Chinese zodiac Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡔⣒⢢⡀⠀⢠⡶⠔⠂⠀⠀⠐⠢⣼⠷⣦⣴⣇⣠⣶⠄⠀⠀⢠⡀⠀⠀⡠⣄⠀⠀⠀⢠⢖⣲⡄⠀⠀⢀⣴⣶⣦
⠀⠀⠀⢣⡀⠁⣓⠚⠋⣀⢀⠄⠀⠀⠈⢻⠡⠄⠹⣿⣿⠏⠉⠀⠀⣠⢇⣉⠹⠟⢀⡘⢦⠀⠀⡗⢹⣻⣟⠉⠉⢙⣧⡲⡉⡆
⠀⠀⠀⠀⢽⠎⠀⢦⡆⠈⢸⡇⠀⠀⠀⢸⠖⠒⠒⠻⣿⠀⠀⠀⠀⡷⠈⠈⠘⠃⠀⠁⢸⠀⠀⠑⠚⡏⠉⠀⡶⠉⠉⡇⠉⠁
⠀⠀⠀⠀⠘⢧⡀⢬⣧⢀⡴⠀⠀⠀⠀⠘⢶⡤⠠⣴⡾⠀⠀⠀⠀⠹⣀⣀⣐⣒⡀⠀⠐⠀⠀⠀⠀⠱⣀⠠⠧⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠼⣷⡤⠤⠤⣼⡗⡀⠀⠀⠀⢀⣞⠉⠉⢱⣄⠀⠀⠀⠯⣱⣶⣿⣿⣶⣦⡸⠂⠀⠀⠀⢠⣶⣾⣾⣶⣶⣦⡀
⠀⠀⠀⢯⣱⠞⠷⢦⠶⠟⢻⠖⠀⠀⠀⢀⣻⠇⠀⠫⣡⠄⠀⠀⠀⡎⡙⠋⠉⠛⠙⠄⠀⠀⠀⢠⣾⡿⠟⠛⠛⠻⣿⣷⡄
⠀⠀⠀⠀⠸⡐⠂⠀⡀⠊⢹⠇⡀⠀⠀⠈⢦⡤⠀⣤⠎⠀⠀⠀⠀⢹⡉⠀⡀⠈⢹⠀⠀⠀⠀⢢⣾⡕⠒⢀⡐⠉⣼⣷⠒⠃
⠀⠀⠀⠫⠤⢽⢶⡏⠙⣶⢾⠒⠃⠀⠀⠀⢠⡉⢻⠀⠀⠀⠀⠀⠀⠀⢹⠣⠈⠝⡇⠀⠀⠀⠀⠀⠻⢷⣄⢼⢷⡰⠟⠋
⠀⠀⠀⠀⠀⠀⠒⠛⠛⠓⠋⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣶⣾⣶⣦⡀⠀⠀⠀⠀⠀⣏⣙⣉⣳⠄⠀⠀⠀⠀⠰⠐⣄⣀⠠⡆⡄⠀⠀⠀⠀⠀⡒⢤⢤⢶⠶⢻⡀
⠀⠀⠀⠀⣰⣏⢠⡀⠀⠌⡿⡀⠀⠀⠀⢠⡉⠀⠀⣡⡀⠀⠀⠀⠀⡤⠡⠄⣀⣀⠥⠰⡀⠀⠀⠀⣰⢁⡈⠈⠁⢠⣄⢹⡄
⠀⠀⠀⠀⠣⣿⠀⠐⡆⠀⠷⠁⠀⠀⠀⢸⢠⡮⢴⡄⡇⠀⠀⠀⠀⡇⠀⠀⢻⠟⠀⠀⡇⠀⠀⠀⠻⣁⢀⡠⠤⣀⡀⡸
⠀⠀⠀⠀⠀⠹⣄⢠⣷⠀⠇⠀⠀⠀⠀⡼⣀⠈⣀⢀⣳⡀⠀⠀⠀⠘⠄⠀⠲⢓⡀⠔⠁⠀⠀⠀⠀⠑⠇⣙⠘⢈⠎
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁

⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣷⡀⠀⠀⠀⠀⢾⣷⣦
⠀⣀⣤⣤⣿⣿⠟⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠀⠀⣀⣤⣤⣭⣿⣿⡷
⢰⣿⣿⣿⣿⡿⣄⠀⠀⠀⠀⠡⡿⣿⠛⠛⠛⠿⣏⠀⠀⠀⠀⢠⢀⣏⣿⢿⣿⠛⠛⣿⠻⣆⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟
⠸⡟⠻⢿⣄⣳⣀⠀⠀⠀⠀⠀⠳⠘⠦⠀⠀⠘⠛⠆⠀⠀⠀⠀⠉⠸⠇⠀⠳⠄⠀⠸⠄⠘⠀⠀⠀⠉⠻⠿⠿⠟⠿
⠀⠈⠉⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⣤⣾⣷⣷⣄⠀⠀⠀⠀⠀⣴⢶⣄⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠻⠆⠀⠀⠀⠀⠀⠀⣀⣠⣿⠟
⠀⠀⠀⠀⢠⣶⣿⣧⣄⡛⠉⠀⠀⢀⡴⣦⡻⣷⣿⣘⣦⠀⠀⠀⢠⡖⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⡀⢠⢤⣸⡇⣸⣿⠙⣿⡆⠀⠀⠀⠳⣬⣷⡌⢹⣿⠁⠀⠀⠀⠾⠁⣹⣿⠛⠛⠻⣿⠀⠀⠀⠀⠀⢸⣿⠿⠿⢿⡟
⠘⠗⠚⠀⢛⠓⠈⠹⡿⠿⡡⠀⠀⠀⠀⠚⠈⠛⠛⠋⠀⠀⠀⠀⠀⠠⡏⠹⡄⠈⠛⠙⡆⠀⠀⠀⠀⣺⡇⠀⠀⢸⡇
⠀⣠⠤⠤⣈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠐⡇⠀⠀⠈⠓⠦⢔⣁⠀⠀⠀⠀⣀⣤⣤⣀⠀⢀⣿⡗⠀⠀⠀⣠⠂⠀⠀⠀⢀⣾⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⣶⣶⣶⣶⣶⣿⠿⠀⠀⠀⠀⣿⣿⣿⣿⣦⣾⣿⣧⠀⠀⠀⢨⣷⣶⣶⣶⣾⣿⠏⠀⠀⠀⣶⣶⣶⣶⣶⣾⣿⣿⣷⣤
⠀⠀⢻⡿⣿⢻⡟⠻⣄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠟⠁⠀⠀⠀⣸⣿⠟⠛⠿⣿⡏⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⣰⠛⠁⢻⡘⡇⠀⠈⠱⠀⠀⠀⠀⠀⠀⠈⠋⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡅⠀⠀⠀⠀⠸⣿⡛⠛⠛⠙⣿
⣶⣿⠷⠶⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣾⣿⣶⣶⣶⣶⣿⣿⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣿⣷⣶⣶

⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀
⠀⠀⢤⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⢶⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡽⡏⢀⢟⡇⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠅⠀⠃⠀⠀⠀⠀⠀⠀⣿⠛⠉⢹⢿⠆⠀⠀⠀⠀⠘⣿⡏⠉⣸⠻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⢀⣴⡷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⣠⣤⣠⡄⠀⠀⠀⠀⢀⡤⠤⣄⢀⣄⣐⡄⠀⠀⠀⠀⡿⣷⣤⣤⣴⣶⣤
⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠐⢟⢋⣙⠿⣭⣛⡿⠃⠀⠀⠀⠀⠉⣠⢤⣼⡼⠀⠉⠀⠀⠀⠀⠀⢠⠟⠛⢏⠉⠹⠂
⠀⠀⠀⠀⠐⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣿⣧⣀⣀⣶⣿⢷⡀⠀⠀⠀⠰⡾⣿⣄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠛⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠟⠸⠇⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⡏⠁
⠀⠀⠀⠀⠘⡇⠀⢀⠁⠀⠀⠀⠀⠀⢀⣾⣹⠀⣿⢙⡦⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠨⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣱⠞⠒⠒⠚⢶⡃⠀⠀⢟⢫⠦⠤⠤⠴⣽⣻⠃⠀⢎⠅⠨⣱⣄⣠⣎⠕⠪⠱⠀⠐⡁⢢⠄⠒⠒⠂⢤⠂⡁
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡀⢫⠀⠀⢰⠁⠀⠀⠀⠀⠀⢣⠀⠀⢎⣲⠋⠀⠀⠀⠀⠑⣖⠰⠀⠰⣚⡇⠀⠀⠀⠀⠈⣟⡲
⠀⠀⠀⠀⠘⢄⠉⠀⠃⠈⢁⠜⠀⠀⢸⠀⠳⣏⣈⡖⠃⡸⠀⠀⠀⣇⠀⠆⢠⠄⠰⠀⣸⠀⠀⠀⠀⡇⠦⠀⠀⠠⠆⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠊⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠣⢄⣀⣀⣀⠤⠃
⠀⠀⠀⠀⠀⣮⢢⡰⠞⣄⠎⣅⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⢀⠤⠤⡀
⠀⠀⠀⠀⢠⢳⠋⠈⣆⠏⠑⡎⡄⠀⠀⢠⠛⢌⠒⣃⠎⢆⠀⠀⠀⢰⣟⠷⣤⣤⠼⣟⣶⠀⠀⠀⢇⢏⣾⡀⠀⣎⣋⢧⠃
⠀⠀⠀⠀⠈⠇⢀⠀⠀⠀⡀⡛⠀⠀⠀⢸⢀⠀⠉⠁⣀⢀⠀⠀⠀⠸⠇⠀⠘⠟⠀⠀⢧⠀⠀⠀⠀⠟⠉⠁⠈⠉⠑⡅
⠀⠀⠀⠀⠀⢇⡈⠀⠀⠀⢁⡀⠀⠀⠀⠨⡀⠀⠀⠀⠀⡸⠀⠀⠀⢿⠀⠂⠠⠆⠐⠂⣺⠆⠀⠀⢼⠀⠤⢀⡀⠠⠄⢰
⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠈⠉⠀⠀⠀⠀⠈⠉⠿⠋⠉⠀⠀⠀⠀⠀⠑⠂⠄⠀⠐⠚⠁⠀⠀⠀⠈⠢⠄⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⣶⡤⠒⠒⢤⣶⡀⠀⠀⢀⠠⠄⡀⠀⢀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⡏⠳⣄⢠⡶⠹⡁⠀⠀⠶⢒⠦⠘⠂⠂⠤⡒⠶⠀⠀⠀⢠⣊⠭⡀⢀⠭⣑⡄⠀⠀⠀⢀⠼⣌⠀⣁⠦⡀
⠀⠀⠀⠀⠀⡁⡀⠈⠁⢀⡀⡇⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⣆⠀⢰⢹⣿⢇⡄⠈⠁⢠⡌⣿⡏⡆⠀⡎⢀⠀⠉⠀⣀⠈⡄
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠈⢀⠇⠀⠀⠈⢃⡈⠀⠚⠀⢁⡸⠁⠀⠀⠉⠙⢄⣀⣈⣁⣀⡠⠊⠉⠀⠀⠣⡈⠐⠭⠃⢉⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⢀⣴⠾⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⣴⣿⣿⠆⠉⠉⢰⠞⠋⠉⠀⠀⠀⠉⠉⠛⠿⣿⣷⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠻⠛⠿⠛⠿⠁⣼⣿⣿⣿⣟⣿⣿⣿⣧⡀
⠀⠀⣼⣿⣿⡏⠀⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⢰⣿⣿⣴⡆⠐⠀⠀⠀⠂⣸⣭⣿⣿⣿⣿⣿⣿⠿⣿⣿⡀
⠀⢸⣿⣿⣿⠁⠀⣠⣾⡿⢱⡶⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣷
⠀⣿⣿⣿⣿⣤⣿⣿⡿⢣⣾⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡝⣿⣿⣿⡄
⠀⢿⣿⣿⠿⣿⣿⣿⣴⣿⣿⣣⣿⣿⠧⡀⠀⠀⠀⢀⡄⢻⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⣼⣿⣿⣿⠇
⠀⠸⣿⣯⣿⣿⢿⣿⡿⢛⠻⣿⣿⣿⣦⡟⠛⣶⣿⣿⠇⣸⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠹⣿⣿⣹⣿⣿⠀⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⠏⣰⣿⣿⡟⠁⠀⠀⠀⠈⢿⣿⣿⣿⡇⠐⣶⣆⠀⢠⡄⠀⡀⠀⢺⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠙⢿⣿⣿⣿⣧⡙⠿⣿⣿⣿⣿⣿⠿⠛⣡⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⠀⣿⡏⢀⣿⠇⣰⡟⢀⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣦⣤⣤⣤⣶⣶⣿⣿⡿⠋⠁⠀⠀⠀⠀⠠⠀⠀⠀⡀⠈⠻⢿⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠁⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⣴⣿⣿⠟⠁⣿⠋⠀⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠂⠀⠀⢀⣴⣿⠋⡻⠛⠛⠛⠻⠏⢙⣿⣿⣻⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⠀⠐⠇⠀⢀⣿⣿⣿⣿⣾⣿⣾⣋⣿⡿⣿⣿⣧⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠀⠄⠈⢿⣿⣿⣿⣽⣯⣟⡿⢿⣿⣇
⠀⢸⣿⣿⣿⣿⣄⣸⣇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣿⣿⠟⠛⠉⠉⠉⠛⠿⠿⠟⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⡿⠏⣰⢃⣦⣤⣄⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⠏⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠘⢿⣿⣏⣠⠞⢁⣾⡿⠟⠛⣁⣤⣤⠀⠀⣹⣷⣶⣿⣿⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⠃⠀⣿⡿⠀⣸⡄⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣷⣶⠛⢉⣡⣤⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣴⣿⣄⣀⣿⣀⢀⣤⣤⣤⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠂⠀⠀⠀⠀⣰⠾⠛⠉⠉⠉⠙⠛⣷⣦⡘⠀⡀⠀⠀⠂⣠⡶⠛⠉⠉⠉⠉⢛⣻⢶⡀⡄⠀⡀⠀⠆⢀⣴⠞⠋⠉⠉⠉⠛⠛⣷⣄⡀⣀⡀
⠀⠀⠆⣠⡿⠋⠀⢀⣶⣿⣷⡄⠀⠈⣿⣿⣦⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⣾⡿⠋⠻⣧⠀⠀⠀⣠⡿⢁⣴⣤⣤⣤⣀⣴⡿⢛⣉⣿⡖⠂
⠀⠀⢠⣿⠁⠀⠀⢸⣿⠀⠸⣿⠀⠀⣿⡿⢹⡆⠀⣿⠃⣀⣤⣤⣄⣠⣴⣿⣿⣿⣷⠆⢹⡇⠀⢰⡟⠀⠀⢻⣿⣿⣿⣿⠿⠿⠿⠛⠉⣿⡆
⠀⠀⢸⣿⠀⠀⢀⣤⣿⠀⠀⢿⣦⣴⡿⠃⢸⡗⠀⣿⣾⣿⠟⢿⣿⣿⣿⡟⠻⠟⠁⠀⢸⣿⠀⢸⡇⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⣹⡇
⠀⠀⠸⣿⡄⢠⣿⠿⠋⠀⠀⠀⠈⠁⠀⠀⣾⠏⠂⣿⣿⣿⡀⠀⠀⠀⠿⠀⠀⣀⠀⠀⣸⡇⠃⢸⣧⠀⠀⣠⣤⣿⣿⣿⣷⣄⠀⠀⢀⣿⠃
⠀⠀⠀⡘⢷⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠄⠈⢻⣿⣿⣶⣶⣤⣶⣿⣿⣿⣿⣿⠟⠀⢡⠄⠙⣧⣾⡟⠻⣿⣿⣿⣿⣿⣷⣠⣿⠃
⠀⠀⠀⠁⠈⠝⠷⢦⣤⣀⣀⣠⣤⡾⠟⠁⠀⠀⠀⠀⠄⠙⠿⢿⣿⣿⣿⣿⣿⡿⠟⣁⠒⠀⠀⠰⠈⠈⠛⠶⣤⣿⣿⣿⣿⣿⠿⠛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣉⣉⣉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⠁⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⣈⣉⣉⣉⠀⠐⢀
⠀⠀⠀⠀⠀⣠⡶⠛⠉⣉⠉⠛⠛⢷⣦⡉⠀⠀⠀⠀⠀⣠⡶⠟⠉⠉⠉⠙⠛⠿⣦⣀⠀⠀⠀⠀⠈⢐⣶⠞⠛⠉⠉⠙⠛⠻⢷⣄⡀
⠘⠀⠉⢠⣾⣿⣶⣿⣿⣿⣷⣤⣤⣶⠎⢻⡆⢀⠖⢀⣾⠛⠀⢀⣀⣶⣴⣶⣾⣀⡈⠻⣧⣀⠀⠀⣰⡞⢡⣶⢀⣀⢀⣠⣤⣤⣄⡙⢷⡄
⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⡍⠠⣾⠃⢀⣀⣈⣻⣿⣿⣿⣿⣿⠋⠀⢻⣇⠀⢰⡟⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⡄⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢸⣿⠀⢹⣿⠿⠿⠟⠿⣿⣿⣿⣿⣿⣿⡟⠀⣹⡇
⠀⠀⠈⣿⣿⣿⠟⠛⢻⣿⡉⢻⣿⣿⡇⠀⣼⠇⠁⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠁⠀⣸⡇⠀⠹⣷⠀⠀⠀⠀⣈⡉⠉⠿⡟⠉⢻⣄⣿⠃
⡀⠀⠀⠘⣿⣿⠀⠀⠀⠿⠋⠀⠛⠛⢋⣼⠟⢀⠒⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣴⡟⠀⠈⠀⠻⣷⡀⠀⠀⠀⠈⠙⠲⢦⣤⣾⣿⡏
⠀⠀⠃⠀⡈⠻⠷⣦⣤⣀⣀⣀⣤⡶⠟⣁⠀⠀⠀⠀⠄⠙⠻⣿⣿⣿⣀⣠⣤⣶⠟⠉⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⣀⣀⣠⣤⠶⠋⠁⠰⠆
⠀⠀⠀⠀⠀⢀⠖⠈⠉⢉⡉⠉⠉⠀⠀⠁⠀⠀⠀⠂⠀⠀⠄⠀⠉⢙⣉⡉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠉⢉⠉⡉⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈⣠⣴⠾⣛⣉⠛⠛⠷⢶⣄⠀⠀⠀⠀⡀⠀⣡⣼⠖⠛⠋⠙⠛⠿⢶⣤⡀⠀⠈⠠⠀⠀⢀⣠⠶⠚⠋⢙⣻⡟⠶⣦⣌
⠀⠀⠀⢀⣶⣿⣧⣾⣿⣿⣷⡀⠀⠀⠙⢿⣄⠀⠀⠀⣾⠟⠀⠀⣴⣶⣶⣶⣶⣦⡙⢿⣆⠀⠀⠌⣠⡾⠃⠀⣴⣾⣿⣿⡿⠀⠀⠙⣷⡄
⠀⠠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⣿⡇⠀⣾⢫⣤⣤⣾⣿⣿⣿⣿⣿⡛⠃⠀⢻⡆⠐⢰⣿⠁⠀⢀⣻⣿⣿⣷⣄⠀⠀⠀⠘⣿⡀
⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠀⣿⠘⠛⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢸⣿⠀⢸⡇⠀⠀⠀⠈⣿⣿⣿⣿⣇⠀⠀⠀⣿⡇
⠐⠀⠘⣿⠻⣿⣟⠙⠿⣿⣿⣿⣿⣇⠀⠀⣸⠇⠀⣿⡄⢀⣾⠿⠟⠻⣿⣿⣿⣿⣿⣷⣼⡏⠂⢸⣧⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⡄⠀⣿⠃
⠁⠀⠀⠹⣧⢿⣿⣶⠄⠉⠛⠿⠿⡿⠗⣴⡟⠶⠀⠘⣿⡈⠉⠀⠀⠰⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⢻⣧⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⠏
⠀⠀⠀⠀⠈⠿⣧⣤⣀⣀⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣦⣠⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠛⠛⠋⠉⡁⠀⠀⠀⠀⠀⠁⠀⠃⠀⡤⠉⠙⠛⡛⠉⠉⠠⡐
⠂⠀⠀⠀⠀⢀⣤⠶⠛⣛⣛⣻⠷⣦⣄⠀⠀⠀⠂⠀⠂⣀⣤⠴⠚⠛⠛⠟⠷⣴⣤⡀⠀⠀⢀⡀⢠⠔⣤⠴⠚⠛⠛⠛⠿⢶⣦⡀
⠀⠀⠀⢀⣶⣿⣤⣤⣿⣿⣿⣿⡆⠀⠙⣿⣄⠀⠀⢀⣼⡟⠁⠀⣠⣴⣶⣶⣶⣦⠙⢿⣆⠀⠀⠠⣀⣾⠃⢤⣾⣦⣤⣄⠀⠀⠈⠻⣧⣀
⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢿⡤⠄⣸⣏⣀⠀⢀⣠⣾⣿⣿⣿⣿⣧⠀⢻⡇⡄⢰⣿⠁⠀⣸⣿⣿⣿⣿⣷⣦⡀⠀⠘⣿
⠂⠠⢰⣿⣿⣿⡟⠉⠈⠙⠛⠛⠛⠃⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣷⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢿⡇
⠉⢠⠈⣿⣿⣯⣴⣶⢶⣦⠀⣠⠶⠶⣤⠀⣸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⠛⠛⠉⢸⡇⢀⢹⣷⠀⢠⣾⣿⠟⠛⣿⣿⣿⣿⣿⣦⣿⠃⠘
⠐⠆⠀⠹⣿⠋⠉⠙⠊⣩⡞⠁⢦⣀⣼⣷⠟⠁⠀⠘⣿⣿⣿⣿⣿⠃⠈⠀⠀⠀⠀⣠⣿⠉⠀⠀⢻⣦⠈⠛⠋⠀⠀⣿⣿⣿⣿⣿⣿⠏⠤
⠠⠀⠖⠃⢘⠿⣶⣶⡾⠋⠀⢀⣈⣹⡿⠋⠈⠀⠃⠄⠈⠻⣿⣿⡇⠀⠀⠀⣀⣠⡾⠛⠄⠁⠀⠀⠀⠙⠷⣦⣀⡀⢠⣿⣿⣿⣿⠟⠣⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⠛⠛⠉⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠤⠄⠀⠒
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠉⠛⠻⣿⠯⠉⢹⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣟⣿⣹⣯⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⣷⡄⠸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣦⡀
⠀⠀⢰⣿⣿⡏⠀⠀⠀⢠⣾⣿⣦⠀⠀⠻⠷⠄⠈⠙⠛⠿⣿⣿⣷⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⢀⣀⡀⠈⠙⢿⣿⣿⡄
⠀⢠⣿⣿⣿⡇⠀⠀⠀⢸⠿⠿⣿⣇⠀⠀⠀⠀⣀⣀⡀⠀⠈⢿⣿⣿⡄⠀⠀⠀⣿⣿⣿⠟⠁⠀⠀⠙⢿⣿⠀⠀⠀⠀⢻⣿⣿⣦⠀⠈⢿⣿⣿⡀
⠀⣸⣿⣿⡏⢇⠀⠀⠀⠈⢠⣄⠈⠻⣿⣶⡶⠿⠟⠋⠉⠀⠀⠈⣿⣿⣿⠀⠀⠀⣿⣿⡟⠀⣠⣶⣦⠀⢸⣿⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠸⣿⣿⡇
⠀⣿⣿⣿⠁⢀⣀⠀⠀⠀⢿⣿⣠⣔⣾⠟⣿⣃⠄⣾⣿⣿⡆⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⣿⣿⡟⠀⣸⣿⢦⣴⣾⠋⣩⣍⠻⣿⣇⠀⠀⣿⣿⡇
⠀⢸⣿⣿⣆⠘⣿⣦⠀⠀⠘⢻⣿⣿⠋⠸⣻⣿⣿⣿⣿⠛⠁⢰⣿⣿⡟⠀⠀⠀⣿⣿⣇⠀⢹⣿⡇⠀⣿⣟⣫⣿⣿⡸⠗⣹⠂⣿⡇⠀⢠⣿⣿⡇
⠀⠀⢿⣿⡟⡀⡈⣧⣤⣤⠲⡿⣿⣿⡀⠄⠸⣿⣿⣟⡂⠀⢠⣾⣿⡿⠁⠀⠀⠀⢹⣿⣿⡄⠀⢻⣷⡀⠘⢿⣿⣿⣿⣿⠿⠃⣰⡟⠀⠀⣼⣿⣿⠁
⠀⠀⠈⢿⣿⣷⣿⣿⣿⣿⣆⡀⣿⣿⣷⣀⣤⣘⣛⣋⣥⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠙⠿⣦⣄⣈⣉⣁⣀⣴⠞⠋⠀⢠⣾⣿⡿⠁
⠀⠀⠀⠀⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠈⠉⠉⠉⠁⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣾⣟⣷⣾⣶⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠉⠻⢿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⢟⣻⣿⣶⣤⡀⠀⠀⠀⠀⠁⢀⠀⠀⠄⠀⢀⣤⣾⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣠⣾⣿⣿⢿⠏⠃⠉⠀⠀⠀⣹⣷⣿⣿⣿⣟⡿⣿⣿⣦⡀⠀⠀⠰⠂⠀⠀⠀⣴⣿⣿⣿⠿⠛⠛⠋⠁⠋⠤⠤⣬⣍⡻⢿⣿⣿⣿⣆
⠀⠀⣰⣿⣟⡛⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⢠⣛⢲⣶⣍⡻⣮⣿⣿⣿⣿⣧
⠀⢰⣿⣿⡥⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⣟⣡⣴⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣇
⠀⣾⣿⡧⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠛⠿⠋⠁⠘⠁⢸⣿⣿⣷⠀⠀⠀⣿⣿⣟⣏⣿⣿⣿⠋⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠉⠀⣼⣿⣿
⠀⣿⣿⣷⣣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣰⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⢻⣿⣿⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠘⣿⣇⠀⢾⣿⣷⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣷⣿⣿⠟⢁⣤⡄⠀⣀⣠⣤⣴⡄⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠹⣿⣦⣈⣿⣿⠀⢸⣿⣿⣿⣿⣿⣶⡄⠐⣶⡆⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⢰⣿⣿⠇⢠⣿⣿⣿⣿⣿⡷⢀⡄⠀⣿⣿⡿
⠀⠀⠀⠹⣿⣿⣿⣿⣇⠀⢻⣿⣿⣿⣿⣿⡇⢰⡿⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⣾⣿⣿⡆⢸⣿⣿⣿⣿⣿⠇⣸⡇⣸⣿⠟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣤⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠠⠀⠀⠀⠀⠀⣬⣿⣦⣄
⠀⠀⢀⣴⣾⣶⣤⣴⣤⡀⠀⠀⠀⢀⣹⣶⣾⣿⣿⣷⣶⡘⠂⠀⢠⣾⢿⣷⣤⣄⣀⣠⣀⡴⠀⠀⠀⠀⠀⣀⣸⣿⡿⠇
⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣀⠀⠀⠈⠉⣿⣿⣿⣿⣿⢿⡇⠀⠀⠀⣈⣠⠾⠛⠛⠉⢭⣿⠂⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠙⠛⠉⠁⣸⠀⠀⠀⠀⢈⡏⠉⡿⠸⠰⠃⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⢹⡇⠀⠀⠀⠘⢿⣿⣿⣹⣧
⠀⠀⠀⠀⠀⠈⡀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⢢⠀⠀⠀⠀⠀⢠⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⡗⠦
⠀⠀⠀⠀⣀⡘⠳⠶⣿⣲⡎⠀⠀⠀⠀⠐⣾⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣷⠀⠀⠀⠀⠀⣶⣶⣤⣶⣾⣿
⠀⠀⠘⣿⡿⠿⣦⣧⣼⣷⡍⠀⠀⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⡼⢿⣿⡟⠛⠩⠟⠃⠀⠀⠀⣹⡿⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠾⠉⠉⠾⠀⠀⠀⠀⠀⠈⠿⣷⠿⠛⠀⠀⠀⠀⠀⠘⠁⠘⢯⠄⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⢿⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀

⠀⠀⠀⠀⣠⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣾⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⡼⠁⢈⣿⣿⡟⠁⠀⠀⠀⠀⠀⢸⣾⣶⣿⣿⠏⠁⠁⠀⠀⣴⠉⠀⠀⢀⣰⣿⢿⠂⠀⠀⠀⢀⣠⣤⣴⣶⣿⣷⡄
⠀⠀⠀⢇⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠁⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⠙⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠈⠲⠿⢿⣿⣟⡁⠀⠀⠀⠀⠀⠀⢨⢋⡃⠀⠀⠀⠀⠀⠀⠀⡞⠿⠃⠉⢹⢧⠀⠀⠀⠀⠀⠀⢿⠇⠀⠸⡷
⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⢄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣶⣾⣿⣶⡄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⠿⠿⢿⣿⣿⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠫⣀⡼⠃⠀⠀⠀⠻⠈⣿⠀⠿⠀⠀⠀⠀⠀⠀⣀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣴⣦⣼⣾⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠣⠸⠁⠀⠀⠀⠀⠠⣎⣀⣷⠦⠄⠀⠀⠀⠠⠔⢻⡿⢿⣿⣿⣷⣤⣐⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠉⢹⡿⠁⠀⠀⢠⠀⠀⠀⠀⢠⢂⡅⢄⡀⠀⠀⠀⠈⡧⢘⠁⡇⠀⠀⠀⠀⠀⢀⠀⠀⢀⣸⠿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠘⠂⠀⠀⠖⢺⠂⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁⠁⠀⠀⠀⢶⠁⠎⢰⠄⠀⠀⠐⠁
⠀⠀⠀⣸⣿⠇⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⠚⠛⣸⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⢀⣿⣿⣿⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⢘⣿⣿⡀
⠀⢠⣾⣿⣿⠉⠉⠋⠀⠀⠀⠀⣰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠶⡀⠀⢻⣿⣿⣅⣤⣶⣶⢂
⠀⢸⣿⣿⣧⣀⣠⠀⠀⠀⠀⡏⢛⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡰⢃⠀⠈⢻⣿⣿⠟⠋⠉
⠀⠀⠙⠛⣷⣉⠀⠀⠀⠀⠀⠐⠁⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠼⠀⠀⢀⠅⠅
⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡠⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⢀⣀⣀⡀⢻⣇⠀⠀⠀⢡⡗⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⣯⠄⠀⠀⠀⢠⣶⣦⣼⠃
⠀⠀⣿⣿⣿⣿⣿⡿⠇⠀⠀⠁⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠁⠂⠀⢀⣠⣿⣿⣿⠁
⠀⠘⠛⠿⠿⠟⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠴⠋⣥⣾⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡤⠀⠀⠀⠂⢜⣁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⢺⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠤⠁⠄⠀⠀⠀⠀⢸⠀⡏⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣴⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠬⠝⠀⣃⣡⠀⠀⠀⠠⠼⠉⠄⣼⠀⠀⠀⠀⠀⣠⣦⣶⣾⣿⣿⣿⠟⢯⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠇⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠛⠉⠻⢿⣿⡋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠧⣦⠀⠀⠀⢠⡶⠋⠁⣀⣌⣳⡀⠀⠀⠀⠀⢰⣶⣶⣆⠀⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠩⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠀⠀⠀⠀⠀⠐⢧⣤⠟⢙⣧⣤⡀⠀⠘⠋⣿⣿⣷⣾⣿⣿⣮⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⣲⣿⣧⣀⡿⠀⠀⣖⣿⣿⠿⠛⠋⠁⠉⣫⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⡀⠉⠀⠀⠀⠋⠳⠄⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⠖⠒⠛⠋⠉⠉⠉⠉⠉⠛⠒⠲⣦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⡇⠀⠀⠉⠑⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣠⣤⣴⣿⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣤⡀⠀⢀⣴⠀⠈⡇⠐⢾⢻⠛⡟⡏⠀⠀⢠⠇⢀⣴⣶⣶⣶⣦⡄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠙⢄⡀⠀⠀⢸⣿⣿⣦⣾⡟⠀⠀⢹⡀⠈⠀⠁⠀⠁⠀⠀⡾⠀⠾⠛⢻⠟⠛⢿⠃⠀⠀⠀⡠⠋⠙⢦⡀
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠑⢆⠀⠀⠁⠙⣿⠟⠁⠀⠀⢘⣇⣀⣤⣤⣤⣄⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠹⣄
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡀⠀⠑⢄⠀⠀⠀⢁⣠⠴⠚⠋⢹⠀⠀⠀⠀⠀⠀⡿⠙⠓⠦⣄⡀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⡼⠁⠀⠀⠁⠀⣴⣿⣋⡠⠀⠀⠙⣤⡴⠋⠁⠀⠀⠀⠈⣇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠙⢦⣤⠊⠀⠀⠐⢿⣶⣶⣶⣆⠀⠀⠈⢇
⠀⣰⠃⠀⠀⠀⢧⣰⣿⡿⡏⠀⠀⢀⡴⠋⠑⢄⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡴⠋⠙⢦⡀⠀⠀⠘⠛⠉⠛⠛⠆⠀⠀⠘⣆
⢀⣯⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠁⠀⠀⠀⠑⢦⡀⠀⠀⢈⣇⣀⣀⣸⡃⠀⠀⢀⡴⠊⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹
⢸⠁⠀⠉⠉⠛⠒⠢⢤⣄⣀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠙⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⠋⠀⠀⠀⠀⠀⠀⠀⢷⡀⣀⣀⣤⠤⠔⠒⠛⠉⠉⠈
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠓⠲⠤⢤⣀⣀⢀⡾⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣀⣤⠤⠴⠒⠛⡏⠉⠀⢀⡀
⡇⠀⣴⣶⣶⣶⣾⡟⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠉⢹⠁⠀⣤⣤⠀⠘⣿⣿⠟⠛⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⣷⠀⠸⢿⣿⣿⣿⣿⣿⠆
⡇⠀⢸⠋⠋⢹⠋⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠉⠀⠀⢻⣿⣷⣶⣿⡟⢀⣀⡀⠀⠀⠀⠀⠀⡿⠀⠀⠈⣿⡟⠋⠉⡿⡖
⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡤⠴⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠛⠒⠶⠤⣤⣇⡀⠀⠀⠉⠀⠀⠈⠀⠁
⢸⡀⣀⣠⡤⠴⠖⠚⠋⠉⠁⠀⢷⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠙⠓⠲⠦⢤⣄⣀⠀⢀
⠈⣏⠁⠀⠀⠀⠀⠀⠀⣠⣶⣷⠈⢧⡀⠀⠀⢀⡰⠋⠀⠀⠀⢠⡇⠀⠀⢸⡀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⠁⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⣻
⠀⠸⡄⠀⠀⣴⢻⣿⢿⣿⣿⠀⠀⠈⠳⣄⡴⠋⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⢦⣠⠞⠀⠻⠻⣿⣿⣿⣿⣿⣧⢀⠀⢠⠇
⠀⠀⢱⡄⠀⠁⡏⢣⡀⢸⠁⠇⠀⢀⡴⠛⠳⣄⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⠞⠑⢤⡀⠀⢀⡟⠹⠀⠨⠏⣹⠀⢀⠏
⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠀⢀⡴⠋⠀⠀⠀⠀⠙⠳⢤⣄⣾⡀⠀⠀⠀⠀⢀⣧⣠⡤⠖⠋⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⡴⠋⢀⡀⠀⡠⢤⡀⠀⠀⠀⢸⠏⠉⠉⠉⠉⠉⠉⢹⡄⠀⠀⠀⢠⡤⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠈⠳⣄⡴⠋⠀⠀⠈⠉⣽⠒⠋⠀⠀⠀⠀⡞⠀⠠⣶⣶⣶⣶⣦⠀⣧⠀⠀⠐⣿⣧⣤⣄⠀⠀⠀⠀⠙⢦⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⠓⣲⠯⠟⠃⢰⠃⢚⣇⣽⣿⣿⠿⠃⠀⢸⡄⠀⠀⣻⠿⣿⣿⡂⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠈⠀⠀⠀⡟⠀⢹⡿⣿⡟⠓⠶⡄⠀⠀⣧⠀⠀⠁⠈⠉⠉⠁⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⢸⠃⠀⠈⠀⠂⠀⠒⠊⠁⠀⠀⢸⡄⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠦⠤⣤⣄⣀⣀⣀⣀⣠⣤⠤⠶⠓⠚⠉⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⡆⢰⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣏⣿⣽⣿⠿⢿⣿⣟⡇⢙⣿⣿⠿⠿⣿⣯⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⢿⣿⢿⣛⣯⣽⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣶⣯⣽⣻⠿⣿⡿⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⢟⣄⢰⣿⡟⠋⠁⠙⠛⠉⠛⢿⣿⡇⢸⣿⠁⠈⠉⠉⠉⠙⠻⣿⣿⡆⣠⡻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣿⡿⣫⣾⣿⣿⣧⢻⣿⣶⣄⠀⡀⠀⠀⠘⢿⡇⣸⣟⣀⣄⠀⠀⠀⠀⠀⣿⠟⣼⣿⠿⢷⣝⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⣵⣿⣿⠿⠿⠓⢾⣷⡹⣿⣶⣶⣷⣶⣿⡿⢣⡇⢸⣷⣾⣿⣿⠀⣼⣷⣶⢎⣾⣿⠅⢿⣿⣿⣿⣮⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣫⣾⣿⣯⠁⠀⠀⠀⢈⡹⣷⡙⣿⣿⣿⣿⣶⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⢃⡾⣫⠅⠀⠀⠙⠿⠿⠟⣷⣝⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣤⣝⠛⢵⣿⣯⣝⠁⠀⠀⠀⣀⢶⣽⣾⣿⡜⠿⢛⣫⣭⣵⣶⡆⠀⠀⠈⠉⠙⠛⠿⢣⣿⣿⣿⣿⣿⠄⠀⢀⣠⢄⣻⣿⡮⠛⣫⣤⡀
⠀⠀⠀⠀⢀⣾⣿⣟⣴⣶⣭⣛⢯⣾⣶⣤⡜⢿⣧⣿⡿⠋⠁⠈⢿⣿⣿⠿⠿⡇⢀⠀⠀⠀⠀⠀⣠⣮⣙⢿⣿⣿⣯⣦⣾⣿⣿⠿⣛⣭⣶⣦⢻⡿⣷⡀
⠀⠀⠀⠀⣼⢻⡿⣼⣿⣿⣿⣿⣷⣮⣝⠻⣿⣾⡿⠋⠀⠀⠀⠀⣈⠱⠖⣛⡉⠥⠬⣍⣛⡳⠆⣀⠿⣿⣿⣷⡝⢿⣿⣿⠟⣫⣵⣾⣿⣿⠿⣿⣯⢿⣿⣧
⠀⠀⠀⢰⣿⣿⢳⣿⣿⣼⣿⡏⣤⣉⣽⢻⣶⡍⡁⠀⠀⠀⣴⢟⡥⠚⠉⠀⠀⠀⠀⠀⠉⢻⣷⣬⠳⣌⠻⣿⡿⠊⢩⣶⣿⣿⠟⠉⠁⢘⣾⣿⣿⡞⣿⣿⡄
⠀⠀⠀⣾⣿⡟⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⡟⣰⣿⣷⢄⡘⣡⠋⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣿⣿⣿⣌⢃⡀⠀⠀⠀⢿⡯⣭⠀⠀⠐⠃⠉⠉⠛⣷⢻⣿⣷
⠀⠀⢀⣿⣿⣧⣿⣿⣟⠀⣴⡈⣿⣿⣿⣿⢇⣿⣿⡟⣼⢱⠃⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣿⣿⣿⣿⡌⣧⠀⠀⠀⠸⣿⣿⣿⣧⣶⣶⣶⡔⣤⣿⣼⣿⣿
⠀⠀⠸⠿⠿⠹⠿⠿⠿⠶⠿⠧⠿⠿⠿⠿⠸⠿⠿⠧⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣧⠸⠄⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠇
⠀⠀⢰⣶⣶⣰⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠐⡆⢃⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⢲⣶⣾⡆⣶⣶⣶⠶⣒⣒⠶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⣿⣿⡟⣿⣿⡟⠙⠿⠟⠛⣠⠼⣿⡆⠀⠀⠀⢿⡘⡄⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⢃⡟⣼⣿⣿⢱⣿⠿⠃⠘⠛⢿⣿⣎⢿⣿⢻⣿⣿
⠀⠀⠀⢿⣿⣧⣿⣿⠄⢀⣀⠀⠀⠀⠀⣻⣷⠀⠀⠀⠈⢡⡙⣄⠀⠀⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⡿⢋⡌⠱⢿⣿⠏⣾⡏⠲⠀⠀⠀⠀⠛⢻⣾⡿⣼⣷⡿
⠀⠀⠀⠸⡿⣿⡼⣯⣤⣸⣿⣧⣤⣤⣤⣿⠿⣃⡠⣾⣿⣧⡙⢮⡓⢤⣘⢿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠋⠀⠀⠀⢀⣘⠿⣷⣿⣤⣈⣤⡄⠀⣀⣿⢧⣟⣿⠇
⠀⠀⠀⠀⢻⡾⣷⣻⣿⣿⣿⣿⡿⢟⣫⣴⢿⣿⣷⣜⢿⣿⣿⣶⠉⠰⠦⣭⣉⡛⢛⣉⣭⠴⢆⡉⠀⠀⠀⠀⣠⣾⠛⠻⣦⣝⡻⢿⣿⣿⣾⣿⣟⣾⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣯⠻⠿⣛⣥⣾⡿⡿⠉⠚⢿⢿⣿⣷⣍⡻⠋⠀⠀⠀⠀⠀⠁⢰⣶⣶⣿⣿⣷⠀⢀⣠⡾⢿⣿⣿⠆⠙⠻⢿⣶⣭⣛⠿⢟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠛⣫⣤⡺⣿⣿⣭⣕⢇⠸⣳⠀⣿⣿⣿⡿⢣⣶⣤⣄⣀⡀⠀⠀⠻⠿⢟⣛⣫⣥⣶⡜⣿⣷⣦⠛⠀⠀⢀⣴⣿⣿⣿⣿⢗⣤⣝⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣝⢿⣿⣿⣞⠆⣹⣯⣔⣹⡿⣡⣿⡿⠿⠋⠉⠛⢿⡇⢸⣿⣿⣿⠻⢿⣿⣿⣌⢿⣿⣆⠀⢠⡿⣿⣿⣿⡿⣫⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡻⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⡀⠀⠀⠈⠃⢠⡇⢸⣿⣿⣿⠃⠈⢹⣿⣿⣎⢻⣿⡧⠀⣱⣿⡿⢫⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢷⣝⢿⣿⣿⡟⣼⣿⣿⡿⠃⠀⠀⠀⣤⣼⡇⢼⣟⣋⠁⠀⣴⣿⣿⣿⣿⣧⢻⣾⣿⡿⣫⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣮⠋⠸⣿⣿⣏⣁⠀⣀⣤⣷⣿⣿⡇⢸⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⠇⠙⣵⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⣾⣿⣶⣯⣟⣻⠿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣷⣿⠿⣟⣻⣽⣶⣿⣷⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣶⣶⣿⣿⡅⢨⣿⣿⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠇⠸⠟⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⣄⣀⣠⡴⣞⢯⢟⢻⡆⠀⠀⠀⡄⠀⠀⣽⣿⣷⣶⠂⢠⣿⣿⣶⣶⡤⣤⢴⣶⣶⢲⣦⡈⠉⠛⢲⡄
⠀⠀⠀⠀⠘⠛⢿⣿⢿⠿⡦⢾⣿⢀⠔⠚⠁⢀⣠⣽⣿⡿⣳⡄⠈⠉⠹⣿⣿⡿⣹⢺⣿⣿⣽⣿⣿⣆⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠙⠒⠚⠉⠈⢒⣶⢖⣿⣼⡩⠝⣿⣷⣷⠀⣠⣴⠿⢿⣿⠛⠿⢻⣯⠋⠹⢿⣮⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠀⠀⠀⣼⠟⣿⡷⣿⣾⣿⣿⣿⠃⠀⠙⠀⠀⣸⡟⠀⢀⣨⠟⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠴⣒⣶⣾⣷⠀⠀⠀⠰⠧⠀⠿⠀⣿⡀⠈⠀⣿⡄⠀⠀⠀⠀⠀⣤⣤⣬⣤⢤⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⢀⠴⠀⠀⠀⠀⠀⠀⠉⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣀⡁⢀⣀⣀⡀
⠀⠀⠀⠀⢀⣠⡞⢻⣷⣿⣯⣤⣠⠀⠀⠀⠀⢠⣿⣿⣿⢿⣧⠀⣴⣶⢶⡟⢷⡄⠸⣟⣾⣿⣿⡝⢻⣬⣿⡿
⠀⠀⠀⣀⣾⣻⣿⠋⠉⠁⢀⣠⠤⣤⠶⣶⡶⣞⣏⢿⣿⠀⠈⠠⢻⣇⣸⣇⣰⡿⠀⠈⢹⣿⡟⠛⠛⢷⣿⣧
⠀⠀⠀⢠⣾⡟⠋⠀⠀⠶⠟⠁⢸⣿⣤⣾⣯⣼⣿⣷⠏⠀⠀⢰⡿⠛⠛⠛⣉⡀⠀⠀⡸⠀⡇⠀⢀⡼⠁⡿
⠀⠀⠀⠸⣇⠛⠀⣄⡄⠀⠀⣰⠛⠁⢸⡏⠀⠀⡞⠹⡆⠀⠀⠸⣷⣐⣯⡷⠋⠀⠀⢈⣁⡈⠁⠀⠈⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠻⠀⠀⠀⠻⠆⠘⠷⠀⠻⠂⠀⠀⠈⠉⡉⠀⠀⠀⠀⢸⣿⣯⠀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣀⡥⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⢷⡶⣶⣶⣶⣧⣤⡄⠀⠀⢸⣛⡾⢷⣿⣿⣿⣿⠋⡇
⠀⠀⢀⣤⣠⣠⡸⣷⣤⡶⢦⣴⠀⠀⠀⠀⠀⠘⢧⣠⣿⡸⣧⢻⣮⡿⠛⠋⠀⠀⠀⠀⠻⣜⣼⣫⣿⠟⡿
⠀⠀⠈⠉⢠⣿⣿⡿⠟⠁⣀⢀⣤⣴⠏⠀⠀⠀⠸⡏⠉⠙⠛⠹⡿⠀⠀⢀⡞⠉⢙⡆⠀⡸⡍⠁
⠀⠀⠀⢠⣿⣻⠀⠀⠀⠐⣻⣯⣿⠷⠲⠆⠀⠀⠀⢁⣀⠀⠀⠀⠉⠀⠀⠈⠑⢀⡼⠁⠀⠀⢠⡶⠆
⠀⠀⠀⠀⢻⣿⣷⣄⣀⣀⣘⣽⣿⠀⠀⠀⠀⠀⢸⣏⣀⣿⣶⣿⣿⣿⣷⡀⠀⢸⣵⢻⣻⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⣦⣴⢶⠀⠀⢻⡿⣿⣿⢟⢯⠉⢩⡟⠃⠀⠀⠙⢿⣿⣏⢹⣿⡅
⠀⠀⠀⠀⠀⢀⣹⠏⠉⠛⠛⠛⠁⠉⠀⠀⠀⠀⠘⣿⡟⠻⣷⣾⣷⡾⠀⠀⠀⠀⠀⠀⡟⠿⠀⡇⠈⠆
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣸⡿⣶⣿⣿⣶⣄⠀⠀⣷⣶⡾⠀⠀⢀⠀⠀⢨⣿⣿⣧⠀⠀⠀⠀⠀⢼⣾⠇
⠀⠀⠀⠀⠀⠀⠋⠰⢱⠙⣿⣿⡿⠀⠀⣿⣿⣿⣶⣤⡀⠀⠀⠀⣾⣿⣿⣷⣦⡀⠀⢠⣶⣶⣤⣶⣶⣄
⠀⠀⠀⠀⠁⠀⠀⡀⠄⠈⠛⢉⡟⠀⠀⢹⡿⣿⣿⣿⠁⠀⠀⠀⣿⢻⣿⣿⣿⡇⠀⠀⣀⣽⠿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠋⠀⠀⠀⠈⠁⠃⠃⠋⠀⠀⠀⠀⠉⠛⠘⠋⠉⠀⠀⠀⠙⠁⠺⠕⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠆⠀⠀⢰⣿⠋⢻⣯⣷⣤⡀⠀⡶⢷⡴⣿⠇⠈⣷⠀⣾⣿⣯⣀⣀⡀⣀⣄⠀⠘⣻⣿⡁⠁⣀⡄
⠀⠀⠀⡒⠀⠀⡼⣿⡏⠳⡉⠶⠛⣷⠀⣇⢸⡀⠀⢋⣇⣿⠀⣽⢿⣿⣿⣿⣿⠍⠋⠀⠀⣿⣿⣿⣿⣿⡇
⠀⡀⠀⠀⠀⠀⠙⠚⢫⠀⠷⣄⡼⠃⠀⠘⠷⣽⣿⡿⠞⠁⠀⠓⠸⡏⠉⠙⡏⡇⠀⠀⠰⠉⡏⠁⢹⢿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠀⡖⣶⡶⠀⠀⠀⠀⣸⡿⠂⠀⣀⡀⠀⠀⣠⢤⠀⠀⢠⣶⣆⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠁⠀⠀⣠⠤⡙⣿⣗⠒⠋⠀⢰⣿⣧⣀⣾⣿⣿⡆⠀⠿⢶⣶⣦⣼⣿⠇⠀⢲⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣇⠣⠂⣸⡿⠗⡶⠀⠀⠻⣿⡿⠟⠛⢿⠃⠀⠀⣼⣿⠛⢻⡿⡄⠀⠈⣻⣿⠿⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠻⠒⠁⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠘⠊⠃⠀⠓⠉⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⢌⠂⠐⠀⠀⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣈⡂⠀⢀⣂⠘⠀⠀⠀⠈⠁⠉⠈⠄⠄⠁⠀⠀⠠⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⣄⡰⠒⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠉⢁⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠃⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠶⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠈⠁⠀⠈⠅⠈⠢⢄⠀⠀⠀⠱⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⢂⡤⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠐⠉⠁⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠁⢀⠀⠄⠠⠎⠀⠀⠀⠀⠠⣍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠁⠀⠀⠀⠀⠐⠃⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠖⠠⠐⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⠣⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣡⠀⠀⢵⣦⠛⠷⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⣀⠀⠀⣀⣴⣢⠝⣦⡄⠠⠤⡀⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⠀⢀
⡔⠂⠄⠀⠤⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠀⠀⠒⠂⠀⠀⠀⠉⠠⠙⠂⠘⠃⠀⠀⠀⠀⠁⠀⡠⠇⠀⠀⠀⠀⠀⠀⢠⠰⡀⠄⠀⠄⢴⠐⠠⠀⠈⠠
⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⢀⠈⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠤⠀⠀⠀⡠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢨⠁⠀⠂⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠜⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢃⠘⠁⣉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⢀⠀⠁⠀⣈⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⣿⠏⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠆⢀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠤⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⣠⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⢣⡀⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠀⠂⠀⠀⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡰⡄⠀⢀⠖⠓⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⠀⠀⠒⠈⠀⠀⡀
⠀⢤⡴⠐⠣⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⠉⢰⠂⢉⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠖⠷⠠⢃⡘⠁⠀⢠
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠌⢶⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠑⡄⠀⠀⢃
⠀⢀⡀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠤⣀⣤⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⣍⠀⠀⢠⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⡌⠡⠉⠁
⠀⠉⠀⠀⠀⠀⠀⠀⠘⢌⡀⠀⠔⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠂⠀⠄⠀⠀⠀⠐⢄⡀⠀⠀⢀⡀⠔⠋⠤⠤⠁⠐⠀⠖
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠁⠀⠀⠀⠀⠀⠀⢠⢄⡀⠀⠀⠀⠀⠀⡠⠃⠀⡀⠆⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢎⡉⠀⠀⢈⡅⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠀⠀⠀⠐⠂⠀⢀⡁⠂⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠔⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⢤⡀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠌⣷⠂⣩⠛⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠚⠋⡰⠀⠈⠙⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⠠⣖⡚⠶⠂⡸⠃⠒⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢀⠀⠚⠃⠋⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠡⠄⠌⠱⠐⠆⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⡘⠂⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠃⠀⠒⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠲⡄⠀⡀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠀⠘⠃⠁⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠈⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠁⠀⠈⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⠘⡃⠨⠅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣆⣐⣂⣐⣉⣒⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠄⠀⢀⡤⠀⠀⠀⢀⠀⠀⠀⢤⠀⠀⠀⢠⡄⠀⠀⠀⠄⠄⠀⠀⠀⡀

⠀⠀⠀⠀⣀⠤⠀⠒⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠂⠠⢄⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⣠⡑⢀⠀⠀⠀⠠⢊⣤⡀⠀⠀⠀⠀⣠⣍⢂⠀⠀⠀⢠⠊⢀⣤⠤⡤⡔⡄⠀⠈⠢⠀⠀⠀⢀⠖⠀⠠⡆⠶⡀⠀⠀⠈⠢⡀
⠀⠰⠀⠀⠀⢀⠠⢸⡞⢭⣿⠁⠀⠀⠀⢀⠁⠘⢦⣴⡿⠛⠹⣶⣤⠟⠀⠁⠀⢀⠃⠀⠀⢹⣶⣰⣾⡁⠀⠀⠀⠡⠀⠀⠂⠀⠀⢑⡦⢴⠁⠀⠀⠀⠀⠱
⠀⠁⠀⢠⣐⣿⣄⠔⡟⡟⠃⠀⠀⢸⠀⠀⠀⠀⣎⠀⣷⠦⣶⡇⢨⠀⠀⢀⠀⢸⠀⠀⠀⠈⢹⣿⠣⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢾⠆⠾⡇⡠⢄⠀⠀⠀⡇
⠀⠠⠀⠸⣧⠣⠄⠳⠚⠣⢀⠀⠀⡀⠀⠀⡀⠀⠐⣀⠈⣉⡁⢀⠜⠀⠀⡀⠀⠈⡀⠀⠀⢀⣸⣻⣹⣻⣆⠀⠀⡰⠀⠀⠄⠀⠀⡀⠹⣟⣒⡷⣊⡀⠀⠰
⠀⠀⠑⣀⠈⠑⠒⠠⠤⠤⠤⢀⠜⠀⠀⠀⠑⢄⠀⠀⠿⠏⠿⠇⠀⢀⠔⠀⠀⠀⠑⢄⠐⠨⠭⠬⠥⠤⠘⢁⠔⠁⠀⠀⠘⢄⠀⠐⠊⠀⠀⠉⠒⢀⠴⠁
⠀⠀⠀⠀⠁⠂⠤⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠂⠤⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡠⠒⠉⠀⠀⠀⠉⠐⢄⠀⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠉⠐⠠⡀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⡀⠔⠈⠁⠀⠀⠈⠑⠢⡀
⠀⢀⠊⠀⣦⣸⣄⠀⠀⠀⠀⠀⠑⡄⠀⠀⠌⠀⠀⣠⡾⢋⣟⣦⠀⠀⠐⠄⠀⠀⡌⠀⠀⠀⠀⢀⣼⠿⡷⡆⠈⠄⠀⠀⡐⠁⢠⢤⣶⡁⣉⣦⡤⡀⠈⢂
⠀⠀⠀⢠⣿⣿⠻⣏⡦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠰⣿⣷⠟⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⢽⡇⢀⣡⣼⠀⠈⠀⢠⠀⠀⢈⣟⡯⣁⣈⢽⢯⠀⠀⠈⡀
⠀⠀⠀⢸⣿⡸⠖⠂⠀⠀⠸⡅⠀⠀⠀⢀⠀⠀⠀⠘⠦⣍⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⡖⠒⠚⠛⡇⠀⠁⠀⢀⠀⠘⠀⠀⢸⢼⠘⢣⠟⢸⠇⠇⠀⢀⠃
⠀⠈⢄⠀⣙⢻⣶⡾⢛⣻⢾⡇⠠⠂⠀⠈⢄⠀⠀⠀⣶⣾⣿⣷⠄⠀⢀⠂⠀⠀⢂⠀⣴⡝⠒⠐⠚⠇⣲⠆⢀⠎⠀⠀⢃⠀⠀⠻⣷⠚⢓⡿⠞⠀⢀⠌
⠀⠀⠀⠓⠌⡀⠀⠀⠀⢀⠠⠔⠁⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠑⠤⢀⠈⠀⠈⠁⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⠤⠐⠒⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠂⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠠⢄⡀
⠀⠀⡠⢊⡠⣀⢀⣀⡀⣀⢄⠙⢀⠀⠀⠀⠠⠊⠀⡀⣠⣀⠀⠀⠀⠀⢢⠀⠀⠀⠠⠊⠀⠀⢠⡤⠀⠀⠀⠈⠢⡀⠀⠀⢀⠞⠁⠀⣀⠀⠀⣀⠀⠈⠢⡀
⠀⡴⠀⠘⢤⢼⡧⠤⢼⠧⡤⠃⠀⠂⠀⠀⠁⠀⠶⣿⡩⡟⢀⡖⠒⡄⠀⠁⠀⠀⠃⢠⠒⣴⣻⣧⣀⠀⠀⠀⠀⠡⠀⠀⠂⠀⢠⡾⠒⠒⢆⠀⠑⢮⠃⠰
⠀⡇⠀⠀⢻⢃⣱⣶⡎⠘⡇⠀⠀⠰⠀⠀⠀⠀⢧⠈⠀⡵⡞⣷⣛⡄⠀⠀⠀⢸⠀⠀⠈⠳⣤⣝⠋⠀⠀⠀⠀⠀⠀⢸⠀⠘⡿⣙⠗⢺⢋⠟⠃⢸⠀⠀⡇
⠀⢷⠀⠀⠀⢸⠛⡽⡏⠉⠀⠀⠀⡀⠀⠠⡀⠀⠈⠣⢜⣶⠕⠿⠏⠀⠀⡀⠀⠈⡀⠀⠀⠀⣿⣧⣑⡆⢀⡀⠀⡠⠀⠀⠄⠀⠱⡘⠃⠚⠁⠀⢀⠜⠀⠰
⠀⠀⠑⢀⠀⠈⠀⠓⠞⠢⠬⢃⠜⠀⠀⠀⠑⢄⠀⠀⠘⠸⠂⠀⠀⢀⠔⠀⠀⠀⠐⢄⠀⠴⠷⠻⠯⠛⠮⢋⠔⠁⠀⠀⠘⢄⡀⠸⠷⠶⠶⠾⠿⢀⠔⠁
⠀⠀⠀⠀⠁⠂⠤⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠂⠤⠤⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣤⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⢹⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣠⣄⠉⠉⢷⣴⣶⣶⣦⡀⠀⠀⠀⣴⢋⠙⠛⢙⣶⠆⢀⣀⣀⡼⠁⠀⠀⠀⠀⢀⣀⣴⠟⣛⣋⣀
⠀⠀⠀⠀⣀⡿⢿⣄⣤⠀⠀⠀⣰⠆⠀⠉⠁⢿⣶⡄⠘⠿⣿⣿⣿⣇⠀⠀⠀⢿⠂⡀⠈⠀⣼⢿⢩⢿⢩⣷⠀⠀⠀⠀⣰⠏⠉⠉⢿⣋⣭⣛⣀
⠀⠀⠀⣴⠋⠀⠀⢹⣯⠤⣴⠚⠁⠀⠀⠀⠀⠸⣿⣁⣀⣠⣤⡤⣾⠸⠂⠀⠀⠀⠙⢛⣾⣛⣁⣉⣈⣨⡴⣏⠀⠀⠀⠀⢿⣀⡀⢁⡼⠁⠀⢈⡏
⠀⠀⠀⠙⠛⣷⣶⣯⠤⣤⣿⡀⠀⠀⠀⠀⠀⢠⣇⣿⠀⠀⢨⣇⣿⠀⠀⠀⠀⠸⠟⣩⡼⠁⠀⠀⠀⠈⠳⣬⠙⠗⠀⠀⠀⣭⢟⡽⠛⠓⢾⡻⢤⡄
⠀⠀⠀⠀⠀⠁⠛⠁⠀⠈⠃⠉⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⣠⠗⠟⢾⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⡤⣄⣀
⠀⠀⠀⢹⡁⣀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠄⢹⠀⠀⠀⠀⠀⠸⣇⢀⣀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠶⢶⣤⣶⣿⠳⠒⠦⣄
⠀⠀⠀⠀⣹⢻⢻⣣⣶⢆⡴⢺⠃⠀⠀⠀⠀⠼⣶⠦⣴⣿⡖⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠻⠷⠖⠒⢶⠛⢿⣦⠀⠐⣇⣠⣀⡼⠃⠛⠛⠀⠀⠀⢙⡆
⠀⠀⠀⠀⡧⢸⡈⠛⠓⠋⢀⡏⠀⠀⠀⠀⠀⣰⡇⠰⣷⡿⢿⣄⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣠⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⣿⢭⣟⣢⣤⡤⢾⠀⠀⠀⠀⠀⢰⡏⠧⣤⣤⡤⠞⣸⠀⠀⠀⠀⠀⢀⣰⠋⡼⠉⠁⠀⢧⡘⢦⡀⠀⠀⠀⠀⠀⠘⢶⢶⡤⠶⠴⢦⢶⡞
⠀⠀⠀⠀⠛⠀⠟⠀⠀⠛⠺⠃⠀⠀⠀⠀⠀⠙⠶⠤⠤⠶⠚⠁⠀⠀⠀⠀⠀⠈⠁⠸⠃⠀⠀⠀⠘⠛⠈⠃⠀⠀⠀⠀⠀⠀⠻⠚⠃⠀⠀⠻⠚⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢠⡞⠙⠛⣦⠀⠀⠀⡼⠀⠀⠀⠀⠀⢻⠀⠘⢧⣀⡴⠶⣆⠀⠀⠀⠀⠀⣠⠾⠉⠉⢻⡟⠃⠀⢰⡄⠀⠀⠀⠀⢀⣼⠛⠛⠳⣶⣄⡀
⠀⠀⠀⣿⢀⡀⠁⣨⣷⣶⣾⡅⠀⠀⠀⠀⠀⠸⡄⢴⠛⠒⣲⢂⡟⠀⠀⠀⠀⠸⣏⣀⣤⢀⡴⠛⠋⠙⣏⠀⠀⠀⠀⠠⣞⣳⠀⠀⢠⠇⠀⠉⡇
⠀⠀⠀⠈⣉⡿⢛⡿⠛⠻⣟⢧⣀⠀⠀⠀⠀⠀⠙⠮⣭⣭⡵⠚⠀⠀⠀⠀⠀⠀⠀⣀⡽⣛⠖⠒⠲⣖⣧⡀⠀⠀⠀⠀⠈⣩⣿⣿⡯⠤⢤⣴⣧⡀
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠐⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⣤⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣐⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣤
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠌⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠃⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⣿⣿⣿⣿⡿⣿⠳⠦⠀⠀⠀⠀⠀⠸⡏⢿⡁⠉⠉⠉⠙⣿⡀⠀⠀⠀⠀⠀⢆⡰⢡⡿⠋⣿⠇⠈⠉⢻⡏⠻⡆⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣟⠛⠻⢿⣄⣈⢆⡀⠀⠀⠀⠀⠀⠀⠻⠀⠳⠀⠀⠀⠚⠉⠳⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⠓⠀⠀⠘⠧⠀⠋⠀⠀⠀⠀⠈⠻⠿⠿⠏⠻⠗
⠀⠀⠀⠈⠉⠉⠉⠁⠠⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⡉⠛⢹⡿⠆⠀⠀⠀⢀⣀⠸⣿⣝⣿⡄⣿⡁⠀⠀⠀⠀⠀⣀⣤⣶⣤⣤⣴⣿⣿⡟⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣷⣦⡀⠀⠀⠀⠀⣿⡛⣷⡌⠻⣿⣷⣼⡗⠀⠀⠀⠀⣾⠃⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢀⢠⡀⢰⢳⡼⣿⠠⣾⣷⡈⣿⣿⠀⠀⠀⠀⠈⠻⣾⣿⣄⣈⣿⡇⠀⠀⠀⠀⠺⠋⠀⣸⣿⡇⠈⠉⠙⣿⡆⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⢻⣿⠃
⠀⠀⠈⠻⠇⠊⠀⠟⠛⠂⠀⠿⠿⠿⢋⡤⠀⠀⠀⠀⠐⠋⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢰⡇⠘⢧⠀⠉⠋⠁⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿
⠀⠀⠀⠀⣀⣀⣀⡉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⣼⠁⠀⠈⠙⠦⣄⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣉⣴⣶⡀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⢠⣿⣿⠀⠀⠀⠀⠀⢯⣀⣀⡀⠀⣀⣴⣿⡿⠃⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣠⣶⣶⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⡿⣿⡉⣿⠋⠻⣦⡀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⡿⠟⠙⠛⢿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⢀⡴⠛⠁⢸⡇⢿⠀⠀⠈⢹⠆⠀⠀⠀⠀⠀⠀⠁⠘⠋⠛⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠰⡿⡟⠛⠛⠛⠋⣿⡏
⠀⠀⠀⠈⠃⠀⠀⠈⠛⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢾⣿⡄⠀⠀⠀⠀⢀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠚⠁⠀⠀⢀⡤⡺⠛⠀⠨⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠢⠁⢂⣼⡍⡞⠀⠀⢀⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉⠄⠄⠀⣾⣿⢀⣿⣆⣀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠤⣴⣹⣿⡈⢿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣦⠬⣅⠽⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣛⣋⠔⠋⢬⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⢁⢌⣁⣴⠮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⠀⠀⠀⠀⢸⣏⠃⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠀⠀⠀⡠⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⢀⣠⣾⠛⠓⢀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠀⠀⠀⢀⣿⡋⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⡠⠀⠀⠀⠸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⠖⠂⠂⠀⠻⠦⠴⠤⠆⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢻⡛⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃
⠀⠀⠀⡀⠀⠀⢄⠈⠃⠀⠀⣾⣶⠄⠀⠀⣾⡁⠂⠀⠀⠐
⠀⠈⠀⠒⠈⠀⠀⢀⠀⠀⠀⠈⠉⠂⠀⢰⠟⠁⣀⣀⡈⠈
⠀⠀⠀⠀⠀⢀⣿⠆⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠀⠠⠇⠀⠀⣠⣷⡤⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠁⠀⣀⣤⠀⢠⣆⠀⠀⠀⠀⢐⣾⣷⡆⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡃⠀⠨⠀⠀⠀⠀⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⢧⠀⠀⡈⠁⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠁⠀⠚⠁⠀⠀⠀⠀⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠸⠀⢠⠄
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠄⠀⢠⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢠⠁⠀⠀⠀⢉⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡴⠑⠂⠁⠀⣀⠐⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⢀⠐⠀⢄⠀⠀⠀⠀⠈⡀⠀⢠⠀⠀⠀⡀⠘⡃⠀⠀⠀⠰⠄⠄⢀⡤⠪⠁
⠀⠀⠀⠀⠀⠈⠀⠀⡈⠁⠉⠀⢠⡀⠈⠁⠀⠀⠀⠀⠁⠀⠈⢷⢄⠀⠈⠀⡀⠀⠀⠀⠀⠅⡠⢊⠂⠀⠀⠸⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠘⠁⠀⠸⠀⠀⠁⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⠳⢐⠄⠀⠇⠀⠀⠀⠀⢸⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠌⠖⠀⠐⠄⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠡⠄⠓⠢⡖⢄⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠈⠙⠲⠄⡄⠀⠁⠀⠀⠀⡐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢅⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠂⠂⠀⠈⠈⠀⠀⠀⠀⠈⠀⣀⠤⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠢⠈⠸⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⠄
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠘⠐⠙⠀⠀⢲⢊⣀⠀⠀⠀⠀⠀⠐⠄⠀⠈⠀⠀⠀⠀⠇⡆⠀⠀⠀⠀⠠⡀⠀⡀⠙⢆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡌⡌⠀⠀⠉⠙⠁⠈⠀⡀⠀⠀⠠⠂⠒⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠣⠄⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠒⠀⠠⢄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠠⠀⠰⠀⠠
⠀⠀⠀⠀⠀⢀⠊⠀⢀⠄⠀⠠⡀⠀⠄⠀⠂⠀⠊⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⢀⡀⡠⠢⠈⠢⠀⠀⠀⡁⠀⠀⠄⠠⠀⠂⢁⠀⡀⠀⠉⠀⠀⠀⠀⠠⡀⠀⢃⠀⠀⠀⡄⠂⠀⠀⠀⠁⠐⢀
⠀⠀⠀⠀⢰⠀⠃⠀⠋⢀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡁⠀⠈⠀⠀⠰⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⢀⠀⠀⠉⠐⠀⠔
⠀⠀⢀⠀⠀⢧⡀⠀⣠⣂⡀⠃⠀⠀⠖⠈⠀⠀⠀⠀⠁⠀⠀⢀⠤⠂⠠⠂⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠒⠐⠀⢄
⠀⠀⠀⠠⠀⠐⠬⢉⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⡆⠠⡀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⢂⠀⠈⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⠀⠈⠂⠀⠀⡠⠔⠒⠒⠒⢄⠂⠀⠀⠁⠀⠀⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⣀⣈⠀⢀⣀⡀⠀⠀⠁⠀⢡⠔⠒⠀⠤⠤⠤⠄⠀⣑⣌⠂⠠⠀⠀⠀⠊⡀
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠂⠐⡁⢄⠨⠁⠩⠀⠈⢙⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠠⡀⢱⠘⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠀⡐⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠡⣀⣆⡂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠤⠔⠀⠀⡠⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠤⢀⣒⣀⣀⠖
⠀⠀⠀⠀⡰⠊⠄⡁⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠊⠀⠀⢀⠘⠂⠀⠁⠀⠀⠈⠊⢔⡤⠠⡀⠀⠀⠀⠈⠒⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠿⢄⡴⠁⠀⠀⠀⠀⠀⠀⠀⣦⣠⠤⠤⢾⠇⡴⣦
⠀⠀⠀⠻⠈⢠⠉⠉⠲⣄⠀⠀⠀⠀⠀⠳⣆⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠠⣄⣸⠁⠀⠀⠀⣼⠁⣡⠄
⠀⠀⠚⠀⣄⠀⠀⡼⠆⠈⢦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⠄⡀⠀⠀⠀⠀⠀⠂⠙⠃⠀⠀⠀⢘⠉
⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⣄⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⣘⣸
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠆
⠀⠀⠀⢰⠋⢉⣉⠓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⡤⠴⠒⠂
⠀⠀⢀⣄⠐⠋⠁⠀⢉⠀⠒⠀⠀⠀⠀⠀⠸⣄⠀⢠⠻⢿⣦⡀⠀⠀⠀⠀⠀⠘⠿⡍⠛⠛⠐
⠀⠰⠉⣞⠀⠀⠀⠀⠀⠉⠙⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠸⠛⠛⠁⠀⠀⡤⣄⠔
⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⣀⠀⠀⠀⠀⠀⠀⠐⠧⡎⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠞⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠐⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⢉⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣠⣂⠆⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠃⢸⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠉⠳⡄
⠀⠀⠀⠉⠀⠀⠈⢹⠇⠀⠃⠀⠀⠀⠀⠀⠀⡴⢢⡉⠁⠀⡉⠀⠀⠀⠀⠀⢀⠀⡰⠦⠀⠀⡀⠀⠈⠄
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⡀⠓⠶⢤⣄⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠒⠉⠢⣀
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠉⠆⠈⠁⠀⠀⡠⠒⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠹⣆⠀⠀⠀⠀⠀⣰⡏⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⠤⣤⠀⠀⠀⠀⠀⢀⢤⣄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⡿⠀⢀⠀⢀⠀⠸⣿⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⢀⡄⠀⢙⣿⣏⠁⢀⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠛⣟⣹⣙⡟⠛⠁⠀⠀⠀⠸⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⢀⢄⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠔⢡⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⡆⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⡀⠀⠆⠀⠠⠃
⠀⠀⠀⠀⢀⠔⠋⠁⡠⠴⠄⠀⠠⡀⠀⡄
⠀⠀⠀⢠⠎⠀⠀⠀⠸⣭⠄⠈⠀⠐⠆⠀⠄⠀⠀⠀⠀⡀⠀⠤⠄⠀⣄⠀⣀
⠀⠀⣠⠟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠅⠀⠨⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⡗⠒⠁⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠘⠔⠒⠤⣀⡼⠀⠄⠀⠈⠁⡠⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠉⠀⠀⠈⠀⠉⠁⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠕⣠⠤⣀⠀⠀⠀⠀⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣂⠀⠀⠀⢀⠌⠌⠀⠀⢀⠙⢆⣀⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⢀⣁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠥⠬⠤⠄⠀⠀⠀⠀⠀⠀⠸⠴⠂⠈⠐⠤⠤⠎⢀⣤⠄⠠⠤⠴⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢧⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⢧⣀
⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣄⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣾⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⡿⠁⠀⣿⠀⠀⠀⠀⠀⣸⠃⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⡇⠀⠀⠀⠀⣰⠏⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢀⣠⡴⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⠀⠀⠀⠀⣼⢃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣁⡤⠶⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡏⠀⠀⢀⡾⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⡀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇⠀⢠⡟⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣷⣦⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⢰⠇⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠙⣮⣳⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣷⣦⣤⣤⡴⠾⠛⠛⠛⠓⠒⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⢀⡀⠠⢠⠀⠀⠀⠀⠰⠖⠆⠀⠀⢠⠤⣄⡠⠤⠠⠀⠀⠀⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠀⣐⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣷⠤⡆
⠀⠀⠀⠣⣤⣤⡴⠂⠀⢐⣿⣿⣿⠀⠀⠀⠛⢦⣼⣶⠆⠂⠀⠀⠈⠙⠃⠀⢻⠁⠀⠀⢿⠿⠟⠟⠁⠀⠈⠿⠿⠋⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⠈⠈⣿
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠴⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⠤⠄⠀⠀⢰⡆⠀⠀⠀⢐⣧⣾⣧⣀⠀⠀⣤⣤⣤⠀⢀⡄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣿⡇⠀⠀⢀⣀⡁⣀⣠⡀⠀⠈⠑⣶⣶⣶⣶⠀⠀⢰⣶⣶⣄⢠⠀⠀⠀⠴⣤⣷⣄⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠙⣷⡾⠟⠁⠀⡀⠁
⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⡀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣈⣩⠧⠀⢷⣀⣀⣴⠃⠀⠀⢦⣀⣤⣤⡄⠀⠀⠀⣤⣤⠀⠈⠁⠀⠀⣀⣐⣊⣿⡷⠄⠀⠛⢧⣬⣿⠁⢀⣤⣤⣤⣬
⠀⠀⠀⠸⠛⠛⠛⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠛⠛⠀⠀⠀⠀⠘⠛⠟⠛⠻⠇⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠰⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣠⠔⠀⠀⠠⠊⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⣀⣠⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠠⣾⣤⣤⡄⠀⠈⠳⣶⣶⠷⠀⠀⠈⠛⠿⠿⠏⠀⠀⠀⠀⠀⠉⠶⠶⠶⠶⠀⠀⠀⠸⠀⠀⠀⠀⠛⠿⠿⠋⠀⠀⢶⣶⣶⣶⣶⠀⠀⡀⠙⠛⠉⠉
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠂⠃⠀⠀⠀⠀⠉⠉⠁⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⣶⡂⠀⠀⠀⠀⠀⣤⣤⣄⠄⠀⠠⠀⣘⣉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣀⠀⢀⡄⡖⠃⠀⠀⣈⣃⡔⠀⠀⠀⣴⣶⣿⣿⣤⡤⠀⠀⠀⠘⡿⠋⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣏⠁⠀⠀⠀⠀⠀⠟⠿⠏⠈
⠀⠀⠀⠙⠛⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠉⠈⠀⠀⠀⠀⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘

⠀⠈⠉⠁
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⡛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⠟⠛⠋⣉⣥⣶⣾⣿⣿⣿⠿⠛⠛⠻⢿⣿⣿⡆⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣀⣤⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⡇⢀⣴⣶⣦⠀⠙⠿⣿⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⣫⣿⣿⣇⠸⣿⠟⠛⠃⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⢛⠁⢠⣾⣿⣿⣿⣿⡆⠈⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢯⣶⡏⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠈⢿⣿⣦⣤⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⡉⣿⣿⣆⣀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠙⢿⣿⣿⣿⣿⣷⡘⠻⠿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⣿⣿⣿⣿⡇⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣷⡄⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣌⠻⣷⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡄⠘⠄⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⠀⠒⢀⠀⠀⠒⠂⠀⠠
⠀⠀⠀⠀⠁⠀⠈⠀⢀⡀⠐⠀⡀⠀⣀⠀⠀⠈⠉⡀
⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠠⠀⠁⡀⠉⡄⠀⠀⠀⢅⠀⠁⠀⠄⠀⠀⠠⠊⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠠⠄⣀⣁⡀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠘⢀⡀⠀⢀⠀⢀⠀⠙⠉⠁⠀⠠⠛⠀⠀⠀⠀⠘⠣⠄⠀⠀⠄⠀⠄⠀⡀
⠀⠀⠈⠀⠁⠀⠀⠁⠁⠤⠤⠐⡀⠡⡔⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠠⠌⠀⠒⠀⠀⡀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠠⢆⠀⡀⠐⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢠⠎⠀⠁⢈⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠄⢗⡌⠀⠐⠀⠰⠂⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣡⠀⠊⠀⢉⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠘⡀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠐⡀⠤⢀⠀⠴⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⣀⠔⠓⠀⠐⠁
⣶⣶⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⣶⣶⠶⢶⡶⢶

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⣠⣤⣤⣤⣀⠀⠀⢀⣤⣤⣶⣤⣀⠀⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⠀⠂⢌⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢢⡂
⠀⠸⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⢿⡛⠃⠀⠀⠀⢼⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⢻⠟⢿⡇⠀⠀⠀⠈⠿⠟⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠋⠀⠀⠀⠀⠛⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣹⣿⣷⣦⠀⠀⠀⠀⣹⣿⣷⣄⠀⠀⠀⠰⣦⣤⣤⣤⣤⠀⠀⠰⣦⡤⢤⠤⠄⠀⠀⠛⣷⣤⣤⣤⡀⠀⠀⠛⣦⣤⣤⣤⡀
⠀⠀⠀⠈⠻⣿⢧⠀⠀⠀⠀⠈⢻⡿⡇⠀⠀⠀⠈⢿⠛⠛⣿⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⢸⠋⠉⠻⡇⠀⠀⠀⢹⠉⠉⢿⡇
⠀⠀⠈⠛⠛⠓⠃⠀⠀⠀⠈⠛⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣾⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⣼⠛⠋⠀⠀⠀⠀⣼⣧⣀⣾⡿⠇⠀⠀⢼⡄⣀⣠⡿⠆
⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠹⣄⠀⠃⠀⠀⠀⠉⢻⣿⣿⠟⠀⠀⠀⠉⢻⣫⣿⠇
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⢨⠛⠁⠀⠀⠀⠀⢸⣿⣦⣠⣿⡇⠀⠀⢾⣿⣆⣰⣿
⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⣲⡀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠸⣶⣄⡀⠀⠀⠀⠀⠉⠛⠿⠟⠁⠀⠀⠀⠙⠛⠿⠟
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⢰⣤⡆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠰⣦⡴⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠉⠹⣶⡀
⠀⠀⢈⣿⣿⣷⡄⠀⠀⠀⠀⠶⠟⢢⠀⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢠⣶⣿⡷
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢐⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⡇⢀⠀⠀⠀⠀⠠⢲⠀⡄⢠⣿
⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠍⠀⠀⠀⡁⠈⣄⠀⠀⠀⢰⡿⠰⡁⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠘⣇⢡⠀⠐⡆⠀⣿⠀⠀⣥⣾⠀⢸⡞⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣾⠁⣠⠿⠃⠸⠿⠟⣄⠘⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠏⡌⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣻⣿⡄⢀⣤⠀⠄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠄⡴⠛⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡔⠁⠀⠀⡸⠀⠈⠛⠀⠀⣰⡿⢷⣀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠈⠀⠀⠀⢰⡅⠀⢠⠇⠀⡠⠂⠀⠀⠛⡄⢰⡜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢹⡃⢀⡄⠀⠀⠀⠘⣷⣆⡈⠄⠀⠀⠘⢶⠷⠂⡠⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠿⣿⣄⡀⠀⠀⠸⠀⠀⢀⡾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢵⣠⣤⣶⢤⡴⠎⠁⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠀⠀⠀⠀⢇⠀⠀⠀⠀⢰⣴⣤⣾⡆⠀⠀⢠⣿⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⣤⣀⣄⣼⠀⠀⠀⠠⣿⠋⢻⣿⡿⠀⠀⠘⠿⠛⠓⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠷⠄⠀⠠⣄⣸⠀⠈⠉⠙⠺⠷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡠⣀⡀⠀⠀⠀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣀⠀⠀⣠⡄
⠀⠀⠀⠀⠱⠕⠃⠀⠀⠀⢆⡼⠁⠃⠀⠀⠀⠠⠡⠁⠐⢀⣀⠤⡠⠀⠀⢬⡳⣀⢇⠇⠀⠀⣖⢆⣸⣿
⠀⠀⠠⠤⡄⠠⠒⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠙⠟⠂⠈⠀⡈⠜⠀⠈⠙⠄⠀⢀⠜⠛⠓⠇
⠀⠀⠈⠲⣈⡶⡤⡁⠀⠀⡀⢸⢓⠀⠀⠀⠀⠀⠔⠀⠠⡄⠀⠀⠀⠀⢀⠆⠀⠀⠠⡂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠤⠀⢸⣄⠀⠀⠂⠶⠄⠁⠀⠀⠀⠀⠸⢄⣀⠀⠘⠀⠀⠘⡀⠀⠀⠎
⠀⠀⠀⠈⠀⠀⠖⠀⡀⠀⠀⠀⠘⣧⠀⣟⠀⠀⠀⡀⢤⢀⢠⡀⣀⡄⢀⠀⢀⠀⠀⠀⠀⠀⠢⠤⠦⠞
⠀⠀⠀⠈⠂⠀⠰⠀⠈⠀⠀⠀⠀⣿⠀⠀⣷⠿⠷⠁⡸⠈⠸⠏⠡⡇⠘⠚⠿⠃⠀⡀⠀⠀⢠⣄⠀⡔⡢
⠀⠐⣄⡀⡤⠀⡀⠀⠀⠀⠀⠀⠀⣿⣄⣰⠟⣼⠃⢸⠁⠀⢠⡐⠘⠁⠀⣼⠀⠀⠀⠌⠔⠀⠹⡨⠷⠤⡅
⠀⢠⢎⣹⠀⠀⠐⠂⠀⠀⠀⢤⡄⠀⠈⡁⠀⢿⠀⢿⡘⠗⡗⠃⠿⢆⡆⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠈⣴⣂⣴⢛⡞⠀⠈⠁⠀⠀⠀⠁⠄⠀⠀⠁⠈⠀⢦⠲⣼⣩⠀⠐⢤⡶⠄⢲⡀
⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠰⠁⠁⠈⠫⠠⢶⠀⠀⢀⠀⠀⢀⡔⢅⠀⠀⠀⠀⠀⠕⠋⠋⢂⠀⠘⠀⠂⠀⠁
⠀⠀⠀⡎⠲⣲⠀⠀⠀⠀⡥⠀⠀⠀⢠⠀⠀⠀⠀⠛⠿⠂⠙⠰⠁⡀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠁⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠁⠸⠴⠀⠀⠀⢔⣁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠁⠀⠀⠀⠀⠀⠀⠁⠓⠀⠀⠈⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠭⠽⠿⢶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣤⠤⠤⠄⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡤⠂⠀⠀⠀⣴⣾⣶⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⣿⣆⣀⠀⠀⠀⢻⣿⠔⠂⠀⣾⢙⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣇⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⢸⣿⣯⣄⣠⣌⣿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⣾⣿⣿⣿⣿⣿⡏⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⣿⣿⠟⠋⠀⠉⠛⠛⠻⣿⣿⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠿⠄⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⣿⣿⣿⣆⠀⠸⡇⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢠⡇⢹⣷⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⡇⠘⣷⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡜
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣧⠀⠀⢀⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡻⣇⠀⠀⢹⡇⠀⠀⠀⠀⠀⣸⠇⠀⡜
⠀⠀⠀⠀⠀⠀⠸⡇⣰⣿⣿⣦⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⢻⡄⠀⠘⡇⠀⠀⠀⠀⠀⡟⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠈⣇⡆⠀⠀⠀⠀⠀⠀⢰⡇⢠⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃⣿⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⢹⡇⠀⢀⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠸⢿⣿⢡⡿⠋⠀⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⡼⠁⠀⠐⢄⠀⠀⠀⢸⠀⡇
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠄⠐⠧⠤⠄⠠⠴⠇⠀⠄⠡⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢀⡠⠄⡀⠀⠀⢰⠀⢀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡔⠁⠀⠀⢰⠀⢀⠜⠊⠁⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠎⠀⠀⠀⠀⡸⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠎⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡄⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠒⠀⠈⠉⠉⠉⠉⠐⠒⠇⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉
⠀⠀⠀⠀⣃⣀⡀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢢⡀
⠀⠀⠀⠰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠁⠒⠂⠤⢀⣀
⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⠀⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⢀⡨⠃⠀⠀⠀⠀⠀⡌⠀⠀⠀⣀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠈⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠀⠤⠔⠊⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠒⠈
⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⢀⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠄⠀⠀⠀⣾⠆⠀⠀⠀⠀⠄⠈⡀⠴⠘⡄⠁⠄⠀⠀⠀⠂⣰⠈
⠀⠀⠀⠀⠀⣠⣤⣾⢿⣧⣤⠜⠀⠀⠀⠀⠀⠠⠀⠀⠀⢘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⣴⠁⢌⡱⢐⠀⠀⢀⢈⠇⠈
⠀⠀⠀⠐⠀⢙⣿⣅⣏⣿⡟⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠈⠀⠀⢄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡉⢀⠒⠢⢀⠀⠀⠀⢀⠚⡆
⠀⠀⠀⠐⠀⠈⠟⠛⠛⠟⠇⠄⠂⠀⠀⢀⡨⠇⠂⢠⡈⢄⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠈⠀⠠⠈⠃⠀⠀⠀⠀⠀⠐⢧⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠐⠀⠀⠀⠀⣤⠼⠀⠀⠀⠈⢁⠀⠀⢰⣿⠛⠁⠀⠁⠀⠀⢄⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢻⡄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⣾⠁⠀⠀⠀⢀⣰⢠⡁⠠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠠⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠁⠦⠘⠘⠸⠁⠀⠀⡀⠀⠀⠀⠀⢀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⢄⠂⠐⣀⢆⠄⠀⠀⠀⣀⣸⣄⠂⠀⠠⠐⣾⡂⠀⠀⠀⠀⠁⠰⡶⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⡀⢀⠀⠀⠀⠀⣰⠇⠂⠀⢌⢄⠂⠤⡩⣚⠌⡄⠀⠈⠔⡷⠾⠉⠂⠀⠀⢸⡇⠁⠀⠀⠀⠄⠀⢰⠀⠀⠀⠀⣠⠁⠀⠀⠀⢀⡴⠂
⠀⠀⠄⢢⠼⢦⠂⠀⠀⠅⠀⠀⠠⠂⢌⠦⠐⡖⠀⢂⡱⡁⠀⠀⠀⠀⠀⠀⠀⠐⠸⣇⣀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡏⠀⢀⠀⠁
⠀⠀⠨⠜⠓⠋⠁⠀⠀⠀⠀⠀⠠⠎⠎⡜⠨⡰⢁⠃⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢏⠀⠀⠀⠀⢠⡆⠀⠈⠁⠘⠁⠀⠄⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠈⠚⠱⠱⠑⠂⠉⠀⠀⠀⠀⠀⡐⠈⡄⠀⡀⠀⡀⠈⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠙⠁
⠀⠀⢀⣀⠀⠈⠋⠈⢠⡦⠀⠀⠀⣤⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢥⢀⡖⠀⠀⡀⠁⠀⢀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠗⣀⠀⢂⠠⠀⢀
⠀⣠⡌⠋⢶⢆⡄⣁⢠⡀⠀⣠⡀⠉⢀⠀⠀⣁⠀⡄⠀⠀⠈⠢⠄⡠⢿⣿⠗⠃⢀⠀⠀⠚⠋⠀⠀⠀⠀⡀⠀⠀⢀⣄⠀⠊⠓⡌⠀⡄⠙
⠀⠀⠁⠨⢄⠈⠁⡄⠈⠳⡶⠙⢁⡀⠀⡔⢆⠉⠀⠁⠜⠠⠀⠀⢠⠀⢀⠋⠠⠀⠀⠄⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⠀⠈⡗⠋
⠀⠀⠘⠓⠀⢒⠀⢼⠀⠸⠆⠇⠘⠁⠀⠀⢀⣀⠤⠤⠀⠀⠀⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡰⠖⠀⠀⠀⠀⠈⢈⠑⡼⠂⣀⢰⠆
⠀⠀⠀⣀⡀⣘⠃⠐⢦⠀⠋⠘⠓⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠴⠏⠀⠀⠄⠀⠟⢠⠀⠀⢹⠄⠒⠉⠀⠀⡀
⠀⠀⢠⡝⠃⡛⠀⠛⢀⢡⠀⠿⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣸⠛⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣇⠄⠀⠀⠀⢀⠇
⠀⠀⠈⠁⠀⣤⠁⠉⠉⠀⢳⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢀⠀⢸⡀⠀⠀⠄⠀⢀⠋⠉⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡸⠧⠀⠐⠀⠀⡘⠀⠀⠀⠈⠸⡟⠋⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢴⡗⠃⠀⠀⠀⠀⠀⢰⣶⡴⣾⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢸⠀⠀⠀⠀⠘⣤⢧⡤⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣾⣯⣝⣿⡷⠤⠂⠀⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠻⢅⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⠶⢧⣄⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⢸⣿⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠛⠉⠀⠀⣤⡀⠀⠀⠀⠈⠛⢾⠟⠉⢹⡇⠀⠀⠀⠀⠘⣿⡄⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠘⠭⡾⠀⠀⠀⠀⠀⠀⠁⠀⣸⣀⣤⣤⣤⣤⣤⣹⣿⣄⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣶⣄⡀⢺⣹⠟⠂⠀⢲⠋⠉⠉⠉⠉⠉⠙⠛⠻⢿⣷⣤⡀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⠀⠤⠔⢯⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠑⢦⣀⠈⠂⠀⠀⠀⠀⣀⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠒⠚⠛⠛⠋⠉⠀⠀⠀⠀⣠⣤⠶⠶⠶⠄⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡾⠋⢀⣠⣴⣶⠿⠇⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣋⣴⠾⠛⠉⠁⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⢁⠞⠉⠀⠀⠀⠀⣠⡶⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠈⢻⣿⣟⣁⣀⠴⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠻⣿⣿⣦⣄
⠀⠀⢸⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⠘⣿⣶⣶⣶⡄⠀⠙⠛⢻⣿⣿⣆
⠀⠀⢸⡇⢸⡇⢸⡆⠀⠈⠛⠛⠛⠛⠓⠀⠀⠀⢨⣿⣿⣿⣿⠇⠀⠀⠀⠀⠸⠀⠀⢹⠄⠀⢠⣀⣀⣽⠟⣹
⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⠇
⠀⠐⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣼⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⡄
⠀⠀⠀⠈⢻⠙⠛⠙⢻⡇⠀⠀⠸⠛⠋⢻⣿⣿⣶⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢹⣿⡟⠃
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡟⣿⡇
⠀⠀⠀⣰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣧⣿⡀⠀⠀⠀⣰⣆⠀⠀⣴⣶⣶⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡿⢹⡿⠃⠀⠀⢐⣿⣿⣷⣦⣿⣿⡝⠃
⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡇⢨⠇⠀⠀⠀⠀⠈⠩⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉
⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣀⡄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣀⣤
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⣴⣶⡄⣰⣿⠄⠀⢸⣿⡇⠀⠀⠀⠀⣰⣿⣷⡀⠀⠀⠘⣿⣷⣤⡀
⠀⠀⠀⢼⣿⡆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠛⠿⣿⡿⠀⠀⠸⣿⣿⣆⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠙⠛⠂⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠈⠛⠋⠀⠀⠀⠀⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢋⠀⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣛⠋⡶⠋⡾⠋⠾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⢿⣇⣤⠀⣟⢱⠟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣳⠎⠉⢈⣿⣟⣣⣿⠿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠚⢿⡿⣢⣬⣌⢛⡫⡉⢠⣟⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⢻⢯⣷⢶⡌⡇⣿⡎⠸⣟⠸⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠁⠸⣿⡀⠬⠶⣿⡟⠘⢛⣤⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣤⣤⣽⣻⢿⣿⣿⠟⢷⡌⢿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⢠⣴⠏⣤⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⢿⣿⣿⣿⣿⣷⣄⡺⣏⢳⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠹⣿⣿⡏⢀⠀⢻⡜⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣇⠀⢀⣿⣿⣧⣤⣷⡀⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⡷⠿⠛⣿⡟⠛⢻⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢻⣆⠀⣿⡇⠀⠀⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⢷⣼⣧⠀⠰⠁⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠈⠛⠻⠿⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠈⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⠿⣿⠋⠈⠁⠀⠀⠀⠀⠀⠀⣄⣠⣤⣤⠶⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⠁⠀⠀⡟⢤⠀⠀⢀⠀⠀⠀⢠⣤⠾⢛⠏⠀⠑⠁⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠹⣿⣿⣿⡿⣯⣶⣶⣿⣇⣀⡤⠊⠀⠀⠀⠀⠀⠈⠁⢀⡞⠁⣴⠁⠓⢄⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠉⠻⣿⠿⠿⢷⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠁⣀⣼⣿⣥⡀⠩⡀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠓⢄⠀⠬⠷⡆⣅⣹⣵⠤⠤⣶⣶⣤⠤⠤⠖⠛⠻⠉⠿⢝⠋⠢⠅⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠃⣴⠔⠛⠀⠀⠀⠠⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⠃⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠞⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠀⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⣷
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⢿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⠲⢤⣀⣀⣀⣀⣀⣤⡶⠀⠀⢀⡇⠀⠀⠀⠀⠐⠛⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠤⣀⣉⣉⣩⠵⠋⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠠⣌⠁⠀⠀⠀⠀⣀⣀⣠⡤⣶⣶⢤⣄⣀⣀⠀⠀⠀⠀⠈⣡⠄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⡷⢠⠴⠊⠉⡉⠁⠀⠄⣿⣿⠰⠀⠈⢉⠉⠓⠦⡀⢯⡁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠘⠀⢀⠀⠀⠈⠀⠀⠄⠀⡟⢿⠀⠠⠀⠀⠁⠀⠀⡀⠀⠃⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⠃⠘⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⠀⠀⠀⠀⠀⠀⠠⠘⡆⢰⠣⠔⠀⠀⠀⠀⠀⠀⠁⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⠀⠀⢲⡒⠒⠶⣄⠀⢄⠀⣷⣿⠀⡠⠀⣤⠶⠒⢒⡞⠀⠠⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣓⠦⠤⠘⢷⣬⣕⠿⡿⣪⣥⡾⠃⠦⠶⣚⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠃⠀⠀⠀⣩⠟⠵⠤⢄⠈⢧⡾⠁⡠⠤⠮⠻⣍⠀⠀⠀⠘⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠠⠀⠀⢀⣠⠴⠒⠀⠀⠃⠀⠀⠘⠀⠀⠒⠦⣄⡀⠀⠐⠄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠋⢰⡀⠀⢰⣟⠓⠂⠐⠛⣻⡆⠀⢀⡆⠹⣆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢲⡿⠉⢦⡑⠂⠀⠙⠳⣄⣠⠞⠋⠀⠐⢊⡴⠉⢿⠖⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡰⡦⠌⡁⠀⠀⠀⠈⠁⠀⠀⠀⢈⠡⢴⢆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣤⣤⡤⠤⠴⠀⠀⠤⠤⢤⣤⣤⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⢆⢸⡄⠈⢈⢀⠁⡁⠁⢠⠇⡰⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠙⠈⠀⠀⠀⠀⠁⠋⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⣄⡀⠀⠀⢀⣠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠦⡃⠀⠀⠀⠀⢘⠴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⡚⠛⠛⠒⠲⠶⢰⠖⠒⠚⠛⠒⠐⣠⠤⢤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⢳⢄⠀⠀⠀⠀⠀⠀⡤⠲⠭⠚⠁⠁⠀⣠⣤⣄⡀⠀⠀⠀⠀⡀⠀⢈⣦⣼⡀⠐⠾⠁⡀⠀⠀⠀⢸
⠀⠀⠀⠙⠓⠆⠀⠐⢀⠍⠠⠒⠒⠀⠠⢐⠏⠉⠈⠛⠉⠉⠒⠒⠢⡈⠻⡅⣀⠙⠣⠌⠠⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠈⠓⠐⠛⠉⠀⠐⠒⠀⠀⠘⠂⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⡀⣉⠀⠀⣠⠀⠂⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠀⠁⠉⢠⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠠⠄⡄⠀⡩⠄⠁⠐⠂⠀⠀⠀⠀⠠⠇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⠳⣄⠘⠀⣠⠄⢒⠀⠁⠀⠀⠀⠀⢀⠄⠄⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡈⠀⠈⡁⠤
⠀⠀⠀⠀⠀⠻⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⢃⠀⠭⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠓⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠠⣀⠰⡌⠁⣀⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡅⠊⠤⡶⡱⢄
⠀⠀⠀⠀⠀⠁⠀⢰⠄⠀⠀⠀⠀⠀⠐⠀⠈⠂⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠐⡈⠛⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠄⠣⠀⠀⠉⠁
⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠢⠀⢠⡀⠀⣀⡀⠤⡀
⠀⡇⠲⣦⡷⠒⡒⢶⠒⡒⡗⣾⠒⡒⢢⠖⡒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠆⠘⠂⠁⠀⠀⠒⢀⡇
⠀⡞⢶⠄⣷⠸⠜⢸⢸⠓⡇⣿⢸⢹⢸⡘⠟⢸
⠀⠈⠒⠚⢃⢰⠚⠉⠛⠀⠙⠋⠊⠙⠺⡍⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣦⡙⣿⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡌⣷⣴⣷⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠛⢿⣿⡿⣾⡆⠀⠙⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠈⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣧⣀⣀⡀⠀⠀⠸⠿⠯⡃⠈⠉⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⡿⠉⠀⢈⡁⠀⠀⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡂⠀⠀⠀⢙⡋⠀⠀⠀⢰⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⠷⢶⣤⣾⣷⣤⡦⢄⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢿⡿⢿⣿⠿⣿⢿⠟⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠉⠀⣠⢡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣇⢀⠀⠀⠀⠀⠋⠺⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣆⢠⠀⣤⠇⠀⠈⠀⠀⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠙⢿⣿⣿⣽⣴⡏⠀⠀⠀⠀⠀⢀⣴⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⣦⣠⣼⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿⣿⣿⣦⠀⠀⠀⠈⢹⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⢿⣄⣀⡀⠠⠼⠿⠛⠛⠲⠔⠂⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⡤⣡⢅⠀⢀⠀⠀⠀⠀⠀⠀⢠⢁⠀⠀⠀⠀⠀⢰⣦⣀⡰⣞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠠⠤⠀⠈⠀⠀⠺⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠱⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠙⣯⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠴⠰⠆⠀⠠⠤⠀⠠⠄⠰⠄⠀⠀⡀⠀⠀⠄⠀⠠⠀⠀⢼⡴⣦⡄⢄⡰⠀⠀⠀⠱⠷⠀⠀⡐⠀⠀⠠⡄⣶⣄⠉⠁
⠀⠀⠀⠀⠀⠀⣀⢒⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡄⠒⣠⠐⢦⡤⠀⠀⠀⠉⠁⠀⠀⡄⠀⠀⠀⠀⢠⢞⣁⠑⠄⠀⠠⠖⠈⠀⠲⠶
⠀⠀⠀⠀⠀⢄⡩⡄⡍⢰⣿⠇⠀⢠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⢀⠰⠠⠄
⠀⠀⢠⣶⡄⠈⠙⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠁⠠⠀⢀⠀⡀⢀⠈⠀⠀⣤⠀⠀⡀⢀⠀⢀⡐⣨
⠀⠀⠀⢂⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠆⠀⠀⡤⣶⠆⢃⠀⣀⡀⠀⠀⠀⠀⠘⠿⠀⠸⡿⠁⠀⠈⠀⠀⠙⠎⠀⠀⢱⠇
⠀⠰⠶⣷⠶⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣬⡄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⡀⢠⣤⡴⠀⠀⠀⡀⢀⡀⠀⠀⠀⣤
⠀⣄⠀⠀⢀⡀⠈⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⡊⠉⢠⣀⡦⣴⡂⡄⠀⢠⠀⢠⠴⣄⠀⢾⢟⠀⠈⠁⠀⠀⠻⠂⠈⠀⠀⠀⠀⠀⠏
⠀⣽⣷⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⠀⠈⠀⠃⠙⠁⠙⠉⠉⠋⠉⠉⠁⠀⠀⠀⠂⠀⠀⢤⣆⠒⢸⠴⣪⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠈⠀⠉⠀⠀⠤⠀⠀⠀⡶⠶⠎⠀⠀⠀⠀⠢⠀⠀⢀⣺⣭⢸⠷⢺⢶⣂⡆⡶⢤⡦⣖⣡⠀⠐⠂⠀⠀⠀⠠⡐⢾⠀⠓⠃⠄⠀⠀⠀⠋⠀⠀⠁
⠀⢠⡈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⣹⣉⡱⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⢰⣿⣿⠂⠀⠀⠀⠀⢠⡄⠀⣠⠂⠀⠀⠱⡀⠀⠀⠀⠾⠂
⠀⢀⣟⣛⣧⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠿⢀⣤⣄⠉⠉⠀⠀⠦⠀⠀⡾⠀⠀⠐⠰⣿⣾⢾⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢷⣄⡄
⠀⠀⠿⠿⠏⠀⢠⣠⠤⣄⠀⢀⣀⡀⠀⢠⣾⣿⣿⣶⡀⠘⣾⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣌⣠⣦⠄⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣀⠀⡄⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢛⠄⠀⠀⠘⠘⠚⠙⠉⠀⠀⢰⣦⣀
⠀⣰⠀⡠⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⣤⣄⠉⠛⠟⠃⠀⠀⠀⠐⠂⠀⠆⠀⡿⣿⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⡟⠀⠃⠚⠒⠂⠀⠙⠑⠂⠈⠛⠉
⠀⣿⢰⠃⠐⢃⠌⠀⠀⠠⣿⣿⡇⠀⠛⠋⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠀⣤⣤⠄⠀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⠃
⠀⠀⠀⢠⡶⠶⢤⣀⠀⠀⠹⣆⠙⣆⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣼⡿⠁⠀⠀⢀⣠⣴⣶⡄
⠀⠀⠀⠸⣆⠀⢤⡈⠳⣄⠀⠸⣆⢸⡄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣸⡿⠁⠀⢠⣶⣿⢟⣿⡿⠃
⠀⠀⠀⠀⠈⢧⡀⠙⣆⠈⢳⡀⢹⠀⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣴⣿⠟⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣦⠈⠃⠀⢳⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣼⣿⣯⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠈⣿⠁⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠤⠄⠀⠀⠀⠈⠁⠀⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠦⠤⠶⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡀⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠈⢳⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠙⠓⠶⠶⠶⠶⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠶⠻⠷⣶⡦⣤⡴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⡿⢛⠉⠛⡿⠛⠛⠳⢶⣤⣠⣤⠼⣷⠉⣻⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⢠⣾⡟⢀⣤⣴⣿⡗⢀⣀⣴⣏⣿⡈⢿⡄⠸⣷⣿⠋⠉
⠀⠀⠀⣀⣠⢔⡥⠾⣿⣏⣤⣿⡻⣿⡿⢏⠘⢿⣶⣿⣿⣿⣿⣿⣿⣾⣷⣷⢉⣿⣧⣀⣀⣀⡀
⠀⠀⢏⣤⢔⡵⠞⠉⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠁⠀⠊⠯⣙⡉⠛⠻⠶⣤⣑⠒
⠀⠀⠛⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠋⠀⠀⠁⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠐⠷⠀⠀⢶⠄⢴⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⠀⠦⠶⠢⡀⣰⢰⣠⠖⣣⣴⠐⢀⠀⢀⣦⠛⠀⠀⠀⠀⠐⣄⠀⣀⣀
⠨⠤⢀⡂⠀⣠⡄⣔⠂⡈⠑⡄⠈⠐⠋⠀⠀⢦⠀⠶⢃⢄⡃⣎⡐⣢⣿⡿⠁⠉⠰⠇⡘⢋⠞⣁⣒⠀⠀⠀⠀⠀⠀⣾⣿⠿⢿
⠀⠠⠈⢃⢈⠛⠃⢬⠑⢱⠊⠈⣀⠂⠄⠔⠄⠞⠬⡋⢽⢀⠂⠜⠀⠀⠈⠁⠀⠠⡙⠓⣕⠡⠔⡛⠇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠃⠀⣀⠀⠘⠆⠁⠀⠀⠈⠈⠀⢢⡴⣒⠎⣠⡂⢌⠂⠟⠛⠛⠛⠹⠗⠶⣶⠤⠄⣑⡀⠘⡟⣧⠄⠀⠀⠀⠀⠀⠿⢿⣶⡾
⠶⢂⣠⡄⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠁⡆⠀⠛⠀⠘⠃⠀⠘⠁⠐⠇⠈⣀⣰⣗⠵⣀⠡⠠⠀⠀⠀⠀⠈⠀⠀⠘
⡦⢤⡷⣾⡆⠐⣄⠀⣀⢀⡀⠀⢀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠟⠛⠀⠛⡽⣦⡐⠰⠂
⠃⣸⠍⠈⠅⠀⣼⣿⠿⣿⠳⠆⠀⠸⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢀⠀⠀⠙⠉⣸⠠⢲⣀⣀⠀⡀⢀
⠀⠈⠀⠀⠀⢀⣿⣇⢀⣠⣵⠶⠆⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠄⠌⠀⣀⡃⢈⣘⣭⢰⠂⠡
⠄⠠⠀⠀⠀⢀⠉⠩⠖⠠⠃⠀⣤⠨⡄⠈⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠃⡀⠉⡃⢉⣂⠘⠤⣀⣀
⠀⠀⠀⠀⠈⠂⡀⠻⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡅⣧⠀⠁⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢤⠎⢯⠛⡟⡄⢚⠏
⠀⢀⠂⠀⢀⠀⠀⢀⡣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣇⢀⡀⣄⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠊⠉⠖⠃⢈⠒
⠀⠀⢐⠐⠂⠀⣶⣾⢦⠙⡡⠀⠀⠀⠀⠀⢠⠂⠀⡀⠀⢀⠀⠀⣀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡈⠻⠀⠀⠘⠇⠲⡉⢀
⠐⢿⢿⠾⠀⠀⠊⠠⢰⣾⣿⡄⠀⠀⠀⠀⠈⢆⠀⠀⡅⢐⡀⠀⠀⠀⠈⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠒⢨⣄⢀⣠⡗⡀⠈⠄
⣴⡡⠀⠀⠀⠁⠀⣄⣶⡊⢢⠠⠀⠀⠀⠀⠀⠈⠀⡏⠀⠈⠀⠀⢀⠀⠀⠘⢀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠠⠆⠩⢀⠜⢛⠁
⠢⢉⡇⠀⠀⢂⠈⠹⠛⢃⡅⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠘⠐⠀⠰⠀⠀⠀⠘⠀⠱⠔⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⡄⠈⠢⡄⠁
⠾⠀⠀⠠⠴⠆⠀⠐⡼⡾⣵⢠⠙⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣤⢄⠉⡠⠠⠎⠈⡀
⢀⡀⠀⠀⠙⠂⠀⠀⠈⠟⠢⠁⣀⣧⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣗⢾⢲⣿⠓⠀⢺⣶⠄⠁⠈
⠈⠉⠀⠠⡦⠴⠿⠙⢈⠡⠤⢂⡚⠙⠙⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢭⡀⠳⠲⠟⠘⠋⠠⠀⠻⠻⢀⡜⠃
⠀⡀⠀⣤⡄⠠⠄⠶⠀⡔⡰⢓⠂⣠⣦⣯⡶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆⠢⡊⢧⠙⡤⠄⢀⣤⣮⡱⠀⢀⣡⡀⠉
⢀⣄⠀⠈⠀⠀⠘⠦⠠⣾⠕⠉⢳⠪⠊⠉⠀⠥⠈⠍⠙⠛⠶⠷⠶⠾⠚⠗⡛⠉⠁⠐⠂⠁⠀⠀⠀⢊⠊⠟⠙⠌⠀⢸⠿⡇⠤
⠈⢋⠀⢠⢤⠔⣀⢀⣁⠁⢀⠾⠑⣆⠁⠈⠓⠀⠀⠀⠀⠄⠐⡀⠠⠂⠠⢂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠄⠀⠀⠐
⠀⢀⣤⡄⠂⠠⠈⠉⠈⠁⢸⡏⡀⠉⠀⣰⠆⠀⠀⠺⠂⠀⠸⣶⣦⡂⠂⠈⠀⠀⠀⠀⠱⣄⣤⣤⣄⠁⠀⠀⠀⠀⠀⠀⠆⠠⠈
⠀⠀⠁⠀⠀⠂⠐⢀⣬⢀⠉⢠⡟⣿⠁⠈⢀⠀⠄⢜⠀⠈⠀⠀⠀⠑⠀⢀⠀⠀⠂⠀⣼⣿⠛⠛⢿⣿⡀⠀⠀⠀⠀⠠⠀⠀⢄
⠀⠀⠨⣶⠒⠀⠀⠀⠩⠀⠀⢠⠛⠇⠀⠂⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠯⠀⠀⡼⡿⠅⠄⠀⠀⠀⠀⡜⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡧⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠈⠁⠀⠀⠁⢠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⡀
⠀⠐⣶⡄⠀⠀⠀⠀⣴⣏⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⣘⡒⠓⢲⠔⢢⠆
⠀⢠⣿⡇⠀⠀⠀⡀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢶⣛⡭⠥⠟⠛⠛⣡⡀
⠀⠸⣿⡇⢀⣴⣿⡟⠀⠀⠃⠀⠀⠠⢈⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠐⠠⠁⠰⠒⠃⣀⣊⡠⢔⠂
⠀⠀⠙⠿⠎⠉⠙⠃⠀⠉⠉⠈⠀⠚⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⠈⠆⠠⠄⢠⠞⠁⢀⣹⣿⣶⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⢂⣤⢀⢐⡠⢀⣠⣿⣿⣿⣿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡤⠜⠁⠀⠀⠙⠛⠋⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⡀⠀⠀⡴⣣⠴⠣⠀⢠⡏⠂⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡴⠉⠀⢸⠀⠀⠈⠒⢼⣿⣅⠀⡼⠀⢸⢀⢃⠔⠉⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡞⠀⠀⣠⢼⣲⠂⠀⠀⢸⣍⠻⠛⣦⡀⡎⢸⣷⠖⠲⣠⠤⢄⠘⣄⠀⠀⠀⢰⢆
⠀⠀⠀⠀⠠⡇⠀⠀⠀⢸⣷⡡⠀⠀⡘⢇⠀⢀⠉⠙⡿⠋⠁⠀⢀⠏⠀⠀⠻⣄⠱⡀⠀⠈⠀⡗⢄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⢿⣿⠖⢩⣷⡮⠭⠉⢷⣦⣇⠀⠀⠀⡏⠛⠁⠀⠀⠈⡶⠃⢐⠀⢸⣷⡬⠖⠋⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠃⠀⠊⠉⢻⣷⣶⠳⠤⢍⠳⡄⢾⠀⡄⢀⣠⣶⠟⠓⠤⢄⣆⢸⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⢀⠄⠚⡝⢻⡠⠀⠀⠓⠛⢻⣿⣿⡿⠋⢻⠀⠀⠀⠀⢿⣬⡝⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⡀⠀⠉⢸⠇⠀⢀⡄⢠⠞⠍⡞⠀⠀⢸⣖⣄⣀⣴⠋⠉⠓⢦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠔⠉⠉⣽⠁⠀⠈⠛⠚⡇⠀⠀⢘⣟⠛⣿⢸⣤⠢⠈⣢⡇⠀⠀⠀⢤⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠬⠷⣄⡀⣀⣧⣸⣄⠀⣾⣷⢰⣿⢸⣤⣟⠻⡁⠀⣀⡄⠀⠀⣧⠴⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠲⢿⡟⠳⡌⢙⠟⠀⠘⣿⠿⣷⢦⣿⣦⡃⣸⣿⣄⣠⡼⠯⠥⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠱⣄⣼⠸⠀⠀⡼⡇⢠⠃⢀⡿⠀⢻⢷⡄⣀⣀⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠷⣹⠙⠻⣿⡁⠀⠾⠞⢿⣟⣉⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⢧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠃⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠰⡄⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⣄⠀⠙⣄⠀⠀⠹⡄⢀⠀⡇⡀⢰⠇⡀⠀⡴⠁⠀⢀⡤⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠈⠳⢄⠈⢢⠐⢄⠱⠸⠄⡇⡇⡞⡜⢠⠎⡠⢀⡴⠋⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠠⡑⠄⠑⠌⠀⠁⠀⣇⠀⠐⡰⠃⢊⠔⠉⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⡀⠨⣑⠈⡀⠀⠠⢀⠠⠤⠤⠄⡈⠢⠀⡀⠔⣋⠤⠂⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠲⠶⠶⠤⢤⣄⣀⣀⡨⠁⠂⠁⢀⠋⡐⠁⠀⠀⠀⠀⠈⢢⠁⠀⠠⠀⠂⠉⣀⣀⣤⢄⣠⣤⣤⠖⠒⠶⠤⠶⠒⠲⠦
⠀⠀⠀⢀⣀⣀⣠⠤⠤⠤⠤⣀⡀⠀⠈⠉⠈⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡖⠉⠉
⠐⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠓⠤⢄⡀⠀⠀⠀⢣⣀⣀⣀⣀⣀⠤⠚⠂⠉⠉⠉⣡⠏⢁⣠⠀⠀⠠⠤⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⡀⠄⠂⢀⡀⠀⠀⠀⠀⢀⠀⠉⠑⠒⣍⣁⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⢁⣐⠊⢀⡀⠀⣠⠤⠒⠉⠉
⠀⠀⣀⡀⣀⡀⢀⠀⠀⡁⢈⣁⣐⢒⣤⣍⣀⣀⣈⣨⣽⣦⠄⠀⠀⠀⣀⡼⠁⠄⠐⣀⠀⠀⠀⠀⡩⠄⣲⠉⠲⡒⡢⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡼⣃⡤⠷⣬⠶⠷⠬⣁⣀⣀⣐⣠⣅⣀⣄⣈⣉⣀⣠⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⢖⡒⠒⢦⠶⠚⢛⡛⠉⠠⠤⠤⠤⠤⠤⠴⠀⠨⠉⠁⠀⠀⠈⠉⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠴⠚⠉⠉⠉⠉⠉⠉⠙⠯⢤⠀⠀⠀⠀⠤⠶⠮⠍⠤⠤⠥⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠒

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢆⢄⢀
⠀⠀⠀⠀⠀⠀⢹⣦⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣣⢯⢯⢿⢾⢸⢰⢠
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠩⣪⠎⢎⢞⡞⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠁⠘⠏⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡶⢦⠞⠛⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣷⣿⢀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⣾⣻⣿⢲⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠙⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠿⠄⠾⠯⠄
⢠⠀⢤⢤⡤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⡟⢻⠆⠀⠀⣀⣶⣖⣀⠀⠀⢰⡟⢹⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⢠⡟⠀⠀⡀⠀⡽⢿⠀⠀⠀⠀⢿⡀⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢂⣾⠃⠀⠀⠙⠂⡇⣸⠔⠋⠀⠀⠘⣷⡐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠇⣀⣠⣤⡁⠀⢹⡏⠀⢈⣤⣀⣀⠸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠙⠯⠉⢼⠆⠈⠁⠰⡧⠉⠽⠉⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⡎⢣⠀⠀⣾⡠⠐⠂⢄⣷⠀⠀⡔⢱⡄⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⣼⣾⡇⠀⣦⣿⡯⠁⠀⠀⠈⠽⣿⣶⠀⢸⣧⣧⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡼⠘⡇⠀⢸⡟⢠⣂⡀⢀⣑⡄⢻⡇⠀⢸⠃⢧⡀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⡉⢁⠀⠈⠀⢸⠃⠈⠵⠏⠹⠮⠁⠘⡇⠀⠀⠀⡈⢉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧⡙⠦⡄⠀⠀⢸⡀⠀⣀⡀⠀⣀⠀⢀⡇⠀⠀⠠⠴⢋⡼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠈⠀⠀⠀⢦⡈⠛⠫⠤⠤⠤⠬⠝⠛⢁⡴⠀⠀⠀⠁⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠖⠒⠒⠶⠟⣻⠀⢀⡀⠀⠀⣀⡀⠀⣟⠻⠶⠒⠒⠲⠦⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡶⣛⣁⠀⠀⢠⣟⠀⠁⡤⠂⠘⢥⠈⠀⣻⡄⠀⠀⣈⣛⢶⣌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢌⡞⠀⠀⠉⠑⢦⠈⠀⠀⠁⡴⠊⠉⠀⠀⢣⡡⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⣤⣍⡑⠢⢄⠀⠳⠀⠀⠞⠀⡠⠔⢊⣡⣤⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡑⠀⠀⠀⠀⠊⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣒⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣿⠟⠛⠿⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠟⢋⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣼⠏⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⢸⡇⠀⠀⠀⠀⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠀⠀⠀⠀⠂⠸⣧⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣿⠇⠀⢠⡇⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢰⡟⠀⠀⢰⡏⠀⠀⣾⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⣰⣿⠃⠀⠀⢸⡏⢀⣾⠇⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣠⡴⠿⠿⠿⠷⡶⣿⣿⠀⠀⠀⠈⣶⣿⡏⣰⣿⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣴⡟⠁⠀⠆⠠⠀⠄⠀⠙⢿⣆⠀⠀⠀⣿⣯⣼⡿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⡃⠀⠀⠈⠀⠀⢀⣀⣀⣈⠉⠻⣾⡀⠀⣿⠟⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠻⣷⠀⠀⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⡀⡹⠆⠀⠀⢠⠀⡄⣠⠀⠀⠛⠃⠀⢻⣆⣼⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠀⢹⣏⠁⠀⠀⠀⠈⠀⢿⣷⠃⣰⣮⡀⠀⠘⣿⣿⡀⢹⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⠀⠁⠘⢻⣿⡄⠸⡟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠰⠀⠰⠀⠁⠄⠀⠀⠀⠐⠿⠿⠍⠀⠀⠀⢀⣽⣧⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⣠⣾⠏⠀⠀⠀⣠⡄⣤⣤⣀⣀⠀⠀⠀⠀⠀⠛⢲⣄⠀⠀⠀⣬⠻⣦⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡿⠀⠀⠀⠶⠟⠉⠉⠙⠛⠛⠁⠀⠀⠀⠐⠊⠛⣿⣦⡀⢀⠀⢀⢹⡆⠀⢀⠂⠀⠘⢿⡆
⠀⠀⠀⠀⠀⢀⣀⠀⠤⣿⣿⢻⡗⠀⢲⠀⠰⣄⡀⢀⣠⡶⠀⠀⣀⠀⠀⠀⠀⠀⢿⡁⠀⠀⣦⢸⣌⠁⠀⢈⠒⠀⠀⢺⣿⣷⡄
⠀⠀⠀⠈⠁⢀⡀⠤⠘⣿⡟⠙⣧⡀⠈⠧⡀⠈⢿⡿⠋⠀⠀⠰⠃⠀⢀⠀⡉⠁⠞⠦⣀⠀⣠⣼⠇⠀⠀⠈⠃⠀⠀⣄⢻⣿⣷⡆
⠀⠀⠀⠀⠈⠀⠀⡀⠀⣿⡏⠀⠈⠿⣆⣀⠀⠀⣀⣷⣄⡀⠀⠀⠀⠢⣄⠈⣳⠇⠀⠀⣘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⡇⠀⠀⠀⠈⠙⠳⢿⣍⣉⣉⣻⣶⣦⣴⣶⣿⣿⣷⣾⣷⣾⡿⠛⠁⠀⠀⠀⠀⠀⣀⠀⢠⠘⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣠⠄⢠⠀⠀⠀⠈⠙⠛⠋⠉⠉⠈⠉⠉⠉⠉⠉⠀⠀⠢⠀⠀⠀⠀⢀⣤⢀⣿⡄⠌⠀⠀⣿⡟⢟⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⢿⣿⣇⢸⡄⠀⢀⠘⠳⡄⣄⣀⢠⠀⣀⠠⠘⠋⠀⠀⠀⠀⠀⠀⡄⣼⣾⣷⣿⡟⠁⠀⠀⠀⣿⣷⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠙⢿⣷⣴⡄⢷⡀⠀⠈⠉⠻⠀⠏⠁⠀⣰⢆⠀⣀⣠⢀⣼⣷⣿⣿⡿⠋⠀⠀⠀⠀⠠⣼⡏⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠈⠙⢿⢿⣬⣷⡀⢀⣀⣀⣀⠀⢀⣴⣿⢟⣴⣿⣧⣾⠟⠋⠛⠋⣀⠀⠀⢠⠀⢀⠀⣿⣷⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠠⠀⠁⢠⠂⠀⡀⠀⣧⣿⣿⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⣿⣿⣿⠻⠛⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⣿⣿⠇⠀⡀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠙⣷⣞⡀⠀⢸⡆⡄⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⢠⢀⡄⠌⠀⠀⣿⠟⠁⣀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠈⣿⣧⠀⠀⠃⠇⢸⡄⢶⡀⠲⡄⠘⠀⠀⠀⠰⠀⡤⢀⣶⣷⡿⣡⢏⡾⠀⠀⠀⣼⣟⣤⣾⣋⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢻⣿⠀⡀⢀⠘⢷⠀⠙⢷⠀⠀⠈⠀⠀⣴⠞⢡⡾⠋⠉⠀⠁⠈⠀⡀⢠⣿⣿⣿⣿⣿⣿⣿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠈⣿⣇⣷⢸⣀⣶⣀⠀⢀⠀⠀⣤⡄⢀⠉⠀⢛⡀⠀⠀⠀⠀⠀⠸⣷⣸⣿⠿⠛⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⣿⣿⠟⠛⠛⠻⣷⣌⠙⢦⢈⣷⡸⣰⢃⡿⢁⡴⠞⠛⠛⢲⣦⣽⣿⡇⡄⢠⡆⣄⣿⣧
⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⢿⡏⠀⡾⠁⠀⠘⣿⣷⣸⣿⣿⣿⣿⣾⣷⡿⠁⠀⠈⣷⠀⣿⣻⣿⣷⣷⣾⣷⣿⣿⣿⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣠⣽⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠷⣦⡀⠀⢀⣼⣿⣿⠗⣀⣀⣀⣀⠀⠀⠀⠰⠷⡞⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡅⠀⠀⠀⠈⢻⡄⣾⢟⣩⠖⠛⠉⠉⠉⠉⣹⡷⠂⠀⠐⡿⢀⣠⠞⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣿⡇⣦⠀⠀⠀⠀⡧⢫⠞⠁⠀⣀⣤⠄⠀⣼⠏⣤⠀⠀⢀⡗⣉⠀⠀⣼⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣇⣿⣧⢰⠀⠀⠃⠀⢀⢦⣾⣫⣤⡄⣴⠋⢠⣿⡆⢀⢈⡾⠃⠀⡰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠻⣿⡆⣇⠀⠀⣄⣾⣾⡿⠟⠁⠊⠀⠀⢸⣿⣿⣼⣨⣾⠁⠈⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠂⠀⠈⢃⡿⠴⠚⣿⡏⠉⠀⠀⠀⠈⠙⠓⢮⣻⠇⣻⣿⣷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⡴⠦⣬⣶⣏⣀⣤⣾⡿⢁⡶⠒⠈⠡⢤⣀⣀⢰⣽⣾⡿⠀⠀⣴⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣤⣀⠤⣬⣽⡿⢿⠟⢃⣤⣭⣍⠉⠉⢠⣛⡳⠦⣤⣤⣶⣾⣿⣿⣷⠟⣠⣤⣤⠸⣷⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⣻⣿⣱⢟⣵⢸⠁⠀⠀⢸⣿⣿⡷⠀⠀⢈⣩⣤⢶⣿⡽⣿⡻⣿⣧⡿⠀⠀⢀⣿⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⡿⣿⡈⠀⠀⣸⠘⠿⣟⣥⠀⠀⢉⠛⠧⣽⣦⡀⠀⠈⠊⣿⡇⠀⡄⣾⢏⣴⡿⠿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡏⢏⣿⣬⣁⣀⣠⠏⠀⣀⣭⠷⠾⢿⡜⣷⡀⠀⠙⠿⣧⠱⡄⡏⢠⣾⠱⠏⢚⣵⡾⠿⠟⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣀⣉⣉⠁⠈⠳⢿⣿⣵⢋⡴⠞⢩⣶⠾⠿⢸⡇⣿⣷⠀⠀⢀⣈⠃⠚⣵⠛⣇⣀⣚⡛⠛⠛⠉⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣍⠉⠉⠉⠙⠓⢦⡀⠈⢀⡞⣴⣧⡞⠁⠀⠀⢸⡇⣿⣿⠀⣴⣿⣿⠇⡀⠻⣤⣿⡟⢷⣤⣴⣆⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣇⣿⡀⣤⡠⣀⠀⢀⠁⠀⠀⣸⣿⢋⠔⠀⠀⠀⣼⢇⣿⢃⣠⣿⠟⢡⣾⡇⣷⢬⡙⠛⠶⡀⠉⠉⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡻⡸⣷⣿⣷⣌⢳⡌⣆⠀⠀⠿⣵⡿⠀⠀⢀⣼⣫⡾⣣⣿⡿⠉⡏⡿⠛⠈⠁⢀⡇⢸⣷⡍⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣙⢮⣟⣻⣿⣳⡿⢋⡴⠖⠶⣦⡙⣦⠖⠋⠚⢩⣌⣛⠋⠀⠀⣧⠀⠀⠀⣠⡞⣷⢈⢷⣍⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣶⢯⡀⢌⣉⣛⡛⠃⣿⡀⠀⠀⢸⣧⢐⠚⠛⠻⠿⢓⣮⣝⢶⣄⠹⠷⠾⠟⠉⢸⣿⣯⡳⣮⣳⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣿⣿⢟⡿⣫⣶⣿⣿⡟⣡⣆⠹⣿⣶⣶⣿⠇⣼⣿⠶⠦⠬⠭⠍⠙⠓⠹⣆⠀⠀⠀⠀⠀⠙⢿⣿⡎⠙⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢫⡿⣁⣠⣤⠞⠩⠞⠁⣸⡷⢠⣭⣭⢡⡈⠻⢦⣄⡀⠀⠀⠀⠀⣀⡴⣛⣤⣶⡟⠀⠀⠀⢀⠙⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡿⠋⠁⠀⠀⠀⣰⡿⢁⡿⠉⣿⢸⣿⣆⢀⢯⣝⣛⣛⣿⠍⠀⠾⠿⣟⣻⣃⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠻⠇⠀⠀⢀⣠⡾⣟⣵⢺⡇⠀⠸⠸⡿⣿⡌⡇⠿⠿⠛⠁⠀⠂⠉⠉⠉⢛⣿⡟⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⢸⣿⣿⡸⣇⠀⠀⠀⠃⢹⢧⡇⠐⢶⣤⣤⣤⣤⣴⣶⣾⠟⠋⠀⣠⡟⣴⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠙⢆⠀⠀⠀⠞⣾⠁⠀⠀⠛⠿⣦⣉⠉⠉⠀⠀⣀⡴⠋⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠳⣦⣠⣾⠃⠀⠀⠀⠀⠁⠒⠤⠤⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠉⠉⠉
⠀⠑⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠈⢿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇⠀⡅⡃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣀⣀⣤⠤⢄⡀⠀⠀⣾⠑⠄⠀⠀⠀⡠⠞⡠⠚⡠⠃⠀⠼⠋⠅⠀⠀⠀⠀⠀⣸⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣨⣭⣈⠉⢢⡈⢆⠀⠉⠀⣄⠀⠀⠎⠁⠀⢀⣴⣁⠔⠂⣠⠔⠁⠀⠀⠀⠀⠀⣼⠃⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠳⠀⡇⢨⡀⠘⠀⠸⡖⠀⢀⠀⣀⣆⣻⠁⠀⢰⡟⡀⠠⢀⠀⠀⠀⠚⠁⣠⢆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠘⢷⣇⠀⣠⠴⣛⣋⡭⠭⠭⠭⣍⣛⠲⢦⣠⠈⠀⠀⠀⠀⠀⡉⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡾⣫⠴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⡳⣤⢨⠾⠭⣀⡀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠉⠱⣀⣀⠠⠁⣴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢳⡀⠀⣀⣠⠀⣁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠈⠙⠓⢸⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢳⡈⢉⠁
⠀⠀⠀⠀⠀⣠⣺⣍⣉⡉⠢⣄⣳⡀⢀⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡗⠒⠠⠐⢄⠀⠀⠀⠀⢀⣤⠶⣤⡀
⠀⠀⠀⢀⡼⠛⣁⣤⣄⡈⠢⠈⠙⠭⢽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠈⠂⠑⠤⠴⠶⠛⣡⡶⠋⠉
⠀⠘⢿⣯⣰⡾⠋⠀⠀⠈⠢⡀⢀⠀⢘⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡿⠚⢿⠳⠄⡑⠦⠤⠤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⢧⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⠈⢣⠀⠉⠓⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠤⠾⠂⠈⢧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⡟⢀⠽⠓⢄⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⠆⠀⠠⠤⣀⣼⣳⣌⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠃⠈⠄⠄⠀⠈⠐⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠒⠛⠓⠀⠈⢱⢮⣙⠲⠤⢤⣄⣠⣤⠤⠖⣚⡩⠖⠛⣬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣡⠒⠉⣁⣀⠀⠀⡂⠀⢀⢈⠉⠙⢲⡶⠖⠒⢚⠋⠉⢄⠀⢀⠏⢧⠀⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⠟⠁⠀⠀⠀⠀⠀⡸⠃⢀⡠⢺⡇⠀⠀⠈⠀⠈⠙⠆⠈⠀⠘⡀⢆⠈⠛⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣿⠀⠀⠀⠀⠀⠀⡉⢀⠀⠀⢠⠉⡠⠀⡡⠊⠀⠀⠰⡀⢸⡆⠀⠙⠢⢟⣶⠶⠴⠦⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶⠟⠀⠀⠀⠀⠀⠀⠓⠋⠀⠰⡃⡞⠀⣾⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣣⡀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣟⣻⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠀⣀⣄⠀⠲⠐⠂
⠀⠀⠀⠀⠀⢀⡀⣐⡻⠿⡄⣥⠃⠀⠀⠀⠠⠄⠐⠀⠀⣀⣠⠼⠶⠖⠒⠒⠐⣺⢿⠉⠁⠀⠀⢀⣀⡄
⠀⠀⠀⠀⢠⣽⢣⣛⠙⠲⠖⠊⠁⢀⠀⡄⣀⠄⠁⠄⠀⠀⣀⠀⡀⠀⠀⠈⠀⠈⠓⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢠⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠠⣞⠇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠡⠀⠁⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠉⠃⠀⠀⠐⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⢸⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠚⢤⠄⠀⢠⠀⡀⡀⠠⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠂⠰⠍
⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠠⠀⠠⠀⠈⡐⠉⠉⠑⡃⠀⠁⠀⠀⠠⠤⠀⠀⢰⣶⡄⠀⠀⢰⣖⡆
⠀⠀⠀⠀⠠⠏⠀⢀⠀⠀⡀⠤⠼⠂⠀⠀⢶⣄⠂⠀⠜⠀⠀⡀⠀⠀⠀⠀⠀⠠⡀⢪⠐⠀⠀⢀⠀⡄
⠆⠀⠀⠀⠀⠀⢀⡛⠛⠁⠚⠒⠀⣤⣠⡄⠀⠹⣤⡠⢀⠄⠀⠛⠀⠀⠀⠀⠀⠋⠜⡄⠁⠀⢐⢠⠙
⠀⠀⠀⠀⠀⠀⠙⠁⠐⠒⠀⢀⢋⠁⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠲⠗⠐⠀⠈⠀⡛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣲⠖⠚⠉⠉⠁⠒⠒⠒⠒⠲⠶⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢠⠞⠁⠀⡠⠴⠶⠤⣄⠀⠀⠀⠀⠀⠈⠓⢝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠁⠀⠀⢰⠋⠀⡠⠋⠀⠀⠀⠀⠈⢣⠀⠀⠀⡀⠀⠀⠀⠊⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠀⠀⠀⠐⡀⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⢠⡀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡆⢠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡴⠚⡇⠀⠀⠈⡆⠀⠀⣶⣾⡇⠀⢠⠇⠀⣧⣴⡇⠀⠀⡀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢠⠎⠀⢀⡇⠀⠀⠀⢳⡀⣀⣈⣉⠀⠀⡞⠀⠀⢀⠛⠁⠀⢀⠄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⣴⠥⠤⠤⠄⣿⡰⠖⠋⠀⠀⠀⠀⠉⠉⠛⠷⠒⠒⠚⢦⡀⠀⠮⣴⡃
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠄⠀⠀⠀⠀⠈⢶⠀⡇
⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⣰⠃⠀⠀⠠⠄⠐⠂⢲⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⣇
⠀⡠⠊⠁⠀⠀⠀⠀⠀⢐⠏⠀⠀⠀⠀⠀⢰⡄⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿
⡸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠹⢦⠂⠀⠀⠀⠀⠀⠀⢀⡠⠮⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣯
⢧⡀⠀⠀⠀⢀⣀⣠⠤⢺⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣠⠀⢈⠛⠒⠒⠒⠒⣞⠁⠀⢀⡏⠑⠂⠒⠂⠳⠉⠀⠀⠀⡇
⠀⠉⠉⠉⠉⠉⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠁⢉⠉⠀⣀⡀⠀⠀⢀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢣⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⠆⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡸⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠑⠀⠤⠄⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⡖⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⢀⠀⠀⠀⢠⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠌⠀⢀⢀⡴⠁⠀⢱⡄⠀⠘⡀⠀⢠⠎⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠣⠤⠴⠚⠁⠀⠀⠀⠀⠉⠐⠀⠓⠚⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣀⡏
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠠⡀⠀⠀⡀⢣⠀⠀⠀⢸⠃⠀⡀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢁⠀⠀⢠⢸⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⣀⣨⣀⡀⠀⠀⢀⣀⣼⣄⣠⠼⠞⠒⠒⠒⠛⠦⠤⠦⣄⣠⡤⢆⣀⡤⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣤⣄⣀
⣿⣿⣿⡿⣿⣶⣦⣤⣀⡀⠀⠀⠀⣀⣠⣤⢶⣶⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⣶⣿⣭⣿⠩⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⡏⢽⠮⢩⣴⣶⣶⣶⣶⣶⣿⡿⣿⣿⣶⣤⣀
⣍⣛⡛⣿⣿⠿⣻⢻⣿⣿⣿⣿⣿⠽⢿⣏⣫⣗⣤⢤⡾⠻⠾⣿⣿⣿⣿⣼⣿⣿⣷⠘⡿⣿⣿⣿⣷⣤
⣿⣿⣿⣿⣿⣿⡾⣿⢻⣿⣿⣻⠧⢉⡥⢞⣝⣔⡿⣿⣶⣶⡜⢿⣯⣟⠿⠿⠛⠍⠛⠤⣼⣿⣿⣿⣿⣿⣷⣤⡀
⣩⡭⠶⠿⠒⢺⣵⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣻⣿⣿⣿⣿⣯⣐⣍⣉⣋⣇⣀⣴⣷⣿⣿⣻⣻⣻⡟⣻⡿⢟⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⡻⡿⢿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡗⢠⠙⠻⣿⡿
⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣏⠞⡠⠰⠀⠀⡸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⣿⣿⣿⣻⡟⠉⢉⡁⣠⣴⡟⣿⣷⣉⡯⣴⣠⢃⣂⡌⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣋⣞⠀⠈⠁⠉⠛⠷⠒⣟⢰⠟⠚⠹⣻⣿⣿⣿⣿⣾⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⠺⢶⠄⡐⠁⠀⠀⠀⠀⠀⣠⡷⠏⠁⠁⡈⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣶⣳⢺⣶⢦⣴⣾⡷⠟⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠌⠿⣿⣿⣿⡿⡿⣣⠟⠛⠿⡏⣿
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢸⣅⣙⣮⠑⠒⠺⡁⠁⢀⡙⣏
⣭⣿⣿⣶⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠁⠈⠤⠄⠀⣀⣵⡇
⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⣳⠀⠀⠀⠀⠀⣌⡻⠁
⣿⣷⣿⢙⡟⣿⠿⠛⠻⠟⣿⢯⡻⣿⣿⣿⣟⡷⣮⢆⡀⠀⠀⣾⡿⠃
⣿⣿⣏⣞⣩⠿⢦⢭⡄⣡⠀⢀⣿⠑⢿⣿⣿⣧⠀⢦⠄⠆⣾⣿
⣿⣿⣿⢻⢜⢷⡄⠆⠂⢈⡘⣆⠉⠃⠸⣿⣿⡟⡛⠃⢀⣴⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⡦⣴⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣯⡿⣿⣟⣻⣿⣯⡅⣨⣭⣿⣟⣻⣿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⡜⢟⣫⣽⡾⠿⢿⣿⣿⣿⣿⡟⢻⣿⠿⠿⢿⠿⢿⣷⣯⣝⡻⢣⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢟⣽⣶⣌⢿⣧⣀⠀⠀⠀⠀⠹⣿⣇⣾⡷⠀⠀⠀⠀⠀⠀⣻⡿⣡⣶⣯⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣿⣿⣿⡿⣹⣎⢿⣿⣇⣴⣀⣤⣴⡌⣷⣽⣅⣸⣶⡆⢠⣤⣄⡙⣱⣿⢫⣯⣿⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⣿⣿⡟⠉⠀⠀⢈⢻⣦⢻⣿⣿⣿⣿⣽⣾⣧⢾⣿⣿⣿⣷⣾⣿⡟⣴⣟⠋⠀⠙⢿⣿⡿⢿⡻⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣝⠿⢳⣿⣿⡛⠁⠀⠀⢀⢼⣻⣽⣧⡻⠿⣛⣫⣭⣭⣥⠈⠉⠉⠙⠻⠿⢟⣼⣿⣿⣿⣦⠀⠀⢀⡄⢹⣿⡞⠟⣫⣄
⠀⠀⠀⠀⣰⣿⡿⣱⣶⣭⡻⢧⣶⣤⣤⢻⣧⣿⡿⠋⠀⠸⣿⣿⠿⠿⡗⢀⡀⠀⠀⠀⢀⣤⣝⢿⣿⣿⣣⣴⣿⣿⡿⢛⣭⣶⣎⢿⡿⣆
⠀⠀⠀⢠⣟⣿⣳⣿⣿⣿⣿⣷⣮⣝⠿⣿⡿⠋⠀⠀⠀⣀⡀⢖⣫⠩⠥⠬⢭⣝⡲⢀⣙⢿⣿⣷⣝⢿⣿⠿⣋⣵⣾⣿⢿⢿⣿⣿⣿⢿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⣦⣭⣏⣿⡆⣀⡀⠀⢠⠞⡡⠊⠁⠀⠀⠀⠀⠀⠈⢿⣷⣌⠳⡝⠿⠟⠀⢰⣿⣿⡟⠁⠀⠈⢿⣿⣿⣿⣿⣧
⠀⠀⢠⣿⣿⣹⣿⣿⠟⠃⢠⣿⣿⣿⣿⢱⣿⣿⢣⢄⠞⠀⠀⠀⠀⠀⢾⡷⠀⠀⢈⣿⣿⣷⡱⡄⠀⠀⠈⣿⣻⣧⣄⠀⢀⠀⡀⢸⣏⣿⣻⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣀⣿⡎⣿⣿⣿⡇⣿⣿⡇⡏⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣧⢹⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣷⣿⣿⣿⡇
⠀⠀⢨⣭⣅⣩⣭⣭⣭⣭⣭⣭⣭⣭⡁⠉⠉⠁⡁⡇⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⢈⢀⣤⣤⢈⣭⣭⣭⣭⣭⣭⣭⣭⣍⣨⣭⡅
⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⠟⢉⣿⡇⠀⠀⠀⣧⢡⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣸⣿⣿⢸⣿⡟⠱⠶⣶⣮⡻⣿⣿⣿⣿⡇
⠀⠀⠘⣿⣿⣹⣿⡇⠀⠀⠀⠀⠁⠸⣿⡀⠀⠀⠘⢊⠧⡀⠀⢿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡟⡐⠣⣿⣿⢇⣿⠣⠀⠀⠀⠈⠿⢷⣾⣏⣿⣿⠁
⠀⠀⠀⢻⣿⣏⣿⣠⣸⣿⣦⣤⣄⣼⡿⢗⡀⣴⣿⣮⠳⣝⠦⣈⠿⣿⣿⣿⣿⣿⣿⠿⣫⠞⠁⠀⠀⢉⡺⢿⣾⣧⣀⣠⡄⠀⣨⣿⣽⣟⡟
⠀⠀⠀⠘⣿⢿⡽⣿⣿⣿⣿⠿⣛⣥⠾⣿⣷⣌⢿⣿⣿⡮⠁⠶⢭⣍⡙⢛⣩⡭⢶⣈⠁⠀⠀⠀⣠⣿⠛⢷⣬⣛⠿⣿⣿⣾⣿⢿⣿⣿⠃
⠀⠀⠀⠀⠸⣿⣿⡽⢟⣫⣴⣿⢿⠇⡉⠻⣿⣿⣷⣬⡛⠁⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡆⢀⣤⡾⢿⣿⠿⠀⠙⠻⣿⣦⣝⡻⢯⣿⣿⠇
⠀⠀⠀⠀⠀⠉⣭⣶⣜⣿⣿⣿⡟⣀⢻⡆⡻⢿⣿⡟⣼⣷⣶⣤⣄⣀⡀⢙⣛⣫⣭⣶⣾⣧⢻⣿⣧⠁⠀⢀⣴⣾⣿⣿⣿⣣⣶⣭⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣮⡻⣿⣿⣭⣤⣿⣶⣼⢏⣾⣿⠛⠉⠀⠄⠙⣯⣿⣿⣿⣯⠙⠿⣿⣧⡹⣿⣧⡀⠾⣻⣿⣿⢟⣵⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣮⡻⣿⣿⣿⣿⢋⣾⣿⣿⠆⠀⠀⠀⢰⡇⣿⣿⠿⠏⠀⣠⣿⣿⣷⡙⣿⣣⣤⣿⢟⣵⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣾⣝⡿⢣⣿⣿⠟⠃⠀⠀⣠⣿⣿⡇⣿⣿⣶⡄⠘⣿⣿⣿⣿⣿⡜⢿⣫⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣱⣾⣽⣛⡿⣶⣶⣿⣿⣿⣿⡏⣾⣿⣿⣦⣴⣿⡿⢿⣛⣯⣷⣎⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣶⣿⣿⣭⣽⣇⣹⣯⣭⣿⣿⣶⣻⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠗⠹⠿⠟⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠄⠉⠆⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣰⠞⠛⠷⣆⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢿⣀⣠⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠀⠀⢀⡟⣠⡴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣏⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢶⣄⡀⢠⡟⠀⠠⣄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠊⢳⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣾⠟⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣧⣀⠀⡄⢻⡄⠀⠀⠀⢀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣱⢠⡅⡐⠁⣸⠁⢀⡤⢞⣫⣥⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⢠⡟⣰⣟⡔⡩⡦⣾⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠙⠏⠈⠀⢣⣿⡟
⠀⠀⠀⢀⣀⣤⠤⠴⠶⠴⢤⣄⡀⠀⠀⠙⢶⣤⣴⠇⠀⠀⠀⣘⣠⡾⠋⠀⠈⠲⡄
⠀⠀⠀⠈⠻⣦⠸⣏⣽⠿⢷⣮⣥⡀⢀⡀⢼⣿⣿⠓⣶⠖⠚⠉⠙⣧⡀⠀⠀⢀⡆⣀⢀
⠀⠀⠀⠀⠀⠈⠳⣌⠨⠳⣄⡀⠩⠃⠀⠀⠀⠹⣇⠸⠇⠀⠀⠈⠀⣿⣷⣶⣶⣟⡃⠉⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡌⠛⢧⡀⠀⠀⠀⣴⠋⠀⠀⠀⠟⠻⢿⣿⣿⣶⣿⣏⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⣦⣶⣾⣧⠀⢀⣀⣀⠂⠀⠔⠒⠉⠛⣟⠛⠳⣄⣠⠾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣷⣾⣟⡏⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠁⠀⠹⠛⠻⢶⣍⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠁⠀⠹⣬⠀⠀⠀⠈⠀⣸⣧⠀⠀⣀⡄⠀⠠⣎⠤⠄⠴⢒⣙⣤⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⡿⢃⣤⠠⠶⠶⣾⣿⣟⣿⠇⠀⠀⠀⠈⠶⣤⢤⣤⢤⣤⢤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢋⣽⣤⢄⣀⠐⡛⠁⠀⠀⠀⠀⢻⣿⢿⣿⡀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣰⢟⢟⣀⣠⡴⣪⡇⠀⠀⠀⠀⢀⡼⠁⠀⠏⠛⡦⢤⣤⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⣏⡼⠭⠍⠒⣈⡴⠿⠁⠂⠄⠀⠚⠀⠀⠀⠀⠀⠐⡂⠻⢿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⡴⠶⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⣀⣴⡳⣄⣃⢳⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢷⣝⢮⣝⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣮⣍⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇