Dessert Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⣀⡸⠀⠀⠀⠀⠀⠀⠀⢎⡀⡱⠀⢀⣄
⠀⠀⠀⠀⠀⠀⡃⠈⡄⠀⠖⠲⠀⠀⢀⠴⡄⠀⠀⢠⠀⠄⠀⡌⠀⡂
⠀⠀⠀⠀⠀⠀⡭⢭⠀⠀⠀⢸⠀⠀⠨⣀⠸⠀⠀⠐⠀⠀⠀⢨⢭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢰⠀⢦⠀⠀⠸⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠊⠉⠑⢄
⠀⠀⠀⠀⠀⢀⡄⠠⠒⠒⠀⢸⠉⠉⢹⠀⢘⠉⠉⠙⠀⠒⠂⢆⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⠀⠸⠀⠀⠈⠉⠁⠀⢐⠀⡇⠈⠢⡀⠀⠀⢀⣤⣤⣒⡀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠷⢀⠀⠐⠚⠀⠀⠀⠀⠀⠀⢸⠀⣨⠀⠀⠀⠀⠀⠀⠘⠒⠃⢀⠴⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠒⠂⠀⠐⠒⠒⠂⡞⠅
⠀⠀⠀⡂⠀⠁⠒⠀⠤⠤⠀⣀⣀⣀⣈⣉⣁⣀⣀⣀⡀⠠⠤⠄⠐⠊⠁⠀⡇⠀⠀⠰⠆⠀⠀⠀⠤⠤⠄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡆
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐⠒⡏⠄
⠀⠀⠀⣉⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢈⠀⠀⠀⢐⠒⠂⠀⠠⠤⠤⠤⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆
⠀⠀⠀⠈⠉⠒⠠⠤⢀⣈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⢉⣀⠠⠤⠒⠈⠨⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠈⠉⠁⡏
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠠⠤⠄⠄⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠆
⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠐⠒⠒⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠒⠒⠒⠒⠒⠒⠂⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣄⠤⢀⡤⠠⢄⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠉⡠⠔⢒⡠⠶⠺⣦⠴⠦⣴⠳⠶⢄⠀⠩⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⠊⡠⠊⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠑⣄⠈⠢⡱⡀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀
⠀⠀⠀⢠⠃⡔⠁⠀⢀⠃⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠈⠆⠀⠑⡘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⢀⠆⡜⠀⠀⠀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠐⡈⠆⠀⠀⠰⡁⠑⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢁⠇
⠀⠀⠎⡜⠀⠀⠀⢠⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠐⡈⠄⠀⠀⠋⢲⡐⠢⠤⠬⣉⣉⠈⠉⠀⠁⣉⣀⣀⡉⣁⠠⠀⢐⡴⠙⠆
⠀⠈⡶⠀⠀⠀⠀⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⡀⠀⠀⠰⡆⠀⠀⠁⠦⠌⣁⠐⠂⠀⠄⠀⠠⠨⠤⠀⠀⠠⠀⠀⠒⣈⣁⣰⠋
⠀⢰⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠀⠀⠀⠈⠒⠤⡉⢈⡁⠀⠐⠒⠀⠀⠀⠈⠉⢉⣀⣁⠤⠒⠁
⠀⠈⠒⠀⠄⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠰⠂⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠂⠉⠂⠤⢀⣀⠠⠔⠊⠁⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠀⠐⠆⠀⠀⠀⢐⠁⠀⠈⣡⠕⠀⠀⠀⠸⢏⣙⡆⠀⠀⠀⠀⠀⢠⡆⠤⠀⠀⢤⠀⠀⠀⢀⡠⠜⠿⢦
⠀⠀⣈⠙⠓⣾⣛⣷⠀⢰⣿⡿⢿⣂⣸⡄⠀⠀⡀⠀⠀⠀⣸⣄⠀⠀⠠⣷⣦⣽⣷⢆⣰⡆⠀⠀⠀⣾⡟⠒⠚⠈⢰
⠀⠀⠦⡀⢘⣫⡭⠶⠀⠈⣿⠂⠂⠈⠉⠀⠀⠀⠰⢸⢸⣿⢯⡇⠀⠃⠀⠀⠹⠿⠀⠀⠻⠆⢄⡤⠀⢸⡷⣦⡄⠀⣼
⠀⠀⠁⠀⠬⠷⠚⠉⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠊⠁⠀⠐⠀⠀⢀⣀⡀⠀⢠⣤⣤⡤⠀⢸⣄⣈⡁⠀⣈
⠀⠀⠀⠀⢀⢀⠀⠤⢀⡢⠀⠀⠀⢠⣾⣛⣛⡿⠞⠀⠀⠀⠀⠀⠀⠀⠤⡅⠈⠧⠺⠖⠶⠶⠶⠖⠀⠈⠉⠉⠉
⠀⠀⠀⠀⢸⣿⣟⡶⣶⢤⣌⡀⠀⠀⠐⠀⠤⢍⣙⣷⠀⠀⠠⠔⠀⠀⣤⣶⣢⠀⠀⠀⠀⠀⢀⣚⣁⠚⠻⢶⢂⣰⠄
⠀⠀⠀⠀⠀⠷⣿⣿⣻⣏⣫⡟⠀⠀⠯⣉⡓⠢⢤⣙⠀⠀⠁⠀⠀⠀⡠⣾⡏⠀⠀⠀⣄⡀⣉⣉⠉⢀⢠⠼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠈⠑⠺⠗⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠙⠂⠐
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠒⠈⠉⠁⢀⣈⣢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡶⠿⠋⣉⣭⠶⠛⣋⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠈⠉⠛⢛⠟⠛⠭⢠⡄
⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⣢⣀⠀⠀⣾⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠓⠒⠦⢤⣄⡀
⠀⠀⢀⠴⠊⠉⠀⢔⣮⡅⠻⢮⣿⣿⣿⡶⢤⣤⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣌⣬⣵⢦⠄⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⣶⣾⣿⣭⠵⠶⠛⠋⣉⣉⣿⠿⠿⢿⣯⣯⣷⣒⣶⣧⣄⣀⣀⣿⠛⠉⠁
⠀⠀⠈⠢⢄⡀⠀⠈⠐⠀⠀⢴⣿⠿⠖⠒⠛⣩⣽⣛⡥⠴⠟⠛⠙⠛⠓⠒⠋⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⣀⣀⣀⣀⣀⠈⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⡒⠒⢄
⠀⠀⠀⢠⠒⡉⢀⠠⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⡃⠀⠀⠠⠀⠈⠀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣰⢦
⠀⠀⠀⠰⠀⠂⠀⠀⠐⠀⠄⢠⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⢇⠀⠉⢀⣇
⠀⠀⠀⠀⢣⠀⢰⠉⠂⠚⠀⠀⠃⠔⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠉⠉⠁⠈⠣⡀
⠀⠀⠀⠀⠀⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⡴⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢔⠞⠝⠀⠣⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠠⠃⡌⠀⠀⠣⡈⠁⠒⢽⢄⡠⡀⣀⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣀⣤⠁⠀⠃⠀⡇⠀⠀⠀⠘⡄⠀⠀⠣⢕⠁⡇⠓⢖⠊⡦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠈⠩⠀⡸⠆⣀⢃⠀⠐⣄⠀⠀⢀⣄⣀⠴⣗⢤⠭⠑⢆⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠬⠉⡾⠥⢓⣄⡠⠢⣉⠔⠫⣉⠜⢅⠔⢀⣀⢴⡤⠵⢤⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⢹⠒⠆⠨⠇⠀⢠⠀⠀⡄⠀⠰⠀⢠⡡⠔⠈⠉⠀⠀⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡆⠘⡀⠸⠀⠈⡀⠀⠄⠀⡄⢀⠊⠀⢀⠀⢠⠦⣔⡑⠤⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣘⠀⢀⡀⢸⡤⠓⠠⣇⠀⠃⠀⠀⠀⠇⡆⠠⠀⡀⠱⡋⠀⠀⠀⡈⠃⠀⢀⡀⢐
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠙⠀⠈⠂⠀⠁⠀⠀⢸⡀⠐⠀⠀⡐⠀⣧⠲⠠⢄⠀⠈⠙⠉⠁⠀⠀⡠⠁⢈⠇
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⠇⣀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠓⠊⠑⠤⠊⠉⠈⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠄⠒⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠
⠀⠀⠦⢄⢀⠘⠂⠀⠀⠐⠀⣈⡭⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⢐⣢⡄⠀⠀⠀⠐⠀⢠
⢀⠀⠀⠀⣙⡀⠈⠀⠀⠀⠂⠀⠀⠀⠃⠀⠈⠀⠁⠀⠀⠀⠀⢚⡩⣥⡶⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⣒⡀
⠙⠛⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠐⠀⠀⢀⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣉⠀⠐⠤⢐⡀⢠⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡿⠀⠀⠀⠀⠀⠂⡤⠀⠀⠀⠀⠉⠛⠓⠲⠾⠭⠦⠀⠀⠘⠤⢄⣀⣀⡇
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⢀
⠀⠀⠘⠀⠀⢈⡁⠀⠩⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣟⣿⣿⡂⠀⢦
⠀⠀⠁⠀⠸⠋⠙⢛⠛⢤⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠘⠛⠉⠁⠀⠀⠀⢸
⠀⠀⠀⡠⢤⣮⣆⣠⠤⡎⠀⢀⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⢃
⠀⠄⠀⡇⠀⠀⠀⠈⢉⢲⠄⠀⠒⠠⠰⠖⠁⠀⠀⠀⠀⣠⠀⡀⠀⠄⠀⢀⠀⠀⠀⠐⡀⢸⠀⠀⠒⠀⠈⠆
⠀⠀⠀⣷⣀⣀⣀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⢌⣉⠙⠰⠂⠦⢤⡀⠀⠀⠉⢺⠀⠀⠀⠀⠀⠀⣀
⠀⠠⣀⡍⠠⠄⣀⣉⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣂⠱⠂⠀⠉⢕⣒⡒⠡⢬⠀⠀⠀⠀⠱⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠴⠬⣄⣉⣡⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⣿⣥⠽⠬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠻⢶⣴⠿⢿⣄⠀⠀⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣶⣦⣶⡞⠁⠀⠀⢈⣷⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠖⠋⠉⠻⢿⣧⣬⣁⣠⣼⣿⠁⢀⣀⣸⡃
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠴⠉⠩⠭⢌⢻⣿⣿⡿⢿⡉⢉⣿⣃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠒⠖⠃⠀⠀⠀⠀⠀⠁⣗⣀⣶⡆⢛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⣸⠂⠀⠀⠀⠀⠀⠠⣿⠅⣿⠇⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠤⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣿⣆⠀⠀⠀⠀⠀⢀⣽⣸⡟⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⠴⢒⣾⡷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⡙⣿⣿⣤⣤⣀⣴⣿⣿⣿⣵⠿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠋⠁⠒⠻⡿⠿⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⣿⣿⣿⣿⣿⣿⠿⠛⠑⠉⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⣄⡀⠀⣾⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⡗⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣗⢶⡖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣴⣷⡄⢛⣿⣿⣿⣖⣦⣼⣀⡀⠀⠀⢹⠀⠀⠀⣥⣾⡗⠆⠀⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠻⠿⢟⠛⠉⣉⣤⣶⣛⣻⡿⣿⠿⢿⣿⣿⣶⣦⣤⡏⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠎⣸⢀⣆⠠⣤⣀⡀⠀⠀⠀⠀⠀⠈⠒⠰⠤⢄⣀⣀⠙⠉⠁⠀⠐⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⣴⣯⣀⢨⣦⢈⣁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢿⣖⠲⣶⡤⣀⡈⠛⠛⣩⣟⣛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠺⠾⢇⣀⣿⣶⡙⢦⠛⣿⡶⢆⠉⠉⠁⠀⡈⠳⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠁⠀⠀⠀⠀⠀⠀⠙⢀⣬⣏⣆⣀⡠⠥⣦⠷⣦⡄⢠⠀⠀⢽⡇⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣁⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣨⣟⣁⡙⣿⣳⢶⣿⢻⠘⠀⠀⢸⡇⠠⡅⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⣭⣿⠛⠿⢾⣿⣦⣭⡿⢀⡁⣰⣿⣿⣿⣧⣌⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣂⣾⣿⣿⡶⠶⣞⣿⠷⠉⢙⣲⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠈⠀⠸⠿⣓⡲⠶⠶⠾⠟⠻⠉⠉⢠⣶⡿⠿⠷⠞⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠄⣀⣀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠒⠒⠂⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠊⡉⠙⠢⣄
⠀⠀⠀⠀⠀⢀⠔⠚⠉⠁⢀⠀⠂⢈⠀⢠⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠀⠀⠀⠀⢰⠁⠀⠆⠀⠂⠀⠠⠄⠈⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠃
⠀⠀⠀⠀⢇⠀⠄⠀⠠⠆⠀⠄⠀⠰⠂⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣩⣯⡢⡀
⠀⠀⠀⠀⠘⡆⠀⠆⠀⠀⠀⠠⠄⢀⡤⢤⡀⠀⢸⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠊⣹⡇⠀⠙⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡀⠠⠀⡠⠤⣀⢀⡼⠀⠀⠱⡀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠛⠦⠔⠤⠎⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠁⠠⠄⠈⢀⠁⠀⠂⠀⠘⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⣀⣴⢶⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣠⢴⠞⡽⠁⣿⡀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠊⡴⠃⡜⠀⠀⢷⠘⠦⢼⡀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡞⠀⠸⠁⠀⠀⠈⠢⡀⠀⠈⠑⢦⡷⣤⠖⡆⢀⣤⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢱⣈⡱⠃⢸⠤⠷⣆⠒⠛⣣⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⠤⠴⡞⠉⠉⣿⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⢅⠑⠂⠺⣄⢀⣈⣳⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⡄⢠⠃⠑⠦⠤⠧⣀⠀⢘⣦⣀⠀⣀⡤⠛⠒⠒⠋⢹⠎⠳⣏⠉⠉⠳⢄⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡪⠵⠐⠸⠿⠭⢍⡲⢄⣠⠖⠲⡄⡨⠛⢦⣀⡤⠒⢦⡰⠒⢄⣀⠴⣦⡤⠶⠳⣖⢢⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠂⡴⠚⢳⠤⣤⡈⣈⣣⠀⠀⢨⠀⠀⠀⡍⠀⠀⢠⠀⠀⢸⠁⣀⠷⠛⠊⠉⢉⠉⡉⠉⢑⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⡀⠀⡆⠀⢹⡄⠀⠘⡄⠀⠀⡇⠀⠀⣾⠀⠀⡾⠊⢁⠀⠁⠂⠀⠀⢀⡤⡀⠀⠀⠠⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢇⠀⢹⠀⠀⢧⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡞⠁⠤⠀⠄⣀⣀⣎⣓⡋⢀⣉⡒⣄⠐⠀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡞⠀⠀⠀⣀⣘⣄⡤⠓⠢⢜⡄⠀⢸⠀⠀⠀⠀⢀⠃⡸⠀⡄⠠⠀⠀⢧⠊⠁⠀⠀⠀⠀⢉⠇⢘⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⠹⠷⠉⠉⠻⡀⠀⠙⠂⠀⠀⠈⣷⠀⠀⡆⠀⠠⠀⢸⠀⣇⢀⡀⠀⣈⠀⠆⡙⠒⡦⠐⠒⠚⡁⠐⠀⢠⠏⠙⢲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠗⠒⠲⠶⠶⠶⠒⠒⠒⠚⢳⡘⠊⠈⢦⠀⢀⣀⡀⠂⠈⣠⣀⣀⠖⠊⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢀⡴⢥⣀⡀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠑⠉⠀⠙⠢⠞⠁⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣄⣀⣀⣀⡠⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⣶⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠁⣀⡀⡱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢸⡧⣟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢷⠋⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡛⢛⡛⠻⠟⠛⢓⠔⢻⠀⢸⣿⣿⠿⢟⢃⣴⣾⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣇⠎⠠⡀⢃⡄⠀⠈⠀⠀⢃⠈⠅⠀⠀⠠⡋⠁⠀⠀⣤⣴⣶⡶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⡀⠐⢻⠀⠟⠦⠬⠤⠿⠧⠖⠁⠁⡀⡏⠦⠤⠦⣤⣤⣥⣤⣄⠐⣨⠒⠠⠦⢠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠁⠀⢠⣶⣾⣿⣿⣷⣦⠀⠀⢀⣀⣀⡷⣧⣀⠔⡿⠘⣿⢄⠀⠊⡗⠐⠒⠀⠠⣼⠐⣅⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣿⣿⠀⣠⣵⣿⣿⣿⣷⡜⣿⣴⣾⣿⣿⣿⣿⢩⠋⢰⠁⠀⠈⣿⡂⣤⢃⠶⠯⠄⠀⣿⣇⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡙⠛⠉⠻⢿⣿⣿⡿⢿⠀⠀⠀⢸⣿⣿⠿⠘⢄⣿⡀⠀⠀⣼⣇⣏⣨⠂⠒⠒⢰⡿⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⣀⠀⠘⠒⠒⠒⠛⠛⠀⠀⠀⠀⠈⠉⠉⠉⢉⣀⣈⠡⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣺⣿⣿⡏⢻⠷⣶⣶⡖⣶⣖⣶⡾⢿⢻⣿⣷⣿⣶⣶⠶⣄⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠟⠿⠿⠿⡟⢹⠀⠈⠉⢓⢺⡀⠀⢹⣏⣿⣟⣃⡒⠠⢸⡇⠀⠉⠉⢏⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⠓⢺⡒⠦⠀⠾⠎⠉⠁⠈⠀⣄⡇⠀⠉⣉⣉⣑⠒⠠⠸⠟⠛⠿⠋⢫⣉⣉⣭⣽⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠃⠀⠉⢘⠛⠛⣀⠀⠀⠄⢀⣀⡈⠓⢛⡿⣿⣿⣿⣿⠛⢷⣶⠛⠻⠿⣽⠿⠟⢻⣷⣾⠧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡀⠀⠀⢨⡾⠤⣀⡀⠀⠀⢀⣀⣨⡇⣸⣼⣿⣿⣿⣿⣷⠈⣿⡃⠀⠀⠰⠀⣀⣆⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠭⢍⣛⣉⠓⠤⢀⣀⠀⣉⣀⣀⡹⠿⠛⢿⣿⣿⣿⣿⣿⡾⣋⡈⣁⣂⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠲⠶⠶⠶⣶⣤⠤⠤⠉⠑⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⠀⢀⡾⠵⠥⠤⠴⠲⡎⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡛⠿⣤⣤⣤⣤⣤⣀⠢⠭⢛⣩⣿⣶⣦⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠀⠐⠒⠒⠛⠛⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⣄⠰⠾⡂⢤⠐⠢⡀
⠀⠀⠀⠀⣀⣠⡖⢯⢤⡜⣼⠳⣤⣄⣀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠼⠁⣤⠇⠘⢁⡤⠃
⠀⣴⠚⠉⠀⡠⢿⠠⠀⠀⠀⠀⣥⠀⠀⠉⠓⢄⠀⠀⠀⠀⠈⢳⣴⠋⠀⠱⢴⠋
⠀⠳⣄⠀⠀⠑⠀⠱⠤⠄⠒⠈⠉⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⢸
⠀⠀⠀⠉⠓⠒⠲⠆⠤⠤⠤⠴⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠚⠘⠂⠀⠚⠘⠓⠀⠀⢀⣤⡖⠖⠚⠙⠛⠒⠒⢒⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡋⠀⠀⠀⠠⠀⢀⡀⣀⣀⡹⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠯⢑⠦⠈⢩⠮⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡻⢦⣀⣀⠀⠀⠀⠀⢀⣀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣼⣶⢊⡉⠓⣷⡘⣄⡸⢧⠀⠀⠀⠀⠀⠀⠀⠘⢦⡁⠪⠸⣀⠍⠉⢸⣉⡩⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⡼⢋⣍⣰⣷⣊⡈⡇⠘⠤⣿⠁⠀⠀⠀⢠⡀⠀⠀⠀⠈⠛⠒⠦⠤⠤⠤⠤⠖⠚⠁
⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⡀⠀⠹⣄⠈⠁⠙⠈⠁⠋⠀⠀⣠⠃⠀⠀⠀⠰⠁⢈⡢⠄⠐⢒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢹⠀⠀⠈⠳⣤⣠⣤⣤⣴⡶⠞⠁⠀⠀⠀⣀⠥⠄⣌⠀⣀⡠⠞⠚
⠀⠀⡔⠒⠀⠀⠀⢀⣀⣀⣸⣦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠁⠀⠐⣄⢢
⣠⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⢠⡀⠀⠀⠀⠀⠀⣀⣿⠀⠓⢄
⠈⠀⠄⠠⠀⢀⣀⣀⣀⠤⠬⠭⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠑⠃⠀⠀⠀⠀⠈⠉⠉⠡⠀⣸⠂
⠀⠀⠀⠀⢀⠴⠛⠁⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⡀⣀⢀⣀⠴⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠂⠈⠠⣄⠀⡄⠀⠀⡚⠁⠀⠐⠀⡨⠀⠀⠀⠀⢸⡟⠙⡆⠀⠀⠀⠀⠀⠀⢀⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣠⢶⣶⣤
⠀⠀⠀⠒⡤⢤⣴⡼⣦⠀⢀⣴⣷⣶⡤⠀⢘⡀⠀⠀⠀⠀⠀⠈⠉⢡⣄⠀⠀⠀⢰⣦⣌⣛⣦⣤⠀⣤⠀⠀⠀⠀⢠⣷⡦⠤⠤⠗⢠
⠀⠀⡙⠂⠀⣴⠿⢞⣛⠀⠀⢿⡟⠉⠉⠓⠚⠃⠀⠀⠘⡄⢠⢠⡔⡾⣿⠃⢰⡀⠈⠛⢻⣿⠟⠉⠺⣯⡂⠀⠀⠀⢘⣿⣀⡀⠀⠀⢸
⠀⠀⢈⠑⠀⠶⢟⣋⡭⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣷⣇⡏⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠑⠊⠀⠘⡟⠙⠿⠀⠀⠿
⠀⠀⠀⠀⠀⠛⠉⠀⢀⠀⣀⠀⠁⠀⠀⠀⢀⣀⣴⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢠⢴⢢⡄⠀⡒⠖⠒⠒⡁⠰⢿⡤⢤⣄⠀⠴
⠀⠀⠀⠀⠀⣄⣁⠀⠰⠀⠄⢂⣀⠀⠀⠰⠾⠯⠬⣝⣛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠖⠆⠀⠓⠊⠛⠙⠓⠛⠛⢙⡁
⠀⠀⠀⠀⠀⢻⣽⣟⡿⢶⠖⣦⣤⠀⠀⠀⠒⠂⠠⠤⢀⣹⣇⠀⠀⢀⢔⠂⢐⢢⢲⣾⣵⠀⠀⠀⠀⠀⠀⣀⣙⣁⡛⠛⠛⣛⣀⣼⠆
⠀⠀⠀⠀⠀⠨⠷⣷⣿⣻⣏⡹⣭⠇⠀⠰⢯⣉⡙⠒⠤⣌⣙⠀⠀⠈⠀⠀⠀⠐⡶⣿⡏⠀⠀⠀⢠⣀⡀⢈⣉⠉⣀⣀⣤⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠋⠁⠀⠀⠀⠀⠉⠐⠲⠭⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠙⠛⠥⠬⠥⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⢰⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣾⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠁⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣸⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⠒⠒⠒⠒⠲⠆
⠀⠀⠀⠀⠀⡀⠈⠢⠄⣀⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣯⣟⣿⠿⠿⢿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣅⠈⠲⢤⣄⣀⣀⣀⣀⡀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠐⠢⠄⣀⣀⣀⡀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠈⠑⠲⠤⣤⣄⣀⣀⣤⣤⡤⠴⡇
⠀⠀⠀⠀⠈⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠋⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⣭⣭⣭⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⢰⠀⡀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⣿⣿⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢣⣸⣀⣃⣸⣀⣇⣨⣁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠷⠿⠸⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠋⠹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⡳⣞⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠖⢿⣤⣉⣉⣤⣿⠲⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⣀⣤⣀⠀⠀⣸⢧⡈⠓⠤⣏⠀⠀⣀⣤⣀⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠐⡇⠀⠈⠉⠉⠹⣆⠉⠲⢤⡏⠉⠉⠁⠀⢸⡂⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡖⠒⠋⠀⠀⠀⠀⠀⠀⢯⣑⣂⡼⠀⠀⠀⠀⠀⠀⠙⠒⢲⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠉⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠙⢦⣄⠀⠀⠈⠧⠤⠴⠒⠒⢦⠀⠀⠀⠀⡴⠒⠒⠦⠤⠼⠁⠀⠀⣠⡴⠋⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣇⠀⠀⠈⠙⠲⠦⣤⣀⣀⠀⠀⠈⠑⠶⠖⠋⠁⠀⠀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣸⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⡟⠓⠚⠙⣆⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⣰⠛⠓⠚⢻⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢿⡀⠀⠀⠘⣆⢀⡴⠛⢦⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⡴⠛⢦⡀⣰⠃⠀⠀⢀⡿⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠉⠁⠀⠘⢧⣀⣠⡞⠁⠀⢳⣄⣀⡼⠃⠀⠈⠉⠁⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⠀⠀⠀⠀⠀⠉⠳⠦⣄⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣰⠟⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠤⠤⢤⣤⠤⠤⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢦⣀⣀⣤⣄⠀⠀⠀⠉⠉⠛⠒⠶⠦⢤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠶⠒⠛⠉⠉⠀⠀⠀⣠⣤⣀⣀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠘⣧⠀⠀⣰⠞⠛⣦⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣴⠛⠳⣆⠀⠀⡼⠃⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠞⠃⠀⠀⠸⣆⠀⠀⠀⢠⡟⠀⠀⢻⡄⠀⠀⠀⣸⠇⠀⠀⠘⠳⠚⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠋⠀⠀⠀⠀⠙⠶⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄⠀⠀⢀⣾⠻⣧⡀⠀⠀⣠⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢈⣿⠀⠀⢸⣇⣀⣽⠇⠀⠠⣿⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣉⣛⣉⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣧⠀⠀⢸⣿⠉⣿⠀⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣯⣀⣠⣼⣿⠀⣿⣤⣄⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⢿⣇⣸⣿⠉⠉⠹⣿⣤⣿⠉⠉⠉⣿⣄⣸⡟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣌⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣡⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣯⠉⠛⠛⢻⣿⠛⠻⠿⠟⢻⣿⡟⠛⠛⢉⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⠷⣦⣴⠟⠻⣷⣤⣀⣤⡾⠛⠷⣦⣴⠾⠃⠈⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⢻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⡈⠉⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⣁⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡟⠛⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠟⠛⢻⣯⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠾⠛⣷⡀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⢠⣾⠻⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⠷⠶⠾⠋⠈⠻⣶⣤⣤⣤⡾⠟⠈⠛⠷⠶⠾⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⢸⣧⣼⠀⣼⣷⠀⣿⢹⡆⣿⢹⡎⢷⣴⠏⠀⠀⣿⣽⡇⣿⢸⡏⣷⠉⣿⠋⢸⣧⣼⡇⡟⢻⡆⣸⣷⡙⣧⡾⠁
⠀⠀⠸⠇⠹⠰⠟⠛⠇⠿⠋⠁⠿⠋⠁⠀⠿⠀⠀⠀⠿⠾⠇⠿⠸⠟⠿⠀⠿⠀⠸⠇⠹⠃⠷⠾⠳⠟⠛⠧⠸⠇

⠀⠀⢀⠂⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⡰⠁⠀⠀⠰⢤⡀⠀⠸
⠀⠀⢀⠐⠄⠀⠤⡰⣄⣀⣀⡸⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠀⠲⢦⠄⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠈⠉⠠⡈⠀⠐⠂⠒⢦⣔⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣼⠆⠀⢀⠄
⠀⠀⠀⢨⠓⠠⠄⠀⠀⠀⠀⠄⠀⠁⠘⠂⠀⠀⠀⠐⠒⢲⠊⣄⣀⣀⠌⠎⠀⠀⠑⠀⠐⡘⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠠⠀⠈⠀⠀⢀⡄⠀⠀⠈⢙⠆⠀⠀⠈⠓⠼⠄⠂⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠰⣤⣀⣠⡞⠀⠀⠀⢀⠎⠀⢠⣤⢤⣤⡀⠀⠀⠀⠀⢠⡄⡠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠙⠳⢦⣤⣠⡟⠀⠀⠞⡕⠙⢹⠿⡀⠀⠀⢰
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠤⠄⠄⠀⠐⠋⠀⠀⡀⠙⠒⠒
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢦⢀⣔⣤⢾⢆⠀⠀⠀⠀⢀⡀⠀⢠⣾⣡⣀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣠⡴⢻⡿⢊⣭⢾⣧⣄⠀⠀⠸⣿⣦⣀⠉⠲⣬⡙⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠘⢧⣾⡿⠴⠿⠛⠿⢷⢶⠏⡀⠀⠀⠈⠳⠍⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢎⠀⢀⡠⠂
⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⢀⡆⠀⠀⠀⠄⠀⠉⠃⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠊⠀⠀⠀⠘⣄⠀⠈⠀⠃⠀⠀⠀⠘⣧⠀⠀⠀⣀⣀⡲⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠠⠀⠀⢜⠳⣄⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠃⠀⠀⠈⠀⠀⠐⠁⠀⠀⠳⢾⠞⠁⠀⠀⠀⠢⠀⠀⠘⠥⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢤⠀⠀⠀⠀⠔⠒⠒⠒⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⡈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠈⢂⠀⠂⠀⠀⠀⢠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⡿⢑⡆⠀⣼⠏⡸⢀⡤⠖⠒⠢⢶⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠉⠀⠈⠠⡀⠈
⠀⠀⠀⠀⠀⠀⠁⠘⠠⠇⠀⠉⠈⠐⠁⠀⠀⢀⡀⠈⢀⠀⣀⠀⠀⠀⠔⢉⠗⠀⢀⠤⡄⠀⠈⠈⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠄⠀⠃⠀⣀⠜⠉⠀⠐⠀⠀⢀⣤⣄⠀⠀⠀⣠⠞⠀⠘⠀⠀⠃⠠⢿⣿⡟
⠀⠀⠀⠀⠀⠀⡔⠂⠀⡿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⠂⠀⡴⠋⠀⡀⠀⠠⠀⠤⠀⠈
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⢊⡖⠀⠀⢠⢔⡉⠔⠁⠀⠀⠞⠀⠀⠀⠀⠉⠉⢱⡆⠀⠐⠢
⠀⠀⠀⠀⠀⡆⡈⠀⣰⠟⡆⠀⢀⠤⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠠⡀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠉⠁⢐⠎⠌⠀⠀⠀⠘⠀⠀⠉⠅⠀⠀⠀⠀⡀⠚⠀⠄⠀⠀⠘⣆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠎⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠁⠀⠀⠠⠀⠙⣆⢠⡀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⢤⢃⠂⠐⠀⠀⠀⠀⠳⣄⠀⠀⠃⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⣶⣧⣀
⠀⠀⠀⠀⠘⠋⠃⠀⠀⡴⠁⣸⠀⠀⠩⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠶⠤⣏⠬⢏⡱⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⠀⠀⠖⠒⢂⣤⣶⣤⣀⠀⠂⠐⠂⣄⣀⠀⠀⠀⠐⠈⠛⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠟⠋⣀⣤⣄⣀⠈⠙⠋⠀⠀⠴⢶⣤⣬⠙⠛⠂⣤⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠄⡉⠛⠛⠛⠁⠀⠀⠒⠀⠀⠀⠀⠉⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠐⡄⠀⠲⠅⠀⠀⠨⢄⠀⠀⠀⠡⡄⠀⠀⠀⡴⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣣⡀⠀⣇⠀⠀⠀⢨⠀⠀⠀⠀⠇⠀⠀⢀⣇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠘⠂⠀⠀⠨⠦⠀⠀⠼⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠦⡄⠀⠀⠀⠀⠀⠀⠙⠲⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⢀⡟⠀⠀⠀⠀⠘⠄⣀⣴⠇⠀⠀⠀⠀⠠⡊⠉⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢠⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣧⠀⠀⢃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢺⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡟⠀⠀⢣⣾⠟⠿⠶⣶⡀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠠⣒⣦⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠢⠀⠐⠂⠉⠀⠀⠀⠀⠀⢁⠤⠖⠛⠁⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠍⠻⠿⡁⡀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠄⠀⠐⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⢸⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠊⠀⡅
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠒⠂⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣼
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠰⡄⡐⠘⡄⠀⠸⣿
⠀⠀⠀⣇⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢸
⠀⠀⠀⢹⣶⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡤⠋
⠀⠀⠀⠈⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⣼
⠀⠀⠀⠀⡟⠔⠀⠀⠂⢦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⣰⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠐⠀⢀⣀⡀⠄⠂⠐⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢩⡟
⠀⠀⠀⠀⠸⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠤⠔⠂⠁⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡏⠛⠐
⠁⠀⠀⠀⠀⠀⠺⠗⣣⠘⢸⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠞⠉
⠀⠀⠀⠀⠰⠀⠀⠀⠈⠛⠢⢄⡁⠙⠛⠳⠶⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠈⠉⠒⠲⠤⣄⠀⠈⠈⠉⠉⠑⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⠤⠤⠤⡀⢠⠔⠒⠒⢤⡀⡠⠤⠠⢤⡀⠀⣀⣀⣀
⠀⠀⠀⠀⣀⡤⢄⡀⣰⣋⣀⡤⠽⠖⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠓⠾⠥⢄⣈⣱⡄⣀⠤⢄⡀
⠀⠀⠀⡞⠁⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⠀⠘⡆
⠀⠀⠀⣧⠒⠋⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠓⢦⡇
⠀⠀⠀⡇⠀⠀⠀⠀⣷⠊⠉⠉⠑⣄⣠⠔⠒⠦⡀⠀⣠⠤⢤⡀⢀⡤⠖⠒⢤⣀⡔⠉⠉⠉⢲⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⣀⣀⠴⣇⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⡏⠀⠀⠀⠈⡟⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡷⢄⣀⣀⠔⠁
⠀⠀⠀⠀⠐⡅⠀⠀⠈⠢⠤⠤⠖⠙⢆⡀⠀⣀⡼⢧⠀⠀⠀⣠⠿⣄⠀⠀⣠⠞⠑⠤⠤⠴⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠳⡄⠀⠀⠀⠉⠉⠁⠀⠀⠉⢒⠊⠁⠀⠀⠉⠉⠀⠀⠀⢀⡴⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡗⠒⠊⠁⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠙⠒⠒⡇
⠀⠀⠀⠀⠀⡟⠢⢄⣀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠖⠉⠀⠀⠀⠈⠑⠦⠤⠤⠴⠊⠀⠀⠀⠀⢀⣀⠤⠚⡇
⠀⠀⠀⠀⣠⡏⠒⠤⣈⡉⠓⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠋⣉⡠⠔⠊⣧⡀
⠀⠀⠀⢰⠁⡇⠀⠀⠀⠉⠉⠒⠒⠦⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠖⠒⠊⠉⠁⠀⠀⠀⡇⢹
⠀⠀⠀⠈⠢⣈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⢊⡡⠊
⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⣉⠙⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠚⢉⣉⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⣖⣒⣶⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⣖⣒⡶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⠭⢿⠉⠉⠉⢹⠯⠭⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠶⠶⠶⠿⠿⠟⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⠎⠉⢳⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢰⠲⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠁⠀⠀⠀⠀⠀⠀⠸⢠⣠⣤⠀⠀⠈⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡉⠉⢳⣎⣠⣃⣀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠃⠀⡁⠘⣉⠉⠳⡒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⠭⠒⠒⣏⠀⢀⡀⠀⡁⠀⣰⠯⠭⣍⣓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢯⠀⠀⠀⠘⠢⠤⠤⠤⠖⠋⠀⢰⣂⣐⠆⣹⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠦⠤⠤⠤⣤⠀⡰⠊⢁⣀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣟⠑⠭⢭⣓⣒⣒⣒⡒⠲⠖⢒⣒⣒⣒⣒⣾⠁⢻⡭⠭⠴⢚⣿⠉⢻⢲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⢶⣤⣤⣀⣀⣀⣀⣀⣀⣈⣬⣭⣭⣤⣤⣬⡇⡿⠴⠖⢒⣩⡟⠀⡇⡼⢠⢷
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⣄⡉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠗⠒⠋⠉⣠⡗⠺⣥⠇⣞⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣗⢾⣉⠻⢭⣟⣒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⢒⣒⣩⣭⣛⣀⣠⣼⠾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠺⣭⣭⣛⣛⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⣻⣿⠭⠭⠭⠿⠛⠛⠋⠉⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣹⡈⣇⢣⢹⠉⣏⠉⠉⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠾⠶⠶⠾⠾⠦⠶⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⣧⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣅⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠭⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠷⠶⢤⣤⣤⣀
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⣀⡀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣧⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⣠⡞⠉⠉⠻⣦⡀⢀⣴⠞⠛⠛⢶⡀⠀⠀⠐⠀⣠⡶⠛⠋⠙⠳⣦⡀⠀⠀⠠⠀⣠⣤⢤⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⠶⠟⠉⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠙⠶⣤⠴⠞⠉⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠛⠉⠀⠀⠈⠻⢦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣤⠶⠛⠛⠛⠉⠉⠉⠉⠉⡉⡉⠛⠛⠛⠓⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣀
⠀⢸⡇⠠⠄⠀⠀⠀⠀⣀⣀⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉⢷⡀
⠀⠘⢷⣀⠀⢀⣀⣴⠟⠉⠉⠉⠻⢦⣄⣀⣤⡴⠖⠛⠲⢶⣄⡀⠀⠀⣀⣠⣤⠤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠖⢀⣀⣀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠙⠷⣤⣤⣤⣤⠶⠞⠋⠉⠀⠈⠙⠳⠶⣶⠞⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣴⠞⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⢦
⣼⣿⠀⠀⠀⠙⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁⠀⢸
⣿⠙⠷⣤⣀⡀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠳⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣠⣾
⠘⠷⣤⣀⠈⠙⠛⠶⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⢛⣩⠞
⠀⠀⠀⠙⠳⠶⣤⣄⣀⡀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⢤⣤⠤⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠋⢉⣉⣡⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠚⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣤⠶⠒⠞⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⣩⡍⣭⢉⣒⣖⣯⠭⠆⡛⠂⠛⠘⠃⢼⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣨⠅⢶⠆⠠⠀⠮⠄⠰⠒⠈⠉⠀⠀⠁⠂⠰⢉⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⠇⠘⠁⠀⠀⠀⠀⠦⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⡠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢆⡀⠠⠀⠀⣀⠔⠋⠁⠀⠈⠁⠀⠐⠃⠈⢁⠤⠒⠒⠒⠀⠒⠂⠤⠀⢀⠤⠔⠒⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠀⠀⠀⠀⢄⠀⣂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⣈⡉⢠⡀⠀⠀⠀⣀⠀⣀⠁⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡦⡤⡀⢀⡀⠀⣀⢠⣤⣦⢤⠁⣤⠀⣠⡄⣠⡌⠤⢼⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣥⣩⣥⣬⣭⣤⠶⢿⣿⣾⡞⣲⡄⢴⣦⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠒⠒⠞⠛⠓⠒⢦⣀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⡠⡀⢠⣀⣦⣴⣀⣀⣟⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡻⠎⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠉⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣮⣄⠀⢀⣀⣀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣤⣶⣽⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⣽⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠚⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣤⣁⡀⠀⢀⢀⣠⣶⠟⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠓⠤⣤⣤⣤⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠄⠀⠾⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⣿⢿⠿⡿⠿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣟⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄⢸⢚⡲⠄⡄⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠙⡟⠂⢠⣀⢂⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣤⣀⣀⣀⣀⣈⣀⣀⣭⡶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠉⢉⡉⠉⠉⠉⠁⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠁⠠⠤⠀⠀⠀⠤⠄⠐⣀⠀⠀⠂⠐⠀⠀⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠈⠁⠀⣀⠀⠀⠄⠀⠐⢄⢖⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣧⣴⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣁⣀⣀⣤⣤⠶⠞⠛⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣤⣤⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⡀⠄⢂⠀⠡⠂⠄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠍⠀⠀⠈⢵⡄⠈⠐⠢⢦⣶⣄⣀⣀⢀⡀⠀⠀⣀⢀⣀⠀⣤⠴⣖⡞⢀⠀⠀⠀⠀⠘⣡⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠂⠀⠱⠀⣀⣋⠀⢠⢙⠃⢠⠀⢼⠉⠷⢶⠀⡀⠠⢐⠃⠀⠊⠙⠂⢠⠀⠐⠤⠌⠀⢀⠀⣀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡷⠧⠉⠒⠄⠁⣄⡀⠂⠈⠀⠠⠀⠀⠂⠀⠙⠀⠁⠀⠈⠀⠀⠀⠁⠀⡀⠀⠀⠂⡙⠃⠢⠐⢉⡂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣇⡀⠀⠀⢀⠐⠒⠄⠈⠁⢀⠐⡀⢠⠀⠀⠀⠐⠀⠐⠀⠀⠀⢤⠦⠀⡄⠀⠀⠀⢐⣤⣶⣶⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠛⠻⣶⣦⣤⣤⣄⡂⠀⠀⠀⠀⡉⠀⠑⠂⠀⠀⠀⢈⠀⠀⠀⣀⣠⣤⣦⣴⣶⣿⡿⠛⠋⠁⡐
⡇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠑⠀⢬⡀⠀⠍⠉⠛⠻⠿⢿⠿⠿⢶⣶⣶⣶⣶⣶⣿⣿⡿⠿⡿⠟⠛⠛⠉⠉⢁⠀⢈⠀⠄⠊⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⣀⠀⠀⠀⠀⠉⠒⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣈⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠤⠾
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣤⡤⣆⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠒⠃⠀⠀⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⢠⡖⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢧⡤⢤⣀⣠⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⢀⣀⢀⡦⣄⡞⠲⠾⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠒⠈⠉⠉⢛⠋⠉⠷⢞⢹⡶⠎⠳⢶⠉⢲⠋⡙⠛⢋⠛⠁⠉⠠⠤⠤⠤⠤⠀⣀
⡇⠀⣀⠠⠀⠀⠒⠈⠉⠉⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⣀⠄⠂⠘⢀⠀⢀⡘⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⢠⢀
⡇⠁⠀⠀⡤⠀⠀⠀⠀⠤⠀⠀⠀⠤⡦⠀⠀⠀⠀⢠⠄⠰⡀⠀⠀⠩⠄⡆⠠⠣⠈⡃⠀⠐⠀⡠⠀⠀⣀⠐⡄⠀⠶⣀⠀⣀⢀⠀⢀⡀⢀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠀⠂⠀⠤⠬⢀⠤⠄⠀⠀⠐⠁⠀⠀⢀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠐⣨⠂⠀⠘⣈⠂⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠶⠀⠘⠠⠒⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑
⠏⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⡀⠦⣤⣀⡴⠟⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠚⠀⠀⠐⣤⢀⣿⠁⠀⠨⣴⣴⣿⣻⣧⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠞⢶⡀⠀⠴⣠⣬⡵⣾⣿⡆⠀⠀⠛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢤⠀⠀⠈⢿⣸⠂⠉⢉⢉⣉⡛⢿⣦⡆⢠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠱⣟⡷⠈⣵⠶⢦⣐⡺⢴⡼⠊⣿⣿⠿⣿⣿⣾⢿⣷⠦⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⠲⣠⠞⣋⣷⣂⠀⣤⣶⡄⠈⢳⣖⣸⠿⢋⢴⣿⡀⠀⠰⢿⣏⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⠃⠀⢠⡟⠉⠉⠈⠿⠀⣼⣛⡓⣶⢨⡟⢡⣾⣧⡘⣿⣿⠷⢦⡐⢻⣅⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢡⡄⢰⣫⡶⣶⡀⠀⣻⠀⡆⢠⣤⣾⣿⣇⢒⡳⢨⣿⡜⡏⠀⣠⣾⠋⣰⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⡆⣡⣈⣧⣅⣉⠳⣤⣥⣾⡗⠩⢍⠿⣿⣿⣭⣶⢟⣼⢃⡏⠘⠿⢋⡴⢫⣾⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⣽⠣⢙⣭⣬⣻⠛⠟⢨⣿⡿⢿⣏⠀⠀⢀⣽⡿⠃⣿⡟⠉⣛⣽⣿⡿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⣀⣼⣿⣡⣰⣴⣿⣧⢙⣋⣉⢀⣀⣭⠉⣀⢀⢏⣯⣾⡿⣻⣡⣶⡿⠟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⣬⡙⠛⠮⠼⠿⡿⣉⣉⣉⢉⡉⠙⣛⣛⣾⣿⣿⣿⠿⠿⠛⢋⣁⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⣇⠈⠉⣰⠲⠶⣦⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢻⡀⠀⢡⠀⠀⠁⠀⠉⠟⡉⢿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣧⢁⢸⣇⠀⠠⢠⣷⣤⢰⢾⣿⣿⣷⣿⣿⢟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⠀⠘⣿⠆⣽⢸⣿⣶⠰⢾⣽⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠂⣿⢆⠚⣺⣿⣿⡉⣿⣿⣿⣸⣿⡟⡿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣷⢸⠻⡀⣿⣿⣿⣦⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡅⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⡗⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣪⠟⠉⠀⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣀⠒⠤⢤⣀⣀⢉⠉⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠄⢀⠒⡢⠤⡀⣀⣀⢀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⣀⡞⠉⠉⠉⠉⠀⠈⠙⠢⣐⠀⡀⠂⠀⠁⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⡴⢃⣀⠀⡤⠿⠿⠖⠂⠀⠠⣄⠀⠀⠀⢀⡬⠷⢶⡤⠤⠤⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠘⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠛⣁⠤⠤⢒⠭⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠸⡘⢄⣀⣀⠀⠀⠀⠀⠀⢀⡤⢚⠔⠁⠀⠀⣔⡁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠑⠄⡀⠈⠑⠒⠶⠂⠉⣩⠔⠁⠀⠤⠤⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠉⠀⠰⠇⠀⠀⠀⠀⠈⠑⠨⡑⠂⠤⠤⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠤⠤⠄⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⣰⢀⠀⠀⠀⡀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠈⠃⠇⠃⠃⠃⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠙⢄⣠⠴⠚⢉⡀⠙⠒⢦⣀⡠⠞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⣄⡞⢁⣤⣶⣄⠀⠙⣦⣾⣿⣿⣷⣆⡜⠁⣠⣤⡀⠈⠳⣀⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢁⡰⠃⠈⣵⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⠞⠉⢀⡴⠈⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢋⡱⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠉⢀⠔⠋⢠⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠒⢋⡔⣈⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠏⡀⢠⠎⢸⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⠤⠖⡉⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⠲⢤⣁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠈⡞⠁⠤⠚⢡⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠁⠀⡇⢲⠀⠈⣷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⡿⡀⣇⠀⣀⡤⠊⠀⢀⠜⠋⡰⠉⠻⢿⡿⠿⣿⡿⠛⢿⣿⢿⡿⣿⠟⠋⠰⢈⠙⢧⠀⠊⢀⡜⠀⠀⡼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠘⢾⡦⣄⣀⢀⣰⠃⢀⡴⠃⡟⠀⠀⢻⠞⠁⢠⠏⠀⠙⢦⡾⠁⢀⡰⠃⢸⠀⠀⣇⣀⠀⣀⡠⠞⣡⠋⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⢬⣁⠹⣄⠀⢀⠔⠁⠀⢀⣾⠐⠚⠁⠠⠃⠀⠀⣧⡀⠈⠀⠰⠃⠀⣠⠏⠉⠉⢁⡠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠠⣄⡀⠀⠀⠈⠑⠺⠷⣶⣶⠖⠒⠋⠘⠦⣀⣀⣀⣀⡤⠞⠀⠉⠒⠲⠶⠖⠚⢁⣠⠤⠒⠉⠀⢀⡠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠶⠶⠶⠶⠶⠖⠒⠒⠚⠋⠉⠁⢀⡴⠒⠒⢀⣞⢀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠈⠙⠒⠒⠒⣞⠉⠉⠓⠦⡀⢀⣀⣠⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠖⠒⠂⠀⣮⣀⣀⡠⠞⠀⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠈⠙⠲⠤⠤⠖⠻⣄⡀⠀⠉⢓⣠⠖⠉⠉⠉⠃⣀⠜⠡⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⢙⣷⣻⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⢻⣦⠀⠀⠀⠀⠀⠀⡰⠋⠠⠤⠈⠒⠋⢈⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣫⡧⠼⢋⡶⠗⡄⠀⠀⠀⠀⠀⠀⢺⣭⣵⣿⣶⠾⣄⠀⠀⠀⠀⠀⢳⡐⠲⠖⠐⠀⠐⢁⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠿⠴⢛⡛⣤⡾⢧⠀⠀⠀⠀⠀⢠⣟⣽⠿⠋⡱⢺⡿⡄⠀⠀⠀⢰⡏⡷⢭⢉⡓⢣⣄⡴⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣖⠾⣛⣛⣩⣴⠏⠀⠀⠀⠀⠀⣿⣍⣠⣤⣶⡿⢫⣼⡇⠀⠀⠀⠀⠹⣦⡶⣦⣥⠵⠿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙⠛⠛⢛⣩⡟⠀⠀⠀⠀⠀⠀⢹⡭⣽⣿⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⢺⣤⡰⠜⣬⠳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡭⠭⠭⠭⣿⠃⠀⠀⠀⠀⠀⠀⠈⢷⡉⣖⣸⣧⣼⠏⠀⠀⠀⠀⠀⠀⠀⣯⠳⣌⠶⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡲⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⡿⡩⠿⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⢪⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣪⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⡿⣶⣄⠀⠀⠀⠀⠀⠀⢀⠾⠄⠒⢘⡠⢢⡀⠀⠀⠀⠀⠀⠀⠀⣮⠉⢼⡸⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠈⠉⣿⠀⢀⣿⡃⠀⠀⠀⠀⠀⠈⣆⡀⠨⣄⠓⡞⠁⠀⠀⠀⠀⠀⠀⣼⠷⠶⣿⡴⠯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢁⣠⣾⣿⠁⠀⠀⠀⠀⠀⢸⡇⡙⠤⡄⠀⢛⠀⠀⠀⠀⠀⠀⢰⡻⠒⠚⢋⣠⡼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⠝⣿⡿⣿⢿⡇⠀⠀⠀⠀⠀⠀⢘⡝⣈⡚⠀⢋⣹⠂⠀⠀⠀⠀⠀⢸⣿⣿⣟⣉⣥⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡟⣳⣏⣿⠀⠀⠀⠀⠀⠀⠀⠺⣞⣂⠈⣵⣦⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣾⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⣿⣇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠭⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢂⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣺⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠉⠉⠉⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⡤⠾⠃⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⡿⠷⠞⠁⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠽⡒⠊⢉⠽⠁
⠀⠀⠀⠀⣼⠋⠙⠓⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠖⠒⠫⠁⠂⡀⠌⢁⣀⠀⠒⢀⠠⢀⡀
⠀⠀⠀⢠⡇⠤⡀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣤⠤⠤⠤⠼⠀⠀⠀⠀⢀⠤⣪⣴⠾⠚⠥⢰⣀⡠⠦⢁⣀⠁⠃⢨⠀⠤⢃⠀⢠⠐⠙⡄
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠈⢉⡇⠀⠀⠀⠀⠀⠀⣠⠵⣚⣿⠍⠂⣀⠤⠀⡀⣈⠙⠂⢠⠤⠔⣃⠠⢄⠀⠋⠀⠀⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣄⣀⠀⠉⠊⠙⣄⡀⠀⠆⡽⠺⠋⢠⣀⠀⠀⠀⠑⠂⣁⠠⣚⠢⠤⣀⠤⠋⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡤⠜⠀⠣⠳⣄⠀⠀⠀⠉⠉⠀⠠⠎⣀⠀⢀⠭⠒⠠⠀⠀⠀⡀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠓⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠔⠒⢦⡀⠀⠀⠀⠀⠈⠉⠛⡶⠦⢤⣀⡀⠰⡟⢋⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠙⠲⠤⠤⠤⠵⠀⠀⠀⠉⠙⠋⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⣀⢛⡦⢄⠀⣀⣀⣀⡳⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠚⠛⢇⣀⣭⠙⠤⡶⠆⠸⢉⣵⡐⠲⠗⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⠖⣂⠄⠀⢁⡀⠀⡀⢀⠀⡀⣤⠄⡤⠤⣢⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠤⢬⡕⣴⣆⡿⠁⢈⢰⠴⡇⣉⣣⣇⣰⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠶⡒⣄⡀⣫⠈⢉⢁⢸⣆⣀⣏⣉⣉⣀⣤⣭⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣁⠒⠤⠤⣌⣉⣈⣂⠉⡙⠉⠉⠈⠉⣉⣉⣠⣥⠴⠞⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣄⣉⡸⣂⠤⣠⡄⢤⣈⡩⣉⣉⣉⢉⣉⣠⢁⠄⢰⡆⠚⢠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⡌⢒⠛⠛⠲⠨⢤⣼⣤⣙⣁⣼⣀⣹⣌⣧⣭⣭⣲⠟⢟⢛⣹
⠀⠀⠀⠀⠀⠀⣀⣠⣸⠧⣄⠉⠃⠙⠓⠖⠶⢲⡤⡬⣀⣩⣭⣥⣤⣤⣽⡴⠶⠶⢛⢋⣩⣀
⠀⠠⡀⠒⠐⢅⠀⠀⠙⠲⢄⣁⠘⠙⠼⠶⠤⢤⣤⣨⣄⣄⡤⣤⣤⣤⣔⣦⢴⠶⠟⣋⣹⠏⠉⢓⠅⣀⠂⠄
⠀⠀⠀⠁⠐⠀⠠⠄⠀⠀⠀⠀⠉⠀⠂⠠⠄⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠶⠒⠒⠉⠁⠀⠀⠀⠁⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⢹⠁⠀⠀⠀⡏⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡠⢄⠀⠀⡠⠤⠀⢀⡀⢀⡀⠀⠀⢀⠀⣠⠲⡄⡠⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣠⠇⢸⠀⢰⠃⠀⠒⠋⡇⡏⠘⠀⠀⡏⠉⠁⠀⠧⠇⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠸⢤⠼⠀⠀⠀⢠⠃⢹⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠘⣤⠎⠀⠳⠴⠃⠀⠀⠀⠀⠀⠀⡏⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢀⣞⠁⢰⠉⡳⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⢞⠉⡦⠈⡳⡀
⠀⠀⠀⠀⠀⠸⡌⠂⠀⠀⠀⠀⠀⠀⠀⠉⠩⡙⠓⠒⠒⠒⠒⠒⠒⠒⢋⠍⠉⠀⠀⠀⠀⠀⠈⡠⠚⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡀⢢⠀⠀⠀⡀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢠⠇⡜⠁⢸⠀⠀⠀⠈⠋⠀⢱⠀⠀⠀⠀⢰⠙⢦⡀⠀⠀⢀⠀⠀⠀⣼⠵⠤⣄⠀⣴⡸⡀
⠀⠀⠀⠀⠀⠈⢲⠃⠀⠸⡄⢀⠆⠀⠀⠀⣼⠀⠀⢀⡀⢸⠀⠀⡇⠀⠰⠛⣆⠀⢀⡇⠀⠀⠘⠚⢹⠃⢣
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠑⠊⠀⠀⠀⠀⢹⠀⠀⣼⠐⠁⠀⠀⢧⣠⠇⠀⣿⠀⠈⡇⠀⠀⠀⠀⠸⣆⡼
⠀⠀⠀⡠⠤⡴⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠀⠀⠀⠀⢈⡗⢦⠤⣄
⢀⣠⠞⠧⠴⠃⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠘⠦⠼⠳⣄
⡏⢯⠀⠀⠀⠀⠀⠀⣙⣒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣒⣋⠀⠀⠀⠀⠀⠀⡽
⠘⢆⡓⢤⣀⠀⠀⠐⢅⡨⠂⠀⠈⠉⠉⠓⠒⢒⣶⠶⠶⠶⠶⠶⠶⣶⡒⠒⠚⠋⠉⠁⠀⠐⢅⡨⠂⠀⠀⣀⡤⢚⡱
⠀⠀⠙⠲⢬⣙⠓⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⢧⣠⠇⠀⠀⠀⠀⠸⣤⡼⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⣋⡥⠖⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⢤⣉⣉⡉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⢉⣉⣉⡤⠤⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣾⡿⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⣾⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⣠⡞⠁⣠⢾⣿⣿⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⢀⣴⠏⢠⡾⠁⢸⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⢿⣦⡀
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡁⢠⡟⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠻⣷⡄
⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠻⣷⡟⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⣿⣆
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠏⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⢿⣦
⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠈⣿
⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠋⣀⣾
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣷⡄⠀⠀⠈⣉⣁⣠⣤⣶⣾⡿⠛
⠘⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⢿⣿⡿⠿⠿⠟⠛⠋⠉
⠀⠀⠈⠙⠿⢷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠆⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠟⠋⢁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡟⢟⣙⠛⣻⢛⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⣶⣶⠂⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠃⠀⠀⠀⣰⡿⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣴⡾⠀⠐⠆⢸⡟⢀⡝⠁⠆⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠈⠀⢠⡟⢛⠓⠒⠤⢄⣼⣷⡼⠃⠀⢀⠀⠀⣴⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣮⣤⢥⠤⠄⣠⣯⣅⣀⣀⠀⠀⠀⣙⡿⠁⠉⠀⠀⢠⣾⠁⠐⡠⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠉⢀⣾⠇⠀⠀⠀⠀⠈⡙⠀⠉⠛⢻⣶⣤⡙⠁⠀⢀⣤⡾⣿⣿⣄⠈⣠⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⡿⠥⠤⠤⠴⣾⣿⡾⠤⢌⣁⣀⠀⠂⠀⠀⠀⡋⠀⡙⠁⠀⠀⣈⣛⠀⠀⣉⣿⡿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢊⣼⣟⡀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠐⣻⣿⡟⠀⠀⠀⣰⣿⣶⣤⣤⣴⣶⣿⡿⢿⣿⣿⣿⣿⡠⡜
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⣐⣿⣿⣿⡻⢟⣳⠒⣰⣶⣤⣁⢤⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⡉⣉⠻⢿⣶⣿⣃⣿⠕⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣭⡶⠻⢏⠍⠹⠉⢀⣠⣿⡿⣿⡏⡡⢸⣯⣷⠟⣿⢋⡅
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢏⣫⣟⣻⣽⣿⣟⡝⠛⡣⢜⣁⣵⡾⠋⣱⡆⠈⢉⣤⣴⣾⠿⠛⠛⠣⣦⣤⣤⣾⠕⠋⢡⣿⢁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⠾⢿⠟⢛⣉⡉⣹⣿⣯⣟⣽⡿⠿⠟⠻⠯⣤⣤⣶⣾⣟⡋⠁⠀⠴⠆⠀⢠⡿⠟⠁⣁⣨⣴⣿⡟⣾
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠷⠶⣟⡛⢋⣭⡴⠛⣋⠍⠉⢀⠔⣶⠂⠁⢸⣟⡅⠰⢫⣠⣄⣤⣤⣦⣶⠾⠿⠿⢿⣿⣋⠍⠥⠾⠃
⠀⠀⠀⠀⠀⠀⣿⢿⣿⡛⠶⠂⣋⣴⣟⣡⡾⣗⣡⣤⣤⣶⣿⡶⠶⠶⠿⠛⠛⠋⠉⠉⠉⠩⠥⠶⠂⠚⠚⠉⠁
⠀⠀⠀⠀⠀⢰⣷⣿⠿⡶⠿⠿⢛⠛⡛⠛⠛⠭⠍⠀⠡⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⢄⠒⠂⠉⣉⣉⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⢀⣀⣀⡀⠤⡇⣰⠉⠀⠀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠚⠂⠦⠤⢤⡚⠠⣏⠀⠀⠁⠀⠐⠀⠀⠈⠢⡤⠤⢀⣀⡀
⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠁⠀⡰⠁⠀⠀⠠⠀⠄⠀⠈⠙⣌⠢⡀⠀⠁⠄⠀⠠⠄⠀⢸⠀⠀⠀⠀⠉⠉⠒⠢⣀
⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢂⠄⢀⠀⠁⠀⢀⡿⠢⣈⠑⠒⠂⠠⠤⠤⠔⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠤⢔⡪⠕⠁⠀⠀⢉⠗⠒⠀⠠⡔⠚⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣔⠤⢑⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢆
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡇
⠀⠀⡌⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡆⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⠔⠊⠱
⠀⠀⢰⡀⠀⠀⠈⠉⠓⠠⠭⢍⣈⡁⠉⠉⠑⠒⠣⠬⠬⢰⣁⣀⣀⣀⣀⡀⠠⠤⠤⠔⠒⠉⠁⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡰
⠀⠀⡎⠈⠓⠤⢄⣀⣀⢀⠀⠀⠀⠀⠉⠉⠁⠚⠒⠐⠒⠒⠒⠂⠀⠤⠤⠤⠤⠤⠀⠤⠤⠤⠔⠒⠊⠉⠀⠀⠀⠀⡠⠒⠉⡇
⠀⠀⡱⣄⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠔⠊⠀⠀⣴⠁
⠀⠀⢇⠈⠑⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠒⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⢊⡎
⠀⠀⠀⣱⢤⡀⠀⠈⠑⠢⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠀⠒⠒⠈⠉⢀⡴⣋⠆
⠀⠀⡎⠸⡀⠙⢕⠢⢀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⠁⠀⠈⠆
⠀⠀⠱⢄⡙⠢⢄⠉⠒⠬⠕⣲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠒⠊⣁⠔⠁⠀⠀⠀⣤
⠀⠀⢠⠀⠈⠑⠂⠬⣑⡂⠤⢀⣈⠉⠉⠛⠃⠭⠄⠈⣀⣀⣀⣈⣀⣈⣉⣉⣉⠁⠤⠄⠐⠒⠒⢂⡠⠋⠀⠀⢀⡀⠔⠊⠀⡇
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠈⠁⠒⠢⠬⣉⡁⠒⠒⠒⠐⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⢀⠔⠁⠀⢀⡠⠞⡃
⠀⠀⠀⠈⠳⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡤⠔⣁⠤⡴⠚⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠹⣑⠒⠩⢄⣒⡒⠠⠤⠤⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠂⣁⠔⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠉⠉⠁⠒⠒⠠⠤⠭⠬⠩⣭⡍⠉⢉⣉⣉⣉⣉⣉⣉⣁⣀⠠⠤⠒⠉⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠐⠒⠂⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠒⠋⠉⠉⠉⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⠀⠉⠒⠠⠀⠀⠀⢀⣠⡤⠖⠊⠁⠀⠀⢀⣀⠼⣃⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠨⣍⣒⠤⢀⠀⠀⢰⠁⠀⠀⣀⠀⠀⢀⣩⠤⡶⡺⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⢀⠢⠈⡏⠂⢀⣀⠚⠖⣋⣡⢤⢤⠞⠁⣈⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷⣦⣀⠀⠀⡇⢸⠋⠁⠀⣤⢋⣠⠴⠿⠞⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢹⡍⠻⡷⠦⣄⣸⣤⡤⠚⠉⠁⠀⠀⠀⣀⣁⣤⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⠀⠹⣄⡿⠢⡈⠰⠧⠄⣀⢀⢀⣀⣠⣤⡤⠴⠒⢒⠖⢩⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⠀⠙⢇⠀⠙⢶⠒⠛⠉⠟⠋⠀⢸⡟⣀⡔⢀⠊⠐⣀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠴⠯⡿⠉⣹⠂⣌⠢⡘⢸⠀⠠⠀⠀⣠⢤⠮⠚⠛⠒⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢱⣟⡆⠀⡐⠀⠀⠑⢆⡜⡇⠉⠈⠉⠉⠉⠀⠠⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠁⠈⠷⠁⠀⠀⠀⠐⠀⠒⠿⠟⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢤⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣋⣭⡤⣍⣍⠐⠀⠤⠤⠤⡤⠶⢖⣛⡭⠷⠒⡛⢛⠛⠦
⠀⠀⠀⠀⠀⣾⠀⠴⢃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠈⢁⣠⢤⢦⡬⢀⠇⠀⠐⣂⣟⣻⡩⢥⠶⠖⠺⡏⠉⠙⠁
⠀⠀⠀⠀⣸⠃⠀⢰⡇⢊⡜⢡⡆⣼⠇⢹⢀⡞⣰⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢣⠤⡤⣤⢤⣤⣤⠤⠤⠼⠤⢷⠓⢒⢒⠂⢨⠉⣦⠀⡀⠄⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠘⣛⡗⣿⠶⣶⣶⡶⠶⣖⣶⣳⣛⣋⠏⠉⡁⡉⢈⠀⣙⣀⣥⡤⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⠙⣿⡂⠀⠀⠀⠀⠉⠛⢿⣯⠓⠓⠓⠚⠤⠧⣍⣀⣉⣉⡉⣩⣀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣐⡒⠒⠂⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠛⠋⠉⠀⠀⠀⠀⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⣦⣄⡉⡑⠒⠢⣄⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠫⠴⠶⢦⣍⡳⣤⣀⠀⠀⠀⠀⣀⣀⣠⡴⢟⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣌⡀⠀⠀⠀⠀⠀⢀⡀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣯⣍⣓⣛⣛⣋⣩⣅⡶⠋
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠑⠢⢄⠀⠴⠃⠀⠀⠈⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠐⠲⢼⡄
⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠉⠙⠲⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠶⢶⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⡾⢉⣔⡉⠀⠂⠀⠄⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠂⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠄⠼⠒⡾⠀⠀⡄⠀⠀⡠⢁⢸⠽
⠀⠀⠀⠀⠀⠀⠁⠒⠢⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠂⠐⠀⠀⠘⠀⠀⠰⠣⣀⡏⠰⡀⢀⡴⠚⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠐⠀⠀⠀⠀⠀⠀⠀⡀⠃⢘⡞⢠⠋⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⣣⣄⠈⠀⡀⠀⠀⢠⠀⢐⣠⠴⠚⠉⣠⡇⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⠋⢀⡜⠁⠀⣠⠋⠀⠀⢂⡤⡨⡿⢁⡴⠂⠁⠉⢠⡀⠀⠀⠀⢀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⡃⠀⣜⠀⠀⣼⠃⡀⠀⢀⡼⢠⣻⢁⡾⠀⠀⠀⠀⠀⢶⡀⠀⢀⡜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣽⠀⠀⡇⠀⣰⠇⡀⢈⡤⢎⣱⢿⠇⡼⠁⠀⠀⠀⠀⠀⢸⣧⡴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⠀⢧⡀⡝⠠⠴⣩⡶⠟⠋⢰⠀⣇⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣢⡄⠣⢈⡴⠋⠁⠀⠀⠀⠀⠂⠻⣤⣤⣤⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉


⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠴⣖⠻⠭⠷⢦⡀⠀⠀⠀⢀⡴⢻⡷⠾⠛⠯⠗⣦⡀⠀⠀⠀⠀⠰⣦⣿⡄⠤⢠⣄
⠀⠀⠀⠀⠀⠀⢠⣇⣀⣡⠖⡀⠴⢃⣷⡀⠀⢀⡎⢘⡼⣶⠶⠋⠂⣾⣵⣧⡀⠀⠀⡀⣤⢙⡯⠶⠤⣍⢸⢷
⠀⠀⠀⠀⠀⠀⢺⣀⠤⣤⠀⣀⣈⣽⣿⠇⠀⠈⢧⣩⡷⠶⣀⡚⠀⣻⣾⣟⠃⠀⠀⣄⢤⣵⡲⡶⢀⡿⢧⣼
⠀⠀⠀⠀⠀⠀⢘⣖⡉⠉⠉⢉⣹⣿⣿⠁⠀⠀⠘⣯⠩⠼⠿⠶⠶⣛⣷⡆⠀⠀⠀⠨⣟⣿⣟⠓⣒⣒⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢷⠤⣭⣭⣭⣥⣶⡟⠀⠀⠀⠀⢯⠭⠝⠛⠛⠫⠭⢿⠇⠀⠀⠀⠀⢿⣿⠻⠝⣉⠳⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣼⣅⣸⣇⣿⡃⠀⠀⠀⠀⠘⠇⣿⢰⣷⢸⡿⣼⠀⠀⠀⠀⠀⠈⣟⣷⡜⠼⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣏⣹⣽⣿⠀⠀⠀⠀⠀⠀⣲⣶⢰⣶⢰⣶⡏⠀⠀⠀⠀⠀⠀⢩⠋⢷⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠽⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠡⣤⣠⣥⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⢹⠨⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⣏⡇⣼⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠿⡽⡄
⠀⠀⠀⠀⠀⠀⠀⣟⣫⣭⡾⢟⡼⢳⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⢀⣻⡹⡄
⠀⠀⠀⠀⠀⠀⢰⣯⣭⣭⡶⢟⣵⣿⡀⠀⠀⠀⢀⣠⠮⣤⢾⡫⠞⢶⣄⠀⠀⠀⠀⠀⡼⠥⠐⠀⣀⢷⢳
⠀⠀⠀⠀⠀⠈⡷⣤⣤⣴⣶⣿⣯⣾⠃⠀⠀⢰⠋⡜⢉⡷⠴⠧⠬⣌⣯⣱⠀⠀⠀⣰⢣⠀⠰⠀⠉⣈⣧⣇
⠀⠀⠀⠀⠀⠀⣿⠶⣒⠾⠻⢟⡻⣿⠀⠀⠀⢸⡜⢨⠯⣆⠠⢈⣴⣿⣿⣟⠀⠀⢠⡿⣦⣤⣄⣘⠀⣨⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣦⣦⣴⣢⣸⣤⡇⠀⠀⠀⠈⢿⡯⠓⣦⡵⢿⣾⣿⣿⠃⠀⠀⠀⠻⠶⢬⣭⣝⣻⠯⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢣⠓⡩⠛⢭⡏⠀⠀⠀⠀⠀⢑⣰⣿⢎⣸⡷⡻⣁⡛⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⡕⡍⣀⣿⠁⠀⠀⠀⠀⠀⠀⠙⠆⠾⣿⢸⠿⠜⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡎⠀⠇⠀⢀⠔⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡌⠙⠒⢼⠠⠐⠛⠠⡟⢀⣔⡿⠒⠒⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠡⡀⠀⣠⠏⠸⠁⢈⡒⢢⡀⠘⣄⠀⠁⠀⠀⠀⠀⠘⢏⠉⠁⢈⡸⠁⠑⣏⠈⣧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣣⣱⡯⠣⢴⡆⠙⡤⠸⠊⢙⣶⣿⣷⣶⣄⠀⠈⠀⠀⠸⠶⣈⣁⠴⡄⢴⠁⠈⢦⢠⡯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢤⣾⠑⢦⢸⠀⠋⢁⡔⠛⣾⡿⣺⣘⠽⢼⡽⣦⣄⢤⡧⢊⡥⢖⠛⠃⠀⠱⠀⠣⡀⣁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⡇⠀⠀⠀⠀⠀⡰⢻⡿⠒⣤⢞⣿⢿⢷⣷⣲⣏⢏⢱⣳⡿⠼⠒⠛⠲⠧⢄⣠⠁⠉⡤⠐⢇⠈⡦⢜⠉⠓⢤
⠀⠀⠀⠀⣠⠊⠁⠐⠀⠱⡀⠀⠀⠀⢠⡀⠀⠀⣼⣯⠮⣗⣏⠿⣻⣟⣿⣻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠓⡄⠸⠣⣀⡸⠈⠃⠀⠱⡀
⠀⠀⠀⡸⠁⠀⠀⠀⣀⡀⠈⠑⠒⠒⠟⠳⢄⣀⣿⣟⣿⣿⢎⣸⣛⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠛⠶⠚⠁⠀⠀⠀⣀⠀⠇
⠀⠀⠀⡇⠀⠀⠃⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⢿⣻⣮⣾⣿⠿⠃⠸⣏⡒⠄⢒⣒⣒⠦⣄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠪⠤⠷⠀⠀⠀⡄
⠀⠀⠀⡼⣄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡈⠉⠉⠀⠈⠀⠉⠢⣕⠢⣄⠀⠀⢠⡇⠀⠀⠤⠄⠀⠀⢀⠔⡇
⠀⠀⠀⡇⠈⠳⡌⠀⠄⠀⠠⠤⠀⠀⢀⣀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠲⠭⠿⠟⠁⠀⠀⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⡇⠀⠀⠈⢦⡀⣀⢀⡀⠀⠠⠌⢉⡀⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠶⠶⠄⠀⠀⠀⡰⠆⠠⠀⠀⠈⠁⣀⠤⠒⠒⠊⠁⣰⢻
⠀⠀⠀⠘⣄⣀⡀⠀⠀⠀⠀⠈⠒⠒⠒⠦⡀⣁⣉⣀⣁⢀⣀⠈⠀⠉⠀⠁⠒⠂⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣴⡿⢦⡀⣰⠿⠛⠛⠉⠨⡇
⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠀⠀⠄⠊⠁⠈⠁⠀⠉⠀⠀⢀⡾⠁⠠⠀⡇⢻⡀⠄⢀⠔⠁⣧⣀
⠀⠀⣠⢴⠈⠣⣉⡇⠀⣀⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⢒⢶⣄⣀⣀⣀⠀⠀⣾⠀⠀⠀⠀⡇⢸⣢⠐⠀⠀⠀⡧⠈⠓⣄
⢀⡔⠁⣸⠀⠀⠈⡇⠠⡟⢘⠥⡂⢙⢿⣷⣄⠀⠀⠀⠀⠀⣠⠖⠏⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡧⠀⢿⠀⠀⣀⠠⡷⡏⠀⠀⠀⢀⣠⠃⠀⠀⠈⢦
⡜⠀⠀⠈⣦⣄⠀⠑⠚⠀⠉⠀⠛⠐⠀⠨⢿⡆⠀⠀⠀⢸⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠃⣿⢠⠞⠛⠉⣻⠀⠀⠀⠀⠈
⡇⠀⠀⠀⡯⠘⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡎⠀⠀⠀⢀⣇⣸⠎⠀⠀⠀⣹
⢳⠀⠀⠀⠓⢦⡀⠙⢿⣦⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⡇⢀⣠⠞⠉⠉⠁⠀⠀⢀⡴⠉⠀⠀⠀⠀⡸
⠀⠳⡀⠀⠀⠑⠉⠢⣀⠉⠛⠿⠷⣦⡤⠒⠶⣾⠀⠀⢷⣤⣤⣤⣤⡀⠀⠀⣀⣤⣤⡤⠞⠋⠉⠓⣏⠀⡸⠁⠁⠀⠀⠀⠀⢀⠴⡪⠊⠀⠀⠀⢀⡔⠁
⠀⠀⠈⠢⣄⠀⠀⠀⠀⠉⡒⠤⣀⠀⠀⠀⠀⢸⡀⢀⡎⠀⠉⠉⠙⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⡀⠄⡪⠁⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠁⠀⠉⠔⠒⠠⢬⡅⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠆⠁⠀⠁⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⣀⡀⠀⠀⢸⠁⠀⡧⠋⠁⠈⠩⠟⠭⠬⠬⠛⠋⠀⠈⠙⠀⠀⠁⠀⠀⠀⢀⣀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⡏⠀⠀⢸⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠜⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠉⡀⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢈⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⡀⠀⠀⢀⠀⠀⠀⢀⠔⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⢁⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⡈⠛⠋⠲⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢩⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠐⠀⢨⠀⠀⠀⠘⢧⣁⣤⣔⠂⠀⠀⠀⢴⡀⠀⢰⡆⠀⠐⠀⠀⠙⠳⠿⠿⠶
⠀⠀⠀⠀⠀⣠⠒⡁⠒⢄⠀⠀⠈⠓⠤⣁⡀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣴⣠⣴⣾⠁⠀⠈⢧⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠳⠤⠤⠄⠀⠀⠀⢀⣴⣄⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡍⠁⠀⠈⣽⡆⠀⠀⠀⠀⡠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⡄⠀⡀⠀⠀⣴⠚⠉⠁⠀⢀⡄⣶⡼⠃⢀⡀⣠⠎⠀⠀⠀⠀⣻⠀⠀⠀⠀⣠⡾⣏⠀⣐⣿⠑⡇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠡⡀⠀⠀⢸⠇⠀⢀⣤⠒⠋⠛⠻⠒⠃⠀⠀⠀⠐⢡⣴⣖⣯⠏⠀⠀⣠⡞⠁⠀⣿⣿⣿⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢁⠀⠀⢸⣾⣴⣿⡧⣂⠀⠀⢀⠀⡄⠄⠀⡐⢞⣣⠿⠟⠃⠀⠀⠀⣿⠀⠀⡠⠏⣫⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠀⠀⠀⠉⠉⠉⠹⣿⣤⣫⣾⠾⠿⠶⠷⠿⠟⠃⠀⠀⠀⠀⠀⠀⡙⠛⢋⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡔⠚⡉⢊⡳⡄⠀⠀⠀⠐⢉⣛⡩⠉⣔⡀⠀⠀⠀⠀⢀⣼⢏⣦⡀⠀⠀⠀⠀⠀⡀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⡅⢩⡀⣀⣸⣷⠀⠀⠀⢆⣷⢍⡐⢀⣿⡟⠀⠀⠀⠀⣼⣗⣫⢞⡗⠀⠀⠀⠠⠋⢺⠭⠮⢜⣗⠄
⠀⠀⠀⠀⠀⠀⠀⣇⡀⠉⠽⣿⡇⠀⠀⠀⠈⠎⠉⣉⠯⢿⡇⠀⠀⠀⠸⡿⣽⣶⣿⡏⠀⠀⠀⠸⣤⡝⢄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣭⣏⣿⣽⠁⠀⠀⠀⠀⠣⠶⠶⠰⢮⠀⠀⠀⠀⠀⢧⣈⣀⣠⠇⠀⠀⠀⠀⣁⣶⣹⣟⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣧⣿⡿⠀⠀⠀⠀⠀⠸⡗⢾⣺⡏⠀⠀⠀⠀⠀⠐⠀⢀⡏⠀⠀⠀⠀⠀⠈⠘⠛⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⠴⠒⢀⢤⡀⠀⠀⠀⠀⢀⣰⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢴⡼⢟⡷⡀⠀⠀⠀⠀⢠⣼⣿⣻⠀⠀⠀⠀⠀⣏⠀⠀⠀⠄⢳⠄⠀⠀⠀⠤⠹⠁⢨⡙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⡥⢞⡇⠀⠀⠀⠀⣼⡿⠟⣁⣷⠀⠀⠀⠰⡵⢢⠙⢢⣜⡍⠀⠀⠀⠐⠾⣮⠇⠘⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡛⠛⢚⣿⠀⠀⠀⠀⠈⣷⣾⣿⣯⣾⠀⠀⠀⠀⠸⣏⠉⠉⠝⠀⠀⠀⠀⠀⢿⡙⠖⡉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣉⣹⠇⠀⠀⠀⠀⠀⠰⣥⣭⣴⠇⠀⠀⠀⠀⠀⠉⢃⠰⠂⠀⠀⠀⠀⠀⠈⡽⢆⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡍⠀⠀⠀⠀⠀⠀⠀⢁⣡⡟⠀⠀⠀⠀⠀⠀⠀⢓⡃⠀⠀⠀⠀⠀⠀⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣆⡀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣾⡷⡄⠀⠀⠀⠀⠀⠀⢀⠾⣧
⠀⠀⠀⠀⠀⠀⠀⡾⠉⢻⠉⣹⠀⠀⠀⠀⠀⢠⡀⠠⢀⡁⠀⠀⠀⠀⠀⡴⢤⣏⢵⡄⠀⠀⠀⠀⠀⠌⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⣾⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⠀⠀⠀⢰⣷⣈⡥⢞⡇⠀⠀⠀⠀⡈⠀⠄⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣳⣿⠇⠀⠀⠀⠀⠀⠲⡤⠱⡮⠀⠀⠀⠀⠀⠈⢿⣷⣾⡏⠀⠀⠀⠀⠰⡳⠶⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣳⡟⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⠓⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⢻⠍⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣤⣤⠀⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢿⣟⣻⣦⡿⢿⢛⣛⣖⣛⣟⣛⣷⢥⠿⠶⢒⡦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⡴⣿⠉⡹⠶⠚⢛⣧⣈⠴⠺⠟⠻⣷⣌⡙⡳⣠⣤⣶⣚⢿⣤⣿⣉⡶⢄
⠀⠀⠀⠀⣰⣿⢿⢿⣿⢁⣀⣠⢴⡿⡟⢁⣠⣴⣛⠻⠻⢿⠛⠋⠉⠉⠛⠦⠍⠁⠉⠻⣧⣤⣼⣶⡄
⠀⠀⠀⣾⣼⣿⣿⣶⠛⠋⢩⣿⣆⢠⣀⠿⠋⠉⠙⢧⡀⠈⢣⣶⣶⣾⠶⠾⣿⠛⠀⠊⠉⢹⣿⢋⢠
⠀⠀⢸⣧⡀⣿⣿⣅⣠⡤⠴⣶⣿⠋⠀⠀⣀⡴⢿⣿⠷⠛⠁⠀⠈⠿⡅⠀⠈⢱⣶⣶⡖⣺⣿⠳⡾⡆
⠀⠀⠈⢻⣿⣩⠛⠻⠿⢷⣤⣈⣙⣾⡤⠾⣭⡉⠀⠀⠒⢀⣤⣴⣶⠚⠻⢦⣀⡈⣡⠴⠾⡟⡉⡰⣻⠁
⠀⠀⠀⠀⠛⠗⢶⣤⡴⢈⠈⠛⢳⠤⠠⣤⣈⣉⣴⣦⣀⡴⣊⣉⣹⠧⣀⠀⠛⠉⣢⢲⣲⡯⢋⠔⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠷⣶⣾⣄⡀⣾⢿⣅⠀⠈⢨⠁⠀⠁⣤⡶⢆⡀⠤⠜⣋⣈⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣿⣿⡿⣙⣟⢚⣛⣛⣒⣒⣳⣤⣭⣭⣥⣤⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠟⠀⣀⣀⣀⡀⠀⠀⢀⡈⠳⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⢀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⡈⠉⠋⠀⠀⠠⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠭⠶⠟⣋⠁⠀⡀⠀⠋⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠂⠀⠡⠀⣀⠀⠂⢀⠀⢬⠇⣀⣀⠋⠀⠀⠘⣷⣶⣴⣶⣶⣶⣶⣿⣧⣤⣭⣄⣠⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠡⠀⠠⠄⢀⡘⢦⠀⡴⢠⠾⢋⣴⣶⣾⣿⣿⠃⠀⠀⠀⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡰⡈⠁⠂⠀⠐⠓⠶⡦⠄⠉⢀⠀⡄⢀⣴⡿⠿⠿⠟⠻⠿⠀⡠⢄⢀⡀⡀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠟
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠓⠀⣀⣤⣭⠍⢀⡀⠈⠉⣁⢀⠠⣤⡐⠲⠶⠶⠤⠀⠀⠈⠁⠀⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠛⠉⠁⡴⠟⣡⣇⠀⣿⠀⢷⣌⠉⠁⠠⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠐⠤⠀⠀⠀⡿⠋⠀⢠⣠⠀⠉⠃⠀⠈⠀⠈⢀⡤⠞⠁⠃
⠀⠀⠀⠀⠀⠀⠀⢯⠷⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠠⠖⠒⠉⢈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠁⠈⠉⠒⠛⠒⠓⠒⠋⠀⠁⠀⠉⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⡀⠀⠤⠤⠀⠐⠂⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡀⠀⠀⢠⣓⡄⠀⠀⢀⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⠇⠀⠀⣿⣩⡇⠀⠀⢿⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣭⡇⠀⠀⣟⢩⡇⠀⠀⣟⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⡇⠀⠀⣟⢥⡇⠀⠀⣟⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇⠀⠀⣿⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣴⡇⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠆⠒⠛⠛⠓⠀⠉⠉⠉⠉⠉⠁⠉⠛⠓⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡖⠉⢢⢀⡴⡄⡎⢸⠀⢸⠁⢸⠀⡇⢡⠀⡎⠀⢱⢠⠒⠀⢴⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠁⠀⢸⠈⡇⠓⠂⢸⠀⠸⠀⢼⠀⢃⠼⠀⡇⠀⣆⢈⠀⠓⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠁⠀⠀⢨⠀⣸⠃⠈⠒⠂⡇⠀⢸⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠤⠤⠆⠒⠒⠒⠚⠛⠉⠀⠉⠉⠈⠁⠒⠚⠒⠂⠤⠤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠒⢤⠀⢀⣀⢀⡊⢦⠀⡰⠋⢱⡀⡎⠱⠀⢠⠊⠑⡄⢀⢤⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⠇⠀⢰⠀⢂⠸⣠⠆⠘⠀⢃⠀⢘⠃⢣⢀⠀⢣⠀⠀⡇⢸⠐⠤⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⣄⡸⠀⠀⠀⡇⠀⠸⠀⠹⣀⡸⢸⠀⠈⡄⠀⠣⠞⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⡄⠀⠀⠀⠀⠀⠀⠀⠳⠤⠚⠀⠀⠀⠀⣈⠀⠀⡇⠀⠀⠀⠀⢠⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⢠⠒⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⡜⠒⡄⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠈⠲⠤⠍⣐⣒⠒⠐⠲⠆⠀⠀⠤⠤⠤⠄⠀⠀⠐⠶⠒⠒⣒⣒⡩⠤⠖⠁⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠬⠭⠍⣉⣉⣁⣀⣀⣀⣀⣈⣉⣉⠉⠭⠥⠤⠤⠐⠒⠂⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠒⠄⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢄⠠⠊⠀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠁⠠⠐⣂⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⡀⢀⢈⠉⣀⡠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠃⠉⠀⠀⢀⢠⢄⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠠⡐⢀⠁⢀⠥⠼⡁⢄⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⡗⡌⢄⡤⢾⢉⠉⠉⠑⡓⠀⠀⠘⡦⢀⣲⠃⠀⠀⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠄⠀⠀⠈⣉⣁⣈⣂⣀⣀⣒⡀⠀⠀⠀⠉⠠⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠠⠤⡀⠀⠀⠀⠠⡀⢀⠀⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠒⠤⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠐⠈⠁⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠐⠒⠂⠀⠤⠤⠀⠀⠀⠤⠤⠀⠒⠒⠒⠂⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⡄⠀⠀⠀⠀⢠⠃⢆⢣⠀⠀⠀⢀⠇⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡆⢱⠀⠀⠀⠀⢯⢸⢸⢸⠀⠀⠀⢸⢠⠱⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⣄⡜⠀⠀⠀⠀⠈⣻⣿⢿⡄⠀⠀⠘⣼⣾⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⣯⢲⡀⠀⠀⠀⡞⠛⢩⣼⠃⠀⠀⡴⠻⠛⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⢸⡇⠀⠀⢸⣴⣿⡗⣿⠀⠀⠀⣷⣰⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢚⣿⡇⣀⡀⢈⣿⠦⣌⣏⠀⣀⠀⢸⣿⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⠶⣤⣶⣾⣿⣻⡏⠭⠛⣿⣿⣭⣿⡯⣑⠢⣿⣿⣭⣿⣿⠿⣋⣿⣷⣶⣶⡴⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⢾⣿⣷⣿⣛⣛⠋⠉⠉⠉⣸⡧⢐⣫⣧⠀⠀⢰⡗⠠⢍⣿⠀⠀⠀⣿⣈⠤⣿⠉⠙⠛⠛⠛⣷⣶⣽⣟⣛⣳⣆
⠀⠀⠀⠀⠀⢾⣷⢻⣽⣶⣿⣭⣭⣿⣶⡞⠲⣮⣩⣭⣭⣥⣤⡶⢮⣭⣛⣋⣭⣴⡶⢦⣭⡿⡿⢭⣴⡞⣛⣛⠺⣿⣷⣴⡟⣿⢻⡿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣯⣚⣿⡸⣿⣿⣿⣿⣾⣽⣶⢶⣾⣿⣿⣶⣮⣻⢯⣯⣭⣿⣶⣿⣯⣿⣷⣭⣿⡿⣿⠿⢿⣿⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡯⣽⣽⠿⣿⣿⣿⣮⣿⣯⣿⣾⣿⣯⣽⣿⣽⣿⣿⣟⣟⣫⣾⣯⣿⣷⣿⣷⣷⣿⡿⣿⣻⣿⣟⡾⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣏⡷⡏⣵⠟⣿⣿⣿⣞⣭⡭⢿⣟⣯⣿⣿⣟⣯⣭⢯⣟⣿⣯⣷⣿⣿⣷⡿⠞⢾⣿⢿⣿⢫⡷⣿⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⣸⠇⠀⢹⡟⣿⠉⣿⡝⣹⠟⠉⣿⣯⡿⠁⠹⣟⣿⣿⠿⣿⡿⢡⡟⠀⠀⠀⢻⣏⢃⡾⠀⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠀⠀⠘⣷⣻⡿⠃⣴⠏⠀⠀⠉⠉⠁⠀⠀⢿⣭⣿⣠⣿⠇⣼⠃⠀⠀⠀⠀⠙⠛⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣘⣻⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣹⠟⢁⣴⡟⠀⣀⣀⣀⣠⣤⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣭⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⣫⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠩⠻⣯⡛⠿⠗⢒⣺⣾⠶⢾⣖⠾⣿⣿⠿⢟⣫⡶⠞⠛⠻⢮⣟⣟⣩⡴⠞⠉⠉⣇⡀
⠀⠀⠀⠀⠀⣀⣀⣠⠶⣻⣄⣀⡀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠉⠓⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣠⡶⠿⣟⣷⣄⣀⡀
⢀⣠⠴⠞⠋⠉⠉⢿⡟⣽⢿⣯⣟⢛⣶⡶⢦⣤⣤⣄⣤⣤⣀⣤⣤⣤⣀⣀⡀⢀⣤⣤⣤⣤⣤⣤⣶⣶⣿⣻⣭⣽⣿⣿⣿⡿⠋⠉⠉⠛⠶⢤⣀
⣾⣧⣄⡀⠀⠀⠀⠈⠻⠾⢿⣽⣿⡿⡷⣿⣿⣷⣷⣿⢿⣿⣫⣴⣶⣮⣿⣿⣛⣿⣭⣴⣦⣽⣷⡾⠿⣶⣾⢿⣻⣿⣵⠟⠋⠀⠀⠀⠀⠀⠀⣀⣽
⠈⠛⠿⣿⣷⣶⣦⣤⣄⣀⣀⣀⠈⠛⠿⠿⠻⠷⡿⠿⣿⣯⣿⢾⣿⣿⠿⣯⣭⣽⡽⠿⣿⡿⠟⠛⠛⠛⠛⠛⠉⠀⢀⣀⣀⣀⣤⣤⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⣉⣩⣭⣉⣤⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣉⣉⣭⣍⣥⣴
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣈⣩⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣈⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⢀⣠⣆⣬⣿⣿⣿⣦⠀⠀⠀⢀⢀⠀⠤⣤⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣓⣶⣤⣤⣖⣲⡄⠀⠈⠮⢻⠿⠟⠋⠉⠁⠀⠀⠀⠘⣜⢷⣾⣿⣿⣿⡄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠
⠅⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⢷⣾⡿⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⡠⠀⠀⠐⠒⠒⠢⢤⠀⠙⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠘⣦⣄⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸
⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⢀⣢⣾⠋⠀⠀⠀⠀⡐⢶⡄⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢠⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠣⡀⠀⠀⠀⠀⠇⠀⠀⡌
⢠⠀⠀⠀⣈⠀⠀⠠⠀⠀⢀⢸⡏⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⢰⣻⣭⣭⣑⠒⠶⠤⠴⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⣄⠀⠀⢰⠀⠀⠀⠇
⠸⣌⠀⢀⠜⠀⢠⠁⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⡇⢿⣦⣀⠈⠳⢤⣿⣿⡿⢷⣶⣶⣿⣿⠀⠀⠀⠀⠄⣸⣷⣦⠀⠀⠀⠀⠀⣸⢿⡇⠀⠘⡄⠀⢸⠀⠀⠸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠙⢦⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⢪⣿⡀⠀⢀⣴⣿⣿⡿⠏⠀⠀⠀⠀⣴⡿⣸⠁⠀⠀⢱
⠀⠀⡇⢁⠀⠀⡇⠀⠀⢀⣼⣿⣿⣷⣵⣄⠀⠀⠀⠀⠀⠈⠙⢮⠻⢿⣾⣿⣶⣄⣀⣀⣼⣿⣿⣾⣿⠟⠉⠁⠀⠀⠀⢀⡠⣺⡟⣡⡟⠀⠀⠀⢸⠀⠀⠀⢠⡇
⠀⠀⡇⢸⠀⠀⡇⠀⠀⢸⣿⣿⡟⢿⣿⠻⣕⡤⣀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⣭⡽⠟⠛⠳⣽⡇⠀⠀⠀⠀⣠⠔⣫⠾⢋⣴⣿⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⡇⠀⡆⠀⢳⠀⠀⠘⣦⣿⣿⡀⠹⣗⢄⡙⠮⣝⡒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣤⠤⠒⣋⡴⠞⣡⢔⣵⣿⠇⠀⠀⠀⠀⡞⠀⠃⠀⢸
⠀⠀⡇⠀⢡⠀⠈⢧⠀⠀⠈⠳⢽⣾⣄⠈⠳⢭⣒⠤⣉⠛⠶⢤⣤⣈⣉⣙⠒⠒⠒⢒⣈⣩⣭⣭⣿⣷⠖⣋⡥⢖⣩⢶⢟⣽⠋⠀⠀⠀⠀⣼⠁⢸⠀⠀⡏
⠀⠀⠀⠀⢸⡀⠀⠈⠳⡄⠀⠀⠀⠹⣿⣷⣖⢤⣉⠛⠲⢯⣕⡒⠦⠤⢍⣉⣉⣉⣉⣉⣉⣉⠿⡿⣻⣿⣯⡴⢚⡫⢞⣡⠞⠁⠀⠀⠀⢀⡾⠁⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠢⡀⠀⠀⠈⠪⠿⠷⢮⣙⠢⢤⣀⡉⠛⠓⠲⠶⠶⠒⠒⠒⠒⠛⢻⣿⣷⣿⡯⠖⣋⣴⡿⠃⠀⠀⠀⢀⣴⠏⠀⠀⢠⠇⠀⢰⡇
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣍⣑⣒⣒⣒⣒⣉⣉⣩⣭⣭⣵⣹⣷⣟⡿⠋⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⡸⠀⠀⢸
⠀⠀⢸⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠑⠈⢓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠗⠚⠛⠉⠁⣀⣠⢴⡺⠝⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⡀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠭⢒⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⢖⡒⠍⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣠⣄⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢤⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⢤⠜⠒⠖⠂⠁⠛⠋⠉⠙⠣⠈⠉⠹⠿⠯⠽⣯⣳⣖⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠛⠁⢀⣤⡶⢾⡄⠀⠙⠲⣶⣦⣤⡦⠒⠤⣄⡀⠀⠙⠻⠛⠃⢒⣶⣤⡀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⡿⠿⢵⣤⡞⠉⠀⠀⠀⠀⠀⠀⢼⣡⣴⠛⠀⠀⢀⠼⠿⠦⠄⠀⠀⣠⠛⠛⠛⠓⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣹⠀⠀⠀⠀⠈⠳⡶⠶⠶⠶⣄⡀⠀⢈⠿⠁⠀⠀⣰⣅⡀⠀⠀⠀⠀⠀⢷⣶⣶⠲⣦⡀⡀⣤⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡟⠓⠒⣦⣤⣤⠌⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠛⠛⡻⠃⠀⢩⣶⣄⠰⠟⠁⢀⣴⡿⢿⣿⣿⡃⣼⡀
⠀⠀⠀⠀⠀⠀⣆⠙⣯⡿⠋⠀⢀⡞⠉⠉⠉⠀⠠⣤⣾⣷⣦⣤⠶⠶⠶⣶⣄⣀⣀⡗⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠻⢶⣿⣿⡿⣡⡿
⠀⠀⠀⠀⠀⠀⠘⣆⡉⠓⣤⠰⣏⣀⡀⢀⣀⠀⠀⠈⠛⢙⠟⠁⠀⠀⠀⣽⠿⠻⠛⠁⠀⠈⠸⢧⣤⣴⣤⣤⣀⣀⣶⣿⠿⢃⣤⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡦⡌⠢⠼⣿⣤⡈⢹⡿⣦⣤⠜⠀⠀⠀⠀⣠⣞⣡⣤⣶⣤⣤⣀⠀⢀⣀⢈⢳⣤⡿⠛⠉⠉⣥⡾⣛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣕⠶⠤⣌⡉⠚⠓⠶⠿⠶⠂⠀⠤⣶⣿⣋⡿⠿⠭⢽⣭⠿⠧⠀⠛⠚⠚⣋⣠⠔⢒⣯⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⢄⣈⠉⠐⠒⠀⠒⠶⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⠠⠄⢀⣐⣉⣅⣠⣶⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠐⡇⢹⡟⠻⠗⢶⡶⣶⢤⡤⣤⣤⣤⣶⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⢿⠀⠂⠀⠀⠸⠀⠃⠘⡏⢹⡏⢻⡿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣻⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢀⠀⡀⠀⠃⢸⣯⣼⡇⢸⣿⢸⣿⢺⣿⣿⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠁⠀⠐⠈⠑⠀⠛⠀⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠻⡿⡿⡿⡿⢹⠿⣿⡏⣟⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣧⣷⣧⣷⣾⣾⣿⣷⣿⣿⣶⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠋⠉⠉⠁⠀⠀⠈⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠶⢤⣄⣀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⣠⠞⠁
⠀⠀⠘⠦⠤⣄⡀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠉⢀⠀⠂⠈⠁⠂⢄⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⡠⠁⠀⠀⠀⠀⠀⠀⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡏⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⢀⠌⢀⡞⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡠⠞⠁
⠀⠀⠀⠀⣰⠃⠀⠰⡀⠀⠀⠀⢀⡠⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣰⠃⠀⢀⣀⣈⠂⠁⠈⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⠃⠠⠊⠀⠀⠀⠈⠑⢄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⣄⠀⠹⣄
⠀⠀⢻⡀⠠⠀⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⣇⠀⠈⠳⢤⣀⠀⠀⠀⣀⡴
⠀⠀⠀⠱⣄⠑⢄⠀⣀⠤⢊⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠓⠦⣅⣀⡴⠋⠀⠀⠀⣠⠖⠛⠛⠛⠲⠦⢤⣤⣀⣀⣀⣀⣠⣤⠤⢤⣤⣄⣀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠸⡄⠀⠀⠀⣀⡤⠖⠉⢀⡤⠖⠋⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠑⠚⠉⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠤⣤⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣧⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡏⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣿⣿⣿⠃⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢸⣆⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣀⣀⣾⣿⣿⣅⣴⣿⣿⠏⠀⠀⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⣀⣽⣿⡟⠁⣴⣿⠀⠈⠙⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢁⣠⣶⣿⠀⣀⠉⠻⣿⣿⣿⣿⣇⡈⠙⠛⠻⠛⣿⣿⣿⣿⣿⣿⠛⠋⢁⣴⣿⣿⠀⠸⢿⣶⣄⡈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠘⠋⢁⣀⡀⠸⣷⣦⣀⠙⠻⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣿⠿⠛⠛⠁⣠⣦⣄⣀⠈⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠉⠻⠿⣦⡈⠙⢿⣿⣶⣿⣿⣿⣿⣿⣿⣉⣉⣀⣸⣿⡿⠋⠉⣁⣠⣾⡗⠀⣿⠿⠛⠉⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠉⠻⢿⣶⣤⣄⣉⠀⠀⠈⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⠷⠶⠿⠟⠛⠋⠀⣀⣠⣤⣶⡿⠛⠉⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⣀⡉⠙⠛⠿⠿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⠿⠿⠛⠋⢉⣀⣤⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣈⣉⣉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣀⣠⣤⣤⣶⣶⣿⣿⡿⠉⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢉⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣉⣁⣀⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⣁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠿⣿⡿⠟⠉⠉⠉⠉⠙⢛⣿⠿⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡾⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣄⣀⢹⣧⠀⠀⢠⣶⠿⠿⢷⣦⡀⠀⠀⣰⡿⠛⠉⠉⣀⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⣧⡈⠉⠛⢻⣿⠀⠀⣿⣧⣤⣤⣤⣽⣷⣤⣴⣿⣷⠶⠾⠟⠛⠉⣁⣼⠟⠿⣶⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⢸⣟⠛⠷⣶⣬⣿⣄⣸⣟⠀⠀⠉⠉⠉⠉⠈⠁⣀⣀⣀⣤⣤⣶⠾⠛⣻⡆⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣶⣤⣀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⣀⣤⣶⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠉⠛⠛⠿⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠛⠉⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡴⠟⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣼⢹⣇⠠⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢫⢢⠳⡈⠂⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠁⠀⠘⢀⠇⡇⠠⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⡠⠔⢁⠜⠁⢠⠄⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠉⠠⠔⠊⢁⠠⠂⡡⠎⠀⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠒⠚⠈⢁⠤⠊⠀⠀⣠⠇⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣀⣀⣨⠤⠔⠊⠁⠀⠀⣈⠀⠁⢠⠇⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⡀⠀⠠⣀⣀⡠⠴⠒⠉⠀⣀⠔⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢤⣁⡀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⢈⡀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠈⠙⠛⠋⠭⠭⠥⠤⠤⠒⠒⠊⠁⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠒⠄⠄⠤⠤⠤⠤⠤⠠⠴⠒⠊⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡑⠒⠀⠤⠠⠤⠬⠤⠄⠐⠊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠤⠤⠤⠤⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠀⠶⠊


⣏⠉⣹⣛⣛⣟⣛⣻⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣛⣻⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠒⢲⣞⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢦⣤⠞⠉⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⢀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠁⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⢠⣶⡾⠟⢋⠩⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⣤⣴⣶⠿⠟⡋⣃⡈⠐⢂⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠄⢀⣀⣤⣶⡿⠷⠜⠋⠡⠀⠈⠀⠀⠘⢷⡄⠈⢂⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣾⠿⢛⠉⠁⠒⠙⣶⡄⠀⠀⠀⠀⠀⠀⠈⢻⣄⠈⢆⣀⣠⣤⡶⠟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠁⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣀⣠⣼⣿⣦⠸⢛⣩⢥⡲⣯⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⢀⡶⠿⣛⠫⢭⣤⠶⣾⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⣤⣶⡾⠿⣛⣭⣿⡶⠿⣞⣗⣥⠬⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡐⢨⣿⣷⣾⣿⣏⡦⠽⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠾⠟⠛⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠑⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠄⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣀⡤⠶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⣀⣿⣤⣤⣄⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠄⠀⠈⠷⣤⣄⠀⠈⠉⠁⠀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢓⠐⠒⠘⢖⠀⠀⢙⡟⣠⠶⢲⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠧⠀⠀⠤⠳⣰⠟⣰⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠰⡀⠀⠀⠑⣄⣠⡞⠃⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠀⠐⡄⠀⠀⠀⠙⠶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡈⢃⠀⠙⠂⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢣⡈⠳⠐⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠙⠁⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⡄⠈⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⢦⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠋

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⠀⠀⣠⣴⠟⠛⠋⠉⠉⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡉⢩⣍⠉⠙⢻⣾⠛⠓⠀⠘⠷⠀⠀⠀⣀⣄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠉⠉⢿⣄⠙⢻⣶⣾⡇⢀⣄⡀⠀⢀⣀⡀⠀⠉⠁⠀⣀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠺⣿⣧⣀⠛⠛⠻⢷⣶⣿⣶⣿⡇⠸⣧⠀⠹⢷⣄⠈⠉⠁⠀⠀⠶⠟⠛⢀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⣰⡏⠈⠙⠻⣶⡾⠛⠉⠀⠀⢿⣅⠀⢠⡄⠀⢀⡉⠀⠀⣴⣦⣤⡄⠀⠾⠛⣿⠃⢹⣆
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣼⣯⣀⠀⠻⠶⠀⠈⠹⣿⠈⠻⣦⠈⠛⠷⣦⣤⣀⣠⣤⣤⠀⣼⠏⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣿⠿⣦⣸⡏⠈⠙⠷⠆⢀⡀⠀⠀⠙⠳⢶⣤⠀⠠⣦⣤⣌⠉⠉⠀⢠⡾⣿⣆⣴⠿⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠈⠙⠷⢶⣤⣄⣀⠈⠛⠛⠿⠛⠃⠀⠙⣷⠶⣦⡈⠻⣷⣀⣀⣤⡶⠿⠛⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠈⠉⠙⠛⠛⠿⠶⠶⠶⠶⠾⠿⠶⠿⠿⠛⠛⠋⠉⠁⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠙⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⠛⠷⢶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⡶⠾⠛⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣬⣍⣉⣉⣉⣉⣩⣥⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢠⡐⠉⠙⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠀⠀⢺⠀⠀⠀⠘⠀⠐⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠘⠒⠊⡁⠀⠈⢹⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣭⣍⣛⣛⡻⠿⠷⠶⠶⠶⠒⠒⠀⠀⠐⠉⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⣈⡉⠉⠛⠛⠿⠿⠷⠒⠚⠛⠃⠀⠀⠀⢴⠿⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣮⣉⡙⠛⠛⠿⠷⠶⠆⠀⢒⠀⠀⠀⠀⠐⠈⠀⡴⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢤⣉⠛⠛⠻⠷⠶⠶⢶⣤⣤⡤⠀⠀⠀⠒⠀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⣤⣤⣤⣤⡤⠤⠄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⣠⠴⡒⠈⢛⡋⢐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⠀⠀⠘⠁⠶⠶⠤⢤⡉⠁⠢⠈⣢⠍⠢⡀⠀⠀⠀⠠⠤⠄⢄⣀
⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠑⠦⣀⠒⠈⠲⣄⠠⠘⠀⠀⠁⠀⡠⠐⠈⢉⢓⡲⣤⣍⡢⡀
⠀⠀⠀⡼⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠸⡄⢤⢦⡀⠳⢤⡈⠀⠑⠂⠀⠀⠀⠀⣷⠒⠋⠉⠉⠉⠉⠉⠙⠛⠦⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠋⠉⠀⠀⠀⠀⠈⢄⠣⢜⠃⢤⠚⠒⢤⣀⠀⠀⠀⠂⢀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠒⠊⠉⠀⠀⢀⡀⠤⠤⠤⣄⡀⠀⠀⠀⠈⠣⡀⠳⢤⡉⠁⣠⠐⠀⠀⠀⣦⢌⠓⠶⠤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠊⠉⠁⠀⠀⠁⠀⠐⠙⠆⠀⠀⠀⠀⠈⠢⣄⣉⠒⠦⠤⠌⣍⠢⣉⠒⢭⠒⢤⠬⠉⠳⣄⣀⣀⣀⡀⠤⠖⡫⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠶⠛⠋⠉⠓⢤⡈⠐⠤⠀⣀⡀⠀⠠⠀⠀⢀⠀⢒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠙⠁⠀⣩⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⠤⠒⠋⣁⣤⠖⠒⠒⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠄⡠⠤⠴⢋⢁⡠⠤⠐⠊⠉⠙⠁⠀⠆⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠖⠉⢁⣀⣀⠠⠀⠀⠀⠂⣀⣬⠵⠖⢒⠛⠿⠅⡈⠳⠤⠾⠀⠀⢳⡄⠀⠀⠀⢀⡔⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢠⠋⠔⠚⢋⣁⡀⠤⠀⣤⠀⠀⢀⣀⡀⠠⣄⠀⣀⠆⠀⠥⠤⠀⢀⠀⠀⡜⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡏⠠⠖⢰⣊⣁⠩⠤⠴⠚⠊⣉⠃⠁⠐⠚⠁⠤⠠⠄⠀⠠⡐⠂⠣⢀⠞⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⢻⡉⠋⠈⣈⣀⣠⠴⠔⠊⠉⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠐⠀⣣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠑⢄⠀⣀⣀⣀⠤⢀⠈⠀⣀⡤⠚⠁⠀⠁⠒⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠼⠤⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠁⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢰⠀⠀⠀⠀⠀⠀⡀⠉⣆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢃⡀⠠⢀⠀⠀⠀⠀⠈⢀⠈⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠆⠈⠙⠲⣄⡀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣰⠏⠀⠐⠀⠀⠀⠀⠀⠙⢢⡀⠎⠠⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠃⢈⡿⠷⠀⣀⠀⠀⠀⠀⠀⠀⢀⡜⢉⡛⢦⡀⠂⢄⠀⡤⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢿⡀⠀⠙⢦⡙⠢⡀⠀⠀⣠⠎⠠⠊⠀⠀⠈⠓⢤⣑⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⠀⡙⢦⡀⠀⠉⠒⢄⣀⡴⠃⠀⠀⣀⠀⠀⠀⠀⠄⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠈⠋⢻⠀⡉⠢⣄⠀⠀⠻⠶⢄⠀⠀⠉⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠔⠉⢂⡀⠈⠉⢹⣠⠞⠳⣤⡁⠀⠀⠑⠂⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⠄⠠⠀⡵⢢⡀⠈⠡⣴⡾⢧⡈⠓⠦⣄⡀⠘⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⠞⣡⠏⡉⣄⡀⠈⠀⡸⠀⢀⣄⣀⣉⡙⠁⡟⠛
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠛⠆⢘⠸⡟⠙⠋⠈⢁⡜⠱⠦⣷
⠀⠀⠀⠀⠀⠀⣬⡁⠠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⡁⢅⠆⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣔⡒⢈⠶⠶⠖⠒⣻⠿⣄⡀⠈⠉⠻⣟⠒⠲⣶⡴⠶⣖⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣥⣄⡀⠀⠉⠛⢦⡀⠀⡴⠃⠀⠀⣹⢦⡀⠀⠈⠳⣞⠉⠀⠀⢈⡷⣄⠈⠑⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠛⣲⠞⢉⡴⠁⠀⢉⡷⢦⡀⠀⠙⢾⡁⠀⠀⣴⠃⠀⠙⢦⡀⠀⠘⣦⠀⣠⠏⠀⠈⢷⡀⠘⣏⠳⢿⡛⠓⣆⣀
⠀⠀⠀⠀⠀⠀⢰⣋⣀⣉⣳⠊⠁⢀⡞⠀⠀⢠⠏⠀⠀⠙⣦⠀⠀⠙⣦⣸⠃⠀⠀⠀⡼⢳⡀⠀⠈⢷⠃⠀⠀⠀⠈⣣⡤⠼⠶⢤⡝⠋⠁⠈⣷
⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠈⣳⠴⠾⠒⠶⠤⣯⡠⠴⠶⠶⢾⡇⣀⣤⠬⣧⣄⡀⣀⣼⣁⣀⠧⠴⠶⠚⠷⡴⠒⠒⠛⢧⣖⠤⠄⠀⣿⢀⣀⣤⠟
⠀⠀⠀⠀⠀⠸⣄⡀⠒⢲⡟⠁⠀⠀⠀⠀⢀⡼⢃⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⢹⡆⠀⠀⠙⣷⠒⠀⠀⢠⡇⠀⠀⢀⣠⣇⣤⡴⠚⠁⣹⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠻⡍⠳⣤⣀⣠⣤⣶⣋⣉⣀⣀⣀⣤⣞⣛⣂⣀⣀⣤⢴⣟⣀⣀⣠⡴⠯⠤⠴⠖⠋⠻⢉⡟⠉⡏⠀⠘⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠀⢳⡀⠁⠉⠉⠉⠇⠃⠈⠉⠉⠉⠀⠈⠀⢸⠉⠉⠘⠃⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠄⠤⠀⢀⠊⡀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠂⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠁⠀⠀⠐⠂⠤⠀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⢁⠘⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠄⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⢀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⡀⠔⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⡄⠀⠓⠀⠀⠀⠀⠠⠀⠈⠁⠀⠈⠁⠀⠀⠀⡐⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠀⡠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠶⠤⠍⢉⡞⢓⠞⠐⠠⠤⣀⡎⢀⠎⠁⠀⠀⠀⢉⠞⢣⠴⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠼⣤⣃⠀⠀⢀⠎⢠⠋⠁⠐⠒⢢⠮⣠⡉⠑⠐⠢⠄⣠⠁⡠⠃⠀⠈⠑⠢⣈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢉⠟⠲⠧⠄⣈⢁⡛⠲⠧⢄⣀⠀⠀⠆⢠⠃⠀⠉⠁⠒⢠⢇⣸⠓⠢⢄⡀⠀⢀⢄⣑⣤⡽⠐⠄
⠀⠀⠀⠀⠀⠀⠠⠊⠡⣀⣀⣀⡼⣄⠃⠀⠀⠀⡸⠉⡟⠢⠤⣀⡉⡝⢢⣮⡀⠀⠀⠀⠀⣇⣠⠟⠯⠦⡤⠬⠖⠁⢀⡀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⣀⣠⣦⠈⠢⠤⠤⠤⠓⠉⠳⠤⠤⠤⠜⠛⠉⢀⠈⠉⠙⠉⠉⠁⠀⢀⣦⣄⡀⠀⢀⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣤⣀⣀⣠⣴⣷⣄⠀⠀⠀⠀⣀⣴⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣷⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⠛⠉⠉⠉⠈⣻⣟⣻⣿⣿⠿⠛⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣟⠾⡅⠓⣶⡿⡯⠽⣏⣹⣟⣥⣬⣶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠲⢭⣽⣼⣉⣉⣉⣩⣽⣿⣶⠿⠛⣩⢜⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣅⣺⣾⡇⠤⠤⠤⠽⠾⠿⣾⣻⣽⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠷⣌⠹⣿⣿⠭⠭⠭⠿⠿⠟⠋⣋⣊⡭⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠤⣈⢹⢿⡷⠬⣭⣭⣙⣭⣬⣭⣷⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⢮⣍⣿⣧⣤⣤⣬⣵⣶⡿⠷⣿⣿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠤⢤⡟⡟⠉⠩⠉⠉⢭⣒⡫⢵⣾⣿⣧⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠐⠁⠀⢀⠴⢾⣿⣔⠝⠒⡿⣯⣭⣭⣉⣭⣥⣴⡾⣛⣭⣿⢿⡋⠒⢄⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣰⢼⡄⠙⠿⣿⣾⡧⠤⠤⠬⢭⣯⣭⣵⣾⣿⣿⣿⣗⡰⣎
⠀⠀⠀⠀⢀⠀⠀⠀⠲⣷⣿⣿⣶⣤⡾⣹⣏⣉⣛⣛⣛⣭⠽⢶⣻⣽⣿⣿⣿⡶⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⢻⢿⣟⠨⠿⢿⣿⣶⣶⢶⣾⡿⢿⢿⡿⠟⠋⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠉⠓⠛⠛⠛⠛⠛⠒⠒⠚⠉⠉⠁⠀⠀⠀⣀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠿⠷⢶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠶⣶⣶⣦⣶⡿⠟⠫⣭⣉⣭⡙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢁⣠⢰⡟⠋⠉⠉⠀⢀⣀⡈⠋⠁⠀⣠⣾⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠿⢀⣾⣠⣴⣶⣤⣾⠟⢛⣥⠀⠰⡿⠋⢁⣤⡬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⢀⣰⣿⠁⣿⣿⡿⠋⠁⠀⠀⠁⠀⠰⠿⣿⢾⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠰⠿⢻⣷⣤⣀⣁⣀⢀⡀⣴⠿⠂⠀⢾⣿⣟⣢⣏⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⣿⠙⠿⠿⠋⢈⣁⣡⡄⠀⠀⣀⣭⣿⣿⣿⣵⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠿⢿⣷⣄⣤⣾⣿⣟⣉⣁⣀⠀⣿⣿⢿⡷⢛⣚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠶⠦⠤⢤⣬⣭⣭⣭⣭⣽⣿⣷⡝⠛⣛⣥⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣈⣉⠉⠉⠛⠛⠛⠉⠉⢉⣉⣁⣤⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠉⢉⣉⡛⠛⠛⠛⠛⠛⠋⢉⣉⠁⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠀⣿⣿⣷⠀⣾⣿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠿⠿⠟⠘⢿⣿⡿⠀⢿⣿⡿⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣤⣤⡄⠀⣠⣤⡀⠀⣠⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠛⠛⠃⠈⠻⠿⠋⢀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣴⣶⣦⣀⣤⣤⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠁⡀⡀⠐⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠉⠀⠀⠀⠀⡠⠈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⡠⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠁⠤⠤⠠⠄⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠈⢀⠀⠀⠆⠀⡰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣈⡂⠤⣀⣈⠀⠀⠀⡀⠈⠀⠀⠀⡊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠓⠀⠀⠄⠄⢀⢀⣱⠀⢠⠴⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠰⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡀⠀⠈⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠐⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⢸⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠺⡷⠿⠟⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⡇⡀⠀⠀⠀⠲⣦⣄
⠀⠀⠀⠀⣿⡆⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡞⠀⠀⡄⠈⣿⣯⣦⠀⠀⠀⢾⣶⣤⣴⡿⠋⠈⠻⢦⣤⣤⣤⡿⠟
⠀⠀⠀⠀⢿⡟⠀⠃⡘⠱⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⣾⠀⠀⠀⢿⠀⢸⡿⡌⡆⠀⠀⠀⠉⠉⠀⠀⢠⣴⣦⣄⠀⠀⠀⠀⠒⠂⢶⡄
⠀⠀⠀⠀⠘⣿⣄⡀⠠⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⢸⠀⢸⡇⣇⠁⠀⠀⠀⢰⣁⡀⢀⣠⣿⡿⠃⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⢠⠠⢊⡄⠀⠀⡰⠁⠀⠀⠀⠀⠠⡇⢽⠀⠀⠀⠀⠈⠀⢸⣧⣿⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠸⣧⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠖⡀⠀⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠈⡏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠈⠱⠎⡀⢡⠆⠀⠀⠀⢄⢠⡆⠀⠀⠀⠀⡟⢰⡟⣿⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣐⣶⠧⠐⢲⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣈⢳⡄⠈⠀⠀⠀⢸⡌⣿⣦⠀⠀⢸⣷⣾⣿⡟⠀⠀⠀⠀⠀⢀⡂⠈⣽⠟⢻⣿⡿⠦⢴⣾⣏⢀⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣈⡒⠀⠀⠸⣧⣹⣿⣧⡄⣸⣿⣿⡿⠁⠀⠀⣠⣤⡀⠈⣙⣿⠇⠴⣾⣿⣦⣶⣦⣿⣿⠄⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⢿⣿⣿⣧⣿⣿⠟⠀⠀⣤⣀⡀⣼⣿⣿⣍⣥⣴⣾⣟⣿⠩⢿⣿⣯⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣄⠓⠒⠒⠒⢢⡐⠶⠒⠒⢲⡌⠉⠛⠋⠁⠀⠀⠀⠀⠈⠙⠻⢿⣶⣻⣶⣶⣽⣿⡿⠿⠿⠿⠛⠛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠬⣿⠠⣤⠤⠤⢼⡅⢠⡤⡤⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠠⣀⠀⠀⠀⠀⠁⠀⡀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⠀⠀⠀⠀⢺⡷⠀⠀⠂⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠀⠀⠀⠀⠀⠠⠂⠀⠐⠁⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣏⣤⣄⠠⣄⣸⣇⣤⣀⡀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡⣿⠈⠉⠁⠠⣽⡆⠉⠉⠴⣿⣇⠀⠀⠀⠀⠀⠐⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⣀⡈⢿⣆⣤⣀⣄⣿⣧⣤⣤⡄⢬⠞⣦⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡎⠀⠘⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⢐⣛⣛⣛⣐⣰⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠇⠠⣄⡽⠁⠀⠀⣠⣄⡀⠀⢈⡛⠛⠛⠛⠓⠈⠹⠗⠂⣀⣄⣠⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣞⣉⣿⣿⣿⣿⣶⣶⣶⣿⣿⡇⠠⣄⡀⣍⡁⠛⠷⠦⡹⣿⣛⠛⠛⠶⠶⢶⣤⣤⡤⣼⡿⣿⣧⡀⢀⣤⣴⣶
⠀⠀⢀⣿⣦⡀⣄⢀⣀⣿⣦⣾⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠮⡛⠶⢿⣛⠲⣤⣀⣀⠉⠁⠐⠒⠐⢛⡋⠁⠀⠀⠀⣠⣴⣿⣿⣿⣿⠛
⠀⢀⠊⠁⠊⠛⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠢⡀⠙⠳⣴⠛⠛⠛⠛⠶⣶⣤⣄⡀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣷⣆
⠀⠠⡼⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠑⠀⡈⠻⠦⣠⣴⣶⣶⣿⣽⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⣠⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⣤⠞⢉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣠⠴⠞⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡴⢞⣫⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⢀⣤⣶⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣫⡥⠶⠶⣦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⣭⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣀⡴⣛⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣰⡟⠁
⠀⠀⠀⣠⢞⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡟⠁
⠀⣠⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣴⠞⣡⠟
⣼⢫⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣁⡴⠋⣡⡾⠃
⣿⠈⠷⣤⣤⣤⣤⠶⠾⠛⠉⣀⡴⠞⠉⣠⡾⠋
⠘⢷⣤⣄⣀⣠⣤⣤⠶⠞⠛⢉⣀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠃⣴⢃⡀
⠀⠀⠀⠀⠀⠁⡀⠘⣇⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠵⣄⠃⣪⢁
⠀⠀⠀⠀⠀⠀⣰⣦⡈⠃⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⢩⣛⣷⡄⣳⢂⠘⠀⠀⠀⢀⣠⡤⠀⠶⢤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⢀⣀⢰⡄⠀⠈⠉⠻⠆⠛⠙⠳⠶⢷⣾⢶⡷⠠⠄⢈⠁⢁⠐⢚⣻⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⡀⣹⠗⣠⣤⡐⠛⠙⠟⠰⠛⠓⣦⠀⠤⢬⡀⠐⠒⠶⠉⡤⠶⡶⠉⣴⢿⣄⡷⠀⠀⠀⠀⡀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⢻⡏⣠⡥⠌⠒⠛⢻⡖⠚⠛⣷⡦⢠⣶⣶⣶⠰⠾⠛⠿⣂⣠⣶⢟⣀⢿⣵⣿⣶⣈⠨⢤⡍⣤⣄⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⣵⡶⢦⣍⠙⠻⠄⢢⣄⠻⠋⠭⠀⠲⠎⢽⣶⣿⣿⣿⣏⣵⣿⣫⣤⣿⣿⣃⣬⣤⣀⢤⣀⣩⣤⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡛⠶⠦⠴⢿⠷⣿⣿⠀⣤⠶⠶⠈⣙⠛⢰⡝⢿⣿⣿⣟⣯⣶⣿⣿⣯⣉⣉⣬⣭⠰⢾⡍⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠋⠉⠉⠉⠉⠛⠶⣶⣶⣿⣿⡏⢰⣷⣾⣷⣿⣽⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠿⠧⣿⣷⣮⡯⠟⠋
⠀⠀⠀⠀⠰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣨⡭⠽⠿⠛⠛⠫⠤⣀
⠀⠀⠀⠈⢧⣝⠢⢄⣀⣀⠀⠀⠀⢀⣀⣄⣀⣤⣶⣿⢿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠈⣷⣤
⠀⠀⠀⠀⠘⡼⣿⣧⣭⣭⣷⣶⣾⣿⣿⣟⣛⣿⣿⣧⣿⣿⣭⣿⣿⡿⠟⠋⠉⠉⠉⠉⠙⠛⢿⣦⣄⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢮⣿⡿⢿⣽⠯⣭⠔⠛⣛⡥⠛⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠃⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⣀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⡴⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠛⠓⢦⡄
⠀⣀⢈⡆⣻⡷⣹⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣝⡎⣿⡆⠀⠀⠀⠀⠀⠀⢠⡞⠃⠠⠴⠂⠓⠋⠁⡈⣗
⠈⢿⢻⣽⣬⠾⣫⣶⠗⣤⠀⠀⠀⠀⠀⠀⠀⢺⣫⣵⣾⣷⣴⢧⡄⠀⠀⠀⠀⠀⠘⣇⠲⠦⠤⠲⠂⠐⠋⣁⡷
⠀⣾⣽⣏⡶⠿⠟⣡⣾⢏⠀⠀⠀⠀⠀⠀⢀⣟⣛⣿⡿⠛⡵⣼⣧⡀⠀⠀⠀⠀⣴⢛⣧⣛⡛⠲⠶⣀⣈⠟⣿
⢠⢷⣦⣤⣶⠾⠾⠛⣹⡜⠀⠀⠀⠀⠀⠀⣞⡝⢫⣁⣠⣴⣷⢟⣸⡇⠀⠀⠀⠀⠙⣮⣃⣮⡉⠏⣺⣿⣯⢶⠋
⠈⣿⠿⠶⢶⡶⠾⠿⣿⠁⠀⠀⠀⠀⠀⠀⣟⡻⠿⣿⣿⣭⢥⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⠁⢟⢹⠁⢎⣼⠋
⠀⢸⡶⠤⢤⠤⠴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠸⣫⠻⢛⡻⣿⢻⣽⡏⠀⠀⠀⠀⠀⠀⠀⢹⡿⣮⠩⡎⡆⢸
⠀⠀⢻⣉⠙⠛⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢷⣿⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣹⠳⡉⣼⠁
⠀⠀⠘⣭⠭⠽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢱⣚⣻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡙⣷⠞
⠀⠀⠀⢸⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣭⠇
⠀⠀⠀⠀⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⢤⡄⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣧⣀
⠀⣸⣿⣿⣿⣿⣾⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠤⣀⠆⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⢻⢆⡿⡇
⣼⣿⠉⠿⢻⡟⠈⠙⣿⡆⠀⠀⠀⠀⠀⠀⠸⡍⠭⡰⠄⢻⣥⡹⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣉⣼⡏⣼⢻⡄
⣽⠁⠀⠀⠸⠇⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⢠⡗⢶⠌⠹⠦⢼⣆⠀⠀⠀⠀⠀⠀⠀⣰⣯⣈⣨⡿⠟⠉⣽⣷
⠙⢦⣀⣀⣤⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⢇⠢⣐⡪⠅⣬⣎⠀⠀⠀⠀⠀⠀⠀⣿⣭⣏⣨⣤⡶⠟⣩⡟
⠈⣷⡖⢲⣿⠛⣿⣻⠏⠀⠀⠀⠀⠀⠀⠀⣨⣆⠭⢌⣈⠉⣤⡞⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠶⢶⣶⣞⡿
⠀⠙⣿⣿⢙⣿⣩⡿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣭⣀⡸⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⢟⣿⣿⣷
⠀⠀⢻⣸⣿⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠫⠤⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣫⣷⣻⣿⡇
⠀⠀⠠⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣳⡙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣿⣯
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⠁

⠀⠀⠀⠀⠀⠀⢀⣔⠒⢢⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠀⠀⠀⠀⠀⠀⣀⠤⣠⣀⢀⠀⠀⠀⠀⣠⣴⠀⢤⣄⣀
⠀⠀⣘⣷⡄⢰⠉⣠⠒⠚⠢⠀⠔⠒⠢⡀⣴⣿⣯⣼⡁⠀⠀⠀⠀⢰⣿⣾⡛⠟⢷⡀⠀⠀⢰⡇⠸⣆⠀⡽⠿⢷
⠀⣾⣛⢧⣿⠜⣲⠧⠀⢀⣐⠆⠆⠀⠤⡇⢿⣯⣿⣿⢁⣤⠀⠀⢰⡸⣿⣽⣇⣉⣬⣷⠀⠀⢺⣿⣦⡈⠉⣳⣾⣿
⠀⢸⣿⡿⡟⠀⠙⣷⣾⣴⠋⠀⢴⣶⡶⠁⠘⣿⣳⡟⢛⣉⣀⣀⣸⠀⠻⣿⣯⣿⣼⠏⠀⠀⢸⣿⣿⣻⣷⣿⣿⡟
⠀⠘⣿⣿⠃⠀⠀⢿⣻⡍⠀⠀⠘⡝⠁⠀⠀⢿⣿⠇⠀⢿⣭⣼⠃⠀⠀⣠⣿⢹⣀⠀⠀⠀⠀⣇⠀⠈⡉⠈⣿
⠀⠀⢹⡇⠀⠀⠀⠈⣭⠃⢀⠀⠀⠑⠀⠀⠀⢸⣻⠀⠀⢸⣗⣿⠀⠀⠀⠉⠁⠈⢉⣴⠖⠂⠀⠻⣄⡀⣀⣠⡿
⠀⠀⢈⠁⠀⠀⣀⣀⡙⠀⣿⢯⣿⠀⡞⢻⡆⢀⣭⣤⣀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣨⣄⠀⠈⠉
⠀⠀⠈⠀⡄⢠⣿⣿⣽⡄⣿⣾⡟⠀⡃⢸⡇⣾⡏⢹⣿⣇⠀⠀⠀⠀⠀⠀⢀⡀⠸⣷⢦⣤⠿⠿⠿⠿⠿⠿⣶⡄
⠀⠀⠀⠀⢡⢸⣿⣿⡇⡁⣿⣽⣿⠀⡆⢸⡇⣿⡇⢈⣿⣿⠀⠀⠀⠀⠀⠀⢙⣇⣤⣶⣭⣁⡄
⠰⡟⠀⢬⣿⣾⣿⣿⡇⡅⣿⣿⣿⠀⡇⢸⡇⠻⣧⣠⣿⠏⢀⣀⣠⢤⣄⠀⢈⢻⣿⣿⣯⣿⠧⠄⣶⠀⠀⡀⡴⡄
⠀⠉⠉⠋⠉⠘⠻⠿⠓⠀⠙⠛⡏⠀⠳⠾⠃⠀⠀⠈⠀⣼⣿⣥⣿⣿⣿⠢⣀⣼⢀⣀⣘⣿⠀⣦⢸⣦⣶⣶⣶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⠇⠀⠈⣻⠿⠿⣿⠁⠀⢸⣯⣻⣸⣟⢻⡬⡀
⠀⠊⠁⠉⢳⠀⢠⠖⠒⢤⡀⡠⠒⠒⢦⢀⡴⠒⠒⢆⢹⣶⣾⣿⣬⣿⡆⠀⠀⠉⡀⣠⠃⠀⠀⠘⣿⣷⠿⣿⢿⣿⠇
⠀⣷⠀⣀⣼⠆⣗⠀⠀⢰⠇⣷⠀⢀⣸⠘⣆⠀⣠⣾⠰⣟⠀⠀⢹⣯⠁⠀⠀⢞⡓⣻⡄⠀⠀⠀⠈⣿⢿⢻⣿⡇
⠀⠈⠉⠉⠉⠀⠈⠁⠉⠉⠀⠈⠉⠉⠀⠀⠈⠈⠉⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⢯⣭⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⢉⣇⣴⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⢀⣀⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⣉⣠⣤⣤⠶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠉⠉⠻⣿⠉⢠⡟⠁
⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋
⠹⢴⣤⣤⣤⣤⣶⣶⣞⣻⠭⠭⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠂⠘⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠑⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⢂⡕⠉⠉⠑⡤⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠀⢀⠤⠂⠁⠀⠀⠀⡠⠁⠸⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡠⠐⠈⠀⠀⢀⡀⠄⠐⠈⠠⠂⢀⠥⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⢁⠀⠠⠶⠠⠬⠄⠒⠀⠈⢉⠰⠂⠁⢀⢆⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⢀⠤⠂⠁⠀⠀⢀⠤⠐⠈⠀⢀⠠⠐⠁⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⠊⢁⠀⠠⠐⠈⢁⠀⠠⠐⠂⠁⠀⠀⢀⠴⠄⠒⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⡀⠄⠂⠁⠀⠀⠀⣀⡀⠤⠀⠀⠁⠀⠀⡠⠊⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠉⠀⢠⣊⠀⠀⢀⠀⣀⢀⡄⠀⠁⠀⠀⠀⠄⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠠⠄⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠠⡤⠀⢄⣄⠀⣀⡀⠀⠄⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠢⣀⠈⠙⠢⢀⡀⠀⠠⠒⠂⢤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠓⠲⣀⠉⠉⠑⡠⠤⠊⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⢤⠄⠑⠒⠄⢀⣀⠄⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⣈⠰⠤⢄⠈⠈⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⠈⠑⠠⣀⡀⠠⠈⠉⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠑⠌⡉⠀⠉⡀⠖⠮⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠚⠛⠃⠀⠒⠂⠉
⣿⣿⣻⣻⣛⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣿⣿⣛⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠲⠋⡡⠞⠋⣩⡘⠂⣀⠰⠖⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⢀⠈⠁⠒⢐⠖⠂⠀⡠⠤⠀⣀⠈⠁⢁⠔⠒⠐⠫⠘⠛⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠐⠛⠋⠀⢰⡧⠤⢄⡀⠀⢠⣊⡀⠀⠈⠛⠓⠶⠂⠀⡀⠒⠂⠴⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠑⠤⠐⣤⡄⠀⢀⠈⠁⠉⠙⠁⠀⠛⠓⢠⠞⠉⡰⠒⠀⠀⠂⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠄⠀⠀⠀⠀⠙⠳⢛⡀⠀⠮⡤⣤⡷⠄⠐⠋⠀⠈⠁⠀⠀⠀⠀⣀⡴⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠛⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠂⡠⠤⠀⠀⠀⠄⠀⠀⠑⡄⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣄⣀⡄⠠⠀⠀⠀⠀⠇⠀⠐⠁⠀⠠⠰⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⢀⠈⡁⠈⠛⠓⠛⠛⠁⠀⠓⠛⠛⠇⠈⠛⠁⠀⠀⣢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠐⠀⠀⠈⠛⠋⠤⢄⣀⡀⠀⣀⣀⣐⣂⣀⡀⠊⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁⠒⠢⠤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠐⣒⣀⠔⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠅⣀⣀⡀⠀⠀⠀⠈⠉⠙⠀⠉⠉⠉⠉⠉⠉⠉⠀⣂⡴⠞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠈⠉⠉⠛⠚⠒⠒⠆⠀⠲⠶⠶⠒⠚⠚⠉⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣤⣀⣀⡇⠀⠀⣀⣀⣠⣤⣤⠤⠴⠒⠋⢁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⣀⡀⠀⠀⠈⠀⡀⢀⣉⣉⣁⣤⣤⡤⠶⠖⠛⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠲⠤⣄⣀⠈⠉⠉⠉⠉⠉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⣤⣀⣀⣀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⠉⠉⣀⣀⣀⡤⠄⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠩⠭⢉⣛⡓⡒⢒⣒⣒⣒⡒⠒⣒⣻⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠞⡀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⡀⠀⢀⣀⡈⠂⠀⠂⢠⡃⠀⠀⠜⠁
⠀⠀⠀⠚⠀⠀⠀⢠⡾⠋⠉⠉⠙⢻⡁⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⠀⠀⠀⠀⡈⣳⡀⠀⡄⣅⣀⡠⠖⠀⠐⠪⣠⡞⠁
⠀⠀⠀⠈⢁⠀⠀⠙⠛⠒⠠⠤⠤⣚⢙⣒⢒⣶⠞⠀⠀⠈⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣰⡏⠉⠉⠉⠙⣿⣁⡀⠁⠀⠀⠀⠀⡅⠈⠂
⠀⠀⠀⠤⠄⠀⠐⠁⠀⠑⠠⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠃⠀⠀⠀⠈⠃⠉⠉⠀⠒⣉⠍⠉⠀⠀⠀⠠⢤⡚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠤⣀⠀⣦⠀⠀⠀⢀⠖
⠀⠀⠀⠀⣀⣠⠤⠤⣤⡞⠉⠀⡀⠄⢹⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⠗⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠉⠐⠄⢠⠈
⠀⠀⠀⣮⠉⠀⠀⡠⠒⠑⠒⠛⠃⠀⠋⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡬⡶⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠚⠁⠀⡀⠀⢀⠀⠀⡼⡯⠀⢀
⠀⠀⠸⠈⠉⢉⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠤⠂⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠊⠙⠀⠐⠂⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠠⠴⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠂⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⢨⠅⠀⢐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠴⢄
⠀⠀⢀⠔⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⡆⠀⡠⠁⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠐⠈⠁⠀⠀⠙⢤
⠀⠀⢈⠀⢀⣤⣤⣠⣴⠂⠀⣀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠑⠦⣄⣻⡀⡧⠴⠋⠀⢤⡄⠀⠀⠢⣄⠀⠀⠸⡄⠀⣀⣀⣄⣀⣀⣀⣤⡀⠐⠂⡄
⠀⠀⢰⡋⠁⠀⢀⡨⠷⠶⢾⠁⠸⠂⠐⠀⠀⠀⠀⢢⠀⠀⣠⠖⠚⠉⠙⢻⣁⣠⡔⠀⡨⠁⠀⠀⠀⡀⠑⠶⣤⣿⠋⠀⠈⠉⠛⡅⠀⠈⠉⢲⡄
⠀⠀⠧⠉⠉⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠈⠀⢠⡻⠤⢤⠔⠒⠋⠤⠘⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠂⠲⠶⣚⠉⣓⠒⣦⡾⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠠⠄⠊⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠢⠄⠠⠐⠁⠀⠉⠒⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡀⠀⡄⠒⠂⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣈⡥⠆⠀⠀⢄⣀⣀⣠⡀⠈⠀⢤⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠄⠀⡀⢀⠄⠚⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⢵⠀⢠⣄⣀⣹⠀⠀⣠⣄⣒⡂⠤⠭⠍⠐⠒⠊⠉⢢⢀⡠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⢀⣀⣀⡀⡉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠒⢟⡈⢀⡨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢀⣀⠁⠀⠀⠀⠈⢱⡀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠑⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢄⣉⠀⠀⠐⠒⠂⠶⡇⠠⠤⠤⠒⠒⠒⠋⠉⠉⠁⢀⣀⡗⠐⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣭⠈⠒⠒⠒⠒⠶⣇⠠⠤⠤⠤⠔⠒⠒⠊⠉⠉⠀⢀⡠⠔⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡓⠤⣀⡈⠑⠒⠒⠒⠂⠆⠀⠠⠤⠤⠤⠤⠔⠒⠚⠉⠽⣧⡀⠀⢀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⢃⠢⢄⡉⢁⠒⠶⠦⠄⣴⠀⣀⣀⣠⣤⠤⠤⠤⠴⡶⠋⠀⠁⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠲⢤⡄⠀⠋⠁⠀⠙⠶⣼⠁⠀⠀⠰⢤⡤⠤⠤⠤⠤⠤⠤⠴⠶⣶⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⣀⠤⣉⠀⠀⣀⠠⢄⣀⣀⠠⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⢦⣤⡠⢤⡤⣄⢠⢴⠀⠀⠀⠀⠀⢀⡤⠔⠛⠈⠢⢀⢀⣀⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠉⠀⠀⠀⠀⢀⡠⠾⠀⠀⣀⣤⠞⡩⠉⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢄⣀⡴⠖⠁⠀⠀⠉⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡼⠋⠀⠀⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⡤⠚⠉⠀⠀⠀⠀⣀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⢀⣀⠀⠀⠀⠀⠀⠈⠢⢀⡀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠙⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠉⠒⠲⢤⣀⣤⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢰⣀⡀⠀⠀⠀⠐⠀⠀⠀⢈⢣
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠉⠓⠆⢭⡠⠆⠒⠐⠰⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠄⠀⠐⠀⠈⠉⠐⡉⠀⢀⡀⣈⠔⠁⠀⠀⠀⠀⠀⠙⠲⡅⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡤⠀⠀⠀⠀⠀⣀⠀⣄⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒

⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠁⠀⣿⠤⡀⡆⠀⠀⠀⠀⢀⣧⢄⠀⡇⣶⡄⢸⡤⠄
⠀⠀⠀⠀⠀⠀⢸⠶⠶⢾⡶⠶⠶⡷⠖⠷⡇⠀⠀⠀⠀⢸⡷⠶⠶⣷⠷⠶⢮⠶⠶⠶
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠷⠈⡟⠁⠀⡇⠀⠀⠀⠀⢸⡇⠉⠀⡿⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢸⡟⠛⠉⡞⡛⠛⢳⣛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠘⢸⡇⣚⣀⣇⣁⣂⢸⢋⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⡟⣭⠁⣙⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣴⣿⣶⣶⣤⡶⠒⠃⠀⠀⠀⠀⣯⠭⠤⡭⠭⠤⣥⡤⠤⢼
⠀⢰⡀⠂⢀⣀⣤⣶⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡠⠀⢳⠀⠀⢸⠀⠀⣠⣔⠂⠁⢀⣀⣀
⠀⠖⠛⠋⠉⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡗⠒⠒⡖⠒⠒⣾⠒⠒⢺⠀⠀⠈⠙⠳⠾⠯⠙⠉
⠀⠀⠀⠀⠀⠀⠹⣿⣭⣁⣀⣀⣀⣀⣀⣀⠄⠀⠀⠀⠀⠘⡇⢀⣀⡃⠀⠀⣻⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⣿⣷⡤⠤⠽⣶⣶⠒⠛
⠀⠀⢀⣀⠀⣶⣦⣿⣿⣿⣿⣦⡀⡀⠀⠀⠀⢀⣠⡄⠀⠶⡀⣀⣹⠶⣶⡔⠀⠀⠀⠀⠘⣷⣰⣤⣤⠬⣃⠶⣶⠛⠣
⠀⠐⢿⣿⣿⣿⣿⣦⣭⣟⣛⠉⢛⡳⢐⣻⣅⡈⠻⣤⡒⠂⠈⢳⠄⣤⡜⠋⠪⣟⢀⣀⣀⠛⠋⠉⠈
⠀⠀⠀⠈⠛⢿⣯⡉⠽⠿⣿⣿⡵⠖⠉⠀⠛⠂⠀⠙⣿⣾⣝⠃⣀⣙⣆⣴⣶⣶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⣒⢅⠀⢂⣐⣂⣄⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢂⡈⠉⠑⠀⠲⣄⠁⠀⠀⠀⠈⢻⠟⠁⠀⠀⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠘⡿⠋⠀⠀⠀⣠⠾⢿⡿⣦⡄⢀⠔⠈⠀⠀⠀⣤⣾⢿⣻⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡃⠔⠊⠀⠀⠀⣠⣾⣃⡺⠚⠳⣿⠟⠁⠀⠀⢀⠰⠋⠀⠉⠢⣍⣬⠏⠀⠀⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣟⠟⠁⠀⠀⠀⡤⠊⠀⠈⠻⣦⣤⠞⠁⠀⠀⢀⠴⠾⢤⡀⠀⠀⠀⠈⠑⢦⡀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠀⡴⣿⣄⡀⠀⠀⠀⠀⠁⠢⡀⢀⣴⠁⠀⢀⡄⣿⣧⡄⡀⠀⠀⠀⠈⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠘⠌⠙⠀⠀⢀⣰⡏⠀⠈⠙⢿⣦⡄⠀⠀⠀⠀⠈⠛⢿⡆⡠⣴⣿⠟⠁⠀⠉⠲⣀⡀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠐⠿⣫⣷⠀⠀⡠⠎⠀⠀⠑⢄⡀⠀⠀⠀⠀⠙⢺⡟⠁⠀⠀⠀⠀⡴⠻⣿⣶⣤⣠⠊⢠⠐
⠀⠀⠀⠀⠀⠀⠈⢛⡀⠀⠀⠀⠈⠛⣦⠞⠀⠀⠀⢀⡰⠛⠛⠢⣄⠀⢀⡰⠃⠀⠀⠀⠀⣠⠆⠀⠀⡼⣿⡟⠁⠀⠞⣠
⠀⠀⠀⠀⠀⠀⡔⢸⣿⣦⣀⠀⣠⠜⠁⠀⠀⢀⡴⡿⠀⠀⠀⠈⣹⡿⠋⠀⠀⠀⠀⡠⠎⠛⢤⣤⣺⡵⠋⠀⠀⢀⣶⠙
⠀⠀⠀⠀⠀⠀⢸⡘⣿⣛⣿⠟⠁⠀⠀⠀⡐⠋⠙⠇⡀⡀⣀⡼⠋⠀⠀⠀⠀⣠⡎⠀⠀⠀⠀⠉⠋⢄⠀⢀⣴⣿⢋⣴
⠀⠀⠀⠀⠀⠀⠀⣧⡌⡿⠃⠀⠀⠀⣠⣞⡀⠀⠀⠀⠈⠻⢏⠀⠀⠀⠀⠀⠞⠙⢹⣶⣄⠀⠀⠀⠀⠀⠙⠻⣿⡇⣺⡜
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⡀⠀⢀⣴⣿⠟⣿⣦⡀⠀⠀⠀⠀⠀⠢⢄⣴⠯⠂⣶⣾⣷⠛⠉⠒⣄⡀⠀⠀⡔⢉⡔⡻
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⡌⣧⣿⡿⠙⢿⡏⢀⠝⠳⠤⡀⠀⠀⠀⠈⠙⠲⣴⣿⠞⠁⠀⠀⢀⣼⣿⠦⠘⡁⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢬⣙⢿⣖⣿⣷⠋⠀⠀⠀⢈⣴⣄⠀⠀⠀⠀⡨⠁⠀⠀⢀⣴⣿⡿⠃⢴⢆⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣾⣌⣧⠙⠛⠓⠢⣄⠀⣠⣾⡖⠿⣿⣦⡤⠂⢀⡤⠀⠀⠞⣫⣶⡶⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣼⡛⢧⣈⢉⣭⣭⣙⠿⢟⣩⣍⡙⢩⣶⠆⠶⢀⣿⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠯⢼⣿⣭⣝⣃⣟⣭⣬⣃⡼⠟⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⠀⢄⠀⠉⠣⡀⠀⠈⡑⠖⠍⠌⠒⠉⠐⠀⠀⢘⡀⠀⠤⡀⠀⠀⠀⠠⠂⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡦⠀⠐⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠈⠑⠂⠒⠄⡀⠀⠈⠢⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⡔⠈⠦⡀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⢀⠴⠋⡄⠀⠹⡄⡔⠁⠈⠒⡠⣤⡠⠐⠃⠀⠀⠀⠀⠀⠉⠢⡠⠟⠉⠑⠂⢤⠀⠊⠁⠙⢆
⠀⠀⠀⣠⠖⠁⠀⠐⠀⠀⠀⠹⢀⠀⠐⠁⠀⠀⠈⠢⠀⡀⠀⠀⠀⠀⠀⠀⠳⣀⡀⠀⠀⠀⠱⡯⠀⡀⠀⢇
⠀⠀⡜⠁⠀⠀⠀⠇⠀⠀⡠⠎⠀⠀⠀⠀⠀⢀⠸⡇⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠪⡂⢸
⠀⡘⠀⠀⠀⠀⢰⣀⡠⠄⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡜
⠀⢇⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠁⠀⠀⢔⠉⣄⠀⠀⠀⠀⠀⣇⢀⣇⠠⢀⣀⢀⣠⠴⢊⠗⠢⢄⡀⢀⡠⠁
⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠄⠂⠠⠤⠒⠁⠀⠀⠀⠈⠐⠠⣠⠉⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠉⠀⠒⠠⠬⢄⡀⡰⠋⢘⣄⣄⠀⠀⠀⠀⠀⡠⠂⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠊⠁⠘⠃⠀⠄⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠁⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⠻⣌⠀⠀⢀⠰⣾⠃⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⣛⣛⣯⣟⣋⣬⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣤⣁⣤⣄⢠⣦⣐⡎⠀⠀⣴⣯⣬⠁⠀⠈⣧⠀⠀⣿⣿⣟⣭⣭⣿⡿⣻⡦
⠀⠀⢀⡴⠖⠛⠲⣤⡀⢛⣿⡏⠙⢉⣹⡿⠁⡴⠏⠁⠀⣀⡵⠶⠴⢿⣀⠀⢾⣿⣶⣤⣿⣶⣾⡉⠁⠀⢀⡿⣦⣀
⠀⢠⡏⠀⠀⠁⠀⢈⣷⠈⡿⣿⡿⣿⣿⠇⢰⡝⠂⠀⣼⠇⠀⠀⠀⠀⢈⡄⠸⣯⣏⣻⡟⢋⣿⡏⠀⠰⣏⣆⢻⣿⢳⡀
⠀⠈⢏⠀⠀⠀⣠⣬⣟⠀⢹⣿⡷⣾⢿⠀⠈⢹⢲⡴⣯⡀⢀⠀⠀⠀⠠⣾⠀⢹⣿⣟⣛⣿⡿⠀⠴⣴⠿⠼⣿⢘⣿⣧⡄
⠀⠀⢋⡀⢀⢠⣀⢢⡼⠀⠸⣿⡷⢮⣿⠀⠀⠺⣼⣗⠀⠣⣠⡀⠤⠤⢶⡋⠀⠀⣿⢹⣿⣿⡇⠀⠼⣯⣾⣱⠚⣜⣿⣹⠃
⠀⠀⢠⡾⣷⡾⡿⡾⠀⠀⠀⠛⠓⠚⠃⠀⠀⠀⠈⢿⣿⣶⣶⣦⣴⡻⠉⠀⠀⠀⢽⢾⣤⣿⠃⠀⠀⢻⣿⣿⣾⣛⣿⡿
⠀⠀⠀⠜⣧⣻⠓⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣧⠃⠀⠀⠀⠀⢸⣿⣩⣿⠀⠀⠀⠘⣟⣿⣹⣯⣽⠇
⠀⠀⠀⢸⣶⢋⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣖⡏⠀⠀⠀⠀⠀⠀⣿⣻⡿⠀⠀⠀⠀⢻⣿⣙⣯⣿
⠀⠀⠀⠈⡟⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣻⣽⣩⠁⠀⠀⠀⠀⠀⠀⠹⢿⠇⠀⠀⠀⠀⠈⣿⡟⣿⠇
⠀⠀⠀⠀⠙⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠈⠛⠤⣤⣤⣤⣤⣀⠀⠀⣠⣤⣶⠏⠁⢀⡤⠞⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⢿⣧⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠂⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⠉⠛⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⠀⢀⠀⠀⠀⠀⠐⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣤⣦⣃⠀⠀⠀⢀⠐⠀⠀⢤⣄⣀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣴⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣼⢄⡀⠉⠉⠛⠶⢤⣿⡀⠀⠀⡦⠭⠉⣝⣓⣦⣄⣀⡦⣆⣀⣄⣀⣠⣤⡶⣞⣻⣯⣵⠶⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠛⠷⣧⣈⠀⠀⠀⠀⠀⠀⠋⠃⠘⠛⠒⠿⠿⣿⣿⣿⡗⣿⠿⠿⠿⠟⠛⠛⠉⢉⣁⡤⡾⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⡀⠈⠉⠛⠻⠷⣶⣀⣾⣤⣤⣀⠀⠀⠤⠀⠀⠀⢠⠀⡀⠀⠀⢀⣀⣠⣤⣼⣿⣽⠷⠾⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⣙⠳⢤⣤⣄⣠⡀⣎⡴⢈⠉⠉⠛⠳⠶⠶⠶⠴⠞⢠⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⢀⣰⣶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠻⠾⣷⡀⠀⠀⠈⡉⢉⡟⠛⠲⠶⠤⢄⣑⣀⡀⠀⢐⠊⡀⣀⣀⣀⣠⣦⡴⠴⠶⠚⠋⠉⣹⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠉⠑⠶⡆⢰⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠀⣀⣀⣀⣤⡤⠶⠛⠉⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⠿⢦⣀⡒⢀⣀⢃⠀⢡⠈⠉⠉⠛⠲⠲⠦⠤⠤⠄⠒⠶⠖⠒⠒⠋⠉⠁⠀⢀⡀⠀⢀⣄⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⡉⠙⠻⠿⠾⢧⣼⣠⣈⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⣉⣀⣠⣶⠶⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣤⣀⣈⡙⠛⢿⣟⢿⣾⢶⣶⣴⣤⣤⢤⣶⠶⠶⣾⡿⠿⢋⣡⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠉⠛⠛⠿⠷⠶⢦⣭⣭⣭⣤⣬⣥⣭⣭⠤⠖⠓⠛⠉
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠾
⠀⠀⠀⠂⠘⠛⠋⣿⣿⣿⣿⣿⣿⡿⠿⠿⠤
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀
⠀⠀⢀⡤⠶⠞⣋⠱⣤⠿⠷⣤⡀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣶⣶⡾⠛⠹⣭⠍⠓⣆⠀⠀⠀⠀⠀⠀⠀⠠⢶⡦⡟⣃⣀⢀⠤⢤⣀
⠀⠀⣿⠀⠘⠋⣁⣠⠄⢀⣰⠎⣿⡀⠀⠀⠀⠀⡼⠁⠶⣸⣁⣤⣤⡶⢖⡀⢴⠞⣫⣇⡀⠀⠀⠀⠀⢀⠒⣉⢟⡛⢋⣀⣀⠘⣛⣭⢳
⠀⣼⠛⠒⠶⠞⠉⠀⠦⠉⣁⡤⣿⣷⡄⠀⠀⢸⡇⣠⣴⡏⠻⠟⠁⢀⣀⠀⣺⣿⢿⢿⣧⡀⠀⠀⢰⠁⠸⢃⡼⠟⠉⠀⠈⢙⣶⣜⠃
⠀⣻⣄⡐⠶⣤⡄⠀⣠⣄⣤⣾⣿⣿⠃⠀⠀⠀⢳⣄⢸⡟⠿⠗⣀⣈⡅⠀⣙⣿⣿⣯⠉⠀⠀⠀⢸⣤⣤⣮⡞⣫⢿⡇⠚⣸⠟⢿⣴
⠀⠻⡮⣍⣙⣛⡛⠛⣛⣛⣻⣿⣿⡟⠀⠀⠀⠀⠐⢯⣙⣛⡒⠾⠿⣥⣤⣄⣿⡿⣫⡭⠀⠀⠀⠀⠀⢙⢺⡝⣿⣷⠾⣤⡤⠤⠶⣾⡏
⠀⢴⡟⠶⠤⠤⠤⠤⠴⠾⠿⢟⣿⡦⠀⠀⠀⠀⠀⡼⠧⣬⣍⣉⣉⣉⣩⣭⣶⡿⢿⣧⠀⠀⠀⠀⠀⠀⢻⡆⣏⣾⠤⣴⠶⡶⢿⡿⠁
⠀⠀⢻⡷⠶⢶⣶⣶⣶⣶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠙⣖⠲⠤⠤⠭⠤⠤⠤⠶⠞⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣟⡉⠳⠃⣠⡙⠺⡇
⠀⠀⢸⣧⡀⢸⣇⠀⢸⡏⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⢰⣿⡆⣸⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⣄⠸⣁⡟⣠⠇
⠀⠀⠈⢿⠟⢻⡿⠿⢿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⡁⣉⡉⠈⣛⠁⢈⣋⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣝⢿⣇⠀⣼⠃
⠀⠀⠀⣼⣆⣼⣧⣀⣼⣧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⢸⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡔⣿⣆⡼
⠀⠀⠀⠸⢯⣽⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣄⢀⣭⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣹⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣝⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⣿⢻⡄⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣯⠳⡄
⠀⠀⢤⣖⣛⣥⠾⢁⣿⠄⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠘⣧⢹⡄
⠀⠀⣾⣉⣩⣤⣶⡿⠏⣰⡟⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠸⠃⢀⠹⣇⢻⡀
⠀⣴⣟⣛⢛⣋⣩⣴⡾⠋⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⣇⢀⡴⠶⢖⡲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡀⠀⡄⠈⠁⢻⡎⢷
⣠⠾⠻⠿⠿⠿⠛⢉⣤⣾⡿⣻⣷⠀⠀⠀⠀⠀⣰⠾⢻⣩⣽⠟⠙⣧⠙⠤⠘⣿⡷⣄⠀⠀⠀⠀⠀⠀⡼⠁⠃⠀⠀⠀⣴⠀⢿⡜⣧
⢹⡿⣶⣶⣶⣶⣿⣿⣿⣯⣾⣿⡇⠀⠀⠀⠀⣼⠃⢰⠉⠁⣸⠷⠾⠿⠶⠶⢦⣽⣧⣘⡇⠀⠀⠀⠀⣸⠃⡄⠀⠸⠃⠀⠀⢀⡌⣷⢹
⢸⣳⡖⡶⢮⣯⡽⣽⡷⠾⣟⣿⡁⠀⠀⠀⠀⣿⡘⠎⢓⡞⣴⠂⣠⠀⠓⣰⣿⣿⣿⣿⠁⠀⠀⠀⢰⣿⣤⣀⣀⡀⠀⠶⠀⠈⢁⣼⡇
⠈⣿⡏⢔⢱⠁⣏⠆⢼⣑⢈⡿⠀⠀⠀⠀⠀⢻⢗⣤⣿⣄⡙⠳⠤⣠⡾⣿⣿⣻⣿⢿⠀⠀⠀⠀⢻⣎⠛⠛⠿⣿⣷⣶⣤⣶⡿⢫⣿
⠀⠸⣟⡿⠵⢮⣦⣎⣿⣥⣾⠇⠀⠀⠀⠀⠀⢠⡻⣿⠃⠉⢹⣤⠞⠛⢿⣛⡯⢿⢷⠃⠀⠀⠀⠀⠀⠙⠛⠷⠶⣶⡤⣤⡭⠽⠶⠛⠁
⠀⠀⢧⣼⠼⣋⡻⢽⠍⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⢰⣿⣧⠉⣾⣿⡎⢿⡛⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡆
⠀⠀⠈⡿⠀⢧⠔⣘⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡛⢡⣾⣮⠋⣴⣦⠹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠹⡍⣆⣸⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠻⠿⠠⠟⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣄⢠⡀⣠⣄⢀⣤⡀⣠⣄⢠⡀⠀⣠⡀⣤⣤⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣼⢿⡇⣿⣾⠈⠃⡇⣿⢸⡇⠀⡿⣇⢸⡇⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⡿⢸⠟⣧⡿⢺⣴⠇⣧⡿⢸⣥⣴⠗⢿⢸⡇⢸⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⠤⠤⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⡽⢇⡠⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡂⠀⠀⠀⠈⠉⠉⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠂⠀⠤⠀⠀⠀⠀⠄⠀⠀⡰⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡞⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢩⠔⠒⠞⡰⠁⢠⣟⠁⠀⠠⠤⠤⣄⢀⣰⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⠁⢠⠇⠀⠉⠁⠒⠐⠊⠀⠠⢙⠥⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡖⠀⠀⠓⠂⠤⠼⠁⠀⡿⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡶⠒⠦⢤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠠⠤⠤⣄⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉