Bow Emojis & Symbols

⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠒⠒⠒⠢⠤⣀
⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⢤⣀⡀⢀⣀⣀⣀⣀⡀⢀⣠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠽⡇⠀⠀⠀⠀⢹⢏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⠃
⠀⠀⡇⠈⠉⠉⠑⠒⠒⠒⠒⠒⠒⠊⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠧⠤⠤⠤⠤⠾⡶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⢀⡎⠀⠀⠀⠀⠀⠀⠹⡄⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣤⠴⠒⠉⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⠉⠒⠢⢤⠤⣄⣀⡤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠲⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠳⡀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣧⣀⣀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⣀⣀⣽⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⢤⣀⠀⠈⠳⡄⠀⠀⠀⠀⢀⡴⠃⠀⠀⣀⠤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠹⣆⠀⠀⢀⡞⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠤⠤⠄⠀⠀⠀⠀⠀⠈⠳⣽⡄⠀⣾⡴⠋⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⢀⡞⠉⠉⠉⠉⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣤⠴⠖⠒⠚⠛⠛⠛⠛⣒⣲⣶⣶⣷⢤⣤⣤⡤⣾⣧⣶⣒⠒⠂⠀⠀⠒⠒⠒⠲⠤⢄⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡞⠉⠀⠀⠀⢀⣀⡤⠶⠚⠋⡽⠋⡠⢫⠃⢠⡏⢸⡇⠈⢏⢦⡉⠋⠁⠂⠀⠀⢀⣀⠀⠀⠀⠈⢑⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⠶⠶⠛⠉⠀⠀⠀⢀⡾⠁⡜⢁⠇⠀⢸⠇⠀⣷⠀⠈⡆⠳⡀⠠⠀⠀⠀⠀⠈⠉⠛⠛⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠀⠸⠀⠀⣿⠀⠀⣿⠀⠀⠾⠀⠑⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡟⠀⠀⢹⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢀⡾⠀⠀⠀⠀⢿⡄⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠁⠀⠀⠀⠀⠈⢻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠈⠱⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⢀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡫⡀⣀⣀⣤⠄⠀⠀⠀⢀⣀⣀⣧
⠀⠀⠀⠀⠀⢿⡶⢦⣄⠀⠀⠀⠀⠀⠀⠠⢿⣾⣦⡀⠀⠀⠀⢀⣀⣠⣾⣿⣿⡟⣫⠅⠀⠀⣠⡶⠞⠋⠉⣩⠟
⠀⠀⠀⠀⠀⠈⠛⢧⣬⣿⣦⣤⣀⣀⣀⣀⣀⣙⣿⣿⣷⡟⠻⠉⠉⢹⣿⣿⣿⣿⣥⣤⣴⣿⣿⣦⣦⣴⡊⠁
⠀⠀⠀⠀⠀⠀⡤⠖⠋⠋⠉⠉⠉⠉⠉⣉⣉⣉⣛⣿⣿⡇⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣩⣿⣿⣿⠀⣀⣧⣠⣿⣿⣿⣿⡶⠆⠀⠀⠀⠀⠉⠉⠉⠹⠁
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣟⠻⡶⠶⠂⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠛⣉⡥⠖⢋⣿⣿⣿⣿⠇⠀⢸⢻⡏⣹⣟⣿⡛⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⡀⠤⠒⠋⠁⠀⢠⣿⠏⡿⢹⡏⠀⠀⣾⢸⡇⠃⢿⠸⡛⠶⢄⣀⣀⣀⣀⣠⠝
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⣰⢿⠏⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠸⡆⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡟⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠧⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠸⡇⠀⠀⡗⠀⠀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡀⠀⠀⠀⠀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡼⠳⠀⠘⡆⠀⠀⢣⠀⠀⠀⡼⠉⠙⠒⠾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢰⠃⠀⠀⠀⠺⠀⠀⠀⠳⡈⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠨⣣⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡲⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⡄
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣷⢤⣀⣀⣤⣠⢾⣿⡏⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⣿⡶⠉⠈⠻⣿⣟⣿⣧⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⡇
⠀⠀⠀⠀⠸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣽⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣧⣀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⠛⢛⣿⣿⣿⢿⡇⠀⠀⣼⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠇⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⣿⣿⡿⢿⠈⠁⠀⠀⣿⣿⢻⣿⣿⠛⠓⠒⠂⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⢸⢹⣾⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⣣⣾⣆⣠⣤⣾⢿⣛⣜⣙⣿⣿⣶⣤⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⡇
⠀⠀⠀⠀⠘⣼⣿⣷⣧⡀⠀⠀⣀⣤⣾⠿⣫⣵⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣶⣿⢿⡯⢗⣿⣿⣿⣿⠏⣰⢻⣿⠁⠀⢸⠃⠉⠳⠝⢿⣿⣏⠛⠿⣿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠈⢏⠛⠿⣟⠿⠚⠁⣠⣿⣿⣿⣿⠃⠀⠃⠘⣿⠀⠀⠘⡄⠀⠀⠀⠀⠻⣿⣷⡀⠀⠉⠻⢿⣿⠃⡸⠁
⠀⠀⠀⠀⠀⠀⠘⠖⠊⠁⠀⠀⡰⣿⣿⣿⣿⠏⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⣿⡟⣆⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⣿⣿⣿⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣿⣿⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡀⠀⠀⠠⣺⢟⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡿⣿⣿⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢀⡫⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣟⢿⣇⠀⠀⠀⡇⠀⠀⠀⠀⢳⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣳⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⣶⣷⣦⣀⣀⣠⣴⣾⡆⠀⠀⠀⢰⣶⣶⣦⣀⣤⣶⣶⡆⠀⠀⢠⣿⣷⣄⠀⠀⣴⣾⣷⡀⢠⣶⣦⣄⠀⠀⢀⣤⣶
⠀⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⢼⣿⣿⣯⠿⢿⣽⣿⣿⡇⢸⣿⣿⣿⡿⠿⣿⣿⣿
⣠⣾⣿⡿⠋⠀⠻⣿⣿⣦⡐⠚⢻⣿⡿⠿⠛⠀⠛⠿⢿⣿⡿⠓⢚⣿⣿⣿⠀⠸⣿⣿⣟⠀⠈⢻⣿⡿⠁⠀⠹⣿⣿
⠀⠰⠏⠀⠀⠀⠀⠈⠻⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠛⠙⠿⠃⠀⠀⠹⠟⠙⠃⠀⠸⠟⠁⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⢾⣿⣷⣦⠀⡀⢀⣴⣿⣿⡇⠀⠀⣿⣿⣷⣄⣀⣠⣾⣿⣿⠀⠀⢻⣿⣿⣦⠀⢠⣶⣿⣿⠇⠀⣿⣷⣦⣀⣀⣤⣾⣿
⠸⠿⠿⣻⡎⠛⣿⡛⠿⠿⠂⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⠀⠀⢼⣿⣿⡿⠛⠛⣿⣿⣿⡆⠀⣿⣿⣿⣿⢻⣿⣿⣿
⠀⢀⣾⣿⠀⠀⠸⣿⣦⠀⠀⠀⠀⠉⣻⣿⡟⠀⢻⣿⣟⠋⠀⠀⠈⢩⣿⠁⠀⠀⢹⣯⠉⠀⠀⢉⣿⣿⡏⠈⣿⣿⣏
⠀⠀⠻⡇⠀⠀⠀⡿⠏⠀⠀⠀⠀⠞⠛⢿⠁⠀⠈⡿⠛⠳⠀⠀⠀⠞⠻⠀⠀⠀⠰⠟⠇⠀⠀⠸⠛⠻⠀⠀⠽⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⢀⡶⠛⠶⢄⣀⡴⠞⠳⣆⠀⠀⠀⢀⣄⡀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣼⣿⣿⡆⠀⣾⣿⣿
⠈⢧⣤⡴⡾⠻⡶⣤⣤⠟⠀⠀⠀⣿⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⢸⣿⣿⣶⣤⣴⣾⣿⣿⠃⠸⢿⡿⢿⡋⠋⣻⠿⣿
⠀⠀⠀⢰⡇⠀⣷⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠉⠻⣿⣿⡿⠀⠀⠸⣿⡿⠛⠀⠈⠻⢿⡿⠀⠀⠀⢠⣿⠃⠀⢿⣧
⠀⠀⠀⠛⠁⠀⠙⠳⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠢⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⠔⣪⣶⢻⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣝⢿⣿⣷⣂⠀⢀⣠⣀⡀⠊⠛⠡⠋⠋⠞⠙
⠀⠀⠀⠀⠀⣖⠒⠾⡆⠘⣿⣿⡹⢯⠻⠿⠦⠘⡿⠛⠛⣶⣿⣿⠿⠿⠛⣉⣱
⠀⠀⠀⠀⠀⢹⠳⡄⠀⠀⠈⠻⠍⠀⠀⣠⣶⣿⣶⡆⠀⠀⠀⢀⣤⣶⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⢣⠙⠆⠀⠀⠀⢈⣉⣩⣭⣭⣭⠔⣶⡄⠀⠀⠿⣫⣿⣿⠏⣰⡦
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠄⣀⡘⣷⣿⣿⢿⣟⡒⠃⠙⢹⡄⠀⣰⢟⣿⣿⣶⠋⣋⠐⠢⣔⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⢻⣿⣿⣿⣿⣝⠀⠀⠀⢰⣿⣿⣆⠀⠀⢺⣿⠇⠀⠙⢿⣶⡒⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢣⠹⣿⡗⠝⠋⠓⠀⡀⢸⣿⣿⣿⣧⡀⢼⣿⠀⠐⣦⣄⡈⠻⢦⡙⢿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⣧⡘⠧⠀⠀⣠⠎⣀⠀⢹⣿⣿⣿⣷⡈⠃⠀⣀⣺⣿⣿⣷⣦⣝⢻⡉
⠀⠀⠀⠀⠀⡰⠃⠀⢀⡠⠀⢻⣷⣀⡤⠊⠁⣲⣿⣷⠈⢿⣿⣿⡟⡟⠀⢐⣛⣛⣯⢽⣿⡯⠭⠝
⠀⠀⠀⢀⠜⣁⣴⠾⢋⣀⡀⠈⣿⡗⠶⣶⣷⣯⣭⣙⡃⠈⠫⡛⢣⣇⣠⣤⣿⠿⠋⠁
⠀⠀⠀⠘⢬⣽⣿⣿⣿⣯⣥⡀⠈⢳⠰⡄⠈⡉⢉⠛⠛⡛⠀⢉⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠠⣇⣿⢠⡇⣾⡆⣴⡟⣰⣿⡟⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣸⣷⣿⣾⢟⡾⠛⡉⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡣⠏⠠⢶⣹⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠉⠀⠀⠈⠀⠽⣻⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠛⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡀⢳
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⠳⢦⣀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⡀⠛⠃⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⣄⠀⠉⠛⠲⠦⣄⣀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣃⣀⣠⠤⠴⠶⠒⠛⠉⠀⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠣⠊⠀⠀⢀⣀⠀⠀⠈⠉⠓⠲⠦⣤⣽⣶⣤⠞⢋⠉⠓⡶⠶⡶⠶⣿⠋⠉⠀⠀⠀⠒⠀⢠⠀⢼⡩⠇⠀⠀⠰⡯⠂⠀⢿
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠻⠼⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠈⢧⠀⠀⠀⠘⠀⠁⠀⢻⡀⠀⢀⠀⠸⣁⠕⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⢠⣖⡆⠀⠀⠀⠀⠀⡀⠐⠯⠃⠀⠀⠘⠞⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠘⠧⠃⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣉⠇⠀⡀⠀⠀⠀⠀⠐⠀⡠⡄⠀⠀⢠⣇⠀⣀⣀⣥⣤⣤⠿⢤⡀⠀⠀⠀⠴⣈⠀⠀⠀⠀⠺⠤⠃⠀⢀⠔⡂⢀⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠘⠀⠠⠀⠀⠀⠀⠀⠀⢰⣙⠀⠀⠀⠒⠁⢀⡴⣿⠉⠉⠉⠉⠀⡼⠁⠠⡶⠉⡷⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡼
⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠀⢀⢤⠀⠀⠀⠀⠁⠀⠀⢈⣠⡾⠋⠀⡇⠀⠀⠀⠀⡼⠁⠈⡀⠀⠀⡇⠀⠀⠈⠉⠛⠒⠦⠤⢤⣤⣤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢀⠘⠚⠀⠀⠀⡀⢀⣠⡶⢛⡿⠁⠈⢸⡇⠀⠀⠀⢰⠃⠀⠸⠬⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠐⢀⣠⡴⠛⠉⢀⡾⣡⢲⠀⢸⡇⠀⠀⠀⢸⠀⣀⡀⠀⠠⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⣼⠁⠈⠁⠀⠈⣧⠀⠀⠀⢸⡀⠑⠃⠀⠀⢀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⣇⠀⠀⢀⣤⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⠄⠀⠀⠀⠀⠹⣆⠀⠀⢹⡄⠀⠰⠜⠀⠀⢀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠓⠃⠀⠀⣊⡇⢀⢹⠀⠀⠈⣧⠀⠄⠀⠠⠀⠯⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠂⠀⠈⣀⡤⠏⠀⠀⠀⢻⠀⠀⡠⢄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠄⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⢸⡆⠀⠓⠁⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣉⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⣤⣤⣄⡀⠀⠀⣠⢤⣤⡄⠀⠀⠀⢀⣤⡄⣤⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣦⣄⡀⠀⠀⢀⣤⣦
⠀⠀⠀⠿⠶⣶⡿⠛⠻⣷⠶⠾⠃⠀⢀⣐⣿⣿⣿⣿⣿⣃⠀⠀⠀⠰⣿⣿⣦⣤⣾⣿⡷⠀⠀⠀⢸⣿⣿⣷⣤⣴⣿⣿⣿
⠀⠀⠀⠀⣴⡿⠀⠀⠀⢹⣷⡀⠀⠀⠈⣻⣿⣿⣶⣿⣿⡟⠀⠀⠀⠀⠛⣻⡿⣿⡍⠛⠃⠀⠀⠀⠈⢋⣿⡿⠋⠙⣿⣏⠋
⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⢻⣟⠙⣿⠋⠀⠀⠀⠀⢀⡾⡿⠀⠻⣷⣄⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠻⠇⠀⠀⠀⠀⠈⠀⡇⠀⠀⠙⠏
⠀⠀⠀⣾⣿⣶⣤⣀⣠⣶⣿⣿⠀⠀⣤⠆⢤⡀⠀⠀⣠⡄⠦⡄⠀⡠⣶⣤⣄⣀⣠⣴⣶⢄⠀⠀⣠⠴⣦⠀⠀⢀⣴⠦⣄
⠀⠀⠀⠛⢻⣿⠟⠉⠹⣿⣟⠛⠃⠀⠃⣀⣨⣿⠦⢾⣯⣀⡀⠉⠀⠿⠭⠝⡟⠉⣷⠫⠭⠇⠀⠀⣿⢀⣬⡿⠶⢯⣅⡀⣿
⠀⠀⠀⢸⡿⠋⠀⠀⠀⠘⢿⡷⠀⠀⠀⢠⡟⠁⠀⠀⠙⣧⠀⠀⠀⠀⠀⢀⡇⠀⢿⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠈⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠓⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠈⢷⡄
⠀⠀⠀⣀⣴⣶⡄⠀⣰⣶⣆⡀⠀⠀⣤⣤⣤⣀⣀⣠⣤⣤⣤⠀⠀⣴⣤⣄⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣿⣿⣿⣧⣤⣿⣿⣿⡟⠀⠀⢻⣿⣿⠏⠋⢻⣿⣿⠏⠀⠀⣿⣿⣿⣷⣤⣴⣿⣿⣿⡄⠀⢸⣿⣷⣶⣴⣴⣶⣾⣿
⠀⠀⠀⠉⠉⣿⡏⠉⢿⣏⠉⠁⠀⠀⢀⣿⣿⠂⠀⢸⣿⣯⡀⠀⠀⠻⢟⣿⡿⠛⠻⣿⣟⠟⠃⠀⠀⠉⢁⣴⡏⢻⣦⡈⠉
⠀⠀⠀⠀⢼⣿⠀⠀⠸⣿⠄⠀⠀⠐⠙⠿⠛⠀⠀⠀⠛⠿⠙⠀⠀⠀⠿⡟⠀⠀⠀⠹⡿⠇⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣄⡀⠀⠀⢀⣤⡀⠀⠀⠀⢸⣶⣦⣤⣤⣤⣶⣾⠆⠀⠀⠰⣾⣿⣆⠀⢀⣼⣿⡶⠀⠀⠀⢸⣯⣷⡀⢀⣾⣿⡇
⠀⠀⠀⣼⣿⣿⣤⣤⣿⣿⣧⠀⠀⠀⠀⢿⢿⣿⢿⣿⡿⡿⠀⠀⠀⣴⣶⣿⣿⠟⢾⣿⣷⣶⡄⠀⠀⠸⠿⣿⡟⢻⣿⠿⠇
⠀⠀⠀⠻⣿⡟⠉⠉⢿⣿⠟⠀⠀⠀⠀⢀⣾⣿⢸⣿⣧⡀⠀⠀⠀⠈⠉⣴⡏⠀⠈⣷⡌⠉⠀⠀⠀⠀⣴⣿⠇⠸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠈⠿⠟⠁⠀⠀⠀⠀⣼⡟⠀⠀⠀⠘⣿⡄⠀⠀⠀⠾⣿⣿⠀⠈⣿⣿⠷
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⠀⠀⠀⠀⠘⠟⠂⠀⠀⠀⠈⠛⠀⠀⠛⠁
⠀⠀⠀⣼⣿⣷⣦⣤⣶⣿⣿⠆⠀⠀⢰⣶⣄⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⠀⣠⣴⣶
⠀⠀⠀⢶⣿⣿⡿⠻⣿⣿⣷⠆⠀⠀⢨⣿⣿⡷⠷⢿⣿⣿⠀⠀⠀⢴⣶⣦⣤⣤⣴⣶⣶⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⠂
⠀⠀⠀⠈⠛⠋⠀⠀⠈⠛⠋⠀⠀⠀⠈⠛⠋⠀⠀⠀⠙⠟⠀⠀⠀⠘⠟⠛⠉⠉⠙⠛⠟⠀⠀⠀⠀⠉⠛⠉⠀⠀⠉⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣶⣴⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡻⠿⠿⣟⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⠕⣰⣿⠏⠀⠀⠹⣿⣆⠪⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⡿⠿⢿⣿⣿⣿⣛⣛⣛⣭⣵⡶⢟⣵⣿⣿⠃⠀⠀⠀⠀⠘⢿⣷⣮⡛⢶⣮⣭⣛⣛⣛⣿⣿⣿⡿⠿⢿⣧
⠀⠀⠀⠀⠀⠡⣾⣿⣿⣿⣿⣿⡿⠿⣛⣭⣵⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣮⣭⣛⠿⠿⣿⣿⣿⣿⣿⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣠⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣤⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇

⠀⠀⠀⢲⣶⣾⣷⣦⣀⣀⣀⣴⣾⣷⣶⠆⠀⠀⠀⢠⣾⣷⣦⣀⣠⣶⣿⣦⠀⠀⠀⠀⠀⣤⣦⣤⣀⠀⣀⣤⣤⣤⠀⠀⠀⠀⣠⣾⠿⢿⣦⢀⣾⠿⠻⣦⡄
⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠸⣿⣿⣿⡽⣿⢿⣿⡿⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⢸⣇⣀⣀⡌⠿⣥⣀⣀⣿⠅
⠀⠀⠀⠚⠛⠻⠟⠋⠀⠀⠀⠙⠻⠟⠛⠃⠀⠀⠀⠀⠀⢰⣿⡇⢹⡿⡧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⣻⣿⠇⠀⢻⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⢠⣾⡿⣷⡀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⣰⣷⣄⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⣸⣷⣦⡀⠀⢀⣴⣾⣇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠻⣿⣄⣈⣳⡋⣀⣸⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⠄⠀⠀⠀⠀⠀⠿⣿⣿⣿⣤⣾⣿⣿⠿⠂⠀⠀⠀⣾⣿⣿⣷⣶⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣩⣿⠏⢻⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠟⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠀⠀⠀⢻⣟⣿⣿⠟⠉⠹⢿⣿⣻⣿
⠀⠀⠀⠀⠀⠾⠻⡏⠀⠀⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⠉⠿⠋⠀⠘⠿⠉⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠁⠈⢿⣿⡆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⢠⣈⡉⠳⣄⠀⢀⠴⠋⢉⣠⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣶⣿⣿⣿⠃⠀⠀⠀⢰⣿⣿⣷⣤⣤⣤⣶⣿⣿⡇⠀⠀⠀⣿⣿⣿⣷⡄⢠⣾⣿⣿⣿
⠀⠀⠀⠈⠙⠉⠉⡿⠛⢻⡍⠉⠙⠉⠀⠀⠀⠀⠀⠘⢻⣿⡇⠀⠸⣿⡟⠏⠀⠀⠀⠀⠘⣻⣿⣿⣿⠛⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⡏⠀⠸⣿⣿⣿⠇⠀⠀⠀⠙⢻⣿⣿⡏⢸⣿⣿⣟⠃
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠹⣿⠟⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠿⠿⠿⠇⢸⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⡀⠀⠀⣀⡤⠔⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⣠⣶⣆
⠀⠀⠀⠀⢠⣿⣿⣆⠀⠀⣼⣿⣧⡀⠀⠀⠀⠈⠓⠶⠶⢾⡿⠺⠺⣿⠶⠶⠒⠃⠀⠀⢰⣿⣿⡻⣧⣀⣾⢿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣧⣼⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⢿⣿⣿⣍⡹⠿⣟⣩⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⡯⣽⣿⣿⣿⠃
⠀⠀⠀⠀⠘⠛⣿⣿⠏⢻⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⢀⣿⣿⣿⡿⠁⠀⠀⢀⣾⣿⣿⣿⠇⠸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⢸⣿⣿⣿⠿⠀⠀⠀⠛⠛⣿⣿⠋⠀⠀⠙⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠘⡿⠋⠁⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡟⠛⠛⠛⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⣧
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⢀⣀⣀⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⣤⣽⠉⠉⠙⠛⠯⣍⠉⢹⣥⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠁⢸⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣄⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢚⣿⣤⣬⣷⣦⣤⣤⣤⣼⣟⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠙⠳⣄⡀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢦⣄⠀⣠⡤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⡶⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣄⣹⣷⠞⠛⠻⣯⡉⣉⣉⣉⣉⣉⣹⣧⣴⠾⠟⠛⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⣿⠃⢀⡀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣄⡴⠚⠛⠻⣏⢹⡆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⡏⠀⠀⣠⣾⡁⠀⠀⠀⣽⠋⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯
⠀⠀⠀⠀⠀⢻⡀⠀⠿⡼⢷⣤⣤⠶⠋⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣤⣤⣄
⠀⠀⠘⠛⠛⢻⡟⠛⠂⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣠⣼⣷⡤⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⢲⣿⠷⠦⠄⢀⣴⠶⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣾⡁⠀⠀⣴⠏⠀⠀⠀⣀⢸⡇
⠀⠀⠀⠀⢠⡴⠞⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠈⠙⠳⠤⣟⠀⢀⣀⣴⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣋⠁⠀⠀⠀⠀⣠⡶⣫⣤⣍⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣤⡾⢋⡟⠉⠉⣿⠉⠉⠉⠉⠉⠁⣿⠀⠀⢹⣆⢙⣷⠶⠦⣤⡾⣫⡾⠋
⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠈⢛⡿⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢹⣿⠁⠀⠀⠘⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⡴⢦⣤⠶⢦⡄⠀⠀⠀⣾⠁⠀⠀⠀⣰⠇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣄⣠⡴⠋⠀⠀⢀⣸⡛⣻⣇⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠻⣄⡀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡶⠛⠉⠙⠉⠉⣙⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣧⡴⠞⠛⠳⠾⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠞⠛⣻⡟⠁⠀⠘⢻⡾⠻⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⣀⣽⣇⣀⣀⣀⣀⣠⣤⡾⠏⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠩⠭⠩⣉⣉⣉⣉⠀⢀⣀⣀⣀⡄⠄⠤⠛⠛⠛⠛⠛⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡷⢿⣿⣿⣷⣶⣶⣦⡿⠏⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⢾⣫⢻⡇⠸⡿⡿⠿⠻⡆⠈⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣡⣶⣶⠶⣬⣿⣿⣄⣄⠘⠛⠟⠁⢀⣼⣿⣿⣷⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡭⠙⢻⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣍⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄⠀⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡀⠀⠘⠻⣿⣿⣿⣿⣿⡟⣿⣮⣹⣶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⢃⠘⠙⣿⣿⣿⣤⣤⣤⣤⡈⢉⣩⣽⣿⢸⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠿⠗⠀⠠⠄⠛⢋⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⣯⡽⣯⣿⣿⣛⣵⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠃⢰⣿⣿⣿⣿⣿⡯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⠁⢀⣴⠟⠃⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢠⠚⢿⣿⣄⣈⣙⣿⣏⠎⠀⠀⠀⠀⠀⠈⠐⠻⢷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡀⠀⠀⠂⠀⠐⠀⠀⠀⠀⡎⢠⣿⣿⣿⡘⠿⠿⣻⣤⣓⣶⣦⣄⡀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠘⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⣷⡆⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢟⠷⣿⠟⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣅⡄⣹⠿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣯⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣶⡌⠁⠀⠀⠧⢀⠀⠀⡀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣁⣤⣴⠖⢢⣤⣤⠀⣀⠀⠀⢤⣅⡰⢟⣷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣅⣀⠀⠀⠀⠀⣩⡏⢁⡀⠀⢠⡟⣿⣿⣿⣷⡀⠅⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣷⣄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⢷⣦⠄⠀⠈⢈⣷⣄⣽⣿⣿⣿⣿⣿⣷⡀⢦⡀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣄⣀⡀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣎⣛⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠐⠛⠛⠉⠁

⠀⠀⢀⣠⣤⡀⠀⠀⣠⣤⣀⡀⠀⠀⢀⣤⣀⠀⣠⣤⡀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⣠⣤⣄⠀⣠⣤⣄
⠀⠀⢈⣿⣿⣿⢿⣿⣿⣿⣏⠀⠀⠀⢺⣿⣿⢿⣿⣿⠇⠀⠀⠀⣿⣿⣿⠶⣿⣿⡟⠀⠀⠈⣿⣀⣉⠶⣉⣀⣹⠁
⠀⠀⠈⠉⠛⠁⠀⠀⠉⠛⠉⠁⠀⠀⠀⢰⣿⡄⠟⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⡟⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠻⣧
⠀⠀⠀⣰⡶⢦⢀⣴⣶⡄⠀⠀⠀⠀⢀⣶⡄⢀⣤⡆⠀⠀⠀⠀⣰⣦⡄⠀⣠⣶⡆⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠙⠷⣤⣯⣤⠾⠏⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⠛⣻⣶⣿⠛⠋⠀⠀⢸⣿⣿⣿⡶⢾⣿⣿⣷
⠀⠀⠀⠀⢾⡟⠀⢻⠧⠀⠀⠀⠀⠀⠀⢾⠟⠘⢿⠂⠀⠀⠀⠀⠀⣴⣿⠋⢻⣷⡀⠀⠀⠀⠛⠛⠉⠀⠈⠙⠛⠃


⠀⠀⢀⡛⠲⡄⠀⡠⠞⢋⡀⠀⠀⠐⣿⣿⣦⣤⣾⣿⡇⠀⠀⠀⣶⣶⣄⣤⣠⣶⣶⠀⠀⢰⣿⣿⣆⢀⣾⣿⣷
⠀⠀⠈⠉⠉⡟⠙⢏⠉⠉⠀⠀⠀⠀⢙⣿⠇⠀⢿⣟⠁⠀⠀⠀⣿⣿⣿⠛⣿⣿⣟⠀⠀⠸⣿⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⢸⠃⠀⠘⣆⠀⠀⠀⠀⠀⣾⡏⠀⠀⠘⣿⡆⠀⠀⠈⢻⣿⠇⠀⠹⣿⡟⠀⠀⠀⢰⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢠⣤⡀⠀⢠⣤⡄⠀⠀⠐⠤⣈⣱⣦⣤⣶⣉⡠⠔⠀⢀⣿⣷⣄⠀⣴⣾⣧⠀⠀⠀⣾⣿⣦⣠⣾⣿⡆
⠀⠀⠀⣿⣿⣿⣶⣿⣿⡿⠀⠀⠀⠀⠀⠞⠁⠀⠈⠂⠀⠀⠀⢸⣿⣿⡛⠿⣛⣿⣿⠀⠀⠀⢿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠈⣽⡿⠉⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣏⠀⠀⣴⣿⣿⡟⠈⢿⣿⣷⡄
⠀⠀⠀⠈⠿⠁⠀⠈⠿⠁⠀⠀⠀⠀⠈⡆⠀⠀⠀⡜⠀⠀⠀⠀⠉⠻⣿⢸⡿⠛⠁⠀⠀⠀⠹⠋⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⡶⠳⢦⡀⠀⠀⣠⠶⠳⣦⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢠⠖⠒⣄⠀⠀⡠⠒⠲⡄
⠀⣧⠀⠀⢈⣶⣎⠁⠀⢰⡇⠀⠀⠀⠀⣸⠇⠈⠳⣀⡰⠋⠀⢿⡀⠀⠀⠀⠀⣟⠀⠀⠈⠂⣤⠐⠁⠀⠀⣻⠀⠀⠀⠀⢻⣀⣀⣈⣦⣴⡁⣀⣀⠇
⠀⠙⠛⠋⣱⠃⢻⡉⠛⠛⠁⠀⠀⠀⠀⠻⣤⡤⠖⡿⡗⠦⢤⡼⠃⠀⠀⠀⠀⠈⠁⣢⡤⠞⠉⠳⢤⣔⠈⠁⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠑⣄
⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣸⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠁⠀⠀⠙⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠋⠀⠀⠀⠈⠃
⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣰⣶⣤⡀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠀⢠⣶⠒⢄⠀⢀⠔⢳⣦⠀⠀⠀⠀⠀⣴⣔⡠⢄⡀⣀⡠⠄⣢⣦
⠀⢹⣿⣶⣤⣶⣶⣤⣶⣿⡇⠀⠀⠀⠀⢿⣿⣿⣽⣶⣾⣫⣿⣿⡟⠀⠀⠀⠀⠀⠸⢿⣷⣶⣶⣶⣶⣿⠿⠀⠀⠀⠀⠀⢿⣿⣿⠿⡿⢿⠻⣿⣿⣿
⠀⠸⠿⠿⢫⡿⢿⡝⠿⠿⠃⠀⠀⠀⠀⠸⠿⠛⢡⣿⣿⡉⠛⠿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠱⣤⣀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣾⡇⠸⣷⡀⠈⠁
⠀⠀⠀⠰⠿⡇⢸⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠋⠀⠀⠀⠘⠻⢿⠇⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣤⣀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢀⣴⠲⣄⠀⢀⡴⢲⣄⠀⠀⠀⠀⠀⠀⢀⣴⠖⠻⡆⣰⠛⠲⣦
⠀⠀⣏⡀⠹⡆⢰⠋⢀⣻⠀⠀⠀⠀⢠⣿⣽⣷⣼⡆⣼⣴⣿⣽⣧⠀⠀⠀⠀⢀⣤⠿⣶⢄⢻⣇⡴⡶⠿⣤⡀⠀⠀⢰⡉⠉⠓⢄⠀⠀⡠⠒⠉⠉⡇
⠀⢰⠋⠛⠓⣷⡶⠚⠛⠙⡆⠀⠀⠀⠙⠦⣝⣻⣿⣷⣿⣿⣛⡥⠞⠀⠀⠀⠀⢿⣇⣀⣀⡁⣿⣿⣈⣀⣀⣸⡇⠀⠀⣠⠛⠛⠓⢒⠷⠶⡒⠒⠛⠛⢦
⠀⠘⠓⠒⢉⡟⢳⡑⠒⠚⠃⠀⠀⠀⠀⠀⣀⡴⠚⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡇⢳⠀⠉⠉⠉⠀⠀⠀⠈⠒⠒⡾⠋⠀⠀⠈⢷⡒⠒⠋
⠀⠀⠀⠠⠟⠀⠀⠷⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⡠⠠⠀⡠⠢⡀⠀⠀⠀⠀⠀⠀⢀⣴⢄⠀⠀⠀⡀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢠⣄
⠀⠀⠀⢣⠀⢀⠀⣰⠁⠀⠀⠀⠀⠀⢀⣼⣿⣆⢳⠀⡴⢡⣿⣿⡄⠀⠀⠀⠀⠀⢀⡞⠙⠓⢦⠀⠠⠚⠙⠳⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠃⣘⡐⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣳⣶⣗⣻⣿⣿⣿⠀⠀⠀⠀⠀⠘⢧⡀⠀⣠⣷⣆⡀⢀⣠⠇⠀⠀⠀⢠⣾⣿⡷⡶⣶⢾⣿⣧
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣼⡞⣷⡉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⣩⠜⠙⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣿⣄
⠀⠀⠀⠀⠀⢦⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣿⣿⠆⠀⠀⠀⣤⣄⣤⠖⠒⢛⠋⢁⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢸⠿
⠀⠀⠀⠀⠀⠈⠳⠮⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⣾⣶⣄⣀⣤⣶⣶⠀⠀⠀⠀⣤⣤⣀⣀⣤⣤⠀⠀⠀⠀⢰⣦⣀⣀⣀⣴⡆⠀⠀⠀⠀⠐⠄⢀⣀⢀⡔⠂⠀⠀⠀⠀⢸⣿⣄⣀⣠⣾⡇
⠀⠀⠿⠿⠋⠉⠛⠿⠿⠀⠀⠀⠀⠛⠛⠋⠙⠛⠛⠀⠀⠀⠀⠸⠿⠛⠉⠛⠿⠇⠀⠀⠀⠀⠉⠔⠈⠉⠐⠦⠁⠀⠀⠀⠀⢾⠛⠉⠉⠉⠛⡿

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⣿⣶⣄
⠀⠀⠀⣴⠟⠋⠉⠙⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢰⣿⠉⢻⣧
⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢠⣿⠏⠀⠈⣿⡆
⠀⢰⣿⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢸⣷
⠀⣸⡇⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⢸⣿
⠀⣿⠁⠀⠀⠀⠉⠻⢷⣦⣤⣀⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢠⠟⣀⣤⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠶⠶⢿⠶⠶⠖⠒⢲⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣿⠀⠀⠀⠀⠠⣿⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡤⢿⡀⠀⠀⠀⢠⡿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡞⠻⣶⡶⠶⠟⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣿⢿⣆⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣿⠇⠈⣿⡆⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠙⢿⣦⣄⣀⣀⣠⣤⠾⠋⠀⠀⠀⠀⣸⡿⠀⠀⠸⣿⡄⠀⠀⠀⠀⠈⠛⣶⣤⣄⣀⣀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠉⠉⠛⣻⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠘⢿⡍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣦⣀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣿⡇⣠⡾⠋⠈⠙⠻⢷⣦⣸⣇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠛⠟
⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⣀⣴⣶⣶⣤⣀⠀⢿⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠉⠉⠉⠉⠛⠷⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⣄⢀⣴⣾⣿⡿⠟⠋⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠙⢻⣿⡿⠛⠁⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⠃⠀⠀⠀⠀⣀⣀⠀⢹⣿⣄⣀⣀⠀⠀⠀⣀⣨⣿⣿⣤⣀
⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠿⠛⠛⠋⣿⡟⠀⠀⠀⢠⣾⠿⢿⣿⡿⠟⠛⠛⠿⣷⣾⡿⠛⠛⠉⠉⠙⢿⣧
⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⡀⣾⡟⠀⠀⠀⠀⠀⠈⣿⣷⣤⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡟⢙⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣀⣀⣀⣤⣿⣿⣦⣀⣀⣀⣠⣾⡿⢿⡿⠟⠁⠀⢠⣿⠏
⠀⠀⠀⠀⠀⢿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠋⠁⠀⠉⠛⠛⠛⠛⢿⣦⣀⠀⠀⢀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠟⠉⢻⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⣶⣶⣶⣶
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠿⢿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢀⣤⣴⣾⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣀⣀
⠈⠋⠉⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣀⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠛⠿⠿⣿⣶
⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟
⠀⠀⠀⠀⠾⠿⠛⠉⠻⣿⣧⣀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉⠻⠿⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠛⠿⣿⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⠛⠉⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠛⣿⡏⠁⠀⠻⣿⣦⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⡟⠉⠁⣴⣿⠟⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠙⣿⣷⡀⠀⠀⠈⢻⣿⣧⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠙⣿⣿⣄⣾⣿⣯⣤⣤⣀⡀⠙⢿⣷⣤⣀⣀⣀⣀⣤⣾⡿⠋⠀⢀⣀⣤⣤⣼⣿⣷⣄⣀⣤⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⠿⠿⠿⣿⣷⣤⠉⠛⠛⠿⠿⠛⠛⠉⠀⣠⣾⣿⠿⠿⠿⠿⠿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣤⣴⣾⣿⡿⠟⠛⠋⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣷⣶⣦⣤⣴⣶⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠾⠦⢤⣄⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⣀⣀⣤⡼⠷⠴⠶⣤⡀
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⠀⠙⢦⠀⠀⠀⠀⢠⠞⠀⠀⡠⠄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⢧⠀⠀⢰⠋⡠
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠠⡖⠒⠀⠀⠀⠀⠀⠀⠀⣈⣾⣧⣤⣷⣋⠀⠀⠀⠀⠀⠀⠀⠒⢲⠆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⣇⠀⠀⠀⠀⣸⠇⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⠛⠉⠉⠀⠀⢀⣀⣤⢴⡾⠟⡻⡿⠛⡶⢲⡞⠻⣟⠛⢧⡤⢀⠀⠀⠀⠀⠉⠉⠙⠒⢄⢀⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣀⣀⣠⡤⠖⠛⠉⠀⢀⡞⢁⠞⡰⠀⢸⡇⠘⡇⠀⢣⠱⡄⠁⠀⠀⠀⠀⠒⠦⠤⣤⣠⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣠⠏⠀⠊⢰⡁⠀⢸⠁⠀⣿⠀⠀⡃⠘⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠈⠁⠀⣸⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣼⠁⠀⠀⠈⣷⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠃⠀⠀⠀⠀⠘⢷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣀⣀⠀⠠⢞⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⡟⣴⣿⡿⠿⠛⠛⠛⠛⠿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠇⠛⠁⣒⣫⣭⣽⣿⣿⣷⣦⣌⢿⣿⡇
⠀⠀⠀⠀⣼⡿⠛⣉⣭⣭⣭⣥⣤⣤⣴⣶⣾⠟⢋⣿⣿⣿⢰⣶⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠹⢠⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⡤⠖⠉⠉⠉⠛⠈⠻⣿⣿⣶⣄⣉⠛⠻⠿⠛⣋⣴⡟
⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⣋⣩⣤⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⢍⣛⠻⠿⢿⣿⣿⡿⠟⡫
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣶⣶⣶⣿⣦⡀
⠀⠀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⣦⡀
⠀⠀⠉⠀⠀⢸⣿⣿⡟⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⣿⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠸⣿⣿⠂
⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠻⣦⣤⣴⠶⠟⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⠿⢶⣦⣤⣤⡴⠶⠶⣿⠃⠀⠀⣀⡈⣿⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣉⡟⠋⠉⠛⣶⠛⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣀⣽⡄⠀⠀⢀⡿⢳⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠓⠶⣟⠉⠉⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠛⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢴⠶⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⣀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡅⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢩⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠻⠶⣦⣀⠀⠀⠀⠀⠀⠀⠛⠦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⣡⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠳⢶⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣀⣀⣤⣤⣤⣤⣤⣶⣶⣿⠿⠋⢁⣠⠴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡌⠳⣤⣀⣤⣦⡀⠀⠀⠀⣸⡿⠉⣽⠋⠁⠀⠀⠉⠙⠁⢸⣷⡚⠉⠀⠀⢸⣿⡀⠀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣴⡟⠁⠉⠻⣶⠶⠟⠋⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉⠉⣿⡏⠁⢾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣦⣤⡴⢋⡀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⢀⣼⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⢤⣅⡀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡞⠋⠳⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⠻⣦⡉⠀⠀⠀⢸⣿⣾⠆⠀⢀⣀⠀⠈⣧⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⠀⠈⠙⠳⠶⢦⣄⠋⠁⠀⠀⠸⣿⡿⠃⢹⡀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⢷⡀⠀⠀⠀⠀⠉⠛⠲⠶⠦⣤⣅⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣽⣦⣀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠤⠤⠤⠶⠋

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⢠⡟⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠈⢷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⡴⠶⠶⠶⠶⢶⣞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣀⣀⣀⣀⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠟⠉⠀⢰⠏⠀⠀⠈⢳⡄⠉⠙⠓⢶⣤⣄⣀⡀⠀⠀⠀⢠⡏
⠀⠀⠀⠈⠳⣤⡤⠶⠖⠛⠉⢁⡼⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠻⣄⠉⠙⠛⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠷⠶⢺⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢸⠟⠲⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣧⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⠢⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠙⠋⠉⠁⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢴⠦⠀⠀⠀⢠⣄⣀⢈⠩⠿⣷⢿⣿⣿⣿⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠩⠉⠒⠀⠀⡁⠈⢹⣿⣿⣶⣿⠈⠉⠹⠛⣿⣿⣿⣿⣿⠟⢙⠛⠁⠀⠀⢀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠐⠀⠀⠀⠤⠲⠶⠿⠻⣿⣿⣠⣤⣶⣷⣿⣟⡟⠿⣷⣿⣶⣠⣀⠁⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣖⡄⠀⠀⢀⡀⢠⣤⣴⣿⣿⠿⠿⠿⠿⠿⠿⢿⣓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⡶⠾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣤⣤⣀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣭⣍⣽⣭⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠋⠁⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠈⠙⠷⣄⠀⠉⠠⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⠀⠀⠀⠀⠀⠠⢴⠖⠁⠀⢀⠎⠀⠀⠐⢤
⠀⠀⠀⠀⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠠⣄⠑⣿⡄⢠⡧⠊⠀⠀⢀⡤⠒⠁⠀⠀⠀⠁⠀⠀⠀⠺⢺
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⣙⣿⡿⠈⠉⠛⢳⣧⣴⣊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⣀⡴⠞⠋⣿⠀⠀⠀⢸⣿⡁⠈⠹⣆⠀⠐⠲⢄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠴⠎⠁⠈⠀⠠⠚⠉⠙⠋⠁⡀⠀⡾⠄⠀⠀⠉⠒⠤⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠶⡄⠀⠀⠀⠀⠀⠱⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢰⣼⡗⠈⠉⠉⠹⢣⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠻⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠸⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢉⠤⠔⠒⢾⣷⡄
⠀⠀⠀⠀⢀⠔⡩⠑⠒⠒⠒⠒⠨⢅⡐⠢⢤⣀⢀⠀⠀⠀⠀⠀⠀⣀⡴⠊⣀⡠⠚⠁⠀⠀⠀⠀⡏⢿
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢚⣷⣈⡈⢻⣼⣿⣟⣛⠛⠫⠥⢀⣀⡀⠀⠀⢀⣴⣠⡞
⠀⠀⠀⠀⠸⡉⡌⠀⠀⠀⠀⠀⠀⠀⡀⠔⢚⡿⢋⠏⠀⠀⠈⠉⠂⠙⢟⠓⠒⠤⠤⠀⣉⡉⠭⠝⠊
⠀⠀⠀⠀⠀⠑⢷⣤⣤⣤⠤⠒⢊⡩⠴⠚⠁⣇⠇⠀⠀⠀⠀⠀⠀⠀⢆⢣
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠻⠷⣶⣤⣄⣠⣶⠟⠋⠉⠙⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⣹⠟⠁⠀⠀⠀⠀⠀⠹⣷⡶⢶⣶⣶⣶⣶⣤⣤⣾⠿⠛⠋⠉⠁⠀⠙⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣧⣤⣀⡀⠀⣠⣴⠶⠿⣶⣶⠟⢷⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠙⣿⠟⠁⠀⠀⠀⠹⣧⣼⠇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡞⢿⡀⠀⠀⠀⠀⢀⣿⣅⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠻⣦⣼⣷⣤⣀⣀⣤⡾⠋⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣻⣧⣀⠀⠀⢀⣠⣴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢸⣿⠉⠛⠻⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠿⠿⠿⣿⡿⠶⣶⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣀⣀
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡆⠀⠀⠀⠀⠛⠛⣻⡿⠛⠛⠛⠛⠂
⠀⣤⣶⣶⣾⣿⠿⠆⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣀⣀⣿⠇
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⠷⠶⡦
⠀⠀⠀⢀⣀⣼⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣼⡟
⠀⠀⠰⠛⠋⠁⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠳⣤⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣧⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣿⠋⠀⠀⠀⠈⢻⣇⡈
⠀⠀⠀⠀⠀⠀⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠙⠓⠒⠀⠀⠀⢘⣿⣿⠆⠠⠀⠀⠀⢿⣿⣟⠀⠀⠀⠀⠒⠛⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠠⠸⣿⡿⠫⠃⠀⠀⠀⠀⠀⠀⠯⠻⣿⡿⠢⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢨⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠓⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠀⠱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠶⠶⠶⠟⠛⠛⣉⣙⠳⢜⣿⣼⢸⢃⡟⢰⡏⢸⡇⢸⡇⣸⠇
⢶⣶⣶⡶⠿⠟⠛⠛⠛⢦⡇⡇⡟⣸⠃⣾⢁⣿⢁⣿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠇⡿⢸⡇⣸⠇⣼⠇⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⣿⢠⡟⢠⡟⢰⡟⠀⢀⡠⣐⢤⡒⢮⡙⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠠⡇⢸⡇⣾⢇⣘⡵⣿⣶⣿⢮⢧⢹⡌⢷⡈⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⡮⣣⣿⡬⣗⡮⢗⡪⠭⣓⣚⣷⢏⡏⣇⢷⠘⣧⠸⣧⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⠯⣝⣚⡭⢗⣫⠵⢚⣫⠵⣒⣫⠭⢯⣽⡜⢧⡙⣧⡙⢧⡙⢧⣀⢤⣒⣒⡺⣝⢯⢢⡐⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠭⠭⠶⣚⣩⠴⢚⣭⠶⠛⠉⠀⠀⢸⢷⢧⢮⡽⠢⢽⡲⣝⡬⢚⣩⡥⠶⢭⡞⡎⣇⢻⡘⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣩⡭⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠻⣼⣸⣖⣺⠵⣊⡵⠞⣋⣤⡶⠶⠶⡅⡇⢿⠘⣇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢇⡯⠖⣛⣡⡴⠟⢋⣡⣴⡶⠶⣆⣿⢸⡀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣛⣭⣥⣴⠾⠛⣉⣤⣤⣤⣤⣿⢸⡇⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⠶⠶⠿⠛⠋⠉⠉⣿⣿⡼⣷⢸⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡏⣇⢻⡌⢿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡯⣳⣙⠬⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣏⣭⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣁⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⢁⣠

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣄⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⠙⠻⢷⣦⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠈⢻⡇
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⢻⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⢀
⠶⠻⠟⢻⣿⠛⠒⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠐⠛⢻⣿⠛⠛⠛
⠀⠀⠀⢘⣿⣄⣀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⣼⣿⣀⡀
⠀⠀⠛⠛⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣉⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠙⠁
⠀⠀⠀⠀⣀⣽⣿⣿⡋⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠈⣿⣿⣷⣤⡀
⠀⠀⠀⠟⠋⠁⠀⠙⣿⡿⠋⠉⠀⠀⠉⠙⢷⡄⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠻⢷⣿⠋⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣿⡁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣠⣤⣀⣤⣾⠏⠀⠀⠀⠀⠹⢷⣤⣤⣤⣀⡀⢀⣴⡟

⢷⡄
⠘⣷⡀
⠸⢯⡙⢦⡀
⠀⠈⢷⡄⠈⠑⠦
⠀⠀⠘⣯⠂⢄⠀⠀⠈⠁⠒⢤⡀
⠀⠀⠀⢣⡄⠀⠑⠠⢄⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠙⠲⣄⠀⠈⠳⡄
⠀⠀⠀⠀⠈⡖⡀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⢆
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⣻
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠗⠋⢹⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⣀⣠⠔⢿⣅⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠛⠉⢀⡠⠴⠛⠉⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠁⣀⡠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣀⣤⠖⠊⢁⣠⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠛⣉⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣡⡴⠞⢦⣄⠀⢀⣠⠴⠚⣉⡤⠖⠻⡏
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡞⠋⠀⠀⢄⢢⡜⠛⣋⡤⠖⠋⠁⢀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠗⢶⣎⡥⣶⠈⡟⠀⠀⠀⠀⠈⡀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⠁⠀⢸⠃⠀⠀⠀⠀⠀⢠⠀⠘⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣠⡴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡰⠀⠀⠀⢀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢃⠀⠀⡈⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠇⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠛⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠛⠉⠉⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡀⠘⢶⡞⠲⠖⢦⠊⣠⠆⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣀⣀⣀⣀⡀⠀⠀⠉⠑⢲⡇⠀⠀⢸⠉⠁⠀⠀⠀⣀⣀⣀⣀⣄⢹⡄
⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠤⣿⠸⣟⠋⠀⠀⠉⠉⠉⠀⣈⣹⠁⣼⢀⣿⣉⡁⠁⠀⠀⠀⠀⠀⠈⠹⠃⡧
⢠⣤⣶⣒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣥⡤⠤⠶⢖⣺⠟⢡⡟⠙⠻⡻⢄⠻⡉⠻⣗⠒⠶⣤⣤⠤⠤⠖⠃
⠀⠀⠈⠉⠉⠉⣻⠆⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣤⠴⠞⠋⠀⡔⣼⠀⠀⠀⠘⣆⠳⣝⣄⠈⠳⢄⡀⠉⠓⠒⠶⠤⠤⢤⣤⣤⠤⢤⡖
⠀⠀⠀⠀⠀⣾⠁⣀⣠⣤⠤⠴⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠈⢇⠈⢮⡳⣄⠀⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⢰⠇⠀⠀⠀⠀⠀⠘⣆⠀⠱⡌⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢸⠀⠀⠀⠀⠀⠀⠀⠹⡄⢱⡙⣆⠀⠈⠙⠲⠦⣤⣄⣀⣀⣀⣀⣀⣀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠌⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢹⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠂⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣠⠞⣇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⠞⠁⠀⢸⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠛⠳⣄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡿⠀⠀⠈⠙⢦⣄⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡇⠀⠀⠀⠀⠀⠈⠓⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⣰⣆⠀⠀⠀⣠⣦⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢘⣿⣿⣿⢆⢀⣰⣿⣿⣿⠀⠀⢰⣿⣿⡳⣤⡴⢿⣿⣷⠀⠀⠀⢿⣿⣿⣄⢀⣤⣾⣟⡿⠁⠀⠀⣿⣿⣿⡦⣤⢰⣿⣿⣿⠁
⠀⠀⠘⠻⠛⣽⡟⠛⠻⣿⡛⠛⠀⠀⠈⢿⣿⣦⠛⠳⣿⣿⠟⠀⠀⠀⢿⣿⣿⡷⠿⣿⣿⣿⡷⠀⠀⠀⠛⣻⣿⠋⠀⠸⣯⣍⠉
⠀⠀⠀⠀⢴⣿⡇⠀⠀⣿⠷⠄⠀⠀⠀⣾⣿⡟⠀⠀⢿⣿⡶⠀⠀⠀⠀⢰⣿⠁⠀⢹⣿⡇⠀⠀⠀⠀⠰⠿⠏⠀⠀⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⢀⣤⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣠⣀⣶⣶⣶⣆⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢸⣿⣷⡀⣀⣠⣿⣿⠆⠀⢠⣾⣽⣻⣿⡽⣿⣿⣟⣿⠆⠀⠀⣰⣿⣿⣆⠀⣠⣾⣿⣿⡀⠀⠲⣶⣶⣦⣀⣤⣶⣾⣿⣿⠃
⠀⠀⢀⣼⣿⣿⡟⠟⣻⣿⣷⡄⠀⢨⣿⣿⣿⣿⣿⣽⣿⡿⣿⡇⠀⠀⠿⠿⠿⣛⠿⢟⠻⠿⠿⠇⠀⠀⠛⢉⣴⠋⠹⣧⡉⠉⠉
⠀⠀⠀⠉⠛⢻⣷⠀⣿⠛⠛⠁⠀⠘⣿⣾⣿⣿⣫⣿⣿⣿⡏⠀⠀⠀⠀⠀⣾⡃⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⣿⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠘⠟⠀⠀⠀⠀⠀⠈⠛⠻⠻⠿⢿⠏⠉⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣴⠖⣄
⠀⠀⠀⣽⣷⣦⣀⣀⣠⣾⣿⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⣠⣤⡄⠀⢸⡏⠈⠿⣦⣀⣠⣾⣡⣤⣿⠀⠀⣴⣶⣤⡀⠀⠀⣠⣴⣶⡄
⠀⠀⠀⣻⣿⣿⢽⣿⢽⣿⣿⠀⠀⠀⢸⣿⣿⡷⣦⡶⣿⣿⣿⡆⠀⠀⠑⢲⣿⣿⠚⠻⣿⣿⣯⠁⠀⠀⣿⣿⣿⡟⠛⢟⣿⣿⣿⡃
⠀⠀⠀⠉⠉⢀⣾⢿⣦⠉⠛⠀⠀⠀⢸⣿⡿⠋⠉⠉⠛⣿⣿⠃⠀⠀⠐⠿⠞⠹⡆⢰⠛⠟⠋⠀⠀⠀⠈⣹⣿⡇⠀⠀⣿⣿⡛
⠀⠀⠀⠀⠀⢸⣿⡞⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠠⠁
⠀⠀⢀⣴⣤⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡾⠐⢦⡀⠀⠀⠀⣠⠴⣦
⠀⠀⠸⣿⣿⡗⣀⣤⢾⣿⣿⡇⠀⠀⠀⣾⣷⡄⠀⠀⣴⣿⣿⡆⠀⠀⢸⣿⣷⡆⢀⣀⢴⣾⣿⠀⠀⠀⣧⠀⣠⣽⠶⢖⡾⣁⠀⣿
⠀⠀⠀⣿⣿⣷⡟⢻⣼⣿⣿⠀⠀⠀⢸⣿⣿⣿⣶⢾⣿⣿⣿⠃⠀⠀⠰⣿⣿⡟⠛⠓⣿⣿⡿⠀⠀⠀⠈⠉⣴⣿⠀⢸⣿⡌⠉⠁
⠀⠀⠀⠈⣻⣿⠇⠀⢹⣿⡉⠀⠀⠀⠘⣿⡿⠋⠈⠈⠻⣿⡿⠀⠀⠀⠀⠙⢻⣷⠀⣰⣟⠋⠁⠀⠀⠀⠀⠀⣿⠙⠀⠉⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠈⠿⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠿⠉⠁⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣿⠄

⠀⠀⠀⢲⣶⣾⣷⣦⣀⣀⢀⣴⣾⣷⣶⡖⠀⠀⠀⢀⣶⣷⣦⣀⣠⣴⣾⣶⡀⠀⠀⠀⠀⣠⣤⣤⣀⠀⢀⣠⣤⣤⡄⠀⠀⠀⢀⣴⡿⠿⣷⡀⢰⡿⠿⢷⣤
⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⢿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⣿⠿⢿⣿⣿⣿⠇⠀⠀⠀⠈⣿⣀⣀⣨⠻⢫⣀⣀⣸⡯
⠀⠀⠀⠚⠛⠻⠟⠋⠀⠀⠀⠙⠻⠟⠛⠃⠀⠀⠀⠀⠀⢠⣿⣇⢸⣿⢿⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠘⢻⣿⡟⠀⠘⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⠀⠀⠀⠙⠿⠆
⠀⠀⠀⠀⢠⣾⣿⣷⡀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⣿⣄⡈⢳⡛⠁⣨⣿⠇⠀⠀⠀⠀⠀⠀⣻⣿⣿⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⠻⢿⣿⣿⣦⣼⣿⣿⡿⠇⠀⠀⠀⢰⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠈⣩⣿⠟⢿⣿⡋⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣷⣄⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠟⠋⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠼⠿⡏⠀⠀⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠘⢻⠟⠀⠈⠿⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠏⠈⢻⣿⣦⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉


⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⢠⡉⠙⠳⣄⠀⢀⡴⠛⠉⣡⠀⠀⠀⠀⠀⢻⣿⣿⣳⣶⣶⣻⣿⣿⠇⠀⠀⠀⠀⣾⣿⣷⣤⣤⣤⣴⣾⣿⣷⠀⠀⠀⢸⣿⣿⣿⣆⠀⣴⣿⣿⣿⡆
⠀⠀⠀⠈⠛⠛⠛⣿⠛⢻⡛⠛⠛⠋⠀⠀⠀⠀⠀⠘⢿⣿⡗⠀⠐⣿⣿⠿⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⢿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠹⣿⣆⠀⠀⠀⠀⠐⣿⣿⣿⡿⠀⠘⣿⣿⣿⡿⠀⠀⠀⠈⠛⣿⣿⣿⠁⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⢾⣿⠃⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⢼⣿⣿⣿⢀⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢀⣴⣄⡀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⢠⣾⣷⣄⠀⠀⣰⣿⣧⡀⠀⠀⠀⠈⠲⠦⠤⢼⣿⠶⠾⣿⠤⠤⠶⠊⠀⠀⢀⣿⣿⡿⣷⠀⢰⡿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣷⣀⣿⣿⣿⣷
⠀⠀⠀⠀⢼⣿⣿⣿⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⠣⠀⠀⠀⠀⠀⢸⣿⣿⣯⣻⠿⢿⡫⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⣿⣫⣿⣿⣿⡿
⠀⠀⠀⠀⠘⠟⣿⣿⡟⢻⣿⣿⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆⣾⣿⣿⣿⠏⠀⠀⠀⣠⣾⣿⣿⡿⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⣿⣿⣿⣿⠆⠀⠀⠚⠛⢿⣿⡟⠁⠀⠘⢿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⣿⠟⠋⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⠛⢶⣄⣀⣤⣤⣤⣤⣤⣤⣤⣶⠟⠛⠛⣷⣤⡶⠞⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣴⠟⠁⢀⣀⣄⣹⣇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣴⡟⢻⡿⠿⠿⢦⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠠⢻⣿⣿⢠⡄⢀⡀⠹⣷⣶⠶⠿⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣿⠟⣀⠁⠻⠟⣸⡟⢻⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣍⣱⣿⣿⡿⣷⡾⠏⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢿⣇⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠻⣷⣶⣾⣿⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠛⣿⠉⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⣘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠔⠃⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⢨⡿⠉⠉⠓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠳⢤⣀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣞⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡶⠾⠟⠛⢁⣿⠃⠀⠈⢿⡿⠏⠉⠉⠉⠉⠉⢿⡿⠁⠀⠘⣷⡝⠻⠶⣤⣄⣀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⢿⣤⢀⠀⣿⠇⠀⠀⠀⠈⠳⣤⡄⠀⠀⣀⣤⠞⠁⠀⠀⠀⢹⣷⠁⢀⠐⣿⡛⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢛⣷⣿⣿⠏⠀⢀⡴⠶⢤⠀⠈⠉⠛⠋⠉⠀⠀⢀⣠⣤⡀⠀⢿⣤⣿⣿⡏⠁⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢘⣋⣿⠏⠀⠀⠸⣄⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⣹⠀⠘⣿⣿⠃⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡐⠁⣸⠏⠀⣠⡴⠾⠿⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣎⠁⣼⠃⢠⡞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠟⠻⢻⣿⣷⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢿⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡠⠊⣿⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⠀⢠⣶⡀⠀⠀⣀⢀⣼⣿⢀⣤⣶⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣿⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣄⡾⠿⢻⣷⣾⠿⠿⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣿⣇⡄⠀⢀⠁⠀⠀⠀⠀⢀⣿⡟⠃⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⢐⠎⢻⣿⣒⡞⢀⠀⢠⢀⣠⡾⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠻⠷⣾⣶⠾⠿⠋⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠛⠋⠁

⠀⠀⠀⠀⣼⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠉⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⠉⠛⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⣤⣤⣤⣤⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⣀⣀⣀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⣀⣀⣤⡴⠞⠋⠁⣠⡿⠉⠹⣯⡈⠙⠷⣦⣄⣀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⢀⣠⡶⠟⠛⠛⠛⠉⠉⠁⠀⠀⢀⣴⠋⠀⠀⠀⠈⢻⣄⠀⠀⠈⠉⠛⠛⠛⠛⠛⠻⣦⣀
⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠉⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⠀⠀⠀⠀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠰⢣⠀⢀⠀⢀⢀⡇⢹⣀⡠⢤⣄⡀⣃⢸⣂⣀⠤⣄⠠⡞⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡼⢋⣉⣙⣷⣷⣈⣃⢸⣛⡷⣯⡋⢙⣥⠤⠻⠥⢤⡟⠉⣫⣤⣏⣀⣸⣃⣹⡶⠟⣫⠝⢷⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠑⠲⢠⣮⣀⣱⡞⢟⡀⣏⣙⣯⣭⣶⢲⣤⣮⣭⣿⣉⣹⡛⣧⣼⣁⣹⣥⠚⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣮⢤⡿⠾⣿⣀⣿⢛⣛⣿⡀⣹⠈⣅⣠⡌⢩⡝⢻⣆⣸⡷⠟⣻⣿⢳⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠛⠓⠒⢢⢯⣀⣽⡞⠳⠤⠌⣽⠉⢹⡥⠞⠛⠛⣾⣅⡸⣧⠚⠀⠀⠙⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡏⣰⠋⢤⣙⡆⠀⣼⠉⡟⣏⡑⣆⠀⢸⢀⠎⠳⣌⡱⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠳⣴⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠈⠁⠀⠿⠞⠀⠀⠉⠳⠀⠚⠋⠀⠀⠀⠉⠀⠀⠀⠀⠈⢷⡴⠋
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠠⢤⣤⣤⣼⣇⣀⠀⠀⠀⠀⠀⢠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⠀⠀⠀⠀⠀⢀⣀⣧⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠸⡏⠉⠁⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠈⠉⢹⡏
⠀⠀⠀⣠⡤⠶⢿⡖⠒⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠐⠒⣾⠷⢦⣄
⠀⠀⠈⠁⠀⠀⠈⢷⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏⠀⠀⠀⠁
⠀⠀⠀⢀⣠⡴⠶⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠳⠶⣤⡀
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠈⠻⣿⡗⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⢞⣿⡟⠋⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠸⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⡟⠀⣠⣶⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠙⠿⣷⣄⠘⠾⠿⠟⠛⠻⢶⣦⣀⣤⣶⡿⠛⠛⠿⠿⠟⢀⣼⡿⠛⠉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⢻⣦⠤⠖⠚⠋⠉⠉⠉⠛⠉⠉⡉⠙⠙⠲⢦⢤⣿⠋⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⣣⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣼⣿⣨⢤⡀⠄⢡⣇⡽⣶⡾⣭⣧⠀⠀⣠⡭⣸⣿⡀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣠⣾⡷⠟⠛⠻⣿⣮⣿⣿⣶⣤⣴⣾⣿⣿⣾⡟⠛⠿⢷⣿⣦⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⠛⠁⠀⠀⠀⠈⠄⠁⠀⠀⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠨⠙⢻⣿⣥⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣶⣶⣄⠈⢦⣐⣀⣠⠤⠖⠒⠒⠒⢦⣀⠀⢀⡤⠔⠒⠒⠲⠦⢤⣄⣀⣰⠏⢠⣶⣶⣿⣶⣄
⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣦⣀⣉⣁⣤⣴⣾⣿⣭⡛⢦⡈⠙⠋⣠⢞⣩⣽⣿⣶⣤⣄⣉⣀⣠⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡾⠾⠛⠋⠉⠀⠀⠈⠙⢷⣽⣒⣻⣵⠟⠁⠀⠀⠈⠉⠛⠿⠿⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⣿⡃
⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠴⠛⠋⠙⣻⡄
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⠀⠀⠉⠛⡀
⠀⠀⠀⢀⣼⣿⣷⠶⠶⠤⠼⠿⢵⣤⣠⣤⣽⣆⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡄
⠀⠀⠀⠠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⡖⠒⠒⠲⢢⣴⠷⣀⣠⣤⡴⠦⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣷⣶⣾⣷⣿⣿⣏⣀⣀⣀⣀⣤⣴⣿⣿⡇
⠀⠀⠀⠀⢸⡧⡀⠀⠀⠀⠉⠉⠉⢈⣿⣿⣿⣿⠉⠼⠩⣽⣿⣿⣿⣿⣿⣭⡉⠁⠀⠲⣯⣽⡆
⠀⠀⠀⠀⢸⣿⣿⠂⠀⣀⣀⣤⡶⠋⢁⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣯⠙⠻⠿⠆⠀⠀⠈⡟⢀⡄
⠀⠀⠀⠀⠘⢹⣷⠛⠋⠝⠋⠁⠀⠀⣾⣿⡟⢹⡄⠀⠀⢸⠟⢿⣿⣿⣙⠂⠀⠀⠀⠀⠀⡇⠀⡅
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡼⠋⢿⡇⠀⡇⠀⠀⡞⠀⠘⣿⣿⡍⠓⢦⡄⣀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠸⠁⠀⡇⠀⠀⣧⠀⠀⠀⠙⢷⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⣦⡀⠀⠀⠀⢣⠀⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠯⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢻⡽⣆⡀⠀⠀⠀⠠⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠿⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⢿⠹⣷⣦⣤⢴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠉⠙⠟⢿⡇⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⠟⠛⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⢿⣿⢿⡛⢿⣆
⠀⠀⠀⠀⠀⠀⣾⠃⣸⠏⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠈⢿⡎⣷⡀⢻⡆
⠀⠀⠀⠀⠀⣼⠏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡘⣇⠈⢿⡄
⠀⠀⠀⠀⢠⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠙⠻⣦⣤⣤⣤⣤⣤⣤⡾⠛⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠀⢸⣧
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣿⠀⠀⠀⠀⠀⢸⣧⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣤⠀⠀⠀⢠⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⣿
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣿⠀⠀⠀⣼⢸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⢸⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⢛⣫⣽⣿⣿⣆⣀⣀⣿⣼⡿⣯⡛⠳⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿
⠀⠀⠀⠀⠈⣿⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣽⡾⠋⠉⠉⠉⠉⠉⠉⠻⢾⣝⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣸⡏
⠀⠀⠀⠀⠀⠹⣧⢹⣧⠀⠀⠀⠀⠀⢀⡠⠴⠚⣉⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣝⡳⠦⣄⡀⠀⠀⠀⠀⠀⣸⡟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⢻⣆⢻⣧⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣍⣁⠀⠀⠀⢠⡿⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣷⠿⢷⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣶⠿⢷⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⢄
⠀⠀⢠⠋⡱⢤⡴⢦⠀⠀⠀⠀⠀⡃⢾⡉⣷⡸⠀⠀⣀⣄⡀⠀⣀⣠⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠮⠧⢴⠃⡘⠀⠀⠀⣠⣤⣌⣒⣁⣀⣀⣤⡞⠁⢀⡙⣟⡉⠀⢈⣧⠀⠀⠀⡞⢉⣹⡤⣤⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⡇⠀⠀⠈⠉⠉⠀⢸⡇⠀⢯⡟⠉⠙⣿⡉⠙⣆⠀⠀⣇⠈⣺⣀⣸⠷⢀⡇
⠀⢀⣀⡀⢀⡴⠲⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠓⠶⠚⠳⠤⣴⠓⠃⣠⡇⠀⠀⠈⠉⠁⠈⠉⠒⠋
⠀⢸⠀⣿⠉⢹⠆⣸⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠦⡬⠋⠉⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⣹⣇⡀⠀⠀⠀⠀⠰⢒⡤⢮⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣟⠃⠀⠰⡿⠀⠀⠀⣠⢤⡀⠀⠀⠙⠁⠀⢀⣸⣯⣷⣶⣶⣤⣄⠰⣈⡷⠚⠒⠁
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣿⣏⣀⠀⢀⣀⣀⡀⠉⠋⠁⢀⡴⠶⠶⢦⣼⣿⠿⠛⠛⠛⠻⢿⣷⣄
⠀⠀⠀⢠⣾⣿⠟⠋⠉⠉⠉⠙⠛⠿⡶⠋⠉⠉⠙⣦⣤⣤⣿⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣤⣤⡿⠿⠛⠛⠷⢾⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠈⣿⣧⠀⠀⠀⠀⣾⣿⠇
⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⡿⠃⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣄⣀⣀⣀⣀⣤⣾⣿⣧⣄⣀⣀⣀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠙⠛⠻⠿⠿⠛⠋⠉⠉⠛⠿⠿⢿⣿⠿⠟⠋⠀⢀⣀
⠀⢀⣀⠀⠀⠙⢿⣷⣦⣄⣀⣀⣤⣶⣿⡿⠋⠀⠀⠀⢀⠎⢉⣲⣄⣠⠴⢤⠀⠀⠀⠀⠀⡔⢒⡖⢪⡄⢣
⢠⡏⣠⡵⢤⡴⠢⡙⠻⠿⠿⠿⢿⡛⠁⡠⢤⠀⠀⠀⠸⡀⠘⣏⠀⣸⠆⢠⠇⠀⠀⠀⠀⠡⣈⡳⠐⠓⠃
⠀⠳⠼⠦⢴⠋⣠⠁⠀⠀⠀⠀⠉⢸⡋⢱⢀⠃⠀⠀⠀⠉⠋⠉⠉⠑⠖⠋⢀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠲⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠘⠈⠈⠉⠀⠀⠘⠈⠠⠀⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⡀⢀⣀⣀⡀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢱⣿⣿⣿⣿⡇⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣻⢿⡿⢿⣿⣿⣿⣿⣶⣽⡻⠿⣿⣿⣿⣿⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⡿⣞⣽⣿⣿⢦⢿⣿⣿⣿⣿⣿⣿⣮⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⠿⠁⠈⢿⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠚⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠒⠒⢤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠒⠚⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⣄⠀⢀⠀⣀⡠⠤⠖⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⣰⠚⠉⠉⠁⠀⡇⠀⣀⣀⣤⠴⠛⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⠒⠦⢤⣀⡀⠀⠀⢸⠃
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠑⠾⠭⠖⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⠷⠒⠒⠲⠦⣄⣀⠉⠙⠒⢾⡀
⠀⠀⠀⢸⣇⣧⣀⣀⣀⣀⣀⣀⡠⠞⠓⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⠭⢭⣥⠄⠀⠉⠣⢄⣰⣆⣀⠀⠀⡏⠉⠓⠂⠋
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠒⢒⣲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⣀⣀⣀⣀⣀⠀⠀⠈⠉⠑⣷
⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡯⠴⠒⠉⠉⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠤⢄⣀
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣄⣀⡀
⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢐⡷⠤⠤⢄⣀
⠀⠀⠀⠀⠀⡧⠀⢀⣠⠤⠖⠚⠉⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠤⠤⣀⢠⠇⠀⠀⠀⠈⢱⠄
⠀⠀⠀⠀⠀⣿⣋⣁⣀⣤⡴⠒⠛⠉⠉⠛⢻⣤⠤⠤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠉⠓⠚⠋⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠉⠉⠁⢰⢃⣀⡤⠤⠤⠖⠚⠁⠀⠀⠲⠶⣒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠉⠉⠲⠤⠤⠤⠤⠤⠶⠦⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤⣀⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⢍⣥⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠁

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠘⠉⠁⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⢀⠀⠀⠴⣦⡀⠀⠢⡀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⡐⠀⠀⣀
⠀⠀⠀⢸⢦⠀⠀⠈⠁⠀⠀⠈⠀⢀⠀⠠⣀⠠⣷⡀⢠⣆⠄⠀⠀⢀⡡⠜⠁⠀⠀⠊⠂⠀⠤⡬⠇
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣿⣷⠘⠓⠺⢷⣦⣤⣒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⢀⣠⡽⠛⣻⠀⠀⠀⣸⣟⠉⠙⢶⡀⠀⠤⢄⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⠈⠑⠁⠀⠞⠉⠙⠃⠉⡁⠐⠷⠀⠈⠉⠒⠤⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢐⣛⣶⡀⠀⠀⠀⠀⠙⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡦⠀⢸⣿⣅⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠶⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠿⣷⣶⣶⠿⠛⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠛⠿⢿⣷⣶⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⢻⣧⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼⡿⠁⠀⢀⣾⡿⢿⡿⠛⠛⠛⠻⢶⣶⡶⠾⠿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣤⣾⠀⠀⠀⠀⠀⠈⣿⣄⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⢀⡟⢹⣷⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶⠟⠛⠻⢷⣶⣶⣶⠿⠿⠿⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠛⠛⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣿⣤⣤⡀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠚⠛⠛⠋⠙⢿⡏⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠈⢻⣿⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣿⠃⢀⣠⣶⣿⠿⠿⠿⢶⣤⡀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠻⠷⣦⡀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠉⠛⢿⣿⣿⣯⣥⣀⠀⠀⠀⠉⢻⣦⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⣤⡾⠿⢾⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢠⣾⡿⠋⠉⠉⠙⢿⡄⠀⠀⠀⠙⢿⡟⠁⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣀⠀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠻⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠛⠉⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⣷⣄⠀⢀⣴⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⢀⣴⠿⠃⠀⠈⠻⣷⡿⠋⠀⠀⠙⢿⣦⡀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠂⠀⠀⠀⡀⡁⡀⠨⠥⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠀⣀⡤⠴⠔⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⢳⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠐⠁⣀⡤⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢃⠠⠁⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢆⡀⠀⠀⠒⠐⡲⠀⣠⠊⢉⠆⣵⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠈⢿⠀⠀⢀⠾⢀⣔⠀⠁⢀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠂⡠⠀⢡⠶⢮⡁⠲⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡅
⠀⠀⠀⠀⠀⠀⠁⢠⡃⠐⣰⡟⠙⠲⢤⣐⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢠⡁⠈⣺⠟⠀⠀⠀⠀⠈⠑⠲⢍⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢠⠁⠐⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢂⠈⣆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠎⣹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣍⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠘⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⡂⢤⣄⠠⠀⣒⣦⣨⣆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠑⠳⢦⣐⠒⢾⣿⣗⡀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢤⡤⢩⡛⣖⣿⣿⠒⠊⠁
⠀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢣⣀⢨⡯⠟⠋
⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⡤⠶⠜⢛⠉⠉⠉⠃
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠖⠛⠛⠉⠁
⠀⠀⠀⠀⠂⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⢤⣀⡤⠖⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠘⡀⡇⠀⠀⢀⣀⣠⣤⠤⠴⠖⠛⠛⠈⠉
⠀⠀⠀⠀⣮⣹⠟⠛⠋⠉⠉
⠀⠀⠠⢊⡴⠋
⠀⠉⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⠀⢀
⠀⠀⠀⠀⠈⢐⠩⠀⠀⠀⠁⠀⠀⠅⠊⠀⠨⠀⠁⠀⠀⠁⠀⠀⠔⠨⠊⠐⠀⠀⠀⠔⠀⠊⠐⠁⠂⠒⠂⠑⠡⠂⠔⠁⠄⠪⠀⠅⠐⠠⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠿⠙⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⣿⠟⠙⠉⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣼⣿⠏⠁⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠛⣀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡀⡀
⠀⠀⠀⠀⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠛⠻⣿⣿⠇
⠀⠀⢠⣾⣿⣿⣿⣿⣦⡂⠀⠀⡀⢀⣀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠸⠋
⠀⠀⠘⡿⢿⣿⣿⣿⡿⠿⠾⠚⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠙⣆⢤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⡁
⠀⠀⠀⠀⠘⡎⠇⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐
⠀⠀⠀⠀⠀⠑⢄⣀⣂⣀⣠⣤⣶⣾⠋⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣠⣀⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢝⠿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠩⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣠⣴⣿⣿⠇⠀⠀⠀⠀⢠⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⠿⣿⣿⣷⣄⣤⢶⣿⠟⠙⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠁⠀⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠉⠉⠉⠓⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠛⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⠚⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠉⠢⠄⣀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡄⣀⠤⠒⠉⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢼⠀⠀⠀⠀⠀⠀⠀⢀⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⡾⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠈⡶⠤⠤⣄⣀
⠀⠀⠀⠀⣀⠤⠖⠋⠉⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠦⣀⣀⣀⣠⠾⢥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠓⠲⢤⣀
⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⣠⠞⠉⠉⠳⣀⠀⠀⠈⠉⠓⠲⠤⣄⡀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⣰⣃⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠹⣄⡴⠋⠁⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠉⠓⢦⣰⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣙
⠉⠁⠀⠀⠀⠀⠀⠈⠉⣳⠀⠀⠀⠀⠈⠲⠤⢤⣤⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⡤⠤⠞⠁⠀⠀⠀⠀⡞⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⡼⠋⠉⠛⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠹⣆
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣠⣤⣤⣤⣤⣄⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢻⡶⠶⠶⠶⣾⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⣇⠀⠀⠀⢀⣀⣤⡴⠞⣻⠟⠀⠀⠀⣾⠁⠀⠀⠀⠘⣧⠀⠀⠈⠙⢿⡓⠶⢤⣤⣀⣀⣀⡼⠃
⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠀⠀⣴⠋⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⢻⣄⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠞⠛⣿⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⡿⠙⠳⣬⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠓⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢸⣿⡿⢟⣥
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⠖⠋⠁⠀⠈⢹⣷⡀
⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⢀⣼⣿⣿⣶⡶⠤⠤⣽⣿⣭⣆⣀⠤⣤⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⢩⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⠤⠤⠤⢤⢤⣀⣼⡤⠀⢀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣶⣦⣄⣠⣤⣟⣥⣶⣾⠿⠋⠉⠉⠀⠀⠀⣀⣤⣾⣿⡅
⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠤⠴⠶⠶⣾⣿⣿⣿⣿⡿⠿⢻⡿⢿⣿⣿⣿⣿⣿⣿⠶⠶⠶⠖⠚⠛⣻⣿⡿⠟
⠀⠀⠀⠀⠀⣽⣗⣆⡀⠀⠀⠀⠀⠀⠀⢀⣰⡿⣿⣿⣿⠆⠀⠓⠐⠦⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠘⢷⣾⡷
⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⣀⣀⣤⣴⡾⠛⠁⢰⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡈⠛⠿⠗⠀⠀⠀⠀⣽⠀⣰
⠀⠀⠀⠀⠀⠘⢩⣿⠿⠛⠫⠟⠋⠁⠀⠀⢀⣿⣿⡿⠸⡇⠀⠀⠀⢰⡿⢿⣿⣿⣯⡛⠃⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⠛⣿⡇⠀⢳⠀⠀⠀⣾⠀⠀⢿⣿⣷⠙⠲⣤⣀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢿⠃⠀⢸⠀⠀⠀⡇⠀⠀⠘⠙⢿⣧⠀⠀⠹⣍⠉⠑⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠀⠀⡜⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⣆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⣿⢧⡀⠀⠀⠀⠸⠀⠀⠀⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⡻⠄⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠹⣾⢿⣤⠀⠀⠀⠀⠀⢸⣇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣿⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⣷⣦⣤⢤⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠋⠻⡿⢾⣿⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠺⢿⡓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠀⠀⠉⠻⣝⠲⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠶⠦⣤⣸⠃⠀⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢄⡴⠋⠀⠀⠀⠀⠀⠈⣧⠠⣤⣤⣤⠤⠤⠴⠖⢲⠿⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢰⡏⠀⠀⠀⠀⠀⠀⢲⣴⣠⡼⠧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢳⣄⡀⠀⢀⣀⣠⡴⠋⠁⠀⢀⣴⠟⠛⠳⠶⣤⣀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⣉⡽⠟⠋⠉⠷⣄⣀⣠⡾⣅⠀⠀⠀⠀⠀⠉⠳⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⠉⣻⠀⠈⠑⠄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣠⠶⠛⠉⢹⡀
⠀⠀⠀⠀⠀⢴⠶⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⠶⠶⢚⣁⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡴⠛⠉⠉⢯⠉⠉⠁
⠀⠀⣀⣠⣤⣼⢶⣿⣦⡤⠤⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢾⣦
⠈⢉⣁⡤⠤⠴⠒⠛⢿⣉⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢷⠖
⠀⠀⠀⢀⣀⡤⠶⠒⠋⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋
⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣈⣹⠶⢯⡉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡳⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠳⣄⡀⠉⠳⢤⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⣀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠖⢶⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢻⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣰⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡴⣰⠋⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠻⣆⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠋⡿⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠛⠳⢦⣄⠀⠀⠀⣀⣠⢾⣥⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⣴⠿⣤⠤⠴⠖⢒⡛⠉⠉⡀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡔⣦
⠀⠀⠀⠀⠀⠈⠀⠉⠐⠒⠢⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠐⣀⣸⣤⣤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠦⣄⠉⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢉⣁⣀
⠀⠀⣠⠴⠒⠒⠒⠲⠦⢤⣤⣀⡈⠙⠶⣄⣹⣦⣤⠤⠤⠤⠄⠀⢀⡼⠃⣀⠔⠀⠀⠀⣏⣁⣀⣀
⠀⢰⠁⠀⠀⠀⠀⠀⢀⣀⣀⡈⣩⣿⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⢈⡴⠞⠁⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠂
⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡏⠀⠀⠀⣠⣼⠀⢰⠀⠀⠀⡄⢸⠲⠶⢦⣤⣤⠴⠻⠶⠖⠒⠒⠒
⠀⠀⢰⡆⠀⠀⠀⠚⠛⠛⡟⠋⠀⠀⠒⡭⢿⣿⣤⣼⣀⣀⢀⣧⣼⣿⣯⣉⡉⠉⠉
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠚⢻⡀⠀⠀⠈⣰⠏⢈⣿⡿⠉⠉⠉⢷⣻⡌⠻⠮⣭⣉⣑⠂⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠈⣇⠀⡀⠀⠀⢀⣠⣇⠀⠀⠀⢻⠀⣼⣹⠃⠀⠀⠀⠘⣧⣧⠀⠀⠀⠈⠉⠉⠛⠒⠶⠤⣀⠀⣼
⠀⠀⠀⠀⠘⢤⠀⠀⠒⠉⠁⠈⢄⠀⠀⠸⢀⡇⣿⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠌⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢸⠁⣿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠒⠉⠓⣄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡠⠒⠠⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠋⠉⠉⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢈⡷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⡞⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠞⠛⠻⣧⠀⢀⡾⠁⠀⠀⠀⢀⣠⠾⠋⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠶⠶⣤⣤⣀⣙⣿⡇⠀⠀⠀⠘⣧⡿⠁⢀⣠⡴⠞⢋⣁⣤⠶⠛⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⣀⣀⣤⡽⠟⠛⣧⠀⠀⠀⠀⢹⣷⣿⣿⡵⠶⠛⠋⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⣸⣷⣄⣤⣴⡟⠁⠉⠉⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠉⠀⢠⡟⠉⠙⠳⣦⡀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠞⢻⡟⠀⠀⠀⣸⠃⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⢠⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⣶⠶⠶⢤⣬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⢹⡆⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⢻⡆⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⠃⠀⠻⣆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠀⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀⣠⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣾⠋⠁⠈⠉⠛⠻⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⢀⢘⣿
⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣀⣠⣤⣴⣿⣥⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠛⣛⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣷⣶⣤⢦⣤⣤⣴⣶⡾⠿⠛⠛⠛⠛⠻⢿⣿⣿⢿⣿⣶⣶⣤
⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠚⠛⢻⣿⣿⠟⠓⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⣸⡟⠉⠉⠙⣿⡇⠀⠀⠀⣠⣤⠀⠀⠀⠀⠲⠶⣿⡿⣿⡿
⠀⠀⠈⢿⣿⣄⣀⠀⠀⠀⠘⢿⡿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠹⣧⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣀⣿⣧⡿
⠀⠀⠛⠛⠻⣿⣅⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⢰⡟⠛⣷⠀⠙⢷⣄⠀⠀⠁⠀⠀⠀⠀⠀⢹⣿⡿⠿⠶
⠀⠀⠀⠀⣀⣼⣿⣿⡛⠒⠖⠒⠚⠛⠉⠁⠀⠀⠈⠙⠛⠉⠀⠀⠀⠉⠛⠲⠶⠤⠤⠤⠶⣶⣿⣏
⠀⠀⠀⠾⠛⠉⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⢩⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⡍⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣷⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣴⡿⠛⢿⣆
⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠈⠻⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣀⣀⣀⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⣠⣴⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠢⣀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣑⢄⣤⡤⠦⠖⠀⠀⠀⢀⣠⣤⣴⣼⡄
⠀⠀⠀⠀⠀⠘⣿⡷⠶⣄⡀⠀⠀⠀⠀⠀⠀⠐⠾⣷⣿⣦⡀⠀⠀⣀⣀⣀⣠⣾⣿⣿⣿⢋⣥⠂⠀⠀⣠⣴⠞⠉⠉⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢧⣬⣟⣦⣤⣀⣀⣀⣀⣀⣀⣈⢻⣿⣿⣶⠙⢿⠀⠁⠘⣿⣿⣿⣷⢯⣠⣤⣴⣾⣿⣷⣤⣦⣴⡊⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⢻⣿⣿⡇⠀⠀⢸⠀⢿⣿⣿⣿⣿⣿⣿⣛⣻⢦⣄⠈⠉⠉⠉⠳⢤
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣹⣿⣿⣇⠀⠀⣿⠀⣾⣿⣿⣿⣿⣥⡍⠈⠉⠉⠉⠉⠛⠛⠒⢒⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣶⣾⠟⠛⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠞⢋⣻⢶⣿⣿⣿⣿⡟⠀⠀⣿⣿⠿⣿⣿⣷⣤⣀⠀⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢠⣾⡿⣿⢿⣿⠁⠀⢸⠃⣿⢠⠹⣧⢻⣆⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠀⠠⠐⠊⠁⠀⠀⠀⣠⣿⡟⠸⠃⢸⡇⠀⠀⢸⠀⠿⠀⠀⣿⠘⣏⠙⠒⠤⠤⠠⠠⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡟⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢸⡆⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣼⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣤⡀⠀⢸⡄⠀⠀⢣⠀⠀⠀⠀⣠⠶⠤⣀⣘⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢰⠃⠈⠀⢄⢳⠀⠀⠈⢧⡀⠐⢠⠏⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡎⠀⠀⠀⠀⠈⠁⠀⠀⠀⠑⢄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣶⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡠⠄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⡀
⠀⠀⠀⠘⣄⠀⠈⠢⢆⠀⠀⡔⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⡐⠉⠁⡄⠀⢀⠦⠊⠀⢀⡘
⠠⠄⠤⣀⠀⠑⢄⡀⠀⠑⠤⡘⢄⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠃⠀⠀⢀⠠⠊⠀⡀⠜⡠⠔⠁⠀⣀⠔⠁⢀⡠⠄⠤
⠠⠀⠀⠀⠈⠑⠂⠌⣂⣄⠀⠈⠐⠬⠆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡰⠁⠀⢀⠔⠁⠀⠀⠮⠔⠊⠀⢀⣄⡪⠄⠒⠉⠀⠀⢀⡠
⡠⠀⠀⠀⢀⡤⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⡤⠄⢀⣀⡀⠠⠄⠂⠁⣀⠔⠊⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠠⣔⢂⠉⠤⠤
⠣⠀⣀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠒⠀⠂⠐⠀⠂⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⢀⣀⠤
⠀⠀⠀⡀⠌⠈⠁⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠩⠀⠀⡀
⠀⠀⠀⠂⠄⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠀⣀⣀⠠⠔⠊⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉⠒⠤⢀⣀⡀⠠⠄⠐⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣤⣤⣀⣀⠀⢀⣴⣾⣷⡀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠀⠀⣰⣿⣶⣄⠀⠀⣀⣀⡀⠀⢴⣾⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶
⠀⣠⣬⡿⣻⣯⣷⣿⣯⣿⡿⣿⢓⣄⠀⠈⠃⠀⠀⣠⣀⠀⠀⠀⢛⣻⣿⣿⠽⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣶⣷⣾⡆⣿⣿⣿⣿⣿⣿⣿
⠀⣌⣷⢂⣛⣛⢻⡟⠛⣿⣿⠛⣫⡥⠀⠀⠀⢠⡼⣿⣿⣷⡀⠀⠙⠿⠋⠀⠀⠙⣿⣷⡈⠀⠈⣿⡿⢿⣛⣿⣭⢽⣤⠛⠛⠛⢡⣽⢭⣽⣛⡻⠿⣿
⠀⠋⣿⢸⣴⣿⣿⣿⣿⣛⣷⣿⠿⢟⠀⠀⠀⣿⣇⡀⠻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢵⣾⠿⠛⢉⣴⣿⠏⠀⠀⠀⠈⢿⣷⣮⠙⠻⢿⣮
⠀⣯⣿⢶⣲⣶⣶⣶⣶⠶⣶⣶⣿⣽⠀⠀⠀⣉⣛⠿⢷⡿⣿⣿⣵⡿⣿⣿⣷⣦⠀⠀⠐⠾⠀⠀⠀⠀⢀⣾⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⣿⣿⢸⣿⣿⡟⣿⣿⣿⣿⣏⣿⣿⠀⠶⣿⣿⠿⠟⠛⠋⠙⣯⢿⣿⣶⣤⣘⣿⠇⠀⡀⠀⠀⠀⠐⢾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣷⡤
⠀⠻⣿⢸⢻⣿⣿⣿⣿⢛⣿⣿⣿⠾⠀⠀⠈⠁⠀⠀⠆⠀⠀⢿⣯⡻⣿⣿⠟⠁⠀⣼⣿⣆⡀⠀⠀⠀⠹⣿⣿⠋⠀⠀⠀⡀⠀⠀⠹⣿⠟⠉⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠸⡿⢉⣿⣾⣿⠂⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⣉⣀⠀⠀⠙
⠀⢠⣶⣶⣦⣄⣀⣀⣤⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣄⠀⠘⠛⠛⠀⠀⠛⠀⠀⣀⠀⠘⠛⠉⠃⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣴⣾⣿⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣶⣠⣦⣰⣿⣿⡿⣻⣵⣿⡷
⠀⠀⢿⣿⣿⢿⣿⠃⠈⠻⣿⣿⡛⠁⠀⠀⠀⣈⣛⣿⣿⣿⣿⣷⣀⣠⣄⣴⣿⣿⣿⣿⣧⠀⠀⠀⢼⣶⣿⣭⣟⠿⢧⣿⡇⣿⢚⣉⣾⣽⣿⣶⣶⣤
⠀⠀⠈⠉⢠⣿⡏⠀⠀⠀⠘⠿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⣀⣽⣷⣿⣿⣿⢎⣛⡃⣦⡺⣽⣻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠛⠁⠀⢀⣴⡄⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣻⣷⡿⠿⣗⣿⣿⣿⣿⣿⡿⠁⠀⣴⣿⣿⣿⣿⣿⣳⣿⢷⡇⠘⣿⡝⣿⣷⡝⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⡄⠀⠀⠁⠀⠀⠛⠋⢉⣾⣿⣿⣿⢧⣿⣿⣿⡿⢿⣿⣿⠇⠀⠀⠙⠻⢿⣿⣿⢷⣿⡿⣿⡇⠀⠘⣿⣮⣻⢿⣉
⠀⣿⣿⣿⣿⣾⡿⠟⠿⣿⣿⣷⣶⣄⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢸⣿⣿⣿⣿⠀⠉⠛⠀⠀⠤⠀⠀⠀⠀⢩⣬⣯⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠸⠿⢻⣿⣿⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡄⢀⣼⣿⣿⣿⡿⠁⠘⣿⣿⣿⣿⡆⠀⢀⡀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⠆
⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣾⣿⣿⣿⣿⠃⠀⠀⢿⣿⣿⣿⣧⠀⣸⣿⣷⣄⠀⠀⠀⣀⡈⠉⠻⣿⣿⠀⠀⠀⠀⠀⣿⡿⠋⠁
⠀⠀⣼⣿⣿⠁⠀⠀⢶⠀⠀⠀⠰⠀⠀⠀⠾⠿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⠀⠘⠿⠟⡣⢉⢸⣟⡿⡇⠀⠀⠀⠉⠀⠠⡄⠀⠀⠘⠁⠀⠀⢀
⠀⠸⠿⠿⠃⠀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡿⠃⠀⢠⣀⠀⠈⣿⣿⠛⠻⠇⠀⠀⣼⠇⢸⣆⠉⠛⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈
⠀⢀⡀⠀⠠⠺⣿⣿⡄⠀⢀⣾⣿⡿⡆⠀⠀⠀⠘⠃⠀⠀⠈⠁⠀⠀⢻⠏⠀⠀⠀⠀⠀⠉⠀⠀⠏⠁⠀⠀⣿⣿⣿⣷⡆⣶⣴⡄⣾⣿⣿⣿
⠀⠀⠀⣠⣿⣿⣿⣿⣷⣤⣸⣿⣿⣿⣷⣆⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣤⣶⡄⠀⠀⠀⣀⣀⢸⣿⣿⣿⠗⠿⠿⠇⢿⣿⣿⡟⣀⣀
⠀⠀⢾⣯⡷⣶⣶⣛⣿⣿⣟⣻⣶⣶⣾⡻⣿⡷⠀⢀⣴⣿⣿⣿⣶⣤⡀⣀⣀⣴⣿⣿⣿⣿⣿⡀⠀⢸⣿⢩⣧⣛⡛⣱⣿⡇⡗⣿⣦⢙⣛⣵⣏⢹
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠸⣿⣅⣀⣈⣭⣽⣿⣿⣿⢿⠟⠋⠁⢀⣿⡇⠀⠀⢻⣿⡟⠛⣱⣿⣿⢻⡇⣿⣿⣧⣻⡟⠛⣟
⠲⢿⣿⢋⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠉⢿⡿⠆⠀⢻⣿⣿⣿⣿⡿⣋⠿⢿⣾⡿⢿⣿⣿⠿⠁⠀⠀⢀⣿⣿⢲⣿⣿⡟⣼⣿⣹⣿⣿⣷⢻⣾⢿
⠀⠈⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠃⠀⠀⠈⠛⠛⣻⣵⣾⠏⠀⠀⠙⠿⣷⣶⣄⡀⠀⠀⠀⢸⡏⣻⣷⣝⣿⡃⣿⣿⡇⢿⣿⢟⣼⣿⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠆⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄⠀⠀⢸⡿⣿⣿⣿⠶⡏⣿⣿⣿⣏⢫⣾⣧⣸⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⣿⠃⠀⠀⠘⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣧⣾⡿⣿⣶⣇⣿⣿⣿⣿⣏⣹⣿⣯⡁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠐⠈⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠁
⠀⠀⠀⢣⡀⢄⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⣀⠀⠀⡹⣿⣠⠙⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⡴⠶⢿⡏
⠀⠐⠔⠘⠁⠁⢇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠉⠙⠛⠛⡁⠀⠀⠈⠁⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠐⠢⡤⠀⠒⠀⠀⠀⠀⢀⠞⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣦⣤⣴⡾⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣴⠋⢣⡞⠋⠉⢢⠀⠀⠀⠀⠀⢘⠉⠉⠓⢦⣤⡤⢴⠋⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠷⢿⢧⠼⡏⠀⠀⡌⠀⠀⠀⠀⢀⡎⠀⠀⠀⠘⠋⠀⢠⡀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠃⠀⠀⠉⠀⡀⡀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠘⠀⠈⡓⠂⠀⠀⠀⠀⠀⢈⠀⢀⠀⠰⠆⠳⠀⠛⡈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠖⠚⠛⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⣸⠀⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⢀⠔⢱⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠤⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠈⠂






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

⠀⣾⣟⡛⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⢛⣻⢷
⢸⡇⠈⠛⢦⣀⠀⠀⠈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠁⠀⠀⣀⡴⠛⠁⢸
⣾⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⢀⣠⡴⠞⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠙⢷⣤⣄⠀⠀⣀⣠⡾⠋⢀⣀⣤⠴⠞⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⣿⣬⣽⣻⣭⣼⡟⠛⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠛⢻⡇
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣛⣛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢉⣉⣉⣉⢻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠁⠙⠋⣽⢿⠉⠛⠉⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⢷⣄⣠⣤⣤⠤⠶⠞⠛⠉⠁⣼⠃⠀⠀⠀⢠⡏⠸⡇⠀⠀⠀⠀⢳⡈⠉⠛⠳⠶⠤⣤⣤⣄⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠁⠀⢿⡀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢰⡏⠀⠀⠘⣇⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⡀⠀⠀⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠶⠞⠛⢷⡀⢰⡇⠀⠀⠀⠀⠘⣇⢀⡾⠋⠛⠲⢮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⢹⠟

⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣿⠿⠷⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⣾⡿⣿⡟⢿⣄
⠀⠀⠀⠀⠀⠀⣰⡟⢁⡾⠁⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠁⠀⠘⣿⡜⣷⠈⢻⣆
⠀⠀⠀⠀⠀⢠⡿⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣇⠈⢿⡄
⠀⠀⠀⠀⠀⣾⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⣀⣀⣀⣀⣀⣀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⠀⠘⣷
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣦⣤⣀⣿⠋⠉⠉⠉⠉⠙⣿⣀⣤⣴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⡄
⠀⠀⠀⠀⢸⡿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⢀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⢿⡇
⠀⠀⠀⠀⢸⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⢸⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣻⣿⣿⡇⠀⠀⣸⡇⣿⣿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣡⣴⢞⣻⡿⠿⠿⠶⠶⠿⠿⢿⣽⡳⣦⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⢹⣇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣩⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣝⡳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⡏
⠀⠀⠀⠀⠀⠀⢿⡆⢿⣆⠀⠀⠀⠀⠀⠐⠋⣉⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣬⣙⠲⠦⠄⠀⠀⠀⠀⣼⡏⢰⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣎⢻⣧⣀⣀⣠⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⣀⣀⣼⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠁⠠⠠⠀⠀⠀⠀⠀⠀⣈⡀⠠⠀⡄⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⢀⣈⡀⠠⣀
⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⡊⠑⠄⣠⣼⢣⠂⠀⠀⠀⡠⠀⠀⠀⠨⣀⠀⠀⠀⠀⠀⣤⢀⡄⢤⠀⣰⢟⡒⠁⠀⣄⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠤⠄⠐⠀⣤⣤⠤⠰⣏⣿⣗⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⠤⡶⠀⣈⢀⡟⠈⠏⠳⣤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⠜⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⡇⠆⠀⢉⠰⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢀⠂⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⢀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠄⠀⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡇⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⢢⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈

⠀⠀⠀⢠⠤⣀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠢⠄⠒⡆
⠀⠀⠀⠀⢀⡀⣀⣁⠉⠙⠲⢄⠀⢀⡤⠒⠃⣀⣤⣴⣶⠖⠀⠁⠀⠀⠀⠀⠄⠄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠒⠄
⠀⠀⠀⠀⠀⠀⠈⠙⠉⢛⣓⠂⠼⠿⠄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣠⣀⠈⠂⠄⠀⠀⠀⠀⣠⠶⠚⠉⣀⣀⡀⠈
⠀⠀⠀⠀⡀⢤⣴⠖⠋⢉⡴⢠⡆⢠⢰⡌⢉⡛⠦⠄⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⢙⠿⡿⢶⠄⠀⢀⡀⠊⢀⣠⣖⢻⣉⠉⠁
⠀⠀⠀⠀⠀⠂⠐⠁⣠⠋⠠⡾⠀⠀⢆⠻⡄⠉⠢⡄⠂⠀⠀⠀⠀⠀⠀⢡⠀⠀⣁⣤⠤⠶⣒⠂⠈⠙⢺⣛⠶⠂⠤⣤⠤⠀⠐
⠀⠀⠄⠁⠀⢸⡀⠀⠁⠀⢘⠁⠀⠀⠈⠣⠍⠀⠀⢘⠄⠀⠠⡀⠀⠀⡀⠴⠟⠛⡩⡴⠻⠛⠉⠀⠀⠀⠀⢉⠉⠛⠳⠈⠙⠲⠄
⠀⠀⠀⠀⠀⠈⠳⠦⠄⠀⠂⠀⠀⠀⠠⠀⢀⠤⠤⠎⠀⠀⠀⠈⢂⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⡳⢦⣤⣄⡀
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⡐
⠀⠀⠀⠈⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢃⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠃⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⣀⣠⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⠀⠀⢀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠉⠙⠻⠛⠛⠋⠉⢩⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢈⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠴⠶⢿⡷⠒⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⢶⣿⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣤⠄⠀⠀⠈⠛⠁⠀⠀⠀⠀⣴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢷⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣽⣟⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⡾⠟⣻⣿⣿⣿⡿⠷⠶⠶⠿⠿⢿⣿⣿⣿⣅⠀⠈⢷⣄⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣁⠀⣰⣿⣿⣿⣿⣿⣦⣄⣀⣤⣴⣿⣿⣿⣿⣿⣆⠀⠀⢻⣦⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⠶⠿⠛⠉⠛⠻⠷⠶⠾⠿⠛⠋

⠀⣠⠖⠋⠉⠉⠓⠦⡀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠲⣄
⡼⠁⢀⣀⠀⢀⣀⡀⠙⡄⠀⠀⠀⡎⠀⢀⡀⠀⢀⡀⠀⢱⠀⠀⠀⢀⠏⠀⣀⡀⠀⢀⡀⠀⢳⠀⠀⠀⢠⠃⠀⣤⡀⠀⣠⡄⠈
⡇⠀⠁⠈⠁⠈⠀⠁⠀⡇⠀⠀⠸⡀⠀⠉⠁⠀⠈⠁⠀⢀⡇⠀⠀⢸⠀⠀⠉⢁⣀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⣉⣀⣀⣈⡀
⠹⡄⠈⠑⠒⠒⠋⠀⡰⠃⠀⠀⠀⢣⡀⠈⠓⠒⠚⠁⢀⡜⠀⠀⠀⠈⢧⡀⠘⠉⠉⠉⠂⢀⠞⠀⠀⠀⠘⢆⠀⠳⣄⣀⠴⠁⢠
⠀⠈⠓⠤⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠚⠁

⠀⠀⣠⠤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠤⣄
⢠⠋⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⡴⠁⠠⡀⠀⢠⠄⠙⣆⠀⠀⠀⢀⡜⢁⠀⠀⠀⠀⠀⠑
⡏⠀⠰⠗⠀⠐⠿⠀⠀⡇⠀⠀⢰⠃⠀⠦⠖⠀⠲⠴⠂⠈⡆⠀⠀⢸⠁⣀⣉⣁⣀⣉⣁⡀⢸⡀⠀⠀⢸⠀⢈⠽⠂⠀⠺⠅
⢧⠀⠐⠒⠒⠒⠒⠀⢀⠇⠀⠀⠈⣆⠀⢠⠖⠒⠢⡄⠀⣰⠁⠀⠀⠘⡄⠳⡀⠀⠀⠀⣠⠇⡸⠀⠀⠀⠸⡀⠀⡠⠖⠒⠦⡀
⠈⠳⢄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠈⠢⣄⡀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠘⠦⣈⠓⠒⢋⣡⠜⠁⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡠⠞
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢀⡴⠊⣡⠀⢠⣉⠳⣄⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠉⠢⡀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠈⠙⢢⡀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠑⢦
⡞⠀⢠⣦⠀⠠⣦⠀⠈⡆⠀⠀⢀⠏⠀⢢⣄⠀⣠⡔⠀⠹⡀⠀⠀⢠⠃⠐⣦⡖⠀⢢⡔⠀⢱⠀⠀⠀⢰⠁⠀⢦⡄⠀⢠⡄
⣇⠀⠀⣀⣠⣀⡃⠀⠀⡇⠀⠀⠸⡄⠀⢉⠀⠀⠀⡉⠀⢀⠇⠀⠀⢸⡀⠈⠁⠉⠀⠁⡈⠀⢸⠁⠀⠀⢸⠀⠀⣉⠀⠀⢀⡁
⠘⢄⠈⠁⠀⠀⠉⢀⡜⠁⠀⠀⠀⠳⣄⠀⠉⠉⠉⠀⣠⠞⠀⠀⠀⠀⠳⡀⠉⠉⠉⠉⠁⣠⠏⠀⠀⠀⠈⢣⡀⠈⠉⠋⠉⠀⡠
⠀⠀⠙⠒⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋

⠀⢀⡤⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⢤⡀
⣰⠃⠀⠖⠀⠐⠦⠀⠱⡄⠀⠀⠀⡞⠁⠐⢒⣀⡒⠂⠈⢳⠀⠀⠀⠀⡞⠁⠰⠤⠀⠴⠂⠈⢦⠀⠀⠀⢠⠎⠀⢠⡄⠀⣤⠀⠘
⡇⠀⠳⠔⠀⠘⠴⠃⠀⡇⠀⠀⢸⠀⠀⣰⠉⠀⠉⣆⠀⠀⡇⠀⠀⢸⠀⠀⠺⠆⠀⠸⠇⠀⢸⠀⠀⠀⢸⠀⠀⠛⠀⠀⠛⠃
⢳⡀⠈⠲⠤⠤⠊⠀⣠⠃⠀⠀⠀⢧⠀⠹⣞⣉⡳⠏⠀⡼⠀⠀⠀⠈⢆⠀⠰⠊⠉⠓⠄⠀⡼⠀⠀⠀⠘⣄⠀⠉⠉⠉⠉⠁⢀
⠀⠙⠦⣄⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠑⠦⣄⣀⣤⠴⠊⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠤⠊⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⢶⣦⣄⠀⠀⠀⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣶⣦⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠈⠻⣷⣶⡿⠿⠛⠋⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠉⠙⠻⢿⡿⠿⠟⠛⠛⠛⢻⡿⠀⠀⠀⠀⣀⣀⡀⢹⣿⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏⢩⣿⠿⠛⠻⢷⣦⣤⣴⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠷⣿⠃⠀⠀⠀⠀⢻⣏⠉⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⡾⣿⣆⠀⠀⠀⢀⣾⠟⣷⠀⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠻⢷⣶⣶⡿⠿⠶⠟⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣀⣠⡾⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢰⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣤⣤⣤⣤
⠀⢠⣴⣶⡶⢿⣿⡿⠷⠶⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠉⠉⢹⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣠⣤⣼⣿⣄⣀⣀
⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠙⣿⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⠻⠷⡆
⠀⠀⠀⠀⠀⠺⠛⠉⠀⠀⠈⠙⠿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⠋⠉

⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠
⠀⢀⣮⡻⣿⣮⡻⣷⣄⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣔⢾⣷⣮⡻⣿⣮⡁
⠀⢈⠻⣿⣮⡻⣿⣬⠿⣷⣝⣿⣮⡳⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣘⠿⣷⣮⣙⢿⣷⣬⠻⣿⣮⡻⣧
⠀⢸⣷⣌⠻⣿⣎⠻⣷⣌⠻⣷⣝⠿⣾⣿⣷⣄⣠⣤⡄⣠⣶⣾⣿⡻⢿⣯⣙⢿⣷⣍⠻⣷⣌⠻⣿⣦
⠀⢲⣝⢿⣷⣌⠻⣷⣮⡻⣿⣮⡻⢿⣮⣛⣷⣿⡟⣫⣾⣿⣿⣮⣝⢿⣶⣝⢿⣷⣝⢿⣷⣌⢻⣷⣌⠻
⠀⠘⢿⣷⣝⢿⣷⣬⡻⣿⣮⣛⢿⣿⣭⣛⡿⠛⠈⠉⠉⠉⠻⣿⣟⢿⣿⡻⣷⣿⢿⣷⣝⢿⣷⣝⢿⡗
⠀⠈⣷⣝⢿⣷⣍⠻⣿⣮⡛⢿⣷⣮⠝⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣝⣿⣮⡻⣷⣝⢿⣷⣝⢿⣷⠄
⠀⠀⠙⢿⣷⣝⢿⣷⣬⡻⠿⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣷⣌⢿⣷⣝⢿⣷⡝
⠀⠀⠀⢳⣝⢿⠷⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠙⢿⣷⡝⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣤⣶⠶⠿⠛⠛⠛⠛⠿⠶⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠿⠛⠛⠛⠛⠿⠶⣶⣤⡀
⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⣿⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⣶⣤⣤⣄⣀
⣿⣠⣴⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠿⠿⣿⡟⠀⠀⠀⠀⠀⣀⣴⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣦⣄
⣿⡋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢙
⢹⣧⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣦⡀⢸⡇⠀⠀⠀⠀⢸⡇⣀⣴⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣤⣶⠶⠿⠛⠉⠀⠈⠻⣿⡇⠀⠀⠀⠀⢸⣿⠟⠁⠀⠉⠛⠿⠶⣶⣤⣤⣤⣤⣤⣶⠶⠟⠋⠁⠀⠀⠀⠀⣰⡟
⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠋⠉⠉⣻⣿⡉⠉⠙⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⡀⠀⠀⠀⣀⣀⣠⣤⣴⠾⠟⠋⣱⡿⠁⠀⢀⣼⠏⠘⢿⡄⠀⠈⢿⣎⠙⠻⠷⣦⣤⣄⣀⣀⠀⠀⠀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⢀⣴⠟⠀⠀⢠⣾⠃⠀⠀⠈⢿⡄⠀⠀⠻⣦⡀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⢿⡄⠀⠀⠘⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣟⠋⠉⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣼⠏

⠀⢀⡠⠔⠒⠋⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠙⠓⢦⣀
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⡏⠀⠠⣦⠀⠀⠀⠀⠷⠀⠀⢸⠀⢰⠃⠀⠀⠀⣄⠀⢠⡄⠀⠀⠀⠈⡆⠀⢠⠃⢀⣤⡄⠀⠀⠀⠶⠖⠀⠘⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢷
⡇⠀⡄⠀⠀⠀⠀⠀⠀⡲⠀⢸⠀⢸⠀⠀⡀⠀⠉⠀⠈⠀⠀⢴⡁⠀⡇⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠀⣿⠀⢸⠀⠘⡇⢠⣦⠀⠺⠗⢈⡇⠀⢸
⠹⡀⠈⠢⣄⣀⣀⣠⠔⠁⢀⡞⠀⠘⣆⠈⠣⠴⢄⡴⢄⠴⠒⠚⠁⢠⠃⠀⠘⡄⠈⢿⣭⣧⢼⡤⢿⡾⠋⢀⡏⠀⠈⢧⠀⠙⢦⣀⣀⣀⡤⠞⠀⢠⠟
⠀⠙⠦⣀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠘⠦⣀⠉⠛⠚⠛⠉⢀⣠⠞⠀⠀⠀⠀⠳⢤⣀⠈⠉⠁⣀⣠⠔⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣀⡠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀
⠀⣠⠎⠁⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠈⠳⣄
⣰⠁⠀⠀⢀⡀⠀⣀⠀⠀⠀⠹⡄⠀⢰⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⢀⠎⠀⠀⢀⡀⠀⠄⠀⠀⠀⠹⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡇⡀⠀⠀⠈⠁⠀⠁⠀⠀⢸⡄⡇⠀⡏⠀⠀⠟⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢳⠀⢸⠁⠀⣠⣄⠀⠀⠀⢀⣤⠀⠀⣿
⣇⢹⡀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⣇⠀⢤⠤⠤⢒⣒⣒⣊⡟⠁⢰⠇⠀⢸⠀⠀⣠⡔⢞⠉⠉⠉⠙⠦⠀⡼⠀⠸⡄⠀⠈⠁⡠⠤⣄⠈⠉⠀⢀⡏
⠈⢦⡙⠦⣄⣀⣀⣀⣠⠔⣡⠎⠀⠀⠘⣆⠈⠻⢭⣉⣀⠤⠊⠀⣠⠏⠀⠀⠀⠣⡈⠀⠈⠉⠀⠀⠀⠀⢀⡴⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠉⠒⠦⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⠈⠑⠦⢤⣤⣤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠈⠒⠤⢤⣀⡠⠤⠖⠋⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠴⠒⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡴⠚⠉⠉⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⢀⠔⠋⠉⠀⠀⠉⠳⢤⠀⠀⠀⠀⢀⡴⠊⠉⠁⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠀⠉⠙⠲⣄⡀
⢠⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⡄⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢠⠎⠀⠀⠀⠀⠀⢀⠀⠀⠈⣧⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⡏⠀⠀⡹⠯⠀⠀⠿⠅⠀⠀⠀⢷⠀⢰⠇⠀⣠⠤⡄⠛⢀⠤⢄⠀⠈⡇⠀⢸⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢸⡄⠀⢸⠃⠀⠀⠀⠀⠄⠰
⣇⠀⠀⢀⣀⣀⣀⣄⣤⡤⠄⠀⡿⠀⢸⡀⠀⠁⠀⠀⣠⡀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡄⠀⠤⣤⡤⡴⠶⠒⠒⠋⠁
⠘⢆⠀⠀⠀⠀⠀⠀⠓⠁⢀⡴⠁⠀⠀⢧⠀⠀⠀⠀⠻⠃⠀⠀⢀⡞⠀⠀⠀⢣⡀⠀⠀⡴⠒⠒⠄⠀⢠⠞⠀⠀⠀⢳⡀⠀⠹⣠⠇⠀⠀⠀⠀⢀⡴
⠀⠈⠑⠦⢄⣀⣀⣀⡤⠴⠋⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢒⠎⠀⠈⠀⠀⡀⠀⣀⠀⠐⠉⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠠⠄⠀⡠⠀⠀⠀⠀⠀⠐⠄⡀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠐⠢⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⢠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⣤⠶⠖⠚⠀⠀⠀⣀⣤⠴⠶⢶⡇
⠀⠀⠀⠀⠀⠘⢿⣙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⡀⣀⣤⠤⠤⢴⣿⣿⣿⣟⡡⠖⠀⠀⣠⣶⠛⠁⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣽⣷⣶⣤⣤⣤⣦⣤⣤⣽⣿⣿⣿⡆⠹⠀⠀⠘⣿⣿⣿⢟⣷⣶⣶⣿⡿⠿⠿⠾⢾⣭⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⣭⣿⣿⡇⠀⠀⡗⠀⣿⣿⣿⣿⡿⠿⠿⠭⢿⡷⢤⣀⣀⣀⣉⣱⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⣿⣿⣷⠀⣠⣷⣠⣿⣿⣿⣿⣟⡒⠂⠀⠀⠀⠀⠀⠈⠉⢁⠃
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢾⣻⣿⣿⣿⣿⠉⠈⣿⣿⣿⣿⣿⡛⠳⠶⠶⠂⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠚⠋⣉⠤⠞⢫⣿⢿⣿⣿⠇⠀⢰⡿⣿⠉⢿⢿⣟⠳⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡀⠤⠒⠋⠀⠀⣰⣿⢋⡟⢹⡟⠀⠀⢸⠀⡧⠀⠸⡇⢿⠷⣄⡀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠒⠀⠀⠁⠀⠀⠀⠀⠀⡼⣿⠃⠈⠀⢸⡇⠀⠀⢸⠀⠁⠀⠀⣷⠘⡆⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡇⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢸⡀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠸⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠁⠀⠀⠀⠀⠸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠈⡆⠀⠀⠀⠀⢠⣄⣀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢠⡏⠑⠀⠀⢇⠀⠀⠘⣄⠀⠀⢠⠃⠀⠀⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡸⠀⠀⠀⠀⠘⠂⠀⠀⠈⠢⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣶⣶⣶⢶⣿⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⣶⣶⣶⣶⣶⣶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⢤⣀⣠⠚⠉⢳⡀⠀⠀⠀⠀⢀⣠⠖⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⠼⣇⡤⡀⠀⢷⠒⠒⠒⠒⠛⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣹⡇⠀⠀⣷⡁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠀⠓⠛⡶⠚⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠑⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣼⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⡀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠒⢺⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠓⠒⢌⣳⡄⠀⠀⠀⠀⠀⠸⠉⠕⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡀
⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠘⣇⠀⠀⠀⣹⣁⣀⣀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⡞⠀⠈
⠀⠀⢰⡏⠹⠦⠤⠼⠋⢹⠀⠀⠀⠈⠳⣄⣀⣼⠈⠉⢙⡏⠉⠑⢮⡷⣶⠖⢶⡖⠚⠛⠉⠉⠀⠈
⠀⢠⠞⠁⣠⠤⠤⢤⡀⠈⢷⡀⠀⠀⠀⠀⢹⠉⠑⢤⣼⣀⣀⡴⢋⣰⠋⠙⡎⢧
⢠⠏⢀⡎⠸⠂⣚⠃⣼⢧⠈⢳⡀⠀⠀⠀⠘⢶⠖⠋⠙⣾⢟⠞⢏⡿⠤⠤⢾⠈⢧
⣿⠀⠘⣏⡃⠘⠤⠃⣤⣼⠀⠈⣷⠀⢰⡒⢦⣾⣀⡠⠞⣿⠈⠓⠺⢥⡀⠀⡼⠀⢸⣇⣀
⢷⡀⠈⠣⠷⠤⠤⠤⠔⠁⠀⣀⣿⣄⣀⣽⠟⢻⣅⠀⠀⣿⠀⠀⠀⠀⠉⠋⠁⣀⡾⠉⠉⢹⡆

⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠂⠀⠁⠀⠄⠈⠲⣆
⠀⠀⢰⣿⡁⠀⠀⠈⠉⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⣈⠀⠀⠀⠀⡀⠀⠀⠀⣼
⠀⠀⠘⣿⣿⣍⠤⠄⠀⠀⠀⠀⠉⠁⠂⠄⡀⠀⣀⣀⣀⣀⣀⣴⣾⣶⣾⣿⣋⠀⠀⠀⠂⠀⠀⠀⢀⣾⣿
⠀⠀⢠⣇⣀⡤⣤⣤⣤⣄⡀⠀⠀⠀⣲⣶⣾⣿⣏⢠⠄⠀⠉⣿⣿⣿⣿⣿⣿⣿⣻⣿⠉⠉⠉⠘⣛⣿⣿⡄
⠀⠀⠘⢿⣧⠀⠀⠀⠀⣀⡵⠿⠛⠛⠋⠉⠩⢍⣽⡎⠐⠀⠀⣿⣿⣿⣿⠿⠿⠃⠀⠉⢡⡒⠦⣼⣿⡿⠿⠃
⠀⠀⠀⠀⠈⠁⡀⠖⢿⠁⠀⠀⠀⠀⠀⠀⠀⣉⣿⣷⣾⣶⣿⣿⣿⣿⣿⠷⠶⢀⠀⠀⢈⠁⠀⣠⣍⠐⣄
⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⡿⣿⡿⢿⣿⣿⣿⣦⡀⠉⠙⠢⢀⠀⠈⠒⢾⣄⡀⠉⢻⡀⢸⡇
⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⠤⠊⠁⢀⡰⠋⠈⣈⠀⠸⠠⡈⢿⣿⣷⣄⠀⠀⠀⠈⠂⢄⠰⣿⣿⣦⡀⢻⡟⠁
⠀⠀⠀⠳⣶⡆⢀⡴⠊⠀⠀⢀⡼⠊⠀⠸⠚⠁⠀⠀⠀⢱⠀⠙⠏⠻⢷⣄⠀⠀⠀⠀⠈⠂⢍⢿⣿⣿⡇
⠀⠀⠀⠀⢿⣡⣟⣃⣀⢀⠔⠋⠀⠀⠀⠃⠀⢠⡆⣶⠀⠀⠳⡀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠈⠙⠿⠃
⠀⠀⠀⠀⠘⠿⠿⠟⠋⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠈⠀⠀⠀⠙⡶⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣠⣧⣿⠀⠀⠀⠀⠘⠀⢰⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⢤⣴⣿⣿⣿⠀⠀⠀⠀⠀⢳⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢇⡟⣽⢿⣏⡿⠀⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢯⣵⣁⣉⡵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠄⢀⡠⠤⠄⢀⡀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⣀⡠⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠉⠈⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⢀⣎⢒⣀⣶⠶⠒⠒⠒⣒⠶⠶⠒⠒⠒⠂⠐⠒⠒⠿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⠛⠓⠒⠒⠒⠒⠒⠶⠶⠦⣖⡒⠒⠒⠶⢦⣤⣀⠱
⠀⠀⠼⠃⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠘⡧
⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⣸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠺⢫⠟⠋⠁⠀⠁⠀⠙⠋⠙⢷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠈⠻⠲⣦⣀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⠤⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠓⣄⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠔⠀⡴⠃⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠘⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣾⠛⠛⠛⠒⠲⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⣷
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⡀⠈⢻⡶⠶⠶⢶⣶⡒⠒⠶⢶⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢺⡇⠀⠀⠀⠀⠉⠓⠄⢸⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠰⣄⡀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⣽⠷⣦⣬⣿⠶⣶⣤⣤⡾⣯⡉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠘⣷⣤⣤⡤⠴⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣾⠃