⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣦⢤⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣽⣞⣿⣻⣶⡴⣽⡯⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡲⣳⡫⡯⡳⣟⡾⡯⣿⣻⢽⡻⣝⣖⢦⣀⡀⠀⠀⢀⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⣀⡠⡮⣳⡳⣝⢮⢮⡫⣝⢮⡺⣝⢗⡽⣳⢻⢮⢮⡳⡵⣝⢖⡞⡮⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢯⢏⣗⡽⣞⣯⢿⣻⣻⡽⡿⣮⢳⢝⣎⢗⣝⢮⡳⣝⢵⡻⣗⣷⣳⡽⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡮⣫⣳⡯⣿⢽⣞⣯⣟⣷⣻⣟⡗⡽⣱⢳⡳⣕⢗⣝⢮⡳⣝⣿⣺⢷⡿⣮⢻⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡯⡳⣝⢮⡺⣫⢿⣽⢷⣻⣾⣳⢿⢵⡹⣚⢎⢧⡫⡮⡳⡕⡷⣝⢽⢿⣞⣟⣿⢽⡳⣕⢯⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢝⢮⡫⡮⡳⣝⢮⡳⣝⢽⡻⣞⢯⢳⡱⡕⣕⣝⡜⣜⢜⣜⢎⢮⢪⣪⣻⣯⢟⢾⢟⣝⢎⣇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⡳⣝⢮⡫⡮⡳⡝⣎⢷⢝⡮⡮⡣⡧⣻⡽⣞⢼⢜⡜⡮⣳⢹⣪⢎⢮⢯⢧⡳⣝⢮⡳⡵⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠝⣎⣗⢝⢮⣫⡺⣪⢿⢝⠜⡜⡸⡘⢆⠣⠪⠸⠨⡊⠪⠪⡙⣟⡯⡻⣪⡳⣹⢼⡱⣳⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡦⣺⡪⡮⡫⡧⡣⣏⢾⠕⡁⢧⢕⢾⣽⡙⢎⢧⠡⢁⠂⢅⠱⣼⡍⢝⢾⢕⣝⢮⡺⡺⣪⡳⣢⡲⡪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣝⢞⢮⢮⡫⡮⣫⡺⣪⢳⡣⠂⡂⠕⣿⣳⡯⠂⠨⠠⠡⠨⢐⢸⣽⡆⠐⡼⣕⢕⡗⣝⣝⢮⢮⡳⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣕⢗⣽⣳⢳⢝⢞⢮⡺⣪⡳⢽⡰⡐⢅⢓⠳⢋⠄⠅⡊⠨⡈⡐⡸⢳⢡⢨⡳⣕⢗⣝⢮⣪⡳⡵⡝⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢳⢽⣕⢏⢯⡺⣺⣪⡺⡳⠝⠞⡐⡐⠨⢐⠠⡁⡂⠅⡂⡂⠝⡐⠕⡳⡽⣺⡳⣕⢗⢵⢝⢮⡫⡞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⡠⣖⢗⣕⢯⡳⣝⢵⣫⢯⣟⡧⣥⣂⠌⠌⡐⡐⡐⠄⡱⢘⢐⢐⠠⣡⡲⣯⣳⣝⢮⡫⡳⣝⢵⢝⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠱⣕⢗⣝⢮⡳⣝⣗⣗⣟⣞⣞⢽⣳⢆⢦⢢⢅⣆⢦⣲⣲⣝⢷⣝⣞⣞⢼⣚⢮⡫⡎⠫⣗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢮⢗⢽⣪⣺⢼⣞⣞⢾⢽⣟⣿⡾⣾⢾⣾⣻⡺⣮⣺⣳⡳⣽⣺⣽⡪⣗⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠄⠑⠝⣟⣞⣾⢽⢯⢫⢳⢟⢿⢻⠷⡹⣹⣺⢺⢞⠙⠑⠑⠑⠉⠁⠀⢠⠢⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠊⢅⠱⢹⣇⠇⡎⡪⡪⡪⢪⢢⢫⡧⡧⡧⣄⣀⢤⢔⡄⠀⣗⠨⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡠⢁⠢⠨⠨⣷⡁⡊⠌⠌⡊⠌⡂⡂⢿⢽⡺⡜⡼⡱⣣⢻⡪⡪⡪⡡⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡁⡂⡂⢌⢐⢁⢚⡧⠠⠡⢁⠂⠅⡂⡐⣹⡽⣽⡪⡳⡹⡜⣕⢷⡑⡕⡆⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡯⡀⠢⢐⠐⡐⢔⣴⣿⣿⣷⣤⡡⢑⢐⣰⣷⣿⣷⡫⣇⢯⢺⢜⠮⡪⠢⣣⣟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⣟⣿⣻⣗⡷⣾⢽⣯⢿⡾⣟⣯⣿⣖⣾⣿⣻⣷⡅⠈⠊⠁⠁⢁⡀⡀⣼⣻⣞⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣟⣾⣳⡯⣿⣽⣻⣞⣯⢿⡽⣞⡷⣯⢿⡾⣽⡞⡣⡢⠪⣗⡪⡪⡩⢹⣺⡷⣯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢾⣳⣯⢷⢿⡽⣞⡷⣯⢿⡽⣯⣿⡽⣾⣟⣿⣽⡗⠡⠠⡁⡚⣧⣂⠪⠑⣕⡿⣽⢯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣗⣯⣟⣯⢿⡽⣯⣿⣽⣟⡿⣞⣿⣟⣾⣻⡾⢈⠌⢂⠂⡂⠸⣶⣟⣯⢿⡽⣯⢿⣺⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣀⠀⠀⠀⠀⠉⣾⢷⣟⣿⣻⡿⡾⣷⢿⣻⣟⣯⢿⣽⣻⣽⢦⠨⢐⠈⣄⡷⣟⣯⡿⣯⢿⡽⣯⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⢞⢶⡠⡀⠀⠐⢿⡽⣯⢿⡽⣽⣻⣽⣻⡽⣾⢽⣻⢾⡽⣯⡿⣷⣢⡷⣟⣿⢯⡿⣯⣿⣻⣽⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣝⢽⢕⢯⡳⡵⡴⣌⣿⢽⡯⣿⢽⣳⣟⡾⣯⢿⡽⣯⢿⡽⣯⣿⢿⣽⡂⠙⢽⣟⣿⣽⢾⣻⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡳⣝⢵⢝⣞⡺⡪⣞⢯⣿⢽⡯⣷⢯⡿⣽⢯⡿⣽⣯⢿⣻⣾⢿⣽⢷⠀⠀⠁⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀