Fasten Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣀⠀⢀⢀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠦⣀⡠⠔⠀⠀⡇⠀⢀⠔⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⡠⠊⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⢀⡠⢄⡀⠀⡀⢀⠞⠁⢀⠴⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⠔⡢⢐⠭⢒⡡⠔⣉⡴⠃⠐⡠⢎⠂⠔⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡪⠐⠥⢊⡡⢒⡡⠐⣩⠊⠠⢀⡞⠣⡀⠰⠄⡂⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠡⠒⡃⢈⠥⣂⠥⣂⠝⠁⢀⣴⠋⠐⢩⡄⠀⡧⢐⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢄⠨⣴⡇⢐⠫⢒⡨⠁⠐⣠⢎⠒⠀⡊⠥⢐⡀⡿⣖⠥⢀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⡨⠔⣸⢻⡇⢐⣪⠉⠠⢀⠞⠃⠄⣂⠡⠀⡂⠥⠀⡿⢙⢮⡐⠢⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⣠⢬⡙⢴⠇⢸⡇⠞⠁⠐⣰⠫⠀⡂⠍⢐⠢⠅⣒⠬⠀⡿⠐⠀⠉⠊⠐⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡈⢀⢤⡀⡐⢁⠵⠋⠈⡶⡮⠏⡐⢀⣼⣓⡢⠁⣒⠨⠐⣂⠭⢒⡠⠀⡯⠉⠂⠀⠀⢹⠅⠀⢠
⠀⠀⠀⠀⠀⠀⢇⠘⢄⠙⢠⠋⠀⠀⡀⡾⡇⣓⢱⠟⢰⠈⠉⡉⢛⠚⡛⠒⠚⠓⠒⠒⠳⠗⠂⠒⠂⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠳⠃⠐⣀⡄⣣⠡⠟⠬⠈⠗⢲⠤⢤⡅⣘⣀⡉⣀⣀⣀⣁⡉⢈⣈⣈⣀⠤⡸⠀⠀⠘
⠀⠀⠀⠀⢠⠀⠀⠀⢨⠉⠉⡟⠫⠀⢘⡵⠁⠀⣀⠬⠐⡈⠅⢂⡨⠔⣊⠥⢒⡩⢕⡀⠘⠷⣒⡁⢜⡠⡆⠀⢀
⠀⠀⠀⡋⠘⡀⢠⣎⡁⡶⡅⢀⠀⢠⠊⠀⢂⡩⠐⣊⠡⠒⡈⠅⠊⠩⠐⠩⠔⣊⠥⢒⡠⠀⠈⢷⡀⠠⠁⣠⠂
⠀⠀⠀⠑⠁⠉⠀⠈⠀⠀⠀⠀⠀⠇⠀⣊⠤⢒⠩⢐⠈⣠⡶⢟⣛⡛⠳⠶⣄⠐⢉⠄⣂⠩⠄⠀⡇⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠔⢊⠡⢂⠡⠠⠏⠁⠀⠀⠀⠒⠉⠭⢷⠀⠭⢔⣊⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢐⠨⠐⣈⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⢐⡊⠥⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢔⡉⢐⠨⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⣊⠡⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⢊⠥⠐⣈⠄⢁⠀⠀⠀⠀⠀⠀⢰⢀⡨⠄⣒⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠄⣐⠩⠀⡂⠈⡆⠀⠀⠀⠀⠀⡠⠐⣂⠭⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠨⢀⡠⠑⠀⢠⠇⠀⠀⠀⠀⢬⠁⠩⣒⠪⠅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⠉⠀⣠⠯⠀⠀⠀⠀⠠⣽⠀⠀⠀⠈⠀⢨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠡⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡠⠴⠞⠓⠂⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⠙⢿⣷⣄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣤⡄⠀⠙⣿⣷⠀⠀⢸⣿⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣴⣿⠟⠁⢀⣴⣿⠟⠁⢀⣴⣿⠟⠀⠀⠈⠉⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠁⢀⣴⣿⠟⠁⢀⣴⣿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠿⠟⠁⢀⣴⣿⠟⠁⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠸⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢀⣴⣿⣟⠁⠀⠀⠀⠀⠀⢙⣿⣷⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠙⢿⣷⣄⠀⠀⢀⣴⣿⠟⠁⢀⣴⣿⠿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣴⣿⠟⠁⢀⣴⣿⠟⠁⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⠀⠀⠀⠀⠙⠛⠁⢀⣴⣿⠟⠁⠀⠀⢀⣴⣶⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⢀⣴⣿⠟⠁⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⣀⡀⠀⢸⣿⡃⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⣿⡇⠀⠀⠻⢿⣷⣤⣘⠛⠁⠀⠀⠀⢀⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠒⠒⠀⠀⠂⠒⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠞⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⢀⡤⠃⠀⠀⡄⠀⠀⠀⡴⠁⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠎⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠁⠀⢀⡐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⣊⠁⢀⠤⢚⡡⠒⢀⡠⠖⣠⠎⠀⠀⡠⠊⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⠤⢚⠡⢊⡡⠒⢉⠠⠚⢉⢠⡞⠁⠄⢀⠼⢁⠢⠔⢃⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡅⠊⡅⠊⢁⠴⢊⡠⠔⢊⡠⠃⠀⠌⣰⠟⠈⠀⠰⡎⠐⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠊⡁⢼⠂⠐⢊⡡⠒⡉⠤⢒⠋⢀⠄⢀⠞⡁⠄⢂⠁⠀⢳⢂⠤⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠐⠁⠊⣼⠃⢈⡥⠊⣁⢰⠞⠁⡀⢀⡴⢋⡁⢤⠑⠊⢁⠀⢸⡴⢈⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡐⠅⢊⣹⣽⠀⠠⢂⣡⡤⠃⠀⠀⣡⡛⠭⢒⡈⠄⠒⣈⠠⠄⢸⠻⡔⢂⠥⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡁⢀⠄⠊⣬⡃⠼⠀⢘⣤⠏⠀⠀⡁⠞⡃⠄⠒⣀⠠⠀⢁⡀⠄⠀⢸⠓⣙⢧⡐⠃⠤⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⢀⣤⡁⢮⣴⠎⠠⡇⠀⡖⠁⠀⠊⣰⠋⠀⠐⠈⠡⠐⠂⠩⠀⠂⠉⠀⢨⡥⠀⠀⠈⠢⠘⠢⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⠀⠀⢠⠋⢂⡽⠂⠹⣄⢄⣳⠎⢠⠂⣀⣞⠡⠄⣐⡀⠤⠐⣀⠡⠐⢂⠡⠄⠀⢸⠖⢒⡀⠀⠀⠐⢦⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠉⠳⣤⠃⡴⠋⠀⠀⠘⢫⡎⠇⣰⠃⣤⠛⠶⠞⢡⣤⡴⢌⣡⣤⠂⣉⣄⢒⣈⡀⢸⣁⣂⠀⠀⢀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣄⠈⠳⣄⠀⡜⠀⠀⡀⠀⠄⠛⣹⠘⣇⠜⠿⠀⡏⢰⠀⡆⢀⠀⢠⡌⡍⡉⢉⡉⡉⡉⠉⠉⡉⠉⠉⠀⠀⢠⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠉⠒⠀⠀⢊⡤⠀⣆⠐⠟⠳⠆⠀⠚⠃⡴⠆⢠⣅⣈⣀⣈⣁⣁⣀⠈⣀⣁⣉⠈⣁⣁⣀⣀⡄⢀⡸⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠉⠉⠉⡛⢩⡥⠄⠙⣢⠂⠀⠀⠒⠈⠥⠐⠈⠡⠐⠂⡉⠄⠒⡉⠤⠒⠉⠡⠀⠀⠻⣭⠍⡉⢁⠊⢀⡴⠀⠀⠸
⠀⠀⠀⢀⠤⢀⡇⠈⢀⡴⠄⣅⣠⠀⠃⠀⠀⢀⠔⠁⢀⠤⠒⢀⠠⠄⢂⡁⠤⠐⣈⠤⠒⣁⠤⠀⣀⠤⠐⣀⠄⠀⠈⠻⣮⣈⠙⠁⡆⠀⣠⠆
⠀⠀⠀⠸⣄⡀⢣⡄⠸⠣⠐⠈⠉⠂⠀⠀⢠⠃⠀⢀⠄⠒⠉⡤⠐⠈⠁⠐⢊⣡⣤⣤⣍⣀⡀⠁⠰⠊⣉⠤⠒⡉⠤⠀⠈⣯⢀⡊⢀⡴⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⡡⠔⢊⡡⠔⣂⠁⢀⡼⢏⣉⡙⠛⠓⠚⠛⠳⣄⠈⠤⢒⡈⠥⠒⡂⠀⢸⡊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠴⠈⣁⠔⢊⡠⠀⠸⠁⠀⠀⠀⠀⠀⠙⠉⠓⠺⢧⠀⣁⠴⠂⢉⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠨⠔⠊⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠔⢂⠩⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⢁⠠⠐⣀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡂⠤⠐⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠡⠐⢩⠔⠊⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠤⠒⠈⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⣉⠠⠐⢋⡀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠤⠐⣂⡁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠈⣁⠴⠊⠁⢸⡄⠀⠀⠀⠀⠀⠀⠰⠀⠴⠂⣉⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⠡⠐⠂⠅⠀⢨⡃⠀⠀⠀⠀⠀⠐⡆⠐⣊⠥⠄⢒⠀⢸⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠐⠆⠀⠀⢠⠦⠄⠀⠀⠀⠀⢀⢸⠁⠀⠀⠒⣁⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣰⢓⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠉⠀⡸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣤⣛⠡⠄⠀⠀⠀⠀⠀⠀⠀⢺⠆⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⣤⣴⠏⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⢀⠼⢦⣷⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡴⠣⠤⣜⡟⣗⣄
⠀⠀⠀⠀⢶⢱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⢧⣵⣾⣗⠈⠀⠈⠁
⠀⠀⠀⠀⠈⢣⠉⠻⡢⣄⠀⠀⠀⠀⢀⣴⣲⠆⠀⠀⠀⠀⠀⢀⣴⠟⢁⠾⠥⣤⣻⢿⢧⡀
⠀⠀⠀⠀⠀⠀⢣⡀⠈⠪⡳⢄⢀⠔⣿⡞⠁⠀⠀⠀⠀⢀⠴⠟⢁⣴⡥⣤⣴⡾⠷⠾⢟⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠂⠑⠮⠿⣤⣴⠶⠒⢛⡿⠋⠀⡠⠀⠈⠀⠀⠀⠀⠀⢶⡿⠋⡜⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠠⣷⣶⣶⣾⠟⠛⠛⠉⠁⠀⣰⡏⢀⡴⡎⡠⡀⡀⠀⠀⠀⠀⠀⣀⣀⠤⣁⡤⠊
⠀⠀⠀⠀⠀⠀⢀⡴⣡⡾⠞⠉⠛⠳⠦⢶⣤⡾⠋⣠⠚⢛⣾⣟⠛⢃⣺⣷⡴⠤⠿⠛⠚⠉⠁
⠀⠀⠀⠀⠀⡠⢯⡼⠋⠀⠀⠀⠀⠀⣠⠾⠋⣴⢾⣻⢷⠯⠟⠻⡵⠛⠉
⠀⠀⠀⠀⠾⠕⠋⠀⠀⠀⠀⠀⣠⠾⠇⣠⠞⠉⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠞⠉⠉⠛⠞⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠔⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⠔⠁
⠀⠀⠀⠀⠀⢰⣯⠭⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣉⣙⣛⣉⣁⣀⣀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⣿⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⡀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠴⣋⣅⣽⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣏⠁⠘⢿⡿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡜⢆⣀⡬⠇⣹⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣌⣡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢠⡟⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⣸⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠁⣰⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣰⡟⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⠟⠀⣼⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠀⣼⠏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣼⠏⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢠⡏⠀⣼⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⣰⠏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⣰⠏⢀⣿⣷⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⣭⡍⣰⠏⠀⣾⣿⡟⢿⡄⠈⢻⣏⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⣤⣄⣀⣈⣼⠏⠀⣼⠏⣿⣧⠈⢻⡄⠀⢿⡆⠈⣿⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣀⣀⣀⠉⢹⡿⠀⣼⠋⢀⣿⣿⣧⠈⣿⠀⢸⡇⢠⡿⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⠟⠛⠙⠛⢿⣾⡇⣸⠏⠀⣼⣿⣿⡿⢿⣿⣧⣸⡇⢸⠃⣰⡟⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣡⣶⡶⢷⣶⠿⢿⣶⡏⠀⣰⠏⣸⡿⣷⠀⢹⡄⢹⡗⢿⣴⡟⣠⡶⠛⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣻⣿⣶⣶⣿⠶⣴⡯⠀⣰⡏⢀⣿⡇⠸⣇⠈⣇⣸⣿⣾⢿⡿⣯⣤⣤⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢾⡿⣥⣻⡼⣻⣦⣼⡇⣰⠏⠀⣼⣿⣿⣤⡿⢀⣿⣿⣿⣿⣿⡟⣹⣧⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠰⡜⣯⣿⠿⣿⣿⣧⡟⠀⣸⠏⣿⡁⢈⣡⡾⠿⣿⣿⣧⣿⣴⠛⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣇⢸⣿⡇⠀⣿⠀⠙⣿⠙⢷⣴⣿⡤⠿⠿⠶⣤⣼⣿⣿⣿⡿⢼⣿⣟⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⢾⣿⡇⠀⢸⡇⠀⢻⡇⢀⣽⣯⣤⣤⣀⣀⡀⣈⣿⣏⣿⣷⣻⣿⡟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠻⠟⣿⠉⢹⣄⠘⣇⠀⢸⡿⠋⠁⠀⠀⠈⠉⠉⢙⣿⣿⣿⣷⣿⣿⣿⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⣶⢡⣶⣾⡿⠿⣿⣤⠾⠷⠛⠛⠛⠛⠿⣿⣿⣿⣏⣻⣿⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣼⣿⣿⡟⣰⠟⢀⣀⣀⣤⣤⣤⣴⣶⣿⣧⣤⣽⣿⣴⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⡶⠛⠛⠉⠉⠉⢉⣹⣿⣿⡁⠈⠉⠁⠈⠉⠁
⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡤⠤⠤⠤⣴⠎⣀⣀⣤⣴⣶⣶⡿⠿⠴⣿⣷⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠀⣀⣴⣿⣡⣤⣤⡶⣤⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣠⣴⡾⢿⣿⣿⣯⣭⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⠟⠋⠀⢀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡸⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⣠⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣋⠔⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠾⠛⠉⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣾⣇⡴⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣾⣿⣿⡀⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣠⡾⠋⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢾⣿⣟⣤⡿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡼⠋⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⣼⣿⣿⡟⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡾⠋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢻⣿⣿⢡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣰⠟⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢧⡘⠧⣿⠆⣸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣬⣴⠏⠀⣰⡿⢿⣆⣀⣀⣤⣤⣄⣀⣀⣀⣀⣤⣤⣀⣤⣤⣀⣀⣀⣀⣀⣠⡼⠋⢤⣝⠶⢾⣰⣿⠁
⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⡿⠀⣼⠏⠀⣼⣏⣀⣀⣀⣉⣀⣀⣀⣁⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣘⢡⠓⠤⠐⢲⣟⣹⡏
⣉⠉⠉⢉⠉⠉⣉⠉⠉⣉⡉⠉⠉⠉⣉⣉⣉⣉⣉⣩⡇⣸⠏⠀⣰⣿⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣏⠀⠶⠒⢂⠉⣿⡿
⣿⣤⠤⣤⠤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⡟⠀⢰⡏⢰⣿⣿⣭⣭⣭⣭⣭⣭⣥⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⡏⠛⠒⠒⣉⣽⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣇⢀⡿⠀⢸⠟⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠾⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠁⢀⡞⠀⣿⠙⣟⣹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣀⣸⣏⠀⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠒⠂⠀⠈⠁⠀⢘⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠀⡀⠀⠸⠀⠀⠀⡠⠋⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠑⠒⠉⠀⠀⠰⠀⢀⠜⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⣀⠁⡠⢔⡩⠖⣀⠴⢂⠌⠁⢀⠔⡉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠔⡪⢐⡡⠒⡡⠔⣈⠄⡾⠁⠔⣠⡋⠑⠈⠁⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⠔⣉⠂⡡⢔⡩⠔⣈⡴⠊⡀⢀⡼⠑⠨⠀⢱⠩⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠐⠄⣤⠀⣐⠍⡒⣩⠖⠀⢈⡴⡋⢀⠠⠒⠀⢸⣄⢁⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠪⢐⣹⡇⠀⢐⣩⡔⠁⠀⣡⡚⠕⣈⠠⠒⣈⠤⠀⡿⣆⠊⠄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⣄⡊⣴⠇⡇⢀⡵⠊⢀⢠⡜⠍⢂⡁⠄⢂⠡⠄⡂⠀⡏⠍⠲⣍⢐⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡰⡶⣌⠳⡏⠨⣇⠘⢁⠄⣡⠏⠀⠈⠀⠀⠄⢒⠠⠀⠀⠀⡗⠠⠀⠈⠀⡁⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠲⣄⡼⢡⠞⠁⠀⡿⡷⢋⢀⢎⡾⢭⣅⣂⣠⠄⢒⡩⠔⣂⠥⠂⢠⡗⠨⠀⠀⠀⢹⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠢⡈⣱⠃⠀⠀⠠⢿⣳⠩⡠⠟⠁⠆⠠⢠⢀⡉⢉⢉⠉⢉⠈⢑⢚⠚⠛⠈⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣙⣃⣐⣠⠄⣸⡈⠚⠓⠀⢙⠃⠶⠠⠬⢘⣥⠌⣈⣤⢈⣠⡄⠈⣀⣤⡤⡤⢄⠜⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⡈⠉⠍⠑⠂⢈⡱⠁⢀⡠⠄⢒⡡⠔⢂⠠⠐⡈⠤⢒⠡⢐⡠⠄⠘⠮⣭⡀⢦⠴⡆⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡀⢢⣏⣁⠷⠇⠀⠀⢠⠊⠀⢈⠡⠐⢈⠥⢒⡈⠄⢂⣁⠐⠈⠔⢂⠥⢐⡈⠄⠀⠈⢷⠄⡰⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⢈⠠⠒⡩⠄⠂⢁⡶⡛⣻⣿⣿⣿⢶⣄⢂⡥⢔⣪⠥⠀⢸⡏⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠡⢒⠡⠀⡞⠁⠀⠀⠀⠀⠀⠀⠜⣦⢈⠁⣀⡭⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡨⠔⢊⠠⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠆⢲⠩⢗⡂⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠔⠊⠅⣂⠤⠀⢰⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⢐⡩⠄⢂⡠⠀⠄⠀⠀⠀⠀⠀⠀⡄⢀⡨⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠡⠐⣈⠤⠒⠀⢧⡀⠀⠀⠀⠀⢀⠠⠔⣂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠡⢐⡨⠀⠀⡼⠄⠀⠀⠀⣐⡾⠀⠥⢐⡊⠁⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠉⠁⢀⡜⠃⠀⠀⠀⠀⢸⡇⠀⠈⠁⠒⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⢞⠈⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣀⣴⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⡴⠒⡏⢸⠋⠉⡏⠉⡇⠀⣰⣶⣶⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⢸⠉⢹⠉⠙⡇⢹⠒⢦
⠀⠀⢰⣿⣷⡀⡇⢸⠀⢸⠀⣰⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣾⠛⠛⢻⣿⣿⣿⣿⢸⣿⣦⠀⡇⠀⡇⢸⢀⣾⣿⡇
⠀⠀⢸⣿⣿⣿⣷⣶⣷⣾⣶⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⣸⣿⣿⣿⣿⢸⣿⣿⣶⣷⣾⣶⣾⣿⣿⣿⡇
⠀⠀⠈⠉⠉⠉⠙⠻⢿⡽⠋⠉⠉⠁⢿⣿⣿⣿⡿⠿⠛⠛⠁⠀⠈⠙⠛⠻⣿⣿⣿⠟⠈⠉⠉⠙⢯⠿⠟⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⡟⠓⢠⣷⠀⣾⠳⠘⣿⠓⣿⠛⢸⡆⣶⠀⢠⡟⠧⢸⠛⠀⣾⡆⠚⡿⠃⠀⣿⢳⡄⡟⠃⢸⠀⠘⣿⠃⣞⠳
⠀⠀⠀⡷⠂⢸⣹⡄⠙⣆⠀⣿⠀⣿⠆⢸⢿⣿⠀⠀⠻⡄⢸⠖⠀⣏⣧⠀⡇⠀⠀⣿⢾⡀⡷⠆⢸⠀⠀⣿⠀⠙⣆
⠀⠀⠀⠇⠀⠿⠙⠇⢧⡼⠀⠿⠀⠿⠤⠸⠀⠿⠀⠸⢦⠟⠸⠤⠸⠏⠻⠀⠇⠀⠀⠿⠼⠃⠧⠄⠸⠤⠀⠿⠀⢧⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠋⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⢠⣤⣤⣤⣤⣬⣿⣿⣿⣷⣄⣀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣷⠀⣤⣀⣀⣀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⡆⢸⣿⢏⣶⣶⣦⢹⣿⠀⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⣠⣾⣿⣿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣼⠃⢸⣿⡟⢿⣿⡿⢸⣿⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀
⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢩⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿⣿⣶⣶⣾⣿⣿⠀⠿⠛⠛⠛⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠘⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⢴⡤⢴⠀⢠⣦⠀⢠⡄⢤⢠⢤⣦⢤⢠⣤⠰⡄⢤⣤⢰⠄⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⢸⡗⠆⠀⢎⣿⡇⡙⠿⣶⡀⢸⣿⠀⢸⣿⠲⠀⡎⢻⣿⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⡀⠚⠛⠀⠘⠂⠚⠛⠙⠂⠋⠀⠘⠛⠀⠘⠛⠘⠃⠓⠀⠛⠀⣠⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠰⡄⣶⠀⢸⡄⣿⠀⣺⣶⢸⠄⣾⠀⣿⠀⠀⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠀⠁⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⡤⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⡤
⠀⠀⠀⠀⢰⠛⡄⠀⠀⠀⢰⠒⠛⡖⠂⠉⠀⠀⠈⡇⡶⢶⠉⠀⠀⠉⠑⢲⡞⠓⢲⠀⠀⠀⠀⡜⠳
⠀⠀⠀⠀⢰⣀⣈⣄⠀⠀⠀⢣⣀⣇⣀⠀⠀⠀⠀⡇⣧⣠⠀⠀⠀⠀⢀⣸⣇⣠⠃⠀⠀⢀⣌⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠗⠃⠀⠀⠉⠁⠒⠈⠁⠀⠀⠉⠂⠈⠉⠀⠀⠀⠓⠧⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⡏⢹⣿⣿⣦⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡏⠉⠙⣿⠀⠀⣿⣿⡏⣿⣿⣿⡍⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⣿⠀⠀⣿⣿⡇⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣶⣾⣿⠀⠀⣿⣿⣷⣿⣿⣿⣶⣿⣿⡇⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠻⠿⠛⠻⠟⠟⠟⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄⡀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠂⠀⠀⠐⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡤⢾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⢈⣵⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⣾⣿⣿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠙⠻⢿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⣠⣴⣶⣄⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⢳⣾⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⠀⠙⠻⣿⣿⣿⣦⣄
⠀⢀⣴⣿⣿⣿⠿⠋⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄
⢠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⢀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣀⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⠀⠀⠀⠀⢠⣤⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣧⣤⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠙⡶⠊⣀⡤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣶⣿⣷⡟⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠉⠘⢶⣟⣺⣿⢦⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⢉⣿⠞⣹⠮⠿⠟⠂⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⣷⠞⢉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠖⠻⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠌⣿⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠻⠿⣯⣟⢻⣿⣾⣿⣤⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠘⠛⠽⣿⢿⣽⣗⣖⣠⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣰⣧⣤⣿⣥⣤⣭⣭⣋⣉⣀⣀⣰⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣝⣛⣩⣛⣻⣟⣽⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⢸⢖⠣⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡸⠀⠠⠄⠘⠀⠈⠀⠀⠀⠀⠳⠈⢁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠁⢀⡀⠀⡀⠠⡇⢀⡀⢠⡀⣀⠀⠉⠄⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⢠⡜⠁⠁⠉⠚⠛⠋⠉⠀⠀⠙⢦⡀⠀⠀⢐⠒
⠀⠀⠀⠀⠀⡀⠈⠠⠌⢁⡀⠢⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠄⠀⠀⠠⠤⠔⠂
⠀⠀⠀⠀⠀⠀⢀⠤⠂⢀⠀⢸⡇⠀⢠⠞⠁⠀⠀⠀⠉⠑⠀⠀⢸⣇⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣷⣀⡔⠒⠖⢤⡀⠀⡴⢈⣝⠰⣾⠃⠀⢀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡟⠅⠀⢀⣽⡦⢁⣉⠘⢳⣦⠀⢼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⡼⠀⢃⠀⠀⠘⡻⢞⠄⠘⡄⠀⠁⠠⡼⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡞⢀⢑⠆⠈⠒⠫⠄⠀⠺⠓⣰⠎⡯⠀⢄⠀⠀⠀⠀⠀⣀⡠⠀⢀⡀
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠈⢲⡀⠀⠒⠤⣄⣀⡴⠞⢠⠞⠁⠀⠀⠀⠀⢀⡔⠉⢠⠀⠉⠉⠠⣆
⠀⡆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⠀⠈⠀⠁⣠⠞⠂⠀⠀⠀⠀⠀⣸⠀⠀⢾⡁⠀⠀⠀⡯
⠀⠘⡄⢳⣠⠎⠁⠀⠀⠀⠀⠀⠀⣀⡠⠚⢢⠙⠒⠛⠛⠁⠀⢰⡟⣀⠀⠀⡼⠋⠀⠀⠀⢱⣀⠀⡀⠁
⢀⡄⠀⡀⠙⠀⢰⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠱⠆⠚⠐⠀⠴⠁⢻⡀⠼⠁⠀⠀⢀⠆⠈⠁
⡤⠀⠀⠐⡆⠀⠈⡄⠀⠀⣿⠀⠀⠀⠀⠘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⡸
⠑⠀⡸⠀⠛⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⠧⣀⣀⠇
⠀⠀⠈⠢⡀⠀⠀⢰⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⡀⠀⢨⠉
⠀⠀⠀⠀⣦⠀⠀⠐⣇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠀⠀⠀⢻⡀⠁⠀⡄
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⠀⢸⡀⠀⠀⠀⠀⠂⠀⠀⠄⠇⠀⠀⠀⢸⡇⠀⢨⠇
⠀⠀⠀⠀⢸⢀⡀⠤⠜⢯⠄⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠸⣇⢀⡟
⠀⠀⠀⠀⠈⠻⡄⠀⠀⠸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠤⣿⠊
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣠⠟⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠏⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠠⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⢠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⢛⡏⠀⢲⠀⠁⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠃⠀⠸⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠷⡄⠀⡸⠀⠀⠀⠀⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠀⠀⠀⠀⡀⢰⠁⠀⣣⠔⠀⠀⠀⠐⠒⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠒⠒⠂⢤⡀⢹⠇⠈⡗⢂⢁⠒⠉⠁⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡀⠀⠈⠉⠉⠢⢠⠟⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣙⠲⣤⣤⣀⡀⣉⠀⠤⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⢿⠿⢿⠿⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣛⣟⣛⣿⣿⣿⠿⠿⠿⠿⡻⠿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠩⠍⣉⡙⠛⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⢉⣿⠋⠉⠀⠀⠀⠀⠀⠀⠃⠦⣀⠀⠀⠑⠢⢄⡀⠉⠙⠫⣀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣤⣶⡤⠚⠉⠻⡇⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠈⠒⠤⣀⠀⠉⠲⢶⣦⣄⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣾⣿⣿⠋⠀⠀⠠⠂⠙⣦⡀⠀⠈⠑⠢⠄⣀⠀⠀⠀⠀⠀⠈⠑⠂⢤⠀⠀⠉⠒⠤⠀⡈⠻⣧⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⡿⠁⠀⠀⢀⠆⠀⠀⢨⠿⢦⢄⡀⠀⠀⠈⠁⠒⠠⠄⣀⡀⠀⠀⠀⠈⠁⠒⠒⢀⣤⠼⠛⣻⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠁⠀⢀⡴⠃⠀⢀⠔⠁⠀⠀⠉⠒⠭⣐⠒⠠⠤⠄⠀⣀⣀⣉⠉⠩⠭⠽⠿⠯⢁⣤⡴⣊⣩⠃⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢀⡴⠊⠀⡠⠖⠁⠀⣠⡴⣁⣀⡀⠀⠀⠀⣠⠶⠦⣀⠲⠶⢦⡄⣤⣤⣤⣤⡤⠤⢦⠄⠀⠈⠉⠀⠉⣛⡆⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⢻⣿⡖⠒⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⢇⠰⢆⣬⡴⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢹⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠃⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠘⢿⣿⣿⣿
⡿⠛⠛⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⡀⠀⠀⠙⠻⠿
⣿⣦⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠎⢌⣱⠎⠁⠒⠤⣄⡛⠿⠆⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⣠⠈⡜⠀⠙⠤⣀⠀⠀⠀⠈⠓⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⢧⠆⡐⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢐⠧⡌⠄⠀⡀⠀⠀⠀⠠⢤⠀⠀⢀⠀⠉⠲⠓⠄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠎⡘⣈⡠⠞⠉⡇⠀⢀⣅⠤⠚⣷⠉⠑⢄⠀⡘⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠒⠚⢴⠛⠉⠀⠀⠐⠁⠀⠄⠙⡶⠎⠽⠃⠀⠀⠉⠹⡦⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠛⡶⠚⠲⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠒⠒⣭⠶⠶⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⡤⠤⠳⠤⡤⢀⡀⠀⠳⠂⠉⠰⠞⠀⠀⠀⣠⣤⠮⡤⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠳⣆⡀⠀⠀⠀⠀⠈⠉⠒⠚⠀⠀⠀⠓⢒⠛⠛⠥⡆⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⠁⢰⡆⠀⢈⣉⠳⡦⣄⠀⠀⠀⡜⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠛⠻⣤⣹⠃⠀⠣⠤⠜⡗⠈⡟⠓⠶⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠦⣤⣤⣴⢻⡇⠀⠀⣔⠀⢰⡇⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢈⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⠉⠙⠉⠉⠉⠘⡀⠀⠉⠀⡇⠀⣷⠀⠀⡇⠀⠀⠀⠀⠛⠉⠉⠉⠉⠈⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⡁⠀⣿⠀⠀⡇⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠓⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠑⠤⠤⠤⠤⠚⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡟⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⣤⣀⣀⣀⣄⣀⣧⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢩⣴⣶⣒⣒⣒⣛⣭⠀⣿⠀⣥⣴⣶⣶⠶⣶⣾⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠜⣅⠢⡀⠀⠀⠀⠀⠀⠀⡡⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠄⠐⠒⢒⡒⣚⣃⣀⣀⣁⣈⣉⣉⣉⣉⣀⣉⣀⣀⣚⡒⢒⠒⠒⠂⠠⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣂⡉⠬⠤⠐⠒⠂⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠐⠒⠂⠤⠥⢉⣐⠂⠤⢀⡀
⠀⠀⢀⠤⠒⣉⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣑⠲⢄
⢀⠔⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠂⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠒⠂⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠢⡙⢆
⡎⢸⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣀⠀⠀⠀⣹⢸
⡟⢌⠳⢄⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⠔⣡⠞
⡇⠀⠑⠢⣉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⠤⠊⠁
⢡⠀⠀⠀⠀⠉⠐⠢⢌⣁⡒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠔⢂⣉⠤⠒⠉⠁⠀⠀⠀⢰
⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠬⢉⣁⡒⠒⠠⠤⠄⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⣒⣒⣈⡩⠤⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢻⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠙⠚⠓⠐⠒⠂⠬⠍⣐⣒⠢⢄⣀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⠀⢀⡤⠥⠤⢤⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠐⠢⢍⡒⢤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠋⠀⠀⠘⠒⡖⠒⠖⠁⠀⠘⠚⠀⠀⠀⢠⡢⠀⠀⠀⠀⠙⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠂⠠⠤⢼⠀⠘⠢⠤⠤⠤⠤⠤⠤⠔⠃⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⣶⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠳⣾⣿⣻⣿⣿⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠛⠛⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣽⣇⣠⣾⣦⣤⠶⠖⠓⠚⠿⢷⣶⣤⡶⠾⠓⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠈⠉⠹⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠛⠛⠿⢻⡿⢿⣿⡿⠋⠰⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠈⠓⣟⣀⣤⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⣤⣀⣸⣷⣷⣿⣷⣾⣿⣿⣷⣷⣿⡿⣿⣿⣿⣾⣿⡿⠿⠿⠿⠓⠂⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁
⣀⣤⣀⣀⣀⣀⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣠⣄⣠⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡾⡧⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡷⣥⡀
⠀⠙⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣽⣯⡧⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡷⠷⢚⠆⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠐⠀⠀⠀⣀⠀⠀⠀⣴⣖⣉⡁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⡿⠉⠀⠠⠤⠴⠦⠄⠀⠗⡆⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⢛⣻⣿⠒⠒⠒⠒⠶⠒⠂⠈⢷⡥⢶⠛
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡤⠄⠀⠀⢀⠀⠀⠀⡼⠓⢘⡆⠀⠀⠀⣶⣾⠄⠼⢾⡗⢀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠨⠇⠀⠀⠈⢉⠀⠀⠰⠛⠃⡠⠷⠀⠀⠀⣟⠻⠀⡈⢿⡅⠸⠁⣇
⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘⢶⡀⠀⠀⢀⡤⢾⠭⣧⢀⠀⣂⣀⡀⡰⠛⢠⣤⣼⣉⢛⣿⢻⠄⠀⢀⡇⠀⣧⠁⡞⣇⠐⡟⣿⠒
⠀⠀⠀⡇⠀⣼⣠⣸⣧⣷⣶⣶⣿⢶⠶⠿⠿⢷⣛⣛⣛⡙⠯⠭⢿⣷⢰⣾⠿⠻⣿⡿⣿⡟⣷⣾⣿⣴⣦⣻⣶⣯⣿⣅⣿⣿⣷⡆
⠀⠀⣠⣧⣸⡭⣟⣹⣿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠓⠦⠬⢭⣍⣀⣚⠛⠾⠿⢿⣷⣿⣿⣿⣿⠍⣿⣿⣿⣷⡽⠿⣷⠃
⠀⢠⠤⣗⣨⠶⣫⠽⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢶⣤⣤⣥⣉⡛⠺⠿⠿⣿⣿⣷⣶⣾
⠀⠀⢘⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⣷⣦⣤⣄⡉⠁
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣀⠀⠀⠙⠳⣦⡀⠀⠀⣠⣴⠟⠁⠀⠀⡄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠻⣦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⡀⠀⠀⡄⠀⠀⠰⠄⣠⡾⠋⠀⠀⠀⠰⠀⠀⠀⠶⠀⠀⠠⠄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠈⠀⠀⠀⠀⠀⠀⣠⡾⠋⣀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠰⠆⣠⡾⠋⣠⡾⠋⠁⠀⠀⠉⠻⣦⡀⠀⠘⠁⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢀⡴⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⣴⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣰⠟⠁⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠦⠀⠘⢷⣄⠀⠀⠀⠀⣠⡾⠋⣠⠞⠁⠐⠃⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⡶⠞⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⣀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⠃⠀⠀⠘⠀⠀⠀⠆⠀⠀⠀⣠⠾⠋⠀⠚⠀⠀⠈⠁⠀⠀⠉⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⠞⠙⢷⣄⠀⠀⠀⠀⢀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠈⠁⠀⣀⡴⠛⠁⠀⠀⠀⠙⢷⣄⣀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡥⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⣩⡼⠷⡄⠀⠀⠀⠀⠀⠀⢠⢞⡭⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡞⢡⠘⣿⡆
⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣼⠛⣉⣥⠴⠾⡇⠀⠀⠀⠀⠀⠀⡏⡞⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠹⣌⠆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣋⠉⣤⠶⡟⠁⠀⠀⠀⠀⠀⠀⣧⢣⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡧⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣤⣿⣤⣧⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣯⣼⣤⣿⡇
⠀⠀⠀⠀⠀⠈⣳⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿

⠀⠀⠀⠀⠀⣀⢀⣤⢠⣄⢀⡀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣶⣤⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣞⠛⣶⡀
⠀⠀⠀⢀⣘⠋⢾⡏⠉⡽⠎⢛⣀⠀⠀⠀⠀⠀⠙⠦⣀⣬⣬⡃⣤⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡿⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡙⠿⠾⠏⡁
⠀⠀⠀⠾⠿⠸⡿⠀⠀⢛⡿⠸⠿⠆⠀⠀⠀⠀⠀⢠⣿⣿⡸⢞⣵⢟⣀⠀⠀⠀⠀⠀⣧⣿⣿⡇⠀⠘⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁
⠀⠀⠀⢶⣶⠰⣿⡶⢠⣾⡷⢰⣶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣧⣌⠡⠿⡋⠀⠀⠀⠀⠀⠹⣿⣿⣷⡗⣰⣿⣿⠇⠀⠀⠀⠀⠀⣠⠀⣾⣮⣩⣶⡆⢠⡀
⠀⠀⠀⠈⢩⣄⢶⣷⢻⣶⢆⣬⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢻⡂⢯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠇⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠻⢦⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⣿⣿⣷⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⢸⠟⢉⣉⠛⣿⠀⢠⢀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄
⠀⢀⡄⣿⣿⣿⣿⢰⣀⣠⣶⢾⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣴⣴⠀⠀⠀⠀⠀⣿⣿⡈⠀⢁⣿⣿⠀⠀⠀⠀⠀⡞⠀⢠⡀⢙⣶⣄⠀⠀⠸⡆
⠀⠘⠇⣿⣿⣿⣿⢸⣿⠶⠶⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣾⣷⣿⠀⢿⢿⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⠀⠀⢀⡇
⠀⠀⠀⣿⣿⣿⣿⠸⣿⠛⢻⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠿⠟⠀⢸⢸⠄⠀⠀⠀⠀⣿⣿⣈⣠⣀⣿⣿⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⡇⢀⡼⠁
⠀⠀⠀⠛⠋⠩⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣄⣀⣠⣎⡞⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⢫⣻⠇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⠰⠚⠋⠁⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⠀⣾⣭⣶⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣤⡤⢸⠩⣭⣭⡭⢻⠠⣤⡀
⠀⠀⠀⠉⠛⠗⣶⣶⣤⣀⡀⡶⠒⢦⠀⠀⠀⠀⠀⠀⢨⡿⢿⡅⠀⣴⡟⠉⢻⣆⠀⠀⠀⣶⡆⠀⢬⡥⠀⢠⣥⠀⠀⠀⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⠀⢠⣶⣶⣴⣬⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⣾⣧⠀⢸⣷⠀⠀⠀⠿⠟⣵⡎⠿⠿⠿⠿⠏⣶⠹⠿⠃
⠀⠀⠘⠿⣟⢙⡿⠿⠿⠿⠿⠿⢿⡟⢙⡿⠿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⢸⣿⣮⣴⣿⣿⣿⣿⣷⣭⣾⣿
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠉⠀⠀⠀⠃⠀⠀⠀⠙⠁⠀⠙⠋⠀⠈⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣀⠴⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠂⠀⠀⠤⠚⠁⠀⠉⠐⠉⠀⠀⠀⠀⠐⠂⢙⠉⢥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡀⠤⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠑⢆⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠈⠀⢀⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢀⡀⣀⣀⣀⡀⠀⠀⢀⠜⠉⠣⠄⠠⠒⠉⠓⠦⣀⣀⠤⠆⠁⠠⠄⣈⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣼⠑⠆
⠀⠀⠀⠀⠀⠀⠸⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸
⠀⠀⠀⠀⠀⡐⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢃
⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⠁
⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠁⠲⠶⠀⠀⣀⣉⣤⡉⠀⠰⠶⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⡙⠋⠀⠒⠒⢠⡄⠀⠀⠄⣤⠐⠒⠒⠂⠛⡍⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠸⠁⠲⣄⠀⠀⢀⡇⠀⠀⠀⣿⠀⠀⠀⣠⠔⢺⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠃⠀⠀⠈⠓⡄⢨⡇⠀⠀⠀⣿⠀⣠⠊⠁⠀⠀⠲⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢠⠉⠀⠀⠀⡠⠎⠀⢸⡇⠀⠀⠀⣽⠀⠈⢢⠀⠀⠀⠀⢡⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⢄⠀⠀⡇⠀⠀⠀⠸⠀⠀⡰⠃⠀⠀⠀⠀⠦⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠳⣀⠇⠀⠀⠀⢀⢀⠎⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⢀⠴⠖⣨⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⣶⢀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠓⠐⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣥
⠀⠀⣀⡤⠊⠀⠈⠑⠢⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠤⠒⠁⠀⠣⢄⡀
⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣶⡟⠁⠀⠀⠀⠀⠀⠀⠉⠓
⠉⢰⠆⢀⡀⠀⠀⠀⠀⣀⡴⢄⠀⠀⣠⡞⠋⠉⠉⢛⢻⣶⡦⠤⢤⡤⠤⠴⠦⠤⠀⠀⠀⠀⠀⢸⠀⢀⡿⠀⡄⠀⠀⠀⠀⢀⠀⠰⡆
⠀⠈⠶⣟⠁⣰⠆⢀⣤⡀⣷⢀⣇⢠⡏⠀⠀⠀⠀⠀⠈⢹⡇⠀⠐⠇⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠀⢨⡏⠀⠇⣀⡀⠠⣄⠈⣷⠶⠃
⠀⠀⠀⠈⠒⠉⠒⠂⠈⢻⡉⠃⢧⡘⣧⡀⠀⡀⠀⠀⠀⣼⡇⠀⠐⡆⠀⠀⡇⠀⠀⠀⡰⠺⣷⣦⠀⣼⠛⡀⢸⠁⠐⠀⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⢸⣦⣬⣷⣶⣤⣤⣤⡾⢁⡦⠀⢠⡇⠀⠀⡇⠀⠀⠘⢤⡀⢘⣿⢠⡏⣭⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⣌⣤⣤⣤⢱⣿⠀⠠⢄⠨⣟⠀⢸⡇⠀⠀⡇⠈⠀⠀⠀⣹⣏⣹⣿⣀⡤⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡈⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠱⠆⠠⠄⠴⠋⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠈⠲⠄⠀⠠⠴⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠐⡇⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠺⢿⣭⣭⣤⣤⣬⣽⡋⢸⡏⠘⢯⣭⣭⣭⣭⣽⡟⠿⠃⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠛⠃⠀⠀⠀⠀⠈⠉⠁⣼⠣⠘⠃⠀⠀⠀⠀⠀⠰⠟⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢭⠴⠆⢀⣀⡀⠰⢦⡁⠞⠓⢨⠴⠆⣀⣀⣀⡀⠰⠮⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠠⡀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠄⠂⠈⣉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠉⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡀⠒⠂⢀⠀⠐⠀⠀⢤⡀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠐⣀⠀⡈⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠨⢀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠆⠀⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⠀⠀⡀⠦⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠒⢄⠀⠸⠂⠊
⠀⠀⣰⡶⠶⠶⠶⠶⠶⠶⠶⢶⣆⠀⠀⢰⠀⠎⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⡱
⠀⠀⡿⠐⠄⠀⠀⠀⠀⠀⡠⠀⢿⡀⠀⢸⠀⠑⠀⡆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢀⠀⠀⡜⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡇⠀⢸⠀⠀⣄⠈⠒⠂⢇⠀⠀⣠⡀⠀⠀⣀⡀⠼⠀⠥⠤⡤⠄⠠
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⣣⠀⠀⠀⢅⠒⠤⣀⠀⠀⠀⠠⠭⠋⠀⠀⣀⠄⠒⡡⠀⡀⠀⠀⣐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡴⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠮⠷⠤⠤⠤⠤⠥⠽⠤⠬⠴⠤⠥⠤⠤⠤⠭⠤⠭⠭⠤⠤⠧⠾⠥⠭⠬⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠥⠥⠤
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠖⣒⣒⣒⣶⡒⠶⠶⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⡴⠚⠉⢩⢅⣠⡀⠀⠈⣆⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⠀⠘⢦⡴⠃⠀⠀⢸⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠄⠀⠀⠈⠲⠤⠤⠤⠤⠤⠤⠴⣋⣀⣁⣠⡤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠒⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠊⠉⠉⠉⠉⣉⡭⠭⠭⠭⢭⣉⠉⠉⠉⠉⠉⢉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠁⠀⡇⠀⣴⣻⣳⡄⠈⡇⢀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⢀⠀⠀⢧⣀⡈⠓⠛⠁⢠⠇⠀⡀⠀⠀⠠⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠶⠶⠤⠤⠭⠭⠭⠭⠥⠴⠖⠒⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠀⠀⠀⡴⠒⠋⢉⣉⠉⠉⠓⡄⠠⠀⠀⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠠⢎⣉⣀⣓⡄⠀⣿⠀⠄⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠄⠘⣇⠀⠀⠀⠀⣀⣀⣀⠏⠀⠄⠀⠂⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠲⠭⠭⠭⠭⠥⠤⠶⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢰⣶⡆⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠸⢿⠇⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡀⠈⢉⡁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣤⡀⣴⣦⠀⣴⣶⡌⠳⣄
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠘⠿⠟⠀⢿⡿⠀⣿⡿⠀⠈⠻⣦⡀⠀⣠⠞⠁⠀⠀⠙⠛⠁⢙⡋⠀⢈⡉⠀⣠⣍⠳⣄
⠀⠀⠀⠘⣧⡀⠀⣿⣷⠀⣴⣦⠀⣴⣦⡀⢠⣤⡀⢀⣤⡈⠻⣾⡁⢠⣶⣦⢰⣿⡇⠀⢿⡿⠀⢿⡿⠀⠻⠟⠁⣹⠆
⠀⠀⠀⠀⠈⠻⣦⡈⠀⠀⢙⣋⠀⠙⠋⠀⠘⠛⠃⠙⠿⠃⢀⣈⡻⣦⡉⠁⢀⣬⡀⢀⣤⡄⢠⣶⣦⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠻⠿⠃⢸⣿⡆⢰⣿⡇⠀⢀⡴⠋⠉⠙⢾⡻⣦⡘⠿⠃⠘⠛⠃⠀⠙⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢀⣤⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢮⡻⣦⡀⢼⣿⡆⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣻⣦⡀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣬⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠛⠛⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣿⣿⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⣶⣦⠀⣿⣷⠀⢿⡿⠈⠻⣿⣷⣄⠀⠀⢀⣴⠟⠁⣴⢦⡀⣠⣶⡄⢀⣤⡈⠻⣦⡀
⠀⠀⠀⠀⣠⠞⠁⡀⠀⢀⣉⡁⢀⣤⡀⠀⣤⣄⢠⡶⣮⠻⣿⡷⠶⠟⢁⣀⠀⠙⣛⠀⠘⠛⠁⠸⠿⠃⠀⠈⠻⣦⡀
⠀⠀⠀⣾⠁⠀⢾⣿⠇⠘⠿⠇⠘⠿⠃⠀⠛⠛⠈⠙⠉⣠⡾⠻⣦⡀⠺⠿⠇⢺⣿⠇⢰⣿⡇⢰⣿⡷⢀⡶⣦⠈⣷
⠀⠀⠀⠙⢷⣄⠀⠀⠀⣴⣶⡀⢰⣶⡆⢼⣿⡆⠀⣠⡾⠋⠀⠀⠈⠻⣦⡀⠀⢀⣤⣄⢀⣭⣄⠀⣈⣀⠀⠙⣩⠞⠁
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠋⠀⣀⣉⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⠋⠘⠻⠋⠀⠿⠿⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠿⠿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣶⣷⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠿⢻⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇⠀⠀⠉⠀⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢸⠁⢿⡣⢰⣶⣶⡿⠁⠀⠀⣶⣖⣶⣀⣀⣀⣀⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⡟⠹⠿⣷⣄⢨⡄⠀⣀⠚⠉⣽⠇⠀⠀⠀⣇⠄⠉⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡠⣰⠃⠀⠁⣈⡽⠖⠒⠢⠽⢳⣀⠛⣠⡞⠁⠀⠀⠀⠀⣇⡩⠆⠀⠀⠀⠀⡇
⠀⠀⢀⡤⠚⢹⠇⢇⠴⡎⠉⠁⠀⠀⡣⠀⠀⣯⡧⠽⡟⢳⣄⠀⠀⠀⠀⣃⡕⠂⠀⠀⠀⠀⣧
⠀⢠⠏⠀⠀⠈⣻⠛⠀⡇⠀⠀⠀⠀⡆⠀⢘⣗⠛⠇⢀⡼⠈⠑⣤⠀⠀⠹⡞⠇⠀⠀⠀⠀⡇
⢰⡃⠀⢀⣀⣾⣇⣀⡼⠁⠐⠀⠀⠀⢁⠀⢸⣄⠒⠀⢨⡆⠀⣼⢿⠀⠀⠘⢿⠃⠀⠀⠀⠀⡇⠀⠀⣀⡀
⠀⠉⠉⠙⠛⠚⠉⢹⡇⠀⠄⢁⠀⠀⠈⡀⠈⠀⡝⣆⠈⠧⣤⣀⣺⠀⠀⡰⢄⡆⠀⠀⠀⠀⣇⡴⠋⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⡤⢼⡇⠀⠈⢂⠂⠀⠀⢀⢸⠁⡏⠈⢦⡇⣸⠉⠛⡆⠀⡁⢎⡁⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠑⠢⠀⠀⢸⠀⡇⠀⠱⡔⠋⠀⠀⢱⡀⡣⡊⡁⠀⡠⠊⣠⡔⠲⣤⣠⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠧⡿⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠹⡄⣼⢇⠀⠀⠀⡼⠃⡅⡜⣃⣈⣓⡉⠀⡇⣸⠁⢸⠀⠸⡀
⠀⠚⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⢲⠋⠀⠀⠀⠹⡗⠒⠚⠒⠚⣧⣤⣧⣉⣿⠀⠰⣷⣶⣷⠗⠒⢺⠒⠒⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⢉⣉⠈⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣤⣤⣶⣶⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣁⠶⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠃⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⣁⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠙⠻⠿⠿⠿⠿⠛⠛⠛⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⠴⠒⢢⡀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣎⣁⣀⣀⡀⠙⢄⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⢋⣩⣽⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠀⠀⠉⠓⠮⡑⢤⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠋⠴⢚⣷⡀⠀⠀⠀⠀⠀⠀⠘⢿⣈⣛⣰⡀⠀⢴⡄⢸⠖⠋⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠹⣝⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠛⠉⠲⠤⠴⠋⠀⣤⣠⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠘⡟⣆⠀⠀⠀⠀⠀⠀⠀⡷⡀⠰⠤⠴⠀⠀⢀⡴⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡿⣼⡄⠀⠀⠀⠀⠀⠰⢇⣈⣓⣶⠴⠶⠒⠋⠀⡧⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣻⣙⣛⡩⣷⠈⠁⠀⠀⠀⠀⣠⠖⣻⠟⢱⠟⠳⣄⣠⠴⠚⣳⠈⢻⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⣿⠀⣴⢲⣄⣠⠞⠁⠾⣅⡀⢸⣄⣴⡿⡅⠀⣰⠃⠀⢸⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣠⡏⠉⣿⠁⠀⠀⢠⠞⠁⢸⠈⢹⠀⠱⡾⠃⢀⣀⣈⡇⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⣧⣴⣛⠉⠙⠒⠾⢧⡀⢸⠀⡟⢀⡼⠁⠀⠻⡄⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢀⣟⡭⠤⠀⠀⠀⠀⠀⠉⢿⡀⣧⠞⠀⠀⣠⠔⠃⠀⣀⠔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣈⡯⠾⣉⣁⣀⣀⣀⣀⠀⠀⠙⢧⡀⢠⠞⠁⠀⠀⡴⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠛⢉⡝⠀⠀⠈⠳⡄⡏⠀⠘⢦⣀⡠⠏⣹⠓⢤⡀⠀⡾⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⢹⣄⡰⠃⠀⠀⠉⢺⡁⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠖⠒⠒⠒⠲⢄⣿⠀⠀⢀⡞⡼⠃⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣸⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢄⣀⣀⠴⠁⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠶⠒⠛⠉⠉⠉⠉⠛⠒⠶⢤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠋
⠀⠀⠀⠀⣰⠋⡠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠟⠀⢴⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠆
⠀⠀⠀⣰⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠉⢠⡀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂⡀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢠⡀⡀⠀⠀⡰⡛⠁⠀⠀⠀⠀⠀⠀⢘⡄⣀⣡
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢆⠀⢁⠜⡝⠁⠀⠀⠀⠀⠀⢀⠤⠚⠛⠛⠋⠀⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠐⠀⠀⠂⠁⠀⠀⠀⡀⠠⠀⠀⠒⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡇⡞⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣤⣴⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⣜⠣⢄⡀⣼⣸⠁⠀⠁⣀⡖⠋⠁⠀⠀⠀⡴⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠂⠈⠁⠀⠁
⠀⠀⠀⠀⠀⣍⠳⠄⢀⠀⠈⠉⠉⠉⢑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⡡⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸
⠀⠀⠀⠀⠈⠆⠉⠒⠢⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⡄⢪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⣀⣀⠀⢀⠔⠒⠀⠀⠀⠀⠀⢐⣉⣶⠆⠀⣀⢬⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡤⣚⡭⠟⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⡠⢚⡥⠋⠁⠀⠀⠀⠀⠐⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠛⣓⡶⠀⠀⠈⠑⣣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠐⢁⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⠞⠁⠀⠀⢀⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠂⠀⠰⠤⠀⠀⠠⠂⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠚⠢⠀⠈⠁⠀⢀⡴⠊⠁⠀⠀⡠⠚⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⡄⠀⠀⢯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠲⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⠋⠀⠀⠀⣀⠬⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡠⠔⠊⠁⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠔⠉⠀⢀⡴⠒⠁⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠐⠋⢹⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣰⣫⠖⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⣀⠀⢨⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡾⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢆⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠉⡄⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢲⡀⠀⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣿⡿⠿⠇





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠀⢀⣀⣀⣤⣤⣶⣶⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠠⣤⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⡎⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠁⠀⠀⢀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄
⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠸⣷
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠀⣠⡴⠟⠁⠀⠀⠀⠀⢿⡆
⠀⣿⠃⠉⠙⠻⠿⣿⣿⣿⣿⡿⠛⠛⠉⢉⡵⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢿⣤⡴⢶⡒⢦⣀⣴⡿⠁
⢰⡿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⣾⠁⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇⣿⠾⠿⠿⣆⢻⡿⢿⡆
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢹⠈⡇⠘⣿
⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠶⠄⠀⠀⠀⠀⠲⠂⠀⠀⠐⠂⠈⡇⣿⠐⠂⠀⣼⢠⡇⠀⣿
⠀⠀⠀⠈⠉⠛⠛⠿⠶⢶⣦⣤⣤⣤⣤⣸⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⣿⣀⣠⣴⣣⣾⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣆⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣡⡟⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣷⣶⡿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣦⣴⣿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⣿⣦⣼⡿⢿⣷⣤⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣀⣼⣿⣿⣧⣤⣿⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣼⡿⣿⣧⣤⣿⣿⣿⣄⣸⣿⣦⡀⠀⠀⠀⣤⣾⣏⣀⣿⣿⣿⣀⣸⣿⣿⣧⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣸⣿⣿⣿⣀⣹⣿⣿⡿⠿⣿⢆⣤⣿⠿⠿⣿⣿⣟⠉⣹⣿⣿⡁⣨⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⠈⣿⣿⣿⡿⠋⠀⢀⣴⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⠉⢹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠋⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢇⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⠟⠁⠀⣠⣾⣿⣿⠋⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⣶⣿⠛⢿⣿⣿⣦⣀⣴⣿⠟⣥⣄⣠⣾⣿⡿⠛⢿⣿⣿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣦⣼⡿⢿⣿⣶⣾⠟⢿⣿⣿⠛⠁⠀⠙⢿⣿⡿⠛⣿⣶⣿⠛⢿⣷⣾⡿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣤⣼⣿⣿⣧⣤⣿⢿⣿⣦⣾⠛⠁⠀⠀⠀⠀⠀⠙⢷⣶⣿⠿⢿⣶⣾⡿⠿⣷⣤⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣧⣠⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣾⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣇⣠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⢉⡠⠴⠒⠂⠈⠁⠀⢀⣉⣉⠉⠁⠉⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠭⢛⣛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣋⠥⠖⠊⠀⢀⡠⠄⠂⠀⣀⠠⠥⠄⠒⠒⠒⠀⠀⠒⠒⠒⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠁⠀⠀⡠⠂⣁⠔⠂⠈⠉⠀⢀⡀⠤⠀⠂⠐⠂⠀⠀⠀⠄⡀⠀⠀⠈⠙⠢⡀⠑⠀⠀⠀⠀⠀⠱⡀⢸⡆
⠀⠀⠀⠀⠀⢠⠏⢠⠊⠀⠀⠀⢀⠌⡠⠊⠀⠀⠀⢀⡤⣊⠕⠂⠂⠉⠉⠁⢀⣀⠉⠁⠐⠦⣈⠂⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡁⢸⡇
⠀⠀⠀⠀⢠⡏⢠⠃⠀⡀⠀⢠⠃⡘⠀⠀⠀⠀⢀⠊⡜⠀⠀⠀⡠⢢⠞⠉⢩⢭⣻⢑⠀⠀⠈⠆⢰⠀⠀⢀⠃⢸⡀⠀⠀⠀⢠⠁⣿
⠀⠀⠀⠀⣼⠀⡎⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⢈⠀⠣⠀⠀⠀⣧⣘⠂⠤⠿⢤⡥⠩⢆⡒⠭⠖⢉⣄⠴⠁⠈⢸⡟⢳⡄⡠⣢⠂⢸
⠀⠀⠀⠀⣿⠀⢃⠀⠀⢀⣼⡀⠀⢆⠀⠀⠀⠀⣼⡇⠄⠈⠒⠤⠶⢤⣤⣤⡭⠭⠉⠀⠤⠀⣒⡂⠭⠀⠊⠀⠀⣘⡷⢞⡯⠚⠁⠀⢸
⠀⠀⠀⠀⢿⢂⠀⠢⡀⣼⠋⣷⢀⠀⠑⠢⢄⣸⣇⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⠀⠀⠀⣂⠩⠄⠊⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⡆⠑⠄⡈⠻⢦⣽⡆⠈⠐⠀⠀⠀⠉⠉⠉⠉⠉⢀⣀⣀⣀⠀⠀⠀⠀⠉⠉⠡⠄⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠑⠢⠄⠉⠉⠑⠒⠒⠀⠠⠤⠤⠤⠄⠀⢀⣀⣀⣀⣀⣠⠔⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠢⡀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⡏⠰⠂⠀⠐⠒⠒⠒⠖⠒⠒⠂⠒⡆⢸⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⣀⣤⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠧⣭⣤⣭⣭⣤⣤⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠳⠾⠿⠿⢧⠀⠢⠀⠀⠠⠤⠤⠿⠶⠶⠶⠿⣃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉





⡿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣉⣠⣤⣤⠶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠶⢦⣤⣌⣉⡙⠛⠶⢦⣤⣀
⠀⠀⠀⠀⢀⣴⠟⢉⣤⡶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄⡉⠛⠶⣤⣀
⠀⠀⠀⣴⠟⣡⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡙⠳⣦⡀
⠀⢀⡾⢁⣴⠋⠀⠀⠀⠀⢀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣌⠙⢦⡀
⢀⡿⠁⣼⠃⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠻⣦
⣼⡇⢰⡟⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠘⣧⡘⢷
⣿⡇⠈⣧⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢉⣁⣀⣀⣀⣀⣀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠸⣇⠈
⣿⣿⠀⢻⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠉⠉⠉⠀⠈⠹⣦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⡿
⣿⠘⣧⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠇⢰
⣿⠀⠘⢷⣄⢙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡾⠋⣠⣿
⣿⠀⠀⠀⠙⣿⠁⠈⠙⢳⣦⣄⡀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠈⠉⠉⠛⠳⢶⣄⡀⠀⠀⠈⠻⢦⣄⠀⠈⠙⠛⠶⠶⣤⣤⣤⠶⠶⠛⠉⣠⡾⢻⡟
⠸⣧⠀⠀⢠⡏⠀⠀⢀⣿⣥⣈⠙⠛⠶⠶⣤⣾⣥⣤⠶⢶⡶⢦⣤⣀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⣀⣀⣀⣀⣤⣤⠶⠛⠉⣰⡟
⠀⠈⠳⣄⣸⡇⠀⠀⢸⡇⠈⠉⠛⠳⠶⢦⣤⣤⣤⣤⣤⠾⣷⠀⠀⠉⣷⠀⠀⢸⡇⠀⢰⣿⡆⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠈⢻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠈⠛⠶⣤⣸⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠛⠛⠛⢾⡷⠂⢸⡏⠉⠛⠶⣤⣀⣀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠙⠛⠷⢶⡶⢶⡶⠞⠋⠁⠀⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠶⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠛⠶⣤⣄⣀⣀⣼⠃⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠈⠉⠉⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣤⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣿⡛⣛⢻⣾⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠛⠋⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢖⡶⣄⣠⢾⠝⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠈⠁⠇⠘⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⢨⣀⡄⢀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢄⠀⠀⠒⣿⣿⠏⠀⠠⠀⠠⠂⠿⣿⢷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⡈⠁⣱⠠⢉⡄⢀⠀⠀⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴⠖⠂⡆⠂⠀⠀⠐⠉⢠⡁⠀⢸⡇⠈⢿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠸⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠡⣀⡴⢖⡠⠖⠲⢶⠦⠈⠆⠀⡘⢻⡿⢀⡴⠋⠀⠀⠀⣀⠀⣤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⣦⣠⣴⣾⣯⣥⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠱⡏⠀⠐⠒⠤⠀⣿⣾⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠤⠼⠿⠛⠉⠉⠉⠉⠉⠉⣷⣄⠀⢀⡀⢀⣄⡎⢧⠀⠁⢀⠀⠀⡀⠘⢻⡇⠀⠀⠀⠀⢀⣀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢟⣧⣵⠸⡆⠐⡇⠠⠄⡀⠀⠸⠃⠀⠈⠉⠑⢏⠡⡀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡩⠍⠉⠡⠞⠛⢻⣄⡇⠀⠑⠀⠀⠈⢰⣶⡶⢤⣀⠀⠀⢸⠇⡵⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠱⡀⠀⠀⠀⠀⣾⣧⣤⠠⠀⠀⠀⠀⣾⣿⠃⠀⠈⠛⠲⣾⢠⣧⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠑⠀⠀⠀⠜⠛⠛⠁⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠁⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢡⡄⠀⠀⠀⡤⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⢤⠀⠀⠀⢠⡌⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠚⠀⢀⡀⠘⠃⠀⣀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠀⠘⠃⢀⡀⠀⠓⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠀⠈⠀⠰⠆⠀⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠰⠆⠀⠁⠀⠦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢧⠀⣤⠀⠘⠁⢠⡄⠀⠋⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⠀⢠⡄⠈⠃⠀⣤⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠘⠂⠀⠀⠀⠓⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠚⠀⠀⠀⠐⠃⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⣻⣿⣿⣿⣛⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣛⣿⣿⣿⣟⣻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣿⡿⢿⣯⣼⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣧⣽⡿⢿⣿⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⣿⣿⣏⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣻⣿⣿⣿⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣿⣷⣾⣟⢻⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡟⣻⣷⣾⣿⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣯⣽⣿⣿⣿⣭⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣭⣿⣿⣿⣯⣽⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⢠⠄⠀⠀⠀⡤⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢤⠀⠀⠀⠠⡄⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠛⠀⢠⡀⠘⠃⠀⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⠀⠘⠃⢀⡄⠀⠛⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠲⠀⢀⠀⠰⠆⠀⡁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⠀⠰⠆⠀⡀⠀⠖⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢧⠀⢤⠀⠈⠁⢠⡄⠀⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠀⢠⡄⠈⠁⠀⡤⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡘⠃⠀⠀⠀⠓⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠚⠀⠀⠀⠘⢃⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⠶⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⡤⡤⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠁
⠀⠀⠀⠀⠀⣴⣾⠟⠁⠀⠀⠀⠀⣠⣤⠒⠈⠉⠀⠀⠀⠀⣿⠖⣋⠟⣓⡻⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠉⠻⣿⣤⡀
⠀⠀⠀⢠⣿⣿⣥⡄⠀⠀⠀⢐⣡⣴⣿⠀⠀⠀⠀⠀⠀⠀⣿⡘⠷⠴⠟⣰⣝⢫⣿⠿⣿⣷⡤⣝⡀⠀⠀⠀⢤⣼⣿⣷⡂
⠀⠌⢠⣟⣿⣟⡻⡇⢀⡔⢪⡙⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠿⢿⣿⣿⣿⠶⢿⣿⣵⡿⣫⡖⢄⠀⠘⢛⣉⣙⡻⣆
⠀⢀⣿⣿⣧⣼⡯⣷⣿⠳⡾⡟⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⢿⣷⠞⣷⡔⠌⣃⠬⠇⣿⡀
⠀⢰⣿⣿⣽⡿⣣⣿⠿⠃⢡⣴⡻⢚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⠘⠿⣷⡀⢼⣏⡉⣿⡇
⠀⢼⣿⣟⣿⣿⣿⣷⡋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⡄⢦⣤⣤⡙⢷⣦⠲⣿⣇
⢀⠸⣧⣭⣿⠟⡻⣿⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠟⠚⢿⡿⢛⠳⣿⣭⣼⠃
⠀⠈⣿⣿⠘⡶⠋⣽⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣴⣧⣷⣿⣶⣿⣿⡿⣴
⠀⠀⠘⣿⡆⠹⠞⠋⣙⣭⣽⣧⣄⠀⠀⣴⣶⡶⠦⣀⠀⠀⠀⠀⠀⠐⠻⠓⠀⠀⠀⠀⠀⠠⠴⣶⣶⣤⣽⣿⣿⣿⣿⣿⡿⣝⠞
⠀⠀⠐⡐⢻⡄⠀⡼⣋⢙⣿⣿⢿⣿⣶⣿⢳⡟⣻⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠰⠗⢘⣭⢺⣿⣹⣿⠻⢻⡿⠳⠋
⠀⠀⠀⠀⠀⠉⠦⡹⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣹⣆⠀⠀⠀⠀⠀⠀⣠⡞⢧⣼⠉⠀⣢⣾⣟⣩⢻⣿⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠠⠉⢻⣯⣿⠟⢻⡏⣛⢿⣿⣿⣿⣾⣷⣾⣇⠀⠀⠀⠀⣸⣢⣴⣶⣶⣶⣾⠟⢛⡗⢶⣫⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠻⠿⣿⣿⣷⣷⠻⢻⣷⡿⣿⣿⣦⣀⠀⠀⠹⣯⠲⡼⠉⢁⣱⣿⢾⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠻⠛⠿⠿⡿⣿⣿⣿⣿⣿⣾⡿⡿⠗⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠁⠀⠀⠈⠉⠛⢿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⣾⣿⡏⠉⠉⠙⣿⣿⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠰⣿⣿⣇⠀⢀⣰⣿⡟⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣤⣤⣤⠈⠛⠿⠋⠁⠀⠀⠀⠀⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢀⣤⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿
⢀⣿⡇⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣿
⢸⣿⠁⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⣾⣿⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡄⠀⠀⠀⠀⢸⣿⡿⠛⠛⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠸⣿⣇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⣿
⠀⢻⣿⡆⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡿
⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠿⠉⠻⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠶⠾⠿⠟⠛⠛⠿⠿⠷⠶⠶⠶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⢀⣀⣠⣤⠴⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣤⡀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⡀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣸⠃⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠛⠃⠀⠀⠰⠶⠀⠀⠀⣀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⡀⠀⠉⠀⠀⠀⠀⣠⢿⣦⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠙⢦⣄⡾⡿⣷⣄⠀⠀⣠⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⢮⡉⠓⠦⣄⣠⣴⠾⠋⢸⡇⠉⠳⣄⣠⣤⡟⣿
⠀⠀⠀⢸⠀⠀⠈⠙⠛⠺⠿⣿⣾⣧⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣛⣋⣉⣀⣀⣀⣠⣤⣿⣶⣛⣋⣁⣀⣀⣠⡼⠶⢦⣤⡿⢷⡄⢻⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢩⡿⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢉⣭⣭⣿⡯⠽⠿⠿⠿⠻⣿⠀⢸⡇⠀⢸⡇⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣾⠉⠉⠁⠀⠀⠀⠀⠉⠹⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⢙⣿⣿⠀⠀⠀⢰⠟⠙⢳⠀⣿⠀⢀⡇⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⡀⠀⠀⠀⠀⢻⡏⢿⠛⠛⠛⠛⠛⠛⠉⠉⢙⡟⢹⣿⠀⠀⠀⠘⠳⠴⠞⠀⣿⠀⢸⡇⠀⣠⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠺⣇⠘⠳⠶⠶⠶⠶⠶⠾⠿⠿⠃⠘⣿⣄⣀⣀⣀⣀⣀⣀⣠⡿⠀⣸⠶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠴⠶⠴⠶⠞⠉⠉⠉⠉⠉⠛⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠑⠒⠄⢀⠐⢂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⢤⡠⠀⠀⠀⠀⠀⠀⠘⠂⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠚⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⠛⠛⠶⠄⠀⠀⠀⠀⠀⠀⠱⠢⠀⠀⠀⠀⠀⠀⠀⠀⢉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⠀⠀⠄⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⡠⠴⢦⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⣤⠀⠠⠤⣦⢸⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣰⠀⢸⣿⠀⠀⠐⠂
⠀⠀⠘⡢⠤⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠒⠂⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠴⠋⢸⠀⠀⣎
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⢤⠔⠚⠁⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠔⠀⡲⠊
⠀⠀⠀⣗⣀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠈⠳⣄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⢄⠀⠑⠲⡤⠤⣤⠤⢠⠔⠖⠚⢙⣫⠀⠀⠀⠀⠀⠈⠳⣄
⠀⡞⠁⠱⣄⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⢀⠊⠀⠀⠀⣡⠀⠀⡖⠁⠀⠀⢞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⢀⠡⠤⣄⠚⢀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠑⠢⡀⠀⠙⣠⠎⢀⣠⠄⠶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢇⡀⢀⣈⠆⢸⡟⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⢧
⢨⡉⠉⠙⡄⢸⡇⠀⡘⠉⢧⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⢧
⠰⡉⠙⡏⠀⢸⡁⠎⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠸
⠀⠈⠉⡇⠀⢸⡇⠀⠀⠀⣰⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠰⢄⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠇⠀⢸⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⢀⣀⣀⡀⢇⡘⠦⡀⠀⠀⠀⢀⣀⣤⠔⠋⠀⠀⠀⠀⠀⡸
⠀⠀⠀⢧⣀⡼⠀⠘⠋⠑⠦⣀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⠄⢀⣀⣀⣀⣀⠈⢽⠀⠀⠙⢏⠉⠀⠀⠀⣂⠀⠀⠀⠀⢀⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠈⠄⠀⠀⠀⠨⡄⠀⣠⠔⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⠀⠀⠠⠋⠀⠀⠀⠀⠀⢰⠀⠀⣀⠠⠖⠈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⣤⣤⣤⣤⠖⢀⠄⠀⠀⢠⠏⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠀⠀⠀⢀⡠⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠤⠤⠤⢤⠤⠄⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠢⠤⠤⢤⡤⠤⠤⠤⠤⠤⢤⡤⠄⠀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠄⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠸⠆⠀⢀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠤⣤⣤⠀⠀⠶⠖⠛⠋⠙⠥⠄⠤⠄⠖⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣢⣤⣤⣤⣤⣤⣚⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⡤⠶⠒⢺⡇
⠀⠀⠀⠀⣰⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢹⠏⠀⠀⠀⢠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢯⡉⠉⠙⢳⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠖⠒⠛⠒⠒⠒⠒⢲⡄⠀⡴⠲⡄⠀⢠⠖⢦⠀⠀⠀⢷⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠳⠶⢶⠶⠶⠶⠶⠞⠁⠀⠓⠖⠃⠀⠘⠲⠚⠀⠀⢀⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⢧⡀⠀⠀⠈⠳⣤⡤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠯⠤⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠙⠒⠲⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣄⣸⠇







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⣀⠀⠀⢀⡤⠴⠖⣚⠛⠋⠩⠍⣁⠐⠒⠠⢀⡀⠀⢀⠤⠤⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡀⣽⣎⡀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠈⠑⠢⢀⠀⠙⠠⢀⡀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⠔⠂⡠⠋⠀⡈⢳⡉⠓⠀⠄⡀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠉⠒⠠⢀⠉⠉⢄⡢⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠜⠀⠀⢀⠁⠀⠙⣦⣀⠀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠁⠒⠠⠄⠈⣀⣢⠼⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠌⠀⠀⡠⠃⠀⢠⠊⠀⠉⠫⢔⠂⠤⢀⣀⠀⠈⠁⠐⠒⣒⢒⣶⠶⢊⣉⡤⠐⡃
⠀⠀⠀⠀⠀⠰⠀⢀⠔⠁⢀⠄⠊⢀⣤⢌⣁⠀⠀⠀⢀⡤⠤⣀⠶⠤⣤⢈⣥⣠⣀⡤⠤⡄⠐⠊⠉⠈⠛⣣
⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⡒⠂⠈⠁⠀⠀⠀⠀⠀⠉⠙⡀⢒⡸⠶⠀⠌⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢰⠀⠀⠀⡀
⠀⠀⠀⠀⡸⠁⠀⢠⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢆⠀⠀⢄
⢀⡀⠤⠒⠁⠀⡠⠃⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠈⢄⠀⠈⠣⢄⡀
⠣⢀⣀⣀⠠⠒⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠢⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠠⢁⣂⠖⠀⠠⢠⡐⠳⠆⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠄⢤⠡⠁⠉⠂⠄⡀⠀⠀⠁⠲⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡰⢲⠢⠁⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⡝⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠠⢑⠣⠃⡠⢄⠀⠀⠔⠒⣄⡠⢄⠀⠉⢉⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⢁⡦⠖⠈⠀⠘⠀⠀⢃⣁⠜⡇⠀⠑⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠤⠶⡤⠧⠄⠀⠀⠀⠀⠈⡁⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠰⠶⣵⠶⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢤⢤⡤⠤⠑⠤⠄⣀⠀⠘⠂⠉⠰⠂⠀⣀⣠⣤⠦⠤⢀⡠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠒⠤⣀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⣍⠍⠓⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠀⡉⠒⠦⣄⡀⠀⠀⠀⠀⣜⠎⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠤⢤⣄⢰⡇⠀⡖⠂⣽⠱⣦⢄⣘⠮⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⠁⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⡮⢱⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠦⣤⣴⣞⣏⠀⠀⡆⠀⡞⠀⢸⠀⢸⠆⠀⠀⠀⠀⠀⣀⢁⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠉⠉⠉⠁⠈⡄⠈⠁⠀⠀⢸⠀⢸⠀⠀⠀⠀⠉⠉⠉⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢆⠀⠀⠀⢀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠆⠀⠀⠀⠀⠀⠐⠃⠀⠸⠀⠀⠢⠤⠤⠤⠤⠤⠤⠄⠤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣤⣤⡄⣬⢀⣤⣤⣤⢤⣤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠨⠷⠾⠟⠯⠭⠴⠄⡿⠠⠶⠾⠟⠛⠲⠾⠅⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⡠⠄⠀⠀⠠⢄⡪⠳⢕⡠⠀⣀⣀⠀⢤⡲⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣚⣻⣭⣭⣽⣶⣶⣶⠯⢽⡳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⣩⠴⣚⣭⠿⣛⣋⣭⡥⠤⠤⣬⠉⣢⠘⡎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⠖⣫⠶⢛⣭⣖⣫⠽⢖⣲⣶⣷⣫⡵⢞⡡⠞⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⡏⢸⠁⢾⣍⣼⣿⣿⣿⣿⣿⣿⠽⠖⢛⣁⣤⠴⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢮⣛⡒⠚⠛⣛⣛⣉⣩⣥⣴⣖⡚⠋⠉⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉⠀⠙⢦⡀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⡴⠒⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⢂⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⢸⣇⣀⢽⡀⠘⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⡉⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⡖⠚⠋⢿⣀⣀⣇⠀⣿⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⠖⠒⢻⡏⠀⠘⠛⠉⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠸⣔⣀⣹⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡆⠀⣴⠛⢧⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⠀⠀⠀⠀⢀⡤⠶⢤⠀⠀⢈⣧⡤⢼⡀⠈⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣌⡇⡀⠀⠀⠈⣇⠀⠈⡗⠋⠉⣿⣀⡤⣧⢀⣻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⣾⢳⠁⠀⠀⠀⢻⠀⠀⠰⠒⠋⢻⠁⠀⠈⠉⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠋⠇⠀⠀⠀⠀⠘⣖⣠⡵⠀⠀⢸⡇⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⣀⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠁⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⢀⠄⠂⠈⠁⠈⠉⠀⠢⢄⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢠⠀⣆⣔⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⢈⠆⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠄⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠊⠄⠀⠀⠀⠐⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠂⠀⢸⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣽⠟⢻⣏⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢁⢠⢴⠊⠐⢀⣨⠉⠕⡢⢦⢬⡉⠁⠀⢁⠅⢤⣀⠀⢂⢤⣄
⠀⠀⠀⠀⠀⢀⠌⢠⠔⠚⣉⡀⠀⢠⡖⠗⢒⣦⣐⣂⣁⢄⣩⡆⣤⢷⠖⠢⡆⣐⣀⢌⠑⢿⣆
⠀⠀⠀⠀⠀⡈⠀⣎⠀⠞⣁⡅⢀⠏⠘⢠⠋⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⡇⢡⠀⢈⠽⠂⢈⠈
⠀⠀⠀⠀⠀⠠⡀⠈⠓⠤⠤⣉⣀⢤⣀⡆⡄⠉⠙⠀⠀⠀⠐⢡⡴⠾⢼⡧⣤⣭⢥⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠀⠼⠯⣈⡻⠓⠲⣒⠲⠶⠴⠲⢾⠷⠶⠊⡀⠤⠄⠰⠼⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠉⠠⠾⠅⠠⠀⠀⠢⠸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄⠂⠐⢀⠀⢠⠀⠀⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠈⠀⠌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠀⢁⣀⡔⢐⠨⡀⠐⠒⠑⠀⢸⠀⣀⡄⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠉⠚⠛⠉⠄⠂⠔⠨⢘⠳⡦⠄⡠⠐⠜⠡⢐⠠⠀⠁⠈⠪⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⢠⣶⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠗
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⢐⣻⢿⣿⣿⣇⣿⣿⣦⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠘⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣵⣾⣿⣿⣷⣯⠟⢻⣿⣿⣿⣧⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣩⣵⣾⣿⣿⣿⣿⡿⣟⣭⣶⣿⢸⣿⣿⣿⢹⣿⣿⣿⣶⣭⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠟⠋⠁⠀⠙⠿⠿⠋⢸⡿⠟⠋⠀⠈⠙⠻⢿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣯⣽⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣺⣯⣝⣭⣭⣩⣽⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢉⣉⣉⣉⣉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠉⠉⠉⠉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⣟⠉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣉⡉⠉⠉⠉⠉⣹⠟⠁⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠉⠀⠀⠀⣠⠞⠁⠀⣠⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡄⠀⣠⠞⠁⢀⣤⠞⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣧⠞⠁⢀⣤⠞⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠞⠁⢀⣴⠟⠁⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡞⠁⢀⣴⣞⣁⣀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠙⣶⣟⣉⣉⣉⡉⠉⠉⠙⠛⠛⠛⠒⠒⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠰⣏⣩⠷⡶⠋⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠓⠒⢶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠈⠳⣌⠁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢷⣄⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢳⣤⠟⠋⠀⠸⣆⡀⠀⠀⠀⠀⠀⠐⠛⠀⠘⠛⠛⠛⠛⠛⢂⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⢀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⣧⠻⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⣿⣾⣿⠃⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠋⠁⠀⠀⠀⠀⢯⠀⣠⡔⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡆⠰⠒⠀⢸⣀⠀⠠⠄⢀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠈⠈⠀⠈⠜⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⣤⣤⡄⠀⣿⣿⠀⣤⣤⡄⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠟⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠾⠿⠟⠋⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⣀⠠⠤⠤⠤⠖⣉⡀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⡁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⠓⢄⠀⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⢢⡴⠂⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢷⣄⠀⢸⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⡇⠈⠉⠁⠈⡆⠑⠢⢤⠠⠤⣄⣀⣀⣠⠤⠄⢤⠤⠃
⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⡄⠀⠀⠀⢀⣘⠤⢼⣤
⠀⠀⠀⠀⠀⠀⠘⣇⡄⠀⣰⢀⢰⠀⣀⡀⠀⠀⣇⣤⡖⠒⠈⠙⢲⢺⡁⠀⡄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠉⠀⠲⡤⡎⠓⠜⠁⠈⢧⡀⠈⣀⢏⠀⠙⠦⠤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠈⠉⠀⠀⠙⢁⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣕⡒⢶⢤⠤⡤⠴⡒⡍⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠙⠣⣤⡓⠐⣇⢧⠀⢸⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠈⠑⡤⠛⠚⠓⠁⠚⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠳⢄⡀⠀⠀⠀⠀⠀⢰⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡏⠒⠲⠤⠤⠰⡾⠋⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⢸⢲
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⠀⠀⠀⠀⢀⠆⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠈⢷⠤⠴⢚⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠸⢄⣀⡀⣀⣀⣰⣀⡠⠤⢤⠒⠊⠉⠀⠀⡖⠒⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⡀⠀⠀⠀⠀⡠⠚⠫⡀⠀⠄⣬⣀⢀⣼⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⠀⠀⡠⠊⠘⢠⣀⡈⢒⡺⠄⣀⠠⠾⣦⡀⠀⠀⢀⠇⠀⠸⢀⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡆⠀⡅⠀⠀⠀⢇⠈⢉⡷⠀⠈⠂⠐⠁⠙⡄⠐⠁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⡀⢣⠀⠀⠀⠀⠉⠉⠀⡇⠀⠀⠀⢀⡰⠁⢡⠀⠀⠀⠀⠀⡘⠀⠃
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠐⠁⠈⠢⢄⡀⠀⠀⢀⡠⠋⠁⠒⠈⠁⠀⠀⠀⠑⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄⣄⣤⣄⣤⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⣀⡀⠀⢀⣀⡀⡀⠀⠀⠀⠀⣧⣿⣿⣿⡟⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣗⣠⠞⣻⣿⢿⣿⢠⠶⠴⠋⠀⠀⠀⠀⠘⣿⡿⢿⠥⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠀⠀⠳⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠸⡇⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢀⠀⠀⠀⠠⣵⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⡄⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣾⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣤⡿⡇⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⣡⣾⠏⠉⢉⠍⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⢉⠉⠁⠀⣠⣤⣤⣤⣶⣶⣟⡻⠿⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡡⠞⠉⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡠⠀⠀⣠⣶⣾⡿⣛⣋⣭⣉⣉⣛⣛⠻⣦⣭⡉⠻⣿⣷⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠟⣡⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠊⠀⣠⣾⣿⠟⣫⠞⠋⣉⣉⣉⣉⣉⡙⠳⣤⣍⠻⣦⣈⡙⠿⣿⣄⠘⢦
⠀⠀⠀⠀⠀⣰⣿⡧⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠂⠀⠆⠀⠘⠁⠀⢀⣾⣿⡟⣡⠞⢡⢖⡛⠋⣉⡭⠉⠉⠿⣷⣦⠙⣷⣍⠻⣿⣦⠸⣿⣦
⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣿⠟⣰⡇⠂⠀⢉⠔⠚⢉⡤⠒⠲⣦⣄⡙⢷⡘⢿⣷⡈⢿⡆⠙⢿⣧
⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⣰⣿⢀⡿⢃⡆⢀⢸⡆⢸⡿⢀⣼⡿⢷⣍⢳⡄⣿⣀⢻⣿⡌⢿⡄⠸⣿
⠀⠀⠀⣿⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠠⠀⠁⠠⣿⡇⢸⡇⢸⡃⣿⠀⣷⠀⢠⡌⠉⠀⠶⠙⢆⠻⠘⣿⠘⣿⡷⢸⣧⠀⣿
⠀⠀⠀⣿⣿⡆⠘⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠘⠁⠀⠀⢀⠀⠀⣈⠀⠃⢸⣿⡇⢸⡇⠸⠇⢸⡄⠹⢃⠈⣡⢞⡃⠀⣴⣾⠁⢸⡿⢁⣿⡗⣸⣿⠀⣿
⠀⠀⠀⣽⣿⡇⡇⠘⠂⠀⠚⢀⠀⣼⠀⠈⠀⣶⠀⢸⡄⠙⠀⠀⢿⠀⠀⠘⣿⡇⢸⣯⠀⢃⠸⣷⡄⠻⣦⡙⠚⠋⢺⣿⠋⣠⣿⣿⡌⣿⣇⢹⡇⢀⣿
⠀⠀⠀⢹⣿⡇⡄⢠⠀⠀⣤⠙⠀⠈⠀⣾⡆⠙⠀⡄⠀⠲⠀⠀⠈⠀⠁⠀⢻⣧⠈⣿⣇⠸⣷⡙⢷⡤⣄⣀⣠⡿⠟⠁⣰⣿⠟⠛⣷⣿⡏⣸⠇⣼⣿
⠀⠀⠀⠘⣿⣿⡇⢤⡀⠄⠙⠃⠀⢶⡄⠘⠁⣶⠀⢼⠃⣿⠀⠀⡞⢦⢠⠀⠘⣿⡄⢸⣿⡄⢈⡻⣦⣷⣶⣾⣤⣥⣤⣶⡿⠃⣠⣾⣿⢋⣴⠇⣰⣿⡟
⠀⠀⠀⠀⢹⣿⣯⣼⡇⠀⣤⡂⠰⠎⠁⡄⠀⢰⢰⡦⠁⢤⠸⣇⠀⠘⠷⡀⠀⢹⣿⣄⠙⣿⣦⠙⢶⣬⣬⣍⣙⠻⠿⢋⣵⣾⣿⢟⣵⣿⠇⣠⣿⡟
⠀⠀⠀⠀⠀⣿⡟⣿⠧⡄⠙⠛⡀⠸⠗⡀⢸⠄⠉⣴⡜⢆⡀⢸⠐⢃⠘⢿⠀⠈⢻⣿⣷⡈⠻⣷⣄⠈⠙⠛⠛⠿⠾⡿⠟⠛⠁⣹⣿⠁⣼⣿⠏
⠀⠀⠀⠀⠀⢸⣷⠹⠇⠁⠰⠆⠁⢠⡜⠛⢀⡀⠰⠉⢡⡘⣿⠀⡀⢿⡆⢠⡓⠀⠈⣿⣿⣿⣦⣌⡙⠻⢿⣷⣬⣷⣶⣤⣴⣿⣿⠟⢁⣤⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣧⠘⢰⣄⠸⣿⠀⢡⣦⡀⠛⡀⠸⢧⠀⢡⡄⣇⠈⡱⠈⣿⡆⠀⢻⣿⣿⣿⣿⣿⣦⣤⣈⣭⣍⣛⠛⠉⣩⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠀⡆⠻⠆⠀⠆⠶⠙⣡⢀⡳⠀⢀⠲⢈⡁⢡⠸⠇⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢻⣿⢀⡀⢃⡀⠻⠀⠠⢷⣌⠁⢃⡄⠈⡷⠀⠆⠈⠀⠃⠀⡄⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠈⠃⢠⡙⠀⠘⠋⣄⠘⠇⠀⡄⠀⣀⠀⠃⠀⠀⣥⠀⠀⠀⣿⣿⠀⠹
⠀⠀⠀⠀⢠⠀⠈⣿⠀⠁⡀⢠⠀⠃⠀⠴⠀⠛⢀⠀⠀⠋⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠄
⠀⠀⠀⠀⠈⠀⢰⣿⠂⠃⠁⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⠐⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠐⠃
⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⣸⣿⣿⠁⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⣰⡿⠁⡴
⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⢀⡠⠆⢀⣾⡿⠁⡼⠃
⠀⠀⣰⣿⡿⠉⠀⠀⠀⣀⠀⠀⠀⣠⡄⢈⣀⠰⠟⡀⠐⢃⣤⠄⣀⣀⣠⣾⡿⢁⡜⠁
⢀⣾⣿⣿⣡⣤⣤⣤⣾⣥⣴⣦⣾⣿⣴⣿⣷⣴⣾⣷⣾⣿⣷⣿⣿⣿⣿⠟⠁⠊
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣤⣏⣹⣏⣼⣏⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠶⠶⣾⣿⣛⣛⣛⠛⠛⠛⠛⠒⠶⠶⣤⣄⣀
⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠳⠶⣤⣄⣀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⡾⠃⠀⠀⠀⠀⣀⣤⡤⠶⠶⠖⠒⠛⢛⣓⣲⣶⣶⡶⠤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠈⠻⣆
⠀⢸⡇⠀⢀⣤⠞⠋⠉⠀⣀⣤⡴⠶⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⢦⣤⣀⡀⠀⠀⠀⠀⠉⠳⣄⠀⣿⡄
⠀⢸⣧⡴⠋⠁⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⠀⠀⠀⢙⡿⣿⠁
⠀⢸⡟⢧⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⡞⠁⣿⡀
⠀⢸⣧⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⣿⡇
⠀⢸⡟⢷⣄⠀⠉⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⣀⡼⠋⣼
⠀⠘⡇⠀⠉⠛⠶⢦⣤⣌⣙⣛⣶⣶⣤⣤⣴⡶⠛⠋⠉⠉⠙⠻⣿⣟⠓⠶⣦⣀⣤⣴⣶⣛⣋⣥⣤⠴⠖⠛⠉⠀⠀⣿
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣩⡽⠟⠛⠳⣦⣀⠀⠀⠀⠈⣿⠀⠀⠀⢻⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠈⠙⢦⣤⣀⡀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⣹⠷⣤⡀⠀⡿⠀⠀⠀⣿⠉⠙⠳⢦⣄⣀⠀⢀⣀⣤⡴⠛⠁
⠀⠀⠀⠀⠀⣀⣭⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣠⣴⠛⠁⠀⠈⢙⣿⠇⠀⣀⣼⠃⠀⠀⠀⠀⠀⠉⠛⠿⣭⣄⡀
⢀⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠙⠛⠳⠶⠶⠿⠿⠛⠛⠉⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣄
⠻⣄⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⢀⡾
⠀⠙⢧⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣰⠟
⣶⣶⠶⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠋⠀⠀⠀⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣾⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⠶⡋⣳⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⢩⡴⠚⠹⣿⣿⡿⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⢳⡀⠀⠘⣟⠹⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⠳⠴⢛⣡⡴⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢉⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠂⠀⠀⠘⠒⠀⠤⡀
⠀⠀⠀⢀⣠⡤⠟⠓⠛⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⢀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⡞⠀⠀⠀⣀⠤⠴⣶⣻⡯⠥⠼⠥⠀⠀⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⢧⠀⣀⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠟⠁⠀⠀⠀⢀⣀⣤⡤⠤⠐⠒⠐⣒⡛⠩⠤⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⢰⣼⡿⠷⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠂⠄⡄⠀⠀⠲⠀⠀⠻
⠘⡷⠖⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⢸
⣾⠁⠀⠀⣀⣠⠤⠶⣖⣛⣫⣭⠥⠦⠔⠂⠀⠈⠑⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⡈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢻⡀⠀⠊⢉⡴⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⢗⣽⠋⠀⠀⢀⣀⣤⠤⠶⠶⣲⠾⠿⠯⢥⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠃⢸⠇⠀⠀⢰⢟⠉⠈⠙⠻⣼⠃⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠈⠢⠤⠴⠋⢸⡀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣼⠄⠒⠚⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⢀⠇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸
⠀⠙⢆⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠂⠒⠀⠚⠀⠀⠀⠀⠤⠧⠠⠴⠒⡶⠁
⠀⠀⠀⢻⠛⠉⠉⠛⠉⠉⠙⠯⠉⠙⠯⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣓⠀⡇
⠀⠀⠀⢸⡆⠀⠀⠤⠤⠤⠤⠤⠤⠀⠄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠠⠤⠤⠤⠤⢤⣤⠶⡶⠶⠀⡇
⠀⠀⠀⠺⡉⠀⠰⠶⠶⠶⠒⣒⣒⣛⡫⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠀⠠⠤⠤⠴⣶⣾⣿⠀⡆
⠀⠀⠀⢺⡅⠐⠶⠶⠶⠚⠛⣛⣚⣓⡭⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⣀⣀⣀⣀⣠⣵⣶⣶⢶⠀⡇
⠀⠀⠀⢾⡅⠀⠶⠶⠶⠒⠚⢛⣒⣛⠻⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⣤⣤⡶⣶⡃⣠⠇
⠀⠀⠀⠈⠳⣄⡀⠒⠒⠒⢚⣓⣛⣾⣥⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠉⠭⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠉⡭⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣛⣛⣿⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⡴⠋⣠⠟⠁⠀⠀⣴⠏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⣠⡞⢁⡴⠋⠀⠀⢀⡾⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣼⠋⢀⡴⠋⣠⡞⠁⠀⠀⣰⠟⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⠞⢁⣴⠋⠀⠀⢀⡾⠃⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣴⠋⢀⡴⠋⢠⡞⠁⠀⠀⣰⠟⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⠳⣾⡁⣠⠞⠁⣴⠏⠀⠀⢀⡾⠃⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠈⠙⢿⣤⡾⠁⠀⠀⣠⠟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠳⢦⣀⠀⠀⠀⠈⠛⢦⣄⡼⠋⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠉⠳⣤⡀⠀⠀⠀⠉⠻⢦⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠐⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⣠⡾⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⡞⠉⠳⢦⡀⠀⠀⣴⠓⠶⠶⠋⠙⠋
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠛⢦⣀⣴⠃⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡉⢁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⣴⠏⠀⠀⣴⣤⡀⠀⠉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣠⡾⠁⠈⠙⠷⣤⡀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣴⠏⠀⠀⠀⠀⠀⠈⢻⡦⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠘⠳⣄⠀⠀⠀⠀⠀⣠⡟⠁⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢙⠷⣤⡀⠈⠙⢦⣄⢀⣼⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣰⠋⣰⠏⠙⢶⣄⠀⠈⠛⠁⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡼⢁⡾⠃⠀⣠⠞⠉⠛⣦⣤⣤⠾⠃
⠀⠀⠀⠀⠀⢠⡾⠁⣠⠟⣠⠟⠀⢀⡴⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⣴⠟⢀⡼⢃⡼⠋⠀⣠⡞⠁⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⢠⡾⠃⣠⠟⢡⡞⠁⢀⣴⠋⠀⢀⡾⠁
⠀⣤⠏⢀⣤⠋⣴⠏⠀⢠⡞⠁⠀⣰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠄⣀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣀⢄⠀⠀⠈⢀⠠⠐⠈⠀⠀⠈⠀⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠈⡗⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠢⠧⣀⣀⠜⠲⣤⠖⠛⠓⢦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢇⠀⠀⠀⢨⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⢵⣊⠈⢉⠛⠑⠒⢤⡀⠀⠀⡀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠰⠀⠀⠀⠀⠇⠐⠐⠁
⠀⠀⠀⠀⠀⢤⢠⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⠠⠀⠠⠔⠁
⠀⠀⠀⠀⢀⠚⠘⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠗⡀⠀⠀⠈⠃⠀⠒⢦⡔⠒⣶⣶⣾⣿⠿⠿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠋⢸⡔⠁⠀⠑⠀⠀⠀⢀⠀⠀⠈⢿⣿⣿⡟⠉⠉⠀⠀⠜⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠄⠀⠀⠐⠀⠀⠀⠢⡀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠔⠓⠒⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⣀⡀⠀⠀⠈⠂⡤⠀⣀⣀⡀⠀⠀⠀⠐⠈⠀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠤⢱⠄⠁⠀⠀⠀⠀⠀⠂⠈⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣆⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠉⡀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⢂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠉⠀⠈⠢⡀⠀⠤⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠉⠂⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⣬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡴⠿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣦⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣶⣶⣿⣷⣶⣿⣧⣴⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠋⠀⠈⠉⠁⠀⠉⠁⠉⠉⠈⠋⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠸⡿⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡷⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⣦⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠛⢿⡟⠁⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⣄⠛⠀⠀⠀⣼⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠸


⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⣤⣀⠀⠈⠀⠒⠀⣀⡘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠤⠦⠀⠆⠀⠀⠀⠀⠀⠀⠘⠂
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⡔⠀⠀⠀⣠⠄⠀⠠⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠰⠂
⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡁⡀⠶⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣦⣼⣿⣿⡻⠦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠻⣿⠛⠛⠉⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⢠⠀⠀⠀⠀⠠⠄⠀⠐⠒⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣶⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠿⠛⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣠⣀⣠⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠶⠿⠿⠿⠶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣴⡾⠟⠉⢁⠤⢖⣒⣒⠒⠦⣄⣀⡀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠟⠛⠛⠻⣿⣾⣿⣅⣀⠀⡰⠁⠀⠀⢀⣽⠿⠿⠿⣿⣿⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣄⣀⢀⣼⠟⠁⠈⠉⠙⠽⡇⠀⢀⣴⡏⠀⠠⡶⢦⠺⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠈⢉⠛⢦⠀⠀⠀⠀⠀⢻⢇⡞⡻⠇⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣿⠷⣦⣄⠀⠀⠀⠺⣷⣤⣄⠀⠀⠀⠀⠉⢿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢱⠀⠈⠻⣏⡽⢷⣦⣼⣀⠈⠑⠶⣶⢶⠖⠋⠀⠈⠻⣕⢄⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡜⠀⠀⠀⣻⣿⠿⣿⣿⣿⣶⣮⣐⠾⣿⡗⠢⠄⣀⠻⠟⠒⠛⠐⠆⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠃⠀⢠⠎⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⣦⣝⡻⢶⣾⣷⣾⠀⠀⠀⠸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠠⠀⡞⠀⢻⣿⣦⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣶⣿⢻⣿⣯⣤⢄⠀⢣⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢿⣿⡿⣿⢿⣿⣷⣄⠀⢸⣿⣿⢿⣿⣿⣿⣿⠉⠻⢿⣿⣿⠘⣷⠈⣆⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡉⠻⣮⢣⢻⣿⣿⣦⣸⣿⣿⡀⠉⠻⣿⣿⣷⣄⠀⠉⠻⢧⣿⡀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠃⠀⢻⣇⣸⢏⣿⣿⣿⡉⠻⣿⣦⡀⠈⠙⢿⣿⣷⣦⡰⠎⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡀⠀⠀⠀⠀⠀⣸⣿⣿⣾⠟⣻⣿⣷⣄⣿⣧⣙⢦⡰⠆⢹⣿⣿⣿⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢇⠀⠀⠀⠀⣠⣿⣿⠿⡇⠀⠈⢽⣿⣿⣿⣿⣿⣷⣝⠶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠴⠚⠛⢷⣦⣄⣧⣀⡈⠒⠺⠿⣿⣿⣿⣿⣿⣷⣌⣀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠉⠁




⣶⡶⢶⢶⠶⠶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡀⠀⢀⠀⠀⢸⣽⡿⠿⠛⠋⠉⠉⠉⡇⠀⠀⡀⠀⢀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠂⢀⠐⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⡀⠐⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠂⠀⠐⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⠂⠀⠐⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⣿⣷⣿⡇⡿⠛⠋⠉⠀⠀⠀⠀⢸⣿⣾⣿⣷⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠁⠀⡈⠀⠒⠠⠤⢀⠀⠀⢂⡀⠀⠀⠀⠀⡀⠤⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⢠⡗⡋⠐⠂
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⠀⠀⠈⠐⠛⠹⠋⠉⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢼⠀⣠⡄
⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠄⠐⡶⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⢬⣁⠀⠀⢁⠂⢄⠀⠀⠀⠁⠄⠀⠀⠀⠗⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⣈⣁⣄
⠀⠀⠀⠀⠀⠀⢐⡠⣤⣙⠁⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠢⢦⡀⠀⠤⠀⠈⢼⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⢀⣀⣨
⡀⠀⠀⠀⠀⢀⡬⣧⢦⠤⠀⢀⣤⡀⠀⡶⠒⠒⠒⠒⠒⠂⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠀⠒⠀⠀⢳⡀⠉⠀⠀⠀⠀⠡⡀⠀⠀⠁
⠈⠐⢀⡀⠀⠘⠛⡇⡰⢈⠏⠁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠠⠀⠀⠙⣄⠀⠠⠀⠀⠀⠑⡀
⠀⠀⠀⠈⠐⠄⣤⠖⠉⠙⠂⠄⣀⠀⠀⠈⠀⠀⠐⠀⠀⠘⠀⠀⠈⣀⡠⠔⠃⠀⠀⠀⠀⠀⣀⣀⠤⢴⡆⠀⠀⠀⠀⠈⢆⡀⠀⠀⡆⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠓⠐⠠⠄⣀⠈⠁⠀⠐⠃⠐⠚⠋⠉⠁⠀⠉⠀⠀⠀⠀⠀⣤⡒⠖⠛⣹⣧⣴⣚⠋⠀⠀⠀⠀⠀⠀⠁⠀⣾⣿⣭⡬⣽⠆⠀⠠⢀⡤
⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠈⠁⠒⠀⠤⠄⢀⣀⣀⣀⡀⠠⠄⠐⠒⠈⠁⢸⠁⠀⠀⠺⠭⠁⠊⠀⠀⠀⢀⣀⠤⢄⡀⠀⡻⠤⣈⡽⣫⣄⣄⢠⡎⢀
⠁⠀⠉⠀⠂⠄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠄⢀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢀⡀⠀⠈⠙⠋⠉⠅⠈⠀⠠⠀⠈⠉⠉⠐⠢⡄⣠⠔⠉⠀⠄⠀⢀⡄⠀⢠⠀⠁⠀⠀⠀⠀⠀⢠⠔⠉
⠀⠀⢡⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⢰⠟⠁⠀⠀⢀⠄⢀⠝⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⡔⠁⠀⠚⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠄⠀⠚
⡀⠉⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠀⠀⠀⠉⠓⠲⠤⡀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⡘⠀⡀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠊⠀⢠⡶⠤
⠈⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⣤⠉⠦⠄⠀⠄⠀⠀⠀⠀⠀⠈⠒⠠⢄⡀⠀⠀⠀⠀⡘⠀⠀⠁⠀⠀⡠⠈⠀⠀⠃⠔⠀⠀⠀⢀⡟⠀⣔
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠇⢠⡄⣠⠔⠃⠀⠘⠀⠐⠀⠀⠀⠀⠀⢿⢆⡐⠄
⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠨⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠮⡷⣮⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⢸⠃⢱
⢠⠃⢄⣰⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠔⡈⠓⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⣠⠋⠈⣻⠴⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢀⡎⣐⠎
⢆⠲⢀⡿⠶⡄⣮⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠁⠋⠀⢠⠆⣆⠀⠌⠁⠀⢀⡐⠀⠠⡀⠀⠀⠀⠈⠻⣅⢀⡐⡀⡑⠀⢴⠒⡤⢀⠀⠀⠀⠀⠀⠺⠔⢩⠦
⢀⠟⠚⢁⠶⠁⠀⠀⠀⠀⠠⢤⣀⠴⠚⠁⠀⠀⠀⣐⡊⢂⠀⠀⠢⡀⠀⠀⠠⣉⠀⠀⢀⠉⠢⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠽⡣⠞⠢⢄⠀⠀⠀⠀⠔⠀⢠
⠾⠀⠀⠀⠘⠂⠀⠀⠒⠉⢉⠉⠠⠤⢄⣤⣖⡀⠄⡀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠁⠀⠉⠁⠀⠀⡀⠁⢀⡀⠀⠀⠉⣐⢾⡤⠙⠦⣠⠆⠇⠀⠀⠀⣠
⠀⠀⠀⠐⣤⠀⠀⢀⠀⢁⠀⠈⠉⠙⠉⠠⠀⠑⢂⠈⠐⣇⠀⡠⠖⠉⠀⠀⠀⠀⠀⡀⡀⠈⠂⢀⠉⠀⠀⠈⠀⠄⠀⠈⠒⠙⠢⠔⠂⠿⠔⠀⣀⠀⠀⠀⢰
⠀⠈⢠⠔⠸⠠⡴⣔⣀⣠⡁⠉⠉⠉⣱⡆⡀⢇⠨⠆⠐⢠⠙⠀⠀⠀⠀⠀⡄⠐⠋⢄⠈⠁⠀⠀⠑⢤⡈⠉⠀⠀⠑⢤⡀⠀⠈⠀⢡⠎⠀⠀⠀⠀⠀⠀⠊
⠀⢠⡀⠀⠈⠀⠀⠉⠛⠛⠉⠉⠉⠉⠙⠂⠉⠀⣀⠦⠂⠀⠀⠀⠂⠀⠀⠀⠐⡀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠂⠀⠐⠆⠀⠘⢆⠀⣰⠃⠀⠉⡀⠀⢀⡔
⠂⠀⣙⣦⣴⣄⣀⣀⡀⠠⢤⣀⠰⣦⠐⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠐⠒⠖⠂⠄⢀⠀⠀⠀⠀⠀⠈⣤⠃⠐⠂⠐⢀⠄⠉
⠀⡈⢤⠐⠂⢪⠝⡲⢄⡀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠤⠔⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣀⠀⠀⠈⠻⠀⠀⡰⠄⠉⠀⠀⠀⢀
⠀⢧⡌⠿⣋⣿⠦⠼⠦⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⣀⠞⠉⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢎⠀⠐⠉⣄⠐⠀⠂⠙⠁⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠘⢟⠶⠁⣭⣈⣴⠄⡀⠃⠀⠀⠀⡂⠀⡴⠐⠈⠀⠀⠠⠀⡜⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡄⠀⡀⠀⠀⠈⠐⠀⠙⠁⠐⡇⢦
⠀⠀⠀⠉⠓⠒⢦⣤⡤⠤⠉⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠂⠀⠀⠆⠀⠐⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃⠆
⣶⣶⣶⣾⣿⡾⣿⣷⣶⣶⣾⣿⣿⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣾⡶⢶⢷⠶⠶⠶⢶⣶⠶⠿⠷⡾⠾⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⢿⣶⡿⠋⠀⠀⢈⣿⠂⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠋⠀⠀⢀⣴⡿⠋⠀⠀⠀⣀⣀⣀⣠⣾⠟⠁⠀⠀⣠⣄⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠰⢿⣯⠀⠀⣠⣶⡿⠛⠛⠛⠛⠁⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣠⣿⣧⣾⡿⠋⠀⠀⢀⡀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⣿⣿⡿⠃⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⣠⣾⣿⣿⠿⢿⣿⣿⣦⡀⠀⠀⠙⢿⣿⡿⠋⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠺⣿⣿⣿⣇⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠉⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠻⣿⣿⣷⣄⠀⢀⣿⣿⣿⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣾⠿⠿⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠻⢿⣾⠟⠁⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠈⠻⣿⡿⠋⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣦⣼⠟⠁⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠈⠻⣿⡿⣷⣄⠀⠀⠈⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠙⠁⠀⠀⣠⣶⣶⣶⣶⣿⣿⣿⠟⠀⠀⢀⣿⡇⠈⠻⣷⣄⡀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⣠⣄⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠁⠀⢀⣴⡿⠋⠀⠀⠀⠈⠛⠿⠛⠋
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠘⣿⡆⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠙⢿⣿⡿⠋⠀⠀⢀⣾⡏⣼⡿⠛⠛⠛⠛⠛⠻⣷⡄
⠀⠀⣠⣼⠟⠁⠀⠀⣠⣄⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠉⠀⠀⢀⣴⡿⠋⠀⢿⣧⣀⠀⠀⠀⠀⠀⣿⡇
⢠⣾⠟⠁⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠐⢿⣿⡀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣿⡿⠃⠀⠀⢠⣿⠇⣼⡿⠁⠀⠀⣠⣾⣧⣀⣠⣿⠇
⠘⢿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢻⣧⣄⣠⣾⠟⠉⠙⠛⠋⠁
⠀⠀⠙⢿⣦⡀⠀⠀⠙⢿⣿⡿⠋⠀⠀⢀⣴⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠉⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⢦⣤⣄⣀⡀⠀⠘⡇⠈⡏⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣀⠉⠙⠛⠲⠮⢽⣿⣛⣺⡷⠶⣧⣤⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡷⢲⣦⣤⣄⣀⡈⠩⢿⡟⠲⠶⢭⣭⣛⡓⠶⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠿⠞⠃⣼⣿⣭⡽⢻⡶⢲⡦⣤⣄⣀⡈⠉⠙⠓⠲⢬⡟⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⢀⡾⠁⠈⠉⠀⢸⡇⢸⡇⣸⣿⣏⢉⣿⣷⡶⢶⣼⣷⠟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣤⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⢸⡇⣨⡟⢹⡏⠛⠋⣠⠿⣿⣶⠟⠉⠓⠶⠦⢤⣤⣤⣤⣤⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⣁⣀⡀⠀⠀⠀⠀⠀⣸⡟⢹⡇⢸⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠉⠁⣿⠫⢭⣉⡉⠛⠲⣦⣴⣏⣹⡇⢸⡇⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣠⡽⠟⠿⢧⣤⣈⡉⠙⣻⡷⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠙⣷⣈⣽⠟⣷⣦⣤⣄⣉⣽⣟⠿⢶⣤⣌⣉⠛⠻⠶⣤⣄⣀
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⢷⣤⠿⠛⣷⡾⣤⣄⣀⠈⠉⠙⠓⠶⢤⣌⣉⡽⣷⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣿⠀⢸⡏⢹⣿⡶⢦⣤⣤⣀⡀⣹⣀⣿⠖⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⢾⡟⠛⠋⠻⣿⣏⣽⠿⢯⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢻⡇⢸⡇⠀⢀⡾⠁⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⣿⠀⢸⡇⢸⣧⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⣿⠀⣸⡧⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⣿⠋⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⠄⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⠶⠛

⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⡿⠟⠉⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠟⠛⠁⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠁⣀⡀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⢀⡀⠀⠀⣐⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⣰⡆⠀⠀⢀⣌⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠃⠀⠘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠈⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠗⠀⠀⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⠀⠀⢠⣶⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣇⠀⠀⠈⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠹⠿⣷⣄⣾⠿⠂⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠀⣼⡆⣹⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⡀⠀⣀
⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠁⢘⣿⣿⣿⣿⣿⣿⣿⡟⠿⢶⣿⠃⠀⣿
⣠⣿⣿⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠋⠉⠉⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠠⠄⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⢀⡀⠉⠲⢄⡀⠀⠀⠀⢀⡴⠁⢀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⠀⢀⠀⠉⠲⣄⣠⠏⠀⠀⠈⠀⢀⡀⠀⠀⠀⠘⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⠁⠀⠀⠀⠈⠀⠀⢀⠀⠙⠦⣖⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⢤⡃⠀⢀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠆⠀⠀⠀⠒⠀⠀⡴⠋⠉⠲⢄⡀⠈⠲⢌⡀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠆⠀⢀⠎⠀⠀⠀⠀⠀⠉⠢⢄⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢄⡀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⡀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⠙⠲⣄⠈⠓⢤⡀⠀⠀⠀⠀⢠⠎⠀⠀⠁⠀⢀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⠀⡀⠙⢦⣀⠈⠓⢄⣀⡴⠃⠀⠃⠀⠀⠀⠈⠁⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠰⠀⠀⠀⠀⠉⠀⢠⠈⠓⣤⡀⠀⠀⠲⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡴⠃⠈⠓⢤⡀⠀⠀⠖⠀⠀⠀⠐⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠀⢀⡄⠀⠀⠀⠘⠀⣠⠞⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠂⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⢤⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄⡀
⠀⠀⠀⢀⣠⠞⠋⠀⠀⢀⣠⠤⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠶⠤⣄⣀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⢀⡞⠁⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠶⠀⠀⢀⣄⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⠀⠹⣄
⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⢤⣄⣀⣀⡀⠀⠓⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⣿
⠀⠀⢸⠙⢦⣴⣻⢷⣄⠀⣠⠔⠒⠉⠉⠀⠀⠀⠀⢀⣀⣤⠴⠶⠲⢭⠙⢲⣤⡤⠞⠋⢸⠈⠓⣤⡤⢾⣻
⠀⠀⢸⠀⠀⠀⠉⠙⠛⠻⠷⠶⠶⠶⣦⣴⠶⠶⠾⠯⠥⠤⠤⠤⠶⠾⢿⣭⣥⣤⣤⣴⡞⠓⡶⠛⢳⠘⣿
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣠⠖⠒⠒⠒⠒⠒⠶⡄⢸⡟⠛⠛⠉⠉⠉⠉⡇⠀⡇⠀⢸⠀⡏
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡿⠶⠶⠶⠶⠒⠒⠒⢻⣾⡇⠀⠀⢰⠒⢲⡀⣿⠀⡗⠀⢸⠀⡇
⠀⠀⠀⠀⠈⠑⠒⠤⢄⣀⣀⠀⠀⠘⡏⢹⠛⠛⠛⠛⠋⠉⢙⡟⢹⡇⠀⠀⠘⠶⠞⠁⣿⠀⡇⢀⣼⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⣈⡙⠛⠛⠛⠛⠛⠛⠃⠘⠷⢤⡤⠤⠤⠤⣴⣏⣠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠓⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡟⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣾⣷⣾⣿⣶⣶⣄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠩⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠣⠈⠁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠰⠆⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⣿⣿⣿⡿⣻⣿⣻⣿⣿⣧⠘⣞⠁⠉⢉⡙⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢃⣼⣿⣿⣿⣮⣿⣏⢛⣿⣿⣿⡿⣧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣾⣿⣿⣻⣿⣟⣿⣿⢬⣿⣿⠿⠠⠋⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀⣾⣿⣿⣿⣯⣽⣿⣴⣿⣌⣿⣿⢰⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠞⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣿⣯⡙⠳⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣷⠼⣯⢳⡟⢹⣴⣿⡆⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⣿⣿⣿⣿⣷⣤⣀⡀⢀⣠⣶⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⣶⣾⣿⣧⣼⣿⣿⣿⣧⡀⠀⠆⠰⠄⠀⠀⠀⠀⠀⢠⣿⠷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢟⣿⣿⣿⣏⢻⣶⣄⡙⢿⣧⡤⢿⡹⢿⣿⣶⣤⣀⣠⠀⢀⣀⣠⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢫⣾⣿⠟⣛⣿⡍⣿⣿⣿⣯⡁⠢⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⠉⣳⣽⣍⠻⡇⠸⠏⣿⡿⠿⢿⣿⡅⢸⣿⣿⣿⣧⡰⠙⣿⠉⠾⣿⣿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣧⣿⣆⣄⣿⢃⠀⣠⡝⣿⣿⠻⠿⠿⣿⣧⡀⢡⢀⣰⣿⠏⣗⣸⣿⣿⣋⣩⣽⣟⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡉⣹⣿⣿⣿⣿⣾⢂⣙⣷⠏⠀⠀⠀⠀⠈⣿⣷⣮⡎⠁⠑⠞⣠⣿⣿⣿⠏⠀⠀⠉⢻⡎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⠏⠉⣿⡭⠿⣭⣟⣟⣥⣤⣾⣷⡦⠀⠀⣿⣿⣿⣷⡄⠀⡤⠻⣿⡿⠁⠀⠀⠀⠀⠈⣷⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢈⣅⡿⠈⠀⣴⣿⡁⠀⢈⡍⢉⡿⣿⠻⣿⠻⣿⠞⣿⣿⣿⣿⣷⣤⣶⡆⣿⠁⠀⠀⠀⠀⠀⢠⡇⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣤⣿⣿⢿⡀⣀⠟⠀⣰⣶⣶⣤⣤⣴⣯⣴⠾⢶⡿⠶⣿⣿⣿⣿⣿⣿⣁⠈⣉⠉⠀⠀⠀⠀⡠⠋⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠿⣿⣿⣿⣿⣵⣄⢠⣾⣃⢉⣹⠯⠞⠥⢸⣾⠇⡄⠟⢹⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⢀⡴⢶⣿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣩⢿⡏⣟⣸⠃⠀⠰⠀⡾⢁⡔⢽⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⠋⠀⠚⢷⣤⣧⣈⣠⡶⠂⢠⡏⢀⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣏⣀⣀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢈⣴⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠈⠉⠁⢀⡤⠟⠀⡸⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣉⣽⡛⢋⣱⡿⣧⡀⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⢫⣿⣿⣿⣿⣿⣿⣷⣾⠿⠻⠛⠶⢾⣿⠃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠹⣿⣿⣛⠿⠁⠘⠛⣻⡄⠹⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢇⡻⣿⡿⣿⣾⡿⠛⠛⠁⠰⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⣿⣿⡷⣿⢿⣷⢀⡘⣿⢧⣴⠆⠀⢁⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇⣾⣼⣿⠁⡤⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⣽⡿⣿⣺⣿⣿⣿⣿⣸⠛⣥⢨⡁⢠⠄⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠘⠣⠾⠤⠤⠾⠛⠛⣫⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠹⣿⣿⠀⠀⠀⠀⠂⡀⣠⢟⡥⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⡗⣿⡆⣠⣴⣶⣿⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣠⣀⣀⢛⣷⡿⢻⡅⣼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⠿⠃⣨⣵⣿⣻⣿⣿⡿⣴⠺⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡷⠹⠿⠟⣿⡇⠘⢻⡄⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠋⣀⣼⣿⣿⣿⣿⣿⣽⣿⣿⠿⡥⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠁⠠⠤⡎⠀⠀⠈⡘⠀⠀⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⢺⣎⡁⡆⢹⣿⣿⣿⣿⠛⡫⣾⣃⠀⡞⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠛⢁⣴⣶⣤⣤⣀⠀⠀⠀⠀⠀⢀⠀⠹⣿⠹⣿⡿⠿⣻⣷⡟⣽⣿⢶⣬⣙⠷⡿⠛⠖⠃⠀⠀⣀⣥⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡈⠸⠋⠘⠁⠉⢱⠆⡀⢀⠀⠈⠁⡞⣤⣾⣿⣤⣦⣟⢿⣶⣿⣿⠼⠋⠙⠛⢜⣃⣰⣤⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡌⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⡟⢫⡿⠝⣿⣿⣿⡆⠻⣿⣿⠀⠘⢛⡛⣦⣿⡿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣠⣴⣶⣾⣶⣶⣦⡀⣄⣤⡦⠀⠘⢣⣄⣴⣶⣿⡷⠀⠉⠛⡶⠶⣾⡽⡿⠓⡄⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠊⠛⠻⠿⠊⠉⠋⠑⡟⠃⠀⠐⠶⠟⠋⠏⠉⠋⠀⠀⠀⠀⠈⠀⠁⠐⠟⠓