋⠀⠑⠉⠢⡀⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣾⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⡇⣿⡟⠀⠀⠁⠀⠀⠀⠈⠢⡀⠀⠀⢦⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⢲⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇⣿⠠⡀⠀⠀⠀⠀⠀⠀⣄⠈⠦⣀⠙⢦⣆⠫⣀⣘⡇⣀⣤⡶⠿⣿⣽⡧⠌⠻⣿⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣇⠇⢀⡄⠀⠀⠀⠀⠀⠀⠈⠓⠂⢀⣂⣈⠟⠂⠈⢉⣘⢉⣴⣒⡉⠙⠋⠉⠀⠳⡽⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣿⡄⢀⠠⠀⢀⣀⣀⠀⠀⠀⣀⣤⣾⣿⣿⣦⣤⣄⠼⢳⠉⠀⠀⠉⠀⠀⠀⠀⠀⢇⢸⣿⣿⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠘⣷⡀⢠⠠⠟⠉⣙⣿⡟⢛⡿⠃⠈⠻⣿⣿⣿⣤⣰⡞⠤⢤⣄⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠘⣿⣄⠣⡐⢾⣿⠿⢻⡧⠀⠀⠀⢠⣧⠈⠉⠙⢤⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠰⡇⣿⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡈⠻⣷⣄⠀⠁⠀⠉⠀⠀⠀⠴⠋⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⣿⠏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡼⢋⡾⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡢⠀⢀⣠⡴⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣄⡉⠛⠻⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣥⣤⣶⠾⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣄⣀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠶⠖⠒⠚⠋⠉

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡤⠷⠖⠚⠛⢸⡟⠓⠲⠶⢤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⢀⠀⠤⠠⠀⢸⣿⡆⠠⠄⡀⠀⠀⠀⠀⠈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠫⠀⠄⠁⢀⠤⠚⠁⠀⢀⣄⠀⢸⣿⣧⣀⠀⠀⠓⠀⠀⠈⠀⠀⠈⢳⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠄⠛⣦⣀⠴⠛⠉⠁⠀⢸⣿⣿⠉⠙⠢⣄⣤⠞⠡⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠚⠀⢀⡼⢻⣷⣄⡀⠀⠀⢸⣿⣿⠀⣠⣶⡿⠻⡄⠀⠘⠄⠀⠀⠀⢻⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠇⠀⠀⠀⡌⠀⢀⡞⠀⠀⠹⣿⣿⠀⢀⢸⣿⣿⣿⣿⠟⠁⠀⠹⡆⠀⢨⡀⠀⠀⠈⣧⠁
⠀⠐⣶⢲⡖⠀⠂⠀⠀⠀⠀⣼⣀⣠⣠⣴⣤⣴⣾⣶⣶⣿⣿⣿⣿⡟⠁⠀⠐⠙⡄⠀⠀⠀⠀⠀⠁⠀⠀⠲⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠹⡆⢷⠀⠀⠀⠉⠋⠉⣉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡍⡄⠀⠀⠺⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⠖⠒⠒⠂⠀⠀⣿⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠈⡄⠀⠳⡆⠀⠐⠀⣤⣤⣾⣶⡭⠔⠚⢿⣿⡟⠛⠛⢋⡏⠋⢩⡍⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⠀⠀⠽⡄⢠⣾⣿⠟⣿⣿⡇⠀⠀⠻⢿⣿⣄⢀⡾⠁⠀⡐⠀⠀⠀⢰⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠹⡀⠀⣽⠿⣋⠀⠀⢻⣿⡇⠀⠀⠀⠀⢉⡿⣿⡀⢀⠜⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣄⠀⠀⠀⠈⠻⡁⠀⠈⠓⠲⢼⣿⡇⠀⢠⠴⠚⠉⠀⢀⠹⠊⠀⠀⠀⢠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣰⠈⠀⠀⠈⠘⠢⢄⣀⠀⣿⡇⠀⠀⢀⡀⠄⠒⠀⠀⠈⠀⢄⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⠲⣄⡀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠨⠝⠲⢤⣄⣀⠀⠀⢸⡇⠀⠀⠀⣀⣀⡤⠶⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠙⠛⡇⠘⠛⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠒⠒⠁⠑⠤⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣉⣤⣶⠾⠿⠿⠿⠿⠶⣶⣤⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⡾⠋⠁⣀⡤⠶⠶⠶⠶⢤⣄⡈⠛⢷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⢠⡾⠋⣀⣴⠋⠁⣀⠠⠼⣧⠄⣀⠈⠙⢦⣀⠉⢷⡆⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠⡿⠁⢰⠏⠀⡰⠊⡡⠐⠃⣿⠂⢌⠑⢆⠀⠹⣆⠈⢿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢠⡏⠀⡚⠀⠊⢀⣠⠤⠿⣆⠀⠑⡀⢃⠀⢹⡄⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠈⠻⣿⠂⢸⡧⠶⡟⣛⠛⠻⡇⠀⠀⢸⣤⣤⣭⣽⠶⢼⡇⠈⣿⠟⠁⠀⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠘⣇⠀⢣⠈⡄⠀⠹⣶⠒⠋⠉⢠⠁⡜⠀⣸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⣷⡀⠹⣄⠀⠱⣄⠐⠤⣿⢠⠄⠒⡡⠎⠀⣰⠏⠀⣾⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⣷⡀⠉⠳⣄⠀⠙⠒⢻⡖⠒⠋⠀⣠⡾⠉⢀⣾⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠁⠈⠻⢦⣄⡈⠛⠓⠶⠶⠶⠶⠚⠛⢁⣠⡶⠟⠁⠙⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢷⣶⣴⣤⣤⣶⣶⡾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⣄⣠⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠚⢻⠁⣿⡟⠓⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⡾⠀⣿⣷⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣉⠀⠀⠀⣠⠴⠖⠛⢹⡇⠀⣿⣿⡉⠛⠲⠦⣄⡀⠀⠀⣉⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠘⢎⠑⠶⣋⠁⠀⠀⠀⢸⠀⠀⣿⣿⡇⠀⠀⠀⠈⣙⠶⠊⡱⠃⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⢧⡀⠈⠓⢦⣀⠀⡏⠀⠀⣿⣿⣿⠀⣀⡴⠊⠁⢀⡜⢧⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡞⠁⠀⠱⣄⠀⠀⠈⢻⠃⠀⠀⣿⣿⣿⡟⠁⠀⠀⢠⠎⠀⠈⢷⡀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⠁⠀⠀⠀⠘⣦⣀⣀⣼⠀⠀⠀⣿⣿⣿⣧⣀⣀⣴⠃⠀⠀⠀⠈⣧⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣠⣤⣧⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⣿⡿⠋⠀⠀⠀⠉⠉⠙⠒⠒⠶⠾⠤⣄⣀⣷
⠀⠀⠀⠶⠶⢾⣟⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣳⢯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠒⡶⠶⠤⠤⣄⣀⣀⠀⠀⠀⣠⣾⣿⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢻⠛⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⡀⠀⠀⠀⢠⠟⠉⠉⢻⣿⣿⣿⠀⠀⠀⡟⠉⠉⠻⡄⠀⠀⠀⢀⡟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⢧⡀⠀⡰⠋⠀⠀⢀⣼⣿⣿⣿⠀⠀⢰⣧⡀⠀⠀⠙⢆⠀⢀⡾⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢳⡜⠁⢀⡠⠖⠉⠀⣿⣿⣿⠀⠀⣸⠀⠉⠳⢄⡀⠈⢣⡞⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢠⢎⡠⠶⣍⡀⠀⠀⠀⢸⣿⣿⠀⠀⡇⠀⠀⠀⢀⣩⠶⢄⡱⡄⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⣉⠀⠀⠈⠙⠳⠦⣤⣈⣿⣿⠀⢸⣇⣤⠴⠖⠋⠀⠀⠀⣉⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⢿⣿⠀⡼⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣼⣿⢀⣧⡤⠤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡨⠅⠒⠒⠨⢅⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠊⠀⠀⠀⠀⠀⠀⠑⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⡀⡔⠒⠒⠒⠒⢢⢀⢇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⡇⠀⠀⠀⠀⢸⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠶⠶⠶⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⢀⣀⡀⢀⣀⡀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⠔⠂⢩⠀⠀⠀⡆⠀⠀⢉⠐⠢⣀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣠⠊⢠⡀⠀⠀⠀⠀⠀⣁⠀⠀⠈⠀⠀⢀⠑⡄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡰⠁⠀⠀⠁⠀⡠⠔⠒⠘⠛⠒⠢⢄⠀⠐⠋⠀⠈⢆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⠀⠱⣄⡀⣠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠒⠈⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡆⠀⠀⠀⢁⣦⣄⡉⠉⣳⠤⠤⡀⡀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠉⠉⠀⠠⡟⠃⠈⠒⠙⢄⡠⡿⠉⠐⠤⣸⠉⠁⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢡⠀⢀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠉⠁⠒⢒⠆⠵⠦⣄⡀⡌⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢆⠁⠀⠀⠀⠙⠢⣀⠀⠀⡀⠀⣀⠔⠋⢀⠀⠀⠀⡰⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠢⡀⠴⠃⠀⠀⠀⠈⠉⠽⠁⠀⠀⠀⠀⠙⢂⠔⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠈⠢⢄⠀⠰⠀⠀⠀⣆⠀⠀⠀⠃⡠⠔⠁⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠈⠑⠂⠀⠤⠤⠀⠐⠊⠉⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢤⣤⣄⣀⣀⣠⣠⡤⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣌⣉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⢿⣿⣿⡛⠛⠋⠛⠛⠛⠿⢿⣷⣦⣙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠋⠉⠀⠀⠸⠇⠀⠀⠀⠀⢀⡌⠙⢿⣷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠁⠀⠀⠀⢤⠀⠀⠀⠂⠀⠀⣤⣴⣿⠀⠀⠀⠙⢿⡷⠈⢿⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⢀⢴⡁⠀⠸⡄⠀⠀⠀⢀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠹⠀⠈⢿⡆
⠀⠀⠀⠀⠀⢠⣿⡿⡷⠀⠀⡠⠁⠀⠹⢷⣤⠗⢠⡅⠀⢚⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠑⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣾⣿⣷⠂⠀⠀⠉⠓⢤⣀⢘⣧⣫⣺⠁⢀⣾⣿⠟⠷⠖⠚⠋⠁⠀⠀⠀⠀⠄⠀⢸⣇
⠀⠀⠀⠀⠀⣿⣿⠄⢠⣤⠔⠀⠀⠀⠋⠁⠈⠉⢹⡟⠛⢻⡏⠛⠃⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⠀⣿⣿⠇⠀⠉⠠⠀⠀⠀⠀⡰⢖⠶⣾⣷⣶⣾⠷⣦⠰⢦⣄⠀⠀⠀⠉⠀⠈⠀⠀⢸⣿
⠀⠀⠀⢸⡆⢿⣿⡄⠀⠀⠀⠠⠒⠒⠋⢱⣯⣷⣿⣿⠉⡏⢪⣙⣆⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠈⣿⡘⣿⣧⠀⠀⠀⠈⠂⢀⡼⠛⠑⠋⣿⡿⠀⠀⢹⡄⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠘⣷⡘⣿⣧⠀⠀⠀⠀⠉⠠⠀⠀⠀⣸⠀⠀⠀⠀⢇⠀⠀⠁⠀⠀⠀⠀⠐⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢷⣌⢿⣷⣔⠀⠀⢀⠀⠀⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣧⣝⢿⣷⣤⡀⠰⠐⠃⠀⠄⠈⠁⠀⠀⠀⠀⠀⢀⡀⠐⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣬⣛⠿⣿⣶⣤⣤⣄⣈⣀⣀⣤⣤⣶⡾⠟⣂⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣮⣭⣍⣛⣛⣛⣋⣩⣭⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⢈⠀⠠⠤⠼⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠋⢀⠔⠈⠀⠀⠀⠀⣆⠀⠀⠀⠈⠙⠻⣿⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠂⠀⠀⠀⠐⢦⣀⠀⠀⢸⡀⠀⠀⣠⠂⠀⠀⠀⠹⣿⣆
⢠⡄⠀⣤⠀⢀⠀⠀⠀⠀⢀⣠⣴⠀⡀⠀⠀⠀⠀⠀⠀⠙⢿⡆⢸⡇⣠⣾⠃⠀⠀⠀⠀⠀⢿⣿⠤⢀⠀⠀⠀⠀⠀⠀⣤⡀⢠
⠈⣿⡀⢿⣆⠀⠀⢠⡴⠾⠿⠿⠿⠀⠇⠀⢤⠤⠤⠶⠶⠶⠆⣴⡏⠹⣏⣁⣀⣀⣀⣀⡤⠀⢸⣿⣇⣀⣈⣀⣠⠄⠀⠀⣿⣤⡄
⠀⠘⠏⠀⠿⠀⠀⠀⠀⠑⠂⠄⣀⠀⠇⠀⠀⠀⠀⠀⢀⣶⠦⢙⢣⣼⡏⠉⠉⠁⠀⠀⠀⠀⣸⣿⣿⠿⠛⠉⠀⠀⠀⠀⠿⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠀⠀⠀⠀⢀⡞⠁⠀⢹⡇⠉⠻⢦⣀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⠀⠀⠈⠀⠀⠀⠸⡇⠀⠀⠀⠈⠁⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠈⠢⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠛⠠⢀⠑⠀⠀⠀⣀⣁⣀⣀⣤⣴⣾⡿⠟⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⣤⣤⡼⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⠀⠀⠀⠀⢰⡇⢠⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣍⠛⠶⣄⡀⣾⠀⢸⡇⠀⠈⣧⠀⠀⣀⣤⠴⢺⡿⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢷⣄⠀⠙⢧⠀⢸⡇⠀⠀⣸⠖⠛⠉⣠⠞⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢧⡀⠈⢇⠸⡇⠀⣰⠃⠀⣠⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠧⣄⡀⠀⠙⣦⡘⣦⡇⢠⠇⣠⠞⠁⠀⢀⣰⣃
⠀⠀⢀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠉⠛⠶⢬⣻⣿⣧⣿⣞⣡⠴⠖⠋⠉⠈⠉⠙⠛⠲⠦⢤⣄⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠈⣿⣼⢷⣼⠋⠀⢠⣴⣶⣯⣭⣤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⣶⣾⣿⣿⣿⣿⣖⠲⠶⠒⠶⠶⠶⠶⠶⠲⠶⠶⠾⢿⣿⣶⠦⠀⠀⣿⠤⠤⠁
⠀⠀⠀⠸⠇⠈⠟⠀⠀⠀⠈⠉⠛⠓⠲⠦⢤⣤⣀⣀⣀⣠⡴⠖⠋⣩⠞⡟⣿⢻⠳⣌⠙⠓⠦⣄⡀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⠀⠀⠀⠻⠶⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠁⠀⣠⠞⠁⡸⠁⣿⠈⢧⠈⢳⣄⠀⠀⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠞⠁⠀⢠⠇⠀⣿⠀⠈⢧⠀⠙⢷⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠁⠀⠀⣀⡞⠀⠀⣿⠀⠀⢘⣧⣄⠀⠙⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣁⣤⡴⠖⠛⠉⣧⠀⠀⣿⠀⠀⣸⠁⠈⠛⠶⣄⡙⢦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠄⠀⢻⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣿⡀⢀⣸⠷⠀⢀⣸⣗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⢡⡄⢸⠄⣷⠀⠘⡯⠛⠁⣰⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⠶⣄⣣⣼⣠⠓⣄⢠⠃⣴⣎⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣧⠈⢿⡟⠹⠿⣿⡿⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣮⣹⣿⣿⣿⣿⣯⣤⣤⣤⣬⣶⣶⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡶⠊⣉⡉⠗⢈⣡⣴⣶⣶⣿⣟⣟⣻⢯⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠞⠁⣠⣾⣿⣿⡟⠋⠀⢸⣧⠀⠀⠉⠐⢩⣟⣧⣷⣄⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⠀⠀⢠⣾⣿⣿⡿⠋⠁⠀⠀⣸⣿⣅⣀⠀⠀⠀⠈⠱⣟⣽⣧⣄⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣹⡆⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⣿⣷⠀⢀⡀⠀⠀⢀⣤⢾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠀⢀⣿⣿⣿⣿⠁⡄⠀⢳⠀⠴⠀⣿⢻⣣⣿⡇⠀⠀⠁⠀⠈⡽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢸⣿⣿⣿⣯⠈⠁⠀⠂⠋⢁⠀⣻⣾⣿⣿⠿⣦⣤⣤⣄⡀⣙⣿⣿⡇⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⣿⣿⡟⠙⠛⠻⠿⣿⣭⣙⠬⣿⣿⣿⣦⣤⣶⡎⠹⠏⠁⣩⣽⣿⡟⠂⠰⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣿⣿⣧⣀⠀⠀⠀⢀⣴⣿⠀⢹⢸⡯⢭⣿⠉⠛⠂⠀⢀⢶⣿⣿⠇⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠉⠁⠈⠀⢸⣿⠃⠀⠈⠡⠀⠀⢀⣈⣿⣿⡿⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡦⡀⠀⠀⠀⠈⠁⢀⢰⡟⠀⠀⡀⠀⠀⣠⣬⣭⡻⡟⠁⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣆⣄⢀⠀⠀⠀⣿⠁⠀⣀⣣⣤⣿⣿⣿⣿⢻⣈⣧⣳⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣷⣽⣷⣿⣝⣻⣿⣿⡎⣼⣿⡾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣯⣿⡿⣿⠿⠿⠿⠛⠛⢿⣾⣯⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠁⠀⠀⠀⠠⠄⠀⠀⣵⠋⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢨⣿⠏⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⢈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⣠⠎⣻⠆⠀⠀⠀⠀⠀⠀⠀⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣀⣻⠆⠛⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠛⠻⠷⢶⣤⣶⠶⠟⠉

⠑⢄⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣢
⠀⠀⠑⣄⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠠⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⢣
⢀⠤⠒⠉⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠁⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⣀⠠⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠒⠄⡀⠀⠀⠀⠃
⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠉⠒⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡤⠤
⠒⠄⠀⠀⠀⠀⠀⠀⠀⢐⣀⠤⠤⠒⠊⠉⠀⠁⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⡀⠤⠐⠒⠉⠁⠀⠀⠀⠑⡄
⣀⣀⢠⡠⠤⠴⠒⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⢃⠀⠒⠂⠈
⠢⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⠠⠀⠀⠉
⠰⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⢀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠈⠁⠀⠐
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢤⠀⠀⢀⠙⡴⠢⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠐⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣶⠒⢂⠀⠀⠈⣠⡴⠂⡀⠂⠀⠀⢤⠄⠠⠀⠀⠀⠐⠀⠥⠤⠎⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣡⣦⢠⡀⢸⠀⢸⠀⠀⡀⠓⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⣀⠄⠰⠫⠄⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣬⣭⣛⠺⡁⠀⡙⠎⠀⠄⠀⢀⡸⠠⣀⠈⠀⠀⠀⠀⡄⠀⠀⠦⠀⠊⢡⣈⠀⠆⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠉⠀⠀⢸⠏⠙⠖⡄⠀⠀⠁⠐⠀⠀⠁⠀⠀⠀⢀⠀⠄⠰⢅⠀⠀⠀⣠⠦⠀⠀⠍⠛⡐
⠀⠀⠀⠀⠀⠀⠀⣼⠉⠄⠘⠵⢦⠋⠀⠀⡀⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠎⠀⠀⠀⠁⠒⣦⠀⣾⢁⠃
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢀⡠⠚⠁⢀⡠⢊⢴⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠂⠀⠋⢊⠃⠄
⠀⠀⠀⠀⠀⠀⡈⠠⣄⡠⠭⠭⠀⢐⣈⢔⣣⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢲⠂⠀⠀⠀⠀⠀⣇⠌⡐
⠀⠀⠀⠀⢀⠎⠘⢆⢤⣉⠭⢿⡯⠭⢀⣻⣹⠄⠐⠂⠀⠁⠀⠀⠀⠀⠀⡐⣱⠒⠒⠚⡆⠀⠀⠰⣿⣆⡼⣰⠃
⠀⠀⢀⠀⠧⠤⢀⣀⣁⠀⠀⠀⠉⠒⡊⢁⠟⡀⠀⠀⠀⠀⠄⠀⠀⡀⣘⠢⣸⣐⣛⣙⣃⠛⠀⡀⣛⣛⢃⠏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⢓⠂⠄⢤⡆⢀⡀⠠⠠⠐⠐⠀⠉⠁⠀⠈⠢⡈⠢⡀⠀⡏⠀⠀⣀⢧⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡈⠢⡂⠀⠫⢄⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀
⠠⡀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠠⠊⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠑⡆⠂⠉⠀⠒⠒⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠂⠉⠁⠀⠈⠐⠤⡀
⠐⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠤⣀
⠀⠐⡄⠀⠀⠀⠈⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁
⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠑⢄⡀⠀⠀⢈⣆⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠑⢄⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣠⠦⠆⠐⠐⠒⠒⠒⠒⠒⢄⡀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡔⠉⠉⠂⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⡠⠊⠈⠀⠀⣀⠠⠀⠘⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⣀⠔⠁⠀⠀⠀⡀⣀⠠⠤⠄⠐⠓⢤⡀⠀⢠⢞⠘⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⠢⡀
⠀⠉⠉⠁⠀⠀⠀⠒⠂⠤⠔⠠⡀⢀⣣⣤⠮⠤⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⢈⠟⠀⠈⠛⠁⡱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠎⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢰⠀⢀⡀⠠⠠⠀⠒⠂⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⡠⠔⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠊⠐
⢀⠄⠀⡒⠨⠆⠒⠉⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠃
⠐⠈⠁⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠆
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⢩⡍⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣿⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢀⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣤⣤⣀⣈⣁⣀⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⢛⣻⣿⣶⣶⣤⣄⡀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢛⣩⣄⢰⣟⣿⠍⠉⠉⠉⠉⠻⢦⣴⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠺⣿⡿⠛⠉⠉⠀⢀⣀⡀⠀⡀⠀⠙⣧⡈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢠⡟⣻⡶⣞⢷⣷⣤⣼⠿⢿⡞⢿⡅⠀⠈⢷⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⠟⠛⠋⠙⢿⣧⣤⣽⣿⡷⠀⠀⠀⠀⠀⠘⣧⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠰⢿⣶⡴⣄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣴⠟⠀⢻⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⢷⣄⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠀⠀⢸⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣆⠀⠀⠈⠉⠀⠈⣿⡀⠀⠀⠀⠀⢠⡶⠋⠀⠀⢀⡿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⣰⡆⠀⣰⠟⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠻⢦⣄⡀⠀⠀⠀⢿⣄⠀⣠⡟⢹⣋⣴⠞⢉⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠙⢶⣄⡉⠛⠳⠶⢦⣤⣿⣿⠿⠶⠛⢋⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠻⠶⠦⣤⣤⣤⣤⡴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣄




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

⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⢄⠀⠀⠑⠀⢀⣀⣀⣇⠀⠂⠁⠀⠀⣠⠀⠀⢠⠃⠀⠀⡴⠁⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢀⡠⠷⡊⠉⠉⠀⠀⣀⣈⣦⣀⣀⡀⡔⠁⠀⢀⠎⠀⢀⠞⠀⠀⢈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣏⡠⠔⠊⠁⠀⠀⠘⡤⠒⠒⠉⠁⠀⠀⠂⠀⠈⢻⠒⡤⣋⠀⣠⠋⠂
⠀⠀⠀⠀⠔⠚⣇⡇⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⡀⠀⠈⠀⠀⢜⣔⠁⢀⡽⠧⣀⡀⢀⢂⠀⢀⣀⡠
⠀⠀⠀⠀⠉⠐⠪⠄⠀⠄⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢄⢀⡴⠋⠈⣱⠮⣄⡀⠀⠈⢸⠁⠉⠄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣠⠜⠫⡀⢀⡜⠉⢣⠀⠉⠓⢦⣞⣀⣀
⠀⠀⠀⠀⣀⣤⢤⠧⠴⠦⡄⢀⣀⣰⠚⡟⠛⠋⠉⠉⠀⠀⠀⡼⠋⠀⠀⢨⡀⠀⢀⡜⠀⠀⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡏⠉⢇⠀⠀⡇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣸⠈⠑⠼⡀⠀⠀⠀⠀⢀⠢
⠀⠀⠀⠀⡀⠤⡔⢳⢊⣹⣇⣠⡼⡤⠤⢵⠴⠶⠶⠖⠚⠁⠀⠈⠠⠀⢠⠃⠀⡀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⣱⠋⠉⠀⠳⡀⠀⢱⠀⢸⣀⣀⣀⣀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠠⠌⠀⠈⠣⡀
⠀⠀⠀⠀⠀⢠⠟⠦⣀⠀⠀⢘⣤⠔⠇⠀⠀⠀⡜⠀⠉⠒⠤⣀⣠⠃⠀⠀⠀⠐⠀⠁⠀⠀⠀⡀⠘⠢
⠀⠀⠀⠀⢠⠏⠀⠀⠈⢢⡔⠁⠈⠣⡘⠀⠀⢸⠀⠀⠀⠀⠀⢠⠋⠑⠢⠄⣀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠳⣤⣄⣀⡙⡇⠀⠈⡄⢀⣀⣀⣠⣃⣀⣠⣤⣤⡤⡭⠶⠤⣤⡶⠖⠒⠛⠉
⠀⠀⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⣏⠉⢹⠋⠉⠉⢁⠏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠢⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠢⠂⠃⠂⠘⠞⠚⠘⠂⢀⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠈⠁⠈⠀⠈⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠼⠔⠀⠉⠁⠈⠀⠀⠐
⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠂⢄
⠀⢀⠄⠪⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⢸⠀⠀⠀⢀⠄⢀⡀
⠀⠈⢄⠀⠀⠀⠈⠐⠢⠤⡀⡀⠤⠐⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠘⡀⠀⠀⠀⢸⠠⠐⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⣰⡤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠠⠀⠈⠢⡀⠀⠀⠀⠀⢀⠤⠐⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⢢⡤⠐⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠂⠁⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠄⠀⢀⠠⠐⠈⠀⠀⠠⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡁⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠁⢙⠁⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⣠⣤⠴⠶⠶⠶⠒⠲⠶⢶⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢿⢻⣤⡀⠂⠉⠀⠀⠀⡀⠐⠂⠄⠀⠀⠍⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⠁⠅⣿⣷⠚⠃⢀⡀⢰⣟⡷⣀⡀⠀⠀⠀⢀⣤⣌⣡⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠅⠉⠀⠀⠀⢸⠟⣷⠦⣼⢳⣞⠙⢚⣉⡽⠂⢀⣁⠈⠉⠙⡆⠙⠶⣤⡙⢦
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⣀⢠⠅⠀⢉⠸⣾⠽⣶⠲⠶⠶⠚⡉⠀⠀⠀⢄⠙⠻⢿⣾⣡⣤⠢⠀⠙⠛⢳⡀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⣼⠁⠜⠀⠁⠀⠀⠀⠀⠀⣼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⢼⠀⠀⠀⠈⢀⣹⡤⠖⠋⠀⠈
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⠀⠀⠀⢣⠇⠀⠀⠘⣆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠴⠞⠉⠀⢳
⠀⠀⠀⠀⢠⡇⠀⡄⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠄⠀⠀⢀⣠⡴⠖⢻⡉⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣀⠽⡞⠓⢦⣀⡤⡶⠚⠉⠀⠐⠀⠀⢧⠀⣀⠀⡀⠀⠂⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠉⠛⠋⢁⣠⠽⡖⠋⠛⢦⡸⡀⠀⠀⠀⢀⠀⢸⠸⡏⡇⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⢀⣀⡤⠶⠛⠉⠀⠀⢻⡄⠀⠀⠹⣧⠀⠀⠀⠈⢀⡏⠈⠷⠃⠀⠀⡄⡏
⠀⠀⠀⠀⠸⡶⠒⠛⠙⣆⢀⣀⣠⠴⠚⠉⠁⠀⠀⠂⠀⠀⠀⠀⢳⡄⠀⠀⠻⣄⡀⠀⠀⡟⠀⠀⠄⠀⠂⠀⢰⠇
⠀⠀⠀⠀⠀⢷⣀⣀⡵⠞⠛⠷⠶⠲⣄⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢳⡀⠀⠀⠈⢳⡀⢰⠏⠀⠀⠀⠀⠀⢀⡟
⠀⣠⡀⣤⠴⠚⣯⠁⠀⠀⠀⠀⠡⡀⠸⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠗⠋⢘⠀⢀⣀⠄⢀⡿⠁
⠙⣿⠷⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠘⠄⢹⠀⠈⠀⠀⠀⠀⠁⠄⠀⠀⠈⢧⠀⠀⠈⡀⠀⠔⠀⠈⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠳⣄⠠⠀⠀⢹⡄⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠈⢷⠀⠀⠁⠀⡀⡰⢒⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣈⠠⠄⢻⡅⠀⠀⠀⠀⣀⠀⠀⠃⠀⠀⢈⡧⠤⠶⠾⢶⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣟⡒⢮⣳⣤⣄⠉⠀⠀⢀⣀⡤⢀⡽⠋⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢧⣤⣬⣥⣬⠤⠾⠗⠚⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣫⠵⠮⣝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣻⣿⣿⣟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠾⠛⠛⠛⠛⠷⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⣁⠤⠀⠒⠂⠀⠀⠐⠒⠀⠤⣈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠔⠉⢀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠢⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⠎⠀⠀⠀⣸⣷⠎⠁⠀⢸⡇⠀⠈⠁⠂⢄⠀⠀⠀⠱⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡰⠁⠀⠀⡠⠊⠀⠀⢂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠁⠀⠀⠌⠀⠀⠈⢢⣄⣠⠄⠚⠓⠠⢄⣠⡔⠁⠀⠀⠣⠀⠀⠈⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠘⠀⠀⠀⠀⢀⠋⣴⣿⣿⣿⣷⣦⠙⡄⠀⠀⠀⠀⠂⠀⠀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⣤⣤⣤⣴⡆⢸⣿⣿⣿⣿⣿⣿⡇⢰⣶⣤⣤⣤⣤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⡄⠀⠀⠁⠀⠀⠀⠀⢡⠘⣿⣿⣿⣿⣿⡿⠃⡌⠀⠀⠀⠀⠀⠀⠀⢠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢡⠀⠀⠈⡄⠀⠀⠀⢠⠷⠌⡙⠛⠛⢛⡤⠾⡄⠀⠀⠀⢠⠁⠀⠀⡌⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⠘⢄⠀⠐⠁⠀⠀⠀⢹⡏⠀⠈⠄⠈⠂⠀⡠⠃⠀⠀⡰⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠢⡀⠀⠀⠑⢄⡀⠀⠀⠀⢸⡇⠀⠀⠈⣦⣤⠊⠀⠀⢀⠔⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠐⠢⡀⠀⠀⠈⠁⠂⠀⠸⠇⠀⠐⠘⢿⠟⠆⢀⠄⠂⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⡈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⢁⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢧⣂⡄⠀⣉⠁⠀⠀⠈⢉⠀⢠⣐⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠲⠖⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⡀⠀⠀⣤⣤⣤⣄⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠒⣀⣀⣀⣀⣠⣶⣤⣤
⠀⠀⠀⠀⢐⣿⣿⣿⣷⣤⣤⡘⣷⣍⢠⢵⣼⠋⠐⢿⣿⣿⣿⣿⡖⠀⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⠿⠿⣿⣿⣿⣿⣿⣿⣯⣼⣬⠟⢷⡀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣀⡀⠀⢂⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠁⠀⣤⠛⠁⠀⣿⠟⠉⠠⠤⠀⠀⢀⣼⢿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⡄⠘⣻⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⠿⠉⠂⠀⠀⠀⠀⠀⠀⠉⢹⣾⣿⣿⣿⠿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⣁⡈⠈⠛⠶⢶⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣷⣶⡿⣿⣿⣛⡿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⠿⠟⠁⠀⠀⢻⡏⠀⠘⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠁⠀⠀⢤⡄⢀⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠃⠃⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⠃⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡄⠀⠀⠀⣇⠀⠀⢠⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⡤⠖⠚⠋⠉⠀⣿⠉⠙⠓⠲⢤⣀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠲⣄⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⠀⠀⢸⡇⠀⣿⣧⢰⣶⣤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⡼⠁⠀⠀⠈⠲⣾⡿⠂⠀⠀⠀⠀⣿⣿⠈⠉⠋⠀⠀⣠⠂⠀⠀⠘⢦⠠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡀⠀⠈⠳⣦⣀⠀⠀⠀⣿⣿⡇⠀⠀⣠⡞⢿⣷⠄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢰⠃⠀⠀⢠⣿⡿⠀⠀⠀⠈⠻⣧⡀⠀⣿⣿⠃⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⡆⠛
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣤⣬⣷⣶⣶⣾⣿⣦⣌⡳⡀⣿⣡⠞⠋⠁⠀⠀⠀⠐⠛⠇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⠇⡧⠀⠀⠀⠐⠚⠛⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⢛⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠒⠂⠀⠀⢠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠸⠖⠂⠀⠀⠀⣠⣴⠞⣡⣿⠀⢳⣌⠙⠻⠛⠛⠛⢹⣍⡉⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⡆⠀⠀⠀⠀⣀⡄⢀⣼⠟⠁⢰⣿⣿⠀⠀⠹⣷⣄⠀⠀⠀⢿⡿⠀⠀⠀⢰⠃⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠻⣿⠟⠁⠀⠀⠸⣿⣿⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠙⣆⠀⠀⠐⠁⠀⢀⣶⣦⡄⣿⣿⠀⢀⡀⠀⠘⣿⠟⠓⠂⠀⠀⣰⠋⠘⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄⠀⠀⠀⠀⠉⠛⠃⢹⣿⠀⠸⠗⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠦⣄⡀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⠉⢛⡒⠶⠤⣿⠀⠤⠶⢒⡛⠉⠀⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢸⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⣀⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⢀⣠⡶⠛⠉⣙⣼⡦⠔⠂⠀⠉⠀⠀⠠⢀⡘⢋⡉⠬⠀⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡴⠛⠁⣠⡴⠛⠡⠞⠁⠀⠀⠂⠀⠀⠀⡆⣀⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡾⠉⢀⡴⠛⠋⣁⠀⠁⠀⠀⣀⠄⠤⢀⢠⠠⠱⠛⢻⠗⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠉⢢⡄
⠀⠀⠀⠀⢠⡞⠁⠀⢀⡴⠋⢀⠔⠁⠀⠀⠈⠁⠀⣀⡤⠊⢡⣂⣸⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠀⠀⠀⠀⠙⣃
⠀⠀⠀⢠⠏⠀⠀⢠⠎⢀⡴⠃⠀⠀⠀⠀⣠⠆⢰⣣⡶⢁⡨⠜⠋⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠫⡆⠰⡀
⠀⠀⢠⠏⠀⠀⣠⠋⢠⠞⠀⠀⠀⠀⠘⣷⢃⠀⠿⠩⠐⠉⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠒⡜⢄
⠀⢠⡏⠀⠀⢰⠃⢠⠏⠀⠀⠀⠀⠀⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠤⡀⢀⡠⠀⠀⠈⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠐⠈⠆
⠀⡼⠀⠀⢠⠏⢠⡏⡀⠀⠀⢠⡶⠖⣣⢦⢿⣠⣤⡀⠀⠀⡠⠲⡿⠾⣅⠀⠐⡁⠲⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠘⡄
⢠⠇⠀⠀⡾⠀⣼⠀⠀⠀⠀⣷⡵⠋⠁⠀⠀⢻⣧⢽⣦⣤⠷⠔⠒⠄⠖⠁⠀⢸⡂⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⢱
⣸⠀⠀⠀⡇⢠⡇⠀⠀⠀⡠⠗⠓⠒⠈⣳⠀⠀⠃⠘⠏⠘⢦⡶⠶⡖⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⠁⠈
⡏⠀⠀⢠⠀⢸⠁⠀⢀⠼⠆⠀⠀⠀⠀⠉⠢⡀⢠⡤⣄⠀⠀⢀⣀⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠳⣤⠀⢸
⡇⠀⠀⢸⠀⢸⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠈⠠⣤⣆⠀⠀⠀⠀⠩⡈⠙⠶⢳⠤⠒⠋⠉⠣⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⢾⠙⠰
⡇⠀⠀⢸⠀⢸⠀⣿⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢶⣀⠀⠀⠀⠘⠉⠚⠀⢁⡆⠀⠀⠀⠀⠙⢦⡦⠐⡇⠀⠀⠀⠀⠐⢠⢸
⢳⠀⠀⠀⡆⠸⡇⣼⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠻⣦⡀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠹⣀⠀⡀⠀⠀⠀⢐⣠⡀
⢸⡄⠀⠀⣣⠀⣧⠘⣿⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢿⣆⡀⣀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠈⢽
⠀⢧⠀⠀⠸⡄⠸⡆⠘⢿⢇⣐⠀⡀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠷⢙⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠸⡆⠀⠀⠱⡀⢻⡄⠈⠻⠞⠻⣭⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠹⡄⠀⠀⢳⡀⠳⡄⠀⣀⠀⠀⢉⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠹⡄⠀⠀⠱⡄⠹⣆⠛⣤⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⡤⡌⠀⠀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠹⣆⠀⠀⠘⢦⠈⢳⣌⠳⣄⠀⠘⢧⣷⣀⠀⠀⠀⠀⣤⡙⠁⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠑⢄⠙⠳⣌⠃⠀⢈⣿⣩⠃⠀⠀⠀⠀⠃⠰⣄⠈⢣⡀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠑⢦⡈⠻⢦⣄⡙⠷⣤⠀⡂⠀⠀⠀⠀⠉⣠⠔⠁⠀⡄⠀⠀⠀⠀⠀⠀⠊⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠓⠤⣈⠛⠶⢮⣿⣧⣄⡀⣀⡀⢰⡇⠠⠄⠈⠟⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠉⠒⠠⠌⣉⡙⠛⠻⠿⠿⠤⠤⠤⠤⠤⠒⠒⠚⣉⡡⠼⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠈⠉⠁⠀⠀⠀⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠠⣿⡇⠀⠀⠸⠄⠠⠤⠤⠄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠁⠀⠀⠀⠠⠤⢤⣴⡿⡇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠂⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣈⠉⠑⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠉⢁⣀⡤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣿⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠟⠛⠛⠛⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⢁⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⡏⢹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣴⣿⣿⣿⣿⣿⠟⢋⣥⣶⣶⣿⠃⠘⣿⣶⣶⣬⡙⠻⣿⣿⣿⣿⣿⣧⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣷⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⣿⣿⣿⣿⠏⣴⣿⣿⣎⠛⢿⣿⡇⣀⢀⢸⣿⡿⠛⣱⣿⣿⣦⠹⣿⣿⣿⣿⣷⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣧⡀⣸⠃⠈⠓⠘⣇⢀⣼⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⣿⣿⠥⠿⠟⠛⠛⠉⣍⡁⠀⡖⣡⣌⠢⠀⡬⠉⠉⠛⠛⠻⠿⠬⣿⣿⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣿⣿⣷⡒⣶⣤⣤⣄⣀⣉⠉⠀⢦⣛⣛⡴⠀⠓⣀⣀⣠⣤⣤⣶⢲⣾⣿⣿⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⡿⠁⢹⡄⢠⡀⢠⡏⠈⢿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣿⣿⣿⣿⣆⢻⣿⣿⢟⣠⣶⣿⡇⠐⠁⢸⣿⣶⣄⡻⣿⣿⡟⣰⣿⣿⣿⣿⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⢿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⡿⠁⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢻⣿⣿⣿⣿⣿⣦⣌⡛⠿⢿⣿⡄⢠⣿⡿⠿⢛⣡⣴⣿⣿⣿⣿⣿⡟⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣇⣸⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⢁⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋⠉



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠀⠠⠀⠀⠜⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⢀⠀⠀⠀⢀⣠⣆⠀⠀⠠⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠠⣶⡀⠀⠈⠀⠀⢸⡇⠀⢀⠂⠀⢐⣤⠈⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⡀⠀⠈⠀⣤⡀⠀⠀⣿⢀⠀⠆⢀⣤⠈⢁⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⡀⠀⠓⠠⢄⡈⢫⠓⢴⣿⢨⡴⢂⠝⠁⠀⠀⠤⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⣀⣄⣠⣤⡀⠀⠛⢓⡞⣭⡙⣿⣷⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠐⠀⠀⠀⠉⠉⠉⠛⠛⢻⡿⣦⣙⣡⠧⡤⠀⠐⠊⠉⠙⠉⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢡⠠⠄⠀⠀⢀⡾⠕⠋⢸⡿⠒⢼⣄⠀⠀⠀⠄⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠠⠀⣀⠀⠋⠀⠀⠀⢸⡇⠀⠀⠉⠃⣀⠂⡀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠛⠏⠀⠀⢀⠀⠀⣿⠁⠀⢀⠀⠀⠹⠋⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠂⠀⠀⠀⠈⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣤⣀⣀⣀⠀⠀⣀⣀⣠⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⠿⠿⠿⠛⠋⠁



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

⠀⠀⠀⠀⠀⢀⡀
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠐⠒⠒⠀⠀⠂⠀⠀⠀⠀⣀⠄⣀⠀⠀⠀⢀⣞⠻⠾⠗⠈⠱⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⢰⠂⠀⠁⠈⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠌⠁⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠰⠀⠀⠀⠂⠀⠀⠒⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠀⠀⠀⣦⠁⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠠⡀⠀⠀⣀⠤⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣺⠇
⠀⠀⠀⠀⠀⠀⢰⣧⣀⣀⣀⣀⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣼⡏⠁
⠀⠀⠀⠢⠀⠀⠀⠀⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠰⠈⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡐⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠘⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⠀⠀⠀⢠⡔⢠⡼⠄⠀⡉⡏⠧
⠀⠀⠀⠀⠀⠠⠀⠀⣤⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⠟⢠⠋⠛⠀⠀⠀⠀⠀⠀⣉⠁⡉⠀⣽⠟⢟⡞
⠀⠀⠸⠀⠀⠀⠈⡉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣩⢇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠰⣄⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⢽⡄
⠀⠀⠀⠑⠄⠀⡀⠀⠀⠀⠀⠀⠘⠦⠦⠐⠋⢁⡀⢀⣀⠀⠀⠉⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣠⣤⣤⣤⡞⠫⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠁⠙⣷⣶⣄⡤⣦⣦⣤⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⣿⣿⣿⣦⣾⡟⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠛⠻⠻⠛⠻⠻⣿⣿⣿⣤⣶⣤⣤⠀⢀⠀⠂⠁
⠀⠈⠛⠛⠛⠛⠛⠛⠓⠚⠒⠒⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠾⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡤⠖⠚⢹⡇⠉⠒⠦⣄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⢷⣄⣀⠤⣾⡇⠠⢄⣀⣴⠟⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⠿⣿⣦⣿⡇⢠⣾⡿⢏⠀⠈⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣘⣁⣀⣈⣀⣈⣉⣻⣇⣼⣿⣷⣾⣦⣤⣼⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠹⡉⠛⢻⠛⣻⣿⠋⣿⣦⣠⡀⢠⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢈⣿⡿⠛⠀⣿⡟⠻⣿⣎⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢟⠉⠈⠙⠀⣿⠇⠉⠈⢙⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠒⠤⠄⣿⠤⠴⠚⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⣆⢀⣀⣀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠉⠛⠛⠋⠩⢕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠑⠌⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⡀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⡀⠈⢊⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠈⠀⠀⠀⠀⡀⠀⣾⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢐⣤⣄⣀⡘⣿⣿⠌⣀⣤⡤⠖⠂⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠀⠀⢧⣧⠠⠤⢬⣤⣬⣭⣴⣾⡋⣻⠿⠿⠿⠶⠶⠖⠀⠀⡧⡃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠈⠀⠀⢈⣠⡴⢿⡟⢋⢽⣿⡇⠈⠛⠲⠤⠀⠀⠀⠁⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢱⠀⠀⠀⠀⠀⠀⠐⢦⡛⢀⣿⢸⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣥⠤⠤⣤⣄⣀⡀⠀⠉⠜⠣⢸⡏⠈⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡉⠁⣦⡀⣶⠀⢀⣶⡀⣉⠙⠒⢒⠶⢶⣶⠒⢒⡚⠛⣿⠟⢛⠒⣲⡾⢥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣯⠳⣿⠀⣾⣿⣧⢹⣠⠃⢸⠰⡏⢬⡄⣾⣷⡄⢸⠀⣼⠸⣅⡅⢰⣎⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣈⣇⣙⣠⣬⣤⣅⣀⣉⣘⠃⠀⢘⣀⣙⣯⣤⣿⣶⣻⡿⠟⠋⠛⣳⣶⣥⡋⢰⠃
⠀⠀⠀⠀⠀⠄⠘⢛⣿⣿⣿⣿⣿⣿⡆⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠸⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠉⠀⠈⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣦⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⣠⣤⣤⣤⣤⣤⡀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⢹⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠛⠋⠁⠰⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣤⡀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⢀⣈⣉⡉⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠉⠉⠉⠀⠁⠀⠐⠂⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢤⠞⠂⢀⢠⢤⢲⣐⣰⣐⣒⣒⣦⡤⣄⣀⠀⠀⠀⠀⠁⠒⠤⣀
⠀⠀⠀⠀⠀⢀⠔⠁⢁⠤⣞⣭⠶⠛⢋⠍⠩⣭⣭⣭⠭⠽⢿⣷⣿⣷⣦⣤⣀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⢀⠔⠁⢀⡔⣭⠾⢋⣤⣶⡯⠷⠒⠊⣍⠍⠉⠉⠉⡽⣛⡿⠭⣙⣻⣿⡿⣦⣄⠀⠀⡈⢂
⠀⠀⢠⠂⠀⡠⣱⠟⢁⠔⡡⠚⠁⠀⠀⠀⠀⠟⠓⠀⠀⣴⢰⠍⠃⠀⢀⠉⠛⠯⣟⠿⡷⣄⣊⢄⠙⢦
⠀⢠⠁⠀⡘⡵⠋⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠼⢸⡃⠀⠀⠀⠀⠀⠠⠈⠳⣌⣢⡳⡅⡀⡈⢷⣄
⠀⠇⠀⣘⣽⠉⡔⡷⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⣪⠅⡾⢀⡀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣷⣵⡨⡪⠈⢯⣆
⠘⠀⢀⢫⡇⣰⣴⠁⠀⠀⢀⠤⡀⠔⠀⠀⠀⢀⠏⣜⣦⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡎⢧⢢⢕⠘⡚⡆
⡀⠀⠸⣹⠂⠿⡇⠀⠀⠀⠀⠈⠀⠀⣦⣤⣠⢮⢚⠳⣝⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢶⠘⣴⡀⢕⠀⠧⣳
⡇⠀⢛⣗⢘⢸⠀⠀⠀⠀⠀⠀⠀⠀⣈⠿⡏⠱⣬⣴⠿⣼⣭⣭⣽⠿⠟⠛⠉⠀⠀⠀⠀⠁⠀⠀⣯⣇⠕⡂⣑⢮
⡇⠀⢸⡇⢸⡿⠠⢄⡀⠀⠠⠴⠾⠿⠶⢿⠁⠀⣠⠄⡴⣿⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⡁⢕⠀⠮⡙
⠇⠀⠸⡇⢸⢻⠐⡿⠇⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⣼⡏⠋⠛⠿⢦⡀⠀⠀⢀⢄⠄⠂⠀⠀⠐⠀⣴⠇⢕⢠⡑⠇
⠸⡀⠀⢻⡸⡀⡆⢀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡾⠉⢿⣰⠀⠀⠀⠀⠠⠂⠑⠁⠀⠀⠀⠀⠂⢠⢿⡲⠅⡶⣸
⠀⢧⠀⠀⠳⣳⡷⡄⠀⡀⠀⠀⠀⠀⢻⠁⢠⡞⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⣦⡬⣎⣜⢣⡇
⠀⠘⡳⡀⠀⠈⠻⣭⠢⡀⠢⠀⠀⠀⡼⢠⡟⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠰⠿⠿⠊⠽⣍⡟
⠀⠀⠘⢞⠢⡀⠀⠈⠻⣾⣶⣄⠊⠠⣷⡣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣠⢞⢟⡰⣊⠶⡑⡞
⠀⠀⠀⠈⠣⣍⠢⣄⠀⠀⠉⠳⠍⣢⣥⣀⠀⠐⠀⠀⠠⠀⠘⣯⠄⠐⠀⠀⣀⡤⣾⡷⢞⠕⢨⡎⢣⠝
⠀⠀⠀⠀⠀⠈⠙⢬⣙⢢⣄⡂⠠⡠⢉⠙⠛⠫⠵⠒⣲⣶⣶⣶⣖⠒⠭⠝⢛⢟⣫⡪⣃⢵⠱⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣌⡑⠒⡤⣥⣁⡊⠒⠔⠢⠂⠄⠠⢂⠕⡪⢚⣕⢥⢼⠺⣙⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠓⠴⠤⣱⣩⣹⡹⡙⡙⡆⢋⢋⢩⣹⣈⠦⠖⠋⠁
⠀⠀⢐⢄⡢⣐⠄⡂⢔⡠⢄⠄⡠⢀⠄⡠⢀⢄⠄⡠⢈⠉⡩⢉⠉⣉⠁⡠⢀⠀⡠⢀⠄⡠⡀⢄⢔⢄⠠⡠⡠⢀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣩⡅⢢⣢⣴⡾⡳⢶⡗⢀⠀⠀⠀⣠⠛⠁⠀⠀⠈⣁⡀⠀⠀⢠⣔⣦⣤⠤⠙⠉⠪⠁⠀⠀⠶⠶⠄⡀
⠀⣀⡜⠛⣁⡀⠀⠀⠀⠀⠈⠉⢀⡝⠚⠿⣧⣜⡛⠅⠠⠖⠃⠁⠀⠀⢀⡴⠋⡠⠀⠹⠋⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠆
⠀⠀⠉⠉⢠⣽⠠⠀⠀⠀⠀⠀⠈⠳⠊⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠸⠃⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣇⣈⣤
⠀⠀⠀⢀⣐⡿⢀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠋⠀⠀⠀⠀⠀⠀⣠⠼⠇⢀⡀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠝⡋⠉
⠀⠀⠐⢻⣿⡌⠂⠀⠀⠀⠀⠀⣠⣟⠋⠀⠀⠀⠀⠀⠀⠀⠐⠒⢛⠆⠀⠈⣝⣁⡋⣦⡤⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠻⠾⢿⠿
⠀⠀⠀⠀⠉⣿⣆⠀⣠⣶⣆⠬⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣘⣾⠋⠀⠀⠀⠀⠀⠀⠉⢧⡀⠀⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠂⠀⠐⠀⢑⠖⠆⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣰⠉⢚⣗⠀⣠⣾⠦⠀⠐⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⣟⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⣄⠀⠀⠛⢢⡟⠓⠒⢊⠀⠟⠉⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⣏⡀⠀⠀⠀⠀⠈⠉⠉⠠⣴⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠑⢶⣦⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢇⠀⠀⠀⠀⠀⠀⠀⣑⡒⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡑⠀⠀⠀⠀⠀⣯⡍⠁⠀⠀⠀⠀⠀⢲⡏⠀⠀⠀⠀⠼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠃⠘⠟⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⡐⠀⠀⢰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠁⠀⠀⠀⠀⠀⠘⣗⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⡇⠀⠀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢤⡜⠉⠀⠀⠀⠀⣀⡤⢶⠶⢤⣀⠀⠀⢀⠻⠀⠀⣴⣦⠀⣀⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣴⠋⢄⡈⠁⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⠲⡧⡼⡁⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⢂⡗⣉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⠁⣀
⠀⠀⢀⠀⠀⢶⠀⠀⠀⣀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣯⠀⠀⠀⠀⠀⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠛
⠀⠀⠈⠁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠶⠶⠶⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⣿⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢸⣿⡇⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⣀⣀⡄⠀⠀⠀⣶⣶⠶⠀⠀⠀⠀⠙⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠉⠀⠀⠁⠀⠀⢀⣀⣠⣤⠀⠀⠀⢰⣶⣶⠾⠿⠟⠛⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⡿⠇⠀⠾⢾⠀⠀⠀⠀⣤⣤⣤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣠⡤⢀⣠⣤⣤⣶⠾⠘⠟⠛⠛⠀⠀⠒⠀⠀⠀⠛⠀⠀⠀⠀⣿⠍⠈⠀⠀⠀⠁⠀⠀⠀⠀⠶
⠀⢰⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠿⠟⠃⠀⠀⢀⡉⠁⠁⠀⠈⠉⠉⠀⠀⠤⠀⢀⣚⣀⡀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠛⠃⠀⢉⡀⠲⠀⠀⠀⠘⠇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛⠛⠛⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠚⢦⣄⠀⠀⠀⢼⡇⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⢀⠀⠀⠀⠸⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠾⠃⠀⠀⠀⠀⠤⠀⠀⠀⡤⠄⠀⠀⠀⠀⠙⠙⠀⠀⠀⣶⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣦⣤⣤⣄⣀⡀⣀⣀⣤⣤⣤⠀⠛⡋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠐⠆⠰⠆⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠚⠀⠀⠀⠀⣄⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣦⠀⣾⣶⣶⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶
⠀⠀⠀⡀⠀⠀⠀⢩⡄⠀⠀⢸⡧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣈⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣶⣶⡆⠀⠀⢠⣄⡀
⠀⠀⠀⠁⢰⠂⠀⠀⠀⣐⠄⠸⠇⠀⣿⡷⠀⠀⠀⢠⠀⠀⠀⠀⠛⡟⠛⠿⠿⢿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠙⠛⠂⠀⢀
⠀⠀⠀⢶⠀⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⡄⠁⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠠⠄⠀⠀⠀⠰⡆⠈⠶⠐⠂⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⢀⡀⠤⢠⢐⡀⡆⣂⢂⣐⣐⢒⡐⠂⡀⣄⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⡀⠄⢂⣡⡴⠼⠋⠉⢀⣀⠀⠠⠤⠌⠉⣍⠹⢓⠶⠬⣓⢤⡀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡠⠢⣀⠶⠚⠓⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠉⠙⠺⣆⣀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠎⠡⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠐⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡔⢀⡶⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠎⣿⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠂⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⡬⠀⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠌⢠⠃⣀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠭⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠁⠨⢁⡫⠤⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠨⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣓⠂⠀⠀⠀⠀⠀⠄⠈⠀⠁⠀⠀⢠⠀⡨⠋⠘⢲⡌⢽⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠥⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢌⣡⡞⠫⢽⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠱
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⣨⡄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⠞⠀⠀⠀⠀⡠⠀⠐⠁⠀⠀⠀⠀⠠⢤⠄⠀⠀⢠⠃⡙
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡯⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⠒⠈⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠐⡶⢯⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⢥⠀⠁⠀⠀⠀⠀⠀⠀⡀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠠⠊⠾⠹⠜⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠢⡀⠀⠀⠑⠀⡀⠀⠀⠀⠀⢠⡧⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠔⢢⠶⠚⠆⡶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⠲⢄⡀⠀⠀⠁⠂⠤⢀⣈⡁⠀⠀⠀⢀⣀⠠⠄⠒⠛⠉⠁⠀⠀⠀⢀⡄⢂⠁⠈⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠙⠂⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠡⠐⠀⡢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠈⠉⠒⠢⠠⠄⡄⡀⢀⠄⡤⡤⠤⠒⡈⠡⠀⠀⠣⠀⣠⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣄⣀⡀⠂⠀⠀⠀⠂⠠⠈⠀⢂⠈⣀⣥⠰⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡉⠈⠐⠒⠓⠒⠒⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠄⠊⠠⠂⠀⠐⢀⠄⠂⠡⠀⡠⠀⠀⠔⠀⠠⠂⠈⠄⠂⠀⠐⠀⡠⠒⢁⠔⠈⠄⠀⠔⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠒⠨⠉⠒⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⡒⠦⡄⣤⣀⣀
⠀⠀⠀⢠⠄⠒⡚⡉⠀⢀⣜⡄⢄⠀⡰⠓⡐⢈⡲⠦⢄⡀⠀⢀⡠⠔⠶⠡⡒⠈⠀⠀⢀⠼⢡⡣⠄⠊⠻⡔⠆
⠀⠀⠀⢐⠀⠀⠱⡐⢌⡀⣀⣀⠾⢶⣀⣀⣐⡀⠇⠀⠀⠈⠉⠀⠤⣸⠀⢧⢊⠲⡀⠀⡆⠀⢼⠁⡠⠐⠒⠁⡇
⠀⠀⠀⠸⠆⢐⠀⠘⢄⠓⢄⠨⠤⠤⠄⠋⠂⢠⢠⡔⠉⠑⡀⢸⠀⢷⣤⡾⠈⣃⠐⠒⠀⡄⢊⣉⠐⠀⠀⠐⠁
⠀⠀⠀⠘⠘⠃⠀⠉⠂⠙⠢⣉⠂⢄⡂⠀⢐⢆⢲⠀⠀⡰⠁⢛⡰⠚⠂⠛⣒⠟⠊⠁⠀⠀⠀⠀⠀⠈⠉⠐⡇
⠀⠀⠀⢀⠀⣀⠀⠛⢠⠋⠲⢄⠉⠢⢌⣰⢎⡎⢇⠠⣪⡲⠍⠲⠤⢍⠢⠊⠀⢀⡠⡐⠤⢌⠙⠔⠃⠐⠂⠤⠁
⠀⠀⠀⢸⣀⠋⠀⣖⠁⠀⠀⠀⢱⠀⠠⢡⠲⢌⠲⡾⢋⠒⠀⢢⢈⠜⠀⠀⡰⠡⠊⠀⠀⠀⠡⠄⢀⠀⡬⢵
⠀⠀⠀⠈⠦⢅⡂⢄⡈⠑⠄⠤⠃⢠⢡⠑⢀⠠⠱⡈⢂⠱⢁⠤⠁⠀⢠⠒⢌⠺⡒⢄⡀⢀⡝⠌⠾⢦⠀⢸
⠀⠀⠀⠀⢞⠉⠊⠶⢌⠢⡤⠐⠒⠇⠈⠉⠀⠒⠚⠼⠂⠑⠁⠀⢀⠔⡕⠀⠪⡒⢌⠂⠍⡩⡟⠠⠀⠰⠀⢸
⠀⠀⠀⠀⢸⣀⣴⠞⠉⢱⢱⣀⡯⠘⡦⢤⠂⣀⡀⡀⣀⡀⢄⠘⢅⢨⠀⠀⠀⠈⢂⠉⠒⠠⢬⣉⣀⢒⡒⡚
⠀⠀⠀⠀⡧⠊⠀⢀⠠⢺⢀⠸⠇⡤⢻⡿⠀⢓⡀⠣⢰⠉⠑⡕⢌⠣⡢⠀⠀⠀⠸⠂⡇⣡⠁⠀⠁⢢⠀⡅
⠀⠀⠀⠀⡇⡠⠒⠅⠀⡘⡸⠾⢻⠳⣦⠛⠒⠚⡞⡌⣀⠣⣀⠈⡢⠱⡈⠳⡤⢀⠀⢱⢁⢸⠤⢐⢜⢫⣍⡇
⠀⠀⠀⠀⠫⠀⠁⠀⡔⡰⣷⡠⠛⠀⡘⠷⡢⠜⡔⠜⠀⠑⠠⡄⠀⠀⠈⡒⢌⠒⠍⡊⠀⡡⡙⢟⢑⠁⠀⡇
⠀⠀⠀⠀⠀⠐⢀⢎⠜⠀⠀⢧⠤⠠⠀⢡⢊⢮⠊⠀⠀⠀⠐⠡⠁⠀⠄⢁⡠⠛⠒⠤⠕⣢⡈⠢⠹⡣⢀⡇
⠀⠀⠀⠀⢸⢔⠕⠁⢀⣁⠤⠘⠒⠒⠴⢥⡃⠣⣀⠀⠀⠀⢘⠀⢂⠠⠂⠁⠀⠀⠀⠀⠀⠀⠉⠁⠓⠶⠭⠇
⠀⠀⠀⠀⠘⠚⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠍⣒⠨⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠉⠉⠉⣽⠉⢉⡽⠋⠙⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠈⣠⠖⠶⠶⣿⠊⠉⠀⠈⠀⣾⠁⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢰⠋⡳⣠⠞⠁⠀⠀⠀⣿⠀⡴⠦⣤⡖⠚⠀⢀⠐⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⡴⠒⢫⢴⡿⠁⠀⠀⠀⠀⠀⣿⡴⢃⣶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⣰⠏⢀⣀⣠⣼⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⢓⣛⣃⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣰⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠋⠈⠉⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⢀⠘⣆
⣄⣀⡀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣿⠒⠦⣜⣿⡄⣠⢳⠖⣋⠀⠀⠀⠀⠀⢀⣀⠀⠘⡆
⠿⠿⠃⣟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠯⣿⠤⣄⠈⠛⠙⠃⠈⢉⣩⠇⠀⠺⠯⣟⠉⠈⠱⣄⣿
⠀⠀⢰⣋⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠏⠀⠀⣿⠀⠈⠓⠚⠋⠉⠉⠉⠱⡖⢤⡀⢀⡜⠀⠀⡠⠻⣼⡇
⠀⠀⢸⡏⠉⠺⣗⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣵⣞⣤⡔⠋⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠈⠑⡆⠉⠓⠲⠤⣄⣀⣀⣰⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣀⣀⡤⠤⠖⠒⠋⠁⢠⡇⠀⠀⠀⢰⡇
⠀⠀⠀⣇⠀⠠⠀⣠⠇⠀⠀⠀⠀⠀⠀⠈⠹⣉⠉⠉⠉⠋⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢻⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠖⣿⠲⢶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠞⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢷⠀⡀⠀⠈⠑⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢀⡴⠋⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷⣀⡴⠟⠁⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣄⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⣿⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠈⢆⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠱⠀⠀⠀⠀⠀⣸⡀⠀⠀⠌⠀⠀⠀⢸⡀⠀⠀⠀⡞⠀⠀⠀⢠⠞⠁⠀⢀⠜⢶
⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢳⣀⠤⠔⠒⠒⠊⠉⠉⠈⢣⡀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡰⠃⠀⠀⣰⠋⠀⠀⠀⠁
⠀⠀⢀⡷⠀⠀⢀⣠⠴⢒⠁⠙⢆⠀⠀⢀⣀⣤⠤⠖⠒⠻⠉⠒⠲⠦⣏⣀⠀⠀⡴⠁⠀⢀⣼⢁⠀⠠⠎⠐
⠀⠀⢸⣠⡴⠚⠉⠀⠀⠀⠀⠀⠈⢷⡊⠉⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢸⡀⢩⠾⢄⡀⢠⡞⠀⠁⢄
⠴⠚⠹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠐⠠⠄⣀⠀⠀⡊⠀⠂⠄⣜⣡⠋⠀⠀⡽⠻⢄⡀⠀⠀⠑⠀⠀⠀⠀⢀
⠒⠒⠦⣞⢁⠀⢀⡀⠄⠀⠀⠉⠀⢸⡀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⣠⠞⠉⠙⢦⣞⡁⠀⠀⠉⠉⠉⡏⠉⠙⢉⠉⠁
⠀⠀⠀⠈⢩⠂⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠂⢙⣦⠞⠁⠀⢀⡴⠣⡈⠙⠳⢦⣄⡀⢠⠇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣀⣀⣀⣠⡤⠖⠋⠀⠱⣄⣠⠟⠀⠀⢱⠀⠀⠀⠈⢙⡿⠶⠦⡧⣤⣄⣀
⣠⣤⠤⠴⠷⠒⠒⠲⣆⠀⣀⣀⣹⠉⢿⢉⢉⠉⠁⠀⠐⠒⢂⠀⢠⡞⠃⠠⠀⠀⢼⣅⠀⠀⢀⡞⠀⠀⢀⠁
⠀⠀⠀⠀⡆⠀⠀⠀⡏⠉⢹⡀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⡵⢏⠀⠀⠀⠀⠀⢸⠀⠑⠢⣞⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⣀⣧⠤⠴⢻⠃⠀⠀⢧⠀⡀⢘⣀⣀⣀⣀⣀⣀⣴⠟⠀⠀⠑⡀⠀⠀⢀⡇⠀⠀⠈⠈⠑⢦⣠⠀⠀⠀⢠⠃
⠒⠉⠩⣅⣸⡤⠶⠻⡖⠚⠛⠛⡏⠉⠉⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠞⠀⢑⠾⢤⣀⠀⢶⣞⠳⢦⣴⠇⠂
⠀⠀⠀⡼⠀⡆⠀⠀⠘⢦⠀⠀⢱⣀⣀⣇⣀⣤⠤⣤⣀⡀⠀⠀⠀⠀⠠⢠⡍⠠⢤⣉⠀⠠⠈⢹⠾⢄⠙⠦⠙⢦⠨
⠉⠉⣹⠛⠶⣄⠀⠀⠀⠀⣳⡤⠔⢇⠀⠀⠀⠀⢰⠃⠀⠉⠑⠢⢄⣀⢀⠎⠀⠀⡀⠀⢀⠂⠀⠁⠀⠀⠐⠄⡀⠀⠑
⠀⣰⠃⠀⠀⠈⠳⣄⡠⠚⠁⠙⢄⠸⡄⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢈⠟⠒⠤⣄⡀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⣰⠇⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⠣⡇⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠉⠛⠢⢤⡀⠀⠀⠀⢀⣀⣀⣤⣤
⣏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠦⢤⣽⣀⣀⣀⣧⣤⣤⡤⠤⢤⡾⠴⠶⠶⠶⠒⠺⡖⠛⠛⠛⠛⠻⢭⡉⠉
⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠉⠰⠇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠑⠦⣀

⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⣤⣀⣀⣠⣄
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⣼⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠟⠛⣿⣿⣿⣿⠟⠛⠒⠀⡀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⢀⡀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣴⣶⣾⣄⢻⡿⢋⣉⣴⣿⣿⣷⣜⣆
⢿⡟⠙⢿⠏⠀⠀⠀⣤⠀⠀⢠⡄⣼⠃⠀⠀⠀⢀⣾⣷⣶⣤⡍⢻⣿⣇⣘⡁⢨⣉⣤⡄⢹⣿⣿⣿⣷⡀
⠘⣿⣄⡀⠀⠀⠀⠀⠉⠀⠈⠈⠁⠁⠀⠀⠀⠀⣼⡿⠟⣡⣤⣴⣿⣿⣿⣿⣿⠰⣿⣿⣷⣆⣈⣥⣌⣿⣧
⠀⠈⠻⠷⠀⠀⠀⠀⠠⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⢃⣤⣌⠛⢿⣿⣷⠄⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⣤⣾⣿⣿⣿⠟⣠⣿⣿⣤⣍⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⢋⣡⣤⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡋⠰⣿⣿⣿⣿⣤⣥⡆⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣶⣌⢻⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣦⡀⠀⠀⢀⡔⠻⠉⠙⠿⣿⣿⣿⣧⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⠿⠟⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣦⣤⣀⣀⣤⣀⡀⠀⠀⠚⠁⠀⠀⠉⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠦⠤⠤⠤⠴⠶⠾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠠⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣤⣤⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣽⣿⠋⠲⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣀⠀⠀⠀⢀⣈⣵⣤⠀⠀⠀⢀
⠀⠀⣠⣤⣤⣀⣀⣘⣛⣶⣦⣻⣕⡿⢶⣄⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⣘⣁⣰⣶⣾⣿⣿⣿⣿⣿⣾⣦⣴⣶⣦⣄⣀⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⣐⡿⠁⠈⢿⠏⠁⠀⠒⠀⠀⠀⣴⡿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⠃
⠀⠀⠈⠁⠀⠈⠻⣿⣿⣿⣿⣿⣦⣄⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⣴⡀⣹⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠍⠃⠀⠀⠀⠀⠀⠀⣐⣿⢿⣿⣿⠟⠻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣥⣄⡀⡉⠛⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣯⡟⠻⣿⣿⡿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⣀⣄⣀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣷⣟⡉⠀⠀⠸⡏⠀⠈⠻⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⣀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢧⠠⠾⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡌⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠻⣿⠏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⢀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⢸⡟⠉⠉⢉⡉⠉⠀⠈⠁⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠈⠙⣿⡏⠀⠀⠀⠀⠀⡜⠉⠓⢤⡛⢈⡲⣈⠑⣶⣏⠀⠀⣰⠃⠈⢲⡄
⠀⢸⡇⢀⡀⠈⣀⠀⠁⣀⣀⣀⡀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣸⠀⠀⠀⠀⠈⠳⣬⠞⣻⢧⣌⡙⢲⠧⣄⣀⡾⠉⠳⣄
⠀⢸⣷⣤⣤⣤⣤⣤⣤⣀⠠⣤⡀⠈⠻⢿⡆⢰⣶⣄⣀⣤⣶⠿⠿⠛⡟⠛⠛⡿⠿⣷⣶⣄⣀⡜⢁⡜⠓⢧⡀⢩⠟⠒⠢⢽⡇⠀⠀⢠⠗⢦⡀
⠀⠸⢿⣛⣫⣼⣿⡉⢻⣿⣷⡈⠻⢷⣤⣾⡿⣪⣿⡿⡟⠉⠀⠀⠀⠀⡇⠀⡘⠀⡼⠁⡹⢿⣿⣦⣎⠀⠀⠀⠈⠛⠶⠤⣄⣼⠀⠀⢠⠎⢠⠏⠈⠲⣄
⠀⠸⠿⠯⠷⠋⠘⣷⡀⠹⣿⣷⡀⠸⢟⣿⣼⡟⠁⠀⠱⡄⠀⠀⠀⢰⠇⡐⢀⡞⠀⠜⢠⠋⣸⠻⣿⣷⣤⡀⠀⠀⣀⣠⠾⠋⣷⢠⠏⢀⡏⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⡜⢀⣷⡞⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠙⢦⡀⠀⢸⡜⣡⠟⢄⠎⢠⠋⡼⠁⠀⡴⠋⠻⢿⣷⣋⣁⠀⠀⢀⡾⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⢠⣴⣟⣴⣿⣦⣽⠿⣿⡟⠉⣆⠀⠀⣠⠤⢄⣀⣹⡶⠖⠻⡅⠀⣀⠷⢧⠞⠀⣠⠞⠁⠀⠀⠈⢿⣷⠴⠼⢶⠾⠤⣧⣸⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣠⣿⠋⠀⠙⢻⡟⠁⠀⢸⡇⠀⡜⠀⢠⡃⠀⠀⠀⡰⠁⢀⣀⡹⣞⠁⠀⢈⣳⡞⠁⠀⠀⠀⠀⣠⣾⠇⠀⢰⠷⣆⣰⡃⡜⠉⠉⢹⡟⠒⢦⡤⢤⡇
⠀⠀⠀⣴⣿⠹⡆⠀⠀⠀⠀⣀⣀⣤⣿⢤⣇⠀⠀⠓⢤⡀⣼⣄⠀⠘⢆⠀⠈⡦⠖⠉⡇⠙⣆⣀⣤⣶⣿⠟⠳⣄⣀⡞⠀⢀⡟⠙⡆⠀⣠⢾⠀⠈⢺⠤⠤⡇
⠀⠀⣼⡟⠁⠀⢳⣤⠒⠊⠉⠀⢸⣿⣿⣦⡀⠈⢲⡄⠀⡼⠛⢯⡀⠀⠐⣇⠀⠱⣄⣠⣴⣶⣿⢿⠟⠋⠀⠀⠀⠀⠉⣳⠖⠺⢯⣓⣿⡊⠀⠈⢇⠀⢸⠀⠀⡇
⠀⢸⠟⠀⡜⠀⠀⢿⠀⠀⠐⣚⣉⣥⠼⣠⠛⠓⠲⣷⠴⣇⠀⠀⠙⠢⣀⣸⣶⣾⣿⠟⠛⠉⠀⠠⣷⡄⠀⠀⠀⣠⠔⠁⠀⠀⠀⠙⡅⠈⠉⢢⡼⠒⢺⠖⠒⡇
⠀⠈⢀⠈⠀⠀⠀⢸⡗⢺⣋⣉⡡⠴⢎⠁⠀⠀⢀⠇⠀⠀⠙⠲⠶⢶⣿⠟⠋⠁⠘⠦⢦⠀⠀⠀⢀⠽⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⡏⠀⠀⢸⠀⢠⠇
⠀⠐⠁⠀⠀⠀⠀⠈⢻⡀⠁⠀⠀⠀⠈⣧⡤⠖⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣾⣷⣄⣰⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣇⣀⣤⣼⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠀⣀⣴⠋⠁⢀⣀⣤⠤⣤⣤⣶⡖⢋⣹⣿⡇⢀⠔⠋⠙⢿⠃⢈⡏⠀⠀⠀⠙⡆⠀⡀⠀⠀⢀⡠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣠⠴⠚⠋⠉⠁⠈⢳⡋⠉⡇⣀⣠⡿⢿⠛⠯⠭⠔⡿⣿⣅⣀⣶⣀⠏⠀⢸⠁⠀⠀⠀⣠⣧⣸⠿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠁⠈⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠀⠀⠈⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⡈⠙⡄⠰⠀⠀⠹⠋⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⡴⡊⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⣛⡳⠇⠀⠑⠅⠠⠈⠀⡠⠀⣀⠀⠀⠀⠀⡀⠀⢠⠀⠈⢉⡐⠠⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⡼⠉⠳⢦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⢠⠄⠀⠀⣙⠃
⠀⠀⠀⠀⠀⢀⡟⠀⠀⢑⠳⠁⠀⡀⣼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⣷⡐⠇
⠀⠀⠀⠀⠀⠸⠈⠀⠢⠅⠀⢀⠊⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡄⠀⠈
⠀⠀⠀⠀⡰⠲⡈⡓⢚⣦⣤⣖⠫⡷⣇⠀⠀⠐⠀⠀⠀⠀⠀⠐⡀⠠⢸⠀⠀⠀⣴⣀⡹⣰
⠀⠀⠀⠼⠄⢀⣀⡈⠁⠀⠓⠠⠊⡹⢁⠀⠀⠀⠠⠀⠀⠀⡚⢄⣇⡀⣐⣐⠖⢀⢚⡛⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⣴⡀⠀⠀⠀⠀⠂⠈⠁⠀⠘⢄⠑⢄⠈⡅⠀⣀⠼⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡐

⠠⠄⢀⠀⠀⠀⠀⠠⠀⢠⡄⠀⣠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠠⡤⠀⢀⣀⡀
⠀⠄⠀⢀⣤⡞⣦⢄⡀⠀⢠⣾⣻⣽⣾⠃⠀⠀⣦⣶⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠧⠞⠀⠀⠠⠄⠀⠀⠘⣿⢿⡦
⠀⣠⣾⣿⢿⣯⣥⡿⣿⣷⣿⣿⣷⠟⠁⠀⠀⠀⠿⠛⠀⢠⢀⠀⠀⢀⢀⠛⠋⠀⢠⣶⠀⠀⠀⠂⠀⣰⣶⣦⠀⠙⣧
⠀⠈⠛⢿⣿⣿⣿⣿⠟⠁⠈⠉⠁⣠⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠈⠉⠁⠁⢀⣦⡀
⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⣀⠀⠀⠀⢸⡏⠀⠖⠂⠖⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢳⣄⡀⠀⠀⠸⡇⠀⢀⡀⠀⠀⠈⠀⠀⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠋⢁⠀⠀⢰⡇⠀⣾⠟⣀⠀⠀⠀⠀⠀⢰⡄⠀⣠⢖⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣠⠟⣠⣆⡩⠀⠀⣚⣵⠟⠀⠀⠀⢻⠀⢿⣍⡽⠁⠀⢀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠖⠛⠋⠁⢈⠙⠛⠒⠺⠿⠟⠁⠀⠁⠀⠀⠈⣧⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠛⣯⠀⠀⣄⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠠⣿⡤⠀⠀⣰⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⣽⢿⠂⠀⣿⠆
⠀⠀⠀⠀⢢⠀⠀⠀⡿⠀⠀⠘⣆⠀⠉⣹⡐⡟⡁⢎⡱⢜⣱⢸⢰⣹⠦⢯⡱⠋⠇⢆⡇⠀⠀⠻⠇⠉⠀⠀⠀⠋
⢀⣀⠀⠀⠈⣧⡀⢰⡧⢀⣤⠆⠈⠉⠉⠀⠁⠈⠀⠀⠀⢈⣽⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇
⠀⠈⠉⠒⢶⠤⣷⣾⣖⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⣀⣀⣤⡀⠀⠈
⠀⠀⢀⣠⣴⡜⢩⣿⡌⠛⠛⠓⠓⠂⠀⠀⠀⢠⡇⢱⠀⠀⠀⢀⣴⡋⠹⣿⣿⡷⣷⣔⠞⠃⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠙⢦⠀⠀⣤⠀⠀⠀⠀⠈⢻⡬⠀⠀⢀⣾⠿⠇⠀⠙⠛⠁⠙⠟⠀⠀⠀⠠⠴⠀⡀⠄⠠⠈⢷⡀
⠀⣾⠿⣿⡆⠸⠁⠀⢰⣆⠀⣀⣿⡆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣤⡰⣶⣦⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀
⠀⠘⣷⣿⠃⢀⣴⣤⣼⣿⠂⠀⠈⠁⠀⠀⠀⢠⣾⠃⠀⠀⣠⡄⠀⣠⣾⣿⣿⢻⣿⣿⠟⠀⠠⠀⠀⡠⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠈⠋⠀⠈⠉⠈⠉⠁⠀⠀⠀⠐⠒⠀⠀⠼⣿⣧⣤⠄⠁⠀⠀⠛⠩⠁⠀⠸⠟⠉⠀⠀⢀⠐⠂⠀⠀⠄⠀⠰⠿⠿⠃
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⠀⠀⠈⠉⠉⢠⣦⣄⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠸⣆⠀⠀⠀⠈⠛⠻⢿⣿⣦⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠦⠠⠀⠀⢻⡂⠀⢰⣴⣾⡇⠀⠀⠉⠁⠀⠴⠾⢿
⠀⠀⠀⠀⠰⠚⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠩⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠾⢿⠟⢿⡟⠁
⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣂⣽⠇⠀⠀⠀⠀⠀⠀⣀⠠⠄⡄
⠀⠘⡀⠀⠈⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣶⣚⣿⢯⡏⣶⣾⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠈⠐⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣯⡧⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢀⡀⠀⠀⠀⡄⠠⠴⣻⠽⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠂⠀⠀⠀⠁⠈⠉⢀⣠⠀⠀⠀⡄⢀⣀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠰⢀⡄⢂⢀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢊⡩⢖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠤⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠔⠀⠠⠀⠀⠈⠁⠀⠀⠀⠀⠌⠀⠉⠁⣠⠴⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⠀⠀⠈
⠀⠀⠀⠂⠁⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⢚⡩⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⢀⡀⠄⠊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢒⠬⠥⣤⠖⡋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⢰⡀⠀⠀⡀⢐⣈⣽⣾⠇⡀⠰⣽⡦⢠⡂⢛⣙⣆⠀⠀⠀⠈⣸⠃⠀⠀⠀⣹⣟⠄
⠀⠀⠀⠀⠀⠀⠂⠉⠉⠀⠒⠒⠒⠉⣕⠻⢃⢬⣻⡿⠟⢸⡏⢑⡞⠍⢷⠽⣾⠻⡟⠋⠓⣔⠀⠠⠭⠒⠒⠒⠚⠉⠋⠓⠒⠚⠛⠛⠛
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⢻⣧⣤⠓⢰⠌⣿⡕⢉⠝⡃⠀⠀⠀⠀⣝⠖⠁⣄⢀⣀⣦⠠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠄⠀⠈⠁⡔⠱⠺⣂⣠⡇⠘⢠⡇⠀⠀⠀⡴⠁⠀⢀⣯⡑⠋⡉⠂⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠈⠛⠋⢒⡊⠀⠘⠛⠒⠉⠁⡉⠁⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣡⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⢀⠔⠂⠐⡀⠀⠈⠐⠄⡀
⠁⠀⠀⠀⢠⠀⠀⠀⠀⠁⠃⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠃⠀⠀⠀⠈⠢⠀⠀⠀⠀⠁⠠⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠢⠀⠀⠀⠠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠠⠋⠁
⡀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠜⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠔
⠁⠀⠀⢀⠀⠀⠦⠀⠀⠐⠒⠀⠘⠀⠈⠁⢀⠊⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠳⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢤⠴⠀⠄⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡠⠀⠐⠂⠙⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢄⠀⠰⠐⣂⠰⡄⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠴⠄⠀⠀⠀⠆⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠂⠉⠀⠀⠈⠂⠀⢀⠀⠀⠰⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⡠⠊⠀⠈⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠡⡀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠠⠊⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠈⠰⣅⠠⣀⣀⠤⠤⠄⠀⠴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡀⠔⠀⠢⠠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⡀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠞
⣀⣠⣀⣀⣀⣀⣀⣤⣠⣀⣠⣤⣈⣀⣀⣀⣤⣠⣀⣑⣤⣠⣄⣤⣄⣤⣌⣴⣄⣤⣄⣈⣤⣄⣀⣠⣄⣀⣄⣀⣀⣀⣠⣄⣠⣄⣤⣦⣄⣀⣠⣤⣠⣄⣄⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⢹⠀⠀⠀⠀⠈⠉⠛⠶⣞⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⡞⠀⠀⠀⠀⢀⣀⡀⠀⠈⠻⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣄⡀⠀⠀⠀⡾⠀⠀⠀⠀⢀⡟⠉⠉⢳⣄⡀⠈⢳⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠲⠶⠚⠁⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⠙⡇⠀⢷⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⢀⡇⠀⠸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡞⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠀⠀⠀⢰⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠸⠦⠶⠶⠦⣄⠀⠀⣠⠶⢦⣄⠀⠀⠀⠀⢀⡾⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠈⠙⢳⣀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠙⠦⣤⣀⡀⠀⠀⠀⠀⣀⣀⡤⠞⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣈⣉⡉⠛⠋⣉⣉⣡⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠀⢸⠀⠀⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣤⣄⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⡬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣋⣭⣤⡶⠶⣤⣭⣍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣩⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⣇⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⣸⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣧⡀⠀⠀⢀⡠⠧⠼⢆⡀⠀⠀⢀⣴⠟⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⢶⣴⠋⠀⠀⠀⠀⠘⣦⡶⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢿⠀⠀⠀⠀⠀⠀⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⡀⢀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠤⠴⠖⠲⠦⠬⠷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⡴⢞⣛⣭⣭⣷⠶⠶⠶⢶⣾⣭⣭⣛⡳⢦⣄⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠾⣫⣵⠾⠛⠉⠡⠀⠀⠂⠀⠀⠀⠀⠀⠉⠙⠛⢷⣮⣝⠷⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⣴⢟⣵⡿⠛⠁⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠠⠀⡀⣠⡄⠀⠈⠙⢷⣮⡻⣦⠀⠘⢦⡀
⠀⠀⠀⠀⠀⢀⠞⠀⢠⡞⣡⡿⠋⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠈⢄⠀⠀⠀⣸⡿⠁⢀⠀⠀⠀⠙⢿⣌⢷⡄⠀⢳⡀
⠀⠀⠀⠀⢀⠎⠀⣰⡟⣼⡟⠠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀⡜⢡⠃⠀⠀⠑⢄⠀⠀⠈⢻⣦⢻⣆⠀⢱⡀
⠀⠀⠀⠀⡞⠀⢰⡟⣼⡟⠀⠀⠀⠌⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⡇⢠⠞⢰⠏⣠⠀⠀⠀⠀⠂⠀⠀⠀⢻⣧⢻⡄⠀⢧
⠀⠀⠀⢸⠃⠀⣿⢰⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢳⣦⡀⢀⣼⡷⠋⢀⣿⡾⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⣿⡄⣷⠀⠘⡇
⠀⠀⠀⣿⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢅⡀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⢹⡆⠀⣷
⠀⠀⠀⡏⠀⢸⡇⣿⡇⠀⠀⠰⠂⠀⠀⠤⠤⠶⣶⣶⣿⣿⡏⠀⠀⢹⣿⣿⣶⣤⣤⣤⡤⠐⠐⠂⡅⠀⠀⢸⣿⢸⡇⠀⣿
⠀⠀⠀⣧⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠓⠦⣤⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡿⢸⡇⠀⣿
⠀⠀⠀⢻⠀⠘⣧⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⢀⣼⡟⠋⠻⢷⣄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣾⡇⣼⠃⠀⡟
⠀⠀⠀⠸⡇⠀⢻⡄⢿⡆⠀⠀⠈⠄⠀⠀⠀⠠⠞⢡⡏⣠⠎⢹⡇⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠁⠀⠀⣰⡿⢠⡟⠀⢸⠇
⠀⠀⠀⠀⠹⡀⠈⢷⡘⢿⣄⠀⠀⠀⠂⠀⠀⠀⢀⠟⣰⠋⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⣰⡿⢁⡾⠁⢀⠟
⠀⠀⠀⠀⠀⢳⡀⠈⢷⡌⢿⣦⡐⠀⠀⠈⠢⠀⣜⡞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢡⡾⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠻⣦⡙⢿⣦⡀⠀⠀⢸⠟⠀⠠⠄⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⢀⣴⡿⢋⣴⠟⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⣦⡙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢋⣴⠟⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠛⠷⣤⣉⠛⠿⠷⣶⣦⣤⣤⣤⣤⣤⣶⠾⠟⠛⣉⣤⠾⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠉⠛⠷⠶⣤⣤⣤⣤⣤⣤⣤⣤⠶⠾⠛⠉⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣷⣶⣶⣿⣶⣷⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣾⣿⣿⣿⣷⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⡉⠉⠉⠉⠉⠁⠀⠐⠒⠒⠢⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢆⠕⠐⢀⣀⡠⡄⢴⢲⠰⡤⡤⢤⣄⣀⡀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⠁⢀⡠⡒⣉⣦⡷⠾⠾⠟⣓⣓⣛⠺⠞⢲⡿⢇⣲⣠⡄⣀⠀⠀⠀⠀⠈⡒⢤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⢆⣩⡶⠛⠋⣁⢥⡞⢁⣨⠿⠽⠯⠯⠭⠭⣽⣟⣗⡾⢟⣿⢭⣗⡦⣀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡴⠁⠀⣠⢎⡱⠞⢃⡤⢾⣯⠟⠚⠉⡁⠐⣶⣹⠀⠀⠀⠀⠞⣯⣝⠙⠲⠤⣽⡷⣿⣻⢷⣤⡀⠀⠕⡈⢦⡀
⠀⠀⠀⠞⠀⢀⡴⣩⡾⠁⡰⢋⡴⠊⠀⠀⠈⠀⠀⠀⠛⠛⠀⠀⢀⣾⢄⡗⠈⠀⠂⡠⠀⠉⠛⢾⡹⠫⡟⢦⣈⢔⡤⠙⣦⡀
⠀⠀⠌⠀⢀⠾⣵⠃⢀⡾⡷⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢯⠴⢸⣓⠀⠀⠀⠀⠀⠀⠠⠄⠉⠢⡈⢢⡛⡬⢊⠕⡈⢷⢄
⠀⡜⠀⠀⡼⣹⡛⢠⢿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠃⣂⡣⣼⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⣜⡄⡪⢒⠈⣟⣆
⢠⠁⠀⢰⠳⡏⢠⢃⡞⠀⠁⠀⠀⢀⠀⢀⠤⠂⠀⠀⠀⣾⢃⠾⠩⠁⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣮⢻⡔⠥⡢⠸⠬⣇
⠨⠀⠀⡧⣹⠂⣾⡿⠀⠀⠀⠀⠐⠩⠒⠁⠀⠀⠀⠀⣰⢡⡶⣚⢭⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡘⡆⢣⠪⢐⡀⡗⢼⡀
⡆⠀⢠⣳⡇⢰⢋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣲⣹⢡⡤⢙⣯⢸⣿⠀⠀⠤⢤⣤⣤⣤⡄⠀⠀⠀⠈⡟⠋⠇⢸⣿⢐⡥⢂⢸⠳⡇
⡇⠀⠸⣽⠅⢈⣸⠀⡁⠀⠀⠀⠀⠀⠀⢀⣀⠤⢛⡇⠘⠷⠶⠿⠛⣾⢶⣶⡲⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡧⣊⠅⢸⣝⡇
⡇⠀⠀⣿⠅⣻⣿⠀⢆⣀⠀⠀⠠⠶⠾⠿⠾⠿⢿⠀⠀⢀⡴⢀⡾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇⡆⢕⠭⠸⢬⡇
⢡⠀⠀⢿⡀⡸⢿⠀⣻⡿⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⣾⡏⡟⠛⠿⣦⣄⠀⠀⠀⠀⣀⢀⠔⠂⠀⠀⠀⠄⢀⣦⢁⠅⡊⡟⢾
⢸⠀⠀⠘⣆⢸⠈⡆⠩⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⣠⡟⢻⣧⢁⠀⠀⠀⠈⠀⢀⠤⠊⠖⠁⠀⠀⠀⠀⠠⠁⣸⡟⢴⡪⣰⡙⡎
⠀⣧⠀⠀⠘⣌⣧⣴⡈⠁⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣰⡓⠀⠀⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⢋⠗⡩⢰⢇⢹⠁
⠀⠘⣧⠀⠀⠘⢾⣯⣷⣄⠠⠀⡀⠀⠀⠀⠀⡟⠀⣴⡓⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠒⣰⣧⢞⣯⢶⠷⣌⡇
⠀⠀⠡⡱⣄⠀⠀⠙⢷⣎⡳⣄⠐⠠⠀⠀⢀⠇⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⢀⣘⣛⠛⣋⢠⡟⢦⡟
⠀⠀⠀⠹⣜⠣⡀⠀⠀⠙⠻⣿⣗⢦⡈⠀⢼⡞⡌⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⢀⡴⢋⢜⡥⡋⣠⠣⣘⠎
⠀⠀⠀⠀⠈⠣⣍⢳⢄⠀⠀⠀⠉⠳⢌⣓⣬⣐⡈⠀⠐⠀⠠⠀⠤⠀⠀⢻⣥⠀⠂⠘⠀⢀⣠⢴⣻⣶⢗⡵⠊⣴⠙⣦⠏
⠀⠀⠀⠀⠀⠀⠈⠺⣌⡝⠦⣄⠀⠄⢀⡈⡉⠓⠻⠿⢶⣒⡒⠂⢤⣤⣤⣤⣤⠰⢒⣒⡩⠿⢚⠟⡫⣶⠕⣫⠚⣌⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣸⡹⠲⡮⣌⣐⡡⠢⠔⡤⣂⠄⡩⢍⡩⢉⠍⣉⠍⣠⢔⡠⣺⠕⢉⡪⡔⢫⠈⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢥⣸⡉⡏⠖⢲⠠⠬⡤⣭⢤⢥⡈⣵⢯⠤⡥⢾⠓⡉⢏⢣⡸⠖⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠚⠚⠂⠦⠵⢼⣌⣆⣣⣘⣄⣧⡴⠼⠖⠛⠉⠁
⠀⠀⠐⠭⢪⢕⡩⢒⡩⢐⡡⢔⡢⢒⠤⢂⠤⣢⠔⡠⢒⠔⣠⠔⡠⢀⠤⢀⠤⢀⠄⣀⠔⣂⠔⣠⠔⡠⢂⠔⡢⢒⠔⡪⣪⢔⡡⣪⢞⠔⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠄⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⢐⡆⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠀⠀⠀⡠⠘⠑⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠐⠐⡂⠂⠀⠂⣀⠀⠀⠒⠐⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⣧⡴⠆⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⣸⣷⡂⠀⢀⡔⠁⠄⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠁⠠⠠⠀⠀⢠⣿⣿⣿⡖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡸⠿⠛⠁⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠋⠀⠐⠀⢀⠠⠀⠀⠤⠀⠄⠠⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⣶⣶⣦⣤⣤⣀⣀⣀⣙⣢⣀⣀⣀⣀⣀⠀⣀⣀⣤⣤⣤⣤⣄⣸⣥⣒⣴⠶⢅⠹⠄⠀⠱⡀⢠⣤⣀⠀⠂⡀⠀⠀⢠
⠉⠙⡟⢿⣿⠛⠛⠻⣿⣿⠛⠛⠻⣿⣿⡻⠿⣿⣯⠉⠹⡉⢻⠿⣿⣇⣈⣶⣦⠝⠂⢤⣛⠁⠈⠀⠀⠄⠀⠠⠀⡀⠠⠄⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠀⠙⢮⣍⠉⠙⠻⣤⡀⠀⠀⠹⣷⠀⠸⡿⣝⠦⠄⠇⠠⠻⠻⣟⠉⠫⠛⠳⣦⣗⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠙⠳⣄⡀⣿⠀⠀⢿⢮⣡⣤⣶⣄⠀⠀⠈⢷⣄⡼⠃⠞⢹⣿⡿⣟⡛⢿⣇⣈⡤⠚⠉⠁⠐⠀⠤⠀⠀⠀⠀⠀⠂
⣀⡠⠤⢀⠀⢸⡿⠲⠀⠀⠀⠀⡰⠿⢁⣿⡑⠀⠾⠘⣵⡏⡆⠈⠳⠾⠿⠒⠚⠓⠛⠛⠃⠈⠙⠌⡻⣞⢹⣧⠀⠀⢀⠀⠀⠀⠀⠳⠆⠀⠀⠀⠀⠠
⠁⠀⠠⠆⠐⠀⠀⠀⠠⠀⢀⠀⠀⠀⠞⣿⠳⣄⠁⠀⠉⡰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠑⠸⢼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠴
⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣿⣿⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣴⣧⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣤⣤⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣿⣿⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠉⠉⠙⠛⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢉⣀⣤⣶⣶⣿⣿⡇⢸⣿⣿⣶⣶⣤⣄⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢠⣾⣿⣿⣿⣿⣿⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠉⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣿⣿⣿⣿⣿⡿⠃⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠙⠛⠿⢿⣿⣿⡇⢸⣿⣿⡿⠿⠛⠋⣁⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠈
⠈⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠈⠆⠀⢠⡠⠴⠂⠉
⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠂⠀⠀⠰⠄
⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⡀⠖⠉⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⢀⠀⠻⠀⠀⠄⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⣀⠈⣦⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢢⡤⠖⠋⠁⠀⠀⢣⠀⠀⠀⠀⠈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣶⣿⣿⣛⡻⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠣⠁⠀⢳⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⡆⠰⠊⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠉⠉⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠉⠘⠄⣀⠤⢚⠉⡑⢆⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠉⠉⠀⠀⣀⣤⣤⣴⣿⣿⠏
⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡠⠤⠤⠔⠂⢡⡈⠀⠀⣠⡅⣈⠆⠀⣠⣤⣶⣿⣿⣿⣧⣴⣤⣤⣶⣶⣿⡿⠿⠿⠿⠛⠋
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⠢⠀⣀⣸⣶⣶⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢐⣤⣶⣾⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠀⣠⣴⣿⡿⠟⠋⠉⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⢠⣾⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣤⣀⣀⡀⠀⠀⣠⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣾⠏⠻⠉⣿⣿⠿⠿⢿⣿⡙⠻⣿⣾⡿⣿⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢻⣏⣩⡒⣏⣰⣷⣶⣿⣶⡶⣿⣿⡀⠀⠻⠯⣤⢾⣽⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⡞⠋⣿⣿⠟⠟⠋⠉⢩⠍⠁⠉⠐⠀⠈⠀⠀⠀⠀⠀⠈⠈⠟⣿⣧⡝⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣼⢿⣿⠃⠈⠁⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣶⣿⣿⢿⣎⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢹⣿⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣯⣻⣿⣆⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣤⣿⣿⣶⣦⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣾⣿⡟⣻⣃⠀⠹⠆⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⣿⣿⣼⠏⠙⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⣿⠷⠾⢻⣿⣶⣧⣙⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⣸⣿⣙⠢⠌⠐⢧⣜⠳⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⣶⣿⣿⠫⣝⠙⢿⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢻⣿⣅⡁⠀⠀⠀⢄⠘⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣾⣷⠿⠟⠠⠇⠙⡛⢻⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠘⣿⣿⠋⢰⡀⠀⢻⡦⢫⡻⣿⣆⠀⠀⠀⡰⠃⠀⠀⠠⣿⢿⣯⠘⠃⠲⠆⠀⡀⢰⣻⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⠀⣿⣿⡇⢺⠠⠀⠐⠀⣀⣀⣿⣿⡄⠀⠀⠀⠐⠀⠀⡀⠉⠻⢿⣷⣦⠀⠀⠐⠀⣾⣿⠁⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠈⣰⣿⡿⢷⣿⣠⣸⣿⣾⣿⠟⠚⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠤⠸⣿⣿⣀⠀⣄⣿⡿⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠲⣿⣿⣷⣼⠷⠟⠛⠡⡍⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣹⣿⠃⢾⣟⡞⢡⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⢿⣿⠀⠂⠀⢀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⢿⠟⠀⢾⠎⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣻⣦⣴⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣰⣿⣿⣆⡔⢁⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣤⣁⡀⠂⠀⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⣤⣶⣯⣽⠝⢉⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⣤⣤⠴⠆⠀⠀⣀⣠⣠⣤⣬⠟⢋⣡⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣤⣍⣙⡛⠛⠛⠚⠛⠛⠛⣋⣋⣥⣴⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣿⣿⣿⣿⣡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⣿⠛⠛⢻⣟⣿⣭⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⣽⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠒⣋⣿⣼⣿⣏⣘⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣛⣭⣿⣿⣷⣶⣤⣬⣿⣿⡿⢟⠉⢙⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠤⠿⠷⣶⣿⠋⢀⣀⡀⢀⣙⣻⣿⣭⣶⢿⣏⠀⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⠁⠀⠀⠉⠉⠉⠙⠋⠛⠉⠛⢄⢀⣁⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠳⠶⠶⠶⠦⠤⠤⠔⠟⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣴⣶⣾⣶⣶⣶⣷⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣷⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⣠⡤⠴⠶⢖⠖⠒⠒⠶⠶⡦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢯⢻⣤⡀⠂⠉⠀⠀⠀⡀⠐⠂⠄⠀⠀⠍⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⠁⠄⣾⣷⠚⠁⢀⠀⢰⣞⡷⢀⡀⠀⠀⠀⢀⣤⣌⣀⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠅⠈⠀⠀⠀⢸⠍⣶⠦⣼⢳⡎⠙⢚⣉⡼⠂⢀⣁⠈⠉⠙⡆⠙⠶⣄⡙⢦
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⠄⠀⠉⠸⢾⠽⣶⠢⠶⠶⠚⠉⠀⠀⠀⠀⠙⠻⢿⣾⣡⣤⠢⠀⠙⠛⢳⡀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⡼⠁⠐⠀⠁⠀⠀⠀⠀⠀⣼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢸⠀⠀⠀⠈⢀⣱⡤⠖⠋⠀⠈
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⠀⠀⠀⢠⠇⠀⠀⠘⣆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠴⠞⠉⠀⢳
⠀⠀⠀⠀⢠⡇⠀⡀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠄⠀⠀⢀⣠⡤⠖⢻⡉⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣀⠽⡞⠓⢦⣀⡤⡶⠚⠉⠀⠀⠀⠀⢧⠀⣀⠀⡀⠀⠂⣧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠉⠛⠋⢁⣠⠽⡖⠋⠛⢦⡐⡀⠀⠀⠀⢀⠀⢸⠘⡏⡇⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⢹⡀⠀⠀⠹⣇⠀⠀⠀⠈⢀⡏⠀⠷⠃⠀⠀⡀⡏
⠀⠀⠀⠀⠸⡶⠒⠋⠙⣆⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠻⣄⡀⠀⠀⡟⠀⠀⠄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢷⢀⣀⡥⠞⠛⠻⠶⠲⣄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢳⡀⠀⠀⠈⢳⡀⢠⠏⠀⠀⠀⠀⠀⢀⡞
⠀⣀⡀⣠⠴⠚⣯⠁⠀⠁⠀⠀⠁⡀⠸⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⠓⠋⢀⠀⢀⠀⠄⢀⡿⠁
⠙⣿⠷⠀⠀⠀⠈⢧⡀⠀⠀⡀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢧⠀⠀⠀⡀⠀⠐⠀⠈⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠳⣄⠠⠀⠀⢹⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⡠⢂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠣⣀⠠⠄⢻⡅⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⢈⡧⠤⠶⠾⢶⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣟⡒⢮⣳⣤⡄⢁⠀⠀⢀⣀⠤⢀⡴⠋⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢧⣤⣬⣥⣬⠤⠼⠗⠚⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⣤⣤⣀⣀⣀⣐⣀⡀⠀⢀⠀⢀⣀⣀⣀⣀⣠⣂⣠⣄⠠⠀⠐⠀⢀⡀
⠉⠛⠻⡟⣛⣻⣿⠛⠛⠿⣟⠛⣿⣍⠉⢙⠻⣿⣤⣶⣔⡂⢆⠃⠀⠈⠀⠀⢀⠀⠀⠀⠄⡀
⠀⠀⠀⠉⣶⠀⠈⠓⢤⡀⢹⡆⠸⣟⣍⣘⡀⠉⠱⣌⢠⠝⠟⣦⣤⠤⣤⠀⠀⠠⠀⠀⢀
⣀⠄⠀⠀⠿⠆⠀⠀⠀⠟⢺⡗⠠⠟⣽⠛⠉⠶⠶⠚⠋⠒⠆⠉⠊⠳⣞⣿⠁⠀⠀⠀⠀⠶⠀⠀⠀⠠
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢹⡟⣦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠗⠀⠀⠀⠀⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⢿⣿⢓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢹⠀⠷⠉⡾⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡆⠀⠀⢧⣰⣧⣽⣠⣇⠀⢠⡀⠀⠀⠀⣀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⣿⡿⣿⠃⡉⢦⣀⣘⣟⣻⣿⣏⠈⢧⠀⠉⢒⣫⢶⣿⡽⡈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣻⣮⣿⣄⢸⢸⠁⠈⣏⣿⣿⣦⣼⠟⠋⠁⠀⠀⣼⣷⣠⣹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⠛⢿⣿⠛⢹⢸⡄⠀⣿⡿⣿⡿⠀⠀⠀⢀⣠⣤⣿⠿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣷⡅⠀⠀⡁⠀⠸⢹⠠⢸⢙⣇⣸⣿⣶⣶⡿⠛⢻⡉⡄
⠀⠀⠀⠀⢠⣤⣀⣀⣤⡦⠙⢾⣿⣦⠀⡈⠉⠹⢸⡄⣼⡿⠘⢿⠿⠏⠇⠀⠀⢀⠧⠈⢄
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⠀⢸⠙⢿⣿⠁⠀⢸⠘⢧⠟⡇⠀⠀⠀⢰⠈⠠⠒⡡⠔⠲⡀⠳⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡴⠋⠀⠀⠀⠐⠀⢸⣾⠫⠛⢧⡔⢂⠈⢁⠤⠲⡼⡀⠀⠀⢸⣀⡲⢚⣽⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⠿⠀⠀⠀⡎⠀⠀⢸⠙⣆⢿⠑⡵⡌⠐⠁⠀⠀⠳⣌⣂⣤⠜⢋⡴⠋⢀⢔⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡴⠀⣠⠂⠀⢠⣾⣆⣸⣿⡀⠘⣈⠀⡂⢄⣀⣤⠔⢋⡠⠖⡁⠦⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⢤⡈⠉⠒⡶⠛⠙⠃⠀⠘⡇⣾⡒⢂⢹⡄⣿⠖⠛⢁⠰⢂⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣞⡀⠀⠉⠉⡇⠀⠀⠀⠀⠀⢣⠹⠍⠉⠜⣡⡧⣐⡴⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡐⠀⢸⡿⠋⠍⠉⠉⠉⠫⠉⢴⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡷⠶⢿⡉⠏⢀⠀⠀⠀⠀⠰⢺⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠭⠤⠤⠤⠤⠤⠠⠀⠼

⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⣚⣶⣂⡀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡬⠵⠚⢛⣿⣉⣤⣯⣤⣉⣿⡛⠒⠮⣅⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡵⢾⣧⠴⠖⠋⠉⠘⣀⣠⢿⣄⣀⠋⠉⠙⠲⢤⣹⣷⢭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡵⢋⡴⠊⠉⣀⢤⣲⡬⠷⠓⠛⢸⡗⠒⠮⢥⣖⠤⡀⠋⠳⢄⠙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣣⣿⣴⠋⠐⣦⣮⠞⠉⠀⢠⠀⠀⠃⢸⣧⠀⢀⠄⠈⠙⠪⣵⣴⠊⠳⣄⣷⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⠃⡼⠉⢠⢮⠞⠙⠷⣄⡀⠀⢧⠀⠀⢸⣿⢀⡞⠀⠀⠄⣠⠞⢣⡳⡐⠙⢏⠙⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⠃⡸⠁⢠⣧⠏⠀⠀⠑⠌⠻⣶⣜⡷⠀⢸⣿⣿⠁⠂⣠⡾⠃⠀⠀⢹⡵⡀⠈⣆⠸⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⢶⡧⠀⣆⡏⠀⠐⠲⢤⣈⣂⠈⠻⣿⡀⢺⣿⠃⣰⣿⣟⣠⡤⠖⠂⠀⢳⡱⢀⣸⣶⣧⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣺⣇⠇⠀⡇⢸⣀⣰⣼⣥⣤⣤⣶⣶⣿⣿⣷⣦⣬⣷⣼⣷⠾⠛⠉⠉⠑⠂⠀⠀⠀⠼⢯⣀⣸⡇⢹⠀⠀⣿⡓
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠋⡏⢻⠉⢹⢯⠍⠩⠭⠉⣉⡉⠉⠉⣉⣭⢿⡿⣟⠿⣿⣿⣿⣿⣿⡿⠿⠿⢿⡿⠛⢻⡏⢹⠩⠀⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⡇⣷⣾⡖⠀⡞⡇⠀⢀⡠⠴⠚⣻⣿⠟⢡⣾⡇⠹⣷⣄⠙⢛⠳⠦⣄⡀⠀⣸⢳⠀⢼⣷⡾⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⡀⢧⠀⠰⡿⡄⠀⠀⢀⣼⠟⠁⢔⣿⣿⡇⠀⣿⠻⣷⣄⠢⠀⠀⠈⢰⢧⠃⠀⡞⢨⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢳⡈⢧⢀⠱⡿⣄⣠⠟⠁⠊⠀⡼⠹⣿⡇⢠⠙⣇⠈⠙⠷⣄⢀⡰⢫⢂⢀⡼⢀⡏⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠳⡿⠣⡀⢈⡾⡷⣄⡀⠀⠰⠁⠀⣿⡇⠘⠀⠘⠄⠀⢀⡨⢿⢷⡁⣠⠿⣷⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢦⡙⢦⡀⢈⠚⠽⣲⠦⣤⣀⣹⡇⣃⣀⣤⠴⣞⠭⢃⠁⣠⠜⢁⡴⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⢦⣾⠗⠦⣀⡀⠉⠘⠃⢻⣧⠋⠘⠂⠁⢀⣠⠴⢿⣧⠴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⢑⡢⢤⣉⣹⣿⠒⠶⡿⠖⢺⣷⣋⣉⡤⢖⡫⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠭⠩⠟⠛⠿⠛⠫⠭⠭⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠀⠀⠀⠈⠙⠻⣷⣦⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣃⣀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠿⠭⠙⢿⠛⠛⠻⠟⡟⠛⢻⠛⠽⢿⣷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠁⠀⢣⡀⠀⠀⡤⠤⠤⣮⢀⠠⠤⠤⠜⠀⠀⡴⢁⠃⢀⡜⠀⠀⠀⢸⠉⠙⠻⡿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣀⠀⠈⠢⣨⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⠄⢘⡰⠁⠏⠁⠈⠑⠁⠀⠀⣎⠀⢸⣉⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣻⠢⠤⡀⢱⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡠⠊⡤⢦⠀⢸⠀⠀⡔⠒⠢⡄⠀⠀⠓⠑⢄⣰⡏
⠀⠀⠀⠀⠀⠀⢠⡿⣧⡀⠀⠈⠁⠀⡐⠒⠄⠀⡇⠀⠀⠀⠀⠀⠣⡄⠺⠤⠊⠀⡆⠀⡜⠀⠀⠀⡅⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣸⣷⠀⢧⠀⠀⠀⡰⠁⣄⠘⢲⢧⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢠⠃⢜⣁⢤⠤⠤⠜⠀⣠⠤⡀⢠⡟
⠀⠀⠀⠀⠀⠀⢿⣝⢤⡀⠓⠤⠴⠁⢶⠄⠀⡸⠸⡀⠀⢀⠤⢄⠀⠀⠠⡇⠀⠸⠀⠀⠀⢓⣋⠀⠀⠀⠈⢱⠈⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣶⣄⡀⠀⠀⠀⠀⣷⢒⠎⠒⠉⠀⢀⠇⠀⡠⠃⠀⡇⠀⠀⡎⠁⢨⠂⢀⠔⠋⠊⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⢄⠀⢀⠋⠉⠀⠀⠀⠀⠸⡀⡰⠁⠀⢀⠃⠀⠀⠘⡄⠀⠉⣁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣾⣤⣀⣐⣒⣒⣒⣒⣛⣓⣶⣶⣾⣢⠤⣀⡔⠃⠀⠫⠤⠇⠀⡠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠛⠳⣿⣷⣶⣤⣴⡁⠈⢡⡿
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠛⠿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠒⢉⠀⠀⠰⣶⣄⠠⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡀⠀⠀⠀⠀⠀⣶⣦⣿⣿⣷⣦⣀⠀⢠⡄⢀⡈⢷⣶⡄⢹⣿⣷⣿⣄⣰⡀⣀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣿⣿⣿⣿⣿⣷⣶⣖⣾⣷⣿⣿⣿⣿⣿⣿⣯⢰⣎⣻⣌⢿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣛⣯⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣷⣶⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣛⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⣽⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⣿
⡿⢿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣦⣴⣷⣾⣿⢿⣤⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣶⣶⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣿⣿⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⣉⣁⣤⣤⣤⣤⣤⣤⣈⡉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢁⣠⣶⣿⣿⣿⣿⣧⡙⣿⣾⣿⣿⣿⣿⣶⣄⡈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⡟⣿⣮⣹⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣆⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣰⣿⣿⣿⣿⣿⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⡟⣿⡟⢿⣟⣿⣿⣿⣿⣿⡿⠋⣁⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⣿⣻⣧⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡘⣿⡜⣿⣿⣿⣿⣿⣿⠃⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⡇⣯⣽⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⣿⣷⣿⣷⣿⣿⣿⣿⡿⠁⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣿⣿⣿⣿⡿⢃⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣜⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡉⠛⠿⣿⣿⣿⣿⣻⣧⣿⣿⣿⣿⠿⠛⢉⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣉⣉⣉⣉⣉⣉⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⢶⣿⣳⣄⡀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡴⢤⣄⣀⣀⣀⣾⣿⣿⣿⣷⠀⣴⣶⠀⠀⠀⠀⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⣇⣀⢀⢘⣿⣿⣷⣞⣿⣿⣿⣿⣃⣿⠩⠻⠇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠘⠫⠉⠉⠉⠉⠉⠁⠹⣿⡟⣿⣿⠃⢛⠁⠀⠀⠀⠀⠙⠇⠀⠀⢰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⣧⣿⠋⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠺⣯⠅⠀⠀⢳⣆⣀⠀⠙⠁⠰⣄⠀⣴⠒⠁⠘⣥⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣿⣆⣀⠀⠀⠹⡇⠀⠀⢀⣀⣼⣿⣿⣭⣉⣉⣻⣛⠛⠓⠒⠿⢽⣿⣿⣽⠿⢿⣿⣵⡂
⠀⣤⡀⠠⣠⣴⣶⣆⡉⡉⣿⣿⣿⣾⡿⣿⣿⣶⣽⣶⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⠻⠷⠶⢶⣦⣴⣿⣷⡷⡶⠿⠟
⠘⠋⠿⠿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠰⠿⠛⠉⠉⠀⡀⠀⢀⡄⠀⠈⠛⠙⠻⠟⠛⠛⠿⣶⣆⠙⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠻⢿⡷⢶⣤⣤⣀⡀⠀⠀⠀⢀⡴⠛⠱⣴⣦⡤⠤⣤⣤⡤⠤⠤⠤⠭⠤⠬⠋⠉⠁

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




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⡟⠉⠻⣿⣿⣧⠀⠀⠀⠈⢀⠀⠀⠠⠀⣀⣠⣶⣶⣦⣤⣀⣀⣀⡀
⠀⠀⣶⣶⣶⣶⣿⣿⣮⡯⠏⢷⠄⢿⡟⠑⠂⠀⢠⣾⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠃
⠀⠀⠉⠉⠉⠻⣿⣿⣿⣦⣤⣷⣦⠀⠀⠀⠀⢘⣶⣷⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠋⠁⠀⠀⠀⢀⣿⣶⣄⣍⣻⣿⣾⣿⣿⣿⣿⣿⣿⡏⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣉⠁⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠛⠀⠹⠏⠙⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣦⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⢠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⡟⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏