⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣺⣞⡶⣶⣻⢽⣺⣒⡦⣤⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡴⣯⣳⢗⣗⢇⠯⡧⡣⡫⡺⣳⣳⣷⡫⣦⡀⢀⢇⠥⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢢⢀⡴⣻⡵⣯⢻⣰⣱⢯⣻⡺⡮⣞⣮⡺⣺⢿⣮⣗⢅⢇⢣⠠⡸⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⢕⢕⢘⢽⣗⡯⣗⣯⣞⡾⣽⣺⢽⢽⣳⣳⡯⣮⡯⡳⣿⣗⢮⢪⢊⠎⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡪⢢⡳⣻⢽⣺⢽⣺⣺⡎⣟⢾⣺⢽⣳⢯⣺⣺⣳⣻⢽⢏⢿⣷⡨⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠍⣖⡯⣟⡾⣝⢞⣾⣱⡟⡞⣽⢽⣞⡯⢗⢿⢼⢽⣝⢽⣺⡻⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣳⢯⢳⣫⣗⢻⣲⣟⣺⢿⣺⢽⢾⡝⣶⣏⣷⢷⣣⣗⡷⡽⢽⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣝⣽⢸⣳⣽⠷⠟⠗⡿⣟⣳⣟⡯⣷⢏⡪⡍⣍⢷⣳⢯⢫⡿⡽⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣞⣞⣞⣚⣗⡧⢿⣏⡯⡲⣽⣷⢻⣾⣵⣼⡪⣿⡿⣺⢵⡻⡽⣺⢯⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣺⢼⢺⣜⢮⢟⢾⣿⣟⣿⣷⢿⣿⣿⣻⣿⣯⡍⡞⣵⣝⠇⣽⢽⡅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠊⡷⣱⣻⡺⣖⢻⣿⣻⣿⡿⣿⢿⣟⣿⣷⢟⢼⢽⣳⢽⠐⢵⠈⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢮⡪⣯⡫⣦⢎⢟⠿⢿⣿⣿⢿⡫⡣⣓⢾⢽⠪⠗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⡝⡷⣔⢟⡮⡕⢕⢍⢓⠕⡪⢓⢩⠪⢲⣫⢳⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢽⣝⢪⣲⢧⢝⣎⣧⣳⢽⢱⣽⣽⣾⣞⣯⣁⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⣟⣖⢯⢞⡷⣱⣮⣽⣿⣟⣷⣟⣯⣯⣿⣾⡾⣿⣿⣷⣿⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣹⣷⣼⣾⣯⣫⢾⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣿⢿⢿⣻⣽⣯⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣷⣷⣷⣴⡻⣻⣿⣿⣷⣿⣿⣿⣾⣿⢿⣿⣿⣟⢿⣿⣿⣿⣿⣽⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⣷⣧⣷⣮⡿⡿⣿⣯⣿⣿⣽⣾⣿⣾⣿⣿⣯⣯⣻⣿⣿⣿⣾⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢾⢿⢿⣟⢾⢎⢞⣞⢷⣿⣿⣿⣽⣟⣿⣽⣿⣻⣯⣟⣾⣟⣟⣿⢮⣶⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣾⣿⣿⠸⣺⢯⢞⡼⡭⣿⣷⣿⣿⡯⣿⣿⢿⣿⡿⣾⣿⣿⣿⣿⠈⢻⣿⣿⣻⣽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⡟⠇⠨⢯⡮⣇⢯⣳⣟⣿⣿⣾⣿⣿⣿⣿⢟⣿⣿⣿⣿⣷⡟⠀⠀⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⣿⡿⠁⠀⠀⣭⡫⣧⣻⢸⣟⣭⣟⠯⣟⣯⣿⣞⢿⡿⣿⣿⣿⡗⡃⠀⠀⠀⢻⣿⣽⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣯⢷⠟⠀⠀⠀⠘⣗⡧⡳⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣬⣿⡣⠀⠀⠀⠀⠀⠹⣿⣿⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣻⣿⣾⣟⠎⠀⠀⠀⣀⡶⡜⣝⡽⣅⣿⣿⡿⣷⣿⣿⣾⣿⣯⣿⣿⢿⣻⣿⠁⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⢀⢐⠄⣄⡮⣻⣸⡪⣹⣾⣿⣿⣿⢿⣿⣽⣿⣷⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⠓⠁⠀⡐⢕⢸⢗⡵⣳⢳⢭⣾⣿⣯⣷⣿⣿⣿⣿⣷⣿⣟⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⢿⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⣰⣪⡞⡎⢌⠎⡍⠕⣽⣿⣿⣿⣻⣿⣿⢿⣿⣾⣿⣷⣿⣿⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣽⡞⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡷⣵⠑⢅⠁⠈⣨⣿⣿⣿⣽⣿⣿⣻⣿⣿⣯⣷⣿⣿⣯⣿⣿⡧⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢽⣺⢸⠱⠂⢠⣿⣿⣿⣻⣿⣻⣿⢿⣟⣯⣿⣿⣿⣻⣽⣿⡿⣿⡀⠀⠀⣠⣾⣿⣿⡿⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡝⣞⣇⠇⡢⠿⣿⣿⣟⣿⣿⢿⣿⣿⣟⣿⣿⣯⣿⣿⣿⣿⣿⣿⡇⢰⣾⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠉⢓⢊⠪⡢⢪⢩⢙⡛⡻⠿⢯⣿⣿⣟⣿⣯⣿⣾⣿⣺⣿⣷⣽⣿⣿⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡨⠢⡃⠇⡆⢕⢌⢢⢑⢜⠸⡐⢆⠆⡎⢝⢛⠝⢷⠳⠿⣿⣧⣿⣿⣯⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢪⢊⢪⠱⡘⡌⢆⠕⡌⢆⢣⠪⡢⠣⡊⡢⡑⡅⢅⢣⢑⠢⢺⡿⣟⢿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢜⢔⢑⢅⢕⠱⡘⡌⡪⡘⡌⢆⠣⡊⡎⡪⡨⢪⢘⢌⢆⠱⡑⣹⡏⣿⠜⣿⠹⡣⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡔⢅⢇⠪⡂⡇⢕⢌⢢⢱⠸⡨⡊⢎⢢⠱⡘⡌⡢⡑⢔⠱⡐⢔⠹⣹⡙⡘⡌⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⡑⡜⢌⢆⢣⠱⡘⡌⢆⢣⠪⡸⡐⢕⠅⡣⡑⡱⡨⢢⠪⡊⡌⡌⢆⠣⠑⢕⠱⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢕⢱⠸⡨⡢⢱⢑⢕⢑⢅⢣⠱⡑⢜⢌⢎⢜⠔⡱⡘⡌⢆⢕⢜⢨⠢⡣⡙⡌⡂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢢⢱⢑⢕⠌⡎⡢⡣⡱⠨⡊⡎⡪⢂⢇⢎⢢⠃⡕⡱⡸⡨⡢⡣⢅⢣⢱⢑⠸⡸⡨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢅⢇⢕⢅⢣⢱⢑⢜⢌⠪⡊⡎⡪⢨⢢⢣⢱⠡⡣⡱⡸⡰⢱⢸⢨⢂⢇⣣⡳⡡⢪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢢⢣⢱⠡⡣⡱⡡⡣⠱⡱⡑⡕⡸⡰⡑⡅⡇⢕⢅⢇⢎⠜⣔⡕⣕⣯⣾⣾⠐⢕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣾⣮⣎⣪⢼⣘⣜⢜⣔⣕⣕⠕⣕⣥⢧⢧⣥⣿⣿⣿⣿⣿⡿⡿⡈⠢⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⣿⡿⣿⣿⣾⣷⣟⡮⣿⣿⣿⢾⡾⣿⣻⣽⣿⣽⣷⣿⣿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⠿⠿⠿⠿⢽⣿⣛⣹⣿⣿⣿⡯⣿⣿⣻⣛⣻⣻⣿⣯⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣿⣿⣿⣿⣿⣿⣿⢿⢸⣿⣿⣷⣿⣻⣿⠿⠿⠿⣿⣿⣟⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣾⣿⣿⣿⢸⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁