Sheep Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠀⠀⠀⠀⣼⣟⡁⠀⠀⠀⠀⠀⠀⢀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⣽⡷⢧⡀⠉⠉⠀⠉⠓⠀⠈⠁⢤⣀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⡈⢸⢿⣧⡀⣤⠀⣾⠛⠀⠒⠒⠀⠀⠀⠀⢠⠀⠀⠀⡐⠈⠣⡑⠂⠁⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⢞⡞⣟⣷⣤⡾⠧⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠨⠀⠀⠀⠀⠹⣴⠁⡄⡀⠁
⠀⠀⠀⠀⠀⠀⠁⢄⡏⡀⢁⣿⠫⠀⠀⢀⢀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠉⣷⠈⣶⣶⡆
⠀⠀⠀⠀⠀⠐⠀⠐⠄⠓⠈⡟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⡄⠀⠀⠀⢸⣾⡏⣟⠄
⠀⠀⠀⠀⠀⠀⠀⡃⢠⠊⠀⢰⡀⠀⣂⢀⠀⣗⠀⠀⠄⣼⡀⠀⠀⢀⠀⡉⣸⡂⠘⠀⢙⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢠⢋⣴⡯⣄⣦⢹⡄⠙⡞⢆⠀⢀⣘⣳⢦⢣⣼⣾⣵⣴⢻⠡⡶⡿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡅⣮⡷⣷⣬⡏⣗⣤⡱⠳⢼⣾⡿⣿⣿⣷⣿⣿⣿⣷⢿⣯⣮⣿⢁⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⣾⣿⣿⣿⣿⣿⣷⡯⣯⣿⣿⣿⣿⣞⣿⣿⣿⢿⣿⣿⣟⣿⣦⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⣿⣿⠿⠟⠛⠉⠚⠛⠋⠉⠉⣿⡏⠁⠈⢟⣭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢣⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠼⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠙⠛⠛⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠰⠟⢁⡀⠀⠀⠀⣀⡴⠚⠋⠉⠉⠙⣲⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠳⣤⠖⠋⢉⣩⠽⠶⠾⠯⣄⡀⠀⠀⠀⠻⠿⠟⠛⣆⠀⠈⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠲⠾⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⣠⠴⠚⠛⠉⠙⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⣠⠴⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⣿⠀⠀⣤⣶⣤⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⢀⡿⣆⠈⢿⣿⣿⠃⠀⠀⠀⠀⢀⡾⢦⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⠳⣤⣀⠀⠀⠀⢀⣠⣴⠋⠀⠀⠙⢦⡀
⠀⣠⠴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣾⣿⠶⣄⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠹⣿⡄⠀⠀⠀⠈⢳⡄
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡇⡏⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣷⠀⠀⠀⠀⠀⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⢿
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⣧⠀⣀⣤⠌⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡆⠀⠀⠀⠀⠀⢸
⠀⣨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣼⠀⢹⡏⡽⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣧⠀⠀⠀⠀⠀⣸
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣀⣠⡼⠃⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡟⠀⠀⠙⢦⣄⣀⣠⡴⠋
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⠒⠦⠤⢤⡴⠖⠋⠁⠀⠀⣠⠟⠀⠀⢀⣠⠴⠋
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⢻⡀
⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠿⠖⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠞
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠒⠒⠛⠁
⠀⠀⠀⠘⠳⣤⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠳⢤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⣀⠀⠀⠀⢀⣠⠟⢧⡀⠀⠀⠀⠀⠀⣴⢷⡀⠀⠀⠀⠀⢀⡼⢷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠛⣻⠛⠉⡟⠀⠈⠙⠶⠤⠤⠶⠛⠁⠀⠉⠓⠶⢶⡞⠋⢷⠈⣧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠀⣿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸⡆⠘⣆⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠁⠀⣿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣇⠀⠘⣆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⡎⠀⠀⠀⠀⢀⡀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠚⠋⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠔⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣠⠖⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⢆⠀⠀⡐⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣾⢿⣿⣤⢠⣁⣷⠀⠀⠀⠀⠆⠀⡞⣄⣠⠀⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠂⢠⠚⠩⠉⠋⠁⠀⠀⠀⠉⠀⠀⠀⠌⠀⣄⠿⣯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡄⠰⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⢤⠖⠛⠉⠙⠲⣤⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⣠⡴⠞⠋⢿⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⣻⠙⠳⢦⣄
⠀⢀⡴⠋⠁⠀⠀⠀⢘⣷⠤⠤⠞⠓⠚⠉⠁⠀⠈⠉⠉⠁⠀⠉⠓⠖⠳⣤⣀⣀⣴⡏⠀⠀⠀⠈⠛⢦⡀
⣠⠏⠀⣀⣀⣤⠴⢚⣽⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⣯⡓⠦⣤⣀⣀⠀⠹
⢯⡀⠛⢉⣀⣤⠶⠋⢈⡇⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⠃⠙⠲⢤⣀⡉⠓⢀
⠀⠉⠉⠉⠁⢀⣠⣤⠞⢷⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⡾⢧⣤⣄⠀⠈⠉⠉⠉
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠳⢦⣤⠞⠉⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⣷
⠀⠀⠀⣠⠼⠃⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⢸⡇⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⣀⠈⠉⠉⠁⢀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⢷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⡼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⠛⠛⡷⠦⠤⠖⠳⠦⠤⣾⠛⠛⡏⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣿⣀⣠⡷⠀⠀⠀⠀⠀⠀⣯⣀⣠⣧⣤⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⡒⢋⠙⠀⠉⠀⢈⠀⠁⠐⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠐⠁⣀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠀⠀⠀⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠉⠀⠀⠉⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡁⠐⣁⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠁⠀⠀⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢼⣰⣰⣄⣀⣆⣿⣿⣿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠐⠂⠀⠂⠀⠀⠀⢀⡐⠀⢀⣄⢀⠀⡀⠀⠲⢶⡟⠻⢟⣛⠉⠛⠻⠿⣿⡙⢿⡉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠐⠀⠀⠘⠀⠀⠘⣆⣠⣴⣶⣮⡿⠛⢲⡾⢶⢋⣩⣤⣤⡺⠿⠋⠀⠀⠀⠀⠈⠁⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠉⠔⠠⠃⢠⠄⠘⠄⠀⣰⣶⡋⢡⣴⣿⣿⣧⣶⡏⢠⠈⠈⠻⠿⠿⠛⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⢀⡀⠀⠰⢀⠰⣤⡀⠉⠀⠀⠀⠈⠉⠻⣿⣤⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠁⠀⠀⠀⠀⢀⠈⠀⠀⠀⠰⠦⠶⣦⠀⢶⡾⣿⡙⢠⠈⣸⣆⡙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⡀
⠀⠀⠀⠀⠀⠀⡄⠃⠐⠀⠀⠀⠀⡀⠀⠁⡄⢠⠀⢠⠁⠈⠀⠀⣀⠙⡙⢦⠑⠌⠳⢬⠁⢃⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠈⠀⠀⠀⠈⠀⠄⠨⡂⠉⠀⠐⠆⣨⡘⢦⠈⠁⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⡀
⠀⠀⠀⠀⣠⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠐⠐⢆⠇⠀⡃⢛⠀⣄⣻⠴⠤⠈⠙⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⢤⣄⡀⢀⡇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠉⠈⠀⢠⠀⡀⠈⠀⡄⢀⠈⠉⢇⣉⠳⣔⡿⡆⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢒⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢠⠀⠈⠀⠀⠊⠀⠀⠀⠄⠀⠔⠑⠀⣅⢀⢶⣍⠙⢿⣿⡿⣿⣿⠟⠛⠒⠒⠂⠀⠀⣀⣸⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠄⢠⣈⣛⠀⣠⠟⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠂⢉⣽⡷⠻⢢⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠐⢠⠀⠀⡀⢀⣠⢶⣮⣜⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠤⠀⠎⣤⠘⢫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠠⡀⣉⢳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡥⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣝⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡃⠀⠀⠀⠀⡰⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣽⠿⡄⠀⠀⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⠀⣸⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢐⣛⣿⡈⣛⡁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⣷⣶⡞⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⢣⣿⣯⡜⢣⣀⠀⠀⠀⠀⠀⣻⡥⠀⠀⠀⠀⡀⠻⣦⡄⣤⠀⢤⠄⣠
⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⣹⣿⣿⣿⣶⣷⣟⣗⣴⣢⣿⣿⣷⠀⠀⠀⠘⣧⣴⠷⡶⣶⠶⠇⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⣛⣿⣷⣷⡶⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡶⣠⣾⡿⠋⢿⠟⠛⠁⠀⠀⠀⠀⠉⠉⠉⠿⣿⣦⣤⣾⠟⠛⢿⡏⠔
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠅⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡀⠀⠀⠀⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠘⠷⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠿⢤⠀⠀⠘⠿⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⠠⠖⠉⠱⠔⠢⠠⡀
⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⢀⠤⠒⠸⠤⣀⡀⠀⠀⠠⠄⠒⠊⠉⠉⠈⠉⠉⠳⠀⡄⢀
⣔⠋⠉⠉⠙⠢⠐⠶⠍⢉⣩⣅⣔⣀⣀⠤⠒⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠖⢢
⠈⠑⠢⢤⠤⢖⢆⡠⠀⠨⣋⠑⢄⢀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⣔⠉⠀⡎⠒⠹⠁⠈⠃⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⣅⠀⠀⢇⠀⠀⠀⠀⡀⢀⢁⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⠀⠀⠀⠀⠀⠸
⠀⢸⠄⠀⠀⠀⠱⡈⠒⠊⣁⢂⢇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⣇⠀⠀⠀⠀⠀⢙⠢⠭⢃⠜⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠐
⠀⢇⠀⠀⠀⠀⠀⠀⠉⠉⡁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞
⠀⢸⠄⠀⠀⢀⡠⠒⠖⠚⠀⠉⠕⠑⠆⠀⠀⠀⠀⠀⠀⠀⠼⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠸⡀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠏⠑⢦⣀⠀⠀⢀⡖
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣀⣀⣄⣀⣀⡤⢶⠖⠉⠁⠀⠀⠀⢉⠁⢊⠏
⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⡔⠙⣄⡄⠀⢀⡰⠤⠊⠁⠀⠀⠀⠀⠀⢠⠁⢸⠀⠀⠀⠀⠀⠀⢸⠂⢨
⠀⠀⠀⠀⠀⠉⠢⠤⡞⢻⠁⢀⠇⢸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⢾⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠠⡇⢸⠀⠈⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠀⢈⡄⠀⠀⠀⠀⢀⠟⠋⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠸⡄⠀⡇⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⠾⠤⠴⠃
⠀⠀⠀⠀⠀⠀⢀⠞⠒⠒⡇⠀⠋⡛⠛⡆
⠀⠀⠀⠀⠀⠀⠸⢾⣀⡤⠇⠀⠓⠧⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⢶⣶⠾⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⣀⣤⣴⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇
⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠙⠛⠷⣦⣀
⠀⣠⡾⠛⠁⠀⠀⢀⣀⣀⡾⠛⠷⠶⢦⣄⣀⣀⡀⠀⣀⣀⣀⣀⣤⢾⡅⠀⠀⠀⠀⠈⠙⢷⣄
⣴⠟⠀⣠⠴⠚⠛⠉⢉⡿⢁⣀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠉⠉⠁⠀⠈⣿⠒⠶⠦⣤⣀⠀⠀⠙⢷⣄
⠻⣦⣀⣀⣀⣀⣀⣀⣼⠇⡾⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡆⢹⡆⠀⠀⠀⠉⠛⠦⡄⠀⢻⡆
⠀⠈⠉⠉⠉⢙⣿⣿⣿⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⢸⡷⠶⢦⣤⣤⣤⣄⣤⣤⡾⠃
⠀⠀⠀⠀⢰⡿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣹⣏⠉⠉⠁
⠀⠀⠀⠀⣸⡧⠀⠸⣧⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣄⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠉⠙⢿⣤⣤⡶⠶⢶⣤⣀⣠⣤⣤⣀
⠀⠀⠀⠰⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠉⢻⡿⠋⠉⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠈⢻⡏⠁⠉⠛⣷⣤⣤⣄
⠀⠀⠀⢠⡿⠃⠀⠀⠈⢷⣄⠀⠀⠀⢤⣄⣼⣇⣤⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠙⣷⣄⡀
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠙⠻⠶⣤⣤⣀⣉⣡⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⣦
⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠻⣦⡀
⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⢿
⠀⣼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿
⠀⠙⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟
⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡏
⠀⠀⠻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠘⢷⣄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣀⣀⣀⣠⣴⢶⡾⠛⠋⠁
⠀⠀⠀⠀⠀⠙⠛⢿⡟⠛⠉⠻⣦⣤⣤⣤⡶⠟⢷⡶⠶⠾⠻⣦⣤⣤⣴⠟⠁⢀⡀⢀⣤⣀⣤⡾⢦⣤⡶⠟⠁⠈⠙⣿⠉⠉⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢸⡏⠀⠙⠛⠛⠛⠛⠛⢹⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣿⠀⠀⠀⠀⢰⡟⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢸⡟⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢰⡟⢠⡄⠀⠀⣿⠀⠀⠀⠀⣼⠃⣠⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢾⣧⣼⣧⣤⣤⡿⠀⠀⠀⠀⢰⣿⣀⣿⣀⣀⣸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢴⡙⠻⠶⠾⠛⠐⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣾⣿⡆⠀⠀⢠⣿⣠⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠸⣯⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢃⣀⣀⠘⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣠⣿⡧⢀⠂⣠⡝⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣶⣿⣿⣿⡷⣶⡄⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣧⣿⣿⢷⡄⠰⠀⠀⠀⠀⠀⢨⡿⣵⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⢳⠰⠐⠀⠀⠀⠀⢺⣿⣟⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⣿⣿⣿⣿⣿⣰⢤⣆⢀⠄⠀⣼⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣏⣉⣱⣾⠆⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡉⠀⢸⣿⣿⣿⠏⠉⠛⠛⢿⣿⣿⣿⠉⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⢀⣾⢻⣿⠋⠀⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢸⣿⣰⡇⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢸⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠻⠁⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢸⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣾⣿⠀⢀⣿⡼⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⡀⢀⠀⠀⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣴⣿⣿⣿⣆⣼⣿⣿⣄⣴⣾⣾⣷⣿⣿⣷⣾⣿⣿⣷⣶⣶⣿⣿⣷⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠁


⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣤⣶⣶⣶⣤⣤⣾⣿⣿⣿⣿⣶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠠⠂⠀⠐⠢⡄⢀⣄
⣿⣿⡿⠿⢋⣿⣿⠯⣠⠰⠛⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⡀⠀⠒⠃⣤⢸⠀⠙⡍⠒⠤⢀
⡟⠁⠀⣠⣿⡿⠃⠘⠢⠶⣣⣾⡿⢻⣿⣿⣿⣧⠀⠀⠀⠈⢹⠑⠼⠗⠒⠃⠈⠢⡈⠂⠀⠀⢙
⣿⣶⣾⣿⠋⠀⠀⠀⠀⠀⣿⣿⣷⠈⣿⣿⠋⠀⠀⠀⢠⠁⠸⠀⠀⠇⠀⠀⠀⠀⠘⢆⠈⠒⠁
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠘⠇⠀⠀⠀⠠⠁⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠘⡆
⣿⣿⣿⣇⠀⣤⠀⠀⠀⠀⠀⣿⣿⣿⣼⡀⠀⠀⠀⠦⡠⠊⠀⠀⡀⠀⠀⠀⠀⣤⠀⡀
⣿⣿⣿⣿⣷⣬⣤⣤⣤⣤⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠐⠦⠤⠊
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠁⠀⠑⡦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢇
⠛⣿⠛⠙⡟⠿⠿⠿⠿⠿⠿⠟⠛⢹⠀⠀⠘⡄⠀⣠⠇⠀⠸⡒⠤⠄⣀⠀⠀⠀⣀⠀⢤⠒⠀⣤⠄⠐⣇⡀⠀⣈⠆
⢠⠃⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⣀⡠⠜⠀⢧⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠳⡀⠀⠀⠀⠁
⣼⣤⣤⣤⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣧⣤⣤⣤⣤⣤⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠐⠂⠀⠘⠧⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠠⠐⠁⠰⠂⠀⣳⣤⢌⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠁⠀⠀⠀⠐⠚⠙⣤⡞⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣂⠔⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⣲⣴⣾⠇⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣼⣿⣴⣦⠀⠀⠀⠀⠄⠄⢠⠈⠀⡁⠶⠀⢠⠀⠀⠀⠀⠄⡂⠀⢘⣿⣴⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⠿⣿⣿⡄⠀⠀⣦⠸⠀⣦⠈⠻⣿⣆⠐⠐⠀⠀⠀⠀⢸⡶⠀⣿⣶⣿⣷⡀⠠⢄⠀⠀⠀⠀⠐⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣧⣄⠀⠀⠀⠈⠓⠀⢰⣿⣿⣷⣤⡀⢈⡀⣾⣾⣿⣿⣿⣿⣿⣿⠿⢧⠸⡄⠀⠀⠀⠀⣠⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⠀⠀⠀⠢⣰⣾⣿⡿⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠉⠀⠀⠀⠉⠛⠤⡀⢀⣠⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣴⣼⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡙⢻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠈⠟⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠞⠀⠀⠴⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⡠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠛⠓⢦⡀⣠⠴⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠺⢁⡴⠋⠳⣄⠀⠻⠁⢀⡴⠦⣄⠘⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢧⣀⠀⠀⠉⠉⢹⠄⠳⢤⡰⠋⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⢀⡔⠦⠖⢳⡄⢫⣀⡤⣄⢀⡏⠀⠀⣠⠒⢦⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⠞⠁⠀⠀⠈⠓⣆⠀⠀⣠⠵⡄⠀⢀⡝⠀⠀⡸⠃⣤⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣘⠒⠲⡆⠀⣰⠒⠃⠀⣏⠉⠀⢉⡇⠘⠦⠦⣠⠇⠀⢹⡏⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⠴⠋⢻⡀⠈⠛⠁⠀⢀⣄⠘⠦⠞⠉⢀⡼⢦⣀⡀⠀⣀⣼⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣀⡶⠋⠉⠉⢙⡞⠁⠀⠀⢸⡷⢤⣤⣤⣴⡋⠈⠛⠲⠶⠒⠋⠀⣀⣈⡉⠉⠁⡿⠙⠶⢤⣀⣀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡀⠀⠀⠈⠀⠀⠀⢠⡏⠀⠀⠀⣠⢾⡇⢀⡴⠋⣹⣿⣷⣄⠀⠀⠀⣰⠚⢩⣿⣿⣷⡀⡇⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠇⠀⡏⠀⠀⠀⠀⠀⢰⡄⢸⣀⣠⠴⠚⠁⠸⡇⢸⠀⠀⢻⣿⣿⣿⠀⠀⠀⡇⠀⠈⢿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⣀⠑⢤⣤⠴⠓⠤⠚⢀⣀⡁⠀⠀⠀⠀⠀⢷⠘⢧⣀⣀⣉⡽⠋⠀⠀⠀⠹⣤⣀⣀⣨⠟⣹⠃
⠀⠀⠀⠀⠀⠀⣾⠀⠀⣎⠀⡀⣸⠂⢠⠄⠀⠀⠀⠀⣏⠀⠸⠂⠀⠚⠉⢳⠘⣆⠀⠈⠉⠁⠀⠀⣴⠒⠒⢲⠀⠉⠉⠀⢠⡏
⠀⠀⠀⠀⠀⠀⢹⡄⣀⠈⠉⠉⠁⠀⠸⣤⡤⠇⠀⠀⠈⠁⠀⢀⠀⠀⠰⡏⠀⠘⢦⠀⠀⠀⠀⠀⠘⢦⡴⠋⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠲⡄⠸⢄⣀⡴⠃⠀⠀⠈⠓⢦⣄⡀⠈⠑⠊⠉⢚⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⣇⠘⠦⡄⠀⠀⣸⠃⠀⠸⡀⠀⢀⠀⠀⣠⠇⠀⠀⠀⠀⠈⠳⡄⢀⡤⢤⡈⠉⠙⠛⡟⠋⠉⠁
⠀⠀⠀⠀⢠⠞⠙⢦⣀⠈⠓⠊⣁⠀⠀⠀⢈⣉⠉⠉⠉⠁⠀⠀⢆⠀⠀⣠⠴⠁⢪⡀⠀⠽⠀⠀⢠⡇
⠀⠀⠀⠀⢿⣀⣠⠞⠉⠙⣷⢸⠁⠀⣀⢠⡏⠀⠁⠀⠀⠠⠤⣄⠈⠑⠋⡠⠒⢤⠀⠀⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢹⣌⠓⠒⢁⡀⠹⣀⣠⠄⠀⡀⠀⣸⠀⠀⠀⠀⣄⡼⠀⣸⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠙⠛⠉⣇⠀⠀⠀⣀⠀⠉⠉⠁⢠⢦⣤⣤⣤⢴⡞⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠙⠛⠋⢹⡷⠦⠤⠖⠋⢸⠀⠀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢰⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡄⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠒⢺⢻⠚⠛⠉⠉⣇⠀⠀⠀⠀⣾⠛⠛⠒⢺⡏⣟⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⢤⡀⣾⠸⡇⣠⠶⣄⣿⠀⠀⠀⠀⣿⣀⡤⣄⢸⠃⢹⣀⡞⠓⠮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠙⠟⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠛⠁⠀⠈⠛⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠲⣤⠤⠤⢤⣀⡠⠖⠋⠉⠉⠉⠓⠲⢄⠀⠀⣀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠚⠁⠀⠀⠀⠀⠀⠉⢦⡀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡀⣠⠊⠒⠉⠉⠑⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠉⠑⠒⢄⡀⠀⠀⠀⠀⠀⣠⠗⠉⠁⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢸⠉⣙⡲⣧⣀⣄⠀⠀⢠⣄⣀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢧⠤⣽⠃⠀⣈⡙⠚⠉⢀⡈⠓⠶⢶⡖⠛⠋⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢧⣀
⠀⠀⠀⡤⠀⣾⢀⣨⠗⠒⢄⡠⠤⠭⡓⠀⠀⣧⠚⠉⢉⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⢸⡀⣠⠞⠉⠸⣔⣧⣼⣣⣦⠀⣹⢠⡄⡿⠉⢉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⣠⠟⠁⠀⠀⠀⠈⠉⠀⠙⠻⠒⠁⢨⡟⠁⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⢸⣇⣰⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠢⡀
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠐⠒⠬⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠹⡄⠀⠘⢦
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠈⡙⢒⠤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈
⣇⠀⠀⠀⠲⡄⠀⠀⢀⣠⣀⠀⠀⣀⡼⠀⠀⠈⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠹⣄⢀⡼⠓⠀⠀⠐⠃⣶⠈⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⠢⢄⣠
⠀⠈⠻⢤⣀⡀⠀⠀⠀⢘⣧⠴⠋⠀⢧⠀⠀⠀⠱⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⡼⠓⣄⣀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⠇⠈⠉⠉⢣⡀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢸⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⠴⣉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠉⠉⠙⡏⠉⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠓⠾⡀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡅⠀⠀⠈⠑⠲⠤⠤⠤⢤⠔⠊⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣸⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣆⣀⣀⣧⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⣧⣤⣤⣤⡾⠀⠀⠀⠀⠀⡼⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡇⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠓⠦⣄⣾⠀⠀⠀⠀⠀⠀⠀⢸⣀⣼⡀⢀⣀⣠⠇⠀⠀⠀⢸⠉⡉⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⣀⡴⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢸⣠⡇⠀⢀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠉

⠀⢀⠀⠤⠂⠄⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠐
⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⠄⠀⠀⠁⠀⠽⠀⠠⢤
⠀⠀⠀⠀⠀⡌⠘⢶⠑⡆⠐⠂⠈⠃⠀⠀⠀⠀⠀⢰⠀⢀⡆⠀⠀⢀⢀⡀⠄⠀⣲⡟⣔⣿⣤⣾⡦⡀⠄⠀⠦⠼⠟
⠀⠀⠀⠀⠀⡁⠠⠦⡀⢠⡶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠐⠠⢝⠦⠄⠠⠏⠀⠁⠀⠀⠀⢘⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⢀⣈⣉⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢸⡗⣠⡴⢰⠦⢶⠀⡆⠰⡫⠀⠀⠀⠀⢽
⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⡄⣃⢀⡲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠤⠠⠄⠀⠂⣀⣢⣘⣜⣃⡠⡠⣷⣤⢴⣷⢶⣤⣴⠤⠠⠀⠄⢒⡀⢀⡀⡤
⠀⠰⢵⣄⡺⢿⣬⠛⠞⠳⠫⠿⠑⠒⠉⠓⠛⠑⠉⡁⡉⠀⠈⠀⠀⠈⠁
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⠰⠾⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⡈⢠⡀⠀⠀⠐⠐⠊⠁⠉⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠠⠀⠀⠤⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠓⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠙⠂⠊⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣀⠠⣤⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠘⠃⠀⠀⠈
⣿⡿⠿⠿⠻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⣀⣤⣖⣰⣤⢤⠤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠲⣄⠀⣷⢸⡇⢰⣶⣶⣿⣶⣶⣆⣀
⠀⠀⠀⠛⠛⠛⠛⠿⠴⠿⢿⣿⣿⣿⣶⣦⣄⠀⠀⠀⢠⣤⡀⢰⣾⣿⣷⣶⣤⠀⠀⠀⠀⢀⣾⢸⣏⣷⢘⠀⢸⣾⡗⣼⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣷⡀⠀⡂⢀⣴⡶⣶⣤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣜⣓⢄⠀⣿⣿⢛⣉⣉⣠⣹⡁⣿⣿⣋⣲⣹⣧⣥⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣂⢃⠈⢿⣿⣿⣯⣿⡉⠀⠀⠀⠉⠙⠛⠛⠻⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢟⠹⣿⣿⣍⢦⣼⣿⣿⣿⠿⢟⠛⠶⠦⣄⣄⣀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢈⠀⣿⣿⣷⢸⣿⣿⣿⣿⣶⣾⣿⣷⣶⣤⣄⡁⠘⠻⠟⠂⢀⣃⡄⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⠀⣽⣿⣟⣼⣿⣿⣿⣿⣿⡿⣃⡿⣿⣿⣿⣿⣷⡄⠀⢸⣿⢽⣿⣿⠿⣷⣌⡆
⠀⠀⠠⠄⠀⣀⠀⠀⠀⠀⢿⣿⣿⣧⣠⣿⣿⣿⣻⣿⠀⠿⠿⠿⢇⣿⣿⣿⡒⠬⣿⣻⣷⡀⣹⣿⣻⣿⢇⠑⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠒⠦⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠘⣿⣛⡃⣛⣉⡴⢿⣦⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⣿⣿⡏⡀⣿⣿⣧⣿⠻⡿⠿⠛⠁⠀⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⠇⠀⣿⢛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢼⣿⣷⠾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⣤⣤⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠒⠒⠀⠀⠰⠆⠀⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠰⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣎⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢊⠀⠀⠈⡄⢀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠘⢄⠀⠄⠀⠠⠄⠀⡀⠀⠀⠀⠀⢀⡄⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠃⠀⠀⠈⠁⠀⠀⢨⠗⢄⠘⠁⢠⡄⠀⡆⠀⠀⠰⢣⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠈⠤⠀⠀⠀⠂⠀⢀⠴⠳⠀⠀⠀⢱⢀⡄⠀⢈⠈⢱⡌⡆⠀⠖⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⡀
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⠀⠀⠀⠋⠴⠀⢀⡄⠀⢨⠀⠀⠀⠉⠀⠁⠀⠀⠀⢰⠁⡇⠂⠀⠀⠠⠀⢰⠆⠴⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⡌⠀⠃⠀⠀⠀⠀⢠⠇⡀⠀⠀⠈⠁⠀⢸⠅⠀⠀⠀⠀⠀⠀⡀⠀⠘⠰⠿⣇⠀⠉⠈⠀⢨⣀⠀⠀⠛⠆⠁⠀⡁
⠀⠀⠀⠀⢠⠃⣆⠀⠀⠀⠀⢀⡇⣀⠀⠀⠘⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡏⢠⠄⢨⠀⣸⠀⠀⣀⠀⠠
⠀⠀⠀⠀⠈⣢⣀⠀⠁⠀⠀⡾⠠⠀⠀⠀⠋⠁⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠗⠂⠀⠂⠀⠁⠀⠀⠘⠀⡀
⢀⠀⠀⠢⣴⠉⠉⠛⠻⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡜⠢⠀⠀⠀⠀⠀⠀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠒⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠈⠁⠒⠀⠤⠤⠤⠄⠀⠀⠀⠀⢱⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠈⠁⠀⠛⠁⠴⠂⡀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠰⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠐⠂⠁
⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣴⣿⣶⣦⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠘⡛⠃⣐⣿⣿⣿⡟⠶⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⠀⠀⢿⣿⣿⣿⣿⣇⢰⡀⠀⢀⣤⠼⠧⠦⣿⣷⣴⣶⡦⠐
⠀⠀⠀⠀⢀⠄⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⠀⠀⢘⣿⣿⣿⣿⣗⣼⣅⠀⠙⣿⣷⣀⣀⡹⣿⣿⣥⣤⣼⡌⣀
⠀⠀⢠⣖⠁⣴⢆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠙⣿⣿⣿⣿⣿⣿⣫⣴⣿⣿⣿⣿⣿⣋⣀⡀⣹⣿⠋⡏⢹⣿⣿⣻⡉⡍⢉⡭⣤⠬⠂
⠀⢠⣾⣟⣓⣈⣤⣴⠄⠛⠋⠋⠉⠁⠒⠒⠂⠙⠓⠿⠿⠿⠿⠓⠿⠿⠟⠻⠿⠟⠛⠿⠿⠿⠿⠛⠛⠀⠂⡛⠛⠛⠿⠅⠰⠈⠉⠉⠈⠁⠈⠀⠀⠀⠠
⠀⠀⠀⠈⠉⠁⠄⠈⡉⠈⡀⠀⠀⠀⣀⠐⢀⠀⠐⢚⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠠⣌⡃⢿⡌⣉⡂⢱⣝⠲⡏⣙⢳⣴⠈⠍⣴⡍⠊⣇⠥⡂⠀⢀⠶⠞⠦⢼⠀⡆⠠⢈⡤⠸⢵⢂⣶⢁⡔⠂⢲⡐⠆⠐⠈⠄⠰⡤⢆
⠀⠀⢀⣿⢙⡆⢸⢫⠁⠔⢿⠇⠘⡸⢻⠛⣤⠗⣾⢈⡝⣺⢶⣧⢡⠙⠤⢠⣐⣦⣾⡅⣶⠻⣍⡆⠞⣀⣽⡆⠶⡏⠋⡍⡉⠁⣀⡛⠃⣷⡤⠖⠣
⠀⠀⢈⣿⢸⣇⣷⡫⠜⢻⣴⡏⠰⣉⢾⣀⣻⡀⢾⠀⠼⢋⢸⡇⡛⠷⢬⡘⢼⡡⡤⢀⣅⠔⢻⡳⢦⡥⠺⢛⠿⠁⠦⣯⡌⠷⡋⠁⠀⣾⣁⣀⠁
⠀⠀⠀⣿⢸⣏⢸⡇⣿⠛⢸⣇⢘⣟⢾⡑⢽⡆⣿⠳⣿⣯⢱⣎⢵⠻⠎⢣⠿⣵⣄⢸⡇⢠⣬⠁⠀⣿⡶⣻⢧⡀⣶⡰⢶⠶⡀⢆⡿⣡⠈⠒
⠀⠀⠉⣿⠘⠷⢸⡦⠯⣀⢸⡇⣀⣉⣻⡓⢺⠇⢹⡘⣿⠣⢸⡅⢒⡼⢦⡡⠾⢼⡇⢸⡏⣈⣭⠗⠀⣿⠒⡯⢿⡁⣽⡵⢿⣷⡅⣬⡝⣡⢌⡉
⠀⠁⠈⠉⠛⠒⠺⠧⠤⠤⢼⣅⣈⣁⣽⡉⠉⠁⢻⠃⠒⠒⢾⠔⠠⠜⢵⣧⠀⠼⣆⢡⡦⠩⡍⠁⠙⣿⠃⠑⢻⡓⣿⡴⣿⣶⠄⢼⡅⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠶⠶⠼⠧⢤⣀⣘⡇⠀⠀⠀⢰⡎⠁⠈⠂⠐⣿⠂⠰⢼⠳⣿⠤⠟⣀⣀⣾⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠤⣤⣄⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠓⠛⠒⠤⠤⠀⣭

⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠋⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣀⣀⣀⣽⣄⡀
⠀⠀⠀⠀⠀⢀⡼⠿⠟⠛⠛⠉⠉⠉⠙⠛⣻⣥⡶⠦⠀⠀⠀⢀⣀⣀⣤⣤⣤
⠀⠀⠀⣠⡶⠋⠀⣀⣀⣀⡠⣤⣴⡖⢺⣿⣿⣧⢤⠀⠶⠚⠛⠉⠉⠁⠀⠀⢸
⠀⠀⠘⠛⠛⠉⢩⠏⠘⠛⠀⠈⡟⣷⠞⠁⢸⠀⣎⡄⠀⠀⠀⢀⣠⡄⠀⠀⢸
⠀⠀⠀⠀⠀⢠⣯⠤⢦⣀⡀⢰⣿⡿⢤⣤⢾⡄⢻⡄⣀⠴⠚⠋⢹⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣾⡿⢳⣄⠀⠈⡇⠀⠙⢦⣀⣀⣀⡟⠀⠀⢠⠃⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢲⣴⡟⠁⠈⣿⠣⡴⠞⠀⠀⠀⠀⠉⣹⣿⣿⣶⣾⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠸⣇⠀⠲⣦⡀⠀⠀⡰⠋⢑⠞⠉⠁⠀⠀⠀⠀⠙⢿⣧⡔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢻⣦⡼⠁⠀⡏⠀⠀⠀⠀⠀⠀⢀⣤⣾⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣿⡁⠀⠀⣇⠀⠀⠀⠀⣠⣾⣿⠷⣫⠑⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢹⢇⣿⠶⣿⣿⣶⣴⣾⣿⡫⠕⡡⣐⡵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢸⢸⣡⡶⡟⢝⢝⡿⢿⣭⢖⠔⣊⠤⢚⣽⣷⡀
⠀⣠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡟⢸⡻⡌⣌⢇⠡⣻⣶⣿⣷⡫⣷⢯⣣⢿⡻⣧
⠀⠙⡟⠳⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⣿⠝⣎⢂⣦⣷⡿⠟⠉⠘⢿⣕⢿⣷⠱⣒⣽⣇
⠀⠀⢧⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢿⠧⡇⣿⣮⣶⠿⠛⠉⠀⠀⠀⠀⠈⢻⣯⢾⣿⡻⣝⣿⡄
⠀⠀⠈⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡹⡀⡟⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢻⣾⣷⡽⣧
⠀⠀⣦⡀⠈⣇⠀⠀⣀⣼⠀⠀⠀⠀⠀⢻⣗⡝⣽⣿⢷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣛⡭⢔⣿⡆
⠀⠀⠘⢏⢳⢸⡄⢀⡍⢙⡖⠀⠀⠀⠀⠀⠹⣮⣪⢿⢸⠟⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⡛⠽⠧⠀⣀
⠀⠀⠀⣼⠈⣦⣧⢾⠀⣯⠀⠀⠀⠀⠀⠀⠀⣬⠇⣸⢸⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⣠⣴⣶⣿⠿⡃
⠀⢳⡲⡈⣇⣹⣻⡼⡰⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⡇⡎⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠋⠉⠀⣿⣿
⠀⢠⣷⣉⢫⣹⡟⡗⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠾⣄⠧⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⣆⠀⢀⣠⢄⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠘⢤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣽⠞⠛⠉⠉⣩⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣀⣤⣿⣃
⠀⠀⠀⠀⠀⠀⢛⣷⠀⡀⢠⠀⢀⠀⠀⣀⣐⣊⣼⣟⡀⢀⣀⣀⡀⠀⣼⠁⣠⡞⣠⠟⠀⠀⠀⠀⢀⣠⠿⠂⠶⠖⠠⢶⡟
⣀⣀⣀⣴⠛⠦⡼⠿⠜⢳⠿⣤⣯⠀⣸⣿⣿⣿⣿⣿⣿⣟⣁⠈⠉⣉⣉⡴⠛⠫⠿⢾⣥⣀⣰⣛⣉⣁⣀⣀⣠⣄⣠⣼⣥⣾⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⣀⣀⠀⡀
⠀⠀⢠⣤⣤⣤⣦⣤⡤⢶⠆⠀⠀⢩⣥⣭⢄⣈⠀⠀⠀⠀⠀⠺⠀⢉⡉⠹⣦⠀⠀⠉⠙⠻⣿⣿⣶⡀
⠀⣠⣿⣿⣿⣿⣿⣿⣄⡞⢡⡀⠀⠈⠙⢿⣾⣿⣦⣄⠀⠀⠀⠀⠀⠈⠁⠸⠤⠂⠀⠀⢳⣌⠉⠉⠙⢿⡄
⠀⣿⣿⣿⣿⡿⣿⠿⡟⠃⣀⣠⠠⠀⠀⠀⠈⠟⠋⠀⠀⠀⢀⣤⣶⣿⣷⣄⣼⣧⡿⠂⠉⢁⠀⠀⠀⠘⠁
⠀⢿⣿⢿⣿⣻⢻⠿⣿⠿⣿⡧⠌⠷⠤⠀⠄⠤⠂⠀⠤⠀⠀⢙⣿⣿⣾⣿⣿⣿⣧⣶⣾⣿⣿⣿⣿⣿⡇
⠀⣾⣿⣧⣮⣭⢠⣰⣿⣦⣴⢃⣀⣉⠋⢽⣶⡄⠈⠁⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣹⢿⣿⠇
⠀⠻⣿⣿⣿⣿⡶⣿⣿⣿⡇⢸⣿⣿⣶⣾⣷⠄⡀⣝⣿⡿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠐
⠀⠐⣿⡿⠾⠿⣷⣿⣿⣿⣧⣠⣭⣭⣡⣬⣽⣷⣩⣿⣟⡀⣿⣶⣶⣶⣤⡼⠛⢻⣧⣸⣿⣿⣿⣿⠏
⠀⠀⢤⠀⡀⢀⠀⣿⡏⡽⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⡛⠳⡤⠜⠛⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠚⢿⣿⣅⣸⣇⠉⢿⡿⢿⣿⣿⣿⣿⡏⣦⠻⠿⠿⠟⠿⠙⠿⢯⣙⡶⠶⠟⡓⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠺⠷⠦⣘⣻⠿⣿⠿⠿⠗⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣶⣾⠟⠉⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⡶⢶⣶⣤⣄⡀
⠀⠁⠀⠀⠠⠀⠀⠀⡀⠀⢛⣈⠀⡀⣈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠫⢦⡀⠀⠉⠻⢿⣿⣶⣤⡀
⠠⠀⠀⠂⠀⡀⠀⠠⠀⠐⠀⠈⠁⠀⠀⠀⠀⡨⢭⡉⠉⠉⠛⠻⠿⢿⣿⢿⠋⣁⢀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠻⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠂⠙⡆⠉⠃⠀⠀⠀⠀⠀⠈⠓⠶⢧⠀⠀⠠⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⣠⠤⠀⠖⠚⠉⠉⠂⠀⠀⣐⢤⢀⣤⣤⡄⠀⠀⠀⢀⣠⣀⣀⣠⣿⣿⣿⡶⠶⣾⣦⣅⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠉⠉⢉⣤⣤⣦⡄⣑⣱⣶⣶⠈⠒⠃⠠⣾⣷⣀⣘⠛⣿⠀⠰⠶⠄⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠛⠋⠉⠁⠁⠀⠁⠁⠁⠀⠈⠈⠀⠀⠈⠘
⠀⠀⠀⠳⡌⢸⣼⣷⣿⣿⠿⠿⠿⠓⠀⠋⠈⠉
⠀⠀⠐⠰⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⡀⠀⠀⠀⢀⡀⢀⠀⢀⡀⠀⣀
⠀⠰⣦⣄⣶⣤⡦⠄⠀⠀⠠⠷⠿⠚⠛⠺⠳⠓⠖⠁⠷⠿⠄⠀⠀⠐⠄⠁⣸⠬⠥⠤⠛⠉⠉⠉⡉⠈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣴⣶⣶⡀⠀⢀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠻⢷⣾⠿⢟⣻⡿⠟⣿⣿⣿⣿⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⠿⢛⡴⠚⠉⠀⠉⠩⣼⣧⠤⠿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣩⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡟⠁⢀⣉⣠⠄⠀⢄⠀⠉⠈⠁⠀⠀⠩⣝⣿⣿⣦⠀⠀⠀⣠⣾⣿⣿⣿⠿⢿⣶⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣋⡵⢋⣹⣶⡆⠀⢘⡼⠂⠀⠀⠀⠀⣀⣸⠟⣛⠻⣷⣶⣿⣿⣿⣿⣿⣶⣤⡤⠀⢠⣽⣶⠏
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣠⣀⠀⠀⠀⣾⣟⠐⠒⢬⣉⠉⠤⡠⢀⠄⠀⠐⠀⠀⠀⢰⠟⢁⣾⣿⡷⢻⣿⣿⣿⣿⣿⣟⡛⠻⠿⣿⣿⡿⠋
⢀⣴⣿⣿⣿⡿⢻⡿⡛⠿⠿⣷⣄⣸⣿⣏⣰⣷⣮⡉⠳⠀⡊⠀⠀⠀⠀⠀⠘⣦⡆⣠⣾⣿⠿⠆⣿⣿⣿⣿⣧⣄⢾⣿⣶⣶⣿⡿⠁
⢸⣯⣼⣿⣟⣿⣟⣭⣄⠀⢐⣞⣿⣿⣟⣨⣭⣛⠟⠛⠂⣀⠤⠀⠶⠰⠤⠀⠀⠹⡗⠟⠉⠹⡄⠀⣿⣿⣿⡿⣿⣿⠿⠿⠟⠟⠃
⠘⢻⣿⣿⣿⣿⣿⣯⢶⣾⢿⣯⣿⣿⣿⣟⣛⡏⢀⡒⢦⣿⠆⠀⠀⠀⠀⠀⠀⢀⠙⢷⡄⠀⠀⠸⣿⡏⠀⠀⠉
⠀⠀⠀⢹⣿⣿⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⡟⠀⣿⠀⢀⣠⡤⠤⠒⠒⢣⣤⣤⣁⠀⠀⢲⣿⣿⡀
⠀⠀⠀⠀⠀⠙⣹⣿⣿⣿⣿⡿⠿⠛⠛⣿⣿⣷⣄⠀⣴⢣⣶⣿⠅⠀⠀⠒⠋⠉⣠⣿⠻⡆⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠈⠁⠀⠀⢀⣿⣿⣿⣿⣸⣿⣿⣧⣤⡀⠀⠀⠀⠀⠸⣿⣷⣄⢹⣦⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡧⣹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠿⠀⠨⣽⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣧⡿⠁⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣷⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠈⣷⡀⠀⠈⡁⣠⣤⠶⠛⠛⠛⠛⢛⣹⠃⣿⣿⣀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣷⣦⣙⢿⣿⣟⠉⠁⠀⠀⠀⠀⣀⣀⣾⣏⣼⣿⣿⠿⠛⠉⠛⢷⡀
⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣷⣾⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠚⣿⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠳⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⢸⡏⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣿⡄⠀⠀⠀⠀⢠⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠃⠀⠀⠀⠀⠘⠁⠘⣿⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⢀⠀⠀⢰⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⠀⠀⢠⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⠛⠀⣼⠀⠚⠁⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠷⢶⡆⠀⠋⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣿⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡝⠙⡇⠘⣿⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢸⡟⢀⣴⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠘⣿⠀⠀⣿⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣺⣗⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣶⣿⣶⣚⣁⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡛⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠞⠶⠟⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⠂⡀⠀⢀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡿⠘⡇⠘⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢰⣿⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣤⣾⡏⢀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⣿⣿⣿⣿⣧⠋⠀⠘⢶⡄⣟⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢽⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁⠞⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⡀⢀⡀
⠀⠀⠀⠀⣀⠀⠀⠈⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠄⠂⠀⠂⠀⠀⠔⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠛⠻⠟⢿⠂
⠀⠀⠀⠀⠀⠀⢀⠤⠐⠁⠀⠀⠀⠀⠀⡀⠀⠈⣀⣠⣤⠤⡤⠀⠐⠀⠲⠛⢻⣿⣿⣿⣾⣿⠷⠶⠶⢦⡀⠠⠶⢖⠉⣀⡟⠁
⠀⠀⠀⠁⠢⢄⡋⠀⠀⠀⠀⠀⠀⠀⢉⡽⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⣦⡀⢰⡆⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡟⢻⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣇⠘⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢹⣿⣿⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠘⣿⣿⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠘⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⡇⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⠇⠀⣹⠿⢿⣙⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⣦⣶⣶⣶⣶⣶⣶⣦⣶⠾⠿⠿⠛⠁⠀⣠⣤⣶⣿⣿⣾⡿⠃⣠⠾⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⣴⣿⣿⣿⢿⠟⠛⠁⢉⣁⡀⣀⣀⣀⣀⣤⣤⠝⠛⠟⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠐⠋⠈⠉⠁⠁⣠⠄⣠⣾⣿⢿⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⠡⠞⠱⠋⠁⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⣶⣰⣶⣧⣤⣀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠈⡱⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⢄
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠘⢳⠊⠀⢠⣖⡀⠀⠋⠧⠞⠃⠀⠀⠀⠀⠀⠎⠀⠀⠄⠀⠀⠀⢁⠀⠐⡦⠀⠀⠰⠷⣠⠄⠀⡐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⢈⡄⠀⠀⠠⡚⠳⡄⠉⠺⣾⣿⣷⢾⣷⢆⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢀⡀⠀⠄⢷⣿⣷⣤⣔⠉⠀⣀⢠⡖⠓⠑⠈⠄⠀⠀⠉⠛⠻⠿⠽⢷
⣿⣿⢿⣿⣿⣶⣾⣿⣿⣿⣿⣯⠉⠩⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣽⣿⣿⣟⣶⣿⣧⣴⡧
⢿⢿⣷⣬⣭⣭⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠀⣤⣤⡄⢤
⣶⣶⣵⠯⣿⣭⣿⣿⣿⣿⣿⣇⡀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣾⣿⣿⣥⣄⣈⣷⣤⣤⣤⣀
⣿⠏⢻⣿⣿⣿⡿⡉⠉⡻⣟⠛⠛⠃⠤⠀⠀⠀⠀⠒⠤⣄⣠⣀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠉⠉⠛⠿⠿⢤⢈⠀⠻⠟⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⢿⣿⣿⣷⣤⣤⣦⣤⣄⡀⠀⢀⡀⡛⠃⠀⠀⠀⣀⣈⡉⠉⢁⡂⠀⣤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⣄
⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⠿⠿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⢿⠿⢯⣭⣿⣿⣿⣿⣿⣿⣾⡿⢿⢷⡿⣷⣴⣼⢿⣦⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⣴⣶⣾⣿⣿⣿⣻⣿⣿⣷⣆⣉⣭⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣶⣿⣿⣿⣿⣟⠿⣿⡿⣿
⣥⣽⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣴⣿⣅⣿
⠹⠟⠛⠟⠿⢿⣭⣿⣽⣤⣤⣤⣠⣤⣬⡽⠋⠉⠁⢸⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠟⠻⠿⣿⣿⣛⠛⢛⣳⣖⣖⠶⠶⠶⠶⠶⠶⠴⠯⠭⠙⠻⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣇⡁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢷⡾⢶⣶⠶⠤⣀⡢⠤⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠃⢀⠀⠈⠛⢻⣈⣁⣀⡈⠙⠛⠛⠛⠛⠿⠿⠷⣶⣶⣿
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠀⠒⠦⢄⣠⣦⠈⠀⠀⠀⠀⣶⡦⢄⣀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠄⡀⢈⡉⠓⠀⠀⠀⠀⠀⠐⠻⢷⣇⡱⣀⡀⠐⠀⠁⣾⣾⡛⠛⠻⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⣀⠀⠈⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢞⣥⠞⠉⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣿⠁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢻⡄⠀⠀⢀⡟⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠙⠶⠶⠛⠀⡏⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢳⡀⠀⠀⠀⠀⠃⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠀⢣⠀⠀⡀⢠⣦⠀⠀⡖⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣼⡇⡠⢷⢂⡍⠉⠓⠛⢾⠤⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⡼⠉⣿⠀⠀⠁⠀⠀⠀⠀⠸⢦⠀⠀⠈⢣
⠀⠀⠀⠀⠀⢀⣾⠃⠀⢲⡆⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⠈⢷⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠙⠢⣄⣸⢀⣀⡀⠀⠀⠀⠀⢸⠀⠀⢳⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⡴⠛⠉⠈⣍⡳⡄⠀⠀⢸⠀⠀⠈⢷⡀⠀⣠⢾⣀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣾⣀⣠⡤⢜⡶⠳⠀⠀⠀⢸⠀⠀⠀⠈⢷⡞⠻⡄⠀⢠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⠛⡶⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡸⢻⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡇⢸⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⡇⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢱⡇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡎⠀⢸⣇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠃⠀⠀⣽⠋⠉⠉⠁⠈⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⡄⠀⢀⠀⠀⠀⠻⠖⠚⠛⠛⠒⠒⠋⠀⢀⡴⢇⣤⢤⡤⢴⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⡠⠀⠉

⠀⠀⠀⠀⣀⣀⣀⣀⣠⣾⡿⣷⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⠁⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠀⢀⣤⣶⣶⣶⣾⣿⠏⠀⠀⣀⣠⣤⣤⣼⡿⠁
⠀⠀⠞⣼⡿⠋⠉⠉⣿⠇⠀⠀⠀⢨⣿⡿⠿⠿⠿⣿⡿⠋⠀⠀⠀⠸⣿⣿⠿⢿⣿⠏⠀⠀⠀⢹⣿⠿⢿⣿⠇
⠀⠀⠀⠇⠇⠀⠀⠀⠹⠄⠀⠀⠀⠸⠍⠇⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠋⠇⠀⠸⠇⠀⠀⠀⠀⠈⠉⠀⠀⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡀⠀⠀⢠
⠀⠀⣤⣶⣶⣶⣶⣶⣤⣤⣤⠀⠀⠀⠀⣿⣭⣥⣤⣤⣾⡿⠛⠀⠀⢮⠀⠀⠀⠀⣠⣶⡄⠀⠀⣾⣿⣦⣀⣰⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣿⣿⠿⠿⣿⡿⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⡟⢿⠉⠉⠉⢹⡿⡅⠀⠀⠀⠀⢸⣟⠁⠀⠀⢸⡇⠀⠀⠀⢰⠛⡏⠀⠀⡏⠙⢆⠀⠀⠀⠈⠈⣿⡿⠛⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⡄
⠀⠀⣀⣀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠹⣿⣷⣶⣿⣿⡇⠀⠀⠀⢀⣠⣿⡄⠀⠀⢸⣶⣤⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⣸⡏⠁⠀⠀⠀⠰⣦⣀
⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠐⢶⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⠟⠀⠀⠠⣤⣶⣿⣿⣿⠇⠀⠀⠀⢀⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠉⠿⠀⠀⠀⠀⠀⠀⠈⠿⠅⠀⠀⠀⠀⠀⠉⠹⠏⠀⠀⠀⠀⠀⠺⠿⠿⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⣴⠆⣀⣤⣤⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢛⣵⠟⠉⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣾⣿⠇⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⢿⡆⠀⠀⠀⠀⢰⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⡀⢀⣠⡞⠀⣿⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣇⠀⠙⡛⠛⠉⠀⠀⠋⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠹⡆⠀⠐⠀⠀⠀⢠⠀⠀⡆⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢸⣇⠀⠁⢠⡶⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣸⡆⠀⢳⣰⡀⠲⠲⢾⣭⣤⣄⡸⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣰⢿⣧⡰⠋⠷⠤⠞⠀⠀⠀⠀⠈⣟⠛⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⢶⣤⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠙⠓⣾⣧⠶⠶⢦⣀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢻⡄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⢠⠞⠉⠀⠀⠰⢦⣌⢵⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿⡄⠀⣠⡴⠻⣤⣤⠶⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣰⣇⡀⢀⣤⢤⣙⣷⠸⠦⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢿⡴⠋⢻⡄⠀⠀⣴⣾⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢦⣬⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⣤⡀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣰⢏⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⡿⠸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢸⡄⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣸⠀⠀⣿⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢰⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣶⢸⡇⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡏⠀⠀⣾⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⣮⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⠃⠀⢰⡟⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢿⠀⠀⠘⢿⣶⠤⠤⠤⠖⠈⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢀⡀⠀⠀⠀⣾⠃⠀⣀⣀⣀⡀⠀⠀⢀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣒⣦⠀⠀⣸⠇⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠛⠛⠉⠀⠀⡀⣴⠞⣤⣴⢦⣶⠖⢾⡤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠔⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠛
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⢿⢿⣿⡿⠿⢿⡿⠿⣿⠿
⣿⣶⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣼⣟⣀⣀⡀⠄
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠶⠶⢦⡀⠀⢰
⠀⠀⠀⠈⠛⠿⠟⢿⡏⠹⠁⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⣈⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠔⢺⡙⢧⣴⡇⢙⡦⡀⠰⢊⡩⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣦⣤⣄⢰⣀
⠀⠀⠀⢠⡎⠀⠀⠈⠙⠲⣿⠟⡉⡀⠙⢲⡎⠀⠀⠈⠀⠀⠀⡀⠤⠀⠀⢀⠜⢡⠤⠈⢻⣿⣿⣿⣿⣄⠀⠀⠀⢄⣀
⠀⠀⠀⢸⣇⠀⠱⣦⣠⣶⠹⢄⣛⡤⠀⠺⠷⢤⡀⠀⣀⡀⠀⠀⠀⠀⠀⡀⢠⣤⠀⣤⠀⣬⠉⣭⣽⣛⠃
⠀⠀⠀⠘⣿⣄⡤⢿⣧⣿⣷⡿⣵⡛⠁⣀⣔⡿⣿⣿⣿⠀⠀⠀⠀⡀⠀⢀⣈⣉⣀⣉⣀⣿⣶⣿⣿⣿⣿⣦⣀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⢻⣿⣾⠟⠓⢦⡀⠀⠈⠁⠈⠉⠹⠛⢛⡏⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠘⢿⣿⣷⣦⣀⡉⠓⠶⠚⠓⢛⣷⣰⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣤⣴⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣾⠿⠾⣫⠀⠀⢠⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⡋⠀⠀⢠⡾⠀⢀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣾⡇⠀⣸
⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠟⠀⠀⠁

⠀⣀⣀⣀⣀⣴⣾⢿⡀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡸⣿⣿⠿⣿⣿⠇⠀⠀⢀⣀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⢰⣿⣿⣿⣿⡿⠙⠂⣀⠀⠀⣠⣤⣤⣾⠋
⠁⢻⠁⠀⠀⣿⠀⠽⠛⡟⠉⠀⠀⡾⡟⠉⠉⢹⡏⠁⠿⢻⠛⠂⠀⢸⠋⠉⠉⡿⠀⠸⢻⠋⠀⠀⠼⠛⠛⡏⢰⡶⡞
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣤
⣠⣤⣤⣤⣿⠃⠀⠀⠀⢀⣤⣤⣤⣶⠖⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⢠⣆⣀⣠⣶⠄⠀⠀⠀⡄⠀⠀⢀⡀
⣿⡿⠛⣿⡏⣴⣶⠃⠀⢸⡿⠿⢿⠟⣀⣠⠄⠀⣿⡿⣿⣿⣿⠿⢆⣀⠀⠀⣸⠟⠛⣿⠃⣀⣠⠀⠀⣿⠿⢿⣿⠁
⠋⠃⠀⠋⠃⠁⠈⠀⠀⠈⠁⠀⠘⠀⠉⠉⠀⠀⠉⠃⠀⠀⠉⠀⠈⠉⠀⠀⠉⠀⠀⠙⠈⠉⠃⠀⠈⠀⠀⠀⠈⠘⠉

⢀⣤⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⢠⡄
⢸⣿⣿⣷⣶⣿⡀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠐⣶⣄⣤⣾⡇⠀⠀⣀⣀⣀⣾⠂⠀⠀⠀⠀⢠⡶⠀⠀⠀⢀⣀
⠘⠛⢿⣿⣿⠿⢃⡀⠀⠰⣶⣿⡿⣀⡤⠀⠀⠈⠻⣿⠿⢁⠀⠀⠹⣿⣿⠟⠀⠀⢠⣤⣶⣾⠇⡀⠀⠀⣠⣽⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣂⣂⣀⣀⣐⠒⠒⠒⢒⠒⠒⠂⣀⣐⠐⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⣄⣀⣐⣂⣒⣂⣀⣐⣒⣒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣤⠤⢶⠶⣶⣶⠾⠿⠒⠀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣶⣿⣿⠿⢛⣂⣶⣾⣽⢋⠙⣿⣻⣿⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣻⠫⢛⢛⢛⣛⣛⠛⣉⠛⠋⣩⣿⣿⣯⣙⡛⠛⡙⠉⡛⢛⡛⠛⢋⣙⠻⣛⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣛⣯⣯⠯⠿⣿⣞⡿⠯⠀⠀⠭⣿⣿⠿⣤⡉⠳⢾⣶⣌⡳⣮⣝⠳⢾⣍⡓⠶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣷⠶⢟⣛⣿⣿⡾⣞⣿⣷⠃⣶⠀⠑⣠⣿⣿⣿⡶⡇⢿⣷⣮⠛⢿⣶⣭⣙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣥⡶⣿⠿⣫⣿⣿⣿⣿⣿⡋⣸⢣⠊⢹⡟⢯⢻⣿⣷⡏⣯⣿⣿⣿⣄⠛⢿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣋⣵⣾⣿⣿⣿⣿⢿⡏⠙⢠⣿⡏⢠⡿⠋⠀⢻⣏⡉⠤⢝⣿⣷⡹⣿⣷⢷⣞⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠟⢹⡿⡏⠦⡸⣟⣼⣿⡇⣾⡇⢀⣧⠰⣿⣟⣇⡚⣿⣿⣧⠩⢻⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⣠⡎⡛⣑⡁⣞⣿⣿⣿⣿⠀⣿⢳⣾⣿⡀⣿⣿⣷⡍⣽⣿⣿⣧⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡿⠧⢭⣬⣷⣿⣿⣿⡿⢻⣤⣿⣿⣿⣿⣿⣿⣿⣿⢧⣳⣿⣿⣿⡆⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠭⢛⡦⡬⣾⣷⣿⣿⠟⠁⢷⣿⣿⣿⣿⣿⡟⣿⣿⣿⣻⣦⣽⢿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠐⠙⠛⠝⠻⠋⠀⠀⠀⠉⠀⠀⠁⠁⠁⠀⠙⠿⠟⠛⠘⠛⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠆⠒⠐⣶⣶⣤⣤⣤⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢤⡤⣰⢆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⢶⣿⣿⣿⣿⣿⣽⡄⠸⠐⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣴⡏⠀⠀⣻⣭⣦⡝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠇⠀⠀⣽⣿⠿⣿⣿⣿⠟⠀⠀⠀⠻⠏⠉⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡀⠀⣿⣿⣿⣿⠎⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⡿⠁⣀⠔⠀⠀⠀⠀⣿⠏⠉⠀⡎⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣦⣴⣻⣿⣿⣿⠉⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⢀⡞⢉⡙⠶⢦⣤⣄⣀⣠⣤⣴⣇⠀⠀⣠⠀⢶⣿⣿⣿⣷⣄⡄⠀⠀⠀⠀⠀⢿⠿⣿⡻⣿⡧⢛⣥⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢡⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠆⠀⠛⠳⠶⠨⣅⢈⣿⣩⠡⠀⠀⠀⠀⠀⠀⠲⣀⡇⠉⢀⡘⢳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⠤⠻⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠠⣤⠀⠀⣿⡀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⢲⡔⣄⢠⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠈⢘⢁
⠀⢀⣤⣶⣶⣶⠲⠶⠦⣤⣄⣀⣀⣀⣠⣤⡤⠤⠠⣤⣄⣀⣀⣀⣀⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⠛⠶⠞⠋
⠀⢸⣿⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⢰⣾⣿⣿⣿⣿⣿⠿⣟⡧⠀⠀⠀⠀⠀⠀⢦⡠⠠⠢⡔⣀
⠀⢸⣿⠛⣿⢱⡆⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠁⠈⠁⢘⣿⣿⣿⣀⣄⡻⢿⣆⡄⠀⠀⠘⠷⢿⡄⠃⢠⣿⣿⣿⣦⠐⠢⣶⣤⣤⣶⣶⣶⣶⣶⣿⢉⡙⢿⣷⣶⡄
⢀⣾⡟⠁⠀⢈⡴⠁⠘⠿⠁⠀⠻⠿⠋⢀⡀⠀⠠⡆⠈⠉⡹⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⣧⣀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣧⡸⠧⠸⢿⣿⣿
⣸⣧⡿⢂⡞⠋⠛⢦⣄⣀⣀⣀⣠⣴⠶⠿⣧⠀⢸⣀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣭⣿⣿⣿⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⠉⠉⢀⠀⠀⢾⣿⣿
⠀⢹⢃⡾⠙⣤⠤⠚⠁⠉⠉⠉⠁⠀⠀⠀⠘⡇⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠆⣴⣶⠐⡌⠉⠉⠁⠉⠁⠀⠀⠘⢣⠀⠀⠘⡇
⠀⢰⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠀⠉⣻⢇⣷⣄⣀⣀⣠⣤⡶⠺⠏⣈⡱⠀⠀⣧
⠀⠘⠶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⠀⢠⠎⠀⠈⠉⠉⠉⠁⠀⠀⠃⢨⡆⢃⠈⡿
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⢤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡈⠀⠀⡀⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢄⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡐⠀⠀⠀⣇⡇
⠀⠀⠀⠀⢹⣿⣿⣿⡛⢛⡃⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠸⣿⣿⣟⠁⠘⠁⠀⣸⣿⣿⣿⣿⡟⠿⣿⠃⠀⢺⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠐⣦⣀
⠀⠀⠀⠀⠀⣿⢻⣯⠀⠀⣼⠀⠈⠛⠉⠙⠿⢰⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠏⢀⠤⣿⣭⣽⣲⡶⠒⢀⣶⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢸⣷⣿⠈⠁⢀⡞⠻⠷⢦⣤⣤⡤⠴⠿⡆⠀⣶⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⢠⣤⣾⠟⣡⣿⣿⡿⠛⠀⢻⣿⣿⣿⣿⣿⣿⣿⠛⠀⠿⢲⣤⣀
⠀⠀⠀⠀⠏⡇⢿⠀⡜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⣼⡿⣏⠰⢈⡟⠛⠋⠁⠀⠀⠀⠈⠓⠀⠈⠹⣿⡟⠷⠀⣶⣿⡿⠿⢿⣿⣿⠍⠀⠀⡄⠀⠻⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠈⢂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢶⡀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⡴⠖⠀⠀⠀⢡⣂⡀⠙⠋⠀⠀⠰⡇⢰⡿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠱⠶⠈⢷⡷⠀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠀⠻⠿⠘⢧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢦⣄⣠⠀⢸⡿⠿⠿⠿⢿⣷⣦⠘⣁⠀⠠⠟⠛⡟⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⢀⣄⣄⣴⣤⣶⣦⣄⠀⠈⠀⠀⠀⢀⢀⡔⠀⠀⣼⣷⣲⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣯⣿⠋⠉⣁⠈⢠⣿⣾⣿⣷⣲⣤⣿⣿⡟⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠑⢤⣄⢠⣿⣻⣿⣿⣯⣿⣷⣯⡷⠆⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⢿⢿⣿⠿⢿⠿⣿⣿⣿⣤⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⠀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢻⠁
⠀⠀⠀⠀⠀⠀⡀⠀⡿⠀⠀⠂⠀⠾⣿⠳⡆⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡄⡜
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠐⠁⠀⠀⠀⠁⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣨⠀⢀⣀⢀⣀⣀⠀⠀⡀⣀⣀⣠⣤⠸⣦⠀⣤⣤⣀⡀⢀⢀⡀
⠀⠀⢤⣤⣤⣤⡄⠲⠶⠆⠸⠷⠶⠶⠿⠘⠟⠂⠘⠛⠛⠉⠉⠁⠈⠙⠛⠙⠛⠉⠀⠉⠀⠉⠙⠛⠛⠉⠛⠋⠀⠁⠁
⠀⠀⠀⠀⠉⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⠉⠉⠉⠉⠉⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠠⠀⠤⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠂⠁⣀⣀⡀⠉⠢⡀⠀⠀⠀⠀⠀⠩⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠉⠀⠀⠏⠱⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠇⠀⠀⠀⠂⠀⠆⠀⠐⠀⠀⠀⠐⠆⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⠀⠀⠀⠀⢰⠀⠀⠃⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠂⠄⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠂⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠷⠦⣀⠋⢙⡂⠲⠤⠚⠋⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡆⠀⣀⠀⠂⠀⢠⠀⠈⠰⠋⠁⠀⠀⠀⠀⠘⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠁⠀⢸⠀⠈⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⡄⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⠀⢁⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠉⠉⠋⠀⠀⠀⠀⠒⠀⠼⡀⠀⡀⢸⠀⠀⠀⡀⠤⠐⠒⠀⠀⠀⠀⠡⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠥⣼⡀⣀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⠣⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠀⠀⠀⠀⠀⠀⠰⠀⠀⢤⣽⣩⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢠⡁⠀⠀⠦⡼⣡⠶⡀⠀⢨⡄⠀⠀⠀⠀⡐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⡀⠀⢀⡤⠀⣀⢢⡀⠀⡤⠁⠘⠄⢽⠒⠀⡏⠒⠥⠔⠁⠈⢄⠀⢀⠊⠀⠀⡇
⠀⠀⠀⠀⡀⠠⠀⠠⠤⠂⠁⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠈⠙⠋⠤⠀⠀⠒⠐⠢⡀⢁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡇
⠀⠀⢠⠊⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⠀⠀⢀⠤⠒⠂⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⢀⢼⡀⢀⠔⠀⠀⠁⠀⠉⠈⠁⠈⢲⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠈⠓⠲⡄⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⡀⠀⠀⠀⠀⠂⠘⠂⠀⠀⠀⠀⣧⠤⠊⠀⠀⠀⠀⠀⠀⠀⡇
⡄⠀⠀⠀⢀⢀⡀⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠒⢤⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠂⠰⠒⠤⡀⠀⠀⠑⢄⠀⠈⠀⠁⠀⣠⠚⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠄⠄⠀⢹⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠑⢄⣀⠀⣀⠠⠤⠤⢀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⢀⠠⠔⠒⠂⠁⠀⠀⠀⢸⠀⢁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⡋⠉⠀⠀⠀⠈⠳⠔⠂⠐⡏⠁⠀⣃⠀⢸⠀⠀⣀⣀⣸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⢸⠀⠀⠈⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠑⠰⠃⠀⠀⠛⠀⢹⡀⠀⠀⠀⢀⣭⣉⡀⢈⡇⢤⣤⣤⣤⡤⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣤⣾⡟⠁⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⢤⣄⣠⣀⣀⡀⠀⢀⣠⣤⣶⣦⢀⣤⣶⡿⠟⢱⠟⣉⣅⠀⠀⢀⣼⡿⠛⠋⠀⣐⠕⠖⠚⠗⠾⢿⣿⣿⣿⡿⠿⠿⠿⠻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠂⠰⠞⠛⠛⠛⠛⠛⠋⠈⠙⠚⠋⠐⠓⠲⠶⡾⠋⠈⠃⠴⠿⠛⠁⠀⠀⠐⠹⠑⠾⠀⣠⣆⡄⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢂⡄⠀⢀⠀⢀⡀⠒⠀⠛⠒⠀⠀⠀⢒⣀⣀⡀⠀⠀⠀⠀⠖⠂⠉⠁⣠⣩⠰⡆⣶⠶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠉⠀⡒⢀⠐⠀⠀⠸⢛⠛⠋⠋⠁⠁⠀⠀⠴⠒⠋⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡃⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠍⠹⠧⠥⠬⠄⣀⡀⢀⣀⢀⣶⣀⣤⢤⡶⣿⢸⣗⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⠋⠁⢨⡇⣿⡨⡛⣼⡦⢿⡆⠀⠀⠀⠀⠀⠀⢀⣄⣂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢹⣿⣿⣿⠽⠉⢿⣷⣿⣇⠖⣿⣇⠸⠷⠀⢀⡀⣀⣤⣤⣴⣿⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠤⠄⢀⣀⡀⢀⠀⣀⣴⣴⣻⣯⣭⣶⣶⣿⣿⣿⣿⣟⣹⢟⢰⣴⣾⣿⣿⣿⣿⠿⢿⢺⠓⠚⠿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠙⣿⠉⣹⠉⠀⣽⠀⣀⡠⠔⠊⣁⡀⠀⠀⡀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⣿⡄⠀⣿⣙⠀⣠⣾⣷⣄⢉⠳⠦⣄⣁⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⡇⠀⣿⠟⠈⠈⠙⠋⠛⠻⢷⡶⠀⠉⠛⣶⡀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣟⠀⣿⡟⠀⠀⠀⠀⠀⠀⡦⢹⡆⠖⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⣤⠤⢤⠤⠿⠤⠌⠁⢀⣿⣿⠄⠠⠤⠄⠀⠴⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠂⠀⠐⠀⣈⠁⠀⣿⣿⡗⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣤⣤⣤⠬⢻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⢀⣠⣤⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠻⠁⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⢀⣀⡈⠁⠉⠀⠀⠀⠀⠀⠂⠄⠀⣰⡖⠛⠣⣸⣿⠟⢀⡴⢦⠀⠤⠄⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡞⣿⡏⢻⠇⠁⠀⣶⣶⣶⣤⣤⣀⣀⡀⠀⠀⢈⣤⣿⣵⣶⣿⣧⠀⠐⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢶⣆⢶⣮⠀⢀⣿⣿⡛⢻⣿⣿⠿⣍⡳⢤⣼⣯⣏⣽⣭⠽⢿⡄⣀⣶⡎⠀⠒⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠤⢀⣸⣛⣨⣏⢠⣭⠁⢸⣿⣿⣄⣸⣀⠠⠰⣿⣾⣿⡏⢁⣻⣿⣶⣷⡰⠛⣉⡄⠀⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠈⣿⠄⣿⠆⣿⠂⣸⣿⣿⣂⣻⣫⣶⡴⢿⣿⡜⣤⣼⡽⢿⣆⢉⡴⠟⠛⠁⡴⠀⠳⡄⢹⣿⣿⠀⢀⡀
⠀⠀⠀⣴⡦⣶⣗⢾⡀⢘⠀⠘⠁⠀⠀⠀⠀⠀⠰⡇⢠⡅⠀⠀⠉⠈⠉⠈⠉⠛⠭⠉⠃⠉⠀⠒⣀⠙⠋⣠⣴⡆⢰⡇⠀⠀⢧⠀⣿⣿⡄⠿⠽⠶⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣷⠀⠇⠀⠀⠀⡀⣄⠂⢴⣲⣶⣶⣝⢳⡄⠀⠙⢷⡾⠋⢁⡄⠘⡇⠀⠀⡟⢀⡿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⣇⣄⠀⠀⠀⠀⠘⢀⣿⠿⠛⠋⢹⣙⣫⣽⣿⠿⠿⣿⣏⢿⠂⢲⣤⣀⣴⠟⢁⠀⢳⣀⣰⠃⣸⠁⢀⣀⣀⣀⡠⠤⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢺⣿⡇⠀⢸⣷⣾⠢⢤⣤⠴⠞⠛⢣⡤⣲⣿⡏⠀⣦⢹⣿⣾⣧⣤⣤⣉⣉⣴⠟⣤⠈⠻⠤⠞⠃⠀⣠⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣩⣿⣿⡶⣿⣿⣿⣶⣶⣶⣶⣿⣿⡗⣚⡽⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣉⣾⢏⣷⣄⣀⣀⣠⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠥⢾⣿⢿⣿⣿⣿⡿⠟⠉⠉⠉⠙⠛⠱⣺⠿⣿⣭⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠾⠯⠟⠋⠋⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⣯⠷⣫⣖⢋⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠈⣿⡆⠀⠀⠀⠀⠀⣸⣿⠀⢨⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⠀⢻⣷⡀⠀⠀⠀⣠⣿⣿⡄⠸⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⠀⠘⡏⠻⢀⣾⣿⠷⠀⠤⣼⣟⡻⢿⣿⡰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡄⠁⢀⣿⡷⠂⠀⠀⠀⣀⣿⣿⣷⣭⡳⣬⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⠇⢀⡾⠋⠀⠀⢠⡶⠿⠿⣿⣿⣿⣿⠿⠎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣴⣾⣶⡏⠀⠀⠉⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠠⠼⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⣶⣶⣦⠀⣿⣿⡆⠀⠀⣠⣤⣌⣿⡗⠀⠀⠀⠀⣼⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣬⠙⣿⡶⠿⣿⠿⡿⣿⣷⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠠⢴⣶⣶⣶⣶⣶⣄⠀⠀⣆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉⠙⢿⣿⣿⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⣄⣿⣦⣤⣀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⠁⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠦⣄⠹⣿⣿⣿⣿⣿⣿⣟⣭⣁⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠈⠻⣿⣷⣀⠀⠀⠈⠛⢾⣿⣿⣿⣿⣿⣿⣿⣟⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣷⣶⣶⣶⣶⡶⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⢤⣶⡶⠶⠀⠀⣤⣤⠄⠀⠀⠀⠈⠈⠓⠄⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⠀⣀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠸⢠⠀⡌⣡⣤⣤⣤⡴⠾⠀⠀⠀⠀⢠⣄⠀⣈⣓⡛⠀⠀⠀⠀⢠⣶⣤⡈⢹⣾⠀⠀⠀⢺⣾⠉⣠⣴⡦⠀⢦⢰⠃⡺⢯⡦⠙⡆
⠀⠀⠙⢧⣀⢁⣈⣉⣹⣿⣿⢶⠁⠀⠀⠀⠿⢿⣿⠿⢿⣷⣴⣶⡀⠀⠀⠙⠿⠿⢾⣿⣆⠀⢀⣾⡿⠾⠿⠞⠁⠀⠀⠀⠈⠷⠉⠀⠈
⠀⠀⠀⠐⢰⡸⣿⣿⣿⣿⣿⠉⠀⠀⠀⣾⣿⣿⣏⠉⠀⠉⣽⣿⣷⣶⡆⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⢤⠀⣠⣾⡆
⠀⠀⢀⡆⢸⡣⣻⣿⣿⣿⣿⣀⡌⠀⠐⠈⠯⠾⢿⣦⣄⣬⣿⣿⡷⠟⠁⠀⠀⠀⠀⠸⠯⡐⡬⡿⠀⢀⣤⡴⠂⣦⣌⠉⠀⠘⠧⠤⠂
⠀⠀⢲⢅⡈⠈⠙⠛⢛⣻⣭⢥⠂⠀⢀⣀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⡶⠃⢀⣫⣾⣿⣿⣭⢿⣧⠀⠀⠀⢀⢀
⠀⠀⠈⠈⠍⢷⣿⣿⣿⣿⠤⠊⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⢄⠀⢀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣽⣿⣼⣟⡄⡿⠀⢨⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣙⣼⠛⢻⣵⣃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⡟⣹⠃⠀⠀⠟⠿⠻⠂
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠇⢞⠄⡀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢻⣿⣄⣾⣿⡿⠄⠀⠀⠀⠀⠀⠀⣼⣱⡟⠀⣠⠾⠁
⠀⠀⢠⡀⠶⢶⣾⣿⣷⡆⢰⣾⣯⠀⠀⠀⠺⡆⠀⠀⠀⠴⠚⣽⡿⢯⣿⡿⣍⠳⠄⠀⠀⠀⠀⠀⠳⠭⠶⠛⣁⣠⡀⠀⠀⢲⠿⠷⣦
⠀⠀⠈⠿⣶⣄⢙⣿⣿⡿⠿⠿⠟⠁⢀⠈⣗⠨⠄⠀⠀⠀⣰⠃⡷⠴⣿⡇⠹⡆⠀⠀⠀⠀⠀⢠⣤⡦⠀⠀⣿⣿⡷⠀⠀⠀⠀⣠⣬⠄
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠻⣴⣿⣦⠞⠀⠀⠐⠁⠀⢹⣩⣿⠇⠀⠈⠀⠀⠀⢀⣸⣟⢧⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣷⣌⣳⡿⠛⠀⠀⠀⢀⣠⡄⢦⣽⣿⠆
⠀⠀⠀⠐⠀⠈⠃⠖⠒⠒⠒⠊⠉⠉⠀⠀⣦⣀⠀⠰⢟⢨⣿⡀⢶⣤⣤⣤⡀⢀⣀⣄⠀⠈⣳⣬⣙⣗⣨⡂⠠⣗⡸⢮⣝⢷⣾⡟
⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⡀⠀⣀⣤⠀⢠⣿⠉⠀⠀⠈⠘⣿⣿⠀⠻⠿⠿⠋⠘⠏⣠⠀⠀⠈⠓⠿⣽⣿⣷⣭⣽⡷⣭⣛⣷⡾⠁
⠀⠀⠀⠀⠈⠙⢿⣯⠀⣿⣿⣿⣿⣯⡀⢾⠏⠀⠀⠀⠀⠀⢀⡀⢀⣤⣀⣀⠀⠲⡀⠋⡀⠀⠀⣀⣀⠙⢿⣿⣿⣷⣿⣿⡿⠋⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠠⡟⠁⡏⠀⠀⠈⢻⠋⢸⠂⠀⠀⠀⠀⠀⠘⡟⠈⠁⠀⠀⠀⠈⢷⠀⠗⠀⠘⢿⣿⠃⠀⠙⡟⠛⠛⣟⠁⠀⠀⠸⠿⠟
⠀⠀⠀⠀⠀⠀⠁⠁⠸⠀⠀⠀⠀⠊⠀⠎⠀⠀⠀⠀⠀⠀⢀⣀⠔⡄⣤⣦⢴⡤⣁⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠳⠈⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣛⡛⣿⣫⣽⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⣻⣧⣶⣍⣽⣿⡟⢿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⢯⣿⣵⣺⣿⣿⣿⣿⣿⣧⡼⢷⣟⡻⡾⣷⡿⣿⣿⡹⢏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣾⢾⣿⣿⢿⣷⣿⣛⣿⣿⣿⣿⣿⣿⣧⣾⣝⣷⡷⣾⣿⣿⣿⣿⣿⠋⣿⣟⢻⣿
⠀⠀⠀⠀⣿⣿⣳⣿⡿⣷⡿⠯⣽⣿⣿⠿⡫⠲⣾⣯⣭⣽⣗⡾⡻⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠻⠻⠿⢳⣷⣿⣿⣿⣿⡵⢶⠊⢀⠀⠙⢿⢿⣿⣿⠟⠁⠀⠉⡟⠿⠿⠿⠟⢿⣯⡿⣿
⠀⠀⠀⠀⠅⠀⠀⠄⢹⠇⡀⠀⠈⠁⢺⢸⠭⡀⠀⠈⣎⣙⠄⡜⣭⣥⠀⠁⣿⣿⠀⠀⢸⠤⠽⠿
⠀⠀⠀⠀⠄⠀⡀⣀⣸⣀⣀⣀⣀⣀⣉⣘⣚⠃⠤⠬⠤⠤⠤⠧⠧⠴⠤⠴⠠⠀⠄⠀⠚⠒⠂
⠀⠀⠀⠀⣦⣤⠤⠤⠤⠤⠤⠤⠬⠥⠤⠤⣤⣁⣀⣀⣒⣂⣀⡀⡄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⢀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠑⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈
⠀⠀⠀⠀⠀⠀⡤⠚⡩⠛⠉⠉⠙⢢⠀⠀⠀⠀⢸⠀⠀⠀⢼⡃⠀⠈⠑⢄⠀⠀⠀⠀⠀⠈⢧⠒
⠀⠀⠀⠀⠀⣸⠀⡘⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⡄⣋⠂⠛⠀⠄⢢⠠⠀⢧⠀⠀⠀⠀⠀⠈⢯
⠀⠀⠀⠀⠀⡗⠀⡷⠶⠆⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⣄⡸⠀⢘⢀⡓⠀⠁⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣤⣗⣾⡥⠜⢴⡂⣤⠄⠃⠱⡒⠚⠛⠋⠡⣍⡝⡏⢨⠉⠢⠈⠀⢸
⠀⠀⠀⠀⠃⠬⠉⠭⠂⠀⠁⠀⠀⠀⠀⠑⠠⠀⠀⠀⠈⠓⢙⡲⣦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠀⠀⠈⠉⠁⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠈⠫⣛⣀⠂⠀⠈⠘⠣⢤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠄⠀⡀⡴⣻⣷⡖⡛⠚⠷⣦⣄⣀⡀⣄⠀⡠⠤⢤⠄⠠⢤⣴⣶⣒⣶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⣘⣸⣿⣿⡆⠈⠀⠀⣮⣕⣁⠒⠐⠀⠀⠀⠀⠀⠀⠁⠁⠾⣫⡥⠀⠀⠉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣬⣿⣳⣿⣟⣡⠃⠀⡀⢀⡺⣗⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⢈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⡍⠄⠀⠀⠁⣴⣿⣾⣅⢷⡀⠀⠀⠀⠀⠀⠀⠰⠻⡿⠂⠀⠀⠀⢀⢨⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣞⡿⣿⡿⣣⡦⠀⠀⠐⣿⣿⣿⣿⣾⠇⠳⠀⠀⠀⠀⠀⠀⠰⢚⡦⠀⠀⠀⠈⣙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡻⣿⡿⣴⢿⣇⠀⠀⠀⢸⣿⣿⣿⣷⢴⣦⡄⢀⠀⢀⡄⠀⠖⣽⠇⠄⠀⠈⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⣿⣷⣾⣃⠀⠀⠠⢾⣿⣿⣿⣿⣿⣟⣿⣶⣶⣇⣴⣤⣾⣡⡶⡔⣰⣺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠄⣿⣯⢀⠀⡠⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣹⣭⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣭⣏⡸⠃⠀⠀⠈⠙⠋⠙⠛⠛⠋⠉⠈⠻⣯⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⢿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡟⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⢤⠶⠶⠷⢦⠞⢡⣿⣿⡶⠶⠂⠀⠀⠀⠀⠠⠴⠴⠴⠶⠰⠤⠶⠎⠺⠿⢿⡿⢺⣿⣷⣶⠖⡴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣦⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠲⠶⣶⣽⣶⣿⣶⣠⣷⣦⣤⣴⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⢀⣀⣠⠄
⠀⠀⢀⢠⡀⠀⠀⢀⠴⢾⣿⣿⣿⣿⣿⣿⣛⣻⣯⣽⡿⡦⣄⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠘⢦⣷⣤⣤⠴⣚⣭⣿⣿⣿⣿⠿⣛⣉⣭⣿⣛⡙⢿⡈⢦⠀⠀⠀⠀⣜⣭⣄⠩⣽⣿⣿⢯⣄⡀
⢠⢶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢟⡽⢛⣭⣭⣶⣾⢷⣷⣺⣇⠀⠀⠀⣽⣟⣿⣧⣻⣿⣿⣷⣜⣧⣀
⢀⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⢣⠊⢩⣿⣿⣿⣦⠀⠀⢻⣿⡀⠀⠀⣿⠋⠈⣿⣿⣿⣿⣯⣿⣿⣿
⢈⡞⣘⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⡁⣰⣿⣿⣿⣿⣺⣷⢤⣿⣿⣇⠀⠀⢿⡦⣾⣻⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⠏⣠⣾⣿⡿⣿⣹⣿⣿⣿⣸⠞⣿⣿⣿⣿⣾⣿⣥⣾⣿⣽⡿⠂⠀⠈⠑⢿⣿⣿⣿⣏⠛⣿⣿⡏
⠻⠛⠛⠛⠿⠟⠟⡴⣟⣿⣿⣿⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⣀⣀⣙⢿⣿⣏⠳⣼⣿⣇
⠀⠀⠀⠀⠀⢀⣴⣿⢿⣫⣿⣿⣿⠟⠛⢛⣿⣾⣶⣾⣿⠛⠁⠀⠀⠀⠀⢰⣫⣿⣿⣿⣷⣽⣿⣶⡿⠿⢿
⠀⠀⠀⠀⠀⢉⣵⣿⣿⣿⣿⡟⠃⢀⣠⠾⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠁⣏⣀
⠀⠀⠀⠀⠠⣻⡿⠋⢞⣿⣿⣃⣤⠄⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢀⢸⣿⡁
⠀⠀⠀⠀⣰⢻⡗⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠹⣶⣷⣤⡀⠀⠀⠀⠀⣀⣤⣴⣿⣄⣠⣿⣾⣿⡅
⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣷⠖⠀⣀⠤⠐⠂⠀⠈⣙⡛⠿⣿⣇⠈⠉⠉⠉⠙⡇⠠⣌⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⠤⡒⠒⠒⠂⢀⡄⢀⣀⣀⠈⠁⠀⠈⠢⣀⠀⠀⣀⠔⠠⢽⣿⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⢡⡞⠁⠀⠀⠀⢸⣴⣻⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⣦⣻⣿⣿⡇
⠀⠀⠀⠀⠀⡟⢸⢿⣿⣷⣿⣴⣿⠂⠀⠀⢸⠛⠜⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⣮⣷⣼⣿⣿⠈⣷
⠀⠀⠀⠀⠀⢿⡘⣆⢻⣻⣿⣿⢧⡎⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⠐⣼⣿⣿⣿⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠙⢮⡂⠻⡟⣿⢸⡄⢻⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡌⣠⡏⣿⠿⢃⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⢾⡌⢦⣷⣷⡀⠀⠀⠀⠀⠀⠀⠐⡊⣠⣤⠞⢹⡿⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠃⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⢠⣧⡿⠋⠀⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠀⢠⡦⠺⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣛⣩⣋⣯⣫⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣴⣾⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠁⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠙⠻⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⢀⣀⡀⠀⠀⣀⣀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠻⣿⣿⣿⣿⣿⡿⣻⠄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⣿⣿⣿⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠿⠋






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⠒⠛⠋⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢏⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡤⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠘⢄⣉⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠶⠆⠀⠀⢰⣶⣄⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠤⠒⠒⢤⡤⠒⠒⠢⢌⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠲⡀⠀⠀⠀⠀⢀⣧⣀⣀⡀
⠀⠀⠀⡞⠁⠀⠀⣀⣠⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⢣⠀⢀⠞⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⡜⠀⡏⠠⠛⠒⠦⣄
⠀⠀⠀⠉⠓⠚⠉⡡⠚⡅⠀⠘⠻⡀⠀⠀⠶⠀⠀⢸⠉⢳⠤⣀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠈⠒⠤⠤⠓⢄⣀⣀⣀⠴⠃⠀⠈⡇⠀⠙⠦⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢦⠴
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⠂
⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢤⠑⡄⠀⠀⠀⠀⠀⠀⠶⡞⠳⠤⠶⠋
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢋⡴⠃⠀⠀⠀⠀⠀⠀⠤⣟
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⣿⠀⠀⠀⠀⢢⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⣙⣡⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⢻⡄⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡤⠞
⠈⠻⣦⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⠖⠃
⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠑⠒⠲⠠⣤⡴⢦⢴⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⣆⠀⢀⣠⠞⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢳⣄⣀⡀⠀⠀⣀⣠⣼⠀⠸⠉⠉⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡏⠉⠉⠁⠀⢸⠀⠀⡇⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸⡀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⡇⢸⠦⠤⡇⠀⠀⠀⠀⢸⠉⠉⡇⠀⡗⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⠇⢸⠀⠀⢸⠀⠀⠀⠀⢸⣀⣀⡿⠀⡇⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⣤⣶⣾⣿⢿⡟⠧⢃⣖⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⣛⡙⣿⣟⠉⠛⢿⣿⣿⡿⣦⡁⠠⣿⣿⡏⢉⠛⠿⣶⣄⠀⠀⠀⠙⠛⢻⣶⣄⠴⢷⡞⣀
⠀⠀⠀⠀⠀⠀⢀⣴⢛⠃⠸⠛⣻⣿⡄⠒⠂⣺⣿⡿⠖⢌⠳⠄⣿⣿⣷⣼⣧⣶⣄⠉⠀⠀⠀⠀⠀⠀⠙⢻⣧⣾⠇⣴⣦⠄⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⢻⢏⣿⣯⣴⠾⢫⣽⣿⡛⢨⣷⡟⣠⣿⣿⣿⡁⢘⢿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣄⠹⢮⣀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⡶⣸⣿⡯⠶⣦⣿⣿⡏⢉⣽⡟⣷⣿⣿⣿⣯⡥⢶⡭⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⣬⣿⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢡⣿⣿⣡⣴⣽⣿⣿⠑⣻⣿⣴⣿⣿⣿⣿⡿⢛⠨⠭⢷⣤⢁⡀⠀⠀⠀⠀⣀⠀⠀⠈⢿⣿⡜⣿⣧⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣯⣾⣿⣿⣏⣼⠿⢁⣿⣿⡿⠿⠛⢀⣠⣤⠶⢦⡀⠈⢳⡀⠀⠀⢰⣿⡆⠀⣀⢸⣿⣧⢸⣿⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⣿⣿⠇⣰⣾⠋⠀⣲⠆⢈⡽⢡⣞⣻⣷⣄⡀⣼⡇⠀⠀⠘⣿⡁⢠⢟⡀⡹⣿⡌⣿⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⠞⣻⣿⣿⣏⣤⣴⣿⣵⣿⠻⢉⡉⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⠿⣧⣸⣿⣇⢈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⢃⣴⣿⣿⣿⢿⣿⡿⢿⡏⠻⣷⣾⣾⣿⣿⠏⠀⠀⠀⠀⠙⣿⣧⣭⣵⣿⣿⣿⡜⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠿⠷⠿⣿⠿⠋⠥⠀⠀⣿⣿⠘⠷⣤⡾⢋⣉⡭⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣽⢿⡇⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡿⠳⠒⠀⢈⣭⡴⠊⠀⣀⣠⡶⠿⠛⠀⠾⠿⠂⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢌⠻⣼⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡌⠀⠤⠞⢋⣠⣴⣿⣶⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠜⣶⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣭⣠⠄⢀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠀⡀⠘⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣰⠏⢀⡔⣡⣤⠀⠀⠐⠄⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣉⡉⢙⡂⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⣺⡿⠁⢖⣿⣿⢟⣡⠆⠀⠀⠀⠓⢄⠀⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⢰⣯⡷⠀⡼⢻⣿⡿⠓⠉⣉⣉⡉⠓⠊⠓⠀⠀⢻⣿⡾⢻⡐⣤⡀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⣿⡿⠁⠆
⠀⠀⠀⠀⠀⠀⢀⡿⢻⠇⠘⢁⣾⡏⠀⣴⣿⠟⠋⠁⠀⠀⠀⠻⣦⡀⠹⣿⡠⣷⡌⠛⢦⣀⣀⣀⡀⡀⠃⢓⣛⣉⣭⠀⣠
⠀⠀⠀⠀⠀⠐⠋⣀⣠⣆⣤⠞⠛⠀⠜⣿⡱⠀⢀⠠⠐⠂⠀⣈⣉⠭⢤⡈⠻⣿⣿⣷⣬⣉⣥⠹⣿⣿⣿⣿⣿⡏⠉⢰⡏⠃
⠀⠀⠀⠀⠀⠀⠈⣾⡿⣩⣧⠇⠀⠀⠰⠃⠁⠈⠀⠀⣠⠔⠉⠁⠀⠀⠀⣭⣦⣈⠙⢻⣿⣿⣙⠷⣌⠛⠿⠛⠋⠀⠀⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠁⢺⣿⣴⣴⣶⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠠⢫⡞⣿⣶⡄⢈⠙⠻⢿⣿⣿⣷⡾⠃⢀⠲⠦⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢸⠉⣿⣧⠀⠀⠶⠲⣶⠦⠀⠀⠀⠀⠀⢑⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⡀⠀⠐⠀⠀⠀⠀⠀⢰⡀⠠⠀⠀⡟⠀⠸⡿⡆⠀⠀⠀⠀⠄⠀⠀⠈⠐⠀⠀⠊⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣰⡆⠀⠀⢰⠀⠀⡇⠀⠀⠈⠃⡀⠈⠙⡀⡀⠀⠐⠤⣄⠀⢀⡀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⣤⢠⠈⠀⠀⠘⠀⠀⢀⠀⠐⠀⠄⢻⡝⠄⢢⠀⡀⠉⠢⡈⡈⠋⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⠇⣄⠀⠀⠀⠀⢸⠀⠑⠀⡘⡆⠁⠀⠀⠄⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣄⣧⣀⣼⡿⠈⡀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣏⡿⣧⠀⠇⢰⢸⡇⠀⠁⠀⡄⠀⠀⠊⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⣿⣿⣿⡿⣿⡳⡄⢀⠸⣾⡿⠢⠆⠰⡗⠀⠀⠀⠤⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠚⠋⢹⣿⣿⣿⠝⣧⣼⣾⣿⣧⡀⠃⠀⡇⣤⠙⠖⣿⣾⢀⣯⢡
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⢻⣬⣿⣿⢿⣦⣭⣿⣿⡿⢿⣿⢀⢀⡷⢻⣎⣖⣉⣀⣀⣇⣠⣀
⠀⠀⠀⠀⠀⠀⢘⠄⠀⠀⠀⣀⠀⠸⠛⠛⢑⣿⣿⣿⣿⣿⣷⠿⢿⣿⡟⠟⢻⡏⠈⣼⡇⠉⣹⣻⠏⠁⢹⡿⠧⠤⠴⠖
⠀⠀⠰⡄⢀⠄⡌⠂⠀⠀⢈⡷⠀⠀⢰⡶⣁⣾⡛⠟⠃⠻⠼⠁⠘⠋⠃⠱⣆⠁⠀⠻⠀⠈⢹⣟⢤⣾⠐⡧⠤⠄⠶⣶
⠀⠀⢀⣷⡎⣴⣇⠀⣄⢠⡟⣠⣤⢔⡿⠃⠛⣛⣧⡴⠃⠄⢦⠀⢦⡄⠀⠀⠉⠀⠐⠘⠀⠂⠐⠀⠀⠃⠀⡷⠴⠃⠐⠂
⣇⠀⢈⣿⠳⣾⣯⣾⡏⣸⡶⠟⠋⣹⡵⠖⠛⣩⠏⠀⣀⠀⢀⠀⠀⠀⠤⠀⠠⠀⢀⣠⠀⡤⢶⠖⢸⠇⠘⡗⠂⠀⠐⢶⣶⠦⢤⡄
⡟⠀⣼⠸⣷⣿⣿⣵⡿⣿⠗⢋⣸⣇⣤⡾⠿⢓⣦⡂⠁⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⠇⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠁
⠺⢀⡷⣠⣿⣾⢣⠟⣤⣾⢒⣿⠿⢋⠝⠀⣀⣾⣏⠉⢀⣈⣠⣶⡿⠤⠍⠒⠒⢀⠀⣿⡿⠀⠼⠀⠀⢀⣰⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢺⣷⣿⣿⣵⣾⠟⣱⣯⣿⣷⠢⣈⣠⣾⣿⣻⣿⡟⢻⣭⣯⣉⣙⣛⣏⡛⠉⣵⣷⡿⠁⠀⠀⠀⠀⠘⠀⠀⠀⢀⠾⡟⢷⣄⢀⣤⣾⣿⣦⣄⡈
⠀⢸⡿⡳⣿⣿⠕⣫⣾⣿⣿⣿⣿⣿⣻⣟⣧⣽⣦⣔⣦⣭⣽⣿⣿⣽⠿⣶⣾⣿⣿⣧⣀⣀⣀⣤⣠⣤⣤⣄⣠⢋⡀⡃⠈⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⢠⣿⢀⣷⡿⣫⣾⢿⣻⡟⢨⠿⠛⢾⣅⣹⣿⣯⣿⠿⢶⣟⡛⠖⠙⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣒⡀⡠⠆⣿⢿⡿⡿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢉⣿⣾⠑⣿⣏⠻⠞⣿⣿⠛⠒⠓⢸⡍⡉⣉⣻⣟⣛⢷⣁⢈⣳⣦⣬⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣛⣿⣻⣳⣿⣾⡓⣗⣿⣾⣾⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⠿⠿⠾⠶⠾⠶⣶⣶⣾⣷⣾⡦⠤⡿⢷⡤⠼⢶⣶⣦⣤⣯⣿⢷⡿⠧⠧⠿⠿⠿⠿⢿⣿⣿⠽⠿⢿⣭⣿⣿⣻⣙⣛⢹⣟⣓⠻⠺⠺⠗⠿⠫
⠀⠀⠈⠙⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠜⠃⠒⠖⢙⢤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠁⠀⠀⢠⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀
⠀⠀⠀⡀⠀⠴⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣧⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣳⣛⣝⣱⣫⣬⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄⢠⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣟⣿⢧⢤⡤⢀⡓⣼⠀⡇⠀⣀⠀⠈⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⢀⣤⣦⣄
⠀⠀⠀⢠⣤⣿⣷⣏⠉⠉⠁⠈⢡⠘⡀⠇⠸⠿⢧⣤⣯⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠦⠴⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠓
⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣟⢛⣛⣠⢻⡤⣾⣀⣠⣤⠀⢠⡀⠀⠀⢸⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⣄⡨⢻⣦⣀⡻⣿⣿⣿⣩⣀⠀⠀⠈⠉⠛⠿⠷⣨⡄⠀⠸⣿
⠀⠠⣀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡶⡀⠲⢧⡁⢘⣧⣁⡨⣿⣿⣋⣰⣶⣤⣀⠀⠄⠀⠀⠀⠀⠀⠀⠛⢀⡀
⠀⠀⠈⠁⠙⠣⠴⢤⣤⣀⡀⣾⠟⠀⠀⢛⢧⠈⣼⣿⣧⣿⡇⣯⣉⠻⣿⣿⣿⣷⣶⣤⣀⡴⠒⠋⠉⣠⡅
⠀⠀⠀⠀⠀⠐⢠⣶⣿⣿⡆⣿⢸⡆⠀⣼⣕⠤⣿⣿⣯⣿⣷⣿⢿⣆⠬⠿⡿⣿⣏⠛⢿⣿⣿⣿⠉⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠹⡟⢀⠀⣿⣾⣷⣿⣷⡴⡔⣖⢧⣘⢿⣇⡀⡏⣿⣿⡆⠈⣿⣿⣯⠀⣿⣷⡶
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣆⠻⠎⣸⣿⣿⣿⣿⣿⣷⣾⣮⣷⠙⡓⠆⠉⢃⡸⢿⡇⡀⣽⣿⣿⠠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘⣷⣶⣹⣁⣼⣟⣿⣠⣭⣼⣿⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⢻⣿⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠉⠛⠉⠉⠀⠉⠻⣿⣿⣿⣿⣿⠀⠓⠘⣿⣿⣿⠉⠉⠀⠈⠻⠿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣠⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠾⠿⠿⠛

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣸⣯⣍⣭⣭⣩⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⡢⡄⠀⢸⡽⣢⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢮⣿⠟⠀⠀⠉⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⣞⢽⡆⠀⢱⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠶⠤⠶⠔⠒⠚⠒⠒⠒⠁⠀⠀⠀⠀⢀⠴⣒⠣⡀⠀⠀⠀⠀⠀⠀⢰⠂⣷⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠣⠵⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⢣
⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⢠⠦⣱⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢶⣳⠀⠀⠀⣴⣒⡄⠀⠀⠀⠀⠀⡸
⠀⠀⢠⠞⡡⢤⡙⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠳⣄⣀⣀⣀⡹⠯⠤⠤⠤⠔⠒⠋
⠀⠀⡎⠀⠑⠾⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⠒⣆⠀⠈⡆⠀⠀⠀⠀⢀⡠⢯⡀⡈⡷⠲⡖⠋⣓⡄
⠀⠀⠘⢄⡀⠀⢀⠤⡲⢄⠀⠀⠀⠀⠀⠙⠮⣥⡠⠞⠀⠀⢀⣠⠊⡉⠂⠙⠁⠉⢁⡀⢉⡀⠳⢻⡖⠲⡄
⠀⠀⠀⠀⠉⠉⠻⣘⠥⠞⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⢰⡅⢌⠉⠱⠤⠀⠈⢡⣀⡀⠀⠠⠄⠈⠁⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⣸⡀⠚⠰⠄⠀⠠⠀⠀⣀⠠⡠⠀⣀⠦⠚⠨⠥⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠇⢰⡀⢠⠶⡄⠖⢺⣼⡆⣤⢨⠉⠧⠴⠇⢰⡓⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠛⢲⡿⠄⢼⣫⡑⠚⠁⠀⠘⠀⠶⣾⣴⢂⠞⢺⡷⠒⠶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠂⠀⢀⡴⢹⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡟⢦⡀⠈⠢⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⠤⠒⠁⠀⡟⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢃⠀⠉⠲⠤⣀⣀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⠚⠱⡀⠀⠀⠀⠀⡖⠋⠉⠉⠑⢲⠀⠀⠀⠀⢀⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣻⠏⢘⠂⠹⣟⠀⠀⠀⠳⣄⠀⠀⣀⠜⠀⠀⠀⢀⢞⠈⢿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⠄⢇⣠⠠⣈⡓⠤⣀⣀⠈⠉⠉⠁⣀⣀⡠⠖⢡⢊⡇⢀⠍⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠖⣞⣳⠷⡀⣏⣠⠀⡀⠀⠰⣀⣨⠉⢩⠉⠉⠉⠐⠂⠈⠀⣏⢱⠈⠙⠏⢹⣢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠻⣀⢰⠙⠛⣀⡉⣎⡇⠀⠙⣀⠠⠂⠢⠀⡀⣉⠁⠢⠀⠈⢁⡀⢈⡧⠒⠃⠊⢰⣂⡼⣡⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣠⣈⠙⠻⣎⢡⡄⡀⢠⠀⡏⣈⢡⢄⠉⡀⠒⠀⠢⡼⠋⠠⢀⠒⢈⣀⠁⣉⠇⠀⠊⡀⢦⣈⣹⡤⠜⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠱⡄⠙⣏⠀⡑⡱⡄⣘⢹⣀⠼⣀⠽⠥⢯⣹⢾⣟⠤⠣⢄⠼⢌⣡⢈⠧⠄⣉⡅⢨⡯⠋⣠⠖⠙⢢⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⢄⣀⡠⠇⣠⣯⡟⠙⣿⠂⣌⢳⡇⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣷⢋⡄⢺⡟⢉⣿⡇⢰⡃⠀⠀⡸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠈⠉⠀⠉⠈⠉⢧⡀⠀⠀⣀⣀⠴⠳⢤⣀⣀⣀⣀⣠⠃⠁⠈⠁⠈⠉⠉⠓⠤⢍⣓⣚⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠊⠉⠓⠶⠊⠉⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠉⠀⠀⠀⠀⡀⣀⣠⣀⡀⠈⠁⠙⠛⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⢀⢤⡴⠒⠒⢠⡴⠋⠉⢡⡴⢦⡉⢧⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣄⠀⠰⣄⣠⠇⠀⣀⣸⣇⣀⡀⠹⣄⣀⣡⠟⠲⣄⠀⠀⠀⠀⠀⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣤⠶⠶⣄⡴⠒⠦⣼⠉⠈⠙⡏⠉⢃⡤⠶⠶⢄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⠀⢀⡼⠁⠀⠀⢸⡆⠻⠶⠃⢠⡟⠀⠀⠀⢀⡷⠋⠉⠙⢦⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣀⡠⠤⠖⠚⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠖⢲⠀⠈⡧⠤⣀⡀⠀⠀⠑⠒⢤⡀
⠀⠀⣰⠋⣡⣴⣶⣿⣿⣿⣿⣿⢦⣀⠀⠀⠀⢀⡴⠋⠲⣤⣀⣠⠴⢷⡀⠀⠀⠈⠛⠖⠋⠀⣸⣇⣀⠀⠉⠙⠲⣄⡀⠀⠙⠓⠒⠦⣄
⠀⠀⡇⣼⣿⣿⣿⣿⣿⣿⡿⠃⢸⠉⠛⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣤⣾⣿⣿⣿⣿⣷⣦⡀⠀⠙⣆⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⡏⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢿⣿⣿⣿⣿⣿⣿⣧⡀⠘⡖⠢⣄⠀⠀⠀⢃
⠀⠀⠀⠀⠈⠉⠉⡴⠚⠁⣰⠃⠀⣠⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⢸⡇⠙⢿⣿⣿⣿⣿⡇⢀⡇⠀⠈⡇⠀⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⢸⡁⠀⢠⠃⠀⠀⠙⠿⠟⠃⠀⢀⣄⣤⡀⠀⠀⢸⣿⣿⡾⠃⠀⠀⠀⠀⠀⣾⡇⠀⠀⠙⠛⠿⠿⠗⠋⠀⠀⠀⠛⠲⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠀⠀⣤⡄⠀⠀⠀⠘⢿⠿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡆⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⡧⠀⠀⠾⣄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠬⠇⠀⠀⠀⢸⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⢀⡠⠢⠄⠀⠀⠀⠀⠀⠀⡰⠴⠋⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⣤⣤⣤⣤⣤⣴⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⢰⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣠⠃⠀⠀⠉⠑⡄⠀⠀⠀⠀⣞
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⢀⡄⠀⢀⣀⡠⠄⠭⣑⠒⠚⠁⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠈⠹⢤⣄⣠⠞⠁⢀⣀⣀⢀⠀⠀⠘⠀⠒⠤⠋⠀⠀⡋⠈⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⣀⡴⠃⠀⠀⠠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠇⠀⢀⡀⣄⣠⠇⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣴⣀⣀⡀⠀⠀⠀⠈⠢⠤⠴⠤⣠⠜⠀⠀⠀⡇⠘⡄⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠙⠶⠚⠛⠋⠀⠉⠀⠀⠀⠀⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠇⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠞⠙⢒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠓⠒⢲⡤⠤⠦⣄⣀⠼⢛⠒⠒⠚⢻⠆⠤⠐⠲⠦⠖⢫⡏⠉⠉⡅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⣸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢠⣶⣶⣾⠀⠀⠀⣧⣤⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⡾⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⡂⠁⠀⠀⠈⣱⠊⠉⠀⠀⢹⠏⠆⠁⠀⢈⡀⠠⠄⡀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠹⠖⣦⡀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⠀⠀⠋⠃⠀⢀⣽⠊⣐⣦⣀⣾⠈⠃⣀⡀⠀⡈⢡⣈⡷⠖⠈⠑⡆
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠆⠢⠀⠀⠀⠀⠀⠀⠁⣞⠁⠀⠀⣈⠟⠉⠀⠈⠙⢶⠉⠀⠈⠰⠖⠒⠁
⠀⢀⠠⠤⣴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⢹⠀⠀⠠⠄⠀⢘⠲⠀⠀⠀⠃
⠠⠆⠀⠀⡝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢨⠦⣀⠀⢀⡠⠊⠂⠤⠀⠘⢆
⢣⠀⠀⠀⢣⡤⠀⠀⡐⣡⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣌⡆⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠑⠄⢀⣠⠘⠀⠀⢰⠀⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱
⠀⠀⠀⢻⡤⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⢲⡀⠔⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠄⠀⠀⠀⠜⠁⣶⠂⠀⠀⠀⠀⠀⠧⠤⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⢳⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠻⠄⠀⠸⡀⠀⠀⠀⠓⠂⠀⠰⡖⠉⠀⢀⡞
⠀⠀⠀⠀⠀⠉⠻⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠘⢠⡶⠀⠙⣦⡀⠀⠀⠀⠀⢠⣁⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⢴⢧⡀⠀⠀⣠⠦⡔⠁⠠⠀⠀⠀⠖⡄⠀⠀⣆⣀⣠⠟⠖⠒⠃⠉⠓⠈⠀⠈⠑⠒⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠈⠏⠋⠀⠀⠈⠐⠛⠣⠤⠤⠜⠳⡶⡎⠁⠹⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢑⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡇⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡷⣵⠀⢱⡏⠷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⢼⡀⠘⣜⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⠿⣿⣽⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⢛⠛⠫⠌⠙⠛⠓⠚⠛⠋⠤⢟⣛⠻⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣻⠿⡫⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠭⣛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⢟⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢝⢿⣿⣿⣿⡄⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⡶⡡⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣉⣭⡤⣶⣾⣿⣽⣶⣿⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣧⢛⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣶⣺⣿⣿⡿⠿⠟⣿⠉⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢴⣶⣾⣿⣽⣿⠿⠿⣿⠉⣿⣀⠀⢰⣆⠘⠀⢿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⡸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣚⣿⣿⣿⣿⠟⠛⢏⠙⡇⢸⡄⠀⢻⡆⢸⣿⣇⠘⣿⣆⠀⢸⣿⣯⣿⣿⣟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠇⠃⢀⣀⣤⣴⣖⣺⣿⣿⣿⡿⠟⠻⣿⣿⠀⢿⣿⠀⣦⠘⣆⠀⠈⡇⠀⡀⢷⠀⢿⣿⡀⢹⣿⡆⠸⣿⣿⣿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⢀⣀⣭⣤⣶⣺⣿⣽⣷⣾⠟⠻⣯⠉⠳⠈⡇⠰⡆⠸⣿⡇⠘⣿⡇⠸⡄⢹⣆⠀⣿⠀⢁⠈⠆⠘⠋⢧⣈⣿⣿⣴⡿⣿⣻⣯⠯⠷⠛
⠀⠠⡖⣲⣿⣽⣶⣿⠟⠛⠻⡍⢀⡈⢻⡆⠀⢻⡆⠀⠀⢹⡀⢿⠀⢿⣷⡀⢻⣷⠀⠷⠀⣿⡄⢹⣆⣽⣦⣴⡶⣿⣿⣿⠭⠟⢚⣋⡉
⠀⠀⢻⣿⣿⣿⣿⣿⡄⠰⠀⢳⠈⠇⢸⡇⠀⡀⢷⠀⡀⠀⢇⠘⠇⢸⣿⣇⠀⣀⣱⣤⣴⣿⣿⣻⣿⠽⠟⠛⠋⠉⠁⠀⡀⣾⣯⣿⡇
⠀⠀⠘⣟⢿⣿⣿⣿⣇⠀⣀⠘⡆⠠⡄⢻⠀⣀⠈⠇⠸⡀⣘⣠⣤⣾⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢣⣼⣿⣿⠁
⠀⠀⠀⢹⣻⣿⣿⣿⣿⡆⠸⠂⢹⠀⢷⣈⣠⣽⣷⢾⣿⣿⣿⠿⠉⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣿⣿⣿⠏
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣤⣴⣿⢿⣿⣿⠿⠗⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⣿⣿⡟
⠀⠀⠀⠀⢸⣿⣿⣻⣿⠿⠿⠛⣋⣩⣅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣡⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠙⢿⣿⣿⣷⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣣⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣮⣕⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢔⣊⣵⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣶⣶⣭⣥⣖⣒⣒⣒⣒⣢⣭⣭⣶⣾⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠄⠒⢩⡥⠤⠀⠋⠀⠀⠀⠀⠈⠑⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⠉⠲⢤
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠐⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠙⠇⣸⡇⠀⠑⢠⣄
⠀⠀⠀⠀⠀⢀⡼⡃⠈⠀⣰⠗⠀⠀⣄⠐⡄⢶⢐⠀⢘⣀⡀⠐⠂⢀⠘⠈⡀⢸⣾⣿⣿⣅
⠀⠀⠀⠀⠀⢿⣿⡏⠚⢠⡏⠀⠬⣟⣿⣦⠘⠀⠈⠀⠘⣟⢯⠗⠀⣀⢳⣠⡔⢺⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢻⠟⣾⢨⣧⠀⠱⣦⠁⠋⡀⠀⠀⠀⠀⠋⠈⣃⣀⠙⠺⢳⣿⡆⠉⣿⡟
⠀⠀⠀⠀⠀⠀⣸⡎⣿⣸⣧⠀⣠⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠾⠀⢸⡏⣠⣼⡿⢧⡀
⠀⠀⠀⠀⠀⣴⠋⠑⡾⢹⠏⢰⡏⠀⠀⠣⢶⠶⠶⡄⠀⠀⠀⠀⠀⣠⠀⢺⠀⢋⠔⠑⢼⣇
⠀⠀⠀⠀⠀⣇⠀⠀⠘⡟⠀⢻⢆⠀⠀⣚⣠⣔⠖⠃⠀⠀⠀⠀⠰⡇⠀⠘⢷⠏⠀⠀⠀⣻⣦
⠀⠀⠀⠀⠀⢻⡶⠀⢠⣧⠆⠉⣨⡄⠀⠈⣭⣭⣄⠐⠀⠀⠀⠀⠃⢠⢀⣠⣾⠃⠀⠀⠀⠙⣿
⠀⠀⠀⠀⣠⣟⠁⠀⠨⠿⠟⠛⠉⠀⠻⠿⠕⠀⠈⠙⠶⠨⠂⠐⢳⣽⣿⣿⠏⠰⡄⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⡵⡀⠀⠀⣴⣢⣀⠀⢀⠀⠀⠀⠙⢿⡇⠀⠀⢠⡀⠀⠉⠈⠙⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⣿⣿⠀⣀⠀⠛⠉⣀⣴⠅⠀⠄⢀⢄⡼⠃⠀⠀⢠⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⣜⢧
⠀⠀⠀⢰⡟⣿⣠⡇⠀⢀⠞⢩⡦⠐⠁⡴⣱⢋⣴⠂⠀⠀⡜⠃⠀⠀⠀⠀⡑⣄⡧⠀⠀⣾⢮⣻⡇
⠀⠀⠀⢸⣷⠘⠋⣿⠀⠎⢠⣏⠎⠀⠸⡷⠟⠁⢺⡄⠀⠸⠁⠀⠀⠀⠀⠀⣧⣿⠇⠀⠀⠘⣷⣿⡇
⠀⠀⠀⢹⣿⣠⣆⠈⠢⡀⢸⠘⠀⠀⠀⠱⡀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⡆⣿⢿⡃
⠀⠀⠀⢸⡇⠙⠙⢦⡀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣶⢟⡿⠃⠀⣀⠀⣰⢷⡿⠈⡷
⠀⠀⠀⢻⣿⢀⣆⡀⠱⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠁⠀⢀⣴⣧⣾⡷⠋⠀⣼⠃
⠀⠀⠀⠀⢿⣾⣿⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⡀⠀⢀⠞⠁⠀⠀⣠⣦⣶⠿⠟⠋⠁⢀⣿⡼⠃
⠀⠀⠀⠀⠀⠘⢿⣦⡾⢄⡀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⣀⣤⠾⠛⠁⠀⣀⣤⠞⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⠞⠉⠀⠀⣠⣾⠾⠋⢡⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠊⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠚⠉⠉⠉⠓⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⣠⠖⠊⠉⠁⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠙⢦⣀
⠘⢦⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣀⠔⠊⠀⠀⠀⢰⠀⠀⠀⠙⠾⠃⠀⠀⠀⠀⢈⣷
⠀⠀⠙⠖⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠞
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣀⣠⠴⠋⠁⠀⠀⠳⣄⣀⣀⣀⣀⣀⡤⠤⠔
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣠⠞⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⡄⠀⠀⠀⢀⣠⡤⠶⠛⠁⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀
⠀⠀⠀⠃⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠙⢄⡀⠈⠑⢢⠀⠀⠀⠀⠀⢀⠆⠀⠀⢀⠀⠀⠀⠈⣆
⠀⠀⠀⠸⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⡄⠀⠀⠀⢸⠒⠒⠚⢏⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠱⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠈⢦⠀⠀⠸⡇⠀⠀⠀⠑⢦⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢱⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠶⣷⡀⠀⢳⡀⠀⠀⠀⠀⠈⡉⠀⠀⡾
⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⠋⠙⣄⠀⠹⣄⠀⠀⠀⢀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠙⡖⠉⣙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢧⡀⢠⠞⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣹⣎⠈⠑⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⠁⠈⡷⠓⠋

⠀⠀⠀⠀⠀⠀⢀⣀⡐⣀⡀⢀
⠀⠀⠀⠀⠀⠈⠟⠁⠉⣦⣾⠿⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠨⠔⠀⠐⢶⣤⡀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠐⡛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢧⡀⠀⠀⠀⠀⠀⣨⡇⣀⠀⢀⡄⣀⠀⣠⣀⣹⣋⠦⠀⠀⠀⢤⣳⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡥⠀⠀⠀⠀⣿⣽⣯⣽⣮⣿⣿⣿⣼⣿⣿⣶⣿⣿⣄⠀⠀⢐⣲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠀⠀⢰⣾⣿⠉⠛⠛⠛⠿⠿⠿⣿⣿⡿⠋⠙⠻⣿⣦⡀⠘⢿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⠀⠑⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⠁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡠⣀⢄⣤⣠⣀⣄⣀⢠⣶⠧⣶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠀⢀⣤⣶⣥⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠋⠉⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠁⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⡄⠀⠀⠀⠀⠀⡴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢹⡀⠀⠀⠀⢰⣧⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣾⠔⠀⠀⠰⢾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠃⠀⠀⠶⠀⠀⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣶⣤⣾⣷⠀⣾⣧⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡆⢾⣿⣃⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣟⣻⣿⡿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⣀⠹⣷⣿⣿⣿⡟⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣳⡝⣆⠘⣇⠉⢀⡖⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠇⠈⠀⠉⠉⠻⠙⠋⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣆⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⡉⠛⠿⣾⣦⣤⣄⡀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠈⠛⣧⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⡿⠀⢰⡇⠀⠀⠀⠀⠀⣠⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠉⠀⠀⠀⠀⠀⣶⢋⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠛⠳⣄⡀⠀⣆⡀⠀⠀⠀⣿⠟⠹⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠆⢈⣿⢷⠟⠋⢿⡄⠀⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠆⠀⡀⠀⠀⠀⢀⣎⣴⠟⢃⡀⡠⠀⠺⣧⠀⠀⠈⠆⠀⠡
⠀⠀⠀⠀⠀⠰⠈⡁⣠⣄⣠⣶⣀⣀⠿⠃⠿⢿⣷⠞⠋⠁⠀⠀⠀⠈⢣⠀⠀⠰⠀⠀⠠⡀⠀⠸⡀
⠀⢀⣠⣤⣶⡾⠋⠙⣿⠉⠉⠉⠋⠀⠀⠈⢂⠀⣀⣳⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠡⠀⠀⢷⡄⠀⠀⢀
⠶⣿⣿⡿⣿⠋⣠⡪⠀⢠⡂⣀⠀⣀⣦⠄⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢀⠀⠙⢦⣤⠀⠁
⠀⠀⠉⠙⠛⠛⠛⠚⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣌⢦⡄⠀⠀⠉⠑⠒⠲⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⠻⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣉⣉⡉⠉⠉⠉⠉⠉⢉⣉⠉⠉⠉⠉⠉⠁⠀⢸⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠱⣬⡀⠴⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠠⠴⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠿⠋⠊⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⣦⠀⠀⠠⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣤⡄⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣤⡤⢈⡁⠀⣘⠠⠘⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠈⠛⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⣾⣷⣦⣬⡙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠿⣿⡿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢤⣅⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠈⣀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢂⣄⡐⢸⡆⢶⣿⡖⠀⢸⡄
⠀⠀⠀⠀⢀⢀⠀⡀⣀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣇⠀⠀⠀⠈⠐⠠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠛⠁⠘⣇⠈⠉⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣭⣽⠁⣀⣬⣭⡍⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠋⠉⠘⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⡴⠇⠀⠚⠖
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡶⣄⡞
⠀⠀⠀⠘⠛⢈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⡿⠀⠁
⠀⠀⠀⠀⠀⠀⢧⣶⣶⡆⠀⠀⠀⣠⣬⣗⣀⠀⠀⠀⠂⠀⠤⠄⠀⠀⠀⣀⠤⠒⠉⣿⣿⠏⢹⢳⡀
⠀⠀⠀⠀⠙⠒⠒⠛⠛⠃⢀⡠⠀⣸⣄⣉⣈⣷⣟⣒⣆⢀⠄⢂⣴⢖⠋⠁⠒⢴⢺⡿⠁⢀⡗⠂⣷⠤⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢫⣿⣀⠤⢼⡇⣿⠋⠸⡯⠿⠽⢾⣿⣿⡖⠒⠚⢁⣄⣀⣿⠃⣠⠚⠁⢸⣿⠀⢠⣼⣿⣦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣹⣾⣿⣧⣶⣤⣧⣤⣴⣁⣇⡈⣇⠀⠀⠀⠀⢸⢃⣴⡏⡀⠀⢃⣿⡄⢠⡏⠉⢹⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⣼⢏⣿⢻⢹⣿⠛⢿⠻⠿⡿⡿⡿⣿⣿⣷⣶⣾⣾⣼⣿⣇⣜⣺⣗⣽⣾⣗⣤⣼⡷⢲⣶⡀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⢷⡞⠈⠋⣿⣀⣌⣧⣖⣞⣎⣆⣆⠇⡇⡇⡏⢹⢸⠉⠏⡟⢻⢻⣿⣿⣿⣿⣿⣟⣽⣿⣏⣀⣀⡀
⠀⠀⠀⠀⠉⠛⠿⠟⠿⣿⣿⡟⠒⠒⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣻⡳⠹⡽⣞⣦⣇⣧⢸⣸⡏⠁⠀⢺⠓⠻⡖⠊⠽⠷⠭⠄⠉
⠀⠀⠀⠀⠀⠀⠉⢁⣐⣶⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠈⠉⠉⠛⠛⠛⠿⠿⣾⣬⣜⣌⣎⢋⢷⡈⠀⠘⠀⠀⠱⠄
⠀⠀⠀⠀⢀⣤⣽⣿⣿⣿⣿⣿⣷⣶⣶⣾⣶⣶⣶⣶⣦⠷⠶⠶⠦⠤⣤⣤⣤⣤⣤⣉⣉⣛⣛⣻⣦⠇⢀⣀⣀⣠
⠀⠀⠀⠀⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣖⣙⣂⣂⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⢆⣉⡀⠀⢉⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⡇⠹⢯⡿⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠀⠆⠔⠲⠄⠤⠤⠤⠤⠦⠠⠤⠤⠴⠾⠋⠀⠳⡄⠀⠀⢄⠈⠀⠈⢂
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣐⣴⣴⠧⠀⠨⠍
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⣾⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⡇
⠀⠀⠀⠀⠀⣷⠀⠀⢀⡀⠀⠀⢀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠨⣽⠇
⠀⠀⠀⠀⠀⣿⠅⠀⠘⣷⠀⠀⡮⠔⣐⠀⡀⠐⡄⡆⠀⠀⠀⠄⠀⡄⡀⠀⠀⠐⠀⢉⣿
⠀⠀⠀⠀⠀⢻⣿⠄⠀⣿⣷⣿⣺⣤⣽⣷⡿⣿⣿⡧⡀⠀⠀⡰⣞⣷⣿⣱⣕⣆⣤⡼⠃
⠀⠀⠀⠀⠀⢸⡇⠷⢂⣻⡿⣿⣿⣿⣿⣿⣿⠿⠿⠷⣊⠡⠔⣒⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠸⣧⣼⣾⠟⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⢹⣿⣷⡟⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⢸⡷⡇⠀⠀⠀⠀⠀⠀⠀⠧⣾⠃⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⢸⠸⡀⠀⠀⠀⣇⢁⠀⠀⠀⠀⠀⠀⠀⡰⠘⠀⠀⠀⠀⠸⠃⡄
⠀⠀⠀⠀⠀⠀⠸⣦⠓⠀⠀⠀⢾⣌⠀⠀⠀⠀⠀⠀⠀⣅⢠⠀⠀⠀⠀⠰⡆⠡
⠀⠀⠀⠀⣀⣀⣤⣽⣿⣦⣄⢠⣼⣿⣷⢤⡶⠾⠿⢶⣦⣽⣿⣦⣶⢴⣶⣶⣽⣿⣧⠶⡴⢦⣠⡄
⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠒⣀⢀
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣀⠃⠀⠀⠀⠀⠄⠀⠀⠈⠐⠂⠁⠀⠀⠂⠀⠐⡀⠀⠀⠀⢁⠴⠊⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠳⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⡠⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠁⠀⢾⠟⣩⠙⣆⠀⠀⠀⠀⠀⠀⢀⣿⣀⣌⢛⠀⣁⡀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠢⡀⣿⣿⢿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⠜⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠐⣄⠀⠀⠀⠀⠀⠀⣀⢩⢁⡀⠀⠀⠀⠀⡠⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠤⣀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠄⠚⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠢⠤⡀⠈⠑⠂⠀⠀⠀⠀⠉⠠⠄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢙⠀⠀⠈⠉⠈⠀⠀⠓⠤⠔⠂⠂⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠈⠡⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠜⠦⠀⠀⠀⠀⠈⠉⠀⠒⠂⠁⠒⠁⠈⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡇⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⡀⠖⢀⡉⠀⠀⠀⡜⠁⢇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠆⠀⠀⠀⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡃⠀⠈⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠱⣄⠀⠀⠢⣸⠀⠀⠀⠀⡆⠀⠀⠀⠃⠀⠀⠀⢰⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⡆⠀⠀⠀⠀⡇⠠⠤⠤⡖⠀⠀⠀⠀⡄⠡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⠁⠀⠀⠀⢰⠀⠀⠀⠀⢙⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⢀⡸⠀⠀⠀⠀⠈⠀⠂⠤⠔⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂


⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣀⠤⠂⠊
⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡤⠋⠀⢀⠴⠃
⠀⠀⠹⠘⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⡀⠀⠀⠀⠀⡀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠒⠒⢦⡄⠷⡀⠀⠀⠀⡸⣿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁
⠀⠀⠀⠙⢷⣄⠀⠀⢸⣿⡏⠀⠙⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠙⣷⣾⣿⣹⠃⠀⠘⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡅⠀⢀⣴⡆⠀⠛⠛⢧⡀⠱⣄⠀⠀⠀⢀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⡃⠀⢊⡥⠐⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢰⣿⠟⠉⠀⠀⠈⢹⣿⠄⠀⠀⠀⠉⠀⣠⠀⠀⠀⠀⠋⠘⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢡⠖⠉⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠻⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠺⡷
⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⣸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣰⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠉⠉⡛⠛⠲⠶⢤⣄⣀⡀
⠀⠀⣰⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠠⠄⣀⠉⠛⠛⠳⠾⠷⠶⠒⠒
⠀⣰⣿⣿⣿⣍⢀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣴⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣤⣤⣀⡀
⠿⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿
⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣶⣶⣶⢤⠀⠀⣤⣶⡶⣶⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⢠⣿⡟⢣⣬⡊⠀⠀⠀⠀⢸⠇⣈⢹⣿⣇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣗⠀⠀⣿⣿⠁⠈⠉⠑⠀⠀⠀⠀⠂⠈⠛⠈⢿⣿⠈⠁⣸⣷⠀⡀
⠀⠀⠀⠀⠀⠀⣠⣽⣿⣷⡴⢂⣿⡿⠂⢀⣶⣾⣆⠀⠀⣠⣯⣤⠀⠰⢾⣿⡔⣮⣻⣿⢷⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣽⣿⣿⠙⣿⡁⣼⣿⣧⣬⣿⡄⢰⣿⣥⣤⣿⣶⢈⣻⡛⣿⣿⣮⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⢸⣿⣿⡟⢟⣿⣿⣿⠄⠠⣿⣿⣿⣿⢻⣿⣿⡇⣿⡟⣟⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⢿⡽⡿⣾⢿⣿⣌⡛⣻⠋⠁⠀⠀⠈⠙⣿⠛⣡⣿⡿⣷⢻⢿⡇⢭⣿⡏
⠀⠀⠀⠀⠀⠸⣿⣿⢯⢰⣿⣿⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⣿⣿⡣⢱⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣷⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢫⠅⣿⣿⣿⣿⣿⠀⠀⢀⣀⠘⠋⣀⡀⠀⠀⣿⣿⣿⣿⣿⠇⠙⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠙⢿⣿⣿⠀⠀⠘⠿⣿⣿⠿⠇⠀⠀⣻⣿⡿⠋⣽⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⠇⢸⣿⣿⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⣿⣿⡇⠸⣿⣿⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡇⠀⠀⠈⢻⡆⠀⢀⣾⣭⣬⢇⡀⠀⢰⡿⠋⠀⠁⠈⣿⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠅⠀⠀⡤⠠⠻⣾⣿⣝⣒⣒⣫⣿⣷⠟⠃⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠜⢻⡼⠀⠀⠀⠀⠀⠃⠉⠙⠋⠙⠋⠉⠀⠀⠀⠀⠀⠀⢤⡙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⠔⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠢⡅⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡆⡀⠀⠀⠄⢨⡲⣖⠁⠠⠆⠀⢀⣰⡿⠊⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠇⠀⠁⠀⠀⠱⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦
⠀⠀⢺⣿⣿⣟⣠⣶⢿⢷⣦⣄⠀⠀⠀⠐⢚⣹⣿⢿⣿⡿⣿⣧⣭⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠄⢾⡗
⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠐⠉⢉⡀⠘⢹⡎⠛⡂⠀⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⢇⣾⣿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠈⠳⠀⣘⣏⠀⠙⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⡷⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠲⠆⠁⠀⠁⠀⠐⠇⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠄⠈⢃⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠿⢉⡋⣻⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⡗⢦⡤⠞⢹
⠀⠀⢸⠀⣾⣻⣟⣻⡆⢶⠤⠀⠀⠀⠀⠀⠀⠀⡽⣧⣌⣡⣿⡛⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⡁⠀⢀⣴⡆
⠀⠀⠘⠀⠹⣿⣿⡯⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢷⡸⣿⠅⠢⢤
⠀⠀⢨⡀⢰⣿⣿⣷⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⢃⣁⣴⠆
⠀⠀⠀⡇⢸⣿⢹⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢎⢱⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣉⡉⢽⣿⠟⠁
⠀⠀⢴⣷⣼⡿⢸⣿⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⢤⣿⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠙⢦⣤⡶⠒⡆
⠀⠀⠘⠛⢹⡇⠈⣏⠙⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡱⣎⠀⠱⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠈⢢⡇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢀⣭⣄⣤⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠚⢻⠙⠸⠻⠻⠻⢻⠳
⠀⠐⣦⣤⣽⣿⣟⣿⣿⣤⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⢀⣿⣟⣿⣿⢿⠀⠀⠀⠀⠀⣀⡘⢱⠀⠉⠸⠈⠈⢱
⢜⣱⣿⣿⣿⣿⣿⣿⣿⣿⣧⣱⠄⠀⠀⠀⠉⠓⢤⣴⣿⣿⣿⣿⣿⣾⣿⡂⠀⠀⠀⠉⠀⢸⣀⡤⣿⣧⣄⣼
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⢻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣭⣗⣶⢶⡖⣶⣪⡅
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠋⢳⣿⣿⣷⠀⠀⠀⠀⠀⠀⢾⣵⣿⣽⣿⣯⣿⠇
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠉⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠳⠿⣾⣽⡮⡷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⣄⣀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣀⣀⣠⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⣉⣉⡉⠉⣻⠏⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠹⣟⠉⢉⣉⣉⡙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠉⢙⡟⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢻⡋⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⢼⡇⠀⣠⢤⡀⠀⠘⣿⣿⣿⣿⠁⠀⢀⡠⣄⠀⢸⡧⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣄⣀⣼⠀⠀⣿⣿⣿⣿⠀⠀⣧⣀⣠⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠛⠛⠋⠀⢠⣿⣿⣿⣿⡀⠀⠙⠛⠛⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢠⠞⠛⠛⠛⠛⠛⠛⠳⡄⠀⠀⠀⣼⢃⣤⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣦⣄⣸⣦⣀⣛⣀⣀⣛⣀⣴⣇⣤⠴⢾⡁⣞⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⣥⣬⠽⠟⠋⠁⠀⠀⢸⣷⣿⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣀⣴⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠉⠉⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠻⠿⠿⠟⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡏⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠙⢿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⠀⠠⠤⣄⣀⡀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢫⣦⡞⠁⠀⢠⣾⡀⠀⠀⠹⣶⣨⡗⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⣇⢸⡟⠀⠀⣤⠈⡟⠀⣤⡄⠀⠘⣿⡇⣸⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⢸⣧⡴⢿⣿⠀⠁⠀⣿⡷⢤⣠⣼⠁⡟⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣇⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣳⢠⡄⢀⡤⢦⡀⠀⢀⡆⢰⢧⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢺⡅⠘⣿⣷⠇⢀⣼⡷⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣦⡹⣶⣿⣿⠦⣮⡟⢀⣴⡏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⢓⠿⣈⣁⣸⡇⠘⡏⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⡈⡋⣋⡂⠀⢀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢖⠀⠐⠻⣄⠀⠉⠁⣰⠒⠀⢠⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠈⠳⡆⠀⠀⠀⠈⠈⠀⠀⣸⠟⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠳⡄⠀⠀⠀⢀⠜⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠉⠳⠴⠞⠁⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠩⣿⠀⡇⠀⢠⠀⠀⠀⡀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢒⣿⠀⢙⣷⡘⡇⠀⢰⢁⡜⠀⢀⣿⡯⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡆⠈⢻⣷⠀⠒⠀⣾⠁⠀⢸⠉⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⣇⠀⠀⠘⣄⠀⣰⠏⠀⠀⣼⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣽⠀⠀⢠⣿⡶⣿⠀⠀⠀⣧⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⣿⠀⠀⢸⣧⣀⣹⡇⠀⠀⣿⠁⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⢿⠀⠀⢸⠟⠛⠛⠇⠀⠀⡻⢄⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠁⠀⠀⠀⠈⠒⠀⠊
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣤⣤⣄⣤⣠⣇⣀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠷⠶⠶⠆⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡇⠀⡇⠀⡇⠀⡧⣄⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠦⠤⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠃⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠶⠶⠶⠶⠶⠶⡄⠀⠀⣠⠤⠤⡤⠤⠇⠀⠀⠀⠀⠀⠀⣸⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡷⠒⠊⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡤⠖⢺⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⢰⠛⠛⠛⠛⠉⠀⠀⠘⠓⠒⠒⠒⠒⠒⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⢰⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⢯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⢯
⠀⠀⠀⠀⠠⠤⠤⠤⠬⠽⣀⣸⣀⣸⣀⣸⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠐⠒⠒⢲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋⠉⠉⠉⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⡆⠀⡎⠐⢰⠀⡆⢐⡆⢰⢐⠐⠀⢀⠀⠀⢎⠀⠀⠀⣁⢢⠂⢠⢠⡔⢰
⣤⣤⣤⣤⣤⣤⣴⣦⣶⣮⣼⣤⣧⣧⣤⣼⣾⣴⣶⣶⣦⣤⣶⣥⣶⣴⣤⣦⣶⣦⣴⣼⣶⣶⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⡄⠀⠀⡤⠤⠄⢀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠐⠀⠀⠀⠀⠀⠷⠄⠄⠉⢀⠠⠀⢈⣄⠈
⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⡂⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⡈⠆⠜⠆⡨⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⣰⡛⢻⣳⢛⣡⠀⠀⠀⠘⠒⢲⣀⠁⠠⠞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⣾⣄⣀⣈⣣⠀⠀⠴⠥⠿⠺⠚⠒⠓⠆⠀⢰⡈⠉⠈⠙⢄⡀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⣆⡀⠀⢄⠄
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡃⠈⠃⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⠤⠀⠈⠀⠠⠉⠀⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⠀⠂⠄⠃⠁⠀⠀⠈⠁⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡄⢡⠀⡆⠀⠤⠄⠀⠰⠆⠁⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠔⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠈⢊⠺
⠀⠀⢠⠀⠐⠃⠀⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠀⣤⠦⠀⠐⠀⠀⠀⠀⠀⠀⠠⠤⠖⠂⠀⠀⠈⠀⠐⠐
⠀⢀⠰⠤⠘⠀⠀⠄⠀⠐⠁⠀⠀⠀⢰⠀⢀⣲⡋⠉⠀⠀⠀⣠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣀⠉⢀⡀⢀⡔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣈⣀⣀⣉⣈⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠲⠋⠀⠋⠀⠀⠡⠔⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠁⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⢠⠄⠃⢐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠒⢦
⠀⠀⠀⠀⠀⠀⡇⠀⣄⣀⣀⣀⡀⠐⠂⠀⠀⠀⠠⠀⠀⠐⠀⠀⢀⡁
⠀⠀⠀⠀⠀⣀⡵⠖⠉⠀⠀⡀⠀⠦⣑⣀⠠⢆⠀⠀⠀⠈⠀⡆⠀⢱
⠀⠀⢰⠊⠉⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠤⠉⠉⠑⠒⠺⡁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠈⠙⠒⡏⠀⠀⠀⠀⢿⡿⠀⠀⠀⣰⣾⠄⠀⠀⠀⠀⠀⠪⠛⢥⠂⠐⠒⠶⠤⠒⣒⠒⠪⠍⠉⠒⢊⡁⠀⠀⠉⠂⠐⠒⠒⠒⠦⢄
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢸⠀⠀⠰⢦⠤⢤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠂⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠤⠀⡠⠀⠀⠀⠀⠀⡸⠀⢀⠠⠎⠀⠈⠉⠈⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠒⠲⢄⠀⠀⠀⠈⢙⣆⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢄⣂⠀⠀⠀⠄⡀⠀⠀⠀⠀⡰⠃⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠘⠓⠒⢄⠀⠀⠀⠈⢣⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⣀⠀⢤⣊⡵⠒⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⡇⠀⠀⠂⢬⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠀⠀⠀⠈⠆⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠑⠀⠀⠀⠀⠈⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢠⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⡀⠀⠀⠢⠀⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⡀⠀⠀⠱⡐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠤⠤⠤⣀⡀⠤⠤⣖⡁⠀⠀⣀⠀⠀⠀⠀⢀⡠⠙⢦⠀⠈⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠂⡰⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠑⢤⣀⣀⣀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣠⠃⠀⠂⠀⢉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢰⠃⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡏⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠃

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠶⠶⣶⣾⣶⣤⣤⣤⣤⣤⡄⣶⣠⣦⣴⡆⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⡧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣾⣿⣿⣷⡧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢹⣿⠟⠉⠉⢩⣥⣿⣿⣿⣿⣿⡇⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⣀⣤⡄⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⢸⣿⣿⣿⣿⣿⣴⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠬⢸⡇⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢸⠰⢀⡁⡏⠇⡇⠆⣌⢸⠱⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣴⡇⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣀⡍⡀⠀⠀⠀⠀⢀⣧⣾
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠀⠀⢈⠏⢻⡶⠟⣗⠇⠀⠀⠀⠠⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠸⢾⠃⢸⠇⠼⡄⠀⠀⠀⠀⢹⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡦⢟⣟⢀⡄⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠓⠬⠈⠘⠁⠚⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣀⣾⣿⣿⣿⣿⡟⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠛⢿⡼⣿⡟⠻⣶
⠀⢸⣿⣿⣿⣿⡿⠏⠉⠁⠙⢿⡖⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠶⠿⠃⠀⠀⠖⠛⠁⠘⠛⠂⠚⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠞⠀⠰⠓⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢰⡺⠿⡱⠄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣷⣇⠒
⠀⠀⠀⣠⣶⣶⣿⣇⡸⣦⣤⢟⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⣠
⠀⠀⠰⣿⣿⣿⡏⢻⡗⠈⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠰⠶⢮⣷⣄⢠⣾⡃
⠀⠀⡄⠛⢁⡯⠓⠀⠁⠒⠃⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣿⣿⣿⣇
⠀⠀⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠏⠉⣹⣎⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠛⠿⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋

⠀⣀⠀⠀⠀⠀⠤
⠀⠿⢿⡮⠀⠀⠀⠀⢰⡿⣿⡇⢤⣄⣀⠠⠀⠀⠤⠀⡀⣄⣤⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣤⠀⡀
⠀⠀⠘⡆⠀⠀⠐⠓⠌⣿⠏⡀⠀⠈⠉⠀⠀⠀⠀⠈⠋⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠙⠋⠁⠈⠙⠛⠿⣦⣄
⠀⠀⡀⠄⠀⠀⠀⣀⣇⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣷
⠀⠀⢿⣿⣥⣤⣶⣿⣿⡽⣿⣿⣿⣷⡟⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⢄⠿⣧⡤⣾⠀⢿
⠀⠀⠀⠀⠀⠀⠙⠛⢿⡄⠙⢿⣿⣿⠅⣤⣾⠀⠙⢷⣿⣦⣀⣶⣆⣧⣄⣾⡆⣴⡀⣠⣾⣿⣿⡑⠀⠉⠋⠉⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠘⠻⢿⣻⣿⣿⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⣤⣠⢈⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⢸⣿⣿⣿⣄⣹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣀⣈⣛⣿⣿⣿⣿⣿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣢⣤⢿⣷⣽⣿⣿⡤⠮⣿⣿⣿⣿⣿⣿⣿⣟⡀⠠⠍⠉⢹⣿⣧⣿⠻⣿⠙⢻⣶⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣶⣿⣿⣿⡟⠎⠀⠀⠀⠀⠈⠉⠉⢻⠟⠉⠀⠀⠀⠀⠈⣿⣿⣿⠄⠈⢶⣿⡟⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣧⠾⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⢻⣷⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠇⠀⠀⠀⠀⢿⡇⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠾⣏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠨⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⡷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠃⠀⠀⠀⠀⠀⠠⣖⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⢻⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⠀⣿⣧⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⠀⣹⣿⣆⠀⠀⠀⢀⣾⣿⣿⣦⠐⢡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁⠀⢹⡇⠋⣠⣿⠿⠟⠀⠒⠶⣯⣽⣛⠿⣿⣆⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⣠⠈⠁⢰⣿⡿⠂⠀⠀⠀⢀⣸⣿⣿⣿⣮⡙⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⡟⠀⢠⡿⠉⠀⠀⢀⣴⠿⠿⣿⣿⣿⣿⣿⡿⠆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣥⣴⣯⣶⠏⠀⠀⠘⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠉⣉⣽⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⣠⣤⣤⡀⢸⣿⣿⠀⠀⠀⠀⢀⣀⢶⣶⡆⠀⠀⠀⠀⢠⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⡈⠻⢿⣧⣤⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠟⠋⠉⠻⣯⣽⡽⠛⠋⠀⠀⠀⠀⠀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠐⠺⢿⣿⣿⣿⣿⣷⣄⠀⠀⣷⡀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⠀⠉⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣄⣌⣅⣀⣀⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠢⣄⡘⢿⣿⣿⣿⣿⣿⣯⣏⣭⣍⣠⣤⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠈⠛⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠦⠀⠀⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠋⠀⠀⠀⠀⢰⠄⠉⣿⠛⠚⢄⣤⣦⣀⣀⢀⠀⠀⠀⠀⠈⠉⠙⢶⡅
⠀⠀⠀⠀⠀⢀⣤⠦⢀⣤⡄⠀⣠⠿⠃⠀⠀⠀⠀⠀⠛⠀⠀⠽⠠⠀⠈⢿⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⣠⣤⣴⢶
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡏⠀⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣀⡀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⠀⠀⢰⡏⢹⡇⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠸⠀⠸⠧⠿⠁⠀⠀⠀⣀⡖⠒⠶⢶⠿⠷
⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⣀⣀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠶⠀⠶⢦⠲⠃
⠀⠀⠀⠀⢸⡿⣤⠊⠉⠉⠉⠸⡶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠖
⠀⠀⠀⠀⠸⠧⠹⠦⡠⠄⠀⢸⡧⠛⠀⠀⠀⠀⠀⣰⠶⣶⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢤⣀⠀⣠⣀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠟⢸⡟⠉⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⠞⠉⢻⠏⠹⡏⠉⠉⠁⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠐⠘⠀⠠⠿⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡀⠀⣠⣠⣶⣻⣽⣿⣷⣀⢺⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢺⣿⣾⣾⣉⣟⣹⣿⣧⠙⢿⠫⠈⣵⡿⣶⡂⠴⠶⠶⠶⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠰⢤⡀⠀⠀⠀⠀⡀⢤⣮⡧⣍⢛⡉⠀⠉⠉⠁⢙⡿⠛⡛⣱⣢⣸⠋⠱⠒⢀⡀⠀⢤⡴⠬⠿⠄
⠀⠀⠀⠆⢀⠰⠀⠛⢻⢷⡀⢨⠗⠚⢻⣿⣿⣷⣄⡒⣢⡠⣽⢃⡛⢿⡇⠐⠀⠐⠂⢿⣴⠀⠀⠉⣻⣿⠀⠷⠀⠀⠀⠆⡀⠐⢂⣠⣄⡀
⠀⠀⠀⠒⢦⡌⠉⠁⠀⠘⠛⠀⠘⠄⠁⠀⠀⠑⠛⠛⠓⠛⠋⠈⠃⠈⠙⠀⠀⠁⠀⠉⢉⠑⠂⠁⠈⠙⣰⣴⡞⠽⠶⡾⣿⣶⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠠⠀⠨⠀⠀⠀⠀⢀⡤⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠠⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠠⠀⠀⢴⠦⠀⠀⠺⠖⠒⠀⠀⠴⠀⢠⡀⠀⠀⣠⡀
⠀⠀⠐⠆⠷⣾⠇⢶⢾⠞⠠⠇⣾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢀⣄⡀⡐⡀⢠⡄⢦⠴⠂⢮⣄⡄⠶⣀⡄⢤⢰⣸⠈⢀⡅
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠓⠈⠁⠃⠀⠘⠀⠀⠁⠈⠀⠀⠉⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠤⣴⣿⣷⡍⠉⠃⠨⠩⣿⣦⡀
⠀⠀⠀⠀⣀⡀⠲⠾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢿⣷⣤⠄⡀
⠀⠀⠀⠀⠀⠈⠉⠉⢉⣿⣿⠉⠀⠀⠀⠀⠀⠰⣿⣷⣶⣶⣦⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⡈⣴⣿⣏⠓⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⣆⠀⢤⣀⠀⣹⡿⣿⣿⣷⣶⢴⡶⠂⠒⠒⠈⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄⣛⢛⣶⣿⣿⢻⣿⠃⠁⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣛⣶⢣⣿⡿⢿⡿⢿⠿⠷⣋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠸⣶⣤⢡
⠀⠀⠀⠀⠀⠀⢸⡿⣽⣥⣤⠦⡑⠿⠴⠂⠒⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠸⣿⡋⢉⠀⠭⠁⣀⡤⠀⠀⠀⠀⠀⡁⠢⠐⢺⠀⠀⣄⠀⠀⠀⠀⠆⢰⡀⠁⠀⣴⣰⡰⠠⠀⠀⠀⠀⠀⢟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⣟⡄⣠⡴⠺⣵⠏⢈⡉⠐⠀⠀⠀⢿⡇⢢⣍⢸⡀⠚⠀⠀⠀⠀⠀⠘⣡⡀⣋⢶⠉⠁⡂⠀⠀⠀⠀⠀⠈⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣿⢶⣩⢀⣼⣿⡴⣭⡤⠄⠀⠀⠀⢸⣧⡄⢻⠈⡑⢂⡈⡇⠀⡴⣄⡀⢜⣿⣿⣇⠀⡀⠁⠀⠀⠀⠀⠈⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣲⣿⣿⣿⡱⢦⠴⠀⠀⠀⣄⣻⣿⣿⣙⣺⡁⡜⣤⠀⠣⣨⣿⣧⣼⣿⣿⣯⣀⠂⢀⠐⠀⠀⠀⢬⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡬⠠⠝⣫⣤⣹⣿⣿⣿⣿⣏⣿⠲⢄⣴⠂⣻⣿⣿⣻⣿⣿⣿⣿⠠⠀⠀⠢⠓⠘⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⠀⠿⠏⣷⣼⣿⣿⣿⣿⣿⣿⣻⣴⣿⣭⣾⣿⡿⠿⠟⢻⣿⣿⣿⣟⡙⢤⣞⠻⢭⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿⡇⠘⠒⣻⣿⡿⠀⠈⠁⠈⠉⠉⠋⠉⠉⠉⠀⠀⠀⠀⠀⠻⣻⣿⣿⣿⣞⣬⣿⣆⠈⠽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡇⠀⠀⢁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠻⠿⣿⣿⣯⠁⠀⠀⠺⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢀⣻⠿⣿⣧⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⡇⠁⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⢠⠏⠈⢣⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡟⠀⠀⠀⡄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⡇⠀⠐⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⢀⠃⠀⢀⡿⠁⠀⠀⠀⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣵⣿⢰⣼⡇⠀⣠⣿⣠⢤⣦⡖⣶⡴⣆⢀⠀⢰⠀⠀⢊⢠⡁⣴⣷⣏⣤⣱⣾⠇⠀⣄⢀⡀⣷⣄⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠠⣁⣸⣷⣾⣿⣿⣿⣿⣷⣮⣿⣿⢿⡿⠿⢏⠛⠻⢼⣝⠓⡚⠛⢛⣾⠾⣿⠿⠿⣿⣿⣿⡿⢶⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠷⠦⣦
⠀⠀⠀⠀⠀⠀⠁⠀⠙⠒⠓⠊⠛⠈⠃⠘⠈⠙⠋⠉⠋⠀⠈⠀⠀⠀⠀⠀⠐⠒⠂⠘⠀⠒⠀⠢⠀⠁⠀⠐⠀⠈⠘⠋⠙⠉⠋⠉⠹⠙⠛⠑⠐