⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡜⠁⡀⢠⠏⠀⢀⠔⠎⠉⢉⣒⠢⡀⣗⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⡄⠀⠀⠀⠀⡜⢇⡞⠀⠠⠀⢼⡴⢞⠃⡘⢀⠜⠊⠃⠑⠹⡯⠀⢳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⢦⡀⠀⠠⡃⠘⡇⠀⠂⠀⡕⠀⡌⠠⠁⠊⠀⠐⠈⠀⠄⠇⠀⣼⡴⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠱⡄⠈⢆⢀⠱⡐⠈⠀⢇⠀⡇⠨⠀⠐⠈⠀⠄⢁⠎⠀⣰⠋⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢀⠹⡄⠈⢑⣤⣘⠦⠤⠵⠒⠗⠖⣒⠚⠤⣐⠔⠁⢀⡔⠁⢰⢹⣀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡀⡼⠗⠋⠁⡀⠀⡀⢀⠠⠀⠄⠠⠀⢀⠑⠌⣢⢐⠎⠀⡀⣟⣪⣢⠪⢬⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠐⡰⠊⠀⠄⠆⠂⢠⠃⠄⠀⡀⢰⠐⠀⠐⢰⠀⠄⠈⠪⣣⠀⡀⠀⠱⣄⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⢡⢐⡃⠀⠂⢰⠁⠠⡘⢐⠁⡀⠄⢸⠀⠠⠈⡎⡇⠀⢌⢢⠙⡄⢀⠐⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢵⠀⠂⠁⢘⠀⡀⠇⢸⠀⠀⠠⢸⠀⢀⢨⠣⡹⡀⠸⠀⡆⠸⡀⢀⠨⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡠⢇⠠⢨⢐⠀⢰⠁⢨⠀⢈⠀⢜⠀⢀⢸⢅⢣⣃⠈⡇⢑⠀⢇⢢⡕⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠁⠌⠁⡹⣉⢭⠙⠯⣛⢝⠜⢺⡲⣘⠶⣝⢶⠳⢾⡑⡇⠓⠀⠘⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢽⡽⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢧⡃⢠⠁⠐⢼⠲⢶⡼⣴⣬⣲⣇⠙⢜⣴⡿⡶⡶⣗⠓⣾⢁⠀⠨⠀⢸⡔⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣺⣳⢽⣳⢦⡀⠀⠀⠀⠀⠀⣸⠀⣰⠀⠂⡝⡗⣼⣽⣺⣪⣗⣽⠀⣭⢟⡯⠟⠞⢗⠚⢼⠀⠠⢸⠀⠐⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣫⣟⣾⠯⠿⠆⠀⠀⠀⠀⢮⠠⣿⠀⠄⢺⡌⠄⢊⠐⠂⢧⡇⠐⡀⠂⠄⡑⢈⠰⠨⡎⠀⠄⢸⡊⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡏⢾⣻⣄⠀⠀⠀⠀⠀⠈⢳⣙⡇⢀⢸⣳⡈⠄⠨⢈⢈⠃⡂⠄⡁⡂⡐⡀⢢⡽⠀⠐⢀⢗⠅⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠈⢳⣟⣦⠀⠀⠀⠀⠀⠀⠘⢾⡀⢸⡉⠳⢮⡋⠲⡀⠮⠭⠭⢑⠠⣀⣖⡯⡎⠀⢨⢎⢢⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⠄⠀⠀⠀⢠⢿⣫⢷⢇⠀⣧⣄⣴⡧⠁⣽⠦⠚⡂⠅⡉⣻⠀⢐⠇⠀⣞⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⠅⠀⠀⠀⣼⣟⢮⡫⣻⠀⢽⣯⣷⠡⢈⠡⢀⠑⠲⠶⣢⣟⣀⢸⠂⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣾⠀⠀⠀⢀⣷⢗⡗⣝⡞⢄⣿⣺⡝⢀⠂⡐⠠⠘⠦⢅⣾⡿⣽⢿⠄⣯⢻⡽⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣺⠇⠀⠀⠀⢸⣽⢵⢝⡞⣞⢾⣻⠞⢁⠐⡀⢂⠡⠘⠲⠼⢿⢽⣽⡳⣧⣟⣝⢞⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣳⡟⠀⠀⠀⣴⡯⣳⢝⡵⣝⣾⠻⠡⠈⠄⢂⠐⣀⣢⢵⣲⢔⣄⡨⣘⣹⡮⡳⡵⣝⣵⠁⠀⠀⠀⡰⠚⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⡽⠀⠀⠀⠀⣿⣝⢮⣳⢽⠏⠅⠨⠀⠅⠨⡰⣯⢏⠯⢯⡯⢿⢽⡽⣽⣳⣻⣝⡞⣮⠇⠀⠀⢀⡞⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⢾⠃⠀⠀⣠⣶⢻⣪⢷⣏⡗⠠⠁⠌⡈⠨⡼⠇⣯⡗⣝⣯⢱⡺⣽⣃⢽⢢⣟⣞⣯⢾⡇⠀⠀⡞⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢽⡟⠀⠀⠀⣹⡷⢽⡳⡽⡮⣺⡌⢈⠐⢠⡽⣿⣗⡿⣧⣿⣵⢾⣮⢿⣼⣎⣞⣷⣳⣯⣯⠇⠀⡼⢡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣿⠀⠀⠀⠀⣾⢝⡽⡮⣫⢞⢵⢝⡶⡬⣾⢯⡷⣿⣽⣻⣾⣽⣻⣾⣻⣗⣯⡿⣾⣟⡞⠀⠀⢠⢇⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣷⡇⠀⠀⠀⢸⣏⢗⣽⡝⡮⣫⡳⣝⢮⣫⡿⣯⢿⣻⣾⣻⣾⣻⣽⢾⣗⡿⣗⣿⣳⣯⢿⣻⣂⡼⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣿⠁⠀⠀⠀⢸⢮⡳⣕⢟⣽⣪⣞⣮⣷⣟⣿⡽⣿⣽⢾⣗⣿⢾⣻⣽⣞⡯⣟⣾⡳⡽⣽⣺⣁⠀⣻⠀⠀⠀⠀⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣯⣿⠀⠀⠀⠀⠘⠷⠽⢮⣷⢿⣽⡿⣯⡷⡿⣞⣿⣳⢿⢽⣺⣝⡯⣿⣽⡷⡯⣿⣺⣏⡝⣉⡗⣵⢧⣟⣦⣀⣀⣴⡪⢣⠩⡓⡖⢖⣲⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣷⡯⠀⠀⠀⠀⠀⠀⠀⣸⢿⣻⡷⣟⡯⡯⡯⣗⣗⡯⣯⣻⣺⣺⢽⣳⣿⣿⠻⠳⢏⣀⢄⠠⡈⠌⠙⢀⠛⠒⠱⠴⠬⢍⣍⡓⣚⠺⢥⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⣷⡇⠀⠀⠀⠀⠀⣠⣾⣟⣿⡽⣯⡿⣽⡯⣟⣽⣺⢽⣺⣞⡽⣞⣿⣽⡾⠉⠦⡪⣤⣢⣬⢷⢄⢂⡡⢀⢂⢑⢐⠔⡐⠔⠠⢉⠒⡓⠥⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⡿⣾⣻⣿⢯⣷⡿⣷⣿⣻⣟⣿⣻⣽⣾⡷⣿⣽⢿⣻⣽⣾⣿⣻⣕⢕⡑⢧⡢⡣⡫⡙⡎⡷⣱⢱⢍⢯⢺⢱⢓⣎⣷⣵⡵⣾⢽