Noodle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣴⠰⠐⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠙⠀⠀⠄⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⢤⡀⠀⣠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡠⡘⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠠⡀⠈⢋⠀⠅⠰⠟⠀⢦⡀⠤⠒⡀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⢀⠁⠀⠀⠀⢠⠀⠀⡵⠐⢔⢂⢱⠇⡌⠐⠠⢱⢶⣤⣀⡀⠐⠀⠀⠀⠀⢡
⠀⠀⠀⢈⠀⠀⢠⢠⢀⠀⠋⠉⠃⡞⠑⠡⠀⠁⡸⠄⠠⠤⠀⠉⠉⠁⠤⠄⠀⠀⠀⡆
⠀⠀⠀⠘⡀⠀⠀⡶⣀⢂⡤⠀⠙⠓⢂⠲⣖⣀⡀⠀⣻⠀⠠⠁⠠⠚⠁⡀⠀⠀⢠⠁
⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⡀⠊⠓⠈⠁⠀⠤⠉⢉⢛⣓⠀⢀⣀⠈⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠙⠄⢀⠀⠀⠁⡀⠀⣄⡈⠁⠀⠀⠸⠃⠞⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⠈⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢷⠀⠀⠀⢠⣇⣀⣀⠀⠀⢀⡀⠀⠀⣀⣼⠀⢀⡿⢀⡞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣇⠀⠀⣸⣏⡤⠙⣇⣴⣏⣹⢀⡾⢹⡏⠀⣼⠇⢸⡏⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢹⡆⠀⡟⢹⠛⣹⠇⣿⣸⠟⠹⣧⠇⣧⡼⡿⠀⠀⢉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⣸⡇⠈⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⣛⣀⣤⠤⠤⠤⠶⠶⠶⠶⠤⠤⢤⡤⢤⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠐⠂⠉⠁⣀⣀⠈⡠⡄⠢⢷⢄⠙⢄⠀⠀⠈⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⢠⠋⢉⠙⠒⢢⣮⠭⣼⠋⢣⡳⡕⢌⠳⡕⡌⢧⠀⢀⠀⠈⣍⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣀⣠⢤⣧⡀⢀⣀⡠⠾⢶⢶⡟⡄⢧⡑⢌⠢⡱⣜⣜⢎⠒⠁⢑⣚⡠⠳⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠁⠀⢾⠁⢀⣠⠟⢙⡭⠒⣒⣚⡡⠤⠼⣜⢆⠳⣀⣱⡝⣄⠈⢆⠣⡀⠙⠦⡄⠀⠘⡄⠀⠙⢦
⠀⠀⠀⠀⠀⢸⡇⡠⠴⣒⣚⣚⣕⣋⡭⢗⣲⣊⡥⢖⣋⡉⠭⢹⡜⡄⠸⣉⠉⠪⡳⣜⢏⠺⡑⢄⠳⢄⠴⡻⢰⡀⠈⢧
⠀⠀⠀⠀⠀⠘⣿⢖⠭⠤⢜⣫⠥⢴⡊⠁⣠⠔⡪⢕⣲⠬⢖⡪⢳⡸⡀⠀⠧⠠⠼⣜⠮⣓⣊⣿⠴⠶⠮⠤⣈⠇⡀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⣖⢉⡽⠋⠀⠀⠀⠘⢦⡃⢮⡀⠻⠤⣄⣑⢄⣻⠱⣌⣓⢒⣒⡨⡽⢋⡟⣹⠁⠀⠀⠀⠀⠈⣇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡹⣾⠀⡰⠛⠳⡄⠀⠘⡗⠤⣌⣉⠓⢦⠀⠱⡹⡗⠚⢉⣍⡤⢬⣐⣪⠴⠓⠓⠒⢦⡀⠀⣠⡏⢀⡾
⠀⠀⠀⠀⠀⠀⠈⣇⠈⠻⣧⡀⠀⣿⠀⠀⣟⠀⣭⣌⡽⣦⣍⣉⣉⡻⢦⡪⢊⠜⣉⣀⡍⠀⠀⢠⡄⠈⢷⠞⠋⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠙⠶⣯⣀⢰⠇⣰⣮⣭⣥⣲⣥⠾⡮⠽⠦⢴⡞⠉⠁⠀⢲⡄⠀⠀⠀⠀⣰⡴⠞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣐⣊⡕⠊⠀⠀⠀⡼⠀⠀⠈⠀⣀⣧⣠⡤⠶⠛⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠶⠶⠶⠖⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠙⢛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣍⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢰⡿⣹⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣳⢏⡟⡷⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡽⣣⣿⣼⣁⢈⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⢫⢞⡽⣡⠞⢛⣿⣷⣮⡳⣞⡛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠲⣌⡓⠎⢿⢸⡸⢥⣷⢳⡼⢪⣷⠿⢯⣚⢫⠷⢂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣿⣯⣬⣷⣭⣻⣿⣟⣵⣏⣽⣿⣶⣼⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣄⣉⣉⣉⠉⠉⠛⠛⠋⠉⣉⣉⣉⣁⣤⣤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⢁⠈⠋⠁⠸⢄⣀⢤⢄⢠⣐⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⠄⠊⠃⠈⠈⢨⠐⡰⠤⣔⠀⠀⠀⠀⠐⠁⢡⠐⠂⠳⠤⡤⠤⠄⣀⡀⣀
⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠉⠁⠀⡀⠐⡱⠋⠉⠓⠄⠀⡔⠁⡀⢀⠀⢈⡠⠒⠒⠠⡠⠂⠉⠁⠆⣀⠉⠉⠢⡄⢀⡀⠈⠉⠒⠲⠤⢀⡀
⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠧⢀⡀⡀⠀⠹⢄⣐⠀⠁⠀⡅⢄⡁⠀⡘⠀⡀⠀⢀⡀⡀⢓⠑⠄⢬⣄⠽⠆⣀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⢫⡀⠀⠀⠀⠀⠀⠀⠄⢐⠄⢀⣀⡅⡀⠐⢄⠀⡤⠊⠀⠈⢉⠎⠀⠀⠈⠲⢅⡤⠐⠉⠀⠀⠀⢠⠀⠀⠄⠀⠙⠦⣀⠈⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠑⠦⣄⠀⠀⢀⠰⠊⠀⠨⠁⠀⠒⠀⠁⠢⢏⡀⠠⠄⠂⣜⣀⠀⠀⡜⠈⠣⢄⠀⠀⠀⢂⠌⠁⠈⠀⠈⢰⠀⠀⠀⠁⠢⢄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⢦⣄⣀⡐⣀⠀⠀⠀⠀⠀⠀⠈⠁⡲⠞⠀⠈⠑⠊⠠⠄⠈⠊⠑⠂⠠⡂⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⡛⠓⠶⠶⠴⠤⢦⣴⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⡤⠤⠽⠶⠒⢚⣟⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠀⠀⠀⠀⠀⣀⣰⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠒⠒⠚⠋⠉

⣿⠿⠿⠿⣿⢟⢿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣟⣻⣻⣿⣿⣟⣟⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣺⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠕⠊⠁⣀⠤⠒⠉⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⢉⡠⠤⠒⣉⠤⠔⣂⠥⠄⠺⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡪⠕⢊⡠⠄⣂⠥⢔⣂⠥⠐⣊⣁⣤⣴⠦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠝⠊⣁⣴⠮⢙⡂⠥⠒⠈⣁⡀⠤⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠮⢛⣩⡴⣞⠫⠕⠂⢉⡡⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣺⣿⣽⢿⠅⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣏⣿⢧⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⡽⣟⢯⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣳⢯⣟⡾⡽⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⣉⣉⣤⣤⣤⣤⣤⣤⣧⣏⣿⣾⣽⠁⢀⣀⣀⣀⣀⣀⣀⠉⠉⠉⠉⠛⠓⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⣠⡴⣞⣿⢽⣷⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡦⢤⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠂⠭⣙⡛⠛⠾⠿⠿⠶⠶⣿⣿⣷⣶⣿⣭⣟⣶⣝⣳⣽⣿⣿⣿⣿⣿⠭⠿⠷⠾⠟⠛⣛⡋⠥⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠬⠅⠀⠀⠉⠉⠐⠒⠒⠂⠠⠶⠤⠤⠤⠤⠀⠀⠠⠆⠀⠀⠠⠆⠀⠀⠐⠒⠒⠈⠉⠁⠀⠀⠨⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡒⠒⣂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⢉⠄⠒⡒⡀⠉⠪⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡬⠐⠈⠀⡜⠁⠈⢣⠁⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⡀⣼⣀⠈⠁⠀⡤⢀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠢⢈⣉⣀⡉⠄⡪⠊⠀⠀⠀⠀⠀⠀⢀⡀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡒⠤⠤⣀⣀⣀⠀⠀⠀⠉⠒⠂⠐⠒⠉⠀⠀⣀⣀⠀⠤⠄⢀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⢭⣭⣭⣭⣭⣭⣭⣭⡥⠤⠤⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠰⠐⠒⠒⠒⠚⠈⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣛⣘⠠⡄⣅⣉⣉⣉⠉⠅⠠⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠦⠦⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠁⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡡⢭⣝⠀⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡏⠛⠀⠠⠭⠀⢔⣍⡐⠠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣌⠄⡘⡇⠐⢤⡤⡀⢴⡤⠈⡦⠄⡈⡴⡂⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣝⣀⡘⡉⠀⠄⠓⠞⠁⠗⢄⠻⢋⠤⠫⡄⠀⣀⣱⠘⢓⣤⡉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠭⢄⠁⠂⡀⠉⣀⡆⣮⠤⠶⠟⢀⣀⡀⠇⠉⢉⣈⠄⢍⡺⢿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡫⡁⠬⣧⡀⢀⣀⡈⠋⢀⠁⣀⡢⠈⠬⠒⠛⠴⡄⡈⢀⠂⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⠛⠶⣾⣤⣃⣀⣚⠂⠊⠀⠀⡐⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⠶⠞⢛⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣬⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣉⣠⡤⠴⠚⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢈⠉⢉⠙⠛⠛⠛⠟⠛⠛⠛⢉⠩⠉⠉⠈⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠡⡀⠀⠈⠅⢒⣤⣀⢤⠀⢐⡖⠖⣦⢀⠀⠠⠀⠄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣊⢐⠀⢀⠀⢸⡷⡿⡿⣻⢯⢫⢣⠍⠹⡅⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢸⡇⣼⡳⣻⣾⣺⣿⠿⠧⠆⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⢀⠠⠩⠄⠍⠈⠡⠓⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠛⠋

⡶⣶⢶⢶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⣶⣶⠶⣶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣠⣤⣴⣶⣶⢄⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⣩⠟⠛⠉⠓⢦⠤⢄⣀⡰⠋⣀⣀⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠈⢧⠀⣰⢃⡞⡱⠚⠙⠲⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣄⣀⠀⡼⠀⣠⠔⠒⠒⢦⡀⠈⡶⢃⡜⡼⢁⡤⠤⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⢁⣀⣤⣤⣈⣙⡇⢠⡇⠀⡖⠀⠀⢳⠀⣷⠞⡴⢃⡞⠀⠀⠈⡇⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠛⠛⠛⠛⠛⠋⢁⡴⢋⡠⠤⠤⢄⡙⡇⠈⢧⡀⠉⠀⣠⠏⠀⡷⠊⡴⠋⡤⢄⠀⣠⠇⡼⡼⢳
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⠏⢀⠔⠊⠉⠉⠛⠋⠁⣠⠎⣠⠤⠤⠤⣈⠙⣄⠀⠉⠉⠉⠁⢀⠼⡃⢠⠛⢚⢥⣀⣉⡀⣿⡱⠁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⡟⢦⡞⢠⠏⢀⠔⠋⠉⣉⠉⠉⣀⠼⣉⠑⠦⣄⣀⣱⣤⣉⣉⠉⢹⣞⡓⣄⢑⡴⡓⢯⡎⠞⢳⢵⢀⠽⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠑⢾⣀⠏⣠⠞⢉⣩⠭⣉⡉⠲⡌⠓⠤⢄⠤⠤⠤⢄⡈⠙⢿⡀⠙⣼⣾⣋⠉⣶⠯⣒⡯⣊⣘⣦⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⠓⠧⣴⣋⡤⠦⢤⡙⢦⡈⠒⠶⠖⠒⠲⢤⡀⠈⢢⡀⠱⡴⣿⠕⢮⡆⠛⠉⢢⣼⠗⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⠦⣀⠀⠀⠀⠈⠙⠒⠦⠽⢤⣉⣙⣛⠉⠑⠲⣄⠉⢦⠀⢳⣀⣱⣹⣤⠤⠷⠒⠊⠁⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⠋⠉⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣍⡒⠒⠶⠦⠤⠤⠤⠤⠶⠶⠒⠒⣩⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠦⠤⠤⠤⠲⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡤⣦⣂⠤⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢖⠐⠢⣀⠀⠀⠀⠀⠁⠀⠈⠙⠿⣽⡻⠶⣭⣑⡲⠤⣀
⠀⠀⠀⠀⠈⠳⣅⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣝⠻⣷⢦⣍⣒⠤⣄
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠓⢦⣀⣤⠤⣤⣠⣶⣶⡦⠽⡻⡶⣭⣚⠿⣷⡦⣭⣑⠲⢤⣤⣤⢄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣼⣦⠀⠀⡤⠸⠿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣭⣒⠭⣛⡲⢤⣍⡛⠶⢬⡯⢲⣦⣤⣀
⠀⠀⠀⠀⢀⢔⡿⠋⠀⠀⡇⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⣀⣉⠛⢶⣭⣑⠪⢝⡓⠶⣬⣁⠒⠭⢙⣿⣦⡀
⠀⠀⠀⠀⣾⠘⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠯⡿⣿⣿⣟⡩⢋⡹⣼⣽⣭⡭⣻⢻⣻⣯⣻⣻⢶⣭⣑⠢⢍⡛⠷⣦⠬⣉⠓⠠⢄⡀
⠀⠀⠀⠀⠟⢾⣦⣀⣠⣔⣣⣀⣻⣀⣀⣀⡀⣼⡳⡆⠎⣷⣶⠿⣿⡽⣟⣡⣷⣿⢿⣿⢯⣽⣿⣶⣾⣿⣿⣶⡬⡉⠒⢭⣶⣬⣕⡢⠬⣉⠒⠢⠤⣀
⠀⠀⠀⠀⢸⠀⠉⠛⠿⣽⣻⡽⠧⢭⣯⣇⡋⣾⢨⠣⣘⢮⡳⣎⢕⡭⣶⣇⢣⢣⢮⢟⢾⡿⣿⣷⢹⣻⣿⣿⠿⢲⣭⡲⠤⣉⠓⠬⣕⡒⠭⢒⡢⠠⢉⠰⡆
⠀⠀⠀⠀⠀⣧⣄⡀⠀⠀⠀⠉⠉⠐⠒⠚⠹⠭⠤⠿⢾⣚⣿⣿⣿⣷⣿⣾⣿⣾⣞⣻⣛⣒⣋⡩⠭⠔⠒⠊⠉⠁⠀⢈⣱⡦⢉⡲⠤⣈⡑⠢⣄⡉⠛⠈⠁
⠀⠀⠀⠀⠀⠸⡄⠉⠓⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⣺⣥⡼⠀⠀⠈⠑⠢⣍⢳⡔⣻
⠀⠀⠀⠀⠀⠀⠹⣾⣛⡛⢷⢤⡈⠉⠙⢓⣲⣶⣦⠠⠤⠤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠤⠤⣴⠶⡿⣍⡉⢁⡴⡿⣫⣴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠙⣶⠬⡳⣕⢌⣑⣊⢕⡪⠍⢭⠳⡂⠤⠤⣤⢶⣺⣙⠿⡒⢤⣀⣀⢔⡯⢚⠭⠭⢝⣮⣉⠞⡴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠮⣺⣗⠾⣲⠷⠊⢩⣐⢽⣯⣶⣦⢕⡩⢄⡠⢽⣯⣖⠀⣒⡩⢖⣥⣮⣽⣿⣿⣿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡦⠤⠴⠯⠭⣍⠢⢍⣛⣛⣋⣱⠾⣓⣒⣻⣿⡯⠍⡿⣿⠿⢓⡭⠒⠒⠛⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣤⣄⣀⡀⠉⠒⠒⠀⠒⠂⠉⠀⠀⠈⠉⠙⠒⠒⢒⢉⣁⡀⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣭⣉⣉⡙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢉⣉⠩⠤⣒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠭⠭⠭⠭⠭⠭⠭⠭⠽⠶⠒⠂⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣶⣽⣤⣦⣭⣵⣥⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⢚⣛⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⡤⠶⠚⣛⣉⣭⡴⠶⠚⠛⠉⠉⣀⣀⣤⣴⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢛⣋⣩⣤⠴⠖⠛⠉⣁⣠⣤⣤⣶⡾⠿⣟⣛⣭⡭⠷⠖⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⣱⢞⣫⣭⢹⡶⣦⣤⡤⠶⢚⣛⣩⣥⠶⠖⠛⣉⣉⣤⣴⣶⠾⢿⣛⣛⣭⡭⠷⠖⠛⠋⠉⠁⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡀⠀⣀⣤⣾⠿⢿⣷⣿⣼⣿⣿⣾⠿⣯⣶⠞⠛⣉⣉⣤⣤⣶⠾⢟⣛⣭⡭⠶⠚⠛⠉⠉⠁⠀⠀⣀⣀⣠⣤⡤⠶⠖⠛⠛⠉⠉⠉
⠀⠈⣠⢞⣥⣾⣿⣿⣟⣿⣿⡿⣯⡻⣎⢳⡙⣏⢿⣿⣻⣿⠿⠿⠛⠛⠉⠉⠀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠉
⠀⣼⣿⣿⡿⠟⢛⣋⣩⣿⣝⣿⣎⣷⢹⡄⣿⣿⢈⣏⣁⣀⣠⣤⡤⠶⠞⠛⠛⢉⠁
⢀⣿⣿⣴⣶⣿⣿⣿⡭⠷⣿⣿⡟⣿⢸⣷⣇⡟⣼⣿⡉⠉
⣸⢿⣿⣿⣋⣉⣤⣴⣶⣾⣿⡿⣱⢇⣾⣽⣿⣷⡟⢿⣻⡄
⣿⣾⠈⠉⠉⠹⣟⣿⣿⣿⣿⣳⢫⣿⡿⣿⢿⣟⣧⠈⠳⠛
⠈⠁⠀⠀⠀⠀⣿⣿⢸⣹⡇⡏⣾⢸⣿⣼⣻⣿⣾⡷⢦⡀
⠀⠀⠀⠀⠀⢠⢟⡿⣯⣿⣧⢷⣿⡿⣿⡏⣿⠉⣿⣿⣷⢻
⠀⠀⠀⠀⠀⢻⣸⣷⣿⣿⢹⣧⡻⣷⣧⢻⡘⣿⠵⠋⡟⣿
⠀⠀⠀⠀⢠⣞⣽⣾⣿⡿⣿⣏⣷⣜⣿⣿⣳⣼⢷⡄⢿⣻⣄⡀
⠀⠀⠀⠀⢸⣽⡇⠈⢻⣿⣿⣿⣿⣽⣞⣿⣿⢿⣷⣻⠀⠙⢾⣝⢶
⠀⠀⠀⠀⣸⣿⠃⠀⠸⣷⣿⣿⡻⣯⣹⣿⡿⣿⣿⣿⠀⠀⠀⠿⠾
⠀⠀⢀⠀⣿⢿⡀⠀⠀⣿⣿⡏⣿⣶⢿⣿⡟⠋⢿⣿⡀
⠀⠀⠘⠆⠘⢯⣷⡀⠀⡟⣿⡇⣿⣷⠞⣼⠃⠀⠈⢻⣝⣦
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢷⡻⣷⣿⢣⡟⠁⠀⠀⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠀⢠⣿⣿⢫⣷⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⢣⢿⣿⣿⣿⣯⣴⣞⢲⡄
⠀⠀⠀⠀⠀⣾⣫⣷⣶⠖⠋⠀⢻⣷⡌⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠛⠁⣿⣿⠀⠀⠀⠀⢻⡿⣦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠙⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠤⠄⠐⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠀⣒⣒⣒⣀⡩⠥⠤⠤⠤⠄⠒⠒⠒⠒⠚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢭⠋⠻⠉⠀⠀⠀⠀⠀⠀⠒⠒⠐⠒⠒⠒⠒⠂⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠀⠒⠂⠀⡖⠂⠀⠂⠠⠤⠄⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠂⠀⢀⣭⠘⢢⠐⠨⠡⢄⢄⠂⠀⠀⠈⠉⠐⢤
⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠐⡒⠐⢤⡀⠈⣧⡀⠨⣕⣈⡈⣓⠤⣅⠁⠄⠈⠒⠂⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠆⠉⠐⠒⠤⠄⣠⣏⣀⣒⣒⣍⢻⣉⡀⣀⣀⣀⡺⣍⠷⠤⠖⠒⠉⡘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠐⠒⠂⠠⠀⠄⠐⠒⠊⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⡏⡟⢻⢻⠛⣿⡷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⡇⢸⢸⡇⣿⡇⠈⠻⣷⣦⣤⣤⣶⣶⣶⣶⠶⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⡇⣿⢸⢸⡇⣿⡇⠀⠀⣀⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⢸⠸⡇⣿⣧⣤⣾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⢸⢸⠀⡇⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣸⣧⣸⣸⣀⡇⢸⣇⣠⣤⣶⡶⠿⠿⠿⢶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢛⣻⣿⡿⠟⢛⣛⣛⣛⣛⣛⠻⠿⣿⣿⣡⠖⢒⣛⣉⣛⡒⠲⢬⣙⣿⣶⣶⡶⠿⠿⠿⠿⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣾⡟⣡⢖⣿⡿⢋⡥⠖⣋⣩⠭⠤⠤⠤⠬⢭⣓⠢⣝⠻⣾⣉⡤⠤⠤⣍⣓⣶⡿⢋⡡⠴⢒⣒⣋⣉⣙⣓⣒⠢⢭⡛⢷⣦⡀
⠀⠀⠀⠀⠀⣸⣟⣰⣃⣿⣏⣴⣋⣔⣋⣥⣖⣚⣛⣛⣛⣓⣶⣌⣹⣎⣷⣽⣿⣴⣶⣶⣤⣿⣏⣴⣋⣴⣋⣡⣴⣒⣒⣒⡲⠬⣙⠲⣍⢢⡙⣿⡄
⠀⠀⠀⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢶⣶⡄
⠀⠀⠀⠘⢿⣦⡘⠛⠀⠚⠛⠃⠀⠒⠒⠀⠐⠶⠖⠀⠰⠶⠖⠀⠛⠛⠃⠀⠛⠛⠃⠀⠛⠛⠃⠘⠛⠃⠀⠛⠛⠀⠲⠶⠂⠀⠶⠶⠂⠴⠶⠆⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠾⠿⢿⣿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠛⠉⠉⠉⠉⠁⠉⠉⠉⠙⠛⠛⠷⠶⣤⣀
⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠶⠚⡡⣚⠒⠤⣒⠲⠄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢃⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⣀⡴⢾⡄⠉⠚⢪⡡⠺⢔⠿⣋⠀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⡀⠖⠐⡾⠠⢂⠅⠆⢤⡔⠚⢀⡐⠒⢬⡀⣍⡛⠓⡀⢰⠤⠄⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡴⠁⢴⣯⣙⠕⠻⠆⢠⠛⡔⠂⠀⢒⠆⡐⢹⠖⡙⡖⣡⡆⠀⠀⠄⠠⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠸⡄⠀⠀⠀⢸⡇⠀⠀⡀⠚⢃⢘⠎⠀⠈⣢⠐⠀⠀⠒⢌⣯⢇⠠⠊⠠⣃⠀⠀⠦⠔⠀⠀⠀⡋⢂⠉⠄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⡸⠁⠀⣠⣏⠀⠸⠀⠄⢀⠀⠴⠀⠀⣨⡀⣳⢀⠠⡴⠄⠀⢶⡌⠔⡦⠏⣐⠃⠩⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠱⡀⠀⢸⣇⡬⢂⢀⡞⠁⠘⢦⣀⠐⠌⠑⠘⢒⡔⡨⠀⠀⣹⠈⠙⢷⣄⣀⠈⡒⠁⠀⠇⠨⡻⡟⠋⠪⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢣⠀⠀⢿⣇⢴⣾⡆⠀⣠⠏⠹⠛⠷⠖⠒⡎⠀⡰⠉⡀⢻⠀⠀⠨⣤⣍⠘⡀⣠⠴⢂⠂⢀⠯⠉⡓⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⡜⢉⠚⣢⡌⢠⣄⠑⠀⠐⠄⢈⣶⡑⠐⠄⠚⢄⠀⠂⠠⢈⣛⡒⢁⢊⠅⠀⠸⠸⡌⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢆⠀⠀⢻⣦⡰⠇⢌⡌⠀⠌⣡⣄⣂⠍⠁⠙⢮⣐⣐⡢⠛⠫⠋⠉⠑⡃⠁⢁⠚⢄⡀⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢆⠀⠀⠹⡻⢶⣔⡀⠐⡶⢊⠀⣀⣥⠀⠀⢠⣠⣠⢠⡀⠭⡄⢢⡭⠔⠠⠃⠀⢩⣵⠟⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⠻⢾⣤⣔⠋⠙⠃⠒⠡⠪⣤⠀⠉⠭⡣⠈⠒⠒⣈⣤⣽⠾⡋⢕⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⡄⠀⠀⠀⠳⡄⠀⡈⠉⠛⠿⠶⢷⣾⣶⣶⣴⣶⣾⣶⡷⠿⠟⠛⠉⢐⠠⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⠀⠀⠀⠀⠈⠳⣀⠀⠁⠀⠀⠀⠀⢀⣂⣀⡀⠀⠄⠀⠀⠀⠀⠌⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠱⡀⠀⠀⠀⠀⠀⣿⣶⣭⣙⡛⠶⠶⠾⠿⠶⠞⣛⣫⣥⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢆⢳⢀⣆⠀⠀⠀⠈⠻⣟⡻⠿⠿⢿⣿⡿⠿⠿⠿⣛⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡝⣆⢣⠸⣆⠀⠀⠀⠀⠀⠉⠛⠒⠒⠶⠶⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡆⢇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠿⠿⡿⢿⣟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣻⣿⣻⣻⣟⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢤⣴⣾⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣿⡽⣷⣿⢿⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⣶⣾⣿⣧⣿⣿⣯⣿⠽⠿⠽⠯⠭⣭⣭⣽⣿⣿⣖⣲⣶⣶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⣖⣺⣭⡷⠶⢛⣛⣭⣿⣿⣿⡶⣿⣷⣞⠳⡳⣭⠽⣟⣛⡿⢿⣾⠿⢗⣭⣻⢭⡛⠛⠷⢾⣭⣗⣲⢤⣀⡀
⠀⠀⠀⢀⡤⢞⣭⣷⣟⣋⣭⣴⢶⣿⣿⠿⣿⣛⣛⣿⢮⣾⣿⣇⣷⣧⡿⣟⣾⡿⡚⢯⡻⣯⣝⣓⢿⡵⠼⣶⣿⣷⣶⣭⡙⠻⣾⣭⡳⢤⡀
⠀⢀⡖⣩⣾⠟⣫⣿⣿⣿⣩⣿⢟⡭⣲⣿⣿⣿⣿⣿⣿⡽⡏⡆⣿⢿⡹⣻⢿⣪⢺⡿⣻⢳⣮⢭⣝⡻⣽⢺⣯⣯⣿⣿⣜⡦⣄⠙⠻⣷⣝⢲⡀
⢠⠏⣼⡿⢃⣾⣿⣯⣭⢟⡿⣗⣽⣞⣿⣿⣿⣛⣿⠿⣏⣿⢽⣇⣏⣾⢷⣿⣇⣟⢬⣚⣾⠿⠵⠿⢾⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⠘⣿⣷⣻
⡿⢸⣿⠳⡋⣯⣽⣿⣕⡻⣿⢟⣏⣾⡿⣿⣿⡸⣿⣿⢿⣯⢺⢸⣴⣿⣯⣷⡏⣿⠵⢿⣯⠲⠷⠷⡿⠷⠳⡛⠿⠟⠟⠟⢿⣿⢿⣿⣿⣿⣼⣿⡇
⣷⢸⣿⡄⠉⠙⣷⣿⣿⣿⣮⢻⣟⣿⢿⣿⣿⣿⡶⣶⢯⣿⣻⣸⣿⣟⢵⣿⣻⣽⣿⣷⣹⢻⢻⠿⣛⣛⣛⣻⣽⣿⣛⣿⣿⣿⣽⣿⣿⢿⢿⣿⡇
⠘⢦⠻⣿⣄⠀⠙⠫⢿⡶⢿⡿⢿⣾⣸⢿⣿⡾⣿⢟⣿⣿⣿⣕⣝⣿⣿⣿⢳⣾⣿⣿⡇⣿⡧⣫⣿⡾⣹⣿⣿⣿⣿⣿⣛⣽⣸⣻⠕⣣⣿⠟⡴
⠀⠈⠳⢌⡛⢷⣤⡀⠐⣿⡿⣟⣛⣦⣿⣻⡿⣽⣿⣾⣿⣿⣿⣿⢛⣟⣛⣻⣿⣿⣿⣟⣽⣿⢷⣾⣯⣶⣿⡿⣿⣛⣻⣿⡿⠟⢁⣤⡾⢛⡡⠞⠁
⠀⠀⠀⠀⠙⠲⢭⣛⠷⣦⣍⣙⡒⠒⠓⠻⢿⣮⣾⣭⣟⣻⣿⣿⡷⡯⢋⣿⣛⣿⣿⡿⡿⣻⢟⣿⣿⢿⣿⣿⠟⢉⣉⣥⣴⠾⣛⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠬⣝⣛⡿⠶⢶⣤⣬⣛⣻⣿⡿⠛⠋⠑⠚⠛⠚⠯⠽⠛⠓⢛⣛⣛⣛⣯⣥⡶⠶⠿⣛⣫⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠮⠭⠭⣭⣭⣽⣛⣛⣛⣛⣛⣛⣛⣛⣿⣭⣭⠭⠭⠵⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⢛⣉⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣖⣛⣩⣭⠴⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⢿⠿⡟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⠃⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⣧⣣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⡇⡇⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢹⢼⡄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡞⢸⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⢀⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⠤⣤⣤⣄⡀⣪⣄⢀⣡⣤⣴⢮⡗⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣮⣭⠮⠽⠯⠭⣿⣯⣥⣬⣭⣽⣶⣭⣇⣱⣄⣙⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡲⠖⠐⠶⠂⠠⠄⠠⠤⠠⠤⠠⣤⠌⣍⠍⣉⠍⣩⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣠⣤⣴⠶⠶⢟⣟⣻⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢶⢾⢿⣿⣿⣿⣿⣿⣭⡿⠶⠶⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⡏⣿⢸⢸⠈⡿⣾⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣟⣟⣛⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⢿⢸⢸⢀⡇⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢸⢸⢸⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⢸⢸⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢸⢸⠈⠘⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⢸⣸⣘⣀⣀⣗⣿⣇⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣭⣩⣽⡿⢛⣛⠋⠉⠛⣛⡛⠿⣷⣖⡪⡙⣷⡀
⠀⠀⠀⠀⢀⣴⠶⣛⡻⠿⠯⠝⣛⣛⢷⣶⣈⡀⢐⣒⡢⢍⣪⡿⢟⣛⣻⠷⣦⡀
⠀⠀⠀⢠⡿⡱⣫⢒⡭⢉⣉⣩⣕⣒⣕⣪⣻⣖⣒⣒⠪⣿⢫⢎⠕⣒⡲⢝⢎⢻⡄
⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣶
⠀⠀⠀⣿⢀⣀⢀⣀⠀⣀⢀⣀⠀⣀⢀⣀⠀⣀⠀⣀⡀⣀⡀⢀⡀⢀⣀⢀⣀⣀⢸⡏
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⠇
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⢄
⠀⠠⠈⠀⠁⠀⢀⠈⠀⣶⠃⠀⠀⡄⠀⣠⠊⡠⠒⠚⠂⠠⠀⠀⠑⠂⠀⠃⠀⠀⠞⢁⣀⡄⠀⠐⠉⠈⠀⠀⠀⢳
⠀⠀⣰⡀⢀⠀⢈⠁⠀⠈⠀⠀⠀⡆⠀⡇⢈⠀⠈⠀⠀⠒⠆⠀⠀⢀⡴⠀⠀⠈⠀⠘⠋⠃⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠇⠑⠂⠀⠀⠀⠀⡀⠀⠀⠀⡠⠁⠀⠁⡘⠧⢥⣄⠚⠀⣠⡀⠀⠈⠁⠀⠀⠘⣄⠀⠀⠀⠠⠀⠢⣄⠀⠀⢀⡜
⠀⠀⠐⢀⠁⠙⠁⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⢠⣀⡈⠁⠀⠉⠀⢀⠈⠀⠀⠀⠀⠈⠳⠤⢤⣄⠈⣀⠠⠤⠴⠏
⠀⠀⠀⢀⠁⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡓⠀⠀⡀⠀⡀⠁
⠀⠀⠀⠚⠀⠚⠆⠋⠂⠝⠓⠂⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠒⠂⠂⠐⠋
⠀⠀⢀⠀⠤⣐⣒⣒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⢀⠔⢁⣼⣿⣿⣿⡌⠱⠖⢦⠹⣷⠀⠀⠀⠀⠰⣶⠊⣱⠀⣀⣀⡄⢀⠀⠀⢠⠘⢧⠀⢀⡠⠤⠶⢒⠒⡒⠠⢄⢀⡠
⡾⢠⢶⡾⣻⠾⢷⣣⣴⡛⢧⣤⢹⡆⠀⠀⠘⠼⠟⢪⡃⢁⠦⣀⡀⡼⠀⠀⠈⠓⠌⣙⠻⢆⣖⣘⣁⠔⠁⠰⡄⠁⠂
⢧⠸⠛⠛⢥⣤⠺⢄⠉⠙⣿⡇⣾⠀⠀⠀⠀⠀⠀⢳⠂⠘⠦⣄⣠⠞⠀⠀⠀⠀⠀⠻⣾⣻⣭⡿⠗⠏⠀⣀⡀
⠘⢦⠀⠀⠀⡁⠀⠀⢄⡀⢈⣴⠃⠀⠀⠀⠀⠀⣀⠘⠦⠤⠖⠋⢀⠠⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋
⢀⣈⣳⣦⣠⣭⣤⣤⡴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠛⠛⠛⣋⣀
⠘⠛⠛⠛⢛⠛⠛⠛⠓⠒⠃⠀⠀⠀⠀⠀⣾⢄⡀⠤⣢⢀⠠⢀⠆⣠⠄⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠐⠛⠈⠉⠁
⠀⠀⠀⠀⠘⠀⠌⠇⠛⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⢛⣉⣉⣉⣥⣤⣤⣤⣤⣤⣤⣤⣍⣉⣉⣉⠛⠛⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠚⣉⣥⠴⠖⠛⠋⠉⠉⠉⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣈⠉⠉⠉⠛⠓⠶⢦⣄⣉⠛⠲⣤⣀
⠀⠀⠀⢀⣴⢟⣥⠖⠋⣉⣀⣀⢀⣀⣤⣤⡴⠞⣋⣩⣥⠤⠤⠶⠦⠤⢤⣭⣍⣙⡓⠶⣤⣀⡀⣀⣠⠿⠿⠻⣤⣽⠳⣄
⠀⠀⢠⡞⣵⠋⠁⣰⣿⣿⣿⣿⣿⣧⡀⠠⠶⣛⣭⡤⠶⣶⣶⠶⠒⣶⣶⣶⢤⣭⣙⠓⢦⢍⣹⡏⣩⠶⡛⢳⡄⢻⣦⡘⣷
⠀⠀⣸⣹⠃⣤⠆⣿⣿⣿⠟⠋⠙⠻⣿⣆⠈⢉⣠⠶⢛⣡⡴⠞⣋⣩⣤⣶⣶⠒⠛⠛⠳⢈⡧⣼⢣⡞⢁⡼⢃⣹⡎⢷⢸⡇
⠀⠀⣿⢸⡟⠁⠀⢿⣿⠃⠀⠀⠀⠀⢸⣿⣴⠟⣡⠾⢋⣥⠶⠛⣩⠶⠛⠉⠉⠉⠛⠳⢦⡙⣷⠙⠈⠓⠋⣠⣼⠉⠻⣾⢻⡇
⠀⠀⢸⡄⠻⣆⡀⠈⢿⣧⡀⠀⠀⢀⣼⡿⠁⣌⣡⠶⠋⠁⢀⡾⢡⣶⠀⠀⠀⠀⢀⣠⠾⠛⠛⠿⠷⣿⣟⠃⠀⠀⣰⠏⢸⡇
⠀⠀⠘⣇⠀⠈⠛⢦⣄⡙⠻⠿⣿⠿⠟⣱⣮⣟⠻⣷⣄⡀⢸⡅⢻⣧⡀⠀⠀⡴⢋⣁⠀⠀⠀⠀⠀⠀⢙⣧⡴⠟⠁⠀⣾
⠀⠀⠀⠹⣶⣄⡀⠀⠈⠙⠛⠶⢤⣤⣀⣀⠉⠻⣷⣬⣟⠿⠇⠻⢦⣙⣃⡤⢾⠁⣿⣏⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠹⣿⣿⣷⣦⣤⣀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠶⠶⠶⠶⠾⠷⠖⠚⠛⠛⠛⠉⠉⠁⠀⠀⢀⣀⣤⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠘⢿⡿⢿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⣒⠶⠶⠶⠶⠶⠶⠒⣛⣫⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠉⠀⠀⢰⣶⡾⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣯⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠐⠳⣶⣿⣿⣶⡄⠨⠙⠻⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣽⣿⣭⣭⣿⣭⣭⣿⣯⣭⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡾⠁⠀⣼⠋⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇⠀⢰⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡆⠀⠘⣇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢿⡀⠀⢻⡄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⠈⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇⠀⢰⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡟⠀⢀⡟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠖⢿⠟⠛⣿⠋⠉⢻⠟⠛⢻⠟⠛⠒⠶⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⣋⣭⣤⣤⣀⣤⡤⢾⡷⠶⢿⠟⠛⢻⣟⠛⢿⡷⠶⠶⣤⣤⣀⡀⠈⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣼⡋⠀⢀⣀⠉⠻⣄⠈⢷⠀⣸⣷⣾⡿⠿⠿⢿⣷⠀⠀⠀⠀⠉⠙⢻⡞⢻⣦⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠞⠋⠉⠙⠛⠿⣿⣿⣆⠹⡇⢠⡞⠉⠋⠁⠀⠀⠀⠀⢿⠀⠀⠀⣀⣠⡴⠟⣰⡏⣽⠛⢦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⣿⠀⠀⣰⣾⣶⣦⡈⢻⡏⢠⣿⣿⣴⣾⣿⡿⢿⣷⣦⠀⢸⣧⣤⠶⢿⣅⣠⠞⠉⣽⠃⣰⠿⣭⢷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠘⣧⡀⠘⢿⣿⣿⡇⠈⣿⣿⠟⠉⠛⠉⠀⠀⠀⠀⢹⣷⣿⢡⡴⠶⣄⢉⣿⣠⠞⠁⣴⢋⡴⢋⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣬⣻⢦⣤⣉⣁⣠⡾⢻⡇⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣟⢾⣡⢟⣠⣿⣁⡴⠾⣷⣫⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣯⣭⣅⣀⣈⠳⣬⣙⣛⣛⣿⣛⣃⣠⡼⠛⠧⣿⣤⣶⣞⣁⣬⣵⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⢿⣿⣿⣟⢻⣿⣿⣿⡛⣿⣯⣭⣍⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣰⣾⣣⣿⣧⣲⣟⣾⣿⣲⣟⣼⣿⣗⣶⣣⣿⣧⣓⣏⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⣀⣴⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢀⣀⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠸⠒⠃⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⢀⣀⣾⡶⠋⠀⢀⣀⡤⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡟⢹⠄⠂⢺⣝⡴⡶⢶⠋⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡇⠀⢣⣀⡟⠁⠱⠀⠨
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠠⠀⠀⠱⡀⢈⠈⡧⡄⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡵⣽⢶⣬⡌⢃⣻⡀⠀⠐⠀⠀⡑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⢺⣁⣀⡤⠽⠿⣶⡅⠑⣦⠤⠆⡺⣻⣿⣶⣦⣴⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⢀⣤⡀⠀⡀⢀⠘⣿⡻⠁⣴⢫⣟⣯⣿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣻⣿⣇⣵⠶⢋⣌⡉⣶⣶⣮⠅⣠⣯⡟⠁⠀⢈⣀⣤⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣾⣟⣈⢄⢜⡁⠤⠠⠻⢛⣝⡶⡄⢀⠠⢄⣛⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠆⠀⠀⠉⠉⠙⠚⠛⠿⠿⠿⠾⠟⠓⠚⠛⠋⢉⢁⠀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣄⣀⡀⠀⣀⠀⣀⣀⢀⣈⣤⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠟⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠃⠀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣧⠀⠀⠀⠀⣿⣤⠤⣄⠀⠀⢀⣤⡀⠀⢀⣤⣴⡇⠀⢠⣿⠀⣾⣡⡼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡆⠀⠀⢸⣿⣡⠆⠘⣧⣴⣿⡀⡿⢀⡾⢡⣿⠁⢀⣾⠇⠀⣿⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⣾⠉⣟⠛⢻⡏⢾⣀⡿⠛⢹⣇⡞⢸⣀⡞⣿⠀⠀⠈⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⣷⢀⡿⠀⠻⠶⠛⠀⠈⠋⠁⠀⠀⠉⠀⠈⠉⠘⠷⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣤⡤⠶⠒⢛⣋⣩⡍⠉⠉⠉⠍⠉⠉⠉⢩⠗⠋⡭⠛⠛⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠀⢀⣀⡀⠀⠀⠀⢀⣀⣤⣀⣠⠒⡏⢹⠛⣟⢦⡈⢢⡀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣰⠋⢉⡉⠉⠓⢲⣯⠔⢶⡞⠉⣦⡑⢌⠲⣌⠳⡙⣄⢹⡀⠀⣀⠀⠀⣧⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣤⣅⠀⠀⠀⢀⣠⠾⠾⠤⣾⢣⡘⢇⠙⢌⠳⣌⢣⡹⣌⢦⠳⠞⠉⠓⡚⢁⠛⢄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠃⠀⣰⠋⠉⠀⢘⡾⠳⢖⡫⠭⠤⠴⠊⣉⣉⣧⢳⡈⢣⠀⠙⢎⢧⠉⠙⣆⠳⡀⠀⠯⣀⡁⠀⠈⢣⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣸⣠⠴⣻⠿⠶⣒⣋⣠⠴⠋⣉⡭⠤⠤⠜⣧⢹⠀⢙⡷⠛⠦⡓⢄⠸⣗⢭⡢⢄⠀⢹⠀⠀⢀⢧⢀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⡔⣋⡥⠤⠤⣴⡾⠕⢒⣊⡵⠒⠒⢒⣫⠥⢖⡒⠂⢁⣹⡆⢧⠈⠓⢦⠀⠈⢢⡱⡌⢆⠙⣦⡱⡌⠑⢒⡡⠊⠘⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠊⠥⠒⠒⣫⡥⠖⠛⠦⣄⠀⣠⠞⣡⠖⣫⠥⠒⢊⡭⠂⣹⡌⢧⡀⠘⠢⠒⠚⣳⣝⡢⣥⣤⡾⠞⠛⠛⠒⠒⢤⡇⠠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡩⢤⡟⠁⠀⠀⠀⠀⠈⢳⡏⠐⣅⡀⠓⠤⢤⣈⡳⣄⢣⠘⢦⣉⣓⣒⣈⡭⢶⠎⣩⠋⣼⠀⠀⠀⠄⠀⠀⠀⢳⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡏⢷⣜⠀⣠⠞⠛⢦⡀⠀⠈⣿⠦⣄⣉⡉⠓⢦⡀⠈⢧⠹⡗⠤⠏⢉⢏⣠⠬⠥⢖⣡⠾⠖⠓⠒⢦⡀⠀⠀⢀⡾⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⣇⠀⠀⠘⡇⠀⠀⣿⠈⣀⠒⠙⢳⣄⡑⠒⠚⠒⢿⣄⡰⣫⠏⡴⠚⠉⢹⠁⠀⠀⠀⠀⠀⠙⣆⡠⠚⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣀⡇⠀⠀⣿⠀⣒⠛⠛⠋⣊⠽⢛⢝⡛⢦⣉⠙⣁⣾⠴⠒⠒⠺⡄⠀⠀⠀⠁⠀⠀⢻⢀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠙⠳⢦⣴⣏⠒⠚⠿⠭⠭⢒⡪⢟⡭⠒⠉⠉⠉⡿⠁⠀⠠⠀⠀⢹⡀⠂⠀⠀⣀⣤⠾⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣶⣭⣥⣖⣉⣀⣀⠀⠀⢰⡃⢀⣀⣀⣀⣀⣨⣧⡴⠞⠛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠠⠄⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⣛⣛⣉⣭⣭⣭⣭⣭⣭⣭⣭⣉⣙⣛⡛⠳⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⣫⣥⠶⠖⠛⠋⠉⠉⢉⣁⣠⣤⣤⣤⣤⣤⣤⣉⣉⠉⠙⠛⠳⠶⢦⣭⣙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠶⣻⣭⠶⠦⣤⣤⡴⡶⢞⣛⣭⠴⠶⠶⠶⠶⠶⠶⠬⣭⣟⡳⢦⣄⣀⣀⣼⠿⡟⠷⣾⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⣼⠏⠀⠀⣀⣀⠙⢦⡞⣻⡽⠶⣺⣿⠿⢛⣻⣿⠿⠶⢶⣮⣝⡻⢦⣽⢏⣴⣿⡙⣷⢘⣿⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⠟⣿⠀⣰⠟⠉⠉⠻⣎⣷⣀⣴⢞⣫⡴⢞⣫⣵⣶⣾⣿⣿⣏⣉⡙⣳⡟⣾⣽⣋⣴⠏⣼⣷⢹⣾⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠹⣧⠀⠹⣇⢿⡀⠀⠀⢀⣿⣹⠟⠴⢟⣥⠾⠛⣡⡾⣋⠁⠀⠀⠈⠉⠻⣮⣿⣭⣈⣭⣵⠾⠏⠈⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⡙⢿⣿⡶⠶⠟⣡⣟⡰⣶⣿⡁⠀⢰⡏⣼⡏⠀⠀⠀⠀⠀⢠⡟⠀⠀⠛⠉⠉⠀⣠⡾⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠈⠙⠳⢿⣿⣛⡛⠉⠛⠻⣶⣝⠻⣶⡼⢷⣘⠿⢄⣠⣤⡴⠶⠟⠁⠀⢀⣠⣤⠶⠛⠉⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡛⠷⣦⣄⡀⠀⠉⠙⠛⠓⠶⠾⠿⢿⣦⣤⣤⣽⣿⣿⣭⣤⠶⠶⠶⠛⠛⠋⠉⠀⢀⣠⡴⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣍⡙⠛⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠛⢋⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⠙⠛⠶⠶⣤⣬⣉⣉⣛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣉⣉⣥⣤⠶⠶⠛⣫⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⣛⣛⣛⣛⣛⣻⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⢍⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠶⠒⠒⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢺⠀⠻⠇⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠐⠀⠀⠀⠃⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⡓⠀⣀⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠠⠁⡅⠀⠈⠈⠀⠀⠈⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⡆⠀⢀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠐⡄⠀⠃⠀⠐⠀⠞⡏⠀⠣⡀⠠⠛⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡌⠀⠀⠀⠀⡠⠀⢀⠘⠂⠂⣂⡀⡻⠠⡅⠐⠠⠌⠶⡦⣄⣀⠀⢁⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠉⠉⠉⣔⠋⠊⠔⡂⢀⡸⠄⠠⠀⠀⠈⠉⠉⠠⠤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⢆⠡⠂⣤⠀⠩⠜⠢⠠⣤⡒⠉⠀⠀⣺⠀⠀⠄⠄⠴⠚⠀⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠘⡄⠀⠘⠈⠉⠈⠀⠔⠐⠀⠂⠁⢈⡍⠓⠒⢾⠳⠄⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠶⡁⠀⡀⠠⠐⠀⠀⠀⡀⠐⠊⠉⠁⢂⠟⢀⡦⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠐⠄⠀⠠⠈⠡⠈⠷⠂⠀⠀⠀⠁⠀⠀⠀⣠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⣤⠒⠒⠶⠶⠃⠉⠁⠀⠀⠀⣀
⠀⠀⠀⣴⣿⣯⡿⣷⣾⣧⣤⠼⢿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢶
⠀⠀⠘⢧⡏⠉⠉⠉⠁⢙⡇⠀⢀⡾⠃⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⡦⠀⠀⠀⠀⠀⠘⣿⢰⣶⣾⣄⢀⠰⡆
⠀⠀⠀⠀⠉⠓⠒⠲⠶⠎⠓⠒⠈⠀⠀⠀⠀⠀⠀⣶⣠⣿⣿⠿⠿⡿⢝⣶⡆⠀⠀⠀⠰⢿⣿⣿⣿⣷⢿⡓⠅
⠀⠀⠀⠀⠀⠰⠤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢿⡟⠿⠁⠀⠐⠱⠀⠛⡗⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⣤⣤⣤⠄⠀⠀⠀⠉⠢⡄⠀⢠⡤⠆
⠀⠀⠀⠀⢠⢜⣧⡟⣼⣻⣶⢶⡠⣢⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠁⠀⢛⣿⡇⣡⣶⠟⢉⢨⡼⠀⠀⠀⠠⣸⣯⠾⡛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠃⠜⠒⠁⠀⠀⠀⣰⢲⣿⡿⡴⠷⡧⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣤⠀⠀⠅⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢺⡿⣩⠖⠜⠁⠀⠀⠂⡀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⠄⠀⠂⠄⠀⠀⠀⠀⣠⡿⢁⡤⠀⠀⠀⢀⠌⠀⣠⣶⣾⢿⣿⣟⣷⣶⣦⣄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣤⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡸⠀⣿⣿⣽⡞⠛⠛⠛⣿⠛⢻⣿⣧⢀⡆⡀
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⠟⠿⠷⠊⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣻⣏⣀⠀⠐⠂⣸⣄⢉⣿⣯⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⣈⣀⣀⠄⠀⠀⠀⠀⠉⢩⣭⣍⡛⠛⠛⣛⣻⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠓⠰
⠀⠀⠀⠀⠠⣐⣢⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢈⣾⠋⠉⠙⢻⣿⣿⣷⠀⠀⠀⢴⣾⡿⠙⢃⡤⠚⢻⣿⣿⣿⢳⣆⠀⠀⢀⠔⠀⢀
⠀⠀⠐⣾⡃⡻⠛⠈⠙⢇⣀⣿⡀⠀⠀⠘⢿⣄⣠⠎⠀⠂⠰⣙⣿⣿⣿⠟⠀⠀⢸⡀⢤⠃⠀⡖⠁
⠀⠀⠀⢺⣿⡇⠀⠀⠀⢸⣿⡿⠄⠀⠀⠀⠀⠈⠙⠛⠒⠞⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠺⢧⣘⣇
⠀⠀⠀⠈⠙⠻⠶⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠨⠙⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⠀⢀⡴⠁⠘⢦⡀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠠⠀⡂⣼⠰⣺⣟⡉⢓⡄⠈⢛⣤⣾⣆⢄⣆⣄⠀⢈⠀⠂⠄⣀
⠀⠀⠀⠀⠀⡀⠂⠀⠠⢂⠔⠂⢚⡟⢻⣿⣬⣢⡭⠄⠀⣥⢋⣿⢿⣿⣿⣿⠁⠄⡨⠄⠀⠀⠀⠀⠤⡀
⠀⠀⠀⢀⠆⠀⠀⠀⠁⠀⠠⠀⣾⡿⣾⣛⠫⠵⠈⠒⠈⠣⢸⠫⣿⣿⣿⡦⡟⠌⡔⢡⠲⡀⠀⠀⠀⢱⡄
⠀⠀⠀⠘⣆⠀⠀⠀⡀⠨⠅⠁⢿⣟⠿⣾⣿⣿⡐⣏⣖⣋⣼⣷⣿⣿⣩⣦⣑⡶⣀⠈⠂⠉⠀⠀⠀⣰⠇
⠀⠀⠀⠀⢻⣦⣄⠀⠀⢠⡄⠐⠀⠀⠻⣿⡿⣿⣶⣴⣭⣽⣻⣿⣿⣽⡽⠋⠀⢠⠀⠀⠃⠀⠀⣠⣼⡟
⠀⠀⠀⠀⠀⠙⠿⣿⣦⣤⣈⠀⠒⠘⠉⠈⠻⠉⣻⣟⣈⠏⣉⢋⡠⢀⠲⣥⠂⠉⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣤⣄⣬⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢧⣯⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⡟⣉⡴⠿⠿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠟⠀⢲⣼⠇⢾⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⡁⡟⣰⠋⠀⠤⣀⠀⠈⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢿⠄⢾⣌⣫⣬⣞⣾⣃⣤⠤⠀⠀⠀⠒⠀⢨⡭⠿⠿⠇⣱⠃⠜⠨⣦⠜⣷⡶⢉⡀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠐⠶⢄⣵⣮⣻⣿⣿⣿⣿⡿⣍⣸⣞⣀⣄⡤⢔⡤⢦⡬⠻⣷⢉⣠⣶⣦⣈⣴⡟⢠⣿⠀⢰⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣝⠻⢻⣿⠫⣷⣿⣽⣉⢁⣐⡚⣛⣳⣾⣿⢿⣷⣿⡦⣝⣯⢳⣮⢁⢋⡙⣀⣯⡈⢀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠙⠷⣦⣝⣷⣽⣿⣿⣆⢶⡲⠬⣽⣛⣛⣛⣿⣛⣩⣴⣷⠶⠭⣶⣯⠟⢿⣉⠉⠠⢟⣫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠉⠉⢋⣵⢿⠧⣝⣾⡿⣟⣓⣋⣭⣶⣤⣴⣬⠿⠋⠂⠼⠿⠓⠀⠩⠴⠾⢟⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠄⠀⠙⣶⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠀⠐⢰⢢⠴⢊⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠮⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⠊⢫⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⣄⣀⣈⡀⠀⣀⣀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠙⠛⠈⠁⠘⠉⠁
⠀⠀⠀⢠⡄⡀⠀⠀⠀⢀⠀⠀⠀⢀⡀⣀
⠀⠀⠐⠲⠾⠷⠾⠿⠷⠿⠾⠷⠿⠿⠿⠿⠶⠶⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠔⠃
⠀⠀⢬⣬⣹⣧⣴⣦⣤⣷⣴⣤⣦⣶⣖⣧⣄⣀⣀⣀⣀⣀⣀⣐⠒⢒⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠟⢛⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⠿⢛⡝⠀⣹⠿⠘⠣⡚⢛⣋⠙⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡿⠋⠀⠀⡤⠿⠶⠟⠁⠐⢋⡴⠛⡚⢛⣡⣷⢈⠻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣥⢤⣤⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⢻⡽⣞⢻⣷⢘⡗⠌⡩⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣿⣿⡄⢃⠀⣾⣿⣿⠖⣲⠖⠡⣤⣤⡄⣶⢘⣧⡻⣟⡉⢚⣽⢪⡻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⡟⠉⠉⡇⠈⣤⠴⠼⢫⠤⠬⣧⣼⡶⢢⣾⡿⣿⠎⢷⡙⢠⢠⣁⢿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⡇⠀⠁⠤⠊⠀⢠⡀⠀⠙⣄⡊⣉⣷⣿⣿⣧⡻⣎⢣⡭⣿⣿⠍⣿⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⣀⠀⡏⠀⢀⠀⠆⣴⢻⣦⡀⠀⣟⢣⡝⢿⢇⡹⣏⣳⡹⣧⡹⡿⠿⢰⣿⣿⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡷⠗⠚⡇⠀⠀⠈⣿⢋⠁⡸⠜⢹⣷⣯⢻⣿⣿⢻⡿⣘⢷⡜⢶⣶⣿⣿⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠋⠀⠀⠀⣡⡀⠀⠈⠁⢀⣴⠇⣻⣿⢿⣻⣟⠋⣭⢿⣷⣿⣎⢿⣎⢻⣿⣿⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡸⠀⠀⡴⠛⠋⠳⠐⣶⣶⣿⠟⠻⣷⠿⢦⣭⣿⣷⡟⠋⢉⠄⠀⢨⣿⡲⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⡆⠀⠈⠓⠔⡀⠒⢐⣮⡙⢀⣾⠃⣼⠦⠈⢉⢻⠋⠀⠈⢀⣿⣿⣿⡿⢷⣟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣶⠠⠰⠀⣮⠍⠉⡄⠁⠰⠒⡖⢠⠊⣿⡄⠀⠘⠉⠈⠉⠁⢃⣸⣿⡏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⠦⡀⠓⢀⣤⣧⠁⡴⠿⠞⡦⣠⣌⡊⣷⣶⣤⣴⠶⣲⡿⠟⠉⢿⣆⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣴⣶⣿⣿⣿⡀⠀⣀⣶⣊⣵⣤⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⢻⣦⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⡡⠔⠋⠀⠀⠀⠀⠀⠀⠻⣧⡀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣄⢠⣞⣵⣤⡿⡿⣿⢰⡿⣠⡾⣿⠀⠀⠀⢾⡀⡴⢻⣧⣴⣢⣇⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣎⠉⠈⠈⠀⠀⣿⠈⠉⠉⠀⣏⠀⢠⡴⠟⠁⠉⠉⠀⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠠⠤⡀
⠀⠀⠀⢠⢤⣠⡄⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠉⠞⢱⠻⡽⠷⠯⠷⠯⠧⠧⠿⠵⠽⠆
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡤⠀⣀⣠⠖⠁⠀⣠⣤⣤⣄⣀⢀⣁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣊⡐⠫⠲⢄⠙⠇⣠⠞⠋⠀⠀⢰⣶⣤⠄⠉⠙⠳⢬⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣴⣯⣝⡙⠻⢹⡃⡀⠸⠀⣼⣟⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢉⡤⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠩⠄⠘⣿⠈⠸⠇⠃⣈⠁⣥⡀⣻⣿⡟⢳⠅⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠗⠄⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠙⡄⠀⡀⠀⣿⣧⡄⢠⣁⠻⠧⣻⢵⡈⣿⡿⣥⣄⢠⡁⣠⡶⣿⣶⣄⠀⠀⠀⣰⠆⠂⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢿⣀⠀⠀⠙⠿⠙⢧⣙⣄⣩⣭⣥⣄⢼⣽⣿⣿⣽⢈⣿⢾⣿⣤⣻⣿⣟⣠⡧⠞⢸⡗⢠⣄⠰⣘⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠻⡏⠁⣀⣽⡆⢠⣬⠟⢩⡿⣿⣿⣴⣿⡏⠈⢻⣿⣿⣿⣿⠟⠋⠉⠉⢉⣀⠔⣾⡅⠀⠛⠶⣡⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣤⠞⠛⣡⣂⡼⢉⣼⣯⢟⠟⣹⡾⠏⢁⡴⣫⣿⣼⣿⢿⡀⡤⠚⠋⠹⡇⣾⢿⣧⣰⢎⠀⠤⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⢲⠉⣿⢁⡼⣵⢿⠏⣾⠟⠁⡰⢋⣽⢏⣼⣷⡟⣼⡟⠐⠀⠀⠀⠳⠽⢁⠋⢀⣨⣄⠓⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣅⠖⠾⢷⣿⢸⣷⣿⣹⢰⣿⠀⢰⣧⣽⠋⣼⢻⡏⣰⣿⣧⡀⠀⠀⠀⢀⡴⠛⣢⡤⠥⠠⠅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣧⡧⠀⠀⣙⣷⢸⣾⣿⢿⠘⣿⠀⣾⠻⣿⢠⣧⣼⢳⣿⢓⣽⢻⠞⠓⠶⠞⠅⡈⠁⣀⡤⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣶⣿⣿⣦⣟⣻⣾⡄⣿⡀⣷⣴⣇⣺⢷⡏⣸⣏⢸⣁⡴⢶⣭⡀⣤⣴⡷⠛⠛⠠⣄⡀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⣏⣿⡿⢿⠿⠿⠷⣽⣇⡜⣼⣿⣿⣻⣇⣿⣿⣴⣿⡇⠀⠙⡟⢯⡙⢍⠛⠲⢄⠀⠙⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⡧⠈⡛⠉⠀⠀⠀⠀⠍⠉⠙⠻⢿⢼⡿⠛⠉⠍⠁⢀⣾⡀⠀⢹⣾⣹⡛⢳⡤⢠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡘⢰⢴⣾⢰⣶⣿⡍⢸⠇⠀⢾⠏⠁⠀⠨⠁⠀⢹⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠿⠿⣷⣦⢤⡀⣰⢿⠶⡈⠻⢄⢶⡎⢀⡿⣀⢂⡞⠀⠀⠀⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠻⠻⠋⠻⣷⣯⣦⠴⠃⡀⠀⣀⡉⣉⣭⡵⠞⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠋⠒⠒⠀⠐⠚⠛⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣀⣀⠀⠀⠀⠀⢀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢓⡂⠉⢐⣒⣛⢛⣉⢉⢉⡉⣁⢀⢀⠀⠀⠀⠄⠄⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢦⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡖⢦⡈⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠲⠉⢘⠇⠀⣖⠀⣈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⣌⠐⠻⠀⢤⡤⡀⣾⠉⡆⠀⠳⠖⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠡⡄⢦⡉⢀⡈⠁⢡⠃⣌⡋⠆⢳⡄⣠⠴⠈⠵⣦⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣔⣴⠂⠠⡆⠐⠛⠙⡍⠩⠐⠂⠨⣤⣠⢖⡃⢰⠙⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢓⠾⣬⣖⡑⠀⠽⢐⠺⢃⢀⣀⠰⡦⠀⢁⣁⣘⣃⣤⠶⢚⡍
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⣉⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⡉⠁⠶⠚⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠠⠨⠉⠉⠙⠛⠋⠙⠉⠉⠉⡉⠑⠅⠉⡀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⣟⣶⣷⣤⡿⡭⡷⡤⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡿⣹⣯⣯⣯⣾⣷⣮⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢙⠉⠙⠉⠉⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⢠⠁⠀⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⣀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠄⣒⣪⡭⠭⠶⠶⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣖⣶⠒⠃⠀⢰⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣒⣯⢯⢺⣤⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡿⢿⣼⢻⡿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣴⣿⢬⣿⡎⠁⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠀⠀⠀⠀⠉⠾⢤⣼⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣀⣀⣾⠟⣉⠲⣤⣠⠞⠊⠑⠒⢣⣤⣤⣸⡏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⡒⠶⣾⠗⠚⠙⣛⡊⠣⡗⠈⠁⠙⠁⠀⠈⠐⠀⠠⢿⠣⢷⣰⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣴⣯⣎⣀⣽⣷⠁⠀⠀⠀⠀⢹⣾⠋⠀⠀⠘⢤⣤⣀⠀⠠⢼⡗⠀⠀⢀⡿⡶⢦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠑⠋⠁⠉⢛⡀⠀⠀⠀⠀⣀⣼⣖⣂⡀⢴⠻⠖⠀⢠⠀⠌⢳⣀⣀⠜⠋⠠⠀⠈⠩⠯⡍⠙⡦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣾⠀⠀⠀⠀⠀⠰⢋⣭⢾⣻⢚⣿⠉⠄⠐⣠⣕⠷⣿⣅⠀⠀⠀⢀⡆⠌⠳⡀⠑⢁⡴⠀⠰⣾⠁⠀⠁⡟
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢹⡤⢀⠀⡁⠀⡇⠛⠟⣱⣭⣥⣿⢷⣾⡄⢀⣰⠞⣻⢧⠙⠙⠾⠍⠂⠐⣦⡈⢁⡿⣄⡊⣁⠈⠀⣠⠞
⠀⠀⠀⠀⠀⠈⢆⠠⣄⠀⠘⣆⠈⠐⠬⠭⠽⢛⣿⡿⠟⢿⢷⣷⡿⢟⣦⡌⠈⠀⠁⢪⣴⣿⣿⠮⣶⠞⠛⠞⠀⠀⠚⢁⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠧⣤⣀⠉⠉⠉⠈⠙⠯⣧⣬⣤⣓⠙⠛⢷⣾⣞⣿⣤⡬⠿⢚⠟⠁⠀⠉⠁⠀⠤⠀⣀⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢏⣉⠀⠓⠂⠀⠀⠀⠀⠉⠛⠒⠒⠉⠁⠀⣀⣀⣀⣐⣀⠤⠤⠄⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⣶⡿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣴⣸⣾⣇⠾⠿⣋⠉⢁⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢟⠽⢩⣶⢩⣶⢡⢺⡿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⡜⡑⡆⡈⣾⡈⡜⣀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠹⠀⢡⡱⠘⡷⡄⢋⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⠇⠈⡌⣰⢡⢣⢻⢨⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⡐⠀⠁⢹⠃⠀⡘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣟⣿⣶⡾⢀⡇⢸⢸⣿⠘⣱⢷⠘⣷⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣿⣿⣿⢟⣋⣍⡅⠈⠱⠸⠆⢿⢀⢻⣠⢀⠿⠿⠮⢿⣿⣫⣤⣮⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢃⣿⣿⣿⣟⣻⣯⣿⣷⡐⠃⣰⢸⡄⠈⠎⣏⣸⣿⣿⣿⣿⣎⣿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡏⣸⣿⣿⣿⣹⢿⢊⣭⣩⣷⠾⠫⢿⣿⣾⢿⣿⣄⣉⣩⣿⣿⠋⣁⣤⣿⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡇⣿⣿⣽⡏⡟⣾⡤⠍⢻⣷⢾⣾⡷⠟⢋⣙⣉⣛⡿⠏⠿⠿⢿⣿⠟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣇⠙⣿⡇⠇⡇⣏⡰⠷⣾⣿⡿⠋⠠⣦⣿⣿⠟⠁⣴⣿⣿⣷⣆⢹⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣧⡈⠳⠀⠃⠈⢃⠿⢀⣆⣨⡟⠀⠀⠀⠹⣿⠏⠀⠘⢿⣿⣿⣿⠟⣸⣗⠭⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣦⣀⣀⢼⡖⢋⣤⠤⠷⣤⠤⠤⠤⣎⠀⠀⠀⠀⣉⣉⡁⣤⣿⣡⡄⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣿⠉⣉⣠⣍⣡⣶⠀⠘⡛⠛⣇⠿⠶⠶⢲⣭⢋⣈⠑⡄⣿⣿⡃⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣾⣯⣀⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣻⣿⣿⣯⣿⣿⣿⣾⡅⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠲⣶⡶⣶⣶⣶⣶⣶⠖⠒⠀⠒⠲⣶⣶⣶⣶⣶⣶⣶⠒⠂⠀⠒⢲⣶⣶⣶⣶⣶⣶⡶⠂⠂
⠀⠀⠀⠀⠹⣶⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢀⣠⣤⣴⣷⣤⣄⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠈⠀⢺⣷⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⡿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠶⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠐⠒⠒⠂⣉⣉⣉⣉⣀⣠⣀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡤⢾⡀⠀⠠⡀⢘⣄⢂⣸⢋⠔⣦⢀⠋⠁⠀⣄
⠀⠀⠀⠀⠀⠸⣿⣍⣈⣁⢂⡀⠆⡀⣈⠂⣹⣶⣼⣀⣘⠚⣡⣴⡿⠆
⠀⠀⠀⠀⠀⠀⠹⡀⠈⠉⠉⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠙⢛⣻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣠⡴⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡏⠀⣼⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⠀⠀⠐⡀⠘⣷⣅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢹⠀⣤⣌⣻⣽⡇⠀⢿⣿⣟⣿⡯⣝⣽⢦⣀
⠀⠀⠀⠀⠀⠀⢠⣮⣾⠋⣉⣐⣿⠛⣧⠀⠹⣤⣿⣿⣿⠺⣬⠟⢙⡛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡿⣼⣒⣫⢟⣿⠠⡍⣶⣾⣿⢩⡾⠛⠉⠡⠭⡍⣛⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⡑⠛⠛⠆⠳⠮⠵⠄⠿⠤⠶⠟⠒⠚⠉⠉⣅⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡈⣿⣷⡔⣿⣷⡆⠀⠀⠀⠀⢠⣶⣷⡆⠀⠀⡸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⢻⣷⣬⣽⣏⣷⣤⣤⡼⠈⠻⠉⠀⣠⣦⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣻⣿⣿⣿⣿⣯⣤⣤⣶⣾⣿⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠩⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⡀⣶⠒⢠⡄⠠⡿⠒⡟⠉⡽⠤⣼⠶⠚⠁
⠀⠀⠀⠀⠀⠘⠋⣹⡥⠚⠁⠀⣿⠄⣸⢹⡀⡇⠀⡇⠸⣇⣠⠋⠀⠀⠀⠀⠀⢀⡴⢀⡄
⠀⠀⠀⠀⠀⠴⠾⣿⠒⣒⡿⠃⣇⣀⡟⠀⢿⡇⠀⠃⠀⣀⣀⡼⣀⢀⡆⠀⢠⢞⠔⡩
⠀⠀⠀⠀⠀⠀⣀⣿⠚⠁⠀⠀⠉⠁⢠⡆⡀⢸⣤⣴⢧⡿⡵⠿⠙⠋⠀⠀⠋⠁⠋⠀⠀⣀⢀⣀⣀⣄
⠀⠀⠀⠀⠀⠊⠁⠋⠀⠀⠀⡀⣤⣱⣿⣷⣧⣼⣨⠌⠈⠃⠀⠀⢀⣀⣤⠤⠄⠒⠢⢄⡰⠛⠋⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣤⣼⣖⠿⠻⡟⡇⠸⠇⠁⠈⢀⠤⠂⠑⢎⣹⡋⢱⡌⢳⣉⠙⢦⠀⠈⠲⢄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡴⢸⠋⢹⠁⠀⠀⠃⢀⣀⣤⠖⢊⠏⠀⠀⠀⠈⠻⡫⣳⣄⡼⡌⡇⢸⢗⡒⢄⠀⠙⠦⣀
⠀⠀⠀⠀⠘⠦⠎⠀⠀⢀⣠⠔⠊⢹⠟⢛⢦⡄⠀⡠⠀⠒⠐⢆⠝⠀⠀⠑⢶⣃⣾⡄⡷⢸⡠⢒⢄⢌⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢀⠨⠅⠌⠰⠌⠒⢦⣶⣆⠀⠌⠀⠀⠀⡀⠀⢏⡲⠷⢧⣟⠄⡼⢁⢮⡀⠤⡉⠲⡀
⠀⠀⠀⠀⠀⢠⡏⠴⢦⠀⠢⣈⠪⠑⠈⠀⠄⠸⠠⠄⠈⠙⢾⡀⠀⠁⠀⠀⢡⠊⠀⠀⠀⠙⢾⡡⡢⢊⡤⡪⢈⡦⢷
⠀⠀⠀⠀⠀⢸⠣⣀⠈⡇⠒⢂⠀⡀⡤⠠⡈⢢⡁⠂⣴⡶⢷⡌⣑⣾⣶⢀⠃⢠⠒⠒⠀⢢⠈⡟⣢⡭⠖⠋⠁⠀⡼
⠀⠀⠀⠀⠀⢸⠀⠈⠳⣇⡈⠈⠱⡈⣆⣧⣌⣀⠤⠖⠳⢄⡔⠉⠀⠉⢎⠛⠤⣸⢰⣰⡦⠼⠒⠉⠡⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠙⠢⣔⡙⠊⠉⡹⠁⢀⣀⠀⡜⢠⠊⠉⣆⣸⡤⠤⠚⠋⠁⠀⠀⠀⠀⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠙⠲⢤⣇⣠⣃⣀⣦⡥⠷⠒⠊⢉⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠤⠄⠀⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠙⠲⠤⣤⡤⠤⠖⠚⠉⠁⠀⠀⠀⣰⣆⡜⣧⢻⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠔⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠟⠋⠁⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠐⠉⡀⠄⠒⣉⠤⠐⡂⢽⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡒⠕⢊⡁⠄⣊⠩⢐⡊⠥⠒⢉⣡⣤⠨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠮⢒⣁⡤⠦⣁⠢⠅⠒⢉⣀⠤⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣨⣴⣞⠯⠓⢂⣉⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢵⢾⣿⣽⡻⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣯⣷⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠒⠒⢺⣽⣽⣻⢷⠛⠒⠒⠒⠒⠒⠲⠶⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠅⢀⣠⡤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⡈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡻⠿⠿⣯⣿⣺⣿⣿⣿⣿⣿⣿⣻⡿⣝⡻⣟⣿⣟⣿⣿⣿⣻⣿⣿⣿⡿⠿⠾⢛⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢐⠈⠉⠐⠒⠰⠄⠠⠤⠭⠭⠭⠉⠩⠉⠉⠩⠍⠉⠉⠭⠉⠉⠠⠀⠖⠐⠒⠈⠉⢐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡱⢊⠄⠒⣂⡀⢑⠕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢨⠀⣎⠀⠈⠂⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆⠺⢤⣀⠀⡠⠂⡔⡜⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⣀⠀⠀⠀⠀⠀⠀⠑⠵⣂⡄⣀⡐⠭⠊⠀⠀⠀⠀⠀⠀⠤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢥⣀⣒⣒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⢐⣒⣂⣀⠥⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣰⠃⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣸⠃⠀⠀⡼⠃⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⠃⠀⢀⡞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⠃⠀⢠⠟⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡼⠃⠀⣰⠏⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡼⠁⠀⣼⠃⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣤⣤⡾⠁⠀⡾⠁⢀⡾⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⡤⠤⣄⠀⠀⡴⠋⠉⠉⠙⠶⠋⠀⣼⠁⠀⣼⡁⢠⡟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠿⠁⢤⡀⠘⣧⣼⣧⡀⠠⠤⣄⡠⠤⣼⠃⠀⡼⠁⣻⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⣠⠤⠀⠙⣄⣀⠀⠈⢻⣄⣀⣀⡠⢴⠇⠀⣼⠗⠺⠧⣴⠟⠉⠉⠙⣦
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⢠⠤⢼⠁⢀⣀⠀⠁⠈⢣⠀⢸⡇⠀⠉⣴⠟⠛⠛⠃⣀⠤⣀⠀⠀⠀⠀⡀⠙⠛⠛⠲⣄
⠀⠀⠀⠀⢀⣼⣇⡠⠒⠒⠃⠀⣀⣰⠃⠀⣰⠞⠋⠛⠷⠞⠋⢳⣾⠃⠀⢀⣀⣰⠁⠀⠈⠉⠙⡆⢠⡈⠑⢆⡀⠀⢹⡆
⠀⠀⠀⣠⠟⠁⠈⠛⠦⣴⡒⠞⠀⠈⠀⢰⡇⠀⢀⡤⣤⠤⠄⠐⠟⠛⠻⣏⠀⠈⠀⡴⠒⠦⠀⠀⠀⠹⢄⠈⠉⡇⣸⡇
⠀⢀⡾⠁⠀⠰⡄⠀⠀⠈⠙⠳⢦⣄⣠⠟⠁⠀⡎⠀⠀⣀⠄⠀⠀⠀⠀⢸⠋⠉⠉⣀⣀⣠⣤⡤⠴⠶⠾⠛⠛⠛⠉⠻⣆
⣴⠋⠀⠈⢆⠣⠸⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣍⡁⡠⠔⠃⢀⣀⣀⣤⠴⠿⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⡠⢀⠀⠘⢧⡀
⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⣟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⢡⠃⠀⠀⠈⠳
⠀⠀⠀⠀⠙⡷⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢹⠌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠒⠛
⠀⠀⠀⠀⠀⣇⠀⠀⠉⠛⠶⢤⣄⡀⠀⢀⡼⠃⠀⢸⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠒⠛⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣤⣀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⢸⡇⠀⠀⠀⠻⣦⡤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⢹⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⣿
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣤⣧⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠘⡇⠀⢠⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣧⠈⠓⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⡰⠃⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⣻
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣿⣤⣤⠴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⣿⠀⠀⣀⣠⡤⠶⠚⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣞⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⢧⢏⡜⣸⠓⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⣿⢸⢸⣏⠀⠀⠀⣩⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢿⣿⡞⣮⢣⢹⡤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣊⡽⡾⣱⢣⢏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣚⣧⢻⣸⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⢈⡇⡷⣹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⠉⠀⠀⠀⠀⠀⡞⣼⢹⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⣽⣆⠀⠀⠀⠀⠀⣳⡹⡜⣎⢣⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣻⠀⣀⣤⣶⣿⣵⢧⢟⡜⣸⡦⢤⣄⣉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⣿⠾⢟⣒⡮⢵⣛⡻⠾⣚⡻⠥⣖⠲⠬⢙⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⣛⡛⠶⠽⠯⠤⠭⠭⠥⠭⠿⠷⢒⣛⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠉⠉⠙⠛⠒⠒⠒⠒⠛⠋⠉⠉⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡓⠲⠦⠤⠤⠤⠤⠤⠤⠴⠶⠚⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠿⡷⣓⣴⢲⣴⢢⣶⢀⣾⣹⡷⠂⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠲⠦⠬⢤⣭⣌⣡⣭⣥⠥⠴⠚⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠤⠤⠤⠤⠤⠤⠶⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⡀⡠⠖⠒⠠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣽⠖⠋⠑⠫⣍⣆⣸⣤⡠⣌⣣⡀⠀⣀⣀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠒⠉⠉⠀⣀⣀⣀⠀⠈⠀⠀⠀⠘⠉⠀⠀⠁⠀⠈⠐⠄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠁⡔⡞⠀⡄⠠⠀⠁⢀⣀⠀⠈⠢⠀⠀⠀⢀⠔⠉⠀⠀⠈⠐⢄⠀⠀⠆
⠀⠀⠀⠀⡠⢂⡁⠳⠀⠑⢆⣸⢤⠁⠀⡴⣿⡟⠛⣆⠀⢳⠀⢠⠃⠀⡴⣻⡿⠷⡀⠀⠆⠀⢈⣁⣈⠑⠄
⠀⠀⠀⢠⠀⡁⠇⢀⠗⠠⡴⢃⢸⠀⠀⣿⣿⣷⣶⣿⠀⢸⠀⢸⠀⢰⣿⣿⣷⣴⡇⠀⢸⠸⣮⣍⠙⢆⡘
⠀⠀⠀⠀⢄⠈⢆⠈⠫⣹⠀⡏⠸⣆⠀⠙⢿⣿⡿⠏⢀⠞⠀⠘⣆⠈⢿⣿⣿⠟⠀⢠⠇⠀⣉⠂⠡⢘
⠀⠀⠀⠀⡠⠛⣉⡡⠌⠙⣦⣡⡀⠫⢗⠤⠤⡤⠤⠖⢋⡀⠀⢀⠈⠓⠤⢤⡤⠤⠖⠁⠈⠻⢳⡖⢁⠮⠄⡈⠑⡀
⠀⠀⠀⡰⠀⠠⠁⡠⠄⢄⠈⡖⠻⢒⡀⣙⡀⠀⢶⣄⡈⠟⠛⡛⠀⠀⢀⢀⠤⡴⠀⣧⠀⢶⣄⠈⢿⢊⣩⣭⣉⠓⢤
⠀⠀⡠⠁⣠⣇⣰⠋⣅⠙⠋⡟⢉⣀⠈⢣⡈⡛⡟⠀⡏⠻⠬⠜⣩⣆⠘⠓⠚⢁⠔⢰⠀⢸⡇⣧⠈⠃⠀⠀⠈⢳⣀⡀
⠀⠙⠒⠂⢹⠀⠙⠲⢿⣇⣨⣱⠁⡴⡇⠀⠻⣿⣄⡼⢹⣖⡖⠈⠁⠨⣿⣷⡛⢉⣠⠟⠀⡼⠀⣘⡤⠖⢛⡇⡆⠀⣆⠡
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠙⠚⠓⠾⠦⠤⠤⣥⣤⣞⣉⣉⣀⣩⡖⠤⡉⠉⠉⠉⠳⠚⠛⠉⠁⠀⠄⢸⠏⡁⢴⢳⠀⡆
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢨⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢤⠉⠰
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠸⠠⠔⠁⠀⠀⠀⠀⠀⠀⣸⠁⠘⠂⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄⠊⠀⡠⠃⢀⠀⠀⠉⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠽⣒⣒⣒⣒⣒⣒⣒⣒⣪⠭⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢠⡎⠉⠙⠲⣄⣠⠤⠶⠶⠶⠶⠤⢤⣀
⠀⠀⠀⠘⣇⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⡆⠀⠀⢤
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰
⠀⢸⠃⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠘⣆⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢀⣠⣞⣠⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠛⠻⠿⣖⡦⣄⣀⡤⠤⠔⠒⣊⣩⡤⠟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⣀⣠⣤⢤⡤⣤⡤⣤⣤⣤⣤⡤⠤⠴⠒⢊⣉⣡⢤⣶⠒⠛⠋⠉⢁⣀⡀⠤⠄⢒⣛
⠀⠀⠀⠹⣄⠀⢀⣠⠖⣋⡤⡶⣻⣿⣿⣻⣟⠿⠷⠛⠛⢋⣉⣡⣤⢶⣶⢞⣿⣹⣧⡤⠤⠟⠓⠒⢈⣉⣠⡤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠈⢹⡻⣀⣾⣿⠥⠷⠛⢋⣉⣡⣤⣶⣶⣿⣻⣿⠽⠿⠟⠛⢋⣉⣉⣤⣤⡴⣶⣶⠟⠉⠉⠁
⠀⢀⣀⡤⠴⣒⣋⣩⣥⢴⣶⣿⡯⠿⠾⠛⣛⣉⣩⣥⣴⣶⡖⣞⡿⢻⣿⣿⣻⣮⢿⣺⡕⠋
⠮⢭⣶⡾⠿⢿⣓⣚⣭⠭⠴⠖⠶⠾⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠛⠛⠓⠋⠉⢸⠃
⠀⠘⠒⠚⠋⠉⠉⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⠒⠦⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠖⠒⠋⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡴⣒⣒⣦⡀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⢀⣔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢽⡮⣖⡒⣝⣷⣄⡀⢀⡠⣪⢔⡯⢍⣪⡳⣄⣀⡠⢺⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡪⠥⢜⢞⠮⣓⠻⠟⣪⢝⡥⣒⣒⢽⡪⢵⣶⣾⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠙⠪⣷⢵⣦⡭⣭⣴⣿⡫⠴⠮⣗⣭⣭⣽⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣄⡀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠉⠉⠉⣀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠉⠉⠓⠒⠲⠶⠦⠤⠤⠤⠤⠴⠒⠚⠉⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠞⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣧⣿⣤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠤⣽⣿⡤⠤⢤⣀⠀⠀⠀⢰⡟⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⠉⠀⠀⠀⠈⢿⠉⠉⢹⣇⣀⣀⡏⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⢉⣡⣤⣤⣤⣤⣤⠀⢠⣼⣇⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡤⠾⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠷⣤
⠀⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⡴⠋⠀⠀⠀⠳⢦⡶⢶⣋⣽⣙⣷⣲⣶⣶⣖⣛⣓⣶⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣉⣈⣁⣈⣉⣀⣉⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣠⠅⣀⣀⠈⠁⠄⠬⢊⡁⠒⠐⠢⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠇⠀⠀⢀⡌⠉⣩⡉⣀⡈⠛⠀⣈⣠⡀⠀⡐⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢠⠞⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠖⠊⢀⠀⠦⣤⠴⡀⢀⡈⣠⣄⠀⠚⣀⠀⠆⠩⢁⡀⠘⠄⠀⠰⠀⡀⢣⢄⠀⠀⡠⠁⣴⠋⣠⠋
⠀⠀⠀⠀⠀⠀⢠⠂⠉⠀⢈⡡⢈⣤⠬⠿⠦⢐⣊⠀⣉⡀⣤⡴⠀⢭⡍⣀⡀⠲⠊⡉⠀⠐⠀⠄⡀⠀⠲⡘⠓⠊⢀⡞⢁⡼⠁
⠀⠀⢀⠀⠀⣠⠈⠡⠆⠀⢋⡴⣡⠄⠴⠫⣂⡄⢁⡄⠉⠣⠈⡐⡠⡆⡀⠄⠀⠤⠙⢾⡀⠄⠐⠚⠛⠐⠆⡛⠐⠀⢫⣠⡎⠀⠀⠀⠠
⡀⣰⠁⢀⠐⣥⠒⠋⠈⣟⠡⣖⡀⠒⣃⠶⠌⡴⢩⡆⠐⢶⠠⠟⣰⠀⠓⠀⣉⠲⠀⠒⠂⠙⠜⠁⠴⠂⢀⠑⡤⢀⠠⠄⣆⠢⢄⠀⠐⡌⠆
⣇⠻⣀⣿⠃⠃⣴⠛⡩⠋⠗⠂⠀⠲⣬⣤⣀⣦⡈⠣⠰⠆⠓⣉⣉⡀⠣⣭⣥⣄⣙⠉⠀⡈⢰⡑⠂⠼⢀⢀⣬⣀⠄⢐⠈⠧⢀⢰⡠⠂⣼
⠹⡳⢌⡳⠶⣈⡉⡙⠲⢥⣶⡁⢦⠀⠦⠤⡀⠄⠉⣤⡐⢋⠴⠶⣦⣍⠂⠐⠂⠢⡀⠁⡍⠹⣌⣉⢉⠉⠟⣠⣶⢀⠳⠘⣚⣶⠥⢊⡡⢺⠇
⠀⢳⠀⠉⠒⠢⢍⣙⡓⠶⠤⢶⣌⡳⣀⠂⠤⣈⡐⠼⠆⠾⢉⡀⢶⡄⠈⠓⠀⠲⢞⣠⡀⠚⠀⣅⢠⣸⣀⡿⠥⠶⠒⣋⡩⠤⠒⠉⠀⡞
⠀⠘⡄⠀⠐⠀⠀⠀⠈⡉⠑⠒⠒⠠⠬⠭⠭⣈⣙⣒⣒⣒⣛⣛⣛⣒⣛⣂⣚⣒⣛⣛⣋⣀⡉⠭⠭⠤⠐⠒⠒⠉⠉⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠢⠤⠤⠤⠤⠄⠀⢀⣀⣀⣀⣀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣉⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠛⠋⠉⣀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠣⡄⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠟⠀⡸⠁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⡐⠀⠀⠀⡀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠆
⠀⠀⠀⠀⠀⠀⢰⡏⠠⣐⡈⠀⠀⠀⢂⠠⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡴⠒⢦⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⡿⢀⠭⢒⡂⠀⠀⠀⠐⠌⠁⠀⠀⠀⠈⠙⢶⣄⠀⠀⢀⣠⡇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣾⠁⠀⢊⣁⣤⡤⠶⠖⠒⠛⢛⣛⣉⣉⣍⣉⠉⢹⡇⠀⠸⣄⡄⠀⣀⠀⢶⠁⢀⣀⣀⡤⠖⠒⠲⢤⡀
⠀⠀⠀⠀⣼⣧⡴⠞⣋⣡⡤⠤⠶⠒⠺⠛⠛⠋⠁⠀⠀⢹⠞⠋⠀⠀⢠⠟⠀⣴⠇⠠⠊⡴⠋⠉⠙⠂⠀⠀⠀⠀⢹⠤⣄
⠀⠀⠀⢰⣟⣇⣰⣟⠩⠔⣂⠨⠀⠀⠀⠥⠐⠀⠀⠀⠀⣸⠀⠤⡀⢰⠋⢠⠎⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠼⢄⡀
⠀⠀⠀⠀⠉⠉⠉⢿⡄⠀⢂⠩⠄⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠐⠃⢠⠀⠀⠀⠀⣀⠷⠒⠁⣀⣠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠂⠀⠀⢻⠀⠀⠒⣈⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠁⠀⠄⣀⡤⠒⢋⡠⠴⠒⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢷⠀⠤⢐⡂⠀⠀⠀⣠⢞⣩⡤⣹⢀⣠⠴⠚⠉⠅⠒⠉⠀⠠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣧⠊⡡⠄⠀⢀⡞⣱⣿⡿⠗⠛⠉⠀⠖⠒⣚⣉⣀⣤⡤⠤⠶⠖⢲⠒⠀⠀⠀⠀⠀⢀⠀⣸⠏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠨⠀⣼⢰⣩⣤⣤⠴⠖⠒⠚⠉⠉⠉⠀⢠⠂⠀⠀⡤⠒⠲⡟⠲⣧⣀⣀⣠⠟⠉⠁
⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⢿⡀⡊⠁⣿⢸⣿⣿⣮⠟⣶⡄⠀⠀⠀⠀⣄⣀⣈⣉⡉⠉⠁⠀⠀⠀⣸⠉⠁⠀⠀⠀⠀⢀
⠀⠐⠁⠄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠐⡀⠘⠓⠒⠒⠛⠋⢹⡇⠀⠀⠀⠀⠀⠐⢁⠉⣽⠋⠛⠃⠀⠈⣧⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠠⠃⠹⠦⠴⠛⠦⠖⠃⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠈⠁
⠀⢀⡤⠴⠶⡒⠂⠀⠀⢀⠤⠤⡀⢹⣦⡤⢄⠀⡀⠤⠤⠼⡇⠀⢀⣀⣠⡄⠀⠀⣀⡠⠤⠤⡄⠀⢀⣀⣀⠠⡄⠹⡆⠀⣠⠔⠐
⠀⢫⡇⠀⢸⢠⠀⠤⠀⢹⠀⠀⡇⢸⣽⠀⢸⡏⡇⠀⠀⠀⠸⡰⣿⠀⠀⡟⢠⡞⠁⠀⣀⣀⡼⢰⡇⠀⢸⢸⡇⠀⡅⠀⣾
⢸⢾⠀⠀⢸⠀⠀⠀⣼⢠⡇⠀⡇⢨⣿⠀⢸⡇⡄⠀⢰⡀⠀⣷⣻⠀⠀⣴⣽⠀⠀⡾⠓⠊⠀⢸⡇⠀⢸⣿⡇⠀⠇⠀⡯
⢸⡞⠀⠀⡇⠀⠀⠀⢸⢸⠀⠀⡗⢻⣿⠀⢸⡇⡗⠀⢸⣇⠀⢸⡿⠀⠀⣿⢸⠀⠀⡇⠀⠀⠀⢘⡇⠀⠈⠉⠁⠀⢠⠀⣿⡇
⡆⡇⠀⢠⠇⠀⠀⠀⢸⢸⠀⠀⡇⣸⡏⠀⢸⠧⢺⠀⢸⣿⡄⠀⠁⠀⠀⣟⢸⡀⠀⠳⣀⣀⣄⡐⡇⠀⢠⡟⡇⠀⢸⠀⣿⢷⣲
⡧⡇⠀⠈⠉⠉⠉⠉⢹⠘⡆⠀⠉⠉⠀⣠⠇⢸⢸⠀⠸⣧⢧⠀⠀⠀⢠⠙⡈⢧⡀⠀⠀⠀⠈⡇⡇⠀⠀⣧⡇⠀⢸⠀⢀⡔
⢯⠓⡖⠒⠶⠖⠲⠶⠞⠑⢝⡦⣤⡴⠞⠁⠀⢸⡾⢴⠶⠉⠿⠛⠻⠟⠟⠀⠈⠒⠬⠽⠿⠿⠟⠻⠟⠶⠾⠻⠟⠒⠞⠀⠺⠷⠤
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠉⠉⠒⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣊⠭⠥⠤⣀⡠⠔⠒⠈⠉⠒⠲⠤⣀⠀⠈⠑⠒⠒⠒⠦⣀
⠀⠀⠀⠀⣠⠔⠒⠋⠀⡞⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠉⠈⠉⠑⠲⢄⠀⠀⠉⠒⠢⢄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢱⡀
⠀⡴⢻⠀⠀⠀⠀⠀⠀⠀⢉⠗⠒⠒⠈⠉⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣗⠒⠒⠦⡀⢱⢄
⡞⠀⢈⡇⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⢀⡠⢤⡀⠀⠀⢸⠀⠀⠘⡶⠴⠀⠀⠀⠀⠉⠀⠀⠙⠂⠀⠈⢏⠀⠙⡄
⣧⠚⠉⠘⠦⠤⠞⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⡇⠀⠀⡈⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇
⡇⠀⠀⡤⡔⠋⠉⠉⠉⠓⠚⠉⣠⠚⠉⢳⠀⠑⠤⠴⠃⠀⠀⠀⠀⠉⠲⠤⣀⠴⠊⠉⢱⣀⡤⠃⠀⠀⢣⡀⠈⠢⡀
⠘⠦⣀⠀⠸⢠⠖⠒⡶⠒⠒⠊⣡⠴⠒⠁⢀⣀⡀⠤⠤⠀⢀⣀⣀⠠⠤⠚⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⡄⠘⢆⠀⡇
⠀⠀⠈⢢⡀⠇⠀⠀⠁⠀⡰⠋⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠚⠁⠀⠀⢲⠀⠀⠀⠁⠀⠘⡄⡇
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠓⠒⢴⠁⠀⢀⠤⠀⠠⠤⠤⠤⠤⠤⠒⢆⠀⠀⠀⡌⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢠⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⠀⠀⠀⣰⠁⠈⢳⠔⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠢⠤⠼⠤⢄⣀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠸⠁⠀⢠⠃⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣀⣀⣀⠀⠀⠀⡱⠀⠀⠀⣠⠔⠒⠉⠓⣖⠒⠒⠉⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠉⠓⠯⣀⡀⢀⡜⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠀⠈⠉⠉⠑⠒⠤⣀⠀⠀⠈⠢⢄⣀⡠⠄⢀⡼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠑⠒⠤⣀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢹⠦⣀⣀⣠⠤⠤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸

⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠐⢰⠀⠀⡀⠁
⠀⠀⠀⢀⣿⢈⣿⣐⣀⠐⣀⣸⣿⣀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣾⠀⠀⡇⡇
⠀⠀⠀⠰⣿⠚⢹⡏⣿⢀⣿⢹⡏⣾⢻⣿⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⢀⠀⠤⣄⢦⠞⠀⠀⠀⠀⠀⢸⠀⠸⠐⡟⠀⠀⠁⠁
⠀⠀⠀⠈⣿⠀⠹⡷⠿⠾⣾⠿⠧⠿⡾⠿⠀⠀⠠⠀⢰⠏⠀⠀⠀⡤⢐⣮⡍⠤⠀⠀⠀⠘⠀⠝⠀⠀⠀⠀⠘⡀⠀⢠⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⢀⡏⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢸⡇⠀⢸⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠁⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⠶⣬⣥⡄⠀⠀⠀⠀⢨⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠈⢉⡣⢤⣌⠢⡀⡁⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠐⠠⣆⠁⠀⠀⠀⢠⡘⢃⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠴⠄⣀⠀⢒⡊⠙⠚⠷⢂⣰⡬⠃⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠈⠉⠁⠉⠒⠒⠓⠚⠛⠀⠀⠀⠀⠃⡞⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠠⠐⡨⠴⣂⠀⠁⠲⡩⠦⡀⠈⠁⠀⠀⠀⠀⢠⣀⠆⠃⡀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡇⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠄⡟⠑⡶⣈⢀⣶⣶⠻⠃⠀⠀⠀⢰⠘⠋⡰⠂⣦⣲⠦⠾⠿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠉⠀⠜⠁⠀⠀⠐⠾⠊⠀⣴⠼⠶⠋⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠐⣁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢋⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠠⣀⠀⢤⡀⠀⠀⠀⠀⠀⠐⠖⠘⠂⠀⢀⠀⣠⡞⠁⠀⠀⠀⣾⠃⠀⠀⠘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢦⡌⣐⡢⡔⢲⣶⡭⠛⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠊⣀⢶⣞⠉⣁⡬⠑⠂⡩⡏⠀⣇⠶⡺⠟⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠌⣛⠚⢂⣈⣛⡖⠀⠉⠄⠱⢤⡀⠐⠀⣉⡱⠋⡍⠃⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠔⣦⡀⠥⠉⢐⢶⣠⡇⠀⣖⡀⠁⠐⢊⠁⠀⢰⡨⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠈⠒⡈⠂⠰⡦⢮⣍⠠⣬⠤⢜⡂⣍⣥⣦⣉⠙⠉⡥⢀⢠⠌⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢸⠈⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠉⠻⠧⢶⣴⡁⡠⣦⠀⣀⠡⠔⢉⠡⣄⠀⡐⢿⠏⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⠀⠀⠈⠁⠉⠉⠉⠉⠛⠛⠙⠩⣯⡿⠿⢂⡀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣶⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⣫⠗⣿⣿⣿⡏⣿⢨⠏⡼⠹⢛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣯⢶⣫⢷⡟⠛⠻⢿⣅⣿⣾⢜⡁⣀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠿⣻⣿⣿⣏⠿⠇⠼⣭⣦⢹⣿⣿⣿⣧⠉⠰⡾⡿⢫⡿⣿
⠀⠀⠀⠀⠀⢀⡠⠖⢉⡠⠂⡻⣿⡇⣞⠱⣧⣼⠂⢮⣡⣾⣿⣿⣿⣿⠇⢀⢱⠃⢼⣷⠏
⠀⠀⠀⠀⠀⠉⣠⠖⠉⠀⠀⠙⣅⡳⣹⣷⡂⢀⡒⢋⠉⡋⠙⠯⠥⣄⠰⢁⠚⠀⢟⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⣝⣓⡀⢄⣠⢤⢀⣠⠄⢒⣛⣹⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⠀⠀⠀⠀⠋⠀⠀⠤⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⡔⠒⠀⠡⠏⠁⠀⠔⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠆⠠⢁⢸⣾⢠⠠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡔⠋⠀⣹⠇⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⡏⣷⠚⡅⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡝⠛⣋⢸⢰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣀⠀⠉⠈⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣜⣰⣇⠙⢁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠜⣡⡏⣷⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⢰⢋⡇⠉⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⣸⣽⡏⢸⣧⢖⣼⣿⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣻⣛⣛⣧⡀⣿⣃⠻⡷⠿⡘⢦⣴⣁⣄⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣀⠀⣾⢉⣗⣄⣽⠵⠼⠁⠈⣁⣢⣍⣩⣮⣙⣻⠉⠿⡤⡖⢊⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢘⢶⣦⣝⡶⠻⣪⡝⣻⣏⡐⠎⣻⢩⠐⡤⠞⣀⠲⠆⣼⠈⣦⣮⣽⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡏⣎⣽⣿⣿⣿⣷⣷⣶⣶⣿⣬⣿⣤⣽⣦⣿⣶⣾⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⢾⣿⣯⣾⣿⣿⣿⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⡉⣯⣿⣿⣿⣟⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⡁⢤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠛⠙⠉⠛⠙⠉⡉⠁⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠁⠀⠀⠀⠀⠐⠀⠀⠡⡄⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡒⠲⠶⠴⠤⠶⠦⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⢶⡶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⣶⣶⠶⣶⠶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⡏⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠉⠉⠉⠉⢩⡿⢹⡿⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⣀⡀⠤⠤⠤⠠⠠⠤⠤⠤⠤⣾⢧⣿⡁⠀⠀⠈⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡙⠢⢄⣀⣀⡀⠁⠘⠈⠀⠐⠈⠋⣛⢇⣀⠭⠟⢋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢉⠐⠂⠤⠡⠍⠉⠉⠉⠉⠉⠩⠍⠄⠐⢂⡉⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠬⠀⡀⡀⡀⢀⠀⠀⠀⠀⢀⢀⢠⠨⠸⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠁⠂⠄⠁⡁⡃⠛⠃⠘⠙⠀⣘⡈⠅⠄⠂⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠰⡦⠤⠀⠀⣠⡄⢠⢤⡖⠒⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣏⠑⣀⣰⠯⣿⡀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠈⢱⠉⡄⡌⡥⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢙⣆⢺⢂⡵⠛⠳⢹⡧⢼⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⠛⠙⠧⠼⠜⠷⢤⠤⠼⠄⠚⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠌⡦⠠⣄⣀⣀⣀⣀⣈⠤⣤⠒⠦⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠠⠙⠘⠐⠘⠘⠘⠈⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⣀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠶⠶⢦⣤⣀⡀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠷⣤⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⡴⠛⠁⠀⢀⣠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠛⠒⠶⣄⡀⠀⠈⠙⢦⣀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣴⠋⠀⠀⣠⡶⠋⠁⠀⠀⣀⣀⣠⣤⢤⣤⣤⣀⠀⠀⠀⠙⠷⣄⠀⠀⠙⢦⡀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⣠⣴⣾⣯⣀⠀⣠⡶⣿⡏⣼⡟⠁⡠⠀⠀⠉⠙⠲⣤⠖⣤⢬⣷⡄⢀⣈⣳⡀⠀⠘⣷
⠀⠀⠀⠀⢰⡏⠀⠀⣾⠁⢸⣿⣿⣿⣿⣿⣿⡁⢡⡈⣷⠜⠓⠚⢳⣾⣿⣷⡀⠀⢸⠀⠃⣜⣹⣿⣿⣿⣿⣷⡀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⣸⠇⢀⣀⣿⡅⠀⢰⠏⠉⠃⣰⡏⠓⢄⠀⠀⣿⠿⠛⠉⠙⣦⣞⡲⣤⣟⢸⣿⢿⡟⠋⠈⣧⠀⠀⢻⡀
⠀⠀⠀⢰⡇⠀⢀⣿⠚⣩⣼⣷⣶⣶⡏⠀⠀⣼⣫⠀⠀⠈⢣⠀⠁⠀⣠⠖⠛⠷⠶⣮⣵⡼⠟⠁⣈⡧⣀⠀⢻⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⡏⣼⣿⣿⣿⠋⢸⣧⣴⠖⣾⠃⠀⣄⢢⣠⠇⠄⠴⣁⡀⢰⡆⠀⣠⠙⠇⡆⠘⣇⠀⣸⣄⣼⠁⠀⢐⡇
⠀⠀⠀⢸⡇⠀⠀⣷⠹⣿⣿⠃⠔⠁⡿⠃⢀⠇⠀⠀⣿⠏⠀⠀⠀⠀⠀⠹⡄⠁⠀⠙⠘⠛⠧⠤⠞⠀⣏⣤⢘⠇⣴⠿⠋
⠀⠀⠀⠈⣷⠀⠀⢻⣀⡤⠘⠧⢤⣀⠿⠒⢺⡀⠀⡰⣿⠀⠀⠀⠀⠀⠠⢂⡇⠀⢀⣾⡋⠂⠀⣀⣤⠜⠸⢃⡜⢠⠞⣉⠙⢇⢀
⠀⠀⠀⢀⣽⡊⣠⠜⢿⠀⠀⠀⣰⠃⠀⠀⣾⡷⠛⠛⠻⡶⠛⠓⢄⠀⠀⣸⣥⡀⣼⠀⠻⠉⢭⠀⣸⣄⣠⡍⠙⠃⠾⠋⣠⠇⢸
⠀⠀⠀⠊⠀⡟⠁⠀⣨⠟⠚⠉⠉⠉⢙⣿⡈⣇⠀⠀⠀⠘⣧⡄⡼⠛⢱⡋⠀⠉⠋⠑⠲⣶⡞⢁⡀⠀⠙⡄⠶⠀⡴⠊⠁⠆
⠀⠀⠀⠀⠀⠁⡀⣸⠁⠀⣴⣶⡿⠿⠟⣛⠛⠉⠑⠠⢀⣀⠈⢿⣧⣤⣤⣃⠀⠀⠀⢰⡆⣽⣧⡘⠓⠀⣰⡇⠀⡰⠃
⠀⠀⠀⠀⡠⢸⠀⡏⠀⠀⣿⠁⠘⠓⠀⠉⢸⡆⠀⠈⠀⠉⠑⢄⠐⠀⠀⠈⢳⠤⣄⡈⠀⠙⢿⣿⡦⠚⠛⠶⣞⡥⠖⠲⡶⣄
⠀⠀⠀⠀⠱⠈⠀⣧⡀⠀⢻⡄⠀⢠⣤⣶⣾⣇⠰⠦⠀⠟⢀⢸⡆⠀⢀⠀⡄⠀⠀⣧⣀⣶⣞⡙⢿⠀⣠⡀⠉⠀⣸⣇⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠾⢁⣄⡈⠧⣤⣶⣿⣤⣐⣲⣿⣡⡼⢇⠀⣸⢀⡇⢰⢀⣿⠟⠉⣍⡙⢮⠁⢉⣵⣏⠹⠿⠛⡉
⠀⠀⠀⠀⠀⢀⠀⢀⣼⣇⣀⣀⣠⡿⣳⣿⣿⣿⣿⣿⠛⣍⠁⢠⠘⡇⣿⣼⢁⡇⣼⣷⠛⡆⠙⠛⢸⡤⠋⠀⠈⣧⣿⠀⠇
⠀⠀⠀⠀⢀⡏⣰⣿⣿⣿⡟⠉⠀⠙⣾⠟⣿⡏⣤⡜⣧⠗⢒⠢⡄⣿⣿⣿⡿⣿⣿⣇⣰⣷⣦⣤⣼⠦⠈⠀⣠⠟⠁
⠀⠀⠀⠀⠈⠇⢿⣟⣻⠏⣰⣾⠡⣯⣉⣇⠉⠐⣦⣤⠿⠀⠈⠑⣇⠛⢉⣩⡿⢯⠠⠀⣿⣿⣿⣿⣿⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢨⠇⣼⣿⣿⠀⠻⡀⠉⠉⢻⠁⠀⣠⡀⠀⢀⡈⣻⢷⣎⠳⡖⣶⠀⡟⠉⠉⢠⣿⣿⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⡿⠏⠀⠀⠙⣄⠀⢸⡀⠀⣿⢇⠀⢠⠋⠁⢀⣿⢀⡇⣯⡼⠃⠀⡰⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣛⠓⢦⣑⠶⠃⠘⠶⠋⠀⠀⡞⡞⢸⠀⢀⣠⡴⢾⣷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠐⣧⣀⠀⠈⠙⠓⠲⠤⠤⠤⠼⣇⠣⣌⣻⠿⠛⣉⣩⣍⢹⣜⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣭⣉⣒⣒⠲⠤⠤⠄⠀⠀⠈⠙⠚⢛⣋⣉⣡⣤⠽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠀⣐⣂⣀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠚⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣛⣚⠀⡄⣇⣩⣉⣩⠩⠅⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠶⠧⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡡⢭⣍⠂⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⡇⠛⠀⠠⠥⠀⢄⣍⡐⡠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣄⠄⠘⡇⠀⢤⡤⡄⢴⡄⠈⡆⠄⡈⡴⡂⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣍⢀⡘⡙⠀⠄⠓⠟⠁⠓⢄⠻⢋⠤⠣⡀⠀⣀⣱⠈⢑⣤⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⠭⢄⠁⠂⡀⠉⣐⡂⣮⠤⠶⠞⣠⣀⡀⠃⠉⢉⣀⡄⢍⡸⢼⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡣⡁⠬⢧⡀⠀⣀⡀⠋⢐⠁⣈⡢⠊⠬⠂⠛⠶⡄⡘⢂⠂⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡛⢷⣾⣤⣃⣀⣚⠂⠊⠀⠀⡀⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⡶⠾⢛⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠳⢦⣭⣉⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣥⡤⠶⠛⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⢉⠉⠉⠛⣿⠟⣿⡿⠟⠿⣿⠿⠿⢿⣬⠻⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠿⣛⣛⣛⠿⣾⣯⣼⢯⣼⢟⡟⠖⣾⠿⣶⣽⣧⣽⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢛⣉⣙⠻⢾⡷⡿⡿⣻⢿⢻⢣⠇⠹⣟⣉⣉⡙⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣉⣙⢿⡬⡇⣼⡳⣛⣾⣺⣿⡿⣧⡞⣉⣉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸⣧⣹⣟⢿⣮⣻⡷⠿⢃⣿⢰⡟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣮⣝⣧⣿⢏⣴⠞⣛⣡⡾⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠿⠾⠿⠾⠟⠛⠋

⡶⣶⣶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣷⣶⣤⡄⠈⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⢄⣿⠿⢩⡼⢟⣩⡭⣤⣤⡯⢉⠉⠉⠀⠀⣤⣤⣥⣤⣍⣙⢻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠔⣸⣿⣿⣛⣉⣹⡿⣾⣯⣶⣿⣴⣟⣳⠾⣯⢛⣿⣥⣦⣴⣿⣭⢻⠉⠉⠙⠿⢻⠿⣻⣿⣿⢿⣶⣄
⠀⠀⠀⠀⠀⡀⢠⣾⡿⠟⢉⣩⣁⡀⠀⠀⠉⠉⠻⢟⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣴⣀⣤⣾⠿⠛⢁⣤⣾⣮⡛⢿⣦⣤⡄
⠀⠀⠀⠀⠐⢠⣿⡟⢁⠀⠀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠀⠀⠀⠈⠁⠛⠿⠿⣿⡿⣿⣿⣿⣆
⠀⠀⠀⠀⡄⣿⡟⠀⢸⣷⠿⢟⣛⡛⠉⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠈⠋⢀⡀⢿⣿⣿
⠀⠀⠀⠀⡇⢿⡇⢠⠊⢠⡶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠉⡟⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠇⠙⠿⣷⣆⣐⣖⡀⣀⡼⣦⣼⣿⡟
⠀⠀⠀⠀⠱⠸⣧⠀⠀⣏⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⡇⠘⢟⣼⠿⠉⠘⠉⠛⠚⢿⡏⠀⠀⠀⠀⠈⠙⡻⢿⣿⠿⣤⣾⣿⡿⢠⠁
⠀⠀⠀⠀⠀⠱⠙⢦⣄⠘⣦⡀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠘⠃⠀⠭⢿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⣉⣿⠟⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣦⠻⣦⡬⢝⣛⣛⡉⠀⡀⢀⣀⣰⡠⠤⣹⠞⢲⠞⠻⢦⣄⡀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⠃⣨⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⡙⠦⣬⣉⠹⣻⣟⡟⡓⠇⢀⡀⠀⢠⠞⠶⠀⠈⣋⠹⠶⢶⣾⣿⣦⣀⡤⠀⢀⣠⣿⡷⢛⣥⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣿⣷⣮⣝⡻⠶⣶⣴⣂⡐⢀⣤⠄⠀⠉⢠⣤⣤⣐⣁⠀⠛⣻⠟⢛⣻⣿⡿⠟⣋⣥⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣶⣬⣭⣉⣙⣓⡒⠒⠲⠾⠷⠾⠿⠿⠿⣿⣟⣻⣯⣭⣶⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢴⠆⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⢘⠀⠀⠈⠁⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠀⠉⠈⠁⠠⡀⠀⠀⠀⠀⠠⠀⠈⠈⠁⠐⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣑⡀⠀⠀⢀⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣠⡤⠶⣾⡿⠿⠛⠋⠀⠀⠀⠈⠙⠻⠿⣿⡶⠦⣤⣄⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣶⡯⠿⠟⢓⣈⠥⠔⠒⠚⠛⠛⠛⠛⠛⠋⠒⠒⠤⢬⣀⠛⠻⠿⣿⣶⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡤⠚⠉⠀⢀⡀⠀⢀⠣⡀⠐⠀⠲⢀⠂⡕⢀⡀⠈⠙⠲⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⠔⠁⣀⠂⠠⠂⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⢠⠂⠠⠙⢦⡀⢀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡴⠃⠀⠀⠀⠁⠀⠀⠀⠰⠆⠀⠀⠀⢈⡃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢧⠄⠙⢆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠈⠁⠀⠤⠁⠠⠀⠐⠿⠂⠀⠀⠀⠀⠀⠀⢧⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡾⠁⠀⠀⣶⠋⣭⡔⢄⢠⠄⢀⡀⢀⢰⠆⠀⠐⠀⠀⣰⠖⢢⠀⠀⢰⠀⣄⡤⢄⠀⠀⠈⢧⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠠⠀⢰⠃⠀⢠⡄⢻⡀⠒⠃⡌⢸⡀⠥⠟⡠⢸⡆⠀⢠⠀⢠⡏⠼⠇⢃⠀⣾⠀⠋⡐⢁⡀⠀⠀⠘⡆⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢀⠀⠀⠀⢸⡇⠚⣳⡌⠺⡇⠺⡌⠀⢸⡅⠈⠀⢀⡾⢀⠀⢻⡀⢀⡏⢰⢦⠑⡈⠁⠀⠀⠀⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⢸⠀⠀⠀⠀⠷⠈⢩⠐⠁⠓⠠⣩⡌⠈⠩⢩⡄⠄⢠⡤⠄⢠⠄⠀⠉⡅⠈⠉⠀⠀⠀⠀⠁⢸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢃⠈⠀⠐⢅⡐⣀⡀⠘⠂⣦⠀⠈⠁⢹⡆⠀⢸⡇⣀⠛⣠⠀⣾⠀⠁⠀⠀⠀⠰⡦⠀⠀⠀⠀⡆⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠸⡀⠀⠀⠀⠔⡦⢙⠆⠀⢿⠰⠀⠀⣸⡇⠀⢸⡇⠻⠒⣿⠀⣿⠀⠂⢁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⠑⠒⠋⠀⠀⠸⠀⠀⠀⠃⠀⠒⠈⠓⠀⠀⠛⠂⠛⠒⠂⠀⠀⠄⠑⠀⠀⠀⢀⠷⠀⠒⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠄⠒⠀⢉⣁⡉⣉⢀⡀⠐⠀⠠⢀⡀⠀⠀⠸⠇⠀⠀⠀⠀⢘⠡⠇⠰⠀⠀⠀⢠⣾⣤⣤⡀⠐⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠑⠀⡀⠀⠀⠘⠛⠒⠒⠀⠀⠖⠖⣢⠄⣤⠈⠀⠀⠂⠠⠄⢀⣀⠈⠑⠒⢁⣀⡠⠴⠿⠛⠉⠀⢳⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢯⣠⣤⣶⣶⣾⣉⠉⠉⠉⠉⠐⠂⠀⠀⠉⢉⠀⠊⠽⠂⠀⠢⣀⡄⠀⣀⠀⡀⢠⠀⠀⠁⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣀⠀⠈⠛⠿⢿⣿⣿⡇⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⣀⣈⢓⠀⠒⠘⠁⢈⣀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⢀⠀⠐⠀⠀⠀⠁⣠⡀⠀⣤⠈⠉⠓⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉⠁⣬⠅⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠩⠤⠤⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄⠤⠤⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠉⢉⠉⣉⣉⡉⠉⠀⡉⢀⢁⠈⡀⢁⠈⣉⠈⡁⣈⢉⢈⡁⡁⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⣀⣀⡀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠀⠁⠁⠈⠀⠁⠁⠀⠀⠀⠀⠔⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⡀
⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡠⠬⢍⣔⠵⢀⠊⢴⡈⠅⠠⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⢠⣀⠜⠫⠛⠁⠀⣨⡗⠁⡙⡉⠒⠤⣶⡍⠓⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⠒⢀⡤⠧⠦⣮⠁⢉⡏⢀⣀⣋⢄⢈⠢⣑⠶⡼⢀⠋⢩⠀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⡀⠀⠀⠀⢸⠃⠀⠀⡠⣮⠀⡞⢋⠓⣄⡌⠀⢸⣌⢲⡂⢒⡡⠂⡎⠀⠀⣅⡫⠂⠈⠠⠊⠑⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣿⠀⠀⡜⠀⠀⣸⠁⢰⠂⡉⠀⠀⡁⠀⠉⡋⣱⠄⠀⣩⡀⠀⣀⣀⢠⣶⠊⠔⠃⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠀⠀⢹⣄⠨⡇⣰⠛⠉⢆⢂⠠⠐⠢⠰⢆⠖⡉⢹⢹⡜⡲⢧⣀⡀⢉⠔⠊⢐⠈⣿⡶⠿⠩⠢⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢣⠀⠘⣿⣨⣵⣏⠀⢠⡟⢹⠛⠲⡤⠀⡇⢀⠞⢣⢸⢄⠀⢸⣭⣉⣊⣀⡀⠴⠀⠈⢸⠑⡂⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⣙⣲⣬⢥⣄⠑⠀⠰⢄⣼⣾⢂⠨⠘⣃⣀⠠⠨⢈⣛⡞⣐⡨⠀⠀⠖⠄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠆⠀⠈⢷⣆⡎⠈⣄⠁⠬⣡⣄⣠⠉⠈⠳⣄⣦⡦⠛⠉⠍⢉⢘⠀⠉⡔⠣⣀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠆⠀⠈⢋⠻⣶⣄⢙⠖⠋⣀⣈⡄⠔⠀⣥⠦⢤⣄⡐⢆⠲⠆⢂⠜⡀⢀⣴⠟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡄⠙⠻⢶⣬⣁⣩⢀⠁⢊⠲⠇⠀⢒⡑⣠⣋⣩⣥⡶⠟⡋⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠉⠙⠛⠻⠿⠿⠷⠿⠿⠿⠛⠛⠉⡑⠂⠅⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠀⠈⠂⠀⠀⠀⠀⣂⡤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⣟⡳⠦⣤⣄⣀⣀⣤⣀⣀⣤⡴⢞⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⢳⠀⡄⠀⠀⠀⠻⣿⣷⣶⣬⣭⣭⣭⣭⣴⣶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⡆⢻⠹⡄⠀⠀⠀⠈⠑⠪⠭⣭⣭⣭⣭⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠐⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣷⠀⠀⠀⠀⠸⠈⠋⠀⠀⢀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠀⢾⡻⣶⣷⣇⠀⠀⣄⠀⢀⡏⠎⢦⣼⡇⠀⠀⢶⡎⡇⠀⠀⠀⠀⢲⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⡙⣿⣿⣷⠀⠇⡀⡞⠁⢠⢀⢿⠙⡆⠀⢛⣿⣸⠀⠀⠀⠀⢸⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠐⡆⠀⠀⢶⡌⠻⣿⡄⠞⢡⢃⠠⠃⠌⠈⣷⠙⡆⢸⣿⣏⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠟⣦⡙⣷⢰⠈⠾⠀⠘⢀⢠⠇⡀⣷⢀⣿⣯⢣⢳⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⣯⣷⣌⢷⡄⠀⠃⢀⠃⠈⣆⢳⢺⣸⣿⢃⣿⣼⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠙⣇⠀⠀⢸⣽⢫⣆⠁⡆⠀⠈⠰⠀⣠⢀⠚⣿⣹⠀⢸⣇⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠈⠀⢹⡀⠀⠈⡷⢸⣹⡧⡨⣦⠀⢠⡆⠁⡾⠰⣾⢿⡆⠰⣿⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠰⠡⠙⢷⡟⢎⣳⡄⠐⢀⢡⡄⠁⡈⣷⠀⡟⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢂⠆⠈⠻⣫⡷⣥⣡⡇⢸⢇⠀⡀⣿⡄⣧⡇⠀⠀⠸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠄⠀⠈⠌⡀⠀⠹⣻⡝⢦⠸⣇⡿⠀⠇⡼⣧⢈⣿⡄⠀⠀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠐⠐⠀⠀⢙⣷⣎⠱⡜⣷⡀⠂⢧⣿⡈⢹⣷⠀⢠⡇⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⠀⢸⢺⢫⣷⡘⢮⡻⢷⠀⣿⡇⠌⡜⡇⠈⢷⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢁⠀⠀⠘⠆⠻⢷⣎⡳⡱⡅⣽⢻⣄⠇⢹⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡄⠐⠘⡆⠀⠘⣿⣷⡽⢮⠺⣣⡻⡐⣧⢇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡘⡀⠀⢁⠀⠀⢻⡷⢳⡎⠱⡙⡷⡇⣿⡼⡇⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠐⡡⠀⠀⡆⠁⡘⣡⢱⣟⢗⠙⣎⠿⢿⣣⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⠃⠀⠁⠸⢸⠸⢸⡝⣾⡂⢋⣧⡈⡳⣽⡇⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⠀⡀⢠⠀⡇⠀⣿⣏⢻⣿⡀⡟⣔⠜⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠄⠀⠀⠘⡄⢠⠀⣼⡜⢿⣿⣿⣤⣿⢆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⡀⠐⡀⣠⠆⢣⡜⡄⠹⣠⠀⡟⡜⢟⢿⠸⠷⢔⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢷⠀⠱⣩⠆⠈⣶⣀⢸⣿⡈⣻⢱⡨⠋⣳⣼⣼⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⡆⠀⠉⣟⢤⢻⣷⢂⣿⡧⠏⡆⣧⣸⡗⠹⣿⠇⠐⢝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠀⠀⠘⡄⢘⢹⡎⣸⣧⢼⣇⠸⣞⢃⠘⣟⢦⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠈⣷⠈⠹⢷⠽⣯⡞⣿⠀⢉⢋⠀⠁⠸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡆⢸⣴⢻⡄⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣈⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠟⢉⣄⡙⢿⣿⣿⣿⠿⠟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠏⣰⣿⣿⣷⡄⢻⣿⡇⢰⣷⠀⣿⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⡿⢀⣿⣿⣿⣿⣷⠈⣿⡇⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢁⣴⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢹⡇⢸⣿⢀⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣧⡈⢿⣿⣿⣿⠟⢀⣿⣷⡈⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⡇⣴⣿⣿⣿⣿⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⣿⣿⣿⣿⠏⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⡇⢸⣿⡇⣿⣿⠟⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⡇⢸⣿⡇⠙⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠛⠛⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⢣⢣⢏⣞⠉⠙⢲⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⢻⣸⡸⡦⠀⠀⣠⠖⠒⠒⠶⠶⠶⠶⠶⠶⠶⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠭⣟⣇⡇⡷⣸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣟⣼⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠛⡞⡞⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⣴⣳⢳⢫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣳⡀⠀⠀⠀⠀⢳⣹⡸⡘⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡇⠀⣀⣤⣾⣭⢇⣇⡟⣸⣤⣌⣉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣷⠿⣓⡺⢽⣛⡺⢾⣚⡯⢴⡒⠦⢉⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠩⣛⣓⡺⠿⠤⠭⠭⠤⠿⢟⣒⣻⠍⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣉⣛⠒⡒⠒⠒⠒⠒⣛⢉⡅⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣘⡛⠣⠟⠽⢯⠵⠯⠼⠗⢚⣁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣤⣀⣀⣤⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠂⠃⠐⠠⠄⠄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠁⠀⠀⠀⠀⠀⠀⢀⠔⣦⣤⣲⣆⣄⡄⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠈⠛⠈⠛⠒⠲⠓⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢱⠄⠀⡄⠀⠰⠀⠀⠋⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣄⣀⣀⠁⠀⠘⠒⠒⠐⠒⠀⠈⠀⣈⣁⣠⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠤⠤⠀⠒⠒⣒⣂⣉⣉⣡⡤⠤⠤⠤⠔⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⠖⣶⣖⣋⣉⣉⡉⠁⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⣄⡀⠁⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠩⠠⠤⠀⠀⠠⠦⣠⡀⢄⠀⠀⠤⡠⡀⠉⠑⠒⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⣀⠀⠐⠠⡔⠛⠠⢜⢲⡨⣭⠅⡍⠦⠀⠀⠄⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠗⠤⢄⡀⠈⠐⠂⢠⡱⠂⠂⠓⢬⣩⠷⡒⠂⡓⠭⢲⠤⢉⠉⢁⣈⡡⠖⡏
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠁⠒⠚⠒⠐⠰⠮⠲⠽⠴⠠⠔⠲⠀⠂⠓⠊⠉⠁⠈⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠉⠙⠐⠒⠒⠒⠂⠉⠉⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠙⡀⡎⠀⠀⢀⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣷⣓⢷⡰⣾⡃⠐⢲⣿⣽⠻⣯⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⢹⡿⣻⣌⢻⡛⣺⣷⣾⣦⣴⣷⣿⣿⣷⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢃⠻⢗⣶⡞⣷⠙⣷⣤⣧⣿⣿⣿⣿⢿⣿⢣⡿⡽⢷⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠛⣤⡠⠤⣼⡛⢓⣾⣧⡇⢾⣧⠝⣫⡟⣻⣼⣾⣟⢼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠢⠶⠖⢋⣑⣒⡊⢷⠹⣾⣿⠎⡿⢃⡍⠀⢛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠞⠳⠀⠛⡃⠸⠇⢀⣀⡀⢒⡋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠀⢀⠈⠇⠐⠀⠀⠀⠀⠀⠈⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠂⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠡⠀⠒⠃⠀⠀⡀⠐⢀⠃⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠀⠀⠄⠀⠈⠀⠀⠲⠀⠀⠀⠄⠈⠠⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠐⢤⠀⠀⠄⠀⡀⠀⠀⠐⠉⠂
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⡀⠁⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡈⠁
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢙⠈⠐⠉⠀⠐⠒⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⡀⠐⠀⠁⠈⠉⠀⠱⠂⠀⡠⠀⠉⠑⠂⠀⠤⠄⠀⠄⠀⠀⠀⠒⠀⠀⢊⠀⠀⠀⠀⡄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⢀⡄⢀⠂⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠠⠒⠠⢤⡂⠀⠤⠀⠀⠂
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠐⠚⠈⠀⠀⠀⠀⠦⠀⠈⠁⠀⠈⠈⠀⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠆⠀⡆⠀⠐
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠂⠀⠀⠀⢰⡃⠀⠀⢀⡀⠒⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⣀⡀⢈⠀⠈⠀⠀⠀⠀⢨⡀⠈⠀⠀⠀⠀⠀⠀⠰⠃⣰⠇
⠀⠀⠀⠀⠀⠈⠣⡀⣄⠀⠀⠀⠀⠀⠈⠲⠀⠐⢤⠀⢠⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢀⠄⠀⢠⠁⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠅⡀⠀⠀⠀⠁⠀⠁⢀⠘⠀⠀⠄⠀⢀⡠⠄⢀⠀⠀⠣⠁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡤⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠨⠅⣑⠀⠀⠥⠀⠠⠙⡄⣁⠃⠈⠀⣀⠀⠀⠴⠁⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠅⠀⠉⠀⠈⠈⠀⠉⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣽⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣉⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣚⡭⠔⠚⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣵⣚⣩⡥⠤⠤⠴⠖⠒⢒⣛⣉⣉⣉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⡿⡷⡗⠒⠒⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣛⢟⠞⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣢⣄⣀⡠⢼⢼⠻⣸⠤⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢞⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠀⠠⠀⢀⠀⢠⠈⠀⠈⢸⠀⢸⠘⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠚⠚⠚⠚⠚⠚⠚⠚⠛⠚⠚⠛⠚⠛⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⡀⠀⣀⣀⠀⠀⠀⠀⣀⠤⠊⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⣀⠀⣉⠉⣁⢉⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠳⡗⢅⠿⣀⠇⠷⠙⠄⠘⢂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡠⠔⠒⠒⠒⠠⠤⠤⠠⠤⠒⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠤⠔⠒⠂⠀⠀⠀⣰⢒⢖⣴⡶⠯⢞⡩⣫⠋⠉⠃⠒⠒⠒⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⠀⠀⠀⠀⠀⡠⣪⢝⠻⣿⣾⣵⣵⢟⡕⡮⠛⠓⠊⠀⣀⠤⠔⠒⠉⠉⢿⣿⠀⢷
⠀⠀⠀⣀⣀⣠⣤⣴⡎⠀⠀⠀⣀⢤⡶⠶⣒⣚⣵⡕⡡⡪⢽⣿⣛⣓⣩⢞⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢸⣿⡆⢸⡆
⠀⠀⠈⣇⠀⠸⣗⢺⡧⠀⠀⣜⡕⡡⣊⣭⣿⣿⡿⠮⢚⠟⠁⢀⣠⠤⠚⠉⢀⠀⠸⣁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠜⣧
⠀⠀⠀⢻⡀⣠⣿⢼⡧⣏⠩⣮⢎⣼⡿⠓⠈⠉⠉⣉⠥⠒⠋⢁⡀⠰⡒⢦⠘⣆⢀⠱⠤⠀⢀⡠⠤⠒⠉⠁⢹⣽⠀⢺
⠀⠀⠀⠨⣷⣟⣽⢻⣷⢳⣮⡵⡿⠋⠀⣀⠤⠒⠉⠀⠠⠢⡀⣇⠙⡆⠱⣠⠇⠈⢁⡠⠔⠊⠁⠀⠀⠀⠀⠀⡼⣷⣷⢶⣇
⠀⠀⠀⠀⢽⣿⣛⡭⣿⣿⡯⠊⢀⡴⠊⠁⢠⣄⠈⣆⠸⣄⢱⠈⠲⠃⠀⣀⠴⠚⠁⠀⠀⠀⠀⡠⣪⠭⣗⡔⣱⣿⣿⡥⣷⡀
⠀⠀⠀⠀⠘⡿⠀⠀⠸⣿⣦⡴⠋⠀⠀⠀⠀⢳⡙⠺⠆⠈⠉⢀⡤⠖⠋⠁⠀⢀⡠⠤⣤⣀⣠⣳⣳⢊⡲⣭⣼⣿⣿⡮⠛⡇
⠀⠀⠀⠀⠀⢻⣄⠀⣠⢿⣽⡃⠀⠀⠀⠀⠀⠀⠁⠀⣀⠴⠊⠉⠀⠀⠀⠀⠀⡎⡔⡩⢛⣻⣷⡥⣣⡿⡫⠭⠿⠿⣹⡇⠀⣗
⠀⠀⠀⠀⠀⠀⣷⡚⠁⢪⣿⢿⡀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⡠⡲⠽⢶⣶⣞⢼⢟⡔⡡⢒⣒⣻⡫⠎⠀⠀⠀⠀⢿⣷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢘⡆⠀⠈⡿⣟⣷⣀⠴⠊⠁⠀⡠⠤⣀⡀⣰⡹⡰⣫⠅⣛⣻⣷⡪⢞⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢪⡄⠀⠀⢻⣹⡇⠀⠀⠀⣞⠔⣊⣙⢓⣚⠕⣱⣵⡯⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢯⡤⠚⠃⣷⣷⣶⠤⢔⢯⢎⢔⡶⣶⣶⣭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⣿⣿⣟⢿⣵⣫⣣⡿⠋⠉⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠤⠤⠒⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠋⠉⠉⠙⠛⠽⠯⠤⠐⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣰⡟⣿⠛⣿⠛⣦⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⡿⣿⣟⣴⢋⣼⣃⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣄⢳⡌⢷⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣛⣉⣴⢿⡆⣻⠈⣷⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣴⠟⣉⡥⣶⠟⣴⢋⡼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠴⠦⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣴⣿⠰⡇⢸⡇⢼⡁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣍⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢻⡌⢷⡈⢷⡀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣻⠈⣷⢈⡇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⢋⣼⢃⣼⠃⠀⠀⠀⠘⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⣿⡷⣏⣳⣾⢹⢾⣏⡆⣏⣀⣹⢟⣐⢿⡅⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣬⣤⣤⣬⣥⣤⣥⣬⣭⣤⣭⣥⣬⣥⣬⣭⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡠⠀⠛⡟⢛⡒⠦⢀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣤⣬⡁⢈⠁⢚⡀⠖⢋⠻⠆⠈⢿⣭⠀⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢤⡀⠀⡀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠸⠖⠃⠐⠦⠀⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣹⡤⠽⠃⢀⣄⣁⠀⠀⠖⠁⠀⠀⠀⠀⢀⣀⣬⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠉⠁⠀⠔⠀⠙⠉⠉⠠⢀⡆⠀⢠⠔⠀⢀⣀⣠⣄⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⠀⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⡁⢀⠁⠀⡴⠉⠀⠀⠀⠀⠲⠹⢆⠻
⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠸⠀⡎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡜⠃⠙⠿⡀⠀⡰⠞⠛⠲⡄⠂⠀⠈⠀⠀⠀⠀⠑⠂⠀⠀⠲⡀⠀⠀⠠⣄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢿⡒⠀⠀⠀⣷⡀⢁⠀⠀⢠⢀⡴⣤⡶⢤⢤⠄⠀⠀⠀⠀⠀⠀⠳⡤⠀⠂⡁⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⢘⣛⠢⠈⠬⢅⡛⣐⠷⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠁⠉⠐⠤⢴⠟⣀⡄⠀⡀⠀⠀⠀⠀⠀⠄⢠⡾⠃⢾⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠈⠀⢨⣩⣭⡄⠰⠓⡄⠀⠀⠒⠋⠠⠟⢠⡆⠘⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⡗⠀⢃⠈⠻⠚⠀⡀⠘⠀⢀⢀⡐⠃⡤⢤⡟⠁⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠑⠀⠀⠉⠀⠀⠀⠀⣠⠀⡃⠜⠳⣤⣤⠈⣃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢀⣶⣶⣒⣠⣇⠙⠂⢡⡔⢄⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠘⠃⠀⠉


⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠶⠶⠶⣶⡶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⡾⠿⠿⣦⣶⠿⠻⢿⣶⡿⠻⢿⣭⣭⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡅⠀⠀⣿⠃⠀⠀⢰⣿⠀⠀⣾⣏⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠠⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣇⠀⠀⢿⡇⠀⠀⣸⡇⠀⢠⣿⣋⣉⣉⣉⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡄⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⡇⠀⢰⣿⡀⠀⠀⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠁⠀⠀⣿⡇⠀⢸⣿⠻⣷⣴⡿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⡿⠁⠀⠀⣠⣴⡾⠟⠻⣷⡀⠀⢿⡆⠈⢻⣧⠀⠀⣰⡿⠁⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢰⡿⠁⠀⢠⣾⠟⠁⠀⣠⣴⣿⡇⠀⠸⣷⠀⠀⢻⣇⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⣾⡇⠀⠀⣿⠃⠀⢀⣾⠏⠁⣼⡇⠀⢀⣿⠀⠀⢸⣿⠋⠁⢀⣀⣤⣤⣤⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⡿⢿⡇⠀⠀⣿⠀⠀⠸⣿⠀⣰⡿⠀⠀⣼⡟⠀⠀⣼⣧⣴⠿⠛⠋⠉⠉⠉⠙⠻⢷⣄⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⣿⡀⠀⠻⣷⣄⣠⣿⡿⠋⠀⢀⣼⠟⠀⠀⣰⡿⠋⠁⣀⣤⣶⠾⠿⣷⡄⠀⢸⣿⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡿⠶⠶⠿⠷⠶⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠶⠿⠿⠶⠶⠾⠿⠷⠶⠾⠿⠿⠶⠶⠶⠿⠷⠶⠿⠿⠶⠶⠶⠶⠾⠿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣀⣤⣤⠤⠴⢤⣀⣀⣀⣸⠦⢤⠤⠤⠐⠒⠒⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠠⣤⡤⠤⠤⠤⠒⠒⡞⢛⣻⣧⣈⡏⠛⠿⣿⡛⣯⡄⠤⠀⠀⠠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠐⠀⠀⠀⠀⠒⠒⢻⠓⢃⡻⠿⢄⡀⠀⠈⠒⠾⣿⠏⠉⠉⠉⠫⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⡀⠀⠂⠀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣶⡎⠀⣀⢐⣨⠀⡀⢧⡑⠰⠂⢜⡿⣣⠴⣦⡀⠀⠗⠒⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢰⢻⣶⠷⡸⠿⡏⣀⠀⠉⣃⢠⠛⠻⡙⠺⣿⣃⠛⢤⠞⠁⣎⣴⣴⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⣤⣤⣢⣥⣀⣦⠤⢋⣠⣀⣴⡈⣰⣾⣿⣿⣷⣤⣤⣬⣭⣀⣶⣭⣷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠉⠉⣉⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠠⣤⣶⣶⢶⣶⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡶⠿⢿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠤⠽⠿⠿⠿⠿⠿⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡞⣶⢿⣦⠤⠤⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⣿⢸⣷⠦⠀⠀⠈⠻⣤⣤⡤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⣿⢸⣷⠦⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⡇⢿⠈⡷⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣴⢿⡘⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡾⢣⡾⣧⢹⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣴⢋⣴⠋⠀⠘⣧⡹⣎⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠈⠳⣬⡳⢮⣝⡓⠶⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣷⣶⣭⣝⣓⠶⢭⣛⠿⣿⣿⣶⣶⣦⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠉⠉⢁⣤⢞⣫⠿⣋⡵⠞⣫⡵⢞⣻⣭⣿⣶⣬⡛⢦⡙⢯⡻⢿⡿⣿⣦⡈⠉⠙⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⣰⠟⣵⠟⣡⠞⢋⣴⣿⡥⠶⢟⣛⣫⣭⡭⠿⠿⣦⡙⢦⡙⣦⡻⣎⢳⡝⣦⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢷⣼⢃⡾⢁⣾⡷⢞⣛⣭⠴⠶⣛⣋⣭⡤⠶⠖⢛⣛⣩⣷⡈⢷⡌⢷⡘⣇⢻⡸⣧⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣾⣯⣽⣶⣞⣻⣭⣵⣶⣖⣛⣛⣭⣭⣴⢾⣧⣈⣷⣸⣧⡿⠾⠛⠉⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠉⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣬⣭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣥⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠶⠖⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠉⠉⣉⡿⠿⠶⠶⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣠⠾⢋⣭⠤⠶⠿⠯⣿⣟⣫⠶⢛⣩⣭⣿⣛⣉⣉⣻⣶⣤⣄⡀
⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⣋⡭⠭⠭⠽⢿⣿⣶⣋⣁⣴⠏⣤⠖⣛⣛⣛⡲⠦⠤⢶⡿⠟⠛⠛⠻⠶⠿⠷⠟⠁⠀⠙⢷⡀
⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠲⠦⠴⣻⠟⢉⣤⠴⣶⣶⣶⣶⣦⡤⣤⣬⠴⢛⣵⠛⠉⠁⠈⠉⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠳⣦⣄⣠⡞⣥⠞⣋⣉⣉⣙⣛⡛⣛⣁⡴⠋⣰⠞⠉⠉⠀⠈⠉⠛⠒⠲⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⣷⣿⣭⡿⠶⠿⠿⢿⣭⣭⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠛⠻⠿⠿⢷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣴⣶⣿⠿⠿⠛⠉⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⣧⡀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠉⠛⠛⠲⠶⠦⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠶⠛⠋⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⣀⡀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠻⣦⡇⢠⡏⢳⡀⣴⢳⡄⢰⣦⡄⢀⡄⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⠀⣶⠀⣆⢸⡏⢙⡆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⢻⡇⢸⡇⣼⠀⡇⢸⡇⣸⠇⣿⢸⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⣯⢠⡟⠋⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠈⠁⠈⠛⠁⠀⢷⠞⠀⢿⡼⠃⢸⡇⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⠀⠻⣤⠏⠈⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠛⠶⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣠⣤⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⠀⠀⠈⠉⠉⠛⠛⠛⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠛⠛⠉⠉⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⣭⡳⣄⠀⠀⠀⢀⣤⠴⠶⢦⣄⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⢀⡤⠤⢦⣄⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⢦⣝⢮⡓⠦⠴⢋⣠⠶⠒⠶⣝⢦⣀⣀⣠⣾⣫⠤⢦⣍⠳⣄⣀⣀⣴⡿⠞⠛⠲⣌⡛⠶⠶⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⠓⢬⡳⣍⣓⢚⣋⡴⠖⣛⡳⢮⡳⠯⣭⠵⣋⡤⠶⠶⢮⣝⠺⠿⠿⢋⡴⠞⣛⣛⠲⣭⣙⣋⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡙⢦⣙⠶⢭⣭⠵⢞⣋⣭⣍⡓⢯⣓⠒⠛⣩⡴⠞⣛⡓⢮⣝⡓⠛⢋⡴⠛⣉⣭⣽⣷⣶⠦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⢦⣉⠛⠒⠒⢛⣩⡴⠶⠦⣍⠳⢯⣝⣋⣥⠶⣛⣉⡙⠳⢮⣙⣛⣋⣠⠞⣡⠶⠒⠶⣭⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠉⠛⠓⠚⠋⠀⠀⠀⠀⠈⠳⣤⣀⣠⡴⠞⠉⠀⠙⠳⣄⣈⣉⣉⡵⠞⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠉⠁



⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣥⣤⣧⣤⣧⣤⣷⣥⣶⣭⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣻⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣲⣾⠿⢿⣛⣿⣿⣯⣿⣯⡭⡭⣍⣽⡿⣿⣯⣽⢿⣿⡿⣟⠿⠿⣷⣶⣦⣤⣀
⠀⠀⢀⡤⢖⣯⣷⣛⣩⣥⣶⣿⢿⣿⣿⣿⣯⣿⣿⣹⣾⡿⣗⣶⢖⢾⣻⣷⣛⣿⣵⡵⣶⣶⣮⣭⡛⠾⣽⡲⢄⡀
⠀⡴⣫⣾⢛⣽⣿⣿⣵⣿⡯⣺⣿⣿⣿⣿⡿⣿⣷⢿⡟⡟⣿⢿⡞⣟⣽⣵⢞⣯⣽⣿⣺⣿⣿⣿⣽⢢⣌⠛⣷⣝⢦
⡼⣱⣿⡱⡿⢿⡶⣿⣿⣿⣟⣽⣿⣿⣿⣯⣿⣿⣿⡜⣺⣿⣿⡞⣿⣻⡿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⢽⣣⣈⢿⣎
⡇⢿⣇⠓⠻⣿⣿⣶⣽⣿⢿⣿⣿⣿⣮⣛⣿⣿⣆⣿⣿⣟⣿⣞⣯⣯⢻⡜⣿⡿⢿⣿⣟⣿⡿⣿⣿⣯⣾⣿⣿⣸⡿
⠹⡜⢿⣄⠀⠻⠿⣿⢿⠿⣯⣿⡿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢗⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⣫⡿⢣
⠀⠈⠢⣝⠷⣤⣀⠻⣿⣭⣷⣾⣿⣿⣾⣿⣿⣿⣿⣟⢭⣝⣿⣿⣿⣿⣽⣿⣷⣿⣿⣿⣻⣯⣿⡿⠟⣋⣤⠾⣫⠔⠁
⠀⠀⠀⠈⠙⠲⢽⣛⠶⣬⣍⣉⡉⠛⢿⣷⣾⣽⣿⣿⣛⣥⣿⢿⣿⣿⣿⣿⣽⣿⠿⢛⣛⣡⣤⠶⣛⡯⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⠭⣛⣓⣺⡿⠿⠶⠦⠤⠤⠤⠭⠥⠶⠶⢶⣖⣛⣛⠭⠭⠓⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢸⣃⠠⠤⡀⣬⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠐⠼⠃⡀⠊⢰⣷⡌⡰⠀⢀⡀
⢀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠟⠉⠁⠁⢰⢈⠆⡴⠓⡆⢜⠩⠀⠁⠀⣆⠠⡀
⠀⠀⠀⡈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠐⠁⠄⠎⠀⠚⢣⣨⠀⠄⠀⢰⣀⡰⢄
⠶⠞⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣀⠀⣀⣦⠀⠃⠃⠞⡀⠄⠺⢃⠔⢤⡓⠄⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡮⢊⠀⠔⠂⠄⢀⡌⡀⠀⠦⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⢤⡞⠁⢀⠀⠁⠀⠁⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠇⠸⠀⡴⠁⠀⠀⠀⣴⣦⢚⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⠋⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢁⠃⠀⠀⠀⠂⠳⠀⠰⠃⢈⣆⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡆⠘⢯⠀⡁⠀⠀⠀⡇⠠⢀⠀⢠⠳⡷⣄⠈⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠉⠁⠠⡇⢀⠆⡼⠀⠀⠀⢈⠆⠔⠉⠂⠈⡟⣀⠀⠆⠰⠈⡛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⠐⠚⢩⠂⠀⠐⢱⠛⠚⡁⠊⡐⠁⠀⡂⠰⡇⠂⢠⢄⠤⢯⡠⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡸⠃⠀⢀⠼⡄⡄⠀⡃⠀⠁⠀⡖⠡⢦⠘⠀⠀⠏⠎⡇⠉⠐⠄⣘⡁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⡀⠀⣁⠆⠐⠉⠀⠍⢀⡐⢧⠰⡀⠀⠀⠘⠺⠀⠀⡜⠠⡄⠈⢶⡄⠸⠑⠐⠡⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⢰⣏⡼⣨⢂⠀⠀⠀⣱⡈⡇⠡⠀⠀⡠⠘⡐⠞⡁⢠⠹⡀⢰⢥⠘⢠⠐⠀⡋⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣾⠤⠊⠐⡉⠄⠀⠀⠈⠽⡇⡃⠅⠀⠈⠀⣆⠉⡞⠀⣠⣜⣀⣎⡤⠗⡄⢚⢸⠄⠈⠂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⡄⢠⣡⡤⣤⠂⠀⢢⠀⠀⠀⡴⠡⠁⠐⠠⠠⢀⡿⠃⣷⢀⣐⡀⠀⡀⡠⠊⠀⠲⣄⣯⡰⠜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠘⢄⢲⠰⡇⠀⢀⢰⠆⠐⠀⠤⠀⢀⡋⠉⢁⡈⠀⠠⠟⢧⡞⠁⠁⣡⣛⠃⠀⠄⠨⣟⠏⣆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⢈⠓⡹⠀⠚⠦⣀⡴⡲⠶⡒⢇⠚⠛⠂⢶⣈⠣⠀⠀⣀⡴⠊⠁⠩⠀⠀⠀⠓⢤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡳⢬⣀⡠⠶⢦⡈⠀⢀⣘⠃⠄⣀⡤⢈⠀⠈⠁⠀⠀⠀⠇⠀⠀⢈⣃⣠⣴⠾⢛⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠉⠛⠿⠶⣶⣤⣌⣁⣀⡀⡀⣇⢀⢀⢁⣐⣦⣤⣌⣤⣴⡶⠿⠛⠋⢑⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⢐⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠐⢂⠨⠀⠈⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⡑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠴⠊⠁
⣿⢿⢿⠿⣿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣟⣟⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠛⣉⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣉⣡⡴⠞⠛⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣛⣩⣴⣾⣋⣭⣤⠤⠴⠶⠶⠖⠛⠛⡛⠉⠉⣉⣉⣀⣀⣠⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣯⣽⣽⣧⡧⠶⠶⠶⠶⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⣄⣳⣜⣦⣑⢤⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠲⢬⡙⡍⢎⢳⢈⡆⠀⠀⠁⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡠⠤⠤⣄⠀⠀⠀⢀⣠⢇⣷⢇⠟⡼⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣯⣖⣿⣯⣲⣍⣉⣿⣉⣼⣁⣏⣼⣸⣟⣭⣺⣿⣷⣬⣱⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⣬⣭⣭⣥⣭⣥⣭⣬⣭⣭⣭⣭⣭⣭⣬⣥⣭⣭⣭⣬⣭⣬⣥⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢘⡀⠇⠈⡀⡇⠸⠀⡇⠈⡀⠇⢸⠀⡇⠀⡇⡇⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣇⣰⣀⣇⣰⣀⣆⣃⣀⣇⣰⣠⣄⣀⣀⣇⣧⣠⣇⣦⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⡴⠚⠉⠉⠉⠙⠒⠒⠤⠤⣀⣀⣠⠤⠖⠋⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⡤⡄⢆⠀⢰⣟⡃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠲⣼⣰⡁⢸⣾⠁⢹⡀⣇⡿⢸⣤⡄⠷⠔⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠊⠀⠈⠁⠙⠋⠈⠓⠊⠀⠈⠀⠀⠀⠀⢀⡤⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠴⠚⠋⠉⠉⠉⠙⠓⠒⠶⠦⠤⠴⠖⠚⠉⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⡤⠴⠞⠁

⠀⠀⠀⠀⢀⠄⠒⢀⢀⠔⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣆⠀⠀⠀⠀⢀⠖⢀⣀⣀⠒⢰⣤
⠀⠀⡂⠁⠀⠄⣀⡤⡏⡷⣶⣤⣄⣠⣶⣿⠑⠀⠀⠀⢸⢇⠀⠀⠤⠴⡲⢍⣅⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⠏
⠀⠄⠻⣿⠖⣛⠱⣀⣍⡀⠎⠸⠫⠽⠟⢉⠔⠀⠀⣴⠃⡼⠀⠀⠀⠸⠒⠛⢄⢂⠅⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠇⠆
⠀⠀⠀⠈⢉⣺⡉⠁⢐⠎⠀⢒⡁⠄⠒⠁⣀⢔⠟⠁⠊⠂⠀⠀⠀⠀⠢⠠⠊⡀⠀⠀⠂⠀⡆⠀⢀⡠⠀⠀⢘⢀⢠
⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠉⠀⠀⢀⡔⠞⠊⣠⠖⠋⠀⠀⠀⠀⠀⠁⠈⠀⠀⠠⠀⠀⡄⠀⢇⢀⡀⠀⣀⣀⣈⡤⠚
⠀⣄⠅⠀⠂⠀⠨⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣰⠣⠀⠄⠀⠀⢀⠌⠎⡩⢥⡭⠆⠀⠀⠀⢀⣀⣀⡀
⠀⡗⠒⠐⠀⠀⠀⢹⠀⢀⠀⠀⠠⠤⠀⢀⠀⠀⠀⢚⠂⠀⠀⠀⠀⠅⠀⢈⣀⣀⢀⡌⠤⠦⠀⠐⠳⣢⠛⢛⣿⣶⣾
⠀⡿⡦⢀⣀⣀⠠⠺⡄⠀⢱⠔⣒⡩⣿⡭⡥⠀⠀⠡⠀⠀⠀⢀⣀⠂⠀⠘⠀⠀⠀⠐⠃⠀⠀⠈⠀⠻⠶⠟⠙⠋⠁
⠀⡇⡇⠀⠀⠀⢄⠀⡇⠀⠈⠎⢟⡿⣿⢯⡇⢡⠀⠀⠑⠠⠄⠘⠃⠀⠀⠀⠀⣐⠂⠀⠀⠀⠠⢀⡀⠀⠀⠀⢀⣴⠎
⠀⠙⢀⠀⠀⠀⠈⢢⢰⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀⠄⠀⠀⠀⡔⡀⠀⢠⡄⣄⠁⠂⠀⡬⠄⡠⠬⠟⡀
⠀⠀⢨⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⢤⣐⣂⣬⠴⠒⡺⠉⠉⢂⠀⢀⡊⠐⠸⠁⠀⠐⠀⠘⠁⠀⠇⢩⠴⠛⠃⠁
⠈⠒⠦⡄⠀⠀⠀⢀⡀⠇⠀⠀⠀⠀⠀⢈⢡⡎⠁⠀⢁⠀⠀⢨⠈⡀⠪⠌⠁⢒⡂⠀⠄⠌⠀⠀⠀⠀⠠⢒⡨⠸⡀
⠀⢀⣀⣀⡀⠄⠀⢀⠀⢀⠔⠀⠀⠠⡀⠀⠈⢷⡀⡐⠁⠀⠀⠀⡄⠡⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠘⠒⠒⠀⠈⠁
⢈⠔⣉⣡⢞⡋⠥⠐⠀⢈⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⡠⠊⢺⠀⢀⠀⢠⡴⠢⠄⠤⡆⠀⣀⣠⣒⠒⢲⣄⣀⣀
⠈⠂⠉⠀⠀⢀⠄⢲⡄⠈⠂⠤⠤⠔⠃⠀⠈⠀⠷⠄⠄⠀⠒⠃⠄⠂⠈⠀⠈⠻⠄⡠⠤⠗⠛⠒⠚⠒⠲⠾⢶⢦⠢
⠀⠀⢀⣴⠆⡖⠀⠀⣰⢒⣶⣤⣴⣖⢄⠀⠀⠀⠀⠁⠀⡠⡄⠤⡤⡀⠀⠀⠀⠠⠀⢀⡓⠀⠠⠌⠀⠀⠠⢧⠔
⢀⠰⣿⢇⣴⣿⣫⣿⡗⠐⡟⠿⣽⢿⢆⠡⡀⠀⡀⠮⠉⢉⠍⠒⠆⠉⠒⣄⠀⢀⠸⣿⣂⠠⢀⢠⠄⠀⠀⠈⠀⠀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡆⠀⠀⠘⢌⡁⡖⢲⠘⡀⡶⠢⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠚⣉⡖⢰⢸⢊⠇⠀⠀⢠⠐⠉⠁⠀⠘⠁⠁⠂⠀⢠⠰⠒⠑⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢐⠋⠈⢄⠳⣘⠾⠾⠞⠂⣩⡠⣌⠰⢤⠠⠔⢢⡤⢐⣃⠠⢊⡁⣀⣀⠨⠍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⢇⠠⠌⣠⡆⣑⠋⠽⠙⣃⡐⠠⠅⡈⠘⠐⠄⣤⠤⠂⠘⡒⠢⠀⠃⠘⠅⠀⠡⡘⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⢆⠉⠛⠋⠭⠔⣒⠲⠒⠎⠐⠖⠖⢠⠳⠄⡰⠁⣎⠀⣀⡴⠁⠪⠄⠴⠦⢂⡠⠄⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⠦⠈⣀⠒⠂⡰⢒⣬⠐⢆⠣⢢⡘⠿⠷⡈⠀⢀⠀⢈⠡⠖⠁⣨⠏⠀⢲⠲⢐⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡋⠰⠂⠀⠥⠤⢒⠃⠞⣉⠄⠊⠑⠤⠭⠭⢥⣈⠥⡴⢞⣩⠔⢂⡋⠆⠰⠃⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠹⢄⠀⠲⠜⠏⡀⠸⠡⡆⡲⢘⠃⡡⠖⡹⣮⣜⣊⠁⠖⣂⣀⡒⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠑⠤⠍⠂⢀⣀⣠⣔⣣⡑⠤⠭⠤⠶⢋⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⠁⣸⡇⠀⠀⠀⠀⠀⢀⣴⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠓⢛⣿⢋⣿⢋⢾⣽⢃⣽⢋⣽⢋⣽⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠘⠿⠛⠿⠛⠧⠿⠋⠻⠚⠿⠊⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠤