⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡽⣯⢿⢽⣻⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢯⡿⡽⡾⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣯⢯⡿⣝⣗⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⣯⢯⡯⣷⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢠⢰⢰⣰⡰⣄⢄⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⢯⡯⣗⣯⡃⠀⠀⠀⠀⠀⠀⠀⠀⣔⣮⢾⢮⣷⡷⡿⣯⣟⣞⡾⣜⣔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⢽⡽⣽⣺⡂⠀⠀⠀⠀⠀⠀⢀⢜⣵⢯⣟⣯⢷⣟⣯⣷⣟⣷⣻⣳⢵⣕⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢯⢯⣟⣞⣞⢎⠀⠀⠀⠀⠀⢠⣺⣽⢾⣻⣽⢾⣻⢽⢺⣳⣻⣞⡾⡽⣝⠾⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡯⡯⣟⣞⡮⡧⠀⠀⠀⠀⠀⣗⣿⣾⢿⣛⢎⢇⢎⢎⢲⢘⡺⣺⣽⢯⡳⡹⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣝⡽⣽⣺⢽⣺⠀⠀⠀⠀⠀⠑⢯⣿⣣⡳⡱⡕⣕⢵⣱⢕⢎⢎⡚⡗⡭⢺⢸⣢⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢵⣫⣞⣾⣝⣚⡢⡤⣄⢆⣤⢄⣅⢌⣞⣮⢷⡝⡎⡿⣹⢹⢸⢰⢱⢱⢱⢱⠱⣝⢇⠂⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢩⢗⡗⣗⢽⢯⠷⡿⡽⣞⣟⢟⣿⣺⡻⠹⠺⡻⣽⢸⢘⢜⢜⢜⢎⢎⢎⢪⢪⢣⣳⢽⠔⢕⢑⠅⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣌⢯⢷⣻⣳⢿⢽⡽⣫⡿⣽⢽⢷⣻⣗⡿⡮⣿⢷⣇⣇⢟⢷⣻⢼⡿⣽⢯⡀⠀⠘⠵⡵⣕⢇⢇⢇⡣⡣⡣⣣⣣⣗⢯⠣⡑⢅⢂⢕⢐⠡⡀⡀⠀⠀⠀⠀⠀⠀⠀
⠁⠉⠉⠈⠁⠉⠁⠉⠉⠉⠉⠉⢹⢯⢯⢯⡻⡩⡕⡕⣜⢬⢹⢵⣿⣻⣻⣯⢣⢣⡀⠀⠹⣮⡳⣝⢕⢕⢕⣕⣗⡷⡫⠢⡣⡑⡑⠔⢄⠅⠕⡨⠢⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣫⡯⣏⢯⢝⣞⢼⡱⡳⡩⡻⣽⣟⢮⣓⢕⢅⡀⠀⠐⠹⢪⢮⣳⣳⣳⡿⡹⡨⡱⡨⢢⠱⡑⢕⠜⡌⡆⢆⢌⠢⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣗⡷⣯⢷⡿⣽⣺⣳⢽⡱⡣⣣⣗⢯⣗⣗⢧⠋⠻⣶⣴⣄⣄⢼⡿⣞⣿⢹⢸⠨⡂⢇⢅⠕⢌⠢⡑⡅⡣⡣⡢⢅⠅⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢽⣞⣟⣿⣻⣞⠺⢹⢾⢽⣺⢜⢜⠜⡮⡃⠀⡀⢌⢞⢿⡿⣿⣽⣳⢯⢞⢎⢎⠪⡊⠔⢅⠣⡑⡌⡢⡑⡌⢎⢆⢣⠡⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡯⣷⣻⣞⣷⣻⠀⠈⠙⡽⡎⡇⡇⡇⡣⠣⣣⢪⡣⣏⢮⡻⣷⣿⣻⡽⡣⡣⠣⡱⡘⢜⣐⠱⡨⢢⠱⡨⢊⠪⡢⠡⡃⢅⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣳⣻⢾⡽⣞⠀⠀⠀⠈⡮⣣⢣⢣⠣⡣⡊⢕⢯⢞⡽⣽⣳⣿⢯⠪⡊⡂⡣⢒⠸⡐⡔⡱⢡⢱⢨⠢⡑⡅⡪⡨⡪⡐⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣗⡯⡿⣝⣇⠀⠀⠀⠀⣟⢮⡣⣇⢏⢆⢇⢕⢔⠩⡙⢞⢽⢿⠇⢇⠕⢌⠢⡡⢑⠌⢜⠸⡸⡰⡡⡃⡇⡣⡣⡣⡪⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣳⣳⣻⢽⣻⢎⠀⠀⠀⠀⣗⢯⣞⡮⣺⢸⡸⡸⡰⡱⡡⡱⢨⠪⡘⡐⢅⠕⡨⢂⠕⢌⠢⡑⡱⡱⡹⣸⢸⢜⢜⢜⠬⡱⠨⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣳⣳⢯⣻⢾⣝⠀⠀⠀⠀⠈⠗⠳⠛⠎⢧⢳⡱⡱⡱⡱⡪⡢⡣⣊⢪⢐⢑⢌⠢⡑⢥⠱⡘⣔⢵⡹⣪⢳⢣⢳⢱⠱⡘⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡽⡾⣽⣺⡽⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢝⣎⢮⢪⢪⢪⠪⡢⡣⡱⡨⡂⢇⢕⢕⢵⢝⡮⣇⢏⡎⡎⡎⡪⡢⡣⢣⢊⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣽⡳⡯⡾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢺⢜⣜⢔⢕⢕⢜⢜⢔⢕⢕⢜⢜⣞⢽⡺⣪⡣⡳⡹⡸⡸⡸⡸⡨⡢⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢾⢽⢽⣝⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠕⣗⢵⡱⡵⣝⢮⡳⡽⣪⡳⣝⢵⢝⢮⢺⣸⢸⠸⡸⡨⡪⡸⡰⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡽⣽⣳⣳⡽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠊⢫⢎⡗⣝⢞⢮⢺⡪⡳⣝⢕⢧⢳⢹⡪⡪⣊⠪⡊⡢⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⣗⣗⣗⡿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢕⣝⢎⢧⢳⢕⡝⡮⡪⡇⣏⢮⢪⡪⡹⡸⡜⡔⡔⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣺⣺⣪⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⢪⡳⡹⣕⢧⢫⡪⣣⢫⢪⢪⢪⢺⢸⢜⢜⢎⢮⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣞⣞⣞⣞⣞⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢕⡳⣹⡹⣜⢮⣣⡫⡪⡎⡇⡇⡇⡇⡇⡣⡣⡣⡣⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⢞⣞⣗⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡧⡳⡕⡧⡣⡫⡺⡸⡪⡪⡪⡪⠪⡪⠪⡊⢎⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡾⣽⣝⣗⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⡣⡣⣫⢺⢜⢵⢹⢸⢸⢸⠸⡸⡸⡨⡊⢎⠜⡌⡒