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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⡊⢉⡅⠂⠈⠐⢀⣯⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠩⠅⠂⠉⠙⣬⢋⠖⠂⠤⡠⠀⣀
⠀⠀⠀⠀⡖⠉⠙⡌⢆⢁⢀⠖⠺⡈⢠⠁⠌⠉⢣⠁⢢⠉⠑⠢⢄⣀⠤⠐⠊⣙⢣⡘⢄⡀⠀⠀⢀⠞⡇⣌⠔⠉⠀⠈⣱⢸
⠀⠀⠀⠀⡇⠀⠀⠘⡌⠢⡄⠤⠤⡴⠓⢣⠤⣴⡄⠄⠸⢰⠀⣀⡀⠄⢀⡔⢢⠗⠈⣗⢡⡙⢄⠀⡸⠠⠁⣏⢀⠔⢊⠉⠄⢸
⠀⠀⠀⠀⡷⡁⡱⠀⠈⠢⡈⠢⣀⠦⠤⠴⠠⠉⢃⠀⡆⣬⠊⠀⠘⡂⢰⠇⠘⣦⣴⡗⠀⣣⠈⠒⣁⣠⠄⢇⢁⣘⡀⠐⠀⠡
⠀⠀⠀⠀⡇⠘⠀⢈⠉⡀⠈⠒⢌⡑⠢⣀⠃⠀⠐⡞⡐⡘⠀⠀⡰⠁⠙⠳⠖⠓⠐⠓⣒⠞⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⢻
⠀⠀⠀⠀⡀⠎⠈⠀⠿⠀⡜⠓⢄⠈⠒⠤⡉⠂⡔⡱⢹⣁⡠⣎⢕⡨⠥⢭⣉⠲⡤⠊⠁⠀⢀⠤⣐⢂⡈⢣⢰⠒⠂⠤⠄⡜
⠀⠀⠀⠀⡇⠈⠏⠀⡤⠊⠀⠀⠀⠉⡆⠀⢈⠏⡴⡉⠢⣀⣾⢛⠥⠀⢅⡠⢀⠇⠀⠀⣠⢞⠕⠋⠀⠀⠙⢈⢰⠀⢠⢁⢂⡄
⠀⠀⠀⠀⢟⡒⠤⣀⠑⠂⠤⡀⠀⢠⠃⢀⠎⠜⡀⠌⠢⡈⠣⡐⢤⠂⠀⡳⠃⠀⢀⣄⠱⡸⣀⠀⠀⠀⠸⡈⡘⣘⣀⠊⠉⡇
⠀⠀⠀⠀⢀⡨⢕⡢⣑⠢⡀⠈⠉⣁⣀⡜⠸⣠⡀⣂⡁⡜⢄⠑⢌⠴⠊⠀⠀⡠⢋⢀⡳⡈⠢⣈⠢⢄⢾⡱⠁⠀⠙⡀⠐⡇
⠀⠀⠀⠀⢘⢣⠀⠈⣉⣗⢌⢳⠉⢀⡌⢢⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⢀⡠⠊⡐⠁⠀⠙⠮⡓⠤⣁⠒⠳⢅⣉⣀⠈⠁⢀⠇
⠀⠀⠀⠀⢸⠸⣴⠿⠋⠀⠈⡆⡧⢴⠇⠘⣗⣲⠉⣦⠤⠤⣀⠤⣔⡒⣄⠱⡀⢇⠀⠀⠀⠀⢸⠐⢠⢁⠓⡲⠤⡤⠬⠭⢬
⠀⠀⠀⠀⢸⠔⠁⠀⣀⠔⠊⡇⡏⠽⢡⠤⢻⣟⠀⢘⣅⢸⠰⢘⠀⠈⢮⠣⡈⢪⣢⠀⠀⠀⠘⡌⢸⢸⡎⠀⠀⠀⣱⠀⣸
⠀⠀⠀⠀⢸⢀⡠⠊⠅⠀⠰⢡⠷⠝⡟⠻⣦⠛⠒⠒⢳⢣⠇⣀⠳⣀⠀⢑⠌⢦⡙⢷⡤⢀⠀⠈⡜⡀⢗⠤⣒⢝⠫⣅⣸
⠀⠀⠀⠀⠘⠃⠈⠀⠀⡰⢣⣾⡄⡘⠛⡄⡙⢕⠦⡠⢃⢎⠎⠀⠑⠤⣉⠁⠆⠀⠙⠦⡉⠢⢍⡒⢡⠘⢌⠒⡕⠱⠋⠀⢸
⠀⠀⠀⠀⠀⡀⢠⢀⡔⡡⢣⠉⢹⣀⣀⣠⠁⠂⡼⣠⡣⠃⠀⠀⠀⢠⢊⠐⢠⢀⠈⠀⣌⠵⠦⢈⡐⠬⡀⠑⢌⢪⢢⠀⡀
⠀⠀⠀⠀⠀⡇⡠⢊⠞⠀⠆⠀⠀⣧⠠⢄⣀⠎⡔⢱⠁⠀⠀⠀⠀⠸⠀⢀⠀⣀⠴⠊⠁⠀⠀⠀⠈⠉⠙⠓⠬⢆⣣⠓⡇
⠀⠀⠀⠀⠀⣏⣔⡡⠤⠒⠒⠉⠁⠀⠀⠀⠈⠉⠒⠢⢅⣒⠒⠤⣀⡰⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⢀⢀⢁⡙⢊⡒⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⢀⣀⣄⢃⣤⢒⠐⣐⠀⠀⠀⠀⠀⡁
⠀⠀⠀⢜⠀⠆⢦⠰⢤⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠛⢘⢼⢀⠽⠒⡽⠊⠰⢛⠆⠀⠀⠀⠀⠂
⠀⠀⡔⡒⠀⡐⡀⡄⠀⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠊⠘⠀⠃⠈⠁
⠀⠀⠁⢠⠁⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠀⠀⠃⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠡⠐⠂⠐⢳⢘⢂⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠁⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠈⢠⠤⠤⠰⢀⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⡦⡦⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣤⡴⢦⣦⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣮⣾⣶⣬⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣿⣗⠦⣄⡀
⠀⠀⠀⠀⣀⡴⠞⠉⠀⢘⠵⢻⡧⡙⠢⢝⡳⢦⣀
⠀⣠⡴⠚⣁⣤⠤⢤⣄⠉⡠⢺⡷⢌⡒⠄⠀⠀⠈⠛⣦⣄⡀⠀⢀⣀⣀⣀⣤⡤⠤⠶⠶⣶⣆
⠀⣿⠀⣰⠋⣴⠲⣦⠈⣧⢀⡰⡏⠒⢍⠂⠀⠀⠀⠀⡙⠪⣿⣩⠍⠉⠀⢠⡀⢀⣠⢄⠌⣽⡌⢷⡀
⠀⢸⡀⢻⡀⠙⠶⠋⢠⠇⠉⠀⣿⠓⠤⠀⠀⠀⠀⠀⡉⠢⣸⢁⣔⠀⠀⢀⡽⢿⡁⢀⠜⣿⡘⠀⠹⣄
⠀⠸⡇⠘⣧⠀⠀⢀⡟⠀⠀⠀⠙⠈⠐⢀⣀⣀⠀⠀⢌⠢⣸⡿⠋⢀⡄⠀⠀⠀⠉⡠⢊⡇⢡⠀⠀⠘⢧
⠀⠀⣧⠀⠈⠳⣤⡾⠀⠀⠀⠀⠀⣠⠆⠈⠁⠀⠐⢶⡄⠑⢼⡇⠀⠟⠀⠀⠀⠀⢀⠔⣹⠳⠀⠂⠀⢀⡏
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠁⢦⡈⠃⢰⡀⠀⠀⠀⠀⠤⠊⣹⡄⠁⠀⠀⣸⠁
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⠀⢀⡴⠖⠲⠶⣄⠈⠳⠄⠈⠁⠀⢤⣀⡀⠐⠉⣿⠘⠀⠀⠀⡿
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢠⡟⢰⣄⠀⢸⡁⣞⠙⣷⢸⡆⢀⠀⢶⣀⠀⠀⠈⠁⢲⣴⣧⠁⠀⠀⢀⡇
⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡈⢀⢼⣯⡳⡜⣇⠉⠛⢁⡞⢀⢸⢀⠀⠙⠀⣀⡀⠀⡼⣿⠀⠁⠀⠀⢸⠃
⠀⠀⠀⢹⢠⡞⢛⣦⡴⠋⠀⡣⣼⡷⢌⠢⠘⢦⣠⠟⠀⡈⢻⠁⣀⢀⠄⠀⠁⢀⣠⣿⠀⠀⠀⠀⣼
⠀⠀⠀⢸⡎⠛⠋⠁⠀⠀⠸⢋⣡⣿⣆⡀⠀⠀⠁⠀⢠⡘⢼⠞⣡⠋⠀⠀⢐⠡⡾⢃⠃⠀⠀⠀⡟
⠀⠀⠀⢸⡇⠀⢀⣠⡤⠖⠛⠉⠁⠀⠈⠙⠓⠲⢤⣄⡀⠑⣼⠖⠁⠀⣀⣀⣥⣾⠳⡀⠂⠀⠀⢠⡇
⠀⠀⠀⠈⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠖⠛⠉⠉⠀⠀⠈⢳⡁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿

⠁⡀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠶⣤⢄⡴⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠵⣢⢄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⠐⠛⠓⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠤⠤⠤⠴⠤⠀⠐⠒⠒⠒⠀⠀⠊⠉⠉⠉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢖⣤⣀⠤⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠙⡵⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠈⠶⡀⣀⡤⠒⠈
⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⢿⡀
⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠊⠁⠀⠀⠀⠈⢷⡄
⠀⠐⠀⠀⠀⠐⠉⠓⠠⣀⠀⠁⠀⠀⠀⢀⡠⠔⠊⠁⠀⠠⠀⠀⠀⠀⠀⡀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡤⠒⠊⠁⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠁⠀⠈⠐⢄⡀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⢀⡠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣤⡤⠋⠀⠁⢀
⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠈⠑⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠢⠤⣀
⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠐
⠀⠀⠈⠙⠶⣄⡀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢈⡝⠾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⣤⣀⠀⢀⠴⠉⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀
⠀⢉⡿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡴⠋⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠓⠲⢤⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠉⠉⠓⡶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡙⠗⠲⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⣀⠀⠀⠎⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠶⢤⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢀⡠⠐⠑⡡⠒⢃⠉⠙⠓⠶⢤⣄⣀
⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡠⠒⠁⠀⢀⠜⠀⠀⠈⣆⠀⠀⠀⠀⠈⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⣀⠔
⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣀⠔⠉⠀⠀⠀⡠⠊⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡶⢮⣄⣀
⠀⠀⠀⠩⠉⠓⢦⣄⡀⠀⢀⠆⡠⠐⠁⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉
⠀⠠⠐⠀⠀⠀⠀⠀⠉⣳⡾⢊⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢀⣀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⢣⠀⠈⠑⠒⣤⠼⣄⠀⠄⠀⠀⢀⠄⠠⡀⡀⠠⠤⠤⠳⡒⠉⠀⠀⣸⠟⠉⠉⠓⠒⠠⠄⣀
⠀⢀⡴⠤⠒⠁⠀⠀⠀⠀⠀⠑⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣧⣴⣤⣦⣶⣴⣤⣤⣴⣶⣶⣴⣦⣴⣷⣶⣶⣦⣶⣦⣤⣤⣤⣦⣤⣶⣶⣶⣶⣦⣤⣾⣧⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠐⠒⠒⠂⠂⠒⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢐⡾⠷⠶⠇⠘⠳⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠄⠉⠁⠐⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⢰⡖⠒⠈⠀⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡌⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠠⠄⠈⠀⠢⠀⠠⠰⠀⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡀⠀⠉⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠁⠀⠠⣾⡀⠀⠀⠀⢸⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⣴⡗⠂
⠀⠀⠀⢀⠀⠀⠀⠙⠛⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⠄⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠞⠛⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡔⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣠⠐
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠂⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠁⠀⠤⠐⠀⠀⠁⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠈⠂⠀⠀⡀⠈⡛⠀⠀⠤⠴⡃
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠙⠀⠛⠉⠀⠐⢲⣅
⠀⠀⠀⠀⠐⠈⠀⠀⡜⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⣤⡄⠴⣤⡀⠀⠀⠀⠀⠀⠠⠤⠀⠶⠀⣾⢿⣆⣠
⠀⠀⢠⠀⡀⠀⠀⢴⣿⡾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠸⠟⠀⠽⡇
⠀⠀⠈⠂⠁⡀⠀⠂⠐⠀⠀⠀⠀⢠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠑
⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢹⣆⠀⠈⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠻⠆
⠀⠀⠀⠈⠃⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣄⣤⣤⣤⠤⢄⣀⣄⣄⣠⡤⢦⠤⣄⣄⣀
⠀⣠⣄⣠⡀⠀⣄⣀⣠⣤⣤⣤⣦⣷⣷⣶⣻⡁⠂⠐⢂⠀⠀⠂⠀⡀⠀⠀⠀⡀⣀⣀⣿⣿⣷⣶⣷⣾⣶⣷⣤⣦⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠿⣿⣿⣿⣿⣿⣙⣀⣈⡀⠀⢀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠉⠉⠉⠉⠙⢉⣻⣿⣿⣷⣾⣷⣶⠖⠀⠠⠂⠁⠀⡀

⠈⢦⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠈⠎⠀⠀⠀⠀⠀⣇⠀⠠⠢⠂⠀⠀⠀⢸⡀⠀⠀⠀⡸⠀⠀⠀⢀⡼⠁⠀⠀⡨⠳⣄
⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⠤⠶⠒⠒⠚⠋⠉⠙⢆⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⣰⠃⠀⠀⢠⠞⠀⠀⠠⠊⠀⠀⡈
⠀⠀⠀⣳⠀⠀⠀⢀⡠⠴⠊⠉⢦⡀⠀⠀⠀⣀⣠⠤⠶⠚⢗⠒⠲⠦⢤⣏⡀⠀⠀⡰⠁⠀⠀⣰⠋⢀⠀⠔⠡⠄⠐⠊
⠀⠀⢸⣁⡠⠔⠛⠁⠀⠀⠀⠠⠀⢱⡖⠚⠉⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⢹⡉⢑⠾⣁⠀⢀⡼⠉⠃⢀
⡠⠔⢺⡁⣳⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠂⠄⢀⠀⠀⠀⠀⠍⠠⠄⣀⡼⢠⠎⠀⠈⣱⢾⣀⠀⠀⠀⠑⠄
⠦⠤⣄⡵⠋⠀⠀⠀⣀⠠⠀⠐⠂⠈⡇⠀⠀⠀⠀⠈⠙⠅⠈⠢⠀⠀⢠⡽⠛⠳⣄⡼⠃⠀⠈⠑⠓⠒⢺⠒⠲⢖⠒⠚⠉
⠀⠀⠀⠙⠚⡦⠜⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠰⠐⠪⣀⡴⠋⠀⠀⣠⢏⠙⠳⢦⣄⡀⠀⠀⡿⠀⢀⠈⠑⢄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠞⠙⢦⠀⢀⡾⠁⠀⢣⠀⠀⠈⠙⠳⣴⣇⣀⣈⣀⠀⠀⠠
⠀⠀⣀⣀⣮⣤⣤⣤⣄⠀⠀⠀⠀⡶⢺⡶⠖⠒⠚⠛⠉⣁⠠⠀⠀⠀⡵⢏⢀⠀⠀⢘⡆⠀⠀⠀⣰⠋⠀⠈⠏⠉⠉⠙⢛
⠋⠉⠉⠉⡇⠀⠀⠀⣹⠖⢲⡚⠉⠃⠈⡇⠂⠀⠉⠈⠁⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⢸⠓⠦⣀⢰⠃⠀⠀⠰⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢳⠀⢀⣠⡟⠀⠀⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠑⢄⠀⠀⠀⠀⡞⠀⠀⠨⠛⢦⡀⠀⡂⠀⠀⠀⠠⠉
⣀⠤⠴⡖⠺⡉⣁⣸⣇⣤⣤⡬⡧⠶⠶⢾⠶⠒⠒⠚⠚⠛⠋⠀⠀⠀⠀⠢⡀⠀⡼⠃⠀⣀⠀⠀⠀⠈⠳⢧⡀⠀⡠⠁
⠀⠀⠀⢳⠞⠋⠉⠁⠳⡄⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠂⠠⣀⠃⠀⠉⠲⡅
⣤⡤⢤⣿⡀⠂⠀⠀⠀⠘⢆⠀⠀⣷⠒⠚⠒⠒⠛⢹⠟⠲⠤⣀⠀⠀⠀⠀⡰⠁⠀⠐⠠⠄⣀⢀⠴⠉⠐⢄⠀⠀⠈⢦⡀
⠀⠀⡾⠁⠉⠳⣄⠀⠀⢀⡰⢳⡉⠙⡆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠉⠒⠦⣴⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠒⢀⠀⠈
⠀⡼⠁⠀⠀⠀⠈⢳⡔⠉⠀⠀⠙⣆⢹⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠑⠶⠤⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⣼⠁⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠈⢻⡄⠀⠀⢹⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠐⢀⣀⣀⣉⣒⣤⣄⣀⣤⣤⣤⠤⠶⠖⠚
⠧⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠶⡧⠤⠤⢼⠶⠶⠖⠒⠒⢺⠛⠛⠛⠉⠉⠉⠉⣯⠉⠉⠀⠁⠉⠓⢦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠛⠫⣿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡘⠀⠀⠀⠙⠫⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⡶⠶⠶⢶⣿⣟⣛⣛⣛⣛⣛⣻⣿⡷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠒⠒⡄⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠀⢹⠀⠀⠀⡇⠹⠀⠀⡏⠀⠉⠑⢻⠽⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠇⠀⠀⠘⠢⡀⠀⡞⠉⠉⢹⠒⠒⠊⠉⠉⠉⠀⠀⡎⠀⡇⡠⢲⢄⣀⣀⣀⡜⠀⠀⢈⠟⠻⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠢⠒⠢⣄⠀⠙⠒⠃⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⠇⠸⠔⠁⡜⠀⠀⠀⡀⠀⠀⠀⠧⣄⣘⠤⠞⣹⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⡏⠓⠲⣄⡸⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠔⢃⡔⢙⠆⢀⠇⠀⣰⠋⠈⠑⠄⠀⠀⠈⠀⠱⢤⣿
⠀⠀⠀⠀⠀⠀⠀⣼⢷⢦⠀⠀⠀⠀⠀⡸⠉⢲⠀⢸⠀⠀⠀⠀⠀⠀⠑⢄⡙⠒⠋⠀⢸⠀⢰⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢧⠀⠀⠀⡠⠁⣤⠀⠓⡗⡄⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡏⠀⣇⠤⣒⡂⠒⠚⠀⢠⠠⢄⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣯⣣⢄⠀⠳⠤⠴⠁⢶⠆⠀⢀⠃⢇⠀⠀⣠⠤⢄⠀⠀⠀⡇⠀⢀⠃⠀⠀⠈⣓⣁⠀⠀⠀⠀⠑⡄⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣿⣢⣄⠀⠀⠀⠀⠀⢸⠖⢲⠑⠚⠁⠀⢀⠇⠀⢠⠇⠀⢸⠀⠀⢰⠋⠀⢸⠀⢀⡴⠊⠚⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡢⣄⠀⠀⡞⠓⠁⠀⠀⠀⠀⢸⠀⢠⠃⠀⠀⡎⠀⠀⠀⠣⡀⠈⠓⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣕⣤⣇⠀⠠⠤⠤⠤⠤⠤⠷⢧⣤⣤⣤⡧⣀⡀⢀⠴⠁⠀⣏⣉⡇⠀⢀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠷⢾⣽⣖⣦⣄⣀⡠⠒⠒⠋⣾⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠷⢿⣷⣶⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣨⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡭⢽⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠇⠀⠉⠲⢝⡷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡸⠀⠀⠀⠀⠀⠉⠻⢟⣷⣦⡀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡇⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣟⣻⡿⠭⠭⣭⠭⠭⠭⣭⠭⡭⠍⠛⡿⠭⣟⣻⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠸⠋⠈⢳⡀⠀⠀⠀⢀⣀⣀⣀⡞⠀⠀⠀⣀⣀⣹⠀⠀⢀⡼⢠⠇⠀⢰⠃⠀⠀⠀⢉⡗⠪⠽⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡇⠀⠀⠀⠙⢢⡀⠀⡏⠀⠀⢀⠏⠉⠉⠁⠀⠀⠀⠀⠀⡎⠀⡸⢀⠖⡻⠤⠤⠤⣄⠜⠀⠀⠀⡼⠉⢛⡯⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠒⠒⠒⢤⡀⠀⠙⠒⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠳⠊⠀⡇⠀⠀⠀⣀⠀⠀⠀⠘⢦⣄⣘⠖⠞⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡏⠛⠒⢆⣀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠴⠋⡰⠚⣣⠀⢸⠀⠀⣠⠋⠀⠙⡦⠀⠀⠀⠉⠈⠳⢄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⣇⠀⠀⠀⠉⠀⠀⡖⠒⢦⠀⢰⠃⠀⠀⠀⠀⠀⠀⠣⡄⠘⠤⠖⠁⠀⡞⠀⢰⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢰⡿⡾⠀⣇⠀⠀⠀⠀⣸⠁⡀⠈⠦⣼⡄⠀⠀⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⢀⠇⢠⠋⣀⠤⢤⡠⠬⠆⠀⣀⣀⡀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣼⡗⣇⠀⠈⢧⡀⠀⡸⢁⠤⡉⠀⠀⡇⡆⠀⠀⠀⢀⣀⠀⠀⠀⠀⣱⠀⠀⢸⠀⠈⠉⠱⠯⠆⠀⠀⠀⠀⠑⢦⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣝⡢⣄⠈⠉⠀⠀⠉⠁⠀⢰⣃⣘⠢⣀⡴⠋⠈⠱⡄⠀⠀⢳⠀⠀⡇⠀⠀⢀⠴⠒⢳⡀⠀⠀⣀⢤⡸⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣝⡦⣀⠀⠀⠀⠀⣸⣇⡰⠁⠀⠀⠀⠀⡰⠁⠀⡰⠁⠀⢀⠇⠀⠀⠸⡀⠀⠸⣄⣠⠞⠁⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣕⠦⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠱⡄⡸⠁⠀⠀⢸⠀⠀⠀⠀⠙⡄⠀⢀⣀⣀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣷⡧⣔⣒⣒⣒⣒⣒⣒⣒⣛⣓⣒⣲⣶⣿⡢⠤⣀⡠⠚⠀⠀⠣⠤⠼⠁⠀⡠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠷⣮⣽⣒⣦⢄⣠⠎⠉⠉⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⢶⣯⣶⣲⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣭⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣤⡶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠁

⠀⠀⠀⠀⠀⣀⣠⣴⣶⣄⠀⠀⠀⣠⡄⠢⠀⢐⣀⠶⢟⣿⣷⣦⣶⣶⡄
⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣦⣺⣛⣼⣩⢶⡇⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠉⠁⠀⠀⢀⠄⠀⢀⣤⣶⣆⣀⣄⣀⣤⣄⣠⣀⣤⣤⣄⣀
⠀⠀⠀⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠛⢋⡡⠾⠃⢀⣾⠿⠟⢛⣀⠀⠀⠀⢀⣤⣦⣄⢀⣁⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠄
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣷⣄⣴⣿⣷⠀⠀⠈⠃⠀⠀⠈⠁⠀⠀⣴⡿⠹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠘⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣟⠀⠀⠀⠀⠀⠀⠀⢤⣧⣠⣮⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠈
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠐⠂
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣁⣀⠙⠻⠹⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠆⠴⠃
⠀⠀⠀⠈⢹⣿⡟⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣶⣦⡴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠿⢧⣞⠀⠙⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣾⠎⠉⠻⣿⣿⡟⠛⣿⣿⡟⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣇⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣛⠁⠀⠀⠀⢹⡏⠀⠀⠈⡟⠿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⣵⠄⢀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣘⠋⠊⠀⠘⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣶⣿⣶⣦⣤⣶⣶⣶⣶⣶⣤⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣤⣶⣦⡀⠀⠀⢀⣠⡀⠦⠀⢐⡀⣾⠾⣿⣷⣶⣤⣶⣆⡀
⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆⣺⠟⣸⢁⢶⣅⠀⠀⣹⣿⣿⣿⡿⠃⠀⠀⠀⠋⠀⠀⠀⢀⡀⠀⠀⣠⣴⣦⣀⣀⡀⢀⣄⣀⣀⣀⣠⣤⣀⣀
⠀⠀⠀⠈⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠛⠏⠡⣼⠻⠀⣴⣿⡿⠿⠋⠀⠀⠀⠀⢀⣠⣄⡀⠀⣀⣀⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⣠⣾⣧⠀⠀⠘⠟⠁⠀⠀⠛⠁⠀⢀⣴⡟⣽⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠸⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿⠿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⢈⡙⣃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠙
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡾⠋⠙⠭⣿⢿⣬⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⡀⣀⡆
⠀⠀⠀⠀⢿⣿⣿⠟⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣾⣦⡀⣈⡈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉
⠀⠀⠀⠀⠀⣿⡏⢀⡀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣯⣛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠛⢿⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⡿⠃⠀⠈⢹⣿⡟⠁⠙⢿⣿⣅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣾⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠀⠀⠀⠀⠀⠻⡀⠀⠀⠨⡉⠛⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⢹⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⠀⣴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⠁⠁⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡆⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣤⣴⣤⣄⢀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⢺⣿⠋⣽⣿⣿⣿⣿⡏⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢐⣢⣀
⠀⢠⣤⣤⣀⣤⣻⣶⣔⣟⡷⢦⡀⢸⣿⣿⠿⠁⠀⠀⠀⣠⣤⣀⢀⣊⣠⣶⣶⣿⣿⣿⣶⣦⣴⣶⣤⣀⣤⡀
⠀⠸⠿⠿⢿⣿⣿⣿⣿⡏⠁⣼⣃⠈⠛⠁⠀⠉⠀⠀⢶⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⡿⠛⠉
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣿⡟⠆⠀⠀⠀⠀⠈⢷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢘⣯⣌⠙⡓⣶⣿⣽⣿⣿⣿⣿⣿⣿⣏⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠙⢿⡿⠻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⣶⣄⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡿⠀⠀⠈⠃⠀⣉⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠈⠁⢀⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠁⠀⠀⠀⠀⠀⢿⣿⠇⠚⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⢋⡴⣿⣭⡛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠴⣉⡴⢻⣇⡉⠓⢦⣉⠛⢷⣦⣀
⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠠⠞⠁⠀⣸⡏⠙⢦⡀⠈⠙⠢⠈⠙⠻⣦⣄
⠀⢠⣤⡾⠋⠁⢀⣴⡶⠶⠷⠶⣶⣄⠀⣠⠞⢹⣟⠳⢤⡈⠓⠤⠀⠀⠀⠀⠀⢙⡻⢦⣄⣀⠀⠀⢀⣀⣀⣠⣤⣤⡴⠶⠶⠶⠿⠛⠛⢛⣿⢷⡄
⠀⠸⣏⠀⠀⣰⡟⢁⣴⢶⣤⡀⠈⢻⡆⠁⠀⣨⣿⠳⢤⡉⠓⠆⠀⠀⠀⠀⠀⠀⠙⠦⣍⢻⡟⠛⣩⠉⠉⠀⠀⢀⠀⠀⠀⠀⣀⠀⡰⠋⣿⡄⢻⣆
⠀⠀⣿⠀⠀⣿⠀⢸⣇⠀⣸⡿⠀⢸⡇⣠⠞⠁⣿⣄⡀⠉⠓⠄⠀⠀⠀⠀⠀⠐⠢⣄⠈⢹⡷⠛⠁⣀⠀⠀⠀⠙⢷⣤⣴⡿⠋⠜⢀⣴⡏⢹⡀⠙⣷⡀
⠀⠀⢿⡄⠀⣿⡀⠀⠙⠛⠋⠁⢀⣿⠁⠀⠀⠀⣿⣀⠉⠓⠦⠄⠀⠀⠀⠀⠀⠠⣄⠈⠳⣬⣧⢠⠞⡁⠀⠀⠀⢠⡾⠛⢿⣄⠀⣠⠋⢸⡿⡀⠃⠀⠈⢻⣄
⠀⠀⢸⡇⠀⠸⣧⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⠃⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⡈⠳⣄⠈⣿⣫⠎⠁⠀⣀⠀⠀⠀⠀⠀⠉⠁⣡⠖⢻⠃⢳⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠘⣿⠀⠀⠙⢷⡄⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠛⠀⣀⠀⠙⠢⡈⠳⣿⠁⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⡴⢃⣠⣿⡄⠈⣇⠀⠀⠀⠀⣸⠃
⠀⠀⠀⣿⡄⠀⠀⠀⠛⢷⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠈⢦⣿⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⡟⠱⡀⠘⠀⠀⠀⢠⡏
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣄⠀⠻⠃⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⡵⠚⢹⣧⠀⠣⠀⠀⠀⠀⣼⠃
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠻⣦⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠉⣀⡴⣾⠟⣇⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⠀⠀⢀⣶⠟⠛⠛⠛⠻⣦⡀⠀⠙⠷⠀⠀⠀⠀⠀⠐⢶⣤⣀⡀⠈⠁⠀⣿⢇⠘⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣦⠀⠀⠀⣾⡃⢠⡾⠛⢻⣦⠸⣷⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠈⠉⠁⢰⣄⢰⣿⡈⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿⠲⣄⠀⢹⡇⠘⣷⣤⣼⠏⢠⡿⠀⢠⡦⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⢀⣿⣾⠇⢳⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣠⠞⣿⡦⣌⢳⡈⣿⡀⠀⠉⠀⢠⡿⠁⢀⢸⡇⣠⠂⠀⠀⠀⠠⣤⣄⠀⢀⣾⠋⣿⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⣿⠀⣴⡿⠷⣦⣀⣠⡾⠋⠀⠼⢁⡠⣿⣧⣈⠳⣄⠈⢷⣄⠀⣰⡟⠁⠀⠈⢾⡟⠁⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡄⣿⣄⣤⡿⠛⠉⠀⠀⢀⡴⣫⠞⢻⣇⡉⠓⠀⠀⠀⠙⠿⠋⠀⠀⠘⣆⢸⡇⣠⠶⢃⡼⠁⠀⠀⠀⣠⠴⢛⣿⠙⡆⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢸⡇⠈⠉⠁⠀⠀⠀⠀⠀⠨⢚⣁⣤⣾⠿⣭⣁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⢺⡿⠁⣠⠎⠀⠀⠀⠀⢀⡡⠖⣿⠋⢆⠈⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⠉⠁⠀⠀⠈⠙⠛⠷⢶⣤⣄⣀⠀⠀⠈⢢⣸⣯⡴⠃⠀⠀⠀⠀⠀⣉⣠⣼⠿⡄⠈⠆⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⣧⢀⣠⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣀⢹⣏⣀⣀⣤⣤⠶⠾⠛⠋⠉⢻⣆⠹⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⢀⢈⠀⠈⠀⠀⠀⠉⠁⢀⡀⠀⠓⠀⠐⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠦⠤⠄⠀⠀⠀⠸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠒⠐⠒⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠀⣀⡈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠐⠒⠠⠠⠦⠦⡄⠠⠄⠈⢀⠀⠀⠀⠀⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢠⠄⠀⠀⠀⠀⡀⠀⠀⠁⠠⠀⠀⡀⠀⠐⠀⠈⠐⢀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⡀⢀⡄
⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠄⠊⡀⡀⠄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⣉⣄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠘⠓⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠃⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠊⠍⠰⠀⡌⢒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣊⠭⠽⢶⡒⡤⢤⢄⣀⡀
⠀⠀⠀⠀⣠⠤⠖⢞⢫⠰⠐⠂⣌⣣⠰⠀⠄⢠⠷⢦⠰⢈⡑⠶⢄⡀⠀⠀⠀⣀⡠⠔⣾⠉⡔⠊⠉⠀⠀⠀⡠⡟⢠⢃⠴⠒⠒⠯⡑⢢
⠀⠀⠀⠀⠇⠀⠀⠘⡄⢣⠐⠜⠀⠀⢂⣌⣤⠂⠀⠀⠣⠀⠸⠸⠂⡌⠙⠒⠉⢀⡀⣺⠀⢷⡨⡓⢄⠀⠀⢰⠁⠇⣾⠃⠀⡀⠤⠔⠃⠘
⠀⠀⠀⠀⢆⡠⠠⡀⠈⢆⠙⢇⠀⠀⡚⠀⠘⡋⡸⢿⠈⠁⡇⡇⡴⠒⢆⢁⠀⡆⠸⣏⠀⣸⡂⢺⡀⢣⣀⠆⠈⡀⡟⠤⢊⠀⠂⡘⢠⠆
⠀⠀⠀⠀⢈⠣⡤⠃⡀⢀⠑⢄⡙⠢⣉⠉⠉⠱⠀⠀⢂⡸⢰⠹⠀⠀⠈⠆⢼⡏⠀⢽⠿⣿⣀⣐⣖⣀⡤⠤⠐⠃⠑⠒⠠⠤⠄⣁⣀⡇
⠀⠀⠀⠀⢸⠀⡁⡀⢄⡠⠃⢀⡉⠢⢄⡙⠢⢌⡉⠉⢹⢃⢦⠃⠀⢀⠜⢀⡈⠭⢉⡉⠁⠁⣩⠝⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠸⣀⡨⠀⠃⢠⠇⠙⠦⣀⠉⠒⢤⡈⡲⢡⡂⠪⡠⢔⣵⡪⠔⡲⠲⡦⡈⠢⠊⠀⠀⢀⡤⣒⠬⠭⣄⠹⠌⡏⡉⢐⠒⠤⡇
⠀⠀⠀⠀⢸⠀⠸⠀⢠⠖⠁⠀⠀⠀⠀⢱⠀⢀⡜⡰⢧⡈⠢⣴⠿⠫⠀⢄⠢⡤⢃⠜⠀⠀⢀⠔⣡⠊⠁⠀⠀⠈⢄⡃⡇⠀⠄⣜⣠
⠀⠀⠀⠀⢸⣉⠒⢄⡀⠑⠒⠢⡀⠀⢠⠎⠀⡞⡰⣁⠀⠈⢢⡈⢢⠐⣤⠊⠀⡱⠋⠀⠀⡠⡈⠢⡣⣀⡀⠀⠀⠸⠸⡰⣠⣃⡈⠀⢸
⠀⠀⠀⠀⠸⣀⠭⣒⠬⡑⠦⡀⠈⠉⢁⣀⡼⠰⣅⣀⣐⣈⢠⠑⡄⠱⣌⠴⠊⠀⠀⢀⠜⣀⣈⢦⡙⢦⡈⠢⢀⢶⢣⢣⠁⠈⠃⠀⢸
⠀⠀⠀⠀⠀⢻⡀⠀⠉⣹⣳⡌⢖⠉⠁⡐⢣⡀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⣠⠔⢡⠊⠀⠈⠳⢍⠢⣌⡑⠢⠫⣇⣈⡀⠀⠉⠀⢸
⠀⠀⠀⠀⠀⡹⣇⣔⠿⠋⠁⠘⡌⣆⢤⡧⠈⠷⠠⡎⢑⡤⠄⣀⢀⠠⠤⠦⡀⠳⡀⢂⠀⠀⠀⠀⠀⢡⠠⢨⢃⠖⣤⣤⣤⠍⣉⣉⡙
⠀⠀⠀⠀⠀⣧⠜⠁⠀⢀⡠⠒⡇⡗⣻⠆⣤⠹⣽⠇⠀⢗⡀⡸⡤⠰⠉⠑⢌⢢⡈⢪⡢⡀⠀⠀⠀⢸⠃⢸⢸⡼⠁⠀⠉⠒⡆⠀⡗
⠀⠀⠀⠀⠀⡇⠀⣠⠔⡉⠠⢠⢁⣧⡲⠶⢶⢄⡹⠧⣤⠽⢥⢃⠇⠘⢆⠀⠈⠳⡝⢄⠙⢷⣄⡀⠀⠀⠳⡸⠈⣧⣀⡠⣐⠾⢓⡒⡇
⠀⠀⠀⠀⠀⡷⠊⠤⠀⠀⢠⠇⣾⡈⢡⠭⡀⠹⣗⢄⠀⢠⢃⠎⡔⠓⢌⡑⠒⠒⢁⠀⠑⢄⠑⢬⡑⠦⡠⠃⠣⡑⢤⢞⠕⡩⠚⠙⡇
⠀⠀⠀⠀⠀⠆⠁⠀⠈⡠⢃⠎⠶⢧⠁⠀⠡⠰⢨⠓⡝⢡⢎⠎⠀⠀⠀⠉⡱⢠⠈⠘⠀⠄⠙⠢⣈⠑⠢⣌⠰⠘⢦⡑⢼⢄⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⢃⠜⡱⠃⡌⠀⠘⠖⠒⠒⢃⢀⠜⡡⡳⠃⠀⠀⠀⠀⢸⠀⠆⠀⠂⠃⢀⡬⠊⠉⠉⠒⠠⠭⣦⣄⠑⡌⢊⢣⣰⡇
⠀⠀⠀⠀⠀⢸⡴⣫⠎⠀⠀⣉⡩⠭⠟⠒⠒⠲⠯⣜⠁⠣⣀⡀⠀⠀⠀⠈⡆⠐⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠧⢭⠂
⠀⠀⠀⠀⠀⠸⠾⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢉⡒⠠⢚⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⣼⠛⠟⠓⠈⠑⢷⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠙⠇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠈⠀⠀⠘⠒⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠐⠀⠀⣰⠂⠁⠀⠈⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣠⡯
⠀⠀⠀⡀⠀⠀⠛⠳⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡴⠿⠂
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⢀
⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠘⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠑⠈⠀⠀⠀⠀⣹⡁⠀⠒⠖⡅
⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣩⣇⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢶⠶⠐⠳⠆⠀⠀⠀⠀⠀⢀⠀⠛⢀⣿⠻⢧⡖
⠀⠀⢀⠠⠀⠀⢘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⣩⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠲⣄⠀⠠⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣤⣤⣥
⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠐⠲⠆⠒⠋⢀⡀⣀⣀⠀⠈⠁⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⢀⣀⣀⠀⢀⠀⢀⣀⣀⣠⣤⣶⣦⡴⡞⠉⠩⠍⠉⠉⠉⠉⠉⠉⠈⠀⠉⠉⣷⣶⣤⣴⣤⣦⣤⣄⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢿⣿⣿⣶⣿⣋⠉⠉⡉⠉⠉⢙⣿⣿⣿⣿⣶⣿⣷⣷⣿⣿⣷⣿⣷⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⢻⣿⣿⣶⣿⣶⡶⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⢶⣤⣶⣶⣾⣷⣦⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣐⡢⡀⣂⢾⣟⡁⠙⠛⣿⣿⣿⣿⡧⠀⠀⠀⠀⠉⠀⠀⠀⠀⡠⠀⠀⢀⣠⣤⣾⣶⣀⣀⠀⢀
⠀⢀⣶⣶⣶⣶⣶⣶⣿⣿⣼⣮⡟⢻⣦⠀⢹⣿⡿⠛⢁⡀⠀⠀⠀⣴⣶⣶⢄⣤⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄
⠀⠐⠻⠟⠛⠿⣿⣿⣿⣿⣿⣁⠀⣶⣭⡀⠈⠛⠀⠀⠀⠀⢀⠀⢾⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣿⣿⠻⠄⠀⠀⠀⠀⠀⠛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢶⣋⡉⠛⠿⠶⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣶⣷⣾⣿⣿⣻⠿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠺⡄⢀⣀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣾⣟⠋⠀⠈⡟⠁⠘⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢦⢠⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠠⠤⠤⠤⠤⠄⠒⠉⠀⡀⢀⡀⠀⠀⠀⢀⡀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠸⠒⠒⡄⠀⣀⣀⣀⠀⠀⠈⠈⠁⢈⡠⠄⣀⣀⡈⠁⠁⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⢤⠀⢣⠀⠀⠈⠀⠀⠀⠀⢀⡠⠃⠀⠀⡀⡀⠘⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠃⢀⠃⠠⠈⠐⠀⠀⡠⠊⠁⠠⣦⠄⠀⠟⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢋⠉⠁⢀⠁⠀⠀⠀⠀⢇⠀⠀⠐⠀⠀⠀⡐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠐⠐⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠠⠁⠀⠀⡟⠃⠸
⠀⠀⠀⠀⠀⠀⠀⡇⡀⢀⣸⠀⠀⠒⠲⠒⠀⠀⠡⠀⠑⠒⠊⠉⠢⣠⣀⡠⢢⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡠⠊⠉⠐⠊⠉⢱⡀⠈⠀⠀⠄⠀⠈⠀⣀⠀⠀⠓⠒⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⣠⠁⠀⠀⠀⠀⠀⠀⢀⣱⡠⠤⠒⠊⠉⠁⠸⡀⠉⠑⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠆⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⡀⢨⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠒⠂⠀⠀⠄⢀⠀⠀⠀⠀⠠⣤⡤⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠀⠒⣀⣊⡓⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠘⠀⠀⠀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠡⠨⠍⠥⠈⠀⠠⢀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⡪⠁⠀⠀⠀⠀⠄⠐⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠈⠰⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢁⡤⣠⡄⣀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠄⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠘⣡⠂⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠠⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠒⠀⠀⠀⠀⠀⠁⠈⠀⠀⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢤⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠤⡄⠐⠂⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⡀
⠀⠀⠐⠒⠒⠒⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠠⠄⠆⢐⢂⣅⣠⣀⠀⠀⣀⠄⠚⠀⠀⠀⡀⠠⠀⠄⠤⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠠⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⡴⠘⠉⠈⠈⠀⠀⠀⢈⠀⠀⠆⣀⠀⠀⠀⢀⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠠⠡⠀⠀⠀⠀⠀⠒⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠐⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠐⠀⠁⠐⠀⠊⠀⠀⠀⠀⠌⠄⠄⠀⠀⠀⠀⠂⠂⠒⠀⠀⣀⠀⢭⡄
⠀⠀⠀⡀⣉⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⢀⡀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠸⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⡴⠊⠰⠶⢙⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⡀⣠⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠉⠉⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠦
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠤⢀⠠⠤⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠭⡥⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣬⡁⣀⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠑⠀⠀⠌⠀⠀⠀⠀⠠

⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⢖⠁⠀⠀⠀⢀⠁⢸⡀⢀⡀⠄⡐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⣠⡟⠁⠀⠀⢠⠝⢶⡀
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢀⣀⣠⣤⡤⠤⠶⠖⠒⠛⢯⠀⠀⠀⠀⠀⠀⠁⠀⣰⠇⠀⠄⠀⠀⡞⠀⠀⠀⠀⣴⠏⠀⠀⢀⡴⠁⠀⠀⢹
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣠⠼⢶⡉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⣄⣀⣀⡂⠐⠀⡴⠁⠀⠀⠀⢀⡞⠀⠀⠀⢀⡾⠁⠀⠀⡠⠊⠀⠀⣀⠜⠁
⠀⠀⠀⣠⠛⠀⠀⢀⣠⠴⠚⠉⢄⠀⠀⠳⡄⠀⢀⣀⣤⠴⠖⠛⠉⠉⠀⢱⡀⠈⠉⠉⠻⣷⠤⣄⡀⢠⠞⠀⠀⠀⣠⣟⡠⠄⠂⠚⠃⠈⠉
⠀⠀⠀⣼⣠⡴⠚⠉⠀⠀⠀⠀⠀⠉⠂⠀⢸⣏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢸⡄⠀⡽⠳⣄⡀⢀⣴⠋⠀⠉⠢⡀
⣠⠴⠚⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠂⠄⣀⠀⠀⠀⠀⠀⠎⠐⠦⢀⣀⡼⢁⡞⠁⠀⠀⣹⡾⣅⡁⠀⠀⠀⠈⠠
⣤⣀⣀⣈⣳⠟⠀⠀⠀⠀⠀⢀⡀⠠⠤⠒⠀⣗⠀⠀⠀⠀⠀⠈⠙⠲⡐⠚⠢⠀⠀⠀⢈⣹⠟⠳⣤⣀⣴⠋⠀⠀⠙⠒⠲⠶⠶⡶⠴⠶⡴⠶⠒⠚⠋
⠀⠀⠈⠉⠳⠤⣄⢀⠖⠊⠉⠀⠀⠀⠀⠀⠀⢺⠀⠀⠂⠀⠀⠀⠀⠀⢁⡀⢤⠃⢀⣴⠟⠁⠀⠀⣠⡟⠳⢦⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⡨⠢⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣏⠁⠀⠀⢀⡴⠋⠳⡄⠀⠈⠛⠲⣤⣄⠀⣼⠁⠀⠰⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⣀⣀⣀⣤⡴⠞⠉⠀⠘⢦⡀⣰⠟⠁⠀⠀⢹⡀⠀⠀⠀⠀⠉⣻⠷⠶⠶⣧⣤⣤⣀⣀⣘
⠀⣀⣀⣀⣠⣯⡤⠴⠶⢤⣄⠀⠀⠀⠀⢺⡟⠛⣟⠛⠋⠋⠉⠉⠉⢀⠤⠤⡀⠀⠀⢠⡿⢯⢀⣀⣀⣀⣘⡇⠀⠀⠀⠀⣴⠃⠀⠀⠀⠇⠀⠈⠉⠉⢹
⠋⠉⠉⠁⠈⡇⠀⠀⠀⠀⣿⠖⠒⣟⠉⠉⠁⠀⣷⠒⠀⠁⠉⠈⠉⠀⠀⠀⠈⢢⡴⠋⠀⠀⠀⠀⠀⠀⢸⠟⠢⣄⡀⣰⠃⠀⠀⠀⠜⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢧⠀⠀⠀⣠⡟⠀⠀⠸⡄⠀⠀⠀⢹⠀⠀⠀⠀⠐⠀⠀⠀⠀⣠⠟⠙⢂⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣹⠣⣄⠀⠀⠀⠆⠀⠀⠀⠀⣠⠑
⠀⠀⢀⣀⣠⢼⡶⠒⠚⣹⡇⠀⠀⠀⢳⣀⣈⣀⣸⣄⣀⣠⣤⣤⣤⣤⣤⠾⠁⠀⠀⠀⠙⣄⠀⠀⠀⣰⢇⠀⠀⠠⠁⠀⠈⠙⢦⣸⠀⠀⠀⠀⡰⠁
⠒⠋⠉⢸⡁⣀⣧⡴⠶⠿⡗⠛⠛⠛⠛⡏⠉⠉⠹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣰⠃⠀⠈⠐⠧⢀⠀⠀⠀⢠⠍⠓⢤⡀⡴⠁
⠀⠀⠀⠀⣽⠁⢡⠀⠀⠀⠙⢆⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠧⣀⠀⠀⠀⠀⠀⠉⠐⢤⡃⠀⠀⠀⠈⢧⡀
⠤⠤⠤⣴⣧⣄⠘⠀⠀⠀⠀⠈⠳⡀⠀⠀⣷⠚⠋⠉⠉⠉⠉⢉⡟⠉⠲⠦⣄⡀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠐⠢⢄⣀⣀⠔⠁⠈⠑⢄⠀⠀⠀⠙⢦⡀
⠀⠀⢠⡟⠀⠉⠻⣤⠀⠀⠀⠀⣀⠽⣞⠉⠙⡆⠀⠀⠀⠀⢀⡞⠀⠐⠀⠀⠀⠉⠑⠲⢤⣴⠃⠀⠔⠀⠄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠉
⠀⢠⡟⠀⠀⠀⠀⠈⠳⣄⡠⠞⠁⠀⠈⢣⡀⢻⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠀⠀⠀⣰⠋⠙⠒⢤⣄⡀⠀⡌⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠑⠠⡀
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠹⣌⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡀⠈⠉⠛⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣀⣀⠈⡇⠀⠀⠀⠸⡆⠀⠀⠀⢀⣀⣀⣠⣏⣀⣠⣤⣤⣤⣤⣴⣤⡤⠶⠾⠿⠶⠶⣖⠿⠛⠛⠋⠉⠉⠈
⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠒⠶⠒⢻⠟⠛⠛⠉⠉⠉⢩⠏⠉⠀⠀⠀⠀⠀⠠⠀⢿⡀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⢦⢄⡔⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠦⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠄⠒⠂⠈⠉⠉⠛⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠊⠒⠒⠉⠂⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠳⠤⣦⡄⠀⠀⠀⠀⣀⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠱⡶⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠁⠀⠘⢄⠀⠀⠀⡠⠄⠂
⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⣶⠊⠁
⠀⠀⠈⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠈⠓⠀⡀⠀⠀⠀⠀⠀⢀⡄⠐⠊⠁⠀⠀⠀⠀⠀⢀⡀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣤⡤⠒⠈⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⣀⠀⠔⠂⠁⠀⠈⠂⢄⡀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⢀⡠⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠴⡉
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠈⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠈⠛⠦⣄⡀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠐⠁⠀⠀⡀
⠀⠀⠀⠀⠙⢶⣤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄
⣀⠀⠀⠀⡀⠊⠈⠉⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⡀
⠉⢓⣶⣎⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡴⠋⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⢈⠙⠓⠦⣤⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⡀⠉⠀⠀⠀⠀⠈⠉⠛⢲⠦⢤⣄⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣤⣀⡀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡉⠛⠳⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⣤⣀⣐⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢈⡽⠛⢲⠧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠤
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠂⠁⣠⠊⠀⢆⠀⠉⠉⠛⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡠⠂⠁⠀⠀⡴⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣄⣀⡄⠂
⠀⠀⠙⠳⠦⣄⡀⠀⠀⠀⡄⠀⠰⠂⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠓⠶
⠀⢀⠄⠀⠀⠀⠉⠓⢦⣼⠔⠁⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢀⠠⠐⠁⢃⠉⠒⠢⢤⣠⣏⠀⣀⠀⠀⣀⡀⢀⢀⠀⣠⣈⠧⠄⠒⠉⢉⡶⠛⠒⠒⠤⢀⡀
⠀⢀⡤⠄⠚⠁⠀⠀⠀⠈⠣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣥⣶⣴⣶⣦⣦⣤⣴⣶⣶⣴⣦⣾⣶⣶⣶⣶⣦⣤⣤⣴⣦⣴⣶⣶⣶⣦⣤⣾⣥⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⢖⠀⠀⠀⠀⠀⠀⣇⢀⠀⠤⡐⠁⠀⠀⠀⢈⡇⠀⠀⠀⠀⡌⠀⠀⠀⠀⣠⡟⠁⠀⠀⡠⠻⣦
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⣀⣠⡤⠦⠤⠖⠒⠚⠋⠙⡄⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣸⠁⠀⠀⠀⣴⠋⠀⠀⢀⠞⠁⠀⢈
⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣀⡤⠞⠻⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⣦⠤⢤⣄⣀⣀⡞⠀⠀⠀⠀⡴⠁⠀⠀⢀⡾⠁⠀⢀⡔⢁⣈⠠⠔⠁
⠀⠀⠀⡞⠁⠀⣀⡤⠖⠋⠁⠐⢄⠀⠙⣆⣀⡤⠴⠖⠛⠉⠁⠀⠀⠈⢄⠀⠀⠈⠙⣷⠒⠤⣄⡞⠁⠀⠀⣰⠟⠄⠂⠉⠉
⠀⢀⣀⡷⠲⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⡧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⢸⠂⢠⠏⠑⠦⣀⡴⠋⠀⠀⠐⠢⡀
⠒⠉⠀⢧⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠤⢀⣀⡀⢀⡁⠀⠉⠐⠠⢯⣴⡃⠀⠀⣠⠞⠑⠦⣀⣀⠀⠀⡜⢀⠀⠀⢀⣀⣠
⠛⠒⠲⢴⡋⠀⠀⢀⡀⠤⠐⠂⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⡂⠀⢀⣴⠞⠁⠙⢳⣴⣋⠀⠀⠀⠀⠉⠉⠉⡏⠉⠙⢋⡉⠁
⠀⠀⠀⠀⠉⢩⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠘⠂⠙⣦⡴⠛⠁⠀⠀⣠⠿⡄⠉⠛⠶⣤⣀⠀⠀⠸⠃⠀⢠⠂⠀⢢
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠱⣄⠀⢀⡾⠃⠀⠘⡆⠀⠀⠀⠉⠛⢶⣦⣄⣀⣄⡀
⠀⠀⠀⠀⢠⣇⣀⣀⣀⣁⠀⠀⠀⠀⢤⡶⠾⡷⠶⠶⠖⠚⠛⠋⢉⣀⡀⠀⠀⢈⣷⡏⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡟⠁⠁⠉⠏⠉⠛⠛⠓
⠒⠛⠛⠋⢹⠉⠀⠀⠀⢹⡦⠤⣶⠒⠚⠇⠀⡧⠤⠐⠒⠒⠂⠂⠁⠀⠀⠀⣠⠞⠁⠀⠈⠈⠉⠉⣿⠦⣀⠀⢠⡎⠀⠀⠀⠰
⠀⠀⠀⠀⠘⡆⠀⠀⢀⣼⠀⠀⠸⡄⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢄⠀⠀⠀⠀⠀⢀⡇⠀⠀⢙⠾⣄⠀⠀⠀⠆⠀⠀⠀⠀⡜
⠀⠀⠀⣀⣠⣧⠴⠒⢋⡟⠀⠀⠀⢷⠀⣀⣀⣹⣀⣀⣄⣠⣤⣤⣤⡴⠋⠀⠀⠀⠑⡀⠀⠀⠀⣼⡀⠀⠀⠌⠀⠈⠓⢦⣰⠀⠀⠀⠀⡜
⠒⠋⠉⣿⣀⣸⡤⠴⠾⣷⠒⠛⠛⠛⣏⠉⠉⢹⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡼⠁⠀⠁⠂⠄⡀⠀⠀⢠⠏⠳⢤⣀⠌
⠀⠀⠀⢀⡟⠀⠆⠀⠀⠈⠣⡀⠀⠀⠸⡄⠀⠘⡇⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠡⢀⠀⠀⠀⠀⠈⠁⠢⣄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⡾⢧⣀⡁⠀⠀⠀⠀⠙⣄⠀⣀⣿⠉⠉⠉⠉⠉⠁⡼⠉⠉⠒⠤⣄⡀⠐⠀⠀⡜⠁⠀⠀⠁⠀⠀⠀⠀⠔⠀⠀⠑⠠⡀⠀⠀⠑⢤
⠀⠀⣼⠃⠀⠉⠳⣄⠀⠀⠀⡠⠞⢯⡁⠘⡆⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠉⠓⠢⣾⣁⠀⠂⠐⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⣸⠃⠀⠀⠀⠀⠈⢳⡴⠋⠀⠀⠀⠑⣄⢹⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠙⠲⠤⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠈⠂
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⢾⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠉⠓⠲⣄⣀⣀⣀⣀⣀⣠⣤⡴⠶⠶
⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠲⠦⣤⣇⣀⣀⣀⣽⣤⣤⣤⡤⠴⠴⡾⠷⠶⠖⠚⠚⠛⠛⢻⡛⠛⠉⠉⠉⠙⠫⢭⡁
⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⡈⠉⠡⣟⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⣶⣶⠿⠶⠶⢷⣶⣷⣾⣶⣾⣷⣶⣶⣶⣶⣶⣿⣶⣶⣶⣷⣿⣾⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣶⣶⣾⣷⣶⣿⣶⣷⣶⣿⣶⣾⣶⣾⣿⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⢶⠁⠀⠀⠀⠀⠀⠸⡇⢀⢀⠤⡰⠃⠀⠁⠀⠀⣸⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⢀⡼⠋⠀⠀⠀⡨⠳⣄
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣀⣀⣱⡤⠤⠶⠖⠒⠚⢻⡀⠀⠀⠀⠀⠀⢀⠀⢀⡟⠀⠀⠀⠀⣸⠃⠀⠀⠀⣠⡞⠁⠀⠀⣠⠎⠀⠀⢈
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣠⠼⢶⡉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⣹⣄⣀⣀⣀⠀⠂⣰⠋⠀⠀⠀⠀⡰⠃⠀⠀⠀⣴⠏⠀⠀⢀⠖⢁⠀⢀⡠⠋
⠀⠀⠀⢠⠟⠀⠀⢀⣠⠴⠚⠉⠦⡀⠀⠱⣄⠀⢀⣀⣤⠴⠖⠛⠋⠉⠀⠘⡀⠀⠉⠉⠛⢷⠤⣄⣀⢀⡞⠁⠀⠀⢀⡾⣃⠤⠐⠒⠋⠀⠉⠁
⠀⠀⠀⣸⣠⣴⠚⠉⠀⠀⠀⠀⠀⠈⠂⠀⠘⣟⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠘⡇⠀⢨⠟⢦⣀⠀⣠⡟⠁⠈⠐⢄
⣠⠴⠚⢻⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠁⠂⠤⢀⡀⠀⠀⠀⠀⠮⠁⠢⢄⣀⣰⠃⡰⠋⠀⠀⢈⡷⢯⡀⠀⠀⠀⠀⠑⠄
⣤⣀⣀⣈⣳⠞⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠂⢻⠀⠀⠀⠀⠀⠈⠐⠲⡀⠐⠣⡄⠀⠀⢀⣩⡿⠳⢦⣀⣠⠞⠁⠀⠈⠓⠲⠶⠦⣾⠴⠷⢤⠶⠖⠒⠛
⠀⠀⠈⠉⠳⡤⢄⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⢤⡊⠀⣠⠞⠋⠀⠀⢀⡽⠛⢦⣄⡀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠕⢄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣵⣞⠁⠀⠀⠀⣰⠟⠙⣆⠀⠈⠙⠳⢦⣤⡀⢀⠃⠀⠀⠆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⣠⡴⠞⠋⠀⠈⠳⡀⢠⡾⠁⠀⠀⠘⡆⠀⠀⠀⠀⠈⢹⡿⠶⠶⣴⣤⣤⣀⣀⣀
⠀⣀⣀⣀⣠⣯⣤⠤⠤⢤⣄⠀⠀⠀⠀⠒⣟⠛⣿⠛⠛⠛⠉⠉⠉⢀⡠⠤⠄⠀⠀⢀⣼⠯⢀⣀⣀⣀⡀⣏⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⠀⠀⠉⠉⠉
⠋⠉⠉⠉⠈⡇⠀⠀⠀⠀⣿⠶⠒⣟⠋⠉⠋⠀⢻⠒⠂⠒⠉⠉⠉⠁⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⡟⠒⢤⡀⢀⡟⠀⠀⠀⠀⠃⠀⠂⠀⠀⢠
⠀⠀⠀⠀⠀⢷⠀⠀⠀⣠⣿⠀⠀⠘⡄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠢⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢉⠿⢄⡀⠀⠀⠸⠀⠀⠀⠀⢀⡎
⠀⠀⢀⣀⣠⢼⡶⠖⠚⣹⡇⠀⠀⠀⢻⡀⣈⣀⣸⣇⣀⣠⣀⣤⣤⣤⣤⡼⠋⠀⠀⠀⠈⢄⠀⠀⠀⢀⣯⡀⠀⠀⠌⠀⠀⠙⠢⣄⡇⠀⠀⠀⢀⠎
⠒⠋⠉⢸⡁⣀⣧⡤⡴⠿⡗⠚⠛⠛⠛⣏⠉⠉⠉⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠡⡀⢀⡞⠀⠈⠑⠪⢀⡀⠀⠀⠀⡜⠙⠦⣄⢠⠎
⠀⠀⠀⠀⣽⠋⢠⠀⠀⠀⠙⢦⠀⠀⠀⠸⡄⠀⠀⢯⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⢀⠀⠀⠀⠀⠀⠈⠐⠢⣌⠀⠀⠀⠈⠳⣄
⠀⠠⠀⢴⣯⣄⢘⠀⠀⠀⠀⠈⠳⡀⠀⠀⢻⠒⠚⠛⠉⠋⠉⠉⡽⠉⠓⠦⢄⡀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠑⠂⠀⡀⢀⡠⠊⠀⠑⠢⡀⠀⠀⠈⠳⣀
⠀⠀⢠⡎⠀⠙⠻⣤⠀⠀⠀⠀⢀⡽⣶⠋⠉⣇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠉⠙⠲⢤⣠⡏⠀⠠⠀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀⠀⠈
⠀⢠⡟⠀⠀⠀⠀⠈⠳⣄⣠⠞⠁⠀⠈⠳⡄⠸⡄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠲⠦⣄⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠈⠢⢄
⢠⡾⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠙⣄⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣀⠀⠉⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣄⣀⣀⠈⣷⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⣀⣞⣀⣀⣄⣤⣤⣤⣤⣤⣤⣤⠮⠽⠶⠶⣶⠶⠞⠛⠛⠉⠉⠉⠁
⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠛⢿⠶⠶⠶⢒⡟⠛⠛⠛⠋⠉⠉⡟⠉⠉⠉⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⣀⣀⣸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣖⣐⣂⣸⣃⣄⣀⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣄⣀⣀⣀⣀⣀⣠⣀⣀⣈⣑⣦⣄⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⢀⣀⣤⢤⠴⠶⢶⠶⠶⢦⣤⣄⣀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣠⡶⠛⠉⠁⠀⠀⠀⠀⣶⣶⠂⠀⠀⠀⠘⠋⠓⢦⣄⡀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠙⢷⡄
⠀⠀⠀⠀⢀⣼⠋⠀⢀⣾⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠹⣦
⠀⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠀⠀⢀⠄⠊⡀⠀⠐⠒⢀⠀⣾⠐⡿⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⣧
⠀⠀⢀⡾⠁⠀⣴⠋⠀⠀⠀⠀⠀⢀⢀⡌⠃⠀⠀⠰⡈⢸⣿⠀⠀⠐⠀⠀⠀⢀⡠⢆⠀⣀⠀⠀⠀⠈⠹⣆⠀⠘⣧
⠀⠀⣼⠃⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⣦⠀⠀⣤⣾⣿⠀⠀⠠⠐⠄⡀⠈⠀⠀⠀⠁⣀⡤⠒⠀⠀⢿⡄⠀⢹⡆
⠀⢀⡿⠀⠀⣟⠀⡀⠀⠀⣸⠃⠄⠀⠀⠀⢉⠈⢷⣄⠸⣿⣿⠀⠀⠀⠠⠞⠀⢐⣠⣤⡶⠟⠉⠆⠀⠀⠀⠀⣷⠀⠀⣿
⠀⢸⡇⠀⢸⡇⣠⡀⠀⢀⡀⠀⠀⣀⠠⠂⠀⠀⠀⠛⣷⠼⢛⠐⢄⢀⣠⣴⣾⣿⣟⠁⣀⠀⠀⠠⠀⠀⠀⠀⢹⠀⠀⢹⡀
⠀⢸⡇⠀⢸⡗⣭⣿⡇⠀⠓⠦⢤⣤⣤⣤⣤⣤⣴⣾⣏⠰⣿⡷⠆⣿⡿⠛⠛⠛⠛⠛⠛⠓⠒⢀⠀⢸⡿⡿⢺⠀⠀⢸⡇
⠀⢸⡇⠀⠘⣇⠉⠀⠀⠀⡃⠀⠀⠀⣂⣼⣿⠿⠟⠋⠉⠀⠈⣠⡜⢷⣤⡀⠀⠀⠀⠄⠀⠀⠴⠈⠀⠀⠀⠀⣸⠀⠀⣼⠁
⠀⠈⣷⠀⠀⣯⠀⠀⠀⠀⠠⣠⣾⠿⠛⠏⠁⢠⠔⠀⠀⠀⠀⣿⣿⠂⠙⢧⠀⢀⠀⠀⠀⢀⠉⠃⠀⠀⠀⠀⡏⠀⠀⣿
⠀⠀⢻⡄⠀⠘⡖⠀⠀⠴⠚⠋⠀⠄⠀⠀⠀⠀⡘⠀⠘⠆⠀⣿⡏⠤⠒⠀⡁⠈⠀⠐⠆⠀⠆⠀⠀⠀⠀⣾⠁⠀⢸⠇
⠀⠀⠈⢷⡀⠀⠹⣤⠀⠀⠀⠀⠰⠀⠀⢤⡀⠀⢠⠀⣀⠀⠀⣿⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢠⡟
⠀⠀⠀⠈⢷⡀⠀⠙⣦⠀⠀⠀⠀⠈⠀⢴⠀⠐⠀⠀⢄⠀⠀⡏⠈⠉⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⡟
⠀⠀⠀⠀⠈⢻⣄⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠄⣴⠟
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠻⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⠏⠁⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠂⠀⠉⠓⠦⣤⣀⠀⢀⠀⠀⢛⠃⠀⢀⠀⢀⣰⡤⠴⠛⠁⠀⢀⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠈⠉⠙⠓⠒⠚⠓⠒⠛⠉⠉⠁⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⣴⣾⠿⠛⠉⠉⣿⡏⠉⠙⠛⠿⣶⣄⡀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠀⣀⠤⠒⢺⣿⢟⠒⠆⢤⡀⠈⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⡰⢎⣀⡴⠚⢻⣿⠸⠛⠲⢦⣈⡷⡄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⡜⢀⡼⠻⣶⣄⣿⣿⠀⡇⣠⣴⠟⣦⠘⢆⠀⠹⣷⡀
⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠀⣾⡇⠀⢸⢀⣾⣀⠀⠹⠿⣿⣿⣤⣿⣿⣧⣤⣸⣧⣘⡄⠀⣿⡇⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠸⣷⢿⣾⠁⠀⠒⠶⣿⡿⠿⣿⣶⣶⣶⣶⣶⣶⣾⡀⢀⣿⠛⠛⠛⠛⣛⣛⡿⠿⣿⡿⠒⠒⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⠀⢻⡉⢉⣽⣿⡏⢹⣿⣿⣿⣍⠉⢠⠏⢰⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠳⡈⢳⣾⠛⠁⠄⢸⣿⡇⠙⢻⣷⡏⢠⠆⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠙⠯⣈⠛⠶⢴⢸⣿⢧⠖⠋⣀⠝⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣷⣄⡀⠈⠑⠒⠼⣼⣿⠴⠒⠈⠀⢀⣴⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠈⠙⠿⣶⣦⣤⣄⣿⣧⣤⣤⣴⡾⠟⠋⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠯⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡇⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣾⡿⠟⠋⠉⣿⡇⠉⠉⠛⠿⣿⣦⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⡄⠀⠀⢠⣿⡇⠀⠀⠀⠰⠋⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⢰⣄⢸⡿⠇⢠⡾⠃⠀⠀⠀⢻⣿⡄
⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⢻⣿⣶⣶⣶⣾⣿⣧⣤⣤⣀⣀⣀⣀⠀⢀⡀
⠀⠈⠉⠀⠀⠀⠉⠉⠉⠉⠙⣿⣟⠛⠻⠿⠿⠿⡄⠀⠀⢀⣀⡀⠀⠀⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⢴⠇⠀⣾⣿⠈⠛⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡴⠀⠀⠀⠀⣿⡏⠀⠀⠀⣳⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠻⣿⣶⣤⣀⡀⣿⣇⣠⣴⣾⡿⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠦⠴⢸⣧⠶⢠⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠗⠉⠁⠀⠀⠀⢸⣷⠀⠀⠀⠉⠻⠶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠞⣉⡲⠄⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢰⠎⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠫⠤⠤⠉⠀⠀⠀⡀⠀⠀⢸⣿⡇⠀⡐⠀⠀⠈⠤⠎⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠁⠀⠀⠀⠀⠈⠓⢤⡐⠀⠀⢸⣿⣷⠀⠀⣠⠋⠀⠀⠀⠀⠀⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣙⣇⠀⢸⣿⣿⣠⠞⠁⠈⠀⠀⠀⠀⠀⠘⣧
⢲⡀⣶⠀⠀⣀⣀⣠⣤⣿⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣦⣸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢰⡆
⠀⠟⠘⠇⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⣻⠋⠋⠉⠁⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⡀⠄⠀⣴⠟⢻⣿⡇⠀⠈⣯⡉⠀⢀⠀⠀⠀⠀⠀⢠⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⠞⢁⠈⢸⣿⡇⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄⠞⡴⢀⠁⠀⠂⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠫⢦⣁⠙⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⣓⢔⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠵⣦⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⡀⣴⡾⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠸⡇⠰⠸⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⠿⣯⣭⣵⡋⠉⠉⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢛⣡⣭⣶⣆⣹⣿⡏⢀⣤⣶⣾⣻⣿⣿⣿⠿⢋⡍⠉⠽⠛⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠛⠿⠋⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⠶⠾⠋⠁⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⣠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠴⠟⠻⣾⣷⣷⣤⡄⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣴⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠁⠀⠈⢆⡀⠐⢦⢤⣀⣀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡛⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣼⡟⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣰⣿⠃⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⢿⣿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣀⣴⣿⠿⠛⠉⠀⠀⠀⣿⣿⠀⠀⠀⠉⠙⠻⣿⣶⣄⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⣀⡤⠒⠚⢹⣿⡟⡋⠑⠒⠤⣀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣤⡊⠁⣠⡤⠶⣿⣿⡇⡟⠷⢦⣄⠈⠑⣦⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢀⠞⠀⣹⣿⣍⡀⠀⣿⣿⡇⢠⠀⠀⢈⣿⣿⠀⠑⡄⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⡎⠀⣼⠃⠈⢿⣿⣶⣿⣿⡇⢘⣤⣾⡿⠃⠈⢷⡀⠸⡄⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⠀⠀⣸⣿⠀⠀⣸⣀⣸⡇⠤⠤⠤⠛⠻⣿⡿⠷⣾⣿⣿⣷⣶⣤⣼⣧⣀⣧⣀⡀⣿⣷⠀⠀⠀⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠈⣿⡾⢿⣿⠃⠀⠐⠒⠶⣿⣿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⡀⠀⣸⡟⠛⠛⠛⠛⠛⠛⠛⣛⣿⠿⣿⣿⠟⠒⠒⠀⢸⣿⣓⣛
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⡀⠹⣏⠉⠛⢻⣿⣿⡟⢻⣿⣿⣿⣶⡆⠉⠉⢹⡏⠀⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢧⠀⠹⣆⣰⣿⠿⠛⡆⢸⣿⣿⠛⢿⣿⣆⣠⡿⠀⡰⠁⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠈⠣⣀⡽⢿⣅⡀⠀⠃⢸⣿⣿⠀⠀⣨⡿⢿⣀⡼⠁⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠙⠦⣀⠈⠙⠳⢾⢸⣿⡿⠾⠛⠉⣀⠴⠋⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣦⡀⠀⠀⠉⠒⠢⠼⣼⣿⡧⠴⠒⠊⠁⠀⢀⣤⣾⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠛⢿⣷⣦⣄⣀⡀⠀⣿⣿⠀⠀⣀⣠⣤⣾⡿⠟⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢻⡟⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠁⠀⠀⣼⠀⣬⠱⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠘⠆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡇⠀⠘⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠸⣿⠀⠀⠀⢀⡄⠊⠀⠀⢸⡀⢻⣀⡎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠰⠈⠀⠀⠀⢻⣄⠻⣄⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢯⣹⡀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣷⡄⠀⠀⠀⠢⡀⠀⠀⠀⠹⣦⠉⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⢀⣠⡤⠞⠋⣁⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢼⠿⣦⣀⠀⠀⠀⢀⣀⣠⣤⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣿⢋⣥⣴⠾⢛⣩⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣸⣶⠴⠞⠛⠙⠛⠻⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣽⠟⣩⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⠉
⢠⡶⠶⢦⡀⠀⠀⢀⡄⣾⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⣡⣴⠞
⢿⣠⡴⠞⢧⡀⣶⢭⣀⠹⣇⠸⣧⣶⢀⠀⠀⠀⢀⣠⠤⠖⣏⢉⡝⠉⢸⡏⠉⢢⣄
⠀⢉⡵⠖⡛⢻⣿⣦⡽⠀⢙⣧⡈⠻⠿⠀⠀⠈⠀⠀⣠⠶⣼⣯⣽⣿⣿⣥⣤⠾⠿⠿⢳⡀
⠀⠈⠷⣦⠶⠞⠛⠲⠶⠚⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣬⣄⢀⣀⣀⣀⣀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⢹⣟⠉⠉⠉⠙⣯⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠞⢻⡷⠤⠴⣶⢿⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⢷⣶⣶⣿⠀⠋⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠇⠀⠀⠀⣾⠀⡄⠀⠀⠀⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡞⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠃⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⣏⣠⣤⣤⣤⡀⣿⠀⣿⠤⠤⡤⠤⠼⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⠛⢷⣶⣿⠀⠀⣿⠀⣿⠀⠀⣀⡴⠞⢿⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⠇⠀⠀⠪⢝⠻⣦⣿⠀⢻⣠⠞⠋⠀⠀⠈⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡟⠀⠀⠀⠀⠀⠁⠂⣿⠀⢸⠁⠈⠀⠀⠀⠀⢸⡆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠶⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⣭⣿⡄
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡀⠤⠤⢸⡤⠤⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⠔⣈⠠⠄⠒⠀⠁⢸⡇⢀⠐⠂⠌⢁⠂⠜⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣠⠄⢀⠈⠠⠀⢘⣠⣌⡄⢸⣧⣠⣄⡊⠀⠄⠉⠀⢀⠄⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠀⢄⠀⡡⣴⠞⠓⠈⠁⠀⢸⣿⠈⠉⠛⠹⢶⢔⠁⣠⠈⠔⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠂⡀⠐⠄⣠⡪⠊⠀⠀⡀⠐⠀⠀⢸⣿⡀⠂⠄⡀⠀⠙⢝⢄⠠⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⠐⠂⣰⠝⠀⠀⠐⢦⡀⠀⠀⠀⢸⣿⡇⠀⠀⢀⠴⠀⠀⠹⣣⠐⠂⠡⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⢠⠁⠒⣰⡝⠀⢀⠂⠀⠀⠙⢷⣄⠀⢸⣿⡗⢀⣴⠋⠀⠀⠀⠀⢸⢇⠐⠀⠆⢀⠉
⡃⠀⢀⡀⠀⠀⠀⠀⠀⡇⣄⢐⣀⣽⣁⣀⣤⣤⣴⣶⣾⣦⣝⠆⢸⣟⠴⠋⠁⠀⠀⠀⠀⠀⠀⠿⠀⠂⢰⠸⠀⠀⠀⠀⠀⢠⣀⣀⣠
⠖⣋⣩⠄⠀⠀⠀⠉⠉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡄⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢶⢶⠖⠒⠒⠀⠀⢀⠀⠄
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠇⢁⠒⠂⢭⡄⠀⠀⠀⠀⠐⣠⡶⢋⣾⡇⠈⢦⡉⠫⠉⠉⠉⡉⠀⢀⣳⠀⠒⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⡈⡄⠂⠘⣶⡀⠀⠠⠀⡴⠋⠀⢸⣿⡇⠀⠈⠙⠶⣄⠀⠀⠀⠀⣴⠃⠐⠀⠁⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⡐⠉⠘⢕⣄⠀⠈⠠⠀⠀⠨⣿⡇⠀⠀⠀⠀⠈⠂⠀⢀⣾⠂⠈⠂⠌⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠈⠁⣨⠳⢷⢄⠀⠀⠀⠀⢿⡇⠀⠀⠈⠀⠀⡠⣴⠟⢁⠈⠁⠀⠐⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠂⢉⠚⠶⢤⣀⣼⡇⠀⣀⣤⠶⠻⡊⠐⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢈⠀⠀⠀⢌⡀⠐⠀⠀⢸⡇⠀⢠⠀⠐⠀⡠⠄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠆⠉⠒⠤⢀⡁⢐⣚⡇⢐⡂⢈⡁⠤⠐⠈⠰⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠒⠂⠒⠒⡖⠒⠲⠶⣶⠶⠶⠤⠀⠀⣀⣤⡴⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠁⠀⠀⠀⠀⠀⢦⣄⠀⠀⢀⡀⠉⠓⠋⠀⠀⠀⠛⠟⠀⠀⠴⣦⠀⠁
⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⣠⠞⠀⢀⠠⣤⣄⣀⣠⣄⡏⣉⣶⠀⠀⠀⠀⣀⣁⣀⠀⠀⠀⠨⠁⠀⣀⠁⢠⡀
⠀⠀⢀⡴⠋⠀⡀⠀⠀⣀⡤⣄⠀⢀⡾⠗⣠⣤⢭⢥⣼⡟⢫⣤⣤⠀⣉⢸⣤⠽⢯⣽⣷⠆⢀⣤⣞⢉⣬⣜⣤
⠀⣠⠟⠁⠀⠠⢝⠀⢸⣿⠺⡥⣴⡈⢉⢁⣱⠾⠛⠢⠚⠛⣒⠶⠶⠟⠀⠀⠈⠉⢑⠂⣤⣶⣿⢛⣵⣿⠟⢘⡿⢷⡄
⡠⠇⠀⠀⠀⣴⠋⠀⠀⣀⣽⣿⠣⠈⠀⣽⣷⠾⠓⠀⠀⠀⡉⢀⣰⠀⠄⠀⢀⣠⣀⠀⠩⠭⡡⣾⣾⡁⠀⠻⠷⣬⣷
⠷⣄⣀⠀⢸⢿⢳⣄⢀⣦⣹⠟⠀⠀⢀⣠⣤⠤⣤⡤⠤⢤⣤⣈⡙⠿⣷⣀⣜⠛⠋⠁⣀⣜⣻⣭⣟⣁⠀⢤⣥⣼⠻⠗⡀
⡇⢠⣿⣵⣾⣿⠿⣷⡿⠋⠁⠀⢠⡾⠋⠁⠀⠀⠀⠀⢀⣀⠀⠉⠙⠳⣄⠈⠙⢧⣦⡘⠊⢻⣟⠛⠓⠈⠀⠈⠉⠉⣀⡾⡇
⡇⠨⣹⡄⢠⣱⠊⠉⠁⠀⠀⠀⠈⢷⣄⠀⠀⢠⡴⠛⠛⠛⢛⠓⣦⠄⠙⢷⡀⠀⠻⠋⠐⣯⣩⡿⠇⠀⢀⣴⣾⡛⣿⠿⠃
⢙⠀⠀⡞⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣝⠂⠐⢣⡞⡅⠀⠀⢀⠸⢻⣷⣆⣸⠇⠀⠀⣿⢦⢾⣿⠀⠀⢠⣼⢟⣟⠟⣁
⠘⡆⠉⠂⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡆⠀⠞⠀⠙⣓⢒⣉⣠⣄⣹⣿⣿⡆⠀⢈⣤⣍⠂⠁⠀⠠⣀⣵⣾⢁⠎⠘
⠀⠀⠀⣀⣀⡂⡀⠺⠖⠠⠄⠀⠈⠀⣩⣽⣦⠀⠰⣾⡿⣯⡉⠁⠈⠙⢷⣟⠛⠿⠛⢋⡹⣶⣠⣾⠞⣁⠌⣽⣿⢷⣶⡆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠐⠾⣿⡿⠋⠀⣶⠻⣷⠸⠇⠀⠀⠀⣸⠏⣀⡶⣶⣿⣿⣿⣇⠀⠈⠛⠁⠀⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠳⡟⣦⠀⠀⠀⠀⠀⠀⣿⣼⢫⡇⠈⢁⠹⣿⣿⣷⡜⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⢻⣇⠀⠐⠁⢸⣿⣌⣹⣧⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠆⠀⠀⠀⠀⠀⣿⠀⣩⠿⠀⠀⠈⠛⢻⣿⣻⣦⡀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⢱⣶⡄⠀⠀⣿⠉⡄⠀⠀⠀⠀⠀⠘⠻⠿⠄⠿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⡇⠀⠀⣿⢸⡃⠀⠀⠐⠃⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡇⠀⠀⢻⡈⠁⠀⣠⠄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⣼⠇⠀⠀⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⡿⠞⢂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢸⣷⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠰⠖⠀⠒⠂⠐⠒⠀⠠⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⢋⡡⠔⠋⠁⠐⢲⠀⠀⠀⠀⠀⠰⠀⠀⡔⠋⠈⠁⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡾⢥⠀⢠⠖⠊⠀⠀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⠠⠄⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢁⠤⠃⣴⡥⠤⠰⢃⠬⢅⠀⠀⠀⢀⡔⠂⠉⠉⠐⣆⡰⠓⠲⠃⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡀⣠⠞⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⣰⠋⠀⣴⢆⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡋⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠹⠶⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠉⠉⠓⠊⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⡀
⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢦⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⣀⠔⠲⠊⠙⠢⡀⠀⠀⣴⣩⠵⠆⠀⠀⠘⢧⡀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⢰⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠊⣁⣀⣀⠀⠀⠀⠙⠤⠴⠉⣇⠀⡠⢤⠀⠀⠀⠙⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡿⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠉⠀⠀⠀⢣⣀⠀⠈⢉⣀⠀⠀⠉⠀⡎⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⣨⢲⡆
⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠋⠀⠉⠉⠉⠒⡌⢂⠀⠀⠀⠈⢳⡉⣠⢭⡉⠉⠲⢤⠀⠀⡄⠀⡇
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⢢⠀⠀⠀⠀⠉⠁⠀⣇⠀⠀⠈⢧⠀⢇⠀⡇
⠀⠀⢸⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠣⡀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠘⣄⡈⡆⡇
⠀⠀⠸⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡈⣆⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠱⢣⡇
⠀⠀⠘⡇⠀⠘⡀⠀⠀⠀⠀⠀⠀⠙⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢳⢀⡇⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⢡⠀⠀⠐⣄⠀⠀⠀⠀⠀⠁⡘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢠⠏⠈⠀⠀⠀⠀⠀⠀⠜⠀⡆
⠀⠀⠀⠀⢧⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠒⠀⠉⠁⠈⠉⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼
⠀⠀⠀⠀⠈⢆⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⢧⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣀⠔⡂⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢎⡔⠁⡰⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⡰⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣄⠀⠀⠀⣀⡤⠚⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣀⠠⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠠⠤⠤⠤⠤⠤⠤⠶⠖⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⢀⡀⢤⣶⣂⡔⠒⢦⣤⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠶⠒⠒⠓⠒⢐⣀⣂⣂⣒⣒⠢⠤⠤⢉⣛⠱⢤⡑⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠠⠔⠊⣍⡤⠤⠐⠒⠉⠉⣰⠀⠀⠀⠀⠀⠈⠉⠙⠒⠤⣈⠙⠦⣤⡀⢀
⠀⠀⠀⠀⠀⠀⡠⢀⠕⠊⢀⡠⠒⠋⠁⠀⠀⡀⢀⠀⢠⡏⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⠙⠢⣈⢻⣦⠉⡀
⠀⠀⠀⠀⡠⢊⠔⠁⡠⠒⠁⠀⠀⡀⠀⠀⠈⠀⠀⢀⣾⡇⠀⣀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⢢⣿⠁⠈⡋
⠀⠀⢀⠊⡰⠁⡤⠊⠐⠀⠀⠐⠛⠋⠠⢤⡀⠀⢀⠞⡛⡇⠚⠁⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣷
⠀⠀⠉⡸⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠰⢈
⠀⢀⣶⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠈⠀⠀⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐
⠀⢸⢿⡜⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂⠀⠀⠀⠐⠀⠠⠌⣠⠇⠀⠄
⠀⠈⢝⡇⠰⠀⠀⠀⠀⠀⠀⠀⢀⢄⠘⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣰⡟⠀⠀⠈
⠀⠀⠐⡟⢄⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠊⡸⠁
⠀⠀⢀⠀⠀⣛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠴⠁⠄⠀⠁
⠀⠀⠈⢳⣄⠀⠀⠁⠻⠗⠄⠀⠀⢀⣀⣀⣀⣁⣀⡀⠤⡤⠔⠒⠈⠁⠀⠀⠄⠀⢀⠠⠐⠁⠀⡀⠊
⠀⠀⠀⠀⠙⢭⡂⢤⢀⠀⠠⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⠈⠀⠀⢀⢹⠏
⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠁⠒⠀⠠⠠⠤⠀⠠⠀⠠⠀⠠⠀⠀⠂⠈⠉⠀⢀⡶⠐⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠒⠰⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⠀⣀⣤⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠀⠀⣀⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠙⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡌⠈⠀⠀⠀⢀⣀⣤⣾⠛⠉
⠀⠀⠀⠀⢾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣨⣷⡄⢠⣤⣶⠟⠋⣡⡇
⠀⠀⠀⣀⣼⣧⠀⠀⠀⠀⠀⠀⣀⠤⣂⣥⣴⠾⠟⠋⠀⠿⠋⠉⡟⠛⠛⠋
⣠⣴⠟⠋⠀⠈⢷⡤⠤⠄⠂⣩⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⡶⠷⠶⠶⠾⢶
⠹⣄⠀⣀⣠⣴⠞⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠘⠿⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⡤⠤⠤⠤⠤⠼⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⢨⠀⠀⠀⢀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⡄⠀⠀⢸⡇⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣼⡇⠀⠀⢸⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠋⡇⠀⠀⢸⠻⣆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣸⠇⢨⠇⠀⠀⢸⡇⢹⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⡏⠀⢸⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⢷⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⢷⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⢠⡿⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠈⢷⡀⠀⢀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢙⡷⠟⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡆⠀⠀⠀⠀⠈⠳⣞⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠒⠒⠂⠐
⠀⠀⠈⠉⠁⣾⣿⣷⣦⣀⠀⠀⢠⠀⠀⠀⣠⣆⣉⣷⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣴⡟⠉
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡗⠀⠀⠁⠀⢰⣿⣿⣿⣿⡿⠛⠈⠹⡟⠉⢿⡅⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠉⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠋⠀⠀⠀⠁⠀⠰⣠⠄⠀⠀⣆⠀⣀
⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣌⣁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠀⠃⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠻⠿⣿⡦
⠀⠀⠀⠀⠀⠀⣺⣧⡶⠆
⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠚⠋⠉⠀⠀⠀⢀⣺⣾⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣿⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣴⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡦⠀⠀⠀⠀⠀⣴⣴⠿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠁⠀⠀⠀⠀⣨⣯⣤⡀⠘⠺⡟⣧⣤⣬⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⣰⣾⣿⣿⣿⣧⣀⣤⡀⠉⢉⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡝⠿⠿⠿⠿⣿⣿⣿⣿⠏⠹⣿⡇
⠀⠀⢀⣠⠶⠂⠀⠀⠀⣴⢤⡀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⡷⠀⠀⠈⢹⣿⡏⠀⠀⠈⢁
⠀⠰⠋⠀⠀⠀⠀⠀⢀⣾⣿⣦⣤⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⡿⠟⠋⠀⠀⠀⠀⠀⠹⠧⠀⠀⠸⡌
⠐⢂⣄⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟
⠀⢸⣿⣿⣦⣄⡀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣟⣿⣿⡿⠋
⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠛⠀⣶⠇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⡇⢠⠹⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢸⡆⠱⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⡇⠀⠀⠀⠀⡠⠀⢳⠈⣧⢀⠻⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠀⠀⠀⠈⣿⠀⠀⠐⢣⠀⠀⠀⢣⡘⣟⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⡻⡄⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠳⡌⠓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡶⠋⣁⣤⠶⢛⣡⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠿⢷⣄⣀⣀⣀⣀⣤⡴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣵⢶⣟⡩⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠼⠷⠚⠋⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⣠⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⢛⣩⠄
⠀⠀⠀⢿⣠⠶⢷⡀⣶⣅⠘⣧⠸⣇⢆⠀⠀⠀⢀⠠⠴⠞⢫⡛⠉⡶⠙⣥⡀
⠀⠀⠀⠀⣭⠶⠋⢹⣷⡬⠇⣈⠧⠙⠻⠃⠀⠀⠀⢠⠶⣾⡯⠿⠿⠧⠶⠿⠟⠻⡀
⠀⠀⠀⠀⠘⠳⠞⠋⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠻⠷⣤⣤⣤⡴⣶⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⢀⣿⣦⣀⣀⣸⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⣯⣭⣽⣧⠻⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠀⠀⠀⠀⠉⠀⠀⡆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠃⠀⠀⢸⠀⣧⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡞⠀⠀⠀⢸⠀⣿⠀⠀⠀⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠁⠀⠀⠀⢸⠀⣿⠀⠀⣀⣸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡿⢯⣽⣏⠉⢹⠀⣿⠈⠉⣩⠼⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡸⠀⠀⠋⢛⠧⣼⠀⣿⡴⠟⠁⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠃⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡟⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠘⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⢣⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠈⣄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⢀⠀⢠⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡆⠀⠀⠀⢀⠀⠀⠀⠈⠀⢀⠂
⠀⠈⠀⠀⣤⣤⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠛⠃⠀⠘⠂⠀⠀⠀⠈⠁⠀⠀⢒⣀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⢡⡍⠀⠀⣾⡍⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠠⣼⣧⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠁⠐⠀⢀⣀⣤⠀⠀⣤⣤⡤⠶⠖⠓⠀⠀⠛⠉⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣤⡄⠶⠶⠀⠈⠁⠀⠹⠀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠄
⠀⠷⠿⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⢠⠍⠀⠀⠉⠁⠀⣆⣤⣥⣤⠀⠀⠀⢻⡄⠀⠈⠁⠈⠄⠒⠀⠀⠃
⠀⠀⠀⠀⠀⢰⣆⠀⠀⠰⣄⡀⠀⢸⠆⠀⠰⠀⠀⠀⠉⢀⡤⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠘⠁⠀⠀⠐⢀⡀⠒⠀⠀⠀⠉⠁⠀⢰⡆
⠀⠄⠤⠀⠀⠀⠀⢰⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠈⠀⠀⠠⠀⠀⢸⡇⢽⠀⠀⠀⠀⠰⠂⠟⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠉⠓
⠀⠀⠂⡄⠀⠃⢀⠸⠇⢽⡄⠀⠀⡀⠀⠀⠷⠷⠶⢶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠟⠀⠀⠙⠲⠆
⠀⠀⠂⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠂⠀⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠒⢲⠈⠉⢹⡍⠉⠁⡖⠂⠠⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡊⠀⠀⠃⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⠀⠈⢱⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢀⡏⣿⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣁⡴⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠁⠈⠻⣷⣄⡀⠀⠀⣀⠥⠐⠂⠉⢹⠇⣿⡉⠉⠐⠂⠭⣀⠀⠀⢀⣤⣾⠟⠀⠈⠁⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠄⠀⠀⠀⠀⠙⢮⠻⣶⣊⠀⠀⠺⠇⠀⣸⠀⣿⡇⠀⠸⠇⠀⠀⣱⠶⣫⡾⠁⠀⠀⠀⠀⠰⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⡠⠳⣌⠻⣷⣤⡀⠀⠀⣿⠀⣿⣧⠀⠀⢀⡴⠚⣡⣾⠛⢄⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⢤⡄⡜⠁⠀⠈⢧⡈⠻⣿⣦⣄⡇⠀⣿⣿⣠⠖⠋⣠⣾⡟⠁⠀⠈⢆⢠⡤⠀⠀⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠀⡸⠀⠸⠇⠀⠀⠙⣄⠈⠻⣿⠇⠀⣿⣿⡅⣠⣾⡿⠋⠀⠀⠺⠃⠈⡆⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⢄⠤⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣆⣀⣠⣤⣤⣤⣤⣾⣷⣶⣿⣷⣄⣿⡿⠛⠛⠛⠷⠤⠤⠤⣤⣄⣀⣰⣀⡀⠀⠀⠀⠀⠀⠁⠀⡄⢠
⠀⠀⠀⢀⡖⣿⠀⠀⠀⠛⠒⠺⠿⠿⠿⢿⣭⣝⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠿⠓⠒⠾⠀⡼⢉⠁
⠀⠀⠀⠈⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠉⠉⠉⠉⣹⣶⣶⢾⣿⣿⠈⢻⣟⠛⠻⣏⠉⠉⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⢀⡘⡀⠸⠇⠀⢀⣴⣿⠟⠁⣸⣿⣿⠀⢸⣿⣷⣄⠘⢦⡀⠀⠺⠆⢠⢃⡀⠀⠀⠀⠀⠀⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡠⠀⠀⠀⠀⠉⠁⠱⡄⠀⣠⣿⠟⢁⡴⠊⠁⣿⣿⠀⣸⠈⠛⢿⣷⣄⠱⣄⠀⣠⠃⠈⠉⠀⠀⠀⠀⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢈⣾⠟⣡⠞⠉⠀⠀⠀⢿⣿⠀⡟⠀⠀⠀⠉⠻⣷⣌⢳⡁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⠁⠀⠀⠀⠀⣰⣟⡵⠛⠣⣀⠀⠻⠃⠀⢸⣿⠀⡇⠀⠘⠇⠀⣠⠜⠛⢷⣝⣆⠀⠀⠀⠀⠘⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠄⢠⡾⠟⠁⠀⠀⠀⠀⣍⡂⠂⠤⢼⣿⢸⠥⠄⠖⢀⣭⠀⠀⠀⠀⠈⠻⣷⡀⠠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠋⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⢙⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡆⠀⠀⢀⠀⠀⢸⡇⠀⠀⡀⠀⠀⢲⣀⡠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⠠⠄⢼⠅⠀⠀⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⡀⠀⠀⣤⣤⣤⣄⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠒⣀⣀⣀⣀⣠⣶⣤⣤
⠀⠀⠀⠀⢐⣿⣿⣿⣷⣤⣤⡘⣷⣍⢠⢵⣼⠋⠐⢿⣿⣿⣿⣿⡖⠀⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⠿⠿⣿⣿⣿⣿⣿⣿⣯⣼⣬⠟⢷⡀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣀⡀⠀⢂⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠁⠀⣤⠛⠁⠀⣿⠟⠉⠠⠤⠀⠀⢀⣼⢿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⡄⠘⣻⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣿⣿⠿⠉⠂⠀⠀⠀⠀⠀⠀⠉⢹⣾⣿⣿⣿⠿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⣁⡈⠈⠛⠶⢶⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣷⣶⡿⣿⣿⣛⡿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⠿⠟⠁⠀⠀⢻⡏⠀⠘⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠁⠀⠀⢤⡄⢀⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠃⠃⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⠃⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁

⠀⢦⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠈⢲⠀⠀⠀⠀⠀⠀⣇⣀⠀⠄⠜⠀⠀⠀⠀⣇⠀⠀⠀⠀⡼⠀⠀⠀⠀⣠⠏⠀⠀⠀⠜⠳⣄
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠳⡄⣀⣠⠤⠵⠶⠒⠚⠋⠉⠹⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢰⠇⠀⠀⢀⡼⠃⠀⠀⡠⠊⠀⠀⡸
⠀⠀⠀⢱⡀⠀⠀⠀⠀⣀⡤⠒⠙⢇⠀⠀⠀⠀⠀⠀⣀⣠⡤⠼⡦⠠⠤⢤⣀⣰⠁⠀⠀⠀⡰⠃⠀⠀⢠⡞⠁⠀⢀⠚⠠⠀⠠⠊
⠀⠀⠀⡏⠀⣀⠤⠖⠋⠁⠈⠢⢀⠈⢣⣠⠤⠖⠚⠉⠁⠀⠀⠀⠘⡄⠀⠀⠀⢹⡟⠒⢦⣜⠁⠀⠀⣰⠟⠂⡁⠁
⠀⣀⠤⡗⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠄⢀⠀⠀⠀⠀⠀⠀⡲⠀⡀⠀⢀⡇⢠⠏⠈⠓⢦⣼⠃⠀⠀⠈
⣉⠀⠀⠳⣸⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢧⠀⠀⠀⠁⠒⠠⢤⠄⠠⢁⠀⠀⠁⢚⣶⢧⣀⠀⣠⠟⠈⠓⠤⣤⣀⣀⣆⣤⣀⣀⣤⠤⠔
⠉⠉⠙⠻⣅⡄⠈⡠⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢁⠀⣈⠂⢀⡴⠟⠁⠀⢉⡿⢧⣄⡀⠀⠀⠀⠀⢸⡄⠀⠈⡠
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠈⠀⣈⣶⡋⠀⠀⠀⣰⠏⠳⡀⠈⠙⠷⢦⣄⠀⡘⠀⠀⠎⠀⠈⠢⡀
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣀⣀⣠⣤⠴⠚⠉⠀⠙⢆⣠⡾⠁⠀⠀⢳⠀⠀⠀⠀⠈⣹⠷⠶⢶⡦⣤⣤⣀⣘
⣀⣠⣤⡤⣼⠶⠶⠶⠶⣄⠀⢀⣀⢈⣿⠉⡏⠉⠉⢉⣉⣁⡀⠔⠐⠀⠀⢀⡼⠛⠤⠤⠤⠠⣸⡀⠀⠀⠀⣰⠃⠀⠀⠐⠀⠀⠀⠀⢩
⠁⠀⠀⠀⢹⠀⠀⠀⠀⡟⠉⠹⡍⠀⠁⠀⢿⠉⠈⠀⠀⠀⠀⠀⠀⠀⣰⢟⠁⠀⠀⠀⠀⠀⣼⠉⠒⢤⣸⠃⠀⠀⠀⠁⠀⠀⠀⠀⣌
⠀⠀⠀⠀⠘⡄⣀⣠⢾⡇⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠈⠢⡀⠀⠀⠀⢀⡇⠀⠀⠰⠉⠳⢄⡀⢸⠀⠀⠀⠀⡔
⣠⠤⠖⡶⠋⢏⣁⣀⣾⣤⣤⡤⠼⣷⠶⠶⢾⡖⠒⠓⠛⠛⠛⠛⠉⠀⠀⠀⠀⠐⢀⠀⢀⡞⠁⠠⣠⠁⠀⠀⠀⠉⠶⣄⠀⠀⡸
⠀⠀⠀⢓⡖⢛⠉⠉⠈⢣⡀⠀⠀⠸⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞⠀⠀⠀⠀⠀⠐⠀⢀⠈⠀⠀⠙⢮⡀
⠀⠀⠀⣼⣃⠰⠀⠀⠀⠀⠙⣄⠀⠀⢳⠶⠒⠓⠒⠒⠒⣲⠓⠦⢄⡀⠀⠀⠀⠀⢀⠞⠀⠁⠒⠠⠀⠀⠀⡠⠊⠑⠠⡀⠀⠀⠙⢄
⠀⠀⣰⠇⠉⠻⣤⠀⠀⠀⠀⣨⢶⡚⠉⡆⠀⠀⠀⠀⣠⠃⠀⠀⠀⠈⠙⠲⢤⣀⡞⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠙
⠀⣰⠇⠀⠀⠀⠈⠳⣄⡴⠊⠀⠀⠱⡄⠸⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠒⠦⣄⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀
⢰⠏⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⢦⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠄⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠦⣤⣀⣇⣀⣀⣀⣻⣀⣀⣄⣤⣤⣤⣾⠤⠴⠶⠶⠶⠶⢶⡖⠲⠛⠛⠛⠛⢯⣏⠉⠉
⠉⠙⣲⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⡉⠉⠉⣟⠉⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀
⣶⣶⠷⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣷⣷⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣷⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠙⠢⠀⠰⠚⠀⢛⣤⡦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⢴⣾⣿⡏⠉⠹⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⣀⡀⣽⣿⣿⣿⣿⣿⣿⣿⣤⣀⠙⠋⠉⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠃⢀⣠⣄⢶⡄⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⠤⠛⠛⢻⣥⣷⣾⣧⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢦⠀⠀⠀⣠⣴⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠁⠀⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡇⠀⠸⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⠈⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠅⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠖⠰⠶⠶⠒⠶⠶⢶⠲⠶⠶⠶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣶⣬⠢⠦⢶⠛⣶⣴⣦⣶⣾⣿⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠙⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣻⣆⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⠿⣷⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢟⣤⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣫⣶⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢋⣤⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢧⠹⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣫⣴⣿⣷⣞⣿⣿⣿⠿⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⠹⣥⣾⣿⣿⣿⣿⣿⡷⣟⡿⢷⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣋⣴⣷⣌⣻⢿⣿⣿⡟⣡⡴⠋⠦⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣡⣾⣿⣿⣿⣿⣿⣾⡿⢛⣿⠟⣫⣴⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣫⣴⣿⣿⣿⣿⣿⣿⠟⠉⠑⣝⢓⠼⣟⡔⢻⡛⠿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢟⣡⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⠋⠻⣿⡄⠀⠙⢦⡀
⠀⠀⠀⠀⣠⣴⣿⠿⣫⣶⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⣷⣶⣦⠀⠙⢦⡀
⠀⠀⣤⣾⡿⢟⣵⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣸⣿⡏⢹⣿⣎⢧⠀⠀⠙⢦⣀
⠀⠈⣿⣫⣴⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡗⡇⢠⡟⢿⡎⢧⠀⠀⠀⠙⢷⣄
⠀⠀⠉⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⡏⣽⡇⣀⡇⠈⢿⡜⣇⠀⠀⠀⠀⢹⣯⡻⣄
⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⠃⣿⡇⣦⡇⠀⠈⢿⡘⣆⠀⠀⠀⠀⠙⠿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⡏⠀⣿⡇⠋⡇⠀⠀⠈⢷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠁⠀⠙⠓⠃⠃⠀⠀⠀⠈⠗⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣸⠁⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠦⣄⡀⠀⠀⣤⣀⡌
⢀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠠⠀⢠⠃⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⢳⡾⡴⣻⠉⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⡀⢠⠞⠒⠂⠀⠀⠐⠊⣒⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠦⠴⠶⢾⠶⠖⠒⠒⠒⠿⣅⡷⠃⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠤⠤⢀⡤⢎⠀⠐⠈⠁⡰⠃⠀⢀⣠⡤⠖⠚⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠀⠀⠀⠀⠀⢀⡞⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠑⠁⠀⠀⠀⢸⢁⡴⠊⠉⠀⠀⢀⣠⠤⠤⢶⡶⠖⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⡟⠉⠉⠉⠉⠉⠛⢶⢤⣀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠎⠀⠀⢀⡴⠚⠉⠀⠀⠀⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠓⠦⣄⡀⠈⠉⠒⢤⣀⡀
⠀⠀⠀⠀⠀⡠⠀⢐⠞⠙⢦⣀⡴⠋⠈⠢⡀⠀⠀⢰⠀⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠉⠑⠲⢤⣀⠀⠉⢙⠒⠦
⠀⠤⠖⠂⠈⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠈⠢⡀⠆⠀⠀⠲⠀⠀⠀⠀⠀⠀⢀⠀⡄⠼⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⢮⣀
⠀⠀⠀⠀⣠⡞⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⢂⠀⠀⠀⠀⠀⣸⠊⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⡾⠻⣄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠲⡀⠐⡀⠀⠀⡠⠊⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀
⢀⣴⠟⠁⠀⠹⣦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣡⡠⠊⠀⠀⡆⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⠤⣄⣀
⠟⠁⠀⠀⢀⡴⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠂⠐⠂⠓⠂⠀⠠⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠙⢲⠦⢄
⠀⠀⢀⡴⠋⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⣠⠁
⠀⣰⠟⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⡐
⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠲⢤⣄⡀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢱⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠘⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠄⠙⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠄⠀⠤⠂⠀⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢦⣀⣀⠀⠀⠀⠀⠀⣀⠤⠘⠳⣄⠀⠀⠀⠀⠀⣀⡤⠐⡚⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠴⠂⠉⠀⠀⠀⠀⠀⢙⣦⣄⡂⠈⠀⢀⠜⠀⠀⠐⡄
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣾⣿⣶⣶⣶⣶⣶⣶⣿⣷⣾⣾⣿⣷⣶⣾⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣤⣶⣦⣄⠀⠀⠀⣠⡄⠲⠄⠀⡂⣼⠾⣿⣿⣶⣤⣶⣦⡀
⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆⣺⡟⣨⢋⢷⣯⠀⠀⢹⣿⣿⣿⡿⠋⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠀⢀⣴⣶⣄⣀⣀⢂⣠⣀⣀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠘⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠟⠏⠩⣴⠿⠀⣠⣿⣿⠿⠏⠀⠀⠀⠀⠀⣠⣤⣀⠀⢐⣀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⡀⣀⣾⣧⠄⠀⠀⠛⠁⠀⠀⠙⠃⠀⠀⣠⣾⢻⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⣿⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣀⡆⢀⡙⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠈⠃
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠙⢯⢿⠿⣡⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣇⢀⣴
⠀⠀⠀⠀⣿⣿⣿⡟⠛⠛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⣶⣦⡀⣈⡈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠈⠁
⠀⠀⠀⠀⠀⣿⡟⠀⡀⠠⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠛⢿⣥⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⠟⠀⠀⠙⣿⡿⠋⠈⠿⣿⣯⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⠄⠀⠀⠀⠀⠹⣇⠀⠀⠀⠏⠛⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡂⢠⣾⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢬⡉⠈⠀⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠅⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⣀⣠⣤⣤⣄⣀⣀⣀⡀
⠀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⢄⣀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣦⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠈⠀⠈⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠛⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠛⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀
⠀⠠⠾⠿⡤⢮⡌⠷⠺⠦⠶⣰⢾⡝⠿⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠉⠉⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠂⠀⠀⠀⠐⠚⠡⠋⠀⠉⠁⠁⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠤⠀⢄⣀⢀⠀⠤⠤⠤⠀⠄⠀⣠⣄⣤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⠀⠀⢀⠙⠉⠉⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢽⣯⡳⠀⠀⠀⠀⠂⠒⠀⠉⠀⠛⠃⠀⠈
⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢛⡷⡆⠁⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣆⠀⠀⠀⠀⠀⠈⣃⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣢⣍⡰⠀⠀⠀⠀⠠⠀⠀⠀⢤⣤⠀⠀⠘
⠀⠲⠄⠒⠠⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠀⠁⠀⠀⢀⡤⠤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠀⠀⠀⠀⠤⠤⠤⠂⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠲⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⢠⡔⠒⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠅⠂⠀⠀⠀⠀⠀⢠⣌⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣤⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠀⠀⠄⠠⠠⢤⢠⠀⠀⠀⡀⡒⡤⠀⠀⠀⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠀⠈⠈
⠀⠀⠀⢀⠠⠄⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⠀⠴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒
⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠄⠀⠒⠦⠤⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⡻⢿⢻⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⣻⡿⠞⢛⠉⠉⠉⠉⠛⠻⢿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⠟⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠓⢌⠳⣽⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠣⠙⣟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠇⡇⢹⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠰⠘⠀⠘⢰⠀⡎⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢏⡎⠁⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠺⣷⡀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠚⣡⠞⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢿⣦⡈⠒⠆⠀⠀⠀⠀⠀⡴⠏⢁⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⠟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡊⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠈⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣇⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣻⠦⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⡍⠇⠨⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣽⣬⣿⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣁⢠⢢⢤⡾⠧⡒⠂⡀⠀⠀⢀⡜⠁⠀⠀⠀⣁⠀⠀⠀⢀⠂⣤⡤⠍⠀⠀⠀⠀⠀⠤⠤
⠀⠀⢀⡐⢋⡀⠀⠀⠀⠀⠀⠉⢀⠙⠚⠼⣧⠉⠨⠀⠔⠂⠀⠐⠀⠀⠀⠊⠀⠀⠐⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠉⠁⠠⡇⠀⠀⠀⠀⠀⠈⠰⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠘⡃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠠⡄
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⡄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠤⠀⠁⣂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠷⠾⠁
⠀⠀⠀⠀⠀⠀⠿⠀⠰⠛⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠒⠢⠀⣀⠀⠀⢀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠔⠁⠀⠨⡀⣴⡁⡙⠀⠘⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠃⠀⠀⢀⡈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠒⠀⡀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠰⠀⠰⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣤⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢀⠜⠈⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡮⠀⠀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⠔⠁⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠌⠀⠀⢄⡀⠀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢩⠀⡆⢨⠙⣆⠀⠀⠀⠀⠀⠀⠰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠤⢭⣉⡃⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣘⠀⠤⢘⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣦⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⢀⣀⣀⣰⣿⣀⣄⣀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⣀⣠⡶⠟⢻⠉⠁⠀⡙⣿⡁⠀⠉⢹⠛⠵⢦⣤⣀⡤⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⣠⡴⠛⢏⠁⠀⠀⠘⠀⣀⣀⠃⣿⣇⣀⠀⠈⠀⠀⠀⢈⠟⠷⡤⢀⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡝⠁⡀⠀⠈⣂⠤⠒⠊⠉⠀⠘⠀⣿⣻⠀⠈⠉⠒⠠⠄⡊⠀⠀⢀⠛⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠁⡤⠜⢉⠔⠉⠀⠀⠀⠀⠀⠀⡸⠀⣿⣿⡄⠀⠀⠀⠀⠀⠈⠐⢄⠙⢷⠄⠑⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢎⠀⠀⢀⠔⠉⠳⣦⡀⠀⡠⣔⡲⠘⡇⠀⣿⣿⡿⠴⣒⡤⡀⠀⣠⣴⠋⠑⢄⠀⠀⢈⢶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠁⢀⠎⠀⠀⠀⢉⡻⣿⣶⡉⠀⢸⠀⠀⣿⣿⣿⠀⠀⣁⠞⣫⣟⠁⠀⠀⠈⢂⠀⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠀⠀⠀⡌⠀⠀⠀⢀⣊⠟⢊⠻⣿⣦⣌⠀⠀⣿⣿⣿⡤⠚⣡⣾⠟⢟⣢⠀⠀⠀⠈⣄⠀⠀⠘⣣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⢘⠀⠀⠀⠀⢾⠃⠀⠀⢡⡈⠻⣿⣆⠀⣿⣿⠏⣠⣾⣿⠃⠀⠀⢣⠆⠀⠀⠀⠸⠀⠈⠉⢩⡄
⢤⡄⣀⠀⡄⠀⠀⠀⠀⡇⠀⢀⣀⣃⣠⣤⣤⣾⣷⣶⣾⣿⣿⣿⣶⣬⡿⠛⠉⠛⣾⠟⠋⠁⠉⠉⠐⠚⠛⠦⠠⠤⣀⣇⣀⠀⢈⡗⠀⠀⠀⣤⠤
⠈⣷⠙⣦⠃⠀⠴⠶⠶⣿⣿⡿⠿⠿⠷⠿⠿⠟⠻⠟⠛⠿⠛⠿⠛⢻⠀⢸⣿⠆⢸⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡷⠶⠶⠆⣹⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡏⠑⠒⠂⠰⣤⠤⠀⢀⣀⣀⣤⣾⢷⣤⣤⢴⣟⠻⢿⣿⣿⡿⠿⢿⣿⠛⠛⠋⠉⡍⠉⠀⢸⡇⠀⠀⠀⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣄⠀⠤⠱⠀⠀⠀⠀⢛⣆⠀⠀⣼⣿⠟⢁⣾⣿⡇⠈⢿⣷⣄⠙⢄⠀⢀⣚⠃⠀⠀⠀⢰⠡⠄⢀⣞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣵⡄⠀⠀⢇⠀⠀⠀⠀⢡⣷⣾⠟⣡⠔⢻⣿⣿⡇⠀⢀⠛⢿⣷⣌⢲⣧⠉⠀⠀⠀⢀⠎⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⡠⠌⢆⠀⠀⠀⣠⣟⣥⡾⣁⡀⢸⣿⣿⡇⠀⠸⠀⣀⡽⣻⣷⣅⡀⠀⠀⢀⠎⠤⣀⡰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢯⡄⠀⢀⠳⣀⡼⠟⠁⠀⠀⠉⠺⢿⣿⣿⡇⠀⣿⠿⠚⠁⠀⠀⠙⠻⣄⡰⠊⠀⠀⣸⠟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠃⢆⡈⠳⢀⡀⠀⠀⠀⠀⠀⢻⣿⡇⢠⠃⠀⠀⠀⠀⠀⣀⠤⠊⢤⡗⣀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢷⣄⠀⠀⠀⡈⠑⠠⠄⢀⡀⢸⣽⡇⢰⠀⣀⠀⠤⠐⠉⡀⠀⠀⢀⣴⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠐⠃⣠⣎⡀⠀⠀⠀⠄⠀⠈⣿⡇⡌⠀⠀⠄⠀⠀⠀⡈⣦⡔⠃⠁⠀⠀⠙⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠐⠋⠉⠉⠓⠦⣾⣤⣀⣀⣹⣧⣁⣀⣠⣼⡦⠶⠋⠋⠉⠓⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠉⢹⣿⠉⠈⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⣀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⢀⡔⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠉⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⡀⠐⠁⠀⠀⠔⠃⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠒⠒⠚⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠠⠄⠐⠈⠀⢀⠠⠐⠉⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠈⠀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠆⣢⣀⡄⣤⡲⠄⠀⠀⠀⡆⠀⡠⠊⡠⠐⢀⠈⢀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡤⠤⠞⠀⠐⠀⠈⠁⠰⠀⣀⡁⢨⠀⠀⢠⠌⠠⠊⠀⠀⠀⠐⢄⠹⠀⢀⡌⠀⠐⡀⡀⠐⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⣨⠶⠛⠓⠶⣕⠉⠀⠀⠀⠀⠀⠘⢀⠁⠀⠀⠀⠀⠀⡐⠙⠰⠒⠆⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠆⠀⠈⣼⠃⠀⠀⠀⠀⠹⡷⠀⠀⠀⢀⠀⠐⣎⠂⠀⠀⠀⠀⠀⠑⣣⠒⠐⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣸⢗⠀⣀⠀⠤⡄⠀⠉⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⣀⣝⣲⠦⢴⠞⠋⠈⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⣀⢀⡁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢏⠁⠀⠁⠈⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡑⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠛⠛⠚⠓⢤⣀⠀⠀⠁⣀⡤⠛⠒⠚⠛⠓⠒⢶⣄⠠⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠈⠈⠀⠉
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣠⣜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⡄⠀⠀⠀⠀⠈⠂⠘⠁
⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠤⠴⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⣁⣀⣀⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣦⣀⠀⠀⠀⢀⣀⣴⢛⣀⢂⡀⡛⣦⣀⡀⠀⠀⠀⣀⣴⣃⣀⣀⠀⡐⡀⢀⡀⣀⢐⣀⣀⣐⣀⣂⡐⣀⡂⣐⣀⣀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠨⠀⠜⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠄⠄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢌⣛⠀⢆⠀⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣐⡐⠀⠀⠠⠅⠀⠠⠀⠐
⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢁⡀⠨⠰⠔⡀
⠀⠀⠀⠈⠀⡀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠋⠈⠚⠁
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢳⠀⠀⠀⢀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⡳⠒⠊⠉⢀⠎⠀⢡⠀⠀⠀⢀⣀⠤⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠕⠒⢦⡀⠀⠀⡜⢁⠤⠒⣤⣏⠀⠀⠀⡵⠒⠉⠁⠀⡰⠁⠘⡄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⡠⠤⡙⢲⡎⠀⢸⠀⡰⠁⡜⠀⠀⡜⢀⢤⡀⠀⡴⠁⠀⠀⠘⣄⣀⠤⠔⠚⠉⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢧⠀⡧⠃⠘⠒⢋⡜⠀⢀⠑⢦⣎⠴⠋⢀⠇⡼⠁⡎⠉⡇⡰⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡠⠔⢁⡜⠁⣀⣀⣀⠞⠯⢇⡿⣁⠎⡤⡀⣠⢋⡜⠀⠰⣇⠀⡙⠉⢱⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠘⠤⢤⠞⠀⡞⠁⢠⢯⠀⡔⠉⢠⠿⡐⠦⠺⢴⡟⠹⡄⠀⠈⡛⠲⠒⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢠⠒⠲⡄⠀⣠⠋⠀⡜⠁⡰⠃⢸⠀⠹⣤⣋⠜⠁⠀⢀⠞⡠⠴⠁⠀⡼⠁⠀⠀⠀⡰⢫⠃
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣀⠞⠀⠘⣅⡴⠳⡀⠀⠳⡴⠁⠀⠻⠤⣶⢧⡀⠀⠀⢀⠎⠀⢳⠀⠀⡼⢁⠤⠒⠤⠚⢁⠏
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡇⠀⠀⢀⠞⠁⠀⠑⣄⣼⠇⠀⠀⠀⡰⠁⠰⢅⠀⢀⠞⠀⡴⠋⢀⡼⠁⠘⠒⠲⠚⢉⠏
⠀⠀⠀⢠⠞⢣⡀⠀⣠⠤⣹⠀⣰⠋⠀⠀⠀⢀⠟⠉⠀⠀⠀⡰⠁⠀⠀⠈⣇⠎⠀⠀⠧⠚⡽⠹⡀⠀⢀⠀⢀⠎
⠀⠀⠀⠹⡄⠀⠙⡆⠈⠲⢋⡶⠛⠉⠒⠤⣤⣏⡀⠀⠀⠀⣼⠕⠒⠒⢤⣾⡟⠀⠀⠀⠀⡰⠁⠀⠉⠉⠉⢱⡎
⠀⠀⠀⠀⠹⡄⠀⡇⠀⢠⢞⣀⣀⠀⠀⣠⠃⠀⠉⠳⠤⡼⢃⠔⠦⠼⢁⠞⠙⠶⠋⡗⡰⢁⣠⣀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠘⢆⠱⣤⠟⠉⢀⣈⠵⠒⡇⠀⠀⠀⢀⡞⣜⠁⢀⡤⢴⣮⠀⠀⠀⣰⣱⢡⠎⠀⠀⣓⠀⡜
⠀⠀⠀⠀⠀⠀⠈⠶⠧⠒⠋⠁⠀⠀⠀⠸⡀⠀⢀⡞⠓⣮⣉⡁⣀⠞⠈⡆⠀⢠⣿⢃⡈⢦⡀⠠⡇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⠏⣸⡠⠴⠒⠉⠀⢆⠀⠹⡤⢣⠃⣇⠙⠒⠃⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠈⡆⠀⢠⠃⠀⣈⣙⠢⣄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠏⠀⠀⠓⠒⠃⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⡤⠆⠀⣀⣀⢤⣤⠤⠤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⢈⡧⠖⣊⣭⡋⠀⠀⠀⢠⣞⣷⠎⡉⠈⢙⠿⡟⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣀⡴⠚⣡⡖⣿⠹⡿⠃⠀⠀⠀⠚⠉⢶⣜⣿⡷⣯⡁⢦⣾⣧⣉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠞⢡⠞⠉⠀⠙⢾⡟⠁⠀⠀⠀⠀⠀⠀⠒⠈⠉⠹⣌⢹⣯⠀⠉⠻⡃⠀⠈⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠞⢡⣴⠎⠀⢴⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣿⣧⣠⡤⠘⠉⠑⠶⠚⠡⠙⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⡴⠃⢠⡈⠀⢀⣴⢉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⠐⠀⠀⠀⠀⢠⡴⠎⢧⡀
⠀⠀⠀⠀⠀⠀⠠⠂⣼⠵⠆⠀⠉⣷⠃⠸⣮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠁⠀⠀⠀⢠⡠⠘⠃⠡⠀⡀⢳⡀
⠀⠀⠀⠀⠀⢠⠃⣼⠁⢰⠀⠀⠀⣠⣤⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠿⣀⣤⡴⠚⠁⠀⢀⠀⢄⢣⡴⠟⢳⡀
⠀⠀⠀⠀⠀⠃⢰⢷⢳⠒⡟⠋⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⡁⠈⢆⠀⢄⣀⢷⡎⠂⢂⠀⣨⣧
⠀⠀⠀⠀⠈⠀⡿⡿⡀⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠰⣹⣀⣬⢷⡏⢱⣤⠟⠺⣤⡞⢯⣻
⠀⠀⠀⠀⢠⢠⣷⣧⣶⣶⣯⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠛⠋⢀⠄⠀⠀⢹⣜⣿
⠀⠀⠀⠀⢸⢸⣟⣧⡁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⢛⢹
⠀⠀⠀⠀⢸⠸⡼⣿⣇⠀⡴⠈⠛⠛⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⢸⣼
⠀⠀⠀⠀⠂⡀⡄⠘⠿⣿⣧⣄⠀⠀⠀⣀⣀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢘⡁⢡⡀⠀⠀⠨⠻⢷⣤⡴⠁⠀⠀⠀⠀⠈⠙⠉⠁⢐⣲⣤⠤⣖⣩⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡟
⠀⠀⠀⣶⢻⣷⠀⢂⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⣤⠴⠖⠛⠙⠁⠀⠀⠀⢀⣤⣤⡶⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠐⠃⢺⡆⢿⣧⡈⠦⠀⠀⠰⠖⠂⢾⠛⠟⠉⠉⠁⠀⠀⠀⠀⣀⣠⡤⠀⠀⠋⠙⠖⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠃
⠀⠀⢠⠞⢷⠈⢿⣆⠈⢆⠀⠀⠀⠀⠈⠢⣀⣄⠠⢤⠴⠞⠂⠩⡉⠀⠀⠀⠀⣀⣤⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠃
⠀⠀⠀⠀⠘⡇⠈⢿⡷⢄⠳⡀⠀⠈⠀⠀⠈⠳⢦⢤⢀⡀⣀⣤⢱⡶⠺⢛⠏⠁⡴⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠹⠀⠸⣷⡀⢡⠈⠢⡀⠀⠠⠀⠀⠀⠀⠉⠳⣍⠀⠀⠱⣀⣱⡼⣦⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠹⣷⡀⠑⠀⠈⠑⢄⣀⠀⡀⠀⠀⠀⠼⢷⣷⣳⡙⡉⠀⠋⠀⠄⠀⠀⠀⠀⠀⠀⢀⡠⠚⢁
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠹⣷⡀⠀⠈⠑⢀⢈⠑⠢⢄⣄⣀⠀⠠⢿⣷⣷⣽⡋⠀⡀⠀⠄⠀⢀⣠⠴⠚⡉⡀
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢻⣧⠀⠀⠀⠀⠀⠈⠒⠤⢀⠉⠙⠒⠒⠛⠻⠿⠿⠟⠓⠒⣂⠉⣁⠸⡗⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠀⠁⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⢿⣷⠠⠖⠊⠙⠂⠒⠉⢉⣿⣩⣿⣿⣿⠉⠉⠛⢓⣿⣶⠒⡶⠢⠤⣄⡀
⠤⠀⠀⠀⠀⠀⠀⠒⢾⡇⠀⠀⠨⢿⣷⠀⠀⠐⠓⠛⠫⠉⣉⣉⡉⠉⠉⠉⠉⠛⠛⠛⠻⠴⠴⡀⢀⠀⣠⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⣶⡁⠙⠻⣿⣇⡀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠒⠢⢤⠀⠀⠀⠀⠀⠀⠀⣈⣨⠟⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠓⠤⡀⢀⠈⡟⣿⡟⣶⣶⣶⠴⢶⠶⠶⠶⠤⠤⠶⠒⠒⠒⢶⠟⡟⡇⢹⡀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⠃⠿⠾⠧⠠⠀⢀⣀⣀⣀⣀⢀⣀⠀⠀⠾⠴⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣩⡅⢢⣢⣴⡾⡳⢶⡗⢀⠀⠀⠀⣠⠛⠁⠀⠀⠈⣁⡀⠀⠀⢠⣔⣦⣤⠤⠙⠉⠪⠁⠀⠀⠶⠶⠄⡀
⠀⣀⡜⠛⣁⡀⠀⠀⠀⠀⠈⠉⢀⡝⠚⠿⣧⣜⡛⠅⠠⠖⠃⠁⠀⠀⢀⡴⠋⡠⠀⠹⠋⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠆
⠀⠀⠉⠉⢠⣽⠠⠀⠀⠀⠀⠀⠈⠳⠊⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠸⠃⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣇⣈⣤
⠀⠀⠀⢀⣐⡿⢀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠋⠀⠀⠀⠀⠀⠀⣠⠼⠇⢀⡀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠝⡋⠉
⠀⠀⠐⢻⣿⡌⠂⠀⠀⠀⠀⠀⣠⣟⠋⠀⠀⠀⠀⠀⠀⠀⠐⠒⢛⠆⠀⠈⣝⣁⡋⣦⡤⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠻⠾⢿⠿
⠀⠀⠀⠀⠉⣿⣆⠀⣠⣶⣆⠬⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣘⣾⠋⠀⠀⠀⠀⠀⠀⠉⢧⡀⠀⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠂⠀⠐⠀⢑⠖⠆⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣰⠉⢚⣗⠀⣠⣾⠦⠀⠐⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⣟⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⣄⠀⠀⠛⢢⡟⠓⠒⢊⠀⠟⠉⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⣏⡀⠀⠀⠀⠀⠈⠉⠉⠠⣴⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠑⢶⣦⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢇⠀⠀⠀⠀⠀⠀⠀⣑⡒⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡑⠀⠀⠀⠀⠀⣯⡍⠁⠀⠀⠀⠀⠀⢲⡏⠀⠀⠀⠀⠼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠃⠘⠟⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⡐⠀⠀⢰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠁⠀⠀⠀⠀⠀⠘⣗⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⡇⠀⠀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢤⡜⠉⠀⠀⠀⠀⣀⡤⢶⠶⢤⣀⠀⠀⢀⠻⠀⠀⣴⣦⠀⣀⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣴⠋⢄⡈⠁⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⠲⡧⡼⡁⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⢂⡗⣉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠈⠉⠉⠉⣩⠸⣶⣶⡤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⡀⠠⠶⠀⠀⢤⣰⠞⠙⠫⢴⣷⣶⣿⣖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⣀⣀⠀⣀⡀⢸⣿⣶⠆⠒⢀⣴⣟⣜⢢⡀⠀⠈⠙⠛⠿⣷⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⠛⠫⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣿⡤⠀⢈⣿⡁⢉⠢⡀⠀⠀⠀⠀⠉⠐⠬⢦⡀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠂⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠂⣀⠛⠿⠋⠀⠰⣎⠢⡀⠀⠀⠀⠀⠀⠀⠈⢑⡶⠈⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠡⠛⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣘⣿⣧⣴⡀⠀⠀⠙⠓⠈⠢⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⡿⠿⢷⡀⠀⠀⠀⠀⠐⢌⠢⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠤⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⡀
⠀⠀⠀⠀⡀⠤⠃⠉⠀⠀⠐⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠱
⣠⠤⠂⠁⠤⠤⠤⠤⠴⠶⠤⠰⠄⠹⢿⣿⣿⣿⠟⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⣦⡡⡀⠀⠀⠀⠀⠀⢡
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡟⢿⣯⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠒⠠⠔⢁⡀⠀⠀⢀⣴⣾⣿⣿⣿⣬⠢⡀⠀⠀⠀⠀⢣
⡇⠀⠀⠀⢀⣤⡴⢶⢶⣤⣀⠀⠀⠀⡇⡇⠀⠀⠉⠙⢶⡀⠀⣀⣀⣀⠔⠁⠀⠀⢀⢎⣴⡿⠃⡠⠒⠉⠻⣿⠿⠿⢿⡇⠀⢈⠢⡀⠀⠀⠀⢃
⡇⠀⠀⡴⠯⣾⠥⣾⠤⢷⡬⢷⡀⠀⡇⡇⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣄⢒⠒⢿⡫⠒⠉⠇⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠑⠌⠢⡀⠀⠀⢂
⡇⠀⢸⣇⣀⣇⣀⣿⣀⣸⣇⣀⡇⠀⡇⡗⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠐⢌⡢⡀
⡇⠀⠘⣇⠀⣇⠀⣿⠀⢸⡇⢀⡇⠀⡇⣇⠘⡄⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡠⠊⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡠
⡇⠀⠀⠙⢯⣹⣍⣿⢉⣟⣩⠏⠀⠀⡇⣟⣠⠴⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠉⠀⠀⠀⠀⠀⢀⠔⠁
⡇⠀⢀⣀⣀⣈⣉⣛⣋⣉⣀⣀⡀⠀⡇⡏⠔⠢⡘⡄⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⡇⣿⡌⠄⠡⠘⡄⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡜⠤⣃⠬⢀⣀⢀⠘⣿⢀⣀⡀⠤⠄⠒⠉
⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇⣏⠥⠒⠉
⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠒⠒⠀⢉⢍⡵⠋⠉⠉⠉⠑⠰⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⡄⠄⠐⠀⠀⢸⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠄⢂⠩⢇⠀⠁⠀⠀⢀⠠⠁⢁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠁⠀⠀⠀⡀⡁⠀⢀⠰⠐⠐⡀⡄⢆⢡⠸⡈⢆⠡⡰⡀⠣⣮⠧⡁⠠⢃⠂⠀⣤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠐⡛⣀⣀⣜⣔⡀⠠⠰⠀⢣⢡⣆⢂⢂⢢⠣⡡⡑⡘⢌⠢⡑⢌⢪⡣⣍⠪⣒⣌⡛⣾⡥⠶⠶⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠈⠙⠛⠛⠛⠉⣿⣏⠙⠳⠷⠿⠷⠾⠶⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠊⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢻⣾⣷⡿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⡾⢿⣿⡗⢔⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⠁⠂⢸⡇⠘⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢠⠇⠀⢀⢸⡇⠀⢁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⠀⠀⢸⢸⡇⠀⠈⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡼⠀⠀⠀⢠⢸⣧⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⠃⠀⠀⠀⢸⢸⣿⠀⠀⢲⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣤⡞⠀⠀⠀⠀⢸⢸⡯⡄⠀⢈⡄⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢋⡟⠃⠀⠀⠀⠀⣸⣸⣿⡇⠀⠀⠑⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⣤⣧⣼⣿⡇⠀⠀⠀⢣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠃⠀⠀⠀⣠⠞⠁⢸⢻⣿⣷⡄⠀⠀⠈⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡞⠀⠀⣠⡞⠁⠀⠀⠈⠚⣿⢼⠻⣄⠀⠀⠅⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸⠁⢀⡾⠋⠀⠀⠀⠀⠀⠀⢹⣿⠀⠙⣥⡄⢲⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⢻⣼⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢐⣭⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠙⡟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠈⠉⢨⣿⣇⠭⢂⡀⠀⢁⠐⠈⣁⠈⠀⡀⠈⠁⠹⠷⠈⢌⠰⠒⡠⣯⣷⡀⡀⠂⡀⠄⠀⢀⠂⢐⠀⠀⠀⠄⢀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠐⠀⠐⠈⠀⠂⠁⠀⠟⠋⠅⠂⠑⠊⠁⠈⠁⠁⠐⠈⠐⠈⠑⠈⠐⠁⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢄⣴⠿⠛⠋⡙⠉⢹⣿⡿⣷⣆
⠀⠀⠀⠀⠀⠀⣯⠸⡇⠀⠀⣾⣷⣶⠞⠛⠓⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢧⣌⣒⠄⠙⠿⣿⣄⡀⣀⣸⣿⠇⢀⣀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠶⢿⣿⣿⣿⠿⠛⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣡⠖⣱⠺⣾⣤⣤⣴⡤⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⣠⣾⣿⣶⢿⡀⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢰⢱⣿⡿⠋⠀⠑⠉⠢⡀⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣾⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⡇⣿⡟⠀⠀⠁⠀⠀⠀⠈⠢⡀⠀⠀⢦⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⢲⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇⣿⠠⡀⠀⠀⠀⠀⠀⠀⣄⠈⠦⣀⠙⢦⣆⠫⣀⣘⡇⣀⣤⡶⠿⣿⣽⡧⠌⠻⣿⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣇⠇⢀⡄⠀⠀⠀⠀⠀⠀⠈⠓⠂⢀⣂⣈⠟⠂⠈⢉⣘⢉⣴⣒⡉⠙⠋⠉⠀⠳⡽⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣿⡄⢀⠠⠀⢀⣀⣀⠀⠀⠀⣀⣤⣾⣿⣿⣦⣤⣄⠼⢳⠉⠀⠀⠉⠀⠀⠀⠀⠀⢇⢸⣿⣿⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠘⣷⡀⢠⠠⠟⠉⣙⣿⡟⢛⡿⠃⠈⠻⣿⣿⣿⣤⣰⡞⠤⢤⣄⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠘⣿⣄⠣⡐⢾⣿⠿⢻⡧⠀⠀⠀⢠⣧⠈⠉⠙⢤⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠰⡇⣿⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡈⠻⣷⣄⠀⠁⠀⠉⠀⠀⠀⠴⠋⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⣿⠏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡼⢋⡾⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡢⠀⢀⣠⡴⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣄⡉⠛⠻⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣥⣤⣶⠾⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣄⣀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠶⠖⠒⠚⠋⠉

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⣿⣇⣀⡀
⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⡿⢩⣙⣛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠇⠀⠀⠉⠉⠛⠒⠶⠭⡝⣿⣇⣀⡀
⠀⠀⠀⠀⣿⠇⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠛⠛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⢸⣿⣀⡀⢀⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣂
⠀⠀⠀⠈⠉⠛⠛⠿⠟⠈⠉⠙⠛⠿⠷⣶⣦⣤⣼⣿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⣸⡟⢻⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣆⠀⠉⠉⠛⠻⠿⢶⣶⣤⣤⣿⣷⣾⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⡏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣿⡇⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣿⡇⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠿⠿⣶⣿⣷⡶⠿⠟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠂⠀⢋⢉⣀⣀⡀⠇⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⢈⠀⠀⠀⠀⡤⠄⠤⢬⠀⠀⢠⠂⣉⢀⠉⡏⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡔⠐⠂⠐⠄⠘⠂⠉⠹⠄⠭⠭⡤⠶⠀⠴⠆⠀⠀⠀⠀⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣠⡀⡤⠦⠐⠒⢲⠀⠀⠀⠀⡇⠀⠀⠀⠛⠭⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡅⠠⠤⠤⠴⠀⠴⠶⠚⠒⠒⠒⠒⠛⠂⠈⠉⠉⠉⠉⠉⠁⠒⠚⠣⠤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠐⠂⠀⠀⠔⠄⢀⡤⢀⡀⡔⡩⠭⠤⡤⢔⣒⠒⠒⠒⢠⠀⢀⠠⠐⠂⠉⠉⢉⠝⡉⠉⡉⠑⡀⢘⠉⠉⣉⠙⠛⠒⠦⡙⠢⡀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠉⣀⡀⡟⠰⠡⠂⢉⣄⣀⡁⠑⢢⢠⠃⠠⢃⠔⠉⠀⠀⠙⣥⠉⠀⡀⠸⠈⢡⠈⣞⠉⡩⠓⡉⢁⠒⢌⢆⠘⣆
⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⣟⠅⠔⠋⠀⢀⢈⡽⠩⡎⠀⡤⢀⠔⠊⠁⠈⠓⢷⠰⠷⠾⢼⠤⠚⡀⢉⡸⡐⠁⢀⠉⠁⡈⠌⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠻⢦⣛⡁⢋⡎⠀⣀⡇⠀⣧⠁⢄⠰⠄⠲⢣⠜⠀⠀⡀⣨⠦⠚⡇⢈⠀⠠⠃⠒⠖⡤⣊⠤⢲⡨⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣐⣞⣋⠈⠁⠨⢋⠔⡆⠀⣧⡳⡤⠜⠀⠐⠗⢤⣀⠀⢀⠉⢰⠉⡇⢸⠂⠀⢂⠴⣋⢭⣔⠬⠜⣃⠄⣈⠧⠤⠄⠐⠒⠢⢄
⠀⠀⠀⢀⡠⠤⠐⠒⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠃⠀⠘⣄⡇⠀⠠⠐⠀⢁⣁⠁⡼⠄⠼⠧⠳⠞⠒⠚⠒⠒⠒⠚⠒⠒⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣰⣅⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠐⠉⢡⠋⠀⣋⠉⠀⠀⠀⢀⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠇⠀⠤⢄⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⠤⢘⣃⣀⡒⠢⣄⠀⠀⠀⠀⢸
⠀⠀⠘⡄⡎⢹⡆⠆⠂⠂⠐⠀⢖⡭⠑⣼⠀⠀⠀⠀⠀⠀⢀⡠⢶⡶⠿⠿⠿⠶⣖⠤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠜⡡⠚⠉⠀⠀⠈⠳⡌⢣⣀⣀⠠⠤⢀⡀
⠀⠀⠀⢹⠘⠛⠀⠄⠀⠀⠀⠀⢜⠠⡢⢹⡆⠀⠀⠀⠀⢠⢋⡞⠁⠀⠀⠀⠀⠀⠀⠑⣜⣆⠀⠀⠀⢸⠀⠀⠀⠀⡈⣼⠁⠀⡠⠒⠲⡄⠀⠸⡘⡇⠀⠀⠀⠀⡁
⠀⠀⠀⢀⠦⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡞⠀⠀⡠⠐⠒⠒⡄⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠸⡁⠀⢀⡇⠀⢠⣇⡇⠀⢀⡠⠞
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⢒⠖⠂⠙⠒⠒⠒⠚⠷⡇⠀⠀⢇⠀⠀⡠⠁⠀⠀⢸⢸⠤⠤⠤⠴⠖⠒⠒⠂⠻⡻⡀⠀⠈⠛⠉⠀⠀⣸⠉⠉⠉
⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⠉⠉⠀⠀⠀⢠⣯⠏⣀⣀⣀⣀⠠⠤⠤⠤⠄⠓⠙⠢⢤⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠚⠦⠤⠤⠤⠔⠒⠊⠁⠈⠳⢤⣀⣀⠀⢀⣀⡴⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡤⠷⠖⠚⠛⢸⡟⠓⠲⠶⢤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⢀⠀⠤⠠⠀⢸⣿⡆⠠⠄⡀⠀⠀⠀⠀⠈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠫⠀⠄⠁⢀⠤⠚⠁⠀⢀⣄⠀⢸⣿⣧⣀⠀⠀⠓⠀⠀⠈⠀⠀⠈⢳⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠄⠛⣦⣀⠴⠛⠉⠁⠀⢸⣿⣿⠉⠙⠢⣄⣤⠞⠡⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠚⠀⢀⡼⢻⣷⣄⡀⠀⠀⢸⣿⣿⠀⣠⣶⡿⠻⡄⠀⠘⠄⠀⠀⠀⢻⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠇⠀⠀⠀⡌⠀⢀⡞⠀⠀⠹⣿⣿⠀⢀⢸⣿⣿⣿⣿⠟⠁⠀⠹⡆⠀⢨⡀⠀⠀⠈⣧⠁
⠀⠐⣶⢲⡖⠀⠂⠀⠀⠀⠀⣼⣀⣠⣠⣴⣤⣴⣾⣶⣶⣿⣿⣿⣿⡟⠁⠀⠐⠙⡄⠀⠀⠀⠀⠀⠁⠀⠀⠲⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠹⡆⢷⠀⠀⠀⠉⠋⠉⣉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡍⡄⠀⠀⠺⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⠖⠒⠒⠂⠀⠀⣿⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠈⡄⠀⠳⡆⠀⠐⠀⣤⣤⣾⣶⡭⠔⠚⢿⣿⡟⠛⠛⢋⡏⠋⢩⡍⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⠀⠀⠽⡄⢠⣾⣿⠟⣿⣿⡇⠀⠀⠻⢿⣿⣄⢀⡾⠁⠀⡐⠀⠀⠀⢰⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠹⡀⠀⣽⠿⣋⠀⠀⢻⣿⡇⠀⠀⠀⠀⢉⡿⣿⡀⢀⠜⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣄⠀⠀⠀⠈⠻⡁⠀⠈⠓⠲⢼⣿⡇⠀⢠⠴⠚⠉⠀⢀⠹⠊⠀⠀⠀⢠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣰⠈⠀⠀⠈⠘⠢⢄⣀⠀⣿⡇⠀⠀⢀⡀⠄⠒⠀⠀⠈⠀⢄⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⠲⣄⡀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠨⠝⠲⢤⣄⣀⠀⠀⢸⡇⠀⠀⠀⣀⣀⡤⠶⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠙⠛⡇⠘⠛⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣤⣤⣶⣦⠄
⠀⠀⠀⣀⣀⡀⢀⠀⠠⠀⠀⠀⠀⠀⠈⢻⣿⠟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣴⣶⣥⣤⣄⣀⣄⣀⡀⢀
⠀⠀⠀⠚⠛⠛⢿⣿⣿⣿⡋⠀⢠⣁⠀⠘⠀⠀⠀⠀⠀⠀⠎⢘⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠑⠾⠉⠀⠀⠀⠀⠀⠀⠀⠔⠒⠟⠚⠟⠻⣻⢿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣥⣦⣀⣀⠉⣗⠲⠿⣶⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⣿⣿⣧⡘⠁⠀⠀⠋⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣷⣤⡄⠀⠀⠀⠀⠀⢻⣿⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠀⠀⠀⠀⠀⠀⠸⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠀⠠⠀⠀⠀⠂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⣀⡄⠂⠀⠀⠐⠀⠀⠀⡊⠀⠀⢆⡄⠀⠀⡇⠀⠢⠤
⠀⠀⠀⠀⠀⠄⠈⠀⠁⡠⡀⠀⠋⠡⠠⠂⠀⡀⠂⠈⠈⠁⠐⢀⣀⡝⠥⠄⣠⣇⠀⠀⠠⠆
⠀⠀⠀⠀⠀⢀⠁⠶⠩⠀⣰⠈⠀⠐⢈⡁⠀⠄⠀⠈⠀⠀⠀⡉⣛⠚⢂⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠀⠈⢁⡈⠁⠨⠐⠂⠈⠍⠉⠭⠅⠩⠤⠀⡽⠁⠄⡠⡎⡅⠺⠏⢛⠓⠀⠔⠀⡀⠠⣠⡀
⠀⠀⠀⢠⠀⠀⠄⠈⢁⣠⠤⠐⢀⠑⠀⠁⠀⣠⡀⠀⠀⠀⠀⡄⢰⠅⢙⣋⠊⠉⠉⠫⡀⠈⠀⠉⠀⠉⠋
⠀⠀⢁⠀⢦⡀⠂⠡⠠⠧⠤⠄⠀⠁⠀⢁⡄⠉⠁⠀⠀⠀⠀⠁⢈⠀⠅⠿⠀⣀⠀⠱⡀⠀⢁
⠀⠀⠀⠐⠀⠈⢀⢰⠛⣻⣧⣃⢀⣚⢁⢈⠁⠐⠲⡉⠀⠈⠙⢓⣂⡖⢀⠀⠐⣩⠀⡲⠚⠊⠀⣀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠠⣀⠀⢀⠀⠀⢨⠀⠜⠨⠄⢣⢠⠁⠈⠈⠁⢆⠀⠠⢰⠅⠀⠘⣤⣧⠃⢡
⠀⠀⠀⡀⠆⠀⠀⠆⠈⠂⠐⠠⡴⠢⠀⢀⡄⠂⠐⠘⠆⢄⠐⠨⠀⠀⠡⣤⠦⠴⠆⠋⠉⣡⣼⣻⡀
⠀⠀⡜⠀⠑⡀⠀⠑⢀⢀⠀⢸⠂⠀⠀⠈⠉⠱⡠⢀⢀⣼⡆⣁⣤⣶⣦⠹⡄⠐⠠⠄⢪⣄⠀⠀⠅
⠀⠀⠇⠀⠀⡏⢰⠌⠆⡚⣎⠠⠀⠀⠀⠀⠀⢀⡀⠐⠈⠊⠑⠘⣷⣙⡼⠀⠹⣄⠒⠀⢀⠉⠵⠖
⠀⠀⠀⠀⠀⠀⠘⡆⠆⢱⡶⢴⠀⣠⡄⣀⠈⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⢆⢠⣀⣬⣱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⢁⡄⢠⣯⠀⡇⠀⠂⠈⠰⠤⢀⠀⠀⠁⢀⣀⣠⡤⠴⠀⠉⠂⠁⠃⠃
⠀⠀⠀⠀⠄⠠⡞⠉⠁⠀⠻⡀⣸⠇⠂⠀⠀⡰⢀⠀⠀⠈⠀⠐⠀⠭⠁⠀⠀⢀⡀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠈⠀⠎⢘⡈⠉⠠⡤⠂⠀⠀⠄⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠠⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⢢⡞⣰⣿⣿⣿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⣠⣶⣄⠀⠀⠀⠈⠀⠀⢠⣴⣾⣿⣿⣾⡇⣿⣿⣿⣿⡀⢈⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⢄⣴⣺⣽⣿⣿⣿⣿⢹⣷⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⣴⣳⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣧⡙⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣶⣬⣍⠉
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠓⠒
⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⣿⠇⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣦⣝⠻⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⡀⢀⣠⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⡇⠻⣿⠿⠟⠻⠿⠛⠁⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸
⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⡏
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢠⣿⡿⠁⠉⠛⠋⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠘⣿⣷⡙⣿⣄⠀⠀⠀⠀⠙⠏⠁⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣼⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣧⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠙⢿⡀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣀⣀⣀⣈⣻⣧⣀⣀⣠⣿⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠠⠀⠰⠆⠰⠆⠰⠦⠀⠀⠈⠀⠀⠀⠀⠀⠈⡀⡀⠁⠈⠀⠀⠅⠀⠁⠀⠀⠰⠆⠀⠆⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠒⠒⠁⠑⠤⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣉⣤⣶⠾⠿⠿⠿⠿⠶⣶⣤⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣶⡾⠋⠁⣀⡤⠶⠶⠶⠶⢤⣄⡈⠛⢷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⢠⡾⠋⣀⣴⠋⠁⣀⠠⠼⣧⠄⣀⠈⠙⢦⣀⠉⢷⡆⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠⡿⠁⢰⠏⠀⡰⠊⡡⠐⠃⣿⠂⢌⠑⢆⠀⠹⣆⠈⢿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢠⡏⠀⡚⠀⠊⢀⣠⠤⠿⣆⠀⠑⡀⢃⠀⢹⡄⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠈⠻⣿⠂⢸⡧⠶⡟⣛⠛⠻⡇⠀⠀⢸⣤⣤⣭⣽⠶⢼⡇⠈⣿⠟⠁⠀⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠘⣇⠀⢣⠈⡄⠀⠹⣶⠒⠋⠉⢠⠁⡜⠀⣸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⣷⡀⠹⣄⠀⠱⣄⠐⠤⣿⢠⠄⠒⡡⠎⠀⣰⠏⠀⣾⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⣷⡀⠉⠳⣄⠀⠙⠒⢻⡖⠒⠋⠀⣠⡾⠉⢀⣾⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠁⠈⠻⢦⣄⡈⠛⠓⠶⠶⠶⠶⠚⠛⢁⣠⡶⠟⠁⠙⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢷⣶⣴⣤⣤⣶⣶⡾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⣄⣠⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⠿⠛⠛⠛⠛⠋⠁⠈⠓⠶⢶⣶⣤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⢿⠟⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠉⠽⠿⢿⡿⠏⢛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣷⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣠⡶⣤⣶⡞⠋⣀⣤⣾⣿⣿⣿⡆⠀⠀⠀⠀⢀⠀⠘⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠀⠀⣠⣴⣾⣿⣿⣿⡿⠟⠁⠀⠀⠶⠂⠀⠀⠀⡄⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⠋⠀⠀⢰⠿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠚⢰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⢠⣶⣿⣷⣶⣄⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠿⢻⣿⠂⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢣⣿⠀⣀⣸⣞⣩⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣸⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠛⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠋⠈⣀⠻⢦⡙⣿⣿⣿⣇⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠿⡿⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣍⣉⣤⣤⣀⣁⠀⠌⠁⠘⡷⠈⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⡀
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⠛⠻⣿⢿⣿⣿⠃⠀⠀⢠⡅
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣶⣾⣿⡿⣿⣿⣿⣿⣿⣶⣄⣩⣦⣤⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠁
⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⠳
⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣘⡿⠟⢋⡀
⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⢦⠤⠶⠒⠐⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣀⠩⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣤⡿⢀⣆
⠀⠀⠀⠀⠀⠀⢰⣿⡀⠸⠆⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠉⠉⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠃⢒⠃⢄⣠⠃⠈⠀⠀⠈⠁⢀⠀⠀⢐⠻⡳⡄
⠀⠀⠠⡄⠀⠀⠀⠉⠻⠿⣮⠼⠘⠁⠀⢀⡄⢀⠀⠀⠀⠀⠀⠈⡇⣃⢱⠀⠰⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡐⠊⠉⢨⢰⣍⣷⢄⢈⣇⠀⠳⡹⣿⣸⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣥⡴⠷⣶⣾⠿⠷⢧⣸⠟⢻⡿⠛⠿⠿⠒⠁⢀⣾⠐⠂⠀⠈⠘⡋⠀⢀⡿⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠉⠉⠀⠀⠀⣀⣄⠀⠀⠙⠓⡗⠁⠀⠀⠀⣀⣴⣾⣯⡀⢀⣠⡴⢞⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠻⠀⠀⣦⣼⠍⠀⠀⠀⢾⢿⣧⠘⠛⣻⡭⠒⠋⠝
⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠙⠳⢶⣄⠀⠀⠀⠀⢉⣼⣄⢦⡀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡟⡈⠁⠀⠀⢀⡴⢚⢧⠀⠀⢖⡎⢙⣓⠂⠴⠂⠀⢠⡾⠁⠀⠁
⠀⠀⠀⠀⠀⣾⠧⠄⢀⣠⣴⡾⠁⠀⢺⠆⠀⢠⡟⢸⣿⠀⢻⠀⠀⠸⣷⣀⠁
⠀⠀⠀⠀⠀⢹⣧⠾⠝⠟⠉⠀⠀⣠⡟⠀⠀⢾⠃⢸⣿⠀⠘⡇⠀⠀⠌⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⢀⣀⡾⠛⠀⠀⢠⠟⠀⠸⡿⢄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣠⡤⠊⠀⠃⠀⠑⢬⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⡀⠀⠀⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⣀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⢸⡟⠉⠉⢉⡉⠉⠀⠈⠁⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠈⠙⣿⡏⠀⠀⠀⠀⠀⡜⠉⠓⢤⡛⢈⡲⣈⠑⣶⣏⠀⠀⣰⠃⠈⢲⡄
⠀⢸⡇⢀⡀⠈⣀⠀⠁⣀⣀⣀⡀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣸⠀⠀⠀⠀⠈⠳⣬⠞⣻⢧⣌⡙⢲⠧⣄⣀⡾⠉⠳⣄
⠀⢸⣷⣤⣤⣤⣤⣤⣤⣀⠠⣤⡀⠈⠻⢿⡆⢰⣶⣄⣀⣤⣶⠿⠿⠛⡟⠛⠛⡿⠿⣷⣶⣄⣀⡜⢁⡜⠓⢧⡀⢩⠟⠒⠢⢽⡇⠀⠀⢠⠗⢦⡀
⠀⠸⢿⣛⣫⣼⣿⡉⢻⣿⣷⡈⠻⢷⣤⣾⡿⣪⣿⡿⡟⠉⠀⠀⠀⠀⡇⠀⡘⠀⡼⠁⡹⢿⣿⣦⣎⠀⠀⠀⠈⠛⠶⠤⣄⣼⠀⠀⢠⠎⢠⠏⠈⠲⣄
⠀⠸⠿⠯⠷⠋⠘⣷⡀⠹⣿⣷⡀⠸⢟⣿⣼⡟⠁⠀⠱⡄⠀⠀⠀⢰⠇⡐⢀⡞⠀⠜⢠⠋⣸⠻⣿⣷⣤⡀⠀⠀⣀⣠⠾⠋⣷⢠⠏⢀⡏⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⡜⢀⣷⡞⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠙⢦⡀⠀⢸⡜⣡⠟⢄⠎⢠⠋⡼⠁⠀⡴⠋⠻⢿⣷⣋⣁⠀⠀⢀⡾⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⢠⣴⣟⣴⣿⣦⣽⠿⣿⡟⠉⣆⠀⠀⣠⠤⢄⣀⣹⡶⠖⠻⡅⠀⣀⠷⢧⠞⠀⣠⠞⠁⠀⠀⠈⢿⣷⠴⠼⢶⠾⠤⣧⣸⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣠⣿⠋⠀⠙⢻⡟⠁⠀⢸⡇⠀⡜⠀⢠⡃⠀⠀⠀⡰⠁⢀⣀⡹⣞⠁⠀⢈⣳⡞⠁⠀⠀⠀⠀⣠⣾⠇⠀⢰⠷⣆⣰⡃⡜⠉⠉⢹⡟⠒⢦⡤⢤⡇
⠀⠀⠀⣴⣿⠹⡆⠀⠀⠀⠀⣀⣀⣤⣿⢤⣇⠀⠀⠓⢤⡀⣼⣄⠀⠘⢆⠀⠈⡦⠖⠉⡇⠙⣆⣀⣤⣶⣿⠟⠳⣄⣀⡞⠀⢀⡟⠙⡆⠀⣠⢾⠀⠈⢺⠤⠤⡇
⠀⠀⣼⡟⠁⠀⢳⣤⠒⠊⠉⠀⢸⣿⣿⣦⡀⠈⢲⡄⠀⡼⠛⢯⡀⠀⠐⣇⠀⠱⣄⣠⣴⣶⣿⢿⠟⠋⠀⠀⠀⠀⠉⣳⠖⠺⢯⣓⣿⡊⠀⠈⢇⠀⢸⠀⠀⡇
⠀⢸⠟⠀⡜⠀⠀⢿⠀⠀⠐⣚⣉⣥⠼⣠⠛⠓⠲⣷⠴⣇⠀⠀⠙⠢⣀⣸⣶⣾⣿⠟⠛⠉⠀⠠⣷⡄⠀⠀⠀⣠⠔⠁⠀⠀⠀⠙⡅⠈⠉⢢⡼⠒⢺⠖⠒⡇
⠀⠈⢀⠈⠀⠀⠀⢸⡗⢺⣋⣉⡡⠴⢎⠁⠀⠀⢀⠇⠀⠀⠙⠲⠶⢶⣿⠟⠋⠁⠘⠦⢦⠀⠀⠀⢀⠽⣤⠤⠞⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⡏⠀⠀⢸⠀⢠⠇
⠀⠐⠁⠀⠀⠀⠀⠈⢻⡀⠁⠀⠀⠀⠈⣧⡤⠖⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣾⣷⣄⣰⠁⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣇⣀⣤⣼⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠀⣀⣴⠋⠁⢀⣀⣤⠤⣤⣤⣶⡖⢋⣹⣿⡇⢀⠔⠋⠙⢿⠃⢈⡏⠀⠀⠀⠙⡆⠀⡀⠀⠀⢀⡠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣠⠴⠚⠋⠉⠁⠈⢳⡋⠉⡇⣀⣠⡿⢿⠛⠯⠭⠔⡿⣿⣅⣀⣶⣀⠏⠀⢸⠁⠀⠀⠀⣠⣧⣸⠿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠁⠈⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠀⠀⠈⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⡂⠀⠀⢀⡤⠀⠀⡐⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠆⠀⠰⠄⠀⠀⢤⠐⠀⠆⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡀
⠀⠀⠠⠆⠸⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠘⠀⠇⠀⠀⠀⠄⠀⠀⠠⠀⠀⠑⠀⠰⠀⠄⠀⠀⡄⠠⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠞⠈
⠀⠀⠀⠂⠂⠀⠀⠰⠠⠂⠀⠀⠀⠀⡄⠰⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠂
⠀⠀⠀⠒⠂⠀⠀⠈⠀⠀⠈⠀⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠂⡘⠀⠘⠋⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⠅
⠀⠀⠀⢄⡀⠀⠀⠐⠎⠀⠀⠀⠈⠀⠀⢀⠄⠂⠈⢁⡄⠀⠀⠢⠆⠐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⢰⠃⠀⠈⢄
⠀⠀⠀⢀⡄⠀⠐⡤⠀⠈⠀⠀⠀⢀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠰⠄⢀⡄⠀⠀⠀⠺⠄⠀⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠐⠦⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⢒⢋⠀⠀⠁⠀⠀⠂⠐⠀⣠⠀⠀⠀⠄
⠀⠀⠀⢰⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⢆⡑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠋
⠀⠀⠀⢀⠀⠁⠁⡜⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠠⠀⠉⠁⠐⠀⠀⠀⢉⠈⠀⠰
⠀⠀⠀⠈⠁⠈⠁⠤⡆⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⢐⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠛
⠀⠀⠠⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠃⠂⠐⠊⠀⠀⠀⠀⠀⠀⠁⠁⠡⠅⠀⠀⠀⠀⠀⠀⠠⠑⢀⠊⠀⠘⠃⠀⠀⠀⠀⠀⠀⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⢠⠀⠓⠃⢀⡠⠒⠠⢀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠑⡀⠀⠀⠠⠄⠀⠀⣠⠉⠀⢀⠀⠀⢁⡄⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠈⠁⠐⠂⠘⡀⠀⠀⠓⢠⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢄⡈⠁⠀⣀⠱⠀⠀⠀⠐⠁⠀⠀⠚⠡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠚⠃⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢂⠐⠀⠀⠀⢀⠀⠀⠀⠀⢀⡅⠈⠀⠀⠀⠀⠀⣀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠞⣢⠀⠀⢈⠑⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠞⢉⠄⠀⠀⡄⣠⠀⠀⣄⠀⠚⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣄⣁⣤⡀⡀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢞⣥⣦⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣤⣄⣠⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣠⣤⢆⣴⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⢿⣿⣿⡿⠿⢿⠿⡿⣿⣦⣄⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣧⣿⡟⠿⣿⣾⣿⣷⣄⣀⠀⠀⣿⡛⠉⠁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣴⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣾⣿⣿⣿⣿⠉⣼⣹⣿⣾⡟⠂⠀⠈⢷⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣹⣽⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⢁⣿⡿⣿⣾⣿⠿⢿⠟⠁⠀⢤⡤⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣿⣿⢿⡿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⢠⣾⣯⣾⣿⣿⣿⣧⠈⠧⠀⢀⣸⡆⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⠃⠙⠃⠃⠈⢁⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣄⣻⣯⣿⣿⣿⣿⣿⣿⠁⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣤⣾⣶⣤⣾⣿⣿⣿⣿⣿⣿⣶⣦⠀⠈⠻⣿⣿⡿⠛⢿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⠟⠃⠀⠀⠀⢿⡟⠀⠀⡘⢻⣿⣧⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣥⡀⠀⠀⠀⠀⠘⢣⠀⠀⠀⢈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢀⣰⡿⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠻⠃⣴⠂⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢫⠈⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠁⠙⢶⣦⡀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⠈⠙⠻⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣷⣴⣇⠀⠈⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢿⣿⣿⣿⠏⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠹⡿⣿⣿⡝⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠚⢻⠁⣿⡟⠓⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⡾⠀⣿⣷⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣉⠀⠀⠀⣠⠴⠖⠛⢹⡇⠀⣿⣿⡉⠛⠲⠦⣄⡀⠀⠀⣉⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠘⢎⠑⠶⣋⠁⠀⠀⠀⢸⠀⠀⣿⣿⡇⠀⠀⠀⠈⣙⠶⠊⡱⠃⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⢧⡀⠈⠓⢦⣀⠀⡏⠀⠀⣿⣿⣿⠀⣀⡴⠊⠁⢀⡜⢧⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡞⠁⠀⠱⣄⠀⠀⠈⢻⠃⠀⠀⣿⣿⣿⡟⠁⠀⠀⢠⠎⠀⠈⢷⡀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⠁⠀⠀⠀⠘⣦⣀⣀⣼⠀⠀⠀⣿⣿⣿⣧⣀⣀⣴⠃⠀⠀⠀⠈⣧⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣠⣤⣧⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⣿⡿⠋⠀⠀⠀⠉⠉⠙⠒⠒⠶⠾⠤⣄⣀⣷
⠀⠀⠀⠶⠶⢾⣟⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣳⢯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠒⡶⠶⠤⠤⣄⣀⣀⠀⠀⠀⣠⣾⣿⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢻⠛⠉⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⡀⠀⠀⠀⢠⠟⠉⠉⢻⣿⣿⣿⠀⠀⠀⡟⠉⠉⠻⡄⠀⠀⠀⢀⡟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠈⢧⡀⠀⡰⠋⠀⠀⢀⣼⣿⣿⣿⠀⠀⢰⣧⡀⠀⠀⠙⢆⠀⢀⡾⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢳⡜⠁⢀⡠⠖⠉⠀⣿⣿⣿⠀⠀⣸⠀⠉⠳⢄⡀⠈⢣⡞⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢠⢎⡠⠶⣍⡀⠀⠀⠀⢸⣿⣿⠀⠀⡇⠀⠀⠀⢀⣩⠶⢄⡱⡄⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⣉⠀⠀⠈⠙⠳⠦⣤⣈⣿⣿⠀⢸⣇⣤⠴⠖⠋⠀⠀⠀⣉⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⢿⣿⠀⡼⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣼⣿⢀⣧⡤⠤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⣀⣰⣶⣶⣄⣀⠀⢤⡀⠂⠀⡀⠾⢻⣿⣶⣶⣶⠄
⠀⠀⠀⠿⠛⢿⣿⣿⣿⣿⣷⣽⣥⠮⢚⣷⠀⢠⣿⣿⡟⠋⠀⠀⠀⠁⠀⠀⠀⢀⣀⣴⣿⣧⣤⣦⣴⣶⣤⣤⣶⣶⡤
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣟⠉⢀⣦⡙⠀⠰⠟⠉⠁⠤⠀⠀⢀⡶⣿⣗⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⠃
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣵⡿⢿⠇⠀⠀⠀⠀⠀⢀⣤⢀⡛⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠈⠃
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠓⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⡀⠑⠛⠷⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⠀⠞
⠀⠀⠀⠘⣿⡏⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣶⣶⣶⢾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠙⠳⣯⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⠟⠀⠈⢻⡿⠋⠹⢿⣯⡁⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠒⣾⣷⣦⣀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠣⠀⠀⠈⡙⠁⣀⠠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⠟⠄⠀⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡄⢉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⣤⣴⣶⣦⣤⣤⣶⣶⣦⣤⣤⣤⡀

⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠳⣄⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠉⠉⠉⠉⠉⠉⠉⢣
⠀⠀⢀⡨⠷⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀
⡤⠒⠉⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⣀⣠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠤⠤⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⢀⡇
⠀⠀⠀⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠄⠂⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⣀⣀⡠
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⣀⡠⠤⠒⠚⠇⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠉⠉⠁⢣⡀
⠀⠁⠁⠀⠀⠤⡀⠀⠀⣀⡀⠤⠴⡖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⣠⡠⠤⠦⠐⠒⠚⠉⠙⠂⠄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠈⡀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠙⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠈⠀⠀⢀⠠⠀⠒⠉
⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⢀⣀⡀⠀⠠⠀⢢⠀⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⡔⠊⠉
⢱⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠁⠉⠀⠀⠒⠀⠰⠀⠀⢄⡀⠀⠀⠀⠀⠀⠆⠀⠀⣀⠤⠀⠂⠉⠁
⢸⠤⠀⠀⠀⢀⡄⠰⠀⠀⠀⠀⠀⠉⠈⠁⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⡰⠆⠁
⢸⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠲⠂⠄⠀⣀
⢸⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠢⠀⡀⠀⠀⠀⠀⠀⢹⡀⢐⠂
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠔⠂⠓⠄⠀⠀⠰⠀⠿⠋⠳⠲⠶⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠊⢿⣦⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡄⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠈⠟⠉⠙⣦⡀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢳⣄⠀⠀⠀⢰⣾⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠇⠀⠀⠀⢀⠀⠈⠀⠀⠀⠹⣷⣄⣀⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡾⠃⠀⠀⢀⠔⠻⣦⠀⠀⠀⠀⠀⠈⣛⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠓⢶⣎⠀⠀⠀⠙⢷⣤⣤⣆⡤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠳⣄⠀⠀⠐⠉⠛⠈⠀⣠⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠙⡦⢀⠐⠚⠛⠀⠀⠀⣰⠇⢰⣿⡿⠀⣿⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋⠀⠀⠀⠀⠀⠀⠀⢀⠋⢱⠆⠸⡟⢻⠁⢿⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠁⠀⠀⠀⢸⠀⠀⢃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣶⣄⣼⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡴⢴⠾⠿⣦⣿⣷⣤⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢦⣴⡠⠤⠒⠉⠁⠀⠀⠀⠹⡽⠃⠉⠉⠁⠉⠐⠠⣿⡎⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⣀⣀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠈⠉⠉⠙⢲⡴⠁
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠾⠽⣛⣹⠗⢛⣦⣶⡿⠛⠓⢶⣾⣿⡖⠒⠲⠶⣶⣶⣾⣷⣶⣶⠺⠥⣤⣼⠿⠷⠒⠒⠶⠶⠶⡒⠢⡀
⠀⠀⠀⠀⠀⢸⡀⡀⢀⡠⠞⢉⠀⠀⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⠋⠀⠰⡀⠀⠀⣀⠉⡆⢱⠀⡇
⠀⠀⠀⠀⠀⢸⠈⡔⠁⠀⠀⢸⠀⠠⠸⠆⠀⠀⠐⣃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠻⠆⠀⠀⠈⠢⡐⠜⠀⢰⣟⠠⡇
⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠘⠒⠒⢺⡇⠀⠀⢀⡀⠉⠉⡀⠒⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣿⠀⠀⠀⠀⠀⠈⢦⡀⣸⢻⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠂⠀⠀⢀⠼⠁⠀⠀⠀⠁⠈⠀⠀⠂⠀⠠⠀⠀⡀⠀⠘⡄⢀⠔⠋⠀⠀⠀⠀⡁⢂⠓⠂⢰⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⠫⠃⡀⢸⢸⠀⠋
⠀⠀⠀⠀⠀⡼⠀⠀⠘⠶⠃⠠⡇⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠁⢸⢸⠀⠸
⠀⠀⠀⠀⠀⢶⠀⠆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⠄⠀⠀⠀⠀⠘⠈⠀⡀
⠀⠀⠀⠀⠀⢸⠀⡂⠐⠂⢢⠀⠁⠀⠀⠀⠐⠂⠠⢁⠁⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠸⢀⢢⣄⠋⠀⣁⠂⣀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢁⠐⡀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠀⠀⢸⠀⠐⠲⢅⡒⣀⡙⠃⡄⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠑⡀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠐⠠⠀⠀⠀⠈⠀⠀⢈⠉⢂⠀⠀⠈⠐⠇⠀⠀⡁⢠⡁
⠀⠀⠀⠀⠀⢰⠀⠐⢄⠠⠂⠉⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠄⠰⡀⠀⢠⡀⠀⢸⠀⠀⠇⠀⠀⠀⠠⠐⠆⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡀⢘⣀⢑⣄⣀⡃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠜⠆⠈⠛⠀⠀⠀⡇⠀⢸⠄⠂⠀⠀⠀⠐⠀⠀⠐⣧⣀⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠁⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠠⠄⠠⠰⣇⣀⣸⣤⠤⢠⡔⠁⠀⠜⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⢀⣔⣵⣿⠟⠋⠉⠀⠀⠈⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⣼⣿⡏⠀⣀⡴⢚⣿⠟⠛⢿⣟⠲⣄⡀⠈⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⣾⡉⠀⢾⡇⣄⣽⡆⣿⠆⠈⣹⡆⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣇⠀⠙⠶⣬⣷⣮⣭⣴⣏⡤⠞⠉⢀⣿⣿⠇
⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⡀⠀⠀⠀⣀⣠⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣮⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠏

⣿⡯⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣛⣛⣛⡛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠟⣉⡥⠀⣠⠀⣄⠀⢤⣉⠻⢦⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⣡⡴⠞⣻⡿⢿⡟⢿⣟⠶⣦⡉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⠏⠀⠚⠟⠁⠾⠿⠀⠿⠷⠄⠻⠗⠀⠙⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⣠⣾⣥⣤⣾⣋⣀⣸⣇⣀⣻⣦⣤⣽⣷⡈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⠃⣾⣿⡿⢀⣶⣶⣶⠀⣶⣶⣶⡄⢺⣿⣷⡘⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⣰⠏⠀⠀⣿⠋⠉⠉⢻⡏⠉⠉⠹⣯⠀⠈⢻⡄⢿⣿⣿⣿
⠀⠀⠀⢸⡇⣸⣿⣿⡇⣸⣿⣿⣿⠀⣿⣿⣿⣷⠈⣿⣿⣇⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⢠⡟⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠈⣷⠘⣿⣿⣿
⠀⠀⠀⢸⡇⢈⣉⣉⡀⣈⣉⣉⣉⠀⣉⣉⣉⣉⠀⣉⣉⣉⢸⡇⠀⠀⠀⢸⣿⣿⣿⣇⢸⡷⠶⠶⢾⡷⠶⠶⠶⢿⡷⠶⠶⠶⣿⡷⠶⠶⣿⠀⣿⣿⣿
⠀⠀⠀⠸⣇⢸⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⡟⢠⣿⣿⡇⣸⡇⠀⠀⠀⢸⣿⣿⣿⣿⠈⣧⠀⠀⠸⣧⠀⠀⠀⢸⡇⠀⠀⠀⣿⠁⠀⢠⡿⢰⣿⣿⣿
⠀⠀⠀⠀⢻⣄⠻⠛⠓⠀⢉⣉⣉⠀⣉⣉⡉⠁⠘⠛⠟⢠⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠹⢃⣀⡀⣿⣶⠶⠶⢿⡷⠶⢶⣾⣿⣤⣤⡾⢁⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣦⡘⠻⣷⡈⠻⣿⠀⣿⡿⢁⣼⠿⢃⣴⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡈⢿⣄⡈⢻⣦⡀⣿⡇⢀⣼⠏⣀⣼⠟⣡⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣉⡀⠈⠀⠉⢀⣉⣥⡶⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣌⠛⠶⢿⣷⣾⣷⣿⠷⠞⢋⣡⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⡠⠐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠈⠀⠀⠂⠈⠉⠉⠐⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢱
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠄⠈⠀⠔⠒⠀⠀⢱⠒⠒⠒⠂⠈⡈⡆⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠣⣀⠀⠀⢀⣠⠏⠀⠈⠉⠁⠀⠠⣇⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠁⠓⠲⠬⠭⢭⣭⠥⣤⡤⠤⠤⠤⠀⠐⢰⠀⠐⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠁⠂⠠⣀⣠⣀⣤⣤⣈⣁⣀⣀⣃⣀⣀⣀⣀⣀⡸⠀⠁⠇⠀⠌⠀⠰⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⠀⠀⠉⠉⣁⣀⣀⣀⣾⡇⢰⣷⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣶⡆⢸⠇⢀⡀⢠⡟⠛⠉⢉⢹⣿⠛⣿⡿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣥⣿⡄⣿⢃⣴⡟⢁⢰⡟⢹⡟⢀⣿⠃⠓⣺⡆⣾⠂⣾⣦⢾⣦
⠀⠀⠀⠀⠀⢠⣄⣀⢀⣀⣀⠹⣏⢸⣧⡻⠴⠛⠧⠿⠜⠧⠼⠧⢺⣿⢀⠀⠿⠄⠻⠤⣿⡗⠾⠟⠒⠶⠤⢴⠶⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇



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