Hot cold Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡔⢒⣒⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡤⠐⠀⠀⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡬⠋⠉⠉⠉⠀⠀⠀⠀⠀⡠⣀⠘⠭⠆⠀⣐⡲⠄⡠⢦⡄
⠀⠀⠀⠀⠀⡏⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⣠⠖⠒⡶⠒⠛⠒⢛⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠀⣀⣀⣀⣀⠒⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢾⠃⠀⠀⠠⡞⣿⣉⣉⣉⣉⣉⣙⣚⡛⠓⢦⠆⠆⠀⠀⠀⠀⠀⠀⢰⡚⠋⠉⠹⡍⠰⢬⣉⠙⣦⢀⡀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⡏⠁⣀⣤⠤⣤⣀⠀⠈⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠛⠒⠲⠶⠶⠶⠒⠊⢹⢅⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⡇⠀⢧⠀⣀⠆⡜⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⣠⢆⠾⢊⣀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢡⢀⠸⣄⣀⡴⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢡⡙⠶⠤⠤⠤⠕⠚⠁⢘⠆⠈⡃
⠀⠀⠀⠀⠀⠘⠦⠤⠄⠵⠃⠀⠀⠀⠀⠀⠘⣎⣃⣀⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⢤⣌⣙⣓⣒⣚⣩⣭⠭⠥⠶⠚⠁
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠤
⠀⠀⠨⠔⠁⠀⠀⠀⠠⠬⠕⠀⠀⡠⢢⠄⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠰⣍⡆⠀⠀⠑⠖⠀⢀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⢜⢢⡀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡴⢤⣷⣿⡇⢀⣀⣀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⣠⢹⡿⠷⠞⠛⠛⠓⠋⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣤⠶⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠛⢶⡟⠁⣰⣋⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡴⠋⣀⣤⡴⠶⠛⠛⠛⠉⠉⠙⠛⣿⠁⢀⣼⠂⠸⡇
⠀⠀⠀⢀⣠⠶⠶⠦⢴⡇⠀⠀⠀⠀⠀⠀⣷⠟⠉⠉⠉⠓⣆⠀⠀⠀⠀⠀⠀⠀⠂⡙⠁⣾⣏⡁⠀⣠⡶⣄⠀⠀⠀⣀⣀⣧⠞⡨⣄⠀⢰⡇
⠀⠀⢠⡟⢡⠖⠒⠂⠀⢙⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠙⠛⠛⠛⡟⠋⠉⠉⠁⠀⢰⣦⣭⣴⠟⠁
⠀⠀⠘⢇⡈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣹⠖⠲⠆⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣼⡴⠶⣦⠀⠠⡄⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣹⢦⡈⠳⠤⠶⠂⣰⣿⣄⣠⡴⣟⣍⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣋⠙⠀⠀⢻⡀⢰⡇⣿
⠀⠀⠀⠀⠈⢳⡦⢤⡶⠞⠁⢸⡍⠓⢶⡶⠛⢡⣿⣁⡿⠊⢡⢇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⢿⣃⣠⡶⠟⠃⢸⡃⣿
⠀⠀⠀⠀⠀⠈⣧⡄⢳⡀⠀⠘⣿⡆⢸⠇⠀⢸⢳⣷⣇⣠⢏⡾⠁⠀⠀⠠⡲⠄⠀⠀⠀⠀⠸⡇⠀⠀⠘⠛⠁⠀⠀⠀⣀⢀⡏
⠀⠀⠀⠀⠀⠀⢹⡷⢸⡇⠀⠀⢿⠃⣼⠀⠀⢸⣀⣾⣄⣡⠞⠁⠀⢼⠛⠻⡆⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠳⣾⡁⠀⠀⠈⠉⠁⠀⠀⠀⣽⢇⠀⠀⠀⠀⠀⠈⠳⢶⣧⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⣤⣤⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠶⠶⠖⠛⢡⠞⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠻⡮⠉⠑⠒⠒⠚⢉⣩⠭⠭⠭⠵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠍⠳⠶⠖⠀⠀⠀⠘⠋⠀⠀⠀⠉⠒⠲⢤⡀
⠀⠀⠀⢠⡄⠀⠀⠀⠒⠂⠀⠉⠀⠀⠀⠀⠉⠁⠀⠛⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠒⠒⠒⠒⠊⠉⠉⠉⠉⠓⠦⠤⠤⠤⠶⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠑⢤⡀⠀⢠⠃⠀⠱⡀⢀⠀⠀⠀⠀⢀⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠲⡒⠤⠤⣀⡀⢸⠀⠀⠀⣱⠀⠣⠤⢀⡠⠇⡟⡄⢀⡠⣤⠃⠀⠘⠒⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠈⡆⢁⡨⠍⠑⠂⠉⠉⠉⠁⠒⠁⠈⠁⠀⡟⠀⠀⠀⠀⠀⠦⠤⡄
⠀⠀⠀⢖⠀⠒⠒⠢⢄⡑⠤⠔⡪⠒⠁⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠳⣀⠀⠀⠀⡹⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢺⠁
⠀⠀⠀⠀⠀⣈⡑⣒⠊⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣃
⠀⠀⣤⣚⠉⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠷⠒⠒⠒⡤
⠀⠀⠀⠀⠉⠢⢄⠜⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠪⡄
⠀⠀⠀⠀⢀⡠⠒⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⡴⠋⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠑⣄
⠀⠀⠈⠉⠉⠁⠒⣋⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠋⠁⠈⠉
⠀⠀⠀⠀⠀⢠⠊⠀⠈⡎⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⠇⣀⡀⠤⢎⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠢⡀
⠀⠀⠀⠀⠘⠉⠀⠀⠀⡎⠀⠈⢢⠑⠢⢄⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠚⠀⠀⠀⡜⠁⣠⠀⠀⠀⢄⠀⠀⡐⠢⢄⣿⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⢣⡞⠁⠀⠀⠀⠀⠳⡴⠀⠀⠀⠀⠢⣀⢨⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠚⢧⣀⡠⠓⢤⠃

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀⠀⣠⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠛⠀⢠⡟⢠⡶⣦⠘⣧⠠⠤⠤⠄⠀⠀⠀⠀⢰⣶⡆⠀⣹⣿⡁⢸⡏⠀⠀⠀⢹⡆⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⠞⠛⠓⢸⡇⢸⡇⣿⠀⣿⠰⠶⠆⠀⠀⠀⠀⠀⠈⠉⣿⣿⣘⡟⠀⢸⡇⠀⠀⠀⢸⡇⠰⠶
⠀⠀⠀⠀⠀⠀⠴⠶⠆⢸⡇⠀⠀⠀⢸⡇⢸⡇⣿⠀⣿⠐⠓⠛⠂⠀⠀⠠⢾⣷⢾⡷⠾⣿⡇⠀⢸⡇⠀⠀⠀⢸⡇⠐⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣄⢸⡇⢸⡇⣿⠀⣿⠈⣛⣃⠀⠀⠀⠀⠈⠁⡈⣶⣾⢋⣇⠀⢸⡇⠀⠀⠀⢸⡇⢘⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣤⠀⢸⡇⢸⡇⣿⠀⣿⢈⣉⡉⠁⠀⠀⠀⠀⢨⡿⡏⠉⢻⣿⠃⢸⡇⠀⠀⠀⢸⡇⢈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇⢸⡇⣿⠀⣿⢀⣭⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⢸⡇⠀⠀⠀⢸⡇⢨⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⣿⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⣿⠠⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⢶⣄⢸⡇⠰⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇⣿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⣿⠀⣿⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣴⠟⠁⠙⢷⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⡾⠋⠀⠛⢶⡘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣇⠀⠀⠀⢀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⡇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡙⠶⣤⡴⠞⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠻⢦⣤⡶⠟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠟⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⣿⢻⢿⠿⠿⢿⢿⠿⢿⢿⢿⠿⠿⡿⢿⡟⡿⠿⡿⣿⣿⢻⢿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣾⣶⣼⣷⣿⣾⣾⣶⣭⣾⣷⣷⣧⣿⣧⣷⣼⣷⣿⣿⣾⣮⣿⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⢸⡆⢀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢋⣿⠀⢸⡿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⠀⢸⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣼⣿⡿⠟⠋⠉⠀⠀⣿⠀⢸⣷⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠟⠁⠀⠀⠀⢠⣿⣷⠀⠀⢰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⣧⣴⣾⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣶⣤⡀⢀⣠⣿⣿⣿⣿⡛⠉⠀⠀⠈⠉⠛
⠀⠀⢀⣴⣾⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠻⣿⣿⣿⡿⠟⠛⠿⢿⣿⣿⠆
⣠⣾⣿⠟⠋⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠈⣿⣿⡄⠀⠀⠀⠀⠈⠁
⠙⢿⣿⣦⣄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠻⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⣤⣿⣿⣿⣷⣦⣤⣶⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠿⠛⠁⠈⠙⣿⣿⣿⣿⣥⡀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢻⣿⡟⠻⢿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⣿⣿⠉⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣦⡀⠀⠀⠀⠘⣿⡿⠀⠀⠸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠙⢻⣿⣷⣦⣄⣀⠀⠀⣿⠀⢸⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⣿⠀⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⢸⠇⠈⠻⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣼⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠙⢿⠀⣇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⢠⣤⡀⠈⠻⣷⣄⣸⠀⣿⡄⢀⣼⠁⠀⠀⣀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⡄⠈⣿⣇⠀⠀⠈⠻⣿⠀⣉⣉⠛⠻⣤⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠿⣿⣦⣼⣿⡄⠀⠀⠀⢹⠀⣿⣿⣿⣶⣄⠙⣿⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣽⣿⣿⣷⣤⣀⠀⢸⠀⣿⣿⣿⣿⣿⣧⠘⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁⠀⠈⠛⠿⣷⣾⠀⣿⣿⣿⣿⣿⣿⡇⢹⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⡀⠀⢀⣠⣶⣿⢿⠀⣿⣿⣿⣿⣿⣿⡇⣸⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠙⠛⣻⣿⣿⡿⠛⠉⠀⢸⠀⣿⣿⣿⣿⣿⡟⢠⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣶⣿⠿⣻⣿⠃⠀⠀⠀⣸⠀⣿⣿⣿⠿⠋⣠⣿⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⠃⢀⣿⡏⠀⠀⢀⣴⣿⠀⣉⣉⣤⣶⠛⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠘⠛⠁⢀⣴⡿⠋⢹⠀⣿⠃⠈⢻⡀⠀⠀⠉⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣠⣾⠀⡏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠙⠢⣄⠀⠀⠀⢠⠃⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⡀⠠⡃⠀⠀⠀⢹⢀⢶⠀⠀⠀⠀⢠⠊⠀⠀⣇⠤⢄
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠉⠒⠦⡀⢣⣀⣀⡠⠔⣃⣀⣈⣙⣒⣒⣋⣸⠀⢣⠤⠒⢹⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡼⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⠓⠒⡶
⠀⢫⡉⠉⠉⠑⠒⠤⣄⠳⠤⠖⡩⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠑⢄⠀⠀⠀⠀⠈⡇⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⡿⠁
⠀⠀⠀⠀⠑⠢⠤⠴⠊⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⣠⠴⠒⠈⠉⠙⡆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⣄⣀⣀⣀⣀
⠚⠲⠤⡀⠀⠀⢀⢇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣨
⠀⠀⠀⠈⠓⠤⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠓
⠀⠀⠀⣠⠔⠒⢢⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣠⠞⠁⠀⠀⢸⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠈⠣
⠰⠗⠒⠒⠦⠤⠞⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠟⠒⠤⠔⠒
⠀⠀⠀⠀⠀⡴⠋⠑⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠁
⠀⠀⠀⢀⡞⠀⠀⠀⠘⡄⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣧
⠀⠀⠀⡜⣀⠤⠤⠤⠚⡡⠤⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢄
⠀⠀⠀⠋⠀⠀⠀⠀⢰⠁⠀⠈⠳⡈⠓⠤⣀⡀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢀⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠴⠃⠀⠀⠀⡩⠋⠀⡀⠀⠀⠀⢀⠀⠀⠀⡠⢤⣀⢸⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⠀⠀⠈⣆⣠⠞⠁⠀⠀⠀⠈⠳⣄⣰⠁⠀⠈⠉⠱⡀⠀⠀⡶⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠈⠓⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠀⠀⠀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⠙⢦⣀⡴⠋⠑⣼
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⢠⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢠⣀⡀
⠀⠀⠀⠀⠀⠲⣾⣇⠈⣿⢰⣬⡙⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⣆⠈⡇⢠⣬⡙⣦
⠀⠀⠀⠀⠀⢈⣉⢈⣻⣿⢸⣿⣿⢸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢈⣳⡇⢸⣿⡇⢸⠦
⠀⠀⠀⠀⠀⠰⢿⡏⠀⣿⠸⠿⣣⡾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⠏⢀⡇⠸⠟⣁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⠸⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡇⠐⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣀⣀⣠⣤⣼⣿⣿⣤⣤⣀⣀⡀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⣠⣶⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣦⣀⣠⣾⠟⠁⡀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠟⠁⣠⣾⣿⣿⣿⠇
⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⠈⠉⠁
⠀⠀⠻⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⣾⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠈⠙⠻⠿⠟⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⠋⠉⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⣷⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣋⠉⠙⠛⠛⠻⠿⠃
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⣀⡀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣏⡀⢀⣴⣿⣿⡿⠋⢀⣴⣾⣿⡟
⠀⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣿⠟⠁⠀⢀⣤⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⠋⢻⣿⣷⠀⠈⣿⣿⡟⠻⣿⣿⣿⣿⣿⣶⣶⣿⣿⣟⡉⠀⠀⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⣿⣷⢀⣼⣿⣿⣿⣶⣾⣿⣿⡄⠘⣿⣿⡍⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⣀⣠⣴⣶⣦⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⡏⢹⣿⣿⣿⣿⢿⣿⣿⠛⠿⢿⣿⡷⠀⢹⣿⣷⡀⠀⠸⣿⣿⡀⠈⠙⣿⣿⣤⣤⣿⣿⣿
⠀⠀⣴⣿⣿⣿⡿⠟⠋⠀⠀⠹⣿⣿⣧⡀⠀⣠⠞⠁⣠⡾⣿⣿⡇⠈⣿⣿⡿⠋⢨⣿⣿⣷⣤⣄⣀⠀⠀⠀⠻⠟⠃⠀⠀⠻⠿⠇⠀⠀⠹⣿⠿⠟⠛⠋⠉
⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠞⠁⣠⣾⠟⠁⢸⣿⣷⠀⠉⠋⢀⣴⣿⣿⣿⣿⠿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⣾⣿⣿⡆⠀⠘⣿⡿⠀⠀⣠⣾⣿⠟⠉⣿⣿⡆⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⣠⣾⠟⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⢀⣼⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⢀⣸⣿⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠿⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣠⣾⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡉⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣿⠀⣧⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠀⣿⠘⠿⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣀⣾⡆⠈⠛⣿⠀⣉⡉⠻⢿⣄⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⢿⠁⣿⣷⣦⠀⣿⠶⢶⣤⡀⠻⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣈⣿⣿⣧⣄⡀⣿⠀⣿⣿⣷⡄⢻⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷⠀⢀⠘⠻⣿⠀⣿⠀⠀⠈⢻⡄⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢉⠁⠀⠈⢙⣿⣿⠀⣿⣿⣿⣿⠀⣿⣦⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⡦⠀⠀⣿⠀⠀⠀⠈⣿⠀⠙⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣶⡾⠛⠉⣿⠀⣿⣿⣿⡟⢠⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠉⢁⣤⣶⠀⣿⠀⠀⠀⣰⡟⠠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠿⠋⣿⠇⠀⣠⣿⠀⠿⠛⢉⣴⠿⠿⠦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣠⢐⠀⣛⣿⠟⠀⣿⣤⣤⡾⠟⣠⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣿⠀⣶⠟⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣥⣤⠀⣯⢉⣭⣄⡸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢂⡤⠤⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⠃⠀⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠶⡀⣇⠴⠁⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⠀⣠⠆⠘⣤⢦⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢸⠀⠀⠐⠓⠚⠒⡺⡿⣒⡒⠗⠺⠂⠀⢸⠇⠀⡇⠀⠀⠀⡔⠁⠤⡀⡀⠈⣇⡀⠀⠀⠀⡇
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠋⢠⣇⠘⠢⠀⠀⠀⠀⡼⠀⠀⣇⠀⠀⠀⡇⠠⠁⣠⠋⠀⠉⡇⠀⠀⢰⠣
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠰⡧⠀⠀⠀⠀⠀⡰⠃⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⢐⠆⡸⠀⠀⢠⢯⠞
⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠈⠉⢀⣠⠔⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⡇⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⣧⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⣦⠀⡇⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠐⠒⠒
⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⣿⠀⣿⣿⠀⡇⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠐⠶⠂⠀⠀⢀⢤⣤⠄⡀
⠀⠀⠐⢄⡕⠚⠺⡥⠖⠀⠀⣿⠀⣿⣿⠀⡇⠰⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠰⠶⠶⠀⡚⠻⣄⣇⡴⠛⣂
⠀⠀⠘⡸⣄⠀⢀⣟⡘⠀⠀⣿⠀⣿⣿⠀⡇⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣄⠀⣿⠰⠶⠆⠀⠟⢛⡿⡿⣟⠛⠿
⠀⠀⠈⠀⠎⢹⠙⠆⠉⠀⠀⣿⠀⣿⣿⠀⣇⠠⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠰⣿⡿⠀⣿⠰⠶⠶⠄⠙⢿⣠⣷⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣤⣿⣿⣄⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠁⠠⢾⡷⠄⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⣿⣿⣿⣿⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣿⣿⣿⡿⠃⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣿⣿⣿⡿⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣉⣉⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣉⣉⣤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠲⢄⡀⠀⠀⠀⠀⢀⡇⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠢⡀⠀⢀⡞⠀⠀⠀⢳⠀⣀⠀⠀⠀⠀⠀⢀⡤⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠢⠤⠤⣀⡀⠀⢸⠀⠀⠀⠀⢸⠄⠘⠦⣄⣀⣀⡸⢂⡏⢇⠀⢀⣠⣤⠏⠀⠀⠐⠓⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⠳⡌⠓⠶⢒⣺⠥⠶⠒⠒⠒⠒⠒⠦⡼⠁⠘⠊⠁⠀⣿⠀⠀⠀⠀⠀⠀⢀⣇⡀⣀
⠀⠀⠀⠀⣀⠤⠄⣀⣀⠀⠀⠘⡄⠀⠀⣠⠔⣡⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠑⠢⡌⠓⢋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⡼⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠈⣓⡲⠒⠋⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢀⡤⠖⠋⠉⠀⠈⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢒⣄⡠⠤⠤⢀⣀
⠀⠀⠐⠛⠒⠦⣀⠀⠀⢀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠤⠞⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⣀⠴⠒⢢⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠈⠫⡀
⠀⠀⠀⠴⠥⠶⠤⢤⣀⡼⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡷⠦⠤⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠘⡌⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡸⠁⠀⣀⣀⣠⠇⣀⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⢤⡀
⠀⠀⠀⠀⠀⠠⠷⠊⠉⠉⠀⠀⡞⠉⠉⢢⡙⠢⣄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡱⠀⠀⠉⠒⢦⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⡖⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠖⠋⠀⠀⠀⠀⢴⠁⢀⣴⠂⠀⠀⠀⠐⣦⠀⠀⡜⠉⠓⠦⠿⡄⠀⠀⢀⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠀⠀⠀⠀⠀⠀⠀⠈⢶⡟⠁⠀⠀⠀⠀⠀⠈⢳⡶⠁⠀⠀⠀⠀⠙⢤⡀⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣠⢶⠀⠀⠀⡴⣄⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠓⠤⠚⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⢿⢉⣛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⢤⣧⠀⠈⣿⣿⣶⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⡛⠘⣿⣷⠀⠀⠙⢦⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠑⣦⣿⣿⣿⣿⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣶⣮⠝⠀⠀⠀⠀⡗⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣶⠋⠁⣿⣿⣿⡟⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⢎⠉⣵⣾⠀⠀⠀⣠⢣⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠾⣟⣛⠯⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⡿⣫⠤⡤⣖⣥⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠟⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣵⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⡦⠀⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠁⠀⢀⡏⠀⢴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⡏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡗⠤⣤⠴⠒⠲⢤⡤⠖⠛⠤⣤⠴⢾⠁⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡤⠶⠦⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡼⠉⠙⣆⠀⠀⢹⡄⠀⢸⡄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⢿⠀⠀⢠⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⣄⣈⣓⡤⠚⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⢸⠇⠀⢰⡄
⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠀⠀⠀⣇
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢻
⠀⠀⠀⣾⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣟⣻⣿⣛⣛⣟

⣿⡿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠃⡀⣀⢀⡀⠀⠀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⡄⠁⠈⢁⠈⣈⠀⠁⠈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠠⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣏⠀⠀⠐⠃⠀⠂⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠄⠒⠂⠓⠂⠠⠾⢿⡟⠋⠉⠉⠉⠉
⣿⠟⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⣿⣿⣷⣿⣿⣿⣶⡆⢻⣿⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡠⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠈⢣⣀⡀⠀⠀⣀⣴
⣿⣿⣟⠻⣿⡟⢿⣷⠈⣿⡿⢿⣿⣿⠿⣿⣿⣧⠀⣠⣼⠽⢷⣄⠀⠀⣀⣀⣁⣀⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣏⠀⠀⠀⠀⠉⠀⠈⠀⠈⠋⠁⠰⣿⣿⠟⠀⣽⠁⡎⢴⢈⠀⠀⠞⡋⢈⠿⣯⠓⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠟⠀⠑⠐⠁⠀⠀⠢⢁⠌⠀⢸⠇⠀⡄⠈⠉⠻⣿⣿⣿⣿
⣿⣿⡏⠈⠓⠂⠈⠒⠀⠐⠂⠀⠀⠀⣿⣿⣿⣿⣶⠀⠀⢠⡤⠄⠐⠒⠀⠒⠀⠀⠉⢀⠲⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣷⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⣿⣛⣻⣟⣻⣣⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣦⡀⠀⠀⣠⣿⡿⢻
⠟⢛⡟⠀⠐⠂⠒⠓⠀⠒⠂⠐⠀⠀⢀⣉⣩⠛⢻⡏⣛⣦⣄⡀⠀⠀⠀⣀⣡⡴⣿⠖⠙⢻⠉⠛⠽⠙⠻⠗⠛
⠀⠰⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⢀⡴⠂⢹⣤⣰⣾⢿⣿⣏⣁⣀⡙⣸⣏⣀
⠀⠀⣆⢄⡠⠒⢂⣀⠔⠒⠀⠒⠢⢄⠘⣀⠎⠀⠀⣏⢶⠞⣽⠿⡿⢟⢏⠉⠉⠉⠁⠀⠑⢄
⠀⠘⣃⣉⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢈⠟⠀⢀⢼⣧⣴⡿⢻⡇⢫⣁⣼⡀⠀⠀⠀⠀⠀⠈⢧
⢰⠋⡀⢤⠈⢆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠐⠂⠞⠁⢚⣙⣛⣩⣔⡇⠀⠀⠀⠀⠀⠀⠈⢇
⢸⠀⠀⡀⠀⠈⠂⠀⢠⣀⣀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠘⠄⠌⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠒⠒⠂⠤⠐⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢣
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡛⠁⣪⠈⣽⣿⣷⣀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢠⠈⠀⠈
⠀⠀⢀⣠⣶⣿⣿⣿⡋⢬⢻⡾⢷⣾⣿⣟⠑⠭⠙⡓⠶⢤⣭⣐⡂⠠⠁⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⠀⣀⠠
⣤⣾⣿⣿⣟⠉⠻⣿⣷⣶⣾⣶⣿⣿⣟⠛⢷⣄⢆⣉⣤⣼⣿⣿⣉⣹⡆
⠛⢏⡩⣀⡻⣦⡐⣼⡟⠛⣻⣿⣿⠿⣿⣷⣄⣹⣛⣋⣉⣹⣿⣿⢙⣻⣿
⠀⠀⠙⠤⢲⣼⠟⣛⢛⣿⣿⣿⣇⠀⡄⢙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡿⠀⠀⡆⠀⠀⠀⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⢁⣀⠤⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⣀⠀⠀⢀⣉⠉⠙⠳⢦⣄
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⣶⡋⠁⠀⠀⠀⠀⠀⠀⢴⡄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠦⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠁⢀⣀⣀⣤⣴⠞⠋⡀
⠀⢠⣭⠿⣿⠷⠦⠤⢤⣤⣤⣤⣄⣠⣤⣄⠀⢀⣀⣀⣀⣀⣠⣀⣤⠤⠤⠤⠴⠖⠒⠚⠋⠉⠻⠋⠁
⠀⠀⢀⡆⠉⠙⠓⠒⠒⠠⠴⠶⠶⡤⠄⠀⠀⠴⠲⠐⠀⠀⠀⠀⠤⠖⠠⠀⠀⠀⠉⠀⠀⣀⣀⣤⡤⠴⡖⠉
⠀⠀⠈⠇⠐⠒⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡴⠶⠶⠞⠛⠋⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⢸⠇
⠀⠀⠀⠀⢿⡔⠀⠁⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⢀⣀⣤⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁
⠀⠀⠀⠀⢹⡟⠦⢤⠀⢀⡎⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠉⠐⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠀⣐⠶⠖⠷⠦⣤⢄⣀⣀⣀⣠⣤⣤⣤⣤⣤⡔⠚⠒⠉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⡀⠈⠀⢀⡴⠋⠁⠀⠀⠤⡄⠀⠉⠉⠲⢄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢻⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣧⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠰⠁⠀⢸⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠂⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠤⠐⠂⠉⣁⣤⠌⠒⠒⠲⠖⠛⠁⠀⠀⠀⠐⠀⠀⠄⡁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⡀⠀⠘⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣠⠟⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁
⠀⠀⠀⠐⢧⣄⣀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⠶⠖⠒⠒⠊⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠟⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠈⢪⠀⠁⠀⠈⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠖⠒⠚⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣾⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢠⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⣸⣿⠃⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⢀⠔⠀⠒⠠⠄⢲⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⡞⠀⠀⠀⠀⠀⣿⡟⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⢹⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⢟⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣧⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢇⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣽⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣤⣄⣉⣓⠀⠀⠀⣚⣁⣠⣤⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⡄⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢀⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⣙⣦⣀⣀⣶⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠒⠦⢤⣽⡿⣾⡏⠀⣾⣯⣤⣤⣠⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣤⠤⠤⠴⠒⠊⠁⠉⠋⠀⠈⠙⠒⠦⠤⣤⡤⠤⠶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠰⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⣾⠀⢀⠌⠀⡇⠀⣸⣇⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⡟⠰⠋⡿⠀⡌⠀⠀⢳⠀⡅⠱⡀⠨⢫⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣼⢠⠃⠀⡇⠐⠀⠀⠀⠀⠄⠀⠀⠑⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⢩⣇⡤⠤⣽⡀⠀⠀⠀⠀⠀⣤⠶⠖⠶⢦⣄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢇⡾⠁⢀⣴⠄⠁⠀⠀⠀⠀⠀⢠⣶⡆⢀⠀⠈⢧⠘⠂⡀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⢠⣿⣿⣾⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠘⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠻⠄⣀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠙⠄⠀⠀⠀⠀⣠⠞⢻⠀⢀⠀⣰⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⢠⡀⠁⡀⠀⠀⠀⣀⡁⠀⠀⢀⣀⣀⣤⣴⢶⠏⣁⠰⡤⠀⠉⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠁⠄⠙⠀⡘⠂⠄⠜⢣⣽⡟⠉⠉⠀⠈⡿⠗⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⡗⣁⡀⠀⢀⣀⣀⣀⡼⠁⠀⠉⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠉⠀⠘⡇⠂⠀⠀⠀⢀⠖⠉⠀⠀⣠⣾⠟⠋⣹⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣤⣠⣤⣶⣖⣳⣤⣦⣤⣀⣤⣤⣾⡿⠋⢠⠊⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⣿⣿⡿⢄⡉⢻⠟⣟⡨⣿⣿⣿⡙⣿⠛⠛⠋⠁⡠⠀⠁⠀⠀⠀⡀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣠⣿⣿⣿⠁⠀⠈⡏⠘⠋⠐⣿⡇⢹⡇⠘⡇⠈⠉⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⣿⣿⣿⡇⠀⠀⠰⣷⣶⢤⣾⠙⣧⣀⣠⣶⡇⠀⠀⠀⠤⡄⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⣾⣿⣿⣿⠀⠀⠀⢸⣈⡃⠀⢛⣿⣸⡏⠉⣻⡗⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠉⠀⠈⠓⠋⠘⠉⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⢿⢑⡐⠢⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢀⡀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆⠀⠒⡉⡠⠄⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠄⠀⢨⡱⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⢤⣄⣤⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡉⣥⣴⣁⠆⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣿⣿⣷⣼⣻⣶⣤⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣨⣶⣿⣿⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⣿⣿⣿⣽⣷⣦⣤⣤⣿⢿⣿⣿⠿⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠊⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠻⢿⣿⣿⣿⢿⡟⣭⣿⣿⣿⣿⣿⠟⣾⣿⡟⠉⣶⣾⢿⡕⣳⢟⣿⣿⡷⠋⠤⠤⠤⠆⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⣯⣿⣟⣯⣿⠾⠀⠤⣉⣏⣸⣟⠉⠉⠿⠟⠫⠊⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⢀⣠⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠚⠛⠃
⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠙⢿⠨⡻⣶⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄⡀
⠀⠀⠀⠀⢠⣶⣷⢦⣹⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠿⠟⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢠⣿⡿⣿⣿⣿⣿⡿⣿⣿⣄⣤⣤⣶⠶⣶⣶⣞⢿⣿⡟⣿⠏⠀⠀⠘⠋⣛⡼⠛⠀⠝⠠⠄⣤⣄⣀⡀
⠀⠀⠀⠀⠈⠿⣿⣷⣮⣭⣿⣿⣿⣿⣽⣿⣯⣽⢛⡟⡝⢘⡺⢽⣇⡈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⢺⣿⠧⠽⠛⠻⠛⠿⣿⣿⣛⡘⢏⡀⢛⢻⡃⡈⣻⢿⡄⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⣉⣀⠀⠀⠠⣈⣁⣼⣿⢿⣿⣿⣷⣿⣿⣿⣵⠷⠞⠁⣠⣾⣷⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠿⠿⢿⢿⣿⣟⠛⢿⢿⣯⣉⣉⣤⣤⣤⣶⠟⣿⣿⢯⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣦⣾⣿⣛⡾⣿⣛⣛⣁⣿⣿⡿⠛⠠⣨⡥⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡫⡿⣿⣿⣳⣻⠤⡈⠉⠛⠛⠉⠀⢀⣴⣿⠿⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣟⡿⣿⣿⣿⣯⣿⡑⣠⠒⠀⠀⢠⣾⡝⠉⠀⠠⡌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣏⣿⣿⡻⢿⣙⣿⢛⠳⡤⠤⢠⡟⠀⢘⠀⠀⠃⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣻⣿⠿⠿⢯⣗⣯⣿⣿⣿⣛⡕⠁⠀⡸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣟⡃⣠⣴⣶⡾⣻⣷⣾⠛⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡽⣿⣿⣯⠼⠟⢛⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢉⠛⢏⠃⠀⠁⠘⡀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠴⠋⠀⠀⠀⠀⠐⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠝⢦⡀⠀⠀⠀⠀⢰⡀⠀⢇⠈⢌⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠂⢳⠀⠀⠀⠀⣺⠙⣄⠈⠒⠄⡁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠱⡝⠀⠀⠀⠀⢸⡅⢊⠕⢦⡀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠀⠀⠀⠀⠀⠻⡄⡪⠔⠉⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡢⠙⣧⠀⣰⠆⠀⠀⠀⠀⠈⢶⡁⡪⢐⠉⢳⡀⠀⠀⠀⠀⢀⡴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢓⠌⡊⠕⣺⠀⡏⣆⠀⠀⠀⠀⠀⠀⠙⢆⠅⡪⢐⢹⡄⠀⠀⢰⠋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⢐⠩⢒⢥⡇⠀⠑⢌⡳⡌⢷⠢⡀⠀⠀⠘⡯⢂⠔⠍⣧⠀⠀⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⠅⢊⡑⠁⣧⠀⠀⠀⠈⠁⢸⡆⠝⣦⠀⢠⡯⢰⡄⠁⣺⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⡣⠅⠈⠱⣄⠘⠧⡀⠀⠀⠀⡼⡐⠉⣼⢠⡮⠀⡌⡇⠀⡊⠀⠀⣟⢯
⠀⠀⠀⠀⠀⠀⡭⢢⢄⡀⠀⠀⠀⢳⠡⠈⡆⠈⠣⡄⠋⢦⢀⡞⠅⣤⠀⡿⠏⢀⠜⢀⠇⢰⠃⠀⢸⠇⡸⡄
⠀⠀⠀⠀⠀⠀⠘⢄⠊⠝⢦⡀⠀⠀⢣⡄⠘⣄⠀⠈⢢⠀⠿⠀⡜⠑⡀⠁⡰⠃⠀⡌⢀⡏⠀⢀⡞⠢⠂⣇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡉⠒⠨⡇⠀⠀⠀⠙⣦⠈⢆⠀⠀⢣⠀⡜⠀⠀⡧⠊⠀⠀⡜⢀⡞⣀⣴⠋⠁⠀⠨⢺⠀⠀⠀⢀⠞⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡒⢤⣀⠀⠀⠀⠘⣅⠀⠆⠀⠀⣶⠀⠀⠘⠁⠀⠀⡘⢀⡿⡛⠉⣀⡠⡖⠀⢨⡇⠀⠀⡴⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⡊⢑⣄⠀⠀⣿⠀⠀⠀⠀⠙⠀⠠⠀⠀⠀⠀⡅⢂⡠⠒⠉⢁⠜⢀⣠⡞⠀⠀⠰⠖⢉⡤⣰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣄⠘⣆⢀⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠋⠀⠀⢠⠋⠀⣴⠎⠀⡀⠀⣠⢞⣉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢀⠈⢆⢹⡾⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡏⢠⡼⢁⣰⣼⠁⡼⣃⣅⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠸⠀⠈⡄⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡁⢩⣃⢛⣫⣿⣼⣟⣧⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇⠀⠀⢇⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣉⠈⣃⢘⡋⣁⡈⣛⣋⡙⣋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢎⢡⠉⢃⠙⢸⠁⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠑⠆⠀⡙⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⣒⠄⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡠⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠈⠁⠤⢀⣀⡀⠀⠀⠀⣀⣀⠀⠀⣀⡐⠩⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠀⠀⠠⠅⠊⠥⢒⠡⢐⡈⠄⢂⠠⢀⡠⢄⡉⠍⡉⠄⡂⠤⣂⠤⢀⠠⢐⡠⣀⠤⣀⠠⢐⡀⢄⠤⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠉⠀⠉⠀⠈⠀⠈⠀⠈⠀⠈⠀⠈⠀⠈⠁⠈⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢹⡏⠀⢀⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣾⠃⠀⣸⣞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⠟⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⣒⣒⠮⠭⠭⠭⠍⠩⠍⢒⣒⣶⠁⠀⠀⣷⠂⠀⡰⢏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⡩⠕⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣿⡏⠀⠀⢸⡟⠀⣴⡧⣸⠁
⠀⠀⠀⠀⠀⠀⡠⢚⡥⣚⡟⠓⠁⠀⣀⣠⠤⠤⢐⠒⠒⠒⢒⠒⢠⣼⣿⠁⠀⠀⣿⠁⢠⣿⢃⣯⡑⢄
⠀⠀⠀⠀⣠⢊⣴⡿⢟⢹⣟⣿⣖⡽⣫⣆⣉⣩⣄⣉⡑⢊⡥⢌⡹⠉⣏⠀⠀⢸⡏⠀⣼⡝⣼⣿⢿⣢⡑⣄
⠀⠀⠀⢰⢣⣿⠟⣾⡯⣾⡫⠒⠁⠁⠀⠀⠀⠀⣸⠯⠋⠉⠩⠥⠬⠈⠛⠓⣂⡻⡁⠰⣿⡳⢿⡿⠎⣹⣷⡌⢆
⠀⠀⠀⠃⣾⣯⣷⠿⣺⠏⠀⠀⠀⠀⠀⠀⠀⣰⠵⠊⡍⠠⠀⠀⠘⢠⠂⠒⢌⠻⡾⠻⢫⠵⢊⡹⢤⣿⣿⣾⠸⣄⣀
⠀⠀⢀⡆⣿⣿⢏⢴⡏⠀⠀⠀⠀⠀⠠⠀⣰⠏⠒⠀⣀⠀⠁⠀⠀⠁⠀⢂⣀⠄⠙⡒⠎⢳⠈⡆⢘⢻⣿⣿⢆⣿⣿⣿⢻⢟⢶⡄
⠀⠀⠈⣧⢻⡿⠃⣾⠀⠀⠀⠀⢠⣤⠆⣰⠇⠀⠒⠀⠈⠑⣢⠁⡡⠄⠊⠁⠀⠈⡄⢣⠘⢌⠳⣜⢪⢻⣿⡟⢸⣿⣿⣿⢎⣄⢈⢻⡀
⠀⠀⠀⢿⣦⠳⣁⠉⡄⠀⠀⠀⢀⠀⡰⣇⠂⡀⠀⠴⠒⢉⠖⠂⡈⠒⠤⠀⡀⢀⠁⡌⡆⡠⢃⠜⣼⣸⡟⠡⡿⠿⠟⠛⠚⠒⠬⢌⣇
⠀⠀⠀⢸⣾⡐⣜⠣⢡⠀⠀⠀⠈⣰⢃⠹⢦⡄⠠⠀⢀⠀⠰⠀⠱⠠⠀⣀⣩⠋⡰⢠⣇⣇⡇⢀⡿⣋⣶⢠⠇⠒⠒⠒⠒⠒⢤⠀⠹
⠀⠀⠀⠀⣯⡟⣯⢱⠮⡓⠦⣀⣠⢿⣏⡑⣮⠣⡲⠀⣯⢀⡀⢲⠠⠴⢞⡏⣙⢟⡲⠟⣹⠭⢂⡠⣾⡽⡟⡜⠀⠀⠀⠀⠀⠀⣾⠇⢠
⠀⠀⠀⠀⢸⢻⢸⠘⡆⠀⠑⠒⠬⠉⠛⠓⠦⠵⠾⢫⣵⣿⣶⣒⣯⣭⡵⠾⠟⢒⠋⠭⠒⠊⠵⣾⡇⣿⢷⠃⠀⠀⠀⠀⠀⣸⡿⠀⡘
⠀⠀⠀⠀⠀⡇⡏⡷⣧⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠦⠴⠤⠀⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⢰⣽⠞⣹⡞⠀⠀⠀⠀⢀⣼⡿⠁⡰⠃
⠀⠀⠀⠀⠀⢹⣷⠸⢸⡙⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠈⡅⠏⡶⢿⠃⠀⠀⢀⡴⣿⠟⢀⡔⠁
⠀⠀⠀⠀⠀⠘⡎⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠦⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⢧⢸⡃⡸⢀⡠⠶⣫⠟⠁⡠⠊
⠀⠀⠀⠀⠀⠀⢳⢹⢷⢹⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⣾⢣⡿⢛⡽⠞⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠸⡿⣾⡎⡇⠈⠙⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⢿⢣⣏⡽⠒⠉⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⢣⣧⢻⢳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠬⣾⡧⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠒⡏⢹⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣷⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣻⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡿⣍⠒⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠈⠁⡶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠒⠤⢄⣀⣀⠀⡀⠀⢀⣀⣀⡠⠤⢒⡪⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠊⠁⠒⠈⠓⠒⠊⠉⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣶⣶⣶⣶⡆⠀⠀⠐⠂⣉⠙⡆⠐⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⣆⣉⠀⠀⠄⠀⠀⢀⡄⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢰⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⢫⠓⠃⡀
⢸⣿⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠙⣌⠃⠅
⠸⣿⣿⣤⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡀⠀⠀⠀⠀⠀⠀⢹⢠⡀
⠀⠉⠛⠻⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⣥⠤⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⣆⠈⣇⣽⠀⠀⠀⠀⣄⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⣴⣧⡂⠉
⠀⠀⠀⢠⡀⠀⠀⣡⠇⠀⠀⠠⡄⠀⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠹⡆⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⢀⡀
⠀⠀⠀⠀⠉⠾⠤⡏⠀⠀⠀⠀⠐⣤⡤⠀⠀⠉⠀⠀⠀⠀⠀⠾⢇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠐⣾⠇
⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐

⣠⣴⣖⣶⣶⣶⣶⣶⣤⡀
⠨⣿⣿⣽⣿⣿⣿⣯⣿⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣿⣶⣶⣾⣷⣶⣾⡦⣤⠀⠀⠀⠀⠀⠀⣾⠛⣶⣶⣶⣷⣿⣶⣿⡦⣤⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⢶⣤
⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢹⡄⢿⣿⣿⣿⣿⣿⣿⠧⠼⠃⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⣿⣿⣿⣿⠧⠼⠃⠀⠀⠀⠀⠈⢧⠐⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠛⠛





⣿⢹⣿⣿⣿⣿⣿⡂⡀⠀⠀⠀⠀⠀⠀⢰⡏⢹⣿⣿⣿⣿⣿⣀⡀
⣿⢸⣿⣿⣿⣿⣿⡏⠉⣦⠀⠀⠀⠀⠀⢈⡇⢸⣿⡟⠛⣿⣿⠉⢹⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄
⣿⢸⣿⣿⣿⣿⣿⣁⣀⡿⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧⣤⣿⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⢿⠈⣿⣿⣿⠛⠻⣿⣿⡏⢲⡄⠀⠀⠀⠀⠀⢸⡇⣠⣤⣠⣤⣠⣼⠒⣲
⣿⢸⣿⣿⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣤⣼⣿⡟⠙⠋⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⠟⠛⠁
⣿⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣇⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏



⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⣠⣴⣦⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢾⣤⣭⣭⣭⣷⠀⠀⠀⠀⠀⢾⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⢻⡆
⠈⣤⢻⣿⣿⣿⣿⠅⠀⠀⠀⠀⢸⡇⠀⣤⡤⣤⢰⡏⠀⠀⠀⠀⠀⠀⣷⠀⣴⣤⣤⣦⣾
⠀⢻⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣇⠀⡄⠄⠛⣸⠇⠀⠀⠀⠀⠀⠀⢻⠀⢿⣿⣿⣿⣟⣀⠀⠀⠀⠀⣤⢤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢠⡶⣶⣶⣶⣶⣶⣤
⠀⢸⡀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⠀⢰⢲⢺⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⣷⣞⣾⡟⢸⠀⠀⠀⠀⣧⢸⣿⣿⣿⣿⡟⣳⠀⠀⠀⠸⡇⣿⣿⣿⣿⣿⣿⣉⣵
⠀⠈⡇⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡆⢸⣟⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣛⣛⣿⠉⠁⠀⠀⠀⠀⢸⣾⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⡞⠁⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⣿⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠻⣧⡀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠙⣷⡀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠾⠤⠤⠤⠾⠧⠤⠤⠤⠾⠥⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⣭⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠶⣦⣤⣍⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠛⠛⠋⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠳⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣹⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠚⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠔⠊⠉
⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⠤⠤⣄⣀⡀
⠀⠀⠀⢰⡋⠁⠀⢰⣶⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣶⡆⠀⠈⢙⡆
⠀⠀⠀⠀⠈⠑⠒⠦⠬⢭⣙⣛⣛⡛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢛⣛⣛⣋⡭⠥⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠪⠭⢭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡭⠭⠕⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡄⠀⠐⠓⣐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣶⢄⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⢷⠾⣀⡤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠘⢫⣛⠉⠀⠀⠀⠀⠀⡄⠀⢀⢀
⠀⠀⠀⠀⠀⠔⢊⣤⡥⠄⠀⣥⡄⠈⣣⣾⣷⡀⠀⠐⣄⡀⠀⠀⠀⠀⠀⠘⠻⠿⠉⢹⡀⠀⣰⠁⣠⠋
⠀⠀⢠⡦⠴⠲⡁⠀⠀⠆⣨⠣⢀⡬⠟⢉⣿⣷⠀⡀⢹⣧⠀⠀⣀⠀⠀⠀⠐⠂⢚⣛⣻⠂⠻⣴⠞
⠀⣰⡏⠀⠠⣔⣀⠒⠃⢈⣛⣒⣘⠷⠤⠾⣿⣿⠀⣰⣶⡟⣥⡎⠿⣏⣄⢀⣤⣙⣸⠄⣙⣶⡤⠠⠆⠀⣄
⠀⣝⠀⠀⣄⡼⠻⠀⣤⣾⡿⠋⠁⠀⣤⣶⣿⣟⣼⣷⣿⣿⢉⢿⣤⡏⢹⣿⣿⣿⣿⠖⠛⠋⠁⠀⠀⠁⠘
⠀⢹⠀⢸⠟⠀⠀⡄⣽⣽⠃⠀⢀⡀⣻⣿⣷⡋⠈⢉⣿⣿⣏⠀⢿⡄⠈⢿⡿⢋⣠⣴⠀⠁⢄
⠀⢺⡇⣷⠀⠀⠀⢸⣿⣿⣾⡿⠇⢡⣿⣿⡿⠀⠀⣼⣻⠿⣿⡦⢈⣆⡀⠀⣶⡟⠁⠘⠆⠢⠀⠳⣶⡀⢸
⠀⠈⢿⠟⡜⢱⡀⢀⣉⣼⣿⣷⠁⡾⣿⣉⠃⡀⡼⢋⣿⣄⠈⢛⡇⠙⣷⣶⢿⡇⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠐⡶⣥⢾⢇⣌⣙⠑⠻⢯⢸⠀⣾⡏⠐⢠⣇⢀⡀⠛⢿⡾⠃⠈⠉⢼⡇⢑⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠰⢧⣷⡾⢹⡟⣿⠠⣶⢊⡾⢠⡟⠀⢠⣿⡇⣾⠟⠀⠀⠹⣄⠀⢀⡼⠁⠛⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠈⠈⠁⠀⠠⣼⠁⢀⣄⣴⣎⠈⢨⣿⡟⠑⠋⠁⣠⠖⣰⣿⡷⠋⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⡷⣤⠀⠀⣹⣦⡶⢋⢴⣷⢠⣿⡉⢤⡤⠤⠚⠁⠀⡅⣾⠷⡀⠀⠹⣆⠀⠀⠀⢠⢠⣄⡀⡠
⠀⠀⠠⠯⣻⠜⣳⣾⠟⢩⣶⣿⡞⣱⣸⣿⣟⢦⡀⠀⢀⣤⣾⣿⣷⡑⠊⢙⡑⠮⠤⢤⣌⣶⣿⣷⠆⠦⠄
⠀⠀⠀⠀⠁⢾⣩⣷⡀⠉⠹⠋⣾⣿⠛⠻⣿⣶⣿⡖⣻⣿⡽⣿⣿⣾⣷⣦⠿⠀⠀⠀⠀⠉⢹⢏⠞⡀
⠀⠀⢄⠀⠀⣈⣝⣛⣸⣧⡠⣾⣿⣿⣆⠀⠛⠉⣉⣀⣤⣄⣦⠼⣿⣿⣻⣿⡀⣷⡶⢶⣲⡴⢀⡎
⠀⠀⠀⠀⠀⡼⢟⣥⣿⣶⣾⡿⣿⣿⣿⣷⡞⠚⠉⠁⠈⠙⠀⠀⠀⠈⡟⢿⡁⠉⣦⣴⣷⡟⡘⠁
⠀⠀⠀⠀⠀⠠⣎⣻⣿⣿⠟⠀⣾⣿⣿⣿⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⣾⣟⣾⣿⣾⣿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠜⢋⣼⡟⠃⠀⠀⠛⢿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⡼⢩⣿⡟⠤⠿⠃⠀⠀⣄⣁
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠘⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⣠⢠⣆⠘⠉⣵
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣿⣿⣾⠏⠀⠰⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠂⠈⣿⠟⢋⠀⠀⣰⡏
⡀⣀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⣀⣀⣠⣤⣴⣖⣋⣉⣛⣀⣠⣶⣉⣀⣾⣀⣰⣟⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⣿⡆⠀⣾⣿⠀⣴⣾⠿⣿⣦⠐⠿⣿⣿⠿⠆⠀⠀⠀⠀⢠⣾⠆⠀⠀⣶⡆⠀⢠⣶⡿⠿⠗⠀⣿⡿⠿⠿
⢸⣿⣧⣤⣿⣿⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⣿⡇⠀⣿⡟⠀⠀⠀⠀⣿⣧⣤⣤
⢸⣿⡏⠉⣿⣿⠀⣿⣧⡀⣸⣿⠇⠀⣿⣿⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⣿⡇⠀⢿⣷⣀⢀⡀⠀⣿⡏⠉⠉
⠸⠿⠃⠀⠻⠿⠀⠙⠻⠿⠟⠋⠀⠀⠻⠿⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠿⠇⠀⠈⠛⠿⠿⠃⠀⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟
⠀⠀⠀⠀⢰⣶⠀⢰⣶⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠘⣿⡄⠈⣿⡄⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠶⠶⠶⠶⠶⣶⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠻⠁⠀⠻⠃⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣭⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣽⣿
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⢹⣿
⣠⣶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡇⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⢸⣿⠿⣶⣄⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢿⡎⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣴⡿⣣⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠈⢿⣇⣀⣀⣀⣀⣀⣀⣀⣰⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡸⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠐⠂⠐⠚⠑⠁⡀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⡇⢀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠋⠩⠶⠂⡀⠀⠀⠀⠄⠠⠭⠥⠤⠄⢀⡁⠂⠈⠀⠉⢠⣖⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⣿⣿⡶⢶⢶⣦⣥⣤⣭⣭⣭⣭⣤⣤⣭⣤⣾⣶⣶⣾⣦⣽⠖⠉⣀⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠍⠉⠿⣿⣿⣿⡿⠅⢺⠅⢸⠅⢀⡤⠖⠒⠶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢁⣭⡭⢭⣿⣖⣾⠅⢐⡡⢸⡴⠋⠀⠀⠀⠀⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⣈⣭⡭⠵⣖⡋⠀⠊⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠥⠀⠐⣒⣛⣩⡭⠄⢈⠁⣾⠀⠀⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣈⢋⣩⡭⠥⠶⣒⠀⠄⠀⣟⠀⠀⣀⠤⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣒⣚⣛⡭⠅⠒⡁⠀⡯⠌⠀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠭⠭⠷⢖⣚⣋⠀⠂⣸⣡⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢘⣫⣭⣭⡷⠠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣌⠠⠄⠀⠀⠀⠀⡀⢀⣀⠒⣀⣐⣲⣿⣿⠯⠀⣩⣾⣁⣀⣀⣀⣀
⠀⠀⠀⣠⡤⠶⠒⠚⠉⠿⠉⠀⠀⠤⠤⠤⠿⣗⣦⢄⠀⠀⠀⠐⠀⢛⣀⣿⣿⣿⠿⠭⣴⣾⡿⠿⠿⠿⣯⣭⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⣠⡀⠀⠰⠅⠀⠀⠀⠀⠐⠄⠀⠈⠓⠤⢤⣄⣀⣀⣀⣉⣍⣱⣶⣶⣦⣴⠶⠋⣠⡤⠂⠀⠠⠀⠀⣈⡭⠆⢀⡄⣈⣷⣽
⠀⠀⠀⠉⠳⢯⣝⣓⠒⠶⠦⠤⠤⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣭⣤⣄⣀⣀⣤⣶⣶⣶⣯⡤⠤⠶⠶⠒⢚⣉⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠷
⠀⠀⠀⡀⠀⠀⠀⠉⠙⠛⣛⣥⡉⠉⠉⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣀⡀⠀⠀⠀⢀⡀⠀⣀⣀⣤⣤⣶⠶⠶⠿⠿⠿⢿⣿⣷⠄
⠀⠀⣼⠇⠀⠀⠀⠀⣴⣿⣿⣿⡇⢠⣾⡿⢿⣿⣿⠃⠀⠀⠀⠠⣿⣿⣿⠛⠛⠻⢿⣷⣦⠀⠻⢿⣿⠟⠉⠀⣿⣿⠶⠶⣶⣦⣄
⠀⢰⣿⡀⠀⢀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣸⣿⡇⠀⣀⣴⣿⣿⣿⠆⠀⠀⢠⡄⠀⠊⣿⣿⠀⠀⢻⣿⣿⣷
⠀⢾⣿⠇⠀⣼⣿⣿⣿⠟⠿⠶⣶⣶⠶⠿⠿⣷⣦⠀⢸⣇⠀⠀⣸⣿⣷⣿⣿⣿⣿⡿⠋⠀⠀⠀⣼⡇⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⠀⢾⣿⡾⠛⣿⣿⣿⣧⣀⠀⢠⣿⣿⣷⠀⠀⠈⠉⠀⠀⠻⣦⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⡀⠀⢸⣿⣦⣀⣿⣿⣿⣿
⠀⠀⠈⠀⠀⠙⢿⠟⠉⢛⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⢸⡟⠛⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠿⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠚⠉⠋⠁⠈⠀⠀⠀⠈⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠈⠻⣦⣀⣠⣴⣿⣧⣤⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣡⣤⣍⢻⣧⠀⠀⠀⠀⠀⣼⡿⠋⠁⠀⠈⠙⢻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⣿⣿⣿⢸⣿⠀⣠⣤⣤⣴⣿⠁⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⠀⠈⠉⠉⠙⣿⡄⠀⠀⠀⠀⠀⢀⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⠀⠀⠀⠀⢀⣼⣿⣦⣄⣀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⠀⠀⠀⠰⡿⠋⠀⠈⠉⣿⡏⠉⠀⠈⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣿⣿⣿⠸⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⢈⣿⣿⣿⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⣾⡇⠀⢠⣿⣿⣿⣿⣷⡀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣿⣿⣿⣿⣿⠇⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠛⠿⠿⠿⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣤⣀⣀⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⣀⢄⡀⢀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠺⢽⠸⣷⣆⡀⠀⠀⠀⠀⠐⠀⠀⢰⡿⢋⣤⣤⡀⢠⡾⣿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢂⠉⠀⢠⣾⣶⡄
⠀⠐⣗⣞⣌⡷⠋⠀⠀⠑⣲⣴⠂⠀⠀⣈⢵⡿⠟⠉⠀⠘⠷⢭⠞⠁⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠻⢿⡿⠞⠀⠀⠈⠉⠉
⠀⠀⠈⡀⠀⠀⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⡠⠄⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⢀⢀⠀⡀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢠⠀⣀⡀⠀⡀⠀⠀⢠⣤⡄⣤⣄⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠔⠀⠀⢙⡄⠀⠸⣿⣿⡿⠀⠀⠀⠀⣾⣟⣟⣡⠀⠀⠁⣾⣝⡇⣽⠏⠀⠀⣾⣿⣽⠤⠴⠒⠈⠀⠀⣶⣿⡏⢞⠈
⠀⠀⠀⢰⡆⠢⠎⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⡿⠿⠿⠋⠀⠀⡀⢻⡎⠉⢡⡊⢠⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠘⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠻⠀⠀⠀⠙⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠠⣄⠀⠀⠉⠳⣄⣀⠀⣀⡔⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠈⠳⢤⣀⣠⡴⠛⠉⠙⠛⠓⠶⣤⣟⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠷⠶⠖⠋⠁⠀⠀⠀⠀⣤⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠈⠀⠀⠆⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⠉⠉⠛⢆
⠀⠀⠀⠀⢀⣀⣤⡶⢺⡿⡟⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⢀⠀⠀⡄⠀⠀⠉⢄⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠐⢏⡀⣀⠩⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡄⠀⠀⠁⠀⠀⢸⠠⢀⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⣿⣿⣧⠀⠀⠀⠀⣿⣷⡈⠀⡧⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⠀⣿⣿⠃⠀⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠈⡇⠀⠠⠀⠀⠀⢘⠁⠈⢷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡴⢯⠀⠐⠀⠀⠀⠀⠀⠘⠞⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠐⠁⠀⠀⠀⠀⠐⠀⠈⠀⡃⠄⣠⠀⣀⠀⠀⠀⣀⠀⠰⠖⣄⠀⠀⠀⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠥⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠉⠀⠀⠀⠀⠋⠳⠀⠀⠐⠂⡀⠉⠠⠁⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠄⠀⠀⠤⠂⠙⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠦⡀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠤⣡⣈⣀⠀⠀⠈⠐⠂⡀⠂
⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠇⠀⠀⢀⠈⢦⡀⠀⠀⣄⠀⠀⠈⡹⠎⠉⣹⣦⠿⠶⣤⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠙⢀⠚⠀⠀⠀⠈⠀⢀⠄⠀⠀⠞⢸⡏⠹⣆⠠⢱⡄⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠑⣄⠀⠀⠀⠀⠠⡀⠀⠾⠀⢸⡇⠀⠿⠳⡄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠙⠀⠀⣴⠀⠀⠈⠓⠀⠤⠜⠃⠀⠘⠀⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣾⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣯⣥⣷⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠆⡀⠀⢀⣾⣷⣿⣷⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⠉⠙⠂⠀⠀⠐⢾⣿⡋⠀⣼⢋⠻⣷⣾⣿⣷⡆⣀⠀⠀⠀⣠⣿⣿⠈⣧⣰⣛⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠃⠀⠀⠀⠀⠠⠀⠀⠀⠙⠿⣿⣷⣎⣡⣾⣿⣿⣿⣷⣦⣿⣤⣴⣿⣿⣿⣿⡿⠿⡟⠻⣍⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣿⣟⠀⣌⢉⡿⠛⠛⢻⣿⣿⣿⣷⡖⢛⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠁⠀⠈⠁⠌⠀⠸⣿⡿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣠⠤⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠
⠀⣟⠀⣿⠃⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⢀⠄⡄⠀⠀⠀⠀⡾⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠈⣼⠀⠀⠀⠁⡘⠀⠀⠀⠀⢀⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣄⡀⠀⠀⠀⠀⠰⣴⣻⠟⢦⠀⠀⠸⣶⡿⡆⠀⢰⣿⣇⠀⢀⠐⠊⡡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢉⣭⣷⡄⠀⠀⠀⠙⠀⠎⠀⠀⠀⠘⠇⠆⠀⢠⠃⠌⠀⢸⠸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢁⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⢂⠴⠃⠀⣀⠠⠀⠂⠀⠠
⠀⠀⠀⢠⣄⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣀⡤⠖⠛⠉⠉⠁⠀⠈⠉⠉⠙⠲⠦⣄⠀⠀⠀⢀⠎⠀⡴⠒⠂⠈
⠀⠀⠀⢰⡏⠀⣞⠁⠀⠀⠀⠀⠀⢸⠀⠢⠴⠶⠦⡄⠀⠀⠀⠀⠀⢀⡴⢊⣥⡶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢾⡝⢦⡀⠀⠠⠞⠁⠀⠀⠀⣀⣀
⢰⡄⠀⣰⠏⢀⡼⠃⢅⠀⠀⠀⠀⢸⡀⠀⣲⣏⠁⠘⣄⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢦⠀⠀⢠⠞⣿⢇⣀⠴
⢻⡄⠀⣹⠂⢈⣷⠠⣏⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠲⠄⠀⠀⡼⠁⠀⢀⠞⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠓⡄⠀⠀⠀⠈⢳⡀⠀⠈⠉⠉
⢻⡄⠀⢻⡄⢘⡧⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣼⣾⣷⠄⡇⠀⢸⠀⠀⠀⠀⠀⠾⠿⠷⣀⣹⠀⠀⠀⠀⠈⢧⠀⠠⠤⣤⣀
⠸⡅⠐⢿⡀⢸⡧⣾⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠂⠀⠀⠀⠀⢠⠞⠉⠀⠐⠀⢀⠀⠀⠀⠉⠑⠣⣄⣀⠀⠀⠀⢸⡄⠀⠨⠟⣋⠁
⣾⡁⠀⠸⠀⠈⠀⡞⠁⠀⠀⠀⠤⡀⠀⢉⠷⠖⠋⠁⠀⢸⠀⠀⠀⠀⠀⣠⠞⠀⠈⠓⠒⠛⠛⠀⠞⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢸⡏⠀⣴⠋⢳⠀⢀⣠⡄⠀⠀⠀⠙⢦⣲⣷⡯⠂⠀⠀⢸⡀⠀⠀⣤⠞⠁⡀⠀⠤⠤⠤⠤⠤⠤⠤⣄⣙⠳⠶⠶⢶⠀⠀⠀⠀⠀⢸⡇⠀⠠⢶⢄⣀
⠀⠙⠂⠙⣻⠿⠤⠮⣅⡀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠹⠀⠙⢦⣀⠐⠒⠀⠐⠂⠀⠀⢀⣠⣽⠟⠂⡎⠀⠀⠀⠀⠀⣼⠀⠐⠲⣤⠀⠁
⠀⠀⢠⡞⠁⠀⡀⠀⠀⠙⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⠉⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⣰⠃⠀⠤⣄
⠀⠀⣿⡀⠀⠀⢿⡄⠀⠀⢱⡀⠀⣈⠉⠙⠳⠦⠤⠄⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠈⠂⢄⣀⣀
⠀⠀⠸⣧⠀⠠⠘⣷⠁⠀⠀⢧⠀⠀⠈⢦⡠⣤⣶⢁⠄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠰⠮⠵⠦⠄
⠀⠀⠀⢹⡆⠀⢀⡸⣷⠂⠀⠘⡆⠀⠀⠀⠉⠛⠓⠉⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠙⠦⢤⣄⣀
⠀⠀⠀⠀⢿⡀⠀⠀⢻⡆⢀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⠠⣔⣿⣃⡀⠈⢡
⠀⠀⠀⠀⠘⣷⠀⠀⠚⢿⡀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣾⣄⢠⠈⠀⠀⠀⠀⡠⠂⢀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⢀⣤⣄⠀⠈⠙⡀⠰⣄
⠀⠀⠀⠀⠀⢹⣇⠀⠀⢘⣧⠤⠀⠘⡆⠀⠀⠀⠀⡜⠉⠀⢻⡄⠀⢠⠂⠀⡠⠐⠁⢀⠔⠁⣠⠃⠀⢀⠀⠀⡄⠀⢰⡀⠀⠀⠻⠶⠈⢳⠀⠀⠑⠂⠂
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢹⣆⠀⠀⢹⡀⠀⡠⢊⣀⠠⠤⠚⠀⠀⣇⠿⠱⡇⠀⢠⢋⣄⣠⡇⠀⢀⢧⣤⣄⢧⠀⠀⠉⠋⠉⠉⠳⡄⢸
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢿⡎⠀⠀⢧⠀⠀⠁⠀⠀⠀⠀⠀⢀⡝⠀⠀⠹⠀⠸⡜⠞⠟⡇⠀⠀⠻⡉⠈⢨⠁⠀⠀⠀⠀⠀⠀⠇⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠈⣷⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠐⠁⠀⠀⠸⠄⢠⠆⠀⠀⠐⢁⠖⠁⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠠⣿⣿⡆⠀⠹⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⢄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠉⣉⣤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⠀⢀
⠀⠀⠠⣾⠿⠧⠠⠐⠀⠐⠀⠂⠀⠀⠀⠠⠸⠻⠫⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠄⣠⣤⡀⠈⠁⠀⠀⣀⢘⣹⣇⢀
⠀⠀⠀⣹⠇⣰⡆⢘⢶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠶⠐⠀⠸⠀⠀⠘⠀⠔⠁⠀⠈⡀⠀⠀⠀⡚⡏⠈⠛⡻⠄
⠀⠀⢠⣏⠲⣿⠡⢾⡋⠛⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣶⣄⢸⡟⣷⢀⣰⣆⣀⣸⣇⣁⡀⠐
⠀⠀⠀⢙⣷⡼⠢⣾⢁⠀⠀⠀⠈⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠸⡷⠟⠼⠹⠿⠏⠿⠿⠽⠷⠀⠓
⠀⠀⠀⠀⠀⠿⢶⡋⡙⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠠⠀⠀⠀⠀⢠⠀⠀⠐⠀⢠⠀⡿⠋⣶⣴⣦⣶⣦⣴⢴⣦⣶⣴⣦⡆⠈
⠀⠀⠀⠀⠀⢆⣹⠷⠁⠀⠀⠀⠈⠁⠛⠁⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠦⢀⠂⠂⠀⠢⠶⠐⠆⠀⠈⠵
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠠⠒⣴⣦⡐⠒⠦⠂⠃⠀⠀⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⡃⡀⠀⠀⠐⠋⠪⠈⠀⠃⠀⠀⡣⢬
⠀⠀⠀⠀⠀⠐⠛⣆⠀⡄⠀⠀⠀⢀⠀⢠⡀⠆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈⠁
⠀⠀⠀⢰⠀⠀⢸⡇⢸⠁⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠸⡆⠀⢛⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡆⢀⠀⠀⡄
⠀⠀⠀⠀⢈⠀⢠⠀⠀⠀⠀⠀⠀⢠⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢀⣀⠀⠀⠀⠀⠀⠘⠀⣰⣄⣵⣀⡙⠀⠀⠀⡃⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⡰⢦⣼⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠹⠂⡀⠀⠀⠀⡇⠀⠀⠘⠒⡀⠀⠀⠀⡳⠂⠀⣐⡃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠼⠿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡂⠀⠁⠀⠀⠀⢧⠀⠀⢀⢠⡀⠀⠀⠀⣇⠀⠀⠸⡇
⠀⠀⠀⠐⣁⠂⠀⢣⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠆⢠⠀⠀⠀⠉⠀⠠⠘⠃⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠁⠀⠀⠀⣹⠖⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠠⣅⣰⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠈⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠁⠀⣶⠀⡂⠐⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡐⣼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠉⠈⠁⠀⠀⠀⠀⡇⠰⠆⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣦⡏⠯⣀⡗⢮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠁⠈⠉⠁⠀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢻⡤⣿⡵⠚⣧⣈⡏⢩⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡾⡧⠵⣤⣀⡀⠀⠀⠒⢧⣸⣍⣹⠖⣧⣽⡖⢻⠒⢋⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠹⣽⠵⢿⡷⣞⡗⠀⠀⠀⠛⠭⠦⠈⠙⠂⠈⠄⠉⠛⠃⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⢠⠃⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠔⠀⢸⡄⠀⢠⠋⠀⠀⠀⠀⠀⠰⡀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣧⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣌⢀⣀⣸⣅⣀⣎⠀⠀⠀⠀⠀⠀⠀⢣⡀⢹⣀⡀⣀⣠⣀⡀⠀⢻⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠴⠾⠉⠉⠁⢸⠁⡹⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⡏⠀⠀⠀⠉⢉⠒⢻⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠏⡅⢰⠇⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠃⢀⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡿⠀⣇⠿⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠈⠁⠀⠹⡄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⣀⣤⡾⠟⠛⣛⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠋⠉⠉⠙⠛⢷⣄⠀⠀⠘⢦⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡿⠋⠀⣰⣿⣏⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⢀⡀⠀⠹⣧⠀⠀⠀⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠃⠀⢰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣾⣷⠀⠀⢹⡆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⣾⢰⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⢸⠀⠀⢻⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠈⠀⠀⠀⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠹⠃
⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠓⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⣤⠴⠂⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣄⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠔⠊⠁⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⢀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠍⠳⣄⠀⢸⠳⣄⢰⣄⠈⠙⣣⠶⠶⠤⣤⣀⣀⣀⡀⠤⠤⠶⠒⠛⠉⠀⠸⠀⢀⣄⡴⠊⠻⢀⡄⢀⡔⢻⠀⣴⠏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠙⢦⡄⠈⠿⠀⠙⢤⡏⠱⣾⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠲⠻⣍⠀⠠⠖⢹⠟⠁⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠃⣀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣤⠋⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠈⢣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠁⠏⠈⠇⠀⠀⠀⠀⠀⠙⠶⣄⣀⢀⣤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠑⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠸⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠉⠓⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢩⡍⠀⠀⠀⡀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡇⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣸⠁⠀⠀⠀⠀⠸⡆⠀⠹⡀⠀⠀⣠⠞⠁⠀⢀⣤⡀⠀⠈⢦⠀⠀⠀⠀⠀⢀⡴⠃⡞⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢵⠀⠀⢇⣠⠞⠁⠀⠀⠀⡏⠀⢹⠀⠀⠈⠳⣄⠀⢀⡴⠋⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢈⡆⠀⠘⡅⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠈⠛⠉⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⢸⢻⡄⠀⢳⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⡤
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠒⠒⠒⠀⠶⠂
⠀⠀⠀⠀⢀⡴⠊
⠀⠀⣠⠖⠋⠀⠈⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀
⣶⠆⠀⣰⣦⣤⣤⡀⠀⠀⠀⠀⠠⡟⠀⠀⠀⣠⠁⠀⢀⡀
⡟⠁⠙⠛⠛⢉⣿⠟⠁⠀⠀⠚⢸⠁⠀⠀⣰⠃⠀⠀⠁
⣿⣶⡶⣄⣴⠟⢷⡄⠀⠀⠀⠀⡈⠀⠀⢠⠇⣴⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⢐⣶⣶⣦⣤⣄⣀
⣿⣿⣿⣿⣡⡥⠌⣿⣤⡀⠀⢀⡇⠀⢠⡿⠀⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣆⡭⣄⣁⢸⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣼⣿⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣏⠀⢀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣄⢈⠐⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣰⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠁⠀⠀⠀⠀⠀⢠⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣁⡤⢔⣂⠀⠀⠀⠀⠀⠀⢀⣀⣴⠶⠒⣏⠀⣠⠄⣴⢠⣶⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣻⣶⣧⣼⣿⣾⡷⣿⡟⣡⢣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⡻⢾⣏⡛⡵⣯⣖⣬⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢾⣛⣶⣼⣿⡿⣾⣽⣯⣵⣾⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠀⠀⣀⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⡋⢀⣀⣠⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣈⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢠⠤⣤⠤⠀⣀⣀⣀⣀⣠⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠲⣟⠛⢿⡉⠛⢍⡉⠣⣈⠙⢄⡀⠀⠀⠀⠀⠀⠉⢉⡟⠀⣼⠛⠢⣤⣀
⠀⠀⠀⠀⠀⠀⣾⢧⣄⣀⡀⠀⠈⠁⠀⠉⠂⠀⠁⠀⠈⠃⠀⠙⠂⠀⠀⠀⢴⣃⣾⣤⡴⠃⠀⠀⣀⣸⣷
⠀⠀⢀⣠⡤⠐⣿⡀⢈⡟⢹⡟⢲⢦⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣄⣉⣁⣤⣤⠴⠒⠛⠉⢀⡿⠓⠦⣄⡀
⠀⣾⠉⠁⠀⠀⠈⠛⢦⣤⣈⣀⣿⢠⠀⠌⠀⡏⠀⡏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠛⠁⠀⠀⠀⠙⣦
⠀⡿⠓⢦⣄⡀⠀⠀⠀⠀⠉⠉⠙⠚⠓⠒⠶⢧⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠖⠚⠋⠉⠁⠀⠀⠀⠀⣀⣤⠔⢻⡏
⣼⠃⠀⠀⢈⡉⠑⠒⠒⠤⢤⣤⡄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤⠤⠒⢲⡏⠉⠀⠀⠘⣇
⠉⠓⢦⣤⣜⣁⡟⠀⠀⠀⡶⠀⣶⠈⠉⢉⠉⡏⠉⠉⠉⠉⠉⢹⡏⡟⠉⠉⠉⠉⢹⡏⠀⠀⠀⠀⠀⠙⣿⣀⣤⣶⠛⠉
⠀⠀⠸⣇⠈⠉⠙⠒⠒⠺⠧⠶⢧⣤⣤⣧⣤⣇⣀⣀⣀⠀⡀⣸⣃⣋⣀⣀⣀⣀⣸⣧⣤⣤⡤⠴⠖⠚⠉⠉⢰⡏
⠀⠀⠀⣿⡓⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⢸⣍⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠸⣍⠙⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣟⠁⠶⢤⣀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣟⠓⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢾⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠈⠉⠓⠶⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣸
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠐⣏⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠟⠙⠲⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢸⠉⡓⠶⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢸⢀⣉⠃⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⢀⣈⠛⠲⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣦⣈⠙⠓⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡏⢀⡙⠛⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⢿⡀⠈⠛⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⢸⡛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠙⠛⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠛⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡇⠘⠳⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠛⠢⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠓⠓⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠲⠤⢌⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠒⠤⣤⠀⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⠀⠒⠢⣤⢀⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⠀⠈⠉⠉⠀⠐⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⠤⢤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠒⠚⠋⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢰⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠤⣶⢀⡄⢠⣀⡿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣿⣞⠉⢻⣄⠼⣭⣶⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣦⣤⡟⠻⣿⡉⢩⣄⣸⣿⣶⣀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⣿⣿⣿⣿⣤⠀⠞⢿⠿⠛⣏⣼⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⣿⣿⣿⣿⣷⣾⣷⣿⣷⠋⢡⣀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣀⣿⡀⢹⠀⣹⣿⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣭⡉⢹⣿⣯⣉⣟⣹⢛⡿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠻⠿⠿⠁⣿⡏⠀⠈⠉⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡰⠁⣀⣴⣶⣶⣶⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡔⠁⢸⣿⣿⣿⣿⣿⣟⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣠⠟⠛⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠁⠛⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⠀⠀⠀⢘⡅⠘⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤
⠀⠀⠀⢠⡀⠀⠀⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⢀⡄⠁⢨⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⢸⡇⠀⡀⠀⡜⠾⠀⢈⣱⡀⠀⣸⠃⠀⠀⡀⡏⠀⠀⠀⠀⠈⠁⠀⠈⠈⠉⠁⠀⣀⠀⠀⠶⣾⢻⠋⠓⠂⠠⢀⠀⠀⠀⢻⣇⣉⠠⣴⣿⠆
⠀⠀⢀⣿⣧⣐⢀⣸⣷⠈⠁⡄⢀⡇⢀⣏⠆⠀⠰⢷⣵⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣀⣀⠨⢍⣀⡟⠿⠿⠷
⠀⠀⣼⣿⣿⣯⣿⠅⠀⠐⢾⡁⠘⢶⣿⣿⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⢀⠽⣫⣾⣿⣿⣿⣿⣶⣿⣿⣶⡄⢀⡇
⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣧⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⢠⣤⣄⠀⣀⣈⣿⡆⠀⠀⠀⠀⠈⠀⠰⡮⢻⣿⠿⣿⡿⢿⣿⣾⣿⣄⡇
⠀⠀⣿⠻⣿⡿⠀⢀⢀⠀⠀⢈⣟⣾⣿⣿⠘⠆⠀⠎⠁⠀⠀⠀⢊⠀⠀⠀⠀⠐⢂⣩⣿⣿⣿⣼⠇⠀⠀⠀⠀⠀⠀⠌⣠⡿⢀⣀⠀⠀⠈⣩⣿⣿⣿⣧
⠀⢀⡏⠀⠡⠁⠀⡼⡈⣆⠀⠀⠹⣿⣿⣟⢷⣄⠀⢀⣀⠀⠀⠠⡟⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠘⠚⢋⢩⣒⣛⣛⣿⣶⢿⣿⣿⣿⣿⡿
⠀⠀⡇⠀⠀⠀⠀⡗⠘⠹⣦⣀⡀⠈⠻⣿⢿⡄⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢿⣿⣿⣿⡇⠀⠀⣠⣶⣿⠀⠉⠉⠁⢉⣡⣴⣶⣿⣿⣿⡿⠛⡇
⠀⠀⣠⠂⠀⠀⢠⠁⠀⢻⣿⣿⣷⣆⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⣷⠈⠙⠿⢀⣾⣿⡋⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⠁⠛⣿⡆
⠀⢠⠏⠜⠀⠀⡌⠀⠀⠸⣿⡇⢿⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⣿⡇⠀⠀⣘⣋⣁⡤⣾⣿⠛⣿⣿⣿⣿⣿⣿⣿⣆⡙⣿⣧⡿
⠀⠈⣰⠀⠀⣰⡇⠀⠀⠀⣿⡇⠞⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣴⣾⣿⣷⣼⣿⣿⡿⡿⣿⣿⣿⣿⣿⡏⠉⠛⢿⣿⣿⡇
⠀⢠⡇⠀⠀⢿⠁⠀⠀⠀⣿⣿⣄⣴⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⣿⣧⠀⢀⡴⠋⠤⠻⠇
⠀⠀⡁⠀⠀⡾⠀⠀⠀⢰⣿⡟⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣿⣿⠙⡃⣾⣿⣿⣯⣷⣾⣿⣷⡻⡅⠒⠉⠉⠁
⠀⠀⢛⣷⡾⠁⠀⠀⢀⣼⣿⣷⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠋⠈⢀⠀⠀⠋⡿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢠⡤⠛⠀⠀⠀⣴⣿⣿⣿⣟⣿⠀⠄⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡔⠂⠀⣿⣿⡟⠿⠠⡄⡀⠀⠀⠀⠰⣾⡿⣛⣫⣿⣿⣿⣿⣿
⠀⠨⡖⠀⣀⢴⣾⣿⣿⡿⠿⢿⡏⠀⠀⠈⠉⠉⠀⠉⠀⢈⣀⡀⠀⠲⢿⠇⠄⠀⣸⣿⣿⣧⣤⡤⠁⠀⠀⠀⠀⣠⣯⣴⣾⣿⣿⣿⣿⣿⡇⢠⠚⢿⣷⡀
⠀⢰⡶⠀⡨⣿⣿⣿⡟⣥⣶⣿⣿⣶⣶⣤⡄⢠⡦⠽⠛⠛⠛⠓⠲⢤⣼⣠⣦⣀⣿⣿⣿⣿⣤⢶⡿⡄⠀⣠⣶⣾⣿⣽⣿⣿⣿⡿⣿⣿⠁⡀⠤⡀
⠀⢘⠀⢰⣗⣖⣶⣿⣿⣿⡿⠛⢻⣿⣿⡉⠁⠀⠀⠀⠀⠀⠰⠴⠖⠛⠛⠋⢉⣀⡿⠿⠛⠋⠉⠉⠛⢻⣮⣻⣿⣿⣿⣿⣉⣀⣒⠂⢸⣧⣤⠀⠲⡀⠂
⠀⠈⠱⣄⣿⣿⣿⠭⢥⣤⣶⣶⣾⣿⣿⣿⣗⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠿⣿⠻⣿⣿⣿⣿⣟⣷⣾⣧⣤⡀⣐⣿⣿⡇
⠀⠀⠀⠀⠀⠰⠚⠖⠞⠒⠚⠻⠿⠿⠟⠁⠲⠄⠀⠀⠄⠲⠄⠴⠦⠄⠲⠶⠖⠲⠶⠖⠀⠀⠀⠰⠀⠀⠀⠲⠆⠀⠁⠻⠿⠿⠿⠾⠿⠾⠿⠿⠿⠿⠛⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⠶⢚⡿
⠀⣠⣴⣒⡋⠀⡠⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢠⠾⠃
⠘⣫⠞⠁⢠⠎⠀⣀⢣⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠉⠓⠀⣀⠀⠫
⣸⢁⡀⠀⠈⢱⡄⢿⠀⠱⣄⠀⣠⠤⢤⣄⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢾⢹⣯⢹⡇
⣷⠻⡆⢠⢠⡆⠙⠦⡀⢰⡌⠳⣷⠀⠀⠈⠙⢦⡀⠀⠀⣠⠔⠚⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⣸⡇
⠉⠀⠙⢯⣸⢳⡼⠑⢝⢮⡁⠀⠀⠳⣄⠀⠀⠛⠙⢦⡮⡴⠒⠒⠒⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣸⡇⣿⠇
⠀⠀⠀⠈⠛⠀⠀⠀⣈⣣⣵⣄⠀⠀⠈⠳⣄⡀⠀⠀⢳⡶⢶⡒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⡆⠀⠀⠀⡏⢹⡇⣿
⠀⠀⠀⠀⠀⠀⣠⠞⡡⢖⠶⠌⠁⠀⡖⠀⠀⠉⠓⠒⢋⠇⠈⠙⠢⢤⣄⣀⠀⠀⠀⢀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣧⠚⡇⣿
⠀⠀⠀⠀⠀⠀⢯⡀⢀⣀⠤⣶⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⡇⠀⠉⠉⠉⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⢏⣒⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⡟⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠸⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠘⠞⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⢴⠇⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⢦⣀⠰⢦⡀⣸⠀⠀⠀⠀⠀⠀⢀⡴⡟⠉⠉⠓⠾⠀⡏
⠀⠀⠀⠀⠀⠀⠸⢍⡒⠈⠙⠲⢤⣀⠀⠀⠀⣼⣠⣧⣤⢤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠤⣀⠀⠈⠉⠒⠦⢜⣒⠚⠁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠒⢤⣀⠀⠀⠈⠉⠓⠲⣤⡀⢸⡬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠰⣬⡑⠦⣄⡀⢀⡞⠁⠳⣼⣅⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠍⠀⡄⠀⠀⠀⢙⡄⠀⠀⠉⠋⠀⠒⠒⠻⡿⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⢮⣀⡀⠘⠓⠒⣶⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⡽⠿⣝⣻⢹⣒⣲⡶⠦⠤⠤⠶⠶⢒⡖⡿⠋⢉⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⡬⠟⢒⡉⢠⠶⠤⠞⠀⠀⠀⠀⢀⠞⡼⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠓⢸⣋⡴⠋⠀⠀⠀⠀⠤⠤⠴⣧⢾⡶⠀⡧
⠀⠀⠀⠀⠀⠀⢀⡔⠊⢻⣻⡿⠗⠶⠤⠤⠤⠤⠤⠤⠤⠴⠿⠤⠤⣴⣿⡓⠒⢦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠩⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣇⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠚⠋⠉⠙⢧⡀⠀⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣡⡼⠶⠚⠋⢉⣀⠤⠔⠒⠊⠉⠱⡀⠀⠻⣄⠐⠌⢂⡤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠙⣦⠀⠀⠀⠀⠑⢎⢆
⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⢀⠂⠀⢀⡔⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠈⢧⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⢀⠀⠀⢠⠂⠀⢠⠎⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡠⠞⠉⡇⠀⠀⠈⢧⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠃⢠⣦⡄⢀⠇⠀⠀⢠⠖⠉⠁⠀⠀⣀⣽⠔⠊⠁⣠⠴⠋⠀⠀⠀⣠⠞⡇⠀⠀⠘⡜⡄
⠀⠀⠀⠀⠀⠀⠠⠊⡠⠂⡰⠁⣀⠀⣀⣹⣼⠓⠦⠤⠼⣙⣒⣒⣚⣫⣕⣫⠤⠒⠉⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠘⠪⢕
⠀⠀⠀⠀⠐⢠⠁⠈⠀⡰⢡⢋⣱⠊⢀⣈⣿⠳⠶⠶⠶⠖⠒⠒⠋⠉⠁⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠂⠈⠁⠀⢠⠃⠸⠬⢿⡀⠹⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠉⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⣀⡤⠤⠧⠤⢄⣀⣀⣙⣛⣛⣛⣀⣄⣀⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠾⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⢠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡸⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣹
⠣⡑⠦⣀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠖⠛⠓⠒⢊⡁⠀⠀⠀⠀⢀⣀⣠⠤⠖⢋⣩⠴⠚⠁
⠀⠈⠑⠢⠭⣑⡒⠲⠦⠤⠤⠤⢤⣤⡤⠬⠽⠯⠤⠶⠒⠚⣋⣉⡡⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⣠⣿⢀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⣿⡧⢿⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠀⠀⣠⣤⡀⠀⣿⣿⣾⣿⢇⢸⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠛⢿⣧⣴⣿⣿⣿⡏⣿⣈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⡇⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠖⠋⠁⠀⠀⠀⠘⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣄⣀⣼⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣶⣿⣷⣷⣶⣯⣦⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⣷⡀⣤⣀⡄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣿⣛⣟⠛⢹⣧⣾⣇⣸⣿⣯⣿⣿⣿⣷⣦⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⡀⣁⣀⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣶⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠂⠁⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣠⡾⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠲
⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠋⠱⠚⠉⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠡
⠀⠀⠀⢌⡾⠟⠉⠀⠠⠤⠶⠶⠶⠿⠛⠉⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣧
⠀⡠⠜⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠄
⢠⠁⠀⠀⠀⠀⠀⠀⠠⢤⣤⣶⡶⠶⠟⠋⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣇⠠⠒⠂⠀⠈⠁⠀⢀⣤⠞⢃
⠀⣆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡄⠈⢷⣶⡶⠶⠶⠴⠞⠋⠁⣀⣠
⢰⠃⠈⠉⠉⠉⠁⠂⠀⠀⠐⣒⠄⠠⠤⠄⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣶⡿⠿⠛⠄⠨⠶⠶⠶⠶⠶⠚⠛⠋⠉⠀⣱⣀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡴⠞⠠⡙⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⡢⠀⣀⡀⢀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⣠⡿
⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠶⠶⠖⠒⠒⠒⠒⠒⠲⠶⠶⠦⣵⣬⡙⡳⠦⢤⣤⣀⣈⣉⣉⣉⣉⣈⣁⣒⣶⡶⠾⢟⣋⡥⠊
⢧⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠾⡷⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠿⡇⠀⠉⠐⠂⠀⠀⠀⠉⠉⠉⠁⠀⠀⠐⠂⠉⠀⢰⡀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⢰⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠿⡿⠢⠄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠠⠄⢾
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠀⠸⠂⠀⠀⠀⠀⠀⠈⢉⡉⠉⠁⠀⠀⠀⠀⠀⠀⡎
⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣖⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣁⣶⠃⠈⡇⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠘⣆⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣴⢡⠀⠀⠈⠁⢀⣤⡀⠀⠀⠉⠀⠀⠀⡷⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⢘⠜⣸⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠰
⠀⠀⠀⠑⡵⡄⠀⠀⠀⠀⠀⠀⡠⠮⠂⠀⢉⡇⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⡞⡆⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠈⢊⣕⠠⠄⠠⠄⣪⠐⠁⠀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢠⠁⠀⠀⠈⠁⠀⠀⠘⡄⠀⠀⠀⢹⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠑⡇⠀⠀⠀⠘⠛⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⢵⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠀⠀⠀⠀⠀⠀⡆⠑⠂⠤⠀⠀⠀⠀⠠⠤⠄⠊⢠
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢆⠉⠀⠒⠒⠒⠒⠂⠈⢁⡌⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⡌⠀⠀⣀⡀⠸⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢈⡟⠁⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠘⠓⠛⠶⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠛⠛⠛⡛⠛⠀⢻⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⢠⡀⣠⣄⣀⣀⣀⣡⡄⠀⠀⠲⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠰⠷⣤⠤⠤⠤⣤⠄⣘⠟⠀⠀⠀⠀⠙⠋⠀⣤⡄⠀⠚⠃⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡄⠒⠀⠀⠀⠛⣀⠏⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⣀⣴⠿⠁
⠀⠀⠀⠀⣀⣈⢙⣛⠛⢻⡆⢀⡟⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⠓⣄⠀⠀⠈
⠀⠀⠀⠶⣏⣉⣉⣉⣉⡈⠁⡼⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣀⡄⠀⢠⣿⡇⠰⠆⠀⠙⠿⣦
⠀⠀⠀⠀⠀⠁⠀⠀⠻⣥⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢠⡿⠷⠀⠀⠀⠀⠠⠄⠘⡀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠁⠀⠀⢀⠔⠉⠀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢇⠀⠀⠀⠀⠀⠀⠃⠚⠛⠟⠋⠀⠀⠀⡘⠃⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣶⠉⢹⡆⠀⠀⠀⠀⢀⡰⠛⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣸⠁⣶⠀⠀⠀⠀⠁⢲⡄⠀⠀⠀⠀⠀⠘⠁⡐⠋⠀⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⣿⠀⠀⠀⠀⣆⠈⢷⡀⠀⠀⠀⠀⣼⠆⠀⠀⣰⠆⠀⣶⠀⠀⠀⠀⠀⠐⢀⡀⠖
⠆⢀⡸⠓⢤⣄⠀⢀⠀⣤⣤⠼⣿⠃⠀⠀⠀⠀⠠⠈⠻⣄⠀⢀⡘⠃⠀⠀⠀⠡⠄⣻⠇⠀⠀⠀⠀⠀⠠⠛⢷⠀⠀⠠⣄
⠀⠈⠓⠶⠄⠀⠀⠉⠀⡤⠶⠾⠉⠀⠀⠀⠀⡇⠀⠀⠀⢸⣷⣿⣷⠀⣤⣀⣠⣤⣬⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠰⠀⡀⠸⠃⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠘⠀⠀⠠⠀⠀⣹⣿⠟⠛⠛⠉⠀⠈⠉⠛⣅⠀⠐⠀⠄⠀⠀⠀⢣⡔⠓⠀⢻⡁⠀⢰⣧⠈⠃⠀⠀⢠⠞⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠱⠄⠀⣀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⢘⠿⠀⠀⠀⠀⠀⠀⡴⠃⢠⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⠈⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢠⠀⠙⠀⠀⠀⠀⠀⠘⠀⠀⠾⢤⠤⢤⣈⣭⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠆⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⣾⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠀⠠⣤⡤⠀⠐⢀⣤⣀⣀⣀⣀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣹⡆⠀⣠⡿⢨⡀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⣹⢇⡸
⠀⠀⠠⠀⠀⠀⠀⠾⠁⠀⠀⠘⠀⠿⠃⠀⠀⠀⢸⡇⠀⢰⣦⡚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠶⠆⡀⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⣡⣀⣠⡄⠀⠀⢀⠶⠦⠴⠦⠀⠰⠋⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄⠀⠀⠀⠂⠀⠀⠀⠀⠹⣄⡀⠀⠀⠔⠀⠀⠀⣴⡄⠀⣦⣀⡴⣷⣴⣿⠟⠛⠁⠀⠀⠙⠓⠒⠛⠚⣷⡂⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣻⠷⠶⠶⣦⣤⡔⠛⠀⠀⠘⠋⠀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠸⣿⠁⠐⠛⠒⠟⠛⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢼⣿⡏⠈⠱⢄⠘⢦⡀⠀⣼⡦⠀⢀⠀⠂⠀⠀⢀⡟⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣿⠇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⡿⠈⠓⠀⠀⠀⠀⠻⣦⠈⠹⣿⣧⡄⡄⠀⣠⡟⠁⠀⠂⠀⠀⡀⢀⠀⠀⠀⠀⠀⣤⡼⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢈⣻⠟⣻⡿⠿⣡⡶⢋⠠⠀⠀⠀⠀⠀⢰⣿⣶⠤⠀⠉⠉⠁⠀⢸⣧⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣿⠇⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣄⠀⠀⠀⠀⠈⠁⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⢀⣀⣴⣾⣷⡄⠀⢀⣀⣀⠀⢰⠃⢸⡄⠀⢀⣸⣄⣀⡀⠀⠀⢠⣿⣿⣿⣏⠈⠉⠲⠶⠶⠶⠚⢋⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠁⠀⠀⣹⡇⠀⠀⠙⢿⣤⣿⢀⡼⠇⠶⠛⣯⡍⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠐⠒⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢸⣦⣀⣾⣆⠰⣦⡴⠃⠀⠀⠀⠀⠀⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠖⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠃⠈⠀⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣦⡀⠀⠠⠄⠐⡀⠚⠙⠛⢦⣄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⡀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣀⣀⡠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡉⠉⣉⣍⠛⠛⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢼⠇⢲⡏⣸⠇⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡾⢠⡟⠀⠀⢸⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣠⡀⢸⠇⣼⠁⠀⠀⣿⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠻⠑⡿⢠⡟⠀⠀⢠⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⡇⣼⠁⠀⠀⣼⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⢠⡏⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⡟⢴⡇⣸⡅⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼⢃⡏⠀⠀⠀⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡿⣸⠃⠀⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⡿⠀⠀⠀⣾⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡗⢠⣾⢸⡇⠀⠀⢀⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⡇⣼⠁⠀⠀⣸⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⣧⡟⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣿⣿⠇⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢴⠆⢰⣿⣿⠀⠀⠀⡿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠁⣹⣿⡏⠁⠀⢸⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⣿⡇⠀⠀⣾⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⢰⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣤⣀⣸⣿⣟⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠃⠈⣿⣿⡏⠀⢠⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⣿⠀⠀⣼⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⣸⣿⣟⣠⡾⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⣿⠛⠉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡏⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⡃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣴⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⠻⢿⠟⠛⠿⠿⡿⠟⢻⣿⢿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣾⣷⣾⣷⣾⣿⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣢⠖⠈⠉⠉⠉⢹⠁⠁⠈⠀⠈⠒⠐⠂⠤⢀⣀⡲⣏⣻⣄⣲⣂⢲⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣾⣿⣿⣥⣤⣶⣶⣶⣶⣾⣶⣦⣶⣶⣦⣤⣤⣤⣀⠀⠈⠉⠛⢍⣻⣍⣿⣍⣻⣛⣛⡛⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡳⢦⣄⠀⣮⣷⣤⣧⣤⣿⣍⣫⣙⡄
⠀⡖⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣟⣿⣿⣿⠟⡹⠏⠀⠙⣷⣿⣿⣿⣆⣶⡤⢿⣤⣭⣉⣆
⢢⠙⡀⠀⠀⠀⠈⢻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣶⣵⣾⣴⢟⠟⠃⢻⣿⣦⣶⡤⢷⣤⣵⣄⣦
⠀⢂⠀⠈⢀⠉⠀⠀⠘⢷⡌⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡛⣭⡾⢃⠊⣠⣾⣿⣿⣿⣿⣶⢿⣄⣩⣄⣱⡀
⠀⠀⢢⠰⠄⠠⠄⠤⠤⠤⣹⣷⣄⠀⠀⠁⠐⠛⠉⠉⠙⠙⠛⠛⠉⠛⠋⠉⢁⣾⣾⡫⠔⠁⠊⠉⠀⠺⣿⣿⣿⡿⣯⣷⣄⣨⣄⣙⣄
⠀⠀⠀⠁⠒⠲⠲⣶⡶⠶⠿⠛⠉⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣋⡁⠀⠀⠀⢿⡄⠀⠤⠿⢟⣫⣾⣿⣄⣽⣤⣈⣄⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠽⡷⢤⣀⠀⠀⠀⠈⠀⠒⠒⠪⠭⣤⣭⣥⣶⣿⡿⠿⠿⢛⣫⠽⠛⡩⠝⠋⠀⢛⣹⣯⠽⣦⠤⢷⡤⢭⣤⣬⣥⣄⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⣠⣤⣤⣤⣴⡾⠛⠛⠉⠉⠀⠀⠀⠀⣀⠥⠔⠂⠉⠀⠀⠀⠀⠀⠣⢿⣷⣿⣷⣾⣷⣶⡾⠦⠴⠷⠶

⣾⣿⣿⣿⡷⢤⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⣄⣀⣂⣐⡄⠄⠀⠀⢰⣦⣤⣴⣆⡀⠀⠀⢠⣀⣉⣀⡆⠀⠀⠀⣾⣿⣿⣷⣄
⢹⣿⣿⣿⡧⠞⠀⠀⢿⣿⣿⣿⣏⡂⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠸⣿⣿⣿⣇⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠸
⠈⠛⠛⠛⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠛⠛⠛⠉⠀⠀⠀⠀⠿⣿⠟⠀⠀⠀⠀⠹⠿⠿⠇

⠀⣮⣭⣭⡆⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣠⣴⣦⣄⢀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣤⣤⣄
⠀⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢲⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿⣿⡗⡂⠀⠀⢸⣿⣿⣿⡧⠊⠀⠀⢸⣿⣿⣿⠋
⠀⠘⠛⠛⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠘⠻⠿⠋⠉⠀⠀⠀⠐⠛⠛⠛⠂⠀⠀⠀⠈⣿⣿⡟⠋

⠀⠀⣤⣤⡄⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⢀⣀
⠀⢠⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡷⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠈⠻⡟⠃⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠉⡟⠁⠀⠀⠀⠀⠀⠙⡏⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠻⡿⠁
⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣂⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⣧⡄⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠉⣉⣁⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⣠⣶⠖⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⢀⣀⣙⡛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣦⣦
⠀⢸⣿⣿⣿⠀⠀⠀⢰⣶⣶⣶⣶⣦⢄⠀⠀⠀⣶⣶⣶⠀⠀⠀⠠⣾⣿⣀⣀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠘⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡧⠘⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⢸⣿⡇⠀⠀⠀⢰⣶⣶⣿⣿⠟
⠀⣀⣙⣏⣀⡀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⣉⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠚⠛⠛⠛⠂
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣼⣾⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠑⠄⠀⠀⢀
⠀⠺⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠈⣿⣿⣿⠀⠀⠀⢸⣿⣷⣤⣄
⠀⠀⠈⡏⠀⠀⣀⡀⠀⠀⢸⣿⡇⠀⠀⡀⠀⠘⢿⣿⣿⡿⠃⠀⠀⢸⣿⣿⣿⡏⠁⠀⠀⣿⣿⣿⠀⠀⠀⠰⣿⣿⣿⣿⠆
⠀⠀⢀⣅⠀⢸⣿⣿⠀⠀⠈⣛⠁⣠⣾⡿⠀⠀⠀⢨⡅⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣶⠆⠀⠀⠈⠉⡋⠁
⣤⣬⣭⣭⣭⣤⣤⣤⣤⣴⣾⣿⣿⣿⣯⣤⣤⣾⣿⣿⣯⣿⣷⣤⣤⣾⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣷⣴⣶⣾⣿⣿⣿⣶
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠻⢿⣷⡄⠀⣀⠀⠀⠀⢸⣿⠀⠀⠀⢀⠀⢀⣴⣿⠿⠛⠻⢿⣷⣄⠀⣀⠀⠀⠀⢸⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⢹⣿⡈⢿⣷⣄⠀⢸⣿⠀⢀⣴⡿⠃⣾⡿⠁⠀⠀⠀⠀⢹⣿⡄⢿⣷⣄⠀⢸⣿⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠇⠀⠙⢿⣷⣼⣿⣶⡿⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠙⢿⣷⣼⣿⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡞⠛⠻⡆⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣾⣿⣿⣆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⣠⣾⡿⠃⢸⡇⠀⠀⡇⠘⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⢸⣿⣿⣿⣿⠈⠻⣿⣦⡀
⠀⠀⠀⠀⣼⡿⠋⠀⠀⣸⡇⠀⠀⣇⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⢸⣿⣿⣿⣿⠀⠀⠈⢻⣷⡀
⠀⠀⠀⢸⣿⠃⠀⠀⡼⠋⠀⠀⠀⠙⢷⡀⠀⠈⣿⣧⠀⠀⠀⠀⢰⣿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⢿⣷
⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢘⡇⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠸⣿⡄⠀⠈⢷⣄⠀⠀⠀⢀⡼⠃⠀⢠⣿⡏⠀⠀⠀⠀⠸⣿⣆⠀⠀⢻⣿⣿⣿⣿⣿⡿⠃⠀⢀⣿⡟
⠀⠀⠀⠀⠻⣿⣄⠀⠀⠉⠛⠓⠛⠋⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠉⠛⠛⠛⠋⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠘⠻⣷⣦⣤⣀⣀⣀⣠⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣀⣀⣀⣠⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠁



⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠀⡁⠀⠀⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞⠁⠀⠀⢈⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢿⣤⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢄⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⢁⣮⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢠⠀⠀⢀⣀⣠⣤⣤⣴⣆⢸⣧⣭⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠜⠙⣇⣘⢦⣜⠛⠛⠿⠿⠿⠯⠭⠿⠿⠿⠟⠛⠓⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠛⠑⠄⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡐⠀⠀⣀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣈⠻⢶⣄⡀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠳⣶⣬⣽⣿⡿⠖⠚⠍⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣠⣾⣌⣙⣿⣿⣿⣿⣷⣦⣬⣤⣤⣄⡀⠤⠒⠁⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠁⠀⠀⠀⢤⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⠁⠀⠀⣀⣀⣀⣰⡬⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⢄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⢀⡄⢸⠀⠀⠀⢀⣀⣀⣸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⡇⢸⠀⠀⠀⠙⠫⣍⠁⠀⠀⠈⣉⡽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⡌⠀⠀⠀⠀⠀⠀⠑⢦⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⢀⡸⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⣦
⠀⠀⢀⡤⠤⡤⠶⠦⡀⠈⠁⠉⠀⢠⠔⠁⠀⠈⠈⡦⠔⠚⠀⡄⣀⡇⠳⠤⣄⠀⠀⠀⠀⠀⠀⣀⡀⠈⠂⠠⠔⠊⠀⢠⠆⠐⠄⠠⡀
⢠⠐⠋⠀⠀⠀⠀⠀⠑⠢⡀⠀⢀⡔⠀⠀⠀⢠⠊⠀⢀⠄⠈⠀⠀⠃⠀⢄⡀⠱⣀⡴⠋⠁⠋⠉⠙⣆⡀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠉
⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠱⡀⠀⠀⡟⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠑⡀⢹⡀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⣇
⢧⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠁⠉⠑⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⠤⠤⢀⣀⠴⠊
⠀⠧⠤⠤⣀⡀⠔⠀⢉⠔⠐⠉⠑⠂⠐⢄⠀⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠀⡜⠀⠀⠀⠀⢀⣀⠔⢁⣠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠁⠸⢆⠀⠈⠢⢀⡀⢀⣀⠄⠊⢀⡼⠉⠛⠉⠒⠊⠁⠀⢰⠁⠀⠀⠈⠀⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠓⢤⣀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢹⠉⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠳⢤⣤⠖⠋⠁⢀⡀⢀⣠⣄⠈⠒⠀⠒⠁⢀⣠⠊⠉⠒⠐⡄⠀⠀⠀⠈⠦⠀⣄⠀⠀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠉⠀⠈⠧⣄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠘⢧⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠂⠔⠤⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡀⠀⠀⣀⠀⠔⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣟⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⣧⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠁⠀⡿⠛⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍⠉⠉⠛⠻⠿⠟⠀⠙⠿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢀⠀⠀⠈⠄
⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡞⢿⡴⠴⣾⡽
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠧⠁⠀⢠⡿⠃⢀⣀⡱⠀⠀⠀⠀⢀⡀⡆
⣿⣿⡿⠿⠀⠀⠀⠀⣀⣀⣤⣶⣦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠰⠟⠁⠚⠿⠋⠀⠀⠀⠀⠀⢨⣇
⣿⣿⠁⠀⠀⠀⠀⠈⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⣶⣦⡄⢠⠿⠋
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣿⣷⢫⣄⣔⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⣺⢟⣷⣶⡶⣶⣶⣖⣿⣷⣿⣿⣿⣿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡄⠀⠀⠀⠀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢺⠿⠥⠼⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⡧⠤⠤⠤⠄⠁⠀⠀⠀⣠⠤⡤⠴⠧⢾⣅⠀⠀⠀⠀⠀⠀⢰⡝⠁⠉⠁⠘⠊⠀⠥⠃
⠀⠀⠀⠀⡇⠀⠀⠀⣆⠀⠀⠠⣾⣗⣚⣉⣀⣐⣚⡒⢢⡄⠄⠀⠀⠀⠀⠀⣖⠚⠛⠯⠩⢍⡙⢢⣀⣠
⠀⠀⠀⠀⡇⠀⠀⢸⡟⠀⠀⠀⠉⠄⡤⠤⠤⢤⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠓⠒⠂⠉⢸⣧
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⠊⡌⢀⠂⠀⠀⠀⠀⠀⠀⢀⡤⠜⢤⣀⠀⢀⣐⠮⠪⡑⠢
⠀⠀⠀⠀⠳⠤⠄⠾⠃⠀⠀⠀⠀⣏⠂⠉⠉⣀⡞⠀⠀⠀⠀⠀⠀⠀⠘⢤⣌⣑⣲⣬⣄⣒⣒⣺⠥⠔
⠀⠀⡀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠄
⠀⠀⠛⠁⠀⣀⠀⠒⠁⠀⢐⠕⠀⠀⠀⠀⢀⣠⡖⠒⠒⠲⣦⠀⠀⠀⠀⠀⠈⠪⠆⠀⠈⠁⢀⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⡛⣿⣧⣴⣶⣦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⣴⠋⣠⣞⡵⠋⠀⠀⠀⠀⠀⠀⣞⡵⢂⣀⣤⠴⠖⠛⠛⠛⠓⢶⠋⢀⡆⠘⡆
⠀⠀⢀⡤⠒⠒⠴⣇⠀⠀⠀⠀⢸⡟⠉⠉⠉⢢⠀⠀⠀⠀⠀⠈⠸⠁⣿⣅⣀⣰⣶⣄⣀⣀⣠⠿⠞⠥⡄⢠⡇
⠀⠀⣟⠰⠉⠉⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠉⠉⣹⠁⠀⠀⠀⢸⠓⠾⠋
⠀⠀⠈⢲⡤⣤⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠉⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣏⣤⡄⠀⣀⢾
⠀⠀⠀⣿⠀⠀⠀⠀⣷⣌⠲⠤⠖⢐⣿⣄⣤⣞⣍⣳⠀⠀⠀⠀⠀⠀⠘⡇⠀⢰⡛⠋⠀⢷⠀⡿⣿
⠀⠀⠀⠈⢳⠲⣖⠋⠁⣯⡙⢲⠚⢁⣿⣴⠟⢡⢣⠏⠀⠀⡀⠀⠀⠀⠀⣷⠀⠈⣯⣤⠞⠋⠀⠇⣿
⠀⠀⠀⠀⠈⣷⠸⡄⠀⢻⡇⣼⠀⢸⡛⣿⠲⣳⠏⠀⡤⢬⡛⠀⠀⠀⠀⢹⡀⠀⠈⠀⠀⠀⠐⢠⡇
⠀⠀⠀⠀⠀⠹⣼⠃⠀⠈⠓⠃⠀⠀⣻⠋⠉⠁⠀⠀⠳⣤⡇⠀⠀⠀⠀⠈⠳⢦⣤⣀⣠⡤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠶⠶⠖⠚⠡⢿⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡌⠛⠀⢉⡍⠉⠡⣔⣒⠒⠛⠃
⠀⠀⠀⡀⠀⠀⢈⠀⠰⠄⠀⠀⠨⠏⠀⠤⠔⠂⠀⢀⡀⠀⠀⠀⠠⠆⠀⠀⠈⠭⠥⠤⠄⠤⢤⡀⣀⣀⣈⠽⠂
⠀⠀⠈⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣟⣻⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠐⣂⠠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣀⠨⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠊⢀⠐⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢉⠀⢀⠘⠂⡄⢇⠔⢂⣶⠦⢦⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡠⠼⢦⡀⠁⠪⠁⢠⠞⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡩⠝⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠘⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠉⠐⠀⠀⠤⠤⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡜⠀⡐⠁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡈⢢⡚⢀⠌⢤⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⡗⡀⠘⣾⠾⢊⠥⠚⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠰⣤⡤⠇⠑⠀⢩⣿⣧⠄⢀⡊⠁⠐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠋⠑⡄⠆⠁⠀⠀⠀⣬⠴⡧⡵⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⡀⢀⠽⠐⢄⠔⢢⣞⣥⠾⢳⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠢⢘⠂⠀⡐⡐⠴⡋⡸⠷⠄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡇⢠⢊⠱⠃⢃⠠⠵⠴⠰⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠠⠓⢊⣄⣀⣠⣼⣿⣾⣷⠤⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠐⠛⠒⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⡤⠶⠶⠶⡶⠛⠛⠛⠻⠶⠛⠛⠛⠶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢳⣄⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⡏⠀⡤⠖⠛⠛⠓⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⠄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠘⠶⠶⠶⠚⠁⠀⠀⠀⠀⠘⠓⠲⠒⠛⠀⠘⣧⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⡤⠀⠀⠀⠀⠀⢠⣿⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣬⣿⣄⣀⣀⣀⡀⠀⠀⠀⠉⣉⣁⣀⣀⣀⣠⣤⣴⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⢙⣷⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⣠⡴⠟⠋⠁⠀⣦⣤⡾⠿⢿⡟⠷⢶⣶⣶⣶⣿⣿⡟⢛⣛⣩⣥⣤⣴⠶⠶⣤⣄⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⡴⠟⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⣆⠀⠈⠉⠛⠛⠉⠁⠀⠸⣇⠀⠀⠀⠈⠙⠳⣦⡀⠹⣿⡄
⠀⠀⠀⠀⣾⣣⣴⠿⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⡀
⠀⠀⢀⣼⡿⠋⠁⠀⠸⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⣀⣼⡃⣿⣇
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⣆⠀⠈⢷⣄⡀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣧⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢷⣼⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢈⣿⡏⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿

⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠻⠛⠛⠁⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣯⣬⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⢰⢇⣾⣿⡇⡾⣿⢻⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢰⣿⣿⣳⠗⣼⣿⣶⣿⢰⠀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣷⣾⣿⣿⣵⣿⣿⣿⣿⣿⣿⣽⡇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠡⠿⠟⠉⠶⠚⢻⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⣛⣡⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢠⣶⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠟⢋⣴⣾⡿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⣠⣿⠟⠉⣰⡿⢋⣿⡏⠀⠀⠀⠙⠻⠟⠛⠛⠉⠙⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡿⠋⠀⠀⣼⠛⢁⣾⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢀⣠⡾⢃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣶⠾⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠿⣟⢟⣿
⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠓⠒⠒⠓⠛⠛

⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠒⠺⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉