⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⠔⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢀⣀⠀⠀⠀⠀⢀⡴⠇⠀⠀⡇⣠⠴⠤⣴⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡀⠀⡾⣿⠃⣴⠋⠉⠁⠀⢀⣴⠋⠀⠀⠀⠔⠋⣀⡤⠄⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠳⠞⠁⠙⠋⠉⠉⠉⢀⠞⠻⣬⡙⠲⢤⡀⠀⠠⠤⠤⢬⣙⣒⣲⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠳⢦⣨⢧⠀⠀⠀⣀⠕⢃⣀⠽⣛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⡤⠀⠀⠀⣧⢸⣠⠴⠋⡟⠈⠵⠒⣏⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠋⠉⢀⡄⠀⠀⣹⠋⠀⠀⣼⠁⠀⠀⠀⢹⡆⠀⠀
⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⢠⣀⣾⣿⠀⠀⣇⠀⣀⡴⠁⠀⠀⠀⠀⠀⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣠⣤⣶⣾⡿⠟⣻⡴⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠿⠟⣛⡭⠗⠋⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀
⠀⢀⣀⣠⡤⢤⣤⣄⡉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀
⣴⡏⢻⡁⠩⠟⠁⠙⡟⣆⠀⠀⠀⢀⣠⡤⠶⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣿⡟⠀⠁⠀⡆⡶⣿⠙⡏⠀⠀⣴⠋⠁⠀⠀⣼⡇⠀⣠⣴⡋⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠈⢿⡄⠀⠰⡹⣅⠙⠀⠀⠀⡼⠁⠀⠀⠀⢀⡿⠀⠀⠀⢸⣠⠀⠀⠀⠀⠀⡀⠀⠸⡇
⠀⢘⣧⠀⠈⠻⠺⣷⠀⢀⣼⣥⣀⠀⠀⠀⡾⠁⠀⠀⠀⠿⠁⠀⠀⠀⠀⢠⣷⢀⣧⡇
⠀⠈⢧⡀⠀⠀⠀⠈⢳⣼⡇⣿⠏⠛⠂⣼⣃⡤⣶⢦⡀⣠⠀⣀⣄⣤⠗⠒⠙⠞⠘⠀
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠹⣄⠈⠀⠀⠀⠉⠀⠀⠁⠀⠳⠗⠋⠉⠉⠛⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

---

⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⣤⣀⣀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡿⠀⠀⠀ ⢀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣗⣤⠞⠋⠁⠉⠉⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣤⣿⣧⠄⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣦⠀⠀⢩⡿⠀⢸⡏⠀⢀⣼⢿⣄⣀⣀⣀⡀⠀⠀⠀⠀⣠⣴⠯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠛⠉⠀⢀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠻⣦⣴⠿⠁⠀⠈⠛⠛⠛⠁⠀⠉⠉⠉⠉⠀⠀⠀⣠⠾⢿⣤⡀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠈⠛⠷⣦⣄⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠓⠒⠶⡶⠦⢤⣤⣤⣭⣭⣭⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋ ⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠈⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢷⡄⠀⠀⢀⡠⠤⠄⠒⣺⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⠞⠋⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⠃⠀⢀⡤⠤⠒⠚⢁⣠⡴⠞⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⢹⡇⠀⢸⡄⠀⠀⣀⡤⠞⠋⢠⠇⠀⠀⢉⣀⣤⠴⣾⠋⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⡟⠃⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⣧⡴⠚⠉⠀⠀⢠⡟⠀⠀⠈⠋⠁⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠚⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⣿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣾⣿⣿⡇⠀⠀⠀⠀⠘⣧⡀⠀⢀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣾⣿⣿⣿⣿⣿⡇⠀⢀⣠⡴⠞⠋⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⡿⠛⠛⢁⣠⣽⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⣼⣿⣶⣶⣿⣿⡿⠿⠿⠏⠉⠉⣡⣴⡾⠿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠛⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠉⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠷⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢿⣿⠀
⠀⠀⠀⢀⣀⣤⣤⣴⡚⠛⠲⠦⣤⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢀⡴⠞⠛⢦⡾⠁⠈⠙⠳⣦⠞⠁⠀⠈⠻⣏⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢨⣷⡄⠀⠈⢷⡀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢸⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀
⠻⣄⢻⠄⠀⠈⠃⠀⠀⠀⠀⠀⢀⣠⡤⢶⡛⠦⣄⣻⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⡜⠀⣼⠙⣧⣼⠀⠀⠈⠻⠁⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡏⠀⠀⠀⢀⣠⣔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀
⠀⢿⣻⡄⠀⠀⠀⠀⠀⢠⠁⣼⠃⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠉⠉⢩⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀
⠀⠈⠻⣷⠀⠀⠀⠀⠀⠸⡀⠛⢦⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠴⣵⣄⣀⣙⣦⡀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⡀⢻⠀
⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠘⠧⠀⠰⠿⣆⠀⠀⠀⠀⡠�

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣼⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⡟⠠⠀⠘⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣷⡘⠿⢿⣿⣿⣿⡿⠁⠀⡇⣧⠸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠞⠿⢿⣿⣿⣿⣷⣶⣤⣉⠛⠟⠁⢂⡆⢡⢸⡅⢹⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⣉⣤⣶⣾⣿⣷⣦⡈⠙⠿⣿⣿⣿⡏⣉⣄⣰⣾⣿⣌⠘⣿⢈⣿
⣿⣿⣿⣿⣿⠟⢋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿
⣿⣿⠿⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⡿⠷⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠟
⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣷⡶⠀⣙⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⢀⣛⡻⠿⠿⠿⠿⡿⠿⣛⣭⣼⣦⡙⠟⣁⣴⣶⣿⣿⠀⢸⣿⣿⣿⠏⢹⣿⣿⣿⣿⡟⣸
⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣂⣁⣄⣤⣿⣿⠀⠸⠿⢋⣡⠂⣿⣿⣿⣿⣿⡧⢸
⡆⣿⣿⣿⣿⣿⠿⢇⢿⡿⢋⣥⣬⡉⠛⣿⣿⣿⠟⠡⡄⠐⣾⣿⠏⣼⣿⣿⣿⣿⣿⢁⣾
⣧⠘⠛⣨⣉⣥⣾⣶⣤⣴⣿⠟⣡⣶⣿⣿⣿⡏⠀⢶⠔⠀⣿⠋⣼⣿⣿⣿⣿⣿⣷⠰⣿
⣿⡆⠙⠋⣩⣴⣤⣙⡋⣤⣤⣾⣿⣿⣿⣿⣿⡇⠳⠠⠎⢐⣡⣾⣿⣿⣿⣿⣿⣿⡏⢤⣿
⣿⣿⣆⡘⠻⠿⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⣛⣩⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⠐⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡼⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠁⠈⢷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣧⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⣿⣿⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣾⡁⣹⡄⠨⣷⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣃⣾⢹⡇⠘⣇⠀⠸⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣤⣀⠀⠀⢀⣼⢋⣿⠃⠘⣷⠀⣿⠀⠀⢿⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡷⠶⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣴⡿⢡⡾⠇⠀⠀⣿⡃⢿⠀⠀⠈⣷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠟⢋⡵⠟⠁⠀⠀⠀⠘⢷⢸⡇⠀⠀⣼⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⢾⡏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀
⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⢷⣦⣄⡐⠒⠉⢻⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣿⣄
⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠈⣙⣻⡶⠄⠀⠘⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀
⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢾⡟⠁⠀⠀⠀⣀⣴⣾⠿⠍⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠀⠀
⣾⠣⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠀⠈⢻⣦⡀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀
⢸⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠒⠀⠀⢰⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠏⢀⣤⢀⣤⢀⣤⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠫⠿⠋⠿⠟⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣤⡾⠋⠁⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢉⡿⠀
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠶⠟⠋⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠛⢶⣦⣤⠀⠀⠀⠀⠀⣴⡞⠋⠋⠻⣧⡀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀
⠀⢹⡆⠀⠀⠀⢀⣤⣀⠀⢀⣠⡴⠟⠛⠷⢿⣤⣤⠞⠉⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀ ⢀⣾⠋⠞⢻⡴⠛⡾⡇⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀
⠀⠘⣷⣤⣄⣼⠏⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢻⠀⠘⢁⣼⢳⡇⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀
⠀⠀⢹⣦⠈⠉⠀⠀⣀⣠⣴⠾⣦⣀⠀⠀⠀⣀⣤⢶⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⡞⠋⠀⣾⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀
⠀⠀⠀⢻⡻⢦⣄⣾⠛⠉⠀⠀⠀⠉⠛⠶⣾⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡟⠀⣠⣾⣥⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠿⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣈⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀