Postman Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠋⠉⠉⠛⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⢀⠔⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣠⡴⠚⢹⣅⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣾⣿⣿⠷⢶⣾⣿⡭⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣧⠀⠠⠶⠂⢷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢀⣦⣤⣞⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢈⠿⢿⠀⠀⠀⠈⠋⠉⢹⠃
⠀⢀⠀⠒⠚⠩⣙⣶⡛⠓⢦⣙⣌⠀⠀⠑⢄⠢⢤⣤⡴⢪⣰⠦⣄⡀
⢰⣱⣿⣦⠀⠀⠔⠓⠭⣓⢦⣈⠛⢆⠀⠀⠤⠸⣦⡼⠁⢹⢿⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠘⢻⠟⠻⠀⡘⠀⠀⠀⠈⠣⡈⠳⢌⠒⢿⡄⠀⢸⣿⡆⡄⢸⡄⠀⢸⠀⠀⠀⠀⠀⠀⡀⢐⠂⡹⠁⠈
⣤⡸⡀⠀⣷⠁⠀⠀⠀⠀⠀⢱⢦⣠⢵⢤⠙⠂⣻⣿⡷⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⢰⡀⢱⠃
⠀⠀⢣⠀⡇⠀⠀⠀⠀⠀⠀⡈⠀⠘⣦⠀⠀⠸⢸⡟⠁⡀⢁⠄⠀⢚⣹⡄⠀⠀⠀⠀⠞⠉⣩⠭⠂
⠀⠀⠈⢻⠁⠀⠀⢐⣂⣤⣤⠧⠤⣤⠇⠀⠀⠀⢱⠇⠌⠀⠉⠁⠉⠀⡯⢳⠤⠤⠤⣤⠄⠀⡜⠑⢄⡀
⠀⠀⠀⢸⠀⠀⠈⠉⡡⠊⠁⠀⢠⡲⣄⣀⡠⣴⠉⣆⢀⠀⢁⠀⢀⡜⣧⠏⠀⠀⠀⢸⣄⠀⣣⣀⡤⠼⠒
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣭⠁⠀⠋⡀⢸⠃⠊⠀⠠⠊⠁⠈⠀⠀⠀⠀⠈⢆⣩⡅
⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⠀⠀⠀⣿⡟⠀⠠⠔⠁⠈⡤⠄⠀⠀⠀⠀⢰⠀⠀⣀⣀⣀⡨⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠛⠻⠭⣄⡀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡈⠏⠉
⠀⠀⠀⠀⠀⠀⠀⡘⠳⢥⣀⡀⠀⠀⢀⠁⠀⠀⠀⠀⢂⠤⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠋⠊⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢣⠀⠀⢀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⣕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠂⠀⠀⠀⠀⣀⡤⠴⠶⢤⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠊⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡵⠁⠀⢠⣴⡀⠀⠀⠀⠀⢀⡀⠠⠤⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⣤⣬⣧⡀⠀⠀⢨⠗⣿⣷⣆⣀⣉⣹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠰⣿⡀⣈⢻⣥⡝⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠝⢯⡛⠃⠀⠀⠐⠿⢿⢿⡐⠋⠓⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⢦⡀⠀⢀⡀⠈⢧⡲⡞⠁⠀⢠⢦⡀⠀⠀⣎⠀⠤⣄⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢔⡿⠷⠛⢀⠄⠳⡤⠀⢣⢹⠀⠀⠀⠀⢻⣶⣄⡀⠀⠀⠀⠀⠤⠄⠈⡧⡠⠤⠤⠤⠤⠄⠄⠀⠀⡀
⠀⠀⠀⢀⠄⢀⠄⠀⠀⢐⣀⠸⡩⠪⣆⠙⣆⡘⠿⠂⠀⢰⡆⠀⠱⣝⠮⣍⠝⣇⠀⠀⠀⡰⠁⡗⣉⠠⡇⠀⠀⣶⣴⢀⡇
⠀⠀⠰⣄⠀⢘⣦⠢⣰⡇⠉⠆⠱⡱⡘⡣⠈⢧⡀⠀⠟⣼⡇⠀⠀⠘⠶⣖⠞⠻⠀⠠⠊⠀⢰⠇⠘⠓⠋⠀⡘⣿⠁⣸⢠⣄⣀
⠀⠀⠀⠈⢓⠀⣈⢳⡀⠈⠁⢀⣀⣘⢀⡁⠀⠀⠳⣄⢸⠁⡓⠤⣄⣀⡀⠪⠄⠀⣰⠳⠄⢡⣼⢰⣖⡲⠤⣀⠙⠃⢠⣇⡩⠃⠌
⢀⠀⠀⠄⣁⣠⠂⠀⡹⠂⠀⠀⣀⣈⣃⠙⠀⠀⠂⠙⢿⣀⢻⢱⣿⠀⣹⢲⣤⣴⣁⠀⡐⠐⠯⠤⢝⣦⣀⠈⠳⣄⡎⠨⠐⠒⡤
⠉⠈⠉⠉⠈⠐⠒⡎⠀⡀⠀⠀⠀⠘⠍⢳⠄⠀⠰⠁⠀⠻⣼⡆⢿⠖⢹⢯⡉⠹⡧⠀⣝⠛⠦⠤⣄⠈⠻⣆⠀⠹⣆⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠴⠶⠭⡬⣄⠀⠀⠀⢀⡠⣖⠁⢹⡈⢷⣎⠠⣶⡀⢻⡐⣮⡛⠦⣄⡈⠳⠄⠀⠠⢢⠃
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣀⠄⠀⣁⠊⢀⡤⠒⠁⠀⠘⢦⡨⠇⠈⠉⠷⣜⢿⠆⢧⠹⣿⡲⣄⠙⠆⠀⠔⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠈⠁⠀⠘⢉⢿⢟⡆⠀⠀⠀⠀⠈⠷⠀⠀⢦⡰⣜⢦⡁⠘⣆⠙⡡⢺⡄⠀⠀⠀⡠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⡀⠀⡟⠉⢹⡨⠘⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡎⠙⣄⠙⠊⠀⢴⡦⣀⡀⡴⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡿⣤⡃⠀⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠙⢦⡈⢧⡔⡱⠀⠙⠒⣠⡤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠒⠀⣄⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠙⠦⣰⠁⠀⠀⠘⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⡸⢊⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡆⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠠⠊⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⢳⡈⠙⠦⣀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡀⠀⠀⠀⠹⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡣⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣷⣦⣶⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠉⣉⡉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠃⠆⠈⠆⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠐⢄⢛⡠⠆⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⡠⠖⠁⠘⠠⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢡⣌⣋⣁⠲⠦⠨⡅⢐⡠⣄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⡠⠊⠀⠀⠀⠀⡀⠐⠈⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡐⠀⠀⠀⠈⠀⠉⠁⠒⠠⠤⢉⠒⠤⣀⠀⠀⠸⡄
⢀⡔⠋⠀⠀⠀⢀⠆⣤⠷⣀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⣀⠔⢣⡜⠙⢢⣴⠶⠆⠠⢤⣤⠤⣤⢤⠜⠒⠠⣁⢪⡤⠂
⠳⡤⠤⠄⠀⢀⠠⠥⠚⢢⠀⠑⠞⠁⠀⠀⠀⠀⠀⠀⠀⢉⠇⢸⠁⡜⣡⣎⡷⠀⠀⢀⡄⡲⢌⠀⠀⠀⢀⣨⠹⠁
⠀⠑⢄⠀⠀⠀⠀⠀⠀⢨⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣌⠀⢣⣙⡡⠃⠀⠀⣾⢸⡟⡎⢠⠀⡴⣀⠉⠀⢣⡀
⠀⠀⠀⠑⢄⠀⡂⡠⢆⡁⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠃⠀⠀⠘⠢⠚⠀⡮⠺⡔⠀⠱⠐⡖⠛
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠦⣀⠀⠀⠀⠉⢲⠎⠐⠂⠠⠄⢀⣳⡄⠀⠙⢅⣘⡃⠞⠀⠀⠀⢀⣀⣡⠴⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⡎⠀⠀⠀⠀⠀⢀⠀⡸⠓⠲⡤⢤⣤⡤⠴⢲⠞⠋⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡌⠀⠀⠀⣄⡎⠘⠒⠂⠀⡻⠚⠢⢄⡰⠃⠀⢠⠞⠁⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢴⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⠁⠓⠀⣸⠃⠀⠰⢣⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢴⡹⠀⠈⢩⢀⠇⣤⠀⢠⡟⠂⢠⠃⣧⠁⠀⠀⠀⠀⢀⡴⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠣⡀⢀⠌⡘⠀⠀⠀⣸⠀⢀⡞⠀⠻⢄⡀⠀⠀⣠⠎⣠⠞⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠇⠶⠀⠀⡋⠀⢠⠇⠀⠀⢸⠙⡶⣞⣡⠞⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⢀⠀⠀⠀⡇⠀⢸⣀⡤⠴⠚⠂⠛⠂⢺⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⠈⠁⠀⢠⡇⠀⢾⡁⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⢸⢰⡂⠀⡇⠉⠀⠊⠑⡄⠀⠀⠀⠀⢀⡸⠀⠀⣖⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠈⠉⢹⠉⠉⠙⡏⡿⠀⠀⠀⠀⢸⡄⠀⠀⢰⡡⠃⠀⢰⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠉⠉⠉⠁⠐⡚⠙⡇⠒⠀⠀⠀⢸⠓⠐⡲⢋⡀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠷⢄⣻⠐⠀⠀⠀⢸⠀⠘⠋⡹⢀⠂⠄⣮⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡟⢆⣀⠀⠀⢸⣀⢀⣤⢃⢆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⢸⢰⠀⠉⠉⠉⠉⠉⠁⠈⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⢸⡀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄⠀⠀⠀⠠⡇⠀⠀⢸⡄⠀⠀⠀⠈⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠵⠆⢤⣄⣠⡤⠠⡤⠖⠃⠀⠀⠈⠅⣀⣀⣀⢀⣀⡀⡴⠾⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠋⠉⢀⣀⣀⡀⠀⠀⠤⠀⠀⠰⣾⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⢀⣀⣀⠀⠀⠉⠉⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⣡⠀⠀⠀⠀⠀⠀⠀⣇⠐⠋⠀⠴⠀⠀⠀⠀⠀⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠉⠉⠉⠉⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠒⠿⠍⠙⠒⠒⠒⠒⠒⠀⠈⢹⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⣄⡠⢄⠓⠈⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢨⡅⢀⡼⠁
⠀⠀⠀⠀⠀⣠⠴⢶⡀⠀⠀⠀⠀⠀⢧⡀⠀⣯⠐⠈⠀⡀⠠⢀⣐⠠⠬⠑⠒⠒⠀⠀⠀⠀⢀⡶⠒⣿⠞⠁
⠀⠀⢀⡴⠞⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠙⠦⣽⠄⣂⣅⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠏⠡⠀⣿
⣠⡾⠋⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⣸⠏⢁⠋⠿⠷⠶⠦⢤⣄⣀⣀⣠⡴⣛⠁⠀⠁⠀⠃⣼⠓⢶⡀
⠹⡅⠒⠠⠄⣀⠀⡀⠀⠀⠹⣆⠀⠀⠀⣰⠛⢙⣆⠌⠀⠀⠂⣴⣖⢂⡄⠀⠀⠀⣍⢰⣿⠈⠃⠀⠀⣿⠀⠀⡇
⠀⠹⣄⠀⣀⡀⠀⡁⠀⠀⠀⠘⢦⡀⠀⣷⠀⠘⢏⠀⠀⠀⠁⠛⠻⠋⡇⠀⠀⠀⠈⠙⠉⠁⠀⠀⢀⡇⠀⢠⠇
⠀⠀⢹⡟⠀⠟⢳⣅⠂⢀⡀⠀⠀⠹⣄⠸⣆⠀⠘⡄⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⣥⡴⠎
⠀⠀⣾⠁⠈⠀⠆⢨⠁⠀⠉⠀⠠⠄⢈⢳⣌⠛⠶⢷⡄⠀⠀⢀⣠⣴⣿⣿⣾⣿⣿⣷⣦⣄⡀⢠⡏
⠀⠐⡇⠀⠀⠈⠀⢸⡀⠀⢀⠀⠀⢀⣠⠞⠁⠀⠀⢀⡽⢦⡘⠻⠿⠿⠿⣟⣛⡻⠿⠛⠛⠋⡱⢻⣦⡀
⠀⠀⢻⡀⠀⠀⠀⠀⠱⣄⠈⣀⡴⠋⠀⠀⢀⣠⠶⢏⠀⠘⡍⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢺⠈⠙⠦⣄⡀
⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⡿⠿⡄⠀⣠⡴⠋⠁⠀⠈⡆⠀⠐⡄⠈⠓⠶⢤⣤⣤⣤⠖⠃⠀⢨⠃⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠸⡷⠤⠒⠒⠊⠁⠀⠀⢱⠞⠁⠀⠀⠀⠀⠀⠘⡄⠀⠈⢂⡀⠀⣬⠃⠀⢣⢀⡀⣸⡇⠀⠀⠀⠀⣦⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⠔⠆⠀⠀⢰⡇⠀⠀⠀⠘⢄⠀⠀⠳⡒⠊⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢳⡄⠐⠒⠀⠉⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠈⠧⡀⠀⠙⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠘⣦⢎⠕⠢⣔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠴⠆⠐⠰⢄⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣀⡤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⠤⢪⠚⢄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠄⠀⠠⠀⠰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠤⠖⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠣⣄⠀⠈⠲⢶⣀⣀⣀⡤⠼⠿⡀⠠⠀⠀⡀⢀⠛⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⠚⠉⠁⠀⠀⠀⠀⠉⠒⠋⠋⠁⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠈⠉⠐⠒⠒⠂⠀⠀⠀⠸⠏⠩⢩⣙⡖⢻⠈⢢⠀⠀⠀⠀⣠⣠⡄⣴⣦⣾⣾⣦⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠒⠂⠠⠤⠤⠀⢀⣀⣀⣀⣀⠘⠻⠿⡇⠠⡇⠀⢣⠀⠀⠀⠛⠛⠋⠉⠈⠀⠀⠀⠀⢀⣀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠸⡀⠈⢣⠀⠀⠀⠀⣠⠀⠐⠂⠀⡙⡏⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠉⠁⠀⠑⠒⠖⠀⠈⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠐⠒⠒⠒⠂⠤⠀⠒⠒⠒⠒⠒⠉⠉⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠃⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠛⠛⠋⠙⡆⠀⠀⠹⣄⠀⠀⠀⣀⣀⣠⡤⠴⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠛⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠤⠤⠄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠠⠤⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠦⠤⢀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⠤⠴⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⡀⣑⣿⡟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⣾⣡⡬⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⠀⠀⢠⣿⠟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣤⣴⣞⠋⠁⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⢿⡀⠀⢠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣯⣭⣭⣭⣭⣭⣽⣯⣭⠅⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠐⠶⣮⣿⣦⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣤⣤⠤⣤⠴⠶⠶⠄⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢬⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀⣼⠃⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠸⣤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣤⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠚⣧⣄⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣤⣾⡗⠀⠀⠙⠾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠈⠣⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠘⠁⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡟⠀⠠⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⣄⢠⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠹⠿⠿⠸⣿⠿⠿⠇⠋⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣙⡛⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠘⢧⣤⣇⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⠴⣒⣫⣥⡴⡾⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⣾⠿⠞⠟⣛⠈⢩⣅⡹⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡃⠀⠀⠒⢙⠀⢡⠴⠁⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⣅⠀⠀⠼⣇⣀⣠⠎⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡖⢚⣭⣥⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠛⣷⢦⠉⢹⡆⠀⠀⠈⠉⠘⡆⠀⠀⢘⣇⠀⠀⠀⠀⢀⣀⣼⠀⢻⣒⡇
⠀⠀⠀⠀⠀⢀⣀⣶⠒⠛⣶⡤⠿⠚⢻⠀⣧⠀⠀⠀⢠⠔⣃⣤⡖⣿⡼⣶⠶⠖⠚⠉⢀⡷⠚⠉⠁
⠀⠀⠀⠀⣀⡿⣅⢹⣀⣄⢿⠒⠚⢻⣘⣧⢿⣒⠒⠶⢿⣉⡽⠻⡏⢻⣳⡼⡤⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠹⣄⠘⣾⡄⣹⡼⠶⣺⠟⠉⠀⠀⠈⠉⠛⠻⣏⡳⣄⢻⣄⢻⡇⣇
⠀⠀⠀⠀⠀⠈⡷⠛⠉⢁⡴⠊⣡⣀⣠⠴⢾⡟⠂⠀⠀⠀⠉⠻⡜⣿⡌⣇⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣦⡀⢘⣯⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣹⣿⢼⣾
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⣹⠉⠀⡟⢲⠘⡇⠀⠀⠀⠀⠀⠀⢸⡧⠶⢚⡟
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⢹⣲⣣⣳⣼⢘⡇⠀⠤⠶⠚⠃⠀⢸⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠉⠉⣉⣭⡶⢻⡇⠀⠀⠀⠀⠀⢠⡟⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⠶⠛⠉⠀⠀⣼⠁⠀⠀⠀⢀⡴⠿⠶⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣠⣤⡶⠛⣉⣀⣤⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⣉⣠⠟⠛⣻⠏⠁⢠⡟⢀⡾⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟⣿⣀⡈⢁⣠⡼⠃⠀⠀⢿⣶⣿⣥⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠉⠉⠉⠉⠀⠀⠀⠀⠘⠛⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢿⠛⢳⠶⠦⣄⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⡒⢉⠉⠉⣉⣉⣉⣉⣽⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣋⣽⣉⠉⡩⠉⢻⡇⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠈⠀⠁⠀⢘⣥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢻⡦⠔⡆⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠋⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠛⡋⢻⡖⢦⣄⣀
⠀⠀⠀⠀⣀⠤⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⢸⣿⢻⡿⢾⠃⠀⠀⠈⠛⠦⣄
⠀⠀⠀⠀⣳⣞⡆⠀⢸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣸⡆⢿⣼⡇⢸⠀⠀⠀⢠⡄⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢻⠉⠀⠀⡼⢋⣈⠙⢦⣄⣀⠀⠀⠀⣰⠁⢸⠋⣇⠈⢷⣧⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠸⡇⠀⠀⠙⢻⠉⢡⠏⠀⠈⠉⠒⠉⠉⠀⣼⠀⢻⡀⠈⢻⡀⠀⠀⠀⠀⣿⠳⣦⠴⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠛⠂⠉⠛⠯⢤⣟⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⢧⠀⠀⣹⣆⠀⠀⠀⣏⠀⡸⠛⠛⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⠤⣤⣴⡇⠀⠀⠈⢷⡋⠁⣈⣿⠛⠛⠛⣟⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠏⠁⢹⠀⣠⠊⠉⠙⠳⣄⣰⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠈⢹⠂⠀⠀⣸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠒⢿⣤⠤⠴⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣄⣀⣠⠿⣄⣧⣤⣄⣀⣀⣀⣀⣀⡤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣸⠈⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡿⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⠁⢷⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⠇⠀⠸⡆⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣞⣡⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡉⠙⠓⠒⠻⣟⠲⢶⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⣄⡈⠉⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠾⠷⠶⠶⠾⠿⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠘⠉⠉⠐⡀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢑⣦⣀⣈⣀⡀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣴⣿⣿⣿⣿⣿⣯⣉⣉⡛⢿⣶⣦⣄⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠘⡟⢿⡷⢤⣤⣬⣭⡭⣽⠟⢿⣿⣿⡷⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⡇⣦⡄⣦⠀⠰⢋⡹⡞⠀⠈⢿⣿⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠁⠧⣴⠇⠀⠘⣬⠑⠁⠀⠀⠈⣿⡋⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣅⠀⠀⣀⠀⠀⠀⠀⠀⠀⡏⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣬⠿⠮⠭⠤⢤⠀⠀⠀⢰⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠢⢄⣀⣀⠤⠊⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡐⠒⠒⠂⠀⢀⡀⢰⡁
⠀⡴⢾⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠋⠀⣸⠙⣿⠳⣄
⠀⠸⡀⢱⡈⢷⠠⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠖⠒⠴⠀⢸⠛⢧⡀⢀⡟⠀⡍⠀⢸⠗⠦⣀⠠⣤⡀
⡇⠐⣄⠀⢱⣤⡷⠙⢄⣀⣴⠋⣩⠇⠀⠀⠀⠀⡎⠀⠀⠀⠀⣠⠃⠀⢸⣠⣴⣿⣿⣄⡾⠀⠀⢸⠀⠠⠁⠀⣰⠋⢙⡦
⠘⢄⠈⠳⢯⠀⠀⠀⠀⠈⠀⡺⠁⠀⠀⠀⠀⣠⡇⠀⠀⠀⢰⡃⠀⠀⢸⠁⡙⢻⠃⢉⠃⠀⠀⡿⠠⠃⠀⡰⠁⠀⠀⢀⡄
⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠉⡇⠀⠀⠀⠈⣿⠁⠀⢸⣄⠃⢸⡅⠎⠀⢠⡶⠃⠁⠀⡰⠁⡐⠀⠀⠈⣇
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⣀⠀⠀⠀⢀⡔⠀⠀⢧⠀⠀⠀⠀⠹⣆⠀⠈⡿⠀⠈⡿⠀⠀⡠⠃⠀⠀⡐⠀⡰⠁⢠⡾⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠉⡤⠶⠚⢁⡜⢀⠀⠀⡀⢤⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⣿⢀⠜⠁⡠⠊⠀⠀⠀⣰⢃⡼⠁⠀⢸⢇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⣠⠎⠀⠈⠳⣶⢧⣀⠏⠙⢷⣶⣤⡐⠐⢢⠀⠘⢿⠏⡠⢊⠤⠤⠄⠀⢠⡇⣾⠁⠀⠀⠈⠢⠭⠖⠸⡇
⠀⠀⠀⠀⠀⠀⣾⠿⠛⠁⠀⠀⠀⠀⠈⢳⠏⠀⠀⠈⠳⡈⠙⢶⣤⣀⠀⣮⠈⠀⢸⠀⠀⠀⢀⡟⢠⠹⣆⣀⣀⣀⣀⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠘⢯⡁⠀⣈⠿⠿⠶⠤⣼⣄⣠⠀⣼⣀⣸⣀⡿⢩⣄⡀⠉⠉⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⢀⣠⣤⡄⠀⠀⠈⠙⢛⠙⣏⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⠃⠀⠀⠀⠀⢀⣀⣀⣀⣠⠬⠿⠞⠋⠸⣧⠝⠀⡀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠷⣾⣍⡉⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣿⠀⣀⠜⠀⢠⣾⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢦⣄⠉⠛⠷⣦⣄⡀⠀⠀⠙⣄⠀⠀⠀⢸⣏⣤⢀⣴⡉⠪⣿⣏⣀⣀⣠⣤⠤⠤⠶⠒⠉⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣬⣉⡓⠲⠄⢉⠛⢷⣦⣄⣈⢧⡀⠀⠈⣧⠶⡟⠀⣷⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠈⠉⠙⠛⠻⠿⠷⢦⣄⣻⢻⡇⡇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠁⠺⠇⠇⠸⣿⢧⣀⠀⠀⠀⠀⠀⢠⡎⠱⡆⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⢸⢰⡀⣿⠀⠙⠳⠶⠤⠤⠤⢼⢣⠤⣿⠀⠒⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⠄⠘⠧⣄⣸⢸⢸⡇⣹⠀⠀⠀⠀⠀⠀⠀⢸⣾⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⠴⠒⠈⠁⠀⠀⠀⠀⠈⡍⢎⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣸⠷⠶⠿⣄⣀⡀⠀⣀⣀⣀⡠⠤⠔⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢉⣉⡉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡃⠴⠈⠂⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢄⡋⠠⠇⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⣀⠔⠁⢀⡄⠗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣡⣍⠉⢁⠲⠢⢉⡅⢐⣠⣄⠀⠀⠀⠀⠁⡀
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⢈⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠉⠁⠐⠒⠀⠤⢈⡙⠠⢄⡀⠀⠀⢹⡄
⠀⡠⠊⠀⠀⠀⢠⢤⡠⢦⡄⠠⠤⢨⠆⠀⠀⠀⠀⠀⠀⢀⠔⢉⡌⠑⠲⣾⠵⠤⠤⠤⡄⠀⣤⢠⠥⠒⠬⣑⢰⡄⠂
⠺⠀⠄⠀⠀⣀⡀⠥⠙⠢⡈⠑⠖⠁⠀⠀⠀⠀⠀⠀⠀⢉⠆⠸⠃⡴⣡⣏⡶⠀⠀⢀⠀⠴⢌⠈⠀⠀⢀⠉⢘⠁
⠀⠑⢄⠀⠀⢰⠀⠀⠀⢠⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣆⠀⢦⣙⡡⠃⠀⠀⠈⢸⡟⡎⢠⠀⣤⣀⠙⠈⢧⡀
⠀⠀⠀⠑⢄⠘⢀⡠⠖⣁⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡀⠃⠀⠀⠘⠂⠚⠀⡮⠺⡖⠀⡃⠰⡖⠓
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠑⠤⡀⠀⠀⠀⠉⢠⠆⠱⠒⠂⠠⠄⢳⡀⠈⠻⢍⠑⠠⠟⠀⠀⠀⢀⣀⣁⠔⠋⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠎⢠⠁⠀⠀⠀⢀⠀⡸⠓⠦⡤⢤⣤⡤⠠⢶⠶⠛⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠆⠀⠀⠀⣀⠂⠐⠦⠀⠀⡳⠒⠄⣀⣠⠋⠀⢀⠞⠁⠘⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⠑⠗⠀⢨⠁⠀⢰⠣⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢠⡝⠉⠉⠙⠂⡆⣤⠀⠀⠇⠀⢤⠃⣣⠀⠀⠀⠀⠀⠀⡤⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⠀⢀⡶⠸⠀⠁⠀⢸⠀⣀⡜⠀⠻⢄⠀⠀⠀⣠⠎⣠⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠁⢰⡆⠖⠀⠀⡆⠀⢀⠇⠀⠀⠨⠙⣦⣜⣡⠞⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣀⡠⠄⠒⠂⠛⠐⠷⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠈⠀⠀⢠⡇⠀⢾⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠈⠠⠄⠀⡇⠑⠂⠚⠑⡄⠀⠀⠀⠀⠀⣤⠀⠀⣘⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡉⠉⠀⠘⠛⠛⠛⠚⡿⠀⠀⠀⠀⢰⡀⠀⠀⢐⡩⠃⠀⠀⡏⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠝⠁⠐⠚⠒⠒⡞⠛⡇⠠⠤⠤⠤⠼⠣⠄⡰⠋⡁⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠷⢀⢳⠀⠠⠤⠤⢼⠀⠘⠒⡙⢡⠀⡀⣤⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣄⠀⠀⠀⢸⠀⠀⣠⠁⠂⣰⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⢱⠀⠉⠉⠉⠉⠉⠁⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠤⢀⣀⣀⣀⠀⣤⠤⠃⠀⠀⠀⣆⠀⠀⠀⢀⣀⠀⣤⠼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠈⠉⠁⠀⢀⣀⡈⠉⠉⠁⢠⠾⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠉⡁⠀⠀⠀⠉⠁⠐⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⣇⠐⢂⡀⠤⠀⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢒⣂⣀⣀⣀⣐⡒⠒⠒⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠋⠄⠐⠒⠒⢒⣒⣒⣒⣀⣀⣀⣀⣀⣒
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣫⣽⣶⣶⣶⣶⣶⣾⣭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢧
⠀⠀⠀⠀⠀⢀⡞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣇
⠀⠀⠀⠀⠀⡾⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠁⠉⠉⠉⠛⠿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⢸⢧⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⡠⠔⠒⠀⠀⠉⠉⠈⠒⠹⣿⣿⡟⡞
⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⡿⠃⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⡿⢵⣅⡀
⠀⠀⠀⠀⠘⣇⣿⣿⣿⣿⣿⢁⡞⠀⠀⠀⠀⣀⣀⢀⠀⠤⠶⠾⠧⡄⢸⡗⠛⠲⣝⢦
⠀⠀⠀⠀⠀⠘⣎⢿⣿⣿⣧⡾⣷⠀⠀⡴⠚⠋⠁⠀⠀⢆⠀⠠⠄⠀⠈⡗⠉⢄⢸⡎⡇
⠀⠀⠀⠀⠀⠀⠈⢳⣌⡿⠿⢿⣿⡆⠀⠀⠔⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠳⡆⠀⣸⢇⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢼⠸⠉⢄⠻⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⡄⠀⠀⠀⠰⣴⢫⠞⠀⠀⠀⠀⢀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣜⢦⣀⠀⠓⠀⠀⠀⢤⣀⠀⠀⠉⠈⠁⠉⠄⡘⠀⠀⠀⣿⢸⠀⠀⢀⡤⢖⣭⠶⠋⢫⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⣟⣛⢷⡀⠀⠀⠙⢷⣦⣤⣤⣤⣴⣾⡇⠀⠀⢰⢇⠏⣠⢖⣭⠞⠋⠀⠀⠀⠀⠻⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣝⡲⢦⣄⡀⠈⠙⠛⠛⠛⠉⠀⠀⣠⢏⣎⡜⣱⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⡞⣧
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠉⠓⣮⣝⢳⣦⣀⠀⠀⠀⠀⣰⢏⣾⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣸⠂
⠀⠀⠀⠀⠀⣠⢟⠽⣝⣚⡭⣝⢦⠀⠀⢠⢞⡵⣶⣿⣿⠁⣢⣭⣗⣻⣿⣾⣭⡞⠁⠀⢀⠤⢄⡀⠀⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⢀⡼⣵⠛⠄⠉⠣⡀⠘⢯⡳⣤⢟⣾⣧⡘⢿⣿⡖⠙⣿⡏⠘⣿⣿⠟⠀⠀⡠⠬⠶⢤⣌⠑⢄⡴⣫⠞⠁
⠀⠀⠀⣞⢾⡱⠀⠀⠈⠢⢌⡢⣌⠻⣴⣿⣿⣿⣿⣄⠻⡇⠀⢸⡇⠀⢻⣅⠀⠀⠀⠋⣀⠒⠀⠀⠀⢸⢸⠃
⠀⠀⠀⣸⢼⡃⠀⠀⢀⠀⠀⠈⠛⠻⣾⣿⣿⣿⣿⣿⣧⡘⢦⣸⣇⠀⠘⣿⣧⡀⠲⠄⣈⠛⠀⠀⠀⢻⣮⡳⣄
⠀⠀⠀⠘⠳⣝⡳⣄⡀⠑⠈⡀⠀⡡⠈⡉⠻⢿⣿⣿⣿⠙⢆⠙⢿⠀⠀⣿⣿⣿⣦⣀⣴⣿⡦⠀⢄⢸⣿⣿⣮⢳⡀
⠀⠀⠀⠀⠀⠀⠙⣾⢹⣷⠈⠁⠰⠓⡀⢉⣀⣾⣿⣿⣿⠀⠀⣷⣄⠳⣄⢸⣿⣿⣧⢩⡛⢿⡇⠀⢀⣿⣿⣿⣿⣧⢿
⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⠀⠀⠀⠀⡥⣦⣿⣿⣿⣿⡟⠀⢀⣿⣿⠷⣈⠻⣿⣿⣿⡼⣍⣧⢧⣤⣾⣿⣿⣿⣿⡿⣼
⠀⠀⠀⠀⠀⢸⢳⣿⣿⣿⠀⢠⠀⢸⢠⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⠀⠈⠢⡈⠻⣿⣷⡽⠚⠶⣤⣉⡛⢛⣛⣫⠴⠃
⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣆⠀⠀⣼⣾⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠀⠀⠀⢹⣦⡈⠻⢆⡀⠀⠀⠉⠛⠿⢯⣟⡲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣽⣛⣿⣿⣓⣻⣁⣿⣿⡿⣿⣿⡇⠀⢀⣿⣿⣿⠀⠀⠀⠘⣿⣟⣦⡀⠙⠆⡀⠰⠀⠄⠰⠞⢛⣾⠿⣏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣼⢹⣟⣿⣾⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣮⡻⣦⡤⡉⠤⠀⠀⠀⠀⠀⠉⠀⠙⢮⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣿⣿⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣾⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣿⣿⣿⣿⣿⠃⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⢿⣿⡿⢤⣄⡀⠈⠿⠋⠀⠀⠀⠀⢀⣠⣿⣿⠟⣫⠖⢮⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡵⠟⠲⢭⣟⣒⡶⠶⠶⣖⣚⣫⠵⠶⠖⠋⠁⠀⠀⠙⠮⣝⣲⠶⠤⣤⣤⠴⣛⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉⠉⠀⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣷⣦⣀⠈⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠿⠛⠛⠙⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠂⠀⠀⠒⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢺⠆⠀⠒⠄⠀⠈⠿⠋⠛⠁⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠉⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠂⠀⠀⠀⠀⠀⣶⡖⣤⡀⠀⠀⠀⠀⠀⠂⠀⠦⠀⣀
⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⣠⣴⣾⣿⣿⣿⣿⣧⢔⠀⠀⡀⠀⢠⡟⣻⣿⣿⣆⠀⠀⢀⣴⡃⠀⢀
⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⠀⠳⠈⣴⣷⣾⠿⣿⣿⣿⣿⣧⣰⣿⣿⣷⣴⡇
⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⠈⢿⣿⡇⠃⣼⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠄⠠⢸⣿⣷⣼⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⡟⠿⠿⠟⠋
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⠀⠀⠀⠀⠀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⣿⣿⣿⠃⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣮⣝⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡄⠀⠀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠘⠂⠀⠐
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡄⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣰⣸⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠙⣿⣿⣿⡇
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⢾⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⠒⠒⠚⠛⠛⠉⠁⠤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⣴⣀⣉⣠⣤⠖⠋⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠛⠻⠯⣭⣉⣀⣀⣀⣠⣴⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣠⠎⠁⠈⢉⣩⣤⣴⣾⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠁⠀⣠⡾⠟⠙⣧⡑⠿⠃⠀⠉⡟⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢀⣾⣛⣷⠀⠀⡇⠉⠒⠒⠖⠋⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠾⢧⣭⣥⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠉⠀⠴⢻⡗⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣷⡀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠉⠙⣿⡟⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⡤⣶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⢀⡟⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠳⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⢸⠇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⣴⣯⣀⣀⡀⠀⠀⣾⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠋⢹⡏⠉⠹⣏⡿⢻⣿⠉⢹⠉⠉⠉⠉⣿⣷⠀⠀⠀⠀⠀⠀⠘⣧⡈⠉⠉⠉⢙⡇⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⡷⠀⠀⠀⠀⣇⠀⠀⣿⢉⠣⣿⠀⣸⠀⠀⠀⢠⡟⣼⢷⣄⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣿⣇⠀⠀⠀⢻⡍⠁⠀⢹⡇⢰⠇⠀⢉⡷⠀⣠⠟⢰⡇⠀⠙⢷⣄⡀⠀⠀⠀⢿⣷⣦⣤⡄⠀⡿⠳⣤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢿⡿⠀⠀⠀⠀⠙⣦⠀⠘⣧⡞⠀⢀⡞⢡⡾⠃⣰⣿⡇⠀⠀⠀⠉⠻⢷⣦⣤⣼⣿⣟⠉⠀⣸⠃⠀⠀⠉⠻⣶
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⢳⡀⣿⠀⢠⣮⠶⠋⣠⠞⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⠀⣰⠃⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⣠⣴⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣹⣿⡴⠛⢁⡤⠞⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣥⡶⠾⠛⠉
⠀⢀⠠⣼⣿⣿⣿⣿⣦⣤⣶⠦⠿⣇⠀⠀⠀⠀⢀⣠⠴⠛⢁⣠⠶⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⢀⡏⠀⢹⣿⣿⣿⣇⠈⠉⠁⠀⠀⣿⣁⡶⠶⠚⠋⣀⣤⡶⠿⠅⢤⠀⠒⠒⢂⣈⡩⣿⡇
⣸⡃⠀⠸⠷⠻⣾⠿⣷⠀⠀⠀⢀⡿⣇⣠⡤⠖⠋⠉⠉⠉⢻⠋⠉⠉⠉⠉⠀⠀⠀⣿⡇
⡿⠑⠒⠒⠒⠒⣿⠀⣿⠒⠒⠚⠉⠀⣿⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⡇
⡇⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣯
⠛⠲⠶⠶⠶⢦⣤⣤⣴⡶⠶⠶⠶⢛⣋⣤⣤⣤⣼⣇⡀⠀⠀⠀⠀⠀⣀⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠈⢂⠀⠀⣉⠻⣆⠀⠀⢠⡞⣉⠉⠉⠉⠩⠋⠛⠳⣦⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣬⣽⣿⣿⣿⣿⣿⣾⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⠛⠛⠙⠛⠳⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣸⠃⠀⠀⠀⠀⠀⠘⣆⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡟⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⣇⣀⣇⣀⣀⣀⣻⣏⣀⣀⣸⣇⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⢉⣉⣉⣉⣉⣀⣀⣠⣤⣤⣬⣤⣉⣉⣉⣙⣓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣛⣻⠛⠉⠉⣿⠉⢁⣀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢹⣏⣩⣭⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣼⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡤⣤⠏⠀⠀⢠⣷⠀⠀⠀⠀⠀⣾⡆⠀⢸⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⢸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡆⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣦⣄⣀⡀⠀⢀⣬⡛⠉⢀⣀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢳⣄⠉⠻⣯⡟⠉⡟⠿⣭⡟⠁⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⢀⣸⠋⢷⣴⠏⠀⢰⡇⠀⢈⡿⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠉⠉⠉⣷⠛⠙⢦⡴⠞⠁⣠⠟⠀⠹⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢠⡇⠀⠀⢈⡇⠀⣴⢿⡄⠀⠀⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣠⡾⠳⠶⠶⠛⠳⠾⣇⠀⠻⣄⠀⣸⠶⠶⠺⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⣿⠟⣦⣶⠛⠋⢻⣄⠀⣀⣼⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⠘⣇⠀⠀⠉⠉⠉⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⠶⠶⠶⠤⠤⠴⣏⠻⣿⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⢿⡆⢹⡄⠘⣇⠀⠀⠀⠀⠀⢀⣠⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡟⠘⡇⠀⠻⣤⣈⠳⠶⠶⠶⠛⠉⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣸⠃⠀⣧⠀⠀⠀⠉⠛⠲⣶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠒⠶⠶⠶⢶⡏⠀⠀⢹⡶⠦⠤⠤⣤⠤⠾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢤⣤⣤⣤⣤⣤⡤⠟⠀⠀⠘⠦⣤⣤⣤⣤⣤⣤⡬⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠦⣄⣀⣀⣀⣀⣤⠴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁⠀⠀⠀⠉⠉⣁⣤⡤⡴⣄⣙⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣸⣝⣥⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣀⣤⣤⠶⡖⣻⣽⠧⠟⣛⣩⠵⠞⠓⠒⠒⠒⠲⢶⣦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣁⣠⡤⣖⣺⣽⠧⢿⢺⣏⣩⠥⠖⠚⠉⠁⣀⣀⣠⣤⣤⠶⣶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣩⣥⢤⡶⣶⣿⣹⠉⠉⠉⠉⠉⠉⠉⠉⠩⡇⠀⢘⡇⠘⠛⠒⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣆⣾⡧⣟⢏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡿⢾⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣤⠴⠶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠳⠘⣆⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠔⠦⠀⢿⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿
⠀⠀⢀⣠⣴⣶⣤⣖⡳⣤⡤⠷⢤⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⣸⣹⠊⠉⡏⠉⠉⣏⣇⡀⣸⠀⣸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢿⣉⣋⡉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⢿⣽
⠀⠺⣅⢻⣇⠀⢷⠀⢸⡟⠢⢴⣿⠟⠛⠛⠲⢍⠻⡗⠲⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⣾⠃⠀⠀⠀⠀⠀⣰⢟⣉⣉⡽⠃⠀⠀⠀⠀⢀⠀⢀⣀⠀⠓⠋⣿
⡇⠄⢼⣏⣛⣒⣻⣖⡚⠒⢶⣾⠟⠀⣠⠆⠀⠈⡆⣷⠀⠀⠀⠀⢆⠉⠲⢄⡀⢀⠞⢹⡷⣼⢻⣷⣄⣀⣀⣴⢾⡇⢸⠉⠁⠀⠀⢋⣯⡉⠭⠬⠤⠰⢂⠀⠀⠀⣿
⠇⠄⢸⡆⠀⠈⠉⠉⠉⠉⣹⠃⠀⢰⡏⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⢣⠀⣀⠙⠣⣄⠈⣷⣼⢰⢻⡀⠉⠉⢛⢆⠹⣾⠀⠀⠀⠀⠀⠀⠓⢉⢙⣉⣉⣉⣀⣀⣀⣿
⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⢀⡴⡣⢋⣤⣀⠀⠀⠀⠀⠀⠉⠘⠦⢄⣈⣑⣿⡿⠋⠘⣧⣄⣀⣨⣾⠟⠻⠒⢒⣚⣛⣛⣛⣉⣉⡉⠉⠉⠉⠉⠉⠀⠉
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣿⠽⠊⠀⡾⠿⣱⠉⠙⠲⣦⠀⠀⠀⠀⠀⠀⠀⠉⢇⡀⠀⢿⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢻⡀⠀⡠⡩⢳⡶⠾⠥⠦⣤⠀⠀⣰⡻⣧⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡆
⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠻⣮⡞⣰⠏⠀⢚⠛⠿⣅⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⢤⡏⠙⠉⣳⢤⣍⣻⣯⠉⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣰⠇⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠻⣦⣄⣀⠀⣼⣿⠷⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢠⣦⡀⠀⠀⠀⠙⠯⣍⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡏⠻⢦⣀⣀⣀⣀⡴⠟⠁⢠⣦⣀⠀⢀⣤⠤⠖⠛⠁⠀⠀⠀⢀⣴⢞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠉⠉⠁⠀⠀⣠⠟⠀⠹⢦⡀⠈⠲⣄⡀⠀⠀⢀⡴⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⡏⠛⠲⠶⠖⠛⠁⠀⠀⠠⡀⠙⢷⣄⠀⠉⠑⠲⠋⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠌⠛⢶⣤⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠒⠒⠲⠶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠐⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣤⠼⠗⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠔⢺⡇⢠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠿⣭⣉⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⡇⢸⠀⠀⢀⣀⣀⣤⣤⡴⠶⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣶⡇⢸⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾⣿⠀⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠋⠉
⣀⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣿⣧⣶⣶⠶⠟⠛⠛⠉⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡶⠿⠛⠋⠉
⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠦⠤⣄⡀⠀⠀⢸⣿⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⠿⠟⣿⡏
⠀⠀⠀⠀⠉⠉⣛⡛⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⢉⣑⣺⣟⣤⣶⣾⡿⠿⠟⠛⠉⢹⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠸




⢀⢀⢀⠀⡀⠀⡀⢀⣀⡀⢀⡀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⣀⢀⠀⣀⡀⢀⡀⣀⡀⠀⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠉⠉⠉⠉⠉⠁⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡖⠐⠒⠢⠤⠄⠀⣀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠖⠲⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⢄⣀⣀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠀⠀⠀⠀⠀⠀⠻⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⡟⠉⠋⠀⠈⠑⠢⣄⠀⠀⠀⣠⠜⢻⠰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⠁⢸⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠇⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀⠠⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⡀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡌⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⠀⡀⢾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢐⣇⢀⡤⠀⠒⠂⢉⡧⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⡶⠶⠁⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢀⡤⠚⠉⠀⠀⠀⣀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⡸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⣄⠀⢀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠁⢀⠾⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡆⣀⡜⠀⠀⠀⠀⠀⠀⠸⡆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⠋⠀⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⠂⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠸⣄⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣠⠜⢧⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠠⠔⠒⠈⠣⣄⡠⠷⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠫⡄⠀⠀⠀⠠⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺
⠀⠀⠈⢦⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⠠⠟
⠀⠀⠀⠀⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⡀⠤⠤⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢀⡀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠑⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢐⣇⢀⡀⠐⡶⠒⠻⡇⠤⠤⠤⣀⣀⣰⠁⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠀⠀⠀⠐⠚⡃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠉⠀⠉⠠⠾⡀
⠀⠀⠀⠴⠀⠠⣠⣌⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠴⠶⠀⠀⢰⣿⠃⠀⠀⠀⢳⠀⠄⠀⠂⡀⣒⡏
⡂⠈⠀⠀⠀⠀⠙⠛⢠⠀⠀⠀⠹⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡠⠂⠙⡰⢁⣈
⡇⠀⠀⠀⠀⠀⠀⢠⡏⡰⠠⠀⡀⠀⠀⠀⠻⠤⠤⠤⠤⠤⠐⠒⠒⠚⢣⣀⠀⣀⠄⠀⠀⠀⢹⡩⠴⠚⠁
⡇⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢲⣠⠒⢴⠼⠫⠛⣩⠏⠁⠀⠀⠀⠀⠈⡇
⡇⢰⠀⠀⠀⠀⠀⠀⢸⠉⠢⣴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠶⢿⡿⠀⠀⠀⠀⠀⠀⠀⢳
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⣷⠁⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡌⠀⠀⠀⠀⠀⠀⠀⡠⠐⢁⣵
⠀⠍⠀⣀⣀⠀⠀⠀⢠⡿⠀⢀⠴⠤⢌⡙⠁⠈⣡⣵⠀⣶⠳⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⣴⢬⡥⠚⠲⠠⠄⣠⠤⢈⡵⢚⠦⣀⢀⡀
⠈⠀⠐⠒⠋⠙⠒⠠⠌⠉⠉⠁⠀⠀⠀⢣⠀⠀⠿⣤⣰⣹⣧⠈⠢⡀⠀⠀⠀⠂⠁⢀⡴⠊⠁⠂⠠⠖⠒⠓⡴⠏⡴⢡⠔⡡⠊⢸⠇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⡗⢄⢀⡜⠈⣦⠀⣠⠖⠉⠀⠀⠀⠠⢀⣽⡶⠶⠀⠀⡴⠲⢎⣀⠔⠁⠀⢹⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢟⠙⠰⡀⠉⠁⠀⠀⠀⠀⢀⡰⡫⠁⠅⡳⢄⡠⠟⣡⡦⠬⠉⠀⠉⠳⡋⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠙⠀⢳⣄⢐⠀⠀⣀⠞⡁⠈⢀⠀⢀⠔⠁⢠⠞⢹⢀⡀⠀⠀⠀⠈⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⣀⠀⠘⡂⠉⠳⣌⡹⣄⠁⢡⣃⣊⡀⣀⠔⠁⠀⡨⠟⠻⢆⠀⢧⢸⠇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠂⠙⠄⠀⡳⠀⠐⠢⠉⠻⠶⣤⣧⠟⠋⠀⠀⡠⠪⢀⠄⠀⠈⢧⣸⠎⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⢶⣤⣭⣤⣤⣤⣤⣤⣼⡄⠤⢯⠇⠀⢆⠀⢐⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠛⠛⠛⠛⠛⠻⡀⠀⢣⠀⠀⢆⠈⢞⠄⠂⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠙⠢⢌⠓⢤⡀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠗⢄⡈⢆⢸⢂⢂⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠉⠲⢌⡓⢄⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠐⣺⣾⡇⡆⣢⣱⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⣀⣩⠶⡙⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠊⠉⣩⠞⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣄⠈⠁⣶⠀⠀⠐⠀⠀⠀⠂⠐⠀⠀⠀⠀⡔⠂⠉⠀⠀⠀⠐⢀⡔⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣋⣁⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠶⠶⠾⠶⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠦⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠛⠛⠋⠉⠛⠛⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⡀⡀⠀⠀⠀⡀⢀⠀⠀⠀⣀⢀⠀⠀⠠⠀⢠⠀⠀⠀⡀⠄⢀⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣁⣤⣤⣤⣄⡉⠓⢦⡀⠀⢀⡠⠖⠉⣁⣤⡤⣤⣀⡉⠳⣄
⠀⢠⡤⠤⠤⠤⠤⣄⠀⡴⠁⡰⠋⠁⠀⠀⠀⠀⠙⠲⣄⠙⠶⠋⣀⡴⠋⠁⠀⠀⠀⠀⠙⢦⡈⢳⡀
⠀⠀⠙⠢⡄⠀⢠⡏⣸⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳
⠀⠀⢀⡴⠃⠀⣿⠙⠳⠤⠧⢤⣤⣤⣀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠙⠚⢦⡀
⠀⠀⠛⠒⠒⠒⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠖⠒⠒⠛⠉⠉⠉⡽⠛⣲⣄⢠⠇⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠙⢏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠁⣿⣁⣀⣀⣉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡈⠑⠦⣄⠀⣀⠴⠚⢁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠉⢁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⡀⠀⢺⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡈⠈⢽⡟⠋⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠿⠿⠿⠿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠓⢦⣀⠀⠀⠀⠀⣠⠞⠙⡇⠀⠿⠿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢈⡷⢦⣀⠔⠧⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣔⣋⣀⣀⣀⣀⣀⣀⣁⣷⠿⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⣡⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⣡⠔⠋⢸⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣛⣡⠴⠋⠀⠀⠀⢸⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⣛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⡾⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣉⣉⣉⡉⢻⣧⣿⡏⠉⠉⠭⠭⠭⠭⠭⠭⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⣽⠀⠒⠂⠀⠀⣼⢻⣿⠃⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠭⠭⠍⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⡿⢓⡿⣿⡇⠈⠉⠁⠀⡿⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠊⡰⠋⡜⣹⡇⠀⠀⠀⢸⡇⣽⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⡉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣠⣾⣥⡾⢳⣿⡃⢀⠊⣄⣼⣵⣿⣇⠈⠄⠐⡰⢠⢀⢀⠀⡀⢀⠂⠀⣀⣀⣀⣄⣀⣄⣀⣀⡀⠀⡀⢠⠤⢤⠤⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⣰⢋⣾⣧⢂⠜⣿⣿⣿⣿⣿⠂⠈⡴⡱⢣⠫⠂⠔⢠⠃⡜⢠⠋⠀⠌⠀⠬⢤⠦⡴⠄⠴⠰⠂⢰⠒⠲⠓⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⢡⠎⣼⡇⠈⡰⣿⣿⣿⣿⣿⠃⡜⡰⡱⢡⠃⠌⠠⢃⠜⢠⠃⢀⠎⢀⠀⠀⠁⠌⢀⠌⡐⠉⡘⠉⣞⣁⡞⣓⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠰⢣⠞⣹⡇⠀⠀⠈⠉⠉⠉⠀⢘⠞⡐⠡⢁⠌⡰⢁⠎⠠⠁⢀⠊⢀⠃⡰⢁⠊⠠⠂⠄⢀⠌⢀⠂⢠⠮⢤⢬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠋⡰⢻⡇⠀⠀⢀⣀⣀⣀⣀⣈⣀⣀⡁⣈⠀⡀⠀⠀⠁⠀⠂⠠⠃⠜⡠⠃⡰⢁⠊⠀⠂⠀⠁⠐⠁⣠⢊⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡼⢁⣾⡇⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠈⠀⠁⠀⡠⠁⡐⠁⠀⢀⠊⢠⠖⣽⣻⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⣆⣠⠃⣿⡿⢋⣽⣿⡆⠐⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠴⠯⡽⢧⢯⠼⢥⠏⣹
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣚⣫⣭⡤⠶⠶⣛⣛⣻⣭⣤⣾⣿⣿⡷⠶⠶⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣯⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣷⣶⠿
⣀⣤⣤⣶⣾⡿⠿⠟⣛⣛⣭⣭⠵⠶⠖⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣧⢻⠎⡿⢹⣿
⢿⣯⡥⠴⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣥⣇⢿⡜⣷⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣏⣾⡰⣧⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡋⢸⢡⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇⣸⢁⡟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⡇⣹⢊⡟⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⡇⢸⠈⡗⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢰⠊⡷⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⡇⢰⠞⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡃⡇⢀⣸⠃⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⡇⢸⢸⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⡴⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢇⢰⢸⣡⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⢸⢁⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⢸⠏⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠁⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣈⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⣤⠀⢀⡠⠤⠭⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠖⠁⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠏⢰⠁⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡎⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⡊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠷⠶⠶⠭⠭⠤⠤⠤⠤⠤⠤⠀⠨⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠠⠤⠤⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠰⣅⠅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠰⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⢤⡤⡀⢠⢄⢠⠂⣄⠤⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠀⠈⠁⠃⠘⠜⡀⡀⠁⡀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠤⠉⠾⣆⢶⣰⣃⣓⢇⣉⡡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠉⠉⠉⠀⠀⡄⢀⣀⣠⠤⠴⠖⠒⢻⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⡇⢸⠁⠀⠀⠀⠀⠀⠸⣿⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠧⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠇⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⢼⣿⣿⣤⡀⠀⠀⠀⢀⡾⣿⣿⠶⠂⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢠⢿⣿⡇
⠀⠀⠀⠀⡸⣿⣿⡆⠉⠃⠀⢀⠞⠀⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣏⠙⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⢠⠃⢼⣿⡛⣄
⠀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⣾⠿⣧
⠀⠀⠀⢠⡞⠁⠀⣿⠀⠀⠀⠀⢀⡾⠁⠀⣷⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⠀⢠⡟⠁⢸⣇⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⢠⡾⠃⠀⢻⡄
⠀⠀⢠⣟⠀⠀⠀⠸⣧⣀⠀⠠⣾⠁⠀⠀⠸⣦⡀⠠⣿⡁⠀⠀⢹⣆⡀⠀⣾⠀⠀⠀⢹⣄⠀⠘⡟⠁⠀⣯⡀⠀⢠⣞⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⡀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠐⡏⠉⠉⠉⠉⠉⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⡇⠀⠀⠀⠀⠠
⠀⠀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡇⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⣤⣤⣄⢀⣤⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⢸⣿⣿⣿⣿⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣾⣿⣿⣿⡁⠙⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡶⠆⠀⠀⠀⣼⣿⣧⡀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⡶⠶⣿⠿⠿⢿⣿⣿⣿⣦⡀⠀⠀⠈⠀⠂⠸⠇⣼⣿⡿⣿⣻⠂⢼
⠀⠀⣇⠀⠂⠠⠀⠀⢸⣿⣿⣿⠇⢀⠀⡃⠀⠀⠀⠈⠙⣮⣍⠓⠄⠀⠀⠀⠐⢶⣾⡇⡿⠁⢸⣄⠀⢹
⠀⠀⡇⠀⠆⠀⠀⠀⠸⢿⠿⣿⢀⡟⢀⡇⠀⠀⠀⣠⣼⣿⣿⣽⣦⣴⣆⡀⠀⠘⣿⠞⠃⣀⠀⣉⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⢡⣼⣷⣄⣶⡀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⣻⡀⣼
⠀⠀⡇⠀⠀⠀⠀⢀⠤⠞⠛⠉⣀⣾⣿⣿⣿⣿⣷⡅⣿⣿⣿⡝⠛⠋⠀⠀⠀⠒⣇⠀⠀⡙⠖⠈⠓⢾
⠀⠀⡇⠀⣰⠇⣤⣶⣶⣾⡿⢻⣿⣿⠟⠻⢿⣿⡟⣿⡟⣿⡿⠷⠀⠀⠀⠀⠀⠀⠸⠀⠀⣷⡾⠀⠀⣼
⠀⠀⡏⡸⠗⠘⠻⠉⠉⠀⠀⢸⣿⣿⠀⣶⡵⣾⣭⠃⠐⠻⠃⠀⠀⠀⣀⣽⠀⠀⠀⠀⠀⠜⠁⠀⠀⢻
⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠘⣿⡆⠈⠁⠐⠉⠉⠛⠻⠛⠻⠯⡁⠛⢧⠀⢠⡄⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢸⣿⣿⡆⠀⠀⢸
⠀⠈⣇⠀⠀⠀⠀⠀⢀⣤⣴⣦⣶⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠘⠛⠋⣹⣷⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢸⠙⣿⡇⠀⠀⢸
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢏⣿⠁⠀⠀⢸
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠈⠀⠀⠀⢸
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢸
⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣩⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣎⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣟⣿⠿⠁⠀⠀⠀⠁⠢⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠂⢦⡀⠀⠀⠀⣴⣿⣿⡿⠉⠂⠁⠀⠀⣀⠄⠀⠀⠀⠀⠠⠀⢃
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣆⣾⣿⡿⠁⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⠿⠃⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⠱⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣳⡼
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠉⠀⠀⠀⠀⢻⠽⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀⣄⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢹⡛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠹⣤⢤⡀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢨
⠀⠀⠀⠀⠀⠹⠦⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠁⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠁
⠀⠀⠀⠀⠀⠀⠣⡀⠉⢠⠀⠀⠀⢀⣀⣤⣶⡟⠛⠁⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⣾⣿⣿⣿⣿⣇⠀⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣌⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣯⠾⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⡛⠿⢟⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠋⠉⠁⢚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⠢⠁⠀⠀⠀⠀⠀⢀⣠⣤⣾⠟⠋⠁⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣄⠀⠀⠀⠀⣠⣶⣿⣿⡯⠀⠀⠀⠀⠀⢀⣤⣾⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠭⠯⠄⠀⠻⢿⣿⣿⣿⣦⡄⢀⣠⣴⣶⣿⠟⡉⠐⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⡿⠋⠙⠯⠁⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⣠⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣶⡆⢀⣠⣤⣶⣶⣤⣤⣀⠀⠀⠀⢘⣿⣿⣿⣴⣿⣿⢟⣎⣀⣀⣀⣀⣀⣀⣀⣐⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣠⡴⢾⣿⡇⠀⠀⠀⠀⣿⡟⠀⠛⠿⠿⣿⣿⡿⠿⢟⣉⠉⠉⠉⢻⣿⣿⣿⣿⠯⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠉⠁
⠀⠀⠀⣠⡾⠋⠀⢸⣿⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢀⡀⠀⠻⣿⡿⠋⢀⡀⠀⣴⡦
⠀⠀⣼⡏⠀⠀⠀⣿⡟⠀⠀⠀⠀⢸⣿⠀⠀⡋⢻⣿⡄⣶⡆⢸⣿⢀⣾⡟⣿⠀⠀⠙⠀⢀⣿⡏⢀⣿⠇⠘⡉⢿⣷⠀⣶⡇⢠⣿⡇
⠀⠀⠙⠛⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡿⠀⣸⣿⠈⣿⡇⣿⡇⢸⡇⣼⣿⣰⠏⢠⠀⠀⠀⢸⣿⠀⢸⣿⠀⣿⡏⢸⣿⢸⣿⠁⢸⣿⠀⢠
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢿⣇⠀⣿⣿⠀⣿⠀⢿⡇⢸⡇⣿⣟⠉⠀⡼⠀⠀⠀⢸⣿⠀⣿⣿⠀⣿⡇⢸⡟⢸⣿⠀⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠸⣿⣼⠻⣿⣸⠏⠀⠘⢧⣼⠃⠘⢿⣤⡴⠁⠀⠀⠀⠘⠿⡼⢹⣿⠀⢿⣇⣼⠃⠸⣿⣸⠻⢿⡼
⠀⠀⠀⠀⠀⠀⠀⠉⠃⣀⣀⠀⠀⠀⠈⠀⢰⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣿⡿⠀⠀⢀⣀⣀⣀⣈⣁
⠀⠀⠀⠦⣤⣤⣴⠾⠛⠛⠛⠛⠛⠛⠒⠦⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⢿⣭⣿⣿⠃⠀⠉⠉⠉⠉⠉⠉⠉⠛⠲⠚⠁
⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⢠⣴⣶⡄⠀⣠⣦⡀⠀⣴⣶⣦⠀⠀⢠⣶⣶⡄⠀⠀⣠⣴⣶⣮⣭⡁⠀⠀⢰⣶⣶⠀⢀⡀⠀⣶⣶⡆
⠀⠀⠀⣨⣯⣍⠀⠀⠀⠀⠀⢸⣿⣿⣧⣾⡿⠋⠀⠀⣿⣿⣿⣆⠀⢸⣿⣿⡇⢀⣾⣿⡟⠀⠹⣿⣿⡄⠀⢸⣿⣿⠀⣾⣷⠀⣿⣿⣷
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣄⠀⠀⢸⣿⣿⣿⡄⢸⣿⣿⠁⢸⣿⣿⠇⠀⠀⣿⣿⣇⠀⣼⣿⣿⣸⣿⣿⣇⣿⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⣿⣿⣧⠀⢸⣿⣿⣿⣷⣸⣿⣿⠀⢸⣿⣿⡀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⢸⣿⣿⠀⢸⣿⣿⡏⣿⣿⣿⣿⠀⠸⣿⣿⡇⠀⢀⣿⣿⡇⠀⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿
⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⢸⣿⣿⡇⠘⣿⣿⣿⠀⠀⠻⣿⣷⣄⣼⣿⠟⠀⠀⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠈⠉⠉⠀⠀⠙⠋⠁⠀⠈⠙⠁⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢀⡄⠠⢠⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢂⡜⠀⢤⠀⡀⠄⠐⠀⣤⣦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡾⠿⣿⣸⣹⣾⡟⠀⠸⠭⠟⠛⠛⠀⠈⡄⠀⠀⢀⠀⠠⠔⠲
⠀⠀⠀⠀⢀⠄⠀⠸⣿⣿⣿⡇⠈⠁⣀⠉⢸⠃⣀⣤⣴⣶⡄⠀⠀⠠⠒⣂⣉⣠⣶⣼⣷⠀⠄
⠀⠀⠀⠀⡌⠐⠫⠖⠀⠉⠉⠛⡀⠐⠶⠂⣨⣀⠘⠛⠉⠀⠁⠀⠀⠀⠘⠿⠟⠃⠉⠁⠀⠀⠘⡂⢲
⠀⠀⠀⠰⠀⠀⣀⣀⡀⣀⣤⢏⡟⠷⣶⡾⣻⠜⢦⡄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈
⠀⠀⢠⠃⠀⠀⠟⠛⠉⠈⢉⡀⢣⠜⢿⠓⡜⣰⡶⣤⡌⠉⠲⣄⣀⣠⣤⣶⣾⣿⣷⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠉⠐⠂⠀⡄⠐⠛⠋⠙⢮⣂⣿⣷⣰⠏⠐⣿⡷⢂⡆⢸⣿⣿⣿⣿⠿⠟⠋⠀⣀⣠⣤⡔⠀⠰
⠀⠀⠀⠀⠀⡟⠀⢰⠀⠀⠀⠀⠀⢍⠻⠟⠁⠤⠤⣽⣷⣼⣅⡀⠟⠋⣁⣠⠤⠖⠚⠙⠛⠃⠛⠃⠀⠀⡆
⠀⠀⠀⠀⡰⠁⠊⡹⠆⠀⠀⠀⠀⠀⠀⠈⣠⡀⠀⢸⣿⢿⠤⡈⠸⣋⣡⡏⠐⡆⠀⠀⠀⠀⠘⡂⠈⠉
⠀⠀⠀⢀⠁⠖⣄⠀⡇⠀⠀⠘⡄⠀⢐⣪⣤⣤⣶⣶⣿⣧⡓⣀⣤⠹⣿⠃⢠⡤⢴⣀⠀⠀⠀⠃
⠀⠀⢀⣾⢣⠏⠙⠛⢻⣦⣀⣴⣿⣿⣿⡿⠿⠉⠄⠈⣉⣽⠋⠈⠑⠓⣯⠀⣿⡭⠭⠒⠂⠀⣀⡸
⠀⠀⣼⣿⣎⠀⠀⠀⢠⣸⣿⣿⣿⣯⢥⣀⣸⣄⠀⣠⣬⣼⡆⡀⠀⣣⡏⠈⣭⣤⡴⠶⠿⠁
⠀⠀⣿⣿⣷⣶⣶⣤⣬⣻⡟⠁⠀⠐⠿⣿⡿⠛⠿⢿⣿⣿⣶⣶⣿⢇⡇⢠⡭⠤⡀⠀⠀⡆
⠀⠀⠸⣿⣿⠿⣿⣿⣿⡇⢤⣤⣀⣀⣀⡂⢀⣠⡤⠴⠘⣿⡏⠁⡔⠀⢃⠸⠛⠉⣁⡠⠤⠞
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣀⠀⠀⠀⠀⢀⡀⢀⣀⣠⣤⢿⣇⠀⠃⠀⠀⢃⡇
⠀⠀⠀⠀⠀⠐⢻⣿⣿⡟⠉⠉⠛⠉⠉⡏⠉⠉⠉⠀⠈⢻⡿⠤⠀⠂⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣧⣀⠀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠋⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠻⡟⡛⢻⡻⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠒⠒⠒⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠁⠂⠠⠊⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⡘⣇⠀⣀⣀⣠⡇⠀⠀⠀⠀⣀⠄⠊⠁⠀⠈⡆⠀⠀⢀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⠤⠚⢀⣹⠋⠉⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⢀⡤⠔⠛⠀⢀⡆⠀⢄⠀⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠚⠁⠀⠀⠀⠀⠀⠀⠑⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠙⢬⡉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⣂⢈⠭⡝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠸⠀⠀⠀⠀⣀⡤⠚⠉⢀⣿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠞⠁⠐⠄⢉⡼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢺⠁⣀⠀⢀⠞⠸⡏⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⣄⢀⣄⠀⡶⡄⠐⠀⡇⠀⠀⠀⠸⠀⢰⠀⠀⠈⢇⠈⠉⢉⣠⡴⠛⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠉⢹⢠⡥⢧⠀⡆⣇⣀⡦⠀⠀⡇⢸⠀⠀⠀⠈⠳⠚⠉⠀⠳⠶⠖⠲⠶⠞
⠀⠀⠀⠀⠀⡇⠤⠤⠤⠀⢀⠀⣀⡀⢸⡇⠸⠀⠀⠘⠏⠀⠈⠁⠁⠉⠀⠀⠀⢀⣃⠜
⠀⠀⠀⠀⠀⡇⠦⠤⠄⠀⢀⠀⠀⠇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠈⠁⠀⢀⡠⠤⠔⠒⠒⢹⢿
⠀⠀⠀⠀⠀⠻⠥⠥⢤⣴⣆⣤⡬⠀⣸⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣀⣠⣔⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⢧⠀⠀⠀⢀⡇⢈⣀⣀⣀⣉⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⢊⣨⠇⠀⠀⠈⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠖⠒⠒⠒⢚⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢄⡀⢀⣠⠒⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠋⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⣀⣀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⠘⠀⢠⠀⠔⠁⣠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣬⣡⡄⠀⠀⠀⠀⠈⢡⠴⠂⠁⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢀⣰⠗⠒⠒⠒⠚⠃⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣿⠿⠿⠿⠛⠛⠛⠋⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⣠⣤⣦⣤⣀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠟⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⡟⠋⠉⠙⢻⣿⣿⠟⠁⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣿⣿⡿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⠶⠿⠛⠃⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⡾⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣶⣶⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣾⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⢿⣿⡿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣀⣀⣤⣴⣶⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠀⠀⢲⡀⢧⠀⢻⡄⢸⡄⠰⡄⣇⠀⡄⠀⡀⠀⣠⠀⠈⠁⠀⡞⠀⣰⠃⢀⠀⠀⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⠀⠳⡄⢳⡀⢳⡀⠇⠀⡇⢹⠀⡇⠀⡇⠀⡿⠀⡔⠀⣸⠁⢰⠇⢠⠏⣠⠎⢠⡞⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢧⡈⢦⡀⠙⠄⢳⡀⢳⡀⠀⠀⢸⡀⠇⠀⡇⠀⠇⠀⡇⠀⠀⠀⠏⢠⠏⡴⠋⣴⠋⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⣄⠳⡄⠳⡄⠀⠤⠁⠀⠃⠀⠐⠘⠃⢀⠀⣄⠀⢀⣤⣤⡶⠶⢶⣤⣄⡈⠁⠞⣡⠀⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡈⠳⣜⠦⠉⣠⣴⡾⠿⠛⠻⠿⣦⣄⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠙⢻⣦⡀⠀⠴⠋⣡⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠳⣝⠳⡌⣠⣾⠟⠁⠴⠀⠀⠐⠂⠀⠻⣦⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣷⡀⠰⠋⠁⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠳⣌⠃⣰⣿⢋⡴⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣧⠀⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠦⠈⢠⣿⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠶⠶⣶⣶⠲⠶⠾⣿⣄⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣦⡀⢸⣿⡀⡿⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡟⠿⢼⣿⣧⣤⡴⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⡷⢸⡇⠷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣿⠓⢒⡲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⠿⠿⠿⣿⣿⠖⢶⣦⣵⡿⠰⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⣲⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡞⠛⠉⠁⠀⠀⠈⠉⠁⢀⣴⡿⠋⠀⠸⣿⣿⣦⣤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠙⠻⠧⣤⣤⣤⠶⠾⠶⢿⣿⡙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⡀⠀⢀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⣠⠄⣠⠀⡀⢀⠆⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠄⣠⣴⠿⠋⡀⠠⡄⠓⠀⠈⢦⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠁⡴⠁⠼⢡⠏⢠⠂⡠⠀⠈⢻⣦⠀⠀⣠⣴⠿⠛⠁⢀⠀⠹⣆⠙⣦⠀⢀⡀⠙⢦⡀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⡞⠁⠀⣰⠃⢠⠏⠀⠀⢀⡇⠀⠙⢷⡾⠋⠁⢠⠀⠆⠈⢧⠀⠘⣦⠈⢷⡀⠹⣦⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣠⠏⠀⠀⣴⠃⢠⠏⢀⡄⠀⣸⠁⢀⡄⠀⣄⠀⡀⢸⡄⠀⠀⠘⣧⠁⠘⢷⡀⠻⣄⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⣰⠀⡼⠃⢠⡏⠀⣼⠁⢀⡏⠀⣸⠁⠀⠁⠀⢣⠀⣧⠀⣦⠀⠸⣆⠀⡈⢳⡄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠃⠈⠁⢠⡟⠀⠀⡿⠀⢸⠃⠀⠏⠀⠀⣶⠀⢸⠀⢹⡄⢹⡄⠀⠙⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡉⠉⢦⣤⡀⠀⢸⣿⡏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠳⡱⡀⢸⣿⡷⠤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢣⢣⠈⡟⡀⠀⠀⠀⠈⢳⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢯⡉⠉⠙⠛⠛⠛⠛⠿⠿⠿⢿⠀⠀⠸⠸⠀⠀⠂⠀⠀⠀⠀⠀⢳⢳⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠢⣄⠀⠀⠀⠀⠐⠛⣿⠂⠄⠀⠀⡄⣴⣿⣭⢀⠀⠀⠀⠀⠈⡇⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠳⢤⡀⠀⠀⠀⠉⢀⡇⠀⠀⠀⠈⠻⠤⠊⠀⠀⠀⠀⠀⡇⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠉⠲⣤⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡤⠊⠀⠀⠀⠀⠹⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠏⣀⣀⡀⠀⠀⠀⣀⣡⡼⠷⢿⠿⠿⠷⠶⠶⠶⠶⠶⠦⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⣿⣶⣶⣶⣶⠶⠶⠒⠋⠉⠀⣀⣤⣼⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⠤⠤⠤
⠀⠀⠀⠀⠀⡠⠔⠚⣉⣡⠤⠜⢛⣛⠿⠿⠛⠛⠉⠁⠀⠀⡀⠠⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠘⡗⠒⠚⠉⠉⠉⠉⠀⠀⠀⣀⣀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏

⣶⣶⠿⣷⠶⢶⣶⠶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠄⠒⠀⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣀⠠⠤⠒⠒⣒⣒⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⡴⢊⣡⣤⡤⠖⠂⠽⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡰⢋⡇⠀⠚⣵⡉⢢⠤⢾⡭⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣺⣄⠀⠀⠛⠓⢣⣀⡸⠓⠢⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⡉⠛⠁⢀⡄⠀⠰⣀⠁⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠀⠀⠀⠀⠑⢄⡀⠀⠈⢉⡽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠈⠉⠉⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠇⠀⢀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠤⠀⠉⠉⠉⠉⠉⠉⠉⡉⢩⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠤⠁⠙⢄⠓⢄⡈⠑⠒⠖⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠈⠢⣀⠄⠳⣼⢾⠧⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠜⡋⠀⠀⠀⠌⠉⠁
⠀⠀⠀⠀⠀⣴⣉⠐⠀⣀⠜⠀⠀⠀⠀⠀⠁⠀⠀⢩⡌⠀⢰⠀⢀⡠⢆⠀⢀⡠⠄⠂⠁⣀⠀⠀⠒⠓⠀⠀⠀⠄⠀⠀⣀⣀⣀⣀
⠀⠀⠀⢀⠞⠁⠈⠳⡄⢀⣠⡄⠀⠀⣮⢭⣄⣬⡄⡟⠀⢀⡏⠀⠎⠀⠀⠋⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⡎⠀⠀⣠⠔⠉⢹⣿⣷⠀⠀⠀⠈⢉⢨⡇⠀⢀⠏⡜⠚⢦⡀⠀⢀⡠⠔⠁
⠀⠀⠀⢁⠀⠀⣧⠀⢠⣿⡟⢸⠁⠀⠈⠉⠊⡍⠀⢀⠎
⠀⠀⠀⢸⠀⠀⡇⢀⣾⡿⠀⢸⠀⠀⠀⠀⢼⡀⣠⠟⠁⠀⠁
⠀⠀⠀⠀⢂⡄⢧⣼⣿⠃⢀⢰⠀⠀⠀⢀⠇⠸⠋⢻⠀⠀⠃
⠀⠀⢀⡋⣙⠃⡖⠒⡤⣍⣉⠚⠳⢆⢠⡎⢠⢁⠀⣸⣀⣀⡔
⠀⡔⢾⣍⠙⠛⠛⠺⢣⡀⠉⢑⠢⡈⡇⢠⣿⠎⠉⠉⠀⠀⢠
⠀⣇⡀⡨⢉⣑⡒⠤⣄⡀⢙⣀⣃⣵⢀⡟⢻⠀⡄⠀⠀⠀⠈
⢈⠙⠁⠘⠳⡆⢉⡃⠀⠀⠉⠉⠀⠉⢈⠀⢸⠀⡆
⢰⢄⠀⠀⠀⠃⢨⢼⠇⠀⠀⠀⠀⡇⠸⠀⢸⠀⢁⠀⠀⠀⢀⡁
⠸⣄⡉⠒⠆⠾⠴⠾⠂⠀⠀⠀⠀⠃⢠⠀⠘⠊⠀⠑⡖⠋⢹⠄
⠀⠂⠈⠑⠠⠤⠤⠦⠄⠀⠀⠀⠀⡄⢠⠀⢨⠀⠀⠀⠇⠀⠀⠇
⠀⠈⢲⣤⣀⣀⣀⣀⡠⠀⠀⠀⠀⡤⣤⣠⠊⠀⠀⢀⡓⠒⠒⠇
⠀⠀⠀⠈⠑⠂⠭⠤⠤⠀⠀⠤⢴⠒⠉⡄⠀⠀⠀⢸⠀⠉⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢷⣦⠧⡀⠀⠀⠀⢠⠺⢴⣿⠶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠪⣵⠾⣡⠤⠔⠁⠀⠀⠀⠘⠐⠶⣏⣹⠒⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠰⠅⣀⠠⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠉

⠀⠀⠀⣀⣤⠖⠒⠒⠒⢒⣶⣶⣶⣶⣶⣶⣶⣶⠒⡒⠒⠒⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⣤
⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣠⠤⠖⠖⠶⠲⣶⣷⣤⣀⣀⣴⣿⣶⠒⠖⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣾⣻⣿⣿⣵⣶⣶⣶⣶⣶⣤⣀
⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠘⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡞⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⢠⡶⠛⢁⠤⠔⠒⠒⠒⠂⠤⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⡿⠟⠛⠛⠻⣿⣿⣶⠏⠀⠈⠁⣀⠤⠔⠒⠒⠒⠦⢄⡀⢳
⠀⠀⠀⠀⣀⣤⣶⠶⠾⠶⠶⣶⣤⣀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠠⢹⣿⠃⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⣸⠇⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⡾⠋⠁⠤⠂⠀⠀⠀⠀⢨⣙⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⣀⠴⠚⠋⠉⠉⠉⠉⠉⠙⠛⢷⣦⡀
⠀⢠⡟⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⣾⢠⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⡇⢸⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠈⠉⠛⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠋⣉⣤⣤⣄⡀⠀⠙⢿⣆⠀⠀⠀⢀⣿⠃
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⣤⣶⣶⣦⣤⣈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣠⡶⠿⠛⠛⠛⠟⣷⠀⠀⣸⡿⠀⠀⡜⣼⡟
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠁⠀⠠⣸⣿⠛⠳⢮⣳⡀⠀⠀⠀⠀⡠⢊⡠⠒⠉⠀⠀⠀⠀⢀⣠⣾⣟⣤⣾⠟⣁⣄⣾⣵⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣷⣶⣶⠶⠶⠶⠿⣧⡀⠈⠿⣖⣢⢉⣭⣿⡆⠀⠀⠈⠻⣶⣶⣶⣶⣦⡤⠤⠤⠤⠤⠴⠶⢾⣿⣿⣿⡿⣿⣿⣯⣭⣭⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣼⡉⠉⠉⠉⠉⠉⠉⠛⠷⣦⣬⣉⣉⣛⣛⣋⡨⠝⠛⠻⠿⢤⠞⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣇⠀⠀⠀⠀⢾⠀⠀⠀⠈⢑⡲⠤⠤⣀⣀⣀⣀⣀⡀⢶⣫⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣆⠀⠀⠀⠸⣿⣄⠀⠀⠀⠈⠉⠉⢓⣶⣾⣿⣭⣟⠻⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⢼⣿⣾⡿⣆⠀⠀⠀⢻⢻⡄⠀⠀⠀⠀⠐⢉⣮⠿⠛⠁⠉⢳⡄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⢸⣿⠋⠀⠹⣧⠀⠀⡜⢸⡇⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠸⠇⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⣾⠃⠀⠀⠀⠸⣇⠀⢅⡼⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⢼⡇⠀⠀⠀⠀⠀⢹⡇⠈⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠘⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠈⠉⠉⠉⠉⠀⣠⣤⣀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣤⡀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⡏⡿⣿⣿⢿⣿⠛⠛⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢰⣿⣇⣿⣿⣿⣿⣶⡶⠶⠿⠿⠿⠿⠿⠇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣨⣯⣧⣿⣿⣿⣿⣿⢀⡐⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢂⡀
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣶⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣡⣾⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣦⣍⠻⣿⣿⣿⣿⣿⠟⣩⣴⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⠓⣠⡙⠻⢋⣤⡘⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠟⣉⣴⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠿⢋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⠇
⠀⠀⠀⠿⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣶⣿⣿⣿⣷⣦⣴⣾⣶⣴⣶⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⣀⠤⢄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢳⠀⠀⠀⢰⠇⠀⠈⢣⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣌⠁⠀⡾⠀⠀⠀⠸⡎⠁⢀⡜⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠑⠈⠀⠀⠈⠉⢦⡈⠒⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣾⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⠉⢈⡻⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠈⠛⠿⢶⣶⣤⣤⣤⣠⣤⣤⣤⡶⠞⢿⣇⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠉⢹⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣄⠀⣹⠆
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡴⠚⠋⠛⠓⣶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⠉⢳⡀
⠀⠀⠀⠀⠀⣰⠟⠀⣴⠏⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢳⡀⠙⠳⢤⣀
⠀⠀⠀⣠⠴⠋⢠⡞⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠳⠦⣄⠉⠛⠶⣄⣀⣀
⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠉⠳⠦⡀⠀⠈⠙⢦⣄
⣶⡋⢀⣴⣄⢰⡏⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣤⣄⣈
⠈⠛⠋⠁⠉⠋⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠶⣦⣤⣄⠀⠀⠀⠀⠀⠹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠠⠀⡈⢿⣆⠀⠀⡀⠄⢀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠻⠷⣦⣄⣸⣿⣶⣀⣠⡶⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠀⠀⣠⡾⠉⠛⠁⠻⠟⠻⣦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠉⠁⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣀⣀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⡆
⠀⠀⠀⡇⠘⠀⠁⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⣀⣠⣿⣿⣿⣷⡄
⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⢟⣅⠀⠻⠟⢻⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢇⠈⠉⠀⠄⢠⠜⣤⡀
⠀⠀⠀⡄⠀⠀⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡾⠖⠂⠀⡀⡀⡉⠈⠓⡦⡶⠤⣀
⠀⠀⠀⡇⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡤⣞⠋⢀⠇⠀⢸⠠⠁⠀⠈⠳⢄
⠀⡠⢶⢃⠼⡒⢓⣼⣶⡶⡶⡤⢶⡖⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡗⣿⡓⠹⠀⢴⡎⠀⡄⠀⠀⠀⠼⣇
⢸⠁⠀⠳⢾⡓⠒⠒⠐⠒⠒⡛⡛⠛⠊⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⠸⠛⣿⣷⠋⠀⠀⡀⡀⠁⠀⠀⠀⠀⠸⣦⡀
⢸⠀⠀⠀⠀⠹⡄⠀⠘⠲⠀⠑⠟⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠀⠀⡄⠘⠃⡀⠀⡼⢃⠁⠀⠀⡀⢀⠀⠠⠈⠛⣦
⢸⠀⠀⠀⠀⠀⡟⠒⠐⠒⠒⠛⠛⠛⠛⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣆⠀⠀⣷⠰⠀⠀⡴⠀⠀⣆⠆⢀⣿⣿⡓⠂⣀⠀⣿⣧
⢸⠀⠀⠀⠀⠀⠁⠀⠐⣶⠶⠲⠶⢶⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⡄⠉⠀⠀⠌⠁⢀⡄⠿⢬⣈⢹⣿⣖⠊⡭⠶⣾⣿⠁
⢸⡀⠀⠀⠀⠀⠐⠀⠴⡇⠘⠚⠘⠨⢠⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠹⣇⢻⣴⢲⠃⢀⠠⢠⣈⣀⣀⠀⠀⠟⠃⠀⠈⣙⣼⣿⠀⠢
⢸⡃⠀⠀⠀⠀⣼⠀⠀⠮⠖⠘⠶⠞⠞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣿⠈⢸⢠⣀⠸⡇⠀⠉⠉⠙⡳⠋⠓⠢⣄⣹⣿⠿⠷⢾⠄⠓⡦⣤
⢈⡇⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢈⣿⠳⢨⣀⠀⢠⣷⠰⠆⠀⡤⢄⣀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⣠⣯⡀
⢀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣇⠀⢀⣀⣀⣤⠚⠁⠈⢣⣀⣈⣉⣽⡿⠀⣸⠉⡉⠘⠉⠓⠄⣠⠀⠀⢹⣦⡾⠋⠀⠀⠀⠀⣠⣴⡞⠉⠀⠙⣄
⠀⠧⠤⣤⣄⣀⣼⣀⣠⣤⣤⠤⠤⢴⣚⠁⡠⠀⡉⢻⢤⣀⡈⢗⠀⠀⣠⣯⡿⠛⠋⠁⢃⣷⣾⠀⡀⠑⠀⠤⡀⠒⢀⡿⠲⠶⣦⣤⣤⣴⠞⠋⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠿⠟⠛⡟⠛⠻⣿⣿⠀⠀⠺⠷⠶⢄⡈⠉⠀⢀⡤⠷⠘⠧⡿⠟⠋⠀⢀⠎⠀⢸⣿⠛⠻⢿⣶⣆⡠⣀⠶⣿⣇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⡆⠀⠀⡇⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠙⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇⠀⠀⠈⠿⣿⣿⡆⠀⠹⣿⣷⣄⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⣠⡿⠃⠀⠀⠀⢠⡏⣻⣷⡄⠀⠘⣏⠛⢻⠇⠀⠈⢆⠀⠀⠀⠀⠀⣤⠞
⠀⠀⠀⣧⠀⠀⡇⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡻⠁⠀⠀⠀⠀⠀⡇⠁⠙⢿⣄⣤⣼⣿⣦⣀⣀⡀⠀⠱⢀⣀⡤⠞⠁
⠀⠀⠀⠟⠀⠀⡇⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠃⠀⠀⠀⠀⠀⣰⠛⡄⠀⠘⡋⠛⠿⣿⣿⣿⢿⣁⡤⠞⠉⠁
⠀⠀⠀⢠⠀⠀⠂⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⣠⣶⠟⠀⠸⡄⠀⢱⠀⠀⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠁⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡌⠀⠀⠀⣰⢿⡏⠀⠀⠀⢱⠀⠀⢧⠀⠀⡿⣿⡆
⠀⠀⠀⠘⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⠁⠀⠀⠘⠃⣾⠁⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⠿⠟⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⢀⣀⣀⠠⣶⡿⠉⠉⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠿⠂⢀⣀⣀⣀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⠋⠈⠙⠻⣦⣤⠀⠀⠀⣤⣤⡴⠛⠋⠀⠀⠀⠀⠈⢻⣿⠿⢿⣶⣦⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠉⠉⠉⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⣿⠇⠀⣿⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢴⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⣿⡇⠀⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠈⡇⠀⢠⣿
⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠛⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⣿
⠀⠀⠠⣾⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⣷⡆
⠀⠀⠀⠉⠁⠙⡿⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣤⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⣤⣤⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡖⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣥⡀⠀⠀⠀⢀⣶⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠀⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠋⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠿⠋⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠁
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠛⠋
⣀⣀⣉⣉⣉⣈⣹⣿⣿⣿⣿⣋⣁⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠕⠜⠉⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠊⠁⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⡄⠀⠀⠀⢀⣔⣉⠀⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠂⠁⠀⠀⠀⣈⣧⣤⣴⣿⡿⠿⠛⠃⠀⠀⠀⠈⠉⠐⣢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠶⣉⡁⠀⠀⠀⠠⣤⣶⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⢀⣀⠤⠤⠒⠒⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠑⠢⢄⡀⠀⠉⠉⠀⠀⠀⣀⠤⠤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠙⠒⡄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡤⠂⣀⠀⠉⡏⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠰⡀⠀⡇⡌⠀⢳⠀⢡⠀⢸⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠃⡇⠀⠸⠀⢸⠀⠨⡤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠀⠱⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⠀⠀⣀⡠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣋⣍⣹⣝⣽⣻⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠐⠓⠐⠲⣄⡈⢳⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⠀⣋⣁⡸⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠈⠉⠲⡄⠀⢀⣴⠾⠟⠛⠋⠉⠉⠙⠳⢄
⠀⠀⠀⠀⢸⣀⠀⡈⢉⡆⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢀⡀
⠀⠀⠀⠀⢀⠌⠋⠁⢁⠇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡠⠤⠔⠒⠊⠉⠑⠞⠉⠉⠉⠀⠀⢉⣹⣦
⠀⠀⠀⠀⢸⠀⠈⠉⠉⠓⣿⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⢻⡟
⠀⠀⠀⠀⠸⠤⣤⣀⣀⡀⢸⣇⣀⡤⠴⠒⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣺⡏
⠀⠀⠀⠀⠀⠀⠀⢈⠟⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣠⣤⣤⠤⠤⠴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠚⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠒⠋⠁⠉⠉⢉⣿⠟⠠⠴⠒⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡵⣀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡖⠒⠂⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣡⠴⠋⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⡔⠃⠴⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠁⠦⠀⡀⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢰⡠⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠟⠛⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣫⣛⣩⣛⣟⣹⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠝⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢤⠒⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡤⡞⢃⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢀⡗⠁⠉⠻⣤⡼⠃⠀⠀⠀⠀⠌⠐⠀⠀⠀⠄⡊⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⡞⠔⠉⢩⠀⠀⠀⠀⠀⠀⠐⢈⡀⠀⡀⠀⠐⡪⢀⠁⢂⠅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢼⠀⠀⠲⠳⠲⠀⠤⠤⠶⠒⠫⡀⠀⠢⠍⢡⡀⠐⠡⠊⠄⠅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣀⣑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣈⣀⣀⡀⣀⡀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠉⠉⠓⠋⠁⠀⠀⠀⠐⠒⠒⠒⠚⠛⠛⠛⠛⠃⠙⠛⠋⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡋⡀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢦⠀⠀⠲⠆⠀⠀⠀⠀⠠⠄⡀⠀⢠⢠⠄⠀⠀⠀⠀⠔⠤⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⠀⠀⠂⣂⢔⠐⠌⠢⢠⠘⠋⠀⢀⣤⡀⠀⢠⡞⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡘⠁⠈⠀⠠⠀⠂⠀⠁⠀⠙⠀⠀⡀⠸⠦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈⠐⠀⡇⠕⢣⣀⡀⠐⠈⠀⠀⠀⠀⠁⠑⠀⠀⡃⠂⣬⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⠁⠄⠀⠰⡀⠁⠨⢀⢀⠀⠐⠀⠀⠁⠀⣔⠀⠀⡁⠈⠶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⠂⠤⠤⠄⠊⠀⠀⠀⣄⣐⣂⠀⠀⣆⣈⣘⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠥⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⠄⡠⠀⡀⠀⠠⢀⠄⡀⠔⡠⠀⠠⠂⠠⢀⠄⡀⠄⠠⢀⠄⠠⠔⠠⠀⠠⠀⠔⡠⣀⢄⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠠⠐⠁⠔⠀⠂⠠⠂⠐⠈⠐⠁⠒⠁⠂⠁⠊⠀⠊⠐⠁⠊⠀⠈⠀⠁⠒⠁⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢤⣤⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡝⠢⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡇⠀⠀⠀⠑⠂⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣙⠀⠀⠤⣀⣀⣀⠀⠢
⠀⠀⠀⠀⠀⠀⣀⠴⠊⡇⡀⠀⠀⠀⠀⠈⠉⠛⡻⠷⡛⠀⠀⠀⠈⠒
⠀⠀⠀⣀⠔⠊⠀⠀⠸⠀⠘⢆⠀⠀⠀⠀⢀⣞⠁⠀⠁⠂⢘⣿⣷⣶⡄
⠀⠀⡈⠀⠀⠀⡀⠀⠁⠀⠀⡺⠆⠀⠀⠀⡎⠪⠥⠬⢆⠀⠚⠁⣠⣿⠃⠀⢀⣀⣀⣀
⠀⠀⠘⢄⠀⢰⠀⠀⢀⠐⠎⠀⠀⠀⠀⠀⠑⢈⡰⠻⠃⢀⣀⠴⡺⢫⠉⠁⣰⠀⠀⠀⠀⠀⠸⠕⠀⠀⡡
⠀⠀⠀⠈⠂⣇⠔⠈⠀⢠⣳⢦⠄⣀⣀⣀⠠⠄⢹⣶⣿⢇⠀⢠⡇⠀⠑⠊⠁⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠠⠀⣷⠀⠀⠀⠀⠀⡘⠁⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠂⠀⠀⢫⢧⡀⠀⠀⠀⢀⣴⠟⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⢣⡱⣄⠀⠐⠿⠉⠀⣰⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠋⠀⠀⠀⠀⠀⣀⡠⢵⣌⠑⠦⣄⣠⠞⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣄⠰⠬⠭⠽⠓⠋⠁⠪⢧⣄⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠙⠶⣤⣀⡀⠀⠀⢀⡀⠀⠉⠳⣖⢶⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡠⠋⠀⣀⡤⠶⠚⠋⠉⠁⠉⠉⠉⠉⠉⠙⠲⢤⣈⠑⢽⣦⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣳⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣮⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠋⠁⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣄⠀⠀⠀⢈⣍⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠻⠟⡀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⠓⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⣶⠒⢚⡿⢿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⣹⢟⡭⠖⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⡇⠀⡏⠀⠀⠀⠀⢤⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢸⡅⠀⡇⠀⠀⠀⠀⢸⠙⢦⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠁⢀⡤⢾⡇⠠⡇⠀⠀⠀⠀⢸⡆⠈⢧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢈⡽⠶⠚⠉⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⢧⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣸⡇⠀⠀⢠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠉⠉⢹⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡶⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡞⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠙⢦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⣰⠃⠀⠀⠳⢴⡎⠙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠉⠀⠀⠀⠀⠀⣹⢶⠞⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣺⣁⣼⠁⠀⠀⠀⠀⠀⠀⡿⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⣀⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⢧⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⣀⠀⢸⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡈⠀⢸⡇⠀⠀⠀⠀⠀⠄⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠸⢤⡾⠃⠴⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⠀⠀⢠⢠⣰⢠⣀⢀⣆⣠⢀⣄⣠⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⣿⣿⣿⡏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣿⣿⠀⠉⠠⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣷⠊⢹⣿⠀⢠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢈⠇⢼⣿⠀⢨⠀⠀⢀⣥⢐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡧⣠⣿⢷⣿⣿⠀⢸⡤⣴⠿⠀⠒⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣱⣿⣿⣷⣿⣿⡇⢸⣿⣧⠐⢦⡄⣤⠈⠀⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡾⢛⠁⠐⣾⣿⠿⠇⢸⣿⡿⠿⠋⠀⠀⣀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠆⣿⠇⠀⠀⠸⣯⣖⠄⠀⠀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠭⢟⣺⠦⢹⡀⠀⠀⠈⠩⣵⣾⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠓⠀⠀⠀⠀⠈⠒⠾⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡶⠶⠶⢦⣤⣤⣤⣼⣟⠉⠁⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⠛⠷⠶⣿⣥⣀⡀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠉⠙⢻⡆⠀⠀⠀⠀⠀⢿⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⡶⠶⠶⠟⠛⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠸⣯⡀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡴⠞⠛⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⠙⠉⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣀⣀⡀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⣀⣀⡀⢀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣇⠀⣾⢁⣾⠉⠉⢿⡌⠉⣿⠉⠁⢸⡏⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣴⠇⠘⣿⠀⠀⣼⠇⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠓⠚⠋⠀⠀⠛⠀⠀⠘⠛⠛⠓⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⢀⣠⣿⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣻⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⠍⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣆⠀⠀⠀⠀⡾⣾⣿⣿⣯
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠺⢸⡿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⢻⣿⣿⠛
⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⠙⠲⠄⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠷
⠀⠀⠀⠀⣼⠟⠈⢿⡄⠀⠀⠀⠀⢠⡿⠁⠙⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣿⠟⣿⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⣸⠏⠀⠀⠈⠻⣤⣤⠀⢠⡿⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢠⡟⠁⠀⢻⡀⠀⠀⠀⠀⣰⡿⠀⠙⣷⡀
⠀⠀⠀⠛⠓⠀⠀⠀⠀⠉⠀⠀⠘⠳⠂⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠛⠓⠀⠀⠈⠉⠁⠀⠀⠐⠛⠁⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠤⢤⡀⢀⠠⡄
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡠⠀⠀⡠⠉⠀⠀⢀⠀⠀⠀⠶⡉⠁⠀⠀⢀⠠⢄⠀⠠⠊⣶⠀⢁⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠰⡊⡁⠀⠀⢀⡔⢀⠀⠀⠀⢑⠍⠅⠐⡆⠀⠀⠠⡠⡀⠀⠸⠆⡀⠀⠈⠒⠀⠐⠆⠀⠐⠒⠁⠈⡀
⠀⠀⠀⠀⢀⠀⢀⠀⠀⠒⠈⠀⠌⢀⠜⡄⠀⠀⠐⢢⠐⠀⢼⠀⡠⠂⠀⠈⠐⠁⠀⢈⡐⠆⠀⠀⠀⠀⠀⠑⡀⢌⠉⠂⠄
⠀⠀⠉⠒⢁⠀⠀⠁⠀⠆⣠⠐⠂⠁⠄⣰⠀⠀⠀⠐⠀⠀⠈⠋⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒
⠀⠀⠁⠐⠁⠀⠀⠡⠊⠤⠁⠐⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢖⠆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠉⠀⠀⠀⠀⠀⠀⠐⢄⡁⠀⡀⠀⡀⠂⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢁⠀⡛⠒⢂⠀⠀⠀⠀⠀⢀⡠⠤⢀⠀⠈⠀⠀⠀⠀⠀⠀⢠⢢⠀⡠⠊⠤⠄⠌⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠾⡀⠀⠁⠤⠁⢀⠸⠃⠀⠐⠁⠀⠀⠀⢀⠀⠓⠠⠂⠁⢔⠀⠀⠀⠀⠀⡴⢄⡒⠓⠔⠉⡀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠆⠑⢆⠡⠄⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠰⠀⠈⠄⡀
⠀⠀⠀⠐⢗⠀⠀⢀⠐⠁⠀⠀⠀⠐⠀⠠⠀⠀⠄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠂⣀⠀⠀⠤⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠀⠀⠊⠀⠀⠈⢀⣀⠴⠋⠀⠀⠄⡀⠀⡠⠀⠀⠀⣀⠀⠀⠉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⡔⠐⠐⠀⠀⢀⠤⠄⠄⠀⠂⠀⠀⢢⠄⠂⠀⠀⠀⠀⠀⣛⣁⠀⠀⠀⠁⠂⡴⠂⡱⠐⢩⠀⠀⣆⢈⠆
⠀⠀⠀⠔⠁⡀⠰⠔⢒⣤⠀⠘⢅⠀⠀⢣⠑⢨⠀⠸⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⣀⠊⠀⠉⠀⠑⣀
⠀⠀⠀⠂⠈⠀⠀⠠⠏⠀⢃⠀⠀⠁⠈⠀⠑⠂⠀⠀⠣⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠀⢠
⠀⠀⠘⡋⠉⠘⠁⠀⠀⢀⠐⠂⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⢢⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⢴⠀⠐⠉⠹⠗⣀⠔
⠀⠀⠀⠐⠤⠥⠐⠀⠈⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠄⠂⡄⠀⠁⠒⠊⠢⡀⢀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠉⠃⠀⢰⣀⡀⠀⠀⢠⡄⠀⠀⣘⠢⡐⠀⣧⠻⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢹⠀⠀⠄⠤⢄⠐⠋⠃⠀⠀⠈⢁⠀⠂⠉⠀⠸⠜⠂⢀⢤⠀⠀⠀⠀⠀⡄⠀⠔⢲
⠀⠀⠀⠀⠰⡉⠨⠀⠀⠀⠀⠀⠈⠔⠀⠒⠉⠂⠄⢐⠀⠀⠀⠀⠉⠉⢀⠟⠛⠚⢅⠀⢸⠀⠀⠀⢐⠁⠁⢈⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠒⠀⠀⠀⠉⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠑⢌⠀⠀⠀⠀⠙⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠀⠂⠈⠙⠲⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠶⠿⢆⡀⠀⡐⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣦⡤⠒⠊⢉⣙⣉⣙⣹⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢼⣿⣿⣿⢿⠿⠿⡿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⡃⠰⢶⡦⠀⢨⣤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠁⠀⠈⠉⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⡀⠀⢈⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠁⠀⠌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠠⢐⠂⠀⡴⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠩⠀⠀⠀⠑⠀⠀⠑⣶⡄⠆⠙⢶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠁⠀⠀⠀⠱⡀⠀⠀⡈⠀⢀⠙⣿⡓⠀⠲⠃⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠑⢄⠀⠀⠀⠙⠊⢉⡁⠀⠈⠰⣿⣿⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠐⠢⢤⣈⠻⢀⠆⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⣀⣀⣀⣀⣀⣀⣨⡙⡎⠀⠰⠶⠒⣦⡬⢓⣒⡈⣉⣅⣤⠴⠙⡄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⢀⠇⠈⠉⠉⠉⠀⠀⠀⠀⢀⡀⠇⠀⠘⠄⢼⠁⡈⠉⠀⠀⠀⠀⠀⠂⠀⠘⡄
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⠁⠀⠀⠀⠤⠀⠈⠀⠀⢠⠀⠀⠀⠀⢤⠀⠁⡀⢲⠄⠀⠀⠀⢀⠀⠼
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢲⢏⠀⠈⠀⠠⣀⣀⣀⣧⠊
⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⢀⡾⠀⠀⠠⡀⠀⣠⠖⠙⠍⠉⠉⠉⠉⠱⢦⠝⠀⢐⠀⠆⠀⠀⠀⠀⡩
⠀⠀⠀⢠⡿⠛⠉⠉⠛⢻⠶⠶⡷⠊⠀⢤⣈⠘⡅⠀⠂⠂⡀⠐⠀⣠⡄⠁⠀⠀⢿⢆⡠⢤⣤⡖⠊
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠓⣀⠶⠂⠀⠀⠂⠀⠊⠀⡆⠀⠀⠀⠀⣀⠀⠈⢠⠇
⠀⠀⠀⢻⠀⠀⠀⠀⡀⢄⠀⠘⢿⣥⠈⠉⠁⠩⠇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣦⣤⣭⠴⠋
⠀⠀⠀⠈⠓⠦⣄⣳⣌⣦⣦⣔⣀⠏⠙⠂⠒⠺⠶⠆⠀⠀⠀⠒⠀⢘⠁⠈⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠆⠘⠒⠒⠊⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠻⠿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⡿⠋⠁
⠀⠀⠀⠠⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠀⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⣀⠔⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢏⠲⣄⠀⠀⠀⠀⠀⡴⡅⡸⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡏⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡈⢧⡇⢹⣄⠀⠀⡼⠀⢹⡿⢃⣾⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⢧⢸⣹⠖⠲⡟⡶⠋⠱⠯⠷⢦⡀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣳⡦⠤⣿⡇⠀⠀⠹⡇⠀⢀⠀⠀⢀⠇⢀⣠⠔⢋⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓⠦⣄⡀⠉⢻⠋⠉⠀⠀⠰⠀⢻⣀⡆⣀⡴⠛⣿⢉⡴⠚⣩⠟⣄⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠓⠮⣷⣤⣤⣤⣠⣬⡏⠌⣻⠅⠀⠀⠈⢧⣀⠔⠁⠀⠘⣏
⠀⠀⠀⠀⠀⠀⢀⠀⢸⠓⢤⠽⠶⠞⢉⡞⠉⠀⢀⠀⠒⣿⠛⠏⠳⢄⣀⡤⠚⠁⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠘⠛⢍⠉⠉⠙⠳⣾⡀⠀⠀⠘⢮⣀⠀⠀⢀⡰⠧⠴⠾⠶⠮⣅⡀⠀⠀⠀⠀⢀⡴⠁⣀⡤⠖⠒⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠈⠳⢤⡞⠉⠉⠳⣄⣰⣠⢋⠘⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡠⠖⠁⡠⠚⠁⢠⠀⠀⠀⠀⠀⠀⠀⣙⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢼⣋⣿⡟⣶⠁⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⠎⠀⠀⠀⢀⣀⠤⠔⠒⠋⠉⠉⠀⠀⠉⠂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣶⠞⠉⡟⢹⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⢉⡀⢠⡖⣤⠀⣀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉⢀⠀⣦⠈⣯⠄⠀⢻⠣⡄⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢊⠉⣿⠋⡙⢦⣼⣛⣇⣀⣀⢀⣀⣀⡤⠤⠖⠚⠉⣁⡀⠰⢾⡁⠨⢷⠛⣆⠘⠖⣂⣈⠤⠤⠤⠒⠒⠢⡤⢼⣤⢤⠖⡲⣷
⠀⠀⠀⠀⠀⠀⡼⠥⠤⠴⡇⢀⡇⣸⠈⢏⠉⠉⠉⠉⠀⠀⢀⣄⣠⡆⢸⢻⠹⡆⠈⠧⠀⢘⡠⠴⠚⢉⣁⡀⢠⡖⠢⠴⠶⣴⠟⢩⣃⠼⠛⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡰⠃⠀⠘⣧⡀⠀⠀⢠⡄⠀⣯⠉⢷⠈⠳⠛⣁⡤⠖⠋⠁⠀⠀⠀⣇⠀⠙⢾⡂⠀⠀⣰⠟⠒⡺⠛⢒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⢉⣀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⢀⠖⠉⠀⠠⢬⠷⠺⣞⡁⠄⠀⠙⣏⠩⠉⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⡻⠯⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⠶⠒⣻⢌⣡⡟⠦⣤⣤⠴⠛⠛⠛⠃⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠉⠒⢦⣄⡀⠀⠀⠀⠀⠀⣠⠴⢾⣁⠀⠀⠀⣸⠀⠉⠀⠀⢹⠓⠒⠒⢲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⡴⠋⠀⠉⠲⣄⠀⢠⠞⠁⣠⠊⢉⠷⣀⣠⠛⣄⢀⣴⡚⠉⠀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠈⠳⠃⠀⢰⡇⣠⠀⢀⡏⢹⠀⠈⡏⠀⢹⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⢁⡴⠛⢆⠈⡀⡸⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡴⠛⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠉⠀⠀⠈⠳⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉⠀⠀⣧⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠉⠉⠙⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠠⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠠⡐⠾⠝⠓⠒⠒⠒⠀⠀⢷⠀⠀⡇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡀⠀⣠⠆⠔⠉⠀⠁⠀⠀⢀⣀⠀⠀⠀⠀⢀⣸⣄⡼⠁
⠀⠀⢀⡴⠊⠉⢳⡀⠀⠀⠀⠳⢄⣸⡀⢀⠀⠀⠐⠈⠁⠀⠀⠀⠀⢀⡤⠊⠀⡏
⣠⠖⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢹⠎⠙⠳⠶⠤⢤⣀⣀⣀⣤⢖⠁⠐⠀⢂⡗⠦⡀
⠹⡀⠐⠀⠄⡀⠀⠀⠳⡀⠀⢠⡞⢉⡄⠀⠀⠀⣶⣄⡄⠀⠀⠬⣸⡷⠈⠀⠈⡇⠀⡇
⠀⠹⣤⠒⠤⡀⠀⠀⠀⠙⢆⠘⣇⠀⢃⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠃
⠀⢠⠇⡀⠀⠉⡆⠐⠂⠀⠀⠱⣌⠲⢬⣄⠀⠀⢀⣠⣼⣦⣴⣷⣦⣄⣀⢀⡟⠉⠁
⠀⢸⠀⠀⠀⠀⡇⠀⡆⠀⢀⠴⠋⠀⠀⡨⢧⣘⠻⠿⠿⠟⠛⠛⠛⢛⡩⢺⣦⡀
⠀⠈⣆⠀⠀⠀⠈⢦⣥⡞⠁⠀⣠⠔⠋⠄⠀⢁⠑⢄⡀⠀⠀⠀⢀⠜⠁⡜⠀⠉⠲⢄⡀
⠀⠀⠘⣆⣀⠤⠔⠚⠉⢻⡤⠚⠁⠀⠀⠈⠄⠀⠢⠀⠈⡹⠛⢟⡁⠀⣼⠁⠀⠀⢀⠀⠙⢦⡀
⠀⠀⠀⢳⡀⠀⠀⠀⢀⡰⠆⠀⠀⡆⠀⠀⠈⢄⠀⠹⡖⠉⠀⠈⠉⠛⠋⠀⠀⠀⠘⡄⠀⠀⠹⣄
⠀⠀⠀⠀⢳⡀⠒⠈⠁⠀⠸⠀⢀⠁⠀⠀⠀⠀⠢⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⡱⣁⠐⢠⡀⠀⠀⠀⠀⠀⠀⢠⠛⠉⢀⠉⢺⣧
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠤⠖⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠈⠂⢤⡀⠀⣀⣠⢿⠀⠀⠀⡀⠸⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡑⢢⡔⠉⠉⠀⠀⠀⠁⠈⠉⠁⡁⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⠀⠈⠉⠀⠀⠀⠉⠀⣤⣾⠋⢃⠐⡀⠀⠀⢶⣾⠾⠷⠟⠛⠂⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠒⠾⠃⠙⡄⠐⡀⠀⠀⠀⡀⠀⠀⠠⢤⠐⠂⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠉⠉⠈⠁⢒⠂⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠉⠉⠁⠀⠒⣀⣀⣈⣉⣉⠀⠉⠆⠀⠀⠀⠀⠀⠀⠁⢚⣀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠘⡄⠀⠈⠢⠤⠤⠒⡶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠈⠉⠉⢻⠀⠀⠀⠀⠀⠀⢨⡏⠉⠉⠁⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⣀⡀⠀⠀⠀⣈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠻⠷⠶⠦⣤⡄⠀⠀⠀⠀⠀⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢿⣿⠶⠶⠦⠀⠀⠀⠀⠙⣷⠒⠶⢠⣤⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⠀⠈⢳⡀⠀⠀⠈⠙⢷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡄⠀⠀⡠⠀⠐⠒⠀⠀⢠⡄⠆⠀⠀⢠⡄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⢶⣶⠄⠈⠀⠀⠀⠀⠀⠀⠀⠒⣾⡶⠀⠀⠀⠀⠀⠈⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠁⠀⠀⠀⠂⠀⠀⠈⠁⠔⠻⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢶⡿⣧⠉⠘⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡿⠁⠉⠉⠁⠈⣰⡆⠋⠀⠀⠉⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠈⢃⣾⠀⠀⠀⠚⠁⠙⠄⠀⠀⠀⠊⣿⣹⠇⠀⠀⠀⠀⠀⣰⠏⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⡟⠀⠀⠸⠃⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⡇⠀⠀⠀⠻⡈⢧⣀⡀⠁⠀⣀⣠⡞⣡⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⣸⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠙⠷⠶⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠋⢻⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣞⣁⡶⢸⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠙⠿⣧⣼⣯⣸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢦⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠶⣶⣄⡈⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣘⣻⣷⣦⣴⣟⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠛⠇⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠒⢊⣉⣉⡉⠉⢉⠙⣿⣿⣇⣶⣼⠟⠁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢠⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠛⠛⠛⣛⣛⣉⣙⣿⣻⣿⣴⣶⢾⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠘⡏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢸⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠤⠤⠤⠤⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠐⠒⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣄⣠⣤⣤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣋⣀⠴⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠊⠉⠉⠀⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢴⣶⣶⣶⡤⠶⠋

⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣧⡬⢉⣥⣤⣍⡩⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⡿⠱⣿⣿⣿⣿⡷⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣾⣿⠛⠙⣿⡆
⣷⣶⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⢿⣇⠀⢹⣷
⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣄⣴⡿⠋⣀⣠⣴⡄⠀⠀⠀⠀⠸⣿⡀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠉⠙⢻⣿⠿⠟⠋⣁⣤⣤⣤⣤⣴⡾⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠹⣷⣶⣶⠿⠛⠉⢻⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⣤⡀⠀⣠⣤⣤⡀⠀⣠⣤⣤⣤⣤⠀⣤⣤⣤⣤⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣆⠀⣼⡿⠀⣼⡟⠉⠙⣿⡄⠈⠉⣿⡏⠉⠀⣿⣏⣉⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⣶⣿⠁⠀⢿⣇⠀⢀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠟⠛⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠿⠃⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠿⠇⠀⠀⠿⠿⠿⠷⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣻⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢻⣿⣿⡄⠀⠀⢠⣿⣿⡏⠀⠀⠀⣿⣿⣛⣛⣋⣉⣻
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠘⣿⣼⣷⠀⠀⣾⣧⣾⠁⠀⠀⠀⣿⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⢹⣿⠽⣇⣸⡟⣽⠇⠀⠀⠀⠀⣿⣷⢿⠶⠶⡶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⢿⣯⠽⡿⣼⡿⠀⠀⠀⠀⠀⣿⡿⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡾⠶⡶⢶⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠘⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⣿⣵⠿⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠾⠃⠀⠀⠀⠙⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠺⠿⠿⠷⠶⠶⠶⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢿⣇⠀⠀⠀⢠⣿⣿⣯⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⣠⣾⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣟⣿⣿⠟⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠉⣿⡟⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⣿⡼⣿⣿⣿⣿⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⣿⣯⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠾⠿⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣌⣿⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠠⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⡄⠈⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⢤⠄⠀⠀⠀⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠷⣤⡀⠀⠒⠒⠂⠀⠒⠲⠶⠶⠶⠶⠶⠶⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠷⣍⡉⢉⣛⣋⣛⣃⣀⣀⣀⡀⠀⣀⣤⠾⠛⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⢶⣌⡉⠉⠉⠉⠉⣉⣵⠾⠋⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⣐⣾⣿⣿⡩⢽⣓⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠂⠈⠁⠠⠒⠒⠉⠋⠉⠀⠁⠀⠁⠀⠁⠀⠉⠲⢄⠈⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⢀⣤⠖⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⡀⢠⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣊⣑⡉⢉⡈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠂⠈⠙⡎⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⢏⣵⣛⣫⣍⠽⠗⣎⣶⣤⡀⠑⢄⠀⠀⠀⠀⠀⢸⠁⠀⢀⡄⠀⠀⠀⠀⠀⠐⠓⠂⡀⠌⠈⠀⠀⠀⠀⢀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡯⡄⣿⠏⣓⣀⢄⣑⣢⣆⡛⡎⡆⠈⠆⠀⠀⠀⡀⢸⠄⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠘⠰⡘⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡥⣐⣟⠛⢭⡀⠀⠀⠢⡢⢂⢦⡇⠀⢸⠀⠀⠀⢐⣼⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢁⢃⠢⢁⢡⣇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣗⢀⣿⠀⠀⠉⠑⠦⣄⣁⡑⣿⠃⠀⠀⠀⠀⠀⢀⢽⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⡈⠈⡄⠌⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣧⣨⣿⣆⠠⠤⠀⠒⠉⠅⡐⠚⠀⠀⠀⠄⠀⠀⢐⡾⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠠⠑⠸⢠⢰⢇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠐⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⢽⢠⢐⠀⡆⠀⠀⠠⣇⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣨⠃⠆⠂⢅⢃⠂⢳⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠉⠑⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⢽⢨⠐⠂⡇⠀⠀⠀⢿⡇⠸⠒⠢⡄⠀⠀⠀⠀⠀⠀⠠⠑⡀⠂⡈⡘⠌⠄⡈⢌⢌⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⠙⠓⠲⢤⣤⣀⣘⣼⠠⠄⡇⠀⠀⠀⠽⣇⠀⠮⠀⡘⠀⠀⠀⠀⢀⢀⢃⢓⠰⠰⠱⠠⡘⠐⠈⠉⠼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠘⣿⢀⡂⠀⠀⠀⠀⢨⡿⢦⣤⡴⡑⡀⠀⠀⠀⠌⠆⢄⢢⢁⢀⢣⢡⢠⠣⠹⠰⢸⣃⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣿⡐⠂⢰⠀⠀⠀⢜⠌⢼⠋⠑⡐⡐⡜⡌⡌⡌⠌⠆⠆⠂⢀⣀⠡⠤⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣼⢿⣿⣆⠀⠀⠀⢀⠎⠀⠀⠀⠬⡀⢿⡇⡁⠸⠀⢀⣀⢂⢊⢂⠠⠰⠐⢐⣈⢀⡤⠄⠐⠂⠉⠉
⠀⠀⠀⢀⣀⣤⣤⡶⡞⡟⠛⡏⣉⣠⣤⣼⠾⢧⡀⢠⠎⠀⢀⣐⣢⡵⢶⣿⠏⡯⠻⣈⣎⠣⠤⡧⢖⠖⡚⡩⡣⠡⢪⡆
⢠⢔⡾⠟⠛⠁⣀⣧⣧⠷⣖⡛⠟⠉⠁⠐⠀⠀⠑⠷⠶⣚⠛⣉⣧⣿⡭⠟⢣⢙⡾⠁⠀⠀⠀⡗⡡⡪⢊⢎⠔⡕⡡⡇
⡇⣏⢀⡴⠿⠛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⡇⣇⡧⠚⠉⠀⠀⠀⠀⠀⡯⣪⢔⡵⡫⢊⢔⠼⠁
⢛⠪⡺⢷⣷⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣒⢭⠗⠋⠃⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢚⠔⡡⡢⢺
⠘⢦⣈⠉⠐⠛⠿⡭⢶⢶⢶⠦⠿⠻⠛⠉⢡⢀⣸⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⡡⢊⢔⢽
⠀⠀⠀⠉⠒⠒⠢⠧⠴⠼⠼⠴⠴⠒⠚⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢒⠵⡫⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠵⡪⢊⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠄⠄⣠⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠉⠲⢂
⠀⠀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⣰⠁⠀⠈⢦⣀⠀⠀⠀⠀⠀⡠⠚⢅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠀⢉⡆⠀⡤⠘⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡤⠼⣄⣀⣀⠀⠀⠀⠀⠀⢀⢠⠄⠀⢤⠔⠏⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡀⢙⠦⣀⠀⠀⠀⠀⡇
⠀⢠⣞⠁⠀⡧⠀⠈⠁⠀⠉⡉⠉⠁⠜⠀⠀⠀⠀⡤⡀⠀⠀⠙⠛⠋⠒⠂⠉⠈⣍⠀⠀⠚⠳⣄⠀⢠⠃
⠀⢀⠃⣀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠇⠀⢀⠀⡠⠁⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡨⣧⠋⠢⡀
⠀⠀⠀⠸⠊⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡫⢛⠁⠀⠀⠀⣠⠀⠡⡀⠐⠆⠂⠀⠀⠀⠀⠀⣩⡕⠙⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⣓⢤⣤⣀⣠⡤⣤⠒⠙⠢⣼⠻⠀⡰⡴⠀⠀⠀⡠⠰⣶⣤⣤⡤⠐⡶⠮⣖⡓⡄⠘⠢⠔⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡷⠀⣠⠒⠦⡙⢍⠈⠚⣶⠏⠀⠀⠀⠀⠀⠁⠀⡿⠀⠀⠀⠀⡇⠀⠈⡂⣹⠆⢠⡄
⠀⠀⠀⠀⠀⠀⡇⡤⠚⠀⠀⡇⠀⠀⠠⡀⠙⡄⠈⢆⠀⠀⠀⠀⠀⡆⣞⠁⠀⠀⠀⠀⡘⠶⡈⠡⠸⠈⣤⠃
⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠑⠤⡀⠀⠀⣠⡶⠐⠈⠄⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⡐⠀⠘⠈⠈⢹
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠳⡤⠤⢤⣤⣤⣀⠀⢀⠃⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠂⠤⣤⣀⣀⣀⣀⡠⣶⡻⣑⠒⡁⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠀⠀⠸⡵⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡟⢀⠇⢐⠉⠙⢮⡀⠀⠀⡀⠀⠀⠀⠀⡠⠉⠀⢹⡄⠊⡠⠉⠑⠠⡀
⠀⠀⠀⠀⠀⡠⡴⠣⣌⡢⠤⡼⠳⣮⠔⠉⡄⠀⠀⢱⠀⠁⠀⡰⠖⠆⠺⠱⢄⣀⡆⠘⡯⠀⠀⠀⠀⢸
⠀⠀⠀⠠⡎⠀⠑⠄⠈⠁⠀⡄⠀⠘⡓⠊⠀⠀⢀⡜⠀⠀⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⠁⠀⢠⠋⠀⠲⠄⠂
⠀⠀⠀⠀⢣⡀⢀⢀⠀⢀⡰⠀⠀⠀⠙⠢⠤⠒⠉⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⠎⠀⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣟⣿⣛

⠆⢿⣦⠄⠀⠀⠀⠤⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣟⢻⡟⣿⣿⡀⠉⣿
⠀⠀⠘⠿⠀⢴⡇⠀⠀⣈⣤⣤⣤⣄⠀⠀⠀⣀⣤⣤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡏⠀⠀⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣴⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠠⠄⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⢤⡀⢸⣿⣿⣿⣿⣿⣿⠀⠸⠮⠛⠀⠀⠀⠉⠁⠀⠀⣿
⡆⠀⠀⠀⠖⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⣼⣧⣀⣀⣀⣠⣤⣤⣤⣀⣿
⣿⡟⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡤⢠⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿
⣏⡀⠀⠀⠀⠀⠐⠓⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣤⣤⣀⠀⠀⠀⠘⠃⠉⠙⠛⣿
⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠂⠃⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠉⠉⠉⠀⠺⡆⠐⠆⠀⢀⣠⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⡂⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠛⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠛⢻⣿⣿⠿
⠀⠆⠀⣠⡄⠀⠀⠀⠀⣠⣤⡀⠀⠀⢠⣤⠀⣤⡄⠀⠀⢀⣄⣠⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⣶⣷⣷⣶⣶⣶⣶⣶⣧⣉⠆
⠀⠀⠀⣾⡇⠀⠀⠀⢰⣿⠉⣷⠀⠀⠘⣿⡄⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⣿⣼⠿⠿⠟⠻⠿⠿⣿⣿⣿⠶
⣶⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⣿⠶⠄⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⣿⡿⣷⣀⣀⣀⣀⠀⠀⢻⣿⣿
⢸⠁⠀⣿⡇⠀⠀⠀⠀⣿⣤⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠰⣿⢠⡄⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠻⣷⣾⣿⣿⣿⢿⠿⣿⣿⣿⠟
⠻⠀⠀⠻⠿⠟⠀⠀⠀⠙⠛⠉⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠿⠿⠗⠀⠀⠀⠀⠀⠛⠋⠀⠀⡀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉
⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣿⣿
⣿⣿⣿⡛⢿⣿⡏⣿⠏⣿⠏⣿⣟⣿⠙⣿⠙⣿⠹⣿⣻⡟⢻⣛⣿⢻⣟⣻⣿⣿⠁⡾⢶⣴⡖⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆
⣿⣿⣽⣷⣬⣿⣇⣿⣇⣽⣆⣿⣟⣿⣀⣿⣠⣟⣠⣿⣻⣇⣠⣿⣧⣸⣏⣹⣿⣿⠈⢧⡌⠁⠐⢨⣿⡿⠿⣿⣿⠿⣿⣷⣾⣿⣿⡆
⣿⣿⡇⢸⡇⢹⣏⢸⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣟⠿⡟⢹⣿⠛⣿⣻⣿⣿⠀⠘⢥⡐⠆⠘⠛⠛⠚⠿⠿⠿⠿⠿⠿⠿⠟
⣿⣿⣧⣾⣧⣼⣯⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⣸⣟⣷⣌⣻⣿⣿⠀⠐⠸⠇⠠⡆
⣿⣿⠁⢾⡇⢻⡅⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠉⣯⣿⠉⣿⡏⣿⡿
⣿⣿⣷⣾⣷⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣤⣿⣿⣄⣿⣧⣿⡿⠀⢠⣶⣶⣄⣠⣶⣤⡀⠀⢠⣾⣷⣦⣠⣶⣦⡀
⣿⣿⣧⠸⣇⣿⡇⢹⡇⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡼⢋⣿⡇⣿⣯⣯⣽⣿⣿⣷⠀⢺⣿⣿⣿⣿⣿⣿⠇⠀⠸⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣦⢿⣿⣿⣼⣷⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣥⣯⣽⣧⣽⣯⣭⣿⣿⣿⡏⠀⠈⢻⣿⣿⣿⣿⠃⠀⠀⠀⢹⣿⣿⣿⡿⠋
⣿⣿⠷⣿⠸⣿⢿⡇⣿⣷⡖⣿⠁⣟⢿⣿⠶⣾⠇⣿⠈⣿⣸⡏⢹⡏⢹⡍⣿⡇⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⢿⣾⡿⠀⠀⠀⢀
⣿⣿⣷⣿⣾⣿⣿⣷⣿⣾⣷⣿⣷⣷⣾⣿⣶⣿⣦⣿⣤⣿⣽⣧⣾⣷⣼⣴⣿⡇⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠸⣿⢁⡤⠀⠈⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠻⢹⣧⡄
⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣷⣶⣾⣾⣿⣷⣾⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⣸⠃⠀⠀⢀⣀⡀⠀⢰⠏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⠀⠀⣴⠋⢉⣿⢀⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡇⠀⠈⠁⣸⠃⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⠀⠻⢦⣀⣀⣿⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⢹⡟⠶⣄⠀⠀⠈⢹⡇⠀⠀⠀⣠⡶⢦⡀⠀⣠⠀⠀⣠⡄⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣸⠃⠀⠙⠀⠀⠀⢸⡇⠀⠀⣼⢻⡟⠶⣿⠞⢻⠀⣰⠟⣧⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠇⠸⣇⣠⠏⠀⢸⣷⡏⠀⠈⡏⠉⢀⡴⠛
⠀⠀⠀⠀⠀⠀⠹⣦⡀⢀⣠⠞⠁⣤⠀⠀⡀⠀⠀⠘⣧⢠⡏⠀⡀⠈⠁⠀⠀⠀⠛⠀⠀⠀⢷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢻⣄⣰⠇⢀⠀⠀⠙⠟⠀⢸⣟⣙⡷⠀⢠⡞⢦⠀⠀⠀⣀⣤⡤⠶⣤
⠀⠀⠀⠀⠀⣰⠞⠛⠛⣦⠀⠀⠀⠀⢹⡏⠀⣯⣿⠀⠷⣴⠇⠸⡏⠹⡆⢠⡟⠀⣼⢀⡴⠞⠉⠀⢠⡄⠘⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠸⠇⠀⠈⠀⣠⠴⠶⡄⠀⢰⠆⠀⣼⠁⣰⡿⠋⠀⢀⣀⡀⠹⣤⡴⠃
⠀⠀⠀⠀⠘⣇⠀⠀⣠⡾⠁⢀⣠⣶⠀⢀⡴⣤⡾⠁⠀⣰⠇⢀⣤⠀⢀⣿⠀⠉⠀⠀⣴⣏⣡⠟⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢧⡀⠀⢀⡴⠛⠁⢹⣠⠏⠀⣿⠁⠀⣰⠏⠀⣾⣿⠀⣼⡇⠀⠀⠀⣰⣟⠉⠁⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣾⣏⠀⠀⠀⠸⣿⠀⠐⠃⠀⢠⡟⠀⣼⠃⢻⣴⠟⣷⠀⠀⣰⠏⢿⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠘⢷⡀⠀⠀⠁⠀⠀⠀⠀⢸⡇⣰⠇⠀⠀⠀⠀⣿⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠘⠷⠟
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠶⠚⣛⣋⡉⠉⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣆⠀⠀⣏⣀⣧⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣿⣿⣿⣿⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠀⠠⠆⠀⠐⣙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠀⠀⣀⣈⠥⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠠⣀⣀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣧⡀⠀⣀⠀⡿⢶⡶⢶⡆⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠻⡄⢈⠞⣡⠟⠀⢸⡇⠀⠀⠀⠀⠀⢸⣽⠀⠀⠉⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡀⠀⠀⣹⠋⣴⠃⠀⠀⢸⣇⠀⠀⠀⢠⠤⠜⢻⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣾⠁⠀⣀⣠⣼⣥⣤⢾⠛⠉⢽⡆⠀⠀⠀⣿⠋⢻⢻⡾⠀⣼⠉⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡟⠀⠀⠁⠀⠀⠀⢻⠸⡄⠀⣹⡇⠀⠀⠀⣿⠀⢸⢸⣥⣠⡏⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣀⣸⣦⡟⠉⠉⠀⠀⠀⠀⣿⠛⠛⠛⠁⠀⠙⠒⠲⠤⡾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⠚⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⡿⢿⡿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡜⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⠞⠀⠀⠀⠀⠀⢠⣶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠋⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⢀⡾⠁⡴⠃⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢠⡟⢁⡜⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣰⠏⢠⠎⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠸⣿⣴⠋⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢻⣤⣴⠆
⠀⠀⠀⠀⠈⠛⠳⣤⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⣀⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠂⠈⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠀⡒⠁⠀⠀⠀⠀⠀⠀⠊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢀⠐⢁⠌⠁⠁⠐⠒⠒⠠⠤⠄⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣂⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠂⡤⠄⠀⠠⣦⠄⠀⠀⠀⠀⠀⠀⠀⠁⠒⣂⠴⢛
⠀⠀⠀⠀⠀⠀⣋⠂⠀⠀⠡⠄⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⢏⠊⢠⡭
⠀⠀⠀⠀⠀⠀⢳⠶⠴⢤⢄⣀⣀⡀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠤⠄⠀⢀⣀⣀⢀⠀⠀⠀⣠⠴⠋⠁⢄⣤⢚⢃⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠉⠉⠉⠒⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢩⠃⣠⢴⠞⠣⢣⢂⢢⠇
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠐⠒⠲⠲⠦⠬⣼⢟⢍⡢⠡⠣⢃⢢⢺
⠀⠀⠀⠀⠀⠀⠘⠀⠈⡟⡇⠀⢠⠏⡗⡠⢒⠢⣄⠀⣄⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣏⢈⡶⡱⡡⠡⠣⠱⢹
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢷⢻⠀⣞⡼⡸⣠⠛⠻⡈⣆⠓⢳⡳⡷⢬⡇⡯⣃⣒⢶⡀⠀⣯⠞⡑⡔⡰⡱⡱⠡⢩
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡚⡾⢱⠃⡇⡇⠀⠀⣗⠸⠀⢸⣒⡇⠀⠀⡯⣇⠈⠉⠁⠀⡟⠌⡜⡔⡐⡱⠡⠡⠫
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣧⢱⠇⠀⣇⢧⠀⢠⡂⡽⠀⢸⣊⡇⠀⢠⢗⡭⢼⠆⠀⠀⡏⠌⠌⡔⡐⡡⠡⠡⡃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠚⠀⠀⠘⢮⣛⣉⡞⠁⠀⢸⡢⡇⠀⢸⣵⡇⠀⠀⠀⠀⢎⠚⠘⠘⡴⡱⡱⡳⡇
⠀⠀⠀⠀⠀⠀⠀⢇⠐⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠾⠃⠀⠸⣬⣩⡻⣿⡄⠠⡏⠌⠜⠜⡜⡽⡱⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠠⢏⠎⠎⢌⣬⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠒⠒⠒⠤⠤⢀⣀⣀⣀⠀⠀⠀⠠⠀⣇⣮⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢂⠡⢊⠄⡊⠔⡡⢂⢔⠅⢈⠔⡁⢂⠔⡠⠂⢔⠔⡡⢀⢄⠄⡠⢀⠄⡠⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠊⠀⠠⠒⠕⡡⠀⡡⡪⢂⠕⢀⠔⡁⡠⢊⠔⡡⢂⠌⡔⡡⢊⠔⡡⡪⢊⢔⠕⡡⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁⠊⠐⠁⠊⠐⠑⠁⠈⠀⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣴⣾⣿⣿⣿⣶⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡀⠀⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⢛⡛⣷⡀
⠀⠀⠀⢀⣼⠿⠿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠈⠁⣸⡇
⠀⠀⢀⣿⠁⠶⠶⠆⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠻⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠃⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⠟⠋⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⡾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠘⠻⣦⣀⠀⠀⣀⣀⣀⣤⣶⠶⢶
⠀⣀⣀⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⣠⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⢈
⣾⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢿
⢻⣦⣀⣀⣀⣀⣀⡀⠀⠀⠰⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⣿⠆⠀⠀⣾⣿⣯⣥⣤⡿
⠀⠸⣿⡉⢉⣩⡿⣿⡄⠀⠀⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⠀⢀⣿⠁⠀⠉⠁
⠀⠀⠈⠛⠛⠉⠀⠈⠻⣶⡶⠟⠁⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠻⢶⣄⡈⠻⣦⡀⠀⠀⠀⣠⣴⠿⢷⡄⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠉⠛⢷⣾⠿⣦⣴⠞⠋⠁⠀⠈⢿⣆⢀⣴⡞⣫⣝⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⢻⣦⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣿⣟⡵⣫⡿⠛⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢳⠀⠀⢹⠟⣹⠟⣺⣿⣉⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣾⡿⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢣⡀⠈⠛⠛⢶⡋⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⣡⢮⣾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣵⡟⠁⠀⢀⣤⡾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠉⣉⠉⡉⢉⡉⢉⣉⣉⣉⠀⠀⡀⣀⠀⣀⣀⣀⠈⣻⣷⣶⡿⠛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣯⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣭⣶⣾⣿⣿⣦⣄⠀⠀⢨⡟⠦⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⣽⡿⣿⣿⣿⣿⣿⣿⣧⠀⠰⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡇⢹⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠉⠉⠛⠻⢿⣿⣿⢸⡇⡆⢸⡇⣿⣷⣶⣤⣤⣀⣀⠈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⢹⢸⡧⣇⣸⠇⣿⢿⣿⣿⣿⣿⣿⣷⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠸⡆⢱⢀⠀⠀⠈⠂⡀⢸⡀⣿⣿⣿⠀⣿⠘⠋⢻⠛⠛⢻⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢷⠸⡜⡆⠀⠀⠈⠙⠋⠁⣿⣿⣿⠀⣟⣤⢀⢀⠀⠀⢀⣃⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢷⣸⣤⣧⣷⣄⣀⣀⣒⣒⣂⣿⣿⣿⣠⣽⣿⣨⣾⣷⣾⣩⣯⣾⣴⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⡿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⠃⠀⣿⡿⡇
⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠃⠀⠀⠉
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠁
⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠞
⠀⠀⠀⠀⠀⠀⠑⠶⢤⣄⣀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠋⠉⠉⠛⠓⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠒⠉⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⢳⣦⣀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠈⢧⠉⠛⠲⠤⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠈⣧⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⠉⠁⠀⠀⠈⠉⠙⠓⢦⣀⣰⠃
⠀⠀⣸⠂⠀⠀⠀⠀⣮⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢌⢳⡄
⠀⡴⠁⠀⠀⠀⠀⠀⣧⡈⠓⠦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⢹⡇
⢾⡁⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢸⡗⠦⣄⡀
⠀⠉⠓⠤⣄⣀⡀⢀⡇⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠙⢲
⠀⠀⠀⠀⠀⠈⠉⠙⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⢚⡿⠛⢲⠒⠦⢤⣈⠐⢼⡇⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠀⠀⢀⡴⠋⠀⠀⢸⠀⠀⠀⠈⠓⢾⡇⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠞⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠚⠋⠉⠁⠙⠲⣄⡀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⣤⠴⠶⠿⠧⠤⣤⣀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣿
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠐⠒⠊⠑⠒⠂⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠉⠈⠁⡀⠈⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢧⠀⠀⠀⠀⢀⡄⠀⠀⠀⡌⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠁⠀⠑⠤⢤⡴⠃⠈⣒⠒⠊⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠃⢀⠀⠀⠈⠒⠤⠤⠤⠤⠚⠁⠀⠀⡠⠈⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⡠⠔⠲⠉⠊⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⡘⢗⠢⢄
⠀⣀⠤⠤⠊⢀⣠⡇⠀⠘⡄⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠡⠀⠠⠀⢸⣄⡀⠑⠤⠤⣀
⣎⣀⢤⡄⠀⡹⠀⢉⠃⠀⠈⠢⢄⠀⠉⠒⠒⠀⠤⠤⣤⢤⡤⣤⠤⠤⠐⠒⠒⠉⠀⠔⠁⡐⠁⠘⡅⠀⢏⠀⢠⡤⣀
⠈⠑⠁⠣⡈⠇⠀⡌⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀⠔⠊⠀⠀⠀⠈⠑⠦⣀⠠⢔⠖⠁⠠⠊⠀⠀⠀⢡⠀⠰⢀⠜⠈⠊
⠀⠀⢀⠄⠒⠓⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⢀⠠⠂⠁⡤⠈⠉⠉⠀⡆⠀⠸
⠀⠰⠃⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠄⠀⠈⡀⠀⠂⠁⢀⠀⠀⠀⢀⠇⠀⠀⡇
⠀⠀⢠⠗⠐⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠂⢁⣠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠀⣀⠀⢼⠷⠒⠒⠀⠀⠀⣀⣀⣠⣄⣒⣀⣀⡈⠰⠆⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣾⠃
⠀⠀⢨⡀⠠⠀⠐⠲⡂⠒⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⠸⠗⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⢋⠏
⠀⠀⠈⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⢇⠀⠀⠀⠀⠀⡸⠈⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠉
⠀⠀⠀⠈⠑⠂⠈⠈⠉⠀⠀⠀⠸⢤⠤⠍⠭⡳⠒⠒⠒⠂⠐⠂⠒⠒⢎⠈⠭⠄⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠞⠒⠂⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠐⠒⠳⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣈⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣹
⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠶⠶⣶⡖⠒⠲⠶⠤⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⠴⠒⠒⠒⠓⠲⢤⡉⠳⡀⢄⢤⡀⢀⠦⠠⣍⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⣆⠊⠙⠋⠁⠐⠀⠿⠌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣬⣤⡤⠤⠶⠖⠒⠒⠒⠒⠓⣆⢇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡗⠛⠛⠋⣉⣉⣉⣉⣩⣭⡿⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠀⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢰⠖⢤⢰⠛⢾⡙⠀⠉⡇⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠄⢸⠆⠚⢸⠀⡇⠓⢆⠀⡇⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡀⠀⠘⠤⠃⠒⠛⠀⢷⠀⢸⡇⡿⠀⠀⠀⠀⣀⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢸⠀⣀⣀⣀⣠⣤⡤⠤⠤⠤⣤⣤⣾⢀⡇⣶⣴⡾⠿⣛⡽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⡯⠟⢒⣒⣒⣛⣛⣉⣉⣉⣉⣩⣭⣽⣿⣿⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢸⡇⠉⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⣤⣿⡀⢸⢸⣄⠀⡴⠀⣦⡀⣤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢿⠷⡼⠳⣿⠻⡟⠳⡟⣾⠉⢷⠠⢻⠇⠙⣿⠳⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⡓⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠆
⠀⠀⠀⠀⠠⠐⠐⢃⠀⠀⠀⠀⠀⠀⠀⢤⡟⠐⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠑⠆⡀⠀⠀⠀⠀⠀⠐⢈⣤⡆⠀⢀⡆
⠀⠀⠀⠀⠀⢠⢨⡟⠆⠀⠀⠀⠀⠒⣲⠁⣹⠁⠀⠀⠀⠀⠀⠀⠙⡉⣇⣀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠈⠁⢌⠂⠀⠀⠀⠀⢈⠋⡆⢨⣐⠕
⠀⠀⠀⠀⠐⢲⠁⠘⠁⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⢦⣀⣋⠑⠑⠀⠀⠀⠠⠦⠕⢺⠀⣠⠀⠀⠚⠃⠀⠀⠀⠘⢾⠀⢠
⠀⠀⠀⠀⠀⢘⠐⠚⠀⠀⠀⠀⠀⠀⢀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠘⢯⠝⠲⠄⠀⠀⠀⠀⠀⠀⠈⠇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣐⠂⠷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⡌⠑⡈⢄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⠀⠈⠫⡂⠀⠀⠀⡰⢁⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠰⠤⠀⠀⠀⠀⠀⠀⠐⠞⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠅⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠀⠀⠀⠀⠀⠀⢸⡁⡍⠀⠀⠀⠀⠀⠀⡎⢻⠟⠀⠀⠀⠀⠀⠀⠀⠞⢻
⠀⠀⢀⡀⠀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣧⠀⠀⠀⠀⠀⠚⢻⠓⠄⠀⠀⠀⠀⠀⢉⠿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠙⠄
⠀⠀⠀⠻⡂⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠂⠟⠃⠀⠀⠀⠀⠐⡆⣛⢤⠂⠀⠀⠀⠀⠐⡿⣠⢧⠀⠀⠀⠀⢉⣅⠘⠇⠐⠤
⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠉⢂⠀⠀⠀⠀⠀⢠⡃⠘⢑⠂⠀⠀⠀⠀⠰⡅⡀⡀⠆⠀⠀⠀⠀⠁⢚⠀⡰
⠀⠀⠀⠀⠀⣹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠐⠇⢰⠈⠀⠀⠀⠀⠀⠀⡗⢀⠂⠀⠀⠀⠀⠀⠀⠈⠘⢳⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡌⢂⠀⠀⠀⠀⠀⠀⠸⡀⠘
⠀⠀⠀⠀⠘⠀⢆⠐⡀⠀⠀⠯⡤⠮⠤⡴⠀⠀⠀⠉⠈⠫⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠙⡌⢆⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⡀⠐⠀⠣⣐⠄⠀⠠⣶⠔⢆⠧⠄⠀⠀⠀⣆⠀⢡⡄⠀⠀⠀⠀⡄⢸⠀⡀⠀⠀⠀⠀⠈⡠⠀⠈⡼⠂⠀⠀⠀⣠⢀⠂⠠⠀⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠉⠁⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⠞⡟⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠎⡘⠀⠀⠀⠀⠀⢆⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠛⢌⡉⢰⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣀⣤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠶⠚⠛⠋⠉⠉⠉⠙⠛⠛⠲⠶⠶⢦⣽⣿⣉⣀⣀
⠀⢀⣀⣤⡴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠠⣾⣋⣉⠓⠒⠦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠶⠶⢦⣤⣄⡀
⡼⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣄⣈⣉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠉⠉⠀⠀⢀⣀⣀⠤⠤⠔⠒⠚⠛⠉⠁⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠸⢦⣴⠦⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠦⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠒⠋⠉⠉⠉⠀⠀⠈⣿⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠤⣤⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⢲⠲⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠈⢃⠘⠃⡹⡶⢋⡑⢾⠉⠒⠦⣄⣠⣄⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠈⢧⠞⠀⢱⣈⣋⡸⡇⢠⠖⢾⠃⠠⡼⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠸⣄⣉⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠙⠲⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⣀⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠉⠉⠉⠁⠀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠉⠉⠉⠁⠀⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠘⡉⠂⠤⠀⠀⢀⣀⣠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠰⡀⠀⢠⠔⠉⠀⠀⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠂⠒⠀⠂⠀⠀⠀⠀⠋⠒⠂⠤⣀
⠀⡰⠁⠀⠧⣀⠀⠀⠀⡁⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢅⡆
⡔⠀⠀⠀⡗⠄⠉⠐⠒⠦⠤⠤⠱⠤⠤⠤⠤⠀⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢄⣃
⠈⠑⠤⢀⡗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡅⠈⠐⠤
⠀⠀⠀⠀⡟⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠠⡤⠄⠒⠒⠒⠒⠒⠀⠤⠤⢈⡐⢤⡇⠀⠀⡰
⠀⠀⠀⠀⢏⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠄⠒⠀⠀⠉⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡇⠀⡜
⠀⠀⠀⠀⠀⠈⠐⠀⠤⠄⢀⣀⣀⣀⣀⠀⠀⠤⠤⠀⠶⡒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡤⠊⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠑⠢⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠸⢿⣿⣿⣿⣿⠵⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠉⠉⠛⠒⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣦⠀⠀⣿⢳⡀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠷⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢸⡇⠀⡟⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⡀⠀⠀⠀⠀⠀⠙⠦⠴⠟⠀⠀⣿⠀⠀⠙⣆⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣀⡴⠞⢧⣀⡤⣾⠋⠁⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠈⠉⠀⠘⣆⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣀⠀⠀⠀⢿⣷⣦⣄⠀⠀⠀⠀⠘⣆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠳⢦⣀⠸⣿⣿⣿⣿⡶⣤⡀⠀⠘⣆⠀⣸⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠉⠓⢿⠛⠉⠁⠀⠀⠉⠳⢦⣹⣦⡏⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣠⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠺⣉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣑⠦⣄⣀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠈⠹⣶⠭⣝⣒⣶⠶⠃
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣤⠈⠑⢲⢴⠃
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣟⢏⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠲
⠀⠀⠀⠀⠀⡠⣾⠦⣄⡀⠘⠶⠄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⢏⡇
⠀⠀⠀⠀⣼⣿⠇⣷⠀⠹⠽⠛⢿⢿⠷⣤⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢸⣿⠀⠀⠀⠀⢸⣾
⠀⠀⠀⢸⣿⣯⣀⣿⡀⠀⣀⡴⡫⠋⠀⠀⠉⠙⠓⠒⢶⠛⠉⠢⠄⠀⠈⡹⠁⠘⣿⠀⣀⣀⣀⣸⣿
⠀⠀⣴⠏⠀⠀⠀⡈⣿⣎⠵⡻⠃⠫⠁⠀⣀⠰⢢⡀⠀⠈⠉⠉⠉⠐⠊⠀⠀⣰⠋⠉⠛⠓⠒⢺⣿⠒⢻⣿⠛⠛⡟⠛⠛⠛⠛⠛⠛
⠀⢸⢻⡀⠀⠀⢠⡼⠉⣹⠻⡇⣒⡀⠉⠅⠀⠠⢫⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⠇⠀⢸⣿⠀⢠⠇
⠀⢸⡀⢿⣷⠒⠋⠀⢀⡏⠀⡵⠐⠀⠉⠁⠀⢲⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣟⣀⣤⡶⠟⠁⠀⠀⠀⣿⠀⢸
⠀⠀⠑⠦⣤⣤⣤⠤⣾⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣥⣸⣴⠏
⣀⣀⣠⣾⣉⣉⣹⣆⣰⣟⣉⣉⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣽⣏⣙⣡⣏⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⣀⡠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⠻⣶⣤⣀⣀⣠⠔⠀⠁⠀⠐⠒⠛⠋⣩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⣤⣀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣯⢻⣧⣤⣴⠶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠉⠓⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠿⠶⢶⣤⣀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣄⣀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄⠀⠀⣴⠋⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠉⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠞⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⢀⣠⣶⠟⣩⠞
⠀⠀⠀⠀⠀⠐⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠁⠀⠀⢀⣠⠴⠚⣹⡿⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢝⡛⠛⠛⠒⠛⠛⠉⠉⠀⠀⢀⣠⡤⠖⠛⠉⠀⠀⣰⠟⣲⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠻⢆⣀⣀⣀⣤⠴⠶⠒⠋⠉⠀⠀⠀⠀⠀⢀⣴⠏⠀⡿
⠀⠀⠀⠀⠀⢀⠔⣡⠄⠀⢘⣿⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠛⠻⠿⠧
⠀⠀⠠⠀⠀⢡⡴⠋⠀⣴⠟⠁⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⡠⠊
⠀⠀⣀⡠⠖⠋⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⠔⠊
⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢼⣿⣿⡍⠉⠉⠙⠛⠓⠒⠒⠶⢦⡀⠀⠀⣾⣿⣷⡀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⢰⣶⣶⣶⣤⣭⣭⣭⣭⣙⢿⣿⣿⠏⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠈⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⡄⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡇⠀⢠⣶⣶⣤⣶⣶⡄⠀⡿⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⡿⠋⠉⠻⠋⠉⠻⣿⡇⣿⣿⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣼⠁⠀⠘⢿⣿⣿⣿⡿⠃⢠⡇⠀⠀⢸⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⣼⣿⣷⡀⠀⠀⠀⠀⣸⣿⢡⣿⣿⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⠿⠋⠀⠀⢸⠃⠀⠀⢸⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣦⣀⣴⣾⣿⡟⢸⣿⣿⡇⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣇⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣀⣠⣤⣤⣤⣾⣷⣦⠀⢺⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡎⣭⣥⣭⡿⠿⠿⠟⠛⠛⠛⠈⠙⣿⣇⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣄⠀⠈⣿⣿⣿⣿⡿⠿⠟⠛⠋⠀⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡸⣿⣿⠆⠀⠀⠀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠤⠒⠊⠉⢡⠋⠁⠀⠀⠉⠱⡄⢀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠠⠈⠁⠀⢀⡀⠀⠈⠧⣄⣀⠲⣶⡴⠁⠈⣷⡤⠚⠁⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⢐⡐⢀⣁⣤⣤⣤⣤⣤⣄⣄⣀⣀⣈⠑⢦⣼⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣲⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢹⣇⠀⠀⡠⠟⠻⠮⠉⠉⠙⠛⢿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⠃⠈⢷⢤⠼⠷⢿⠂⠀⣠⣆⡘⢲⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠤⡔⠢⠀⠀⠀⠀⠀⠘⡆⠹⣦⠀⢃⠀⠀⠁⠏⢘⠟⠻⣿⣚⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠠⠁⠘⠀⠀⠀⢄⠀⠀⠀⠀⠳⣾⡉⠀⢀⡠⠰⢠⠀⢸⡀⠀⠁⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⣿⠀⠀⠈⠤⣄⡀⠙⠋⢠⢁⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠆⠀⠀⡿⡆⠆⠀⠀⠀⠀⣈⡓⢡⣯⡟⠉
⠀⢀⠊⠰⠆⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⡠⡇⠈⠂⠀⠀⠀⠀⠀⢀⣿⣿⣷
⢀⠔⢷⡄⣡⠀⠀⠀⠀⠀⠀⢀⡾⠱⢴⡖⢫⢃⣵⣄⣄⣲⣏⣲⣶⣾⣿⡿⢿⣿⡄
⠉⢳⡀⠙⡅⢀⠀⠀⠀⠀⢀⣾⠝⡿⠋⢠⣿⣿⣿⣿⠉⠀⠀⢀⠤⠒⣁⡴⠞⠿⣧
⠱⡄⠙⢄⣹⠀⠀⠀⠀⢀⣾⢇⠜⠀⣰⣯⣿⡟⠀⠰⢀⡠⠂⠁⣰⠞⢁⡤⣴⣶⣼⣧⡀
⣧⣜⣦⡬⠋⠀⠀⠀⢀⣾⡁⠀⠀⢰⣿⣿⣿⢃⡤⠚⠁⢀⠠⠊⠁⣠⠴⢊⣵⠋⠉⠙⠻⣷⣤⣄
⣎⠫⡙⢄⠀⠀⠀⠀⡾⢻⡆⠀⢀⣿⣿⠟⠋⠁⠀⠀⠞⠁⣠⣷⡽⠁⢠⡾⠁⠀⠀⠀⠀⠈⢿⣿⣷⣄
⣻⢷⣜⢦⠢⡀⠀⣼⠗⠉⣇⠤⠊⠉⠀⠀⠀⠀⠀⠘⠟⣻⣏⡞⠦⣤⡿⠀⠀⠀⠀⢀⠔⠊⠁⠘⣿⣿⣧
⣶⣿⣿⣦⡱⡔⢺⡇⢀⣔⡁⠠⠴⠒⠒⠀⠀⠀⠀⠀⠀⢠⣏⣽⠀⠀⠀⠀⣠⢀⠔⠁⡀⠀⠀⠀⠈⢿⣿⡇
⣿⣿⣿⣿⡝⢮⣫⠗⠋⠁⢀⣀⣀⣤⡤⠴⠶⠒⠛⡃⠂⣼⠁⠀⠀⠀⢀⣴⣯⠋⠀⠀⢠⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⡟⠁⣰⣵⡶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⢰⠀⢰⣿⠀⠀⠀⢨⣿⣿⠇⠀⠀⠀⠀⢣⠀⢀⡀⠀⢸⣿
⣿⣿⡿⠀⣸⠋⠥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⠞⠃⠀⠀⠀⢼⣿⣿⠀⠀⡄⠀⠀⠀⠱⣄⠹⣶⣾⣿
⣿⣿⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⡵⠀⡔⠀⣠⣿⡀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠁⠀⠀⠀⡀⠈⠳⣌⢻⣿⣇
⣿⣿⡇⡾⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠸⠋⣻⡿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢿⣆⣄⠘⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡦⡄⠀⠀⠀⡟⠀⡠⣞⢁⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⠅⢧⠈⣆⠀⠀⣧⣞⠀⣹⠟⢚⡇⠀⠔⠲⣔⠒⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠘⣏⠢⣄⡏⠑⣾⢄⡸⣼⠉⠓⠊⢩⢿⣴⡥⢒⠏⠀⠀⠀⠀⠘⡆⠀⢱⡀⣸⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣤⢄⢸⢋⢢⠀⡼⣏⡙⠛⢦⣇⠀⠙⢿⡄⣠⢠⠞⠉⠀⢿⡁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠖⠁⠀⣼
⠀⠀⠀⠀⠀⠀⡠⠬⠨⠼⡆⢆⣿⡀⣓⠙⢮⡟⠁⠀⠐⢦⢛⡙⢧⠔⠀⠀⠀⡷⠀⠀⠀⢀⡴⠃⠁⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⠞⠊⠀⣿⠉⠓⢠⠑⣒⠶⠋⠩⢳⡔⢯⠙⠛⢮⣋⠵⢄⡤⠒⠁⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠚⠥⣀⢀⠾⠓⠦⣴⠷⡦⠴⠾⢴⣥⠴⠚⠓⠚⠓⠢⢄⣀⣠⣿⡕⣢⡦⠖⠚⠉⠉⠙⠒⢶⡋⠀⣀⠀⣀⠤⠤⡀
⠀⠀⠀⠀⠀⣀⢠⡤⠭⣦⠀⣠⠟⠙⢣⡠⢴⠞⠁⠀⠀⠁⠂⠀⠀⠀⠉⠢⣀⡴⠋⠀⠀⢁⠀⠀⠀⠀⠀⠱⡞⢸⠞⠁⢀⡀⢹
⠀⠀⢀⠔⠛⣤⡤⢬⢖⡾⠋⢹⣤⢤⣎⠉⠉⠉⠉⠉⠑⠒⠦⢤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡮⡶⠀⡏⠃⡜
⠀⠀⠺⠮⢩⣋⠴⢧⣮⣤⢶⠃⡎⡸⣏⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣰⣉⡩⠿⣸⢰⢡⡟⠋⢸⠇⠀⠀⠀⠈⠀⢰⠏⢛⠇⢀⢴⢄⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠚⢉⡼⠁⠛⡦⠒⢲⠆
⠀⠀⠀⠀⠀⠀⢋⠜⣇⠞⢹⣟⡥⠾⠷⢾⣇⣀⣤⡤⠤⢤⣌⣀⡘⠀⢧⣧⠞⢠⡟⠶⡖⢀⠀⠀⠀⠀⠀⣰⡟⣝⡷⠚⢻⠔⠋⣑⠆
⠀⠀⠀⠀⠀⠀⠛⠈⠰⣎⡉⠉⣳⠀⠀⣉⡷⡞⣷⣄⠀⠀⠀⠀⠉⠒⠢⢄⣀⠈⠀⠘⠀⠀⠀⠀⣀⣠⠞⢙⡟⠟⠓⡦⠼⣋⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⢃⣽⠖⢚⡏⢀⠀⢸⠳⠽⠢⢄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠒⣒⡶⠟⣿⡃⠀⠈⢻⢓⡟⢋⡷⠢⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣅⣿⠥⠞⢺⠁⠹⢩⠓⠋⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⣠⡖⠋⠝⢑⡟⠃⠉⠻⡞⠻⢽⣯⡉⣻⣭⠽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢢⣎⣸⣥⢤⡀⢸⣇⢀⡴⢶⣛⠒⠲⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⠤⣯⣠⡎⢳⡠⡻⢛⠹⡉⠀⢸⡌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣋⡷⠚⢱⠁⣠⠀⠉⠲⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⠦


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⢺⣯⣍⣭⣭⣍⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⢶⡀⠀⠀⠀⠀⢀⣴⠞⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠿⢸⡇⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡏⠀⠷⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠳⠐⠋
⠀⠀⠀⠀⠀⣴⢶⢳⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠘⢦⡼⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠥⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠚⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠛⢦⣀⠠⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⣀⢀⡀⠐⣒⠒⣛⠀⠀⢀⡴⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠳⣄⡀⠀⠀⢀⡀⣀⡀⠀⠀⠈⢯⡽⠃⠀⢫⣹⢏⣠⠞⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⢦⣄⠈⠻⡽⠁⠠⢄⡀⠀⠀⢠⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⢀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠛⠉⠁⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠸⢿⣥⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⠟⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡋⢉⣉⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠳⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⡀⠈⠁⡶⠆⠄⡀
⠀⠀⡠⠄⠤⢤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⡦⠂⢠⠀⠀⠀⠹⡆
⠀⠀⠀⠘⠻⡉⠀⡀⠉⡍⠈⠻⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠛⡿⠁⢠⠄⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠓⠀⠀⠒⠒⠒⠒⠇⠀⠀⠀⠀⠀⠀⠠⣄⠀⡦⠀⣠⠀⢠⠀⣶⡖⢀⡀⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡟⠭⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣥⡀⡄⠀⡀⠈⠁⠘⡿⠑⠻⣻⡇
⠀⠀⡆⠀⠀⢠⠀⠒⠒⠒⠂⠀⡄⠀⠀⠀⠀⠀⡰⠶⢾⣇⠑⠤⢲⠀⠉⠙⠊⠒⢤⡿⠛⠒⠘⠶⠄
⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⣀⠐⠣⡀⣐⣲⡇⡹⠂⠚⠊⢀⠁⢲⣼⠟⠁⠀⣀⣤⠚⠣⡀
⠀⠀⠑⢲⣶⣾⣶⣶⡖⠐⣎⡀⠠⠝⠀⣄⣆⣠⡾⠋⠁⢈⣿⣦⣀⡀⠀⣠⡟⠙⠲⠒⠚⠉⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠈⠑⠖⠋⠀⠀⠁⠀⢠⠀⢸⡇⠀⠹⣿⣆⠈⢿⣦⣀⠀⠄⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡟⠀⠀⠀⠂⠻⣆⣈⣧⣍⠀⠈⠂⠀⣠⠞⠁
⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⣠⠗⡀⠘⠙⢻⣿⡿⠥⠚⠉
⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠁⠀⢂⠈⢿⣇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠚⠀⠀⠀⠂⠘⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠇⠀⠀⢸⠀⠀⠈⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠱⡀⠀⢸⠀⠀⣼⢠⠀⡇⠐⢀⡎⠀⠀⡠⠂⠀⠀⠀⠠
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⡬⠶⠾⠶⠳⠶⠶⣧⣆⣿⠀⢸⣡⡧⠾⠶⠷⠶⢥⣄⣀⡠⠊
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⡀⠄
⠙⢦⡀⠀⠐⢄⠑⠤⣄⣀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠈⠢⣅⠒⠠⠥⠦⠀⠈⠉⣡⣶⢀⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣧⠤
⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⢰⡏⣩⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⢿⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⡇⢹⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠋⠁⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⡆⠈⠉⠛⠛⠶⠦⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⢶⣿⡉⣀⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⣲⠆⢠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠄⠀⠀⠂⠈⠉⠁⠀⢀⣴⡿⠶⢴⡿⠿⠖⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠐⢡⠜⢙⢿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠡⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠔⣡⠎⡐⢹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠑⢄⠁⠀⠒⠂⠐⠂⠒⠠⡀⠙⡆⠨⢶⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡴⠁⠀⢠⠃⠌⢹⣆⠀⠀⠀⠀⢀⣠⢾⠋⠀⠱⡀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠈⢄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⠇⠀⠀⡆⢹⠀⢀⣤⠖⡏⠁⠈⢇⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⠁⠘⠁⡇⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠖⠒⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆⠈⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣰⡀⠹⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢼⠀⢆⠀⢰⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢣⠘⠀⡇
⠀⠀⠀⠀⠀⢀⠁⡀⢀⠔⢀⠠⡄⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠈⠒⠂⠛⢠⠉⢆⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⡦⠀⠈⠀⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠰⠆⠀⠀⠁⠀⠠⡀⠀⢀⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡤⠀⠀⢠⡄⠀⠸⠆⠐⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⡀⠚⣧⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠊⠀⢰⡀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⡤⠄⠲⢧⡀⠀⠀⠀⢀⡙⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⡠⠤⠺⠛⠭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣃⣝⠀⡀⠀⠀⠠⡀⠀⠀⠓⠊⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠁⠀⠰⠀⠀⠐⣄⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⠜⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢄⢰⡉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠰⠠⣀⠀⠀⠀⠀⠀⠀⡀⡔⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⠈⠀⠀⠐⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡄⠰⠦⠄⠀⠀⠀⢠⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢠⡀⢄⡒
⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⢤⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡐⠲⠰⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠈⠂⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⢀⠎⠀⠀⠀⠟⢠⠂⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢒⣒⡲⢤⠄⠀⢀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠏⠀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢇⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢐⠄⣸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣸⠀⠀⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⢀⠃⠡⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠄⠀⠀⠡⡀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣟⠳⢄⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠌⠀⠀⠀⠀⠱⡀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⠀⠀⠙⠢⣄⢸⠀⠀⢀
⠀⠀⣀⠠⠤⢄⡀⠀⢀⡴⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⣎⠉⠀⠀⠈
⠀⠘⢃⠺⢦⣄⠙⠐⠧⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠈⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢸⠀⠀⠀⠀⡿⠀⠀⢀⠆
⠀⠀⠀⠑⠀⠑⠳⣄⠀⠀⠀⠑⢆⠈⠀⠀⠀⠀⠀⡠⠔⠚⠀⠀⠀⢠⣢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠀⠃⠉⡆⠀⠀⡼⠁⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠐⠀⡉⣂⡀⣠⠋⠀⠀⠀⠀⠀⠸⣠⣤⣂⣀⣡⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠑⠲⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠙⡆
⠀⢀⡤⢄⣾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠴⠊⠁
⠀⣏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠖⠒⠒⠒⠛⠳⢤⡀
⠀⠘⠦⠤⠤⣴⠇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⣠⡤⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⡷
⠀⠀⠀⠀⠀⣀⣉⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃
⠻⣏⠉⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸
⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢦⣀⣀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠚⠉⢧
⠀⠀⠀⢸⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠊⠀⠀⠀⢸⡀
⠀⢀⣠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡇
⢰⠋⠁⠈⠁⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠸⣆⠁⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠛⠓⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣰⡥⠤⠖⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡴⠶⠶⠛⠳⠶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡰⠀⡐⠁⠖⠀⠀⢀⣾⡶⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡘⠁⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⠀⠀⣠⡿⠿⠟⠁⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⡱⠶⠒⠛⠁⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠃⠐⠛⠛⠻⡏⠙⢧⠀⠀⠀⣠⠏⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣾⣧⡤⠞⠁⠀⠀⢀⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣷⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⢠⣿⣿⣷⡶⠞⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⢀⣿⣿⣿⣿⣟⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣶⣶⡶⠶⠶⠶⢶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡋⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣠⠴⠚⠉⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠐⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⣶⣭⣥⣤⣤⣤⣤⣤⣤⡀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠙⠻⢷⣦⣄⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣿⡏⠙⢿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣿⡇⠀⠀⠈⠻⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢶⣶⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢤⠀⡠⢠⠆⢦⡀⢶⠆⢰⠆⠀⠀⡆⢤⢀⡴⠰⣄⢤⣠⠄⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣤⠁⢣⣀⣠⠃⢸⠀⢸⣁⠀⠀⡇⣹⠘⣄⢀⠟⣠⠻⡀⠀⠀⣀⣀⣤⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⡠⠄⠐⠈⠉⠁⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠆⠀⠀⡐⢂⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⢄⠐⠒⠒⠠⠤⡄⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠣⡀⠀⠀⢀⠈⠐
⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠬⠵⠶⡶⠾⢷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠀⠀⠁⡀⠂⠀⠀⠠⣄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠂⠀⡀⠃⠀⠐⢁⠄⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠉⠀⠠⣤⠄⣀⠁⡌
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⣠⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠐⠀⠑⠂⠐⣀⠀⠈⠑⢄⠈
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠈⢳⠒⠤⠀⠐⡀⠀⠀⠀⠀⡄⠈⠢⣀⠀⠀⢀⠀⠂⠁⠘⢀⡀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠉⠉⠈⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⠀⠠⠂⠀⠘⢦⠀⠀⢠⣁⠀⠀⠈⠑⠀⠀⠢⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⢤⠄⠐⠀⠁⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠘⡀⠢⢄⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠠⢄⡘⡄⠀⠀⠁⠀⠈⠢⡀⠀⠂⠄⠑⣄⠀⠀⠄⠀⠀⠑⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡆⠀⠈⠉⠙⠒⠒⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠂⡀⠸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠑⠴⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠤⠄⢤⠀⠀⠐⠀⠀⠀⠀⠉⢤⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠨⠁⠒⠤⡀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⠂⠤⠄⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠒⡥⠴⢀⣀⣀⡀⠀⣀⡀⠤⢄⡀⢀⣀⣀⣀⠑⢀⠄⠀⡄⠀⠀⠀⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠢⠀⠀⠀⠈⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠈⠲⣄⠀⠀⠉⠁⠒⠂
⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠤⠄⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠠⠄⠠⠈⠴
⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠀⠆⠁⠀⠂⠀⠤⠜⠀⠁⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢀⢡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⡄
⠀⠐⢄⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⠰⡀
⠀⠀⠀⠑⠢⢤⣀⠠⠘⠀⠄⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀
⣤⣤⣤⣤⣬⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿