⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠶⠶⠶⠦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠉⠙⠳⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⠦⣤⣄⡀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⠙⢷⣄⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢹⡿⣦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⠈⠇⠈⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⣹⣧⠄⢻⡿⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⢸⡇⠘⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⣀⣤⣴⠞⠋⠉⠉⠁⠀⠀⠈⠉⠙⠒⠧⣌⠀⠀⠀⠀⢀⣴⠇⠀⢠⡟⢠⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣻⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⢀⡠⢟⡟⠀⢀⣼⣧⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣇⠀⠀⡀⠀⠀⠀⢀⣴⣊⡵⠿⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠯⣤⣴⡾⣋⡁⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣆⠀⠉⠛⠯⣭⣉⣉⣀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⢷⣤⣀⣞⡥⠞⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⡄⠀⠀⣸⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣠⣴⠾⣛⣉⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⡿⠃⠀⠀⢀⣼⠁⠀⢠⡇⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠸⠿⣿⣛⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⣠⠔⠋⠀⠀⢀⡤⣿⠃⠀⣠⠟⡇⠀⢰⠐⡄⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠛⠷⣶⣤⣄⣀⣄⠀⠀⠀⠀⠀⠐⠒⠊⠉⣀⣠⣤⠶⠋⠁⠀⡀⢄⣪⠟⡻⠃⢀⡴⠃⠀⣧⠀⢸⠀⢣⠀⠀⠀⠻⢧⣄⣀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⣴⣿⠀⠀⣼⣩⡏⣛⣓⣲⠖⠶⡖⣺⣭⣛⣉⣠⣴⠶⠬⠟⠚⠉⢀⡜⣁⠴⠋⠀⠀⠀⢹⠀⣾⠀⠸⡗⢤⣀⠀⠀⢈⣉⣛⣿⠶⠆
⠀⣀⣾⠛⠳⡈⢻⣦⣰⠋⢠⡇⠀⠀⠀⣤⣸⠃⠀⠀⠀⠀⠀⠈⠲⡀⠀⢀⡪⠞⠊⠁⠀⠀⠀⠀⠀⢸⢠⢿⠀⠀⢳⠀⠘⣿⠛⠛⠉⠁⠀⠀⠀
⢰⡿⢉⠘⢆⠈⣾⢿⣯⡀⠀⡇⠀⠀⠀⠇⣸⢶⣦⣤⣀⠀⠀⣀⠀⠈⠳⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠘⡇⠀⠘⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠸⣧⣀⡑⣴⡿⣡⡾⣿⡱⣄⢷⠀⠀⠀⢇⣿⣀⣈⣙⣛⣿⣶⣬⡑⠢⠀⠀⠁⠀⠀⠀⠀⢠⢤⠤⢔⡿⠁⠰⣇⠀⠀⠸⡄⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠈⣻⢯⡁⠉⠙⠁⢸⠄⣿⣾⡄⠀⠀⢸⣟⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠀⠀⢰⣿⡇⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⠀⣿⣦⣀⠀⡾⠀⣿⡽⣳⣄⠀⢸⣽⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠿⠟⡾⠀⠙⣦⡀⠀⠙⣷⣄⡀⠀⠀⠀⠀
⠀⠀⣿⢀⣿⣿⣿⣷⠃⣰⣟⢯⣳⢻⣦⡀⣿⠀⠀⠀⠀⠀⢀⣠⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣴⠇⠀⢸⡇⣏⠗⣦⢤⣭⣿⣿⠿⠀⠀
⠀⠀⠹⣏⢿⡿⠟⢁⣴⠿⣜⡯⣞⡳⣽⢻⣾⣤⣀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠉⠙⢲⢤⡀⠀⠀⠠⢄⠥⣏⠀⢀⣿⢹⠻⣞⢥⣾⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠛⠷⢶⣯⣷⣫⢽⡞⣵⢻⣜⣿⢻⢿⣶⠻⢭⣑⡾⢆⠀⠀⠀⠀⠀⠀⢈⠞⠁⠀⠀⠀⠀⠀⡏⢀⣼⣿⢸⣦⠿⠛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣾⣵⠋⣼⢸⣷⣦⠀⠀⠈⠑⣦⣤⣤⣴⠒⠉⠀⠀⠀⠀⠀⠀⢰⠃⣾⣿⡏⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡷⢟⣾⠃⢿⣻⣷⡲⠤⣄⣀⣀⣀⠀⢀⣀⣀⣄⣠⠤⠞⢉⣾⣿⣯⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣾⡛⠁⢀⣸⣷⣹⣷⠦⡰⢻⣿⣯⡿⣿⢻⣇⣯⠇⣠⡴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⣏⣁⠼⣷⣻⠀⢈⣿⣿⡟⣠⣿⣜⣿⣧⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠚⠉⠀⠀⠈⣿⣤⣾⢿⡿⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⣶⣶⢾⣯⣞⡀⠀⠀⠀⠀⠀⣯⣸⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣾⢯⣉⣞⣙⣶⣧⣼⣿⣶⣶⣾⣷⣤⡀⠀⠀⢹⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣧⠖⠋⢙⣿⣿⢿⣫⢟⡽⣷⣿⡹⣟⡿⣦⣄⠈⣇⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣙⣳⡶⣿⡻⣝⢮⡳⣏⢾⡱⢧⡻⣜⡳⣭⢻⣿⣿⡀⠀⠘⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⠀⣀⠀⠀⢀⠠⠒⠁⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠔⢪⢁⡠⠯⣴⣆⡐⠤⠉⢉⠠⠀⠈⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠁⠰⠚⣭⠛⣛⢶⢐⠀⢄⠀⠀⠀⠀⣀⡠⠴⡺⢄⠀⠀⠀⠀⠀⡐⠉⠒⠁⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⢡⣾⠟⡭⠀⠀⠈⡀⠀⠈⠈⠁⡉⠂⠀⠀⠀⢎⠓⢄⠀⠀⠀⡱⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠀⢠⢗⠛⢃⡬⠀⠀⢠⠁⢣⢠⠐⡀⠐⠠⢁⢢⡀⠀⠀⢷⡊⣧⠀⠈⠀⠀⢀⡖⢐⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⢄⣀⢠⡈⠔⣧⢸⡅⣖⠀⣀⢠⡦⢀⠟⡈⢀⠀⠀⠀⠈⢀⠕⠀⢰⠬⣇⠀⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⠍⢁⢅⡇⢮⠱⣚⠣⡇⢰⠀⣼⠀⢂⠚⢐⠀⠀⠀⠀⠀⠌⠣⡑⠌⡌⠒⠞⢉⠄⠀⠀⠀⠀⢀⠆⠑⠀⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠅⠐⢡⢺⡀⠈⢧⣼⠛⠀⡀⢠⡧⠊⠁⠀⠤⠳⡀⠀⠀⣜⢊⣤⣓⣷⡴⡠⢔⠊⠀⠀⠀⠀⠀⡚⠁⠀⢠⠼⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⢀⠃⣮⢟⢪⢶⡉⠀⠰⣷⠁⠈⠗⢡⡾⠟⣹⣿⣷⠀⡪⠋⢹⣿⢿⠗⢸⠈⠑⠳⠀⠀⠀⠀⠀⠀⠉⠂⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣸⠃⠀⢳⡣⠈⢩⠩⣙⡢⠀⠟⠄⠉⠢⢫⠽⠏⠊⠀⠀⠈⠻⡼⢱⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠙⢷⢒⡅⢅⣈⢌⡆⠠⠁⡡⡐⠰⠀⠀⠀⠀⠀⣑⢈⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠔⣃⣾⢂⠆⣒⣏⠴⠀⠂⠃⠀⠀⠀⢠⠚⠚⠇⢀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⢧⢾⣸⠼⡆⠀⠀⠄⣀⠀⠀⠈⠒⠁⣔⠿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⡧⣄⠀⠀⠀⠉⡒⡲⡆⠊⠁⣀⠴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠑⠒⠤⣄⠀⢰⠀⠀⠀⠉⠉⠱⣲⡏⢂⠸⣀⣇⠋⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠢⣠⡘⠎⠣⠎⠄⡀⠀⠀⠀⣷⣬⣀⣼⣧⣩⠉⠀⠀⠀⠂⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⢁⠁⠢⣄⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠰⠂⠀⠘⠀⠁⠊⡀⠐⡀⢈⢢⣀⡸⢛⣿⣿⣷⣿⣿⣿⣦⡤⡁⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢀⡀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠒⡠⢜⣧⡰⠁⠀⠀⠀⢇⠀⡐⡖⠠⡳⡀⠈⢆⣠⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⠉⠉⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠮⢧⡧⡘⡖⠁⡇
⠀⠀⠀⠀⠀⠔⠉⢀⠖⠀⡐⠉⠇⠀⠀⠀⠀⠠⡒⣦⡄⠀⠈⢣⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢂⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠌⠉⠀⠰⠀
⠀⠀⠀⠀⠀⢾⡀⠉⠀⠘⠅⢰⠃⠁⠀⠀⠀⠀⠀⡆⡇⠀⠀⣘⡀⠀⠸⣿⣿⣿⣧⢻⣿⣿⣷⣻⡀⠈⢁⠄⣡⠔⡆⠀⠀⠀⠀⠀⠀⠀⠟⡆⣧⣠⡤⠃⠀
⠀⠀⠀⠀⠀⠈⢦⠌⠂⠄⠀⣀⠀⡖⠄⠀⡀⠀⢨⠫⡅⠄⠑⣁⣁⠀⠀⣿⣿⣿⢧⣟⣿⣿⣿⢣⡇⠀⡁⣄⣷⡴⠃⠀⠀⠀⠀⠀⠀⠀⣹⣾⣥⡼⠓⠀⠀
⠀⠀⠀⠀⠀⠀⠸⡙⢄⠀⠀⠀⢁⣷⣶⣞⡌⠱⡨⣣⣔⠀⠒⠰⣸⠀⠀⣿⣿⣿⡿⣯⣿⣿⣳⢯⡇⠀⡇⠈⣿⣷⣦⡀⠀⠀⠀⢘⡑⢷⣿⣿⣿⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣙⣶⣼⣖⣄⣻⡟⣾⣿⣿⣿⣶⣽⣿⠀⠈⠀⢉⠁⡀⣾⣿⡿⣿⣿⣻⣾⣿⣯⣿⠀⡓⠆⣿⣽⣻⣽⡆⢀⠔⠁⠁⠼⡿⣿⢧⠄⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡲⣿⣿⣿⣿⣿⣿⣽⣳⣿⣿⣿⣿⠉⠛⠒⠀⠉⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠆⠈⣿⣯⣿⣿⡗⠁⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣆⡜⠻⢿⡿⠋⢩⣿⣿⣿⣿⡏⢿⣯⣀⢀⠀⠜⡋⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⣿⠇⠱⡄⠀⠒⣷⣾⣿⣿⣿⣿⢇⣾⠿⠾⠆⠀⠈⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢤⠉⣾⣿⣿⣿⣿⣾⠄⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀
⠀⠀⡰⣽⢿⠏⠀⠀⠀⠉⠈⢻⣿⣿⣿⣿⣿⠸⣿⣧⡀⠀⠈⣹⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣃⢸⣀⣠⣿⣿⣿⣿⣿⣘⢀⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀
⠀⢰⣿⠋⢾⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠉⢫⠉⠉⠕⠀⠀⠉⠉⠉⠉⠉⠙⠛⠁⠐⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣷⣃⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣿⠠⢹⡆⠀⠀⠀⠀⠀⠘⣻⣿⠁⠀⠀⠀⠀⠀⠀⣧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⢿⡷⣾⠀⠀⠀⠀⠀⠀⠀
⢠⣸⣿⣣⠂⢿⣆⠀⠀⠀⢀⣾⣿⡏⠀⠀⢀⠔⢒⠢⢜⡅⠐⡀⠀⠀⠀⠀⠀⠀⡀⠐⠁⡠⠀⠀⢀⠽⣑⠄⠸⢿⣿⣿⣿⣿⣿⣿⠏⡧⡇⠀⠀⠀⠀⠀⠀
⢰⡇⣿⣧⣻⢸⣿⣧⣄⣴⣿⣿⣿⠁⠀⢀⠎⠀⠀⠡⣠⡎⠑⠚⠢⢀⡀⡠⠐⠈⠀⠀⠄⡀⠔⢂⠀⠰⠀⠀⠈⠈⠛⠻⣿⣿⣿⣿⠇⢟⣳⠀⠀⠀⠀⠀⠀
⠸⡞⣿⣷⢯⡽⣿⣿⣿⣿⣿⣿⣿⠀⢠⠋⠀⠀⠀⢠⠀⢀⠠⠂⠀⠀⠈⠈⠁⢲⠀⡀⠈⠀⠀⠀⠈⠂⡀⠈⢃⠳⡀⡀⣿⣿⣻⣿⣇⡶⣼⠀⠀⠀⠀⠀⠀
⠀⠹⣼⣿⣯⢷⣻⣿⣿⣿⣿⣿⡇⠀⠈⡧⠩⠜⠄⠒⠑⡀⠐⠀⠀⠀⠀⢀⠐⡠⠂⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢘⢒⠀⠿⠿⣷⣟⣯⣷⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠹⢿⣿⣯⢷⣯⢿⣿⣿⣿⠃⠀⣠⣿⡄⠀⠐⠀⡄⠀⠀⠀⠀⠀⠔⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠸⡿⣿⠞⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠸⣿⣿⣻⣞⣿⣿⣿⠃⣠⣾⡿⡛⠛⠀⠀⢈⡃⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⡀⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠙⢿⣷⣯⣗⣿⢋⣾⣿⣿⡇⠀⠈⠁⠌⡠⡚⡂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⢒⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⣨⣿⣿⣿⣿⣿⠂⢀⠈⠈⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢈⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆⢸⠁⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠄⡀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣟⣿⣻⢿⣿⣏⠀⠀⠐⠁⠈⢀⠀⠀⠐⠔⡀⢀⠀⠀⠀⠀⠀⠐⢰⠀⠀⡄⠀⠈⠀⠀⠈⠢⢠⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⡟⣯⣛⣧⣟⡾⣽⣻⣿⣿⣦⣄⠀⠀⢠⡊⠅⠀⠠⠄⠀⠁⠲⠀⠑⡂⢆⠄⠂⠀⠀⠀⢄⠀⠀⠀⢀⣴⣿⣟⡻⢾⣽⢣⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⡧⣝⡲⢏⣞⠾⣽⣳⢯⣿⡽⣿⣿⣷⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⡄⠀⢀⣠⠄⣄⢀⣴⣿⡿⢿⢏⡒⠳⢚⢸⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠹⣦⢹⡙⣮⣛⠶⣭⢻⣷⢻⣭⢟⡿⢿⣄⠀⠀⠀⠀⠀⠀⠚⣒⣾⣶⣶⢨⠔⠲⣼⣿⣿⣼⣴⡿⣿⢣⡝⢮⡘⣈⠁⣊⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡴⣩⢛⡜⣻⢇⡟⣬⢫⡜⢫⢿⣿⣶⣶⣶⣦⣴⣶⣾⣿⣿⢿⣷⣶⣿⢻⡟⣿⠿⢯⠵⣧⠓⡜⢢⠑⣤⡝⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣯⣜⣹⢣⠚⡔⢣⠘⢥⡚⢬⢫⡝⢯⢻⠷⣭⠿⣿⢏⡻⢽⡛⣎⠳⡜⢦⠛⡬⢻⢆⡙⠌⣡⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙