Basil Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠤⠛⠢⣤⠮⠑⠨⣹⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠄⢺⠖⠳⢦⣀⡉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⣾⣾⠋⠁⢀⢠⡟⠂⠀⠀⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠟⠃⠙⢷⡄⢼⡟⠀⠀⠀⠀⠀⠐⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⡉⠀⠀⠀⢲⣿⣌⡇⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡗⠀⠀⡀⢨⣿⡿⡛⠤⠀⠀⠀⠀⠀⠸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠃⠀⠀⢸⣿⡿⢠⠀⢀⣷⠳⡀⡀⠤⢸⡮⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⣸⡿⠁⠀⢳⣼⠟⠀⡜⠻⣄⡾⢳⠀⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⡇⢡⠀⡠⠞⠁⠀⠸⠀⢸⡇⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⠶⠓⠞⣶⣶⣄⣀⠰⢧⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⣠⠔⠁⠁⠀⠈⠈⠻⠷⣿⠀⠀⠀⠀⠀⠐⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠹⢴⡆⢀⡀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⣄⠠⡀⠑⠿⣅⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠻⣉⠀⠀⡡⠂⠠⠄⠀⠠⠀⠀⡇⠀⢙⣡⢾⡄⣀⡈⠙⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢀⠐⠀⠁⠀⠊⠀⢀⠰⠀⠀⠀⠱⣄⠁⡖⢠⠀⡀⠉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⡀⢠⠆⣼⠇⠀⠀⠁⠠⠀⠀⠀⠀⢰⠀⠰⠃⠆⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣰⡷⠧⠞⠉⠆⠀⠀⠀⡄⠀⠀⣠⣆⣾⠁⠀⠈⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⡿⠋⠀⣀⣀⡀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢀⣤⡾⠟⠀⡤⠈⠀⠰⠄⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⠉⠁⠀⠀⠀⠀⡄⣠⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠈⠳⡀⠀⠀⣰⠍⠀⠀⠀⠒⠲⡀⠉⣉⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢷⠂⠀⠈⠀⠀⠀⠴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⢇⠀⡄⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠇⠀⠀⢀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⡠⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⠄⣼⣀⠀⠀⠀⠀⠀⢣⠀⡀⠄⠒⠋⠉⠉⠉⠒⠒⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠈⠉⠑⢄⡀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠸⡅⠀⠈⠲⡄⠀⠱⡌⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠤⠤⣈⣢⣀⣣⡀⠀⠀⠘⡄⠀⢱⠀⠀⣀⣤⠴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⠀⠀⣷⡀⠬⢶⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣬⣶⡖⠋⠓⢦⡀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⡠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠼⢧⡀⠀⠀⠹⡄⠀⣗⠀⠀⠀⠀⠀⣀⡠⠐
⠀⠀⠀⠀⠃⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠹⠀⠀⠈⠀⠂⠀⠈⠁
⠀⠀⢀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢛⣰⡴⠁⠢⡀⢠
⠀⠀⠀⠀⠈⠉⠒⠀⠠⠄⡀⣀⣀⠀⢀⡠⡤⠤⠐⠁⠀⠞⢁⠆⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠃⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⠀⢰⣰⢰⠀⢦⢰⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠈⠈⠀⠈⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠒⣒⣶⢰⠴⢎⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖⣀⣠⠟
⠀⠀⠀⠀⠐⢦⣄⣀⣩⠿⠟⠋⠄⣀⣀⣀⡤⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠏⠉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⠚⢠⡀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣤⡔⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⢉⣔⡔⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⡁⠀⢠⠞⢀⡼⢋⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣰⠋⠀⢰⠃⡠⠚⠒⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣲⢤⣀⠀⠀⣰⠃⠀⢰⠃⠀⢀⡼⠊⠀⠀⣠⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡐⡎⢉⠳⣇⠀⠀⢼⢀⡴⠯⠤⠤⠖⠊⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⠬⠧⣨⠇⠈⣇⣠⠞⠁⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠙⢖⡲⠚⠓⣶⠋⠉⠳⣖⠒⣒⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⣠⠤⠤⠤⣍⣹⢿⢸⡛⣯⣍⣈⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠵⠋⠁⠀⠀⠀⣠⡞⠁⠈⠳⣅⡙⠌⢳⣦⣽⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⠤⠖⠒⣦⠞⠁⢻⠀⠀⠀⠀⢉⡟⠙⠓⠿⡾
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠚⠁⢀⣴⡎⠀⠀⠀⡞⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⡴⢃⡤⢤⡤⠞⢉⡼⠁⠀⢀⠜⠁⠀⢀⡴⠋
⠀⠀⠀⢀⡜⣁⣥⡾⠉⠀⠰⠋⠀⠠⠖⣉⣠⠴⠚⠁
⠀⠀⠀⠿⠯⠥⠟⠚⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢀⡀⠀⠀⠀⠀⠀⢦⠀⢰⡄⣀⠙⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠑⡄⠀⠀⠀⠀⠙⢦⡀⠈⠀⡀⢹⡄⠀⠀⠀⡠⠤⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣤⡀⠀⢀⠈⡀⠀⠀⠀⢶⡘⠻⣷⣤⠴⢾⣦⣮⡴⠚⣙⡶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠈⢿⠀⡈⠀⠑⠀⠀⣠⠤⠷⣿⣶⡿⠶⢞⣿⣿⣧⣶⡧⠴⣏⣙⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢱⣤⣾⡟⠁⡀⢀⣂⠞⠁⠀⣠⡾⢻⡿⣿⣿⢿⠉⣷⣮⣭⡳⣌⠫⡑⢤⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣹⣷⣰⠿⣿⢧⣼⢇⡼⠃⢀⣀⡴⢏⠳⡄⠃⠀⢹⣇⠀⢃⠈⡶⣄⣈⡑⢾⠢⣽⠄
⠀⠀⠀⠀⢀⠴⣒⠄⠠⣿⠛⣟⠛⠙⣿⣿⣿⡀⣶⣸⣧⣿⠁⠀⠠⡟⠀⠀⠀⢹⠀⠀⢸⣿⣆⠘⠄⢹⡌⢃⡀⡠⣣⡀
⠀⠀⠀⡄⣡⠞⠉⢀⡴⠃⠠⡃⢀⣸⡿⢾⣿⣿⢿⣿⣿⠁⠠⢀⡜⠀⠀⠀⠀⣽⠂⠀⢹⣿⣿⣦⠐⠀⠳⠈⢧⡄⡈⢳
⠀⠀⢰⣶⣃⢀⡀⢭⡤⠴⠔⢺⠛⣡⠜⠉⠏⢀⣺⣿⣗⣤⠒⣿⣣⠀⠀⠀⢈⡇⠀⠀⣮⣿⣿⣿⣻⣕⠀⠀⠘⢃⠈⢺
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⡰⠁⡴⠃⠀⠀⣷⣿⣿⣿⡧⠁⣴⠃⠀⠀⠀⠀⠸⡇⢠⡾⣷⣶⣤⣍⡛⢿⡦⠘⡆⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡎⠀⠂⢀⣶⣫⣿⣿⢻⡇⠀⣿⠣⠘⡀⠀⠀⢰⣷⣿⣷⣿⠀⢹⣮⣉⠓⠵⠍⠤⣁⠳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣇⠀⣤⡾⣳⡟⠛⠁⣼⡇⣀⣽⠃⠀⠧⢀⣴⣿⢿⣿⣿⡘⠰⠈⡀⠙⡁⠢⢌⢆⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣿⣠⢏⡴⢻⠅⠀⣰⠏⣿⠹⡏⠓⢀⣴⡿⣿⣿⣾⣿⣿⣿⡰⠀⣧⠀⢘⢀⠀⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⣱⠊⠀⠈⣇⠀⠋⠀⢹⣷⡥⠾⠋⠀⠀⢿⡃⠿⣿⣻⣿⡟⢦⡟⢤⢸⡄⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢀⡟⠀⠀⣠⡇⠈⠀⠀⠈⣷⢈⠳⠄⠀⠀⢸⣿⣷⣼⣿⠿⣀⠈⠳⣶⣨⢷⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⣿⣸⡇⠀⠀⠴⢒⡿⣾⡀⠦⠀⠠⣿⣿⡿⠀⣹⣎⡈⠑⢄⠀⠉⠳⢽⣄⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⠛⢋⣿⢿⣆⠀⡀⠈⠀⢹⠌⠂⠀⠀⡇⢹⡇⠸⠁⠙⡟⠓⠂⠱⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⢤⡞⠙⣧⢻⡄⢁⣀⡀⢰⠆⠐⢄⠸⠁⠀⣇⠀⠀⣼⢿⣦⢦⣄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠚⠡⢺⣇⠀⠹⠀⡿⠉⣬⣄⠃⢷⡴⣡⡶⣄⠀⢸⡇⢠⡯⠀⢻⡀⠙⠧⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡌⢻⡄⠀⠐⡷⠛⣻⣿⣿⡟⠻⣟⠀⠘⠄⠸⡇⠸⠁⠀⡇⢱⣄⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⠋⣧⠀⠁⠀⣰⣷⡞⢿⠈⣿⢀⠊⣸⣟⠲⠼⡄⢧⠀⠠⠀⠀⠈⡟⠢⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠰⠈⡄⣀⣴⣛⠋⡇⠘⢠⣿⡆⠠⠀⣿⡀⠀⢹⠘⢧⣀⡀⠀⠀⢿⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⣴⠟⢹⢋⣀⠘⢃⡘⢉⡏⢳⡶⠀⢠⡈⠀⢸⠇⠈⠣⣤⣄⣂⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠸⣎⣬⠟⠋⠀⢰⠁⠈⢧⣠⠎⠳⡐⣼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⠀⠀⠀⠳⢄⡧⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣀⣀⡠⣾⣿⣀⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⠀⠉⡀⢃⡟⠄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⢸⠷⠀⠀⢀⠈⢲⣄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⢸⡀⢸⠛⣆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⠱⣄⠀⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠀⠀⠀⠀⠀⠀⠈⠚⠦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠄⣠⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⠎⠀⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣢⣥⣄⣏⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢇⡀⠰⡙⣦⠆⠀⠉⣉⠔
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⢈⠻⣤⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄⢁⠘⣼⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡃⠒⢰⢾⡟⠀⠀⠑⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⠴⠷⠢⣤⣖⡍⠁⠀⠈⠀⠡
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠘⠀⠄⣀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡻⠦⣄⠈⡗⢶⡆⡴⠚⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠘⡄⢸⡜⢦⠙⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⡀⠀⠀⠀⡇⠸⠁⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣤⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠑⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⢯⣻⠍⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠏⠐⠳⡀⠳⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠆⢀⡼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢆⣴⣤⣶⡶⠴⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡷⠁⠄⠰⢇⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⠂⠆⠘⠛⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⡝⠲⢊⣛⠳⠶⠄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣠⣸⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡸⢉⡟⡸⠀⠔⠈⠣⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⢱⡋⢈⠀⠀⠀⠀⢶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠞⠰⡇⠸⠀⠠⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠗⠁⣌⠠⣀⡀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⢈⠇⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⡇⠀⠹⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢘⣊⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⣈⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠼⠀⠂⠀⢁⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⢘⣹
⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠤⠤⠀⠈⢩⠅⠀⠃
⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⢠⠾⠂⡀⠀⠀⠃⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⣀
⠀⠀⠠⠀⡄⠀⠀⢈⠀⡦⠀⠐⠖⡂⠱⠀⠘⠆⠀⡂⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣸⠻⢴⡇
⠄⠀⠂⠀⠂⠁⠆⠈⢧⡤⠊⢠⣆⢣⢄⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣿⠻⡇⠀⠚⠃
⠀⠀⠀⠀⢀⠰⡞⠀⣠⠀⠈⣁⡀⣶⠈⠘⠂⠛⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⣶⡎⠃
⠀⠀⠀⢠⡍⠀⠘⢶⠁⡠⠞⠉⠀⡇⠈⠙⠓⠄⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠐⣀⢾⡃
⠀⠀⠀⠁⠀⠀⣰⢚⡏⠛⢀⠠⠢⠁⠀⠀⣠⣖⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠁⠁⠀⢸⡇
⠀⠀⠀⠀⠀⡼⢁⠀⡇⠀⣤⡱⣇⠠⠀⣠⢁⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠈⢲⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠄⣁⠀⢰⢀⣴⠆⠉⢥⢠⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⣼⣷⡄
⠀⠀⠀⠀⠂⡜⠁⠀⣸⣏⠁⠀⢀⡀⠸⣾⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣽⣯⣿⡀⢀
⠀⠀⠀⠀⢎⠃⠀⠈⠘⡋⡌⠃⡉⣑⠄⣘⢁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⣿⢏⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⢻⠀⠋⠘⠈⠡⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡆⠀⠀⣿⣷⣍⡃⢲⠒⠂
⠀⠀⠀⠀⠠⠐⠀⠀⠀⢠⡀⢀⠀⣼⣠⠦⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠈⠻⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⡤⣖⣒⠢⢤⡀⢀⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣤⣀⢀⣴⣦
⠀⠀⠀⠀⠱⡄⠛⠳⣶⡈⣃⢃⠷⣪⠍⢉⡑⢄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⠦⣈⣡⠜⡟⣏⡉⠻⠍⠻⠓⠪⣧⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡇⣈⠴⠁⠉⠒⠒⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⡧⠋⠈⠙⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⢹⣌⠻⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠺⣿⣷⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢤⣾⣦⡌⢻⡀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢈⢿⣿⣿⡶⢀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⣶⣿⣿⣮⡹⠇⠈
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠐⠂⠀⠁⠀⠁⠉⠁⠑⠶⠄⢀⠈⠀⢰⣌⠻⣿⣿⣥⢀⣆⣀⣤⠶⠤⠄⠀⠀⣤⠄⠀⠆⠖⠠⠀⠀⣀⡀
⠀⠀⠀⠀⠠⠞⠉⠉⠺⠤⣤⣴⣦⡰⣦⣶⡄⢤⣀⡀⠀⠀⠁⠀⣽⣿⣷⣶⣿⠿⠟⠋⣉⣀⡤⠂⠘⠋⠀⣤⢀⣶⣿⠃⡾⠶⠒⠁
⠀⠀⠀⠀⠀⢳⣶⣶⣶⠄⠀⠈⢡⣤⡈⠉⠙⠈⠻⣷⡀⣤⣄⠀⠈⢿⣿⠟⣁⢰⣶⣾⡿⠉⠀⠀⣹⣦⣿⣿⠘⠋⠁⠀⠀⢀⡀⡤⠁
⠀⠀⠀⠀⠀⠀⢧⣲⣾⣿⣷⡦⢈⠉⠰⣿⣷⣄⣠⡀⠁⠛⢿⠘⣷⣼⠃⢰⣿⡘⠟⠁⠀⠀⠀⢸⣿⡇⠙⠀⠒⠀⠀⡀⠶⢇⡼
⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⣿⣾⣿⣷⡦⣀⡉⠛⠻⠻⣦⣴⣤⡘⢿⡇⣿⡄⠉⠁⠀⠀⠀⠀⢠⡾⠋⠀⠈⠀⠀⢀⣘⠿⣇⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠿⢻⣿⣿⣿⣿⣣⣶⣤⣿⣿⣿⠿⢿⣷⡹⣿⠁⠀⠀⠀⢀⡰⠋⣡⣴⣂⣤⣤⣼⠿⢏⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣉⡩⣾⣿⠟⠋⠉⠀⠀⠀⠀⠀⠈⠛⣶⣦⣶⠄⠺⠟⠛⠛⠛⠿⡙⠛⠋⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⣀⣀⡀⣀⣠⣠⣠⢟⣇⢴⣶⣒⣠⣤⡤⠐⠒⠊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⢰⣿⡟⢠⣿⠷⣿⡏⣾⡟⢚⣿⣿⣿⣿⣿⣯⣤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠾⠿⠙⢻⠇⣀⣤⡄⣰⣷⣾⠟⠈⠟⣼⣿⣿⣿⡿⠿⣛⣋⣭⡤⠄⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⠘⠛⠛⠐⠛⠉⠁⠀⠀⠰⡿⠃⣨⣵⣾⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠢⠙⡻⣿⣿⣾⣿⣿⠅⠀⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡏⢰⣦⣈⡛⠻⠟⢻⣿⣿⣶⣶⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⣿⠀⣶⣤⠈⠙⠛⣫⣭⣥⣭⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠘⠿⠀⢄⠈⢻⣿⠿⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠒⠀⠀⠀⡀⠀⠀⡐⠒⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠀⠀⠀⢠⡆⠁⠀⠀⠀⠀⢀⣀⡀⠷⣀⠀⠀⣘
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠎⠉⠒⢆⣤⠊⢸⠋⠀⠐⠀⠀⠑⠀⠀⠀⠈⠑⠂⠁⠤⠶⠊⠀⠀⠀⠀⠀⠁⠀⠠⠷⢤⠈⠧⠋⢉⠄⠀⡀
⠀⠀⠀⠐⡀⠀⠀⠈⠀⠀⠀⠁⠈⠤⣀⡠⠇⠀⠶⣆⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⢠⣆⠛⠃⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠄⣰⡃⠀⠀⠀⠠⠀⢱⡀⡠⠄⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡇⠀⠀⠀⠀⣀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⠁⠀⠈⣁⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠠⠀⠈⠈⠁⠸⠦⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠦⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠚⠀⢀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠒⠐⢁⢀⠁⠀⠀⢀⠀⠀⠀⠀⠙⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⢀⠀⡸⣆⢱⠟⢤⣀⠀⠀⠀⠀⢀⠀⡙⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣀⣀⢐⣀⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠈⠉⠉⠁⠐⠓⠐⠒⠉⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠴⠂⠣⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠂⠀⠀⠀⡶⠠⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠀⠒⡁⠈⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣤⡄⡨⠰⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠠⠞⠁⠀⠀⠒⠰⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠉⠹⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣠⣀⠛⠀⠀⠐⠤⠀⠀⡀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⠀⢰⡉
⠛⠉⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⢀⣚⡀⠀⠀⡑⠂⠀⠀⠀⠀⠀⠠⠀⠐⡀⠈⡆⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠉
⠀⠀⠀⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⡬⡉⠀⠀⠔⠂⠀⠀⡐⠀⠀⠀⠀⠐⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣄⠀⠀⠀⠡⠀⠤⠤⠬⠄⠀⠁⠂⠀⡠⠀⠀⠾⠓⢠⣀⠀⠀⠀⠣
⠒⠂⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠞⣄⠚⡙⣨⠀⠀⠀⠀⠀⠠⡀⠀⠛⠀⠀⠀⢀⠀⠀⠁⠃⠈⠛
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡅⠀⠀⠈⠛⠉⠯⠋⠀⠀⠆⠐⠒⠾⠇⠀⠀⠠⠀⢄⠠⣧⠀⠠⠃⠀⠀⢀⠀⠀⡀⡴⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⣔⡄⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠄⠀⣄⡀⢤⡈⠃⠀⣀⠀⠀⠾⠀⠘
⠀⠐⠀⠀⢁⣄⠀⠀⠄⠀⠀⠌⠀⠈⠫⡃⠀⢀⠂⠀⠀⠀⠀⡌⠁⠀⠂⠀⣤⣌⠉⠐⠄⠀⠀⠀⠀⠀⠙⠀⣀⠀⠀⠊⣄⠁
⠂⡐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⣤⣤⣦⠀⠁⠀⠀⠀⠈⠒⠀⢀⠀⠈⠁⠉⢋⡂⠀⠀⠀⣤⠀⠀⠐⠉⠀⠀⠈⢀⢠
⠈⠀⠀⠀⠀⠀⠐⡬⢠⡀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠤⠈⠷⠀⠀⠀⠆⠀⠀⠀⠤⠄⡀⠀⢄⣀⠀⠑⠇⠀⢀⠀⠀⠐⠀⠀⡀
⠤⢀⠐⠋⠁⠀⠀⢰⡴⢣⠆⠀⠈⠀⠄⠃⠀⠊⡠⡑⠜⠀⠔⠉⡢⠄⠀⠀⠈⠀⡀⠃⠀⠾⠉⠀⠀⢀⠴⠈⠀⠀⠀⣠⣾⠏
⠀⠀⡀⠄⠄⠀⠀⠙⠞⠁⠀⠀⠂⠉⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉
⢀⡀⠿⠀⠁⠠⣀⠀⠀⢀⢤⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠤⠀⢠⠒
⠿⠛⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠴⠀⠀⠀⠢⠆⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠊⠒
⠀⠀⠀⠀⠂⠀⠀⢀⡁⠀⠀⠀⠈⠉⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠴⠁⣐⠒⠁⠀⡡⠄⠀⠀⠠⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢂⢀⣀⣀⣐⡀⠀⠃⠅⠀⢀⠀⠀⣠⠦⣁⠀⠀⠀⠀⢆
⠤⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡴⡀⠽⠛⢐⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠀⠀⠁⠀⠀⠲⠆⠘⠿
⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡌⡂⠀⠀⠐⠿⠒⣗⠟⠀⠀⡄⠠⠤⣴⡅⠀⠀⢀⠀⡀⠈⣆⠀⢀⠆⠀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⢠⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⠘⠆⠈⠀⠀⠀⢴⠀⠸⠀⠀⠁⠁⠀⠈
⠀⠠⠀⠀⠂⡀⠀⠀⡀⠀⠀⡐⠀⠐⢟⠇⠀⠀⠄⠀⠀⠀⠈⡑⠃⠀⠀⠀⢀⡙⠓⠨⣁⠀⠀⠉⠀⠀⠲⠀⠀⠀⠀⠐⠀⠃
⠄⢠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠈⠂⠀⠀⠀⠐⠥⠀⠀⠀⠐⠋⠛⠖⠄⠀⠀⠀⣀⠀⠀⠠⠒⠉⠀⠐⠀⢉
⠐⠁⠀⠀⠀⠀⠠⣘⢀⠀⠀⠀⠁⠀⠉⠻⢿⠀⠀⣀⠐⣦⠀⠀⠀⡌⠀⠀⠀⣀⡈⠁⠀⡀⠀⠉⠢⡆⠀⠀⠀⠀⠠⠈⠈
⣀⠀⠠⠖⠂⠀⠀⢡⣨⢇⡄⠀⠐⠀⠀⠆⠀⠄⢀⠠⡲⠀⡠⠒⢄⡀⠀⠀⠐⠀⠀⠃⠀⣬⠛⠀⠀⠀⣠⠘⠁⠀⠀⢀⣴⡗
⠀⠈⠀⠀⡀⠀⠈⠺⡱⠊⠀⠀⠄⠒⠀⠀⠀⠢⠡⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠍

⠀⠀⠀⠀⢀⣀⡠⣠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡻⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠄
⠀⠀⠀⢀⡄⣡⣾⡉⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠁⠀⢀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠲⢄⡀
⠀⠀⠀⠀⠀⠘⢿⠁⢀⣀⡀⠀⠛⢋⣤⣾⠓⠀⠀⢾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠰⣟⣧⢿⣀
⠀⠀⠼⢷⣤⣀⠈⢠⡿⢿⡿⠆⠀⠛⢉⣰⡜⣆⠀⢸⣿⣿⣿⢿⣺⣧⣄⠀⠀⠀⡰⣟⣯⠝⠚⠩⠖
⠀⠀⢀⠊⠛⠋⠀⢰⣿⣷⣾⡇⠀⠀⠈⠳⠛⠋⠀⠀⠋⣋⢀⡈⣿⣿⣿⡆⠀⠠⣟⣫⢽⣚⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠸⢟⡿⠿⡃⠀⠀⠀⠀⠀⣀⠀⠀⠐⢽⣶⡿⢛⡿⠁⠀⠀⠀⠈⠙⠚⣭⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣾⠋⠀⠀⠀⠀⢠⢄⣀⠀⡀⠀⠀⠀⠒
⣠⠆⠀⣀⣀⠀⠀⠈⠉⠉⠉⠁⠀⢰⣤⣀⣴⡄⠀⠀⠀⠀⠉⠛⠛⢿⡿⠀⠀⠨⣟⡾⠇⣰⡏⣰⡆
⠁⣠⣾⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠘⢙⣭⣥⣠⢾⢿⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡟⠀⣿⠃⢀⡀
⠀⣉⣠⣈⡿⢾⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣧⡘⠇⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⢀⣄⠀⠉⠀⠀⠁
⠈⠛⠉⠉⠀⣠⡀⠀⠀⠀⠀⢸⣿⣿⡿⠿⢿⣛⡇⠐⣢⡄⠀⠀⠀⠀⠴⠂⠀⠁⠀⠀⣾⣿⡧
⠀⠀⠀⠀⠸⠿⢷⣤⠀⠀⠀⠿⠙⣋⡲⣖⣋⣥⣴⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⣬⣯⣤⠀⠀⢀⡀
⠀⠀⠀⠀⠠⣶⠟⣫⡿⠀⠀⠀⠀⠘⢿⠿⠟⠛⠉⠁⠀⠀⠀⣐⡀⠀⠀⠀⠀⠻⣿⠧⠀⠉⠉⠀⠀⠿⠿
⠀⠀⣴⣶⣿⣧⣴⢿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣻⣶⣤
⠀⠀⢻⣿⣿⡋⠉⠘⠃⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⢛⡛⡛⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⠷
⠀⠀⠐⣷⣷⣿⣞⣾⣾⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⡀⠀⠈⠉⠛⠿⠂⠀⠂⠁
⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀
⠀⠀⠀⠘⠛⠛⠛⠛⠃⠠⣶⠸⣷⡀⠀⠤⠀⢰⣾⣶⣀⡄⠀⠀⠀⢠⢯⣆⡿⡻⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠠⣖⢻⡄⢹⣇⠙⠿⠆⠀⠀⢀⠈⠁⠉⠁⠀⠀⢰⣣⢏⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠆⠛⠀⠉⠀⠀⠀⠀⠀⢸⣷⣶⠠⠀⠀⠀⠀⠘⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⢿⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠐⠒⠲⠬⠉⠛⠛⠛⠛⠛⠋⡙⣛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡛⠀⠀⠀⠀⢂⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⢳⡘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠈⠓⢺⣤⣞⣁⣀⣀⣀⣀⣤⣤⠾⠋⠀⠀⠀⠀⠀⢣⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⠶⢶⣶⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢶⣶⣶⠿⠖⠚⠃⠀⠀⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⠀⠀⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠙⠻⣿⡦⠿⠦⠀⠀⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡇⢀⣀⢾⣿⣶⣷⡿⠶⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡤⠀⠉⠉⠉⢻⠏⣿⡟⠟⠓⢶⣬⡁⠀⢹⣷⣿⣏⠉⢀⣀⡴⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣿⣷⣄⢻⣿⣿⣷⣯⣥⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣴⠶⢶⡄⠘⣿⡿⣷⣿⡏⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⡇⠀⠉⣩⣭⣭⡟⠀⠀⠘⠽⠿⠛⠛⠁⠀⢹⡇⣼⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡅⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠀⠀⢀⡴⢋⠀⠀⠀⣀⣝⠢⠄⠀⠀⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⣀⣈⣴⣿⣿⣿⣿⣿⣿⣷⣤⡶⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠈⠛⠛⠿⠿⠭⠭⠭⠿⠛⠉⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣤⡀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⠈⠙⠻⣶⣶⣶⣶⣿⣿⣿⣿⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣯⡛⠿⣿⡀⠀⠀⠈⠙⠿⣿⡿⠿⠋⠁⢠⡿⣿⣿⣏⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠋⠉⣥⣈⠛⠷⢾⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣸⣿⡾⠟⢉⣼⠋⠙⠛⠿⢿⣶⣤⣄⣀
⠀⠀⠀⣰⣾⠿⠛⠋⠁⠀⠀⠀⠀⢰⣿⠛⣿⣿⣶⣮⣷⡁⢀⠀⠀⠀⢀⢈⡁⠀⣴⣯⣵⣶⣿⠟⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⢿⣦⡀
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠉⠻⣮⣙⠻⢷⡄⠈⠁⠀⠀⠉⢠⠾⢛⣫⣿⠿⡃⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠀⠀⠀⠀⠙⠻⢿⣿⣤⣤⣄⣠⣶⣷⣿⠟⠋⠁⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⣸⡿⠀⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠠⠀⢀⣭⣿⣿⣿⣿⣋⠉⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⠀⠘⣿⡆
⢀⣿⠃⠀⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⠇⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⢹⣿
⢸⡏⠀⠀⣿⠀⢰⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⣿⣿⢿⡟⠀⠀⢿⣿⣿⣿⣦⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠰⡆⢸⡇⠀⠀⣿
⠛⠁⠀⠀⢹⡀⡆⠀⠀⣿⣿⣿⣿⣿⣤⣀⣀⣰⣿⣿⣿⣷⠏⠀⠖⠀⠈⢿⣿⡿⢿⣧⣀⣀⣀⣠⣼⣿⣿⣿⣿⣇⠀⠀⠀⢿⣿⡇⠀⠀⠘
⠀⠀⠀⠀⠈⣷⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠤⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢒⣁⡤⣠⣦⣤⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠛⠛⢻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣤⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡲⣿⣿⡆⠀⠀⠀⣠⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⣀⢰⣿⡄⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠟⠁⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⡀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣇⠂⠀⠀⠀⠁⡴
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⡻⣿⣿⣷⣄⡀⠶⣾⡁
⠀⠀⠀⠀⠀⢀⠖⠛⢢⡀⠙⢮⡛⠿⠿⢿⣿⣿⡻⡢⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠹⡄⠀⠱⡄⠀⢰⣿⣿⣿⣇⢡⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⣄⠀⢹⣿⡿⣿⣿⣇⠀⠀⠀⠀⢾
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠈⠉⠈⢿⣷⠘⠿⠸⡄⠀⠀⡰⠛
⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢱⠀⡸⠁⠀⠘⢿⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⠁⠀⣤⠀⢺⡏⠁
⠀⠀⠀⡇⠀⠀⠀⢀⠀⢾⣣⡟⠀⠀⠀⢀⣀⣠⠀⠁⠀⠈⢁⡠⠂⠀⠙⠛⠛⠀⢀
⠀⠀⠀⢡⠀⠀⠀⡡⢖⢛⡷⠒⠘⠉⠉⠁⠀⢰⠀⠀⠀⠀⢸⣽⣇⠤
⠀⠀⠀⠘⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠖⠒⠉⢸⠐⠄⣾⣛⠛⠛⠛⣛⠛⠃⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠏⠀⢀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠲⠆⠶⢶⠒⢚⠋⠍⠅⠤⢨⡧⠷⠶⠶⠾⠶⠶⠞⠣⠶⠀⠀⠀⠀⠀⠠⠀⠼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣶⣦⠀⠀⠀⠀⣠⡞⣻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⠃⠀⠀⣰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣀⣀⣠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣩⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠠⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣩⣿⣿⡿⠟⠛⠉⢉⠄⠀⠐⢒⡲⣄⠉⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠛⡉⠀⠐⠒⢄⠀⢷⠐⠄⠀⠀⠁⠘⣦⠀⠹⣿⣿⣇⣀⣠⣤⡶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉⠀⣰⠚⠁⠀⠀⠀⠈⢧⠘⣧⠈⣟⠒⠀⢄⠘⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⠀⠤⠤⡀⠀⢷⠐⠀⠶⢤⡤⠄⣸⠀⠈⠛⠒⠒⢋⣩⠤⠤⠀⠀⠹⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⡴⠋⠀⠀⠈⠁⠹⡄⠘⢦⣁⠈⠵⠒⣋⣁⡀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⢰⡹⣿⣿⣿⣿⣿⣿⣿⣿⣟⠒⢶⡄
⠀⠀⠀⠀⠀⠀⣰⢟⡏⢰⠀⢻⡀⠻⣦⡀⠠⣀⠗⠀⠀⢁⡤⠀⠉⠀⠀⠁⠀⢰⠾⣷⣧⡀⠀⠀⠀⡦⠀⢳⠹⣿⣿⣿⠟⠻⠿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⢰⡟⢨⡇⢸⡀⠀⠙⢈⠥⠖⢋⣁⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣶⣿⣽⠇⢠⠴⠚⢁⠀⠈⢧⢹⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⢸⡇⠊⣽⡌⢧⠀⠀⠀⡤⠖⠉⠀⠀⣰⠿⣷⣦⠀⠀⢰⣭⡇⠀⠀⠈⠉⠀⢀⡤⠆⠈⠁⠎⢣⡈⣧⢻⣿⣿⣶⣬⣷
⠀⠀⠀⠀⠀⢸⡇⠘⢁⣷⡘⣧⠀⠈⠁⢂⡀⠀⠀⠳⣾⣿⡾⠁⠀⠀⣉⠤⠔⠂⠠⣄⠀⣏⠀⢀⠀⠀⠀⢈⡇⠘⣆⢿⣿⣿⣿⣃⣀⣤⡴⢶⣄
⠀⠀⠀⠀⠀⠈⣿⡀⠊⡸⢷⠘⣧⠀⠐⣆⢩⠛⢂⣠⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠀⢹⡄⠘⠆⠊⢛⣢⡤⠞⠁⠀⢹⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢁⠞⣧⠘⣧⠀⠈⠉⠉⠁⠤⠒⠨⡝⢦⠀⠹⡔⡶⠦⠄⢀⠀⠃⠀⠀⢉⡩⠴⠒⠮⠉⢦⠈⣧⢹⣿⣿⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢠⠟⣧⠘⣧⠀⠀⢺⠐⢀⣀⠀⠀⠀⢣⠀⠉⠓⠒⠉⣁⠤⣒⢦⡀⢻⡁⣄⡀⠀⠀⠈⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠜⢹⣇⠈⠃⠀⠈⢧⡘⠉⣉⠠⠴⢋⡀⠀⠀⡤⠀⠀⠀⠀⠀⢳⡀⠳⣜⣉⡠⠀⠀⢀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠃⡼⣧⠀⢶⡀⠀⠁⢀⡤⠀⠀⠤⠈⢳⠀⢳⠨⠗⠲⠀⠆⠸⠃⠀⠀⢀⣠⡴⠞⣹⣿⣿⠛⠉⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⢀⠞⣧⡀⢳⡀⠀⢺⠀⠰⠀⣀⠀⠀⡇⠀⠛⠛⠉⠀⢀⣀⣤⠴⠚⠉⣁⣴⣾⣿⣿⣿⣿⢶⣄⠀⠀⠀⠻⣿⣝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢀⠞⢷⡀⠳⣄⠈⢧⢀⣠⡬⠗⠋⠀⢀⣠⠤⠖⠚⠉⠁⣀⣤⣶⣿⣿⣿⣿⠉⠙⠻⣿⣷⣬⡇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠏⢻⣄⠙⠂⠀⣀⣠⡤⠴⠒⠋⠁⢀⣀⣤⣴⣾⣿⣿⣿⢿⣿⣿⣿⠟⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠃⡹⠛⠛⠉⠉⢀⣀⣤⡴⠶⠟⠛⠉⠉⠛⠻⠿⠛⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣥⡴⠶⠞⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡶⠒⠒⠒⠒⠒⢲⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⡒⠒⠒⠒⠒⠒⠚⠃⣠⠗⠒⠒⠒⠒⠒⠚⢧⡀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣠⡽⣛⡛⢦⡀⠀⠀⣞⢓⣒⣒⣒⠒⣒⣒⡒⣒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠹⡟⠁⠉⢻⠇⠀⠀⠉⠉⠉⠉⠉⣩⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡶⢶⣏⣀⠀⠀⠀⠀⠀⢯⠀⢸⠆⢸⡄⠀⠀⠒⠒⠲⡖⠒⢲⡖⠒⢶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⢘⣀⣤⣄⣘⠁⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡴⠀⠀⡶⢳⢄⣿⡀⠀⣀⣰⠯⠥⠤⠤⠭⢷⣀⡀⠀⠀⠀⡇⠀⣸⣇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣅⠀⠘⠳⢦⣠⣟⡏⢠⠈⠓⠶⠯⣿⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⡞⢹⣆⡏⢹⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡶⠴⣇⣹⠇⢺⠶⠶⠶⠶⢾⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠙⠚⠙⠓⠚⠈⠓⠋
⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠳⠶⠶⠶⠶⠞⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⣶⠶⠶⣶⣶⡶⠶⢶⡶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣿⠸⠭⠇⠸⠭⠭⠿⠀⠯⠟⢸⠀⠘⠚⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢛⣛⣛⣛⣛⣛⣛⣛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣈⣉⣉⣉⣉⣉⣉⣉⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⠐⢀⠠⣄⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠈⠁⠈⠃⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⢠⡄⢠⡄⠀⠀⠀⠀⢸⠀⡯⠀⠀⠀⢀⣤⣤⡤⢤⡄⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣤⣤⣤⣤⣬⡇⠀⠀⠀⢸⣇⡇⠀⠀⠀⢿⣿⣯⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢫⡄⠀⢼⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⣾⣿⣿⡟⢳⡆⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⢸⣇⡀⣸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣇⣸⣇⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠿⠃⠘⠇⠀⠻⢧⡀⠀⠀⠀⣿⠀⠀⠀⢠⣿⡿⠛⠇⠘⠃⠀⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⢸⣿⠀⠀⣰⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⢀⠀⣄⣀⠀⠀⡔⢹⠀⢸⣿⠀⢰⣹⣷⡄⠀⠀⠀⣴⡄⠀⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠱⣼⣷⣿⣿⣴⣧⡄⠈⠄⠸⣿⠀⢨⣼⠛⠁⠀⠀⠀⡀⢽⠈⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠊⢛⣛⣿⣿⣭⢃⣿⣷⡀⢠⡁⠈⣿⠀⠘⠏⣶⢶⠀⠀⢀⡴⠈⠢⣤⠐⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡽⢿⣿⡿⣿⣿⢯⠇⠀⠀⡅⠀⣿⡂⠀⣴⣿⣤⣤⡖⡉⠀⠀⠀⠡⠉⢠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⠀⠈⠘⠃⢊⢹⠰⠀⠀⠀⠇⠀⣿⡇⠀⡼⣿⣾⣾⡿⣮⣴⣦⡤⣀⣵⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣶⣤⡄⢀⡀⠈⡈⠶⠁⠀⢀⣄⣤⣿⣦⣤⣄⣿⣿⣻⣶⢟⢥⣰⡌⢸⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡟⠻⣿⣿⣿⣿⠋⢻⣿⢉⣿⡿⠉⠀⣿⠉⠙⡟⣿⣿⣿⣻⡛⠛⡿⠙⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣷⢾⣟⢻⣿⣿⣟⣽⣴⣿⣿⠁⠄⠀⣿⠀⠀⡆⣿⣿⡾⠿⣶⠡⡀⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⣿⣿⣤⢿⢿⣿⣿⣿⣸⣿⣿⡿⠀⡆⠀⣿⠀⠀⣹⡿⠃⠀⠀⠘⠇⢠⡙⣿⣿⠟⢿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣹⡟⡇⠈⠑⢿⣿⣿⣿⣿⣿⠋⠀⡇⠀⣿⠀⢨⢿⡇⠀⠀⠀⠀⠀⠜⠀⠀⠉⠀⢸⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣷⡅⠀⠐⠠⡙⢿⣿⠋⠈⠁⠀⣦⣤⣿⣦⣴⣸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⣦⣤⣤⣼⣾⣷⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠚⡿⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠺⠋

⠀⠢⣀
⣀⠀⠀⠑⠒⠄
⣿⣶⣦⣤⠆⠀⠀⢦⣀
⠀⠀⠈⠹⣮⠀⠀⡀⢈⠑⠄
⠀⠀⠀⠐⣿⣆⠀⢡⠀⢢⡄⡀⡄
⠀⠀⠀⠀⣾⣿⠀⡞⠠⣿⠀⣿⠸⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠻⣯⣼⣇⣰⣿⡀⢿⡄⣇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡛⢄⣇⠘⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⡀⠇⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢅⠀⠀⠀⠉⠺⡀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠚⠀⠀⠄⠀⡄⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣦⣄⣿⣦⣴⣶⣶⣶⣷⣤⣤⣴⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠿⠤⠽⠿⠿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠒⠚⠛⠉⠉
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡐⠀⠀⡀⢠⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⢻⣤⠁⢾⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠳⠒⠒⠒⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠒⠲⢤⡓⢿⠆⠹⣼⢗⡤⠤⢶⡤⠄⠐⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠙⣦⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⣈⣙⡶⠶⠀⢶⡆⢼⣛⣷⣩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢂⣠⠶⣦⠞⡉⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⣒⡿⡧⠀⡄⡴⣯⢍⡀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠟⠄⢂⡡⠂⢀⠻⢈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠴⠛⢚⠩⠈⡇⢸⡇⢰⢎⡣⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⡟⠀⠀⡸⢠⡼⠀⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠃⠀⠁⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣿⢿⡃⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠑⠄⣻⠀⠑⠦⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠶⠨⡅⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠗⠀⠀⠀⠀⠀⡶⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠄⣀⠤⠐⠊⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⢹⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⣁⣤⠴⠂⠊⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⠀⠀⠀⢀⡄⠢⡄⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢶⠚⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠁⠀⠀⠀⠀⠀⠋⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢺⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡲⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⢐⣪⣦⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣯⠀⢀⣀⡤⠶⠒⠋⠁⠀⠀⠀⠤⠖⠒⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣏⡹⠶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡘⢇⡛⢉⠀⠀⢀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢃⡼⠒⠉⠀⠀⠀⠀⠀⠰⠀⠤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣤⣀⣀⣀⣤⣴⡖⠚⠉⠉⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣏⠀⠀⠉⠉⢢⡉⠉⠓⣶⢭⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣹⡛⠋⠉⠩⠽⢿⣄⠀⠙⠀⠈⢻⣏⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠉⠀⢲⣿⣷⡶⠉⠀⠀⠀⠘⢧⡀⠀⠀⠈⠀⠱⡦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠐⠁⠂⠔⠋⠉⠉⠙⠀⠀⠈⠀⠀⠈⠻⣶⣄⠀⠀⠀⠙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠰⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠲⣤⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢸⠸⡇⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡞⠫⣝⠛⡿⠧⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⣂⡤⠞⠛⢏⡀⠘⢒⣲⡶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠎⡰⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠈⠑⣛⣽⢟⠀⢶⢹⢡⡑⠠⣹⣭⡓⢦⣄⡀⠀⠀⠀⠔⢁⠔⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢈⡶⣻⢇⡣⠛⡱⢸⡜⢎⢧⠳⢄⢢⣩⢟⡾⣿⠀⠀⠀⠔⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣴⡛⡱⢺⠷⠇⠃⢘⣨⠒⠼⢛⣵⠟⢁⣿⣤⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠉⠛⠓⠛⠃⠛⠚⠚⣁⡤⠖⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠈⠉⠉⠁⢀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⢿⡿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⢀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⠳⡖⢋⠕⢊⣾⠟⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⢀⣿⠃⠀⠉⠉⢠⣿⠇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢧⠀⠀⠀⣾⣿⣀⣀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⣧⣀⣼⡇⢀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣄⠀⣀⣠⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣟⢶⡀⠀⠀⠀⡼⠁⠀⣰⠋⠉⠁⢸⡇⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⣽⢫⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⠿⠀⢹⡆⠀⡰⠁⢀⡜⠁⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣇⢀⣠⠗⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣦⡿⣄⢀⣀⡼⢀⠔⠁⠀⡎⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⢷⡿⡆⣤⠀⢁⠀⡀⢰⠃⠀⠀⠀⠀⠀⣰⠃⣬⣿⡿⠟⠋⠉⠀⠀⣀⣤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⡿⣔⣿⡻⠛⢡⡀⠀⠀⠀⠀⡴⢁⣾⣿⣿⡄⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠀⠈⠁⠙⠉⠳⠤⠏⢹⣄⣀⢀⣜⡡⢟⣫⣥⣾⣤⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠀⢀⣺⣷⣤⣦⡀⠀⢀⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⣿⣏⠀⠉⣷⢀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠓⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣙⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠉⠉⠀⠀⠁⠀⠒⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⣀⣀⣠⡶⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣉⣀⣀⣀⣠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠉⠉⠝⢻⡇⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠙⠛⢶⣄⠀⠀⠈⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡇⠀⠀⠠⠙⠿⠷⢶⣄⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠶⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠾⠶⠶⣶⣶⡶⠶⠶⠟⠛⠉⠉⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠁⢰⣿
⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣖⣤⣤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠦⠀⠀⠀⡄⠀⠀⢀⡎⠀⠁⢀⡖⠀⠀⠀⠀⢸⠃
⠀⠀⢸⣶⣤⡀⠀⠀⠀⠀⠀⠉⠛⠒⠳⠶⢢⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠄⠀⠉⠀⠀⠀⣀⡤⠖⠉⠀⢀⣤⢸⡂⠀⠀⡘⠋⠀⠀⠀⠐⣿
⠀⢀⠉⠉⠛⠛⠽⠆⣶⣶⠀⢸⡗⠒⠛⡿⠟⠁⠀⠀⠀⢀⣀⣀⠈⠉⠑⠂⠀⠀⠀⢀⣀⠦⠚⠉⠀⠀⠀⠀⢰⠃⠉⠀⡻⢣⡞⠁⠀⠀⠀⠀⠜⠁
⠀⠈⠻⢿⣶⣤⡄⢀⣀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠉⠀⠀⣀⡠⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⢁⡌⠁⠀⠀⠀⢶⣾
⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠁⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⢀⠠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⣇⡄⣴⠉⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣀⣤⡤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣿⣤⠍⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠉⠉⠓⠶⠤⣀⠀⠀⠀⠀⠀⠀⡾⣿⡁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠛⠒⠒⠚⠛⠋
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡄⡀⠀⠀⠀⠸⣿⣿⣿⢷⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠤⠦⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠉⠛⠿⣿⣮⣭⡙⠳⢦⣤⠔⠒⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠠⡶⠋⠉⠋⠉⠓⣆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣷⣀⣠⡦⠔⠋⠉⠀⠀⠀⠀⠠⠎⠈⠐⢦⣄⣀⣀⠀⠀⣸⠷⣦⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠛⠙⢸⡟⢛⣏⠀⠀⠘⡇
⠀⠀⠀⠑⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⣠⠀⠀⠀⠀⠈⡿⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠈⠙⠿⠿⢿⣏⣀⣚⢛⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⡀⠑⢤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠲⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠔⠒⣒⠀⠈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡀⣀⡠⢄⡠⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣚⡡⠴⠖⢚⣉⠩⢀⠔⢀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⡉⠤⠄⠄⢠⣴⣄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣡⠶⠋⣁⡴⣖⣿⣍⡤⢊⠁⡠⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⡄⠐⠈⠀⢠⣶⣿⣦⠻⢿⠿⠤⠮⠱
⠀⠀⠀⠀⠀⢠⡶⠋⣠⠖⣫⠄⣠⢞⡭⢞⣩⠞⣋⠔⣡⠞⡕⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⠀⠀⣀⣤⣤⣄⠘⠿⠿⢋⠔⢰⣾⣿⣿⣶⠃
⠀⠀⠀⢀⡴⢋⡴⢞⡥⢊⣡⠞⡡⣊⡴⠋⣡⠞⣡⡾⣡⢞⡌⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⢔⣵⣶⣦⡠⣿⣿⣿⣿⠀⠀⠀⢁⣙⠸⣿⣿⣿⡟⢠
⠀⢀⡴⠋⡤⢊⡴⠋⣴⠟⠁⣠⡾⠋⣠⠞⢁⣾⢋⡴⢫⡞⡤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠠⠉⢿⣿⣿⡿⣡⣌⠛⠛⢡⡮⠇⢸⣿⣿⣧⠀⠉⠔⣰⠋
⢀⡞⣡⢎⣴⠋⢠⡾⢁⣤⡾⠋⣠⠞⢁⣴⠟⣱⠟⣵⠏⡼⣡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠄⣤⣶⣾⣍⢉⣾⣿⣿⣧⠜⠙⠆⠀⣈⡛⠋⢄⢉⠀⡼⠃
⢸⠱⢣⡟⠁⣴⠏⣰⡿⠋⣠⠞⢁⣴⠟⣡⡾⢣⡾⢃⡞⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡊⠀⢿⣿⡿⠟⠘⠿⠟⡛⠕⠀⠁⣾⣿⣿⣿⡆⠄⢡⡞⠁
⡏⢠⡿⢁⣼⠏⡼⠋⣠⠞⣡⡴⠛⣡⡾⢋⡴⡫⠐⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⢡⣾⣿⣶⠀⠀⢠⠼⠉⣴⣿⣷⡄⠻⢿⠿⠫⢂⡴⠋
⡇⢸⠇⣸⡏⡾⣡⠞⣡⠞⣩⣴⠟⠁⢀⣥⠞⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⢿⣿⠿⠋⠖⣙⣭⣄⡀⠻⢿⠟⠅⢠⠔⢋⡴⠋
⢷⢸⢀⡿⢈⡴⠃⢚⣥⡾⠋⢀⣤⡾⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⢢⡔⠊⠺⠰⣿⣿⣿⠧⠐⣄⠠⠂⣡⠶⠋
⢸⡄⠈⠃⠞⣠⡾⠛⣡⡴⠞⠋⣁⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠧⠄⠀⠤⠯⠉⠑⢊⣁⣤⠴⠛⠁
⠀⠱⣴⣄⡈⠁⠠⠬⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠖⠁⠙⠒⠲⠶⠶⠶⠒⠛⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒⡀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠁⠀⠀⠐⠉⠀⠘⠀⡇⢠⠀⠒⡂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠔⠠⠀⠀⠀⠀⠀⢀⡴⠂⡀⠀⡇⠀⠄⠁⠀⠁⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡁⠩⠀⠀⠀⠀⠀⣀⠃⡀⠁⠀⠀⢀⠀⠠⠇⠀⠰⠀⠡⡀⠠⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠢⡀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠄⠀⠂⠌⠰⠂⠠⠒⠨⠄⡀⢠⠐⢀⡌⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⠀⢌⠀⠀⠀⢀⣀⠈⢌⢀⠑⠈⠐⠈⠀⠀⠅⠈⠀⠂⠁⡀⢁⡔⠁⠀⠂⠄⢂⡀⠐⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡀⠀⠠⠁⠆⠀⠀⣵⠀⠑⢅⠣⠀⠐⢀⡁⠀⠰⢅⠐⠅⠈⠀⡐⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠐⠐⠄⠀⡆⠆⢀⠀⠀⢃⠀⠀⠁⠀⠡⠎⠀⡠⢀⠠⠂⢊⠄⠠⠁⠀⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠄⢱⠐⡄⠈⡔⢠⠘⠀⠀⠀⠀⢀⠇⠀⠀⡁⠐⠀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⡀⠰⠈⠁⣇⢀⠀⠀⠀⠀⠜⠀⠐⢈⡤⠐⠎⢀⣀⢀⣀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⠀⢠⠠⠘⡘⠀⠀⠀⠀⠁⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠡⡑⠁⠀⠀⠀⠀⡀⠄⣀⠀⠀⠐⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠀⠀⢀⡠⠂⡁⠂⠁⠀⠀⠠⠐⠂⠄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣰⠛⠻⡆
⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠉⢻⡏⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣠⣏⡸⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⠦⠶⠛⢻⠶⣤⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠓⠲⢤⣤⠶⠛⠉⠀⢹⡆
⠀⢀⡾⣧⣀⣈⣇⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡿⢀⣴⣋⣙⣦⣭⣹⡟⢷⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠘⣇
⣴⣯⣤⡟⠁⠈⠙⢷⣄⠀⠀⠀⠈⠉⠛⠲⢶⢺⠀⠀⠀⠀⣾⣼⣶⣫⡟⠉⠉⢳⣤⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢰⡏⠀⠀⠀⠙⣧⠀⠈⠛⢦⡀
⠀⠀⠘⣇⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣤⠤⠚⠾⠅⠀⠀⠀⢹⡀⠈⠉⣉⠛⠋⡿⠃⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⠟⠀⢸⡆⠀⠀⠀⠀⡿⠀⠀⠀⠈⣷
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠉⠉⠙⢲⡴⣷⠀⠀⠀⠀⠀⢳⡀⠀⠉⢀⡼⠁⠈⠁⠀⠀⠀⢷⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠙⠶⠤⠶⠚⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣤⣤⣤⣤⠴⠿⣄⡇⠀⠀⠀⠀⠀⠀⠳⣄⢀⡞⠁⠀⠀⠘⠃⠀⠄⠘⡆⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠓⠒⠚⠋⠁⠀⠀⠀⠈⠉⠛⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠘⠃⠀⠐⠂⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠁⠀⡀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⡛⠽⠟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡄⠀⠈⠛⠶⠶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣧⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⢳⣤⡏⢻⣆⣠⣯⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠘⢧⣄⠀⠀⠀⣿⢠⡆⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⣀⡈⢳⡏⠉⣻⣾⠙⣧⠈⢿⣹⠆
⠀⢀⡶⠒⠒⠒⠒⠚⠛⠛⠛⢛⣛⣛⣻⣆⠀⠀⠀⠀⠀⠀⣟⣉⣉⣉⣉⣹⠂⠀⠀⢿⠘⡇⣠⡤⣤⣴⣿⠟⠁⠀⠀⢸⣏⣉⡙⠳⢴⡃⣿⢰⡿⣧⠀⠻⡄
⠀⠘⠹⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠀⠀⠀⠀⠀⡴⠟⠉⠉⠉⠉⠉⠲⣄⠀⠸⠀⠀⣿⡀⠁⢹⡇⠀⠀⠀⠀⠀⢀⣀⡿⢦⠀⠙⢧⣸⠃⣼⣆⣀⣽⡦
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠆⣰⠛⠛⠻⣿⢦⡼⠃⠀⠀⠀⠀⠀⠈⢿⠙⠛⠓⠒⠚⠛⠚⠛⠉⠁⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣄⣹⣦⡀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡏⠀⠉⢻⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⠶⠛⠶⣿⣶⣦⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡀⠀⢸⡇
⠀⠀⠀⠀⠈⣇⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⡀⢁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⣿⣿⠷⠮⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⢀⣾⡭⠽⠧⠬⣍⡉⣿⣋⣽⣇⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⡴⠶⠶⢶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⡀⠀⠀⣤⠀⠈⠙⠳⢦⡀
⠀⠀⣠⠞⠁⡀⠘⠀⡀⠈⠙⢦⡀⣽⡛⢯⡉⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⠰⢦⠀⣤⠈⠁⣠⠌⠛⢦⡀⠀⠀⠀⠀⠀⠀⢸⣧⣌⠉⠛⠻⢧⣄⣀⡄⠀⠀⠙⢦⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠉⠀⢠⠀⠙⣦⠙⣿⢛⣷⠘⡆⠀⠀⠀⠀⢾⠀⠞⠀⠀⠀⠈⣠⠄⠀⠴⠖⠈⢷⠀⠀⠀⠀⠀⠀⢸⣇⠊⣷⠀⠀⠦⠈⠙⠳⣄⣠⠆⠀⠹
⠀⠀⠹⣄⠉⠀⠠⠄⠀⠀⠀⠀⢠⠌⣧⠉⢻⣽⡆⡇⠀⠀⠀⠀⠈⡿⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⣾⠇⠀⠀⠀⠀⠀⢸⣏⡴⠏⢠⠖⣛⣉⣽⡳⣬⣻⣦⡤⠶
⠀⠀⠀⠈⠳⢦⣀⠀⠀⠰⠆⠀⠀⠀⠘⡇⢸⢉⣷⠇⠀⠀⠀⠀⢰⡟⠖⠛⠛⠳⢶⠶⠛⠉⠳⠶⠞⣩⡷⠀⠀⠀⠀⠀⢸⡁⠀⠀⣟⢾⣙⣧⣷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠐⠀⠀⢻⠟⢫⠏⠀⠀⠀⠀⠀⠀⢻⡗⠛⠛⠛⠛⠻⢿⣝⡛⠛⠋⣹⠃⠀⠀⠀⠀⠀⣾⠛⢳⣀⡽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣸⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣀⣨⡿⢦⠾⠃⠀⠀⠀⠀⠀⠀⠿⠖⠋⠁
⣶⣶⣶⠶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣷⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠀⠀⠀⠀⠀⠸⢿⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⢺⣿⡇⠀⠀⠀⠀⢀⠀⣒⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⢶⢦⠄⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⢯⣿⡇⠀⢀⡀⠀⢰⡄⣿⡷⠖⣘
⠀⠀⢀⣤⠀⠀⠀⠀⠈⡢⣀⡀⢊⠄⠒⠰⠾⠿⢿⡿⠿⠶⠀⠐⠀⡀⠀⠀⢾⣿⡇⠀⢄⠠⡄⠄⣿⡥⠶⠄
⠀⣰⠏⠉⡇⠀⢴⡞⠟⠟⡇⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠿⢧⠀⣀⠁⢸⡞⠁⡔⠁⢀
⠸⠯⠀⠤⡀⠀⠀⣑⣦⢜⠀⢰⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⣈⠐⠌⢳⣏⡡⣍⢉⡃⣴⠂⣠⠄
⠀⣠⠾⣆⠑⣀⡾⣋⡩⠈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠸⢏⠑⢶⠄⣶⣠⡾⢿⣿⣀
⢸⠏⠁⢸⣤⡿⠳⣄⠐⢄⠘⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠣⡄⠈⠛⠀⠊
⠀⠀⠀⠈⠛⠖⡚⠂⠉⢲⡷⠟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠐
⠀⠀⠀⠀⠀⡴⠿⢷⣤⣤⡶⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⢠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣵⠤⠠⠀⠚⠐⠃⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⡿⢿⠙⠶⠖⠀⠀⠈⢛⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣌⣀⠠⠀⠀⠀⠀⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⣄⠤⠀⠠⠀⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⠆⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⢷⣶⠾⠛⣉⡤⠞⣡⣿⡿⠛⠷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢡⣾⡟⠁⠀⠿⢤⡼⠥⢤⣴⣿⡿⠁⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⣾⣿⠁⠄⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⡀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠈⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠈⢷⡀⠀⠀⠀⣸⡿⠛⢿⣶⣤⣤⣤⣤⣤⣼⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣷⡀⠀⣠⣿⡇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣀⣀⣀⣀⣿
⢀⠀⠀⠀⠀⠀⠀⣾⣆⢠⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠞⠿⠷⠛⠉⣿⠃⠀⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣷⣴⠞⠋⠉⠉⠉⠙⣿
⣼⡄⠀⠀⠀⠀⠀⢻⡜⠛⣧⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣻⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⡿⠹⡆⠀⠀⠀⠀⠀⢿
⣿⣧⠀⠀⠀⠀⢠⡜⡇⠀⠸⣷⡇⠀⠀⠀⡼⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⡧⠚⠁⠀⠀⠀⠀⠀⣸
⣿⣿⣆⠀⢰⡄⠸⠀⠀⠀⠀⣸⠇⠀⢀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡿
⢹⡟⢿⣆⣺⢷⡜⢦⣀⢀⣠⠟⢀⡴⠋⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠈⣧⣈⣿⣯⣠⣿⡆⠈⢉⠁⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠛⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠘⣿⣇⠙⢿⠙⢿⣀⣿⡀⢀⠘⡴⢀⣴⢆⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⣶⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡏
⠀⠀⠈⣿⡄⠀⠀⠘⣿⡿⣇⢸⣷⡙⡟⠼⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢠⣾⣿⣿⣿⣡⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀
⠀⣧⠀⢹⣷⡀⠀⠀⢻⡇⠹⣾⡏⢿⡆⠀⠀⣼⠿⣆⠀⠀⠀⠀⠀⢀⡾⠁⣤⣿⡿⠿⠛⢉⡝⠀⡔⣠⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⢳⣾⣿⡇⠀⠀⠀⠀⡀⠈⠀⠀⠙⠳⠛⠋⠀⢹⣧⣀⣀⡀⢠⣞⣡⠶⠋⣁⣤⣶⣿⣿⣤⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⡀⠈⠋⠋⠀⠀⠀⢦⣿⣦⡀⢠⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠘⣧⠀⠀⠀⠀⠀⣷⣾⣿⡯⠻⢿⣿⣦⠀⠀⠀⢸⣿⣿⡿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠘⣧⡀⠀⠀⠀⢻⣿⣿⡁⠀⠀⠈⣿⣿⣀⣤⢿⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⢷⣤⣤⣀⠹⠿⠿⠶⣖⣒⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⢻⡉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣾⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣶⣯⡙⢧⡐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢘⣏⡛⣿⣶⡶⣄
⠀⠀⠀⠐⠛⠋⢻⣿⣎⡷⢸⡇⠀⠀⠀⠀⠀⡀⢀⡞⣼⣿⡄⠈⠧⣍⣛⣷⣷⠹⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠏⡿⣷⣷⠂⠀⠀⢰⢏⢁⣎⣾⠏⣾⠇⠀⠘⠦⣭⣍⣿⣧⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⡿⠀⠀⢠⣿⡄⢸⣿⡯⠜⡹⠀⠀⠀⠈⢶⡤⢬⣽⣇⣼⠁⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⣟⣷⣼⣷⣶⡾⠁⠀⠀⠀⠀⠀⠙⠓⠒⠛⠣⠀⠀⢀⢴⢣⡏⣰
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣦⣴⣤⠀⠀⠈⢿⣟⡢⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⣾⣾⡟⣹
⠀⠀⠀⠀⠀⠀⡞⣿⣿⣻⣄⣙⠍⠷⠄⠀⠀⠀⠀⢀⣠⣠⣾⣶⡃⠀⠀⠀⣀⣀⣀⠀⣀⠸⣾⢁⢻⠶⡻
⠀⠀⠀⠀⠀⣠⠛⣩⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⡻⢟⣤⣤⣄⠀⠀⢓⠯⢿⣷⣮⣷⣿⣾⣾⡿⠶⢤⡀
⠀⠀⠀⠀⠠⣣⣴⣯⡹⠇⠀⡀⠤⣤⡀⠀⠀⠀⠘⠟⠂⠀⢠⣨⢸⢹⣧⠀⠀⠑⠒⠲⣿⡝⡹⣿⢏⠻⠳⣶⠾⡀
⠀⠀⠀⠀⡼⣿⢻⣆⠃⠀⠈⠳⣏⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠘⠈⢸⠀⠀⠀⠀⠀⣹⣷⣿⣿⣦⣕⠀⣾⡷⠦
⠀⠀⠀⢸⠇⣸⠈⠎⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⣇⣿⣿⡌⣧⠟
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣬⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠚⠻⡇
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⣀⣸⣻⠷⠋⠀⠀⢀⠠⠶⡒⢀⣴⣮⠤⠾⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢀⠔⠁⠀⠀⠁
⠀⠀⠀⠀⢪⡿⠛⠁⢀⡀⣁⢿⣼⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠧⣝⣋⣀⣀⡴⡂⢀⠀⠀⠀⠄
⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⣀⣨⣶⣾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢼⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⣦⡤⠀⣀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⠀⠁⠀⠀⠤⡀⢀⣀⠤⠞⠉⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⣀⣀⣀⣠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠀⠤⠁⠂⠠
⠀⠀⠀⠀⠘⠒⠒⠀⠀⠀⠀⠻⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠉⢀⠘⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡴⢊⡴⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣴⠇
⠀⠀⠀⢀⠰⠛⠟⠙⠃⢀⣠⠴⢤⣶⡆⠀⣴⣂⣄⠀⠀⠀⠀⠀⠲⢤⣤⡦⠞⠁⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳
⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠰⣦⣌⠈⠩⠝⠡⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠿⠂⠀⠀⠀⠀⢣⠀⠀⢶⣶⣧⣀⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣠⣉⡵⠶⠒⠉⠉⠉⠀⠀⣠⡶⠖⣒⡒⠉⣉⡩⠴⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠂⠀⠈⠉⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡵⠿⠏⠁⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣂⠤⣄⣀⠜⠁⠀⠀⠈⠂⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⡃⠀⠀⠀⣉⣉⣉⣀⠀⠀⠀⡴⣿⡶⢶⣤⣀
⠀⠀⠀⠀⠀⡀⡦⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢉⢋⡙⡉⠀⠀⠘⣿⣿⣇⣀⡉⠛⠃
⠀⠀⠀⠀⠉⣸⣷⡀⠀⠀⠀⢘⣫⣝⣛⡅⠀⠀⠀⢸⣹⣏⡇⠀⠀⠀⠨⠛⠋⠉⠁⠀⠀⢠
⠀⠀⠀⠈⠃⠒⠛⢇⠀⠀⠀⢾⣿⣿⣿⡧⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⡜⠉⠉⠁
⠀⠀⠀⢠⣤⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⢈⠀⠀⠀⠀⣀⣄⣣⣀⣤⣀
⠀⠀⠀⠀⠋⣛⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⣼⠣⠛⠀⠃
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣭⠯⡴⢦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⣾⣿⣿⡷⠚⡟⡟⢦⠀⠀⠀⠀⠀⠀⠀⣹⣏⣿⣹⣏⡷
⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡿⢿⣿⣿⣷⠀⠓⠀⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠸⠿⢿⣤⡀⠀⠀⠀⠀⠁⢉⣙⢦⢤⠜⠯⠏⠀⣂⡈⠀⠀⠀⠀⠠⠆⠀⠐⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣴⠿⢫⣿⠀⠀⠀⠀⠀⠈⣿⣿⣾⣶⣶⣷⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢾⣿⣷
⠀⠀⠀⣤⣴⣿⣧⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⡿⢿⣬⣟⡁
⠀⠀⠀⢿⣿⣿⠏⠙⠘⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣦⡄⣀⠀⠀⣤⣀⣄⣤⠀⠈⠛⠓⠀⠘⠙⠃
⠀⠀⠀⠨⣯⣯⣷⡶⡶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠀⠀⡏⣅⢹⢸⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣿⡿⡿⣪⡮⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣯⣭⡍⠀⠀⡇⣿⢸⢸⠀⠀⠀⣿⣷⣶⣦
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠀⠘⠛⢳⡶⠒⠚⠀⠀⠀⠀⠈⠍⠀⠀⠀⠓⢻⠞⠚⠀⠀⠀⠀⠈⠁⠈

⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀
⠀⠀⠀⢙⣽⣿⡗⠀⠀⠀⣠⠀⠠⢤⡼⡇⠀⠀⠀⠀⠀⠀⠀⡗⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⡷⣦⠄
⠀⠀⠀⠈⢻⡿⠀⠀⠀⠘⠁⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⣰⣧⣿⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢟⣃⡌⣵⣽⣶⣿⠃
⠀⠀⠀⢸⣿⡇⠀⠀⢰⣷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡾⢯⡏⠸⡄⢻⡀⣿⠇⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⠟⢁⣀⠔
⠀⠀⠀⠘⣿⢧⠀⠀⢸⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠃⠀⠿⢀⣧⡇⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣤⣶⣶⣯⡇
⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⣤⡄⠀⠈⠤⢦⣅⣠⡀⡇⠀⠀⠐⣿⠀⠀⠀⢠⣿⣿⣿⡽⣿⠃
⠀⠀⠀⠀⢻⡇⠀⠀⠐⣿⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠀⢰⣿⣿⣯⣽⣿⡿
⠀⠀⠀⠀⢹⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⠏
⠀⠀⠀⠀⠺⠹⠀⠀⢰⡇⠀⠀⠀⠀⠀⡆⠀⠀⠑⠾⠂⠀⠀⠀⠀⢸⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠁
⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⣷⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⡿⡏⠀⠀⠀⣶⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣶⣇⡈⠋⠉⠉⠁⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⢸⡇⡇⠀⠀⠀⣿⠀⠀⠀⣿⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⣼⣧⣼⡁⢻⡇⠀⠀⠀⠀⠀⢴⣾⠇⠀⠙⠉
⠀⠀⠀⠘⢰⡇⠀⢠⡤⣿⡄⠀⠀⢿⣿⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢸⡇⣾⣅⠛⠛⠋⠟⠁⠀⠀⠐⣷⣿⣁⢀⣦⡀
⠀⠀⠀⠀⢸⠁⠀⠘⠃⠛⠂⠀⠀⠉⢹⡇⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠘⠃⠈⠙⠳⠀⠀⠀⠀⠀⠀⠘⠿⠋⠁⣾⣿⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⠀⠒⠀⡀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠰⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢻⠆⠀⠀⣀⠃⢀⠆⠀⠀⢰⡟⠀⠀⠈⠃⣿⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠠⣴⠇
⠀⠀⠀⠀⠈⠳⣄⡄⢻⣧⢚⣶⢀⣠⠞⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠠⠀⢄⣀⠀⢀⣠⣀⣤⠼⠷⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣤⣤⠀⠀⠈⣽⣿⣿⣿⣛⠛⠃⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⡀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡛
⠀⠀⠉⠉⠉⠉⠉⢢⢆⢤⡕⠌⠓⠒⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠒⠀⠀⠀⠀⠀⠀⠀⠐⢰⣄⣀⣰⠆⠴⠁
⠀⠀⠀⠀⠀⣀⣠⠋⠘⠸⡏⠂⢠⡔⠀⠀⠀⣴⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠉
⠀⠀⠀⠀⠼⠟⠁⠀⠰⠔⠃⠀⠈⠀⠀⠀⣀⣍⠁⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠛⣿⣰⣿⡏⠀⠀⣤⡀⠀⠀⠀⠚⠋⠁⠀⣤⣤⣀⠀⠙⡾⠂⠀⠘⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣾⡇⠀⠀⠀⠀⣠⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⣠⡄⠀⠀⠈⠀⣠⡄⠀⠀⠿⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⢲⠟⠃⠀⠀⡀⠀⠿⠃⠀⠀⠀⠀⣤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⣪⠀⠀⠀⢼⡇⠀⠀⠀⣠⡀⠀⠸⠟⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠠⡵⡱⠀⠀⣀⢄⠀⢀⡄⠀⠿⠃⡠⡀⢰⡄⠀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⠀⠉⠀⠈⠤⠂⡄⠀⠀⡰⠁⢰⠘⠈⠒⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠠⠤⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠈⠚⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢩⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⠤⠬⢆⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡑⢄⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠄⠠⠤⠤⠤⠤⠄⠄⢰⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⣾⠲⠷⣦⡉⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⣿⣄⠀⠀⠙⣦⠀⠀⢹⠋⠉⢱⠒⠒⠀⠀⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠉⠷⢄⡠⠇⠀⠀⠸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⢈⡳⢦⣀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠊⠁⠀⠀⠙⠻⢶⣦⣤⡶⠼⣻⣀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⡤⠔⠀⠉⠉⠉⠉⠉⠁⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣀⢀⣀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢸⠀⠀⠀⠀⠠⣄⡀⠀⠀⠤⠄⠒⣩⠏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⠀⠀⠀⠀⠀⢀⠶⢆⡀⣀⠰⠚⠁⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠖⠓⠀⠤⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⢌⡦⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠻⣦⠈⠡⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢶⠶⠶⠶⠆⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠔⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢋⡆⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠏⠦⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠰⠀⠳⣄⠀⠀⠀⣀⢠⠔⠒⡝⠁⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠈⠷⠐⠊⠁⠀⢀⡠⠟⠲⠦⣤⣀⡀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⠱⣤⡜⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⡠⠛⠻⠉⠉⠉⠁⠀⠀⠈⠒⢄⡀⠀⢀⠞⠀⠈⠙⠃⠤⠤⠌⠿⠿⣧⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⠓⠤⣀⣀⣀⣀⣀⣀⣀⡀⠀⡨⠛⢷⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢈⣳⣤⡚⢠⡄⠀⠀⠀⠀⠀⣤⣤⣤⡀⣀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⣒⣖⡚⠛⣻⣿⡛⢻⣿⣖⠒⠒⠖⠲⠚⠷⠲⣶⣶⡶⠶⣶⣶⡿⢵⣶⣤⢶⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⠋⠉⠃⠈⢉⠉⠃⢹⣏⣹⠃⠀⠀⠀⠀⠀⠰⣿⣽⣿⢺⣿⣹⡿⢺⣯⣽⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡦⠤⠴⣦⠤⠤⠤⣤⡤⣥⣤⣤⣤⠤⠤⣤⣤⣤⣬⢭⣥⣤⣭⣭⣥⣬⣭⣉⣠⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⢠⣀⣀⣠⣘⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣻⣻⣯⢸⡇
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠐⠒⠚⢦⠔⡄⠤⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠰⣆⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠍⢹⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⢘⣄⠀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠥⠤⠬⠤⠤⠤⠤⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠓⢤⣤⣇⠀⠱⢦⣄⠀⠠⣄⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠓⠒⠚⠛⠛⠛⠒⠛⠓⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃

⠀⠀⠀⡠⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢀⠠⢐⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣀⣢⢠⣈⣀⣥⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⡿⠂⠀⠀⡐⢁⢲⡀⠀⠸⠀⠀⠀⠀⣠⠤⠶⠶⠿⠿⢶⡄
⠀⠘⠈⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠾⠀⠀⠀⠀⠀⠛⢾⡡⠀⢁⡆⠀⠀⠀⢹⠀⠀⠀⠄
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠠⠀⠀⠀⠀⣶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠐⠀⠀⠀⠐⠒⠘


⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⠀⢀⣠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢘⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠰⣆⠀⡀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⡄⠄⠀⠀⠀⠀⠸⠁
⠀⠀⠠⠂⠀⠀⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⢎⠣⣄⠀⠀⠈⠀⡃⠈⠠⠈⠀⠠⠀⠑⠠⠀⠀⠀⢀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⢙⡧⠄⠀⠀⠀⠀⠐⠨⠤⠒⠀⠡⠀⠀⠀⠀⡁⢈⠀⠠⠋
⠀⠀⠈⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠊⠀⣀⣤⣤⣀⡈⠀⠀⠀⠀⣿⢿⡄⢸⣟⠖⠁⠀⠀⠀⠀⢀⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠐⠀⠂⠀⡰⢛⡅⠈⠀⠁⠀⠀⠰⠀⠀⢨⠉⠃⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⢋⣀⣠⡠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⡀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠠⡬⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⢀⣠⠄⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠂⠈⠂⠀⠀⠀⢠⡏⠃⠀⠀⠀⠙⡆⠀⠀⠀⡀⠔⠒⠂⠀⡀⢀⠠⠀⠀⢀⠔⠀⠀⠀⠑⠄
⠀⣀⠀⠀⠐⠚⠛⠛⢲⠇⠀⠀⢀⠈⠀⠂⠂⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢈⠀⠀⠒⡄⠶⠀⠒⢄⡜⢂
⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠤⠁⠔⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘

⠀⣀⠀⠀⠀⠀⡀
⢀⠀⠀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣰⣤⣤
⠀⠀⡀⠙⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃
⠀⠀⡇⠀⢣⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⡜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠦⡀
⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠶⠒⠃
⠀⠀⢀⠀⠸⠀⠀⠠⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠄⠄⠤⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⢀⡀⠀⢀⣳⡀
⠀⠀⠰⠀⠊⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠘⠃⢀⠄⠀⠀⠀⠑⠦⠴⢆⡂⠒⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⣄⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠴⢁⢖⠀⠀⠀⠀⢹⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠑⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡠⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠢⢄⠄⠐⢅⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠘⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⠰⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠊⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠀⢣⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⣀⠀⢰⠆⠁⢀⠀⠑⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡔⢄⠀⠘⢀⠀⠀⠀⡀⠀⢠⡀
⠀⠈⠸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠
⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢈⡁⠂⠀⠃
⠁⢀⠀⠑⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⢀⠀⠀⡀⠀⠀⣀⠄⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠠⠀⠀⠁⠤⡀⢀⠃⢁⠀⠠⡄
⠀⠀⠈⠄⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⠂⠀⠀⠈⠀⠁⠀⠄⠀⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⠄⣈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠖⠀⠀⠈⠂⠀⠀⡀⠀⡠⠤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⢀⡀⠠⠤⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⢤⠄⠊⠈⠀⠀⠑⠠⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠈⢢⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀⠀⠠
⠄⠀⠀⠀⠈⠁⠢⠀⢀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠑⠠⢄⠀⠐⠀⠠⠂⠀⠀⠀⠀⢀⠀⠀⠂⠀⠐⠂⠂⠄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣈⣀⣀⣀⣀⣈⣀⣄⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠛⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿

⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣶⡆
⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇

⣿⣿⡟⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠊⠁⠯⠽⠭⠇⠤⠀⠿⠿⠼⠵⠸⠥⠁⠸⠿⠫⠼⠯⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠁⠀⠀⠀⠀⠈⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠂⠀⠀⠀⠢⣤⣀⣠⣴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠙⠢⢄⣀⡀⠀⠀⢀⣀⣠⠟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠅⠠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢁⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡊⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠒⠂⠀⣀⠄⠀⠀⣀⡄⣀⡤⠞⣹⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠣⠀⠈⠉⠁⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⣉⠙⠛⠛⠉⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢅⣀⠤⣤⢤⣤⢤⡤⠼⢑⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⡄⠀⠜⣀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠼⡇⠂⠀⠒⠒⠒⠒⠂⠀⠐⡁⠀⠠⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠣⠜⠉⡤⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠂⠉⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠢⣀⡠⠆⠀⠀⠀⠀⠀⠀⡠⢶⠁
⠀⠀⡀⠀⠠⠀⠀⠀⠀⠇⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⠀⠀⢀⣀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⣀⣀⣤⣤⡲⠯⠖⠋
⡀⢀⡍⠀⠐⠒⠒⠂⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⢀⠀⣤⢯⡑⢢⣀⠀⢀⡔⠙⣏⡙⠉⢹⠏⠀⠀⠙⣄⠀⠀⡀
⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡦⢀⢀⡀⠀⠀⡀⠀⣀⠠⠤⠌⠀⠠⠀⠘⢯⣻⠄⠈⠒⠚⠳⠆⠀⠀⠉⢹⠀⠀⠀⠀⡇
⡇⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠠⠀⢀⠀⢀⡀⠀⢀⠤⢄⠀⠀⢀⡤⠴⠖⠊⠷⠀⠚⠀⠘⠒⠲⠤⢄⣸⠒⠤⣀⡴⠁
⡇⠀⠈⠦⠤⢤⠤⠤⠴⠏⣄⠀⠀⢩⠀⠀⠀⠁⠀⡯⠉⠐⠅⢤⣀⠀⠀⠘⡄⠀⠈⠁⠠⠠⠠⠄⠀⠀⣀⡤⠞⠋⠀⢠⣧⡴⢲
⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣟⠀⠀⠀⠐⡬⠤⣈⡤⠼⡆⠁⡀⠀⠀⠀⣺⠌⠉⠀⠀⢀⣠⠔⠉⢹⣤⠶⠥⠤⠤⠤⠤⠤
⡇⠁⠈⠁⠀⠉⠁⠉⠉⠉⠉⠁⠈⠈⠉⠁⠁⠁⠁⠉⠐⡀⠀⡌⠁⠉⠉⠁⠉⠁⠐⠠⠄⢀⣀⠳⣊⣤⠤⠚⠉⠀⠀⠀⠸⡎⡅⠉⠁⠉⠉⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖
⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⢹⠃⠀⠒⠂⠒⠁⢧
⡇⠀⠀⠀⠀⠀⠈⠉⠒⠂⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄
⠧⠤⠤⠤⠤⡌⠀⠀⠀⠀⢨⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡔⠉⠑⠲⠤⢤⡼⠀⠀⠀⠀⠀⠀⠀⢣⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢣⡀⠀⠀⠀⢀⠷⠤⣄⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠐⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⣄⠀⠀⠀⠉⠉⠐⣒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠉⠁
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⡯⡖⣿⣿⣿⣶⣶⢶⣰⣶⡶⣶⣦⣤⠤⣠⣤⣤⣤⡤⣀⣀⠠⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⣟⢸⣿⣶⡘⢿⣿⣿⣿⣿⡿⣻⣿⣿⡿⠩⠿⢍⢪⠁⣭⣽⣻⢤⣄⠀⠀⠀⠀⠀⠀⠀⠿⣿⢳⣤⠃⡘
⠀⠀⢸⣷⡿⢿⡻⠿⢿⣿⣟⣯⣽⣞⢿⡟⣾⡃⢾⣧⡤⠀⠀⢠⣭⡇⢿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠊⢃⣿⠥⣷
⠀⠀⢸⣿⣟⡿⡷⣿⣾⣿⣏⢹⣿⠤⠈⣾⠁⠒⣨⣌⠮⢚⣃⠈⠋⠃⠹⠇⠛⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹
⠀⠀⢸⣿⣿⠷⣏⣿⣿⠟⠋⠀⣿⢠⡼⠃⢈⣶⡹⠟⣂⡟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢮⡬⠀⠀⠀⠹
⠀⠀⠸⣿⡇⠳⣿⣿⡟⠠⠀⠀⣿⡿⠀⢲⣦⣿⣥⡘⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠀⠹⢷⢶⣯⣀
⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣟⣻⣿⣿⢿⡿⣯⠖⠀⠀⠀⢀⠀⠀⣻⡾⢰⡟⠛⠀⠂⡐⣽⣿⣿
⠀⠀⢸⣿⣿⣿⣻⣆⣤⠀⢽⣈⣿⡟⡿⡿⣬⣟⣿⡿⡼⣷⣂⡀⠀⠀⣠⣸⢃⣴⣿⠿⠿⠷⢺⡧⣴⣾⣟⣟⠿
⠀⠀⢸⣿⣿⣿⡻⢹⡿⣦⡬⠍⣛⠳⢿⣿⣶⡧⣏⡳⠀⠀⠙⢅⣀⣦⠀⡀⠀⠉⢠⣾⣷⣠⠌⣥⣹⣿⢿⣯⣽
⠀⠀⢸⣿⣿⣿⣧⢸⣶⡾⠃⠀⠀⠙⠲⡌⢳⣿⡛⠀⠀⠠⡀⠀⢹⣿⣞⠁⠀⡄⠈⣡⣹⣿⣾⢰⣼⣇⡿⠌⠺
⠀⠀⢸⣿⣿⣷⠋⢸⠟⣟⡍⠉⠙⠒⠦⡄⢀⣿⣿⣇⠀⠀⠈⢀⣼⣿⢀⣰⣾⡷⠂⢉⣛⠿⣽⡂⣿⣼⣶⣿⡿
⠀⠀⢸⣿⣿⣧⠓⠘⡷⠩⣤⣤⡶⢶⢶⣄⢿⣅⠄⠿⠲⠤⣤⢸⣿⣿⣿⣿⣿⣿⣧⢠⡴⠞⠋⠙⠧⠻⢿⣿⠁
⠀⠀⢸⣿⣿⡇⢲⡄⢙⣾⣿⣷⣤⠶⢶⣿⣿⣿⣷⣤⠀⢰⡅⠈⠙⠹⠛⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⢸⣿⣿⣷⣾⣿⣾⣿⡁⠉⠁⠀⠉⠉⠿⣿⣿⡿⠀⠀⣿⣀⣀⢀⡀⠀⠰⠙⢻⣭⠷⡴⠤⠶⠾⠞⠃⡀⢸
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡀⣤⣀⣀⣤⠇⠀⢠⣤⣿⣿⣿⣿⣿⣷⣶⣿⡧⢈⣦⣾⣖⣠⣄⡢⣄⡀
⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⣀⠈⠻⢿⣿⢿⣛⢿⣿⣿⣌⠱⢿⡿⢿⣿⣿⣿⣿⣼⣿⣾
⠀⠀⠘⡿⠿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⡈⡻⣿⣿⣧⣿⣿⣿⣿⢷⣤⣸⣿⣿⣿⡿⣾⣿⠿
⠀⠀⢰⣧⣄⣤⡀⠈⣇⠀⠀⠘⣿⣿⣿⡿⠿⠿⠋⠙⠛⢿⣥⣷⣾⡿⠟⠿⠿⣿⡟⣼⣿⣿⣿⣿⡿⢿⠟⠁⣰
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠏⣸⣿⡿⠟⠉⠀⢀⣴⣶⣶⣿
⠀⠀⠘⡿⠿⣿⠿⠻⠙⠙⣧⣀⠀⠀⠙⠻⢷⣤⣄⢀⠀⠀⠀⠀⠀⣀⠠⠚⢀⣴⣿⠟⠀⣀⣀⣠⣼⣿⣤⣌⣹
⠀⠀⠀⡧⢤⣤⣾⣴⣄⠀⠈⠙⠿⣦⣤⣄⡀⠀⠀⠉⠉⠉⠉⠉⠁⣀⣠⣴⠿⠛⠁⢰⣿⣿⣿⣿⣿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⡏⣼⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⣿⡿⠿⠋⢩⣩⠆⢀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⡏⣋⠄⢠⣿⣿⣷⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡰⣿⠺⠟⢃⣀⣿⣌⣡⣿⣻⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠇⡀⠃⠀⢰⣾⣿⠟⡻⣿⣿⠿⠟⠻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣎⠀⠟⡋⣻⣿⡁⣼⣷⣝⣿⣶⣿⠆⠀⠒⣶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠲⣄⡀⠀⠀⠀⠈⢙⠿⠉⢠⣄⣻⣾⠛⠾⠿⣟⣿⣿⣿⡧⠒⠀⠀⠀⠀⠉⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢹⣿⣪⣟⣵⡾⠗⠀⠒⣼⣿⣿⣿⠶⣶⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢠⣾⣿⡏⠉⠀⠀⠀⠀⠰⣿⣿⣿⣿⣾⣿⣿⣶⣶⡀⣀⡀
⠀⠀⠀⠀⡠⠀⠀⠿⣿⣿⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠛⠛⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠦⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⡔⠊⠉⢹⡢⠄⠀⠀⠀⣀⡴⠁⠀⠉⢦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⣹⠃⠰⣤⣀⣀⢀⠉⠓⠈⠉⠉⠉⢀⣀⣤⡢⠃⠀⠻⢪⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡬⢙⡲⠤⠌⣛⣛⠋⣉⠑⠢⠉⢁⣐⣛⣋⣉⡠⠤⢖⣛⠥⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⣉⠁⠒⠒⠒⠚⠁⠀⠀⠈⠛⠒⠒⠒⠈⣉⣡⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⡠⣢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣭⠴⣠⣄⣀⣀⠤⢼⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢊⠈⠁⠁⠀⠀⠀⠉⠉⠀⠀⠉⠂⠒⠢⠄
⠀⠀⣀⠬⠀⡁⠃⠀⣰⠁⠛⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠠⠤⡀
⠀⢁⠀⢠⡈⠀⠒⠊⡉⢀⣠⠀⠀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⡼⣉⣘⠛⠒⠀⠀⠀⠀⠆
⠀⠐⣝⢦⠀⠈⠉⠉⠁⠄⡉⡀⠐⡂⡤⢠⠀⠈⠁⠀⠀⠀⢀⠀⢀⢀⡀⠄⠄⠀⠀⠀⠩⠉⠒⠒⡆⠀⣾⠀⠠⡀
⠀⠀⠨⣦⡑⢀⠱⠀⠄⢰⣀⣀⣼⣘⠶⠇⠀⠀⠠⠞⠁⡀⡔⡂⡀⠘⠸⢳⠲⠒⠂⠀⠀⠀⠀⡜⠀⡔⢸⠀⠀⠁
⠀⠀⠀⠀⠙⠓⠒⠂⠀⠑⠋⠉⠉⢀⠀⠀⠀⠀⠐⠀⡎⣘⠂⠁⠁⠀⠒⠈⠀⠠⠀⠀⠀⣠⠞⠠⠉⢀⡞⠀⢀⠁
⠀⠀⠀⠀⢀⡀⣀⠤⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠐⢨⠧⡜⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⠟⡡⣾⣀⡴⠋
⠀⠀⠀⢠⣀⠀⠉⠑⠺⣯⠭⠱⢖⠀⠀⠐⠀⠀⠆⠘⡄⠟⠂⠠⠤⠴⠖⡒⣺⠻⢋⡀⠀⣀⡉⠥⠴⠂⡊⠁
⠀⠀⠀⠀⠉⠂⠀⠐⠀⡄⠳⡠⠤⣴⠁⠀⠀⠀⠘⠑⣧⠀⠈⢤⡤⠔⠋⠀⣀⣔⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠘⠀⠀⣀⠀⠀⠀⠀⣧⠈⡖⠈⠀⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠖⣏⠁⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⣤⠈⢦⢅⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⢀⠁
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣌⠹⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⡤⠤⢤⡀⠀⠀⠀⠙⠆⣰⠿⠃⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢱⠀⠀⢀⡀⠀⠀⢠⡞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠀⠀⢸⣧⠀⢰⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢧⡉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⡴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠃⠀⠀⠀⠚⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠲⠤⠞⠋⠛⠿⠿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠈⢧⣄⣀⣠⡞⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢀⣤⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⡂⠀⠀⢸⣧⣤⣤⣤⣄⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣄⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⢸⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠛⠻⠛⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⣽⣿⣿⣿⡇⠀⣀⣀⣀⢀⣀⣀⠀⠨⣿⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡿⠁⠀⢨⡟⣱⣿⡇⠀⢿⣿⣿⣿⣿⣿⠁⠈⣿⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠙⢿⡇⠀⠀⠛⠻⠿⠛⠁⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣾⡇⠀⢀⠀⠀⠤⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡇⠀⢈⠀⢁⠀⠆⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢠⣄⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠀⠙⠛⠟⠋⠁

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⣖⣫⢿⠿⠃⠀⣀⣴⣦⡀⠀⠀⠀⡴⠋⠉⠀⠀⠀⢀⣴⣛⣶⣄⠀⠀⠉⠙⠒⠂⠀⠀⠀⣀⣀⡆⠠⢄⢀⣴⡿⠁
⠀⠀⠀⠰⠋⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⢔⠿⣓⢿⡯⣟⣶⣄⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⠀⢖⣻
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠻⠡⣗⢫⠬⡝⢧⣚⠽⠀⢁⠀⠀⢠⡿⡿⠻⠟⢸⡟⢰⣯⣿⣦
⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢴⠸⣹⣷⠀⢹⣿⣗
⠀⠀⢠⢷⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣖⡰⡠⢀⢀⣀⡆⠐⣒⣦⠀⠀⡀⢀⢀⠁⠀⢀⡀⠀⠸⠈⢹⡏⠘⡿⣿⠏
⠀⠀⠠⡿⠠⠃⠰⡟⠟⠛⠓⡄⠀⠀⠰⠯⠒⠓⠛⠻⠚⠁⠀⠃⠻⡾⠻⠚⠋⠙⠀⠀⠨⠽⠃⢀⣤⡈⠇⠀⠈⠿⠄
⠀⠀⠀⠀⠀⠀⣴⢰⡄⠘⠻⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠁⠀⠀⢀⡀⠀⠀⢠⡖⢄⠈⠿⠇⠀⣶⣦⠀⠀⢀
⠀⠀⠀⠺⢿⣿⣿⣿⣧⡄⠀⠘⠀⢠⠀⠀⠀⠈⣀⣀⣀⠀⠀⣼⠀⠀⠀⠀⢿⠶⣿⠇⠈⠙⠉⢀⣠⣤⣤⣄⣀⡀⠀⠻⠆
⠀⠀⠀⠐⠿⣿⣿⣿⣿⣧⠀⠀⠀⠈⢰⠀⠀⣿⠉⠉⠁⠀⠀⠁⢨⣽⣶⠀⣤⠀⣤⣄⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⡀⠀⠀⣿⠀⣽⠉⡏⠀⠀⠀⢰⠎⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⣭⡇⠀⠉⠉⠉⠁⠸⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠰⡟⠀⣿⠀⡇⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣴⣿⠇⠀⣁⡀⢀⣀⠀⠀⠙⠉⠉⠀⠀⠃⡏⠙⡿⠁
⠀⠀⠀⠀⠀⠇⠀⣿⠀⠇⠀⢰⡀⣰⣆⡀⠀⠀⠑⠀⠀⠀⣀⠰⠾⠉⠀⠀⣿⢹⣿⣯⠀⠀⠰⢄⣀⠀⠀⠀⢀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠄⠠⢈⡐⠁⠷⢀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠡⠄⢘⠑⠂⠗⢸⠁⠀⠀⠀⠁⠛⠋⠘⠛⠛⣮⣂⠀⠀⠠⠀⣀⠀⠀⠀⣀⠀⠴⠃⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠘⠁⠘⢆⡈⠡⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡇⠰⢆⡆⢠⠀⠤⢤⠤⡄⢠⡄⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠂⠌⡃⠛⠉⡀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠈⠙⠻⠶⠧⠴⠦⠛⠊⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢄⡘⣆⠀⠀⣠⣿⣿⡿⠀⠀⠀⠠⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣞⣿⣿⣿⠃⠀⠀⠰⠀⠉⠉⣠⠀⠀⡘⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠰⠊⠹⣶⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⠊⠀⣰⠁⠀⢰⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⢰⢦⠀⠀⠀⠀⣶⣀⣀⣠⣶⢀⡀⠀⠀⠀⠐⠛⡇⠀⠀⠀⠑⢶⣴⣿⠾⠟⠁
⠀⠀⠀⠀⠃⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⢟⣣⠞⠁⠀⢀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⢸⡏⠀⢠⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⣠⡀⢰⠀⢠⣦⡏⢀⠄⢦⣶⣹⣧⣤⣄⠀⠀⠀⣸⠁⠀⠘⢷⣀⠀⠀⠀⣠⣀⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠙⠛⠛⠋⠁⠀⠄⠊⠙⢻⠉⠀⠈⠻⣧⠀⠁⠹⡄⠀⠀⠀⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠀⠀⠀⢤⡆⠀⠈⠢⣀⣀⣀⣶⣶⠾⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠰⠶⠛⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣄⣀⣠⣶⠀⠀⢰
⠀⠀⠀⠀⠀⣴⣾⣶⣶⣶⣶⡊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⡸⠀⠀⠀⠀⠠⠛⠃⠹⠀⠇⠠⠄
⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⡛⠀⠀⠻⢦⣄⣀⣀⣀⣀⣀⣦⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⡀
⠀⠀⠀⠄⠠⣾⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⡀⣩⠴⣺⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡴⠟⠏⡹⢿⣿⣦⣤⡀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠞⢁⠞⡽⢠⣿⣿⣿⠀⠀⠀⣠⣶⠃⠀⠀⠁⣴⣣⣤⠿⢿⣿⣷
⠀⠀⠀⠀⠄⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⠁⡰⠃⡜⡰⠋⣷⣿⣿⠀⠀⠀⠻⢯⣄⣀⡀⠀⢀⣿⣥⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠹⢛⡿⣿⠿⠋⠀⠀⠀⠀⠀⣧⠞⣀⠴⠀⠀⠀⣻⣿⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⢰⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⣿⣟⡛⢚⣭⠀⠀⠀⠀⠀⢀⣤⡤⣿⣿⣿⡿⢀⣠⣄⡀⠀⠀⠀⠀⢀⠰⢶⣶⣶⣾⡀⡰⠶
⠀⠀⠀⠀⠀⠀⢸⣟⠒⠈⠋⠀⠀⠀⠀⠀⢿⣿⡿⠋⠉⠁⠀⢾⣿⣿⡶⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⠄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣶⡄⠀⠉⠉⠀⠀⠀⠀⠀⠈⢿⡷⠆⠛⠛⠁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠃⢀⣾⣿⣦⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣟⡷⠾⠿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⢠⡛⠋⠉⠁⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠳⠿⠿⠵⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠨⠇⠘⢛⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⣀⠀⠈⠉⠁⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠁⠀⢠⣤⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠐⠀⠀⠀⢠⠀⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣠⠴⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠂⠈⠈⠀⠁⠀⠀⠠⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠀⠀⠺⠂⠾⠀⡠⢂⠀⠀⠀⠄⠁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠐⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⢀⠀⠀⠀⡀⣄⣀⣙⠄⡀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⢠⠀⠀⠀⠀⠀⠠⠃⠀⠈⠻⢿⣷⣶⣤⣬⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣰⠀⠐⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣬⣐⢏⣁⣀⠠⠀⠀⠐⠀⢰⣤⣽⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢨⠁⢠⣄⢈⡼⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢆⠀⠠⢤⡀⠀⣮⠻⡀⢠⣈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⡑⢳⢸⠂⡇⠾⠢⠜⠔⠆⡱⠡⣈⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢊⢍⡉⠱⣎⡹⠰⣈⢌⡀⢘⠀⠄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢫⡀⡠⡺⡆⠀⡬⣽⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⠚⠤⠁⢢⢤⢒⢃⣄⡠⣀⡑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⢸⡽⡋⡷⣇⡞⣇⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠻⠇⠱⠙⠫⠆⠃⢜⠘⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⢷⡗⣑⠗⣒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⠀⠰⡀⡜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡟⣆⡖⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡀⢈⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡠⠀⠀⠋⢧⡏⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡐⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠇⠀⠐⠨⡯⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢌⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⢐⡈⠜⢁⡀⠀⠀⠀⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⣿⠔⠻⠁⠀⠀⠀⠀⠀⠀⠸⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼


⣿⠿⠿⠿⡿⢛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣻⣿⣟⣿⣿⣿⣿

⣇⠁⠒⠻⡚⠂⠀⠐⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣅⣠⠾⠷⠦⢄⡀⠙⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠀⠚⢷
⢆⢻⡶⠖⠙⡚⢦⣁⠉⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠇⢀⣯⣤⣄⡀⠀⠀⠈⠑⠐⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⢀⡴⠋
⢀⡞⠀⡆⠀⢰⠀⢣⡑⢬⣀⠀⠀⠀⠀⠀⠀⠦⡘⠳⡀⠀⠀⠀⠀⠀⠀⢸⡧⡄⠀⠈⠉⠓⠀⢠⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡠⠺⠁⢠
⠃⠈⠢⢼⡀⣈⣄⡠⠥⠄⠈⠀⠀⠀⠀⠀⡀⢧⡀⠀⢧⠖⠀⢱⠀⠀⠀⠂⠀⠈⠑⠐⠲⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡠⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠉⠁⢸⢆⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠣⡐⠀⠜⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠠⠤⣦⡀⠁⠢⠸⡇⡰⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡦⢅⡀⢄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠤⠌⡀⠢⠀⠉⠦⠀⢸⡃⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠂⠀⠙⢾⡈⡆⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣂⣤⠤⠿⠓⡚⠚⠛⠚⠒⠺⠤⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠒⠒⠂⠹⣔⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠝⣡⠎⢀⡔⠀⡰⠁⢀⡀⡀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠐⠂⠉⣆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠡⠤⠚⠡⠴⠛⠈⠀⠀⠀⠀⡆⢀⠁⢠⠀⢠⡔⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡒⠒⢀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠈⡆⠘⠀⡸⠀⢠⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣙⡄⠀⠁⢀⣼⡤⠓⠀⡔⠀⡀
⠀⠀⡤⠤⢄⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⣀⡀⠀⠀⠺⣟⢸⠀⡸⠗⠂⠀⠀⣀⠆
⡕⠀⠀⢰⠏⢀⡴⠋⢀⡔⣉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠒⠚⢷⡌⠣⠸⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⣠⣿⠋⠑⠲⢦⣈⠑⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⠀⠈⠙⣷⣶⠞⡟⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡷⠞⠉⠉⠉⠑⠢⢄⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢀⣀⡴⠚⠋⠈⡟⣄⠀⠃⠟⠯⣀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⢂⠀⢄⡀⡁⠠⠀⡀⠀⠸⡆
⠛⠛⠳⠶⣄⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠵⠂⠁⢰⠀⠀⣱⠟⡙⠢⠷⣄⢀⡉⠳⠀⠀⠀⠀⠀⠀⠀⢡⣉⡉⠙⠲⢧⣘⡄⢀⠈⡆⠀⢩
⠢⠄⠀⠀⢀⣀⠽⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣠⡁⠀⠐⠉⡄⢸⠁⢀⠆⢸⢆⠈⢠⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠐⠂⠈⠿⣄⠀⢘⠀⠘
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⣸⠀⢸⠈⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠖⠂⠉⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡔⠻⠁⡥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡰⠀⠀⠀⢀⠀⠀⠀⠀⡤⡆⢉⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡃⠘⠀⠐⡰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣌⣉⣳⣼⡆⡆⢱⠠⡇⢸⠀⠃⢸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡇⠀⣠⠞⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢤⠄⠈⠳⣼⠀⡀⢱⠈⠄⢰⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⠞⠉⠒⡉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⠤⠀⠙⣆⠁⢀⢀⣴⠣⠔⣁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠊⠁⠈⣉⡠⠖⠀⠀⡀⠀⠀⠀⠄⠠⡐⠆⣄⡀
⠀⢰⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠲⠤⣀⡀⠀⠘⣆⣠⠞⠡⠐⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠒⠀⡃⠀⢦⠀⣳⠀⠸⡈
⡀⢸⠎⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠻⠁⠠⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⢦⡀⠀⠈⠀⢸⣿
⡁⡸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢉⡴⠁⠀⡠⠊⢀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⠭⠓⠀⠀⠘⠻
⣱⠷⠦⣀⣤⠚⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⡇⠀⠀⣼⡧⢤⣋⡀⠰⣇⡒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠴⣢⢶⣌⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢹⡌⢿⡀⠀⠀⠀⢀⡔⠂⠁⠀⠀⠀⠀⠀⠀⢠⡾⣣⠶⣤⡀⠀⠀⠀⠀⠈⠉
⠀⠀⣀⣠⣄⠁⣤⢀⡀⡄⠀⠀⠂⣤⠤⠄⠐⠀⠀⠀⠀⢸⡀⢸⡧⠈⠣⡄⠀⠠⠉⠶⠀⠀⠀⠂⠀⠀⢀⡰⢿⣜⡱⢶⡌⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤
⠀⠀⠉⠉⠉⠀⠸⠼⠃⠇⠀⠀⠺⠶⠒⠺⠛⠁⠀⠀⠀⠘⡗⠈⠡⠀⣰⠇⠀⠀⠤⠼⠒⠈⠤⠀⠀⠀⠚⢵⢏⣺⡷⠋⠀⠀⠀⠀⠀⠂⠀⢺⣗⣤⠟
⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠖⠛⠁⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠰⣦⠙⠃⢰⠀⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠀⠃⣀⠈⠓⢄⣤⠀⠀⠀⢠⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢞⠁⣽⠆⠉⠀⠀⠀⠀⠛⠇⠀⢀⠴⠋⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢠⣴⠏⢠⣤⠀⠙⠃⠀⠀⠀⠀⢠⠙⠃
⠀⠀⠀⠀⡀⠻⠀⠀⠀⠀⠀⠧⠀⢠⣠⡤⠀⠁⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠙⣁⣠⣤⣤⣤⡀
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⣦⠤⡤⠀⠀⠀⠀⠀⠙⠇⢠⣄⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢘⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡉⠀⠀⠈⠀⣀⠀⠐⢦⡀⠀⠛⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⡷
⠀⠀⠀⠀⠀⢾⡧⣴⠄⠀⠀⠧⠀⠁⠀⠈⠀⠀⠛⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡆⠀⠀⠀⠒⠃⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠘⢷⠜⠃⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⢀⣴⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠉⠟⡄⠀⠀⠀⠀⢰⠄⠸⣀⡴⠆⠀⡀⠀⠀⢄⠀⢰⠀⠀⠀⢠⣴⣶⣶⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⡇⠀⠠⣠⡁⣠⡤⠘⠋⠀⠀⢈⡁⢀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠉⢀⡰⠶⠃
⠀⠀⠆⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠀⠀⢸⠁⠀⠀⠀⠀⠈⠁⠈⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⣩⡁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⡗⠀⠀⠰⠟⠆⠀⠀⢀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢤⠄⢨⠀⡀⠚⠁⠀⠀⢀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠈⠀⢀⠟⠀⠀⠀⠂⢀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣟⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠐⢣⠀⠈⠒⠂⠀⠀⠀⠀⠠⢠⡄⢠⣄⢹⡆
⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠟⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⠜⠃
⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣴⠶⠿⣷⣶⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣦⠀⠀⢠⣶⡄⠀⠀⠀⢦⠀⠘⢇⠀⠀⠀⠀⠀⢀⣤⣤⣠⡄⠠⠶
⠀⠱⣤⣤⣄⣄⣰⣤⠆⠀⠀⠙⢿⣤⣿⣦⡼⠇⠀⠀⡀⠀⠋⠉⠃⠀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠰⠀⠀⠸⠃⡰⣿⡟⢶⡏⠓⣤⠀⢿⠯⣍⡼⠇
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠴⠶⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠈⢱⣏⣠⠼⠷⠞⠁⠀⠀⠀⠈⠁

⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡤⣤⡤⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣄⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠀⠀⠀⠀⠆⣠⠆⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠺⠀⠀⠠⠀⠀⠀⠈⠻⠶⠶⠶⠟⠁
⠀⠀⠀⠀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠂⠠⠀⠀⢑⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠀⠀⠀⢠⣬⡑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠄⠀⠄
⠀⠀⠀⠼⠃⣀⠴⢟⠀⠀⠠⠖⠐⠒⠀⠈⠆⠀⠀⠀⢴⣬⣿⣷⣤⡈⠀⠀⠀⠀⠀⣰⡀⠊⣱⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠒⠶⠐⠿
⠀⠀⠀⠲⠀⠙⢄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠋⣹⡄⢣⡷⠀⠀⠀⠀⠂⠙⠁⡟⠁⠀⠀⠀⠘⠛⢃⣀⡄⢶⡄⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⢠⣤⣤⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠐⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢱⡄⠀⠀⠀⠀⣀⣤⣴⠾⠟⠂⢀⠀⠀⡤⠈⠀⡀⠺⠆⠀⠀⠀⠀⠘⠛⠛⠂⠀⡄⠀⠀⠀⢈⠛⡀⠀⠰⣺⠉⣧
⠀⠀⠀⢶⣿⠿⠃⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⣶⠟⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠦⣿⡶⠀⠀⠀⢄⣠⡀⣀⣸⣀⡲⠐⠀⠀⠀⠀⢀⡀⠀⠀⢑⠊⠁
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠐⠋⠀⠀⠀⠀⣼⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠺⢀⢈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠙⢻⡋⠉⡟⠛⢻⡙⠏⣙⡿⢛⡋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣷⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠌⢁⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣿⠱⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠂⠨⡉⠓⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢧⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⢀⠚⠃⠄⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠆⠀⠀⠀⠀⠀⠀⠢⠀⠇⠀⠀⠀⠀⠀⣸⠀⢠⠐⠿⠤⠛⢢⡔⠤⣔⣓⠴⣷⡼⠲⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡄⠀⠀⠀⠀⠀⢀⡇⠸⣄⠀⠀⢠⡀⠛⠀⣨⠐⢁⠈⢄⢙⣋⣉⣉⠙⡘⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠧⠀⠀⠀⠀⠀⠈⠉⠁⠙⠓⢶⡞⠀⣕⡤⢤⢆⣁⣒⣁⡶⠴⣷⢚⣖⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠧⣄⣠⣄⣀⠀⠀⠀⠀⠀⠀⣀⣰⡾⠓⠒⣏⣀⣼⡟⠶⠾⠶⠮⠭⠽⠿⠿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠲⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⣽⣟⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣿⣿⣿⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⡄⢿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⣿⢟⣫⣽⣶⣾⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠉⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣯⣿⣽⡟⣿⣿⣿⠟
⠀⠀⠀⠈⠂⡀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⢀⡄⠀⠀⠀⠲⠾⣿⣿⣿⣿⣯⣿⣧⣿⣿⣿⡿⠟⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠘⣷⣿⣷⣦⣀⠀⠀⠀⠉⠉⠹⠋⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠙⠭⠽⠯⠿⠿⠛⠋⠁⠀⢀⣀⠻⠿⣿⡷⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡄⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠰⣶⣬⠙⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⡆⠐⠀⠀⠀⣀⠰⠀⠆⠀⠀⠀⠀⠈⠳⣌⠳⢄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠐⠦⠈⠀⢀⡁⠀⢀⠄⠀⠀⠀⠀⠀⠱⣤⣿⣦⣹⣿⣃⣠⣴⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣟⠻⣅⠀⠀⠀⠀⠈⠛⠛⠛⣻⡿⠿⡟⠻⠏⠙⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠉⢰⠈⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠻⣝⠻⣷⣜⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⣿⠿⣿⣿⡿⡿⠟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢧⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿
⠀⠀⠀⠀⡀⠀⠈⡌⠀⣸⣯⠟⠁⠀⠀⠀⠀⠀⠀⣴⣾⢿⣿⣧⣡⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣴⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠏
⠀⠀⠀⠀⠈⢼⠀⡇⢰⣟⢕⣡⡄⠀⠀⠀⠀⠀⠀⠉⢡⣾⣏⢻⣿⠻⠗⠀⠀⠀⠀⠀⠀⣿⣻⣟⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠸⡆⣷⣿⣟⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣞⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢠⣤⣶⡾⠿⠿⢛⠻⢿⠋
⠀⠀⠀⠀⠀⠀⣿⡿⠝⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⣽⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣿⣷⡆
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠗⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠃


⠀⠀⠁⠁
⠀⠐⠒⠃

⠀⢀⡀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠤⠄⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⠄
⠀⠘⠀⠀⠀⠀⠀⠈⢄⠀⠰⠁⢀⠀⠉⠆⠀⠀⠌⠠⠊⠠⠀⠟⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠈⠀⠡⣈⢀⡄⠐⠰⣲
⠀⠀⠐⠀⠐⠀⠀⢉⠣⠀⠀⠣⡀⠈⠀⠈⠆⠈⢠⠴⡀⠊⠀⠀⠀⠂⠀⠠⠀⠄⠀⠁⠀⠀⡠⠁⠀⠒⠄⠀⠉⠉⠀⢐
⠀⢀⠀⣋⡀⠀⠀⠀⠈⣀⢀⢀⠈⠒⠈⠄⢀⡴⠀⡯⢚⠁⠀⠀⢠⠐⠂⡐⠠⢀⢀⠀⢀⠜⠀⢣⢄⣀⡀⠀⠀⠀⠤
⠀⠘⠀⠀⡆⠀⡠⠊⡁⠀⠃⠂⢌⡑⢄⠀⠸⠋⠀⢸⡵⠀⠀⢸⠃⠀⢀⡅⡀⠘⠄⠐⠁⢀⠀⠠⣀⡀⠀⡀⠠
⠀⢠⠖⢦⠀⢄⣠⠒⡂⢜⣊⠼⠓⢊⠁⢠⠒⠒⠤⡀⠃⠀⠀⠀⠁⠨⠉⠁⠀⠀⠀⠀⠠⠂⢤⠀⠠⠀⠀⠁⠀⠈
⠀⠀⡆⣈⢂⠀⠀⠀⠀⠒⠒⠒⠈⠁⠀⡌⠀⠀⢰⠀⠰⢠⠀⠀⠂⠆⠀⢀⣴⣶⢻⣟⣳⡆⠀⡀⠀⢀⠀⢀⡰⠉⡘
⠀⠀⡄⠂⢠⠀⠀⠀⢖⠒⠀⠠⠀⢲⠆⠑⠀⠲⠈⠀⠀⠁⠃⠀⠀⠀⠀⢀⢀⠽⠉⠉⠉⠉⠀⠀⡀⠐⠀⠀⠉⠉
⠀⠀⠈⠉⠁⠔⢄⠀⡄⠉⠀⠒⠀⠁⢀⠀⣴⡢⡄⠀⠠⠀⠀⠀⢀⠖⡆⠀⠀⠀⠀⠍⠀⠁⠀⡀⠃⠀⠀⡔⠉⢆⠁⢠
⠀⠦⠀⠀⠈⠀⠀⠀⠀⢀⠀⢈⠈⠡⠏⠓⢉⠠⠣⢶⠠⢅⢤⡤⠀⠀⠀⡀⠈⣁⠀⠈⠈⠀⠂⠀⠀⠀⠨⠀⡠⠊⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠂⠈⢀⣐⣃⡠⠤⠗⢉⠀⠀⠀⠀⠰⠀⠀⡀⠼⢑⠀⠉⡂⠲⢀⠑⢀⡱
⠀⠀⠐⠀⠘⢀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠈⠉⣤⠀⠀⢘⠭⠁⣀⠀⠐⣁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠎
⠀⠐⠂⠀⠀⡠⠄⠀⠀⠊⢟⠀⠀⢠⠶⠛⠉⠐⢹⣶⠄⠈⡠⠀⠀⠀⠀⠈⠀⢠⠄⠀⠀⠀⠀⠀⠀⣡⠀⠀⠘⠀⠂⠄
⠀⠰⠠⠍⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢈⠠⠒⠀⠀⠀⠀⠉⠉⠉⠈⠐⠡⠄⢀⣀⠀⠂
⠀⢠⠀⠀⡀⠄⠈⡆⡕⠀⠈⠀⠐⠀⠀⠠⠒⠃⠀⠀⠀⢰⠀⠀⠀⠌⠀⠀⠄⠠⠏⠀⡀⠀⢀⠠⠀⠀⠙⠇
⠀⠀⠀⡀⢀⡄⠀⢡⠁⢀⡀⠌⠀⡄⠂⠂⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠒⢅⡀⢀⠰
⠀⠠⠐⠊⠸⠈⠆⠈⢀⠡⠀⠀⠦⠠⢖⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⡄⠀⢠⠂⠉⠀⠀⠀⠈⠀⢀⠜⡀⠀⠀⢻⠧⠂⠁
⠀⠀⠠⠨⠄⠀⠂⠀⠀⠒⠈⠀⠀⠀⠀⠄⠀⠀⠀⠠⠒⠃⠀⠀⣀⠀⢌⢀⠢⠤⠬⠔⡀⠤⠔⢊⠈⠃⡀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠠⠂⠀⣀⡀⡔⠒⠒⡆⠀⢢⠀⠐⠀⠀⠁⠀⠈⠀⠠⡈⡀⢄⠀⠠⠀⠈⠀⠀⠀⠀⠐⠁
⠀⢀⡀⠐⠀⠀⠀⠀⠂⠄⠁⠀⠀⠘⠄⠂⢘⣀⣸⠀⠀⡀⠀⠐⠄⢤⠆⠀⠀⠈⠚⠁⠡⠊⠈⠠⠁⠀⠤⠀⠐⠀⠒⢤
⠀⠀⣠⡶⢀⡶⠀⢀⠀⠃⡀⠂⠈⠠⠀⠀⡀⠄⠠⣀⠀⠀⠘⠀⢀⠀⠀⠀⠃⠀⠠⣀⣠⣤⠄⠀⠀⠀⢄⠀⠀⠀⠀⠔
⠀⢸⣆⠅⠊⠀⡀⠁⠀⡂⠀⠀⡀⢊⠀⠌⡠⠊⢉⡉⠠⠀⠈⠠⠊⠀⠀⠀⠁⠀⢀⠜⠈⠡⡀⠀⠀⠀⡈⠀⠈⠁⠁⢠
⠀⠀⠁⠂⠀⠀⠃⠀⠀⠈⠀⠉⠀⠈⠀⠋⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠒⠤⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠈⠀⠈⠒⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⣤⢆⡀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠃
⠀⡀⠄⢀⣀⡄⠀⡀⠀⠠⢄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣂⣀⠀⣪⣥⣔⠢⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠁⠀⡁⠉⡇⣼⣻⡀⠸⡇⠀⠀⠀⡀⡀⡀⣀⣀⣀⢀⠂⠀⠈⠰⠋⠀⠀⠿⠟⡃⠀⡀⠀⠀⠑
⠀⠑⠈⠀⠀⠀⠉⠛⠁⠀⠀⠐⠂⠀⠃⠐⠛⠓⠂⠛⠐⠀⡀⠈⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠐⠁⠀⠀⠀⠀⢑⠀⣄⠀⠀⠀⠀⢀⡀⠉⠂⡀⠀⠀⠀⢀⣤⣶⣶⠂⠀⢐⣿⡻⡝⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠂⠀⠆⠀⠀⠁⠀⢈⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣿⣵⡣⣄⠂⣹⢿⣿⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢢⠐⠋⠀⠀⠀⣀⠀⣶⠀⠀⠀⢀⡕⠒⣤⡄⠀⠀⢀⠀⠠⣤⠿⡟⠗⠩⣴⣟⣅⣴⠀⣰⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠈⠠⠈⠀⢀⣼⣇⠀⣿⠃⢐⣾⡏⢀⣾⠋⠀⠐⠀⠈⠉⠁⠐⠀⠈⠉⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠰⠀⠀⢡⡐⠀⢻⣿⠀⣿⠀⢸⣿⠀⣼⣃⣤⣶⣾⡆
⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢦⡀⠹⠀⢿⡀⢸⣟⠀⣿⣿⠿⠿⢿⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠄⠈⠓⠘⠁⠀⠹⣷⠆⠠⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡤⠀⠀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡒⠸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠀⠂⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢡⠀⠠⠀⠀⠈⠡⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠌⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢤⢠⠀⠀⠀⢈⡈⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠃⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣧⣴⡆⠀⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⠿⠏⠀⠀⠀⢰⣿⣿⣿⣦⣤⣄⡀
⠀⠈⠀⡀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠊⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠸⠆⠀⠀⠀⠀⠀⢠⠑⠶⠃⠀⠀⣳⠖⠀⣠⡄⡐⠀⠴⠉⠒⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠈⠁⠀⠀⡤⣻⠀⠀⠊⠳⣼⠀⢀⡴⢡⠞⠁⡜⠒⠁⠙⢡⣀⠀⠒⢀⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣀⡄⠀⠐⠀⠀⡔⠀⠀⣰⡃⠀⢸⠀⠠⠀⠀⡘⢰⠃⣠⠞⠀⣀⡆⠂⠀⠀⢀⣀⠈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠀⠁⠘⣆⡰⠀⣼⠀⠀⠀⢱⠃⠀⠀⡇⠀⢂⡔⠀⠈⢉⣥⠔⠈⠁⠀⠀⠀⠀⠀⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠄⠀⢦⠈⠀⠴⠁⠀⠀⣠⠏⠀⠀⠀⡟⢠⢎⣠⠴⠊⠁⢠⡖⠒⠒⠒⠀⠤⣤⣀⣙⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⢀⠀⠂⢀⠗⢀⡴⠊⠀⠀⠀⠀⣀⡴⠟⠋⠩⠍⣀⠐⠂⠩⠉⠉⠩⠿⠥⢤⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠱⡈⡞⡰⠋⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠈⠢⣄⠈⠐⢀⠀⣀⣀⠀⠀⣠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠃⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⡎⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡠⠊⠀⠀⡠⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠀⢀⡴⠊⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢤⣀⠀⠀⠐⢂⣠⣥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢈⠑⠢⢄⣈⠁⠀⠈⢉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠸⡳⡀⠀⠙⠂⠠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⣄⠁⠙⠓⠤⢄⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⣿⠙⣷⢀⣀⡀⢀⡀⣀⢀⣀⠀⣿⡇⢀⣀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⠃⣸⢼⣿⢸⣿⠁⠹⣧⡁⣿⡇⣿⠜⠃⠹⣇⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠚⠛⠘⠛⠀⠐⠚⠃⠛⠃⠙⠓⠂⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣋⠉⣉⡩⣙⠝⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣙⠔⣚⣭⣬⡽⠶⠣⠬⢌⡴⠃⠀⢠⡖⢲⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠩⠳⠀⠀⠀⠙⢶⡻⡦⣄⠀⡀⣔⢩⣄⢼⠁⠀⠀⢸⠯⢌⣃⠣⢢⢉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡅⢛⠥⢞⢦⡀⠀⠀⠨⣅⣅⢉⡭⠑⡨⡇⢈⡞⠀⠀⢀⣿⡶⢮⡮⣝⣾⣄⠱⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡎⣉⠋⣴⣇⠠⢘⠇⠓⣄⠀⠀⠘⣎⠑⠩⠙⡅⢴⣶⠇⠀⢰⠏⡉⡳⣤⣽⠶⠷⠛⣞⠪⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣜⣵⣉⡐⢁⠙⡃⠸⡎⠘⣎⢳⡀⠀⠘⡌⠑⢨⢟⡄⣾⠀⢀⣾⡙⢿⣮⣕⡙⢾⡒⠉⠈⣦⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢟⡶⣰⠑⢂⠖⠦⢀⠒⣒⢰⣃⡀⣂⠀⠘⡅⢀⣄⡅⡆⢀⣞⣦⠟⣮⡀⣼⣋⣿⡷⠖⠉
⠀⠀⠀⠀⠀⠀⣀⡛⠒⠋⠒⠶⣄⣀⣷⡂⡀⢱⠙⣒⠍⢷⡀⠱⡾⣘⢷⠀⡼⣏⠻⣯⣼⣏⡷⠊⠁⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠓⠀⢨⠲⠤⣀⠲⣤⡀⠉⠑⠒⠯⠤⣧⣂⣠⠛⣄⠳⣀⡌⣼⠓⣽⡦⠐⠉⠁⠀⣀⣠⣄⣤⡴⠶⠖⣾⠡⢙⢧
⠀⠀⠀⢀⣄⢃⠸⠃⣀⢀⣁⣠⠺⢧⠋⣻⠿⣔⣶⠀⠄⢍⡙⠛⠂⠙⠱⠗⣋⣡⣖⡲⣶⣟⠛⣹⢶⣈⡶⣷⠋⣄⢹⡆⣄⠸⡄
⠀⠀⠀⢸⠈⡈⢸⡄⠈⠀⠈⠃⡁⢐⡡⠖⠂⠀⢉⣓⣄⠿⠴⠀⠀⠀⠀⣙⠣⢜⣿⠷⠏⣿⣼⣃⣿⣽⢧⡿⣿⣿⢹⡇⠊⢃⡇
⠀⠀⠀⠈⣦⢁⢰⡗⣰⠮⣔⡈⠩⢈⣅⡠⠀⠾⠉⠉⢀⣀⣤⡄⢀⡀⠀⢯⢑⠦⡌⠙⠳⢮⣝⣣⠿⠛⢺⣵⣏⢏⢹⡇⡸⢐⠇
⠀⠀⠀⠀⢣⡈⢉⠛⢦⢥⣰⡤⠐⠉⠁⢀⡠⠂⢲⡚⣯⠠⡞⣠⣟⣯⠀⢸⡞⡧⣈⢲⣄⡀⠀⠉⠓⠤⢞⣍⠚⢃⡟⢀⠋⡼
⠀⠀⠀⠀⠀⠻⣐⣈⡍⠁⢀⣤⣠⠴⠺⢛⠃⠒⢁⡀⢸⠖⢻⣏⣿⣽⠀⠀⢿⡌⠙⡳⢾⠓⢦⡀⠀⠀⠀⠈⠉⠻⣥⣀⡜⠁
⠀⠀⠀⠀⠀⠀⢨⠷⠦⣔⣷⣞⡗⠀⠝⣚⠆⡓⠀⢡⠋⢠⣯⣿⣹⣧⢧⠀⠸⣌⢀⡀⠘⠬⠂⠬⠶⣄⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠸⡌⠇⣱⡘⠠⢀⢀⠀⡍⠓⠡⡴⠃⢀⢮⡴⡤⢿⣥⣿⠀⠀⢷⠈⣡⠭⠦⠁⣠⠅⣬⡳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠙⣄⠈⢆⠫⡴⠑⡼⠁⢀⡞⢞⣿⣟⣋⣻⡇⡆⠀⠘⡇⣨⣐⠩⠇⠁⠁⢉⡜⢡⠁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠓⠌⠳⠤⣉⢀⡞⠀⢀⣼⡱⢯⣿⣽⣋⣗⣁⢣⠀⠀⢻⣲⠇⠡⢐⣦⠶⡛⠁⣛⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣨⡄⠁⣹⠀⣀⣼⣶⣿⠿⠶⢯⣿⡏⠉⢸⡄⠀⢈⡧⠗⠚⠛⠀⠨⣀⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⣇⠈⠩⠙⢛⠛⠳⠶⣒⠚⠻⠻⡄⠸⣆⣁⣀⡬⠥⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠄⣁⣈⣉⣋⣈⣠⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠁⠀⠀⠀⠀⠉⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢰⠶⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠠⠄⡐⠂⠠⠈⠀⠈⡉⠞⠁⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠒⠀⠀⠀⠀⢀⠀⣠⣤⡤⠴⠊⠉⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⢸⡀⠰⠂⠺⣟⠁⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠁⠈⠀⣨⠀⠀⠤⡀⢤⠀⢤⣇⣤⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠀⠀⠀⠰⠁⠁
⠀⠀⠀⠀⠀⢰⠀⠃⠀⢹⠀⠀⠁⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠁⠠⠀⠀⠀⠀⠠⠂
⠀⠀⠀⢀⣠⢈⠀⠰⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢐⣦⣀⠀⠐⠂⠀⠤⠀⠠
⠀⠀⠀⠸⡕⠪⢄⡀⡆⠀⠠⠀⠀⠐⢄⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠠⠃⠀⠀⠀⠄⢃⡀⠸⡆⠈⢠⡈⢂⠐
⠀⠀⠀⠀⢀⠀⠀⠘⠐⠢⠀⡁⠀⠀⠀⠑⠂⠀⠀⣀⠠⠀⠀⠠⠄⡤⠀⢀⢠⢀⠂⠀⠑⢰⡇⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣦⠄⢀⠀⠀⠐⠉⠉⠐⠀⠀⠁⠀⠀⠰⠀⠀⢐⡂⠀⠀⢰⠓⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣆⠨⡇⠀⠀⠠⣤⠀⠀⠀⠄⢦⠀⠀⠀⠲⡇⠀⠈⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠄⠀⠀⠙⡅⠀⠀⠀⠐⡀⠀⠀⠀⢿⠀⠀⠀⠸⡄⠀⠀⢨⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⡀⠀⠀⢣⠀⠀⠀⠐⢁⠀⠀⠀⢸⡄⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠇⡀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠚⣆⠀⠀⠀⠃⡀⠀⠀⢯⠀⠀⠀⠈⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣯⣦⣧⣾⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠤⠔⠢⢤⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⣤⢤⠀⡠⢠⣄⢀⣄⢀⣀⠀⢠⣤⡀⣠⠄⡤
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠁⠉⠉⠉⠉⠈⠁⠁⠈⠉⠀⠉⠉⠁
⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠒⠒⠊⢳
⠀⠀⠈⠓⠒⢷⠒⠂⠸⠤⠖⠉⠛⣄⢀⡮⣀⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡨⠋⠀⠀⠉⠓⢤⡀
⠀⠀⠀⢀⠎⠈⠙⠒⠒⠒⠒⠋⠁⢀⠀⠀⠀⠀⠀⠀⠘⢉⡆
⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠠⠀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢳⠤⣀⣈⡠⢄⣀⠈⣀⣤⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠈⡄⢀⠀⠀⠀⠀⠀⢧⠀⠤⠀⠀⠈⠉⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠠⠤⣌⠢⡀
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠉⠉⠉⢨⠀⡠⡖⠁⠀⠀⠀⠀⠀⠈⡖⢄
⠀⠀⠀⡇⢠⠀⠀⡆⠀⠀⡇⠀⠀⢀⡀⢀⡠⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡌⠉⠭⠭⠭⠭⠍⠋⠥⠎
⠀⠀⢴⠃⠠⠀⠀⠀⠀⢀⣇⣀⣀⠀⠈⣉⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡀⠀⠀⠉⠳⣄⣠⣄⡀⢸⠉⠙⢏⠙⠛⢡⡟⢦⡀⠀⣀⠄⢢⢧⠄⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⣤⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠘⡏⠉⠀⠰⠚⠂⠀⠀⢴⠋⠂⠄⡀⠀⠀⠐⠂⠤⠔⠒⣁⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠶⣼⣄⣀⠤⢞⡥⠒⡂⠉⠁⠙⠁⠀⠊⠓⠲⠦⣅⠀⠀⠁⠀⠀⠀⠈⠀⠈⠗⠀⣤⠀⠀⣸⡀⠀⠀⠀⠀⠀⢸⠗
⠇⠀⠀⠀⠀⠀⠈⠂⡤⠏⠡⡀⠈⠑⠨⠑⣒⠀⠀⠀⠀⠒⠒⠂⢁⠅⠀⠀⢠⣀⣀⣀⣠⠀⠰⡀⠀⠇⠀⡀⠩⣷⣦⣤⣤⣤⣶⡧
⡆⠀⠀⠀⠀⠀⠀⡔⢸⠁⠀⠘⠢⠄⡀⠀⠀⠰⢁⠀⠀⠀⢀⡠⠂
⠁⠀⠀⠀⠀⠀⠀⣗⠃⣀⠀⠀⠀⠀⠈⠐⠒⠚⡳⣶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡬⠤⠭⢭⡭⠄⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⢄⠀⢀⡾⠄⡀⢥
⠁⠀⠀⠀⠀⢀⠞⠁⠉⢦⠤⠉⣽⠂⠠⠐⠚⡦⡀
⠁⠀⠀⠀⠀⠘⡀⠀⠀⠘⢄⣀⠼⡄⠀⠉⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡳⠄⣀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠚⠁
⣶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣰⠋⠉⠉⠉⣗⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣧⣄⠀⢀⣀⡈⠲⣀⣤⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢘⡷⡆⠀⢠⣄⠁⠀⠋⠀⣹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⣄⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⢰⡬⢤⡤⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⣆⠓⠊⠀⠀⣠⠔⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⠒⠉⠀⠀⠹⣷⣄⣀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢧⡀⠀⠀⢀⡏⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠓⠢⠞⠀⠀⠀⠀⠀⠘⢆
⣠⠶⠿⣽⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠁⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠦⣤⣀⣤⠖⣫⡥⢭⣑⣶⠚⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠔⠋⠀⠀⠀⢯⠙⢷⡄⠀⢧⡀⠀⠀⠀⠀⠀⠀⣸⠀⣞⢇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠻⣦⡀⠱⣄⠀⠀⠀⠀⠀⡇⠀⠘⢾⣷⢄⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⡓⠾⢦⣄⣀⣀⠀⣷⣄⠀⠀⠻⣮⠳⡄⠀⢀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⠈⠓⠒⢶⡙⠲⣝⢷⠈⠓⠦⣄⣹⡷⡈⠢⣼⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠑⢄⡀⠀⠹⢤⣈⣻⡆⠀⢀⣀⣩⢃⣷⣄⣈⣳⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠛⠛⠛⢻⠿⠆⠀⠀⠉⡿⠯⢭⣉⣽⣟⣋⣉⣉⣉⣉⣉⣉⣉⣩⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⠒⢾⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠙⠲⢤⣄⡀⠀⠀⠉⠳⣶⣦⣤⣤⣤⡤⢤⣤⣤⡶⢶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠒⠤⢄⣉⠉⠉⠁⠀⠀⢀⣤⣤⣭⣭⣥⣤⡴⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣤⡤⠶⠴⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣼⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠉⠓⠲⢤⣉⡙⠲⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡇⣰⡄⣄⢠⡀⣼⡇⢸⡇⢢⣰⢠⢠⣠⠋⡿⡶⣤⣬⣙⡶⡦⣶⣀⣠⠂⣼⣦⣾⣾⢦⠀⠀⡀⠀⠀⢉⠧⡎⡦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠁⠈⠀⠁⢹⠉⢹⡏⠉⠈⠈⠈⠈⠀⠁⠁⠋⠉⠉⠛⠧⢬⣉⠉⠛⠻⠇⢿⢹⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣦⣄⣸⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⢀⠀⣀⣀⣀⣀⡀⣀⣀⣠⢀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠠⠄⡀⢄⠀⢀
⠀⠀⠀⠀⠠⠬⠥⠤⠖⠒⠩⠭⢍⠀⠈⠲⠖⠋⠠⠀⠆⠀⠉⠓⡜⠒⣾⣔⡾⠄⠀⠀⠀⠀⠀⠀⡀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⡄⠂⠈⠄⠁
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠀⢀⡀⠀⠀⠀⠀⢠⠤⠔⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠈⠟⠛⠁⠀⠀⠀⠠⠼⠀⠉⠀⡀⠣⠀⠐⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠋⠀⠀⠀⠀⠒⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠀⠠⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠘⠰⢄⣀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠠⢀⠀⠀⠀⠀⠈⠀⡀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠈⠉⠱⠣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⠀⠀⠀⠀⠀⠀⠠⠽⠛⠀⠄⠀⠁⠈⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢄⡆⠀⠀⢘⠀⢀
⠀⠀⠠⠀⠠⢂⣀⣀⣀⠀⠈⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠀⣀⠀⠀⠀⠈⣑⡀⠇⠀⠀⠠⠄⠠⡰
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⠀⠀⠀⠀⠉⠄⣊⣁⡄⠀⠉⠀⠀⠀⠁⠀⠈⠀⠀⠀⢀⡂⠁
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⠙⠒⠈⠀⠀⠀⠈⠁
⠀⠀⣿⣿⢿⡿⣿⣿⡿⣿⢿⣿⡇⠀⠀⣠⣄⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⡄
⠀⠀⠘⣿⣶⣾⣷⣿⣷⣾⣾⡿⠃⠀⢿⣿⣿⣿⡿⠟⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠛⠁⠀⠀⠈⠂⠀⠀⠂⠑⠀⠁⠀⠀⠀⠀⠀⠠⠄⠤⠁⠀⠀⠠⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠸⠧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⣀⣀⢀⣁⡦⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⡁⠝⠂⠠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠰⣬⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠒⢠⡂⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⣡⡂⠀⠀⠀⠀⠀⡣⢠⡀⣄⣤⢀⡄⠘⣂⢉⣿⠁⣤⠲⠆⡾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢺
⠀⠀⠀⠀⠀⠉⣽⡍⠓⢀⠀⡀⠀⠀⠀⠤⠀⠠⠆⠀⠽⠇⠀⠀⠀⢀⠎⡀⠸⠶⠿⣯⣊⣁⣈⡁⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠁⠴⠤⠼⡄⣠⠄⠀⠀⡀⠀⠤⣡⠉⠁⠀⠀⠀⠀⠀⠈⠓⠛⠁⠒⠯⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⡀⠤⢦⢀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⣀⣀⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠸⠂⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠠⠂⣀⡀⠔⠀⠍⠀⠂⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠁⠀⠉⠑⠚⣶⣷⡞⠤⠀⠀⠀⠀⠀⢀⢾⣁⠉⡀⠜⠁⠁⣠⠀⠀⠴⡌⡀⣢⢴⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠘⡷⣤⣤⡀⡀⢠⣴⣶⠿⣾⣷⡀⠄⠀⠀⠀⠀⠈⠀⠈⠑⠐⠇⠤⠤⠤⠤⠼⠡⢚⡫⠕⠈⠓⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠘⣟⠁⠀⠀⠟⢱⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠀⠉⠁⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⣄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠓⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⠐⠒⠈⠁⠀⡀⠄⣀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠿⠗⠀⠀⢠⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢠⡰⠃⢀⣀⡀⢠⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠛⠛⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⣬⠂⣠⠅⠀⠀⠀⠀⠀⠀⠀⢺⠀⢀⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⠠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢤⢥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠖⠂⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⠁⠈⠀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢀⣦⡀⠀⠀⠀⣼⢂⡀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠆⠀⠀⠀⠈⠳⠾⢄⣗⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠤⢤⠤⣀⠄
⠀⠀⠀⠀⢠⡀⠰⣎⡀⠂⠻⠶⡶⠚⡋⡿⠥⣠⡆⠀⠀⠀⢴⠋⠁⠀⠀⠀⢠⠌⢁⣆⠄⠀⠀⠈⣭⡩⣷⡆⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⠀⠘⢦⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠃⠍⠄⠐⠊⡝⠀⠀⠀⠀⠱⠒⠄⢐⣀⢁⠀⠉⠉⠉⢀⣀⣰⣦⠴⠟⡟⠀⠀⠀⠀⠛⠀⠰⠆⠀⠀⠀⠀⠀⠘⠙⢿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⡠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣄⠀⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⣀⣤⠤⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

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

⠀⠀⠀⠀⡀⣀⡀⠤⠒⠈⢈⠇⠀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⠀⠐⠀⠐⡄
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠷⡤⠸⢗⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⡇⠝⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠐⠈⠁⠀⠀⠈⠀⢀⠄⠀⠀⡇
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠀⠀⡴
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣼⠗⡯⣄
⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠁⠀⠀⠀⣤⣷⣖⠅⣅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⠈⠀⠀⠀⠿⡿⡔⠈⠀⠀⠀⠀⠀⠀⠠⠄⡄
⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣰⢧⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⠀⠁⠇⠀⠀⠀⠀⠀⢸⣟⣿⣷⡃⠀⠀⠀⠀⠐⠈⠉⠊⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡅⠀⢘⡀⠀⠀⢰⡌⣽⣾⣿⣿⢫⢡⠀⠀⠀⠆⠀⠀⠀⠀⠐⠈⢠⡔⠠⢄
⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⠀⠄⢸⣻⣿⢿⠏⠀⠀⠀⠰⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠸⣯⡿⡟⢠⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⡀⠹⡷⠃⢸⠀⠀⠀⠈⠀⢠⠁⠀⠀⠁⠼⣄⠀⣀⡀⣀⠖
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠁⠀⠀⠅⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⠀⠏⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠛⠋⣩⢄⡼⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⢀⡀⠀⣸⣵⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣤⠀⠀⢀⡴⠋⠀⣠⠟⢁⣴⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⢠⡞⠀⣠⠾⠷⠖⠋⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣰⠏⠀⠀⠀⣼⣠⡞⠁⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⣩⡙⠳⢤⣄⡀⠀⣼⠃⠀⠀⢀⡏⠀⠀⠀⣠⠟⠁⠀⠀⢀⣠⠖⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⣄⠸⡆⠈⡙⠻⣧⡀⠀⠀⢸⡇⢀⣴⠾⢥⣤⣤⠴⠖⠋⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢠⣉⢳⣇⠀⢻⠀⠈⣷⠀⠀⣸⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠈⡉⠉⠳⣿⡀⠀⣸⣶⣿⣁⡀⠀⠀⠀⢀⣀⡤⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⠳⣍⡛⠛⠉⢉⣷⣏⠀⠀⠉⠻⣟⠛⢋⣉⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⢀⣀⣀⣀⣀⣈⡙⠛⣻⣿⠁⡟⠻⣯⡙⠓⠘⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⠴⠚⠉⠉⠀⠀⠀⠉⣹⠟⠁⠹⣄⢷⠀⢿⡙⠻⣝⡓⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠚⠁⠀⠀⠀⠀⠀⠀⣠⠾⣇⠀⠀⠀⠈⠳⢤⣀⣙⡀⠻⡝⢷⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⣠⠴⠶⠒⠒⣶⠞⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⣽⠉⠙⠛⠶⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⠶⠋⠁⠀⢀⣴⠞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣠⠶⢋⡇⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣴⢃⣠⠴⠖⣶⠶⠋⠁⢠⡞⠁⠀⠀⢀⡴⠋⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⢀⡾⠁⢈⣥⣶⠋⠁⠀⢠⠞⠋⠀⠀⣠⡴⠋⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⣰⣯⠴⠚⢹⣟⣡⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠚⠋⠉
⠀⠀⠀⠀⠙⠓⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣄⡜⢧⡄⠀⡄⢠⠈⡛⢠⢖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣏⠙⠁⡏⠁⢰⣄⢰⠃⠀⠀⠛⠃⠀⠰⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠁⠀⠚⠃⠀⠀⡄⣸⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣌⠀⠀⠀⠀⠀⢰⡥⠃⠂⠀⠀⣾⠽⠂⢁⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡀⠀⠀⠀⡇⠀⠀⡴⠀⠀⠀⠀⠀⠀⠨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⣤⡀⠀⠀⠀⠀⠀⣵⠀⠀⢠⠀⡠⠊⠁⠀⠀⢾⡭⠄⠂⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣆⣘⡅⠠⠆⠙⣤⡀⠀⠀⠀⠈⠀⢀⣦⠎⠀⠀⠀⠒⠀⠀⠀⠀⠰⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢏⠁⠘⢍⢡⡀⠀⠘⠃⠀⠀⠀⠀⠀⣰⢏⡀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠲⠼⠂⠀⠀⢈⢁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠘⠁⠢⣀⢀⠀⢰⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⠀⠀⠀⠀⠀⠀⠀⠈⡉⠘⠋⠓⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⡍⢩⣠⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡘⠂⠀⠀⠀⠀⠀⠉⣠⣄⢰⡀⠀⠀⠀⢠⡤⣄⡀⠀⠀⠰⡔⠉⣤⠇⠀⡠⢫⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡦⢚⡷⠀⠀⠁⢸⠁⠈⠀⠃⠀⠀⠈⢋⡄⠀⢈⠙⠓⠦⣤⣄⠀⠀⠀⠉⠀⠀⠁⠘⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡯⠀⠀⢁⡄⡆⢰⠇⠀⠀⠀⠈⠙⢤⠂⠀⢸⠃⠀⠀⠀⠁⠈⠑⠒⠀⠀⠀⠋⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣶⠴⡶⠋⠿⠀⠘⣻⠀⠀⣴⣄⢰⠄⠀⠀⠰⠀⢀⣬⠂⠀⠦⠉⠂⠀⢠⣤⣅⢠⣤⣤⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠶⡆⠈⠃⠀⠀⠀⠀⣄⠸⣼⠀⠀⠀⠐⠆⠀⠈⠛⠿⠧⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢹⣰⡄⠀⢞⠰⢶⠶⡶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⣀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠁⣼⡄⣿⡇⣤⢰⣄⠠⣦⢰⣦⢰⣿⠐⣴⡤⢆⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⡇⣿⡃⣶⣸⣿⢀⣿⢀⣿⢸⣿⠀⣿⠀⠸⣿⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠂⠈⠀⠉⠀⠉⠁⠉⠀⠉⠘⠁⠀⠉⠀⠉⠀⠀⠈⠉

⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⠘⠻⣷⡀⠘⠀⠀⠘⠈⡽⠃⠀⠉⠁⠂⠀⠀⠀⠀⠜⠀⠠⠀⠀⠀⣄⠀⠄⠀⠀⠀⠀⠈⠆
⡀⠁⠉⠐⠰⠦⡀⠀⠀⠀⠀⢠⣎⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⡤⠀⠘
⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢀⣼⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⠛⠒⠀⠠⠀⠀⠉⠁⠀⠀⠂⠐⠂⠀⠀⠀⠀⠤⠀⢷⡀⠄
⠠⠀⠀⢀⣠⠟⠑⠑⠞⠚⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠃⡸⣇
⣴⣶⠶⠋⠁⠀⠀⠀⠀⠀⠀⣀⣄⣠⠀⣀⡀⠀⠀⠀⠀⠀⠜⣦⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⣌⡀⡖⣢⠄⠀⠀⠀⠈⠳
⡀⠀⡀⠀⠀⠄⠀⠀⠀⠠⠜⠁⠀⠀⠀⠉⣿⣄⣀⡀⠀⠀⠀⠜⠳⣥⡄⠀⢀⡰⢃⡄⠀⠀⠀⠀⠛⠐⠬⡗⠻⣖⣺⣇
⡴⣒⠟⠳⠶⣄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⠁⠀⠀⠀⠀⠐⠶⠝⠳⠏⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠈⣿⡧
⡾⡃⠀⠀⠀⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠙⠉⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⣤⠀⠀⢀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⣦⣠⣤⠶⠂⠁⠀⠠⣰⣤⣟⠒⡀⠒⠋⠀⠀⠀⠀⠡⢠⡟⠈⠁⠀⠀⢠⡀⠑⢤⠀⠀⠀⠀⠀⢃⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡶⠟⠁⠠⠃⠀⢈⡅⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⢻⡆⠀⡤
⢀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠃⠀⠸⠂⠀⠀⠀⠀⠀⠈⠲⢦⣤⠤⠶⠖⠛⠡⣈⠀⠀⠀⣱⣶⠊⠧⠄⠀⢸⠀⢀⣀⠁⠐⠂
⠈⠀⠀⠀⠀⠀⠛⠥⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⢾⡿⠉⠃⠀⠀⠀⢸⡆⠀⠈⠉⠀⠐
⣀⣤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠈⠀⠀⠀⠐
⠟⢠⡈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⢷⣦⣀⠀⠀⠠⠠⠒⠦⣄⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣔⣀⣀⡤
⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⠀⢰⣡⢌⣠⣿⣷⡖⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⣤⡀
⠀⠀⠄⠠⣤⠇⢤⡰⠂⠀⠀⠈⢿⣂⠀⢡⡴⡷⣶⠋⠀⠚⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡖⠊⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠽⡃⢀⡀⠀⠀⠀⠀⠹⠿⣿⡶⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⣤⠿⠃⠀⠀⠀⠀⣀⣾⡇
⠀⠀⠀⠀⠀⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡦⠴⠋⠋⠀⠀⠀⠀⠀⠉⠁⣼⢆⡀
⣀⢀⡀⡤⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠃⠺⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡝⠀⠀⠀⠀⠀⠀⠡⣄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⢀⠈⠁⠲⠉⠀⠀⠀⠀⠘⠃⣰⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⠴⠂
⣀⣀⣐⣀⣀⣀⣀⣀⣀⣠⣄⣠⣀⣒⣀⣉⣛⣷⣠⣬⣃⣀⣁⣀⣀⣀⣀⣀⣀⣸⣋⣀⣃⣀⣂⣐⣃⣀⣀⣀⣀⣂⣸⣋⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣿⣿⣿⣿⣿⡟⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⢸⣿⣿⡿⣻⡟⠀⠐⠿⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀⠀⣼⡿⠋⠐⠋⠀⠀⣠⣶⣶⣾⣿⣿⣿⠿⠿⢿⣿⡿⠟⠛⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⠙⢿⡿⠛⣿⣿⣿⣿⣿⣿⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⣽⣧⣾⡟⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠨⡟⠳⢮⣷⡄⢙⣿⣿⣿⣿⣿⣶⣶⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⣿⠋⠙⠗⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠛⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣧⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⡜⠛⠻⠟⠻⠛⠉⠉⠛⠉⠛⠛⠋⠙⠋⠉⠉⠉⠉⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠈⠉⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣶⣶⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠿⠿⠿⠿⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⣻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠀⠊⠑⠀⢙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⣀⠄⡀⠀⠀⠀⠀⠀⠘⣰⣤⠔⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠋⠐⠲⠟⡩⢆⠈⠀⠀⠀⠀⠀⠀⢇⣀⠀⢠⡴⠂⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣢⣬⣤⡀⠀⡄⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡇⠀⠙⣄⠀⠀⢨⡛⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⢣⡀⠀⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠈⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠱⣤⠁⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⣀⣤⣤⣄⣀⠀⣀⣀⡄⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⣴⣻⠃
⠀⠀⠀⠀⢀⣠⣴⣶⣆⠈⣿⡟⢳⡄⠈⢹⣿⠿⠟⠋⢉⣭⣅⣀⡀⠀⠀⢰⢻⣆⡀⠀⠀⠀⡠⠀⠁⠀⠀⠀⢀⣴⡾⢡⡏
⠀⠀⠀⠀⠈⢛⠻⣿⣿⣶⣿⣭⣬⠍⢠⡀⢲⣿⠻⠇⠈⠳⣮⣛⣄⡈⠀⣿⠘⠿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠁⣼
⠀⠀⠀⠀⠀⠈⢦⡹⣿⣿⣿⣿⣿⠦⣈⠁⣼⠿⠀⠀⠀⠀⠈⣿⣾⢁⣀⡏⠀⠀⣿⣿⣦⣄⠀⢀⣤⣶⣿⡿⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⠛⠛⠛⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⠇⠀⠀⠈⠙⢿⣿⣿⣿⡿⠛⠉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣜⡻⢿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⢰⡿⠿⠷⠿⠶⣿⣇⣴⣷⣤⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⣶⣶⣤⡀⠀⣀⣰⣶⣶⣶⣴⣶⢾⣷⣦⡀⣴⣶⣶⣤⣴⣦⣤⣄
⠀⢸⡇⠀⣄⡀⠀⢸⣿⡇⠘⣿⡇⠀⠀⠀⠀⢼⡟⠀⠈⣿⢿⡀⢿⡇⠀⣻⠏⠀⡿⡿⠁⣀⣀⠈⠻⣦⣿⡏⠀⢀⣀⠀⠉⢻⡷
⠀⢺⠁⠀⡿⣻⣿⣾⣿⠃⠀⣯⠁⠀⠀⠀⠀⣾⠃⠀⠀⢸⣿⣇⠈⣿⣺⡟⠀⢠⣿⠃⢰⣿⢻⣷⠀⢻⣭⡇⠀⢸⣟⡗⠀⠀⣿⡆
⠀⣿⠀⠸⠟⢹⣿⢸⣿⠀⠀⣯⡀⠀⠀⠀⣰⡟⠀⣼⡄⠀⣿⣿⡀⠸⣿⠁⠀⣼⣿⠀⢿⡿⢸⣿⠀⢸⣿⠃⠀⠈⠁⠀⠀⢠⣿⡆
⢈⣿⠀⠀⣦⣬⡿⢻⡏⠀⠀⣿⣇⣀⠀⠀⣿⠇⢠⣿⠷⠀⠸⣿⣇⠀⠉⠀⢠⣿⠿⠀⠸⡶⣾⡿⠀⢸⣿⠆⢠⣤⠀⣴⣶⠿⠟
⣸⡇⠀⠀⣿⡟⠋⣻⡇⠀⠀⣿⠿⢻⣷⣠⡾⠀⢤⣤⣀⠀⠀⢻⣿⡄⠀⠀⣼⡽⢷⠄⠀⠀⠈⠀⠀⠿⠿⠀⠸⠿⣇⠈⢿⣶⣄
⣿⡇⠀⠀⢹⠇⢰⣿⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⢸⡟⣏⠀⠀⠘⣿⣷⠀⢠⣿⡁⠿⣷⡀⠀⠀⠀⣼⣿⣿⠀⠀⣿⣿⣆⠀⠙⢿⣧⡀
⠙⠋⠹⠿⠿⠗⠈⠻⠿⠛⠿⠿⠿⠿⠛⠿⠿⠿⠻⠟⠿⠿⠟⠿⠿⠋⠿⠿⠟⠀⠀⠹⢿⡷⢶⡟⠛⢻⣿⣶⣶⣿⠟⢿⣿⠟⢶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⢉⡇⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡀⠐⠀⠇⢀⠀⠀⢀⡀⢠⠉⠒⢁⡀⢈⡼⡃⢀⠴⣀⠂⠐⠃⠈⠁⣁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠔⣸⠀⠀⠀⠙⠾⠀⢠⠊⡴⠋⢠⠋⠁⠀⠁⠄⠄⠀⠀⣤⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢢⠴⡀⠀⠃⠀⡞⠀⠀⠴⡇⠀⠀⡆⠀⡀⢀⠌⢸⣠⠴⢁⣠⠼⠊⠀⠀⠀⠤⣄⡠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⡀⠙⠤⠀⠰⠁⠀⠀⢰⠃⠀⠀⣧⠀⢡⠊⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⢀⣀⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠚⠀⠈⠂⠀⢘⠃⠀⢀⡴⠃⠀⠀⠀⠟⣰⣥⠶⠋⠁⣀⠸⢭⣤⣤⢌⣭⠒⠶⠤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠎⢀⠖⠁⠀⠀⠀⣠⠴⠚⠉⠀⢐⡀⠀⠀⠈⠁⠀⠀⠘⠑⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠱⡜⠐⠁⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢖⠦⠤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⠔⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠔⠁⠀⣠⠖⠁⠘⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠐⠊⠀⢀⡠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⠠⢀⠀⠀⠚⠉⠁⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣦⡀⠉⠲⣄⢀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠡⡀⠀⠱⢤⣀⡀⠈⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠢⠖⠢⠒⠚⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠀⠟⢘⡩⣿⠈⣿⠉⠸⣯⠀⣿⢠⣾⠙⠋⢿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠻⠗⠿⠀⠟⠀⠀⠾⠃⠿⠈⠻⠆⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈

⡿⢿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡄⠘⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⠈⠘⣟⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣹⣿⣾⠿⠿⣀
⠿⠶⠒⠒⠒⠀⣠⣤⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⢿⣇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⣀⡀⣀⡀
⠀⣶⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠲⠆⣠⡶⠆⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣙⠓⢀⡉⠀⠀⠀⠀⢀⡀⣾⡿⠂⠀⠀⠉⠉⠀⠀⠙⠻⠆
⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠲⠄⢀⣀⠀⠀⢀⣥⠀⠀⣴⡿⠀⠈⣿⠀⠀⠀⠀⠈⠀⠀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠘⣿⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠀⠘⠟⣠⡴⠙⣷⠀⠀⠉⠀⠀⠘⠀⢀⠶⠂⠉⠙⠿⠃⠈⠉⠹⠷⠀⢰⣦⠀⢠⣽⣶⣄
⠀⠀⠀⠈⢠⠀⠀⠀⠘⠷⠦⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠁⠀⣿⠀⠀⠀⠀⠺⣄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠉⠉⠁⣶⣤
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠉⠉⠀⠈⠓⠦⠀⠀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣿⠀⠀⠀⢰⡦⠀⠀⠀⠰⠶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠇⠀⠀⠀⠀⡟⠀⠀⣠⣿⡶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⡿⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢻⣤⠀⠸⠇⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣤⣤⡤⠀⠘⠛⠁⠀⠀⠀⠉⠙⢿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢨⣼⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃⠀⠉⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠒⠺⠿⠛⠂⢀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⠂⢛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⣠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠃⣀⣀⠀⠀⠈⢿⠀⠀⠀⠀⠙⠛⢀⣼⡆⠀⠒⠶⠂⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠉⠃⠀⠀⢨⣷⠀⠀⢀⣀⠾⠟⠉⠁⠶⢶⡆⠀⢀⣈⡁⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⢀⣤⠀⢸⣿⠀⣀⠀⠀⠀⠀⠰⠆⠈⠙⠃⠠⢄⣉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠘⠋⣴⣿⡿⠀⠉⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠻⢆⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠆⠀⠀⠀⠀⠈⠛⡄⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠼⠛⢻⠀⠀⠀⠘⣦⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡎⠁⡄⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡷⠀⠀⠁⠀⠀⢱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠙⠋⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⣿⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣤⡄⠀⠀⠀⠀⠈⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⡀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢤⡄⠀⠀⠀⠀⣤⡤⠈⠛⠛⢳⣤⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠀⠛⣿⣦⣴⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠛⠁⢀⠔⠂⠀⠀⠉⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⢀⡀⠚⠀⠀⠀⠀⠀⠘⠀⠀⠻⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⢀⡄⠛⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣤⣄⠛⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠒⠶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀⠀⢀⣤⡖⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠶⠄⣶⣇⠀⠀⠀⠸⠿⠀⠘⢋⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠶⠀⠶⢶⣤⡈⠻⠆⠀⠀⠿⣯⠀⢰⡄⠀⠀⢿⡶⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠞⠋⠀⠀⠀⠀⠀⠙⢣⠀⠀⠀⠀⠀⢿⠀⠸⣿⡆⠀⠈⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠳⣀⠀⠀⠀⢸⠀⢀⡾⣧⣄⠀⠀⠀⡀⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠈⠀⠈⠀⠘⠻⣶⠶⢾⣧⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢈⡀⠘⣷⡆⠀⢴⡶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢺⡿⠁⠹⠃⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⢠⠀⢸⠇⠀⢀⠀⠀⠀⠀⢱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠛⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷⣀⡀⠀⠀⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠿⠀⠀⠀⠀⣀⣤⠛⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⠙⠛⠛⠃⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠀⠀⠀⢰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣴⡿⠃⠠⠘⢿⣗⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⠿⢀⣴⣄⢰⣿⣿⣦⡤⣿⠁⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣼⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠁⠀⠈⠉⠀⠀⠉⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⣤⣤⣄⣤⣤⣤⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⡀⣀⣄⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⣠⠶
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡸⠆⠦⠾⠬⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⠀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⡖⠀⡐⠂⠂⢰⣭⡥⠴⠂⡍⢀⡜
⠀⢀⣄⠀⣀⣀⠀⣀⣀⣀⢸⠠⠄⠄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠑⢾⡄⠀⠀⠀⠀⠀⢀⠾⣉⠠⠄⣐⡤⣬⠖⠊⠉⠀⠀⢀⡨⠞⢡⡟
⣠⠋⠖⣻⠛⢄⢀⠟⠓⢹⣟⡀⣀⡀⠀⠕⠠⡄⠀⠀⠀⠀⠀⠀⢗⠐⠆⠂⢒⠇⠀⠀⠀⠀⠀⢺⡀⠻⣷⣟⡱⢲⠁⢀⠤⠇⣚⣼⡿⠖⢚⡵⠃
⠀⠀⠓⠊⠁⠉⠉⠉⠉⠩⠈⠉⠁⠉⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣲⣐⡀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣈⠉⠓⠻⣤⠷⢞⣋⡥⠚⠓⠋⢁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠦⠤⠤⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡇
⠀⠀⠀⣀⣤⣤⠀⣠⠖⠡⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣰⢲
⠀⠀⠀⠙⣝⡼⠋⠉⠂⠄⠀⡟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣠⠖⠒⠢⣤⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠶⠧⠬⠷⢄
⠀⠀⠀⢀⣼⣄⣀⣁⠀⢃⣀⣄⣀⣧⡀⠀⠀⠀⢤⠆⠲⠢⣄⣜⣧⠛⠒⠒⣾⣡⣀⣧⢠⢔⠞⡄⠀⠀⠀⠀⠀⢸⠁⠁⠆⠀⠀⢠⡇
⠀⠀⠀⠈⣷⠒⠤⠭⠭⡭⠭⠥⠴⢾⠃⠀⠀⠀⢈⢦⡬⠷⣗⣾⣀⠀⢀⡠⢊⡗⣻⣉⠽⢿⣴⠀⠀⠀⠀⠀⠀⠘⣯⢭⣿⣿⣿⣽⠃
⠀⠀⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⣸⣾⣡⣫⠖⣩⣹⣶⠶⢯⣉⡀⣉⡶⠏⣽⠀⠀⠀⠀⠀⠀⠀⢰⡠⠀⠀⠀⠘
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠃⠀⠠⢀⡟⠀⠀⠀⠀⠘⣟⠲⢬⣈⣉⣙⣦⠀⠀⠉⣽⣛⣡⡴⢚⠟⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠙⣶⢤⣤⣤⡤⢶⡞⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣉⡉⠙⢦⣤⠞⢉⣉⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣀⣄⣺
⠀⠀⠀⢀⡠⠴⠶⠶⢶⣤⣤⡆⠀⡶⣢⠶⡀⠀⠀⠀⠀⠀⠰⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⡴⡋⠨⡁⣀⣠⣤⠯⠞⣤⣾⠞⣵⠾⠃⠀⠀⠀⠀⠀⠀⢀⣤⠟⠛⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⢰⠁⢀⣦⣴⣿⠋⢀⠴⠄⡷⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⢯⢀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠈⠉⠳⡀
⠀⣿⣶⡫⠝⡳⣆⢂⠳⢥⣠⠴⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢰⢻⠛⠛⠛⠛⠛⡟⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠄⠀⠠⠠⠅
⠀⣿⠳⢽⣿⣷⣮⣷⣶⣶⣶⣶⣿⡿⢟⡇⠀⠀⠀⠀⠀⠀⢸⠈⣦⠠⠐⠀⢀⡗⢸⠀⠀⠀⠀⢀⣀⡜⢉⠉⠉⠫⠍⠉⠉⠉⠉⠩⠋⠀⠈⢱⣀
⠀⠘⣷⣤⣈⠉⠉⠉⠉⠉⠉⠉⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠘⢦⣈⣓⣶⣖⣋⣠⠞⠀⠀⠀⠀⠙⠒⡆⢨⣷⣦⣤⡄⠀⠀⠀⠀⠀⠈⡂⠀⢰⠂
⠀⠀⠈⠳⢬⣉⡛⠒⠀⠀⢀⣠⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣉⣹⣁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣋⣙⣋⣀⣀⣀⣀⣀⣈⣉⣁⣠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢢⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢠⡀⣶⡾⡄⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠙⢻⡑⣌⠃⢀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠠⡐⡇⠈⡄⠈⡆⣰⠃⠀⠀⡠⠀⢀⡩⠄⠚⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠐⡧⠀⢹⡆⡽⡇⢀⡴⠋⢀⠔⢁⡤⠒⠀⢀⠀⠄⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢾⣿⣮⣷⡀⢈⠏⢴⠛⡊⢀⡴⠁⣐⣁⠤⠤⠶⠧⠀⢐⠒⢂⣂⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠀⠀⠀⠀⠀⠀⢹⠃⠘⠦⠘⡃⣱⣿⠿⠿⠿⠷⢖⡀⢀⣀⣤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⣀⡀⢀⣀⡉⠉⢑⣦⣬⡦⠤⡬⡀⣻⣋⠓⣄⣀⣀⣠⣴⠶⠚⠉
⠀⠀⠀⠀⠀⠔⠋⡉⠭⢄⣀⠀⣀⣤⠟⠋⢉⡏⡘⢱⣶⡀⠀⠳⠤⠀⠺⡯⢄⡀
⠀⠀⠀⠴⠀⢀⣁⡀⢀⣀⡴⠛⡱⠃⢀⡠⢋⣴⡷⢿⢿⠹⣶⡄⡂⢀⡀⢻⡄⠌⠢⣄
⠀⠀⠼⠀⠉⢀⣠⠾⠋⡍⣠⠊⠀⣠⣮⣾⣯⣍⣀⣤⣧⠜⠓⠉⠷⣦⣴⡀⠙⡖⠀⠀⠁⠢⡀
⠀⡴⢀⡴⠒⠉⢉⡤⣊⣴⡧⠔⠋⠁⣿⠀⠠⠺⠟⠢⢸⡄⠀⠀⠀⣈⢳⣌⠑⠸⠒⠠⠀⠈⠙⠂
⣠⣔⡯⠴⠤⠒⠛⠉⠉⠀⠀⠀⠀⠀⡇⠰⢀⢼⠢⠀⠀⢳⡀⠀⠀⡟⢸⡟⢷⡒⠂⠤⣐⠀⠀⠀⠳⡄
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⠀⠀⣷⣄⠀⠁⠈⠀⠀⠹⣦⣄⠀⠁⠂⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠸⠈⡀⠀⠀⢀⡟⠈⠳⣔⡀⠈⡀⠀⡇⠙⢍⠁⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣴⠀⠀⢸⡇⠀⠀⠈⠳⡄⠐⠀⠘⠀⠈⡟⢏⠑⠢⡁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⠋⠀⠀⡽⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⠀⢳⠈⠳⢄⠈⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠢⡈⠂⠀⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⢄⡀⠀⠘
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣒⣊⠩⠍⡀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠒⠂⠒⠒⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣨⠔⢊⣭⣵⢒⡠⠊⡰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⠁⠀⠀⣤⣬⡸⣿⣏⣽⠆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣡⢖⠋⣠⢞⢽⣫⠟⡡⣢⢞⡀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠂⢁⣤⣤⡘⠿⠟⠡⢲⣷⣿⣦⢀⠃
⠀⠀⠀⠀⠀⠀⣠⠞⡡⢊⠕⣡⠞⢁⡵⢋⡴⣫⡞⡱⣫⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⢤⣾⣿⡦⠿⣿⡿⠃⡄⣠⣬⡜⠿⠿⢏⡞
⠀⠀⠀⠀⠀⡼⣡⣪⠞⣡⠞⣡⡴⢋⡴⢋⡴⣫⢞⡵⡱⡚⠀⠀⠀⠀⠀⠀⠀⢀⡜⠠⣈⣬⣿⢟⣿⣷⣄⡰⠏⠁⠻⠿⡃⠤⢁⠎
⠀⠀⠀⠀⢸⠕⣵⠃⣴⢋⡾⢋⡴⢋⡴⢋⡴⣫⢞⡴⡵⠁⠀⠀⠀⠀⠀⠀⠀⡞⠀⠁⢿⡿⠟⠘⠿⠛⠁⠁⣰⣿⣿⡆⠄⡰⠋
⠀⠀⠀⠀⡏⢸⠃⣾⣣⢏⡴⣫⠴⣋⡴⠫⢞⠅⣨⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢡⣾⣿⡇⢠⠴⠋⣼⣿⣷⠘⢛⠟⣡⠞⠁
⠀⠀⠀⠀⢧⢾⢸⠃⣡⠋⢚⣵⠞⢁⣤⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⡛⡩⡎⣵⣿⣶⠈⠉⡁⠐⡡⠚⠁
⠀⠀⠀⠀⠸⡀⠈⠔⣡⠞⣋⠴⠞⣋⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠣⠡⠤⠝⠛⠁⣃⡥⠴⠋
⠀⠀⠀⠀⠀⠑⠗⠂⠐⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠓⠒⠒⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠐⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⢀⠄⠀⠂⠀⡀⠀⠑⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠈⠀⢀⡀⠀⠀⠤⠀⠀⠀⠁⠀⠘⠀⢺⠉⠈⠋⠊⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⢜⡁⠉⣈⣙⣉⢩⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠈⠠⠂⠐⡄⠀⠐⠄⠠⢀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠃⢠⢃⠀⠀⠈⠀⠀⠀⠐⡂⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⢰⠀⠃⡆⠂⠀⠀⢰⠁⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢃⠀⠀⠀⠁⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⠃⠀⠀⠀⠄⡀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠸⣯⡉⣿⣿⣿⣿⣿⣿⣿⣿⠋⣩⡿⠀⠀⠀⠀⠀⢿⣉⢹⣿⣿⣿⣿⣿⣿⣿⡟⣉⣽⠇⠀⠀⠀⠀⠀⢺⣍⡉⣿⣿⣿⣿⣿⣿⣿⣿⢉⣩⡿
⠀⠀⠀⠀⠀⣠⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣧⡀⠀⠀⠀⠀⠀⢀⡼⢻⣿⣿⣿⣿⣿⣿⣿⡟⢿⣄⠀⠀⠀⠀⠀⠀⢀⣠⠟⣿⣿⣿⣿⣿⣿⣿⣿⠻⣄⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⡇
⠀⠀⠀⠀⣿⢶⣦⣤⣴⡶⣶⡴⡖⣶⢦⣆⣴⣼⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⣾⡇
⠀⠀⠀⠀⣿⢺⣏⣿⡴⣿⠾⡾⣭⣻⣻⣟⣿⢺⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣇⠀⠈⢢⣄⠀⠀⠀⠀⠀⣴⣯⣿⢻⠃
⠀⠀⠀⠀⣿⢹⠻⣾⢫⣸⡓⣻⢚⣾⢿⣼⢿⢻⠇⠀⠀⠀⢸⣷⣛⣶⣮⣼⠃⣿⢻⣟⢰⣧⡿⢶⡟⠀⠀⠀⠀⠀⣏⠻⣦⣘⣿⣷⣤⣴⣶⣿⣿⡿⠃⣸
⠀⠀⠀⠀⢸⣔⢿⠟⢛⣿⣿⣙⣞⣛⣆⢷⢏⡽⠀⠀⠀⠀⠀⣿⣯⣉⣿⣾⣯⣽⣯⣤⣿⣺⣟⣿⠇⠀⠀⠀⠀⠀⢻⡀⠈⠛⣻⣿⣛⣻⣟⣛⡉⠀⢀⡟
⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠙⠉⠉⠋⠙⠋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠙⠉⠋⠉⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠉⠉⠉⠉⠉⠉⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⡤⣤⠀⠀⠀⠀⠀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣧⣤⢤⡄⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄
⠀⠀⠀⠀⠈⠓⣦⣿⣿⣿⣿⣿⣿⣿⣿⣦⡖⠋⠀⠀⠀⠀⠀⠛⢲⣼⣿⣿⣿⣿⣿⣿⣿⣇⣴⠚⠁⠀⠀⠀⠀⠀⠘⠳⣦⣿⣿⣿⣿⣿⣿⣿⣿⣴⠞⠋
⠀⠀⠀⠀⢠⡶⠋⠙⠛⠛⠛⠛⠛⠛⠛⠉⠛⣦⠀⠀⠀⠀⠀⣴⠛⠙⠛⠛⠛⠛⠛⠛⠛⠃⠙⢶⡄⠀⠀⠀⠀⠀⣠⠞⠃⠛⠛⠛⠛⠛⠛⠛⠛⠈⠳⣄
⠀⠀⠀⠀⣾⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⣋⣤⡶⣋⣿⡛⠿⣿⣫⣽⣿⡇⢹⡇⠀⠀⠀⢸⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⣿⡟⢿⣿⡏⣽⣿⣦⣹⣿⣏⢻⣧⣾⡇⠀⠀⠀⢸⣿⣭⣽⣷⣞⣿⣯⣿⣯⣹⣾⢿⣽⣿⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⣿⣿⣮⣿⣿⣿⢰⣿⢟⣽⣿⣶⢿⣿⡇⠀⠀⠀⢸⣿⣻⠿⣾⢿⣿⣵⢟⣽⣛⣿⣇⣿⣿⠀⠀⠀⠀⢘⣿⣧⠿⣶⣿⢿⣷⣾⣛⣿⣠⣼⣷⣿⡇
⠀⠀⠀⠀⣿⣿⢋⣽⣧⣹⣤⣿⣾⢿⣷⣶⡾⣿⡇⠀⠀⠀⢸⣿⣿⣻⣷⣿⣽⣿⡿⢟⣿⣧⣿⣹⣿⠀⠀⠀⠀⠀⣿⣿⣾⣛⣿⣿⣥⣼⣟⣻⣿⣧⣿⣿⠁
⠀⠀⠀⠀⢻⣿⣿⢟⣿⣿⣛⣉⣻⣧⣿⠟⢠⣿⠁⠀⠀⠀⠸⣿⣍⠻⣿⣾⣯⣙⣿⣿⣯⣽⠾⣿⡏⠀⠀⠀⠀⠀⣿⣿⣹⣿⣛⣿⣉⣽⡿⢿⣼⡟⣻⣿
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠏

⠀⢂⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠲⠶⠒⠉⠀⠀⠀⠀⡇
⠀⠠⠀⠀⠈⠂⠄⡀⠀⠀⠀⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⢠⠊
⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⣠⠶⢦⡄⣠⠶⢦⡀⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⢿⣀⣠⠇⢧⣀⣠⠇⣿⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⠀⢀⢤⡀⣀⡌⣠⢤⠀⣬⡀⣠⢄⠀⡀⡀⣄⢀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠠⠛⠇⠳⠤⠀⠟⠄⠳⠼⠂⠟⠅⠣⠴⠃⠧⠇⠏⠻⠸⠤⠃

⠀⠠⠤⠄
⠀⠈⡉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠁⠀⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠐⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀
⠐⠈⠀⠀⠀⠈⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠊
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⡪⠂⠀⠀⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⠁⠀⠀⠈⠹⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⢀⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠑⠄⠀⠀⠀⣀⠠⠂⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡀⠙⣦⡠⠄⠒⠒⠒⠒⠒⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢰⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠋⠀⠀⡴⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠁⣀⠰⡟⠃⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢠⣼⠀⠀⠀⠀⠀⠙⠋⣀⠀⠉⠉⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠋⠉⠲⠄⠒⠉⠀⠀⠀⠀⡄⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⣀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡰⠋⠁⠐⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠤⡱⢄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠹⣦⡈⠋⠴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⣤⣤⣤⠤⠤⠐⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠘⢷⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠠⠄⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠸⢡⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⠢⠄⠀⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠂⠘⢆⠀⠀⠀⠀⠀⢀⡀⡤⢸⠇⠀⠂⠀⠀⠀⠀⠀⡸⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠓⣤⠠⠔⠒⠈⠁⠀⣸⢤⣄⡀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠦⡀⠀⡴⠁⠀⠀⢀⡠⠔⠋⠀⠀⠈⠉⠛⠲⠦⠴⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢟⡤⠤⠤⠤⠴⠀⠉⠳⢄⠀⠀⠀⠀⢀⡜⠛⠦⢤⡄⠉⠀⠀⢦⣤⣄
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡄⣠⠊⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠉⠑⠒⠒⠒⠀⠐⠒⠒⠢⣤⠔⠁⠈⠻⣦⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠵⠾⠅⠴⠂⠀⠀⠀⠐⠀⠒⢾⣶⣦⣠⡄
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢐⠋⢉⣉⣉⡙⢛⣛⣛⡛⢛⣛⣋⡉⠉⠉⠉⠀⠉⠛⠛⣛⣛⣛⠛⣛⣛⣛⠁⣀⣦⣀⢰⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂⠿⠉⠉⠁⠘⢋⣉⡟⠸⣏⡉⠟⠀⠀⠀⠀⠀⠀⠸⣿⣩⣿⠺⣿⣭⣿⠺⣿⣭⣿⢿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠦⠤⠴⠦⠤⠬⣭⠤⢤⣤⢤⣤⠤⡤⢤⣤⣤⣤⣤⣬⣭⣥⣤⣬⣭⣥⣤⣈⣉⣅⣸⡇
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡧⠀⠀⣴⣀⣀⣂⣆⣀⣀⣀⣠⣀⣀⣒⣲⣒⣒⣒⣚⣓⣚⣛⣛⣛⣛⣛⣛⣛⣿⠍⢹⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⡤⠤⠄⣠⡀⣀⢀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⠁
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠂⢰⠒⠒⠒⠒⡖⠲⠒⠒⠒⠒⠋⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠧⢤⡤⢤⣤⣤⡤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠙⠲⠤⢧⡀⠘⠒⡤⢄⡀⠠⢄⡀⢀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠒⠛⠛⠋⠛⠓⠛⠛⠃⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢄⡀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠻⡀⠀⠀⠘⠀⠁⣄⠀⠀⠀⢠⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠐⣦⣀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠹⣆⠀⠀⠀⠀⢻⠀⠀⠀⢁⡀⠀⢰⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠤⢄⡀⠀⠀⢸⠀⠀⠘⣦⠀⣀⡀⠀⣤⠌⣠⠛⠁⠀⡆⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠅⠀⠀⠈⣿⠏⠹⢦⠍⢰⠇⠀⠀⢰⠁⢀⡾⢁⠀⠀⠀⠀⣀⠤⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⠀⠈⠳⣶⠀⠀⢀⡌⠀⠆⠈⢻⡇⠀⡆⠀⣎⣰⠏⠀⠌⠀⢠⠄⠂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠉⠳⢤⣽⠀⠀⣼⠁⠀⠀⠀⠀⢱⡀⠇⢠⡿⠁⠀⣾⡄⠈⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠰⢶⣶⣦⣄⢀⡀⠈⠳⣄⠀⠈⠳⢦⡟⠀⠀⢀⡀⠀⠈⣷⣶⠎⠀⢀⡾⠋⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢌⡋⢤⡉⠠⠙⣦⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠃⠀⣰⣏⣀⣀⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡙⢦⣀⠙⢷⡀⠀⠀⠀⠻⣤⡁⠀⣰⡦⠚⠃⠉⠉⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⡀⠀⠙⢀⠹⣶⠘⠛⠂⣤⣄⡀⠘⣧⠞⠁⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠙⠦⡀⠳⣄⠀⠀⠈⠉⠲⣦⡀⠀⠀⠀⢀⠀⣤⣼⠀⠒⠒⠀⠀⠉⡡⠿⠟⠁
⠀⠀⠀⠀⠀⠈⠓⢂⡀⠶⠄⠘⠂⠀⠛⠶⢶⠀⠹⠆⠀⠀⠀⠀⠀⠻⡄⣀⡴⠞⠋⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⡖⠒⢀⠤⢄⣀⠳⠀⠸⠦⢄⠀⣀⠀⠤⠾⣁⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠻⠖⡄⠐⠀⠉⠛⠻⣦⣀⠀⠛⠁⠀⠀⠀⠈⠳⡆⠁⠀⢀⡄⢀⣀⣀⠠⠐⠒⠚
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠐⠦⡄⢠⣄⠢⡔⢄⠀⠀⠙⠲⣤⡀⢀⣤⠄⠒⠚⠒⠂⠰⢿⣦⠈⠉⠉⠐⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠼⣿⣦⣤⣄⠀⠀⠈⠃⣀⡀⠆⠀⠀⠀⢀⣹⣧⠀⠀⠀⠀⡀⣤⣥⡴⠶⠖⠈⢉⠉⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠛⠷⢶⣋⡉⠿⠶⣉⣀⠀⠀⠀⢀⣀⠴⠟⣤⠤⠞⠋⠀⣀⡀⠀⠠⢤⡤⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡒⠂⢠⠈⠳⠖⠠⢀⠈⠛⠷⡁⠀⠀⣀⣀⡈⠁⣤⠶⠋⢉⡄⠐⠀⠉⠁⠀⢠⡄⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣶⣶⠞⠿⠓⠚⠓⠨⢷⣄⠀⠹⠒⠀⠀⠈⠙⠛⢠⣤⢀⡈⠀⠈⠁⢴⣀⠤⠌⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢈⣧⣤⣤⢠⡶⠒⠂⢠⡈⠣⣄⠘⣀⠀⠀⠀⢠⡴⠒⠻⡷⠖⠂⢠⣄⡉⠀⠀⠱⠃⡀
⠀⠀⠀⣀⠀⠀⡌⠁⠲⡀⠸⡇⠀⡤⠁⠤⠀⠀⠀⠘⠆⠙⣄⢉⡷⠉⠹⢦⣤⠈⠀⠈⠀⠁⠄⠘⡉⠀⠂⢲⠀⠑
⠀⠀⠀⠈⠓⢰⡿⠀⠀⣾⣶⠷⠦⣤⠄⣰⡆⠀⢰⡷⡤⠚⢻⡶⢀⣤⣀⠁⠀⠈⢠⡌⢱⣣⣄⠘⠗⠀⠈⠀⠶⡈
⠀⠀⠀⠀⣠⡋⠀⠀⡰⠋⠃⡀⠀⠀⠀⠙⡛⠃⠀⠛⠆⠀⠀⢳⣀⡄⢩⣐⡦⠈⠛⠀⢀⠍⠛⢳⡆⠀⠀⠀⣀⠙⠀⠀⠰⡆
⠀⠀⠀⡐⠿⠃⠀⠰⠁⠀⠀⠑⠀⠀⣀⠊⠀⣀⡤⢴⣤⣴⣞⠉⠁⡧⠈⠁⠐⣦⠄⠛⢫⡤⠀⡀⠘⣂⡀⠀⠁⠀⠀⢠⠄⠃
⠀⢰⡤⠃⢀⢀⡼⢤⡀⠀⢀⣤⠲⡖⠙⠓⢾⡋⠀⣸⡆⠀⠙⢇⠁⠀⠀⠀⡌⠁⠀⠀⢸⠀⠀⢹⡞⠉⠉⠖⠂⠀⢠⡦⠀⣤⠰
⠀⢸⠇⢀⡼⠁⠄⠀⢹⡟⠉⢀⡴⠃⠀⠀⢀⠉⠈⠉⠳⠀⠀⠺⣷⠶⠶⢤⡿⠀⠀⠀⠊⠀⠁⠰⡇⠀⠀⠋⣀⣼⡏
⡀⠘⢀⡌⠳⠖⠀⠀⠀⠛⠰⠂⢡⠀⠀⠀⠸⠂⠀⠀⠀⠀⣠⠞⠁⠀⡀⠀⠘⢣⡀⠀⠀⠀⠀⠀⢧⡀⠀⡀⠀⢸⡇⠀⠀⠰⡆
⠈⣄⡿⠁⠀⠀⠣⠄⢀⣀⠀⠀⠀⠀⢀⡼⠟⠛⡟⠓⠦⣠⡏⠀⠀⠀⠁⠀⠀⠀⣿⣠⠤⠴⣶⠴⢎⡁⠀⠀⠀⠸⣇⠠⠎⠠⡇
⠀⠉⠰⣧⠈⠳⣄⢀⡜⠋⠘⠓⣄⣠⡟⠀⠀⠀⠁⠀⠀⠹⠇⠀⠐⠀⠀⠀⠒⠂⣼⠃⠀⠀⠀⠀⠀⠙⢦⡤⡖⠢⡀⠀⠀⣠
⠀⢠⣾⣹⠆⠀⠘⣿⡀⢀⠀⠀⠈⠿⠀⣤⡴⠀⣆⠀⠀⠀⡃⠀⣀⣤⣤⣤⣄⢀⣿⠀⠀⠀⡀⢠⡤⣴⠟⠀⢀⡄⠀⣦⠐⢠⡦
⠀⢮⡄⠉⠀⢄⠀⠛⠑⠞⠀⠄⠀⠘⣇⠀⠀⠀⠈⠁⠀⠈⣡⠞⠁⠀⢄⠀⠈⠛⣏⠉⠀⠉⠁⠀⢠⡏⠀⠀⡞⠐⡿⠁⠘
⠀⠾⢫⣴⠶⢄⡀⠰⣄⣤⣄⣀⠀⠀⢙⠶⠉⠙⠋⠁⠶⣤⡏⠀⠀⠀⠈⠀⠀⠀⠈⣣⠶⠆⠀⢶⡜⠃⠀⢉⡀⠼⠇⣀⠀⡟
⠀⢠⠸⡇⠐⠀⠙⣶⠋⠀⠉⠈⠛⢶⡎⠀⠀⠀⠃⠀⠀⠈⢷⠀⠙⠛⠁⠀⠙⢠⠟⠁⠀⠀⢰⠆⢱⣆⣰⠋⢥⡀⣀⣥⣴⠇
⠀⢸⣧⠀⠛⡀⠀⠘⣷⠴⢀⡀⠀⠀⢻⡀⠢⣤⠀⠠⠄⠀⠠⣧⣠⣤⣤⣄⣰⠇⠀⠀⠀⡰⠃⠤⠋⢨⠋⠠⠜⢁⠙⠁⢼⠆
⠀⠸⠙⣧⡀⠑⢤⡀⢡⠀⠈⠑⠂⠀⢀⣧⠀⠀⠀⠀⠀⣠⡴⠋⠁⢠⠇⠈⠳⣆⠈⠁⢈⣤⠒⠶⢦⡀⠀⠀⠀⡮⠠⠂⣠
⠀⠀⣠⠀⠑⣄⠀⣨⡽⠶⣤⡀⣠⠖⠉⠉⠉⠉⠱⢦⠾⠁⠀⠀⣠⠘⡀⠀⠀⢸⣤⠖⠉⠀⠀⣷⠀⣴⣶⠶⢢⡀⢀⠾⠋
⠀⠀⠙⣶⡆⠀⢻⡁⢰⡄⠀⠹⣧⢀⣀⢀⡀⠀⠀⠈⣦⠀⠈⠉⠀⠀⢉⠁⠀⢀⣏⣀⣀⣀⡴⠛⠻⠃⠈⡇⢀⡼⣾⠃
⠀⠀⠀⠘⢷⠄⠈⠁⠙⢠⠀⠀⠙⡄⠀⠀⠑⠠⠄⠠⠛⢀⡠⠤⠛⠋⠁⠀⠙⠧⡀⠀⠀⠀⠀⠀⠇⢀⡀⠁⢨⠟⡀
⠀⠀⠀⠀⠀⠂⣰⣧⣀⡀⢑⡀⠠⣧⡤⠒⠒⠲⠒⠢⣴⠏⠀⡀⠀⠀⠀⣠⠀⣀⡿⠒⠒⠛⢶⣞⠓⠀⢀⡘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠙⣯⠙⣇⠉⠁⠺⠏⠀⠀⠀⠀⠀⠀⠈⢣⠈⠉⠐⠀⠁⠉⠸⠋⠀⠀⠀⠀⠀⠛⠓⠫⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠂⠀⠀⠐⠄⠁⠂⠀⠶⠐⠂⠐⢷⣀⠀⠀⠀⢀⣇⠀⢀⠀⠒⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢌⣀⣸⣆⠀⠀⠀⠀⣠⠆⠉⠉⠛⠀⠈⢙⡟⠉⣀⠤⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣾⣿⣶⣶⣦⣤⣤⣤⣯⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⡴⠚⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣣⡉⠀⠀⣼⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣖⣋⣟⣉⣭⣿⣯⣤⣶⣛⣒⡆⢀⣘⡏⠿⣍⡉⠙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣛⣄⠀⣡⣾⠋⣻⣉⡇⠑⠾⡼⠿⡶⠋⠙⣿⣿⡋⢳⠀⠹⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⡁⣠⡏⣰⠳⣄⢈⣿⣥⣀⡴⠃⠀⠉⠉⠉⠻⣌⢛⣿⣤⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⢤⣼⡵⠛⠁⠀⠀⠁⣸⢇⣠⣴⣤⣄⣀⣠⣤⣀⠈⠉⠀⠈⢦⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⢰⠃⠀⠀⡆⢠⣤⡴⠿⠯⠖⣿⠀⢉⡽⠿⠯⢭⣷⠦⠀⠀⠀⢷⡀⠷⠴⢤
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣿⠀⠀⢠⣿⠞⢛⣛⣶⡦⠀⠹⠶⠋⢠⠖⠒⠢⣽⣷⠀⠀⠀⢸⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⣿⡄⠀⣿⢸⠀⡏⠀⠈⣳⠀⠂⠀⠀⠻⣄⣀⣠⢞⢻⡀⠀⢀⡞⣷⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣆⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡹⠀⢹⢷⡀⠀⣬⣇⠻⠖⠚⢁⠤⢄⣀⣀⠤⢤⡀⠒⠋⠀⢳⡤⠎⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⢧⠙⢦⡀⡇⠘⠚⠁⠹⣀⠀⠀⠀⠀⠀⠇⣀⣠⠴⢊⣱⣄⢀⡾⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⡿⠧⣿⡟⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⢧⠀⠉⢳⡢⠤⣤⣤⢭⣤⠤⠴⣶⠚⣏⠉⢹⡗⠾⣏⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⢭⣛⣁⣀⣬⠗⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢼⣹⡾⠉⣿⡠⣿⣒⣊⣹⣶⣾⣆⠘⡇⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣧⠀⡿⠛⠛⠉⠉⠀⠀⠀⠸⣷⠗⠒⢋⡵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠙⠏⡿⠀⣰⠒⠒⠒⠚⠙⠧⣈⢷⠀⢸⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢾⣃⣀⣤⠤⠤⠶⠒⠋⠀⢧⠀⢧⣤⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⢨⠴⠒⠛⢣⣠⡤⠖⠛⡇⠀⠣⠄⠀⠰⣍⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⡷⠛⠀⠀⠀⠀⠀⠙⠢⠤⠃⠀⠒⢾⣻⣹⢻⠗⣯⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡶⣦⣀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⡀⠀⢀⣼⢧⣹⠤⠷⠧⣏⣻⢲⣻⣽⣥⠚⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢯⣱⠴⣥⣏⣟⢛⣤⠟⠁⠀⠀⠀⠀⠀⠉⠀⠀⠈⣯⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢿⣹⢙⡟⡶⣤⣀⣀⣠⡾⢍⡻⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠧⡽⠋⠉⠉⠛⠳⢯⣸⣏⣧⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣛⣿⡧⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣟⣦⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⡜⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣟⣯⡽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠛⠉⠁

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⢰⡔⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⣗⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣿⠾⠛
⠈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣷⠉⣷⠀⢠
⠀⠢⡀⠀⠄⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣿⡰⢻⢁⣦
⠀⠀⠐⡞⠀⠀⠀⠑⢦⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⢀⣿⢷
⠀⠀⠀⠑⠀⠀⠀⠀⠈⣇⠀⣀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠣⣬⣿⡾⣫⣾
⠣⠀⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⢀⠀⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⣷⣿
⠀⠁⠀⠀⠀⠱⣝⠆⠈⠀⠀⠐⠁⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⢠⠾⠁⣜⠅⠀⠂⠉⠳
⠀⡖⠀⠀⠀⠀⠘⠋⠘⠓⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠁⠀⣸⠐⠀⠀⠀⠀⠦
⠀⡄⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣟⠀⠀⠀⠀⠀⢠
⠀⠃⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⣾
⠀⢽⢸⠀⠀⠠⠀⠀⠀⢀⠄⠀⠘⡆⠀⠰⠀⠃⠆⡄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠠⠌⠠⡆
⠀⠸⠀⠆⠀⠘⠄⠀⠂⠀⡃⠐⢆⡀⠀⠣⠄⠠⠊⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⠁
⠀⠈⠇⢠⠀⠀⠀⠀⣀⠒⠂⣶⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣏⠀⠀⠀⠀⡏
⢦⡀⠐⠈⡇⠀⠈⠀⠘⠗⠀⠈⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴⠻⠀⠀⠀⡸⠇
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢀⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠀⢀⠸⣦⣴⣷⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⢼⣞⡀⠀⣿⡇⠀⡇⢸⠈⣩⠀⣠⣴⣦⣀⠾⠿⠗⣻⡎⠁⠀⠲⠀⡀⠈⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣌⠛⢾⠆⠀⡇⡇⢸⠁⡟⢠⠇⣶⣿⣿⣿⣿⣧⡄⢼⣧⣴⡇⢀⠂⠀⠁⠀⢠⡻⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣿⣿⣿⣿⣥⠏⠘⣿⣿⣿⣿⣿⣿⠁⢸⠃⢏⣴⢛⣀⡀⡀⠀⢺⠟⠀⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⢸⣿⣷⣿⣿⡏⠀⠀⠈⠻⢿⣿⠛⠁⠀⠉⠈⠀⢰⣿⣿⣿⣿⣿⣤⣀⠾⢦⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠈⠻⠿⢿⠿⠃⠀⠀⠀⠀⠸⠟⠀⡔⢰⢂⠆⠀⠘⠿⣿⣹⡿⣿⣿⣿⠀⣴⡷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠘⣆⢰⠀⡀⢦⠀⢸⢐⠰⠁⣔⢸⠸⠀⠏⢸⢀⠴⢶⠀⣿⣿⡴⠛⢽⡿⠛⠫⣶⣾⣷⠆
⠀⠀⠀⠀⠀⠀⢱⣶⠆⠻⠿⠷⢹⡜⡆⢷⡘⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡟⣠⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠉⢠⣶⠄⠠⣀⢳⣿⡌⣷⢹⡄⠀⢤⢀⠄⣠⠀⡀⢸⠁⠀⢨⣤⣍⡛⠁⠄⠀⠀⠀⢠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⡴⡆⠈⠙⣿⢷⡸⣯⣿⡀⢫⠈⢂⠉⠀⢃⠁⠀⢰⣿⣿⣿⣿⡆⣠⡄⠀⠰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠇⢀⣴⣼⡏⣷⣿⡞⣷⡀⠀⣿⣧⣶⣿⡇⡀⢸⣿⣿⣿⡿⠃⠙⣡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣉⠉⣿⣿⡿⠷⠘⢳⣶⣿⣿⣿⣿⣏⡿⢸⣿⣿⢋⣴⣶⡞⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣇⠈⠀⠀⠀⠀⠻⣿⣿⣯⣿⣿⡿⢇⣬⣍⠱⢿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡈⠉⢿⡿⠛⠀⣘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣄⠎⠰⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢒⡦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠤⢊⡡⠒⠉⢀⡀⠀⠈⣧⠀⠀⠀⠀⣀⡤⠒⠉⣁⡀⠂⣀⠈⢌⠆⠀⠀⠀⣀⣤⣶⣾⡯⠥⢌⡉⠱⢦
⠀⠀⠀⠀⠀⡴⢣⠔⠁⠀⢀⣄⡤⠟⠒⡈⢠⠃⠀⣠⠮⠤⠤⠤⢀⡠⠔⠋⢀⠀⠸⠀⠀⡠⠊⠁⠀⠉⣉⠒⢄⣠⣽⡂⠀⡇
⠀⠀⠀⢀⢼⡶⣁⠠⠞⣊⣩⡠⢂⡠⠒⠂⠁⠀⢸⣔⠂⠈⣡⠖⠉⠀⡀⢀⠬⠔⠁⠀⠀⡧⣀⡀⣠⣊⠤⠒⢙⣿⣿⡧⠊
⠀⠀⠀⠃⡪⠔⠉⠑⠒⠐⠂⠈⠁⠀⠀⠀⠀⠀⠘⠤⣉⡁⠐⠀⣀⠭⠓⠁⠀⠀⠀⠀⠀⠳⣄⠀⢆⢀⣀⡤⠌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠤⣄
⠀⠀⠀⠀⠀⢀⡖⢻⠔⢯⠔⠺⡖⢻⣑⡄⠀⠀⠀⠀⣰⠋⡰⠋⠳⠀⢱⣄⣀⠀⠀⠀⠀⠀⠀⠔⠀⠷⢆⡈⠩⠛⣿⡶⠄
⠀⠀⠀⠀⠀⣇⡜⠛⡖⠛⠲⡎⠉⢃⣀⠇⠀⠀⠀⠈⣿⣀⣤⢲⠮⣛⣹⠁⠉⠑⠀⠀⠀⠀⢾⣓⣦⢔⡲⣖⠭⣚⣛⣲⣹⡄
⠀⠀⠀⠀⠀⠈⣿⣶⣷⣶⣾⣿⡒⠋⡽⠀⠀⠀⠠⠸⡀⢦⣉⡆⡔⢻⡅⠈⡵⠀⡀⠀⠀⠀⠋⣿⣿⣿⣿⣿⣿⣻⠙⢿⠀⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠇⠀⠀⠀⠀⠐⠄⡇⠉⠰⣗⡯⠔⠚⢁⠴⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠷⠒⠁⠀⠀⠀⠀⠀⠀⠙⠢⠤⠀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠋

⠀⡠⡄⡄⢀⣤⣆⠀⠀⢀
⠀⠟⠇⠛⠊⠉⠁⠀⢦⣼⣀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡇⠸⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⣠⠄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠈⠛⠿⡋⢄⡀⠀⡀
⠀⠀⠀⠀⠀⢿⣿⣻⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⡠⣶⣿⣿⢿⣷⣾⣷⣾⢤⣆⡀
⠀⠀⠀⠀⠀⠀⣿⠘⠙⠇⠈⠃⣿⣿⣽⣿⡽⠋⠀⠀⢀⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⡥⡄
⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⢀⢸⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠿⠑⢲⡾⣋⠿⠇⠀⠀⠀⠀⠀⠘⠏⣿⣿⡇⠘⢻⣿⣿⠫⠿⣿⣿⡟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠿⣿⡤⣤⠆⡀⠀⠀⠀⠀⠼⣿⣿⠛⠛⠃⠀⠉⠛⠛⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣭⣿⡞⣋⣯⠉⠀⠈⢦⠀⠀⠀⠠⣹⣿⢳⡤⡄⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣾⣿⡏⠀⠀⠀⠘⠀⠀⠀⢀⠈⣫⣏⠔⣹⣖⡻⣿⣿⣿⡅
⠀⠀⠀⠀⢀⡾⣿⣿⣾⣿⣿⣿⠇⠀⠀⠀⠀⠆⠀⠐⢂⡻⡇⢈⡮⢥⠐⢁⣀⣤⣽⡿⣤⣀
⠀⠀⠀⠀⠈⠢⣿⣿⣿⣿⣿⡿⡤⢤⣄⣀⡄⠀⠀⠀⡽⣠⢻⣿⡙⡆⠘⠉⠕⠟⣽⠃⣰⠏⡻⠖⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠙⠛⢫⡇⠀⠀⢠⡟⠰⠞⡻⠃⠔⠀⠈⠋⣸⠇⢰⡿⠆⠀⠀⢸
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⡼⠀⠀⠄⡈⠳⠤⠶⠶⠶⠟⠀⠿⣵⠶⠒⠒⠈
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⡆⠀⠀⠇⠀⢀⠁⠀⢤⣌⣿⡀⠀⠀⠀⠀⠀⠀⠀⡁⠚⠉⠀⢀
⠀⠀⠀⠀⠀⠈⠿⣿⢻⡿⡿⡇⣿⡇⠀⠀⠀⡴⢃⣀⢀⡆⢳⠞⠓⠒⠂⢀⡀⠀⠀⠀⡗⠒⠚⠛⢺
⠀⠀⠀⠀⠀⠀⠐⠀⠘⠁⠀⠁⠙⠇⠀⠀⢰⣞⠥⠠⠄⡏⠉⠉⠙⠛⢛⣼⡇⠀⣠⠴⣛⡿⢿⣿⢿

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⢠⣶⡄⠈⠙⡄⠀⠀⠀⠀⠠⣶⠾⣫⣵⣾⠇⠀⠀⠀⠀⢰⣿⣧⣴⡾⠀⠀⠀⠠⠀⢄⢄⠄⠠⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠹⣄⠴⠆⠀⠀⠀⠀⠀⡈⠀⢀⢹⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⠄⠱⢲⠂⠀⠀⠀⠀⢀⡄⡜⠲⠄
⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠐⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢘⣛⠂⠀⣛⣛⠀⠀⠀⠀⠀⠀⣀⣀⣻⣷⣷⠀⠀⠀⠀⢴⣛⡿⣿⣶⡀⠀⠀⠀⣀⢠⣶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⣿⣿⠇⠀⠟⠻⠀⠀⠀⠠⣾⢟⡫⠿⠃⣫⠉⠀⠀⠀⠀⠀⠈⣿⣿⠿⠁⠀⠀⠀⣿⠅⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠿⠿⠇⠸⠀⠀⠇⠀⠀⠐⠉⣑⣐⣋⣩⠴⠃⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⢰⠺⠓⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉

⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠸⢿⡿⠀⢠⣶⡄⠀⢀⣤⣤⣄⢠⡿⢋⣵⣿⡇⠀⠀⠀⣾⠁⠀⢠⣤⠀⠀⠀⠀⣠⣤⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⠿⣷
⠀⣠⣤⣴⡆⠀⡈⠀⠀⠀⠻⣯⡿⢟⣵⣿⡿⠛⠀⠀⠀⠀⠘⠂⢠⡟⡹⠀⠀⠀⠀⣿⣿⣿⣧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢩⠉
⠀⠛⠛⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠒⠶⠚⠫⠒⠁⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁