⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⡦⡣⣓⢖⢆⣆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡪⡺⣜⠮⣝⣜⢎⡗⣎⢮⡫⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢝⢎⢂⠂⠔⡀⠅⠌⠄⠅⡹⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⢵⠩⣐⡬⣔⣄⠅⢊⣄⣅⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢩⢺⠨⠐⡶⡒⡐⡈⠦⣵⢨⢱⠀⠀⠀⠀⣀⣠⣠⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡊⠆⠂⠅⠌⡐⡀⡂⡇⢂⠂⡧⠀⣠⢴⡯⣷⢯⡷⣷⣻⣞⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠨⡈⠔⢐⢀⠢⠊⠄⡢⠁⡴⡯⣯⢟⠝⠫⠛⢷⣳⢯⡯⣗⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠣⣂⠌⠄⠅⠅⠃⠥⠂⣜⣯⢿⠍⡂⠌⡨⢈⢄⢯⣟⡾⣽⣺⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡎⡈⠢⡃⡇⢬⢨⠌⠁⢠⣟⣾⣋⢭⡡⠡⠠⠩⣡⣩⣺⣽⣻⣞⣷⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣗⣮⡐⠨⠨⠢⣱⣧⠀⠸⣳⡿⡎⠸⢟⠨⢈⠌⠺⡞⠅⢷⣗⣷⡻⣞⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⡯⣟⣞⣯⣗⡷⣻⣳⣥⠨⢪⡯⣷⢤⣙⣽⡇⠅⡂⠅⡂⠂⠅⡐⢨⠨⣪⢷⡿⡽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣟⣞⡾⣝⣗⡷⣳⣻⢞⣯⢷⢯⣧⢿⡽⡯⡿⣽⣺⣽⣆⡂⠕⡲⠡⢁⢢⣳⣿⣻⣯⢿⢯⢿⢺⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣗⣗⣯⢾⢽⣳⢯⣞⡯⣞⣟⡾⣽⣳⢯⢿⢽⢽⣫⣷⣻⣗⣿⢷⣥⡔⢜⠜⠌⣿⣾⣻⣞⣯⣟⣿⢐⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣗⣯⢾⢽⣻⢾⢽⣞⣽⢽⣺⢽⣺⣞⣯⢿⢽⢯⣗⣿⣺⣽⡾⣿⢷⢏⠢⢁⣱⡽⣞⣗⡿⣾⣺⡽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣗⡯⡯⣟⡾⣽⣻⣺⢾⢽⢽⣽⣺⢞⡾⡽⣯⣻⣺⣺⣗⣷⣻⣽⠣⠡⠨⢴⣗⣿⣽⣫⢟⡾⣝⣯⢿⡽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⡷⡯⣟⡽⣯⣯⢷⢯⢯⣟⣽⣺⣺⡽⣽⣫⣗⣯⢾⣞⣷⣳⢿⣺⠈⢌⡴⣻⣞⣷⡳⡯⣻⣺⢵⣟⣿⣝⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣗⡯⣯⢷⣻⣳⡯⡿⡽⣽⣺⢞⡾⡵⡯⣗⣷⣳⢯⣗⣿⣺⡽⣽⣺⡸⡮⣯⣟⣾⣳⢯⢯⢷⢽⢵⣟⣷⣏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢗⢯⢯⣟⡾⣺⢿⢽⣫⣗⡯⣯⢯⢿⢽⣽⡺⣞⡽⣞⣾⣳⢟⣗⣗⢯⡻⡮⣷⣻⣞⡯⡯⡯⡯⣳⣿⣳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢑⠐⡁⡂⠭⣟⡿⣽⣳⣳⢯⡯⡿⡽⡽⡮⣯⢯⡯⣟⣾⣳⣻⣺⣺⢽⢽⣝⣗⣷⡳⡯⡯⣫⢯⣿⡽⣏⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⠈⠄⡂⠆⢹⡽⣗⣟⡾⡽⡯⡯⣟⡽⡯⣗⣯⢿⡽⣽⣷⣻⣺⣽⢽⣳⣗⡷⣗⡯⡯⡯⡯⣟⣷⢻⣟⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⠨⢐⠠⡃⡨⡿⣯⢷⣟⡯⡿⡽⠽⣝⠫⡟⡾⣯⢿⣽⢾⣳⣻⣺⢯⡷⣯⢿⢽⣺⢽⢽⡽⣿⠡⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡂⢂⠂⡂⣕⣿⣫⣟⣮⣟⣓⣚⣘⣐⠡⠈⠌⣻⢽⣞⢿⣝⣗⡯⣯⣻⡺⣝⣟⡾⡽⣽⠸⡍⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢐⠠⠡⢐⢸⣞⣷⣻⣺⣞⣶⡲⡒⣒⠨⡈⢔⢽⢵⣫⣗⣗⣗⡯⣗⣗⣯⣗⣷⣫⣿⢧⡫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⠨⠠⠡⢸⣞⡷⣽⣺⣳⢗⣯⢿⣝⡯⣟⣯⢿⣻⣽⡎⣹⣾⣻⣯⡿⣞⡿⣞⡿⣞⡿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢂⠡⢑⡼⣾⢽⣳⣳⢯⣻⣞⣽⣺⡽⡯⣯⣟⡷⡷⣷⣻⡾⣯⣷⢿⣻⡽⣯⢿⡽⡽⡯⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⡈⠄⢻⠽⢽⣺⢾⡽⣞⡾⣺⣵⣻⢽⣳⣗⡿⡯⣟⡷⣟⡿⡾⣿⢽⣽⣳⢯⢿⢽⣻⡽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠨⠘⡌⢄⠐⠄⠅⠅⠩⢁⢅⢑⠩⢐⠐⡈⡂⣯⢿⡯⣿⣻⡽⣯⢾⣺⢯⢿⢽⣳⢿⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠡⠡⢈⢒⠨⠠⠡⠨⠈⠄⡂⡢⢁⠢⢂⠂⢅⡿⣯⣟⣯⣯⢿⡽⣽⢾⣻⡽⣯⢯⢿⢽⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠨⠠⢑⠠⠅⡇⠨⠐⠡⢑⠡⡕⡊⡪⠨⠐⡈⠔⣟⣷⢯⡷⣟⣯⡿⡽⡯⣗⣯⡯⡿⣽⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡘⡐⡌⢮⠂⠅⠡⢑⠠⢱⢑⠔⢐⢈⠐⠄⡑⣻⡽⣯⣟⣯⣷⢿⢽⢯⣟⡾⣽⣫⣷⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠹⡘⡊⠅⠡⢁⠂⡂⡵⡑⡨⢀⠂⠌⡐⢨⠸⣽⢷⣯⢷⡿⡽⡯⣟⡾⣽⡳⣗⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⢐⠠⠁⠅⡂⠌⢄⢓⠌⠔⠠⠨⢐⢈⢰⠈⣯⣿⣺⣯⢿⡽⡯⣷⣻⣳⣻⡽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀