Doodle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠹⢀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣀⡎⠀⠀⠋⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⢃⠀⠀⠀⠀⠈⠑⠄⠀⠀⠛⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⠀⠄⣀⠜⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢂⠀⠀⠘⠪⣀⠠⠀⠒⠒⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣰⡁⠈⠁⠀⠀⢠⠞⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠐⡒⣠⠚⢠⣮⣩⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⡠⠒⠉⠉⠑⠒⢄⣀⡴⠚⠉⠀⠀⠈⠉⠲⢄⠔⠁⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢸⠀⢞⢁⡈⠦⠤⠇⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡜⠀⠀⢠⡀⠀⠀⡰⠋⠀⠀⠀⠀⣶⡆⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠃⡴⠊⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣠⠃
⠀⠀⠀⠰⡁⠀⠀⢀⢳⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡲⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠰⡄⠀⠀⢀⠼⠃⠎⠠⠈⢰⠀⠀⡀⠂⠄⠀⡆⠀⠘⠳⣄
⠀⠀⠀⠀⢣⡀⢤⠼⡸⠀⡇⠀⠺⠷⠀⠀⠀⢠⡶⠁⠀⠀⠑⠀⠀⠠⠀⠀⠀⠀⠀⢣⡀⠀⣀⠸⠖⠚⣡⠤⠼⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠠⡆⠀⡘
⠀⠀⠀⠀⠀⠉⠒⠚⡡⠐⢺⡄⠀⠀⠀⠀⠐⢸⠀⠠⠀⠀⠀⠀⠀⠀⣱⠤⠒⠈⠉⠈⠉⠈⠉⠑⠒⠾⣀⠀⡧⠀⡌⡀⠀⠁⠐⠒⠖⠀⢰⠀⢀⡝⠉
⠀⠀⠀⠀⠀⠤⣠⠊⠀⠐⠁⡜⢦⡀⡰⣉⠑⡸⡀⠀⠀⠀⠀⢀⡴⠊⢀⠤⠒⠂⠈⠉⠉⠉⠀⠒⠠⢄⠀⠉⠳⣴⠁⠀⠀⠀⠀⠘⠀⢠⠃⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⡃⣱⠈⠈⠢⢀⡀⢸⣈⠁⢀⢀⠗⠉⠒⠂⠒⡸⠋⢠⠞⠁⣀⠔⠂⠉⠀⠀⠈⠉⠐⠤⡀⠈⠢⡀⠈⢇⠐⡂⠀⠀⠀⠀⠀⠋⠐⠀⢀⣀
⡖⠤⣀⠔⠉⠀⠈⢢⡈⠀⢡⠙⣄⡹⠄⠳⠶⠤⢄⠀⠀⡸⠁⠠⠁⡠⠊⢀⠄⠒⠉⠁⠀⠀⠉⠑⠢⢌⠑⢄⡈⠆⠘⠃⠀⣠⣀⡔⣁⠀⢠⡀⠀⠈⠑⢀⠜
⠑⢤⠃⠀⠀⠀⠀⠃⠉⠒⢢⠆⠁⠀⠀⠀⣀⠀⠀⢱⢀⡇⢠⠃⡰⠁⡔⠁⠀⡠⠔⠊⠉⢙⠉⠐⠤⡀⢱⠄⠣⡀⠀⠀⠀⠁⠹⡎⢠⠀⡀⣉⠀⢀⡴⠋
⠀⢀⠀⢰⠓⠀⡀⣂⠀⢠⠁⢀⣤⡀⠀⠈⠛⠁⠀⢐⢀⡇⣆⠀⡆⠘⠀⡀⠊⡀⠀⠀⢀⠜⠀⠀⢀⢹⠁⠀⠀⠀⠀⠀⠀⢃⠀⠇⠼⠃⢀⡄⣠⣎
⠀⠈⡄⠀⡐⠁⡂⠀⠈⠹⡀⠀⠉⠁⠀⠠⠤⠀⠀⠾⠌⠀⠀⠉⠁⠃⡴⠁⠀⠀⠁⡩⠕⠒⠒⠢⣎⢀⠆⠀⠀⠀⠀⡀⠤⠘⠂⠀⠤⣀⠉⡀⠇⠀⢳
⠀⠀⠘⠄⡇⠀⠀⢄⠀⢀⣑⣄⣀⠀⣀⣀⠤⠚⡇⠀⠀⠀⠀⠀⠀⠙⢂⡑⠄⣠⡊⠒⠉⠉⠉⠐⢌⢺⠤⠤⣄⡔⠃⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⡜
⠀⠀⠀⠈⢎⢦⣀⣴⣿⣿⣿⣿⠟⢙⣤⡀⠀⡃⠀⠀⠀⠀⠀⠀⣀⡠⡞⠀⢨⢱⠁⠀⠀⠀⠀⠀⠀⢡⢃⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠊
⠀⠀⠀⠀⠈⢳⠛⠛⠛⢋⡉⢠⠶⣩⠎⢙⠄⠐⠤⠄⠴⢢⡀⠰⣧⠸⡃⠰⢘⢀⠀⠀⠀⠀⠀⠀⢀⠀⠿⡂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠆
⠀⠀⠀⠀⠀⡿⠤⠈⠡⠬⠷⠚⠈⡠⢐⠡⠊⠀⠀⠀⠀⠀⡨⢢⠀⠣⡈⠲⡒⡚⠒⠒⠒⠂⠈⠉⠀⠀⢀⠄⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠆
⠀⠀⠀⠀⢰⠀⠀⠄⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⠰⠁⡀⠀⠀⢠⠰⡌⡉⣃⣄⡀⠀⣀⠀⣤⢄⡀⠐⡇⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⣠⣊⡀⢠⠎
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡸⢁⠠⠤⠴⠁⡠⠐⠁⠀⠀⠀⢨⡇⠈⡉⠑⢿⠈⠉⠀⠈⠐⠒⠂⠈⠉⠐⣂⠾⠀⠀⠀⠀⡰⠚⠉⠀⢀⠀⠑⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⠃⣀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⡟⠠⡀⠡⡀⡗⡠⠆⠀⠂⠒⠒⠁⠈⢡⠄⠘⠁⠈⠓⠢⢤⠀⠀⠀⡏⠀⢀⠃
⠀⠀⠀⠀⠈⡇⠀⢀⡠⠤⡖⣾⠁⠀⠀⠀⠀⠀⠀⢀⠠⠦⠜⠠⠀⠀⢲⠓⠄⣅⢹⢀⢀⣁⠂⠀⠤⠬⢤⠁⠀⠂⠂⠀⢀⡀⠀⠱⡄⠀⠈⠉⠉⠒⠈⢰⡀
⠀⠀⠀⠀⠀⢹⣆⣩⠖⠋⡀⠤⣇⠀⠀⡀⠀⠀⡈⠤⠔⡀⠶⠆⡈⠁⠀⠃⠀⠀⠻⠀⠀⠤⠤⠄⠀⠒⠚⡀⠀⠀⠀⠀⠀⠀⠁⢀⡇⠀⠀⠀⠀⣀⠔⠘⠁
⠀⠀⠠⠄⣉⠀⠙⢧⣴⠊⢀⣴⣿⣎⢁⠤⠄⠜⠀⡠⠀⠀⡠⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠒⠳⣄⠀⠀⠉⠁⢀⡤⠚⠠⣀⠠⠒⠉
⠀⠀⠀⠰⢁⣀⠀⠀⠈⠙⠛⠛⠛⠛⠛⢤⣀⡀⠊⣀⡠⢴⠃⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠞⠀⠀⠈⠈⠛⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠚⠁⢈⣀⠔⢂⠀⠤⡄⢠⡀⠀⣄⠀⠈⣉⠀⢀⣄⡇⠘⠂⠐⡂⡀⠤⠒⠒⠒⠒⠂⠐⠢⢌⣱⠈⠤⡑⠠⠐⢋⣐⠉⠉⠀⠒⠒⠒⠀⠤⠤⠴⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠀⠳⠃⠹⡴⠀⢳⣠⠃⠙⡴⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠈⠀⠰
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⢐⠐⡲⢄⣠⣄⠀⠠⠁⢀⠈⣀⠀⢆⢶⠸⢆⣴⡄⢠⡀⠀⡶⣖⣀⡄⠀⠀⢦⣤⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢠⠀⠘⠊⠠⠐⠒⠁⠘⠁⠀⡁⠛⣙⣶⣰⣷⣴⠾⡍⣷⣂⢁⡀⡴⠉⣰⢥⣠⣤⡀⣠⡆⠀⠀⣀⠀⠀⠈⠣⠴⠂⣠
⠀⠀⠀⠀⠉⠹⠊⣔⣠⡆⠀⠀⠀⠀⠁⠠⠴⣿⡿⣾⡿⢌⠙⣿⣍⣨⣭⡁⢀⣹⡼⠟⠟⠁⠉⢡⣡⣄⠁⠴⠑⠀⠒⠂⢈⡉⠀⠂
⠀⠀⠀⠀⡀⠀⣾⣶⠽⠁⢀⡀⠠⡾⢰⣾⠊⣽⣷⠋⢀⡀⠀⠀⢀⡀⠀⠈⠒⠀⢣⠰⠠⣤⣴⡶⢤⠀⠘⡆⠀⢃⡀⢀⠤⣤⣆⡀
⠀⠀⠀⠀⠀⠐⠃⠇⡿⢋⠀⠀⠘⢿⣌⣉⡄⠋⠀⠀⠛⣀⠤⡂⢌⠉⠔⠀⠌⢠⠀⠀⠸⠉⢁⣀⠈⠀⢈⢛⠶⠾⠓⠁⠈⠋⣛⠃
⠀⠀⠀⠀⠃⠀⠀⠀⢀⠈⠂⠛⠋⡠⠠⡟⠄⠂⣀⠀⣼⡾⠟⠛⢻⣷⡄⣰⡖⣄⣀⣠⠔⡤⠋⡹⢷⠷⠓⠀⠰⠛⠂⣀⡄⠺⠓⡻
⠀⠀⠀⠀⠠⡀⢀⣀⣤⠝⠒⠒⡚⠛⡼⠿⠀⢟⣉⠀⠈⠀⢀⡴⠟⡋⡑⠋⠉⢀⡁⡿⠃⢡⠀⠉⠃⠈⣀⣐⣒⠊⠉⠠⢞⣗⣛⠁
⠀⠀⠀⠀⠀⠠⡮⢨⣅⠀⠠⣦⣥⢀⣔⣀⠀⠀⠀⠱⠀⠀⣉⢳⣄⠈⠋⠓⣻⠟⠉⣁⠛⣳⠘⠃⠸⠪⠋⠛⡑⣚⠀⡛⢨⠥⠟
⠀⠀⠀⠀⠀⢸⡶⢧⠀⠀⣠⣏⠉⠈⢉⣥⣈⡉⠇⢀⡂⠚⠟⣸⣝⠢⣤⣴⣧⢤⡞⠟⢬⢻⣃⠹⠆⠀⠀⠺⢁⡈⣭⣅⠰⠻⠻⠁
⠀⠀⠀⠀⠰⡼⣻⣿⣿⡇⣈⣹⣾⡯⢟⠀⢣⣀⠁⣍⠀⠀⠈⢸⣿⠀⣿⣻⡏⠀⠉⢿⡇⠘⠓⠂⠀⣿⢀⡀⢈⡗⢩⢛⡄⢲⢲
⠀⠀⠀⠀⠀⡼⣻⡩⣵⡹⠇⠉⠁⠀⠒⢩⠝⢡⣌⠉⣉⣡⣴⣿⣷⠚⠋⠁⠀⣡⡀⠰⠤⣤⣴⣦⠄⠨⣭⣂⢉⠃⣴⣌⣧⢪⠌
⠀⠀⠀⠀⠀⠲⡯⣈⠛⡃⠐⢀⠯⣀⣿⣷⣿⣴⣿⣾⣿⡯⣥⡄⢛⢓⡶⠶⠞⠉⢹⢦⣀⡀⠀⠀⠺⠂⣀⠋⣢⡾⣮⠉⠀⠠⠠
⠀⠀⠀⠀⠀⣈⣋⣵⣶⣆⠁⠉⠀⠘⣿⡀⠈⠻⣡⣶⣤⠀⢰⠟⢙⡁⠂⠀⠀⠀⠈⢷⣛⠙⠛⠓⢀⡘⠫⠃⠑⠠⡴⠄⢀⣶
⠀⠀⠀⠀⠀⠉⢙⠉⡉⢹⡇⢀⡀⠀⠙⡷⠄⣀⣉⣿⣿⠇⠀⠈⠙⢛⡇⠀⣀⣀⠀⢠⠈⡗⠄⠆⡀⠀⠀⠘⠾⢫⡵⣻⡏⣿⣧
⠀⠀⠀⠀⠷⠶⠦⠴⠶⢯⡔⡋⢉⠈⣦⡀⠸⠌⣩⠀⠀⢀⣤⢀⡀⠁⠀⠘⠛⠶⠄⢀⣰⠣⠤⣤⢁⠀⣄⣢⡷⠌⣼⠃⢀⡌⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠉⠉⠀⠈⠁⠒⠃⠁⠓⠻⡣⡴⡢⠀⠤⠀⠸⣧⡄⠨⠙⠈⠀⠀⠙⠻⠈⠉⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣻⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡇⠼⠃⢀⡀⠀⠀⣾⣿⣷⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠐⠂⠀⠀⠀⢀⡩⠤⠤⡜⢸⡤⠒⢉⣉⡉⠉⠻⣿⣿⣏⠀⠀⢀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠤⠴⠅⠀⠀⢈⠰⠋⠱⠀⠹⠿⣿⢲⣄⣸⣿⣿⣿⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣿⠆⠈⢑⠊⠁⠐⠂⢀⡀⠔⠀⠃⠐⡤⢌⠉⠁⠘⡄⠀⠀⣠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡴⡯⠒⢉⣈⡑⡄⠈⠈⢹⠤⣀⡠⠼⣀⠇⣸⣄⠀⡘⠀⠀⠰⠏⣩⠿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣰⢱⣶⠞⢿⣧⣸⡆⠀⢈⠖⠁⠀⠀⠀⠀⡀⠙⠑⡌⠀⠀⠚⠛⢿⣿⡎⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⢸⠀⢉⢻⢻⡿⠟⡓⠒⠚⠤⣀⠒⡶⠀⠘⢑⠀⠀⠈⠢⠤⠀⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠸⠦⠓⢾⡀⠀⢡⠊⠀⠘⢷⣇⣤⠖⠈⡣⠄⡲⣾⡟⣠⠊⠑⠆⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢡⠳⢄⣠⡠⢠⠼⠋⠹⠀⠀⣀⠧⠤⢏⡁⣤⣨⠑⢢⠃⢗⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⢂⠩⡔⠸⣄⡀⠈⠀⠀⢀⠀⠓⣄⠎⠁⡀⢀⣀⣈⠫⡥⠒⣧⢡⢨⠋⢔⠀⡂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠤⢊⢓⣾⡅⠘⠃⠀⠘⠁⠀⢸⠀⠈⠃⠚⠛⠋⠀⣇⠜⠀⠀⡿⠐⠠⣄⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠨⠴⡈⢹⢁⠀⢶⠾⡿⠀⠀⠀⡄⡂⢐⢀⣾⠄⠀⠈⠀⢀⠀⣯⠀⠅⠈⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⣼⡦⠤⠤⠤⠤⠤⠤⠤⠧⠬⠥⠤⣥⣤⣤⣧⡤⠤⠤⠿⡐⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠹⢿⡿⠟⠁⠀⠀⢀⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣄⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡀⠀⠀⠀⠀⠀⣠⣴⣦⣄⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠈⠉⠉⠉⠉

⠀⠀⠀⠀⡠⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⠂⢀⣀⣀⠀⠀⣀⠤⣀
⠀⠀⢰⠉⠁⠀⣀⠈⢉⠆⠀⠀⢀⣆⠀⠀⠶⣉⣩⠤⢦⣠⠏⠀⠀⠀⠀⡠⠄⠀⠤⣀⠀⠀⠋⠀⡀⠰⣣⣤⣄⡱⠊⠀⠀⡈⢂⡀
⠀⠀⢅⣠⡀⠀⢉⣀⡸⠀⠀⠶⣾⣿⣿⠃⣀⡀⠀⠑⠤⠇⠀⠰⠄⠀⢸⢠⠀⠀⠰⠌⡇⠀⠀⠐⠆⣴⠇⠀⠈⠁⠀⠢⠰⠚⢉⢽⡇
⠀⠀⠢⡀⣭⡖⢡⡀⠀⢦⠀⠀⠸⠉⠙⠂⠈⠁⠀⢀⡶⡀⠀⠀⠀⠀⢸⠀⠢⠤⠴⠀⡇⠀⢀⣠⡌⠩⠗⠢⡀⠀⠀⠀⡠⠒⠈⠁
⠀⢃⠀⠑⠈⠁⠈⠃⠀⠈⠀⠀⠳⡄⠀⠐⠖⠀⣰⢻⠇⠹⡀⠘⢇⠀⠀⠉⠂⠤⠄⠊⠀⠀⠀⣛⣙⠛⠛⠃⠈⠉⠀⠉⠀⠂
⠀⠀⠓⠀⡄⠈⢷⠀⠀⡀⢶⡀⠀⠀⠰⣆⣠⠞⡏⢻⠖⢦⣵⣈⡾⠐⠀⠀⠀⠀⠀⠀⣠⠀⢮⣨⣈⣉⣯⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡤⠎⢇⠀⠀⠀⢠⠇⠀⢁⡤⡴⠀⠓⠚⠀⠡⣸⣢⠎⠀⠀⠀⣀⢨⣄⠀⢀⠀⠀⠁⢁⡸⠂⠀⠈⠀⠀⠀⠡⠐⢀⠞⠉⠣⡀
⠀⢠⡮⣄⠞⠆⠀⢀⣸⠀⢠⠋⠀⢇⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⡎⢁⣶⡦⠵⡀⠀⢠⠊⠁⠀⠀⠀⢰⡇⠀⠀⠂⢀⡼⣾⣰⣇⣅
⠀⠀⠀⠛⠀⠀⠀⢸⠁⠀⡆⠀⠙⠰⠉⠀⠀⠪⠽⠻⠟⠀⠀⠀⢿⣀⡟⠁⣶⠃⠀⠀⠉⠀⠊⠀⠁⠉⣀⠀⡀⠀⢸⠟⠻⠯⠛⢾
⠀⢀⣄⣰⣿⣷⠀⠈⠛⡆⢡⠘⢍⢈⡁⠀⠀⠀⡤⣷⡿⠂⠀⠀⠀⠉⠙⠍⠁⢀⡤⡥⢄⠀⡜⠢⡀⠰⡇⣉⡇⠀⠧⢤⢄⡠⣄⡠⠂
⠀⠸⣟⡛⠃⡿⠀⠀⠀⢀⡈⠢⣀⠀⠱⡀⠀⠀⠙⠁⣌⣿⠫⣀⣇⠆⠀⠀⢘⣥⡗⢳⡪⡇⠀⢠⠁⢀⣈⠁⠀⢀⠀⠸⣀⣀⡅
⠀⠀⠈⠙⠛⡅⠀⡎⠑⣃⢸⠀⠀⠉⠉⠁⠀⢀⣤⣤⡄⠀⠺⠽⠑⢆⡠⡀⠘⠶⣉⡍⡲⠂⠀⠈⠀⠘⢄⣋⣦⣇⠤⡒⡆⢀⡂⠀⡀
⠀⠀⠀⠀⠀⠈⢆⠘⠢⢁⠎⠀⠀⠀⣀⣴⡖⢱⠉⣏⠃⠀⠀⠘⠜⡛⠑⠃⡐⠐⠄⠈⠀⢀⣀⣰⢦⠀⠀⠉⢹⣅⡠⠜⠀⠀⡀⣁⡀
⠀⠀⢰⢲⠀⠀⠈⠢⠀⠀⠀⠀⠀⠸⠿⡟⣠⡞⠉⣼⠀⠀⠀⠶⢶⠯⠵⣠⡞⠂⢀⡴⠊⠁⠂⡼⠒⠃⠀⠀⡜⢣⠀⠀⠀⢨⡈⠁⡅
⠀⠀⠈⡅⠀⠀⢀⡀⠀⠠⠤⠀⢄⡠⠀⠸⠻⢼⣾⠟⠀⠀⠀⠀⢸⠀⣼⡿⢰⡂⢻⣀⡠⠖⠋⠀⠀⠀⢀⠴⠤⣠⠓⠒⡄⠀⠉⠐⠃
⠀⠀⠫⠼⡖⣭⣶⠶⡾⣟⣛⠻⣷⣱⠀⠀⠀⠘⠋⢀⡤⠤⠄⡀⠀⠼⣿⡇⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⢟⠀⢀⠿⣄⣠⠇⣴⣿⢦⡀
⠀⠀⠀⠀⢅⣿⣕⠁⠀⠀⠀⠱⠛⠋⠉⡆⠀⠀⠀⡎⠠⠠⠀⢹⠀⠠⣄⠀⠀⠀⣰⠆⠀⢀⢤⣠⠲⢄⠀⠀⠀⠤⡄⡀⠀⢿⣼⣿⣷
⠀⠀⠀⣔⠁⠀⠀⢃⠀⠀⠀⢨⣀⣜⡠⠊⠀⠈⠀⢱⣤⢀⣠⠞⠀⠀⠀⢳⡄⢰⡁⡀⠀⢀⠾⠁⠀⠀⠑⣄⠐⢠⠁⠁⠀⠈⠛⠋⠉
⠀⠠⠀⠈⠢⠤⠒⠁⠀⠀⠀⠀⠀⢀⠖⠑⢆⠀⠀⠈⣆⣀⡼⠀⠀⢸⣽⣿⣿⣿⣷⡄⠘⡈⢉⠩⠛⡅⠉⠈⠀⠀⠀⢰⠁⠒⣲⡤⠄
⠀⠂⡀⠀⠀⠀⡠⠲⣤⠒⠒⢤⡰⢃⡀⢀⠈⣂⠀⠀⠘⠖⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣁⠇⢸⣸⣀⣹⡸⠀⠀⢀⡠⣾⢯⡀
⠀⠂⠐⠀⠀⡜⠀⠀⠈⢦⣀⡜⠃⠁⠓⠉⠋⠈⢆⠀⠀⡠⢤⡔⠆⠘⠌⠛⠛⠻⠿⠭⠵⠀⠀⠀⠀⠠⠀⠀⣠⣮⣤⢹⢸⣹⣄⠀⡀
⠀⠀⠀⠃⣜⢀⣄⢀⣤⣀⢻⡀⠀⠀⡤⠄⠀⠀⠘⡄⠀⢱⠔⢳⠣⡀⠀⢀⣤⣖⣲⢦⣄⠀⠀⡰⠬⠂⠀⣜⣉⣁⠉⣾⡜⠉⠈⢣⡀
⠀⠀⠀⢰⣿⣿⠛⠛⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢸⠀⠘⠒⠚⠚⠀⠀⣾⠃⢨⠈⠄⠙⡆⠀⠈⡅⠀⠀⡄⠠⠤⠬⠽⠭⠭⠭⠭⠵
⠀⠀⠀⠿⠿⠿⠷⠾⠛⠛⠛⠿⠿⠆⠀⠀⠒⠒⠒⠊⠀⠀⡰⢶⠀⠀⠀⠹⡄⢰⠀⠐⣠⠃⠀⠀⢲⠀⠀⠈⢦⣌⣀⣀⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠁⠈⠁⠈⠀⠁⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⠀⠉⠀⠁⠈⠀⠀⠈⠀⠈
⠀⢐⡀⠂⣠⡀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠘⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠐⢀⣄⠀⠀⢐⠀⠀⠀⠐⠀⠀⠀⠣⠀⠀⠠⠀⠀⠀⠀⠀⠀⢐
⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢁⠀⠀⠐⠦⠈⢀⡀⠀⠀⢰⠆⠀⠀⡄⠠⠤⠀⠆⠀⠀⠄⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠲⠄⠁⣐⠀⠀⠀⡶⠀⠀⢠⠀⠤⠄⠠⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⡀⠀⠀⠐
⠀⠰⠬⠀⠐⠀⠤⠄⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠢⠅⠀⠂⠠⠤⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⡀⠈⠀⠠⠀⢠⠀⠀⠀⣄⠠⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠁⠀⠄⠀⡄⠀⠀⢀⡀⠄⢠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠙
⠀⢠⠀⠂⣀⠀⢀⠀⠄⠀⠄⠀⠂⠀⠀⠐⠀⠂⠀⡄⢀⡀⠠⡀⠀⠀⡰⠀⠀⠀⡄⠐⢀⡀⠀⠀⢠⠀⠠⠀⠐⠀⠀⠀⠂⠐⠀⢀⠀⣀⡀⢄⠀⠀⢀⠆
⠀⠈⠁⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠘
⠀⢂⠀⠀⡰⡀⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠄⠀⠆⠀⠀⠐⡀⠈⢀⣆⠀⡂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠢⠀⠀⠠⠂⣀⣀⢰⠀⢀⠀⡀⠀⠀⠀⠆⠀⠐⠂⠀⠀⠐⠀⠀⠀⠁⠀⠀⠐⠄⠀⠀⠔⢀⣀⡀⡄⠀⡀⢀⠀⠀⠀⠰⠀⠀⠒⠀⠀⠀⠂⠀⠀⠈
⣶⣶⣶⢶⣶⣶⣶⣶⣾⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠈⠉⠉⠉⠒⢢⡤⠤⠒⠒⠒⠒⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡴⠋⠀⠀⠀⢀⣀⡀⢠⠞⠁⠀⠆⠀⠀⠀⠀⠠⠤⠆⠙⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠤⡎⡡⡒⢤⠏⠀⠀⢀⠴⠊⠁⢀⠠⡎⠆⠀⠀⠀⠈⠡⣤⡌⠁⠀⠐⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⢈⡧⠼⠫⠼⣀⠀⡴⠁⢀⠴⡉⠄⣀⡇⠃⠀⠈⠂⠀⠀⠈⠀⠒⠀⠁⢀⣈⡀
⠀⠀⠀⠀⠰⠁⠌⡰⠁⢀⠔⠒⢆⠈⢻⢀⣠⠬⠴⠶⠮⣈⡏⡅⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⢀⠉⣀⣩⣑⣦
⠀⠀⠀⠀⣇⣸⣠⠇⣀⣈⢆⡘⠊⠀⢀⡏⠀⠀⢀⠀⠀⠈⠹⡄⠀⢀⡀⠤⡄⢀⠞⠀⠀⢄⣠⠟⠁⠀⠀⠈⠁
⠀⠀⠀⡔⠁⣠⠤⢄⡀⠀⠉⢫⡭⠭⠵⠧⣀⠀⢠⡄⠀⡀⠀⢸⣦⣳⠄⡰⢁⡜⠀⠤⠀⠀⡧⠖⠉⠉⠗⢆
⠀⠀⢰⠁⢸⠁⢰⠢⡱⠀⡔⠁⠀⠀⠀⠀⠀⠱⡍⠁⠀⠀⠀⣸⡿⠻⢤⣉⡽⠀⢀⠀⢀⠞⠁⠄⡈⡢⠔⠚⠂
⠀⠀⠘⡄⠘⢄⣈⢉⣁⣼⠀⠀⠀⠀⢀⡀⣀⡀⠸⡄⠀⠀⢠⣿⣷⣾⡿⠋⠠⡀⢀⡜⠉⠀⠂⠄⣸⠁
⠀⠀⠀⠘⣦⡤⠤⠥⠤⠼⢄⡀⠀⠰⡁⠐⢢⠈⡆⣇⠤⠒⠛⠉⠉⠉⡵⠒⠲⣄⣯⠤⠤⡌⠐⠀⡇
⠀⠀⡔⠉⠀⣀⠀⠀⠀⢀⡀⠙⣄⠀⠉⣈⣡⡴⠋⠀⢀⡀⠄⠴⢴⡴⠣⠼⠀⠘⠃⠀⠀⢸⣄⣔⠁
⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⣿⣿⠋⠀⡠⢎⠁⠐⣈⡈⢼⠇⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⢱
⠀⢰⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⣿⣿⠃⠀⡰⠁⣢⠕⣉⠤⠒⢺⣄⠀⠘⠁⠀⣤⠀⠉⠀⠀⠀⠪⡀
⠀⢸⠀⠠⠈⡄⠀⠀⠀⠀⠠⠂⠀⡟⡛⠒⡒⠛⠛⠓⠚⣇⡠⠤⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⢇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢰⡃⠄⢊⠄⠀⡠⢂⠔⡇⠀⠀⠀⠉⠉⠉⠀⢲⠒⢒⡈⠉⠉⣥⠀⠈⡆
⠀⠈⠢⢤⡀⢀⣀⣀⣀⠀⢠⢤⠾⠤⠶⠅⠐⠂⠒⠒⢲⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠘⠀⠀⢸⠀⢸⠀⢇⠐⠂⠒⠀⠒⠂⠀⢸⠀⠛⠂⠃⠀⠈⠛⠈⢸⠀⠀⠠⣀⠉⠀⢀⢠⣇
⠀⠀⠀⢀⠇⠀⡇⠀⠘⠀⢸⠀⠘⡆⠒⠂⠀⠂⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠑⠤⠬⠂⢂⣈⣙⠶⠃
⠀⠀⠰⣉⣀⡰⠀⠀⠸⣄⣀⡩⠂⠘⠤⠤⠤⠤⠤⠘⠀⠈⠉⠉⠉⠁⠐⠈⠁⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆⠒⠀⠀⠀⠀⠒⠀⠀⠀⠀⠔⠂⠀⠂⠤⠀⠀⠀⠀⠒⠒⠀⠴⠂⠀⠀⠀⡀⠀⠀⣄⣾⠁⠀⠀⠀⠱⠤⠤⣀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⣠⠀⣠⠀⠀⢠⣤⢴⣿⠘⣿⠀⠀⣰⣶⠄⠀⢀⡀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠃⡤⠀⠀⠃⠀⠀⠀⢠⣤⣴⣿⣤⡀⠈⠛⠘⠿⠃⣤⣢⢰⣿⠿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠂⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠸⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠷⠀⠿⠖⠀⠄⠈⠉⠀⠀⢀⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠠⢆⣀⡀⠀⠉⠈⠁
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢸⣥⠀⠀⠀⠀⠙⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠁⠀⣶⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠦⠀⠀⠀⠿⠇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⣴⣶⡄⠀⠁⠀⡀⠈⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⠖⠈⠀⠀⠀⠺⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡄⠠⠆⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⢀⣠⡀⢀⣀⠄
⠀⠀⠀⢡⡄⢤⠀⠀⠀⠀⠀⠈⠿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⠀⠀⠀⠀⠀⠀⠿⣿⡧⠀⠈⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠒⠤⠈⡔⠀⠓⠀⠠⠒⠂⡀
⠀⠀⢀⠐⠀⠀⠘⠻⠂⠀⠀⣠⡄⣠⡄⠀⠀⡄⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡆⠀⠀⢰⠀⠀⠀⠰⡆⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠘⠀⢀⡠⠤⠴⡆⠀⠀⠟⠀⠀⠛⠀⠀⠳⠠⢾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠇⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣤⣄⠈⣆⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠲⠤⠀⢹⣷⠄⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣿⣿⠇⠀⠀⠀⢸⣿⡷⠀⠀⠀⠸⣿⣾⠂⢸⣄⠀⠀⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⢰⣶⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⡗⠀⠀⠀⠈⠉⠁⠀⠀⣤⠀⠉⠀⠀⡿⠛⠂⠘⠻⠆⢤⡄⠀⠈
⠀⠀⠐⠦⠀⢠⣤⣷⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠐⠐⠿⠶⠿⠀⠀⢠⠄⠀⠒⠀⠀⠀⠀⠀⠰⠁
⠀⠀⢀⠀⠀⠀⠈⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣄⣀⠉⠛⠀⠀⢴⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠀⠱⡀
⠀⠀⠀⠘⠃⠙⢃⡐⠃⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣙⠛⠻⠂⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⣾⠂⠀⠀⠀⠘⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⡇
⠀⠀⢠⠘⠀⠀⠘⠏⠀⠠⠀⠀⠀⠀⣰⡆⢈⣁⠀⠙⠻⠿⠋⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠒⢨⡄⠀⢸⣧⠀⣇⠀⠀⠸⡴⠋⠀⢀⣤⣖⣱⣦⠀⠀⢠⠃
⠀⠀⠰⢄⡀⠀⠐⢀⣰⠄⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠐⠀⠀⠒⠈⠀⠀⠀⠀⢀⡈⠁⠐⠻⣷⡤⠄⠛⠀⠈⠓⠂⠘⠃⠀⠀⣮⣿⠋⢩⠇⠀⢠⠎
⠀⠀⠀⠀⠛⠁⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠂⠀⠛⠲⢶⣌⠁⠀⠀⠀⠁⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠈⠁⠘⠀⠀⠰⢧⡀
⠀⠀⢀⢀⣦⣀⣀⣠⠄⠀⠈⠀⠈⠀⠀⠀⠀⠈⠐⠦⣤⣤⠞⠁⠀⣀⠀⠀⠀⠀⠙⣿⡶⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣇⠀⠀⠀⢀⡀⠀⠁⠀⠠
⠀⠀⠈⢁⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠛⠁⠀⠀⠀⠀⠀⢹⠃⠀⣾⢿⣷⡄⠀⠀⠀⠙⠿⠟⠃⠀⠀⢀⡿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⣀⡀⠀⠀⠀⢀⣄⡀⠀⠀⠏⠀⠀⠀⠤⣄⠀⠀⢰⡇⠀⣴⠀⠀⠻⣿⣿⡇⠀⠀⠤⠒⠁⠀⠀⠀⠀⠸⠃
⠀⠀⢀⠀⡀⢀⣤⠀⠀⢸⣿⣿⠆⠀⠀⣿⣿⡷⠀⡘⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠰⠿⠇⠀⠀⠀⢀⣄⣀⠀⠀⠀⠠⠀⠀⠀⢀⡄⠀⠀⠄⢀
⠀⠀⠐⠈⡁⠀⣤⡆⠀⠀⠉⠁⠀⠀⠀⠀⠉⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⠤⠀⠠⠀⣀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠣
⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⢴⡖⠉⠳⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣀
⠀⠀⠠⠒⠸⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠐⣶⣾⠏⣀⡐⢄⣀⡀⠀⣤⡄⠳⠄⠐⡴⢿⠁⠹⠦⣀⠀⠄⠛⠢⣤⠤⠆
⠀⠀⡇⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠻⠃⠘⠁⠁⠀⠛⠃⠀⠀⡀⠀⠈⠀⠀⢠⠄⠀⠀⠀⢠⡄
⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣀⠀⠀⠂⠀⣀⣤⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⣈⢁⣶⡀⠀⠀⠀⠀⣠⣀⣀⣀⣠⣤⣄⠀⠀⠀⢀⠀⠀⠠⣿⣷⣤⣤⣤⣶⣾⡿⠀⠄⠀⣠⡿⠿⠆⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⢀⡜⠓⠂⢂
⠀⠀⠊⠁⠀⠈⠁⣄⠀⠀⠈⠋⠀⠀⠀⠙⠛⠁⠀⠀⠀⠾⠃⠀⠀⠘⠋⠀⠀⠀⠈⠉⠀⣠⣤⣶⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⠂⠀⠉⠃⣤⣠⡜
⠀⠀⠦⢄⣀⣀⠀⠋⡉⠀⠀⠀⠀⠀⠤⠀⠀⠒⠐⠛⢇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠤⠀⠈⣉⠁⠀⠀⠀⢠⣤⣤⣤⣴⠂⠀⠀⠙⣷⣤⣤⣤⣶⠁⠀⢈⡀
⠀⠀⠀⠀⠀⠉⢄⠀⠀⠐⠀⠰⠀⠁⠀⣀⢀⣀⠀⢀⣼⠿⠗⠂⠀⠀⢦⣤⡤⠤⠤⠶⠶⣿⡟⠛⠀⠰⢶⠊⠀⠀⠀⠀⠱⠆⠀⠀⣄⠀⠀⠈⢉⠀⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⣨⡄⠀⠀⠀⠜⠣⠄⢀⣁⠀⠘⠣⠄⡀⠀⠀⠀⠀⠈⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⣀⠀⡀⡀⢀⠀⠀⢀⡀⠀⠀⡀⠀⣀⠀⣀⣀⠀⠀⠀⠀⢀⠀⡀⣀⠀⣀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⢀⣃⠐⣏⠉⠉⣉⡭⠟⠀⡴⠲⣍⢿⢳⡈⣟⢳⡹⣤⡽⠥⠟⣴⢻⢸⡷⠇⠠⢤⣀⠉⣼⣿⣟⡼⠉⢱⡆⢾⠀⣿⣈⣟⣉⡇⣿⠘⠷⠆
⠀⠀⠀⠀⠀⢀⣼⣇⡘⣦⠀⣇⡴⢚⢶⡻⣿⣼⡈⣧⢧⢸⡌⣧⢠⡖⢶⠀⡟⣾⢸⡇⠺⠀⣤⡄⣼⡿⡿⢹⣳⡟⢳⡙⢮⠳⡌⣩⠤⣌⣠⢏⡾⣻
⠀⠀⠀⠀⠀⢘⣩⡴⢋⣬⢉⣥⡙⠮⢾⣁⣾⣩⡿⢸⡌⣧⡇⠘⠶⠃⠛⣛⣱⠇⠈⣩⡟⠦⠭⠾⠯⠼⣃⣬⠵⢖⣲⠘⠺⠃⠈⠱⣞⡳⣍⢾⠀⠛⠂
⠀⠀⠀⠀⠀⢨⡼⣽⠞⣛⣉⣛⠡⣞⠵⠙⣮⣱⢦⢸⢧⡏⢛⣒⠻⣆⣿⠉⣖⣷⣼⢹⢃⣴⡗⣠⡄⠺⢷⡒⠋⢉⣯⡀⣰⠚⣶⠒⣦⢳⢸⣿⡆
⠀⠀⠀⠀⠀⠐⢃⣴⡿⣻⣛⢮⡳⡝⢦⢴⡥⢬⡁⣿⠘⣄⢯⣩⡇⣠⠶⢦⠙⣿⢣⣿⢋⣾⡿⡿⣡⣾⠾⣷⡛⢉⣏⠀⣧⣠⡟⠲⡟⣼⢸⣋⣳⣶⡆
⠀⠀⠀⠀⠀⢰⣿⣿⡫⠗⠈⠳⡹⡝⢦⣿⡶⡄⠹⡞⡇⣸⣰⠖⠲⠇⠀⢘⡧⣇⠈⠁⡿⢋⣾⢟⣿⣣⣶⠊⣿⢻⠏⣴⠶⣜⠧⠴⠳⣏⢻⡿⣄
⠀⠀⠀⠀⠀⢠⡻⣿⣽⣄⢀⡴⣣⠏⢾⣿⡿⡇⢰⣷⡇⡏⣿⠀⠀⢀⣴⣿⣥⣸⠀⠀⠀⠈⠁⠾⢋⡿⣱⢿⣾⢸⡆⡿⣄⣼⠂⣠⡦⢸⠇⡿⣈⢳⡄
⠀⠀⠀⠀⠀⠈⢳⡝⣎⢯⡉⠀⣭⡝⣦⢾⣼⠁⠘⣎⢣⡙⣎⠳⢞⡿⣫⡞⣹⠋⢀⡴⠦⢤⣀⣰⢏⡼⠁⢼⣿⣿⡇⣧⡜⣡⢾⡟⡵⢋⣾⡽⠉⣏⠇
⠀⠀⠀⠀⠀⢰⣶⣿⢸⢸⣽⠻⡜⣿⡜⡎⣯⢧⠀⠘⡧⠑⢾⡴⡎⠛⣿⠲⡜⣦⡞⡴⠋⠓⠦⣄⡘⢧⣀⢺⣿⣇⡇⢩⣞⡵⠋⢾⠁⣏⠺⡶⢦⣽
⠀⠀⠀⠀⠀⢐⡯⣶⠸⣬⣿⢶⡹⣿⣿⣿⠸⡞⢷⣊⣥⡴⢮⡛⡗⣶⠹⣶⣃⠀⠸⣵⣦⣆⣤⣠⡙⠲⡌⠳⠟⢛⡃⢹⠷⡄⣶⣾⣇⢹⡼⡏⠙⠳⠆
⠀⠀⠀⠀⠀⠠⠽⠯⢤⣄⡻⠏⠳⣙⢃⣭⠼⢿⠋⠁⠀⠀⢰⣇⡟⣈⣭⣝⡛⣡⠖⣦⢤⡈⢉⣭⣉⢉⡥⠖⣶⢸⢿⢸⡄⡇⣿⢸⠻⡄⢷⠹⠖⢚⡁
⠀⠀⠀⠀⠀⢠⠞⠛⠳⣆⣹⣆⡤⣌⢩⢻⣿⡎⡇⠀⠀⠀⢸⡟⣇⡇⣿⠉⠉⢻⣄⢀⣼⡃⢸⠇⢸⣯⡝⣋⣅⠘⣛⣸⡇⠷⠟⣸⢇⣙⣋⢴⣿⠘⠇
⠀⠀⠀⠀⠀⠠⠊⠙⢦⣈⣉⢿⣓⡿⣸⢸⡇⡇⠳⠤⠶⢒⣚⠁⣨⢦⣙⢦⡀⣀⣉⠉⢀⡿⣼⠀⣠⣾⡁⣿⢸⡇⣏⠰⠆⠻⡶⣱⠋⠀⠙⣟⠛⢹⡇
⠀⠀⠀⠀⠀⢠⣀⣀⣸⣿⠉⣇⡽⠚⣁⣯⣾⣁⢿⡙⢦⡟⢹⠇⡟⢶⣉⡦⢉⣥⠭⣿⣋⠀⢛⡋⣽⠁⣿⢸⡄⣇⢹⡀⠂⠈⢧⢳⣠⡖⠀⠹⠄⠈⡇
⠀⠀⠀⠀⠀⠐⠯⠭⠞⠁⣸⡿⠀⢸⣷⣯⠖⣮⣷⣿⠀⠀⣿⣬⣛⠉⢁⡴⠋⠀⠀⠀⠿⠞⣻⢟⡼⠂⢸⣼⡇⢿⠸⡇⢸⠆⣸⠇⡿⠀⠀⢀⣠⢴⡃
⠀⠀⠀⠀⠀⠈⢉⣛⡒⠒⣋⠳⢤⠟⠟⠻⣽⡽⠃⣯⣤⡼⢻⣗⣏⠳⣄⢳⡄⠀⣠⣤⣼⠳⡼⡎⢷⡀⣸⠃⢧⣸⠀⣇⣠⠞⣡⣌⣧⠀⠘⢋⡏⢸⡃
⠀⠀⠀⠀⠀⢸⡄⢸⢿⡸⣏⣿⣠⠤⣤⢠⡖⠒⠒⠦⣤⠴⡲⣍⢉⡭⠵⢟⣤⡸⣇⠈⢯⢧⠹⡽⣆⠉⣿⢹⡎⢉⣀⡉⢱⠋⣰⡟⣿⡤⢖⣾⠥⣌⠁
⠀⠀⠀⠀⠀⠸⠃⣾⠘⣇⠉⡞⠁⣤⠞⣠⣍⢳⡀⠀⡀⠿⠟⣸⠘⠷⢶⣞⣁⠹⣟⣆⠘⣏⢧⠹⡜⣦⠹⡆⢷⣜⣧⣽⡎⢳⣻⠃⠀⠀⡏⣴⢳⠘⡇
⠀⠀⠀⠀⠀⠘⢦⠿⠤⠟⠀⣇⠈⣹⣃⣿⢸⡼⡇⠘⣏⢹⠿⣥⠰⣯⡭⣤⢈⣳⡘⣾⣦⠘⣮⣧⠙⠟⢀⣟⡴⣿⠿⡿⢡⠖⠛⠛⢙⡗⣧⡿⠸⠶⠃
⠀⠀⠀⠀⠀⠘⣮⠓⠚⣻⢀⣼⡍⣡⢿⣠⠼⠃⣹⠿⢧⣼⣆⡷⣠⢶⢦⣿⡿⣿⡇⠘⢿⡇⣨⣯⢾⣟⡞⡼⣡⣿⡏⣡⣭⡿⣆⡾⣹⠷⡆⢴⠛⢹⠆
⠀⠀⠀⠀⠀⠈⢯⢙⣿⡏⢸⣿⢁⣹⣉⣟⣦⡾⢁⡀⠀⢷⠀⠠⣯⡿⣾⡿⠀⣿⡷⠀⣠⠝⢋⣄⣿⢼⠱⣇⢧⠸⡇⣿⠀⠻⠴⠖⠋⠂⢿⢸⡇⡟⡀
⠀⠀⠀⠀⠀⠘⠾⠸⣷⡇⣿⣿⠻⡼⢇⡍⣼⣁⣸⣁⡇⢈⠳⣄⠀⢸⡿⢁⡀⢉⣵⣿⣴⣷⢸⣿⣿⠈⣷⡼⣼⢣⡏⢸⣨⣴⡾⡾⠗⢇⢸⡌⠿⣳⠃
⠀⠀⠀⠀⠀⢰⣼⣷⢛⣤⣿⡃⢺⣙⣋⣻⢛⣩⣭⣽⣟⣛⣛⣣⣤⠀⣴⣿⣿⣬⡁⢸⣿⢿⣿⢻⣿⠸⣇⢷⡹⡞⣇⢸⡇⠵⠶⠒⠚⢉⠀⣧⠀⣻⠃
⠀⠀⠀⠀⠀⠈⢸⡟⣇⢳⠙⣿⣌⢋⡬⣍⡼⢡⡴⠤⣬⣭⣭⠥⠞⠉⠻⢥⡀⣞⠁⠘⠋⠸⡿⠸⡿⠀⠿⠾⠃⣧⣽⡌⡇⠀⡴⢳⡀⡟⣧⢹⡼⡟⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠉⠈⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠁⠈⠀⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⣤⣶⣶⣦⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⢹⡩⠆⠀⠐⣿⣧⣯⣽⡛⣿⣿⡆⠀⠐⠀⠀⠀⡀⠀⣰⣯⣟⢿⣷⠆⠀⢀⣀⡀⠘⠚⠀⠰⣄⢳⡸⣆⣇⣾⣸⣰⢇⠖
⠀⠀⠀⢰⠛⠲⣄⠀⢀⣀⡤⢤⠈⠹⡿⠛⠛⠛⠁⠀⠀⠀⠈⠀⠀⣰⣳⣯⡻⠟⠁⢀⡞⠻⣬⡙⢶⠀⠐⠢⣜⣶⣿⣿⣿⣿⣿⣿⣿⡴⠒
⠀⠀⠀⢸⡄⠀⠈⠓⠋⠀⠀⣸⠁⠰⠃⠀⠀⠀⠀⠓⢦⠖⠻⠛⠛⠻⢿⣎⠁⠀⠀⢾⣀⢄⡞⢀⡏⠀⠘⣒⣺⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⡀
⠀⢀⣠⠞⠃⡀⠀⠀⠈⠀⠰⣇⠀⠀⣼⠡⠭⠭⣧⠀⢾⠀⢠⣶⡄⠀⠀⠨⡟⣦⣄⡀⠈⠙⠉⠉⠀⢘⣩⠝⣻⣤⣿⣿⣿⣿⣿⣏⣹⡷⠤⠄
⢠⡏⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠑⢄⠈⢧⣀⡴⠁⠀⠘⣦⠀⠉⠰⣿⡇⢀⣾⢿⣲⠉⠀⣄⢠⡄⠀⠀⠴⢋⡽⡿⣿⣿⣿⣿⣿⣿⣿⡙⠲
⠀⠙⠒⠶⠶⡄⠀⠀⠀⣠⠤⠴⠶⠚⠀⠀⠉⠀⠀⣠⣞⡿⣷⣤⡀⠀⣰⣟⣮⠓⠀⠀⣴⡄⣤⠀⣀⠀⠀⠋⢰⠃⡎⡏⢹⢹⠙⡍⢦⠙
⠀⠀⠀⠀⠀⢹⡀⠀⢠⠇⠀⠀⠀⠀⠘⠂⠀⢀⣶⣯⣿⣿⡞⠀⠉⠛⠻⣟⠏⠁⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠸⠁⠁⢾⠘⠀⠓⠈
⠀⠀⡆⠠⠀⠀⠳⠴⠋⠀⠀⢀⡴⣚⠦⣀⠰⣿⣾⣽⢿⡞⠀⢀⢀⠀⠀⠀⠀⢼⣉⡇⠀⠀⠀⠀⠀⠀⠶⠂⠀⣠⠗⠤⠀⠀⠀⠾⠂⣠⠖⠠
⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⣰⢋⣸⣿⣷⠈⡧⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣌⡀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⠀⠀⠀⢴⣯⡄⢀⡵
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⣘⠿⠟⠋⢰⠃⠀⠀⢀⣤⣤⣤⣤⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠂⠀⢸⡇⠘
⠀⠀⢀⠴⠚⡉⢉⠩⠹⠲⣄⠈⠓⠖⠋⠛⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣤⡄⠀⠀⡇⠀⢠⣾⣶⢸⡇
⠀⢰⠃⢲⣆⠳⠼⠦⠸⠄⢸⡆⢠⣶⡆⢀⣶⣶⣿⡿⠟⢿⣿⠟⢿⣿⢟⢻⡟⣙⢿⢻⣿⡟⣿⣿⣿⣿⠃⠀⠈⠁⢠⠖⠛⢦⡀⠉⠁⢸⡇
⠀⠘⣆⠀⠀⢀⢠⡇⠆⠀⣸⠃⠀⠉⠁⢸⣿⣿⣿⣷⢹⡧⣻⢸⡇⣿⢸⢎⡇⣿⢸⡏⠿⣇⠽⣿⣿⣿⣷⡄⠀⢠⠯⠄⠀⠠⠹⡄⠀⠈⠛⠶⠂
⠀⠀⠈⠛⢲⡤⠅⣀⣀⢘⣳⢄⠀⠀⠀⠀⢹⣿⣿⣿⣬⣶⣿⣷⣾⣿⣷⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠛⡃⢀⡟⢀⣶⣿⣇⠀⢹⡀⠀⠀⠀⠶
⠀⠀⠀⡴⠹⡜⡿⣅⡘⣾⢿⢈⡇⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠁⢸⠇⠰⣿⣿⣿⡄⠈⡇⠀⢰⡂⠢
⠀⠀⠀⢳⣀⠏⠛⠓⠂⢛⣠⠎⠀⠀⠀⠀⠀⠙⠛⢉⣀⠸⣿⣿⣿⣿⣿⡿⡙⠿⠿⢿⠿⠟⣀⠌⠥⢤⠀⠀⢸⠀⠈⠼⣿⡿⠃⠀⡗⠀⣴⠃⢠
⠀⠀⠀⠀⠈⠙⢻⢂⠞⠉⠀⠀⠈⠀⢠⣶⣿⣿⣶⡈⠁⠀⠈⠙⠛⠛⠉⠀⡿⠁⡶⠀⠀⠨⢄⠀⢠⠈⠆⠀⠸⡇⠀⠀⠀⠀⠀⠀⡧⠀⢨⡗⢈
⠀⠀⢀⣀⠀⠀⠾⠋⠀⢀⡞⡳⠐⠀⠘⣻⣿⣿⣟⠇⠀⠀⢀⣀⡤⠤⠤⠴⠒⠒⡶⠲⠤⠤⢤⠬⣥⣀⠀⢀⡴⢿⠓⠒⠒⠒⠚⢛⡟⢦⠀⠙⣤
⠀⠀⠻⣌⠳⠴⠒⠛⠙⠉⠈⢷⡀⠀⢀⣉⡛⠛⠉⠀⡴⢊⡍⡖⠒⠊⠉⠉⠹⡄⣷⠀⠀⠀⠀⠀⢹⢸⡆⢸⡀⡼⠓⠒⠒⠒⠖⠚⣆⢠⡇⠀⠈
⠀⠀⢀⡞⠀⠀⠀⣄⡄⠀⠰⠾⣷⡀⢻⣿⡿⠀⢠⢞⠞⠁⣿⡇⠀⠀⠀⠀⠀⡇⢻⠀⠀⣸⡀⠀⣸⠀⡇⠀⠟⠁⠀⠉⠠⠉⠂⠀⠈⠛
⠀⠀⠸⡇⠀⠀⠀⠈⣡⣶⣿⣿⣿⡇⠀⠀⠀⣰⢣⠏⠀⠀⢸⣇⣀⣤⠤⠤⠴⠗⣼⠆⠈⠛⢋⣀⡘⣷⣷⣄⠀⠀⣀⡤⣴⣶⣶⠦⢤⡀
⠀⠀⠀⢷⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⢀⣴⠁⠛⠉⠉⠉⠉⠀⠀⠀⠒⠂⢻⡇⠻⠾⠛⠋⠉⠉⠉⠉⢼⣿⠀⢸⣛⠻⠿⠿⠿⠟⠛⡯⣝⡲⣄
⠀⠀⠀⢸⣄⠰⠶⢲⣤⣤⣿⣁⠀⠀⠀⠸⣼⡆⠀⠀⢠⢞⣭⣓⢦⠀⠀⠀⢸⡇⠀⢠⣾⣭⣝⢦⠀⠀⢸⠁⠀⠘⡇⢰⣶⣴⣿⣦⠀⡇⠀⡷⢸
⠀⠀⠀⢸⠛⠲⣞⠋⠙⣳⣶⣚⣠⢆⠀⠀⣸⣧⣶⣤⣼⢾⡀⢹⡎⣇⣠⣤⣼⣧⣀⣼⢿⣄⡿⣸⢿⣿⢿⠇⠀⠀⠱⣌⠻⠿⠿⠡⣰⣟⣩⠤⠋
⠀⠀⠀⠸⢦⣀⣈⣉⣉⣁⣽⣤⣤⠼⠀⠀⠛⠛⠛⠛⠘⢮⣙⢋⣠⠇⠀⠀⠀⠀⠀⠈⠳⠤⠴⠃⠀⠀⠀⠀⢠⡞⠉⠭⣽⡶⠶⠾⠭⠏⢓⣦
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⣀⠀⠀⢄⠀⠀⠀⠄⡀⠀⠀⣦⠀⢀⡀⠀⠀⠠⠂⣄⠀⠀⠂⡀⠀⠀⠀⠀⣄⡀⠀⠠⠀⡄⠀⡀⠀⡀
⠘⠟⠃⠰⠂⠀⠀⠤⠀⠈⠂⡀⠁⠀⠩⣠⠀⠐⠂⢀⣹⠀⡇⠀⠀⠀⢤⠀⠈⠉⠀⠔⠁⠈⠀⠈⠃⠪⠙
⠀⢀⡠⡀⠀⠀⠀⣀⠠⠤⠤⢁⠀⢀⠀⠈⠀⠀⠀⠹⠽⠠⠃⢐⠀⠂⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠶⠀⠹⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋⠉⠃⢠⣶⡀⠀⠀⠘⠀⠐⠀⠠⠒⠀⠀⠄⢀⠀⢉⠑⠒⠂⠐
⠀⢅⠙⠀⠀⠋⠉⠐⠊⢱⠒⠂⠂⣉⡃⠐⠔⠀⠀⠉⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⢁⠐⠂⠀⠆⡁⡁⠠
⠀⠠⡉⠀⠀⠹⠁⠁⠀⢸⠀⣄⠀⠉⡀⠀⠀⠱⠒⠀⠀⠄⠀⠐⠀⠉⠅⠒⠤⠀⢀⢈⠀⠈⠁⠛⠁⠀⠄
⠀⢀⠀⠀⡀⠠⠀⠀⠀⠸⠀⠀⠀⠐⡇⠀⠀⠀⢠⡄⠀⢠⠀⠀⡾⡄⠀⠀⠀⢄⠈⠉⠀⠀⠀⠀⠌⡐
⠀⢈⠀⠸⠀⠀⠀⠐⠘⠋⠠⠂⠀⠀⢃⠀⠀⠀⠀⠀⣉⠀⠀⠺⠀⠂⠀⠚⠇⠈⣢⡄⠀⠀⠀⠀⠀⢰
⠀⠰⠀⠀⠀⠀⢴⠄⠀⠆⠀⠈⠀⠀⠀⠡⠐⡀⠀⠀⠀⢀⠀⢠⣀⣀⠈⠉⠛⠀⢀⣤⡀⠋⢒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠀⠀⠀⠀⠠⠀⠠⠀⢠⠉⠉⠃⠀⡆⠀⣌⢿⣾⣒⣝⣿⣷⠀⣀
⠀⢀⠀⠀⡁⡀⠀⡆⠀⠀⠀⠀⠨⠀⠀⠀⠀⡁⠀⢀⠂⠀⣄⠐⠀⠀⠀⠀⠁⠈⠉⠁⠸⢹⠉⠈⠈⠃⠉
⠀⠖⡂⠀⢲⠀⠤⣶⠀⠀⡀⡀⠀⠀⢀⢀⠀⠐⡅⠀⠀⠀⠀⠀⠄⡈⠀⢀⠀⠴⣶⠄⠀⠘⠀⠀⠂⠀⠐
⠀⠈⠜⠀⠀⠀⠀⠁⠠⠀⠄⠠⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⡠⠀⠙⠣⠀⠀⠀⠀⢸⣀⢠⡀⠀⠰⠆⠂⡀
⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠘⢂⠈⠑⠊⠀⣀⠐⠂⠀⠀⠀⠒⠀⠝⠀⠄⠀⠀⢀⠃⠅
⠀⡆⠀⠀⠆⠀⢠⣶⣦⡀⠢⠌⠀⠀⠒⠶⠀⠁⠈⠏⠀⠀⠀⠑⠁⠀⠂⠀⡀⠀⢅⠉⠆⠀⠀⠀⠈⢉⡀
⠀⠀⠁⠡⠀⠀⠈⣩⠉⡁⠀⠀⠠⠀⠀⢠⣀⠈⠈⠀⡀⠀⠁⠀⠀⠀⠀⢀⠀⢀⠘⡁⢊⠀⢠⠀⠀⠈⢅
⠀⠀⠂⠀⠀⠸⠭⢡⠤⡇⠀⢐⡂⠀⠀⠀⠛⠁⠐⠂⠀⠄⠀⠌⠈⠇⠉⠀⠀⠈⠀⠀⢖⠀⠈⠀⠀⠀⠁
⠀⠐⠄⣆⠀⠰⠆⠀⠀⠠⠀⠘⠋⠁⢒⡽⠯⢕⣆⠈⠁⠀⡌⠱⠀⠀⠀⢐⣏⠓⠀⢁⠁
⠀⠀⠣⢵⠀⠘⠀⠀⠀⠆⠀⠄⠀⠐⢿⣄⣀⣼⡧⠀⠐⠂⠀⠀⠀⠂⠀⢀⣗⠨⠀⠀⢀⠈⠉⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠁⠂⠐⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠔⠀⠐⠄⠀⣠⣤⣄⠀⠀⠀⣰⣾⣿⣶⡄⣀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣶⠄⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣾⣿⣿⣿⣿⡟⣷⠀⠘⡬⠁⠄⠸⠀⠙⠿⠋⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⠀⡰⠊⠉⠒⣄⠀⠀⡴⠋⠀⠀⠀⠉⠳⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⠌⢀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡇⠐⠀⠀⣸⠀⡘⢀⠔⠚⠓⢦⡀⢀⢹
⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⡌⠀⠨⡔⠈⠈⠐⠄⠀⠛⠾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠘⢆⠀⠀⣇⠠⠁⣏⠀⠸⣇⡸⠃⠈⣸
⠀⠀⠀⠀⠙⠻⢿⠟⠋⢀⠀⠀⡇⠀⠀⢄⠀⠀⠀⠈⠄⠁⠀⡀⠉⠉⠀⡙⠓⠛⢈⡉⠁⠀⠻⠃⠀⠱⡀⠈⠢⠄⣘⣦⣀⣀⣀⣠⣞⠁
⠀⠀⣼⣿⡆⠀⠊⠀⠀⠻⠃⠀⠈⠢⣀⣀⣉⡰⠤⠀⠘⠐⠀⠀⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⡇⢀⣠⡀⠀⣨⢭⣉⣁⠤⠴⠃
⠀⠀⠈⠋⠁⠀⡄⠀⠀⠀⠀⠊⠀⠀⠀⠈⡇⢂⠀⠀⠁⢀⡀⠀⠁⢠⠀⠈⠀⡀⠘⠀⠀⠀⠀⢀⢠⣄⠈⠉⠀⠈⠚⠁
⠀⠀⠋⢀⠐⠄⢡⠀⠀⢀⠂⠀⠐⡀⠤⠤⡧⠤⠀⠠⢶⣿⣟⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⢠⣄⠻⣆⢻⡆⠀⡰⠊⠉⠑⠢⡴⠊⠉⠑⢦
⠀⠀⡰⠸⠅⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⣾⡟⠉⢠⣤⠀⢀⠈⠀⠀⠀⠀⢢⡀⠙⢷⡜⠗⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠀⢀⠀⠀⠉⠉⠐⠒⠒⠆⠀⠃⠀⠠⠞⠛⠉⠀⠈⠉⠀⠠⠈⠀⠁⠄⠠⠀⡅⠀⠀⠀⢰⣶⠀⠙⢄⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠇⠀⠀⡄⠈⠀⠈⠀⠀⠀⣀⢀⡠⢰⣶⠀⠀⣀⣤⡔⠂⣤⣤⡀⠘⠂⠀⠀⠀⠀⠠⠁⠀⢠⠖⡶⣄⠀⢀⣄⠙⠢⡀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⡀⠐⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠨⢀⠖⢦⠀⠃⠀⠀⢻⠀⡃⠸⠇⠀⠘⢋⣤⣀⣤⣄
⠀⠀⠀⠩⠇⠀⠀⠀⠁⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠋⠁⣼⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⠄⢸⡄⠁⢰⢀⣤⡀⠸⣿⣿⣿⡿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠈⠀⠀⢘⣷⣦⠀⠀⠠⠀⠃⢢⡔⠀⡀⢘⡇⣀⠜⠀⢉⣠⣤⣌⡙⠏
⠀⠀⠾⠆⠀⠀⠀⠁⠀⠀⠸⠀⡀⠀⢀⡈⠛⠿⠷⠶⣶⣶⣶⣶⠶⠾⠿⠟⠋⠀⠀⠀⠢⠀⢸⡇⠀⡇⠀⡏⠀⢀⣼⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣀⣀⠀⠀⠀⠀⣀⣤⣶⠖⣷⣄⠀⠄⠀⠀⣤⣄⠀⡇⠀⡇⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠟⠀⢠⣾⣿⡿⠁⠀⣿⣿⣷⡀⠀⠂⠈⠁⠀⡆⠀⡇⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⠁⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⢰⣿⣿⣿⠃⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⡆⠀⡇⡇⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⠀⠀⠀⠀⡐⠀⠀⡤⠀⠘⠋⠁⠈⠙⠃⠀⣾⡿⢿⣿⣀⠤⢄⣸⣿⢿⣿⢀⣀⣀⡆⠀⡇⠀⠁⢷⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠂⠀⠀⡄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⠀⠀⠈⠉⠀⡈⠁⠀⠑⠦⠔⠃⠾⠆⠸⣿⠛⡿⢹⡏⠿⣿⡇
⠀⠀⠀⡆⠀⠄⠀⡐⠁⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠐⡀⠋⠀⠀⢠⠀⢸⠀⠀⠀⠀⢰⣷⠀⢀⣀⣤⡄⠀⠠⠀⢀⡀⠘⠃⠈⠁
⠀⠀⠀⠃⠈⠀⠌⠀⠀⠀⠀⠀⠀⠈⢿⣿⠗⠀⠀⠀⠈⢄⠀⠠⠬⠀⢸⠀⠙⢿⣶⣼⣿⣷⣿⡿⠃⠐⠀⠀⠄⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠈⢀⠀⠀⠒⢸⠀⡀⠈⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⣀⡀
⠀⠀⠀⠠⠤⠤⠄⠀⠠⠤⠤⢀⣀⣀⣀⣈⣦⠀⠀⠀⠀⢀⡀⣂⠀⠠⠼⠘⠟⠀⠈⠉⠉⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠙⠃

⠀⠀⠄⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢀⡻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠁
⢰⠆⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠨⠭⠀⠀⠀⠀⠀⠐⠄
⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⠀⠀⠀⠀⠢⡌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡄⢀⠀⠀⠠⠄⠀⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢂⠈⣟⠠
⢀⠅⢀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙
⠀⠁⠁⠀⠀⠀⠀⠂⠀⠀⠀⡄⠁⠐⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄
⠀⠃⠀⠃⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⠀⠒
⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠡⡀⠀⠀⢀⢀⠀⠀⡀⠁⠀⠀⢀
⢀⠰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡛⠺⢈⠀⠀⠀⠀⢀⠀⠀⠀⠈⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠐⠀⢂⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠉⠁⠈⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠁⠀⠄⠀⢁⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠈⠀⠠⣠⠀⠀⠀⠘⡿⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠻⣄⠀⠀⠀⠈⠠⠤
⠀⠀⠀⠀⠤⠾⢎⠁⠀⠀⠀⠈⢃⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣾⣧⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠲⠀⡖⠠⠀⢤⠀⡄⠂⣶⠂⠀⠶⠶⡒⠒⠴⢄⠀⠀⠖⠲⢆⠀⢶⠂⡀⣄⠀⠀⠀⡐⠀⠀⠀⡀
⠀⠀⠀⠈⣱⡦⢤⣀⠈⠀⡉⠄⠀⠰⠖⠀⠀⠀⠈⡀⠀⠎⠠⠂⠂⠈⠀⣁⠈⠁⢁⠠⢀⠀⠠⠊
⠀⠀⠀⠀⠉⠀⠈⠀⠁⠀⠁⠄⠒⡴⠈⠙⠂⣠⠀⠀⠀⠎⠀⡀⠂⠀⠁⠊⠀⠠⠄⠀⠀⠀⠰⢆
⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⠉⡀⠀⠤⠚⠓⠀⢘⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⢄⠀⠑⢀
⢸⠃⠀⠀⠀⢀⡀⡀⠀⡄⠀⠮⠾⠴⠗⠛⠁⠀⠀⠀⠘⠂⠀⢃⠀⠀⡀⠀⠁⡂⠀⠈⠃⣤⣜⠓⠀⠀⢐
⠘⠀⠀⣤⠋⡀⠀⠁⠀⠀⠉⠀⠚⠉⠁⠀⢄⠐⣄⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠈⣛⠃⠀⠀⠘
⠰⢀⠀⠈⡅⠀⠀⢀⠀⠀⡀⠀⠀⣴⡶⠄⢀⠀⠀⠀⡀⡀⡀⠀⠓⠠⠀⠀⠀⠃⠀⠁⠐⠀⠀⠀⠰⡀
⠀⠋⠈⠀⠁⠀⠀⠘⠀⠇⠁⡰⠴⠉⠅⢠⠀⠁⠺⡍⡃⢄⡀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠭⠎
⠀⠈⠀⠀⡀⠂⠐⠆⠀⠀⠀⠀⠓⠀⠁⠈⠀⢀⠜⠉⠉⠶⠁⡀⠉⠀⠲⠀⠀⠀⠀⠀⠀⡄⡐
⠀⠀⠀⢀⣘⡱⢦⠀⠀⠈⠀⠀⠂⠁⠀⠘⠀⠘⠁⠀⠀⠀⡀⠀⠀⠐
⠀⠀⡀⠄⣀⠀⠀⠀⠀⠀⠈⡄⠀⠤⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢤⠀⢶⢄⣀
⠀⠀⠀⠀⠉⠐⠀⠀⠀⢀⠀⠉⠉⠀⡃⠐⢀⠆⠀⡅⢀⠀⠄⠐⠀⠉⠉⠀⠀⠀⠀⠀⠠⠂
⠠⠂⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠒⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢈⠀⠀⡀⠀⠀⠀⠀⠁⠀⠛⠠⠃⠀⢦⠀⠀⠀⠀⢠⠄⠐
⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⡀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⢰⠀⠀⣰⣔⣀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠂⠈⠇⠀⠈⠐⠀⠀⠘⠀⠘⣂⠀⠘⠋⡀⠀⠀⠈
⠈⠀⠀⠀⠀⣉⢱⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⢀⠈⠀⠬⡧⠀⠀⠀⠀⠀⠀⠠
⠈⠓⠐⠈⠀⠂⠄⠀⠀⠠⠄⠀⠀⠈⡘⣁⣀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠐⠚⠃⠀⠈⠀⠀⣰⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⢀⠝⠩⠁⢆⢟⡀⠀⠀⡀⠀⠀⠀⠀⠀⣯⡽⠁⠠⠆⠀⠀⠁⢸⣇
⠀⠦⠤⠼⠀⠀⠤⠤⠤⠀⠀⠀⠴⠂⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠾⠧⠤

⠀⠀⠀⢠⠄⢀
⠀⠀⠀⠀⡀⢀⣎⡗⠈⠆
⡀⠀⠀⡀⠈⠈⠉⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⡀⣆⡀⠀⠀⡅⣲⠁⣠⠎⡀⠘⠃⠇⢀⣤⡔
⠈⢁⡇⠀⠀⠀⡇⠐⠁⠰⣷⠦⢄⣼⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡄⢘⡠⠀⠀⣸⠕⣀⢐⣿⣫⡴⠛⢉⣀⡀⠌⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁
⠀⡜⠀⡠⢰⠿⠯⢀⡜⣽⠋⠀⢰⠋⢀⢙⡆⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⣸
⢸⠂⠀⢡⠏⠀⢀⡞⣰⠃⠀⠀⢰⠀⠺⡋⠠⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⠋⠀⠧
⡏⠀⠠⠗⠠⠀⠾⢇⢻⡀⠀⠀⠀⠱⠀⠀⡤⡶⣄⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠺⡄⠀⠉⠀⠉⢺
⠇⠀⡀⡬⠀⠀⠂⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠤⠤⢀⠉⠉⢾⠶⠯⠄⢀⠀⠀⠻⡿⠀⠐
⣦⠨⠀⡁⠀⠀⡄⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢀⣭⣇⡀⢠⡁⢳⡔⠁⠀⠀⠀⠀⠑⢦⠀⠈⠁⠐⢄
⡇⠀⡎⠀⠀⠀⣷⢀⠀⠀⠤⡘⠤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠃⠉⠀⠀⠘⠧⠐⣨⡇⠀⠀⠀⠀⠙⠀⠊⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠸⣇⠅⠀⠀⠀⠀⠈⠉⠀⠀⠐⠶⠚⠉⠉⠀⠀⠀⠀⣀⠀⢀⡈⢳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢳⠖⠀⠀⢐⠀⠀⠘⣦⡐⠂⠀⠀⡀⠀⠒⠤⣀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠠⢄⠙⣄⠀⠐⢑⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠆⠀⠐⠈⠀⠀⠀⢠⠙⠦⣀⡘⠅⠐⠀⢀⡠⠉⠀⠀⡀⢔⠅⠀⠀⠀⠀⠈⢷⣿⠀⠀⠀⠱⣀⠀⠐⠂⠈⠉⠀⠀⠀⠠
⠄⢀⡆⠁⠀⠀⠀⡀⠀⠢⡀⠈⠙⠳⠶⠶⠤⠴⣲⠖⡁⠐⡀⠀⠀⠀⠀⠄⠀⠈⣻⡇⠀⠀⠀⠀⠁⠀⠀⠤⠦⠀⠄
⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠠⠃⠌⠁⠀⠀⠀⠀⠀⢀⠠⠀⠈⠉⠛⠀⠀⠀⠀⠤⠘⠃⠀⠉⠉⠉⠓⢄
⠀⠀⠀⠈⢷⡀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠀⠀⠠⡀⠑⡤⣀⠀⣀⣀⡀⠀⠀⣀
⠤⠄⢀⠀⠀⠙⢦⡀⠘⢆⠄⠢⠀⠀⠀⠀⠀⠀⠀⠈⣩⠝⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠄⠉⠉⠁⢀⠀⠀⠈⡆
⠀⠀⠀⠑⠐⢄⠀⠙⠒⣬⣷⣄⣀⣉⣀⡭⠴⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢰⣾⠀⠐⠁
⢀⣀⡀⠀⣈⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⡀⠀⢀⠀⠀⡀⠀⣀⣀⣀⣂⣀⡀⣀⡀⣀⢀⣀⣀⠀⣀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢡⣼⠿⠛⢶⣤⣼⣗⣤⣼⣿⣟⣙⠟⡿⢿⣦⠆⠸⣿⣤⣿⣏⠀⢸⣿⠟⠀⠉⠒⠈⠻⠟⠋⠁⢀⣿⣿⣿⣄⠀⣀⠀⢿⣉⠂⠠⠉⠀⠀⣿⣿⣯⣶⠀⣤⠀⠰⡟
⣿⡁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡟⢀⣿⠿⠀⠋⠉⠩⠿⠀⠿⣇⡀⠀⠀⠀⢀⡆⢀⣠⣶⣿⣿⣿⣿⡿⠶⠋⣴⡿⠛⠷⣤⣀⠀⠀⣀⠆⠘⣿⠐⠙⢦⣀⣀
⠉⠁⠀⠀⠀⠀⢿⣤⣵⣭⣶⣤⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡄⢀⡴⠋⠁⠉⣽⠿⠟⠛⠻⣿⣇⠀⠀⣼⠀⠀⠔⠉⣫⡟⡆⠀⠀⠀⣿⠈⢰⣶⣤⡽
⠀⠀⠁⠀⠀⠀⠈⠙⠛⠻⠿⠻⣿⡠⣿⣦⠀⠀⠀⢠⣶⣶⡆⠀⠀⠀⢹⣏⠎⠀⢠⡴⠾⠓⠶⠿⠷⢿⣿⣏⠻⣿⣿⣤⣖⠀⣼⣟⡱⠀⠀⢀⣠⣿⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠹⠄⡀⠀⠈⠠⠠⠄⠀⠀⠀⢸⣿⠄⠈⠀⠀⠀⠀⠀⠀⠘⡶⠼⠿⢷⣦⣿⡤⣉⢶⠞⠋⠁⠀⢠⠈⣿⠃⢠⡟⠀⠣⣄
⠛⠀⠀⠀⠀⠚⠀⠀⠀⠀⣀⣀⣤⣟⣃⣘⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⠋⠰⠖⠁⡀⢀⡜⢰⣿⠶⠿⠷⣦⣄⠈
⢦⣤⣴⠀⠀⢠⡀⠀⢠⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⢣⡟⢠⣏⠀⠀⠀⠀⠉⢣⡄
⡇⠈⠹⢶⡶⢷⣤⡴⢻⣿⣿⣿⣿⣿⣿⠿⠋⠀⣤⣀⣤⣤⣤⣶⣾⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠸⠌⡟⢦⡀⠀⠀⢹⣷
⡈⢠⣤⠀⣰⡾⠁⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠀⠿⠿⠿⠟⠛⠋⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠉⠀⠀⠀⠉⠻⢾⣄⠈⠳⣄⠐⣿⠁
⡇⠈⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣾⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠸⣿⠁
⣿⡀⢸⡿⢷⣾⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢈⡋⠉⢉⣉⣀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠⡟⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣮⣤⡀
⠛⠛⠺⣇⡀⠀⡤⠠⠀⠀⠀⠀⠀⠀⠘⠿⠏⠀⣾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣾⢋⣰⣀⣠⠤⠶⡟⠉⠖⠂⠠⢤⣀⡀⠀⠀⢸⣧⣈⢻
⠀⢠⣼⣿⡷⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠛⢿⡗⢺⣿⡀⡷⠄⠶⠷⠴⠶⠶⠇⠤⣭⣤⠓⣺⠀⣧⠈⠻
⣦⣀⠉⡍⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠻⠿⠟⠛⠛⠀⠀⠀⠀⠀⠀⣰⡶⣤⣤⡴⢾⣷⡖⣰⡿⡀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣭⡄⣿⠀⠦
⠙⠿⠇⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⡀⠈⠠⡀⠀⠀⢀⣀⣀⣼⠛⣃⡠⠉⢀⣼⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⢿⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣦⣦⣶⣤⣦⣤⣾⡿⣿⣷⣠⡶⣿⣤⣶⡾⠛⢩⡿⠞⠡⡀⠀⠙⢿⣿⣿⣿⠀⣿⣥⣤⣈⣿⡇⢀⣀⣀⡀⠀⣿⣟⣉⣼⣿⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠐⠚⠉⠛⠛⠂⠉⢀⣿⣿⡏⢀⣼⣿⡇⣤⣤⣿⣀⣤⣶⣄⣀⣀⠀⢿⣿⡻⣦⣻⢿⣿⡿⠋⠀⠈⢻⡟⠁⠀⠘⢿⣿⠟⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⠀⠀⠀⡉⠉⢐⣠⣾⣿⣿⡇⢠⡟⢹⣿⡇⢿⣿⣿⢁⡀⡟⠉⠙⣷⠘⣿⡟⢹⣿⣷⣬⠓⠤⣤⡄⠸⠇⠀⣠⡤⢚⠡⣾⡁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠛⠛⠛⠛⠛⢿⣿⢹⡟⢧⢸⣷⣾⣿⡇⠘⢿⣿⠀⣴⠃⠀⠀⡿⢷⠸⣷⠋⠈⡿⠋⠀⠀⠈⠙⣢⠤⣌⠉⠀⠀⠀⠹⣷
⣿⠀⠀⠀⠀⠀⠹⣮⠏⠀⠀⢀⠀⠀⠁⠀⠀⠀⣿⣇⠘⣧⣿⡿⠿⠛⠳⢾⡟⢱⡟⠀⠀⠰⠙⠺⣧⢹⣤⠾⣧⡾⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⣽⡄
⠄⠒⠀⠀⠒⠀⢀⣿⡄⢀⡀⠠⠀⠐⠂⢀⣠⣤⣼⢿⣷⡿⠋⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⠁⠀⢀⢿⣆⢋⣴⣿⡅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣄⠻⣄
⠀⢀⠀⠀⣀⠜⣩⠶⡇⢾⡛⠀⠀⠀⠀⠀⢿⣬⣟⢢⠈⠀⠘⠲⠀⠀⠀⠄⡀⠉⠙⠛⠃⠀⣠⡿⠀⣿⢿⣙⡛⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣷⣼⠷⣄
⣼⡟⣳⡼⠃⠠⢻⡦⢻⠶⠋⢲⣶⣶⣶⡿⠀⠀⣼⠋⠓⠀⠀⠀⠁⠀⠀⠀⠀⠢⠈⠁⡀⠀⠸⣶⠶⢿⣟⣿⣷⣦⣤⣀⣀⠀⠀⣸⡀⢀⣀⣤⣴⣾⣷⣻⡟⠀⠘
⣿⢿⣟⣇⣀⢀⢠⣔⢉⣷⡤⠤⡌⠉⠀⠐⢲⣿⠋⠀⠀⠀⠀⠂⠈⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⢠⡴⢤⠟⢻⡏⠙⡿⠿⢿⣿⣿⣿⣿⣽⠿⢿⠟⢻⣿⣋⣰⡀
⡋⠈⣿⣿⠿⣿⣛⣿⣿⢏⡐⠀⠿⣀⠄⠀⠀⢾⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣄⣤⡏⠀⠀⠀⣸⣷⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⣀
⠀⢃⢈⠟⢁⠿⠋⠁⠀⠀⠀⠠⢀⣸⣄⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠠⡎⠰⠦⢾⣻⣿⣥⠟⡌⢻⠗⠈⠈⢿⠀⠀⢤⣀⣀⣀⣀⣀⣀⣠⡄⠀⢸⡇⠈⠻⠿⢿
⣌⣉⠁⠀⠀⠀⠀⠀⠙⠀⠀⢀⠈⣿⠙⣧⠀⠀⠈⠃⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠐⠀⠛⣿⢷⡥⢚⢿⡗⠀⣿⡀⠀⠀⣟⠛⠋⠉⠙⠻⡇⠀⠀⣸⠇⠀⠁
⡹⠉⠰⠀⠠⠄⠀⠀⠀⠠⡆⠠⢉⡿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⠟⠃⠀⢀⠀⣀⡘⣿⣶⣶⣦⣴⣻⡄⠛⠀⢺⡇⠀⠀⣟⠗⠠⠄⠠⠴⣷⠀⠀⣿
⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⡾⠛⠺⠏⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣙⠋⣩⡿⠛⣉⣍⣉⡙⠢⣄⣨⡅⠀⢸⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⣤⡤⠤⠶⠶
⠁⠀⠀⠤⠀⠀⠀⠀⠀⡈⠉⠀⠀⠀⠀⠀⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠐⢻⡛⠛⠋⣡⣤⣭⣭⣍⡙⠿⣷⡜⠻⣧⠀⢸⡇⠀⠀⠀⠀⢀⣿⠀⠀⣿
⠲⠶⠿⠐⠃⢀⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠋⣉⠀⢹⠉⠀⠈⣿⡟⢷⣌⡛⠻⣿⠀⢸⡷⡤⠴⠒⠛⠉⣿⠀⢸⡏
⢧⣀⣄⠠⣴⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣨⠆⢱⣦⡘⢧⣿⠛⣿⡙⣿⣧⣾⣇⠀⠀⠀⠀⠀⣿⣶⢾⡇
⣀⣝⠉⡙⢻⣷⣤⣤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠁⠀⠀⠂⠰⠀⠇⢉⠇⠈⣿⣷⣄⠉⣽⠋⢴⣿⣤⣾⡏⠀⠀⠀⠀⠀⣾⣶⣾⡇
⠁⠈⣷⣿⣿⡽⣧⣠⣿⢶⡟⢤⣀⣀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣤⠃⢸⡆⢂⠸⣧⠈⠑⠢⣕⣦⡿⠡⢼⣿⡀⠀⠀⠀⢸⣷⣶⡚⢧⡀
⠀⢠⣿⣿⡏⢰⡈⢻⣿⣷⣼⡿⣏⣈⣯⣿⡦⢠⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣿⢠⣿⡿⣦⠳⠈⢷⡐⢀⡴⠿⢡⣴⡿⢻⡇⠀⠀⠀⢸⡟⢿⣷⡆⠙⢳⣆⣀⡀
⠃⣘⣿⣼⣷⣤⡕⠚⢻⡇⣹⣔⣿⣿⣭⣹⣶⡀⠻⣶⠴⢀⣠⣤⡀⢀⠃⠀⠀⡁⣮⣼⡃⠘⠳⣦⢠⡷⠚⠓⢠⣾⣿⣥⡿⠃⠀⠀⠀⠸⣧⣐⢹⣿⣷⠀⠀⠀⢹
⢽⣿⠋⠙⠟⣀⠀⠀⠈⠀⠘⠿⠋⢿⡷⢿⡛⣿⣿⣌⠀⠀⠹⣇⢽⠛⠰⠗⠀⠿⣼⢧⣻⣄⡼⣡⣾⡀⠀⠀⠈⣻⡵⠿⠆⠀⠀⠀⠀⠀⠀⠙⠓⠦⢍⣀⣀⣤⣴
⠀⡇⠀⠛⡃⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⠾⠿⠌⠀⢿⣤⣤⣶⣿⣦⠁⣴⡇⢸⣶⣇⢦⣃⡽⡛⠉⠀⠉⠛⠓⠋⠁
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⢡⡏⡜⢦⡰⣧⠻⣝⣎⣾⠗⡇
⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡧⡟⠎⢰⡞⣷⡹⣿⡿⠷⠃
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⡐⠀⡄⡏⢱⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣣⢸⠀⠀⠙⠁⣼⣸⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⢒⣿⡀⠤⠤⡇⠀⠀⠀⢀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣳⣤⣤⣷⣴⣶⣾⣿⣿⣿⠁
⠀⠀⠈⠀⢀⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⣯⣟⣿⣆⠀⠀⠀⠀⠐⠦⠄⠐
⠀⠀⠠⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣏⣽⣿⠟⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠉⢉⡉⠁
⠐⠚⠛⠿⠷⠶⠾⣟⠛⠛⠹⠛⠿⣷⣿⠟⢻⣿⣿⡿⠿⠽⠛⠛⢛⣖⣿⢿⣿⡟⠉⠉
⠀⠀⠉⠛⠛⠀⠰⠦⢭⣍⣉⣁⠀⠈⠛⠋⠬⠽⣿⠅⠀⠒⠒⠒⠒⣂⣉⡭⠉⠉⠀⠒⠒⠒⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠁⠀⠈⠉⠑⠒⠀⠀⠀⠉⠉⠓⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⣿⡂⠤⠤⣯⠀⠀⠀⠀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣦⣤⣼⣴⣶⣾⣿⣿⣿⠏
⠀⠀⠈⠁⢀⣀⣀⡀⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣋⣯⣽⣻⣇⠀⠀⠀⠀⠀⠲⠄⠐⠒
⠀⠈⠡⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣟⣽⣿⠟⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠈⠍⣉⠁
⠒⠚⠛⠿⠷⠶⠾⣟⡛⠛⠫⠛⠻⣷⣾⠿⢛⣿⣿⣿⠿⠯⠟⠛⢛⣷⣛⡿⣿⣿⠉⠉⠁
⠀⠀⠉⠛⠛⠀⠐⠦⠭⣍⣉⣁⠀⠈⠛⠋⠥⠽⣿⡧⠀⠐⠒⠒⠒⣒⣀⡭⠉⠉⠀⠐⠒⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠃⠀⠀⠉⠑⠒⠒⠀⠀⠈⠉⠙⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⣀⠉⠁⠀⠀⠀⠀⠀⣈⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠦⣈⠛⢦⠀⠀⢀⠴⣶⠶⢤⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠁⡎⣸⣿⣿⣆⠙⣿⣞⡾⣥⣿⣍⣉⣙⣒⠲⢤⣤⣄⢂⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⢫⠙⣿⣟⠛⢋⢿⣝⢿⣿⣶⣌⠙⢷⣶⣖⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⣹⢿⣿⣷⠘⢧⡀⠛⠃⣸⡼⣯⡼⡟⢻⣿⣿⣦⣮⡻⣿⣃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣱⣾⣃⣏⣻⣿⣇⢀⣹⣧⠀⢡⠄⡏⠃⣿⠇⠟⢉⣿⣿⣿⣮⡻⣤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠴⠶⠶⠦⢬⡻⠿⢿⣶⡶⠚⡈⠀⣼⣿⡿⠄⠀⠉⢹⣿⣷⣽⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣤⣒⠢⡀⢀⡀⢄⠒⠤⠄⡴⢺⣿⠟⠋⠀⢀⣴⢻⡿⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⡑⠛⠦⠕⠒⠊⠀⠀⠈⠓⠂⠒⢫⡍⣨⢁⣾⢻⣿⣯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡏⣼⢿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡞⢆⠰⣤⣄⣀⠀⠀⠀⠀⠀⣉⣾⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢧⡀⠳⣜⠿⣮⠙⢶⣄⠀⠀⢈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣻⣷⣦⡈⣆⣼⣿⣷⣬⡛⠿⣄⢼⣿⣷⡄⠈⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠿⢿⣿⣿⢿⣿⣿⣿⣾⡿⡟⠿⣹⣷⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⡀⢀⡀⠀⠀⠛⣦⣹⣿⣷⠻⣿⣿⡈⠁⠉⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⣷⠀⠀⣿⣿⣿⣿⣯⠈⢉⣟⣿⡶⠒⢹⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢀⣮⣿⠀⣷⣿⣿⣿⢿⡟⢰⠿⣫⣟⣡⣤⣼⣶⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⠏⣼⣿⢇⣿⡟⣼⡠⠴⣺⢿⣻⣿⣿⣿⠻⣿⣟⠀⠙⠆
⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠶⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣰⡿⢋⣾⠟⡼⠉⠀⢘⣵⢿⣿⣿⠿⠏⠙⠈⠛⣿⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢂⣼⡿⣛⣽⣾⣿⠁⡈⠉⣴⡿⠃⠈⠀⠀⣰⠟⣵⠿⠟⠉⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠫⣾⣿⠿⠋⠛⠛⠁⠸⠋⠀⠀⠀⠀⠈⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠊⠀⠀⠀⠀⠂⠨⠀⠀⠀⢀⠀⠰⠂⠀⠄⡀⠌⠀⢚⠀⠀⠂⠀⠀⠀⠐⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠂⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠐⠀⠂⠀⠠⠂⠀⠀⠀⠊⠀⣀
⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⣠⠄⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠈⠁⠁⠐⠀⠀⠄
⠀⠀⠀⠠⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡀⠰⠄⠀⠀⠀⠉⠀⠀⠀⠀⠃
⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠅⡂⠀⠈⠂
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⡀⠈⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠃
⠦⠀⠈⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁⠀⠘
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠀⡀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠤⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⢄⠀⠀⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠒⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⠐⠈⠂⠀⢂

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠂⢐⢧⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⢸⡌⢀⣤⡞⠐⠃⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⡂⢀⠔⣸⣿⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⡉⣰⡿⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣤⠾⢂⣾⣿⠀⡟⣧⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⢈⠘⠁⠀⣾⣏⡏⠀⡇⢿⡀⠀⠀⠐⠒⠀⠀⠀⠸⠆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⠁⠀⠀⣾⣿⣿⡇⠀⡇⢸⡇⠀⠀⠀⠀⠑⡀⠀⢠⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⢎⡀⠀⢀⡿⣿⣿⣿⠄⠠⡇⠀⣿⡂⠀⠀⠀⠀⠈⠆⢀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⣉⣀⠀⡀⠀⡇⠀⢻⣏⠀⠀⠀⠀⠀⠀⠸⡷⡶⢦⣤⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣤⣼⣶⣤⣤⣤⣃⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⢠⣴⠶⠾⠿⡭⣤⣒⡚⠛⣛⠛⠛⠻⠿⠟⠛⠻⠿⠿⠿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⠁⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠃⠀⠤⠖⠊⠉⠩⠵⠖⠛⢳⡖⢀⣤⡴⠶⠶⢟⣩⠴⠤⣤⣴⣶⡆⠨⢤⡤⠉⠀⢀⢀⡤⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠴⠤⠈⠭⢉⣉⡥⠴⠶⠒⠒⠒⠲⠄⠈⠉⠉⠀⠀⠀⠉⠛⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣾⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⣾⣿⣿⡿⠿⠿⢃⣀⣀⣀⣀⣤⠴⠶⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⢟⣩⣴⣶⣶⣾⣿⡿⣻⣿⡟⠡⡜⢿⣼⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀⠸⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢷⢹⢸⡿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣮⡉⣸⣿⡿⣿⣿⣿⣿⣿⣿⡟⠛⠀⡟⠀⡟⡾⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⡇⣿⣿⣧⣈⣉⡙⢿⣷⣶⣤⣀⡀⠀⢙⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣛⣻⣽⡔⢮⣅⣈⣛⣛⣫⣥⣾⣷⣮⣝⠻⢿⣿⡿⠿⠛⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠿⡿⣦⣝⡛⠿⠟⠛⢋⣛⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⢉⣴⠟⠉⠉⠀⠙⠃⠀⠀⠀⠐⠳⣶⣶⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡿⠿⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣤⣤⢀⣤⣤⣤⣤⢠⣤⡄⣤⣤⡤⣤⣤⢠⣤⣄⢠⣤⡄⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡌⠁⢸⣿⠘⢿⣷⣬⣉⠀⣿⣦⣼⣿⠀⣿⡇⢸⣿⣿⣦⣿⠀⣿⡏⢈⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣾⣿⠼⡿⢶⣿⠿⢰⣿⡆⢼⣿⠤⣿⣧⢼⣿⠌⠿⣿⠀⠻⣿⡶⢿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢒⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣆⣀⠀⣀⣤⣶⣿⣟⣛⣿⣿⣿⣿⣿⡟⢿⣷⣶⡅⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣼⣷⡘⢷⣽⡿⠿⠿⣛⡋⡉⠛⠛⠻⣾⣿⠏⣴⣿⣶⣄⠀⠀⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣞⠉⠀⣴⣿⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠈⣉⣉⣉⣡⣄⣜⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡁⢈⣧⣾⣷⣾⣿⡯⣄⡀⠀⠀⣠⠴⡒⡋⠉⠉⠀⠈⠠⠀⠀⣉⣻⣿⠿⢿⡿⢿⡿⣿⡿⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣉⣿⣿⡿⠋⠀⠀⠉⣽⠟⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠞⠐⣛⠛⠈⢀⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⢠⠀⣱⠟⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠠⣄⡈⣿⣦⣾⣥⣇⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⣒⠿⡈⠠⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣛⠶⣄⠈⠁⠉⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⣄⡠⠀⠀⠀⣤⠀⠀⣰⡦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢖⣛⣿⡦⠖⢻⣿⣿⣿⠦⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⠉⣹⠃⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢰⢖⡓⢲⠶⠚⠁⠀⢰⠇⣰⠏⢇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣿⣿⣿⣷⣀⠀⡀⡠⠚⠙⡟⡇⡿⠀⠀⠀⠀⢸⢞⣅⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣷⣾⣄⡜⡇⢀⣿⠇⠠⣀⠀⣀⡄⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡱⢄⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⢹⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡐⠀⡄⡌⠙⢻⣶⣯⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⠿⠛⡿⠲⠁⠀⡠⡀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢳⠮⠥⡘⠃⠀⠀⠀⠙⢿⣷⣆⠈⠚⢿⣿⣿⣿⡿⠁⠀⡰⠃⠀⠀⠀⠈⠛⣀⠭⢶⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⣤⠀⠀⠀⠀⠙⢻⡄⠀⠈⠹⣿⣿⠀⠐⡏⠀⠀⠀⠀⢀⡔⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠿⠒⠦⣤⣿⢟⠁⠀⣧⠀⠀⠀⠀⢤⡁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠻⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠑⢶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣠⡤⠼⠗⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣠⡤⠤⠶⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠖⠛⠋⠉⠁⢸⠃⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣻⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡏⣄⣀⡠⠤⢲⠚⠀⠈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⡤⠶⠒⢺⠏⠁⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡆⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠈⠀⠈⠆⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⢀⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⢀⡠⣆⣴⡟⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣠⠴⡶⠛⠭⠠⢠⣏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⡾⡔⠚⠋⡀⠟⢀⣠⣴⣶⣖⣾⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠉⠀⢀⡧⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣗⣊⣥⣴⣾⣿⣿⣿⠿⠛⡿⣿⣿⣿⣋⢀⡀⣀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠈⣛⠛⡛⠋⣋⣯⣍⢓⣻⣷⡎⢿⣻⣯⠙⣿⣿⣿⡟⠛⠶⣶⣿⣷⣞⣉⣀
⠀⠀⠀⠀⠙⠉⠀⠀⡠⠙⣉⣠⣿⣭⣽⣿⣿⡞⢛⣯⣥⣶⣿⣟⡛⠟⢛⣩⣶⡈⠀⣍⣛⣛⠛⠻⣿⣿⡿⢶⣿⣯⣟⡛⠁⡤
⠀⠀⠀⠀⠀⠒⠒⠂⠉⠉⢭⣥⠄⢘⣿⣛⡉⠉⠥⠼⠛⠋⠉⠩⠔⠂⣸⡿⣟⠁⠀⢶⣭⣹⡧⣠⠤⢴⣶⣶⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣦⠄⠀⠉⠁⠀⠀⠀⠉⠙⠉⠀⠀⠒⠬⠭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⢀⣠⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⡀⡀
⠀⣴⡞⠍⠚⡉⠀⠀⠋⠀⠁⠀⠀⠀⢠⠂⠙⠃⠐⡀⢠⠀⠀⠆⠀⠀⠀⠠⠀⠀⠈⠀⠈⡇⠀⠀⠁
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠐⠀⢀⠀⠐⠐⠐⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠏⠀⠀⠀⢀⠐⠁⠀⢀⡀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⡀⠚⠀⠀⠀⠠⠎⡬
⠀⠀⣢⡶⡤⠄⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠣⠩⣶⣄⠀⠀⠀⠂⠀⡀⠀⠀⢀⠴⠀⠁⠐⠀⠀⠀⠀⢀
⠀⠀⢟⣿⢿⡃⠀⠀⢼⣚⣭⠀⠀⠀⠀⡀⠀⠀⢌⠔⠊⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠉⡅⠉⠀⠀⠘⠉⠒⠁⠠⡀⠀⡀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄⠂⠀⠀⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢠⡄⡀⠀⠀⣀⣀⠈⠁⠐⠀⠐⠁⠀⠀⢀⣀⡠⠀⠀⡀⢀⠐
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠁⠂⠄⠀⠀⠐⠈⣡⠁⠠⠞⠀⠈⠉⠀⠀⠀⠢⠀⠀⠀⠈⠈
⠀⠀⠀⠀⢀⠀⢈⠀⠀⠄⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠘⠀⠅⡈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠤⡀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠐⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣪⢯⡔⠆⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠒⢈⠷⠃⠀⠀⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⠦⠤⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⢡⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠎⡐⢁⠞⡠⣊⢔⡁⢒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⣘⣃⣃⣊⣐⣡⣊⣊⠕⡰⠲⠕⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣲⠟⢹⣿⡿⡪⠋⠛⣽⣽⡝⡟⠹⣷⣷⣯⣀⣠⡾⠪⣂⢐⢌⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⢀⣤⠔⠛⠿⠱⠑⡀⡢⠙⣿⡇⡄⠜⣾⣿⣟⣄⣜⣿⣿⣻⡟⠊⠻⣿⣷⡶⢵⣯⡇⠀⠀⠀⠀⠀⠀⣠⣶⣫⡂
⢀⣠⠖⣁⠣⠖⠇⠀⠀⠀⠀⠍⠉⠎⢅⢄⠌⢜⣿⣞⢀⠈⠜⣿⣽⡟⠔⠴⣽⣽⣟⣔⢜⣿⣿⢁⣦⣿⣷⣘⡮⣿⣥⠂⢀
⠫⣁⠑⢬⡁⣧⢠⠆⠀⠂⡙⠂⠃⣀⢂⡎⠊⠊⠻⠏⠔⠐⡠⣻⣿⠏⡊⡀⣾⡿⣫⠎⠨⠛⠋⠉⣉⢉⢩⣉⡡⣿⣆⠡
⢀⠈⣢⢤⠳⡈⠜⣠⠤⠈⠀⡠⢠⡾⠛⠇⠀⠀⡐⠄⡸⡈⢀⡿⠋⠈⢁⠁⠙⠈⢄⢠⣢⣥⢾⣉⠀⠀⠀⠀⠙⠯⣍⠒⡠
⠀⠀⠈⠁⠉⠐⠲⠤⢤⡄⠀⠐⠰⡋⠅⠲⡀⠄⡠⢑⠀⠀⠎⡈⣖⣱⣕⡤⠴⠚⠻⠛⠈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⣷⠖⣚⣒⠛⠒⠓⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣲⡽⡟⢌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠲⡆⢠⠒⠠⠄⢤⡀⣀⠀⠒⢶⠐⠀⠰⡶⢦⡆⠰⠢⡤⠀⠀⠄⡴⠰⣤⡀⠀⢰⠖⠀⢠⠀⠀⠀⠄⢀⠆⠀⠀⠀⢀
⠀⠀⢀⡠⠐⢢⣄⣀⠁⠈⠓⠐⠛⠁⠀⠀⢠⣴⠁⠀⠀⠀⠃⡀⠀⢄⠀⣰⠅⠄⠘⠃⠙⠉⠙⠂⠲⠀⡈⠀⠀⠀⠤⠁⠀⡀
⠀⠀⠀⠀⠀⠿⠉⠉⠏⠸⠄⣀⠢⠁⣀⣀⡨⠱⢦⡄⠀⠀⠀⠃⠀⢀⠎⠀⢀⠠⠀⡒⣒⠒⠀⠀⠈⠀⠈⠂⠄⢀⡀⠀⢀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠀⠈⠈⣼⡇⠀⠀⠀⠠⢧⡀⠀⠀⡈⠀⠀⢇⠀⠀⠂⠀⡀⠀⠈⠉⠀⠂⠀⡀⠉⠱⠈⠀⠈
⠈⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⡮⡟⢠⠐⠀⡙⠏⠉⠁⠀⢸⠄⠀⢀⠀⠀⠀⠀⢀⢡⠀⠀⠀⢐⣄⠀⠁⣠⡀
⢠⠃⠀⠀⢀⠂⠀⡀⢀⠀⠠⡆⠀⠶⠹⠓⠼⠛⠏⠅⠀⠄⠀⡀⠊⠃⠀⠈⢄⠀⣀⡀⠀⠀⠀⡠⠀⠀⠉⢡⣤⣜⠋⠀⠀⠈⣱
⠘⡆⠀⠀⡀⠋⡈⠀⠘⠀⠀⠐⠉⠁⠐⠛⠋⠃⠀⢤⠐⢢⣀⡃⠀⠀⠀⠀⢰⢂⠂⠀⠀⠀⠀⠄⠀⠈⠀⢠⠼⠛⠧⠀⠀⠀⠉
⢠⡅⠀⠀⠙⡆⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⣤⣀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⠰⠆⠀⠖⠀⡤⠀⠉⠂⠠⡄⠄⢐
⠀⢵⠂⠖⠸⠃⠀⠀⠀⠆⠘⡄⠆⢂⡀⡸⠿⠆⠁⢀⠲⢄⣤⢐⡇⠪⡚⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢳⢄⠔
⠀⠀⡀⠀⠂⠀⠀⠀⣒⠀⠀⠁⠀⠐⢉⡉⠀⡇⢸⡇⠀⢀⣠⣇⣶⠘⠴⠀⠀⡤⠄⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉
⠀⠀⠀⠀⠠⣄⣁⠀⠙⠀⠀⢀⠀⠀⠀⢁⠀⠀⠀⠀⠈⣔⠃⠀⠀⠘⠃⠠⠀⠀⠀⠘⠁⠀⠀⠀⠤⠀⠀⠰⠀⠎
⠀⠀⠀⠁⢁⣘⡺⠶⡆⠀⠀⠀⠀⠀⠂⠁⠀⠀⠈⢀⠀⠉⠀⠀⠀⢀⢀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⡀⠀⠀⠁⠀⠀⠀⠀⠈⡄⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠁⠀⠀⠲⠀⠄⠔⠂⠀⡀⣠⡄⠠⣤⠀⣀⡀⠀⠀⢀
⠀⠀⠈⠀⠈⠛⠀⠄⠀⠀⠀⠀⠀⠙⠒⠂⠰⠆⠠⠀⢄⠀⠐⣂⡀⠀⣉⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠁⣠⠀⠈⠂⠀⠀⡀⠄
⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠐⠀⠃⣀⡀⠣⠄⠁⠀⠀⠀⠰⣀⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠂⢀
⠀⡀⠠⠴⠖⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠙⠃⠀⡀⠀⡆⠚⠘⠀⠀⠀⣯⡸⠀⠂⠀⣛
⠀⢉⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠆⠀⠀⠆⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⢘⠁⠂⢡⠐⠸⠆⠠⠀⠹⢽⠐⠿⠐⢛
⠀⡌⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣤⡀⠀⠀⠀⠀⠀⡀⠄⢀⠄⠀⠰⠐⠀⠀⡀⠈⣀⠀⢀⣈
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣦⡀⠀⠀⢀⠂⠀⠀⠀⡀⠀⣀⠂⠀⠂⠈⡳⠀⠂⠘⡰⠂⠀⡠⠸⠀⠘⠂⡀⠀⠛⠋⡀⠀⠀⠀⠙
⠐⠆⠀⠀⠀⠀⠒⢳⡀⠈⠛⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣿⡁⠀⠀⠀⠁⠀⢠⠀⢀
⠀⣥⢀⡠⠠⠀⡘⢘⠁⠀⠀⠀⠀⠀⠀⠀⠴⢠⠄⠀⠢⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠱⠂⢠⣁⡏⠀⠀⠰⠀⠀⠀⣄⣀⣼
⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⣤⠶⠶⠖⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣠⡄⠀⠀⠀⠀⠀⡄⠰⣅⠁
⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⡰⠚⠀⠀⠘⣴⠡⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡾⠎⠀⠀⠟⠀⠀⢀⠀⠀⣿⠄
⠀⠠⠄⠠⠜⠀⠀⠰⠴⠦⠦⠀⠈⠑⠲⠒⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠼⠻⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠤⡄⠀⡤⠄⢄⠀⢠⡄⢀⢠⠀⠀⢴⠤⠀⠀⢰⣤⣤⠤⠤⠆⣤⡄⠠⠀⠄⣠⠄⣤⣤⠀⠀⣠⡤⠠⠀⣄⠀⠀⠀⠠⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠠⠌⠁⠑⠤⢀⡀⠀⠉⠈⠙⠆⠰⠾⠂⠀⠀⠀⣀⣨⠏⠀⠀⠈⠸⠆⠀⠀⣀⠂⠈⣩⣃⡀⠸⠗⠐⠗⠓⠲⠀⢤⠀⠀⢁⠀⠀⠀⡂⠞⠁
⠀⠂⠂⠉⠁⢀⣾⠟⠛⣶⠄⡄⠀⠀⣄⠒⠐⠀⠀⠛⣋⣀⠀⠀⢈⠀⠠⠆⠀⠘⢠⠘⠁⠀⠀⣀⣀⢠⢤⡄⠀⠀⠠⠀⠁⠢⠀⠀⠒⠁⠀⠀⠈
⠀⠄⠐⠀⠀⠀⠈⠀⠀⠉⠀⠈⠐⠶⠈⠠⠒⠮⢶⠏⠈⠉⠛⠀⢠⠄⠀⠀⠀⠰⠀⠀⢀⠔⠉⠀⠒⠐⠂⠈⡀⠤⠤⠀⢁⠀⠈⠀⠶⢗⡀⠔⠖⠄
⠀⡶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠚⠃⠀⠀⠀⡴⡻⠃⠀⣀⣀⡀⣏⣶⠄⠀⢀⠆⠀⠀⠊⠣⠀⠀⠈⠆⠠⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⢠⠇⠶⣉⣀⣀⠉⠀⠀⠀⠀⢰⡆⠀⠀⢄⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣶⡤⠀⠀⣜⣀⠀⠀⢀⣀
⠀⡇⠂⠀⠀⢀⠂⠀⣄⠀⡀⠀⢰⡆⠀⠸⠇⠛⠒⠽⠛⠭⠥⠀⠀⠀⠀⡀⠂⠉⠈⠀⠈⣄⠀⣀⣀⠀⠀⠀⢉⠤⠀⠀⡈⠉⣴⣶⣜⠉⠁⠀⠀⠠⣱
⠀⡇⠀⠀⢀⡠⠏⠈⠀⠈⠓⠀⠀⠑⠋⠁⢀⠚⠓⠋⠣⠀⢀⠐⠲⣄⢀⡃⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠈⠐⡀⢀⠀⠡⠀⡠⠽⠻⠗⠀⠀⠀⠀⡋
⠀⡗⠀⠀⠘⠣⠄⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⠀⠈⠁⠒⠚⠋⠁⢀⠀⠀⡀⠀⢠⣆⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢀⡁⠈⠛⠆⠀⡀⢀⠀⢰
⠀⠣⡄⢂⡀⢰⠇⠀⠀⠀⢠⠀⢰⠀⡄⠃⠀⠀⣾⣿⡙⠋⠐⢤⡄⣀⣀⣀⡆⠤⢨⠆⠀⠀⠀⠐⠢⠀⠀⠐⠀⠀⠀⠀⠄⡈⠁⠀⠀⡀⠀⠹⣀⠀⣈
⠀⠙⠃⠈⠀⠈⠀⠀⠀⣠⡈⠃⠘⠆⠁⢈⣦⠰⠍⠙⡄⢰⡄⠊⠉⠐⢗⡔⣣⠰⡁⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡧⠬
⠀⠀⠰⠀⠀⠀⠀⡄⠀⣠⡄⠀⠀⠈⠢⠈⠠⠦⠀⠀⠇⠰⠇⠀⡀⢤⡖⠒⠛⢠⣈⠅⠂⠀⠒⠀⠠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐
⠀⠀⠀⢀⢀⠐⡦⢤⠀⠀⠁⠠⠈⠢⠀⠀⠀⠀⢂⠀⠀⢀⡀⠀⢰⣭⠂⠀⠀⢈⠉⠀⠆⠀⠀⠀⡀⠁⠀⠀⠀⠐⠒⠀⠀⠀⠛⠰⠇
⠀⠀⠀⠀⠈⣀⣜⡒⠿⣶⠀⠀⠈⠀⠀⠀⠂⠁⠀⠀⠀⠀⠃⡀⠈⠁⠄⠀⠀⠀⠀⣀⠰⠀⠀⠀⠣⠄⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠂⠀⢀⠑⠀⠀⠀⠀⠀⠘⢁⠀⠀⢀⠠⠀⠀⠀⠁⠀⠀⠀⢀⠀⡀⠐⠈⠀⠀⠀⠢⠆⢠⢸⠐⠀⡀⠀⡀⣄⠀⢠⣤
⠀⠀⠐⠂⠈⠐⠶⠀⡀⠈⠀⠀⠀⠀⠀⠸⠶⠶⠈⣠⡄⢠⠀⢠⠀⠀⠰⠀⠀⠀⠐⠂⠀⡀⠀⠴⠶⠆⠀⠀⠀⠀⠁⠁⠙⠀⡈⠍⠘⠏⠃⠀⠀⠀⡈
⠀⠄⠀⠈⠀⠄⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⠄⢀⠀⠬⠃⠀⠠⡊⠁⠔⡈⠁⠀⠁⢄⢀⠀⠀⠀⠐⠀⠀⠐⠀⢠⠎⠁⠀⠀⠀⠀⠀⠨⡀⢄
⠀⡞⠀⠀⢀⡀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠀⠈⠀⠄⠂⢀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡔⢠⠀⠀⠀⠠⡦
⠀⠤⣐⣊⡩⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⢁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠰⣷⠀⠸⠇⠀⢠⠀⠀⠀⠀⠁⢌⠂⣁⡀⠀⡤
⠀⠀⣤⠋⠀⠀⢀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠆⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠓⠀⠀⠀⠈⢈⠀⠛⠁⠀⣠
⠀⠌⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⠠⢂⢤⡀⠀⠀⠀⠀⠀⠀⢀⠰⠀⣂⠄⠀⠀⠏⠡⠀⡀⡀⠉⢀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⣆⠄⠀⠀⠀⡐⠀⠀⠀⠀⠠⠂⢀⠱⠀⠀⠆⠈⢻⡦⠀⠂⠘⢠⠆⠀⣀⡀⠟⠀⠈⠧⢀⠄⠈⠛⠋⠀⠀⠀⠀⠈⠻
⠀⡄⠀⠀⠀⠐⠀⠤⠷⡄⠀⠚⠟⠙⠓⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⠤⢦⠀⠀⠢⠆⠀⠀⠀⠁⠀⠀⠀⢠⠂⠀⠀⣦⠃⠀⠀⠀⠐⠃⠀⠀⡀⠈
⠀⡃⠀⠀⢀⠀⠀⠴⠄⡏⠀⠐⠀⠀⠀⠀⠐⠀⠀⣀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠒⣳⠛⠀⠀⣀⠀⠀⠀⠀⠁⠀⣻
⠀⠛⠃⠒⠁⠈⠀⠆⠠⠀⠀⠀⠠⠤⠤⠀⠀⠀⠈⢁⠘⣃⠀⣀⠁⣠⠀⠄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠓⠟⠀⠀⠀⠀⠀⠀⠀⡼⠲⠚⠛
⠀⠀⢀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡄⠀⠀⠀⠀⠈⢻⠙⠋⢉⡀⢻⡖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⡺⢾⠖⠀⢀⡀⠀⠀⠀⠂⠈⢻⡄⠀⠰
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡔⢆⡀⠀⠀⢀⠞⠈⠀⠁⠀⠱⡇⠰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠘⠀⠀⠀⣠⠀⠀⣾⡗
⠀⠺⠷⠶⠶⠟⠀⠀⠰⠖⠶⠲⠖⠀⠀⠙⠒⠒⠋⠁⠐⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠛⠻⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⣀⣀⠀⣀⠀⠀⠀⠀⠀⡀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠠⣀⡴
⠀⠀⠀⠁⠀⠀⠔⠁⠀⠀⠀⠂⠐⠈⠀⠀⢀⡇⠶⢹⠋⣤⡘⡏⣠⡌⢻⢰⠇⠀⠀⡄⠗⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢬⣀⠧⠱⣀⡄
⠀⠠⠀⠀⠈⠀⠀⠀⠀⠄⠀⠒⠂⠀⠆⢀⠀⣇⠛⣨⣄⠛⣡⣆⡛⣁⡦⡾⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⢠⣶⠄⢰⣦⠀⠀⢁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⡈⠉⠉⠉⠛⠋⠈⠉⠉⠙⠁⠀⠀⠀⢠⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡄⠀⠀⡘
⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠀⠦⠁⠀⠀⠊⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠙⠟⠂⠁⠠⡀⠀⠈⠀⢂⣈⡛⣛⡁⡄⠐
⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠈⣀⠠⡀⠀⠀⠀⢀⠠⠌⣤⡄⢠⣤⠱⠀⠐⠀⠀⠁⠀⠄⢀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡁⣀⠄⠀⢀⡀⢰⢀⠀⠀⠀⢤⠡⢈⠠⠌⣁⠔⠠⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠢
⠀⠀⠀⠂⣀⠀⣀⠈⠀⠀⠰⡄⠻⠃⠀⣛⠃⡠⠂⠀⠀⠐⠂⠐⠉⠐⠐⠀⠈⢆⡘⠀⠐⠁⠀⠀⠀⠈⠠⠀⠀⠀⠀⣄⠀⠀⠒⢡
⠀⠀⠀⠂⠍⠡⠉⠔⠀⠀⠀⠀⠒⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠁⠠⠀⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠁⠀⠀⠑⢀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠂⠀⠀⠀⠲⠆⠀⠀⠠⠀⠀⠀⢀⠂⠀⠀⢠⠂
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠇⠎⠀⠀⠀⠥⠠⠀⢀⣿⠉⠟⣭⠛⣭⠛⣽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⠀⠜⠀⠤⢤⠌⡦
⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣶⣴⣶⣵⣮⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡰⠀⠑⠀⠤⠥⠙⠮⠬⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠂⢀⠀⠀⠀⠐⠀⢸⡿⠉⠉⠉⠁⠀⠀⠀⠑⠁⠀⠀⡂⠀⠂⠀⠆⠁⢂⢀⡀⠀⠀⠀⠄⠀⠨⠒
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⢀⠰⠀⠀⠠⠦⣀⡀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠁⠠⡒⠂⠀⠀⠈⠀⠀⠀⠐⢐⠀⠀⢀⡀⢨⢘⡏⢄⡀⠀⠀⠀⠐⠕⠀⠀⠢⠀⢀⠀⠀⡠⠉⡀⠒⠂⠀⠲
⠀⠀⠸⠁⠈⠀⠄⠍⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⢀⠊⠔⣩⣅⠀⣨⡉⠁⠀⠀⠀⢈⠀⠀⠀⠀⠁⠀⠀⠀⠄⠘⠟⠀⠀⠓⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⣁⠂⣙⠁⠀⠀⠀⠀⠢⡘⠌⠤⢂⠔⠀⠀⠀⠈⠢⠀⠀⠀⠠⠊
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢀⠂⠀⠀⢀⠀⠀⠀⠻⠿⠏
⠀⠀⠧⠀⠀⠀⠈⠁⠀⣉⣀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠄⡀⠀⠀⡠⠂⠉⠐⠄⠀⠀⠢⠠⠔⠀⠀⡀⠈⠁⠀⢄
⠀⠀⢸⠀⠀⢀⡴⠚⠛⢿⣿⣿⠗⠀⠀⠀⢀⣤⣶⣶⣶⣤⠀⠀⠀⠓⠒⠀⠒⠚⠈⠀⠀⠀⠀⠠⡆⠀⡀⢐⠀⠀⠀⠀⠰⠈⢀⠀⠠⠄
⠀⠀⠠⠁⢀⡎⠶⡀⡄⢸⣿⡟⠀⠀⠀⣀⠏⣁⠈⠋⣀⡉⢧⡀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠠⠈
⠀⠀⠀⣇⣸⣷⣄⢁⣤⣾⣿⣷⣤⡄⠠⢼⠀⠛⣁⣀⡛⠃⠰⠄⠀⠀⠀⠸⡀⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠈⢻⠛⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠁⠀⡝⠛⠅⠂⠁⠀⠀⠀⢠⠔⠀⠔⣤⠠⠄⠀⡤⠀⢠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⠀⠀⡇⢸⡉⠃⣉⠀⠸⡇⢰⣬⠙⣿⠸⡏⠀⠀⠀⠠⡀⡜⠄⢡⡠⠠
⠀⠀⠀⠀⢃⠀⠈⠻⣿⣿⢿⣯⠻⢷⠀⠀⠀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠹⠦⠴⢷⠾⠶⠦⠴⠶⠤⠶⠿⡴⠃⠀⠀⠀⡁⠀⠒⠒⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⣀⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠠⡀⠀⠑⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⠉⣶⣄⣻⣯⣿⠿⢟⣻⡦⠀⠀⠰⡆⠀⠄⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⢿⣿⣿⣏⠀⢸⣽⣽⠇⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡰⣿⣿⣦⣀⣬⣙⣛⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⣿⠟⠛⠋⣀⣤⠞⠋⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣀⣤⠤⠴⢶⣯⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣿⣿⣿⣷⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣿⣿⣿⣿⣿⡟⣻⠷⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣿⣿⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢁⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡌⠀⠹⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠐⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣆⣶⣶⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣧⠺⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣆⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠿⣿⡿⠿⠿⢯⣭⣿⣿⣾⣿⣮⣜⡿⣆⠀⢉⣦⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣒⠚⠓⠈⢀⣀⣀⡼⢥⣿⣿⠼⠋⠛⠻⢿⣼⡄⠀⢀⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠒⠒⠢⣤⣤⠰⢿⣿⡦⠌⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⡾⠙⠏⢀⣼⠅⠀⠀⠒⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣀⡔⢿⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠣⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⡀⠀⣿⠽⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢂⡀⠀⠀⠀⡜⠀⠀⠠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣭⣿⠉⣷⣦⣅⣀⣀⣀⢀⠀⢘⡇⠀⠀⠈⣀⠀⠠⣀⣑⣤⡶⠂
⠀⠀⠀⠀⣀⣒⡒⢶⣶⣦⣤⣤⣦⣄⣸⣿⣿⣿⣿⡶⠶⠿⠤⠿⠤⠬⠥⠼⠷⣶⣾⣟⣛⠉⣛⠽⣁⡀
⠀⠠⠄⠀⢻⡛⠛⣿⣯⣽⠟⢛⣻⣏⣙⠛⠃⣿⣿⣿⣍⣻⣶⠿⠀⣲⣿⢗⠓⠿⠯⠀⠿⢿⡋⠤⠟⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠈⠀⠰⠴⠂⠨⠤⢈⣭⠉⠷⠾⢤⣀⠈⠑⡊⠁⢤⠘⠻⠿⠓⠂⠈⠁⠐⠂⠀⠠⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠁

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢀⡼⢃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢀⠀⠀⣠⣾⠿⠋⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢠⡟⣀⡾⠋⠁⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡟⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡜⡿⠁⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡼⠁⠈⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⡼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⡼⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡾⠁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⣠⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠉⠉⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⢀⣴⢟⣁⣤⠀⠀⠀⠀⢀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠸⣦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠂⠀⢀⣴⣿⣿⣿⣭⣤⠴⠶⠖⢻⣿⣯⣤⣤⣤⣀⣀⣀⣀⣤⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠲⠿⠻⠭⠉⣡⣀⣀⣀⣀⣴⣁⣀⣀⣠⠤⠤⢤⠤⠤⡌⣿⡓⠒⠂⠀⠤⠠⠤⠀⢀⣠⣤⣤⣤⣤⣤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣛⡛⠻⠿⣯⣭⣄⣀⡈⠃⠸⠧⣬⣍⣀⣀⣀⣀⣀⣀⣀⣀⣐⠆⠀⢉⡿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⡿⠛⠛⠛⠻⢿⣿⣟⠶⠶⠶⠶⣬⣭⣭⣭⣤⣤⣤⡶⠶⠚⠋⣁⣠⡾⠀⠠⠔⢒⣋⣥⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠁⠀⠀⠀⠙⢿⣷⣤⡀⠀⠚⠁⠀⠉⢻⣿⣿⣿⣿⠿⠛⠛⠉⣀⣤⡶⠾⠛⠉⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠀⠀⠀⠀⠉⠻⢿⣟⣓⠛⠀⢀⣠⡟⠈⠙⠛⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢄⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠀⠀⢀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⡟⠀⡀⠀⠠⠴⠀⠀⠀⢀⡀⢀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢀⣀⣠⡄⠀⠠⢄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⣀⣤⢾⡟⣿⠛⢛⡀⠀⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣩⠿⠼⠓⠋⠻⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠁⠀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢀⡀⢴⡶⠚⡿⠇⠀⠀⠸⣗⠒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡨⠀⢀⣤⣾⡯⣶⠚⣿⡾⣤⣼⡦⠴⠶⠾⣶⡿⠦⠴⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⣷⠾⢛⣉⡥⠷⠞⠒⠛⢟⠉⠉⠀⠀⠀⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠚⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠠⣄⠸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣀⠀⢰⣶⣶⣿⣿⣿⣿⣛⠿⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣤⠶⢺⡿⢸⣶⣞⢫⣾⢿⣿⣿⣿⣙⡻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠛⡉⠀⢸⣷⢞⣵⣞⣷⣾⣿⣿⣿⣿⣿⣿⡦⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣟⣷⡿⠺⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⠇⠀⠁⠀⠀⠈⠹⡿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠴⠦⣤⡶⠶⠖⠒⠲⢶⣶⣤⡽⠿⠿⠉⣤⣦⡀⢀⣀⣰⣶⣇⠀⢠⡀⠃⠐⠶⠖⠀⠀⠀⠀⠒⠶⠖⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡗⣰⡿⣶⣶⣶⣤⠀⠐⠒⣯⣷⡷⠈⣽⣾⣿⡏⣀⡘⠻⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠡⣶⣾⣍⠙⣃⣠⡔⢛⡿⠃⠀⠚⠿⣻⣥⠙⢿⣷⣄⠀⠀⠀⠀⠈⠁⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠙⠛⠚⠃

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡎⢠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⢀⠀⠀⠐⠀⣀⠤⠀⠀⢀⠀⠀⠀⠘⠄⠀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠉⠠⠀⠘⠓⠁⠀⠀⡐⠂⠾⡐⢄⡀⠀⢠⡘⠀⠄⢼⣷⡤⠅⢷
⠀⠀⠀⠀⠀⠠⠄⢰⡀⠀⠂⠠⠀⠀⠀⠀⣰⡿⣣⣾⣧⣁⣈⠀⣇⠉⠀⠀⠀⡀⠱⡄⠈
⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢀⠀⠈⠆⠠⠄
⠀⠀⢳⡀⠀⠈⠛⠿⠿⠷⠾⠲⠶⠶⠶⢶⣶⣒⣛⣛⣛⣛⣛⣉⠉⣿⠀⡸⠃⠀⠀⠀⠀⠂
⠙⠛⣉⣉⣐⣛⣛⣛⣋⣁⠀⠀⠈⠛⠿⠯⠭⢥⣤⠀⠀⠀⠀⠀⠀⢹⡗⠰⠀⠀⠀⢱⠀⠀⠁⠀⠀⣠⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣦⣬⣭⠉⠁⠀⡀⢠⠀⠀⠀⠬⣤⣴⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⡉⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠀⠀⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠛⠁⠀⠀⢰⡀⠐⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⠁⠠⠄⠀⠀⠀⠀⠀⠀⠁⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡓⠀⠀⠀⠄⠈⠛⠉⠓⠄⠀⠠⠀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣀⣀⣤⣤⣶⣦⣀⣀⣀⣓⣠⣤⣤⣤⡀⠀⠀⠶⠿⠿⠛⠛⠉⣉⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠉⠈⠛⠁⠉⠋⢹⠙⠋⠈⠙⠉⠉⠁⠀⠀⢸⣿⣿⣿⣷⣾⣷⣴⣿⣶⣴⣶⣦⣾⣷⣦⣾⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣧⣿⢿⣦⣄⣬⡴⠶⢦⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡉⠉⢠⣈⠉⠻⠟⣋⣉⣛⠿⢿⣿⣿⣿⣿
⠀⠀⢀⣀⣞⣀⣉⣛⣾⣭⣵⣞⣛⣛⣾⣯⡀⠀⠀⠀⢸⣿⣿⣿⠿⢱⠿⡶⣬⠑⠒⠋⣡⢶⣬⡉⠚⢿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠷⠾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢦⣄⠀⠀⠠⡇⠀⠀⢹⣦⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⣯⠑⢦⣀⣇⠀⠀⠸⣿⡷⡑⢢⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡃⠸⠀⠀⣹⣿⠄⠀⢸⠇⡇⠀⠀⠻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢼⣿⠀⠀⠸⠁⠀⠀⠀⠀⡼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿⢇⠀⡚⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣷⢸⠀⠇⠀⠀⠀⠀⠀⠀⢹⣜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣻⡋⡞⠀⠀⠀⠀⠀⠀⠀⢸⣻⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⡄⠀⠀⠀⢀⠀⣸⢾⣶⠃⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⢂⠀⠀⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣅⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡅⢣⠀⠀⠀⠀⠀⠈⣀⠚⠉⠃⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⢀⣠⣤⣽⣿⡧⠖⣂⣉⣀⠀⠀⠀⠀⠸⣿⣧⢸⠀⠀⣀⡀⠀⢈⣵⡖
⠀⠀⠀⠀⠀⠀⣰⣿⠁⣀⠔⠀⠀⠀⢀⡈⢻⣷⣶⣭⣿⠭⠍⣡⠄⠀⢀⣈⣿⣿⣬⠁⠀⣁⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠨⠷⠩⠿⣁⣈⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⣃⣤⣶⣿⣿⣿⣿⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡏⠉⣿⣷⠂⢀⣠⣤⣾⣿⣿⣿⣿⠿⠋⠉⠙⠿⢀⠴⠃⠀⠄
⣠⣄⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣴⡿⢓⢀⣷⠀⢸⣿⡇⢀⣆
⣿⣿⣿⠿⢿⠂⠉⠚⠦⡀⣶⣬⣀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣇⠉⢈⣉⣶⣾⣿⣿⣿⡀⣿⡿⠛⢸⣿⣦⣤⣀⣀⣠⠤⢢
⠉⠉⠉⠛⠢⠀⠀⠀⠻⣦⣤⣈⠉⠛⢿⣷⣾⣶⣷⣿⣿⣷⣾⣿⡿⠿⠿⠿⣾⣟⠿⠿⣿⡿⠿⠁⠹⠆⠀⢿⣿⣿⣏⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⡤⠌⠉⠉⠉⠛⠶⡿⠍⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⢿⣿⡯⠀⠘⢷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠳⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⢱⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⣿⠃⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⢀⣄⠀⢁⣤⣄⡀⠀⣤⡀⢠⣄⠀⣤⡀⢠⣄⠈⢩⡍⠁⢀⣤⣀⡀
⠀⠀⠀⢰⡟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠋⠀⣿⠀⣿⠀⠘⠇⠀⣿⠀⢸⡇⠀⣿⠁⠸⣿⡄⢸⡅⢠⡟⠀⠘⠇
⠀⠀⠀⠸⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣴⠀⠀⣿⠀⢿⣦⣀⠀⠀⣿⣀⣸⡇⠀⣿⠀⢀⡻⣇⢸⠇⣾⡷⣆⣀⣀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠛⠀⠀⣿⠀⠈⠙⢻⣷⠘⣿⠉⢹⡇⠀⣿⠀⠰⡆⢻⣼⡅⢿⡇⠀⢸⡏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣆⠀⠀⣿⠀⣿⠀⢸⡇⠀⣿⠀⢰⡇⠘⣿⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣃⡀⠀⢐⣟⣀⠛⣳⣖⠁⢀⣿⣆⢸⣷⡀⣿⣇⣸⣧⡀⢹⡀⠀⠙⣳⣞⢿
⣿⣿⡿⠿⣿⡿⠿⠿⡿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⢀⣿⠋⠁⢹⠀⠀⠀⠀⠀⠀⢀⣏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣾⠀⠀⠀⠀⠀⠀⡜⠹⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⢀⣾⡿⠀⠀⢀⣠⣄⡰⠀⠀⠉⠀⠀⠀⠀⠀⡀⠠⠆⣤⡄⠰⠖⢀⡆
⠀⣀⣀⣀⣀⣀⣷⣿⣿⣁⣀⣀⣻⣿⣿⣃⣀⣀⣀⣀⣬⠠⠤⠤⠬⠞⠀⠀⠀⢉⠠⠄⠃
⠀⠉⠉⢉⣽⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⡖⣄⠰⠆⠈
⠀⢀⣰⣿⣿⣋⣹⣆⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⢤⠦⠤⠒⠒⠚⠿⡃⢈⡤⠄
⠀⣹⣿⠟⠉⢉⣽⣿⣶⣦⣄⠀⠀⠀⠀⡰⠚⢛⡎⠀⠀⠀⠀⢀⣦⠏
⠀⣿⣏⣀⣠⣿⣟⣻⣛⣿⣿⣧⣤⠤⠤⠬⠿⠿⠒⠒⠒⠋⠉⠙⡏⡆
⠀⢻⡏⠉⢹⡏⠉⠸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡤⠜⡇
⠀⠀⢳⣄⣀⣛⣄⣀⣧⣠⣼⡿⠤⠤⠴⠖⠒⠒⠛⠉⣉⣁⡀⠁⢀⡆
⠀⠉⠉⠉⠻⠭⢥⣤⢿⠖⠋⠀⠀⠀⠀⣶⣿⡆⠀⣐⣿⡿⠷⠒⠉
⠀⠀⠀⣀⣀⣀⣀⣀⣼⣧⠤⠤⠴⠶⠒⠛⠛⠋⠉⠁
⠀⠉⠉⣹⣿⣿⣇⠀⠀⣧
⠀⠀⠀⢻⣿⡿⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⣤⣴⡶⠶⠿⠃
⠀⠀⠀⠀⠉⠛⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣇⠠⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠀⠀⠀⠀⠀⡀⢀⣀⣧⠜⠖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠦⠶⠒⠚⡟⠉⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢬⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡢⠐⠀⠀⠈⠉⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠚⠋⠉⠁⢸⠇⠀⠀⠀⠀⠰⡆⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢻⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡃⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡇⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣴⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠷⣾⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⢤⠖⠻⠟⠁⠂⡂⡾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣀⠦⡜⡾⠉⠉⡠⠘⢁⣤⣴⣾⣿⣯⣷⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⡀⠿⠐⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣬⣥⣶⣿⣿⣿⣿⣿⣋⠻⢿⣿⣿⣿⣁⣀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⢀⡖⢮⣙⣛⡛⠉⢩⣽⣭⣙⣋⣿⣶⠳⣿⢿⣯⢙⣿⣿⣿⠝⣛⠷⣴⣶⣶⣾⣍⣀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠠⠌⠋⣁⣬⣿⣭⣿⣿⣿⡞⢛⣯⣥⣶⣿⣿⣛⠛⠛⣉⣵⣆⠁⢨⣉⣛⣛⠙⠻⣿⣿⠿⠶⣿⣯⣟⡛⠁⠠⠄
⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠉⣭⣥⠀⢼⡿⠛⠋⠁⠤⠼⠛⠋⠉⠩⠕⠂⢀⣿⠿⡋⠀⠰⣮⣍⣻⠥⠤⠤⢴⣶⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠁⠁⠀⠀⠀⠘⠻⢿⣷⣶⡄⠀⠒⠉⠀⠀⠀⠈⠙⠛⠉⠁⠀⠢⠭⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢦⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⣟⣷⣼⣿⣿⣿⡿⢟⣯⣽⡟⠟⡴⢋⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⢷⣶⣿⣿⣿⣿⡿⠋⠑⢋⡽⢛⡅⣠⢿⠟⡉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⠿⣿⡘⣿⡇⡄⠀⠀⠀⢉⡞⠟⢸⣴⣷⣦⣿⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠀⣘⣶⣶⣝⠻⡧⢳⣄⣀⢀⣼⣤⡄⣨⡭⠭⣂⣀⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠽⣿⣿⠀⢷⣆⠛⠛⠛⢛⡭⠀⢰⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣷⣤⠙⠿⠶⣟⣫⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠩⢭⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⣠⠴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣣⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⡏⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠈⢻⣷⣄⠀⠀⠰⠿⣧⣀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡇⠀⠀⠀⢿⣿⣷⣦⡀⠀⠀⠉⢿⣿⡟⠋⣉⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠀⣦⡄⠈⠿⠟⠙⠻⣷⣤⣤⣾⣟⠳⣌⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⢻⡝⢳⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⣧⠀⢻⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠠⠐⠒⠁⠈⠉⠀⠀⠀⢸⣧⣼⠟⢻⡄⠀⠰⡀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⣄⠀⠀⢀⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⢿⢸⡆⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣶⡾⠿⠯⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡀⠀⢠⡾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠛⠃⠒⠶⠶⠦⢤⣤⣴⡇⠀⣠⣾⣦⣴⣯⣀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠉⠙⠻⠿⢯⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⠸⢿⢰⣶⡤⣬⣤⣤⣀⣄⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣬⣵⣾⣿⣿⡏⠉⠉⠀⠀⠀⣀⡀⣀⣒⣒⣴⠺⣥⡿⣿⠶⠞⠛⠫⠉⠑⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⣦⣦⣤⣤⡤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠬⠽⠷⠶⣿⣿⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡼⠛⠛⠁⢠⣤⡤⠤⠄
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⣉⡉⠀⢀⣀⣀⣀⣉⣉⣉⣀⠀⠄⠀⠀⠀⣉⡥⣤⣄⣓⡙⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣶⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⢋⡄⠀⠀⠀⠀⢲⡴⠋⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⢃⠀⠿⢿⣿⣿⣦⡾⠁⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⣿⡥⢾⠀⢠⣶⡇
⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠈⠳⠀⠘⠟⠁⣿⠇⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠙⠻⢿⣧⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣷
⠀⠀⠀⠀⣰⠿⠁⠀⠈⠉⠀⠀⠀⠀⠈⠈⠓⢀⡀⠤⠤⠄⠻⠀⠀⠀⠸⠹⡤⠀⠀⡐⠀⠔⠈⡇⠀⠀⠀⠀⠀⠲⠈⠀⢀⣀⡀⢻⡆
⠀⠀⠀⢠⠏⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢰⡆⢛⠶⠄⠀⠀⠀⠀⢀⠑⠀⠀⣶⣄⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⠠⣍⠡⣦⡁⣀
⠀⠀⠂⠈⢢⡈⠛⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠰⠋⣠⡎⠀⠀⠀⠘⢿⡀⡠⠖⣁⡠⣴⠊⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣶⣶⠟
⠀⠀⠐⠗⠤⣅⣀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⢈⣿⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⣰⣿⡿⠁⣠⡾⠁⠀⠉⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⣿⠏⢀⣴⠏⠀⠀⢀⣼⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⣿⡟⠁⣠⡾⠁⠀⠀⣠⡟⠁⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣾⣿⠋⢀⣴⠏⠀⠀⢀⣼⠋⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣿⣿⡟⠁⣠⡞⠁⠀⠀⣰⡟⠁⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⢀⣼⠋⠀⠀⢀⣼⠋⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠟⠁⠀⠀⣰⠟⠁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⢀⡾⠋⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣰⠟⢁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⡾⠋⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣴⠟⢁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⢠⡾⠃⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⣴⠟⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⣿⣦⣄⣠⡾⠃⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠛⢿⣿⣦⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣀⣤⣶⣾⣿⠿⠛⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋