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