Consume Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠂⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠖⠉⠉⠀⢀⢔⠀⠔⢁⠀⠈⠦⣀⡀
⠀⠀⠀⡴⠂⢉⠄⠉⠔⠁⡔⠀⠀⠈⠀⣄⠈⠢⠀⠀⠀⢍⢢
⠀⠀⢸⠠⠊⡡⠂⠀⠖⠊⡤⠒⠁⠀⠀⠈⠙⠐⠢⡀⠂⡌⠀⡀
⠀⠀⡜⠁⠀⠀⠐⠒⠒⠊⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠲⠐⠀⢠
⠀⠀⣤⡊⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠣⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠂⠤⣀⡀
⠀⠀⢠⠈⠋⠀⠀⠀⠈⡃⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢂⢧⡺⣖⡇⣦⡄⡀⢔⠄
⠀⠀⠈⢂⠸⣄⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢧⡄
⠀⠀⡈⠀⠀⠈⠳⣄⠀⠀⠙⠒⠀⠀⠐⠁⠀⠀⠀⣠⢎⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠈⠀⠉⠉⠈⠀⠉⠒⠠⡤⡤⠤⡤⡠⠄⠒⢭⠤⣨⠡⠃⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠂⠀⠀⡠⠂⢰⠪⡕⣒⡥⠇⡄⠀⠀⢀⣄⠀⠀⡔⠉⠐⡄⠀⠀⠀⣄⠀⠀⡰⠂⠀⠀⠀⠀⢀⡳⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡀⠤⠈⠀⠀⣌⢀⠀⠀⠀⠁⡆⠀⠀⢨⠻⠂⠀⠀⠀⠀⢸⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⣘⠘⠚⠀⠐⠂⢉⠘⠀⠀⠄⡆⢇⠀⢀⡎⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⠀⣀⣀⣠⠴⠊⠁
⠀⠀⠀⠰⡅⠀⠁⠂⠀⠀⠃⠠⠤⣤⣀⡤⠬⠶⠎⠀⠀⠤⠤⠤⠤⡀⠀⢠⡄⠀⠀⠀⠀⣀⠀⣤⠏⠀⠋⠇⡠⠊⠉⡄
⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⠧⠈⣉⣉⡁⠀⡔⠀⢀⠀⠀⠘⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠠⠀⣀⠴⠀⠦⠜⢤⠷⠀⠀⠀⢀⠔⢂⠀⠰⠀⠘⠉⣑⠠⡐⠏⢀⠒⠂⠸⣅⠴⡋⢠⡤⣴⢏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠠⠀⠪⠈⠁⠀⠀⠀⠘⡠⠃⠀⢹⡤⠭⠤⠤⠤⠤⠤⠤⠯⠤⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠉⠐⠄⠈⠀⠀⠀⠠⠄⠉⠀⠐⣔⠳⠀⠤⠠⠔⠒⠒⢒⠶⠶⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢠⠀⠠⠊⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⠃⡜⠁⠐⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⠂⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣈⡂⣰⠁⠈⣉⠉⠉⢈⡇⠰
⠀⠀⠀⠀⠀⢐⠀⠴⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⡆⠀⢤⠀⠀⠀⠖⠸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠈⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠉⡀⢠⡤⠄⠀⠠⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠀⡀⢄⠀⠀⠀⠀⡀⠀⠘⠀⠀⠢⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⢦
⠀⠀⠀⠀⢀⡀⠠⠄⠶⣒⣀⣠⣤⣬⢬⠭⠩⠉⠉⠉⠉⠉⠉⠁⠈⠉⠩⠭⠿⠿⣿⣦⣄⣠⣄⣰⣀⣀⣼⠤⡄⠀⢀⡀⠈⠁⠘⣦⠈⢧
⠀⠀⠀⠀⢸⠙⠋⠀⠐⡒⠶⣤⣤⣤⣤⠴⠶⠌⠉⠉⠀⠩⠬⣍⣁⣄⣥⠬⠥⠥⠴⠖⠒⠀⠀⢠⣤⣴⣶⣾⡇⠛⠉⠀⠀⠀⠀⠈⡁⢘⡄
⠀⠀⠀⠀⢸⣤⣦⠀⠀⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣠⡽⣿⡇⠀⠀⠀⠀⠄⠀⠀⠤⠀⠇
⠀⠀⠀⠀⢸⠋⡇⠀⠀⠀⠀⠉⠉⠙⠛⠀⠂⠀⠀⠀⠀⠚⠷⠖⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡏⠉⠉⠉⠀⠀⠈⢭⡄⢸⠇
⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣦⡀⠀⠁⠀⠀⠰⠀⡼
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡖⢤⣀⠐⠁⣼⠃
⠀⠀⠀⠀⢸⢏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣢⣲⡲⠂⠀⠀⠀⣿⡄⠸⣿⡇⠀⠑⢤⡾⠁
⠀⠀⠀⠀⢸⠓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢰⠄⣿⠃⠀⣿⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠠⣄⣈⡁⣨⠤⠀⣿⡇
⠀⠀⠀⠀⢸⡂⡇⢐⡀⠀⠀⠀⣀⣤⠤⠄⠀⠀⠀⠀⠈⠉⠉⠁⠰⣦⣀⢀⡉⠉⠉⠉⠀⠉⠀⠈⠉⠈⠁⣿⣸⠀⣿⡇
⠀⠀⠀⠀⢸⡀⡇⠸⣷⣄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣠⠂⠀⠀⠀⠀⠀⠀⣐⣳⡀⠀⣿⣿⠀⢿⡇
⠀⠀⠀⠀⢸⡁⡇⠀⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢡⠁⠀⠀⠀⠀⢤⡀⠐⠮⠄⠀⢻⣿⠀⢹⡇
⠀⠀⠀⠀⢸⠁⡇⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢽⡂⠀⠀⠀⠀⢒⠀⠀⠢⣀⠀⢸⣿⠀⢼⡁
⠀⠀⠀⠀⢸⠅⡇⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡗⠈⠀⠀⠀⠀⢀⣺⣗⠄⠀⠀⠀⢀⡀⠀⠠⢽⠀⠈⢸⠀⢻⡇
⠀⠀⠀⠀⢸⡇⡇⠀⢸⡿⠂⠤⠀⣀⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⢰⣯⣼⡧⠀⠀⠀⠲⠄⠈⢹⣯⠀⠀⢀⠀⢼⠂
⠀⠀⠀⠀⢸⡇⡁⠀⠀⠁⠨⢁⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣠⣅⠀⠇⠀⠀⠀⣀⠀⠀⢈⣿⠀⢸⣿⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠛⣉⣽⠀⢸⣶⣷⣤⣀⣀⠻⠿⠟⠁⠈⠛⠛⠁⠀⠂⠀⠀⠉⠠⠄⣿⣿⠀⢸⣿⠅⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⠀⣸⠀⣈⣛⣆⠀⠀⠀⠛⠗⠤⠄⠀⠀⠖⠒⠀⠀⠤⠀⠀⠀⠀⣿⣿⡂⢸⣿⣑⣿⡄
⠀⠀⠀⠀⢸⡇⡀⠀⠉⠛⠿⠿⣿⣿⠀⣿⣿⣿⡾⣿⣥⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⡇⢸⣿⢀⣿⠅
⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠐⣂⠙⠿⠀⣾⣿⣿⣇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⢻⣿⠃⠀⢽⠐⣿
⠀⠀⠀⠀⢸⡷⣾⠉⠀⠀⠒⢻⡿⠀⠀⠷⡿⣼⡁⠀⠀⣉⣉⣁⣀⣀⣀⣠⣀⣀⣀⣀⣤⠄⠄⠀⠈⢁⡀⠀⠸⢠⣿⡆
⠀⠀⠀⠀⢸⡗⢺⠃⠀⠀⣀⣬⣥⣤⢀⢤⣤⣾⣶⣶⣶⣦⣄⣀⣀⣀⡀⠂⠒⠒⠶⠶⢦⠤⠀⢀⣠⣼⡗⠀⢠⣼⣿⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠡⠸⠿⠷⠿⣛⠿⣷⣿⣿⣿⣷⣿⣿⣶⣿⣏⣩⣭⣥⣤⣤⣴⣶⣤⣶⣾⠿⠿⠿⠷⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠛⠛⠛⠛⠛⠋⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⢠⡶⠀⠀⠀⠀⠶⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⣿⡏⠀⠀⠀⠐⠋⠀⠀⢸⠈⠉⠀⠀⢸⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡙⠿⠀⠀⠀⢀⠀⠢⣄⣼⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⠀⠀⠀⠴⢿⡀⠀⠀⠁⠀⣠⠀⠀⡾⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡓⠛⠲⠼⣧⡀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⢠⠇⠀⠀⠀⢸⣗⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠛⣶⡾⢻⠛⣦⡀⠀⠀⠠⠀⡆⠄⢀⣴⠋⠀⠀⠀⢔⢺⡿⢾⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⡾⠏⠀⢸⠀⠀⠙⣿⠲⡶⠶⡖⠛⢏⢻⠷⡄⠀⠀⢠⠺⣔⡈⠱⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠦⠗⢶⣬⠀⠀⣼⠏⢠⠃⠀⢵⢀⠈⠈⢧⡙⣦⡀⠀⢱⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡙⢸⠀⡇⣿⢘⠷⣼⠋⠀⠀⡤⡀⠀⢎⡷⠹⡎⢧⣧⠟⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⡘⡟⢇⢸⠀⡇⣿⢸⠀⣿⡀⠀⠀⢳⠱⠞⠈⠁⣱⠥⡄⠱⡄⠀⡞⠀⢸
⠀⠀⠀⠀⠀⢠⡇⣗⢸⣮⠀⡃⣿⢨⣤⡾⠃⠀⣼⢈⡇⠀⣀⡤⠖⠋⠀⢠⣇⢰⠁⠀⢸
⠀⠀⠀⠀⠀⢸⡇⣿⢸⢻⣄⣥⡟⠋⠀⢳⣀⣠⡿⣶⠁⣼⣁⣠⢾⠻⣖⠋⠈⠛⠀⠀⣼
⠀⠀⠀⠀⠀⣿⡇⢿⢸⢸⡏⡟⢷⡀⠀⠘⣧⣁⡟⠁⡸⢹⢹⢱⡟⠀⠈⠳⡤⠔⠊⠈⠉⠓⠒⠠⣄
⠀⠀⠀⠀⠀⢻⣇⢸⢸⢸⠇⡇⠘⢷⡀⠀⠘⠋⠀⢰⡇⢸⢸⣼⠇⠀⠀⣰⢥⣉⣉⡉⠤⠬⠒⠈⢸⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢸⡟⠛⠿⠾⠶⣦⣼⣬⣿⣄⠀⠀⢀⣯⣤⣼⣾⣿⣴⣾⣿⠋⢧⡀⠀⠀⠀⠀⢀⣠⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣀⠉⠓⠒⠒⠊⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠠⠖⠺⠷⠶⢤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⣦⠈⠙⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣿⣧⣀⡀⠀⠀⠈⢳⡀⢹⣀⣨⣿⣿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⢦⡀⠀⠀⢻⣾⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣄⣴⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⡿⢛⣛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⣂⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡔⣥⠒⢺⠀⡄⠈⢢⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣤⠀⠙⠉⠀⢸⣼⠀⠒⢦⡗⢰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⢿⡅⠀⠡⠶⠒⠒⣺⡈⣶⡎⢡⢰⡹⡁⠀⠜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢙⠣⠤⠄⠤⠤⠒⠓⠶⢿⣟⠘⢁⠐⠒⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣤⠀⠀⠙⠂⣀⣿⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡂⢀⠀⠠⠟⠗⠀⠤⣤⣤⣿⢟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢻⡈⠀⠀⠀⠐⠒⠒⠀⠈⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⡞⠀⠒⠂⠐⠒⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢸⡇⠀⠀⠀⠀⠀⠏⢰⡥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠾⠳⠦⠤⣤⣴⠤⠆⠘⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣤⣷⢶⣶⠀⠀⠀⠀⢀⢀⣤⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡠⠖⠉⣤⡺⠭⢝⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣁⣀⠤⠒⠉⠀⠀⠀⢸⣽⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡟⠴⠞⠂⠀⠚⠒⢃⠞⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠇⠠⠆⠀⠀⠠⠀⠹⢋⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⢄⠀⠉⠀⡨⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠀⡄⢄⢀⠀⠀⠐⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢱⡄⢀⠷⠀⠀⠀⠀⠀⢀⡜⡀⡸⢀⠀⠉⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢘⡾⠄⠀⡀⠀⠀⢸⠀⠠⠀⣠⣴⣶⣦⠀⠐⠀⢫⠉⠂⠂⠚⠓⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⠬⡤⣞⣒⣒⡺⢮⣠⣿⣇⢈⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠄⠄⠈⠁⠄
⠀⠀⠀⠴⢳⠀⠀⠀⠐⡁⢘⣶⡄⣶⣶⣶⡄⠉⠉⠉⠀⠀⠈⠀⠄⠀⠀⡀⠀⡀⣉⠀⠀⠄⠁⠀⠘⢦⡀
⠀⠀⠂⠒⠺⠷⠀⠤⠀⠰⠏⠻⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠧⠉⠉⠀⠀⠀⠰⠃⠞
⠀⠀⠀⠀⠀⠀⠰⢴⢒⣲⡂⠀⠉⠛⠛⢁⣀⣀⣀⡤⡤⣤⣀⣀⣀⠠⠤
⠀⠀⠀⠀⠀⢰⣶⣿⠫⠈⢱⡆⠐⢸⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡷⠁⠄⡀
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠙⢦⡄⡘⠉⠉⠉⠋⠋⠋⠋⠉⠉⢉⣁⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠢⠬⠤⠥⠜⠉⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⡼⠃⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⣱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠀⠀⠀⠀⠀⡐
⠀⠀⠀⡀⠀⠢⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣠⣤⣆⠏⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠈
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠀⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡀⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠆
⠀⠀⠀⠈⠂⠈⠉⣉⢀⣠⡤⠆⠂⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠐⠀⠀⠀⠀⢀
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⠴⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠂
⠀⠈⠆⠁⠆⠀⠤⠄⡔⠂⠉⠁⡤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⠅⠀⠀⡇⠀⠀⠒⠂⠙⠋⠘⠃⠁
⠀⠀⠈⠐⠓⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⠀⠔⠀⢐⠀⡇⠀⠀⠁⠀⢠⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠌⠀⠀⠢⠀⠠⢍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠠⠀⠀⠘⠀⠐⠊⠉⠀⠀⠠⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⡇⡁⢇⡡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠁⠀⠉⠁⠘⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠓
⠀⠀⢠⠦⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⢠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠁⠀⡀⠀⠀⠀⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⢆⠀⠾⠡⠄⠆⡱⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠤⠤⠛⠀⣀⣥⠀⠀⠀⠀⠀⠀⠐⠛⠐⠘⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠂⠀⠐⠒⠁⠀⠀⠀⠀⠀⢰⠀⢈⠈⠉⠉⣉⢉⡉⠀⢹⠀⠀⠀⠀⠀⠀⡀⢠⠀⠐⢬⡀⠀⠀⠘⠀⠀⠀⠀⢀⠚⠉⠀⠀⠆⠔⠀⠁⠓⣄
⠀⠀⠀⠀⠥⣬⠈⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠊⠜⠃⠨⠀⠄⠂⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠒⠂⠀⠆⠐⠔⠂⣸
⠀⢰⠀⠀⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⣉⡁⠘⠋⠁⢀⣀⠔⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠃⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢰⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠋⠧
⠀⠀⠀⠀⠐⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠰⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠁⠄⣲⣦⣴⡔⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣠⠟⠛⢻⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢁⣿⡯⠛⠏⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⣸⣋⠀⠀⣸⣼⠀⠀⠀⠀⠀⠀⠐⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠸⢿⣷⣭⣤⣿⠿⡀⠀⠀⠀⠀⠀⠀⠈⣐⣷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣰⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢈⡿⠤⠤⠤⠼⠾⢿⡆⠀⠀⠀⠀⠀⠀⠀⠘⡃⡏
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠳⠄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⢠⡄⢤⣤⣤⣄⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⢸⠀⠈⠿⠿⠡⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠲⠤⠄⠰⠆⠀⠀⠀⠀⢀⡁⠀⠀⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⣜⠀⠀⠀⠀⠀⣾⣆⣀⣀⣄⣈⣉⣩⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢙⣿⡗⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢰⡠⠀⢤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⡂⠀⠄⠀⠀⠀⠀⠈⠉⠉⢫⠍⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣏⠀⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⢸⡿⡿⠧⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠦⣄⣒⡛⠯⠁⢀⣀⠤
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠰⠛⠉⠀⢁⠐⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⣠⠂⠀⠀⠀⠀⡀⠀⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡀⠀⢰⠃⠀⠀⡀⠀⠘⠿⠂⠌⠻⠃⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⡇⠀⢠⠶⠦⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣭⣉⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⢸⠀⠀⡞⠰⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠋⠀⢀⠎⠀⢰⣆⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⣯⠀⠀⠀⢿⣷⣦⣬⣂⠠⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠹⡀⠀⠀⠈⠁⠀⢉⠋⠁⠠⠀⢀⠀⠀⠀⢀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠀⠈⠀⠀⠈⠑⠒⠒⠂⠁⠐⡄⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⡀⠀⡀⠀⡠⠀⢄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠐⢯⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡼
⠁⠀⠀⠨⣀⣤⠀⡀⠀⢇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡸⠀⠀⠀⡏⠰⠖⠂⠛⠬⠉⠙⠦⠐⠂⠈
⠀⡄⠀⠀⠀⣄⠀⠁⠐⠈⠻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠀⠀⢀⠔⠠⠤⠀⠀⠒⠒⠁
⠀⠀⠀⠀⢰⡀⠳⣀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⣿⣄⠀⠉⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠹⡁⠀⠂⠄⢈⠉⠒⠢⠤⠤⣀⣀⣀⣀⣀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁⠙⡤⢀
⠀⠀⠀⢀⠀⠂⠉⠛⢄⠈⠑⠠⢀⠀⠀⠀⣸⠤⠄⠄⠐⠂⠀⠀⢀⠈⠁⠈⠔⢄⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢀⣹⠄⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡱
⣀⣀⣈⣑⣂⣀⣤⣄⣄⣊⣁⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠭⣟⣛⣛⣛⣛⢛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢡⣶⣄⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⢠⣴⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡀⠀⠀⢉⣿⣥⣄
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⣿⠿⠋⣿⠿⣿⣶⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⣤⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⣧⡀⢀⣤⣿⡇⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠰⣶⣶⠆⠀⠀⠀⠀⠈⠻⡿⠟⣿⣿⣿⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⣿⣷⣶⣿⣿⡷⠶⠶⠶⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠷⠶⢶⡶⠾⠷⠶⠶⠶⠆⣠⣤⣤⣤⣤⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⢐⠟⠛⠛⠒⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠿⠿⠿⠿⠿⠅⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣶⡤⠀⠀⠀⠈⠿⠿⠃⠥⠄


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⣀⣤⣾⣿⠇⠶⢄⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄⠔⠃⣾⣿⡷⠞⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠷⣾⣽⣿⣿⣶
⠀⠀⠀⠀⠀⠘⡟⣿⣿⣿⣿⣟⢛⢛⣛⣛⠛⣤⣤⣬⣿⣥⣤⣤⣤⣤⣧⣤⣤⡤⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⣽⣿⣿⣷⣬⣯⣭⣭⣤⡄
⠀⠀⠀⠀⠀⠀⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠹⢷⣿⣿⣿⣿⡯⠉⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⡿⠀⠀⠀⢀⣠⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣿⣿⣿⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⠇⠀⠀⢐⡃⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⠀⢀⣀⣀⠀⠀⠀⠒⠛⠂⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⡄⠀⠀⠘⠿⠟
⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠐⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠰⠀⠀⠀⠀⢀⠀⡴⠴⢦⡖⣖⢲⢒⡆⢲⠂⣰⣴⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠐⠒⠒⠒⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠈⠀⠓⠓⠚⠀⠁⠘⠒⠓⠘⠀⠉⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡤⠀⡀
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢰⣶⣜⡆⠀⠀⠀⣦⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⢿⣿⡿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢐⡀⣂⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⣠⣭⣄⠀⠀⠀⠀⠀⠈⣙⣋⠉⠹⡄⠀⠀⠀⣠⡄⠀⠿⠂⠀⣤⣀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠸⠿⠇⠿⠿
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢿⣿⣿⠿⠀⠁⠀⠀⠀⠈⠁⠰⣶⠀⠀⠉⠉⢶⡦⠀⠀⠀⠸⣿⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡠⢤⣄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣧⡴⠁⠀⠈⠻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠐⠃⠀⠀⠀⠀⠆⠈⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣄⡀⢀⣀⣀⣀⣀⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣧⣄⣛⣛⡻⢿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠛⡯⠀⢀⠁⡄⠏⠸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠈⡇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⡆⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⣽⢛⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣹⣿⠿⠋⠉⣀⣖⣩⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣴⣶⢶⡶⠛⠛⠃⠀⢠⣄⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠏⠁⠀⠀⢀⣤⣴⣶⣤⡀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⡐⠀⠀⠀⡂⠀⠀⡀⠀⠀⠀⠀⠀⣴⣶⣶⣿⠇⠀⠀⠀⠀⠸⠿⠩⢨⢹⠇⢶⣿⣿⣿⠟⠟⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠀⣾⣿⣷⠀⠀⠀⠰⠿⢿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠲⠔⠂⠘⠛⠋⢁⣶⣿⣤⠘⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⡀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣉⠁⠀⠀⠺⢿⣶⠟⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠉⠉⠻⢨⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠃⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⢨⣿⠇⠂⢠⡄⢰⠗⠀⢿⡿⠂⠀⠀⠂⠀⡀⠀⠻⣿⠿
⠀⠀⠀⠀⣠⣶⣶⣶⣤⡖⠀⠀⠀⠀⠀⢀⣀⡀⢀⣘⠁⠀⠀⠀⠀⠀⠀⠐⠂⢸⣿⡄⢀⡀⠐⠀⠀⣰⣦⣤⣤⣇⡀⠀⠀⣀⣄
⠀⠀⠀⢰⣿⣿⣿⣿⡿⢁⡄⠀⠐⠚⢿⣿⣿⡃⠚⢡⣾⣿⣧⣄⠀⢠⡀⠶⠂⠀⣤⣶⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠠⣿⣿⡇
⠀⠀⠀⠈⢿⣿⡿⠟⠡⠿⠀⠀⠤⣤⡀⠀⢩⣷⡀⢻⣿⣿⣿⣿⠀⢸⣿⡆⢈⡀⢸⣿⣧⠲⣤⡀⢸⣿⣿⣯⣴⣶⡆⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⢀⠌⢀⡀⠀⣦⣤⣀⣀⣀⣈⣻⣃⣀⡛⠛⠟⠃⢀⣾⣿⡇⠛⠁⠀⠛⢉⣴⣮⣅⢸⣿⣿⢹⣿⣿⠇
⠀⠀⢀⠄⠀⠠⢂⠐⠙⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡷⠟⠀⢰⡟⠋⠀⣿⣿⣿⣿⠈⣿⡟⠀⣿⡟
⠀⠀⠀⠀⡀⠀⢀⠀⠘⢀⠉⠛⡛⡻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⣠⣥⣤⡈⠙⠛⠛⠁⠠⠭⠁⠀⣍⠥⠤⠾⢿⡆
⠀⠀⠀⢠⣀⡀⠀⠀⢠⣁⡀⢸⣩⠟⢀⣰⣦⣶⣿⣿⣷⣄⠀⠀⠀⢾⡆⠀⣿⡉⢿⡅⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠈⠁
⠀⠀⠀⢤⡭⠀⠀⠀⠀⠉⠉⠺⠿⠶⠈⣿⣿⣿⣿⣿⣿⣿⡄⠘⠳⠌⠁⠀⢿⣿⣦⠅⠠⠎⢠⣾⠟⣋⣽⣿⠇⢠⣾⣧⣼⣷⣆
⠀⠀⠈⣛⡆⠛⠒⢤⡴⠂⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠠⠶⢀⠀⠀⠺⠿⠿⠀⠀⠀⠘⠿⠿⠛⠉⠁⠀⠈⢹⣿⣿⡟⠃
⠀⠀⠀⢡⢞⣡⠄⣿⡇⠀⠀⠀⠀⠁⣿⡀⠉⠋⠉⢉⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢁
⠀⠀⠀⠀⣭⠵⠆⠘⠃⠀⠀⠐⠂⠀⢛⠛⠀⢠⡆⢸⣿⣿⠀⠀⠀⠀⠠⡶⠶⠶⠶⢶⣴⠀⠀⠀⠈⠁⠀⣄⠀⢠⣭⣥⣶⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⠄⠸⡄⠀⠘⠁⢸⣿⣿⠘⠃⠀⢠⣤⣤⡀⠀⢠⣦⡶⠶⢶⣶⣶⣤⡀⠀⠀⢸⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣴⣇⠀⠀⠁⠀⠰⠇⠘⣿⠛⢀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣶⣾⣿⣿⣿⣿⡇⠀⣿⡇⠀⢹⣿⣤⡶⠛
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⣿⠀⠈⠀⠀⢿⣿⣿⠅⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠰⠾⠉⠉⠉⠙⢿⡿⠋⠁⠠⠼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⢀⣴⣿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⢹⣿⠀⠀⠀⠀⠰⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠦
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠃
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣄
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⣄⣀⣠⣾⡿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠋⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣤⣄⣀⣀⣤⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠛⠋⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⢀⣠⣶⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡖
⠀⠀⠙⢿⣿⣿⣿⣷⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⢀⣶⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡀⠀⠀⠀⡼⠂⢀⣀⡀⠀⡀⣤⣴⣿⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣼⣷⣶⣠⣴⣶⣾⣷⡖⠀⠀⢷⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⠹⣿⡿⢢⣬⣶⣤⣀⣴⣡⣼⣟⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡰⠋⠀⠈⠛⠛⠛⠛⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡧⠛⠀⠀⠀⠀⠀⣈⣁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣟⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠆⠉⠂⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⡹⠀⣈⠀⠘⠀⢀⠈⠳⠸⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⡀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠐⢚⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣀⠀⠀⠠⠇⠀⠀⣤⢀⡤⡶⠆⠈⠐⠀⠄⠨⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠌⠈⠀⠀⢰⠀⢂⢠⡀⣄⣸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠰⠛⠧⠕⢶⣤⣈⡀⢇⠃⠘⢀⣳⣌⣶⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠁⢙⣻⣶⣶⣿⣿⣿⣿⣿⡿⡛⣃⠉⣜⢧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⠋⣙⢿⣿⣿⣟⡟⠛⠙⠃⢀⠈⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣴⡿⠿⠿⠟⢉⠛⢿⣿⠆⠀⠀⠀⠀⠀⠄⣸⣿⣿⣾⣿⢧⠘⣯⣹⣿⡄⠀⠀⠈⢣⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠟⣀
⠀⠀⢰⠋⢶⣴⡶⣷⣿⡧⠒⣼⣧⠀⠀⠀⠀⠀⠘⡏⣴⣿⡟⠿⣿⣷⣿⣿⣿⣷⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⢺
⠀⠀⠀⠀⠀⠀⠀⡉⣁⠀⢃⡼⣿⡆⠀⢀⠀⡀⣠⣻⣿⠛⠛⠟⢿⣿⣿⣿⡯⣅⢙⡛⣲⣿⣿⣿⣿⠟⢋⣿⣿⣟⣾⠛⠉⠉⠈
⢠⣤⣤⣶⣶⣤⡄⠀⠀⠀⠀⠐⢿⣷⠀⠞⢸⡷⠟⠛⠻⢶⣴⣶⣿⣉⣠⠄⠰⠈⠹⣟⣿⣿⣿⠿⡉⠐⣌⣿⣿⣿⣇⠱⡄
⢸⣿⣿⣿⠟⠋⠁⠀⠀⠀⠌⠠⠟⠠⠤⠴⠂⠂⠀⠲⠰⠬⢻⠋⠘⣉⣀⣀⢒⡂⠀⡀⠙⣿⣿⡄⠙⣆⠈⢷⣾⠛⢻⣦⣿⣾⣿
⠺⠿⠟⠡⡞⠀⣠⠐⠁⠀⢐⠁⠶⠆⠐⠒⠒⠀⠓⠚⠒⢶⣌⡀⠀⠉⢉⡁⠈⣉⣠⡉⠛⣿⣿⣿⣦⡈⠁⢾⡁⠀⠈⢻⣿⠿⣿
⠀⡎⠀⢠⠁⠚⢁⣠⣤⣦⠞⠛⠂⠐⣀⠀⠉⠀⠉⠁⢘⡋⠋⣴⣄⠀⠀⢠⡄⠉⣡⡶⠞⠋⠉⠉⠉⠙⢶⣤⣤⡴⠚⡉⠁⢀⣾
⢰⠴⣂⣼⠛⠉⠉⢁⡀⠀⠐⠀⠐⠂⠾⢷⣶⣤⣶⠾⠿⠛⠶⠵⠾⠿⢷⣾⣿⡿⠁⠐⠒⠒⠒⠀⠄⢀⡀⠙⢿⡀⠋⢀⡄⠉⠠
⢨⢐⣿⠃⠐⠒⠀⣁⡀⠄⠆⠒⠚⠃⠀⡸⠟⠉⡀⠠⠀⠤⠀⠠⠄⠀⢀⡀⠉⢷⡀⠀⠲⠠⠄⢀⡀⠀⠉⠲⠄⠑⣎⠁
⠁⢻⠇⠀⠀⠘⡋⢉⡠⠤⠔⠚⠓⠀⣰⠁⠀⠈⢀⡤⠠⠀⠠⠀⠄⢤⡄⠀⠀⢈⣷⡀⠀⠀⠀⠀⠉⠉⠁⠠⠀⠀⠈⢷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠒⠞⠛⠓⡖⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠈⠢⠄⣁⣉⠠⠔⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⢰⠁⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠐⢪⠀⠀⠸⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠇⠀⠀⢇⠀⠈⠛⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠢⢀⠀⠀⢀⣠⠃⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⠴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⢀⣀⠀⠀⠈⠢⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡠⢋⡆
⠀⠀⠀⠀⠀⠀⢣⠭⠅⢀⠀⠀⣠⠤⢄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠂⠉⠁⠀⠐⣀⣀⠠⠄⠂⠀⠉⢧⣀⠅⡆
⠀⠀⠀⠀⠀⠀⠀⠒⠮⡠⠔⠊⠀⠀⠀⠈⠁⠐⡄⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⡀⠄⠂⠑⠒⠚⢶⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠀⠐⠒⠒⠀⠤⠄⢀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⢀⠔⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠈⢣⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠀⠀⠑⡔⠁⠀⠀⠀⢀⣀⣱⣀⡠⠒⢤⣄⢀⣐⣀⠀⠀⠀⠀⠀⢣⠎⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⢀⣔⠉⠡⢭⡖⠚⠋⠉⠁⠀⠙⢛⢤⣬⡠⠍⠑⡄⠀⠈⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡘⠀⠉⠐⠖⠂⠠⠤⠤⠤⠤⠤⠤⠀⠐⡖⠈⠁⠈⢆⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠠⠀⠔⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⠈⠁⠀⠀⠀⠀⠛⠀⣠⡀⣼⣟⢋⠁⠀⣤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⣿⣷⣄⠀⠀⢸⣿⣿⣿⠋⠉⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣰⣿⣿⡿⣷⣾⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠘⢲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⣿⣿⣿⣿⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡴⠚⣏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠀⠀⠈⠛⠻⠋⣰⣿⠿⠿⠿⣿⣿⣀⣀⣠⣤⡀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⠀⠃⠀⠘⠀⠀⠀⠀⠀⡞⠈⠙⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠻⠿⠿⠛⠛⠻⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡍⠐⠶⣦⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠐⠶⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠉⠀⠤
⠀⠀⠀⠈⠁⠀⠀⠹⣷⣤⣤⣬⣿⣿⣶⣶⣦⣄⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣥⣤⣤⣶⣶⠿⠃⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⠁⠘⣿⠀⠀⠈⠛⣿⣧⠀⠀⠐⣾⣿⣿⣿⣿⡟⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢤⣿⣀⠀⠀⠀⠸⣿⣦⣴⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⣿⣿⠇⠀⣽⣿⣿⣇⠀⠀⣿⣿⣿⡏⠉⠛⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠢⣴⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠁⠀⢿⣿⡄⠀⠙⠿⠿⠃⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠻⣿⣧⣤⡤⠀⣀⠴⢻⠋
⠀⠀⠀⠀⠰⣦⣿⡿⠋⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⢠⢤⣆⡠⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠁⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⣤⠀⠀⠀⠀⠀⠉⢻⡿⠂⠀⠀⣠⠟⢆⠀⠀⠀⢿⣿⣤⣴⣶⣿⠿⣿⣤⡀⠀⠙⠣⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠈⠁⠀⢀⣶⣿⣿⣿⣷⣦⣼⣿⣿⣿⠆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢼⡷⣾⣿⠿⠟⠹⣿⣿⣿⣿⣿⣏⣛⣁⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⣦⠔⠀⠠⠄⠀⠦⠀⠀⠀⠀⠀⠀⠹⡿⢿⣿⡟⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣿⡇⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠿⠁⠀⠀⠘⣿⣷⣤⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣁⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠖⠖⠛⠓⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⢨⣇
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠑⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣠⣄
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠂⠔⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⣞⣰⢺⡤⢶
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡼⠉⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠁⠈⠛⢾⡄⠛⠲⡆
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢀⡠⠔⠂⠁⠀⠀⠀⠀⠀⢀⣀⣀⢳⣦⠴⠃
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡤⠞⠀⣻⠉⠁⠚⠛⠛⠋⠙⢻⠛⣏⠉⠉⢥⣿⢣
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⣀⡀⢹⠐⡀⠄⠀⠀⠀⠀⡞⠀⣸⠀⣐⣿⠃⠘⡇
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠂⢿⣧⣭⠵⠶⠶⡾⠀⠀⢿⣿⠿⠋⠀⢀⡇
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⣄⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣀⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠉⠉⠉⠙⠓⠲⠾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⢤⡄⠀⢀⡜⠁⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⢶⡟⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⠒⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠶⠤⣄⣠⡗⠲⠶⠦⠤⢤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⣃⣠⣤⣴⣶⣿⠶⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠮⠋⢨⠆⢫⡀⢻⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠙⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠛⠛⠒⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠠⢤⠌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⢀⠀⠐⠿⣷⣄⣀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡇⠈⠀⠾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢸⣾⣡⣼⠙⣿⣷⡄⠘⠿⣷⣄⢀⣈⣱⡄⠀⠀⣼⠶⣶⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘⠀⠀⠀⠲⠀⡀⠈⣿⡁⠀⠘⣿⠷⡤⢾⢻⣿⣂⣿⣃⣁⡀⠀⣱⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⡐⠀⠀⠙⣶⣿⣿⠀⢀⡟⠀⢰⡄⣿⠋⠀⠡⠿⣿⠛⠋⠀⣤⠙⠛⠛⢦⢄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠻⢆⠂⠺⣿⣿⠀⠻⡇⠀⢸⣿⡏⠀⠀⠀⢰⣿⡇⢤⠸⡏⠀⠀⠀⠄⠈⣟⣋⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠈⢣⡀⠈⢿⠀⠀⣷⡄⠀⢻⣇⠀⢀⣴⣾⣯⣿⣦⡸⣓⠀⠀⠀⠄⠀⠈⣉⣸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠦⣄⢳⣴⠏⠀⠒⢬⣿⠉⠉⢻⣟⠀⣀⠈⣿⠉⣉⠑⢶⡄⠀⠀⢘⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡀⠀⠀⢀⣁⣤⣷⣼⣁⣰⣿⡟⣭⣶⣿⣄⣿⠃⣤⣄⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠘⠃⢠⡀⣿⡀⠁⣮⣽⣯⢻⡏⠁⠀⢻⡀⠸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢷⣀⣷⣤⠖⠀⣇⢰⣂⣉⣀⣈⣹⣉⣀⣠⠽⣽⣿⣾⣷⣿⣤⣾⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣴⢙⣿⠃⠀⠀⠀⠐⠀⠀⠀⠀⠈⢼⣿⣿⣷⣤⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠓⠦⠀⠀⠀⠀⢠⠀⠢⣄⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣤⣆⢱⣾⡋⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣠⣤⣿⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣴⣆⡹⠷⠋⠉⠉⠛⣙⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣿⣿⡋⠙⠻⡏⠉⠉⠉⠉⠋⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡀⠿⠇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⡿⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠍⢉⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠈⣟⠘⢿⡗⠶⠰⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡧⠀⣿⡇⢈⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣼⣧⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠰⠀⠸⠁⣀⠰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⣿⣿⠋⠻⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠶⣧⣤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣿⣦⣤⣤⣤⣄⠀⠲⠆⠀⠀⠀⠀⠉⠛⢦⢤⣤⣤⣤⣤⣶⢠⣤⣤⣤⣠⣄⣀⣤⣤⣶⠟⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠹⣿⣷⠒⢤⣤⣤⣤⣀⣠⣼⠷⣟⠛⠛⠛⠉⠋⠙⠛⠿⠿⠻⠿⠿⠿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⣿⢿⡿⠟⠛

⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⣄⢠⠀⠀⡀⠀⣀⢠⡄⠀⣠⢲⡄
⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⣿⢰⢄⡇⣠⢂⡟⢸⠀⣶⡇⣿⢰⠀⣾⠁⣴⢧⠄⣷⢸⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⡜⢋⣿⠁⢸⣇⡾⡷⠻⠺⣿⡾⠞⣿⡼⠿⠊⠸⡧⠚⠃⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⡏⢸⠇⠀⢉⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠠⠔⣾⠆⢀⡀⠀⠀⣴⡾⣷⢻⠇⣷⣸⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣇⣿⠀⣾⠏⠀⠀⠸⠇⠛⢸⡼⢉⡝⣻
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠆⠀⠀⠀⠉⠉⠛⢻⣧⠋⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⢠⣟⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣮⢰⠄⢠⣏⠀⠀⣀⢀⢄⠀⢀⢄⢀⡀⠀⢰⡆⣠⣏⡰⣏⠀⢸⡟⡤
⠀⠀⠀⠀⠀⠀⠠⣿⡇⣾⢠⡟⠹⢲⡟⣿⠇⣾⢀⣿⠞⣸⡇⣿⢈⢻⡏⠨⠁⣼⠃⣼⣿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠇⣿⠸⢧⠄⢸⡇⣿⠄⣿⣘⠙⠚⠉⠛⠙⠋⠈⠛⠁⠀⣿⠜⠀⠛⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠈⠁





⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠉⠉⠀⠀⠀⣀⠀⠈⠉⠙⠻⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠐⠿⠀⠀⠀⠛⠃⠀⠀⣶⡆⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠐⠿⠂⠀⣀⣠⣤⣴⠶⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠙⣿⣷⣤⣤⣶⣶
⠀⠀⠀⠀⠀⠀⣼⣿⠃⢠⣤⡀⠀⢀⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⢻⣶⣤⣤⣴⡶⠿⠛⠋⠉⠀⠀⢻⣇
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠈⠉⠀⣰⠟⠁⠀⠀⠀⠀⣀⣠⣤⣴⡶⠿⠛⠛⠉⣀⠀⣤⡀⣾⡆⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⣸⣏⣀⣤⣤⡶⠾⠛⠛⠉⡁⠀⣤⠀⢰⡄⣿⡙⠃⢻⣷⣹⣷⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢸⣿⣀⣠⣴⡶⠾⠟⠛⠉⣁⡀⣤⠴⢶⡟⢸⣿⡄⢻⡄⠘⣷⠈⠛⢷⡜⣯⢻⡞⣧⠀⠀⠀⠀⣿⡄
⠀⣤⣶⠾⠟⠛⠋⢁⣠⡀⢠⣶⡀⢻⡏⣿⢹⡆⠘⣇⠈⣧⣿⡌⣷⣀⣹⡎⠻⠾⠃⠉⢀⣁⣤⣴⣶⡾⠟⠛⠁
⠀⢻⣇⠀⠀⠀⠀⢿⡉⠟⠀⣿⢷⡘⣿⠋⠀⣿⠀⢻⡄⠿⠈⠛⠙⢉⣁⣤⣤⣶⡾⠟⠛⠉⠉⣹⣿
⠀⠈⣿⡀⠀⠀⠀⠸⣧⢰⣆⣿⠛⣷⠹⠆⠀⠘⢁⣀⣠⣤⣴⠶⠟⠛⠋⠉⢠⡿⠁⠀⠀⠀⠀⣽⡟
⠀⠀⢹⣧⠀⠀⠀⠀⠙⠞⠃⠈⣀⣠⣤⣶⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⣤⠀⠀⣼⡿⠁
⠀⠀⠀⣿⣄⣀⣀⣤⣴⡶⠿⠛⠋⠉⠙⠷⣦⣄⣀⡀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠉⢀⣾⡿⠃
⠀⠀⠀⠘⠛⠛⠉⠙⢿⣷⣀⠀⠀⠀⠀⣀⡀⠉⠉⠛⠛⠛⠛⠉⠁⠀⠀⠺⠗⠀⢀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠈⠛⠃⠀⠀⣶⡆⠀⠀⠸⡗⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣷⣦⣠⣀⣀⣈⣀⢀⣀⣀⣠⣤⣶⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⣤⠀⣤⡀⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠙⠒⠚⢳⠱⠀⢸⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠴⠃⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡇⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠉⢹⠀⠀⢰⡃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠛⠉⠙⣆⠀⠀⠀⠀⢸⡇⠀⣠⣄⣀⣀⣤⢸⢠⠀⠀⠑⠦⠤⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⣆⠀⠀⠀⣀⡧⠾⣷⠒⠛⠓⢾⠃⠞⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣪⠵⢚⡠⠖⠋⠁⠀⢀⡼⠦⠤⠶⢺⠀⠀⠀⠀⠀⣠⠋⡥⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣠⠶⡟⠢⣀⠀⣠⠛⡶⢤⡀⠀⠀⡇⢸⢠⠎⡥⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⠤⠖⠉⠀⠀⠙⢆⠈⠉⢁⡜⠀⠀⠈⢦⠀⠁⠈⠈⠁⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⠀⠀⠀⠀⠀⠑⢤⠎⠀⠀⠀⢰⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠉⠀⠉⠁⠉⠈⠀⠀⠀⠈⠉⠀⠉⠈⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠒⠀⢄⣀⣀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠈⠁⠀⠁⠀⠒⠴⠀⠁⠀⠀⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠴⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠀⠐⣚⢀⠀⠀⠀⠀⣈⠘⠁⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⣀⠀⠀⢠⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠀⠉⣀⡀⠀⠄⠈⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣠⡔⠂⠈⡉⢤⠀⠀⠀⠀⠀⣀⣉⢀⢀⠀⣄⡀⠈⠆⡘⢄
⠀⠀⠀⠐⠈⠤⠶⠶⠄⠤⠀⠉⠀⠀⠲⠂⠀⠀⠀⠉⠀⠈⠁⠀⠀⠁⠁⠤⠁⠀⠦⠨⠽⠯⠩⠴⠆⠀⠰⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢤⣀
⠀⠀⠀⠀⠀⢰⣶⣷⣀⣤⡿⠿⠃⠀⠀⠀⢀⡏⣉⣁⣠⣤⣶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠍⠀⠂⠀⢀
⠀⠀⢀⠀⡄⢺⣿⣿⣿⣿⡇⡀⢠⠀⠀⠀⠘⡇⠈⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠶⠀⣶⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠠⣾⣾⠇⣠⡄⠻⠇
⠀⠀⠉⠛⠋⠀⠈⠁⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⠛⠁⠉⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠁⠉
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢰⣦⠀⣾⡷⡆⠀⠀⠀⢀⢸⣧⡤⢠⣽⣄⢠⣿⡄⢀⣈⣯⢀⠀⠀⠀⢴⣶⡶⢧⣤⠀⠀⠀⢤⣴⣦⠀⠀⡀⠀⠀⠀⠰⢷⣄⠀⢠⣤
⠀⠀⠈⠿⣿⣿⣷⡆⡶⡇⠀⠈⠹⠿⣿⣾⣿⣿⣿⣿⣿⣿⡾⠿⠃⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣾⣷⡆⠀⠀⠀⣾⣿⣶⣿⣿⠆
⠀⠀⠀⠈⠀⠁⠈⠀⠁⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀
⠀⠀⠀⠁⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣙⣿⠁
⠀⠀⠀⣴⣾⣿⣿⡷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⢖⣀⠀⢀⠘⣧⣆⢀⡾⠃⠠⣐⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠟⢹⣿⣿⣧⠀⢀⣿⣿⣧⣴⣿⠀⠀⠀⢿⡛⠛⠀⠒⣿⣿⣿⣿⣶⡆⠀⢸⣿⠃⠀⠀⢀⣿⣿⣧⣾⡖⠀⠀⠀⠀⠀⣸⣿⣿⣤⣶⡆
⠀⠀⠀⠀⠸⠿⠿⠿⠀⠘⠻⢿⡟⠛⠻⠀⠀⢀⡼⢹⡄⣤⡈⠿⠿⣿⠿⠿⠇⠀⢸⢹⠀⠀⠀⠸⠿⣿⠿⠿⠇⠀⠀⠀⠀⠀⠿⢿⡿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠒⠋⠐⠄⠤⡄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠐⠎⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠠⠀⠀⠀⠀⠀⠒⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⢀⡐⠒⠀⠀⠀⠒⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠈⢠⡄⠀⠀⠀⠒⣀⠀⠀⠁⠇
⠀⠀⠀⢀⡀⣀⣀⣠⠂⠀⢈⠠⠀⠀⠀⠀⢀⣉⠀⣀⠀⣄⠀⠱⣘⠄
⠀⠀⠀⠓⠑⠒⠒⠀⠀⠈⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠛⠃⠐⠒⠈⠒⠐⠒

⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⠀⢀⢀⣀⣀⢀⠀⢀⢀⡀⣀⡀⣀⠀⣀⢀⣀
⠀⠀⠀⠀⠈⠁⠻⠾⠧⠟⠀⠃⠜⠻⠇⠸⠃⠯⠫⠀⠸⠟⠁⠚⠣⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⢮⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠁⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠐⠐⢷⠆⢀⠌⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠃⠈⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠂⠀⣰⣴⣤⡀⠀⠀⠈⢂⣠⠁⠄⠀⠔⠁⢀⠀⠀⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣀⡀⠀⠈⣢⠠⠄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣰⣿⣿⣿⣦⠀⠀⠀⠹⡄⠈⠀⠀⠀⡜⠀⠀⢠
⠀⠠⣀⡄⠀⠀⠀⠰⠁⠀⠀⡞⠁⠉⠂⠀⡇⠀⡠⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠉⠹⢿⠀⠀⠀⠸⠀⠀⠀⠀⡰⠁⡰⠀⠄
⠀⠀⠘⣥⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠀⠀⠀⠀⠈⣠⠞⢁⠌
⠀⢘⠀⠀⠠⡗⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠈⠢⠀⠀⠀⠀⠀⣠⢞⠀⠀⠀⠀⠀⣈⢠⠐⠁
⠀⠈⡰⢖⠀⠈⠐⠀⠿⠦⣄⠀⣀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡠⠀⠚⠀⠲⠶⠒⠋⠁⠀⠀⠉⠀⠙⠉⠃⠄
⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠊⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢬⣊⣀⣠⣀
⠀⠀⠀⠘⢦⡈⠒⠤⢤⣀⠐⠒⠀⠀⡀⠁⠁⠀⢄⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠋⠀⠀⣀⣀⣠⡤⠤⠤⠶⢶⡡⠄⠈⠢⠝⠊⠁⢹
⠀⠀⠀⠀⡀⠙⠒⠦⡤⠤⠤⠔⠂⠀⠀⠀⠀⠀⠀⠑⣀⣤⣀⣤⠤⠽⠶⢦⣤⣄⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠁⠀⠀⠀⢃⢋⡂⣠⣤⠤⠴⠶⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢴⠛⠍⠣⠀⠝⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠘⡄⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠀⢀⠀⠀⡎
⠀⠠⡀⠀⠀⠱⡀⢂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⣊⠆⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠊⠀⠂⠀⡸⠠⠂⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢳⡄⢆⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⢇⠐⠠⠀⡀⠀⠄⠁
⠀⠀⠀⠀⢠⠂⠂⠀⠀⠰⠀⠄⠀⢈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠁⠀⠁⠂⡄
⠀⠀⠀⠀⡊⠀⠀⠀⢰⡎⠁⠀⠂⠀⠀⠀⠀⠀⠈⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠞⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠐⠤⠊⣿⠀⠢⡈⠢⡀⠠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⠔⠈⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠂⢈⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣨⠔⠁⠀⠀⡀⣁⢹⠇⠀⠀⢀⠀⡀
⠀⠀⠰⣄⡽⢄⡀⢀⣠⠀⣀⠀⠠⠤⠵⠦⣌⡀⠀⣁⡀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡨⠔⠛⠀⠀⠤⠤⠠⠝⠙⠁⢄⠀⢀⣠⠀⠀⠈
⠀⡀⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠈⠀⠈⡁⠠⣖⣒⣒⣒⣀⡀⠤⠔⠒⠉⡡⠤⠈⠐⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠄⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠄⢆⣀⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠀⠀⢈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠀⠈
⣿⢿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣛⣿⣟⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠉⠉⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⠈⠀⠄⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⡶⢻⣿⠀⠀⠀⠘⣦⠀⣴⠃⠀⠀⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠻⠶⣽⢀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣿⣿⠋⢙⣻⣿⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⣿⢼⣿⡆⠀⢤⣾⣿⠹⣿⣦⣄⣀⠀⢀⣀⣤⣶⣿⣿⢿⣿⡆⠘⠻⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡔⠙⠛⠁⠀⠁⣼⣿⣷⣿⣿⣿⣿⡀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠋⢑⣿⣿⣿⡀⠀⠀⣿⣿⣿⠿⡿⣷⣄
⠀⠀⠀⠀⠀⡼⠋⠀⠀⠰⣎⡻⣇⣸⣿⣿⣿⣿⣿⡟⣇⠀⠀⢉⡠⢖⣦⡙⠛⠁⠀⣸⡟⠻⢿⣿⡀⢼⣿⣿⡟⠁⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠰⡇⠀⠀⣤⣤⣬⣿⣾⣿⣿⣿⣿⢿⣿⡋⠙⣦⡀⠀⢿⣿⣿⠗⢀⣀⡾⠉⠈⣠⣿⣿⣿⣾⣿⣿⣟⣀⣤⣤⣤⣽⣿⣷⡄
⠀⠀⠀⠀⠀⢹⡦⢤⡼⣿⣿⣿⣿⣿⣿⠋⠡⡈⠛⢿⢦⣌⡳⣤⠀⠉⣀⣴⣿⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠲⠦⠀⢿⣧
⠀⠀⠀⢴⠻⣾⣿⠮⣾⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣾⠿⠟⠩⠤⠟⠛⠉⠭⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⡷⢒⣶⣿⡟
⠀⠀⠀⠸⡇⢠⣿⣆⢺⡥⠿⢛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⠿⣷⣦⠠⣿⣼⣿⣿⡀
⠀⠀⠀⠀⢴⣼⣿⣿⣷⡶⠋⠹⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⠀⠀⠀⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⢀⠀⢰⣾⣿⣿⣿⣿⠚⠁⢴⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⡄⠀⠀⠘⠛⠿⣶⣤⣀⠀⠀⠀⠘⠃⠃⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⡶⡀⠀⠀⠀⠆⠉⠙⠛⠟⠳⠶⠧⡤⠴⠶⠶⠛⠛⠛⠉⠉⡻⢻⢻⣿⣿⣿⣿⣿⣿⡟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡛⣸⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣿⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣦⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣶⣶⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠈⠁⠩⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠂⠀⢲⣾⣿⣿⣿⡿⠀⢻⣿⣿⣿⡿⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⢀⠀⣽⣿⣿⣿⣿⠁⠀⢸⡿⢾⡋⠙⠉⠛⢙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠎⠰⣴⣿⣿⣿⡟⠀⠀⣼⠿⠟⠀⠀⢠⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⢸⡇⣿⣿⣿⠃⠀⠀⢿⣆⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⢾⣿⣿⠏⠀⠀⠀⢸⣿⠆⠀⠈⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠦⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⣳⣦⣤⣀⣀⣶⡿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡍⢠⣤⡼⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠆⠶⠀⠀⠠⠦⠤⢠⠄⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠛⠛⠛⠛⠛⠋⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⢠⡶⠀⠀⠀⠀⠶⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⣿⡏⠀⠀⠀⠐⠋⠀⠀⢸⠈⠉⠀⠀⢸⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡙⠿⠀⠀⠀⢀⠀⠢⣄⣼⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⠀⠀⠀⠴⢿⡀⠀⠀⠁⠀⣠⠀⠀⡾⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡓⠛⠲⠼⣧⡀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⢠⠇⠀⠀⠀⢸⣗⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠛⣶⡾⢻⠛⣦⡀⠀⠀⠠⠀⡆⠄⢀⣴⠋⠀⠀⠀⢔⢺⡿⢾⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⡾⠏⠀⢸⠀⠀⠙⣿⠲⡶⠶⡖⠛⢏⢻⠷⡄⠀⠀⢠⠺⣔⡈⠱⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠦⠗⢶⣬⠀⠀⣼⠏⢠⠃⠀⢵⢀⠈⠈⢧⡙⣦⡀⠀⢱⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡙⢸⠀⡇⣿⢘⠷⣼⠋⠀⠀⡤⡀⠀⢎⡷⠹⡎⢧⣧⠟⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⡘⡟⢇⢸⠀⡇⣿⢸⠀⣿⡀⠀⠀⢳⠱⠞⠈⠁⣱⠥⡄⠱⡄⠀⡞⠀⢸
⠀⠀⠀⠀⠀⢠⡇⣗⢸⣮⠀⡃⣿⢨⣤⡾⠃⠀⣼⢈⡇⠀⣀⡤⠖⠋⠀⢠⣇⢰⠁⠀⢸
⠀⠀⠀⠀⠀⢸⡇⣿⢸⢻⣄⣥⡟⠋⠀⢳⣀⣠⡿⣶⠁⣼⣁⣠⢾⠻⣖⠋⠈⠛⠀⠀⣼
⠀⠀⠀⠀⠀⣿⡇⢿⢸⢸⡏⡟⢷⡀⠀⠘⣧⣁⡟⠁⡸⢹⢹⢱⡟⠀⠈⠳⡤⠔⠊⠈⠉⠓⠒⠠⣄
⠀⠀⠀⠀⠀⢻⣇⢸⢸⢸⠇⡇⠘⢷⡀⠀⠘⠋⠀⢰⡇⢸⢸⣼⠇⠀⠀⣰⢥⣉⣉⡉⠤⠬⠒⠈⢸⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢸⡟⠛⠿⠾⠶⣦⣼⣬⣿⣄⠀⠀⢀⣯⣤⣼⣾⣿⣴⣾⣿⠋⢧⡀⠀⠀⠀⠀⢀⣠⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣀⠉⠓⠒⠒⠊⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠠⠖⠺⠷⠶⢤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⣦⠈⠙⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣿⣧⣀⡀⠀⠀⠈⢳⡀⢹⣀⣨⣿⣿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⢦⡀⠀⠀⢻⣾⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣄⣴⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⡿⢛⣛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⢀⡤⠾⠶⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠶⠖⢶⠈⠧⠀⠀⠀⠀⠤⣾⠀⠠⠀⢸⡦⠤⠀⠀⠀⠸⠁⠀⢀⡤⡿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠴⠋⠀⠀⢦⠀⠀⠀⠀⠈⠳⣤⡤⠞⠀⠀⠀⠀⣰⠀⠀⠀⠸⣄⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⡽⠞⢳⡀⠀⠀⠀⠀⠈⠃⡴⠚⠉⠉⠙⠓⠒⠶⢶⡖⠃⠀⢸⠋⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⢳⡀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡟⢉⣉⣉⣉⣉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣀⣬⢷⡀⠀⠀⠀⣇⣼⢳⡞⠳⢤⣤⠼⣿⢻⡀⠀⠀⠈⠛⠛⡇⠨⠭⠍⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡭⠖⠋⠀⠀⢀⡀⠸⡇⢸⡇⠀⠀⠀⠀⡏⢸⡇⢀⣀⠀⠀⠀⠛⠛⠓⠚⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠤⠀⠶⠒⠀⠛⠋⣁⣨⣥⠴⣇⣸⠇⠀⠀⠀⠀⣧⣸⠧⢤⣭⣀⡉⠙⠁⠘⠒⠂⠰⠶⠤⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⣠⣾⠿⣄⠀⠀⠀⣠⠿⣿⣦⡀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⠴⠋⠁⠘⢧⡈⠙⠲⠖⠋⣰⠃⠀⠙⠲⣄⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡓⠶⠚⠋⠁⠀⠀⠀⠀⠈⠳⣄⠀⢀⡴⠃⠀⠀⠀⠀⠀⠉⠓⠲⠖⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⣤⠀⠀⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⠀⠀⠀⣠⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠖⠲⠂⠖⠲⠒⠓⠚⠲⠲⠒⠲⠂⠒⠢⠺⠒⠒⠒⠒⠒⠒⠂⠐⠺⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣃⢸⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⡏⠀⠀⠀⠀⢀⡠⢄⢴⠊⠉⢱⡀⡏⠉⠀⣁⢀⠖⠤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠰⡁⠀⠈⢎⡆⠀⠀⢣⠡⠀⠀⢠⡌⠀⠀⡰⡰⠁⠉⢢
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⠀⣧⣤⣤⣤⣀⣱⡀⢺⠁⠘⡀⠀⠀⣾⠀⠀⢸⠁⠀⢠⠛⠳⡄⢀⠎
⠀⠀⠀⢰⡉⢖⠉⠉⠀⠀⠀⠘⠤⠃⠀⠀⠀⠉⢁⡣⢉⣇⠀⢇⠀⠀⢹⠀⠀⡍⠀⠀⠸⠀⡸⠀⡎
⠀⠀⠀⢸⠪⠭⢉⣛⣒⣛⠛⠚⠶⠚⠛⣛⣚⣛⡉⠭⠑⡟⡀⢸⠀⠀⢸⠀⠀⠇⠀⠀⡇⠀⠇⣼
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠧⣀⡇⠀⠸⠀⢸⠀⠀⠀⣃⡴⠊⠸
⠀⠀⠀⠀⡇⠀⠀⠀⣼⢷⡀⠀⠀⠀⢠⡾⣦⠀⠀⠀⢸⡀⠀⠈⠉⠒⠓⠒⠚⠒⠒⠉⠁⠀⠀⡇
⠀⠀⠀⠀⢡⠀⠀⠈⠁⠀⢁⣴⣶⣦⡉⠀⠈⠁⠀⠀⡏⡇⠀⠞⠋⠳⠀⠀⠀⠀⠞⠉⠳⠀⢰
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣿⡟⣻⠁⠀⠀⠀⠀⢠⠄⢻⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⢀⣀⠀⣀⠀⢼⢀⣸⡀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⡝⡤⠖⢀⣉⣛⠫⠽⠿⣓⠒⠶⣶⣶⣶⣶⣤⣬⡿⣍⡓⠛⣉⣐⡶⣚⠭⠭⠓⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡴⠛⢒⡒⢛⡒⠢⠤⠤⠒⢬⠲⣦⣄⣢⣶⠖⠥⠦⠶⠶⠛⠛⠛⠚⠲⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⢲⠋⠀⠀⠰⠃⠈⠁⠺⡄⠘⣀⠘⢣⡄⠰⠂⣰⠂⡤⢸⡇⣨⡃⠘⠁⠀⠀⠀⠱⡖⠁
⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠆⠤⠤⠤⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠤⠤⠄⠒⠒⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣥⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⣠⠄⠘⠦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢺⣁⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠂⢀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠉⠀⠀⠘⡅⠉⠁⠉⠀⠀⢀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⢴⡟⠃⠀⡀⡀⠀⠀⠰⠌⠢⡀⠀⢦⣤⣾⣯⣿⣿⣿⣿⣾⣄⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⡾⠗⣤⣤⢄⡸⠀⢀⣠⠀⠀⠈⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣝⠙⠻⠾⢶⡷⠄
⠀⢠⡀⠀⠀⢛⠩⠽⠿⢤⣽⣿⡟⠛⠃⠀⢀⣱⣟⣿⣭⣽⣿⣿⣿⠘⣏⣷⣤⡆⠀⠈⠃
⠀⠸⠷⣀⡴⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⢉⣉⣉⡙⠛⢿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣠⣤⣾
⠀⠀⢠⠟⠂⣄⠀⠀⠀⠀⠀⠀⠀⣰⣠⠤⠤⢴⣿⣷⣭⣤⠤⠾⠿⠿⠿⣿⣿⣷⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⢴⣄⣀⣠⣿⡇⠀⠀⠀⠀⠀⠀⣾⣛⡋⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣼⣯⡁⡀⠄⠀⠘⣿⣿⣿⠁
⠀⠘⣻⣿⣿⣿⣿⣤⣀⣀⣤⣤⡀⠾⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣹⡏⠀⠀⠀⠀⣽⣿⡟
⠀⠀⢐⣨⣽⠏⠜⠀⡟⠈⠉⠀⠀⠀⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠛⠻⣇⠙⠿⣷⡄⠀⠀⠀⠛⠉
⠀⠀⣿⣿⡿⠀⠀⣰⠀⠀⡠⠂⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣷⣶⣿⠃⠠
⠀⠀⣿⣿⣇⠀⢸⣿⣀⠴⠁⠀⠀⠀⠀⢀⡜⢀⣾⣿⣿⣿⣿⣿⣿⣧⠀⣿⡏⠙⠙⠓
⠀⠀⣿⠋⣈⠉⠛⠛⠿⠲⠲⠂⠰⠒⠛⣉⢩⢸⣿⣿⣿⣿⣿⣿⣿⢟⣠⣿⣿⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢿⠃⢹⣷⣶⣶⣷⣶⣶⣶⠖⣳⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⠋⠋⠛⠍⠋⠉⠉⠈⠉⠁
⠀⠀⢸⠆⢸⣿⣿⣿⣿⣿⣿⡧⢀⠘⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡄
⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠇⠀⠠⣟⢻⣿⠿⡻⠻⠿⠿⠿⠓⠂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠈⠉⠉⠁⠀⠰⢀⡂
⠀⠀⠀⠀⠀⢀⡔⠈⠲⡖⠛⠉⠉⠉⠒⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄
⠀⠀⢀⡀⢀⠐⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⢠⠈⠀⠀⠢⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⡀⢀⣀⠀⠀⠀⠀⠉⠢⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡱⢄⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠈⠀⠀⠀⠀⠠⠁⢠⣀
⠀⠀⢠⠌⠈⠁⠀⠀⠑⠠⠀⠀⠀⠘⠦⠀⠀⠀⠀⠉⢉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡒⠀⠃⠄⠠⠀⠒⠇⠐⠁⠀⠂⡄⡀⠠
⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠂⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢀⠀⠀⠐⢄⠀⢀⡀
⠀⠀⠣⠀⢀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠄⠀⣀⣤⠁⠔
⠀⠀⢠⢳⡏⣄⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢇⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉
⠀⢠⠁⠀⠐⢨⡀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡞⠄⠀⠘⠒⠒⠙⠲⣄⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃
⠀⠀⠀⠀⠀⠀⠁⠁⠤⠀⠀⠀⠀⣀⡀⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⣶⠀⡐⠐⠒⠋⢁⠐⠉
⠀⢀⠀⠀⡀⢀⠜⠚⠢⡢⠑⢀⣠⠝⢃⠘⠀⠀⢀⠀⠀⡀⠀⠀⠠⠤⠐⣁⠀⡀⠠⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠙⠒⠊⡡⠀⠀⠁
⠀⠀⠁⠑⠸⢏⠀⠀⠀⠀⢀⠀⠀⠀⠘⡛⠒⠄⠠⢴⠀⠒⠀⠀⠔⠚⠋⡟⠁⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⡄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠑⡄⠀⠀⠁⡀⠀⢀⠀⣀⠀⣀⠀⣀⠀⢀⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⠆⣌⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠌⠃⠀⠠⢤⣍⠉⠀⠑⠠⡔⠉⠀⠀⠀⢨⠬⡃⠈⠅⠂⣨⠿⠭⡼⢭⠥⡓⢉⡙⠄⠀⠠⠎⠁⠢⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠠⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠣⢀⡀⠤⠒⠉⠉⠂⠀⠀⠀⠀⠀⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠢⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠠⠤⠀⠂⠉⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⡄⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠄
⡇⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀
⠁

⣷⣷⡶⢾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣚⠉⠉⠉⠉⠉⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠛⠀⢖⠀⢸⡁⠉⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠙⢦⠀⢈⣑⣸⠀⠀⠇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡞⠁⠀⣳⠀⡀⠸⣧⡠⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⣤⣼⣀⣀⣠⡃⢠⣷⠀⠙⡴⠟⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⠋⢱⠆⠉⠁⠀⠀⣾⣿⡇⠀⠀⠈⢹⠉⡗⠂⠤⣀
⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠁⠀⠀⢀⡴⠁⠀⠀⠀⠀⡼⠀⡠⡇⠀⠀⠀⡜⠀⡇⠀⠀⠀⠉⠒⠤⢢⡀
⠀⠀⠀⢠⡎⠁⠀⠀⠀⢀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡜⠉⢹⡅⡁⠀⠀⠀⠁⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠓⠦⢤⣥⠥⠔⠚⠳⢄⡀⠀⠀⠀⠠⠊⠀⠀⠀⠑⠒⠤⠤⢀⣀⣀⡠⠎⠀⠀⣄⠀⠀⠀⠀⠀⠀⡼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡇⠀⠀⠀⠀⠑⢢⠀⠀⠀⡰⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢤⡄⣀⡏⠉⠁⠁⠠⡀⢀⣀⠠⢾⣄⣀⣰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡅⠈⡹⠀⠀⠀⠑⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡎⠉⠱⡀⣼⣆⠀⣠⣀⣇⠀⠀⠀⠠⣄⣀⣀⠀⢀⠎⠀⣀
⠀⡀⢀⠄⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙⠻⢷⣄⠀⢀⡈⢣⠹⣧⡀⠑⠷⣄⠀⠀⢸⡱⢌⠲⣁⣶⣙⠈
⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠒⠦⡀⠀⠀⠀⠀⠀⠉⠢⣄⢉⠧⡀⢙⠉⠓⣌⢻⠢⣀⠈⢳⡀⢸⠀⠀⠑⠄⣀⡤⠔
⠀⠈⢄⠀⠀⠀⣠⠆⠀⠁⠀⠀⠀⠀⠀⠈⠂⢀⠔⠀⠀⠒⠀⠀⠉⠂⠼⢾⠀⠀⠈⢢⠑⢄⠙⠣⢽⣾
⠀⠀⠀⠉⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠱⠎⣰⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣀⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⠿⣿⡟⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⣰⡿⠋⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⠈⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡏⠉⠙⠓⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⢀⣿⡿⠀⠀⠀⣰⣿⣿⣿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡁⠢⠛⣖⠁⣼⡭⠭⠛⠓⣒⠿⣿⣿⠏⣴⠁⠀⢀⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠏⠑⠋⣻⡚⠲⠶⢦⣤⠤⠭⢭⠭⢿⣴⡃⠀⠀⡞⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠋⢀⡀⠀⠊⢹⠃⠀⠀⠀⢀⠈⠃⠀⠀⢈⠑⡌⢻⣄⣾⢦⣄⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣤⣬⠁⠀⡏⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⣼⣟⡀⢨⣿⣿⣦⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⣽⣿⣆⢖⠛⠉⠀⠀⠈⠀⣀⣠⠀⠀⢠⡴⠋⠓⠮⢍⣛⠿⣆⢀⣻⣆⣀⣴⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢿⣿⣿⣿⣿⣷⡖⠋⠁⠲⠖⠚⠁⢀⠞⡽⠷⠂⠀⣤⠀⠈⣝⢮⣝⢿⣿⣿⣦⠳⡀⠀⠈⠉⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⢿⣿⣿⣿⣿⡷⢲⣤⡤⠄⣰⠃⣸⠀⢠⡆⠠⠼⡟⠁⠈⢳⡝⣯⣿⣿⣿⡇⢳⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠬⣉⡛⢛⡷⢉⣏⠙⠒⠁⠀⣿⠀⠀⠀⠀⢠⡁⠤⠤⠘⠛⢦⠙⣿⣿⡏⡼⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠓⠛⠚⠿⢦⣀⡀⠀⣸⠆⠉⠲⡆⠀⠉⣳⠆⠀⠀⠘⡇⢹⣏⠜⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡃⡠⠞⠳⠤⢶⢛⡩⠀⠀⢀⣴⣷⣿⡥⠒⠒⠛⠛⠃⣀
⠀⠀⠀⠀⢠⠦⣤⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⢰⣧⣄⡀⢄
⠀⠀⠀⠀⠀⠢⣌⠻⢿⠀⠀⠀⠀⠀⠀⠀⣀⠴⠤⠔⠢⣀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠛⠛⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢀⣤⣄⡀⢀⠴⠋⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠈⠑⠦⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣛⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠤⠴⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣶⢟⣫⣽⣭⠻⡿⠿⢿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢉⡠⠖⠛⠲⣄⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣭⣶⣿⣿⣿⠟⢼⣿⣶⣿⢿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠖⠋⠀⠀⢀⣠⣇⣈⣉⣀⡀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⢟⣫⣤⡀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣠⠖⠻⠿⣿⣿⣿⣿⠃⢈⣿⣿⣿⣿⣿
⠀⠀⠀⡖⠚⠛⠓⠛⠛⠛⠛⠒⠒⠚⠛⠛⠛⠒⢲⠀⠀⠀⢸⣿⣿⣿⢹⣷⣶⣿⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣿⡝⣿⣿⣿
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⣤⠛⠛⢳⡀⠀⠀⠀⢀⣴⢶⠆⠀⢤⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⢠⣶⣯⠀⢀⠀⠀⠀⠀⠀⢠⣬⣯⣴⣦⠀⠀⠀⠀⢠⣿⠀⠀⢰⠄⠀⠀⠀⠺⠿⢾⠀⢤⣸⠀⠀⠀⠀⠀⣠⣬⣯⣤
⠀⠀⠸⠿⠻⠿⠶⠀⠀⠀⠀⠀⠻⠿⢿⠏⠉⠀⠀⠀⠀⠘⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠿⠤⠤⠾⠆

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⠒⠢⡀⠄⠀⠀⠀⠀⠀⠀⢸⣯⠃⠀⠀⠀⠀⠀⢠⡏⠀⢨⠂⠀⠀⠀⠀⠀⡶⠉⠉⢀⣄
⠀⠀⢠⣶⣷⣾⣷⠀⠀⠀⠀⠀⣼⣴⣰⣾⡆⠀⠀⠀⠀⢀⢈⣀⣾⣷⡄⠀⠀⠀⠀⠸⢷⣾⣼⡀⠀⠀⠀⠀⢸⣗⣀⣀⣿
⠀⠀⠘⠿⠿⠂⠀⠀⠀⠀⠀⠀⠻⠃⠀⠻⠇⠀⠀⠀⠀⠀⠾⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠰⣿⡷⠀⣛⠿⠀⠀⠀⠀⠐⠿⣿⠿⠀⠀⠀⠀⠀⢀⠒⠺⣿⠟⠀⠀⠀⠀⠀⠀⢘⣿⣋⠀⠀⠀⠀⠀⢰⣦⠘⢿⠟
⠀⠀⣴⣾⣶⡿⠋⠀⠀⠀⠀⠀⢰⡞⠛⢷⡄⠀⠀⠀⠀⠸⠦⣼⣿⣷⡄⠀⠀⠀⠀⢰⣿⠛⢿⣧⠀⠀⠀⠀⠙⠉⡻⠿⡴⡂
⠀⠀⠙⠙⠛⠃⠀⠀⠀⠀⠀⠀⠘⠓⠒⠘⠃⠀⠀⠀⠀⠀⠀⠛⠛⠙⠃⠀⠀⠀⠀⠈⠛⠀⠙⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛

⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⠠⡀⠀⣠⣄
⠀⠀⢀⣶⣹⣟⠀⠀⠀⠀⠀⠀⣠⠄⡀⣶⣶⠀⠀⠀⠀⢀⣽⣇⣹⣫⡄⠀⠀⠀⠀⢠⢄⡀⡀⣄⠀⠀⠀⠀⠘⠋⣁⢛⣟⡀
⠀⠀⠚⢻⣿⣿⣷⠀⠀⠀⠀⠀⣿⡾⠓⠿⠿⠀⠀⠀⠀⢈⣿⣧⣽⣿⡆⠀⠀⠀⠀⣿⡿⠓⢻⣿⠀⠀⠀⠀⠀⢈⣤⣿⣿⣧
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠰⢖⠶⣠⡀⠀⠀⠀⠀⠰⠀⠿⠀⠆⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠄⢰⣗⠀⠀⠀⠀⢀⣠⠀⣀⣀⢀⣀
⠀⠀⣴⠂⣺⢀⠾⠦⠀⠀⠀⠀⠀⡔⣒⠢⠀⠀⠀⠀⠀⢠⣼⣟⠁⣘⠀⠀⠀⠀⠀⠸⢦⣤⢿⣦⠀⠀⠀⠀⠹⣦⣽⣟⣹⣟
⠀⠀⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢸⣧⣭⡼⠀⠀⠀⠀⠀⠸⠿⠿⠰⠿⠇⠀⠀⠀⠀⢠⣾⠻⢿⡟⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠦⠚⣋⣭⣥⡤⠶⠶⢦⣭⣭⣙⡓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⢖⣡⠴⠚⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠛⠶⣍⡻⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡶⢋⠀⠀⠀⠀⠀⠠⣶⡂⠀⠀⠀⠀⠛⠏⠀⠀⠀⠀⠀⠈⠛⢦⡙⢶⡀
⠀⠀⠀⠀⠀⠀⣠⢏⣴⠋⠀⠀⠀⠀⠀⠀⣀⢤⠴⠐⠀⠖⠶⠦⢤⣀⠀⠀⢴⠦⠀⠀⠙⣦⡹⣤
⠀⠀⠀⠀⠀⣤⢃⡾⠁⠀⠰⡾⠀⢀⠴⢺⠩⠐⡀⡀⠄⢂⢠⠀⠀⡈⠟⢦⣀⣤⣴⣾⣿⣿⣷⡘⡦
⠀⠀⠀⠀⢰⠇⠜⠁⠀⠀⠀⠀⣰⠹⠘⡈⡄⠆⢃⠱⠘⡈⣄⣦⣧⣶⣿⠿⢿⠛⢻⡀⢹⣿⣿⣷⠸⡆
⠀⠀⠀⠀⡞⢰⠀⠀⠀⠀⠀⡘⠁⠀⡀⠁⢀⣀⣠⣴⣷⣿⣿⢿⣿⡉⢇⠐⢤⣇⠀⠁⠀⢻⣿⣿⣆⢻⡀
⠀⠀⠀⢘⡇⣿⠀⠐⡾⠆⢠⠃⣃⣥⣴⣾⣿⡿⠛⢛⡇⠈⢻⡄⢻⣧⠈⡷⠢⠈⣆⣠⣰⣴⣿⡿⣿⢸⡅
⠀⠀⠀⢸⡇⣿⠀⣀⣤⣴⣾⡿⠿⣟⠉⡉⢣⠸⣷⠈⣧⠀⠀⢳⡈⠛⣣⣼⣶⣾⣿⡟⠛⠉⠀⠀⣾⢸⡅
⠀⠀⠀⢨⡇⣿⣿⣿⡟⢉⠙⣇⠀⠹⡆⠉⣸⡆⢹⣇⢸⣇⣿⣶⣿⡿⠿⢛⢩⠱⣸⠀⠨⠿⠀⠀⣿⢸⠆
⠀⠀⠀⠀⣷⢹⣿⣿⣷⠈⣿⢻⡈⣀⠹⣄⣿⣿⣶⣿⡿⡟⠏⠏⢡⠰⠘⠈⡄⢆⠏⠀⠀⠀⠀⢰⡏⡾
⠀⠀⠀⠀⠸⣆⢿⣿⣿⣧⣈⣠⣷⣿⣿⠿⢛⢻⠉⠀⡁⡀⢀⠀⠀⠀⡀⠁⡤⠋⢀⠀⠀⠀⢀⡿⣰⠃
⠀⠀⠀⠀⠀⠛⣎⢿⣿⡿⠟⠛⠉⡉⠑⢼⡌⡄⠆⢃⢁⠰⡈⡄⠆⣃⡵⠊⠁⠀⠺⠃⠀⠀⡘⣰⠋
⠀⠀⠀⠀⠀⠀⠉⣦⡻⣦⠀⠀⠸⠿⠀⠀⠀⠈⠚⠒⠒⠒⠓⠓⠉⢁⠀⠀⠀⠀⠀⢀⣴⢞⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡳⢤⣀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠻⠃⠀⠀⣀⡴⢟⣵⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⡳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⢟⣩⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⡭⣍⣛⣛⣛⣛⣛⣛⣭⠭⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡀⠀⠠⠅⠀⠀⡠⠂⠈⠢⡈⠁⠠⠄⠄⠀⠂⠀⢀⠴⠀⠐⠛⠻⠀⠸⠇⠄⠄⢄⠀⠀⠊⠅⠀⠀⡠⠊⠉⠠⢀⠇⠘⡄⢀⠀⠀⠀⢀⠴⠀⠐⠛⠳⠁⠸⠔⠠
⠀⠀⠈⠀⠐⠀⠂⠁⠀⠀⢀⠜⠀⠀⠖⠀⠨⠀⠠⡀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠑⠄⠀⢀⠘⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀
⠺⠂⠳⣀⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠠⠀⠠⠃⠀⠈⠠⠈⠀⡔⠓⡜⢂⠀⣾⠃⠺⢆⠽⣠⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠂⠔⢂⠀⣺
⠀⠘⠂⠁⠀⠀⡠⠐⠀⠁⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠈⠲⢪⡖⡛⠀⢙⠒⠂⠛⠛⠏⠀⠀⠠⠐⠀⠀⠐⠢⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠈⠶⢢⠔⡛⠀⢛
⠀⠀⠀⠀⠀⠀⡶⠤⣂⢴⡤⢴⡆⠀⡏⠀⠄⢩⠂⠀⠀⠆⠐⠈⡧⢾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠄⣂⢠⡆⢶⡆⠀⡍⠀⠈⢩⠂⢰⠀⠀⠀⠈⡳⢮⠃⠀⠈
⠀⠀⠉⠀⠀⠀⠉⠉⠷⠘⣣⠍⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠆⠀⠐⠂⣀⢀⡀⠁⠀⠀⠉⠀⠀⠀⠉⠉⠳⡜⣫⠝⠁⠀⢠⠀⠀⡀⠀⠀⠑⠀⠆⠀⠐⠂⢀⢀⡀
⠀⢀⠊⠀⠈⢢⠀⠀⠀⠈⠁⠀⠀⠀⠸⠄⠀⠄⠀⠀⢀⣲⠦⠀⠀⡁⠀⠀⡀⣥⠀⠠⠈⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⣶⠦⡀⠰⡉⠀⢀⡀
⠀⢨⠀⠀⡠⠂⢀⢀⣑⠄⢀⣴⠤⠂⠀⠀⠀⣦⣡⡾⠟⠁⠀⡀⠀⠀⠂⠀⠀⡀⠀⢀⡀⠀⡀⠂⠀⢂⡐⠄⢀⡕⠴⠠⠀⠀⠀⣤⣱⡾⠟⠋⠀⠀⠀⠁
⡄⢢⠘⠈⠀⠀⢈⠠⡔⠲⠵⠷⣄⠒⠀⠀⠈⢜⠁⠀⠀⠠⠊⠀⠀⠀⠒⠐⠀⠀⠄⠡⠚⠈⠀⠀⠄⠀⡴⠲⠾⠶⣄⠂⠀⠀⠀⢎⠀⠀⠀⠠⠊⠀⠀⠈⠂
⢀⡁⠀⣀⠀⠀⠈⢐⠂⠠⠄⢂⠄⢀⠀⢠⣂⠀⠀⠀⠀⠀⠘⠀⠇⡈⣄⠀⠀⠀⢀⡀⠀⣀⠀⠀⠐⢐⠂⠠⠄⢂⠄⠀⠀⣘⣀⢀⠀⠀⠀⠀⠸⠁⠇⡘⡀
⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡯⠇⠀⠀⢀⠀⡰⠂⠺⠀⠀⠠⠀⠌⠢⠰⠁⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⣠⡿⠇⠀⠀⢀⠀⡯⠒⠲⡀
⠀⠃⢀⠀⠀⠀⣠⠄⠒⠀⠲⠀⠐⠂⠁⠂⠈⠁⠐⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢀⠀⠀⢀⠀⠃⠀⢠⢄⡒⣀⢶⣄⠐⠂⠑⠐⠈⠁⠐
⠀⠂⠁⠐⠊⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡀⡀⡀⠀⠄⢀⡀⠀⠀⢀⠀⠀⠀⠒⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡆⣀⠠⠀⠒⣀⠁⠀⠦⢷⠀⠘⡉⠁⡙⠀⠀⠀⠉⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⣂⣀⠠⠀⠒⡀⠁⠀⢨⠷⠄⠘⠃⠉⠛⠀⠀⠀⠉⠀⠀⠉
⠉⠀⠉⢉⡄⠀⠀⠀⠀⢀⠨⠀⠁⢀⣀⡀⠄⠀⠠⠀⠀⡀⠴⣀⢦⠀⢀⡁⠉⠀⠉⠀⠋⢉⡆⠀⠀⠀⠀⠀⡤⠀⡃⠀⢌⡄⣄⠀⠃⠀⠂⠠⠴⣐⢦⠀⢀⡀⠉
⢉⣑⠈⢛⠄⠀⠀⡠⠊⠈⠐⡈⠃⠀⠖⠀⢀⠂⠀⣀⠖⠐⠒⠋⠺⠀⠰⡃⣀⣄⢢⣄⠨⢋⠖⠀⠀⡀⠊⠈⠀⢀⠃⠐⡔⠀⠀⠀⠀⠀⠔⠐⠒⠉⡳⠁⠘⡏⣑
⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⢀⠘⠀⠀⠔⠀⠰⠀⠨⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢋⠐⠀⠀⢀⠜⠀⠀⠆⠂⢀⠀⠈⡀⠀⠀⡠⠈
⠿⣆⡵⣠⠀⠀⠈⠃⠈⠃⠀⠀⠀⠀⠂⠀⠀⠃⠀⠈⠂⠊⠀⡐⡁⣜⢂⠀⣷⢃⡿⢂⠵⣠⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠂⠂⢀⡜⡁⣄⢂⠀⣾
⠒⠛⠈⠁⠀⠀⡀⠂⠁⠀⠁⠢⠀⠀⢀⣀⣃⣀⠀⠀⠤⠀⡀⠒⠢⢔⡚⠁⠙⠐⠓⠛⠋⠇⠀⠀⠀⠀⠉⠉⠁⠢⠀⠀⢀⣀⣂⣀⠀⠀⠤⠄⡈⠖⠢⢔⡺⠀⢋
⠀⠀⢀⠀⠀⠀⡷⢒⣒⢒⡒⢶⡇⠐⡅⠀⠀⢨⠂⢀⠀⠆⠀⠈⡗⠺⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣶⢒⣂⢒⡒⢶⡆⠐⠍⠀⠀⢨⠂⢰⠀⠀⢐⠈⡷⢮⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠌⠶⡜⣣⠍⠀⠀⢀⠀⠀⠈⠀⠀⠁⠒⠒⠀⠐⢂⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠌⠷⠜⣯⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⠒⠂⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⣐⣶⠶⡄⢠⣁⡀⣀⡀⡤⠀⠠⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣶⠶⡀⠰⣁⠀⣀⣀
⠀⢠⠀⠀⡀⠂⢀⢀⡠⠄⠠⡔⠴⠂⠀⠀⠀⣤⣦⡿⠟⠁⠀⠈⠀⠀⠂⠀⠀⡀⠀⢠⠀⠀⡠⠂⠀⢀⡈⠄⠨⡔⢤⠢⠀⠀⢠⣔⣤⡿⠟⠁⠀⠄⠀⠁
⡔⢢⠊⠀⠀⠀⠀⠀⠌⠒⣋⠶⡴⠒⠀⠀⠈⢊⠀⠀⢀⠄⠊⢀⡀⢈⠁⠂⠉⠀⣔⠠⠈⠁⠀⠀⠁⠀⡌⠒⣟⠧⢴⠒⠀⠀⠀⢎⠀⠀⠀⠠⠊⣀⡀⢈⠁⠂⠉
⢀⡁⠀⣀⠀⠀⠈⠜⠂⠰⠄⢀⠄⠀⠀⢠⡇⠀⠉⠁⠀⠀⠘⠒⠃⡌⣄⠀⠀⠀⢀⡁⠀⣀⠀⠀⠀⠈⠊⠡⠤⢈⠄⠀⠀⢘⡄⠀⠉⠉⠀⠀⠘⠒⠂⡬⣄
⠁⠈⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣨⠯⠇⠀⠀⢀⠀⡸⠂⠸⠀⠀⠠⠁⡀⠃⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⡀⠀⠀⣨⡿⠅⠀⠀⢠⠀⡯⠂⠒
⠀⠁⠀⠀⠀⠄⣠⢀⡔⠭⣳⠄⠘⠄⣉⣀⢈⡁⠜⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠃⠄⣠⢆⡔⢭⣳⢄⠘⠆⣉⣀⣈⡁⠜⠀⠀⠀⠁
⠀⠂⠀⠐⠀⠀⠁⠂⠀⠀⠀⠐⠇⠀⠀⡀⠂⠀⣀⣀⡀⠀⠄⢠⡀⠀⠀⢰⠀⠀⠀⠂⠀⠑⠀⠀⠡⠀⠀⠀⠀⠈⡆⠀⠀⠠⠂⠀⣀⣀⣀⠀⢀⢠⡀
⠀⣀⠀⠀⠀⠀⣎⠀⠄⠀⠀⢈⠀⠀⠲⢿⠀⠘⠅⠀⠜⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⣂⠀⠄⠐⠀⣀⠀⠀⢰⡷⠀⠸⠁⠀⠘⠀⠀⠀⠀⠀⠈⢀
⣁⣀⣉⣈⡀⠀⠀⠀⠀⠀⡈⣀⣁⢀⣀⣀⣄⠀⠠⠠⠀⣀⡒⣑⢢⡀⢀⣀⣁⣀⣉⣀⣃⣀⡂⠀⠀⠀⠀⢀⡤⣀⣃⢀⣌⣄⣀⠀⠃⠀⠂⣀⠐⣑⣦⡀⢀⣀⣁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣧⠇⠀⢈⣦⡤⠤⠖⠒⠒⠒⠒⠲⠦⠾⣅⣀⠀⢳⣷⣀⡀
⠀⠀⠀⠀⢀⡴⠚⠉⣽⠻⣧⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⡟⢹⣍⣓⢦⡀
⠀⠀⠀⢠⣿⣖⣺⠽⠛⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠙⠪⠭⣿⡆
⠀⠀⠀⢸⠘⢧⠀⠀⡼⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⡴⠃⣿
⠀⠀⠀⢸⠀⠀⠑⣶⠃⠀⠀⠀⠀⢸⠳⣄⡀⠀⠦⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣏⠀⠀⡿
⠀⠀⠀⠸⡆⣀⡴⠃⠀⠀⠀⠀⣠⠏⠀⠀⠉⢙⣛⠉⠉⣛⡒⠒⠚⠉⠓⢤⣀⠀⠀⠀⢈⡽⢠⡇
⠀⠀⠀⠀⢳⠘⣶⡤⠤⠴⠒⠋⠁⠀⠀⠀⠀⠉⣤⠀⠀⣤⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⡏⣇⡼
⠀⠀⠀⠀⠘⣿⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⡇
⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⡄
⠀⠀⣦⠤⠞⠁⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠳⠤⣤⠄
⠀⠀⠈⠳⣄⡀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢰⣖⣶⣶⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠒⠶⠶⠛⠳⣄⡀⠀⠀⠀⣸⣟⣛⣿⢽⡇⠀⠀⠀⠀⠀⣀⡴⠚⠲⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡦⠤⣇⠛⠛⠛⠋⣇⣀⣤⠤⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢚⡟⠲⠼⢦⡀⢠⣞⣡⣤⠤⠶⠻⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣸⠁⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢧⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣤⣏⣠⠤⠖⠚⡇⣇⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢨⡇⠀⠀⠀⠉⣩⣷⡀⠀⠀⠀⠀⠀⢧⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⣴⣃⠀⠀⠀⠀⠀⢸⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠦⠤⢧⣤⠶⢦⡴⡏⠀⢸⣧⣤⣤⣤⣤⡀⣧⣠⠞⣿⡄
⠀⠀⠀⠀⢠⡤⠤⠤⣤⣟⣳⣿⣖⡾⠏⢧⣀⣸⠇⠉⠛⠉⠀⣾⠉⠉⡇⠉⢿⣷⣴⣧⣿⡤⠤⠤⠤⣄
⠀⠀⠀⢠⠏⠀⡠⠞⠉⣈⣿⡽⢯⣀⣀⡤⡄⠀⠀⢀⣀⡀⠀⠈⢓⣚⡁⣠⠼⢃⣾⣷⢬⡙⠲⡄⠀⠹⡄
⠀⠀⢀⡞⠀⢸⡁⠀⠀⣟⠻⢿⣛⠯⠵⠾⣷⣤⠴⠯⠭⠿⠶⠖⠋⢉⣉⣥⠖⢫⠿⡤⠞⠀⠀⢹⠀⠀⢹⡀
⠀⠀⡾⠀⠀⠀⠳⢤⣀⠈⠓⠦⣬⣉⡛⠒⠒⠶⠶⠶⠶⠖⠒⠚⢛⣉⣡⠴⠚⠉⠳⠇⠀⣠⠴⠋⠀⠀⠀⢳
⠀⡼⠁⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣬⣉⣙⣛⣒⣒⠒⠒⠚⢛⣋⣉⣁⣀⣠⡤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠈⢧
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡡⠂⠏⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠁⠀⠢⠂⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠙⠉⢉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠒⠀⡠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄⠄⠚⠦⠎⠀⠀⠸⣔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡠⠃⠀⠀⠀⠰
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⢀⠞⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡁⠀⠀⢈⠀⠀⠀⠀⠈⠀⠀⢘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠠⣤⣤⣀⡠⡄⠘⢦⢀⢠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠉⠉⠐⠠⣤⣄⡤⠖⠋⠉⠉⠉⠀⠀⢈⡟⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠃⠀⢀⢼⣀⠄⠀⠀⠀⢀⠄⢀⢀⡀⠀⠀⠀⠀⡘⡀
⠀⠀⠒⠀⢀⠀⢀⠀⠀⣀⢀⠃⠉⠀⢰⣀⡦⠐⢠⣴⡎⠁⠑⠀⠋⠀⠉⠈⠉⠈⠀⠀⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠈⠀⠀⢈⡀⠈⢀⣀⣈⡡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠊⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⡆⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠠⢄⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠈⠙⠤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠡⠂⠀⠀⠀⠀⣔⠃⠋⠓⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠠⠀⠂⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⢈⠉⠈⠀⠀⠀⠀⡌⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠈⠂⡀⠀⠂⡀⠈⠁⠁⠄⠀⡁⠀⢀⠠⡄⢀⠈⠀⠀⠠⠄⡀⣀⢀⠀⣼⣇⢄⠀⠀⠀⠈⠀⠀⢀⡀⠠⠲⠀⢀⡀⢀⠀⠀⠐⠁⢁
⠀⢰⡀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠁⠐⢁⡄⠀⡀⠈⠑⡽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⣤⡠⡤⣤⡴⡀⠈⢤⡀⠔⢀⠠
⠀⠀⣦⣄⡠⠆⠀⠒⠁⠉⠀⠀⠀⠀⠡⠄⠠⡁⠀⠀⠁⠀⠀⠀⢀⠌⠀⠀⢨⡕⠀⠀⡀⢠⠀⠀⠀⠠⠀⠀⠠⠀⠦⠄⠘⣄⢋⠀⠤⠂⠀⠀⠀⠀⠆
⠀⠀⠀⡀⠂⠀⠀⢀⡀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠠⠄⡀⠀⠀⢘⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡌⠡⠒⢈⠀⠂⠀⠀⠀⠀⠐⠀⢀⣀⠈⠀⠀⠣⠊⠁⠆⠀⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠅⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠂⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣄⣄⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠊⢀⡐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⣶⣿⣿⣿⠛⢛⡛⡻⡟⣿⣿⣿⣶⡄⠀⠁⠀⠉⠀⠠⠸⡈⣈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⡐⠂⠠⠁⠀⠀⠀⠀⠀⠑⠄
⠀⠘⡒⠐⠈⠀⠀⢀⠀⢀⣿⢛⡟⣿⢻⡛⢛⣿⢛⡛⠻⠟⠟⢻⣿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⢀⠀⣂⢀⣀⣀⣀⠀⠀⡀⡂
⠀⠀⢡⠀⡄⠀⠀⠀⠀⠘⣿⢨⠁⢹⡜⣿⢸⣿⢈⡁⡇⠀⠀⡆⣿⠃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⡂⠀⠀⡁⠀⠁⠀⠀⠇⠀⠍⠛⠋⠀⠛⠀⠐⠈⠁
⠀⠀⠀⠆⠃⠀⠀⢀⠀⠀⠹⣼⣼⣮⣭⣿⣼⣿⣼⣧⣡⣬⣤⣴⠏⠀⠀⠀⠀⠰⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⠀⠐⠀⠀⠀⠉⠛⠛⠛⠻⠛⠛⠛⠛⠛⠉⠀⠀⠀⠂⠀⠀⡀⠀⠠⠄⠁⠀⠀⠀⠀⠈⠀⠀⠀⢐
⠀⠀⢅⠀⠀⠀⠂⠀⠒⠠⠀⠀⢠⠐⢀⠀⠀⠠⢀⠀⠠⡀⠀⠀⠀⠀⠄⠀⠀⠠⡈⠀⡀⠔⠀⠄⠀⠀⣀⣀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢉⠀⠸⠀⣕⠐⠘⠂⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⢀⠗⠁⠀⠀⠉⠄⠀⠀⠂⠈⠂⠀⣈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⡀⠀⠐⢀⠀⠀⠘⠕⠀⡀⠀⠀⢘⡂⠀⠀⠆⠀⠀⣀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠑⠂⠀⠘⠀⠰⣀⣀⠬⠃⢈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠁⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠐⠬⠄⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡄⠀⠀⠀⠀⠀⠀⢀⠰⠸⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠁⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠖⠃⢀⠄⢀⠀⠒⠑⢀⠀⠀⠠⠀⠀⠀⠃⠁⠦⠀⠐⠀⠅⠀⢸⠀⠁⠈⠀⠂⠀⠐⠀⢴⣊⠖⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠠⠤⠆⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⢀⠀⠀⠨⠂⠀⠀⠠⠊⠀⠀⢄⠀⠀⢀⡈⠀⠂⢰⣀⡠⡰⠱⢡⠄⠀⠀⠀⠀⡆⠀⠠⠀⠠⠀⠁⠒⠈⠀⠤⠀⠀⠀⠀⣠⠀⢀⠠
⠀⠈⠀⠀⠀⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠠⢇⡀⠀⢸⢆⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠐
⠀⠢⠀⠠⠄⠂⠁⠀⠀⡀⠀⠀⡠⠔⠊⠁⠁⠀⠀⠈⠍⠀⡄⠀⠀⠈⠚⠚⠁⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⡄⠀⠀⡀⠀⠀⡄⠀⠄
⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠑⠂⢀⠉⠀⠐⠀⠄⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠁⠀⠠⠤⠄⠀⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠓⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠸⡇⠀⢠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠂⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⣶⠀⢰⣶⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⠋⠉⢀⣤⣀⠀⢠⣤⣀⣀⣠⡾⠀⢸⣿⠁⠀⠀⠀⠀⠀⠠⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡿⠶⢶⡆⠁⠴⠆⠀⠀⡿⠿⡀⢹⠿⠀⡘⠛⠀⠦⠀⠀⠀⠀⠈⢬⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠉⠇⠀⣀⡀⠀⠀⠀⠈⣃⣾⡀⠈⠃⠀⠀⠀⡀⣀⠃⠇⢰⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣴⡄⠀⠿⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⠛⢠⡏⠀⣀⠉⠃⢀⡤⠖⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⣨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠹⣶⠃⣝⠁⠀⠀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢒⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠛⠖⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⠁⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣄⠀⠀⠀⠀⠀⠀⣠⠔⠂⠁⠀⠀⠀⠀⠠⣀⣿⡇⠀⠀⠄⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⢀⡤⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣿⡴⠒⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⣦⡈⠛⢀⣼⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠡⡀⠀⠀⠘⣇⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠦⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠖⠛⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⢋⡟⠁⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢳⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠁⠀⠀⠀⠀⢎⠀⣾⠗⠀⠰⠿⠻⢿⡤⣄
⠀⠀⠀⠀⠀⠀⠀⡷⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣁⣴⣧⣀⣀⠀⠈⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⡾⠿⠿⠆⠠⡾⠃
⠀⠀⠀⠀⠀⠀⠈⢠⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣎⣉⣳⡀⠀⠀⠀⠀⡤⡤⠒⡶⣠⠒⡦⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⢠⠮⠭⠭⠭⠽⠄⠀⠀⠀⣸⠥⠄⠬⠦⠨⣆⠀⠀⠀⣜⠁⣼⡙⣧⢎⢫⣾
⠀⠀⠀⢠⡗⡒⡒⢖⠠⣒⡲⣶⠾⡀⠀⠀⠀⠘⡎⠉⠉⠉⢹⠁⠀⠀⠀⠙⣟⢻⡭⡛⢹⠁⠀⠀⠀⢹⠙⠓⠓⠚⠚⢉⡇
⠀⠀⠀⠀⢹⠒⠾⡉⣘⢿⠓⠞⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡟⢤⣺⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠉⠁⠘⠳⣰⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡁⡝⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⢒⣂⡉⣐⠲⡄⣠⢤⠀⠀⠀⠀⠀⣀⠤⢤⠒⠢⡄⠤⡀⠀⠀⢀⡠⠔⠒⠒⠠⢄⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠀⠀⢠⢋⠖⠧⠎⢀⡠⡫⠞⠹⡆⠇⠀⠀⢠⠞⠁⠀⠀⠣⡀⢣⣀⣈⠆⠀⣎⣀⣀⣀⣀⣀⣠⡕⠀⠀⠀⢠⠋⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠈⠆⠏⠋⠻⠒⡿⢭⢅⠙⠂⠀⠀⡜⠣⡀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠈⡄⢠⠈⡄⠰⢠⠁⠀⠀⠀⠸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢆⠳⢖⢐⢢⣀⡰⢋⡞⠀⠀⠀⠀⡗⢄⣈⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢡⣘⠀⢃⣃⠎⠀⠀⠀⠀⡼⢑⣤⠴⠊
⠀⠀⠀⠀⠀⠑⠢⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⢭⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⢠⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⢖⣂⠴⡒⣒⣄⠀⠀⠀⠀⠀⡤⠤⠤⠧⠧⠤⣄⣀⡀⠀⠀⠀⢸⠀⣇⠾⠼⠄⠀⠀⠀⢀⡔⡢⣄⡠⢚⠦⡀
⠀⠀⠀⠀⡎⡡⠒⠉⠒⠒⠒⢝⢦⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣯⣗⠅⠀⠀⠀⡸⠀⡧⠋⠈⠣⡄⠀⢠⠊⠀⢱⠈⢳⠸⡄⠈⠆
⠀⠀⠀⢸⡴⠁⠀⠀⠀⠀⠀⠈⢿⠂⠀⠠⡆⠀⠷⠤⠄⠖⠋⠃⡲⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠸⡀⢀⠇⠀⢸⠀⡇⠀⡄
⠀⠀⠀⠸⠤⠤⠤⠀⠒⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠑⢤⣃⠴⠛⢄⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠤⠤⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⣀⠀⣞⣵⣵⠗⠺⡄⠀⠀⠀⢠⠃⠘⡀⠀⠀⠀⠀⠀⠀⣠⠊⠱⢼⢌⢷⠀⠀⠀⣠⣀⡠⠤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⢰⣊⠭⣔⡺⣽⠝⠁⠀⠀⡇⠀⠀⠀⡞⣚⠓⠙⡆⠀⠀⠀⠀⣴⠥⠔⠚⠉⢉⡏⠀⠀⢸⠀⠘⠀⣀⠈⠀⠀⠋⠀⢹
⠀⠀⠀⢎⣪⠛⠊⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠹⠀⡇⢰⠃⠀⠀⡠⠊⠀⠀⠀⠀⡠⠋⠀⠀⠀⣞⡋⢛⡓⠒⡞⠉⣖⡛⢳⣋⡢
⠀⠀⠀⠈⠡⡀⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠸⠋⠁⠸⡄⠀⠀⠣⢄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠘⠬⠅⠈⠫⠬⠍⠤⠬⠥⠜
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡄⠠⠃⠀⠀⠀⠀⠈⠢⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⡽⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⢉⣀⠤⣄⡰⠶⠶⠦⠤⠤⠤⣤⣌⡑⠀⡀⠀⠀⠀⠰⣖⠛⠉⠀⠀⡠⠊⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣤⠶⢚⣉⣉⡤⠄⠀⠠⠀⢤⣄⣀⣀⠀⠠⢍⡁⠘⠦⣄⠀⠀⠙⣆⠀⠀⠞⠀⣀⠀⣠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡶⠿⣛⡷⣄⠐⠒⠒⠂⠠⠤⢄⣀⣀⠠⠌⡉⠒⠿⣶⣬⣷⣄⠀⠸⢀⡄⠀⡔⣡⠞⢡⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣶⡾⠋⠡⡀⠀⢣⡐⣄⠳⣌⢢⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠤⣄⡀⠨⣙⠛⣠⣄⠸⡤⢞⣋⡠⠤⣤⡙⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣡⢔⣿⣿⢿⠣⡌⣆⠡⡈⢆⠱⡌⢣⡈⠷⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡳⣨⡟⢿⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣠⠞⡁⡸⡱⢣⠏⢀⠹⡌⢆⠱⡈⢦⠘⠆⠁⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⠙⢻⠃⢸⣿⡗⠺⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡼⢣⠎⠌⡐⢁⢇⡀⡌⢆⠙⡌⢆⠙⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⡞⢠⠏⠀⠘⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡟⣡⢃⠌⡼⢀⠎⢸⠁⠑⠘⠂⠀⠀⠀⠀⠀⢀⣠⢄⡀⠀⠀⠀⠀⢀⡴⣯⠏⠀⠀⠀⠀⠀⠀⡼⢡⠎⠀⠀⠀⣷⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠧⡻⢠⠁⠎⡜⢀⡎⠀⡎⠀⠀⠀⢰⣿⣷⡄⠀⢰⣋⣀⠴⠃⠀⠀⣠⠴⣏⠀⠘⣆⠀⠀⠀⠀⢀⡼⠱⣇⡠⠴⠚⠋⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢰⠁⠆⠈⡸⠀⡼⠀⡸⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠁⠀⠀⢀⠴⢞⠀⠀⠸⠀⠀⣿⠀⠀⢀⡠⠋⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⢠⣄⣄⣼⢸⠀⠀⠁⢠⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠃⠁⢸⠀⠀⠀⠀⢠⠏⣀⣀⣫⠤⣄⡴⠒⠋⠉⠁⠀⠉⠂
⠀⠀⢀⡴⠞⠁⢸⣿⣿⡯⢸⡀⢀⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠘⢦⣤⣤⠴⣋⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⢠⡿⠛⢷⡄⠀⠀⢿⣿⡇⣿⠁⡞⡐⠁⠀⠀⠀⠀⠰⡔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⣡⠞⠁⠀⠀⡇⠸⠀⠀⠀⠀⠀⢸⠀⠸
⠛⠳⣄⠀⠻⣦⡀⠀⢻⣓⡿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣸⠟⠛⡛⠏⠀⠀⠀⣀⣠⣁⡆⠀⠒⢒⣒⠒⠣⠞⢸⠶⠶⢤⣀
⠀⠀⠈⣙⣦⣈⣳⣴⣋⡿⢹⡿⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠚⢛⡇⠀⠀⢀⣀⠠⠂⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⢤⡀⠈⠳⣦
⢀⡴⠚⠉⠀⠈⠉⠙⠻⠄⢉⣤⡀⠀⠉⠛⠒⠦⠤⠤⠶⠖⠛⠉⠉⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣀⣀⡘⣠⠸⠤⠚⠀⠀⠀⠈⢧
⠘⢧⡀⠀⠀⢀⣤⠖⢻⠶⣼⡌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡇⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⡇⡇⠀⠠⠊⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⡿⡇⠀⠀⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⡅⠀⢀⣀⡴⣣⡄⠀⠤⠶⠒⠚⢣⡇⠀⠹⠦⠌⢀⣲⠴⠟⠋
⠀⠀⠀⠱⡄⠘⡇⠀⠀⢰⠃⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣀⣀⡷⠞⡉⠉⠉⡹⠀⠀⠒⠚⠂⠉⠁⠀⠀⠀⣔⠉⠱⡄⠀⠀⠀⡠⠆⠀⠀⠀⢰
⠀⠀⠀⠀⠘⢦⣽⣄⡴⠋⠀⢿⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠉⠁⠀⠀⢙⡠⠊⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠈⠓⠒⡶⠖⠒⠉⠀⢀⠄⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣠⠴⠚⢋⠉⠉⣻⠀⠀⠀⠀⠀⣬⢸⠀⠀⠀⠤⠄⣎⠀⢈⣂⡤⠖⠃⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣀⣬⣤⣞⢡⡀⠐⢛⡛⠉⠽⠚⠀⠀⠒⠒⠀⠉⠉⠉⣍⣀⡸⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠚⠉⠉⠖⢀⣸⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠚⠛⠻⢭⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡘⢄⣀⡿⡄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣠⠇⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠉⠉⠁⣰⠇⠀⠠⠤⠤⠴⠚⠥⣆⠆⠈⠭⠿⠅⠐⢁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠁⣈⠁⠘⠛⠀⠉⣉⣉⣡⣤⡤⠤⠴⠒⠚⠋⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣶⣶⣶⣾⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣏⠀⠀⠈⠉⠉⠉⠙⠛⠿⠻⣿⣿⣉⣫⠍⠟⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡄⠈⣿⣿⣿⡇
⠿⣿⣿⣿⣿⣶⡶⣶⣦⣤⣀⠀⠀⠀⠘⢿⣥⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⡇
⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠳⠶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⠋⠣⠋⠳⣼⣿⡃⣯⣭⣿⡿⠿⣿⣿⣿⣿⣷⣤⣀
⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠤⢛⣿⡃⠀⠀⠂⣿⣿⣿⣿⣾⣿⣇⠀⣿⣿⣿⡇⠀⠉⢻⣷
⣿⣷⣿⣶⣦⡀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠰⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣼⣿⣷⣿⣿⣿⡟
⠁⠀⠙⠿⠿⣿⣦⣄⠀⠀⠀⢻⡇⠀⠀⠀⠘⣿⣶⣶⡿⢿⣇⡴⢻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣸⡇⠰⠟⠛⠚⠻⠿⣿⣷⡀⢻⣇⣸⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣟⠿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢰⠀⠀⢀⣁⣀⡀⠀⠀⢩⠙⠻⢿⣶⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡏⣽⡇⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣶⣦⣤⣳⣴⣦⣉⠛⠃⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⣛⡛⠿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣇⣠⣤⣶⣾⢟⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣿⡦⠀⠺⣿⣿⣹⣿⣟⠟⡛⠛⠻⣿⣿⣿⣦⠀⠻⠏
⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠻⣿⣿⣿⣟⣣⣴⣿⣿⣿⢿⢦⣆⣾⡟⢻⣽⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣦⣤⣴⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠛⠿⣷⢻⣿⣿⣿⣿⣿⣾⣸⣿⣿⣦⡙⠯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣶⣴⣦⣴⣿⣽⠁⣿⣿⣿⣿⣿⠟⣿⣿⣿⠿⢿⣦⣝⡀⣹⢿⣿⣿⣿⣿⣿⣤⣄⣿⣿⡇
⣿⣿⣿⣷⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⡃⣿⣿⣿⣿⢃⣿⣿⣯⣰⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣌⣉⡉⣈⠁⢈⣁⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⡈⠩⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⢶⣴⢮⢬⠧⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠉⡙⣿⡄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⣡⠟⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⣾⣦⣄⠘⠒⠒⠒⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠘⡏⡀⠓⠀⣀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣵⠿⠋⢉⡍⠁⠉⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠁⠀⠀⠛⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢏⠀⢄⠐⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡆⠀⡇⢀⣀⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠘⠛⠻⠿⣿⠿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠠⠄⠁⠻⠛⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⡴⠇⢠⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⣿⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⣀⣠⠀⠀⠀⠀⣀⢓⡼⠛⣞⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⡀⠂⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡴⠖⠻⠟⠉⠉⠣⢤⣀⠀⠀⠀⠈⠀⠐⢐⡀⠀⠀⠀⠀⠀⠁⣠⣖⠓⢮⢀⠀⠀⢃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠿⠀⠀⠘⠀⠀⡀⠀⠀⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠰⠥⠊⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠁⠀⠀⠇⠀⠀⠀⠀⠈⠃⠘⠆⠀⠀⠀⠂⠒⠄⠀⠀⠀⠀⠀⠀⡐⠁⠂⠀⢔⠀⢾⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠸⠄⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣸⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠓⢀⢀⡀⠳⠋⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠰⢳⢄⡀⢀⠀⠋⠁⢣
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢹⡗⣶⠦⡤⠠⣄⠀⢠⠾⡅⡸⢄⡠⠛⠠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠘⠀⢩⢁⠔⣷⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣯⡠⡟⠁⣿⠬⢺⠟⠈⠉⡄⠀⠀⢀⠘⠀⠀⡀⢨⠀⠀⠀⠀⠀⠀⠀⠚⢻⠷⢴⠀⠀⠈⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⣷⠀⡧⠀⣾⠁⠀⠀⡇⠀⢀⡈⠀⠀⠀⡇⠈⠀⠀⠀⠀⠠⣄⠟⠈⠁⠀⢘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠇⠿⠺⠧⠸⠯⠠⠖⠊⠙⠆⠈⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣆⡀⠈⠧⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⢰⠂⠀⢀
⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⠨⡄⢀⣇⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢰⠀⠀⡇⠀⠈⠉⠀⠒⠊
⠀⠀⠀⢀⠀⢠⣤⡶⠾⠷⠾⠷⢟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡧⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣦⣼⣄⣰⣦⣐⡀⠀⠀⢀
⠀⠀⠀⠈⣆⠈⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⠁⠈⠁⠹⠿⠋⢠⠆
⠀⠀⠀⠀⠘⠳⢶⣄⣀⠀⠀⢀⣀⣰⠀⢀⡆⠀⠀⠀⠀⠀⢀⠀⡰⠾⠉⠀⠀⠀⠀⠀⠀⠀⣱⡡⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠛⣉⠉⠲⠈⠁⣀⡄⡀⣀⣓⡈⠘⠙⠀⠀⠠⠿⡃⠀⠀⠀⠀⠀⠈⠓⠂⡐⢦⠄⠀⠠⠸⠆⣈⠟⠶⢦⣴⣦⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢹⡇⠀⠀⠀⢸⡋⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂⠀⠀⠀⠀⠉⠀⠈⠀⢈⡰⡚⠀⠀⠀⢠⡔⠀⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠈⡷⡄⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣁⠀⠀⠀⣼⠃⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⠐⠀⠀⠀⠀⣩⣍⣑⣶⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢛⠢⠄⠉⠙⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠚⠛⢻⣿⣿⣿⣿⢋⢣⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⢿⣶⣾⣷⣦⡤⠤⠤⣤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢹⣿⣿⣿⣿⡶⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⣠⣿⣿⣿⣿⡧⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣁⣹⣿⣿⣿⣿⠧⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⣴⠀⣷⢀⡀⠀⢠⣤⣰⠇⣤⠤⠤⠤⣤⣬⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣤⠖⠋⠁⣸⠀⣼⠟⠓⠒⠒⠙⢿⣄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡟⠞⠁⠀⠀⠀⠀⠀⠀⠙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣠⡀⡒⢀⣤⡄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠔⠋⣴⠏⠁⠀⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⡡⠁⡌⠙⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢸⠇⠀⠂⠀⠁⠀⠀⠀⠀⠀⢸⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢾⠀⣠⡴⠞⠛⠉⠛⠓⠶⣤⣿⠈⠁⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠘⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⣿⣿⣿⣿⣶⣮⣭⣿⡆⠀⡀⣿⠤⠤⠒⢋⣝⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⣻⠂⠀⠀⠰⢤⣷⡆⠀⠀⢠⡃⠀⠘⠛⠉⠸⣿⡿⠉⠛⠋⠀⠀⠁⠀⠀⠀⠀⠈⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⣿⠀⠀⠀⠀⠀⠈⠉⠙⣲⠦⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠙⢿⠀⠀⠀⡏⠉⠉⠁⠀⠀⢀⠈⢄⣀⠀⢴⠘⣡⠋⠀⠂⢠⢀⣠⣉⡀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠒⠂⠉⠀⠀⠀⠀⠑⢀⡜⠠⠴⡀⡀⢀⠁⠀⠀⠄⣻⡓⠚⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⠀⠈⡄⠀⠠⠀⠀⠀⠀⠀⠀⣀⠘⡁⠀⠤⠐⠈⠤⠄⠀⠀⠀⠈⠉⠃⠒⣢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡰⠉⠀⠛⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡠⠴⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢢⠀⠀⠀⠀⠀⠘⠀⠰⠁⠀⠀⢨⡄⢘⣀⠀⢀⣀⡔⠒⠊⢁⡀⠈⠈⡳⣦⡀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⣸⢷⡆⠀⠀⠉⠀⠰⠇⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⣀⠋⠀⠀⣴⠙⢮⡻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣠⠞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠃⡀⠘⠉⣀⠈⠁⠀⠀⣀⢃⡀⠀⠀⠈⣿⡿⠿⢿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣠⡴⢚⠡⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠂⢀⠉⠀⠊⠀⠉⠈⠀⣼⠋⠈⠁⢤⡤⠾⠁⣤⠀⠀⠀⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠆⠀⠀⠙⠓⠒⠂⠂⠀⠀⠀⠈⠦⡀⠀⠀⣠⡇⠀⠘⠦⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠠⠤⠤⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣠⣤⠀⣀⣸⠏⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡁⠠⠀⠀⠀⣀⡤⠖⠚⠉⢁⠀⠈⠁⠁⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈


⠟⠂⠀⠀⠐⠂

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠚⠛⠛⠋⠙⠛⠛⠓⠲⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⣠⣶⣾⣿⠿⠟⠛⠛⠛⠋⠙⠛⠛⠛⠿⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⡿⠋⠙⢿⣿⣆
⠀⣰⠇⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢂⣴⣿⡿⠋⠀⠀⠀⠈⢻⣿⣆⠀⠸⡆
⢠⡏⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⠀⠀⢀⣀⠀⠀⠀⢿⣿⡆⠀⢹
⣸⠁⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⠀⠀⠸⣿⠀⠀⠀⠘⣿⣿⠀⠈
⣿⠀⢠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⠀⠀⠀⡿⠀⠀⠀⠀⣿⣿
⣯⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢹⡄⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡿⢋⣴⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⠀⠀⢰⣿⡿⠀⢠
⠈⣧⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠁⠀⠀⢀⣿⣿⠃⠀⣼
⠀⠸⣆⠀⠸⣿⣷⡀⠀⠀⢀⣴⣿⡿⢋⣴⣷⣶⣶⣶⢓⡚⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⣰⠃
⠀⠀⠙⣆⠀⠙⣿⣿⣄⣴⣿⡿⠋⠰⠿⠿⠿⠿⠿⠿⢬⠥⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⣠⣿⡿⠃⠀⣰⠃
⠀⠀⠀⠘⢧⡀⠈⠻⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⠛⠉⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡶⡆⢶⢰⢢⡖⢲⡔⡆⡖⠲⡄⢰⡖⢲⣴⡆⣖⡂⢲⡖⢰⣒⡆⣰⣦⢠⡖⠒⢲⡖⢰⢠⡖⢲⣴⣦⣰⠰⣖
⣼⣭⣿⣬⣯⣬⣷⣾⣥⣧⣷⣾⣥⣼⣷⣾⣼⣧⣶⣯⣼⣧⣼⣬⣧⣯⣽⣮⣷⣦⣼⣧⣼⣬⣷⣾⣭⣥⣽⣴⣾
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣧⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠀⠀⠀⠀⠘⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⡀⠀⠀⠀⠨⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠻⡿⣿⠋
⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢇⣿⣿⣾⣝⠿⣹⣧⡝⢿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠿⠏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⣿⣧⣿⣿⡹⡷⢿⣿⠖⣷⣶⣯⣽⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⡎⣉⣙⠟⣳⣿⣿⡿⢳⣾⣷⣿⣿⣷⣯⣻⣂⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣭⡍⣭⣙⣻⣿⢻⡿⢧⣽⣿⣟⣿⡰⣤⣯⠸⣿⣿⣿⣩⣝⡻⣦⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣟⣽⣶⣶⣦⠻⢿⣻⣿⣷⣿⣿⣯⣿⣿⣿⣦⣙⣛⣻⣶⠆⣾⣿⣿⣿⣭⢀⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣆⣿⣥⣾⣶⣿⣿⣗⣯⣛⣻⣿⣧⢹⣿⠿⣛⣋⠚⣿⣿⣿⡷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⢿⢟⣣⣬⣽⣯⣿⣷⣯⠿⣿⣻⣟⣸⣿⣿⣿⣿⢿⣮⣿⣯⣟⣿⣶⣯⡝⣿⣿⡦
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣯⣿⣟⣟⣛⣽⣿⣶⣶⣾⣿⣿⣶⡉⣿⣿⣿⣷⡹⣿⡿⢿⣿⣿⡏⠛⠟⢻⣼⣭⣌⣿⣛⣿⣙⢿⣿⡿⣻⣧⣻⣷⢶⣄
⠀⠀⠀⠀⠀⢻⣟⣿⣍⠟⣿⣿⠿⣿⣦⢟⣿⣿⣎⣿⣿⣿⡟⢰⣿⠿⣿⢹⡿⣿⣿⡿⣏⣟⡿⠿⣬⣿⣿⣿⣻⣿⣿⣿⣷⣶⣿⣭⣻⣿⣟⣿⣿⣿⣿⡆
⠀⠀⢠⣄⢾⡿⣿⣟⣿⣿⠿⣮⣿⣿⣿⣿⣧⣽⣟⠻⣿⣷⣾⠿⢸⣿⣻⣾⣿⣿⣷⡞⣿⣿⣿⡟⢶⣿⣿⣷⣞⣿⣿⣿⣿⣿⡉⢿⣿⣾⣿⠈⢿⣿⡿⣷⠄
⠀⠀⢨⣋⣤⣾⠟⢿⡿⠟⠈⠛⠻⠿⠿⠃⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣛⡲⢤⡀
⠀⠀⠀⠀⢠⠖⠋⡍⠉⡉⢉⠙⠳⣄⠀⠀⠀⠀⠀⠀⣴⠶⠾⠷⠶⣦⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⢚⣩⣭⡇⡿⢟⡙⢦⣹⡄
⠀⠀⠀⠠⣏⣀⣃⣀⣁⣀⣁⣘⣀⣸⡆⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⡇⠀⠀⠀⠀⠀⢠⡟⠒⠻⡆⠀⠀⠀⠀⢠⡟⣴⣋⣐⢺⡇⣷⡯⢟⣫⣍
⠀⠀⠀⠀⣯⠤⠤⣤⡤⠤⣤⡤⠤⣽⠇⠀⠀⠀⠀⠈⣯⠈⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⣾⠛⢳⠀⣿⠀⠀⠀⠀⣾⢰⡇⢯⡽⢸⣧⠵⣚⠉⡇⢹
⠀⠀⠀⠀⢳⠶⣶⡒⠒⠒⠒⣶⠶⣾⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡶⠛⠀⡟⠀⠀⠀⠀⢹⡌⣇⢰⣷⡌⠄⠺⠿⣰⢇⡿
⠀⠀⠀⠀⠘⠳⠾⠷⣦⣠⠾⠷⠖⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣇⠀⠀⠀⠀⠀⠻⣌⠳⢭⣴⣏⡷⠞⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⢴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⣴⠻⡌⢿⢸⡟⣸⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡅⠻⣜⣳
⠀⠀⠀⠈⡏⠛⠛⠚⠛⠒⠛⠋⣿⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣦⣀⡀⠀⠀⠀⢠⡞⠉⠠⠈⠠⠉⠄⠈⢳⡄⠀⠀⠀⠀⢀⡞⠰⠀⠄⠠⠈⡽
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⢁⡿⠀⠀⠀⠀⢹⡏⠹⡟⠉⢹⡏⢩⡟⠁⠀⠀⠀⠀⣸⣇⠐⠀⢂⣠⠞⠁
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣠⣤⣼⣧⣤⣤⣤⣤⣿⣭⣤⠀⠀⠀⠀⠀⢷⠀⣿⠀⣸⠁⣼⠁⠀⠀⢀⡤⡼⣃⡼⠓⠛⠉
⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠹⢦⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠀⠀⠀⠘⠧⠼⠤⠿⠴⠃⠀⠀⠀⠈⠳⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣳⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⣄⠀⠀⠀⠀⣾⣁⣈⣹⣄⣙⣦
⠀⠀⠀⠀⢀⣾⣩⣥⣀⣠⣩⣧⠀⠀⠀⠀⢀⡞⠉⠉⢹⠏⠉⠉⡟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠲⢾⡀⠀⠀⠈⢷⠖⠲⠖⣶⣶⡞
⠀⠀⠀⠀⢻⣥⣄⣠⣤⣀⣤⣬⡇⠀⠀⠀⣼⢶⠶⠶⡶⠶⢶⠶⠶⡶⠶⢶⢾⡆⠀⠀⠀⠀⠀⢀⡴⠋⠳⢤⡀⢨⡇⠀⠀⠀⠘⣷⠖⢋⣡⣼⠁
⠀⠀⠀⠀⢸⡄⠈⡇⠀⣿⠀⣸⠃⠀⠀⠀⣿⣯⣛⣛⣟⣛⣋⣛⣫⣿⣚⣉⣿⡇⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠙⣿⠀⠀⠀⠀⠀⢹⡞⢋⣠⡇
⠀⠀⠀⠀⠀⢷⠀⣷⠀⡟⢠⡏⠀⠀⠀⠀⠘⠲⠾⠭⠭⣬⣭⣭⣬⠭⠽⠷⠚⠀⠀⠀⢿⣉⠹⡄⠀⠘⣦⣠⡾⠃⠀⠀⠀⠀⠀⠈⣿⠉⡿
⠀⠀⠀⠀⠀⠘⠓⠛⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣻⠙⠖⡦⠀⠀⠀⠀⣠⠞⠳⣄⢀⡴⠛⣦⠀⠀⠀⠀⠀⣯⠭⣽⡇⣴⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣏⡽⠷⣤
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⣷⠛⣯⣿⣹⠆⠀⠀⣰⠋⠐⣦⠈⣯⠀⢶⠈⢳⡀⠀⠀⠀⢸⠀⢸⠀⣹⠏⢹⣇⠀⠀⠀⠀⠀⠀⣠⢞⡵⣫⡞⠁⠀⠈⡇
⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡟⠛⢻⡾⠃⠀⠀⠀⢿⠀⣰⠃⠀⣽⢠⡏⠀⢸⠇⠀⠀⢀⡾⠀⠸⣾⠁⠀⠀⢸⡇⠀⠀⢠⡖⣫⡵⣫⡴⠋⠀⠀⢀⡼⠁
⠀⠀⠀⢰⡋⠀⠀⢀⣴⠏⢀⡴⠋⠀⠀⠀⠀⠀⠈⢧⡈⢁⡴⠳⣌⠃⣰⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠸⣏⡽⠛⠁⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⣴⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⠛⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠉⢳⣄⣀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠓⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠛⠋⣉⠉⣉⠉⢉⠙⠛⠿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⣤⡀⠾⠀⠛⠀⠉⠀⠛⠀⠶⢀⣤⠉⠻⢿⣿⣦⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⣀⠰⠆⠈⣀⣤⣶⣶⠿⠿⠿⣶⣶⣤⣀⠁⣰⣦⣤⣿⣿⣿⡿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⣀⠛⠀⣠⣾⠟⠉⠀⠀⢀⣀⣠⣤⣶⣿⣿⠿⠿⠛⣛⡉⠩⣤⡄⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⣤⠉⢀⣾⣿⣥⣴⣶⣾⡿⠿⠟⠛⠋⣩⠁⠀⢶⡄⢸⡏⠻⠄⢿⣿⡄⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣧⣶⣿⣿⠿⠿⠛⢛⡉⣥⣤⠆⠐⣿⣧⠀⢻⡇⠀⠸⣧⠈⠻⢷⣄⠘⣯⢻⣿⠸⣇
⠀⠀⢰⣾⡿⠿⠟⠛⣋⣉⠀⢰⣶⢶⣆⢹⣇⠀⢿⡆⠀⣿⠹⣆⠈⣿⡀⠀⢿⡆⢺⣆⣹⡇⢻⠄⠻⠀⢛⣀⡀
⠀⠀⠀⠀⣴⠶⣦⠀⢹⣿⡆⠈⣿⣀⣿⠀⣿⡄⠸⣷⠀⢻⣦⢿⣆⢹⣧⣤⠘⠗⢀⣉⣩⣤⣴⣶⣾⣿⠿⠟⠃
⠀⠀⠀⠀⢿⡆⠛⠃⢸⡇⢻⡄⢸⣯⠁⠀⢸⣧⠀⢿⡆⠸⠃⢀⣉⣤⣤⣶⣶⣿⡿⠿⣿⠛⣹⣿⡟
⠀⠀⠀⠀⠘⣷⠀⣿⠈⣿⠛⢿⡄⠿⠀⠀⣀⣩⣤⣴⣶⣾⣿⠿⠟⠛⠋⣩⣿⠟⠀⣤⠉⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠙⠷⠟⢀⣉⣤⣤⣶⣶⣿⡿⣿⣿⣛⡉⠉⠀⠀⢀⣀⣤⣾⠟⠁⠀⣶⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⢶⣾⣿⠿⢿⣿⣿⡉⠹⠇⣠⡀⠉⠛⠛⠿⠿⠿⠟⠛⠉⢀⣄⠸⠇⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢿⣿⣶⣄⡉⠀⠿⠀⣶⠀⣤⠀⣶⠀⠿⠀⢉⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧
⠀⠀⠀⠀⠈⣛⣟⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣻⣟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠻⣿⣶⣼⣿⣿⡧
⠀⠀⠀⢲⢸⣿⣿⣿⢿⣿⡿⢿⣀⣐⣿⣿⣗⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣋⣠⣀⣻⡿⠿⠟⣿⣿⣿⣯⡶
⠀⠀⠀⢸⡸⣿⣿⣿⣶⣾⣦⡀⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣶⣶⣿⣿⣿⡟⡇
⠀⠀⠀⠈⠛⠟⠉⢛⠋⢻⣿⣿⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡈⠀⠀⠀⡀⢸⣿⣿⣿⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⣿⣿⣿⠀⠁⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠈⠁



⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣘⡿⠿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⣀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣀⣠⣶⣿⣷⡄
⠀⠀⠀⢶⠀⣼⣿⣿⡿⠛⢿⣿⣿⣂⣐⣖⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣐⣾⣆⣈⣙⡻⢛⣻⣿⣿⣿⣆⠀⡶
⠀⠀⠀⢸⡘⣿⣿⣿⣷⣷⣶⡄⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄⠀⠀⠸⣿⣿⣿⣿⣿⡿⢀⡇
⠀⠀⠀⠈⠛⠚⠛⠛⠋⣿⣿⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢹⡀⠀⣸⣿⣿⣿⡟⠛⠓⠛⠁
⠀⠀⠀⠀⡈⠀⠀⠀⠀⣿⣿⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢣⢠⣿⡟⠙⢿⣿⡆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠴⠿⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣌⠹⢿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠠⠐⠂⠀⢺⣿⢟⣭⣭⡻⣿⣦⡙⢧⠔⠊⠁⠀⠉⠲⣄⣀⣠⡤⢄⣀
⠀⠀⠀⠣⡀⠀⠰⣘⠃⠀⣼⣿⢸⣷⡘⣿⢸⡿⠿⣿⠀⠀⠀⠀⠀⠐⣾⠏⠁⠀⠀⠈⠑⣦⣀⣀⢤⣄⡀
⠀⠀⠀⠀⠙⠢⢄⡈⠀⢠⣿⣿⡌⣿⣃⡿⣸⠇⠀⡎⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣏⣿⣿⣦⣭⣴⣯⡀⠼⠇⣀⣀⣀⡀⣀⣸⣇⡀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠭⣛⡻⠿⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠂⠙⢆⠀⠀⣼⠁⠀⠀⠀⠀⠀⣼⠏⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠉⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡆⢸⠀⢀⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡍⡁⠀⠑⠘⢧⡀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣴⠟⣠⠎⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠳⠀⠺⢦⣄⣀⣀⣀⣀⣠⣤⠾⠁⠀⢀⡿⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣰⣧⣬⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠒⢻⣿⣿⣿⠿⢿⣿⣭⣤⣤⣾⣯⣒⣀⣤⣼⡏⠀⠀⠀⣠⣾⢻⣿⠻⣯⡹⣿⣿⣿⠟⠃⢁⣀⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠱⢻⡿⢹⠇⠀⠋⠉⢿⠙⣿⣿⡟⠛⠙⠻⡿⢶⠶⢾⣷⣿⣸⣿⢆⡿⣱⣿⡟⠀⡰⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠐⠉⠀⠉⠀⠀⠀⠀⠈⠃⠈⠙⢿⡀⠀⠀⠉⢌⣷⠀⠹⣿⣷⣭⣩⣴⣿⡿⠀⢸⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠚⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢻⠱⡄⠀⢿⢿⣿⣿⣿⡧⠀⣽⣿⠃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⡄⠀⠀⠈⠀⠘⣄⠀⠑⢟⡿⡿⡄⠀⡿⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡀⠙⢆⠀⠀⠇⠀⢸⢣⡀⠀⠉⠚⠀⢀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⣤⣤⣄⣀⣀⣤⣤⠾⠿⠾⠁⠀⠀⠀⠀⣸⡄⠀⠓⠉⣃⣤⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠲⢬⡉⠙⠛⠛⠀⠀⠖⠶⣦⣤⣀⣠⡾⠃⠀⠀⠀⠀⠁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠨⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⢀⣴⠖⠛⠛⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠶⢤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠲⣦⡀
⠀⣴⠏⠀⠀⠀⢀⣤⡀⠈⢻⡄⠀⠀⠀⢀⣤⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⣤⡀⠀⠀⠀⢠⡞⠁⢀⣤⡀⠀⠀⠀⠹⣦
⣼⠃⠀⠀⣠⡴⣿⢻⢿⣧⡀⢿⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⡿⢀⣴⡿⡟⣿⢦⣄⠀⠀⠘
⢿⣤⡴⠚⢁⡼⠃⣾⢸⡏⣳⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣞⢹⡇⣿⠘⢧⡈⠓⢦⣤
⠀⠘⠷⠶⠿⣤⡾⠛⢋⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡝⠛⢷⣤⡿⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠋⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠙⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠳⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠞⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡂⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨⡇
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⢳⡞⠉⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠃
⠀⠀⠀⢀⣹⡆⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠘⣧⣀⠀⠛⢃⡼⢧⡈⠛⠀⣀⣴⠇⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢰⣟⡀
⠀⠀⢰⡟⠉⢿⡄⠀⠀⢻⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡿⠀⠀⢠⡿⠉⠻⣆
⠀⠀⢹⡆⠀⢈⣿⡴⠂⠈⢧⣀⣀⣠⠶⠋⠀⠠⣄⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣠⡄⠀⠙⠷⣄⣀⣀⡼⠃⠐⢦⣿⣁⠀⢰⡏
⠀⠀⠀⣿⠞⠋⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣿⣷⣦⣄⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣠⣴⣾⣿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⠳⣿⠁
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⢀⣤⣾⣄⠀⠀⠀⠀⠀⠈⠻⣿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⣿⠟⠁⠀⠀⠀⠀⠀⣠⣷⣦⡀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠈⢻⣤⣀⣤⡾⠋⠁⠀⠉⠛⠶⣤⣀⡀⠀⡠⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢄⠀⢀⣀⣤⡶⠛⠉⠀⠈⠙⠷⣤⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⢈⡭⠷⢶⣄⢀⣀⠀⠀⠀⠀⠉⠙⠛⠿⠾⣯⢭⣿⣍⡉⠉⣩⣿⡭⣽⠷⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⢹⠋⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠧⠞⠀⠈⠉⠉⠁⠀⠳⠼
⠀⠀⠀⠀⢠⡶⠞⠷⠀⠀⠀⠀⠠⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢳⡄⠀⣀⣀⣠⡶⠛⠶⠟⠉⠉⠻⠶⠛⢶⣄⣀⣀⢠⡟⠁⠀⠈⠻⣶⢶
⠀⠀⠀⠀⠀⠀⠉⣹⠷⠦⠄⠀⠀⠀⠀⠰⢿⣤⣼⡏⠁⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠈⠹⣿⠀⠀⠀⠀⠀⠿⠚⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠸⢧⣤⣤⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣸⣝⡆⠀⠀⠂⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣤⣤⣤⣄⣀⣠⣼⠟⢠⣄⠀⠀⠀⠀⠀⢀⡾⠉⠉⠉⠙⣦⠀⠀⠀⡀⠈⣷⣀⡀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠁⠘⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡼⢧⣄⣠⣤⣄⣸⠀⠀⠈⠋⠀⠈⠉⠙⡶⠶⠚⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⢀⣠⡀⠀⢀⣠⣄⠀⠀⢀⣤⣀⠀⠀⠉⠁⣀⣤⡉⠀⠀⣠⣄⡀⠀⢀⣄⡀⠀⢀⣠⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣯⣭⣽⣿⣿⣭⣭⣿⣿⣯⣭⣽⣿⣿⣿⣿⣯⣭⣽⣿⣿⣭⣭⣿⣿⣯⣭⣽⣯⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠒⠀⠘⠓⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠹⠟⠀⡀⠀⠲⠆⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠪⣄⠀⠀⠢⠄⢳⠀⠈⢩⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⣆⡤⠒⠲⡞⠀⠀⠘⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡟⠀⠀⠀⡟⠀⣤⠀⢻⡶⢊⡥⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⠰⠄⢣⣤⠀⠤⠀⢀⣿⡆⠀⠁⠃⠂⢜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠂⠁⢀⠞⠀⠀⠀⠀⠀⡾⣿⣧⠀⠀⠀⠀⠀⠀⡋⠉⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠉⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⡔⠀⣀⠼⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡜⠁⠐⣇⠘⠀⠀⠀⠐⠃⠀⠓⢤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠠⣄
⠀⠀⠀⠀⠘⢦⣄⣐⣀⣊⣀⠥⠲⢍⡀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠓⠤⢄⣀⣀⠀⠀⠀⢀⡠⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⢹⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⡰⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠠⠀⣀⠀⠀⠀⠀⠇⠀⠤⠼⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠠⠀⠂⠀⠀⠀⠀⠣⡀⡄⠀⠈⠉⠁⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡄⠀⢀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡌⠀⠀⠀⠀⢶⣧⡀⠀⠤⣔⢀⠀⠀⠀⠀⢦⣀⠀⡀⠀⡠⠋⣀⣠
⠀⡀⢀⠔⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠑⠿⣦⡀⠀⢠⣈⠣⡘⢿⣄⠀⠑⠷⣄⠀⠀⠈⡧⠷⢌⠻⢠⣞⣍⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⢙⢦⡀⠙⠉⠙⢆⠙⡗⢤⡀⠈⢳⡄⠀⡇⠀⠀⠳⢄⣀⣤⠴
⠀⠀⢃⠀⠀⠀⠀⢀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀⠄⠒⠒⠂⠄⠀⠉⠃⠀⢵⣬⠀⠀⠀⠳⡈⠢⠈⠛⠯⣙⣦⡇
⠀⠀⠀⠁⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠄⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⢤⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣁⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣇⠈⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠿⠿⢦⠾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠰⣦⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣸⣇⠀⠀⡿⠒⠒⠂⠒⠒⠤⣰⡏⠀⣱⢀⣠⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠶⠻⢀⣨⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⡋⠈⡆⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠳⣄⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣤⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣸⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡤⠚⠉⠉⠁⠈⠒⠉⠉⠉⢦⠤⣄⠀⠀⣧⠀⠀⠀⠀⢀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⡀⠀⠀⠀⢸⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣧⠀⠀⠀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀⡘⣆⡴⠒⠃⠀⠐⠋⠙⠂⠀⠀⠀⠰⠋⠙⠆⠀⠉⢳⡇⠀⣀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠈⠉⠳⠽⣇⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⢀⡸⠓⠈⠁⠀⠀⢰⡇⣼⣡⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⢦⠀⠀⢀⣀⣀⣠⣈⣳⡀⠀⠀⠀⠰⡄⠀⠀⢠⣶⠄⠀⠀⠀⣏⠤⠤⠤⣤⠀⢀⠇⡟⢸⢏⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⣠⠷⣄⠀⠀⠀⠉⠒⠶⠚⠁⠀⠀⢀⠜⠙⡆⠀⠀⡏⠀⢸⡀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⢺⠀⠀⠀⠹⡄⠀⢸⠁⠀⠈⠑⠲⠤⣀⠀⠀⠀⣀⡤⠖⠁⠀⢀⠟⢢⡼⠁⠀⠀⡹⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠗⢢⠀⠀⠈⠳⡞⠳⢄⡀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⣠⠋⠀⢀⡇⠀⢰⠒⠷⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡤⠖⢉⣆⠀⠀⠀⢇⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡼⠀⢰⣏⠙⠓⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠋⢁⣼⡁⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢺⣈⠉⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠥⡞⢇⠀⠀⠀⠀⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⢱⢬⣭⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠬⠃⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⢱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣭⠟⠛⠻⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠐⠂⠀⠀⠈⡉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠟⠀⠁⠁⠀⠀⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠃⠀⠀⣀⠀⢀⣊⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡅⠀⠀⠘⠢⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⡞⠀⠹⣦⣀⠀⠀⠀⠉⠀⠔⢧⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡊⠉⠀⠀⠀⠀⢄⠀⠀⠀⠉⠲⠶⢆⠉⠀⠀⣰⣭⣎⣋⠉⡂⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠢⡀⠀⠀⠀⠘⠢⠔⠁⠀⢨⣇⣈⡍⠉⠁⠁⠀⣌⣈⣉⠀⠐
⠀⠀⠀⠀⠀⢀⠴⣋⠠⠦⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠀⠀⠀⢠⠒⠂⠄⠀⠀⢀⣄⡀
⠀⠀⠀⠀⡰⢃⣤⣬⠷⠦⢤⣶⣴⣷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠀⠀⠀⠀⠉⠑⢦⠀⠀⡸⠘⢄⣆
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠡⡈⠉⢳⠀⢀⣠⡬⠴⠤⠦⠀⠠⠄⠤⢄⣨⠿⠊⠀⠀⠈⡏⢆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⡂⠀⢼⠀⠀⠰⠀⠃⢠⠀⠃⠘⡄⠀⠀⠀⠉⠉⠉⠀⠀⢸⣹⣆⣡⣀⠤⠂⠃⢸
⠈⠉⠹⡀⠀⠀⠀⠀⢀⡴⠳⠊⠉⠑⠂⠆⠲⠈⠉⠀⠀⠙⠢⠤⠤⠤⠤⠤⠤⠒⠋⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡖⠭⠒⢀⡍⡁⣈⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠃⠢⠀⠒
⠀⠀⣀⠁⠀⠀⠀⠈⠀⠀⠀⢀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠀⠖⠀⠀⠊
⠀⠀⠁⠀⠀⠀⠒⠒⠒⢒⣈⣩⠴⠚⡅⠀⠀⠀⠀⠀⠀⡠⠞⠍⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠐⠒⠒⢀⣀⡠⠤⠖
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⡀⠠⠔⢻⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⢠⠊⠁⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⣠⠄
⠀⠀⢸⠄⠀⡀⠀⠀⠀⠀⢀⣀⡤⢾⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⠘⠀⠀⠠⠤⠐⢊⠁⠁⡔⠊⡥⠀⠀⠀⠀⠀⠀⢩⠝⣀⢋⠼⡟⢣⠉⡄
⠀⠀⠘⠀⠄⠀⠈⠉⠉⠭⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠞⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢼⣤⣦⣼⣤⢧⢿⠁
⠀⠀⠀⡇⠀⠀⡀⠄⠐⠀⠀⠙⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢜⣕⢕⡵⡡⡏
⠀⠀⠀⠘⠀⠀⠀⣀⣈⣀⡠⠴⡺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⡪⣾⢮⡇
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⡆⠀⠀⠀⠀⠀⠀⣀⠄⠘⠀⠀⡚⢀⡀⠀⠃⢦⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠞⠁⠀⠀⢀⣠⠀⠀⠀⠀⣀⢄⠈⢱⠚⠇⢸⠁⠞⠂⡜⠓⢀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐
⠀⠀⡠⠖⠂⠒⠉⠈⠁⣄⣀⣠⠞⠁⠘⠉⠀⠀⠀⠀⠁⠈⠃⣰⠸⠀⠟⠲⠀⣼⠇⠀⠈⠀⠀⠀⠀⡸⠤⠠⠄⠤⢀⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠂⠅⠧⠼⠿⠏⠀⠀⠃⠀⠀⠀⠀⠀⣺⣁⣊⣐⣂⠐⠋⠻⠐⠠⢀⠤⠀⠈⠆
⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠌⢀⡘⡈⠀⠀⠀⠀⢄⣁⣠⠎
⠇⠀⠀⠀⠀⠀⠀⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢨⠆⠀⢀⡈⠉⠀⠒⠃⠐⠂⠐⠀⠜
⠀⠀⠀⠀⠀⣀⣴⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠤⠌⠀⠉⢐⡋⣘⠁⢀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⢀⣠⠖⠊⠉⠄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⢸⠁⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢧⡀⠀⣀⡀
⠀⣠⠋⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠂⠀⠀⡄⠀⡄⠀⠠⠀⠀⢀⡀⠀⠀⠀⢸⠁⠀⠈⠃⠪⡁⢟⣠⣄
⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣡⠞⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡒⢠⠉⠂⡠⠀⠀⡄⠀⠘⠀⠀⠀⠀⠈⠓⠀⠤⠘⣤⣗⣲⣄
⠈⢙⡁⠶⠄⠤⣤⠤⠤⡬⢄⠒⠚⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⡁⠂⡠⠊⠀⢀⡰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⡢⡑
⠀⠈⡀⠀⠉⠀⠐⠈⠀⠐⣁⠤⠚⢀⡁⠴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠕⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠘⠀⠘⠀⠤⠶⠒⠒⠒⠀⠚⠩⠉⠂⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠂⣒⣨⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢦⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⣿⣿⠉⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣵⣦⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⣾⣿⠻⢿⣿⡿⠂⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣄⠀⣼⣿⣇⣄⣠⢀⠀⠀⠀⠀⠀⣀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⢰⣦⣤⣴⣿⣷⣿⣿⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠛⣛⠿⠿⠿⠿⠿⠿⠿⠂⠁⠀⠛⠿⢿⣿⣿⣿⣿⣧⣿⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠏⢸⣿⣻⣷⣾⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠛⣿⡦⠀⠀⠀⠀⠀⠐⣽⡖⠀⠀⠀⠀⠘⠇⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⢻⣿⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⢤⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡤⠤⠤⢤⠄⠀⠀⠀⠀⢠⣿⣿⡭⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣦⣀⠀⠀⠀⠚⠛⠁⠀⠀⠀⣀⡴⠛⣿⣿⣦⣾⣿⣿⣿⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣞⠛⠖⠒⠂⠠⠶⠶⡟⠋⠉⠻⠿⠟⠋⠛⠶⠤⣀⣀⣀⠠⠐⠛⠉⠀⢠⠏⠙⣿⣿⡿⠟⣉⣤⡾⡿⣉⠿⣄
⠀⠀⠀⠀⠀⠀⣸⣧⠟⢂⡜⠙⠒⠒⠚⠁⠀⠓⠲⣤⡴⠊⠉⠉⠀⣤⠀⠀⠀⠀⠀⡀⠔⠁⠂⠀⠀⠀⠀⠀⡇⢈⡷⢤⡈⠑⠾⣦
⠀⠀⠀⠀⠀⠀⣿⣇⡀⢻⠀⠀⠀⠠⠤⠂⠀⡠⠆⠘⠦⣥⠤⠖⠊⠠⣄⢐⡒⣩⠥⠀⠀⠀⠀⠀⠀⠀⢀⠜⣀⣹⠗⢀⣌⠛⢢⢞⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣉⡿⠀⠀⣀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠠⢱⢀⣷⠹⡀⠀⠀⠀⠀⠀⠃⡖⠛⠤⣬⠋⠠⠦⢀⣘⣣⣶⣻⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠟⠁⠀⠠⡈⠳⠄⠀⢀⡔⣃⠀⠀⠀⠀⠀⠐⡡⢸⠆⣸⣧⠱⡀⠀⠀⢀⢜⣼⣡⠛⢶⣿⣹⢷⣤⣞⣿⠿⢤⣿⠁
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢾⣦⣤⡴⣯⠛⢻⣷⣵⣒⣒⣫⠞⡷⡇⢘⠿⢿⣷⣬⣁⣩⣵⠟⡶⢉⣱⣿⣿⠉⣾⠇⣸⠛⢠⢾⣏
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣌⡹⠷⣷⡋⢱⠿⣤⡟⠛⣽⣋⣾⢧⡇⠈⣺⠮⣀⠟⢫⡼⢉⠾⢤⡌⠙⣿⣧⡟⠃⣤⣁⡴⠧⣼⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣜⣿⣶⣿⣿⠓⣤⡀⢈⣿⠏⠁⠉⠙⠦⢼⠁⣴⡉⢱⠆⢤⡋⡲⢦⣀⠛⢺⣿⡴⠛⣤⡟⢙⡶⣇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢳⣯⠟⠻⣿⣿⣿⡶⢔⠛⠛⠃⠀⠀⠀⠀⠀⠀⢣⡄⣠⣤⠀⠁⠺⣓⡶⠅⣱⢿⡾⠷⣶⣿⡟⠩⠑⠂⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡔⢳⡿⠋⠀⠀⢹⣿⣿⣷⣬⣉⠉⠁⠐⠛⠛⢴⡋⣈⣉⠠⠤⠀⠀⠀⠀⠈⠳⠴⣍⣩⣥⠤⠴⠟⠓⠂⠀⠈⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠞⣴⠋⠀⠀⠀⠀⠀⢻⣿⣿⣦⣝⠓⡶⣶⣶⣶⡦⢠⣤⣤⣤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣀⣈⣈⣁⣀⠀⠀⠀⠀⠀⢈⡿⠿⠛⠚⠛⣒⡀⢤⣼⠆⠚⠂⢸⣛⣋⣩⠤⣄⣀⣀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠀⠀⠀⠰⣏⣀⡀⠀⢈⣀⣈⣱⠂⠈⠀⠀⢰⠿⠷⠿⠿⠶⠦⠭⠿⠿⠷⠶⠤⠤⠤⣤⣤⡾⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⡈⠉⠉⠉⠉⠉⢀⣀⣠⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠁⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⣶⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⠀⡀⠀⠘⣿⠷
⠀⠀⠀⠀⢠⡄⣠⣤⠀⣄⠙⢦⣭⣭⠾⠃⠀⠀⠀⠀⠀⠀⠀⣰⡋⣠⠠⢠⡀⣬⠻⡄⠀⠀⠀⠀⠀⣰⣆⣀⣈⠹⡇⠾⣿⠆⣼⠃⢀⣴⣄
⠀⠀⠀⠀⠸⣆⣻⣟⣡⡟⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣥⣉⠀⠀⠁⣸⡷⠀⠀⠀⠀⠀⠙⠻⠉⠙⣧⠿⣤⢯⡼⣧⣾⠋⠿⠋
⠀⠀⠀⠀⠀⠈⣿⣿⡇⢀⣀⣼⠏⢻⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⢰⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣿⢻⣇⠙⠉⠁⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠸⣧⡀⠛⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⡎⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⢠⣄⠀⢀⠈⠁⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣤
⠀⠀⠀⠀⠀⠀⢨⡰⣿⣇⣾⢿⣤⡅⠀⠀⠀⠀⠀⢸⡇⣿⣿⢂⡇⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣤⠤⢘⣿⡟⣿⣼⣷⠻⡃⠤⣤⠀⠀⠀⠀⠙⣶⣶⠏⠀⠘⢿⠿⣀⠀⠀⠘⢿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠈⢹⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠸⡿⣶⣦⣿⣛⡻⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⡀⠶⠞⢻⡆⢀⡀⠶⠞⢻⡄⠀⠀⠀⠀⠀⠀⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠠⠞⠉⠛⢋⠙⠛⠉⠳⠆⠀⠀⠀⢀⣤⣼⠇⢹⡧⠀⠀⣾⠷⠾⠃⠀⠀⣾⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠟⠁⠀⠀⠿⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⡀⡀⡀⡆
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⣀⣀⠀⡀⠀⣀⡀
⠀⠀⠀⣰⠖⣶⣯⣉⠀⡀⣠⡀⡤⢤⢠⣠⣤⡤⡄⠀⠀⡰⠒⢒⠒⠒⡌⢃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢿⣿⠇⢹⡄⢿⡿⢢⡟⠸⣿⡿
⠀⢀⠀⠉⣼⣿⡟⠙⠛⠑⠋⠓⠓⠚⠘⠘⢛⠓⠃⠀⠀⢇⠀⠈⠀⠀⡇⢸⠠⠄⠄⠄⢸⠀⠀⠀⠀⢰⣿⣿⡆⠀⢹⣿⣿⡏⠀⢰⣿⣿⡆
⠀⠈⠛⠾⠁⠘⣧⠀⠀⠐⠿⠿⠿⠷⠿⠿⢿⠿⠂⠀⠀⠀⠁⠀⠀⠉⠃⢸⠔⠀⠂⠂⠊⠀⠀⠀⠀⢒⣭⣽⡕⠒⢺⣯⣯⡗⠒⢪⣿⣭⡓
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡇⠀⢸⡏⢹⡇⠀⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠈⠁⠈⠁⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠰⠶⠆⢄⢠⠢
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⠤⠂⠀⠀⠀⢀⠄⠈⡠⠋⠉⠑⢆
⠀⠀⠀⠀⢀⡀⣠⢃⠤⠇⡠⠀⢀⠀⢠⠁⡎⢸⡀⢰⡀⠀⠈⠑⠲⠤⡀
⠀⠀⠀⡴⠋⣀⠤⠂⣀⠔⠁⡰⠃⠀⡠⠀⠀⠘⢦⣀⠉⠀⠠⡀⠁⢢⠙
⠀⠀⠰⡇⡞⠁⠔⠉⠀⠚⠋⢀⠖⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⠈⠑⡄⠀⠆
⠀⠀⢠⠛⠀⠎⠠⠐⠀⠒⠛⠉⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠡⡀⢸
⠀⠀⢻⡔⠚⢠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠿⠁⠀⠀⠀⠘⢆⣃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡠⢠⡠⡀
⠀⠀⢰⡍⠻⡜⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠬⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡧⢲⡃⢾⡤⡞⣹⣀⠀⠀⡠
⠀⠀⠈⢷⠀⢣⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠈⠉⠉⠉⠁⠛⠾⢟⣾⡃⡊⠁
⠀⠀⢠⡀⢂⢸⠳⣄⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⡠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅
⠀⠀⠠⡈⠉⡠⣄⢈⡳⢤⣀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣀⠤⠎⢐⡡⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠒⠶⡤⠤⢤⠤⡖⠒⠉⠳⠔⠦⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⠔⠋⣠⠋⣝⢒⢫⠼⠀⡀⠀⠀⠀⣠⢀⠀⠀⡔⠉⠉⠲⡀⠀⠀⠀⢡⡀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⣹⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⡄⣀⠐⠁⠀⢀⡃⠀⠈⠁⠉⢸⠀⡇⠀⠀⠀⡙⠧⠀⠀⡸⡀⠀⠀⠃⠀⠀⠀⠀⠫⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠮⠱⠴⡄⠤⠴⠖⢋⢸⠀⠀⠰⢸⠀⡇⠀⠀⢰⠇⠀⠀⠀⠐⠔⡂⠀⢂⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⢠⠇⢀⠀⣰⠀⠀⢀⡄⠀⠀⠀⠀⠀⡆⠹⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡛⢲⠶⠒⠶⣖⠋⡩⠤⢤
⠀⠀⠀⠀⠸⠄⢠⡀⠉⠀⠀⠀⡆⠀⠉⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡀⠀⠽⣇⠠⠤⠤⠤⠤⠂⠀⡗⠋⠀⠀⠀⢸⡜⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀⢠⠁⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣳⡀⠀⠓⢂⠤⠴⠖⠒⢂⣾⠀⢀⡄⠀⠀⠈⡤⡁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠞⠀⠆⠴⠵⣬⢳⠀⠀⠀⢀⠔⠉⣠⠀⢒⠃⡉⠁⠼⣍⠢⡄⠓⢒⡒⠛⣃⣛⡦⠴⠫⠤⢤⠤⢾⡗⠀⢌⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⡠⠂⠀⠀⢠⠘⢄⠀⠧⠋⡁⢀⣀⢄⡠⠞⣠⠃⠀⠈⣧⠤⠤⠤⠤⠤⠤⠤⠤⠴⠾⠶⠦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡴⠊⠘⠓⢤⠉⠐⠒⠀⠀⢀⣀⡒⠛⠀⠑⣄⢶⠤⠤⣤⠄⣤⠤⠤⠤⠤⣤⡤⠤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢇⠀⠀⠀⡰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣠⠃⡰⠧⠤⠤⠶⠛⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣭⣁⣈⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠃⢰⠇⠀⠀⠀⠠⠤⠠⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠒⠒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠉⠉⠉⠀⠐⠂⣤⠞⠀⠀⠒⠒⠒⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⡟⠉⠉⠉⡟⢁⡎⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠡⡆⠀⠀⠀⠀⠀⠐⠶⠞⠒⠒⠺⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣴⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢹⣿⠏⠙⣿⡍⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡏⠀⡄⢹⣧⡄⣤⠀⠀⠀⠀⡄⢠⣤⡤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠉⡦⢁⣟⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠈⢸⣷⢸⡿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣿⣀⣀⣸⣿⣌⣁⣇⣻⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢩⣩⠏⡈⢀⡹⣿⠁⢹⣯⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⢀⠋⢸⣵⣿⠃⢹⠀⣾⣆⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣾⣿⠀⠀⢈⣿⣿⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢿⡄⢀⣸⣿⣿⢀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⠀⠐⠉⠀⢡⠐⠒⠒⠛⠛⠲⠒⠒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⣈⣤⡔⣤⣤⣄⠀⠀⠀⠀⣴⣷⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠂⠀⠀⢶⣶⣍⣉⣈⡍⠋⣼⣶⣦⠀⠀⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⣳⡀⠰⢾⣏⣿⣿⣿⣿⣿⣻⣿⡇⠀⢠⡟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣉⡻⠟⠛⠑⠀⠚⠛⠋⣿⠟⠛⢻⡟⠛⠒⠒⠿⠛⢛⣿⣟⡲
⢑⣄⡠⡀⠀⠀⠆⢠⡄⠀⠁⠀⠈⠉⡅⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⢳⡇⠲⠤
⠀⣿⣍⠬⣷⣦⣿⡏⢳⡀⠀⠀⣤⣤⣦⣀⠀⠀⠃⠀⠀⠀⠀⠀⠐⢦⡤⠀⣩⠉⠈⠉⢻⠇⣸⣿⠇
⢙⢿⠿⡄⢻⡿⣿⣷⣾⣤⡀⠀⣿⣿⣇⠀⠀⠰⣤⡆⠀⠀⠀⠀⢀⣽⣄⣴⡿⡧⠀⠖⣻⣶⣿⣿⡇⠀⣶
⠉⠀⣠⣶⣿⣷⣿⣿⣿⣟⣧⣰⣿⣿⣿⡁⢀⠀⡿⢷⢠⡀⢀⢠⡿⣿⣿⡟⣿⣀⡀⣸⣿⣿⣿⣶⣅
⢐⣻⣿⣿⠟⠋⠈⠉⠉⠛⠙⠛⠛⠿⠻⣿⠽⠿⢷⠾⣧⣤⡄⢻⣿⣿⣿⡿⠿⡿⠿⣿⢿⠿⠋⠋⠉⢠⡄⣸⠁
⣿⣿⣿⡿⣠⣤⣦⣴⠀⠀⣠⣀⡀⠀⠀⡀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣠⡤⡤⣶⣻⣤⣛⣴
⣿⣿⣿⣿⣿⣿⣿⢏⡐⣾⣿⣿⣿⣴⣟⣹⣿⣿⣯⢛⣛⢃⣾⣻⣿⢳⡇⡂⢉⠉⣿⠿⢾⣿⣿⣇⣿⣿⣿⣿⣿
⣾⣷⣿⣿⣿⣿⣳⣾⣿⣿⣿⣟⣷⣿⣿⣧⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣡⡇⠹⠇⣿⡷⢾⣿⣿⣿⣿⣿⠿⣿⠛
⠙⠉⠉⠙⠛⠛⡛⠛⣻⣿⡿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠆⠆⣿⣇⢘⣿⣿⠿⠛⠉⣀⡉⢀
⠀⠉⠉⠁⠉⠉⠉⠸⠏⠐⣿⠉⠹⠛⠀⠀⢿⡛⠋⠉⣸⠿⠿⣿⠿⣿⡿⠷⠖⠚⠋⠉⠉⠠⣶⣶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠻⠶⠤⠤⠦⠴⠤⠤⠴⠠⠤⠤⠦⠅⠤⠤⠴⠰⠦⠤⠤⢤⠤⠄
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠶⠀⠄⠿⣿⣀
⠀⠀⠀⠀⠀⠀⣻⣗⠀⠀⠀⠀⠀⠀⠉⢩⡌⢐⣿⣷⡘⠿⠿⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⣛⠓⠀⠀⠀⠀⠀⣦⣸⣿⣿⣷⢰⠄⠀⠀⠀⠀⠀⢠⢿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠛⠻⠛⠛⠟⠛⠀⠀⠀⠀⠀⠘⠛⠟⠛⠻⠛⠃⠀⠀⠀⠀⠀⠘⠻⠛⠛⠻⠛⠀⠀⠀⠀⠀⠀⠛⠟⠛⠻⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣄⣸⡿⣤⣄⡀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠀⠛⢋⣭⡉⠋
⠀⠀⠀⠀⢀⡛⣿⣿⣿⠄⠀⠀⠀⠀⢰⣿⣿⣷⡀⠈⠉⢿⣿⡾⠀⠀⠀⠀⠙⠾⣿⣿⣷⡞⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡃⢰⡟⢿⡁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠿⠿⠟⠿⠿⠿⠀⠀⠀⠸⠻⠟⠛⠻⠛⠃⠿⠀⠈⠻⠄⠀⠀⠀⠸⠟⠛⠛⠛⠿⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣾⡆
⠀⠀⠀⠀⢠⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⢸⣿⣿⣡⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⢦
⠀⠀⠀⢿⣿⣿⣿⣷⣷⠀⠀⠀⣶⣶⣶⣶⣶⣶⠀⣻⡿⣯⡇⠀⠀⠀⢈⣽⣿⣿⣿⣷⣶⡆⠀⠐⣶⣶⣶⣶⣿⡟⢻⡄
⠀⠀⠀⢸⣿⡿⠿⣿⣿⠀⠀⠀⡿⠿⠿⠿⠿⡿⣰⠟⢀⣿⣿⠆⠀⠀⣾⠹⡿⣿⡿⠿⢿⡇⠀⠀⠿⠿⠿⣿⡿⠇⠀⣱⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠛⠗⠊⠀⠀⠀⡯⠥⠠⠄⠢⠴⠇⠀⢀⠂⠁⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣤⢀
⠀⠀⠀⡌⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⡤⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠁
⠀⠀⠀⠐⠄⠀⠚⠀⠀⠤⠀⠀⡰⠆⡅⠠⠃⢡⠀⢀⡞⠉⠀⢀⠀⠙⠄⠀⢠⡤⠀⠀⠄⠠⠤⡀⠀⠰⠭⠜⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈
⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠮⠀⢰⡂⠰⢀⡀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡕⠀⠐⠀⠀⠘⠀⠂⡀⠀⠺⠀⠐⠈⠣⠐⠂⠀⠀⠀⠀⠠
⠠⠒⠂⠀⠀⠀⠀⢤⢢⣤⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠱⠆⣀⣀⡀⠔⠃⠀⠀⠈⠠⠄⢃⠀⠁⠲⠀⠀⠓⠀⠀⠁⠀⠀⠁⠂⠀⠀⠄⠈⠁⢀⡄⠀⠰
⠀⠊⠁⠀⠀⡪⡿⠑⣯⢹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠠⠀⠰⠀⠀⠀⠀⠘⠒⢂⠉⠀⠀⠠⡄⠀⠀⠀⠀⠠⡮⠁⠀⠐
⠀⠀⠂⠀⠀⠀⠖⠴⠛⠁⠀⡄⠀⠀⠀⠰⡖⠀⣀⠀⠰⣃⠂⠠⠀⠐⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠊⠀⠀⣄⡐⠀⣚
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠄⠁⡤⠁⠠⠀⣤⠇⠀⠤⠀⠀⠀⡔⠁⣀⢄⡀⠀⠄⠀⢠⠀⠴⠘⠀⠀⡄⢀⡁⠦⡀⠀⠈⠚⠁⠐⠊⡀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⠠⠒⠀⠀⠠⠠⡀⣠⡀⠀⡠⠀⡄⠀⠀⠀⠐⡅⠌⠃⡐⠁⠠⠆⠀⠀⠀⠀⣀⠀⠀⠆⣠⠄⠖⢛⠀⠀⠀⠀⠀⠀⠁
⠀⣀⠀⠀⠀⠀⠀⠀⠠⠂⢉⠀⠢⠀⠀⠀⠙⢺⢃⢹⡏⡀⠇⡏⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣻⡯⠎⠃⠀⠩⣄⡠⠌⠚⠀⠀⣀⠀⠀⠀⢀
⠁⠀⠉⠀⠡⠀⠀⢠⠂⠄⠒⠀⠀⡃⠀⠀⠀⠀⠐⠀⠀⠊⠀⠙⠈⠀⠀⢀⣀⣀⡀⠀⠀⠀⣴⡿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣰⠨⣜⡿⣃⠀⠁⢇
⠀⠒⠀⠈⠀⠀⢠⠀⢰⡀⣀⣀⣸⠀⠀⠀⠀⠀⠤⠄⠠⡀⠀⠀⠀⢀⠞⠁⢀⠀⠈⠑⠀⠀⠑⠛⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠈⣇⢀⡹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠉⡍⠀⠀⠀⣐⢢⢤⣌⢁⠀⠐⠀⠀⢰⡀⠰⣀⠠⠀⠀⡆⠀⠀⣀⠦⣀⠀⢀⡄⡠⢄⠀⣨⢆⠀⠀⠀⢀⠠⠐⠀⠀⢶
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠹⢮⣄⣞⣯⢋⡿⠀⠀⠀⠳⠆⣀⣀⢀⠔⠁⠀⠀⠰⠂⠂⠀⠨⣧⣤⣝⣤⢤⡜⠀⠀⠀⠀⡠⠤
⠀⠠⠄⠀⠀⡀⠈⢠⠀⠀⠀⠀⡆⢐⠀⠀⠘⣿⡎⢟⣽⠉⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⣰⡄⠀⡀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠐⡛⠂
⠀⠀⠉⠀⠀⠄⢀⠀⠀⠀⠀⠠⡁⢈⠀⠀⢰⡞⢞⡽⠀⠀⠰⠀⠀⠀⠒⠀⠀⢰⡚⠓⢀⠤⣀⣹⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠧⠀⠰⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⣟⡌⠀⠀⠀⠄⠐⠂⠤⢀⡅⠀⠈⠥⠨⠀⠀⠀⠘⠀⠀⠀⠀⠆⣉⠡⠘⢑⠀⠈⢢⠄⠀⠤⢼⠁
⠀⠀⠀⢀⠄⠀⠀⠀⡾⠀⠀⠠⠤⠤⠄⠀⠈⠊⠀⠀⠀⠒⠂⠐⠐⠀⠂⠘⠀⠀⠀⠀⣀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡈⢹⠀⠀⢸⣒⡚⠛⡟
⠈⠀⠀⠈⠀⣠⠀⠘⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⣀⠁⠸⠀⠈⠃⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠁
⠀⠸⢄⠀⠀⠉⡏⡉⠉⢹⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠄⣀⢑⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠂⠑⠀⠀⢠
⡀⠂⡘⡂⠀⠀⢠⠀⢀⡞⠀⠀⢀⠈⠀⠒⣈⠀⠘⠄⠀⠄⠀⠀⠠⠀⠃⠀⠃⠙⠀⠛⠀⠀⢴⣉⡤⣉⠂⡧⠀⠀⠀⠓⠠⠓⡀⠀⠀⠀⠒⠂⠀⠀⣎
⡶⡶⢺⠃⠀⠀⠈⠋⠉⠁⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⢏⠝⣬⠀⠀⠀⢘⠀⠀⠁⡁⠀⠀⠀⠀⠘⢀⠀⢙
⠊⢶⡍⠀⠀⢀⣠⠤⢦⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⢐⣀⠀⠀⢰⡁⠀⠰⠄⠀⠀⠀⠱⣍⣫⡖⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢘
⠓⠊⠁⠠⣞⠿⠖⠓⠋⠈⠈⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠂⠀⢰⣀⣪⠐⠜⢓⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠀⡀⠇⠁⠀⠀⠈⠀⠀⠁
⣀⠀⠀⠀⠈⠃⠠⠄⠀⠀⠀⠀⢀⠰⠁⠐⠄⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠈⠡⣀⡤⠐⠋⠠⠄⠀⠀⢐⣦⠃⢶⢱⡥⠀⠀⠀⠀⢪⠰⡄⠀⠀⠀⡤
⠛⠂⠀⠀⣀⠀⠀⠀⣠⠀⠀⢰⠢⣔⠈⣉⡃⠀⠙⠿⠟⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠪⠤⠴⠚⠁⠃⠀⠀⠀⢨⡀⠼⡀⠀⠀⠃⢀
⠉⠀⠀⠀⠙⠁⠀⠤⠀⠀⠀⠸⠤⣄⠈⣸⠃⠀⣀⠠⢀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠈⠃⠀⡃⠈⢤⠀⠈⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠑⠒
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠁⠀⠠⡅⠁⠈⠷⠀⠀⠀⠀⠀⠰⠗⠂⣀⡀⠀⠀⡉⢄⠰⠖⠂⠀⠀⠀⣰⠶⠂⠑⠀⠀⠑⠒⠀⠁
⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⢄⡀⠛⠛⠛⠓⠃⠀⠀⠃⠑⠒⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⠤⣧⠀⠀⡆⠘⢵⠶⠟⠸⠿⠆⠗⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⢠⠀⡇⢀⠄⣤⡀⣰⢠⡄⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠀⠃⠃⠇⠸⠣⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠾⠁
⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣀
⠀⠔⢂⠶⠘⣆⣁⡀⠔⠒⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠖⣒⡒⠒⠤⡀⠀⠤⠒⠀⠀⠀⠀⠀⣠⠖⢻⡀
⡏⠙⠳⣄⣀⣭⠤⣽⣄⣀⡼⡄⠀⠂⠀⠀⠀⠀⣯⠀⡀⠀⠀⠀⠀⠀⠈⢶⠼⡄⢌⢑⣦⠄⠀⠠⠟⣩⠾⠛⠁
⠇⠀⠀⠀⠀⠀⠀⠀⡜⢻⠾⠃⠀⠀⠀⢀⠀⢀⣧⣠⣀⣀⣌⣠⣄⣦⣤⣴⣾⠇⠀⠄⡉⣁⠴⠚⠉
⢰⢰⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⠄⠀⠜⠀⢧⡈⢙⣛⣚⣛⡓⠋⠍⢝⡪⠟⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠋⠢⡀
⠘⠤⢤⣤⣤⣠⣦⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠚⠃⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠆⠀⠀⠁⠰⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠯⠄⠉⠙⠲⣄⠀⠀⠀⢹⡀⠀⠀⠀⢠⠏⠀⠀⠁⢀⣤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⣀⡒⠁⠀⠂⠀⠀⠀⠀⡠⠁⠂⠀⠀⠀⠢⡀⠑⢄⠀⠀⠃⡀⠀⠀⠘⠀⠀⠀⠈⠍⠉⡉⣝
⠀⠀⠀⠀⠲⠅⠀⠀⠀⠀⠀⠉⠁⠀⠀⢄⠀⠀⠀⣰⠁⠀⠀⠆⠀⠀⠀⣈⣦⠄⠆⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠐⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⢀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠃⠀⣠⡴⠚⠉⠀⠘⠀⠀⠀⠀⠀⣸⠀⠀⠀⠠⠀⡆⠀⠀⠀⠀⢰⣼
⠀⠀⠀⠀⠀⠠⠠⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠛⠁⢀⣠⣤⡦⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠔⢉⠉⢳⠀⠀⠀⠀⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⣠⠞⠙⠠⠁⠉⠉⠀⠉⢱⡀⠀⣀⣠⣁⣉⣡⡀⡀⠀⠀⠸⣽⣋⣦⣄
⠀⠀⢆⠀⢸⡄⠀⠀⣴⠟⢪⠟⠈⠐⠒⠣⣄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣥⣾⣽⠀⠀⢉⣏⠉⠹⡟⡇⣧⠄⠀⠀⠈⠉⠀⠈
⠀⢰⡃⠀⢸⠃⠀⣺⣛⠔⠁⠀⠀⠀⠀⠀⠈⡀⠸⣄⣀⣤⠔⢲⠟⣿⣥⡴⣚⣡⡆⠀⡇⠢⢴⡽⠿⣏⠀⠀⠀⠀⠀⠀⢀
⠀⡇⡅⠀⢸⠀⠀⠈⠓⠢⠤⠤⠀⣀⠀⠀⠤⠁⠀⠈⣭⣄⣀⣋⣭⣾⡟⠋⢱⡜⠀⠀⢇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠁⡠
⠀⢹⠁⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢅⣀⣉⣉⡠⠴⠒⠉⠀⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠣⡜⠁⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠂⡀
⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⣀⣤⣴⠟⠛⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢶⠈⠢⡀
⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⡙⠉⠉⠛⠉⠑⡉⠀⠠⠐⠗⠛⠉⠉⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡿⣼⢄⣶⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠆⣀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡸⢰⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠
⠀⠀⠀⠀⠀⠀⢸⡃⣿⣿⣿⣿⠀⠀⠀⠈⠛⠛⠉⠁⣸⣿⠿⣿⡿⠿⢿⣿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠁⣿⠇⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⢿⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠄⠘⢿⣦⡀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠈⠉⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡘⣀⠁⠐⠲⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡖⠒⠀⠈⠁⠙⠛⠿⠿⠿⠯⠝⠉⠉⠁⠀⣤⣶
⠀⠀⠀⠰⣄⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠖⠒⠒⠂⣀⠠⠤⠄⠀⢤⠤⢤⡤⠴⠶⢛⡛
⠀⠀⠀⠀⠻⡄⠀⠀⠀⢸⡀⠀⠀⠣⠀⢠⠀⠀⣦⡎⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⡀⠁
⡀⢀⠀⠀⠀⠂⠀⢠⠀⠀⠁⠀⠀⠀⡄⠀⡄⠀⠈⢠⠄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠌⡀⠀⠀⠀⠀⡀⠼⣇⠀⠀⡇⠀⠰⠃⡄⠃⠀⠰⣧⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠠⠟⠀⠀⢀⣀⣠⣀⣉⣀⣀⣇⠀⠀⠀⠁⠈⢀⠠⠄⠤⠤⠤⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣑
⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⢀⠀⠀⠒⠂⠀⠀⠀⠒⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⡀⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡆
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⢹⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠋⠉⠉⠉⢉⠉⠛⠙⣻⠦⠐⡒⠀⠁⠀⠀⠀⢀⠀⠀⠐⣻
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠀⠀⡀⠀⠀⠀⣀⡄⢀⠖⢀⡀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠈⣾
⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣄⠀⡀⠀⢀⣀⣠⣀⠀⠀⢀⠀⠄⢀⡈⡉⠀⠀⠐⠂⠂⠀⣀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢸⠚⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠖⠙⠆⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠶⠦⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠛⠳⡶⠲⠶⠓⠔⠂⢀⠀⠀⠖⠒⠂⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⡐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠿⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠑⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠇
⠀⠀⢀⣾⠿⠳⡀⠀⠀⠀⢀⡀⠀⠰⠁⠊⠗⠢⠐⠈⠹⡄⠀⢀⣰⠁⠀⢗⡀⠀⣀⠀⠀⠈⢁⢀⡠⢦⡀
⠀⠀⠀⣹⠀⢀⢦⠀⠀⠀⠉⠃⡰⠁⡤⠀⠀⠀⠀⠀⡀⢻⢠⣿⣯⡣⠥⣺⣿⠘⢡⠁⠀⠀⢀⡸⠀⢤⣳
⠀⣠⣾⠁⠑⠘⠈⡃⢠⢂⠊⠘⠤⠶⠑⠄⠀⠀⠐⣲⠀⡼⢀⠃⣉⠩⢛⠉⢹⠀⠀⠚⢳⣀⠀⠁⠀⣸⠸⣦⡀
⣼⣿⡇⠀⣤⣴⡿⣅⠁⡜⠒⠃⠄⢌⠤⢗⣒⡶⢖⠓⠒⠓⠺⡀⠘⡢⠮⢤⡘⠤⠤⢰⠈⠏⡀⠀⠀⣿⠀⣿⣿
⢻⣿⠧⠀⣟⠋⡴⣇⣽⡶⡀⠀⠀⠣⣛⡟⠽⣛⡠⠐⢻⠀⠀⠀⣙⠝⠚⠃⠈⡔⣡⣔⢓⠰⡧⠄⢰⡏⠀⣿⣿
⠀⢻⣆⠈⠋⢛⠡⢥⠋⡙⠀⠀⠀⠀⡀⣈⣉⡀⢀⠀⠉⠀⠺⣯⠭⢭⡟⠀⠀⠣⣋⠩⣨⠋⣀⣤⡏⣀⣠⣿⡟
⠀⠈⢟⠟⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⡺⠁⡈⠊⠉⠀⠘⢿⡿⣿
⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣭⣤⣤⣤⣤⣤⣤⣽⣎
⠀⠀⣷⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣼⣿⣯⣿⣿⣿⢿⣯⣿⣿⣯⣿⣿⣽⣿⣛⣿⣏⣽⣿⣿⣿⣿⣏⣿⣿
⠀⢀⣀⡠⠬⢭⠉⠉⠉⠉⣉⣉⣭⠉⠉⠉⠉⠉⣉⣉⠉⠉⠛⠛⢛⣛⡛⠛⠛⠛⠛⠛⢛⣛⡛⠉⠉⠛⠉⢙
⠀⣮⣀⣀⣀⣙⠒⠒⠚⠉⠁⠀⠻⠤⠴⠒⠚⠉⠁⠻⠤⠴⠖⠋⠉⠰⢇⣠⡤⠴⠒⠋⠁⠐⠧⠤⠤⠖⠋⠙
⠀⠙⠚⠓⠿⠿⠽⠯⢿⣽⣿⣿⣿⣿⣖⣶⣲⣖⣾⣲⣲⣖⣶⣲⣶⣖⣶⣲⣖⣶⣲⣶⣖⣶⣲⣖⣖⣿⣿⡿⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠠⢤⡈⣷⡀⠀⣀
⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⢀⢀⣴⡞⢹⣧⣿⣱⡡⣥⢾⡼⡆⠀⠀⠀⠀⠀⣠⠀⢤⢀⣆
⠀⢰⠁⠀⠀⠠⠀⠀⠄⠀⠀⠈⡆⠀⠈⠢⣇⣧⡞⣼⡟⣸⣸⡕⡯⢁⡯⠃⠀⠀⢀⠈⠀⢱⣶⣿⣷⣀
⠀⢳⣂⡂⠀⠀⠀⠀⠀⣀⣀⣈⢗⡄⠀⠀⢹⠙⠃⠷⠴⠿⠧⠶⠜⢊⠁⠀⢠⠔⠉⠁⠈⢁⣉⣉⣀⠀⠩⠶⢤
⠀⣶⣦⣽⣭⣭⣿⣹⣷⡤⣼⢵⡞⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠖⠢⠖⣄⠖⣤⠒⠤⠲⡔⠶⣾
⠀⢫⣽⡯⠹⢖⣛⣻⣷⣾⣿⢿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡻⠶⠶⠶⠶⠖⣖⣒⣒⣲⣟⡇
⠀⠀⠑⠂⠀⠀⠀⠀⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣀⣀⡜⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⣀⢀⣀⠤⠔⠒⠉⠒⢄
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⡠⠤⡴⠁⠘⠉⠀⠀⠔⠠⠕⠀⠐⠁⢁⡈⠀⡼⢒⡶
⠀⠀⣀⣔⣊⣁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠬⠵⠦⠀⠀⠑⢢⣘⣌⣀⣉⣀⢀⣀⣐⣒⣒⣂⣈⣡⣴⡊⣤⠋
⠀⠀⢒⣒⣶⣶⣶⣶⣶⣀⣈⣏⣉⣻⡿⠯⣿⠏⡹⡄⠀⠀⠓⣖⠪⡭⢿⡶⢲⣲⣒⣶⣿⣿⣿⡿⢔⣵⡿⠂
⠀⠀⠁⠐⠚⠒⠚⠒⠒⠂⠀⠀⠉⠀⠈⠈⠁⠀⠉⠁⠀⠀⠀⠈⠁⠒⠀⠉⠉⠓⠂⠐⠒⠓⠒⠉⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⣀⠔⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡔⢣⠃⠀⠀⠀⠀⠀⢰⣍⡠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡃⢸
⠀⢸⠁⠘⠎⣓⡦⠀⠀⠀⠀⢠⠒⢶⡀⠲⢥⠇⠀⠀⣠⠤⠀⠐⢆⡹⠲⠶⢦⣀⠀⢀⢔⢒⣄⢠⠚⡟⣿⠛⠛
⡀⠮⡝⣉⣉⣩⠖⠳⡀⢠⠋⡀⢁⠉⣈⢢⠜⡄⠀⠀⢿⡓⠒⠒⠒⠒⠒⢒⣚⠏⠀⠰⠧⠄⢉⠎⣣⠿⠉⠘⠂
⠑⠶⠤⣤⢤⠾⣶⡚⠀⠘⣍⠙⠈⢉⡠⢯⠴⢸⠀⠀⠈⣦⣒⣙⣭⣋⡑⠠⡼⠀⠀⢀⠀⣰⣯⣾⡋⣦
⠀⠘⣮⡱⢪⠼⣱⠃⠀⠀⠡⢍⡉⠉⠉⣀⠔⠁⠀⠀⠀⢰⠄⣰⢀⢀⠀⢰⠃⠀⠀⡗⠀⠨⠝⠉⠀⣺⡗⠉⡄
⠀⠀⠘⡾⠿⠾⡾⠀⠀⠀⠀⠀⢀⢙⣉⣀⠀⠀⠀⠀⠀⠀⠄⢽⠈⠤⠀⡏⠀⠀⠀⠹⡉⠛⠛⠓⢈⡽⠧⠕⠁
⠀⠀⠀⠐⡄⢠⠃⠀⠀⠀⠀⢈⡀⠀⠀⢀⡹⠀⠀⠀⠀⠀⢠⣞⡘⢚⣾⠃⠀⠀⢠⡖⠚⠒⠂⠒⠋⠙⢱⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⣀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⡿⣧⣤⣧⣙⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣾⠟⠛⢿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⣧⡤⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣈⠛⢳⡌⠁
⠀⢀⠃⠀⠀⠐⠠⠀⠀⠀⠀⢠⣭⡿⠶
⠀⡈⠀⠀⠀⢀⡍⠀⠀⠀⢠⣄⣈⠙
⠀⢃⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣉⠁⠀⠀⢀⣄⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢂
⠀⠀⢂⡠⠄⠁⠀⠀⠀⠀⢤⣇⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠰⢼
⠀⡄⠈⠀⣀⣴⣷⡀⠠⣎⣹⡎⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⣱⠊
⠀⠀⢱⣾⣿⣿⣿⣷⡀⠈⠩⣀⠆⠀⠰⣶⡾⠿⠿⠿⠿⢿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠢⡔⠉⠁
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⠘⠛⠛⠠⠀⡄⠀⢀⣧⠄⠄⣤⠀⠀⠠⠀⠀⡀⢀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣤⠘
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣁⣂⣀⡀⢂⣐⡂⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣉⠻⠯⠉⠙⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠗⡤⢈⢲⣶⠶⢶⡦⠀⢸⣿⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⡟⡀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠁⠐⠂⠠⠀⡀⢀⡀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠤⢂⣶⣄⠀⠀⠀⠀⠀⢈⣡⣾⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣶⣿⣿⣿⣷⣤⣤⣴⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠛⢋⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣀⣠⡴⠞⢁⣠⣴⡴
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⡈⠉⠥⠶⠾⠟⠋⠁⢀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣤⣶⠖⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣈⡁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠠⠴⠾⠛⠉⠉⠙⠛⠓⠒⠒⠀⠀⠀⠘⣿⣿⣿⣿⣿⡃⠀⠈⠟⠛⠋⠙⢋⣈
⠀⠰⠶⠶⠶⠾⠛⠻⠿⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡛⢄⠀⠀⠀⢀⡀⠆⢈⡀⡀⠀⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠨⠜⠓
⠀⠀⠀⣠⡯⠠⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡴⠠⡞⠁⠀⠀⠘⠢⢐⠉⢉⡀⣐⠒⠴⠆⣐
⠂⠀⠀⠸⣄⡀⡴⠀⠀⢀⡄⠀⠀⠀⠀⡐⣙⠀⠀⠀⣐⣉⠉⠀⢠⠀⠘⢤⠀⠀⠀⠀⠀⠀⠉⠐⠀⢈⡄⠒⠀⠀⡀⣀
⠐⡄⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⡈⡇⡀⠔⠁⠀⢦⡉⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠊⠒⠀⠂
⠀⠈⠀⠠⣤⣤⣴⠴⠋⠁⠀⠀⠀⠀⠀⠀⠶⣉⡄⠐⠤⢂⢡⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠤⠒⠁
⠀⠀⠀⠀⢠⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⢠⡀⢀⣤⣄⢠⣤⡄⣠⣤⠀⣠⣤⡀⣠⣤⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⣿⡇⠘⣧⡉⠀⣿⠀⣿⣀⠀⡏⢸⡇⣿⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠉⢰⡷⣿⢠⡌⣷⠀⣿⠀⣿⠁⠀⣯⢸⡇⣿⢸⡇⣿⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠐⠴⡀⠅⠠⠄⠀⠀⠀⢂⠀⢀⡤⠁⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠉⠉⠀⠉⠀⠉⠀⠀⠉⠉⠀⠈⠉⠀⠉⠉
⠀⠀⣀⢲⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠔⠭⠭⣷
⠀⠉⠒⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡮⠄⢀⣀⠴⠚⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⣀⡈⡔⡆⠚⠰⡤⢤⠀⡀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⢤⡉⠉⠉⠁⢠⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢃⠈⠀⠀⢀⠀⠄⡈⢿⠀⠀⠀⠀⠀⠙⠛⠛⠛⠓⠾⠿⢃⣔⡉⠸⠤⡡⢤⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠂⢀⠀⠘⢀⠄⢚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣹⣼⣿⣿⡿⠋
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠑⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠴⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠉
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠡⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⠁⠐⠀⠀⠴⠀⠀⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⢠⢖⠢⡔⠀⠀⠠⠄⢒⣤⣀⡐⣪⡰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⣀⢐⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠸⣄⠈⠛⠓⠀⠈⠲⠄⠀⣀⠄⠚⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⢉⣸⠘⢢⡌⡐⠺⢏⠀⢀⠀⠈⡦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠘⢤⢀⡀⠀⠀⠔⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⡀⠼⠄⣀⡀⠀⠈⠁⢉⡈⠓⡢⠊⡡⢶⡄⢂⠶⠀⠀⠀⠉⢄⡀⢀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠐⠬⣑⠂⠒⠐⢃⡁⠶⠊⠀⠀⠀⠀⠀⠀⠓⠠⠤⠶⢆⣀⠈⠉⢠⠜⠉⠀⠣⠀⠀⠁⠆⠀⣲⡄

⠀⠀⠀⠀⠀⠀⠀⣠⢶⠒⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⡏⣽⢒⠛⢛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⣦⠀⠀⠀⣠⣶⡟⠉⣩⡉⢷⡄
⠀⠀⠀⠀⠀⠀⣰⠃⣆⠈⢁⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠾⠞⠻⣇⠀⠀⢹⠁⢲⣟⠉⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢻⣶⣾⠶⢬⡾⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣃⠂⢀⠁⡞⠀⠀⢸⡗⢀⡹⢀⡀⣼⠁
⠀⠀⠀⠀⠀⢠⣴⠛⡇⠓⣤⣧⣵⢠⠙⢆⠀⠀⠀⠀⠀⠀⠀⣈⠷⣛⡾⢤⣄⡀⠈⢧⣰⣤⣔⣒⣛⣷⣤⣤⣄
⠀⠀⠀⠀⢀⡏⢘⡤⠷⢾⣤⣅⠈⠘⡆⠈⠀⣀⣠⠤⣤⣀⣼⣹⣀⠀⠀⡼⠀⠱⣠⠶⠚⠋⠁⠈⢳⠀⠀⠀⢈⢳⡀
⠀⠀⠀⠀⠸⣇⡏⠀⠀⢠⣰⣿⠇⠤⠒⢺⡉⠁⠀⢀⠀⠲⠀⠀⠀⡉⠉⠉⢲⢺⡇⠈⣇⠀⠀⠀⠈⡇⠀⠀⠈⡆⢳⠲⣦
⠀⠀⠀⠀⠀⣏⣂⣔⣔⢈⡷⡇⠠⠤⢔⡎⡠⢰⠓⠒⠒⢠⢀⠂⠀⠁⠀⢀⣯⣼⡇⠀⡝⠢⣀⣠⡤⠴⠖⠒⠒⠻⢿⠶⣿
⠀⠀⠀⠀⣠⣽⡿⠿⢟⣯⠜⠁⠀⠀⠀⠀⠐⠈⠑⠀⢀⡈⠀⠂⠀⢀⡀⣿⠀⣟⠀⢸⠓⢺⠋⠁⠀⢀⣀⣠⣤⣀⡞⢀⡇
⢠⣤⣄⣾⠃⠐⢋⣉⠛⢧⡤⢤⣀⣡⣌⠉⠆⠀⠀⠠⠁⠀⠀⠠⠄⠀⠀⢿⡀⣟⠀⣼⠀⢸⠴⠚⠉⠉⠀⠀⣀⣟⠀⢸⠁
⢁⠀⠀⠙⢦⣀⣄⣸⣦⢀⠀⢀⣿⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣈⣷⡀⠀⣿⠀⢸⠀⢤⡴⠦⠍⠓⠒⢘⠇⣸
⠘⡦⢄⡀⠀⠹⡏⠉⡟⢦⣜⡠⢧⢀⡟⠀⠐⠒⠤⢤⣀⡀⠀⠤⠖⠚⠉⠁⠈⠳⠤⣿⠀⡟⠈⢀⣀⡤⠴⠒⠋⠉⠀⣿
⠀⣃⠀⠈⠑⠢⣧⠀⢣⢸⠟⠦⠴⠛⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⣿⠀⡷⠊⠉⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠙⠓⢦⣄⡀⢸⠀⢸⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣀⡀⠀⠉⠻⡆⠈⣇⣠⠤⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⡀
⠀⠀⠁⠈⠙⠒⠀⠧⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠤⠤⠤⡆⢠⣀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⣀⣀⣀⣀⣠⡄⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⣀⣸⠀⠀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⣸⠀⢉⠉⠉⠙⡋⠉⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠆⢀⠠⠀⢠⠃⠀⠀⠀⣀⢰⡇⠀⠲⠌
⠀⠀⠀⠀⡀⠈⠀⡇⠀⠀⠃⠐⠀⡀⢀⠀⠀⢀⡡⠷⠓⠚⠉⠉⠉⠉
⠀⠀⠀⡞⢲⡺⣄⢡⠀⢀⠆⢠⡇⠁⠈⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡀
⠀⠀⠀⣇⢸⡍⠳⣆⣰⠆⣠⠋⣰⡠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⡢
⠀⠀⠀⢻⠘⡇⠀⠈⠉⠛⠛⠛⠉⠀⣾⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣿⣿⣷⣾⣶⣤⣀⣳⣄⣦⣄⣠⡴⡊⢁⣀⣤⣖⣁⣤⠾⠁
⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⢃⠠⠉⡛⡛⠛⠿⣿⠿⢿⡿⢿⣿⣿⡿⠿⢿⣿⠛⠙⠻⡤
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⡈⢀⣌⣉⡒⢒⢫⣀⡴⠶⡾⠵⠶⢥⠤⠤⠤⢤⡤⣮⣭⣵⡿⠃
⠀⠀⠀⠀⠀⢹⣇⠀⢀⣀⣀⣀⣀⣈⣍⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⢈⠀⢀⣴⣿⣿⡟⢁⠇
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠓⢤⣀⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠦⠤⠤⠀⣀⣀⡤⠴⠒⠋⠁

⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⢄⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣏⠉⠉⠉⠁⠤⠤⠄⠠⣿⣿⣿⣿⣿⠇⠉⠉⠉⠉⠉⢹⡇
⠀⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⠆⠀⠀⢀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⡗⠀⠀⠀⠀⠒⠂⠀⠀⠀⠈⡃⣸⣿⣿⣷⢠⣆⣥⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣿⣿⣿⣿⣾⣾⣶⣷⣾⡟
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⣿⣿⠏⡉⣭⣩⣭⣿⣯⣭⢻⡄
⣀⣷⣹⣿⡶⣦⣤⣀⣀⠀⠀⠀⠀⠉⠁⠶⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢹⡇⣿⠉⢸⣿⣿⢝⠺⠟
⢿⡿⢹⣯⡇⢈⣿⣿⠿⢿⣿⣶⣶⣤⣠⣶⣶⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⡩⠂⢸⡇⢠⢀⣤⣿⣿⣾⣿⣿
⣴⡷⢾⣿⡇⣴⣿⣼⠲⢛⣿⠿⣿⣿⣿⣷⣮⣄⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⡣⠐⠈⠀⠈⠁⣀⣀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣶
⠠⠇⠸⡷⠧⢴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⣼⡇⠀⠀⠀⠀⠨⠿⠿⣿⣾⣇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⣀⣀⣀⡀⠀⠀⠈⢿⠿⣿⣿⣿⣿⣿⡿⠷⠶⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣬⠤⠀⠠⠀⠈⠧⠀⠤⠂⠀⠀⠀⢠⣾⣿⣭⣿⠀⢨⣾⣾⣿⣶⣄
⠀⠀⠀⠁⠀⠀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠠⠀⠉⠁⠀⠀⠀⣀⣉⣥⣦⣀⣀⡛⠻⠿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠉⠉⠉⠉⠙⠒⠶⠷⠧⢮⣥⣤
⠀⠀⠀⠀⢼⡿⠿⠿⠿⢿⡇⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡄⠀⠀⠀⢲⣤⣤⣄⡀⠀⠀⠀⠐⠲⠄
⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⣷⣶⡾⠿⠛⠛⢛⣩⣿⣿⣿⠿⠛⠛⢛⣿⣿⣿⠟⠛⠛⣿⣿⣿⠛⠀⠀⠀⢨⡟⠛⠛⢿⣶⣤⣀⠠
⣿⣿⣿⣿⡟⠃⠀⡀⠀⠀⡏⠀⠶⠒⠚⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⣰⣶⣶⣶⣶⡿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠛⠉⠁⢰⣿⣿⣿⣿⣏⠁⠀⠘⡍⠉⠉⠉⠉⠓⠒⠦⢄⡀⠀⠀⠈⠈⠙⠛⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠠⠊⠀⠀⠘⣿⣿⣿⡟⠁⠀⠀⠀⢓⡌⠉⠉⢢⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠱⢄⡀⠀⠀⡈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣅⣤⡆⠀⢖⠣⣀⢀⡴⠄⠂⠉⠒⠢⡂⠀⠙⣄⠀⠀⠈⢉⠀⢄⠀⠑⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠀⠀⠡⠁⡰⠁⠀⠀⠀⠀⠀⠸⡐⠘⡆⠑⠀⠀⠀⠀⠀⢷⣄⠀⠠⡀
⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠒⠓⠀⠀⠀⠋⠀⠀⢠⢃⠀⢱⡀⠠⠀⠌⠀⠀⠘⡌⣆⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠁⠉⠉⠙⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣐⣺⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣸⣓⠠⡅⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠈⠍⢢⠁⣀⠝⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⡌⢱⠁⠂⠈⠦⠀⢢⡀⠂⠀⠀⠀⠀⢠⣄⣀⠠⠍⠀⠀⢰⢉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠋⠈⠁⠁⠀⠀⡀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠂⠑⠀⠀⠂⠀⠂⠐⠀⠀⠀⠒⠀⠘⠀⠀⠀⠀⠀⢟⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⡤⠀⠀⠀⠀⠔⠤⠜⠡⡗⠒⠊⠩⠭⠭⠄⢶⠒⠒⠒⠒⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠝⠒⠉⠏⠑⢲⠀⡒⠂⠊⠉⠉⡏⠁⠀⠀⠄⠂⠀⠀⠀⠑⠒⠤⠤⣈⣉⣙⠒⠒⠠⠤⠀⢉⠗⢖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠄⠁⡀⠠⡤⢀⠀⠀⠂⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠂⠤⠤⠤⢀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⣐⡦⠬⠮⠤⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠒⠂⠄⡀⠐⠂⠄⠈⠉⠉⠀⣉⡀⣀⠀⠩⠑⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⡞⢒⠀⣓⡄⠄⣠⠕⠋⠀⠀⠀⠀⠀⠀⠀⠑⠨⡂⠀⢰⡓⡒⠀⣳
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠐⢐⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠂⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠈⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⡀⡁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠄⠐⠃⠄⣠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⡀⠀⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠡⠀⡁⠀⠁⠁⠀⠀⡀⡠⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⠦⠒⠀⢦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠁⠀⡂⠁⠀⠀⠁⠀⠀⠀⡠⠀⡀⠱⣖⠂⠒⠐⠒⢒⡖⠂⠒⡐⠒⢐⢂⠒⣶
⠀⠀⠀⠀⠁⠂⠀⠤⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡌⠠⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠠⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠋⠉⠉⢩⠁⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠐⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠂⠈⡀⠅⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠊⠀⡃⠠⠀⠀⠀⢀⡂⠐⢈⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣁⣃⣀⠤⠔⠚⠁
⢀⡾⠷⡀⠀⠂⠐⠀⠀⢈⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢤⠠⠂⠀⠀⠀⠁⠈⠈⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⡇⣿⠀⠀⠀⢘⠀⢀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣯⢹⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣿⣸⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣶⣿⣧⣼⣿⣷⣇⣠⠀⠀⠀⠀⠀⠀⠀⣸⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣛⣛⣛⣛⣋⣉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠟⠁⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠻⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠟⠙⠿⣿⡿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣄⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⢸⣀⣨⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠔⠊⠉⢣⠀⠞⠉⠉⠈⡆⠀⡤⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⡔⠋⠁⠘⡌⠆⠀⠀⠈⡇⢸⠀⠀⠀⠃⡸⠀⠀⠀⡹⡰⠋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠋⠉⠄⠀⠀⠀⠐⠀⠠⠌⠉⠓⠲⡄⠀⣀⠼⠚⡄⠀⠀⠸⣀⡄⠀⠀⢸⠃⠀⠀⢠⣡⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠁⠤⠀⠒⢀⠀⠁⡀⠀⢀⠀⠀⠈⠀⠴⡀⠘⡄⠀⢱⠀⠀⠀⢻⡇⠀⠀⡜⠀⠀⠀⡇⠀⢩⠂⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠁⠀⠀⠀⡀⠒⠀⠀⠀⠀⣀⠁⠀⠐⠀⢱⡀⠰⠀⠀⡆⠀⠀⠸⡇⠀⢀⠃⠀⠀⢸⠃⠀⡎⠀⡎
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡆⠀⠀⠀⣾⣯⡷⠀⠀⠀⠀⣧⠀⠇⠀⢣⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⢰⠁⣴⠁
⠀⠀⠀⣀⣀⠠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⣾⣿⣶⠈⠉⠁⠀⠀⠀⠀⢡⠑⢼⡀⢸⠀⠀⢠⠁⠀⣸⠀⠀⠀⡇⠀⡬⠊⠹
⠀⠀⠀⠑⢄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣹⡟⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠉⠚⠦⠤⠼⣀⣀⣿⡀⠤⠴⠓⠉⠀⠀⡇
⠀⠀⠀⠀⠀⡽⢦⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⢰
⠀⠀⠀⠀⢸⡀⠀⠙⠢⡉⠑⠒⠠⠤⠤⠄⠀⢀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠖⠒⠛⢳⠀⠀⠁⠀⠈⠀⠀⣀⠀⠀⠁⠀⠀⠁⠀⡜
⠀⠀⠀⢀⡤⢟⠑⠢⠤⠜⠦⡀⠀⠀⠀⣀⠠⠤⠒⣲⠪⣉⠁⠀⠀⠀⢀⣀⡀⠠⢼⠄⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⡞⠹⡋⠛⠋⠋⠉⠛⠋⠉⠙⠙⠉⠋⠁⠑⠚⠛⠒⠚⠋⠁⠒⠒⠒⠘⡇⠀⠀⠀⠀⠀⠘⢿⠏⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠙⢢⠍⣂⣀⣀⣀⣀⣀⣀⠠⠤⠶⠶⠄⣀⠐⠂⠐⠂⠀⠐⠀⠂⠀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⢀⣀⡠⠽⠦⠤⠔⠚⠋⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠤⠤⠤⠴⠒⠉




⡿⠿⠿⠿⡿⠻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿