Preschooler Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠄⠒⠂⠤⢤⡴⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠹⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣅⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⢰⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠖⢮⡀⠈⠙⠲⢄⠀⠀⠀⢀⠔⠁⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠀⠀⡻⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡄⠒⠳⡤⣀⣀⡀⠀⠀⠀⣀⡀⠀⣀⡠⢶⠀⠀⠱⠴⠚⢣⠀⠀⠀⢑⡤⠀⠀⠀⠀⠀⠀⠀⣀⣤⢀⣀⠽⠦⡀⠀⠀⠀⠀⠀⡾
⠀⠀⣠⠞⠁⠀⢸⠤⠄⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⡎⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠛⠻⠛⠓⠒⠒⠋⠁⢉⡉⠀⠀⠀⡏⠳⣌⠀⠀⢀⠇
⠀⠀⢹⠋⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠇⠈⠳⣄⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⡇⠀⠈⢣⣀⠞⢄
⠀⠀⠓⠤⠲⢖⢽⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⢄⣀⣤⣆⣠⠎⠀⠀⠀⠈⢧⠀⢰⡀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠣⠤⠶⢹⡏⠀⠀⠱⡄
⠀⠀⠀⠀⠘⠋⠀⠳⣄⠀⠑⠒⠚⠀⠀⠀⠀⠀⠀⠀⣀⡥⠊⡸⠀⠀⠀⠀⢀⣬⠆⠀⠑⢄⡀⠙⠋⠁⠀⠀⠀⠀⠀⠀⢀⡠⡖⠁⠘⣄⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣢⣄⣀⣀⣀⣀⣠⣤⠔⠚⠑⠢⢄⡧⢿⠀⢀⣦⡸⠀⠀⠀⠀⣀⠬⢵⣦⣤⠤⢤⣤⠔⣲⠮⢥⡒⠓⢄⠀⠙⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⣿⠉⠈⢁⡤⣇⡙⢖⠒⠒⠮⣷⣠⡷⠋⠀⠀⠀⠀⠀⢠⠃⠀⠐⡇⠉⠉⠋⠀⠀⣏⠀⠀⠙⢍⠙⣗⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣰⠏⠀⢠⠮⠀⠈⢱⠀⠑⢦⡀⠈⠻⡁⠀⠀⠀⠀⠀⠀⢸⡀⠀⣴⠃⠀⠀⠀⠀⠀⠻⣄⣀⣀⢼⠀⡏⢡⠜⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠋⠁⠀⠀⢸⠀⠀⠀⠀⠣⡀⠀⠱⡀⠠⠹⡀⠀⠀⠀⠀⠀⠀⢇⢻⡝⠀⠀⠀⠀⠀⠀⢄⠙⠻⡄⠀⡆⠹⢀⣰⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡟⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⣴⠗⡐⠄⢡⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠇⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⡹⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⢼⣀⣀⠀⠀⠀⠀⢀⡠⠊⠀⠀⡜⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢥⣀⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠍⠉⠉⠙⢰⣢⢀⡴⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢲⡶⠤⠤⠤⣤⠤⢤⠲⠿⣜⡢⠜⢁⠦⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢆⣠⠤⢄⡀⠀⠈⡷⢫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⡟⠀⣸⠀⢐⠂⠈⠑⢭⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢹⡤⠊⠀⡴⠡⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡉⠢⣄⡀⢀⡼⢧⡔⢁⡴⠻⡀⠀⠀⠀⢳⠤⠔⠃⠐⠤⣤⣤⣄⡤⡞⢄⠀⠀⠀⡎⠱⣖⡋⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢀⠖⠉⢇⠀⠈⣳⠆⠀⠹⠉⠀⠙⣧⣀⡠⠃⠀⠀⠀⢠⠂⠀⠀⠀⡜⠁⠀⠀⠀⠉⠢⠉⠒⡺⠀⠉⢙⠀⠰⠀⠀⠀⠀⡾⠀⢀⡀
⠀⠀⠀⠠⠀⡏⠀⠀⠀⠉⠈⠁⠀⠀⢸⠀⠀⠀⢳⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢫⠀⠀⠀⠀⢀⡞⠁⠈⡀⠙⠢
⠀⠀⡂⠀⠀⠹⣄⠀⢀⠀⡀⠀⠀⣠⠇⠀⠀⠀⠀⠙⠶⠶⠖⠚⠁⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠑⠦⠤⠴⠋⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠠⢀⠀⠈⠙⠈⠁⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣌⠀⠀⠀⠀⠀⢠⡀⠐⢠⣴⣎⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠻⢦⡀⠀⠀⢀⣾⣶
⠀⠀⠀⠀⠀⠀⠛⠂⠁⠊⠁⠀⠀⡄⢺⡔⢿⣅⡖⢰⠐⡖⠉⠁⠀⠋⠀⠈⢤⠀⠀⠀⠀⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠐⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⢿⢇⠈⡄⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⢘⡘⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⠀⠀⠀⠰⢠⣴⣦⡠⡄⠀⠈⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⠄⠀⠂⠀⠀⠀⠃⠠⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠊⠁⠈⠁⣤
⠰⡶⢄⠀⠀⠀⣶⣶⣦⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⡺⠆⠰⠶⠀⠀⠀⠀⠀⠂⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠄⣤⣦⡄⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⢠⠀⠀⠀⠀⠐⠂⠤⠀⠀⡭⠅⠀⠀⢀⣤⠀⠀⠈⠋⠃⠙⠉⠀⠀⠀⠀⠀⠀⢠⠐⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠂⠀⠄⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⣴⣂⠂⢠⣄⠦⠤⠂⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡦⡴⡶⢃⣴⠂⠀⠀⢀⠀⠀⠻⠋⠀⠘⢋⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣷⡤⡍⠀⠀⠀⠿⠃⠀⠀⠀⠚⠳⠖⠛⠾⠀⠀⠀⠀⣀⠰⠤
⠀⢀⣀⠀⠀⣐⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⣠⣤⣄
⠀⠈⠉⠀⠀⠈⠉⠀⠀⡻⠓⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⡉⠛⠁⠉⠁
⠀⠀⠀⠨⣶⡴⣶⠘⠿⠃⣡⠜⠓⠀⠀⠀⣀⢀⡀⠀⠀⠄⡀⡀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠀⠈⠁⠀⠌
⠀⠀⠀⢁⡈⣡⣄⢀⣀⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠉⠀⠐⠀⠀⠀⠀⣨⠴⠚⠃⠘⠂⠀⠀⠀⡦
⠀⠀⠀⢘⠋⠌⠛⡌⠛⠀⠀⠀⠀⠀⣤⣶⣀⣀⣀⢀⠀⠀⣼⡵⠾⠇⠀⢀⠀⠀⢠⡄⠀⡌⠁⠀⠀⣀⢠⡄⠀⠀⠀⡎
⠀⠀⠀⠫⠅⠀⠀⠄⠂⢠⠀⠀⠀⠈⠛⢹⣏⣹⣿⢛⣤⣾⠃⠀⠒⠘⠀⠘⠇⠀⠀⠁⡄⠁⠀⠀⠀⠈⠀⠀⠀⠀⡔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣀⣠⠀⠀⠘⣿⣉⣿⣿⣏⡝⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢁⣐⡀⠀⠀⠀⢀⠠⣂⠵⡈⠲⢂
⠀⠀⠀⠀⠀⠠⠤⠴⠦⣄⠈⠋⠁⠀⠀⣘⡇⠿⠀⠸⠇⢿⡀⠀⠀⠀⠀⣀⢠⡀⠉⠀⠈⠁⠀⠠⣄⠀⠁⠀⢐⣖⡰⢱⢔
⠀⠀⠀⢸⡂⠀⠘⠀⠀⣻⠀⠀⠀⠀⠀⠉⠉⠛⠀⠙⠃⠉⠁⠀⠀⠀⠠⠝⠉⠀⠀⠀⠀⠀⢠⢀⠉⠀⠀⠀⠔⢐⠇⠀⠰⡀
⠀⠀⠀⣠⣧⡔⡻⠛⣪⣴⣄⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢺⠰⠆⣞⠀⡅
⠀⠀⠘⣭⢪⠕⣇⢸⡹⡖⡁⠀⠀⠀⠀⠀⠀⠈⠀⢰⡶⠶⢶⣆⠀⠠⠄⠀⠀⢠⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣭⠛⠓⣭⡔⠷
⠀⠀⠀⠀⠉⠉⠙⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉
⣿⣿⠛⢿⠛⡟⣿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣄⢸⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣦⢠⣀⡄
⠀⠀⠀⠀⠀⡻⠛⠘⠿⠿⣾⡞⠀⠀⡔⢉⡻⢽⠗⠛⠶⢽⢎⡉⢱⠀⠀⡨⠴⢛⣷⣦⣀⠋⢷⡤⣤⡾⣽⣟⡿⣿⣤⣤⣀
⠀⠀⠀⠀⡼⠁⠀⡀⠁⠀⠀⡇⠀⢰⠀⡞⠠⡀⠀⠀⠀⠈⣆⡇⡎⠘⡏⢠⠀⢌⠈⠘⠹⡷⣿⣿⠞⡀⠀⠘⠛⠈⡿⣿⣿
⠀⠀⠀⠘⢧⠘⢄⠀⢀⡴⢀⡿⠃⣸⡼⠱⡀⠑⠤⠤⠞⢁⢾⠐⡇⠀⢣⠸⣄⣀⠴⠀⢠⡧⠸⢿⠀⠣⣀⣀⡴⠁⡏⢿⡙
⠀⠀⠀⠀⠈⢳⣤⣉⣁⡴⠋⠀⠒⠉⠀⠀⢙⣦⠤⠤⡖⠁⠈⢦⡱⠂⢀⣹⠦⢤⡤⠖⠁⠀⣶⡄⠓⣤⣀⣁⡤⠚⠀⠀⠁
⠀⠀⠀⣀⡞⢹⢣⢀⢤⡗⣄⠀⠀⣠⡤⠚⢩⠯⢰⠀⡟⢄⢀⣄⠠⡎⢩⠏⠀⢰⡏⢹⠦⠚⠁⢀⣸⣁⠴⣀⣽⡀
⠰⢶⠉⠸⣶⠃⠈⠁⠀⣷⠟⠢⣀⠈⠁⢠⠋⠀⠘⠀⢹⠈⢳⣥⠀⡽⠁⠀⠀⠀⡏⠚⠀⢘⠟⣩⠎⠀⠃⠀⢇⠈⢳⠦
⠀⠀⠀⠀⢣⣄⣀⠀⣀⣸⠀⠀⠛⠀⠀⠑⡦⢄⣀⣀⣈⡇⠀⠀⠀⢱⠒⠀⠀⢠⠇⠀⠀⠀⠸⢷⣤⣤⡤⣤⣼⣆
⠀⠀⠀⠀⡜⠀⢀⡄⠉⡇⠀⠀⠀⠀⠀⢰⠃⡸⠙⡆⢹⠀⠀⠀⢠⣇⢀⡴⡀⣈⡇⠀⠀⠀⠀⠀⢡⠏⠉⠙⡅
⠀⠀⢠⡊⠁⢀⠂⣻⠄⠘⡄⠀⠀⢀⠤⠾⣴⠃⠀⢳⣴⣇⠀⠀⣀⢼⠉⠀⠀⢹⣀⣀⠀⠀⣀⡼⢭⠂⠀⠀⢯⠧⣀
⠀⠲⠍⠩⠖⠊⠀⠱⢖⣈⡳⠄⠀⠙⠒⠒⠋⠀⠀⠈⠃⠒⠓⠘⠒⠚⠁⠀⠀⠘⠤⠒⠃⠈⠑⠒⠊⠀⠀⠀⠘⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠰⢷⣄⡀⠀⠻⣿⣿⣿⡿⠃⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣠⣴⣤⡄⠙⠻⢷⣶⣿⣿⣿⣷⡶⠾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⢀⣠⠾⠃
⠀⠀⠀⢴⣄⠀⠹⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢨⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠻⢦⣤⣤⣌⣩⣿⣿⣿⣠⣶⠏⠁
⠀⠀⠀⠀⠉⠛⣶⣾⣿⣿⣯⣤⡀⠀⠀⢀⣽⣿⣿⡁⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠟⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⡶⠿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠹⣿⡆⠈⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣿⣿⠀⢰⣿⣿⣿⣿⣿⠀⠈⠁⢠⣿⡟⠈⣿⣿⠀⠀⠀⠀⠀⣼⠟⣵⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠛⢻⣿⡟⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢸⣿⡄⠻⣿⠀⢸⣿⡟⠀⠀⠀⢸⣿⣧⡀⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⠛⠛⠛⢿⣿⡆⠀⠀⠀⠀⠀⠈⣿⠁⢻⡇
⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢸⣿⣿⠈⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⢿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⢻⣿⠀⠀⠀⠀⠘⠻⡇⠀⠀⠀⠀⠀⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠸⠇
⠀⠀⠀⠀⠀⠠⢾⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠶⣶⠀⣿⡿⠆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣨⣷⠀⢶⣏⣀⣠⣽⡇⠀⠀⠀⠰⣿⡏⠀⣿⣧⠀⠀⠀⠀⠀⢀⠙⢄⣴⣶⣶⣶⠍⠀⠀⠀⠀⠀⠀⢀⣷⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡳⠀⠀⢿⣿⣿⣿⡷⢠⣴⠀⠀⡻⣷⣴⣿⣟⠀⠀⠀⠀⠀⠘⢷⣤⣿⣜⠟⠁⠀⠀⠀⠀⠀⠀⠘⣶⣟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣿⠿⠆⠀⠈⠻⠽⣿⣿⠟⠁⠀⠀⢰⣡⢾⣿⡏⠀⠀⠀⠀⠀⠀⠀⣹⣿⣽⠋⠐⠒⠆⠀⠀⠀⠀⠀⠈⣽⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠉⣴⣾⣿⣷⡄⠀⠀⠀⢠⣾⣿⣷⡆⠀⠀⠀⠘⣯⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠋⠀⠀⠀⢠⠤⠀⠐⠛⣚⣻⣿⢟⠓⠦⣀
⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⡿⠁⠀⠀⠀⠀⠉⠉⠋⣀⡤⠶⠚⢋⡻⣟⣷⡛⠓⠒⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠿⠿⠏⢓⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠉⠁

⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠈⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣟⠫⠭⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣿⣿⣿
⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⣿⣿⣿
⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿
⣿⣿⣿⣿⣿⣷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣯⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢀⣳⣿
⣿⠻⡿⢿⢻⢿⣿⡄⡰⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⢄⡱⡄⠀⠀⣐⠂⠀⠀⠤⠤⡄⠀⣖⠀⠀⠀⠀⡈⠀⠀⢿⣿
⣿⠀⠷⡟⣶⢸⢙⢁⡆⢁⠀⠀⠀⢀⠀⠁⠁⡄⠀⠉⠻⠟⠓⠚⢛⣛⢘⢘⡏⣃⡞⡅⠀⡀⠉⠄⠀⠀⠀⡡⢌⣿⡀⣈⠀⠀⠄⠀⢠⣸⣿
⣿⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠨⠅⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠭⠀⢠⠇⠀⠈⠀⠈⠀⠀⠀⢠⣶⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠂⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠁⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⡿⣄⣀⠀⠀⠀⠀⠠⠐⠀⠻⠛⠳⠟⠻⡛⠟⠋⠉⠀⠁⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠀⠂⠀⠙⠀⠀⠨⡑⠀⠒⠀⠂⠚⠛⠛
⣧⡄⠀⣀⣀⣀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣀⣀⡀⠀⣤
⣿⡇⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⣻⣿⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡤⠄⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⣿⣿⠀⢸⣿⠁⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠤⣼⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡄⠂⠀⠀⠀⠀⠀⠂⠰⠀⠀⠀⠀⠀⢐⢠⣿⣿⡇⠀⣿⣿⠀⢸⣧⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⣼⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡗⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⡂⢸⣿⣿⡇⠀⣿⣿⠀⢸⣿⠀⡂⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⡇⢀⣤⠀⢀⣀⢀⣿⡅⢠⠄⠀⢤⣀⠠⢸⣿⣿⡇⠀⣿⣿⠀⢸⣿⠀⠄⣠⡄⠀⢤⠀⣽⣇⢀⣀⠀⢠⣄⠀⣿⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⣇⠐⠈⠉⠉⠑⢸⣿⡇⠈⠉⠉⠉⠒⣰⣸⣿⣿⡇⠀⣿⣿⠀⢸⣿⣰⡐⠊⠉⠉⠁⠀⣿⣿⠀⠉⠉⠉⠐⢀⣿⣿⣿⠀⠀⣿
⣿⡇⠀⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⡇⠀⣿⣿⠀⢸⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠙⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠠⠐⠀⠉⠀⠉⠁⠙⠓⠒⠠⢄⠀⠀⠀⠀⠀⡀⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠱⢦⣤⠖⠁⠀⢂⠀⡠⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠠⠔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠞⢋⠟⠀⠀⡆⠈⠛⠓⠶⠶⠦⠤⠤⠤⠤⢴⡆⠀⠀⢴⡏⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⢀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⡤⢺⠀⠀⠀⡏⠐⠀⠤⠀⠀⢀⡀⠀⠀⢤⡾⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠞⠳⣿⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠳⢤⣀⣉⣉⣁⡴⠚⠁⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠄⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⡁⠛
⢠⡀⠴⢊⢃⣀⡠⠄⠀⠀⠀⠠⢄⣀⡀⠒⠂⠤⣄⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⣀⠄⢀⠼⣄
⠀⠀⠰⠪⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⢌⣉⠒⠠⠤⣉⣓⣦⣤⣄⣀⣀⣀⣀⣀⣤⡴⠞⢉⣀⡀⠠⠤⠐⣒⣩⠤⠚⠁⠀⠈⠳⢤⠀⣰⠀⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣌⢀⡝⠩⣉⣩⡽⢋⢉⣉⡀⠠⠤⠤⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡁⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠌⠀⠀⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡆⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢲⡟⠀⢌⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢌⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠶⣤⣤⣤⣄⣀⣀⣤⡤⠾⠋⡁⢄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠠⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡃⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠡⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠢⠆⠁⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠂⠄⢀⠐⠂⠀⠂⠀⠜⠃⠀⠀⠀⠄⠀⠠⣏⡴⠂⠈⢀⡈⠅⠂⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠋⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⢀⠌⠘⠁⠀⠈⠅⠀⠀⠀⣐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⢀⠘⠟⠀⠂⠀⠀⠀⢰⠁⡀⠀⠀⠀⠀⣷⠀⠀⠀⠻⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠀⠀⠀⠀⠀⠀⡀⠙⠀⡈⠀⠀⠀⠀⠸⡀⠉⠄⠀⠀⠀⠀⠀⠀⠈⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠠⠀⠀⠀⠀⠚⠁⠠⠀⠀⠀⠀⠀⠀⠑⡱⣼⠁⠄⠀⠀⠀⠀⠉⢀⠠⠂⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢰⢦⣀⣐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠄⠘⠁⠀⠒⠉⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠢⡡⡰⠴⠺⠤⠁⠀⠭⠄⠀⠀⠠⡀⠀⠀⠀⠤⠄⡘⠀⠀⠀⠀⠀⠀⠈⠈⠂⠂⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠠⠭
⠀⠀⠈⠀⠀⠀⠀⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠟⠓⠒⠒⠒⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⡘⠃⣤⠅⠘⠇⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⠋⢠⡄⠀⠗⢀⡈⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠚⠂⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠤⠤⠅⠀⠀⠀⠀⠠⠤⠔⠺⠁⡀⠀⠀⠀⠀⠀⠀⠠⢙⡀⠀⠀⠀⠀⠀⠤⠉⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⠴⠀⠀⠀⠀⠐⠒⠐⠒⠉⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀⠀⠘⠺⢝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⣽⣶⡶⢾⡥⠤⠤⠤⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡹⠊⠛⠉⠉⠀⠤⣄⠀⠀⠀⠀⠀⠈⣧⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⢠⢆⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠂⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠞⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⠈⠀⢀⡠⠖⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠄⢀⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢔⠀⠀⠀⠀⠰⡒⠊⠉⣀⠄⡇⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⣌⡉⢅⡾⠀⠀⠀⠀⣠⠏⠀⠀⠠⢐⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠔⢁⣀⠴⢳⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⢴⠒⠋⡍⡉⠀⠤⠒⠊
⠀⠀⠀⠀⡀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⢊⣠⠀⠑⠚⠁⢡
⠀⠀⠀⠀⠠⠀⠀⢁⣀⡠⠖⠀⠤⠤⠔⠚⠤⠐⠒⠉⡜⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡴⠤⢍⡑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⢄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣇⣀⠀⠈⢑⠖⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡭⢷⡀⢀⣀⣀⣈⡑⠲⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢸⢀⠇⠉⠉⠈⠁⠈
⠀⠈⠂⠀⡀⠀⠀⡴⢢⠜⠁⠀⠀⠉⠉⠀⠀⠀⠈⠐⠢⢤⣠⠀⠉⠉⠒⡖⠀⠀⡠⣇⠜
⠈⠢⠠⡄⠈⠑⠲⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣸⢅⠀⠥⠒⠉
⠀⠀⠀⠪⡲⢄⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣤⣤⣤⣤⣬⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⣶⣿⣶⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⣴⣾⣷⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠾⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⠃⠉⠀⠀⠀⠸⢸⣿⣿⠗
⠀⠀⠀⠀⠈⣯⡇⠀⠀⠀⠀⠀⠀⢹⣸⠃⠀⠀⠀⠀⢸⡏⣿⡁⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⢸⡏⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⢯⠀⠀⠀⠀⠀⠸⡇⢹⠇⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠸⠇⣿

⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⢀⣿⣿⠲⢤⡀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢘⣿⣿⡏⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠚⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢻⣿⡿⠛⠆
⠀⠀⠀⣴⠟⠈⢻⡄⠀⠀⠀⣼⠏⠈⢻⣆⠀⠀⠀⠀⠾⣿⣿⡿⠆⠀⠀⠀⠠⣾⠟⢻⠀⠀⠀⠀⠀⣿⠿⣷⡀
⠀⠀⢠⣏⠀⠀⠀⠹⠶⠂⢰⣇⠀⠀⠀⠹⡦⠆⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⣼⡁⠀⠸⣦⠀⠀⠀⣰⡟⠀⢸⠧
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠐⠒⠢⠄
⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⠀⠠⢀⠀⠀⡀⠈⠀⡠⠤⠤⠒⠒⠢⢄⠀⠈⠤⡀
⠀⠀⠀⡠⠂⢁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈
⠀⠀⠈⢀⠔⠈⠉⢢⠀⠀⠀⠀⠀⠀⠀⢣⠀⡐⢁⠀⣀⣀⣠⣀⣀⡀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰
⠀⠠⠔⠁⠀⠀⠀⠀⠣⠠⠤⢀⡀⠀⠀⢸⢐⠀⡈⠉⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡠
⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⠈⠑⢤⡜⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰⢓⠅
⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠠⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢪⠦⣀⠀⠀⠄⢐⠈⠀⣀⠔⠁
⠘⠣⢄⣀⡑⠄⣈⡐⠒⠒⣁⡠⠔⠁⡠⠶⠆⠢⠠⠼⠤⡀⠀⠁⠘⢢⣤⣀⣀⣀⡠⠊⠀⠈⠙⢒⠶⠒⠶⣋⠁
⠀⠀⠀⠠⢠⠜⠁⠉⠉⠘⠉⢦⡈⡉⠀⢀⡦⠤⠤⡄⠀⠋⠉⠉⠋⠟⠐⢤⠤⠄⠒⡣⠤⠎⠀⠁⠐⠄⠀⠈⢆⡑⠒⠢⢤
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡒⠁⠀⠘⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠬⢁⡀⠂⠠⣤⠤⠤⠄⠤⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢇⡀⠀⠀⠄⠠⠤⠤⠤⢼⣀⡀
⠀⢀⣳⡀⠀⠀⠉⠈⠈⢂⡔⠐⠒⠂⠀⠀⠀⠚⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠀⠀⠀⢰⣀⣀⣀⣀⣦⠤⠔⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠊⠉⠈⠳⣄⡴⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠈⠁⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⢉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠫⡉⢢⡉⠢⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⢄⠀⡄⠲⡈⠣⠑⢲⠤⡄⠀⣼⠖⠀⠀⠀⡠⢺⠋⢍⠒⠀⠀⠁⠀⠁⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡀⠣⠘⠀⠁⠀⠄⠈⣏⣥⠾⠃⠀⠀⠀⠀⡔⡏⢀⡀⠀⠀⠀⠀⡀⠀⠀⢸⡱
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠂⠀⠀⠀⢀⡄⠀⡏⢀⣀⠀⠀⠀⠀⠀⠈⢇⠀⠁⡀⠀⠀⢀⡁⠀⠀⡬
⠀⠀⠀⠀⠘⣷⣀⣈⡽⠳⡀⠀⠒⠒⠒⠉⢀⡴⠡⡋⠈⠃⢀⠐⠠⡆⠀⠈⠢⡀⠙⠂⠐⠊⠀⢀⠔⠁⢠⡤⢤
⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠈⢓⢶⡴⠦⣚⣉⣠⠔⠉⠁⠀⠈⠓⠚⠳⢄⣠⢤⡌⠓⡴⢤⣤⠖⠥⣤⠴⠚⢦⡼
⠀⠀⠀⠀⠸⣉⡹⠒⠒⠾⡉⠉⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠀⠀⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣀⣀⣀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⡀⠀⢀⡱⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠒⠒⠒⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠃⠑⠘⡀⣠⣻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠰⢏⣈⡹⠃⠀⠰⣛⣃⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠁





⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣿⡟⠀⠀⢸⡧⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠀⠐⠒⡄⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠃⠀⢄⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣂⡔⠁⠐⠤⠸⠁⠀⠀⠀⠈⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡤⠀⠀⠀⢠⠖⡉⠂⠠⠀⠀⢀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠄⡽⠀⡀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠉⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠀⠀⠀⠙⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠒⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢄⡀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠚⠣⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠐⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃
⠀⠀⠀⠀⡄⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡀⢀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠚⠛⠋⢉⣉⠉⠁⠲⠦⢤⣀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⠀⠀⠈⠉⢲⡤⠒⠉⠉⠈⠈⠉⠉⠒⠢⢄⡀⠈⠙⠒⠤⣀⠀⠉⠒⠶⣉⠁⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⣀⣀⣠⠖⢻⣷⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠑⠦⣄⠀⠈⠓⠤⣤⠤⣀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠊⠁⢠⠎⠀⠀⠸⠘⣧⡀⠈⠑⠢⣤⣀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠉⠲⣤⣀⠈⠑⢤⡱⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠞⠀⣰⠃⠀⠀⠀⡇⠀⠈⢻⣦⡀⠀⠀⠉⠑⠂⠤⣄⡀⠀⠀⠀⠀⠈⠉⠒⠠⠤⣀⣈⣑⢒⣤⣽⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠆⡰⠁⠀⠀⢀⠞⠀⠀⢀⠞⠉⠻⣳⠦⢀⡀⠀⠀⠀⠀⠙⠐⠦⠤⠀⣀⣀⣀⣀⣤⡴⠎⠛⢁⡤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠉⠒⠤⢉⡁⠒⠒⠂⠀⠀⠀⠒⠒⠒⠈⠉⢁⣀⣴⡿⢥⣴⣏
⠀⠀⠀⠀⠀⠀⢰⠀⠠⣴⣉⣀⠠⠐⠉⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠙⠛⠒⠒⠒⠀⠀⠀⠐⠒⠲⠦⣉⣀⣀⣀⣀⣀⣽⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠋⢹⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠘⡀
⠀⠀⠀⠀⢀⡜⠀⠀⢀⠆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⡀⠀⠀⢣⡀
⣀⠤⠤⠔⠋⠀⠀⢠⡌⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢱⡄⠀⠀⠙⠦⡄⣀
⠣⣀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⠄⣀⠀⠀⠀⢈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠶⠤⢀⣀⣀⣀⣀⣀⣀⠤⠶⠐⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⡨⠥⠶⠶⣖⠒⠒⠒⠈⠁⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣰⠋⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⠑⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢱⣤⣴⣶⡶⠿⠿⠛⠛⠋⠉⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠛⠛⠋⠉⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢧⠠⢤⣤⠤⠤⠤⠤⠀⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡤⣤⣤⠃⠠⣤⣄⣐⡲⣦⠼⣿⣟⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⢠⡄⢰⡆⢹⡿⠉⠙⠃⠐⡏⠈⠃⠛⠦⠞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⢶⣾⠛⠛⠒⠛⠒⢋⣈⣷⢀⡛⢻⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⡎⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠀⠀⠀⠀⠀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⠙⣧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡇⠀⠀⠀⠀⢀⣀⠠⠆⠚⠁⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⠒⠒⠲⠶⠒⠀⢤⣤⣤⡼⢧⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣀⣀⠤⢤⣉⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣤⡀⢀⣀⡀⣀⣰⡟⠒⠛⠛⠛⠛⠉⠉⠉⠁

⠃⠀⣠⠞⣁⢖⡡⠀⠀⠉⠱⡆⠀⠀⠀⡠⠞⠁⠀⣰⠧⠀⡰⠁⠀⠀⢠⡇⠀⠀⢀⡿⢃⡴⠟⠋⠁⠀⢀⣀⣤
⣠⠞⢁⡶⠟⠉⠀⠀⠀⠀⠀⡀⢈⢀⠚⠁⢌⢀⡾⠁⠈⡸⠁⠀⠀⠀⡾⠀⠀⢠⣿⡶⠋⢂⣤⡤⠖⠛⡉⣂⣤
⢁⡔⡹⠁⠀⠀⠀⠀⠀⠀⣰⣡⡞⢩⣼⡱⠶⠟⠓⠒⠒⠳⠤⢄⡀⠀⠁⢀⣴⣿⠿⠞⠋⠉⢀⣀⡤⠖⠋⠉
⠎⢰⠁⠀⠀⠀⠀⠀⢀⡼⣏⡾⠗⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⣋⣀⣄⠤⠤⠖⠚⠉⠁⠀⠀⣠⠴⠛
⠀⡇⠀⠀⠈⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⡀
⠀⡇⠀⠀⠀⠀⢸⡟⠡⠊⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⡠⠴⠟⢁⢀⣴⠚⠃
⠀⣇⠀⠀⢀⣠⠏⠀⢀⡠⠂⠁⠀⠀⠀⠀⢀⣀⣀⠀⠤⣤⠴⠊⠀⠀⠀⠀⢀⡤⠞⡃⠀⠀⠀⠈⠓⠆⠍⢹⡀
⠀⢸⡇⠀⣾⡟⠀⠔⠁⡠⢄⣂⢂⣤⡩⢅⡶⠦⠄⠐⠈⠀⠀⠀⠀⣠⠴⠺⠉⠀⢀⣷⡀⠀⠀⠀⠀⠀⠑⢌⣷
⠀⠀⣿⣔⣿⣧⢀⣴⣽⣾⣿⠿⠿⠽⠷⣮⡤⣄⠀⠀⠀⢀⣤⢖⢋⠀⠀⣠⣴⠚⠋⠀⠲⡀⠀⠀⠀⠀⠀⠈⠛
⠀⣼⠀⣟⣿⡇⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠚⠉⠀⠈⠀⠀⠚⠁⠀⠈⠀⠀⠁
⢠⢻⡥⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣤⣀⣀
⡆⢸⢰⣯⠏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉
⣃⣾⠗⣁⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣯⠴⠚⣩⣼⡃⠀⠀⠐⠀⠀⠄⣀⠀⡀⠀⠀⢀⠂⠀⠀⠀⢀⠀⠠⠤⠄⢀⣠⡄⠀⠀⢀⡀⠀⠀⠀⠁⢐⣶⠛
⢗⠈⣴⠞⢍⡀⠤⡼⠇⠤⢀⠘⠁⠚⠉⠠⠤⡾⠀⠄⢠⠀⢡⠀⠀⠀⠀⠀⠁⡄⠀⠐⢁⡇⠀⠀⠔⠀⠈⠓⡀
⡰⣞⣠⣴⠞⠁⢠⠃⠀⣀⡾⠀⠀⠄⢸⣠⣾⠁⠀⣠⠇⣐⣝⠰⣂⣀⡀⡄⠀⢇⣠⢃⣯⡀⠆⠀⠀⠀⢀⠜⠁
⠘⠛⠳⠶⠚⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⠊⠀⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠄⠀⢠⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢸⠀⡇⢠⠘⠀⡇⢠⣘⠚⡂⣈⠑⠁⠀⠀⠀⠀⠀⠀⢠⡾⢁⡼
⠀⠒⠶⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣧⢊⡴
⠀⠤⠀⢠⣝⣧⣤⣤⠄⠀⡄⠀⠀⠀⠀⠀⠤⣀⢠⣄⠠⠀⠀⠀⡄⠀⠠⡄⢤⣠⠀⠀⢀⠄⣠⠔⣝⣷⠟⡩⠊
⠀⠀⣀⠔⠚⠛⠋⠀⠀⠀⡇⠀⠈⢀⠀⠀⠀⠘⢷⣿⣆⠡⠀⠀⢰⠀⠀⡅⣜⣿⠀⠀⡆⣴⢋⠈⠞⠁⠈⠀⣀
⠼⡀⠤⠄⢤⠤⣄⣀⣀⠀⢰⡀⠀⠀⠂⠀⠀⠀⠀⠈⠻⣦⠱⡀⠀⢣⠀⢣⣿⣽⡇⡰⣰⢁⣨⣅⢀⠠⡂⠭⣒
⠢⠀⢄⠀⢀⠠⠴⠒⠘⠋⠛⣷⠀⠀⠀⠑⠄⠀⠀⠀⠁⢜⢷⣌⠢⡀⠣⡈⢎⣿⣷⣟⣧⣞⣡⡾⠷⠛⠛⢋⠙
⠂⢦⢘⡖⠁⠀⠀⠀⠀⠀⠁⠈⢦⡀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢝⢷⣬⣂⣙⢌⠻⣿⣾⣿⠟⠿⢂⣈⢩⢩⠌
⠈⣖⡏⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠻⣦⡁⠀⠀⠀⠀⠉⠂⠤⣀⠀⠛⠿⣿⣿⣿⣾⣿⣧⡴⠯⣤⠒⢂⠋⠀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠠⣂⠭⣄⣀⠘⠈⠙⠶⣄⣀⢀⠐⠀⠠⠤⢬⣭⣥⣴⠾⠻⣻⢻⣿⠙⢳⡌⠻⡍⠀⠔⠁
⠀⢹⠀⠀⠀⢀⠔⣥⣮⣍⡂⠈⡉⠑⠛⠲⢤⣠⡈⠉⠙⠛⠛⡛⠛⠯⡁⠀⠀⠀⡹⣆⡿⢊⠥⢛⡆⢱⠂⠀⢠
⠀⠸⡇⠀⠠⣵⡷⠒⠉⠉⠀⠀⠁⠉⠐⢀⠠⠀⠈⠙⠲⣤⣤⡀⠀⠀⠙⢲⣄⠀⢋⢸⡔⢁⢠⡎⢸⡌⣇⡸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣆⢻⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢀⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠉⠁⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⣠⣄⠀⠀⠀⠀⠀⣉⡙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢰⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠉⢁⡶⠀⠀⠀⠈⠛⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⠓⠀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠛⠛⠁⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣤⡴⠶⢾⣿⣿⡻⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣿⣤⠶⢛⣡⣬⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠳⣼⠓⠞⠋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢰⡏⠀⢠⡇⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟⢷⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠸⣇⠀⠻⣆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣏⠁⢀⣀⣴⠞⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠄⣠⡿⠞⠛⠉⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠇⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠷⠶⠶⠶⠶⠶⠛⣿⠁⠀⠀⠀⠙⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠏⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⡞⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⣿⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡾⠁⠳⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⣸⠃⠀⠀⠙⢧⣤⡾⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⢳⠞⠃⠀⠀⠀⠀⠀⠁⠹⣦⣀⡿⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠏⢩⣧⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀⠀⠀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣥⢿⣆⠸⠿⠋⠀⠀⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⢿⣷⣄⣠⣴⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⣀⣀
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢦⡀
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⠀⡀⠀⠀⠀⢀
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠠⢀
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠹⣿⣿⣿⠟⠉
⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⣤⣄⣀⠀⠀⠀⠀⢀
⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠈⠉⠛⠻⠷⠶⠶⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣴⠏

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠁⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠀⠚⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⡿⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣷⠀⠀⢀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⠀⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⡄⠰⠿⠿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣬⣉⡉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠋⠉⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣷⠀⣿⠉⠛⠛⢿⣿⠿⠟⠒⠀⠀⠀⠀⠀⠈⢻⡞⠛⠂⢤⣤⣀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⣄⠁⠀⠀⠀⠀⠀⠙⠛⠶⣤⣄⠀⣀⣀⠀⠀⠹⣆⠀⠀⠀⠀⠙⠻⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠶⠿⠷⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡿⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡄⠀⠀⡀⠂⠀⠁⠉⠀⠀⠰⡶⠲⠆⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⢻⡇⠀⠀⠃⠀⠀⠺⣿⣿⠂⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠻⣿⡷⠀⡄⠀⠀⠀⠀⠀⣿⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⢾⠿⠋⠀⠀⠀⢸⠀⠀⠀⠀⠃⠤⠘⢿⠟⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢀⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⢿⡈⠁⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⢹⡿⠉⠉⠉⠉⠐⠒⣤⠶⠶⠀⠈⠉⠉⠉⢀⣼⢿⡼⠃⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢸⠇⣤⠀⠀⠀⢀⠴⠋⠲⠀⠀⠀⠀⢀⠔⣭⢻⣼⠃⠀⠀⠀⠀⠀⢀⣼⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⡆⠀⠈⣰⠿⡀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⢀⣰⣉⡿⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⣿⠀⠀⠀⠙⣠⠿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠀⠀⠛⣄⠛⢦⣀⡈⠀⢀⣀⡰⠾⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠃⠀⠀⠀⠀⠀⠙⢷⣄⠉⠛⠛⠋⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠘⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣰⣿⡝⠓⠚⠛⢢⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠌⠛⠿⠓⠀⠘⠃
⠀⠀⠀⠀⠀⢠⣿⡤⠙⠛⠛⢳⣄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⡞⠉⣤⢀⡁⢰⡿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠘⠷⠾⢷⣤⠁⢺⡇⢿⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⣀⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠙⠟⢿⣇⣼⣀⣿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣀⡀⠛⢷⣤⣾⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠋⠙⠀⠻⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠀⢸⡟⠉⠛⠛⠻⣷⣆⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣀⡀⠙⠲⠶⣦⣄⣩⣿⣿⣷⣴⣿⣃⡀⠀⠀⠀⣠⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠈⠛⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠙⠻⢶⠒⣀⣉⣉⠉⠉⠉⠉⢻⣿⣿⣦⣾⣾⠿⠁⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⣦⣦⠀⣶⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠐⠚⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣉⣛⣻⣿⣿⣤⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⣿⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⡛⢻⡏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢨⠇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠒⠒⠶⠖⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣠⣄⣶⣤⣤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣀⠤⠄⠈⢛⣀⢠⣤⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠉⠉⠉⠉⠉⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠴⠆⠉⠉⠉⠁⠶⠿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⣀⡀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⣼⡟⢻⡆⣿⣷⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣇⣸⡇⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠋⠁⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠄⠊⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠠⠐⢁⠀⠀⠀⠀⠂⠄⡈⠂⠄⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢱⡇⠀⡔⠁⢀⠄⠒⠒⢤⡀⠈⢂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⢸⠀⠀⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⢸⠀⠘⣀⣤⠂⠐⠄⢰⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣸⡇⣈⣷⣾⣿⡟⠀⠀⠘⡇⢀⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⡁⠄⠂⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⣿⡀⢻⣮⣿⠦⣀⠀⠀⣴⡇⠀⢀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⠛⠶⠦⠤⠤⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⢺⣿⡷⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⢠⡟⢰⣶
⠀⠀⠀⠀⣤⣤⢶⣿⡇⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠠⣿⣯⡳⡄⠀⠀⠀⠀⠀⠀⣧⣸⡿
⠀⠀⠀⠀⠀⣟⠛⡛⠀⠀⠀⠀⠀⠀⠀⣯⠟⠀⠀⠀⠀⠀⡐⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠧⠓
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⣩⣯⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡶
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢻⣿⣭⡟⠂⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⢀⣾⣿⡧⢄⣀
⠀⠀⠀⠐⠚⠋⣷⠀⠀⠀⠀⠀⠀⢀⢄⢐⡇⠀⠀⠀⠀⠀⣤⣄⢘⡇⠀⠀⠀⠀⢀⣘⣿⣿⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢸⣟⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣠⣆⣀⡀⠀⠀⠀⠀⠀⠀⣰⣿⣆
⠀⠀⠀⠘⣿⣿⣾⣄⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠙⡟⡟⢻⡿⣧⠀⠀⠀⠈⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠛⠛⠻⡀⠀⠀⠀⠀⢀⡜⢹⠀⠀⠀⠀⠀⠀⠀⣷⠏⠀⠘⢧⠀⠀⠀⡇⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢠⣄⢀⣿⡀⣀⡄⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠠⠤⢿⡿⠤⠄⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠘⠛⣿⡟⠃⠀⠀⠀⠀⠀⢀⣹⣷⡄
⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠈⠋⢁⣾⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠲⠶⣆⡟⢸⡆⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣝⢿⣿⣿⣿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣛⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣯⣿⣿⣿⣿⣷⡄⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣄⣠⣴⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⣩⣭⣭⣭⣭⡭⠭⠽⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⢛⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠄⠀⢠⠀⠀⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡏⡇⠀⠀⠀⠸⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡆⠀⠀⣿⠀⢰⡄⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡄⡼⠀⠀⡄⠀⡄⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠊⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠻⣗⠠⣤⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠰⠒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠲⣄⣝⠙⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⠠⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡟⡤⠃⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠰⠀⣀⣀⠀⡇⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠛⠀⠘⠇⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠃⠘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣶⣶⣶⡆⢰⠒⠀⢸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡆⠀⣠⢂⡴⠠⠄⠀⠠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠸⠤⣄⣸⠀⠀⠀⠀⠀⠀⠈⡇⢀⡠⠀⢻⡔⢡⠞⠀⠀⠀⠀⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠈⠧⠃⠀⠠⢶⣖⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⢄⠁⠀⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣫⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣟⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠙⠁⠻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠇⠉⠉⠀⠀⠀⠀⠀⢰⣦⠘⠿⠿⢿⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⣠⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢛⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠓⠒⠛⣻⠆⠀⠉⠉⠉⠉⠙⠚⠓⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠛⠲⢤⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢠⠏⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡼⠁⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡰⠋⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠖⠋⠀⣀⡴⠞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣀⢦⢠⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⣀⠀⢀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠛⠓⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡛⠛⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⢠⣶⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠇⠀⠀⣠⡾⠛⢦⠀⠀⠀⠀⠀⢠⡏⠀⠀⢰⠟⠻⡄⠀⠀⣤⠞⠀⢀⣴⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠴⠛⠁⢀⡴⠛⠁⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⠀⠀⣯⠀⠀⡇⠀⠀⢳⣀⣠⠾⢀⣆⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣀⠀⣀⣀⡤⢾⠃⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠱⡀⠀⠀⠀⠀⢀⡠⢦⢾⣄⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠉⣉⣥⣤⠞⠀⠀⠀⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⠙⠓⠶⠖⠛⠉⠀⠀⠀⠉⠓⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣴⣟⠥⠖⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡆⢠⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⠉⠁⠀⢀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠛⠟⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢳⡄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⠀⢸⠂⠀⠀⠀⠀⠀⢰⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢿⡀⡴⠖⢶⠀⣖⠲⢸⣧⡶⠂⣴⣖⣦⠘⢿⠋⢸⡶⠲⣄⣴⠖⢶⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠾⠃⠳⠤⠾⠐⠮⠽⠚⠇⠹⠆⠻⠭⠧⠀⠸⠀⠸⠷⠴⠋⠻⠦⠾⠻⠇⠿

⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠉⠉⣿⣿⣿⣿⡏
⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃
⠀⠀⠀⠈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣄⡉⠀⠈⠛⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠲⢤⡀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠉⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⣀⣴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠒⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢮⢋⠚⢁⠆⢠⠀⠀⠀⢈⠒⢌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠁⠈⠠⠇⠘⣰⠀⠀⠃⠀⠄⢣⡈⠃⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣆⠃⠦⠀⠐⠀⡈⢦⡐⠄⢠⡈⠁⠁⠀⢘⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢆⠘⠤⢆⠃⠈⠂⠉⡀⠀⠙⠲⠎⠰⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡈⡀⠀⢀⠠⠄⠀⠀⠀⠴⠶⠂⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡔⠃⣿⡀⠁⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠉⣛⠷⢤⣄⣀⣀⠀⠀⡴⠋⠀⢠⡐⢮⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠘⢦⠀⠀⠀⢀⣤⠶⠀⠀⠀⡼⠃⠀⠀⠀⣻⠀⠀⠉⠉⣉⠷⡞⢀⠟⠛⢶⣬⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠳⢦⣀⡀⠀⠀⣀⡤⠞⠁⠀⠀⠀⢸⡁⣔⠁⣠⠊⠁⠀⠑⠉⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⢧⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⡖⢟⠀⠁⠀⡇⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⢦⡤⣄⠀⠀⡀⠀⠀⣀⡀⣀⡴⠓⠛⠁⠈⢦⡄⠀⠙⠦⠤⠤⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⡤⠀⠀⠀⠀⢸⡇⠈⠛⠋⠙⠲⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡤⣤⡴⠏
⢀⣀⡀⢰⠦⣄⣀⣀⡠⠄⠒⠊⠉⠁⠀⢧⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⣬⠶⠤⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⡄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢈⣜⣋⡤⠈⣁⠀⣀⣆⣀⣀⣀⣀⣀⠤⠤⠒⠊⠙⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢨⡴⠋⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠚⠁⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠛⠛⠓⠲⠶⠦⣤⣤⣤⣀⣀⣤⣤⣠⠄⠠⠶⠒⠛⠋⠉⠉⠉⠉⢁⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠖⠒⢶⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⠋⠉⠉⠉⠙⢻⡏⠉⠀⠀⣇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀⠀⣀⣤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣿⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣀⣼⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠀⣇⠀⠀⠀⠀⢀⣾⠷⣭⣽⠋⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀⣤⠞⠙⣆⠀⠀⢀⡞⠁⠈⠉⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢿⣥⠟⠃⠀⣠⡞⠀⠀⢸⣀⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠁⠀⣠⠞⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠠⠶⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣆⢻⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢀⣈⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠉⠁⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⣠⣄⠀⠀⠀⠀⠀⣉⡙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠰⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠉⢁⡶⠀⠀⠀⠈⠛⠋⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠛⠛⠁⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣤⡴⠶⢾⣿⣿⡻⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣿⣦⠴⢋⣡⣬⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠳⣼⠓⠞⠋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡏⠀⢠⡇⠀⠀⠀⠀⠀⢠⣄⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟⢷⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠸⣇⠀⠻⣄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣏⠁⢀⣀⣴⠞⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⠄⣠⡿⠞⠛⠉⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠷⠶⠶⠶⠶⠶⠛⣿⠀⠀⠀⠀⠙⠶⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠏⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⡞⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⣿⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡾⠈⠳⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣸⠃⠀⠀⠙⢧⣠⡞⠛⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠙⢳⠞⠃⠀⠀⠀⠀⠀⠀⠹⣦⣀⡿⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⠏⢩⣧⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⣤⣴⣼⣷⣤⣄⣤⠀⡀
⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠛⠻⢿⣄⠀⠀⠀⠀⠀⠀⣸⣟⣁⠀⠀⠀⠀⠈⣉⡛⣷⣯⠀⠀⠀⠀⠀⠀⢠⣄⢠⡄⠀⠀⠀⠀⠀⡀⣀⠀⣀
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⢰⣿⠎⠁⠀⠀⠀⠀⠀⠉⠋⠈⢻⡄⠀⠀⠀⠀⠀⠀⠻⣾⡿⠁⠀⠀⠀⠀⣿⣿⣾⠏
⠀⠀⠀⠀⢸⣷⣶⣶⣤⣄⣀⠀⠀⠀⢘⡆⠀⠀⠀⢈⣿⠀⠓⢀⣾⠃⠀⠀⢶⠄⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⣹⣿⡟
⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡟⠿⠶⢦⣾⡇⠀⠀⣴⣿⣇⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣦⣾⣿⣷⣄⠀⠈⢻⣿⠀⠠⣿⣇⣀⠀⠀⠀⠀⠀⣨⡟⣿⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣴⡿⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⠀⠉⠛⠻⠋⠀⠀⢹⡆⠀⣶⣿⣿⣷⣶⡄⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣹⣿⣿⣮⣷⣤⣀⣀⡀⠀⠀⠀⠹⢶⣌⣉⡈⠉⣉⣁⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⢾⣷⠟⠋⠀⠀⠀⠀⠀⠉⠛⣳⠀⠀⠀⠀⠀⠀⠙⣿⡟⢹⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠟⠛⣷⣤⣀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⣿⡷⠛⠶⠀⠀⠀⠶⣦⠀⠀⣿⡆⠀⠀⠀⢀⣄⠀⣽⠄⠸⣆⣠⣶⣦⠀⣠⣾⡿⠛⢉⡛⢁⣾⡏⠈⡛⠛⠿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⠁⠶⠀⣾⠀⠠⡦⠈⠀⠀⣿⣿⣄⠀⢠⣿⣿⣿⣋⠀⣀⣿⣿⣿⣿⣿⣿⣩⣴⠶⠋⠁⠈⠉⠛⠳⢦⣘⠻⢿⠁
⠀⠀⠀⠀⠀⣾⣇⠀⠀⢰⣟⣀⡀⠀⠀⠀⠀⠙⠋⢿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋⡤⣄⢀⡄⠀⠀⣀⠴⢦⠈⠉⢻⣶⢶⡄
⠀⠀⠀⠀⠀⠘⢻⣆⠀⠀⠉⠉⡁⠀⠀⠀⠀⣶⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠃⠀⠖⠙⢠⣷⠀⠀⠁⠺⠈⠀⠀⠀⢻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⡿⠀⠀⢀⣼⣿⣿⣿⡿⢱⣡⢻⣿⣿⣿⣿⣿⠘⣿⠸⣧⡀⠀⠀⠀⣾⣧⣴⠆⠀⠀⠀⠀⠀⠀⣸⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣦⣀⠀⠀⣠⣿⠿⣿⣿⣿⡅⠸⠇⣿⣿⣿⣿⡝⣿⡇⠛⠀⣿⡷⡶⢶⢰⡟⠟⣻⠶⣶⠶⢶⣶⣤⢾⣿⠟⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠻⣿⠛⠛⣻⠇⠀⠀⠈⠻⣿⣄⣼⠟⣿⣿⣿⣿⠉⠀⠀⢀⣿⡁⠀⠊⣼⡇⠀⠋⠀⠋⣻⠿⠛⠁⣴⠟⠀⠀⠀⣼⠻⡄
⠀⠀⠀⠀⠀⢰⡟⠀⡀⠀⠀⢻⣧⣴⠟⠀⢀⠀⠀⠀⠙⣿⣀⣾⣿⣿⣿⣿⣧⣀⣀⣬⣿⣿⡿⣷⣿⣷⣶⣶⢶⣤⣿⠇⠀⣴⡷⠀⠀⠀⠀⡀⠀⢻
⠀⠀⠀⠀⠠⣿⣄⣠⡇⠀⠀⠀⠉⠀⠀⠄⠈⠈⣷⣤⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⠛⢷⡀⠻⣿⠆⣸⡿⢀⡀⠀⠀⣼⡇⠀⠈
⠀⠀⠀⠀⣴⠏⢹⣿⠀⢀⠀⠀⠄⠈⡀⡀⡄⠀⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⢿⡀⠈⣿⡄⠹⣷⣿⣿⣿⢿⣆⢠⣼⡇
⠀⠀⠀⣰⡏⢀⡞⠛⠀⠀⡀⠀⠀⢀⢀⢃⠀⠀⢹⣿⣿⠀⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠁⠀⠿⠁⣸⣧⣾⣿⣿⡄⠹⣿⡿⠁⢘⣿⣸⣿⣶⣦⣤
⠀⠀⢸⠏⢀⡿⢁⡿⠀⠐⠃⠀⠚⠺⠀⠸⠂⠃⣸⣿⣿⡄⠈⣿⡿⣹⣿⣿⣕⣿⣿⣿⠇⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⠀⠙⠃⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⣿⠁⣸⣿⣿⢿⣶⠶⠿⣶⣶⣶⣶⣿⡿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣛⣛⣿⣿⣿⣆⠀⣠⣿⣿⡿⣿⣿⣿⣿⣿
⠀⢰⡿⢠⡇⠀⣻⣿⣡⣾⣿⡷⣾⣿⣿⣯⢻⣿⡇⢻⣿⣿⣧⠀⢻⣿⣿⡿⣱⡿⣫⣿⣿⣿⣿⣟⢿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣝⡳⣾⣿⣿⣿⣿
⠀⢸⠇⢸⡀⠀⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠈⣿⣿⣿⣿⡾⠳⣻⣿⣿⣿⡿⠟⠛⢳⡌⢿⣿⡋⠹⢿⣶⣮⠻⣿⣿⣿⣿⣿⡿
⢠⠏⠀⢈⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⡟⠋⣁⠀⠈⢿⣿⣿⣵⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⢸⣿⣿⡀⠀⢿⣿⡧⣿⣿⣿⣿⠟
⣠⣶⣧⣿⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡇⣠⢰⣬⣿⣿⡷⣫⣿⣿⣿⣿⠃⢠⣾⣿⠿⠿⣿⣿⣷⡀⠘⣿⣿⡿⠟⠛⠁
⠋⠿⠛⠏⠀⢰⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡌⣿⣷⡿⣷⣿⣾⣿⣿⡿⣪⢿⣿⣿⣿⣿⠀⣼⠅⠀⠀⠀⠀⠀⠹⣷⡀⢿⣏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧⣿⣿⡿⢿⣿⣿⣿⣿⣡⣴⢟⣿⣿⣿⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣿
⠀⠀⠀⠀⠀⢿⣿⣧⣾⢿⣿⡿⠃⠸⢿⣿⠿⢿⣿⢿⣿⣿⣬⢿⣿⣿⣿⣿⢟⣥⣷⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⡇⠀⠀⠈⣿⠀⢸⣿⢸⣿⣿⡷⢻⣿⣿⣿⣧⠟⣡⠿⣿⣿⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡇⠀⠀⠀⣿⡀⢸⣿⢸⣿⡃⢚⣼⣿⣿⣿⣟⣯⣵⣾⣿⣿⡏⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⠘⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡁⠀⠀⣀⣿⡇⢸⣿⣴⣿⠶⠿⢿⣿⣿⣿⣿⣿⣿⢫⣶⢼⣿⣿⣿⠃⠀⠀⠀⠀⠀⣤⣶⣾⣿⡿⢶⠏⢉⣿⠂
⠀⠀⠀⠀⣠⣤⣴⣿⡿⣿⣻⣿⡾⠿⢿⣿⠻⣿⣿⠀⠀⠀⠀⢸⣿⣿⡏⣩⣥⣾⣇⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⢋⣿⣿⣿⣼⣷⡿⣿⡄
⠀⠀⠀⣾⣏⣡⣬⣿⣅⣿⣿⣩⣤⣠⣶⣮⣤⣿⣿⣤⣶⣶⣤⣾⣿⣿⣷⣬⣭⣭⣭⣍⣩⡬⠿⠇⠀⠀⠀⠀⠀⠻⠷⠤⠤⠍⠠⠤⠴⠦⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢋⠁⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠏⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡑⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢁⠀⠀⠀⠀⢀⠄⠐⠀⠚⠺⢦⡀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⣀⡀⠀⡀⠀⢀⠀⠎⠐⠸⠀⠀⠀⠰⠁⠐⠀⠀⡀⠀⠀⠙⠄
⠀⠀⡀⠀⣶⠆⠅⠂⠀⠈⠉⠁⠉⢋⠀⠁⠀⠈⠊⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠈⢌⠒⠂⠁⠀⠀⠀⡡⠂
⠀⠀⠀⠀⠉⢄⠀⠀⠀⣀⠀⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡄⠀⠀⣠⡖⡤⠠⠖⡈
⠀⠀⠀⠀⠀⠀⠁⠂⠀⢄⡤⠀⠣⠐⠛⠒⠒⠀⠀⠀⠀⠀⠀⠐⠘⠛⠋⢰⠈⠀⠸⠆⢈⠂
⠀⠀⠀⠀⠀⠠⠂⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢀⠀⢸⠈⠁⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⠀⠖⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⣀⡜⣴⠀⡠⠄⠀⠀⠱⠚⠢⡜⠀⠀⠀⠀⠁⢬⠀⡄⣪
⠀⠀⠀⠀⠀⠀⠀⠨⠼⠤⠤⠀⠀⠀⠀⠀⠀⠀⠘⠚⣝⡏⡘⠩⠀⠘⠉⠋⠀⠶⠃⠀⠀⠀⠀⠀⢀⠀⢌⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠐⠀⠀⠠⠆
⠀⠀⠀⠀⠀⡀⠀⠀⡆⢀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⣴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⣤⠄⠀⠀⠀⠀⠀⢀⣾⡷⠿⣷⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣂⠸⠛⣤⣾⣀⣀⠀⣸⣿⠁⠀⠰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣀⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠙⠿⠹⣿⡝⠿⠿⠖⠀⠉⢀⠀⣰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠢⣤⣐⠏⠀⠀⠀⢀⣤⡴⠦⡀⠀⠀⠀⠀⠀⠀⢿⡤⠉⡅⠁⠙⠧⢀⠀⠠⠀⢠⠞⢀⣁⣾
⠀⠀⠀⠀⠀⢰⠷⣠⣴⡶⢦⡴⠞⠀⠈⠉⠀⠀⠀⢀⡿⠑⢆⠀⣷⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⢴⠂⠀⠀⠀⠀⠀⠀⢘⣯⠁
⠀⠀⠀⠀⠀⣰⣾⢧⣏⠀⠀⠴⠄⠀⢠⣠⣠⣤⠀⣾⣿⣤⣉⣵⣇⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠐⠀⠀⠘⠓⠀⠀⠀⢁⣀⠁
⠀⠀⠀⠀⣼⡿⢏⠀⠹⢗⠀⠀⠈⢻⠀⢀⣿⣷⣾⣿⣾⣷⣭⡿⣿⣿⣦⠀⠀⠀⠀⠃⠀⠀⠱⠇⠀⡀⢰⣦⠀⠀⠀⠈⢻⠀⠄
⠀⠀⢰⣧⡈⢷⡄⠙⠛⠋⠁⠀⠀⠘⠀⣤⣿⣿⣿⣿⣶⡇⢹⣿⣿⣿⣿⠃⠀⠀⠀⠰⣄⢀⡄⠀⠀⠀⢸⠃
⠀⠀⢸⡟⠁⢀⣿⣄⠀⠀⠀⠀⡄⠀⢸⣿⣿⣿⣿⡛⠉⣧⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢈⣤⡄
⠀⠀⠀⣷⣿⠏⠉⠉⠀⠀⠀⣘⣃⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠁⠀⠻⠟⠛⣿⣿⣿⣿⢿⣿⣿⣿⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⡄
⠀⠀⠀⣼⣿⣿⣿⣿⠏⠛⣿⣿⡇⠀⠀⠀⠀⠀⡿⡿⠋⠉⠀⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠟⢻⢃⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⣷⠶⣶⠶⠶⢿⠷⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡿⢋⣿⣿⡏
⠀⠀⢸⣿⣏⢀⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⡟
⠀⠀⠘⠿⢿⡿⠿⣿⣿⡿⠋⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⣹⠿⠂
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣀⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣐⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠙⡏⢻⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣽⡿⠗
⠀⠀⠀⠀⠀⠀⢠⠇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠁⠀⠸⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⠉⢷⡄⠀⢾⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠹⢾⣿⣯⣿⣿⣠⣤⡀⣰⡟⠀⠈⢿⣄⠈⠷⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⢶⣛⣽⣟⣀⠀⠀⠈⠻⢧⡴⠋
⠀⠀⠀⠀⠰⣾⠊⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠈⠁⠲⠀⠀⣾⣟⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⠀⢢⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⠋⠉⠀⠈⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⢦⣀⣐⣠⣤⡼⣃⡤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠉⢳⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⢿⠇⠀⠀⠰⡿⠀⠈⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠵⠛⣶⢤⢄⢀⡤⡔⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠋⢠⢀⠋⠑⢶⡴⠒⢣⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠗⢄⡼⣘⠀⠀⠈⠀⠆⠻⣼⡦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢾⣧⠇⠀⠀⠀⠀⠂⠚⢿⣇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣾⢄⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡨⢠⣧⣀⠀⠀⠀⠀⠀⣀⣤⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⠈⠉⠉⣉⡋⠉⠉⠊⠀⠩⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⠿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠀⡿⣚⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⢑⣤⣷⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣶⣶⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉⢻⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠂⠸⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⡀⠈⠁⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠐⠢⡑⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠄⠠⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠂⠄⢀⠀⠀⠀⢀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠀⢀⠀⠈⠁⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⠦⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⡀⠀⣀⣠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⠛⠻⠋⠐⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⢀⣀⠄⢆⠁⠀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠘⡀
⢻⡀⢲⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢡
⠀⢳⡈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠆
⠀⠀⠻⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢼⣄⣀⡤⠒
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠛⣋⣤⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⢀⠠⠠⢄
⠀⠠⠁⠀⠀⠀⢂
⠀⠁⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠠⠁⠀⠀⣀
⠀⠢⣀⡀⠄⠁⠀⠀⠘⠤⠅⠀⠀⠂
⠀⠀⠢⠓
⢀⠀⠀⠀⠀⢐⢶⡄⣆⣀⢠
⠚⠃⠀⠀⠰⠚⠉⠁⠈⠉⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⢀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠠⢬⠄
⠀⠀⠂⡀⠀⢀⠀⠀⠁⠀⢰⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡀⠐⢄⠀⠂⠄⠀⠄⠁⡠⠊⣀⡀⠀⠀⠀⠂⠀⢀⠼⠀⠀⠠⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⢁⠀⠀⠉⠐⠒⠒⠂⠁⠀⠀⠉⠠⢀⡌⠀⠀⠀⠁⠀⠐⠄⢐⠄⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠐⢀⠇⣀⣀⡃⠀⠀⠀⠀⡘⡎⠀⠀⠀⠁⠀⠀⠀⠆⠀⠑⠠⠁⠀⠀⠐⢝⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⣂⠀⡀⢀⣁⠀⠀⠐⢢⣀⢃⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐⡌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡠⠂⠀⠀⠈⠜⠒⠲⢂⣤⠄⢀⠀⠡⠄⠑⠤⠠⠔⠀⠀⠀⠀⠈⠀⠦⡀⠀⠀⠀⠀⠁⢤⣹⣠⡢⠀⠀⠀⠀⠀⢀⠄⠊⠙⠳⠟⠦⡤⣆
⠀⠀⠋⠀⠠⠀⠈⠉⠒⠨⠴⢤⡤⠔⢉⠢⡄⡑⠀⠁⠒⠒⠀⠀⠭⡉⠉⢀⠂⡠⠂⠀⡄⠀⠀⠀⠈⠉⠛⢶⡅⡀⠨⠁⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠠⠄⢃⡀⢀⠠⠀⠆⠀⠂⠁⠀⠀⢰⠎⡀⠀⠀⢠⠀⠀⠀⣱⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣇⠐⠀⢄⠀⠀⠀⠀⠀⠀⢀⡀
⠰⠆⢀⡀⠔⢀⠁⠀⠀⢀⠀⠀⠠⠀⠀⠀⢀⠁⣇⣀⣀⠈⠤⠠⠐⠁⠀⢸⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠀⠑⣤⣀⠀⠠⠐⠁
⠀⠀⢈⢃⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠠⠸⠈⠁⠀⢆⣀⣀⠀⢀⠀⠄⢆⠀⠉⠰⠆⠒⠁⠀⠀⠀⢀⠆⠀⠀⣔⠀⣀⠀⠈⠀⡀⠀⢠⠀⠠⠶
⠀⢠⠀⠉⠑⢄⠀⠈⠀⠀⠀⠒⠁⢀⠠⠊⠀⢀⠘⠀⠀⠀⠀⠉⠉⡀⠐⠀⠀⠀⢱⡄⡀⠀⠀⠀⠀⠀⣀⡠⢊⡤⠂⠀⢰⠀⠀⠁⠀⠈⠁
⠀⠙⠐⠢⢄⣀⣈⣀⠀⢰⠲⣖⡚⠱⡐⠀⠐⠋⠀⡀⢤⠀⠀⢐⡇⠀⠀⣷⠀⢀⠃⠐⠉⠑⡖⠀⢈⠩⠅⠊⠁⡤⣄⠀⠀⠀⢀⠀⠀⠀⠀⠺⠿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠹⠄⠀⣀⣠⣤⣤⣀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⢤⣄⡀⠀⠀⠀⣰⢞⣡⠄⠀⠀⠈⠙⢦⠀⢀⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠉⠓⠀⢸⢣⡟⠁⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠸⡘⠃⠀⠀⠀⠀⠀⠀⠀⡟⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠤⠶⠖⠒⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢿⠙⢹⠇⠀⡴⠃
⠀⠀⠀⠀⣠⠄⠀⢳⠀⠀⠀⠀⠀⡾⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⡇⡏⠀⣼⠁
⠀⠀⢠⠞⠁⠀⠀⠸⣇⠀⠀⠀⣠⡇⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⣷⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢬⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠒⠒⠂⠸⠿⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠄⠀⣠⠦⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣔⠤⢊⣉⠤⠚⠁⠀⠑⢦⣘⣳⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⠒⠉⠉⠂⠀⠀⠀⠐⣋⡉⠙⠀⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⣰⠀⠈⣴⣶⣷⠀⠀⠀⢸⣷⣦⡉⠀⢰⡶⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⡇⠀⠀⠻⠿⠏⢀⠀⠀⠘⠿⠿⠃⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⡤⠈⣁⠀⠀⠀⠀⣤⡠⠤⠤⢠⡆⠀⠀⠀⢀⡍⠠⠴⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢾⡓⣛⠮⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡤⠄⠤⢤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡖⠒⠒⠒⠒⠂⠈⠉⠁⠢⠔⠈⠉⠁⠐⠒⠒⠒⠒⢲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠛⠚⠛⢻⠒⠂⠀⠀⠈⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⣛⡓⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠃
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠠⠤⠤⠴⣈⢃⣊⡸⠤⠤⠄⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⡖⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠤⢤⡤⣤⠤⢤⡤⣤⢠⣤⣀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠤⠤⠤⠤⠤⢤⣄⣀⣀⡤⢠⡤⣤⠤⢠⡤⣤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠠⠤⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⡤⠤⠄⢲⠊⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣦⠾⣇⠀⠀⠀⠀⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⣶⡌⣌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⢱⣶⡈⠳⣄⠀⠀⠀⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠀⢀⠞⠉⠐⢤⠟⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠹⡄⠂⠉⠳⠀⠀⠘⠉⢱⠀⠸⠀⠂
⠀⠀⠀⠀⢀⠦⢦⠀⠰⠀⠄⠀⠀⠸⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠇⠀⠠⠄⠼⠀⠰⠦⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⡆⠀⢘
⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⣶⣄⣴⣦⣤⣀⣠⣶⣾⣿⣿⣿⣿⣿⣾⣶⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠀⠒⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠠⠤⠄⠀⠀⠠⠤⠄⠀⢹⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣟⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡟⠀⠈⠁⡆⠀⢀⠈⠓⠉⠀⠀⠉⢰⢉⢬⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠠⣀⣈⣉⣁⣀⡀⠀⠀⠀⢠⠤⠵⣾⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⣀⣠⡴⠚⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠟⡏⠉⡀⢙⡯⢹⡟⠳⣲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡇⢠⠉⣁⡰⠁⡔⠁⠈⢢⠙⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣠⡏⡜⠉⠉⢹⠛⠣⠜⠓⠄⠀⠘⣌⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡘⠋⠙⠿⣿⡜⠀⠀⠀⠀⢹⠀⠀⠀⠠⣄⠀⣿⠇⣮
⠀⠀⠀⠀⠀⢠⢷⡏⠐⠒⠠⢼⡇⠀⠀⠀⠀⠘⠤⢤⢤⣤⠛⡀⡿⢰⠁⡇
⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠸⠀⢿⡄⠈⢱⠼⡇⠸
⠀⠀⠀⠀⠀⣈⢀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢣⠘⡇⠀⠈⢲⡧⠃
⠀⠀⠀⠀⠀⠙⢾⡶⠶⠹⢺⠁⠀⠀⠀⠀⢀⡀⠀⠀⡷⠹⠤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⠲⢤⣀⣀⣀⣀⣀⣤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠲⠯⠀⣤⢠⡤⠎⠱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡔⠠⠤⠀⠉⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡌⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡜⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠃⢰⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡁⠹⡀⠀⠱
⠀⠀⠀⠀⠀⠀⡀⠀⢠⠊⠀⠀⡞⠁⠀⢁⠀⠀⠳
⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣀⢰⠀⠀⠀⠈⣇⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⢠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⢔⣫⣝⢓⣀⠀⣀⠀⠀⠀⠀⠀⢀⠰⠶⠤⢀⠀⠀⣾⣀⡀⠀⢀⠀⣤⣀⠀⣠⠼⠇⢀⣠⠶⠒⠒⠲⢦⡀⠀⢄⠀⠆⣠⠤⠤⠤⡀⣷⠎
⠀⠉⡱⡣⡠⠀⠢⡂⠕⡆⠀⠀⠄⠀⠀⠀⠀⠀⠈⠆⠁⠀⠙⠁⠈⠉⠉⠀⠈⡆⠀⠀⠟⠀⠀⠀⠀⠀⠀⠓⠐⢈⣺⠊⠁⠀⠀⠀⠀⢻⡓
⠀⠠⡉⡉⠀⠀⠀⠀⡀⡰⠀⠺⠀⠄⠀⠀⠀⢀⠆⡘⠄⠐⡁⠀⠀⠀⠀⠀⡄⡘⠀⠚⡀⠂⠀⠀⠀⠀⡐⡠⠁⠘⢋⢀⠀⠀⠀⠀⡔⢀⠁
⠀⠀⠑⢄⡀⠠⠠⡐⠕⠀⠀⠀⠑⠠⣀⣒⣂⡡⠊⠀⠀⠀⠈⠢⠐⣀⣐⠪⠐⠀⠀⠀⠈⠢⢤⣐⣂⠬⠊⠀⠀⠀⠀⠃⢅⡂⢐⡊⠔⠁
⠀⠀⠀⠠⠎⠉⠉⠆⠠⢀⣀⠠⠀⠲⠈⢀⠁⠋⠀⠂⠆⠂⠦⠐⠂⠈⠁⠂⠂⠀⠤⢤⠔⠊⠅⠈⠁⠈⡈⠂⠴⠠⠐⠊⠨⠉⠉⠐⡦⡀
⠀⠀⠈⠀⠁⠀⠀⠈⡄⠀⠀⠀⠀⢸⠂⣒⠐⠃⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⡄⠀⠀⠀⠀⠘⡤⠠⡤⢤⠄⠀⠀⠀⠀⢀⠃⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⡆⢰⠒⠁⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⡄⠔⠒⠀⠀⠀⠀⠀⠀⠈⡎⢁⠈⠀⠀⠀⠀⠀⠸⠐⡄⠄⠀⠊
⠀⠀⠀⠀⠒⠃⠈⠓⠀⠀⠀⠀⠀⠒⠃⠀⠘⠒⠀⠀⠀⠀⠀⠀⠒⠃⠚⠂⠀⠀⠀⠀⠀⠐⠚⠁⠈⠓⠃⠀⠀⠀⠀⠀⠒⠃⠙

⣿⠿⣿⡿⠋⠀⠀⡠⠁⠀⣼⡿⠁⠀⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣤⠘⠀⠀⠀⠀⠀⢠⢸⠟⠀⠀⠀⣿⣿⡏⣰⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⠀⣠⡀⠀⡀⠘⠂⢀⠀⠀⠠⠀⠤
⠛⠁⠐⠀⠀⠀⠀⣰⡿⠀⠀⠀⣠⣷⣿⣿⢼⠟⢻⣿⠟⢁⣾⣿⣿⣿⣿⣿⢰⣾⢪⣆⣧⣤⣄⣟⣀⣤⣤⣤⣤
⣀⣀⠀⣾⡄⢀⣄⡭⠁⠀⢠⣾⣿⣿⣿⣿⢸⣿⣛⠁⣤⣿⣿⣿⣿⣿⢿⣿⠀⠀⠋⠉⠉⠋⠟⠸⠛⠛⠛⠟⠛
⣿⣿⣿⡟⠻⠋⠉⠁⠀⠈⢿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢤⣠⣾⣼⡷⠀⣄⣰⣀⣼⣶
⣿⠿⠿⣾⣷⣿⣤⠀⣴⡄⠶⢿⣝⡋⠿⠿⠝⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢐⣿⡄⠈⠹⠿⠿⠿⠿⢿⣿⠿⠿
⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠆⠀⣿⣿⣿⠻⠿⠿⣟⠃⠀⠉⠉
⣿⣿⣿⣿⣿⣏⣁⣭⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⡿⢿⠟⢿⢒⣖⣂⡀⠐⠻
⠿⣿⣿⣿⣿⣉⣤⣡⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⢀⣀⡿⣾⣷⠉⠀⠈⠉⠀⠠⢻⡿⡅⠲⠦
⣿⣷⣍⣉⢉⢻⡿⢿⣿⣿⣿⣿⣿⡿⣾⣿⠛⢻⡟⢛⢛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣇⣀⠀⠀⠀⢸⡇⢸⣿⣴⡿
⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣭⣽⣧⣸⣿⡿⣿⣶⣶⣶⣿⣥⣀⡀⠀⠀⠀⣿⣿⣿⣿⣄⣄⢀⡡⠤⣞⡽⢛⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠉⠭⠽⠯⣛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⢀⣠⡄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⠤⢘⡉⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⡨⠟⠀⠘⣄⡥⠠⢄⡀⡠⠚⠉⠑⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡝⠁⠀⠉⠢⣄⠀⠀⢸⡀⠀⠀⠙⣶⠂⠀⢀⡇⣀⠴⠚⠉⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⢳⡴⠋⠓⠢⣕⣄⠀⢣⡖⠀⠀⢹⠄⠒⠋⠁⠀⠀⠀⠀⣐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠑⢒⣒⠃⠀⠀⠈⠒⠂⠐⢶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠴⠶⠆⠀⠉⢇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠤⠒⠳⡀⠈⠣⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠙⠄⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠴⢧⣀⠀⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢣⡜⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣠⠿⡄⠀⠀⠀⠀⠀⠉⠈⠹⡂⠀⠠⠤⣀⠀⠀⠀⢀⡇⠓⠳⣃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⢀⠔⠉⣠⠤⢷⣀⡠⠶⠆⠀⠀⠀⠐⠷⠀⠀⢰⠛⠦⠔⠒⣉⠠⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⠔⢡⣔⠉⠀⠀⠈⠉⠀⠀⡄⠀⢀⠐⡀⠠⢳⡤⣸⡟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣬⠇⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠃⠐⠀⠳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⢈⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⠐⠒⠒⠋⠙⢍⠣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠳⠻⣇⠀⠀⣀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣄⠀⠀⠀⠀⠀⠑⢤⣯⡗⣊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠻⢿⣷⠦⣀⠀⠀⠀⠀⢿⣟⠓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠛⣤⣀⠀⢠⠏⠀⠈⠑⠤⣀⠀⡾⣿⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡢⠛⡿⠛⠉⠢⡀⠀⠀⠀⠀⡽⠺⢼⣷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠡⠤⢴⡾⠃⠀⠀⠀⠀⠁⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠐⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠃⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠕⠄⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⡀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠀⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠰⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢡
⠀⠀⠉⠐⣆⠄⠀⠀⠀⠀⠀⠂⠂⠂⠀⠈⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢃
⠀⠀⠀⢠⠇⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠖
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡟⠉⠉⠉⠉⠛⠛⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠆⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⡤⠖⠋⠁⠾⠥⠤⠤⠔⠚⠉⢡⣞⣁⣠⠤⠖⢫⡇⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣤⡀⣠⡀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠃⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⡀⠀⢷⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡀⣠⣄
⠀⣤⣼⡏⣿⠉⣇⣤⡄⠀⠀⠀⠈⡷⢦⣼⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠿⡇⠀⠀⠀⠀⣸⠇⣤⣤⡏⢻⡏⣿⣴
⣴⡽⣮⠓⠘⠀⡟⢸⡇⠀⠀⠀⢸⡇⢢⣟⠀⠀⠀⠀⢸⣷⣿⠀⠀⣠⠃⠀⠀⠀⠀⣾⣽⡆⠀⠀⠀⠀⠹⣀⡴⠢⣠⡟⠀⣿⠘⡇⠈⠐⢫⡾
⠈⠛⢶⡀⠀⠜⠁⣸⠇⠀⠀⠀⠀⣧⠀⡟⠀⠀⠀⠀⠀⠉⠁⠀⢞⠉⠁⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠛⡴⢫⡏⠀⠀⢾⡀⠹⠆⠀⣠⠞
⠀⠀⠈⠳⣦⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣭⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣼⠃⠀⢠⡴⠋
⠀⠀⠀⠀⠹⡆⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⡟⠒⠂⠤⠤⠤⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⢀⣧⡼⠃⠀⠀⡼⠃⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣠⡞⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠙⢦⣀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣠⠞⠃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⣽⣷⣶⣤⣄⣈⣳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣡⣴⣶⣿⣇⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠲⠤⣀⣀⣀⣀⡀⠤⣴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢧⡀⠀⠀⠀⠀⢀⡰⠋⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠓⠴⠶⠒⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠫⠭⣉⡉⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠛⢉⣉⠭⠝⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠉⠓⢦⣙⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣋⡴⠚⠉⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠞⠁⠀⠀⠀⣀⣤⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣶⣤⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣤⣶⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠄⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⠤⠞⠉⠈⠉⠉⠀⠙⠙⣮⡓⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣙⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⢰⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⣃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⠔⠉⢉⡁⠀⠀⠀⠀⠀⠀⠀⠰⠃⠰⠃⠾⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠏⡆
⠀⠀⠀⠀⠀⢀⠠⡀⢣⡆⠀⠀⠄⠀⠁⢀⡐⣀⣐⣄⣀⠀⠉⠛⠿⠄⠀⢀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠘⢷⣔⣈⡁⠀⡀⠀⠰⠂⡌⠙⠛⢿⣿⠯⢵⠀⠀⠀⠀⠀⠉⠉⠉⢉⠃
⠀⠀⠀⠀⠀⠀⠋⠙⣿⠃⡾⢡⣴⠀⠀⠀⠙⠳⡏⠀⠀⠀⠋⠉⢡⠀⠶⣾⣝⢳⣦⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢁⣯⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⠞⠒⢾⠁⠀⠘⣙⡆⠀⠁
⠀⠀⠀⠀⢠⡀⡀⠀⢀⡀⢿⠙⠀⠀⠀⠀⠀⠀⠹⡄⢀⡠⠖⠣⠀⠀⠀⠀⠀⠀⡰⢃
⠀⠀⠀⠀⢀⠙⠦⢤⣌⣿⠾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⡤⠴⠊⠀⡎
⠀⠀⠀⠀⡄⠀⠀⠀⠈⢻⡄⢢⡀⠀⠀⠀⠀⠀⠀⣤⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢷⣼⠙⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣼⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⢙⠓⠦⡟⢀⠀⡀⠀⠀⡌⠀⠀⠀⠀⠀⢠⣌⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣼⡟⠁⠙⢄⠀⠀⠀⠀⠨⡁⠸⣇⠔⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⡄⠀⠀⠀⢠⡛⢳⣼⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠲⣿⡄⢸⠇⠀⠀⠀⠀⢷⣾⢁⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣤⡾⡀⠀⠀⠀⠀⡠⢉⣉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢩⡄⠡⡄⠀⠀⡌⡀⡆⢰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠓⡀⠸⡆⢸⡟⣸⠇⡸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢠⠀⠉⠈⢳⡟⢠⢣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢪⣿⡀⢸⠀⠰⣇⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠘⠌⠁⠀⠀⠀⠀⠈⠃⠐⠂⠁⠀⠀⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠦⡀⠀⠀⠀⠀⣆⡙⢄⠐⠀⡀
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠡⠚⣁⠀⠀⠀⠀⠄⣢⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣶⠢⢹⡆⡴⢋⡩⠉⠻⣆⠴⢋⣽⡄⠠
⠀⠀⠀⠀⡅⠀⠈⡀⠀⠠⠀⠀⢠⣿⡄⢀⡀⠀⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢧⡸⢟⣀⡼⢻⡇⣾⡿⠀⠀⣿⣤⣉⠈⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠑⠚⣢⠅⠀⠀⠨⠟⠁⠀⠉⡷⠀⠀⢀⠀⠀⠀⢐⡀⠀⠀⠀⠐⠀⠀⠀⠙⠙⠃⠀⠀⠳⢬⣀⣀⠴⠟⠀⢹⠆⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢅⠈⠥⡀⠀⠀⠀⠀⠦⢠⡦⠀⠀⠀⠌⠀⠀⠀⠘⣭⡀⠀⠀⠐⠠⠒⠃⠤⠀⠀⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡤⢢⠴⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠤⡀⣀⣀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢯⣤⣬⠦⠄⠠⠠⠀⠰⠀⠀⠀⠘⠭⠅⠀⠀⠀⢠⣀⡀⠀⢀⣤⢠⣅⠀⠡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⢀⠋⡍⢸⠈⣿⡄⠀⠠⠆⣼⢃⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⣷⡋⠀⠤⠀⠀⠛⢧⢊⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠀⠴⠀⠀⠋⢀⢨⡇⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠤⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡭⠥⠤⠤⠤⡯⢤⠌⠰⠇⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠈⢀⢄⡰⢀⣀⣻⣤⣤⡀⠀⠀⠶⢒⣲⣤⠴⢶⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢥⡐⠅⠀⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⣀⣀⣀⠐⠨⠋⠉⠉⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠩⠴⡮⢄⢀⡀⣂
⠀⠀⠀⢀⡄⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠈⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠒⠀⠂⠀⡁⠈⠑⠈⡋⣂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠠
⠀⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣾⠉⠉⠉⠙⠛⠛⠛⣠⣤⣄⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣸⠿⠿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠉⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣤⣤⣤⣤⡜⠛⠻⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣤⣤⣄⡿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠉⣤⣤⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣀⣤⣤
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠉⠉⠉⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣦⡜⠛⠻⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠉⠋⣤⣀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⡟⠻⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠿
⣤⣤⣶⣶⣶⡎⠉⠉⠉⠙⠛⠛⠛⠿⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣱⣦⣤⣤⣤⣶⡎⠁
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣤⣀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠈⠙⠛⠿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⣿⡿⠿⠿⠟⢧⣤⣤⣤⣤⣤⣤⣀⡸⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣷⣦⣄⣀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠛⠻⠇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠉⠉⠛⠛⠛⠛⣶⣶⣾⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⡀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡞⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣴⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣶⣤⣰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠙⠛⣁⣀⡀⠀⠀⠀⢸⣿⣿⡿⠿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠉⠛⠿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢡⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⡿⠟⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢻⡀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣄⠾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⡀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⢳⣶⣶⣶⣶⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠛⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⣀⣠⠛⠉⠉⠀⠀⠀⠀⣼⣿⣿⣿⣶⣦⡤⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠛⣃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠻⢿⡏⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶⣦⣤⡼⠛⠛⠛⠛⠛⠛
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣴⣦⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣇⣀⣤⣤⣶⣶⣶⡏⠉⠉⠉⠉⠛⢟⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⠼⢿⣿⣿⣿⣿⣿⡇
⠟⠋⠉⣸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠛⠛⠻⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠉⠻⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠙⢣⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⠟⠛⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⠉⠙⠛⠿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⣤⣄⡸⠿⣿⣿⣿⣿⣿⣿
⣶⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣤⣄⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣄⠿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⣀⠸⠿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠉⠙⠛⠿⢿⡇
⣿⣿⣇⣤⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣦⣤
⠉⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠙⠻⠿⣿⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿