Larva Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⡴⠒⠛⠉⠉⠉⠛⠒⢶⠶⠶⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣷⠤⠶⠤⠤⢤⣄⣀⣄⣤⢦⣷⠀⢰⣻⠄⢀⡇⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠈⢻⡛⠛⠒⠲⠤⣼⠻⣿⣏⣸⣧⣀⠀⣀⣼⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢻⠀⠈⣿⣅⣠⠏⣯⡉⠈⣿⣶⠶⠶⢾⡀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⣀⠞⠛⠀⣿⠀⡇⠉⠛⡇⢸⢦⡀⠀⢹⢶⢶⠖⠋⢡⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢹
⠀⠀⣀⡤⢤⣄⡀⠀⠀⠀⠀⠾⠷⠚⠁⠀⠀⠀⡇⠀⣇⠀⢸⡇⢸⡄⠉⢓⡏⣸⠈⢧⣀⠈⠋⠁⠀⠀⠀⢀⣀⣤⠴⠋⠁⠀⠀⠀⠀⠀⠘
⢀⡞⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⣤⠤⠷⣤⡿⠀⠀⡇⠀⣧⠀⣸⠁⣿⣄⠀⢸⠉⢉⣉⡉⠉⠉⠉⠁
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢰⡻⣤⠖⠚⠉⢀⡴⠒⢛⡶⠟⠀⣿⠀⢿⢸⡀⠈⣧⡘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠀⠀⣀⡤⠖⠛⠉⠉⠙⠛⠳⣄⠀⠀⠙⠓⠀⠀⢰⣻⣤⠞⣫⣶⠛⡉⣛⣶⠞⠈⡷⢰⠃⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺
⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⠋⠀⠀⠷⠛⠚⠉⠉⠀⠀⢰⡇⣸⠐⢯⠇⠈⠙⠲⠦⢤⣤⣤⣤⡤⠴⠒⠋⠁⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⣿⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣽⢰⣳⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠈⣇⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⣳⠧⢶⣿⠒⢒⣦⢶⣒⠛⣳⠞⣟⡷⠀⡏⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠸⡄⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠘⠞⡵⠋⠀⠻⠞⢰⠃⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠙⣶⠋⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠤⣤⣤⡜⠁
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢷⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠓⠦⢤⣤⣄⣀⣀⣀⣠⣼⠦⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣳⣦⡀⠀⠀⠀⠀⢼⢰⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⣿⣿⣦⠀⠀⠀⠈⡞⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⡀⠀⠀⣇⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣆⠀⡿⢿⣽⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠿⠛⠿⠿⠛⢶⡁⠀⠀⠀⢀⡀⢀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡐⠿⢃⡆⠀⠀⠀⠀⠀⠈⢶⣾⣿⡿⠟⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡶⠛⡟⠛⣠⠄⠀⠀⠀⠀⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢰⡟⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⠰⠀⣴⢀⣼⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠿⢿⠠⠴⢷⣼⡿⠧⠄⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⢀⠘⠏⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⣄⢠⡁⠀⠀⣾⣴⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠻⢿⣿⠦⠶⢶⣿⡟⠃⠀⠀⠀⣠⡿⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠉⢹⠋⠀⠀⢸⣿⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⢠⡇⠀⠀⢠⣿⡟⠀⠀⠀⠀⠈⣷⣴⣶⣦⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡿⣿⣶⡇⠰⠞⣻⣿⠃⠴⠀⠀⠀⣰⠃⠈⠻⣟⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡀⠀⡿⠁⠀⢠⣿⡇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣾⠀⠀⣀⣀⣿⣧⡀⠀⠀⠀⠀⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⠇⠘⠛⣿⠟⠉⠉⠁⠀⢀⡾⠉⠙⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⢧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⢸⣤⡴⠦⢴⣿⠷⠶⠄⠀⠀⠀⡿⠻⣿⣿⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⢿⡟⠁⠀⠀⣾⡿⠀⠀⠈⠳⠀⡈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣑⣄⣁⢠⠖⢀⣿⠠⠶⠀⠀⠀⢀⣿⣦⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠻⡏⠀⠠⣾⡟⠀⠀⠀⠀⠀⠞⢻⠙⢿⠆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠴⠃⠖⢲⣶⠇⢠⡄⠀⠀⢰⠇⠀⠀⠀⠁
⠀⠀⠠⠶⢦⣌⠳⣀⣠⠖⠃⠘⣿⣆⣙⠈⠀⢋⠁⠀⠀⢷⣄
⠀⣤⣤⡶⠶⢿⣷⡟⠡⠀⠀⠀⣀⣼⣿⣷⡄⠈⠀⠀⠈⠣⠉⣷⠞⢷⣶⣶⣶⣤⣄⡀
⠀⠀⠀⢀⡔⠋⢹⠁⠀⠀⠀⠈⢛⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠻⣆⠀⠉⠛⠿⠿⢿⣿⣆
⠀⠀⠠⢶⣿⣿⣿⠀⠀⠀⠀⠀⠙⠿⠋⠁⠁⠀⠀⠀⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⠀⠉⠛
⠀⣤⣾⠟⠋⢁⣬⣶⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⣿⣿⣿⣶⣤⣀
⠘⠉⢀⣤⣾⣿⡿⢋⣀⠀⣠⣤⣄⡀⠈⢀⣠⣤⣤⣤⠶⣾⣏⠀⠀⠘⠣⠙⠿⣿⣿⣆
⠀⠈⣽⠿⠙⠉⣴⡿⢋⣴⢿⣿⣿⣿⣿⣿⣿⣷⡿⠿⣦⠸⠗⣷⣄⠀⠀⠀⠘⠄⠛⠿⡄
⠀⠀⠀⠀⠀⠞⠁⢠⡾⠁⣿⣿⡟⠛⣻⣿⣿⣿⣷⣦⣄⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠈⣿⣧⡘⠻⢿⡇⠀⠸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠟⢿⣷⡶⠀⠀⢤⣴⣆⠴⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⡴⠞⢠⡄⠀⡀⠈⠟⠋⠀⠰⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠀⠀⠈⠃⠘⠛⠀⠀⣤⢸⡷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⢀⡴
⠀⠀⠹⣿⣶⣤⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉
⠀⠀⠀⠹⣿⣿⡿⠗⢠⣀⠀⠔⢙⣤⠀⠀⠀⢶⣀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠈⠛⢡⣤⠉⠘⠃⠘⠁⢸⣀⣀⣠⣴⢿⣄⣀⡤⠸⠀⠀⠆⠒⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠐⠒⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣠⣬⠶⠚⠉⠉⠙⢥⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀⠀⠀⠀⠀⠀⢀⡤⠐⣈⠠⠄⠱⢇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠦⠤⠄⠾⣒⠬⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠛⢳⡄⠠⠠⠤⠀⠀⠀⠔⠒⢿⣦⡀
⠀⠀⣀⣴⣞⣿⣙⠠⠉⠓⠢⠄⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⢰⣦⠄⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠊⠙⢶⣛⠛⣟⢱⠻⠄⢈⠁⣳⠀⣼⡈⢸⡿⣽⠖⢦⡀⠀⠀⠀⠴⠀⠚⢱⣎⠀⠹⠆⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⠀⠈⢨⡍⢋⡱⣉⣎⢃⠂⠉⢲⡍⢤⡈⠣⡞⠣⠾⠣⢞⣹⡆⠀⠀⠀⠀⠀⠀⠙⡟⠰⢿⣷⣤⣤⣼⣿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠁
⠀⠀⠀⠑⠃⠀⠀⠁⠉⢦⡄⣧⡤⢦⣤⠞⣭⡄⠒⠐⢮⡽⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⠏⢻⡄⠉⠉⠉⠉⠋⠉⠙⠷⢦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠁⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⡄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋
⠀⠀⠀⡀⢀⣤⣉⢂⡠⣀⣲⣿⡄⣦⣀⠀⠘⠛⠷⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣸⢃⣤⣄⠀⢼⣿⣿⣉⢹⠿⡿⠹⡷⣧⡀⢤⠀⠛⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣤⣤⣭⠀⠚⣻⢧⢫⣤⢲⣾⣷⣶⡝⠋⠉⠂
⠀⠀⢯⠈⡿⠛⠀⠘⢻⣿⠿⠋⠚⠳⠴⠥⢘⣻⣿⠁⠘⢆⡀⠱⠀⠀⠀⢀⣤⣾⣿⢿⡆⠙⣿⣷⣦⢰⣼⢽⡶⢰⣶⣟⠩⠟⣯⡿
⠀⠀⠀⠙⡀⠀⣀⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢃⠀⠀⠈⠀⠀⠰⠿⣿⡿⠿⠿⠇⠽⠓⠉⠩⣼⠵⠻⠔⡋⠀⠉⠳⡀⠘⢃
⠀⠀⠀⠀⠀⠺⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢄⡀⠀⠈⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠠⡔⠂⠐⠲⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠐⠪⠀⠀⠀⠀⠁⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠚⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⠀⠀⠀⠒⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣴⡆⢀⡛⠙⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⢠⡀⠀⠀⠀⠀⣰⠋⠉⠉⠛⢦⡀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⣠⡶⠋⠀⠀⠀⠀⠀⢻⡄⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠐⠦⠀⠀⠀⠀⠀⠈⠐⠤⢠⣀⣀⡌⠀⠀⠀⠀⠀⠀⠘⡏⠁⠰⢄⠀⠀⠀⠀⢠⣿⢋⡠⠞⠁
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠛⠿⣄⠀⠀⠀⠀⣀⡾⠁⠀⠀⠈⢷⣀⣠⠔⠉⠙⢫
⠀⠀⠀⠀⡠⠊⠁⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣾⣏⠀⠀⠀⢀⠀⢸⣿⢡⣠⣥⠼⠃
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠐⢠⡄⣀⠀⠀⠀⣀⢠⡀⠀⠀⠀⠀⠀⢀⣿⡟⠻⣷⡀⠀⣈⡵⠉⢛⢟⠛⠉
⠀⠀⠀⠘⢄⣀⣀⣀⠠⠴⠒⠈⠀⠀⠀⠀⠀⠴⠀⠌⠙⠢⣄⠀⠀⢀⣾⡟⠀⢀⣿⣿⣿⣥⣴⣁⣦⠾⠃
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠛⢿⡿⠦⣦⣾⣿⣿⣿⣿⠏⠋
⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣾⣿⣽⣯⣿⠇
⠀⠀⠀⠀⡸⠶⠦⠤⠤⠀⢀⣀⡀⠀⣀⠀⠀⠀⠈⠒⠶⠶⠤⠤⢾⡿⠶⠛⢻⣟⠛⣋⣹⣷⡀
⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⢈⠀⠀⠀⠀⠀⠀⣿⠀⢀⣸⣿⣾⠟⠁⣸⣿⢦⡀
⠀⠀⠀⠀⠱⣄⡀⣀⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣠⡿⢷⣞⠉⠈⠙⣦⣼⡿⠋⠀⢹⣶⣀⣀⡀
⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠛⠋⠁⠀⠀⠹⣦⣤⣴⠞⠋⠓⢦⣄⣿⠏⠀⠀⢉⢳⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡤⠀⠶⠀⢄⠀⠀⠀⠀⠀⠀⣼⠟⠓⢦⣄⡀⠀⣿⠛⠓⠶⣤⣀⡿⠀⠀⠉⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢣⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣤⣴⠟⠁⠀⠀⠀⠈⢹⡿⠧⣄⣀⡀⠈⣿⠉⠉⠒⠶⣿⢃⡚⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣇⠀⠀⠀⠀⠀⠀⢈⡾⠃⠀⠀⠉⠙⢿⠿⠶⠶⢤⣀⣼⡀⠉⠺⣟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠁⠃⠈⠓⢦⣄⣀⣀⣴⠏⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣯⠉⠓⠦⣼⠆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⣀⠄⠂⠁⠀⠐⠀⠀⠀⠀⠀⠈⠹⠛⠳⢤⣦⣬⣥⣽⣌⣾⣁⣄⣐⣁⣀⡴⢇⠀⠀⠀⢻⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡔⠋⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⠴⠁⠉⠀⠀⠀⠀⠀⠀⣤⣏⡉⠉⠉⠉⠁⢰⢼⡟⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⠔⠁⠀⠀⡀⢀⠔⠉⠀⡅⠀⠀⠀⠀⠀⠀⡴⠃⣿⠁⠀⠀⡀⣰⢇⣾⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠉⠻⣄⣀⠀⣠⠜⠁⠀⠀⢰⠀⢀⠀⠀⠀⢀⡞⠁⡀⣿⢀⣀⣴⠿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣄⣠⣀⣃⣭⣀⣣⣀⣠⡶⣿⣤⡼⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢰⣿⢹⢽⣽⡿⢳⣶⣤⣀⠀⠀⠳⢦⣠⡔⡀⠀⠀⢀⣦⢀⡶⠟⠀⠀⣀⣤⢶⡺⢿⣏⠭⡯⣿⡄
⠀⠀⠀⠈⢻⣾⣻⣿⡏⡽⡾⣿⡏⢹⡦⣄⠀⡹⡏⣽⣿⣿⡏⢹⡍⠀⣠⢴⡋⢹⣿⣷⣏⢹⣯⣟⣳⠟
⠀⠀⠀⠀⠀⠙⢷⣟⣿⣹⢿⡿⡆⢸⡡⣾⡛⢧⣷⡟⠛⠛⢷⣎⡿⢻⣶⢸⡇⣴⢻⡿⣭⣿⣻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡶⣻⣆⣿⠳⠼⣷⢿⠑⣾⣿⣷⠢⠲⣿⣿⡷⠊⡿⣼⠥⢿⣟⣸⢟⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣿⠿⣿⡟⠛⡟⠯⠿⢿⣷⣾⡿⢯⠟⣻⠛⢓⡿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⣿⣷⣄⡙⠣⣈⠚⣷⠿⣿⣿⢿⣾⠒⣡⠞⢉⣤⣾⣟⣯⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣿⣻⣶⣾⣽⣿⣿⣗⣺⣿⢻⣯⣷⣾⣟⣽⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⠿⢻⢹⣷⣯⣽⣺⠸⡝⠿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡿⣗⣺⢿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣤
⠀⠀⠀⠀⠀⠀⠻⣩⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⡿⣁
⠀⠀⠀⠀⠘⡍⣹⠛⡏⢩⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⣟⡧
⠀⠀⠀⠀⠀⡿⠁⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⠄⠠⡀⡌⢀⠀⠀⠀⠀⠀⠀⣾⣥⣄⠄⢹⣯⠉⠁
⠀⠀⠀⠀⣵⠁⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⡀⠠⣤⣿⣶⣄⡡⢃⠀⠀⠀⠀⠀⠀⡛⣿⢾⣆⢨⢯⢿⡀
⠀⠀⠀⠀⠸⡍⠀⠀⠀⣹⠃⠀⠀⠀⠈⠐⠌⠍⣿⡛⣠⣘⢙⣿⡊⢡⠂⢁⠀⠀⠘⡇⣯⡽⢻⠾⢸⣾⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢼⠀⠀⠀⠠⠄⠀⢝⡲⢿⣧⣍⣀⣰⡿⢉⡫⠥⠤⠄⠀⠀⢿⡌⣿⡼⡖⣺⡧
⠀⠀⠀⠀⠀⢳⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢈⠧⡉⠽⢻⠿⠏⣃⡜⠁⠀⠀⠀⠀⠀⠘⣮⢯⡹⢲⡯⠁
⠀⠀⠀⠀⠀⠈⢧⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⠀⠐⡅⠀⠙⠦⠀⠀⠀⠀⠀⠀⣿⢿⡽⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⠿⠿⠿⠟⠿⠿⠿⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢦⣴⠞⠛⠛⠓⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣀⡤⢴⣯⠁⠀⠀⠀⣿⡇⠀⠀⠀⣸⡿⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢻⣧⠀⠀⠀⢻⡇⠀⠀⣰⡿⠁⠀⠀⢀⣾⠟⠳⢦⡀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⣠⠴⠿⣦⣀⠀⠀⠀⢻⣧⣠⠴⠞⢳⣶⡶⢿⣅⡀⠀⢠⣾⠋⠀⠀⠀⣽⣄⡀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠻⣷⣄⡴⠞⠁⠀⠀⠀⠈⣿⡇⠀⣿⣿⠳⢿⡁⠀⠀⣠⡾⠋⠀⢹⡄⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢠⣷⣤⣀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢿⡇⠀⠸⣿⡷⠾⣟⠁⠀⢀⣼⡇⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⢰⠏⠀⠉⠛⠿⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠻⣇⠀⠘⠳⣴⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣹⠿⠶⠶⢶⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢿⡀⣀⣠⣤⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠈⣿⠛⠋⠉⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣠⣾⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠳⣤⣶⠿⠋⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⣼⠟⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠈⠳⣤⣰⣿⠋⠀⠀⠀⣼⡟⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⢰⣿⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠶⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣷⣶⣶⣶⣾⣿⠿⠿⠟⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⢀⣀⣀⠀⢀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡟⠛⠉⠙⠛⢿⣧⣀⡀
⠀⠀⠀⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢁⣤⣬⡻⣦⠀⢀⡾⢋⣭⣭⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠂
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠷⠾⠃⣿⠀⢸⡇⠸⠷⠿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣤⡾⠋⠀⠈⢷⣄⣀⣀⣤⡿⠁⠀⠀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⢀⣴⠟⠉⢙⣻⣷⣶⣶⣶⣶⣟⡛⠉⠻⣦⡀⠀⢀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣿⠁⠀⢠⠏⠀⠈⡏⠀⢹⡇⠀⠙⡆⠀⠈⣿⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⠚⢦⡤⠞⠁⠀⠀⠳⢤⣠⠟⣄⣠⣾⠃
⢠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣉⣻⣿⣶⠶⠶⠶⠶⠶⠶⣶⣿⣟⣉⠙⣷⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤
⠀⠀⠉⠙⠛⠳⠶⠶⣤⣤⣤⣸⣧⠸⡀⠀⠀⠈⢱⡄⠀⠀⢠⡏⠀⠀⠀⣨⠇⣼⣧⣤⣤⡴⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣷⣬⣑⣒⠒⠋⠀⠀⠀⠀⠑⠒⣒⣫⣵⢾⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡤⠛⠻⠿⣛⠛⠛⠛⠛⢛⡿⠟⠛⠦⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣦⣀⠀⠀⣨⠇⠀⠀⠰⣏⠀⠀⣀⣠⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠟⠻⢿⣯⣥⣤⣤⣤⣤⣬⣽⣿⠿⠻⣿⣶⣤⣀⡀
⠀⢀⣀⣤⣤⣤⣴⠶⠟⠛⠉⢁⣿⢡⢶⡀⢸⣠⠴⢤⡀⠀⡠⠶⢄⣯⢠⡶⡜⣷⠀⠉⠙⠛⠶⠶⣦⣤⣤⣤⡀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣌⣉⣠⠞⠳⣄⡴⠃⠈⠧⣤⠼⠙⣄⣉⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣩⣍⠛⠛⢻⣶⡶⠶⠶⣶⣶⠾⠛⠛⡫⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣝⣋⠀⠀⢏⣀⣸⠂⠰⣅⢀⡵⠀⠀⣓⣫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠽⡛⠛⢻⣿⡷⠶⠶⣾⣿⠿⠛⢻⢯⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣛⠁⠀⣏⢀⡹⠀⢸⡁⢈⡇⠀⠈⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡛⣿⠿⣿⡶⠶⢶⣾⠿⢶⢟⣛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡼⢮⡁⣹⠀⣏⠈⡷⢧⡘⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣿⠻⢿⣷⠶⣾⡿⠷⣿⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣄⣧⣼⣘⣥⣗⣠⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠍⡽⣏⡹⢯⠩⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡶⣿⣥⣼⣿⢶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣻⣧⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⣿⣿⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠋⠁
⣿⢿⠿⠿⡿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣿⣛⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠁⠈⠀⠁⠈⠏⢻⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠶⢻⡋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣧⡠⠀⠁⠀⣀⢀⣰⣤⣤⣾⣄⣠⡆⠀⠀⠀⠀⠠⠛⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠾⢿⠏⠹⠁⠀⠁⠀⠙⠙⢷⣤⠎⠀⠀⠀⠁⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣾⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠶⣏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣦⣤⣤⣀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡾⣯⡾⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠙⠁⢠⣄⠈⢻⡏
⠀⠀⠀⠀⠀⠀⣴⡞⢫⠉⣇⣸⣀⣏⢙⡟⠷⣦⣦⣤⣴⡶⢟⣹⣶⠋⠁⠀⠀⠀⠀⠀⠀⣠⠟⠁⠙⠀⢠⣄⠀⠀⣉⣠⡿
⠀⠀⠀⢀⣠⣼⣏⣳⣾⠛⠋⠉⠙⠉⠛⠓⢾⠦⣿⣼⣤⣷⠾⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠀⠀⣀⠀⠀⠛⠂⠀⣸⠏
⠀⠀⠀⠀⠀⡹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠁⠀⠙⠷⠀⠀⢶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠀⠀⠻⡆⠀⢤⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠦⠀⢤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠛⠁⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠻⠉⡄⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡏⢀⡀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣅⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶⠟⢉⠀⠀⠀⣁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠈⠙⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠃⡀⠀⢨⣃⡵⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⣀⣤⡴⠛⠉⠙⢦⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣶⠛⠛⠟⠛⣿⠉⠉⢻⡆⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣤⣟⣀⣀⣠⣧⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡦⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠉⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠃⠀⠀⢾⣷⣶⢴⣾⣿⣿⣶⠀⣀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣾⠟⠀⠀⢹⣿⣿⣿⣿⠏⠀⢸⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⠀⠀⢠⣿⣿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⢼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⡰⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⣄⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣴⠛⠁⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠐⠶⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡋⣷⣶⠀⠉⠀⠀⢀⠀⠰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣶⣄⣀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁⣀⣠⣶⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢹⣧⡀⠰⠶⠶⠛⠒⠶⠾⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⣟⡁⠀⠀⠀⠀⠀⠀⢀⡀
⣤⣤⣤⡄⠠⣶⣄⣀⣀⣀⣀⣤⣤⠈⢿⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣿⣶⠛⠛⠛⠛⠛⠛⠛⠂⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠿⠿⢦⣄⡀⠀⠀⠀⢠⣿⡀⠀⣀⣠⠶⠖⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⣷⣼⣿⣧⣴⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡀⢰⣿⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⣀⠰⢿⣃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠑⣾⣿⡀⠀⠀⠘⣿⣇⡀
⠀⠀⠀⠀⣀⠜⠋⠀⡠⠟⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠸⢧⠀⠉⢿⣆⡀
⠀⠀⣠⡾⠁⠀⢀⡾⠃⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠑⣄⠀⠙⢿⡆
⠀⠐⠟⠁⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠈⠀⠀⠈⠻
⠀⠀⠀⠀⠄⠁⠀⠀⠀⢠⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡟⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⡜⠛⣤⠀⠰⣶⣆⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣧⠀⢀⠀⠀⠀⠈⠛⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢂⡀⠈⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠃⢠⣤⣶⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣶⣶⡾⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠈⠉⠉⠻⠟⠀⠀⠀⠀⠀⣾⠋⠁⠀⣀⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠏⠀⠀⢳⣶⣶⣤⡶⠆⠀⠀⠀⠀⠾⠀⣰⣷⠀⠀⠀⢰⣿⠇⠀⠴⠾⠻⠷⠀⠀⠘⠷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⣀⡀⠀⠁⠀⠀⠀⠈⠋⠀⠀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣶⣤⡀⠀⣿⢿⣿⠋⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠘⠓⠊⠙⠀⠀⠈⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠃⣀⣀⣀⣀⣴⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢤⡤⠛⠿⣿⠿⢿⣿⠟⠛⠛⠛⠉⠉⠉⠁⠈⠉⠙⠋⠁⠙⠿⣿⠇⠀⠀⠆⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⡇⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣠⣼⠉⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⢠⣿⡀⠘⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠛⠛⠃⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢹⣷⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣧⣤⣄⡀⠛⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⢷⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⠘⠻⠗⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣤⣤⣀⡀⠠⣶⣶⡄⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⡏⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠔⠶⠟⠃⠀⠀⠘⠟⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠁⠘⠋⠀⠀⠀⣴⠶⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠀⢀⣤⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠟⣄⡀⠀⠘⠉⠛⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠉⠛⠀⠘⠛⣦⡈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠛⠁⠘⠿⠃⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠦⢀⠈⠁⠀⠘⠋⣉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠓⠀⠀⠀⠀⠀⠀⠸⠶⠟⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⣀⣿⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣿⣿⣁⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠗⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠙⠉⠉⠁⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠈⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣤⣀⣠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⣴⣇⠀⠀⠆⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣁⣀⠀⠉⢉⣉⠿⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⢷⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠙⠛⠶⠄⠈⠛⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠋⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠿⠛⠃⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣤⣄⠀⣰⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠏⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⡏⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣀⠀⠀⠀⠀⠀⠀⣠⡿⢧⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⠛⠀⠀⠀⠀⠀⠀⠛⠃⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠻⠃⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣎⠀⣴⣦⠀⢰⣿⣆⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠉⣿⣿⡏⠀⠀⢨⣿⣇⢠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣾⠛⢻⣿⠀⠀⣿⠏⢠⣄⠈⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⢰⣾⡏⢠⣾⠃⠀⠀⣿⣆⠘⢿⣇⠀⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⢱⣾⠏⠀⠀⣿⠆⠀⠈⣿⡟⠀⣰⣿⠇⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠒⣿⣷⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠹⡆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⣿⠿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠃⠀⠀⠀⠀⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⡀⠀⠀⣼⢸⢀⡄⠀⠀⣦⢸⡏⡄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠀⣏⢸⣸⡇⠀⠀⢼⣾⡇⡇⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣿⢸⣿⡇⠀⠀⣾⣿⠇⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣿⣿⣤⣤⣿⡿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠁⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠠⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⢀⣼⣿⣷⣶⣤⡴⠶⣿⣧⣀⣤⢶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣿⡿⠁⠀⠀⡄⠀⠀⠀⢹⣿⡾⠟⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠻⢶⣶⣤⡀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠁⠂⠀⠀⣠⣿⠀⠀⠀⠀⠀⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⢆⠀⠀⣠⣾⠛⠛⠛⠛⠛⠛⠛⠋⠉⠻⣧⠀⠀⠀⢀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⣻⣿⠃⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠹⣷⣛⢳⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣔⠂⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣄⣀⣤⣤⡀
⠀⠀⢀⣀⣠⡿⠷⠾⠋⠙⠓⠛⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠶⠛⠛⠻⠶⠛⠷⣦⣤⡄
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢰⡏⣽⠟⢻⠿⠛⠿⠛⠛⠛⠛⡟⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣀⡘⢀⠀⠀⢀⣀⣀⣀⣀⣀⣼⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠛⡟⠛⠛⢛⠉⠉⠉⠩⠉⢹⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣼⣏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⢩⠉⠉⠉⠉⠉⠉⠉⠉⠏⢹⡍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠶⠤⣦⣤⣤⣤⣤⡤⠤⢤⣴⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡅⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⢸⡏⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠛⠛⡖⠒⠿⠺⠷⠿⠿⠿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠃⠀⢀⢸⠀⠀⢀⡆⠀⣾⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⢛⠛⠛⠛⠛⠛⠛⢛⢻⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣨⣀⣀⣠⣀⣀⣀⣀⣴⡟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢉⠉⠉⢉⠉⠉⣽⠉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⣀⣀⣈⣀⣀⣀⣴⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠋⠉⠁⠉⢹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣤⣤⣤⣤⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⣍⠀⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠠⠤⠀⠀⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠑⡀⠀⠠⠫⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡎⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢿⣿⠀⠀⠀⢀⣀⠰⠄⠚⠻⡃⠀⠈⠁⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⢀⠄⠀⢾⡿⠢⡟⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠚⠀⠀⠈⠀⢰⡇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⢀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢨⠓⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠄⠁⠀⠀⠱⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⡠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠔⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧

⠀⣼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠄⠀⢀⡀⢀⠀⡀⠀⠐⠂⢀⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠃⠐⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡀⠀⠀⠂⠘⠂⠒⠒⠊⠓⠘⠀⠃⠚⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⠀⠠⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡃⠀⠀⠀⠀⠀⠿⠇⠠⠭⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢶⢤⣤⣿⣼⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⢀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣦⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠆⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⠖⠒⠒⠂⠒⠆⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠠⠋⢻⠀⠀⠀⠀⠀⡒⠺⠋⠂⠰⣶⢤⣀⣶⣶⢚⣶⣛⣦⣤⣄⡄⠀⠙⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛
⣤⠤⠦⠤⢤⡀⠸⡧⣾⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⢸⡇⠀⠸⡳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢹⡄
⣿⠀⠀⠀⠀⠙⣦⡟⢳⣠⠞⠁⣤⡏⢀⡔⠋⠁⢸⣷⠶⣦⣰⠲⣆⣞⡉⢩⡿⢾⠈⣛⣷⣰⠋⢻⣤⠖⢶⠖⠒⢦⠀⣠⠴⠟⠉⢻⠀⡇⢰
⣿⠀⠸⠟⠀⢠⠙⠃⢸⠛⠦⠤⡄⠈⣾⡀⠻⠃⠀⣿⡀⠉⠁⠀⡏⡴⠚⠀⢙⠸⡄⠳⠶⣤⡀⠈⡇⡄⢰⢲⡀⠸⡏⢧⠀⠛⠀⢸⣠⣟⣿
⢿⣀⣀⣀⡤⢋⠀⠧⠼⢃⡏⠉⠁⣀⢇⠙⠚⡆⠀⡇⠙⠦⡤⠞⠀⠓⠒⠒⢋⡀⠙⠒⠚⠃⠙⠉⣀⠉⢉⣈⠙⢛⣁⣌⠉⡇⠀⠘⡏⢣⡼
⡼⠮⠭⠤⠔⠋⠲⠶⠞⠦⠭⠭⠭⠕⠋⠙⠒⣇⠀⢻⠓⠦⠤⠴⠋⠑⠒⠒⠋⠙⠒⠖⠊⠉⠙⠋⠉⠋⠋⠈⠉⠉⠀⢸⠀⡇⠀⠀⡟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⣼⣁⡇⠀⢸⠃
⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡾⣏⠀⠈⠉⠀⢀⣾⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⢠⠟⣗⠦⠤⠤⢞⣿⡀⣠⢄⠀⠈⢠⣄⠀⢀⠀⠀⢀⢦⣀⠀⡄⠉⢀⣀⣈⣃⣈⣉⣃⣒⣾⠿⢹⡇
⢸⡀⡠⣄⠀⢸⠁⡇⠀⢰⠻⡄⢸⠀⠀⠉⡟⠋⢉⡇⠓⠃⠸⡆⠄⡏⠈⠓⠋⠹⡆⢸⠀⠀⠀⣿⠀⢸⠉⠈⠉⢸⡇⠀⣿⢹⡀⢨⠇
⠀⠉⠀⢸⡀⣸⠀⢹⣀⡟⠀⠳⠎⠀⠀⠀⢳⠀⢰⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡀⡸⠀⠀⠀⠘⣇⠀⡏⠀⠉⠉
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠎⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠧⠏⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣟⡋⠀⢀⣸⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠠⣐⢾⣿⣿⣿⣿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡎⠉⠉⠁⠈⠛⠛⠋⠉⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣿⣿⠀⠀⠀⠀⠴⠶⢶⣶⣤⡄⠀⣤⣴⡶⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠻⡆⠛⠀⠀⠀⠀⠁⠚⠋⠉⠉⠀⠀⠋⠉⠉⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀⣀⡠⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠘⣋⠐⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣄⠀⣸⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡆⠀⠀⠀⠀⠀⠈⠻⡿⡿⠀⢀⡽⠋⢉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠉⣠⠖⠉⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠚⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠘⢄⠀⠀⠀⠙⠲⢤⣠⡏⠀⠀⠀⠀⢠⣴⡇⣠⠏⠑⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠞⡿⠀⠀⠀⠀⠀⠼⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢆⠀⢰⠃⠀⠀⠀⠀⠈⣴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⢀⠞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡴⠁⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡼⠀⣴⢁⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠃⢰⠃⠀⡇⢠⣇⣈⣁⠀⠀⠀⠀⡠⠀⣀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢷⠀⢸⠀⢸⣄⣬⣷⡾⢿⡛⢷⡍⠀⢀⡼⡅⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡀⠈⢦⠀⢹⡀⢻⣤⡾⢿⡟⠻⠁⠈⣦⣀⣻⣛⣩⣽⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢦⡀⠈⢦⣀⣱⣶⠿⡟⠀⢁⣀⣰⣦⣼⠳⠮⣍⣉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢄⡙⠢⣀⣙⣶⠶⠋⠉⠠⣀⣸⠴⢶⣝⠛⠠⠬⠭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠤⣭⠾⠟⠉⢣⣀⣀⣴⢶⠻⡉⠲⠤⢄⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⡂⠠⢭⢶⡟⠑⣄⣀⣥⡖⢚⢹⠘⡌⠳⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣁⣽⣵⣀⡹⣶⡾⡉⡆⢇⠘⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⢩⣿⡀⢘⣿⣿⠋⢇⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠰⢋⡿⣻⠷⣿⠉⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠉⠀⡟⠐⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡀⠀⠀⠀⠀⠀⢀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠎⣈⡴⠿⢿⣄⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣡⢂⣾⡋⠀⠀⠀⣿⡄⠀⠀⠀⠀⠠⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻⣿⠑⣿⡙⠛⠤⠻⠗⣾⣿⡲⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⢀⡾⢹⣶⣿⡙⠣⢀⣾⣾⣿⡿⣿⣦⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣧⣤⡼⠱⠊⣿⣿⣷⠤⣽⣿⣿⣿⣽⣿⣿⣷⡀⡀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠈⠀⢻⣥⠹⠹⡿⠀⠸⣿⣏⣀⣿⣻⣿⣿⣿⣿⣟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢇⣰⠟⢿⡆⠾⠿⠛⠀⠀⠉⠿⠿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢘⣧⣦⣬⡗⣿⣿⣿⣷⣠⣶⣦⣤⡴⡤⠤⢆⣼⣿⣿⣿⣿⠘⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠓⣿⣿⣿⣿⢹⣿⣿⣿⢿⣻⣽⣿⣶⡟⢿⣿⣿⣿⣿⣿⠇⣿⢀⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣁⣾⣿⣫⣰⣿⣿⣿⣸⣿⣿⣿⣿⣧⠰⣼⣿⣿⣿⣯⣾⣧⡞⣡⡿⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣰⣾⣿⣿⣷⣿⣟⢿⠿⣿⣿⣿⣿⣿⡿⣿⣧⠈⠛⠿⠿⠿⠿⠛⡟⣵⣶⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣙⠛⣻⣿⣎⠻⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⡜⣿⣧⠀⠀⢀⡀⠀⠀⢘⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠙⠛⣹⣿⣦⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⣇⠀⠫⠠⢰⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⢨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣩⣭⣾⣿⢿⣿⣿⣦⡀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠶⠋⠁⣹⣿⣿⣿⣿⣦⡈⠀⠀⠉⠛⠿⣿⣿⣿⣿⠃⠀⠀⠄⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠁⠉⠙⡖⠲⠞⠺⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠚⢠⠂⡠⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⣩⣿⣿⡿⠟⣼⣿⡟⣾⣿⣿⣿⣿⣿⣿⠟⠑⠀⠒⠋⣠⣅⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣼⣿⡟⢠⣿⣿⣿⢛⣿⡿⠋⠀⠀⠀⠀⠜⠃⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⣫⣿⠟⠑⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡑⠀⠀⠀⠀⣰⣷⣶⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⣼⠇⠀⠈⢻⣎⢨⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣚⣿⡞⡛⠠⢙⣹⡏⣽⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⣠⣾⣿⣿⣿⣿⡖⢋⣼⣿⣏⣻⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠀⣥⣿⣟⣷⠈⠌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⣿⣿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠁⠶⠿⠶⣽⡿⣄⣰⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿⣿⣏⣦⢤⣤⣤⣴⣦⣴⣿⣿⣶⣯⣴⣿⡃⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⢿⠸⣏⢿⣿⣿⣿⠿⢻⣿⣿⣿⣟⢿⣿⣿⢸⣿⣿⣟⠫⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣛⢮⣷⣿⣿⣿⣿⡿⢏⣼⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣤⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⣷⠋⠈⠉⠉⠉⢀⣾⡿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⢟⣿⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣴⣈⡇⠀⣾⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣿⡟⠓⠒⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⢁⣴⣿⣿⣿⣷⡾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢂⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠉⢀⣴⣿⣿⣿⢿⣷⠯⡭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⢀⣘⡿⠿⠿⠋⠁⠀⢀⣴⣿⣿⣿⣿⣷⡀⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠉⠀⢁⠱⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢨⣀⠷⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠮⠀⠀⠀⠙⣿⣟⢿⣿⣿⡘⢿⣷⡀⠉⠀⠉⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠙⠿⢅⠀⠙⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠟⠵⠀⢀⣽⠟⠛⠛⢷⣄⣀⣸⡟⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣟⢠⠄⣰⡟⠁⠀⠀⠀⠀⣿⠋⠙⢿⣤⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣿⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡿⠷⣦⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡄⠀⠰⠄⢠⡿⠁⠀⢀⣾⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⠶⣶⡟⠰⠂⣠⡿⠁⠀⠀⠀⣸⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠀⣼⠷⣶⡟⢋⡀⠀⠀⣠⡾⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡿⠀⢸⣷⣤⣤⡶⠟⠋⠀⠀⠀⠀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠈⣿⣀⣓⣀⣤⣤⠶⠟⠋⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣍⠁⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⣦⣤⣀⣀⣀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠞⠻⣦⡘⠂⠈⠉⠉⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣾⠋⣄⢹⡆⠒⠈⠻⣦⣀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠙⠀⣿⠀⠀⠈⢿⡀⠀⠀⠈⠙⠻⢶⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢿⡀⠀⠀⠈⠻⣦⣀⣀⣀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⣀⣀⣨⣿⣤⣤⣤⡾⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⢻
⣀⣠⣤⣤⣀⡀⠀⢸⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⢸⠧⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡆⠀⢸
⢿⠀⠀⠀⠀⠉⢳⣈⣷⣾⡀⣠⠞⠋⠙⡗⠀⢀⠴⠞⠛⢳⣀⣤⣄⢠⣤⣤⢀⡖⠛⠛⣶⣸⠇⠻⢤⡄⢸⠷⢿⡀⣠⣤⣄⡤⢤⣀⠀⠀⢀⣠⡖⠚⣷⠈⡇⠀⣿
⢸⠀⠀⣖⡄⠀⠈⠿⡇⠈⡷⣇⠀⠘⠛⠳⣤⡏⢠⣤⠀⢸⣿⠈⠻⠼⠀⢸⠞⣓⣢⠈⠳⡸⡇⢴⣒⡓⠛⠀⠈⡏⡇⠈⢁⡀⠀⢹⡶⢾⠉⠡⡆⠀⢻⠀⣧⣤⡏
⢸⠀⠀⠉⠀⠀⡼⠀⡇⠀⡇⢈⣍⣩⠆⠀⡟⢧⣀⣁⠀⢸⡹⣆⠀⠀⠀⡜⢸⣁⡀⣀⡠⠃⢣⡀⠉⣹⠿⣄⣀⠇⠧⢤⠇⢹⣀⣈⡇⠘⠦⣤⠀⠀⢸⣶⡧⢾⡅
⢸⡤⣤⠤⠴⠋⣠⠀⠙⠛⡁⢾⣀⣀⣀⡴⢣⣄⠀⢹⠀⠈⣇⡈⠛⠛⠋⢀⣀⡈⠉⠉⣠⣄⠀⠉⠉⣩⣀⣀⣀⣤⣀⣀⡰⢦⣀⣠⠤⠶⡄⢸⠀⠀⢸⣇⣨⠟⠁
⡞⠓⠲⠶⠒⠋⠈⠓⠚⠛⠓⠶⠶⠶⠖⠚⠁⠈⠙⢻⡆⠀⣿⠙⠲⠶⠖⠋⠀⠉⠉⠉⠁⠈⠙⠓⠋⠁⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡏⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡇⡼⠀⠀⢸⠁
⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⠾⠁⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢠⢞⡏⠀⠀⠉⠀⠀⢀⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢠⠿⡷⢤⣀⣀⣠⢞⣏⠀⢀⣄⠀⠀⠃⢀⡀⠀⠀⠁⠀⠀⣠⡄⢠⢤⡀⠐⠀⡀⢀⡘⢦⣙⠒⠦⠤⠤⣒⡿⢹⡄
⢸⠀⠀⡀⠀⠀⢰⠛⡆⠀⠀⡴⣄⠀⢸⠀⠉⠒⣦⠥⠶⢻⠉⢧⡞⠈⡆⡀⢰⠏⠙⣤⡼⠶⡆⠀⡟⠈⠁⠀⢷⠀⠀⡟⠚⠙⠚⢹⡏⠛⢻⠻⡇⠀⢸⡇
⠘⠧⠞⠉⡆⠀⣟⠀⢧⠀⢸⠀⢸⣀⡿⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠙⠦⠼⠁
⠀⠀⠀⠀⢳⣀⡇⠀⠘⠳⠞⠀⠀⠉⠁⠀⠀⠀⠘⣇⠀⡾⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⣟⢀⡇⠀⠀⠀⠘⣦⡼⠁⠀⠀⠀⠀⠳⣤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⢴⠒⣒⢫⡉⠡⢶⠲⢤⠤⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⠱⠏⣆⠹⡌⢇⠘⣆⢱⡀⠈⡇⠈⡆⠀⢠⠏⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⣗⣿⣿⣦⡘⣆⠹⡌⢦⠘⡆⢧⠀⢳⠀⢿⠀⣸⠀⠀⡞⠀⠙⠂⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣯⠁⣿⡀⠸⡷⣌⠻⠿⣮⠳⡽⣌⠓⠙⠈⠂⠀⠀⠈⠀⠿⠀⣸⠁⠀⠀⠀⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⣼⠻⣄⠙⢦⡘⢷⣄⡈⠛⠆⠀⠈⢧⠀⠈⠃⠀⢀⠀⡀⡄⢰⡄⠀⠠⠇⠀⢀⡴⠊⠁⠀⠈⠱⣄
⠀⠀⠀⠀⠀⢀⣼⣛⠶⣬⡙⠶⣍⠓⠏⡉⠳⡄⠀⠀⠈⢰⡄⣄⡇⣸⠀⣷⢇⣾⡇⣰⠀⠀⠀⠋⠀⠀⠀⣀⣠⣤⠘⡆
⠀⠀⠀⠀⢠⡟⢄⣉⠓⠦⠭⣓⢬⡳⣦⣻⣷⣟⡲⠄⢧⢸⣧⣿⣷⡇⣼⡿⣾⠞⣠⢋⠞⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣧
⠀⠀⠀⠀⣸⡛⢦⣈⠙⠛⠓⠮⣽⡛⢶⣭⣛⢯⣟⠦⢸⣿⡿⠳⠛⠉⠉⠀⠈⢳⡗⣫⠞⢁⡴⠃⣀⠀⠀⠀⣤⣤⣀⣿
⠀⠀⠀⣸⣇⠙⠲⢬⣹⡦⣀⠀⠙⠿⣦⣍⢻⡙⣝⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡖⢋⡴⠊⠁⠀⠐⢦⣀⡀⠀⠈⣇
⠀⠀⢠⣏⠻⣷⣄⡀⠈⠓⠦⣭⣆⠀⠈⠙⢧⢳⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠷⠒⠚⠋⠁⠀⠀⠀⠉⠓⠦
⠀⠀⢿⣷⣤⡈⠛⠿⣗⠢⣄⡀⠈⠳⣵⢄⢸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠬⠭⣟⠛⢻⣯⣁⠀⠀⠤⡀⠀⣼
⠀⣶⡞⠉⣿⢿⣗⣦⣄⠙⠲⣝⣆⠀⢹⣌⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⢲⡏⠀⠉⢻⠻⠷⣿⣆⡏
⠀⣿⢤⡄⢉⡞⡇⠉⠛⣷⣄⠀⢹⡧⣼⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⡇⢼⡇⠀⠀⣿⡄⠀⠀⡿⠁
⣴⡇⢠⣿⣦⢀⣷⡆⠀⣹⣸⠀⢀⣿⣇⠈⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠁⢠⣸⠀⠘⠃⠀⠀⠈⠃⢀⡼⠁
⠹⣦⣾⣿⠃⣸⠉⠳⣶⣧⡽⠲⡏⠉⢻⡆⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣆⣄⠈⢻⣷⡤⠤⣤⣤⡤⠖⠋
⠀⠀⠙⠓⠋⠁⠀⠀⠀⢸⠀⢰⡇⠀⠀⡗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠸⣿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠁
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⣤⣀⠀⢀⣠⣴⣾⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠉⠁⠀⠉⠛⠿⣿⣿⣿⡿⠟⠉⠁⠀⠉⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣦⣄⡀
⣠⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦
⣿⣿⡉⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠈⣻⣿
⠘⢿⣿⣶⣄⠀⠀⠀⠻⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠃⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠙⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣶⣶⣿⣿⠀⠀⠀⠀⠀⢻⣿⣧⣤⣴⣶⣶⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⣿⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⡿⠋⠉⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢻⣿⣧⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣶⣶⣿⣿⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⣤⠤⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣛⠛⠋⠉⠙⠢⠉⠳⠀⠀⠄⠙⠛⠳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠓⠦⢀⣀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠶⠶⣤⡀⠨⠈⠇⠀⠀⠴⠀⠀⠰⠆⠁⠀⣀⠀⠆⠀⠀⠐⠈⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⣀⢄⡌⠻⣦⠀⠀⢡⠀⠀⡂⠀⢀⡎⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣾⣦⠀⠸⡆⠀⢸⡂⣰⠇⢀⡼⢀⡠⠊⠀⣀⡄⠙⠁⠀⠀⠀⠀⠀⠬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠶⣄⡹⣿⣿⣿⣿⣺⣯⣿⣶⢿⣿⡿⣿⣿⢿⣿⣶⡶⠋⠁⣀⡀⠀⠶⠀⠀⠀⠀⠀⢌⣷
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⠻⣿⣯⣺⣿⣿⣟⣩⢯⣞⣝⣼⣟⡠⣣⡾⢻⣿⠚⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⡙⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢧⡙⢿⣿⣿⡿⢫⢽⣿⡿⢛⣼⣏⣘⣻⣿⣷⣞⠛⠉⠉⠀⣄⠀⡀⠀⠁⠀⡈⢻⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢈⣷⣬⡉⠉⠙⠱⡭⠔⠈⢹⣷⠒⠰⢾⣿⣿⠈⠙⠓⠒⠀⠀⠀⠀⠀⡀⠀⣈⣦⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠉⠀⠈⠛⢶⣄⣀⠀⠀⢀⣤⣿⢬⣛⣛⣿⣿⡛⠒⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢸⡟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠛⠛⠻⢿⣿⣓⣦⣽⡿⣿⠙⠲⠀⠀⠈⠁⠠⠀⠀⠠⢀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢹⡿⠛⠏⠀⠻⠀⠀⠰⠄⠀⠀⠀⠀⠀⡀⠈⣧⡏
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠰⠄⠃⠀⠀⢤⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⢱⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⡆⠀⢰⣇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⣾⣤⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣶⣶⣾⣯⣤⣤⣴⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⣾⡇⠀⣐⣻⣶⣧⣤⣏⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⡶⡄⢲⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢀⡄
⠀⠀⠀⠀⢀⡀⠄⢸⣾⣿⣿⣿⣿⣿⡏⠉⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢡⡤⠒
⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣶⣼⣿⡇⠫⠀⠢⠀⠈⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⣤⣶⣿⣿⣟⣿⣿⣿⣿⡿⣿⣿⣷⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣱⣏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⣀⠛⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣿⣦⡄
⠀⠠⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⢿⣶⣿⣇⡀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠿⠟⠛⠁
⠀⢠⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣸⣿⣿⣿⣧⡀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢸⣿⣿⣾⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡿⠷⠘⠟⢉⡅⢀⠀⣾⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠠⢀⣉⡉
⠀⢸⣿⣿⣿⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣌⢛⢠⣾⣿⡌⢁⠈⣶⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠸⢿⡇
⠖⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠀⣿⡿⠷⠀⠰⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⢀⣀⣺⣿
⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣧⣴⣦⡀⢙⡟⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠿⠿⠛⣿
⣰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⣿⣛⣻⣿⣿⣿⣳⣾⣿⣿⣿⣧⣤⣴⣾⣷⣦⡶⢀⠀⢿
⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⢸
⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⡿⢿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣰⣿⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣷⡀
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⠀⠀⠀⠈⣿⣧
⣸⣿⠁⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣶⣿⠟⢿⣷⣾⡿⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠦⡀⠀⠀⢠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢲⠀⢹⠀⠠⡇⠀⢄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⢈⠀⠀⠙⠒⠚⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠾⢄⡀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⢿⡿⠛⠛⣛⠟⠛⠛⠛⠛⠒⣶⠋⠀⠀⠀⠀⠀⠈⢲⠋⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣾⠟⠋⠉⡠⠊⠀⠀⡴⠋⠀⢀⠜⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⡆⠀⣇⠀⠀⠤⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠋⠁⡴⠁⠀⢀⡞⠁⠀⢀⠞⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⢦⣄⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⡽⠁⠀⠀⡜⠁⠀⢀⡞⠀⠀⢠⠏⠀⠀⡰⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⣀⣴⠟⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣶⠋⠀⠀⣸⠃⠀⠀⡼⠁⠀⠀⡜⠀⠀⠀⡟⠀⠀⢰⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⡴⠋⡆⠀⠀⢠⡏⠀⠀⢰⠇⠀⠀⢰⠁⠀⠀⣸⠀⠀⢀⡏⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⣰⡟⠀⢰⠀⠀⠀⢸⠀⠀⠀⡾⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠹⣇
⠀⣰⡟⠀⠀⡀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢻
⠀⣿⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢸
⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡟⠉
⠸⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢿⡀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⡀⠀⠀⠨⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣷⡀⠀⢷⠀⠀⠀⣇⠀⠀⠈⡇⠀⠀⠸⡆⠀⠀⢸⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⢷⣄⠘⡀⠀⠀⠸⠀⠀⠀⢻⠀⠀⠀⢧⠀⠀⠀⣇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠛⠛⢿⠛⠛⢿⣿⣿⡄⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⣦⣧⠀⠀⠀⣧⠀⠀⠘⡇⠀⠀⠘⡄⠀⠀⠸⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣡⡄⠀⠀⠆⠀⠈⣿⠋⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠈⠛⣷⣄⠀⠘⡆⠀⠀⠹⡀⠀⠀⠱⡀⠀⠀⢡⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣦⣽⡄⠀⠀⠐⡀⠀⠀⢳⡀⠀⠀⢣⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⡴⠋⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⢙⣶⣦⣄⡙⣆⠀⠀⠱⣄⠀⠀⠳⡄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠟⠙⠻⢷⣶⣤⣌⣦⣀⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠉⠙⠛⠛⠛⠿⠷⢶⣾⣷⣶⣶⠶⠶⠟⢻⣿⣿⠏⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠐⠂⠀⠀⠀⠂⠀⠒⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠂⠀⠀⠐⠒⠒⠒⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⣾
⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⣸⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⢶⣤⣤⣀⣀⣀⣀⣀⣤⡴⠶⠶⠖⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠇⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣠⣴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣾⣋⣁⣀⣀⣤⣤⣴⠶⠶⠶⣤⣤⣀⣀⡀⠙⣷⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣓⣒⣦⣔⣶⣤⣴⣶⣞⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⠉⠒⠠⡀⠀⣀⠄⠒⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡔⠁⠀⠀⠀⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠒⠒⠢⢄
⠀⠀⠀⠀⠔⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠠⠤⠤⡀⠀⠀⢀⣀⡀⢀⡀⠤⠤⢼⣠⢚⠉⠙⠒⠲
⠀⠀⠀⢱⡀⠀⠀⠀⠢⢄⠞⠁⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠘⠁⠀⠀⠀⠀⠈⢂⠀⠀⠀⡸
⡠⠚⠣⠚⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⣠⡄⠀⠀⠀⠀⠃⠀⢠⠃
⢣⠀⠀⠀⠀⢰⣉⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢀⠃
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⢠⠄⠀⠀⡜
⠀⠈⢧⠀⠀⠀⢠⣤⠴⠒⠳⡀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠘⠐⠒⠒⣟⣀⣠⠼
⠀⠀⠈⢇⠀⠀⠈⡀⠀⠀⠘⣌⠂⠄⠤⠀⠀⡰⠻⡑⠀⠀⠀⠠⠀⢨⡇⠀⠀⠀⠀⠀⠀⠠⠞⠉⢱
⠀⠀⠀⠈⢆⠀⠀⣱⠀⡄⠀⠈⠑⠒⠀⠒⠊⠀⠀⠑⠢⠤⠤⠤⠒⠱⣇⠀⠀⠀⠀⠀⠀⢰⠼⠚⠁
⠀⠀⠀⠀⠈⠶⣉⡤⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠜⠁⢠⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠔⢦⡀⠀⠀⠀⣀⠔⠳⣄⠀⠀⠀⣠⡀⠀⣀⠜
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⠀⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣄⣀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣠⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠿⠿⡋⠈⠛⠟⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠀⠀⠀⠀⠀⡼⠛⢛⠓⢦⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢰⣟⠻⢏⠿⠶⠀⠀⠀⠀⠙⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢸⢹⣳⣈⣘⣿⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠂⢠⣧⣀⣷⣤⣿⡿⡈⢢⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⢻⠿⡻⣿⣿⢛⡳⠳⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⣀⣈⠻⢿⣿⣷⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⣿⢹⣿⣷⣄⠙⠿⠻⢯⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⡇⣿⣯⡻⣧⡐⣄⣀⠈⢻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⡟⡏⠘⣷⢹⣿⣷⡈⢿⡻⠿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣆⣿⡇⠃⡀⠹⡎⢿⣿⣿⣆⠑⡕⠶⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣼⣿⡇⡇⢠⡇⡀⢻⣌⢻⣿⣮⣱⣜⡲⢾⡏⢻⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣿⢿⡇⠀⢸⡇⣷⠀⢻⣦⡻⡟⣿⣿⢦⣺⣧⠀⣧⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⡿⣻⢸⡇⠀⣼⡇⢹⣧⠀⢻⡟⠦⠘⣿⣷⣻⣿⢀⣿⢠⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠈⠀⠀⣿⣧⠀⢿⣇⢠⡙⠆⠀⢹⠄⠉⢁⣾⡟⡎⣇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢹⠀⠀⢰⣿⢿⡄⠘⢿⡄⢻⣄⠀⠸⠀⣰⡟⢣⠀⠀⠸⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡟⠀⠱⡀⠘⣷⠀⠹⣦⡀⢠⣿⣿⣄⠀⠀⠀⢿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠙⠀⠸⡆⠀⠈⠛⠆⠙⡏⢿⣦⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠐⠲⠦⠐⠈⠁⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⡦⠀⠀⠀⠎⠔⣹⡶⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠀⠒⠒⠒⠒⠒⠒⢼⣀⣧⣄⠀⢸⠸⣸⠋⠀⠀⠈⠱⢄⠀⠀⠀⠀⢱⢀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⢀⡀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡤⡤⠼⠀⡆⢠⠎⣠⣤⠀⣤⣤⣍⣧⠀⠀⠀⠀⢿⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣀⡤⠴⠶⠒⠒⠋⠉⠉⠉⠉⠐⠒⠚⠹⠀⡄⠀⡰⠀⡘⣾⣿⡿⠀⠻⣿⣿⠟⡇⠀⠀⠀⣾⠀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡥⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡇⠀⢣⣈⡁⠰⠿⠆⣈⣉⣠⠇⠀⠀⢀⡏⠀⠙⢦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠦⣑⢽⣿⣁⣀⣀⣸⢰⠏⠀⢀⢀⡞⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⡙⢧⣤⡤⢴⠟⢀⡴⢵⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠀⠙⢿⠏⢀⠞⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⣠⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢳⣷⣦⣠⢠⠞⣹⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡿⣇⠀⠇⢱⡏⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⡇⢹⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⠿⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⣶⠃⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢠⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡯⠔⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡇⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⢸⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠶⠦⣄⣸⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣼⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡇⠀⢀⣤⣄⣀⠀⠀⣀⣤⣄⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠧⠔⠁⠀⠀⠉⠉⠉⠁⠀⠈⠳⢦⣀⣀⣠⠖⠛⠛⠓⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢠⣦⣄⣿⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠛⣿⣿⠇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⣽⣿⣿⣷⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⣰⣾⣿⡉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠙⠿⣿⣿⣷⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⣿⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣾⣋⠀⠀⠀⠀⠀⠀⣀⣨⣍⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⠿⠿⠿⣷⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣀⡀⢀⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⣬⣷⣵⣿⠏⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣙⣿⠛⠛⠛⠷⠷⠟⠛⠛⠂⠀⠀⢠⣶⠾⠶⠷⠿⠿⠿⣿⣷⣄⠀⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⡀⡀⡄⢀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠶⠛⠛⠛⠛⠻⠿⡾⠾⠷⠶⠷⠴⠶⠾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⣬⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣶⣶⣶⣴⣤⣀⡀⢀⠀⣀⣠⣠⣀⡀⢀⢀⢀⣀⣤⣤⣀⡀⣀⣠⣠⣤⣤⣶⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠆⠀⠘⢯⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣧⡀⠀⠀⠀⣹⣿⣅⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣷⠀⠀⠘⣿⣿⣦⣤⠀⠀⠀⢸⢿⣿⣿⡇⠀⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣏⠀⠀⠀⣻⣿⡋⠙⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⣻⣿⣿
⠀⠀⣰⣿⣷⠀⠀⠀⢠⣿⣷⡄⠀⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⢻⡿⢿⣇⠀⠀⢰⣿⣿⣷
⠀⠀⢨⣿⣿⠀⠀⠀⢈⣿⣿⡁⠀⠀⠀⢸⣿⣿⡄⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⣿⡅⣼⡏⠀⠀⠹⣧⢸⡿
⠀⠀⠸⣿⣿⠗⠂⠐⢺⣿⡿⠀⠀⠀⠀⢈⡏⢹⡀⠀⢀⡏⢹⡁⠀⠀⠀⠀⠀⢹⡅⠸⣿⠇⠀⠀⠷⢨⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣄⣠⠴⢲⠳⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡕⡈⠦⠶⢷⡁⠃⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢣⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢀⢾⡤⢤⣧⡀⠀⠀⠈⢆⣀⣀⣀⣀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠁⠃⠒⡡⢻⠘⠆⡠⠖⠉⠁⠀⠈⠱⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠜⠉⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡜⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⣀⡀
⠀⠀⠀⣀⠔⠊⠉⠑⢒⣊⡟⢲⣶⢤⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⣀⣀⡀⠉⠢⡀
⠀⢠⠜⠁⠀⠀⠐⠟⠀⠀⠀⠘⠈⠀⠹⠀⠀⣠⠖⡿⠁⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠏⠀⠀⠈⢦
⠐⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠁⡔⠁⠀⠀⠀⢸⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘
⢸⠁⠨⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⠤⠤⠤⠀⠀⠀⠀⢠⠇⠀⠀⡛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡃
⢸⠀⠀⠀⠙⠛⣄⠀⣀⢀⠀⠀⡐⠁⠀⠀⠀⠀⠀⡀⠀⢠⠏⠀⠀⠀⠃⠈⠳⣄⠀⣀⢤⣀⡠⠤⠶⢊⠖⠀⠀⠀⢠
⠈⣇⠀⠀⠀⠸⣋⠋⡉⡰⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢰⠀⠀⠀⠉⢅⣀⡀⠀⠀⠐⠁⠀⠀⠀⠀⡜
⠀⠙⢆⠀⠀⠀⠉⠛⠫⢤⣄⠀⢆⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠑⠦⣀⡀⠀⠀⠀⣀⡽⠞⠓⠤⢖⣉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⣂⣀⡀⠀⠀⠀⢀⣀⣠⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠰⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⣦⣤⣤⡘⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⡿⠿⠛⠉⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⣠⡴⠛⠛⠻⢶⣶⠀⠀⠀⣠⣴⣶⡿⠛⠛⠒⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡗⠡⣤⡶⢶⡶⢶⣌⣷⠀⠀⢼⣷⡶⢶⡿⢧⣄⡀⠸⢣⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢘⣇⠀⠀⠀⠈⠙⠀⠉⣿⠀⠀⠀⠨⠀⠈⠁⠀⠈⠁⣀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠰⡄⠀⠀⠀⢠⠞⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡝⢀⠀⠀⠀⠀⠈⠓⠲⢦⡴⠞⠚⠛⠀⠀⠀⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⣠⡀⢄⣠⣄⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⠋⣽⠀⠈⠉⠀⠉⠑⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣶⠀⢰⠆⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⠀⢰⠃⠀⠰⠀⠀⠀⢀⣴⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⡇⠀⢰⠀⢀⣠⣿⡟⠁⣼⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢩⣿⠁⠿⣿⣿⡇⠀⠀⣶⡿⠿⠋⠀⢀⡷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⢻⡇⠀⠀⢹⠇⡀⠄⠉⢠⣀⢀⣀⣀⡁⢒⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⠹⣷⠛⠀⢠⡋⢀⠀⣀⠴⠋⡉⠉⠉⠙⠃⢉⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⢦⠀⠀⠀⠈⠑⠢⣼⡗⠀⠉⠁⢀⣴⠟⠛⠛⠲⠦⠀⠀⠀⡼⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢸⣷⣄⠀⠒⠋⠀⢰⣶⣶⣴⠂⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⢀⣠⡤⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠈⣿⣿⣦⠀⠀⢻⡿⠿⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⡀
⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⣹⢿⡆⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣦⠂⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢷⠀⠠⣀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠈⣿⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠘⣴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠤⠤⠀⠀⢠⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢬⡅⢣⠈⢧⠰⠀⠸⡆⢾⠀⠹⠉⢧⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡚⠣⡀⠙⢆⠣⠀⠑⣆⠀⢃⢸⢀⠀⠀⡏⢰⠃⡹⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡈⢆⠈⠲⡀⠳⢣⣀⠨⡀⢰⠎⢻⣀⡄⠁⢀⠔⠁⡀⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠂⠈⠈⡃⢀⣈⠀⢡⠉⡓⢲⡏⢀⡀⣿⠀⠈⣯⠀⠐⠁⢰⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡚⠍⠐⠢⠈⠒⢄⣈⡲⣜⢆⠉⣭⢺⣧⣉⣠⣿⡌⣥⠿⠰⢆⡏⣀⠀⠀⡤⠀⠰⣿⣷⡀
⠀⠀⠀⠀⠀⢀⢴⣃⠠⡀⠑⠢⡀⠀⠀⠣⡀⢨⣍⢍⢢⣬⡿⠋⠉⠁⠉⠉⠲⣏⣡⣻⡣⠄⡸⠀⠀⠀⢙⣿⣇
⠀⠀⠀⠀⢀⠎⢠⡌⠑⠠⣌⠑⠺⣷⢞⡛⢛⡓⡰⠦⠛⠋⠀⠀⠀⠀⠀⠀⠀⠸⡅⡸⢃⡰⠷⠶⢲⣆⠀⢸⣿⡄
⠀⠀⠀⡤⣴⣿⣇⠀⠁⠦⡀⠑⢦⠈⠑⣦⣀⠷⣧⡺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⣇⣿⡐⢡⡄⠀⢩⣷⢸⣿⣧
⠀⠀⠈⣼⣿⠿⣿⣵⣄⠘⢌⠳⢄⣱⡶⢣⠉⠖⢴⣟⢦⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣾⣥⣤⣾⣿⣼⣿⣿
⠀⠀⣿⣯⡀⠈⢿⣿⣮⣤⡨⣌⢃⣿⠶⣼⣷⣤⡀⠙⣶⣌⠲⡄⠀⠀⠀⠀⠀⣿⣿⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⡿⢤⢼⣿⣿⣿⠥⠨⣴⡟⢢⡈⠻⣷⣌⢳⣌⠻⡶⢻⡄⠀⠀⠀⣴⣿⣿⣼⣷⣿⡿⣿⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠛⢿⣞⣿⣿⣿⡏⠀⢠⣿⠁⣀⡅⠀⢸⣿⠉⣽⢸⡧⣹⠇⠀⠀⣢⣿⣿⠿⢋⣵⣿⣷⣿⣏⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⡟⡇⠐⢸⠀⠀⡿⢟⠃⣏⣽⣇⡿⠀⠀⢼⣿⡿⢡⣶⣿⣿⣿⣿⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣽⢻⠿⠇⠀⢻⢷⡀⡞⠀⠀⣷⡻⢴⡏⢻⢁⠆⠀⣼⣻⣿⠀⣼⣿⣼⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠺⠴⠇⠀⠀⠙⠃⠲⠃⠈⠉⠀⠀⣿⣿⠟⢰⣿⣿⣏⣼⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⠿⠛⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠉⠲⡄⠀⠀⡴⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡈⣳⠀⢻⠀⠸⡁⠐⣆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣸⠀⠀⠉⠚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣠⠴⠒⠒⠛⠦⣄⠀⣀⣀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⢶⠾⠛⠛⣻⠟⠋⢉⡽⠋⠉⠉⠉⠙⠛⡿⠁⠀⠀⠀⠀⠀⠈⢿⡁⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⡿⠋⠉⢀⡴⠃⠀⢀⡞⠁⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠀⠀⠀⠀⢸⣧⡘⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⡿⠋⠁⢀⡜⠁⠀⢀⠞⠀⠀⢠⠏⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⠷⣤⡟
⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⡼⠁⠀⠀⡞⠀⠀⢀⡎⠀⠀⢠⠃⠀⠀⡼⠁⠀⣠⣤⠄⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⣠⣴⠞⠁⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⣠⣾⠏⠀⠀⢰⠃⠀⠀⡼⠁⠀⠀⡞⠀⠀⢀⡏⠀⠀⣸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⢠⡾⠋⡞⠀⠀⢀⡇⠀⠀⢰⠇⠀⠀⢸⠁⠀⠀⡼⠀⠀⢀⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠻⣆
⠀⣰⡟⠀⢰⠇⠀⠀⢸⠁⠀⠀⣼⠀⠀⠀⡏⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡆
⢰⡟⠀⠀⣸⠀⠀⠀⡟⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣿
⣾⠃⠀⠀⣿⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠞⠋⠀⠀⠀⠀⢸
⣿⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣧⠀⠀⢹⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣦⠀⠸⡆⠀⠀⢸⠀⠀⠀⢳⠀⠀⠀⣇⠀⠀⠘⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠙⢷⡀⢧⠀⠀⠈⡇⠀⠀⠸⡆⠀⠀⢸⡀⠀⠀⢳⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠁⠀⠉⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠻⣾⣄⠀⠀⢹⡀⠀⠀⢳⠀⠀⠀⢧⠀⠀⠈⣆⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣼⣿⣿⠿⠟⠋⠁⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⢀⣽⣷⣤⡀⢳⡀⠀⠈⢧⠀⠀⠘⣆⠀⠀⠸⡄⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠶⣷⣀⠀⠈⢧⠀⠀⠘⢆⠀⠀⠹⣄⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣦⣷⣄⠀⠈⠣⡀⠀⠘⢦⡀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠈⣹⣿⣿⣶⡿⣦⣤⣄⣙⣦⣀⣈⣳⣄⣀⣀⣀⣀⣀⣤⣤⣴⡶⠞⠛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠈⠉⠉⠉⠙⣿⣿⣿⠟⠉⠉⠉⠹⠿⠟⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣷⣶⣶⣷⣾⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠐⠃⠈⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡈⠁
⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠈⠠⠀⡰⠿⡠⡶⡟⠀⠀⠀⠀⠀⠀⠀⢀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠁⠀⠐⠀⢀⣇⡈⢁⢠⡶⣄⡀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠠⣀⡀⠈⠆⠙⠻⠿⠋⠙⢠⠇⠀⠑⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠂⠀⢀⣀⠀⠜⠀⢠⢀⡴⡋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⡇⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠘⠄⣸⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠀⠀⠀⠘⢧⡆⠀⠀⠀⠀⢀⠔⠀⠀⡰⡛⠿⣼⡝⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠊⠀⠀⠎⢀⡸⣿⠌⢀⠄⠀⣠⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠈⠁⠁⠐⠀⠀⠀⠀⢀⠀⠈⠇⠉⣀⠼⠓⢉⠰⠆⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠆⠀⣀⡀⡸⠟⠻⠷⠂⠀⠀⣄⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣧⠿⢁⡐⢉⠒⢶⣀⣀⠈⠀⠄⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣘⠁⡀⠀⠁⠀⣄⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡆⠀⠀⠉⡀⠁⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⡀⠀⠚⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⡀⠑⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣠⣤⣤⣤⣄⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢭⡙⠲⣾⠀⠀⢸⡀⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡘⠓⠶⣧⣦⠙⠆⢦⠈⢷⣌⠳⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣁⠀⠀⠀⠀⠛⠂⠈⠳⢤⣈⡉⠀⠀⠙⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⢀⡴⠋⠁⠠⠒⠉⠉⠉⠁⠀⣀⡤⠖⢒⣠⡤⠤⠄⠀⠀⠉⢙⡛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠙⠛⠓⠂⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡰⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠲⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣸⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⣿⡄⠀⢸⠁⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡼⣡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡀⣀⢸⡇
⠀⠀⠀⠀⠀⣿⠃⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⠴⠖⠒⢾⡆⠉⠛⠋⠛⠁⠀⠀⣄⣀⣠⡴⢻⣿⣷⣦⣄⣀⠀⢹⡏⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠠⠟⠀⠀⠠⠤⠤⠔⡂⠔⠈⠁⠀⠀⡀⢠⡀⠀⠀⠈⣹⠟⠓⠾⠿⣿⣧⣄⣉⣷⠘⡇⢰⠇
⠀⣠⡴⠶⠶⣿⣤⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢠⠊⠀⠲⢤⣤⣤⣤⡄⠀⠀⠀⢸⣇⡾
⡾⠁⣀⣀⣀⠀⠈⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠠⠋⠀⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠛⠳⢦
⠇⠚⠉⠉⠙⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⠶⣤
⠀⡀⠀⠀⠘⠻⣿⢀⠀⠀⢀⠄⠀⠀⠀⠀⣰⠟⠁⠸⣆⠘⠿⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠂⠀⠀⢀
⠀⠹⣆⢀⠀⠀⢿⢸⡀⢠⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⠚⠋⠉⠉⠘⣿⠀⠀⠀⠀⠀⢸⠀⢀⡼⠀⠀⣀⠔⠁
⠀⠀⠈⠻⣆⠀⠼⢸⡇⣾⠀⠀⠀⠐⠁⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⡇⠀⠋⠀⠀⠉
⠀⠀⠀⠀⢹⡄⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡴⠟⠛⠹⣄⣴⠉⠳⣄⡙⠒⠤⠤⡀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣾⠃⢀⡤
⠀⠀⠀⠀⠀⠙⠶⠶⢻⡇⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⡴⠿⣄⠀⠸⡟⠒⠤⣤⣤⣀⠀⠀⢸⠇⠀⠀⠀⣾⡟⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡞⠁⠀⠀⡴⠛⠉⠀⠀⠀⠀⠉⠀⠃⠀⠞⠙⢿⣿⡆⢀⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣻⠖⡼⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⠁⢀⡼⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠞⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⠶⠛⠉
⣿⣿⣿⣿⣷⡾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⡿⡿⡿⠿⠿⣿⣷⠿⠿⣿⠿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣿⣿⣻⣿⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢴⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣚⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠐⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣏
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤
⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠂
⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⠄⠀⠙⢷⡄⠀⠀⡴⠾⠛⠛⠛⠓⠄⣿⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠃⠰⠶⠀⠀⠀⠀⠀⠀⠐⠶⠂⠈⠄⢀⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠰⠒⣟⣻⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠶⠁
⠀⠀⠀⠀⠀⠀⠐⢎⠻⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣸⣿⣟⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠩⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡎⠩⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡿⣦⣄⠀⠀⠀⠀⠀⢀⣴⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢛⢒⠀⢐⢛⣋⠁⢸⡀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⢸⣿⢨⡉⣅⢿⣿⢿⣿⡿⣿⣿⢸⣿⡿⠋⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⡏⠀⠉⠀⢸⣿⠱⢄⠦⢹⣿⢸⣿⢷⡀⠘⣿⣾⣿⣿⣻⣿⣿⠀⠀⢫⣩⣶⣾⣿⣷⣶⡉⡗⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⢸⣧⣦⣀⣰⣾⡇⠀⠀⠀⠈⣿⢸⣷⣈⢷⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⢛⣿⣿⡿⠿⣿⠀⠀⢸⣿⡟⠋⣀⠋⣤⡈⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠷⢗⣀⣘⠾⠿⢸⣿⣟⢈⣴⣿⣿⣵⡿⣾⣻⣿⠀⠀⠩⢿⣬⣿⣻⣯⣤⡿⠂⠀⢸⡿⢰⡆⠉⣴⠙⠁⣿⣾⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢸⣿⣿⣦⣻⣿⣿⣾⣛⣻⣾⣿⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠠⠉⠉⣉⠉⠉⠉⠉⡉⠉⠉⠀⢀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢈⣙⣿⣻⣿⣟⣛⣟⣛⣛⣋
⠀⠀⠀⢸⣿⡇⢀⣠⣾⣿⣷⡀⠀⢸⣿⣿⠟⠿⣩⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⢸⣿⡿⠋⠉⢻⠟⠉⠙⢿⣿
⠀⠀⠀⣤⠳⣤⠈⠁⣴⣿⡿⣇⠀⢸⣿⡟⣂⡀⠀⣬⣼⣿⣿⣿⣿⢸⣿⣯⣟⣫⣷⣿⣿⣿⣿⣿⣿⢸⣿⣾⠟⠀⡆⢆⠀⢻⣶⣿
⠀⠀⠀⠸⣒⠇⠀⠐⣿⡏⣼⠿⠀⢸⣿⣿⣿⡗⢸⣿⣿⡟⠻⣿⣿⢸⣿⣿⣿⣿⣷⣷⣿⣷⣟⣿⣿⢸⣿⣿⠀⣾⢀⡸⡇⢀⣿⣿
⠀⠀⢰⣶⣿⣶⠆⠀⠈⢻⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠵⠀⢸⣿⢸⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⢸⣿⣿⣷⡅⢸⡇⣡⣾⣿⣿
⠀⠃⠀⠿⠻⠿⠀⠰⣤⠾⠋⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣀⣠⣾⡿⠸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⢸⣿⣿⣻⣴⣿⣷⣬⣻⣿⣿
⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⢀⠀⣉⠉⡉⠉⠉⠉⠀⠀⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄⠀⠈⠉⢉⠉⠉⠉⠉⡉⠉⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⠀⠀⠀⢸⣿⣿⢿⡀⠀⠀⠀⠀⢰⣿⣿⡹⢿⠟⣟⢿⠿⣽⣿⣿⠀⠀⠀⣸⠷⣶⡶⢿⡇
⠀⢸⣿⣿⡿⠿⠿⣿⢋⣠⣾⣿⣿⠀⠀⠀⠈⣹⣿⡏⠀⢀⣀⠀⠀⢸⣿⣯⣥⣾⣿⢿⣿⣦⣭⡿⣿⠀⠀⢠⣿⣿⡼⣿⣧⣿
⠀⢸⣿⣿⣥⣴⣾⡟⣽⣿⣿⣿⣿⣤⠀⠀⢠⣿⣿⣿⡀⠛⠛⠀⠀⢸⣿⡿⣿⣵⣿⣿⣶⣻⡟⣷⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⢀⣼⣿⣿⢿⣿⠏⢠⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⡇⡀⠀⠀⠀⢸⣿⣿⣿⣻⣿⣿⡿⣹⣿⣿⡇⠀⠀⢻⡿⣿⣿⣿⣿⣿⠇
⠀⠻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⠀⠀⠛⠿⠿⠟⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣾⣿⣿⣿⠇⠀⠀⠈⠀⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣉⠑⡄⠀⡔⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠜⠀⠡⠀⠣⠬⠂⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⠤⠤⠧⣄⠀⣀⣠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⡶⠛⢛⡽⠋⢩⠟⠉⠉⠉⠉⡟⠁⢀⠀⠀⠀⢻⡁⡀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⠏⠀⢠⠊⠀⡠⠋⠀⡔⠁⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⢠⠷⣭⣀⡼
⠀⠀⠀⠀⢀⣠⠞⢋⠏⠀⢠⠃⠀⢠⠃⠀⡰⠁⠀⡜⠀⢀⣀⠀⠀⠀⠀⠙⢦⣀⣀⣀⡴⠏⠀⠀⠙⢧⡀
⠀⠀⢀⡴⡟⠁⠀⡎⠀⠀⡎⠀⢀⠇⠀⢠⠃⠀⡸⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦
⠀⣠⠞⢡⠃⠀⢸⠁⠀⢸⠀⠀⡜⠀⠀⡎⠀⢀⠇⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠈⣧
⢠⠏⠀⠸⠀⠀⡎⠀⠀⡇⠀⠀⡇⠀⢀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠸
⣿⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⢠⠁⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣤⣄⣀⣀⣀⣠⡤⠖⠁
⣷⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠸⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⡀⠀⢃⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀
⠀⢳⡀⢸⠀⠀⢱⠀⠀⢣⠀⠀⠇⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⠀⢻⡋⠉⢻⡿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼
⠀⠀⠹⣦⡇⠀⠈⡆⠀⠘⡄⠀⠸⡀⠀⢱⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⢻⣿⣿⣿⠿⠃⠀⠀⡼⠃
⠀⠀⠀⠈⢻⣦⣀⠹⡀⠀⢱⠀⠀⢣⠀⠀⢇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠉⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠻⢿⠿⠳⢷⣄⡀⠳⡀⠀⢣⠀⠈⢦⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠗⠷⣤⣄⣑⣄⠀⠱⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠉⠛⠛⠛⠓⠲⣿⣶⡶⠒⠚⠻⣿⠏⠁⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠠⠄⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠃⡖⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱⢀⠞⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠱⠒⠉⢱⡨⠀⠀⠀⠀⣠⡀⠀⠘⡆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠰⠄⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠠⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⢀⠀⠂⠀⠤⢀⣀⡄⢄⣸⠛⣯⡲⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠔⡁⠀⠉⠱⠇⠀⢀⣀⠈⠀⠀⠈⠓⠈⠁⠁⡀⠈⣀⠀⠈⡀⠠⠠⠀⡄
⠀⠀⠀⠀⠀⢸⠀⢠⡰⠀⠀⠀⠂⠄⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠙⢠⡄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠱⡌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠂⢀⠒⠁⠀⠀⠀⠀⢠⠀⠀⢀⠀⢤⢀⢲⠀⠠⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠰⢸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⣀⡀⢙⢚⢀⣀⢘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢙⢛⡟⠛⠛⠿⣛⠛⠻⢿⣉⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⣴⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⡰⣯⣍⢲⣤⣀⡀
⠀⠀⠀⠀⠀⣰⣿⡟⣥⠎⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣌⣄⢺⡦⣀
⠀⠀⠀⠀⢰⣿⣿⢃⠎⠀⠀⠀⠀⠀⠀⣸⢻⡃⠀⠀⠀⠙⢿⣦⡙⣌⢱⣄
⠀⠀⠀⠀⣿⣿⣧⠎⠀⠀⠀⠀⠀⢀⣴⣿⣯⣡⣤⣤⠀⢴⣿⡿⣷⣜⢮⣯⣷⡄
⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⢠⢋⣼⣿⣿⡉⢉⡁⠛⠂⣠⢤⣿⡌⢰⡿⣙⠀⡈⠩⡄
⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢿⣧⣂⣶⡟⢩⣿⣧⣿⡏⢻⡊⠀⠀⢸⠀⠄⠄
⠀⠀⢀⣿⣿⠏⠀⠀⠀⠠⣪⣿⣿⣿⣿⡟⣿⣿⣿⣿⣗⣀⣴⣿⣿⣿⣟⠳⣌⣷⣠⢀⣾⠇⠄⣀⠀⣡
⠀⠀⢸⣿⣿⠀⠀⢠⣾⣾⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡍⢹⣿⣿⡿⠋⠰⣤⣬⢿⡿⠛⣾⡦⠠⠄⢙⡄
⠀⠀⢸⣿⣿⠀⢠⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣷⡴⣿⠿⠃⣀⣀⣿⣟⠘⡗⣿⣿⡇⠀⠀⣼⠀⢙
⠀⠀⢸⣿⣿⣴⣿⣿⣻⣿⣿⣿⣿⣿⣿⠿⠟⢻⠛⢻⣿⣿⡏⠈⠢⠀⠘⠛⠻⠿⠿⣷⡿⠟⢷⣇⣰⠇⣀⢸⡅
⠀⠀⠸⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣟⠃⣦⣀⠀⣤⣀⣀⣀⣤⣤⡀⣼⣿⣿⣿⣿⣷⣽⠃⣠⣬⣻⡇⠘⢀⡾⠃
⠀⠀⢰⣿⣿⣿⡇⢸⣿⠿⣿⣿⣿⣿⣷⣾⣭⣭⣿⣟⣻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣄⣸⠣⣝
⡀⢴⣾⣿⣿⠸⣧⠘⣿⡄⢿⣿⣿⣿⣿⣿⣿⠿⠋⢻⣿⣿⣿⣿⣮⣍⠻⣿⣿⡿⣿⠀⣿⣿⣿⢿⡿⡋⠱⣤⠟
⠙⢾⡿⢻⣿⣅⠙⣧⣽⣿⣮⣝⣻⣿⣿⣿⣧⡞⢠⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡆⡞⣿⣿⣿⠀⣹⠟⠁
⠀⣿⣿⢠⣼⣍⠻⣿⢿⣿⣿⣿⣿⣿⣿⠿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⣿⣻⣷⣶⠶⠂⣀
⠀⠈⠻⣌⣿⣿⣧⠞⠀⠀⠈⠉⠉⠁⠀⠀⢠⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⢉⣛⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣥⡶⠁
⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⣠⣴⡄⠀⠀⢠⣿⢹⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣿⣿⣯⣉⣁⡄⢀⡄
⠀⠀⠀⠀⠈⣿⣿⣷⣾⣯⣤⣌⡟⠀⢠⣿⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣼⣿⣿⡉⠉⠋⠉⠀⣸⡇
⠀⠀⠀⠀⠀⠸⣿⣿⠋⠁⠀⠀⠀⠀⣼⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠀⣠⣾⣿⣿⣿⣿⣶⣶⣶⣾⣿⡥
⠀⠀⠀⠀⠀⠀⢻⣿⢇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⠔⢁⣴⣿⣿⣿⣿⣿⡻⠿⣿⣯⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡌⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⡠⠖⣡⣶⣿⣿⣿⣿⡟⠻⣿⡻⢷⣶⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠘⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁⣀⠔⠈⣠⣾⣿⣿⣿⣿⣿⣿⡌⠀⠉⠻⣶⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡱⣄⣠⣤⡬⠟⠛⠉⠛⠉⠀⡀⠔⠈⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡉⠡⡀⠈⠀⠈⢧⠀⢀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠱⣆⠘⣧⡀⠁⠀⠀⠠⣰⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⣀⠼⢷⡶⠾⠓⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣷⠙⢿⣿⣿⣿⣿⣛⠛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠺⣇⢹⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⣿⣿⣿⡇⠻⣿⣿⣧⠀⠉⠉⠉⠉⠉⠳⡄⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠙⣿⣿⣿⢷⡀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠻⢿⣷⣍⠀⠀⠙⠻⢿⣦⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠘⡇⠐⣾⠠⠄⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⡾⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢺⡄⠀⠁⣀⠿⠰⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⢿⣧⠘⢀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣶⣦⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡵⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⢀⠀⢀⡀⢀⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⣴⠀⠲⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⣧⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⠀⠀⠘⡕⢊⠉⠀⠈⢻⠿⠿⢿⡿⠈
⠀⠀⠀⢌⠻⡄⠒⠀⠄⠐⠒⡻⠴⠀⠀⠀⠀⠘⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣼⣿⠏⠀⠀⠀⠙⡍⠴⠚⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢸⣿⣿⡿⢻⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡟⢁⣤⣿⣿⣿⣿⡇⠀⣿⣿⣿⣇⠿⠟⠛⠛⢬⣿⣿⣿⡇⠀⣿⣿⣿⣿⠟⠋⢰⠟⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠘⣿⣿⣿⣿⣿⡇⠀⣿⣿⣷⣟⡀⠀⠀⠀⣿⣽⣿⣿⡇⠀⣿⣿⣿⣧⡘⠛⣿⡀⠂⣙⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠙⠻⣿⣿⡇⠀⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣦⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⢹⣿⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠙⠁⠀⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣾⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⣿⣿⡿⣿⣿⢏⢉⣿⣿⠀⠀⣿⣿⠏⠀⣠⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⡇⠀⣿⣣⡾⠁⠀⠀⢹⡿⣿⠀⠀⣿⣿⣿⣿⢷⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣗⡀⠀⠀⣰⣇⣿⠀⠀⣿⣿⣿⢧⠀⠀⢸⣿⣿
⠀⢸⣿⣿⣿⣉⠤⠀⡀⠠⢥⣭⢿⣿⠆⠀⣿⣿⣾⣧⢲⣿⣿⣿⣿⠀⠀⣿⣿⡁⠲⠃⢀⣾⣿⣿
⠀⢸⣿⣿⣶⣷⣤⣾⣿⣿⣾⣷⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣤⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠹⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠹⠿⠟⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠂⠀⡀⠀⠀⠀⢠⣙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡴⢶⣿⠿⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢈⣉⠠⠤⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠀⠀⠀⠀⠀⠀⣷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡆⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡄⢾⢿⣿⠀⠀⠿⣷⡶⠀⠄⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠔⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣠⣤⣶⣀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⡶⠉⠁⠀⠈⣩⣀⣤⡄⠀⠘⣧⠸⣆⠒⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣾⠈⠁⠀⠀⠈⠀⣸⡇⠀⠀⣤⡶⢿⣿⣉⣱⠀⢻⡿⠂⣽⣦⠀⠀⠀⠈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⣠⣿⣿⣀⣶⡿⠀⠘⣿⢻⣏⠛⢻⡟⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⠿⣿⣿⡿⠁⠀⢀⠈⣠⣿⣦⡀⠀⠆⢰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⣾⡅⠀⠀⢠⣿⣿⠇⠀⢰⡟⠀⣿⣿⣿⣿⣶⠀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⣀⡉⠀⠈⠉⠛⣯⣷⣶⣾⣷⣄⣠⣿⣿⣿⡀⢀⣿⡿⠀⣻⣿⣿⣿⣷⣄⡀⢀
⠀⠀⢀⣄⣀⡴⠀⠀⠀⠈⣛⠻⣿⣿⣿⣷⣾⣧⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣇⠠⠟⢁⣼⣿⡏⡿⣿⣷⣿⣻⣿⣿⣆⠀⢰⣶⣄
⡤⠞⠛⠉⣉⣷⣤⠿⠃⠈⠉⣻⡿⠉⠀⠙⣿⠉⠁⠙⠿⠀⠃⠙⢻⡟⠋⠁⠙⢀⣴⡿⣿⣿⡟⠃⠈⣿⣿⣿⡿⠃⠈⠀⠀⠩⣿⡆
⠀⣤⣴⡾⠏⠉⢳⡄⠀⠀⣠⡿⠃⣀⡀⢠⡀⠀⠀⠀⠐⣦⠀⠀⢸⣿⣿⣷⠀⠈⣩⣾⣿⣯⢴⣤⣼⣿⣿⣏⠀⠀⠀⠀⠐⢦⣼⠃
⣿⣿⠃⠀⠛⠀⠏⣠⣈⣿⣿⡇⠀⠻⠷⠀⠀⠀⠀⠀⠶⠀⣾⠶⣿⣿⣿⣿⡀⠀⣿⣿⡿⣷⣤⢙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠉
⣿⢋⣀⣤⠀⠀⣀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣷⣾⣿⣿⣿⣿⡇⠠⣿⣯⣥⣌⣠⣾⣿⣿⠟
⣿⣈⣿⣧⣤⠞⠁⣀⣀⣈⣿⣶⣾⣿⣿⡗⠶⢢⣶⣾⡿⣾⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣿⠻⣿⣿⠟⠻
⠻⢏⠿⠋⠀⢀⣴⣿⣿⡿⠿⣿⣿⣿⡿⠀⣀⢸⠿⠤⠂⣿⣿⣿⣿⣿⣿⣿⣷⣦⣠⣽⣿⣷⣾⡟⠀⠾
⣦⣬⣄⣀⣴⣿⣿⣿⣿⣀⡸⣿⣿⡟⠁⠀⢠⣾⢀⣤⢈⣿⣿⣧⠀⢹⣿⡛⣿⣷⣬⣿⣿⣿⢋⣤⣀
⠈⠙⠻⠿⣧⣙⣿⣿⣿⣿⣿⣿⣿⡇⢠⣴⣿⡏⢸⡟⣤⣿⣿⣏⣠⣾⠏⣠⣿⡿⢿⡿⢿⡏⢈⣿⣟⠋⠠⠀⣀
⠀⢀⣴⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣧⣌⡁⣿⢃⣼⣿⣌⠻⠿⠿⣿⣿⣶⣿⣿⠻⣿⡙⢛⣀⣉⣿⠟⠁⠀⠀⣽
⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣏⣱⠿⣿⠿⢋⡙⠻⠿⠟⠷⣶⣶⣿⢿⣿⣿⣿⣿⡟⢉⣿⢹⡟⣿⡀⠀⠀⠀⠀⠀⣤
⠂⠙⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⠠⠄⢛⣿⢦⣶⡔⣾⣿⡟⢛⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡄
⠀⠀⢸⡿⠉⣤⠛⣿⡿⣿⣿⣿⣿⣿⣷⣄⠈⠁⣸⡟⢻⣿⣇⣠⡾⠇⠀⣿⣿⢻⣿⡿⣿⣿⡀⠹⣦⠀⡀
⣿⣷⣾⣿⣄⣀⣠⣿⡇⠘⣿⣿⢿⡿⠿⡿⠃⠀⠉⠀⠈⣿⣿⡍⠁⠀⣠⡿⣤⣾⣙⣷⣿⣿⣷⣶⣥⣀⣁
⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⢙⣿⣤⣤⣴⡆⠀⠀⠀⠀⠀⠈⠛⠃⢀⣼⣏⠀⠸⠿⣿⣿⣉⣾⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⢢⡶⢾⣿⣿⣿⣿⣿⡄⠀⠙⠦⡀⠀⠀⠀⠈⠛⣿⡟⠂⠀⣰⣾⣿⣿⣿⣼⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⡛⠉⡝⣿⣿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠿⠿⠛⣿⣿⢿⠅⣈⡹⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣁⠀⢁⣿⣿⣿⣿⣧⣀⠀⠀⣀⣰⣾⡷⣿⣾⣿⡄⢀⡀⠀⠈⣷⣤⣤⣿⣷⣿⣿⣿⣿⣳⣤⡀
⣿⣿⡇⢸⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣇⣿⢿⣿⡟⣿⣧⣽⠃⠀⣴⢻⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⠃
⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠻⣿⣞⢿⣿⡿⣿⣾⢧⡈⢉⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡀⢴⠄⣹⣿⣤⡿⣷⣿⣿⣮⣻⣶⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⣤⣿⠄
⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣧⡀⢀⣸⣿⡇⠀⠒
⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣉⣈⣿⡿⢧⡀⠀⣁⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣅⣴⡆⣿⣿⣿⣿⠛⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣾⣿⣿⣿⠏⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧⠐⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣧⣙⣋⣿⣿⣿⣿⣿⣿⣿⢀⡀⢠⡄
⣿⣿⡿⠿⠿⣿⠿⣿⣿⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⢻⣿⣿⣿⢿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣾⢓⣾⣿
⣿⣿⣿⣶⡞⠛⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣇⣀⣴⡿⠟⠂⣘⣿⠿⠿⣿⡟⠛⢻⣿⣿⡄⠙⠻⠟⠁⠈⠉⠀⠙⢿⣿⣿
⠋⠱⣿⠿⠿⠿⠦⠀⠀⠀⠈⠻⢿⣦⣤⣤⣴⣿⡿⠿⠇⠘⠀⠤⠀⠉⠉⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣀⡀⠀⣠⠤⠤⣄⠀⠀⠀⣀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠞⠁⠀⠀⠀⠈⠟⠁⠀⠀⠀⠹⠉⠹⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣀⣀⡀⠀⠀⠠⠃⠀⠀⠀⠀⠟⠒⠀⠀⢄⡀⠠⢀⡀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠃⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠙⠀⠀⠀⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⡆⠀⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡄⠂⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠣⡄
⠀⠀⠀⠀⠀⠀⢨⠦⠄⠘⣄⠀⠀⡀⠀⠀⡵⠦⠖⠃⠀⠈⠓⠒⠳⣄⠀⠀⢀⡼⠤⠔⢣⠔⠁⠀⠀⠢⡀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠸⣄⡄⠀⠁⠀⠀⢉⡖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⢀⠇⠀⣴⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠉⢻⠀⠀⢠⠔⢊⣰⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠒⣄⠘⣄⣀⡔⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢆⡀⡀⠀⠀⢀⡞⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢹⡀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⠀⠉⣗⠀⠀⠐⠢⠉⠔⢄⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠄⠀⢸⠓⠒⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⢒⣿⠀⠀⠂⠸⠶⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠀⠐⠀⢸⡤⢤⡀⢄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⡆⠁⢹⠀⠀⠀⠂⠀⠂⠁⠀⠀⢀⣄⠀⠀⠀⠑⠂⠀⠐⠁⠀⣼⠒⠀⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣸⡀⠀⠀⠀⠀⠀⠀⢀⠔⡞⢸⠢⡀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠘⢎⡇⢸⣉⠏⠀⠀⠀⠀⠀⠀⣠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⡴⠶⠤⠤⡇⢸⡅⠀⠀⢀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⢒⡀⠀⠀⠀⡏⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⢤⠀⠀⠀⡧⠤⣼⠦⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠙⢲⠃⠀⠀⢠⡏⠉⠁⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⣆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⡀⠀⠀⢸⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢳⡇⠀⠀⠈⡇⠀⣀⠴⠋⢳⡃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢿⠉⠀⠀⠀⠀⢿⣅⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⠀⠀⠸⠀⠃⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈

⠀⡴⠚⠓⣶⣾⠟⠻⣟⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢶⣲⠟⠛⢯⠛⠲⣄
⢸⠡⣂⠢⢸⢁⢀⢄⢸⡆⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢄⡀⢘⡇⢄⡠⠈⡇⠀⠈⢧
⣸⢦⣉⣠⠾⣔⠉⢈⡼⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣈⣉⠜⢇⠊⠑⡰⠁⠀⠀⠸⡆
⣟⠀⠈⠀⠀⣨⣽⡉⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⣉⡉⠀⠀⠀⠀⢀⡇
⠹⣄⡀⠀⣠⣼⢏⣣⣦⢠⢴⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣠⠎⣁⣤⠀⠀⠀⢼⠁
⠀⠈⠙⠛⢻⠟⠉⣱⣧⣠⣶⣦⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⡟⠋⢁⣯⣀⠔⣒⢦⢷
⠀⠀⢀⣴⣿⢠⣾⣿⡿⢿⣿⣿⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⠁⣔⣁⡤⠛⢦⣉⠼⣸
⠀⠀⠘⠛⣿⠀⠁⢸⡇⠀⠩⠕⠮⢟⢦⣀⣀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢻⠀⠀⠀⣇⠀⠀⠀⠀⠉⠳⣄⣀⣀⣀⣠⠤⢤⣀⣀⡀
⠀⠀⠀⢀⣿⠀⣠⣾⣿⡷⢠⣾⣿⣦⢹⡏⠉⠉⠿⠊⣀⣘⢫⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⢀⠔⠋⡱⠀⡰⢂⡒⡄⠈⠀⠀⠉⠀⠀⣀⠈⢠⢽
⠀⠀⠰⠿⠿⡄⠛⠋⠹⣆⣘⢿⣿⠟⠈⣴⣿⣷⠀⠀⢿⡿⢢⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠮⠼⣇⠘⠚⠉⣇⢀⠳⢌⡥⠃⠀⣞⠟⡆⠀⠺⠭⠃⡬⠿
⠀⠀⠀⠀⠀⢻⡄⠀⢠⣿⣿⣇⠀⢀⠀⠈⠛⠋⣰⣦⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠀⢀⠜⢉⡇⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⣿⠟⠢⣟⡛⠁⠙⣻⣿⡷⠴⣶⡶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠼⠳⢼⡒⠁⠈⢳⠎⣱⠤⢤⣴⠞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠒⠻⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡿⠒⠚⠞⠓⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣠⠄⠛⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⣠⡇⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣹⠆⠀⠀⠰⣏⠀⠰⡏
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡁⠀⠀⠀⠀⠙⣦⠀⠉⠢
⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠋⠀⠀⠀⠀⠀⣠⢸⡆⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠀⠀⠀⢀⣠⠞⠁⠀⠳⢤⣄⡀⠀⠀⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣹⠦⢤⣄⡀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠻⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣠⢿⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠤⠤⠤⠶⠶⠚⠛⠉⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢸⡆
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠄⠘⢀⣾⢁
⠀⠀⠀⠀⠀⠠⡂⢻⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⡱⢃⠞⣠⠂
⠀⠀⠀⠀⢀⡩⠒⣁⠜⠛⢳⠶⣦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⠴⢶⠞⣻⠟⡱⢣⠎⡔⠁⠴⠁⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠐⠁⠘⠁⠈⠀⠋⠀⠉⠀⠋⠐⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣤⣤⣤⣀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢠⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣙⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠁
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉

⠀⠀⠀⠀⢀⠤⠒⠒⠉⠉⠉⠙⠋⠙⠢⣀⠀⠀⠀⠀⠀⠐⢿⣒⢄
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣢⡄⠀⠀⣀⡋⢱⠤⣁⠲⠄
⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠃⠀⠀⣾⠀⠀⠀⠀⠉⠒⠬⣐⢤⡀
⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡰⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⠈⠙⠪⢐⠤⡀
⠀⠀⠀⢀⡔⢡⢆⡄⠀⡠⠀⠀⣰⠁⢀⣀⢀⣼⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠪⢔⢄
⠀⠀⠀⡎⣰⠝⢉⣠⠎⠀⠀⣰⣛⠀⢛⣋⠁⣼⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠑⠑⢄
⠀⠀⠸⠋⣠⣴⡏⠀⠀⠀⢠⡟⠈⠈⠋⠄⣰⡿⠇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠠⠘⠀⠀⠈⠁⠢⠀⢠⣿⡿⢆⣤⣤⣤⠟⢀⣆⡲⣧⣷⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠂
⠀⠂⠀⠈⠁⠀⠀⠀⠀⢼⣿⡿⠋⠹⠸⠇⢰⡏⠀⢹⡏⣩⣹⡷
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢧⡌⠀⠀⠀⣟⣷⣲⡟⠀⠀⠀⠠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢀⣀⠀⠀⠀⠀⠑⢤⠴⠚⠏⠙⠻⠙⠓⠛⠁⠀⠀⠠⠇⡗⠋⢠⠆⠀⡴⠣⡼⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢸⡏⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢸⠇⠒⢃⢜⡼⢨⠘⣳⣣⠃
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡀⠀⠀⡀⠀⢸⡟⢐⡷⣰⢏⡴⣣⢾⠜⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⢴⡈⠦⡀⢱⣄⠀⣧⣸⣾⢁⢎⢙⣳⣫⠼⠃⠁
⠣⡀⠀⠀⠀⠀⠀⠀⠒⠂⢀⣹⠿⢿⡶⣛⡛⠋⠉⢳⡎⢉⣲⠟⠁
⠀⠹⣆⠀⠀⠀⠀⠖⠀⠀⢁⠄⠀⠀⠈⡝⢁⡀⢀⠘⠟⠉
⠀⠀⢿⢷⡀⠀⠄⠀⠀⢀⡎⢠⠀⠀⠜⠀⢿⠃⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠞⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛
⣀⣤⡤⢤⣄⡀⠀⢻⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⠧⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻
⣿⠀⠀⠀⠀⠉⢧⣨⣶⣇⠀⡴⠋⠉⣹⠂⢀⡤⠞⠋⠹⣆⣤⣤⠀⣤⣤⡄⡾⠛⠛⣷⣼⡇⠻⠶⡄⢸⠟⢿⣀⣤⣤⣤⠤⢤⡀⠀⢀⣀⡴⠚⢻⡇⢸⠀⢸
⣿⠀⢠⣶⠀⠀⢸⢿⠀⢸⠾⣁⣀⡉⠙⢦⣼⠀⣴⡄⠀⣿⡇⠘⠶⠃⢨⡟⣙⣲⠀⠳⡄⡇⢰⣚⣛⢻⠀⠘⡏⡇⠈⣡⡀⠀⢹⠖⡟⠁⠴⠄⠀⡇⢸⠦⣼
⣿⠀⠀⠀⠀⢀⠎⢸⠀⢸⡄⡨⠤⠽⠀⢸⠛⢤⣠⡀⠀⡏⢷⣀⠀⠀⡸⠀⣇⣀⣀⡴⠃⠹⣄⣈⣹⠟⢤⡤⠇⠷⠤⠇⠹⣤⡼⠀⠙⠤⡆⠀⠀⣷⢾⡖⣧
⢹⠦⠤⠔⠚⢁⣤⠈⠉⢉⠐⠧⣄⣀⡤⢊⡤⣄⣀⡇⠀⢳⣀⠉⠛⠋⢀⣤⣀⣀⣀⣠⢦⣀⠀⢉⡠⣤⣀⣀⢶⣀⣀⡖⢦⣠⠴⠶⢲⠀⣧⠀⠀⢹⣀⡽⠃
⠀⠙⠓⠚⠋⠁⠈⠉⠉⠉⠙⠒⠒⠒⠊⠁⠀⠈⠉⣷⠀⢸⡇⠙⠒⠒⠉⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢾⣣⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠒⠃⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⠤⢤⡤⢞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣌⣙⡒⢒⣒⡽⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣓⣀⢀⣰⢾⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⠶⣶⣾⡿⠿⠿⠿⠿⠷⠾⠿⢿⣽⣷⣶⣾⣶⣶⣤⡴⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⣯⣵⣶⣶⣶⣶⣶⣾⣯⣯⣽⣟⣶⣮⣭⣛⡿⣿⣦⣄⣀⣤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⠟⠛⠟⠛⣹⠿⠽⠛⠛⠿⣿⣿⣿⣿⣷⣿⣽⣻⢿⣿⣷⣽⡿⡿⠃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀⣴⣲⣶⣶⣶⣶⣶⣾⣿⣦⣤⣄⡀⠿⠿⣿⣿⣿⣷⣮⣙⣿⣁⣦⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠻⣶⣶⣦⡉⠉⠉⠉⠉⠛⢿⣿⣶⣶⠀⠈⣏⣿⣫⠴⣶⠆⣶⣬⣿⣿⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣿⠇⠀⣷⡀⠀⠀⠤⣄⡀⡀⢈⠻⣷⣿⡿⠋⣠⣻⣏⣼⣿⣿⠿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⢿⣷⡌⠿⢮⡿⣷⣮⣽⣿⣿⣷⡤⠊⢀⡀⢾⣿⠟⣿⣯⣿⣷⣿⣿⡏⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣂⣀⣾⣶⣿⠻⡎⣿⣿⡿⣿⣮⠶⣾⣿⣿⣘⣿⡷⣬⣍⠛⠿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⣿⡟⣿⣿⣿⠟⣩⣪⢃⣿⡿⣿⣟⣮⡷⢿⣿⣟⣻⠿⣷⣾⣯⣍⣠⣴⣌⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣿⣬⣿⣿⢁⣴⣿⣿⡿⢋⢴⣽⣿⣍⡙⡆⠙⢿⠿⣿⣿⣿⣽⣿⣷⣼⣦⣥⣨⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⢹⣿⣿⢻⣿⠏⢉⢕⣽⠿⠩⢿⣿⣽⣷⢀⠈⠁⠈⠏⠛⠯⡙⠿⣷⠝⢿⣏⣿⡿⠃⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡟⢻⡿⠃⢾⠇⡠⣠⡿⠛⢜⢃⣣⣿⡟⣿⡈⣿⣦⡰⣄⢀⠀⠈⠂⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣴⣿⣿⡣⢟⡴⣪⢨⣽⠃⣤⡌⣽⣿⢿⡷⣿⣷⡈⢻⣿⣿⡆⣇⢳⣶⣶⣾⡟⣼⣿⠟⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣾⣿⣟⢮⠱⣿⣿⣄⠀⣿⣵⢆⡼⠿⠃⣾⣧⠀⣿⡇⣷⣹⣿⣧⣿⢸⣿⣿⣿⣇⢿⣿⣦⣠⣼
⠀⠀⠀⠀⠀⠀⠀⣶⣏⣽⣿⡷⣝⡶⣿⡿⢿⣿⣷⣆⠉⠉⣀⡞⢸⡿⣿⣧⠻⣿⢹⣿⣿⣿⣿⣼⣿⡿⠏⣉⠉⣩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⣧⣽⠟⢡⣿⣾⣿⣾⣣⠈⡛⢿⣿⣷⣦⣈⠐⠋⠜⡡⣿⢀⣿⠉⣿⣿⢿⣿⣿⡿⣇⠘⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⢰⣿⠟⢁⣴⠟⣿⣿⣿⣿⣿⡆⢹⣦⢻⣿⣿⣷⠤⡤⠞⣱⡿⡜⣿⡄⣿⣿⣼⠿⣿⡄⣼⣄⣾⣿⣿⣇⡀
⠀⠀⠀⢀⣴⡿⢋⣴⣿⡿⠻⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣿⣿⠀⠠⣬⡟⣸⣿⣌⢷⡙⢧⢾⡄⣌⠣⠙⢿⣿⡿⠟⠁
⠀⠀⣀⣾⢿⣵⢿⣿⣭⠄⢸⣿⣿⡏⣿⣿⣿⡇⢸⣿⣿⣿⣧⣀⡲⣶⡶⢟⣽⣿⣾⣧⡀⢠⣝⠘⢷⣷⣼⣿⡁⠀⢸⡆
⠀⣸⣿⡷⣿⠁⠈⠛⠉⠀⢸⣿⠻⠃⣿⠻⣿⡇⢸⣿⡟⢁⣿⣧⠌⣉⣴⣿⣿⣉⣹⣿⣿⣮⣿⣦⣤⣻⣿⣿⣷⣶⡟
⡴⠿⠿⠋⠁⠀⠀⠀⠶⠂⢼⣹⣧⣄⢹⡄⣾⡇⣾⡿⢁⣿⣿⣿⠟⣿⡿⠁⢀⣼⣿⠟⠙⠻⠿⠟⠛⠿⢿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢙⣧⣿⡆⡇⣿⣇⢹⡇⣾⢻⡏⣿⡞⣛⠁⠀⣿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣿⣷⠿⣿⣿⢀⣸⣿⡾⠇⣈⠿⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣿⣿⡸⣿⣿⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⢿⣿⣿⣤⡴⠒
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠑⠄⠀⡔⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠃⢸⠀⠳⠬⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠔⠒⠛⠤⣀⡠⠴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⢛⠟⠉⣩⠋⢁⠔⠉⠀⠀⠀⢹⠁⢀⡄⠀⠀⠹⡄⡄⠈⣧
⠀⠀⠀⠀⠀⠀⣀⣤⡞⠋⢁⠎⠀⡠⠃⠀⡔⠁⢠⠊⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣰⠛⠲⣤⠏
⠀⠀⠀⢀⣤⠞⠉⡜⠀⢀⠃⠀⠰⠁⠀⡜⠀⢠⠃⠀⢰⡆⠀⠀⠀⠀⠙⠲⠤⠤⠞⠁⠀⠀⠈⠳⣄
⠀⠀⣴⢛⠃⠀⠈⠀⠀⡌⠀⢀⠃⠀⢰⠁⠀⡎⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⢀⡾⠁⡘⠀⠀⡇⠀⢰⠁⠀⠸⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⣼⠁⠀⡇⠀⢰⠀⠀⢸⠀⠀⡇⠀⠀⠃⠀⡸⠀⠀⠀⠀⠀⠀⢸⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⢸
⡏⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⡇⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⠉⠀⠀⠀⠀⠈
⢿⠀⠀⡇⠀⢨⠀⠀⢸⠀⠀⡇⠀⠀⡀⠀⢣⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠘⣧⠀⢃⠀⠀⡆⠀⠸⡀⠀⢡⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠸⡿⠛⠿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡾
⠀⠈⠣⣘⡀⠀⢱⠀⠀⠇⠀⠘⡀⠀⢰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠙⢄⡀⠁⠈⢹⣿⣿⣿⡿⠆⠀⠀⡼⠁
⠀⠀⠀⠙⢷⣄⠈⢆⠀⠘⡄⠀⢱⠀⠀⢇⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠛⠋⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠿⣿⠿⠾⣆⡀⠙⡄⠀⠣⡀⠈⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠷⠺⢦⣄⣑⣄⠀⠳⣄⠘⢆⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠉⠉⠛⠛⠓⢲⣿⣶⠖⠛⠛⢿⠟⠁⠀⠙⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⡄⠀⠀⠀⠀⠘⠚⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣀⡤⣄⠀⣠⢤⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡞⠁⠀⠈⠛⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢀⣠⠤⠖⠒⢦⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢳⡴⠛⠉⠀⠀⠀⡤⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠉⠁⠀⠀⠀⣁⣠⣤⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣇⠀⣠⡤⠖⠋⠉⠀⣤⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢿⠁⠀⠄⠀⠀⠀⢀⣀⣘⡟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡄⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠹⣄⣀⣠⠋⠀⠀⠀⠠⠖⠀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣇⠀⠖⠀⠀⠀⣠⠞⠁⠀⢙⢦⡶⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⢦⣄⠀⣠⡞⠁⠀⡀⠀⠈⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠈⠉⢹⡀⠀⠘⠁⢀⡴⠋⠀⠙⢷⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣀⣠⡏⠀⠐⠂⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡷⢤⣀⡤⠞⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⡛⠛⠛⠳⠶⠶⠶⢤⣤⡴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢳⡄⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢴⣿⣿⣛⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠁⠀⠀⠀⠀⠰⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁
⠀⠀⠀⢠⡟⣷⠀⠀⠀⠀⠘⠦⣤⡀⠀⠀⠀⣀⣀⣤⠇⠀⠀⠀⠀⢀⣴⡦⠶⣦⣤⣤⣤⡀
⠀⠀⠀⠈⠛⠋⠀⣠⣤⣤⣤⣴⠾⠁⠀⠀⠀⠙⢯⠁⠀⠀⠀⠀⠀⠈⢳⣤⣄⠀⠀⠀⣩⡗⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⡄⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⢸⠋⠛⠛⠁⠀⠀⠙⠛⠓⠶⠶⠶⠶⣦⠀⠛⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⡷⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢧⣀⣀⣀⣤⠶⠶⣄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠾⠋⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠘⢧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠙⣧⣄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣧⠀⠀⠀⠀⠀⢸⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⣯⡿
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⠞⠛⠀⠀⠀⠀⠀⠀⠀⣠⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠒⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⡀
⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠿⠿⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆
⠀⠀⢸⠁⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠶⠶⠶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣤⡀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⣇
⠀⠀⣹⡀⠀⠀⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⠶⠶⠶⠾⠿⠶⣦⣤⣀⠀⠀⣽
⠀⡼⠋⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣟⠛⢷⣄
⢸⠃⠀⠀⠀⢀⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢹⡄
⢼⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡇
⠸⣆⡀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⢈⣻⣶⡶⠶⠾⠿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⡿⠛⠻⠶⢶⣶⣤⣤⣤⣶⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠷⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣴⡾⠿⢷⣤⣀⣀⣠⣤⡶⠟⠙⠻⠶⠶⠶⠾⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⢠⣄⠀⣴⠶⠶⢦⡀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣄
⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡄
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢤⡀⠀⠀⠀⠀⢸⣿⡇⢸⣿⠇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢻⣿⣶⠇⠀⠀⠀⠀⠀⣿⣿⣾⡿
⠈⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠲⡄⠀⠀⠀⣴⠟⠛⠻⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⢠⣾⡿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠁⢀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠉⠉⠀⠈⠙⣿⠀⠀⠀⠀⠀⢠⣿⡿
⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠻⣿⣦⣤⣤⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠈⠉⠛⠛⠉⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣶⠶⠾⠛⢩⣤⣤⣶⣿⡿⣿⣧⣄⢀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣶⣦⣤⣴⣶⠿⠿⣶⣶⣦⣤⣶⣦⣀⣀⣤⣾⠿⠻⣿⣧⡀⠀⠙⢻⣿⡟⠋⠉⠙⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⢉⣿⡟⠁⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣏⡉⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣀⣀⣰⣿⣟⣤⣶⣶⣄⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠻⠿⣿⠏⠁⣸⣿⣤⣀⡀⠀⠀⠀⠀⣠⣾⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣩⣉⣉⣻⣷⠀⠀⠀⠀⠙⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣤⣉⠛⠿⣿⡅⠀⠀⣤⣤⣀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣦⣤⣤⣤⣶⣿⣷⣶⣶⣶⡀⢀⣀⠙⠻⣿⣶⣿⠏⠀⠀⣽⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢻⣷⡘⣿⣿⣶⣽⣿⣦⣤⣤⣾⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⣤⣶⣶⣿⣶⣿⡷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⣾⡟⢻⣟⣿⣿⣿⣿⣿⣿⡉⠹⣿⣿⣟⠉⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⢿⣿⣧⠈⢻⡿⠛⠛⠛⠿⠿⠇⠀⢈⣿⣿⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠟⠋⠻⣿⠟⠻⣦⢻⣿⠀⠀⢀⣾⣦⠀⢠⡀⠀⠀⢸⣿⣿⣿⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⣤⣦⡀⠀⠈⣿⣶⣿⡇⠀⢸⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⡆⠀⣠⠀⠀⠀⠀⠀⠀⠉⠿⡇⠘⣟⣿⣿⣿⣿⣿⡄⠀⢿⣟⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠛⠁⠉⠀⠀⠀⠀⠀⠀⣄⠀⠇⠀⣿⣿⣿⣿⣟⠃⠇⠀⢹⢿⣿⠃⠀⠀⠙⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡃⠺⠀⠐⠋⠛⠋⠀⠀⢰⣿⣵⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣟⠻⣿⣠⡤⠀⠙⢿⣿⣿⣿⣿
⣭⡿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠛⠻⠀⠀⠼⠿⠏⠘⢿
⣿⣿⣿⣶⣭⡓⠶⣤⣀⠀⠀⢄⢠⠥⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠿⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣿⣿⣶⣌⠙⢿⣧⣄⡄⠀⠀⠀⠤⠖⠁
⢿⣿⣿⣿⣿⡿⠛⠁⠀⣼⣿⣿⣏⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⠟⠀⠀⢀⣰⣿⣿⣟⣭⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣧⣽⠖⣥⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣾⣧⣲⣦⣤⣄⡀
⣿⣿⣿⣿⡗⠀⠀⢸⣿⣿⣯⣷⢟⡹⡇⡩⢬⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠀⠀⠈⠁⠔
⣿⣿⣿⣿⠂⠀⠀⠀⢙⣿⣯⣾⣊⣵⣆⢓⣫⢴⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠶⠂⠀⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⡽⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠀⠀⡀⣀⣤⣶⣶⣤⣤⠤⠀⠀⠀⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⣀⣠⣴⣿⣿⣿⣿⡿⠛⠻⠄⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣠⣴⣿⣿⣿⣿⠟⠛⠒⠲⠖⠒⠒⠒⠚⠿⢿⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣴⣾⣿⣿⣿⣿⣿⠿⠛⠛⠦⠀⠀⠀⠀⠀⠠⠤⠦⠀⠀⠀⢛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⢄⣀⣄⣤⠐⠛⠛⠉⠀⠴⢟⣽⠋⠍⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣼⣫⣼⣮⣿⣽⣶⣿⠿⠻⣷⣦⣄⣀⣀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣟⣹⡟⣷⣿⣟⣿⣩⡔⣶⢦⠙⡟⣿⣿⢛⣻⣷⣤⣀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣞⢻⣧⣭⣿⣿⣿⣿⢷⣏⠲⣾⣦⣧⡿⡟⢿⣿⠛⠉⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣣⣿⣿⣿⣿⣿⣿⣿⣿⣌⡾⣾⣷⡾⠋⠛⠛⠾⣿⠉⠙⠒⠄⣼⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣷⡿⠋⠀⣤⣶⣦⣶⡏⠀⠀⠀⣰⣼⣍⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠉⠃⠀⠀⢰⣿⣯⣟⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⣿⣯⣿⣿⣿⣿⣵⣿⡗⠁⠀⠀⠘⠛⠛⠛⠁⠈⠁⠀⢮⣿⣯⣻⡷⢚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣽⣿⣿⣿⣿⣿⣻⣿⣞⠀⠀⠀⠀⠠⣤⠤⠆⠐⠀⠀⠀⣸⣯⣿⣟⣫⣻⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻
⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⡈⢰⠁⠀⠂⡄⠐⡄⠈⠀⠐⠀⡦⠀⡆⡆⠀⢰⠀⠢⠐⠂⠀⡁⠀⡆⠀⡆⡄⢠⠀⡆⠐⠀⠆⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠁⢸⠀⠀⠀⡁⡘⠀⠀⠀⠀⠀⡇⢃⡇⡇⠀⢈⠀⠇⠀⡀⠀⡅⢠⠐⡜⠃⡇⢸⠀⡇⠀⠀⠐⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠈⠒⠀⠒⠁⠃⠀⠀⠂⠐⠂⠁⠘⠁⠃⠘⠀⠂⠀⠀⠁⠀⠃⠘⠀⠁⠀⠐⠊⠀⠇⠐⠂⠐⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢠⣤⣤⣀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠘⠿⠟⠀⠻⠿⠃⠀⠀⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠐⠃⢀⠀⠚⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠃⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠛⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⢀⢀⡀⢀⠀⡀⡀⠀⠀⠀⠀⢁⠀⠀⠀⢠⠀⠀⠀⠀⣶⠉⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠈⠈⠁⠀⠀⢁⡁⠀⠀⢀⡀⠛⠀⠀⠀⠘⠁⠂⠀⠀⠛⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠙⠁⠀⠀⠀⣿
⣿⣿⠀⠀⠀⡍⠙⠦⠞⠉⡅⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣥⣤⡶⠛⢳⣤⡾⠛⠳⡖⠛⢻⣦⣀⠶⣦⡀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⡶⠛⠻⡏⠀⢼⠁⠀⠼⠀⡇⠀⢠⡆⠀⣸⠀⢃⠰⠈⡟⠙⠉⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣇⠀⠘⠀⠀⢷⠀⢸⠀⢰⠆⠀⢀⠀⢸⠃⣤⠃⠀⢸⠀⢰⡇⠀⡠⠁⢠⡟⠉⢻⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢸⡇⠀⠀⠀⠈⡇⢸⠀⠂⠀⠀⠘⠀⡜⣰⠇⠀⠀⢸⠀⣼⠀⡜⠁⠀⡜⠀⠀⡸⠙⡳⡆
⠀⠀⠀⠀⠀⢠⣿⣯⠀⠀⢸⣧⠀⠀⠀⠀⢹⣸⡾⠀⠀⠀⠀⢀⢱⡏⠀⠀⠀⡸⠰⣡⠏⠀⠀⢀⠃⠀⡴⠁⠀⡠⢿⣤⣀
⠀⠀⠀⠀⢠⣿⣿⠋⠁⠀⢸⣿⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⡱⠃⠀⠀⠀⡜⢀⡼⠁⡠⠊⢀⠞⠁⠈⠁
⠀⠀⠀⢀⣿⣿⠁⠀⢠⣆⢸⣿⠀⠀⢀⡠⠄⣿⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⣡⠞⡠⠊⠀⢠⠃⠀⠀⢀⡴⠦⢄
⠀⠀⢀⣼⣿⣿⣤⣶⣿⣯⣿⣿⠓⢤⡏⠀⠀⣿⠃⡤⠒⠈⢉⣿⠀⠀⣀⣠⣴⠁⠀⠀⠀⢀⣾⠕⠁⠀⠀⢠⠂⠀⠀⠄⠁⠀⠐⣚⠁
⠀⠀⢸⣿⣿⣽⠋⣿⣷⣿⣿⣿⣿⣿⣿⠿⠋⣿⣧⠶⣦⣴⣿⣯⣷⠚⠁⢸⣧⠀⠀⠀⢠⡿⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠠⠚⠉⢳⡀
⠀⠀⣼⣿⣿⡏⢠⣿⣿⡇⠙⣿⠟⠟⢿⣏⠀⢻⢿⣆⠀⢈⣿⣿⡿⠷⣶⣿⣿⠛⠋⣻⡏⠀⠀⠀⡠⢂⠄⠀⡀⠀⠄⠀⠀⣀⡤⠒⢈⠿⣆
⠀⠀⢹⣿⡞⢠⣿⢹⣿⠃⡼⠛⠒⠤⠄⣿⢲⠟⠀⠻⡶⠾⢤⠽⢿⡄⢹⣿⣿⣶⣤⣿⠇⢹⡆⣼⠏⠀⠀⠁⠀⠀⢀⡠⠞⠁⠀⠀⠀⢈⣑
⠀⠀⢸⡿⣰⠟⠛⣼⡿⠁⠀⠀⠀⠀⣰⢃⡞⠀⠀⢸⢴⡆⠀⠀⠘⣧⣀⡅⣽⡿⠠⣿⣦⣼⡟⣁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡀⠖⠉⠉⢣
⠀⠀⠈⠓⠀⠀⣾⠋⠁⠀⠀⠀⠀⢰⢃⡞⠁⠀⢠⡟⣸⠇⠀⠀⠀⣾⠉⡕⠋⢷⣠⣿⣿⣿⠀⠈⡿⠀⢀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⡼⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⠀⢀⡞⢰⠏⠀⠀⠀⣰⠏⣴⠁⠀⠀⢻⠃⠸⣿⣷⢞⣣⠔⠁⠐⠒⠒⠒⠒⠀⠀⠉⠉⠉⠉⠠⣴⠢⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣾⡟⠃⠀⠀⠀⣰⠏⣰⠇⠀⠀⠀⠌⣳⡤⣼⣿⠟⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣏⡴⠋⠀⠀⠀⠀⠀⢻⡞⣭⣿⡄⢀⡤⠋⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⣀⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢹⣜⣿⣿⣟⣋⣤⠤⠶⠶⠶⠶⠒⠒⠛⠋⠉⠉⠉⠀⠀⠩⢿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡿⠉⢄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠀⠠⠠⠴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⣿⣷⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⣿⡿⠉⠌⠉⠉⠛⠛⠿⢟⣛⠉⠉⠠⠤⠤⠄⣀⣀⡀⠀⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣻⣿⣾⣿⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄⡀⠀⠀⠀⢉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣶⣿⡿⠋⠀⠛⢶⣄⢀⣀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⢶⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞⠋⠉⠉⠻⣿⣿⡏⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋⠉⠁⠈⢿⣽⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⢤⠀⣀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠛⢾⠻⠏⠀⠀⠀⠀⠀⣀⠄⠙⠀⠀⠀⠀⡠⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⡀⠀⠀⠂⠀⠀⠄⠀⣇
⠀⠀⢀⣴⣿⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣀⠀⠀⣰⠀⠓
⠀⠀⣿⣿⠁⠑⠄⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠃
⠀⠀⣿⣟⠛⠛⠛⠀⠀⠰⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠇
⠀⠀⢹⣿⣶⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⢷⣄⠈⠠⡀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠻⣿⣇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠲⢄⡀⢀⡴⠋
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⣨⡟
⠀⠀⠀⠀⠀⠉⠻⣶⣀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢂⠀⠀⠀⠀⠘⢠⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⣄⣀⡀⢠⡤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠛⠲⠶⠶⠦⠶⠶⠶⠶⠶⠞⠛⢷⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⣴⠖⠛⠉⠉⠉⠙⠓⠲⡶⠶⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢻⠶⠶⠶⠤⢤⣤⣀⣤⣠⢦⣽⡀⠰⣿⠆⠀⡿⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠈⢻⡛⠛⠒⠲⠦⣼⠛⢿⣯⣡⢧⣀⣀⣀⣸⡃⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⠀⠈⣿⣧⣠⡏⣽⡉⠀⢹⣷⠖⠖⢾⡁⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡶⠋⢀⡼⠛⠁⣼⠀⣇⠉⠛⡿⢸⡷⣄⠀⢉⡷⡶⡖⠛⢁⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠈
⠀⠀⣠⡤⠤⣤⡀⠀⠀⠀⠀⠾⠷⠚⠉⠀⠀⠀⣿⠀⣿⠀⠀⡇⠘⡇⠈⠙⡿⢰⠃⠳⣄⡀⠉⠁⠀⠀⠀⢀⣀⣠⠴⠚⠉
⢀⡞⠁⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⢀⣤⠤⠿⢦⡼⠀⠀⣷⠀⢿⠀⢸⡇⢸⣦⡀⢸⡏⠉⣩⡉⠉⠉⠉⠁
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢠⡾⣧⠖⠚⠋⢀⡴⠒⠛⣳⠞⠁⢸⠀⢸⡸⡇⠀⢳⣄⠓⠋
⡇⠀⠀⣠⡴⠒⠛⠉⠉⠉⠛⠻⣄⠀⠀⠉⠛⠀⠀⢰⣟⣦⠞⢫⢴⠛⣉⣛⣳⠾⠁⢹⢀⡏⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖
⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠋⠀⠀⠿⠚⠛⠉⠉⠀⠀⠀⡏⢸⡇⠿⡽⠀⠉⠓⠶⠤⢤⣤⣤⠤⠴⠖⠛⠉
⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠋⢧
⢸⡄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠛⣿⣷⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢹⠷⣿⡞⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣧⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣳⠿⢶⣿⡓⢒⣶⢶⣖⠛⢛⡞⢻⣽⠀⣾⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠸⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠓⣳⠋⠀⠘⠞⢁⡏⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠘⢶⠋⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⡴⠋
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣄⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠤⢤⣤⣤⣤⣤⣤⡬⠷⠤⠤⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡏⢢⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠀⢀⠇⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡠⠚⠉⠀⠀⢠⡇⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠸⡀⠀⠀⠀⠀⡇⢀⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡤⣄⢰⢻⠄⠀⠈⠓⠒⠃⠀⠀⡤⣄⠈⠉⠀⠀⠀⡤⣦⣠⢤⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣟⡏⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⣇⡀⠀⠀⠀⠉⠀⠈⠙⣸⣾⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠉⠉⣦⡀⠀⠀⠀⠀⠀⠙⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠒⠀⠀⠀⣹⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⣀⡴⠋⠉⠐⠒⠒⠒⠒⠒⠂⠉⠁⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣜⣁⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠑⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠄⠐⠐⠚⠛⠉⠓⢦
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⡤⠶⢄⣀⣀⡤⣄⣀⣀⡴⠦⠤⠤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠿⣿⠿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣠⣤⣀⠀⠀⢸⣿⣿⣿⣟⣟⣿⣿⣿⣿⡇⠀⠀⠀⡀⢀
⣿⣿⡤⠿⠴⢯⣼⣛⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣧⠀⢸⣿⣿⠛⠋⠀⠛⠛⣿⣿⡇⠀⢀⣀⣿⣾⣷⣇⡀
⣿⣿⡀⠀⠀⠀⠀⢹⣿⣿⡇⠀⢸⠿⠟⠛⠉⠉⠉⠀⢸⣿⡏⠀⠀⠀⠀⠀⢹⣿⡇⠀⣬⣿⣏⠀⣹⣯⣀
⣿⣿⣧⠀⠀⠀⢀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣸⣿⡇⠀⢠⠾⢻⡟⢿⡏⠉
⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣦⣄⣀⣠⣴⣿⣿⡇⠀⠀⠀⠘⠁⠀⠁
⠀⠀⠀⣠⣤⣦⣤⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⡏⣿⠿⣿⢻⣿⣿⡇⠀⡟⣦⣤⣦⣤⣔⡳⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿
⠀⠀⣿⣿⡿⡿⡿⣿⡄⠀⣿⣿⠿⠁⠈⣀⠈⠘⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⠁⠀⠈⣿⣿
⠀⢸⢸⠀⡆⡆⠁⠐⢀⠀⣿⠇⠀⠁⠀⣟⡀⠀⠁⢿⡇⢸⣿⣯⠀⠋⢀⣿⣿⡆⢸⣿⣿⣿⠀⣥⠂⣿⣿
⠀⠆⠠⠀⠅⠇⠠⠐⠀⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣷⣤⣾⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣩⣭⣍⢉⠉⠉⣷⣾⣿⡿⠟⠛⠻⢿⣿⣷⡎⠉⠉⢉⣭⣯⣍⠉⠉
⡟⠈⠉⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣸⡇⠀⣿⣿⣿⠁⠀⠀⠀⠈⣿⣿⡇⠀⠀⣾⣿⣿⣿⡆
⣿⠤⠂⠨⢽⣿⣿⣿⣿⣿⡇⠀⢰⡆⣿⣿⣿⠉⠀⠀⣿⣿⣿⡀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠸⣿⣿⡿⠃
⣿⣿⠁⣹⣿⣿⣿⣿⣿⣿⡇⠀⠘⠿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣷⣤⣀⣤⣾⣿⣿⡇⠀⠀⠀⠈⡏
⣿⣿⣦⣬⣭⣿⣿⣿⣶⣶⡇⠀⠀⠀⢿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡧
⠉⢉⠉⠉⢉⡉⠉⢉⠉⠉⣷⣶⣶⣶⣶⣶⣶⣶⣶⡆⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠻⠟⠀⠈⠁⠀⠿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠛⠋⠉⢻
⠀⠠⠄⠀⢶⣶⠀⠰⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡄⠀⠀⠀⠀⠀⢸
⠀⣀⣄⠀⠀⠀⠀⣠⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣦⡀⠀⢀⣴⣿
⠀⠘⠛⠀⠘⠃⠀⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣶⣿⣿⣿

⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠒
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⡻⠵⠿⠿⢟⣿⣿⠟⢋⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠌⠀⠀⠠⠊⠛⢋⡄⠀⣶⣾⣟⡶⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠈⠁⠀⠀⢜⣿⣿⣥⣽⣿⠿⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⣀⢠⣤⢨⣽⣷⣡⣤⠤⠜⠁⠀⡠⠀⠀⢁⡇⣀⠀⣸⡿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⣿⣿⣯⣤⣿⣿⣿⣟⠓⠉⢁⣜⠓⠒⢲⣯⣾⣥⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⢿⣿⣿⣿⣹⠛⠃⠀⠘⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠻⠋⠿⣿⣿⣯⣼⣿⠟⠁⣠⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⣿⣿⠿⠛⢿⡆⠈⡇⠀⣤⣼⣿⣿⡄⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⠟⠛⠀⠸⢿⣾⣤⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⢴⡄⢀⢀⣿⠀⠀⠀⢀⣼⣿⣿⢋⣩⠿⣿⣿⣧⣷⡀⠀⠀⠀⣀⠀⠀⢉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣀⠆⢺⣹⣿⣿⣿⣶⣶⣿⣿⣿⣏⡼⠻⣄⣠⡼⠻⣏⠻⠋⠑⡘⢾⣇⣠⡖⠁⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⣞⣁⡤⠎⢻⡟⢹⣿⡿⢿⣿⣿⣿⣿⣀⣴⠟⠣⣀⣀⡼⠳⡀⢡⡘⣿⡆⢮⢯⠐⠀⠀⡀⠀⠀⠀⠀⠉⠛⠿⣿
⡇⣤⣴⠞⠉⠀⠀⠀⣼⡇⢸⣿⡇⠀⠙⠻⢿⣿⣿⣇⣠⣅⠜⠋⠀⠀⣽⣿⣏⣿⣇⠀⠣⣄⡄⠘⠇
⡇⠜⠀⠀⠀⠀⠀⠀⢹⣧⣼⡟⠀⠀⠀⠀⠀⢻⡻⡙⢿⣿⣤⣀⣤⣿⣿⣿⣿⣿⠈⢢⠀⣀⡉⠢⡀⠀⠈⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⡄⠈⠀⠈⢀⣡⡾⢿⢿⣶⡾⠛⢿⡿⠿⠿⣿⣿⣿⡠⡀⠱⡄⠳⡄⠈⠀⠀⡀⣷
⡇⠀⠀⠀⠀⠀⢸⡰⠈⠃⣸⠇⠀⠀⣠⠟⠉⠀⠘⣷⣿⣧⣺⡄⢉⠢⠄⠘⣧⡉⠻⣷⣤⣀⣀⠘⠀⠐⠀⠙⢿⣧
⡇⠀⠀⠀⠀⠀⢀⣇⣠⡾⠋⢀⣠⠎⠁⠀⠀⠀⠀⠘⢿⣝⢀⡙⠻⡦⢤⡀⠈⠂⠀⠈⢿⣿⣿⣦⣇⠀⠸⣦⠀⠉⢤⡄
⠃⠀⠀⠀⠀⠀⣸⣡⡟⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⣀⠈⠃⡙⠛⠾⣿⡓⠀⢢⠼⣿⣿⣷⣦⣦⠀⣦⣀⣥
⠀⠀⠀⠀⠀⡰⢿⡿⠑⠋⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠂⢀⡼⠛⠻⢿⣶⣿⣶⣶⣤⡉⢩⣽⣿⣿⣿⣿⣿⣿⣷⢾⣿⠟
⠀⠀⠠⠂⢸⣶⣦⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣠⠟⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠋⠀⠈⠉⠙⠒⠛⠿⣷⡄
⠀⠀⢠⠔⡿⣿⡿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣤⡌⠁⣈⣷⣄⣈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣃⣀⣀⠐⠂⣤⠄⠀⠀⠀⠀⠀⠀⠀⠹⠀⠄⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⢷⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠳⠈⠁⠀⠀⠠⠀⠠⠀⢀⠀⠘⣾⣶⢋⣄⢸⡖⠀⠉⠀⠀⠈⠁⢀⣀⣀⣺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠠⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⠀⣀⣠⡴⣿⣿⣿⣿⣿⡞⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣈⣉⣌⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣤⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠰⣤⡄⣠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⣤⠏⠀⠀⣠⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣯⣙⣷⣶⣶⣿⣾⢛⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣰⣯⣌⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⢿⡿⢀⣿⡀⢀⡀⠀⣸⣿⣤⣼⣿⣿⢿⣿⢿⣿⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⢟⠿⣿⣿⡇⢸⡿⠀⣿⣿⣶⣿⣿⡟⣿⡻⣷⣎⠻⡶⣿⣿⣿⠟⣹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇⢿⣆⠘⣿⣿⣿⡇⣸⠏⣡⣶⣿⣿⣿⣿⣏⠛⠹⣿⣿⣿⣿⣧⣼⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣎⣼⣿⣿⣷⣿⣶⣿⡿⢾⣿⣿⣆⠙⠇⠈⣿⣿⡌⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣯⠻⣿⣿⣿⣿⡟⣻⣿⠿⣿⣆⠀⣿⣿⡏⠘⣿⣿⣿⣿⣷⣿⣿⣿⡿⣿⣦⣤⣼⣿⣿⡛⣉⢋⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠙⠋⠛⠉⠉⠙⠛⠻⠛⠋⣤⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣯⡌⠙⠂⢳⠿⣇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠻⣿⢫⣿⢿⣿⢛⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢻⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠏⠀⠛⠚⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⠟⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⡆⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠸⣿⣿⣿⣿⣏⢹⡆⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡉⠀⠀⣄⣾⣿⣿⣍⣿⡾⣧⡸⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⢄⣀⣼⣿⡿⣿⣿⣿⣦⡸⣇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⣻⣌⣿⣿⣿⣿⣿⣿⣇⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢃⣀⢀⣀⡉⠻⢿⣿⣿⣿⣿⣿⣷⡄⠠⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠚⢻⣋⣁⡀⠀⠈⢲⣿⠛⣿⣿⣿⣷⣠⡇⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣷⣀⣸⣶⣿⣿⣿⣿⣿⠿⢿⣿⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣯⣿⠿⢋⣿⠿⣷⣾⣻⣿⡟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢰⣿⣿⣿⢻⣿⣯⣿⣿⣏⣴⡿⠋⠀⠹⠟⠃⠈⠀⠈⣿⣧⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡜⢩⣿⣏⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣠⣤⣼⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠾⠋⠞⠃

⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠸⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢁⠇
⠀⠀⠀⠀⠀⠀⠑⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣦⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠠⢤⡏⠀⢸⡤⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢄⠀⠀⠀⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⢀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⢸⠀⠀⠸⣆⣀⣀⡴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠙⠶⢤⣼⢀⡤⠞⠉⠀⠈⢧⣀⡀⠸⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠒⢹⡏⠀⠀⠀⠀⠀⢀⣴⣚⠭⠔⠒⠉⠉⠙⢻⠁⢘⣦⠙⡏⠓⠦⣄⡀
⠀⣀⡤⠖⠋⠉⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠉⠉⠙⢺⣄⠀⠀⢹⠃⠀⠀⠀⠈⠙⠲⢤⣀
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢾⠀⠀⠀⠀⠀⠀⢧⣠⠼⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⡉⠈⠀⠏⠸⠈⠹⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢠⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡆
⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠈⠂⠄⠀⡸⠀⠀⠀⠀⠀⡀⠀⠀⢠⢦⣄⡄⠐⡆⢈⣙⡆
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣾⠁⠀⠀⠀⡀⠀⢠⢴⣟⣿⣯⢡⠤⡴⠀⠋⠐⢻⡀
⠀⠀⢠⠚⠀⢀⡆⠀⠀⠀⠀⢰⣀⣠⠤⠴⠤⠴⠒⠊⢯⠀⠀⠀⠀⠀⣸⢳⠞⡩⣿⣿⣿⣾⣤⣠⠂⠀⢸⠇
⠀⢠⡧⠠⠄⠊⠀⠀⠠⣸⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠂⠀⠀⠀⣹⡿⢰⣿⣿⣿⣿⡟⢻⣿⠖⣤⡞
⠀⡌⠀⠀⣂⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⠈⠀⢹⣽⡈⣿⣿⣿⣿⣿⣾⣿⡿⠋
⠀⡇⠀⠈⠅⠀⠠⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠘⠶⡽⣮⣿⡿⠟⠋⠉⠁
⠀⠀⢀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡀⠀⠀⠀⠀⠹⠀⢈⡿⠛⠲⠦⠤⢀⣀
⠀⡖⠀⠀⠈⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢀⠠⠀⠁⠀⠀⠁⠀⢠⡀⠀⠀⠈⢱⡀
⠀⢱⠀⡤⠢⣤⣤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣥⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⡀⢾⡿⠁
⠀⠈⠻⣧⠀⠀⠙⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⣄⣀⠀⠀⣼⣰⢂⡠⠖⠃
⠀⠀⠀⠙⢧⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠓⠎⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⣹⠦⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠉⠉⢉⣱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠶⢾⣦⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠓⣶⠶⠒⠉⠉⠉⠻⠶⠶⠒⠊⠙⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⣄
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠸⡇⠸⠃⠀⠀⠀⠀⠀⠀⠈⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⡀⠀⣿⡆
⠀⠀⠀⠀⠀⠈⣿⣧⠀⢰⡟⣠⠇⢠⣤⠤⠀⠀⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⡿⡷⠀⣹⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠂⠄⠀⠼⢯⣤⣄⠐⠀⠀⣦⡀⠃⠀⠁⠀⠈⠀⠀⠀⠸⠋⣀⢀⣠⣶⠞⠋⠁⣧⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢿⡁⠀⣀⢤⣐⠉⣡⡼⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⡟⠙⠋⢡⡄⠀⣴⣷⠈⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠉⠁⢀⣀⠉⠁⠉⠒⠒⠙⠦⡀⠀⠀⠀⢰⣤⠞⢉⣥⡁⠰⠇⣀⣤⠶⢺⠿⣿⠀⣿⣯⣿⠘
⠀⠀⠀⠀⠀⠀⣿⣿⠃⣤⣤⡴⢦⣉⠀⠾⠇⢸⣿⠇⡀⢹⡇⠀⠀⠘⣿⠀⣈⣟⣁⣤⣼⣿⣿⡴⣯⡼⠃⢠⣿⢰⡟⢰
⠀⠀⠀⠀⠀⠀⡇⣿⡄⢿⣿⣷⣿⣿⣷⣤⣤⣴⣾⠃⠀⣾⡇⠀⠀⠀⣿⢻⣿⡈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⢠⢸⣧⠀⠹⣿⡷⣿⣿⣿⣿⣿⣧⣾⣿⡿⠀⠀⠈⠀⣿⣾⣿⣿⡶⢮⣿⣶⡿⢯⣿⠀⠀⠀⠀⣿⣡⡌⣦
⠀⠀⠀⠀⠀⠀⠈⢸⣿⡀⠀⢻⣿⣾⣝⣋⣙⣽⡟⣱⣿⠃⠚⠀⠀⠀⢻⡘⢿⣿⡄⠲⣬⣭⣵⠾⠃⠀⠀⠀⠀⠈⣿⡟⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠈⠙⠛⠋⠉⠁⢰⣿⡏⡨⠇⠀⠀⠀⠈⠳⠾⢿⣷⡄⠀⠀⠀⣠⣾⣷⡆⠀⠀⠀⢸⡧⢸⡇
⠀⠀⠀⠀⠀⠀⠸⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⢀⣀⠀⢈⡿⠁⠀⠀⣸⣿⣿⣿⡇⡆⠀⣠⣟⡇⢨⠃
⠀⠀⠀⠀⠀⠀⠂⢸⣿⠀⠀⣰⣿⣷⣦⡀⠀⠀⠀⠣⠄⣤⣾⣷⠀⠀⣾⠿⠛⠋⠀⠀⠀⢘⣿⣿⣿⣿⠯⠁⣠⣿⣽⠃⡟
⠀⠀⠀⠀⠀⠀⢸⡼⣿⠀⠀⠹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢰⣷⡀⠀⣽⣷⡀⠀⢀⣠⣤⣾⠟⣹⡿⠟⠁⢰⡏⣿⠏⡜
⠀⠀⠀⠀⠀⠀⠀⢴⢿⣷⡄⠀⠸⢿⣿⣹⢯⣷⣦⡤⣄⣀⣘⣿⣷⣛⣻⣿⣧⣾⣿⣿⡏⠁⢰⠋⠀⠀⢠⣿⢱⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⣿⠤⢀⠀⠉⠻⣆⢹⣿⡛⢿⠿⣭⣋⠛⠛⡛⣿⣿⡿⠏⣀⣷⣿⠃⠀⠀⢀⣼⢿⠘⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⡇⠘⣷⡄⠀⠙⠋⠙⣿⣷⣤⣉⠋⠛⢿⠟⠛⢩⣴⠿⣿⠿⠁⠀⠀⠀⢠⡏⠸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⣿⣿⣄⠀⠀⠀⠘⢷⣌⠙⠷⣦⣴⣤⣶⣿⣥⡿⠋⠀⠀⠀⣠⣶⡿⢧⠀⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡁⠙⠉⣉⠻⢶⣄⡀⠀⠛⢷⣴⣫⣤⣇⣰⣾⡿⠁⣀⣤⣶⣿⡿⢋⡔⠁⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣅⠀⠂⢉⠀⠈⢻⣿⣿⣶⡄⠈⠙⠛⠛⠋⠁⠀⢰⣿⣿⡿⠏⢠⣾⡁⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⢼⡄⠀⠀⠀⠘⠻⣿⠇⠀⠀⠀⠀⠀⠀⣀⠈⢿⡏⠀⠀⠈⠻⣿⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣼⣿⣦⡀⠀⠀⢠⡎⠀⠻⠇⢀⠀⠀⠸⠿⠀⣸⠃⠀⠀⠀⠀⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⡄⣼⢿⣿⣿⣿⣦⣄⣀⠈⠦⣤⣤⡰⠶⠾⠶⠶⠖⣿⠀⠀⠀⠀⠀⣠⣤⠞⠡⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣧⡟⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⣀⠀⠻⣤⣀⠀⣴⣿⣟⡁⠈⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⢻⣴⣴⣶⣅⠀⠀⣠⠄⠀⠀⠉⠉⠛⠿⢿⣿⣿⣶⣤⡙⢠⠻⣿⠉⠁⠒⠉⠉⠀⠀⠈⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠉⢿⣿⣇⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠙⣿⡄⠀⠹⣷⡀⠃⠀⠀⠀⠀⠀⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⣿⠿⠋⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢺⣇⠀⠀⠘⣷⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣀⢀⣀⣠⣤⣴⣶⣾⡿⠿⢿⣿⣿⣶⣶⣤⣸⣿⣆⠀⣠⡌⠀⣐⠀⠤⠀⠀⢘
⠀⠀⠀⠀⠀⠠⠔⠀⠐⠀⠀⠀⠀⣀⡀⠀⠈⢛⣟⣿⡿⠛⠛⠷⠟⠀⠀⠚⡉⠀⣠⡿⡿⢻⣿⣿⠟⠋⠀⠀⠉⠙⣀⠠⠶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠂⠀⠀⠈⠉⠀⠀⡀⠃⠈⡟⠁⠲⠄⠀⠀⠀⠀⠤⠀⠀⠙⠧⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡆⢀⢄⣸⡿⠖⢁⡀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣘⣵⣾⡿⠛⠙⢛⡛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢻⣿⠀⣰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⣦⣤⣴⣾⣿⣿⠿⠛⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣹⡿⠟⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠲⠶⣶⣶⣂⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣉⣿⣶⣶⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣤⣤⡀⠀⠀⠈⠀⢀⡼⠁⣿⣿⣷⣦⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣶⣄⣠⡞⠈⠺⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣺⣯⣛⣿⣋⣋⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⣰⠷⣄⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢠⡟⠙⢷⣄⠀⣰⡏⠀⠈⠻⣦⠀⠀⣼⠃⠈⠻⣦⡀⠀⠀⠀⣴⠟⠁⠙⣧⡀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⢦⣄⢀⣾⠁⠀⠀⠙⢷⡟⠀⠀⠀⠀⠘⢷⣾⠃⠀⠀⠀⠈⠳⣤⣠⡾⠁⠀⠀⠀⠘⣿⣤⣶⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣿
⠀⠀⠀⢻⡟⠛⠲⠶⠼⠇⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣶⠒⠒⢻⣿⠂
⠀⠀⠀⠘⣧⣠⡤⠤⢤⣀⠀⣿⢼⡇⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⢿⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡆⠀⣿⡏
⠀⠀⢀⣠⣿⡇⠀⠀⠀⠈⢧⣼⠾⡄⡴⠋⣉⣽⠀⡴⠚⠋⠹⣧⡶⣦⢴⠶⣆⣞⠉⢹⡷⡇⠘⢲⣆⣼⠛⣧⡴⢶⡶⠒⢦⡀⢀⣤⠼⠉⢻⠀⡇⢠⡟
⢶⡟⠋⠉⠀⡇⠀⠿⠂⠀⡘⠃⠀⡟⠧⠤⢄⠉⢷⡇⠺⠇⠀⣿⡀⠙⠋⢀⡏⡬⠕⠈⢓⢡⠘⠭⢭⡉⠀⠋⢡⠀⡴⡄⠀⡟⢻⠀⠚⠂⠸⣂⡟⣺⠛⣷⡦
⠀⠙⢦⡀⠀⣇⠀⢀⣀⠔⠁⠳⠤⠇⡜⠀⠈⢀⡄⠙⠒⡆⠀⡇⠳⢦⣤⠜⠈⠓⠶⠒⢁⠈⠳⠤⠞⠉⠒⢁⠈⠉⣁⡙⠒⣃⣀⠉⡇⠀⠀⡟⢫⣽⡶⠟
⠀⠀⠀⢹⣦⠿⠭⠭⠤⠖⠳⠤⠴⠦⠭⣅⠬⠭⠚⠙⠲⣧⠀⢻⠒⠤⣤⠴⠋⠒⠒⠒⠉⠓⠦⠐⠋⠛⠒⠋⠓⠒⠁⠉⠉⠉⠈⡇⣏⠀⠀⡟⠻⣿⣁
⠀⢀⣴⠟⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣇⡏⠀⢠⡇⠀⠀⠙⢷⣄
⢠⣾⣋⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⡴⣟⠀⠀⠉⠀⠀⣼⠁⣤⣤⠶⠾⠛
⠀⠈⠉⠛⠻⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⡟⢗⡲⠤⠤⢞⣿⡀⢠⢄⠀⠀⣠⣄⠀⡀⠀⠀⣰⢆⡴⢦⠀⠁⣄⣠⣙⣦⣍⣒⣒⣒⠿⠋⡇⢿⡄
⠀⠀⠀⠀⠀⣇⢀⠤⡀⠀⡏⢸⠀⢠⠛⡆⠘⠃⠀⠉⡟⠋⠉⡇⠓⠃⢘⡄⢀⡇⠈⠒⠋⢻⠀⡇⠀⠀⠸⡇⠀⡏⠁⠈⠀⡇⠀⢹⣹⡄⢀⡇⠈⢷⡀
⠀⠀⠀⠀⣼⡏⠁⠀⣇⢀⠇⠈⣆⡸⠀⠙⠞⠀⠀⠀⢹⠀⢸⠅⠀⠀⠸⡇⢸⡇⠀⠀⠀⢸⠀⡇⠀⠀⠀⣇⢠⠇⠀⠀⠀⢧⡀⣸⠛⠛⠻⠶⣦⣼⣷
⠀⠀⠀⠀⠛⠛⠛⠛⣻⡟⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠳⠎⠀⠀⠀⠀⠳⠼⠀⠀⠀⠀⠘⠶⠃⠀⠀⠀⠈⣹⡿⠶⣤⣀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣀⣤⡶⠟⢻⣿⡇⠀⠀⠀⢠⣶⡄⠀⠀⠀⢠⣾⠷⣦⣀⠀⠀⠀⣠⡾⠷⣦⡀⢠⣿⠀⠀⠀⠉⠛⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠉⠁⠀⠀⠈⣿⡅⠀⠀⠀⣸⡏⠙⢷⣄⣰⡿⠁⠀⠀⠙⢷⣄⣴⠟⠁⠀⠈⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠁⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢹⡇⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠲⣾⡇⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣠⣤⣀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣰⡟⠉⠙⠻⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣄⣀⢀⣀⡀⢀⣴⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⢸⣿⡟⠋⠉⠛⠟⠻⠿⠟⠉⠛
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠟⠉⠉⠀⠀⠈⢻⣿⣿⡏⣿⣿⠀⠀⢰⣿⠋⠹⠋⠙⣿⡆⠀⠀⢸⡿⠀⠀⠀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡟⠀⠀⠀⠀⠀⡀⠀⠿⠟⣰⣿⣿⠀⠀⣾⡇⠀⠀⠀⠀⠘⣷⠀⠀⢸⣷⣀⣀⠸⣿⠀⠀⠀⠀⢀⣀
⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⠃⠀⠀⣷⣀⠀⠀⠀⠀⠀⣶⣦⣶⣿⣿⣿⠀⠀⠻⡟⠀⠀⠀⠀⢰⣿⠀⠀⢸⣿⣿⣿⣆⠀⠘⠃⠀⣠⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⣦⣠⣤⣤⣀⣤⡀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣤⣤⡀
⣿⣿⡟⠋⠁⠀⠈⠉⠻⣿⠉⣉⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠻⣿⡇⠀⢹⣿⣿⣿⣂⠀⠀⠀⣠⡄
⣿⡟⠀⠀⠀⠀⠀⠀⠠⢀⣼⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⢰⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⣷⣶⣾⣿⠐
⣿⣷⡄⣀⢀⣀⣄⢠⣄⣾⣿⣿⣿⠀⠠⣏⣀⣹⣿⣦⣿⣏⠉⣿⠀⠸⣿⣟⣛⡛⣋⣉⣉⣋⠀⠀⢀⣿⡇⠀⠀⠈⠻⢿⣿⣿⣿⠟⠁
⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣦⠀⢀⣤⣄⠀⠀⠀⣿⣿⣻⢛⣟⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢻⣷⣤⣴⣦⣤⣤⣿⠀⠀⢸⣿⣧⢳⣹⣿⣿⣿⣿⠋⠉⢿
⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⠛⠉⠈⠉⠻⣿⣿⣿⠛⠿⠻⣷⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⣿⣿⣮⢇⣻⣿⣿⢣⢲⣶⣤
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡏⠀⠀⠀⠀⠀⠻⣿⣿⠇⣠⣾⡿⠀⠘⠻⣿⢿⣿⣿⣿⢿⣿⠿⠀⢸⣿⣿⣿⡏⠁⠉⠁⠱⣿⣿⣿
⠀⠀⠙⠻⢿⣿⣿⣿⣧⠿⠛⠀⠀⣷⣀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⠀⠀⠀⠘⢿⣿⣿⣿⠞⠁⠀⠀⢸⣿⣿⣿⢀⠀⠀⠀⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣆⣀⠈⠀⣡⣾⣿⣿
⣴⣦⣤⣶⣶⣦⣶⣤⣴⣶⣶⣦⣶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣶⣦⣴⣶⣦⣴⣦⣶⣦⣶⣴⣦⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠿⢿⣿⡿⠿⠿⠿⢿⣿⠿⠻⣿⠀⢀⣤⣄⡀⠀⠀⠀⢀⣤⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣀⣄⣀
⣿⣶⣶⣦⠀⠀⠀⠀⠀⢼⣶⣶⣿⠀⢿⣏⣈⣿⣴⣶⣤⣟⠁⣹⡇⢈⣿⠿⣿⣿⡿⠟⠋⠉⠉⠛⢿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⡟⢸⣷⠀⠀⠀⢸⣿⣿⣿⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⣿⣷⡾⣋⠀⠀⠦⠀⠀⠀⠀⣹⡇⠀⢸⢟⣿⣿⣿⣿⣿⣿⣙⡷
⣿⣿⣿⡇⠀⠀⠀⠀⠐⢀⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⢠⣿⣏⣠⣿⣷⣤⣄⣀⣀⣤⣶⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠆
⣿⣿⣿⣷⣄⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠙⠻⠿⠛⠃
⡿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣄⢠⣾⠆⠀⠀⠀⠀⣿⣿⠋⠻⠟⠋⠉⠙⠛⢁⣙⣿⣏⠀⣠⣤⣄⣤⣴⣶⣤⣀⣤⣤⡀⢸⣿⠟⠛⠁⠀⠀⠀⠈⠀⠙⣿
⠀⠀⠀⠈⠛⢿⣾⡏⢀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⢸⡏⢠⣿⣿⣿⣿⣿⣿⣧⠈⣿⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣰⡖⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⠀⢠⡄⣠⣤⡄⢤⠄⢸⣿⣷⠀⠙⢻⣿⢿⣿⣿⣿⢿⡟⠛⠁⢸⣏⠀⠀⢀⡄⠀⠀⢠⠀⠀⢠
⠀⠀⠀⠹⠧⣿⣿⣿⣿⡿⠃⠀⠀⣿⣿⣦⡀⠀⠛⠛⠃⢀⣰⣿⣿⣷⠀⠀⠀⠹⢿⣿⣿⡿⠟⠁⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣤⣴⣶⣤⣶⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣟⣛⡛⢛⣛⣛⡛⣛⡛⣛⡛⣛⣋⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣘⣛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣉⣉⣭⡁⠛⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⡿⣿⡻⣍⣯⢿⣻⣞⣿⣻⣯⡿⣻⣿⣻⣿⢻⣿⡿⣿⣿⣿⣿⣿⡿⢡⡤⠛⠁⠀⠀⠉⢷⣄⠀⣿⣿
⣿⣿⣿⣿⣯⣿⣿⣿⣟⣿⣿⣿⣻⣧⣿⣿⣿⣿⡛⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣸⣿⡿⣻⣿⣻⣿⣿⠀⣼⠄⠀⠀⠀⠀⠀⠀⢸⣆⢹⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿⣿⣿⣻⣵⣫⣿⣷⣿⣟⣿⣿⣿⣧⣝⣧⣿⣏⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣟⢻⣿⣿⣿⣧⣏⣼⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣵⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⡧⠀⠀⠑⢌⠲⡝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢋⠜⢡⡙⠎⠁⠀⢩⡛⣨⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⢂⠔⠊⠻⠃⢀⠀⠀⠛⠙⣷⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡱⢣⣏⠀⠰⣕⣠⣁⢅⣀⣠⣖⢠⣷⢦⠡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⠁⣞⠈⠳⢷⡝⠿⢣⡘⢛⠣⣤⣿⠋⠘⡅⣮⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡎⠀⡇⠀⣴⣻⣟⢿⡘⠓⣛⣣⣾⢡⡄⠀⡇⢽⣿⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡴⠃⢰⣾⣏⣵⠿⣿⠿⣿⣿⣿⢿⣿⡔⠉⢚⠻⣿⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡷⣿⣞⠳⣿⣻⡇⣯⣿⣿⡿⠋⣼⠀⠘⣶⡗⠐⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠛⡁⠻⣿⢾⣟⣛⡇⣿⣿⣿⣅⢰⡏⢧⠀⢹⠁⡆⠘⡌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢰⡀⡈⢧⢻⣿⣷⣿⣿⡿⡘⣸⠆⠈⡆⠀⠀⡏⠀⢃⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣇⠀⠀⡇⢷⠈⠋⣿⣿⣿⣿⢃⢇⡇⠠⠀⡃⢀⡇⡇⠀⠈⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠹⣆⣀⡇⣸⠀⣹⣿⣿⣿⣿⠎⡌⠃⠀⠀⡰⠀⡼⠁⠀⠀⢸⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⣿⡇⣿⠀⡏⢹⣿⣿⠟⠊⠇⠀⠀⠀⣁⡴⠃⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢁⣿⢲⢻⣼⡟⡠⠂⢠⣄⡀⠀⣜⡽⠁⠀⠀⠀⠀⢸⣷⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡜⢯⣈⣧⡟⠁⣠⣔⣿⠟⣠⡼⠋⠀⠀⠀⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣁⣼⣁⣔⣛⣩⣅⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⢹⣿⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠷⣾⣿⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠶⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⢋⣁⣤⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⢤⠒⠊⠉⠉⠉⠉⠐⢲⡖⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣠⡀⠀⣸⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠙⠁⢀⡏⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣤⣤⣤⣴⠃⠀⠀⠀⠀⢀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡿⠟⠉⣿⣿⠙⠛⢿⣿⠿⣿⣿⣷⣤⣤⠴⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠴⠿⠟⠋⠀⠀⠀⣿⣿⠀⠀⢸⣿⡆⠉⠛⣿⣿⣿⣦⡀⠙⠋⠀⠀⠀⠀⢀⣀⡤⠞⠁
⢀⣾⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⠆⠀⢸⣿⣧⠀⢰⣿⣿⣿⣿⣿⠉⢙⡛⠛⠋⠉⠉
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢠⣾⡿⠛⠋⠁⣠⣴⣾⣿⠟⠀⢸⣿⣿⢸⣿⣿⣆⡙⠋
⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠛⢄⠀⠀⠉⠃⠀⠀⣾⡿⠟⢉⣴⣾⣿⣿⡿⠛⠀⣿⣿⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠀⠀⠘⠋⠋⠀⠀⠀⠀⢰⣿⣿⠐⠿⠀⠈⠙⠓⠶⠦⠤⠴⠶⠒⠋⠁
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠏⠸⡆
⢸⡀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⢰⣯⠑⠢⣀⠀⠀⠀⠀⢀⣀⢀⡴⠊⠁⡟⠺⠗⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢧⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣠⠶⠚⣿⡍⢉⠽⢋⣭⠉⣩⠏⠸⠷⠀⡇⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠈⢦⣠⠏⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠈⣴⠋⠀⠈⠉⢠⡟⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⣀⠀⠀⠀⣠⠏
⠀⠀⠈⢯⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠁
⠀⠀⠀⠈⢣⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣿⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠳⠤⣤⣄⣀⣀⣀⣀⣼⡦⠤⠤⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠋⠉⠉⠉⠉⠉⠋⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡞⢿⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⢹⡓⢄⠀⠀⠀⠘⠈⠂⠀⠐⠒⠃⠀⠀⠀⠀⠔⣻⠇⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⠄⠀⣀⣴⠀⣀⡀⠀⢄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠁⠀⠀⢒⠃⠀⡗⠂⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠈⠀⢈⠋⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠑⢤⡀⠰⠀⠀⣸⠀⠀⠀⢀⡠⠞⠁⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠈⢷⣄⢙⠇⠀⠀⣴⣤⠴⣞⠋⠀⢠⠞⠩⡙⢦⡀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢸⡇⠀⢀⡹⠿⠒⠚⠉⠁⠀⠀⠘⢦⡴⠃⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡇⠀⠀⠀⣀⠴⢚⡿⠁⠀⠀⠘⡇⢰⠋⠀⠀⠠⠤⠤⠤⠤⠀⠀⠈⣷⠠⡀⠀⡇⢀⠀⠀⠙⢦⠀⡀⠀⠀⢸
⠀⠀⠀⠀⣧⠴⠒⠋⠁⢀⡞⠀⠀⢠⠇⠀⢹⡏⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄⠸⣤⠃⠀⡇⠀⢦⡀⠀⠈⠳⡈⠓⢄⣸
⠀⠀⠀⠀⣧⠀⠀⠀⢀⠏⠀⠀⢀⡞⠀⠀⢸⠁⠀⠀⠀⠈⠉⠉⠉⠓⢆⠀⢸⡄⣧⡀⠀⣧⠀⠀⠙⢦⡀⢀⠝⠀⠀⣼
⠀⠀⠀⠀⢸⡀⠀⠀⠈⠒⠦⢄⠞⠁⠀⢀⡏⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠂⠀⡃⠁⠈⣦⡿⠀⠀⠀⠀⠙⢧⡀⠀⢀⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢸⡿⢿⠋⣀⠀⣠⢸⠅⠀⠔⡧⠀⠀⠀⠀⠀⠀⠀⠈⣆⡼
⠀⠀⠀⠀⠀⠈⢧⠀⢰⡄⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡏⠀⢀⡴⠋⠀⡇⢧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢳⡀⢳⡀⠄⢠⠞⠁⠀⠀⠀⠀⠀⠀⣇⡴⠋⠀⠀⠀⢻⠸⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡴⠃⡇⠀⠀⠀⠀⠈⠀⢷⠀⠀⡇⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠱⡄⢀⣀⡤⣞⠁⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣈⢢⠊⠀⠸⡄⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⢸⣁⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⠤⣷⣠⣇⣀⣀⣀⣀⣠⠤⠼⠓⠋⠁