Loot Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠐⠒⠴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠘⠷⠶⠶⢤⣤⣶⣬⣩⢷⠤⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⣲⠶⠒⠶⣾⣿⣾⣿⡟⢃⣋⡣⠏⠙⢿⠉⠲⠶⠒⣲⠀⠀⡀
⠀⠀⠀⢀⣸⠦⠗⠒⠒⠒⢒⡒⠒⠲⣶⡥⢀⠀⠀⠀⠀⡠⠊⡰⠚⢥⠤⠖⠚⠋⠙⠫⢭⣲⡿⠁⠀⠀⣠⣸⣤⠀⠀⢠⠏⠉⡠⠃
⠀⠀⠀⢿⣇⣰⣶⠯⢿⣛⣿⣿⣭⣴⣋⣀⣦⣳⣀⣀⣰⠁⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⢩⣽⠁⠀⠀⢾⡷⠄⠀⣇⢠⡟⢀⡜⠁
⠀⠀⠀⠀⠘⠞⠋⠉⠁⣿⡇⠸⣅⣤⠞⢛⣧⢿⠄⠙⠏⠃⣠⣷⡀⠀⠀⠰⡀⠀⠀⠀⠴⡯⠀⠀⠀⣸⠇⢀⡽⠟⠚⢷⡋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠡⠟⠙⢯⣥⣾⡯⡿⠚⠒⠛⠋⠁⠙⠢⢠⣀⣷⡀⠀⠀⣴⣗⣤⣀⣦⡟⠀⢠⢻⠀⠀⠠⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡶⡇⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣷⡀⢸⣿⣿⣟⣷⣾⣵⠄⠘⢾⡄⠀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠁⠀⠀⠀⠀⠀⠀⡠⠻⠿⠾⣿⣷⡾⢿⣟⠝⠛⣛⠿⠒⠋⠉⠙⠙⠉⠉⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⣤⠤⠤⠒⠲⣦⠞⠁⠀⠀⠀⣠⣾⡟⢿⣶⣤⠀⢀⣈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⣠⠶⠚⠉⠀⠀⠀⠀⢸⠁⠀⠀⠀⣀⠀⢙⡻⣿⢾⣷⡄⠀⣿⠁⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣀⡄⠁⠀⠉⠡⣭⣬⡽⠋⠀⠙⠛⠋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣤⣄⠀⠀⢀⣎⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣿⣶⣶⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⡶⣶⣶⣶⣶⣶⣶⣶⢛⣬⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⠏⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠞⠯⠽⠧⠎⠶⠸⠞⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠳⠼⠀⠀⠨⠖⠦⠔⠦⠶⠔⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠬⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣀⣀⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣼⣿⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣄⡀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠐⠨⠤⠥⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣿⣶⡦⠄⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⠈⠒⢺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠢⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠉⠛⠛⠛⠉⠁⠀⠀⠠⠁⠈⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⡀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⡀

⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠸⣿⣽⠟⠈⠀⠀⠂⠀⠀⡀⣰⣄⣤⣤⠂⠀⠤⠂⡀⢠⠉⣽⠛⠃⠀⠀⠀⠐⠀⠀⠀⢀⢀⣀⣀⣀⡀⠂⠀⠀⠀⠠⠄⠠⢶⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⣤⢤⣤⢌⠉⣱⡠⠀⠀⡤⢼⡁⠿⠛⠛⠀⠀⠀⠀⠀⢀⢠⢈⣵⠿⣼⡽⢩⡉⠡⢆⣀⠀⠀⠀⠀⠀⠀⠿
⠀⠐⠀⠀⠀⠀⠀⢠⡀⠀⠀⢾⡻⠠⡬⠖⠒⠀⠙⠚⢷⣄⠀⠀⠀⠀⠀⠀⠀⠛⠻⠾⠆⠰⢯⡀⠶⠀⠀⣎⡰⢧⠄
⠀⠀⠀⠀⠀⠀⢠⠚⢃⡀⠀⢨⡇⢠⣷⣒⣿⡌⠀⣾⣏⡑⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⣶⠀⠸⢧⣄⢈⡃⠀⡔⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡦⢣⣾⣄⠈⠈⠁⠉⠉⠀⠸⣷⠒⠀⠀⠀⠀⠀⣴⣷⣷⣦⣘⡋⠃⠀⠈⢼⡇⣿⠂⢁⢷⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢣⠋⠈⠏⢀⣰⣢⣀⢀⣀⠀⠈⠧⠀⠀⠀⠀⠐⡁⡀⢲⡄⢹⠃⣈⠁⢀⡺⢣⡫⠂⠀⣏⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢼⠀⣷⣄⠈⠉⠉⢉⣼⣿⣶⡆⠀⠈⠀⠀⠀⢀⣿⠀⢪⡴⠎⣘⡿⠧⠞⣰⡟⠀⠀⢀⡿
⠀⠀⢀⠆⠀⠀⠀⠀⠐⡎⢧⡑⢬⣑⠀⠀⠛⠻⣿⠏⠀⠀⠀⠀⠀⠀⠭⠊⠀⠀⠉⠉⠁⠈⠙⠉⣉⠀⣢⠠⠞⠁⠀⠀⠀⠀⡄
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⠂⠀⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠃⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠂⠉⠀⠀⠀⠀⠀⠀⢀⣹⠄
⠀⠀⣾⢟⠥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡁
⠀⠀⣿⡄⡀⢠⣿⢧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢊⢀⣀⣀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠀⠀⣺⠅
⠀⠀⣿⡃⠁⣴⡯⠀⢀⢠⠖⠁⠀⠀⡤⢘⡟⡡⠃⢀⢀⠘⢈⣒⣿⡁⡤⢉⠭⣁⠈⠓⠦⡕⢔⢦⠐⠰⠌⣗⠳⠦⢄⡠⢔⣢⠇
⠀⠀⣸⡿⠂⠉⠉⠙⠾⠾⢻⠅⠀⢃⠉⠉⠈⠁⠀⠀⣡⠞⠛⠁⠐⠟⣇⢸⠿⠃⠀⠀⣶⠿⣆⢣⠃⢠⠀⠈⢃⠐⠦⠥⠴⠍⠁
⠀⠀⠁⣤⢰⠄⢀⣀⠀⠀⠀⢊⠂⢤⣤⠒⡄⠀⠋⠜⠋⠨⠺⠀⠀⠀⢸⠿⠒⠶⠆⠀⠀⠴⢃⣾⠄⢐⡤⠀⠂⠀⠀⢘⠶
⠀⠀⠘⣿⣄⣀⣛⣒⡻⠀⢠⣤⡀⠈⢁⠀⠀⠀⠀⠈⣶⠤⠴⠾⠓⠐⡶⠂⠀⠠⠦⠉⠁⠀⠀⠁⠀⠈⠀⠀⣾⣷⠀⢘⣳⡀
⠀⠀⠀⠉⠿⠟⠛⠙⠻⠇⠀⢨⠀⠾⠿⠀⠠⠴⠟⠀⣴⠀⠴⠚⠉⠀⠷⠆⠠⠾⢶⣤⠄⠠⠤⠶⠓⠀⠛⠶⠚⠋⠀⠞⠿⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠑⠉⠉⠉⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⠐⠒⠂⠀⠀⠈⠉⠉⠀⠀⢀⢀⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⠀⠀⣀⠀⣀⣀⣀⣈⣁⣀⣀⡥⡤⠶⠀⠈⠛⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠀⡈⠑⣁⠜⠀⠀⠀⠱⡔⠁⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠌⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠠⠆⠀⠀⠀⠒⠒⠀⠒⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠧⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠊⠉⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡠⠂⠉⠉⠓⡦⣄⠀⠀⠀⠀⢹⠙⠓⠶⡄
⠀⠀⠀⠀⠀⢀⡤⠊⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⣀⡀⠄⠊⠁⡀⠔⠁⠀⠀⠀⢀⠞⠀⠀⠙⢦⡀⠀⢸⠀⢀⡞⠁
⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣉⡉⠀⠤⠤⠀⠂⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⠋⠓⢾⠀⣾⡀
⠀⠀⣠⠊⠀⠀⠑⠠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⠤⠼⢤⣀⡀⠀⠀⡄⢂⠀⠈⠛⠀⠘⡆
⠀⢈⠷⠂⠀⠀⠀⠀⠀⣱⠶⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠑⠾⠉⢢⡏⢀⠴⠆⡞⠁
⡜⠁⠀⠀⢀⡀⢱⡦⠞⠁⠀⡮⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠧⠉⠈⣽⠁
⠑⢎⣠⡀⢘⠛⠞⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢴⡇⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⡉⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠒⠀⠈⠉⠉⠀⠀⠀⠀⠸⣇⠀⠀⠀⢨⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⣄⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠂⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣡⣶⣿⣷⣶⣄⡀⢛⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠛⢿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⣤⠴⠶⠶⠶⣾⣿⣿⡿⠆⠀⣼⣿⠟⠛⠛⠲⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠐⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢷⣦⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⠕⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠸⠄⠒⠋⠉⠁⠐⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡀⠀⠂⠀⢀⠔⠈⠀⠀⠀⠀⠁⠀⡠⠂
⠀⠀⠠⢖⠀⠀⠀⡔⠁⡤⠶⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠄⠐⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠁⠸⠀⠀⢸⠀⠰⢥⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⠀⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠦⠀⠈⡇⠀⠀⠀⠀⡠⠐⠐⠃⠀⠀⠀⠐⠂⢧⡀⠀⠀⠀⠠⠄⠂⠁⠂
⠀⠀⠀⠀⠀⠠⠀⠡⣀⡜⢀⡜⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⡔⠁⠉⠀⠀⠐⠎⣀⡠
⠀⠀⠆⠀⠀⠀⠢⠰⣭⠆⠋⠀⠁⠀⠀⠀⠀⡏⠀⡀⠐⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠠⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⢀⠀⡐⠀⠀⠐⢦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢡⠀⠀⠀⠀⢀⠴⠚⠉⠁⠈⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠁⠀⠀⠀⠀⠀⡀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⡀⢠⠄⠚⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠄⡀⠀⣀⠀⠀⠀⠀⣠⣾⡿⠁⠀⣠⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠈⠀⡐⠀⠀⠉⠀⠀⡤⠊⠸⠋⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢁⠀⠀⠀⠀⠀⢠⠀⠀⠆⠁⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠂⠀⠘⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠢⡤
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣌⣥⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠠
⣛⣙⣛⣛⣃⣈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣃⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣤⣄⢀⣀⡀⠠⠄⡀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣄⣨⣙⣭⣬⣽⣮⣧⣾⣶⣾⣶⣶⡶⣿⡿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣽⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⢉⣻⣶⢟⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⣿⣉⢀⡀⠠⠀⠤⢄⣐⢒⣒⣺⣿⣭⢭⠽⢏
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣷⣒⣒⣲⣾⣿⡿⠛⠋⣿⣷⢼⣷⣾⣿⣽⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡙⠿⣯⣷⣼⣷⠎⣿⣉⣈⡈⠸⠿⠿⣻⠛⠋⢉⠉⠉⣿⣄⡂
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⡍⠉⣛⣂⡯⢃⠠⡖⠠⣴⢞⠃⠒⠋⢻⡈⠉⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡛⣿⣿⣼⣿⣷⠐⠁⠀⡄⠀⠀⡧⣀⣹⣿⡃⣄⠠⠼⠴⠒⢒⣻⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣯⣿⣷⡀⠀⠐⢅⣃⢈⣷⣿⡇⡘⢩⠗⠀⢻⠉⠁⠈⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⣿⡷⠉⠉⠈⡃⠁⠀⣌⢿⣅⣡⡖⠀⠀⠈⠀⠀⢀⡦⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣿⣾⣿⡿⠀⠀⢰⠁⣀⣠⡇⠤⠭⢾⣄⠓⣄⢳⡎⠩⠭⠷⢫⣾⣿⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣌⣩⠍⠗⠴⠆⠛⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠋⠦⠀⠈



⣿⣿⠛⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡀⣤⡀⢀⣀⣀⣀⣀⡀⣀⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⣠⠄⢀⢀⣀⢀
⠀⠀⠀⢸⣼⣿⣴⣶⣾⣟⣻⣿⣣⠎⠉⢙⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣛⣻⣿⣿⣿⠿⣮⣷⣿⣿⡿⠿⣿⡿⠆⣠⡞⣁⠀⠀⠘⠛⢷
⠀⠀⠀⢸⠟⠛⠛⠿⠗⠋⠉⠁⣀⣠⡴⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢸⣿⣯⣻⣿⣿⡿⢷⣞⣶⣿⣿⣷⣄⣀⣠⣄⠀⢹⢬⣥⣤⣀⣴⣀
⠀⠀⠀⠀⠀⡀⠀⣀⣄⣠⣤⣾⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⡿⠻⠿⠛⣩⣩⣿⠿⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣠⣿⣿⠟⣻⢟⡃
⠀⠀⠀⠀⠀⠀⠸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣟⠀⠀⣰⣿⣦⣐⣤⣝⣻⡍⣤⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⢴⠀⠐⣦⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣻⣏⣹⣿⣶⣤⣤⣄⣶⡬⣿⠁⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣾⣿⣿⣏⢿⣿⡇
⠀⠀⠀⣨⡶⠾⣿⣼⣧⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠿⠟⣿⣍⣽⢭⡷⠊⣴⡆⢿⡁⢸⡆⡶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⣿⣿⢣⣤⠖⠋⣺⣷⣿⣿⣿⣿⣿⣿⡿⣱⣾⣿⣿⣿⣿⣟⠻⣿⣽⣿⣯⣽⡜⣇⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣰⣿⣿⣿⡇
⠀⠀⠀⢨⣙⣿⣿⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣋⣉⣉⡛⡿⠃⣸⣷⡿⣟⣆⣻⣿⣷⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣷⣿⣿⣿⢿⣿⣿⣿⣟⣩⣟⣿⣿⣧⡜⣫⣩⣿⣿⡿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⡿⣿⣿⣿⣿⡇
⠀⠀⠀⢿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⠿⣿⣿⣧⣼⣿⣿⣿⣿⣧⣼⡿⣹⣿⣿⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⢯⣀⡉⢭⣿⣿⡟⢋⣿⡟⣿⣿⣿⣿⠇⠀⢦⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⢉⠁⠸⣿⣿⣿⣯⣿⣟⣻⣿⣿⣿⣺⡀⡈⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⢰⠂⣠⣶⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣮⡁⢀⣽⡿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣷⣦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣦⢽⣿⣿⣿⣦⣾⣿⣡⣄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣇⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣻⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣧⣽⡿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣼⣧⣽⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⣻⣿⣇⣤⣄⠀⠀⠀⢀⣾⣿⣿⢟⣿⣿⣿⣿⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⢩⡝⣻⣿⣿⣿⣿⣿⣿⣿⢿⡷⢼⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⠟⠀⠀⠀⣿⣿⣿⣿⡈⠃⢿⣿⣿⣷⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣨⣁⣭⣭⣿⣿⣿⢿⡟⡽⠛⣷⣿⣟⡻⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⣀⣿⣿⠛⢺⣿⣆⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢘⠀⠀⠀⠀⢈⠉⠽⠆⠀⠀⢾⣿⠧⠤⠀⠀⠀⠻⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⡟⣧⠛⣆⠠⣿⣿⣿⣿⣶⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣬⢀⡀⠀⠈⠣⠖⠛⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢏⠁⠀⠀⠀⢸⣿⡿⣿⠀⣷⣿⣷⣾⣿⣿⣿⣿⣿⣀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠀⠀⠀⠀⢀⣀⡀⣠⣤⣤⣀⣀⣀⡀⠀⠀⠀⢰⣿⣿⣶⠇⠀⠀⠀⢸⣿⠴⡌⣶⣿⣿⣿⣟⠋⣩⡏⢽⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣻⣶⢷⣿⣾⣏⡛⣱⠞⠋⢶⠏⣁⡭⢀⣰⣧⣾⣿⣿⣯⠀⠀⠀⢀⣿⣭⡁⣿⣄⣙⣿⡿⢻⣿⣿⡥⠀⢸⠛⡛⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣾⣿⣿⣽⣀⣭⣴⣾⣇⡶⠿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡗⣿⣿⣄⠀⠹⣿⣿⣿⣷⣤⣾⣶⣷⡂⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣮⡁⠸⣿⣿⣿⡿⠙⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⢿⣿⣿⣿⣿⠆⠙⢹⣿⣿⡟⠘⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣧⣀⣀⣀⣀⣉⣭⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣬⣹⣿⣿⣿⡻⣿⣿⣿⣿⣧⠀⢸⣿⣿⡏⠀⠙⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣧⣤⣼⠋⣬⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⡿⠀⠀⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡻⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣻⣿⣿⣭⣿⣿⣿⣿⣿⣼⣿⣿⣿⣷⣿⣿⣿⣬⣷⣄⣀⣤⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⣼⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⣤⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠃⠈⠀⠀⠐⠒⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⣷⣾⣷⡾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣿⣷⣶⣷⣾⣶⣿⣿⣷
⣿⣿⣿⣀⣀⣤⣤⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠉⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣀⣀⣀⣀⣀⠀⠀⠠⠤⠄⠀⢒⡀⠙⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠃⠀⠐⠆⠐⠢⠖⠰⠒⠒⠒⢉⡑⠀⠀⠀⠀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠀⠀⠀⠀⠁⠐⠈⠀⠀⠀⠀⠈⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⢀⠔⠈⠈⠙⠶⢤⡀⠀⠀⠀⡏⠉⠒⠆
⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠀⠀⠀⠀⠀⠠⠐⠈⠀⠄⠊⠀⠀⠀⠀⡜⠀⠀⠈⠳⣀⡀⡇⠀⡐
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠤⠤⠤⠤⠤⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡐⡁⠉⢇⡘⠣⡀
⠀⢠⣎⠀⠀⠀⠀⠀⢀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠙⠓⠒⠦⢄⣠⠦⡁⠄⠀⢁⢀⡇
⢠⠖⠁⠈⠐⠠⣀⣠⠖⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢾⠀⠀⠀⠀⠀⠀⠈⠀⢸⠢⠆⠁⡜
⢧⡄⠀⠀⠶⣄⠏⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⢣⡀
⠀⠙⠋⠒⠋⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠙⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠠⠤⠀⠀⠐⠂⠀⠀⠈⠉⠀⠀⠀⠀⠘⡆⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡠⠤⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⢤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣚⣁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⣀⣀⣀⣀⡀⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠹⢧⠈⢉⠉⠉⠁⣘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢤⡄⠀⠀⠀⣠⢀⡀⠀⢀⣀⡼⠽⣖⡀⠀⡰⣂⢯⣯⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠀⣼⠀⠀⠀⠀⠀⠀⠀⣨⣥⣬⣏⠟⠀⠠⠐⠀⢹⣼⣿⡆⣖⡏⠁⠀⠀⠋⠛⠛⠻⢊⠉⠁⠋⠐⣋⡳⡄
⠀⠀⠀⠀⢀⠏⣠⢦⡏⠁⠀⠸⠀⠀⠀⢀⠀⠀⣸⣿⣻⣿⣿⣧⠀⣴⣦⡀⢀⣹⣿⣗⢨⣃⡀⠀⠀⣶⣦⣴⡟⠍⠀⠀⠀⢀⠸⢲⡀
⠀⠀⠀⠀⠸⡃⠸⣯⠇⠀⠀⠀⢤⣤⣀⣀⣈⣀⠛⠋⠉⠠⠠⠀⢰⣿⣿⣿⣿⣿⣿⡟⠀⠠⠊⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⣆⣎
⠀⠀⠀⠀⠠⣷⡀⠱⣯⡖⠀⠀⢽⣿⣟⣻⣿⣿⡇⠀⠈⣀⣡⡀⢾⣿⣿⣿⣿⣿⣿⠀⢀⡌⠠⡄⢸⡿⣿⣿⣿⣷⡀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠁⡐⠛⠛⠛⠛⠿⠿⠩⠾⠕⠚⠛⠉⢰⣿⣿⣿⣿⣿⣇⠀⠚⠁⠁⠀⢸⣿⣿⣿⣿⣿⣿⡖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢎⠱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣬⣷⣿⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣦⣤⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠐⠛⠚⠁⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣽⣿⣾⣦⣤⣤⣴⠾⠛⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⢶⣶⣭⣽⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠠⣅⣭⣷⣾⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣼⣿⡿⠿⠿⠿⠛⠙⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠉⠙⠛⠲⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢛⣲⣶⡦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠉⠉⠙⠛⠓⢲⡶⠦⢤⣤⣄⣀⣀⣤⠞⠋⠁⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠉⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠹⣆⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠳⢦⡄⠀⠀⠀⣀⣤⣄⣀⣤⣽⣻⣷⡦⣤⣤⣤⡤⠤⣤⣤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⣴⢻⣧⣀⣀⣠⣭⣷⣾⣇⣀⠀⠀⢀⣀⣽⣦⣀⣀⠀⠀⠘⢷⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠠⡶⣿⣿⣁⣠⠞⣱⡿⣶⡶⠶⠶⣶⣿⠾⣾⡿⣿⡿⢿⣿⡟⠓⠿⠿⣽⣿⠲⠶⢿⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠉⠛⠛⣿⣭⣟⣛⣛⣿⣷⠾⣿⠿⠶⠾⢿⠿⢶⣿⣄⣤⡴⣿⠉⣙⣻⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⢀⠉⢹⡟⠿⢧⣼⣇⡀⢠⡟⣴⣿⣤⣤⣴⣶⠟⠋⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡆⢸⠄⢸⡇⠀⠀⠀⠉⠙⠛⢿⡿⣿⠋⠁⠐⡷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠘⠧⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢻⡂⠀⢈⣟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⠀⠰⡯⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⣭⡙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠷⣴⠞⠃⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⢸⣗⡶⠶⠶⠶⣖⣿⠀⠀⠈⠉⠓⠶⣤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣤⠶⢛⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣸⣇⣤⠶⠛⠉⠀⢰⣯⣍⣉⣉⣉⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠶⠚⠋

⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿
⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⡀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⣼⠏⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⢷⣄⣰⣿⡤⢤⡗⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠁⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⠒⠰⠀⠀⠀⠉⠂⠀⠘⠀⠽⠂⠀⠀⡀⠀⠂⠀⠀⠁⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⢠⠦⣄⣀⣀⠀⠀⠈⣀⡠⠤⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⢀⠔⢂⣌⠔⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⡇⢀⠀⠉⡉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠱⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠄⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢆⠀⠀⠀⣁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡷⠀⠀⠀⢀⠒⠻⠶⢄⡀⠈⢳⡶⢦⣤⣄⣀⠀⠀⠀⠀⠀⣿⣿⠀⠡⠾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠠⠄⠒⡾⠋⠲⠴⠇⠘⠀⣄⣀⣀⡠⠾⢶⣔⡀⠀⠈⣸⠉⢊⡃⢲⠈⣦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣆⣀⠀⠄⠁⠰⠛⠲⠴⡤⣄⢀⣤⡝⢉⡉⡀⠈⠀⠄⠻⠟⡛⠋⠈⠉⠁⢀⡄⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠀⠀⠠⢄⡐⠈⠁⠀⠀⣰⠟⠣⡀⢀⠀⠘⠉⠉⠉⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠢⠀⢀⠱⠔⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠂⣁⣴⡾⠋⠀⠀⢤⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠂⠀⡇⠀⠀⣀⣤⣶⠟⠋⢁⠀⢦⣀⡀⣈⣳⣦⣤⣀⣀⣠⠄
⠀⢀⠀⠀⠀⠀⠈⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠂⠀⡇⠀⠟⠛⢉⠠⠀⠈⠀⠀⠀⠉⢿⣛⣋⣉⣠⣽⠏⠁
⠀⠈⠓⠶⠾⠿⠷⠛⠀⠀⠀⠀⠀⠀⠉⠉⡆⠉⠁⠂⡀⠀⠂⠀⠀⠀⢀⡃⠠⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢶⣶⠾⠾⣧⣶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠳⢦⣤⣤⣴⡶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠷⠛⠁

⠀⠀⠀⣀⡀
⠀⣶⣛⣿⠿⠿⣶⣴⠖⠛⠋⢳⡆⢀⣄⡴⠲⣦⣶⠉⠉⡙⢒⠖⠖⠖⠚⠀⡠⢄⡀⠠⠠⠒⠀⠂⠀⢀⡤⠀⠀⠤⣤⣤⣀⣀⡀
⠀⣹⣏⣋⠄⠀⢀⣈⣀⣀⡀⠈⠛⢁⣈⡀⠀⠀⢸⠃⢸⠦⠾⠆⠀⠀⠠⠠⠀⠀⠁⣀⣤⣰⣫⡄⠀⠉⠀⠀⠠⠂⠒⠉⡀⠆⣿
⠀⡏⣀⠍⠀⠀⠈⠂⠈⠈⢹⠫⠂⣈⡧⠤⠄⠙⠄⢧⣄⠀⣰⠦⣤⠀⠀⠀⠴⠴⢦⡏⠁⣷⡋⠨⠍⠀⣰⠉⠁⠀⠌⠀⠩⠐⠳
⠘⢧⡈⠁⠀⠀⠀⠀⠀⠀⠘⡧⠀⣦⠤⢴⣈⠀⣠⣖⠛⠰⠧⣀⡈⠁⠀⠀⠀⠀⢀⣠⡀⠈⣧⣀⠐⠂⠈⢡⣀⢀⡄⠀⠀⠨⠂
⢰⡟⠁⠀⠀⠀⠀⢰⣇⡤⣰⣃⠀⠁⠛⠙⠙⠀⠘⣯⡤⠀⠀⠀⠁⠀⢠⣴⣴⣤⠸⠟⠃⠀⠀⢹⣧⢳⢀⠀⣳⡨⠀⠀⠀⠀⣞
⠀⣇⠁⠀⠀⠀⠀⠀⡀⡔⠉⢞⠀⢀⢤⠀⠀⠀⠀⠉⢦⡂⠀⠀⠀⢰⠟⠉⠩⡉⢹⡇⠐⠒⠀⢰⡏⠜⡐⠀⡵⠁⡀⠀⠀⠀⡟
⠀⣿⠀⠀⠁⠀⠁⠉⢰⡇⢰⡀⠀⠛⠛⠛⢡⣿⣦⡄⠀⠙⠂⠤⠀⠍⣰⠎⠰⣃⡼⠁⢺⢇⡴⢋⡴⠋⠐⠀⢸⠀⠁⠈⢀⡠⠇
⠀⠘⣷⠀⠀⠀⢀⡤⡄⢧⡐⣍⠓⠀⠀⠸⠿⣿⡿⠁⠀⠀⠉⠁⠲⢚⡡⠀⠀⠙⠒⠋⠁⠒⠒⠙⠁⠤⠀⡴⠋⠀⠀⠠⢁⣤
⠀⠸⣧⡄⡀⠐⢀⠋⠉⠈⠑⡦⣈⠀⠢⠄⠀⠉⠀⠀⠀⠰⠴⠤⣦⡶⠀⠀⠀⠀⠇⠀⠒⠀⠀⠊⠄⠘⠀⡁⠀⠀⠀⠀⠀⠹⡄
⠀⣠⡿⢋⣩⠂⢀⠇⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣐⠀⠆⠀⠂⢀⡀⠀⠴⠀⠀⠀⠀⠀⠀⠠⣥⡃
⠀⢸⡇⠁⠁⣈⣟⣶⣤⣄⡄⠘⠂⡀⠀⠒⠇⠀⠀⠈⠀⡁⠈⠁⠀⣀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⢠⠤⠌⢴⢦⠀⠀⢸⡇
⠀⣼⡟⠁⢠⣞⠀⠁⠉⡥⠋⠉⠀⣨⠙⣶⢚⠄⠀⠀⠀⠄⠣⢲⠋⣩⠙⣒⠋⠉⠳⢤⡒⢖⢄⠀⠠⡘⢢⠶⣆⠀⠀⣀⠤⡙⠁
⠀⠈⣿⡆⠙⠓⠀⠨⣥⣇⠀⠀⡄⠑⠚⠁⠃⠀⠄⢀⡴⠖⠋⠀⠳⢖⠘⢷⠞⠀⠀⢠⣹⢆⠡⠁⠀⡀⠈⢃⠀⢬⣁⣀⡁⡃
⠀⢹⣿⠀⡄⠀⠀⠀⠀⠁⠀⢢⡈⢀⡠⠀⠀⠀⠀⠚⠀⠀⡀⠀⠀⢘⡎⠐⢤⣄⠀⠈⣩⡞⡀⡀⢸⣧⠌⠈⠁⢦⡀⢀⡀⠃
⠀⠻⣿⡀⠀⠈⠉⠐⠢⠄⣀⡉⠃⠈⠉⠀⠀⠀⠳⣠⣠⡠⣤⠳⠦⣤⠔⠁⠀⣈⠛⠊⠑⠚⠁⠀⠚⠁⠀⠴⢦⠀⠃⠠⡐⠋
⠀⠀⠛⣿⡿⠛⠛⠳⠶⠟⠁⣿⣧⠾⣿⣀⣠⣴⣤⣤⡤⣤⡄⠓⠈⣻⣤⣤⡴⣦⣀⣠⣄⣤⣤⠶⠤⠖⠤⠤⠶⠦⠾⠳⠿⠇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠔⠊⠉⠉⠒⠊⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣾⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠤⡀⠀⣤⠦⢤⡘⠦⣄⣀⣀⣀⣀⡤⠶⠦⣤⣤⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠈⠃⠀⠀⠻⠶⠛⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠽⠉⠉⡍⠉⣩⡍⡭⠭⢥⠤⠲⠀⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⣤⣀⡀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⣀⣤⣖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡟⠐⠤⡀⠀⢘⠀⠈⡕
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣾⣿⣿⣿⣿⡟⠀⠀⠀⠘⠖⢼⠀⣜
⠀⢀⠄⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠤⢄⡀⠀⣈⢀⡀⠙⠀⢱
⢀⠜⠃⠀⠀⣀⢀⠜⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠉⠈⣷⢠⠈⣱⠁
⢧⡀⡄⠰⣦⠏⠁⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠳⡀
⠀⠉⠁⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⢻⡟⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠁⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣰⠂⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟
⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠂⢄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠈⠀⠀⠉⠂⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⠂⠀⠂⠀⢄⠀⠀⠘⠀⢈⢤⠀⠀⣀⠠⠀⠁⠀⠓⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠘⠀⠀⢄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠢⠀⠁⠘⠚⡀⠀⠀⠀⠀⠀⠄⠄⠈⢂⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⡄⠀⡀⠀⠀⢃⠅⢂⠀⠆⠀⠀⠀⠀⠈⠀⠀⠀⠰⠂⠀⠘⠉⠀⠐⢀
⠀⠀⠀⠀⢰⠄⠀⠈⠀⠰⠀⠀⠂⠀⡈⠀⠀⠃⠀⠃⠰⠂⠈⠖⠢⠀⠔⢀⠠⡀⢀⢈⡖⠀⠐⠃⠀⠐⠃⠀⠚⠀⠘⠌⠀⠀⢀⠀⠀⠀⠀⠘⡡
⠀⠀⠀⠀⠈⠀⠐⠀⢀⠁⠀⡈⠀⢀⠁⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠤⠆⠀⠴⠂⠀⠶⠀⠸⠀⠈⡆⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠂⠀⠤⠀⠀⠀⠀⠀⡎⢀⢠⠁⠀⠔⠀⡰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠅
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠁⠀⠄⠀⠀⠂⢀⡴⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⡀⠀⣂⠀⠀⠆⠀⢡⠀⠀⠆⠀⠀⠀⠀⢀⠆⠀⢤⡀
⠀⠀⠀⠀⠠⠠⠀⠈⠚⠠⠤⠂⡀⠤⠁⠴⠈⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡡⠀⣀⠢⠠⡀⣤⠀⢀⡄⣀⠠⡄⠐⠃⠉⠁⣿⡄⢰
⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠐⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠘⠁⢀⠁⠆⡱⣄⠅⠈⢀⡌⠁⠀⠀⠀⠈⢀⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⠘⠂⠀⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⡆⡁⠀⠀⠀⢢⠉⠨⠁⡉⢀⣄⡤⠶⠞⠛⢣⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠉⠉⠀⠑⠒⠐⠓⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠤⠤⠤⠠⠄⠰⠀⡤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠄⠀⢀⡀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠄⠈⠀⠀⠤⠄
⠀⠀⠀⠀⢀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠂⠈⡅⠀⡊⢖⠀⠂⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⡄
⠀⠀⠀⠐⠁⠀⠀⠐⠀⠈⠂⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⡷⠀⠏⠋⠀⢰⠜⠀⣴⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢩⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣈⡁⠉⠈⠀⠐⠀⠐⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠀⢐⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⢈⢡⠀⠂⠁⢁⣻⠀⢱⠃⢈⠐⠘⠀⣠⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⡠⠙⠁
⠀⠀⠀⢀⡀⠈⠀⠀⢀⠈⡀⠀⡄⠀⠀⠀⠀⣿⠀⡐⡄⠀⣇⠀⢸⠀⢈⠚⡄⢀⢠⠀⠀⣶⠃⢉⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠐
⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠀⠈⠁⠈⠁⠃⠘⠙⠁⠤⢅⡃⠈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠎⡤⠂⠉⠉⠐⠣⢤⣄⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴
⠀⠀⠀⠘⠘⠅⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡴⢠⠈⠁⠀⠀⠀⠀⠀⠘⢠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠁⠀⢀⠀⠀⠀⢀⠁⢀⡟⠘⡖⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠔⠒⠁⠘⠋⠀⠈⠁⠰⠤⣀⡀⠀⢀⡠⠔⠊⢩⠤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⠂⠀⠈⠄⠀⠀⠀⠂⠄⠁

⢁⢶⡶⡀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣲⣤⠀⠀⠀⠀⠀⡄
⠉⣛⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⡟⢉⠆⠀⠀⠀⠀⢰
⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⡟⣦⠉⢢⠀⠀⠀⠀⠀⠀⢠⢸⣦⡱⠁
⢄⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⢑⠤⡀⡹⠊⠀⠠⠀⠀⠀⠈⠀⠁⠟⠓⠀⠀⠀⠀⠀⢀⠄⠀⠦⠤⢄
⠔⠋⣸⣹⡗⡄⠀⠀⠀⠀⠀⢠⣀⡆⢾⡹⠆⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⣛⠷⠠⠜
⠢⡖⠉⡩⡎⠀⠀⠀⠀⠀⠀⠀⠀⡁⢨⡇⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠂⠠⠄⠠⠂
⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⣀⢔⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡴⣌⣀⣱⠀⠀⠀⠀⢠⢁⡿⢘⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⣂⣡⢄
⠀⠀⠀⠀⣀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⣀⠀⠸⠀⠀⠀⠀⠀⠙⠠⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠻⢤⠈
⠀⠀⠀⢸⢲⠷⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⣎⠱⡄⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢆⢬⣷⠁⠀⠀⣠⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⡿⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⠁⣀⠾⢣⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⡤⢋⡝⠂⠀⠀⠀⢀⢀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢿⣬⠏⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠈⠀⠀⠀⠀⢃⠀⢥⡀⠡⠄⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⣀⠔⣢
⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠊⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡐⢦⡿⡨⠀⠀⠀⠀⠨⠤⠼⡀⠀⠀⠀⠀⢠⠊⢀⡿⠹
⠀⠉⠀⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⡌⣹⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠐⠀⠀⠀⠌⣓⣺⢿⠀⠀⠀⠀⠀⠑⢏⡠⠐
⠀⠀⠀⠀⠀⠀⠀⢊⢳⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⠉⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠊⠙⠝⠋⠀⠀⠈⠀⢀⣀⡀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢈⠤⢐⡀⠀⠐
⠀⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⣉⠓⠮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠤⣠⠃⣮⡒⣷⠀⡇⠀⠀⠀⠀⠀⠐⢾⢿⣃⠜
⠀⡿⡝⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡾⢆⡡⢄⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠹⠇⠀⢸⢀⣀
⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢌⡀⠋⠉⠐⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣛⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⠢⠄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡼⣷⡔⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣎⠉⠢⡔⠃⢀⠀⠀⠀⠁
⠀⠀⢈⠆⠈⠖⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣷⢠⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⣤⠂⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⣙⡛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣹⣿⠅⠀⠀⠀⠀⣨⠖⠉⠀⡱⡄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣨⠴⢛⡆⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢎⢰⡓⠻⠃⠀⣸
⠀⡀⡀⠀⠀⠀⠀⠀⠈⠉⢙⠀⣛⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡙⣋⠔⢎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠊⠉⠉⠑⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⣀⠠⠤⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⡀⠀⣀⢈⡉⣉⢩⢠⡤⠤⢒⠂⠀⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠠⠀⠀⠀⠐⠠⢀⡠⠔⠁⠀⠀⠈⠲⠁⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠐⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠓⠒⠢⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠊⠉⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢀⠔⠈⠀⠀⢩⠓⠦⣀⠀⠀⠀⠃⠈⠉⢱⠄
⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⠀⠄⠀⠠⠄⠂⢁⡀⠀⠊⠀⠀⠀⠀⢠⠃⠀⠀⠈⠓⣄⣀⡶⠀⣰⠃
⠀⠀⠀⡠⠚⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠤⠤⠄⠀⠤⠤⠤⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡐⡀⠈⢃⣰⠣⢄
⠀⢀⡞⠀⠀⠀⠀⠐⠄⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠓⠛⠒⠲⠤⣀⢠⣄⡁⠤⠀⢉⠂⢸⠂
⢀⡴⠋⠉⠒⠠⣀⠀⣠⠞⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⢹⠧⠤⡈⣹⠃
⣏⡀⠀⠀⢠⣄⣸⠉⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠐⢣⡀
⠈⠳⠼⠤⠴⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢹⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠤⠄⠀⠐⠒⠲⠖⠀⠀⠀⢰⠆⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠉⠒⢤⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

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


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

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⢀⣴⣶⠞⠭⠰⡖⠨⠝⢶⣶⣄⠀⠀⠀⠀⢰⠶⠽⣯⡀⠶⣀⣿⡧⢶⡄⠀⠀⠀⠀⠀⢀⣾⠟⢫⣽⣿⣍⠻⣷⡀
⠀⠈⠋⢸⠟⠲⢦⣤⠶⢺⢯⠉⠋⠀⠀⠀⠀⠈⠋⢁⣼⡼⣶⠇⣧⣉⠙⠀⠀⠀⠀⠀⣴⡾⠟⢰⣯⣻⣿⣽⡆⠹⢷⣦⡀
⠀⢀⢀⡾⠇⠀⠀⠀⠀⠐⠿⣆⢀⠀⠀⠀⠀⣠⢊⠉⣁⣼⣟⣦⡈⢈⠳⡄⠀⠀⠀⢼⣏⣠⣼⣆⣸⣿⣿⣇⣠⣧⣄⣹⣧
⠀⠈⣮⡄⠀⡀⠀⠀⠀⡀⠀⣼⡞⠀⠀⠀⠀⠀⡋⢰⠏⣠⣿⡀⠻⡆⢹⢀⠀⠀⠀⠸⣯⣾⣿⠋⡋⣿⣿⢙⠉⣿⣷⢿⠇
⠀⠠⢹⣵⣿⣿⣧⢰⣿⣿⣶⣽⡁⠀⠀⠀⠀⠛⠓⢾⡾⠉⡀⠉⢶⠷⠛⠛⠀⠀⠀⠀⠈⠛⠛⠻⢿⣎⣱⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠟⣯⠈⠽⢿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⢿⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢼⣿⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣷⢾
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣈⣉⣅⣰⡶⡦⠀⠀⠀⠀⠀⢐⣓⣿⣧⣿⣾
⠀⠀⠀⠀⠀⠀⣼⣿⡼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣩⡹⠋⠻⠶⠖⠀⠀⠀⠀⠀⠸⠇⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠸⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢠⡶⣍⠲⣖⠢⣄⠀⣀⣀⣤⣤⡠⢤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠻⠧⣌⡣⠝⠻⣭⡉⠉⠉⠉⠉⠉⣟⢼⠉⠓⣯⡷⣸⡀⠀⠀⠀⣀⠉⢛⣭⠠⣝⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⣀⠤⠬⣅⡀⠀⡙⢦⡈⠓⣛⣒⣂⣀⠠⠤⠽⠶⠽⠿⠭⠶⠭⠭⠭⢉⣀⣛⣓⣒⣀⣠⠬⢤⠆⡻
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢉⠞⠁⠀⠀⠀⠙⢿⠋⡠⢋⠤⠤⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⢹⠁⠁⠀⠀⠀⠀⠀⣠⢋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⡟⠀⡜⠀⠀⠀⠀⠀⣠⢋⡜⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⢡⠎
⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠀⠀⡴⢡⢎⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢻⠈⢱⠀⠀⢀⠞⡰⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡰⢃⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢆⡠⢋⡞⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠈⢡⠎⡴⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠠⠤⠤⠀⠀⡀⠶⠶⠒⠒⡋⣥⣔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣃⣘⣓⡴⢾⠛⠏⠙⠋⣳⠶⠤⠤⠔⣒⣒⣺⠿⠭⠭⠥⠾⠖⠓⠚⠛⠉⠓⠀⠈⠉⠐⠀⠶⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠒⠚⣳⡾⠾⠿⠛⠛⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠱⣖⣉⡹⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣍⠓⢤⣉⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠠⠤⢦⣐⣟⠫⠭⠧⠒⠒⠋⢩⠥⠶⡆⢀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⡇⠙⠲⢬⡙⠲⢤⣉⠓⠦⠄⠐⠒⣒⣂⣉⠭⠥⠴⠖⢒⣚⣋⣍⠡⣴⠄⠶⡛⠉⡏⠐⠝⢠⠀⠀⡇⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⡇⠀⢔⠶⣬⠑⠢⡌⠙⡏⠉⠉⢩⠤⠶⠲⢲⠀⣴⡄⡏⢠⣾⢁⣄⠀⢰⠴⣇⠀⣿⠀⣀⠸⠀⠀⡇⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢜⣿⢾⡜⣇⠀⡇⡴⢧⠀⠀⢸⠀⠀⠀⢸⠀⠉⠨⡇⢸⡟⠈⠁⣠⠀⢀⠈⢦⢿⠘⠋⢠⠀⢀⡇⢀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⡸⣇⢀⢇⡏⠀⡇⠙⡏⠀⠀⢸⠀⠀⠀⢸⠀⣶⠶⣃⣿⠁⣰⢖⣿⣖⡻⣧⠎⢸⡠⣶⢸⠀⢸⣿⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠳⣄⡨⠭⠞⠁⠀⡇⢰⢳⡀⠀⢸⡀⠀⠀⢸⡇⠀⠐⡏⠻⢶⠣⣯⣽⣋⠵⡘⡆⣸⡇⠀⠸⠦⠞⠛⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠓⠌⡙⠶⣄⠀⠀⡇⠈⣟⠀⠀⢸⡇⠀⠀⣾⡇⠿⠃⡇⠀⠈⣆⠣⣌⣉⠞⡰⠋⢉⣀⠤⠔⠒⠂⢉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⡈⠢⡈⠳⣄⣳⠰⣉⡆⠀⢸⡇⠀⠀⣿⡇⠀⠀⠷⠜⠒⣊⡱⠶⠶⠟⠓⣊⣡⡤⠴⠒⠊⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⢌⠛⠀⢿⠀⠀⠸⠷⠶⠟⢛⣁⡠⠤⠖⠋⠉⠀⠀⠤⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡉⠂⢸⠀⠤⠤⠒⠒⢉⣁⡠⠤⠔⠒⠋⠉⠛⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢸⠀⠀⣠⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠚⠋



⣶⣶⣶⣶⣾⡷⢿⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⣷⠿⠿⠿⡷⢶⣶⠶⠶⠷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⣿⠔⠒⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠥⣄⣀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⣠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠈⠉⢒⠒⠤⡥⣀⣀⠀⠀⠀⠈⡙⠋⠀⠀⠁⠐⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢂⠀⠀⠀⠀⠑⠀⠀⠉⠉⠓⠒⠤⠄⣀⢀⠀⠀⠠⠄⠉⠲⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠁⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⢒⢶⠁⡄⠐⠒⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⣀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢸⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⢋⣬⠝⣦⢜⡣⠄⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡀⠄⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡼⢴⠚⣥⢻⡞⢰⢃⠴⡶⡟⡏⡯⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠁⠐⠀⢸⠀⠁
⠀⠀⠀⠀⢀⡠⠤⠒⠐⣻⠃⣷⣀⠡⢭⠟⢶⣈⡓⠲⠐⢨⡖⣏⣴⠂⢫⠓⢆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠠⡞
⠀⠀⢸⡗⠈⠁⠀⣔⠋⣹⣻⠁⣏⣩⡊⣏⣀⡿⣀⢼⠋⡴⠙⣡⣯⢐⡂⣠⣡⣷⣢⡤⠛⠏⢰⡄⢸⡄⢀⢡
⠀⠀⠠⠀⢲⠂⠄⠉⠉⠭⠳⢴⢳⣴⢫⠛⣛⢜⠧⡤⢿⠰⢚⣛⣺⢁⣍⣩⣉⡵⢷⠛⣤⣀⣠⢗⣤⡺⣈⠄
⠀⠀⠐⠀⠸⠀⠀⠀⢘⠁⣤⣀⡁⠉⠛⠻⣭⣭⣧⣤⢯⣉⣎⠍⠳⡴⠷⠶⣏⠒⠀⠀⢻⣬⡹⠂⠉⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡔⠋⢠⣶⣻⢝⡿⠹⡩⠧⢤⠟⣒⠚⢳⠳⣲⠚⡶⠛⢫⠍⠀⠀⠄⡖⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣯⠝⠁⠈⠉⠀⠰⡈⡓⠴⣯⣉⡷⡤⠞⢋⣭⠀⠘⠃⠀⠀⠀⡇⢸
⠀⠀⠀⣄⠤⠂⠀⠀⠀⠀⠀⠀⡾⠷⠀⠀⠀⠀⠀⠀⠴⠊⠀⠐⠀⠨⠠⠀⠩⠄⠀⠀⠀⠀⠀⢀⡟⠇
⠀⠀⠀⠈⠀⠄⠁⠀⢐⠀⡀⠀⢻⣷⠀⠀⠀⠀⠀⠀⢰⢃⡆⠀⠀⠀⠀⠀⢸⠩⠀⠀⠀⠀⠀⢸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠸⡅⠀⠀⠨⢯⣵⠀⠀⠀⠀⠀⠀⢭⠇⠀⠀⠀⠄⠀⡏⡘⠀⠀⠀⠀⠀⣨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⡆⠂⡘⣿⢲⠄⠀⠀⠀⠀⠰⣊⢇⡀⠃⠀⠀⣂⡀⠀⡀⣠⠴⡺⢅⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠙⢋⣩⣿⢽⠅⢠⣤⣤⡀⠀⡗⢋⡗⡒⡆⡤⠗⢢⣖⡟⠓⡿⢾⠗⠻⣟⡼⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣉⣾⡗⠘⣶⠊⢰⣄⡩⠖⢋⣽⠀⣥⠠⣍⣠⣗⠕⠹⣖⡳⠾⢫⠹⠏⠵⠶⠾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣏⣠⢉⣉⠍⣳⣙⡤⢼⡶⠛⡗⠐⢳⡶⢻⠛⡾⡗⠐⡿⠹⣖⠛⣢⡘⠓⠖⠦⠤⠵⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣍⣉⡴⠚⣰⣉⠖⠔⢓⡶⢧⣤⢿⠒⢤⠇⢥⡞⠛⢷⠤⢯⢹⢻⡍⡟⡷⣂⠠⠛⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠭⠓⠒⠑⢋⠚⡭⢥⠒⠢⣾⣞⣷⠈⢠⢴⠛⠖⢫⡈⣋⣈⡿⣇⢉⠗⠣⣿⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠦⠤⠋⠈⠀⠙⢿⣿⣡⣨⡧⣾⡟⠧⠤⠅⡀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠃⠀⠀⠀⠀⠀⠀⠁⠛⠀⠉⠀⠀⠀⠀⠀⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠑⢄⠛⠄⡀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⡁⠁⠀⡠⠾⢁⠠⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢖⡀⠳⡈⠓⡘⡠⣛⠩⠟⢩⢠⡀⠀⠀⠀⠀⠀⠀⠀⠢⡆⠙⠣⣙⠆⠸⢠⠖⢡⡔⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠴⠉⢠⡍⠡⠤⣷⣷⡒⠀⠶⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⢳⠈⠦⣽⣄⡂⠵⠤⠌⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣤⠜⢂⣽⡏⠳⣙⢌⠓⠘⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡩⠙⣉⢜⢩⠟⣷⠈⣔⠲⡀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣀⠐⠟⡸⠉⡆⢹⣌⠇⢬⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠆⢲⣡⠎⢠⠰⡧⢷⠌⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢸⣼⢿⣧⣌⣻⠆⣚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡘⢰⡰⠠⠐⢖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠯⣬⡑⣽⣯⣅⡪⠑⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠁⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡸⠻⠖⣩⣾⢻⣻⠺⣝⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠼⠛⠸⣱⣯⠇⢧⡧⣰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⠈⡐⠉⠀⠖⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠥⠥⠄⠀⠒⠒⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠐⠒⡆⠀⡦⠄⣀⣀
⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠦⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢀⠁⠀⠀⠀⣙⠓⢀
⠀⠀⠀⠌⠁⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⢐⡉⠭⠀⠀⠁
⠀⠀⠸⠀⠀⠀⠀⠈⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢦⡈⠀⠀⠀⢀
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢘⠀⠀⢀⡀⢀⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠀⠀⠀⠀⠈⠒⠄⡀
⠀⠀⠀⠀⠈⠁⠒⠀⠤⠀⠈⠀⠀⠀⠀⢀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠠⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠢⠄⡀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡄⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠒⠔⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠤⠠⠤⠒⠂⠊⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⢠⠋⠀⢀⠀⠀⠀⠀⠙⡄⠀⠀⡇
⠀⠀⢠⠇⠀⢠⠆⠀⠀⠀⠀⢀⡇⠀⢀⡇
⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⢀⡼⠁⠀⣼⣁
⠀⠀⣿⠀⠀⠘⢧⣄⣠⡴⠛⠀⢀⣼⡁⠈⠙⠢⣄⡀
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠙⠲⢤⣀⠀⠉⠓⠤⣀
⠀⠀⠀⠈⠳⠦⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⡜⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢌⠞⠠⢂⢄⠌⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠎⡠⢁⢊⠎⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡱⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣉⣭⣍⣁⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡴⠛⠉⠁⠀⠈⠉⠙⠷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⣀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣏⣁⣤⣤⣀⠙⠶⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣤⠿⣿⢯⡀⠀⢙⣧⡀⠈⠳⣶⣲⠖⢲⡖⢦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣧⣀⣘⣷⣝⡲⠿⣿⠇⠀⠀⣼⣁⡴⣋⡤⠞⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠎⠁⣰⠳⡄⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡆⠀⢸⡀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⢠⡇⠀⠀⠀⣰⠏⠹⣄⠀⠀⢀⡀⠀⢳⡀⠀⢻⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃⠀⡾⠀⠀⣾⠛⠁⠀⠀⠈⠓⠚⠉⠉⠛⢦⢳⡀⠀⢻⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⠀⢰⠇⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⣼⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡄⠀⢳⡄⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠀⢠⡏⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢻⡄⠀⢻⡄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠇⠀⡾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡼⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠀⣸⠇⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⢀⡟⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆⠀⢻⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⣼⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣔⣺⡐
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠐
⠀⠀⠢⢰⡆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⡧⠅⠒
⠀⠀⠀⠰⠀⠀⠀⠸⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣄
⠈⠀⠐⠂⢠⡌⠁⠄⠀⠀⠀⠀⠐⠒⠶⠶⠖⠢⠄⠐⠒⣶⣶⣶⠶⠖⠒⠀⠖⠒⠒⠀⠤⢤⠀⠂⠈⠉⠋
⠈⠁⠀⠈⠃⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡉⠀⠐⠒⠀⠀⠀⠀⠀⠀⠁
⠠⠄⠛⠳⠤⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⣋⣁⣤⡄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠒⠂⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠉⣀⠀⠀⠀⠀⠀⠶⠄

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⠀⠀⠀⠀⣠⠄⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⠄⣄⣀⣀⣀⡀
⠀⠀⢠⠞⡤⠚⠋⠉⠓⠀⡉⢿⣛⣛⣛⠻⠿⠯⠉⠉⠉⠉⠉⠙⠓⠒⢲⣶⡶⠤⠤⠤⢤⣄⣀⣀
⠀⠀⡏⡰⠀⠀⠀⠀⠀⠀⣙⠤⠛⣖⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⢤⣤⣻⣟⣁⣀⣀⣀⠀⠤⠤⢬⣽⣗⣦⣄
⠀⠀⠇⣧⡄⠀⠀⢀⡤⠚⠁⢀⠴⠋⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠮⡭⠭⠭⠭⢭⣿⣟⣛⣻⣧
⠀⠀⡆⢻⣿⡾⠖⠁⢀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⣀⡠⠞⣡⠴⠋
⠀⠀⢸⡜⠋⠀⣠⣴⣯⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣉⠴⠊⠁
⠀⠀⠀⠱⣤⣊⣉⣁⣤⣤⣤⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠿⠿⠛⠿⠿⢿⣿⣾⣷⣮⣁⡀
⠀⠀⠀⠀⡏⡛⠿⣶⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣋⣛⣉⣛⣲⣶⣤⣄⣀
⠀⠀⠀⠀⡇⡏⢳⣬⡙⠻⣦⢄⣀⣀⣀⣠⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠖⢚⣛⣛⣛⣛⣭⡭⠭⠭⠭⠭⠩⠭⢵⡦⢸
⠀⠀⠀⠀⡇⡇⡶⠈⠋⠓⡀⢸⠉⣭⣥⡤⠤⠤⠶⠖⠒⠒⠂⠘⠉⠉⠉⢩⣥⣦⡄⠀⠀⠀⠀⠈⠉⠁⠐⠒⢾⡇⢸
⠀⠀⠀⢠⣇⡈⠳⠦⣿⠈⡃⣼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⡀
⠀⠀⠀⠀⡇⣧⡀⠀⠀⠀⡇⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠇⣸⡇
⠀⠀⠀⠀⠈⠪⡑⢄⠀⠀⡇⣷⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠒⠒⢋⣉⣩⠤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠲⣕⢤⡇⢻⢸⣇⠀⠀⠀⠀⣀⣀⣠⠤⠴⠆⢀⣉⣉⡥⠤⠆⠐⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢈⠘⠟⠒⠒⢋⣉⡭⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⣴⠖⠁⣼⠋⢠⡟⢠⡌⢀⣄⠀⠀⣀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⣀⠀⢀⣀⣀⣀⣠⣤⣶⣾⣿⣧⣶⢿⣥⣶⣿⣥⠾⠋⣠⣿⢳⡿⠙⢀⣠⡄⣍⣈⣛⣿⣷⣦⣤⣤⣤⢤⣤⢸
⠙⠚⠂⠉⠩⠥⠤⠭⠉⠭⠛⠻⣛⢓⡟⠛⠻⠭⠿⠤⠚⣿⡃⣈⣁⣹⡿⢾⣾⠷⠿⠿⠿⠿⠷⠶⠓⠒⠂⠉⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⡒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⠄⠠⠠⠼⠟⠁⣸⠋⣁⠀⠈⠚⠃⠀⠉⠐⠀⠀⠀⠠⠀⢀⡀⢀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠰⠟⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠬⢬⡀⠲⠄⠀⠀⠀⠀⠞⠁⠪⠀⠀⣏⠀⠀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⡀⠀⢠⣿⣿⣇⣡⠀⠀⠀⣈⠍⠀⡼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢠⣟⣿⣿⣿⣾⠿⠃⠈⠁⠘⣀⣾⠎⠀⠜⣷⡮⠁
⠀⠀⠀⠐⠘⠈⠘⠁⠀⢀⡀⠀⠀⠀⠀⠀⠊⠀⠾⢣⣤⣶⣆⣀⣠⣤⣤⠀⠀⠀⠀⠀⠄⠀⢀⣤⣿⡿⠏⢂⠐⠀⡀⠠⣤⣛⣻⠎⠀⡴⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣅⠀⠀⠀⠔⣚⣓⠠⠴⠤⡤⣬⣉⣉⣙⡛⠛⠃⠿⠿⠿⣦⣤⣄⣾⡟⡋⠂⢀⡀⢥⠉⠀⢠⣿⣿⠏⠀⠜⡟⢠⠃
⠀⠀⠀⠀⠀⠀⢠⣵⡉⠖⠋⣭⣦⢻⣾⡟⣶⠄⠀⢀⣀⠀⢀⡬⠋⣁⣀⣀⡀⠐⠀⠠⠤⢠⣬⣀⣉⣴⢈⠉⣦⡔⠠⠾⠿⠋⠀⣤⡀⣡⠃
⠀⠀⢀⠠⠀⠀⢡⣭⣹⠆⠀⠘⣯⡆⠈⣥⣠⡾⢦⢼⣣⣤⣀⠀⠀⣠⣿⣿⠿⣿⣟⡷⣶⣃⢀⡤⣭⠁⠁⠰⠗⡳⠖⠀⠤⢤⣾⣿⣷⠃
⠀⠀⢸⠐⣦⢠⣤⠀⠀⠈⠁⠚⠓⠟⠾⠿⣯⣿⣃⣶⣻⣿⢧⠷⣾⣿⠥⣻⡀⡇⢉⣷⠛⣛⡋⠀⠠⢿⡃⠵⠫⠅⣀⡤⣶⣿⠿⠛⠁
⠀⠀⢘⠀⡿⢸⣿⠀⠀⠀⠀⠉⣶⣶⠀⠀⠄⢀⠉⠁⠘⠛⠘⠪⠥⢀⣠⣍⢿⣉⣴⡨⢒⣂⣀⠰⠎⠁⢀⣤⣴⣿⡿⠛⣿⣿
⠀⠀⢸⠀⡁⢸⣿⠀⠀⠀⡅⠀⣿⣿⠀⠀⢀⡄⠀⠀⠀⠀⠂⢠⣤⣀⣀⠈⠈⠉⠓⠛⠋⠁⢀⣠⣴⣾⣿⣿⣿⣿⣧⣦⣿⣿
⠀⠄⣬⠀⠁⢸⣿⠀⠀⠀⡇⠀⣹⣿⠀⠀⣿⡅⣦⠀⠀⠀⠀⡜⠛⠷⠀⠈⢡⣷⢲⡖⠒⢲⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣤⡤⠤⢤⠤⡤⠄
⠀⠀⡿⠀⠀⢸⡿⠀⠀⠀⣿⠀⢉⣿⡄⠀⠈⠀⠸⣧⡀⠀⠀⠅⠀⠀⠀⠀⠸⣿⢻⡇⢸⡆⠻⣛⣩⣄⢠⣿⣿⣿⣿⣷⣻⣿⣿⣿⡷⣶⣗⠖⠒⠀⢀
⠀⢀⢃⠀⠀⢸⡇⠀⠀⠀⣿⠀⠈⢻⣷⠸⣤⣴⣶⣿⠀⠀⠀⢉⡃⠀⠀⠀⠀⠀⢸⡇⢸⣧⡖⣿⣿⣟⠠⢈⣭⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣛⣋⣉⠉
⠰⠬⢽⣿⣖⣾⣦⣤⣀⣀⣿⠀⠀⠘⣿⠀⠉⣿⡇⠈⠅⠈⠀⠿⡀⢸⡇⣤⠀⠀⢸⣿⠈⣿⣿⣹⣿⠿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣻⡭⣭⣤
⠠⠠⢾⢯⣿⢽⣿⣽⣯⣿⣿⣿⣶⣶⣿⣄⣀⡈⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⠀⢸⡇⠀⣿⣿⠋⢛⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⠘⠀⠲⠶⠿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠸⠇⠟⣑⠀⢸⡇⠀⣿⣿⣠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣟⣒⡚⢛
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠯⠤⣜⣇⣰⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⣉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠚⠋⠉⠉⠉⠉⠙⠲⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣉⣉⣉⣉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠘⣷⣨⣍⣁⣒⠻⢭⠩⠭⢛⠲⣶⣶⣄⠻⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣪⡶⠛⣧⡈⣷⡆⠀⠉⠻⡆⢻⣶⢄⠑⢌⢿⣾⠇⠙⡆⠀⠨⣳
⠀⠀⠀⠀⠀⢂⠼⠉⠀⠀⠈⠉⠡⠤⠀⠐⠀⡄⣼⡯⡀⠀⠀⠈⢫⠀⠈⠁⠀⠠⢸⡄
⠀⠀⠀⠀⢸⠸⣅⠀⠀⠀⣀⣈⣐⣈⠏⢀⠜⣰⣟⠥⡂⠄⠀⡆⣿⡴⠃⠀⠐⠀⠀⢱⣄
⠀⠀⠀⠀⠈⡆⠀⠉⢁⣲⡬⢵⣂⡠⢐⡡⣾⠞⡣⠑⠂⢀⡰⣱⠁⠐⠀⠀⠄⡠⣠⡳⢦⣍⡒⠤⡀
⠀⠀⠀⢠⡊⢀⣾⢣⢶⢌⣤⣴⣶⣾⢷⣿⣫⠵⠬⠒⠊⢩⡔⣇⠂⣁⣁⣁⣪⣜⠉⠛⠶⡬⣛⠷⣮⣑⠦⣀
⠀⠀⠀⢈⢏⢾⣮⢧⢣⣿⠃⠀⠈⠻⠿⣶⣖⣦⢶⡜⡿⡔⣱⡸⠛⠛⠋⠉⠉⠙⠳⢶⢗⡮⣙⣷⣦⣍⡛⢶⣭⣒⠤⡀
⠀⠀⠀⢃⣾⣿⢣⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⠐⡹⡽⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣗⠉⣛⢷⣮⣙⠳⣮⣕⡢⣄
⠀⠀⢰⣿⣾⢣⢣⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠱⢱⡹⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠿⡻⣦⣿⣿⣗⣷
⠀⣔⣽⢯⢣⣳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷⣧⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠋
⡴⣿⢯⡷⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢿⡆⢷⡱⡀
⢻⣳⡿⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣿⡘⣧⣧
⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁
⠀⣀⡀⠀⠀⡀⢀⠀⡀⢀⠄⡀⠀⢀⠠⠀⢄⠀⡢⠐⠖⠠⠀⠔⠠⠂⢄⠠⢀⢄⠠⡀⢄⠄⡠⡀⠔⡠⢀⢄⡠⣀⠄⡀⡀⢀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡫⠿⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⣰⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠘⢠⡧⠀⠀⠸⡿⢢⠄⠀⠀⠀⠀⠀⣠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠃⠀⡴⣿⡁⠀⠀⠈⠀⠘⠒⡆⠀⠀⠀⠚⠿⣿⣶⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⡋⠀⠀⠀⠀⠂⠀⠻⠞⢂⠀⠀⣀⢀⠀⠀⢠⠀⣄⢀⠀⢹⢡⠀⠀⠶⠿⠶
⠀⠀⠀⠀⢀⣀⡀⠀⠈⠙⠛⠃⢀⢠⠀⠀⢠⠀⠀⣼⣼⡶⠾⠿⣼⠞⠻⠿⠿⣿⣾⠀⠀⣼⡂⠀⠀⣰⢀⡄⣀⠀⢸⡄⢀⣀
⠀⠀⠀⠀⠉⠙⢳⣶⣶⡄⠀⠀⠺⢾⣤⠷⠾⠇⠀⠉⠉⠁⢀⣀⣀⣄⣀⠀⠀⠀⠋⠁⠈⠉⠛⠀⠐⠟⣿⣿⡿⠄⡙⢁⣤⣿⣿⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣦⡀⠀⠈⠛⠀⠀⠀⢀⣈⣠⣿⣿⣿⣿⣿⣿⣷⣶⣶⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⢶⣿⡿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢶⣦⣴⣶⣿⢿⠿⠉⠉⣘⣿⡭⠉⠡⠿⠿⢿⢿⣿⠿⢿⣿⡟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠐⠠⢶⠛⠓⢀⡤⠀⠩⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⣦⠀⢀⠐⢝⡂⠀⠀⠀⠀⢀⡠⠄⠀⠠⢄
⠀⠀⠀⠀⠐⠁⠀⠀⠈⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⡄⡙⣷⡀⡠⠂⡡⠂⠰⠶⠶⠢⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⠂⠠⢤⢀⠳⠸⢏⣰⣁⠧⢔⣋⣦⡀
⠀⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡄⠇⠎⠤⣺⣰⣀⣄⣑⣿⠬⠛⠋⠉⣉⠉⡀⡉⠁⠀⠀⠄⡀
⠀⠀⠀⠀⠁⠈⢦⣀⡀⠀⠀⠤⠂⡄⠀⠀⠀⠀⠀⠈⠀⢀⠤⣶⣿⢟⣯⣿⣿⣿⡿⣞⠻⠋⠹⣍⡻⡰⢝⢎⢶⢠⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠁⠀⠀⠀⠀⠀⠀⢰⠇⠘⠹⠟⢞⢙⡷⡏⣽⢁⢻⡆⡀⠢⠀⠑⠄⠀⠈⠁⠙⠀⠤⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢱⠏⢀⢔⡵⠃⢸⡅⡟⢨⠋⠘⡁⠐⡳⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢠⢢⠎⠀⠀⠐⣼⡇⡶⠀⠀⠑⢾⡄⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠏⠀⠀⠀⠀⠺⡻⠁⠀⠀⠀⠀⠢⣎⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠐⠁⢹⠂⠀⠀⠀⠀⠀⠰⠖⢒⣒⣒⣒⠒⣲⣤⣴⡶⠶⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢈⠩⠑⠂⠀⣺⠀⣸⡦⠄⠄⢀⣀⡀⠀⠂⠉⠀⢉⣉⣩⣭⣭⣭⣍⡭⠥⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠈⠀⠁⠉⠀⠀⠁⠀⠀⠌⠉⠓⠢⢀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⡞⢁⠀⠁⠀⠀⠀⠀⠀⠀⡀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠂⢐⠠
⠀⠀⠀⠀⠀⠅⠠⠀⠠⠁⠀⠅⠀⠀⠤⠜⠶⢋⣡⠬⠀⠄⠨⢀⢀⠈⣁⢀⡀⣀⣠⠴⢆⠠⢀⡀⠐⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠁⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠂⠁⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠠⠂⡀⠀⡀⣀⠤⠆⠀⠀⠀⣀⠀⢀⡤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠒⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠑⠂⠀⠉⠘⠂⠑⠠⢄⡄⠚⠀⡚⡄⠀⠠⠀⠈⠉⠻⠄⠀⡨⠀⢤⡀⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠀⠀⢀⠀⢀⠠⡀⢎⠀⠶⠶⠇⡀⠀⠀⠀⠀⠀⠀⢀⣸⡆⢆⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢠⠀⠂⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠖⠈⠀⠀⠀⠀⠤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠺⠀⠀⠂⠀⠀⠙⠀⠀⠀⠀⠀⢀⠈⠦⡄⠀⠀⠀⠀⠀⠀⠀⠈⣄⠘⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢋⠀⡠⠀⠀⢀⣀⠀⠀⣀⡄⠀⡀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠵⠀⠀⢀⠀⢳⡀⢸⡃⢹⢠⣁⠀⠀⠀⢀⠔⠀⠄⠀⠀⢀⠀⠀⢠⡀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠉⠑⠂⠀⠚⢄⠀⠈⡡⠔⠀⠀⠀⢲⡄⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⠈⠛⠘⢊⠁⠀⠀⠀⠀⠀⠀⢾⠀⠂⠀⠐⠀⠀⠀⠙⢂⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠁⠄⠀⠀⠀⠀⠀⣀⠀⠀⠆⠰⠄⠤⢉⠀⠀⢀⠀⠀⠙⢇⠀⠀⠀⠁⠀⠈⢁⡉⠀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠐⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣉⠀⣀⡀⠀⢀⠀⠀⠀⢄⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠘⠀⠀⠀⠀⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠰⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠅⠀⠀⠐⠠⡄⠀⢄⠄⠀⠀⠀⠀⠀⠀⡈⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠾⢧⣼⣷⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⡼⣤⣈⢁⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠄⠃⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠠⠈⠀⠀⠠⠂⠀⠁⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⡀⠀⠀⠐⠀⠘⠀⠁⠀⠁⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣑
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠖⡍⠃
⠀⠀⠀⢀⠀⣈⡻⣴⣿⠃⠀⠀⠀⠀⠀⠀⠞⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠄⠀⠀⠁⢀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⢀⠀⣀⣀⠀⠀⢀⢀⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣿⡿⢿⣿⣭⠟⠋⢿⣿⡿⠿⡿⣿⡿⣿⣿⡟⡹⡟⠿⣛⣛⣉⣙⡛⣛⣛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡋⣿⢿⣿⣿⣿⣟⡵⣾⣷⣭⣿⡶⠍⠗⠤⣜⣿⢷⡾⠰⣜⣟⠻⣿⡽⢿⡿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣾⣿⣷⣶⣿⣿⣏⣿⣿⣯⣟⣿⣾⣿⡶⠾⣿⣇⣠⠓⣾⣿⣿⣷⣬⠙⣋⠴⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠈⣧⣣⣟⡛⠛⠛⠻⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟⢹⡟⠆⠙⢛⣛⣓⣛⣛⣛⣋⣉⣭⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⠏⠉⠉⠛⢳⣼⣴⣦⣶⣤⢤⣤⡤⣤⣽⣍⣹⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣆⡀⢠⣴⢪⣿⡏⣿⢾⡾⠷⣠⡾⣿⣿⡟⢻⣿⣦⣿⠿⠿⣛⡩⠷⠺⢟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⠾⢾⡿⡶⣿⡇⣿⣏⠟⠂⠀⠁⣶⣾⣷⣡⣿⣿⣿⠋⣁⣤⣤⣴⣷⣾⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⣿⣿⣖⣛⣏⣷⣷⢸⡇⣀⣤⣿⣿⢿⣿⡿⢙⢿⣿⣿⣾⣶⣚⡽⠻⠟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠈⠉⠉⣛⢸⠘⠻⠿⠭⢿⣿⣭⣝⣿⣦⣾⣿⣿⣿⣧⣶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠤⠄⣹⣹⣇⣪⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠄⠱⠀⠀⠂⠀⠀⠀⠀⠉⠁⠶⠾⠀⠀⠄⠉⠀⠉⠠⠉⠀⠧⠝⠀⠀⡀⠀⠨⠖⢂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠐⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⢘⠠⠄⡆⠀⠀⠀⠤⠄⠀⠀⠠⠠⠠⠀⠠⡀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⢐⡴⠄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠠⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠀⠉⠁⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⢀⠄⡭⠂⡀⠀⠂⠀⠇⠀⠠⢀⡈
⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠠⠋⠘⠇⠀⢐⠀⠀⠀⠐⢤⠀⡠⠊⠀⠀⠀⠅⠐⢈⢀⠐
⠀⠀⠐⢐⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⠀⠀⢄⡤⢰⡆⢄⠄⢰⠄⣄⠄⠠⠀⠀⠀⠘⠃⠀⣉⠉⡁⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⡅⠀⠐⠉⠁
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠤⠠⠠⠤⠤⠤⡄⠀⠌⠡⠸⠄⠌⠅⠸⠀⠌⠄⠀⣀⠁⢰⠀⣋⠀⣁⠈⠃⠀⠘⠀⢠⣤⠀⡀⠀⠀⠀⠀⠀⠀⡂⢀⡀⢸⠷
⠀⠈⠁⠉⠀⠀⠀⠀⠀⣒⣘⣒⣒⣒⣒⡃⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠠⣿⠐⠽⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁
⠀⠀⠀⡀⠀⠀⠀⠉⠀⠰⠉⠂⠞⠶⠛⠂⠸⢾⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⢸⠀⠁⠀⠁⠀⢀⣀⣠⠀⠀⠰⠲⠒⠒⠖⠲⠲⠒⠲⠲⠀⢀⣀
⠀⠀⠀⠁⠀⠀⠀⠰⠰⠀⠆⠀⠰⠐⠀⠀⠠⠀⡆⠀⠀⠀⠠⠀⠀⠤⠀⠀⢂⠄⠘⠀⠀⠞⠀⠄⡇⠰⠆⢳⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠀⠖⠀⠂⡆⠒⠀⠀⠠⣀⡀⠀⡀⠀⣀⠀⠀⠀⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠐⣖⠒
⠀⢀⣈⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⢀⠀⠀⢰⠀⠀⢀⠀⠁⠈⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁
⠀⠈⣈⣀⣁⡁⠀⠀⠀⠀⢀⡄⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠉⠔⠀⠀⣤⡄⠠⠀⠀⠠⣿⠆
⠀⠀⠉⠋⠛⠃⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠈⠀⠁⠀⠈⠁⠉⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠶⠚⢛⣡⠶⠞⣛⣉⣉⣛⡛⠶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠁⠀⣴⠟⣡⠾⠛⠉⢉⣩⣭⣝⣷⣯⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠃⠀⠀⣼⢃⡾⠁⠀⠀⠀⣾⣁⣴⠟⣷⠈⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠻⣥⣤⣴⠏⠀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣌⢷⣄⠘⣧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣦⣙⠳⠘⣧⡙⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⢁⣽⠗⠂⣼⠻⣦⣍⡛⠶⠶⠶⠶⠞⣋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⠁⠀⡟⠁⠀⣼⠃⢠⡟⣽⠛⢻⡖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⢟⣴⠟⠁⢰⡟⠛⠃⠀⣰⠏⢠⡟⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢋⣵⠟⡁⠀⣾⠛⠃⠀⠀⣰⠏⢠⡟⣸⠃⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⣿⠙⠉⠀⠀⠀⣰⠏⢠⡟⣸⠇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⢰⡟⢀⡟⣰⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⣱⠏⠐⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣏⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⢿⠿⡿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣻⣿⣛⣿⣿⣟⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢻⣿⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠋⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠀⠀⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡿⠁⠀⣁⠔⠀⠐⠠⡄⠀⠀⠘⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⣀⠀⣂⠖⠀⢀⠁⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⡔⠂⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⠀⠠⠀⡤⠆⢠⠀⠄⡀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⠀⢀⠀⠈⠀⠐⠐⠁⠠⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⡿⢿⠿⠟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠶⠶⠶⠿⠿⠿⠿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄
⡇⠀⡠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⣀
⡇⠰⠚⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠄⠊⠀⠀⠀⠀⠀⡀⠈⡱⠿⠏⠲
⡇⠀⠀⢻⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠘⠀⠆
⡇⠚⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡃
⣿⣧⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠦
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⣠⣾⡿⣿⡏⠉⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⡆⢻⡇
⠀⣰⡿⢿⡧⠈⢾⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢻⠀⢸⡇
⢠⡟⢀⣾⣧⠀⠈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄⢸⣧
⣾⠇⢸⠏⡇⠀⠀⣯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢸⡇⢸⣿
⣿⠀⣿⠀⡇⠀⠀⢿⠀⢸⣿⣿⣿⣿⣭⣯⣭⣭⣭⣭⣭⣭⣉⣽⡇⠁⠘⠀⣿⠘⡇⠸⣿
⢻⡇⢹⣧⡇⠀⠸⡧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣼⣿⡀⠀⠀⣿⠀⣿⠀⣿
⠸⣧⠈⢿⣷⠀⢦⣷⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⢛⠻⣷⣄⠀⢿⡄⣿⠀⣿
⠀⢻⣆⢸⣿⠀⢈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠀⢿⡆
⠀⠀⠙⢿⣿⠀⢦⣻⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠉⠉⢉⣉⣉⣁⣀⣈⣉⣉⣁⣉⣹⡇⢸⡇
⠀⠀⠀⠀⠙⢿⣷⣿⡄⠘⠛⠛⠛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣭⣭⣥⣤⣤⣤⣤⡤⠤⠤⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠛⣿⣏⣭⣭⣭⣭⣭⣭⣭⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠈⠻⣿⣿⣿⣿⣿⣿⠟⣉⣉⣹⡿⠋⠉⢉⣻⣿⣭⣭⡿⠛⢿⣆⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣄⡈⠻⣿⡏⢍⣿⠟⠩⠉⣙⣿⣿⠫⠍⠉⢻⣦⣀⣹⠶⠶⠿⢷⣄⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⠻⣿⣦⡈⠻⣷⣿⡐⠂⠐⣢⡾⠷⢔⣒⣒⣿⡿⠛⣷⣤⣀⣈⣼⣿⣷⡈⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⡙⢿⣿⣦⡀⠻⠿⠿⠿⠷⠶⠿⠛⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⡻⢷⣍⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣤⣿⣷⣅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠆⢠⣶⣶⣶⣶⣶⠶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣎⢻⣦⣽⣷⣼⣿⡿⠿⢿⣿⣿⠿⠿⠿⣿⡇⠀⣸⣧⠀⢸⣿⣚⣛⣭⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡙⠳⣭⣹⣾⡟⣿⡇⠀⢸⣿⣶⠬⠭⢥⣾⡇⠀⠛⠛⠃⢸⣿⣉⣽⢽⣿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣬⡙⢿⣧⣿⡇⢰⢸⡿⠿⠷⠶⠦⠿⠷⠶⠶⠒⠒⣺⣛⣲⣶⣻⣿⣄⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⣿⣿⣾⣾⣿⣿⣿⣯⣽⣯⣤⣬⣭⣭⣭⣽⠶⠶⠶⠾⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⡟⡀⠀⠀⠂⠀⠀⠀⢄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠒⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠓⠒⠒⠒⠒⠂⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠉⠛⠒⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠁⠐⠠⢄
⠀⠀⠀⠀⠀⣧⠀⠠⠶⢤⣤⣄⣀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠤⠴⠶⠶⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠆⡈⠀⠄⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢈⡕⠒⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⠋⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠄⡀⠀⠀⠀⠁⠀⠀⢀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡀⠀⠀⠀⠰⠒⠒⠒⠄⢛⣿⣾⣷⣿⣦⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠉⠉⢠⠀⠀⠰⠶⢦⡄⠀⠈⣿⣿⣿⣿⣿⣧⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡰⠃⠀⠀⠀⠈⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠡⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢀⡯⣍⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⡆⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⣶⣴⠖⠁⠀⠀⠀⠀⣠⠟⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠚⠋⠉⠀⠀⠀⠀⠀⠙⠻⠿⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⡏⠉⠉⠛⠛⠻⠛⠉⠉⠛⠛⠟⠟⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠛⢿⡿⠿⠟⠋⠉⠙⠉⠉⠛⠛⠛⠛⠛⠙⠛⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠿⠿⠛⠋⠉⠉⠉⠙⢹
⡇⠀⠀⠀⠀⠀⠠⠂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠒⠋⠀⠈⠁⠉⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⣿⠆⠀⠀⠀⣀⡉⠀⠂⠒⠈⠀⠀⠀⡔⠈⠀⠀⠀⠠⠁⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⣿⠀⠀⠀⣸⠁⠈⠀⠉⢱⡀⠀⠀⢠⠃⠀⠀⠀⠀⠆⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⠀⠀⠀⣇⠀⠀⠀⠀⠀⣩⠀⠀⡇⠀⠀⣰⠆⣀⢰⡆⠀⠈⠑⡀⠑⢄⠀⠀⠀⠀⢀⣀⡀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠘⠤⣄⡀⠄⠴⠀⠀⠸⠀⠀⣠⠉⠀⠿⠀⠁⠄⠀⡰⠀⠀⠀⠁⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠈⠁⠀⠀⠄⠀⠀⢀⠀⣀⡀⠀⢀⡠⠒⠂⠀⠀⠠⠄⠀⠀⢸⣿
⣿⣿⡀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠂⠈⠁⠀⠀⠀⠀⠀⣀⢀⣠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⢄⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠢⠀⠤⠔⠁⠀⠀⠀⢹
⣿⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠊⠈⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠂⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠄⠐⠁⠈⠁⠈⠁⠀⠉⠂⠀⠛⢻
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡄⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀⠀⠀⢸
⣿⡇⠀⠀⠂⠀⠀⢄⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⣠⠀⠘⠛⠀⠀⢸
⣿⠤⠀⠀⠀⠀⠀⠈⠂⠒⠄⠀⠈⠂⠄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹
⣿⠀⠀⢁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⠠⠄⡀⡀⣀⡜⠀⠀⠀⠀⠠⢀⡄⣀⠁⠂⠠⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣼
⣿⠀⠀⠠⡀⠀⠀⡠⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢍⡀⡀⣤⣠⠄⠀⢤⢀⡀⠀⣀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣘
⣿⣤⣠⣄⣼⣷⣤⣤⣶⣾⣦⣤⣤⣤⣤⣤⣤⣶⣾⣤⣤⣤⣤⣤⣤⣴⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣥⣤⣤⣀⣀⣤⣀⣿⣦⣄⣤⣀⣀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣰⣀⢶⠶⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣨⣟⣿⣿⡯⠴⠶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⣻⢿⢽⡋⠻⣶⡄⣀⡀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢼⣿⠸⣄⢿⣄⠀⠑⠈⢂⠀⠀⣤⣲⣾⡏⠩⡍⠀⣀⣼⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣾⠸⣿⠀⠌⢻⣟⠇⡄⠀⠀⠀⢀⣾⠑⠻⡿⣦⠈⢸⢯⡾⢻⡄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠛⠀⠀⡈⠳⣄⠀⠀⠀⠀⠘⠀⠀⠘⠙⣿⠀⠀⠀⠁⠸⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠭⠨⠡⠤⠄⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⡀⠀⠀⠈⠙⠦⣄⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⠮⠥⠼⡭⠭⠭⠭⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠈⠑⢤⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠙⠲⣤⡀⢈⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡈⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢷⣆⠀⠀⠙⡃⢬⡁⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠦⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⢤⠗⢣⡹⣷⢶⠷⣤⠘⢙⡲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠋⠁⢀⣀⠐⠋⠉⡗⢷⣬⡾⡏⠙⠲⠿⣧⣘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢫⡄⢀⣰⢁⡤⠏⢁⣆⣇⣴⣷⣦⣿⣿⣿⣷⣼⡀⢾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣯⣥⣶⣯⣼⣿⣗⣤⡆⣼⣼⡟⠛⠛⠉⠙⠉⢋⠉⠉⡉⠻⠦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠉⠛⠉⠉⠉⠉

⣶⢶⢶⢶⡾⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⣶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣄⣀⣀⣀⣀⣀⣠⣀⣄⣀⡀⢀⣀⣀⣀⣠⣄⣀⣠⣄⣀⠀⢠⣤⣀⣄⣀⣀⡀⣀⣀⡀⢀⣀⠀⣀⣀⠀⣀⣀⡀⣀⣀⣀⠀⠀⣀⣤⣤
⠀⠀⠀⣻⣿⣿⣿⣿⡈⢿⡿⠿⣿⠇⠁⠀⠁⠉⠉⠻⣿⣿⣿⣿⠃⢀⣤⡟⠛⠋⠻⢎⡻⣦⠚⡅⠈⠋⠀⠘⡿⠀⠁⡿⠡⣿⠿⠁⢀⣼⣿⣿⣿
⠀⠀⠀⣿⣝⠿⠈⠉⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠈⠉⠀⡀⠀⢐⣠⣷⡈⠳⠀⠂⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠊⠉⣾⡿⣟
⠀⠀⠀⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠻⠷⣦⡙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⢞⣿
⠀⠀⠀⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣶⣶⣤⣤⣦⣴⣿⣷⡶⠤⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢛⣉
⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⢿⣷⣄⠰⣿⣟⠛⠛⠛⠛⠛⠻⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡼⠷
⠀⠀⠀⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢡⣧⠀⣄⡀
⠀⠀⠀⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢠⣷⡶⠶⢶⣖⣲⣖⣨⠽⠯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀
⠀⠀⠀⣿⣿⣧⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣼⡟⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠛⠻
⠀⠀⠀⣿⡏⢙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣱⣾⡟⠋⣉⣀⠀⠀⣀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠈⠑⠳
⠀⠀⠀⣷⣤⡈⠹⣿⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣾⣿⣶⠾⣋⣥⣶⠶⢃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤
⠀⠀⠀⣦⣄⠀⠀⣿⡛⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣶⠿⠛⠙⠋⠀⠸⠁⠛⢀⣴⣞⠃⡀⢰⣦⠀⠀⡄⠀⢠⠀⢀⢠⣤⡀⠐⣿
⠀⠀⠀⣏⠑⠆⠀⢲⣾⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡏⢰⡇⠘⣿⣶⢀⣷⠈⣄⣧⡀⢺⣽⢷⣄
⠀⠀⠀⠿⠿⠓⠀⢈⠙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢻⠀⠀⠀⠀⠛⢀⡿⠀⠀⢿⣿⣿⣿⡆⢹⣎⣷⡀⢻⣿⣽⠿
⠀⠀⠀⣦⣀⣀⣄⠿⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠁⠀⠀⣸⣿⣿⣿⣷⢸⣿⣿⣿⡀⠉⣠⣴
⠀⠀⠀⣿⡟⣽⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣰⠗⠄⠀⠀⠀⠀⠸⣦⠀⠀⡏⠘⠻⣾⣿⣷⠛⠋⠛⠁⠀⠻⢿
⠀⠀⠀⣬⡁⠉⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡆⠀⠀⠀⠋⠀⠀⠀⠀⢰⡇⢿⣿⠀⣤⠀⠘⠀⢘⣿
⠀⠀⠀⣿⠃⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡀⠀⠈⠀⠒⠀⢀⠀⠘⣹⠀⠀⣴⡶⠀⠀⠀⠀⢿⡇⠘⣿⠈⠉⠀⠀⠀⠛⣉
⠀⠀⠀⣶⣖⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠸⣇⣀⣠⣾⢶⣿
⠀⠀⠀⢿⣯⠁⣲⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠠⠔⠛⢧⡀⠀⠀⠂⠀⠀⠀⠀⡰⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⣿⣿
⠀⠀⠀⡦⠀⣀⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀⠑⡀⠀⠀⢠⠀⠀⠹⡦⠀⠀⠀⠀⠀⠠⠗⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢹⣿
⠀⠀⠀⣤⣴⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⣷⣾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⢀⣠⣆⣀⠀⣰⣬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⡀⠐⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢃⠄⠀⠊⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣿⣿⣿⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠼⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠟⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⡄⠀⠀⢠⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢰⠀⠀⠀⠀⢰⢧⠌⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡐⠀⠈⠀⠀⠀⠨⠀⠀⠀⠀⢀
⠀⠀⠀⣿⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡛⠀⠀⠘
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢠⡤⠤⠀⠐⠠⢖⠒⠒⠒⠒⠒⠚⠉⢉⡉⠐⠢⢤⡀⠈⠂⢄
⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠒⢖⠚⠋⠉⣉⣉⡉⠉⠉⠡⠤⠤⠄⠀⠀⠀⠘⠲⣄⠀⠀⠀⠑⡌⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠬⠷⣄⠀⠑⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠈⠢⡀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠒⠒⠘⢆⠀⠀⠀⠘⢆⣀⢉⡁⠀⠀⠠⠤⠄⠀⠐⠒⠊⠀⠠⠈⠄
⠀⠀⠀⡘⠀⠀⠉⡩⠓⢄⠀⠀⠀⠀⠐⡄⢀⣀⣀⣀⣀⣤⠤⠤⠤⠄⠲⠖⠒⠒⠒⠛⢠⠀⠀⠀⠨⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠀⠠⣠⠖⠁
⠀⠀⠀⠁⠀⠀⡴⠁⢀⠜⢄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠴⠆⠐⠗⠀⠋⠀⢈⠏⠉⠀⠀⠀⠀⠀⠠⡠⠞⠁
⠀⠀⠀⠀⣀⠞⠁⢠⠎⠀⠀⠰⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠶⠖⠶⠒⠒⠺⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠋⠀⣠⠏⠀⡀⠀⢀⡊⣀⣀⣠⠤⠤⠤⠤⠒⠒⠒⠂⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⣀⣠⠤⠤⠤⡆⠀⠀⠠⣀⠔⠁
⠀⠀⠀⠆⢀⣰⠃⠀⠜⠀⠠⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠴⠖⠒⠚⠛⠉⠉⠉⠁⠀⠀⢀⣀⣀⢠⡇⢀⡠⠊⠁
⠀⠀⠀⠰⠘⡃⢀⠎⠀⠠⠦⠔⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⠐⠛⠒⢖⡊⡉⢉⣡⡄⢀⠢⢷⣋
⠀⠀⠀⠀⠂⢣⠞⠀⡰⠁⣀⣀⣀⣤⡤⠤⠶⠖⠒⠒⠋⠉⠉⠁⢀⠄⠀⠋⠈⠀⡀⠈⠀⡤⠛⠙⠛⠦⡀⠀⢹⡆⠠⠄⠤
⠀⠀⠀⠀⠐⡈⠄⠐⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤⠤⠴⢶⣖⣻⠝⠃⠀⠠⣞⢡⡍⠉⠉⢘⣗⠀⡤⠐⠒⠓⠠⢎⡀⠂⠉⠤⢻⠅⠀⠒⢢⡐⠂⠄
⠀⠀⠀⠀⠀⠠⣤⣤⡤⡶⠏⠠⡌⠟⠩⠽⠓⠚⠂⢤⡋⠀⠀⠀⠀⠤⢈⠭⠙⠋⠉⢩⣝⠁⠉⠉⣶⠤⠄⠀⡸⠂⠛⠚⢣⠛⠀⠀⠀⣈⡀⠤⠖⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢼⡁⠀⢠⣂⡀⠉⣦⠊⠁⠒⠉⠓⢦⠀⠒⠢⣄⠰⠤⠤⠐⠈⠀⡡⠔⠠⠤⣈⠀⠀⠙⠂⠀⠀⠂⠥⠖⠒⠉⠁⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠙⣏⠀⠀⠘⠀⢴⡄⠀⠀⠀⡐⠚⠀⠠⠶⡉⠀⡠⠀⠀⠀⠘⡳⠀⣀⣀⡟⢠⠔⠒⠉⠉⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉⢹⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⢀⡀⠘⠁⠀⠀⢱⠗⠂⠠⠤⠄⠚⠀⢁⡒⠒⠂⠈⠘⠉⠁⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠀⣀⠀⠁⠀⠀⣀⠤⠆⠐⠂⠉⠁⠀⠀⠀⠀⢀⣀⡤⢤⠀⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠤⠒⠚⠉⠀⠀⠀⠀⠀⠀⢀⠀⠠⠰⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠠⠐⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊⡁⠀⠀⠠⠀⠀⢀⡀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠠⠀⠀⠈



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

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⡏⠉⠉⠛⠛⠻⠛⠉⠉⠛⠛⠟⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠙⢿⡿⠿⠿⠛⠉⠙⠉⠉⠛⠛⠛⠛⠛⠙⠛⠋⠉⠉⠙⠛⠛⠛⠋⠉⠙⡿⠿⠛⠋⠉⠉⠉⠙⢹
⡇⠀⠀⠀⠀⠀⢠⠒⠀⠃⠲⡀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠐⠋⠀⠈⠁⠉⠀⠐⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⣿⠆⠀⠀⠀⣀⡉⠀⠒⠒⠈⠁⠀⠀⡔⠊⠁⠀⠀⠰⠁⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⣿⠀⠀⠀⣸⠁⠈⠉⠉⢱⣄⠀⠀⢠⠃⠀⠀⠀⠀⠆⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⠀⠀⠈⣧⠀⠀⠀⠀⠀⣹⠀⠀⡇⠀⠀⣰⠆⣀⢰⡆⠀⠉⠑⡀⠑⢄⠀⠀⠀⠀⢀⣀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠘⠤⣄⡠⠄⠴⠁⠀⠸⡀⠀⣠⠉⠀⠿⠈⠁⠄⠀⡨⠁⠀⠀⠉⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⢀⡀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⢈⠁⠀⠢⠄⠀⠀⢀⠀⣀⡀⠀⢀⡤⠒⠂⠀⠈⠰⡄⠀⠀⢸⣿
⣿⣿⡁⠀⠀⠀⠈⠣⠀⠀⠁⠀⠀⠀⠁⠈⠁⠒⠋⠀⠀⢀⠤⠀⠂⠀⠀⠀⠀⠒⠒⠈⠁⠀⠀⠀⠀⠀⣀⠂⣠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠙⠋⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠀⢄⠠⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡤⠄⠀⠁⠀⠀⠀⠀⠀⠀⢰⠁⠐⠢⠄⠤⠔⠉⠀⠀⠀⢹
⣿⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠈⠁⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡄⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠊⠉⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⢤⡖⠆⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠤⠐⠁⠈⠁⠈⠁⠀⠉⠂⠀⠛⢻
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡆⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀⠀⠀⢸
⣿⡇⠀⡠⠊⠐⠀⢤⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠉⠀⣤⡀⠘⠛⠀⠀⢸
⣿⡤⠀⡖⠀⠀⠀⠈⠂⠒⠀⠀⠈⠂⣄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠤⠀⠀⠁⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻
⣿⠀⠀⢁⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠘⠠⠤⣀⡀⣀⡞⠀⠀⠀⠀⠠⢀⡄⣀⠁⠂⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣼
⣿⠀⠀⠡⡀⠀⠀⡠⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢭⡀⡀⣤⣠⠄⠀⢤⡀⡀⢀⣀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣈
⣿⣤⣠⣤⣼⣷⣤⣤⣶⣾⣦⣤⣤⣤⣤⣤⣤⣶⣾⣧⣤⣤⣤⣤⣤⣴⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣥⣤⣤⣀⣀⣦⣀⣿⣦⣄⣤⣀⣀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⠿⠛⠉
⠀⠠⣤⣶⣶⣿⡿⠿⠛⠛⠟⠛⠋⠉⠙⡩⠀⠀⠀⠉⢉⡉⠁⠀⡄⢀⡀⠛⠛⠋⠁⠀⠈⠁
⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠔⢀⠀⠉⢩⠛⠀⠀⠀⠒⣄⠀⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⠀⠀⠀⠐⠒⠀⢀⠀⠋⠀⠀⠀⣠⢀⣀⢀⡀⠀⣤⡀⠀⠂⠈⢀⠈⡀⠀⠀⠠⠀⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠶⠒⠒⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠘⡆⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠋⠈⠙⠲⢤⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢨⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

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

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⠄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⢢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠷⣵⣦⣤⡤⠀⢀⣀⣈⣡⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢄⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⠈⠣⡱⣌⠨⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡄⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣌⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢪⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠔⠒⠒⣂⠙⠛⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠐⠒⠉⠉⢈⠉⠍⠙⠿⣿⣿⣅⠀⠀⠀⠀⠀⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠛⠛⠛⠿⢿⣿⣶⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠙⣿⣧⡄⠀⠀⠀⠁⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠄⠒⣄⠉⠉⠉⠒⠪⢍⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⠀⢀⣀⡠⠤⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⠀⠀⠈⣦⣀⠀⠀⠀⠀⠑⢜⢿⣿⡄⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⡭⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠊⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣹⣿⣿⡶⣾⣿⣿⣿⠟⢻⣿⣿⡏⠷⣠⠘⢹⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⢴⣤⣤⣶⣷⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⢟⣿⡿⣫⠀⠶⡏⢶⠷⣿⣼⠐⢿⣓⣬⣟⡥⠬⠶⢌⡆
⠀⠀⠀⠀⠀⠀⠀⣬⣿⣯⣭⣻⣛⣌⣼⡯⠍⢴⠚⠛⢣⣁⠤⠥⢼⣧⠀⣿⢆⣠⣤⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢻⠞⢿⣿⣿⣥⢄⣐⠈⠙⠧⢤⠩⡄⣒⣿⠿⠻⣽⠉⠀⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠈⠧⠎⠉⠉⢿⣧⢹⣿⣶⣦⣶⣲⡍⢠⡏⠀⠀⠟⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⠰⣦⡄⠀⠀⠀⠉⢶⠮⢍⣻⣿⢯⣍⡹⠀⠀⠀⣿⡄⠀⠀⢹⣿⡅⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢨⡇⠀⠀⠀⢠⣤⡓⠖⢳⣿⣷⡆⠤⠐⠀⠉⢽⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠂⢼⣧⣤⣤⠀⢨⡛⠁⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⡇⠂⠀⠀⠀⠀⠀⣿⠤⡄
⠀⠀⠀⠀⠀⠀⣀⢹⡟⠀⠀⠀⠈⠻⠿⠻⢞⡥⠀⠀⢸⣿⣿⡇⠀⠀⢀⠀⠜⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢀⡠⠿⠋⠁
⠀⠀⠀⠀⠀⠁⠉⠸⠷⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢂⠉⢀⡀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠳⣦⠤⣀⡀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⡀⠤⠒⠉⠁⠀⣀⣀⠲⠂⠀⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣼⣿⣿⡇⠤⠔⠒⣉⣀⠀⠀⠀⠀⡀⠿⣙⣷⣿⣆⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣒⣀⠅⢀⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣿⡛⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⠖⠒⠒⠒⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠶⠶⠶⠦⠤⠤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠉⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡹⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠠⣠⠄⢰⠒⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢠⠧⠀⡒⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠆⠈⠀⣯⡔⠃⠃⡀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠴⠒⠄⠀⠀⠀⠀⠀⠀⢸⠋⠛⠳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠐⠒⡴⠦⠀⢾⣿⡟⠳⠁⣀⢀⠀⠀⠀⢶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣈⠔⠃⡼⢀⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠠⠉⠀⠀⣴⡶⠶⣤⣤⣤⣀⣠⡶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠿⠇⠀⡀⠀⠈⠉⠉⠀⠙⠿⠿⠶⠶⠶⣦⠀⠀⠀⣤⣶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⣶⠿⠋⠁⠀⠀⠤⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣰⡟⠁⠀⠀⠀⠰⠈⠀⠀⠀⠐⠠⠀⠀⣻⠞⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢻⣇⠀⠀⣸⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⢋⣠⡬⠇⠀⠀⠀⠀⣀⣀⣀⣠⣤⡶⠿⠛⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⡖⠀⠀⠀⠀⠀⠀⠀⢴⣏⡁⠀⠀⠀⠀⠀⣾⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⢶⣦⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠀⠀⠀⠈⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⣿⡷⠶⢦⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗
⠀⠀⠀⠀⠀⠹⣑⣒⣀⠐⠂⠍⠏⠩⠑⠒⠒⠒⢒⣒⠒⠲⠦⠤⠤⠤⠤⢤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠦⣤⢤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣥⣭⣬⣬⣤⣧⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒