Numeric Emojis & Symbols

⠁⠐⠓⡆⠀⠀⠀⠀⠘⠋⠀⠀⠀⠠⠠⡄⠀⠀⢠⠖⢋⠒⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡷⠀⠀⠘⠦⠞⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢼⠁⣹⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⡇⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠩⠟⠀⠀⠀⠚⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢖⠢⠀⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⠃⠀⠀⠈
⠀⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⠀⠀⡐⢦⠀⠀⠘⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀
⠀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠂⠀⠀⠀⠁⠀⠀⢞⢈⡆
⠀⠀⠀⡀⠀⠀⠀⡇⠑⠒⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠚
⠀⠀⠲⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠨⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⡴⠲⢦⣄⠀⠀⠀⢸⡙⢦⣀⣸⠀⠀⠀⠀⠀⠱⠄⠀⠀⣀⣀⣀⣀⠀⠀⠀⠐⠋⠀⠀⠀⣰⡍⣰
⠀⠀⢧⣤⠜⢸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⢄⠀⠀⠀⠈⠣⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁
⢀⣠⠀⠀⠀⠀⠀⠯⡄⠀⠀⠀⡀⠀⠀⢇⣘⠉⠳⠀⠀⢀⠔⣀⠀⠀⠀⠀⠐⣀⠀⠀⣀⡀⠀⠀⡖⠀⡆⡆
⠈⠐⠄⠀⠀⠀⠀⠁⠁⠀⠀⠸⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠊⣉⠇⠀⠀⠳⠉⠁⠀⠀⠒⠉⠀⠀⠙⠚⠁
⠀⠀⢀⡀⠀⠀⠀⢀⠀⠤⠀⠀⢀⢤⡀
⠀⠀⠈⢣⠀⠀⠀⠀⠓⠀⠀⠀⢳⡀⢹⠄⠀⠀⠀⠀⠐⡉⠀⠀⠀⠄⣞⡀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠀⠋⡡
⢀⡴⡀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⣰⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠃⠃⠀⠀⠤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠼⠊⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡽⠀⠀⠀⡔⠒⠂⠀⠀⠀⡀⢤
⠀⠀⠀⠀⠀⠠⠉⠀⠀⢀⠀⠀⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⣀⡖⠀⠀⠘⠒⠚⠁⠀⠀⠀⠑⣖⢀⡄⠀⠀⠉⠈
⢠⢤⡀⠀⠀⠀⠀⠀⠀⡟⡠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠐⠒⠛⠛⠛⠓⠛⠛⠛⠃
⠀⠀⠀⢀⠆⡆⠀⠀⠀⢀⠤⠶⢦⣅⠀⠀⠀⠀⡤⠀⠦⠾⡤⠀⠀⠀⠀⡠⠹⢡⠀⠀⠀⠀⠀⠄⠴⠤
⠀⠀⣂⣁⠀⠆⠀⠀⠀⠃⠔⠒⠀⢸⠀⠀⠀⠀⠁⠐⠂⣔⠀⠀⠀⠀⡰⠁⠖⢠⠀⠀⠀⠀⠀⠀⠎⢁⠈
⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠌⢀⠎⠀⠀⠀⠀⠈⠒⠆⠀⢡⠀⠀⣀⡀⠈⠀⠈⣰⠀⠀⠀⠀⠁⢢⠀⠱
⠀⠀⠀⢨⣀⡄⠀⠀⠀⠰⣁⣀⡀⠀⡱⠀⠀⠀⣏⣉⣁⡠⠎⠀⠀⠀⠀⠀⢁⢸⠁⠀⠀⠀⢸⣉⣁⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣤⡉⣄⠀⠀⠀⡀⠀⣠⡌⢢⡀⠀⠀⠀⣀⣀⣃⠀⠀⠀⠀⢀⡀⣠⢀⡀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠞⠀⣔⡒⠂⠀⠀⠠⠤⠤⡄⠀⣠⠇⠀⠀⡌⢠⣀⠀⡃⠀⠀⢠⠃⢠⠒⡆⢸⠀⠀⢀⠜⠡⠐⠢⡈⢆
⠀⢘⠀⠠⠀⡀⢳⠀⠀⠀⢀⠎⢀⠞⠁⠀⠀⢀⠜⢀⣀⠐⢅⠀⠀⠈⠦⣈⡉⠁⢀⠀⠀⢸⠀⡇⠀⠀⡇⢸
⠀⠘⡄⠐⠔⠃⡸⠀⠀⠀⡌⠀⡎⠀⠀⠀⠀⢸⡀⠡⠤⠃⣰⠀⠀⠀⡤⠐⠉⣀⠌⠀⠀⠘⢆⠈⠒⠊⢀⡜
⠀⠀⠈⠁⠒⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠐⠒⠊⠁⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⣤⣤⣤⣤⣄⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠳⡄⢤⡀⢀⠈⣆⣠⠖⠊⠀⠀⢠⣴⡆⠐⢶⡄⣤⡀⢀⠘⣆⡤⠶⠃⠀⠀⢠⢸⠀⠔⢷⡌⣤⡀⣀⠘⣂⡤⠖⠃⠀⠀⢠⢼⠀⠔⢶⡀⣄⡀⣀⠰⣀⡤⠖⠃
⠀⣃⡈⠉⠉⠤⢥⡄⠀⠀⢀⠄⠊⠀⢰⢀⠀⣇⡈⠉⠁⠤⢥⡄⠀⠀⣀⠀⠈⠀⢠⣀⠀⣇⠈⠉⠁⠤⢥⡀⠀⠰⡀⠀⠉⠀⠀⡀⠈⣇⠈⠉⠁⠤⣥⡀⠀⠐⡀
⠐⠉⢇⠄⢀⣀⡀⢀⠀⠀⡿⠀⠘⢳⡜⠁⠐⠉⠇⠀⢀⣀⠀⢰⢀⠀⡟⠀⠘⢷⠘⠁⠈⠉⢃⠀⢀⣀⠀⢠⠀⢀⠇⠀⠛⢷⠘⠁⠈⠙⢃⠀⣀⣀⠀⣀⠀⢨⠇
⡶⠖⢸⣦⠀⠈⠷⢸⠸⠣⣆⠀⠀⡸⠁⢶⡶⠖⢸⣦⠀⠈⠷⢸⠸⢠⡆⠀⠀⡜⠀⢶⠶⠖⢹⡄⠀⠉⠇⣾⠺⢠⡆⠀⠀⠜⠀⣶⠶⠆⢾⡄⠀⠙⡇⣷⠿⢠⡆
⠃⢠⡤⠛⢒⣦⣤⡈⠠⠴⣿⠛⠁⣀⣃⡈⠁⢠⡬⠛⢒⣤⣤⠈⠠⠶⣿⠛⠁⣀⣃⡘⠀⢠⠜⠛⣒⣤⣤⠈⠠⠶⣟⠋⠀⣀⣃⡘⠀⢠⠜⠛⣒⣤⣄⠉⠠⠶⣟
⠆⠀⣤⡴⠉⠀⠹⢷⠀⠀⢸⡤⢨⡁⠈⠹⠆⠀⣤⡖⠁⠀⠹⣶⠀⠀⢸⡄⢨⠁⠈⢻⠀⠀⣤⡖⠁⠀⠹⣶⠀⠀⢸⡄⢨⠁⠈⢻⠀⠠⣤⠆⠁⠀⢹⣶⠀⠀⣻
⣠⠴⠚⣇⠀⣀⠀⠘⠷⢶⡄⢰⠈⠳⠶⣂⣤⠴⠚⣇⠀⣀⠀⠘⠷⢶⡀⢸⠈⠳⠶⣂⡤⠖⢺⣆⠀⣀⠀⠘⠷⢶⡀⣶⠘⠳⠶⣂⡤⠖⢺⡄⠀⡀⠀⠹⠶⣶
⣀⡀⠀⠙⣻⣥⣰⣄⠀⠈⢳⡜⢧⡐⠿⣭⣀⡀⠀⠙⣫⣥⣸⣄⠀⠈⢱⠘⣦⡐⠿⣭⣀⠀⠀⠙⣫⣅⣸⣀⠀⠘⢳⠸⣦⡰⢿⣥⣀⠀⠈⠛⣫⣄⣾⡀⠀⠈⢲
⠉⢛⠀⠰⣄⠈⢳⠉⠉⢀⠀⢀⣿⡛⠃⠀⠉⣛⠀⢰⣄⠈⣷⢉⠉⣀⠀⢀⣛⡛⠁⠀⠉⣋⠀⢰⣄⠈⡇⠉⠉⡀⠀⣀⣟⡛⠀⠀⠉⣣⠀⢸⡀⠈⡇⠉⠉⡀
⠚⠁⠶⠀⠽⠁⠈⠼⠓⢸⡄⠀⠀⢻⠀⠰⠚⠁⠶⠀⠝⠁⠀⠼⠓⢹⡄⠀⠈⢳⠀⠴⠊⠁⠶⠀⠝⠁⠁⠼⠓⢹⠀⠀⠈⢳⠀⠴⠊⠑⠶⠀⠝⠀⠡⠾⠓⢻
⠖⠋⠀⠀⢠⣴⡇⠐⠻⡆⢴⣀⣀⠀⣇⡤⠒⠃⠀⠀⢠⢼⡄⠐⢻⡄⣶⣀⣀⠈⣇⠤⠒⠃⠀⠀⢠⢸⠀⠘⢳⡄⣦⣀⣀⠘⣆⠤⠚⠃⠀⠀⣠⢼⠀⠚⢳⡀⣦
⠀⠀⣀⠀⠈⠀⢀⣀⠀⣇⡀⠉⠁⠴⢶⡄⠀⠠⣠⠀⠈⠀⢀⣀⠀⣁⡀⠉⠡⠴⢦⡄⠀⠀⣀⠀⠈⠀⢈⡀⠀⣇⡈⠉⠁⠴⢦⡀⠄⠀⡀⠀⠁⠀⢀⡀⢀⣁⠈
⠁⠀⠿⠀⠈⢻⡎⠁⠀⠉⢃⠀⢀⣄⡀⢠⠀⠄⠏⠀⠉⢳⡎⠁⠈⠉⢣⠀⢀⣄⡀⠀⠀⡄⠏⠀⠉⢳⠈⠀⠈⠉⢇⠀⢀⣄⠀⡘⠀⠠⠇⠀⠉⢷⠈⠀⠊⠙⣇
⠸⠃⣆⠀⠀⠘⠁⢻⡶⠒⢸⣆⠀⠈⠗⢸⠸⠠⣆⠀⠀⠘⠀⢿⠶⠊⢸⡆⠀⠈⠇⢿⠸⢰⣆⠀⠀⠞⡀⢿⠶⠂⢸⡆⠀⠈⠇⣧⠻⢰⣆⠀⠀⠎⡀⣷⠶⠂⢿
⠠⠒⢿⠉⠀⣠⣇⣈⠁⠠⠖⠙⣛⣧⣤⠈⠠⠲⢯⠉⠀⣠⣇⡈⠀⠠⠞⠙⣻⣧⣤⠈⠠⠲⣯⠉⢀⣠⣅⡈⠀⠠⠞⠙⣻⣦⣤⠈⠐⠲⣏⠉⢀⣸⣅⡈⠀⠠⠎
⡀⠀⢸⡤⢨⡁⠀⠹⠆⠀⡴⡖⠁⠀⠸⢷⡀⠀⢸⡄⢨⡁⠀⢹⠀⠀⠶⡆⠁⠀⢸⢷⠀⠀⢸⡄⢨⠁⠀⠹⠀⠠⠶⠎⠀⠀⠹⣦⠀⠀⢨⡄⣭⠀⠈⠻⠀⠠⢶
⠳⢶⡆⢸⠈⠛⠖⣂⡤⠖⠛⣇⣀⣀⠀⠘⠳⢶⡄⢰⠈⠛⢖⣂⡤⠖⠳⣇⣀⡠⠀⠘⠷⢶⡀⣷⠈⠳⢖⣀⡤⠒⢻⣇⢀⡀⠀⠘⠷⢶⠀⣇⠈⠳⢒⣠⡴⠚⢹
⣀⡀⠱⡌⢷⣐⠻⢧⣄⡀⠀⠙⠩⢥⣸⣄⣀⡀⠱⠜⢧⣐⠿⢥⣄⡀⠀⠙⠯⣤⣸⣄⣀⠀⢱⠘⣦⡐⠿⣤⣄⡀⠀⠙⠭⣤⣺⣄⣀⠈⢳⠘⣦⡘⠿⣤⣀⡀⠈
⡉⢀⠂⢀⣭⡙⠁⠀⠈⣉⠀⠸⣄⠀⠷⢈⡉⢠⠀⣠⣭⡉⠁⠀⠈⣩⠀⢸⣄⠈⠇⢉⠁⣀⠀⣠⣭⡙⠁⠀⠈⣩⠀⢰⣄⠈⡇⢉⠁⡀⠀⣠⣭⡉⠀⠀⢉⣡
⠛⠸⠆⠀⠀⢹⠀⠴⠊⠁⠲⠀⠜⠁⠀⠜⠛⢹⠄⠀⠀⢹⠀⠴⠋⠁⠳⠀⠜⠁⠀⠜⠛⢹⠄⠀⠀⢿⠀⠔⠋⠀⠷⠀⠝⠀⠠⠞⠋⢻⠀⠀⠈⠇⠀⠖⠋⠐⠷
⠻⡆⢰⣄⣀⠈⣧⠴⠚⠋⠀⠀⢠⢼⠇⠘⠻⡆⢶⣀⣠⠈⣦⠴⠚⠁⠀⠀⢠⢼⠀⠘⠻⡄⢶⣀⡠⠘⣧⠴⠚⠁⠂⠀⢰⢾⠀⠚⢻⡄⣦⣀⡠⢐⡦⠔⠛⠁
⠀⣡⡀⠈⠀⠐⠶⡄⠀⠀⣠⠀⠈⠀⢰⣠⠀⣧⡀⠈⠀⠔⢶⡄⠀⠀⣤⠀⠀⠀⢰⡠⠀⣧⡀⠉⠀⠒⢶⠄⠀⠐⡠⠀⠁⠀⢀⡀⢈⣥⡀⠁⠀⠒⢶⠄⠀⠀⡤
⠈⠀⢃⠄⢀⣤⡀⢰⠀⡀⠟⠀⠈⢻⡎⠀⠀⠀⢃⠄⣀⣤⡀⢰⢀⠀⠇⠀⠈⢻⠈⠀⠈⠈⢣⠀⣠⣤⡀⠰⢀⠠⠇⠀⠈⢻⠈⠀⠊⠈⢣⠀⣠⣄⡀⡤⢀⠰⠇
⡷⠊⢸⣶⠀⠀⠛⢸⡘⠡⣆⡀⠀⠜⡀⢻⡖⠋⢸⣶⠀⠀⠟⢸⠘⠰⣇⠀⠀⠜⡀⢿⠖⠃⢸⣆⠀⠈⠇⢷⠛⠰⣇⠀⠀⠞⡀⢿⠖⠃⢻⡆⠀⠈⠇⣷⠛⢰⣇
⠁⠠⠖⠉⣫⡵⣦⡀⠀⠒⢿⠉⢁⣠⣥⣌⠁⠠⠖⠉⣫⡵⣦⠈⠐⠒⢯⠉⢀⣤⣥⡈⠀⠰⠞⠉⣫⣵⣦⠈⠐⠒⣯⠉⢀⣬⣥⡈⠀⠰⠎⠙⣫⣷⣦⠀⠐⠲⣯
⠇⠀⠶⡞⠀⠀⠘⢿⡀⠀⢸⠤⢨⡀⠀⠙⠆⠀⠶⡆⠀⠀⠘⢿⡀⠀⠸⢄⢨⡀⠀⠹⠀⠀⠶⠎⠀⠀⠸⢾⠀⠀⢸⠄⢨⡀⠀⠹⠀⠠⠶⠆⠀⠀⠹⣷⠀⠀⠹
⠴⠞⠋⣧⣀⡠⠀⠘⠻⢾⡆⢸⠀⠛⢚⣣⠴⠚⠉⣧⣀⡠⠀⠘⠳⢾⡄⢺⠈⠛⢚⣡⠴⠚⠹⣆⣀⡠⢀⠘⠳⢾⡀⣷⠈⠛⢚⣡⠴⠚⢹⣆⣀⡄⢀⠘⠳⢿
⣤⣀⠀⠉⠭⢦⡼⣆⣀⠀⠹⠎⣷⣜⡻⢦⣤⣀⠀⠉⠽⢦⡸⣄⣀⠈⢹⠈⣷⣜⡻⢦⣄⣀⠀⠉⠽⢦⣸⣄⣀⠈⠹⠘⣧⣘⠻⢦⣄⡀⠀⢉⠽⣦⣺⣄⣀⠈⠳
⠀⣩⠄⠸⣦⡀⢻⢈⡁⢠⠂⡠⢭⣍⠁⠀⠀⣩⠀⠸⣄⡀⠷⢈⡁⣀⠀⡠⢭⡉⠁⠀⢈⣩⠀⠸⣄⠀⡇⢈⡁⣄⠀⡨⢭⡉⠁⠀⢈⡩⠀⢺⣄⠈⡇⢈⠁⣐⠁
⠋⢀⠻⠀⠜⠁⠀⠚⠛⠸⠆⠀⠀⠹⠀⠔⠋⠀⠳⠀⠞⠁⠀⠚⡋⠸⠆⠀⠀⢻⠀⠖⠉⠀⠓⠀⠎⠀⠀⠚⡋⢹⠀⠀⠀⢻⠀⠖⠉⡈⠗⠀⠋⢀⠠⠚⠋⢹

⠀⣀⣀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⣀⣤⣤⠄⠠⣤⣀⠀⢀⣤⣄⠀⣀⣀⣀⣀⣀⣀⣤⣤⣀⡀⠀⣀⣤⣤⣤⣤⣀
⠀⣿⣿⠀⠰⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣷⠀⠸⣿⣿⣿⠀⠘⠿⠿⠃⠘⡧⣿⠀⠿⢿⣿⣿⣟⣛⣿⡟⠛⠃⣾⣿⠁⠀⠀⠈⢻⣷⡀
⠀⣿⣿⣶⣶⣿⣿⣷⡆⠀⣶⣶⣶⣾⣿⡟⠀⢀⣤⣿⣿⠛⢀⣠⣀⠀⠈⣙⡉⠀⠀⠀⠀⣀⠉⠉⠉⠁⠀⠀⣿⣧⠀⠀⠀⠀⢸⣿⠇
⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠙⠉⠉⠉⠁⠀⠀⠈⠻⠿⠿⠄⠹⠿⠛⠀⢸⣏⣛⠀⠀⡴⠋⠉⡉⠳⡄⠀⢀⡀⠙⠿⣦⣤⣤⣤⡿⠏
⠀⡖⠛⠀⠾⣿⣦⣾⢻⣦⣴⡟⣷⣄⡖⠲⣆⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠶⠟⠀⠘⣇⠀⠀⢹⠆⣿⠀⢸⡇⠀⣾⠀⠀⠀⠀⣠
⠀⠓⠲⡄⢠⣿⢡⣿⢿⣧⢿⣧⣿⣿⡇⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⣦⠀⠀⠈⠛⠛⠁⢠⠏⠀⢸⣇⣀⣿⡀⠀⠐⠛⣿
⠀⣤⠴⠃⠸⠟⠘⢿⣼⠟⠲⣦⡿⠋⠧⡤⠏⠀⠛⠛⠛⢻⣿⣿⣿⠀⠀⠿⠇⠀⠀⢀⣀⣠⠴⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⢀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠠⡦⠴⠶⠀⠀⣸⣿⣿⣿⠀⣀⡀⣴⢦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⣿
⠀⠀⢠⠈⠀⠀⣾⣿⠁⠀⠀⠀⠉⣿⣧⢀⡷⣔⣓⠀⠀⣿⣿⣿⡟⠀⠈⠙⠷⠾⠁⠀⢸⡟⠲⣄⡀⠀⠀⠀⠈⠙⣶⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣴⡾⠛⠁⠀⠷⠨⠧⠀⢰⣿⣿⣿⡇⠀⢠⣤⣤⣤⡄⠀⢸⣧⡀⠀⠉⠓⠶⠦⠤⠶⠛⠁⠀⠀⣿
⠀⠀⠀⠜⠀⠀⢠⣾⡟⠁⠀⠈⢻⣷⡄⠈⠻⣙⡋⠀⢸⣿⣿⣿⠃⠀⢸⣿⣿⣿⠇⠀⣠⣠⣀⣤⡄⣠⢀⡀⣠⠤⣀⠀⠀⠀⣿
⠀⠀⠀⠊⠀⠀⠀⠛⠿⠶⠶⠶⠟⠋⠀⠐⠿⠒⠙⠀⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⠀⠈⣿⢾⣡⡾⡁⠻⠬⢸⡁⠀⢈⠆⠀⢀⣿⡀
⠀⠐⢲⡶⠂⠀⢀⣠⣀⠀⣀⣀⢀⣤⡀⣀⣀⠀⠀⢀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡟⠀⠀⠉⠉⠈⠛⠁⠘⠛⠀⠙⠒⠋⠀⠀⠉⠉⠉
⠀⠀⢸⡇⠀⠀⡏⠀⠈⡇⡊⢥⡈⣽⠏⣿⢼⡁⠀⢸⣿⣿⣿⠇⠀⢸⣿⣿⣿⡇⠀⣤⡶⠶⠆⠀⢀⣤⡴⠶⠶⢶⣤⣀⠀⠀⢆
⠀⠀⢸⡇⠀⠀⠙⠒⠚⠀⠋⠚⠘⠛⠂⡛⠘⠁⠀⣼⣿⣿⣿⠀⠀⣸⣿⣿⣿⠁⠀⡤⣭⠦⡀⠀⠺⣿⣄⠀⠀⢀⣿⡿
⠀⠀⢸⡇⠀⠀⢠⣶⠾⠛⠲⠶⣤⡀⠀⠈⣿⠀⠀⠛⠛⠛⠛⠀⠀⣿⣿⣿⣿⠀⠀⣳⡀⢶⠀⠀⣀⣬⡿⠟⠻⠿⣧⣄⠀⠀⠀⡀
⠀⠀⢸⡇⠀⠀⠘⢧⣀⠀⠀⠀⠀⠉⠓⢤⣿⠀⠀⣰⠟⣶⣄⡀⢀⣿⣿⣿⡇⠀⠀⠥⠍⢾⠀⠀⣿⣯⠀⠀⠀⠀⣨⣿⠇
⠀⠀⢸⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⣃⠈⠁⢸⣿⣿⣿⠃⠀⠀⠻⠟⠛⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⢸⣇⠀⠀⠀⣸⣧⣤⣤⠀⠀⢀⡴⠚⠉⠉⠀⠀⢰⢿⡇⠀⢸⣿⣿⣿⣤⣤⣤⣄⢠⠖⢲⣄⣶⠶⣦⢰⡶⣶⡆⢰⡖⢀⡔⠚⠂
⠀⠀⢸⡿⠋⠀⠀⢸⡇⠀⣿⠀⢀⡏⢀⣠⣤⣀⠀⠀⠈⠻⠃⠀⢸⣿⣿⣿⣿⣿⣿⣟⢸⠀⢸⣿⣿⢻⣷⢸⣷⣿⡇⣿⠇⠘⠧⢤
⠀⠀⠀⠀⢀⣀⠀⠘⠃⠀⣿⠀⢸⡇⢾⠀⠀⠈⡇⠀⢸⣿⡁⠀⠘⠛⠛⠛⠛⠛⠛⠋⠸⢦⡼⠋⢿⣼⡿⠸⣧⣿⠿⣿⣤⠄⣤⣼
⠀⣴⣾⠛⠉⠉⠛⣷⣆⠀⠀⠀⠀⠳⣌⣁⣀⡴⠃⠀⢰⣼⠇⠀⢠⣶⣾⡀⠙⣿⣶⣦⠀⠀⣀⣠⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢠⣤⣤⣀⣉⡀⠀⠀⠀⢠⡶⡄⠀⠀⠉⠉⠠⢴⣿⣟⠋⠀⣼⣿⡿⠿⠿⠿⠀⠘⠿⣿⣿⠿⠿⣿⣿⡇
⠀⢿⣷⡀⠀⠀⢀⣾⡿⠀⣤⣼⣿⣯⣭⣿⣿⣿⣿⡆⢸⣿⡇⠀⢴⣶⣶⠄⠀⣿⣿⣿⠀⢿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⠇⠀⣿⣿⡇
⣀⣀⣈⣉⣛⣛⣉⣉⣀⣀⣉⣉⣛⣉⣉⣉⣉⣉⣉⣀⣀⣉⣀⣀⣀⣉⣙⣀⣛⣋⣉⣁⣀⣀⣈⣉⣛⣛⣛⣛⣛⣋⣉⣁⣀⣀⣉⣉⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⡞⡠⢄⠈⡆⠀⠀⢀⠞⠁⠀⡇⠀⠀⢀⣀⡐⢂⢱⠀⠀⠀⠾⣤⠒⡀⡤⠀⠀⠀⢠⠎⠀⢸
⠀⠀⠀⢸⠅⡀⢸⠇⡇⠀⠀⠘⠋⡇⠀⡇⠀⠀⠀⠉⢀⡼⡸⠀⠀⠀⠀⠰⢞⡻⡅⠀⠀⢠⢃⡞⢻⣸⡀
⠀⠀⠀⠸⣆⠳⠞⢠⠃⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⡴⠛⠒⠠⡄⠀⠀⡔⠲⣀⠇⡐⠀⠀⠿⠬⡭⡁⣴⠇
⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠚⠒⠒⠒⠒⠃⠀⠀⠈⠒⠢⠒⠁⠀⠀⠀⠀⠀⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠐⠀⡀⢀⠄⠀⠀⢀⠞⡁⣉⡂⠀⠀⢚⣋⡛⠋⠙⡇⠀⠀⡜⢁⠤⡍⢢⠀⠀⢠⡞⣋⣉⠲⡄
⠀⠀⠀⠀⣇⣰⡆⠄⠀⠀⠀⢾⠸⠄⠥⡄⠀⠀⠀⠀⢨⠃⡄⠀⠀⠀⣻⠌⠒⢃⡜⠀⠀⡟⢸⡃⣽⠀⢱
⠀⠀⠀⠀⣀⡁⢘⡆⡆⠀⠀⠘⠀⠀⠂⢹⠀⠀⠀⢀⠇⡀⠀⠀⠀⠀⡇⠰⡁⡇⢹⠀⠀⠙⣮⠿⢷⠀⡜
⠀⠀⠀⠈⢧⡁⢋⠐⠁⠀⠀⠐⢀⣁⣠⠎⠀⠀⠀⢸⣠⡇⠀⠀⠀⠀⠷⣄⠈⣀⠞⠀⠀⠘⢍⡑⢋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠀⢀⡀⠀⣀⠀⣀⣀⡀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣟⡋⠘⢸⢶⡏⢻⡎⣓⡄⠉⣹⠁⠀⣇⣘⠉⣿⡁⠄⣶⠄⢱⣜⡧⠤⣿
⠀⠀⠀⠀⠀⠀⠝⠛⣿⢠⢸⡆⣴⣏⣠⡿⣦⢰⡯⠀⠰⡇⣲⢐⠶⡃⡅⠛⡈⣸⡏⣓⣶⡋
⠀⠀⠀⠀⠀⠀⣾⣇⣿⢠⢌⠨⠉⠉⢹⡧⣬⢰⠛⡆⢀⣅⣹⠹⢼⣧⣭⣟⠻⡟⠁⢉⣠⡿
⠀⠀⠀⠀⠀⠀⠨⢭⣡⠚⡏⡆⠀⠐⠸⣧⣿⢨⠛⡅⢀⣻⠙⢁⠈⠁⣾⢘⠋⠑⡄⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠒⡴⢁⠮⢡⡶⣦⠛⣼⠠⣤⠘⠤⠃⠀⣇⡕⠘⠰⢐⡃⣿⣔⡊⡀⢿⣼⢧
⠀⠀⠀⠀⠀⠀⠈⠉⢸⣹⢸⡷⣿⢰⣶⠈⣳⠰⢯⡁⠀⣮⡇⡘⣴⢸⡇⣿⢸⡅⠗⠊⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠛⠘⠋⠓⠋⠙⠚⠁⠈⠓⠁⠀⠛⠁⠈⠙⠛⠘
⠀⠀⠀⠀⠀⠀⡖⡂⢖⡶⣶⡖⣢⡞⣶⣶⢐⢴⣲⠀⢠⠂⢲⢰⡆⠀⠀⠆⠴⠶⣦⢰⡂⢴
⠀⠀⠀⠀⠀⠀⣇⡃⠿⢹⡆⣴⣃⡴⠇⡈⣐⠒⣿⠀⠀⢱⠎⢀⣷⣖⠲⣷⣾⣛⣋⠂⣉⠘
⠀⠀⠀⠀⠀⠀⡄⣿⣀⢻⡆⢿⠟⣹⢋⢅⢡⡆⣻⠃⠰⣿⡦⣷⡄⣍⡇⣯⣩⡏⠉⠴⢣⢦
⠀⠀⠀⠀⠀⠀⠓⠋⠙⠟⠱⢐⠘⣿⣮⣜⠈⡙⣿⠀⠀⠩⣲⡮⢶⡟⣶⣼⣌⠀⡂⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠺⣿⣼⠞⡇⠚⢸⢁⠙⢿⡇⢙⠡⠀⠠⣠⠿⠸⢠⡞⣷⡛⠳⣷⠀⢣⡶⣧
⠀⠀⠀⠀⠀⠀⢲⡶⠾⠗⢿⣽⠈⠁⡀⢭⡄⢀⢈⠁⠰⢳⡆⡇⣾⣷⢳⡅⣰⣹⢔⠘⠧⣿
⠀⠀⠀⠀⠀⠀⠚⡇⢮⡹⠸⡽⠀⠀⠀⢤⡇⠄⠠⠀⠰⣯⠄⡟⢯⡼⠿⠰⠈⠁⠸⠟⠷⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣍⠄⢰⠀⠀⠀⠀⢠⠒⣠⡆⣅⣀⠀⣄⡩⣛⡸
⠀⠀⠀⢀⡤⠶⠴⢾⢎⡎⠀⢀⡀⣠⠺⣁⣀⣀⠉⠛⢯⠀⠀⠀⠀⡎⠀⡴⠮⠭⢠⡟⠀⡤⠀⡇
⠀⠀⡰⠋⢠⠔⠒⠪⡉⢹⡤⠊⠑⢺⠀⠈⠀⣸⠃⠀⡼⠀⠀⠀⠰⢥⣀⠑⠢⣀⠞⢀⡎⡑⠀⡇
⠀⢰⠃⢠⠇⠀⠀⠀⡷⡏⢀⡄⠐⢸⣠⠤⠊⠁⣠⠞⠡⢀⠴⣀⣀⠀⢈⠆⠀⢹⡀⠀⠀⠁⠀⠉⢹
⠀⢸⠀⠘⢆⡀⣀⡴⠁⢹⠃⣧⠀⢸⡀⠀⠀⠛⢳⠔⠳⠚⠿⠻⡀⠉⠁⠀⢀⡜⠉⠉⠉⡇⠀⡏
⠀⠈⠣⣀⠀⠉⠁⠀⣠⠏⠀⢸⠀⠘⡏⠓⠒⠤⣾⠀⠸⣁⠕⠀⡟⠛⠒⡋⠁⣀⠀⠀⠀⠓⠒⠃
⠀⠀⠀⠈⠉⠓⠚⠉⠁⠀⢀⠸⢤⠤⠇⠀⠀⠀⡨⠓⠀⠀⠀⠺⢄⢀⣠⣤⣃⣹⣀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⡖⠤⠤⣡⡖⡍⢃⠀⠀⠀⢸⠀⠠⡏⠉⣱⠀⡼⠉⢀⡠⠬⣥⣏⣄⡿⠦⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠸⣅⣀⣀⡀⠈⠒⠁⡧⢄⡌⡼⣒⡇⠈⠉⠁⡜⠀⠀⠈⠉⠲⢤⠇⠀⡖⠉⢢⢀⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠧⠄⣈⠉⠚⡭⠋⠉⠉⡇⠀⡜⠉⢱⠀⢸⣇⠀⠓⠔⠋⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡴⠁⠀⠀⡔⠊⠁⢠⡊⠀⠀⠀⠀⢧⡀⠙⠒⠊⠀⡸⠉⠲⠤⠴⢶⠀⠀⡞
⠀⠀⠀⠀⠀⢀⡎⠀⢠⠏⠀⠀⠀⠀⠓⠒⡄⠀⠘⡆⠀⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠚⠀⠀⠀⢰⠓⠒⠒⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠒⠉
⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⠀⠀⠀⠀⠀⠀⡠⠤⠤⠠⢄⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⡤⣤
⠀⠀⠀⠀⠀⢈⡭⢾⣋⣻⠀⠀⠀⠀⠀⠘⠴⠚⠒⢦⡀⢐⠀⠀⠀⠀⠀⠣⠒⠓⢲⡀⠱⠀⠀⠀⠀⠀⠀⢀⣞⣿⡍⢹⠀⠀⠀⠀⠀⠀⡷⢾⠗⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣒⣺⠀⠀⠀⠀⠀⠀⢤⢤⠺⣁⠜⠀⠀⠀⠀⠀⢠⠯⡼⠁⠥⢼⠀⠀⠀⠀⠀⢠⣟⣟⢒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠀⠀⣠⣞⣒⡶⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠑⡄⠀⠀⠀⢰⠯⣿⢤⣴⢯⢽⠦⠀⠀⠀⠀⠈⠁⠉⠙⢳⡀⢘
⠀⠀⠀⠀⠀⠀⠀⢰⠠⡆⠀⠀⠀⠀⠀⢠⢮⢵⠶⢧⠤⢤⠄⠀⠀⠀⢀⠦⣤⡤⡴⠏⢀⠇⠀⠀⠀⠘⠛⠛⠛⠛⡟⢺⠚⠀⠀⠀⠀⢠⢤⣀⣠⡾⠁⡔
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠚⠚⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠓⠃⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠘⠚⠒⠓⠛⠉


⠀⠀⠀⠀⠀⠀⣀⡤⡤⡖⠀⠀⠀⠀⠀⢰⢶⡖⡖⢲⢲⠲⠀⠀⠀⠀⠀⢀⠤⡒⣒⠦⣄⠀⠀⠀⠀⠀⢀⢄⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⢤⡀
⠀⠀⠀⠀⣰⠫⣤⠞⠛⠉⠀⠀⠀⠀⠀⠘⠚⠓⠓⣻⣾⡾⠀⠀⠀⠀⢰⠅⣼⠉⠉⡯⠨⡅⠀⠀⠀⢠⠍⢀⡟⠙⢣⠀⢓⠀⠀⠀⠀⢰⠁⡼⠋⢻⡁⠱
⠀⠀⠀⢠⣇⣷⢟⣏⡉⠐⢄⠀⠀⠀⠀⠀⠀⠀⣰⠯⡴⠁⠀⠀⠀⠀⠀⢣⡨⢷⣞⣭⣞⠀⠀⠀⠀⠸⠀⠸⣆⣀⢼⡭⢸⠀⠀⠀⠀⡇⢀⡇⠀⠀⡧⢩⡇
⠀⠀⠀⢸⡻⢿⠁⠀⢹⣂⢈⠀⠀⠀⠀⠀⠀⣰⠿⣻⠁⠀⠀⠀⠀⠀⠰⠫⣶⠋⠙⣦⠈⢢⠀⠀⠀⠀⠉⠂⠤⠐⡾⠋⡹⠀⠀⠀⠀⡧⢐⡇⠀⢠⡇⢸⠇
⠀⠀⠀⠀⢃⡘⠣⡤⠾⠁⡜⠀⠀⠀⠀⠀⢰⣋⣹⠁⠀⠀⠀⠀⠀⠀⠨⣀⠹⢤⢴⠟⢁⡆⠀⠀⠀⠀⢠⢠⠶⠋⢁⠘⠁⠀⠀⠀⠀⠹⡀⠹⠦⠞⢁⠜
⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠊⠁⠀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠊⠈



⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣚⣭⣽⣒⠦⣄⠀⠀⠀⠀⠀⢀⡤⢖⣚⣭⣝⣒⠦⣄⠀⠀⠀⠀⠀⢀⡤⢖⣒⣭⣽⣒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⠟⢻⣿⣿⣮⢳⡀⠀⠀⢰⢫⣾⣿⠟⣛⡛⢻⣿⣮⢳⡀⠀⠀⣰⢫⣾⣿⠟⣉⡙⢿⣿⣮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⣿⣿⣥⡆⢸⣿⣿⣿⡇⣧⠀⢀⢇⣿⣿⣷⣾⣿⡿⢀⣿⣿⡇⢇⠀⢠⣇⣿⣿⣿⣾⡿⠟⣸⣿⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣿⣿⣿⣿⡇⢸⣿⣿⣿⡏⡿⠀⠘⡌⣿⣿⣿⡿⢋⣴⣿⣿⣿⡏⡞⠀⠸⡏⣿⣿⣿⢿⣿⣷⡌⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⢿⣿⣿⣇⣸⣿⣿⠟⡱⠃⠀⠀⠳⡙⢿⣏⣀⣛⣛⣛⣿⠟⡱⠁⠀⠀⠳⡙⢿⣿⣌⣛⣋⣴⣿⠟⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢍⣛⣛⣛⣋⠵⠊⠁⠀⠀⠀⠀⠘⠲⢍⣛⣛⣛⣋⠵⠊⠁⠀⠀⠀⠀⠙⠲⢍⣛⣛⣛⣋⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⣀⡤⠔⣒⡲⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠴⣒⡢⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣒⡲⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣒⡲⠤⣄⡀
⠀⠀⢠⢞⣵⣾⣿⣿⠿⣿⣶⣝⢦⠀⠀⠀⢠⢞⣵⣾⠿⠿⠿⢿⣶⣝⢦⠀⠀⠀⢠⢞⣵⣾⡿⠿⠿⣿⣶⣝⢦⠀⠀⠀⢠⢞⣵⡾⠿⠿⠿⠿⣶⣝⢦
⠀⢀⡏⣾⣿⣿⡿⣡⢈⣿⣿⣿⡎⣇⠀⠀⣏⣾⣿⣿⠰⠿⠿⣿⣿⣿⡎⣇⠀⢀⡏⣾⣿⡟⢰⡿⢷⣼⣿⣿⡎⣇⠀⢀⡏⣾⣿⣿⣿⣿⢃⣼⣿⣿⡎⣇
⠀⢸⡸⣿⣿⠏⢴⠿⠸⣿⣿⣿⡿⣿⠀⠸⡳⣿⣿⣷⣴⣾⣦⠘⣿⣿⡗⡸⠀⢸⡔⣿⣿⡇⢠⣶⣦⠘⣿⣿⡷⡾⠀⢸⡿⣿⣿⣿⣿⠃⣾⣿⣿⣿⡿⣸
⠀⠀⢧⠻⣿⣶⣶⣶⢠⣾⣿⡿⢣⠇⠀⠀⢧⠻⣿⣧⡙⠿⠛⣰⣿⡿⢣⠇⠀⠀⢧⠻⣿⣷⡘⠿⠟⣰⣿⡿⢣⠇⠀⠀⢧⠻⣿⣿⡏⢸⣿⣿⣿⡿⢣⠇
⠀⠀⠀⠳⢭⡻⠿⠿⠿⠿⣋⠵⠃⠀⠀⠀⠀⠳⢍⡻⠿⠿⠿⠿⣋⠵⠁⠀⠀⠀⠀⠱⢌⡻⠿⠿⠿⠿⣋⠵⠃⠀⠀⠀⠀⠳⢌⡻⠿⠿⠿⠿⣋⠵⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣥⣶⣶⣶⣦⣝⠢⡄⠀⠀⠀⠀⡠⢚⣥⣶⣶⣶⣦⣝⠢⡀⠀⠀⠀⠀⡠⢚⣥⣶⣶⣶⣦⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣴⣿⣿⢡⣶⣦⢹⣿⣷⡜⡆⠀⠀⡜⣵⣿⡟⢡⣶⣦⠹⣿⣷⡝⡆⠀⠀⡞⣴⣿⣿⢁⣶⣦⠹⣿⣷⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣿⣿⣿⠆⣉⡁⢾⣿⣿⡇⣿⠀⢰⢣⣿⣿⣧⠸⠿⠟⠀⣿⣿⣧⢳⠀⢸⡇⣿⣿⡇⢸⣿⣿⠀⣿⣿⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢿⣿⣇⠸⣿⣿⢀⣿⣿⢇⡏⠀⠈⣇⢿⣿⣿⢻⣶⡿⢠⣿⣿⢇⡇⠀⠈⡇⢿⣿⣧⠸⣿⡿⢠⣿⣿⢇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡻⢿⣶⣤⣴⣾⠿⣫⠞⠀⠀⠀⠘⢮⡻⢿⣶⣤⣶⣿⠿⣫⠞⠀⠀⠀⠘⢮⡻⢿⣷⣤⣴⣿⠿⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠭⠽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠾⠭⠽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠾⠭⠽⠖⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣨⠴⠒⠛⠓⠲⢬⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢄⣨⠶⠚⠛⠓⠶⣽⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠄⠒⠒⠒⠦⣅⠂⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢸⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⢰⠖⣦⠀⠀⠹⣌⠄⠀⠀⠀⠀⢀⠂⠌⠀⠀⢴⠛⣆⠀⠈⠱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣠⠟⠀⠀⠀⢹⠘⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⣤⡗
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⢸⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⣀⠀⣇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠁⡱⣄⠀⠀⠘⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠪⣷⣄⠀⠘⠒⠖⠀⣀⡼⡱⠁⠀⠀⠀⠀⠀⠂⠡⣀⠀⠘⠛⠁⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠓⠲⠒⠛⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⢶⣶⣶⠿⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠲⠶⠖⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡵⠊⠉⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠛⠉⠉⠉⠉⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣨⠞⠉⠁⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡇⢸⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣘⡟⠀⠀⠀⣿⠉⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠐⣼⠁⠀⠀⣾⠙⠇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⣇⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠿⠲⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠄⣇⠀⠀⠀⣿⠶⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠜⣧⠀⠀⠀⠀⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢙⣧⠀⠀⠀⣤⢘⡇⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⣿⢀⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠪⡳⣄⣀⠀⠘⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠨⡳⣄⣀⠈⠉⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠐⢝⢦⣄⣈⠉⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⢐⡬⠶⠶⠶⠮⣔⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠤⠴⠶⠬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠭⠴⠯⣤⣢⢄
⠀⠀⠀⠀⠀⢄⡴⠋⠀⠠⠤⣤⠀⠈⠹⣌⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⡤⣄⠀⠈⠑⢄⢀⠀⠀⠀⠀⠀⠀⣼⡾⠉⠀⢀⡤⣄⠀⠉⢷⡁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢱⠇⠀⠀⢸⡇⣸⠀⠀⠀⠘⡆⠄⠀⠀⠀⠀⣘⣿⠀⠀⠀⢸⠀⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⢰⡏⢻⠀⠀⠀⢠⡇⠂⠀⠀⠀⠀⢉⣧⠀⠀⠀⠈⠓⣿⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠊⢣⡀⠀⠠⠇⠀⠀⠀⣠⢏⠀⠀⠀⠀⠀⠀⠡⠱⣄⠀⠘⢧⠾⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠈⢜⢦⡀⠀⠸⠦⠟⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠑⠻⣷⡦⣤⡤⢶⡚⠕⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⢓⠦⢤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣤⢶⡫⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⡤⠶⠶⠶⢬⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠞⠁⠀⢠⠶⣄⠀⠈⠳⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡏⠀⠀⠀⢸⠀⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⣄⠀⠀⠸⠦⠟⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢷⡦⣤⣤⣤⡴⣚⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠉

⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣾⣿⣿⣿⠀⠀⢀⣾⣿⢿⣷⣄⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣤⣶⣶⠶⠆
⠀⠀⠈⠛⢻⣿⣿⠀⠀⠈⠛⠛⣼⣿⡏⠀⠀⠀⢰⣿⣿⣯⡀⠀⠀⢠⣿⣟⣴⣶⡄⠀⠐⠿⠛⠻⣷⣤⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠈⠉⢻⣿⡷⠀⢠⣿⠿⠿⣿⣿⠇⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⣸⣿⣿⡄⠀⢴⣿⣿⣶⣿⠃⠀⢠⣄⣀⣴⣿⡿⠃⠀⠀⠀⠀⢠⣿⣿⡂⠀⢴⣷⣦⣶⡿⠟
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠈⠉⠉⠉⠉⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠈⠉⠁
⠀⠀⠀⣀⣤⣀⠀⠀⢀⣀⣀⣀⣀⣠⣄⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣀⣀
⠀⢀⣾⣿⠿⠛⠁⠀⠈⠛⠛⢻⣿⣿⠏⠀⠀⣾⣿⡋⣹⣿⠂⠀⠀⣰⣿⠿⣿⣿⣦⠀⠀⣰⣿⡿⠻⣿⣷⡀
⠀⢸⣿⣿⣶⣶⣤⡀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢨⣿⣿⣿⣿⣷⡄⠀⢿⣿⣄⣼⣿⣿⠀⢰⣿⣿⠁⠀⢹⣿⡇
⠀⢸⣿⣿⡟⢈⣿⡇⠀⢀⣾⣿⣿⠁⠀⠀⢠⣿⣿⡇⠀⣹⣿⠇⠀⠀⡉⠋⣹⣿⠟⠀⠘⣿⣿⣆⣀⣿⣿⠃
⠀⠀⠛⠿⠿⠾⠛⠀⠀⠈⠛⠿⡟⠀⠀⠀⠀⠻⠿⣷⠶⠛⠉⠀⠀⠘⠻⠾⠛⠉⠀⠀⠀⠘⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢻⣿⡄
⠀⢀⣠⣿⣧⣤⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⣶⣶⡦⠀⠀⢀⣠⣽⣥⣄⠀⠀⢴⣶⣶⡶⠀⠀⠹⠇⢀⣼⣿⠏
⠀⠈⠉⢿⠏⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠙⠛⠋⠀⠀⠈⠉⣽⡍⠁⠀⠀⠛⠛⠛⠋⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡍

⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠠⣾⠟⠛⣿⡆⠀⠀⠀⠀⠀⠾⠟⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠈⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠰⠿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣏⣀⡀⠀⠀⠀⠀⠀⣤⣄⢀⣼⡷⠀⠀⠀⠀⠀⠀⠀⢶⡦
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⢀⣿⣟⣛⠛⠁⠀⠀⠀⠀⢀⣾⡏⣉⠻⠃⠀⠀⠀⠀⠀⠻⣿⣶⣿⡿⠃
⠀⠀⠀⢠⣾⠋⣿⣿⠀⠀⠀⠀⠀⠸⠿⠛⠻⣷⡄⠀⠀⠀⠀⢸⣿⡿⠛⣿⡆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠻⠿⠿⣿⣿⠗⠀⠀⠀⠀⢰⣶⣤⣴⣿⠇⠀⠀⠀⠀⠈⢿⣧⣠⣾⡟⠀⠀⠀⠀⠀⠙⠋⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠸⠿⠿⢿⣿⡇⠀⠀⠀⠀⢠⣾⠿⢿⣦⠀⠀⠀⠀⠀⠀⣴⡿⠿⣶⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠘⣿⣷⣾⣟⠀⠀⠀⠀⠀⠘⣿⣇⣠⣿⣷⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⡆⠀⠀⠀⠀⠀⣈⡛⠛⣻⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁

⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⢰⣿⡏⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀
⠀⠀⠀⢸⣿⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠟
⠀⠀⠀⠘⢿⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣠⠐⠒⠲⣶⣄⠀⠀⠀⣠⠐⠒⢶⣦⡄⠀⠀⢠⣶⣶⡀⠀⣀⡄⠀⠀⢠⣶⡶⢶⣶⣤⣤⠆
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⣿⣄⣤⠀⣹⣿⡇⠀⠀⣿⣤⠄⢈⣿⡿⠀⠀⠈⠀⢹⠇⢻⣿⡇⠀⠀⢸⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⠛⠋⣰⣿⡿⠁⠀⠀⠈⠁⠀⠾⣟⡁⠀⠀⠀⠀⠌⠀⢸⣿⡇⠀⠀⠸⠛⠓⠒⢶⣤⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⣀⣀⠀⠀⢹⣿⡆⠀⠀⡈⠀⠀⢸⣿⡇⠀⠀⠀⣀⣀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣜⣋⣀⣀⣀⣀⠀⠀⢼⡿⠛⠃⠀⢸⣿⡟⠀⠀⠁⠀⠀⢹⣿⡏⠃⠀⢼⡿⠻⠃⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⠄⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠈⠣⠄⠀⠤⠿⠋⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠈⠣⠄⢀⠴⠿⠋

⠀⠀⠀⠀⠀⣠⡶⠊⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⣴⣾⠉⢿⣦⡀⠀⠀⢠⣴⠏⠉⢳⣦⡀⠀⠀⠀⠀⣤⡾⠉⠹⣶⡄
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⠏⠀⠀⠸⣿⣿⠀⣸⣿⠇⠀⢀⣿⣿⠀⠀⠘⣿⣷⠀⠀⠀⣾⣿⠃⠀⠀⢻⣿⡆
⠀⠀⠀⣸⣿⣿⠋⠈⣷⣦⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⢙⣻⠦⣿⣋⠀⠀⠈⣿⣿⠀⠀⢰⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⢰⣿⡟⠀⠀⠀⠀⣴⣿⠃⠀⠈⣿⣧⠀⠀⠈⠻⠦⠠⠾⣿⡿⠀⠀⠸⣿⣿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠸⣿⣇⠀⠀⣸⣿⠇⠀⠀⣿⣿⡇⠀⠀⠀⠀⢿⣿⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢻⣿⡄⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠙⠻⠦⠤⠟⠋⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠈⠛⠷⠄⠼⠛⠁⠀⠀⠀⠀⠤⠞⠋⠀⠀⠀⠀⠀⠀⠙⠷⠄⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠄⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⣴⡶⠦⠀⠀⢠⡄⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⢶⣦⣤⣤⡤⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⣻⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠀⠀⠀⣻⠀⠀⠂⠀⠀⠀⠉⠀⣠⢞⠟⠁⠀⠐⠂⠀⠀⠀⠈⠲⠶⢿⠄⠀⢀⡴⡻
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⡀⠀⠀⢸⢸⠀⠀⠐⠁⠀⠀⠀⡰⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠉⢦
⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⣼⠃⠈⠂⣠⢜⡇⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⢐⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⡇⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠑⢤⣤⣤⣠⣤⡦⠈⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡜⡬⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠱⣄⠀⠀⠀⠀⣠⡮⠀⠀⠀⠀⠀⢂⡀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠐⠶⠶⠷⠇⠀⠀⠀⠀⣤⣤⠀⠀⠹⡦⠀⠀⠈⠉⠓⠒⠚⠋⠁
⠀⠀⠀⠀⠤⡴⠂⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⢰⡆⠀⠀⡠⠀⠀⠀⠀⠀⠀⣤⣠⣼⡜⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣿⠄⠀⠀⠈⠲⠾⠾⠇⠀⠀⠠⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣀⡀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠙⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⢭⡉⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⢀⣿⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⣦⠀⠀⠀⠀⠀⠈⠀⣀⢀⣀⢤⣄⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠈⠈⠋⠉⠁⠀⠀⢿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠀⠀⠀⠀⠰⢤⣤⠤⠤⠤⡤⣴⣿⠀⠀⠀⠀⠋⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠁⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⡀⠈⠢⠀⠀⠀⠀⠀⢿⡉⠉⠙⠀⠀⣷⠀⠀⠀⠀⠀⠀⢰⡿⠛⠂⠀⠀⣹⠀⠀⠀⠀⠀⠑⠦⣤⣤⣐⠷⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠏⠈⠁⠀⢸⡇⠀⠀⢣⡀⠀⠉⠀⠁⢀⠰⠏⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⣔⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⡄⠀⢸⡇⠀⠀⠀⠙⠳⠶⠶⠾⠗⠊⠀⠀⠈⠀⠀⠀⠀⢀⡤⢤⣄⠀⠉⣄⠀⠀⠀⢠⠂⠁⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠇⠀⣸⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢯⡎⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠈⠉⠀⠀⠀⡠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢠⣄⡲⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠖⠋⠊⠁

⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡶⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠙⠛⠿⣦⣄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠈⢻⣦
⠀⠀⠀⠀⢀⣿⠁⠠⠂⣁⠀⠐⠄⠀⠀⠈⣿⡀⢀⣿⠁⠀⠀⢠⠞⢁⠤⡤⠌⢢⡀⠀⢻⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠐⢀⡈⠁⡄⠀⠀⠀⣸⡇⢸⣿⠀⠀⠀⢄⠀⢃⠀⠘⡃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⣿⡄⠀⠀⠘⢤⠀⠉⠁⢀⠜⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠘⢿⣄⠀⠀⠀⠈⠁⠈⠁⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣠⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⠉⠉
⢿⡿⠛⠛⠻⣿⠟⠛⠛⠛⠛⢿⡿⠛⠛⠛⠛⣿⡿⠋⠀⠀⣸⡟
⠘⣿⡄⠀⠀⠿⠀⣀⡀⠀⠀⠸⠇⠀⠀⣠⡾⠏⠀⠀⠀⢠⣿⠁
⠀⠈⢿⣄⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠻⢷⣤⣀⣿⣇⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⠶⢶⣶⠂
⠀⠀⠀⠀⠀⣽⡟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠈⣿⡍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⡶⠾⠿⠿⣿⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⣿⠶⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⠶⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠶⠾⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⡴⠶⣄⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢀⡿⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠞⠛⠁⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⣾⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⢀⢄⠨⡂⠀⠀⠀⠀⠀⠀⢀⠀⢅⠀⠀⠀⢀⣌⡰⢂⢳⠀⠀⠀⠀⠀⠀⠄⢀⠤⡆⠀⠀⠀⠀⠀⢠⠹⡶⠯⢡
⠀⠀⠀⣠⠞⠚⠛⠪⠕⠳⡄⠀⠀⠀⠀⡠⠝⠬⡿⠀⣠⠊⠁⠉⠉⠛⠛⣆⠀⠀⢰⠒⠒⢜⠧⣻⠗⡶⠀⠀⠀⢀⠞⠉⠉⠒⡇
⠀⠀⣴⠁⢠⠖⠉⠙⢢⡀⠙⡄⢀⡴⠊⠀⠀⡀⣷⠀⠳⢤⠔⠛⢲⠀⠀⣿⠀⠀⠘⠒⠂⡴⠀⢀⡞⠁⠀⠀⢠⠏⠀⡼⡇⠀⡇
⠀⢠⡇⠀⡞⠀⠀⠀⠀⡇⠀⢷⠈⠧⠖⢺⠀⠀⣿⠀⠀⠀⠀⢀⠞⠀⢠⠏⠀⠀⠀⡴⠋⠀⠀⠾⢄⠀⠀⢠⠋⠀⠮⠤⠇⠀⣇⣀
⠀⠸⡇⠀⢻⡀⠀⠀⣰⠃⠀⣾⠀⠀⠀⢸⠀⠀⣿⠀⠀⢀⡴⠋⠀⣰⣋⣀⣀⠀⠀⠙⠒⠲⡄⠀⠀⢧⢰⣃⣀⣀⣀⣀⠀⠀⣀⣸
⠀⠀⢳⡀⠀⠉⠛⠋⠁⠀⣰⠇⠀⠀⠀⢸⠀⠀⣿⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⠀⣰⠤⠤⠶⠃⠀⠀⡼⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠙⠲⠤⠤⠤⠴⠚⠁⠀⠀⠀⠀⠸⠤⠤⠟⠀⠀⠛⠛⠒⠒⠒⠒⠒⠋⠀⢯⣄⣀⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠘⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⡰⢄⠤⢍⠀⠀⠀⠀⠀⠀⠐⠂⠔⠳⠀⠀⠀⢀⣀⣀⣴⢡⡞⡥⢱⣄⡀⠀⠀⢀⡀⢠⡄⠀⠀⠀⠀⠀⢠⢂⣄⡚⠅
⠀⠀⠀⢰⡋⠪⠓⠼⠞⠉⡇⠀⠀⠀⡠⠖⠟⠻⠖⢹⠀⠀⢸⠀⠀⠀⠁⠈⠒⠩⢸⠇⠀⣀⣧⣼⣤⣼⡀⠀⠀⢠⠞⠉⠒⣛⠵⠛⢦
⠀⠀⠀⢸⠀⠀⡞⠋⠉⠉⠁⠀⢀⡞⠀⢀⣔⣚⠛⠛⠀⠀⠙⠒⠒⢒⠖⠀⠀⣠⠞⠀⡜⠁⢠⠄⢬⠀⢹⡄⠀⡏⠀⢰⠋⠉⢳⠀⣸⡇
⠀⠀⠀⢸⣀⣀⠉⠙⠲⣄⠀⠀⣸⠀⠀⠈⣀⡈⠉⢢⡀⠀⠀⠀⣰⠋⠀⣠⠞⠁⠀⠀⢧⡀⠘⠒⠊⢀⡼⠁⠀⢧⠀⠈⠓⠒⠋⠈⠁⡏
⠀⠀⠀⠀⠁⠉⢹⡄⠀⠘⡆⠀⣧⠀⠀⡎⠀⠙⡆⠀⣧⠀⠀⣸⠁⠀⣰⠋⠀⠀⠀⣠⠞⠁⢀⣀⣀⠉⠓⣄⠀⠈⠓⠶⠤⡤⠀⠀⢠⠇
⠀⠀⠀⡰⠒⠒⠊⠀⠀⢠⠇⠀⠸⡄⠀⠓⠦⠞⠁⢀⠇⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⡇⠀⠰⣇⠀⣸⠆⠀⢸⠀⢰⠶⠒⠊⠁⠀⣠⠏
⠀⠀⠀⠳⠤⣤⣤⠤⠖⠁⠀⠀⠀⠙⠲⠤⣤⠤⠖⠋⠀⠀⠘⠓⠒⠚⠁⠀⠀⠀⠀⠳⣄⠀⠈⠉⠁⢀⣠⠎⠀⠸⢤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣴⣯⣿⣿⣿⣧⣼⣿⡼⣿⣿
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡂⠀⠀⠀⠀⠀⠀⠀⢦⣤⠆⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⡔⠓⢹⠀⠀⠀⠀⢀⠔⠉⠛⠳⠲⡄⠀⠀⠀⠀⠀⡃⠉⠒⠘⢫⠃⠀⠀⠀⠀⠀⡜⠒⠑⡆⠀⠀⠀⠀⠀⢰⠁⢚⢸⠊⠁
⠀⠀⢐⢁⣀⠀⢸⠀⠀⠀⠀⠈⠢⠚⠙⡄⠀⡇⠀⠀⠀⠀⠀⢉⠝⠁⢠⡃⠀⠀⠀⠀⢀⠌⢀⠞⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠄⡀
⠀⠀⠈⠁⢸⠀⢸⠀⠀⠀⠀⠀⠀⢀⠜⠀⡼⠁⠀⠀⠀⠀⠀⠙⠖⢦⠀⠈⠆⠀⠀⢀⠎⠀⠛⠊⠀⠘⢲⠀⠀⠀⠀⠘⠓⠲⡀⠈⢦
⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⡔⠁⠀⠈⠁⠘⡆⠀⠀⠀⢠⠖⠶⠊⠀⢀⠆⠀⠀⠈⠁⠉⠀⢳⠀⠘⠉⠀⠀⠀⠀⠔⠂⠚⠀⢀⡌
⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠃⠀⠀⠀⠘⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⢀⣀⣈⢁⡀⠀⠀⠀⢀⣀⣀⢀⣄⡓⣄⠀⠀⠀⠀⠀⡐⢀⠒⠀⠀⠀⠀⠀⠀⠀⡐⢠⡄⠂⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐
⠀⠀⢀⠔⠉⢁⠭⠤⠇⠀⠀⠀⢈⠀⠀⠁⠁⠘⢸⠀⠀⠀⠀⠔⢉⠉⡉⠱⡄⠀⠀⠀⢠⠊⢀⠬⢅⠉⢢⠀⠀⠀⠀⡔⠉⡨⢭⡉⠳⡄
⠀⠀⡎⠀⠐⠓⠒⠦⡀⠀⠀⠀⠈⠉⣩⠋⢀⡴⠊⠀⠀⠀⠐⣆⠈⠒⠃⡠⠃⠀⠀⠀⢸⠀⠰⢤⠼⠀⢹⠀⠀⠀⡜⠀⡎⠀⠀⢸⡀⢱
⠀⠀⡇⠀⡆⠉⢣⠀⢷⠀⠀⠀⠀⡴⠁⢠⠎⠀⠀⠀⠀⠀⡜⠁⡠⠤⢄⠈⢢⠀⠀⠀⠈⠳⠤⣤⠀⠀⡸⠀⠀⠀⡇⠀⢇⠀⠀⣨⠃⢸
⠀⠀⢳⡀⠑⠒⠃⢀⠎⠀⠀⠀⢰⠃⠀⡞⠀⠀⠀⠀⠀⠀⢇⠀⠑⠠⠊⠀⡼⠀⠀⠀⠀⡖⠂⠁⢀⡴⠃⠀⠀⠀⠹⣄⠈⠉⠉⠁⣀⠞
⠀⠀⠀⠉⠓⠒⠊⠁⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⠀⠀⠓⠂⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠊⠁






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣭⣭⣭⣭⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⣽⣿⣽⣿⣿

⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⡀⠀⠀⠀⢀⣀⣀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⣀⣀⢀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠛⢻⣿⠟⠛⠃⠀⠀⠀⠛⢿⡿⠻⣿⠟⠃⠀⠀⠘⠻⣿⡟⢿⣿⠛⣿⡟⠃⠀⠀⠀⠛⣿⡿⠻⣿⡟⠛⠛⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⢹⣧⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣷⣿⠇
⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⡆⠀⠀⠀⣴⣿⣷⣾⣿⣶⡄⠀⠀⢠⣶⣿⣷⣾⣿⣶⣿⣷⡆⠀⠀⠀⣴⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠈⠹⣿⡏⠉⠉⣿⡿⠉⠀⠀⠀⠈⠉⣿⡏⠉⠉⣻⡟⠉⣿⡏⠁⠀⠀⠀⠉⠹⣿⠉⠉⠉⣿⠏⠙⣿⠉⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⢿⡇⠀⣸⡿⠀⠐⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣸⡟⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠃⠀⠸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⠀⠀⢀⣿⠀⣸⣇
⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠶⣶⣶⡶⠶⢶⣶⣶⢶⣶⣶⣶⣶⢶⣶⣶⠆⠀⠀⠀⢶⣶⣶⢶⣶⣶⢶⣶⣶⠶⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢀⣿⠇⠀⣿⠀⢸⡏⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢻⣧⣠⣿⠃⠀⠀⠀⠀⠀⠀⢻⣇⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣸⡟⠀⠀⣿⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⠃⠀⠀⣿⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣆
⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣀⣀⣠⣿⣄⣼⣧⣠⣿⣇⡀⠀⠀⠀⣠⣿⣧⣠⣿⣧⣨⣿⣦⣀⠀⠀⠀⢀⣠⣿⣧⣸⣿⣆⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⡀⣀⡀⣀⣀⢀⠀⠀⠀⣀⣀⢀⡀⣀⢀⡀⣀⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡟⢻⣿⠟⢻⣿⠟⠃⠀⠀⠛⢿⣿⠛⢿⣿⠛⢿⡿⠻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣾⠏⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⣦⣿⠃⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣧⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠛⣿⡄⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣾⣿⣷⣾⣿⣶⡄⠀⠀⣶⣾⣿⣶⣿⣿⣶⣾⣷⣶⣿⣷⡄

⠀⠀⠀⠈⣿⡏⠀⠀⠀⢹⡟⠀⢹⡏⠀⠀⠀⠘⣿⠁⢹⣿⠁⢹⡟⠀⠀⠀⠈⣿⠁⠈⣿⡁⢸⡟⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢹⣧⣿⠃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢿⡏
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢹⣏⠀⣿⠏⠀⠀⠀⠙⣿⠁⣹⡟⠀⢹⡏⠀⠀⠀⠘⣿⡁⢹⡟⠁⢹⡟⠀⢹⡏
⠀⠀⠀⠀⠀⠀⣿⣴⡟⠀⠀⠀⠀⠀⢻⣧⣿⠁⠀⢸⡇⠀⠀⠀⠀⢹⣧⣿⠃⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠈⣿⡇⢸⡿⠁⢹⣿⠁⢹⣿⠁⢹⡏⠀⠀⠀⠘⣿⠁⠉⢿⡇⣿⠏⠁⠀⠀⠈⢻⣇⣸⡟⠁
⠀⠀⠀⠀⠸⣧⣾⠇⠀⢸⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢿⡟⠀⠀⢸⣿⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣴⡖⣶⡄⠀⠀⠀⠀⣰⡿⢿⣆
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠁⠉⠉⠁⠀⠀⠈⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣨⣿⠋⠈⣿⡋⠀⠀⠀⠈⢿⣇⣹⡟⠁⢹⣿⠁⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡟⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣧⠀⠀⣿⠇⠀⠀⠀⠀⣰⡟⢿⡄⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠈⠉⠉⠀⠀⠀⠈⠉⠁⠈⠉⠁⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⢹⡏⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠀⢸⣿⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠀⠘⣿⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠦⠀⠀⠀⠀⠀⠠⠾⠷⠤⠴⠿⠦⠀⠀⠀⠀⠠⠾⠷⠤⠾⠿⠤⠴⠿⠦⠀⠀⠀⠀⠠⠾⠷⠄⠀⠀⠾⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠂⠀⠲⣶⠖⠀⠀⠀⠀⠐⢲⡶⠂⠀⢲⣶⠒⠒⣶⠒⠀⠀⠀⠀⠐⢲⣶⠂⠀⢲⣶⠒⠒⣶⠖⠒⢶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⡏⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣾⠃⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡃⠀⠀⣀⣿⣀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⠁⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢰⡿⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣸⡇⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢘⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠁⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠷⠀⠀⠰⠾⠷⠶⠾⠿⠶⠶⠛⠖⠀⠀⠀⠀⠰⠾⠷⠖⠾⠿⠆⠀⠶⠻⠶⠀⠀⠀⠀⠀⠾⠻⠆⠀⠶⠻⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⡖⠀⢐⣶⠒⠒⢲⡖⠂⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠀⠐⣶⠖⠒⢲⣶⠒⠒⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣼⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣸⡟⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠹⣧⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣃⠀⢈⣿⣄⣀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⡀⢀⣻⣆⣀⣸⣿⣀⣀⣿⣀
⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡶⠒⣢⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣿⠀⠻⠿⠛⠁⠘⢿⣿⣿⡿⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣥⣤⠀⠀⢀⡀⠀⣾⣷⡦⠀⣀⡀⠀⠀⣴⣿⡇⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠉⠉⢠⣾⣿⣿⡆⠀⠁⢠⣾⣿⣿⡆⠀⢀⣤⣤⣀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⣀⠉⠛⠋⠁⠀⢀⣸⣿⣿⣿⠇⠀⢿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠀⠀⢿⣿⡇⠀⣾⡷⠊⠁⠀⠀⠉⠣⣄⣠⣄⠙⠛⠋⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣤⣤⡀⠉⠁⠀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣇⣀⣈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠹⣿⣿⡇⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣀⣉⡀⠀⠙⢟⣀⠤⠲⣾⣿⣿⡟⠉⠁⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⣴⣿⣿⣿⣿⡄⠀⣨⣤⣤⡀⠀⠉⠋⠀⣴⣷⠀⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢿⣿⣿⣿⣿⠇⠘⠿⣿⡿⠇⠀⣿⣷⡄⠈⢉⣀⣀⡈⠉⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣦⠀⠉⠛⠛⢋⠀⠀⢀⣄⡀⢀⡤⠿⢿⠁⠀⣿⣿⣿⣿⣷⣦⠀⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⣴⣿⣧⠀⣾⣿⡿⠁⠀⠀⠀⠱⡀⠈⠙⠛⠛⠋⠁⠀⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣶⣿⣶⣄⠙⠛⠋⠀⠹⢿⠃⠀⠀⠀⠀⠀⢻⣦⠀⠀⣴⣾⣷⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠛⠛⠛⠋⠀⠰⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⠿⠃⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⡄⠀⢀⣤⣀⠙⠻⠟⠀⢇⠀⠀⠀⠀⢀⠏⢁⡀⠀⣠⣶⣶⣄⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⣿⣷⡄⢠⣶⣮⡢⣀⣀⣠⣾⠀⠿⠇⠀⠻⠿⠿⠟⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠹⢿⣿⠿⠁⢸⣿⣿⡇⠀⣠⣬⣀⠀⠀⣤⣦⣄⠀⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢠⣤⣦⡄⠉⠉⠀⢸⣿⡿⠟⠀⠘⣿⣿⠿⢀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠀⣼⣿⡄⠀⣰⣾⣿⣦⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠛⠷⠤⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⡀⠀⢀⡴⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠸⠸⠀⠂⠘⠂⠀⠀⠀⠀⢻⢦⠈⢇
⠁⠀⠉⠀⠇⢠⠏⠀⠀⢀⣀⣀⣀⣠⠤⠀⠀⠀⠀⢸⠞⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡄⠳⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠛⠒⠲⠤⢤⠀⠀⠀⡴⢻⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢤⡀⣧⠈⢣⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣞⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡀⠀⢳⡀⠀⠀⢀⡀
⡀⠀⠀⠖⢚⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠳⠀⠉⢀⡀⠢⡙
⡙⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣩⡽⠶⠆⠀⠀⠀⠀⠁⠀⠀⠐⠢⢷⡀⠙
⠀⠀⠀⠀⠀⠀⣠⠞⢀⠀⠀⠀⠀⠀⠀⣠⠜⢂⡀⠀⠀⠀⢠⣤⣤⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠑
⠀⠀⠀⠀⢠⣞⡥⠖⠋⣡⡀⠀⢀⡠⠞⢁⠴⢋⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠈⠋⠁⢀⡴⠊⠁⠀⠀⠈⠡⠞⢁⡴⠋⠀⠀⣀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣉⠭⠟⠉
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⢦⠘⣆⠀⠀⠀⢸⠃⡏⠘⣄⠆⠀⠘⣋⣀⡤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣦⠀⠀⢸⠀⡇⢀⢿⠀⠀⠀⠉
⠀⠀⠀⣀⣠⠤⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠐⠃⠚⠘⠂⠀⠀⠀⠀⠀⢻⢀⡟
⠀⠀⢋⣡⠤⠖⠚⣉⡄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡤⠖⠚⠉⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣸⠁
⠀⠀⠀⡶⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣧⠇
⠀⠀⢠⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⠰⠦⢤⣀⣀
⠀⢀⡞⢠⠇⠀⠀⠀⠀⠀⠀⢲⡀⠑⢼⠀⠀⠀⠁⠉⠙⠒⠦⢤⠀⠀⠀⠀⠀⠀⠀⠶⠖⠒⠒⠛⠛
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠓⢄
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠈⡧⠸⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⡀⡀
⠀⠀⠀⠀⣿⡴⠋⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⠀⢇⡇⠀⠀⠀⠀⣄⠀⢦⠹

⠀⠀⠀⢈⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⢀⣿⡹⣯⡀⠀⠀⠀⠀⠀⣤⣤⣼⣷⣿⣂⠀⠀⠀⠀⠀⣿⣴⣂⣀⡦⡄⠀⠀⠀⣴⣿⣿⠿⢻⣿⠶⠀⠀⠀⠀⢰⣿⣿⣿⢿⡀
⠀⠀⠘⠿⠶⠿⠃⠀⠀⠀⠀⠀⠛⠛⠷⠶⠋⠁⠀⠀⠀⠀⠤⠿⠿⠽⠿⠧⠤⠀⠀⠀⠀⠙⠿⠾⠿⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠁

⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⢀⣠⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⢸⣿⣿⣷⢲⡄⠀⠀⠀⠀⠾⢀⣞⠹⣿⠓⠀⠀⠀⠰⣟⣿⠀⠀⢸⣏⡷⠀⠀⠀⠀⣶⣒⣚⣛⣓⡂⠀⠀⠀⠀⠀⣰⣟⢽⣦
⠀⠀⠈⠙⠛⠙⠚⠁⠀⠀⠀⠀⠦⠌⠏⠠⠟⠁⠀⠀⠀⠀⠈⠁⠳⠶⠂⠉⠀⠀⠀⠀⠀⠀⠻⠿⠿⠯⠇⠀⠀⠀⠀⠀⠙⠿⠾⠋

⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠀⠠⡀
⠀⠀⢾⣿⣾⣿⡧⠀⠀⠀⠀⠀⢰⣿⣿⣤⣤⠀⠀⠀⠀⠀⢀⡴⣤⡐⢀⠇⠀⠀⠀⠀⠀⣿⣿⣽⣭⣿⡛⠋⠀⠀⠀⢰⠀⣺⠀⢨
⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠘⠯⠿⠃⠁⠀⠀⠀⠀⠘⠛⠛⠒⠚⠓⠛⠁⠀⠀⠀⠀⠀⠂⠤⠄⠙⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢨⡟⠀⠀⠀⠀⠀⠈⣿⡀⠀⢠⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⣾⠃⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣤⡿⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠃⠀⢹⡇⠀⢹⡇⠀⢹⡇⠀⠀⠀⠀⠀⢹⡇⠀⠈⣿⡀⢠⡿⠁⠀⠀⠀⠀⠀⠈⢿⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⡿⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡇⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠁⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⢻⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠀⠈⢿⡀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣸⠏⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣸⠏⠀⠈⣿⠁⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠹⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠹⣇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠛⠙⠋⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠚⠋⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⣳⣧⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⢉⡱⠀⠀⠀⠀⡀⠀⠦⠄⠠⢒⣼⠒⢐⢚⣺⠀⣻⡃
⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠥⠤⠄⠒⣒⢲⡂⢎⢫⣻⡦⠭⢯⠟⡖⠋⠉⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡐⠀⡄⠀⠂⠀⣀⠀⠀⠀⠀⠓⠂⠋⠉⠉⠉⠁⠹⠀⠚⠚⠁⠀⠀⠖⠀⠀⠐⠨⢍⣀⠀⠀⢄
⠀⠀⠀⠤⢶⡤⠀⠀⠀⣠⠐⣲⣾⡞⠨⢵⣶⠅⠐⣖⣈⢥⡀⠶⢁⠁⠀⠀⢀⠀⠀⠀⢈⡐⢁⡢⣀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣈⣤⠴⠧
⠀⠀⠑⠤⡐⠀⠐⠀⢄⠙⠒⢋⣜⣿⣀⢘⣻⠤⢀⣰⣏⡿⣥⡀⠀⠁⢂⡐⠀⠌⣃⢠⠐⠀⠦⢄⠠⠑⠤⣏⣀⣈⣟⣶⡞⠉
⠀⠀⠀⠀⠈⠁⠠⡀⠀⢢⠄⠀⣠⠤⢉⣉⠩⢅⠀⠒⠾⣟⠉⠤⠴⠂⢀⡌⠀⢠⠈⠃⠄⢀⣀⡀⣰⡄⠴⠄⠵⢤⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢡⣒⡶⠃⠀⠐⠁⠄⢬⡼⠿⣾⠷⣀⠀⠀⢉⡁⠀⡀⢀⣀⡀⣀⣠⣠⣜⣠⠒⠡⢶⡄⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡛⠉⠁⠀⠈⣿⣦⠟⠋⠛⠛⠋⠉⠀⠉⠁⠈⠀⠁⠀⠀⠌⠠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣢⣀⡰⠆⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠴⠖⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⡀⡀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠲⠠⠑⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠗
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠢⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠑⠀⠒⠀⠈⠘⠐⠀⠀⠀⠀⠦⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡲⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠢⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠙⠀⠄⢀⠀⡀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢑⣀⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡤⠄⡃⠀⠀⠀⡴⢀⠂⠀⠀⠀⣀⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⢠⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡴⣀⡀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰
⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠐⡂⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦⠀⠈⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⣶⠀⠀⣶⠀⠀⠀⠀⠀⠀⢴⡆⠀⣶⡆⠀⣶⠄⠀⠀⠀⠀⠀⢰⡆⠀⢴⡆⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣷⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⠁
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸
⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢠⡄⠀⢤⡄⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⣤⠀⠠⣤⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢠⡄⠀⣤⡄⠀⣤⡄⠀⣤⡄
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣼⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⡞⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⠁⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡇⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⠀⢰⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⡎⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⡸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢰⠇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠁⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠁


⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⢸⡇⠀⠀⢣⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠈⢇⠀⢠⡿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠈⢇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⠁⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢳⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⢧⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⣼⠃⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡿⠀⠘⡆⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠚⠃⠀⠘⠛⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠘⠂⠀⠘⠓⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠘⠂⠀⠚⠇⠀⠚⠃

⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⢸⣿⣷⣒⣒⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠘⢻⣏⡟⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠛⠛⢿⣿⣿⡆⠀⠀⠀⠀⠈⠛⠛⠛⣻⣿⣿⣷⣿⠗⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⢸⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣫⡁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⢀⣤⣠⡀
⠀⠀⠀⠀⢸⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⡄⠀⠀⠀⠀⠀⢸⣯⣯⣏⣀⣀⣸⣿⣸⡇
⠀⠀⠀⠀⢸⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⣤⣄⣤⡀⠀⠀⢘⣯⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⢸⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣾⣷⣧⣄⣴⣤⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⢸⣿⢸⡇
⠀⠀⠀⠀⠸⠿⠇⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣿⣿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠼⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣻⣟⣛⣛⣿⡇⣿⣿⣹⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣽⣷⡄
⠀⠀⢸⣟⣟⡛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⠛⢁⣿⣾⣺⠀⠀⠀⠀⠀⠀⣿⡿⠻⣻⣿⡯⣷
⠀⠀⢸⣿⣿⢨⣟⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⢀⣵⣾⣷⣿⣿⣿⢇
⠀⠀⢸⡿⣿⠽⠛⠛⢿⣿⣿⣦⠀⠀⠀⠀⣸⣿⣿⡟⠚⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⢀⡿⣿⣿⠟⠋⠻⣵⣿⣷
⠀⠀⢠⣄⣤⡀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢸⣷⣷⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣟⣿⣇⠀⠀⠀⣽⣯⣿
⠀⠀⠘⣿⣿⣿⣶⣴⣾⣿⣿⠏⠀⠀⠀⠀⠸⣿⣿⣷⣦⣖⣽⣿⡿⠃⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣾⣿⣿⠏
⠀⠀⠀⠈⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⠷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠻⢿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⡿⠛⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⠀⠀⠀⢈⣿⣿⠗⠀⠀⠀⠀⣿⣿⣿⠁⠀⠈⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⢆⣾⣿⣿⠃⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣏⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⢀⠄⢀⠄⠀⠀⠀⠀⣀⠠⠌⠒⠈⠁⠀⡀⠀⠀⠀⠀⡉⢠⠉⠈⠀⡄⠀⠀⠀⡇⢄
⢀⠔⣲⠦⣄⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠈⠠⡄⠀⠀⠉⠀⠀⠀⠀⠀⢁⡀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠕⣄⠀⠈⠁⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠄
⠒⡘⣢⣤⣔⣓⣂⣂⣄⡀⠀⠀⠀⡠⠜⡩⠃⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠐⠀⠀⠀⠀⡠⢒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢆⡠⠊⠀⢀⠴⢳⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡀⠌⠗⡠⡔⠁⠬⠒⠚⠃⠀⠀⠀⠐⠀⠤⠀⠘⠀⠀⠀⠀⢀⡌
⠀⠄⡙⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠄⠁⠀⠀⠙⠣⠔⠁⠀⠀⠀⠀⠈⠰⠀⠁⠀⠀⣀⠀⠤⠀⠀⠁
⠀⠪⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠛⠛⠒⡄⠀⠀⠀⠂⡤⠤⠤⠩⠒⠑⠙⠒⠒⠓⠂
⠀⠀⠈⠉⠉⠀⠀⠀⡀⠀⢀⠠⣀⡀⠀⠀⠀⠀⢀⠋⠧⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠠⠂⠀⠀⠀⠠⣄⣀⣀⠀⠀⠀⠀⠠⠄
⠀⠠⢄⠀⠢⠴⠔⠐⠊⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⠀⠀⠉⠉⠒⠐⠌⠤⣄
⠀⠀⠀⡀⠀⠀⢀⣀⠠⢤⢐⣒⣭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠤⠐
⠀⠀⠠⢒⠈⠬⠃⠀⠈⠉⠁⠀⠀⠀⠠⠀⠀⡐⡩⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠄⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠔⠘⠲⣈⡀⠀⢰⡰⠂⠙⢢
⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠒⠀⠩⠲⢂⣋⣀⡐⠤⡄⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠘⣿⠀⠀⠀⡝⡂⠀⠘⢧⠀⠀⢠
⠀⠁⣀⠀⠒⡠⠜⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⣀⠀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⣀⡀⠀⠀⠁⠒⠈⡇⡂⠀⠠⠀⠉⠃⠁⠀⠀⣐
⢄⠀⠈⠂⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡽⠙⠓⠢⠀⠀⠀⠀⢀⡀⠀⠐⠀⢄⠀⠀⠈⠂⠊⠉⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠹⠂⠀⠀⢔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠠⠁⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⢀⠤
⠀⢀⣤⡀⠀⠡⡀⠀⠀⠀⠈⠩⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠖
⠂⠀⠀⠻⣄⡀⠐⠀⠀⠀⠰⣀⠀⠀⠈⠑⠂⠤⡄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡲⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠈⢧⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡄⠄
⠀⠀⠀⠀⠀⠂⡀⠀⣐⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⠌⢠⠎⠀⡀⠀⠀⠀⡀⠀⠈⠉⠈⠉⠐⠒⠂⠤⠤⠀⠀⠀⠱⠀⠀⠸⡀
⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠂⢤⠀⡰⠁⠀⡔⠀⠀⠀⠂⣀⣐⢢⠤⠐⠀⡤⠄⢀⣀⣀⠀⠀⠀⠀⢣⠀⠀⢡⠡
⠀⠒⠀⠀⠈⠀⡁⠀⠀⠀⣖⣠⣄⠀⠀⠀⠀⠁⠺⢦⢄⠀⠀⠈⠀⢀⠎⠀⠀⠀⠀⠠⡀⠀⠈⠉⠁⠑⠛⠑⠒⠒⠤⠀⠀⠀⠀⠀⡂⠀⠘⡄
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢩⠆⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠀⠀⡸⠂
⠀⠀⠰⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠊⡴⠁⠀⢠⠆⣦⡀⠀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢄⡀⠐⠁⠀⠈⠙⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠀⠀⠠⡀
⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣒⣂⣀⣀⣀⣀⣀⣀⣁⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠖⠃⠀⠀⠇⠀⠀⠀⠀⠔⢐⠐⡀⡂⠀⢐⠀⠂⡂⠂⢐⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⡄⢼⢿⡧⡿⣿⢼⢾⣧⡧⣿⢿⢼⡇⢠⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠂⠐⠀⠐⠀⢺⡇⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠂⠓⢀⠀⠀⢄⣀⠀⠁⠀⠀⠀⡀⢸⢽⡇⣿⣿⢸⢸⡇⡇⣿⢺⢸⡇⠰⠀⠀⣿⡇⠈⠃⠘⠀⠙⠀⢻⡇⠀⠀⠘⢿⣶⣿⠟⠁
⠀⠀⠀⠀⠼⠑⠦⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⡇⠈⠁⠈⠀⠈⠀⢹⡇⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢰⡖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠉⠀⠂⠠⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢰⡟⡩⠉⠉⠘⠩⠛⠉⠙⣷⠀⠀⠀⠙⢦⣄⠀⢀⣠⡴⠖⠊⠁
⠀⠀⠒⠚⠛⠛⠃⠀⠀⠁⠒⠐⢲⡆⠀⠀⠀⠘⠶⠔⡥⠤⠒⠁⣀⡼⠀⠀⠀⠀⠈⠃⠀⠀⠀⠤⣤⣤⣤⣼⠿⠀⠀⠀⠀⣠⣽⣿⣿⠉
⠀⠀⠀⠀⢠⣄⠀⠀⢀⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠸⣇⣀⡾⠛⠉⠀⠀⠀⠀⠀⢠⡶⠞⠒⠒⠒⠂⠒⢒⣶⣦⠀⠀⣠⣾⣿⠟⠉⢻⣷⣄
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⢀⣀⣐⣿⠀⠈⢻⠟⠁⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⣠⡾⠋⠈⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⢀⣤⠤⠤⠤⠙⠻⠠⠤⢤⣄⠀⠀⠀⠀⠰⣄⠀⠀⠉⠀⠀⣠⠌⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⠂
⠀⢺⡃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡞⠁⠀⠀⠀⠈⢦⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠁⠀⠀⠀⠁⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⡾⠿⠛⠛⠋⠉⠀⠀⠀⣾⡋⠀⠀⣠⠾⣄⠀⠀⢙⡦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡴⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠞⠁⠀⠈⠳⣴⠋⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⢀⢀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠠⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠠⠀⠠⠄⢁⠖⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠃⡄⠀⠀⠀⢀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠠⠀⡀⠆⠀⠀⠑⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠐⠈⡀⠠⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠐⠀⠈⠠⠈⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⡄⠀⠐⠀⠀⠀⠀⠀⠀⠄⠁⣤⣲⣿⡟⠐⣦⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⣿⢸⢻⡇⠀⡥⣿⢱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⢉⠸⠀⣿⢹⠉⠁⠑⡇⢻⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⡀⠀⠀⠂⠋⠚⢢⡀⠀⣇⣴⠈⠁
⠀⢰⡟⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠀⠀⠐⠀⠁⠘⠘⠁⠀⠉⠁⣀⠐⢄⠀⠀⠈⠐⠤
⠀⢸⠶⣾⠄⠀⠀⠀⡀⠀⠀⢁⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢸⣦⣍⣀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⡤⠊⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂
⠀⠈⠀⠀⠀⠈⠀⠀⠀⠠⡀⠀⠀⡣⣠⠄⠁⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠉⠁⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⢰⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠈⠂⣄⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⢦
⠀⢰⠀⠀⠀⠀⠀⠘⡄⠺⡦⠀⠀⠀⠈⠀⠉⠢⢄⠀⡀⡁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡴⢢⡆⢸⠆
⠀⠈⠀⠀⠀⠀⠀⠀⠈⠲⠶⠖⠓⠀⠀⠀⠀⠀⠀⠈⠉⠂⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠤⠬⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⣠⣄⣀⡀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣀⢀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⢀⠀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢦⣏⣉⡤⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣛⣉⡼⠡⠤⡤⠀⡀⠘⢋⣐⣥⣤⣄⣠⣄⣀⣀⣲⣖⡲⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠚⠙⣷⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠖⠛⣷⠂⡀⠛⠻⠁⠀⠘⠉⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⡿⡃⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⡟⠐⠃⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣶⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⢀⡀⠠⡆⠀⠀⠀⠈⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⣿⣸⣧⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠡
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡙⣿⣄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣆⢻⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⡷⠈⠉⢀⢐⠂⠀⠀⠀⠀⢀⠀⠺⠇⠀⠀⠂⠀⠀⠀⠈⠉⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢈⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡔⠀⠀⠀⠀⠆⠀⠀⠀⠺⢠⣠⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣐⠂⠁⠐⠚⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⠂⠂⠐⠒⠁⠀⠀⢀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠂⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠒⠠⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠳⠜⠜⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠲⠒⠂⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠄⠀⠠⠶⠀⠀⠀⠀⠀⡀⠀⣠
⡿⠋⢁⡄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠜⠀⠀⠀⠐⣠⣴⣿⣿
⠁⠀⠺⡤⠀⠀⢸⣿⣿⣿⡿⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡄⠀⠀⠤⠄⠀⣼⣿⡿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣿⣶⣤⣤⣴⣾⡿⠋⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠄⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠈⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⡐⠠⠈⠀⠀⠀⠀⠀⠸⣿⣿⣿
⣿⡿⢋⠀⠀⠀⠑⠀⡀⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⠀⢀⠄⠀⠀⠀⠀⠀⢿⣿⣿
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠲⠄⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢄⣀
⡠⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢨⠖⠐⠀⠁⢀⠤⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲
⠀⠀⠀⠁⠀⠀⠀⡀⠀⣄⠀⠈⠀⠀⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⢓⡆⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡖⠀⠘⡅⠀⠀⠀⠀⠀⢉⠂⠐⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠋⠀⠉⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠒⠢⡁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠿⢤⡀⣀⠠⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠫⠀⠀⡀⣀⡴⡃⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣠⡾⣿⢻⡿⢷⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠛
⣇⠀⠀⠐⠀⠀⠀⠀⣀⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣀⣿⢸⣷⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⠀⠄⠀⠀⠀⠀⢠⣿⣯⢿⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣾⣿⣶⣄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢲⡆⠀⠀⠀⠀⠀⠀⠀⡐⢻⠁
⠀⠀⠀⠀⠀⢐⣂⣤⣤⣈⠀⣠⣤⣤⣔⠉⣠⣀⠀⠠⡶⠤⠀⢸⡇⠀⠤⣤⡀⠀⢀⠾⢤⣿⡀⠀⢠⣤⠅
⠀⠀⠚⠶⠒⣛⠴⣭⣥⡆⣀⣋⣀⣀⣻⡇⠀⠀⡩⠶⠉⠁⢀⣺⣁⠀⠒⠒⠛⠁⠠⣀⣲⠷⢄⣜⠒⢚⣂⣀
⠀⠀⠠⠤⡤⠭⠶⠃⣈⠑⠲⠒⠀⣠⠏⢀⠀⢈⣲⠾⢖⣀⡈⢉⣁⣴⠃⢀⠀⠈⠀⠚⢺⠀⣀⣀⢠⣚⣛⣂⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠉⢁⠆⢠⢾⣧⣤⣤⠀⠀⠂⠀⠉⠲⠶⢁⡀⠛⠻⠘⠁⣈⣛⣛⣧⣴⠿⢀⣐⠀⣀⡀
⠀⠀⠀⠀⠀⠐⠶⠖⠒⠁⢰⡚⠢⠐⠒⢉⡐⠛⠓⣀⡛⠛⠋⠉⠐⠀⠐⠚⠋⠀⠀⠀⢠⣬⣤⡈⢻⣧⣬⣉⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉⠉⢀⣷⠶⠀⠾⣿⢇⣨⣩⠔⠀⠰⠛⠛⢛⣷⣃⠀⠀⠙⠶⢤⣬⡥⠎⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣠⠟⠁⠀⢂⣤⣤⣌⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠢⠥⠤⠶⠛⠁⠀⠀⢀⠛⠉⢉⣡⡜⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣦⡄⠀⠀⢸⣿⣶⣶⣤⡄⠀⠀⢸⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣠⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠴⠦⣤⣄⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣿⣆⠀⠀⠀⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣿⠟⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠒⠛⢐⣤⣀⠀⠀⠀⢠⣴⣾⣿⣿⣶⡤⠀⠀⠀⠀⢀⣤⠤⢤⠠⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢻⣾⡿⣷⣿⠃
⠀⠀⠀⢾⣹⡇⠀⣿⣧⣿⠆⠀⠀⠀⢹⠀⢸⠀⢸⠀⠀⠀⠀⢰⣿⣷⣬⣿⣠⡾⠀⠀⠀⠀⢰⠾⣧⡿⢷⠀⠀⠀⠀⠀⢸⣯⡿⣯⣿
⠀⠀⠀⠀⠙⠄⣄⠬⠋⠉⠀⠀⠀⢀⣾⣄⣾⣄⣼⣄⠀⠀⠀⠘⠟⠛⠻⠻⠋⠀⠀⠀⠀⠘⠛⠒⠛⠓⠚⠛⠂⠀⠀⠀⠺⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣴⠹⡍⢩⠏⢦⡀⠀⠀⠀⢾⢉⢹⡄⢤⡄⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠈⣃⣜⢉⢟⡖⢤⠀⠀⠀⠀⢠⣤⠯⢧⣀
⠀⠀⢺⡇⣻⡇⣿⣿⠀⢸⡧⠀⠀⠀⢨⣧⣯⣵⣼⡄⠀⠀⠀⠑⣺⣿⣿⠀⣿⡒⠁⠀⠀⢺⠋⠀⢹⠺⠤⡎⠀⠀⠀⢶⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠉⠻⣰⣣⣞⣆⠞⠀⠀⠀⠀⢸⣿⣿⣿⣽⡇⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠈⠒⠐⠻⣄⡛⣤⠀⠀⠀⠀⠙⣿⠿⢿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣄⠀⠀⠀⠀⠀⣸⣷⡆⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⣦⠀⠀⠀⠀⢠⠻⡈⢹⠐⢒⡆⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤
⠀⠀⠀⣯⣽⣿⢺⢺⠾⡷⡇⠀⠀⠀⠂⠙⠛⡉⠛⠶⠄⠀⠀⠀⢘⠗⠀⠐⢟⠀⠀⠀⠀⠙⠒⠃⢸⠀⣎⣘⡄⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠛⠛⠛⠛⠿⠿⠋⠀⠀⠀⣠⣶⣶⡄⠂⠀⠀⠀⠀⠀⠠⣿⣿⣶⣾⣿⡧⠀⠀⠀⠀⠀⣀⣸⣀⡈⠁⠀⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠻⠛⠻⠂⠀⠀⠀⠀⡇⢴⡦⠀⣿⣻⣿⡇⠀⠀⢠⢝⢿⣿⡿⣆⡀⠀⠀⠀⢀⣀⣴⣒⣲⣆⣀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣷
⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⠀⠀⠀⣥⣤⣤⡄⠛⠛⠛⡁⠀⠀⠇⠀⠑⠀⠛⢩⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣦⢸⡇⢿⣼⡿
⠀⠀⠀⠠⣾⣿⣾⣼⣿⡆⠀⠀⠀⣿⣯⣸⡇⠐⠟⠀⡇⠀⠀⠈⠢⢤⡄⢴⣿⣿⠂⠀⠀⠀⠿⠿⠷⠿⠿⠇⠀⠀⠀⠀⠉⠈⠁⢿⣾⠟
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣴⢳⡆⠀⠀⠀⠀⣴⢱⣎⢱⡆⠀⠀⠀⣴⢹⡎⢱⡆⠀⠀⠀⠀⢠⡞⢱⡆⠀⠀⠀⠀⠈⢱⡍⣶⠙
⠀⠀⠀⠀⠀⠘⠴⢇⡁⠀⠀⠀⠀⠸⠏⠀⢀⣿⠀⠀⠀⠐⢃⡰⢧⡟⠀⠀⠀⣀⢿⡼⢇⡄⠀⠀⠀⠀⡸⢧⡤⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⡀⢰⣾⠋⠀⠀⠀⠀⠸⠷⠞⢳⡄⠀⢀⠛⡞⢠⡞⠃⠀⠀⠀⠀⠱⠞⠒⠋⣶
⠀⠀⠀⠀⠀⠀⠀⣱⡇⠀⠀⠀⢀⠶⣸⠟⠁⢀⠀⠀⠀⠄⡀⢀⡰⢇⡇⠀⠀⣿⣸⣏⡸⣇⡂⠀⠀⠀⢀⠀⢀⠾⣠
⠀⠀⠀⠀⠀⠀⣬⢧⣼⡇⠀⠀⢚⣼⣣⣜⣣⣿⠃⠀⠈⠤⣻⣼⡧⠎⠀⠀⠀⠀⠀⠘⣣⡄⠀⠀⠀⠐⠜⣣⣟⠴⠃




⠀⠀⠀⠀⠀⠀⠀⣠⢦⣤⠀⠀⢠⠒⣤⢲⡔⢢⡄⠀⠀⠀⣤⠒⣴⢢⡀⠀⠀⠀⣠⠒⣤⢢⡀⠀⠀⠀⠀⡄⢠⡄⣀
⠀⠀⠀⠀⠀⠀⣶⠹⠊⠉⠀⠀⠈⠉⠉⢉⢱⡾⠁⠀⠀⢰⣉⠋⢉⡎⣱⠀⠀⠀⡉⡞⠉⠊⢱⡄⠀⠀⢘⣱⠏⠙⣏⡆
⠀⠀⠀⠀⠀⡾⣁⣿⣰⢆⠀⠀⠀⠀⠀⣼⠿⠀⠀⠀⠀⢀⠿⣤⢿⡼⠃⠀⠀⠀⠿⣤⢀⡼⢧⡄⠀⠀⣼⠻⠀⠀⠛⣤
⠀⠀⠀⠀⠈⣧⠏⠀⢠⡜⠃⠀⠀⠀⢰⢹⠇⠀⠀⠀⠀⠈⣶⠉⠀⢱⣞⠀⠀⠀⠀⠛⣶⢱⡾⠀⠀⠀⢱⡖⠀⠀⣶⡙
⠀⠀⠀⠀⠀⠉⡿⣀⢾⡹⠁⠀⠀⠀⣀⢿⠀⠀⠀⠀⠀⠘⢀⠶⣰⢇⡜⠀⠀⠀⣀⠾⣸⠟⠁⠀⠀⠀⠀⡸⢇⡼⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁

⠀⠀⣀⠀⢀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⣩⠇⠘⡆⠀⡴⢋⠀⠸⡥⣷⠀⣾⠉⣿⠀⠈⣹⠁⢀⢣⠀⠐⠛⡿⠂⠀⠘⢠⠀⠀⢼⡋⠉⣷⡄
⠀⠀⠳⠒⠀⠀⢸⠟⠉⢻⠀⠀⠋⠀⢷⣀⠟⠀⠀⠃⢀⡄⠀⠀⠐⡻⠉⠀⠀⡈⣁⡀⠀⢨⣿⠿⣿⣄
⠀⠀⠈⠀⢀⠀⠈⠓⠒⠃⢠⠆⠠⡧⠀⢀⣤⣤⣤⡄⠘⠀⠀⣄⠀⠁⠀⠀⣚⣯⣿⠅⠀⢿⣅⣀⣸⣿⠇
⠀⢰⠁⣰⠏⣿⠀⠀⣤⠀⠁⠀⠀⠀⠸⠀⢀⣾⡋⠀⠀⠤⢾⣷⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⢉⡀
⠀⠈⠀⣿⢠⡏⠀⢸⣿⠀⠀⢰⣷⡆⠐⢠⡀⠀⠉⣳⡄⠀⠈⠛⠀⣀⠀⠀⠒⡆⠠⢻⠀⢰⠂⠀⣠⠊
⠀⠀⠀⠙⠚⢁⠀⢸⣿⣀⣀⣾⣿⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠐⠳⡇⢠⣞⡁⠀⠘⠀⠀⠠⠤⢤⡤
⠀⠠⠀⠀⠀⠸⠀⠘⠛⠋⠉⣿⣿⠀⠀⠭⠅⠀⠀⠰⠶⢶⡶⠀⠀⠈⠁⠀⡭⠀⣀⣴⠾⠁⠀⠰⠿⣄
⠀⠀⣴⣋⣀⠘⠀⢀⡀⢀⠀⢿⡏⠀⠀⣠⣄⡀⠀⠀⢀⡞⠀⠀⢤⣆⡄⠀⠀⣰⣟⣁⣀⠀⠶⣤⣤⠿⠁
⠀⠐⣯⡠⠟⠀⠀⠈⠓⡛⠀⠘⠃⠀⣾⠃⢈⡇⠀⠀⠞⠀⠀⠈⠿⡿⠏⠀⠰⣿⠋⠁⢹⣧⠀⣠⡤⠤
⠀⢀⢀⠀⡰⣶⡴⠀⢈⣠⣄⠀⠀⠀⣯⣀⣼⠃⠀⠀⠀⢀⣠⡄⢀⢴⠀⠀⠘⡿⢤⡤⠿⠋⠀⠚⢦
⠀⢈⣁⣀⠈⠀⠀⣰⡟⠉⢹⣧⠀⠀⠙⠛⠁⠠⠀⣠⣶⣋⣸⡇⠀⠘⠀⠓⠀⠈⠉⠀⣀⣀⡀⠛⠁⣠
⠀⠠⠤⠤⠀⠀⠀⠙⠷⠾⢿⠇⠀⣠⣤⠖⢦⠀⠀⠈⠉⠉⢹⡇⠀⢀⣴⠶⢶⣤⠠⣾⣏⣠⣿⠀⣈⢿
⠀⠀⠀⣴⠶⠖⠀⠀⠀⣠⠟⠀⠐⣿⣁⣠⡾⠇⠀⠀⣠⡄⠘⠇⠀⠰⠂⠀⣺⡟⠀⠙⠛⢛⣿⢀⡟⣀⡄
⠀⠀⠀⠿⠒⢦⠀⠠⠞⠋⠀⠀⢀⣼⡿⠛⢧⡆⠀⠚⢹⡇⠀⢀⠀⠀⢀⣾⠟⠁⠀⠐⠶⠿⠁⢠⡾⠋⠁
⠀⠀⣤⣄⣀⡾⠁⢠⣶⠶⣤⡀⠘⢿⣤⣴⠟⠁⠀⠀⠸⡇⠈⠛⠉⠠⣿⣷⣶⡶⠦⠄⠀⠀⠀⣾⠟⢳
⠀⠀⢀⣀⣀⠀⠀⠻⠁⣴⣿⠃⠀⡀⠀⠀⣠⠞⠛⢢⡀⡤⣒⡂⡀⠀⣤⡄⠀⢀⡄⠀⣀⣀⠀⠘⠳⠋
⠀⠀⠠⣴⣧⠀⠀⢀⣾⡟⠁⠀⠸⣫⠆⠀⢙⣦⣴⣟⠸⡕⠧⠷⠃⣀⠉⠁⣀⠈⠀⢘⣧⣾⠀⠀⣠⠶⡄
⠀⠀⠀⠟⠀⠀⠠⣿⠿⠶⠶⠶⠀⠀⠀⠀⣏⣁⣠⡿⠀⠉⠀⠀⢰⢳⠀⠒⠟⠫⠀⢾⣁⣠⠃⠀⣇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⠿⠿⠇⠀⠀⠀⠻⢿⣿⠿⢿⣿⠿⠂⠀⠀⠀⠿⣿⡿⠿⣿⡿⢿⣿⡿⠇⠀⠀⠀⠻⢿⣿⠿⢿⣿⠿⠿⠿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠰⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⡄⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢐⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣧⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⢀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣄⣿⠇
⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣄⣀⡀⠀⠀⠀⣀⣸⣿⣀⣼⣿⣀⠀⠀⠀⠀⣀⣿⣧⣀⣿⣇⣨⣿⣄⡀⠀⠀⠀⣀⣸⣿⣀⣀⣀⣽⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠿⣿⣿⣿⣿⠆⠀⠀⠀⠾⣿⣿⣿⠿⢿⣿⣿⡿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣾⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⣿⡏⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢠⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣿⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣿⠇⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⣿⣇⢀⣿⡇
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⣿⡇⠈⣿⡅⢸⣿⠁⠀⠀⠀⠀⠀⢹⣿⠀⠹⣿⡅⣸⣿⠃⠀⠀⠀⠀⠀⠀⠹⣿⡀⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⣸⡿⠀⠀⣿⡇⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢀⣿⠇⠀⠀⣿⡇⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⣿⡇⢀⣿⡃⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⠋⢹⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢻⣧
⠀⠀⠀⠀⠀⠰⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣟⠉⣿⡟⠉⢹⣿⠉⠁⠀⠀⠉⠹⣿⡏⢙⣿⠏⠉⣿⡟⠉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣿⣾⡏⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠙⣿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⡟⠹⣷⡀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣶⣿⣿⣶⣾⣿⣶⡄⠀⠀⣶⣾⣿⣷⣾⣿⣷⣶⣿⣷⣶⣿⣷⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠖⠒⠚⠉⠉⠉⠉⠁⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⢧
⠀⠀⠀⣠⢴⠊⠉⠀⠀⠀⠀⠀⠀⠀⡴⢲⠀⠀⡴⡿⠛⣍⢳⠀⠛⠉⠑⡜⡆⠀⠀⠘⡆
⠀⠀⣰⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⢞⣴⡾⡄⠀⠓⠃⠀⣸⡞⠀⠰⣦⡔⢡⠇⠀⠀⠀⢱
⠀⢀⡇⡜⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⣇⠀⢀⡤⠔⠵⢧⡄⢠⠤⠜⢨⠇⠀⠀⠀⠀⢇
⠀⢸⣰⠀⡈⠈⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠓⠛⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⢏⠈⢠⠣⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⢸⡀⠇⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠘⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢳⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠈⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣵⡆⡏⡇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⡀⠀⠀⡖⢇⢀⣀⠀⠀⠀⠀⠀⠓⠽⢻⢳⠀⣗⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢇⠀⣤⠛⡈⣥⠼⠀⡞⠒⠢⠤⡄⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⡄⠘⠋⢣⣸⠄⠀⠉⠓⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠒⠒⠊⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠱⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠉⠀⠀⣀⣀⠠⠤⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡧⠒⠚⠉⠉⠉⠀⣀⣀⣀⡀⠤⠤⠒⠒⠒⣈⣉⠁⡉⠤⠤⢄⣒⣒⡶⠭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡜⠀⠒⠀⠈⠩⠭⠤⠤⠤⠄⠒⣒⣀⣨⠭⠤⠔⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠧⠤⠤⠤⠶⠖⠒⠒⠊⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⢀⣀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡾⢋⡗⢀⣾⠏⠀⠞⣩⠇⠀⠒⣿⣃⠀⣰⠏⣼⠃⣸⣋⠉⠁⣴⣋⠀⠈⢙⡿⠋⠰⣯⡿⠂⣼⢋⣽⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣁⡾⠃⢀⡟⠀⣤⡶⢏⣴⠃⢘⣋⣸⠂⠉⣹⠏⢀⣉⣸⠟⠸⣟⣹⡇⠈⡿⠉⠀⢾⣋⣷⠀⠉⣹⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠂⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⡄⠀⠀⠀⠀⠀⠀⠀⡠⠱⠃⠀⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠴⠖⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢀⠔⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡜⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⡀⠀⠀⠀⠀⣰⠀⠀⣇⣇
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠈⠀⠀⠀⠠⠀⡤⠄⠀⠀⠀⢰⢰⠀⠀⠀⠀⠀⠀⠀⡀⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⠁⠠⣤⠀⠀⠀⠀⠈⠘⠀⠀⢀⢌⠆⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠲⠗⠀⠐⠀⠀⠀⠀⠀⠀⠠⠿⠄⠀⠀⠠⠤⠠⠤⠤⠭⠈⠀⠀⠀⠀⠀⠠⠄⠤⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠄⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠠⢄⣀
⠀⠀⠀⠀⢠⢤⡄⠀⠐⠊⠀⠀⠀⠀⠀⠀⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠂⠀⠀⠀⢠⠆⠀⠀⢑⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠩⡠⠀⠀⠀⠀⠀⠠⠮⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠊⠀⠀⠀⠀⠛⠂⠀⢀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢀⣀⡀⠀⠈⠠⠢⠀⠀⠀⠀⠀⠀⡌⡄⠀⠀⠀⠀⠀⣀⡀⠀⠈⠻⡢⡀
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠃⠅⠀⠀⠀⠀⡘⠐⠀⠀⠀⠀⠀⠸⠘⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠈⠁⠀⢀⢔⠜⠀⠀⠀⠀⠈⠒⠒⠀⠀⢠⢂⠆⠀⠀⠀⣰⡰⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⢀⠂⠃
⠀⠀⠀⠀⠒⠒⠚⠓⠂⠀⠀⠀⠀⠀⠂⠀⠠⠴⠖⠓⠁⠀⠐⠒⠐⠓⠃⠀⠀⠀⠀⠀⠀⠐⠒⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠈⠆⠄⠀⠀⠀⠀⠠⠖⠃⠀⠀⠀⠀⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢡⣀⠀⠀⢀⢁⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⢀⠀⢤⠀⠆⠀⠐⠇⠓⠙⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠺⠇⠈⠧⠘⠀⠘⠂⠁⠀⠛⠈
⠀⠀⠀⠀⠀⠃⠈⠀⠍⠁⠀⠀⠁⠉⠈⠰⠋⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠈⣠⠂⢀⠀⠀⠀⠰⠁⠶
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢑⣄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠁⠀⠀⡀⠡⡑⠬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢡⣀⠰⢸⣷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⢠⠈⠂⢄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠄⠀⠀⢡⠀⠌⢰⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⢸⣿⢸⢸⣿⠠⠀⠀⠀⣶
⠀⢀⠀⠀⢠⠀⠙⠓⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠦⠒⢁⡀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⢸⣿⢸⢸⣿⠀⣿⡏⠠⣿
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⢠⡤⠠⠀⠑⠄⡰⠃⠠⡀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠀⢠⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⡉⠉⢉⠉⠉⣭⠉⠉⡏⠉⠛⠀⠃
⠀⢀⡔⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⣠⠊⢀⡴⢂⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠼⠎⠁⠔⠀⡰⢆⡰⠆⠀⠀⠀⠀⠡⢄
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠰⠋⢒⣈⠀⠨⠆⠀⠀⠄⠒⢀⣉⣠⡤⠴⠴⠂⠀⠈⠁⠀⠀⠀⠀⡥⡀⠨⡤⠈⠚⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠾⡰⠁⠀⠠⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠘⣆⡄⠀⢸⠖⠀⠀⠁⠀⠀⢠⠂⡀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡀⠀⠐⡄⠀⠀⠁⠁⠀⠀⠀⠉⠐⣦⣠⣄⡀
⠀⠀⢰⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢢⠤⢤⡀⠀⠀⠘⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡟
⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⢀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡆⠀⠢⠙⣻⠷
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠛⠲⢞⡷⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡸⠂⠓⠐⣠⣤⣼⣤
⠀⠀⠀⠀⠀⠀⡀⠢⣀⠀⠐⠒⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠐⢮⢁⡥⠀⠀⠀⠀⠀⠀⠘⢅⣀⣩⠼⠁⠛⠛⠛
⠀⠀⠀⡌⠀⠤⠴⠀⠉⠀⠀⢸⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠁⡀⠀⠉⠶⠀⠀⠀⠀⠀⠀⢠⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢣⠀⠀⠀⢀⠠⣰⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠘⠣⠀⠀⠧⠐⢾⡄⠦⠡⢹⠠⠄⢟⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆⠈⠀⡇⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡅⠒⣨⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠇⡄⠀⠀⠁⠀⣠⡨⢦⡀⠀⢀⡔⠔⠦⡀⠀⠀⠀⡠⠊⡴⠋⠡⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠒⠢⡀⠀⠀⠀⠀⢀⠛⠃⠀⠀⠄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣆⣇⢀⣿⠀⠀⣿⠀⠀⠑⠀⣮⣤⣤⣴⡟⠀⢀⠈⡠⠊⠀⠰⠀⡆⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠂⠀⡀⠴⠆⠀⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠔⢡⠞⠀⠀⠀⠀⡆⠇⠀⠀⠀⠀⠀⠈⠢⡀⠀⢠⠚⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⢃⢸⠀⠀⠀⠀⠀⠀⢰⣯⣭⣽⠃⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⡀⠀⢀⣀⡀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠛⡼⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠃⠈⠁⠈⠛⠂⠀⠀⠘⠂⠈⠀⠁⠀⠉⣠⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠉⠀⡤⠤⠃⠹⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢢⣱⡀⠀⠀⠀⠐⠈⠀⠀⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⣠⠄⠞⠉⠀⠀⠀⠂⠀⠐⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠈⠱⠀⠀⠀⠀⠄⠒⡀⠀⠀⠀⠀⠀⠀⡂⠀⡀⠀⠀⢀⡔⠊⠀⠃
⠀⠀⠀⠀⠰⠄⠀⠀⠤⠄⠀⠄⠀⠀⠀⠀⠐⠏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠁⠀⠀⠀⣀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠁⢀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠤⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⢸⡷⠀⢾⠆⠀⠀⠀⠀⠐⣿⠂⠰⡷⠀⠰⡷⠀⠀⠀⠀⠀⢾⡇⠀⢾⡆⠀⢐⡿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⢧⠀⢸⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⣯⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⣄⡏
⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⣸⣇⠀⣸⣆⠀⠀⠀⠀⢀⣿⡀⢀⣧⠀⣠⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⢀⣄⠀⢀⣄⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⠀⣠⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⣼⠁⠀⠀⠀⠀⢹⡄⠀⢸⠋⠀⢸⡇⠀⠀⠀⠀⠈⢿⠀⠀⡿⠁⠀⢹⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠇⠀⠀⠀⠀⠀⠀⢧⠀⡏⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⡆⢰⠃⠀⠀⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⠀⠀⠀⠸⣾⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠿⠂⠐⠿⠂

⠀⠀⠀⠀⠀⢶⠂⠀⢰⡧⠀⢰⡆⠀⢾⠆⠀⣾⠆⠀⠀⠀⠀⠐⣷⠀⠰⢷⠀⠀⢰⡿⠀⠀⠀⠀⢿⡂⠀⢀⡾⠆
⠀⠀⠀⠀⠀⠘⡇⠀⡾⠀⠀⢸⠇⠀⢸⠀⠀⢹⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢧⣠⠏⠀⠀⠀⠀⠀⠀⠻⣄⡞⠁
⠀⠀⠀⠀⠀⠀⢻⣰⠇⠀⠀⢸⡀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⢷⡀⠀⠀⠀⠀⠀⠀⢠⠿⣄
⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⣸⣇⠀⣸⡀⠀⣿⡀⠀⠀⠀⠀⢀⣿⡀⢀⣴⠃⠀⢻⣄⠀⠀⠀⠀⣠⠏⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⢠⣄⠀⠀⢀⣄⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢠⠟⠁⠈⢿⠁⠀⠀⠀⠈⠻⡄⠀⡼⠋⠀⢸⡏⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢙⡞⠁⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢳⡀⠀⠀⣸⠀⠀⠀⠀⠀⢀⡞⠙⣆⠀⠀⢠⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⢿⠆⠀⠿⠆⠀⠀⠀⠰⠿⠀⠀⠸⠷⠀⠺⠗⠀⠺⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠈⡀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠅⠀⠀⣠⠂⠀⠀⠠⠎⠀⠈⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⡠⠀⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠄⠀⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢄⡤⠀⡀⠈⠀⠀⠀⠀⠐⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠀⠀⡀
⠠⡀⠀⠀⠀⠐⠈⠁⠀⠄⢠⠀⠀⠀⡄⢰⣉⣃⣘⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⡄⣶⣹⣁⣛
⠀⠀⠀⢀⠤⢠⡄⠀⠀⠀⠈⠃⠀⠀⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡃⠀⢠⠤⢠⠄⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠉⠀⠀⠌⠀⠼⠀⢠⡄⠀⠂⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠰⠃⠉⠁⠀⠀⠌⠠⠼⠀⢠⠄⠐⠂⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⡀⠀⠀⠀⠀⠄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⣀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢢⡀⠀⢠⠄
⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠂⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⡀⠄⠀⠀⠀⠀⢀⡐⠒⠢⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⢂⠐⠀⠂⠀⠀⠀⠀⠀⢀⠐⠒⠂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠄⢂
⠀⠠⠠⠞⠀⠀⠀⠈⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠉⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠈
⡀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟
⠀⠀⠀⠀⠉⠁⠐⠂⠀⢀⡀⠀⠀⠀⡀⢠⢰⡄⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠔⠀⠀⢀⡀⠀⠀⠀⠀⣠⢰⠄⡦
⠈⢢⠀⠀⡀⠀⡀⠀⠀⠀⠘⠆⠀⠀⠃⠜⠃⠒⠒⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠚⠄⠀⠀⠃⠟⠐⠒⠂
⠒⠊⠀⠠⠉⢩⠁⠠⠔⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⢀⡂⠒⠒⠀⠀⠠⠉⢨⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⢂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠀⠰⠐⠈⣄⠃⠀⠘⠁⠀⡀⢀⠀⡄⠀⠀⠀⠀⠁⢁⠀⠀⠈⠁⠀⠀⠔⠀⠀⡌⠁⠀⠈⠀⠄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠶⣶⠲⠀⠀⠀⠰⢶⡖⠆⠖⣶⠶⠀⠀⠀⠲⣶⠶⠀⠶⡶⠖⠰⢶⡶⠂⠀⠀⠰⢶⡖⠂⠖⢶⠶⠐⢶⡖⠆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⣾
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣰⡇
⠀⠀⠀⣀⣿⣀⠀⠀⠀⢀⣸⣇⡀⣀⣿⣀⠀⠀⠀⣀⣿⣀⠀⣀⣟⣀⢀⣸⣇⡀⠀⠀⢀⣸⣇⡀⠀⠀⣸⣿⣀
⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠀⠈⠁⠁⠀⠈⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠈⠹⡏⠁⢉⡟⠉⠀⠀⠀⠉⢹⡏⠀⢹⡟⠉⠈⢹⡏⠁⠀⠀⠉⢹⡏⠁⢹⡟⠉⠈⢹⡏⠁⠉⣿⠉
⠀⠀⠀⠀⠀⠀⢿⠀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⣼⠁⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣧⡏⠀⠀⠀⠀⠀⠀⠀⢹⣶⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠤⠿⠥⠀⠀⠀⠀⠀⠀⠀⠤⠿⠤⠀⠀⠠⠼⠧⠄⠀⠀⠀⠀⠤⠿⠥⠀⠀⠠⠼⠧⠄⠤⠿⠤

⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⣀⣀⣀⢠⣀⣀⡀⣀⣀⣠⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⢀⣀⣀⡄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠇⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⡄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⡿⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠀⠀⠐⠚⠓⠂⠒⠛⠒⠐⠚⠛⠒⠀⠀⠀⠀⠀⠒⠛⠒⠐⠚⠛⠂⠐⠛⠓⠂

⠀⠀⠀⠰⠲⣶⠆⠲⣶⠲⠀⠀⠀⠀⠰⠲⣶⠆⠲⣶⠶⠀⠶⣶⠲⠀⠀⠰⠲⣶⠆⠰⣶⠶⠀⠶⣶⠲⠰⢲⡶⠆
⠀⠀⠀⠀⠀⠹⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠹⣆⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦⣼⠋⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⢀⣀⣾⡁⣘⣧⣀⠀⠀⠀⠀⢀⣀⣾⡁⣘⣧⣀⠀⣀⣿⣀⠀⠀⠀⢀⣾⡃⢘⣧⣀⠀⣀⣿⢀⢀⣸⡇⡀
⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠈⠈⠉⠀⠁⠉⠉⠈⠉⠉⠁
⣿⣿⣿⣿⣛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣻⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⡀⠀⠀⠀⢠⠈⠀⠂
⠀⠀⠀⠀⠀⠀⢠⣼⣦⣲⣦⠀⠘⠆⠀⣤⠼⢿⣷⣦⣄⡀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⢈⠝⠉⣉⣀⡄⠀⠀⢀⠠⠐⠛⠉⠁⣀⣀⡠⢈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠐⢀
⠀⠀⠀⢀⠁⠀⠄⢾⠆⠀⠘⡷⠞⣡⠀⠁⠀⠀⠊⡭⠽⢷⠷⠞⠂⢀⡄⠀⢤⣴⣶⠿⠷⠀⡀⠀⠀⢰⢣⣀⣠⡛⣿⣷⠀⠀⠣
⠀⠀⠰⣚⣧⡀⠀⠀⠀⠀⠀⢠⣺⣷⣤⡔⢁⠔⠁⠀⠀⠀⠀⠀⠀⣁⠼⡄⠈⠁⣀⣀⣀⠀⠀⠠⡀⠘⠿⠿⠿⠟⠠⠟⠀⠀⡄
⠀⠀⠀⠐⠺⠿⣯⣿⣬⠥⠖⠀⠉⠀⠈⠙⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠓⡻⣴⣿⣾⣿⠿⠇⠀⢀⠣⠀⢀⠄⠂⠀⢀⣀⣴⡺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠿⣿⠆⠀⠒⠀⢈⡽⢶⣶⠀⠀⠀⠀⠀⠐⢋⡃⠀⠀⢀⣀⢴⣫⢃⡴⠁⠀⠐⠚⠿⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢳⣤⣀⡈⠀⣠⣖⣻⣵⡄⠘⡆⢬⡏⠀⠀⠀⠀⠀⠀⠈⠝⠿⠿⠟⠞⠋⠁⡮⣇⣀⣀⣀⣀⣀⣀⠀⠀⡈
⠀⠀⢠⠄⠀⠀⠀⠃⠑⠛⢻⠀⠀⢀⠋⠉⢡⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⢺⡿⠿⠿⢷⠶⠼⠿⠿⢿⣿⡷
⠀⢼⠁⠀⣴⣒⣷⡄⠀⠨⢍⣦⣀⠀⠁⠀⢀⡤⡀⢢⡏⠀⠀⠀⡀⠠⣷⡀⠈⢿⡿⠁⠀⣀⡀⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⡀
⠘⢸⠀⠀⠈⠋⠁⠀⣀⣀⠀⠹⡿⢭⣿⣿⠥⠞⠁⢺⣿⣿⣾⣤⠀⠈⣔⣣⡀⠀⠀⠀⠀⣈⢿⣷⣄⠈⠀⢤⣶⢿⡷⣦⠀⠈
⠈⢋⢧⠀⠀⠸⠀⢘⠟⢿⠇⠀⠑⠀⠀⠀⢀⠠⠀⠂⠀⠁⠀⠀⠂⠀⠭⣄⣻⣿⣲⡀⠄⠉⠉⠉⡇⠀⢈⣈⡉⠀⠠⣝⣖⣦
⠀⠘⢒⢷⣦⣀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠐⠁⠀⣤⢤⣶⣶⣶⣤⠤⢤⡀⠈⢯⠉⠛⠁⠀⠀⠀⠐⡆⠀⠻⠿⠿⠆⠀⡙⡉⠉
⠀⠀⠀⠈⠙⢻⣿⣽⣾⣍⣥⠗⠂⠀⣀⡸⣄⠀⠈⠛⠛⠛⠙⢉⢻⠿⠃⠀⠈⠀⠀⠀⠀⠀⠀⢁⢧⡀⠈⠀⠀⢀⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⢴⡀⠀⠀⠀⠀⠀⠀⠘⠍⣻⣷⣶⣩⣵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣽⣯⣿⣽⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⡞⠡⡼⣹⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣽⢿⡍⣹⣿⣷⣸⣆
⠀⠀⠀⠀⠀⡠⢚⠙⢿⣿⡇⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣇⠉⠈⢿⣪⠟⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠃⣚⡠⣼⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡲⢍⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣔⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠠⠤⠞⢡⣤⠀⠀⠀⠀⠀⠀⢰⡶⠒⡲⣒⣒⡲⡒⠲⣶⠀⠀⠀⠀⠀⢀⣾⠃⢸⣿⡦⠄⠛⠉
⠀⠀⠀⠀⠀⢠⣅⠐⣿⡇⣶⡆⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠶⡹⣯⢜⡷⡱⠆⢸⠀⠀⠀⠀⠀⠘⠛⠃⠉⠈⡆
⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠸⠷⠤⣱⡯⠭⠷⠥⠴⠿⠀⠀⠀⠀⠀⠀⠀⡔⢪⠖⣩⡒⣴⢢
⠀⠀⠀⠀⠀⠈⠩⠤⠉⠡⠉⠡⠌⠩⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡺⣄⡞⣠⣟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠽⣋⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣶⣄⠁⣠⠟⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠦⡝⢻⣿⣿⠃⠀⠀⠀⠀⣠⡞⠁⡘⠷⡧⣯⠞⠁⠻⣧⡄⠀⠀⠀⠀⣿⣿⣿⣿⡀⣫⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣿⣿⠃⠀⠀⠀⠀⠀⠈⠳⣄⣁⣠⠞⡅⣿⣦⣴⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠤⠤
⠀⠈⡆⠀⠀⠀⠐⠄⠀⢸⠤⠭⠀⠀⠀⠠⠀⠠⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣙⢦⡀⠐⠀⠰
⠀⠀⠄⡄⣤⢠⠄⠀⠀⠀⠀⢨⠀⠀⢂⠠⠉⠁⠂⠀⠀⠀⠥⠭⠀⠀⠀⠀⠀⠀⠀⠀⡜⢨⠇⠙⢦⡀⠙
⠀⠀⣭⣯⣿⣹⠋⡄⠀⠠⡤⢤⠀⠀⠀⠀⣷⡿⠀⠀⢠⡀⠈⠄⢀⡄⠀⠌⠀⣠⠀⣜⣱⠋⠀⠀⠀⠙⢦⡀
⠀⠨⠐⠓⠛⠊⠒⠡⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢌⠂⠁⠀⠀⠈⠁⣀⠀⠀⠁⠀⠛⢉⠔⢉⠈⢈⠓⢄⠘
⠀⣤⣒⣒⣒⣒⣲⠄⠀⡠⠂⠙⣿⣹⣦⡀⠀⠀⠀⢮⠴⡀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠌⠠⠁⠀⠀⠡⠈⣆
⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠿⠾⠷⠇⠀⠀⠜⠒⠳⠃⠀⠀⡐⠀⠀⠈⠄⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠔⠀⠀⠰⠊⠀⠀⠐⠆⠐⠷⣶⣶⣶⠾⠀⠀⠀⢀⡄⠠⢠⠄⠀⡀
⠀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠈⢀⠀⠁⡀⠀⢀⠀⡀⢠⠀⣀⠀⠀⠀⠀⠠⣤⡤⠄⠤⠤⡀⠀⠀⠀⠀⣀⡀
⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣄⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡐⠈⠉⣿⣽⣳⡀
⠀⢼⠉⢍⡩⣉⢍⢉⡆⠀⠀⠀⢀⢐⠀⠀⣠⠂⡡⠀⠄⠁⢆⠀⠀⠀⢲⣵⣤⣆⣶⡄⠀⢰⠀⠀⠀⠟⠛⠺⠃
⠀⢸⢈⢈⠡⡉⣈⠈⡇⠀⠀⡇⢰⢐⠀⡠⠆⠨⠤⠤⠤⠅⠈⠤⠀⠀⣱⢺⠲⡗⣯⠀⠀⠀⠢⡀⠀⠀⢀⠌
⠀⡸⠑⠜⠣⠛⠜⠘⡇⠀⠀⡀⠀⣲⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁⠈
⠀⠠⢍⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠞⠭⠭⠭⠝⠆⠀⠈⠉⠁
⠀⠀⠀⠳⣄⠀⠀⠀⠀⡔⣈⡐⡀⠀⠈⠀⠈⠁⠀⠁⠀⠉⠀⡀⠀⠀⡉⢎
⠀⠀⠀⡦⡈⠳⣄⠀⠠⢡⠀⡐⠁⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢠⣹⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄
⠀⠀⠀⠃⠈⠆⠈⠳⢄⠈⠿⠃⠀⠀⢀⡀⠉⠁⢀⡀⠀⠉⠁⡀⢀⡁⠁⠀⢄⡄⠀⠃⠄⠀⠠⠀⠃⠀⡈⠛⠁
⠀⠀⠉⡉⢭⣥⣍⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⡀⠤⠐⠀⠀⠀⡌⡓⡀
⠀⠀⠊⠀⢸⣿⣿⡧⠀⠀⠚⠀⠁⠛⠀⠐⠀⠀⠀⢀⠠⠤⢤⢤⢤⢤⠠⠠⡄⠀⠀⡀⠙⢀⠀⠀⡐⠀⡇⠐⡀
⠀⠀⢄⠀⠀⠀⢀⠁⠀⠀⢴⣙⠀⠀⠀⠈⢸⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⢶⡆⠀⠰⣶⣶⣳⣶⠶
⠀⠀⠀⠁⠒⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⠏⠀⠈⢿⣆⠀⠀⠀⠀⠤⢴⣿⠀⠀⠀⠀⠀⢀⣎⠀⠀⠙⣯⡃⠀⠀⠀⢸⣧⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⢰⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠻⠗⠀⢠⡿⠃⠀⠀⠀⠀⣉⣀⣰⠿⠃⠀⠀⠀⠀⢀⡼⢡⣿⠂⡀
⠀⠀⠀⢸⣿⡄⠀⠀⣼⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠰⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⣾⠀⠀⠀⢠⣞⣀⣸⢿⣒⡀
⠀⠀⠀⠀⢻⣇⢠⣀⠿⠏⠀⠀⠀⠀⠀⠸⢻⠀⠀⠀⠀⠀⠠⢁⡝⣀⣀⣠⠃⠀⠀⠀⢾⠿⠀⢄⣉⠷⠒⠀⠀⠀⠀⠀⠨⣦⠁
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠛⠁⠀⠀⠀⠀⠁⠐⠋⠉⠉⠀⠀⠀⠀⠈⠑⠒⠛⠉⠀⠀⠀⠀⠀⠀⠘⠋⠉⠛⠃


⠀⠀⠀⠀⢠⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣀⣀⣈⣁⡆⡄⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢠
⠀⠀⠀⠀⣼⠛⠛⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⢼⡇⠀⠀⠀⠀⡟⠉⠉⠉⢉⠋⠀⠀⠀⣼⠋⠀⠀⢸⣷⠀⠀⠀⠀⣾⠃⠀⠘⡷⢆⡀
⠀⠀⠀⠀⢨⡔⠒⠲⣦⡀⠀⠀⠀⢸⣿⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠹⣷⣤⣄⣸⠋⠀⠀⠀⠀⣿⡀⠀⠀⣘⡃
⠀⠀⠀⠀⢀⡀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠁⠀⢸⣷⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠠⠆⣴⠋⠉⠙⠻⣷⠀⠀⠀⠀⠈⠓⠒⢫⣿⡇
⠀⠀⠀⠰⣿⠛⠀⢀⣿⠃⠀⠀⠀⠸⠟⣠⠀⣸⡟⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⣋⣦⠀⠀⢀⡿⠀⠀⠀⠀⣾⠇⠀⣸⠟
⠀⠀⠀⠀⠈⠙⠚⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠀⠀⠀⠀⠀⠈⠑⠊⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⠔⢸⡆⠀⠁⠀⠀⢀⠐⠈⠢⣀⡴⠚⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⢸⢀⣀⣀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⢄⠶⠋⠀⡠⠃⠀⠀⠘⢷⣄⠀⠀⠁⠀⣠⠌
⠀⠀⠀⠀⠀⣴⠐⠊⠉⡀⠀⠀⠀⠀⠐⠂⡇⠀⠀⠀⠻⠄⠀⠈⠤⠀⠐⠒⠀⠁⠀⠀⠀⠀⣾⠆⠀⡴⠉⠀⠀⠀⠀⢀⡴⠋⠠⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠁⠀⠀⢀⡠⠤⠤⠇⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⣬⢄⢎⠀⠀⠀⠀⠀⠀⠻⢦⡀⠠⠚⢷⣄⠀⡩
⠀⠀⠀⠀⠀⠙⠒⠊⢹⣿⠠⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠙⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄
⠀⠀⠀⠀⠀⢰⠟⡖⣄⢀⣴⡟⠡⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⠉⢡⣠⡤⡄⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠘⠦⠴⢋⣾⠏⠠⠀⡠⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠶⠶⠶⠦⡀⠀⢺⡃⠉⠉⠀⠈⠁⡠⠆⠀⠀⠀⠀⢹⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡰⡳⠃⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⢈⣿⠆⢀⣀⠘⢄⠀⠀⠀⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⢀⢜⡝⠁⠐⣠⠊⠀⡞⠩⠲⡀⠀⠀⠀⠈⠓⠐⠒⠒⠀⠀⠐⠀⠈⠉⠁⠀⠀⠀⠑⠴⠋⠻⠷⠊⠀⠀⠀⠀⠀⠀⠛⣒⡒
⠀⠀⠀⠀⠀⠘⢏⡀⠀⡔⠁⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣩⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣤⠤⠄⣀⠀⠀⠀⠀⠀⠰⠈⠀⠀⢠⣶⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠫⠙⡄⠀⠀⠰⣿⠉⠐⠢
⠀⠀⡐⡹⠁⠀⢤⡀⢱⠀⠀⠀⢸⣷⠀⡅⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡺⠟⠁⣀⠴⠋⠀⠀⠀⠀⠈⠁⠒⢴⣀⡀⠉⣠⠂
⠀⠀⣥⠃⠰⢡⣿⠇⢀⠇⠀⠠⡾⡋⠰⠀⠀⠀⠀⣴⡄⠒⡄⠀⠀⠀⠰⢿⡋⠀⡀⠰⢥⣀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⢁⠖⠁
⠀⠀⢿⡀⠘⠿⠋⠀⡼⠀⠀⠀⢰⣴⠈⡇⠀⠀⠀⠋⣁⠸⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠉⠓⡦⠀⠰⡿⠋⠀⣁⠜⠁
⠀⠀⠀⠙⠢⠤⠖⠋⠀⠀⠀⠀⠸⣿⠈⠃⠀⠀⣾⠗⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠁⠀⠀⠙⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⣤⡶⠦⠀⠀⢠⠆⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⢶⣦⣤⠤⡤⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄⠀⠀⣻⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠀⠀⠀⣻⠀⠀⠀⡀⠀⠀⠈⠀⣠⢞⠝⠁⠀⠐⠂⠀⠀⠀⠈⠻⠰⢿⠄⠀⢀⡴⡻⠁
⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡀⠀⠀⢸⢸⠀⠐⠐⠁⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠙⢦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⣼⠃⠈⠀⣠⢘⠇⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢐⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⡇⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠑⢄⣤⣤⣠⣤⡦⠈⠀⠀⠀⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⡌⡨⠀⠀⠄⠀⠀⠀⠒⢤⡀⠀⠀⠱⣄⠀⠀⠀⠀⣀⡮⠀⠀⠀⠀⠀⢂⡀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠶⠶⠷⠇⠀⠀⠀⠀⣤⣤⠀⠀⠹⡦⠀⠀⠈⠉⠓⠐⠚⠋
⠀⠀⠀⠀⠤⠰⠂⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢰⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⣤⣤⣼⡜⠀⠄⠀⢀⣽⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣿⠁⠀⠀⠀⠲⠶⠼⠇⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠙⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣠⡴⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠁⢭⡉⠉
⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⣀⢀⣿⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⠀⣀⢀⣀⢤⣄⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠁⠀⠀⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣔⡟⠋⠀⠀⠀⠀⠰⠤⣤⠤⠤⣤⣤⣴⣿⠀⠀⠀⠀⠋⠀⠀⠀⠉⠛⠁⠀⡠⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠐⠈⢉⠁⠁⠀⢀⡀⠀⠀⠀⠀⢠⢪⡁⠉⠉⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⡀⠈⠢⠀⠀⠀⠀⠀⢿⡋⠉⠑⠀⠀⣷⠀⠀⠀⠀⠀⠀⢴⡮⠑⠦⠀⠀⣿⠄⠀⠀⠀⠀⠙⠶⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠎⠈⠁⠀⢸⡇⠀⠀⢣⡀⠀⠁⠀⠀⢀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣔⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⡄⠀⢸⡇⠀⠀⠀⠙⠳⠶⠶⠾⠗⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⠀⠉⣄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠁⠇⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢯⡎⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠉⠀⠀⠀⡠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢠⣤⠲⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠆⠋⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢠⠔⢿⣿⠀⠠⠖⣿⡇⠀⠀⠀⣠⠖⠉⠀⠀⠉⠻⣦⠀⠀⠀⢀⣾⠉⠀⠀⠉⢷⣄⠀⠀⠠⠚⠛⠛⠛⠛⠛⡻⠋⠀⠀⠀⠀⠀⠀⣠⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠿⠆⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⣿⡇⠀⠀⠈⢿⣶⣤⠴⠂⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⣤⣖⠋⠀⠀⠀⠀⠀⠀⢀⡴⠓⠒⢲⣄⠀⠀⣠⠞⠁⠀⠀⠀⣿⡇⢀
⠀⣠⡤⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⢀⣠⠄⠀⠀⠀⠈⣿⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠛⠛⠛⠛⠛⠛⣿⡟⠻
⠀⢿⣇⠀⠀⢀⡼⠋⠀⠀⠀⠀⣿⣧⠆⠀⠀⣀⠤⠚⠋⠁⠀⠀⣀⠀⠀⢿⣯⠀⠀⠀⠀⢀⣿⠇⠀⣼⡏⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠉⠙⠊⠉⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠚⠛⠛⠛⠛⠛⠻⠿⠿⠀⠀⠀⠙⠳⠦⠤⠶⠛⠁⠀⠀⠈⠛⠲⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⠷⣦⣤⠤⠀⢠⣿⢃⠤⠒⠛⠳⢶⣄⠀⠀⠀⡴⠿⠶⠶⠶⠶⠶⣶⠖⠀⢀⣴⠞⠉⠉⠓⣦⡄⠀⠀⠀⠀⣴⠞⠉⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣾⡗⠁⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢸⣏⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠋⠉⠙⣷⣆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢙⣢⠄⠀⣴⣛⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣷
⠀⢠⡴⠂⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢹⣷⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠘⢧⡀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⣠⠞⠋⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⡸⠋⠀⠀⠀⠘⢧⣀⠀⠀⡠⠊⣾⠃
⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⠤⠚⠁

⣿⣿⡟⢛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡔⣤⣐⢦⣄⡰⣥⣄⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠⠤⠄⠐⠊⠀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣣⠶⣿⡦⠾⣷⠶⣿⣤⡽⠞⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠁⠀⠆⠀⠀⠀⣁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢠⡿⠀⣼⠃⣰⡟⠀⠀⢀⣀⠀⠈⢙⣟⣂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⣼⠇⣼⠇⣰⡟⠀⠀⠀⢟⣿⡇⠀⠀⢳⣍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠆⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⢴⣿⣴⡿⣶⡿⠐⢀⣀⠓⠀⡀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣿⣵⣿⣶⣿⣴⣎⡍⠉⠙⢦⠈⠄⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠀⣤⠤⡤⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣽⢿⠿⣿⡇⠻⠿⠻⣿⣸⡇⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠄⢠⣠⣨⡜⠂⠑⠀⠀⠀⠀⠀⠀⢠⣻⣟⡯⣿⣿⣿⠏⠀⢻⣿⣦⣀⣘⣩⠟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⡄⠈⠀⠇⢀⠐⠀⡄⠀⣀⠀⠀⠀⠒⢿⡿⠿⠿⣿⠃⠀⠀⣈⠉⠃⡄⠄⠀⣸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠊⠀⠀⠀⠀⡇⢠⠈⠀⢠⡆⠀⠀⠘⢉⠍⠉⠈⠱⠴⠅⠉⠉⢈⣭⠍⣭⡍⠉⠈⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠙⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⢠⢃⠀⠠⠀⡀⠀⠀⠀⠀⢠⣎⢊⣄⠈⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠨⠀⠆⠐⢠⡄⠀⢠⢠⠀⠀⡀⠀⠀⠀⡄⣄⠀⢠⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠰⡄⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⡄⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠂⠡⡂⠀⠀⠀⢘⠀⠀⢂⠀⣰⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⡇⠀⠀⠀⢈⡀⠀⠀⠀⡆⠀⠀⠀⠤⠀⠀⠤⠀⠄⠀⠀⠀⠠⠠⠀⠤⠀⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠁⠡⠀⠀⠀⠁⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠒⠒⠂⠒⠀⠂⠒⠐⠂⠐⠒⠒⠒⠀⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⣉⢀⡉⠀⠈⡀⢈⠈⠉⢈⠀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠀⠀⠀⠀⢀⣆⣀⡀⠔⢠⣀⠀⠀⣀⠀⡀⠀⢀⡀⢀⢀⠀⣀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠔⠀⠀⠀⠀⠘⠂⠀⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠


⣾⣿⣿⣶⣶⡷⢿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡿⠿⡿⢶⣶⣿⡿⠶⠶⢶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡷⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠄
⠀⠀⠀⠀⠀⠀⠀⡄⢤⡐⣮⢱⣆⡲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⠹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠰⡠⠀⠀⠈⠱⢾⠿⠉⠯⡉⠁⠀⠸⡄⠀⠀⠀⠄⠀⠀⠋⠀⢉⡇⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⣀⠘⣷⠀⠀⠀⠠⠆⠁⠀⢀⠀⠆⠀⠀⠀⠀⣼⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠄⠟⠁
⠀⠀⠀⠀⠄⠐⠀⢿⡀⢂⠉⠀⠀⠐⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⣠⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢡⠏
⠀⠀⡸⠀⠀⢹⠀⡁⢱⠀⣿⠁⢸⣿⡆⠀⠀⠀⠀⢀⢊⣾⣷⢢⣤⡤⠔⠀⠄⠄⠔⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠈⠀⡠⠋
⠀⠀⢻⡀⠀⠘⠀⠐⠋⠀⠷⠖⠘⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢵⣢⡀⢀⠜⠁
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣠⠋
⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣠⢄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠅⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠂⡐⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠠⡀⠀⣖⠀⠹⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⣏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠁⢠⡖⠆⢱⣀⡉⣡⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣶⣦
⠀⠀⠀⠀⠀⠈⠣⣌⠁⣀⠞⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠻⠃⠀⠀⢸⣿⠁⠀⠈⠿⡶⠿⠋⠀⢀⣾⠟⠛⣦⡈⢻⣶⡟⠛⢿⣆⠃⠀⠃⠀⢼⡏⠉⢹⡧⢠⣿
⣶⣦⣤⡄⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⢀⣶⡄⠈⢿⣦⣶⡿⠁⠀⠉⣠⣤⣼⠏⠀⠀⣀⠀⠘⠻⠿⠟⠁⠈⣁⣀
⠀⢀⣽⠄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣾⢻⣷⠀⢀⡀⠀⠀⠹⠀⠀⠈⢉⣡⣤⣄⡀⢻⣇⠀⠺⠋⠛⠀⠘⠉⢙
⠀⡿⠃⠀⣶⡶⠶⢶⣶⠀⠉⠙⡃⣸⣯⣬⣿⡶⠈⢀⣴⢶⣦⡀⢦⠀⢸⡏⠉⠉⠛⠿⠿⠆⠀⠀⠀⢴⡇⠀⠻
⠰⠄⠀⠀⢈⣿⣾⠟⠋⠀⢀⣾⠏⠈⠉⠁⠹⠇⠀⢸⣷⡀⠙⢿⣦⠀⠈⠻⠿⠀⡀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤
⣀⠀⠀⠀⠙⠉⢛⠵⠂⠀⣼⡟⠀⠀⢀⡀⠀⣤⠀⠢⠙⣿⣤⣈⣿⠀⠀⠠⣴⠿⣁⠴⣶⡄⠂⠀⠀⠉⣩⡿⠋
⠄⠀⣴⣶⠿⠇⠐⣾⠇⠀⠀⠀⠀⠀⣈⣀⡀⠀⠀⣴⡶⠆⠉⠋⠁⠀⠀⣀⠀⠸⣯⣀⣼⡟⢰⠀⠀⣰⡟⠁
⠀⠀⠛⢷⠀⢀⣀⣀⡀⠀⢀⡄⠛⠛⠉⠉⠁⠀⡀⢹⣷⠾⢶⣄⠀⠀⢀⣿⣷⡀⠉⢹⣿⠃⠸⠄⠀⠿⠁
⣤⣤⣄⠀⢠⣿⠋⠙⠿⠀⠈⠁⠀⠀⠰⠆⣶⣿⣁⠈⡁⠀⣸⣿⠀⠀⠘⠃⠙⣿⣄⠈⠁⣀⡄⠀⠀⣀⣀⣀
⠃⣨⡿⠀⠼⡿⠛⢻⣦⠰⢠⣾⠟⢷⡆⠀⠀⢉⡿⠃⣀⡀⠉⠁⠀⠸⠆⠀⠀⠈⠻⠃⢀⡉⢹⡄⢸⡟⠉⣻⡇
⠀⠉⠁⢠⣤⣤⣀⣈⣭⠀⠠⠿⢶⡾⠃⠀⡀⠀⠀⠘⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠀⢠⡿⠁
⡾⠆⠀⠈⠻⣯⡛⣻⡿⠷⠂⢀⣠⠀⠀⣀⡀⠀⠀⣶⣶⠾⠋⠀⠃⣰⡿⠛⠁⢀⡀⠀⠀⣤⠰⠷⠷⠀⣿⡷⠟
⣴⠦⣤⣀⠀⠈⠿⣿⠁⠀⠀⠘⠋⢠⣿⠛⢻⣧⢰⣦⠀⢰⣷⠀⠇⢹⣷⠀⣼⡟⢻⣷⠀⠏⠀⠀⠀⣂⠀⠀⣤
⣃⣀⣨⣿⣀⣹⣷⣀⣀⣀⣀⣀⣀⣘⣿⣤⣾⣿⣀⣁⣰⣿⣧⣀⣀⣾⣏⣀⣿⣇⣀⣿⣆⣀⣀⣀⣀⣻⣷⣾⣟
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠤⠀⢠⡆⠀⠀⠀⠀⡠⠤⠤⠀⢠⠄⠀⠀⠀⠀⡠⠠⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼
⠀⣀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⣠⠤⠀⠀⠙⡆⠀⠀⢀⠀⢀⡄⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⡏
⠅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠇⠀⠀⠀⠐⠈⢀⠠⠁⠀⠎⠀⠀⡀⠀⠀⣠⠟⠀⢀⡇⠀⢠
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠀⠀⠀⡀⠀⠀⢨⠀⠀⠘⠀⠀⠀⠂⠘⠛⠛⠛⠀⠘⠀⠀⣸
⠀⢀⠀⠀⠀⠀⢼⢀⡀⠀⠀⠀⢠⣾⠋⠀⠀⣀⢲⠂⠀⠀⠀⢰⣉⢀⠀⠀⢸⠀⠈⠀⠀⠀⠀⡀⠀⣴⡆⠈⠁
⠀⠁⠀⠀⠀⠀⠀⠼⠃⠀⠀⠐⠤⠡⠀⠀⠀⠡⠟⠀⠀⠂⠀⠀⠉⠀⠀⠠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠈⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠁⠀⠀⣀⡀⠪⠂⠀⠀⢀⡀⠈⠀⠀⠁⠠⠰⠀⠀⠀⠀⠁⢀⡄⠀⠹⡄
⠀⠀⠐⠿⠿⠿⠟⠶⠏⠀⠀⠀⠁⠀⠾⠃⠐⠀⠉⠀⠀⠀⠀⠚⠉⠐⠂⠀⠀⠀⡾⠀⠀⠀⠀⠀⠺⠇⠀⢀⠇
⠐⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠀⢀⡄⠀⠛
⠀⠀⢀⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⣷⡀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠉⠃⠐⠀⢀⠜⠁⠀⠀⢠⠀⠀⣾⢁⡄⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⢀⠆⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠐⠂⠀⠀⢴⡄
⠀⠀⢀⡖⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⡟⠀⠀⢹⡄
⠀⡀⠘⠳⠶⠶⠂⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢘⡇
⠀⢈⢀⠂⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⣄⠀⠀⢈⠇
⠀⠀⠛⠛⠛⠋⠀⠠⠎⠀⠀⠀⠀⠀⠈⠃⠀⣠⠎
⠀⠒⠠⠄⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⢶⡶⠦⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡋⠉⠀⠀⠀⠈⢀⠈⣁⣀⠁⠀⠀⠀⠉⢙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠓⠀⣄⠀⠀⠀⠀⠀⠸⡆⣿⣿⠀⠀⠀⠀⠀⠀⢀⠚⠈⠓⢤⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠙⢦⠘⣆⠀⠀⠀⠀⠀⢧⣿⣿⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠋⠀⠁⠀⠀⠀⠀⠀⠳⣜⠦⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⡴⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⣨⢦
⠀⠀⡼⠡⠀⠐⠶⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⣀⡤⣚⠽⠂⠀⠈⢧
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⠁⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢮⠉⠀⠀⠈⠗⠋⠁⠀⠀⠀⠀⠘⡆
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣟⠂⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠻⠷⣄⡀⠀⢀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡷⠄⠤⠖⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣟⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⣭⡏⠠⠶
⠀⡇⠄⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠠⢸
⠀⡧⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠸⡄⠀⢀⡤⣖⡭⣚⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⢠⠇
⠀⠀⢳⠐⠀⢫⠗⠋⢁⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠮⠝⠀⠂⡞
⠀⠀⠀⢳⡚⠀⢀⡴⠋⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡞
⠀⠀⠀⠀⠱⣌⠀⠀⠀⠀⠀⣠⢫⠏⢠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⣡⠎
⠀⠀⠀⠀⠀⠈⠣⣀⠄⠀⠰⣳⠃⠀⡞⠀⠀⠀⠀⢹⠀⣶⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠠⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⣤⠀⠸⠃⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣤⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⡀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⢀⣨⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠤⠼⠧⠤⠖⠒⠒⠋⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣻⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⣤⣤⣤⣿⣿⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠙⢿⣷⣾⣿⠟⠁⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠿⠿⠿⠿⠿⠿⠿
⠛⠛⠛⣿⣿⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣠⣾⣿⢿⣷⣄⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣴⣶⣶⣶⣶⣶⣶
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠁⠀⠙⢿⠗⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⣴⡟⢻⣆⠀⢀⣴⣦⠀⠀⢰⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣧⠀⠀⣀⣀⣀⣀⣿⣷⣀
⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠙⠷⠾⢃⣴⣿⠟⠁⠀⠀⠀⠉⠛⠻⠿⣿⣷⣦⠀⠀⣤⣶⣿⡿⠿⠛⠉⠁⠀⠀⠿⠿⠿⣿⣿⠿⠿
⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⠟⣵⣶⣦⡀⠀⠀⢀⣠⣤⣶⣿⡿⠿⠀⠀⠻⢿⣿⣶⣦⣄⣀⠀⠀⠀⣶⣶⣾⣿⣿⣶⣶
⠘⢿⠟⠁⠀⠀⠀⠀⠀⠀⠐⠿⠟⠁⠈⢿⣤⡿⠃⠀⠸⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⡟⠀⠀⠉⠹⣿⡟⠉⠉⠉

⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⣠⣶⡆⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⣠⣤⣤⣿⣿⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠙⢿⣿⣾⣿⠟⠁⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠻⠿⠿⠿⠿⠿⠿
⠙⠛⠛⣿⣿⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣠⣾⣿⢿⣿⣦⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⠀⠙⠿⠃⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣿⠆⠀⠀⣾⡏⣹⡧⢀⣴⣾⡇⠀⠀⠸⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠀⠀⣠⣤⣤⣴⣿⣿⣤
⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠈⠛⢛⣡⣿⡿⠋⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣷⠀⠀⢰⣿⣿⡟⠛⠉⠀⠀⠀⠀⠙⠛⢛⣿⣿⠛⠛
⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⣿⡿⢃⣾⠟⣷⡄⠀⢀⣤⣶⣾⣿⠿⠟⠋⠀⠀⠈⠛⠿⣿⣷⣶⣤⣄⠀⠀⢾⣿⣿⣿⣿⣿⣿
⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠻⠶⠟⠁⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠈⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠉⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⢛⣿⡟⠛⠛⠛⠛⣿⡟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠤⠠⠠⠄⠄⠠⠄⠄⠠⠄⠄⠤⠄⠀⠤⠀⠀⠤⠀⠄⠤⠠⠄⠦⠀⠄⠆⠀⠀⠠⣶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⢠⠏⠦⠾⠀⠈⠀⠀⠀⠀⠶⠂⠀⠀⠤⢠⢶⡞⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢁⣦⢀⠬⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⢀⠀⠀⠀⢀⠀⠊⠉⣍⠑⡄⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠁⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠢⢔⠂⠀⠀⠐⠙⠁⡹⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⡀⠈⠀⠑⢄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠠⠀⢀⡐⠀⠀⡀⠀⠀⡑⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢈⠶⡆⠀⠈⢂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠱⡀⠀⠀⠀⠀⠀⠄⠠⠤⣤⣄
⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⡀⠀⠲⠀⠀⠀⠀⠀⠀⡀⢀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢰⣆⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠉⠀⢄⠈⠠⠈⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠄⠀⢀⠠⠀⠈⠀⠀⢀⠀⠀⠐⠀⠀⠀⠹⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⣴⢾⡴⢸⡟⠏⠐⢀⢁⠀⠤⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⣄⠀⠄⠤⠀⠂⠐⠛⠍
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⢨⣀⠀⡄⡯⣿⢸⠆⠤⠀⠀⠀⠐⠀⠃⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡏⣿⣸⢿⢸⣗⣗⣿⠰⠃⠒⠀⠀⠀⠀⠀⠐⠀⠀⠇⠀⠀⡇⠀⠀⠘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⠑⢿⢼⣚⣸⣿⣯⣿⠓⠀⠀⡀⠁⠀⠀⠄⠘⠠⢠⢂⠀⢠⠃⠀⠀⡀⠁⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁⠀⠀⠀⠄⠀⠀⠁⠘⠹⠟⠛⠈⠁⠐⠁⡀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠘⠋⠀⠀⢤⠄
⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠂⡀⠁⠔⠁⠀⡀⠈⡠⢲⠂⠂⣀⠀⡀
⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⠀⠂⠡⡀⠀⠀⡘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⡀⠂⢀⡰⠀⡀⢀⠀⠀⠔⢈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠃⡠⠊⠴⠃⢀⠈⣠⠆⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢨⠀⠊⢀⠔⣁⡎⠉⠀⠀⠀⠀⠀⠀⠠⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡈⠡⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣨⠄⡐⠡⠆⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠠⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠉⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠰⠒⡄⠄⠀⠀⠀⠓⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠩⠀⠀⠀⠀⠤⢤⠤⣤⡶⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠁⠁⠓⠢⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣦⣤⣶⣿⠋⠁⠀⠀⣀⣀⣀⣒⣂⡀⢄⣾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣚⣛⣚⣛⣛⣛⣛⣚⣒⣚⣛⣒⣚⣓⣛⣛⣛⣛⣛⣛⣛⣛⣋
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣄⠀⢀⣤⠖⠚⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠚⠋⠉⣷⡀⣾⠁⠀⠀⠀⠸⣤⠎⢹⡆⠀⠀⢀⠀⠀⠀⠀⢹⣤⢶⠲⠦⢤⣀⡀
⠀⣀⡤⠶⢶⣄⡀⠀⠀⠘⡆⠀⠀⠀⣠⠠⠂⠹⡟⠀⠀⠀⠀⠀⢱⡀⢸⡇⠀⠀⠫⠇⠀⠀⠀⢸⠃⡏⠀⠀⠀⠀⢹⢀⡶⠶⢤
⢻⠁⠀⠀⠀⠄⠙⠦⣄⠀⢧⠀⠀⠀⢠⡀⠀⠀⢻⠀⠀⠿⠂⠀⠀⣧⠈⠀⠀⠀⠀⠀⢀⡤⠖⠁⢀⠇⠀⠀⠀⠀⢸⡞⠀⠀⢀
⠈⣧⠀⠀⠀⢠⠀⠀⠈⠳⢼⡄⠀⠀⠀⢉⠇⠀⡇⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠸⡆⠀⠀⠀⢢⡀⠀⠤⠐⣧⠀⠀⠀⠘⠂⠰⡇⠀⠀⠀⣀⠀⠀⠀⢹⡇⠀⠀⠰⡄⠀⠀⢣⠀⡦⠀⠀⢀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣧⠀⠀⢠⠍⣴⠊⣉⠳⣧⢀⣀⡇⠈⠦⣀⣈⣶⡇⠀⠀⡎⢣⠀⠀⠀⢀⡏
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⡀⠷⠄⠤⠤⠔⠒⢛⠁⠉⠁⣠⣥⡘⠀⢂⡷⣤⣀⣀⣴⣄⣀⣀⡄⠉⠑⠚⢁⠀⢆⡀⠀⣼
⠀⠀⠀⠘⣆⢀⡀⠄⠒⠉⠁⠀⣀⣶⠤⠴⠖⠒⠋⠛⠛⠉⠁⠀⡯⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠛⠳⣄⣉⣹⠏
⠀⠀⠀⠀⠈⠳⣤⡤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠛⢠⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡅⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡟⢳⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠇⠀⢇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠀⠀⠘⡄⠀⠈⠆⠀⠀⠀⠀⢀⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣉⡒⠠⠄⣀⣀⠃⠀⠀⡇⠀⠀⠀⢱⠀⠀⠘⣄⠤⠔⢊⣡⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⡌⠀⠀⢘⡛⡛⢛⢛⣙⡁⠀⠀⢤⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡎⠈⠉⠉⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠙⠒⠦⣀
⠀⠀⠀⡴⠋⢀⠴⠚⠉⢉⣩⠙⠢⡄⠈⢣
⠀⠀⡼⠀⣰⠃⠀⠀⢸⡉⠉⠂⠀⠘⡆⠀⣇
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣚⣋⣉⣉⣉⣛⡒⠲⢤⡀
⠀⠀⢣⠀⠹⣄⠀⢳⠖⠚⠁⠀⣠⠎⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⡞⠭⣴⣀⣠⠖⠛⢳⠾⢙⡆⠘
⠀⠀⠈⢷⣄⡈⠙⠒⠲⠖⠒⠋⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣈⡙⠒⠒⠒⠒⠒⠒⠚⣉⣤⣾
⠀⠀⠀⠀⠙⠿⣷⣶⣶⣶⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⠤⠤⠴⠶⠶⠤⠤⢤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⠔⣒⡏⠉⣩⠭⠿⣗⣲⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡈⠺⠭⢿⣟⣁⣀⣤⠥⠴⠚⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣀⣀⠉⠙⠿⣿⣿⣿
⠀⢀⣠⠤⠖⠒⣒⣒⡒⠒⠦⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡴⠋⠁⠀⠀⠉⠲⣄⠈⢿⣿
⡔⠁⡔⣊⣩⠉⢉⡽⠛⢯⡭⠖⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⣠⠄⠀⢀⣠⣀⠈⢆⠈⢿
⣷⣄⡙⠒⠮⠭⠭⠤⠤⠤⠖⠚⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠈⠹⣄⣠⠎⠀⣸⠶⢸⠀⢸
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠣⡀⠀⠀⠀⠀⠁⢀⡞⠀⡸
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠉⠉⠉⠛⠋⠉⠉⠉⠀⠘⢿⣷⣄⠉⠒⠤⠤⠤⠔⠋⠀⡴⠃
⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⠴⠋

⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢹⡏⠙⡟⠀⠀⠀⠀⠀⠘⣟⠉⢻⠋⢹⡟⠀⠀⠀⠀⠀⢹⡏⠈⢿⠀⠀⢸⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⣧⢀⡎
⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣸⣇⣀⣇⠀⠀⠀⠀⠀⢀⣿⣀⣼⣀⣸⣇⠀⠀⠀⠀⠀⣸⣇⠀⠀⠹⡞

⠀⠀⠰⣶⠂⠀⠐⡶⠒⣶⠂⠀⠀⠰⣶⠀⠀⢰⡖⠒⣶⠂⢶⡖⠀⠀⠲⡖⠀⠀⢲⠖⢲⡶⠒⣶⠖⢲⡖
⠀⠀⠀⠸⣇⠀⡼⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⡜⠀⠀⣿⠀⢸⡇⠀⠀⠀⢻⡄⢀⡎⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢻⡴⠁⠀⠀⣿⠀⠀⠀⠀⠀⢻⡼⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠈⢷⡜⠀⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠠⢤⡄⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢤⠄⠠⣤⠄⠀⠀⠀⠀⠀⠀⢤⡤⠀⢠⠤⠠⣤⠤⢤⡤
⠀⠀⠀⠈⢳⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠋⠀⠀⣿⠀⢸⡇
⠀⠀⠀⢀⠎⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠹⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⣧⡀⠀⣿⠀⢸⡇
⠀⠀⠐⠛⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠚⠓⠒⠛⠂⠀⠀⠀⠀⠀⠀⠚⠃⠀⠘⠓⠒⠛⠃⠚⠓