Waffle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣒⠴⠲⣲⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡂⠀⡴⠆⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠲⣶⣶⣻⣿⡿⣷⣏⣋⠶⣿⣤⣴⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣤⠀⣞⢂⡹⠥⠖⠙⣨⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢛⣾⣿⣿⣿⣻⣿⣾⠿⣟⣻⠾⢿⢿⡧⣀⢔⠤⢀⠀⠀⠀⠀⠀⠀⠚⠧⠒⠈⣥⠄⠘⣟⣀⠘⢦⣂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠠⡿⠉⠛⢿⣋⢷⣟⠟⣛⣧⣿⢿⣯⣖⠏⠀⠀⠁⠢⠀⠀⠀⠀⠀⠈⢏⣠⠜⢒⣃⠈⣤⠀⡘⡥⠖⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⡤⢄⣀⠀⠀⠉⠀⠋⣀⣵⣿⣿⡫⠵⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⡸⣧⠔⠙⣭⠄⠘⣏⣠⠸
⠀⠀⠀⠀⠀⠀⠀⠐⣾⣶⣶⣌⣧⡙⠈⢛⣛⢢⢤⣐⣿⡷⢿⣿⣑⡭⢥⡦⢀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⣩⠕⠰⣟⣠⠙⢞⣃⡬⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣛⣿⡤⣙⣑⣦⣤⣤⣲⣶⣿⢻⡷⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⡻⣿⣿⣯⣽⣷⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣾⣿⣿⣯⣟⣓⣬⠿⠟⠛⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢡⣭⡏⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⡠⠒⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡇⠒⠂⢩⠀⠀⡰⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢆⡃⠾⠃⡀
⠀⠀⠀⢀⠄⠴⠷⢄⡐⢚⠀⡠⠏⠲⢄⠀⠀⡠⠃⠀⢀⠤⡆⠀⠂⠣⣕⠂
⠀⠀⠀⢪⠰⣽⣑⠢⡘⡿⣏⣄⠖⣲⢘⠛⠉⠀⠀⢀⣤⢀⢡⠀⣀⠔⢉⡄⡀
⠀⠀⢀⠜⠢⢌⣓⢪⠟⣃⠯⣋⣩⠄⠊⢀⠤⢦⡼⠀⢀⠀⠤⡉⢠⠁⠀⢱⠮⡫⠵⣄
⠀⡠⠁⠀⠀⣀⡘⢁⠈⠁⢣⠘⡄⠑⢄⢊⡙⢁⣖⢢⠀⠀⠀⠀⠉⠑⠂⠡⣒⠡⠒⢩
⡐⠀⠀⢠⢊⢴⢀⠎⠀⠀⠀⡆⡅⠀⢨⢏⠙⠪⣄⠅⠖⢴⢀⡀⡀⢠⠞⠆⢡⡠⠒⣉⢷
⡇⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠑⠁⠀⣸⣄⠉⠂⢼⢸⠢⠌⠈⠉⢙⢠⡤⢆⠘⠔⠊⣀⡼⣢
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠦⣑⠢⡆⡜⠙⠠⢏⠂⣘⣄⠗⠉⣑⠰⢚⠥⠊⢁⡇
⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⠷⢤⣀⡽⡰⠭⣒⢄⡀⠀⠀⢀⠤⣊⠽⢌⠣⣀⠔⢻⢬⠑⢆⠀⠀⠀⠀⢀⡀
⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⡏⠠⡁⠈⡄⠀⠀⠦⣀⠈⠉⠚⠥⠭⠕⠊⠀⣀⠜⠀⠀⢠⠎⠀⠅⢀⡆⠀⢀⠔⠁⡠
⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠈⡷⡀⠁⠀⣈⠂⠤⠤⠐⠙⠢⢄⣀⣀⣀⠔⠚⠀⠠⠤⠔⠡⠔⢀⡠⢪⠃⡠⢃⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡑⠲⠤⢌⣉⣁⣒⠒⠒⠒⠒⠓⠒⣂⣀⣀⣀⡠⠤⠔⢊⡡⠖⢁⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⢉⠡⠄⠂⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡠⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢖⣒⡿⠉⠉⠉⢉⠠⣀⣉⣦⡤⠶⠿⢷⣶⡶⠶⠶⠮⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣀⡨⠉⡀⠒⣈⣦⠵⠖⠛⠉⠁⣠⠤⠴⠶⠖⠒⠒⢲⣦⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⣴⣥⠴⠶⠚⠋⣉⡤⢖⡾⣿⠀⢼⠓⠐⠒⠲⠿⢲⣾⣫⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠶⠋⠁⠀⡔⣠⢖⡿⢥⢖⣣⡐⡏⢰⣻⢀⢀⢢⡲⡗⡜⠁⣿⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⡡⢖⣥⣶⣸⠀⠛⠘⠒⠶⠷⣿⣿⡇⠈⣇⠠⡱⠳⠑⢰⠇⠀⡏⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⢵⡿⠁⢊⡴⡿⣿⠇⠀⣴⣶⡖⠲⠶⠶⠶⠄⠇⢀⣬⠓⠋⠍⠉⠘⠒⠚⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⠞⡀⣰⡫⢊⢎⡼⠀⢠⣷⠇⠈⢉⡉⠐⢺⠀⠀⣮⣌⣉⣙⠛⣛⡻⣳⡆⠂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⢋⠞⢼⠹⣥⣀⣉⡇⡀⢸⢹⠀⠐⣁⠤⠂⡸⠀⢠⡇⣠⠔⢉⠉⠓⡞⣿⢟⢠⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣰⠣⢃⣦⣬⣭⣉⠙⠛⢰⠀⢸⣼⣰⣊⣁⠄⢀⡇⡇⢸⠋⣡⠔⠁⠀⢠⠃⣿⡌⡎⡇
⠀⠀⠀⠀⢠⡞⢋⡽⠃⣠⢏⡿⠭⣭⣉⣙⡆⠌⣠⡴⠶⠦⣤⣍⣉⣉⠀⠀⠘⠾⠷⢮⣅⣀⣼⣀⢧⠁⣇⡇
⠀⠀⠀⣰⠏⢴⡟⠁⣰⢋⠟⢀⡤⢀⡀⢰⠁⠀⣏⡯⠉⠒⢒⠂⠠⢽⠀⢰⠛⠓⠒⠦⣤⣬⡭⢥⡈⢰⣿⠃
⠀⠀⢰⠃⢠⠏⠠⣰⢣⢋⠔⣩⠖⢉⡤⣼⠀⢰⣿⠀⢀⡴⠃⡀⠀⣿⡀⣼⠉⠙⠒⠦⣭⡭⣴⣞⡇⢸⣿
⠀⠀⣿⢀⡾⢀⢧⣿⠃⠀⠈⢀⠔⠉⠀⡇⣿⢺⢹⠐⠋⡠⠊⢀⠀⣻⠇⡇⢀⠤⢒⡩⠖⢁⣿⢻⠃⢸⡏
⠀⠀⡇⢰⡇⠸⠈⠻⠶⠶⠶⠶⠶⠤⠴⠁⣽⡸⣞⣠⠞⠀⠐⠃⢠⢿⠀⡇⠀⢀⡥⠚⠁⣸⠏⣸⠀⣿⡇
⠀⠀⠻⠼⠿⢿⣿⣟⣓⣲⣶⣦⠤⠼⢿⣭⣤⣃⣉⣙⣛⣛⣛⣛⡉⠀⠀⢧⣚⣁⣀⣀⣀⣿⢀⡇⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠒⠲⠶⠦⠤⣬⣭⣍⣉⣉⣉⣉⠓⠂⠀⠈⠉⠯⠤⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢠⠄⠄⢤⡄⢸⠀⠁⢠⡄⠀⠀⣬⠁⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡈⠄⢾⡂⢸⠉⠉⢹⣿⠀⠀⡆⢸⠁⠈⢉⣿⠀⢰⠁⠂⢕⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠁⠀⠉⠁⠈⣁⣐⣚⣁⠀⠀⠃⢈⣐⣒⠚⠋⠀⠈⠉⠀⠈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⡄⢸⠀⠀⢰⣿⠀⠘⠒⠒⢲⣿⠀⠀⠀⢸⠂⠐⢲⡏⠀⡛⠀⠀⣶⣽⠀⡶⡑⡀
⠀⠀⠀⠀⠀⠀⢠⠃⡔⣿⠇⢸⠀⠀⣸⣽⠀⠦⣀⣀⡸⠿⠀⠀⠀⠨⠀⠠⠼⠇⠀⠧⠀⠀⠧⠏⠀⠇⠗⠐⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣀⡀⣀⣀⣀⠀⡠⠤⠤⠤⣤⠀⠀⠀⡤⠤⠤⢤⡄⠀⡤⠤⠤⢤⡄⢀⠤⢄⡀⢁
⠀⠀⠀⠀⠀⡸⠀⡜⢰⣿⠀⡓⠒⠒⡖⣺⠀⡏⠉⠉⢹⣿⢀⢀⠀⡏⠀⠀⣿⡇⠀⡏⠁⢐⣿⠇⢸⠁⠈⣿⡔
⠀⠀⠀⠀⠀⡇⠸⢀⣼⠟⠀⠇⢀⣀⣿⠇⠀⠧⠠⠄⠼⠟⢸⢐⠀⠃⠀⠀⠛⣃⡀⠓⠒⠚⠛⠀⠘⠀⠀⠉⠃⡇
⠀⠀⠀⠀⠀⡭⠀⠀⠀⠒⠂⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠩⢥⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⡄⠀⢠⡄⢠⠤⠤⠤⣤⢀⡄⠀⠀⣤⠀⢸⢸⢠⡆⠀⠒⣲⡆⢠⡖⠒⠒⣰⡄⡴⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠈⡀⢇⠀⡯⡇⢸⠁⠀⣻⣾⢸⠏⠀⢘⣿⠀⠸⢸⢸⠀⠀⠀⢿⡇⢸⠀⠀⠈⣿⠃⡇⠀⢈⣿⢳⠁
⠀⠀⠀⠀⠀⠀⠠⠈⠂⠛⠃⠘⠒⠒⠛⠋⠘⠒⠒⠚⠿⠀⠀⢸⠈⠂⠀⠀⠛⠁⠈⠀⠂⠈⠉⠀⠉⠁⠉⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠱⡀⢤⣦⠀⡄⠀⣂⣲⡄⣐⣒⣒⣲⡆⠀⠘⠀⡆⠀⠉⣽⡇⢰⠍⠉⢭⢿⠄⠶⢲⢖⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠻⠀⡇⠀⠀⣿⠀⡄⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣿⡇⢀⠀⠀⣸⣽⠀⠆⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡉⠈⠉⠉⠀⠉⠙⠉⠉⠀⠀⠀⠀⢉⡉⢉⡉⠀⢀⠀⢈⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠹⢿⠀⣾⠀⠰⣿⡇⠀⠀⡆⠛⠐⣾⡇⠀⠯⠑⡋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠨⠭⠟⠀⠤⠤⣄⣓⣊⡉⠀⠀⠂⠁



⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⢄⠀⢀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣌⠳⠄⠀⢀⣴⣿⣦⡉⠒⠅⣠⣾⡿⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠴⢤⠄⣀⠻⣿⡏⠙⠂⣰⣌⡛⠿⣆⡀⢂⣼⣆⡙⠻⠄⢀⣾⠄⣀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠢⢁⠠⢾⣿⠷⠤⡉⢀⡘⢿⣟⠛⠂⢀⣤⡛⠿⣍⠁⢀⣴⡿⢃⡘⠶⣿⡏⠀⠀⣤
⠀⠀⠀⠀⠀⠀⡀⢠⣾⣷⣦⢐⣦⠉⠲⢬⣑⠢⢠⣿⣷⣦⠈⠉⠡⣿⣽⠟⠂⠈⣡⣿⡏⢐⣿⡿⠶⠄⠉⡀⠾⣿⡇⢦
⠀⠀⠀⠀⠀⠀⢰⣆⠉⢉⡰⢿⣿⡟⠢⠠⢈⡝⠲⢍⡛⠷⢫⣾⣷⢄⡉⠓⢂⣴⡿⠋⢰⣌⠛⠦⠤⢀⣾⣿⠦⣄⠉⢁⡀⣶
⠀⠀⠀⠀⠀⠀⠈⢻⢢⣾⣷⣦⣈⠉⠂⢴⣿⡟⠓⠦⠌⣳⣦⣌⡙⠠⠀⣠⣾⠏⣄⠐⢿⣏⠁⡀⢰⣎⠙⠻⠤⠀⣡⣼⣧⣌⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡛⠻⢿⣿⡿⣱⣿⣦⣌⠑⢂⢠⣾⣷⣤⡀⠉⠳⣾⣿⡃⠠⣿⠒⠄⠀⢉⠰⢿⡟⠉⠁⠀⣮⡙⠻⣅⣈⠧⢈⡀
⠀⠀⠀⠀⠀⡄⠈⢿⣿⡦⡄⣠⡄⠻⢿⣿⡅⣠⣾⣦⡈⠙⠦⣨⣾⣿⠟⡉⠙⠲⢄⡀⠀⣠⣌⠷⠄⠀⢁⠴⢿⣿⡛⠲⠄⣤⡐⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⠎⣶⣿⣿⣶⠀⢈⣕⠿⣿⡀⠈⠀⢀⡻⡿⠋⠀⢾⠋⠀⠀⠉⠲⢤⡙⠀⠀⣀⣼⣷⣤⡉⠓⢡⣾⣿⡷⠄⡧
⠀⠀⠀⠀⠀⢆⠀⠒⣁⠀⡙⠻⠿⢷⣤⣾⠟⠒⠀⠀⠀⠐⢿⠏⢀⣼⠢⠀⠀⠀⠀⠆⠷⠤⡈⠑⠶⣌⡛⢧⡀⢀⣴⣧⣤⡝⠃⢀⡇
⠀⠀⠀⠀⠀⠰⠀⠀⠉⠳⡀⠀⣠⣾⡙⠻⢤⡀⠀⣠⡾⠇⢡⣄⠛⠷⠀⠀⢀⣸⣦⡀⠀⠀⠀⠀⣿⣤⣉⡳⢦⣙⡻⠿⣈⠉⠐⣾⠇
⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠛⣴⣿⣿⣟⡳⠄⠁⣼⣽⠃⢀⣿⡝⠉⠀⠀⣤⠛⠿⣁⠀⠀⣠⣆⡙⠻⠿⠿⣡⣾⣯⡛⠷⢮⣕⣶⡏
⠀⠀⠀⠀⠀⠀⠂⠀⠣⠀⠀⠀⢈⣿⣿⣚⣥⣾⠟⠁⣦⡈⠛⠧⠀⢠⣾⣿⠿⠦⠄⢁⡺⢿⡏⠁⠀⢀⢸⣍⠛⡿⠿⢠⡉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣽⣿⡿⢃⣠⣾⡿⡟⠓⠀⢠⣍⠛⠿⢴⡴⢡⣾⣷⣄⡈⠃⠰⠿⠟⠛⠛⠁⢀⣿⣷⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠑⢶⠀⠓⢼⣾⣿⣿⡿⠁⠈⠁⠈⠙⠷⠎⣴⣿⡿⠛⠲⠄⠰⠟⠉⠉⠉⠉⡶⠂⠀⣠⣆⡀⢀⣾⢿⡟⢛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⣿⣿⡟⠃⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢠⣿⣿⡿⠿⠻⣾⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠛⠢⣤⡴⠿⠷⣀⠀⣀⣴⣿⣦⣀⣀⣀⣼⠿⢷⣦⡶⠟⠃⣿⣤⣤⠾⠋⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠉⠙⠋⠁⣸⣄⠀⢀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠻⠦⠤⢤⣤⠤⠾⢦⢤⡤⠤⠴⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⡀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠿⠃⠀⠈⠒⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣪⣏⢉⡢⡀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⢊⣉⣥⠰⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡋⠸⣿⣍⣐⠇⣈⣢⠄⠤⣒⣋⡉⢰⣶⠎⠁⡘⢿⠉⢑⡴⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⣾⠓⠀⡮⢉⠵⢚⣩⣤⠆⠸⣿⠛⠀⠈⢷⣤⡴⠆⠈⢋⣭⡔⠲⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⣎⠀⡙⢷⠞⡑⢁⠰⡘⣿⡉⣀⡤⠹⠶⠚⣁⡀⢠⣶⡿⠉⠄⢿⡉⠀⣐⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣪⡤⠘⢿⣄⣹⠗⢀⣄⡛⠌⢄⠑⠈⢋⣉⡄⠄⠰⣿⠖⠀⠀⢻⣦⣈⠦⠈⢛⣉⣥⠠⡙⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡋⣾⣟⠁⢤⣠⠉⢡⣼⡇⠰⣿⡌⢆⠂⠀⡘⣟⠉⢀⣀⠹⡤⠾⠒⠀⢡⣤⣂⠑⡀⢿⡁⣠⡴⠘⢆
⠀⠀⠀⠀⠀⠀⠀⡃⡙⢎⠻⠶⠟⣡⣞⠀⠙⢷⡶⠛⣿⣎⢂⠀⠐⡘⠛⢋⣁⡀⢠⣾⠧⠈⠀⢿⣁⣸⠖⠈⢋⣭⣔⠆⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢹⣌⠀⠑⢄⡘⢿⣥⣈⡾⢂⣀⠺⣿⣙⣧⠣⠀⠐⠀⢿⠏⠀⢀⢻⣦⡴⠇⠀⢡⣴⡖⠀⡀⢿⡇⣰⣵⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⡀⠑⢄⠙⠋⣴⡧⠂⣄⣌⠛⢩⣧⠀⠀⠀⠘⠷⢖⣋⠄⢠⣶⡿⡅⠀⢿⣀⣲⡾⠈⢙⣉⣥⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡢⠀⠐⢦⡙⢷⣤⡟⢁⣖⠂⠙⢧⢀⠀⢡⠰⣿⠛⢀⣀⢻⣄⣼⠖⠀⣉⣤⢴⣾⣛⣯⣽⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⠀⠉⠢⣀⢰⣿⣁⠐⣾⠞⣆⢂⠂⢣⡙⠷⠞⢋⣀⣥⠴⢖⣻⣭⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠈⠑⢬⡛⠟⢁⣼⣿⡎⠆⢀⣳⣶⡟⣛⣥⡶⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢷⣄⠀⠈⣾⣿⡿⠋⠈⢦⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡿⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⣠⣤⣤⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀⣀⣄⢠⣤⡄⠀⠀⣠
⠀⢠⡖⠶⠒⢒⢰⣿⣿⣿⣿⣿⣿⣿⠋⢽⣏⣻⣆⣀⣹⣙⣛⣛⣛⣻⠃⡿⠾⠶⠾⣿⠀⢠⣶⣿
⠀⢸⡟⠚⠛⠙⠸⣿⣿⣿⣿⣿⣿⣿⠛⢻⣋⣹⣿⣛⣿⢻⡂⣐⠶⡟⣎⣗⠭⠈⣾⣿⢰⣿⣿⣿⣷
⠀⠘⡵⠓⠋⠋⣿⣿⣉⣉⣉⣹⣿⣿⣾⣿⣿⢻⢿⣿⣿⣯⣭⣴⣦⣭⣹⡇⠉⠉⢸⣿⣿⣿⣷⣿⣿
⠀⠀⣅⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛⣛⢋⣽⣿⡍⢻⠛⠿⠛⢹⣷⣶⣶⠾⣿⣿⣿⣿⡿⠟⠂
⠀⠀⢸⣿⣛⣛⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣵⣿⣿⣿⣿⣶⣶⡅⠾⠁
⠀⠀⠙⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⠋⢫⣬⣴⣶⣶⣦⣤⣀⡉⠉⠀⢀⢻⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠹⣾⣿⣯⣉⣉⣿⣯⡕⣦⢢⣾⢿⣿⣼⣿⢟⣿⣷⡙⣸⠗⠰⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢌⡛⠿⠿⠿⠿⢿⣛⠵⠟⢸⣿⣾⣟⢿⣷⣵⣟⡿⢡⢇⢀⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⢿⣿⣶⣷⣿⣿⣿⡂⠸⠆⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⣴⣾⣿⣿⣿⣿⣷⣶⡂⡄⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡴⢶⣤⣤⣀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠛⢿⠿⣿⢿⣿⠾⣿⣿⣿⠟⠀⠀⠰⡾⠸⡇⢻⡄⢫⠂⠀⠀⠀⠀⠀⢰⠘⠋⢠⡾⣿⠾⢣⣾⠋⣰⠆⣠⠀⠀⠀⠀⢨⢚⣠⡰⡜⢆
⠀⠀⠀⠀⠀⠀⠒⠨⠤⣀⠈⠈⣫⠾⡏⠁⠀⠀⠀⠀⠺⠤⢯⡈⣇⢙⡌⡄⠀⠀⠀⠀⠸⣷⣾⣿⢷⠟⣴⣿⢅⣼⣻⠃⢞⢠⠄⠀⠀⠈⣯⢙⢣⡔⠮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢯⠚⣇⢹⡆⢽⣂⠀⠀⠀⠀⠐⠿⣿⣷⣿⣾⣦⣾⣩⣿⠋⣾⣢⡇⠀⠀⠀⠀⠈⠧⣤⠛⣴⢦
⠀⠀⠀⠀⠀⠀⢀⡀⣿⣶⡄⣠⢰⣄⠺⡉⠀⠀⠀⠀⠀⣿⣎⣳⣬⣭⣴⡷⠂⠀⠠⠀⠀⠀⣨⣿⣞⠻⢋⣤⣝⢻⣏⣿⠟⠁⢀⠄⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⢀⣠⣎⠛⡁⠸⢇⠙⡥⠸⡆⠻⣔⠄⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠻⠷⠚⠋⠀⠀⠔⠀⠀⠀⣤⡶⠂⢧⢄⡄
⠀⠀⠀⢀⠻⢾⡄⡷⢻⡄⠙⣲⠩⣆⠸⣅⠱⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣍⣁⠏⠇⡼⡍⣠
⠀⠀⠀⠀⠑⢄⠙⢷⣼⡹⡄⠘⣿⡸⣧⣙⣢⣽⡿⠀⠀⠀⢠⠀⡄⣤⠀⠀⠀⡴⣴⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡍⣀⠫⠍⡴⠮⢁⣛⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡙⢛⡽⢀⣿⠷⠟⠋⠉⠀⠀⠀⠀⠀⣼⢸⣿⣏⣾⣿⣰⣇⣿⣻⠃⣴⢗⣶⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠣⠯⡋⣴⣯⣁⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠿⠈⠉⠉⠁⠉⠀⠀⠀⢀⡀⢜⣈⠓⣾⣇⡖⢀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢎⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡷⣈⣵⣾⣿⣿⣿⣿⣴⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣧⢾⣟⢀⣸⣿⣿⣿⣿⣹⡟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⠾⢿⡄⣰⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⢿⣿⣿⣯⣼⣿⣏⣿⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢷⣿⣿⣈⣿⣦⣼⣿⡿⣿⣷⣠⣇⠀⠀⠀⠀⠀⠀⢀⡤⠴⡆⢲⡇⢠⡀⠀⠀⠀⠀⠈⠛⠛⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⠚⠈⠙⠫⠿⢏⡭⢷⣿⡿⠿⠃⡀⠀⠀⠀⠀⢸⢜⡃⣑⣛⢈⣝⡈⡙⠃⠀⠀⠀⠀⠀⠀⡀⡀⢤⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⣈⢿⡻⠱⠶⣄⡀⢠⣶⣿⡿⢗⣋⡀⢈⡂⠀⠀⠀⠈⢣⣿⣹⡜⣁⣉⣃⣹⡃⠀⠀⠀⢴⠦⠰⠲⡧⠓⣠⡐⠮⣁⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⣞⣣⣭⣭⣿⣾⣻⣿⣿⡶⠛⠀⠀⠀⠀⠀⠈⢣⡟⢛⣍⢟⡿⢙⡛⠀⠀⠀⠶⠲⠸⠳⠻⣾⠉⡋⢴⡝⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠉⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠻⠤⠔⠉⠀⠀⠀⢀⣶⣺⠘⣏⠇⣻⠓⠶⣶⣡⣾⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡴⢋⡉⠙⠒⠄⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠐⠁⠀⢹⡦⣄⠀⠀⠀⠀⠀⢈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣠⠛⠠⠤⣀⠉⠐⠒⠀⠀⠐⠢⠴⣡⠂⠀⠀⠔⠉⡻⠃⡀⠈⢄⠐⠀⠄⣈⡀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠂⠀⢁⣺⣗⡦⠀⡀⠀⠀⠀⠠⠌⠙⠢⢄⣀⠊⢀⠌⠀⠀⠨⠵⣀⣤⣶⣽⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡔⢀⠀⠀⠁⠀⠋⠀⠂⠘⠢⡀⠀⠐⠤⢄⠀⠀⠉⠲⢮⣤⠄⠀⠀⠀⠀⠉⢺⠋⠀⠀⠉⣉⠒⢄⠀⢄⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠮⣀⢀⠔⠁⡰⠁⠀⠀⠐⠛⠤⣿⢿⡿⠀⠀⣀⢀⠀⠈⠒⢦⣄⠀⢀⡰⠁⢀⠀⠀⠠⠳⢿⣷⣷⠈⢯⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢒⣤⡀⠁⠀⠸⠷⣖⡀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠠⣠⣩⣢⡀⢦⣈⠛⡍⠀⢠⢂⠔⠀⠀⠀⠀⠙⡟⢀⡟⢀⡎⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⠀⠀⠀⠉⠙⡟⠁⠀⠒⠒⠤⡀⠀⠑⠢⣀⡠⠊⠀⡀⠀⠀⠈⠙⠻⣿⣿⠀⣷⠀⠸⠂⠈⠓⢌⠀⠀⠀⠀⡜⠀⠈⠀⠂⡋
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⠓⠄⡀⢀⠌⠀⠐⠀⠀⡠⢄⡀⣝⣶⣄⠀⠀⠀⢸⡠⠂⠀⠀⠀⠀⢸⠃⢀⠁⡰⣥⠋⠐⠢⢄⠈⠐⠤⠞⠀⡔⠀⣰⣇⠇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠉⠉⠢⣄⠈⠁⠀⢠⠡⠂⡀⠀⠀⠈⠈⢿⠃⠀⢀⢀⡀⠉⠂⢄⠀⠀⢠⠃⠀⠀⢀⣄⡇⡠⠒⢄⣰⣹⡆⠀⠀⠈⠀⠰⢃⠋
⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠒⣾⠃⠀⣠⠤⣉⠐⠢⡀⠀⠀⡠⠃⠀⠀⠀⠀⣈⠑⢢⡀⠉⠒⠁⣠⠃⢀⡼⣹⠋⠀⠀⠀⢉⠞⠁⠀⠀⠀⠀⡡⠃
⠀⠀⠀⢀⠆⡆⠀⢁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢉⡢⣌⠑⠊⠁⢀⠄⠀⠠⠊⠉⠢⣿⣾⠆⠀⠀⠀⠀⠎⡴⠑⠠⢄⡀⣠⠊⠀⠀⢠⠎⠀⠔⠁
⠀⠀⠀⠘⣇⠙⢄⡈⠀⣀⡠⠁⠀⠀⠀⠀⠊⠙⠺⣿⣺⠃⠀⠀⠸⢄⡐⠀⠀⠀⠀⡰⠁⠀⠀⡀⠀⢀⡾⣄⣹⣦⡀⠀⠀⠀⢀⠔⠁⢈
⠀⠀⠀⠀⠙⢷⣄⡈⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⡨⠁⠀⠐⠉⠘⠢⣈⠑⠢⠤⠔⠁⠀⢀⠔⠁⢈⠏⠀⠀⠉⡿⠁⢀⠄⡰⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⣧⣄⠀⠀⠀⠀⠉⠣⠄⡀⢀⡔⠀⠀⠀⠀⠀⠔⠦⣼⣷⠀⠀⠀⠀⠠⠁⠀⣐⠳⠤⣀⠤⠊⣀⠔⢁⠎⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢭⣙⠀⠀⠂⠀⠀⠀⠀⠁⢀⠀⢄⠀⠀⠀⠀⠀⡠⠋⢀⠔⡡⠊⠀⣀⠊⠩⢛⡒⠒⠒⠉⠀⣠⢏⠂⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠴⣀⢀⠀⠀⠀⠈⠑⠢⣄⣉⣒⠐⣊⡡⠔⠁⡰⡁⠀⠔⠃⠤⣀⠄⢈⡁⠀⠀⠻⠁⡣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⣀⠀⠀⠀⠫⠍⣉⠁⢀⣠⠞⡑⢁⠊⠀⠀⠀⠀⠁⠒⠒⠀⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣐⡢⠄⠀⠀⠓⡨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⡇⠀⢹⠀⠀⠀⢸⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣠⠃⠀⠘⠦⣀⠀⢸⠀⠀⣹⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⡙⢙⣻⠆⠀⠠⠄⠀⠹⠧⠀⠀⢝⣦⠴⠴⠃⠀⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⠉⠉⠀⠀⠀⢀⣀⢀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠹⢹⠀⠐⠋⠉⢙⡇⠀⠘⠉⠁⠈⢷⠀⠐⠉⠙⠙⠟⡇⠀⢱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠀⠠⡀⠀⢘⡇⠀⠀⡀⡀⣀⣶⠀⠀⣀⣀⣀⣈⡇⠀⢘⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢲⣆⠀⠀⠾⠒⠒⣦⠀⠸⠙⠛⠖⣆⠀⠸⠙⠙⠟⢳⠀⠀⠈⡹⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⡀⠀⠀⣿⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠑⠋⠀⠀⠑⠀⠐⠋⠀⠈⠀⠐⠚⠉⠀⠀⠉⠉⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢦⡀⠀⢠⠤⣔⠶⡄⠀⢠⡶⠶⠄⣄⠀⠰⠋⣲⡶⠶⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠞⠀⠈⠹⡅⠀⠀⡠⠤⠤⣟⠀⢀⡞⠁⠀⠀⠣⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢸⠇⠀⠀⠹⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⡤⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣒⣭⣾⣿⣶⣵⣲⠿⣿⣋⣿⣆⡠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣭⠄⣿⣿⣿⣿⣿⣽⡟⣿⣷⣶⣭⡙⣓⣻⣾⣙⠿⣱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠂⠭⣿⣿⣿⣿⣾⣽⣿⣿⣿⢿⣫⣥⣿⠿⢿⣫⡿⣎⡴⣊⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⢀⠀⠠⣽⡏⠙⠻⢟⡗⠹⣿⣟⡿⠛⣻⣥⣿⢿⣏⡵⣺⠏⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⢳⣦⣅⣙⡤⢉⡂⠠⠈⡙⠒⠍⣾⠋⢙⣿⣿⣾⠷⢛⣅⠀⠀⠲⡄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⢻⣎⠻⣍⢻⣾⣧⣔⠂⠄⢀⣤⢖⣫⣿⣿⣟⡯⠭⠐⡄⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣿⣦⡈⠠⡥⠴⠚⠛⡓⠶⠿⠿⣛⣿⣿⣴⣖⣚⣿⣉⣠⣿⠇
⠀⠀⠀⠀⢄⠀⠀⠀⠙⠻⣿⣿⣽⣿⣿⡦⣫⣭⠵⣶⣶⣶⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠉⠛⠻⠿⣿⣷⣷⣿⣿⣿⠯⠽⣗⣂⣽⣿⠿⠟⠋⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠑⠀⠠⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⡠⠉⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠠⡄⠀⡀⡰⣇⠀⠀⣀⠁⠀⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠁⡅⠀⢠⠀⠐⠀⠐⠉⠀⠺⠖⠈⠁⢀⠀⠀⠀⠤⠀⢂⡀⠀⠀⠡⡄⠀⠑⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠨⠀⠀⠀⠈⠀⠀⠒⡀⠀⠂⠉⠀⠀⠈⠳⡀⠀⠈⠂⠀⠀⠈⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠂⠀⠤⠀⡗⠤⠈⣀⣀⣀⠀⠀⠀⠀⠤⢰⠀⠀⠈⠀⠀⠀⢁⡀⠀⠀⠠⡀⠀⠈⠂⠀⠀⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢰⠃⠀⠀⡀⠀⠉⠉⠩⠀⠀⠃⠈⢂⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⢀⠀⠀⠰⣄⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠶⣶⢡⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⡀⠠⣦⢀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠈⠳⠀⠀⠈⠁⣀⠀⠐⠢⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠱⠄⠙⠷⠶⣀⣀⣀⠠⠤⠤⠐⠂⠲⢭⡉⠣⡡⡀⠀⠀⠀⠀⠀⠥⢤⡀⠀⠀⠢⠀⠀⠈⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠐⠲⢤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⠀⠀⠀⠀⠉⢄⢀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠰⣆⠀⠀⠀⠀⣀⠀⠀⠈⠋⠑⠲⠤⣤⣄⣀⡀⠀⠀⠸⠀⠀⢀⣀⠴⠞⣓⣕⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⣄⠠⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠓⠒⠁⠒⠂⠴⠶⠀⠀⠉⠛⠓⠒⠚⠛⠛⠉⠀⠉⠀⠀⠀⠳⠤⠴⠶⠆⠛⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⢈⣀⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⢶⣾⣧⣴⠶⠆⠤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⢠⣄⣐⢈⣡⡤⠄⠀⠀⠀⢀⠉⠻⠟⠃⣀⣤⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣒⠀⠂⠀⠈⠛⠻⢿⡍⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠉⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡉⠉⢀⠠⠔⠀⠀⠉⣁⠀⠴⢶⣶⣶⣿⡖⠒⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⣀⣤⣤⣴⣶⠖⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠂⠀⠈⠉⠈⠀⣤⣤⣶⣄⣀⡀⠁⠀⠀⠈⠉⠛⠿⣆⡀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠉⠛⠿⣿⣭⣠⣄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⡾⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠁⠀⣀⣀⣠⣀⡈⠀⠀⠀⠈⠛⠿⠶⠤⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣷⣤⣀⠀⠀⠢⠈⠑⠄⠀⠀⠈⠁⠀⠂⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⣤⣔⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠾⠟⠀⠙⠳⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⠿⣿⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠿⠟⠷⠒⠒⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈⠉⠋⠁⠈⣉⣛⣛⣶⣤⡁⠀⠀⠀⠀⠀⣤⣾⠫⡭⠁⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢿⣿⢻⢷⣶⣤⣴⣾⡿⣿⣏⣛⠁⠈⠉⠉⠀⠀⠀⡈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠞⠃⠙⠒⠀⠀⢈⡉⠉⠀⠛⠓⠈⠁⠈⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⣤⡶⠤⠤⠦⠀⣀⡀⠴⠶⠖⠒⠀⠦⠄⠀⠀⠀⠀⠀⠠⠤⠤⠤⠔⠀⠀⠟⠛⠢⣤⣀⢀⣠⣤⣤⣀⣄⣀⣄







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣋⠌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣡⢞⡡⠔⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠃⡾⢱⠋⡴⠀⣿⡒⠒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢰⡇⢰⠁⢿⡀⠱⣄⣈⣉⣀⡀⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⢸⡀⠘⣆⠐⢌⠢⠤⣀⣀⣀⣀⡀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠈⢧⠀⠈⠲⣄⣩⣴⣷⣦⣤⡀⠉⢱⣨⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠳⡀⢣⣄⣸⣷⣤⣌⣿⠉⠉⠈⠛⣿⣿⡄⢸⣬⢷⣤⣽⢶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠡⢾⣯⣹⣿⣷⣼⡏⠀⠙⣿⣿⡴⠀⠀⣠⣿⣿⣷⣿⣿⣮⣿⡻⡶⣋⠔⠊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠙⠒⠶⢿⣿⣿⣿⣿⣄⣠⣿⣿⣷⣯⣶⣿⣿⣷⣾⣛⡻⣿⣏⣼⣿⣗⡴⢛⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣀⠂⠠⢄⡘⣿⠋⠻⠿⣿⠿⠟⢻⣿⣿⠿⣿⠻⠿⣿⣩⣿⣿⣽⣏⣵⠞⣫⠇⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⢦⣍⡒⠢⠽⣋⠐⠦⢤⢈⡉⠓⠶⣭⣙⣛⣡⠶⢛⡿⣯⣿⣧⣿⣫⠴⡟⠁⠠⣠⡀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢻⣯⡻⣲⡶⢭⣑⡢⠤⣀⠈⠂⠤⠀⠀⠻⠒⣡⡶⣿⣿⣿⣛⣥⠴⠚⠆⠀⠀⠻⢷⡀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠤⣤⣤⣀⠻⣷⠙⠿⡄⢻⣿⡿⠶⣬⣙⠂⠀⠴⠶⣛⣥⣾⣿⣿⣿⠧⠷⠒⢈⣹⠀⠀⢀⣀⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣿⣤⣿⣷⣄⠈⠢⣤⠤⠖⠒⠟⠛⡓⣾⠿⠿⠟⣛⣿⣿⣶⠶⢟⣛⣽⢛⣔⣠⣯⡿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣟⡻⢿⣿⣷⡀⠿⠚⣚⣃⣤⣥⣤⡶⢲⣾⣿⣿⡯⢹⣿⣿⣿⣿⣿⣿⣿⠟⣡⠂⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣍⣟⣫⣩⣽⣿⣷⣿⠿⣿⣿⣋⠹⢧⣾⣿⣶⣿⠿⢛⡫⠔⠈⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣟⣛⡋⢉⠭⣤⣚⣋⠩⠍⠒⠋⠁⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢶⣦⣤⣤⠴⣻⠃⠀⠀⣾⣿⡀⠀⢀⣤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣏⠆⠀⠀⠈⠛⠛⠶⠋⠀⠀⣾⣿⠁⠙⠛⣿⠄⢀⢀⡉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⣿⣿⣷⣿⣶⣦⡄⠀⠀⠀⠈⠛⠿⠶⣤⣶⠏⠀⠀⣾⡧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣼⣿⣿⠌⠀⢉⡟⠁⠀⣴⣶⣄⣀⣀⠀⠀⠁⠀⠀⠸⣿⣿⣹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠛⠷⢦⠞⠀⠀⣸⣿⡛⠋⠛⣿⣿⠀⠀⢀⣀⠀⠀⠈⠉⠃⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⠿⣧⣄⣀⠀⣴⠟⠀⢀⣿⢿⣷⣶⣤⣀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⣿⡿⠙⠛⠛⣿⡇⠀⣰⣶⣦⣄⠀⠀⠉⠛⠋⠀⠀⣿⡟⠂⠁⠈⣹⡿⠀⠀⢰⣶⣌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠻⢿⣿⡀⢀⣴⠏⠀⢠⣿⡟⡿⠿⢷⣤⣤⠀⠀⠀⠀⠙⠓⢶⣄⣠⣿⠃⠀⠀⣹⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠈⠙⠛⠋⠀⢀⣿⣿⠀⠀⠀⠀⣰⠟⠀⠀⣰⣦⣤⣀⡀⠀⠙⠋⠀⠀⢠⣿⡯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣷⣦⣄⠀⠀⠀⠀⠀⠙⠿⢦⣄⡀⢸⡟⠀⠀⣴⣿⠻⠛⢿⣿⣷⡄⠀⠀⠀⠀⠙⠻⠾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢻⡿⠀⠀⢀⣠⣄⡀⠀⠀⠈⠉⠉⠡⠀⢰⣿⠧⠄⠀⠙⢉⣿⠁⠀⢠⣷⣶⣦⣄⣀⡀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠁⠀⢀⣾⣿⡿⢿⣿⣦⣄⠀⠀⠀⠀⠈⠻⢶⣦⣄⣀⣾⠅⠀⠀⣾⣿⠿⠟⠛⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⡟⠁⠀⠛⢻⡏⠀⠀⣠⣦⣄⠀⠀⠈⠉⠉⠉⠀⠀⢠⣿⣿⣶⡶⠶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣄⣀⣿⣿⣃⣀⠀⠘⡏⠀⠀⢀⣿⡟⠻⣿⣷⣶⣀⠀⠀⠀⠀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠙⣿⡌⠀⠀⢀⣿⡟⠀⣀⣀⣨⣬⣷⣿⣄⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣸⡿⣡⡾⠛⠋⠉⠁⠈⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣼⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠱⠲⡄⡀⠀⠠⠀⠇⠄⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢶⣮⡤⡄⠀⠀⠄⠐⠈⠄⠀⠀⠀⠀⠔⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠚⠀⠈⠉⠢⠴⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠆⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡉⠐
⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠀⢴⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠰⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢀⠀⣄⣀⡀⠠⣤⡄⠀⣤⣄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠑⢷⡀⠁⠀⠀⠁⢤⣄⠀⠢⣀⡄⠃⠈⠀⠂⠜⠳⢤⡄⠀⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⣄⠘⠏⡀⠙⠆⢈⠲⢄⡀⠑⢤⣈⠄⠀⠀⠀⠲⠛⠑⠀⡀⢉⠠⡴⠛⣈⣅⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠁⠈⠀⠀⠀⠀⠀⠛⠳⣦⠙⠣⣤⠀⠀⡀⠘⠃⠀⡀⠀⠀⡔⠃⣀⡀⠘⠉⢉⡣⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠆⠀⠂⢀⣀⠰⠟⠶⠀⠀⠀⣀⣤⣐⢠⡐⠣⠄⡑⠀⣨⠔⠋⣀⠀⠊⠉⣛⡀⠀⠀⠀⠴⣆
⠀⣄⡰⠇⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣄⠈⠁⢠⠄⠀⠀⢀⣀⠙⠁⡄⠀⠉⣀⡴⠇⠺⣄⠀⠈⠙⠃⡀⠀⠈⢀⠤⠶⣤⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠁⠰⠀⠀⠀⢠⠖⠋⠉⠳⢀⡠⠖⢉⣠⣴⠤⢴⣄⠑⠢⣄⠈⢀⡴⠶⣄⠀⠀⠀⠈⢀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⣠⡀⠤⣤⠀⠘⢀⡤⠔⠉⠀⠀⠈⠁⢄⡀⠀⠈⠛⣠⠈⠻⢤⡀⠀⣤⠑⠀⣀⠐⠏⠁
⠀⢠⡈⠛⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠛⠉⠀⠀⠈⠁⠖⠉⠀⡤⠾⠋⠑⠺⢦⡄⠈⠁⢤⠘⠉⠛⢄⡀⠙⢦⡄⠀⠚⠛⠃
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢤⠴⣠⡀⠠⢄⠀⠀⠀⣠⣔⣤⠀⠀⠠⡄⠈⠀⢀⢀⡈⠢⣄⠉⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠖⠋⠀⠘⠁⠀⠀⠀⠉⠂⠀⠀⠰⠊⠁⠀⠉⠺⠀⠀⠀⠀⠆⠀⠀⠙⢷⣌⡑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢏⠀⠀⠀⠈⠲⠴⠂⠀⠀⠰⠞⠳⢆⣀⠒⠆⢀⣤⠤⠀⠶⡆⠀⠀⠀⠀⠀⠉⠁⠀⢹⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⡠⢉⡴⢦⡈⠀⡄⠤⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠐⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠀⠀⠂⠈
⣶⣶⣶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣾⣤⣿⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢶⣖⣒⡒⠀⢠⡬⢁⡐⡒⢲⣒⣂⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⢒⠛⣩⣅⠼⠶⢌⠃⠄⠄⣊⣥⠞⠉⠀⠈⠣⡈⠫⡉⠉⠉⡨⠉⡒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡁⠺⢿⣦⣤⡄⠀⠀⢀⢴⡃⠀⠶⠿⣵⠆⡀⡖⠀⠀⠀⠠⠬⠒⢈⡴⠋⠇⠀⠐⠩⢓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡦⢤⡈⠁⠈⠓⢄⠙⠃⡠⠐⡈⠴⠊⢫⣓⣴⣦⣨⣞⣀⣖⣈⡩⠵⠃⡀⠀⠀⣻⣶⠊⠙⠂⠀⠀⢉⠎⢙⣦
⠀⠀⠀⠀⠀⢀⡴⣮⣷⣯⣙⢦⡉⠢⣀⡠⠛⣠⣤⡄⠧⠀⠀⡔⠁⠀⠠⠀⠉⠉⠙⠛⠛⢪⡑⣾⠔⢤⣤⣿⣦⡑⢄⡀⠄⣊⣴⠖⠉⠛⠇
⠀⠀⠀⠀⣠⠞⣴⣟⢍⣿⡇⠑⠍⠂⢈⠀⢉⠝⠉⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡂⠀⠀⠀⠈⠹⣦⣤⣾⣫⡄⠀⢀⡠⢚⡭⢑⢤
⠀⠀⠀⣴⡗⢂⠙⢾⡟⠁⠀⢀⣤⢞⡋⢁⣈⣮⠀⠀⠀⠀⢸⠁⠂⠀⠀⠀⠀⠀⢀⠒⣀⠈⠀⠀⣿⣀⣄⡀⠀⠀⠈⠻⡿⢿⡷⢚⣩⡾⠫⠶⠀⠡⠱⡄
⠀⠀⡼⣽⢿⠀⠑⢀⠁⢤⢺⣰⠖⠛⠁⠄⢙⡁⠀⠀⠀⠠⢼⣀⢀⠀⠁⠀⠀⡀⢀⣀⠤⠐⠀⢨⠇⠀⠀⠈⠀⠀⠀⠀⠹⠨⢜⠻⣷⡷⣾⠅⠀⠀⠸⠼⡄
⠀⢰⠇⡿⢛⣠⣼⣿⣤⠞⠉⠀⡄⠀⠀⢠⠋⠀⠀⠀⠀⡀⠀⢉⢠⠀⠀⠀⠀⠀⠠⠈⠀⠁⠀⡾⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠐⠈⣿⢅⠀⢀⣠⠖⢡⢱
⠀⡏⢠⡾⣿⠷⠊⢪⡆⠀⠀⢀⠀⠐⠀⢹⠀⠀⠀⣠⠁⠀⠀⠈⠳⡀⠂⢀⣶⢀⠀⠠⠁⢄⡼⠁⠀⠀⠀⠀⠀⠀⢹⣴⢿⣄⡤⠤⠶⠡⢀⠛⢉⡤⢿⡿⠆⣇
⢸⡇⡦⡙⢦⡹⠀⠈⠓⠙⠉⠒⢰⠞⠻⠿⣆⠀⢀⠇⠀⠀⠀⠀⠀⣨⠦⡀⢻⣄⡀⣄⡥⡞⠁⠀⢀⡆⠀⣀⡠⠀⠀⠿⠈⣷⣤⡄⠤⠀⢀⣱⢦⡙⠷⡁⢀⢸
⡘⠆⣄⠈⠣⡈⠂⠀⠄⢘⡀⠀⠀⡇⠀⠀⡈⣿⡿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⡎⠀⠈⠀⠀⠀⠀⡄⠀⢽⡀⠀⢤⣰⡿⢯⡀⠙⠢⡐⢾⢸⡇
⡇⡀⢿⣃⡠⢊⣤⡀⢾⡛⠇⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣼⡁⠀⠘⣿⡾⣟⣋⣴⠶⠳⠀⠀⡸⠃⣰⡇
⠀⢁⠈⣡⡾⢫⣗⠙⢦⡉⢳⠀⠀⠀⠀⡰⠁⠀⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣄⠀⠀⢀⣠⡎⠁⠁⢀⠀⣿⣧⣈⠻⡁⠀⣀⠴⣊⢾⠇⠉⡇
⠀⠀⡆⣧⣴⢟⠯⣅⠀⠙⢎⣄⠀⣠⣴⠃⠀⠔⠃⢻⡀⠀⠀⣤⠀⠀⣀⣠⣤⣠⣤⠤⡴⠟⠙⢠⠋⠉⣿⣿⠿⡷⣄⠀⠘⠄⠀⣿⣍⡧⠈⠋⣵⢊⡁⢀⢀⠃
⢸⠀⠐⡈⡇⠁⠀⠀⠀⠠⢊⡽⠋⠙⣿⠀⠀⠀⠔⣯⣽⣿⡻⡉⠙⠝⢯⠈⠻⡓⢀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠃⠐⠂⢑⣦⣤⣴⠿⣋⡥⡺⠣⡈⠃⢀⠌⠘⢠
⠈⢧⠀⠐⢌⠢⠀⢐⠀⠀⠁⠀⢀⣨⡿⠶⣤⣶⣿⣿⡿⢋⡁⠺⢦⣄⡀⠀⠀⢸⠳⣭⡂⢤⡀⠈⠀⣰⠆⣀⣤⣤⣴⡟⠙⢋⣥⠞⠕⣊⡄⠀⠀⢀⡀⢠⢀⡾
⠀⠈⢇⠀⠈⢡⡴⠁⠀⠀⢀⡴⠛⣉⢴⡂⢬⡹⡿⠁⠀⠀⠀⠀⠀⢀⡉⠛⠓⡂⢀⡿⠿⣻⡿⠟⠖⠛⢋⠻⣿⣿⠍⠛⢢⡘⠳⣥⣾⠿⠓⠀⢀⡼⢠⠇⣼⠇
⠀⠀⠈⢧⣀⠀⠃⠀⡠⢚⣩⠖⠁⡉⠘⡃⠀⠑⠄⠀⠀⠀⠀⠂⠀⢈⣀⣄⣠⣶⣟⢷⠛⠁⠀⠀⠈⡂⠞⣃⣬⡙⢦⠀⠀⠈⠢⡈⠁⠀⢀⡠⢊⣴⠋⡠⠋
⠀⠀⠀⠈⠻⡄⠢⣼⠐⠊⢙⠺⣵⡕⠋⠉⠀⠀⢀⡬⢋⡉⣿⡟⡿⠟⢉⣻⠁⠉⠉⠲⡑⢄⠀⠐⣈⢴⣾⣿⠏⠙⠦⠑⢦⡀⠀⠈⢢⠈⡩⠖⠘⢁⣴⠇
⠀⠀⠀⠀⠀⠙⢶⡑⠀⠀⠀⠀⠒⢍⡒⠤⠄⠊⣩⡶⡏⠈⠀⠙⣦⡾⠛⠁⠀⠀⢀⠠⠞⡀⡐⢾⣏⡿⢋⣾⢆⠀⠀⠈⢢⡙⢄⠀⠀⣾⠁⠀⣠⡾⡋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢀⠀⠀⠀⠀⠁⠂⠤⢌⡙⠓⠤⢤⣀⣄⠳⣄⠀⠄⢊⡤⣐⣽⡟⠈⠂⠙⢾⡋⣀⣀⠤⠴⣒⣩⠄⠚⠳⡴⠃⣠⡾⣟⣯⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣅⡂⠀⠀⠄⢀⠀⠀⠀⠀⠐⠒⠀⠠⠀⣀⣐⣚⠓⠛⢛⣚⡓⢂⣈⡅⠀⢉⡓⠐⠚⠋⠉⠀⠀⢀⣠⣴⣿⡿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⡈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠛⠃⠐⣀⠀⠀⣀⣠⣴⡿⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⡒⠃⠀⠲⠤⠄⠀⣀⣀⣀⣀⣀⣠⣒⣀⣤⣥⣶⣶⣶⠿⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠄⡀⠀⠀⡀⠐⡀⠀⠈⢁⠀⠋⣀⠀⢈⡀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠁⠀⠁⠀⠀⠡⡀⢠⠕⡀⠀⠀⠈⡐⠐⠡⣀⠀⢀⡅⠊⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠈⠁⠒⠉⢄⠀⡤⠊⠁⠀⠀⢁⠒⠊⢀⡀⠀⢀⠁⠈⠠⠁⠀⣠⠀⠀⠀⠀⢀⠄⢠⡆
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⢁⠒⠚⢁⡀⠀⢀⢼⠧⣀⠁⠂⠄⢁⠄⠀⠀⢈⠐⠀⢄⠀⠀⢀⠈⠁⠢⠄⣀⠔⢀⢔⣵⠃
⠀⠀⠀⠀⠀⢠⡓⠪⢔⡄⠋⡀⠀⠀⡀⠉⠁⠪⠻⣦⣍⡓⠢⢄⡉⠐⠴⢁⡀⠀⠀⢀⠁⠁⠤⡀⠀⡀⠀⡰⣣⡿⠁
⠀⠀⠀⠀⠀⠀⠛⢕⡢⢌⣁⠚⠥⣊⠀⠄⠀⢀⠄⠀⠈⠛⠷⣦⣬⣙⠒⠤⣈⠐⠒⠁⣀⠀⠀⣠⠉⣠⢊⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠽⢶⣤⣌⡐⠪⠅⡒⠠⢀⡰⣢⠞⣹⡿⠻⢶⣤⣉⠲⢤⣀⠁⠒⣀⠔⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠯⣶⣤⣍⡒⢊⡵⡾⠋⠀⠀⠀⠈⠙⠻⢶⡬⣉⢉⡥⡺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⢀⠤⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⡄⠆⡔⢰⡆⡾⠒⡗⢸⢁⡖⡶⢰⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠏⠀⠳⠘⠃⡇⢠⡃⠸⠈⠧⠄⠠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠈⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠂⠀⠊⡄⢈⠕⠊⠀⠀⠁⠀⠊⢀⠀⠀⡀⠊⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡐⠐⠡⡀⢠⠖⠁⠀⠀⠁⠂⠉⣀⠀⠀⡀⠊⠀⠄⠀⡠⠀⠀⠀⢀⠔⢀⡆
⠀⠀⠀⠀⠀⠠⢂⠀⠀⢉⠐⠊⠤⠀⠀⠤⣕⢤⣈⠐⠀⢁⡀⠀⠀⡀⠈⠀⠀⠀⠀⠈⠀⠂⠀⠠⢀⣔⣵⠃
⠀⠀⠀⠀⠀⢎⡑⠢⢄⡁⠄⡀⢠⠄⠁⠐⠉⠳⢦⣉⡓⠢⢄⡐⠊⠀⡀⠀⡠⠈⠀⠂⠀⡠⠂⡠⣫⡞⠁
⠀⠀⠀⠀⠀⠀⠑⠱⢂⢤⣁⠒⠤⢀⠠⢀⠎⠀⠀⠈⡛⢷⣦⣌⡑⠢⢄⡉⠀⠠⠀⠀⠄⢀⢊⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⢌⡐⡂⠅⡂⢊⠔⣡⡾⠋⠙⠳⢦⣬⣑⠢⠤⣡⠔⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⠖⡀⢐⡺⠋⠀⠀⠀⠀⠀⠈⠑⠣⠴⢒⠬⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠰⢀⠤⡄⣴⢠⡇⡙⢀⢤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⢷⠁⢧⠸⠅⡇⢸⠀⡇⢺⡈⠠⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠤⠤⢤⠀⠀⣤⠠⠀⠀⠠⣐⠲⣖⠒⠒⠒⠒⢒⣶⡒⠒⣀⣢⡒⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣞⠟⣀⢀⠦⠀⠀⣀⡤⣉⣉⠻⠯⠉⡀⠀⠠⢍⡰⠤⠤⣤⡤⠰⠏⠉⢐⠢⣔⠃⣀⠠⢴⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⢝⡶⣝⡪⢐⢨⠛⠒⠒⠒⠁⠂⠀⠉⠛⠒⠂⠒⠒⠓⠈⠓⠐⠂⠀⠉⠁⠀⠀⠀⠤⠄⠀⢎
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣜⠻⢳⣬⣓⠫⠓⢦⣥⠤⣤⡤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠄⠒⠒⠒⠒⠒⠒⠒⠙⠉⡉⣽⡀⢀⣀⡀
⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢪⣟⢦⣄⠉⠓⢦⣤⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡤⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⢺⣄⣀⡀⠀⠉⠉⠒⠠⣄
⠀⢸⠀⠀⠀⣠⡋⡀⢒⢲⣾⣦⣍⡲⢭⢒⢤⣈⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⣆⠒⢼⣿⣦⠀⠀⠀⠈⡇
⠀⠈⠷⡢⣄⣀⠙⠢⠄⢈⠈⠛⠻⢿⣶⣮⣑⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠀⠀⠛⠁⠀⣀⣤⡾⠁
⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠶⣖⣒⣀⣬⣬⠭⣭⣭⣄⠀⠀⠀⠉⠉⠀⠁⠀⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠥⠤⠀⢒⣒⡒⠊⠩⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠺⠯⠭⠭⠥⠤⠤⠤⠤⠶⠞⠓⠒⠲⠖⠖⠦⠤⠀⠠⠄⠀⠀⠤⠭⠭⠭⠉⠉⠉⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠒⢒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⠥⠤⠔⠉⡍⠓⠋⡍⠑⠤⠤⢍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⠤⢃⣀⣀⣸⠀⣇⣀⣀⡇⢸⣀⣀⣀⠣⢄⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾⠀⡆⣠⠤⠤⢤⠀⡤⠤⠤⡄⢠⠤⠤⢤⡀⡆⢸⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠖⠁⠀⡇⣿⠀⠀⢸⠀⡇⠀⠀⡇⢸⠀⠀⢸⡇⡇⠀⠑⢦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢘⡆⣉⣉⡁⢈⣉⣉⣉⠀⣉⣉⣉⡁⢈⣉⣉⣉⠀⣉⣉⡁⣞⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡞⠀⠀⠈⡇⣿⠀⠀⢹⠀⡇⠀⠀⡇⢸⠁⠀⢸⡇⡏⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⡄⠴⠴⠃⠹⠴⠦⠞⠀⠷⠴⠶⠇⠸⠦⠶⠼⠁⠳⠴⠄⡜⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⡀⠉⠉⡆⣼⠉⠉⢻⠀⡏⠉⠉⡇⢸⠋⠉⢹⡄⡎⠉⠁⣸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠉⣶⠀⡇⢿⣀⣀⣼⠀⣇⣀⣀⡇⠸⣄⣀⣸⠇⡇⢠⡎⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⠦⣅⡠⠴⠤⢦⠀⡤⠤⠤⡆⢠⠦⠤⠤⣀⡥⠜⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠣⣀⣀⡘⠀⡇⠀⠀⡇⠘⣀⣀⡠⢃⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣈⣓⠚⠉⠑⠒⣋⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⠆⠀⡂⠀⡆⠀⡄⠀⡆⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠃⠊⠙⠀⠀⠀⠁⠀⠃⠀⠃⠀⠀⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⠄⠀⠀⠀⠀⠔⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡈⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⣷⣤⡄⢣⣀⢠⣴⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣛⣁⣿⡇⣸⡇⢰⣿⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠰⠿⠁⠼⠿⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⠉⠩⠭⠭⠤⠄⢶⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⠀⠀⠀⢀⠀⠄⠤⠤⠤⠆⠀⠒⠒⠂⠚⠉⣩⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠐⠒⠒⠂⠀⠐⠒⠒⠀⠐⠃⠀⠀⠐⡟⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣠⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⣿⠋⢹⡏⢈⡟⠉⡉⠙⡿⠈⠹⡏⢈⡏⠉⣀⣩⠁⠹⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡏⠀⣈⠀⣸⠀⢸⠇⢀⠇⢀⠀⠀⣸⠁⠀⠉⣹⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣸⣇⣠⣿⣄⣀⣠⣾⣀⣼⣆⣠⣿⣄⣀⣈⣿⣃⣰⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣠⣴⣶⣷⣦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠾⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠉⠀⠘⠟⣿⠏⠉⠙⠷⣦⣴⠾⠛⠉⠹⢿⡿⣿⣿⣿⢷⣤⣤⠶⠛⠉⠀⠀⠀⢶⣮⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠿⡀⠀⠀⢀⣩⠠⣄⠀⠀⣿⡇⠀⠀⠀⢀⣠⣀⠀⠈⠉⠃⢹⡇⠴⠋⠛⠐⠤⣈⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⡆⠈⠁⠀⠀⠈⠱⠂⣿⠀⢠⠔⠊⠉⠘⠈⠛⠂⢀⠀⢸⡇⠀⡐⣡⠠⠀⢈⠙⡜⠋⠻⢉⡏⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣸⣿⠀⢸⠀⠀⠀⡀⠂⠀⠂⣷⠀⣸⡇⣻⠊⠇⠇⡠⡄⡄⡇⢠⣄⣿⣸⣿
⠀⠀⠀⢀⣠⡴⠾⠿⢿⣟⠿⣶⣌⡘⢤⠀⣀⣤⣶⣿⣻⣯⡛⢦⣅⡢⣄⠀⠠⠃⣂⣥⣶⣿⣿⣾⣟⡷⠶⣆⣇⠧⠇⢾⣿⢿⣿⣿⣦⡀
⠀⣠⣾⣿⣥⡀⠀⠀⠀⠀⢀⡉⠹⠻⣷⣿⣿⣿⠁⠈⠙⢛⣉⠁⠈⠙⠶⣤⣶⡿⣟⡿⠟⠻⠿⣶⡍⠛⢷⣦⣀⣀⣤⡴⠞⠋⠉⠩⢛⣛⠷⢦⣄⡀
⢀⣿⣿⣿⣿⡷⢀⠔⠂⠀⠀⠈⢳⢰⣿⣿⠿⢃⡀⠴⠀⠸⠭⣥⣄⡀⠀⣿⡇⠛⣉⡤⠶⠲⢤⣀⠀⠀⠐⣿⣿⡟⢁⡤⡶⠄⡀⠀⠼⣿⣿⣿⣿⡿⣦
⢸⣿⣿⣟⠟⠁⡟⠀⠀⠀⠀⠀⡀⢸⣿⠋⠃⢸⠁⠀⠀⠀⠄⢄⣉⠿⠂⣿⣧⡋⠉⠁⠐⠈⠙⠻⠻⣦⠀⣿⢻⣇⢈⠄⠐⠀⠈⠙⠲⠬⠛⢟⠿⡅⣿
⣿⣿⣿⡏⠀⠠⠄⠀⠀⠀⠀⠀⡃⣾⣿⠀⠀⣠⠀⠀⠨⢠⣰⡼⠛⢰⠀⣿⠹⡇⠀⢀⢬⠺⣘⣸⣤⣿⢰⡇⢸⣿⢰⣾⣠⠤⠀⢀⡀⢻⠀⠸⣀⣿⣻
⠙⠿⣿⣧⣤⠘⠷⣀⠀⢘⣠⡴⠚⠛⠻⣦⣄⢯⣀⣀⠨⢠⣢⡤⢒⣡⣼⢿⣦⣈⢇⣀⠀⢠⣢⢵⠼⠿⠚⣀⣼⣿⣜⣶⣄⣀⣏⢡⣠⢸⢸⣷⣿⣿⣿
⠀⠀⠸⣟⣿⣶⣤⡴⠟⠋⣁⠤⠋⠀⠀⠀⠙⠻⣦⣝⢓⣨⣵⣾⠟⣉⡀⠀⠈⠙⢷⣮⣗⠨⠀⣀⣤⡴⠞⠋⠁⠀⠈⠙⠿⣯⣟⡙⠋⠉⣐⣿⣿⠿⠋
⠀⠀⠀⠈⠙⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠫⣠⡴⠾⡇⠄⡄⠲⢳⡄⡘⣝⢿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⣿⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢀⠀⠃⡁⢰⢀⡉⢛⠛⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣂⠠⡆⡇⡇⢼⢰⡇⡿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⣿⣄⠀⠁⠅⣼⠘⡇⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣝⠎⠙⢿⢻⡦⢷⡌⠚⢉⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⢨⣛⣂⣤⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠿⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢢⡔⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⠞⠗⢺⡧⠠⢧⣤⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠉⠀⠩⠨⡥⠄
⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣀⣤⣐⣒⣤⣄⣀⣂⣠⣄⣤⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠤⠤⡤⠄⠵⠦⠌⠏⠉⠉
⠀⠀⠀⠀⠀⢠⠈⠙⠒⠚⠛⢛⣻⣿⣿⣉⣉⣉⣥⡬⠋⠉⠉⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡇⠀⢨⠫⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠂⠤⠤⠴⢞⡉⣉⣹⠍⣉⡁⠈⠉⠉⠉⠉⠘⠋⠀⠀⠀⠀⠀⠀⢀⠀⢄⣈⠈⠉⠈⠁⠀⠨⠉⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠘⡶⠤⠤⠤⠶⠾⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⡇
⠀⠀⠀⡀⠠⠀⡧⢭⣤⣭⣿⡿⠥⠿⠭⠭⠭⢭⣙⣓⣒⡒⠂⠀⠛⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⣀
⠀⠀⢰⡀⠀⠐⠮⣥⣐⣒⣂⣀⣉⡄⢩⠤⠴⡦⠤⠶⡶⠤⣔⣒⡀⠈⠉⡀⠴⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠩⣑⠂⠄⠀⠀⠈⢉⣉⣙⠓⠙⢦⡴⠋⣉⣉⡵⠶⠉⠀⠀⠒⠁⠔⠁⣠⠖⠢⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠤⠄⠀⠀⠠⠤⠤⠤⠤⠄⠒⠒⠒⠂⠈⠀⠀⢀⠔⠁⠀⠐⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⢀⡔⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⢀⠔⢉⣤⠞⠋⠥⠐⠀⠀⠒⠦⢄⡑⠢⡝⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⡠⠁⣠⠟⡡⠊⠀⠀⠀⠀⠀⠀⠐⠒⠀⠠⠈⢆⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉⠁
⠀⠀⡰⠁⢰⠇⠜⠀⠀⡘⠀⠀⠀⠀⠀⠀⠠⠈⠀⡀⠀⢠⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⡀
⠀⢰⡇⠀⡞⢸⠀⢀⡆⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠈⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⡀⠀⠀⠀⣠⠞⠁
⠀⢸⡇⠀⠁⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠐⢀⠜⠁
⠀⠈⡇⠀⠸⡘⡀⡏⠀⠠⠔⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡠⠃
⠀⠀⠘⠀⠀⠳⠘⠁⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠆⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠊⠀⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠉⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠉⠀⠤⠐⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢀⣀⣀⣀⣀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠁

⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢠⢀⣤⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠘⠿⣷⣦⣆⠠⠂⢠⣤⡴⠆⢀⣴⣿⣿⣷
⠀⠀⢺⣿⣶⣦⢈⡙⠂⠤⠀⠂⠀⠀⠀⠀⠁⠀⡀⠀⠘⠀⠀⠀⠀⣸⣀⠀⠀⢿⡿⢄⣱⠦⠤⣀⠠⠘⣿⡟⠻⠋
⠀⠀⠘⠿⣿⣿⠀⠨⠉⣓⠲⠮⣙⡲⢰⣶⡂⡆⢀⣈⠀⠀⠂⢀⣦⣴⡇⠈⠉⠉⣉⢛⠿⠃⢄⣤⡐⢂⣬⣤⣤⡀
⠀⠀⠀⢀⣼⣿⣧⣤⣄⣀⣿⡶⡴⠋⡪⢥⡽⠲⠛⠲⢉⠛⢛⣭⣿⠘⡇⠀⣸⠆⣡⠀⡀⣀⠈⢻⣧⡘⠛⠉⠅
⠀⠀⠀⢸⣿⡿⣇⠀⠙⡷⠶⠶⠾⠖⠋⠉⠀⠀⠈⢻⣮⣛⣯⣍⣿⣟⠣⡤⠋⠀⠈⠁⠈⠉⠀⠈⠟⠠⣄⠀⢀⠀⡇
⠀⠀⠀⣘⡩⠞⠹⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠰⠀⠈⠳⣍⠙⢿⣷⣜⣠⣷⡤⠄⠤⠄⠀⡄⠰⡇⠀⠐⣿⣀
⠀⠀⠀⣶⣶⠀⠀⣤⣦⣤⣤⡀⠰⠃⠘⠳⠶⠿⠶⣶⠄⠀⢨⣹⢹⣽⣓⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⡟
⠀⠀⢼⠟⠋⣄⠀⠁⠀⠙⠹⡇⣿⡄⠀⠀⠀⠀⠀⠀⣴⡞⢻⠛⠡⡘⢀⠃⠀⠈⠻⠓⠀⠀⠀⠣⠤⠒⠻⠟⢀
⠀⠀⠀⠀⣄⡾⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠘⠷⠀⠀⠇⣀⡚⢳⡄⢀⠀⢀⡄⡀⠀⠀⣆⢀⠠⠄⠀⣼
⠀⠀⠀⠀⠘⣏⢧⡀⠀⠀⢠⡀⢸⣟⣠⣴⣶⠦⠀⠀⠀⠀⠀⠈⠛⣿⣿⣇⡖⡰⠛⠁⠿⢿⢾⠞⢠⠁⠂⣾⣿
⠀⠀⠀⠀⠙⢿⣿⣿⠀⣤⣴⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢻⠀⢠⣯⣿⣿⣾⡿⣌⣓⣠⡄⠀⠀⠀⠃⠀⣾⡏⣾
⠀⠀⠀⠀⠰⣼⣿⣿⣿⣿⣟⣋⣹⡏⢧⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⡯⣿⣏⠀⠀⣉⠛⢻⠧⣤⣄⣀⠀⠐⠿⠁⠈
⠀⠀⠀⢠⠀⣿⣿⡿⣿⣿⠂⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣼⣫⣿⣿⣿⡗⢰⣿⣶⣾⣿⢾⠊⠉⠀⡀⣸⠀⣀
⠀⠀⠀⢄⡀⠼⢿⣿⣿⣿⠂⠀⠃⠀⠀⠀⠀⢀⣠⡶⢿⣷⠋⣿⡝⣿⣿⣿⣧⣼⣿⣿⣿⡟⠀⠀⣽⡛⠛⣿⣴⣿
⠀⠀⢀⣀⠩⣤⣤⣯⣻⣏⠀⠀⠀⣀⣠⣤⠶⣿⠟⣴⣿⣷⠀⣼⣿⣷⣩⣿⡿⢿⣿⣿⣿⣷⣤⣀⢸⡇⢸⠿⠟⢩⡀
⠀⠀⢰⣿⣿⣿⠺⣿⣿⣿⠿⠿⠿⠿⠗⠋⠉⠁⣼⡿⡦⠉⠀⢹⣿⣿⠈⣿⣿⣬⣿⣿⣿⣿⠚⠹⠼⣿⡌⠠⡙⣼⡇
⠀⢠⣾⣿⡿⠟⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⠇⠠⠤⠤⢤⡼⠿⠿⣷⣿⣿⣽⠿⠿⠿⣭⣀⣤⣼⣿⣿⣿⣿⣇⠃
⠀⠘⠛⠋⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠠⠤⠂⣩⠕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⡠⠀⠀⠔⠈⠐⠁⢄⣼⠑⠀⡀⠘⠫⢷⠘⠠⢄⡀⠔⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠒⠄⠂⠀⠁⠀⡀⠘⠗⠒⠊⠉⠑⠢⡀⠀⠀⠀⠀⢀⡴⠋⣿⣱⣤⣀⠐⠠⠊⢀⣴⢺⠇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣀⠀⢻⠀⠀⠀⠀⠂⢁⡀⠐⣮⠉⣄⡦⠅⠛⠳⠀⣀⠐⢯⣢⡴⠋⠪⣹⠾⠃⢀⡀⠙⠾⡷⠂⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡒⠋⠀⣁⠂⠀⠁⣀⡤⡖⣾⣿⣿⡄⠘⣿⡁⠀⢀⠈⠀⠀⠀⠘⣆⡉⠂⠄⠔⠁⣀⠞⢿⣿⣗⠤⡉⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡠⡾⡉⠀⢮⠤⠐⠀⢀⣠⣄⠀⢻⣿⡿⠟⠋⠘⣿⡄⠈⠓⠈⣁⠤⠖⢡⠀⠀⢹⢯⠆⠀⠀⠐⢧⣵⠟⠈⠈⠂⠉⠀⣀⠐⢿⣨⡀
⠀⠀⢀⠔⠁⠀⡰⠘⠀⠀⢠⣶⢿⣿⡯⡿⣆⠀⢻⢇⡀⠀⠂⢉⣀⣄⠀⢿⣖⠔⠻⠉⠀⠀⡟⠀⣠⠺⣗⠢⣄⠈⠂⠤⠐⢁⡤⢞⣿⡙⢄⠙⠌⠑⢄
⠀⡰⠁⠀⠠⠊⠀⣈⢻⡄⠈⢿⠞⠋⠀⢀⡼⠃⠀⠀⣤⡔⢚⣭⣾⣿⣄⠈⢿⠁⠀⠀⢈⡄⢸⣿⣽⡷⠛⢃⣿⡭⠂⡀⠀⠉⡻⠟⠣⡛⡈⢀⠖⠀⠁⠑⡀
⠀⠀⠀⠀⢃⠀⠀⠻⠈⢿⡄⠈⠚⠒⢊⣠⣴⣾⣧⡄⠘⣿⡿⠛⠉⠸⡙⠦⠈⠓⠂⣁⣀⡀⢻⠘⠫⠄⠀⡨⠋⣠⣾⣿⢶⣄⠈⠢⡀⢨⠓⠃⠀⠀⠀⠈⠜
⡆⠀⠀⠀⠈⠰⠀⡀⠀⠈⢁⡀⠀⢾⣿⢿⣿⠷⢿⣷⡀⠙⢷⡀⠰⠂⢉⣠⣆⠀⠻⣾⣿⣯⣽⣿⢢⡀⠁⠠⣾⡻⢿⠟⢌⠛⠑⡄⠈⠈⡀⠀⠀⠀⠀⠀⠐
⡇⠄⠀⠀⠀⡰⠁⢀⠄⠒⡉⡿⡄⠈⢷⡏⠀⠀⠀⠥⠐⠀⠀⣤⢖⣲⣡⢾⣿⡆⠀⠳⢼⠀⠈⡻⠁⡠⢰⣶⡀⠙⢤⠀⠀⢁⠀⠁⢀⠜⠀⠀⠀⠀⠀⠀⢀
⠠⠀⠀⠀⠀⡄⠠⢀⣷⡷⠞⠻⣿⡄⠈⠓⠒⢁⣀⠤⣲⣱⡀⠘⣿⠟⠋⠁⠀⠙⠀⠀⠈⣧⠊⠀⡔⢀⣼⠀⠳⡀⠀⠱⡈⠀⠉⠉⠁
⠀⠁⡀⠀⠀⠈⠢⠤⠄⡀⢀⠄⠈⢁⡄⠀⢲⣽⣮⡻⠏⠻⢷⡀⠸⢦⠀⠀⠀⠜⡡⠤⠴⠋⠢⡘⢀⠨⠀⠀⠀⡃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐
⠀⠀⠈⠢⡀⠀⠀⠀⠀⢱⠄⢰⠀⢸⣞⠀⠀⢻⡅⠀⠀⠀⠀⠃⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡡⠊
⠀⠀⠀⠀⠈⠀⢄⠀⠀⠘⠤⠼⠤⠉⢽⡄⠀⠀⠛⢆⣀⣀⠀⢀⣀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠀⠀⠉⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠒⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣟⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡆
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠛⠛⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⣄⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣻⣾⣿⣿⣿⣿⣯⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⡀⠚⠛⠁⠀⠀⣰⡶
⠀⠀⠀⠀⠀⢀⣻⣾⣿⣿⢟⣛⢿⣿⣿⣷⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢁⠉⠉⠑⠒⠒⠂⠉
⠀⠀⠀⠀⠀⢈⣽⣿⣿⡏⣿⣿⣷⠿⣿⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠶⢿⣿⣿⣯⢐⣮⣜⣹⡿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠋⡿⢻⡿⣾⣿⣿⣮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣿⣿⣿⣿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⠟⠁


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

⠀⠀⣀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⢀⣿⣿⡿⠿⣿⣿⡇⠀⠀⠀⢠⣶⡚⢓⣦⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⡤⡘⠠⠀⣤⠄
⠀⢸⢿⣿⠁⠃⢹⣿⠥⠀⠀⠀⡿⠿⠣⠻⠿⢿⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠋⢿⡆⠀⢸⠟⠃⠀⠀⠸⣷⣤⣶⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠈⠙⠛⣛⡃⠀⠀⡀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⣭⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠁⠀⠈⡀⠈⠉⡉⢉⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣍⠁
⠀⠀⠀⠀⢤⣦⣂⠀⠰⠦⠉⠀⠀⠈⠆⠇⠘⠇⠿⠇⠿⠸⠿⠿⠁⠿⠸⠀⠇⠸⠇⠿⠇⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠆⠄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠰⢆⡀⠀⣀⠶
⠀⠀⠰⡦⢠⣤⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣆⣶⡄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⡿⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣾⠇⠀⠀⣌⣻⠋⢁⠈⠻⣋⣄⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⣿⡇⠉⢸⡇
⠀⠀⠀⠍⠉⠐⣚⡃⠀⠀⠘⡧⠀⠀⢰⡟⠀⠀⠈⣭⣟⠀⠀⠀⠀⣯⣭⠀⠀⠀⠀⠈⣽⣿⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇
⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⣽⡶⣶⣶⢾⡏⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣷⣤⡎
⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢋⡛⠈⠀⠀⠀⢠⣶⠆⠀⣩⣭⡀⠀⠶⠶⠀⠀⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡀⠀⠀⠀⣤⣿⡛⠙⠋⣹⡶⣤⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣄⡀⢸⣿⣦⣧⠀⠀⣿⣶⣿⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣬⣿⣿⣿⣧⣶⣿⣿⣯⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⠁⠀⠀⠀⢩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠻⠿⣿⣾⣿⣿⣤⣀⠀⠀⡀⣼⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⠞⠁⣠⣤⣾⠟⠛⠛⠳⠛⢹⣷⣥⡀⠘⣧⡀⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣠⠞⠁⠀⢸⡿⣻⣧⣦⣤⣤⣴⣶⣾⣿⡿⢿⡆⠈⢳⣄⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣸⣧⣏⠛⠛⠛⠛⠛⠟⠛⣋⣽⢸⣧⡀⠐⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠁⠀⠀⣠⣿⠃⡟⢿⣿⣶⣶⣶⣾⣿⣿⣿⠀⢻⣧⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⣴⠏⠁⠀⢹⣦⣤⣤⣤⣤⣴⣶⣶⡿⠀⣄⠙⠳⣄⠀⠀⠈⢣
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⢳⣯⣭⣭⣭⣭⣷⣾⠇⠀⠈⠀⠀⠈⠳⡦⠀⠀⢁
⠀⠀⠀⠀⠀⠧⣤⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⣉⣉⣉⣭⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢠⠀⠘⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⣴⣦⣌⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠈⢙⢰⠦⣀⠌⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⢬⣿⢷⣿⣦⡟⠉⠒⠀⠀⠀⠀⠢⠀⡀⢑⠦⣽⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣹⣟⣏⠻⣌⢧⡿⡉⡙⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣃⣿⣿⣿⣿⣿⠆⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡻⣾⣷⡟⠁⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⣠⠇⢿⠛⠿⠛⣫⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢹⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠟⠀⡆⠀⢡⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠈⠅⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⢠⣴⣄⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣐⡀⠀⣉⡈⠀⢈⡄⠁⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠾⠓⠚⠁
⠀⠀⠀⠀⠀⠌⠉⠙⠀⠋⠙⡂⠋⠙⡐⠛⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠋⠁
⠀⠀⠀⡴⠟⠻⠸⠛⠳⠀⠛⠻⡤⠞⠻⡵⠟⠻⠞⠃⠀⠀⠀⠀⢀⣴⣶⡞⠛⣿⠁⣿⠐⠀⠀⣀⣠⠤⠖⠚⠋⠁
⠀⠀⠀⢿⠠⡆⢶⡀⠶⢦⠀⡶⠶⡄⡶⢶⡄⢰⠦⠀⠀⠀⠀⣰⠟⣿⠀⣿⡀⢻⣆⣸⣷⣶⠋⠉
⠀⠀⠀⠸⣤⢀⢠⣥⠀⣤⣤⡀⢠⣤⣄⠀⣄⠀⢀⠀⠀⠀⠀⠘⣤⣼⣦⡼⣷⣛⣿⣿⣿⠉
⠀⠀⠀⠀⣄⡰⠀⣈⢃⡁⠈⣥⠘⠠⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠈⠙⠳⠖⢊⠙⠒⠉⠀⠠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢠⣐⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠉⠙⠛⠶⣤⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣀⠤⡝⠃⠀⣛⠁⠆⠈⣝⡻⠶⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢼⡏⠀⠀⠀⠀⠢⣭⣺⣧⣤⣄⣤⣭⣶⣦⡟⣠⣾⢟⣿⣿⣽⣶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⢀⢀⣻⣿⣛⣿⣿⡟⣿⣿⡿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣦⣴⣦⣤⣤⠤⡤⢤⣴⣿⠆⢐⣠⡶⠶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠄⠡⢤⡤⠄⠠⣟⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣌⠙⣲⠾⣵⠟⠋⣿⠦⠛⠁⠀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠛⠷⠶⢦⣄⣠⣤⣴⣾⡶⢾⡷⠛⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠲⣤⣀⠀⠀⣿⢠⢀⣤⡚⠽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠚⣀⠀⠴⠄⢀⡉⢀⣈⢸⣯⣮⣥⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠉⠛⢻⠛⠒⠋⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠊⠀⠈⣿⠂⠀⠀⠀⠙⢬⣴⣇⠿⡷⠻⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣶⢦⣀⠢⠤⠀⢈⠀⠀⠐⠁⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢀⠀⠀⠀⠖⠉⠙⣷⡾⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠍⠉⠛⠻⠷⠖⢻⠖⠚⠋⠁⠀⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠁⠀⢠⠄⠀⠻⡟⣃⠈⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⠀⠀⠀⠘⠀⠀⠀⢀⣀⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠧⣦⣽⠂⢀⢳⣾⣾⠿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⠀⣬⡄⢄⡟⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⡀⠀⣀⠀⠀⠀⠀⠀⠀⢺⡒⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠐⠃⢢
⠀⠈⠘⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⡠⠀⠀⠈⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠺⠶⠐⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠚⠃⠀⠀⠀⠚⠖⠳⠂
⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢠⠅⠂⠑⠀⠀⠀⠀⡀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⢀⡐⢀⡀⠄⠀⠀⠀⠀⠀⢠⢲⣷⡆
⠀⠀⢢⡴⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢈⡰⢾⡇⠀⠰⠀⠀⠀⠰⠃⠀⠁⠃⢠⡶⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢠⡝⠄⠀⠀⠀⠣⠤⠄⠂⠀⢂⠀⠀⠀⠀⣠⡆⠀⠈⠉⠂
⠀⠀⢠⣮⣤⣤⠀⠀⠀⠀⠠⠄⠀⠢⠈⠀⠀⠀⠈⠀⠜⠅⠃⠀⠀⠀⠠⠆⠀⠴⠈⠃⠀⠀⠀⠈⠋⠙⠚
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠤⠄⡀
⠀⠀⣊⣉⠑⠀⠀⠀⠀⠐⢀⣅⠀⠀⠀⠀⠀⠀⠀⣤⣧⢡⠀⠀⠀⠀⠐⠉⠀⣅⠀⠀⠀⠀⣀⠀⠒
⠀⠀⠙⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⠀⠂⠚⠐⢠⠀⠀⠀⠀⣄⡠⠈⠁
⠀⠀⠀⢀⠁⣀⠈⠀⠀⠀⠀⠀⡀⠀⢀⡀⠁⠀⠀⠀⠌⠬⠱⠸⠂⠀⠀⠀⢠⡌⢤⡌⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠋⠂⠀⠀⠀⠀⠀⠀⠞⠂
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠹⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠁⠀⠀⠀⠀⠴⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⣤⣴⠶⠾⠟⠛⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⡶⠒⣻⠋⢉⡅⢀⣠⡀⢠⡆⣠⡄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠉⠀⠀⢻⣇⢸⠇⣿⠈⣿⢸⡷⠻⠧⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢻⡿⠀⠙⠛⣁⣀⡤⠴⠒⢚⠉⡇⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠟⠁⠲⡎⢉⢁⠄⡄⠈⠠⠄⣀⡿⠉⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢰⠇⢀⡴⣶⠜⣦⡥⠤⠔⢒⣋⢻⣿⣃⡤⠤⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢠⣿⢃⣾⣣⣯⠀⣤⢶⡆⠀⠈⠁⣈⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠸⠏⠸⠟⣽⠋⢼⣿⣾⠁⠀⢀⣾⣋⣾⡿⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⣀⣿⠷⠶⠷⠶⠶⠶⠎⠛⠋⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠠⢤⠒⠰⡐⠀⡄⠀⠀⠀⠦⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⠖⠑⠑⠀⡀⠀⠀⡀⢄⡀⣀⢀⠀⠀⠄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⢸⡀⠀⢄⠩⠘⠀⠳⠌⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⢩⣥⡶⠖⠛⠋⠉⠉⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣾⡆⢀⡴⢚⣥⡁⢲⣶⢲⣆⠀⣴⡟⣷⠀⣠⡶⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣼⣿⢄⣿⠃⠀⣿⠇⣾⣇⣼⠟⠀⠻⣷⡅⢰⣿⣤⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢠⣿⠃⠈⢿⣦⡴⠋⢰⡿⠈⢿⡶⠸⣧⣽⠇⠹⣿⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣤⣤⣤⣄⡀⠀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣧⠀⠀⢀⣀⣀⡀⣨⣤⣌⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣠⣤⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⢴⣟⣉⣄⠀⠀⠀⠀⠙⠛⠛⠛⠳⢤⣽⠚⠃

⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⣦⣄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣀⣀⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠤⠤⠤⢤⣀⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠏
⠀⠀⠘⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠋
⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⢀⠇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣩⠶⣶⣦⡙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⠶⣶⣮⡹⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢸⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡷⣿⣶⣿⣿⡷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⡷⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸
⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⡙⠿⠿⢟⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣝⠻⠿⠟⢡⣿⣿⣿⣿⣿⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⡤⣶⣶⣶⣦⡄⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠿⣿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠠⠴⠄⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣵⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣯⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⠟⠛⠛⠻⠿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⠟⠛⠛⠻⠿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡀
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠛⢿⣦⣄⣀⠀⠀⠀⣀⣠⣴⣿⠟⠛⠻⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠛⠻⣷⣦⣀⡀⠀⠀⠀⣀⣠⣶⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠉⠉⠑⠲⢄⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠴⠚⠉⠉⢩⠉⠉⠉⠑⠲⡮⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡞⠀⡠⠊⠀⠀⣀⠤⠄⠠⢄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠇⡞⠀⠀⠀⠜⠁⣠⡤⠀⡀⠑⡄⠀⠠⠒⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡼⡸⠀⠀⠀⢸⠀⢸⣿⣄⣀⣻⠀⢰⠀⠁⣰⣿⠁⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢰⠃⡇⠀⠀⠀⠸⡀⠘⣿⣿⣿⠟⠀⡈⢀⠀⢿⣿⣿⡶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠯⠤⣇⠀⠀⠀⠀⠑⢄⡀⠉⠀⣀⠜⠀⠀⠢⡀⠉⠉⢀⠔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢓⣶⠦⠤⠤⣀⣀⣠⣾⣆⠀⠘⣄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣦⡀⠀⠀⠈⢻⣿⣿⣷⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠲⢦⣤⣤⣤⠶⠂⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠹⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣦⣌⠓⠤⣀⡀⠀⠀⠀⠀⠉⠉⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣤⣌⣉⠑⠒⠒⡖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣷⣄⠀⠀⠈⠻⢿⣿⣿⣷⣦⣀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢻⣿⣿⣷⣤⡀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⠈⠻⢿⣿⣿⣷⣦⣄⡀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣶⣤⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠓⠢⢄⣉⠛⠻⢿⣿⣿⣷⣦⣤⣀⣀⠀⣀⣩⠽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢰⣿⣿⣆⡀⠀⣀⣤⣤⣼⣿⡇⠒⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣀⡄⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣷⣾⣿⣶⣶⣦⡀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣹⣿⡇⠀⠘⠘⢻⣿⣿⣿⣿⣿⣿⣿⣇⠈⠙⡟⠋⠛⠿⠿⠿⣿⣤⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣄⠀⠀⠿⠋⠈⣿⣿⣿⣿⣿⣿⡄⠀⢰⣿⠷⢰⣶⡄⡿⢸⣿⡿⣿⠰⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣄⣸⠟⠋⠉⢉⣙⣷⣀⣤⣤⣤⡈⠻⠛⠛⠿⠿⢷⠈⣶⣶⠶⣌⣙⣡⣦⣬⣿⣁⣼⣷⣮⠆
⠀⠀⠀⠀⠀⠀⠐⣿⡿⠟⢛⠻⣿⡷⣾⡿⠀⠛⠛⠛⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⣻⣿⡟⠿⣿⣿⣿⠋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣶⣿⡿⡗⢠⣴⣟⣳⣶⠊⠉⠉⠉⠁⠀⣀⠀⠀⠀⠀⠀⢠⣀⡀⢀⣀⣙⠛⠥⠀⢠⣉⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢋⣿⣦⢂⣧⣬⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⢉⣴⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⡇⠀⠀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⡾⣱⣿⣿⠿⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠉⠻⣿⡏⠙⢁⣀⠀⢸⣿⡉⠘⠂⠀⠒
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣏⡁⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⡤⠀⣿⣷⣿⣿⣿⣿⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠹⠟⠋⠙⣿⣿⣿⠃⠀⠀⢀⣸⣿⣇⠀⠀⢠⣾⣿⣿⣦⠀⠀⠈⠁⠀⠀⠀⠉⡟⠀⠉⠉⠉⣿⣷⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠀⠻⡇⠀⣰⣿⣿⣿⠿⠛⢡⣀⣿⠏⠁⠀⠈⢻⣿⣤⣴⣶⣶⣶⣤⣇⢉⣀⡈⢃⡏⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠀⠀⠁⣠⣿⣿⣿⡿⠠⠄⠲⠤⠭⠤⠀⠀⠤⠘⣿⡏⠛⠛⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⣿⣿⡇⠀⠀⠴⠶⠶⣶⣾⣶⣤⣶⣿⡇⠀⠀⢻⣿⣿⠿⠁⠉⠉⠧⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠘⠛⠛⠛⠛⠻⠟⠛⣴⣿⣿⣦⣄⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣾⣟⠋⠉⠉⠰⣶⠂⠉⠉⣉⣿⣿⣤⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢄⠀⠀⠀⠀⣀⢔⡑⢤⡀⠀⠀⣠⣴⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⢊⣥⣞⣲⣍⠢⡤⡪⠷⠛⠛⠶⣍⢢⢾⣛⠥⠬⣳⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠁⠀⣷⠋⠠⠄⠀⠈⢻⡇⡁⠀⠀⠂⠀⠈⢿⢾⠀⠐⢢⠀⢘⡿⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⡽⡀⠀⠈⠂⠀⣸⣇⠧⡀⠀⠀⢁⣴⠽⣿⣄⠀⠈⢀⣴⣷⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠪⢕⣢⣤⡾⣫⡔⣷⣮⢗⡶⢛⣥⣒⣦⣙⠻⣾⠏⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⡏⡎⠁⠀⢆⠈⣗⠐⡏⠀⠀⢢⠈⠇⡏⠀⢀⡜⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⣄⠀⠀⠀⠑⠢⢄⣧⡲⢤⡀⢀⣴⣫⡦⣓⢄⣀⡠⣴⠧⣷⢠⠊⠀⠀⠀⣰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠈⠑⢭⡝⠋⠀⡇⠈⠑⢾⠊⠁⠀⡿⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠲⢄⣸⡇⢀⠤⠓⢤⣀⢿⠄⡠⠔⠃⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠐⠀⠀⣀⣀⡠⠾⠃⠀⠀⠀⠀⠀⠘⠦⢤⡀⠀⣀⡀⠀⣀⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣾⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠩⠁⠀⢀⠀⠀⠰⠦⠔⠂
⠀⠀⠀⠀⠘⠆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠈⠀⠈⡄
⠀⠀⠀⠀⠀⠀⣧⢸⠂⡆⢸⠇⡇⣞⠱⣿⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡄⠀⠀⠀⠀⣤⡤⣤⠤⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⢠⢐⣒⡂⣀⠀⠀⢀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣰⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠃⢄
⠀⠀⠀⠀⠀⠀⠴⠓⠒⠔⢲⢄⠀⠀⠀⠀⠀⠈⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣳
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⣠⠞⣿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢸⡧⣾⢁⡤⢰⡆⣶⢠⡞⢂⣴⠂⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠀⠀⠀⠀⠀⣼⠀⢠⢾⡇⠛⠈⠂⠈⠉⠃⠈⠃⠀⠉⠀⢺⠗⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠉⠘⠁⠀⠀⠄⠀⠀⠀⠀⠔⠁⠀⠀⣿⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⢩⠗⠀⠀⢸⠀⠀⢀⣼⠃
⠀⠀⠀⠀⡞⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⠂⠀⠀⠀⠀⠀⠰⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠈⠑⠄⠈⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡩⠃
⠀⠀⠀⠀⠀⠀⡤⠤⠄⠠⠤⡔⠀⠀⠀⠀⠀⠀⠐⢤⠀⠂⢤⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢲
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠠⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⢀⣜⠀⡠⠄⣴
⠀⠀⠀⠀⠧⠄⢀⡀⠀⡀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢷⠀⠴⠃⠛⠀⠑
⠀⠀⠀⠀⠆⠐⠀⢀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢦⡀⠀⠀⠀⠀⠀⠀⠻⣀⡇⠀⠀⠀⣀⢠⣠⡆⣾⡈⣇
⠀⠀⠀⠀⠀⡀⠀⠈⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡆⠇⠘⠘⠏⠃⠀⠁⢹⠈⢦
⠀⠀⠀⠀⡄⠀⣀⡀⠀⡀⡀⠂⣁⡄⠀⠀⠀⠰⡛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⣿⣶⣶⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠉⠀⠉⠉⠻⢧⡐⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣻⠃⠀⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠻⣿⠀⠇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠠⠁⠀⣿⠿⢳⡿⡹⠛⢣⠘⠻⣿⠊⠛⠿⠻⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⡐⠀⠀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢈⠐⠉⠀⠁⠀⢠⡆⢶⡌⠃⠁⠒⠒⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⢠⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠚⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠞⠀⢸⣿⣶
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣴⡿⢋⠹⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⣿⡇⠀⠀⠘⠆
⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢰⠀⢸⡿⣷⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢀⡜⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠘⣷⣿⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠁
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠿⠁⠀⠈⠀⠀⠀⣀⠀⣀⡀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⣿
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡛⢀⠀⠀⠀⠀⠐⠀⢀⠇⠈⠈
⠀⠳⡄⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⡧⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎
⠀⠀⠈⠢⡀⠀⠀⠀⠀⡈⠀⠃⣰
⠀⠀⠀⠀⠘⢄⠀⡀⢶⣇⠠⠐⡛⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠠⢄⣿⡇⠐⢁⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠀⠋⠄
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⢾⡝⣿⣷⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢃⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢛⣿⣿⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢃⠁⠠⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣿⡯⠀⠀⠀⠀⣰⠋⣹⣿⣯⠟⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠨⠑⠠⢀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⡇⠀⠀⠀⣤⣷⣦⣻⢹⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠻⣿⠱⠀⠰⠀⠨⣻⠆
⠀⠀⠀⠀⠀⠂⠀⠠⡞⡷⡄⠘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⣿⣿⣿⣇⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⡠⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠷⣿⣿⣿⣿⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠤⠂⠁⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡶⣻⣿⡟⣿⡇⠀⠀⠸⠛⠋⠉⠉⠙⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣾⣿⣿⣿⣿⣿⡿⢟⡁⠀⠀⠀⣼⣿⣿⡷⡻⣿⣿⣹⣧⠀⠀⢰⣤⣀⣀⣀⣤⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠽⢛⠅⠀⠀⠀⠀⠀⢠⣿⣟⡛⠷⠭⣿⣿⣾⣿⠀⠀⢸⣿⣿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⣿⣿⡿⣿⣯⠿⢋⠁⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⠸⠿⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣝⡻⢋⡴⠊⠀⠀⠀⠀⠀⢀⡤⠦⠀⠀⣿⣿⣿⣭⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢿⣿⣷⠈⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢸⡦⠶⠶⡐⡇⣾⠈⢻⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠐⠮⠀⠂⠀⠓⠇⣾⠀⢸⡄⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠓⠈⡀⢄⣄⡈⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠱⠾⠓⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠐⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣥⣤⣉⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣤⣿⣶⣆⢠⡀⠻⣿⣶⣶⠆⢀⡄⣴⣿⣽⣿⣿⣟⠀⢀⢀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⢿⣿⣿⣿⡩⣅⠸⡿⢊⡴⣤⣦⡁⢾⠇⣡⠌⣿⣿⣿⣞⣿⣿⢯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠚⠛⠿⠿⠷⠎⠳⠀⣸⣏⠻⣿⣇⠀⡞⠱⠞⠻⠻⠛⠛⠛⠉⠈⠀⢆⡀
⠀⠀⠀⡀⠀⣠⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡇⢀⣼⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠲⡦⠅⠂⢄
⠀⠀⠐⠀⠀⡘⣧⣕⡠⠀⠀⠀⢀⠀⠠⠄⠀⠀⠸⣿⣾⣿⣿⣿⡇⠀⠂⠠⠄⠀⣀⠀⠀⠀⠄⣊⣶⣄⣠⣰⡌
⠀⠀⡄⠀⢸⣿⣯⣿⣿⣶⣦⡄⠀⠠⡴⣓⢤⣦⠤⢿⠏⠿⣿⡿⠤⣰⡦⠘⢦⡄⡀⢀⣠⣶⣾⣿⣿⣿⣿⠟⠃⠆
⠀⠀⢄⠀⠼⣿⣛⣿⣿⡿⢟⣁⣀⠪⣵⢿⠇⡼⢶⣥⣴⣶⣾⣯⡴⢧⠘⣿⡦⠑⣀⣬⣙⠉⢿⣻⣿⣿⡤⠀⡀
⠀⠀⠀⠙⠂⠈⠛⢋⣥⣶⠿⠛⠉⠑⠢⠉⣐⠃⣼⣁⣠⣶⣶⣿⣅⠀⣃⠉⠀⠀⠉⠛⠿⣿⣦⡈⠉⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⡌⠉⠁⠀⠀⠀⠀⠀⠀⡟⣴⢹⣥⣤⣴⣾⣿⡍⣤⢻⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡠⠏⡄⣿⡀⢧⣀⣬⣿⡦⠰⣷⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡃⢟⣮⡌⠻⢿⢟⣡⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡈⠛⠷⠞⠛⠋⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⡠⠚⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡑⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠰⠿⠋⠁⠀⠀⠀⠊⠀⠇⠋⠛⠙⠑⠓⠁⠋⠘⠁⠈⠀⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣬⣿⡛⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠌⠂⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⣷⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠐⠛⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠊⠉⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠰⠉⠉⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣶⣶⣶⣆⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⠘⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⣶⣶⣦⣤⣶⣶⡖⠃⠀⠒⠃⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠉⠙⠒⢦⡀⠀⠀⠀⣀⠤⣤⡀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⡮⠥⠶⠶⠿⠤⣰⠁⠀⣀⣀⡀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⣀⣀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠙⠾⣁⣀⠈⢢⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡔⠁⢠⠋⠣⠀⠀⠀⠀⡠⠴⠤⡀⠀⠀⡴⠉⠀⠙⡆⢣⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⡼⠀⢠⠇⠀⠀⠀⠀⠀⡼⠁⠀⣠⣜⡄⠀⡇⠀⠸⣹⣿⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠣⢾⡇⠀⠳⣄⢀⡠⢇⣀⣀⣙⠢⠤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⠤⠴⠊⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⠦⣀⡠⠞⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢠⠇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠑⢤⡀⠀⠀⠀⠈⢣⡀⠉⢶⣤⣀⡀⠀⠀⣀⡠⠔⣋⡴⠋
⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠙⠦⡀⠀⠈⠓⠦⣀⠀⠀⠙⢄⡀⣹⠹⡙⡿⠛⠛⠛⠉⠁
⣰⠁⢹⡀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠙⠒⢤⣀⠉⣣⠀⠃⡿
⡇⠀⠀⠳⣄⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣀⡀⢹⣿⠷⠚⠓⠲⢄
⣇⠀⠀⠀⠀⠉⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⠔⠋⠉⠉⠻⢧⠀⠀⠀⠀⠀⢣
⠸⡄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠯⠇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡠⠴⠚⡹⠀⠈⠑⢤⣀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠃⠀⠀⠀⢸⠀⠛⠒⠶⠓⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠉⠉⠀⠀⠀⠀⠠⠶⠭⢵⣒⠒⣾⠀⠀⠀⠀⠈⠑⠒⠶⣶⢬⠳⡙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠦⡉⠓⢝⡜⢤⡀⠀⠀⠀⠉⢢⠀⠈⣧⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣹⡠⠜⠁⠀⠈⠑⠒⠤⠤⠼⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⢺⣿⣷⣦⠀⢺⣯⡵⠄⠀⢀⡏⢁⣲⡤⣄⢠⠖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣾⣩⣿⣿⢟⠋⢉⠀⠀⠈⠀⠠⡷⢿⣿⡷⣗⠁⢠⢿⣷⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠛⢻⣿⣿⣏⠉⠈⠑⠛⠉⠀⢴⠋⠀⠼⠀⠉⠙⢩⣁⡄⠀⠀⠞⠛⢈⣀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢈⡑⠀⠀⠀⠀⠉⢀⣀⠀⠀⠀⠛⠁⠈⢻⠒⠐⢦⠀⣤⠀⠀⠁⠀⠀⠀⠀⠘⠉⠲⠀⠀⢰⠦
⠀⠀⠀⠀⠀⠀⠀⣀⠉⠉⠡⣾⣩⣿⠗⠤⠀⠀⠰⡏⠉⢈⡷⠀⠀⠀⢰⠋⠀⡴⠋⠀⠉⠀⠀⠀⠀⢦⡀⡹⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠳⣟⠘⣦⡼⠓⠢⠖⠀⠀⠀⠈⠁⠀⠀⠀⢸⠁⡀⢊⢡⡀⠀⢠⣤⢀⠀⠀⠀⠰⣤⢤⣀⠀⠁⠀⠀⠀⠀⠀⢠⠏⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠸⠄⠸⡀⣴⠆⠀⠀⠀⢠⣾⣦⠀⠀⢰⠃⢘⠃⠁⠀⠀⣰⠿⢁⡘⠁⠀⢀⡀⠈⠀⠈⠐⣄⡀⠀⢀⣼⠋⠉⠗⠂
⠀⠀⠀⠀⠀⣀⣈⣉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣬⠉⠀⠀⠀⠸⠃⠀⣀⣄⠀⠀⠁⠀⠀⠀⠀⠀⠚⠃⠠⠄⢠⠀⠀⠄⠀⠰⡌⠙⠃⢀⣤⣀
⠀⠀⠀⠀⡞⣡⣦⡈⢧⠀⠀⠀⠀⠁⠀⠠⠆⠀⠾⠃⠀⠊⣯⠀⢠⣤⣤⣟⣓⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⡀⠰⠀⠀⢠⣖⣙⠛⣁⢠⡄
⠀⠀⠀⠀⠐⢬⠍⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⣠⠆⠀⠀⠀⣿⠉⠀⠀⠀⠀⢹⡟⠓⠀⠀⠀⠀⠀⡋⠀⠈⠀⠀⠉⠓⠀⠀⠀⠙⠀⠠⡏⠳
⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠈⣾⣟⣿⡿⠿⠁⠈⠀⠈⠃⠀⠀⠑⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠐⠚⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠜⢛⠀⠀⠉⠀⠀⡏⠻⠛⠳⠄⢀⣠⡴⠶⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠈⠀⠀⢀⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠘⢄⠀⠀⠀⣀⣀⣤⣀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠰⣿⣅⣧⠀⠀⣠⣌⠑⠀⠀⠲⠆⠀⣴⣣
⠀⠀⠀⠈⠁⢿⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⢿⠉⠁⠀⢸⣿⣿⣷⣷⣤⡀⠀⠀⠀⠀⠀⣸⠋⠀⣠⣤⠄⠈⠉⠁⠠⣔⣿⣌⡷⢡⢢⠄⠀⠀⠈⠻
⠀⠀⠀⠁⠀⠈⠀⣀⣁⡀⠐⣏⠉⠂⠠⢤⠀⠀⠀⡀⠀⢀⠘⠛⠿⠿⠿⠿⠷⠤⠄⢄⠐⠚⠋⠀⠀⠀⠀⣤⣽⣀⠀⢠⢬⠽⣯⣷⡾⣾⠣⡄⠀⢢⠚
⠀⠀⠀⠀⠠⠚⢻⠁⢠⡆⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⢀⣶⣶⣤⡄⠀⠀⠛⠛⠳⠀⠀⡀⠈⠥⠄⠀⠀⠈⠀⠀⠀⢉⡽⢴⣇⡟⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠲⠞⢓⡚⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⡀⡀⠁⠲⠀⣿⠗⠀⠀⠀⠀⢤⣠⣤⡶⠀⠉⠀⠀⠀⠀⠀⠈⠋⠃⠀⡀⠹⠉⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠴⠤⠈⠀⣀⡤⣠⢤⣄⡀⠀⠀⠀⠠⢾⣼⣅⠂⠀⠁⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⣄⠀⢀⠠⠆⠀⠀⠐⠆⠸⠃⠀⠀⠀⠐⠻⠁
⠀⠀⠀⠀⠀⠀⠠⠇⡴⠁⠀⢠⠀⡰⠝⡄⠀⠀⠈⠀⢈⠙⠂⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣀⠀⠁⠀⠀⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣗⢀⡸⠧⠟⠤⢤⡇⠀⠀⠻⠄⠜⠁⠀⠐⠀⠀⠴⠁⠀⠀⠀⠀⠀⠙⢧⠀⠀⠐⢋⠀⡁⡀⠀⠀⠀⣀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡘⠁⠨⣀⠖⡠⠤⠔⠒⠘⠉⠃⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣠⡀⢴⣷⣶⡿⠶⡄⠉⠂⠤⠴⠶⠞⠋⠁⢀⡗⠀⠀⠀⢀⠀⡀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠛⣷⣿⡟⠸⠗⠒⠁⠀⠀⣆⢀⡀⠀⠀⠀⢸⣿⠀⠀⢠⣿⠀⣼⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡿⠀⠀⣠⠞⠋⢻⣆⠈⠹⠿⠆⠀⠀⢸⡗⠀⠀⠘⠷⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⣿⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠘⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⣄⣤⡞⡵⠊⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡈⠿⣿⣷⡾⡔⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⠈⠁⠃⠘⠛⠃⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⢀⠀⠆⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠣⠴⠒⠒⠴⠤⠒⠤⠌⠉⡫⠦⣄⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⠈⠀⢠⡤⢀⠐⢅⣀⣂⣀⣀⣰⣦⣤⣤⡘⠐⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠈⠀⠀⠛⠇⠈⠻⠸⠿⠿⢿⡎⠿⠿⠟⠛⢛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⣤⢤⣤⣀⠀⣀⣀⣀⣀⡀⣀⣀⣤⣴⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣢⣈⠀⠀⠸⠛⠉⠳⠿⠿⠿⠿⠇⠛⠿⠛⠉⢉⠀⡇⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠹⡟⠿⠠⢰⠹⣤⡄⢠⣄⣀⣀⣀⡄⡀⣤⣤⣶⣿⡇⡀⠙⣄⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡴⡟⠀⣀⡐⡩⠅⠘⠛⠛⠛⣛⠃⣧⠌⠉⠀⣭⡅⢳⠈⠴⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠠⠤⠖⠒⢦⣥⣄⠀⠐⠀⠋⢁⠀⠠⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠘⠀⠠⠤⠀⢠⠀⠀⠠⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠻⠿⠛⠛⢻⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠈⢻⣿⡟⠛⠛⢿⣿⣿⠿⣿⡟⠁⠈⠛⠛⠛⢻⣿⠁⠀⢹⣿⠛⠛⠛⠛⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⢿⣿⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣁⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠙⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣶⣿⣶⣴⣾⣷⣦⣾⣿⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣻⣽⡿⣿⣯⣿⢿⣯⣄⡀⠀⠀⣾⣿⣿⣶⣿⣿⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠀⠀⠈⠛⣿⣿⣿⣻⣷⣿⡛
⠀⠀⢶⣶⣶⣶⣶⣶⣶⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⡿⠇
⠀⠀⠈⣿⡟⢻⣿⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠿⠃⠀⠀⠀⠀⢠⣴⣿⣟⣿⣾⣟⣿⣿⣿⢻⣿⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢸⣿⡄⠀⣿⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣿⣿⣟⣿⣿⣿⣻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠘⠿⠁⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠈⠙⠋⠁⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⣹⣁⣀⣀⣀⣈⣀⣈⡏⠀⠀⠀⠀⠀⠀⢀⡄⣾⣷⣤⣶⡿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠁⠙⡥⡙⢛⠽⢋⣴⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠘⡄⠘⠋⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠤⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⢿⠦⡀⠀⠀⠀⢀⠃⠀⠀⢀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣀⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡗⠀⠀⠰⣫⣄⠀⠀⢈⡀⠀⠰⡖⠀⡰⡇⠀⠀⠀⢀⡠⠀⠀⠲⣡⠴⢾⣿⣄⣀⠁⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠐⠊⢡⣘⠦⡄⢯⣔⢬⣀⠀⠠⠿⠟⠀⠀⠀⠐⣆⢠⡄⢰⠖⠀⠀⠀⠀⠘⠟⠀⠀⠀⠓⠀⠸⠦⠜⢙⣦
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠓⠶⣭⡓⢦⡒⢮⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠦⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠀⠀⠠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢀⢀⠀⠀⣀⡄⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡇⠸⠀⠰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠛⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠳⠇⠀⠘⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠒⠒⠈⠀⠀⠀⠀⠀⢠⡠⢦⡀⠀⠀⠀⠀⠀⠀⠈⠰⠆⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⡴⢘⠃⠀⠀⠀⠑⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡚⢋⡙⠃⡄⢠⠈⠁⠀⠀⠀⠀⢰⡀⣆⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠰⠎⢀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠘⢡⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢀⡴⠂⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠛⣠⠀⢙⠃⠚⠀⠤⠘⠐⡆⡃⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠃⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡍⠀⢨⡤⠀⠀⠰⠇⠀⠰⠇⠀⠀⠀⠀⢸⡇⠀⠸⣿⠇⣠⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡀⠀⠘⠛⣠⠁⠈⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠸⠗⠀⠙⠀⠀⠀⠀⠀⡀⠀⠉⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⢠⠴⠲⢤⣀⣠⠖⢀⠀⠖⠀⠀⠀⠀⠀⠀⡀⢃⡀⠀⠀⣬⠱⠆⠀⠀⠀⠀⡀⠀⠀⠀⠁⠘⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠜⠁⠐⠢⠉⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠛⠷⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣤⡴⣶⢶⡶⣶⠲⣶⠞⠛⠒⠀⠀⠠⠄⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠈⠁⠀⠀⠁⣀⢀⠀⠄⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠠⣤⠀⠀⣤⠬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡘⣿⡿⠃⠀⠀⠀⢀⣤⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣦⣄
⠀⠀⠐⠶⠶⠒⠒⠒⠒⠚⠛⢿⠷⠤⣤⣀⣀⣀⣀⣀⣀⣺⡟⠁⠀⠀⠀⠘⣿⣿⣟⣻⣿⡿⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⣀⠀⠀⠀⠈⠉⠛⠀⠂⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⡄⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⡿⣿⣦⣄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠂⠀⠀⠀⠀⠀⢤⡉⠇⠀⠁⠀⠀⠀⠀⢰⣎⣿⠭⡏⠋⠹⠍⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠐⠚⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢿⣛⡟⠓⠻⡛⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠌⡀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠒⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠰⠤⠂⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠖⠊

⣶⢶⢶⢷⢷⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣖⡇⣶⢰⡅⣶⠀⢶⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣩⡍⢡⣍⣌⣆⡄⡄⠉⢨⣦⢥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡦⠤⣀⡀⠀⠀⠀⠈⠁⠈⠀⠀⠐⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⣷⡿⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠒⢦⡀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣶⣤⡼⡿⠀⠀⠸⠟⠛⠻⣷⣭⡆
⠀⠀⠀⠀⠀⠀⣀⠾⣉⣤⠤⠤⢤⣤⣤⡽⠗⣀⠀⠀⠀⠀⠀⣾⡛⠷⣝⣿⣻⠃⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⣰⡿⢝⡲⢲⢶⡶⠂⠐⢐⠊⠅⣻⣄⠀⠀⠀⠠⡇⡽⠮⢬⢯⡇⠀⠀⠀⠀⠀⠀⣀⢠⣴⣀
⠀⠀⠀⢀⡞⠁⠁⠀⢴⣴⠶⢿⣷⡤⠀⠀⠀⠈⠙⢦⠀⠀⠀⣇⠟⣿⡤⠋⠀⠀⠀⢀⣀⣼⢫⣯⣾⣾⢩⣿⣠
⠀⠀⠀⢸⣇⡀⠀⠀⢀⡧⣦⣼⡏⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠇⣼⠋⠀⠀⠀⣐⣿⣿⣷⣵⣟⣼⣱⣗⡟⣽⢹⢿⣠
⠀⠀⠀⢸⡟⠻⢶⣶⣾⣤⣾⣿⣇⣀⣀⣠⣤⠖⢉⡄⠀⠀⠐⢰⠃⠀⠀⣐⢻⣿⣻⣿⡟⣿⣿⢫⡯⡾⣵⣧⡟⡼⠻⡇
⠀⠀⠀⠀⢿⠀⠀⠀⠉⠙⠛⠛⠟⠿⠏⢀⠀⠀⡼⠀⠀⠀⢠⠇⠀⠀⠀⡄⠀⠀⠹⣾⣻⣯⡿⢶⣙⢯⡿⠎⠰⠈⢰⡟
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡜⠀⠀⠀⠀⢇⠀⠀⠀⠈⠹⢿⣟⢿⠿⡊⠷⠇⣶⣲⠿⠃
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠐⠀⠀⠀⡴⠁⠀⠀⡀⢰⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⠀⠌⢹⡄⢀⢥⣄⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣤⣤⡤⠤⠴⠚⠁⠀⠀⠀⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⣀⡀⠸⣶⡾⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠋⠉⠉⠉⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣠⠴⠒⠛⠛⠛⠓⠶⣄⡀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣠⠞⠁⠀⠀⠀⠀⣠⠞⣧⠀⠙⢦⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣠⠞⠁⠀⠀⠀⠀⢀⡞⠁⠀⢸⠀⠀⠈⠳⣄⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡼⠀⠀⠀⠀⠈⠙⢦⡀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢀⡞⠁⠠⣄⣸⣤⣧⣼⣀⡤⠀⢳⡴⠋⣀⡤⠶⠒⠶⢤⣀⠀⠹⡄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⠀⠀⠐⠋⢹⡁⠀⣽⠙⠂⠀⢀⡴⠚⠧⣄⡀⠀⢀⣠⠼⠛⠀⢻⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⣾⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠸⡄⠀⠀⠀⠀⢀⡴⠖⠒⠒⢦⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠙⣄⠀⠀⠀⠙⠒⠶⠶⠒⠋⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡴⠚⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡄⠙⢦⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⣠⠋⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡈⠳⢤⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠈⠙⠓⠒⠒⠋⠉⢀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠋⠉⠋⠛⠃
⠀⠈⠁⠈⣢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣶⣶⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠘⢤
⠀⠀⠀⠰⠃⠀⠀⢀⡜⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠋⣦⠴⠋⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠓⢦⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢀⡞⠉⠐⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠑⡀⠀⠠⠄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢇⠀⠉⠀⠀⠀⠀⣴⠁⠀⠙⠀⠈⢧⠀⠀⢰⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡇⠀⠈⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡌⠋⠛⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂
⠀⠀⠀⣀⣀⠀⠀⠠⠌⣄⢸⡇
⠀⠘⠻⠻⠍⠀⠀⠀⠀⠈⠘⠀⠀⣤⣶⡶⠶⣶⠦⠤⣤⡀
⠀⠀⠀⠀⠈⢂⠀⠰⢏⠀⠀⢀⣐⣉⣉⢀⣰⣋⣍⢉⣙⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠱⡀⠂⠒⢠⣿⣿⣿⣿⠁⠀⠀⠀⢛⡿⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠐⠞⣮⣲⣔⣾⣿⣿⣿⠇⠀⠀⠀⠀⠘⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⡍⠁⠉⢹⣿⣿⣿⣶⠀⠀⠀⠀⣰⣿⠷⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⣀⣀⣿⣿⣀⡼⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠛⢻⣿⣿⣿⡿⠟⠀⠀⠡⠜⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃
⣿⣿⠟⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢰⣤⣖⣷⠀⠀⠀⠀⣠⣾⣷⡄⠀⠀⠀⣖⣀⣲⠁⠀⠀⡐⢺⢲⣷⣆⠀⠀⢠⠞⠚⠛⣄⠀⠀⢀⢴⠚⠗⡄
⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠾⠤⠟⠛⠛⠀⠀⠀⢿⣿⠿⠀⠀⠀⠙⠓⠛⠛⠃⠀⠀⠒⠒⠒⠒⠚⠀⠀⠚⠒⠚⠒⠚⠂

⠀⠀⠀⢠⣊⣉⣢⠀⠀⠀⠀⣴⡿⡦⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣤⣶⡄⠀⠀⢐⡠⠄⢴⠂
⠀⠀⠀⠸⡸⢸⡸⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢬⠓⠛⠚⡥⠀⠀⠰⠶⠶⠶⠆⠀⠀⠀⣠⠿⠛⠁⠀⠀⠀⠀⢹⠋
⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠚⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠠⠼⠿⠤⠀⠀⠀⠐⣶⣶⣶⡄⠀⠀⢣⠀⡔⠢⢤⠀⠀⢠⣿⠛⣿⡆⠀⠀⠰⠿⠶⠦⠄⠀⠀⠀⣉⣛⣈⡀
⠀⠀⠀⠨⠭⠭⠭⠀⠀⠀⠠⠽⠿⠥⠀⠀⠀⠈⠙⠅⣀⡸⠀⠀⠈⠻⠾⠟⠁⠀⠀⠨⠭⠭⠭⠍⠀⠀⠀⠻⠯⠿

⠀⠀⠀⢀⣤⣦⣄⠀⠀⠀⡠⠒⡀⠂⡄⠀⠀⢀⠤⠀⠤⠀⠀⠀⠀⡠⠒⢄⠀⠀⠀⢠⣠⣀⣄⣄⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢺⣯⣨⣿⠀⠀⠀⠑⣂⠁⢚⠃⠀⠀⠸⣴⣶⣴⠇⠀⠀⠀⢧⣤⡬⠂⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠛⠒⠋⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢰⣴⣴⣤⠀⠀⠀⡎⣭⡢⠠⡀⠀⠀⢀⠒⠒⠒⠀⠀⠀⢰⣿⡗⢻⣦⠀⠀⢰⠁⡤⡈⢂⠀⠀⠀⢰⣶⡾
⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠈⠈⠣⠫⠆⠀⠐⠾⠤⠤⠤⠷⠀⠀⠈⢿⣿⡿⠁⠀⠀⠘⠄⣉⡠⠃⠀⠀⠀⠀⣙
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠖⠒⢲⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⡠⠈⢂⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⠹⢉⠉⢙⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠉⠁⠙⡇⠀⠀⠀⠳⣿⡿⠃⠀⠀⠀⣠⣴⡿⠁
⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⢀⣀⣀⣒⣒⢂⣀⣀⣠⣀⣀⣤⣤⣤⣤⠤⠄⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⣿⣾⣿⣿⣿⣿⣿⠇⣠⣾⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣯⡌⠁⠠⠄⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⢋⣵⣾⣿⣿⣿⣿⡍⠻⠗⠚⠟⠻⠋⠉⠸⠋⠙⠉⣀⠀⠀⠀⠀⠈⠿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡌⠹⠿⠟⢿⢿⡀⠸⣿⣿⣿⣿⡏⠀⠀⠠⠴⠆⠀⢤⠄⠀⠀⠀⣴⣿⣿⣷⣤⣶⣦⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣠⡶⠾⣷⣦⡘⣷⣄⠙⠿⠃⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠹⣿⣿⣷⣿⠇⠈⣻⡷⠀⢨⣿⣾⣿⣿⣿⣿⣿⢻⣿⣶⣦⠀⠀⠀⠀⡈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢻⡟⠉⠁⠀⠘⠛⣣⣴⠀⡍⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⠀⠀⠀⠈⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠃⣤⡄⢀⣴⣿⣿⠿⢤⠇⠀⣿⣿⣿⣿⣿⢾⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⠏⢿⣿⡟⣭⣤⣤⣤⣾⣿⣿⣩⣯⣉⣘⣻⣿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠆⠀⠹⣽⣿⠟⠉⣁⠀⠠⢬⠻⠉⠉⠙⠻⣿⣷⣶⣤⡀⠀⠀⠀⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢾⠁⢠⡇⢀⣿⡏⠀⠩⠠⣤⡀⡀⠀⠀⠀⠐⢂⡄⣉⠛⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢹⣆⢘⣳⡿⠋⠀⠢⠀⠀⠀⠀⡠⠀⠁⠀⠀⠈⠀⠻⠓⡦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣉⣣⣴⠇⡔⠀⠨⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢀⣂⠚⠁⠀⢀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⡇⠁⢠⣄⣀⠀⠀⠑⠴⠀⠀⠀⠀⠒⠀⠀⠄⠀⠉⠀⠀⢀⠀⣖⣾⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍⣹⣯⠀⠸⣿⣿⠦⡀⠀⠀⠠⠱⢶⣄⡀⠀⣀⣀⣀⣴⣦⡀⠈⠀⠹⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣽⣿⠀⠀⣿⣿⢠⡈⠀⠀⣀⠀⠀⠈⠙⠛⡋⠁⠈⣿⣷⢻⡄⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⠀⣿⣿⠇⠻⢴⣦⣴⡦⣄⠀⠀⠀⠂⢹⡿⢿⣿⡆⠀⢠⠋⡄⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣶⣿⣿⣿⣷⣾⣿⣧⡀⣘⣿⡷⢶⣿⣿⣇⣴⣿⣷⠇⡁⢼⡇⢋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣟⣸⣽⡏⢿⣿⣿⣿⣿⣾⣯⣶⣮⣤⣤⣜⣛⣿⣿⣿⣿⢻⡆⢩⠤⠃⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠉⠛⠛⠟⠛⠉⠁⣿⣏⢟⣿⣋⡁⠆⢀⣽⣿⣻⣿⣿⡿⢻⡇⠀⠌⠩⠽⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠰⣶⠶⣶⡆⢀⣏⠀⡲⣅⠠⠀⠀⠀⠉⠉⠉⠙⠛⠃⠸⡧⠐⣤⣀⣉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠛⠛⠉⢁⠜⠛⠢⠙⠓⠀⠀⠛⠛⢲⡄⠀⠀⠐⠀⠀⠐⣻⠎⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣤⡶⠀⠈⡄⠀⠀⠀⠀⠀⠀⡀⠀⢾⣥⣄⣀⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠠⠼⢶⣽⣿⠿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠃⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⣶⣦⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⢶⠛⠉⠉⠁⢀⣘⣾⣽⣿⡿⢟⠛⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣛⣧⠛⣿⣿⣿⢿⣯⣽⣯⣥⣴⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢙⡲⢼⡾⣯⣭⣭⣥⣼⣿⣾⢟⣉⡱⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢶⣤⣇⣿⣐⣀⣤⣿⣾⣟⣻⣽⣾⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠙⠦⣌⣿⣿⠒⣒⣚⣛⣛⣋⣩⠭⢥⣺⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣑⠢⢀⣏⡟⠓⠲⣶⠶⢶⣶⢿⣻⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⠲⢾⣯⣵⣶⡶⠾⠿⢛⣻⠿⣻⣿⠿⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⣼⢝⢭⣒⣯⡯⠔⢖⠂⠤⠬⠕⣲⣿⣿⣿⣧⣀⠀⠉⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡟⢳⣥⣬⣍⡟⠒⠒⢲⣶⣿⠿⢍⣵⣿⣿⡿⡀⠉⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣡⣷⣄⡉⢛⣿⡯⠭⣭⣭⣽⡿⢿⣻⡯⣿⣿⣿⣖⣡⠄
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠉⠻⢿⣿⣿⡏⣾⣷⣾⣖⣒⣺⣭⣭⣵⣾⣿⣿⠿⠟⠁⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠻⠿⣾⣭⢿⣿⣿⠭⠭⠭⠭⠭⠛⠁⠀⠀⠀⢀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⢦⡀⠀⠀⠀⡇⠀⡼⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⠂⠀⠀⠁⣰⠃⢠⣶⣤⡀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⢀⣵⣶⣮⡻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠲⠄⢀⣀⢸⣿⣿⣿⣿⡌⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢾⣶⣧⡀⠹⣧⣝⠛⠛⠋⠙⠁⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⢀⣄⠀⠻⣿⣿⣦⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠈⠻⣿⡿⡃⢳⡄⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠿⣿⣷⡤⠀⡼⠁⢸⠺⡄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⡜⠁⠀⠀⠀⠈⠓⠦⠤⠤⠤⠏⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘





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

⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣶⣶⣤⡆⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⡀⣶⣦⡾⢷⣤⣴⡄
⢴⣆⠈⠀⠈⢀⣶⠆⠀⠀⠀⠀⠀⢠⡟⠀⠘⠉⠘⣇⠀⠀⠀⠀⠀⠀⢉⣴⡆⢠⣶⡌⠀⠀⠀⠀⠀⣀⣿⠋⠀⠀⠀⠈⢛⣷⡦
⠸⣒⠀⠶⠆⢀⡺⠀⠀⠀⠀⠀⠀⠸⣷⣶⣶⣶⣶⡿⠀⠀⠀⠀⠀⠀⡘⠋⢡⣤⠙⢛⠀⠀⠀⠀⠀⢙⣿⠀⠀⠴⠦⠀⢀⣿⡀
⢶⠉⡄⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⣦⣤⣤⣴⠁⠀⠀⠀⠀⠀⠈⢹⣷⣤⣄⣤⣴⣿⠉⠁
⢸⣰⠇⡀⣀⠸⡀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣻⣛⣧⠀⠀⠀⠀⠀⠀⠀⢠⠁⣭⢩⡄⢂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠐
⠐⠿⢀⣇⣺⠠⠿⠀⠀⠀⠀⠀⠀⢠⣟⣓⡛⣛⠻⠦⠀⠀⠀⠀⠀⠀⠿⢿⣿⣾⡿⠿⠃⠀⠀⠀⠀⠀⠰⠾⠿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠚⠛⠈⠛⠲⣷⠆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣤⣄⣀⠁⠀⠀⠀⠀⣰⢉⣿⡿⠿⢿⣦⡀
⠐⠁⢀⣀⣀⠀⠱⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⢰⣟⣿⠁⠀⠀⠀⠈⣟⣻⡆
⠐⡀⠘⠿⠟⠀⡀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠸⠹⠀⠀⠀⠀⠀⠀⠀⠹⣍⣢⣀⣀⣀⠀⢉⣽⣦
⠀⠈⠂⠀⠀⠊⠀⠀⠀⠀⠀⠈⠛⠛⠳⣆⠀⢀⡞⠛⠛⠋⠀⠀⠀⠀⠀⢀⢪⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡿⣜⡟⠶
⢀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⣾⣧⠀⠀⠀⠀⠀⠀⠀⢠⡨⡄⡛⠚⣤⠄⠀⠀⠀⠀⠀⢠⣿⣧⣀⣼⣿⣾⡀
⠑⠲⣔⣦⣖⡦⠒⠁⠀⠀⠀⠀⠀⠀⠿⠶⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠶⣦⢥⠍⠖⠀⠀⠀⠀⠀⠀⠈⠉⣿⠯⢿⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⠤⠤⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⡴⠛⠉⠁⠀⠀⠉⠉⠛⠲⠦⣄⡀⠀⠀⠀⠀⢠⡾⠙⠃⠀⠀⠘⠋⢳⡄⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠁⠀⠈⠉⠛⢦⡀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⣀⣀⣤⣤⣤⣤⣀⣀⣤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠿⠿⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⣻⠟⠀⠀⠀⠀⠻⣟⠻⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⣻⡿⠟⠛⠉⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠉⠛⠻⢿⣟⠒⠒⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⣆⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣼⣧⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣧⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡴⠋⢻⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⡏⠙⢦⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⣧⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣼⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁




⣶⣶⣶⢶⣿⠾⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⡶⣶⢶⣶⣶⢶⣶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⣾⠅⣪⠢⣄⠀⠀⢀⣠⠤⢖⣊⡩⠀⣴⠾⢏⢀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣩⡆⢈⠻⠶⣋⡴⠞⣃⢉⣵⣶⠂⢻⡟⢠⣰⠘⠧⠴⢛⣀⢣
⠀⠀⠀⠀⠀⠀⡠⠚⡥⠘⢿⣤⡼⠣⠊⡁⣾⠿⠃⠈⢿⣄⡴⠀⠛⣋⣡⡤⡄⣿⠗⠀⢂⢣
⠀⠀⠀⠀⡠⢞⠱⣿⡋⢢⡆⢉⠠⠇⢂⠹⡘⠷⠒⠋⡀⢠⣶⡋⠀⠹⣿⠁⣀⠘⠷⠖⣋⡈⢧
⠀⠀⣰⢪⣾⠗⢄⣈⠛⢋⣴⡌⢽⣦⢢⠁⠐⡀⣿⠓⠀⡈⢷⣠⡧⠇⠙⣉⣡⠲⡈⣿⠛⢀⡎⢣⡀
⠀⠰⢈⠪⡻⣧⡼⢋⣴⠈⠻⣦⡼⢻⣧⡡⠀⠐⡜⠶⠚⢋⡀⢠⣶⣄⠂⢻⣏⣐⡴⠈⢛⣩⡥⠄⠳⡀
⠀⠀⢧⡁⠙⢦⡐⢿⣧⣀⡾⢈⡁⢾⣛⣷⡑⡀⠘⠀⣿⠏⠀⡌⣷⣤⠶⠀⠉⣥⣖⠂⡈⣿⠉⣮⡆⢱
⠀⠀⠈⠻⣦⠀⠈⠢⡙⠋⣴⠗⢢⣤⡝⠋⣷⡀⠀⠀⡘⠿⢚⡫⡀⣴⡾⢏⠀⢻⣄⣶⠷⠈⣋⣭⣤⣾
⠀⠀⠀⠀⠈⠳⣤⠀⠈⠢⡙⠷⣾⠏⣴⡗⢈⢳⡀⠀⠠⡸⣿⠉⣄⡘⣧⡼⠟⢀⣉⡤⢶⣟⣯⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢔⠀⠈⠳⢄⠻⣯⣄⣹⠇⣷⠡⠁⢳⡘⢛⣋⣡⡤⢖⣛⣭⣷⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⡀⠙⠪⡙⣡⣾⣿⠇⠁⡠⣽⣿⣯⡵⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣽⠟⠁⠀⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⢀⣀⣺⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⢰⢡⡴⢹⠈⠁⠀⠀⠀⠀⠀⣼⠟⢻⡟⠻⣧⠀⠀⠀⠀⠀⠀⡶⢷⢶⢶⢶⡄⠀⠀⠀⠀⠀⢠⣿⡿⠛⣿⣷
⠲⣾⡗⠒⠁⠀⠀⠀⠀⠀⠀⠀⢹⣶⣟⣻⣶⡏⠀⠀⠀⠀⠀⠀⢠⠰⠶⠖⠰⠀⠀⠀⠀⠀⠀⠈⢏⡛⠶⢛⡜
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⣀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠥⢅⡀
⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠠⠹⠉⠉⠏⠄⠀⠀⠀⠀⠀⢠⠂⢃⠀⢠⠓⢢⠀⠀⠀⠀⠀⢀⠵⠶⠶⠶⠧
⣴⠟⠁⠀⠀⣬⡆⠀⠀⠀⠀⠀⣤⣧⣤⣤⣼⣤⠀⠀⠀⠀⠀⢏⠱⠼⠦⠦⠴⠉⠆⠀⠀⠀⠀⢨⣿⣾⣾⣿⣿


⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⢸⢾⡷⣿⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠓⠢⢄⠀⠀⠀⠀⠀⠀⢶⡆⢿⡇⢸⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⠗⠄
⠀⠘⠻⠻⠿⡀⠀⠀⠀⠀⠀⠀⢧⣤⣭⣤⣴⡼⠀⠀⠀⠀⠀⠀⢸⡀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⢦⣀⣤⣤⡜
⠀⠠⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠸⠃⠸⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋


⠀⠀⡰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⢲⢢⡀
⠀⠠⣉⣩⣿⠀⠀⠀⠀⠀⠀⠀⣠⣔⡧⠤⣀⡀⠀⠀⠀⠀⠀⠀⢠⢤⣥⣭⣤⣀⠀⠀⠀⠀⠀⠀⠰⣊⡻⣭⠇
⠀⠀⢲⣶⠆⠀⠀⠀⠀⠀⠀⠀⣷⠶⡖⢲⢶⡞⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⠝⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁

⠀⢀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣱⠀⣀⡡
⠀⠛⢛⣝⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣲⠥⠿⠌⢒⡂
⢠⠈⠈⠈⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠴⠉⠉⠙⠈⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⣀⠤
⠀⠐⠀⡄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠠⡀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠈⠂⠀⠐⢤⡄⠀⠀⠀⠀⠀⠘⡀⠀⡤⠀⢄
⠀⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠈⠙⠀⠂⠀⠀⠀⠀⠀⠉⠐⢤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠓⠁⠂⠚⠛
⠀⠀⠀⠀⠉⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠣⣀⠁⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣⠀⠀⠀⠀⢰⢇⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠈⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⡠⠴⠁⠀⠱⡀⢸⠧⠃⠀⠀⠀⠀⡷⠆⠉⠑⡀⠀⠀⠀⠀⠀⠡⡄⠀⢠⠄⠤⠠⠠⠀⠤⠠⠔⠄
⠀⠀⠀⢀⣄⢀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⠀⠀⠀⣰⠃⡄⠀⠀⠀⠀⠀⠀⡄⣠⢛⠀⠙⠊⠀⠀⠀⠀⢰⣦⣤⣴⠀⠀⠀⢼⣶⣦⣤⡄
⠀⠀⠀⠈⠛⠋⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠉⠂⣂⠒⠲⣄⠀⠀⠀⠀⠀⠀⢤⠬⡹⢍⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣀⣒⠈⠁
⠀⠊⢠⠌⠑⠛⠐⡀⠀⠀⠀⠢⡀⠀⠀⠀⠀⡇⡂⠋⢉⣆⠉⡃⠀⡈⡇⠀⠀⠀⠀⣀⠼⠜⠛⠠⠴⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠷⠌⠧
⠒⡀⠀⣀⡤⣤⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⡇⠈⠻⣿⣿⠇⠀⠛⢀⠇⠀⠀⠀⠀⢇⠀⠀⢀⡀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠹⡄⠂⠄⠀⢀⠠⠀⠸⢄⡀⠀⠀⠀⠀⠈⠀⠀⠰⠠⠓⡾⠀⠇⢲⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⡆
⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣆⠀⠀⠀⠈⠑⠒⠰⡆⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠘⡆⠈⢡⢀⠀⠈⣅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠐⠂⠀⢀⠀⢀⠀⠀⠰⠈⠀⠀⠀⠀⠀⢀⡤⠃⠸⡒⣲⠆⠀⠽⠷⠀⠀⠀⠀⠀⠀⣸⣈⡀⢸⣀⡀⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠐⠁⠓⠆⠂⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠒⠁⠉⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⠛⠃
⠀⠀⠀⢀⡀⢀⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⣠⡄
⠀⠀⠀⣁⠊⠚⢀⡀⠀⠗⢄⡀⠀⠀⠀⠀⠀⠀⢀⢤⠄⠒⠶⠖⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠚⢃⣎⢛⢠⠂⠀⠀⠀⠀⠀⠀⢀⠸⡁⠼⠄⠙⠂⡀
⠀⠀⡠⠇⡠⠧⡈⠀⠀⠀⠀⠹⢤⠀⠀⠀⠀⠀⢸⠁⡤⡄⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⡀⠌
⠀⠀⣷⡈⣁⡤⠣⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠃⠀⠠⠀⠄⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⢺
⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⡀⡟⠀⠀⡖⢲⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⡀
⠀⠀⠀⠀⢠⠉⡀⠀⣤⠀⢤⠒⠀⠀⠇⠁⠀⠀⠀⢣⣷⠀⠀⡇⢸⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠠⠐⠀⡀⠀⠀⠀⠀⠐⢄⢀⡀⠰⠃
⠀⠀⠀⠀⠀⠙⢿⡔⠁⠀⠸⠷⣶⠖⠀⠀⠀⠀⠀⠈⡿⡧⢀⡹⠸⣦⡄⠤⠄⣈⡱⠤⠀⠀⠀⠀⠀⠀⢠⣦⣤⡀⣼⣴⣄⠀⠀⠀⠀⢠⣾⣴⣤⣮⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠂⡀⠀⠠⠁


⠀⠀⢀⣴⡖⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⢤⡤⣢⠀⣀⡀
⠀⠀⡈⡯⣿⢾⠇⠀⠀⢀⣀⡀⠀⠘⣥⡤⣄⢀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠐⣤⠀⠀⠀⠀⡰⢂⣡⣮⣴⣃⡴⢋⡕⢀
⠀⠀⠉⠑⠛⠋⠁⠀⠰⡀⠁⠀⠄⠀⠙⠲⠶⠮⣌⣛⣉⠙⡓⠲⠶⠤⢦⠀⠸⠁⢀⡀⣏⣰⠏⢨⣟⡽⠋⢰⢋⡼⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⠅⡡⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠚⠃⠀⠀⠀⠑⣩⣀⢀⡸⠛⠓⠀⠼⠏⠔⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠁⣁⣀⣀⣈⡀⠤⠚
⠀⠀⠀⠀⠀⢀⣀⠔⠋⠉⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⡻⠖⠀⠉⠀⠄⣀
⠀⠀⣠⢤⡞⠉⠃⠀⠀⠀⠀⠊⠉⠉⢢⡀⠀⠀⠀⠀⡦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠐⠒⠤⠄
⠀⠀⠁⢄⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠓⠀⠀⠀⡷⠶⠤⢤⣁⣉⡀⠉⠀⠃⠃⠈⠉⠀⠀⠀⠀⣤⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠈⠉⠀⠤⢀⠠⠇⠄⠐⠓⠒⠉⠉⠀⠀⠀⢧⣀⣀⣀⡈⠀⠈⠉⠉⠛⠉⠛⠃⠀⠀⠀⠀⠉⠍⣉⣉⠩⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠒⠒⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠲⠖⠒⠢⣄
⠀⠀⡴⣾⣻⣻⢹⡗⣟⣺⣭⡶⡖⢠⣀⠀⠀⠀⢀⠊⠐⠂⠀⠒⡀⢠⣌⡀⠀⠀⠀⠀⢀⡘⣫⡟⣡⡟⢃⡾⢃⣰⢳
⠀⠀⠂⠀⠈⠙⠛⠽⠗⣾⣻⣏⠯⠝⠩⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠶⠈⡖⠋⠴⠏⣰⠻⠠⠞⠁⡸
⠀⠀⢁⡐⠀⠤⢀⡀⠀⠀⠨⠁⠀⠀⠔⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⢀⣘⢋⣐⠋⣁⡚⢁⡚⢁⡘⠁⢀⠞
⠀⠀⠀⠀⠀⠒⠤⠀⣈⠀⠲⠂⠡⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⠨⠁⠈⠀⠀⠉⠀⠈⠀⠘⠫⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠁⠘⠀⠀⢀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠁⠘⠀⠀⢀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⣼⠛⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠋⠑⠀⠁⡠⠀⠀⠀⠀⣼⠛⠛⠻⢦⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⣠⠋⠁⠂⠁⡠
⠀⠀⠘⠆⠠⠬⢸⠀⠀⠀⠀⠀⠀⠘⠛⢄⠀⠀⠀⠀⢰⠨⡤⣅⡀⠀⠀⠀⠀⡁⠘⠆⠠⠌⠨⠀⠀⠀⠀⠀⠀⠘⠛⢄⠀⠀⠀⠀⢀⠸⠤⣅⡀⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢢⠀⠀⠀⠀⠀⢀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀⠀⠀⠀⠀⠈⢄⠀⢈⡴
⠀⡈⠃⠀⠀⠀⠀⣴⢆⣀⣄⠀⠀⠉⠉⠀⢠⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣴⣖⣀⣄⠀⠀⠉⠉⠀⢠⣤⣶⣤⣀
⣼⠁⠀⠀⠀⠀⢘⠢⠀⠄⠐⠊⠀⠀⠀⠀⢻⡀⠀⠉⢹⠀⠀⠀⠀⠀⠀⢠⣼⠁⠀⠀⠀⠀⠚⠢⠀⠄⠀⠊⠀⠀⠀⠀⢻⡄⠀⠈⢙
⣀⠰⠀⠀⠀⠀⠘⠈⠀⠲⢂⠃⠀⠀⠀⠀⠀⠉⠉⠁⠈⠂⠀⠀⠀⠀⠀⢁⣀⠑⠀⠀⠀⠀⠈⠈⠁⠰⢂⠀⠀⠀⠀⠀⠀⠁⠉⠁⠘
⠁⠘⠀⠀⢀⠀⠀⠐⠀⠐⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠁⠘⠀⠀⢀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⢼⠛⠛⠻⢦⠀⢀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⠋⠑⠀⠑⡠⠀⠀⠀⠀⣼⠛⠛⠳⢦⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⣠⠋⠑⠂⠑⡠
⠀⠀⠘⠶⢠⠌⢸⠀⠀⠀⠀⠀⠀⠘⠛⢄⠀⠀⠀⠀⢰⠨⡠⣅⡀⠀⠀⠀⠀⠁⠘⠶⠠⠤⠸⠀⠀⠀⠀⠀⠀⠜⠛⢄⠀⠀⠀⠀⢐⢸⡀⣅⡀
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢂⠀⠀⠀⠀⠀⢀⠀⣈⡴⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀⠀⠀⠀⠀⠈⢄⠀⢈⡴
⠀⡈⠁⠀⠀⠀⠀⣴⢆⣀⣄⠀⠀⠀⠁⠀⢠⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⠆⣀⣄⠀⠀⠉⠀⠀⢠⣤⣶⣤⣀
⣼⠀⠀⠀⠀⠀⢸⠢⠀⠄⠐⡊⠀⠀⠀⠄⢻⡀⠀⠉⢹⠀⠐⠀⠀⠀⠀⢠⣼⠁⠀⠀⠀⠀⠚⠢⠀⠄⠐⠊⠀⠀⠀⠀⢻⡀⠀⢉⢹
⣀⠱⠀⠀⠀⠀⠘⠈⠂⠲⢂⠁⠀⠀⠀⠀⠀⠁⠉⠁⠘⠀⠀⠀⠀⠀⠀⢁⣀⡱⠀⠀⠀⠀⠈⠈⠒⠲⢂⠀⠀⠀⠀⠂⠀⠉⠉⠁⠘⠂
⠁⠘⠀⠀⣀⡀⠀⠐⠀⠐⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠈⠁⠘⠀⠀⢀⡀⠀⠁⠀⠐⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⢼⠛⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⣀⡛⠓⠂⠱⡠⠀⠀⠀⠀⣼⠛⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⣠⡋⡓⠂⠱⡠
⠀⡀⠘⠦⠠⠤⢸⠀⠀⠀⠀⠀⠀⠘⠛⢄⠀⠀⠀⠀⢰⠨⠤⢅⡀⠀⠀⠀⠀⠀⠘⠶⠠⠬⢸⠀⠠⠀⠀⠀⠀⠞⠛⢄⠀⠀⠀⠀⢨⠨⡀⢅⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢠⠀⠀⠀⠀⠀⢄⡀⢈⡴⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠖⢀⠀⠀⠀⠀⠀⢀⠀⢈⡴
⠀⡀⠁⠀⠀⠀⠀⣴⢦⠀⣆⠀⠀⠉⠈⠀⢠⣤⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⣴⠦⣀⣄⠀⠀⠉⠉⠀⢠⣤⣶⣤⣀
⣼⠀⠀⠀⠀⠀⢚⠠⠠⠤⠐⠊⠀⠀⠀⠀⢻⡀⠀⠉⢹⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠚⢢⠤⠄⠐⠊⠀⠀⠀⠀⢻⡀⠀⠉⢹
⣀⡰⠀⠀⠀⠀⠘⠈⠒⠲⢂⠃⠀⠀⠀⠀⠀⠁⠙⠁⠘⠂⠀⠀⠀⠀⠀⢁⣀⡑⠀⠀⠀⠀⠘⠉⠒⠲⢂⠀⠀⠀⠀⠀⠀⠁⠙⠃⠘
⠁⠘⠀⠀⢀⡀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠁⠐⠀⠀⢀⡀⠀⠐⠀⠐⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⣼⠛⠛⠳⢦⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⣀⡋⠑⠂⠑⡠⠀⠀⠀⠀⣼⠛⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣠⡋⠑⠂⠡⡠
⠀⠀⠘⠦⠠⠬⠸⠀⠀⠀⠀⠀⠀⠜⠛⢄⠀⠀⠀⠀⢠⠨⠤⢅⡀⠀⠀⠀⠀⠀⠘⠆⠀⠬⠸⠀⠀⠀⠀⠀⠀⠜⠛⢄⠀⠀⠀⠀⢠⠨⠀⢁⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀⠀⠀⠀⠀⠀⢀⠀⣈⡴⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠖⢀⠀⠀⠀⠀⠀⢄⡀⣈⡴
⠀⠀⠃⠀⠀⠀⠀⣴⣆⣀⣆⠀⠀⠉⠈⠀⢠⣤⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⣴⣆⣀⣆⠀⠀⠉⠉⠀⢠⣤⣶⣦⣀
⣼⠀⠀⠀⠀⠀⠸⠢⠀⠄⠐⠊⠀⠀⠀⠀⢻⡀⠀⢉⢹⠀⠁⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠚⠢⠀⠄⠀⠋⠀⠀⠀⠀⢻⡀⠀⢉⠹
⣀⢱⠀⠀⠀⠀⠘⠈⠀⠲⢂⠃⠀⠀⠀⠀⠀⠉⠙⠃⠘⠂⠀⠀⠀⠀⠀⢁⣀⠱⠀⠀⠀⠀⠘⠈⠓⠳⢂⠀⠀⠀⠀⠀⠀⠉⠙⠃⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢰⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠐⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢀⣡⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠲⠶⡴⣅⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣷⠟⠛⢵⣦⡈⢿⣿⡁⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠷⢠⡉⠻⣷⡝⣿⡀⣿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⢀⠙⢻⣦⡿⣿⣾⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠰⡀⠀⡁⠄⢻⣿⡟⢁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠳⠴⠾⢫⡖⠁⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠂⢉⣉⣁⣀⡉⠀⠘⣁⣀⣉⡉⠉⠉⠉⠀⠐⠒⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⣡⠶⢿⠋⡏⣙⣈⣏⡀⠈⠂⣍⣁⣀⣀⣀⣉⡉⠁⠲⣄⠀⠀⠈⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢁⢬⡛⠛⠛⣋⠉⠄⠉⡁⠀⠀⠈⠉⠉⠉⠉⠈⠙⠀⠠⡀⠇⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠉⠙⠓⠀⠈⠁⠒⠂⠈⠉⠁⠀⠀⠉⠁⠀⠀⡐⠂⠉⡀⠠⢂⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⡷⣾⡇⢠⣶⠀⠀⠀⠀⣤⣄⠠⣴⣤⠀⠄⡀⡇⢐⢀⣣⡔⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡍⠀⠳⠪⣻⠀⠀⠠⠀⡿⠿⢾⠵⠶⠷⠳⠟⠛⠊⠏⠇⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠐⠏⠀⠀⠀⠀⡀⠀⠀⠐⠀⡗⢈⠂⠀⡐⠠⢠⠀⠀⠀⠸⢠⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠀⠀⠀⠀⢀⣤⠀⢀⡅⢠⣿⣦⣶⢄⠀⠀⠂⠄⠀⠀⠃⡈⠀⠁⠘⢢
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⣴⠟⠁⠑⠦⠶⢳⣀⡄⡀⠘⣵⠀⠀⠠⠀⡘⠀⠃⠆⠀⠰⢠⠱
⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⡾⠋⣁⠀⠄⠤⠤⣠⣾⣤⣹⢀⡤⠘⡵⡀⠀⠀⠁⠸⢰⠖⠓⢦⡘⠀⢡
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠄⠀⠈⠁⠤⠀⢀⣶⣿⣍⣸⡟⠀⠀⢀⠘⢟⣄⠀⠀⠃⡼⠀⠀⠀⠠⡆⠀⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡐⢀⡠⠤⠤⣀⠾⣷⣬⣿⢻⡇⢀⡴⠀⠀⠜⡄⠀⢀⢀⠹⠀⢦⠀⠀⠀⠠⢰
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠈⡀⠐⠀⡿⢿⣮⡿⡟⡞⡇⠀⢠⠄⡜⠜⠀⠀⠸⡘⢠⠷⣤⡕⠺⠐⠀⢺
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢂⠙⡘⠚⡻⠆⢋⡴⠄⠡⠀⡐⠁⣈⠎⠀⠀⠀⢂⠃⡌⡌⠀⠀⠇⠆⢡⡜
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠁⡰⣑⡔⠄⠈⠀⠀⠀⠘⢀⡼⠂⠀⠰⠀⠀⠈⡐⡰⠀⠀⡌⡐⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⠑⠖⠚⠛⢥⣤⢤⢴⠋⠀⠀⢠⠁⠂⠀⠠⢡⢡⠃⡔⠠⠡⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠄⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢂⠌⢀⠄⠡⢃⢃⠜⠐⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠨⠀⠩⢶⡬⢀⡁⠒⠠⠂⠈⠈⠀⠀⠠⠠⠂⠂⠊⠀⠁⢃⣀⣄⡾⢋⠔⡡⢊⠐⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⣀⠔⢉⠔⢛⠰⠂⠴⠦⠠⠀⠤⠠⠄⠰⠖⠀⠂⠈⡩⢊⠔⡡⠊⡀⢀⠔⢡⠎⡰⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠚⠁⠔⠡⠔⠁⠐⠀⠀⢀⠔⢀⠐⡁⠈⠐⢁⠌⡔⢡⠊⠠⠂⡐⠁⠄⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⢱⡖⣆⠀⠀⡾⣿⠄⠀⠀⠀⠀⠙⣂⢱⡟⣦⠀⠀⣾⣻⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣴⢶⡜
⠀⢈⡯⠁⠀⠀⡨⠋⠀⠀⠀⠀⠀⢾⣽⣧⡭⠁⠀⠀⡨⠉⠀⠀⠀⠀⠀⢺⣽⡗⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠈⠀⢠⠒⢦⠣⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢣⠑⠀⢡⡶⢦
⠀⠀⠀⠀⠙⠋⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠀⡀⣀
⠀⠀⢀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣨⣶⠶⠀⠀⠀⠀⠀⡄⠀⠐⠳⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢦⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢴⣂⠀⠀⡆⠀⠀⠀⠐⠀⠤⣀⡀⠀⠄⠂⠠⠄⠀⠀⠀⠀⢀⡿⠛⠒⠒⠂⠀⠀⢻⠅
⠀⠀⠀⠀⠞⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠐⠁⠀⠀⠀⠐⢼⠄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠠⠀⠀⠺⣦⠀⣀⠀⠀⢀⣾⠉⣇⠀⠀⠀⠀⠀⢰⣤⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣦⣂⣀
⠀⠀⠀⢠⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡞⠀⠀⠀⠀⡽⢴⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡤⣄⠠⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⢋⠎⢁⣤⣰⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠇⢨⠀⠺⡿⠛⡄⠀⠀⠀⢀⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⣦⠀⠀⠀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠈⠢⢀⣀⣀⠸⡖⠒⠂⠤⠬⠵⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠻⠟⠀⡀⠸⣿⡿⠀⠀⡄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠢⣰⠤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⡑⢁⣤⡈⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣀⣀⣤⣜⣁⠦⢖⣘⣿⠀⢡⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠘⠿⠃⠀⠀⠊⢉⡡⠁⠀⠀⠀⠧⢠⡀⢀⣄⠸
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠃⠈⠁⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⣀⠀⠄⠒⠂⠐⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⢀⢀⡀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢀⠀⢂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠢⡀⠀⠀⡜⠀⠀⠀⠑⡀⠂⠃
⠀⠀⠀⠀⠀⢰⠁⢑⡤⠒⡈⠁⠈⠉⠒⠤⡀⢀⠀⡀⠀⠀⡀⡁⠢⠂⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠁⠀⠉⠀⠀⠡⠄⠄
⠀⠀⠀⠀⠀⠀⡹⢋⡅⠀⠠⠀⡉⠀⠰⠀⠈⢇⣀⠄⠀⠀⠀⠈⠂⠄⡄⠀⠀⠊⠀⡠⠄⠀⠀⠀⠀⠐⠡⣓⣠⠔⠊⠀⠀⠀⠒⢤⣔⡘⠐⡀
⠀⠀⠀⠀⠀⢠⠇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠐⢈⠇⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⡰⠈⠀⡪⠀⠀⠀⠐⠁⠈⠸⠡⠀⠀⠀⠀⠀⠀⠠⠹⠈⠀⢇
⠀⠀⠀⠀⠀⢸⠀⠐⣿⣿⡗⠀⠀⣶⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⣴⣦⠀⠀⢠⣶⡄⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠈⢄⠀⠈⠉⢹⣶⡶⠘⠟⠋⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢂⠀⠛⠛⣤⣤⠌⠛⠃⢀⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠑⠬⣀⣀⣀⣀⡀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⠐⠤⢀⣀⣀⣀⠁⠔⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡠⢒⠃⢀⠀⠀⠀⠱⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡤⢄⢀⣀⠀⠀⠠⡀⠌⠉
⠀⠀⠀⠀⠀⣐⣪⠵⣾⢀⠨⡀⢠⠄⠀⣶⠮⡕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⢦⡄⠀⡂⠈⣁⠀⢧⠞⡢⡀
⠀⠀⠀⠀⠀⠙⠣⠄⠹⠄⣤⠼⢥⡄⠠⠇⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠧⣀⣠⢯⣠⠀⠏⠀⠜⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⠀⣤⠤⠤⠤⠤⠴⣶⣾⣟⣉⣉⣀⣀⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠞⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠷⢀⡀⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠁⠀⢀⣀⠀⠐⠡⠄⣠⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⣼⡟⣿⣷⡄⠀⢈⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠥⠀⠀⠀⠤⢑⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⢿⣿⡟⢖⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⣧⠠⠌⠀⠈⠉⠩⠥⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣡⠄⡀⠙⣯⢻⣍⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠥⢀⣤⡠⢬⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛⠁⠀⠀⣶⣸⠮⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⣩⣭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟




⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡄⡀⠄⠀⠀⢀⡀⠌⠳⢦⣀
⠀⠀⠀⠀⠀⠀⣿⡟⢤⠀⠀⠈⠠⠐⠢⠐⢈⠨⢓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⡋⠉⠉⢙⡍⢓⠶⢤
⠀⠀⠀⠀⣖⡚⠹⣟⢮⣝⠳⠶⢲⠚⣷⢺⣛⣛⣻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣬⣛⣺⣭⣤⣷⣦⣾⣇
⠀⠀⠀⠀⣿⣧⣂⠘⢳⣝⣛⣛⣛⣛⣟⣿⣭⣭⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢿⣟⣯⣟⣹⣿⣿⣽⢿⣿⣿⠂
⠀⠀⠀⠀⠸⣷⣝⢳⢧⣬⠿⠾⢿⠟⣛⣛⣭⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠪⣻⠶⠶⢶⣟⣛⣻⣯⠭⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⠀⠀⠀⣴⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣿⣿⠃⠹⣿⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠁⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠒
⠀⠀⠀⠀⣠⣾⣿⣿⡷⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⣈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠂
⠀⠀⠀⠀⠿⣿⣿⠏⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠄⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢰⣦⣤⠀⠀⠀⠀⠀⠀⠈⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠙⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠚⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⢀⠢⠄⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⣀⡀⠢⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⡄⠃
⠀⠀⠀⠀⠀⠀⠀⣦⡾⠿⠛⠃⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⣾⣟⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⢀⣿⠀⠀⠀⠀⠠⠀⠀⠤⢠⡀⢴⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠾⠗⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣮⣷⡄⠀⠀⠀⠀⠀⣴⡶⢂⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠛⡀⠀⢀⠔⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣈⣷⢠⢤⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣴⣿⡘⠀⠀⠀⠩⣿⣷⡄⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⡄⠀⣿⣿⣧⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⢀⠔⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⠀⠹⢿⣽⣼⣼⣦⣾⣟⣩⠛⠀⠀⢰⣋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡶⠾⠿⠻⠏⠿⡛⠿⢟⣿⣷⣶⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢬⡕⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠿⣋⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢡⢠⣰⣿⣴⡴⠂⢀⡀⡀⠂⡔⢖⣴⡿⠿⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣖⣓⣶⣊⡽⡿⠁⠐⢲⠂⠷⠿⣿⣿⣿⣿⡷⣷⣿⣿⣿⣿⣟⣋⡥⢦⠀⠀⠙⢿⣾⣳⡶⢤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢠⠋⣬⠤⣴⡟⠛⠘⡙⠃⠫⠷⠋⠙⣶⣶⠤⢤⡈⢆⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠈⠀⢶⠰⣷⠛⣿⢆⣀⣀⠀⠀⠀⣀⣴⣲⠷⠙⠛⠒⠆⠈⠄⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢀⠀⠀⢶⠈⣧⠀⠈⠁⠉⠈⠉⠉⠉⠀⠀⠀⠀⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⠏⠀⠀⠈⣧⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠀⠘⠀⠀⡆⠀⢸⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣴⡃⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⠟⠈⠉⠀⠐⠂⠀⠐⠒⠐⠁⠁⠀⣈⡃⠈⢧⢢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢘⣱⠃⠸⣆⣠⡀⠀⠀⠀⠀⠀⠀⠀⣄⡾⣿⠀⠀⠈⠳⣭⠈⠀⠠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠛⠛⠳⠄⠀⠀⣀⠀⡤⢤⠶⠊⠋⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠢⠤⠤⠤⠴⠾⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣀⣀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⡄⠀⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣾⠋⢣⠀⢱⣄⢀⠴⣩⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢱⣾⣿⣧⢞⣚⣚⢂⡀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣧⢡⣿⣹⢭⣭⡅⠀⠀⠀⠀⠀⠀⣌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⡌⢿⣿⠸⠟⠇⠀⠀⠀⠀⠀⠀⠈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣼⡞⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣿⣥⠇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣛⠄⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣧⣴⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠄⠀⠘⣟⠂⠐⠆⢠⣐⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠎⠀⠀⣄⠀⡆⠀⠠⡷⣢⡴⠆⠜⣻⣟⡻⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠃⠀⠀⠀⠸⣄⣃⡎⠈⢀⡜⠋⠙⠃⠛⢤⣴⣿⠿⣉⣱⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⠔⠃⠀⠀⠀⠀⠀⢀⡝⠁⠘⠁⠀⠀⠀⠀⡐⠀⠀⠹⣟⠉⠉⠩⡃⠀⠠⠄⠄⡀⡁⣀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣤⡀⠀⠀⠀⡠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠉⠀⠀⠀⠁⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠰⢦⠄⠠⠀⠤⠄⢠⠤⢀⠢⠄⠐⡀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⡄⣤⣤⣀⡀⢠⣶⣾⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⡳⣿⣿⢃⢷⣾⡟⢿⣦⡝⢿⣶⢫⣶⣶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⣯⣹⠷⢟⣵⢿⣟⣿⡿⣿⢌⣡⣬⣭⣝⢾⣾⡆⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣿⠿⣙⣿⡇⡿⣿⣙⣿⡿⣿⡏⣾⣿⣿⡿⣿⡇⠛⣠⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣷⡙⢷⣾⣿⡟⢿⣿⣿⣿⣥⠹⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⠿⣻⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣯⣟⡻⢿⣿⣿⣶⣭⣙⣛⣛⣛⣛⣋⣥⣶⣬⣭⣭⣴⣿⣿⠿⠛⣩⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣭⣿⣟⣿⣿⢶⣯⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣭⣤⠶⣶⡿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣅⡀⠙⢋⠛⠻⢷⣮⣝⣛⠿⡿⣿⣿⣾⡿⢿⣿⡷⠋⠅⢰⣾⣦⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣿⣿⣶⣦⣄⡻⠊⠙⠻⣿⣿⣿⣿⠟⢻⠍⣁⣤⣶⣾⡿⢿⢛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣶⠌⠙⠿⢿⣿⣿⣶⣦⡘⣿⣼⢃⣤⣶⣿⢿⣛⠉⠒⣀⣀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠠⠀⠰⠂⠁⠈⡙⠓⢸⣿⡘⠋⣥⠀⠘⢟⣣⣶⡿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⢿⣷⣤⣀⣀⠻⠃⠀⠲⢸⣿⣷⠀⣠⣴⡾⠿⠛⢹⣦⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣶⡌⠙⡟⠛⠿⣦⣤⣀⠸⣿⣻⠃⠛⣥⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⢤⣤⣄⡻⠟⠻⢷⡌⡁⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣼⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⣍⠙⣯⣯⣯⡟⣯⣶⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣜⣞⢻⣧⣟⣀⠽⠽⠼⣶⣻⢻⢩⠏⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⠮⠛⢓⠉⠫⣔⢆⢤⠤⠀⠆⡉⠋⠉⣅⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⣥⣤⣉⣑⠘⢈⡤⢤⡀⢢⢞⢄⣾⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣤⢀⡞⢁⡴⢃⣀⠘⢁⣓⡏⠁⣊⡃⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢡⠟⢛⡿⢱⠃⣸⣠⠏⣡⢿⡾⠑⢋⡞⣠⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠴⠋⠀⠙⣠⣌⣈⣡⣧⢨⡅⢨⠇⣡⠈⣹⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠨⠇⠿⠟⢸⠼⠥⢸⠀⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⡏⣧⠈⡏⢹⣼⣿⣍⠀⠸⣍⣸⠉⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠧⠏⠀⠇⠸⠸⠿⠤⠀⠠⠼⠻⠤⠟⠧⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠞⠛⠂⠀⠀⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠹⣧⣀⠀⠀⢀⣴⣟⣉⣉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⣄⡀⠀⣀⣀⣉⣹⣧⣀⠚⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⣉⣉⣉⣁⣠⣤⣤⣤⡞⠉⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠉⠉⠉⠉⠀⠀⠀⠀⠀⡁⠀⠀⠈⠉⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⢲⣯⡀⢿⡛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠩⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢐⠈⠹⢿⣇⣸⡇
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠸⠀⠀⠀⠹⣿⡄
⠀⠀⠀⢀⣴⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡠⢤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢿⣄
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢀⡤⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢿⡄
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡰⠂⣤⠃⢰⢋⠉⢣⠀⠈⣿
⢀⣿⠁⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⡇⠀⡰⠢⡀⢠⠍⢧⠀⢠⠃⠀⠀⠀⠈⠉⠑⠁⠀⠀⢹
⢸⡇⠀⠀⠀⠀⠀⠀⡎⢰⠁⡏⣸⠀⡜⠁⠀⢸⡀⡇⠙⢄⡇⠀⡿⢻⠑⡎⠉⣉⡶⡯⡔⠐⢄⣠⠚⡇
⣿⠃⠀⠀⠀⠀⠀⠀⡇⠘⠦⠆⣿⢠⠁⡎⢱⠀⡇⡇⢀⡀⠛⢴⣡⠀⡄⡇⢸⣀⡀⠀⠘⢆⠀⢀⠔⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⡀⠹⠘⡀⢇⡸⠀⡇⡇⠘⠙⢦⠀⠙⠀⡇⡇⡠⠶⠃⠀⠀⢸⠀⡇
⢻⡄⠀⠀⠀⠀⠀⠀⢃⠘⡈⡇⢘⠀⢇⠀⠀⢠⠅⡆⢰⠀⠀⠳⡀⠀⡇⠁⠘⠂⠒⡄⠀⢈⢠⠁⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠈⠲⠇⠑⠬⠆⠈⠓⠴⠊⠀⠙⠋⠀⠀⠀⠙⠚⠁⠐⠒⠲⠔⠃⠀⠸⠌⠀⠀⠀⠀⠀⠀⢀⡿
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠻⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠸⣯⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠟
⣶⣶⡶⡶⠶⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠊⣠⠖⠒⠢⢄⡀⠀⠀⠀⠀⢠⠖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⣀⣐⠢⡉⠢⣄⠀⢀⣼⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣾⠁⠀⠠⠬⠩⡙⠲⢼⢇⢻⡙⠛⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⢤⣠⠴⠛⢲⣶⠶⡖⢂⣸⢟⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡙⣷⣏⠹⠁⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⢶⠶⠶⡶⠀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⡟⢹⠈⠆⠀⠀⠀⠀⢀⣾⣥⣊⣉⠉⠀⠀⠀⠀⠀⢠⠞⠛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠈⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⠉⠍⠛⠛⠛⠛⠛⠉⣧⠀⠀⢴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⣾⣶⣶⣶⣤⣤⣤⣤⣤⣴⣿⣆⠀⠽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⡀⠀⠛⠂⠀⠀⠁⠀⠀⠀⠐⠀⠙⢛⡇⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡃⠁⠀⠀⠀⣠⠖⠋⠉⠉⠙⠲⣖⠀⠀⠙⣛⡭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣶⡾⠁⢀⣴⣿⣷⣦⡀⠈⣧⣤⣤⡯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⡿⣿⡇⠀⣿⣿⣿⣿⣫⣿⠀⢸⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢊⣙⢛⢷⡀⡹⠿⣿⣿⠿⢃⢴⡿⠛⣛⣛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠦⠀⠀⠙⠿⣤⣤⣤⣤⠷⠋⠀⢉⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠟⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⣀⡀⠀⠀⠀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣤⣁⡐⠠⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⠿⠿⠿⠿⠶⠿⠿⢿⣿⠀⢤⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣉⣙⡓⠦⣭⣓⠦⣄⡀⠀⠀⠀⠉⠛⠛⠶⠶⠶⠶⠶⠶⠞⠛⠉⠀⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠛⠶⣮⣕⡀⢀⣀⣀⡀⠀⠉⠉⠀⠉⠉⢉⣉⣁⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢻⣦⡀⢍⠙⠛⣷⢦⣄⡀⠀⠀⠀⠒⠲⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠑⣄⡀⠀⠘⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣊⣍⣙⠁⢀⣡⣝⣈⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣻⡿⠛⢋⣩⣄⡉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⠷⠚⡶⠖⣶⡶⢤⣄⡀⣰⠋⠁⠀⢀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠄⣸⣰⠃⠀⢰⡇⠀⠙⠁⡠⢬⡙⢧⡀⠀⠈⠆⠘⢀⠷⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠉⢙⡲⢽⡦⠴⠋⠉⡱⠀⠀⠙⠒⠁⠀⠙⣆⠀⠀⢀⡃⠀⠀⠀⢰⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡼⠉⢳⡞⠉⠘⠂⠀⠘⠔⠀⢔⢢⠘⣦⠒⠉⢁⠈⠑⢦⠀⠁⢸⠉⠉⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢺⣀⡤⢾⠹⣄⢈⠃⣀⠀⠮⠭⠶⢀⠀⡀⠁⢠⠁⠀⠀⠃⠀⠐⠀⢧⡤⠃⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠈⢧⡨⣿⡄⠁⢁⣀⠀⠰⠈⠀⢠⢶⡼⡄⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⢨⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠘⠆⠗⠀⠀⢀⣀⡥⠶⠛⠦⣤⣀⡠⠄⠊⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢁⡼⠁
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠉⠉⣀⣠⣤⣄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⡟⠷⠒⠚⠶⣤⣤⣤⣀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠠⠴⠊⠉⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠈⣹⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠲⢤⣄⡉⠓⠲⠤⡤⠖⠀⠀⠀⠀⠀⣀⣀⠤⠞⠁⠀⠀⣀⡠⠖⠚⣩⠸⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⢤⣀⠈⠉⠛⠲⢄⡀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⣠⠖⠛⠉⠁⠀⣀⡴⠖⠛⢋⣁⡤⠖⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠙⠲⢤⡀⠀⠙⠓⠒⠶⠤⢤⣈⣷⡀⠀⣸⠃⢀⣠⠤⠖⠋⣁⡴⠖⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⢤⣀⠀⠉⢷⠀⣿⠈⣁⡤⠴⠒⠋⠁⠀⠀⠀⢀⣠⠤⠐⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠺⡇⣿⠞⠁⠀⠀⠀⠀⠀⢀⡠⠖⢉⡤⠔⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠦⣀⡉⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢦⠀⠀⣀⠤⠒⢋⣉⡠⠖⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠓⠦⢄⣀⠉⠛⠲⢄⡀⠀⠀⠀⠀⡇⠀⠈⡷⠊⣁⠔⠚⠉⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⡀⠀⠀⠀⠈⠉⠉⠓⠢⢍⡓⠒⠠⡤⢧⡀⣠⠗⠋⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⢄⣀⠀⠀⠀⠀⠉⠢⠤⢥⠔⠉⠀⠀⠀⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⣀⡀⠀⠀⡆⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠧⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡴⠋⠉⡿⠛⢛⢿⡛⠳⠶⣤⡀⣴⠋⠀⠀⢠⣆⢀⡈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠠⢤⣀⢸⣧⡟⠀⠀⢠⡇⠀⠈⠓⠋⢀⠤⣌⠙⢿⣄⠀⠀⠈⠯⡉⠞⢀⡷⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠸⣤⠴⠒⠓⠶⣭⡛⣇⣀⣠⠞⠓⠒⣤⠀⠀⠙⠦⠬⠇⠀⠙⢧⡀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢠⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢹⠟⢯⣀⣠⡤⢞⣥⡀⠀⢠⡶⢄⠀⢀⣄⠈⢷⡀⢀⡤⠾⠦⠤⢤⣀⠀⠹⠕⠈⣿⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣾⠀⠀⢹⡇⠀⠀⠉⣠⣀⡀⠉⠉⠀⠸⣌⡇⠈⡟⠁⠀⢀⡴⡒⠀⠈⢳⡀⠀⢀⠇⠀⠀⠀⠉⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢀⣠⠴⠚⣇⣠⠴⢿⠈⢷⡀⢸⠁⣠⢄⠀⠧⠤⠽⠂⣠⠤⡄⠀⠀⢸⠀⠀⠀⠘⢌⡁⠢⠃⠀⣷⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠘⣧⡀⢹⢧⡀⠙⠋⢀⡀⠀⢠⣦⠙⠊⢁⡴⢤⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡛⠃⠀⠐⣏⣙⣣⠀⠀⠀⠀⠀⠛⣋⣤⠷⣄⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣸
⢠⠇⠀⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣤⣤⣤⣤⡤⠴⠖⠛⠉⠁⠀⠈⠙⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠋⠁⠀⢸⡏
⣿⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⢸⡇
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠉⠉⠉⠁⣀⣠⣤⣤⣀⠀⠀⠀⢸⡇
⠀⠉⠓⠶⠶⣶⢷⣦⠴⠶⢶⣤⣀⣀⣀⡀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠀⠀⠙⢻⣆⠀⣼⠁
⠀⠀⠀⠀⢠⠿⠦⣤⡀⠀⠀⠈⠉⠉⠉⠙⠳⢦⣄⡀⠈⠙⠳⢦⣄⡀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⢀⣠⡤⢾⡹⡆⣿
⠀⠀⠀⠀⠸⣆⡀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠈⠙⠛⠶⣤⣀⠀⠉⠙⠛⠉⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠋⠁⢀⣀⣀⣤⡴⠞⠋⠁⣠⡼⠃⠛⠋
⠀⠀⠀⠀⠀⢸⠛⠛⠶⣤⣀⠀⠀⠈⠉⠓⣦⣀⠀⠀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⣀⣤⠾⠛⠉⢉⣀⣤⡴⠖⠋⢹
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠛⠲⢤⡀⠀⠀⠙⠛⠓⠶⠦⢤⣄⣀⡙⢷⡀⠀⠀⣼⠁⠀⢀⣀⡤⠖⠋⠁⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢦⣤⣄⣀⣀⣀⠀⠈⠉⠛⣷⡀⠀⡿⠒⠛⠉⢁⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⢏
⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠶⣤⣀⣸⣇⠀⣷⣠⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⣀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⢯⡉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⢀⡤⠖⠋⠉⢹
⠀⠀⠀⠀⠀⠀⢨⡟⠲⣄⣀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢷⡀⠀⠀⣠⡴⠞⠋⢁⣀⡤⠖⠋⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠙⠳⢤⣀⠀⠉⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⣧⠔⠋⢁⣠⠶⠋⠉⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⣤⣙⡳⠦⣄⣀⡀⠀⣿⡀⠀⠀⣿⡤⠞⠉⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠉⢹⠈⣳⡤⠞⠁⠀⠀⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⣄⣀⡀⠀⠀⠀⠀⠈⠛⠒⠚⡞⠁⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⣤⣀⡀⠀⠀⠀⢷⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀⣸⣠⠴⠛⠁

⠄⢣⡶⢦⡀⠀⢠⡾⣷⠀⠀⠀⠀⠀⠘⠃⠈⣾⠻⡆⠀⠀⣾⢿⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡜
⠀⠈⣓⡋⠀⠀⠈⣻⠛⠀⠀⠀⠀⠀⢰⣿⣿⣝⣟⠁⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠘⠁⠀⣠⣤⡸⣧⡰⠂⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⠶⣦⠳⣤⠇⣤⣤⡀⠀⠀⠈
⠀⠀⠀⠀⠳⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠞⠀⠀⠈⠻⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡙⠈⢳⣄⣀⣀⣀⣀⣀⣀⡀⢀⡀⣠⣀⡀
⠀⠀⠀⢆⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⣄⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠒⠛⠓⡉⠤⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣧⣀⣀⣀⣀⣀⡀⣀⣤
⠀⠀⠀⠀⠰⣯⣄⠀⠘⡃⠀⠀⠀⠈⠉⠑⠒⢤⣄⠀⠊⠁⠀⠆⣀⠀⠀⠀⠀⠢⣟⡇⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⢀⠜⠁⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁⠀⠀⢈⠁⠀⠀⠀⠁⠉⠿⠂⠀⣀⠀⠀⠀⠊⠁
⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐
⠀⠀⠀⢠⠄⠀⠀⢻⡥⡀⣄⠀⠀⠀⣷⡏⢹⡄⠀⠀⠀⠀⠀⢀⣮⣼⠗⠀⠂⠀⠀⢀⠀⠀⠀⠀⣰⣿⣿⣼⣀⡀
⠀⠀⠀⢰⡄⠀⠀⠀⠉⠁⢹⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⣞⡞⠀⠀⠀⠠⢫⢿⢝⣯⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠀⠘⠀⠙⠘