Coins Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣺⡭⠵⠖⠚⠛⠋⠉⠉⠙⠛⠛⠲⠿⢭⣗⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⡽⠞⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠙⠳⢯⣳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠉⠀⠀⢀⣠⡴⠖⠛⠉⠉⠉⠁⠈⠉⠉⠉⠛⠲⠦⣄⡀⠀⠀⠉⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠈⠳⣝⢦⡀
⠀⠀⠀⠀⠀⢠⣞⡽⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢛⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⢯⣳⡄
⠀⠀⠀⠀⢠⣟⡾⠁⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢷⣻⡄
⠀⠀⠀⠀⡞⡾⠁⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⣨⡷⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⢷⢳
⠀⠀⠀⣸⢱⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠈⠙⢦⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠸⡏⡇
⠀⠀⠀⡿⣾⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣷⢿
⠀⠀⠀⡇⡏⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⢸
⠀⠀⠀⡇⣇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣸⢸
⠀⠀⠀⣷⢿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿⣾
⠀⠀⠀⢸⡼⡆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠻⣦⡀⠀⠀⠀⣸⠇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⣧⡏
⠀⠀⠀⠈⢧⢷⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⡾⡼⠁
⠀⠀⠀⠀⠘⣯⢷⡀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡾⣽⠃
⠀⠀⠀⠀⠀⠘⢯⣳⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣬⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⣞⡽⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⣀⠀⠀⠈⠙⠳⠦⣤⣀⣀⣀⡀⢀⣀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢯⣳⢦⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⡴⣞⡽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢽⣓⣶⠦⢤⣤⣄⣀⣀⣠⣤⣤⠴⣶⣚⡯⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠒⠒⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡈⠉⠛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⢀⣾⠛⠛⠉⠉⠉⠉⠉⢉⣉⣙⠛⠛⠛⠿⢿⣿⣶⣦⣄⡀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⣰⡿⣿⡀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠿⢶⣾⠟⠋⠛⠀⢀⣀⣠⣤⣤⣍⡙⠛⠻⢷⣦⣀⠀⠙⢷⣤
⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠋⠉⣉⡛⣻⡇⠘⢿⣦⡀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠿⣷⣶⣶⠾⠟⠋⠉⠉⠉⠻⣿⣆⠀⠀⠉⢻⣧⠀⠀⠹⣧
⠀⢀⣤⣾⠟⠋⠁⠀⠀⣠⣶⡾⠟⠛⠛⠛⠻⣧⠀⡎⠉⠻⣶⣄⡀⠀⠀⠀⠈⠙⠛⠿⢶⣤⣤⣀⣀⠀⠀⠀⠀⠾⠟⠛⠻⠃⢀⣸⡟⠀⠀⠀⣿
⢀⣾⡟⠁⠀⠀⠀⢠⡾⠛⠁⠀⠀⢀⣤⣤⠀⠙⣿⣅⠀⠀⠀⡝⠻⢷⣤⣄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠶⠶⠶⠾⠿⠛⠉⠀⠀⠀⢀⣿
⣼⣿⠃⠀⠀⠀⠀⣿⡃⠀⠀⣤⣤⣿⡏⠀⠀⠀⠈⢻⣷⣄⢠⠃⠀⠀⠀⠉⡟⠛⠷⢶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⣿
⣿⢿⣧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠹⢷⣦⣤⣤⣶⠟⠋⠙⢿⣶⣤⣀⡀⢰⠃⠀⠀⠀⠀⠀⡟⠉⠉⠛⠛⠛⡛⠛⠛⠛⠛⠛⡟⠉⠉⢀⡇⣰⡟
⣿⡀⡟⣷⣄⠀⠀⠀⠈⠙⠿⣦⣤⣀⣀⡈⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣤⣀⣀⣇⣀⣀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣃⣀⣤⣼⡿⠋
⢹⣷⡇⠈⠙⢷⣦⣄⡀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠷⠶⠶⠶⠶⠞⠛⠋⠉⠉⠉⠛⠛⠛⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠻⣿⣦⡀⢰⠉⠙⠛⠷⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠋⢹⢸⣷
⠀⠀⠈⠻⣿⣾⣄⡀⠀⠀⡇⠀⠈⠉⠙⠛⡟⠻⠷⠶⠶⢶⣶⣶⣶⡶⠶⠿⠛⡟⠋⠁⠀⢶⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣧⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⣀⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣶⣷⣶⣤⣤⣤⣤⣤⣷⣴⣶⣶⣿⣿⣿⠿⠟⠋⠁

⡈⣠⣬⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⡉
⠀⣿⠀⡀⢀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⡇⢰⠰⡀⠀⠇⠀⡏⢱⠈⠁⣷⠀⠀⡁⠀⣤⠃⡃⠀⢸⢠⠃⢸⡜⠀⢀⡖⠀⠀⢱⣧⠸⠀⠀⢸⠀⠉⢸⠁⠀⣿
⠀⣿⠀⡇⢸⠀⠣⠀⡆⠀⡇⢸⢘⠀⡇⢡⠀⢰⠀⠄⡁⡅⢀⠧⡸⡀⢸⢃⠀⢈⢠⡏⠀⢸⠿⡇⢈⢁⢸⠀⢸⢸⠀⠀⣿
⡄⣿⠀⠈⠀⠀⠁⠀⠉⠁⠈⠁⠈⠁⠁⠈⠀⠁⠀⠉⠀⠉⠈⠀⠁⠉⠈⠈⠀⠀⠈⠁⠀⠈⠀⠁⠀⠀⠈⠁⠀⠈⠁⠀⣿
⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠂⠀⠀⣿
⠀⣿⠀⠀⠀⠖⠂⠀⠠⠄⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠐⠈⠀⠄⠀⠀⠊⠑⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⢀⢑⠓⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠄⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠠⠄⠄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣉⡉⠉⠀⠐⡄⠀⣿
⠀⣿⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠂⠀⠀⠸⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠤⠆⠒⠁⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠜⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢁⣝⠛⠛⠛⠋⠉⠙⠛⠛⠛⠛⠛⠛⠙⠋⠙⠋⠛⠋⠋⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠋⠛⠙⠋⠋⠋⠛⠛⠛⠛⠛⠛⠋⡁

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⣟⡏⠙⡟⣿⠲⠒⠲⠶⣶⢶⠶⠶⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢷⠳⠶⢤⣤⡤⢤⣤⣤⣤⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠳⠾⠤⢤⣤⣼⣇⣸⠟⠋⠁⠀⠈⣉⠙⠻⠶⢄
⠀⠀⠀⢀⣠⣤⣴⠶⠾⠶⠖⠲⣦⣤⣤⣄⣸⡟⡄⠀⢀⠄⠊⠈⠢⡀⠀⠀⢿⣿⠛⠷⠶⣦⣄⣀
⠀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣇⠀⠉⠂⠀⠀⠀⠀⢄⠀⠀⠙⣷⡀⠀⠀⠈⠙⠻⣦⡀
⠀⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡉⢿⣆⠀⠀⠁⢀⠀⠀⠀⠑⢄⠀⠸⣷⠀⠀⠀⠀⠀⠈⠋
⣾⡹⢿⣁⣾⠭⡷⠒⣀⣐⣶⣶⣦⣤⣶⡾⠛⢳⡈⢫⡣⡀⠀⠀⠑⠄⠀⠀⡠⠓⠀⣿⣿⣤⡶⠶⠽⣷⣾
⢿⣙⠷⠛⢿⣿⣷⣶⣤⣤⠤⠤⠶⣶⣗⣓⣊⣭⣿⣄⠙⢮⣲⢄⡀⠀⠆⠊⠀⠀⣠⣿⠿⠿⠤⠤⠴⠟⠛
⠀⢻⣿⡶⠶⢤⣤⣀⣀⣀⣉⣹⣉⣀⣀⣀⣽⣿⣿⣿⠙⠿⣟⢙⣾⣿⣤⣶⣶⣼⡿⠋
⠀⠈⠛⠳⢤⣀⣀⣀⠈⢉⠈⠉⠉⠉⣇⣀⣸⣿⠿⠟⠀⠀⠀⠉⠛⠛⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⡉⠤⠤⠤⠄⣉⡉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣈⠻⣦⡀⠀⣀⣤⡴⠶⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠊⠀⠀⠀⠀⠀⣀⡖⠒⣄⡀⠀⠀⠀⠈⠣⣈⣷⡾⢋⡠⠔⠒⠈⠁⠒⠢⢬⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢠⠁⠀⠀⠀⠀⢠⠊⠁⣀⣀⣀⢈⠇⠀⠀⢀⡜⣹⠏⡰⠋⠀⠀⢀⡀⠀⠀⠀⠀⠈⠳⢌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⠀⠀⠀⠀⣟⠀⠸⢅⡀⠀⠉⠀⠀⠀⡼⢰⡏⢰⠁⠀⠀⠐⡏⠙⠂⢑⡀⠀⠀⠀⠈⠳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠈⠙⠢⡀⠀⢀⠃⢸⡇⡌⠀⠀⠀⢸⠀⢠⠚⠉⠁⠀⠀⠀⠀⠀⠱⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡄⠀⠀⠀⠀⠀⢀⡀⠀⠉⠳⡄⠀⠃⠀⠸⠅⢸⡇⢡⠀⠀⠀⠈⢆⠈⠓⠂⠀⠒⢤⡀⠀⠀⠀⢣⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⡀⠀⠀⠀⠀⣏⠀⠉⠀⠉⠀⣠⠃⠀⠈⡇⠈⣷⠸⡀⠀⠀⠀⠈⠑⠒⠒⠒⢤⠀⢱⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⣄⠀⠀⠀⠀⠉⠑⣆⣠⠋⠁⠀⠀⠀⢻⡀⠹⣧⠱⡀⠀⠀⠀⠀⢀⡤⠤⠚⠀⡸⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣷⣈⣳⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠴⠊⣳⡀⠘⣧⡑⢄⠀⠀⠀⠀⠣⠤⠰⢤⠕⠀⠀⠀⡜⢸⠇
⠀⠀⠀⢀⡤⠒⡾⠛⠛⠋⠉⠉⠉⣭⣉⣽⣿⣿⣯⣭⡉⠉⠉⠙⠛⠓⢟⠳⢶⡟⠁⠱⣄⠈⠻⣬⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠞⣠⠏
⠀⠀⠀⢸⠛⠶⢭⣷⣶⣦⣤⣤⣤⣭⣍⣙⣟⣛⣁⣭⣥⣤⣤⣶⣶⣾⠵⠶⠟⡇⠀⠀⠈⠳⣄⠈⠻⢦⣍⡓⠒⠤⠤⠤⠒⢊⣡⡶⠋
⠀⠀⠀⠘⠦⣤⣀⣀⣀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠁⠀⢀⣀⣀⡤⠶⠃⠀⠀⠀⠀⠈⠑⠢⢄⣍⣋⠛⡳⢖⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠚⠚⠚⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠤⣄
⠀⠀⢸⡷⠤⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠼⡇
⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠙⠉⠙⠛⠒⠓⠒⠛⠉⠙⠉⠉⡉⠉⠁⠀⠀⠀⣇⡀
⠀⠀⠘⠧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣴⣃⢽⠆
⠀⠀⠀⠀⢸⠉⠉⠋⠓⠒⠚⠒⠲⠒⠶⠖⠲⠖⠒⠶⠓⠒⠛⠛⠋⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣆⣠⠴⡋⠲⢄
⠀⠀⠀⠀⠀⢸⠻⢿⣶⣒⣒⣤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⢒⣒⣚⣋⡭⡔⠋⠀⠀⢧⠈⠀⠱⣄
⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠈⠉⠉⠋⠉⠙⠉⠙⠋⠉⠋⠉⠙⠉⠉⠉⠀⠁⠀⢷⠀⠈⠀⠈⣆⠀⠀⠈⠳⡀
⠀⠀⢰⣋⠡⠚⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡬⢇⡀⠀⠀⠘⣆⠀⠀⠀⠙⢆
⠀⠀⢸⠈⠉⠒⠒⠶⠭⠭⠭⠭⢵⠷⠶⡶⠶⡶⠶⠶⠶⠶⠞⠒⠛⠛⠉⢹⠁⠀⠈⢦⠀⠠⠐⠉⢆⠀⠀⠀⠊⢣⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡀⠀⠈⢧⠀⠀⠐⠈⢣⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠙⡖⠶⢤⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣤⣤⣶⣶⡿⠥⠒⡇⠀⠀⠀⢣⡀⠀⠄⠈⠳⡔⠀⠀⠀⠘⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⡄⢀⠄⠀⠙⢆⠀⠀⠀⠘⣆
⠀⠀⠀⠀⣧⢤⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣀⣨⠴⣧⠀⠀⠀⠀⠀⠘⢆⠀⠠⠀⠈⢳⡄⠀⠀⠘⡆
⠀⠀⠀⠀⡏⠉⠒⠒⠯⠭⠭⢭⡵⣚⣓⣒⣒⣒⣚⣒⣚⡭⠭⠭⠭⠵⠖⠒⠉⣿⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠁⠈⢦⡀⢀⠸⡄
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠐⠀⠀⠜⠢⣤⠇
⠀⠀⠀⠀⠈⠓⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢀⣠⠴⠖⠒⢚⣛⣋⣉⣉⣉⣉⣙⣓⣒⠦⢤⣀
⣴⠋⢀⡴⠚⠋⠩⣭⡴⠆⠀⢀⣀⣤⣬⡉⠙⠲⢬⡙⢦⡀
⣿⡀⠘⢦⣀⠀⠀⠛⠓⠒⠛⠉⢀⣀⣴⣿⡀⢀⣨⠇⠀⣿⡀
⣧⠛⢦⣀⠈⠙⠒⠶⠤⠤⠤⠤⠤⠴⠶⠖⠚⠉⠁⣀⡴⢻⡇
⢸⣧⣄⠈⠛⠶⠤⣤⣤⣀⣀⣀⣀⣀⣤⡤⠴⠶⠛⢉⣤⠞⢳⡀
⢸⡌⠙⠿⣶⣦⣀⣀⣀⠀⠀⢨⠀⠀⢀⣆⣤⡴⠞⣋⡥⠶⢻⡇
⠈⢻⣦⣀⡆⠈⠉⡟⠛⠻⠿⡿⠿⠿⠿⠿⠗⠛⡏⠁⡇⣠⣾⣧⡀
⠀⢸⡈⠙⠻⡶⣤⣇⣀⠀⠀⡇⠀⠀⣘⣀⣀⣠⣧⡶⠛⢋⣅⠴⠷⠖⠒⢒⠒⠲⠶⠤⢤⣀⣀
⠀⠀⠙⢦⣤⡃⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠁⠀⣴⠞⢉⡴⠞⠛⢛⣩⠉⠉⠉⠛⡛⠶⠶⣭⣝⠲⣄
⠀⠀⠀⠀⠉⠛⠛⠶⠶⣦⣤⣼⣤⣤⣤⣤⣷⣶⢾⣿⠀⠘⣇⠀⠉⢻⣧⣠⡴⠞⠉⠙⣷⣄⠀⠉⢻⡌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⡟⠳⣄⠈⠙⠲⠦⢤⣤⣀⣀⣀⣙⣉⣤⠤⠖⠋⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡨⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠞⢉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣄⣈⡀⠀⢉⡁⣀⣀⣠⣤⣤⠶⠛⠁

⠀⠀⠀⠀⢀⣠⣤⣤⡶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢤⣤⣀⡀
⠀⣀⣴⠞⠋⠁⢀⣀⣤⣤⠴⠶⠶⠶⠒⠒⠒⠒⠒⠒⠓⠶⠶⢶⣤⣌⡙⠻⢶⣤⡀
⢸⡏⠀⠀⣴⠞⠛⠉⠁⠐⢶⣶⡶⠟⠓⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠉⠛⢷⣄⠈⠛⢦⡄
⣿⡆⠀⠀⢿⡀⠀⠀⠀⠀⠘⢿⣷⣶⣶⠶⠞⠋⠉⠁⠀⢀⣹⣿⠀⠀⠀⠀⢹⡇⠀⠀⣿⡆
⣟⢻⣄⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠟⠛⠛⢃⣀⣤⡾⠛⠁⠀⢀⣿⣇
⣿⣸⠙⢷⣤⡀⠀⠀⠉⠙⠛⠷⠶⠶⣶⣶⣶⠶⠶⠶⠶⠾⠛⠛⠋⠉⠁⠀⠀⢀⣤⠾⡃⣇
⠘⣿⣄⢸⠉⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠋⠁⢀⣷⢿⣄
⠀⡟⢿⣿⣦⣄⡀⢸⠈⠉⠙⠛⢻⠛⠶⠶⠶⢶⠶⠶⠞⠛⠛⡛⠉⠉⠀⡇⣀⣤⣾⠟⠁⣀⣿⡆
⠀⣧⣰⠈⠛⠿⣿⣿⣤⣄⣀⣀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢐⣇⣤⣴⣾⠿⠛⢉⣡⣴⠾⡏⢸⡇
⠀⠻⣿⣄⠀⠀⡄⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣥⡶⣶⠛⠋⣩⣀⣀⣧⣾⣅⣀⣀⣀⣀⣀⣀
⠀⠀⣿⠻⣷⣦⣇⡀⠀⠀⢰⠁⠀⠀⠉⠉⢹⡏⠉⠉⠉⠉⣩⠉⠀⠀⢀⣿⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⣦⣤⣀
⠀⠀⣿⣸⠀⠉⠛⢿⣶⣦⣼⣀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⢀⣀⣰⡟⠁⠀⠀⠀⣠⣶⠶⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣦⣤⣄⣈⠉⠛⠷⣦⣄⡀
⠀⠀⠈⢿⣦⣀⠀⢸⠀⠈⠉⠛⢻⠿⠿⠶⣾⣷⣶⣶⣾⣿⡿⠿⣿⣿⡇⠀⠀⠀⠸⣯⡀⠀⠀⢠⣤⣤⡶⠶⠶⠀⠀⠀⠉⠙⠛⠿⠿⣶⣤⡀⠉⠻⣦⣄
⠀⠀⠀⠀⠉⠻⢿⣾⣤⣀⡀⠀⢸⠀⣀⣠⣤⡴⠷⠶⠶⠶⠶⠶⣿⠘⢷⣄⠀⠀⠀⠙⠻⣦⣄⡀⠈⢿⣦⣤⣤⣴⠾⠟⠛⠛⠿⣷⡄⠀⠉⠻⣶⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⠿⠋⠉⠀⢀⣤⣴⠶⠿⠿⠿⣿⡀⣸⠙⢷⣤⣀⠀⠀⠀⠉⠛⠻⢶⣭⣍⣁⣀⠀⠀⠀⠀⣴⡿⠿⠶⠄⢀⣼⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⣴⠟⠉⠀⠀⢀⣤⡄⠈⢷⣇⠀⠀⠈⡟⠳⣦⣄⣀⡀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⠶⠶⠶⠶⠿⠟⠋⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸⣏⠀⠀⣤⣤⣿⠁⠀⠀⠀⠙⣷⣄⣰⠃⠀⠀⠈⢹⠋⠧⠦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⠀⠀⠀⠀⠻⣦⡀⠀⠀⠙⢷⣶⣶⡶⠟⠉⠙⠻⣶⣦⣄⣀⡏⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⢛⠛⠛⠛⠛⢻⠋⠉⠀⣜⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⡟⢷⣄⡀⠀⠀⠈⠛⠷⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⢀⣉⣛⣿⡿⣷⣶⣶⣤⣼⣥⣤⣤⣄⣀⣼⣀⣀⣀⣀⣼⣤⣴⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠉⠻⢶⣤⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠀⠀⠀⠈⠉⠉⢉⣹⠟⢻⡟⠛⠛⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡇⠀⠈⠉⠻⠛⠷⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠛⠋⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣀⡆⠀⠀⠀⠀⢸⠁⠉⠉⠉⠉⠙⡟⠋⠉⠉⠁⢸⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣦⣤⣼⣄⣀⣀⣀⣀⣀⣇⣀⣠⣤⣴⣾⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⡀⠄⠂⠈⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠐⠀⡠⠲⠛⣹⣿⣿⡿⠶⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢁⣀⣤⣀⣄⣀⡀⠐⠠
⠀⠀⠀⠀⠀⠀⣏⠐⠒⠿⠽⠧⢶⢚⡇⠀⡀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠷⢉⣸⣿⠿⢾⣟⣳⡄⠉⠄
⠀⠀⠀⠀⠀⠀⣎⠁⠒⠖⠛⠒⡟⣻⠇⡀⣠⠤⠶⢂⣰⣒⢀⠄⠁⠂⠀⠀⢘⢦⣄⣨⣭⣭⡭⠥⢿⠁
⠀⠀⠀⠀⠀⠀⡏⠒⠒⠳⢷⢾⠿⣽⣷⠣⢄⣊⣛⣛⣛⣋⣀⣵⡇⠀⠀⠀⠘⣀⠀⡉⢙⢻⣟⢹⣭⠂⠀⠀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⡟⠲⢶⣶⣾⣾⡶⣿⣄⣧⡤⠤⠬⠭⠿⢷⣞⣿⣷⠞⠻⣿⣾⣀⠀⠉⠙⢿⣿⣙⣧⣇⣠⠀⣀⡄⢀⣀⠈⠤
⠀⠀⠀⠀⠀⠀⡟⠂⠾⠾⢦⠶⠏⣹⣿⡧⠌⡉⢙⣻⣫⣽⣭⡾⣿⣓⠒⣿⣿⣉⠛⠺⢿⠿⠭⡽⣾⡋⠘⣿⣻⡿⠉⣡⡇⠐
⠀⠀⠀⠀⠀⢠⠷⣤⣧⣼⣯⡉⠀⠻⣷⡶⠢⠤⠴⠶⠿⠗⡐⢚⣿⠭⠈⠿⢿⡽⢛⡒⢻⡦⣶⡟⣿⣝⠛⠉⠉⣙⣭⣿⠁
⠀⠀⠀⠀⠀⠘⣄⣨⣉⣛⣫⡉⠙⢻⣻⡗⠀⠤⠶⡬⠽⡧⠲⢻⡇⣸⢿⠗⣋⣀⣼⣧⡼⢅⡤⠴⣿⡏⠛⠈⠉⣿⣇⣾⣀⡀
⠀⠀⠀⠀⠀⠈⣄⠀⣏⢹⡏⠛⣜⣛⣓⣇⠉⠢⠤⢴⣶⠷⢶⣩⣿⣄⢀⣀⣯⣤⣾⣥⡦⢠⡶⠶⣿⣍⡙⠁⠉⢿⣹⣿⣭⣭⠿⠦⠤
⠀⠀⠀⠀⠀⠠⣄⢈⠙⢻⣟⠒⣶⣦⣽⢧⠙⠛⠻⠿⢿⡿⡿⢷⣷⣏⠀⢠⣃⡼⠿⣿⡦⢼⣷⠞⣿⡍⠛⠶⠾⠛⠻⠗⠳⣷⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⡿⠳⣿⣾⣦⣄⠈⠑⡞⣷⠖⡟⠟⢻⣿⣟⠉⠁⡖⣚⣿⣿⣇⣼⡾⠾⠛⠢⢤⣤⣤⣠⣀⣄⣠⠀⠈⠉⠉⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣭⣿⣏⠛⠭⣗⣛⣲⣾⡿⢿⣏⣹⠉⢉⣟⣫⣿⣿⣍⣽⡭⠭⢽⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⠓⢶⣶⣶⣶⣶⣖⣯⣿⡙⠋⣫⣭⣭⣤⣤⢛⡋⢓⣚⣢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⣾⣿⣿⣴⢾⣾⣛⣂⣴⣒⣂⣩⣄⣠⣀⣀⡛⡙⠻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠛⠚⠛⠛⠛⠛⠛⠓⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠞⠉⢁⣠⣤⠶⠶⠖⠛⣛⡛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣍⡛⠷⣤⡀
⠀⠀⠀⠀⠀⣿⠁⠀⣼⠋⠁⠀⠀⠙⣿⡟⠉⠁⣀⣠⡴⠶⠶⠶⣦⡄⠀⠈⠛⢦⠀⠙⣧⡀
⠀⠀⠀⠀⠀⣿⡄⠀⠙⢷⣄⡀⠀⠀⠙⠛⠛⠛⠉⠁⠀⣀⣀⣴⣿⣧⠀⠀⣠⡼⠃⠀⢸⣧
⠀⠀⠀⠀⠀⣇⡟⢦⣄⠀⠈⠛⠳⠶⣤⣤⣀⣀⣀⣀⣠⣭⣭⣤⡴⠶⠞⠛⠉⠀⠀⣠⡟⣿
⠀⠀⠀⠀⠀⢻⣇⠀⡟⠳⢦⣄⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⡃⣠⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣧⣄⡀⢉⠉⠛⠛⢲⠶⠶⠶⠶⠶⠶⠶⠖⢻⠛⠋⠉⣇⣀⣴⡿⠋⢀⣿⡄
⠀⠀⠀⠀⠀⢸⣇⡆⠙⠻⢿⣿⣤⣤⣀⣸⣀⣀⣀⢰⡇⠀⢀⣀⣸⣤⣶⡾⠟⣋⣡⡴⠾⢋⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣧⣄⠀⡆⠀⠉⠙⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠽⠶⠞⡟⠋⠁⡇⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡏⡟⠿⣷⣄⡀⠀⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⡇⠀⠀⠀⣇⣀⣠⣷⠿⢿⠋⢉⣷
⠀⠀⠀⠀⠀⠀⠈⢷⣇⠀⠀⢙⠙⠿⠷⣦⣤⣤⣤⣯⣤⣤⣤⣤⣷⣶⣶⡾⠿⠛⡏⠀⠀⢸⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣾⣀⠀⠀⢸⠀⠈⠉⠉⡟⠉⠉⠉⠉⢹⠀⠀⠀⠀⣇⣀⣤⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣾⣦⣤⣤⣤⣧⣤⣄⣀⣠⣼⣤⣶⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠉⢀⠀⣠⣂⣸⣦⣴⡼⢧⠤⠠⣤⣦⣤⣆⡄⠀⢈⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⣀⣴⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢦⣄⣠⠦⠌⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣀⠀⣤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⠂⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢁⠀⣠⣾⠟⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⡄⠤⢰⡆⠀⠈⠛⢧⡔⠀⡙⠷⣄
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣨⡿⠋⠀⠀⠀⠀⠀⢀⣀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⢶⠛⠆⠈⢳⣄
⠀⠀⠀⠀⠀⡴⠋⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⢰⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠁⢦⡷⠀⠙⣧
⠀⠀⠀⢀⡾⠁⢀⣰⠟⠀⠀⠀⠀⣀⠀⠀⠀⢀⡴⠀⠀⠀⢠⡏⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢀⣀⡛⢦⠀⣀⠀⠀⠀⠀⢠⠀⠀⠳⡋⠀⠈⢷⡀
⠀⠀⢀⡞⠁⠁⣼⠇⠀⠀⠀⢠⣬⡧⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⠀⠁⠈⣿⡄⠀⠐⠃⠀⠘⡆⠘⠇⣠⠈⣷⡀
⠀⠀⡾⠁⠀⣼⠏⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⢧⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠁⠁⠀⠀⠸⡀⠀⠁⠀⢀⣌⡷⠀⠀⣨⣧⠘⣧
⠀⢸⠃⠈⢰⡟⠀⠀⠀⠀⠀⠲⠖⡀⠀⠒⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠸⠤⠇⠀⠉⠀⠀⠀⠀⠀⠖⢦⢹⣆
⠀⡟⢠⠒⣾⠀⠀⠀⠀⠀⠘⠃⠁⣻⡶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡄⣿
⢸⠇⠈⢸⡟⠀⠀⠀⠠⣇⠀⠀⠀⠉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠛⡇⢸
⣼⠀⠘⣿⠁⠀⢀⣀⣴⠊⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠟⠁⠀⠠⠀⢢⡀⠀⢏⣱
⣿⠀⠀⣹⠄⠈⠁⠉⠃⠀⡸⠂⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠀⡘⢀⡀⠀⠀⠁⢄⣹⠆⢈⣽
⣿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠙⠛⠁⠀⣠⡶⢾⠁⠀⠈⣿
⣿⠀⢀⣸⡇⠀⠀⠀⡀⠀⠀⡀⠈⠈⢻⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠇⢀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸
⢸⡆⢈⢸⡇⠐⠀⠂⠹⢄⡀⠙⠦⣄⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃⠀⠀⠀⠀⠀⠀⠀⢻⡏⣸
⠈⣧⠀⠘⢿⠀⠀⠀⠀⠘⡇⠀⡀⠈⠀⠡⠼⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣥⣤⣿⠀⠀⠀⠀⠙⠚⢀⠀⢐⡛⠁⠀⣄⡀⠀⠀⠀⠀⠘⣾⢃⣿
⠀⢹⡆⢠⡘⣧⠀⠴⠋⠉⠁⠀⠓⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸⢿⡞⠃⠀⠀⠈⠡⠿⡄⠀⠀⢠⡟⣸⡟
⠀⠀⢻⡀⢳⢙⣆⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⡄⠀⠀⠊⢳⠀⠀⠀⢐⣡⡾⠁⠀⠈⣠⡄⢰⡶⠟⠀⢭⣾⢷⡿⠁
⠀⠀⠈⢷⡈⠃⣼⣆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠓⠀⠀⢀⡀⠀⠳⣦⣀⣀⠀⠉⢉⣉⡽⠛⠁⠀⠀⠀⠀⠀⠀⢐⣿⠉⠀⠀⠀⠀⠉⠀⠘⠁⠀⢦⡼⢫⣿⠃
⠀⠀⠀⠈⢷⡀⠈⠙⢦⡄⠀⠀⠀⠀⢀⣴⠀⠙⠶⠶⠚⠋⠙⠛⢃⣴⡿⢉⡴⢀⣀⣀⡐⠀⠠⣶⡛⠛⠳⠶⠟⠋⢁⢠⡀⠀⠀⠀⠀⠀⠀⡀⣰⡞⢡⡿⠁
⠀⠀⠀⠀⠈⢳⣄⠀⡈⠳⣄⠀⢀⡐⢨⡽⠂⠀⠀⠀⠀⣴⠧⢀⠞⣀⣀⡞⠀⠈⠉⠉⢹⣆⡀⠀⠉⣛⡟⠀⠀⠀⠀⠸⠃⣠⠀⠀⠀⠀⢀⣽⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢧⡌⠢⠈⠳⣦⡈⠈⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⢤⠀⠀⠘⠢⠀⢀⣼⣏⠀⠀⢀⣴⠟⢋⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠐⠉⢱⠆⡀⠀⠀⠀⠀⠘⠀⠀⠈⣣⠄⠀⠀⠀⠀⠀⠀⡠⠉⠀⣤⠁⠀⠀⠀⠀⠀⠛⠋⠀⣀⣈⡶⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠢⣏⣳⠶⣄⡀⠀⠲⣀⡤⠿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣹⡄⠀⠀⠀⠀⢀⠀⣰⣴⠿⠋⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣉⠀⠋⢩⡒⢶⠥⣤⣤⣀⠀⠀⠀⠀⠀⠘⠋⠁⠚⡁⠀⣠⠀⣀⣤⠶⠛⣉⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣉⠀⠂⠀⠃⠼⠇⠘⠁⠲⠀⣀⣈⡹⠗⠂⠐⣊⣩⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣦⣤⣤⣤⣤⣤⣤⣶⠶⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠛⠛⠲⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣯⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠞⠛⠛⢻⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⡟⢻⢱⣣⠟⠁
⠀⠀⠀⠀⠀⡜⡑⡝⠁⠀⠀⠀⠀⢸⣜⢔⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⡟⢛⠹⡀⣇⡰⠜⠋
⠀⠀⠀⠀⢠⣷⠱⠣⠀⠀⠀⠀⠀⠀⢿⣿⢻⡿⡿⢿⠷⡶⢶⢶⡶⡶⢾⠟⡟⢛⠻⠉⠏⢹⢈⣄⡧⠼⠒⠉⠁
⠀⠀⠀⠀⢸⣷⢳⢔⣶⣄⠀⠀⠀⠀⠀⠉⠛⠾⠤⢮⣄⣧⣸⣀⣇⣰⣨⠤⠧⠼⠒⠓⠊⠉⠉⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠻⣌⠎⢟⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠻⢜⡘⠀⠟⢻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⠾⢄⡆⢰⠉⡙⠻⡷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡻
⠀⠀⠀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠈⠙⠲⢇⣸⠀⠎⢨⠋⡻⠻⡶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠪⢜⣼
⢀⢴⢟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⠤⣇⣸⠀⡎⢠⠋⡹⠙⡟⢻⠿⡿⠻⡿⢿⢿⢿⢿⡷⣺⣿⣿⢢⡀
⡏⡎⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠲⠧⠼⠄⣎⣠⣃⣸⣀⣎⣌⣌⡮⠷⠛⠉⡇⡇⡇
⣧⡀⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢞⠔⣱⡇
⣿⠟⡦⣕⡻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣫⢵⢻⣷⡇
⠈⠧⣇⡇⡍⡟⡟⡿⢿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⢿⠟⢻⠙⠉⠀⢸⡸⠋
⠀⠀⠈⠑⠣⢧⣇⡇⠀⠀⢸⠀⢹⠉⡟⠙⡟⢻⠛⢻⠛⠻⠛⡟⠛⡟⢻⠋⢻⠉⢹⠁⡇⠀⡇⠸⠀⣸⡠⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠤⠼⠠⡇⣀⣇⣈⣀⣸⠀⣘⠀⡇⢀⣇⣸⣄⣸⢀⠼⠤⠷⠒⠓⠊⠉
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⡴⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠚⠉⠉⠉⠉⠉⠟⠉⠉⠩⠉⠛⠉⠉⠙⠒⡾⢕⡇⠀⠀⢤⡀
⠀⠀⠀⠀⠀⡠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡬⡧⣤⣶⣶⣶⣿⣴⢶⢴⣶⡶⠏⠽⣐⠸⢸⠀⠀⠙⣆
⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠠⡀⠀⠘⢧⣧⣧⣤⣸⣸⡸⡨⣤⣴⡴⠦⠳⣗⠬⣺⡆⠄⠐⠈⢷⡀
⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢼⣸⣨⣀⣇⣅⣅⣥⣤⣤⡄⡒⡼⡭⡒⢽⡄⠀⠐⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⣏⣛⣿⣹⣹⣹⣸⣨⣤⠇⢃⡻⡂⠽⣿⡄⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠻⢻⢻⢻⠻⠉⠛⠛⡋⡃⡇⡇⣆⡷⢕⡺⣿⡌⠀⠀⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⡟⡾⠾⡾⢾⢻⢻⢻⠛⡖⣇⡇⣧⠃⠱⡪⢕⡿⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢶⢶⢶⢶⣶⡾⡾⡾⡾⠟⡽⠑⢸⡸⠀⠀⠘⢕⡪⢎⠢⡀⠘
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣼⣼⣤⣤⣤⣧⡶⢷⣖⢓⢩⢵⢻⠀⠀⠀⠈⠪⣑⠱⣄
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢩⣈⣈⣉⣁⣅⣇⡇⡇⠥⠼⠸⠜⠚⠈⠀⠀⠀⠀⠀⠀⠀⠊⠐⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠉⠛⢷⡦⡀
⠀⠀⠀⠀⠀⠰⠀⠀⣰⣋⣉⠁⠀⠀⢹⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣤⢤⠀⠀⠙⣞⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠻⡛⠛⠀⠀⠀⢸⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⢷⠢⡀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠁⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠒⠻⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠠⠤⠶⠛⠁⠀⠀⠀⠀⣠⠴⠒⠊⠃⠒⠢⣄⠀⠀⠈⠂⠄⢀⡀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⣰⠟⠙⣦⠀⠀⠈⡯⣇
⠀⠀⠀⠀⢀⠂⠀⠀⡄⠀⠹⡢⠀⠀⠀⠀⡇⠀⠀⢀⣠⠟⠓⠀⠈⠀⠀⠀⡯⣿
⠀⠀⠀⠀⢀⠀⠈⣷⣧⠀⠀⢹⡄⠀⠀⠀⢧⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⣰⣑⠎
⠀⠀⠀⠀⠈⠄⠀⠀⠁⠀⢀⡟⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⠞
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣤⣤⣴⣾⡯⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢉⣉⠉⠛⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠴⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠸⢻⠆⠀⡀⠀⢳⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠐⠃⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠖⠈⢃⣤⡔⠀⠈⣯⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠱⠀⡈⠀⡩⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠀⠐⠊⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠋⣉⣁⣤⡤⠦⠄⢤⣤⣬⣉⡙⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⣠⡤⠞⠋⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠲⢦⣈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢅⡴⠛⠁⠀⠀⣀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠐⠒⢆⠈⠙⠄⣨⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠛⠀⠀⠀⠁⢥⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠞⠀⠀⣰⠀⠀⠐⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠈⠡⠔⠙⠂⢢⡀⠀⠂⠠⠡⢀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⡏⢀⡏⠀⠀⢀⡁⠀⢀⣲⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡈⠁⠀⠀⠼⠀⠀⣆⢳
⠀⠀⠀⠀⠀⠀⡘⠀⣼⠀⠀⠀⠄⠐⣤⠀⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠰⡎⣇
⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⢀⡃⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠅⢹
⠀⠀⠀⠀⠀⠀⡇⠸⡇⠈⠉⠀⠂⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠄⢀⡤⠀⢀⣬⠂⣼⢸⠂
⠀⠀⠀⠀⠀⠀⡇⢈⡇⠀⢀⠀⢀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠆⢰⡄⠀⠉⠁⠀⣸⢸
⠀⠀⠀⠀⠀⠀⢧⠀⢇⠀⠈⠑⠀⠑⠀⢻⠄⠀⠀⠀⠀⠀⠀⣀⣀⣈⣂⣠⠀⠀⠀⠄⠀⢠⠞⢀⠀⠀⠀⢨⡟⡿
⠀⠀⠀⠀⠀⠀⠘⡄⠘⡆⠚⠉⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠉⢀⡀⠀⠀⢶⠞⠀⠈⢑⣇⠀⣸⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡜⢹⡄⠀⠀⠀⠀⠲⠀⠀⢀⡀⠠⣄⣀⠐⢚⡶⠇⠀⠀⠁⠀⡴⠋⠀⠀⠋⠘⠁⣬⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⢆⠀⠀⠐⡆⠉⠉⢉⡌⠡⠚⡱⠃⠶⢤⠀⠋⣭⠍⠉⠀⡦⠀⠀⠀⠀⡴⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⡮⡈⠀⠀⠀⡀⠠⡄⠀⠀⠀⠀⠈⡁⠠⠬⠀⠐⢀⡴⠇⢀⡤⢛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠡⣲⢤⡀⠀⠀⠼⠀⠀⠀⠀⢀⢀⣧⠀⠀⠀⠀⣤⠶⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⣘⠻⠓⠂⠀⡄⠀⡀⠉⢨⡄⠐⠤⣚⣭⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠦⠤⠤⠶⠶⠒⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣟⣛⠛⠛⠛⠛⢿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡶⠤⠬⠭⠭⠭⠭⠽⢛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠌⠭⠤⠠⠭⢩⠤⠿⣿⣿⠀⣄⡒⣶⠶⠶⠶⠶⠶⠶⠶⣆⣠
⠀⠀⠀⡀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣟⣫⣃⣉⣑⣑⣀⣨⣿⣿⣠⣿⣿⣦⣤⡤⣤⡄⣤⠮⠤⢿⣿
⠀⠀⠀⡿⠯⣤⣤⣄⣠⣤⣤⣤⣤⠼⣿⣿⣯⣒⡛⠟⠿⠥⠤⡭⢭⣿⣗⣶⣿⣿⣶⣤⣤⣠⣬⠥⢼⣿
⠀⠀⠀⣿⣷⣶⣒⠲⠒⠒⠒⠂⠈⣻⣿⣿⣿⣟⡖⡦⠤⠤⠤⢤⠤⣬⣯⣽⣿⣿⣿⣶⣖⡒⠚⠛⡚⡫
⠀⠀⠀⣿⣓⠲⠦⠶⠤⠤⠤⠶⠶⢾⣿⣿⡯⣍⡛⠻⠟⠛⠛⣚⣚⠛⢻⠿⢽⣿⣿⣭⣯⠀⠄⠖⢐⣺
⠀⠀⠀⣿⢿⣿⣷⣿⣵⡾⠛⣲⡎⠿⣿⣿⣗⣒⡒⠶⠴⠤⠴⠆⠶⢦⣒⣚⣽⣿⣿⣿⣿⣭⣭⠍⠭⢼
⠀⠀⠀⣿⣿⣶⣶⣦⣤⣬⣼⣴⣷⣿⣿⣿⣿⢭⠽⢽⣿⣿⣛⣻⣯⣭⢭⠷⢺⣿⡶⣿⠧⠒⠦⢆⣛⣺
⠀⠀⠀⢸⡷⠊⠩⠉⠉⢉⠉⠉⠉⠉⠫⣳⣾⣷⣶⣒⣂⠀⠀⠀⡀⣚⣊⣉⣽⣿⣿⣿⣿⣭⡭⠩⠍⠸
⠀⠈⠁⢼⡿⠿⣛⣛⣛⣛⣓⣛⣛⣛⣩⡭⣽⣿⣿⣿⡿⠿⢭⢭⡯⡿⠷⣶⣾⣷⣷⣶⣿⣶⣶⣷⣶⣯⡀
⠀⠀⠀⢸⣿⣓⣒⣒⠒⢲⠤⡶⠶⣶⡛⢻⣻⣿⣿⣿⡿⣿⢿⣿⡧⠴⣶⢌⣲⡶⠶⠾⠿⠿⠿⢿⡿⣿⣗
⠀⠀⠐⢺⡿⠿⠿⠿⢛⡛⠛⠛⢒⣚⠛⣉⣥⣿⣿⣭⣽⣿⣽⣿⣿⣯⢽⡓⠒⠤⠭⠭⠭⣭⣥⣼⣿⣿⣿
⠀⠀⠀⠀⠙⠛⠳⠶⠶⠶⠶⠶⠶⠶⢾⠟⣻⣿⣿⣿⣷⣶⣿⣶⣶⣶⡼⡿⣿⣿⣿⣞⣒⣚⣲⣓⣺⣉⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣷⣟⣙⡛⠉⠛⠋⠙⠉⣉⣿⣶⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠷⠴⠬⢭⣩⣭⡭⠽⠽⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠖⠲⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⢀⣠⠤⠖⠒⠒⣛⣛⠒⠒⠲⠤⣄⡀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡤⠚⣉⠤⠒⠊⠉⢉⣉⣉⡉⠉⠑⠒⠤⣉⠓⢤⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⢁⡴⠋⠁⠀⠀⢀⣠⠿⠉⠉⠿⣄⣀⠀⠀⠈⠙⢦⡈⠢⡀⠀⠈⢦
⠀⠀⠀⠀⠀⡜⠀⠀⣠⠋⡴⠋⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠙⢦⠙⣄⠀⠀⢣
⠀⠀⠀⠀⡼⠀⠀⣰⠃⡜⠀⠀⠀⠀⢰⠃⠀⠀⡴⠋⠉⠉⠓⢄⠀⠀⣈⡇⠀⠀⠀⠈⢣⠘⣆⠀⠀⢧
⠀⠀⠀⢠⠃⠀⢠⠇⡸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠣⡀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⢇⠸⡄⠀⠘⡆
⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⣿⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⡷
⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⠀⠀⠈⢣⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠘⡄⠀⠘⡆⢱⠀⠀⠀⠀⠀⢀⣀⠤⢦⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⠆⠀⠀⠀⠀⡎⢰⠃⠀⢠⠇
⠀⠀⠀⠀⢳⠀⠀⠹⡄⢣⡀⠀⠀⠀⢯⠀⠀⠀⠓⢤⣀⣀⣀⠴⠃⠀⠀⡸⠀⠀⠀⢀⡜⢠⠏⠀⠀⡞
⠀⠀⠀⠀⠀⢣⠀⠀⠙⣄⠳⣄⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣠⠞⣠⠋⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠢⡈⠳⣄⡀⠀⠀⠉⠓⣶⣀⣀⣴⠒⠋⠁⠀⢀⣠⠞⢁⠔⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠓⢤⣉⠒⠤⢄⣀⣈⣉⣉⣁⣀⡠⠤⠒⣉⡤⠚⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠈⠙⠒⠦⠤⠤⣭⣭⠤⠤⠴⠒⠋⠁⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣯⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠊⠀⡴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣖⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢏⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠒⢤⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡠⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣊⡤⠚⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡁⢂⡠⣣⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢨⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠣⠈⢆⣀⠔⠊⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡈⠁⠐⢀⡀⢀⡜⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠉⠈⠉⢁⠔⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣒⠬⠖⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣾⣿⣿⡷⣦⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣻⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣽⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣾⣟⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⠿⠋⠀⠀⠀⣼⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣂⠀⠀⠀⠀⣰⣟⣽⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⢶⡄⠀⢀⣰⣶⣷⡻⣿⣿⢻⣿⣿⢣⣿⣶⣀⠉⠱⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⣾⡟⢠⣿⣿⣿⣿⣿⣿⡋⣷⣻⣿⣿⣿⣿⣿⣷⣀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢹⣿⢹⣿⣷⣿⡟⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⢻⡇⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣜⣿⣿⣿⣸⡇⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⢽⣿⣿⣿⡇⠘⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣛⣯⣭⡍⠉⣽⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⠷⠶⢾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢨⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⣿⣟⣻⣂⣀⣿⣿⣿⣿⡆⠀⠀⢀⣴⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⠉⠁⠉⠉⢩⠝⢿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣿⣿⣿⠋⠀⠀⠴⠒⠒⠒⢶⠂⣠⡶⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠢⡈⢻⣿⠀⠀⠀⠴⠞⠋⠑⠀⠰⡿⠛⠴⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⣇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠳⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⡄⠀⠀⠀⠀⠰⠠⠤⠤⠤⠤⠂⣶⡂⢠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠙⠦⠀⠀⠀⠀⠀⠀⠀⣀⢀⡨⠋⠁⡝⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⢿⡷⢹⣆⠀⠀⠀⠀⠀⢀⠀⠀⣀⣼⠛⠿⠛⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣾⣿⡿⠛⠋⣉⣁⠈⣠⣿⣿⣦⠀⠀⠀⢀⠈⠿⠿⡿⠃⢀⣿⣿⣿⣿⡿⡿⠷⣆
⠀⠀⠀⠀⠀⣠⣿⣿⣷⣆⠉⣉⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⢴⣬⣥⣶⣦⣶⣿⣿⣿⣿⣿⣿⣶⣴⣌⢷⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠤⠐⠒⣦⠈⢿⠿⠟⠋⠙⠛⢻⣿⡿⣿⣿⣿⣿⣿⣧⢻⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⡿⠟⠛⠛⢱⠈⠀⠀⠀⠈⠂⠰⠀⠒⡄⠐⠀⠘⣿⣿⣿⣭⣻⣿⣿⣿⣿⢻⡀
⠀⠀⡼⢹⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢧
⠀⢰⠃⣿⠏⣿⣿⣿⣿⣿⣿⢰⣿⡿⠟⠊⠀⠀⢀⢠⣵⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⡄
⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⢸⣿⣷⡄⢇⠀⠀⠀⢀⡘⣇⠀⠀⠄⠀⢸⣇⠀⢠⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡌⣿⣿⣿⣦⡀⢠⡀⠀⠁⣹⡄⣤⠀⠀⠀⣿⠀⢸⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡇⢻⣿⣿⣿⠋⠀⢠⡴⠊⠀⣠⢿⡔⢤⡄⢻⡀⢸⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⢿⡇⢸⣿⣿⣇⠀⠘⡟⠀⢀⣾⠟⠁⣠⠏⣡⣈⣁⣾⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣾⣿⣿⣿⣷⡄⠃⠀⡾⠃⠀⡴⠃⣰⣿⣿⣿⣿⣿⣿⣿⠿⠆⠘⠋⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠋⢉⣧⣀⠀⠀⠀⡜⠀⣸⠇⣻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣄⡀⠀⠈⠃⠀⠟⠛⠿⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢿⣿⣿⣿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣚⣮⢢⡀⠀⠀⠀⠀⠀⢸⡏⠨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠈⠘⠁⠀⠀⠀⠀⠀⠀⠠⢀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣶⠀⠀⠀⠀⠀⠀⢀⡴⢾⣧⠀⠉⢷⣦⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⡠⢉⣱⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⢹⣦⣠⡄⠀⠀⣀⠀⢈⠁⣿⣿⣆⠀⢸⣿⣷⠰⠶⣆⣲⣶⣤⣦⣦⣶⣾⠏⢰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣾⣯⣿⣿⣶⡾⠛⢸⣿⣿⣿⣿⣿⣷⣼⣿⣿⣷⣾⣿⠏⠈⠉⠉⠉⢉⣿⣠⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⠻⣿⡿⠁⠀⠀⣿⣿⠇⣿⣿⣿⣿⣿⣿⠙⠻⠟⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣧⣽⡇⠀⠀⠀⣿⣿⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⣼⣿⣾⣿⣿⣧⣼⢻⣾⡄⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣆⠀⠀⠀⠀⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣿⣿⣼⣿⢻⣿⣷⠀⠀⢫⠉⣿⣿⣿⣿⣿⣿⣿⣿⡐⢒⣲⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣿⡿⢿⡹⣿⣿⣟⣿⣿⡇⡉⠿⠻⢿⠋⠉⣹⡿⠉⠉⣿⡇⣿⣿⡄⠀⠠⠀⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⠷⠠⠠⠀⠄⢨⡅⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⣿⣟⢳⣎⢹⣿⣧⣿⣿⣿⣏⢠⣙⣳⣿⡇⠀⢸⣧⠀⠀⣿⣣⢸⡟⢹⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣟⣿⢼⣿⣿⣿⣿⣧⠀⠈⠉⠀⢻⠀⡽⠿⠀⠀⣿⠉⠈⣿⠈⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠠⠤⠘⠀⢩
⠀⠀⠀⠉⠉⠁⠘⠿⡆⠙⢿⣿⣿⢻⣿⣺⠿⣿⣿⣿⣿⠀⠀⠀⠀⢠⠘⠃⠁⡠⠊⣿⠀⠀⣼⠀⠀⠀⢀⠀⢩⠀⣿⣿⡟⢻⣿⣧⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⢈⡟⢿⣿⢿⣿⣿⣿⣿⠟⠀⠀⢀⠴⠃⢴⣴⠈⣀⣀⣘⣧⡀⣸⠀⠀⢀⣽⡿⠁⢸⣿⣿⡇⠘⣿⣿⡆⠆⠒
⠀⠀⠀⠀⠀⠀⠰⠛⢡⡆⢀⣿⡟⣿⣿⣾⣿⣿⣿⣿⠒⠒⠛⣧⡀⠴⠢⢿⠁⠉⠉⠉⣿⠋⣷⡴⠚⠛⠛⠁⠀⣼⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣾⠇⣸⣿⣿⣿⣿⣿⣿⣭⣿⣿⣆⣤⣤⠼⣾⣆⣎⣡⠤⠝⠛⢻⣯⠅⠒⠀⠀⠀⠀⠀⣠⣿⣿⡇⢀⡤⣿⣿⡗⣒⡢⠴⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠛⢿⣿⡿⣿⣿⣿⣼⣿⡿⠏⠉⠀⠀⠈⠋⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⣠⢿⣿⡏⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣚⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣿⠉⣿⠉⠉⠉⡏⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⣿⣀⣿⣀⣀⣠⣧⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⠀⠀⠄⠀⠀⠆⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠷⠿⠿⠿⠿⠿⠿⠿⠀⠀⠦⠤⠤⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠭⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠔⠒⠻⠿⣷⣶⣤⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⣯⠉⠉⠐⣒⣛⡿⠿⣿⠶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣵⠀⠀⠀⠀⠀⢭⣿⣿⣿⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⣀⡀⠀⠀⠀⠾⠟⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⠞⠀⢀⡠⠔⡿⠀⠀⠀⠀⠠⢂⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠊⠉⠀⢈⠕⠀⠀⠀⠲⠒⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣀⣀⡀⠀⠀⠀⣠⢴⣯⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⡛⠉⠙⠛⢦⠞⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢙⠛⡳⠀⢀⡾⢆⠛⠃⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢩⡟⠘⠧⠀⣸⣧⣿⣗⡫⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⡶⢧⢻⣶⣿⣿⣿⣿⣂⠀⠀⢹⢹⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠀⠀⠙⢿⣿⣿⣿⡿⠒⠚⢷⠾⣷⣿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣴⡞⣟⡂⠀⠈⢳⣬⡿⠁⠀⠀⠀⠀⠀⠉⠻⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢉⣯⣽⡽⢀⡴⢊⡾⠁⠀⣸⣷⣿⡒⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⢃⡵⠋⠀⣾⡇⠀⠀⠙⣻⣙⣭⡇⠀⠀⢀⡏⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣴⠋⠀⠀⠀⢹⣇⠁⠀⠀⠉⠏⠃⠀⠀⢠⡞⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣀⣽⢷⣍⣀⣀⣀⣀⡤⠾⢧⣤⣀⠀⠀⠀⠀⢰⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⡼⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠙⠻⢶⣦⣤⢿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠳⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡼⠛⠶⠋


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀
⠀⠀⠀⣀⣀⣀⠀⠀⣀⠀⠀⠀⢠⣤⣿⣿⣤⡄⠀⠀⠀⠀⠀⣰⡏⢹⣆⠀⠀⠀⠀⠀⠀⣀⣺⣗⣀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠇
⠀⠰⣉⠀⠀⠀⣹⠶⡁⠀⠀⠀⠘⠟⣿⣿⠻⠃⠀⠀⠀⠀⠺⣤⡅⢨⣤⠗⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⠠⣄⣏⣨⠄
⠀⠀⠀⠉⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣀⣼⣧⣀⡀⠀⠀⠀⠀⢠⡶⠋⠀⣀⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣀⣠⣔⣤⠀⠀⠀⠀⠀⢰⣶⢻⣟⣶⡆
⠀⠀⢈⣿⠀⠀⣿⡁⠀⠀⠀⠀⣿⡇⠀⠀⠙⠁⠀⠀⠀⠀⢘⣿⠛⠛⣿⡇⠀⠀⠀⠀⠀⠩⣼⡴⠋⣆⠀⠀⠀⠰⣿⢶⣯⣿⠶⡷⠆
⠀⠀⠉⠉⢻⡟⠉⠁⠀⠀⠀⠀⠘⠿⣦⣤⠴⠃⠀⠀⠀⠀⢸⡿⠀⠀⢻⡇⠀⠀⠀⠀⠘⢦⡼⠙⠾⠃⠀⠀⠀⠀⠸⠿⢼⡯⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⡀
⠀⠀⣠⣾⡟⢻⡗⣄⠀⠀⠀⠀⢀⢼⡧⣿⣄⡀⠀⠀⠀⠀⣠⡴⢻⡟⢦⣄⠀⠀⠀⠀⠀⠄⢸⡇⠤⠀⡇⠀⠀⠀⠀⠀⣀⣶⢠⡾⠂
⠀⠀⣿⣿⠟⠛⠀⢸⠀⠀⠀⠀⢸⠘⣋⡙⢺⠀⠀⠀⠀⠀⢷⡱⣼⣧⢎⡾⠀⠀⠀⠠⠶⠶⠾⡷⠶⠶⠇⠀⠀⠀⠿⣻⣿⣿⣏⡀
⠀⠀⠙⢧⣘⣃⡤⠋⠀⠀⠀⠀⠘⡸⣿⡿⡸⠀⠀⠀⠀⠀⠈⢻⣾⣷⠟⠁⠀⠀⠀⢰⠀⠒⠐⡇⠛⢀⠀⠀⠀⠘⠛⠋⠾⡗⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠐⡶⠜⠣⢶⠂⠀⠀⠀⠀⠀⠀⣯⣽⠀⠀⠀⠀⠀⢀⣀⠘⢻⡟⠃⣀⡀⠀⠀⢀⡔⠉⢸⡇⠉⢢⡀⠀⠀⢀⡔⢿⢤⡤⡿⢢
⠀⠀⢻⡅⠀⠀⢨⡟⠂⠀⠀⠀⠈⠉⢹⡏⠉⠁⠀⠀⠀⠸⣿⠖⢺⡗⠲⣿⠀⠀⠀⢸⠖⠒⢺⡗⠒⠲⡇⠀⠀⢸⣠⣾⣏⣹⣷⣄⡇
⠀⠀⠛⢻⡴⢦⡏⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⢠⣼⣧⡄⠉⠁⠀⠀⠀⠣⣀⢸⡇⣀⠜⠀⠀⠀⠀⠣⣀⢹⡏⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⣀⡀
⠀⠀⠀⣠⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⣄
⠀⠀⠀⣿⠲⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠖⢻
⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠛⠛⠓⠛⠒⠓⠛⠙⠙⠋⠉⢉⠉⠁⠁⠀⢀⠀⢸⣀
⠀⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡼⢉⣳
⠀⠀⠀⠀⠈⢻⠓⠒⠶⠦⠤⡤⠤⣤⣤⣄⣀⣠⣀⣀⣤⣠⣤⣤⣤⡴⠦⠶⠶⠞⠟⠛⠛⠉⢸
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠀⠀⠘⢦⣤⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⢚⡷⠞⠉⣇⠀⠝⠢⡀
⠀⠀⠀⠀⠀⠀⢸⠉⠒⠶⠯⠭⣭⣭⣛⣒⣚⣓⣒⣒⣒⣒⣚⣒⣒⣛⣚⣉⣩⣭⡥⢤⠶⢲⡏⠀⢀⠀⠚⡄⠀⠑⢘⢦⡀
⠀⠀⠀⠀⢀⣀⣼⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠁⠉⠀⠀⠅⠀⠀⠀⠀⠰⠀⠘⢷⠀⠀⠀⠀⢹⡔⡀⠀⠁⡙⣄
⠀⠀⠀⣞⠉⠠⠚⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⣠⡼⣆⡀⠔⠁⠀⠱⡀⠀⠀⠈⢈⢳⡀
⠀⠀⠀⡏⠉⠓⠒⠦⢬⠭⢭⣉⣛⣒⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠖⠛⠛⠉⡏⠁⠀⠘⣆⠀⠀⡠⠀⠹⡄⠀⠀⠀⠡⡹⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠘⣖⠁⠀⢀⠄⠛⣄⢀⠀⠀⠀⠜⢦
⠀⠀⠀⠳⢤⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣋⣽⡄⠀⠀⠈⢶⠊⠀⠀⢀⠘⢦⠢⠀⠀⠈⠈⢣⡀
⠀⠀⠀⠀⢸⠀⠉⠙⠛⠛⠷⠶⠶⢶⠶⠶⡶⠶⢶⡶⠶⣶⠾⠷⡶⠿⠿⠿⠛⠛⠛⠉⠁⢸⡇⠀⠀⠀⠈⢳⡄⠂⠀⠀⡀⠷⡐⠀⠀⠀⠂⢳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠹⣄⠄⠈⠀⠀⠹⢎⠂⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠈⣷⡦⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣃⣀⣠⣤⠤⠖⣚⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠄⠈⠀⠈⡳⡄⢄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡇⠉⠙⠓⠒⠯⠭⠭⢭⣭⣙⣛⣛⣛⣛⣛⣛⣛⣉⣩⠭⠭⠭⠭⠲⠖⠛⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠔⠈⠀⠈⢦⡀⠀⠀⢧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠠⠐⠀⠀⡙⠦⣄⣸
⠀⠀⠀⠀⠀⠳⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣠⡴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠐⠁⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠛⠒⠒⠓⠒⠒⠓⠒⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠁

⠀⠴⣶⣦⡀
⠀⠀⠈⠛⠿⣆⠀⢠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠸⡄⢺⡟⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠸⡇⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢳⣸⡀⠈⣒⡁⣀⠀⢀⡀⠀⠀⠀⠉⢿⣿⣦⠀⠀⣴⣦⣼⣿⡄⠀⠀⠀⣀⣸⣷⣶⣷⣿⡁⠀⣇⣀⠠⠄⠚⠉
⠀⠀⠀⠀⢀⡠⠤⢿⣷⣤⣼⣿⣿⣷⣿⣧⠀⠀⠀⠰⣼⣿⡟⠀⠀⣿⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣣⣶⣾⠇⢀⡒⠄
⠀⠀⠀⢰⡁⢤⠀⠰⣾⣿⣿⣿⣿⡛⠉⠉⣷⣶⡄⠀⠈⠙⢧⠀⣿⡿⠋⢀⡀⠠⠶⢦⣭⣭⣽⣿⣼⣿⣿⣿⡇⠀⠀⡑⠤⠃
⠀⠀⠀⠀⣩⣼⣶⣾⣿⣿⣿⣿⣏⣿⡦⢀⣽⣿⣿⣦⡀⠀⣬⣿⣿⣤⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣼⣿⣧⢀
⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣿⣿⣟⠁⠀⠀⠉⠀⠘⠻⠿⢷⣿⡁⢙⣿⣿⣥⣀⡀⠀⠉⠀⠀⠉⠛⠛⠿⣤⣤⣼⣯⣿⣿⡏
⠀⠀⠀⠚⠻⣿⣿⣿⠿⠛⢻⣿⣿⣿⣶⡤⣾⣶⣶⡶⠂⠽⢿⣿⣿⡟⠉⠙⢿⣷⣶⡿⠂⠠⠀⡽⠀⠀⠉⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⢧⠾⠿⢿⣭⠀⠀⠀⢀⣾⣿⠁⢤⡀⠀⠀⠻⣿⣿⠆⠀⣘⣥⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡷⠻⠁⠀⠀⣠⣶⣿⣿⠉⠀⢸⣿⠢⣄⠀⠀⠈⠛⣝⠻⣅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠄⡞⣀⠤⡀⠀⠀⢻⣿⢿⣿⠄⠀⢿⣿⣆⠈⠀⠀⡀⢤⠈⡆⠈⠳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠉⠀⢰⡆⠳⣘⣦⡏⠀⠀⢸⢃⠜⠃⠀⠀⠀⠉⠻⠻⠀⠀⢇⣀⣠⢃⣀⠀⠈⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣽⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⣄⠛⠀⠃⠀⡆⠐⣶⣀⠎⠤⠁⠀⢡⠀⣀⠆⠀⠀⠁⠀⠒⢁⠂⠈⡀⠛⠀⠃⠀⢢⠀⢶⣀⠼⠠⢀⠀⠀⡀⢀⠀⢀⠃⠁⠀⠒⢃⡀
⠀⠀⡄⠤⠇⠃⣀⡓⠤⠄⠀⡄⠀⡄⠀⠧⠤⠀⡄⠙⣀⠈⠟⠤⣆⢠⠀⡄⠡⠗⠃⢀⡛⠤⠤⠀⡄⠀⡀⠀⠬⠤⠀⢤⠛⣀⠀⠿⠤⣄⠀⡄
⠀⠀⢰⠀⣦⠀⠧⠴⠀⣶⠂⣅⠆⠁⡄⠂⠀⡆⠀⠠⠤⠀⢰⠀⠤⡜⠀⡕⠀⠀⠀⠧⠤⠀⢰⠂⢅⠆⠀⡔⠂⠀⡆⠀⠠⠤⠀⢰⠀⠤⠼
⠀⠀⠉⠉⠁⣦⠀⠠⠂⠏⠒⠀⠀⠐⠁⠃⢠⠁⠖⠀⠀⡄⠈⠂⡄⠐⠚⠛⠁⠀⢰⡄⠀⠖⠊⠓⠀⠀⠐⠙⠂⢠⠈⠘⠀⠀⡄⠀⠂⣄⠐⠒
⠀⠀⠝⠀⢸⠉⡀⠘⠒⠛⠀⢉⡆⢸⠀⠀⡟⠀⠚⠋⠐⡇⠘⠲⠋⢡⠀⣖⠀⡧⠈⢁⠀⠒⠛⠂⢉⡇⢸⠇⠀⣿⠂⠚⠋⠀⡇⠐⠒⠋⠁⠀⣧
⠀⠀⡊⠁⡆⠀⡀⡆⠀⠀⠉⠉⠀⠀⢠⡄⠀⠛⠓⠈⢙⡇⡆⠀⠙⡃⢰⡏⠉⠀⠀⢀⣶⠀⠂⠉⠉⠀⡄⠀⡄⠂⠓⠳⠊⠑⡇⣦⠀⠙⣋⠀⡄
⠀⠒⠘⠛⠒⣤⠀⢅⢠⢸⠀⠁⢘⡀⠸⠗⠁⢦⡘⡛⠋⠑⡇⣺⠇⢀⢽⡛⠛⠓⢠⠄⢯⢠⣸⠀⠁⠈⡇⠠⠟⠀⢢⡐⢛⠋⠙⣿⢸⠏⢀⠱⡇
⠀⢲⣀⣀⣶⣦⣀⡜⣪⠙⠂⠙⠋⠐⠄⠐⠖⢯⣀⣈⠀⡆⡇⠋⣘⡃⠀⣱⠀⢰⢰⣀⣠⢻⠙⠃⠛⠋⠑⡤⠘⠶⢻⣀⣈⠁⡦⣯⠋⣘⣇⠀⢨
⠀⢻⠁⠀⡉⠉⠉⠉⠉⡇⡦⠀⠀⣾⢠⠀⡔⡃⠀⢹⠚⡗⠓⡎⠀⢘⠛⠛⠲⠏⠉⠉⠉⠉⣧⣶⠀⠀⣾⢠⡄⠔⢊⠀⢈⡓⠻⠛⢸⠀⠈⡟⠛⠒⡄
⠠⠚⠓⠚⠚⢦⠀⠀⡇⠧⠗⠀⢀⡿⢼⣠⠗⠚⠒⠓⠢⣇⠔⠚⣒⡃⠀⢰⡆⢀⡇⠀⠀⡇⠷⢿⠀⠀⡽⢼⣇⠖⠚⠓⠋⠒⢦⡰⠚⢓⣚⠀⢠⡆⡇
⠆⣤⠀⡔⠉⠙⠆⠀⠧⠀⠀⢰⣏⠀⠀⣿⢠⠀⠀⢀⡄⣿⢠⡎⠀⠈⡃⢸⣇⣴⣇⡀⠀⠧⠄⠸⠐⣾⠀⠀⣹⢠⡄⠀⠀⣤⢸⠀⣼⠁⠀⣳⠸⣇⠇
⠁⣿⣀⣑⣤⣴⣷⣀⠀⠀⢠⠜⠂⠓⠚⠓⠾⡀⠀⢸⡇⢸⠼⠯⠶⠾⠧⢾⣸⠁⠀⢹⠀⠀⠀⢰⠜⠣⠓⠚⠛⠺⡇⠀⠀⣿⢸⡤⠯⠶⠶⠿⢤⡁
⠘⠙⢰⠋⠉⠳⡀⡬⠶⢤⣸⢀⡀⠀⠀⢠⡴⠧⣄⢠⡏⢹⢀⡀⢀⠤⠤⣤⠯⠷⠤⠾⢤⡤⠤⢼⢀⡄⠀⠀⢀⡄⣇⡤⠤⡏⢻⢀⡀⠀⡤⠤⢤⡇
⠀⠀⢹⣄⣀⡴⢻⡀⠀⠀⣿⢸⡇⠀⠀⢨⠀⠀⣸⠀⡇⢸⢸⡇⢇⠀⠀⡕⠀⠀⠀⠀⣸⠀⠀⢈⢿⡇⠀⠀⢸⡇⣏⠀⠀⢸⢸⢸⡇⢸⠀⠀⢀⡇
⢰⠋⠉⠉⢉⡭⠭⠵⠶⠮⠽⢾⡇⢠⠴⠚⠛⠚⠛⠲⢆⠸⣼⠓⠒⠛⠛⠚⠳⡄⢀⡤⠿⠷⠶⠿⠾⢧⠀⠀⢸⠷⠯⠷⠶⠧⠼⣼⡧⠤⠵⠶⠮⠧⢤
⣹⢠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡀⠀⠀⠀⣀⢸⠀⡂⢠⠀⠀⠀⢀⡀⡇⡎⠀⠀⠀⠀⠀⠀⠀⣷⡆⡇⠀⠀⠀⠀⠀⠀⢘
⣸⢸⡇⠀⢸⢸⡇⠀⠀⠀⢸⡇⡇⢸⢸⡇⠀⠀⠀⣿⢸⠀⠇⢸⠀⠀⠀⢸⡇⣇⡅⢸⡇⠀⠀⠀⣿⠀⣿⡇⡇⢸⡇⠀⠀⠀⣿⢸⢸⡇⠀⠀⠀⢸⡇
⠹⠼⡇⠀⢸⢸⡇⠀⠀⠀⢸⡇⡇⢸⡼⡇⠀⠀⠀⡿⡼⠀⢡⡾⠀⠀⠀⢸⡧⠿⡇⢸⡇⠀⠀⠀⣿⠀⣿⡇⢇⣸⠇⠀⠀⠀⣿⣸⢸⡇⠀⠀⠀⢸⡇
⠀⠀⡇⠀⣌⠋⡅⠀⠀⠀⢸⠙⠉⠁⠈⡇⢀⡄⠀⡇⠀⠀⠀⠙⠀⢠⡄⢸⡇⠀⣿⠋⡇⠀⠀⠀⡏⠛⠁⠈⠉⠉⡇⠀⡀⠀⡏⠉⠛⡇⠀⠀⠀⢸⠙
⠀⠀⡇⠀⣿⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⢸⣇⣀⢿⣀⡇⠀⣷⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⡇⠀⣿⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠈⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⣇⣠⢿⣀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠧⡸⠧⡠⠇⠀⠀⠀⠀⠦⠞⠣⠜⠀⠀⠀⠀⡇⠠⣿⠀⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⠀⠀⡇⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠴⠛⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠛⠤⠇⠀⠀⠀⠀⠀⠙⠋⠈⠋⠀⠀⠀⠳⠴⠛⠦⠞

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⢤⡀⠀⠀⠀⢀⡤⠔⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠢⣄⠴⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠱⡄⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣆⣀⡰⢦⣴⣁⣴⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠀⠀⣀⣀⡀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠋⠉⢀⠛⡅⠉⠓⠲⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⠖⣿⢻⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⢸⡇⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⣻⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⣿⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣦⣼⣡⣯⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠴⠚⠢⢤⣀⣀⣀⣀⠤⠴⠮⠤⠤⠴⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣉⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠰⠏⠁⠈⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠂⠠⣀⣄⡀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⠙⠣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠤⠆⠉⠀⢡⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⢠⡀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠑⣶⠀⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠸⡅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⡨⣠⠤⢠⢃⠀⠑⢺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠊⠁⠀⢄⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠁⢰⡇⠀⠃⡀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡆⠀⡔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠌⠀⠉⠑⠉⠁
⠀⠀⠀⠐⠲⡌⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠑⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠄⡠⡈⠀⠀⠀⢠⠈⠀⠯⠶⠀⠀⠀⠀⠀⠈⠒⠂⠙⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠂⠐⠁⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣀⡀
⠀⠈⠙⡦⠤⠈⠉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠄⣀⡀⡀⠀⢀⡀⣤⠔⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠀⠀⠠⢢⠔⠉⠈⠀⣄⣀
⠀⠀⠈⡠⠰⣀⣴⡀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠐⡀⠀⠀⠀⠀⠀⠀⢃⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣔⠉⠀⠈⠱
⠀⠀⠀⠐⢂⠠⢋⣷⣂⡂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣶⣶⣶⣶⣶⡞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⠔⠀⠁
⠀⠀⠀⣀⠀⠀⠈⠛⣁⡿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠕⠓⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢉⣀⣀⡁⠀⠙⠂⠉⢀⠀⠀⡀⠀⠠⢴
⠀⠀⠀⠀⠀⠀⠀⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠘⠄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠉⠢⠴⡀⡠⣁⠈⡀⠀⠀⠀⠢⠤⣀⢀
⠀⠀⠀⠀⠀⢠⣥⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠄⠀⠑⠀⠸⠤⠄⣀⠀⢀⢀⠚⠄⠼⠀⠙⣠⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⡴⣆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠳⣀⡄⠀⠠⠴⠴⣠⠤⠁⠘⢲⡄⡠⠀⠀⢰⠂⠁⠈⠱⢄
⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠔⠁⠈⠊⢐⡀⠤⠐⠳⠄⠋⣨⠖⠄⣀⣠⡝⠂⠀⠀⠀⣺⠀⠀⠐⠰⣲⡎⠀⠈⠉⡛⠳⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠉⠀⢻⡤⣀⠀⣠⡿⠀⠀⠀⢼⡆⡐⠁⠀⠀⣙⣿⠟⠁⠀⠑⡄⣁⣀⠀⠀⣄⣾⠑⠀⢒⠈⢣⡤⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⢀⣈⠃⢛⠃⠀⠈⡵⠀⠁⠃⠠⣻⣦⣀⡠⠋⠘⠳⠄⢀⡌⠊⠁⠀⢈⢠⠀⠽⣀⣀⡀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠿⢢⣠⠦⠀⠈⠡⡤⣶⠖⠁⠁⢺⣑⡀⠀⠀⠄⠔⠛⠀⢀⠀⠀⡤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡰⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠉⠉⠀⠤⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣐⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⢸⣦⣀
⣼⣿⣿⣷⣤⡀
⣹⣿⣿⣿⣿⣿⣶⡄
⣿⣿⣿⣿⣿⣿⡟⠁⡠
⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⠃⢀
⠻⢿⣿⡿⠄⠛⢢
⠀⠀⠉⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠀⠀⠀⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⢀⣀⣀⡠⠄⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠹⣟⠁⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣴⠛⣤⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠈⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠪⠀⠻⡀⡓⢤⠀⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢸⠀⣀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠀⠀⠄⠀⠠⠀⠀⠀⢹⠀⡴⠁⠀⠀⡇⢠⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠂⠂⠀⠰⠀⡀⠀⠈⠀⠂⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠂⠀⠰⠀⠀⠀⣸⠀⠇⠀⠀⠀⡇⠐⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⡀⠀⢐⠀⠀⠀⢸⠀⡃⠀⠂⠀⡇⢐⡁⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁

⣶⣶⡶⠶⠶⢶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢲⢃⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠨⠀⢐⣋⣻⠀⠀⡞⢀⣖⡆⠀⠀⢸⠰⠯⡇⠀⠀⠀⠀⢘⡀⣟⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠖⢺⠀⠀⡇⢬⠵⠇⠀⠀⢸⠌⠽⡇⠘⠅⢀⣿⠀⡆⢴⡂⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⡀⠠⠅⢈⣉⣭⠀⠀⡓⠭⢽⠃⢀⠄⠀⡯⢭⡇⢐⡅⢐⣻⡇⡣⠽⣇⣀⠀⢸⡀⢽⡀
⠀⠀⠀⠀⠀⠏⢑⣺⠀⠒⠒⡛⠀⠀⢩⠹⢽⡀⢒⣚⣆⣮⣽⣆⣿⣧⣰⣾⣧⣬⣭⣥⣧⣰⣶⣿⣿⡁
⠀⠀⠀⠀⠀⢱⣐⣺⠄⣉⣩⠵⡀⠀⠈⠉⠙⣇⢀⣐⣷⣯⢈⣁⣀⡁⣉⡛⠋⢁⣩⣕⠾⡟⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠔⠚⠠⠼⠋⡁⣀⡄⠈⠡⠿⣮⣋⣍⣉⣹⣭⣍⣀⣀⣏⣡⣈⢉⣉⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣤⣴⡀⠐⠒⠒⠁⠀⠀⠠⣀⠭⠍⠉⢉⠽⠯⠜⠋⠉⠁⠀⠈⠡⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣻⣽⣿⣿⣿⡷⠖⠛⠉⠉⠀⠀⠀⠈⠉⠙⠓⠶⣿⣿⣿⣻⣽⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠘⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⢠⣶⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣷⣤⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣴⣿⣿⠟⡻⣻⣿⡏⠀⠀⠀⠈⣿⣿⡻⡛⢿⣿⣷⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⣿⣿⣸⣷⣿⣿⠁⠀⠀⠀⠀⢹⣿⣷⣿⣸⣿⣿⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢻⣿⣿⣿⣾⣯⣵⠏⣴⣶⣶⣶⣶⡌⢷⣯⣽⣾⣿⣿⣿⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠻⣿⣿⣿⡿⠋⠀⠈⠉⢻⠛⠉⠀⠈⠻⣿⣿⣿⡿⠋⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠉⠁⠀⠀⠀⠲⠴⠚⠲⠴⠂⠀⠀⠀⠉⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢤⣤⣄⣀⣀⣤⣤⡤⠴⠖⠛⠉





⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣻⣻⣟⣻

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⣤⠤⠶⠶⠤⣤⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠋⠀⢀⡴⠛⠁⠀⠀⣀⣀⣴⠂⠈⠛⣦⠀⠀⠹⡄
⠀⠀⠀⣸⠃⠀⢠⠏⠀⠀⠀⠀⢼⠋⠈⠙⢷⠀⠀⠈⣧⠀⠀⢻⡀
⠀⠀⢀⡏⠀⠀⡟⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠘⡇⠀⠀⣧⠀⠀⠀⣀⠀⠀⠀⣻⠄⠀⠀⠀⢀⡟⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢉⣁⣤⠤⠤⠶⠶⠶⠶⠤⢤⣤⣉⠙⠓⢦⡀
⠀⠀⠀⢻⡀⠀⠘⣦⠀⠀⠙⣷⠶⠞⠋⠀⠀⠀⣠⠞⠁⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣾⢋⣥⣴⡆⠀⢀⣴⠟⠛⠛⣳⠶⠎⠙⡆⠀⢹
⠀⠀⠀⠈⣷⣄⠀⠈⠛⠦⣤⣁⣀⣀⣀⣤⠴⠚⠁⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠙⠳⠦⣬⣝⣛⣉⣀⣀⣀⣀⣭⡤⠶⠚⠁⢀⣼
⠀⠀⠀⠀⠈⢿⣷⣤⣀⠀⠀⠈⠉⠉⠀⠀⢀⣠⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣷⣦⣤⣀⣀⣀⣉⡉⢉⣁⣀⣀⣠⣤⣶⣾⣿⢿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣯⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣷⣝⡻⢿⣿⣷⣶⣾⣿⣿⣿⣿⣶⣶⣾⣿⠿⣟⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⣉⣉⣉⣀⣀⣄⣈⣉⣉⡉⠛⠲⢿⣮⣛⢿⣿⣶⣾⣭⣭⣭⣭⣭⣭⣭⣽⣷⣾⣿⢿⣫⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣴⠞⠋⣩⡍⠀⠀⣠⡴⠶⠶⣭⣭⡟⢶⡄⠈⢻⣿⣷⣯⣭⣽⣿⣻⣛⣛⣿⣿⣭⣭⣷⣾⣿⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠳⢬⣉⣙⠶⠶⠞⠁⠀⠀⠀⢛⣀⣠⠴⠃⠀⣸⣿⣭⣟⣻⣿⡿⠿⠿⠿⠿⢿⣿⣛⣿⣭⣶⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⣈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⣀⣠⣴⣾⣿⣯⣟⣻⡿⠿⢿⣿⣿⣿⣿⠿⠿⢿⣟⣯⣵⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣛⠿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⢿⣫⣾⣿⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣛⣯⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡻⣿⣶⣿⣭⣭⣿⣟⣛⣛⣿⣿⣭⣽⣷⣾⡿⣟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⢿⣻⣵⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣝⢿⣷⣾⣭⣟⣛⣛⣛⣻⣟⣛⣛⣛⣯⣭⣷⣿⣿⣿⠋⠀⠀⣠⠶⠚⠛⠛⠓⠶⣤⡀⠀⠙⢿⣿⣫⣾
⠀⢀⣠⠴⠖⠛⢉⣉⣉⣉⣉⣉⣉⣉⡉⠛⠓⠶⣿⣿⣿⣾⣭⣟⣛⣿⠿⠿⠿⠿⠿⣿⣟⣛⣻⣭⣾⣿⠃⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠻⣟⣽
⣴⠋⠀⡴⠞⠋⣩⡍⠁⠀⣠⡴⠾⢯⣍⣙⡓⢦⡀⠙⢿⣝⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⡟⠀⠀⡟⠀⠀⣤⠀⠀⠀⠀⣀⣀⠀⠀⢻⡄⠀⢻⡿
⣿⡀⠈⠳⣜⣛⡛⠶⠶⠞⠁⠀⠀⠐⠋⣀⣠⡼⠃⠀⣸⣟⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⡿⣿⣿⣇⠀⠐⡇⠲⠾⣇⠀⠀⢀⡾⠋⠉⢷⣀⠀⣷⠀⠈⣧
⣿⣿⣦⣤⣀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⣀⣠⣴⣾⣿⡿⢿⣿⣶⣶⣿⣿⣭⣭⣿⣷⣶⣾⣿⢿⣿⣿⡀⠀⢻⡀⠀⠙⠦⡴⠟⠁⠀⢀⡾⠉⠁⣿⠀⠀⣿
⣿⣽⡻⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⢿⣫⣾⣿⣿⣶⣾⣯⣭⣿⣿⣿⣿⣭⣽⣷⣶⣿⣿⣿⣧⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⠇⠀⢀⡏
⣿⣝⢿⣿⣾⣯⣭⣿⣛⣛⣛⣛⣛⣿⣯⣭⣷⣾⡿⣟⣽⡏⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠈⢿⣿⣷⡄⠀⠙⠶⣄⡀⠀⠀⠀⢀⣠⡴⠋⠀⢀⡾⠁
⠙⢿⣿⣶⣭⣽⣛⣛⣿⣿⣿⣿⣿⣛⣛⣿⣭⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⡀⠀⠉⠙⠛⠛⠉⠁⠀⢀⣠⠞
⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣤⣤⣤⣤⣴⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⠠⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠔⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠤⡐⠊⠐⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡠⠀⠀⠀⠐⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠆⠀⠸⠃⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠢⡀⠀⠀⠀⠀⠐⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣀⠈⠢⢀⠄⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢠⠔⠀⠀⡀⠀⡀⠀⠼⢠⠐⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠠⠀⠀⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⠄⠀⠄⠀⠀⠀⠴⠀⠢⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠬⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁⠘⠡⢀⠀⠀⠀⠠⠀⡆⠠⠀⠀⠀⢀⡠⠚⠀⠉⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠖⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠁⠀⠀⠠⣀⠀⠀⢀⠉⠀⠀⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠂⠀⠀⢀⠁⠢⡄⠁⠀⠂⠀⠀⠀⠈⠤⠆⢈⠀⠀⠀⠐⠀⠀⠈⠳⣱⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠠⠃⠀⣀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⢀⠀⠘⠠⠈⠀⠀⠀⠀⡐⠀⡀⠀⡻
⠀⠀⠀⠀⠂⡀⠀⠀⠀⢈⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠄⠁⠀⠁⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠘⢠⡄⠀⠀⠠⠂⠊⠀⠁⢀⠀⡀⠀⠐⡀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠂⠈⢀⠀⠄⠀⡠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⡆⠃
⠀⠀⠀⠀⠀⠀⡈⣁⠀⠀⠀⠀⠀⡱⠊⠂⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⡀⠀⠉⠀⠂⢀⡀⠀⠀⠁⠀⠐⠒⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⡀⠂⠀⠀⠀⠊⢛⣄⠘⠸⠀⣬⠛⠃⠀⠀⠀⠀⠠⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⢀⠔⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠈⠢⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠈⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠐⠀⡀⠀⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠀⠀⠈⠀⠈⠄⠁⠘⠀⠀⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡨⠉⠀⠔⠀⠎⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠔⢀⠐⠁⠀⠀⠀⠀⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠑⠀⠀⠐⠀⠐⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠒⠡⠄⢠⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣋⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⢤⣀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠉⠀⠀⠀⡏⠛⠋⠀⠈⡳⢒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⠀⠀⠀⣄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣠⣦⣤⣭⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡼⠀⠘⠂⣸⡟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠸⠃⠘⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⣿⣿⡿⢩⣿⣿⣿⠈⠻⣿⣿⣷⠀⠀⠀⠀⡀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣸⣿⣿⡇⠀⠀⠹⢿⡿⠁⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠄⠂⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠈⠙⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⢸⣿⣿⡇⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⡁⠄⠒⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠑⠄⠀⠀⠀⠀⠈⢿⣿⣿⣦⣸⣿⣿⣧⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠂⡉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢧⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⢉⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠀⠢⠄⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⠢⣀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣀⠄⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢧⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⠞⠁⠀⣸
⠀⠀⠀⠀⡀⡀⠤⢤⠀⠀⠀⠀⡠⠊⠘⢳⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡞⠁⠀⠀⠈⢠⡀⠀⠀⠀⢀⠜⠒⠒⠺⡀⠀⢠⠃
⠀⡠⠒⠉⠀⠀⢻⠆⡆⠀⠀⣜⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠱⡆⣰⠇⣀⠤⢽⣦⡀⠀⠣⢄⣤⠴⠚⠤⠔⠧⢄⡀
⠀⢁⢳⠄⢀⠠⠐⠁⠁⠀⠈⠻⣙⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠎⠀⠀⠀⠙⠈⠋⠀⡀⠀⠻⢿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠸⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⣀⣀⡀⠴⠐⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠛⣠⠀⢰⣲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡤⢚⠥⠀⠀⠀⠐⠀⡿⡄⢂⠀⠀⠀⠀⠀⠀⠑⠒⠊⢸⠀⡌⠇⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⢈⠠⡆⠉⠉⠉⠠⣆⢠⠀⠀⣮⠞⡡⠔⣀⠔⠊⢩⠟⠀⢸⠘⡔⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢟⠁⠀⠀⠈⡧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠸⣈⣃⣀⣀⣀⠠⠍⠘⠀⠸⣥⠊⡠⠞⠁⠀⠀⠸⠾⣴⠋⢣⠐⡵⠀⠀⠀⠀⠀⠀⠀⡌⢸⢹⠀⠀⠀⢰⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠋⠈⢆⠘⢆⠱⢃⠀⣀⡀⠀⠀⢠⠃⣼⡾⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⢆⠈⠀⠇⡬⠳⣀⡞⠀⣿⡇⠑⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠢⣤⠋⢢⣠⠎⠀⠀⡛⡷⡄⢰⠁⡠⡄⣀⡠⣶⣲⣒⠖⣂⣲⠂⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⢠⠉⡆⠈⢣⡀⠀⠀⠉⠀⡱⠗⣢⣒⣹⠤⢷⠤⡽⠍⣃⠬⡄⢻⢢⠍⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⡎⠀⣣⠀⠀⠙⢦⡀⡠⢾⣤⢚⡯⢿⠁⡟⢛⣫⣙⡜⠉⠛⣹⠗⠕⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠲⠊⠃⠀⠀⢠⣁⡀⡿⣄⡤⠒⠙⢉⠗⡎⢹⡯⣥⡽⠦⡧⠈⠁⠸⢀⠤⠚⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡃⠀⠀⡠⠳⢊⡜⣰⣁⢁⣘⣐⣋⣧⠠⠔⠋⠀⠀⠀⢀⠟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢸⡠⠊⢀⡶⢻⠊⣿⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⣤⡤⠊⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢄⡈⠈⣧⡄⠅⠂⠁⠀⢳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣠⠎
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⡏⣠⠞⢁⡔⡏⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠉⠐⢺⢻⠎⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠖
⠀⠀⠀⠀⠀⠔⠁⢀⣠⠔⢹⡁⠀⡌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠤⠄⠠⠖⢉⠈⠀⠀⠀⠀⠀⠀⢀⣼⣀⣀⠀⠀⠀⠀⠀⢠⢆⣀⡠⠃
⠀⠀⠀⠀⠀⠙⠚⠉⠀⠀⠀⠑⢄⡁⠀⠀⠀⢀⣀⡠⠤⠐⠚⠛⠑⠻⢄⠀⠀⠀⠀⠈⠉⢹⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⡆⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁⢀⣀⣀⣀⣀⠤⠴⠲⣿⢆⠀⠀⢀⡠⠤⠚⠁⠉⠂⢶⡀⠀⠀⠀⠀⠀⠀⢀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⣀⡤⠤⠒⠉⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣀⣤⣞⠊⠉⢉⣀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠂⠈⠈⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⢀⣀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠙⠋⠉⠐⠉⠉⠁⠂⠉⠉⠐⠂⠉⠉⠐⠉⠉⠀⠀⠍⠀⠈⠩⠚⠛⠓⠋⠒⠖

⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠀⢠⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡷⠀⠀⠀⢀
⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃
⠀⠀⠙⠟⢁⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⠘⠁⢠⠔⣢⣤⠀⠀⠀⠀⠀⣠⣶⡠⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⢰⣏⣴⣿⣿⡆⠀⢀⣾⣿⣷⣙⣆⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⡇⠀⠀⠀⠰⣿⣿⠿⡧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠾⠟⠛⠋⠀⠀⠀⠙⠛⠿⠧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⡴⠚⣻⣶⡄⠀⠀⠀⠀⣰⣾⣟⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠀⠻⣿⣿⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠸⢿⡇
⠀⠀⠀⠀⢠⣿⡿⠇⠀⠀⠀⠘⠛⢿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢫⣶⣶⡄⠀⠀⢀⣴⡶⣆⠀⠀⠀⠀⠀⠀⣿⣫⡵⣤⣄⠀⠀⠀⠀⠀⠀⣠⣶⠮⡙⣿⠃⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⢀⣤⠤⣀
⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠈⠛⠻⠃⠀⠀⠀⠀⠀⢀⣰⡷⣾⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣷⢿⡉⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⠀⠀⠀⣾⣿⣧⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡿⢿⠿⣿⣿⣿⠃⢀⣤⣤⡀⠘⣿⠿⠿⠷⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠰⣿⣭⣽⡷⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠐⠀⠙⠿⠟⠁⠔⠀⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠙⠛⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠋⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⣼⣿⣿⣧⣀⣄⠀⠀⠀⠀⠈⠻⠿⠿⠶⠄⠀⠀⠀⠀⠠⠴⠿⠿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⢶⣯⣼⡿⠟⠃⠀⠀⠈⠛⠿⠿⠦⠿⠛⠀⠀⠀⠀⠀⠀⡴⢂⣐⣦⡄⠀⠀⢠⣴⣶⡒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢻⣿⣿⡟⠀⠀⢸⣿⣿⠟⡻⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣄⠀⠀⠀⢀⣴⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢰⣯⣿⠆⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⠇⢻⡀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⡏⠀⠀⢧⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢰⠃⠀⠀⠀⠀⢹⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⣇⣀⣀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡏⠀⠈⡇⣼⠁⠀⢹⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⡇⣿⠀⠀⢸⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠀⠀⢀⡅⢨⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠏⠀⠀⢸⠃⢸⡄⠀⠀⠧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⣠⠖⠦⣄⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠴⠦⡄
⠀⠀⠀⡰⠃⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡠⠞⠋⠀⠀⠀⠙⣄
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠙⠫⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠋⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⠉⠀⠉⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⠉⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣿⠉⠉⠉⠙⣿⡏⠉⠉⠉⠉⣿⡟⠉⠉⠉⢹⣿⠋⠉⠉⠉⠙⣿⡏⠉⠉⠉⣿⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡄⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⣤⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⣄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠇⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⢿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⡿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣿⡇⠀⠀⠀⠀⣿⣷⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠉⠉⠛⠳⡄⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡆⠀⣀⣀⣀⠤⠚⠁⠀⠀⠀⢀⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣢⣤⣤⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠛⠁⠀⢰⡍⢏⣠⠖⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢖⡚⠛⠶⠲⢢⣄⡀⠀⠀⢀⣴⣶⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣥⣀⣀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣿⣿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠢⠒⠈⠐⡀⠀⠉⠭⠓⡲⢦⣤⣀⠀⡄⠐⣿⣿⣿⣟⢟⣀⣮⡇
⠀⠀⠀⠀⠀⠀⡀⠤⠐⣈⠗⠺⠆⠀⠀⠀⠀⠈⠓⠶⠶⠒⣂⢽⣵⣾⠈⢎⣞⣿⣟⠓⣠⢿⣿⢿⢻
⠀⠀⠀⠀⠀⣸⡦⣠⣉⡁⢆⣀⠒⠀⠀⠀⡀⠀⠀⣀⣤⣪⣶⣿⣿⣿⠀⣾⡾⠛⠒⠛⣿⣿⡿⣵⠋
⠀⠀⠀⠀⠀⢸⠉⠲⠤⣭⣛⠲⣭⡒⠂⣩⠗⣛⢬⣲⣽⣾⣿⣿⣿⡿⠞⠋⠁⠀⢀⡤⣻⣾⣼⡫⠵⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠾⣀⠀⠉⣌⣒⠯⠝⠻⣷⣶⣯⣿⢿⣻⣿⣿⡟⠻⠋⠁⣐⡶⣶⣜⣮⢶⣿⣟⣭⠽⣇⠀⠈⠠⠬⣱⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢨⣝⣒⠶⢤⣐⢒⣿⣿⠿⠺⠏⠀⠒⠒⢉⢜⣾⣿⢩⢃⣰⠋⠁⠀⠆⠙⠫⢓⣶⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠲⠟⢂⠉⡁⠲⠤⡾⠻⠟⠀⠉⠀⠀⢀⡴⣵⣿⣿⣿⢨⡜⠃⣀⠤⠊⠀⠀⠀⠐⠂⠈⢹⠋⠓⠠⠤⣀
⠀⠀⠀⠀⠀⠀⣾⡃⠐⠥⠤⠒⣂⠠⠭⣒⡂⠀⢀⠄⢂⡲⢒⣰⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣁⡂⡀⠠⡀⠑⣊⠴⣚⣄
⠀⠀⠀⠀⠀⠀⢻⡀⠬⠥⠉⣁⢐⣒⠀⠠⠌⠉⠑⠒⣒⠒⠿⣫⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⢑⣫⢕⣪⣵⡟
⠀⠀⠀⠀⠀⠀⠸⣖⡦⠩⠥⠤⠔⠒⣀⣀⡾⢏⣉⣘⣓⣒⣺⣿⣿⠟⠻⢿⣦⡉⠻⢄⡲⠤⠤⢀⣂⠕⣖⣬⣴⣾⣿⡿⠿⢛⡝
⠀⠀⠀⠀⠀⠀⠀⡼⣕⢍⡫⣛⠖⠖⡒⡂⢚⠞⠱⡳⠂⠖⠖⠿⣕⢤⠀⠀⠉⢾⡭⣒⣲⣶⣝⣓⣯⣿⣿⠿⠛⡋⠥⣐⣫⡵⠃
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠈⠑⠤⣤⣀⣁⡁⠈⣀⡁⢁⣡⣮⣼⣿⣄⡀⠀⠀⠙⠀⣾⣿⠿⣟⣋⢭⣔⠪⡑⠦⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⢿⣦⣔⠒⣋⣬⡿⣿⣿⣿⢿⡫⠛⠐⠤⡀⠀⣷⣾⡯⠾⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄⡀⠀⠘⢴⠄⠙⣯⣿⢿⣿⠷⠋⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⣃⢸⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠢⢀⠄⡨⠀⠀⡀⢀⠀⠈⢀⠔⠁⢀⠄⡀⠊⠐⠀⠀⠀⠁⠈⠈⡀⢀⠔⡠⢊⢐⠁⡠⢂⠔⡡⡨⡪⢢⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠕⠡⠊⠄⠨⠂⠕⠠⠊⠔⡁⢊⠔⠡⠪⢂⠀⠐⠠⠀⠔⠁⠁⠊⠐⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⢻⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠴⢿⡄⠀⠀⠳⠻⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⢀⡴⠶⢤⣒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣴⠋⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⠑⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⢫⠀⠿⡀⡛⢦⡁⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠐⠃⠀⠀⠀⠀⠀⠀⠀⠉⠆⠈⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠁⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠐⠀⢲⠐⠀⠀⠀⠠⠄⢲⠀⢀⡄⠀⢀⠀⠀⣀⠐⢀⠀⢀⢀⡆⢀⡄⣀⠀⠀⠀⣠⡐⢀⡆⢠⠐⠐⠀⠀⠂⡒
⠀⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⡀⡄⢀⢀⢰⠀⠀⢸⠀⡼⠀⠀⠀⣇⠈⡇⠀⠀⢠⠈⠀⣁⠈⠀⠠⠀⠀⠂⠀⠀⡆
⠀⠀⠀⠸⠀⠀⡇⠀⠀⠃⠀⡄⡇⢀⢸⢸⠂⠀⢻⠀⡟⠃⠀⠘⣏⢀⡇⠀⢸⢸⠀⡀⢸⠀⠀⠪⠀⠀⡞⠀⠘⡇
⠀⠀⠀⠐⠒⠒⠃⠀⠀⠒⠒⣀⡇⢀⣸⠘⠒⠒⠚⣀⡗⠒⠒⠂⢻⡀⠓⠒⠒⢸⣀⡀⣸⡐⠒⠒⠀⠀⠒⠒⠒⠃
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀⡇⢀⢰⠀⡁⠀⢀⣀⡇⠃⠀⠂⣹⢀⠀⠀⢈⢸⣀⡀⢸⢀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡄⡇⠀⢨⢰⠀⠀⢺⠀⣇⠀⠀⠀⣯⠠⡖⠀⠰⢸⠀⠀⢼⠀⠀⠠⠀⠀⡐⠀⠀⡆
⠀⠀⠀⠸⠂⠂⡇⠀⠀⠂⠐⠆⡇⠀⠰⢸⠂⠀⢻⠀⣟⠀⡄⠀⡷⠀⡗⠀⢸⢸⠠⠄⠴⠀⠀⠺⠀⠀⠚⠀⠐⡇
⠀⠀⠀⠐⠒⠒⠃⠀⠀⠒⠒⠒⠇⠐⠚⠘⠒⠒⠚⠒⠏⠀⠀⠀⠓⠒⠓⠒⠚⠸⠂⠀⠘⠒⠒⠒⠀⠀⠓⠒⠒⠃
⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠐⠒⠀⠓⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠐⠚⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠶⠿⠿⠿⠿⠿⠿⠶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⠿⢿⣯⣤⣤⣾⠟⠉⠀⣹⡿⠿⠂⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡟⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⢻⣿⣿
⠀⠀⠀⠀⠀⢹⣿⡇⢸⣿⠉⠛⣿⠿⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⣿⠛⠉⣿⡇⢸⠿⠋
⠀⠀⠀⠀⠀⣿⣟⠻⣿⣿⣀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⠀⣀⡿⠃⠀⣠⣴⡾⠿⣿⣿⡿⠿⠿⠿⠿⢶⣦⣀
⠀⠀⠀⠀⠀⣿⣿⣦⣀⠉⠛⠿⠿⣶⣦⣤⣿⣧⣤⣤⣼⣿⣤⣶⣾⠿⠿⠛⠁⣠⣾⣏⣁⣠⣾⠟⠁⣠⣴⣶⣶⣤⣄⠉⠻⣷⡄
⠀⠀⠀⠀⠀⣿⣿⡏⢻⣿⣶⣤⣄⣀⣀⠈⠉⠉⠉⠉⠉⠉⠁⣀⣀⣠⡄⠀⣼⡟⠉⠉⣽⡿⠁⣰⡿⠋⠁⠀⠀⠈⠻⣷⣄⠈⢿⣆
⠀⠀⠀⠀⠀⣸⣿⣇⣸⣿⠀⠉⣿⠛⠛⠻⣿⡿⠿⠿⢿⣿⠟⠛⠛⡟⠀⣸⣿⣶⣶⣼⡟⠀⣼⡟⠁⠀⠀⢠⡀⠀⠀⠈⢿⣆⠈⢿⣆
⠀⠀⠀⠀⠀⣿⣯⡙⠿⣿⣤⣠⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠐⠃⢠⣿⠃⠀⢠⣿⠁⢸⡿⠀⠀⠀⢀⣾⣧⡀⠀⠀⠈⣿⡆⠘⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⡉⠛⠛⠿⠿⠶⣿⣷⣶⣶⣾⡿⠶⠿⠿⠀⢸⣿⣤⣤⣼⡏⠀⣿⡇⠀⠀⠀⣿⡏⠉⠋⠀⠀⠀⢸⣷⠀⢿⡇
⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⠻⢷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡤⠀⢸⡏⠀⠁⣿⡇⠀⣿⠀⠀⠀⠀⠘⢿⣤⠀⠀⠀⠀⠸⣿⠀⢸⣧
⠀⠀⠀⠀⠀⣾⡿⣷⣼⣿⠀⠀⣿⠀⠉⠉⣿⡏⠉⠉⢹⣿⠉⠉⠀⠀⢹⣇⣀⣀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⠀⢸⡟
⠀⠀⠀⠀⠀⣿⣿⣌⠙⠻⢷⣶⣿⣄⣀⣀⣿⡇⠀⠀⢸⣿⣀⣀⣠⠀⢸⣿⠛⠛⢻⣇⠀⣿⡇⠀⠀⠀⢶⣤⣼⡿⠀⠀⠀⢸⡿⠀⣾⡇
⠀⠀⠀⠀⠀⣿⣿⡟⢿⣶⣤⣀⡉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠈⣿⡄⠀⠘⣿⡀⠸⣿⡀⠀⠀⠀⢹⡏⠀⠀⠀⢀⣿⠃⢰⣿
⠀⠀⠀⠀⠀⠹⣿⡇⢸⣿⠉⠛⣿⠿⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⣷⠀⠹⣿⡛⠛⠻⣷⡀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠹⣷⣤⣤⣽⣷⡄⠙⢿⣦⣀⠀⢀⣠⣴⡿⠃⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣶⣶⣤⣿⣧⣤⣤⣼⣿⣤⣴⣾⠿⠿⠃⠀⠙⢿⣯⣉⠘⢿⣦⣄⠉⠛⠛⠛⠛⠉⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢶⣿⣿⣿⣶⣶⣶⡾⠿⠛⠁



⡿⠿⡿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠛⠉⠙⠛⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⢿⣿⣿⠟⠋⠉⠉⠉⠉⠳⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠻⢿⣀⡀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⡄⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠒⠒⠲⠶⠒⠒⠉⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⢀⠀⠐⠊⡽⢿⣝⢓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠼⠤⢶⠾⠃⠉⠛⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠛⠻⠶⠶⠶⠶⠶⠾⠿⠟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠰⠤⣤⣤⣤⣶⠆⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣤⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠒⣳⣿⡶⠲⣤⡀⠀⠈⠉⠉⠉⠉⠁⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡚⠛⠟⣃⣀⣀⣤⣾⣷⠈⠛⠛⠛⠻⠿⠛⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠉⠉⠛⠛⠛⢛⣋⠉⣿⣿⠓⠶⠶⠶⠶⠶⠖⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣉⣉⣉⣉⡉⠀⣿⣿⠀⠶⠤⢤⣶⡶⠶⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⣀⣉⣉⣩⡅⠀⣿⡏⠀⠦⣤⣤⣤⣤⣶⠄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⢄⣀⣉⣠⡄⠀⣿⣧⠀⣀⣀⣀⣀⣀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣄⣀⣀⣠⡤⠀⣿⣿⠀⠈⠉⠉⠙⠛⢉⣀⣿⣧⣤⠴⠆⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣠⣤⣀⣻⣿⣤⠴⢶⣶⡛⠛⠉⠉⠁⠀⣀⡰⠤⠄⢀⣉⡛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣤⠤⠴⣾⡛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠶⣶⣬⣁⠀⠉⠛⠒⠛⠀⠤⣐⡀⣈⣩⡛⠻⠷⣶⣤⣤⣀
⠀⠀⠤⠤⠶⠒⠚⢋⠩⢥⡀⠀⢒⣊⡭⠽⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣄⣀⠀⠀⢀⣙⣦⣭⡽⠆⠀⣀⣀⡉⠉⣅
⢰⡆⠠⢄⡀⠀⠰⣌⠓⣲⣟⣯⡁⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⣾⣩⣤⠴⠶⠒⣛⣩⡭⠴⠒⢚⣿
⢰⣇⠀⠀⠙⠦⣄⠈⠙⠛⠻⠥⣌⣵⠶⠶⣤⣀⢀⣀⡤⠄⠈⠑⠢⠄⠀⠀⠀⢀⣀⣤⡤⠶⠞⠛⠉⢹⡅⠀⠒⣉⡩⠥⠶⠒⠊⢉⣼⣿
⠀⡏⠓⢦⣀⠀⠈⠙⠦⣄⡀⠀⠈⣅⠻⣲⣆⢻⡏⣠⠴⠋⠉⣁⣠⣤⡶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣠⡤⠶⠖⠛⠉⢁⣿⡏
⠀⣗⠠⣀⠉⠳⣤⡀⠀⠀⠙⠲⢤⣀⠙⠛⠛⠋⣉⣥⣶⠾⠟⠛⠉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠉⢉⣀⣤⣤⠶⠶⠛⢛⣿⡇
⠀⣿⢤⡈⠳⢤⡀⠙⠶⣄⠀⠀⠀⠈⠁⣴⠾⣛⣩⠥⠖⢂⣁⣠⠤⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⣉⣠⣤⠶⠶⠛⠛⣿⣷⣤
⠀⠉⣀⠙⠲⣄⠉⠲⢄⠈⠙⠦⣄⠀⠀⣿⣭⡥⠶⠖⢛⣩⡥⠶⠖⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣚⠫⠭⣔⣒⣬⣵⣾⣿⡿⠟⠋
⠀⠀⠈⠳⣦⡀⠁⠀⠀⠀⠀⠀⠈⠙⠲⣿⣤⡶⠶⠛⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣶⣾⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣀⣤⣴⡶⠞⠁⢸⡇⢀⣀⣤⣴⣶⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⣿⡧⠖⠚⠽⠛⣋⠁⢀⣠⣤⣾⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⠀⣿⣇⣤⠴⢞⣫⣵⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⣩⠴⠞⠋⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡾⣿⢣⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣇⣦⡀
⠀⠀⠀⢀⣠⣀⡤⣮⡾⣫⠞⢡⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣾⢏⣿⢠⡀
⠀⢀⣴⣿⣿⡗⣺⢫⡾⠋⣰⡿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡤⣄⣴⣾⣿⣿⣿⣷⣿⣼⣇⣾⡿⡇⡀
⣴⣿⣿⣿⣿⣗⡿⣸⡇⣾⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣧⢻⡆⣤
⣿⣿⣿⣿⣿⣿⡇⣿⢱⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⣝⠷⣝⡻⣿⣿⣿⣿⣿⣏⣾⣿⢻⣮⣻⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⡣⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⢷⣍⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⡇⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⣿⣿⣿⡆⠀⣿⣿⣿⠀⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⢹⣿⣿⡇⢀⣿⣿⣿⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣸⣿⣿⣷⣼⣿⣿⣿⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⡏⠈⣿⣿⡿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⣿⡇⠀⣿⣿⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⢿⣿⠀⢹⡇⠀⢻⡏⠀⣼⣿⣿⡇⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠈⢿⡆⠀⡷⠀⢸⠃⠀⣿⣿⣿⣷⣄⣠⣶⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠁⠀⠁⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⢉⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⣯⣥⡀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣷⡤⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⣴⡀⠀⠀⠀⠀⠀⢀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢤⡀
⠀⠀⠀⠀⣿⢹⠿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⠏⣿⣿⣾⣿⠛⠃⠀⠀⠀⠀⠀⠟⢃⣤⡍⣹⡄⠀⠀⠀⠀⠀⠀⣀⣷⣙⣼⠁
⠀⠀⠀⢠⡿⣤⡼⣧⠀⠀⠀⠀⠀⠀⢰⠻⡦⢴⠟⡆⠀⠀⠀⠀⠀⠀⣤⡛⠁⠈⠻⣠⡄⠀⠀⠀⠀⠀⢷⡎⣙⣁⡹⠁⠀⠀⠀⠀⠀⢟⢷⢌⣾⢫⠛⡇
⠀⠀⠀⢸⣙⣻⣓⣹⠀⠀⠀⠀⠀⠀⢸⣀⣈⣁⣀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⡿⠃⠀⠀⠀⠀⠀⢰⣿⡆⠀⠰⣿⡆⠀⠀⠀⠀⠀⢀⣀⣀⣄⣴⣤
⠀⠀⣴⣦⣻⣾⣶⣿⣆⠀⠀⠀⠀⠀⢀⣀⣻⣿⣿⣦⡀⠀⠀⠀⠀⠰⣿⡿⠳⠚⢿⣿⠂⠀⠀⠀⠀⠸⠭⠇⠀⠸⠥⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠙⠙⠿⠃⠿⠻⠋⠃⠀⠀⠀⠀⢿⣿⣿⠏⠻⠿⠃⠀⠀⠀⠀⠀⡞⠛⣄⣰⠛⢳⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠛⢿⣿⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠘⠒⠋⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢰⣆⣤⣤⣀⣀⣶⠀⠀⠀⠀⠀⢰⠒⠚⠛⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠰⣦⣸⣇⣨⡁⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⣿⡇
⠀⠀⢸⡧⠷⠾⠤⠤⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠡⡾⢿⢿⣭⠀⠀⠀⠀⠀⠠⣿⡷⠶⠶⢶⣿⡇
⠀⠀⠸⠗⠟⠒⠚⠗⠿⠀⠀⠀⠀⠀⠘⣲⣖⣒⣶⡚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠠⠉⠀⠀⠀⠀⠀⠈⣿⣇⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⡆
⠀⠀⠀⠀⢰⡿⠿⠿⡇⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡢⢛⣁⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠑⠢⡀⠀⠀⠀⠀⠀⠀⢀⣤⡛⢀⡄
⠀⠀⠀⠀⢸⡇⣠⣴⡇⠀⠀⠀⠀⢸⠀⡑⢄⣀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⣵⣦⣄⣤⣾⡆⠀⠀⠀⠀⠀⠠⣼⣍⣙⡷⠂
⠀⠀⠀⢿⣿⠇⠙⠛⠁⠀⠀⠀⠀⠀⠑⠤⣁⣒⣀⣡⠜⠁⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠝⠋⠀⠀⠀⠀⠀⠀⠓⢮⣙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠺⢦⣄⣀⡆⠀⠀⠀⠀⠀⠀⠠⠂⡀⠔⠀⠀⢀⡤⣴⣞⡷
⠀⠀⠀⠀⠀⠀⢀⠤⠢⠈⠁⠀⢰⣮⣛⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠠⠀⠛⡉⠁⠀⠰⣯⣯⣷⡆
⠀⠀⠀⠀⠀⠘⠔⠀⠀⢰⠀⠀⠛⢻⣿⠆⠀⠀⠀⠀⠀⠀⢀⠈⢠⠀⠂⠀⡤⠀⠃⠀⠀⠀⠀⠙⢿⣿
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠁⠀⠐⠂⡨⠋⠀⠀⠀⠀⠀⠀⠀⠞⠀⣶⣿⣇⡎⠀⣀⠀⢀⣰⠁⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠧⠤⠤⣄⠀⢀⣤⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠸⣿⣿⡇⠀⠹⢿⡿⠏⠀⢀⠕
⠀⠀⠀⠀⠀⠆⠉⠆⢠⢧⠾⢛⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠈⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⢀⣦⣿⣿⡁⠈⠻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠐⡇⢀⣠⠬⠿⣿⠏⢴⣸⣷⣶⣶⡆
⠀⠀⢀⠖⢠⣶⠀⠙⣛⡿⠁⠐⡶⣿⣿⣉⠁⠀⠀⠀⠀⠀⣸⡇⠁⠀⠀⠀⠈⠀⠀⢹⣿⣿⢿⣿⣦⡀
⠀⠀⢸⠀⠘⠛⠁⢠⡏⠀⣴⣶⡀⠈⢋⠉⠀⠀⠀⠀⠀⠀⠉⠅⠐⢲⠀⡀⠔⠚⠛⠦⢻⣿⡄⠻⠻⠁
⠀⠀⠈⢑⠖⠀⠀⠸⣆⠀⠘⠚⠁⢀⠌⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠛⢇⣄⣀
⠀⠢⡄⠁⠀⢀⡴⠚⠉⠓⠀⠀⠐⠳⢤⡀⠀⡑⠄⠀⠀⠀⠀⠀⠈⠩⠤⠐⣇⠀⢀⡠⢠⣔⣫⣶⡁
⠀⠀⠰⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠀⠀⠀⠀⠒⠃⠀⠀⠢⠀⠈⠉⠁⠀⠺⠼⠛⠋
⠀⠀⠀⡀⠀⢀⠒⠐⠀⠀⡀⠀⠀⠀⠀⠀⢠⣂⣐⣖⠶⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠈⠳⠆⣀⣀⠠⠖⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣵⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⢠⠉⠠⠀⢈⡀⠀⠀⠀⠀⣤⠞⠁⢀⣤⣄⠀⠈⠑⢤⡀⠀⠀⠀⠂⠀⠀⠀⠀⠎⠀⠅
⠀⠀⡀⢈⡦⠈⠃⠁⠀⠀⠀⠀⠀⣾⡋⠀⠀⠛⢳⣳⡶⠀⠀⠀⠗⡄⠀⠀⠀⠰⣆⠀⡂
⠀⠀⢧⡘⠃⢀⣠⢀⠠⣀⠀⠀⠀⢿⣶⡢⠀⠀⠉⠃⠁⡀⠀⠀⡠⠃⠀⠈⠆⠂⣁⡀⠑⠠⡀⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠉⠛⠛⠶⠶⠚⠉⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠤⣭⣶⡾⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⢀⣤⣴⣶⣶⣶⢦⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣏⣳⣰⣿⣷⣿⣿⣷⣿⣿⡿⠁⠀⠀⠀⠀⠀⡘⣡⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠿⠿⠿⠿⠯⣿⢻⡇⠀⣠⣤⣴⣶⣶⣴⣭⣩⣹⣍⣱⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⢮⣿⣁⠀⠀⠀⠀⠀⠻⣾⠇⠀⠙⠋⠹⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁
⠀⠀⠀⠁⠀⠰⠄⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣄⢀⣠⣄⢀⣤⣶⣿⣿⠁
⠀⠀⢀⣴⣾⣷⣄⢲⣰⢀⣴⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠛⠛⠛⣷⣿⠛⠛⠟⡟⠁
⠀⠐⠛⣿⢿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠛⠀⠀⠀⠀⠀⠀⣦⣀⠀⢀⣿⢆⣴⣦⠰⣿⡇⠀⠀⠉⠑⠀⠀⢻⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠉⠉⠙⠿⠏⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠘⣿⣷⣿⠿⡊⠛⠉⠘⠓⠂⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⠈⠛⠿⠟⢍⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡐
⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⡀⡊⠀⠀⠀⠀⠈⣿⠶⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠉
⠀⠀⠀⠈⠀⠰⠆⢀⡀⢲⠆⠀⠀⠀⠀⢀⠀⠈⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⢠⣴⡇
⠀⠀⠀⠐⠀⠀⢄⣀⣤⣤⠖⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣉⠁⠀⠀⠀⢀⠀⠀⠀⠘⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠳⠴⠶⠶⠴⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⡀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠐⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢧⣄⣸⣁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠄⠘⠀⢀⣠⣿⢻⣷⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠃⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⢿⣿⣿⣶⣖⠚⠁⠈⢿⣿⣜⣿⡧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⣶⡏⠀⢻⣿⡜⢿⣆⠀⠀⢸⡿⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠉⠛⠂⠛⠛⠶⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠲⢤⣞⠙⡗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠃⠈⠉⠉⠁⠀⠓⠒⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠈⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⣴⡆⢸⡍⠲⣿⠀⡟⠲⣄⡀⠀⣠⠞⠁⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣄⡇⢸⠄⠀⣿⠀⡇⠀⣿⠉⠲⣅⠂⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⠇⢸⠄⠀⣿⠀⡇⠀⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⢧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠓⢶⣄⠀⡀⠀⠀⠀⠀⠀⠘⡎⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸⡁⠀⣿⠀⡏⣦⠀⠀⠀⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⢗⢦⡀⠀⠀⡇⢸⡄⠀⣿⠀⡇⢸⠀⠀⠀⠀⠀⢸⠃⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⢵⣄⠀⡇⢸⠆⠀⣿⠀⣧⠏⠀⠀⠀⠀⠀⣸⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡐⠀⠀⠀⠀⠀⠀⠉⠛⠃⠸⠥⠤⠟⠀⠁⠀⠀⠀⠀⠀⢠⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⢈⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⡄⢠⣤⡤⣄⠀⡶⣏⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠄⢙⣯⠀⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣷⣤⡏⠻⢽⣦⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣈⡉⠙⠻⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢰⡟⠉⠉⠁⣀⣀⣀⣤⣭⠉⠉⠙⠛⠿⢿⣶⣤⣄⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣼⡏⢷⡄⠀⠀⠀⠻⣦⣄⡀⠉⢻⣯⣀⣀⣠⣴⠶⠶⠶⣶⣄⠉⠙⠻⣦⡀⠈⢻⣆
⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠉⢁⣠⣤⣴⠶⠶⢾⡇⢀⠛⢷⣄⡀⠀⠀⠉⠛⠷⢦⣭⣛⡋⠉⠀⠀⠀⠀⣼⡿⢶⡤⠀⢸⡇⠀⠀⢻⡄
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣴⠿⠋⠁⠀⢀⣤⠈⠻⣾⠀⠀⠉⡿⠶⣤⣀⡀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠛⠁⠀⠀⣼⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣏⠀⠀⣤⣴⣿⠁⠀⠀⠙⣷⣤⣰⠃⠀⠀⠉⡻⠛⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠾⢻⠃
⠀⠀⠀⢰⡟⢷⣄⠀⠀⠀⠻⣦⣀⠀⠀⠻⠷⣶⡶⠿⠋⠉⠻⢷⣦⣤⣰⠃⠀⠀⠀⠀⡌⠈⠉⠉⠉⠹⠉⠉⠉⠉⡏⠉⠀⢘⣠⡏
⠀⠀⠀⠘⣧⡈⠻⣦⣄⠀⠀⠈⠙⠻⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣈⣽⡿⠿⢿⣷⣶⣶⣧⣤⣤⣤⣤⣧⣤⣤⣤⣤⣧⣶⡾⠟⠉
⠀⠀⠀⠀⠹⣷⣄⠀⢹⠻⢶⣤⣀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣈⡽⢛⢹⡏⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠘⢿⣷⣾⡀⠀⠀⡍⠉⠛⠛⠷⡶⠶⢶⣤⣤⣤⣤⣤⣤⡶⠶⡶⠛⠋⢹⢀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣧⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣇⣠⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣷⣶⣶⣶⣦⣤⣷⣶⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⢛⣙⢛⡛⣛⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣤⢀⡀⣶⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣶⣾⣿⣿⣿⣿⣶⣾⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⡀⠉⠛⣿⣿⣿⡿⠛⠉⣀⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⠂⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠐⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣼⣿⠏⢀⣤⣤⣤⣐⢄⠀⠀⠀⠀⡠⣀⣤⣤⣤⡀⠹⣿⣧⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡿⠋⠭⣽⣶⣿⣿⡾⠛⠋⠉⠛⠻⢷⣷⣤⣤⣾⡾⠟⠛⠉⠙⠛⢷⣿⣿⣶⣯⠭⠙⣿⡄
⠀⠀⠀⠀⠀⢸⡇⠘⠻⣿⣿⣿⠏⠀⢀⣤⡶⢶⣤⡀⠙⣿⣿⠋⢀⣤⡶⢶⣤⡀⠀⠹⣿⣿⣿⠟⠃⢸⡇
⠀⠀⠀⠀⠀⠘⣇⠀⡞⢛⣿⣿⠀⢠⡟⣡⡀⣳⣦⢹⡀⢸⡇⢀⡟⣅⢘⣷⣎⢻⡄⠀⣿⣿⡛⢳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣦⣇⠈⢾⣿⣆⠘⣇⠻⣟⣿⡿⢡⠁⡸⢇⠈⡆⢿⢻⣿⡟⣸⠇⣰⣿⡷⠁⣸⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⣿⣿⠷⡌⠳⠬⠭⢖⣡⠾⠗⠺⠷⣌⡲⠭⠥⠖⢁⠼⣿⣿⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣧⣄⣀⣀⡤⠋⠄⣀⠀⠀⣀⠠⠙⢦⣀⣀⣠⣴⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡟⠀⠀⠀⠉⠃⠘⠉⠀⠀⠀⢹⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡉⠓⠒⠒⠒⠒⠚⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣤⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡷⢚⡶⢻⡗⢻⡧⠽⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠆⠸⢣⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⢁⡞⢠⡟⠀⢀⣤⣄⠀⢹⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⢠⠂⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⣯⣾⣥⡏⠀⠀⠂⢈⠋⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣾⣽⣷⣯⣴⣋⢉⡑⣌⠂⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠖⠰⢈⠲⠄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⣿⣇⠛⢙⢿⡟⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠘⠚⠉⠀⢀⠀⠀⠀⠀⠤⣿⣿⣿⣿⠋⠀⠘⠻⣓⠒⠉⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⢴⠀⠀⠙⠁⠀⠈⠢⠆⠉⠀⣤⠉⣥⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢄⠀⠀⠀⠊⠀⠀⢀⠀⡀⠀⠐⡓⡘⠃⡀⡀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠨⠀⠀⠈⠄⠁⠁⠀⠁⠁⠄⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⠀⣂⠀⠀⠀⠄⠀⠁⠰⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠁⠀⢀⠀⠀⠀⡁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⠀⠀⢐⠀⠐⢁⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠟⠃⠈⠓⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡒⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠓⢤⠀⠀⠀⠀⡀⢠⠀⠀⠜⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠓⠚⠉⠀⠁⠁⠀⠁⠂⢀⡤⠀⠀⡔⠢⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⡀⠀⠠⠔⠚⠀⠀⠄⠫⠤⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠁⠀⠀⠀⠀⣰⠶⠡⠀⠀⠀⠀⠀⡀⠀⠶⠦⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠅⢐⡠⣌⠀⠤⡀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠠⢭⠐⠂⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⡌⠙⣆⣆⡤⠞⣓⠒⠢⢄⠀⠀⣀⠀⢠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⢉⢉⣩⠞⠁⢰⠒⣏⠑⡢⣠⣇⡎⣡⠤⠶⣐⠒⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⡴⣴⠋⡜⣡⢋⢧⠀⠸⡯⠤⠴⣂⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢠⡿⣌⣛⡻⢢⠤⠻⠂⠂⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠳⣄⢀⣠⣾⣷⣦⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠏⢰⣈⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣬⣁⣤⡈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢉⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠁⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⠀⠙⢿⣷⣶⡾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⠞⠲⠶⠖⠋⣳⣶⡿⠛⠁⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢠⡾⠁⠀⠀⣠⣴⡿⠟⠉⣀⡀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠉⢀⡴⠞⠉⠙⠓⠒⠚⢳⡀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⣁⣴⣾⠟⠋⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⣄⣤⣾⠿⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣤⣾⠿⠋⢁⡀⠀⠺⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣡⣶⡿⠛⠁⣠⡴⠋⠁⠀⢰⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⣠⣼⡿
⠀⠀⠀⠀⠀⢠⣾⣿⣷⡿⠛⠁⣠⡴⠛⠁⠀⠀⠀⠀⠘⢿⣶⣶⠿⠿⣶⡄⠀⠀⠀⠀⠀⢀⡤⠞⠉⣀⣴⣿⠟⠉
⠀⠀⠀⢀⣴⣿⣿⡟⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣤⠞⠋⢀⣴⣾⠟⠋
⠀⠀⠀⣾⡟⠀⢿⣧⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡾⢿⡅⠀⠀⠉⢀⣤⣾⠿⠋
⠀⠀⠀⢿⣧⣄⠀⠻⣷⡄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⢀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠙⠻⣷⣄⡙⣿⣆⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢁⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣮⣿⣷⡀⠘⢧⡤⠤⢤⣀⣠⡴⠛⠁⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⠀⠀⠀⠙⠁⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⣀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠠⠄⠂⠀⠀⠈⠀⠀⠀⠀⠋⢤⠀⠀⠀⠀⠗⠂⠀⢀⡀
⠀⠀⠀⣤⠀⠠⠀⠂⠀⠁⠀⠀⠈⢀⠄⠤⠀⢄⡀⠀⡀⠀⠀⣀⡀⠀⠀⠸⡁⠀⠀⠰⡢⠤⡄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠰⠁⢐⣢⠀⠀⠑⠀⠀⠀⠀⣧⡱⠀⠀⠀⠃⠀⠐⠀⠙⠁
⠀⠀⠀⠀⠄⠀⠀⡖⡇⠀⠀⠀⢡⠀⠀⠀⡆⡂⢀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠓⠀⠸⠂⢠⠘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠑⠄⣉⡮⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣀⠦⠒
⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠈⡀⠤⠀⠤⠤⠒⠀⠀⠀⠈⠀⠀⠀⠀⠁⠘⠁⠀⠀⠒
⠀⠀⠀⠀⠈⠧⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠐⠀⠀⠤




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢿⣀⣮⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠛⢻⠋⣷⣄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢣⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣄⠙⣿⣦⡄
⠀⠶⣿⣼⣿⣿⣿⠟⢀⣠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⠁
⣶⣶⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠛⢯⣿⣿⣿⣿⣿⣿⣧⠀⣦⣾
⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠏
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣤⣀⣀⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣶⣶⢈⠉⠉⠙⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⣉⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣧⡸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠿⢿⣿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣷⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠈⣿⣿⣿⠈⢿⣿
⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿
⠀⣾⣿⣿⠟⣿⣿⡿⢿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡆⠀⠸
⠸⡿⠻⣿⢰⣿⢰⡇⠈⢿⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣸⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢰⣿⣿⣆⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⢻⡀
⠀⠀⠀⢀⣿⠘⣻⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⣾⣿⣿⣥⣄⢸⡇
⠀⠀⠀⢸⡏⢀⢼⣿⢿⣿⡿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣯⡥⢼⡇
⠀⠀⠀⣾⠇⠀⠈⠁⢸⡏⠀⠈⠃⢀⣤⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠰⡛⠛⢹⢿⡏⠙⠿⠀⣷
⠀⠀⢀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡾⢿⡿⠶⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⢴⢾⣿⣿⡅⠀⠁⠀⠚⠈⠁⠀⠀⠀⣿
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠉⡍⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠁⠀⠀⠉⠋⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠉⡍⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠀⠀⠁⠁⠉⠱⠄⢀⡀⠤⠀⠀⠀⠀⣀⣀⣐⠋⣡⣴⣚⣀⡀⠀⠐⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠐⠲⢒⠈⢷⠈⡶⠍⠁⠐⠒⠉⠉⠀⠀⠀⠀⠀⣀⢀⠉⠩⢕⠢⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠀⠀⠤⣠⠤⢿⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠲⠛⠻⠋⠁⠀⠠⠾⠶⠒⢤⣄⣀⠈⠑⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⡀⠤⠤⠀⣠⠀⡀⠀⡇⠀⡀⠀⠀⣂⣀⠼⠀⠀⢀⢀⣠⠀⣠⠭⢒⡩⠥⣆⣬⢆⣂⠤⣄⣐⠒⠀⡐⠦⠐⠒⡐⣀⠥⠔⠂⢛
⠀⠀⠀⠀⠀⠆⠀⠀⢀⣀⡇⡆⠀⣤⠀⠀⠀⠀⠀⠀⢠⠏⠙⠆⠠⣙⡿⢒⠈⠭⢒⡓⠶⣈⠉⠓⠛⠥⠒⠂⠔⠀⣒⠬⠳⠚⠓⠀⠀⠀⢠
⠀⠀⠀⠀⠠⣆⠀⠀⣠⣤⠶⢒⡲⡌⠀⠉⠙⠓⡚⡋⠄⣏⠀⠀⠃⢈⣇⣠⢀⣀⠀⠀⠀⣀⡈⢁⠈⠁⠀⠀⠁⠀⣀⢠⣢⠥⠲⠜⣀⠤⡾⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠉⠉⣏⠁⠀⡈⡈⣍⠀⠐⠂⠀⠐⠖⣨⣷⠆⣠⣌⠵⡋⢀⣉⣉⡊⠉⠙⠛⠚⠿⠿⢗⣢⠀⠀⢋⣭⠥⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠒⠓⠋⠑⠒⠒⠐⠖⠚⠉⠀⠀⠀⠀⡾⠩⠁⠂⠠⠤⡀⡄⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⣀⣀⢀⢂⠤⠇⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠷⡶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠿⠗⠛⣩⠍⠀⠀⠀⠊⠁⠀⠀⡤⠀⠠⠄⠀⠀⢦⡄⠀⠀⢠⡀⠀⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⡋⡁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣤⠀⠀⠐⠦⠀⠀⠾⠉⠳⢦⣄
⠀⠀⠀⠀⠀⣴⡶⠛⠁⠀⠈⠂⠀⠀⣴⠄⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⡀⠀⠀⢤⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⠦⠙⢳⡄
⠀⠀⠀⠀⣴⠋⠤⠀⠀⡠⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠔⠀⠀⠀⠀⠶⠙⢦⡀
⠀⠀⢀⣾⢃⠈⠁⠀⠀⠈⠀⠀⠘⠁⠀⠀⢀⠴⢄⠀⡀⡀⠀⡀⠀⠠⠠⢀⣀⠠⠂⠀⠂⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢀⡀⠀⠈⢷
⠀⠀⢸⡧⡃⢄⠄⠀⠀⠀⠀⠀⢀⠠⡀⠠⠊⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢄⡀⠀⢀⠀⠀⠀⠉⠀⠀⠘⡇
⠀⠀⣸⣏⣾⣦⢀⡤⢤⡄⠀⠀⢀⣴⢠⠀⣮⣧⣧⣦⣼⣬⣼⣿⣼⣾⣶⣾⣶⣶⣴⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⣿
⠀⠀⢸⣿⣿⠿⢿⢳⣰⣿⣿⣣⣰⣾⣾⣿⣿⣿⠿⠿⠛⠛⠛⢋⡉⠉⣉⣉⣉⡉⠉⠉⠉⠉⣩⣯⣽⣛⠛⠛⢳⣶⠶⠶⢦⣤⣤⣤⣤⣤⣀⣀⣤⠏
⠀⠀⠘⣿⣿⣷⣿⣷⣿⣿⣿⣿⠿⠛⠋⠉⢀⣤⠶⠛⠛⠓⠲⠋⠙⠛⠉⠉⠈⠉⠙⠛⠲⠚⠁⠀⠀⠉⠉⠙⠉⠉⡓⠛⠶⣄⡀⣀⣈⣉⡉⢉⣉⠳⣄
⠀⠀⠀⢈⡽⠿⣟⣛⣽⣥⣤⣤⣤⣄⣀⡴⠋⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠻⠏⠉⠙⠲⢿⢻⡞⠛
⠀⠲⠛⠋⣶⣿⠿⠁⠀⣔⠀⠀⢀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⡤⠀⠀⣿
⠀⠀⠀⠀⣿⢀⣨⡀⠀⠈⠀⠀⠙⠀⠀⣠⡾⣛⣛⣛⣯⣤⣤⣤⣤⣤⣤⣤⠀⢀⡀⠀⠀⢀⣀⢀⣶⣿⣷⢶⣶⣦⡄⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠘⠻⣿⣿⣷⠀⣴⢦⠾⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⡄⢠⣿⣿⣿⣁⣀⡀⠀⠈⣻⣧⠀⣴⣦⡾⠛⠟⠻⠶⢶⣾⠋
⠀⠀⠀⠀⠀⠘⣿⣻⣿⣠⠇⠀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠶⠶⠶⠒⠚⠛⠛⠛⠿⠿⠞⠛⠋⠀⠀⠈⠉⠉⠙⠛⠛⠚⠷⠦⠤⠤⣄⣀⣀⣠⣟
⠀⠀⠀⣠⣴⠿⢿⡿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⣼⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠴⠄⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣦⣠⣶⣤⠀⠀⠀⠀⢠⣼⣧⡀
⠀⢨⡟⣉⠩⠷⣶⠖⠉⠈⠙⠛⢻⣶⣤⡀⠀⠀⠀⠸⣇⡄⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠾⣇⠀⢰⢦⡍⠀⠹⣷
⠀⠘⣷⣦⣤⣼⣷⠆⠀⢠⡀⠀⠀⠈⠙⠻⣷⣄⡀⠀⠉⠀⠀⠀⣠⣼⠏⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠟⠀⢀⡀⢠⡇
⠀⠀⠈⢻⣿⣧⣥⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⡄⢀⣾⠋⠁⠀⠀⣈⣳⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣉⣀⣀⣀⠀⢀⣀⣰⣦⣀⣀⣀⣀⣴⠟
⠀⠀⠀⣿⣿⣿⡛⠙⢹⡿⣟⣿⠟⠋⠻⣿⠽⠿⣿⣿⣿⣿⣾⡿⠛⣿⡟⠛⠉⢉⠉⠉⢹⡏⠁⣽⠋⠉⢻⠟⢿⣍⣛⣻⠟⠛⣿⠋⠻⠛⣩⡿⣯⣽⠇
⠀⠀⠀⠙⢹⣿⣷⣤⠏⠀⠈⠀⠀⠀⠀⢻⣄⣠⠋⠉⠛⠛⠋⠀⠀⢸⡇⠀⢸⠛⢷⡴⠛⠀⠀⣻⣦⣤⠏⠀⠀⠉⠉⠁⠀⠀⣿⡆⠀⢀⡟⠀⠉⣿
⠀⠀⠀⠀⢸⣿⣽⣥⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣧⣬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠖⠒⠛⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⢿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⡿⠿⠿⣿⣿⣿⣶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⢸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠙⢿⣿⣿⣿⣦⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⢶⣤⣤⣶⠾⠿⢿⣿⠆⣀⣤⣤⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⠥⠤⡶⠶⣶⣶⣶⣦⠀⠀⣴⣛⣓⠋⠉⠀⠀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⢺⣿⣁⣠⣽⣶⡾⠛⣻⠟⠛⠚⠛⢛⡿⠋
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠻⣯⡉⠉⠀⠀⣰⠃⠀⠀⢀⡴⢋⣀⣤⣤⠴⢶⣤⡄
⠀⠀⢰⡟⣿⣿⡿⠿⠛⠛⠋⠛⣉⠉⠀⠀⠀⠀⠀⢀⣠⣝⣻⠶⣬⣿⣄⣀⣀⣁⣤⣤⢴⣿⣿⣯⣥⣴⣶⡶⠿⠛⠁
⢀⣤⣾⣿⡟⠁⠀⠀⠀⠀⠀⢰⣾⣷⡄⠀⣀⣤⣾⠟⠉⠈⠉⠛⠛⠛⠻⠿⠿⠿⠾⠟⠋⠳⣦⣄
⣿⢋⣁⠀⠀⠀⢠⣤⠀⠀⠀⢠⣤⢀⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⢻⣾⣿⣦⠀⠐⠿⠃⣀⠀⠀⢈⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀
⠀⠙⣿⣿⡁⠀⠀⠀⡴⠋⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠲⡄
⠀⠰⡇⠒⠷⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⠜⣷
⠀⠀⠙⠷⠶⠾⣇⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⢛⣽⣇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣶⠿⢿⡛⠛⠻⢿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿
⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⢿⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣈⣧⣤⣴⣾⣷⣶⣶⣶⣤⣀
⠀⠀⠀⣤⢎⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠿⡿⠛⠛⠻⣿⡇
⠀⠀⣸⠃⠈⣛⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡀⠀⠀⢻⡀⠀⣠⡿⠁
⠀⠀⢿⡄⠀⣛⡻⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⡀⠘⣇⣀⣀⣬⣷⡾⠛⠁
⠀⠀⠈⠳⣤⡩⠭⣿⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⡟⠛⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣿⣭⡏⡛⢻⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⢿⣛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣼⠿⣿⣯⣟⣿⣳⣿⠏⠀⠈⠉⠙⢿⣧⣭⠿⣽⢾⣻⣾⣻⣽⣾⢷⡀⢀⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢯⣝⢣⣟⣳⢦⣾⣽⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⡿⣿⣾⣻⣽⠛⢹⣷⠟⠁⢀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣾⣳⠾⣽⣟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣿⣿⣿⣿⣿⡿⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣽⣿⠾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠃⢹⣿⣿⣏⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣽⣦⡀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡘⢿⣏⢁⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣥⣌⡉⠙⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⠿⠷⠒⠾⠿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣠⣶⣴⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⣄⣀⣤⣶⡒⠒⠄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡛⣿⠂⠘⠻⢟⠙⠛⠓⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠴⠂⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣇⠀⣀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢷⣾⣶⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⠦⠾⡿⢿⠭⠥⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣸⣿⣦⣬⣤⣌⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢶⣦⣤⠔⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢠⣿⣿⣿⣿⣿⣿⣿⣶⣄⠈⠛⠂
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣿⣿⣿⣿⣿⡟⠀⠻⣍⠤⠤⠀⠀⢀⣠⠔⠋⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⡴⡾⢋⣭⣭⣽⣿⣿⣿⣿⣿⣾⣽⣶⣼⣷⣶⣶⣶⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡆
⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢀⣾⣿⣿⣿⡿⣿⠟⠁⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣄⣿⣿⣴⣦⣿⣣⣜⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠟⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣛⠛⠓⠂⠀⠈⢻⣿⣿⡟⠁⠀⠐⠚⠛⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⠀⣾⣿⢃⣠⣽⣿⣿⣶⣌⢦⡀⠀⠀⢀⡴⣡⣶⣿⣿⣯⣄⡹⣿⣷⠀⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⣾⠏⢠⣶⣮⣿⣿⣿⡿⠋⠉⠀⠉⠉⠙⠷⣿⣴⣧⣿⠾⠛⠉⠉⠀⠉⠙⠻⣿⣿⣿⣷⣶⡄⠹⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠉⢛⣿⣿⣿⠏⠀⢀⣴⡶⠷⠶⣤⡀⠘⣿⣿⠋⢀⣤⠶⠶⢶⣦⡀⠀⠹⣿⣿⣿⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡄⢀⠟⢙⣿⣿⠀⠀⣿⢣⣆⢙⣿⣮⢳⠀⢸⡇⠀⡞⣡⠈⣿⣷⡌⣿⡀⠀⣿⣿⡋⠻⡄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣬⡆⠈⢿⣿⣇⠀⣿⠸⣿⠿⣿⡿⢸⠀⡼⢧⠀⣇⢻⡟⣿⣿⠇⣾⠁⣰⣿⣿⠁⢠⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⣿⣿⣷⣌⠳⢬⣍⡭⠔⣡⡾⠵⠮⢷⣌⠢⢬⣉⡥⠞⣁⣼⣿⣿⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣦⡀⠀⠀⢀⡰⠋⡁⠀⠀⠀⠀⢈⠙⢆⡀⠀⠀⢀⣴⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠈⠻⠆⠰⠟⠋⠀⠀⠹⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠓⠤⣄⣀⣀⣀⣀⣠⠤⠚⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣶⣶⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⣿⠀⠀⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡆⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⢀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣫⠵⠖⠒⠒⠶⢭⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⠋⠁⠀⣠⡶⣦⣀⠀⠈⠳⣜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠃⠀⠀⣾⠁⣾⣆⣹⡆⠀⠀⠹⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠘⢦⣈⠙⢷⡀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⠀⠀⠀⢿⠉⣿⡇⢠⡇⠀⠀⢠⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢷⡀⠀⠈⠳⣦⡶⠛⠀⠀⣠⢞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⣿⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡶⢤⣄⣀⣀⣤⡴⣞⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⡀
⢀⣴⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⠀⢠⣿⠋⠉⠙⣿⡆
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣿⣦⣀⣠⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⠁⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠉⠻⣷⣤⣶⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⠀⠈⢻⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⣤⣤⣾⠏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⣄⣀⣀
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⡆
⠙⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣇⠀⢀⣼⡟
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⣉⣉⡉⠉⠉⢉⣉⣉⣉⣉⣉⠉⠉⠉⣉⣉⣉⣉⣉⣉⠉⠉⠉⣉⣉⣉⣉⣉⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣛⣛⡛⠛⠀⠀⠛⢛⣛⣛⣛⠛⠃⠀⠀⠛⢛⣛⣛⣛⠛⠁⠀⠘⠛⣛⣛⣛⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣥⣄⠀⠀⣠⣬⣭⣭⣭⣥⡀⠀⠀⣠⣭⣭⣭⣭⣤⡀⠀⢀⣤⣭⣭⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣉⣉⣉⣀⣀⣈⣉⣉⣉⣉⣉⣁⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣈⣉⣉⣉⣉⣉⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠢⠀⠀⠀⠀⠀⣠⣿⣿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠴⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⡀⠀⠀⠀⢀⡀⠄⡁⠛⠉⠀⠉⠢⡀
⠀⠀⠀⠀⠀⣰⠃⡠⠄⣉⠉⠀⠐⠀⢀⠀⠈⠀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢠⠄⠀⠀⠀⢨⡁⠈⡁⠀⠘⢿⠂⢠⡇
⠀⠀⠀⠀⠀⠑⠀⠒⡾⠁⠀⠀⣀⠄⠘⠓⠂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠸⠋⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠸⡀⢦⣴⡾⠁⠀⢀⠌
⠀⠀⠀⠀⠀⢠⠏⠀⠠⠂⠀⠀⠀⢟⠀⠠⠀⠄⣠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⠀⠄⡀⠉⣢⡄⡀⠀⣼⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣠⠛⠀⠀⢀⠠⠀⢀⠄⢚⠤⠀⠂⡀⠀⠀⠀⠄⠲⠀⠀⠃⠄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠂⣄⢸⣶⣿⣿⣷⣤⣾⣿⣿⡿⠻⣿⣿⣿
⠀⠀⠀⣰⠃⠀⢠⡼⠋⠘⠀⠄⠀⠈⠃⠀⠌⠄⠈⠀⢀⡤⠴⠒⠀⠈⠃⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⣿⣿⣿⠏⢉⠒⢾⣿⡿⠃
⠀⠀⢠⠏⠀⢀⡝⠀⡀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⢁⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠀⠚⠉⠉⠙⠿⣇⣩⣮⣴⡄
⠀⠀⡎⡀⢠⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠒⢉⠵⠷⠶⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠉⡑
⠀⣴⠟⠁⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠛⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢠⠃
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠋⠃⠀⠀⡀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡀⢀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠤⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠋⣁⣵⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⠀⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣶⠿⢛⠉⠐⣨⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡴⠃⢠⣤⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡬⢔⣮⣵⡄⠀⠁⢠⠈⢆⠀⢀⡀⠀⠀⠀⠀⠀⢈⠇⠀⠨⠣⣀
⠀⠀⠀⠀⠀⠘⢷⣀⠀⠛⠁⠀⠀⠂⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠛⠻⠃⠀⠀⠀⠀⠀⠎⢠⠷⠀⠀⠀⠀⡀⠉⠀⠉⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠈⠀⠀⠈⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⢳⡙⠤⠁⠀⠀⠀⠐⠂⠀⡂⠈⠑⢌⠀⡰⠂⠄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⣀⠀⠀⡀⠐⣁⡀⠀⠀⢨⡇⣶⣤⣤⡆⠀⠀⠔⠀⣀⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡷⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣼⣦⡀⠘⠀⠁⠀⠀⣼⠀⠈⠀⢠⢀⣨⠄⢀⣾⣭⣭⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡏⠩⣭⠅⠘⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⢻⡀⢨⣤⠈⠁⠀⠀⠀⢤⠴⠞⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣌⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢄⠀⠠⠤⠀⣽⠁⣾⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠛⠉⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢰⡙⠛⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠁⢊⠀⠀⠀⠁⠀⠀⢀⣿⢹⡏⣹⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢻⠇⠘⣦⠀⡀⠶⠷⣥⣾⠖⠛⠁⠀⠀⠀⠀⠀⠀⢀⣸⣥⡤⠂⠹⣤⠴⠤⠀⠀⠀⠀⠀⠀⠀⢠⡴⡴⠂⠀⠀⠱⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠾⠛⠇⠈⠁⠉⠀⠟⡻⠖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡚⠤⣀⠀⠀⠀⠀⠀⠀⡠⠼⢚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣾⢷⠯⣭⡑⠄⠳⡀⠀⠀⢠⠊⡠⢊⡭⢗⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠋⠙⠄⠻⠿⣦⣀⣇⡀⢀⣨⣠⡴⠿⠋⠸⠉⠛⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⢄⡈⠉⠀⠀⠉⠉⠀⠀⠀⠀⠉⡉⠀⠈⠀⣉⢄⡾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣷⡚⢷⣄⠀⢤⢀⠀⠠⠀⠀⣈⠄⢀⣴⡞⢳⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢿⣻⣷⡀⠹⣷⣌⠀⠁⠀⠀⠀⠀⣡⢿⠋⣠⣾⣻⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣿⣿⣆⠙⣟⣧⢀⣀⡀⡀⡴⣻⠏⣰⣿⡿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢯⣽⣦⡈⠚⠺⠺⠧⠓⠓⠁⣰⣯⡝⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠛⠿⣿⣦⣤⣤⣤⣤⣴⡾⢟⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠻⠿⠿⠿⠿⠞⠛⠉

⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⢤⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠃⠠⢻⡧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠉⠙⠛⠈⠉⢧⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠼⠁⠀⠀⠀⠀⠀⠀⢻⣥⠤⠖⠶⠚⠋⠉⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠖⠒⠒⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠞⢱⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠠⣿⣦⣄⡀
⠀⣀⣀⠠⠤⠀⠂⠈⠁⠀⠀⠀⡰⠁⠘⣿⣿⣿⡆⠀⢀⠃⠀⠀⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣿⣿⣿⣿⡄⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠹⣿⣷⡂⠀⢸⡄⠀⠀⡀⣀⠀⠀⣿⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢸⣿⣿⣇⠀⣽⡿⠀⠀⠈⡇⠀⠀⢱⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣾⣿⠇⠀⠀⠀⡇⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠙⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠙⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⣿⣿⠆⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣇⡼⣿⣿⡇⠀⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⡇⠀⠀⢀⢸⢃⠀⠀⢸⣿⠙⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣞⣿⣿⣿⢹⢃⣿⣿⣿⣴⣿⣶⠀⠀⠀⢸⣿⣿⣿⠁⠀⠈⢳⠀⠀⢸⡆⡞⠀⠀⠼⠋⠀⠙⣿⣿⣿
⠀⠀⢠⣿⣿⣿⡄⣿⣿⣿⠀⠀⣿⣿⢉⣿⣿⡿⡀⠀⠀⢸⣿⣿⡟⠀⠀⠀⢻⠀⠀⢸⣿⣿⣀⢠⣷⣶⠀⣶⣿⣿⣿
⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⠀⠀⢿⡿⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡗⣤⣽⣽⠀⡄⢨⣽⣿⣤⢸⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠸⠃⠇⠀⠟⠉⠳⠀⠀⠞⠳⠀⠘⠛⠇⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣤⣷⣼⣿⣿⣵⣌⣿⣧⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿