⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⢀⡴⠋⢿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⡹⣦⠹⣿⣆⢳⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲⣲⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡻⣰⠟⢀⣴⠟⠋⠀⠀⢀⣠⡤⠶⠚⠓⠋⠉⠉⠉⠉⠉⠉⠛⣳⣶⡤⣬⣿⡘⣷⠹⡉⢣⡿⡄⠀⠀⠀⠀⠀⠀⢀⠔⣿⠀⢸⢿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢱⠏⢀⣾⠃⠀⣀⣤⣶⣏⣡⠤⠐⠀⠀⠀⣀⠀⠤⢀⣀⠀⠀⠀⠉⠙⠓⠟⠷⡾⢄⣷⠀⢿⡇⠀⠀⠀⢀⡠⡾⠁⢀⡿⠀⢸⠸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⡟⠀⣿⢃⡴⣚⡩⣤⢴⢯⣀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⡩⡦⣀⢄⠀⠀⠀⠐⢧⡀⠙⠢⣼⡆⠀⠀⣾⣇⡰⢧⡶⠋⠁⠀⣆⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⡿⡻⢧⠎⣿⢸⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢵⡯⠂⠀⠀⠀⠈⠓⠀⠀⠹⣶⣤⠿⠿⠾⠋⠀⣀⣤⡮⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢸⣧⡾⢡⠖⠁⠁⠀⢻⢸⠀⠀⢰⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⡘⢈⠇⠉⠒⠠⣄⠀⠀⠀⠀⠀⠀⠈⢺⣷⣄⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣶⠟⠀⡎⠀⠀⠀⠀⢸⠀⠀⢀⡾⡌⡝⡦⡀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠋⠀⠙⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⡏⠀⢰⠁⠀⠀⠀⠀⡎⣀⠀⠜⠀⡇⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡶⠖⠫⣉⠀⠀⠀⠀⠈⢻⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢸⠀⠀⠀⠀⠀⣷⣧⢀⠜⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣈⠁⠀⠀⠀⠀⠀⠙⠙⢿⣦⣠⣤⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢘⡀⠀⠀⠀⠀⢿⢸⢣⠀⠀⠀⠀⣗⣆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠁⠒⠀⠀⠒⠒⣒⡢⠟⠹⣷⣤⣀⠀⠀⠀
⠀⠀⢠⡤⢤⣤⣀⣰⠃⠀⠀⠀⡌⢧⠃⠀⠀⠀⠈⣧⠈⠆⠀⠀⠀⣿⠻⣾⣦⣀⠀⢤⡀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢄⣀⠙⠻⢯⣁⢉⣉⣉⠀⠀⣀⠄⠀⢑⠉⠻⣦⠤
⠀⠀⠈⢻⣆⠱⢤⣿⠀⠀⠀⠀⡇⠈⣆⢀⠀⠀⠀⠹⣧⠸⡄⠀⠀⣿⠀⣠⡿⠿⠟⣯⢍⣶⣄⡀⠀⠺⢕⠦⣀⡀⠀⠀⠀⠀⠉⠛⠲⢤⣭⣕⣲⠾⠭⠤⢤⠖⠁⠉⠒⠢⠥
⠀⠀⠀⠀⠹⣧⠀⢹⠀⢸⠀⠀⠁⠀⢸⣯⣣⣄⠀⢄⠈⢃⠱⡀⠀⡏⠜⠉⠀⣠⡴⣞⣷⣶⣿⣿⣶⣤⣠⠤⢌⣉⣒⡲⠤⣒⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠛⢿⠀⠸⠀⡆⠸⠀⢸⣧⠻⢿⣽⠤⡑⢄⢣⡱⡀⡇⠀⠀⠰⣫⠿⡿⢿⡿⡿⣻⢩⣿⠋⡗⣦⣤⣀⣀⠀⠀⠉⠓⠶⢶⣶⣾⣉⠁⠀⢀⠀⠠⠴⠾⡀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣞⡆⡆⢠⠇⠀⠀⢸⣿⠾⠋⠙⢗⣌⡺⢅⠁⠙⢄⠀⠀⠀⠁⠀⢯⣯⣙⣿⢽⠞⠁⠀⠇⢹⡸⠋⠛⠯⣍⣀⡀⣾⣿⣿⣻⢿⡿⣿⣉⠒⠲⠤⠴⠿⠗⠒
⠀⠀⠀⠲⢖⣊⣁⡇⡿⠀⡸⠀⠀⠀⢻⡇⢀⣴⣿⣾⣿⣿⣳⡭⣦⡌⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠀⠀⠰⠁⠀⢠⣇⡎⢯⣻⡏⣿⣿⠿⣿⣾⣷⡦⡄⠀⠀
⠀⠀⠀⠀⠀⠀⣼⡞⢁⣴⠃⠀⠀⠀⡾⣴⣿⢿⣟⠻⣻⣽⡉⠓⣮⠙⠻⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⠇⠀⣰⠃⠀⠀⣼⡽⣇⠈⢿⢁⣿⣿⠶⢿⣿⣿⢿⠷⣤⣄
⠀⠀⠀⠀⢀⣴⠟⢴⣵⠇⠀⠀⡆⡰⢿⣿⣯⣀⠻⣷⠜⠊⠀⡰⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⢞⡟⠀⣴⠃⠀⠀⢀⡟⠃⣿⢀⠞⣹⢻⣿⣷⡄⣹⣿⣦⡻⡏⠃
⠀⠀⠈⠓⠽⡶⠛⢁⣿⠀⠀⢀⢣⣻⣼⡿⣎⠉⠋⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⢤⡘⠀⣼⣿⠀⠀⠀⣸⢰⠀⣿⣎⡴⣟⣾⣼⣿⣿⣉⣻⣿⡫⣷⠀
⠀⠀⠀⢀⡜⠀⣠⣪⡟⠀⢀⣾⣿⠇⡄⡇⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡿⠋⡜⣡⡾⢋⠃⠀⠀⢠⠃⡞⠰⢷⣅⠀⠀⠉⠙⣟⣛⣻⢛⣟⣶⣋⠀
⠀⢀⠴⠃⢠⡾⠛⢹⠇⢀⢾⣽⢋⠀⠁⡇⠀⢸⣆⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢘⣧⡮⢋⣼⠃⡞⠀⠀⠀⠂⢰⠁⠀⢿⠉⠫⣍⣉⠀⠀⢩⢏⣟⠟⠻⣷⠏
⠨⠥⠒⠊⠁⠀⢀⡿⢀⣵⡟⡽⣸⠀⠀⠀⠀⢸⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣔⡵⠋⡘⡸⠀⠀⠀⠀⢠⡇⢸⢰⢸⠀⠀⠈⠲⣻⡑⠛⠾⠾⣴⣿⣿⡒
⠀⠀⠀⠀⢠⣴⢿⠎⠉⢱⡱⠁⡏⠀⢸⢰⠀⢸⠀⠀⣿⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠴⠋⠀⢠⢳⠃⠀⠀⠀⢠⣾⠀⡇⡞⢸⡄⠀⠀⠀⠈⠛⢦⣠⣾⠋⡇⢣⡈
⠀⠀⠀⠀⠀⢀⡟⠀⡰⡷⠃⣸⠁⠀⢸⡏⢇⡎⣇⢠⢣⡿⢻⣿⣷⣶⣄⣀⡀⠀⠀⢀⡠⠖⢁⠀⠀⠀⣬⠇⢠⠆⠀⢀⣾⠇⢀⢃⣧⡇⠈⢆⠀⠀⠀⠀⠀⠙⠿⡛⠷⡖⠛
⠀⠀⠀⠀⢠⢾⠤⡜⠘⣡⣴⣿⠀⠀⡷⢧⠸⡄⣿⣿⣿⢷⣾⣿⣿⡷⣧⠈⢹⢳⡒⣉⣀⣠⠎⠀⠀⣼⠏⢠⠇⠀⣠⢫⠆⠀⢸⡼⣇⠷⡄⠈⠃⠀⠀⠀⠀⠀⠀⠈⠓⠍⠀
⠀⠀⢀⣰⡳⣃⡾⠯⠶⣿⣿⡎⠀⣸⠧⠘⢧⠱⣾⣿⣏⣁⣨⠿⠿⠛⠛⠉⠉⠉⠀⡇⠀⢈⣀⣀⡜⢉⣴⠏⢀⢾⣣⠋⠀⠀⣿⣳⡾⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠉⠉⠁⠀⣠⠞⢱⢻⠁⠀⠇⠀⠀⡸⢣⣙⣅⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣧⢟⣴⡿⣳⠞⠓⣾⠋⠀⠀⢸⣿⣿⠃⠘⠆⠀⠈⠢⡀⠑⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⠞⠁⡰⠃⣼⠣⣾⣀⡴⠛⠙⢦⡙⡙⠓⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣼⣵⠿⠟⠉⠀⠀⢠⠇⠀⠀⢀⢯⣿⠿⠖⠒⠛⠉⠒⠤⡈⠒⢄⠑⢄⠀⠀⠀⠀
⠀⠀⠀⣠⡾⠋⠆⣴⣁⣜⣽⡀⢹⠉⠀⠀⠀⠀⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⣤⣶⡞⠀⠀⠀⡌⣾⠏⢀⠔⠀⠀⠀⠀⠀⠈⢢⡀⠑⢦⡑⢄⠀⠀
⠀⢀⣼⣿⠃⢀⢞⣿⢟⣩⣬⣷⡘⣦⣤⣤⠤⠼⣿⣗⣂⣤⡄⠀⠀⣀⣤⠴⠶⣒⠤⣼⢀⣶⣾⡿⣛⡽⠏⢡⡇⠀⠀⡼⡼⠃⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠈⠀⠀⠀
⢠⣿⠃⡇⠀⣾⡟⣱⠋⠁⠐⢦⡹⣧⣀⣠⣴⣎⣭⣿⡿⢿⢣⣽⠶⠬⢭⡿⡆⢨⠇⡽⣼⣹⡟⠀⠀⠀⠀⡌⠉⢦⣴⡟⠁⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀
⣿⣟⢠⣧⢸⡁⣷⠃⠀⠀⢀⣤⣽⢞⣿⣿⣛⣉⢿⣶⢷⡚⠯⢽⣶⣿⣿⢿⣵⡿⢊⣴⡿⡟⠀⠀⠀⠀⡸⠁⢀⢴⠟⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀
⢹⠻⢸⠉⣧⣱⡻⡀⢀⡠⣿⢿⢱⢻⣿⢛⣉⣩⣿⣿⡉⣤⣴⣿⣿⡽⢿⣛⣵⡾⣿⣿⡟⠀⠀⠀⠀⣴⣡⢖⣱⡏⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠂⠀⠀