⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡕⠂⠰⠾⣥⣖⡤⣀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣄⣶⣽⡳⠶⢶⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠈⠉⠐⠦⣉⠻⣷⣷⠶⣽⡿⠖⢶⡛⠻⠶⣶⣿⠡⢶⣽⣾⠛⡩⠔⠊⠉⠘⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣨⠟⠉⠀⠀⠸⡖⠉⠁⠀⠈⠑⠀⡽⠀⠀⠀⠙⠯⣄⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⠻⡢⡀⠀⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠞⠁⠀⠀⠀⠀⠀⠀⢠⣶⠀⠙⠦⣀⡤⠞⠁⢰⣦⠀⠀⠀⠀⠀⠀⠀⠈⢢⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠃⠀⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⠀⠀⠀⠀⠀⠸⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⣤⡤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣀⠴⠀⠀⠠⡤⠒⠉⢁⣀⣤⡤⠤⠴⠶⠒⠒⠶⠤⠤⣤⣄⣀⣈⡑⠲⢄⡠⠔⠒⠤⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢭⠃⠀⠀⠀⠀⠀⣨⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣷⣶⡋⠀⠀⠀⠀⠈⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⡏⠀⠀⠀⣤⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣬⣿⣿⣦⣄⠀⠀⠀⠈⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢾⡁⢫⠑⠦⠂⠐⢄⡰⠡⣀⠒⡞⣨⣟⢻⠥⡄⢠⣤⣒⣒⡈⢹⢯⠽⢽⢿⡿⣏⣳⡛⣓⣲⡼⢼⡿⢿⠟⢁⣕⡄⢀⡔⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠀⠉⠀⠀⠀⢀⡜⠁⠀⠀⢉⣧⣳⢸⢃⣇⣿⡿⣻⣿⣿⣻⣮⢯⢆⠀⡻⡷⡇⣼⣿⣿⣻⡿⣿⢠⣷⣸⢹⡟⣸⠁⠀⠀⠀⠀⠀⠀⠀⡰⣄⠀
⠀⠀⠀⠀⠀⠓⢄⣀⠀⡰⠋⠀⠀⠀⠀⣸⡶⢃⢼⠸⢺⣶⠱⢻⣽⢏⣽⡇⠈⢻⣢⠈⠱⡝⢻⣽⢯⣭⠷⣻⡿⣀⡎⡈⡽⣣⡆⠀⠀⠀⠀⠀⢠⠊⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⢼⠚⡡⣿⢸⡤⢆⣻⢆⠀⡳⠔⠊⠀⠀⠀⠉⠳⢆⡱⣌⠓⠶⢚⡼⢀⡴⡻⠀⡇⣧⢳⢮⢄⠀⠀⠀⠀⠀⠑⣢⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡰⠕⣁⡴⢊⣽⣿⣧⢏⢷⣕⢧⡁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠁⠀⡘⣵⠁⣸⣋⣾⣿⡿⣷⣃⠱⢵⡀⠀⠀⢠⠟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣰⢚⣻⣽⣻⣻⣯⠀⠳⣵⣺⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣯⣇⠜⣱⣽⡟⣯⣷⣿⢿⣟⡒⠿⠍⣅⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡡⣿⠿⢃⣿⢿⣧⡄⠈⢿⣷⣄⡀⠀⠀⠈⠁⠀⠀⣀⣤⡾⡛⠄⢜⣽⣿⣦⢹⡩⣷⡏⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠊⢻⠟⠿⡼⠻⣦⠘⡿⢿⣿⡇⠒⠠⠐⠒⣟⣿⠿⡿⣁⣔⣾⠏⠟⣟⣖⣛⣵⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠨⠗⠀⢠⣵⡇⣨⡢⠇⠀⠀⠀⠀⠱⢟⣘⣭⣄⡀⠉⠀⠀⠈⠙⠲⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢆⠒⡒⡠⡀⠝⡲⡲⢿⠽⣏⡁⠤⠀⢀⠀⢀⡀⠄⢄⣈⡿⣿⢓⢖⠆⢔⢦⢚⠙⡡⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣤⣄⣄⣀⠀⣀⣠⣀⡀⠀⠀⠀⠀⢮⠳⠘⡄⣇⢻⣨⢪⡞⣀⣼⣴⣏⡟⠛⡿⠉⠲⠁⠻⡛⢏⢹⣇⡷⣑⢥⣹⣇⠻⢸⢀⠃⡜⡾⡀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡕⠂⠰⠾⣥⣖⡤⣀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣄⣶⣽⡳⠶⢶⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠈⠉⠐⠦⣉⠻⣷⣷⠶⣽⡿⠖⢶⡛⠻⠶⣶⣿⠡⢶⣽⣾⠛⡩⠔⠊⠉⠘⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣨⠟⠉⠀⠀⠸⡖⠉⠁⠀⠈⠑⠀⡽⠀⠀⠀⠙⠯⣄⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⠻⡢⡀⠀⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠞⠁⠀⠀⠀⠀⠀⠀⢠⣶⠀⠙⠦⣀⡤⠞⠁⢰⣦⠀⠀⠀⠀⠀⠀⠀⠈⢢⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠃⠀⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⠀⠀⠀⠀⠀⠸⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⣤⡤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣀⠴⠀⠀⠠⡤⠒⠉⢁⣀⣤⡤⠤⠴⠶⠒⠒⠶⠤⠤⣤⣄⣀⣈⡑⠲⢄⡠⠔⠒⠤⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢭⠃⠀⠀⠀⠀⠀⣨⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣷⣶⡋⠀⠀⠀⠀⠈⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⡏⠀⠀⠀⣤⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣬⣿⣿⣦⣄⠀⠀⠀⠈⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢾⡁⢫⠑⠦⠂⠐⢄⡰⠡⣀⠒⡞⣨⣟⢻⠥⡄⢠⣤⣒⣒⡈⢹⢯⠽⢽⢿⡿⣏⣳⡛⣓⣲⡼⢼⡿⢿⠟⢁⣕⡄⢀⡔⠃⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⠀⠉⠀⠀⠀⢀⡜⠁⠀⠀⢉⣧⣳⢸⢃⣇⣿⡿⣻⣿⣿⣻⣮⢯⢆⠀⡻⡷⡇⣼⣿⣿⣻⡿⣿⢠⣷⣸⢹⡟⣸⠁⠀⠀⠀⠀⠀⠀⠀⡰⣄⠀ ⠀⠀⠀⠀⠀⠓⢄⣀⠀⡰⠋⠀⠀⠀⠀⣸⡶⢃⢼⠸⢺⣶⠱⢻⣽⢏⣽⡇⠈⢻⣢⠈⠱⡝⢻⣽⢯⣭⠷⣻⡿⣀⡎⡈⡽⣣⡆⠀⠀⠀⠀⠀⢠⠊⠀⠀⢱ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⢼⠚⡡⣿⢸⡤⢆⣻⢆⠀⡳⠔⠊⠀⠀⠀⠉⠳⢆⡱⣌⠓⠶⢚⡼⢀⡴⡻⠀⡇⣧⢳⢮⢄⠀⠀⠀⠀⠀⠑⣢⠒⠀⠠ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡰⠕⣁⡴⢊⣽⣿⣧⢏⢷⣕⢧⡁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠁⠀⡘⣵⠁⣸⣋⣾⣿⡿⣷⣃⠱⢵⡀⠀⠀⢠⠟⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣰⢚⣻⣽⣻⣻⣯⠀⠳⣵⣺⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣯⣇⠜⣱⣽⡟⣯⣷⣿⢿⣟⡒⠿⠍⣅⡏⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡡⣿⠿⢃⣿⢿⣧⡄⠈⢿⣷⣄⡀⠀⠀⠈⠁⠀⠀⣀⣤⡾⡛⠄⢜⣽⣿⣦⢹⡩⣷⡏⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠊⢻⠟⠿⡼⠻⣦⠘⡿⢿⣿⡇⠒⠠⠐⠒⣟⣿⠿⡿⣁⣔⣾⠏⠟⣟⣖⣛⣵⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠨⠗⠀⢠⣵⡇⣨⡢⠇⠀⠀⠀⠀⠱⢟⣘⣭⣄⡀⠉⠀⠀⠈⠙⠲⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢆⠒⡒⡠⡀⠝⡲⡲⢿⠽⣏⡁⠤⠀⢀⠀⢀⡀⠄⢄⣈⡿⣿⢓⢖⠆⢔⢦⢚⠙⡡⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣤⣄⣄⣀⠀⣀⣠⣀⡀⠀⠀⠀⠀⢮⠳⠘⡄⣇⢻⣨⢪⡞⣀⣼⣴⣏⡟⠛⡿⠉⠲⠁⠻⡛⢏⢹⣇⡷⣑⢥⣹⣇⠻⢸⢀⠃⡜⡾⡀⠀⠀⠀⠀⠀⠀⠀