⣠⣤⣤⡤⠤⢤⣤⣀⡀⠀⠐⠒⡄⠀⡠⠒⠀⠀⢀⣀⣤⠤⠤⣤⣤⣤⡄
⠈⠻⣿⡤⠤⡏⠀⠉⠙⠲⣄⠀⢰⢠⠃⢀⡤⠞⠋⠉⠈⢹⠤⢼⣿⠏⠀
⠀⠀⠘⣿⡅⠓⢒⡤⠤⠀⡈⠱⣄⣼⡴⠋⡀⠀⠤⢤⡒⠓⢬⣿⠃⠀⠀
⠀⠀⠀⠹⣿⣯⣐⢷⣀⣀⢤⡥⢾⣿⠷⢥⠤⣀⣀⣞⣢⣽⡿⠃⠀⠀⠀
⠀⠀⠀⠀⠈⢙⣿⠝⠀⢁⠔⡨⡺⡿⡕⢔⠀⡈⠐⠹⣟⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢼⣟⢦⢶⢅⠜⢰⠃⠀⢹⡌⢢⣸⠦⠴⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣿⣇⡬⡌⢀⡟⠀⠀⠀⢷⠀⣧⢧⣵⣿⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢻⠛⠋⠉⠀⠀⠀⠀⠈⠉⠙⢻⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣯⠳⢷⠾⢻⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⠀⡀⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢄⣀⠀⠀⠙⣶⣿⠆⠀⠀⠘⠿⣦⠀⠀⠀⠀⠀
⠀⠀⣠⠴⠒⢲⡦⣄⣰⠀⢰⡄⢸⢃⡄⡿⣸⢀⣠⠤⠦⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⢦⠀⠀⢾⣴⣶⡆⢀⣶⠶⠚⠃⠀⠀⠀⠀
⠀⢰⣧⣤⢦⠀⠀⠆⢉⣦⡘⢟⠣⢿⠃⡇⡿⢁⡈⣀⣀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⣿⣦⢠⣴⣦⠜⡆⠀⠀⢹⡈⢻⣾⠃⠀⠀⠀⠀⠀⠀⠀
⢠⠯⠀⠈⠉⠀⠎⠑⢶⣾⣷⡞⢆⣟⡼⣠⠩⡯⠛⠅⠀⣀⡤⠴⠦⠇⠀⠀⠀⠀⠀⠙⠋⠀⠛⠧⠀⡇⠀⣀⣈⣇⣸⠡⣴⣴⠀⠀⠀⠀⠀⠀
⠸⠿⠹⠿⠒⣢⣄⣠⠄⣈⡿⣷⣽⡿⣷⣟⣞⠔⠳⡒⠾⠣⢄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⡯⠭⠖⢲⡧⣏⠑⢮⡀⠀⠀⠀⠀⠀⠀
⠀⠀⣠⠞⠉⠈⠁⢼⡭⠽⣟⣽⣿⣷⣛⠻⠏⡤⡦⢤⣠⠀⠀⡙⢿⠸⢷⡀⠀⠀⠀⠀⠀⣴⡿⠋⢹⠁⠀⠀⢸⣸⠉⠙⡄⠘⢦⢀⡀⠀⠀⠀
⠀⢰⢁⡔⠢⠖⠛⠆⠸⠌⠉⣽⣿⣿⡯⠤⡨⠬⠥⣀⣀⠀⠀⢨⠃⠀⢸⡟⣆⠀⠀⢠⡾⠁⠀⢀⠇⠀⣄⡄⣼⠇⠰⡤⣼⣤⣼⠏⣧⠀⠀⠀
⠀⣄⣎⡞⢀⡤⠟⣀⣬⡤⡞⢗⠁⢻⣰⣪⢄⡀⠀⠉⠀⠙⠢⡏⠀⠀⠀⠀⢸⠀⢠⡟⠀⠀⠀⡜⠀⠈⠟⣹⠏⠀⠀⣽⡄⠉⠁⠀⠿⣧⣀⠀
⠀⠹⡟⣰⣿⠒⠉⠁⠀⠀⡏⠈⢀⢸⡅⠀⠉⠾⡀⠀⠀⠀⠀⢱⠀⣰⠀⣀⠗⠒⠉⠉⠉⠉⠻⡁⠀⠀⣰⠋⠀⠀⢘⡿⠃⠀⠀⠀⣤⠔⠛⠃
⠀⠀⠱⠿⠃⠀⠀⠀⠀⠀⢧⠀⠘⠸⠙⣄⠀⠀⠀⣤⡠⠀⠀⡾⣲⠛⡸⢛⡀⢀⡀⣀⡠⠔⠒⢱⡄⡴⠃⠀⠀⠀⠚⠒⣻⠿⣦⣰⠇⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡄⠀⠀⠈⢦⣀⡀⠈⣻⢣⢲⣱⢁⠴⣡⢔⣮⠟⠊⠁⠀⠀⠀⣸⠟⠁⠀⠀⠀⢀⣠⡾⠁⠀⠈⠹⢰⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡈⢪⡀⢄⣧⠶⢛⡢⠆⣶⣯⣾⣷⣷⣿⡾⢇⣤⠆⠀⠀⢀⣤⠞⠃⠀⣴⣒⣺⠿⠟⠋⠀⠀⠀⠀⠀⡞⣰⣤⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⢺⣷⡋⡽⠋⢁⣠⡼⢷⣾⣿⣿⣿⡿⣽⡝⡉⣑⡢⠽⠔⠚⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠄⡼⠀⠚⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠶⠗⠈⣑⡶⠋⢀⡀⠈⠀⠀⠄⢢⡋⠁⠹⡧⠀⠉⢻⡙⠓⠢⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡜⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⢾⣟⣧⣷⣋⣁⣀⣀⠤⠔⠚⡟⠐⠀⢀⡇⠈⠀⠰⣇⡀⠀⠀⠀⢄⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⣋⢀⣰⡴⠈⢆⠀⠀⠀⡇⠀⠀⢠⠇⠀⠇⢠⠗⠋⠒⠦⣄⠈⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠿⠁⠛⡩⠚⠉⡟⠲⡄⡇⠀⠀⠀⠀⣾⢠⠎⣀⡤⠴⠾⠭⠶⢍⣉⡉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⣀⠀⢸⡠⠼⣇⠀⡴⠓⢦⣿⣯⢮⠥⠄⠠⠤⣄⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⡇⣱⣧⡴⠙⡧⣾⠟⠻⣶⣶⣯⣿⣵⣫⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣉⡡⡟⠢⣧⡜⡦⠜⣿⡯⣬⣽⣱⠟⡠⡄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠻⣯⡍⠁⢉⣳⠟⢿⠇⠉⢳⡀⢿⡇⠀⠀⠀⠀⢠⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⣶⣜⡟⡔⠙⠑⠢⡀⣑⠤⠹⢦⠤⠤⢒⣵⣿⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⣎⢿⠀⠀⠀⢠⣏⠀⠀⠀⠀⢱⡐⠺⠿⠿⠟⠁⠀⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢧⣀⡈⠻⡦⣀⡠⣷⡿⠇⠀⡀⠀⢈⠼⣑⣒⡤⠤⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣢⣠⡎⡴⠎⠋⠽⢿⣿⣿⣽⣿⣵⢆⡼⢤⣄⠀⠀⠀⠀⢠⠞⣭⡝⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⢣⡀⠀⠀⠀⢺⣿⣿⣿⣾⠉⠈⠁⠀⢳⠀⠀⠀⠻⠀⢈⡵⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⡂⢤⣠⠚⠙⠉⢻⠀⠀⠀⠀⣽⡟⢵⣈⣀⣤⡶⠶⠊⠀⠠⢼⣆⡴⣋⡴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠫⡇⠀⠀⠀⠈⠓⠤⢠⡔⢫⡘⠄⠀⠀⢱⢀⣀⠀⢀⣴⢠⣾⠿⠧⢤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⠉⠩⠵⢶⣿⣰⡀⠀⠀⣀⠀⢨⣿⡦⡉⠒⠒⣒⣯⢘⣏⠛⠋⣸⣋⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡉⠀⠀⢀⡤⣶⣻⣿⣷⣶⣿⣶⣶⣅⣿⣿⣌⢦⣼⣠⣤⣸⣋⣀⠀⡤⠭⠓⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣶⣳⠀⠀⠀⠀⠀⠙⠪⠬⣵⠅⠀⠤⡞⣛⣿⣿⠏⢱⡉⠙⢝⣖⠿⣿⡝⢮⣭⡤⣴⠟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣰⡿⣻⣛⢳⡀⠀⢀⣀⣤⣤⣄⡇⠀⠀⢀⡀⠁⠹⡼⠀⠀⢸⢀⢀⣼⠀⠀⢙⣀⣤⢚⣯⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢾⣿⣿⣿⣿⠿⠦⢄⣘⣟⣿⣿⡷⣄⠴⠋⠀⢀⡴⣇⠀⠀⢸⠉⠁⢈⢷⠭⣉⠘⠟⣺⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⠽⠛⠁⣠⢶⡶⠋⠀⠉⠉⢀⠙⠢⢶⣒⡽⠹⣮⣵⡒⠋⠀⠀⠸⣿⢏⢗⢷⠀⢸⡠⣶⠂⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣦⠟⢀⣾⣿⣿⠇⣇⠻⣝⣻⣆⡀⠀⠀⠹⣼⡧⣄⡁⠀⠛⠦⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡏⠀⢘⢟⡫⠥⣴⡿⡿⣢⠤⣄⠉⠀⠀⠀⠀⠉⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⢺⠯⡾⠳⣀⠔⡞⣾⡿⢿⣝⢼⠋⢀⣧⠤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣀⠀⣀⡄⢰⠇⠀⠀⠀⣟⣚⣷⣀⡰⣷⣠⠟⣃⢀⡞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣏⠛⢹⡇⢸⢠⢀⠀⢀⣟⣁⣀⡩⣾⣾⣿⡿⢯⣥⠭⢭⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠾⣥⡄⣰⠾⢻⡿⠟⣔⢿⣿⠋⠈⢊⡿⢻⡿⣿⡀⠙⡽⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠙⠏⠀⠘⣇⢰⣿⠿⠿⠠⠶⢻⠂⢸⡧⠈⢷⠢⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣎⣡⡄⠀⠀⠀⠸⣆⡜⠳⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢹⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀