Drum Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣉⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣉⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣋⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢉⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠾⠋⣠⣶⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢉⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠖⢛⣛⣛⠉⣩⠿⠋⢁⣴⠟⠉⠉⠙⠛⠛⠛⠛⠛⠶⠶⠶⠶⠶⣦⣤⠞⠋⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠛⠉⠁⠀⠀⣴⠚⠉⠉⠉⠻⣇⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⠶⠋⠀⣠⡶⠋⠛⠛⠶⢦⣤⣀
⠀⠀⢀⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠙⢧⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿
⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⢸
⣿⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⣸
⣿⢷⡄⠀⠙⠳⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠞⠛⠉⠀⠀⠀⠀⣠⡾⣿
⢻⠀⠙⢧⣄⠀⠀⠀⠉⠉⠛⠛⠳⠶⠶⢦⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠶⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⢠⣿⡀
⢸⠀⠀⠀⠈⢻⣶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠁⠀⠀⠀⠀⣸⣻⠁
⢸⡄⠀⠀⢀⡏⢸⠀⠈⠙⠛⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⠒⠛⠋⠉⢰⠂⢧⠀⠀⠀⠀⠀⠀⢀⡏⣿
⢸⡇⠀⠀⣼⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⣧⠀⠀⠀⠀⠀⢸⠇⣿
⢸⡇⠀⢰⠇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠘⣆⠀⠀⠀⠀⣾⠀⡿
⢸⡇⠀⡾⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠸⡆⠀⠀⠀⣿⢘⡇
⠈⡇⢸⠇⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡄⠀⢀⡏⢸⡇
⠀⣧⡟⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡼⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡇⢸⡇
⠀⣿⢧⡀⠀⠀⠀⠀⠀⢷⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠁⢸⠃
⠀⢻⡄⠻⢦⡀⠀⠀⠀⠸⡆⢠⠏⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⢀⣿
⠀⠈⢻⣄⠀⠙⠳⣄⡀⠀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢾⡟
⠀⠀⠀⠙⠳⣤⡀⠀⠙⠓⠶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠉⠙⠓⠶⢤⣄⣀⡀⠀⠘⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡞⣀⣀⣠⡤⠴⠖⠛⠉⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⡀⠀⠀⠈⠉⠉⠛⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⠋⠉⠉⠀⢀⣀⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠞⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⠏
⢠⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣰⠋
⠈⠳⢤⡉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡼⠃
⠀⠀⠀⠙⠶⣄⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⠞⠁
⠀⠀⠀⠀⠀⠈⠙⠦⣄⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣦⣝⠲⣄⣀⣀⣀⣀⣀⣀⣀⣀⡼⣡⣏⡀
⠀⠀⣀⡤⠤⢔⣒⣺⣿⠿⠭⠿⠿⠿⠛⠛⢿⠉⠙⢿⡟⠛⢛⡿⠋⢻⠿⠿⠿⠭⠿⣿⣗⣒⡢⠤⢄⣀
⡴⠋⠁⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡽⠀⡾⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⠈⠙
⡟⠲⢤⣙⡲⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⢖⣋⡤⠖
⡇⠀⠀⠈⠉⠉⠛⠓⠺⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⠿⠿⠿⠿⠓⠚⠛⠉⠉⠁
⣿⡄
⡟⢳⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣾
⡇⠈⢷⠀⠉⠉⠙⠛⠒⠲⠶⣶⠧⠤⢷⡤⠤⣤⣤⣤⣤⣤⡤⠤⣤⡯⠤⢷⠶⠶⠖⠒⠛⠋⠉⠉⠀⣰⠃
⡇⠀⠈⣧⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣰⠇
⣇⠀⠀⠘⣆⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢠⠏
⡟⠓⠦⣤⣸⣄⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣠⣏⣤⠴⠚
⡇⠀⠀⠀⠀⢹⡍⢹⠟⠒⠒⠶⠶⠶⠦⠤⠤⠼⣧⠤⢤⡾⠤⠤⠤⠴⠶⠶⠶⠒⠚⣿⠋⢩⡟
⣇⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡞
⠈⠓⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣠⡤⠴⠚
⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣒⣒⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣒⣒⣛⣛⣉⣉⣁⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣈⡀⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣧⣾⣿⡣⣿⣿⣾⣿⣿⢸⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⢸⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣻⣶⣿
⠀⠀⠀⢻⣿⣷⣤⣤⡠⣶⣶⣶⣾⣿⡇⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣼⣿⠏⠛⣛⢻⡟⠿⠙⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⠉⠋⠙⠛⠟⡻⢻⣿⣷⡍⠛⠃⠀⠀⠀⠀⠈⡟⢻⠛⡟⠃
⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣷⠀⠀⣀⡴⠷⠛⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⣇⡾⠀⡇
⠀⠀⠀⠘⠛⠟⢿⣿⣿⣿⣿⣿⣿⡗⢀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⡥⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⣹⡁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⡇⡟⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠈⠁⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡟⠓⢄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠐⠂⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈
⠀⠀⠀⠰⠊⠁⠐⠀⠀⠀⠀⠀⢠⠃⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠙⠶⣤⣄⣀⣀⣀⣤⠴⠚⠁⠀⠀⠸⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣈⣁⣀⣀⣀⣐⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⣆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⡿⠯⠽⠿⠛⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣬⡙⠛⠿⣿⡷⣶⢶⣦⣤⣤⣤⣤⣤⣤⠴⠶⠚⠛⠉⠁⠀⣡⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣄⣒⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⢹⠛⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣗⡋⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣟⡺⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢻⡿⣿⡏⠀⠀⠄⠸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⠃⠀⠀⠀⠸⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⡇⠊⠀⣿⣍⠐⠂⠠⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠘⣿⣷⢶⣤⠄⠈⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣀⡴⠛⡅⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠸⣤⣉⠛⠿⣾⣤⣽⣧⣄⣀⣀⣀⣀⣀⣀⣹⣧⡴⠶⠛⠉⢀⣴⠇⠀⠀⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠷⣧⣀⡑⠈⠉⠙⠩⠍⢉⡉⠉⠉⠉⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⣠⢄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⠤⠤⠶⠴⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⣠⣾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠃⢹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢧⠠⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⣰⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠄
⠀⠀⠀⠀⠠⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠄
⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠘⡄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠘⡄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠠⡀⠀⠀⠘⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⡀⠀⠘⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠦⡈⠢⢤⣘⡄⠑⣖⣒⣂⣀⣀⣀⡀⠤⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠨⠑⠒⠀⠉⠁⠈⢆⠀⠾⢧⡐⢿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠊⠥⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠣⣀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠒⠈⠥⠖⢂⣀⢀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⣐⠒⠤⠍⠒⠈⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠡⢒⡀⠤⠀⠀⣀⣀⡀⡉⠉⠉⠉⠉⠉⠁⠁⠀⣀⣀⠀⠀⠤⢀⡐⠠⠑⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠜⢂⢀⠔⢄⠉⠰⢆⠐⢐⠶⠂⠂⢐⡆⠒⠐⣀⠒⠀⢀⢀⠉⠀⡠⢄⢀⠒⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠈⠁⠀⠀⠑⠁⠀⠑⠐⠁⠈⠢⠊⠀⠀⠒⠁⠀⠑⠊⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡟⡀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣇⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⠀⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠀⣿⠸⡀⠀⠀⠀⠀⠀⠀⣾⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢰⡏⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⠀⣾⠁⠀⢱⠀⠀⠀⠀⠀⣼⠃⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢹⡆⠀⠀⠀⢰⡏⠀⠀⠘⡄⠀⠀⠀⢠⡟⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣷⠀⠀⠀⣼⠃⠀⠀⠀⡇⠀⠀⠀⣸⠃⠀⠀⢸⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡇⠀⠀⠀⢻⡄⠀⢀⡿⠀⠀⠀⠀⢸⠀⠀⢀⡿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠘⣧⠀⢸⠇⠀⠀⠀⠀⠈⡆⠀⣸⠇⠀⠀⠀⠀⣧⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⢿⡀⣿⠀⠀⠀⠀⠀⠀⢃⠀⡿⠀⠀⠀⠀⠀⢿⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠸⠛⡇⠀⠀⠀⠀⠀⠀⢸⠊⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠳⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉⠀⠄⣀⠀⠀⢀⠠⠐⠉⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣷⣼⣼⣧⣿⣤⣼⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣿⣶⣿⣿⣶⣶⣶⣶⣾⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠒⠤⡀⠈⠀⠐⠢⠀⠀⠀⠤⠤⠀⠀⠀⠔⠂⠀⠀⢀⠤⡒⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠄⢃⠃⠀⠀⠀⠤⠤⠀⠀⠀⠘⡘⠨⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡐⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠅⢐⠐⠂⠀⠀⠐⢐⡈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢁⠀⠁⠈⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢂⠀⠀⡈⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡐⠀⠀⠀⠀⢂⡐⠀⠀⠀⠔⢀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠄⠁⢈⠁⠡⠄⠀⢈⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣂⠄⡀⠀⠠⠀⠀⢐⠠⡄⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠐⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⠊⠄⠀⠉⠢⣄⠄⣀⣀⡀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⢀⢀⡉⡳⠆⣈⣿⣧⣭⣿⡷⠶⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣸⡙⠀⠐⡆⠀⠀⠉⠉⠩⠥⡤⣴⣶⣾⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠙⣶⢴⢧⣀⡀⠀⣀⣀⣼⣧⣿⣿⣿⡷⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠅⡿⡎⠋⠛⠛⠛⢻⢻⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⡆⢸⠃⢣⠀⠀⠀⢀⠇⠈⣿⡧⢻⣿⣿⣞⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⡞⠀⠸⡀⠀⠀⡼⠀⠀⣹⣿⣎⢿⣿⣿⢿⣿⡄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢇⠀⠰⠃⠀⠀⢹⣿⣿⣎⠻⣿⠘⢿⣿⡆⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠸⡀⡎⠀⠀⠀⢸⣿⣿⣿⣷⡝⢶⣤⣻⣿⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⣿⡏⣽⠒⠂⠀⠠⢿⡄⠀⠀⠀⠴⣿⣟⣽⣿⣿⣯⣿⣟⣻⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠻⠿⢁⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣡⣤⣷⣛⣯⣭⣭⣭⣭⣿⣽⠾⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠉⠲⣄⢀⡤⣄⠀⠀⣀⣀⣀⣀⣀⣰⣲⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠶⣄⡀⠙⠿⣍⠉⠉⠁⠀⠀⠀⠀⡴⠛⡌⠉⠉⠉⠑⠚⣳⣾⢤⣀⣀⣠⠤⠶⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣍⠁⠀⢀⣀⣉⡶⢤⡈⠓⢶⡒⠒⠒⠒⠒⠛⠛⠓⠶⠶⠤⢤⣼⡧⠼⠒⠉⠁⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣧⠼⠒⠋⠉⠀⠀⠀⠀⠈⠓⢦⣉⠳⢤⡀⠀⠀⣀⡤⠴⠖⠋⢉⣀⣠⠤⠴⠒⠛⠿⣥⠟⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢮⡵⠶⠛⣋⣉⣤⠤⣖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢠⠖⠚⠉⠙⣆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡇⠀⠀⠈⠙⠒⠶⠤⡤⢤⣀⣈⡓⠚⠉⠀⠀⠀⠀⠘⠦⣤⡔⣣⣄⣀⡠⠤⠴⠒⠋⠉⠀⠀⠀⣇⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⣆⠀⠀⠀⠀⠀⠀⡇⢸⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡏⠉⠒⠦⢄⣀⣀⣇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⣀⡤⠔⠚⠉⢸⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢷⠀⠀⠀⠀⠀⢻⠀⠀⣹⠦⠤⠤⠤⢤⡤⠤⣤⠤⠤⠤⡞⠉⠉⡗⠚⠉⠁⠀⠀⠀⠀⠀⡞⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠀⠀⠀⠀⡸⢦⣴⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣀⡴⢣⠀⠀⠀⠀⠀⠀⠀⢠⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⠀⢠⠇⡼⠹⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠏⠹⡄⢳⡀⠀⠀⠀⠀⠀⣸⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⢹⠀⠀⠀⡞⢰⠁⠀⢣⠘⡆⠀⠀⠀⠀⠀⠀⠀⡸⢁⡏⠀⠀⠘⣄⢳⡀⠀⠀⠀⠀⡇⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷⠸⡆⠀⣸⢁⡏⠀⠀⠘⡆⢳⠀⠀⠀⠀⠀⠀⣰⠃⡞⠀⠀⠀⠀⠘⣄⠳⡀⠀⠀⢸⠃⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⣇⢠⠇⡼⠀⠀⠀⠀⢹⡈⣇⠀⠀⠀⠀⢠⠇⡼⠀⠀⠀⠀⠀⠀⠘⣆⠱⡀⠀⡼⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⡞⢰⠃⠀⠀⠀⠀⠀⢧⠸⡄⠀⠀⢀⠏⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣄⡇⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢹⣸⢁⡏⠀⠀⠀⠀⠀⠀⠸⡆⢳⠀⢀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠹⣄⡇⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢾⡋⣻⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣇⡼⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣉⠷⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠉⠛⠦⢄⣀⠀⠀⠀⠀⠀⠈⣇⣸⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠊⠁⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠈⠉⠓⠲⠦⠤⠤⣼⣉⣿⣀⣀⣀⣀⣤⠤⠤⠴⠒⠚⠉⠁⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢄⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢵⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢷⣤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠲⠟⠉⠹⣏⠉⠉⠉⠉⠍⠛⢛⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠷⢧⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣮⡭⢷⡏⣿⡴⢚⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡟⠋⣉⣉⣉⣤⠤⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠀⠐⠀⠉⠉⠭⢹⢻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠽⠇⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠰⢖⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣵⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣐⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⡭⠷⢿⣀⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣭⣩⣿⣿⣇⣸⢟⢛⣋⣉⣉⣥⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⢠⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠈⠓⠲⠤⣄⣀⡉⠛⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⣌⣉⡓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡬⠽⠿⠒⠦⢬⣍⣛⠻⢯⣩⣿⣝⠛⠒⠒⠲⠦⠤⢤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣙⣦⣀⣈⣇⣀⡀⠀⠀⠀⠀⢀⣈⣩⠷⠶⠒⠋⢉⣁⣠⡤⠴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣤⠴⠖⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣩⠭⠟⠛⣋⣉⡤⣤⡶⠒⠛⠹⡍
⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⣒⣫⠭⠵⠖⠚⠋⠉⠁⠀⠀⠀⠈⠙⠢⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣀⠈⠙⠒⣦⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⢤⣖⠚⠉⢁⣠⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠲⢾⣉⣇⣀⠀⠈⠉⠉⠉⠉⠉⢉⡽⣏⠉⠉⠉⠉⠉⠉⠀⢀⣀⣏⣉⠷⠒⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢹⡇⠉⠉⠉⠛⠒⠒⠒⠲⠾⡦⡾⠶⠒⠒⠒⠒⠛⠉⠉⠁⢹⢹⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⣠⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠙⠲⢼⠾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢼⡶⠚⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣽⣴⠋⠉⠉⠙⠓⠒⠒⠒⠒⢲⠗⢳⠖⠒⠒⠒⠒⠛⠉⠉⠉⠳⣴⣧⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠦⠤⠤⠤⠬⠷⠯⠤⠤⠤⠤⠴⠶⠖⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢀⣨⠦⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠴⣅⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢋⡂⠔⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠦⢐⠝⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠐⠠⠜⢠⠖⢢⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡔⠲⡄⠣⠄⠂⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠐⣡⡜⠋⠀⢑⣦⢀⠀⠀⠀⠀⡀⣴⡈⠀⠙⢷⣌⠢⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣔⡁⡎⠀⠀⠀⠀⠀⠀⠀⠐⠋⠋⠻⡀⣀⠴⠋⠀⢁⣥⠀⠀⣨⡁⠀⠙⢦⣀⢠⠟⠋⠙⠆⠀⠀⠀⠀⠀⠀⠀⢱⢈⣢
⠀⠀⠀⠀⠀⠀⠘⠁⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠀⠀⠊⢄⢀⣴⣏⡱⠂⠐⢎⣹⣦⡀⣀⠑⠁⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠳⣔⠠⡀⠙⣋⡠⠓⢱⠀⠀⡮⠚⢄⣹⠋⢀⠄⣪⣊⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠤⠀⠀⠀⣀⢀⠀⠀⠠⠔⢢⡝⠋⡀⠄⢾⠒⠒⡷⠠⢀⠙⠌⠐⠟⣀⠤⠤⠤⠤⠤⣄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠂⠀⠀⠖⠂⠀⠀⠛⠉⠀⠖⣡⡴⠖⠒⠉⠉⠒⠲⢆⢌⠢⢀⠀⢸⠀⠀⢹⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⡇⠀⠀⠀⠀⡴⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣌⢧⠞⢂⣀⣚⣀⣀⠺⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠸⢅⡓⠂⠀⠀⠓⠂⠀⠀⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠶⠓⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠘⡛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⡐⡀⠀⠁⠀⠀⠀⠀⠀⠠⢛⢛⢃
⠀⠀⠀⠀⠀⠀⡐⡐⠀⠘⠸⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡁⢰⡔⠀⠀⠀⢠⡄⠀⠀⠀⠀⠃⠀⢂⠂
⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠟⢦⡄⡀⠀⠀⠀⠀⢀⣠⠴⠋⠑⡐⡈⠀⠀⠀⠀⠀⠁⠀⠀⣴⠀⠀⠀⠀⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁⠊⠉⠉⠙⠈⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣢⣄⣀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢂⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⣄⣬⣄⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣭⣿⣯⣭⣿⣿⣩⣭⣬⣿⢭⣽⣿⣿⣯⣭⣿⣷⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠿⠿⣿⡿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢼⣿⡀⠈⣿⡡⠒⠁⢸⣿⠀⢹⣿⠬⣿⣿⣿⣿⣾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠸⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⢈⣿⢋⣩⠴⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣿⠀⠀⢿⠀⠀⠀⢸⡇⠀⢀⣿⠀⢴⣄⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⣿⠒⠶⠀⣿⡟⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢿⡄⠀⢙⡇⠀⠀⢸⡇⠀⠈⣿⠸⡅⣸⣿⠃⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣇⠀⢸⡇⠀⠀⢸⡇⠀⢸⣧⢶⣿⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⡄⢸⣧⠀⠀⢸⡇⣄⣿⡟⢛⣻⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⣧⠨⣿⠀⠀⣸⣷⣰⣿⣷⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣷⣿⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⣴⡿⠹⣿⣿⣿⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠐⠂⢈⡿⣛⣿⣿⣿⡕⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣄⣀⣀⡀⣿⣽⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠉⠉⠉⢉⡛⣽⣿⡿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣿⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⣔⣻⣿⡅⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣅⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣷⢚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣶⣤⣄⣀⣀⣀⣀⣀⣰⣴⣿⣿⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠽⣿⣷⡆⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⡤⠐⣠⣤⣼⣾⣿⢗⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⢤⣤⣄⣀⣀⣡⣤⣾⠷⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠠⣿⠉⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠉⣿
⠀⠀⠀⠀⠉⠛⢶⣤⣀⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⣀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⢁⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠉⠻⢶⣤⣠⡶⠟⠉⣀⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠽⣿⣶⠾⠋⢁⣠⣶⣿⠯⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⢀⣤⠴⢒⣋⣭⣭⣶⣶⣾⡿⠟⠉⣠⣴⣾⣿⣧⣄⠉⠻⢿⣷⣶⣶⣭⣭⣙⡒⠦⣤⣀
⠀⠀⣠⠞⣩⣶⣿⣿⡿⢿⣿⠟⠛⣉⣤⡶⠛⠉⠉⠉⠉⠉⠉⠛⢶⣤⣈⠛⠻⣿⡿⢿⣿⣿⣶⣍⠳⣤
⠀⣼⠁⣾⡿⠛⠉⠀⠀⠸⣧⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣼⠇⠀⠀⠉⠛⢿⣷⠈⣧
⠀⣿⣄⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣠⣿
⠀⢿⡈⣳⣮⣙⡲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⢖⣋⣵⣞⢁⡿
⠀⠀⠙⡿⠛⢻⡏⠓⠲⠮⢭⣝⣛⣒⣒⠶⠶⠶⠦⠦⠴⠶⠶⠶⢶⣒⣒⣛⣫⡭⠵⠖⠚⢹⡟⠛⢿⠋
⠀⠀⠀⣿⠛⣿⣷⣦⣤⣄⣀⣀⢸⡿⠿⢿⠉⠉⠉⠉⠉⠉⠉⠉⡿⠿⢿⡇⣀⣀⣠⣤⣴⣾⣿⠛⣿
⠀⢀⡴⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢦⡀
⠀⣾⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⣷
⠀⣿⢦⣹⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣏⡴⣿
⠀⠻⣄⣽⡀⣿⣙⡻⠿⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⠿⢟⣋⣿⢀⣯⣠⠟
⠀⠀⠙⣟⡛⣹⡏⠉⠙⠒⠲⠮⢭⣿⠀⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⠀⣿⡭⠵⠶⠒⠋⠉⢹⣏⢛⣻⠉
⠀⠀⠀⠈⠉⠉⠙⠓⠶⠦⣤⣤⣸⡟⠛⢻⣀⠀⠀⠈⠁⠁⢀⣀⡟⠛⢻⣇⣤⡤⠴⠶⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⡀⠀⠀⠀⠀⢀⠀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠠⠤⠌⠀⡒⠁⠂⠤⣁⠂⠠⠀⢐⢀⡉⡡⠄⠐⠒⠐⠊⠓⠦⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠐⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠒⢀⠀⠀⠀⠀⠀⠀⠺⠤⣀⠐⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠈⠀⠀⠀⠤⠤⠭⠐⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠠⣥⣁⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⡘⡀⠤⠀⠀⠒⠀⠡⠤⠀⠀⠀⠈⠐⠲⠄⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠐⠈⠉⠓⠒⢠⠇⠀⠀⠤⠄⠀⡥⠐⠒⠒⣂⠈⠉⠡⠐⠃⠀⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣄⣀⣀⡀⠁⠒⠠⡄⠤⠐⠂⠀⠒⠒⠒⠢⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄⠂⠂⡃⣀⠠⡥⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠞⠁⠠⠖⠋⠁⠀⠈⠁⠐⠠⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠤⠘⠢⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡂⠩⠠⠀⢐⠀⠈⠅⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⢀⢤⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⠀⠒⠒⠀⠀⠌⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⡄⢁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠔⢀⣤⡄⠆⡀⠀⢠⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠎⠐⠀⠀⠀⠀⠈⠂⣉⠀⢖⠁⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠉⠁⠀⢐⡢⠀⠀⢚⠀⡀⣉⠂⠀⠍⠁⢀⣀⣀⠈⠤⠶⠚⠋⠀⣀⠀⠠⡀⠀⠀⠠⠤⡐⡄⠤⠄⠠⠆⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠠⠄⠃⠂⠐⠀⠀⠀⠀⠀⠀⠂⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⣤⣤⣴⣶⣶⣿⣿⣶⣶⣾⣿⣷⣶⣶⣤⣤⣄⡄⣀
⠀⠀⢶⠈⠀⡆⠁⠀⠀⠀⠀⡀⣔⣾⣿⠿⠟⠛⠋⠉⠉⠀⣠⣶⣶⣶⣶⣦⡀⠈⠉⠉⠙⠛⠿⢿⣿⣖⡤⡀
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢜⣾⠟⠉⠀⠀⠀⠀⠀⢀⣤⡀⠘⠳⢶⣭⡶⠚⠃⠀⣀⣀⠀⠀⠀⠀⠈⠙⠿⣎⠄
⠀⠀⢸⠀⠀⡇⡆⠀⠀⠰⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⣦⣤⣤⣶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠰⡀
⠀⠀⢸⠀⠀⡇⡇⠀⠀⢧⢷⡄⠀⠀⠀⠀⠀⠀⠀⣶⣶⡶⠞⠋⠉⠀⠀⠉⠉⠛⠲⢶⣶⠄⠀⠀⠀⠀⠀⣴⢧⡅
⠀⠀⢸⠀⠀⡇⡇⠀⠀⠸⣄⠀⠀⠀⠀⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣼
⠀⠀⢸⠀⠀⡇⡇⠀⠀⢸⢹⣿⣦⣄⣀⠉⠛⢻⡿⠶⠶⢶⣶⣶⣦⣤⣶⣶⣶⠶⠶⠾⣿⠛⠋⢁⣀⣤⣶⣿⣿
⠀⠀⢸⠀⠀⡇⠁⠀⠀⢸⢸⣿⣿⣿⣿⣿⣷⣾⣷⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣾⣶⣿⣿⣿⣿⣿⣿⢸
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⡼
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⠟⡃
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠐⢄⠙⠻⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⠛⢁⠔⠁
⠀⠀⢸⡀⠀⣇⡀⠀⠀⠀⠀⠀⠑⠢⣄⡈⠙⠛⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡟⠛⠉⣁⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠷⠤⠤⠤⣤⣄⣀⣀⣀⣤⡤⠤⠤⠴⠗⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡇⠀⣀⣤⣀⢀⣀⣀⡀⢀⣄⣤⢀⣤⣤⢀⣤⣤⣄⢤⣤⣤⣤⣤⣼⣤⡽⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢙⣾⣿⢹⣿⣿⣷⣾⣿⣿⠸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠈⣷⣾⣿⣞⣉⣠⢠⣤⣤⣼⣤⡼⣿⣿⢸⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⢸⣿⣿⠻⠿⡿⢿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡛⣿⢸⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⣟⠛⠛⠙⠛⠁⠀⢠⣷⣾⣿⣶⣵⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠁⣌⣡⣿⣿⣿⣧⣀⡄⠁⠀⠀⠀⠀⠀⠀⡇⢸⠁⢸⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠃⣀⣀⣤⠾⠛⠋⠉⠉⠉⠉⠛⠻⢶⣄⣣⠀⠀⠀⠀⣧⣿⠑⣸⠃
⠀⠀⠀⠀⠘⠛⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢀⣿⡾⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠛⠛⠛⠛⡟⠉⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣴⣿⡋⣿⠉⠙⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡤⠖⢻⠓⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⢯⣿⡀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠍⣀⠀⢘⡠⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣸⠀⠀⣙⠢⢀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⠔⡋⠀⠀⠀⠀⠈⠉⠀⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⠋⠉⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠋⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⠃⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠉⠳⣦⣄⣀⡀⠀⠀⣀⣀⣤⡴⠟⠉⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠀⢰⠎

⠀⠀⠀⠀⠀⠘⠻⣿⣿⣷⣶⣤⣄⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⠛⠿⠿⢿⡿⠛⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣴⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠿⣿⣿⠿⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤⣤⣤⣤⣤⣴⣯⡈⢢⡀⠀⠀⠀⣰⣿⣿⣾⣦⣤⣤⣤⣤⣤⣼⣀⠑⣀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣃⣀⣀⣀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⠙⡏⠀⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⡷⠾⣿⠿⢿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣯⠀⢸⠀⠀⠀⠀⣽⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⡀⠀⡇⠀⠀⠀⠀⡿⠛⠉
⠀⠀⣀⣤⢲⠒⠀⠈⢸⣿⣿⣧⣿⣤⣴⣶⣤⣤⣤⣿⡀⢺⣄⣀⣀⣀⣾⣿⣿⣼⣦⣤⣶⣦⣤⣤⣼⣁⠐⣧⠀⠀⠀⠀⡇
⠀⣼⣿⣿⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠭⠭⠭⠭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣿⣍⣠⡄
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡿⣻⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠂⠈⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣾⢫⠎⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠙⢎⢿⣿⣿⢿⠿⠿⠿⠿⠿⠿⣧⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⡱⠁⠀⠀⠀⠀⠀⠀⢰⣼⣷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢣⢻⣿⣾⣷⣶⣿⣷⣶⣶⣿⣦⣼⠇
⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⢯⣿⢧⠁⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⢣⣿⡟⠛⠛⣿⣿⠛⠛⠛⡏⠁
⠀⣸⣿⣿⣸⣄⣤⣤⣤⣀⣰⣿⠚⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⢸⣼⡇⠀⠀⣿⣿⡀⠀⢀⣧⡀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⢈⠿⢿⣿⣿⠿⣉⠀⠀⠀⠀⠀⠀⠀⢸⢿⡏⠀⠈⣿⣿⠁⣰⠏⡏⠳⣄
⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⢋⣿⣹⠀⠀⠀⠀⠀⠀⠀⠘⠂⢺⣏⣿⠲⠞⠀⠀⠀⠀⠀⠀⠀⡸⣾⣏⠢⣀⣿⣿⡞⠁⠀⠇⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠁⣏⠔⠁⠹⣧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡿⠀⣱⡌⠛⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⣯⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡿⢁⣾⠟⠃⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢿⣮⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣪⡾⠋⣴⡿⠋⠀⢂⡁⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠉⠻⢶⣭⣒⡤⠤⠄⣀⠀⠤⢤⣐⣮⣵⠿⠋⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣦⣄⣈⠙⠛⠶⣤⣄⡀⠀⣀⣠⡴⠶⠛⢉⣁⣤⡴⠞⠁
⠀⠀⢀⣀⣤⡴⠶⠖⠛⠛⠛⣉⣉⣉⠛⠳⣦⣤⣈⠙⠛⠿⢤⣤⣶⣾⣯⣭⣍⠛⠛⠓⠶⠶⢤⣄⣀
⣠⠞⣋⣥⡴⠶⠚⠛⠋⢁⡾⠋⠉⠉⠻⣟⢉⣠⣭⠿⠳⠦⣤⣈⣿⠋⠀⠀⠈⢷⡀⠙⠛⠲⠶⣤⣍⡛⢦
⣿⣄⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿⠋⠁⠀⠀⠀⠀⠀⠉⢿⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢁⣼
⣿⠉⠛⠶⣤⣤⣀⣀⠀⠀⠛⠦⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠁⠀⢀⣀⣠⣤⡴⠞⠋⢹
⣿⢦⣄⡀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⠋⠉⠉⠁⠀⠀⣀⣤⢾
⣿⠀⢈⣽⣿⠳⠶⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠶⠶⣿⣿⣍⠀⢸
⣿⠀⢸⣇⣸⡇⠀⣀⡀⠀⠀⠈⣩⣭⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣭⣍⡉⠀⠀⠀⣀⠀⠀⣿⣀⣿⠀⢸
⣿⠀⠀⠉⠉⠀⣰⠟⢷⡀⠀⠰⣏⠀⣹⠇⠀⠀⣠⠟⢧⡀⠀⠀⣿⡀⢈⡗⠀⠀⣰⠟⢷⡀⠈⠉⠁⠀⢸
⣿⣤⠀⠀⠀⣼⠃⠀⠈⢷⡀⠀⠉⠛⠉⠀⢀⡼⠋⠀⠈⢻⣄⠀⠈⠙⠋⠀⠀⣰⠏⠀⠈⢳⡄⠀⠀⢰⢾
⣿⠸⡇⢀⡾⠃⣴⠛⣶⠈⢷⡀⠀⠀⠀⣠⠟⠁⣠⡴⣤⡀⠙⣦⡀⠀⠀⠀⣰⠏⣰⠟⢷⡀⢻⡄⠀⡿⢸
⣿⠳⢦⣾⣁⠀⠻⣤⠟⠀⠈⢷⠀⢀⣼⠋⠀⠀⢿⣀⣸⡇⠀⠈⢷⡄⠀⢰⠏⠀⠹⣦⠾⠁⢀⣀⣤⠶⢻
⢿⣀⠀⠈⠉⠛⠛⠲⠶⠶⢤⣤⣤⣤⣥⣤⣄⣀⣀⣉⣁⣀⣀⣤⣤⣤⣤⣤⡤⠴⠶⠶⠚⠛⠋⠉⠀⢀⣸
⠀⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠄⢴⣧⣤⣤⠄⠀⠈⢓⣒⣶⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠐⠀⠀⣵⣟⡈⠀⠀⠀⣀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠄⠀⠀⠀⡘⠀⣰⣿⠟⠀⠀⠀⠀⡅⠀⢸⣶⣖⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁⠀⠙⢻⠗⠁⠀⠀⢰⡄⣤⣿⣿⠂⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⢹⡟⠒⠀⠀⠀⠐⠉⢹⢃⠀⠀⣸⣶⢲⣶⠶⠆⠀⠀⠀⣹⣿⣃⡀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠀⢀⣼⠴⠒⠟⠿⢿⡞⠃⠀⠀⠀⠀⣤⢿⠉⠛⠒⣶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡦⡄⡀⠴⠂⠀⠈⢄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠻⢧⠄⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠀⢸⣆⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣯⠈⢂⣦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⣾⣿⢉⡤⡾⡿⠲⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠣⠏⢱⢳⣇⡴⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡋⠀⠀⠀⠀⠾⠀⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠛⠉⠁⠀⠙⠦⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠝⠋⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠵⠊⠁⠀⠀⠀⠀⠀⣠⠔⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡽⠚⠁⠀⠀⠀⠀⠀⢀⡤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⢖⣩⢾⣥⣀⣀⣀⣀⣀⢀⡠⢖⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢉⠔⠛⠓⣯⠔⠋⠀⠀⠀⠀⣀⣠⣠⠝⢋⠴⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠸⣄⠀⣀⡜⠀⠀⠀⠀⠀⣼⠁⠀⠈⡟⠁⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⡇
⠀⠀⠀⠀⢸⡷⣌⠙⠲⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠚⠉⢀⣠⠔⡇
⠀⠀⠀⠀⠀⡇⠈⢑⡦⢄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⡠⠤⡖⠊⠉⠀⢠⡇
⠀⠀⠀⠀⠀⡇⠀⡆⢡⠀⠀⠉⠉⢉⠗⠒⠒⠒⠒⠒⠒⡚⢏⠉⠉⠉⠉⠀⠀⠂⢱⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠘⡆⠀⠀⢀⠎⠀⡇⠀⠀⠀⠀⡰⠁⠘⡄⠀⠀⠀⠀⢸⠀⠀⢣⠀⠀⣇⡇
⠀⠀⠀⠀⠀⣧⠇⠀⠀⢣⠀⠀⡌⠀⠀⢃⠀⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⢀⠇⠀⠀⠈⡇⢰⢹⡇
⠀⠀⠀⠀⠀⢻⢦⡀⠀⠘⡄⡸⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⠀⢳⠀⠀⡘⠀⠀⠀⠀⠸⣸⢸⠁
⠀⠀⠀⠀⠀⠘⢦⡙⠢⣄⣁⠁⠀⠀⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⣀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣉⠑⠲⠤⢄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⠤⠤⠒⣉⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠐⣧⡈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠈⠛⢦⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠙⢦⣤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⢲⣶⠋⣠⣾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠑⠦⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠞⠁⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠑⠦⣙⠲⣄⡀⠀⠀⠀⢰⠟⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡙⠓⣆⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠉⠛⠓⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠙⠳⢦⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠶⠟⢩⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡄⠀⢀⠌⠉⢟⠛⠒⠶⠶⠶⣶⠶⢶⡶⠶⠶⠶⠶⠶⢻⠟⠛⢍⠀⠀⢀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⡴⠁⠀⠀⠈⢦⠀⠀⢀⡔⠁⠀⠀⠑⢄⠀⠀⠀⡰⠃⠀⠀⠈⠳⣄⡼⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠊⢣⠀⠀⠀⠀⠈⢧⡠⠋⠀⠀⠀⠀⠀⠀⠱⣄⡜⠁⠀⠀⠀⠀⠀⣸⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣀⠀⢧⠀⠀⠀⣠⠞⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⣰⠃⠀⢙⡶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⢦⣧⣴⠊⠀⠀⠀⠱⣄⠀⠀⠀⢀⡴⠁⠀⠀⠙⢆⠀⠀⣰⣁⣤⠶⠋⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠉⠙⠳⠶⢦⣤⣬⣦⣀⣠⣎⣀⣀⣀⣤⣤⡤⠷⠞⠛⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡞⢁⡄⢀⡀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡿⣱⣯⠀⠼⠷⠆⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠨⠝⠁⠀⢡⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢤⠀⡄⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⣾⠀⠀⠀⠐⣆⠀⠀⠂⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⢀⣼⠃⠀⠀⣿⣦⣼⠀⠈⠁⢤⠀⠀⠀⠈⠧⠈⠀⠀⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠈⠓⠀⠠⡄⢀⢠⣾⣿⣿⣷⡄⣾⣿⣿⣿⣇⠀⠀⡄⠠⠀⣀⣤⢀⡄⠒⠒⣦⣶⠀⣇⠀⠀⠀⢀⣠⣴⡧⠴⠋⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢏⢻⣿⣿⠟⣴⣿⣿⠿⠟⢻⣦⣀⣤⡂⠈⣿⣿⣿⣿⠐⠒⠻⣿⠀⣟⠰⢶⡏⣼⣟⣿⡇⠀⠀⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠟⠀⢀⣿⣿⣿⠀⣴⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⡇⠀⢺⡇⠀⠛⠓⠺⢣⣿⣿⣿⡇⠀⠀⢻⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢀⡀⠀⠀⢲⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣷⠀⠈⠃⠀⠀⠀⠀⠈⠙⠿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⢂⠀⣸⣿⣿⣿⣿⡀⠛⠛⠉⢀⡀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠃⠀⠀⢨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣶⣶⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣤⣤⢻⣿⣿
⠀⠀⠀⠀⠀⣠⣴⣶⡖⢸⣤⣤⣦⠀⠉⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠖⠀⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣀⢸⡿⣫⣤⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⡆⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⢀⡀⠀⠰⠿⣿⣦⠀⠀⠀⠀⠀⠀⢰⣽⣟⡉⠉⠨⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢸⣟⢿⣿⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⠘⡇⠀⠀⠀⠙⣿⠀⠀⠀⠀⠰⢂⠸⣿⡧⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⢸⣿⣰⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⢸⠀⠀⠀⠀⠀⢀⡼⠀⠠⠀⠤⠀⢸⡇⠋⠃⠀⣀⣀⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⡿⣷⣤⡿⠀⠀⢀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠘⡟⠙⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣡⣾⣿⠿⠇⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢞⣃⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠻⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢩⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠠⢈⠉⠉⢓⣶⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢾⠷⠐⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡾⠁⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣈⡑⠒⠒⢺⠒⠒⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⣾⡹⠀⠤⠄⢻⢈⡆⠀⢸⣀⠠⠤⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣧⣤⣤⣾⣼⣧⠀⡚⡂⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣤⢤⡮⢤⡤⣤⣤⣄⠀⠀⠀⠀⠀⣸⠀⠹⡁⠀⠀⠒⣿⣿⠟⠈⣯⡇
⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⡟⣸⡗⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣾⣏⢙⡃⠀⠰⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣄⣿⡇⠀⢘⠿⠫⠉⠀⠉⠹⢆⠀⠿⠿⢿⣿⣿⡟⣿⠅⠀⠙⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⡾⠉⠀⠀⠀⡀⠀⠀⠀⠈⢲⠀⠀⢉⢿⠀⢹⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⢹⣟⠋⢺⠋⠋⠛⡿⠁⠀⠀⠐⢧⡀⠠⡂⠀⠀⠀⢃⠀⢀⣏⠀⣿⢆⡔⣾⠂⠱⡀
⠀⠀⠀⠀⣠⠴⢻⢻⢧⣸⠀⠀⠀⣇⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⣸⠀⠀⠏⡸⢹⠞⢳⡟⠂⠒⠙⡄
⠀⢀⡤⠚⠣⣠⠎⢸⡸⣿⣄⠀⠰⢅⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⡦⠤⣶⣤⣸⣀⡼⣿⠁⠀⠀⠈⢄
⡴⠉⠀⠀⠘⠀⠉⠙⠀⠈⠾⢢⠃⠀⠙⠈⠂⢄⣀⣀⣀⣤⡄⠘⠋⠀⢱⠈⣹⠛⠼⠛⠒⠛⡆⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢲⣶⣶⠖⣂⡂⣄⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣁⣸⣿⡏⠃⡟⠉⠟⡛⢗⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢤⠀⠀⣘⣡⣶⠟⢹⣿⣷⠀⠃⡠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠙⠀⠃⠀⣿⣿⡇⠀⣸⣿⣿⡖⣣⣾⣋⢙⣿⠀⠀⠀⠀⢠⠔⠿⢦⠀⠀⠀⠀⠿⠭⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⢲⠀⠀⠸⣿⣷⢀⡿⠻⣤⣾⣿⠹⣿⡞⠃⠀⠀⠀⠀⡏⠐⢀⡼⠀⠀⠀⠀⣳⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣾⣿⣿⢉⣷⣌⣅⠀⠀⠀⠘⣆⣙⣤⢞⣩⣴⣾⣄⠽⠀⢀⠀⠀⢀⣀⣧⣘⢃⠆⢠⣶⣶⣯⡹⡷⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⢒⢉⣴⡾⣿⣿⡌⢩⢁⣾⣿⣷⠄⠀⢸⣉⣻⣭⣿⣵⣬⣿⣿⢀⡈⣧⢸⡾⣟⢡⠾⢶⡧⣀⡻⢿⣧⠉⣷⢴⠺⡟⡋⠙⢻⠳⢤
⠀⠀⠀⠀⠀⠐⢿⣿⣯⡚⠛⠛⣻⡞⠋⢙⣿⡿⠄⣀⠾⣿⣿⠛⡿⣧⣾⣏⡛⣡⣾⣾⣿⣿⠀⠈⢰⠁⠆⢛⣿⣏⠹⠿⣇⠻⡇⣿⡇⠆⣏⣆⡿
⠀⠀⠀⠀⠀⢠⣦⡹⣿⡿⠆⢴⢥⡽⢁⠞⢰⣿⣯⣯⢜⡫⢾⣿⣿⣿⣿⣿⢟⣫⣍⢿⣿⣿⡆⠀⢸⠈⠀⣼⣿⣼⣤⠟⣻⡄⣉⢸⠀⠘⠡⠙⠷
⠀⠀⠀⠀⠀⢸⣿⣿⣾⣾⢦⣤⣦⣴⣿⡖⠚⠻⠯⠟⢁⡐⢿⡿⠸⢿⣤⣶⣋⣉⣉⣙⣛⣿⣷⣃⠏⣸⣾⣿⣿⢠⠼⠛⢉⠃⢛⣋⠤⠤⣶⠤⠞
⠀⠀⠀⠀⠀⠀⠙⠿⣭⣽⡿⣿⣯⣽⣿⣶⣦⣔⣶⣞⣛⣛⣛⠿⠗⣼⣿⣿⡇⠨⡇⣽⣿⣿⣿⣿⣿⣿⡟⠿⣿⣶⣿⠖⣚⣿⡾⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⢘⣿⠿⣿⣛⠛⣻⣿⣿⣾⢿⡏⢀⣀⡿⣿⣿⡁⠀⡟⣛⣿⣿⣿⣿⣿⣆⡀⠰⢤⠉⠉⡼⠉⠈⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⣿⡏⣉⡌⠉⠛⣻⣿⢏⣎⣡⣿⣿⢲⣿⣿⢿⣶⣇⠠⢶⣿⡿⣿⣿⡿⠿⠓⠛⠛⡻⠛⣳⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⡿⠇⠉⠃⠀⠘⢋⣤⣿⣷⣿⣿⣶⠚⢓⣦⡶⣅⣉⣣⣸⣿⣴⡛⠁⠀⠀⣀⣀⠤⣔⣯⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣟⡻⣿⢿⣻⠽⡛⢛⡋⠙⡙⠛⠫⢟⠈⡿⣿⠏⢩⣶⣾⡞⣿⡿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⡝⠻⡶⠤⠛⠊⠙⠹⠞⠀⠀⠀⠁⠉⠉⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛


⣿⣿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣶⣴⣧⣾⣯⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠀⠤⠴⠒⠒⠋⢀⣀⣀⣠⡄⣾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡍⠉⠋⠋⠉⠑⠤⢄⣀⠠⡄⠀⠰⠤⠤⢤⡤⠖⠒⠀⣀⠀⠀⠈⢻⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⠀⢀⣤⣇⡀⠀⠀⠀⠀⠈⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠛⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣤⣀⣀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣿⣄⠀⠀⠈⠸⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠀⢹⠀⠙⠢⢄⠀⢂⡤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣤⢒⡟⣉⣧⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠃⠀⠀⠙⠁⠀⠀⠀⠉⠺⠇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⡨⢠⣃⣤⠟⠫⠄⡇⠝⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠳⢦⣀⣀⣀⣸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠰⣄⠀⠀⠀⠈⢷⠊⣁⢵⡄⠀⠉⠀⠀⠀⠀⠀⠀⣠⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠃⠀⠀⢠⡞⣳⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠀⢀⡤⠶⠖⠛⣦⠀⠀⠿⠉⠀⠀⡙⢦⠀⠀⠀⠀⠀⠀⠀⡼⠳⣄⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢠⡋⠀⡶⢒⠊⠉⣧⠀⣰⢡⡄⠀⢶⢜⡀⢀⡀⠲⣄⠀⠀⣷⣀⠘⢧⣀⠀⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⡇⠨⡤⢀⠿⣸⢩⡆⠁⣁⠀⢨⣹⠀⠙⢦⡀⠀⢰⡏⠉⡉⠓⣭⠀⠀⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠠⣶⣈⠀⡇⠙⣇⠙⠟⢡⡞⠀⠀⠀⣄⠈⠀⣤⠹⡄⠛⢠⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⣠⠀⢤⠀⠀⠀⣀⣀⠴⣁⠀⠘⠦⠴⠋⠀⠀⠀⠀⢀⣀⠀⣿⠰⠄⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢠⡾⠁⡼⠓⠒⠒⢋⡉⠀⢀⢼⣇⣴⡍⣀⠀⠀⠉⢹⡄⠀⣤⠀⣰⠀⠉⠉⠉⠈⠙⠞⠑⢦⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠻⡶⠁⢠⡄⠀⡾⠀⠀⡿⠖⠉⢧⠇⢸⡀⠀⠀⠀⣧⡄⠈⠳⠞⠀⠀⠀⠀⡤⢲⠂⠀⢰⠒⠛⢰⠶⢤⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢸⡇⢀⡿⠄⠀⡇⠀⠀⡜⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠁⠻⡄⢀⣼⠀⢀⡟⠀⢸⠁⠙⡄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⠀⠀⢯⠀⠀⠀⢳⠀⢰⠃⠘⢯⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡠⠋⣾⠈⣆⡞⠀⢻⠋⠀⠀⠰⡀
⠀⠀⠀⢀⡼⠃⠀⠀⣴⠋⠀⠀⠸⣇⠀⠀⠸⢅⡜⢀⣠⠎⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢳⡀⢘⣠⠋⡠⠖⠁⠀⠀⠀⠀⠹⡄
⠀⠀⢠⡾⠁⠀⣠⠞⢹⡆⠀⠀⠀⠹⡄⠀⠀⢸⡇⢺⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢳⣼⠇⠊⣀⣀⡀⠀⠸⣧⠀⠀⢻⡀
⠀⢰⣏⠀⠀⢺⡇⠀⣸⡇⠀⠐⢦⡀⢙⣤⠔⠋⢀⡎⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⢨⡇⠗⠀⠀⠀⠀⠀⠐⠈⢧⡀⠀⢱⠆
⠀⠀⠹⣦⠀⠀⠻⡄⠉⠛⢦⡀⠀⣠⣎⣤⠀⠀⣾⠁⠀⠀⠀⠀⠀⢹⡆⢰⠀⣶⠀⣧⠈⠇⠻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⢳⡀⠸⢣⡀
⠀⠀⠀⠙⣧⡀⠀⢹⣆⣄⠀⠙⣄⢻⠈⠋⠀⠀⣻⣀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣿⠀⠇⠀⡦⢀⠇⠀⢸⡟⠛⠀⠀⠀⠀⠀⡍⠀⠀⠀⢻⡄⠰⢧
⠀⠀⠀⠀⠈⢷⡴⠻⣌⢻⣇⠀⠈⢻⡄⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⣼⠓⢶⣤⠶⠶⠖⠚⣇⣼⠀⡀⢸⡇⠀⠀⠀⠀⠠⠴⡇⠀⠀⠀⠀⠙⢆⠀⣳
⠀⠀⠀⠀⠀⠀⠀⢼⠸⠋⠙⢦⡀⢸⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡏⠀⠀⡇⠀⠀⠀⠀⣟⢹⠉⠀⢨⡟⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⣿⠋⠠⠈⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⢺⡇⠀⠀⠀⣀⣸⡆⠀⠀⠀⢠⡇⠀⢠⠄⠀⣠⠀⠀⡏⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢾⠓⠒⠉⣿⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⢠⠁⢘⣀⠀⠠⣧⣄⣀⣀⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⡆⠀⠀⡏⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⠀⠀⢾⠀⡼⠀⢸⠉⠁⠀⠀⢠⡈⠉⠛⠛⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⣧⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⠀⠀⣤⠀⡆⠀⢰⠀⠀⠀⠀⢸⣷⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠀⡎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⢀⡌⠀⡇⠀⢠⡀⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⢠⢿⠀⠀⡇⢀⠁⠀⠘⡄⠀⠀⠀⡇⠀⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡺⠘⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠇⠀⠸⠇⠀⡀⢀⡇⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠈⢇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢹⡀⠀⠀⢹⡄⠀⠀⠀⠀⣼⠀⠀⡿⠀⠀⡇⢸⡇⡘⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢺⡀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣧⠀⠀⠈⣇⠀⠀⠀⢰⡇⠀⢸⠂⠀⠀⡇⣘⠁⡇⠀⠀⠀⠘⡆⠀⠀⢇⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡆⠀⠀⢻⡀⠀⠀⣼⠁⠀⡜⠀⠀⠀⠁⠛⡀⡇⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⢷⠀⠀⠈⢧⠀⠀⡏⠀⢰⠃⠀⠀⠀⢸⡈⠇⠁⠀⠀⠀⠀⢡⠀⠀⢸⠀⠀⠀⢸⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢉⡹⣗⠘⡆⠀⠀⠙⡄⢸⡅⠀⡞⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠾⢦⡦⣴⠇⠀⠀⢸⠓⢟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠏⢐⠙⢦⢹⠊⢻⡟⠣⠜⠀⣀⠁⠀⠀⠀⠀⢸⣦⣾⡄⠀⠀⠀⣀⠔⠋⠉⣹⠀⠀⠀⢸⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠚⠀⢀⠽⢦⣴⠞⠁⠀⠀⠀⠀⢰⡀⢀⡞⠁⠀⠀⢾⣁⣠⠴⠀⠁⠀⠀⠀⠀⠙⠛⠶⠤⠭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣮⣈⡹⠂⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⢀⠀⠄⠀⠀⢤
⠀⠀⠈⠀⠢⡀⠀⠀⠒⠀⠀⠩⢂⠐
⠀⠀⠀⠀⠀⠈⠙⠋⣩⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠒⢲⣾⣿⣿⡟⢻⡆⠠⢤⣶⣶⣾⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣌⠀⠀⢸⣿⣿⠛⡇⠘⢓⣢⠄⠠⠻⣿⡇⡆⣿⣿⠀⠀⠀⠀⢠⣶⣒⣚⠡⠤⢤⣼⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⠀⠀⠈⣿⣿⠀⡿⢸⣾⠁⣾⢠⣿⣿⣇⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢀⣀⣸⡿⣧⠿⠼⢿⡇⣿⣠⣿⠿⢟⣚⣾⡿⢃⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⣀⣈⣃⣈⡑⠛⠒⠚⠓⠚⠉⠀⣿⣈⣯⡉⠀⠀⢸⠷⣷⡶⠶⠶⠶⢾⢿⣿⣾⡇
⠀⠀⠀⢸⣿⣿⡟⣿⢉⡏⠩⢭⣭⣿⡇⠀⠀⣠⠤⠒⠒⢚⡷⠞⠛⠛⠒⠲⢤⡀⣿⡇⠀⠀⣇⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⣿⢸⣇⣸⠘⠻⣿⣇⣴⣯⣄⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⣻⣻⠟⠛⠋⢉⡧
⠀⠀⠀⢸⣿⣿⡇⢿⢸⣿⣿⣰⠀⠏⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣷⠀⢸⣿⣷⣼⡆⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⢀⠴⡷⣄
⠀⠀⠀⠈⣿⣿⣿⣀⣸⣿⣿⣿⣇⣿⣿⣿⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⢠⠏⠀⡇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣰⠯⢤⣀⣇⣠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣷⠃⠀⠀⠈⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⣴⢃⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⣀⣀⡠⠤⠖⠋⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢀⡄⡄⣀⡠⠀⢠⣀⣄⣀⠄⢀⣄⣀⢀⢀⠄⠀⣤⡀⣀⢀⡄⢀⣄⣠⣀⣠⠀⣀⣠⣀⡤
⠀⠀⠀⠀⠐⠓⠙⠉⠛⠃⠘⠓⠋⠘⠃⠘⠉⠋⠛⠙⠛⠀⠓⠓⠛⠘⠁⠘⠋⠙⠋⠋⠀⠟⠋⠉⠛⠃
⠀⠀⠀⠀⠀⡴⠤⢄⣀
⠀⠀⠀⠀⠀⠉⠒⠢⠤⣉⣑⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣒⣮⠭⣒⡢⠤⣤⡤⣤⠤⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠀⢀⣀⣀⣈⣉⣑⣯⣀⣵⣀⣀⣀⠀⣈⣉⠷⠖⢒⣊⣉⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⢰⠁⡠⠴⠒⠋⠉⠁⠀⠀⠀⠀⣀⢤⣀⠤⣔⣒⡩⠭⠔⠒⠺⢭⡉⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠳⢄⡉⡱⢶⠢⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⢲⢖⣉⣁⠔⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢱⡞⠒⠦⠤⠤⠤⠤⢼⣱⠤⠤⠤⠤⠤⠖⢺⢺⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣌⠑⠢⣼⢇⣀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⣸⢼⡤⠒⢉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠵⢎⣀⣉⠉⠉⠓⠒⢾⣻⠒⠒⠋⠉⢉⣁⡙⠾⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠛⠓⢤⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠲⣿⣿⡧⠀⠀⠀⠀⠀⠀⠺⣿⡷⠀⠀⠀⠀⠀⢀⣸⢟⣠⣶⠞⠉⠂⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⣿⣿⠀⠀⠀⠰⢦⡀⠀⢀⣿⣷⣆⠀⠀⠀⣾⣻⣷⣏⣹⣧⢀⣀⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠹⣦⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠏⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⠛⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⢷⣌⡛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣠⡾⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠙⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠈⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⢰⣶⡟⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⡶⠟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣈⡙⠻⠷⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠟⠛⣉⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢰⣶⣶⣦⣤⣤⣤⣍⢩⣥⣤⣤⣤⣶⣶⡆⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠿⠿⠿⠿⠿⠸⠿⠿⠿⠿⠟⠛⠃⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠛⠂⠒⠒⠋⠉⠁⠠⠤⠤⠴⠆⢿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢤⣀⣀⣠⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢾⠀⠈⠈⠀⠀⠀⠒⣶⡆⠶⠀⢄⡓⠒⠒⡖⠉⠉⠀⠀⠀⠀⢠⣽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⣀⡤⠶⠓⠠⣀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⠈⠣⠀⠀⠀⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢺⡆⠒⠒⠒⠆⡄⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡴⠚⡟⠢⣄⠀⠀⠁⠀⠀⢁⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⡀⣼⢾⡇
⠀⠀⠀⠀⠀⢀⡠⠚⠇⠀⠢⠧⠀⠀⠉⠲⢼⣾⠁⠈⢦⠀⠀⠀⠀⠀⠀⠀⢘⠀⠞⢋⣴⡗⢺⡟⣇⡠
⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠨⣧⡀⠀⠀⠀⢠⣦⡋⠊⠘⠋⠀⠈⠘⠉⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠰⣀⡠⢤⣰⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠠⠄⠀⠀⠀⠈
⠀⠀⠐⣯⠋⠀⠀⢀⡁⡀⠀⠀⠀⠀⠀⠀⢊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠌⠁⠂⠒⢀⠐⠀⠀⡀⣐⣄⡀⠆⡄
⠀⠀⡀⠻⡀⠼⠂⠛⠁⠀⣀⠀⠀⠐⠛⠀⠉⠀⠀⠀⠀⠀⠠⠄⠰⠀⠛⡀⠾⠁⠨⠁⠀⠀⠈⠉⢋⠙⠃⠘⠉
⠀⠈⠀⠀⣠⢴⢠⡴⣖⠀⠛⠀⠀⠀⡀⠲⣔⡁⠀⣀⡌⢳⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⠀⢀⢰⡅⠁⠈
⠀⠀⠀⠀⠀⠁⠴⠀⠈⠀⠀⠀⠀⠀⢀⡺⢧⠟⠉⠉⠁⠉⠀⠀⠀⠀⠀⠠⠠⠄⠸⠅⠀⠀⢰⣾⡁⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣸⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡗⠂⢰⠒⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠁⠀⠀⠈⠐⠂⠀⠐⠊⠁⠀⠀⠀⠀⠒⠚⠀⠀⠃⠒⠀⠀⠀⠀⠈⠀⠀⠈⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⢶⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣀⣸⠁⢈⣀⣉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⢁⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⢱⡞⠁⠂⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡾⡯⠀⠻⠛⡳⠆⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠤⢼⡇⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡴⣿⣅⣀⣀⣠⢤⡶⠞⣍⠉⠉⠛⠻⢲⡧⣭⠤⠀⣀⣐⣒⣒⣤⣤⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣸⠀⠀⢠⠀⠀⡇⠀⣿⠀⣀⠄⠀⠨⡇⠀⠀⢨⠀⠀⠀⠀⠉⢹⠇⢠⣼⣃⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⢸⠀⢠⡅⠀⢉⣀⣻⣀⣟⢸⠋⠉⠀⣸⠀⠀⠀⠀⡇⢸⡧⠤⠼⡧⠀⠀⢒⣛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⠀⠀⢠⡆⠀⠙⠿⣷⠿⢿⣿⣂⣀⣀⣤⣤⣀⣠⣤⣼⠁⠀⠀⣇⠀⠀⠁⢘⡂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣄⠀⢸⣇⡀⠀⠀⣇⢀⣀⣘⣀⣠⡇⠀⠀⠉⠉⠙⠁⠉⠀⠈⡬⠒⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡄⢀⡾⡿⡄⠀⠉⠙⠛⣸⢿⡍⠉⠁⠀⡨⠙⠓⣀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⢀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣼⣁⣧⣨⣆⣀⡀⠀⠑⠚⠷⣂⠀⠀⠀⠀⠀⠙⢿⣆⠰⡄⣇⠀⢀⣟⠀⠀⠀⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⢉⣿⡷⠾⠟⠯⠭⣤⣀⠈⠁⠀⠀⠀⠀⠀⣠⠆⢻⣇⠢⣿⡀⢈⣉⣀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⡿⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⡑⣄⢀⡤⠀⠀⠀⠈⣿⡄⠬⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠹⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠎⠀⠀⠀⠀⠀⡿⠃⠀⠀⠴⡆⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⢐⠃⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢃⠀⣀⠴⢁⡟⠀⠂⣠⣘⣻⠈⠉⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡄⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⠉⠀⠈⠃⢠⠚⠁⡹⣷⠤⠧⠂⢀⡀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠏⠀⢠⠿⠛⠒⠶⠫⠾⠒⠀⠀⠉⠓⠤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⢀⠴⠁⠀⠸⠆⠀⠀⢠⣦⠀⠀⠀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠱⢿⣶⠀⠀⠀⡀⡀⢀⣴⠟⠉⢀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠛⠊⠈⠀⠀⠀⠘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀




⣿⣟⣛⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⠁⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⣀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⣅⠀⠀⠀⠒⠂⠀⣀⡀⠐⠂⠀⠀⠀⠀⠈⣿⣟⠃⠀⠐⠈⢉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣿⣷⣶⠴⠒⠛⠛⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠴⠗⠂⠀⠀⠀⢀⣀⣉⣈⣿⡁⠀⠀⢀⡴⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢐⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠋⠉⠉⠙⠫⡲⠛⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⣶⠀⠀⠀⠀⠨⢻⡆
⠀⠀⠰⣦⣄⠀⣀⡀⠐⠗⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠿⢷⡟⠀⠀⠀⠶⠚⡛⠁
⠀⠀⠀⠉⢙⡭⢉⢳⣶⡤⡀⠀⠀⠀⠀⠀⣼⢣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠙⠻⢷⣄⣀⣀⣠⠞⠁⠀⠱⠀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠀⢠⡂⠉⠈⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣔⠈⠁⠙⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢔⡂⠉⠈⠉⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠟

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣶⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠴⣲⢶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠐⠂⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡤⠊⠀⣼⣝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠐⠆
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⡠⠾⠔⠁⠀⣠⣾⣭⣻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⠞⠉⠀⠀⣠⣾⡻⣷⣿⣿⣿⡿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⣀⣴⡾⣷⣽⣿⣿⣿⣿⣟⣿⠞
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣽⣿⣿⣿⣿⣿⢿⣯⡿⠋
⣤⣤⣴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣯⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⢿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠉⠁⢸⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣧⣤⣤⣤⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⣦⣤⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⣀⣉⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦⣀⣈⣁⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣄⣀⣈⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦
⣿⣿⠋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⢏⣉⡉⠉⠉⣉⣉⠟⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠸⣿⡿⠀⠸⣿⡿⠀⠀⠀⠀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣍⣙⠒⠲⠶⠤⠤⠤⠤⠤⠴⠶⠒⢚⣉⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣶⣶⣶⣶⣶⣶⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⠏⠀⢻⣿⣿⣿⣿⠃⠈⢿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⣿⠏⠀⠀⠀⢿⣿⣿⠇⠀⠀⠈⢿⣿⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠤⣤⣄⣀⣈⣿⣏⣀⣀⣤⡤⠼⠟⠋⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠶⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠠⣽⡉⠀⠀⣠⢀⣀⠤⢾⣥⡤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣬⡳⠤⣀⡀⣄⠀⠀⢉⣿
⠀⠀⠀⠀⠈⢉⡟⠁⠀⠀⠀⡌⠀⠀⠀⠸⠉⠙⠓⢦⡀⢠⡴⠚⠋⠉⡅⠀⠀⠀⢡⠀⠀⠀⠈⢻⡉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠤⠄⠀⣀⡇⠀⠀⣾⣆⠃⠘⣸⡇⠀⠀⢰⣀⠀⠠⠬⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡸⡄⠀⡘⠛⠛⠒⢦⣤⣀⠈⠁⠒⣯⡟⠀⠀⢯⡵⠒⠉⠀⣀⣤⡴⠚⠛⠛⢂⠀⢠⢇
⠀⠀⠀⠀⠀⡇⡇⠐⣃⠐⠂⠠⢎⡀⠈⠙⢯⡀⢻⡁⠀⠀⢸⡇⢀⡽⠋⠁⢀⡡⠄⠐⠂⡸⠂⢸⢸
⠀⠀⠀⠀⠀⢹⠁⠀⠀⠑⠂⢴⣆⡈⠑⠢⡜⠀⢸⡃⠀⠀⢸⡇⠀⢢⠔⠊⢁⣰⡦⠐⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⣠⠞⠛⠉⠉⠙⠳⣤⡀⠉⠉⠁⠘⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠰⢣
⠀⠀⠀⢀⡴⢳⠷⣄⠀⢀⡴⢿⠷⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⢶⡿⣆⠀⠀⣠⠾⡞⢦⡀
⠀⢀⡴⠋⠀⠘⠀⠈⣷⡞⠁⠘⠃⠈⢳⣄⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⣠⠞⠁⠘⠃⠈⢳⣾⠁⠀⠃⠀⠙⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡿⠉⢉⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠈⠉⠀⣤⣀⣀⠀⠀⠀⠀⠀⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⠿⠋⠀⠀⠀⠀⠉⠻⣿⣿⣷⣶⣤⣤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠀⢘⣛⣻⣿⣿⠟⠋⠁⠀⠀⢀⣤⣦⡄⠀⢰⣿⠛⠛⠿⠛⠉⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠙⠛⠀⠀⠀⠀⠀⠘⠿⠿⠟⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡘⠿⠀⠀⠀⠀⠀⠀⣙⣫⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⠂⠀⠀⠀⠐⢾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⣉⢉⣁⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⢰⣶⣦⣤⣤⣤⣀⠀⠉⠛⢿⣿⡿⠟⠈⠉⠛⠿⠟⠋⠉⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⢐⣡⣤⣤⣤⣦⣤⣄⡀⠶⠚⣿⡗⣿⣿⢺⣿⢿⣿⡗⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⣇⣶⣿⣰⣿⣶⣷⣆⣾⡆⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢵⣯⣭⠛⠉⠀⠉⠁⠉⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⢃⡉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⠀⢀⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢠⢠⣄⣀⣤⠀⠀⠐⢄⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠂⠘⠻⢿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⢀⡈⠓⠈⠉⡄⣰⡆⠀⠀⣤⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠃⠀⠰⠀⠀⠃⠃⠀⠰⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠁⠈⠀⠀⠀⠛⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠸⣇⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣆⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣆⠀⠈⢦⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠀⠀⠀⣻⢿⣿⣶⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠈⢧⣀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣾⣅⠀⠀⣾⠁⢸⣿⣿⣿⣇⣠⣤⣀⠈⣿⣿⣿⣿⠀⠀⠀⠘⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠻⣷⣿⣿⣿⣿⡟⠛⣿⡇⢰⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⠁⠀⠀⠀⣨⣿⣿⣿⣿⣷⣖⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠋⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠟⣿⡟⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡏⠀⣿⡇⢀⣿⣿⣵⣿⣿⣿⣿⣿⣿⡏⣿⣇⣽⣇⢸⣿⣷⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⠃⣼⣿⠛⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠧⠈⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⣀⣰⣶⣶⣶⣶⣿⣷⣾⣿⣾⣿⣿⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣆⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⣴⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⣤⠖⠒⠊⠉⠉⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠛⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠉⠙⠛⠛⠂⢠⣾⣿⣿⣿⣿⣦⣤⣶⣾⣿⣿⣿⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢉⣁⣿⣿⣟⠿⠛⠛⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠘⠛⠿⢿⠿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢸⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢀⣼⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⡟⢢⡀⠀⡴⠋⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠜⠁⡏⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⢿⣷⡧⢤⡽⢎⠈⠢⣸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠔⠉⢦⠀⣟⡴⠙⣄
⠀⠀⠀⠀⠛⠉⠉⠁⠀⡜⠁⠀⠙⠓⢿⡀⠀⠑⣄⠀⠁⡀⠙⠿⣿⣿⣿⣿⣿⡿⠋⢰⠏⠀⠀⠀⠉⠛⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠒⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⣀⡀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⡄⠠
⠀⠀⠀⠀⠀⠀⠰⡇⠀⡏⠀⢸⡤⣤⣏⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣄⠘⠀⢠⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⠊⠀⠈⠀⡀⠘⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠓
⠆⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠠⡀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠔
⠈⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠲⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡆⢀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣈⠀⡄⢀⠊⠉⠁⠀⠀⠀⠈⠁⠐⢄⠀⡁⠀⠀⠀⠀⡀⠀⠀⠰⣤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠂⠌⡐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡇⠀⠀⠐⠣⣇⡐⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠄⠃⠀⠀⠀⠙⠌⠑⢄⠀⠀⠀⠀⠀⠀⠰⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠁⠀⠀⠸⠳⡀⢁⠀⠀⠀⢀⠀⢐⠐⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣡⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠰⠖⠣⠐⠂⠀⢀⡀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠈⢡⠀⠀⠹⡀⠀⠀⠀⠀⠀⡙⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠀⠀⠘⠀⢃⠀⠀⢱⡄⠠⠔⠾⡆⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠠⠍⢆⠀⠘⠻⠶⠤⣀⣀⡠⠀⣮⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠇⠈⣄⠀⠒⠚⡀⠀⠀⠀⠀⠀⣀⠐⠀⠰⠀⠀⠀⠀⠐⡀⠄⢁⢀⡀⠂⡐⠁⠀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠐⠀⠠⠘⡐⠅⡠⢀⠀⠡⠙⢉⠉⠁⠀⠀⠀⠖⠀⠀⠔⠀⠀⠀⢗⠛⠋⠈⠝⠎⢈⢁⣶⣦⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀⠐⡀⠀⡄⠄⢠⠰⡁⠐⠀⠀⢁⠊⡀⠈⢉⠢⢀⡀⠠⠄⠀⠄⢓⠴⡀⢇⠀⠀⠀⢀⣀⣄⡸⠘⠻⠟⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⠠⠈⠘⠁⡓⠂⠀⠠⣾⣿⡆⢈⣩⠤⠄⠀⠲⠀⠑⠈⠀⡄⢸⠨⡫⡉⠀⢁⣠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠐⠀⠘⠀⠇⠐⠀⠀⠙⣋⠉⡀⡀⠀⠀⢀⠀⢀⣠⠀⠀⠀⠁⠌⠀⠀⠀⠬⠢⢰⠀⠶⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠓⣤⣀⠤⣦⠦⠚⠉⠁⠀⠠⠂⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠆⠁⠀⠉⠊⠀⠀⠀⠀⠁⡀⠀⣀⣀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠠⠤⠤⠠⠄⠀⠶⠤⢀⠀⣀⠀⣀⣐⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠓⠂⠀⠦⠤⠤⠤⠞⠀⠀⠉⠰⠂⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣿⣇⣿⠖⠂⢁⣼⣟⠀⠀⠀⠀⠳⢤⣤⣿⡀⠀⢀⠀⠀⣈⣿⣀⠀⠀⠀⢀⣀⣿⣇⠀⠀⢰⣶⠄⠀⠀⠀⠀⢀⠀⢀⣴⡄
⠀⠀⠀⠻⣿⣿⡟⠀⠀⣼⣿⣿⠃⠀⣀⠀⢦⣠⣿⣿⣷⣶⣦⠤⣾⣿⣿⣿⣧⣴⠟⢯⣿⣿⣿⠆⢴⣸⣿⣦⠀⠀⣸⣷⣧⣴⣾⣯⡀
⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⠛⠁⠉⠀⣿⣿⣿⣿⣿⣿⠿⠀⣿⣿⣿⣇⠀⠀⠀⠀⢹⣿⡿⠀⠘⠋⣿⡟⢄⠐⠿⠳⣾⣿⣿⡿⠁
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡟⠀⠀⠀⠀⠙⠛⣻⣿⣿⣿⣷⠀⠀⣿⣿⣿⡀⠀⠀⠀⢸⣿⡅⠀⠀⢸⣿⡗⠀⠉⠀⢠⣿⣿⡿
⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢹⡇⣀⣤⣤⣥⡀⠀⠉⠙⠛⠛⠁⠈⠀⣿⡇⢻⡇⠀⠀⢀⣨⣿⣇⣀⡀⣿⢿⡇⠀⠀⠀⢿⡏⢻⣧
⠀⠀⠀⠀⢿⡿⠧⠀⠀⠀⠈⠹⣿⣿⣿⣿⠗⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⢸⡇⠀⠀⠸⣿⣿⣿⣿⡇⠘⣿⡇⠀⠀⠀⠈⣿⠄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡗⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿⣿⣷⣶⣶⣦⣤⡀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠉⠉⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡧⢴⣶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿⣿⣻⣏⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣶⣿⡿⠋⠁⠀⠀⠀⠈⠑⠢⡀⢈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⡻⣿⡿⢿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⡿⡉⢤⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠈⢲⡀
⠀⠀⠀⠀⠀⠀⡠⠖⠁⠈⠑⠦⠧⠌⠀⠈⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠘⠆⠀⠳
⠀⠀⠀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠙⠛⠷⠄⠀⠀⠀⢀⡀⠴⠊⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠒⣈⣉⣉⡉⠭⠥⠤⠤⠉⣉⣉⣁⣒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡡⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⢍⡢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢹
⠀⠀⠀⠀⠀⠀⢀⠏⢸⡈⠓⠮⣓⡢⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢚⡵⣿
⠀⠀⠀⠀⠀⠀⢸⠀⡾⡏⠓⠤⣀⡭⠵⣖⠒⠤⠬⢭⣭⣉⣉⣉⣉⣭⡭⠭⠤⠒⠒⢻⠟⠹⡿
⠀⠀⠀⠀⠀⠀⢸⠀⢧⢱⠀⠀⢸⠀⠀⣿⠒⠒⠲⠤⠤⠄⠀⢾⡏⠉⢳⠤⠄⠒⠂⡏⢸⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡸⡆⠀⠘⡆⠈⡇⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⢠⠃⡟⢀⠃
⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⣏⠢⡀⢳⠀⡇⢣⠀⠀⠀⠀⠀⠀⢸⠆⠀⢸⠀⠀⠀⢸⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⢸⡀⠙⡾⡆⢰⢸⡀⠀⠀⠀⠀⠀⢸⢰⠀⢸⠀⠀⠀⡇⢸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⡆⢇⠀⢳⢳⠀⡆⣇⠀⠀⠀⠀⠀⢸⢸⠀⢸⠀⢀⠴⠇⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⡸⡄⠸⡈⡆⢃⢫⠙⠲⡤⣤⡤⢼⠈⠀⡴⢶⢹⡘⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣱⣳⠀⡇⢠⢸⣸⡀⣀⣀⣰⣀⢀⢸⠀⡇⡌⢇⢧⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡼⠚⡇⡂⡍⠀⠀⠀⠀⢨⡎⢸⠛⢾⣾⢧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠤⣀⣰⠁⡇⠀⠀⠀⠀⢸⠇⡎⢀⣼⢯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⣧⡞⠒⠒⠒⠒⠺⠤⠉⠙⠛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢉⡰⠒⡶⠤⠤⢤⡤⢤⠤⠶⠲⣎⠻⣀⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣁⣀⣔⣋⠴⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠢⣄⣑⣢⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣻⣿⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠶⠶⠶⣶⠦⢠⣄⣀⣀⠀⡁⢀⣠⠴⠶⠖⠐⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠈⠀⡼⢫⡍⢧⡮⡙⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢷⣬⡽⠛⠁⡟⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢻⠀⠀⠀⠀⠀⣠⠀⠀⠀⠸⠀⠀⠀⢸⡦⢀⠀⣷⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⢼⡿⢿⣇⢀⡇⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠤⠰⠶⣿⢶⣶⢶⣶⣶⣷⣤⣤⡄⡰⠀⢀⣤⣼⡇⠚⠐⠋⠀⣿⠄⠐⠶⠶⠒⠒⠶⠤⣬⠀⠀⠀⢷⠀⢿⣿⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡇⠀⠀⠀⡇⠀⠇⢀⡋⠛⣥⣤⣄⡀⠀⠀⠈⠙⠛⢿⡆⠀⠀⠀⠙⠀⠀⠀⣄⣤⠀⣶⠀⠁⠀⠀⠀⠈⢼⣾⠛⡇⣸⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⠄⠀⣶⣦⣦⢼⡅⣤⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⢃⠀⠀⣀⣀⠀⠀⢻⣿⡀⢸⡇⠀⠀⠀⠀⠀⣼⣽⡿⠓⢻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣭⣩⡀⠀⡠⠀⠘⠉⢁⣠⡄⠠⠀⠀⠀⠈⠛⢛⣉⡉⠉⠽⠿⠷⢿⠸⡇⠈⡧⠀⠀⠀⠀⠀⢠⢿⣷⡦⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⢮⣁⠐⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄⠀⠀⠀⠈⠄⠣⢠⣯⠀⠀⠀⠀⠀⠚⠛⠛⠋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠼⡧⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡤⠆⠀⠚⢹⡜⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣇⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢈⣷⠶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢿⠀⠀⠀⡀⠄⠀⠘⠻⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠘⠛⡿⠙⣇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢖⡶⠄⢠⣈⣀⡴⢎⠀⠀⠀⠀⠀⢀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣴⠘⣧⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⣟⠙⠃⣰⠟⠁⠀⠀⠉⠂⢄⣀⣡⡌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⢶⣠⣌⣼⠛⢼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⡄⠀⠀⠀⣷
⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠐⠚⠁⢸⠛⠁⠘⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⣀⣴⠆⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⣠⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠙⠛⠙⣷⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠉⠛⠋⠁⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⢠⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⢰⡟⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣾⣿⣿⣿⣿⣻⣿⣟⣶⣿⣿⣿⣿⣽⣿⣾⣿⣷⠦
⠀⠀⠸⣧⢿⣿⣿⣿⡇⢸⣿⣿⣿⠟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠹⣎⢿⣿⣿⠇⢸⣿⣿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠲⠶⠤⠤⠤⠤⠴⠶⠚⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣶⣶⣶⣾⣿⠃⢻⣷⣶⣶⣶⣶⣶⣾⡟⠀⠀⠀⠀⠈⠃⠶⠴⠦⠆⠀⠤⠆⠲⡆⠙
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣴⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠛⢾⡷⢰⣷⢰⢿⠚⡿⠈
⠀⠀⠀⠀⠀⠀⠉⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣾⣯⣾⣽⢬⣿⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠲⠆⠶⢴⠠⠴⠆⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣽⣻⣍⣣
⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⣀⢀⡀⣀⡀⣀⢀⡀⠀⠀⠀⠀⣨⣟⣿⣿⢿⣿⣟⣯⡀
⠀⠀⠀⠀⠀⣈⡦⡤⠤⠒⠂⠠⠴⣄⡀⠀⠀⠀⠀⠰⣿⣾⣷⣿⣷⣿⡾⢁⣨⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠛⠿⢿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⡍⠀⠀⠈⠂⡀⢀⠀⠀⠀⠈⣃⠀⠀⠀⠀⢻⣿⣿⣿⣿⡞⣥⣾⡿⢿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⣿⣷⣦⣤⣤⣤⣤⣤⣤⣶⣾⡇⠀⠀⠀⠀⠘⣿⣿⣿⢻⣾⡿⣫⣷⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢹⡿⣵⢟⣽⣾⣿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣛⣀⣀⡀⠀⠀⠀⠀⠛⠻⠿⠿⠛⠋
⠀⠀⣶⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⣴⣿⣿⣷⣦⠀⠀⣀⣀⣀⣄⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣿⣿⣿⣿⣿⢷⣴⣿⣿⣿⣿⣿⡄⢈⠁⠉⠉⠋⠉
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣼⡏⠻⢿⣿⣋⣠⣤⣤⡄
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠜⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟⠛⢻⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢘⠂⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠾⢰⣶⡆⠧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢈⡃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⡽⣦⡐⢈⡉
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⣿⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠈⠀⠈⠻⠿⣿⣿⡿⠿⠋⠁⠐⠆⠘⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⣀⣤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⣠⠔⠈⢶⡯⠭⠤⠤⠤⠄⠐⠂
⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠶⠶⠒⠒⠒⠓⠾⠭⠤⠤
⡔⠒⠛⠉⠩⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⠤⠤⠄⠀⠀⠀⡄
⠈⠙⣒⡒⠒⠒⠒⠒⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣔⠒⢀⠅⠀⢀
⡆⠈⠀⠁⠀⠀⠈⠹⠄⠀⠀⠀⠊⠙⠇⠀⠀⠀⠘⠉⠃⠀⠀⠧⡤⠂⢸
⡇⠀⣦⡀⠀⠀⢠⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⢠⠀⢸
⠇⠀⠛⠃⠀⠀⢸⢸⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⢸⢰⠀⠀⠀⢸⣾⠀⢸
⡇⠀⠓⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⢸⠀⠀⠀⠸⠈⠀⠸
⢻⣠⡀⡦⠤⣄⢀⢤⡀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠈⠈⠀⣠⡤⠀⢲⡾⠁
⠀⠀⠉⠙⠒⠀⠀⠐⠁⠭⠿⠏⠉⠈⠹⠹⠿⠟⠳⠶⠂⢉⣉⠤⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣐⣲⣶⣶⣶⣶⣶⣶⣶⣶⣖⣋⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠁⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡡⠼⠲⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢄⡀⢀⠔⠁⠀⠀⠈⣳⠤⠄⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⢴⡉⢀⡠⠔⠊⠁
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡠⠊⠀⠀⡀⠲⠆
⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠈⠹⠿⠋⠀⢀⡈⢓⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠄⠘⠀⠀⠀⠀⠉⠛⠒⢢⠤⠖
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⣤⣬⡍⣱⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⣂⣤⣾⡙⢟⡫⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠲⠤⢀⠠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⡄⣶⣿⠿⣫⠕⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠐⠰⠿⣫⠵⠋⠁
⠀⠀⠀⠀⠘⠧⣄⡀⠀⠀⠀⣉⠉⠐⠂⠤⣀⡀⠀⠀⢱⠀⠈⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⡶⠧⠄⣀⣀⣀⣀⣨⠆⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⡐⠉⠀⠈⠉⠐⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡚⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⠃⠄⠀⢰⠼⠔⠃⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠎⠃⢀⣠⣤⣬⣀⠀⠐⠢⢄⠀⠀⠀⠇⠚⠀⢸⣠⠖⠛⢽⡍⣩⣤⣯⣤⣤⠴⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠁⠀⠀⠤⠭⠶⠤⠴⢋⠤⠐⠂⠐⠒⠤⣉⣤⡭⠥⠤⠤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠖⢁⡠⠤⠀⠀⠄⠀⠀⠱⡀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣹⠹⠏⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠛⠒⠖⠶⠒⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠀⠈⣣⠀⠀⡄⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⠈⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠔⠀⠀⠁⠃⠐⡄⠀⠀⢠⠃⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢸⣖⠄⢼⣹⠀⠢⠀⠀⠀⠀⡠⠀⠘⡇⠠⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠰⠁⠀⠀⠈⠓⠤⣤⣀⣀⣤⠴⠾⠁⠰⠃⠘⠂⠼⠴⠀⠀⠑⠀⠀⠀⠑⠲⠶⠶⠶⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢿⣤⠤⠤⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠶⠟⠉⠻⣏⠉⠉⠉⠉⠍⠛⢛⠚⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠿⢦⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣾⡭⢷⡏⣿⡴⢛⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡛⠋⣉⣉⣉⣤⠧⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠩⠭⢹⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢞⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⢂⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣽⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣔⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⠭⠷⢿⣀⡷⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠥⣭⣩⣿⣿⣇⣸⢟⣛⣋⣉⣉⣤⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣩⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⢋⣵⣚⣁⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣩⠴⠋
⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⡩⠭⠭⣴⢋⣡⠞⠉⠀⠉⠉⠉⠉⠉⠙⠛⣻⠖⢋⣠⢾⣅⣀
⠀⢀⡤⠖⠛⠉⠀⠀⠀⠀⡎⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢠⠔⠚⠓⢾⣁⠴⠋⠀⠀⠀⠈⠙⠲⢤⡀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⡟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⣽
⣷⡈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁⠀⣠⣾
⣧⠘⠢⣀⠀⠉⠙⠛⠒⠲⠦⠤⠤⠤⣤⣤⣤⣤⣤⣠⣤⣤⣤⠤⠤⠶⠖⠒⠛⠉⠁⠀⠀⣀⡠⠔⠊⢁⣿
⣿⠀⠀⢨⢛⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠖⡞⡉⠁⠀⠀⠀⢸⣿
⣿⠀⢀⡏⠸⡀⠀⠀⠀⠉⠉⢩⢻⠉⠉⠉⠉⠉⠉⠉⢉⠟⡍⠉⠉⠀⠀⠀⠀⠀⣸⠀⢱⡀⠀⠀⠀⡇⣿
⢹⠀⡸⠀⠀⣇⠀⠀⠀⠀⢠⠃⠈⡆⠀⠀⠀⠀⠀⢀⡎⠀⠸⡀⠀⠀⠀⠀⠀⠀⠏⠀⠈⢷⠀⠀⢸⣧⡟
⢸⢀⠇⠀⠀⢸⡀⠀⠀⢠⠇⠀⠀⣧⠀⠀⠀⠀⢀⠞⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⢇⠀⢸⢁⡇
⢸⡞⠀⠀⠀⠀⣇⠀⢀⡎⠀⠀⠀⢸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡇⠀⠀⠀⠀⠸⡄⣸⢸⡇
⠘⡟⢦⡀⠀⠀⢸⠀⡼⠀⠀⠀⠀⠘⡄⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡸⠀⠀⠀⠀⠀⠀⢣⡟⢸
⠀⠹⣄⠈⠳⢄⡀⢠⠃⠀⠀⠀⠀⠀⢣⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⠃⠀⠀⠀⠀⠀⠀⣈⠷⡿
⠀⠀⠈⠙⠦⣄⡈⠙⠲⠤⣄⣀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡌⠀⢀⣀⣠⠤⠒⢋⡡⠚
⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣀⡈⠉⠙⠒⠓⠲⠦⠤⠤⠤⠤⠤⠶⠒⠒⠚⠋⠉⣉⣠⡤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠙⠛⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⡟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠴⣶⡞⣲⣶⠄⠀⠠⣶⣶⢲⣶⠦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⣿⡟⣿⣿⢶⣀⡴⣿⣿⢻⣿⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣶⣿⠿⡃⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠒⢹⠐⠉⠉⠉⠙⠳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣴⣿⣷⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢬⣛⣟⣙⣟⣧⠄⠀⣇⠀⠀⣀⣤⣶⠶⠾⠷⢶⣶⣤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⠁⢀⡼⡟⣦⠞⠛⠁⠀⠀⠀⠀⠀⠀⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡞⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢠⠏⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⠏⠀⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡇⠀⣰⠏⠀⠀⣇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡀⠀⠀⢿⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣾⠧⣄⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠴⠖⠋⢹⠃⠉⠓⠦⡌⢷⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣴⠏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠻⣦⣤⠉⠛⠒⠒⠒⠋⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠙⠛⠶⠶⠤⠶⠶⠞⠋⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠚⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⣤⡶⢲⣈⢠⢘⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⣸⣿⡏⠑⢸⠋⠛⠛⠛⣎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⢐⣁⣴⠞⢹⣿⣿⡀⠀⢀⢔⣻⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠗⠂⠨⠀⠀⣿⣿⡇⠀⢠⣿⣿⣧⢀⣰⣞⠉⢻⡗⠀⠀⠀⠀⢀⠐⢶⣄⠀⠀⠀⠀⢰⡏⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⢤⠀⠀⢹⣿⡇⠀⣿⠿⣋⣥⣾⠟⣽⣧⠿⠁⠀⠀⠀⢀⠎⠠⠀⢨⠀⠀⠀⠀⠘⡄⢠⢅
⠀⠀⠀⠀⡠⣄⣶⣦⡆⢘⣨⠐⠁⠀⠀⠀⠙⡀⠺⡁⡴⢟⣩⣤⡇⢘⠍⠀⠀⠀⠀⠀⠀⡆⠀⠌⡀⠀⢤⣠⣴⠈⣧⣔⣦⣀⣀
⠀⠠⠖⢋⣤⣾⣿⣏⠓⠚⢉⣹⣷⣦⠀⠀⠰⠋⠾⣛⣦⢊⡑⣿⣿⡇⠀⢒⡀⢠⣦⡜⢁⣿⣡⣌⠀⠼⣾⣯⠛⢳⣍⡬⢯⠿⠛⠲⠶⢄⡀
⠀⣶⣶⣟⠫⠿⢿⠿⣦⠷⠚⠿⣿⣿⠀⠀⣼⣿⣿⣿⣿⡏⢙⡟⡿⠗⢉⣮⣷⣾⡏⠁⠈⢀⠈⡃⠿⣶⡜⠛⣦⣎⢲⡥⢁⣰⠀⠀⠀⢀⠃
⠀⣌⢻⣿⣷⣄⣀⡐⠙⡖⠂⠺⢩⣭⣑⡀⣰⠖⡻⣶⣶⣽⣿⣿⣷⠶⢟⠛⣿⣿⣿⠀⠀⢸⢰⠀⢰⣿⣉⡀⣩⣽⣆⠘⡀⠇⠀⠜⠑⠸⡴
⢸⣿⣷⣽⣓⣄⠀⢋⢈⣡⣎⣀⣼⣟⡿⡯⠖⠊⣼⣿⠛⣿⡟⣛⣥⠴⠿⠶⠽⣿⣿⣆⠃⠜⠠⣠⣿⣿⡊⢋⣷⠶⢧⢁⢢⠡⠀⠀⡀⠀⡐
⠈⢯⣻⠿⠿⣢⣿⡿⠿⣿⣟⣀⡀⡀⣉⠠⣴⣦⣹⣯⣠⢉⣿⣿⡖⠒⢲⠒⣶⣾⣿⡿⣤⣥⣿⣿⣿⣿⣑⣩⣄⣀⣥⡄⡶⠎⠁⠀⠛⠉⠁
⠀⠀⠉⢹⡿⠟⠙⣿⣿⣿⣿⣿⣿⣶⣿⣥⣰⡶⡶⠈⠁⣾⣿⣿⡏⠀⢸⣰⣿⣿⣿⣿⣿⣿⠛⠁⢉⠛⠿⠿⢇⡤⠿⠿
⠀⠀⠀⠀⠱⡄⣠⣼⣯⡽⠖⠦⣀⣾⣿⣿⡟⡹⠇⣠⣖⡛⣹⣿⣧⣄⣸⠠⠿⠿⣿⣿⣿⣿⣷⣤⣀⣙⣌⣠⣾⣀⡀⠀⠄
⠀⠀⠀⠀⠀⢷⡙⣽⣿⠐⢤⡀⠘⢠⣾⠛⣵⢇⣲⣗⠟⠀⣟⠓⠍⡺⠿⣦⠈⣽⣿⢛⣿⠿⠛⠋⠁⠀⠀⠘⣁⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠙⠀⠀⠀⠀⠀⡠⣪⣺⣽⣿⣿⣿⣉⢬⢭⣭⣳⣦⣤⠷⣿⣏⢯⢥⣀⣠⡤⠤⢴⣒⣯⣷⡉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣯⣝⣾⣛⠽⠚⣭⣉⣤⣠⡠⡈⠉⠑⢧⠈⡟⣿⢫⠘⡿⠿⠇⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣫⠞⠑⠟⠲⠋⠁⠀⠁⠓⠁

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠙⡃
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠲⠆⠁⠁⠀⠀⠀⠀⠄⡀⠀⡀
⠀⠀⠀⠀⠠⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⡀⠀⠀⠀⠠⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠠

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⢀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣛⣩⣵⣾⡿⠿⢿⣿⣿⣿⠿⠿⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠉⢹⠛⠒⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡄⠀⠀⢸⣿⣷⣶⣤⣭⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⣃⡉⠁⣶⣶⣶⣶⣾⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠿⠷⣾⡏⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣴⣤⣤⣤⣠⣾⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠙⠛⠛⡻⠋⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢼⣦⣷⣄⠘⣿⣿⣿⣿⣿⣿⠋⠀⣰⣧⣼⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠈⣀⠩⣷⡄⠀⠀⠀⠀⠀⢠⣿⠿⡙⠟⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠈⠏⢨⡟⠁⠈⠉⠉⠁⠈⠳⡎⠘⠋⠈⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣶⣦⣶⢊⣾⣿⣷⣶⣾⣿⣦⡹⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢸⣿⡟⠄⠶⠸⠼⢮⡑
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠐⣡⡼⠛⣿⣧⠀⠀⢐⣲⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡸⠓⠀⠀⠀⣾⣿⠀⢠⣿⣿⠄⣠⣞⠙⣿⠀⠀⠀⠀⡀⠲⢆⠀⠀⠀⢰⠏⠉⠇
⠀⠀⠀⠀⡀⠀⠀⠑⠀⣐⠂⠀⠹⡿⠀⡾⠛⣤⠾⠏⢺⡗⠋⠀⠀⠀⢘⠁⢀⠀⠀⠀⠀⠈
⠀⣀⠤⢊⣾⣿⠧⠰⠶⣤⣀⠀⠀⡰⢤⡦⣊⠱⢾⣷⡔⠁⠠⢀⢀⣠⢘⡤⠀⠀⣀⣶⠿⢄⠟⢺⣶⣶⣀⣀
⢠⣤⡼⠟⢿⣿⣆⡤⠾⣿⣿⠃⠀⣷⣾⣾⣿⡞⢺⠿⠇⠂⣸⣼⡟⠃⠘⠉⠛⣶⣭⡿⣇⡜⢉⡉⠅⠀⠀⠈⠡
⢨⡻⣿⣦⣀⣀⠊⡆⠀⠊⣭⣁⡀⡹⢛⣧⣾⣿⣿⣷⠴⠟⢻⣿⣿⠀⠀⡇⠀⢰⣿⣁⣉⠽⡀⠓⠐⠀⠀⠀⢴
⢹⣿⣮⣵⢄⢈⣈⣴⣥⠴⠿⢟⠟⠀⢴⣿⠻⣿⣩⣥⠚⠛⠚⢿⣟⣦⠀⢁⣤⣿⣿⢉⠵⠚⠃⠘⢐⡀⠀⣄⣠
⠈⠻⢍⣩⣾⣿⣿⣿⣯⣦⣄⣤⠔⠛⣚⡿⠖⣸⣿⣿⠀⣿⢸⣿⣿⣿⣿⣿⡿⠿⣷⣤⡶⠐⣶⡼⠛
⠀⠀⠈⢯⠀⣘⣿⢿⣟⡛⣻⣿⣿⡾⣿⢀⣀⠿⣿⣟⡀⣿⢙⣿⣿⣿⣿⣧⣀⠐⠌⠉⣡⠋
⠀⠀⠀⠘⣗⢛⣿⠁⢄⠈⢋⣽⠟⡽⣄⣮⠿⢰⡋⠝⡳⢿⡄⢲⣿⠿⣿⠟⠛⠀⠀⠀⢋⣩⡆
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣠⣮⣺⣿⣿⣷⣩⣭⣽⣱⣶⣸⡾⣯⢮⡥⣤⠤⣤⣒⢪⣵⡌⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⢽⡯⣊⠉⠦⠖⠤⣲⠄⠈⠑⠈⠟⠏⠈⠛⠛⠋⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣫⣋⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠙⠻⢦⣤⣀⠀⠀⠀⠀⠀⣶⢶⡶⠛⠉⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠈⠙⢻⣟⣳⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⣛⠳⠶⠶⣤⣤⣤⣤⣍⣉⣀⣀⣀⣠⣤⣤⣤⣤⠶⠶⠞⣛⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⢛⢻⣿⠶⠶⣶⣤⣤⣤⣤⣵⣮⣤⣤⣤⣤⣶⠶⠶⣿⣟⡛⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣾⢸⣿⢸⣷⡖⢲⣶⣰⣤⢸⡇⣶⣦⡶⡦⣴⣿⠍⣿⡇⠃⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡏⣼⣿⣌⢿⡿⠿⣿⢿⢟⣸⣇⡻⠟⣵⡟⢻⡿⣡⣿⣧⢣⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣸⠤⣿⢸⣿⢸⣿⡾⠟⢡⣿⠋⠙⣿⣜⡞⠛⣿⠃⣿⡇⣿⠓⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣦⢹⣿⢣⣟⣿⣿⠿⢿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠻⢻⣿⢸⣇⣿⣿⠀⠀⡀⠸⡉⠠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡛⠿⣿⣶⣯⣽⣿⣫⣛⠛⠛⠋⣻⣟⢛⠛⢛⠻⠿⠟⠟⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣬⣭⣤⣬⣭⣽⣯⣭⣭⣤⣤⣴⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⡤⣶⣶⣶⣶⣶⣶⠀⣿⣶⣾⣿⣾⣾⣿⣶⣷⣷⣷⣾⣾⣶⣾⣿⣿⣶⣾⣿⣶⣶⣷⣷⣷⣶⣾⣾⣶⣶⣶⣿⣾⣷⣶⣶⣿⣷⣾
⢸⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡏⠀⢹⠈⠉⠉⠙⠛⠟⢿⢿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⢸
⠀⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠙⠛⠛⠛⠀⠀⠀⠙⠛⠛⠋⠀⠀⣼⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠻⠿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⢾
⠀⠀⠀⠀⢠⠞⢧⡀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠙⠋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⢠⠏⠀⠀⠳⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠟⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⢨⠀⢠⠏⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣶⣦⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠻⠇⠀⠀⢸
⢸⢠⣏⣀⣀⣀⣀⣀⣀⣘⠃⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠟⠉⠁⠀⠀⠉⠛⢿⣦⡀⠀⠀⡶⢶⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀⠛
⠀⠀⠀⠀⢈⣉⣭⣭⣍⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠆⠀⡄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠸⣿⠆⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⣿⡇⠀⡇⢠⠀⠰⣦⣭⣛⣛⡛⠛⠛⣛⣋⣭⣴
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠸⣿⣲⣿⠇⠈⠉⠉⠉⠁⠸⢿⣖⣿⠇⠀⡇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢐⢒⣒⣒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⣒⣛⠒⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⡇⠀⣿⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣮⠓⢄⠀⠀⠀⠀⢀⡴⢫⣶⣿⣿⣿⣿⣾⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃
⠀⡇⠀⣿⠀⠀⢀⣤⣾⠀⠀⠀⠀⠀⠀⢠⡿⠻⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⣀⣀⣾⣿⣿⣿⣿⣿⡿⠿⠻⣷⠀⡦⠀⠀⠀⠉⠹⠉⠀⠀⠀⠰
⠀⣷⠀⣿⠀⠀⢀⢿⠃⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣬⣙⡋⣿⡿⠁⢹⣧⣠⣿⠁⠘⣿⡟⢛⣩⣴⣶⣶⣷⣶⣦⣀⣤⣀⣀⣀⣀⣀
⠀⣿⠀⣿⠀⠠⠃⢸⠀⢀⣀⣤⣤⣤⣄⣙⣿⣿⣿⣿⣿⣿⣿⣮⠁⠀⠈⠁⣿⠉⠀⠀⢩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢀⣒⣂
⠀⣿⠀⣿⠀⠁⠀⢸⠀⣬⡍⣛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⣀⣤⣤⣿⣤⣤⣀⠀⠙⠛⠛⣛⣉⣉⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠐⡿⠀⣿⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⠀⣠⣶⣿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣟⢻⣿⡿⠿⠿⢿⠿⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣿
⢠⣷⠀⣿⠀⠀⠀⣸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⢟⣩⣴⣶⣿⣿⣿⣿⣿⣶⣮⣝⠿⣿⣿⣿⣿⣿⣟⣣⡿⣗⠀⠀⠀⠀⠀⠸⣿⣿⣿
⢸⣿⠀⣿⠀⠀⠀⢸⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⣈⣉⡥⠚⠁⠀⣿⠀⠀⠀⠀⣤
⢸⣿⠀⣿⠀⠀⠀⢸⠛⡶⢬⣝⣛⣛⠛⠛⣛⣛⣋⣍⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢠⣶⣶⣶
⠈⠿⠀⠿⠀⠀⠀⣼⡆⡗⠀⠈⠉⢹⠉⡋⠉⠉⢹⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠊⢸⡏⣧⠀⠀⠀⢸⠀⠀⠀⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢀⠞⣿⠳⡀⠀⠀⣿⠈⣿⣿⣿
⠀⠀⠀⢀⠜⠁⠙⠺⢟⠯⠱⡄⠀⠈⠀⡀⠀⢀⡿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣧⢣⣠⠊⠀⣿⠀⠙⢆⠀⠛⠀⣿⣿⣿
⠀⠀⡠⠋⠀⠀⠀⠀⠙⠀⠀⠘⢆⠀⠀⠀⠀⡞⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡛⠛⣷⠙⢄⠀⠛⠀⠀⠈⠣⡀⠀⠻⠿⠿
⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⣼⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⠀⢀⠞⠹⡄⠈⠣⡀⠀⠀⠀⠀⠙⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣾⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢿⣿⠟⠈⡿⠏⠁
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⡄⠙⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠻⠁⢠⣶⣦⣄⢀⠀⠀⠀⢠⣿⣿⡿⠻⣦
⠀⠀⠀⠀⢻⣿⣿⣯⡿⣿⣿⣿⡿⠟⠀⢰⣶⡌⠀⠀⠉⠙⠃⣙⣿⣿⡆⢸⡿⠖⣘⡋⢻⡄
⠀⠀⠀⠀⠈⣿⣿⡟⣶⣿⣿⢿⣆⠀⠀⠛⢿⡁⠀⠀⠀⠀⣸⡽⠋⠻⢏⣻⣷⣾⣻⣿⣿⠁⠀⠀⣴⣦⣤⣤⡀⠀⡠⠀⠐⢲⡄
⠀⠀⠀⠀⢰⣿⠏⣴⣿⣿⣿⡿⠛⠳⣄⠀⠘⣷⡀⠀⢠⣆⡏⠀⠀⢀⣤⡽⠯⣻⡛⠋⡇⠀⠀⠀⠀⠀⠀⣹⡄⠀⠁⣠⣚⣉⣇
⠀⠀⠀⣠⣿⠇⣸⣿⣿⣿⣿⣷⡀⣑⡬⢳⣄⣿⣷⣀⣨⣿⣀⠀⡰⠿⡀⠀⠀⢻⢇⣀⣱⣷⠀⠀⠀⠀⠀⠙⠇⠀⣿⡄⠙⢿⠇
⠀⠀⣼⣿⢿⣷⣿⠟⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣻⣿⣿⡇⠀⠀⠀⠈⡆⣀⠀⢸⣿⣿⣿⣰⡀⠀⠀⠀⢠⣤⣴⣿⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠉⠉
⢀⣼⣿⣴⣾⡿⠁⠀⠘⣿⠛⣛⢿⣿⣿⣿⣿⢫⣾⣿⣿⣿⠃⠀⠀⠀⠀⠶⠓⠲⡖⢻⣿⡟⣿⠁⠀⠀⣴⣿⣿⣿⣿⣤⣀⣼⣿⣷⡀⠀⠀⠀⢀⡀
⣼⣿⣯⣿⢿⡀⠀⠀⠀⠘⣿⣿⣇⣿⣿⡿⣡⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⢦⡀⠀⠇⠘⣿⣷⠾⡄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣀⠀⢀⡽⠿⠇
⠻⢿⣿⣿⣷⣟⢦⣀⠀⠀⢹⣿⣯⣾⢟⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢋⠄⡀⠀⣿⣿⣶⡷⠸⣿⣿⣿⣿⣿⣟⣿⡿⢿⣶⡿⠋⣠⣶⣿⡧
⠀⠀⠉⠻⠿⣿⣿⣮⣷⣦⣤⠟⠋⠻⣾⣿⣿⣿⣿⡿⠗⠀⠀⠈⠢⢄⡀⠀⠀⠀⠉⣱⠛⠻⣿⣁⣀⣤⣼⣿⣿⠿⠿⢻⣿⣿⢋⡴⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⠀⣀⠀⣠⣽⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠤⣿⣷⣾⣿⣿⣧⡄⠀⢺⡿⠿⠛⠉⣍⣴⣯⣀⣨⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⣾⣻⣿⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢛⣛⠋⣷⣤⠀⠀⢼⢻⢿⣿⣿⣿⣏⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣡⠟⡿⠛⠛⠛⠛⠿⠿⣿⠿⠧⠀⢀⡠⠖⢻⣿⣿⣿⡟⢹⡛⡛⢃⣼⡉⠉⠀⠀⢸⠀⠌⠛⠛⢩⡅⠀⠀⠠
⠀⠀⠀⠀⢠⣤⣤⢄⣴⡇⠀⣗⠀⠀⠀⠀⠀⢀⠖⢀⡀⠈⢣⣤⣤⣼⣿⣿⣿⣿⣶⣷⣶⣿⣿⣷⣤⣤⣤⣼⣤⡇⠀⠀⢸⣧⡓⠀⠀⢀
⠀⠀⠀⠀⢺⡿⢣⣾⣿⣧⡄⢻⠀⠀⠀⠀⠀⡾⠈⡉⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠛⣱⣿⠿⢿⣿⠄⠸⠀⠀⠀⠀⢠⠁⠄⠉⠁⠀⠀⠀⠘⢿⡿⠿⠿⠿⣿⠿⣿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠀⠀⠈⡿⠿⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠺⢽⣖⣒⠒⠒⢶⣗⠺⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠚⠓⠒⠒⢒⣶⣒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢢⡄⣀⣀⣉⠉⠒⠢⢄⡀⠀⠀⠀⠀⢀⡤⠖⠊⢉⣀⣠⣄⡤⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡡⠽⠀⠀⠈⠉⠉⠉⡡⠴⠢⢄⠀⠈⣩⣉⣁⣀⠀⠈⠦⣙⡦⣀
⠀⢠⣄⣀⣀⠀⣰⠉⠙⡿⢩⢤⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⣀⠙⢎⠁⠀⣀⠀⠑⢦⠀⠀⠈⠱⡟⠩⣱⡀
⠀⠀⣈⠝⣂⠭⢌⣙⠠⠷⣛⠻⢄⠀⠀⠀⢠⢊⡉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⢧⠀⠀⠀⣇⡜⠙⣮⢕⠢⢄
⠀⠀⠑⠊⠁⠀⠀⢨⠹⢇⡆⢠⠤⢍⣐⠢⠼⠸⣿⣷⡄⠀⠁⠘⣁⣀⡀⠂⠀⠀⠀⢀⡏⠀⠀⣠⠞⢇⡾⢾⣀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠞⣧⢽⢀⡴⠦⠬⠭⣵⡍⠛⠛⠂⣐⡀⣻⣿⣿⣷⣄⠁⢠⠎⠀⠸⠿⣷⡗⣐⠦⣶⣶⣾⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣄⢳⢠⢿⠀⠀⠀⠀⣩⣶⠖⠋⠉⠉⠁⠈⠳⣦⣍⣿⡇⠉⡉⠓⠒⠦⣤⡇⣿⠛⡻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⢸⠘⡈⠀⢀⠆⣼⠟⢁⡤⠒⠒⠲⡄⠀⠀⡈⠺⠁⠀⠀⠀⠀⠀⠁⢰⡄⣸⢈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⢸⠀⠄⠀⠈⢸⠇⠀⡎⢠⣄⣠⡀⢸⠀⠀⢈⠀⠀⠙⣿⣷⣶⡙⠀⣻⡇⣧⢰⠟
⠀⠀⠀⠀⠀⠀⡠⠼⡤⡷⢃⠔⠗⠀⢀⢸⡀⠀⢹⠌⠁⢩⠤⠃⠀⠀⠾⠀⠀⢠⠈⡼⠋⠉⢉⠼⢡⠤⠚⠢⣄
⠀⠀⠀⠀⡠⠞⡢⣄⠤⣖⠮⠇⢢⡄⣉⠀⢣⠀⠸⣶⢶⠞⠀⠀⠀⣸⠂⢀⠀⡦⠄⡇⢀⠴⠁⠂⠠⡀⠄⠒⠚⢢
⠀⠀⢀⣴⣥⢊⣝⣈⣈⣏⣤⣽⣆⣤⣿⣉⣠⡠⡀⠀⠀⠀⢀⠀⢜⠁⠀⣀⣞⣵⣦⣌⣥⡴⣦⣱⡄⣁⣠⣴⣤⣦⣧⡀
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠉⠁⠛⠛⠻⠿⠛⠛⠛⠒⠒⠒⠚⠋⠙⠛⠛⠻⠛⠓⠛⠛⠛⠓⠛⠚⠛⠛⠻⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣾⣯⣭⣭⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠰⣶⣶⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶
⠀⠸⡇⠀⣀⣠⡜⡿⡗⣒⣲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣀⣠⣤
⠀⡀⠀⠀⢀⠁⠘⠐⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠆⠀⠈⠑⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⣂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⣀⢀⡴⠿⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠀⠐⣸⣿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠑⢀⠀⡀⠒⠀⠉⠀⠀⠆⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣿⣿⠃⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀⠀⠀⠀⠀⠀⠸⣀⣀⣀⠇⠀⠀⡤⠠⠄⠉⠱⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⡷⠃⠂
⠈⠉⠒⠒⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⠠⣀⠀⢰⠇⠉⠘⠆⠀⠀⠀⠀⠀⠖⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠐⠠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠠⢀⠀⠀⠏⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠐⠉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢠⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣀⡤⠴⠒⠋⢉⣀⣤⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠙⠢⣄⠀⠀⠀⠀⠀⢸⢡⠞⠁⠀⠙⢶⣋⣉⣤⠤⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⠑⢦⡀⣀⣀⠀⢸⡀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⠏⠁⠈⠙⣆⠙⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⢻⡀⠀⠀⢀⡾⠉⠉⠉⠉⠉⠛⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠙⠶⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠋⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠲⢤⣀⡈⠉⠙⠓⠚⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠴⠶⠒⠚⠋⠉⢁⣀⡴⠞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢧⠀⠈⠉⠛⠒⣶⡤⢤⣤⣤⣀⣀⣀⣀⣀⣠⣄⣤⠤⠶⢶⠾⣏⡉⠁⠀⢠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⠀⠀⣠⠞⠁⠹⣆⠀⠀⠀⠀⣰⠏⠙⢦⡀⠀⠀⢠⠏⠀⠀⠙⢦⣀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣧⠞⠁⠀⠀⠀⠈⢧⡀⢀⡾⠁⠀⠀⠀⠙⢷⣴⠏⠀⠀⠀⠀⢀⡿⣦⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠋⠙⣇⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣴⠛⢦⡀⠀⠀⠀⡜⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠸⡆⠀⠀⢀⡴⠋⠀⠹⣄⠀⠀⠀⢠⡞⠁⠀⠈⢻⣄⠀⡼⠁⠀⣀⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠓⠲⠤⣿⣤⡞⠉⠀⠀⠀⠀⠸⣆⢀⡴⠋⠀⠀⠀⠀⢀⣹⣾⠥⠖⠋⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⡀⠉⠉⠙⠓⠒⠒⠒⠛⠛⠒⠒⠒⠛⠉⠉⠉⢀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠶⠤⠤⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⣻⣿⣿⣿⠀⠀⢢⣀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣶⣦⣈⣿⣿⣿⣤⣶⣾⣷⣶⣾⣿⣿⣿⠿⢷⡦⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠐⢶⣶⣦⣤⣄⡀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣶⣤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠉⠙⠛⣛⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⠀⢀⣴⣾⣿⠿⠿⡟⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠉⠁⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡄⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⣿⣿⣿⠛⠛⢛⣻⣿⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠁⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣅⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡾⢫⡏⠳⣄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠀⠑⢄⠀⠀⣠⠚⡷⣦⡇
⠀⠀⠀⠀⠀⢠⠞⢤⢺⡇⢀⠜⠻⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠂⡴⠋⣳⣾⣁⣀⣧⣿⣷⣀
⠀⠀⠀⠀⣰⠋⠀⠈⠻⠓⠁⠀⠀⠈⢢⡈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠛⠉⢀⠔⠁⠈⣿⡤⣿⠋⠀⠻⡍⠉⠛⠒⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠙⠛⠛⠙⠁⠀⠀⠟⠀⠀⠋⠀⠀⠘⠁⠀⠀⣤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠆
⠀⠀⠀⠀⢀⠰⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢵⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣴⠄⠀⠀⢄⣀⣀⣀⣀⣀⣀⣀⠌⠁⠀⠀⣈⠣⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡇⢀⡄⠐⡆⡿⠿⠿⠿⠟⠻⠿⡄⠀⠀⢈⠞⣰⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣷⡛⠉⠉⠑⠤⡁⠨⠜⠀⠀⠀⠀⠀⠀⠀⠈⢐⡀⠀⠊⠀⠀⠉⠑⢽⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣙⠧⢄⣀⠀⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⢠⠤⢒⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠉⠒⠀⠈⠉⠁⢀⣒⣒⣒⣒⣒⣒⡒⠀⠀⠉⠉⠉⠀⠀⠂⠉⠈⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣠⠴⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠉⠉⠒⠒⠦⢤⠤⠤⠠⠀⣀⡀⠠⠥⠤⠤⠤⠒⠚⡉⠀⡘⢥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠁⢣⠀⠀⠀⠀⢀⡎⠆⠀⠀⠀⠀⠀⠀⠀⡚⡀⢕⠁⡨⠐⠀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢄⠐⠀⠀⠀⢀⠊⠠⠈⡄⠀⠀⠀⠀⠀⠌⠠⠲⡀⢊⢔⠁⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠐⠄⠄⠀⠀⢎⠴⠀⠂⠠⠀⠀⠀⠀⠌⠰⢄⠀⢡⠀⡴⠀⠧⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢐⠄⢰⠀⠌⣐⠄⡀⠀⡀⠄⠀⠀⠀⢀⠢⢑⠢⡀⠵⡀⢀⠡⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠷⣤⣈⠖⠠⠐⠠⠀⠄⡠⠘⡄⢠⠀⢁⠪⠐⠡⣐⠄⢣⡌⡈⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⠙⠒⠻⠶⣤⣆⣥⣠⣁⡰⠆⠈⡀⠨⣒⣀⣠⣷⢿⣶⢟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢀⡀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠉⠉⠉⠥⠄⣼⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠠⠄⠀⠀⠀⠐⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠐⠁⠀⠀⢐⠠⡂⠔⡠⢒⠠⢂⠄⠀⠔⠀⠄⠠⠂⠀⠂⠀⠀⢂⠔⡠⢂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠊⠔⠀⠐⡡⢀⠡⢊⠄⠈⠀⠠⠀⠡⠂⠄⠠⠈⠐⠊⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣀⣆⡠⠐⠲⡶⠀⢠⢠⠀⠀⠀⠯⠉⠀⠀⠀⠀⣰⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣹⡇⡀⠰⢲⣀⣭⡷⢾⡏⢀⠻⠉⠀⠀⠀⣠⣴⡏⠀⠀⡀⠻⡮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢛⣃⣥⣔⠚⠋⢡⣤⡀⠈⠁⠃⠠⣦⡀⡀⣐⡩⠛⠑⠡⣛⡅⠈⠉⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣀⠉⠁⠀⠀⠙⠅⢀⠀⢐⡐⣷⡜⠛⣳⡃⠠⠄⣀⡀⣿⠐⠋⢹⣄
⠀⠀⠀⠀⠀⠐⠒⡙⢿⡃⠀⡾⠒⠀⠀⠀⢠⣤⣞⠐⠀⠉⠊⠰⢈⣡⠷⠚⠋⠉⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠆⠀⠀⠀⠀⠀⢧⠝⢁⣀⢀⡀⠅⢰⣦⠀⠀⠠⣿⠓⠀⠀⢰⢻⡆⠀⠀⠰⠯⠥⢀⡀
⠀⠀⠀⠀⡿⠖⠀⠀⠁⠀⡰⢶⣤⡠⢄⠤⡄⠈⡉⠌⣛⠄⢀⢀⠀⠀⠀⠀⢈⢠⣾⣄⠿⣏⡃⡘⠠⠀⠋⣧⠴⠞⠛⠃
⠀⠀⠀⠀⠉⢀⡤⠀⢠⠀⣷⡘⣿⠃⢈⡟⡃⠂⢀⣘⡏⠀⠛⡌⠀⠀⠀⢀⠛⡿⠃⠳⡍⠑⠀⠀⠀⠈⠉⢀⡀
⠀⠀⠀⣀⠾⡿⠊⠀⠈⠉⡚⠳⠇⠘⣨⡤⢷⠋⢯⠀⠀⠂⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠛⠙⠾⢿⣶⠀⠀⢸⡛⠁
⠀⠀⠀⠈⠋⠀⠀⡀⡀⠐⠇⢲⣶⡄⠀⠀⢴⠿⠆⠀⠀⠀⡤⣶⡶⠀⣔⣲⢀⣀⢤⣠⠄⣴⣐⠀⠉⠀⠀⢀⣀⡴⡂
⠀⠀⠀⠀⠀⠀⠡⣛⡅⠀⠀⡀⡀⠀⠀⠀⠀⠁⣠⠄⣂⢐⣷⢹⠇⡀⠔⠧⣈⣽⣶⣿⡎⠉⠀⠀⡀⠀⠠⣌⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣐⣃⠉⠀⠀⠀⡀⠿⠗⠁⠻⠉⠚⢅⣀⡀⠒⠉⠃⠈⠡⠇⠁⣠⣄⠐⠧⠷⢁⠈
⠀⠀⠀⠐⠃⠘⠃⠀⡄⢼⡏⣿⠀⠀⢀⠀⡠⢀⠀⣀⣀⣀⠐⢿⠋⠁⠀⢀⣀⠀⠀⠀⢰⣧⢿⢐⣃⡧⠴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⢿⣺⣶⠄⠀⠈⣓⣚⠣⠥⣙⡣⢠⡇⠆⠀⠀⠘⡏⠁⠀⠠⠪⠽⡘⠉⠁⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⡹⣿⡄⠠⣀⣥⠷⠚⠋⢹⢣⠀⠘⠀⠀⠀⠀⣄⣼⡁⠀⢠⠤⢖⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠲⠄⠈⠀⠈⣿⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⢨⣀⣀⠀⠂⡹⠋⠀⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⢀⢿⡇⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣋⣉⣉⣉⣻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣿⣖⠋⠉⠉⠀⠀⠀⠁⣽⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠍⢊⠇⠘⠶⠆⠀⠀⠀⠀⠀⠀⠈⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣄⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢷⡘⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣯⠿⣿⡾⡗⠐⣤⡆⠀⢀⡀⣤⣤⣶⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠚⢻⣾⡷⣿⣾⣿⣿⣿⣪⡿⠛⠋⠉⠉⠉⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⢻⡟⢿⣻⣯⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⠷⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣻⣿⠏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠾⢷⠄⠆⠀⠀⢀⠀⠀⠉⠉⠛⠒⠒⠶⠶⢤⣤⣤⣦⣼⣦⣤⣿⣧⣁⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠋⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠲⣶⣾⣿⠿⠿⠛⠋⠉⠉⠙⣛⣮⡙⢦⣀⣠⡤⢶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢠⡽⢦⡙⢶⡊⠉
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⡇⣰⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣬⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡
⠀⠀⠀⠀⠀⠀⠀⣇⣟⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠖⣾⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡷⣷⢉⠐⠦⣴⢠⣀⣀⣀⠀⠀⠈⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣷⣵⣟⠗⣶⣿⢾⣿⣿⢈⡖⡀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣦⣌⡃⠈⡟⢊⣷⣿⣮⣷⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠮⠷⠀⢉⡾⠠⣬⡻⢿⣿⣿⣆⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡼⠁⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠓⢿⣿⣿⣗⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣣⠟⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣶⣤⣤⣀⣠⣤⡴⣟⡭⠞⠁⠀⠀⠀⠀⠀⠀⢳⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣄⡜