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⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠷⢛⠩⣡⠦⢂⢂⢂⠂⡂⢂⢐⢀⠂⠄⡂⠄⢂⢐⠠⢀⢂⢐⢀⢂⠄⡂⡂⡂⠢⠡⡂⠪⡐⢅⠪⣙⢛⡻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣿⣯⣿⣯⣿⣯⣿⣽⣿⣺⡫⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣽⢾⣻⣵⢞⣴⠟⡡⡊⠄⠢⢐⠨⢐⢐⢐⠠⢁⠅⡂⠌⠔⡐⡨⢐⠐⡐⡈⡢⢑⢐⢅⠪⡘⢔⠌⡪⢐⢅⠇⡎⡪⡪⣚⢮⡻⡾⣿⣾⣿⣽⣯⣿⣾⣿⣾⣿⣽⣿⣽⣯⣿⣽⣯⣿⣽⣯⣿⣽
⣿⣻⣿⣽⣿⣽⣿⣻⣿⣽⣞⢽⢽⣿⣟⣿⣿⣻⣿⣟⣿⣿⣻⣿⣟⣿⣟⣿⣻⣯⣯⣷⣿⣻⣚⣞⠵⡩⢂⠊⠌⠨⡐⠨⢂⢂⠢⠨⡐⢔⠨⡘⢌⢐⢐⠔⡡⢂⠢⠪⡨⡢⢱⠨⡊⢆⢣⢪⠢⢅⠣⡣⡣⡱⡑⡕⡯⣿⣾⢿⣾⣿⣻⣿⣯⣿⣾⣿⣯⣿⣟⣿⣟⣿⣿⣻⣿⣟⣿⣟
⣿⣿⣟⣿⣽⣿⣻⣿⣻⣿⣻⣿⣻⣯⣿⣿⣽⣿⣯⣿⣿⣽⣿⣯⣿⣿⣻⣿⢿⣻⣿⣻⣿⢻⡝⡪⡑⢌⢐⠁⠌⡂⡊⢌⢂⢂⠅⢕⠨⡂⢕⠨⡢⡑⢔⢑⢌⠆⢕⢑⢌⢪⢊⢎⢎⢎⢪⡸⡸⡱⡱⡌⡎⣞⢼⢸⢪⢮⡻⣿⣿⣾⣿⣯⣿⣿⣽⣷⣿⣟⣿⣟⣿⣿⣽⣿⣯⣿⡿⣟
⣿⣯⣿⣿⣟⣿⣟⣿⣟⣿⣟⣿⣟⡯⣿⣯⣿⣯⣿⣯⣿⣯⣿⣯⣿⣟⣿⡿⣿⡿⣟⣯⡾⢣⢑⠌⡐⡐⡐⡐⠅⡂⡪⠐⡐⠔⡅⢕⠨⢌⢢⠣⡢⡱⢡⢃⢎⠜⡨⡊⡆⢎⢪⡪⡎⣞⡜⡼⣸⠸⣜⢼⢸⢬⢳⢳⡱⡱⢕⢇⠯⣷⣿⣯⣷⣿⣟⣯⣿⣿⣻⣿⣿⣽⣿⣽⣿⣽⣿⣿
⣿⣿⣽⣾⣿⣟⣿⣿⣻⣿⣟⣿⣷⢿⣟⣿⣽⣿⣯⣿⣯⣿⣯⣿⣯⣿⣻⡿⣟⣟⣯⠷⢩⢂⢂⢊⢐⢐⠰⡈⡂⡪⠠⡑⡌⠕⡌⡢⢣⢑⢅⢇⢪⠸⡰⡱⡘⡌⡆⡣⣪⢪⢪⡪⡎⣗⢵⢝⡮⣣⡣⡫⡺⣜⢵⡓⡽⡸⡸⡱⡹⡘⢾⣽⡿⣯⣿⣿⣟⣿⣟⣿⣾⣿⣯⣿⣿⣽⣿⣷
⣿⣾⡿⣟⣷⣳⣻⣿⣻⣯⣿⣿⣻⣿⣿⢿⣟⣯⢯⢯⣿⣿⣽⣿⣻⣿⣻⣿⣿⢿⢫⢘⠔⢐⢐⢐⢐⠔⠡⢂⢒⢌⠪⡢⡑⡅⢇⢊⢆⢣⢱⢸⠸⡘⡌⣆⢳⢨⢪⢪⢪⡪⣣⢫⢞⢜⢕⢗⢝⢮⢺⡸⡜⢎⢮⢺⢜⢜⢜⢜⢜⢌⢇⢟⣿⣿⣟⣷⣿⡿⣟⣿⣿⣾⢿⣻⣾⣿⣷⣿
⣾⡿⣿⣿⣿⢿⣽⣿⣟⣿⣿⣽⣿⣽⣾⣿⣿⣿⣽⣽⣿⣽⣿⣽⣿⣟⣿⣯⠟⢕⢡⢑⠨⢐⢐⢐⠔⠡⢁⠢⡱⢰⢑⠱⡐⡕⢅⢕⢱⢱⢱⢱⠱⡱⡱⡱⡕⡕⡵⡹⣜⢸⢸⢸⢱⢕⢕⢕⢭⢳⡱⡕⣝⢜⢜⡜⡎⣎⢎⢎⢜⠜⣜⢜⢷⣿⣻⢿⣻⣿⣿⣿⣾⣿⣿⣿⡿⣿⣾⣿
⣾⣿⣿⢿⣾⣿⣿⡽⣿⣻⣽⣿⣾⣿⢿⣻⣽⣾⣻⣿⣽⣿⣯⣿⣯⣿⠽⡂⡣⢑⠐⢄⠕⠡⡐⢌⠌⢌⠢⡱⠸⡐⡅⡣⡱⡸⡐⡕⡱⡑⡅⡇⡇⡣⡣⡫⡣⣪⡎⡇⣇⢣⢱⢱⢱⢱⢑⢕⢕⢕⢕⢕⢕⢕⢕⢜⢎⢮⢢⢣⠱⡑⡜⡵⡱⡻⣿⣿⡿⣿⣾⣿⣾⣿⣾⡿⣿⣟⣿⣾
⣻⣯⣿⣿⢿⣻⣽⣿⢿⣿⢿⣻⣗⢯⡻⡻⡿⣿⣿⣻⣯⣷⣿⡷⣟⣍⣪⢴⡜⡐⢌⠢⠨⡨⠨⡐⢌⠆⡕⢜⠡⡱⠨⡢⡣⣊⢢⢱⠱⡱⡱⡱⡡⡣⡣⡣⡱⣯⢿⡜⡔⡕⡅⡇⡇⡕⡕⢕⢅⢣⢣⢣⢣⢣⢣⠣⡣⡣⡣⡣⡩⡊⢆⡻⣽⡜⣽⣷⣿⣿⣷⣿⣷⣿⣷⣿⣿⡿⣟⣿
⢯⣞⣮⣻⡻⣿⣿⢿⣿⢿⣿⣿⢗⢧⡫⡮⣻⣿⣽⣿⡿⣟⣿⣿⣻⣽⡛⢕⣨⡬⢂⢐⢑⠠⠡⡘⢔⠱⡘⠔⡑⢜⢨⠪⡒⡔⡸⡰⡱⡱⡑⡕⡌⡎⡜⡰⡽⣯⢷⢯⡪⡪⡸⡨⡪⡸⡸⡸⡸⡘⡜⡌⡎⡜⡔⡕⢕⢕⢕⠕⡌⠌⡆⢇⡻⣟⣎⢷⣿⣷⣿⣷⣿⣯⣿⣯⣷⣿⣿⣿
⢕⣗⣵⣳⢽⣟⣿⣿⡿⣿⣿⣽⣏⢗⣝⢮⣳⣿⡿⣷⣿⣿⣿⣻⣯⢗⣮⡏⡃⡂⡂⠔⡡⠈⡔⢌⠢⡑⠅⢅⠕⠅⡢⡣⠣⡊⢔⠕⡜⢜⢜⠜⡌⡎⡊⣞⣯⡯⡿⡽⣎⢎⢆⢣⢱⠸⡰⡱⡸⡐⡌⡊⡎⡆⡕⡜⢌⢎⢆⢇⢇⠕⠨⢪⢪⢻⢗⣟⣿⣯⣷⣿⣯⣿⣯⣿⡿⣟⣿⣽
⡳⡵⣳⣳⣻⣿⣻⣷⣿⣿⣿⣽⣾⣿⣾⢷⣿⣟⣿⣿⣟⣷⣿⣽⢾⢛⠣⡑⡐⡐⡐⡐⠄⡕⢌⠢⠑⢌⠊⢔⠡⢡⢃⠎⢕⢑⢸⢘⢜⠜⡔⡕⢅⠇⢮⣻⣞⣽⢽⢯⢷⢕⠥⡑⡅⢇⢕⢜⠔⡅⡣⢕⢌⠊⢎⢜⠔⢅⢇⢕⢅⢣⠡⠡⡓⣝⢽⣗⣿⣟⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿
⢜⢽⡺⣿⣿⣻⣿⣟⣿⣷⣿⢿⣻⣽⣿⡿⣿⣽⣿⢷⣿⢯⣞⣪⡦⣱⢕⢐⠅⡊⠔⢀⠣⢊⢠⢂⠅⡂⡑⡐⠅⡪⡂⠪⡪⡨⠢⡣⡱⡑⡕⢜⢸⢨⢛⢚⢚⠚⠫⠫⠫⢓⢅⢇⢊⢪⠢⡱⡑⢌⢊⠔⡐⡱⡑⠜⡜⡌⢎⢪⢊⢆⠣⡑⡌⡌⡺⢜⡽⣻⢯⣿⣯⣷⡿⣿⣾⣿⣾⣿
⡸⣜⢼⢹⡫⡿⣟⣿⣿⣽⣾⣿⣿⣟⣯⣿⣿⣿⣽⣿⣯⢿⣾⢟⢅⠫⡐⢅⠪⡌⡐⠄⡃⡢⡑⠌⢊⢂⢎⠌⡢⠱⠀⢱⠨⡢⠱⡱⡘⡌⡎⢜⠰⡸⣝⢞⢜⠎⢎⢪⢸⢰⢱⡸⡨⡂⢇⢪⢘⢌⠢⠁⠈⡀⠁⠑⠨⡊⡪⡸⡐⢅⠪⢰⣱⡽⡌⣮⣮⣎⡯⣞⣷⣿⡿⣿⣿⣾⣿⣻
⡺⣪⣷⣷⣯⣮⢇⣗⢽⣿⣻⣿⡷⣿⡿⣿⡿⣾⣿⣻⣞⣟⢏⢊⡢⢱⢘⢌⢎⠆⡊⡐⡐⢌⢊⢊⠊⠌⠂⣊⡤⢥⢌⠐⢕⢌⠪⡢⢣⢱⠸⡨⠊⠊⠂⠁⠈⠈⡁⠐⢡⢳⢵⢝⡦⡣⡣⡑⡢⠪⡀⡊⢔⢌⡮⢀⢐⠸⡰⡱⡑⡰⡁⡣⣑⡻⣻⣜⡽⡿⣿⣷⣮⣗⣿⣿⣷⣿⣟⣿
⢽⣽⣻⣯⣿⣟⣮⣺⣺⢿⣻⣯⣿⡿⣿⡿⣟⣿⣾⢿⢝⢅⢧⠱⡱⡑⡕⠌⡎⠔⡐⡐⢌⠂⢅⠢⠠⠁⢰⠣⡪⡪⡨⠣⡑⢔⠱⡘⡌⢆⢣⢊⠀⡐⠠⣑⠡⣱⢸⢼⡰⡹⣮⣻⣺⢽⣎⣎⢌⠇⡎⣌⢫⠜⡊⡔⠄⢕⠨⡪⡢⠢⡱⡘⡔⣷⢹⣷⢿⣿⢿⣷⣿⡿⣷⡽⣷⣿⡿⣿
⢿⢿⣾⣞⢿⣻⣯⣿⡿⣿⣿⣿⢿⣿⢿⣟⣿⡿⡞⣍⣾⣻⣣⢗⢕⢱⢡⠑⠌⣂⠢⡨⠂⠌⠄⠌⢀⠈⢸⢨⢲⡙⠌⠌⢌⢆⢣⠱⡘⡌⡆⠕⡠⡈⠣⡙⡝⡪⡞⣏⢮⣞⣗⣗⡽⣽⣺⣗⢷⡱⡸⣘⢮⡳⡕⡕⡅⢕⢸⢸⢐⡑⠴⡑⡜⠽⣗⣿⣽⢿⣿⣟⣷⣿⣿⣿⡽⣿⡿⣿
⢿⣽⣳⡯⡯⣿⣟⣿⡿⣿⣿⣾⣿⣿⡿⣯⣿⠏⣮⣾⣳⢯⢣⢣⠣⡱⢡⠩⡊⢔⠡⠨⠨⡈⠄⡁⠠⠐⡀⠓⡜⡎⢕⢁⢂⢪⠰⠡⡣⢪⠨⡊⢜⣜⢖⣔⢦⣳⢽⣺⢽⣺⣺⢮⣻⡺⡮⡯⣗⣟⣜⢮⢎⡞⡮⡪⡂⢕⢸⠰⡐⣯⢯⢾⣼⣮⣮⢺⣿⢿⣻⣿⣟⣿⣯⣿⣟⣿⣿⣿
⢸⢹⢺⣻⢭⡻⣟⣿⡿⣿⣷⣿⢿⡾⣟⣯⢏⢮⠳⡣⢣⢪⢪⠢⡑⡕⡡⢑⢌⠢⡊⠌⡂⠂⠅⠄⡑⡐⡈⡐⠈⢝⢆⢅⢂⠂⡣⡃⡪⡂⢇⠪⡸⣪⡳⣕⡯⣺⢽⣺⢽⣳⢯⣻⢮⡫⡾⢽⡻⡺⣺⣝⡯⡯⡯⣎⢂⠪⡪⡸⢐⢿⣿⣻⣾⣯⣿⡕⣿⣿⣿⣟⣿⣟⣿⣽⣟⣯⣷⣿
⢱⢨⠢⡊⡪⡻⣯⣟⡿⣿⢷⡿⡯⡿⡳⡫⡪⡊⡎⡎⡎⡎⢆⢣⢣⠱⡈⡢⡊⡢⡑⡁⡂⠅⠅⢂⠂⡐⠐⠨⠈⠄⠃⢆⢐⠄⡱⠰⡐⢅⢣⠑⡌⡞⡮⣗⢯⣫⣗⡯⣟⡾⣽⣺⣳⡱⡡⢕⢱⢡⣷⣳⢯⡯⣯⠪⡐⡱⡱⡸⠨⣟⣿⣿⣻⣿⣻⡧⣿⣿⣾⣿⢿⣻⣿⣟⣿⣿⢿⣿
⢸⠸⡸⡨⢢⢊⢢⢑⠍⡝⢝⠞⡪⡚⡜⡜⡜⡜⡜⢜⢜⠨⡪⢸⢐⢕⢘⢌⠢⠢⡱⠐⠄⠅⡊⠄⢂⢠⡑⠈⠐⡈⢀⣆⠔⠈⢂⠇⡊⡢⡑⢅⠪⡪⡫⡾⣝⣞⡮⡯⣗⡯⣗⡯⡾⣝⡮⡧⣳⢯⢾⣺⢯⣻⡪⡁⠢⢪⠪⡊⡪⢿⣯⣿⣟⣿⣯⢿⣟⣿⣾⣿⣿⡿⣟⣿⣯⣿⣿⣿
⠰⡑⡅⡣⠣⡊⡢⢣⢑⢜⢔⢕⢕⢱⠱⡑⡕⡱⢨⢃⢆⠣⡘⡌⠆⡂⡢⡑⠌⢌⠢⡑⡁⡊⠄⠅⠢⢘⣦⡁⡂⠐⠠⢈⠣⠐⠀⠎⡪⠰⡘⡌⡪⡪⢯⡺⣵⡳⡯⡯⣗⡯⣗⡯⡯⡳⠙⠜⣘⢸⣙⣞⡽⡺⠐⡀⡣⢱⢱⢱⡪⣹⣿⣯⣿⣿⣽⣿⡿⣟⣿⣾⡿⣿⣿⡿⣟⣯⣿⣿
⠨⡢⡑⢜⢘⠜⡸⢘⢌⠆⡕⢔⡑⡔⢕⠡⡂⡊⡢⢑⠔⢅⢃⠢⢑⠐⠔⡠⢑⠄⡃⢆⠕⡨⡘⣬⠌⡂⣳⣷⠄⠅⡘⠐⡀⠂⡁⠌⡊⢎⠔⢕⢌⢪⢳⡹⣪⢞⡽⣝⢮⣻⡺⣜⣖⢮⢏⠯⡪⣺⢺⡪⠎⠠⢐⠐⡌⡪⡢⣻⣯⣒⢿⣯⣿⣾⣿⣷⣿⣿⣿⣟⡿⣿⣷⣿⣿⣿⡿⣿
⢌⢂⠪⠨⡂⠕⢌⠢⡢⡑⢜⢐⢑⢌⠢⡑⢔⠔⢜⠰⡑⡑⠄⠕⡠⠡⠑⠄⠅⡂⡪⡂⡣⣫⢷⢹⣣⡊⠔⣿⢺⣲⣄⡡⠀⠅⠠⢂⠪⡨⡊⡪⡪⡊⡆⢝⢜⢵⢝⢮⡳⡳⣝⡞⡮⡣⡣⡣⣣⡳⡝⢼⠀⢂⠂⢅⢣⢣⢹⣽⣿⢿⣞⣿⢿⣽⣾⣯⣿⣿⣾⣿⣿⢿⣽⣿⣽⣿⣻⣿
⠔⡨⠨⡨⢂⠣⡑⡑⢔⠨⠢⡑⢔⠢⠱⡘⢔⠡⠅⢕⠨⡐⢡⢑⠠⠡⠡⠡⠨⠢⡱⡱⡽⣽⢿⡷⣝⢗⢅⠫⢫⠳⡙⡪⢀⠠⢑⢀⢂⢒⠌⢆⢣⢣⢝⠰⡐⢔⠩⡑⠹⢹⢪⢞⢽⡹⣝⣞⢞⢎⢎⢽⣷⠡⡈⢔⢕⢕⢽⣷⣿⣿⣿⣽⣿⣿⢿⣻⣿⣾⣿⣾⣿⣿⡿⣿⣽⣿⣜⣮
⢈⠢⡑⠔⡡⠊⠔⡨⠠⠡⠡⠨⢐⠨⠨⡐⡐⡡⠡⡁⡢⠨⢂⠂⢌⠌⢌⠌⣜⣼⣽⣺⡝⡾⡹⡹⡸⣝⢜⢮⡲⣕⢬⠢⡂⠆⡂⡂⠢⠀⠕⡑⡌⡪⢪⢱⢡⠣⡑⡌⡊⣂⠂⠉⠘⠊⠃⠓⣡⢇⣗⢽⡗⣕⠀⡎⡎⡦⣿⢿⣷⣿⣯⣿⣿⣾⣿⣿⡿⣷⣿⣷⣿⣷⣿⣿⡿⣯⣿⣿
⠠⡑⠌⢌⠢⠑⢅⠢⠡⠡⠡⡑⠄⠅⢅⢂⠢⠨⢂⠢⢊⠌⢄⠅⠅⢌⢂⢣⡮⣷⡿⡎⡎⡞⡜⣜⢕⢗⡽⣕⢗⡕⣗⢽⡹⡵⣢⢦⢅⠅⢌⠐⢅⠇⡇⡎⡪⡪⡊⡆⢕⠔⠀⠀⠈⠌⡨⢐⠨⢙⢎⣟⣯⠇⢌⢎⢂⢻⣽⣿⢿⣷⣿⣿⣾⣿⣯⣿⣿⢿⣯⣷⣿⣯⣿⣷⣿⣿⣿⣻
⠨⠢⡑⡡⠨⡊⠔⢌⢊⠌⢌⠔⠌⢌⢂⠢⡡⡑⡐⢅⠅⡪⢐⠡⡑⢕⢨⣾⡿⡟⢟⠕⡕⢕⠝⠜⡜⣕⠧⡓⡯⡺⡵⡫⣞⣝⢮⡳⣝⣝⢔⢅⠅⢇⢇⢇⢇⢎⢪⢪⠢⡃⠀⠀⠁⠀⠂⠢⠡⡑⡰⡐⠝⢄⠕⢌⢂⢂⢳⣿⣿⡿⣷⣿⣷⣿⣯⣿⣾⣿⣿⣿⣻⣽⣿⣽⢯⣾⣿⣿
⠨⡈⡂⡪⠨⡂⠕⡁⡂⡊⢔⢬⠮⣒⣥⢷⠻⡐⠅⢕⠨⢂⠢⡣⢑⢡⣟⣿⠘⠜⡌⣆⢮⢢⡕⣗⡬⣢⣣⡣⡧⣕⢕⣍⡪⡨⡣⡫⣞⢮⡳⡕⡅⡣⢣⢣⢣⢣⢣⠪⡒⡄⠀⠁⠀⠁⡀⠈⠈⠰⡐⢜⠨⡢⢃⢅⠪⡐⠼⣿⣷⣿⣿⣿⣾⣿⣽⣿⣻⣽⣷⣟⣿⣿⣟⣿⣿⣿⣯⣿
⢐⠰⢐⠐⢥⣶⣷⣶⣾⡾⣾⡾⡝⡫⠨⠢⡑⢌⠪⢐⢈⠢⡃⠪⣐⣵⣟⠭⣊⢎⢇⢧⢳⢱⢝⢮⡺⣕⢗⡝⣞⢮⡫⣞⢵⡻⣮⣲⡡⣓⡝⣎⢧⢑⢅⢇⢇⢇⢇⢇⢕⢜⢄⢑⠈⠀⡀⠄⠀⠄⠌⡢⢱⢡⢑⠔⡑⢌⠪⡹⣿⣽⣷⣿⣷⣿⣟⣿⡿⣟⣿⣿⢿⣽⣿⢿⣽⣾⣿⣿
⠐⠌⡂⢅⢹⣯⣷⣿⣷⣿⠟⢕⢼⢐⢑⢑⠌⡐⠨⢐⠐⢅⠊⢌⣾⣟⣧⡣⡣⡱⢹⢸⢸⢜⢮⡺⡜⣞⢼⡺⣪⢞⠮⣎⢗⢝⡞⣎⢯⡳⣝⢮⡳⡥⣱⢰⢑⢕⢕⢕⢕⢌⢆⢇⠅⢅⠀⡀⠄⠀⢁⢊⠢⡑⢅⠕⡅⢕⢱⠐⡙⣿⣿⣽⣾⣿⣟⣿⣿⣿⡿⣿⣿⢿⣿⢿⣿⣟⣿⣾
⠨⠨⢂⠅⣞⣿⣻⡽⡳⣡⣳⠫⡑⡐⡡⠂⠢⠨⡈⡂⠅⢅⢊⣾⢿⢍⢇⠇⢇⢓⢕⠇⡗⢵⢱⢕⢵⢝⡜⡜⡌⡇⡏⡎⡮⡳⡹⡪⡳⣝⢮⢧⡳⣝⢮⡳⡅⡎⡜⢜⢜⢜⢌⢆⠝⡔⡡⡀⠄⢀⠀⠑⡅⢕⡐⢅⠣⡣⡱⡈⡢⢊⠻⣟⣯⣿⣟⣿⣷⣿⡿⣿⣻⣿⣯⣯⣿⡿⣟⣿
⠨⠨⢂⢊⠝⢝⣱⣵⣾⢯⣦⢊⢐⠔⠠⠡⠡⠡⢂⢊⢌⠢⣹⡷⡹⡐⡅⢕⠅⠕⠔⡑⢌⠢⡃⢕⠸⡘⢜⠑⠅⡃⠣⡑⢌⠪⡊⡣⢫⢪⢎⡇⡧⡳⡕⣝⢮⢳⡑⡅⢇⢧⢱⢪⣪⢲⣳⢸⢨⠢⡐⡀⠘⢔⠜⢌⢊⢎⢎⢆⠊⡆⡇⡻⣿⢿⣟⣿⣽⣾⣿⣿⡿⣟⣿⣟⣿⣿⣿⣿
⢈⢪⣲⣳⣳⣿⡾⣿⣾⡟⡓⡐⡐⠌⡊⠌⢌⢊⠔⣰⠢⣱⣿⣷⡑⢌⠌⡂⠁⠄⠡⠈⠄⡁⠂⡁⠅⠨⠠⢈⢐⢀⠡⡐⡐⢔⢐⠌⡐⢐⠐⢍⢊⠇⠏⡎⡮⣣⢳⡘⡌⢎⢖⠅⢗⡯⣞⢵⡪⣊⢆⠠⠀⠂⡣⡱⡘⡌⡎⡎⣮⣔⢕⢕⢎⢿⣿⡿⣿⣻⣯⣷⣿⡿⣿⣿⣻⣷⣿⣿
⢐⢽⡿⣿⣟⣯⣿⠿⢑⢐⢐⠌⡐⡡⢂⢑⢐⢐⢌⡯⡨⣾⣿⣯⣿⣆⠁⠄⠁⠌⢈⠠⠁⢀⠂⠄⠌⢌⠌⠆⠕⠌⠪⠐⠅⠃⠆⠕⡐⡐⡈⠠⢀⠈⡈⠐⠩⢊⢇⢳⢸⠨⢪⢪⠢⡫⣞⢵⡫⣞⡔⡅⡂⠀⠐⠜⡌⡪⢪⢪⢺⣿⣳⡱⡱⡕⣕⢿⣿⣿⡿⣟⣯⣮⣫⣿⣿⣻⣽⣾
⢷⣗⣟⢿⡻⢏⠅⡑⡐⠔⡐⢌⢐⢐⢐⢐⢐⠔⡘⠅⣾⣿⣯⣿⢾⣿⡆⠀⢁⠠⢀⠠⠐⡐⢈⠌⠨⠐⠈⢈⢀⠅⡢⢒⠔⡢⡂⡂⠄⠠⠈⢈⠂⡂⡂⡡⠐⢀⠐⠡⢊⠪⡢⢑⢕⢕⢽⣕⢯⢞⢮⣣⢅⠨⡀⠀⢕⢜⢸⢸⡸⣿⡿⣟⣎⢮⡪⡪⢝⢷⣿⣿⣯⣾⣿⣿⣻⣿⣿⢿
⣿⣿⢿⡯⠃⢅⢂⢂⢊⠌⡐⡐⡐⡐⡐⡐⡐⡐⡰⣵⣹⣳⣷⡩⠛⠓⠠⠈⠄⠂⠄⠂⠡⠐⠀⠐⢀⢐⠨⢂⢑⠌⡢⢑⢌⠢⡑⠕⡅⢅⠀⢂⠠⠠⢀⠂⡁⢂⠐⠠⠀⠌⢌⠢⡑⡕⡕⠵⡯⣯⣳⡳⡳⠀⢕⠅⠄⢣⠱⡱⡱⣻⣿⣿⣽⣯⣿⣮⡪⡪⡎⡿⣿⣻⣽⣾⣿⣿⣾⣿
⣿⣾⡟⡐⡡⢁⢂⠢⠂⢅⠂⡂⡂⡂⡂⣂⢆⢢⣿⡿⣿⡿⣿⣿⡄⠈⠄⠁⠌⠀⠂⠁⠐⠈⡀⠅⢂⢐⠨⡐⡐⠅⢌⠢⠢⡑⠌⢌⠌⡢⢃⠢⠨⠨⢐⠠⠂⠄⠌⡀⠡⠐⢀⠁⠊⢜⢌⠇⡏⣞⢮⣞⡝⠀⡧⣃⢇⠌⡪⡪⡪⣙⢯⣿⣷⣿⣷⣿⣻⣮⡪⡪⡪⡝⢿⡿⣟⣷⣿⣟
⣿⣻⢂⢊⠄⢅⠂⢅⠑⠄⠅⡂⡂⡂⡢⣿⡂⣾⣟⣿⣿⢿⣟⣿⠂⡈⢀⠈⡀⢈⠀⠁⢀⠁⠠⠐⡀⠂⠌⡐⡨⠨⢂⠅⠕⡨⠨⢂⠑⠄⠡⢑⠁⠅⢂⢑⠈⡈⠢⡐⡠⢈⠠⠀⠅⡕⢌⠪⡌⡲⡙⠎⠎⢠⡻⣪⢦⢑⠈⢎⢜⢔⢽⣿⢿⣾⣿⣾⣷⣿⢿⣮⢪⢪⢪⢪⢻⢻⢿⣟
⣿⢯⠐⠄⠕⡠⠑⠄⠅⢅⠕⡐⡐⠔⣽⣿⢊⣿⣟⣿⣽⣿⣟⣷⠀⠠⠀⠠⠀⠠⠀⠐⠀⠄⠁⠄⢂⠡⠑⡐⠄⠕⡠⠡⡑⠠⠡⡑⠨⠨⠐⠠⠡⠈⠄⡐⢀⠨⠂⠢⡈⡢⠨⢂⠅⡇⢌⠪⡸⡰⡘⡔⡈⡮⡺⣝⡵⡣⡃⢅⢣⢱⢸⣿⣿⡿⣷⣿⣯⣿⣿⣿⣷⣕⠧⣇⢇⢇⢇⢏
⡿⡯⡈⠪⠨⠠⠡⠡⢡⢑⠰⡐⡌⢌⣿⡿⣸⣿⡿⣿⣟⣯⣿⣿⣿⣶⣌⡠⠐⠀⠐⠈⠀⠄⠂⠐⡀⢂⠡⢐⠈⠌⡐⠡⡂⠅⠅⠂⠅⠡⢂⠐⢀⠈⠐⠀⠄⢊⠢⣁⡐⠄⠅⠅⡢⠅⢑⠱⡈⢎⠪⡊⡎⡮⣪⣪⢺⡘⡼⣅⠪⡸⡨⢿⣷⣿⣿⣯⣿⣯⣷⣿⣯⣿⣿⣵⣯⣺⢮⣪
⣏⠢⠨⡊⠌⢌⠌⠌⡂⢆⢕⢸⣷⢁⢻⡯⣺⣿⢿⣿⢿⣿⣻⣷⣿⣷⣻⣿⣶⣌⠄⠐⠀⠂⢀⠁⠄⢂⠐⡐⠨⢐⢈⠢⠨⡘⠨⡈⠌⡈⠄⡀⠄⠀⠁⢂⠀⠀⠄⠁⡘⠸⢸⢨⠊⠄⠄⠁⢊⢪⢑⠅⡇⡗⣕⢗⣳⡹⣝⢮⢆⢊⢆⢑⠻⣯⣿⣯⣿⣟⣿⣯⣿⣟⣯⣿⣟⣿⣷⣟
⠆⢅⠕⡐⡡⠡⡈⡂⠪⡐⢴⣹⣿⣧⢂⠫⣺⣿⣿⢿⣿⢿⣟⣿⣽⣿⣻⣿⡍⠁⠀⢀⠐⠀⠠⠐⠈⠄⠂⠄⠡⢂⢐⠨⢐⠨⢐⠨⠐⡀⡂⠄⠠⠀⢈⠠⠀⠠⠐⢀⠠⠀⠀⡀⠌⠢⠡⠡⡀⢐⠱⡁⢧⢇⢯⢳⠵⣝⢮⣫⢳⡢⢊⠔⠱⡨⠻⣿⣽⣿⣟⣯⣿⣿⢿⣿⣻⣿⣯⣿
⡍⡂⡢⢂⢊⠔⡐⠌⡂⡊⢾⡮⣾⣿⣷⣌⢺⣿⣻⣿⡿⣿⣿⢿⣟⣿⣟⣿⡇⠈⠀⠠⠐⠀⠂⢀⠁⢂⠁⠌⠐⡀⠂⠌⠠⠨⢐⠨⠠⢀⠐⢈⠐⠀⡀⠐⠈⢀⠀⠢⢐⠀⠅⡀⢀⠀⠈⠨⢐⠀⡘⢌⢪⡳⣹⢪⡫⣎⢧⢳⡣⡇⡕⡌⡌⠢⠨⢌⢛⣯⣿⣿⣟⣿⣿⢿⣻⣯⣿⣿
⡒⢼⣂⠢⠡⢂⠊⠔⡐⢌⢺⣷⢹⣷⣿⣿⣅⡻⣻⣯⣿⣿⢿⣿⢿⣿⣻⡟⠀⡈⠀⠂⡀⠂⠈⢀⠠⠐⠀⠌⡀⢂⢈⠈⠨⠐⡐⠠⢁⢂⠐⡀⠌⠀⠀⠈⠠⠀⠄⠨⠂⢅⢂⠐⡐⡐⡀⠀⡂⡒⠢⠢⡡⡫⡎⣗⢝⣜⣎⢧⢳⢕⣝⠮⡪⠠⢁⠑⡔⣿⣿⣯⣿⣿⣻⣿⣿⡿⣟⣿
⢌⢽⢧⠨⢊⠔⢡⡑⡐⢄⠹⣿⣧⡻⣿⣽⣷⣝⣷⣽⡻⡿⣿⢿⣿⢿⡏⠀⠄⠀⡀⢁⠠⠀⡁⠀⠄⠐⠈⠀⠄⠠⠀⠌⡀⠂⠐⠈⠄⠂⡂⠂⢄⠀⠄⠁⠠⠀⠠⠈⠄⢅⢐⠠⠐⢐⠨⠠⠀⠕⠊⠡⡂⢇⢟⢮⡳⡕⡧⣫⢺⡸⡸⣕⢕⡕⢄⠐⢌⢻⣽⣿⣽⠻⣟⣯⣷⣿⣿⣿
⢐⠌⢿⣕⢐⠸⣔⢿⣔⠄⠅⠽⣾⣷⣝⢿⣿⣻⣞⣿⢿⣿⣾⢿⣽⣟⣿⣆⠐⠀⠠⠀⠠⠀⠔⠈⢀⠁⡈⠠⠐⠀⠂⠐⠀⠅⠨⠐⠈⠠⠠⠑⠠⠂⠄⠂⠀⠀⠄⠨⠀⢕⠐⡀⠂⠀⠁⢅⠑⠄⠂⠨⢨⢊⢗⣕⢗⡝⣎⢮⢣⡣⡯⣪⢳⡱⢅⠂⡐⠝⣿⣟⣯⡣⣿⣿⣿⡿⣟⣿
⢐⠅⠕⡻⣕⠌⣻⢷⢽⢮⡨⠂⠝⣿⣽⣮⡻⡿⣿⣻⣽⣿⣽⣿⣿⣻⣿⠝⠀⠐⠀⡈⠠⢈⠐⢈⠀⠠⠀⠂⠐⠈⠀⡁⠌⠀⠅⠌⢐⠀⢂⠈⠨⠈⡀⠀⠀⠐⠀⡂⠡⢐⠠⠀⠅⡈⢀⠂⠁⠅⠂⠌⡐⢅⢇⢗⡕⡧⡳⡱⡣⡣⡳⡱⡱⡑⢅⠂⠠⢑⠐⢝⢿⣿⣿⢿⣷⣿⣿⣿
⠐⢌⠌⡢⠙⢜⢈⢿⣻⣷⣵⡡⠑⠌⢟⣿⢿⣯⣽⡻⡿⣷⣿⣷⢿⣻⣿⠀⠄⠀⠁⡀⠠⠀⠂⡀⠂⠁⠄⠁⡈⠄⠁⢀⠀⠡⠐⠈⠠⠈⠄⢂⠀⡁⠠⠀⠀⠀⠂⠠⡁⠐⠄⠨⠐⡀⠐⠠⡱⠡⡁⠌⠄⠕⢜⠸⡘⢜⠨⢌⠪⡘⢌⢌⠢⡑⠔⠈⠀⠌⢌⢐⠕⣿⣿⢿⣟⣯⣿⣿
*/
#include <bits/stdc++.h>
using namespace std;
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("popcnt,avx2")
/*
⣿⣿⣟⣿⢯⣿⣺⡴⡵⡝⣞⢵⢫⡳⣕⢧⢳⢱⢣⢳⠱⡱⢱⠱⡝⡹⡘⢌⢎⠎⡪⣻⠌⢮⢻⢺⣝⢕⡏⡎⣞⡝⠔⢅⢫⢚⡜⡮⡪⡪⡯⣺⢝⣮⡳⡑⠌⠢⢂⢂⢂⠂⡂⡐⡐⢐⠐⠠⢂⢂⢂⢢⠢⢃⡂
⣿⣿⣽⣿⣻⣯⣷⣟⡷⡽⣜⢕⢏⢮⢣⢳⠱⡱⢑⠅⡓⢘⠈⠌⢜⢜⣐⢅⢪⡂⡎⡜⠨⡃⣇⢷⢱⡹⡜⡎⡢⣪⠮⣞⡮⢷⢽⡚⠯⡯⡯⣗⡯⣞⡞⡌⠨⡈⠄⢂⠂⠌⡀⢂⠈⠄⠨⠐⡐⢔⢐⢄⠝⢄⠇
⣿⣿⣿⣽⣿⣽⢾⢹⢸⢨⠢⡃⢇⢇⢃⠢⡁⠢⠐⠠⢐⢐⢨⡸⡸⣱⢃⠳⡯⣚⢽⡯⡳⡄⢣⢳⢣⢫⢺⠨⣾⣯⡫⣟⢮⢣⢏⢜⢮⢯⣫⣗⡯⣗⢯⢢⢡⢢⢨⠠⡈⢄⢂⢢⠡⡑⢌⡂⢎⠰⢂⠇⡬⣢⡣
⣿⣿⣿⡯⣟⢜⢜⢜⢜⢜⠜⡌⡆⡕⠈⠰⣐⠡⠨⡂⢅⠍⠆⠏⡊⣗⢵⡣⡝⠧⢯⢺⢺⢯⢸⢰⢣⣣⢳⣱⣿⢝⠙⡊⣍⢕⣕⢯⢏⣗⢵⣣⢯⡳⣝⠌⠌⢊⠪⡊⠎⡢⠑⠔⢅⣂⡢⢪⢘⠸⡐⠅⡟⡎⡢
⣿⣿⡿⣹⢜⡎⣇⢧⢣⡣⡇⣇⠎⡂⣕⡜⡆⠥⡑⡌⡔⡌⡪⡊⡆⡖⡱⡙⡜⡹⡰⢐⢜⣟⢮⡓⣝⢼⢕⣗⢿⢵⡡⠕⡕⡵⡪⡮⣣⢗⡽⣪⡻⡺⡸⢬⢪⡔⡤⡣⢣⢢⢌⠌⡔⡢⢂⠡⢂⠡⡈⡆⡪⢈⠔
⣿⡿⡽⡵⣫⢞⢮⢎⢗⢜⢜⢌⢌⢮⢣⢣⢣⢫⢊⢎⢌⢎⢪⠪⡚⡘⢜⢎⢎⢔⠨⡰⡕⣝⣗⣝⢮⡳⣯⡯⡗⡳⣹⣪⡪⢮⢺⢸⢪⢳⢝⢵⢹⢸⡸⣱⡑⡗⡕⡔⠄⠕⠕⡱⢕⣑⠡⡊⡊⠢⡸⡐⠌⠢⡑
⡿⡽⡽⡽⡵⣫⢯⡺⣝⠮⠱⡰⢍⢮⢪⡪⡪⡪⡪⡊⡆⡣⡕⡕⣕⢳⠡⡑⠱⣐⠨⡚⠜⡈⢾⣪⢗⡯⣗⡯⢈⠚⡎⡎⠮⢕⢕⢕⢜⢔⢕⢑⢅⢇⡞⡕⣕⢬⢹⡘⢌⠆⢕⠨⠐⡈⡪⠌⡜⠬⡒⡐⢕⠅⡎
⣿⣯⣟⡮⣯⣳⡫⡯⣞⣝⡽⣸⣺⡱⣱⢱⢕⢕⢕⢜⢜⢜⢜⢜⢔⡵⡪⡠⠑⠄⢅⢈⠂⡂⢀⠹⢵⣻⢋⠠⡂⢍⢪⠪⡍⡧⡹⡸⣐⢢⢅⣕⣕⢯⢪⢎⢮⡪⡎⣎⢦⡱⡒⠜⢔⠔⡐⡱⡐⠅⡢⡪⢨⠪⡈
⣿⣿⡿⢏⢚⢪⠯⡻⣺⣺⢽⣺⡷⣝⢮⢮⡳⣝⢵⡳⣕⢧⣳⣝⣵⡳⣍⠮⢨⠨⡂⡢⢨⠐⢔⢼⢼⢷⡵⡵⡵⡳⡱⡕⡇⡇⡇⡇⡧⣳⢝⡮⣎⡮⣳⣫⢳⡱⡱⡸⣲⠱⢨⢪⢐⢌⠤⡱⢑⡑⡅⠜⡐⠕⠨
⢇⢇⢣⢣⢱⢡⢱⡑⡅⡇⡏⡭⡍⡍⡏⡳⣛⢺⢹⢺⢝⠟⠞⡚⣺⢽⡪⡎⡐⢕⢜⢜⢔⢍⡆⡮⣌⢪⡊⡎⡎⡮⡪⡳⡹⣪⢪⡺⣜⢮⡳⣵⡳⡯⡳⡣⡳⡸⡸⡜⣮⣣⢑⠔⠨⣐⠱⡈⡎⠆⡌⢌⢂⠣⡡
⢇⢧⢳⡱⡕⣇⢧⢣⡣⡣⡣⣃⢏⢞⠮⡺⣜⢼⢱⢱⢱⢡⡣⣳⣳⣳⡽⣕⢵⢱⡪⣪⢪⡪⣎⢗⡵⡳⣝⢼⡪⡮⡪⡣⡯⢮⡳⣝⡮⠗⢯⢗⢏⢏⠮⡪⡌⡮⣪⢞⣞⢦⠱⡐⠅⡌⣊⢀⠃⠇⠪⢐⢅⢕⠠
⡏⡮⡪⣪⢺⢜⢎⢧⢫⢎⢗⢜⢼⢜⡜⣕⢕⢝⠺⢵⢳⣗⣯⣷⢿⢵⢯⡲⡑⡱⢙⢎⢗⢽⡺⣽⡺⣽⢺⣳⡻⣪⢯⡣⣏⢗⢯⢳⡹⠀⢂⢁⢇⢇⢗⢕⢝⢜⢮⡺⣼⢳⠱⡁⢇⢆⢕⢰⠑⡡⠃⠅⡒⠐⢅
⡺⡜⡮⡪⣎⢮⢺⢸⢸⢪⢎⡗⡕⡧⡫⣪⢞⢜⢜⢜⢔⢎⡪⡎⣏⢝⢵⢱⡢⣕⢌⢎⢜⢢⢣⢣⢕⢵⢹⢔⢝⢎⢕⢕⢕⢕⢕⢕⢵⢡⢦⠀⠫⡪⡪⡎⣇⢗⢗⣝⣞⡑⢅⠪⢑⢡⠡⠢⡘⢄⢃⠅⡡⢑⢔
⢼⡹⡪⣇⢧⢳⡱⡕⡧⡳⡱⣕⠵⡕⡽⣜⡜⣎⢎⢎⢮⢪⢮⢺⡪⡣⡇⡗⡝⡜⣕⢗⢮⠮⡦⣣⢇⢧⢣⡣⡳⡹⠜⢜⢸⢨⡪⣑⠵⠳⡕⣕⠀⠣⡣⡣⡣⡯⣳⠍⠄⢎⢔⠪⡂⠢⡡⣃⢣⠪⡡⠪⡘⢬⢪
⢮⡪⡳⣕⡝⡮⡺⡹⣪⡫⡫⡎⣗⢝⢎⢗⢝⢚⢎⢗⠵⠵⠵⣕⣵⡣⡇⡇⡇⣇⢧⢣⢳⢹⠪⢊⢝⢮⢧⡣⡺⡸⡸⢌⠜⢆⢂⢎⠍⢕⠵⡡⡃⢅⠨⡊⢇⠯⡣⢪⢥⠡⡑⣕⢜⢜⡰⢰⢁⡖⣎⠕⡌⡌⡢
⡗⡝⣞⢜⢮⢳⢝⣝⢼⡪⡯⣺⢸⢪⢺⢸⢸⢨⢢⢣⢣⢫⢱⢱⢲⢱⠹⡹⢺⢪⢮⢮⣣⢗⡭⣢⣪⡺⡇⡇⡣⡃⢇⠣⡃⠕⢔⢐⢜⢌⢪⠂⢇⠣⡂⡈⢆⢑⠨⢡⠕⣕⡕⡕⡭⡣⡪⠊⠜⡮⡕⠌⢆⢊⢆
⡮⡫⡎⡗⣕⢇⣗⢼⡸⡜⡎⡮⡪⡣⡳⣱⢱⡱⡱⡱⡕⣝⢎⢧⢣⢇⢇⢕⢕⢱⢹⢸⢪⠫⣓⢕⢗⢝⡜⡬⢪⢸⠰⡱⡸⡸⢊⢎⢆⠆⡕⢌⠢⡒⠅⢄⣂⢆⣎⢂⠑⠅⣃⠳⡒⣕⢘⢬⢊⠄⢍⠪⢐⠡⢃
⡮⣫⢮⡻⣜⡕⣗⢵⡹⡜⡵⡹⣚⢝⢞⠮⣗⢗⢽⣜⢮⡪⡞⣕⢇⢧⢱⢸⢰⢱⢱⡱⡱⡑⡕⢕⠡⢣⠓⠛⠚⠲⢵⢱⠜⠒⠓⠲⣐⠗⠓⠒⠓⢲⢍⠒⢒⠒⣌⠱⢕⡗⠓⣖⢸⠚⡪⡒⠚⠚⠓⠣⡀⠕⡁
⣇⢗⢵⢹⢸⢪⢎⢧⢳⢹⢕⢽⢸⡸⡸⡸⡸⣸⢪⢮⠳⡝⡾⡮⣓⠳⢝⢼⢪⠮⡮⣪⢎⢮⡪⡲⡨⡸⡀⢪⢻⠀⢸⡇⠄⡏⢵⠁⢸⡊⠈⢋⠛⡯⣄⠨⠉⡒⢮⡊⢆⡇⠄⠉⠄⠂⡅⠆⡈⠛⠙⡧⢂⢣⢐
⡪⣫⢳⡹⡪⡇⡗⣕⢵⡱⣕⣝⢼⡺⣜⢎⢮⢳⡣⡏⡎⡎⡪⢪⠪⡙⡎⡦⢕⢕⣘⢪⢪⢣⢳⢽⡸⡪⡂⢈⢉⣄⠞⡣⣀⠍⢉⣠⢎⠇⢈⠉⡉⢹⢄⠌⠋⣀⡼⡸⢨⡇⠐⣏⢩⠀⡎⡅⠈⡉⢉⢱⡕⢜⠲
⡹⡜⣎⢮⢳⢱⢝⢜⠜⡜⢝⢺⢳⢯⢮⣗⣝⢜⢮⢎⢇⢇⢇⢇⢇⢇⢇⢎⠪⡪⠢⢑⠱⠥⡱⡢⢇⡇⡎⡂⡂⠪⡢⣘⠄⠝⠰⠰⠐⢕⠅⢇⢜⢅⢅⠫⡉⡢⠢⢃⠣⢨⡂⡎⡜⡆⡑⠬⡑⡜⢔⠐⡍⡎⣜
⢞⣎⢮⡪⡮⣪⢪⡢⡣⡪⡸⡨⡪⣊⢳⢓⣗⢯⣗⣗⣭⡳⣝⣜⢮⣪⢮⡪⣪⢪⢪⢂⢏⣽⠎⣂⢢⡃⡕⢅⠢⡑⢴⢑⠄⡃⢅⢂⢅⢕⠸⣊⢂⢁⢆⢊⡢⢭⡘⢆⢍⠆⢧⢑⠔⢬⡡⠃⡎⢢⠡⠱⡂⠮⡇
⢵⣳⡳⡱⡣⠧⡧⣳⢱⡱⡕⡵⡱⢢⢑⠕⡱⡹⡰⢕⡷⣝⢮⢎⢎⡎⢎⠇⡳⡙⡪⡚⡙⢎⡪⡘⢄⢃⠌⡢⢱⠚⢳⡵⠚⢚⢶⠓⠢⣢⠓⢳⢬⠒⠃⠒⠣⣳⠚⢳⠞⠚⣶⠓⢳⠓⢊⠚⢔⡱⡨⢊⡰⡨⠨
⢗⢗⡕⢕⢕⠕⡕⣳⢹⠪⡕⡝⡼⡌⡆⡕⡱⡨⡊⡇⡇⢕⢏⢮⢣⡺⡰⢕⢲⢑⠔⡅⢊⠢⢃⢂⠢⠡⡨⡈⣲⠀⡘⠠⠐⡕⢼⠀⡂⡘⢀⢺⡇⠠⡏⣳⠁⢸⡔⠘⡀⡆⢉⠀⣟⢲⠋⢂⠮⡒⡅⡅⡪⠘⡨
⡇⢏⢞⠢⣕⢜⠜⡎⢎⡪⡪⡚⡜⡜⡜⡜⡜⡔⡕⡜⡌⠊⡌⣝⢾⡸⡱⢅⠝⣔⠱⢨⠘⠌⢢⢐⠕⡣⡑⡐⡸⣀⣰⠵⣈⣘⣽⣀⡯⢢⣠⣸⠱⣄⡩⣑⡨⢎⢣⣐⣰⢣⣀⣼⢰⢺⣉⣹⠰⢡⠪⡐⠄⠕⡆
⡊⡪⣪⢺⠰⡅⣣⠪⣸⢰⠪⡘⡌⡎⡪⢪⢺⢸⢸⢑⡕⡕⡕⡗⡕⡑⡜⠬⠨⡢⡑⡑⡅⢅⠢⠑⠌⡠⡒⠩⡂⠢⡰⢑⠡⡑⡅⢆⠪⢈⢂⠪⡐⠄⡂⡂⢌⠌⢑⠼⡰⢱⠑⢎⢇⣗⢅⢇⠊⠬⡘⠌⡌⡎⢜
⠆⢕⢕⢵⢱⢑⠢⡹⣰⢱⢱⢱⢑⡑⡅⢕⠅⡇⡅⡇⢇⢫⠪⡂⡕⢅⢢⡡⢃⠇⠌⡢⢅⠑⡌⢌⢢⠡⠨⡊⢌⢣⠸⡐⡂⡣⠣⡨⠊⢜⠆⢕⢌⠌⡢⡉⡆⡕⡀⣂⡊⡎⠎⢎⢌⢺⡨⡂⡎⠎⠄⢇⢒⢢⠣
⠩⡒⢕⢹⢘⢌⢎⢎⢎⠮⡸⡨⡲⡱⠡⡣⡑⡌⡢⡹⢨⠪⡊⢔⢑⠅⢅⢊⠔⡌⢌⠂⢅⢪⢐⢢⠢⢱⠱⡐⡐⠅⡣⠅⢆⠜⡘⡐⢩⢃⠝⡠⢪⡪⣊⠜⢅⠆⡊⡆⡢⡹⡎⡦⠑⠥⡱⡌⣂⢣⠩⡐⡘⠔⢵
⠢⢊⠌⡎⢎⢎⠎⡎⡢⢇⢕⢱⢨⠪⡱⢑⠇⢕⢲⢘⢌⠪⡨⣂⢊⠢⡑⠔⡡⡂⠕⢌⠜⡐⢌⢃⡙⢄⢂⠅⡖⢑⠰⢡⠃⢎⡐⡊⡢⠂⠇⠆⢕⢘⠆⡕⡑⡑⢅⠢⡪⡨⠝⡪⠊⠔⡰⠡⡐⠡⠁⡒⣥⢪⢵
*/
using ll = long long;
#define int long long
#define forn(i,n) for(int i=0; i<(n); ++i)
#define pb push_back
#define pi pair<int,int>
#define f first
#define s second
#define vii(a,n) vector<int> a(n); forn(i,n) cin>>a[i];
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
const ll inf = 1e9;
const ll mod = 1e9+7;//998244353;
/*
⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⢿⢂⢂⢂⢂⠔⡐⡐⢔⢔⢡⠪⣛⣿⣿⣿⣿⣿⣿⣿
⣿⡯⣿⣯⣿⣯⣿⡿⣾⢝⢑⢐⢐⠔⢔⢱⢘⢌⢆⡧⣣⡣⡳⡜⡽⣾⣿⣯⣿⣷
⣽⣿⡿⣿⢽⣾⣿⡻⡋⡂⢆⢑⢜⠸⡸⡘⡜⣜⢜⢜⢜⢜⢕⢕⠱⡽⣿⣾⣿⣽
⢵⣻⣿⣯⣳⣿⣟⡯⠪⠐⠌⡂⢅⠇⡕⢕⢱⡿⡜⡸⡰⡑⢅⢇⠣⢝⡿⣯⣿⣿
⢼⣽⢻⣿⣻⣷⡷⡩⡱⢁⠣⢘⠄⢕⠜⠜⢒⢂⢇⣎⢆⢌⡄⢕⢁⡳⣫⣟⣯⣿
⠿⣽⣿⣿⣯⠧⡗⡕⢔⢐⢈⠐⢍⠢⡃⡣⡭⣽⣺⣺⡳⣜⢮⠸⣰⣮⣻⣿⣿⢿
⠪⡸⣘⢓⢕⠕⢕⠅⠕⡐⡐⢌⢈⠔⢌⢢⡫⣞⡾⡲⡪⡯⠇⢕⢼⣿⣾⣿⣻⣿
⠨⡂⠆⠕⡡⢊⠢⠨⡈⡮⡾⢜⠦⠂⡘⡰⡩⡚⡺⢝⢼⢡⢨⢺⣯⣿⣷⣿⢿⣟
⠡⡨⠨⢂⣢⠡⢊⢔⡽⡣⣋⢞⡝⢵⢲⡐⡕⡜⡄⠐⠨⡋⢔⢹⣿⡿⣷⣿⣿⣿
⠐⣼⢿⠫⡐⠌⡂⡾⢜⠜⡕⡽⡪⢏⡗⡧⣣⢣⢲⠠⡀⢜⠰⡡⠻⣿⣿⣯⣷⣿
⣸⣶⠟⡑⢄⢱⣸⣅⠅⠨⠈⠄⠌⡢⠨⢑⠑⢣⠱⡹⣪⠄⠑⢕⣯⢝⢿⣽⣻⣿
⡟⡐⡐⡐⣰⣰⣿⡍⠀⠂⠁⢌⠊⠔⡑⠄⠌⠄⡈⠪⡱⠏⣇⠕⡺⣿⣮⣛⢻⣿
⢇⢂⠢⢢⢯⣿⣿⣦⣄⠈⠐⠠⠡⢑⠠⠁⠈⠢⠨⠔⢘⢕⢵⣱⢑⢿⣿⡿⣷⣖
⢐⡐⠡⡿⣦⢿⣷⣿⠇⠀⢁⠨⠀⠅⢂⠁⠈⠀⡂⠌⢀⠎⣮⡪⡇⡆⠽⣿⣿⣿
⢘⢌⣧⠚⣿⢿⣷⣿⠂⠈⢀⠠⠐⠈⠠⠈⡀⠠⠐⢀⢁⠈⡎⠮⡺⡘⠐⠻⣾⣿
*/
const int N=1e6+5;
vector<int> adj1[N], adj2[N];
const int K=20;
int w1[N], w2[N];
int to1[N][K], to2[N][K];
int p1[N], p2[N];
int d1[N],d2[N];
int tin1[N], tin2[N];
int tout1[N], tout2[N];
int t1=0, t2=0;
void dfs1(int u) {
tin1[u]=t1++;
for(auto&v:adj1[u]) dfs1(v);
tout1[u]=t1++;
}
void dfs2(int u) {
tin2[u]=t2++;
for(auto&v:adj2[u]) dfs2(v);
tout2[u]=t2++;
}
int isa1(int p, int v) {
return tin1[p]<=tin1[v] && tout1[v]<=tout1[p];
}
int isa2(int p, int v) {
return tin2[p]<=tin2[v] && tout2[v]<=tout2[p];
}
int lca1(int u, int v) {
if (isa1(u,v)) return u;
for(int j=K-1; j>=0; --j) {
if (isa1(to1[u][j],v)) continue;
u=to1[u][j];
}
return to1[u][0];
}
void solve() {
int n,k,q,t; cin>>n>>k>>q>>t;
forn(i,n) cin>>p1[i];
forn(i,n) cin>>p2[i];
forn(i,n) if (p1[i]!=-1) adj1[p1[i]-1].pb(i);
forn(i,n) if (p1[i]==-1) dfs1(i);
forn(i,n) if (p2[i]!=-1) adj2[p2[i]-1].pb(i);
forn(i,n) if (p2[i]==-1) dfs2(i);
vector<pi> v; forn(i,n) v.pb({tin1[i],i});
sort(all(v));
forn(i,k) cout<<v[i].s+1<<' '; cout<<'\n'; //cout.flush();
forn(i,k-1) {
cout<<"? "<<v[i].s+1<<' '<<v[i+1].s+1<<'\n'; //cout.flush();
}
cout<<"!\n";
cout.flush();
forn(i,k-1) {
int a,b,c,d; cin>>a>>b>>c>>d;
w1[v[i+1].s]=b;
}
for(int i=1; i<n; ++i) {
d1[v[i].s]=d1[p1[v[i].s]-1]+w1[v[i].s];
}
forn(i,n) {
if (p1[i]==-1) to1[i][0]=i;
else to1[i][0]=p1[i]-1;
}
for(int j=1; j<K; ++j) forn(i,n) to1[i][j]=to1[to1[i][j-1]][j-1];
vector<pi> ans;
while (t--) {
int a,b; cin>>a>>b; --a, --b;
int l = lca1(a,b);
//cout<<"? "<<a<<' '<<b<<' '<<l<<'\n';
//cout<<d1[a]<<' '<<d1[b]<<' '<<d1[l]<<'\n';
int z = d1[a]+d1[b]-2*d1[l];
ans.pb({z,z});
//cout<<z<<' '<<z<<'\n'; //cout.flush();
}
for(auto&x:ans) cout<<x.f<<' '<<x.s<<'\n';
cout.flush();
}
int32_t main() {
ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0);
int t=1;
//cin>>t;
while (t--) solve();
return 0;
}
Compilation message (stderr)
Main.cpp:4:1: warning: multi-line comment [-Wcomment]
4 | // \ __ __ \
| ^
Main.cpp: In function 'void solve()':
Main.cpp:100:19: warning: this 'for' clause does not guard... [-Wmisleading-indentation]
100 | #define forn(i,n) for(int i=0; i<(n); ++i)
| ^~~
Main.cpp:183:5: note: in expansion of macro 'forn'
183 | forn(i,k) cout<<v[i].s+1<<' '; cout<<'\n'; //cout.flush();
| ^~~~
Main.cpp:183:36: note: ...this statement, but the latter is misleadingly indented as if it were guarded by the 'for'
183 | forn(i,k) cout<<v[i].s+1<<' '; cout<<'\n'; //cout.flush();
| ^~~~
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