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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⣄⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⣯⣯⣤⣤⣀⠀⠀⠀⠀⣿⣅⣽⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢺⢀⣿⠉⠁⠀⠉⢳⡄⠀⠀⠻⣿⡿⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⠞⠛⠁⠀⠀⠀⠀⠀⠙⢦⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⣀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⢵⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀
⠐⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣤⣧⣤⠤⠤⠶⠶⠲⠶⠶⠶⠶⠶⢦⡴⠶⠖⠲⠒⠒⠒⠒⠖⠲⠶⠚⢋⣥⣤⣽⣂⣤⣤⣤⡀
⠀⠙⢳⣤⣀⠀⠀⠀⢀⣊⣭⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡶⠿⢿⣦⣷
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢸⢩⣿⡇⠒⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣯⣄⣀⡀⠀⠀⠀⣀⣠⡴⠛⠀⠀⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⡟⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢻⣷⠀⠈⠉⠉⠛⠛⠛⠛⠋⠁⠀⠀⢀⣴⠃⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⣿⠁⠀
⠀⠀⠀⠀⠀⠀⣠⢞⡁⠀⠀⠙⢦⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠤⠚⠁⠀⢀⡏⠀⠀
⠀⠀⠀⠀⢀⡾⣳⣿⠃⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠶⠛⠋⠁⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠰⢸⠃⠀⠀
⠀⠀⠀⠀⡸⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠋⡎⠀⠀⠀
⠀⠀⠀⢠⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⣼⠃⠀⠀⠀
⠀⠀⠀⢸⣧⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⡸⠁⠀⠀⠀⠀
⠀⠀⠀⢸⡿⣿⡀⠀⠸⡄⠀⢲⡀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡴⢀⡜⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠈⢧⢿⣧⠀⠀⢳⡀⠘⠇⢀⠀⠀⢄⠀⠈⢿⣄⡀⠀⣰⠏⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⠟⣥⠏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣯⣿⣧⡀⠀⠀⠀⠀⠈⠆⠀⠈⠳⠄⠈⠹⠇⣰⠟⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣣⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢧⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠳⣜⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣝⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⣫⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣭⣝⡻⠿⢶⣶⣶⣶⠾⠛⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣦⣤⣀⡴⢛⡡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣲⣶⣶⠂⠤⠤⠼⣿⣿⠿⣿⣿⣒⠒⠒⠲⣿⠇⠀⠀⡿⣋⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⠀⠀⠀⠀⠀⠀⠘⠃⡏⠉⠉⠙⣿⡿⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠁⠹⣄⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡼⠋⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⣏⣀⣀⣠⣤⡴⠋⠁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣰⡇⠀⠀⠀⣤⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢿⠄⠀⣰⠏⢷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠙⠒⠛⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣶⣶⣿⣿⣷⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⡿⠿⠉⠉⠉⠉⠹⠿⣿⣿⣿⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠙⣿⣿⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⡇⠀⣀⣀⠀⠀⠀⠀⣰⣦⣤⣠⣤⣤⣤⣄⡘⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⣿⡇⠘⣿⣿⠇⠀⠀⣴⡿⢁⣉⣭⣥⣤⣼⣿⠇⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢺⣿⣧⠀⠀⠀⢀⣴⡿⢏⣴⣿⠟⢉⣩⣽⠟⠁⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⣿⣿⡆⠀⢀⣾⣏⣴⡿⣛⣥⣶⠿⠋⠁⠀⠀⠀⠀⢸⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣷⠀⣾⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣶⣄⡙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⡍⠛⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣾⣿⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣾⣿⣿⠀⠀⠀⠀⠈⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀
⢐⣿⣿⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀
⢸⣿⠀⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀
⢸⣿⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀
⢸⣿⡄⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀
⠹⣿⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⡀⠀⠀⠀⠀
⠀⢻⣿⣆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⡇⠀⠀⠀⠀
⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠃⠀⠀⠀⠀
⠀⠘⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠘⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣶⣧⣄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⡿⡟⣿⣿⣶⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠈⢟⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣟⣟⡻⢏⣳⣽⣷⢎⡽⢻⣿⣷⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⠿⣿⣮⡷⣋⢾⡻⣝⣮⣼⣿⣿⡿⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠻⠿⡿⠾⠿⠿⠿⠟⠋⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣇⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠿⠻⠛⠋⠁⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠒⠒⠒⠶⠦⠤⠴⠒⠚⠉⣰⠟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡿⣤⣄⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⣰⠞⠁⠀⠀⠀⢠⣤⠀⠀⠀⠀
⠀⠀⢠⡟⠸⣿⡿⢀⣤⣄⠀⠐⣷⣿⣿⡷⠀⠀⢻⠀⠀⠀⢀⡴⠋⡼⡇⠀⠀⠀
⠀⠀⢸⠈⠲⡬⢠⣿⣿⣿⡷⠀⠨⣭⠭⠖⣇⠀⣾⠀⠀⡴⠋⠀⡞⠁⣧⠀⠀⠀
⠀⠀⢸⣷⠞⠁⠀⠻⣿⠿⠇⠀⠀⠙⠶⠶⠃⡼⢿⣠⠎⠀⢀⡖⠁⠀⣿⠀⠀⠀
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⣰⠃⠀⡟⠀⠀⠀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠎⠁⠀⢠⠃⠀⠀⠄
⢀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠉⠉⠉⠉⠲⢄⡀⢀⣠⡷⠀⠀⢠
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠥⣀⠀⠀⠀⠀⠀⠀⠈⠓⠧⠤⢾⢀⠏
⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠉⠉⠉⠉⠉⠒⠒⠒⠚⠀⢨⠏⠀
⢸⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀
⠈⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠉⠉⠉⠉⠉⠉⢁⣤⡏⠀⠀⠀
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠶⠖⠊⠁⣴⠇⠀⠀⠀
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀
⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⣠⣎⡀⡀⠀⠀⣀⣰⢶⠶⠚⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⠛⠩⡝⠛⢛⣷⠖⠒⢖⣾⠟⢛⠛⠺⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠚⠉⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠿⣄⠀⠀⠀⠀⠀⣠⣤⣤⠶⢶⡀⠀⠀⠀⠀
⠀⠀⠀⢀⣤⡤⣄⠀⠀⠀⠀⠀⠀⣶⠿⠁⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⢸⣗⠆⣮⣽⢂⣟⠀⠀⠀⠀
⣀⡾⣻⢍⣧⡙⢎⡷⠀⠀⠀⠀⣼⠟⡁⠂⠁⠘⠀⢀⣤⡀⠀⣤⣦⣴⣴⣤⠀⣴⣔⡀⠀⠀⢿⡄⠀⠀⠀⢹⣚⣎⣹⣌⣾⠃⠀⠀⠀
⠀⢷⢯⡞⢿⣉⢮⣻⠀⠀⠀⠀⣿⡇⠎⠀⠌⠀⠡⠘⠏⠁⠈⠛⠛⠚⠛⠉⠀⠘⠉⠁⠀⠀⠘⣯⠀⠀⠀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠀
⠀⠘⠷⢭⡶⠾⠖⠋⠀⠀⠀⠀⣿⡇⠀⢂⠠⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠾⠛⠉⠐⠂⠹⣷⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢴⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⡿⠇⠀⠀⢀⠀⠀⠀⢀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠠⠐⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡿⢁⠰⠀⠌⠠⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠠⠈⣴⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣏⣞⣠⣁⣬⣦⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢁⠸⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠛⠟⡛⣯⠁⠆⠡⢀⠠⠀⠠⠐⠈⠀⡀⠄⠀⡀⠀⠐⠀⠐⠀⠀⡀⠐⠀⢀⠀⠂⡔⣿⡄⠀⡶⠛⠲⠋⠉⢹⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠄⠃⠄⡐⢂⠠⢀⠀⠠⢀⠐⠠⢀⠡⢈⠐⠠⠁⢂⠀⡐⠈⠄⡌⢒⣸⡏⠁⢈⣧⡀⠀⣾⡶⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣍⠲⡐⢄⢃⠰⢠⠉⣆⠡⠒⡄⢂⠔⢢⠘⡠⢌⠰⢠⠐⣈⠒⣌⣲⡿⠉⠀⠀⢼⡁⠀⢈⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠦⢌⠢⢅⠓⠤⡙⣘⠰⡉⡜⢂⢣⢑⡊⢜⠡⣊⣴⣭⠾⠋⠀⠀⠀⠀⠈⠓⠲⠚⠓⠛⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣻⣾⠾⠷⠷⠞⠛⠛⠛⠋⠛⠛⠛⠛⠿⣿⣏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢀⢀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⢀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠠⢀⠰⢀⢂⡐
⠄⢂⠐⡀⠀⡠⠀⣇⢆⢇⠀⠂⠀⠄⠀⠁⠀⠄⠀⠀⠀⢀⠀⢀⠀⠂⡈⠀⡐⠀⠂⡈⠀⠠⢀⠐⠈⠐⠠⠀⠁⠈⢀⠀⠀⠂⠐⢀⠠⠐
⠈⠀⢂⡐⠡⠐⠀⢮⣻⣎⠔⠀⠁⠠⢁⠈⠐⠀⡈⠀⠐⠀⠀⠀⢈⠠⠐⠀⠠⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⢁⠂⠌⡁
⡀⠉⠠⠄⡑⢈⠐⠈⣽⣿⣿⣄⠈⠄⠠⢀⠠⢀⠀⠀⠄⠐⢈⠀⠂⠄⠐⢀⠠⠐⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠈⠐⠠
⠄⡊⠀⠌⠐⢀⠈⠄⡈⢿⣽⡿⣷⡄⠁⠄⢀⠠⠈⢀⠂⢀⠀⢁⠀⠠⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠄⠀⠀⠄⠡⠁
⠂⡀⠄⠂⡈⠠⠈⠄⠀⠙⣯⡟⡽⣻⣶⣄⠀⠐⢀⠂⠠⠀⡀⠂⠈⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡡⠀⠐⡀⠄
⠄⠀⠈⠀⠀⡐⠈⡀⠐⡀⠹⣿⣛⣽⣻⣼⣷⡄⠀⠀⠀⠐⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⡫⠂⠀⡁⠄⠂
⠂⠀⠀⠀⠄⠀⠁⡀⠄⠀⠀⠙⣶⠖⣪⢽⣾⣿⢦⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣵⡿⠛⡄⠐⡡⢐⡂⢡
⠀⢁⠀⠠⠀⢀⠂⠀⠄⡈⠀⢀⠸⣯⣹⠾⣻⣻⣯⢟⣦⡀⠀⠀⣀⣂⡉⠐⡄⠄⠀⠀⣀⣤⣤⣶⣶⣿⣻⣿⠟⠋⠄⡃⠔⡡⢜⢒⡸⢡
⠀⠄⠀⢀⠀⠂⠀⡁⠠⠀⠐⠀⠀⠈⢷⣾⢋⡿⣽⣯⢻⣿⣷⡜⠀⡉⣹⠿⢷⣤⣶⣿⣿⢻⢟⢯⣱⠞⠋⢄⠰⣉⠘⡨⠘⣌⢒⠬⣐⢣
⠄⠠⠐⠀⡀⢁⠂⢠⠀⡠⢐⠠⠀⠄⠀⠙⢿⣱⢧⣻⡹⣾⠇⠀⠂⣤⡛⡞⢼⣿⡫⡗⡞⣽⡺⠚⠡⢌⠒⠌⠒⣄⠣⡅⠯⣀⢏⠒⡥⢣
⡀⠀⠄⠂⢀⠀⠑⠂⠰⠀⡂⠀⡐⠈⠀⠀⠀⠙⠿⣿⣷⡿⠐⡠⡰⡢⢅⣃⣿⣵⡽⢛⠉⣡⠠⢁⠃⢎⡘⢌⡡⢌⡱⡘⢆⠧⣎⢳⣡⠳
⡀⠆⠀⠆⣄⠰⡘⠀⡄⠰⠀⢃⠀⡀⠘⠀⠀⢀⢀⠛⣿⣇⢀⣴⣻⣼⣼⠞⠛⡄⡰⡄⢣⡄⢦⢇⡟⣆⡼⢠⡳⣞⣴⣻⡼⣧⣞⣷⢞⡷
⡅⠃⡰⠘⠈⠆⡅⣣⠘⡄⢘⠈⠆⠁⡀⠰⢠⠈⠎⢀⣿⣞⣮⡷⣳⡿⠋⢠⠃⡄⢡⠀⠞⣇⢮⣜⡸⢮⠹⣼⠹⡜⣮⣳⣻⣯⣟⡿⣿⣿
⠄⠀⠁⠂⠐⠠⢀⠃⠄⠠⠈⠀⢀⠂⠁⠄⠀⠀⠀⡸⣷⠟⣧⣴⠋⠀⠀⠄⢂⠐⡀⢂⠐⣈⠔⡌⡃⢌⠒⣡⢒⠱⣠⢣⡙⠴⣙⠶⡱⣎
⠀⠁⢀⠀⠁⠣⢌⠂⠄⠂⡈⢀⠂⠠⢈⠀⠄⢀⣴⣚⣴⢛⣮⠇⠀⠐⡉⢠⠀⠐⡀⠄⠂⠄⡈⠰⢁⢎⢂⠧⡬⡑⣐⠢⣜⡹⣜⠼⣱⢎
⠄⠡⠀⠈⠀⡁⢀⠒⠠⠁⠀⠂⠄⠂⠀⠀⣠⣾⣿⣿⣿⣿⡿⡇⠂⡁⠂⠥⠈⠁⠀⠀⡐⠄⣁⠚⡤⢚⢬⡚⡴⣃⢧⢫⡔⡳⣬⢳⡱⣎
⡂⢀⠀⠈⠀⠄⠐⠀⠄⠠⠈⢀⠠⠀⣡⣾⣿⡿⠻⠿⣿⣿⡅⡇⠠⠀⠉⢀⠁⠠⠐⠡⢀⡑⠤⡙⠴⣋⠦⣑⠧⡹⢬⠳⣜⡱⢎⢧⢳⡜
⠂⠀⢀⠁⠐⠀⠠⠁⠒⠠⠁⠀⢐⣼⠿⠋⠁⠀⠠⠀⠑⣿⡼⡇⢀⠂⠌⠄⢂⡁⠆⡡⢂⠌⣒⢩⢣⠝⣮⠱⢮⡱⣋⠷⣬⢓⣏⢮⡓⣞
⠂⠄⠀⠈⠀⠃⠀⠂⠈⠀⣀⣼⡟⠋⠀⠂⢁⠈⠄⢁⠀⣸⠗⡇⠈⠐⡈⠠⢃⡘⠰⡐⠢⢍⠲⣌⠲⣙⠶⣍⢧⠳⣍⠞⣬⣓⡞⢶⡹⣜
⠄⢂⠐⠀⡁⠀⠂⠐⣨⣶⠟⠁⡄⠀⠠⠐⠀⠐⡀⠄⠀⣿⢺⠁⠀⡐⠠⠑⠢⢌⡡⠍⡒⡌⢒⠬⢣⢣⠞⣘⢦⡛⣬⢛⡴⢣⡞⢧⡻⣜
⠀⠀⠄⠂⠐⢀⣠⡾⠋⠀⡀⡈⠀⠀⢁⠠⠁⢂⠄⢠⢰⡿⢏⠄⠀⠄⠡⣁⠊⢄⠢⡑⢨⠰⣉⠞⣥⢣⢛⡜⢦⡙⢦⡛⣼⢣⡝⣧⢛⡼
⡁⠂⠄⠀⡵⠟⠉⠀⠄⠀⠅⠐⠀⠠⢀⠁⠀⠔⡈⠄⠈⠫⣏⠀⢠⠈⠡⠄⣈⢂⠒⡅⢪⠱⢌⡚⢤⢓⢮⡸⢣⡙⢶⡙⢶⡹⡜⣎⡻⣼
⠄⠂⠈⡀⠀⠂⠀⠁⠈⠀⠠⠀⠁⠀⠀⠄⠈⠀⢀⠠⠀⣷⡟⠈⡀⠈⢅⠒⡠⢊⠜⡨⢡⢋⠦⡙⠬⢎⠦⡹⢅⡻⢬⣙⣎⡳⡝⢮⡳⢧
⡂⠌⠐⠠⢁⠠⠸⠀⠐⠁⠀⠀⠀⠀⠐⠀⠈⢀⠀⢀⢀⣿⠐⠀⠌⠁⡄⢊⠱⣌⠯⡔⢣⠊⡖⢩⡑⢎⠼⡱⢫⡜⣣⠳⣬⠳⡝⣧⣛⠷
⠅⠂⠁⢀⠂⠀⢀⠄⠈⡀⠁⠀⠀⠁⠀⠀⠈⠀⠈⠀⠸⡟⠀⠈⠂⢡⢀⠣⢱⣊⢮⡙⢤⠓⡸⢀⠞⣈⠖⣡⠓⡼⣡⢛⡴⢛⡼⡱⣎⣛
⡀⡁⠄⡀⠂⠀⠄⠐⠠⠀⠀⠀⠁⠈⠀⠂⠀⠂⠀⠀⢄⠀⢈⠀⡌⢂⡐⡘⠤⢂⠦⣑⠢⠱⡡⢍⠚⡤⢋⠴⣉⠶⣡⢋⠴⣋⠔⡣⣕⢪
⣁⠄⠒⠄⠐⡀⢂⠐⢁⠂⠈⠀⠀⡀⠐⢀⠁⢀⠐⠨⠐⠈⢀⠂⠔⢀⢁⢘⠢⡁⠎⢤⠃⠧⡑⢎⡱⢢⠉⢦⠡⢚⠤⣉⠞⡤⢫⢱⣌⢣