⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⣒⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠃⠁⡍⢹⡏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⣇⠇⠑⣸⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣻⣿⢛⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣟⣬⣎⡇⢎⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⠶⣬⣜⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠛⡛⢓⠛⡲⣛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢇⠒⡐⠈⠴⠡⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡅⠢⠄⠡⣄⡓⡜⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡴⡗⠾⢒⠮⠽⣶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣾⢟⣹⣟⣳⠾⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢸⣾⣿⣿⣯⣽⣦⣿⣽⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢟⡻⢝⣻⣿⣿⣏⡟⣿⣿⣚⣽⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡙⢡⠹⠃⠄⠂⢈⣿⣯⣋⣝⣛⠛⡙⢻⡟⣾⢀⣤⠶⡒⣶⢲⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢆⡑⢪⠐⡉⢀⠂⠤⣿⣸⡏⣑⣮⡛⠑⡢⣛⣿⡿⠀⠌⢙⡚⣻⢿⣭⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢈⠶⡽⡱⣰⠃⠄⠀⣘⣿⣽⡞⡍⣂⠔⡨⠔⣹⣿⠲⠀⠈⠂⢌⠡⢣⣍⣓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢯⣷⢗⣽⠕⣇⢂⡴⢸⣿⣷⣚⠯⡄⢎⠰⣉⠼⣿⠐⠠⠁⠐⠈⣄⣷⡼⢢⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⣿⠿⣣⢓⣯⣴⣋⣼⡿⣟⡳⡭⠝⢢⡑⢢⢻⣿⡉⠄⠀⠸⢟⣾⢋⣷⢸⣿⣿⣻⡝⡶⢤⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣞⣿⣭⣟⢺⣿⣿⣿⢧⣸⣽⣿⣢⡌⢣⢳⡻⡷⡌⢠⢣⣏⣻⣦⢟⣸⣿⠡⠀⠈⠝⣿⣿⣦⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣼⣿⣽⣿⣿⣮⣿⣿⣯⡿⣿⣿⣯⣻⣷⣿⣈⣷⢿⣿⣔⡨⢽⣮⡇⡽⣼⢸⣿⠐⠀⢈⠰⣸⣽⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣾⣿⣿⣿⣿⣿⣿⣧⣽⣿⣷⣿⣿⣟⣿⣿⣾⣷⣼⣹⣿⣿⣻⣿⡿⣏⢣⡳⠾⣿⠠⠀⠠⢡⣼⣿⣶⡿⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⢃⣾⢿⣿⣼⣿⣿⣿⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⣿⢿⣮⣹⡧⣿⡤⢘⣄⣷⣿⣿⣯⣿⠀
⠀⠀⢀⣠⡴⠒⣛⢁⠞⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣿⣿⣿⣮⣿⣿⡿⣿⣿⣍⣿⣿⣿⣿⣿⡷⠀
⢀⣴⣟⡖⢪⠥⠤⠒⢤⣀⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣟⡯⡿⣿⣿⣻⡿⣽⣿⣿⣿⣝⣿⣿⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢻⠿⣿⠀
⠘⡭⣅⠒⡀⠂⠄⠀⠀⠈⣿⢻⣭⣿⣽⢺⣻⣿⣿⣽⣫⠔⣧⡚⣷⢿⠟⣿⣿⣿⣾⡻⠷⠽⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣟⣿⢿⣻⠀
⠀⣷⢌⠳⢈⠁⢀⠀⠀⢰⠿⣋⠧⢾⢏⢾⣻⣽⣿⣶⣩⡟⣸⡍⡝⢻⢛⣿⠟⣯⢱⡐⢩⣲⣿⣿⢾⣿⡧⠮⣏⣿⣿⣿⣿⣾⣿⣿⣞⣿⡆
⠀⢸⡮⣍⠣⠄⡀⠤⠢⣿⡁⠆⠅⠆⠻⠮⣿⡻⣿⣷⣻⣷⠟⡽⠒⣥⣫⡼⢟⡡⡏⣞⢠⣷⢾⣓⠾⣻⡥⡿⠽⣿⣿⣿⣿⣿⣽⢿⣽⣾⠁
⠀⠀⢿⡯⡽⠵⣫⠥⣼⠣⠔⡉⡐⢌⠣⢁⠫⢶⣿⣿⣿⣿⡽⣿⠣⡵⠯⢵⢮⢶⣭⠹⡶⢎⡗⣌⡣⣱⠹⡎⣷⠹⡿⣿⠿⣧⢝⣿⣿⣿⠀
⠀⠀⠘⣿⣷⣟⣥⡞⣟⡸⡐⠠⡱⣨⡄⡼⡜⢹⣡⣿⣿⣿⢶⢣⣾⣥⠷⣝⡛⣫⠴⣻⢏⢷⡎⡴⢹⡌⢷⣙⠬⣳⢟⣫⣽⢚⣿⣿⣿⣿⠀
⠀⠀⠀⢻⣿⣿⣏⢧⡷⢴⠉⣄⡿⢡⣸⣱⢋⣟⢳⣴⣿⣿⣿⣏⣱⣌⠶⣋⡽⣯⡹⢟⡋⣅⢊⡵⢫⣜⡧⡜⣶⣋⣾⣿⣳⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠘⣿⣿⣿⣿⢳⡎⣗⣺⢜⡟⣳⣻⣘⣿⢛⣿⣿⣿⣟⡭⢖⡋⠶⣥⡿⣉⣑⡪⢵⠤⢫⠸⢳⣉⣧⣿⢟⣿⣟⣾⣫⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣯⡿⣾⡽⣳⢯⣿⢾⣭⢻⣾⡿⣏⡛⣆⡟⢩⠒⡷⢡⣥⢞⡐⢎⠂⣏⡝⢪⡼⣴⡟⣻⣿⣾⣿⣿⣿⣿⣿⣿⠀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣸⣿⢷⣏⣿⣷⣿⡿⣱⠞⡼⢭⡒⣻⢒⣠⣎⣁⡯⢢⡼⠟⣽⣁⡟⣥⣿⣿⣿⣻⣿⣿⣿⡿⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣏⣿⣿⣿⢻⡽⢓⣮⣷⡯⢖⣳⣓⣩⢷⡯⣶⣿⣷⡟⣵⣿⡻⠟⣱⢿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣷⢿⣻⣞⣶⣯⣿⣾⢭⢾⡽⣺⡿⡶⣏⣽⣿⠏⣇⣚⣼⣻⣿⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣮⡵⣿⡿⠞⣿⣩⣽⠟⣺⢝⣷⣷⢫⠟⡥⢋⣼⣶⣿⣿⣿⣿⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠿⣏⣿⣿⣿⣿⣭⣷⡿⣫⢖⡽⠧⡾⣧⣾⣟⣾⣿⣿⣿⠇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣷⣾⣳⣯⣽⣷⣵⣞⣿⣿⣷⣿⣿⣽⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣛⣯⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⡾⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣟⣽⣿⠿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⢾⣿⣿⣟⣻⣹⣞⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉⠉⠉⠁⠈⠀⠀⠀⠈⠈⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣏⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀ꎇꀎꉓꀘ ꌩꂦꀎ
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀ꎇꀎꉓꀘ ꌩꂦꀎꎇꀎꉓꀘ ꌩꂦꀎꎇꀎꉓꀘ ꌩꂦꀎꎇꀎꉓꀘ ꌩꂦꀎ