Inner ear Emojis & Symbols

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇⠀⠀⣀⠀⢰
⠀⠀⠀⢀⣀⠀⢩⢦⣀⣀⠀⠀⣿⡆⠘⣧⣄⡀⠀⠀⠀⡀⢀⡀
⠀⠀⢰⠏⠉⣳⡁⠀⣠⠁⠀⠀⢿⣃⠀⢙⠿⠿⣷⣆⣂⣀⠒⠀⠀⠀⢀⡀⢀⣀
⠀⠀⢸⠀⠀⠾⢿⡀⠅⠀⠀⠂⠸⣿⡆⠈⠻⣾⣞⡏⣭⡻⢿⠶⠶⠎⠁⢰
⠀⠀⢸⠀⠀⠉⢩⣥⠀⡈⠀⠈⠀⠙⠳⠄⠀⠈⠚⠱⢟⠹⣶⠦⣴⣦⣄⠘⢄⣀⠀⣀⣐⣲⣦⡀⠀⡀
⠀⠀⠘⠄⠀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠶⠶⣄⣐⣂⣠⣸⡷⣿⣿⣿⣿⡿⢹⡿⣋⠤⢌⠙⣧⠀⠐⠂
⠀⠀⠀⠀⠰⣿⣿⡿⠻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠛⠛⠻⣿⡾⣿⣿⣿⣷⣤⡹⣷⣴⣦⣇⠃
⠀⠀⠀⠀⠀⠹⣿⡃⠀⠘⣿⠿⠛⠉⠋⠉⠀⠀⠸⠛⠒⠒⠒⠬⣷⠿⠿⠿⢿⣿⡍⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡿⢄⠐⡎⠐⠛⠉⠀⠀⠀⠘⠀⠀⠂⠀⣠⡆⡂⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠂⠀⠐⠀⠀⠀⠰⠆⠀⠾⣿⡏⠀⠀⠀⠀⠈⢿⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢆⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠘⠁⠙⣦⡀⠀⠀⠀⢀⣅⡀
⠀⠀⠀⠀⠀⠐⠄⢀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣡⠾⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣿⡿⡂⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠆⠀⠀⠲⠘⡇⠀⠈⢿⣷⡄⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⡇⠀⢀⠈⢿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣿⣿⠟⠉⠁⠠⢤⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠻⠟⠋⠁⠀⠀⣼⣿⣦⠀⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⡠⢾⣄⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢞⣧⣠⣄⡀⠉⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡿⠛⠋⠀⠀⠀⢀⣀⣠⠴⠶⠖⠢⠙⣿⣶⡾⠘⡁⢰⢹⡆⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⢲⣶⣏⠁⣀⣀⣀⣀⡀⢀⣑⣌⡻⠦⠈⣥⠾⢿⣆⠀⠲⣦⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⢠⢸⣿⠟⠉⠹⠯⠙⠁⡙⠃⠀⢘⣺⣿⣿⣟⢛⡶⠈⢻⣖⡚⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠂⠀⠄⠀⠙⠋⠉⠁⠈⣉⡻⠀⠀⠙⠫⡮⣔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣉⣁⡀⢀⣀⠀⢹⣮⠛⠮⠷⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠉⢆⠀⠀⠀⠀⡀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠸⡄⠀⠀⠠⡇⠆⣿⣧⣏⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⢷⣠⠐⣰⠠⢰⣿⣿⣿⣿⣿⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⡇⡠⣙⠡⢸⠈⠈⠿⣿⣿⣿⣿⣿⣿⣇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣼⣷⡟⠈⡠⠨⠴⠘⢗⡈⡩⠉⣛⢻⣿⣿⣷⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠰⠛⣿⣄⢸⢾⢠⣊⣥⠀⢥⢠⠀⣠⠿⢿⣿⣿⣿⣄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢀⣶⣿⡿⠛⢚⠭⠉⣮⡭⣤⣬⣼⣿⣟⣓⣩⢿⡿⠿⣧⣀⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢾⣿⡿⣯⣄⣄⣀⣴⣿⣿⣿⣿⣿⣿⣇⠹⣮⠹⣄⡹⢸⣿⡐⠜⣂⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⢿⡀⠈⣿⣿⣿⣿⣿⣿⣟⢛⣻⢿⠿⠓⣛⣳⣶⣶⡻⢿⣷⡔⢦⣤⣤⢀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣷⡄⣿⣘⣿⣿⣿⠟⢁⣺⣷⣿⡾⣟⣿⣿⣿⡟⢻⣿⣿⡴⣘⣷⣄⠻⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡙⠻⠿⠛⠁⠀⠨⠟⠛⠉⣽⣿⣼⡿⠉⠀⠘⣍⣿⡃⠙⢿⣿⣧⣑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠘⠃⠁⡀⠀⠁⠀⠀⠀⠄⠀⢹⠀⣿⠁⠀⠀⠄⠈⠩⠀⠀⠄⠹⣿⣾⠳⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠅⠸⠀⠀⠀⠀⠀⠸⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠡⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠢⣖⡔⠂⠀⠀⠰⣠⢆⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠎⠀⢠⠊⠉⠉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⣼⠀⠀⢸⡞⠀⢰⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠚⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠐⠞⠀⠀⡾⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠇⠜⠊⠁⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠛⠁⢰⡷⠀⠸⠎⠀⢸⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⡠⣴⣖⣋⢣⣍⢀⡀⠀⣐⠀⢸⡇⠀⠀⠀⠰⠇⠀⠀⠀⠀⠇⠀⠀⢠
⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⠀⡀⠀⣈⣥⣤⣾⢏⣍⢻⣿⣿⣿⡇⡈⠀⣸⠉⠀⢨⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠐⠁⠘⠘⠀⠀⠀⣸⣈⣛⣛⠃⠉⠏⠸⣟⡙⣶⣽⣯⠑⠀⣂⡤⠀⢠⠟⠀⠀⣀⢴⢱⡿⠉⠁⢦⠀⠈⠸⠀⢸
⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢉⣙⡛⢇⡀⡀⢠⢾⣿⣿⢿⠀⣀⣀⣠⣶⡱⣞⣀⠀⠀⠈⠘⠇⠀⠀⢘⡆⠀⡈
⠀⠀⠇⠰⠶⠂⠀⡶⠞⠂⠉⠀⠀⠀⠳⢾⡿⢀⡩⠤⣠⣼⣵⣿⣽⣛⣛⣭⣾⣻⣿⠛⠒⢶⠆⢰⠀⠀⢀⡼⡁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠐⣡⠖⠉⠀⣤⣾⣿⠟⠟⠛⠛⠋⠙⠛⠋⠀⠀⠀⠸⠀⣾⡀⡴⠁
⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⠄⠀⣀⠔⠋⠀⠀⠀⢀⣠⣬⣭⣀⣈⣀⠒⠀⠀⠒⠒⠠⢄⡀⠸⡀⠛⠛⠁⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠸⠟⢁⡠⠞⠁⠀⠀⠒⠆⠐⠟⠉⠀⠀⠀⠀⠈⠉⠉⠉⠒⠒⠢⣀⠙⠢⣅⠀⠀⠀⠀⠀⠏⢐⠂⠼⠋⠘⠷⠃
⠀⠀⠀⠀⠀⠠⠊⣠⡖⠋⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠚⠁⠀⠀⠀⠉⡻⠁⢰⠀⠀⠀⡸⠀⠆
⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⠀⢠⡄⠀⠀⠐⢁⡲⠁⠀⠈⠀⣠⠋⠀⠀⠀⠀⠀⠀⢴⡰⠁⣰⠃⠀⠀⢀⠇⠰
⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠄⠀⠊⠀⠈⠃⠀⠈⠀⠠⠊⢀⠔⠁⠀⣀⡄⠀⠀⣀⢀⠄⠁⠆⡎⠀⠀⢀⠎⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡔⠉⠀⠀⠀⠎⠐⠆⠐⠘⠞⠀⠀⡁⠱⡀⠀⢈⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠐⠁⡂⠉⡇⢨⣷⣄⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⡆⠁⠀⠃⢸⡯⣽⡿⠖⠂⠄⣀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠘⢧⣀⠆⡇⠀⡀⠀⢸⣿⡁⠀⠀⠀⣧⡛⢤⠀⠀⠀⢀⡆
⠀⠀⠀⣼⡯⠟⠁⠁⠀⠀⠀⠈⢙⢀⡇⠀⠀⠄⠘⣿⡀⡀⠀⠹⢻⢂⡜⠄⠀⠀⢸⠃
⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⢐⠀⠘⣿⣀⡀⠀⠆⠈⠋⠀⠀⠀⣼⠀⠀⠀⠠⡀
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠸⣷⡀⠀⢓⡀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠁
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⣿⣷⣶⡼⠒⠀⠀⠀⢠⡃⠀⣀⠈⠇⠀⠘⠇
⢀⡀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⠿⡀⠀⠀⡃⠀⠀⢿⣿⣿⣅⣤⠔⠁⠀⢨⠉⣿⣭⣹⠁⣼⡁⠀⠀⠀⠀⡀
⠀⠀⠲⣜⡇⢀⠄⠀⠀⣠⡄⠀⣠⣿⠀⠱⠀⠀⠁⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡙⢸⢿⣶⣽⣅⠃⠘⠦⢄⣀⠀⠃
⠀⠀⠀⢠⢿⠤⣀⣾⡿⣛⣧⣼⣿⡟⠀⠀⠁⠀⠢⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⡷⣛⣚⢿⣏⣿⡄⠀⡠⠞⢩⠙⢲⢄⡀
⠀⠀⠀⢸⢸⠀⣿⣿⣿⣿⡿⠟⡻⠁⠀⠀⠀⠀⠀⠠⡀⠐⠁⠀⠀⠀⠀⠀⠻⣿⢿⡿⢟⡻⣿⣿⣯⠶⠶⣤⡀⠀⠈⡳⡒⠄⣀⡀
⠀⠀⠀⠈⢸⡆⣿⣿⣿⠙⢧⡀⣤⣀⣀⡀⠀⢀⣠⣤⣤⠶⠿⠛⠛⠋⠀⠀⠠⢌⣺⣾⣏⡾⠓⢸⠣⢂⣰⡌⣿⠈⠃⠉⠨⠀⠀⠀⠁⠒⠢⡀
⠀⠀⡀⠀⣸⣟⣸⣿⣏⣀⣀⣨⠨⣾⣿⣿⣿⡿⠿⠑⠃⠀⠒⠐⠀⠀⠀⠀⠀⠈⠃⠈⠻⣷⡄⠈⠻⠾⠞⣳⣿⢧⠤⠄⠤⠄⠤⠤⠤⣄⠀⣤⠀⠠
⠀⠀⠀⠀⡍⠝⣇⠙⢿⣦⡀⠀⣀⣠⡉⣿⣿⣿⣶⠤⡴⢒⣴⣶⣾⡶⢦⣭⣓⡠⠦⢤⠶⠝⣿⢒⢒⣒⣞⠛⠷⢭⣆⢀⠀⣙⣶⠆⠀⠀⣶⣶⣄
⠀⠀⠀⠀⠸⡀⠘⠦⠄⠈⠛⣷⡼⢿⣣⣿⣿⠟⠁⡂⠐⢸⠳⠒⠀⡑⠁⠉⠉⠉⠉⢹⢿⠋⣾⣿⣿⡿⠿⠻⢦⣄⠙⣷⣦⠤⣬⣷⣤⣀⣘⣿⡿
⠀⠀⠀⠀⠀⠱⡄⠌⢷⡦⠀⢻⣻⣾⣿⠟⠃⢠⡤⠀⠠⡢⠤⠤⢄⣘⠁⠀⠀⠀⠀⠀⣼⢿⣙⠿⠋⢹⠶⠶⣾⠁⠀⡈⢿⣿⣦⡙⢍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠁⠀⠀⡄⣀⠀⠐⠐⠶⠡⢴⠋⠀⠀⠀⠀⠀⠈⠃⠓⠀⠀⣠⣮⠘⠋⠀⠀⢸⣀⣀⣻⡆⠀⠘⠄⠙⢿⡙⠀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠸⠘⠀⠋⠂⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡇⡇⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠐⣌⠳⡄⠐⢧⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⡇⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⣦⣤⠽⣷⢬⡵
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢸⡁⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⣆⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠴⠶⠗⠉⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡙⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⢠⣏⣤⣄⠀⠀⠀⠀⠀⠀⢸⠠⢀⠀⢸⡆
⠀⠀⠀⠀⠘⣾⡟⣿⣧⠀⠀⠀⠀⠀⢸⠀⢸⡄⠈⡇⠆
⠀⠀⠀⠀⠀⢯⡇⢹⣿⡄⠀⠀⠀⠆⠸⡄⠀⡁⠀⢳⠈⠢⡀
⠀⠀⠀⠀⠀⠈⡄⣸⣿⣿⣄⠀⠀⠀⠠⠀⠀⠇⠀⠀⠁⢠⣭⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠘⡿⢷⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠂⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠰⠀⠀⠙⣿⡀⠀⠀⠸⡀⠀⠀⠐⠈⡠⣥⣤⣀⠀⠀⢦⣿⠀⢦⠅⠰
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠑⠄⠀⠀⠘⠿⠾⣟⠻⣷⣷⣤⢉⠀⣀⠀⠀⣠⠔⠀⢄⠐⠦⢤⡀
⠀⠀⠀⠀⠀⠀⢀⢹⡆⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⢀⣴⣶⠶⠿⠿⠿⠟⠻⢿⡛⠈⣏⣿⡻⠻⣤⠙⠉⠀⣰⣶⣶⠬⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢠⣿⣿⡿⣷⣀⣀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠻⣿⣷⣤⣨⣷⣄⡘⠻⣍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⢿⠀⠸⣿⣿⣿⣿⣿⣿⣿⡃⠀⠈⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠷⠾⠛⣿⣏⠉⠉⠙⠲⣌⠙⢮⣿⢲⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢿⣿⣿⣽⣿⣿⣿⣧⣤⡤⢔⠲⠾⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠂⠈⠑⢦⡈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡽⣧⡀⠉⠻⣿⣿⣿⣿⠟⠁⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣇⠀⠀⣈⡿⠟⠁⠀⠠⠚⠄⠀⠀⠀⠘⢦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⣷⣾⢯⡀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠖⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣽⠿⠲
⠀⠀⠀⠀⠀⢀⠀⣠⣤⣤⡀⠀⠀⠠⡀⠀⡇⠀⠀⠀⢿⣿⡆⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠈⣸⣿⡟⠉⠁⠀⠀⠀⠀⠁⡏⠀⠀⠀⠈⣿⣿⣴⡸⣿⣿⡆
⠀⠀⠀⠀⠀⡆⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⣿⣿⡧⣽⣟⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠘⣿⣷⣾⣿⡅
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢀⠀⠀⠀⠀⢄⣂
⠀⠀⠀⠀⠀⠀⠈⢟⠀⠀⠀⠀⠀⠀⠀⢀⡏⠱⡀⠀⠀⠀⠀⠘⠿⠟⠋⠀⠀⠀⠀⢠⣿⣯⢷⡀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣠⡴⡇⢠⣾⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⢿⡎⢷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⢵⣶⣿⣿⡿⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢛⡉⠘⢷⡶⠖⠶⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣗⣿⡟⠋⠀⠈⠁⠀⠀⠀⢀⣠⣤⠤⠶⠛⠛⠋⠀⠂⠈⡽⠾⣋⡾⠂⠘⠀⠀⡰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢟⠿⠁⠀⠀⠈⠛⢿⣿⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⢂⣠⣾⡃⠄⠀⠀⢠⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠳⢦⠀⠀⠀⢀⠀⣿⣿⣶⠆⠀⠂⡴⠖⢻⡟⠒⠠⠤⠐⠒⣚⡂⣩⣥⡔⣴⣦⣌⡉⠳⢤⡀⠀⠓⣄⠀⠀⠹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠉⠳⣄⠺⢀⣿⡟⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⢰⡿⣿⠿⢫⡀⣙⡟⠁⠈⢻⡆⠉⠙⠓⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠓⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠷⠁⠀⠀⠀⠈⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠲⠟⠛⠀⠀⠀⠀⠀⠙⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠠⠀⠛⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⢀⠀⠀⠀⠀⠀⢀⡼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⢸⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣀⡂⠀⣀⠀⠸⡧⠀⠀⠀⠈⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠀⠤⠀⢻⣶⠟⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡎⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠀⠀⠀⠀⠀⠁⣀⠀⠚⠋⠀⠀⣤⠤⠄⠀⠀⡀⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠇⠀⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⡴⡋⠀⠀⠀⢠
⠀⠀⠉⣹⣆⣀⣀⣀⣤⠴⠖⠒⠀⢀⣼⣿⠀⠀⣥⣄⠄⢀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⡤⠂⠰⠈⠳⢿⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠉⢁⠔⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠠⠁⠀⠀⠀⠰⡆⠻⡆⠀⠀⣴⠀⠠⠢⡀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠸⠀⡆
⠀⠀⠀⠀⡀⠁⣰⡟⠁⡀⠀⠀⠀⠀⠀⠀⠇⣀⣷⠀⠀⡟⠀⣰⠀⠈⠒⠚⠁⠀⠈⡄⠀⢠⠀⠀⠀⡇
⠀⠀⢀⠀⣠⡾⠋⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡸
⠀⠀⠸⠔⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⡤⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⠀⠀⠙⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠲⠌⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⢅⣀⣶⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠢⠮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⠓⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⣀⣤⣤⣶⣶⣶⣤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣿⣿⣿⣿⣿⣟⣟⣻⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢛⣩⣶⣿⡿⠟⠛⠉⠩⢤⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢉⣴⣿⡟⠋⠅⠀⣀⡤⣢⡤⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠋⣰⣿⣿⣿⣮⡵⠷⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣴⡟⢭⡿⠁⣾⣟⠿⣿⣿⣧⣶⣶⣾⣿⣿⣿⣶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢱⣿⠁⠶⣈⣭⣿⣷⣍⠻⣿⠿⠛⠛⠛⠿⢷⣄⣀⠈⠑⠦⣀
⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣿⢿⠇⢠⣾⡿⣛⣻⡾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈⠙⣷⣠⠰⠛⣧⡀
⠀⡴⠊⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣿⣭⢠⣿⣯⣾⣿⣿⡟⠈⠻⣿⣿⣧⡑⢄⠀⠀⠀⠀⠀⠻⣦⠀⠘⢷⡀
⢰⢿⡀⢨⣿⣿⣿⣻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣥⣿⢘⣿⣏⢸⣿⡟⠀⠀⠀⠈⠻⣿⣿⡦⡑⠄⡀⠀⠀⠀⠘⣧⠀⠸⣷
⠀⠀⢣⠹⡿⡿⣿⣷⣝⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣼⠟⡟⣠⡿⠛⠸⢿⡿⣴⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⣌⡲⡁⢄⠀⠀⠀⢸⠆⠀⣿
⠀⠀⠀⠱⡄⠙⣜⢿⣿⢮⡻⣿⣿⣶⣄⡀⠀⠄⠀⠀⠀⠀⠀⢀⣴⠞⠋⡠⠓⠁⠀⠀⠀⠈⠳⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⠲⡑⡀⠀⢸⣔⣣⣿
⠀⠀⠀⠀⠹⣆⠈⠈⠻⣷⣼⣽⠿⣷⣌⠙⠓⠶⠦⣤⣤⣤⣬⣭⣤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡜⣜⢤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠈⢻⣿⣗⢌⢻⣷⣄⠀⠀⠃⠠⠴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠻⣞⢷⡄⠀⠀⠀⠀⠀⠀⠀⢿⣷⡜⡞⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠦⡀⢻⣿⣶⣅⠛⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢒⣷⣽⣮⣿⣷⣦⣤⡀⢁⠀⢰⣾⣿⣧⢸⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠳⠢⣿⣿⣿⣦⣰⠝⣿⣿⡷⣦⣤⣤⣤⣀⣀⠀⠒⠈⠈⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣛⣿⣷⣀⣀⣸⢿⣿⡁⢨⡈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢩⡛⠙⣿⣿⣿⣮⣬⣛⠲⢮⣿⣛⠻⠿⢿⣿⣦⡀⢀⠀⠀⠀⠀⠀⢰⣾⣿⠿⠛⠉⢉⣉⡛⠿⡏⣵⣿⢳⢃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣼⣯⠀⠸⣿⡜⣿⣿⣟⣛⣳⡮⠭⣿⣿⣷⣦⣬⣭⣤⣤⣤⣤⣤⠤⣄⣀⣠⣀⢠⣴⣿⠿⠿⣷⣌⣉⡴⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⣽⣿⣿⡯⠐⠘⢝⣿⣿⣿⣵⣭⠉⠉⠠⡔⢦⣬⣉⣛⣛⣳⠶⣶⣶⣶⣶⣶⣬⣝⣻⠟⠁⠀⠀⠈⠛⢋⣴⡿⠃
⠀⠀⠀⠀⣤⣠⡾⠋⠐⡀⣤⣾⣿⢿⣿⢣⡀⢸⣿⣿⣿⣿⣿⣷⠀⠀⣿⡸⠿⢿⡿⠿⠟⠻⠿⠿⠿⠿⠿⠛⠛⠿⠿⠶⠦⠤⢶⣺⠛⠋
⠀⠀⠀⠀⣼⡟⢠⣥⡾⣿⣿⣾⣿⢢⠻⣇⣷⡈⠻⣟⢿⡿⠿⠋⠀⣠⡿⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢠⣿⢀⣠⣴⣿⣿⣿⣿⣿⣞⠇⣿⠛⢁⠀⠙⠙⠀⣀⣤⣾⣿⠁
⠀⠀⠀⠸⣿⡌⠻⢿⣿⣿⢿⠛⢁⠀⠂⠀⠃⣀⣠⣤⣶⣿⣿⠟⠙⣿
⠀⠀⠀⠀⠹⣿⣷⣦⣤⣬⣤⣤⣤⣴⣶⢾⢿⡿⠟⠛⠋⢉⠀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠹⣿⣤⣌⣂⣀⣰⣤⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⢿⢿⣿⢟⠻⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣠⠔⡧

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⢷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠈⣷⡀⠀⣼⣿⣿⠰⣂
⠀⠀⠀⠀⠀⠀⠀⣼⡏⢡⠀⢹⣧⠀⢿⣿⣿⡆⣿⣆⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢡⡿⠀⢸⣇⠸⣿⡄⢸⣿⣿⡇⢸⣧⣅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⡃⠀⠛⠻⡀⢸⣷⣼⣿⣿⣯⠠⠻⣿⣷⠈⣼⣿⣿⡿⢛⠲⡄
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⣷⠀⢻⣿⣿⣿⣿⠌⠢⠤⠘⠀⢹⡿⡡⡇⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣸⣿⣧⠀⢿⣿⣿⣿⣆⠀⠁⠀⡀⣹⠗⡷⠛⣎⣥⢉⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⣿⣷⠘⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣘⣧⣴⣮⣝⢾⣏⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣤⣿⣿⣿⣿⢿⠟⠿⠋⠋⠙⢮⡻⣿⣿⣿⣿⡿⣿⡄⠐⠂⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠹⣿⣿⣿⡿⠛⠉⠀⢀⠠⠄⠀⣐⣬⣽⣷⣿⣿⡏⠙⢿⣮⡙⢦⡙⠧⣍⣳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⢸⣿⣏⠀⠄⣤⠚⠉⢄⣴⣾⣿⣿⣿⡿⢙⡿⣇⣀⣀⡙⢿⣦⡈⠒⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⡄⢾⡿⢹⣿⡟⢆⠀⠀⠈⠉⠻⠿⣭⡁⠀⠈⠀⠀⡟⢹⡇⠈⠈⢿⣦⣀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠈⢻⣧⠸⡆⠀⠀⠀⠀⠠⠹⣿⠀⠐⠀⠀⠧⠸⠇⠀⠀⠀⠻⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣼⣿⡀⡇⠀⠀⠀⠀⠀⠡⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⣾⣿⠟⠃⠇⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⣤⣶⡦⢄
⠀⠀⠀⣼⣿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⢀⠀⠀⠀⢀⡃⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢡⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⢏⡴⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⡆⠀⠁⠀⠠⠀⡀⠀⠉⠀⢸⡎⣻⡘⣯⠋⠀⠀⠃⠀⠀⠘⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⡄⠘⠃⠠⠠⠉⠐⠀⢾⣷⠶⠿⢧⣈⠀⢀⠀⣀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠳⡀⠓⠀⠈⠀⠁⠀⡲⣴⡤⢦⢰⣿⡿⠲⣿⣗⡒⠀⠱⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣦⣛⢷⣦⣄⣶⡿⠿⠿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣠⣾⣿⣿⣿⠿⠿⠁⠀⠘⢻⣿⣿⣾⣿⣿⡉⠙⠓⠯⣟⠷⠖⠦⣀⠐⠦⣴⣤⣤⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠟⠛⠀⠀⣀⠤⠔⠒⠚⠋⠉⣉⣉⣉⣇⢀⣀⡀⠈⠑⢦⡀⠈⠛⠷⣎⡻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⡟⠁⢀⡴⠂⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡌⣿⢿⠷⠶⢦⠙⢷⣄⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠙⢿⣿⡿⠓⢀⣠⠀⠀⠀⠀⢠⣾⣿⣿⠿⠻⠛⠛⢋⠘⠉⠀⠀⠀⠀⠀⠀⠙⢤⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣀⠀⠀⡀⠴⠟⠁⠀⠀⢀⣼⠛⠛⠻⣷⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠐⠀⠀⢀⡔⠧⠀⡀⠀⠀⠹⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣄⠀⠀⠀⣾⠃⠈⠀⠀⠀⠀⠀⢹⣧⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⠀⠀⢿⡁⠈⠀⠀⠀⠀⠀⠀⠙⢷⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⠀⠘⡇⢀⠟⠠⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠃⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢰⢠⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣤⡀⠈⢦⡀⡆⠀⠸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⡇⠀⠀⠻⣿⣿⣿⣿⣆⡄
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣧⠀⠰⠀⢹⣿⣿⡿⠛⣹⣦⣄
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⡟⡄⠀⠀⠘⠿⠟⠋⠀⠀⣼⣿⡆⠀⢀
⠀⠀⠀⠀⠀⢨⠀⣰⣾⣧⣾⡇⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣆⣤⣈⠙⢢⠄
⠀⠀⠀⠀⠀⠈⠰⣿⣿⠋⢋⣁⣀⣀⣤⡤⠶⠚⠓⠒⠭⡿⣿⡟⢻⣀⡎⣧⡁
⠀⠀⠀⠀⠀⠀⠠⠹⣯⡀⠀⣟⣿⣿⣥⡤⣤⣴⣶⣴⣆⣼⣮⣛⣲⣶⣞⡛⢧⡄⠸⣶⡀⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠙⢷⣿⣿⠏⠀⠈⠙⠿⣈⠂⠀⠀⣿⣿⠿⠻⠿⣿⠁⠙⢿⣍⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁⢰⡏⠉⠀⠈⠓⠛⠀⠀⠀⠙⣗⢯⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⢀⡴⠊⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡞⢠⣶⣦⠀⢹⡆⠀⠀⣿⡇⠀⠀⠾⡄⣿⣷
⠀⠀⠀⠀⢸⠁⣾⣿⣿⡄⠀⣷⠀⠀⢸⡇⠀⠀⢳⠰⣿⣿⡄
⠀⠀⠀⠀⢸⠀⣿⣿⣿⣇⠀⠘⣦⠀⣸⡇⠀⠀⠈⢸⣿⣿⣿⣄
⠀⠀⠀⠀⡿⠀⣿⣿⣿⣿⣧⡀⠀⠙⠋⠀⠀⠀⣶⡸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⡇⠀⣿⣿⣿⠋⢻⣧⡀⠀⠀⠀⠀⠀⠭⠳⣤⣍⠻⠿⠿⠿⠷⢆⣦⣤⣤⢤⡀
⠀⠀⠀⠀⢳⠀⣿⣿⡇⠀⣾⣿⠃⠀⠀⠀⠀⠀⣫⣭⠀⠬⠁⣨⣙⢛⠁⢀⣿⢹⣮⢻⣿⢖⡤⢀
⠀⠀⠀⠀⠠⠀⣿⣿⠀⡼⠟⢻⡀⠀⠀⠀⠀⠀⠘⢟⠬⠄⡄⢫⢘⡧⢘⡭⣿⠘⣿⡎⢿⣿⠀⠎⠷⠂⡀
⠀⠀⠀⠀⠀⡀⢸⡏⠀⠀⣠⣾⣷⠀⠀⠀⠀⠀⠀⠈⠊⠿⣐⢒⡛⢲⣏⣾⡟⣿⣚⡱⢿⣿⣷⢶⣶⣄⡘⢻⣢⣄⡀
⠀⠀⠀⠀⠀⡇⢸⡇⢠⣾⣿⣿⣿⠀⠀⠀⠀⠂⠀⠀⣺⣽⣿⣶⣶⣤⣤⣽⠎⢿⣽⠶⠦⡏⣢⢴⡈⣿⢧⠀⣽⠯⢉⣈⠐⠒
⠀⠀⠀⠀⠀⡇⢸⡇⢸⣿⣿⠏⠻⣄⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣿⣷⣄⡙⠻⢛⣽⣟⢘⢷⡄⢽⣋⠁⠑⠀⣀⠀⢶⣶⣦⡄
⠀⠀⠀⠀⠀⢠⠘⣷⠈⢿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠙⠛⢛⡙⡉⢉⣩⠉⠛⢾⡲⡄⢿⣿⣿⣯⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠈⡄⢻⣧⠀⠙⢷⣄⣠⣿⣿⣿⣿⣿⣿⠿⠛⠻⠋⢀⣀⡤⣴⣶⣀⣠⣄⣩⡉⠥⢟⠙⡷⠄⠀⢻⣦⡠⠙⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢱⠈⣿⣷⡄⠘⣿⣿⠟⣱⣿⣿⠟⠁⠀⢠⣶⣯⠉⡠⠗⡏⠂⠘⠙⠛⠛⠀⠀⡋⠉⣿⡇⠙⢄⠙⢿⣧⡈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠸⣿⣿⢦⠙⠿⠿⠿⠛⠁⠀⠀⠀⠉⠉⠉⠀⠃⣾⣧⣸⡀⠀⠀⠀⠀⠀⢷⣶⡿⠁⠀⠀⠑⣄⠹⣿⠓⠤⣍⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣖⡿⠃⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢿⣷⡀⠈⠓⣤⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⣷⡀⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠛⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠛⠛⠛⠻⠿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠁⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⢀⣾⡿⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣀⣀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣤⣤⣴⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⠀⡑⠀⠐⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠒⠶⠒⣒⣶⡶⠂
⠀⠀⠀⢀⠅⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣢⣶⣛⣦⣀⣐⠲⢆⣒⡈⢹⣿⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠼⠀⠀⠋⠐⠃⠀⠀⠀⢠⠿⣋⡵⠚⣉⡾⢟⣵⣾⠏⢰⣿⡿⢛⣫⡙⣷⠀⠀⠀⢾⡄
⠀⠀⠀⠀⠈⠿⢆⣀⣀⠀⠴⢶⠠⡾⢟⣫⣠⠞⢋⡔⢿⠟⢁⣴⣿⠟⣠⣾⣿⡇⣿⡆⠀⠀⠘⢷
⠀⢀⣼⠷⠓⡀⢀⡀⠀⠠⠀⢩⡶⠿⠛⢛⣥⣶⣿⠞⡁⠆⣸⡿⡁⢚⢸⣿⣿⢡⣿⣧⠀⠂⠀⢸⠁
⠠⣄⢰⡆⠸⠁⠠⡄⢀⡄⣰⣿⣿⣿⣿⡿⠟⠋⢀⣰⣜⡿⢋⡴⢷⣾⣿⣿⢃⣿⣿⣿⢀⣼⠗
⠀⢻⣦⠡⠀⠂⠀⠳⣼⠻⣿⠿⡟⣉⣤⣴⣿⣿⣦⡽⢋⣐⡿⣼⣿⣿⡿⣡⣿⣿⣿⣿⣾⠃⡀⠀⣀
⠀⠠⣿⣧⠰⠀⡠⠈⣡⣶⣶⣿⣿⣿⣿⣿⣿⠟⢉⣴⠿⣻⣿⣿⡿⠋⠀⣿⣿⣿⣿⡟⢉⡐⠀⠐⠉
⠀⠀⠹⣿⡞⠲⣁⣾⣿⣿⢿⣿⣟⡿⠟⣋⣴⣾⣿⣿⣿⣿⡿⠋⠀⢀⠀⣾⣿⣿⡟⡔⠀⠀⡀
⠀⠀⠀⢻⣧⠘⣿⠟⠛⢻⠿⢐⣩⣴⣾⣿⣿⣿⢿⠟⠛⠉⠀⠐⠀⠀⠀⣿⣿⣿⠳⠁⠠⠚⠁
⠀⠀⠀⠀⠹⠀⢋⣀⣡⣤⣶⣿⣿⣿⣿⣿⣿⡷⢎⣤⣶⣄⡼⠄⠁⡄⢸⣿⣿⠏⠂⠀⣠⡶
⠀⠀⠀⠀⠀⠊⠿⣿⣿⣿⣿⣿⣟⣿⠿⠛⣩⣴⣿⣿⣧⣿⡗⠀⠒⠀⣿⣿⡟⡈⠀⣾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⢘⠟⢣⣿⣿⣿⣿⡿⠫⠀⠀⣸⣿⠿⠱⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⢀⢸⢰⢸⣿⣿⣿⣿⡷⠀⠀⠀⣸⣟⠂⡠⢀⣯⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢸⣿⣿⣿⣿⣿⡄⠀⠳⣠⣟⢂⠄⢠⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢰⠊⠰⣀⣼⡄⠜⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⠁⠊⠀⠠⣾⣳⡏⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⣽⡟⠋⡀⠀⢰⡷⠁⣴⢻⣾⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡤⡤⠇⢠⣿⠗⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣟⣖⡘⠁⣿⣿⡟⣠⣿⣿⡟⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⢿⣿⡿⠿⠟⠛⠋⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣄⡀⠈⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢹⣿⢹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠙⣛⣭⣭⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣝⠟⣨⣭⡸⢟⣭⣿⣿⣇⣠⣤⣤⣀⢠⣾⣿⣿⡿⠸⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⢸⣿⢸⣿⣿⣿⣿⢸⣿⣿⢇⣿⣿⣿⣿⣿⡿⠿⢟⣟⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⡸⣿⣾⡿⡘⣿⣿⣿⡏⡾⠿⠿⠞⣿⣶⣰⣿⡏⠾⠇⣿⣿⢇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣵⣿⣶⣶⣾⣷⣽⣿⢟⣼⣿⣿⣿⣷⣝⢿⣿⡟⣵⣿⣿⢰⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⡿⠿⢟⣫⣵⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣬⣯⣵⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⣿⣿⣿⣿⣿⣿⣷⣍⢿⣿⣿⣿⣿⣿⣿⢹⣿⣿⢟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⢹⣿⣿⣿⢸⢟⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠇⢻⣿⣿⣿⣿⣿⣿⣛⣋⣙⣛⣘⢿⣿⣿⣄⡻⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢎⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠀⠀⠀⠀⠀⠈⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡿⠀⠀⠀⠀⢠⣶⣿⣿⣯⣭⣻⣷⣤⣄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣰⣿⠿⠋⠉⠛⠟⠗⢱⣾⣿⣷⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⢿⣷⡄⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⢿⣿⡄⠀⠀⠀⣸⣿⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢸⣿⣷⠀⠀⢰⣿⣿⠃⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣀⠄⠈⣿⣿⠀⠀⣸⣿⠃⠀⠀⣼⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡰⡸⣻⡏⠀⢀⣿⠃⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠻⠿⢋⡖⠀⣰⡛⠁⠀⣼⠃⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣬⡽⠾⠟⠋⠀⠀⣼⠃⠀⣠⣿⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢠⡟⠋⠀⠀⠀⠀⠀⠐⠃⠀⣴⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣿⠧
⠀⠀⠀⠀⠀⠀⠀⢜⡟⡿⢫⠟⠁⠩⢛⡻⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠴⣣⠞⠁⠁

⡀⣀⢀⣀⢀⡀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⡀⢀⢀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣶⣤⣴⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡕
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣟⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠐⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣻⡿⠿⣿⣿⣿⣿⣿⣿⡏⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣶⣿⣿⣿⠿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠏⠁⠀⠀⢸⣿⣿⣿⣧⣤⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣶⣿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠷⠟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡾⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢄⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡼⢇⡌⠃⠀⠀⠀⠀⠀⢀⣤⣾⣷⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡏⢿⡼⢿⡄⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢰⠈⠀⢸⠀⠀⢀⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠰⠄⣿⢉⡛⠆⠇⢸⠀⠀⣴⡾⢉⣿⣿⣿⡏⢛⠋⢸⣿⣿⣿⡿⣟⠃⢸⣷⣆
⠀⠀⠀⠀⠀⠀⢀⠾⣽⢸⡇⠘⠀⠘⢀⣴⣿⠃⣼⣿⣿⡿⢃⠁⣠⢞⣿⣿⣿⡇⠀⠀⣾⢇⢻⣧
⠀⠀⠀⠀⠀⢠⣾⢣⡟⢻⡀⠁⣶⢠⣿⡏⠙⠀⢾⣿⣿⠅⠘⡀⠉⢰⣿⣿⣿⣇⠐⠀⠁⣼⣿⣿⡆
⠀⠀⠀⠀⢰⣼⣿⣾⣇⠘⢃⡀⣿⢸⣿⡗⠀⠀⣿⣿⡇⠠⠀⠃⢀⣾⣿⣿⡟⠃⠀⠈⠠⢿⣿⣿⣷
⠀⠀⠀⠀⣾⢻⣧⠀⠅⡰⢼⢄⣿⣸⣿⠇⠀⢸⣿⣿⡃⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠘⠂⠾⣽⣿⣿⣧
⠀⠀⢠⡘⣿⣾⡄⢀⠀⣿⣤⣿⠁⣿⣿⡀⠀⠺⣿⣿⡇⠶⠀⠐⣿⣿⣿⣿⠀⠀⠀⠘⡃⣰⣿⣿⣿⡿⣆
⠀⠃⢸⠀⣿⢿⠃⣼⠀⢸⣿⣿⢳⣿⣿⡇⠀⢸⣿⣿⣧⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣿
⢰⡇⢸⠀⢩⣾⢠⡏⢀⠸⣿⠟⢘⣿⣿⠃⠐⢾⣿⣿⠃⠀⠀⣿⣿⣿⠉⠃⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿
⠈⣅⣀⠀⣶⣅⣦⡴⡉⢳⣿⡏⢀⣿⣿⣇⠀⢸⣿⣿⡆⠀⠀⣽⣿⡟⠀⠀⠀⠀⢰⢟⣾⣿⣿⣿⡟⣧⣟
⢸⡇⡟⢰⡟⣹⡟⣥⠁⠄⢻⡷⠀⣿⣿⣿⡄⣸⣿⣿⣄⠀⢠⣿⣿⠇⠀⠀⠠⡔⢂⣼⣿⣿⡯⠏⢠⡻⠋
⢠⠀⣰⣿⣿⢿⣵⠆⡄⠀⠨⣷⡄⢻⣿⣿⣷⠻⣿⣿⣧⡄⢸⣿⣿⠃⠀⠀⠀⣰⣿⣻⣿⡿⠁⠀⢠⣼⠃
⠌⠀⢸⣧⢿⡜⣇⢠⠱⡀⠀⠙⣧⠘⣿⣿⣿⣆⢻⣿⣿⣃⣾⣿⣿⡄⠀⠀⢰⣿⣿⣿⡉⠀⡄⢠⣿⠃
⠐⠀⢾⣿⣸⣿⣿⣇⡰⣵⡄⠘⡙⢧⡼⠿⣿⣿⡄⢹⣿⣷⣿⣿⣿⠇⠀⠀⣿⣿⡿⠁⠁⠀⢠⣿⠓⠀⢀
⠀⠓⡘⣿⣷⣿⣿⣿⣁⠐⢻⡄⠱⠘⣧⡀⠚⡾⣿⣿⣿⣿⣿⣿⣿⣢⠀⣴⣿⡏⠀⠁⢄⣰⡿⠁⢀⠠⠈
⠀⡀⢇⢿⣿⣿⣿⣿⢻⠳⣎⣞⡠⣷⣌⢻⣦⢀⠙⣿⣿⣿⣿⣿⣿⣿⣼⣿⡿⠁⠀⠀⣾⡿⠀⠀⢠⡇⡤
⠀⠀⠌⣼⣿⣿⣿⣹⣧⣁⠘⣼⠃⠻⡽⢇⣻⣧⡗⠀⢻⣿⣿⣿⣿⣷⣿⣿⠇⠀⠀⣼⡏⠀⠀⠀⣼⢛⡆
⠀⠀⠰⣰⣿⣿⣿⣷⡞⡏⢳⡸⡇⢲⢼⣮⠻⣿⡧⡸⠀⠈⢻⣿⣿⣿⣿⣿⡆⠀⣾⡏⠀⠀⠀⠀⢯⠋⠁
⠀⠀⠀⢹⣆⣿⣿⣿⣷⡰⣌⢿⡿⣦⢫⣟⣥⡘⢷⡧⡀⢀⡇⢿⣿⣿⣿⣿⠇⣸⠋⠀⠀⣀⠆⠠⢦⠃
⠀⠀⠀⠐⣿⡏⣿⣿⣟⣷⣝⡎⢳⡝⢧⡘⢿⡘⢻⠇⠀⢠⡅⡏⠰⠀⡌⢛⣴⠋⠀⠀⣠⠏⠰⢁⠎
⠀⠀⠀⠀⠙⡿⣼⣿⣿⡜⢿⣷⡘⢿⣬⢷⡼⣷⣄⣚⠀⠀⠀⢁⡀⠚⠀⠴⣿⡗⠘⣢⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠸⣻⣿⣟⣿⣆⠻⣵⢈⢻⣏⢳⡘⢦⠍⠐⠀⢠⡌⢡⠄⠀⢚⢁⠉⠈⡡⡞⠁⠁
⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⡄⠹⠎⣦⣻⡄⢳⡜⣇⣺⠀⣸⡷⠀⢀⠀⠈⡸⠆⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⢹⡞⢻⣿⣿⣿⣧⡻⣌⢫⡹⣄⢿⣿⢻⣟⡌⠁⠀⠀⠠⠾⠁⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣝⣷⣻⣧⣋⠞⣳⣿⡄⢻⢻⣿⠀⠆⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠿⣉⣎⣿⡿⣿⡆⢹⢃⣿⡄⣸⠋⠀⠀⠀⠀⠠⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢹⣿⣿⣿⣬⡵⢂⡈⣴⣻⡇⡀⠀⠆⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢼⣻⣿⣾⡙⢧⡅⠘⢫⡟⡄⣇⠰⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡝⣯⠱⠄⠙⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣿⣷⣶⣤⣀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣾⣿⣿⣻⡿⢋⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾⣿⡿⠟⡱⠋⠁⢋⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣉⣉⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⣾⣿⡿⠁⠀⠀⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⡿⠁⠀⠀⠀⣀⣠⣄⣠⣵⣶⣶⣶⣾⡿⠿⠟⠋⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣏⠁⠀⠀⢀⣦⡽⠛⠛⣩⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣬⠋⠀⠀⣸⣿⣿⣿⡿⠀⠀⠀⢀⡴⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠃⠀⠀⢰⡇⠀⠀⢰⣿⡿⠛⠋⠀⠀⣀⠤⢘⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠀⢈⠄⠀⠐⠛⠁⡀⠀⢀⣀⣠⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡂⠀⠀⠀⠘⣰⣿⠿⠉⠉⠉⠛⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⡆⠀⠀⠈⣷⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢻⡀⠀⠀⠈⢧⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⣿⣀⡀⠀⠀⠉⠠⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠘⢿⣿⣦⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⠀⠀⠙⢿⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠂⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⠤⢤⣤⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⣠⣴⣶⣿⠿⠿⠿⢿⣶⣦⣀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢠⣾⣿⠟⠁⣀⠀⠀⠀⠀⠈⠙⢿⣷⡄⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢠⣿⢫⡟⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⠙⣿⣦⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⠟⣻⣧⢴⡇⠀⠀⠀⠀⠀⠀⢧⠈⣿⡆⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠁⠾⠄⠒⠀⠀⠀⠀⠀⠸⣇⠘⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠻⣄⠀⠈⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡀⠀⠈⠳⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⢸⢁⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠷⢤⣀⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⡎⢸⡇⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠉⠀⠀⠀⠀⢻⡄⠀⠀⠀⠁⣿⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⠃⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⡼⠀⠀⠀⣰⠋⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠁⠀⢀⣀⣀⣀⣠⣤⠴⠚⠁⠈⠀⠀⠀⡴⠃⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⡾⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⢀⣀⣠⣤⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣦⣽⣤⣶⣤⣼⣦⣴⣿⣵⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠦⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⣤⣤⣀
⠀⠀⠀⠀⢀⠴⠋⠁⠀⣀⣀⣀⡀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⣀⣀⣀⡀⠀⠉⠻⣦⡀
⠀⠀⠀⠔⠁⠀⢠⡶⠋⠉⠁⠈⠉⠳⣆⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢠⡶⠛⠉⠉⠉⠙⠳⣦⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⠈⢧⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⠘⣷⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣇⠀⠸⣇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢹⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢻⡀⠈⣿
⠀⠀⠀⠀⠀⠹⡛⠲⠤⡤⠤⠤⣄⠀⠀⠹⣌⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠲⠦⠤⠤⠤⣄⠀⠀⠹⣾⡇⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢹⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹⡇⢸⡇
⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⢸⠄⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣠⡀⠀⠀⠀⠀⠀⢸⠆⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠇⠀⠀⠀⠀⣠⠟⠀⠀⣸⠁⣼⠁
⠀⠀⠀⠀⢰⡿⠀⠀⣀⣤⠴⠋⠀⠀⠀⠀⠃⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⣀⣤⠶⠋⠁⠀⠀⠀⠃⢰⡟
⠀⠀⠀⠀⠸⣇⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢦⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠿⢿⣿⣿⣷⡄⠀⠀⠈⠀⣀⠀⣀⣀⠀⢸⣿⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⢀⣿⠛⣿⣿⣦⣰⡀⠘⢿⣷⣄⡉⠃⠈⣿⣿⡿
⠀⠀⠀⠠⠶⣿⣿⣿⣶⣤⣤⠴⡿⠟⠋⣀⡠⠤⣾⠃⢠⣿⣿⣿⣿⣷⣤⣀⡘⠛⠛⠁⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⣿⠀⡀⣾⣿⣿⣿⣦⡀⠀⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣦⣤⣽⣷⣤⣤⣄⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠏⡿⠛⠛⠻⢿⣷⣤⣽⣛⠡⣿⣿⣛⠛⠳⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣾⢿⡏⠀⠀⠈⢆⡔⠀⠀⠐⠒⢠⡀⠉⣷⣤⡤
⠀⠀⠀⠀⠠⠒⠒⠢⠤⣀⡀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣝⣇⠀⣀⠀⢸⡇⢠⡀⢀⠁⠀⢱⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠄⣴⣄⣀⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣽⣼⡿⠃⠀⠻⣄⠑⠒⠊⣠⠞⠀⢀⡟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠉⠉⠙⠒⠦⣌⠻⣿⡿⣿⣆⡀⠀⠀⠀⠉⠉⠉⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⢳⣌⠛⢦⡉⠙⠒⠠⠄⢀⠀⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠙⣆⠀⠙⠳⣄⠀⠀⠀⠀⢄⠠⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠖⠛⠛⠋⠉⠙⠋⠳⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣶⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣟⠛⠛⠒⠒⠒⠒⠚⠛⠻⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠚⠻⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣡⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢇⢃⡞⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠩⢽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⡞⢠⠇⣼⠁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⢐⡊⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⡵⢟⡵⢃⡿⢠⠁⠐⢠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⢩⠔⠉⠰⠋⠡⠎⢠⠀⡌⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠂⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠐⠠⡐⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢌⠀⣿⡀⠀⠀⠀⠐⢄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⢄⠱⢸⡇⠀⠀⠀⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⣿⡿⣿⣿⣦⡄⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠑⢈⣿⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣫⠿⠋⠞⢱⢋⡿⣿⣷⣀⠁⠰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠈⣿
⠀⠀⠀⠀⠀⢀⢀⠀⡄⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⡀⠀⠀⠁⠊⡴⢣⡿⣿⣦⡄⢠⢀⠈⠀⠀⠀⠀⠀⠀⠀⠑⢤⠐⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⠀⠀⠀⠈⠻⣿⣷⣒⡀⠀⠀⠀⠀⠉⠼⢋⠟⣷⣾⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄⣿⠀⠀⠀⠀⠀⢡⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠛⢿⣝⣒⠀⠀⠀⠀⠀⠉⠼⢻⠟⣷⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⣿⠀⠀⠀⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⢿⣅⣀⠀⠀⠀⠀⠀⠉⠰⢉⣿⣄⠀⠀⠀⠀⠀⠀⠀⠢⡙⢨⡿⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠊⡿⣿⣆⠀⠀⠀⠀⠀⠘⠀⠱⣸⡇⠀⠀⠀⠀⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠠⡐⢀⢢⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠫⡙⢷⡉⠖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠁⠌⣼⡏⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢆⠀⠙⣄⡀⠉⠢⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠙⢠⡟⠀⠀⠀⠀⠀⡆⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⣠⣾⣿⣿⡿⢉⢂⣠⣶⣼⣾⠷⠷⢷⣶⣦⣵⣄⣀⠀⡀⢀⣤⠟⠁⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠁⢸⡟
⠀⠀⠀⠁⠀⠂⠀⣠⣶⣿⣿⣿⣿⣷⣱⣾⠟⠉⠀⠀⠀⠀⠀⠀⣀⠠⠈⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠆⢠⣾⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣸⠀⡟⢠⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠁⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠎⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⢸⠀⠀⢀⡠⠟⠁⠀⠀⡀⠀⠀⠀⠀⣦⡾⠁
⠀⠀⠀⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠃⡄⣠⠶⠋⠁⠀⠀⠀⠀⡇⠀⠀⢄⣰⠟
⠀⠀⢀⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠇⠘⢠⡿⠋
⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣾⠋
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠐⢀⣾⠛⠁
⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣦⠈⡀⣄⢀⠀⠀⡘⠀⠐⣠⡾⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⡙⢀⠁⢋⠸⠀⢀⣧⣼⠟⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣷⠀⢇⠓⡘⢰⣤⣶⠿⠛⠉
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢰⢀⠃⣘⣬⡴⠟⠋⠁
⠈⣷⣠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣸⢰⠁⣻⣼⡾⠛⠉
⠀⠈⢷⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⢁⣁⣤⡾⠛⠁
⠀⠀⠀⠙⢷⣤⣀⡀⢰⠀⠘⢀⠀⠠⢁⣘⣤⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠛⠋⠉⠙⠿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠋⠀⠀⠀⠀⢠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣠⣿⣿⡟⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣴⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣰⣿⣿⡟
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇
⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿
⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣤⣄⣀⣀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣦⣙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣷⡂⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣀⣙⣿⣿⣿⣿⣿⣿⣷⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣾⣽⣿⣿⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣈⣉⣉⣹⣍⣟⣙⣋⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⢀⣠⣴⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡿⠋⠁⢸⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠛⠁⠀⠀⠀⠀⠈⠉⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡿⠋⢈⡇⣸⠇⣿⣿⣿⡿⠛⠿⠀⠀⠀⠀⠀⠠⠶⠛⠉⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⣠⢎⡴⣡⠞⢁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⣶⠦⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⢀⠙⠫⣼⡵⣛⣙⠻⠿⠿⠦⠀⠀⠀⠀⠀⠀⠉⢿⣛⠟⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠈⢧⡀⣸⡄⡇⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠄⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢻⢧⣷⡬⠶⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⢀⠤⢖⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡧⠀⠀⠀⠈⠁⣀⢾⡄⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣰⠃⠀⠀⠀⡰⠛⠻⣆⠻⣆⠀⠀⠀⠀⠈⢹⠖⠶⠤⣄⣀⣀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⠏⠀⠀⠀⠀⡇⠀⠀⠹⣦⠙⢷⣄⠀⠀⢠⣷⣄⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⡴⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠳⣦⣉⡛⠛⠛⣡⡿⠳⣄
⠀⠀⠀⠀⠀⢰⣏⣴⠶⡞⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠻⣏⠎⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⠀⠀⣠⣤⣶⣶⣦⣴⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⣾⠛⠻⠛⠛⠛⠻⠟⢿⣿⡿⣦⢄⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢉⢝⣿⣿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣿⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠃⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢂⣱⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢠⣿⣯⣷⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⣿⢿⣿⣿⢿⠆⠀⠀⠀⠀⠀⠀⢸⣿⡷⡛⣤⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⢠⣿⠋⠉⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠙⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠐⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡘⣿⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⠈⡠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⡀⣈⣉⡙⢋⠛⣿⢿⣿⣿⡇⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⣿⣄⣙⠡⣌⡩⠓⢽⡾⣿⣿⣿⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣶⠈⢻⣿⣿⣶⣥⡴⣱⣀⢏⢿⣿⣿⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠚⠀⣹⠀⠀⣿⣿⡿⠛⢷⣾⣿⡻⣾⣿⠏⠀⠀⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢡⣠⠖⡉⠀⠀⣿⠁⠀⠛⢫⣷⣿⡿⠿⠃⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠑⡡⠎⡱⠀⣰⠏⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢄⠖⠁⣼⠋⠀⠀⣾⠁⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡀⠀⢀⠿⠀⠀⠀⠀⠀⠈⠛⣹⣯⡟⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣦⡶⣶⣾⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⣄⢀⠔⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠒⢤⠀⡜
⠀⠀⠏⠀⠀⠄⢀⠀⠤⠀⠀⢀⠀⠀⢀⡇⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠤⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠚⢀⠠⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠇⠀⠀⢸
⠀⠀⠀⢠⠖⠁⠀⡠⠤⡄⢀⣠⠤⠒⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠎⠀⠠⢄⡤⠔⠀⠘⢀⠀⡀⢀⡠⠃
⠀⠀⠀⠊⠀⠀⠀⠅⠀⠀⠈⡆⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣠⠊⠁⠀⠀⠘⠄⠀⠀⠀⠀⠀⠈⠀⠁⠠⠄⢠
⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⡠⠠⠀⠴⠁⠀⠀⠀⠀⠀⠐⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⢁⡒⠢⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠠⣸⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡎⠀⢀⠔⢒⠈⢳⠈⠀⢳⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠠⠀⠂⢂⠠⠐⠒⠐⠠⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠨⠀⢸⠃⠀⢸⠀⠀⠀⠀⠘⠀⢀⠄⠀⠀⠐⠁⠐⠁⠀⢀⠄⠉⣇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠀⡀⠸⡄⠀⠐⠄⠀⠈⠁⠀⢀⡞⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠑⠀⠀⠠⠤⣤⠤⠖⠋⠀⠀⠀⠀⠈⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⠤⢤⠐⠋⠀⠀⠀⠀⢀⢄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢀⠔⠬⡑⠀⠒⠒⠋⠐⠦⠤⢅⣀⣀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠲⠤⠤⣤⣤⣄⣄⣀⣠⠜⠀⠀⢀⡀⠀⡰⡆⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣀⣤⠤⠤⢤⣄⣀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡞⠁⠀⠀⠀⢀⡾⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⢀⠀⠀⠀⠀⡼⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡹⠞⠛⣇⠀⠀⢰⡇⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⢳⣹⡀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠒⠶⠆⠀⠀⠀⣾⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣤⣀⣸⠛⢦⣄⠀⠀⠀⠻⣧⠀⠀⠇⠘⠛⠉⠛⠓⠶⣄⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣍⣸⠓⠾⢷⣦⡀⠀⢙⣠⠴⠶⠶⢤⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡾⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⢠⠞⣡⢖⣛⣛⠲⡌⢷⡀⠀⠀⠀⣿⠀⠀⣸⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠲⠦⢤⣤⣀⣀⠀⣿⢰⡇⡏⠀⠈⣷⢹⠈⡇⠀⠀⢀⡟⠀⢰⠏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠐⠶⢶⣶⣖⣛⠐⣟⠉⠉⠉⠉⠁⠹⡌⢧⣙⠶⢖⣣⠟⣸⠃⠀⢀⡾⠁⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠙⠶⣬⣉⣩⡥⢞⣁⣤⠶⠋⠀⠀⠉⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠙⢻⣧⡀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⢹⣷⡀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⣴⣾⠟⠛⠉⠉⠙⠛⠿⣷⣄⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⢻⣧⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠹⣿⡀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠘⣿⡄⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠐⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⡏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢰⣿⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠋⠀⠀⠀⢀⣤⣾⠏⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢀⣾⡟⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠸⠿⠿⠟⠋⠁⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⢀⣤⣥⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⣠⣾⡏⠀⠀⠀⢀⣤⡿⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⠟⠉⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢰⣾⣦⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⢀⡇⠀⠠⢀⣿⣿⣿⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⡼⠀⠀⠂⢸⣿⣿⣿⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠁⠀⢸⣿⣧⣀⠔⠁⠀⢰⠇⡘⢿⣿⣿⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⢿⣿⠁⠀⣀⣠⣿⢠⠃⠀⢻⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⢿⣿⣾⣾⣿⢇⣾⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠈⢻⣿⣯⣵⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠗⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠁⠄⡀⠀⠀⠀⠀⠘⣿⣷⣆⡀⠀⠀⢻⡆
⠀⠀⠀⠀⢀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠔⠊⠕⠁⠀⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠸⣿⣿⣧⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠘⡆
⠀⠀⠈⠀⠠⣢⠤⡀⣈⢀⠀⠶⠈⠈⠙⢈⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠀⠀⢀⣠⣴⣿⣿⣿⣿⡷⠀⠀⠀⢻
⠐⠉⠀⠰⣿⢀⡴⠾⠈⢄⡀⠀⠀⠀⠇⢀⠀⠈⢀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠎⣣⠾⢴⡟⠀⡈⠂⢠⠄⠈⠂⠸⠠⣲⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠂⢀
⠁⠀⢀⡊⠀⠴⡄⢹⠀⠒⡠⠈⣌⠢⣄⢖⠀⣧⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣠⠆⠀⠄
⠀⠀⠸⡇⡀⠤⠤⢞⢀⡜⠀⠀⠇⡸⠀⢁⡀⣸⣿⠿⠛⠉⠁⠀⠀⠀⠀⠐⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⡏⠀⠘
⠀⠀⠀⠙⢄⣀⣊⡤⠏⠁⠀⢠⠀⡰⠉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⠝⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡰⡋⢂⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠔⠋⠄⡠⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⣿⡿
⠀⠀⠈⠀⠀⠀⠐⠊⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠃⠀⠀⡄
⠀⠀⠀⠀⣀⠎⠀⠀⠀⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠔⠃
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡸⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡰⠡⢠⡶⢶⣿⡀
⠀⠀⠀⠀⠀⠀⢈⣛⡛⣦⠘⠀⠶⠄⡁⠀⠀⠀⠀⠀⢀⣀⢀⢀⡀⣀⣀⣀⣤⣴⠞⠀⠀⠀⣿⣿⠘⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠘⣷⣀⣰⣞⢀⠀⡶⢿⣿⣾⠗⢒⠬⠌⡉⠙⠹⠛⠉⠁⠀⠀⠀⢰⠿⠿⠁⢿⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠉⡁⠄⠀⢀⣤⣿⡟⠋⠉⠓⠢⡀⢤⠁⠂⣄⠀⠀⠀⠀⠀⣺⣷⡀⠀⣈
⠀⠀⠀⠀⠀⠐⠻⡿⠋⠀⠀⠁⠀⢀⡴⢟⣿⣿⣧⠀⢀⣤⠄⡀⠾⠳⣄⠘⡿⠲⠦⣤⡾⢿⣿⠇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡊⠀⡀⣠⢾⡿⠁⣾⣿⣿⣿⠀⣛⡟⡃⠁⠀⠀⠀⠙⢧⣀⣴⣿⣅⠈⢁⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢈⣴⢏⠿⢓⣾⣿⡇⣹⢩⢸⣿⣟⣿⡄⠸⣆⠀⠀⠀⠘⠟⠛⠉⢀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢳⣿⢾⡆⢸⣿⣿⡿⢣⡇⢸⡿⢸⡿⣝⠆⠈⣻⣦⣤⠤⠤⠤⠤⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠘⣿⢹⡇⢸⣿⠊⣡⠏⡇⣿⢇⡗⢷⢙⠧⠀⣺⠋⠀⠀⠀⢒⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠈⠃⢸⣿⣿⢻⡾⢣⢻⢸⡍⢹⣀⣿⢠⡟⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⣄⠿⣿⣧⢻⣟⢾⡇⣇⢸⣽⡇⣼⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⡇⠀⠀⠠⠐⢯⡻⣿⣆⡅⢿⣼⢻⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠁⡀⢻⣎⢻⠇⡟⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢉⡇⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⣴⣶⣿⣿⡿⢛⠋⠛⢿⢿⣿⣶⣤⣀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠈⠂⢸⣿⣿⣿⣿⣧⡀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⡰⣫⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⡏⠿⠁⠀⠀⠀⠀⠀⣠⢾⣹⣿⣿⣿⡿⠟⣿⡆⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡸⣿⠙⠀⠀⠀⠀⢀⣴⠎⣣⣽⣿⣿⡿⠟⠁⢠⣿⡇⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠋⠀⠀⠀⠀⣰⠛⠮⣿⠿⠋⠉⠉⠀⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡟⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⢟⡿⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣴⣿⢿⡻⡿⠋⣉⣴⡿⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⡀⠀⠀⠀⢀⣾⡿⡟⡁⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣷⠃⠀⠀⠀⣼⡧⡘⠀⠀⠀⠰⢾⣷⣷⣶⣄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⡼⠀⠀⠀⢿⡇⠀⠀⠀⢠⠎⠀⢿⢯⣿⣿⣧
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢻⣧⡁⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠻⣧⡫⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠈⢿⣯⣷⣄⠀⠀⢰⣆⣀⠰⣤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠈⠻⣿⣇⡀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠈⠻⣿⡄⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠈⠻⠀⠀⠀⠙⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣕⠦⣀⡀⠀⠀⠀⠀⢀⢄⣤⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣾⣭⣭⣭⣿⣶⣿⣿⡟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⢻⣻⢹⣿⡌⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⢻⣿⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠹⣿⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⠰⢶⣶⣤⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠋⠉⠛⠉⠛⠁⠀⠀⠀⠈⠙⠻⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⡀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⢀⣬⡃⢀⡄⠀⠀⠀⠀⠀⢀⡀
⠀⣀⣤⡀⣀⡐⠒⠠⢥⡀⠀⠀⠀⢠⣶⠿⠯⠚⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣦⠀⠀⠀⢀⣠⡥⠖⢒⣆⣤⣤⣄⡀
⢰⣿⣿⣿⣿⣿⣿⣷⣶⡿⣦⡐⠚⢁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⡙⠖⣰⡟⢛⣿⣷⣿⣿⡿⢿⣿⡧
⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃⠈⢻⡆⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡋⢈⣻⣿⣿⣿⠃⢠⣾⣿⠃
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣶⣼⣶⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⢈⡛⢿⣿⡿⣿⣿⠿⠻⡿⣿⠿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⢰⠿⡿⠻⢿⣿⢿⣿⣿⠟⣋
⠀⠀⠀⠀⠀⠈⠙⠒⠋⠉⠁⠀⢰⣷⣿⡇⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢃⣿⣶⣷⠀⠀⠉⠙⠑⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣙⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⠟⢁⣼⣿⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣹⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣭⣷⣫⣥⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⢸⣿⣿⣿⡿⠀⠀⠀⠀⢸⣽⣿⣁⠉⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣡⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣧⣀⠾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⡿⢳⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⣀⣀⡀⠀⠀⢀⣀⣀⣹⣀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⣁⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢿⢿⡎⠀⠀⠀⠫⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⣤⣤⣤⣤⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡉⠙⣦⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⠏⠐⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢱⡄⠀⣴⠃⠀⠂⢀⠌⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣴⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢀⡇⠠⠀⠀⡌⠀⢠⡴⠋⠩⠙⠛⢿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡇⡼⠐⠀⠀⠀⠀⠀⡞⠰⠀⠂⠘⢷⡀⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⡅⠀⠀⠈⣿⠁⠀⠀⠀⡘⠀⢀⡇⢶⠀⠀⠀⠀⠂⠀⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆⠀⠀⡇⡀⠀⠀⡏⢀⠀⠀⢠⢁⠀⢸⡇⠀⣠⠴⡄⠀⣸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⢰⣧⠟⠀⢀⠛⡄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⣿⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢆⠀⠀⠈⠀⣹⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⡀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⢳⡈⠳⠦⣄⢰⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⢧⠀⡇⠀⠀⠀⠀⠀⢀⠀⠠⠴⠀⠀⠂⠂⢻⣄⠀⠀⠁⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠘⡄⡇⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⢹⠛⢷⣀⣀⣲⠡⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡀⠄⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠁⠀⠀⠈⠉⠉⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡃⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡗⠀⠄⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠠⠀⠤⠄⠀⠀⠠⡄⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠉⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣠⠖⠒⠛⢯⡙⠒⠲⢤⡀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠋⠀⠙⠢⣀⠀⠈⢢⡀⠀⠙⢆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⠀⠢⢤⣀⡈⢳⡀⠀⠹⡄⠀⠈⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠢⣀⠀⠀⠀⠀⠉⠀⠀⠀⠹⡀⠀⢱⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢧⠀⠉⠑⠒⠒⠒⠢⣄⠀⠀⢧⠀⡸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠣⣄⡀⠀⠀⠀⠀⠘⡆⠀⢸⢀⠇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠐⢤⠉⠉⠉⠑⠀⠀⣹⠀⠀⡞⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡠⠂⡼⠀⠀⠀⠀⠀⣠⠃⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡴⠋⣠⠞⠀⠀⡀⠤⠔⠊⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠱⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⡆⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣠⠤⠤⠄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠻⣄⠀⠀⠀⠸⡇
⠀⠀⠀⢠⠃⢠⣶⣶⣄⠀⠀⠀⠙⢷⠂⠀⢀⣷
⠀⠀⠀⡏⢠⣿⠋⠀⠹⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢸⢀⣿⡏⠀⠀⢸⣿⠀⠀⢀⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠇⢸⣿⠁⠀⣀⣼⣿⠀⠀⡞⠁⠀⠀⠀⣿⡧⠈⣦⣀⣀⣠⣴⡖⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠷⠁⢁⡄
⠀⢸⠀⣿⡏⠀⠀⠙⠻⣿⠀⠀⡇⠀⠀⠀⠀⠻⣿⠀⠈⠻⢿⣿⣷⣦⣤⣴⣿⣿⠿⠿⠛⠒⠓⠿⠿⣶⣷⠞⢒⠚⠓⢉⣀⠀⣿⣯⣯⣠⣾⣅⣀
⠀⢸⣰⣿⠁⠀⣠⠀⢀⣾⣆⠠⡇⠀⠀⠀⠀⠘⣿⡀⠀⣤⣲⣶⣿⣶⠏⢽⡻⠷⣶⣦⣀⡀⠀⢀⠀⡩⢝⡂⠄⣠⠾⠉⠉⢹⣿⣿⣿⣿⣿⣿⡋
⠀⢸⣿⡏⠀⣾⣧⢀⣾⣿⣿⠀⠁⠀⠀⠀⠀⠀⠈⠙⠖⢶⣮⣭⣽⣿⣧⠾⣯⣟⣻⠿⢯⣽⣿⠻⣦⣤⣤⣶⣿⣿⢶⡶⣾⡿⠃⣻⣿⣿⣿⣿⣶⡶⢂⣀
⠀⠀⣿⠃⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠗⠸⣿⣿⣿⣟⡑⣭⣃⡾⢿⣿⣿⣿⣷⠀⠀⠈⠀⠀⠈⣿⣿⣿⣿⢟⣴⣿⠉
⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⡀⠀⠢⢴⣾⣿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠻⠿⢟⣭⣉⣁⠙⣻⣿⣻⣿⡀⠐⢬⣙⠛⠀⠀⠀⠀⠀⠀⣴⣾⢺⢏⣾⣿⡏
⠀⠀⠈⣇⡀⢿⣿⣿⣿⣿⣿⡧⠀⠂⠈⢿⣿⣾⣻⣿⠿⢫⠉⣀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣦⣤⡙⣦⣝⢷⣆⡀⠉⠓⠤⣀⠀⠀⠀⠮⠓⠍⣼⣿⣿⠇
⠀⠀⠀⢳⡀⠀⢻⡿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⠏⠁⠀⠘⣿⣿⣿⣿⣿⡛⠉⠈⡈⠛⡟⢿⣿⣝⣿⣿⣷⣄⡙⢷⣄⠀⠀⠉⠳⢶⣦⣤⣨⣿⣿⣯⡅
⠀⠀⠀⠘⡅⠀⠈⠁⠈⢻⣿⣿⣷⣀⣤⡾⠛⠀⠀⢂⣤⣾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠆⠖⠩⠋⠈⠳⣿⣿⣿⣿⣶⣌⠙⢲⣄⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⣿⣿⡿⢋⡁⠀⣀⣤⣾⣿⣿⣿⣿⡿⠋⠉⠳⣄⡁⠀⢀⡇⡀⠊⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣦⡈⠻⢦⣀⠀⣻⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠈⡄⠀⠀⠁⠐⠋⠛⠀⠈⠡⡶⠙⠃⠀⠈⣩⣩⣽⠃⠀⠀⠀⠘⡿⠳⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠻⣿⣿⣿⣿⡟⠁⠀⠐
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢱⣾⠟⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣟⣿⣦⡈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣤⣹⡿
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣴⡟⠛⠛⠛⠛⠛⠛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢀⣾⢋⠀⠀⠀⠀⠀⠀⠀⠠⠛⡿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣿⠿⠿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⣿⠏⠉⠐⠃⠀⠀⠀⠀⠀⠀⠻⠇⢋⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃⢿⣷⣤⣤⣤⣄⣀⣤⣀⠀⠀⠀⠈⠑⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠈⠉⣉⡙⠋⠻⣿⣿⣿⡇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢿⣷⣌⡉⢠⡍⡻⢧⣿⣿⣿⡇⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠀⢻⣿⣿⣷⡔⢳⣼⣜⣿⣿⠃⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⢀⠼⠀⠀⡿⠛⠠⠹⣟⣳⣽⡿⠁⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡡⢂⡄⣠⠇⠀⢠⡾⠛⠛⠉⠀⠀⠀⢰⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠃⣴⠋⠀⢰⡏⠀⠀⠀⠀⣀⣤⣴⣿⠿⠃⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢀⡼⠃⠀⠀⠀⠀⠙⣩⣯⠏⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠻⠿⠷⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⣀⣄⣠⣤⣨⣉⣙⠒⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⣵⠋⠩⠞⠉⠚⠝⠫⠛⠻⢞⠷⢶⣬⣭⣷⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡿⣦⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠛⠻⠛⠿⠛⠿⠻⠛⠋⠁⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⢝⣻⣷⢝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠼⢦⠞⠛⠻⠗⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢛⠉⠁⠀⠀⠕⡩⣪⠕⣫⠖⣣⠀⣈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⠔⠀⠀⠀⠀⠀⡫⢊⡥⣊⠔⢋⡴⢊⠡⠚⣡⠂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⡊⠀⠀⠀⠀⠀⢈⠔⡣⠚⡡⠚⡡⠖⡡⠔⢋⡤⢚⡁⠀⠹⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡭⠀⠀⠀⠀⠀⠀⡡⢊⡴⢊⠴⢊⠴⢋⡤⠚⣡⠔⣉⠤⣂⠄⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠤⠂⠀⠀⠀⠀⠀⠀⠴⣫⠔⡡⠚⡡⠖⣡⠖⡩⠔⣋⠵⠊⣡⠔⠀⢻⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢐⡫⠀⠀⠀⡀⠀⠀⠀⢚⡵⣋⠔⣉⠔⣫⠔⣉⠔⢊⡠⠒⡩⠔⠋⣠⠸⣟⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⡨⠖⠀⠀⠈⢓⠀⠀⠀⡡⢚⠵⢊⠔⢊⠴⢊⡠⠚⡡⠔⢉⡠⠔⡩⢔⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡠⡊⠀⠀⢐⡕⡂⠀⠀⠔⡡⠚⡡⢚⡡⠒⣡⠔⢉⡠⣒⣉⣴⣩⣖⡥⡂⢹⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠚⠓⠒⠒⠂⠮⣐⡀⠀⢋⠔⡉⠔⣉⠔⢉⠤⢂⡕⠋⠀⠀⣀⣀⡈⠉⠳⣝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣤⣴⠞⣩⣋⠶⠀⠈⠑⣄⡡⢊⠤⢊⡥⢊⡡⢺⠋⠀⡴⡪⢝⡫⢜⣹⣷⣄⠘⣶⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣼⠟⡡⣪⠞⡡⠂⠀⠀⠀⠘⣖⡩⠒⡡⢒⠥⢚⠇⢀⡞⡫⠒⡩⣶⢟⠝⣩⢜⡇⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⢞⠟⡡⢊⡴⠀⠀⠀⠀⠀⢸⡤⡪⢐⠥⢊⣹⠀⢸⣫⢔⠽⠊⠐⠁⠊⠕⣽⡇⢸⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢷⣭⢊⡴⣫⢞⠄⠀⠀⠂⠀⠀⣏⠔⡡⠊⡡⠾⠀⡈⠗⠁⠀⠀⠀⠀⠀⠀⡿⢁⣟⡥⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⢧⣔⡕⣫⠆⠀⠀⠀⠀⠀⠵⢋⠔⢋⡴⢚⠇⠀⠀⠀⠀⠀⢀⡠⠀⠀⢀⡎⠤⣒⠪⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣈⠉⠁⠁⠀⠀⠀⠀⠀⠘⢡⠞⣡⠞⣡⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⢐⡩⠔⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠍⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡠⠚⡡⠞⣁⠔⡀⠀⠀⠰⢶⠯⢒⠩⠕⣂⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⡀⠀⠀⡰⠛⠉⠙⠀⠀⠈⠠⢊⡤⢾⣵⣊⠴⣊⠔⠁⣾⡸⢝⣊⠥⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⣧⠐⡄⠀⢠⢚⣾⣳⠀⠀⢀⠀⣩⡶⢧⣌⡡⢚⡡⠊⡽⣛⣖⣩⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣘⢧⡈⠀⠈⠻⠿⠋⠀⠀⡫⠀⠠⢿⣿⣧⠔⢋⠤⡼⢿⣯⠟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠒⠶⠀⠤⠤⣭⣷⣀⠀⠀⠀⠀⠀⠀⠃⠀⠰⢪⢴⠷⢚⣧⣼⠶⣻⡶⠖⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⡀⢀⣩⣟⣏⣙⣒⠂⠀⠖⢒⢲⠀⠰⣎⢉⣭⢯⡤⣾⠷⠗⠒⠶⠶⠖⠒⠂⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢉⣤⠿⣤⡔⠂⠻⠿⠃⣀⠒⣲⡾⢷⠷⢾⡴⠦⢤⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠐⠊⣁⠝⠒⣧⠤⣤⢖⡾⡿⢛⡥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠊⠵⠋⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣏⠉⠉⠉⠉⠛⠿⣶⣄⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠸⣿⡆⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⢻⣿⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡶⠾⠿⠿⠷⣶⣦⣄⡀⠀⢀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣟⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼⡿⠁⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠋⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⣠⣿⠟⠁
⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣴⣿⠏
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⡿⠃
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣴⣿⠟⠁
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠂⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠉
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠏
⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠄⠠⣧⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣾⡆⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⣿⡴⠊⠁⣄⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠟⠁⠶⢐⢿⢁⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡁⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⣿⣿⠌⣴⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣾⠿⠧⠀⠀⢰⣷⠾⢇⣀⣿⠛⠚⢧⠀⣀⣀⣀⣀⣀⡀⠀⠀⣰⡿⠏⣿⣿⡇⣸⣾
⠀⠀⠀⠀⢀⡂⠀⣸⠟⢁⠔⠒⢄⠀⠸⠟⠂⠈⠉⣴⣾⣿⣿⠿⠟⠛⠛⠋⠛⠻⠿⣿⠋⠁⢀⣾⡿⠰⢏⡏
⠀⠀⠀⣰⠃⠀⠀⣿⡀⢸⠰⠤⠜⠀⠀⢀⡀⠀⣾⣿⣿⣿⠏⢀⡀⠀⢀⠐⠀⠀⠀⢠⣠⡖⠉⠉⠀⢀⣾⠃
⠀⠀⠈⠁⠀⠀⢠⠹⣷⣄⠉⠓⠉⠀⣀⣼⡷⠀⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⢠⣾⣿
⠀⠀⠀⠀⠀⢠⠃⠀⠈⡙⠿⠿⠿⠛⠉⠀⠔⠂⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠡⣄⣠⡴⠏⢸⡿
⠀⠀⠀⠀⢠⠋⠀⣀⠞⠁⠐⢤⢄⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⡴⠁⠀⡀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠷⣤⠾⠛
⠀⠀⠀⠊
⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⢀⣀⡀⠀⠀⡇⠀⠀⠀⢣⠀⢰
⠀⠀⡠⠊⠉⠉⠉⠙⢦⢻⠀⠀⠀⠸⡄⠈⣇
⠀⢰⠃⣠⣶⣄⢠⠖⠚⣿⠀⠀⠀⠀⢧⠀⠸⡄
⠀⢸⠀⠁⠃⠀⢻⡀⠀⢸⡁⠀⠀⠀⢸⡄⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣀⢀
⠀⠘⡇⠀⠀⠀⣉⣽⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⡎⡞⠁⢠
⠀⠀⣧⠇⠀⠊⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠒⠒⢚⠉⠉⠀⡠⢐⠚⠑⣝⠢⢄⡜
⠀⠀⡟⠀⠀⠀⠀⢀⡷⢄⠀⠀⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⡄⠙⣮⠄⠀⠘⠂⠈⢧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠒⠛⠲⠦⢬⣉⡓⠲⠤⠤⢄⣀⣀⣀⣠⠤⠤⠤⠴⠒⠂⠧⣀⢧⢣⠈⢲⠮⣝⡄⢀⠞⠉⠉⠉⠐⢶⠂⠀⠒⠒⢀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⠀⠸⠉⠪⡳⡑⠀⠋⡟⠁⠀⢸⠎⣁⣉⠑⠀⠀⢣⡔⠈⠁
⠀⠀⢹⡀⠀⣄⠀⠀⣠⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢱⢱⠀⠀⢇⢒⠀⠸⡜⠄⢀⣆⠄⠀⠸
⠀⠀⠀⢳⡀⠀⢳⠐⡅⠀⠈⠙⠲⠤⢤⣤⡤⠤⠔⣒⠫⠭⠒⠒⠛⠃⠘⠓⠺⠭⢷⣄⣚⡴⠀⠀⠀⠁⠲⡦⢬⣉⣉⣁⣀⠔⠁
⠀⠀⠀⠈⣧⠀⠸⢄⠹⡀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠈⠢⣀
⠀⠀⠀⠀⢟⢦⠐⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠣⡂
⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠡⡀
⠀⠀⠀⠀⠀⢧⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠳⢤⣤⡬⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⠛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣤⡀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠋⠉⠙⠛⢿⣦⡀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠰⢿⣷⣄⠀⠙⢿⡄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠈⠻⣷⡀⠈⢷⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡉⠉⠛⠷⣦⠀⠀⠀⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⠳⡄⠀⠀⠈⢳⠀⠀⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠈⠀⠀⠀⠀⠃⠘⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⡴⠟⠁⠀⠉⠉⠁⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤⣴⠿⠋


⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣻⣿⣟⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⣠⣾⣿⠿⠛⠛⠛⠛⠻⠿⣿⣿⣦⣄⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣍⠛⠿⢿⣿⣷⣤⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⡀⠈⠻⣿⣿⣦⣀⠈⠻⣿⣷⡄⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣄⠈⢿⣿⣿⣷⠀⢹⣿⣧⠀⠀⠀⢰⣿⠇⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠛⣿⣿⡆⠈⢿⡿⠏⠀⢸⣿⡿⠀⠀⢀⣿⡏⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠻⠟⢁⣀⣀⣀⣀⣤⣿⣿⠇⠀⢀⣾⠟⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⢀⣼⣿⡿⠿⣿⣿⠿⠟⠋⠀⢀⣾⠏⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠉⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⠀⠀⠀⣀⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠲⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣀⣤⣀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠊⠙⠋⢀⣽⠟⠳⣄⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢰⠁⠀⠀⣠⠋⠀⢀⡜⠈⢳⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣏⢠⢀⠜⠉⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃⢡⠋⠀⠀⠀⢀⣠⣤⠴⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⢢⡏⠀⢀⡴⠋⠉⠡⠀⠐⠁⡺⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⠀⢠⡟⢾⠁⠀⠀⢀⡤⠊⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⣿⡀⢸⣇⡄⠀⡶⠛⠁⢤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⣧⠘⢿⠟⡀⠀⠀⠀⣼⠛⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢷⡳⡌⠳⢾⡦⠀⠀⢻⣶⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢳⣌⠢⣀⠉⠀⣀⣀⡈⠻⠟⣦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠙⣄⣌⡂⢀⡏⠈⠙⢶⡴⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⣤⡶⠞⠛⠋⠉⠉⠁⠀⠀⠀⠈⠉⠉⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⢀⣼⠋⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⢿⡄
⢀⡾⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⣷
⠛⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣀⣀⣀⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣄⣀⣉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠈⠉⠉⠙⠛⠛⠷⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠛⠷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠐⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡠⠂⠀⠙⢦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⡴⠒⠤⠤⣄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠤⠀⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣰⠳⢤⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠓⠠⡀⠈⠁⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠓⠢⠤⠴⠃⠀⠀⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣠⣤⣤⣤⣀⠘⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠘⢿⣿⣿⣿⡿⠀⠸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠔⠁⢐⡀⠀⠀⡇⠀⠀⠀⠙⡟⠉⠀⠀⠀⡇⠀⣀⢂⠑⠜⠢⢄⣀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢘⠀⡈⠁⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠈⡀⢘⠀⠀⠀⠀⠉⠁⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠀⢀⣇⠀⡀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣉⠁⢈⣉⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣟⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣿⣟⣛⣟⣛⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠖⠬⢍⡉⠁⠒⠠⠄⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣢⠟⠁⠀⠀⠀⠀⠈⠑⠒⠀⠀⣠⠆⠄⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣨⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣉⠤⡤⠄⠠⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠄⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠖⢋⠀⠀⡇⠀⢠⠁⠀⡘⠲⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡴⡇⠀⠸⠀⠀⡇⠀⠘⠀⢠⠁⢀⢷⠀⠠⠀⠀⠀⢠⠟⠋⢉⡩⠝⠳⣄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠎⡼⠀⠰⡀⠀⡇⠀⠃⠀⡄⠀⡆⢀⠎⡘⠀⢢⣤⠤⠖⡫⠔⠈⠀⠀⣀⡀⠼⢦⠈⡄
⠀⠀⠀⠀⠀⠀⡘⢰⠛⠄⠀⠡⠀⠸⠀⢰⠀⠇⢀⠀⡜⡰⠁⠀⣰⡣⠒⠁⡀⠄⠐⠈⠁⠀⠀⠀⠈⣧⠐
⠀⠀⠀⠀⠀⠀⠁⡾⡀⠈⠢⡀⠱⡀⠁⠀⠀⠀⠘⢀⡗⠁⠀⢀⠋⠠⠀⢁⠀⠀⠀⠂⠈⠁⠁⠀⠀⠘⡆⡃
⠀⠀⠀⠀⠀⢠⠀⡇⠈⠢⡀⠑⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠖⠠⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢀
⠀⠀⠀⠀⠀⢸⠀⡏⠒⠄⡈⠑⠤⡑⠢⣐⢄⣀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⣽⢸
⠀⠀⠀⠀⠀⠸⠀⡗⠢⠄⣀⠁⠒⠤⠁⣒⠡⢾⡇⠄⠀⠀⣿⣿⡀⠀⠀⢀⠂⠠⠀⢀⠈⠀⠂⠠⢀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⡀⢳⠠⠤⠄⢀⣉⣁⣐⠂⠠⠛⢿⣄⡀⣠⣏⠛⠂⠀⢅⠂⢌⠐⠠⢀⠈⠐⠠⢀⠀⢈⡇⠄
⠀⠀⠀⠀⠀⠀⢡⠘⡦⠤⠄⠀⠀⠀⠒⢀⠨⢀⠕⡉⡉⠉⠀⠀⠈⠂⠠⡈⠢⠈⠢⡀⠁⠢⡀⠀⠁⡾⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢂⠹⡄⠀⠒⠀⢀⡁⠄⠊⡠⠊⠔⡐⠘⠀⠐⠀⠀⢂⠀⠄⠐⢄⠈⠢⡀⠀⠁⣴⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠙⣆⠂⠈⠀⢀⠄⠊⡠⠊⡐⠁⠆⠀⠀⠀⠀⠀⠀⠈⠄⠀⢢⠀⠈⢀⡴⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠳⣔⠈⠀⡠⠊⢀⠄⠀⠌⠀⠃⠀⠀⠀⠀⠐⠀⠘⡀⠀⢡⡴⠋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠓⢧⣀⡔⠁⢀⠌⠀⡈⠀⠘⠀⠀⠀⠀⡀⣀⡴⠚⢉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢈⡙⠒⠶⠤⠴⠤⠤⠦⠤⠶⠒⠛⣉⠠⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⣠⣾⣿⡿⠿⠿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠻⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⣀⣀⡀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⡶⠟⠛⠛⠛⢷⣄⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⣤⠾⠛⠋⠙⠛⢶⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢰⡟⠀⠀⣼⠟⠀⠀⠀⠀⣀⣀⠙⣷⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣾⠃⠀⠀⠀⢀⣀⠀⠹⣆⠀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡿⠀⠀⢠⡾⠛⠋⠛⢻⣿⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇⠀⠀⣠⠞⠋⠉⠙⢷⣿⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠀⡟⠀⠀⠀⡤⢤⣿⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡇⠀⢰⡏⠀⠀⠀⠀⣸⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣇⠀⠘⣿⠀⠀⠀⠀⢀⣠⣿⡟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⣇⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠹⣧⠀⠀⢰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢻⣄⠀⠀⣰⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠘⠧⠀⠘⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠹⠆⠀⠘⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠉⠉⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠉⠉⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠷⠄⣿⠳⣤⣴⡟⠷⠉⢻⣴⣯⣿⣼⣧⡶⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡆⢠⢦⡀⠀⣤⣄⣤⣍⠀⣶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠞⠈⠛⠀⠉⠀⠠⠾⠃⠈⠉⠁⠀⠀⡀⢠⡄⡀⠤⠦⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⢰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠉⠉⡠⠀⠀⠀⠀⠀⠠⡀⠀⡀⠀⢀⡟⢸⢸
⠀⠀⠀⠀⢇⠀⢠⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⡀⠀⣒⠁⡀⠀⠀⠀⠀⣯⣹⡵⠤⠠⠊⣄⣠⠀⠀⠀⠀⠰⣧⡤⠽⠶⣼⡇⢸⢠
⠀⠐⠀⢀⠸⠂⠸⠀⠀⠈⠇⠀⠀⡐⠀⠚⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡧⠉⠀⠀⠀⠀⠈⢳⢠⠀⠀⠀⠆⠁⠀⠀⠀⠙⠙⠳⢲
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠃⡐⠀⠀⣷⠀⠀⠀⠀⠀⡀⠀⠀⠦⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⡄⠀⠀⠀⠀⢁⠀⠀⢀⠌⠀⠀⠡⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⠶⠒⠀⠈⠐⠀⠃⠀⣀⠀⡀⠈⠑⠲⠖⠚⠉⠁⡀⠀⠀⠀⠠⠁⠒⠦⠤⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠶⣴⡤⠖⠀⠀⠀⠀⠀⠒⠤⣤⡀⢀⣀⣠⡴⢛⡆⠀⠀⠀⠀⡇⠈⠢⠀⠀⠠⠖⠀⠂⠈⡀
⠀⠀⠀⣀⣀⣠⡔⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂⢄⡇⠀⠀⠀⠠⠘⠁⠀⠀⠀⠀⠀⠈⠱⠠⡇
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⢀⣀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠙⢇⠀⠀⢠⠀⠀⠀⢠⡆⢀⠔⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠄⣠⣀⡤⠜⠛⠋⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢠⣿⠂⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢀⣠⡶⠿⠻⢷⣦⡀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢠⣾⠋⠀⠀⠀⠀⠹⣷⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢀⣴⠄⠀⢸⡟⢀⣿⠃⢀⣤⣶⣶⡶⣶⣿⠀⠸⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣾⠏⠀⠀⢸⡇⣸⡏⢠⣿⠋⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⣿⡇⢸⡇⠀⠀⠈⠋⠁⣠⣄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣧⢻⣇⠘⣿⣄⡀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡆⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⣿⡆⠀⠀⠈⣿⡌⢿⣄⠈⠙⠻⣷⣄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠸⣿⡀⠀⠀⠹⣷⡀⠙⠿⠆⠀⠈⢿⡆⠹⠗
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠈⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆




⣿⣿⠿⢿⡿⢿⠿⢿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⣿⡟⢀
⠀⠀⠀⠀⠀⣾⣿⣿⣄⡄
⠀⠀⠀⠀⠀⣿⢿⡟⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢙⣼⡷⣿⣿⣆
⠀⠀⠀⠀⠀⠰⣹⣷⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⣯⢰⠨⡆
⠀⠀⠀⠀⠀⠀⠸⢈⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⢀⣀⣀⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⡟⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠟⠉⠋⢰⣴⣤⢰⡤⣒⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⡇⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠐⠄⢀⣀⣠⣤⣤⣤⣼⣶⣎⣇⠸⣵⣅⠈⣾⡅⠀⠀⠈⣦⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠹⢳⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠟⠛⢻⣿⡭⠛⠂⠷⠻⢷⣬⡻⢶⣤⠾⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢩⢱⠔⣿⣿⣿⣿⣿⣿⡀⢀⣠⣴⣾⣿⣯⠟⠛⠋⠁⠀⠀⠀⡈⢻⣦⣍⣡⣷⡀⠀⠉⠑⠦⣐⠀⠐⢤⣈⠙⠻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢋⠟⡊⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⠀⢀⠀⠐⠂⠀⠀⣀⣤⣤⣤⣤⣤⣇⢠⣤⣄⡀⠈⠳⣤⡀⠈⠙⠢⡍⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⡀⠁⠙⢻⣿⣿⣿⣿⠋⠀⠀⡠⠊⠀⠀⠀⢀⣴⣿⣿⣿⡟⠙⢽⡿⣿⠀⠘⠉⠉⠙⠠⠀⠹⣦⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣏⠀⠀⠀⠈⠻⠛⠉⠁⠀⠊⠀⠀⠀⠀⡸⣋⢊⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣀⡀⠤⠔⠀⠀⠀⠀⢀⠀⠈⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⠸⡳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇⠀⠀⠁⠀⠀⡐⠈⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢊⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠄⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠙⢦⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣿⣿⣿⣆⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣿⣿⡿⠟⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⡒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠁⠀⠋⠀⠀⠍⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⡀⢶⠼⢥⡒⣀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠀⠀⠀⠈⢩⡄⢈⡠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠰⣀⠐⠂⡀⠀⠀⠀⠶⠀⠬⠁⠀⠐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢍⠠⠀⠁⠚⠀⠁⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠉⠈⠁⣀⠀⠴⠞⠀⠀⢤⠄⠀⠈⢀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠈⠃⠁⠀⠀⠀⠈⠀⠀⠈⣋⡀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢀⠈⠠⠌⠰⠈⢆⠀⡄⠀⠈⠁⠀⠀⠉⠀⠂⢀⠂⠐⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣤⠈⠀⠃⠀⠒⢀⡂⠀⠀⡴⢓⠒⠀⠠⢀⠀⠈⠀⠠⠀⠀⠐⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠐⣄⠀⠄⢈⠀⠰⠂⠀⠀⠀⢀⡉⠤⠘⡄⠀⣄⠀⠦⠒⠀⠃⠓⠈⣄⢙⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠀⠆⢈⠀⠐⠆⡀⠀⠀⢈⠁⠀⠀⠠⠆⠠⠀⣛⠘⢁⠐⠂⢀⠰⠂⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠆⢈⠀⠀⠆⡀⠀⠀⢀⠀⠁⠀⠐⠆⡀⢀⠈⠀⠄⠠⠀⠀⣀⠘⠀⠀⠀⠀⠈⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠆⠀⠂⢤⠀⠀⠀⢄⡀⠰⢀⡀⠀⠀⠀⠰⠁⡄⠀⠀⠀⠐⠀⠀⠠⠈⡉⠀⠨⠀⢄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠀⠀⠀⠀⢀⠐⠂⠀⠰⠂⠀⠐⠀⠀⠁⠸⡄⣀⠐⢀⠁⠐⠆⠀⠀⠀⡣⠀⠃⠐⢀⡠⠄⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠐⡈⠁⡄⢀⡀⠘⠀⠠⠠⠄⠀⠁⠂⣤⠉⠐⠀⠒⠀⠀⢀⠈⠀⠀⠈⢄⠰⠀⠀⠀⠃⠈⣀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠙⠒⠀⠀⠔⠂⠤⠀⠶⠀⠠⠔⠂⠠⠤⠠⠉⠀⠀⠈⢨⠀⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⢁⠀⠀⠂
⠀⠀⠀⠀⠀⣞⢙⡆⠆⠀⠠⠀⠈⠀⠀⠀⠐⠀⢘⠂⢤⠀⠒⠰⢂⡀⠃⡄⠀⠄⠒⠀⠁⡄⠐⠂⡀⠂⣀⠀⠀⢡⡌⠀⠐⠀⠀⣄
⠀⡐⢀⠀⠉⠀⠉⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠜⠁⠒⠀⠀⠀⠀⠀⠰⠀⢀⠓⠰⠀⠈⠀⡀⠁⠀⠁⠀⠀⠀⠁⠀⠂⠀⠀⠀⠤⠀⠀⠀⠀⠄
⠀⠀⢊⠀⢀⠀⢀⡈⠁⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⠁⠈⠀⠀⠄⠀⠀⠄⠐⠀⡈⠀⠀⠀⠀⠀⠀⣠⠀⠂⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠈⠀⠀⠂⠡⠈⠉⠁⠂⠀⢙⠲⠪⡄⠐⠂⠈⠁⡄⢀⡀⠤⠈⠠⢄⠀⠀⠀⠀⠀⠀⠄⠲⠤⠔⠃⠐⠀⠘⠀⠀⠀⠀⡄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠒⠀⠳⢄⡦⠀⢠⠀⠀⠒⢀⠁⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡉⠁⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉
⣀⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⣿⣿⣏⣉⣩⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡀⡀
⠀⠀⠀⠀⠀⣼⣿⠟⠀⢹⣦⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣼⠻⡇⠀⠠
⠀⠀⠀⠀⠐⣟⡹⠀⠀⢸⣿⡆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣶⡆⢘⣯⢨⣷⡇⠀⠀⠀⠠⠀⡤⠊
⠀⠀⠀⠀⠀⢿⡇⠀⢀⣾⣿⣧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠋⣵⣮⢿⡿⢻⣿⠀⠀⠐⢁⠞
⠀⠀⠀⠀⠀⠈⠁⠀⢸⣟⣠⣾⡆⠀⠀⢀⣀⣤⣴⣦⣀⣀⡀⠀⡿⢶⣿⣿⣮⣷⣟⣉⣈⣤⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⠶⢾⣿⢿⡿⢿⡛⢻⣿⣿⣷⣯⣮⣿⣿⣏⣻⡟⢙⢃⣼⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⠀⣸⡆⠐⡯⠿⠿⣯⣁⣰⣿⣬⣿⣿⠿⢿⣽⣏⢿⣷⠬⡿⠉⡴
⠀⠀⠀⠀⠀⠀⢳⡀⢻⣿⣿⡿⠟⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⢸⣿⣿⣆⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠻⠦⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠉⢩⡉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠉⠛⠛⠿⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⢀⣴⣿⣿⠿⠛⠛⠛⠛⠛⠿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣿⣷⣌⠛⠻⠿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⡀⠀⠙⢿⣿⣷⣦⣀⠈⠙⣿⣿⣦⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⠀⠹⣿⣿⣿⣷⡄⠘⣿⣿⡆⠀⠀⠀⢠⣿⡟⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢿⣿⣧⠀⠘⣿⣿⡿⠀⠀⣿⣿⡇⠀⠀⠀⣾⡿⠁⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⢸⣿⡿⠀⠀⠈⠉⠀⠀⣰⣿⣿⠃⠀⠀⣼⣿⠁⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⣠⣶⣷⣶⣶⣶⣿⣿⡿⠋⠀⠀⣼⡿⠁⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄⠀⢀⣾⣿⡿⠋⠛⠛⠛⠛⠉⠀⠀⠀⣼⡿⠁⠀⠀⢠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣤⣤⣠⣤⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠋⠁