Sorcery Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠘⠁⠀⠀⠀⠀⠀⢦⠀⠇⢚⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢰⡀
⠀⠀⠀⠀⠈⠻⠆⠰⣤⣠⢸⣼⣶⠀⠀⠀⠀⠘⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⡀⠃⡴⠀⠐⠗⠀⠀⠀⡶⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣉⢣⡎⠀⠀⠀⠀⠀⣠⣌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠀⠀⠀⠀⠉⠀⠾⠀⢀
⠀⠀⠀⠀⠀⢠⡀⠀⠈⠙⠄⠀⢠⡆⢠⣿⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⣱⠟⠀⠘⠋
⠀⠀⠀⠀⠀⠘⣿⡆⠘⠃⠀⢀⠉⠑⠈⠡⣀⠀⠆⠒⠀⡀⠀⢰⣌⡥⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢸⡆⢀⡆⠀⠀⠮⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⡄⢦⢀⠆⠘⠋⢰⡶⣷⡀⣧⠔⢀⠏⠀⠀⠀⠐⡆⠐⠀⣠⠀⠀⠀⠶⠀⠄⡀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠈⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠇⠀⡜⡈⢧⡀⠐⠚⡎⣿⡷⣾⠊⠀⣀⠀⢀⠘⠁⠀⠘⠀⠘⠀⠀⠀⢀⣠⣤⡄⠐⠃⠀⠀⠀⢄⢀⡔⢠⡆⠀⠀⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⡄⡆⠁⣤⡄⠻⡆⠈⣄⢀⢠⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢂⡀⠀⡻⡗⠀⠀⠀⠠⠿⢹⠸⢠⡿⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠑⣄⠀⢿⡿⠀⠀⠀⠹⠎⢸⠻⠒⠒⠂⢀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠃⠋⠀⣀⡁⡾⠃⠀⣠⣄⠈⡠⢈⣤⡤⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⢧⡈⣄⢸⠆⠀⠀⣈⠛⢄⠀⣶⡄⠀⠀⢯⠀⠀⠀⠀⠀⠀⢢⣶⢠⣾⡞⢻⠟⠀⠀⠀⢀⣿⠏⠀⡥⢋⡡⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠄⡀⠻⡀⠐⠁⡀⠈⠢⣤⠁⠸⢇⡀⠀⠀⠑⢀⣀⠠⠄⠘⠶⢿⡟⣀⠻⣠⡶⣏⠀⠀⡀⠛⢠⠈⣀⡧⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢤⣞⠲⠀⢀⠀⠀⣀⠀⠀⢈⠰⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠁⠋⡱⠋⠀⢤⠁⢠⡟⠈⢁⠠⣘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠙⠃⠀⠁⠀⠈⠓⡀⠀⠁⢡⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠈⡨⠈⠁⠀⡈⠃⢀⠀⢸⣧⡀⢈⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠢⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡤⠐⠁⠀⠞⠁⢀⠂⠀⠘⠛⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠐⠀⠀⠶⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⣄⠈⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣣⡀⠉⠀⡀⠀⠀⠂⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢃⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⢦⣀⠀⠓⢤⠀⠤⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⣠⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠒⠊⠑⠤⢤⣥⡀⢀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⢁⡔⠓⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠤⠤⠤⠀⠀⠁⢃⡠⢞⡁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠛⠉⠁⠠⡤⠐⠉

⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠰⠓⠄⠀⠄⠀⠀⠀⠀⠀⢘⣍⡁⣧⠀⠀⠀⠠⡄⠀⠀⠰⠿⠿⠶⠶⠦⠴⠶⠒⠲⠶⠶⠶⠒⠒⠒⠂⠐
⠀⢂⡙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠁⠂⠐⠂⠁⠀⠠⠄⣇⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣾⣁⣯⣦⡄⢆⠀⠈⢰⣆⢨⡄
⠀⠀⠀⢠⡀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠛⠋⣠⠏⠀⠀⠘⠟⠸
⠀⠀⠀⠠⣿⠀⠀⢠⣄⡴⠀⠀⣤⣁⠉⠀⣀⠐⢆⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⣻⣿⡷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣽⠀⠀⠀⠋⠁⠀⠀⠈⠁⠀⠀⠈⠂⠈⠉⢀⠆⠸⡟⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠇⠀⢨⣇⣀⣀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⡀⠁⠄
⠀⠀⠀⣼⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠉⢷⣀⣠⣀⣠⣀⣈⣛⣉⡉⠉⠀⠀⠀⠀⠘⠏⠀⠀⣀⣠⢤
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⢀⠃⠀⠀⢀⣴⣿⣿⣿⡭⣌⣏⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠾⠃⠀⠀⠈⠟⠙⡛⠛⣛⠬⠊⠉⢙⡻⠉⢉⡀⠀⠀⠀⠀⠀⠰⣗⡄⠄⠄⠐⢈⣹⠁
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠔⠋⢠⣯⣶⡂⠀⠘⠃⠀⠈⠁⠀⡀⠀⠀⠄⠀⠘⠇⠒⠀⠁⠈
⠀⢀⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢰⣦⣞⢭⣿⣿⡏⠀⣾⠀⠀⠀⢰⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠈⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣿⡏⠀⠀⠉⠀⠀⢹⡆⠀⡃⠋
⠀⢀⣾⣿⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠇⠾⠀⠖⠀⢰⡀
⠀⢸⣧⣿⣿⡆⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠣⠂⠀⠱⠚⠀⠘⠃
⠀⢸⣿⣾⣿⠇⠀⠀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠓⠀⠃⠂⠓⠀⠉⠉⠠⠶⠀⠀⠰⣶⣟⣛⢓⣙⣋⣓⣶⠆
⠀⢸⣿⣿⣿⠀⠀⣼⡟⠁⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠘⠃⠀⠀⣼⣿⡿⠛⣿⣿⣿⣿⣿⣧
⠀⠀⣿⣿⡏⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⣧⣷⣶⡾⠀⠀⢟⡆⠀⠀⣰⢫⡽⠀⠀⢸⣿⣿⠀⠠⠘⣹⣿⣿⣿⣿⡇
⠀⠀⠘⠿⠁⠀⠀⠀⢠⣿⢿⡿⠃⢹⣿⣿⡿⠛⠁⠀⠀⠈⢋⡀⣄⠁⠛⠁⠀⠀⠘⣿⣿⣄⠀⢣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⣄⢣⡀⠘⡇⢹⡀⠀⠀⠀⠀⢰⣄⠘⣿⠀⠀⠀⠀⠀⠀⠀⢛⡋⡔⢀⠀⢨⣜⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⢽⡯⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠘⠀⡀⠀⠀⠈⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠔⣁⠀⡠⠊⠀⠀⠈⠀⠀⠇⡀⡱⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡅⠀⡠⠋⠁⣒⠤⠶⠒⠀⠀⠈⡄⠠⠁⢀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠀⡠⠊⠀⠀⠀⠀⢀⡔⠀⠀⢀⢀⡄⣀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣆⠈⠀⠀⠀⢠⡄⢀⠨⠃⠀⡠⠠⡈⠀⣁⠀⢀⠐⠸⡤⢠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠘⢷⡟⣿⢀⡤⠈⠒⠀⠰⠋⠁⠀⠀⠆⠀⠇⠀⢃⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡍⠉⣩⡻⡵⢀⠈⠁⠀⠀⢀⡀⠀⠀⠘⠀⠘⡤⠚
⠀⠀⠀⠀⠀⠓⢀⠘⠤⢀⡁⠀⠐⠁⢀⠄⠉⠀⠀⢏⣱⣄⠀⠀⠀⠁⠀⠀⢀⡠⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠪⠔⠀⢀⡤⠀⢅⠈⡔⠀⠀⠀⠙⠮⣢⡀⠀⠀⠀⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠑⢊⠀⢠⠟⠁⠒⠀⠀⠀⠀⠘⠍⠆
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠒⠠⠤⠀⠀⢀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠧⠤⠀⣀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⡀⡐⣾⢖⡀⡀⠀⠠⢼⣀⣀⡠⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠂⠀⠀⠀⠀⠀⠳⠋⠈

⠐⠂
⠈⠀⢀⣄⣀
⢸⣏⣑⡀⠌⢳⡀
⢨⢹⣿⣿⣿⣾⣟⣤⡤⡄⢀
⠈⠳⢹⣿⡿⠿⣿⣿⣉⡀⢼⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⠟⡟⠳⠚⠉⠟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠰⠁⠀⠘⣆⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢨⣗⠀⡀⠠⠈
⢀⠀⠀⠀⠈⢧⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠊⠀⣀⡹⠚⠁⠀⠀⠀⢀⡀
⠘⠀⠀⠀⠀⠈⢣⡀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣀⢶⣀⣾⣿⣦⣆⣀⣀⣹⠃⣀⠀⠀⠀⠀⠈⡁
⢀⠀⠀⠀⠀⠀⠀⣷⣾⣇⡀⠀⠀⣀⡛⠄⠊⠀⠀⠀⢭⣾⠿⢿⣭⣹⣿⣽⣿⣙⡛⠛⡷⠀⠀⠀⢰⡇
⢨⣄⣠⣀⣤⣤⣤⣤⣤⡾⣵⣶⣮⡁⠀⠀⠀⠀⠀⠀⢦⢻⣾⣿⣿⠿⢷⣲⠿⠿⣿⣿⡄⠠⠀⠀⣸⠃
⢸⣟⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣟⣷⠀⠀⢺⠀⠀⠲⢊⣾⣿⣿⡟⢰⠋⣽⣿⠀⢻⣿⣷⠀⣶⠟⠁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⡋⣁⣀⣀⣾⣰⡀⠀⣾⣿⣿⣿⡇⠘⠿⠿⠛⠀⢸⣿⣿⣇⢰⣦⣤⣤⣴
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡎⣿⣿⣿⡿⣛⣿⣟⢰⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢸⡿⣿⣿⠀⣿⣿⣿⣿
⢸⡿⣽⣿⣿⣿⣿⣿⢋⣾⣧⢿⢹⡋⣯⣞⣿⣿⡇⠸⠿⢿⣿⣿⡇⠀⠰⣆⠀⠀⢸⡿⠋⡀⢨⣿⣿⡇⢸
⢸⣿⣿⣭⢻⡿⣿⡋⣿⠟⣿⠘⡾⠇⣿⣿⣿⣿⣿⢈⣶⣦⡈⠛⡇⠀⢀⡟⠀⠀⠀⢀⣼⣡⡾⣿⣿⣷⠘
⢸⡿⢻⣿⣿⣿⣿⡿⣶⡛⢙⡇⢷⣤⣿⣿⣿⣿⣿⣿⠿⠷⠛⠂⠢⠀⡉⣿⡦⠀⠈⠘⠫⠻⠀⣿⣿⣿⣦
⢸⡶⠟⢛⣿⣇⣿⣿⠉⢳⢾⣯⣼⣿⣿⣿⠾⣿⣿⣿⣶⣾⢷⣦⢀⣨⣤⣿⣶⣷⣶⠂⠲⣿⠀⣹⣿⡿⣾
⢈⠱⡿⢙⣛⣿⣿⣟⡾⢟⢻⣥⣿⣿⣥⡌⠀⣿⣿⡿⠻⣿⣢⡟⣿⣿⣭⡿⠹⣾⣩⣶⣿⠏⠐⠛⠛⢳⣿
⢸⡿⣿⣿⣿⣿⣿⡻⡿⠈⣍⣳⣿⡀⡟⠀⡄⢿⣌⡁⣈⣫⣭⣸⡿⣿⣯⣀⢾⣬⣹⣿⣅⣴⣶⢾⡇⣻⣿
⢸⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣥⢹⣨⡟⢿⣿⣤⣟⠛⠹⣯⣿⡌⠻⠛⢻⡟⣙⣿⣿⣷⠄⡼⣝⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠘⣏⠿⣾⣿⣍⢿⣧⣾⣬⣿⣿⣿⠀⣶⡌⣿⡛⠛⢿⣦⢡⣿⣿⣿
⠨⡄⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡄⢻⠿⣓⣭⣧⢀⡿⣍⡐⢫⡅⢻⣷⣾⣿⢿⠀⢀⣿⡏⢹⣿⣿⣿
⢸⡿⠞⣾⣿⣉⣼⡿⠉⠁⢈⣻⣿⣛⣿⠸⣿⣿⣿⣿⣿⣶⣬⡛⠀⠑⡆⢹⣿⣯⢯⣶⢿⡿⣜⣿⠙⣿⣿
⠠⢱⣾⣿⣿⣿⠻⣇⣤⢿⣿⠟⢳⣿⣿⡇⢿⣿⢻⣿⣿⠇⠙⣿⡈⡏⢰⣨⣮⢱⣶⣿⢿⣧⡿⣿⣤⠉⣿
⢘⣸⣿⣿⣿⡿⢿⣿⣤⣀⢘⡰⢿⣿⣛⣯⠸⣿⣛⣽⣿⠁⣲⣿⣷⣻⡦⢤⣴⣾⡿⣁⣌⣻⠃⣿⣗⠘⣻
⢸⣿⣿⡟⢋⠐⢈⢹⣿⣿⣿⣿⣾⠿⣿⣿⣤⣿⣿⣿⡏⠸⣿⣿⣿⣿⣷⣿⣿⣿⣗⢠⣿⣿⣿⣿⣿⡄⠻
⠀⣸⣿⡾⣦⣴⣯⣿⣿⡟⢽⣯⡟⣰⡿⣿⠋⠙⢿⣿⣙⡛⠿⣿⢿⣿⣍⣻⣿⡿⢿⢿⣿⣿⣿⣿⣾⣝⣢
⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣤⣌⡛⢿⠋⠀⣴⣿⣿⣿⣿⣿⡁⢀⡝⢿⣿⣷⣦⡌⢉⣽⣿⣿⣿⣿⣿⣽
⢸⣿⣿⣿⣿⣩⡶⢟⣛⣿⣿⣿⠿⠟⠛⢤⣶⣟⣿⣿⣿⠟⢿⡛⢫⣄⣶⣿⣿⣿⠇⣸⣼⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡿⠛⣿⣿⣷⣿⠉⠀⠀⡀⢠⣾⣿⡿⢸⣿⡏⠐⠀⢀⠶⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡇⠰⠿⠛⣩⣿⣄⣠⣷⣼⡿⠋⣸⡼⣧⠏⡀⣶⣶⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣧⣴⣦⣶⣿⣿⣿⣿⣿⣏⣠⣾⣿⣷⣦⣶⣀⣿⢿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠒⠂⠓⠒⠠⣇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡚⣅⣦⣴⣶⣶⣶⣶⣴⣥⣉⠾⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣼⣷⠿⠟⢉⠋⣦⣤⣤⡀⠉⠙⠿⢷⣽⣧⢂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡤⠀⠀⠀⠀⠀⠀⣸⠋⠁⠀⠀⠠⠄⣿⣿⣿⠇⠀⠀⠀⠀⠈⠛⣗⠨⠀⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡞⠓⡀⠀⠀⡀⠔⠂⠈⠑⠠⣀⠐⠄⠑⠤⠏⠀⠀⠀⡠⠀⡀⠔⠈⠀⠐⠢⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⠈⠀⠀⠀⡄⠀⠀⠀⠙⠘⠷⠦⠠⠤⠴⠺⡂⠉⠀⠀⠀⠀⠀⠔⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⡀⠀⠰⣱⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⠊⠀⠀⠀⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢆⠀⠃⠀⠀⢀⠠⠇⠒⠀⠈⣷⡆⠐⠠⠆⡀⠀⠠⠁⠀⠀⠀⣳⣛⣞⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠠⣾⡀⠀⠀⠀⠀⣰⡟⠀⠀⣠⡴⠟⠐⠣⡀⠀⠀⠀⠁⠿⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠁⠢⣀⠀⠀⠀⠀⠀⣰⠉⠀⠈⠻⣦⡀⠀⠰⣿⠀⠀⢰⡏⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣀
⠀⠈⠉⠉⠉⠉⠣⢉⠉⡍⠉⠉⠉⠉⠉⠉⠋⢍⡉⠉⢹⡇⠀⠀⠀⠘⣿⡄⠀⣿⠀⠀⣼⠇⠀⢀⣴⠏⠀⠀⠩⠉⠉⠉⠉⠉⠉⠉⠍⠋⢋⠉⡁⠙⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⡀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠈⠻⢦⣤⣄⡀⠸⢧⣤⣿⣤⣴⡏⣠⡴⠟⠁⠀⠀⠀⠀⢂⡀⠀⠒⠀⠀⠀⠀⢀⠴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠑⠄⡀⠈⢣⠋⠀⡇⠀⠀⠀⠀⠀⠈⠙⠳⡾⠋⠀⠀⠙⢿⡏⠀⠀⠀⠀⠀⠀⠰⢸⠀⠀⠀⠀⢀⠤⠊⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠒⠀⠈⠒⢌⠀⠀⣦⣄⣤⣴⠶⠿⠷⠶⠶⣧⠀⠀⠀⠀⣼⣷⣤⣄⣀⠀⠀⠀⣠⣤⠀⢀⡠⠂⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠜⢄⡃⠀⠀⠀⠀⠀⠀⠀⠉⠂⣅⠉⠉⠀⠀⠀⣀⣤⣤⠾⢷⣦⣴⡾⠻⣦⣄⠀⠉⠛⢿⣟⡉⣨⠔⠉⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢢⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡾⠋⠁⠀⢀⡾⠁⠘⣷⠀⠀⠹⣷⡀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠰⢌⣶⠏⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆⠀⠀⠘⢷⣄⡠⡢⠄⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠇⠘⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣀⡤⠯⠆⣉⢠⡄⠀⠀⡀⠀⠉⢑⠄⠀⠀⢰⡟⠀⠀⠀⢸⡇⠀⠀⢀⠠⡙⠛⠢⢤⠀⠀⠀⠇⠀⠀⡀⠀⠀⡘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠔⠊⡓⠁⠀⠀⠀⠈⠲⠀⠀⠰⠀⠀⠀⠀⠱⢦⣴⣏⠀⠀⠀⠀⠈⢷⣀⣠⣕⠈⠀⠀⠀⠀⠆⠀⠀⠀⢀⠊⠀⢠⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠲⢼⠀⠀⠀⠀⠀⠀⠀⡇⢶⠃⠀⠀⠀⠐⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠘⡀⠀⠈⢃⠀⠀⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⡀⠀⠀⠀⢀⠴⠀⠎⠀⠀⠀⠀⠠⠑⠉⠻⠿⢿⣿⣿⣿⠿⡿⠿⠛⠍⠊⠄⠀⡀⠀⠀⠡⠀⠀⠈⢄⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠑⣖⠚⢌⠙⢓⠀⠀⠀⠀⢀⡠⠒⠄⣀⠀⠀⠀⢀⣀⣀⣀⠤⡔⠪⠓⢄⠀⠐⢄⠀⠀⠂⠀⡠⠊⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠦⡀⠠⡁⠔⠁⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠉⡁⠀⠀⠀⠀⠉⠢⢌⠢⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠜⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠄⠛⠡⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⢀⡸⡀⠈⠑⠂⠤⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠄⠐⢈⣹⡄⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠀⠀⠐⠈⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣦⡀⣀⣀⡀
⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣛⢿⣿⡻⣿⣮⣻⡛⠿⠟⠃
⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⣟⣿⣭⠝⡻⣿⣷⡶⡆⠈⠻⢿⣿⣷⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣆
⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠈⠉⠅
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠉⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄⠀⠀⣮⣿⣷⣿⡛⢽⠿⣟⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣻⣿⣿⣦⡲⢶⣝⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡸⠂⠂⠀⢺⡿⠂⠀⠀⡀⠄⢒⠞⠳⠒⠤⢀⠀⠀⠀⢿⣿⠀⠒⠐⢿⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢈⠠⠐⠈⠁⠀⢀⠎⠀⠀⠱⡀⠀⢈⣗⠂⠤⡉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡱⠀⠖⢨⡀⡀⠀⠀⠀⢠⠂⢀⣄⠀⠀⠑⡄⠘⡞⠀⢀⢀⡄⠲⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢃⡉⠀⠀⠀⠀⢓⠐⢤⠅⢠⣼⣿⣦⠠⠄⠨⡄⠃⠘⠈⠀⠀⠀⢉⡀⠀⠰⡇⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡠⠀⠀⠀⠀⠀⠀⠄⠂⠀⢸⡿⠙⣿⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⢸⡁⠀⢸⡇⡀⢹⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⡆⠀⠀⡈⠀⠘⢄⠀⠀⠀⠀⣈⣃⡀⠈⠱⠷⠯⠄⠀⢀⣀⣇⠀⠀⠀⠀⠀⢨⡄⠃
⠀⠀⠀⠀⠀⡤⠛⡄⢀⠁⠀⠀⠀⠱⡄⠠⠊⢁⣤⣤⣤⣶⣶⣶⣤⣤⣄⡠⠉⢢⠀⠀⠀⢤⡼⢧⡼⡆
⠀⠀⠀⠀⠀⠀⡆⠀⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⣆⡄⠀⠠⠾⢷⡞⠳⠴
⠀⠀⠀⠀⠀⠀⠁⡘⠀⠀⠉⠢⢄⡀⠀⠲⣤⣤⡀⢠⣶⣶⣶⣦⡀⠀⣾⣷⠀⠄⠁⠀⠀⠀⠘⠃⠀⠀⢃⣀⠠
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⢟⡤⢠⣿⣿⣇⣸⣿⣿⣿⣿⣿⣾⣿⣿⠀⢰⡀⠀⠀⠀⠀⡠⠄⠂⠉⡀
⠀⠀⡖⠒⠒⠒⠞⠒⠒⠒⡖⠀⠀⠟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢸⣤⠄⠀⠀⠀⢰⠒⠂⠒⠳⠒⠒⠒⢲
⠀⠀⠈⢄⠀⠐⠀⠀⠀⠞⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠨⠀⠀⠀⠀⠀⠣⠀⠀⠀⠂⠀⠠⠁
⠀⠀⠀⠈⢛⠛⠓⢓⠛⠓⠛⠒⠚⠓⠚⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠚⠓⠓⠚⣷⠛⠓⠛⡚⢛⡛⡛⠛⠒⠚
⠀⠀⠀⠀⠀⢳⡀⠂⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠳⠀⠀⣠⠛⣄⠀⠀⣰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⡀⠀⠈⣴⠉⠀⠰⢿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠆⠂⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠒⠄⡟⠀⠀⠠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⠀⠀⢀⠖⢁⡜⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⡿⠛⢡⡈⠒⡀⠀⠈⠐⢄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣇⡤⠀⣙⠠⣎⠀⢀⠃⠀⠀⡇⠀⠀⢸⠀⠀⠘⡀⠀⢱⠄⣋⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⡾⠦⠀⠈⠓⢤⣉⡘⠒⠠⠤⠤⠤⠤⠤⠤⠀⠐⢃⣉⡤⠚⠱⣄⢰⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠟⠁⠀⠀⠀⢸⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⠎⠉⠙⠻⠿⣿⣿⣿⣿⠿⠿⠋⠁⠡⡀⠀⠀⣤⠁⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢱⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠒⠀⢀⠠⠄⠙⠂⠄⣀⡠⠄⠀⠀⠠⢄⣀⠠⠀⠻⠠⢄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠋⠀⠉⠠⢀⢀⠀⠀⠀⠀⠣⡀⠁⠉⠁⡜⠀⠀⠀⠀⠃⡀⠄⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠈⠂⠆⣄⡀⠱⡀⠀⠜⢀⢠⠰⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣥⡤⠤⠤⣼⣷⠄⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⣴⣿⠤⠤⢤⣴⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣟⣙⣹⣿⣥⣉⣿⣏⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣽⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣀⠀⠀⣠⣷⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣸⣿⠟⢣⡴⣿⣷⣾⣿⣾⠛⢷⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣧⡤⢘⣟⢻⣍⣻⣽⠃⡞⠈⠀⣨⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠛⢻⣿⣿⣷⣾⣋⣾⢯⣿⣿⣿⣷⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⢀⡤⠴⠋⠀⠀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⡟⠀⠀⣼⣿⠿⢿⣇⣹⠻⠛⠻⠟⠫⠌⢻⣿⡟⠳⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠐⠀⠘⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣆⠀⠀⣼⣿⠲⣾⣿⠋⠉⠃⠰⣶⠀⣤⡸⢻⣆⠦⢼⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠏⣿⠄⠀⢼⣽⣿⣿⣿⣄⠀⠀⠀⣥⣴⠟⠀⠈⣿⣷⠈⣀⠤⠄⠒⠺⣷⡤⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡶⢦⣴⣿⣿⣿⣿⣶⢠⡀⠲⡏⠀⠀⢀⣨⡧⢻⡇⢀⣘⠗⣤⢦⣄⡛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣛⣁⠈⠡⠄⠘⠫⣿⣿⣿⣝⢧⣭⣿⡆⠀⣰⣶⣿⢗⡽⢛⣫⣴⣺⣡⡾⠛⠀⠀⠘⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣿⣧⡀⡀⠑⠢⠌⠙⠛⡿⠾⠿⣿⣷⢿⡛⠭⢂⣡⠴⢛⣹⣿⠋⠁⠀⠀⠀⠀⠀⢀⣠⠠⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣿⡿⠿⢿⣾⣶⣦⣶⣌⠀⠀⠀⠰⠞⣉⣴⠾⣿⣿⣿⣿⠿⠃⠀⠀⠀⡀⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⣾⣿⣿⠿⣧⡀⠀⠀⢠⣛⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢂⣾⠟⢉⣤⣔⣻⣇⣴⣦⣿⡿⠛⣇⣹⡄⢀⣠⢴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣯⣄⣿⣾⠿⢿⡟⠛⠋⠛⠉⠄⢛⣿⣡⣟⣛⣁⣾⢿⣿⣦⣀⠀⠤⠀⠴⠶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣽⣾⣿⡏⠶⢄⣘⣿⠗⠀⠈⠁⠀⢨⠸⠟⣻⠿⢿⣯⣼⣿⣿⣿⣷⠀⠟⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠃⠈⠛⣿⠋⠀⠊⢀⣸⣿⡇⠀⠀⠀⠀⣸⡄⠀⠁⠀⠾⣿⣿⣴⣿⣿⣿⡔⠈⠁⢲⣄⣴⡟⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡎⠀⠀⢸⣿⡻⢿⣶⣆⣴⣾⣿⣷⣄⠐⢀⣠⡿⣿⡿⢿⣿⠁⠀⠀⠀⠈⢻⣿⣴⣾⣿⡀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢸⣿⠁⠀⠀⢸⡿⠁⠀⠘⠛⠿⠟⠈⠙⣧⠀⠉⠉⠳⢶⡾⠋⠀⢴⠀⠀⠀⢀⣼⣿⠿⢿⡉⠙⠳⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠐⠃⣄⠀⠀⢿⠁⠀⠀⠁⠴⠖⠋⠁⣿⠀⠈⠛⠿⢷⣄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢶⠇⠀⣠⠂⣀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠛⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣇⣾⠏⢹⣿⡆⠀⠀⠀⠀⠀⠀⢠⣀⣆⣀⣠⣤⣶⣶⣶⠿⠟⠁⢠⣍⣶⣿⣿⡷⢀⡀⠀⠀⡀⠦⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⣺⣿⠏⠀⠀⢻⣿⡴⡶⣶⣀⠀⠀⠀⠛⠛⠋⠀⢰⣿⢿⡏⣾⣇⣤⣿⣿⣿⣿⠟⣢⣮⡁⠀⠀⣱⠖⣠⡮
⠀⠀⠀⠀⠀⠰⠄⡤⢀⠚⣿⠀⣿⡿⠀⠀⠒⠛⠉⢶⡄⠀⠀⠀⠀⠀⣷⡄⢄⠀⢸⡇⣜⢳⣿⣿⠿⢿⣛⣋⣵⠞⣻⣿⣷⠀⠀⠀⠈⠉
⠐⢀⠀⠀⠀⠀⠀⠀⣸⠶⠁⣾⣿⡇⢀⣄⡀⢀⡀⠛⠿⡶⠀⠀⠀⠀⠈⠗⠀⠀⠸⣿⣿⣷⣿⣙⠳⡟⢫⣵⣴⣾⣿⠿⣿⡆
⠀⠀⠀⠀⠤⠀⠀⠀⣾⠀⣸⣿⠏⠀⠚⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠹⣦⠄⠈⢳⡿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠿⡿⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⣿⡏⠀⠀⠀⠀⠀⠀⢤⣴⣶⣶⠶⠓⠀⠀⠀⣽⢓⠆⠀⠀⣿⣷⣻⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⣀⣠⡤⠤⣾⣿⡟⠀⠀⠀⣠⣶⣿⣿⣇⣀⠐⠘⠃⠘⠻⣿⡟⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠾⣿⣷⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⠿⣿⡿⢿⣍⡀⢉⣿⣦⠀⡄⠀⠀⠉⠀⠀⣉⠀⠰⠀⠀⣼⡇
⠀⠀⠀⠀⠂⠁⠀⠀⢠⠁⠀⢟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⢷⣦⣀⡿⣷⡄⠈⢿⠆⠁⠂⠘⠥⠀⠈⢫⣀⡀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⠩⢶⣿⣇⢠⡴⠆⣤⠶⠟⠀⠀⠀⠈⠿⠁⠀⠉⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠷⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣌⡻⡎⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⣁⣤⣾⣿⡿⠿⠿⢲⣦⣤⣤⣴⣶⣿⣿⣷⣶⣿⡿⣿⣿⣷⣦⣤⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⣿⣿⣿⣿⣿⠇⠿⠿⠿⠿⠋⠀⣼⡿⠀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⡌⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠿⣋⣤⠀⢷⠀⢴⣼⣿⣿⣿⠟⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⣠⠀⠐⠺⡿⠀⠐⠂⠀⠀⢀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠤⠀⠀⠀⢀⣀⣠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢷⡶⠤⣶⡖⠂⠀⠐⠐⠲⠾⣿⠿⠛⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠐⣻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣷⣕⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢍⣿⣿⣿⣿⣿⣶⣄⠀⢀⣝⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢿⣿⣿⣿⣿⡏⣕⢸⣷⣴⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⢹⣿⣿⣿⣿⡧⠦⢹⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣮⣧⣇⢰⣿⠬⢾⣿⣿⢻⣿⡇⠿⢼⣿⣿⡏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⢩⢹⣿⡓⣭⣿⡇⣟⢸⣿⣿⣿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢨⢸⡯⡹⢫⣽⡇⣭⢼⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠤⢼⡇⠀⠀⢿⡇⠲⢸⣿⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣹⣹⡇⠀⠀⠰⡇⠲⠞⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡧⢶⢾⡅⠀⠀⠀⢁⢰⡆⣿⣿⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢇⣶⣸⠁⠀⠀⠀⠘⢀⣀⣿⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣄⢘⠀⠀⠀⠀⠠⣩⣭⣹⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢍⠈⣿⠀⠀⠀⠀⠀⠤⣧⣼⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢶⡶⠋⠀⠀⠀⠀⠀⠰⣶⢸⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⠾⠿⢻⠀⠀⠀⠀⠀⠴⡶⢼⣿⡕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠃⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⢤⣀⡤⢶⠆⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠈⠀⢀⡞⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⢾⣅⣀⠀⠀⠀⠰⣽⣄⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠖⠀⠀⠀⠀⠈⣧⣔⡞⠛⠛⠛⠃⠀⠀⠀⡾⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⢀⣀⣴⠇⠀⠀⠓⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠐⢿⣉⠁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⠁⠀⠘⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⣇⣠⣠⠴⠧⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠀⠀⠀⠈⢉⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠒⢦⣼⣿⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡼⠁⢀⡴⠖⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠠⢾⣿⡷⠄⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢠⠞⢀⡴⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⡴⠋⢠⠞⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠛⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⠀⣠⢞⣡⠞⠛⠂⠀⣠⠏⠀⠀⣀⡤⠖⠚⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⣳⡀⠀⠀⠀⠀⢀⡴⣯⠀⠀⠀⠀⠙⠷⠋⠀⠀⠀⣠⠞⢁⡤⠖⠋⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠁⠹⣄⠀⢀⣠⠏⠀⠈⠳⣄⠀⠀⠀⠀⠰⣄⡴⣋⡵⠚⠉⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠻⡉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠋⠁⠀⠀⣀⡤⠖⠋⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣯⠵⠖⠛⠉⠉⠉⠉⠉⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⢀⠀⠀⣀⣠⡤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢳⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣸⣎⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣖⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⢠⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣦⣈⣉⠛⠲⢤⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠃
⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿

⢸⣿⣷⡄⠀⠠⠀⠀⣐⠁⠠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡀⠀⠿⣿⠇⠀⠀⢀
⠈⠻⠿⡇⠄⠀⠀⠀⢿⣿⣷⡶⠀⠀⠐⠤⡀⠀⠀⠘⠁⠀⠀⢀⠀⣀⠀⣤⠀⠀⠀⠠⣿⣷⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⠓⠀⡤⠒⠀⠁⠀⠀⠀⠐⠆⠀⠀⠈⠁⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠠⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠲⠀⠀⠀⠈⢄⢀⠀⡄⠘⠀⠁⠀⢀⠀⠀⣀⣴⣿⡀
⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⢰⡀⢏⡀⢀⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠄⠂⠀⠀⣐⠈⠃
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣆⣛⣩⠀⢠⡁⠀⢀⣀⣀⠀⠀⠀⣀⠂⠸⣿⠆⣐⠀⠺⠃⡀⠀⠀⠀⢠⠃
⠻⠟⠋⠒⠄⠀⠀⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠰⣌⣉⣠⠄⠀⠀⠋⢡⡅⠈⠦⠀⠘⠛⣤⡀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠄⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠠⣌⣉⣠⠆⠀⠀⠀⠈⢇⠀⠄⠀⠀⠀⠉⠀⠉⢺⠑⡀⣊⠀⠀⠀⠀⠠
⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢐⠈⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣠⣤⣀⣀⠀⠀⠀⠸⠇⠄⠀⠴⠂⠄⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⢸⣿⣿⣿⡷⠂⠀⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⣽⢿⡄⠀⠉⠙⠶⣄⠐⠶⠦⠀⠀⠁⠠⠀⠀⠀⠀⣘⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠛⠁⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⣠⠟⠁⠀⠀⢰⡏⠈⢷⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡉⠀⠀⠀⠀⢰⣧⣤⣤⣤⣴⣾⠦⠤⠼⣷⡶⠶⠶⠶⣶⣷
⠀⠀⠂⠀⠀⠂⠄⠀⣰⣾⠋⠀⠀⠂⠀⠀⡟⠙⠷⣦⣀⣰⡇⠀⠀⠀⠈⣷⣀⣤⠞⠁⢸⠀⠀⢁⣄⠀⠀⠀⠐⠀⠀⢀⠀⠀⢄⣠⣦
⠥⠐⠁⠀⡂⠀⠀⠠⣿⣿⠀⠀⠀⠀⠃⠀⣇⠀⠀⠀⠉⣿⢶⣄⡀⠀⣠⡾⣿⡁⠀⠀⣼⠀⠈⠸⠁⠀⠀⠀⠀⠀⢀⣾⣷⡀⠈⠙⠻
⣀⠀⠀⠀⡀⠀⠀⠀⠻⣿⣆⡐⢁⠔⠀⠀⢹⡄⠀⠀⢰⠇⠀⢈⣿⠿⣯⣄⠘⣧⠀⣰⠃⠀⠀⠄⣲⠀⠀⠀⠀⠀⠛⠛⠛⠓
⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠹⣄⢀⡿⢀⡴⠛⠁⠀⠀⠙⠻⣿⡿⠃⠀⠀⠀⠷⣌⠀⠀⠀⠀⠂⠐⠀⠂⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠒⡡⠀⠀⢠⠀⠀⢀⠀⠀⠈⠙⠿⣯⣀⣀⣀⣀⣤⠴⠚⠉⡀⠀⡁⠀⢠⠁⢸⠀⠂⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢇⡤⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⠀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⠁⡀⠀⠀⠀⠂⠀⠀⠀⠀⢲⣶⣶⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⡁⠀⠀⠀⠀⠃
⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⡀⠞⠁⠀⣀⣄⡀⠀⠐⠀⠀⠀⠙⠻⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠋
⠈⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠔⢠⣿⣿⡿⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡤⠄⠀⠄⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠔
⢀⡀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠁⠀⢀⡔⠀⠀⠀⠀⠀⠠⠖⠒⠛⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⢤⣄⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠾⣄⡀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢍⣿⣦⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢰⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⢹⣿⣏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠈⠻⣶⠀⠀⠀⠀⠀⠀⢠⠞⠯⣉⠹⠏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⡄⠀⠀⠀⠀⠘⡄⢠⠈⣉⠙⢒⢎⠁⡇
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠛⣉⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⡇⠧⠀⠁⡆⡸⠈⠄⡇
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣤⠜⡁⠀⠀⢈⠴⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⠀⠗⠀⡃⠇⠀⠀⠁
⠀⠀⠀⡠⠊⠠⠜⢣⡄⠀⠉⠀⠀⡰⠁⠀⠀⠀⠑⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⣟⠀⠀⣀
⠀⠀⢰⢅⠀⠀⠓⠚⠂⠀⠀⠀⡌⠠⡖⠃⠀⠐⠈⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠈⡀⠑⠠⡀⠀⠀⠰⠴⢆⠣⠘⠈⠀⠈⠀⠀⠰⡤⢇⠀⠁⢠⡀⠀⠀⠀⣧⠁
⠀⠀⠀⠑⠀⡀⠀⢁⡂⢘⠓⠎⠁⠀⠀⠀⠴⡀⠀⠘⠓⠎⠁⠀⣘⡾⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠑⠔⠂⢀⡀⠢⢀⠀⠀⠀⠢⠉⠚⠀⠀⠀⡠⠊⠁⠐⣄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠉⢴⠁⠠⡀⠀⠀⠀⠀⡠⠊⠀⢀⡤⠞⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⣠⠐⢠⡤⠚⠁⢀⡴⠋⠀⠀⢰⡡⠤⠄⠤⠤⠤⠤⠤⢤⣀⠠⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣇⡠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣤⠄⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠀⢁⣀⠉⠤⠈⠉⠃⠰⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⠁⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣅⣀⠈⠀⠂⠀⠐⠀⠀⠀⢐⡤⣀⣽⡷⠀⠀⠀⠀⠀⠈⠉⢘⡺⢿⠮⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠒⠉⠀⠘⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣳⡾⣏⣽⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡜⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣿⣿⣿⣽⣿⣿⣿⣿⣿⠟⠀⠐⠈⠐⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡿⠿⠟⠛⠀⠀⢠⣾⢿⣿⣷⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⣟⠁⠈⠙⠛⣿⣿⢿⣷⣤⠀⠀⠀⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣶⣿⣿⠷⣤⣤⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠋⠉⣥⣾⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠁⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⢠⣾⣦⣶⣶⢒⣷⣿⡿⠛⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢋⡙⠛⠋⠉⠐⠚⠶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡟⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⣬⡜⢣
⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⢠
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⢀⠀⢠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠐⠙⠄⣾⣦⠀⠀⠀⠀⠀⠀⡰⠲⡃⣬⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠁⠀⠀⠀⠀⠀⠈⢡⠓⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠇⢫⣠⣄⣀⡠⡎⠀⣀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠁⠀⠀⠀⠀⠀⠀⣈⠀⠴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⣏⠀⠀⣷⣄⣁⣠⣿⣆⢸⡄⠀⢿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠁⠀⣸⣿⣿⣿⣿⣿⡟⣸⠁⢀⣾⡄
⠀⠀⠀⠀⠀⠨⢁⠤⠛⣠⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣠⣀⡀⠀⣿⣿⣿⣿⣿⡿⠁⣡⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⢀⡏⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣾⣿⢿⡿⢻⣿⣧⣼⣿⣿⠏⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠉⡿⠉⢻⡟⠁⠀⠃⠈⠉⠿⠁⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⢺⣷⡶⢛⣶⣶⠀⠈⠁⠈⠉⢀⣶⠂⠒⠆⢳⡶⠚⠛⠒⠶⣀⠒⢀⣀⠂⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢳⣄⣈⣙⣷⠤⠆⠀⠘⠦⠈⠁⠀⠀⠾⣄⠀⣄⢀⠀⡉⠠⠛⠁⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⢿⡏⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡅⠀⠀⠈⠁⡘⠁⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠈⢻⣼⣷⣄⣀⡠⠂⠀⠀⠀⠀⠀⠀⠁⠌⠃⠀⣠⣀⠞⠀⠀⠀⠀⣀⡾⠀⠀⡄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠈⠀⠖⢻⡏⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠳⣦⣟⠉⠳⠀⢀⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠂⡀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠁⢤⡀⠀⠉⢰⠄⠀⠉⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⣰⠏⠀⠒⢺⠿⠀⠀⠀⠀⠃⠀⠀⠀⠙⠷⠀⠈⣤⠀⠛⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠰⠂⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠠⠘⠀⠀⠀⠀⠀⢀⠖⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠁⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣠⣤⣄⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⡠⠉⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠁⣴⡄⠀⠒⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠉⠈⠁⠀⠘⠗⣼⡟⠁⠀⠀⣠⠂⠀⠀⠀⣠⠦⠭⢙⠉⠉⠓⣤⢠⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⠟⠟⠀⠀⠀⠂⠀⠲⠷⠋⠀⠀⠀⠂⢠⠀⠀⠀⠀⠁⠀⠀⠈⢰⡄⠀⡀⡄⠀⠚⠛⢻⣦⣀
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢠⡄⠀⠀⠀⢻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⣦⠰⠿⣀⡀⠀⣀⢀⣀⠤⢹⣯⡄
⠀⠀⠀⠀⠀⢺⣷⣾⠁⠰⢦⡁⠀⠀⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠈⠶⢮⣴⡄⢶⠀⣿⠛⣿⠉⠀⠈⡄⠀⠀⢡⡀
⠀⠀⠀⠀⠀⣿⣟⣿⡆⠀⡀⠰⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠸⠃⠀⠀⠀⠈⢷⣍⣿⣿⣆⣿⠀⠀⢸⡃⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⣧⠿⣏⠁⠀⠀⠀⢰⡆⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠰⠶⠄⠀⠀⣰⣦⣿⣿⢟⣿⣿⡶⠀⠈⠃⠀⠀⠀⠀⢻⣦
⠀⠀⢸⢦⣸⣿⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⠟⠀⠤⠀⠛⠚⠂⠀⠀⠀⠲⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⠏⣿⠁⠀⠀⣿⣄⣀⣤⠀⠀⢹⡆
⠀⠀⠹⢟⣿⣿⣴⠇⠀⡀⠀⡠⠛⠤⢤⣤⡤⢶⡆⢰⠄⠀⢠⣤⠀⢾⣄⠀⠺⢷⠀⢀⠀⢸⡿⣿⣿⡧⠙⠃⢀⣀⣾⣏⡀⠁⢀⣰⠿⣠⠆
⠀⠀⢸⡟⣿⠻⣟⠻⠇⣠⣲⡴⠃⠀⠀⠀⠃⢀⠴⡃⠀⡷⠀⣿⠀⠸⣻⠦⠐⠠⠿⠉⠻⠟⠁⠀⢏⡀⡠⠔⢛⣿⣏⠉⠛⠻⠟⣛⣂⡇⢀⠀⠀⠠
⠀⠀⣼⣿⣿⡞⣩⠳⣤⠀⠹⣀⢠⠤⠄⠀⠰⠇⠀⠙⠀⠁⠀⣿⣆⠚⠉⣴⢲⣦⠀⠂⠀⢠⡇⣠⢼⡁⠀⢠⣿⣿⠁⡌⠀⠙⠺⣯⠈⣧⠈
⠀⠀⣿⣿⣿⣿⣤⢂⠐⠀⠀⠙⢧⠴⠀⠀⠀⠂⠀⠀⣄⠀⠀⠘⣯⠈⣀⢘⣿⢻⣦⠰⠾⠛⠀⠀⠸⠇⠀⣼⣿⢇⡿⠀⠀⣿⢂⡜⠀⠹⡇
⠀⠀⣿⣿⢺⣿⣽⢾⣆⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢦⣀⣤⡁⢈⣡⢸⠏⣸⣿⠁⠶⠂⡀⠀⠀⠀⣴⢿⠟⢦⡄⠲⠀⣿⣿⣛⠀⣠⡇
⠀⠀⢸⣿⠸⣧⠼⢈⠛⠆⠀⠀⠀⠀⠐⠖⠀⢠⠀⠀⠀⠀⣻⣿⡗⢸⡟⠂⣠⡿⢋⡀⠂⠈⠁⠀⠀⢠⠛⣸⠃⡸⠀⠀⠀⣿⡟⠉⡍⣿⠁
⠀⠀⢸⣿⠀⡟⠀⣿⡆⠀⠀⢀⠀⠀⢀⡀⢠⡎⠀⠈⠀⠀⠋⣼⣿⠃⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⣿⠀⡇⠀⠀⠀⢸⡇⠀⢡⠿⡄
⠀⠀⢸⣿⡄⢉⠀⠏⠠⡄⠀⠀⠀⠀⠈⠁⠰⠄⠀⠀⠀⢶⣤⡀⡻⠀⢸⣧⢰⣶⣷⣄⠘⠃⠀⠀⢈⣧⢠⣿⢰⡇⠀⠀⠀⠀⠀⠘⠿⢰⡇
⠀⠀⢸⢃⣇⢸⡆⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⣋⣡⣄⠼⠁⠈⡿⠛⠋⠀⠀⠀⠀⣾⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠇
⠀⠀⠀⢸⣿⠈⠃⠀⠀⠀⠈⠁⠄⡀⠀⠀⠀⠀⠀⠰⢃⣀⣿⠃⠉⠀⠀⠀⠀⡤⣴⡶⢶⡆⠒⣶⠿⠀⢸⣿⢸⣧⡀⠀⠀⠀⠠⡘⢧⡟⢸⡀⡀
⠀⠀⣀⢸⣿⡂⢀⠀⢸⠀⢠⣴⠄⠈⠳⢄⡀⠀⠀⠰⢀⣼⠏⠀⠀⠀⠒⠀⠀⠐⣾⠃⠀⠀⠐⠋⠄⠀⣸⡯⢸⣇⠻⡀⠀⠀⠀⠀⢸⣷⣮⠀⠁⣼
⠀⠀⠈⢸⣿⡇⢸⠀⠘⡆⢸⣿⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣲⡀⢀⠀⢀⠀⠀⠀⣠⠀⠀⠀⡼⢣⠂⠀⣿⡧⠀⣿⠀⡇⠀⠀⠀⠀⠈⠉⠸⣇⣼⡇
⠀⠀⠀⢸⣿⡇⠸⠀⠀⢧⠀⣿⠀⠀⠀⠀⠁⠀⢀⠀⣿⣿⣿⢇⠛⠀⠉⠚⠃⠁⠈⠀⠀⣠⣿⡃⠀⠀⢻⡇⠀⠘⡆⢸⡆⠀⠀⠀⠀⠀⠀⢉⣠⣿
⠀⠀⠀⢸⣿⠟⢠⡜⠁⠈⢠⡏⠀⠀⠀⠘⠀⠀⠸⢇⡼⣿⣟⠈⡿⠀⠀⠁⠈⣹⠇⠀⢰⣉⣀⣀⣀⣀⠘⠗⠀⠐⢧⡸⣧⠀⠀⠀⠀⠀⣴⣿⡇⠙
⠀⠀⢠⣸⣿⢠⡄⡉⠀⠀⠈⣹⠀⠀⠀⠀⠛⠀⠐⢮⣷⣜⠛⣷⢴⡆⠐⢶⠀⠻⣄⠆⢸⡿⠛⠛⠙⠉⠴⢿⣤⣴⣎⢀⠻⣆⠀⠀⣠⢼⣿⣿⡏⠁
⠀⠀⢸⣽⢿⡜⠋⣅⠰⠆⠀⣿⠀⠀⠀⠀⠑⣄⠀⡜⣫⠿⢿⣵⡾⣿⣶⡋⠰⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠋⢠⡽⠓⢮⠁⣾⡿⠛
⠀⠀⢸⡇⠴⡿⠂⠉⠀⠈⣰⡟⢀⣠⣴⣶⣶⣏⠀⠁⠀⠊⢀⡜⠉⢳⡟⠁⠦⣤⠀⠁
⠀⠀⢿⣹⠂⡀⢀⠀⠀⠊⠁⢛⣉⣵⢨⡝⠛⠛⠁⠀⠀⠰⠋⠀⠀⠻⠚⠀⠀⠀⠈⠶⠋
⠀⠀⠈⠀⠈⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠀⠀⠀⠀⠀⠸⠻⠟⠛⠉
⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡯⠁⠈⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣷⣶⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢏⡉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢠⡢⢀⡀
⠀⠀⠘⢿⠍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠣⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠴⠞⡁⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠙⠚⠿⣷⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⣼
⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⡠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠆⠀⠀⠀⠀⠀⠀⢀
⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢊
⡦⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠶⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠂⠀⠀⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠠⠍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠙⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡅⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⣔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣦⣴⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠂⠀⠀⡰⠉⠀⢀⣠⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠖⠒⠢⡀
⠀⠀⠀⣞⣷⠃⠀⠀⠀⠀⡆⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⣏⣁⣀⣀⣠⣴⡄⠀⠀⠀⠀⠀⡀⡅⠀⠀⠀⡅⡀
⠀⠀⠀⣅⣧⠀⠀⠀⠀⢀⡷⣶⠀⠀⠀⠄⠀⠀⠐⠀⠀⢂⣸⠛⠷⠤⠴⠒⠺⢧⠄⠀⠀⠠⠲⣅⠀⠂⠤⠀⠀⡝⢄
⠀⠀⠀⠙⢇⡀⣀⢲⠄⠉⠤⠀⠀⠀⠀⢦⡀⠀⠀⠄⠀⠀⠟⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠰⠃⠈⠳⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⢱⠦⠤⠶⡶⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⡤⡄⠘⢳⡄
⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣙⠃⢉⣄
⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠤⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠰⠄⠀⣤⣀⠀⠁⣷⣠⡄⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠲⢢⣁⠄⡅⠉⢫⠀⠚⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠈⠀⠀⠀⠊⠀⠀⠰⠀⣧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣵⣀⡠⠀⠀⠀⠀⡀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡐
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢶⡀⣟⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢾
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⡀⠀⢀⡇⠀⠀⣤⠀⠀⠀⣠⡖⠛⠋⠀⠉⠀⡀⠀⠀⠀⣤⠀⠀⢀⠜⠁
⠀⠀⠀⠀⢘⢡⡞⠁⠀⠀⠀⠀⠂⠀⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣖⡄
⠀⠀⠀⠀⠀⣐⣤⠂⠀⠀⠰⠀⠀⠀⢸⢺⡂⠀⠀⠀⠀⠀⢀⠇⠀⡅⠐⠠⠀⠀⠀⠀⢀⣠⣆⠀⠀⠀⠀⠀⠐⢿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠈⠻⡷⠀⠀⠀⠀⠰⠏⠀⠀⠃⠂⠀⠀⠀⠀⠐⠛⠿⠿⠇⡀⠀⠀⠀⢸⢻⣟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠒⠒⠒⠛⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠤⠤⡀⠀⠀⢀⡀⠀⠀⠀⠀⡰⠁⢷⡀⠌⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣠⣦⠵⠒⠀⢢⡀⢠⣄⣩⡽⠁⢳⡿⢿⣿⣾⠇⡄⣁
⠀⠀⠀⠀⢀⠠⠀⠨⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢣⢬⣬⣽⠃⣾⡟⠀⠀⣼⣿⣾⣴⣈⣩⣶⣤⣀⠀⡀
⠀⠀⡤⣊⠐⣖⡂⠙⠂⠀⠀⡖⠨⠆⠀⠀⠀⠀⠀⠈⢯⣽⢇⣶⢻⣿⠶⠙⠫⠟⢿⡟⠙⠻⣿⠟⠻⠇
⠀⠀⢧⠉⡘⡎⢉⠀⠀⠀⠀⡀⠀⠂⠔⠈⠀⠐⠀⠀⠘⡧⠿⢿⡞⠁⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⡶
⠀⠀⠈⢆⠁⣽⡌⠀⠀⢀⠠⡀⠀⠄⠓⠐⠂⠔⠁⠈⠀⠹⠀⢰⡆⠀⠀⠁⠀⢤⢄⠀⣿⣶⣄⡀⡼⢁⡀
⠀⠀⠀⠈⠆⢟⣷⠠⠀⠀⠀⣈⠁⠀⠀⠀⠀⠄⢉⡀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠺⣿⣿⣿⣿⠿⣿⣆⡀
⠀⠀⠀⠀⠘⣤⢿⣧⠀⠢⠄⠤⠀⠀⠀⢀⠀⠀⠠⠄⠁⠂⠈⢦⣀⠀⠂⠄⢠⣤⣾⡅⣀⡞⣽⠿⢿⣷⣿⣿⠷
⠀⠀⠀⠀⠀⠘⡜⣿⣆⠀⠀⢐⢃⡀⠀⠀⠀⡀⠓⢄⠠⠀⠀⠘⣟⡳⣦⣤⣸⣟⣹⣿⠏⣼⠳⣶⣾⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⡌⣿⡄⠀⠀⠂⠀⠉⠈⡁⠀⠀⠄⠀⠀⠀⢀⡹⣠⠏⠛⠻⠻⢻⡿⣼⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡘⣷⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣠⠀⠀⢈⣀⠡⠈⠉⠉⢈⡁⣹⣏⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧⠀⠀⠀⠀⠤⠰⠀⠀⠈⠀⠀⠰⢀⣩⣄⣠⠄⣤⡈⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠛⡯⠙⡛⠹⢗⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠢⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠈⠨⠋⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⢸⣿⠀⣾⢋⣤⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⢹⡇⠀⠰⠂⠀⠀⡿⠀⠃⠈⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⠀⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠀⣠⡶⢶⣿⠁⠀⠠⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⡾⢉⠇⣰⡿⢠⠇⣴⡟⢠⣿⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣧⡼⣾⡏⢻⡾⢾⡟⣠⢿⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣤⣴⣶⡄⣼⡏⢀⡞⢠⣿⣠⠏⠀⠸⡷⠋⣾⡇⠀⢀⡾⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⢁⡼⠃⠹⠷⢋⣠⣬⡿⠷⠦⣄⠀⠀⠀⣿⡇⢀⡾⠁⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡶⠛⠉⠀⣴⠂⠀⡼⠀⠀⠀⠘⠳⠛⡁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⣿⣄⠀⢀⣠⡾⠋⡀⠀⣼⠏⠀⠀⠀⢰⡏⣠⠞⣱⠆⢀⣾⠀⣰⠟⣹⠇⠀⠀⠛⣥⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠛⠛⠛⠁⠀⠀⠁⢸⡏⠀⠀⠀⢀⣿⢿⡅⢠⣿⡴⢻⡏⣸⣿⠞⠁⣠⠄⣀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣷⣄⠀⠀⣼⡇⠀⡟⠀⠉⢀⣾⠷⣿⠁⠀⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡼⠁⢀⡴⢻⡟⠀⠻⣶⠾⠋⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠻⠟⠁⣠⠏⠀⣾⠁⢠⣶⡆⠀⠀⠀⢰⡿⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⢀⣼⠃⢠⡞⣯⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⢁⡀⠀⠉⠀⢁⣀⣠⣿⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⣠⡞⢹⠄⠀⣴⠋⡷⠀⢀⣾⠟⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⣿⢃⣾⣿⠀⣜⣠⣾⢿⣴⣃⣠⣿⠁⣠⢿⡟⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣼⠇⣼⡇⠘⡿⠉⢸⡇⣴⠋⠉⢸⣇⡴⠃⢸⡇⠀⣠⡟⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⣀⣀⣤⣿⣿⠀⣿⣠⡞⠁⠀⠸⠟⠁⠀⠀⠀⠉⠀⠀⣼⠛⠞⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⢁⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡆⣸⠇⢀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⣦⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣼⠀⣰⡇⢀⣾⠃⠀⣴⣿⣾⠿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⢇⣼⣿⠀⣼⠇⣠⢞⣿⣿⡁⠀⠀⠀⢀⣾⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾⡿⠀⠀⠀⢀⣀⡀⣿⠟⢸⣧⡾⠻⠶⠃⣸⠇⠘⢷⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣿⠷⠶⠿⠿⠟⠛⠁⠀⠀⠀⢉⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⢹⡇⠙⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣉⣛⣋⣩⣉⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⣿⣆⢀⣀⣠⣤⣤⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡿⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧⡉⢧⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢄⠻⣿⣿⣟⣿⣈⣶⡿⣦⣴⣶⠄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣏⣠⣀⣠⣄⣹⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣣⣘⣿⣿⠿⢋⣡⢀⡁⡛⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡘⢻⣿⣿⣽⣿⣿⣿⣿⣿⣷⣾⣇⢀⣀⢠⡷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⠇⣼⣿⣿⣿⣿⣿⣿⡣⢿⡎⠁⢮⠄⣡
⠀⠀⠀⢿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⣿⣿⣿⣿⣿⠏⠀⢺⡅⠀⠈⣞⠃
⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠘⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡿⢄⠈⠿⠁⣾⣗⡆⠀⢀⣀⡀⠄⠈⢢
⠀⠀⠀⠀⠻⣿⣿⣾⣿⣷⣤⣄⣠⣤⡘⠛⢃⣀⡸⠿⢿⣀⣈⢣⡀⢸⣿⠟⠿⣿⠟⣿⣶⢰⠀⠈⡆
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣴⣼⣿⣿⣾⣿⣦⡿⠀⠄⠸⣿⣏⡠⢒⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣥⣀⣀⣠⠞⠁⠀⡌⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣿⡿⠿⠿⠿⠛⢫⣿⣿⣷⣿⠍⣇⠀⠀⢠⣁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⡄⠿⠀⢠⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠏⠙⢇⣿⣿⣿⣿⣿⣶⢦⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢻⠟⠛⠛⠛⢻⣮⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡠⢺⠀⠀⠀⠀⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡠⠊⠀⢸⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⢃⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⠀⢣⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠀⡀⠲⢤⡀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡾⠉⠀⠀⠸⠂⠹⡗⢦⡙⢦⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠹⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⡼⠃⣸⠀⣿⠘⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢦⣄⠀⠀⠀⠀⢠⠞⣁⡴⢋⣠⠌⠁⢠⢟⡄⠀⠀⠐⢾⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠽⢗⣢⡤⠹⣇⠀⠀⠀⡞⡼⢋⡴⠋⢀⣠⣴⣫⠎⣠⡄⠀⠀⠘
⠀⠀⠀⠀⣀⣠⣴⣶⣻⠿⠗⡋⠩⠖⠛⢩⣴⣾⣷⠿⠶⣤⡀⢷⣧⣟⡴⠚⡉⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠾⠂
⢀⣴⢶⣿⡿⠟⣛⡭⠶⣚⣯⣷⣾⣿⠶⠛⠋⠁⠀⠙⠳⣌⢷⡀⠘⠟⢠⠞⣡⠖⢋⣉⣓⠢⠀⠀⠀⠀⣠⠤⠤⣄
⣿⠁⣬⠴⣚⣭⣵⣾⣿⠷⠟⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⢹⡜⣧⠀⠀⡏⠀⢡⢞⣩⣤⡈⠳⣄⠀⢠⢎⡴⢒⠒⣦
⢹⣷⣤⣬⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠻⠏⠀⠀⠀⠈⠃⢹⡆⠀⠁⠀⣿⣯⠤⢤⡀⢤⡹⣄⣡⢞⡅⡴⠒⠾⠀⢀⡀
⠀⢿⡦⠤⢿⡆⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣌⢿⡀⠀⠀⠛⡥⠦⡄⢻⡄⠀⠤⠤⠀⣸⣣⠚⠁⠀⠙⠏
⠀⠘⣿⠛⠋⢿⡀⠀⠀⠀⠀⡈⠃⠀⠀⠀⣠⢦⠀⡆⢸⣀⡄⢹⡜⣧⠀⠀⠈⠀⠀⠘⢄⠳⣄⣀⣠⠴⢣⠇⠀⡀
⠀⠀⢹⡆⠀⠸⣧⠀⠀⢾⡄⣿⠀⣼⢻⠀⢯⣾⡟⠙⠋⠉⠀⠀⢷⢻⡆⠀⠀⠀⢤⣶⡶⠦⣄⢠⡀⠐⠋⠀⢀⡿⣤⡄
⠀⠀⠀⢿⡀⠀⢹⡆⠀⢸⣧⣿⣤⠟⠛⠛⠀⣟⡇⠀⠀⠀⠀⠀⠘⣏⣷⠀⠀⠓⡄⠙⣿⣦⠈⣧⠹⡄⠀⠀⠉⠙⡇⠀⠀⠀⢿
⠀⠀⠀⠘⣧⠀⠀⣿⡘⠛⠀⠀⠀⠀⠀⠀⠀⠉⣁⡀⣧⡄⠀⠀⠀⠉⠸⣇⠀⠀⢹⠀⡿⠸⠀⢸⡆⢇⡀
⠀⠀⠀⠀⢻⣟⣫⡽⣧⠀⠀⠀⢰⣆⡀⢰⠶⣾⣇⣹⠻⠛⠆⢀⣄⠀⣦⢻⡄⠀⠉⠀⡇⠐⣇⠀⠳⠤⠵⢿⡆
⠀⠀⠀⠀⠈⣿⡀⠀⢻⡆⠀⠀⠀⣿⣹⠞⠷⠋⠈⠁⠀⠀⠀⠈⠉⠀⣸⠈⣷⠀⠀⠀⠻⣄⠈⠁⠐⠃⣠⠞
⠀⠀⠀⠀⠀⠸⣧⠀⠈⣷⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠋⣀⡾⠃⠀⡀⠀⢮⣙⡓⠚⢋⣡⠞
⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⣧⠀⠀⠐⣤⠄⠀⠀⢀⣠⡴⠖⣋⣭⡴⠾⠛⠉⠀⠀⣀⣿⣀⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⢿⡄⠀⠈⠉⠃⠀⣀⣩⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠈⣷⣀⣠⣴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠐⢠⠀⠀⠀⠊⠉⠉⠀⡀
⠀⠀⠹⣿⣿⣿⡷⠀⠀⠄⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡾⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⣿⠂⠀⢀⡔⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠦⠀⠀⠀⠼⣿⡆⣠⣿⢉⢓⣀⠀⠀⠀⠀⠀⢠⣴⡦⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠀⠀⠀⠀⣸⣧⢋⢇⣼⣿⠇⠀⡀⠠⠤⠶⠛⠉⠀⠀⠀⠀⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⣿⣿⣿⣾⡏⠁⠀⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⡿⣿⣿⠃⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣇⣹⣾⣿⣿⡀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⢀⡴⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢘⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣄⣀⣠⣀⡀⠀⠀⠀⠑⠾⠋
⠀⠀⠀⡇⠂⡒⢤⣄⣀⣠⡀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⢿⣤⣮⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠂
⠀⠀⠀⢸⡉⠉⢉⠻⠋⠁⠀⢀⣤⢤⣖⣋⠉⠀⠀⠀⠀⣀⢐⠀⢠⣾⡿⠄⢠⣾⣏⣀⣀⢀⢲⢩⣵⡗
⠀⠀⠘⢾⣷⣄⠘⢿⣷⣤⣾⣿⠿⠿⣿⠋⠀⠀⠀⣀⡐⠉⠸⢀⣄⣿⣿⣗⡟⠘⣿⠁⣹⡿⠛⢷⣆
⠀⠀⠀⣺⣷⣿⣿⡺⢿⣿⣿⠋⢾⣿⣿⣦⣴⡂⠀⣿⣛⠛⠂⠀⣿⣟⣏⡙⣿⣽⣿⣿⣴⣧⣤⢶⡌
⠀⠀⢀⣡⡼⣿⣿⣿⣾⣿⡿⢿⣾⣿⣟⣿⠋⠀⠀⠻⢿⣷⣌⣾⣿⣿⣿⣿⣋⣙⣿⡮⣤⣯⡶⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷⠟⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⣤⢤⡔⠀⠀⠀⢠⣶⣶⣿⣦⠀⠠⠤⠀⠚⠂⠂⢀⠀⠀⠀⠀⢀⡠⠴⠧⠜⠀⠂⠘⠁⠀⠀⠃⢤⠀⣀
⠀⠀⠀⠀⡶⠀⠀⠀⠀⢸⣷⣄⡻⡿⠦⠠⠀⠈⠊⠁⠀⠀⠀⠀⡜⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉
⠀⠀⢀⡐⠁⠀⠀⠀⠀⠚⠛⠻⠷⠀⠀⠠⠄⠨⣡⠄⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠟⠁⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠖⢢⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠇⠀⠘⠀⠀⢠⡏⠁⢀⣦⣀⣠⣾⣿⡄⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⣿⣶⠾⣯⡟⠛⠏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⢹⡏⠁⠀⠿⣀⠀⠀⠘⠁
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠐⠁⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⡀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠃⠆⠀⠚⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⣠⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠
⠀⠀⠃⠀⠀⠢⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀⠐⠶⠚⢿⣿⣷⣾⡿⠟⠶⠶⢀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠠⠒⠉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡤⡀⠀⠠⠀⠀⢤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢀⢀⡆⠀⣈⢡⠀⡟⠀⠀⠀⠀⠀⠄⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠓⢄⠈⢀⣾⣶⣥⠐⠈⠀⠀⠂⠤⢡⡀⢸⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠅⠎⠁⠂⠈⠀⠀⢶⡟⠀⠀⠀⠘⡢⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⡘⡉⠶⢀⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⡠⢋⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠁⠙⠳⠈⡄⠦⣄⡂⠀⠀⡀⠀⠀⠘⣒⣠⠤⠶⢫⣿⠀⠀⠀⡐⡩⠕⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⣤⠒⢺⠀⠀⠀⠀⠀⠀⠉⠉⠛⣉⣀⢠⠤⡶⢮⠻⡍⠀⠀⠀⢫⠂⠈⠀⠈⣡⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠘⠚⠃⠀⠀⣡⠄⣀⡥⠔⠒⠚⣍⠉⠁⠀⠀⠀⡆⠀⠀⡶⢠⠐⣆⣂⡄⠀⠈⢷⠙⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⢀⢴⣞⡁⠀⠀⠘⠅⠙⠀⣰⣀⢀⣰⣁⣦⠚⡗⢃⠏⡘⠉⢀⢀⠠⢊⣴⡂⠩⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠿⠾⢣⡎⢎⠫⣉⠇⣤⡀⠆⣠⣏⡔⠯⠈⣯⡉⠁⣌⢉⠎⢊⡕⣀⠷⣉⣴⠋⣁⡔⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡁⠈⠢⢿⡿⠿⠃⠀⡇⠡⠈⠻⠀⢯⡇⢀⠹⠎⣛⡫⠥⠖⠻⢝⢻⢖⠽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣾⡓⠄⠀⠀⠀⢀⠄⠀⢉⢰⣀⠎⠀⣚⣤⠶⠚⠛⠁⠀⠈⠀⠀⠈⠁⠁⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⡾⣽⣶⡀⠀⠀⠐⡄⠀⡸⣡⠴⠟⠩⢅⠀⠆⣀⠀⠀⠀⠀⠀⢀⣀⡼⢴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⠀⠒⢃⡺⢋⠞⠁⠀⠀⠣⢀⠉⠀⠀⡁⠻⣤⡤⠖⠋⣁⣠⣾⠟⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣆⠠⣀⡀⠠⠀⢈⡐⠻⢮⠵⠶⠀⠀⠀⠐⣷⡴⠶⣾⡷⢿⣿⣾⡿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠜⠂⢭⢁⢃⡎⠀⠀⠁⢄⣀⡀⠄⠀⣈⠥⠒⣋⣽⡾⠟⠋⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠛⠘⣣⠀⠀⢠⡿⠉⠀⢀⣀⣤⡶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢤⣿⣧⣶⠶⣿⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢁⣤⣤⣶⣶⠦⡄⣼⣿⠋⢱⡆⣠⣾⡿⣀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡟⢼⣿⠟⣻⣿⠃⠀⢸⣿⠃⢀⣿⢰⣿⠏⢠⣿⢰⣿⠏⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣴⣿⠃⠀⠀⢿⡟⢀⣼⠃⣿⡿⠀⣾⠇⢸⣿⣠⡟⢹⣿⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣇⣠⣾⠟⠁⠀⠀⠀⠈⠛⠚⠁⠀⢿⣧⡼⠋⠀⠀⠉⢡⣾⡆⠙⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢀⣾⣿⠀⢀⣀⠀⠀⣤⣾⣿⣧⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⣾⣿⡏⣠⣿⡟⢀⣾⡿⠫⣿⡟⣠⣾⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⣟⣴⣿⣿⠁⣼⣿⠁⠀⠈⠀⣼⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢈⠻⠛⠁⣿⡇⠀⢿⣇⣠⡴⠂⢠⡿⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⠏⠀⠀⠀⠘⠓⠚⠁⠈⠉⠀⠀⠘⠃⠀⠹⣿⣧⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⡆⠀⠀⠀⠀⠀⠀⢰⠧⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⢰⣌⠍
⠀⠀⠀⠀⠀⣠⡀⢀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠀⠀⡀⠀⠀⠀⢠⠋⠈⠁⠈⠇⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣬⣿⣄⣸⣧⠘⣇
⠀⠀⠀⠀⠀⠉⠁⠄⠠⠤⠸⢀⠀⠀⠀⠀⠀⠠⡁⢀⣁⣀⣤⡟⠀⠀⠀⠈⠀⢠⣤⢀⠇⠀⠀⠀⠈⠁⠚⠀⠁⠀⡀⠀⠀⠀⢸⣏⠹⣿⡛⠻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠀⣻⡏⡀⠀⠁⠀⠀⠀⠀⠑⢮⣿⣿⣷⡇⠀⠀⢠⠁⣀⠀⠁⡘⠀⠀⠀⠁⠀⢸⡈⢠⠄⠏⠄⠀⠀⣤⣄⠙⢷⣯⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⡅⠀⠀⠈⠢⠉⠀⠠⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠁⠈⠳⠾⠏⠈⠉
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠐⡌⠆⠷⠂⢸⠇⠖⣀⡀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⡀⠠⡀⠀⢰⣴⠆⢠⠄⠤⠈⠓
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣘⠦⠉⠀⠀⣶⣄⣀⡀⠀⠉⣁⡤⠀⠀⠀⠀⠤⠀⢄⠀⠀⢀⡤⠤⠤⡈⠀⠁⠀⠁⠘⡀⠁⠂⠀⢸⡇⡀⢠⣄⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢮⠅⠀⣰⢋⣇⠼⢤⠟⣲⠋⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠁⠠⠆⠀⠀⠠⠌⠓⠄⢸⣧⠀⠈⠘⠲⠈
⠀⠀⠀⠀⣠⡄⠀⠀⠀⢚⡃⠀⠛⡯⣤⡃⠀⢈⢎⢩⠀⠀⠤⣤⠀⠀⠀⠐⠒⠴⠀⠀⠀⠈⠀⠀⠠⢰⣆⠀⠉⠀⠒⠂⠐⠀⠀⣿⠀⣴⣾⣿⣿⡆⣀
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠙⢉⠀⠀⠑⢤⢉⡕⠓⡬⣯⠀⢴⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⡤⠤⠀⠚⣛
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⢁⣄⡀⠸⠋⠉⠉⠀⠀⠳⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠘⠺⠟⠋⠉⠈⠁
⠀⠀⠰⢀⠀⢠⡀⠹⣧⡀⠀⠈⠋⠀⠆⡘⠀⣗⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠒⠃⢁⡀⣄⡀⠀⠀⠀⠀⠀⡴⢆
⠀⠀⠈⠀⠀⡈⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠱⠍⢾⣿⢺⡀⠀⠀⠀⠀⠁⠀⠀⢀⢤⠐⠛⢊⠉⠀⠀⠀⠀⠀⣀⡬⠧⠉⠦
⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣀⠀⠛⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠼⠲⡁⠋⡀⠀⢸⣯⠐⠒⠒⠒⠀⠄
⠀⢀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠈⢳⣄⠄⠀⠀⠀⠀⠀⢸⣿⠏⢿⣿⠇⡀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠰⠀⠀⣠⡶⢶⣶⣟⡢⠀⣰
⠀⠈⠓⠒⠚⠀⠀⣠⣄⣀⠀⠀⠚⢀⡴⢤⡄⣠⠀⠀⣾⣿⣷⣿⡟⢁⠰⠃⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠠⠁⠀⠻⠆⠀⠂⠁⣬⣷⣷⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠀⢣⡜⠛⠛⣤⠀⠈⠁⠊⠝⣑⣠⠗⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⣿⣿⢿⣿⡿⡿⠿⠿⣿⢇⡤
⠀⠀⠀⠠⣄⢰⣿⣿⣷⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⢀⣀⠀⣓⢀⣀⡀⠨⠇
⠀⠀⠐⠂⠈⢘⡟⢁⡴⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠈⣩⡄⠀⠀⠀⠀⠠⠭⠭⠭⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣶⡏
⠀⠀⠀⣠⣤⣿⣿⣿⣷⣿⣿⣿⠿⠀⠀⠀⠀⠸⠞⠀⠘⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠃⠀⠀⠀⠈⣍⠉⠈⠉⠁⠉⠡
⠀⠀⠀⠈⠉⠈⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠄⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⡆⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⠋⠁⣠⣄⣧⠤⠀⢄⠁⠈⣉⢐⣂⡞⠰⣴⣾⣷⣶⣶⡀⡉⠂
⠀⠀⠀⠀⠀⠀⡀⠀⡂⠁⣀⡴⠶⠛⠉⠁⠀⠀⠀⠀⢇⢘⣉⣻⡟⢠⣶⠏⠀⠉⣷⣧⣞⣨⢛⢂⣤⣀⡀
⠀⠀⢀⠤⠂⣥⠤⠐⡎⠉⠀⢀⠤⢰⡀⠀⠀⠀⠀⠀⠈⣞⢻⡟⢀⡼⣿⣦⡄⠾⢿⡿⣽⡿⠻⣿⣿⣿⣿⡦
⠀⠀⢇⠛⠂⣇⠺⠂⠀⠀⠀⠈⠁⠀⠀⡠⠐⠀⢀⠀⠀⠘⡟⣥⣿⣥⠟⠁⠀⠀⠁⠀⠑⠁⠀⠈⠛⠀⠀⢡
⠀⠀⠘⡆⠳⢸⡄⠆⠀⠀⠀⠐⠀⢀⠀⣅⢀⡀⣁⠆⠠⠄⢻⡀⠈⡍⠀⠙⠳⠀⠀⠀⠀⠐⣤⣀⠀⠀⢠⠯⠀⢀
⠀⠀⠀⠘⡄⣻⣷⡀⠀⠀⠂⠉⢠⠊⠁⠀⠀⠀⠑⠤⠀⠀⠄⢆⠀⠱⠀⢀⠀⠀⠀⠙⠂⠀⣿⣿⣷⣦⣯⣴⣷⣄
⠀⠀⠀⠀⠸⡜⢺⣧⠁⢀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠈⣄⠀⠀⢈⠀⠀⠀⠀⣠⠂⠈⣻⢿⣿⣿⣏⣻⣿⣶⡠
⠀⠀⠀⠀⠀⠹⡝⣿⣗⢀⠉⠈⢣⠀⠀⠀⠠⠀⠀⣈⠁⠀⠀⠀⡘⣷⣄⡀⠈⠀⢶⣿⣿⣴⡴⢣⣟⣉⣹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠱⡘⣿⡄⠀⡀⢂⡰⠂⡄⢤⠄⠂⠀⠱⠈⠀⠀⠀⢹⡦⠹⣿⣶⣿⣦⣾⡿⢡⣿⣨⣹⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠱⡸⣷⡀⠀⠁⠄⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⡆⢣⠘⡄⢄⠉⠩⠿⢧⣿⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⢴⠌⠀⠀⠀⠀⢴⢀⣞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣇⠀⠀⠀⠀⠐⠘⠀⠈⠀⠀⠀⢐⣃⣤⣶⣤⡖⠀⣶⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢝⠀⣩⠁⠛⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠢⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣂⡄⡀⡔⠒⡀⠀⢀⠀⡤⠔⠒⠠⡀⠀⠀⢀⠀⡤⠔⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⡨⣜⢰⢅⠃⡰⢁⠊⠅⡄⠂⠄⠑⠀⡰⢡⠊⠁⡄⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠡⡀⡁⡘⢹⠁⢀⠃⡌⠀⠃⡈⣒⠽⠀⢠⠁⡀⠀⢆⡈⣒⢽⠀⠇⠀⠀⢀⠠⠤⢤⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠊⠀⠀⠆⢘⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠇⠀⠀⠀⠀⢸⠀⡀⢀⢊⠔⠁⡑⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠇⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⡘⠀⠿⠀⢠⠀⠀⠀⠀⡌⠀⠃⠁⢌⠀⠀⠇⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠄⠄⠀⢣⠀⠑⠤⠤⠊⠀⡔⠀⢣⠀⠑⠤⠄⠊⠀⡌⠀⡀⠐⡀⠀⠈⢻⠧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠤⠊⠀⠀⠀⠑⠠⠀⠠⠄⠊⠀⠀⠀⠑⠠⠀⠠⠔⠊⠀⠀⠑⠢⠄⠁⠩⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⢈⠀⢡⠀⠠⠪⢪⢢⠀⠀⠀⡠⡊⠥⢈⠑⢄⠀⠈⡀⡅⢀⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢤⠀⠀⠈⡄⢸⠀⢀⢠⠌⡘⠀⠀⠌⠔⠀⠄⡂⠂⢘⠀⠀⢠⢠⡌⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠈⠎⢀⠁⠀⡘⠘⠀⠀⠈⠢⠤⠊⠀⠀⢠⠌⡞⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠆⠁⠀⠌⠀⡌⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⡆⠃⠈⢢⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠧⠊⢀⠌⠈⠀⠆⠁⠰⠀⠀⠀⠀⠀⠠⠀⢰⠀⡇⠀⠀⠀⢃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡇⣗⠂⠠⠤⠤⢲⡆⠀⠀⢀⠌⠀⠀⠀⢰⠐⠄⠑⠄⣀⠀⢠⠖⠀⠠⣀⡣⠁⠀⠈⢌⠒⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⠀⠈⠁⠒⠂⠁⠑⠤⠐⠁⠀⠀⠀⠐⠊⠀⠀⠑⠒⠒⠂⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⠴⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⣛⣉⣩⣤⠤⠤⠤⠶⠖⠒⠒⠒⠒⠆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⡇⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠈⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⣧⠀⠀⠀⠀⣀⠤⠖⢒⢾⡓⠒⠢⢄⡀⠀⠀⢷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠏⠀⠀⢠⠞⠁⠀⠀⡜⠀⠳⡀⠀⠀⠙⢦⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⠀⢰⠃⠀⠀⠀⣸⣁⣀⡠⠵⡴⠒⣒⠞⢣⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡿⠾⢉⣉⢹⠃⠀⠀⠀⣀⠼⣎⠀⠀⠘⡇⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⢧⠀⠀⢨⠏⠒⢦⣤⣊⠁⠀⠈⢆⠀⢠⠇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠘⣆⢀⡎⡠⠞⠁⠀⠈⠉⠒⠤⣈⣦⡞⠀⢿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡇⠀⠀⠈⠺⣋⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⠀⡇⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠴⠚⠉⠀⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠗⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢰⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠁⢸⠀⠀⠰⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⡟⠉⠭⠭⠭⠭⠭⠍⢉⣉⣉⣉⣉⣉⠉⠉⢉⡉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⣀⡀⠈⠙⡍⠉⢏⠩⠥⠤⠴⠶⠒⠒⠒⣂⣀⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣀⣀⣈⣉⣣⠀⢸⣉⣉⣉⣉⣉⣉⣭⣭⡥⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛


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

⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣴⠇⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣏⣾⣧⣀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠛⡟⢯⡛⡛⢏⠂⠀⠀⠀⠀⠀⠀⠀⠘⠙⣾⣷⡋⣿⠀⠀⠁⠀⠠⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣶⣨⣆⡓⠄⠸⠀⠀⢠⠀⠀⠀⠀⢀⢰⣶⣏⣿⡷⣶⡤⣄⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣼⣿⣿⡿⢩⡇⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⣆⢻⣿⡿⠁⣾⣧⡹⢧⡀⠀⠠⠀⠠⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡇⢙⡿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⣶⣿⠙⣿⣿⡷⠀⠀⠀⠀⠈⢹⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢨⣿⣿⣿⣷⣤⣴⡼⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢲⣶⣤⣿⣿⣏⠀⠈⢻⣷⡀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠈⠀⣿⣿⣿⣿⡟⣅⣾⣿⡇⠀⠀⢀⣀⣄⠀⠈⡘⡆⠀⡸⠞⢿⣿⣽⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⢠⣀⡄⠀⠠⠀⠀⠉⠀⠀⠠⠟⠉⠀⠡⢂⣽⣿⣿⡇⠀⠀⢸⣉⠉⠄⠀⠸⡀⢀⠃⡘⠚⠉⢩⢿⡟⣷⠀⠀⠘⢿⠄⠀⠀⠀⣠⡍
⠀⠀⠀⠀⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⠉⣿⣷⠀⠀⠀⠇⠀⣶⡆⠀⣅⢨⠀⣵⠈⢰⢘⡭⡇⡟⠀⠀⡉⠈⢻⠂⠀⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠞⢩⡿⠿⡿⡟⠀⠹⣷⣄⠀⠀⡀⠀⠀⢠⡀⠹⡇⠀⠹⠼⣒⡉⣈⠇⡇⠀⠀⣉⣹⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠱⣄⠱⣾⣦⠀⠘⠻⣴⣄⣁⠈⠀⣾⠳⠀⣿⡄⡀⢸⠖⢹⠉⢸⠁⠀⠀⢹⣿⢿⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⠈⠳⠿⣿⡇⠐⠂⠘⢿⣿⣿⠒⠻⡿⣶⢿⣿⣴⣾⠀⠘⠴⠇⠀⢸⡀⠉⠇⢈⠀⠉⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣷⣿⣿⠷⣄⡀⠈⣿⡀⠀⠀⠀⣷⡄⡀⠀⢱⡜⣾⠛⠛⠃⠀⠀⠀⢸⣦⡼⠏⢠⠀⢈⡀⢠⣷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢘⠁⢹⡟⠀⠈⠙⣦⢨⣷⠀⠀⠀⢹⡻⣷⡄⠘⣿⣏⢰⡀⠀⠀⠀⠀⠈⢿⣿⣦⣶⡤⠾⠀⠈⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠊⠀⡞⠀⠀⠀⠀⣘⡃⠏⠀⠀⠀⠀⣿⣻⣷⠀⠉⢉⡜⣿⡌⠁⠀⠈⠀⠈⣿⣿⠁⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⡤⠖⠚⠉⠀⠀⠀⠀⠀⢸⠃⣷⠈⠐⠛⠚⠋⠻⢁⣀⡴⢿⡿⠃⣯⣦⣬⠉⣷⣄⠸⣽⣿⣇⣀⡀⣿⣧⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠿⠿⠛⠻⠿⠏⠀⣿⣿⣿⣷⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀

⣠⣤⣤⣤⣤⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣠⣤⣤⣀⣀⣄⣀⣀⣀⣀⣤⣤⣤⣀⣠⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣀⣄⣄⢀⡀⠀⠀⣲⡀⠀⠀⠀⠲⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠀⠀⢠⣄⠀⠙⠂⢀⠈⠀⠁⣉⠋⣷⡆⠀⠀⣬⠖⠠⣇⠀⠈⡲⠄⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣄⠀⠀⣀⠀⠀⠄⣆⢬⡼⠓⠀⢠⣰⡧⣀⣀⣻⣇⡀⠀⠀⠀⢀⠉⢀⠙⣄⠈⠉⡒⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠀⣧⡄⡈⣥⢾⣜⡓⢉⣁⡀⠀⡀⠈⠁⠀⠀⠀⠈⢛⣁⠐⠤⠤⢀⡉⠁⠳⢽⡄⠀⡇⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⣶⣿⣦⣿⠿⢋⠁⡠⢈⣨⠍⣁⠐⠦⠀⠀⠀⠀⠀⠈⢥⡆⢀⠀⠨⡇⠢⣀⠀⣿⠢⣅⠀⠀⠹⣿⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠎⠀⣠⣿⡏⢱⢋⣴⠥⠐⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢀⡆⣷⢀⠈⠀⢹⠆⢺⣷⡀⢣⢹⣦⠞⢹⡆
⠀⠀⠀⢸⡄⠀⡄⠀⣸⠿⣸⠷⣿⠁⣀⢂⡂⣔⣬⣰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⠓⠾⢬⣤⡈⢧⡑⣼⢠⣿⣦⣿⠁
⠀⠀⠈⡀⢧⣠⠓⣤⣿⢟⡟⣸⡿⠬⠋⠣⠑⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠌⢳⣄⠸⡋⠙⢫⣷⣼⠁⢸⣿⠁⢰
⠀⢢⠀⢧⠸⣧⠑⣾⢧⡾⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠙⣿⡗⢾⣿⣊⣋⠄
⠀⠈⣧⢨⠃⠸⣿⣣⡾⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣄⣿⣧⠸⣿⠈⣯⠆
⠀⠀⠘⣷⣀⠀⢻⣿⡡⣺⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⢻⣿⠀⣿⣇⡳⠂
⢦⡀⢰⡛⣿⣦⣾⣯⣠⣿⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠹⣿⠳⣿⣧⣟⠁
⠈⢷⡌⡾⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣱⣿⣧⣿⣿⠃
⠀⠈⢷⡸⡇⢸⣿⣿⠁⣀⡀⠀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⣿⠇
⠀⠀⠘⣿⠷⣸⣿⣿⠘⡿⠛⡸⡯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⠀⠈⠣⣼⣿⣿⣿⣆
⠠⣼⡇⣹⣯⢸⣿⣿⡏⠀⠐⣮⣿⢅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢷⣫⣴⣶⣼⣿⣿⡇⢸
⠀⢨⣟⡵⢻⣿⣿⣿⣇⠀⢐⡫⢷⢝⡂⠀⢄⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⣷⡋⠡⣼⣿⣿⡇⢸
⠀⠀⢻⣫⠌⢻⣿⣿⣿⡆⠀⢩⠽⣝⣃⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⡂⢀⣿⣿⣿⡃⣾⠀⢸⠁
⠀⠀⠈⠧⣞⢄⢹⣿⣿⣿⣄⠈⠬⣯⢲⣵⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⢽⣿⠒⣾⣿⣿⣿⣶⠃⠀⠅
⠀⠀⠀⠀⠻⢎⢔⣿⣿⣿⣿⣦⠨⣧⣾⣿⢓⣂⣤⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣯⣄⠀⡄⠀⢀⣦⣄⢺⣿⣃⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢏⣿⣿⣿⣿⣿⣷⣾⣿⠻⠛⢻⣟⣿⣿⣤⣤⠀⠀⠀⢀⠀⠀⠐⡩⣻⣽⠇⡇⣇⡼⣻⠏⡽⢷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠿⣿⣿⣷⣬⣧⣼⣽⣽⣿⣿⣀⠀⣈⣄⣄⣀⣈⡴⠖⠓⣚⡛⠛⣿⣾⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣤⣀⡀⢹⣿⣟⠁⠀⠀⠀⡀⢀⠌⠋⠙⠻⣧⡊⠁⢀⣅⠚⢻⣛⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠿⢿⣯⣭⣭⠾⢿⣤⣀⢀⠠⠊⢀⣱⡤⢶⣿⠖⡛⠛⠋⠛⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣧⣼⣾⣾⠾⠷⣦⠾⡉⠐⢣⡏⠀⠀⠀⣈⡥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣙⣇⣀⣜⣁⣀⣈⣠⣎⣈⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢆⢀⠔⠈⢁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠸⠃⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⢶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠞⠦⣦⡀⠠⣤⠀⣾⡁⣉⣥⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠘⠂⡀⠐⠈⡽⢍⠀⠀⠀⠀⠀⠀⡦⠘⠇⠐⠀⠀⠀⠰⠟⠁⠀⠸⠀⠀⢰⠉⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠀⠀⠂⠀⠙⡻⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡶⢀⠁⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⡊⡀⠀⠀⠁⠂⢀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠠⠄⡶⠚⠊⠁⢸⡠⠋⠀⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⢘⠁⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⡀⠛⠢⠄⠈⠀⠤⢀⠁⠀⠈⢄⡀⠀⠀⠈⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡦⢠⠤⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⠂⠀⠀⠀⠀⣄⠎⠉⠉⠁⠀⠀⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡰⠳⢤⠄⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠻⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⠁⠀⠀⠈⠉⠓⠦⣄⠀⣰⠁⠀⠀⠀⠀⠚⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡰⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡛⢄⣀⣀⠀⠀⠒⢭⡀⠁⠀⠈⠹⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠰⠆⡀⠀⠀⢰⠋⠀⠀⠀⣾⡤⠒⠂⠶
⠀⠀⠀⠀⠀⠀⠀⡎⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣄⡤⠂⠊⠇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⡼⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⠒⠒⢒⠃⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⢰⠃⠐⢀⠶⠒⠒⠠⠤⠤⣄⣀⠀⠀⢠⡁⠀⠁⢀⣀⣀⠠⠤⠤⠀⠈⠄⢳⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⢀⡏⠀⣰⣁⠤⠀⠀⠀⠀⠀⠀⣈⣹⣦⢿⠄⠠⠭⠤⠤⠤⠀⠀⣀⣐⠒⠥⢼⠀⠀⠀⢀⡠⠊
⠀⠀⠀⢸⡀⠀⠘⠓⠂⠂⠀⢀⣀⣀⣀⣀⢠⠀⠀⡕⢀⣒⣓⣈⣉⡡⠤⠤⠖⠒⠒⠛⠒⠐⠈⠁
⠀⠀⠀⠀⠓⠶⠒⠛⠓⠒⠒⠒⠒⠒⠉⠙⠦⣈⣉⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣤⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⣿⣿⣿⡟⠃⠀⠀⠶⠒⠀⠘⣿⠉⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣟⣼⠋⠈⠓⠀⠀⣦⣄⠀⢰⣦⡼⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⠿⢿⣿⡀⠉⠧⡀⠀⠀⠀⠘⠆⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠐⢦⡀⠘⠆⡀⡆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⠁⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣦⣤⣿⠀⡄⠀⠀⠀⠄⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡆⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣄⠘⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢠⠞⠋⣧⣴⠧⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣦⣀⢻⣦⠀⠀⠂⢀⣤⠖⠉⠀⠀⠀⠀⣤⠈⢹⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣷⣾⠾⠋⠁⠀⠀⠀⠀⣠⡀⠀⠀⣸⡄⠀⠀⠀⠀⢀⠰⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣶⣿⣷⣤⠀⠖⡀⣿⣷⠞⠀⠀⢢⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⠀⠐⠼⠿⣯⡡⠖⠂⠉⠀⠀⣀⣀⠈⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠀⠀⠀⠀⠀⢀⡀⢠⠐⠂⠈⣿⣿⣿⣧⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⠀⠀⠠⠆⠀⠀⠆⠀⠀⠀⡀⠀⠁⢠⠄⣀⣴⣿⣿⣇⠀⠀⠀⠀⣿⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣧⣤⣾⣷⣾⣿⣿⣿⣿⢿⡆⠀⠀⢀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣰⡞⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⡾⢿⣿⣿⣷⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢸⡇⠀⠀⠘⠆⠀⠀⠀⠀⢠⠀⠀⠹⣿⣿⣷⠸⣿⣿⣿⡆⢰⡄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⡿⢸⠇⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⠀⠀⠈⠿⣿⠃⠘⣿⣿⣿⣾⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣧⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠘⠃⠀⠘⢻⣿⠘⠃⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠀⠀⠀⠀⡸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⡗⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣷⣿⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢻⡏⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢹⣿⣼⠇⠀⠀⠀⡟⠀⠀⠀⣠⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢸⡀⠀⠀⢸⠁⠀⠀⢠⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠘⣿⡇⢸⣧⣼⡇⠀⠀⢸⢳⣶⡄⣼⣿⣆⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣿⢿⡇⠀⢠⡇⢸⣿⢀⣿⣿⡍⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠐⡀⢀⢰⣿⠀⣿⣿⣿⡇⠀⢸⡇⣼⣿⣾⣿⣟⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠥⡤⠈⣙⠋⢸⡟⠀⣿⢿⣿⡃⠀⢸⣿⣿⣿⣿⣿⡿⠃⢸⣿⠁⣀⠀⠀⣀⣀⣀⣠⣤⢸⠀⢠⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠒⠛⣭⡁⣾⡇⢸⣿⢠⣿⡇⠀⠀⠘⣿⣿⣿⣿⣧⣤⣾⣧⣀⣀⣤⣤⣿⣿⣿⣿⣿⡄⠀⠈⠀⠀⠂⠐
⠀⠀⠀⢰⠒⠛⣾⠛⣿⡇⠈⣿⠘⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠀⠡⠈⡀⣀
⠀⠀⠀⢠⣿⣿⣾⣿⣿⣇⠀⣿⠀⣿⣇⣀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡟⢠⣴⠆⠐⠒⠀⠒⠘⠁⠡⠌⠐⢣
⠀⠀⢳⣾⡿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⠿⠿⣿⣷⣬⢻⣀⠀⠂⠀⠀⠀⠀⠁⠀⠒
⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡢⠀⠀⠉⢋⡘⠻⠿⠓
⠀⠀⠀⠀⠘⠛⠛⠛⠻⠟⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⢷⣴⠀⣾⠟

⠀⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠁⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣠⣦⡀⠀⢀⠀⠀⠀⠈⠉⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠙⢉⡟⠁⠀⠀⠀⠀⢤⠀⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⢰⣤⣠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⡇⠀⠀⢠⠀⠀⠀⣀⣂
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⠛⡚⣅⠀⠀⠀⠀⢸⣿⡯⠀⠀⡞⠃⠀⠀⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢶⣀⠀⠀⠀⠀⠹⡿⡛⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠄⠈⢆⡀⠀⠀⠐⢍⣣⡀⠀⠻⠿⠿⢷⣿⣽⠄⠀⠘⠀⠶⠀⠀⠀⠄⡠⠀⠀⠢⣄⣠⡆⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠉⠁⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠖⠛⠇⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠀⢈⣴⠛⣹⣤⠀⠀⠀⣠⠖⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡩⠘⠁
⠀⠀⢠⠀⠀⠀⠀⠀⢼⣦⡄⠀⠀⠀⠀⠈⢹⡿⣿⡇⠀⠀⢀⣒⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠀⠀⢰⠶⠶⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠶⠦⠄⠀⠀⠀⣅
⠀⠀⠀⠀⠰⣛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠛⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⢛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣖⡒⠶⠶⣤⣀⣾⢁⠀⡀⠀⠀⢉⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡒⠂⠨⣀⡢⢍⡻⣿⣆⠞⡴⢀⠀⠀⠔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⣀⡒⠭⣓⠯⣺⢎⡞⠐⢁⠄⡀⠐⡿⠀⣀⣤⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠀⠀⠬⣕⡪⢝⠊⢳⠀⠀⠀⠈⣠⡾⣡⠞⡙⠀⠁⠐⠀⠋⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣤⣙⢦⣤⣙⡒⠍⠀⠀⢠⡆⡔⠀⣠⠏⡴⠏⠰⢁⠄⡠⢀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠁⠀⠂⠈⢢⡙⢷⣮⡙⠺⣄⡀⢀⢸⡏⡴⡴⢃⠞⠁⠀⠀⢈⠜⡡⢃⢎⡄⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠢⣈⠢⠁⠀⠀⠉⡳⢤⡙⠦⣙⣏⣞⡵⠳⢂⣞⡰⠤⠤⣞⢼⣱⢿⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠐⠬⣑⠮⣑⠀⠀⠠⣑⣊⣑⣉⣓⣻⣿⣿⣖⠻⡛⠉⠀⠀⢤⡲⢝⠮⡓⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣽⣶⣹⢦⢟⢀⠀⠁⠉⠱⢫⠞⢩⠿⡽⢍⡓⢦⣀⢀⠢⡑⢌⠢⡑⢌⠢⠁⢈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⡵⡱⢣⢣⢎⠄⡄⡀⣴⠋⣰⢏⠜⣷⠒⠝⢦⣌⠳⣵⡘⠂⠁⠉⠀⠀⡐⣌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠰⢡⢣⠏⡎⡜⣰⡼⢡⢞⠁⠀⠀⠺⣏⠣⠀⠈⠳⣮⡙⠦⣷⣼⣤⣦⠽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⢀⠁⠀⠈⠀⠀⢠⣟⣵⠟⣩⢖⡄⡠⠀⣐⢀⠀⣀⠒⠤⢉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣴⣠⣥⡷⠞⠉⣾⠅⢈⠔⣫⢞⠔⡱⣯⡢⣕⠢⣉⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠰⡷⠂⠀⢊⠵⣫⢎⢜⣵⢝⠮⡑⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⣿⡥⢀⠁⢊⠔⠁⣾⠿⣧⠁⠀⠀⠀⣈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡟⠀⠀⠀⠀⠀⠘⢶⣥⣔⣡⡴⠞⠁⠀⠈⠛⠶⠾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠰⠯⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⢤⠤⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠛⠝⢟⠻⠋⠙⣷⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⢄⣀⣀⠠⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⡿⢿⣄⣽⡄⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣇⣿⣿⠘⣌⣹⢋⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠙⢻⣿⣷⣤⣤⣾⣿⠋⠋⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⠀⣤⣠⠀⠉⠛⠛⠛⠋⠁⠀⠀⢀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢤⠈⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠐⠀⠀⠦⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢆⡁⠄⠤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠋⠀⠀⢀⣀⣤⣤⣴⣦⣾⡿⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⣾⣿⣿⣿⣿⣶⣶⣋⣽⡿⣾⣿⢷⠀⢀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⣻⣿⣿⣿⣶⡶⣞⣿⣿⣿⣿⣿⣯⡻⠿⣾⣉⡿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣧⣪⣾⣝⣻⢻⣿⣻⣿⣿⣿⣽⣿⣿⣿⣯⣿⣮⣻⣿⣿⣿⣿⡉⠂⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣾⡋⠁⢿⣿⣿⣷⣾⣯⡟⣿⣧⡾⣿⢻⣯⢿⡼⣶⡻⣿⢿⣬⣻⣿⣿⣿⣧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⡍⢀⣾⣿⣿⣿⣿⣯⣷⣟⢥⣽⣽⣞⣿⣾⣿⣯⢾⣿⣾⣿⣿⣻⡽⣻⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⡷⣮⢳⣿⡻⣯⣵⣿⣾⡿⣿⣻⣿⣷⣷⢻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢿⣿⣯⣿⣿⣼⣿⡷⢿⢿⣟⢿⣾⣇⣿⣷⡿⣾⣿⣿⣿⣇⢟⢹⣷⠀⢰⣡⠄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢀⠖⢿⣿⣿⣿⣿⣿⣆⣾⣧⣿⣿⠟⡛⠙⠿⠷⠋⣻⣿⣿⣿⡼⣿⢿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠁⡘⠚⣿⣿⣧⣿⣿⣿⣿⣿⡏⡉⠀⠀⣀⣀⠠⠜⣿⣿⢿⣿⣷⢿⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠦⠀⠈⢇⠰⢡⣤⡽⣿⣾⣿⣿⣿⣿⣿⣤⡛⠀⠋⣉⠀⡀⠀⣯⠛⢟⣿⣿⣼⣷⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧⠠⠋⢿⣯⣞⣿⢮⣿⡟⢿⣇⠀⠀⠀⠀⣽⣝⢃⡾⣻⣿⣿⣇⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡐⣅⢸⠟⢿⣿⣿⣎⢿⣿⣦⣌⣚⡛⠲⣿⣷⣯⣭⣾⣿⣿⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⡅⠠⠂⣿⣿⣿⣦⣛⠿⣿⣿⣷⣿⣷⣯⣻⣿⣿⣿⢟⣼⠿⣦⠡⢽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡀⠟⣸⣿⣿⣿⣻⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⣶⣰⣇⣼⠋⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢦⣠⣾⣿⣿⣿⣿⣯⣮⣭⡿⠿⢿⣿⣟⡿⢯⣷⡟⣯⣭⣿⣿⣿⣿⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣜⣿⠟⢿⣿⡌⠍⢿⣤⣿⣷⣬⣷⣿⣿⣿⣿⣿⣿⠿⡿⠟⠛⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻⣿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⣧⣾⣁⣐⣒⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣋⣻⣿⡿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠿⣿⠿⢿⡏⡟⠋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠘⣧⡄⠀⠄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠷⣿⣿⣿⣿⣿⡿⢋⣩⣴⠖⣦⣬⣙⠻⣿⣿⣿⣿⣿⣇⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣀⣤⣾⣿⣿⣿⠋⣴⣿⣿⡏⣴⣘⣿⣿⣷⡜⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⣿⣿⡏⣸⣅⠰⣞⢰⣷⣶⢲⠆⣠⣷⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⢻⣿⣷⢬⣙⡿⢛⡄⢿⣿⡿⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⡌⢿⣏⣈⣥⣶⣯⣁⢸⡿⢡⣾⣿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⢿⣿⣿⣿⣿⣶⣍⣛⠛⠛⠛⣛⣩⣴⣿⣿⡏⠀⢴⡆⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣶⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠘⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⢀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⡇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣮⣥⣤⣄⣀⣀⣀⣀⣼⡏⠀⠀⢹⣯⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣇⣀⣀⢸⡏⠉⠉⠉⠉⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠠⠁⣀⣠⣂⠀⡤⠃⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢀⠈⢸⡆⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠃⠄⠀⠁⠀⠁⠏⠀⠀⡤⠂⣠⠾⠅⠀⠀⠊⠭⠋⠀⣈⠉⠛⢤⡀⣧⡀⠀⠘⣿⢃
⠀⠀⠀⠀⠀⠀⠀⢠⢦⡟⠀⢁⠞⠁⠀⣢⠎⠀⣂⣴⡤⠊⠁⠀⠈⠈⠁⠀⠀⠀⣸⣿⣿⡧⠀⠙⢿⡄⠑⢀⣿⡈⣞
⠀⠀⠀⠀⠀⠀⡰⠃⡿⠀⢸⠃⣠⡶⢋⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⡄⠘⠀⠉⣷⣾⣿⣧⡿⡄
⠀⠀⠀⠀⠀⣰⠁⢰⡇⠀⣿⣟⠁⣰⣿⠡⠄⠀⠀⠀⠀⠀⠀⠒⠠⠤⣭⡽⠦⠤⡿⣿⣿⣿⣧⠀⠀⠀⣌⣿⣿⣿⡇⠘⡄⠀⡀
⠀⠀⠀⠀⢠⣿⠀⢸⣧⣰⡿⠈⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⢦⡀⠀⠀⣿⣿⣿⡇⠀⢸⣠⠃
⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⢻⡄⠀⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣼⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⠃⠀⡊⣉⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣧⠀⢀⠀⠀⠀⠀⠀⢈⣽⡇⠀⣿⠀⠀⣄⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠿⡿⠶⡷⠶⠶⣤⣬⣾⣿⣸⣿⣶⡶⠿⡿⣿⠟⠋⠉⢙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣧⣤⣤⣴⣷⣤⣧⣦⣀⠠⠬⣿⣿⠛⡲⣗⡂⢠⢠⣿⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣾⣿⠁⠀⠀⠴⠿⠿⢭⣭⡭⠀⠼⠿⠷⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠶⠒⠾⠖⠛

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

⠀⢀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠀⢀⡄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢮⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⢣⡄⠀⠀⠀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⢀⠀⠈⠁⠠⠂⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣀⡴⡼⠁⠀⡠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⢀⠊⢹⣷⠻⡆⠸⣴⠏⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⡄⠰⠏⠀⠀⡀⠀⠀⠀⢀⠀⠀⣿⢷⣿⣿⣱⢿⣾⢷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⢀⠈⠀⠀⡹⣾⣿⡿⠁⣾⢃⠌⣰⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⢀⡀
⠀⢸⣀⡆⠀⡀⠀⠀⠀⢀⠤⡀⠀⠑⠀⠀⠁⢿⣿⡇⢠⠛⡏⣴⠿⠒⠀⣴⠁⠀⠄⢠⠈⠀⠀⠈⠙
⠀⠐⠀⠀⠈⠇⠀⢀⠔⣋⡀⠄⠁⠷⠛⢷⡉⠷⣼⣷⡶⢻⣠⡿⣢⠴⠆⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠖⠈⣩⠖⠈⠀⠀⠀⠀⠀⠀⠚⠳⣦⣴⡬⠛⠛⠿⣵⠄⠀⠀⠛⠓⠚⣁⡴⠆⠰⢤⠤⠄
⠀⠰⡄⢰⠀⡔⠁⠀⠐⣦⡠⡄⠀⠀⠀⠀⠐⠉⣙⡕⠊⠉⠑⣖⠓⣞⡤⡶⠐⣊⠁
⠀⠀⠄⡌⡀⠀⠀⠀⠀⠈⠻⣶⣃⡀⠀⠀⠀⠀⡿⠀⠴⠀⠛⠹⡆⣉⢷⡄⢀⢈⠙⠢⣀⠀⠀⠀⣠⠄⠶⠀⡄⠀⠀⠀⠄
⠀⠰⡎⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⢀⡝⠳⡄⠰⠟⠀⢿⣸⠎⠷⠆⢘⣋⡄⢹⠗⠂⠉⠐⠖⠒⠈⠀⢠⠾⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠉⠒⣰⣷⡶⠟⠊⣠⣔⣻⣧⢈⣁⣙⣧⠳⣄⢦⣄⠠⠀⠉⣸⢇⣀⡀
⠀⢰⠆⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣼⠆⣠⣤⡀⢠⠞⢢⣴⢇⣼⠿⠿⡄⣉⡻⣹⡆⠈⠴⣉⣦⡰⠞⠁⠀⠈⢉⠆
⠀⢨⡂⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠴⠇⠀⠳⢶⣢⣶⣶⣿⣷⠄⣀⠿⢿⠙⢦⣛⡶⡘⠗⣤⠸⣄⠀⠰⠆⠈
⠀⢠⠍⠀⠀⠈⠷⢀⠀⠀⠀⠀⠀⠈⢉⠅⡲⣓⢛⠶⠹⠃⠌⢹⡄⣀⠙⠶⡫⠄⠰⢇⠠⡏⢳⠚⣤⠀⠙⢿⡓⠳⣄
⠀⢠⠀⣆⠐⠀⣄⠀⠙⠙⠢⠄⣠⣖⣁⡴⢉⠋⡽⢛⢄⣼⢳⣷⡾⠙⠥⠀⡿⣤⡚⠁⠀⠿⠀⠃⢻⡉⠩⣈⠣⢤⣁
⠀⠈⠀⠉⠠⡀⠈⠀⠀⢀⠀⠉⠉⣠⠞⣴⠋⣼⠳⠟⣘⡣⠙⠁⢀⡐⠂⠤⠭⢙⣡⡴⠂⠀⠀⠙⢮⣹⠶⠺⢿⠂⣉⠓⡀
⠀⠀⢀⠀⠰⠁⠀⠀⠴⣿⠀⠀⠀⠀⠘⠀⠢⠙⠒⠎⠁⠴⠣⠾⣦⡀⠀⠀⠀⠈⠳⠶⢞⡉⣤⡄⡀⠈⢷⡒⠦⠀⢉⡜
⠀⠀⠈⠂⠀⠀⠀⠀⢠⣄⠙⠆⣰⡆⠀⠂⠀⠀⠈⠀⠀⠀⢰⡞⢻⣿⡧⠀⠀⠀⠀⠀⠈⡡⠟⠐⣻⣿⡀⠘⢦⠁⠈⠓
⠀⠈⠀⢀⠄⢰⡆⠀⠰⣿⠀⠘⠉⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠⡿⢇⣈⣿⣀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠏⠿⣄⠈⠼
⠀⠀⠀⠈⢀⣈⠹⠀⡀⠁⣾⠀⠀⠃⠐⠀⠀⠀⠃⠀⠀⠀⢷⣄⣤⠙⣟⠻⡶⣦⣤⣄⣰⡁⠀⢸⣿⣿⠛⡇⢠⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠟⠀⢠⡁⠢⠘⠄⠀⠀⠐⣦⠠⡀⠀⢀⠀⠀⠀⠀⢠⡁⣘⣇⠠⡏⡹⢏⡿⢻⢦⡏⠀⠀⠀⡀
⠀⠀⠁⢀⡈⠀⣀⣾⠑⠊⠀⠀⠀⠀⠈⠈⡠⡏⠀⠘⠀⠀⠀⠀⠀⣉⣹⠟⢻⡷⠇⠈⠱⡟⢬⣿⣶⠟⣿⣀
⠀⠀⠀⠸⠓⠁⠙⢉⠀⠀⠠⢦⠀⢠⡀⠄⠡⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⣤⣼⢁⢦⡀⠆⢻⡆⣿⠻⣿⣽⣯⣶⡖⠀⠠
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⣇⠄⠻⢠⠀⢈⡇⠀⠐⠀⠀⠀⠀⠀⠀⠤⠜⣡⡜⠀⠈⠀⡌⢛⣦⡁⠈⠉⢻⡎⣠⠀⠇
⠀⠀⠀⠀⢳⡤⠤⣄⡈⠀⠀⠈⠀⡀⣦⠈⢀⣊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠁⠻⡄⠀⠀⠸⡇⠙⠻⣦⣡⣤⢹⡗⠀⡀
⠀⠠⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠞⠉⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠙⠄⠀⠀⣷⢀⡀⠟⣀⠀⠀⠀⠀⠁
⠀⢰⣤⠐⠒⣋⠟⣚⣡⡾⠲⣟⣵⠞⣍⣓⣶⣤⣼⡵⠦⠠⠀⢠⠀⠠⠄⠠⠤⠆⠤⠤⠔⠉⣉⣠⣾⠷⢦⠟⠷⣦⡄
⠀⢸⡟⠛⠛⠟⠛⠉⠏⠑⠛⠋⠉⠀⡁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠙⠋⠴⠁
⠀⠸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀⡀⢘⡀⢀⠐⠀⡀
⠀⢠⡀⡀⢀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠁⠁⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣋⣴⣿⣿⣿⣿⣿⢟⣿⣯⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣗⣿⣽⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣒⣸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⣟⠋⡽⠋⠉⢻⣿⡿⠿⣿⡿⠛⣿⣿⣿⣿⣿⡟⣭⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣓⣽⣿⣿⣿⣿⣇⠀⠀⣼⡿⠁⠀⢈⡇⢀⣼⡷⠿⢿⣿⣿⣶⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠋⠈⠙⣿⣷⣾⣿⣷⣿⣿⠿⠿⠿⠋⢀⡴⢊⣽⣿⣿⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⣠⠴⠚⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠋⠉⠉⠁⢿⣿⣿⣆⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡳⠀⠀⠇⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠀⠀⠁⠐⢀⠉⠃⠀⠉⣉⡀⠀⠀⠀⠐⠈⠠⢶⡷⠆⠀⠀⠉⠁⠉⠉⠏⠉⠛⠳⣶⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡎⡀
⠀⠀⠀⠁⠀⢀⡤⠒⠩⠀⠀⠀⠒⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⡅
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⠀⠈⢮⣐⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠘⠁⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡤⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠖⢻⠇⠀⠀⠀⠐⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠋⡁
⠀⠀⠀⠀⠤⣾⠁⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣓
⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⠋⠀⠀⠁⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⣉
⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣹⡄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢉⠁
⠀⠀⠀⢠⠼⠀⠀⠀⠀⢀⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠐⣀⣀⡀⠀⠀⢀⡞⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠄⠰⡇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⣤⣶⠟⠋⠉⠉⡙⠛⣦⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢈⣭⣶⣶⣦⣤⣄⠁⠀⠀⠀⠀⢀⡇⠀⠀⣀⢀⣰⠏⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢧⠀⠠⠴⠀⠐⠏⠋⣈⡠⠟⠋⠀⠀⠀⠀⣼⠀⠀⢐⣭⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠷⣼⣶⡄⠀⣀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⣠⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣾⡇⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⡿⣽⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡟⢇⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣅⣅⠙⢃⠰⡄⡄⠀⠀⡀⢠⣾⡏⠀⣁⣼⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣒⣀⠡⢘⣄⣠⠁⢃⣙⡀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠁⠈⠛⠲⠈⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣦⡭⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢀⠾⠒⠑⠀⠙⠳⣦⡀
⣿⣿⣿⣿⢟⣏⣴⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢐⣴⡟⠃⠀⠀⠀⠀⠀⠙⣏⡠
⣿⣿⣿⡿⣻⣽⢿⡿⣿⣿⣿⣿⣿⣃⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣬⠼⣠⠀⡀⠀⠀⠀⠀⢠⣿⢰
⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⠟⣻⣿⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠁⡂⢘⣔⢀⡀⠀⢀⣤⠊⢜⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠌⠘⠈⠀⡯⠥
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠥⢌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠂⠐⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⢀⠀⠐⠀⠀⢠⢀⠀⠀⠀⠀⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢟⣻⣙⣯⣍⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠋⠃⠀⠘⠛⣛⠛⠋⠀⠀⠀⣠⣄⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣷⠄⣀⣴⣾⠿⠿⠿⠿⠿⣷⣦⣈⠻⣿⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣯⣿⣿⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣊⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠚⣿⣿⣿⣻⣿⢯⡍⢉⣉⡉⠛⠒⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⣿⣿⣷⣺⣇⣻⡆⣏⠀⣹⢰⠟⠳⡄⣠⣌⡉⠓⠶⢤⣀⡀⠀⣿⣯⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⡿⠛⣿⣿⣷⣤⡤⣌⣉⣁⡘⠦⠾⢡⡏⣀⡗⠀⢠⠆⠈⠙⠳⢿⣟⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣋⡀⠙⣿⠁⢠⣿⣟⡺⠿⢠⣿⡏⠉⠉⠛⠓⠦⣌⣉⡀⠠⢯⡀⢠⡟⢀⡆⢈⡙⠧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡴⠋⠀⢸⣿⡒⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣘⠲⠟⣴⠋⠙⠆⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣾⣿⢭⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣌⡓⠃⡾⣷⠒⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠧⣿⡟⢳⣤⣙⠀⠀⣰⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢾⣿⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⢠⣿⠃⠙⢳⡞⠁⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⢣⣅⠀⣠⣿⠃⠀⢀⣾⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣲⣿⠏⣴⣿⠁⠀⠀⢸⡟⣫⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠍⠈⠻⢃⣾⡿⠁⠀⠀⠀⠀⣿⠋⠉⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣯⡀⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⣛⣫⣭⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠯⣥⣶⠿⠟⠛⣽⡛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣛⡯⣙⠶⠽⠓⢛⠫⠥⢬⣖⣚⣩⠽⠴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠉⣉⣀⣤⡤⠤⠶⠒⠚⣛⣉⣩⠥⠴⠒⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠨⠉⠉⠀⢠⠴⠒⠒⠚⠉⠉⠁⠀⠀⠀⢀⠀⠈⠂⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡼⠀⠠⡄⠰⢠⣤⣠⡄⠆⠰⠃⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⢀⢠⠚⠁⠀⠤⣀⡶⢤⣽⣯⡻⠄⡇⠠⠖⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡀⡤⢀⣰⣦⠀⠀⠠⠾⡃⠠⡴⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠄⠀⠀⠀⠀⣫⠴⠞⢿⡓⠄⠈⠂⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠂⠀⠀⠁⠞⢳⣘⢋⣸⠃⢐⣀⣽⡧⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠁⠀⠁⠵⠻⣶⣭⣾⠵⠶⠿⣜⠝⠶⠀⠀⡃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⠀⠖⠁⠀⣾⣴⣶⢿⡒⠻⡏⠑⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⠀⠴⢁⡔⣹⠿⣿⣟⡻⡇⠳⠀⢀⡤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠈⣀⠀⠀⠀⠈⠀⠉⠀⠈⠙⠀⠀⠀⣀⡎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⡆⢀⣀⣀⡤⠤⠖⠒⢒⠛⠉⠉⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣀⣀⠬⠭⠐⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡄⢸⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠁⠂⠑⠈
⠀⠀⠀⠀⠱⢵⣰⣄⣀⡀⢐⣁⣹⠀⢀⣀⣴⣰⠵⠅⠀⠀⠀⠀⠀⠀⠁⠘⠀⡔⠁⠀⠈⡆⠀⠀⢀⠄⢸⠀⠀⡄⡠⡀
⠀⠀⠐⠀⠀⡐⠥⡝⢹⠃⠆⢀⠀⠦⢻⠝⡥⠕⡀⠀⠐⠀⠀⠀⡀⠁⠀⠐⠎⡢⠀⠀⡰⠀⠀⠀⠸⠺⠂⠸⠇⠀⡀⠄
⠀⠀⠀⠀⡧⠀⠀⠈⠒⢻⠀⠼⡀⢸⠓⠊⠀⠀⠂⡅⠀⠀⠀⠘⢀⠀⠀⠀⣯⣆⢡⠎⠀⠀⣀⠀⣰⠺⠀⠀⠁⠀⡃⢄⣀⣀
⠀⠀⠀⠀⠈⢶⠮⢒⡤⠊⢂⠛⢃⠈⠣⡔⠢⢶⠎⠀⠀⠀⠀⠀⠀⠀⠰⠀⣉⡁⠰⠀⠀⠀⠡⡀⠉⠀⠀⢀⢀⠀⠑⠉⢀⠌
⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠐⠂⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠈⢄⠘⠀⢠⠍⡡⠃⠀⠀⠀⠀⠱⡀⢠⠚⢳⡖⣛⠆⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠉⠀⠁⢠⠊
⠀⠀⠀⠀⣠⠥⠺⢤⡐⣀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣴⠀⣀⠀⠀⠀⠀⠑⠂⠐⠊
⠀⠀⠀⣻⠁⣒⡥⠒⠛⢠⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢀⠺⣤⠚⢵⣼⡆
⠀⠀⠀⠋⠍⠁⠈⠀⠐⢸⠀⠀⠀⠰⠄⠀⠀⣀⣀⡀⠀⠀⠶⠀⠀⠀⡄⠘⣜⠀⡜⣀⠅⡂⠀⠀⡐⠄⢡⠤⢤⣊⡠
⠀⠀⠀⠀⠚⠀⠀⠀⠃⡘⠀⠀⠀⠦⠀⡔⢉⠄⠂⢌⠑⡄⠠⠆⠀⠀⢃⠀⠈⠉⠉⠁⡐⠀⠀⠀⠀⠀⢠⣘⣣⠀⠃
⠀⠀⠀⠈⠢⠄⢉⠠⠊⠀⠀⠀⠀⠀⠚⠂⠒⠀⠀⠐⠂⠚⠒⠀⠀⠀⠀⠑⠤⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⡇⠀⠀⢸⠀⠀⢸⠀⠀⣰⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠘⡆⠀⡄⢡⠀⢀⠈⢰⠀⡆⠰⠀⠇⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠘⢆⠀⢦⠐⠀⠱⠈⠀⠸⠀⠸⠀⠁⠇⣠⣤⠚⠠⠃⢀⠀⣠⠎
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⡀⠑⣄⠂⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡟⢠⡇⠁⡰⢃⠜⠁⡀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠑⠦⣀⠈⠢⡈⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⣀⣿⡷⠛⠀⠈⠀⠀⡀⠊⡠⢚⡁⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⢌⡑⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡇⠀⠀⠀⠀⢰⠏⠀⡠⢸⡟⠠⠖⠋
⠀⠀⠀⠀⠀⠉⠓⠢⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣇⠀⠀⠀⣠⠟⠀⠉⠀⢸⡇⡠⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡞⢹⣿⢻⠟⢦⣠⠼⠋⠀⠀⠀⠂⢸⡇⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣤⠀⠀⢠⣶⣤⠞⣹⣿⠀⣿⠘⠻⠟⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣿⡄⠀⣾⣿⠈⡇⠸⢿⠴⠋⠀⠀⠀⢸⣧⠀⣀⠀⠀⠀⠀⠀⢸⠇⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⣿⡇⢰⣿⣷⣼⢿⣿⣰⡇⠀⠀⠀⠀⢀⣠⡀⢸⣿⣼⣿⠃⠀⠀⠀⡆⢠⡀⠠⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠈⢿⣧⡞⠘⣿⣇⠀⢬⡉⠀⠀⠀⠀⠀⣾⡟⠷⣸⣿⣿⡃⠀⠀⠀⢸⡇⠘⠃⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⣿⠀⢸⣷⠀⣶⡄⢰⡀⣿⡇⢀⡟⣿⡏⢿⡄⠀⠀⡾⠠⢄⣀⠁⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⡏⢸⣿⡆⢸⣿⠀⢿⣇⢸⡇⣼⠛⠊⠀⠈⠋⠀⠻⣆⣼⢃⠀⠀⠌⣉
⠀⠀⠀⠀⠀⠀⣀⡤⠒⡡⠖⠁⡾⢀⠈⣿⡇⠸⣿⡇⣼⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠈⠙⠲⠄
⠀⠀⠀⠀⠀⠊⠁⠀⠈⢀⡤⢰⡇⠀⢠⣿⠁⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡑⢦⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢸⡇⠆⣼⠃⠀⢠⠈⠛⠁⠀⠀⠀⠀⢀⠀⠀⢠⠀⢀⠱⡄⠘⢦⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠛⠚⠁⡰⢀⠇⠀⠀⡆⠀⠀⡆⠀⠀⡆⠀⡄⢧⠀⢣⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣰⠁⠀⠀⡜⠀⠁⢸⠀⠃⠀⡇⠁⠀⢱⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⢷⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠈⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢤⠄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠹⠉⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⣶⣦⢰⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⣝⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠙⢻⣿⣶⣶⣿⠏⠋⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠈⠉⠉⠀⠀⠀⠴⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋⠉⠀⠄⠀⡄⠀⠀⠀⠈⡀⠉⡀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡾⠿⠿⠟⠛⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⠶⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢹⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠉⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢻⣿⣦⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠄⠀⠀⣸⣿⠛⡉⢩⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠂⠠⠤⠮⠂⠀⢀⣀⡾⣫⣿⠀⠁⣈⣰
⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⢀⠀⠀⠀⣀⠄⠀⠀⢰⣻⡿⣻⡿⡿⠁⢂⣬
⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⣘⠤⠒⠉⠀⢀⣀⣠⠤⠝⢋⡩⠂⠀⠀⢿⠃⠀⠉⠑⠢⢄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⠔⠒⠋⠉⣀⣠⣴⣶⡿⠁⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣧⠀⠈⠐⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠉⠉⠉⠙⡿⠷⡀⠐⠦⣈⠐⠠⡀⠀⠀⠘⡄
⠀⢀⣿⣿⣿⣿⡿⠁⠀⠈⠻⣿⣿⣿⣿⣿⠟⠉⠀⣼⣶⣤⣤⠞⠀⠀⠙⠦⣀⡈⠙⠒⠬⣐⠄⠀⢸⣠⡴⠖
⠀⢸⡿⠿⡿⠟⠁⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀⡈⠉⠐⢠⡀⡠⠜⠚⡅
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠤⡤⠙⠙⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⠍⠀⠀⠀⠀⢀⡀⠀⠀⢠⠔⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢉⡻⠿⠒⠒⠚⢋⡝⠁⠀⢀⠔⠁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⣣⡤⠖⠋⠁⠀⠀⠀⠀⢠⠎⠀⣀⠔⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⣀⣀⣠⡴⠖⠚⣋⣠⣬⠷⠿⠀⠀⢄⠀⠀⠀⠠⢾⠿⠃⡠⠊⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡭⠩⠞⡥⠖⠋⣸⣿⠟⠁⠀⠀⠀⠀⠀⠉⠢⠀⢰⠅⡡⠈⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠴⣚⠯⢀⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢴⠗⢀⠜⠁⠒⠒⠒⠒⠉⠉
⠀⠀⠐⠛⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢧⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⣟⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⢠⠎⣠⠞⣶⠑⠦⠤⠬⠭⠽⡶⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⢟⣵⠟⣮⣟⣛⣛⣒⣾⣍⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡴⣡⠞⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢏⢞⣵⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠼⠧⠧⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⣴⣶⣶⣶⡄⠉⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣰⡀⠻⠿⠿⠟⠃⠸⠂⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⢿⠟⠀⠀⠀⠰⠆⠀⠀⠘⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⣿⠟⠀⠃⠀⠈⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠻⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⢰⠄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠘⠋⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠺⣿⠗⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣭⣥⣤⣶⠾⠃⠀⠀⠀⠈⠻⠿⠋⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠒⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢠⡾⠛⣷⠀⠁⠀⠀⢠⣿⠇⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⣀⠀⢀⣾⣿⣴⣾⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣿⠁⢀⣠⡀⠀⠀⣿⣇⣼⠁⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠤⠀⠀⠀⠀⢀⣿⡏⠀⣠⡿⠃⢠⡟⢁⡟⠀⠀⢻⣿⡁⠀⢀⡴⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⡾⠛⠁⢀⣿⣡⠞⠁⠀⢠⡾⣿⣧⣴⠏⠀⣿⣿⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠙⢳⣄⣀⣼⡿⠁⠀⢀⣴⠋⢀⣿⣿⠁⠀⢸⣿⡟⠀⠰⠂⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠙⠿⣷⣦⣤⠆⣸⣿⠇⠀⠀⣿⠁⢸⣧⣀⣴⠟⠁⣠⡿⢻⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⢠⡿⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⠀⠀⠙⢋⣁⠀⣴⠏⠀⣼⣿⠀⠀⢿⣿⡇⠀⠀⠀⣴⡿⠁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠀⠀⣼⠇⠀⠀⢀⣾⠏⣼⠃⠀⢀⣿⡏⠀⠀⠘⣿⣧⣤⣴⣾⠟⠀⠀⠉⠴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⢻⣿⣀⣼⠟⠀⡤⣄⡼⠁⢰⡏⠀⢀⣾⠟⠀⠀⡀⠀⠈⠙⢛⣿⠃⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠛⠛⠉⠀⣼⣳⢯⣧⠀⠈⠷⠶⠟⠉⠀⠀⠀⠁⠀⠀⠀⣼⡏⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠀⠀⠀⢰⡏⠀⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢰⣿⠁⣠⠏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⡄⠀⠀⢀⣿⠇⠀⠀⠀⢿⠃⠀⠀⠀⠉⠀⠀⠀⢠⣾⠟⠙⢃⣿⡏⣰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣿⠀⢠⣿⠋⠀⠀⣹⡛⢳⡻⠟⠁⠀⢰⡆⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠾⠇⠀⢀⡾⠀⢠⣿⡇⢀⣾⡏⠀⣠⣾⣻⣻⡛⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣾⠇⢠⢿⣿⢀⣾⣿⠁⣴⠏⢸⣿⡟⢿⣦⣀⠀⠀⢀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣰⣿⣠⡟⢸⣿⡾⠁⠻⠟⠁⠀⢸⣿⠁⠀⠙⠻⠿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢀⣤⣶⠿⠛⣿⠟⠀⠈⠛⠁⠀⠐⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣾⠿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⡿⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠚⡿⢰⡤⣦⠸⡓⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠈⠀⠀⠀⣀⣀⡐⢾⠦⢴⣅⠀⣄⠙⢿⠰⣸⠞⢁⡀⢠⣷⠴⣿⠖⣀⣀⠀⠤⠀⠁⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠣⢠⡤⠐⠚⠐⠊⣕⣤⣔⡀⠈⠁⠻⡄⢻⡷⣹⠃⡼⠋⡁⠀⢲⣤⣔⡍⠂⠐⠒⢠⣤⠼⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠂⠶⠄⠀⠈⠑⠂⠛⣮⠿⢷⣾⣿⠈⣹⣿⣶⠿⣭⠟⠂⠒⠉⠀⠠⠶⠂⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠨⠖⣺⡿⣿⡄⣿⢿⣯⡒⢄⠀⠀⠀⠀⡀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡘⢀⣴⠂⠀⠀⠀⠀⡽⢠⢹⣷⡿⠥⠹⡄⠀⠀⠀⠀⣾⣀⠘⣤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢈⡻⠁⠁⠄⠀⠀⠀⠁⠀⡾⢻⢻⠆⠀⠃⠀⠀⢀⠌⠁⠹⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⣠⠤⢀⠀⡀⠘⡇⠀⡀⠀⠀⠙⣠⠘⠀⠁⠀⡀⠨⠃⠀⡀⢠⠠⣤⣠⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡻⡧⠘⣾⡣⠒⠄⣀⠐⠀⠉⢲⣀⢤⡭⡥⢆⣴⠉⠁⠰⠀⠀⠕⢦⡻⠊⠰⣿⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣗⠡⠈⠈⣡⠀⣀⠂⠐⠀⠀⠀⢠⡎⢻⠛⡄⠀⠀⢀⠐⠐⢃⠀⢀⡉⠈⡰⢱⡷⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡈⢹⣶⠳⠄⠨⢦⠙⣇⣰⢠⠌⡑⢳⢤⠟⢦⡔⢀⡀⣤⢰⣀⡎⣴⢮⠠⠞⣳⡟⢁⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠶⣌⡓⢼⣦⡀⠀⢿⡻⣾⠓⠙⠋⠀⠀⠀⠈⠻⠗⢻⡾⣻⠆⢀⣀⣦⠘⣣⡴⠿⠧⠞
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠈⠩⡈⢁⠙⠳⣌⣛⠻⠆⢃⡤⣾⣋⣲⠦⣀⠋⠟⢙⣡⠴⠛⠁⠉⡄⠁⠐⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠂⠀⠌⢀⡀⠠⢙⣧⠴⣿⣷⣾⣷⣿⣾⣿⡶⣴⣟⠁⠀⠀⠀⠀⠐⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⠚⠓⠒⠀⣀⣀⣦⣶⡿⠞⠛⢻⣿⣿⡀⣿⣿⣿⠛⠓⢿⣷⣴⣀⣀⣀⡒⠘⠓⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⠾⠛⠛⠛⠛⠛⠛⡻⠛⢷⣄⡀⠀⢍⢻⣿⣽⢻⠀⢀⣠⣴⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠁⠀⠐⠂⠀⠂⡸⠁⠈⠅⣸⡿⢯⣴⡿⣿⡻⣦⣽⢿⣯⡠⠍⠀⢳⡐⠀⠀⠂⠀⠉⠉⠙⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⡤⠀⠠⠘⠛⠆⠛⢁⠜⣡⣾⠋⣀⡞⠳⠾⣿⡷⠶⢹⣎⠈⢻⣦⡁⢀⠙⠢⠚⠃⠠⠀⠠⡄⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠄⣠⠂⢀⠀⢰⡟⠮⠀⠸⠁⠀⠞⠛⢛⠂⠊⠙⠀⢹⠞⣷⠘⣤⠀⢦⡀⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⣿⡄⢃⣠⠀⠀⠀⠀⣲⡁⠀⠀⠀⢠⣜⠃⣿⡀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢷⣤⣭⠭⢤⠀⠆⣿⠃⠀⣤⠼⢭⣥⣾⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠾⠟⠛⠛⠀⠀⠀⠁⠀⠘⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⢟⠛⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣞⣩⣷⣶⣦⣴⣶⣶⣦⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⠀⠤
⠀⠘⠻⠇⠀⠀⠂⠂⠀⠀⠀⢠⣶⡄⠀⠀⠀⢠⣠⣠⠀⢠⠀⠉⠟⠓⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⡀⡀
⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⢠⡈⠉⠹⠂⡄⠀⠀⠀⠀⣀⠀⠀⠀⠈⠋⠉⠀⣀⠉⠀⠉
⠀⠀⠉⠀⠉⠈⠉⢀⠒⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⡏⠠⠖⠀⡠⠉⠀⠀⠀⠀⠀⢠⠀⠏⠀⠀⠉⠁
⠀⠀⠐⠀⠰⢀⠄⠸⡇⠰⢩⣿⣷⡄⠀⠀⠠⠄⠤⠤⠸⠂⠀⠀⢸⠯⡋⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠁⠘⠃⢠⡈⠻⡿⠃⠊⠅⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⢀⠀⠀⠀⠠⠂⠲⠀⠈⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠁⠰⠄⠞⠁⠀⠀⠀⠘
⠀⠀⠘⠀⠀⠀⠀⠀⡦⠀⠀⠀⠂⠀⡀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠂⠀⠈⠀⠀⠈⡛⠛⡉⠃⠈⠉⠈⠉⠁⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⢀⠀⠈⠁⠀⠀⠀⢸⣶⣾⡇⠀⠀⠀⢈⠀⠀⠀⠀⠀⡇⠀⠐⠒⠒⠂
⠀⠀⠸⡇⠀⠀⠀⠀⠈⣿⣿⣿⠀⠈⠀⠀⡀⠀⠈⢿⠏⠀⠀⠤⠻⠈⠀⠀⣐⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀
⣿⣯⣭⣭⣭⣽⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⡥⠒⠉⠀⢀⡤⢲⠎⢀⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⣟⠀⠀⠙⣆⠠⡇⡇⢀⡔⠁
⠀⠀⠀⠀⡀⠀⣸⠃⢀⠀⢰⠁⢀⣀⣰⣋⣠⣤⡴⠧⠒⠀⠀⢀⡠⠤⠒⣀⠀⠀⠐⠂⢀⡂⠄⠀⡀⠀⠀⣿⠀⠀⠀⠘⣿⣷⢷⣞⡀⠀⢀
⠀⢘⡐⢰⠃⡰⠃⢀⠈⡠⠃⠀⠘⠁⡝⠁⠀⢀⣄⠤⠀⢀⡴⣋⠀⠀⠀⠰⠋⣩⠟⠉⠉⠉⠉⠛⠧⣄⠀⢸⡇⠀⠀⠀⠸⣿⡾⡁⠀⣠⠋
⠀⢸⡇⣿⠞⠁⠶⢉⡼⠀⠀⣈⠀⡼⠁⣄⠐⣠⠃⣀⡶⠫⠤⠅⡦⠤⠖⢊⡉⠀⠀⠀⣰⣷⣶⣶⡊⠈⠳⣌⣇⠓⠠⣄⠈⣿⡇⢧⡴⠁
⢠⡞⣸⡏⠀⣀⡰⠋⠀⠀⣀⡵⠊⠁⣀⣭⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⠃⠀⠀⡈⠻⣦⣀⠉⢠⣿⣧⠘⡇
⠏⢰⡿⠀⠀⣽⠁⢀⣴⠟⢉⣴⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠈⠉⠐⠀⣹⣷⣿⣿⣿⣆⡿⣄
⠀⢸⡇⠀⢰⣿⡾⠋⠀⣰⣿⢋⣤⡁⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣉⣉⣉⣙⣳⣤⣀⣘⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡈⢹⣿⣿⣿⣿⠇⠙⣆
⠀⢸⡇⠀⣼⣿⠱⠆⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣤⠤⠤⠤⠞⠋⠉⠁⠈⠉⠉⠛⠻⣿⣿⣿⡀⠀⠀⠀⠉⠹⣿⣿⣿⣿⠀⠀⠸
⠀⢸⣿⣰⣿⡏⢀⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠙⢶⣄⠀⠀⢠⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⣿⣿⣿⣿⡆⠀⣸
⠀⢸⣿⣿⣿⣿⡏⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣇⣰⡟
⠀⠈⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⡄⠀⣿⣿⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡀
⢷⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠍⢸⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣃⡤⠾⡟⣻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⠀⠀⠀⠀⢀⡞⠀⠀⡇⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢛⣻⣧⠀⢰⣯⠄⠀⠀⡞⣼⠃⣠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣧⣤⣦⣤⣀⣀⠀⢀⢒⣻⣷⡒⣶⣻⣧⣤⣤⣿⣿⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⠟⠉⢹⣿⡏⣿⣿⡃⠀⠈⠉⠉⡿⡉⠉⡏⢹⠛⠛⠻⠷⠭⣽⣯⣷⣿⠿⣿⠟⠛⠛⡛⢻⡿⠁⠀⠀⠀⣠⣿⣿⠿⠛⠛⠛⠻⠿⠿
⠛⠉⠀⠀⠀⠀⢿⣷⣜⢿⣷⣦⣤⣤⣴⣷⣷⣶⣷⣾⣤⣀⠀⠶⠶⣿⣿⡛⢛⡛⣟⣋⡁⢰⡇⣼⣇⠀⠐⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠀⠀⣿⣷⣾⣿⡏⠀⠈⠉⣉⣽⣿⣿⣝⣋⣛⣏⡁⢀⣼⣷⢿⣿⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡆⠀⣴⡏
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⢸⣿⠃⠀⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⢠⠿⠟⠉⣿⡜⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣷⣦⡀
⠀⠀⠀⠀⢀⣤⣶⢷⣄⢻⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⠀⢲⣾⣶⣄
⠀⠀⠀⠀⠾⠛⠁⠘⢿⣦⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡿⢡⣿⣶⣤⡀
⠀⠀⠀⠀⢀⣴⠶⣶⣤⡙⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⢸⡿⠈⣉⠓
⠀⠀⠀⠀⣸⠟⣀⡀⠉⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠃⣸⡟⠻⣷⣄
⠀⠀⠀⠀⠁⢸⣿⠻⢷⣆⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠃⠁⠀⠀⠀⠀⠀⣿⠃⣤⡀
⠀⠀⠀⠀⠀⠈⡿⠀⣦⣌⠀⠀⠀⢠⠄⠀⠀⠈⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⣴⣦⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣾⠟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⢴⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣤⠀⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣄⣀⠰⢶⣦⣄⡈⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣻⣿⣦⡈⠻⣿⡦⣸⣿⠃⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⢁⣴⣿⠇⣼⡿⠃⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠰⠄⣼⠛⢛⢶⠲⠶⠶⠤⠤⢤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⡄⢀⡾⠛⢦⡝⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⠻⡇⢬⠉⠓⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⠰⠆⣠⠶⠋⠀⠀⢨⣿⠇⠀⠀⠀⢀⢠⠀⡀⠀⣚⠃⠀⠀⠀⠀⠀⢻⡄⡀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣩⡤⠖⠋⠁⠀⠀⠀⠀⠘⠸⣇⠀⠀⠀⠀⡄⢠⠁⣸⠉⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠀⠀⠀⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢰⠀⠀⠀⠀⠀⠀⢀⢤⡄⡿⠒⠂⠀⠛⠷⡮⢀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠤⢰⣿⣾⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠆⠀⠀⠀⠀⣠⠞⠁⠀⢀⡧⠀⠀⠀⠀⠀⠀⣨⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠦⣼⣿⡟⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⠀⠀⠀⢀⡼⠃⠀⠀⠀⡮⠀⠀⢂⠀⠀⠀⣨⡇⠀⣰⠛⣄⠀⠀⠀⠀⠀⠀⢸⢱⡟⣸⣿⣿⢢⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠄⠀⠀⠀⣜⠀⠀⢀⣀⠚⠃⠀⠀⠀⠱⢆⠀⣍⠀⡴⠃⠀⠈⠄⠀⠀⠀⠀⠀⠈⠸⢣⣿⣿⣿⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⠃⢠⡤⠋⠀⢀⣤⠀⠀⠀⠀⠀⠀⢇⡜⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡀⣼⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⣀⠠⠚⠋⢁⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⢐⠎⠑⢄⠀⠀⠀⠐⠀⠀⠀⠀⡐⢠⢠⣿⡿⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠾⠛⠋⠁⠀⠀⢳⠁⠀⠤⠄⠀⢠⡄⠀⠀⣀⣀⣞⡃⠀⠀⠳⡄⠀⠀⠀⢀⣴⡟⠛⠛⡟⣿⣱⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣄⢀⣄⣠⣏⣸⠉⠁⠀⠒⠚⡶⠀⠀⣸⣿⢠⣤⢠⣷⢀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⡦⠀⠀⠀⠀⠀⢀⡨⠷⡔⠒⠛⠁⢩⠁⢡⠏⠀⣿⠀⠀⠀⣠⠞⠁⠀⠀⣿⡟⣾⡟⢸⡏⡀⡾⢈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣇⡿⢰⡇⢀⣀⡶⠜⠋⠁⠀⠀⠈⠐⠠⠀⠘⣤⠋⢀⣠⣻⣶⠔⠋⠀⠀⠀⠀⠀⢻⣧⣏⣁⠻⠃⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠓⠻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠸⣆⠀⠀⠀⠀⠀⣠⣶⣽⣿⢹⣿⣦⡐⢦⡾
⠀⠀⠀⠀⠀⠀⢀⣿⡟⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⢿⣄⡠⠄⠀⠚⢹⠋⡇⡄⣾⣿⣿⣿⡾⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣀⣤⡶⠚⠃⠁⠀⠀⠀⠈⠠⢰⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢴⠒⠋⢹⡉⡏⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣼⠻⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠹⣆⠀⠀⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡏⠈⠁⣈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠹⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠃⠂⣀⣤⢤⣼⡟⣿⢿⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢤⢳⡀⠀⠀⠀⠀⣤⡞⠁⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠘⠓⢸⣾⣇⡏⣿
⠀⠀⠀⠀⣾⣀⣀⠀⠀⠀⢤⢳⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢸⣿⣸⣿⡇
⠀⠀⠀⠀⠈⠾⣝⡿⣿⣴⣄⣨⣶⣶⣿⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⢱⡀⠀⠀⡔⢰⠄⠀⠀⠀⠀⠀⣸⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⣿⣿⣿⣟⣛⣛⣛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣷⣶⣾⣜⣓⣶⣶⣶⣦⠤⣤⣿⢿⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠫⠟⠋⠛⠛⠛⠒⠲⠶⠶⠴⠦⢤⣼⣛⣛⣋⣙⠛⠛⠻⠿⠿⠟⣿⣿⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠀⡠⢀⠤⠀⣀⠄⣢⠄⢀⡠⠂⡾⠁⠀⠀⣼⠁⣠⠄⠀⣠⠄⢀⡤⡰⠄⣰⣏⠤⣶⠄
⠀⠀⠀⠀⠀⠻⢷⣤⣀⠠⢴⡞⠁⢾⡤⠜⠓⠸⠧⠤⠛⠂⠻⠖⠈⠃⠀⠀⣰⠇⠐⠿⠖⠈⠻⠄⠺⠧⠄⠸⠋⠐⠮⠅
⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⠤⠒⠚⠛⠓⠒⠶⢤⣄⡀⠀⢀⣾⠋
⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠿⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠈⠉⠛⠓⠲⠶⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠤⠀⠀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠎⠶⠂⠈⣠⢢⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣆⡄⢟⣁⢠⡱⠄⠀⠀⡀⠀⠀⠀⠴⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠁⠰⠀⡈⠠⠀⣄⠀⡙⢉⡄⢚⠁⠈⠀⠊⠈⡄⢀⠌⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⠀⠀⠀⠂⠁⠈⢀⠀⢩⡈⠓⣘⠣⠤⠂⠸⠀⠐⠂⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠧⠢⠁⠀⠸⠧⡃⠀⠠⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⢠⡖⠰⠶⠀⠁⠶⠁⠀⠐⠀⠀⠀⠠⠄⠁⠀⡂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⢄⠀⡀⠰⠄⣀⠄⠒⠓⢤⣵⢶⠉⠁⠀⠴⠀⢢⡤⡴⠋⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣀⠀⣆⠃⠀⠘⣫⠀⣠⠔⠚⠃⠸⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⢲⠀⠀⠀⠀⠀⠘⡆⠉⠀⠉⠀⠻⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡘⢄⡇⠀⠀⠀⠀⠀⠸⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣈⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠡⡤⠀⣠⡔⠰⡼⠂⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣄⠄⠛⠏⠀⡁⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠀⡴⡋⠀⠀⠐⢠⠇⠀⠃⠅⠠⡃⠶⡄⠀⠀⠀⠈⠻⡄⠀⠀⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠱⢌⠁⠀⠀⠀⠄⠀⠠⠀⠀⠀⠁⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣽⠧⣤⢆⣶⢶⡄⣻⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⡇⢀⡃⠀⠈⠠⣯⢈⡇⢰⠀⠐⠀⠀⠀⠀⠠⣤⣀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⢰⡇⢹⣧⡀⠀⠀⢽⡆⣄⢘⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠚⠆⠀⢳⡀⠀⠀⣿⡗⢘⠁⠀⠀⠀⠐⢂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢺⡄⢠⡍⠀⠀⡏⡻⢸⡇⠀⠀⠀⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡠⠂⠀⠂⠀⠀⣽⡾⠀⡹⠁⠀⣼⠃⣳⣾⠀⠀⣤⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠐⠁⢀⡤⠶⡠⢚⡡⠂⣸⠁⢀⠌⠉⢴⠿⡟⡀⢄⠳⢄⠀⣴⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠐⡀⠢⠟⢢⡄⠀⠄⠀⠊⠀⣠⣇⠀⠘⡆⠀⢺⡀⣟⡳⠀⠀⠀⠀⠰⣆⠐⠀⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣃⡀⠀⢤⣈⠶⠀⠁⡑⠲⡆⠰⠖⠎⠁⠙⠀⣴⠿⠄⠀⠑⠾⠥⠑⠂⢀⡮⠕⠘⠧⠀⢀⡂⠀⠀⠀⢀⡀
⠀⢁⠀⠀⠀⢀⣤⠀⠀⠉⠁⠐⠊⢡⠖⢻⠀⠁⠈⠳⠀⢢⡀⠈⠠⡇⠸⢿⣮⡀⢤⣤⠤⠶⢾⡉⠀⠀⠒⢦⠘⠚⢁⠁⡄⠒⠊
⠀⠠⠀⠀⠀⠀⠀⣀⠀⠊⠀⠀⠀⠀⠋⣸⠃⠸⠄⠀⢲⣤⣈⡀⣀⡤⢰⠀⢾⠁⢀⡈⠀⠀⠄⠈⠏⠀⠃⠀⣄⠀⠉⠴⣁⠀⠐⠂
⠀⢠⠀⠀⠀⠀⠀⠤⠀⠀⠠⠆⡀⢀⠋⠀⠀⠁⣰⡞⠈⠉⠈⠁⠋⠀⠉⠀⠘⢴⢻⡁⠀⡀⠃⠀⢰⠄⠅⠲⢄⣦⡀⠀⠉⠉⠶⠓
⠀⣼⠀⠀⠀⠀⠦⣐⣇⡠⠄⠀⢀⠌⠈⢓⠜⠲⠉⡴⠂⢀⠀⠒⠄⠑⠐⠚⢰⠀⠀⢆⠸⠧⡀⢃⡄⠐⠂⠴⣬⠈⠥⠤⠀⠀⡀⢠⣤
⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠴⠉⣁⠀⠈⢠⡄⠀⠀⣄⢀⠌⠈⠔⢃⠀⢰⡊⠠⡾⠄⠫⠀⠈⠧⠀⣁⠀⠀⠀⠐⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉
⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⢐⡄⢠⡤⠞⣣⠖⢓⡖⠀⠀⠀⠀⠀⠀⢀⠾⢷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⣻⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠜⠀⠀⠀⠘⠧⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣦⣀⣇⡄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⣁⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠁
⠚⠉⢉⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⡟⠀⠀⠀⠀⣀
⡆⠀⠸⠁⠉⠉⢳⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣰⠁⠀⠐⣶⣾⠟⠀⠀⠀⠀⠠⡠
⢹⢀⠇⠀⠀⠀⣸⡇⠀⠀⢰⣿⣿⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢰⡞
⠈⣾⠀⠀⠀⢀⣿⠀⠀⠀⣾⡿⣿⣿⣽⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠋⢳
⢇⢦⣀⡀⠀⣼⡏⠀⠀⠀⠘⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡰⠂⡜⠁⠉⡠⠀⠀⠀⠀⠀⠀⢳⡀
⢿⣿⠟⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠠⠄⡀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠹
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠐⠏⠀⠀⠀⢠⠀⠀⠈⠒⠒⠁⠀⠀⠀⠸⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰
⢆⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠤⠀⡀⠀⠀⠀⢀⡿⠿⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀
⣤⡄⠀⠀⠁⠀⠀⢀⡀⠁⠀⠶⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢧⣀⣠⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠈⠡⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣶⡄⡉⣁⣤⡄⢀⠀⠀⠀⠀⣠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡈⠉⣰⡇⠻⠽⠃⠄⠀⠀⢠⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢀⠙⠃⢰⠄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿
⠁⠀⠀⠀⠀⠄⠄⠊⠀⡠⠒⣾⡗⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⡟⠁⠉⠀⠀⠀⠀⠀⠦⠀⠀⢀⣀⣳⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⡅⡀
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣏⡛⠦⡜⣦⠀⠀⡀⠀⠀⢤⣅⡤⠖
⠀⣀⣀⣐⢀⠈⣻⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣤⡀⡀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠋⣶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣆⣰⡀⣰
⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣇
⠀⠀⢶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠘⣆⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⠀⠈⢣⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠙⠢⣌⣆⡀⣀⡤⠔⠒⠚⠉⠉⣉⠷⠂⠀⠀⢹⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡽⠓⠒⠒⠒⠒⠚⠉⠀⠀⢀⣠⠴⢺⣤⡾⣍⠙⠓⠒⠲⠤⣄⡀⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠀⢀⠎⢨⠁⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⡴⢋⡇⡇⠈⠙⠢⢤⣀⠀⠀⠉⠓⢶⣿⣶⢿⡒⠢⢤⡀
⠀⠀⠀⠀⠀⢀⡏⠀⡟⢸⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠊⠀⢸⡇⡇⠀⠀⠀⠀⢈⠝⠓⣒⠶⠲⠾⢿⠀⠙⠲⢄⣈⠓⢤⡀
⠀⠀⠀⠀⠀⣸⠀⡸⡇⢸⠀⠀⠀⢀⠞⠁⣠⠴⠋⠀⠀⠀⢸⠃⠇⠀⠀⠀⠰⠷⠒⠋⠁⠀⠀⢸⠈⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⣇⡼⠁⡇⢸⠀⠀⠀⠻⠒⠉⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠉⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠐⠚⠓⢦⣀⣤⡀⢸⠀⠀⣧⣀⣀⡀⠀⢀⡤⢤⣄⣀⠀⠀⡾⠀⢸⣀⢤⣀⣀⠀⠀⣀⠤⣄⡀⡆⠀⣥⢤⣀⣀⠀⠀⣀⠤⣄
⠀⠀⠀⠀⠉⠀⠙⢹⠀⠀⡇⠈⠀⠉⠛⠋⠀⠀⠈⠉⠛⠉⡇⠀⢸⠁⠀⠈⠈⠙⠛⠁⠀⠈⠉⡇⠀⠀⠀⠁⠈⠙⠋⠁⠀⠈⠁
⠀⠛⠛⢛⣻⠿⢟⣿⠀⠀⣿⣟⡿⢟⡻⣿⢿⡿⣛⠟⡻⢛⡇⠀⠘⡟⣻⢛⡟⢻⢻⡟⢟⠛⣛⠃⠀⢸⢿⡛⢟⡻⣟⠻⣟⠻⠇
⠐⢚⣩⠥⠒⣋⡥⡇⠀⠀⢸⡥⣚⠽⣪⠕⣩⠞⣡⠊⡰⢻⡇⠀⠀⣇⠇⢸⡇⢸⠀⢧⠘⣆⠸⠇⠀⢸⠧⡙⢦⡙⠦⡙⠦⣙⠂
⠈⣡⠴⠚⣩⠴⠚⡇⠀⠀⢸⣊⠵⢊⡠⠊⣡⠞⢁⠞⠁⣼⠀⠀⠀⣿⠀⢸⠀⢸⡆⠸⡄⢸⠖⠂⠀⠸⠷⡈⢢⡈⠢⡈⠳⣌⠃
⢀⡤⠖⢋⣡⢴⠛⠓⠒⠒⠚⠣⡴⠋⣠⠞⠁⡰⠋⢀⡞⢸⠀⠀⠀⢿⠀⢸⠀⠀⡇⠀⢷⠈⠙⡏⠉⢻⡉⠙⢆⠙⢦⡈⠳⣌⠁
⢀⡠⠖⢉⡤⠞⣲⠖⠒⣂⠖⠚⣡⠞⠁⢀⠞⠁⢠⠏⣀⣼⢀⠀⣀⣸⡀⣸⠀⠀⡇⠀⠘⡆⠀⠹⡄⠀⠱⡄⠈⠳⡄⠙⢄⠈⠃
⠀⡠⠖⢉⡴⠊⠀⣠⠞⠁⢀⠜⠁⠀⡰⠋⠀⣰⠃⢸⣉⣉⣉⣉⣉⣉⣉⡏⠀⠀⢹⠀⠀⢳⠀⠀⠱⡀⠀⠙⣄⠀⠘⢦⡀⠱⡄
⠀⡠⠞⠁⠀⡠⠚⠁⢀⡔⠁⠀⢀⠞⠁⠀⡴⠁⠀⢀⠏⠀⠀⣸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠘⡆⠀⠀⢳⡀⠀⠘⢦⠀⠀⠱⣄
⣀⣀⣀⣀⣈⣀⣀⣀⣉⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣈⣁⣀⣀⣈⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠫⠛⠑⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠡⠐⠁⠀⠀⠀⠈⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠠⣀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣯⡙⠄⡈⠉⢲⣤⡆⠀⠀⠀⢄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣵⣄⠢⠈⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢹⠀⠁⢨⣶⣯⡟⠈⢂⠀⠀⠀⠢⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢈⠢⡀⠂⠀⢀⡀⠀⠀⡿⠀⠀⠀⠀⣇⡴⠋⠀⢿⠓⠦⣄⡆⠀⠀⠀⠈⢄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠁⠘⢄⠀⠘⡇⠀⠀⡇⠀⠀⣠⣶⣿⠶⠖⠺⡟⠂⠀⠾⢿⠶⠤⣀⠀⠀⠑⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠄⣆⠀⠈⢢⣷⡀⠀⠀⠀⢿⡴⠛⠁⠀⠸⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢃⠐⣶⣿⣿⣦⠀⠀⠀⠃⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⣿⣿⣿⣿⡷⣄⠀⠀⠈⠀⡀⠀⢸⣰⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⢀⣿⣦⡀⠀⠀⠀⠐⠺⣿⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⢨⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠁⣼⣿⡿⢱⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠀⢀⣠⡤⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⢁⣿⣿⣧⡀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠠⣀⡨⢖⣫⠵⠚⠁⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⣾⣿⣿⣿⣿⡦⡀⢀⣠⡴⠖⠛⠉⠀⠠⠒⠀⠀⣈⠭⠒⠉⠖⠋⢀⠠⢒⣩⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⠿⡯⢁⣀⠠⠖⠀⠀⣀⣤⣴⣾⠀⠀⢀⡠⠔⣊⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢸⢸⡋⠉⠁⠀⠔⠲⢿⣿⣿⣿⣿⣔⣊⡥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⣌⡏⢀⠀⠀⢀⣠⣼⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣝⣤⢔⣨⡵⠛⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋⠁⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠋⠉⠉⠉⢉⣫⡿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⣀⣤⠞⠉⠁
⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠰⢻⡇
⠀⠀⠈⠻⢤⣌⡉⠹⣟⡓⠶⢯⣀⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⠲⢮⣟⠳⢤⣀⠀⠀⠑⠤⣄⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⢠⣭⣛⠦⣄⡀⠈⠙⠋⠓⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠐⣴⡮⡙⠚⡟⠛⠷⢦⡦⢤⠦⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠁⠀⠀⢹⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⣰⠟⣷⢁⡀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠦⣤⣤⠤⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⡾⠛⠛⠋⢀⣴⠏⠀⠀⢠⡀⠐⠲⠬⠍⠛⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠀⣀⡴⣾⣿⠃⢀⠀⣢⠞⢷⠾⠶⠒⠒⠓⠶⣝⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢰⣻⠇⣠⠷⠚⠁⠀⠈⢳⡄⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⠟⢷⡀⠠⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⣷⠀⠐⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣯⠀⠀⢠⡟⠀⠀⠀⢰⡏⠈⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠘⠈⢿⣰⠟⠁⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣤⠾⢧⡀⠀⠀⣠⡾⠋⠀⠀⠢⣀⣺⠁⢀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣤⣤⣤⡴⡆
⠀⠀⠀⠀⠀⣸⠛⠛⠳⣽⣤⢿⡍⡀⠀⠀⠖⣤⣼⠃⠀⠁⠀⠀⠀⠈⠀⠈⠀⡤⠀⠁⠀⠀⣼⠃
⠀⠀⣏⣉⣉⣳⡞⡛⢛⠛⢳⡆⠙⢦⣱⣄⣤⣾⡃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠇⢀⠀⠀⣰⠋⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠈⠉⢯⣿⣽⣯⣿⣉⡄⢀⢀⣿⡟⠟⠉⠀⠀⠁⠀⠁⠀⠀⠐⢌⡴⠋⠁⠠⠐⢿⡷⠦⠤⠤⢤⡿⢛⣾⡞⠉⢛⡉⠉⠉⠉⠉⠉⠉⠙⠶⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⣽⠀⠀⠅⠀⠀⢀⣨⣠⣤⠴⠾⠷⠶⠶⠷⠦⠄⠀⣽⣷⣶⣶⡶⣇⢰⣿⡇⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠄⠁⠝⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⡀⠛⠉⠉⠉⠛⢷⣤⣀⣀⣲⣴⣋⣤⡤⠤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⣀⣂⣤⠀⠀⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠛⠚⠛⠛⠛⠯⣯⡉⠉⠁⠐⠀⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⢼⠛⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣾⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣿⣿⡉⢻⣿⠙⠦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣋⠁⠀⠘⣿⡿⣿⣿⢿⡟⠀⠀⠴⠓⠽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠂⠠⡄⠀⠀⠀⠋⠤⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠡⡑⠀⢸⡀⠐⡄⢀⠀⠄⡆⠰⡰⢀⠁⡠⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⢈⠂⡐⢌⢜⢄⢳⣀⣿⣼⣴⣥⣇⣅⣁⠆⡔⢡⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢜⢄⣩⡾⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⣞⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡈⢱⡟⠁⢀⣖⡽⠿⣿⣿⣿⣿⡿⣿⣿⡉⠙⢻⣅⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠐⣰⠏⢀⣞⣿⣷⣷⢞⣙⣭⣭⣟⠻⣿⣿⣿⣦⠀⠹⣆⠡⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⢠⡟⢀⣿⣿⣯⡋⢽⣧⣿⣿⣿⣿⣿⡎⣿⣾⣿⣧⠀⣿⣆⠉⠡⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⢼⣧⣼⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡆⣿⣿⠤⠄⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢼⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⡿⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠄⠒⠊⠉⡀⠀⡺⣿⣿⣿⣿⣿⣿⣿⣬⣻⣛⣛⣿⣿⣿⣿⠿⠛⢿⣿⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠐⠙⣿⣿⣿⣿⣿⣋⡛⠻⠟⠿⠟⠛⠋⠀⢀⣠⣾⣿⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣽⡟⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣒⠠⠭⢒⣀⠀⣀⣀⣀⣐⡀⠠⠄⠒⡺⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠐⠒⠒⠒⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣹⣷⠀⠀⠸⡦⠀⢿⣿⡷⠆⠀⠀⠀⢻⣧⠀⠀⠀⢀⠐⢻⣷⠄⢠⡀⠛⢀⣤⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⢯⠀⠀⢸⠛⠍⠀⠀⠀⠀⠀⢀⡈⠀⠁⠀⠀⠞⣷⡈⠁⣠⣤⡄⢉⡀⠛⠋⣠⣿⡄⠀⠸⠻⠄
⠀⠀⠀⠀⣦⠀⠀⢹⣷⠆⢠⡀⢀⠀⠀⢀⡀⠀⠐⠠⢼⡏⠄⠰⠟⣿⠂⠈⠁⣤⡘⢷⡡⠄⠀⠈⣄⠚⠉⠀⣠⡇⢹⡇⠀⠀⠀⢀⣴⡇⠀⠰⠃
⠀⠀⠀⠀⠁⠀⠀⠛⠙⢣⠀⠁⠈⠃⠀⣿⡻⠀⠀⢰⣶⡿⠀⠀⠀⠐⢷⡇⠀⣿⣷⣌⢣⠀⣀⣤⣾⣦⣤⣼⡿⢷⣼⣷⠀⢠⡖⣀⣿⠇⢀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⣶⠀⠐⠀⠀⠉⠁⠀⠢⡜⠋⠀⠀⠀⠐⣀⠀⡀⠀⢱⣏⢽⣧⣶⠋⠉⠋⢻⣿⣟⣷⠀⣿⣿⡴⢋⣶⣿⡿⢶⣾⡿
⠀⠀⠀⠀⠀⣿⡏⠁⠘⠇⠈⠒⠂⠒⠻⢄⠰⡂⠘⠛⠆⠈⡧⠤⣄⡀⠁⠙⠲⡤⣿⠀⣻⠋⠘⢦⡶⢸⣿⣿⡇⠀⣿⣿⠁⣼⣿⣿⡇⠀⠁
⠀⠀⠀⡄⠀⠙⠓⠀⣀⡀⠀⢠⡄⠀⣰⡾⢎⠁⠀⠀⠀⢷⠄⣴⣿⣧⣀⢰⣾⡛⣹⣶⡏⠀⠀⠂⠀⢸⣿⣿⣇⣠⣿⣿⣰⣿⡿⠟⣻⣿
⠀⠀⠀⠀⠀⠀⣀⠙⠋⠃⠹⣦⠉⠀⠙⠋⢀⠀⠸⣦⣤⣿⠀⢻⣿⣿⣿⡟⠋⠉⣿⣿⠁⢀⡄⠀⢠⣾⠛⠉⢉⣿⣿⡿⢻⣿⡦⢀⣿⣿⢀⣴⡇
⠀⠀⠀⠀⠀⣼⡟⠃⠀⠀⡈⢻⡆⠀⠀⠀⢨⡄⢰⡌⠛⣿⣧⠀⠿⢻⣿⡇⠀⢠⣿⣿⣷⣾⠅⢠⠋⣽⣆⢠⣾⣿⣿⠀⢀⡿⢴⠞⣿⠏⠙⣭⡄
⠀⠀⢀⡀⠀⠘⠃⡀⠸⠒⠃⠀⢃⢰⣾⣿⣦⠑⢼⣷⣴⡾⢿⣧⣶⣾⣿⡀⠀⠀⡿⠋⢹⣿⣷⡏⢠⣿⣿⣿⣿⣾⣯⠀⠊⠀⣠⣿⣿⣶⡄⠉
⠀⠀⠀⠉⢀⠀⣿⡟⠀⠀⠀⠀⠈⠺⡗⡨⠀⠁⢀⠈⢻⡆⠈⢿⣿⡟⣿⣧⠀⣶⠀⠀⠈⣿⣿⡆⠈⣿⣿⣿⣿⣿⡷⢤⣤⣸⣿⣿⣴⣿⣿⡆
⠀⠀⢠⡶⠟⠁⠀⠀⠀⢢⡀⠀⠀⠐⠋⢀⡀⠀⠈⠀⠘⣿⣦⡟⠻⣿⣿⣿⣷⠿⣏⠀⠈⣿⣧⠄⢰⣿⡿⠟⠀⠀⠀⠰⣿⣿⣏⠈⣿⣿⣿⡇⣀
⠀⠀⠉⠀⠀⠀⣠⣤⠀⠀⢱⡀⠀⠀⠐⠞⠓⠂⢶⣀⠾⠋⠙⠳⣤⣾⣿⣿⣿⢀⣼⠛⠹⣿⣿⢣⠜⠋⠀⠀⠙⣦⣶⣿⡿⠿⣧⣶⣿⣿⣿⠛⠋
⠀⠀⠀⣤⣶⡄⠙⠁⠀⠀⡌⠛⠶⣤⠤⠀⠀⠀⠀⡏⣰⡏⠙⠷⠿⣡⢙⡿⣿⠛⠉⠛⡷⢿⣿⣁⠔⠀⠀⣠⣾⣿⣿⣋⣤⣾⡿⢿⣝⠋⠉⢳⡶
⠀⠀⠘⡙⠀⣁⠀⢀⣀⡙⠆⠒⢀⠃⠀⠀⠀⡻⠎⠸⣿⣤⣶⣶⡄⠀⢨⣄⠀⢠⡀⠀⠀⠶⠈⢷⠤⠴⠋⣿⣿⡿⠛⠛⠀⠉⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠁⠀⠀⣀⠀⣭⡙⠓⠂⡀⢀⣤⡄⠀⣽⠇⢰⣿⣿⣿⣿⠇⠀⠈⠛⢠⣟⡿⢀⣾⣿⠀⠐⢠⡆⣰⡞⢿⠉⢷⠊⠹⡝⠛⠛⠛⠏⠉⢻⡄
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠑⠀⠀⠛⢾⢿⣶⠞⣡⣶⣴⣏⣉⠉⠁⣰⣶⠀⠀⠨⣯⣿⣿⣿⣿⣶⡶⢈⣀⠉⠈⣶⡄⠈⠻⣄⠀⠘⣶⣦⢠⡀⠸⣷
⠀⠀⠰⢦⠈⠃⣶⡷⢀⡀⠱⣄⡖⠠⠘⣣⣿⣿⣿⣿⣿⣿⣧⣨⡿⢿⡶⠆⣀⣈⣈⣋⣉⣍⠛⣴⡟⠻⠧⣀⠀⠉⠂⠀⠻⡷⠀⣿⣿⣿⣭⡀⢸⡆
⠀⠀⠀⠠⣦⣤⡴⠁⣨⠃⡷⢌⡑⠂⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⡙⠋⣡⣼⣿⣿⣿⣿⣿⣿⣄⠘⠓⢀⣤⠄⠀⣂⠀⠈⠁⠀⠊⠙⠛⡟⠛⢀⠈⡇
⠀⠀⠀⢀⣈⠙⠀⠀⠀⠈⠁⠀⠉⢩⣿⡁⠙⠻⠿⣿⣿⣉⠀⠐⠈⠎⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⣨⣴⠖⠸⣿⣄⣀⡀⢠⡀⠀⠀⠁⠀⠉⠠⠁
⠀⠀⠀⢸⣯⡤⠖⠀⠀⣀⡀⠀⢲⣾⠀⡀⠘⡷⢰⣿⣿⣿⣦⠈⣤⠓⠻⣿⣿⣷⣿⣿⣿⣿⣿⠟⠀⢿⡷⠖⣻⣷⣿⣿⣤⣘⣛⡂⠀⠀⠋⠀⣾
⠀⠀⠀⣾⡟⠃⢀⣤⠐⠈⠙⠉⣹⣧⣶⠀⠀⣴⣿⣿⣿⣿⣿⠸⣿⣤⡄⢉⢉⣉⡛⠻⠿⠛⢁⡰⠀⣸⣿⣿⠿⠛⠛⠛⠛⠛⣿⣿⣿⠂⠀⢀⡎
⠀⠀⠀⠙⠧⠀⠘⠇⠀⠈⡁⢠⣿⣿⠟⣀⢈⠛⠙⢿⣿⣿⡿⣞⢻⣽⣦⣜⠉⣹⢃⣄⡀⡄⢀⣤⠶⢿⣿⣷⡶⢶⣿⣿⡿⠟⡛⠉⠉⠰⠆⢸⡇
⠀⠀⠀⠳⠆⠀⢠⠈⠐⣤⡈⠀⣿⠿⢿⣯⣮⣤⣴⣂⡟⠲⠞⠈⠀⣿⠹⢿⣿⣿⣿⣿⡗⡘⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢠⣴⣤
⠀⠀⠀⠀⣶⣴⣗⣀⠠⠏⢙⠹⣇⣆⡘⢁⠟⡸⠟⣿⣴⡗⠂⡌⢰⠏⡄⢺⣯⡽⠛⣿⡏⠛⠃⠀⠛⢻⣿⣧⣉⣿⣿⣿⣃⠀⣚⡤⠒⠀⠉⣟
⠀⠀⠀⣾⣈⠿⠋⠁⣤⣦⠀⠰⠞⣿⣿⣿⣶⣿⣴⠃⡼⢻⡟⣠⣧⣸⢁⠛⡏⣀⣾⣿⣶⣋⠁⠞⣃⣉⣉⠻⣿⣿⡟⣿⡯⠙⡿⢷⡗⠀⡰⠟⠛
⠀⠀⠀⠉⠉⣤⡶⠖⠛⢭⢤⣴⣞⡏⢹⣿⢿⣿⡏⢾⣧⣞⣠⠏⢹⡟⢿⣤⣇⣯⣿⣿⣿⣿⢿⣿⡿⢋⡍⣷⣿⣿⣧⣿⣿⣷⣤⡏⢁⣾⠇⣠⣷
⠀⠀⠀⢀⠀⠘⢛⡄⠀⢀⠀⠀⢹⠉⣂⠈⢾⣟⠛⠛⣿⢿⣿⣶⢛⣴⣦⣸⣯⣇⣿⣿⣿⡇⠀⣤⠀⠈⣰⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡟⠀⠛⠁
⠀⠀⠀⠀⢴⣶⣾⠃⠐⢾⡖⠀⢸⠀⢻⠀⠀⣬⠀⠸⠿⣶⣿⣝⠋⢹⣿⣟⣿⣿⣿⣿⣿⢁⡾⢁⣶⠿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘⡉⠀⢙⡿
⠀⠀⠀⠀⠈⠉⠁⣤⡈⠀⢸⠖⠘⢒⡈⠑⡒⠧⠀⢃⡀⢀⢀⣼⠀⣾⡋⠛⠋⣿⣿⠛⢇⡾⣠⡎⠁⠀⠀⠈⠙⠋⠉⡉⠉⠀⠀⠰⠎⠉⢷⢀⣦
⠀⠀⠀⠻⠆⠀⠀⢿⣧⠀⠸⢦⠰⠾⣧⡶⠿⣤⢀⠀⠑⣌⠉⢯⡤⢿⡿⣤⡼⢿⣿⡟⠸⣱⠿⠇⠀⠐⠀⠀⡀⠀⠀⠘⠢⠀⠀⠀⠀⣾⠇⠘⠋
⠀⠀⠀⠀⠺⢿⣆⠀⠀⢠⢴⡾⢁⠀⢀⣤⠄⣿⠘⣿⠀⠛⣶⢈⡔⠻⠖⠾⠶⢞⡾⠁⢀⡏⢠⣤⢱⠆⠁⠈⡏⠀⢰⣆⣠⣧⠄⠀⠀⠁⢀⣴⡆
⠀⠀⠀⠀⠀⠸⢻⠀⣠⠃⣸⣇⠙⢃⠈⢁⡄⢤⡴⡇⠹⠶⢩⡏⠁⠀⠘⡓⠀⠈⠐⠆⠈⠀⢿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⢡⣤⡄⠀⠀⠚⠛⠁
⠀⠀⠀⢾⣤⣦⢀⣨⠷⠂⠀⠙⣶⡿⢶⣦⣄⠈⢻⢃⢄⡀⠙⢁⡉⠀⠈⢀⣴⠾⠀⢀⣧⡀⠈⠉⢐⠀⠀⠠⠀⣷⡄⢀⣀⣰⡿⠛⠃⠘⣷⡆
⠀⠀⠀⠀⠉⢀⣤⠤⣤⠄⢶⠞⢩⢅⡀⢁⡄⠛⠋⠙⠃⠀⡄⠙⠻⠶⠿⡮⢥⢴⠖⠛⠏⠙⠀⢀⡆⠀⠰⠂⠿⡄⠀⣌⠁⠀⠿⢁⣄⠠⡾⠃
⠀⠀⠀⠀⢠⣦⡔⠈⠛⢢⡄⠠⠅⠊⠀⣤⣶⢀⡀⣼⣇⠘⠀⠀⣤⣀⣤⡀⠀⠾⠟⢢⡶⠋⠀⢸⣿⣿⣧⡄⠀⢀⣤⣾⠀⠀⠠⣤⡿
⠀⠀⠀⠀⠀⠻⡶⠂⠀⢿⡄⠀⠀⠀⠀⠈⠛⠛⠁⠘⠛⠃⠀⠀⠛⠁⠚⠁⠀⠀⠀⠘⠛⠁⠀⠀⠉⠛⠛⠁⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠃⢄⢉⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡉⣠
⠃⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇
⠀⠀⠀⠈⢻⣦⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣴⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⢀⠀⠀⠀⣴⡃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠙⠁⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⢙⣶⡆⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢀⣀⣤⡀⠁⠐⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢁⡀⠀⠙⠒⠄⢈⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣬⠿⠃⠒⠀⠃⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠓⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⢀⣀⢀⡀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⡟⢡⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠀⣟⠉⠀⡸⠃⠀⣠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠁⠀⣴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣺⠔⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀
⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠳⠄⢀⢀⠸⢘⠌⠀⠁⠇⠀⠀⠄⠸⠈⠁⠇⠀⠀⠀⠀⠹⣿⣿⣦
⠄⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠻⣧⡀
⡀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⡄⠊⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠑

⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣰⠀⢰⠀⠀⠀⢀⠀⠶⢀⡀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⢰⡀⠀⠀⢠⡀⠀⢠
⠀⠀⡅⠉⡄⠈⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠃⢠⠞⠉⠂⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢉⠀⠠⡐⠋⠘⠃⠈⠀⠏⠀⠈⠒⢀⠀⢨
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠾⠒⠲⣄⠈⠳⡀⠀⣠⡄⠀⠀⠀⠀⠀⣄⠀⠀⠉⣤⣀⡂⠐⢤⣄⠀⠀⠘⠋⠁⠘
⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⡾⠃⠀⣤⢀⠁⠀⠀⢿⡆⠘⠀⠀⠀⠀⠠⠒⠉⠀⢠⠻⠋⣉⣹⣶⠀⠙⣷⠀⠀⠋⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠐⣏⠀⠀⠀⢱⠀⠀⠀⠀⠈⢡⠃⠀⠀⠰⠃⣠⠜⠀⠀⠚⠀⠷⠀⢷⡀⠀⢠⠀⠰⠛⢐⠤⠀⠀⠀⠀⠉⠀⢸⣧⠀⢸⡇⠀⠖⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⡤⠂⡲⣘⠛⣗⣤⠘⠣⠀⠙⠀⢀⣄⠙⠃⠀⠠⡀⠀⣠⡄⣠⡏⠀⣸⠃⠀⢤⡄⣩
⠀⠀⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣀⠀⠘⣠⠂⣠⠟⠁⠈⢁⣈⠁⠐⢀⡆⠀⠀⠛⢁⠑⡶⠀⠀⣿⡛⠿⠿⠋⢀⣰⠏⠀⠀⣀⡸⢿
⠀⠀⡇⠉⡆⠀⠀⠀⠰⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣥⠀⠐⠃⢰⣯⠀⠀⠀⠀⠀⢀⣶⡄⣥⠀⠀⠀⠈⠣⡀⣀⠀⠀⠑⠒⠲⠚⠛⠁⠀⠤⡀⠸⣀⣈
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠦⠀⡄⠀⠀⠀⢀⠀⣀⣾⠇⠀⠛⠇⠀⠐⠿⣿⡇⢸⣇⠀⣿⢆⣠⠀⠉⣀⠴⡂⢀⠂⡄⠀⣷⠀⡜⠋⠛⠉⡸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡖⠁⠀⠀⠀⠀⣰⣧⡟⢰⠄⠀⠀⡀⠄⠠⣿⢠⡇⣿⠀⠑⠈⠉⠂⡀⠋⣤⠈⣸⡇⣧⡜⠀⠐⠶⣠⣶⣾⢷
⠀⠀⡇⠀⠀⣰⠃⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠀⠘⠀⠀⢠⡟⠁⢃⣼⡀⠀⠀⠀⠀⣸⣷⡌⢃⠘⡆⢻⣏⣀⣀⡀⠚⠛⣦⠀⣤⡈⠁⣦⣽⣦⠟⢻⡇⢘
⠀⠀⡅⠀⠀⠈⠃⠀⠀⠀⠘⠄⠀⣠⣶⡏⠀⠀⠀⠰⡄⠁⣰⣶⠚⢅⠱⢦⣀⣤⣴⡉⠏⣛⠉⠴⣣⠀⢉⠉⠀⣄⢹⠀⠘⢦⠛⡧⠘⠃⠙⠃⢴⣬⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡶⠀⢻⣿⡌⠛⠂⠀⡀⢾⠀⣀⠀⠘⠋⢼⣿⠀⣉⣤⢷⡆⢸⡍⠛⣀⠙⢆⡀⠁⠀⢿⣥⡄⠀⡀⠀⠇⠘⠀⠀⢀⠾⢋⡀⢸
⠀⠀⣯⠀⠀⠀⠀⢈⡷⠀⡴⠀⠀⣀⣤⠐⠒⠋⢤⡈⠀⡀⠉⠀⠀⠀⢻⡇⠀⣨⡼⣷⣾⠕⠊⡙⠰⠚⠉⢂⡀⠀⠀⠰⣶⡅⡀⠀⠀⠀⠀⠈⢰⡾⠃⢸
⠀⠀⠉⠀⠀⠀⠀⠈⠁⠸⢧⠄⠒⠂⢠⠀⠳⢄⠀⠹⣆⠀⠀⠀⠈⠂⢸⣇⠀⢈⠀⢻⡃⠀⣹⡍⠂⢀⡤⠚⠡⣄⠀⠘⣶⣴⡍⠆⠀⠀⠀⠀⢩⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡼⠖⠀⠀⠀⢋⣶⡄⠓⡀⠉⠳⢦⡁⢀⠀⢸⡟⠀⠈⠀⣸⠁⡂⢹⣇⡀⠂⣠⡈⣰⠛⣀⠀⠿⣤⠀⢄⠀⠀⠀⣠⠞⠁⡄⣠
⠀⠀⡄⠀⠀⠀⠀⡾⢠⣾⠃⢲⣶⣟⠀⠈⠁⣿⠀⠀⣠⣤⣄⠙⢦⡀⢨⡗⠂⠔⠸⢷⣈⣱⡞⢋⠀⣤⠼⠈⢉⡴⢇⠀⠂⠀⠀⢸⠁⠀⣐⠁⠀⠈⠓⠹
⠀⠀⡧⠤⠔⠀⠤⢇⣼⠟⠀⣿⢿⣿⠚⣤⠀⠙⣧⡀⢀⠋⠀⠛⠲⠷⣌⡛⠛⠛⠛⠂⡉⢡⡀⣹⣶⠏⠠⢠⡾⡇⠀⢣⡾⠦⣀⠈⣦⠀⡉⠠⠤⠤⠄⣸
⠀⠀⠃⠀⠀⠀⣴⣾⠇⠀⠀⣹⢿⣿⠐⢸⡄⠀⠈⢷⡀⠻⣷⣄⡀⠀⠈⠙⠛⠧⠼⠶⠷⠾⠿⠿⠟⠓⣐⠿⡄⠈⠒⡋⠀⠞⠛⠛⡤⠟⠀⠀⠀⠀⠀⢿
⠀⠀⡆⠀⠀⠈⢻⠋⠀⣤⠀⠇⢸⣿⠀⠈⠀⠀⠀⠈⠳⣄⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠚⠟⣇⠈⡖⠀⠀⣾⠀⠘⣾⠤⠀⠦⠀⠀⠀⠀⣸
⠀⠀⡇⢠⠀⠀⠇⢀⡴⠉⠀⠃⣼⠇⠀⠀⢸⡀⠀⡀⠀⠻⣆⣀⠈⠛⢿⣶⣄⡀⠀⠀⠤⠄⢀⣀⣀⡀⠀⡈⠓⠀⡀⢸⣿⣧⣀⡀⠀⠋⠀⠀⠀⢀⠀⢨
⠀⠀⡁⠘⠆⠀⣀⡌⠁⠀⠁⠀⣿⠀⠀⠄⠀⣷⠀⠹⡀⠀⠙⢾⡦⠀⠈⠻⣿⣿⣷⣤⣤⣤⣴⡾⠋⣡⡞⠃⣠⡾⠃⢸⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⢼
⠀⠀⠟⠃⠘⠃⢸⠀⢀⣿⣠⡄⢩⣷⡄⠀⠀⠘⣧⠀⠀⡀⠀⠈⠷⡼⢆⡀⠈⠙⠷⠾⠟⠉⠁⢀⣼⡟⠀⣠⠁⠘⠃⠸⣿⣿⠏⠠⠖⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠎⠀⠈⠁⠀⠶⠾⢻⣿⢀⡄⢻⣷⠀⠀⠀⢹⣄⠀⠘⠀⠀⠀⠈⠺⢏⠂⡀⠀⠀⠀⡀⠀⢠⡿⠀⡄⢙⡂⠀⠀⠁⠈⢡⣤⡀⢀⠀⠀⠀⠞⠁⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣹⣿⣆⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠈⠀⠉⠀⠈⠀⣈⣡⠏⠱⠀⠀⠀⣠⣌⠟⠻⢮⣧⣆⣰⠇⠀⢀⣿
⠀⠀⠇⠀⠀⠀⠠⠀⠀⣰⠊⠙⢿⣿⣿⡟⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⡏⠀⢠⠀⢠⡎⠁⢈⣅⢠⣼⠋⠙⢿⡎⢉⠉⢻
⠀⠀⠄⠀⠀⢠⠆⣠⠎⡉⠁⠀⠀⠙⢿⡇⠀⢆⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⠒⠄⠀⠠⠒⠋⣠⣼⠟⠀⢠⡞⠂⠺⣁⠀⠸⢧⣸⣧⡀⢀⠸⢅⣸⣇⣤
⠀⠀⠆⠀⢸⡅⠀⠹⡄⠁⠤⠀⠘⠇⣠⠀⠀⠘⢄⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠠⡾⢂⠀⠛⣷⣤⣤⣠⡄⠀⠀⠀⢤⡄⠀⠆⢸
⠀⠀⠂⠀⠃⠰⣦⡄⠈⢑⣆⣠⣤⡄⢸⠀⠀⠀⠀⢂⠀⠀⠀⠄⡀⠀⠀⠀⠀⠈⠉⠿⠿⠛⠉⠀⠀⣠⠀⣷⠈⡆⠀⠸⠁⢘⡁⢀⠀⠄⠀⠀⠉⡀⠀⢘
⠀⠀⠁⠐⠂⠄⠘⢷⠟⠛⠋⠉⠀⠀⢸⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢱⡄⠀⣦⠎⠁⠐⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢾⣧⣀⡀⣀⣄⠀⠀⠸⢂⡀⠀⠀⢮⠀⣤⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⢀⡜⠀⠀⢸⠖⢲⠆⠀⠐⠛⠁⠀⠀⠀⣦⡀⠀⡤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠉⠁⠀⢠⡆⠈⠷⠀⠀⠆⠀⡏⠑⣄⣀⢀⠀⠀⠀⠢⠀⠀⠀⣠⡞⠀⠀⠀⡞⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⡄⠀⠀⠸⠁⠀⠀⢆⠀⠐⠀⡇⠸⠟⠉⠉⠻⠷⣤⣀⠀⠀⠈⠀⠀⠀⢀⡰⢿⠄⠀⡅⠀⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⠐⡀⠀⠀⣀⣴⡄⠀⡇⠀⣿⡟⠳⠦⣤⣤⡾⠛⠙⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⣁⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⢰⣿⣟⠀⢱⡄⣠⠀⠀⢹⣿⠀⢀⠀⠀⠿⣿⠀⠀⠘⠁⡇⠀⢸⡆⠘⣷⠀⠀⡀⠤⠛⠀⠓⠀⠀⠤⠤⠤⢀⡀⠀⢨
⠀⠀⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠈⠁⠈⠁⠀⠦⠀⠀⠛⠀⠸⠀⠀⠂⣿⠀⠀⠀⠀⢱⠀⢸⡇⠀⢻⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠄⠀⠀⠀⢳⣄⠀⠀⠄⡇⠀⠀⢹⡀⠀⠀⠀⢸⡀⠈⣷⠀⠀⢻⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠒⣰⣤⡀⠀⠀⢸⣿⠟⠉⠀⣷⠀⠰⣿⣧⠀⠀⠀⢸⠃⠀⣹⣦⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠞⠛⠁⠙⠁⠶⠚⠛⠦⠤⠴⠿⠀⠀⠿⠗⠀⠀⠐⠚⠓⠘⠛⠹⠶⢶⡴⠆⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⡴⠶⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠐⣦⡄
⠀⠀⠀⣿⢿⡇⠒⠒⠢⠤⠤⣉⣉⠉⠛⠒⠲⠦⠤⢤⣄⣀⣀⠀⠀⢀⣀⡤⠤⠔⢒⠒⣄⣉⠉⠀⡄⢐⡀⠈⠁⣿⢻⣿⣿
⠀⠀⠀⡿⠚⡇⠭⢍⣙⣓⣒⠐⠀⠌⠁⠀⠈⠉⠁⠀⠀⠈⠉⢹⣿⣿⠀⢠⠀⢀⡞⠀⣓⡂⠈⢍⣓⣀⣰⠨⠀⣿⢸⣿⣿
⠀⠀⢠⡇⣤⡇⠒⡒⠂⠀⠤⠈⢈⠁⠀⠀⠠⠀⠀⢀⡀⠀⠀⢸⣿⡇⣠⣿⢷⡄⠀⠀⢺⠭⠭⢍⣂⡀⠀⠭⢠⣾⣿⢸⡟
⠀⠀⢸⡧⡟⡇⣂⣢⠤⠀⠉⣀⠂⠀⠀⡀⠀⡈⠡⠀⠀⠀⠀⢸⣿⡇⡹⠟⡟⢉⠡⠞⣬⠭⠭⠥⢒⣒⠋⡁⢸⢸⡇⣿⡇
⠀⠀⢸⣇⢗⠠⠄⠭⠍⠁⢀⣐⣂⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠼⣿⠃⠀⠀⠿⢸⠃⠆⣒⣛⠩⠭⠭⠥⠤⠄⡇⣿⣧⣿⡇
⠀⠀⢸⣿⣸⢀⣒⣒⡒⠒⠒⠂⠶⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣁⣀⣤⡏⠰⠒⣒⣒⡒⣒⣒⣉⠉⠀⢧⣿⣿⣿⠇
⠀⠀⢸⣿⡟⠀⠀⣈⠉⠉⠀⠉⣍⣀⣒⣒⠐⠒⠒⠠⠤⣤⡀⣿⡇⠦⠍⠿⠹⢷⡌⢨⣉⡁⣀⡀⢠⡀⠀⠀⢿⡿⢹⣿
⠀⠀⢸⡿⡇⠀⠰⢹⠀⠀⠀⣀⠐⠒⠂⠠⠬⠭⠥⠬⠉⠉⢸⣿⡇⢉⣘⣒⣒⡒⠠⠌⠉⠉⠉⠁⠀⢿⠀⢸⢸⡇⣿⡟
⠀⠀⢸⣧⡇⢀⡇⢄⡀⣰⠈⠀⡀⠠⠀⠬⠤⠭⠤⠍⠀⠒⠈⣿⡷⠶⠤⠤⠤⢭⣭⡁⠼⣄⡆⠀⠈⠻⡗⣸⣼⠀⣿⡇
⠀⠀⢸⣿⡇⣬⡇⠈⠀⣾⡇⢸⡁⡀⠀⠭⠉⠭⠉⠥⠈⠁⣴⣿⡧⣖⣳⣒⣒⣒⠒⠶⠀⠨⢵⣄⡠⡀⡇⣿⣿⠈⣿⡇
⠀⠀⢸⣿⡇⢸⣷⡄⢠⣿⣠⣿⠃⠀⠐⣒⠒⣒⡒⣺⠄⠀⡏⠛⣉⣉⡒⠓⠶⠭⢭⣍⣉⣛⠀⠠⣑⠹⠁⣯⡇⣃⣿
⠀⠀⢸⣿⡇⠈⢿⡃⠘⠟⠛⠁⠘⠉⣉⣉⡥⠤⠄⠀⠒⠒⠉⠓⠢⠤⠉⡙⠛⠒⠒⠶⣤⣌⡀⢤⠉⡿⢸⣿⠀⣿⡟
⠀⠀⠸⢿⡧⠤⠴⠶⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⠄⣉⡁⠁⠇⢸⠏⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠈⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⢩⡍⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠀⠈⠁⡘⢻⣿⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣿⣿⣦⠘⠀⡁⣾⣿⣿⣿⣿⡿⢦⣴⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⢿⣿⣿⡇⣿⣿⠿⢿⣿⠋⠉⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢰⣿⣾⣿⠋⠸⣿⠋⠀⠀⠐⢣⣄⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢷⠋⢸⠁⡆⠈⢿⡇⠀⣀⣠⣿⣿⣷⡀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣸⠀⠳⣤⣼⣇⡔⣿⣿⣿⣿⣿⣿⠆⡀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡟⠀⠀⡸⣿⣏⣰⣿⣿⣿⣿⣿⠷⡷⣎⠁⢤⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢡⠌⢸⣿⣿⣻⣿⠿⣯⡀⠈⠹⠟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⢀⣿⣿⢿⡟⣽⢸⡇⠃⠀⠠⣤⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢄⣴⠟⢻⣷⠜⠿⠡⠎⡀⠀⠀⠀⢿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡿⣻⢯⣿⠀⠀⠀⠸⢁⠀⠀⠁⠸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡙⢈⣷⠇⣾⡇⠁⠀⠀⠀⠀⠀⠀⠀⢢⣘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣰⡟⢛⣿⠰⣏⡇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡍⡿⡇⢧⠸⣡⠀⠀⠀⠀⠀⢰⡀⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣿⠛⣇⣃⢈⠁⣼⠀⠀⠀⠀⠀⢀⠡⡈⠀⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣽⣼⠀⣿⠀⠀⠀⠀⠀⢸⠀⠹⡄⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡀⣸⣿⣿⡆⠘⠻⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⠀⠘⢾⣿⢹⣿⡏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡾⠁⠙⢻⡿⠃⠀⠀⠠⠸⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡿⢸⣿⢇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠧⠚⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣶⣶⣶⠂⠘⡀
⠀⠀⠀⠀⢠⣤⣄⠀⣿⣷⣺⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣡⣬⢾⡿⠿⠟⠌⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣠⢠
⠀⠀⠀⠀⠀⣿⣇⡈⢻⣷⣿⣦⡈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣠⣼⠂⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡉⠁⠻⣧⡀⠀⠀⠀⠀⠀⠀⠼⠿⠴⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡦⣽⡿⣿⡇⢸⡇⣼⣷⣄
⠀⠀⠀⠀⠀⠐⠇⠋⢰⣿⡇⣸⣃⣿⡿⢿⡿⠄
⠀⠀⠀⠀⠀⠀⠘⠿⠋⠙⠿⣿⣿⡟⠀⣋⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠋⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⣠⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠺⣄⣉⣉⣉⣉⣉⣠⡤⠟⠀⠀⠀⢰⠇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⢠⡏⠀⠀⠀⠀⠀⡿⢀⣀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣷⣶⣾⣿⣶⣄⡀⠀⠀⢧⡼⢶⡃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠻⣝⠻⠿⠻⠭⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⣀⣠⡤⠶⠚⠉⠉⠳⣾⣿
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣀⡤⠴⠖⠒⠒⠲⠶⢤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣀⡀⠀⠀⢸⣿⣠⠞⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⡴⠋⠉⠀⠀⠉⠙⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⠁⠀⠉⣷⣠⡾⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣏⡉⠓⠶⣴⣿⢹⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣷⣀⡟⠉⢳⣦⠾⡏⡿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢹⢸⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁⠀⣼⣽⢀⣇⡇⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡏⣇
⠀⠀⢀⡠⠤⣰⣶⠯⠋⢙⣿⠁⣸⢸⢿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢿⡀
⢀⡜⣩⠖⠋⠉⠀⠀⠀⢸⣿⡀⡟⣸⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⣇
⣾⡼⠁⠀⠀⠀⠀⠀⠀⠀⠱⣿⡇⡿⡿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠖⠛⠛⠉⠙⡇⢻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⡇⣾⡏⢀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣤⣤⣄⡀⠀⠀⢸⠀⠀⠀⠀⢀⣠⡤⠤⢾⣹⢷⡀⠀⠀⠀⠀⠀⠀⣧⠸⡆
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⡇⢯⣭⠭⠿⢛⣛⣩⡥⠤⠶⠶⠶⠶⢬⣭⣛⣶⣄⢸⡇⠀⣠⠞⣩⡤⠶⢒⣚⣿⡇⣷⡄⠀⠀⠀⠀⠀⢻⠀⢻
⠀⠘⠷⣤⣀⠀⠀⢀⣀⣤⡿⢿⢸⡷⣝⣲⣶⣖⣻⣭⣥⠶⢒⣛⣫⣭⣭⣭⣍⣙⣛⠿⣿⣿⣇⣞⡵⣫⠵⠚⣋⣉⣉⣉⡿⢿⣟⣳⠶⢤⣄⡀⠈⣇
⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⢸⡄⠓⠾⠿⢿⢿⣭⠿⣮⣻⡉⠁⠀⠀⠀⠀⠀⢀⣉⣙⣻⣿⣿⣿⣿⠵⠚⠉⠉⠉⠀⣨⣧⡀⣸⣿⣯⣉⣉⣹⣆⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢿⢠⣼⠷⢾⣼⠗⠒⠚⠛⠛⠋⠉⠉⠉⠉⠳⠶⠶⠛⠛⣿⣿⢭⣭⣿⣋⣁⣤⣿⠛⠛⠻⠯⣭⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠧⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣛⣓⣲⡿⠯⠽⣿⠀⠀⠀⠀⠀⣈⣉⣳⠄

⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢲⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿
⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣴⣿⣿⣷⣾⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣼⠟⢒⡆⠀⠀⠀⠀⣀⣀⣄⣀⣤⣾⡷⢿⣿⣿⡿⠟⠉⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣄
⡟⠀⣿⣵⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣄⣨⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⢿⡿⣷⡀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣟⠛⠛⠛⢙⡃⠀⠀⢁⣿⣿⣿⣶⣶⠶⠆
⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣟⠁
⣴⡾⡗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀
⣽⠟⢿⣶⡆⠀⠀⠀⠀⠀⠀⢸⣟⣹⣿⣟⠁⠛⣿⡏⠉⠻⣿⣿⣿⣧⣀⡴⢾⣿⣷⣴⣾⢿⡿⣿⣿⣿⡿⠛⠹⢿⣟⢾⣿⣿⡇⠀⠀⠀⠀⠀⢦⡻⠄
⡟⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣾⣽⣶⣤⣭⣳⣠⣼⠺⣿⡟⣹⣿⡦⣴⣿⣴⠿⠇⣯⣿⣿⣫⣯⣤⠠⣤⣶⣛⡮⣿⡻⠀⠀⠀⠀⠀⠀⢾⣷⡀
⠤⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⢠⣽⣿⣿⡿⣏⣾⡟⢿⣧⣿⣻⣾⡀⢈⣛⢿⣿⣿⣿⣿⠿⢫⣙⢿⣿⡿⢛⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⡿⠂⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣟⣺⡿⢧⣌⢁⡀⠀⢿⢿⣿⣿⣶⣾⣿⣿⣟⡍⣤⣴⣿⣿⢛⣅⣀⣿⣧⣆⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠓⠀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡽⠧⢥⠡⣟⢢⣙⣿⣿⣿⣿⡿⣿⣿⣿⠿⠯⢽⣿⣿⣿⣿⣿⣈⣮⣿⠵⠼⣿⠵⠶⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣌⣸⣶⣠⣿⡶⠿⣿⠿⠛⢛⣉⡉⢰⣿⣿⣿⣿⣿⡷⠀⠀⠙⠻⠿⠷⣷⣿⣾⣮⣷⣾⣁⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣿⣿⡿⢀⣴⣇⠀⠀⠀⠀⢙⣿⣿⣋⣉⣁⣸⣟⡏⠀⠘⣛⡏⠀⢀⣉⣭⣿⣿⣿⣿⣿⣶⣄⢠⣀⣈⣿⣿⣻⣿⡿⣿⣾⣿⣿⠓
⢸⣿⣿⠇⣾⣿⡿⠀⠀⠀⠀⠀⠉⠛⢿⣟⠻⣿⣷⣶⡶⣶⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡾⠿⠋⠀⣭⡉⠉⠉⢁⣘⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣿⣿⣸⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⢿⣿⣷⣗⣷⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣤⣤⣤⣤⣤⡿⣿⣿⠋⠟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⣿⢿⣯⢹⣿⣿⣿⣿⣿⣿⣶⣴⣤⣴⣾⡿⠿⣿⡟⣹⣶⣿⠟⠁⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⣿⠀⡙⢾⣿⠟⣿⣿⠁⡘⣿⢿⠋⣼⣯⣴⣷⡟⠩⣿⣿⣿⠀⠐⠋⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⣠⠀⣿⠀⣀⣼⣿⣿⣿⣿⣾⡇⢹⠘⡀⣸⣽⣿⣿⣿⡿⠋⢹⣿⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠤⠀⠻⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣱⣿⣿⣿⣿⣿⠄⠀⢸⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣶⣿⡄⠈⢹⣿⣿⣿⡿⣿⣿⣿⠀⢿⣿⣿⠟⠉⠃⠀⠀⣸⣿⠈⠛⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿
⣿⣿⣿⣿⡇⢀⣴⣦⡀⠀⠀⠀⠀⠀⠃⠀⠀⠻⡿⠃⠀⠘⣿⣿⣿⡇⠘⣿⣿⣶⣾⡿⠁⠀⠀⢀⠀⠀⣻⡿⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟
⣿⣿⣿⣿⣧⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⣿⣿⣿⡇⠀⣿⢿⣿⠏⣹⣷⣷⡞⠁⠀⠀⠿⠁⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⣿⣿⡇⠀⣿⡇⠁⠀⠹⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠉⠉⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠛⢈⡉⠉⡈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⢀⡉⢁⡀
⡆⢈⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣟⠁
⠀⠈⠉⠙⠋⣠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣄⠛⠋⠉⠁
⠆⠀⠀⠀⠀⠙⢿⣷⣤⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠠⠀⠁⣠⣠⣾⡿⠋
⡄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠁
⡀⠀⠀⠀⠀⠀⠂⠈⠻⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⠁⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⠘⢿⣿⣿⡿⠃⠀⣾⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⣿⣿⣦⡀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢿⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣴⣶⡶⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⣦⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠈⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠴⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⠀⢸⣷⠀⣠⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡿⢿⣿⣿⣿⣿⣿⣿⡿⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⢹⡿⠁⠈⠛⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⣤⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡄⠀⠀⠀⡆
⠄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠄⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠄⠀⠀⠀⠸⣷⣦⣀⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣠⣴⣾⠃
⠆⠀⠀⢀⣀⣬⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣥⣀⡀
⠄⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣀⣀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛
⠂⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣉⣽⣿⣿⣿⡆⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣯⣙⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠉⠛⠿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⠟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠿⠋⠉⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⢼⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢻⣿⣄⠀⠀⠀⠀⣰⣿⠏⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣾⣿⣿⣦⣤⣤⣾⣿⣿⣷⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠃⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⠐⢆⠀⡄⢤⡀⠀⠰⠠⡆⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠻⠰⠆⠀⠙⠀⠛⠀⠻⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠃⠀⠀⠀⠀⣴⡿⠛⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠻⣿⣦
⡀⠠⣤⣴⣦⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠁⢶⣦⣤⠄
⠃⢠⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⣄⡔⠋⠙⢦⡴⠊⠉⠙⠶⠁⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠀⢀⣤⣌⠀⢠⡶⣄⠀⢠⡶⡄⠀⣴⠿⣶⡿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠆⠁⠀⢠⣿⠋⢹⡄⠹⣿⣸⣠⡾⣧⣇⣼⡏⠀⣿⡇⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣾⡇⢀⢸⣧⢾⣏⡝⠸⣇⡜⠉⠹⠔⠁⢿⡇⠀⣼⠃⠀⣠⠟⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣷⠀⠘⠛⢁⣼⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠋⠀⢊⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡟⠀⠀⠀⡞⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣼⣿⠞⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⣠⣤⡀⠀⢰⡿⣠⠖⣳⠄⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀⣤⣿⡇⠀⠀⠀⢀⡄⠀⠀⣼⠀⠀⣼⡏⠘⠃⢀⣿⣿⣡⠾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢤⠀⠈⠉⠉⠁⣿⡇⠀⠀⠀⢸⡇⠀⣼⣿⠀⢠⣿⠀⠀⢠⠎⣿⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⡆⠀⢻⡇⠀⠀⡰⢻⣇⡼⠸⣷⣠⠏⠻⣦⠴⠋⠀⠿⠀⠙⠿⣶⣦⣤⣤⣶⠿⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠻⠶⠞⠁⠈⠉⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⣠⣷⡀⠀⠈⠉⠉⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣤⠞⠷⣤⣠⣤⠞⠙⠶⠶⠚⠉⠓⠒⠚⠉⠈⠙⠶⢦⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠶⠷⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣆⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠊⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠌⣉⠇⠈⠈⠉⠐⠀⠀⢀⡀⠀⠀⠂⠀⢫⠀⠀⠀⠀⠒⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢲⢄⠰⠘⠂⠀⠠⠄⠈⠁⠄⠁⢀⠬⠋⠀⠀⠀⠀⡒⠀⠧⡄⠀⠀⠀⠀⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠐⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⢀⠃⠰⠀⣰⠂⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢖⠲⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⠀⠀⡀⠠⠀⠻⠀⠀⡄⠓⠁⠀⠀⠂⠁⢀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠊⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⡖⠀⠀⠄⠀⢡⡨⠀⠀⠀⢀⠀⡮⠑⠋⠀⠀⠁⠀⡀⠀⡀⢊⠍⠒⠚⠒⠠⠌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⠇⠀⠀⠀⠀⠀⠤⡀⢀⠄⣸⠀⡆⠀⠁⠈⠀⠀⠀⢠⠄⠠⠃⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⡌⠀⠀⠀⠀⠄⡇⠈⠃⣼⡄⠀⢠⠀⠙⠐⠀⠀⠀⠆⠁⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡘⢆⠀⢠⠀⠂⠳⢄⡊⢀⠃⠀⡼⠀⠠⢀⡄⠁⠈⠀⣤⢶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠒⠤⠀⠀⢷⠋⠀⠀⡧⠐⠂⣀⠄⠒⠀⣸⠃⠀⣯⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡟⠀⣸⠃⠀⠀⡀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢆⢀⠀⠀⠀⠀⣁⣀⣀⣀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣼⠃⠐⣯⡀⠀⠀⡁⠀⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠂⢒⠀⠱⠄⣠⠆⠐⠛⠋⠉⠙⠛⠻⢶⣤⣤⣄⣠⡴⠛⠁⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠃⠄⠘⠀⠀⠀⠀⠀⡀⠈⢳⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠊⢉⠓⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⠁⠀⠘⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⣠⣴⠶⠞⠛⢛⡇⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⠁⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⡤⠤⢀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠊⡇⠀⠈⡀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢩⣀⠠⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⠞⢀⠁⠀⠄⠀⠀⠈⠁⠀⠀⠐⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠀⡴⠃⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⠴⢦⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠛⠋⠀⠉⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢄⠀⠀⠐⡀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⣠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢣⡈⠳⡀⣀⠐⠀⠐⡡⠐⠀⠀⠀⠀⡀⠀⠀⠦⠤⠔⠀⠀⢀⠀⢀⡀⠠⠀⠀⠀⣀⣀
⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠘⠖⠉⠛⠤⠀⠐⠁⢛⠀⠀⠀⠀⠀⠌⠠⢀⣠⠀⠀⠀⡁⠀⠀⠀⠐⠂⠀⠘⠁⠀⢁⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠁⠚⠁⠰⠖⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠁⠔⠂⠀⡄⠀⢠⠇⠒⠀⠈⠰⠀⢀⡀
⠀⢠⠀⠀⠀⠀⠠⠄⠀⠒⠁⠔⢁⠠⢦⠀⠀⠖⠀⠆⠈⠀⠀⡥⠀⠀⠑⠠⠀⠁⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠶⠀⠀⠀⠀⠀⠀⠀⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠤⢄⠑⠂⠀⠐⡁⢈⠠⠀⠒⠚⠶⠈⠁⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠌
⠀⣶⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⠄⠂⠀⠀⠀⠀⠀⠈⠀⢀⠐⠈⠀⠁⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢤⣀⠈⠐⠤⣀⣼⠁
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠠⠃⢀⣀⠀⠀⡞⠀⠀⠹⡆⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠢⠤⠀⠀⠀⢰⡁⠈⣧⠀⢷⡄⠀⠀⡇⢀⡔⠛⣿⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⡄⠀⠀⠀⢸⣷⢠⡟⠀⣾⡛⠶⣴⠗⠺⣧⡼⠻⣆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢘⣿⣀⣀⠔⠁⣉⣉⡀⠀⠘⠳⠒⠁⠀⠀⠀⣠⠀⠉⠓⠦⠄⠊
⠀⠀⡠⠒⠉⠉⠉⠉⠉⠉⢹⡏⠁⠀⢠⠏⠀⠀⢹⠀⠀⠀⠀⣠⠤⡀⢰⡇⠰⣦
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢄⣿⣀⢀⠀⠊⠀⣄⠀⣼⠃⠀⠀⣸⣇⡴⠃⠀⠀⡠⠔⠒⢦⡀
⠀⠸⣆⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢸⣏⠀⠀⢸⠇⢰⣏⠀⣿⣆⣀⠔⠹⠏⠉⠓⡶⣮⣀⣀⣀⡼⠁
⠀⠀⠙⠳⠦⠤⠤⠖⠋⠀⠀⠀⠀⠀⢿⣄⣀⠾⠧⠜⠛⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢷⠃⡏⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢢⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⠀⠁⣤⠀⠀⣤⣠⠄⠀⠰⠬⠀⠀⠄⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠄⡟⠀⠀⠀⡄⠀⠀⠀⠀⠠⡄⢛⠂⠘⠴⢆⠙⠆⡀⠀⠁⠀⠀⠀⠇⠀⠀⠘⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠘⠀⡒⠙⠂⠀⠀⠀⠄⠠⡙⠛⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⡄⠘⠀⠄⡀⠀⠻⣨⠆⠀⠴⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠁⢀⡄⠀⠀⢂⠀⠀⠀⠁⠂⡀⠀⢈⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣘⡣⠀⠠⣅⠀⠒⠀⠂⢰⡫⠀⠀⢈⡅⠌⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⠈⠁⢀⠀⠑⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠄⠀⠂⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣩⠀⠀⡄⠀⡀⠀⠞⠀⠀⠰⠀⠀⠁⠀⠂⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠠⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠶⠚⠚⠀⡂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠀⠀⠀⠀⠂⠀⠀⠐⠰⠄⠀⠀⠀⡀⠄⠀⠀⠈⠀⠀⢀⡠⠁⠂⠀⠀⠀⢰⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡠⠀⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠃⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢄⡀⠀⡐⠀⠀⠄⣀⠴⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⡠⠀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢀⠈⠘⠧⣄⠀⠘⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡽⣀⠀⠀⠀⠀⣠⡴⠂⣀⠄⠁⠀⢩⠄⢀⡀⠁⠄⠀⠈⠰⣄⡈⢷⡄⠀⠀⠀⣰⠠⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢠⣦⠀⠀⠀⣿⠠⡈⢿⣄⡀⠤⠀⠀⠀⠁⠀⠀⠀⠒⣤⠋⠁⠨⣇⠀⠀⠠⠔⠋⠀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⠀⠀⢹⡀⠈⠀⣈⡗⢂⠀⠀⠀⠀⠀⠀⠠⡀⠒⠖⠀⢸⠇⢀⣴⠎⠁⠀⠀⠀⣀
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡶⠀⠀⠃⢀⣀⡬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠑⣀⡔⠋⣁⠐⠀⠀⠀⠀⠀⠀⠁
⠀⡘⠀⠀⠀⠰⠁⠀⠀⠀⠀⡘⠀⠀⠁⢐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢚⠣⠀⠴⠫⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠒⠂⠀⠀⠀⠂⠀⢀⠀⣀⠀⠀⠂⠢⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠀⠀⠈⠀⣀
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠂⡃⠀⠀⠀⠀⠀⢀⡅⠂⡱⠀⠄⠀⠀⠐⠑⠀⠀⢀⡀⠈⠑
⠀⢉⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠲⠁⠈⠀⠠⠉⢀⠀⠀⠸⠀⠀⠀⠁
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⣀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⡀⠈⡀⠀⠁⢠⢤⣶⣶⣦⢰⢠⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠈⠁⠃⠀⠀⠀⠋⠉⠉⠉⠈⠉⠛⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠘⠛⠀⠟⠀⡴⠀⢠⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠈⠁⣄⠀⣀⡀⣠⡄⠘⠦⠀⠿⠀⠸⠀⠀⣶⠀⢠⣤⠀⠀⡉⠁⠰⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢶⠆⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠃⠀⡶⠀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠀⠦⠀⠆⢠⠄⠠⠀⢤⠀⠆⠀⡄⠠⡄⠠⠄⠀⠀⠀⠀⠘⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⡇
⠀⠀⠀⠀⠀⠈⢴⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣆⡇
⠀⠀⠀⠀⠀⠀⠤⠤⠬⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠬⠤

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⡇⢂⣀⣀⣘⣃⣃⠸⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢻⣿⢉⣿⣏⣽⣿⣉⣿⡏⢛⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢿⣧⣨⣿⣶⣿⣿⡿⢿⣿⢿⣿⠟⣿⡿⣿⣿⢿⣿⣾⣿⣵⣿⣃⣹⡿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡿⢿⣏⢹⣿⣀⣿⣧⣼⣿⣶⣿⣿⣿⣷⣿⣿⣤⣿⣧⣈⣿⡿⢻⣿⢿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢩⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⣾⣷⣿⣷⣿⣦⣿⣾⣿⣼⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⢸⡷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣦⣼⣿⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⢼
⠀⠀⠀⠀⠀⢸⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠾⠾⠶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠁⠀⠀⠀⠔⢂⣈⡭⠥⠤⢤⣀⡐⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡷⠋⠀⠀⣀⣤⠴⠄⠒⠋⠉⠁⠀⠀⡄⠀⠀⢉⠛⠆⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡴⠖⠒⠉⣁⣀⣀⣤⣴⣦⣾⣴⣿⣿⣿⣷⣷⠟⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠯⢱⣤⠐⢠⡉⢹⣿⠟⠉⠉⠀⠉⠉⠋⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣷⠂⠀⠀⡜⠁⠀⣀⠤⠴⠤⢄⠀⠀⠒⢓⣛⣒⣒⣚⣛⡒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣒⣸⡆⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠛⠋⡩⡭⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⢿⠞⢫⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠦⠰⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡿⠀⢀⡀⢀⣠⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⡄⠀⠀⠀⠀⢀⡾⠀⢰⣧⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣿⡿⠟⠿⠷⠷⠼⢼⠟⠁⠈⢨⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠂⠀⠀⠀⠀⠀⠘⠀⠈⠀⠁⠁

⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠛⠒⠞⠛⠳⡢⢐⣟⠛⠁⣠⡴⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⠇⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠛⢒⡿⣿⣿⣷⣾⣿⡷⠯⠥⠄⠀⠀⠀⠀⠀⠀⢸⡄⠀⣼⠀⠀⠀⣀⣤⠞⠀⠀⠀⡀⣀⣀
⡯⠌⠀⠀⠀⠱⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠥⢖⡿⠉⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣷⣾⣇⢠⣶⣿⠟⠃⠀⠀⢠⡶⠋⠛⠁
⠀⠀⠀⢀⡤⣀⡝⣝⢶⣤⢤⡤⠞⠃⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠈⢋⠁⠀⠀⠐⠒⠄⣤⡭⢽⣿⣿⣻⣿⣟⠿⣿⣽⣷⣶⠶⠋
⣠⣖⣤⡷⠖⠷⠶⠖⠛⠻⠶⠀⡉⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⣄⠀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠈⠉⠻⢿⣿⣿⣯⣶⣶⣄⣀⡀
⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣀⣀⡀⠀⠀⠀⠀⠀⣆⢰⡇⠀⠀⣹⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣶⡄⠀⢹⣿⣷⣤⣴⣿⣾⠃⢀⣀⣤⡤⣴⣾⠟
⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠀⠀⣻⣿⣿⣿⣿⡿⢰⣿⣿⣻⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠻⣛⢲⡀
⢠⣟⣏⣏⣱⡀⠀⢀⣠⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⢻⣿⣽⣿⣆⠀⣿⣿⣿⣿⠟⠁⣻⣿⣿⣿⡿⠇⠀⡞⠻⣿⣷⣆⠀⠀⠀⢄⠉⠀⣽⢿⣇
⣿⢸⣺⠉⢹⠗⠀⢸⡟⡋⠤⠌⡧⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠃⠀⠺⣯⡉⠀⣿⡟⠀⠀⢠⣝⣀⣴⢿⣬⡉⣷
⣿⣿⣿⡷⠬⠆⣀⣠⣷⣶⣄⣈⠷⠀⠀⠀⠀⣤⣤⠀⠀⠉⣹⣿⣿⣿⣿⣿⡯⣿⣿⣿⠟⠃⠠⠶⡄⠠⣏⠀⡴⣻⣖⠀⣵⡀⢀⣴⡿⠓⣆⡙⣏
⣿⣿⡿⡻⡆⢰⠛⣫⣭⠽⠻⣿⠀⠀⠀⣀⣀⣷⣽⣀⣤⣾⣿⣿⣿⣯⣷⣼⣷⣿⣿⣿⢀⡠⣴⣶⣧⡀⠀⢰⣏⡿⡉⢿⡿⣷⡿⣿⣁⣴⣿⣿⡇
⡃⠹⣿⡇⢠⣶⠊⠀⢰⣶⣀⢈⣳⡄⠙⠋⣩⡟⢸⡏⣿⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣌⠁⠸⡶⣧⠟⢑⣿⡟⢁⣾⣾⢁⣽⣿⢋⣡⣟⣡⣿⡇
⡚⢀⣿⠿⣿⢦⡰⢤⢼⡟⠻⣾⣁⠜⡓⠾⠿⣝⣻⠀⠉⠀⠀⢰⣿⣿⣹⣿⣿⣿⡛⠻⣿⣷⣷⣇⢭⡴⠟⢱⣏⣿⠛⠁⢸⢿⣿⣿⣿⣿⣦⣿⡅
⢾⣿⠁⣰⢃⡯⣿⢳⡞⡄⠀⠈⠙⠛⠋⠛⠓⠋⠁⠀⠀⠀⠀⣾⣿⢛⣽⣿⣿⣿⣧⠀⠀⠙⠻⡷⠶⠶⠶⠷⡟⣯⣥⣀⠀⣼⣧⣿⣽⣿⣿⣿⣷⡀
⣾⣿⣦⣀⡀⠃⠃⢸⡇⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⢿⣿⡆⢒⠻⠀⠀⠀⠀⠀⢠⠇⠉⡏⣷⣆⣿⣿⣿⣿⣟⣿⣾⣿⣷
⠊⡿⠁⠿⢛⠿⣶⢿⠿⠟⢧⢀⣠⣤⣶⣶⣶⣶⢶⠶⢶⠤⠤⣿⣿⣿⣿⣿⡏⠈⣿⡇⠸⣦⡀⠀⠀⠀⠀⠋⠀⢨⢑⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⢻
⢲⣷⢰⣦⣦⡑⢿⣛⢦⠀⠀⣿⣟⠏⠉⠃⠑⡃⢀⠁⠀⠀⣼⣿⣿⣯⣿⣿⣧⣤⣿⣧⣤⣿⣧⣠⣤⣤⣼⠀⠀⣿⣿⣿⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⠾⣿⣿⣻⣾⣏⠦⠀⢠⠏⠉⠀⠀⠀⠀⠈⠈⠁⠀⠀⣿⣿⣿⠈⢿⣿⣿⡟⠑⠙⠿⠯⠉⠿⠐⠀⢸⡇⠀⠈⣿⣟⣿⠈⠻⣿⠁⢿⣿⣛⣷⣿
⣿⣿⠃⡿⢾⣷⡿⠻⠀⠀⡾⠀⣀⣤⣴⣶⣶⣾⡿⢿⡿⣻⣿⣿⣇⣀⣸⣿⣿⣷⣴⣦⣤⣤⣤⣤⣤⣤⣼⣿⡀⠀⠹⣿⣿⣄⣀⣩⣤⣈⣻⣿⣿⣿
⣟⣵⣶⣳⡿⣿⣷⣦⣤⣴⣷⣿⣿⣿⣿⣿⣿⡷⣝⢦⡤⣾⣿⣿⠿⠟⣿⠈⣿⣿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣔⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡾⣟⠧⢽⣿⡟⣿⣿⣿⣟⡻⣿⣿⣿⡿⣿⣯⠻⣯⣿⣾⣿⣶⣿⠻⣶⣏⣽⣽⣿⣿⣿⣿⢟⢯⡻⡿⣿⢯⡻⣝⢿⠿⡟⢏⠋⡙⢿⣿⣿⣟⣿⣿
⣽⣯⣴⣧⣬⣥⣽⣦⣭⣿⣭⣭⣿⣮⣿⣽⣮⣯⣽⣿⣿⣿⣭⣷⣽⣮⣬⣯⣿⣿⣿⣿⣿⣿⣷⣵⣽⣽⣮⣷⣽⣮⣧⣵⣼⣦⣤⣼⣯⣷⣷⣽⣮⣮
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⣀⡁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀
⠀⠠⠖⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠐⠘⣼⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣟⣫⡾⣶
⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠐⠀⠠⠀⠀⠀⠀⠀⠑⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠈⠒⠛⠁
⠀⠀⠃⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⢀⠀⠀⡅⠀⠀⠀⠀⠂⠀⠀⠾⠆⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⢸⢀⠀⡄⠀⠃⠀⠀⠀⡇⠀⠀⢤⠄⠀⠃
⠀⠀⡄⠀⠨⠀⢐⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠉⢸⢸⠀⡄⠀⠀⠀⠀⠀⠁⠀⡀⠶⠁
⠀⠀⢿⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠱⠀⠐⠀⠈⠀⠁
⠀⠀⣾⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⣶⠀⠀⠀⠄⠀⠤⠤⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢈⠛⠿⠞⠃
⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠊⠀⢠⣀⠀⠈⠋⠀⠀⠀⠀⠀⠰⠀⠀⢀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠘⠟⠐⠀⠀⠈⢀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢐⢠⠄⠀⠀⠀⠀⠀⠆⣰⠂⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠅⠀⠀⢈⠈⢠⡀⠶⠿⠂⢠⡄⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⡄⠀⠘⠀⠀⠂⠀⠀⠂⠀⠀⡀⢀⠹⠇⠀⠀⠈⣆⣨⣤⠴⠶⢤⣤⡄⠋⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⢀⢀⡃⠀⠀⠀⠒⠀⡈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠈⠉⠁⢀⠀⠀⠀⠈⠉⠉⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢀⠀⠀⠀⠐⡠⡶⠀⠀⢀⡀⠀⠠⠀⢠⡤⣤⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⢱⣴⠆⠈⠁⠀⣀⠸⠡⠀⠀⠀⠛⣰⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠁⠀⠈⠀⠀⠀⠀⡔⠈⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠃⠈⡴⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⡀⠀⡀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⣀⣷⡀⠀⠐⠢⠀⠀⠀⠀⠀⡴⣦⡀⠄⠀⠀⠀⠘⠀⢛⡑⢄⠀⠀⠀⠠⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠉⠓⠀⣀⠀⠀⠀⡀⠀⢰⡿⡁⠀⠀⣠⠀⠀⠤⠛⠱⢀⠑⢄⡀⠄⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡐⢋⠁⠃⠀⠀⣤⠂⠀⢀⠀⢳⠈⠳⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠄⠀⠃⠀⠀⠧⠀⠀⠀⢄⠀⠠⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠀⠒⠒⠚⠁⠀⠀⠎⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠐⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⠂⠀⠠⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢀⠁⠀⠱⣈⠓⢁⣠⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣀⡀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⠀⠈⠃⠠⡤⠀⡞⠁⠀⠀⠀⡿⠀⠀⠀⣽⠆⠙⠇⠀⠀⢠⡀⢀⣄⡀⠀⠀⠀⣀⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠀⠀⠇⠀⠷⠁⠀⠀⠀⠀⠀⢶⠀⠀⣠⠇⠀⠀⠀⡄⠀⢸⠟⠉⠃⠀⠀⠀⠀⠀⠀⠤⠄⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠰⠀⢠⠁⠀⠀⠀⣀⢿⠀⢐⣆⣶⣤⣤⡶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠏⢉⡿⠂⠀⠀⠀⠀⠇⠀⠀⠀⣠⠤⢾⠆⠉⠀⠀⢀⣀⣁⠀⠀⠀⣀⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠀⢠⠴⠋⠀⣼⠃⠀⠀⠀⠀⢰⠀⠀⢀⠀⠥⠂⡠⠟⠋⠀⠀⠊⠀⠉⠀⠀⠀⠁⠀⢀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡀⠀⠚⣁⠴⡔⠿⠋⠀⠀⠀⠀⠀⡌⢀⠀⠄⠀⠀⣽⠆⣩⡟⠛⠇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣐⠁⠀⠉⠀⠀⠠⠄⠐⠁⠀⠀⢀⠀⢲⠀⢠⡥⢤⠤⠤⣶⣞⣋⣀⡗⢡⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣤⡀⢀⡄⠀⢴⡀⠁⠀⠀⠀⡀⢀⡆⡙⠀⣴⡀⠀⣤⣈⠉⢨⣿⠉⠁⠘⡄⠈⠄⠡⡀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠄
⠀⠀⠀⠀⠈⠀⠉⠀⠤⠐⠃⠈⠁⠂⠀⠀⡀⢡⡌⠀⠁⣠⠿⠁⠀⣍⡙⠷⣼⣇⣴⡶⠂⢙⡤⠐⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠙⡄⢠
⠀⠀⠀⠀⠀⠀⢤⠀⠢⠐⠈⠰⠴⣤⣄⣠⠄⠙⠳⠀⡉⠁⠀⠐⠓⠀⠈⠂⠀⠉⠹⡆⠀⠢⠄⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠰⢀⡒⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠔⠛⡓⠆⠀⠀⣠⣴⡣⠖⠁⠀⣥⡀⠀⠘⠀⢲⠄⠀⠀⠃⡀⠀⠀⠀⠀⠰⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠄⠈⡀⠀⠈⡜⢀⣠⠀⠀⣉⣴⠔⠊⠸⣿⠶⠶⢷⣦⡀⠉⠰⠀⠀⠞⣿⡀⠄⠀⠑⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⢠⡞⠉⠀⠀⠀⠀⠀⠂⠀⠈⣻⣿⣦⣄⡀⠀⡀⠈⠻⡄⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⡞⠁⠀⠐⠂⠀⠀⠀⠉⠀⠀⢡⣄⠺⣿⣯⣻⠁⠀⠀⠈⠁⠒⠂⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠠⠄⠀⢀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠦⠠⠀⣄⣀⣀⡀⣀⣀⣀⠸⠷⣾⠛⣻⠀⠀⠀⠀⠀⠀⡀⠀⡄⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⡀⠂⣤⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣰⠋⠀⡏⠀⠀⠀⠀⠀⠀⡄⢓⡀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⢘⡧⢤⣀⠀⡄⠀⢦⡀⣠⠀⠀⠀⠀⠀⠀⣠⣿⣤⡧⢠⡾⠀⣀⢀⡀⣄⠾⡁⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠝⠦⠀⠠⠆⠙⠤⠀⣀⠀⠀⠀⡠⢉⣰⠟⠴⠃⠰⠠⠀⠴⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢠⣄⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠋⢀⣀⡙⢛⣗⡒⢒⣺⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣰⣗⣿⣙⡿⣿⣓⡷⣏⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⣏⣀⣍⠀⠈⠁⠀⣼⢦⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⠁⢋⡙⣷⠶⣶⢾⡟⣁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣟⠀⢸⣇⣏⣾⡉⢹⡷⣏⠀⢛
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⢮⣍⣉⠈⠉⠈⠁⠹⣧⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⢠⣿⡿⢛⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡛⢿⣿
⠀⠀⢸⣿⣷⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣾⣿
⠀⠀⢸⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⢠⠀⠰⠜⠘⠃⠈⠁⠈⠈⠈⠈⠈⠁⠚⠀⠷⠀⡤⠀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠦⠈⠋⠀⠀⠀⠀⣤⡀⢤⡄⢰⠀⡆⠀⢰⠀⢠⡄⠀⡀⠀⠀⠙⠃⢰⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⢶⠂⠃⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⡆⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠁⠘⠀⠃⠐⠀⠃⠐⠀⠙⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⡇
⠀⠀⠀⣿⡿⢛⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣿⣿⡛⢿⣿⡇
⠀⠀⠀⣿⣷⣬⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣥⣾⣿⠇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢀⠀⠤⠈⠑⠒⠒⠦⢆⡀⠘⢂⡀⣤⠈⠫⡒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣆⠀⢀⠀⢈⣡⣴⡴⠂⢀⣠⡅⣈⣳⣀⠉⠀⠈⠐⠐⢆⠀⠲⠿⢦
⠀⠀⠀⠀⠀⠈⢧⡃⣿⢂⣡⣶⠾⢁⡥⣄⠀⠈⠀⠀⠀⠈⢥⠀⠒⠠⡄⠈⠺⣆⠐⠀⠨⣷⡇
⠀⠀⠀⠀⠀⡸⠂⢻⡿⢻⢋⣀⡠⠁⠒⠐⠆⠉⠀⠀⠀⠀⠈⢣⡆⣀⣧⠐⠂⢼⠙⣦⠀⡼⣇⡰⣆
⠀⠀⢸⡀⢠⠀⡸⢟⡷⣟⢩⣀⣀⣠⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠴⢼⣄⢹⡁⡄⣿⣧⡟
⠀⠀⡄⢧⣊⢲⡟⡿⣼⠟⠉⠑⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⠊⠦⣘⡁⠻⢷⣿⣸⣿⢠⠄
⠀⢣⠰⠞⣧⣟⡽⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⡀⣿⠹⣿⢲⡂
⠀⠈⣷⡀⢸⣿⡡⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣙⣿⡇⣿⣔⠇
⢶⡀⣏⢿⣾⣿⣼⣯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⢹⡟⣿⣿⠅
⠀⢻⣸⡎⣿⣿⠋⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⢾⣿⣿⠏
⠀⠈⣿⣧⣿⣿⢸⠟⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠈⠫⣼⣿⣿⡄
⠠⣿⡼⣧⣿⣿⡅⠀⣻⣟⡆⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡶⠷⣾⣿⡇⣷
⠀⢺⣿⡹⣿⣿⣧⠀⢊⣿⣷⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡿⢈⣿⣿⡇⡇⠀⠄
⠀⠀⠩⣾⠘⣿⣿⣧⠘⣿⡏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣽⡗⣾⣿⣿⣷⠁⠈
⠀⠀⠀⠙⣧⣿⣿⣿⣷⣜⣾⣿⣭⣶⣼⡆⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⣰⣶⢽⡟⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣯⣄⣻⣺⣿⣿⣦⠀⠀⠀⠀⠈⣻⠻⠾⠷⢧⣧⣯⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⢻⠿⢄⣀⢹⣿⣋⡉⠉⢉⠉⠙⠛⠻⣥⠋⢁⡔⢺⣓⣰⣿⣿⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢿⣶⡞⣛⣶⣤⡔⣀⡤⣷⡺⠛⠉⠉⢉⣉⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠻⡟⠉⠝⠉⠂⠨⠃⠀⠐⠒⠉