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