⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢯⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢌⣗⢘⠣⡙⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠊⣤⢀⠊⡕⡈⢔⡷⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢸⡀⢿⡗⡘⡎⣾⣿⠏⣠⠎⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢌⢿⣦⣴⡀⣶⣨⣴⣾⡟⢪⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣣⢵⡙⠿⢟⣛⢛⡳⣫⡼⣗⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⣬⢎⢮⡙⣙⢡⡍⣯⠩⢶⣕⣧⠟⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈⣉⠒⠛⠙⠓⠎⠚⠛⢋⣈⣤⢬⠱⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡌⣫⢿⣝⢟⡟⣟⢽⣟⢿⣽⢧⣳⠢⢻⣿⣿⣿⣿⣷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⡿⣻⣧⠘⡮⣻⡯⡾⠽⡮⣞⠽⣷⡳⣭⠇⢺⣯⢿⣿⣽⡿⣿⣮⢗⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣗⣿⣻⣿⣆⠑⢿⡳⣾⣟⣟⣟⡿⣫⡿⣪⠇⣺⣝⡿⣯⣿⣟⣿⣷⣻⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢟⣽⢿⣟⣿⢿⣾⢿⣞⣷⣻⣇⠷⠠⣏⢾⡵⡩⢮⢿⢼⠁⡏⢴⡿⣽⢻⠿⢛⣭⡫⣖⢳⢆⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢞⣯⢷⣻⢟⣿⣟⣿⣟⣿⡾⣵⢿⡞⣗⢾⣝⠯⡽⠷⠟⢁⢎⢭⣿⡻⡧⣳⢳⡀⢧⡫⣮⡳⣫⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⣻⡳⣯⠟⣛⡻⣫⢻⣝⢙⢩⣟⡽⣪⢷⣋⣤⣦⢶⢾⢶⢿⣫⡿⣽⣿⡸⣣⣗⣼⣯⣾⣯⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡁⢂⢿⡃⠀⡯⣪⢕⢧⡺⡄⢸⣎⣿⡳⣯⡻⣮⣳⢯⣻⣝⢷⡻⣞⣷⣝⣿⣻⡿⢟⡻⣍⢷⢲⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⡈⠊⡜⣘⢗⣌⣺⣮⣳⣗⣽⣧⣾⣮⢷⣻⡵⣟⣵⢯⢷⢯⡾⣫⡿⣝⣾⣵⡏⡶⣲⠈⢗⣝⢮⡳⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡢⠌⠔⠤⠳⡌⡦⡹⡾⢟⣟⢻⣛⠛⢛⣯⢷⣯⣻⣞⣯⣻⢗⡿⣽⢞⣿⡺⡷⣿⢸⢧⢏⣼⣣⡿⣾⢕⡝⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡒⣍⢚⢖⡹⡱⡪⡎⢀⣏⢳⢕⡝⣼⠈⢸⢮⡷⡿⣾⢵⣯⢯⣟⣽⢯⣿⣝⣿⡻⣛⢟⠘⡯⢮⡺⢮⡆⢱⡝⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡪⡢⢝⢌⢇⠷⣸⠪⡎⢾⢄⣳⣝⣮⣞⣮⣧⣹⣮⣿⣻⣽⢷⡿⣽⣯⣻⡷⣿⣞⣿⡧⣫⢝⡇⣸⡳⣝⢮⡏⣜⢞⡼⡱⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡌⡖⣕⢣⡫⣪⢳⡱⣫⢪⡳⡵⢭⡫⢖⡅⢰⡳⣝⣿⢷⣿⣽⣯⡿⣷⣯⣿⣽⡷⣿⣽⣿⣜⣵⣧⣾⣽⡿⡿⣸⢕⢯⡺⣕⢥⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢌⢞⠼⣌⢇⡗⣕⡵⣱⡣⡳⣕⢗⡵⣝⢮⣃⠜⣵⡣⣿⡿⣾⢷⣿⣽⢿⡾⣷⣟⣿⣻⣾⣯⣿⣿⣿⣿⡿⣿⢯⡺⣕⢗⣝⢎⣯⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢨⡣⣫⡺⣪⢗⡽⣬⠺⣆⢟⢼⣶⣽⣾⣾⡷⣹⢳⡝⡮⡯⣝⡭⣏⣞⢝⣏⢟⣝⢽⡝⡯⣷⣻⣿⣿⣿⣻⡿⣿⣷⢝⡮⣳⣝⡽⣮⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢣⡏⡾⣜⢧⣫⠞⣞⡳⣝⡵⣳⣯⣿⣝⣿⡗⡽⡵⣹⡻⡾⣮⢺⢮⡺⣕⡯⣷⢝⡮⣞⢵⡳⣿⢿⡾⣯⣿⣻⣯⡿⣳⡽⣺⢮⢷⢪⢋⢄⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡪⣗⠽⣪⠗⡙⣘⢊⠗⢽⣞⣷⣻⡾⣽⡗⣽⡪⣯⣻⢽⣎⢷⡫⣞⢵⣟⣽⢳⡝⣮⢳⣻⣿⣟⣿⣻⡾⣯⠿⢃⢩⢈⢅⣡⣢⣥⡳⡯⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢘⣈⡤⡢⣈⡲⠫⣳⢳⡝⣦⡙⣺⡯⡿⡻⣝⢮⢞⣵⣛⢷⢕⡯⣺⢕⢿⣪⢷⣫⢞⡵⣫⢾⣿⣽⢯⣿⣻⠃⠘⡯⣫⡻⣕⢯⢶⣕⢟⣆⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡗⣵⠉⢸⣎⢧⣑⣜⢧⣫⢮⣝⢿⢾⡫⡳⣝⢮⡳⡵⣝⢮⡳⣝⢮⣫⡳⣝⢵⡳⣝⢮⡳⣗⡵⡺⡧⣿⣝⣷⠆⠹⣪⡻⣮⢳⢷⢭⣏⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⢮⡲⡯⣪⢗⡽⣪⢗⡗⣷⣹⡯⣿⢾⣿⢪⢗⣝⢞⢮⡳⣝⢮⡳⡵⣝⢮⡳⣝⢮⡳⣽⠊⢞⢽⡺⢷⡽⣞⡏⠀⢝⠞⡷⡫⠷⠻⠎⢮⠠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡧⡻⠘⢅⢩⡉⡙⡫⢾⣧⣻⣞⡗⢿⣻⣷⡿⣾⣯⣷⣽⣮⣷⣽⢾⣎⣷⣹⣎⣷⣹⢞⡯⣳⢝⡗⢼⡻⡯⠇⡐⡭⡪⡪⡪⡙⢕⡩⠢⣅⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡡⠐⠑⠄⠡⠈⠎⡰⢂⠝⣮⣯⢏⣻⣿⣳⣿⣟⣾⢷⡿⣾⢷⣿⣻⣯⣿⢯⣿⣻⢯⢧⡻⣎⢯⣜⣚⢽⠀⠀⠚⢔⢌⠪⡢⢑⢌⠢⣃⠪⠂⠀⠀⠀
⠀⠀⠀⠀⠀⡀⢖⢜⣤⡵⣳⢿⣦⣔⣁⡊⠙⣞⢵⢹⣯⣿⢾⣽⣻⣟⣿⣽⣟⣷⣿⣳⣿⣻⣯⢟⡵⣫⣞⢵⡫⣮⢳⡗⠀⠀⠀⠈⣻⡳⡷⣾⢶⣟⣿⠀⠀⠀⠀⠀
⠀⠀⢀⢔⢮⡉⣾⢯⢷⣝⢷⢯⡾⣝⣽⡻⣷⡨⣗⣝⣿⣽⢿⣯⣿⣽⡾⣷⢿⡇⢙⣯⣿⣽⡳⣳⣝⢷⣹⡝⣷⡹⣮⠁⠀⠀⠀⠳⢏⣯⣻⣞⢿⡮⣷⠄⠀⠀⠀⠀
⠀⠀⢸⡱⣣⢫⣟⣵⡻⣮⣻⡳⣯⢟⡾⡽⣗⢿⡜⠢⠟⣽⡿⣷⢿⡾⣿⣻⡟⢠⠙⠻⠫⣮⢯⣳⣝⣧⡻⣺⢷⣝⡇⠀⠀⠀⠀⡠⣜⡮⡷⡿⣵⣻⣽⠂⠀⠀⠀⠀
⠀⠀⠘⣜⡵⣍⣾⣳⣽⣳⡽⣽⣝⢯⣾⣹⣿⢯⢧⢾⠠⡸⣿⣿⣻⣿⣻⡟⡠⣳⢯⣎⣦⣼⡳⣳⡵⣳⡽⣳⣟⢾⠀⠀⠀⠀⠀⠈⠉⠚⣛⢾⡯⣺⡎⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢞⣵⡻⣮⣷⣝⢿⣮⣻⣧⣻⣜⢾⡳⣝⡯⠀⣟⣞⢾⡽⣽⡽⢁⠼⠯⠻⠞⠷⡽⣝⡷⣽⣳⡽⣫⡾⠇⠀⠀⠀⠀⠀⠀⠀⣾⠽⠋⢨⡿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠑⠿⠞⠿⠳⠛⢪⡵⣫⠚⢉⠘⡃⢨⡷⣝⣯⢾⣳⢃⣔⣱⣈⣎⣪⣒⣔⢌⣌⠕⡙⠾⣝⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣺⣝⢷⢵⢾⠈⣜⣯⢞⡷⣫⢿⣝⡷⣽⣝⣽⢽⣝⣽⣫⣟⡽⣯⡲⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠙⠳⠹⡯⣷⠏⠢⣽⢮⣻⣽⣫⡷⣻⢞⣵⢯⣞⡷⣻⢾⡵⣟⢾⢵⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣮⣐⢄⡉⡃⢱⣯⣳⣟⡾⣪⣯⣫⢿⡵⣟⡾⣽⣫⡷⣟⡝⡱⡡⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⣯⡻⣶⣕⣸⣗⢿⢾⣑⠆⢮⢑⡿⣺⣯⡻⡷⣽⣫⣟⣿⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⣝⢿⡮⣿⣵⣟⣯⣿⣫⡿⣷⢮⡻⡷⣽⣫⣟⡷⣽⣳⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣮⣟⣯⢿⡾⣞⣾⣗⣿⣺⣟⡾⣯⢿⣽⣳⣟⣮⣟⢷⣯⢾⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡾⣽⢯⣿⣝⣿⣳⣟⣾⢷⣯⢿⣽⣳⣟⢾⣳⢯⣾⡻⣞⣯⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡿⣾⡽⣾⢯⣯⣻⡷⣻⡿⣮⢷⣯⣻⣝⡷⣯⢿⣽⢯⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀