⣟⣯⡿⣯⢿⡯⣿⢯⡿⣯⢿⡯⣿⢯⣟⣯⡿⣯⣟⣯⣟⣯⢿⣻⣽⣻⡽⣯⢿⣽⣻⣽⣻⡽⣿⢽⣻⣽⢿⡽⣿⢽⣯⢿⡽⣯⢿⡽⣯⢿⢽⢯⢿⡽⡯⡿⡽⣯⢯⡿⡽⡯⡿⡽⡯⣯⢿⢽⢯⣟⣽⢽⡯⡿⡽⣯⢿⢽⢯⢿⢽⢯⢿⢽⢯⢿⢽⢯⡯⡿⣽⢽⢯⡯⣯⢯⡯⣯⢯⡯⣯⢯⢿⢽⡽⡯⣯⣟⡽⡯⡿⡽⡯⡿⡽⡯⣟⣽⢽⢽
⣟⣷⣟⣯⣿⡽⣯⡿⣽⣯⢿⣽⢯⣿⢽⣷⣻⢷⣯⢷⡿⣽⣻⣽⢾⡯⣿⢽⣻⣞⣷⣻⢾⣻⣽⢿⣽⢾⣻⣽⢯⡿⣾⣻⣽⢿⡽⣯⢿⣽⣻⣻⡯⣿⢽⣻⡯⣿⢽⣽⣻⣽⣻⣽⣻⡽⣯⢿⡽⣾⢽⡯⡿⣽⣻⣽⢽⣻⣽⣻⢯⣟⡿⣽⣻⢯⣟⣯⢿⡽⡯⡿⣽⡽⣽⡽⣽⢾⡽⣽⣳⣻⡽⣯⢯⡿⣵⢷⣻⢯⣟⡯⡿⣽⣫⣟⢷⢽⢽⢽
⡿⣞⣷⣟⡾⣯⢿⣽⣟⣾⢿⡽⣯⡿⣽⡾⣯⡿⣾⣻⣽⣯⢿⣾⣻⣽⣟⣿⣽⢾⣻⡾⣟⡿⣞⣿⢾⣟⣯⣟⣯⣿⡽⣷⣻⣯⢿⣯⢿⣳⣟⡷⣟⣯⡿⣯⢿⡽⣯⡷⣿⣺⢷⣯⢷⣟⣯⡿⣽⢯⡿⣽⣻⣽⡾⣽⢯⣿⣺⣽⢯⡷⣟⣟⡾⣯⢷⣻⣽⣽⣻⡽⣗⣿⣳⢿⡽⡯⣯⢷⣻⣞⣯⢿⡽⣽⡽⣽⣞⣟⡾⣽⢽⣺⢞⡾⡽⡽⡽⡽
⡿⣯⡷⣯⣟⣯⣿⣳⢿⣞⣿⣻⣽⣻⣽⣯⡷⣿⢯⡿⣾⡽⣟⣾⣻⣞⣷⣟⣾⢿⣽⣻⣽⣟⣯⡿⣯⢿⣞⣯⣷⣯⡿⣽⢷⣻⣟⣾⣟⣯⣷⠻⠯⣷⣟⣯⣟⣯⣿⢽⣻⣾⣻⣾⣻⡽⣗⡿⠟⠛⠉⠫⢿⣺⣯⢿⣻⡾⣽⣾⣻⣟⣯⡿⣽⣯⢿⣽⢾⣞⣷⣟⡿⣾⢽⡯⣿⢽⡯⣟⣷⣻⣞⣯⣟⣷⣻⣗⣿⣺⣽⢽⡽⣞⣯⢯⡯⣯⢯⣻
⣿⣳⣿⣻⣞⣿⣺⣽⣯⡷⣟⣷⣻⣽⢾⣳⣟⣯⡿⣯⡷⣿⢯⡯⣷⣟⢷⠻⠾⣟⡷⣟⣷⣯⡷⣟⣯⡿⣯⡷⣷⠈⠈⠉⠉⠋⠓⠻⠺⠿⢾⣻⣦⡀⠀⠉⠉⠳⠿⣻⣽⣾⠽⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⣾⣻⢷⣟⣿⣺⣗⣯⣷⣟⢿⢞⣿⡽⣯⢿⣺⣷⣻⣯⢿⡽⣯⣟⣿⡽⣗⣿⣺⣗⣿⣺⣗⣿⣺⣗⣯⢿⣽⣻⣞⣯⢯⡷⣯⢷
⣿⡽⣾⣻⣞⣯⡿⣞⣷⣟⣿⣽⣽⡾⣟⣯⣟⡷⣟⣯⡿⣽⣻⣽⣟⣾⣯⡷⣧⣄⡀⠀⠀⠈⠉⠉⠉⠙⠉⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠁⠁⠁⠀⠀⠀⠀⠈⣞⣯⣿⣻⣽⢾⣳⣟⣯⣿⣻⣞⣷⣟⣯⣷⢿⣞⣷⣟⡾⣗⣿⣞⣯⡿⣞⣷⣻⣞⣯⡿⣽⢯
⣷⢿⣯⡷⣟⣷⣟⣯⣷⢿⣞⣷⢯⣟⣯⡿⣞⣯⡿⡮⠯⡩⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣗⡿⣞⣯⣿⣻⢾⣳⣿⣺⣽⢾⣳⡿⣞⣯⣿⣺⢷⣟⣯⡷⣯⡷⣟⣯⡷⣟⡷⣯⢿⣽⣻
⣟⣯⣷⢿⣻⣗⣿⣽⢾⣻⣽⡾⣿⢽⢷⣟⣿⣳⣟⣦⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠚⠛⠛⠻⠺⠻⣞⣿⢯⣻⣯⠷⢯⣿⢽⣻⣞⣿⣽⣻⣽⢯⣟⣯⡿⣯⣟⣷⣻
⣿⣽⡾⣟⣯⡿⣾⢽⣻⣽⢷⣟⣿⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠁⠀⠀⠀⣬⢿⣽⣻⢾⣳⣯⡷⣟⣯⣟⡷⣿⡽⣾⢯⡿
⡿⣞⣿⣻⣽⡯⣟⣿⣽⣯⠛⠓⠛⠙⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⡀⢀⠀⢄⣐⣠⣠⣤⢦⣔⡴⡴⡔⡴⣔⣐⢔⣐⢄⣂⢔⢐⠄⡂⡠⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣻⣯⢷⣟⣿⣽⢾⣻⣽⢷⣻⣽⢷⣟⣯⡿⣽
⣿⣻⣽⣽⣾⣽⣦⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⣠⣰⢴⡼⡮⣽⡺⣲⣗⣿⣽⣾⢷⣿⣟⣿⣺⣽⣮⡿⣾⣽⣺⡾⣽⣜⡮⣫⢵⡝⡬⣓⢇⡳⣱⡑⢦⢑⡐⠄⡀⠀⠐⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⣚⢷⣻⣽⣟⡾⣾⣻⣽⡾⣿⡽⣾⣻⣽⢾⣻⣽
⣯⣟⣷⣟⣾⢷⣻⣽⢷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣦⣷⣻⡾⣯⡿⣯⣿⣻⣯⣿⢿⣽⣯⣷⣿⢿⡷⣿⣯⣿⣽⣾⣟⣿⢾⣻⣽⣵⢯⣻⣯⡻⡾⡹⣞⢷⣝⠮⡪⣣⡣⡣⣒⠌⢌⠢⡀⠠⠀⠠⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠈⠉⢉⣝⣷⢯⣟⣷⣟⣯⡿⣾⣻⣽⢯
⣿⣽⢾⣻⡾⣟⣯⡛⠛⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⢾⣻⣽⣾⡯⠟⣏⣿⢿⡾⣯⣷⣿⢿⣯⣿⢷⣿⢿⣻⣿⡾⣿⢾⣯⣿⢽⢿⣻⡽⣽⢻⢞⣾⣫⡮⣟⣮⣻⣞⡯⢷⣗⢾⢽⢮⣣⡁⠢⡊⡪⡈⠠⠈⠐⢀⠂⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠯⣿⣽⣾⣻⣽⢾⣳⣯⢿⡽⣗⣿⣻
⣷⣟⣿⡽⣟⣯⣷⣭⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣻⡿⣾⣟⣯⠟⠊⡠⣽⢯⣿⣟⣿⢿⣽⣾⣿⣿⣾⣿⡿⣿⣿⣯⣿⡿⣟⡿⡺⡻⡻⠝⠯⢏⣮⣛⣊⣓⣝⣙⢘⢈⢈⡁⠉⠊⠉⠉⠋⠃⠓⠑⠄⠓⠕⠠⠌⠄⠠⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⠽⢾⣻⣻⣽⢷⣻
⣷⣟⣷⢿⣻⣽⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢿⣻⣟⣯⣿⠷⡋⡒⢠⣪⡾⣟⣿⣯⣿⣽⣿⣿⣻⣽⡾⡿⡳⣻⣻⢳⣹⣺⣯⣿⡾⣿⣻⣿⣵⣷⡿⣷⣟⣯⣿⢯⣿⢽⣭⣱⡹⣯⡕⢍⡩⡨⢍⢆⡃⡒⠔⡀⠂⢄⠀⠄⠀⠀⡀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣴⡶⣶⣦⣦⣦⣦⣴⣬⣯⢿⣽
⡿⣞⣯⡿⣯⣟⣿⣻⣿⡻⠏⠀⠀⠀⢀⢖⡿⣾⢿⣯⣿⣽⣮⠃⡢⠠⣿⡾⣽⣿⣿⣽⣾⡿⡳⣫⣣⣡⣷⣟⣿⣿⢿⣿⣻⣿⣯⣷⣿⣿⣻⡾⣿⢾⣿⣻⣽⢿⡾⣟⣿⢯⡷⣯⣏⣷⣿⢵⣕⢔⢜⡜⣞⢜⢢⢣⢕⢔⢅⠌⡄⠡⡂⠡⡑⡰⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣺⣷⣻⣾⣳⣟⣯⣿
⣿⣻⣽⢿⣽⣻⣾⣫⣁⣤⣤⡴⠀⢠⠃⣜⣿⣻⣟⣷⣟⠞⢐⠎⡺⡿⣵⣏⢯⢷⢟⠹⢈⢨⣲⣵⣿⢿⣻⣿⣻⣽⣿⣿⣻⣯⣿⣽⢷⣯⡷⣿⣟⣿⢽⣿⣽⣟⣿⢯⡯⢯⢿⡯⡮⣷⠿⡵⡎⣎⢷⠸⡸⠲⡱⡱⢝⠵⠱⡑⠨⠠⠨⠠⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠻⣾⡽⣯⡷⣿
⣿⣽⢾⡿⣽⢷⣻⡾⣯⣷⣯⡯⣠⡏⡊⣾⣟⣯⡯⣗⡏⣢⢇⡧⢯⡻⡔⠝⡈⡆⢢⡡⡺⣺⢽⣷⣿⣿⣿⢿⣟⣿⡿⡽⣏⢯⣛⣝⣏⣞⢝⣝⢪⣉⢋⡉⣊⢉⠊⠊⠃⠫⠑⢍⢓⢕⠫⡊⡓⠵⡿⠅⠌⡈⠢⠩⢂⠑⡁⡪⡊⠔⡁⠐⢀⠀⠈⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⠈⣒⢶⣴⣴⣴⣔⣔⢟⣝⣿⣽
⣷⣟⣯⣿⣻⣻⣽⢿⣽⢾⣷⣻⢽⠊⡆⢷⣹⣫⣻⡎⠀⡗⡕⢕⣵⠻⢈⡼⡹⡪⣕⢵⣳⢞⢿⢺⣾⣯⣿⣿⢿⣟⣿⣾⢿⣻⣟⣿⣟⣿⣿⡿⣿⣻⣿⣟⢿⣻⡿⣿⢿⣗⣶⣦⣦⢤⣌⣄⣈⡈⠊⠀⠄⢄⠂⠌⡢⡐⠰⠨⢂⠅⠀⠐⠀⡀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⣗⣟⡾⣞⣷⣟⣾⢷⣻⣽⣯⡾⣾
⣷⣻⣗⣯⡿⣽⡾⣿⡽⣿⣽⣻⡇⠀⠸⠘⠵⡯⡷⡣⠂⠨⡨⣪⠂⢐⣑⢸⢜⠼⠊⣡⣐⡏⡟⣻⣛⣝⣗⣟⣛⣛⣙⣙⣙⣙⠙⠉⠫⠙⠎⠯⢻⢹⠳⡻⢻⡫⡿⣝⡽⣯⢷⢷⣹⣝⣕⢗⡕⣟⡯⢗⢖⡦⣄⣁⠀⠈⠁⠁⠂⠐⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣯⣿⣻⣾⣳⣟⣿⡽⣷⢯⣿⣻
⣯⡷⣟⡷⣟⣯⡿⣷⣟⡿⣾⠽⣿⡄⠀⠄⠂⠘⢝⢽⡀⢁⠑⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡻⠿⠿⣟⣿⡿⡿⣟⣯⣿⣿⣿⣾⣶⣶⣴⢤⣠⡀⡀⠀⠁⠌⠙⠘⠭⡫⡣⡻⢸⢺⢜⢲⠨⢐⠨⡊⢔⢌⢏⢜⠄⠄⠄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⢷⣻⡷⣯⡷⣟⣷⣟⡿⣯⡷⣿
⣯⣟⣯⣿⣻⣽⢯⣷⣟⣿⣻⣾⢿⣿⣤⠀⠀⠀⠈⠁⡛⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢶⢶⣤⣭⣝⡘⠙⠝⠽⡓⣟⣯⣿⣞⣿⣾⢿⣷⣳⢢⢄⢄⠀⠀⠀⠐⠈⠈⠢⠢⠤⡢⡢⣃⢢⡲⠕⡑⢑⠈⡀⠂⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⢾⣟⣯⡿⣯⡿⣯⡷⣟⣿⣽⣻⣽
⣿⣺⣟⣾⣽⢾⣟⣷⢿⡽⣗⠹⠽⢿⣾⡷⣦⡠⣈⠠⢑⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠐⠐⠁⠑⢰⢯⣦⡈⠪⢻⢷⢯⣯⣶⣄⡀⠀⠊⠪⠯⣻⢺⣹⢓⢏⢧⣙⠢⠅⡑⠄⠄⠀⠀⠠⠐⠐⠈⢐⠐⠁⡁⠂⡈⠄⠠⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣯⣿⡽⣿⣽⣻⣽⣟⣯⡷⣿⡽⣾
⡿⣞⣿⣺⣽⢿⣽⡾⣿⣻⡿⣄⠀⠀⠀⠙⡉⠻⡚⢮⣢⣌⢆⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣔⣯⣙⢿⣳⡥⠀⠑⢍⠯⡿⣟⢿⠵⣄⠁⠀⠀⠂⡑⠽⣚⠮⠨⣂⢅⠥⠊⣀⡂⣂⠀⠀⠀⠀⠀⠀⡀⠄⠂⡂⡊⠌⠄⠂⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡿⣾⣻⣽⣾⣻⡾⣽⣾⣻⡷⣟⣿
⡿⣯⢿⣽⡾⣟⣷⢿⣯⢿⣻⣟⠲⣀⠀⠀⠀⡈⡀⠑⠑⢕⢹⠳⠻⢻⠿⣿⢶⣷⢷⡷⣿⢻⡵⢩⢾⡿⡝⠩⣻⣻⣇⠄⠠⠑⢱⢊⢪⣺⢯⢳⢤⠀⠀⠀⠀⠄⠡⠃⠕⡐⡨⡪⡇⡪⡐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣯⡿⣯⣿⣳⣟⣷⣟⡿⣞⣷⣟⣯⣿
⣿⣻⣟⣷⣟⣿⣽⢿⣽⢿⣯⣿⡵⣿⡪⡀⠀⠀⠀⠈⠀⠠⠐⢀⠉⠂⡃⠊⠓⢉⠩⠈⠄⠁⢄⡣⣯⠛⠀⡡⡳⡽⡺⣇⡀⠈⠀⠠⠠⠈⢂⡂⠧⡣⠂⠀⠀⠀⠠⠐⠈⠀⠁⠅⠌⡂⠇⠂⠁⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣟⣷⢿⣻⡾⣯⡿⣾⣽⣻⣯⡷⣟⣷⣟
⣯⣷⢿⡾⣽⡾⣯⣿⣽⣟⣷⢿⣽⣧⡑⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⢔⢳⡽⡝⠌⠀⡊⡎⣧⣳⢓⠂⠀⠀⠀⢀⠠⠐⠈⠏⡋⣇⡂⠀⠀⠀⠀⠐⠈⠀⠀⠐⠀⠐⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⡑⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⢿⣾⣻⣟⣯⣿⢯⡿⣷⢯⣷⣯⢿⣯⢷⡿
⡿⣾⣻⢿⡽⣯⡿⣞⣷⢿⣽⣟⣿⢾⣟⡀⠀⠀⠀⠁⠀⠀⠠⠀⠠⠠⠂⠄⠀⠊⠌⠃⠌⠌⠊⠈⠀⠀⠂⡢⡪⡢⡓⠌⠂⠀⠀⠈⠀⠀⠀⠀⠂⠁⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣑⠐⠀⠀⠀⢀⣤⣴⣴⣷⢿⣻⣯⣿⢾⣯⣿⣽⡾⣿⣻⣟⣯⡷⣟⣿⢾⣻⣟
⡿⣯⢿⣻⣻⣽⣟⣯⣿⣻⡷⣿⣽⣟⣯⣷⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠂⠀⠀⠀⢀⣾⣿⣽⣾⣻⣟⣯⣷⢿⣻⣷⣻⣾⣻⣯⣿⡽⣷⣟⣯⡿⣯⡿⣾
⣿⣻⣟⣯⣿⣳⣟⣷⣟⣷⢿⣻⡾⣯⣿⡽⣯⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⣽⡿⣾⢷⣻⡷⣟⣿⢾⣟⣿⢾⣻⡾⣯⣷⢿⣽⣟⣾⢯⣿⣳⣟⣿
⡿⣾⣽⢾⣷⣻⣽⢾⣳⣟⣿⣽⣻⣽⣾⣻⣽⡾⣟⣷⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣟⣿⣻⣯⣿⣻⣽⡿⣽⣯⣿⣻⣟⡿⣾⣟⣷⣻⣽⣯⡷⣿⢽⣾
⡿⣷⣻⣟⣾⣽⢾⣟⣯⣿⣺⢷⣻⣗⣯⡿⣞⡿⣯⢿⣾⣻⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣿⣽⣻⣾⡯⣿⣳⣿⣻⡾⣯⣷⣟⣿⣳⡿⣽⣽⢾⣳⣟⣯⡿⣾
⣿⡽⣾⣽⢾⣽⢯⣿⣺⢷⣻⢿⣽⢾⡯⣟⣯⢿⣽⣻⢾⡽⣽⣗⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢷⣟⣿⡽⣯⣟⣿⣽⢾⣻⡾⣯⢿⣽⢾⣟⣯⣯⢷⡿⣽
⣷⣻⣗⣯⢿⣺⣟⡾⣽⢯⡿⣽⣞⣯⢿⡽⡾⣯⢷⣻⡽⣯⢷⣻⣞⣷⣻⢷⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣽⢷⡯⣿⢽⣾⣳⣟⣯⢿⡽⣯⢿⡽⣯⢷⣟⡾⣯⢿⡽
⣷⣳⣟⣞⣯⢷⢯⣟⣽⢽⣽⣳⣻⢞⣯⢿⢽⢯⣟⢷⣻⡽⣯⢷⣻⣺⣞⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⡽⣽⢽⢯⣟⣾⣺⣗⣯⢿⡽⣽⣽⢽⣽⣻⢾⡽⣯⢯⡿
⣗⣗⣗⣯⢾⢽⣳⢯⣞⣟⣞⡾⡽⣽⣺⢽⢯⣻⣺⢽⣳⢯⢯⡯⣷⡻⡾⣽⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡽⣽⣫⣟⣞⣗⡷⣻⣺⢯⣟⣗⡯⣟⣞⡷⡯⣯⢯⡯⣟
⣗⡯⣾⣺⢽⣻⣺⢽⣺⢞⡾⡽⣝⡷⡽⣽⣻⣺⢽⢽⣺⢽⣫⢯⣗⣯⢟⣗⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢞⣗⣗⣗⣗⣗⢽⡳⣝⣗⢗⣗⣟⣽⣺⢽⣫⡯⡯⣯⢯
⣗⡯⣗⣯⣻⣺⣺⢽⣞⡯⡯⡯⣗⣯⢯⣗⡷⡽⡽⡽⣞⡯⣯⣻⡺⠺⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠺⢺⢮⣺⢵⡫⣗⢗⡽⣺⢺⣺⡺⡽⣳⢯⣻⢵⣻
⣗⡯⣗⡷⣳⣳⢯⣗⣗⡯⣟⣽⣳⣽⣳⣳⢯⣻⠽⠏⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠓⠛⠚⠷⠵⢯⣻⣪⢯⢞⡽⣺
⢗⠯⠿⠺⠽⠺⠓⠓⠓⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀