Foraging Emojis & Symbols

⠄⢀⡤⠘⠁⢀⡀⣼⣺⣄⢍⣼⢹⣿⣆⠀⠀⠘⠄⢳⡀⠉⠉⠉⠋⠀⠀⢀⠄⢀⡤⠜⠃⢀⡀⣰⣿⣥⢍⣱⣩⣿⣦⠀⠀⠘⠆⢲⡀⠉⠉⠉⠀⠀⠀⢀⠔⠀⡤
⡠⠊⣠⠔⣮⢛⠀⣼⣿⢿⢯⠾⣟⣿⣿⣦⠀⠀⣠⡄⣉⠀⠀⣀⣀⠀⠐⠃⠠⠊⢠⠔⣊⢚⠀⣼⣻⢿⢻⡶⡟⣿⣿⣦⠀⠀⣠⠤⢈⠀⠀⢀⣀⠀⠐⠃⠀⠈⢀
⠀⠀⢷⣿⡴⠃⢸⣿⣿⣿⣿⢿⣿⣻⣿⡿⠀⠂⠁⠀⢻⣭⠁⣉⣋⡹⣓⡄⠀⠀⢷⣾⡴⠃⢰⣿⣿⣷⣿⣷⣟⣻⣿⡿⠀⠈⠀⠀⠹⣭⠅⢉⣛⣫⢳⡤⠀⢀⠶
⠀⠀⣀⢀⡀⠀⠈⢿⡿⣿⣿⢻⣿⣿⣿⣷⠀⠀⠈⠂⠈⠻⠟⠛⣿⣥⣮⠁⠀⠀⢀⢀⡀⠀⠀⢿⣿⣿⣿⢻⣿⣿⣿⣷⠀⠀⠈⠀⠀⠻⠿⠛⢿⣧⣮⠁⠀⠀⢀
⡄⠀⠥⠀⢻⡆⠀⠈⠶⢿⣿⣿⣿⣫⣿⠋⠀⠀⠀⠀⠀⠈⠘⠛⢮⠙⣿⠄⣄⠀⠳⠀⢹⣆⠀⠈⠲⢯⣿⣿⣷⣯⣿⠋⠀⠀⠀⠀⠀⠈⠈⠛⠮⡙⣿⠤⢀⠀⠏
⠁⠀⠐⠄⠈⠳⠀⠀⠀⢀⡆⠀⣜⠉⠀⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⣀⢀⠠⠀⠀⠀⠀⠈⠹⠀⠀⠀⠀⡴⠀⢼⠉⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠈⢣⡀⣤⡀⢠⠟⠁⠂⢹⡆⢠⠀⡄⢠⠧⣉⣰⡤⠂⠀⠀⠀⠁⢀⡀⠀⠀⠀⢣⣀⣠⣀⠀⡞⠁⠂⢘⡆⢠⠀⠠⢀⡼⣈⣲⡤⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣏⠁⡘⠀⠑⠃⣤⠞⠀⢸⠐⡃⠀⠈⠁⠀⠀⠀⠐⠶⠚⡰⠟⠁⠀⠀⠀⠀⣏⠀⡘⠀⠙⠃⣤⠞⠁⢠⠁⡀⠀⠈⠁⠀⠀⠀⠐⠺⠚⡠⠿⠃
⠀⢀⡀⢀⡀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠻⠞⠁⢞⡁⡀⢀⠀⣀⠀⡀⠀⠤⢴⠂⢀⡀⠀⡀⠠⠄⠁⠀⠀⠀⠀⠀⠀⠸⠼⠁⢺⣁⠠⢀⠀⣀⠄⡀⠀⠤⠤⠆⢀⡀
⠸⠟⠂⠀⢭⣠⡤⠀⠀⠀⠄⠖⠲⠂⠀⢠⠀⠺⢅⠲⡦⠐⣰⡄⠀⠀⠁⠓⠸⠟⠂⠀⢭⣠⡤⠂⠀⠀⠄⠖⠒⠆⠀⢀⠀⠸⢅⠒⡮⠐⣠⣆⠐⠀⠉⠛⠨⠟⠇
⠀⢀⣀⣠⢋⡉⠀⠀⠀⠰⣤⣤⣤⠖⠚⠋⠀⠀⢀⣀⠇⠖⠻⠲⠚⢅⢲⣆⠀⠀⣄⢐⠊⣉⠀⠀⠀⠰⣤⣴⣤⠖⠚⠋⠀⠀⠀⣀⠵⠐⠸⠲⠞⢍⡲⣆⠀⠀⣀
⣤⡟⠉⠊⠁⠀⢠⡀⠀⢀⠈⠛⡉⠐⢆⠀⠀⠀⣞⣀⠠⠈⠠⠷⣀⠈⠀⢩⣤⡿⠋⠉⡁⠄⢠⡄⠀⢀⠈⠛⣉⠐⠖⠀⠀⠀⢾⣀⠠⠀⠡⠴⣄⠀⠁⢉⣄⡼⠋
⣀⠠⠛⠉⠐⠶⠀⠁⠀⠁⠀⠀⠹⣤⡀⢄⠀⢀⠀⠈⠀⠒⢀⣾⡛⠡⠈⠁⢠⠠⠟⠉⠑⠶⠀⠁⠀⠈⠀⠀⠙⣄⣀⢠⠀⢀⠀⠀⠁⠔⢀⣼⡟⢧⠐⠁⢈⡄⠛
⠁⠀⢀⠘⢣⠀⠀⠀⣠⣶⣶⣶⣦⡈⠑⠋⠀⠈⢦⠀⢀⠔⢋⡀⣀⠀⠀⡁⠁⠀⢀⠐⢳⠀⠀⠀⣀⣴⣶⣶⡦⡈⠉⠉⠀⢘⢧⠀⠀⠔⠋⡈⣀⠀⠀⡀⠀⠀⢀
⡠⠚⢁⡠⠇⠀⠀⢰⢿⡋⠻⢬⢿⣷⡀⠀⠀⢸⡈⣆⠘⠒⠒⠀⠤⠂⠋⠀⡠⠘⢁⣠⠎⠀⠀⢠⣯⡋⡷⢥⢿⣷⡀⠀⠀⠸⡘⣧⠈⠒⠒⠂⠤⠀⠉⠀⡠⠀⢋
⢀⠔⠉⠀⡄⣾⠁⢻⢾⣿⡾⣫⡽⣿⣿⡄⠀⠀⢁⠈⠢⠀⠀⠀⠀⠀⠠⠞⠁⠐⠋⠀⡄⣶⠁⣺⢾⣿⠾⣻⡼⢾⣿⡄⠀⠀⠉⡈⠳⠀⠀⠀⠀⠀⠀⠜⠁⡐⠋
⠀⠀⣎⠤⣉⠆⢠⣷⣿⣼⣾⣬⣷⣿⣿⣿⠀⠔⠋⠉⢷⠶⡒⠺⠿⢶⢤⡐⠀⠀⣞⣤⣊⡔⢠⣿⣿⣞⣾⣧⡧⢿⣫⣿⠀⢐⠊⠉⢷⡶⠒⠚⠿⢶⢤⡀⠂⠀⣸
⠀⠀⠉⠛⠁⠀⠘⣯⣿⣿⣿⣾⣿⣿⣿⣏⠀⠀⠠⠄⠘⢿⣴⣿⣿⢃⣚⠃⠀⠀⠈⠛⠉⠀⠘⣷⣿⣿⣿⣿⣷⣿⣿⣯⠀⠀⠰⠀⠘⢿⣵⣾⣿⣆⣸⠋⠀⠀⠈
⠀⠠⣒⠐⢦⡀⠀⠙⣯⣿⣿⣾⣿⠿⣿⠟⠀⠀⠀⠈⠀⠠⠴⢤⡙⠿⣿⡀⠀⠀⢖⠀⢶⡀⠀⠙⣭⣿⣿⣾⣿⠿⣿⠟⠀⠀⠀⠈⠂⠠⠴⠤⡝⠿⣿⡀⠀⠀⣔
⠃⠀⢀⠀⠘⢧⠀⠀⠈⠉⡛⠉⣻⠛⠉⠀⠀⠀⣠⠄⠀⠀⠀⠀⠈⠀⠉⠁⠛⠀⠀⠀⠘⢷⠀⠀⠈⠉⡛⠉⢻⠛⠉⠀⠀⠀⣠⠤⠀⠀⠀⠀⠈⠀⠉⠁⠙
⠀⠀⠀⠀⠄⠈⠀⠀⢀⡴⠃⡄⢻⡀⠀⠀⡀⠀⢏⠶⣀⣀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢄⠈⠀⠀⠀⡴⠃⡄⠹⡀⠀⠀⠄⠀⢃⠴⢄⣀⡀⠀⠀⠀⠓
⠂⠀⠀⠀⠈⡖⠫⠱⠘⠤⠄⢀⣸⠁⢸⢢⠇⠈⠁⠓⠘⠉⠀⠀⣤⢰⢩⣽⠇⠀⠀⠀⠈⡖⠛⢱⠀⠧⠄⢀⣠⠃⢘⠂⠐⠈⠁⠒⠚⠉⠀⠀⣤⣰⢀⣽⠆
⠀⠀⠀⠀⠀⠓⠐⠂⠀⠀⠀⠉⠁⠀⢼⣡⠇⣤⠀⡀⢀⡀⠀⠀⠀⠈⣉⢀⡀⠀⠀⠀⠀⠛⠀⠊⠀⠀⠀⠉⠁⠀⢸⣠⠆⣤⠀⢀⠀⠀⠀⠀⠀⢈⣉⢁⡀
⢢⣴⡀⠀⢒⠈⢁⡀⠀⠀⢀⣠⢀⡀⠈⠁⠀⣰⠊⡅⣐⠀⢨⠀⠀⠀⠡⣍⢠⡴⡀⠀⢒⢈⢁⡀⠀⠀⢀⣠⡄⡀⠀⠁⠀⢸⡒⣉⢀⠀⢈⡌⠢⠀⠍⣭⠰⡴⡂
⠁⠀⠀⢩⠴⠛⠋⠀⠀⢀⡀⣀⢀⣠⡥⠼⠀⠀⠁⠀⣍⣈⣿⣭⣠⡄⣀⡀⠈⠀⡐⠩⠴⠛⠋⠀⠀⢀⠀⢀⠀⣀⣤⠼⠂⠀⠉⠀⣩⢀⣻⣭⣤⡄⣄⡀⠈⠀⡀
⢠⣴⠖⠿⠖⠀⠀⠀⠀⠈⠙⡿⠿⢠⡀⠀⠀⠀⡰⠚⢁⠀⢂⣄⠀⠐⠈⠓⠀⣶⠖⠾⠖⠀⠀⠀⠀⠈⠙⢿⠻⠀⡠⠀⠀⠀⣠⠘⠁⠀⢂⣠⠀⠐⠊⠛⠀⣰⡖
⠋⠁⡤⠐⠢⣀⠙⠇⠀⠄⠀⠀⢷⠀⠁⠀⠀⠀⠁⠙⢀⡠⠤⣠⣷⢦⠀⠈⠛⠁⡤⠐⠢⣄⠈⠇⠀⠐⠀⠀⢳⠀⠀⠀⠀⠀⠉⠘⠀⡠⠄⣠⣷⠆⠀⠊⠙⠁⡤
⠁⠈⠀⠠⢄⠀⠉⠀⠀⣠⣤⣤⡀⠛⠎⠿⠀⠐⡄⠀⠀⣠⠞⠛⠁⠈⠀⠄⠔⠀⠀⠠⣄⠀⠉⠀⠀⣠⣤⣤⣀⠙⠞⠹⠂⢸⣄⠀⠀⢀⠔⠙⠁⠈⠀⠄⠈
⠀⠠⠖⢀⡎⠀⠀⢠⢾⠿⢿⢉⣿⣆⠀⠀⠀⢠⠹⡄⠰⣀⣬⣀⢒⡀⠔⠁⠀⡠⠖⢀⡼⠀⠀⢀⣾⡿⣟⡙⡿⣧⠀⠀⠀⢸⢸⢆⠠⣅⣬⣀⡒⡈⠴⠀⠀⢠⠖
⠃⢠⠖⠉⢀⣠⠄⢾⣽⣤⣠⢽⡼⣿⣧⠀⠀⠈⠒⠳⣄⠀⠀⠀⠁⠀⠀⡠⠊⢠⠖⠈⢁⣠⠄⢸⣿⣶⣡⢿⡼⣿⣷⠀⠀⠈⠃⠙⣄⠀⠀⠀⠀⠀⠀⡠⠊⢀⠖

⠈⠠⢒⣩⡾⠖⠁⠀⠀⠐⠉⠉⠐⠁⣀⣀⡆⠀⠙⠪⢙⡛⠠⢾⣖⢀⣈⣀⠈⡁⠘⠁⠁⢒⢘⣴⠷⠋⠀⠀⠐⠉⠉⠀⠁⢐⣀⡦⠀⠙⠢⠅⣚⠠⢰⡷⡂⣀⣀
⣀⣄⢴⣡⠄⠠⠀⠀⠀⠫⢶⣾⢴⢉⠉⠁⠀⠀⠀⡠⠴⠉⠩⠍⠉⠣⠡⣌⠻⡤⠀⣀⣥⣴⣁⠤⠠⠄⠀⠀⠊⢶⣾⢶⡏⡉⠉⠀⠀⠀⢠⠤⠪⠊⠍⠉⠃⠉⢤
⠋⠀⢉⠐⠂⠠⡤⠀⠀⡄⠀⢫⣌⠈⠃⠀⠀⠀⠰⠧⣎⠐⠀⢐⠚⢦⣀⠀⠀⢤⡮⠟⠁⢀⡀⠂⠀⢤⠀⠀⠀⠀⠉⣤⠁⠃⠃⠀⠀⠀⠯⢤⡐⠂⠀⠒⢱⣀⡀
⠘⠋⠁⠈⠳⠥⠈⠀⠈⠀⠀⠀⠹⣤⣀⠤⡀⠀⡄⠀⠀⠀⠒⢁⣼⡿⠻⡅⠁⡄⠤⠄⠋⠉⠈⠓⠦⠀⠁⠀⠁⠀⠀⠘⢤⣄⠠⣄⠀⣄⠀⠀⠑⠐⠈⣴⡿⠟⠥
⠀⠀⠐⢳⠀⠀⠀⢀⣠⣶⣶⣶⣤⡈⠑⠚⠁⠀⡹⡆⠀⢀⠔⢋⡀⠀⢀⠀⢘⠕⠁⠀⢀⠐⢻⡆⠀⠀⠀⣠⣶⣶⣶⡦⡄⠉⠚⠉⠀⡎⣆⠀⠀⡔⠛⡀⠀⢀⡀
⠐⠁⢠⡟⠀⠀⠀⣾⡽⠋⡽⣸⢿⣷⡀⠀⠀⠀⡇⢻⡄⠸⠤⠶⠤⢌⠔⠒⠃⠀⣀⠴⠋⢀⡾⠁⠀⠀⣸⣻⠋⡯⢑⣿⣻⡄⠀⠀⠀⢷⠸⣆⠈⠦⠴⠤⢤⡡⠒
⡴⠋⠀⡀⢤⡆⢸⣧⣧⡶⡴⣫⡲⣿⣿⡄⠀⠀⠈⠂⠳⢄⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠛⠁⢀⢤⣶⠀⣯⣾⡴⣴⢺⣘⢾⣟⣆⠀⠀⠈⠓⠙⢦⡀
⠀⡖⣉⠵⣩⠁⣼⣴⣏⣏⣷⣹⢇⣿⣿⣿⡄⠀⡤⠾⠒⣤⣀⡠⢤⣶⣤⣀⠈⠀⠜⠀⡰⢊⡹⢪⠅⢰⣣⣟⡻⣱⡏⣇⣿⢿⣿⡆⠀⡠⠶⠆⢦⣀⡠⠤⣴⣦⣄
⠜⠻⡽⠞⠁⠀⣏⢿⣽⣿⣷⣻⣯⣽⣿⣿⠃⠀⠆⠄⠀⢹⣦⢆⣠⣭⣠⠫⣵⠂⠀⠠⠻⢯⠶⠋⠀⢿⢻⣿⣿⣿⣿⣟⣽⣿⣿⡇⠀⠂⢀⠀⠙⣦⠦⣠⡬⣡⡎
⠀⣀⣄⡀⠀⠀⢹⣿⢿⣿⡿⣻⣿⣿⣿⣿⡄⠀⠀⠈⢄⡀⠹⢟⠛⢿⣵⣶⡆⠀⠀⠀⣀⣀⣀⠀⠀⠘⡿⡿⣟⣿⢫⣿⣿⣿⣿⡇⠀⠀⠈⠢⡀⠙⠟⠛⠻⣿⣶
⠸⠐⠀⢻⣆⠀⠀⠱⠾⣽⣿⣿⣿⣩⣿⠏⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⣍⢻⣧⠤⡄⠰⠑⠀⢹⣧⠀⠀⠘⠶⢿⣿⢿⣿⣏⣽⡟⠁⠀⠀⠀⠀⠀⠈⠉⠙⠓⢌⠹
⠈⠢⡀⠈⢻⠀⠀⠀⠀⢀⡇⠀⣾⠉⠀⠀⠀⠀⢠⣖⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠁⠄⠈⠙⡇⠀⠀⠀⠀⡥⠀⢸⠉⠁⠀⠀⠀⠀⣶⠃⠀⠠
⠀⠀⠀⢆⠀⢄⠀⠀⣰⠟⠀⡆⢹⡄⢀⠀⢰⠀⣘⣄⠎⢤⣤⠅⠀⠀⠀⠉⢁⡀⠀⠀⠀⠀⠢⡀⠠⡀⠀⢠⠞⠁⢄⠘⡇⠀⡀⠀⡄⢀⣇⠾⢠⣤⡤⠀⠀⠀⠉
⠀⠀⠀⠈⡖⠫⠙⠇⠹⠤⡀⢀⡼⠁⢸⠃⠘⠀⠉⠐⠋⠋⠁⠀⠀⣤⡖⡦⣴⡿⠀⠀⠀⠀⠀⣵⠛⠍⠷⠘⠦⢄⠀⣠⠇⠀⡗⠄⡇⠈⠐⠋⠙⠁⠀⠀⢤⡴⡢
⠀⠀⠀⠈⠛⠠⠒⠀⠀⠀⠈⠉⠀⠀⢸⣀⡜⢠⡤⢄⣀⢀⡀⠀⠀⠀⠀⢉⠉⢀⠀⠀⠀⠀⠀⠙⠣⠔⠁⠀⠀⠀⠉⠁⠀⠰⣇⡴⠁⡤⠀⣀⢀⡀⠀⠀⠀⠀⠉
⣖⠂⠰⠢⠈⠁⢀⠀⠀⠀⢀⣀⡀⡀⠈⠉⠀⢘⡶⣒⡁⡰⢀⠚⠈⢂⠐⡨⣩⡍⢦⣴⡂⠀⠶⠄⠉⠀⠀⠀⠀⠀⣀⡀⡀⠀⠉⠀⠀⣳⢖⡀⣐⢀⠘⠊⠐⡠⠠
⠈⠀⡒⣩⠾⠖⠁⠀⠀⠈⠉⠁⠈⢁⣀⣈⡆⠀⠙⠪⠙⣛⠠⢾⣓⣀⣈⣀⠀⠁⠊⠁⠁⢐⢈⡾⠷⠋⠀⠀⠐⠉⠉⠀⠁⣐⣀⡷⠀⠉⠃⠍⣛⠀⢸⣗⣂⣀⢀
⣼⣤⢾⣥⠄⠠⠀⠀⠀⠻⢶⣾⣶⢉⠁⠁⠀⠀⠀⡤⠴⠕⠩⠍⢉⠃⠩⣌⠻⡄⢀⣀⣥⣴⣥⠤⠀⠄⠀⠀⠚⢶⣾⢷⡏⡉⠈⠀⠀⠀⢠⠴⠮⠊⠭⢉⠓⠉⢤
⠋⠀⣁⠐⠂⠠⡤⠀⠁⡀⠀⠣⣌⠈⠂⠀⠀⠀⠸⠥⢆⠒⠀⢐⠚⣢⣀⠀⠀⠤⡶⠟⠁⢈⡀⠂⠀⢦⠀⠀⡀⠀⠉⣤⠁⠓⠂⠀⠀⠀⠯⢤⡐⠀⠀⡚⢳⣄⡀
⠘⠋⠁⠈⠓⠤⠈⠀⠀⠀⠀⠀⠹⣤⣀⢤⡄⠀⡄⠀⠀⠐⠊⢁⢾⡯⠳⠄⠁⡄⡀⠄⠋⠁⠈⠛⠮⠄⠀⠀⠁⠀⠀⠘⢤⣄⠤⣄⠀⣄⠀⠀⠐⠒⢉⣴⡿⡙⢆
⠀⡀⠘⢱⠀⠀⠀⢀⣴⣶⣿⣷⡦⡈⠙⠛⠁⠀⡟⡆⠀⢀⠔⠋⡈⡀⡀⠀⢈⠔⠁⠀⢀⠐⢹⡆⠀⠀⠀⣠⣶⣾⣿⡦⡄⠉⠚⠁⠀⡌⣦⠀⠀⡔⠋⣀⢀⢀
⠚⠁⣠⠟⠀⠀⠀⣞⡽⠋⠾⣸⣿⣷⡀⠀⠀⠀⡇⢹⡄⠸⠤⠶⠤⢌⠔⠒⠁⠀⡀⠐⠃⢀⡼⠀⠀⠀⣸⣹⠋⡷⣀⢾⣿⣄⠀⠀⠀⢷⠸⣇⠘⠦⠖⠦⢤⠡⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡀⢀⣀⠀⠀⠋⠀⣤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⠹⣷⣿⢏⡢⡄⠀⠀⠀⠀⠷⡤⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣠⣴⣷⠿⢻⠘⠄⠀⠀⠀⠀⠀⠀⣁⢠⠴⣆
⠀⠀⠀⠀⠀⠀⢀⣶⣴⠛⠋⠀⠀⠀⠀⠀⠀⠿⠟⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠎
⠀⠀⢀⣠⡴⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁⠐⠲⠠⡆⣀
⠀⠀⠉⠀⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⣀⣄⣰⠀⠀⢀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠻⢿⣿⣶⣎⣏⢀⣞⡈⣆⡀
⠀⠀⠀⠀⠀⠀⢀⠤⣠⠞⠐⣤⣤⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠆⠀⠀⠉⠉⠛⢿⠿⢿⣿⣷⢸
⠀⠀⠀⠠⠤⠐⠉⣈⣥⣼⣴⣿⠿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⠀⠀⠀⠈⢘
⠀⠀⠀⠀⠀⠀⠐⣿⠿⠋⡸⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢶⠶⠖⠒⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠐⠠⠀⠀⠠⠤⣖⣯⠍⢛⣃⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢐⢠⣼⣡⣴⣾⡆⠀⣤⣈⣠⣉⠉⠹⠄⠠⠍⡀⢤⡠⠀⠠
⠀⢀⡀⡔⠀⠀⠀⠀⠀⠀⡀⢒⠐⠚⠙⠛⠟⠿⠿⡿⠿⠦⠘⠛⠙⣙⡓⠦⠄⠀⠀⠀⠈⠓
⠀⠀⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀
⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠐⢌⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠠⣤⡦⣤⣶⣾⣿⣤
⡀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣤⠶⠖⣠⠾⠛⠛⠿⣿⡟⡻⡟⠟⠓⢀⣭⣿⠉⠋⠛⠟⠋
⠭⠓⠀⠦⠤⠶⠀⠀⠀⣉⠉⠤⠀⠀⠀⠀⠀⢀⣀⣴⢦⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠒
⣴⣤⣤⣀⡉⠁⠀⠀⠉⡴⠶⠦⠀⠛⠋⡈⣉⡉⠉⠉⠁⣐⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⣽⡟⠀⠀⠀⠀⠂
⠽⠯⠁⠀⠀⣀⣂⣀⣅⡀⠀⠠⠀⢠⡖⠚⠐⠐⢀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣧⡀⠀⠀⠐⠗
⠓⠂⠘⠈⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠤⡤
⡷⠒⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠂⠀⣀⠀⠀⠀⠀⠀⠀⢘⣀
⠀⠀⠀⠀⠀⠀⣸⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢓⠀⠀⠠⠀⠀⡼⠓⡔⠠⠀⠀⠀⠀⢸⡟⢦⡀
⠀⠀⠀⠀⠀⡂⠉⠈⠀⠀⠀⠘⣻⡆⠀⠀⠠⠾⠆⠀⠀⣏⠰⣬⡄⠀⢆⠠⢀⡀⠈⠛⣍⠿⠀⣄⣞⣿⠏⢿⢁⣀⣀
⠀⠀⠀⠀⡀⠀⠗⠀⠂⠀⠀⠀⣮⠁⠂⠀⠘⠓⡧⠀⠀⠀⠀⡏⡃⠰⡄⠁⠒⠛⣄⣠⠾⠉⠀⠈⠸⡏⢸⠀⠛⠹⣧⡈⠀⣄⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⡖⠀⠀⠁⢹⣶⣇⠀⣳⣝⣓⢠⠀⠀⠀⠁⣘⡂⢰⢤⡄⢷⠘⠫⣴⠿⠀⠀⢀⡗⣼⣀⣀⣁⣿⠉⢠⢿⡇
⠀⠀⠀⠀⠀⢸⠀⠀⣉⣤⡄⢀⣾⡏⠈⣧⡝⠉⠉⠀⠀⠀⠀⠀⠑⢡⡸⣄⢤⣄⣠⡄⠟⠛⡆⠆⡘⠷⡈⠉⠁⠹⡿⣤⠋⢸⣅
⠀⠀⠀⠀⢀⡈⠑⢱⣿⣯⠇⠚⢹⡅⠀⠁⣥⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⡅⠘⡀⢹⡣⡄⠲⣔⢘⠀⠀⠁⠀⠀⠀⢳⡌⢶⠀⢿
⠀⠀⠀⠀⠀⠁⠀⣾⡁⣴⠶⣦⣾⡀⣃⠀⣯⡀⠱⠄⠀⠀⠀⠀⠀⠀⡅⣄⠀⢠⡘⣇⢸⡄⢸⢼⠂⠀⠀⠀⠀⠀⠹⠁⠈⠧
⠀⠀⠀⠀⠀⠘⠡⣤⣷⠀⡀⠻⢻⡇⢸⣇⡇⠀⠀⣯⠀⠄⠀⠀⠀⠀⠂⢿⠀⢨⢵⢺⡄⡿⣌⡀⡠⢀⣀⣀⣀⣰⠇⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠠⠶⠭⠛⠛⠀⠘⠳⠶⠸⠁⠠⠤⠿⠀⠁⠓⠒⠒⠦⣀⡚⢢⣼⠃⢼⠃⠰⠈⢣⡁⢀⠀⠈⠉⠉⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⢠⠃⠀⣰⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠃
⠀⠀⠀⠀⠀⠸⠀⠀⠛⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⠃⠀⢠⡀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠛⠛⠵⠋⠠⠀⠀⠀⣀⡬⢿⡀⠀⠀⠁⣠⣶⠃⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⠠⠐⠶⣶⠿⠛⠛⠉⠁⠀⠈⠓⠀⠀⠚⠛⠉⠢⡐⠚⠀⠀⠰⠀⠀⢀
⠀⠀⠀⠀⢀⡄⢿⣿⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣲⠀⠀⠄⠑
⠀⠀⠀⠀⠀⠁⢸⠁⠀⠀⠐⣾⣋⣲⡄⠀⠀⠠⠖⢀⣾⢐⣶⣀⣤⣀⣀⠀⠀⠀⣴⣦⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⡌⢻⠏⠀⠀⠠⣶⣶⣿⡿⢿⣶⣭⣙⠓⢬⣦⠀⠀⠒⠛⠿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠙⠁⠀⠀⠠⣶⠻⣿⣿⡡⣬⣿⣿⣿⣷⠀⠀⡶⣦⡀⠀⢀⠀⠀⠀⠀⠉⠄⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠢⣄⠀⢳⣄⠀⠀⢿⣾⣿⡿⣿⣿⣿⡿⠋⣟⡄⠀⢠⡈⢁⠘⠻⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣾⡙⣿⡿⡇⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠈⠃⠀⠀⠀⠀⠉⠀⠈⠒
⠀⠀⠀⠀⢠⡖⢶⣦⠀⢀⡸⠟⠛⠁⠱⡄⢤⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠸⣼⣿⣿⠇⢀⠀⠀⢀⡐⢲⣄⣼⣿⣿⠆⡀⠀⠀⢰⣤⣄⣀⣄⠀⠀⠀⠀⢀⣴⡠⢿⣿⠃
⠀⠀⠀⠀⠀⠻⠟⢙⣇⣈⡂⠀⠀⠻⣬⣿⠿⣿⡁⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⠑⠉⠁⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡟⠋⠛⠦
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⢹⠏⠹⠻⠃⠀⠀⠀⠀⠀⠀⠀⣴⣰
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⣤⣤⣶⣶⣤⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣀⣠⣄⣠⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣹⣿⡿⠿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣼⣻⢟⣿⣿⣿⣷⣗⠀⠀⠀⠀⠀⠀⠜⢳⠶⣶⣻⣿⠟⠁⠀⠀⠀⠉⠉⠀⠷
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠿⡿⠿⠿⠿⠿⡽⠏⠉⠛⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣁
⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⣀⣸⣿⣿⠆⠀⠀⠀⣸⢿⣿⣿⣿⠙⠿⠇⠀⠀⢀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠒
⠀⠀⣧⣿⣿⣿⣷⠀⠀⠀⡿⣾⣿⣿⣿⡀⠀⡆⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠐⣟⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⡘⡇⠀⢀⡷⣿⣿⣿⣿⡄⠀⡇⠀⠀⠀⢸⣿⣿⠛⠀⠀⠀⣠⡀⠀⠀⠀⠙⢋⣿⣿⡏⢷⠀⠀⠀⢠⣦⢄
⠀⠀⠀⢸⣿⣿⠇⠃⠀⠻⠆⣿⣿⢹⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⢸⣿⣿⡇⠸⠀⠀⢠⣾⣷⠞⠃
⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⣿⡏⠈⣿⡆⠀⠀⠀⠀⠈⢹⡏⢻⡀⠀⠀⣯⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡞⣼⣿
⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⢻⡏⠀⢹⡇⠀⠀⠀⠀⠀⠘⡇⢸⠁⠀⠀⠀⡟⣇⠀⠀⠀⠀⢸⡏⢸⡇⠀⠀⠀⠀⣿⢻⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣶⣶⣶⣶⣾⣧⣼⣧⣤⣤⣤⣴⣇⣘⣇⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢀⡄⠐⠛⢡⢀⣀⣶⡾⢏⣼⣧⣀
⠀⠀⠀⠐⠶⢤⡤⠘⠂⠀⠠⠾⠇⠈⣰⡈⠌⣿⡌⣷⣿⣿⣿⣿⣿⣄⣠⡄
⠀⠀⠀⠀⣀⣹⣿⣶⣶⣶⣶⣶⣦⣤⣬⣥⣤⣽⣷⡞⢻⡇⢹⣿⣿⣿⣿⣯
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⣶⣶⣿⣿⣿⣾⣿⣧⠀⣿⣤⣄⣠⣀⣐⡆
⠀⢀⣼⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡀⢀⡿⣿⣿⣻⣿⣿⣿⣿⣷⣿⣿⣙⣛⡛⠛
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡯⠉⢁⢠⡙⠹⡏⠀⢠
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡟⢻⢿⣦⢾⣿⡻⠍⠛⣀⡀⠸⣿⣿⢸⣿⣏⣺⣇⡀
⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣿⣿⣾⣿⣿⣶⣸⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢺⣿⣿⣿⣿⣿⣽⠿⠿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣝⣿⣿⡇
⠀⠠⢿⣿⠿⠿⠿⠿⣿⡿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣷⣮⣴⣿⣿⣿⣽⣿⢳⣦⣟⡋⠟⡻⣿⣿⣾⣿⣿⡇
⠀⠀⠀⢀⡀⢀⣤⠄⢀⣉⠉⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣷⡍⢻⣿⣏⣽⣿⡿⠿⠟⣿⠻⣿⡇
⠠⣀⣴⠙⠿⣛⣋⡀⣿⣁⣠⡄⠀⠠⢴⣿⣿⣿⣿⣿⡛⠛⠟⠃⠉⠉⠓⠼⣿⡿⠟⠉⠀⠀⠴⠷⠖⠿
⠀⠉⠉⠐⠀⠀⠀⣄⣹⣭⣿⣳⣄⣀⡄⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠶⠿⠿⢿⣶⣶⣾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⢠⡍⠛⢮⡋⠃⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⣀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢴⠂
⠀⠸⡇⠀⢬⣣⡼⠃⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢀⠀⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙
⠀⠀⠀⢀⣼⠌⠀⠀⠀⡀⠀⠠⡷⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⢾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠂⢠⡂⠉⠃⠀⠀⠀⠐⠐⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥
⠀⠀⡀⠀⠀⢸⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠂⠀⠀⠀⠀⠀⠁⠀⠀⠹⣀⡀
⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡎⢀⠀⠀⠀⠈⠀⠀⠀⠠⠂
⠀⠀⠀⢄⡆⢸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠂⠤⠤⣦⣤⡤
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠸⡽⠀⠀⠚⠋⡟⢺⡃⠀⢀⣄⡀
⠀⠀⠀⠀⠘⣀⡃⠸⠁⠀⠀⠀⠀⠂⠀⢸⠀⠧⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠈⢼⣿⡁⠀⠀⠀⠀⣿⠘⠇⠀⣞⣿⡗
⠀⠀⠀⠀⢀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⠉⣸⠁⠀⠀⠀⣠⣧⣄⠀⢀⣸⣿⣅
⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠀⠀⠲⠀⢠⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠂⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⣠⣤⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⡇⠀⢤⡀
⠀⠀⠀⠘⡶⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡈⠀⠈⠀⠀⠠⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⢰⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⡄⢄⠀⠀⠀⠀⢲⡀⡖⢡⠀⠀⣷
⠀⠀⠀⠀⢁⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⡐⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠁⠀⢉⠇⠰⠆⠀⡿
⠀⠀⠀⣤⡄⠀⠟⠊⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⣾⠀⠐⠤⠄⠁⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⡷⠀⠘⠉⠐⠀⠐⠀⠀⡄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠙⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠐⠂⢀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡄⠄⠀⣦⡔⠀⠀⠀⠈⠁⠈⠁⠰⠶⠀⠀⠀⠀⠀⠀⠁⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⣠⠀⠀⢠⡅⠀⠀⠁⠀⠀⠸⠒⠈⠀⣠⠀⠒⠄⠠⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⢃⠄⣄⠀⠀⠀⠀⠀⠂⡀⣤⡄⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⣿⠀⠀⠀⠆⠀⠀⠐⠤⣤⠄⠀⠀⠀⠙⠀⠀⢠⠀⠀⡵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⣿⡍
⠀⠀⠀⠉⠀⠀⣀⠀⠀⣀⠀⢀⠀⠀⠁⠉⠀⠀⠐⠂⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⡀⠀⣀⣀⡀⣼⣿⢟⣽⠀⠀⠀⠀⠐⠂
⠀⠀⠀⢀⠀⠀⠈⠀⢰⠏⠀⠘⠳⠇⠘⠃⢤⠀⠀⠀⠀⢑⠀⠋⠀⠀⠀⠀⠀⠈⡤⡁⠀⠈⠁⠀⠈⠉⠀⠉⡟⠚⠁⠀⣀⠀⠀⠀⠔⠡⠄
⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠀⠀⠀⠰⠾⠶⠂⠀⠀⠀⠀⠈⠋⠀⠉

⠀⠀⠀⠀⠀⢰⠒⠲⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠖⢢⣶⠆⠀⢀⡀⠀⠀⠐⢲⡄⠦⣠⣄⡀
⠀⠀⢀⠀⠀⠸⠀⢷⣶⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⡇⠀⢸⣿⠶⣤⣼⣿⣶⣶⡀⡠⡦⢀⣘⣿⣥⣠⡆
⠀⠀⠸⢿⣷⣼⣶⣘⢷⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣇⣤⣾⣿⣶⣿⣿⣿⣿⣿⣿⣦⢷⣾⣿⣷⣾⡙⡇
⠀⠀⢀⢘⣿⣿⣿⣿⣿⣿⡫⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣥⢸⣿⣿⣿⣿⣿⣷⣍⣿⣿⣟⢫⡅
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧⡄⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⢛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣴⢹⡿⠉⠉⢻⣿⣿⠢⠙⠻⢿⣿⠏⠙⣿⠟⠉⠛⠋⢩⡇
⠀⠀⠘⠁⢸⣿⢿⣿⠟⣿⡻⣿⣿⣿⣿⣿⣷⣈⠰⡦⠀⡀⠙⢋⡁⠀⠀⠀⠀⡀⢠⠀⢧⣄⠀⠀⠄⠃
⠀⠀⠀⠀⠈⠉⠀⣀⡈⣠⣤⣄⣁⡤⢤⡿⣯⡟⠻⣧⡈⠀⠀⠈⠁⢂⠀⠀⠈⠛⣤⡀⣀⡁
⠀⠀⢸⣴⢴⣄⣞⣹⡇⢿⣻⣿⣁⣴⣦⣉⣿⣤⣔⣿⡖⣀⠀⠀⢰⣿⡇⠀⡀⠤⣿⣿⣿⠃
⠀⠀⢸⣿⣧⣉⣹⣿⣿⣿⣦⢿⣿⣿⡿⣿⣿⣿⣿⣸⠉⢹⠗⣤⣬⣭⣧⠡⠀⠘⠿⠿⠿⠓⠀⠀⢀⡄
⠀⠀⠘⣵⣦⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣶⣾⣿⣧⡈⠉⠀⠀⠸⣻⣶⣿⣶⣄⠀⢈⡄
⠀⠀⢸⡽⠛⢛⣿⣽⣿⣿⣿⡿⠟⡿⣿⣏⣿⣿⢿⣿⡿⡿⠿⡿⠻⣿⡦⢄⡠⠖⣻⢟⣩⣿⣿⣧⣾⠁
⠀⠀⢘⣠⢶⣿⡋⠈⣋⠋⢱⣧⣞⢹⣇⣱⣦⣄⣨⣤⡤⠘⣅⡁⠀⠙⠀⣾⣿⠄⠁⠘⣿⣄⢹⣿⣾⡇
⠀⠀⢈⣃⠿⢟⣷⡤⠈⠛⠁⠀⠀⡸⣿⣭⣉⣻⣿⣯⣀⡀⣿⣷⣿⣟⢛⣟⡉⢻⣼⣿⣿⣿⡶⠿⠿⡇
⠀⠀⠈⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠩⠿⠿⣿⡿⠿⠿⠿⠿⢿⢿⠿⢿⣿⣿⣿⠿⠟⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠉⠉⣭⣭⡍⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠙⠀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣶⣶⣿⣿⣷⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⢄⣠⣤⡴⡠⢄⣦⡀⠀⢸⡄⠀⣀⠹⣿⡀⠀⣤⡀
⠀⠀⠀⠀⠀⣷⣾⣦⠀⣠⣄⠨⣿⣤⣽⣯⣽⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣇⣬⡟⣿⣿⣵⣿⣿⣿⣌⢸⣅⣸⣿⣷⣅⠈⠀⢻⠏
⠀⠀⠀⠁⠀⠀⣨⣿⣿⣿⣿⣶⣮⠻⣿⣿⠟⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⡯⠯⠙⢳⣿⣿⣿⣿⣿⣿⣧⡙⠟⣿⡿⣿⣾⡷⠬⢄⡄
⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⢿⣧⢜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡀⡖⣷⡖⢸⣿⣿⣿⣿⣿⣿⣿⢻⣦⣼⣿⣏⣧⣀⢲⣞⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⢿⣿⡧⠤⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣻⣟⣭⣷
⠀⠀⠀⢀⠙⡿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⣠⢉⣿⡿⠛⠛⠻⣿⣿⣶⡂⠈⠙⠻⣿⣿⣿⡟⠉⢿⣿⡿⠟⠻⠟⠛⠋⣹⠿⠻⠆
⠀⠀⠀⣴⡾⠂⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣟⢸⠋⣀⡀⠀⠀⠙⣿⠟⠃⠀⠀⠀⠈⠉⠉⠁⠀⠈⠋⠄⠀⠀⠀⠀⡀⠻⣿⣒
⠀⠀⠀⠈⠁⠀⠀⠰⣿⡗⠛⠟⠋⠁⠷⠄⠻⠻⡿⠿⠛⠻⣿⣿⢬⣄⢉⡗⠀⠤⠀⠈⠀⢴⠀⡀⠀⠀⠀⢀⣤⠀⠧⠀⠛⢧⠄⠀⠀⠃⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡄⢀⠴⢀⣀⡤⠤⣸⠗⣷⡾⠛⠒⣧⡄⠀⠀⠀⠀⠈⠐⢂⡄⠀⡀⠈⠉⢡⣤⠀⣀⡀⠀⠀⠀⠉
⠀⠀⠀⠘⠋⣗⡠⣐⣆⢀⡞⠙⣿⠰⣿⠰⠄⡂⠁⣠⣤⡈⠈⣿⣅⣀⠰⡿⣵⠂⠀⠀⠀⠀⣤⣿⣷⠀⠀⠀⢀⢸⣿⣇⢿⡏
⠀⠀⠀⣰⣴⡟⣿⡌⠹⡺⢿⣾⣿⣦⣿⠀⣀⣽⣾⣟⣿⣿⣿⣿⣿⣿⣷⢛⡇⠞⢿⣤⠀⠀⠛⠻⠇⠠⠀⠀⠀⣾⣿⣿⣮⣴
⠀⠠⣴⠏⠈⣸⠿⠟⠲⣶⣾⣿⣿⣿⣿⣷⢠⣿⣿⣿⡇⣽⣿⣿⡿⣿⣿⠸⢀⡁⠚⢁⣼⣯⠛⠿⣿⠀⠀⠀⠀⣈⡉⢛⣉⡁⠀⠀⠀⠀⡾⠂⢻
⠀⠀⠀⢀⡀⢁⣶⣷⠆⢉⣻⣿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⢛⣹⣿⣿⣿⣦⡀⠀⠀⣰⣶⣏
⠀⠀⠀⠀⣠⢞⢱⠟⠓⢛⣿⡋⣿⣟⣾⣿⣿⣿⠟⣿⠿⢿⠇⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⣄⠀⠠⠴⢞⣽⡟⣉⣻⣿⡿⣦⢰⡟⠉⢻⡄
⠀⠀⡀⢀⠀⡸⠀⣀⣴⣾⠏⠁⠻⢿⠗⢛⣩⣀⠾⠷⡞⠛⢞⡉⠉⠉⠘⣋⡁⡀⢦⠀⠲⠀⠀⠫⠈⢠⣿⣧⠀⠚⠋⢾⣿⡏⠉⣿⣿⣸⣶⣿⡇⠁
⠀⢀⠀⠀⠘⢱⡛⢩⡽⣶⡀⠀⠐⣦⣀⠤⠞⠻⠒⠰⡿⣦⣾⢿⣷⣦⣼⣿⠛⠃⠀⣷⡤⣀⡀⣤⣄⢼⡿⣿⣅⠀⣠⣤⠿⣿⡄⠙⣿⣿⣿⡿⠈⠂
⠀⠀⠀⠀⠀⣀⣁⠼⠛⢉⣯⠤⠄⠀⠀⠀⠀⠀⣀⣘⠻⢶⣤⣤⣈⣻⣿⡿⣤⣀⣐⣿⣵⣿⣧⣭⡉⣼⣃⣀⢸⣤⣿⣿⣦⣿⣿⡿⠋⠫⠻⣧⡴
⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣽⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⣠⡤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⢠
⠀⠀⠀⢀⠀⠀⠀⡀⢂⣀⢸⣿⣥⣼⣯⡷⣾⠀⠂⠀⠀⡀⠀⡆⢴⣢⢐⡚⢀⣠⣶⠟⠁⠀⠀⠀⠀⠀⢀⠼⢿⣿⣿⣭⡇
⠀⢠⡤⢧⣠⣦⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣗⢐⣭⣶⣤⠉⡆⢰⠸⡿⠻⣠⣿⣿⡏⠀⠀⣠⣾⣿⣿⣷⣦⢶⣛⣿⣿⣿⣷⣀
⠀⢸⣿⣾⣿⡟⠈⠉⣉⠙⣿⣿⣿⣿⣿⣿⡋⠘⠉⠙⠛⢡⡘⠘⣠⣅⣞⢿⣿⣿⡄⠀⢠⡿⠛⢿⣿⣿⠟⠀⣉⡉⠹⢿⣯⢉
⠀⠈⠉⢿⡿⣿⣀⢸⠟⣲⠿⣿⣾⣿⢿⡟⠁⡈⣁⣭⣦⠃⣹⣶⣿⠂⣿⣼⠛⣿⣿⣻⣿⣶⣤⣾⣿⣧⠾⣾⣿⣏⡿⢙⠕⡃⣄
⠀⣀⣞⠀⠓⠙⠛⢸⡏⣿⠲⠧⠏⢿⣾⡿⠿⠟⠉⣿⣷⢠⣿⡿⠿⠷⣾⣯⣷⢿⣿⣿⣽⣾⣿⣿⣿⣿⣷⣿⣿⣿⣟⣤⠖⡻⠇
⠀⣿⣿⣦⣴⣄⣀⣾⣿⣿⣤⡀⢰⣶⣿⠙⣾⣶⣾⡏⢿⣶⣾⣿⣶⡌⣧⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣦
⠀⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⡿⢛⣙⠉⠉⢉⣉⠸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣾⣿⣿⣿⣿⣿⢿⣷⠮⣻⣿⡿
⠀⡻⣿⣿⣿⣿⣁⣽⣿⣿⣿⣧⣿⠀⢿⣿⣷⣶⣿⣿⣼⣿⣿⣿⣿⡿⣿⣻⣿⣧⣿⣿⣀⣾⡇⣿⣿⣿⣿⣏⡄⣼⣯⣷⢼⠏⠑
⠀⣿⣿⣯⣁⠹⣿⣿⣍⣽⢿⣿⣿⣾⣏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⠛
⠀⠛⠿⣿⣿⡇⡟⠛⢿⣧⣢⣿⣿⣿⣿⠲⢛⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⢻⡿⠿⢿⣿⣿⣧⡾⣿
⠀⠛⣧⣼⣿⣿⣷⣄⣠⠋⡹⢿⣿⣿⣿⣶⣿⣛⠉⠉⠛⠛⠉⠉⠛⠛⠓⠀⠉⠛⠉⠉⠀⠄⠴⢖⡒⠉⠈⠛⠻⠟⠛⠯⡑
⠀⠀⠈⠛⠾⢿⢿⣿⣿⡞⣷⣷⣩⡾⢠⡟⣹⣏⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⢚⢛⣙⣻⠁⠠⠄⣼⡿⠅⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠈⠉⠙⠓⠉⠙⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⡹⣷⣶⡿⠟⣿
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢳⣴⡟
⠀⠀⠀⠐⠷⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⡠⣀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣻⢹⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠠⣷⠻⠿⠇⣰⣿⣿⣿⣿⣿⣿⡘⠘⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠐⣿⣿⣿⣿⣿⣿⢰⡆⠒⠀⠀⢀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠚⠛⠛⠓⠺⢿⡿⢂⣀⣀⣠⡁⢤⡄⠙⠂⢀⡀
⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠓⠰⣿⡿⠽⡧⠄⠁⠀⠀⠹⣁
⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠈⠇⠀⠀⠀⠴⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟

⣿⣿⣿⣿⣿⣿⣧⣿⠦⢦⣼⣭⣽⣿⣿⣿⣿⡆⡀⠀⡡⡄
⣿⣿⣿⡛⠿⢻⣿⣿⡎⢠⣿⡿⣿⣾⣿⣿⣿⣿⣯⡗⣿⣿⡀
⣿⣿⣿⣷⡈⠸⡿⣿⡿⠀⠘⠀⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡿⢿⢿⣿⠃⠀⠀⠉⠀⢠⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣷⣿⣿⠻⢤⣄⣨⣤⡀⠈⠍⠽⣿⣿⣿⣿⠿⠿⡿⢿⣿⣿⣧⡀⡀
⣿⣿⣦⡄⠀⣤⣤⠬⠉⠋⠐⡶⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠐⠆
⣿⣿⡿⠆⢠⡈⢀⠤⡖⠴⣾⡎⠈⣉⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡷⠚
⣿⣿⣿⣦⣆⣿⣾⢷⣴⣤⣿⣷⡶⢛⣻⣿⣻⢏⣈⡯⣿⠀⠈⠛⠡⠈⠘⠋⡀⢤⠀⢠⡄⠀⠲⠁⠀⣀⠀⠀⢀
⣿⣯⣶⣠⣿⣯⡜⠘⢿⣿⡿⣝⠀⣦⣼⢠⣾⡍⠉⡵⠿⣖⢾⣦⠁⠐⠀⠘⣃⡈⣸⠯⣽⣃⡆⠀⠈⢡⣀⣠⠀⠠⢂
⣻⣯⣿⣻⣷⠶⣄⠀⣾⣿⣟⡇⠀⡍⣛⢸⣿⣿⣿⠇⠀⠛⣴⣇⣠⠁⠀⠀⡙⠻⣝⣀⣹⣏⠃⢀⠀⢙⢻⠃⠀⢀⠇
⣿⣿⣿⣟⣿⠀⠀⠀⣿⡿⣿⡁⠀⠀⠁⠈⣿⣥⣽⡇⠀⠠⢯⣿⣿⣀⡀⠆⠀⠀⠛⣼⡿⠀⠀⣤⣤⣤⠘⢤⠀⢹⡄
⣿⢿⣿⡏⣿⠀⠀⢰⡟⡁⣸⣇⣠⣀⡀⠀⣾⣿⢿⣷⠀⠰⣮⡘⠻⣿⣷⠤⣤⠠⠀⣿⡇⠀⠀⢃⠋⠩⣆⠀⠀⠀⠁
⣿⣁⣙⣂⠻⠇⠂⢻⡋⠿⠿⠿⡟⢻⣿⣿⣿⣿⠨⣿⠇⢀⣠⣻⣾⣻⣿⡁⠀⠀⠀⣺⣧⠄⠀⣸⠀⠀⠈⠀⠀⠀⠇
⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠐⠌⠩⢿⠟⣏⢀⣿⣾⣿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⢽⠂
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠢⠀⠀⢀⠀⠀⡰⠂⠄⠀⠀⠀⠀⣼⠦⡀
⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⢰⣦⠀⠀⠀⡴⠄⠀⠀⠂⢄⣀⠀⡀⠀⠐⠁⠸⢖⢶⠀⣀⡶⣿⠒⡟⢀⡀
⠀⠀⠀⠐⠀⠶⠀⠀⠀⠀⢠⡏⠐⠀⠰⠳⣇⠀⠀⠀⣸⣃⠀⠀⠀⠈⢣⠀⡬⠆⠀⠈⢹⠏⡆⠹⠛⣯⡉⠀⣀
⠀⠀⠀⠀⢠⠀⠀⡖⠀⠀⠀⣷⣞⡀⣰⣝⣃⠀⠀⠀⠈⠀⡆⢰⢰⠜⢆⠛⢵⡾⠀⠀⢨⡇⣏⣀⣐⣻⠉⣠⢻⡆
⠀⠀⠀⠀⠀⡇⠀⣭⣦⡆⣰⣿⠀⢱⡞⠉⠁⠀⠀⠀⠀⠀⠀⡜⠆⣦⢄⣦⠻⠉⠇⠂⠌⠳⠉⠁⠈⢿⢲⠉⢸⣅
⠀⠀⠀⠀⢤⠘⢸⡾⠗⠃⠋⡶⠀⠀⢰⠀⡗⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⣟⠒⡀⢾⠈⠀⠀⠀⠀⠀⢸⡆⠱⡀⠳
⠀⠀⠀⠀⠀⡀⠘⡄⠚⠻⣿⣷⠠⡄⡸⠂⠈⡄⠀⠀⠀⠀⠀⡂⣇⠀⠷⢘⡀⢓⠘⠳⠄⠀⠀⠀⠀⡸⠁⠀⠁
⠀⠀⠀⠀⠀⠀⣛⡿⣤⠆⠁⡿⣀⣓⡟⡀⠀⣷⠠⠠⢀⣀⣀⠀⡝⡀⢨⠋⡗⢚⡘⠆⠆⠀⠀⠀⠰⠅⠠⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠈⠁⠀⠉⠁⠀⠀⠀⠀⠈⠑⠂⠙⠓⠈⠁⠈⠀⠈⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣄⠀⠀⡄⠀⠀⠀⠀⢀⡠⠤⠴⣶⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢶⣶⣦⠀⠀⠀⠈⣇⠀⠀⢰⡟⢛⡄⢰⠁⠀⠀⠐⠉⠁⠀⠀⣰⠇⢀⡤⣴⠾⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣟⡆⣴⢋⣉⣤⣼⡙⡅⠀⣰⣿⡀⠸⡄⠀⠸⡄⣠⣷⢸⠀⠀⠀⠀⠀⣠⣤⡀⠉⠀⠘⠋⠁⢀⣼⠗
⠀⠀⠀⠀⠀⣀⣠⣽⣾⣯⠥⢈⡟⠻⠿⠟⠛⣿⣃⠞⢁⠅⢧⠀⢡⠰⣶⣧⠙⠛⠀⠀⣠⣶⣿⣿⠿⠿⠟⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠛⠋⢹⡘⢇⣠⠞⠁⠀⠀⠀⠀⡧⠋⠀⢸⣀⣘⡆⠈⣆⠀⠁⠀⠀⣠⣤⠟⠉⠁⠀⠀⣰⠞⠂⠀⢀⡟⠋
⠀⠀⠀⠀⣀⠀⢀⠎⠉⠁⡽⠀⢠⡀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣷⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⡶⠛⡀⠹⣄⠖⠱⡄⠙⠛⠛⠲⣄
⠀⠀⠀⠀⠉⢁⣾⣶⣿⣿⠇⠀⠀⠙⠂⢀⢞⣭⠖⣀⠀⢀⡞⢉⡠⡀⠀⠀⠀⠀⠀⡹⢿⣇⠀⠉⠁⢠⡌⣣⠀⠘⡄⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⡌⠁⠸⠫⠆⠀⠳⡀⠀⠙⣆⠀⠀⠀⠀⢳⡀⠉⠀⠀⡄⠈⠉⣹⡄⡀⣷⠀⠠⠞
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⢈⡿⣄⠀⠀⠀⠀⠒⡊⢐⡆⢹⡀⠀⠀⠀⠀⠉⠛⢧⡀⢣⡀⢿⡿⠂⣧⣯⣤⣤⣤⣤⣤
⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠘⠒⠁⠀⠘⢦⡒⠀⠀⡇⠲⢯⣶⣿⡏⢸⡇⠀⢀⡀⠀⢀⣴⣶⣏⠒⠬⣤⠶⠾⠋⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣤⠀⠀⠀⠀⠀⠀⠈⠛⠳⣜⠦⣀⠉⢉⠀⣼⣃⣴⣿⣿⣶⣿⣿⣿⣿⣆⢠⣌⣢⣘⣴⣿⣸⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣦⡀⠀⠀⣹⡉⠓⠶⠶⠛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⠛⠿⣯⣿⡿⠿⣟
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⡄⢳⣦⣼⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠿⠃⠀⠠⣎⠹⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡵⠶⠶⢦⡀⢹⣿⠿⣿⣿⣿⣿⣿⡄⢻⡟⠉⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣤⠈⠩⢭⣉⠓⡆
⠀⠀⠀⠀⢠⣦⠀⠀⣾⡡⠔⠀⠀⠀⣡⡿⠋⠉⠲⣍⢻⣿⣿⣿⡄⠁⢰⣢⠀⠻⠉⣩⡍⠉⠻⠿⢿⣿⠟⠻⠂⠀⠁⠀⠀⠀⠀⣱⠇
⠀⠀⠀⠀⠀⠁⢠⡞⠉⠀⠀⠀⠀⠘⠁⢀⡀⢠⠖⠚⣦⡻⣿⣿⣿⣆⠀⠀⠀⣀⣀⣉⣁⣐⣟⣀⣀⣀⣀⣀⠀⢠⡤⠀⣀⡴⢋⠏
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢠⣾⡫⠀⠀⠀⣾⠟⠤⣈⠹⣿⣿⡆⠀⠀⠙⠣⣟⠒⣦⠐⠒⢒⣶⠟⠉⠀⠀⣠⠞⣉⠔⢁⡀
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣰⠯⢟⣀⡀⠀⠀⠉⠀⠀⠚⠁⠙⠛⠛⠀⠀⠀⠀⠈⠑⢯⡠⠶⣯⠀⠀⠀⠀⣾⡣⠊⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠀⡏⠙⠻⠀⠀⠀⠀⢠⣞⣃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣗⡴⡼⠀⠀⠀⣸⣏⠀⠀⠲⢤⣄⡀
⠀⠀⠀⠀⢠⡀⠀⠻⣿⣄⠀⠀⠀⠀⠙⢦⣤⡴⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⢀⠎⣠⠿⠿⠀⠀⢀⠀⢿⡘⠳⣤⡀⠀⠀⠉
⠀⠀⠀⠀⠈⢿⣶⣄⠹⣿⣦⠀⠀⠀⠀⠀⠀⠲⣤⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⢀⠎⡼⠁⠁⢘⡀⠀⠛⠁⠀⠙⠢⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠈⢧⠙⠇⠙⣿⣷⡀⠀⠀⡄⠀⠀⢸⣷⢸⠁⠀⠀⢀⣠⠴⠒⠂⠀⡎⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠓⢤⡑⠄⣿
⠀⠀⠀⠀⠠⣶⣾⣧⠀⢀⣘⣿⣷⡄⠀⣧⣄⣠⣼⣿⠀⠆⠀⡴⠋⠀⠀⠀⠀⠀⢷⠀⠀⡟⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⢻⣿⡀
⠀⠀⠀⠀⠀⠈⠛⠋⣴⡿⠃⠈⢿⣿⣆⢷⡈⠉⢉⣿⠀⠈⣾⠁⢀⡤⠚⠉⠙⢷⣌⣇⠀⢣⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣠⡿⡋⠙
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⣠⣤⠀⠻⢿⠦⠙⠾⠿⠋⠀⠀⢹⣾⠋⠀⠀⠀⠀⠀⠙⣿⡀⠀⠂⠀⠀⠠⣶⠀⠀⠙⣤⣾⠏⡰⠁
⠀⠀⠀⠀⠀⣠⡄⠘⣿⣧⣿⢸⡆⣰⣶⣆⣀⣿⡆⠀⠀⢀⣿⠷⣄⠀⠀⢄⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⣀⣤⠶⢋⢳⡎
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠛⠻⣿⣿⣿⠋⠉⠛⠉⠀⠀⠀⢸⣿⠀⠈⠛⠶⢤⣵⣄⣀⣀⣀⣁⣤⡤⠴⠖⢛⡉⠄⠊⠁⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠾⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠑⠓⠗⣷⢻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠠⠐⢀⡠
⠀⠀⠀⠀⠀⢁⣴⡶⠦⢤⢼⠿⡿⠿⢿⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⢯⣀⠀⠀⠀⢀⡠⠔⠊⣁⠴⠂⢀
⠀⠀⠀⠀⢴⣙⠛⠙⢲⠖⡶⠘⡇⠙⣎⣻⣟⣿⡿⠀⠀⠀⠀⠀⢶⣾⣟⠉⠤⠌⢹⣦⠐⠈⡁⠄⠂⠁⣀⣤⠊⠁
⠀⠀⠀⠀⠀⠈⢼⣥⣭⣀⣤⣬⣧⣤⣼⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡉⢀⣠⠴⠛⠉⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠙⠁⠀⠀⠀⠀⡹⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁⠀⢀⡠⢿
⠀⠀⠀⠀⠀⠀⢸⠁⠘⠴⠀⠀⠀⠀⠀⠸⡯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⢀⣀⠀⣿⠤⠊⠉⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⢀⠄⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⢝⠀⠒⠠⠦⢤⣀⠀⠀⠀⣇⣀⡤⠞⠉⢹⠑⠄
⠀⠀⠀⠀⠀⠀⠈⢶⣈⠉⠉⠁⠀⠀⢀⠠⢼⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠀⠰⣟⠁⠀⣠⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠠⢿⣽⠇⣿⡏⢹⠟⡴⣼⠃⠀⠀⠀⠀⠀⠐⡽⣠⢠⣐⠲⠦⣤⡀⠀⠀⣿⢇⣰⣾⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣙⠙⣻⡴⡞⠭⠁⠀⠀⠀⠀⠀⠀⢸⢼⣟⣃⡟⢷⣦⣤⣌⠘⢻⡿⡻⢿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠙⠿⢳⡿⢖⣿⢿⡷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠉⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠐⢎⠂
⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠁⠀⡁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠂⠢
⠈⠐⡆⠀⠀⠀⠁⠀⠀⠀⠦⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⢠⡄⠀⢅⡂⠀⠀⠀⠐⢠⠀⢀⡀⠐⠠⠀⠚⠲⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⣼⢢⡄⢀⠀⠀⠀⠂⠜⢦⠀⣤⠤⠇⠰⠃⣅⠀⢠⠀⠀⡦⠍⠈⢀⡖⠀⠀⠀⠀⢹
⠀⠀⡎⠀⠀⠀⡀⠄⢁⠰⠉⠉⠀⠨⠂⠂⠀⠘⠋⠐⠃⠀⠀⠀⠈⠄⠐⠖⠊⠑⠀⡇⠀⠂⢀⣿⠧⢄⠀⠀⠘
⠀⠌⠀⠀⠀⠀⠡⢄⢀⠀⠀⠀⠀⡦⣂⠍⠂⢀⠀⢀⠀⠀⠀⠀⢀⠔⠀⠁⡔⠁⡎⠑⢄⠀⠋⢛⠟⠛⠉⠦⡀
⠀⢐⠀⠀⠀⠀⡄⠈⢢⠑⠀⢀⠈⡠⠊⠀⢉⡟⢀⠿⠀⠠⠒⠀⠀⣠⡆⠀⢀⠌⠁⠀⠈⡀⠀⠀⠀⠀⡀⠀⢃
⠁⠈⠉⢲⠀⠀⠎⠢⠀⠂⠔⠡⠊⠅⠀⡐⠀⠀⠀⠀⢆⡀⠈⠐⠂⠁⠩⡀⣀⣀⣀⠀⢰⣾⠀⡀⣀⠀⡀⠀⢸
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠂⠐⡁⠀⢀⠎⠀⠐⠤⠤⠔⠃⠀⠤⠀⠀⠀⠀⠑⠒⠳⠤⠤⠤⠋⠙⠃⢀⠀⠓
⠀⠀⠘⠀⠀⠀⠂⠐⠂⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠒⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⢄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠄⠀⠆⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠈⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠆⠀⠀⠀⠀⢂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⢆⠀⠀⠐
⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣀⠄
⠀⠀⠛⠀⢦⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⢀⢈⠛
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⠶⠶⠶⢶⢶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠊⠉⡇⠁⢐⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡴⠋⠘⡆⠀⢉⢶⠲⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣧⠔⠁⠈⣇⢀⠔⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠋⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡰⠁⢀⣀⣰⢇⠞⠉⠳⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠋⡜
⠀⠀⠀⠀⢸⠁⠀⡸⠉⣀⡜⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠤⡄⠈⠙⢦⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⡼
⠀⠀⠀⠀⡌⠀⠀⡇⠸⣅⡀⡗⢤⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠳⡀⠀⠀⠈⣧⣀⣠⠴⡆⠘⢣
⠀⠀⠀⠀⣇⠀⠀⢳⠀⢸⡇⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠙⢆⠀⢠⣄⡀⣀⡔⢰⠚⠁
⠀⠀⠀⠀⠘⢦⡀⠀⠉⠚⠀⠀⠀⠉⠀⠧⢄⠀⠀⠀⠀⢰⢢⠀⠀⠀⠀⠈⠢⡀⠀⠉⠋⠀⠉⠁⠀⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡞⠘⡄⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠃⠀⠸⠀⠀⣀⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⡜⢆⠀⢳
⠀⠀⠀⢀⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠠⠤⣄⣀⡰⠃⠀⢇⠀⡸⠀⢸⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⠇
⠀⠀⠀⢸⠀⠀⣽⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⠒⠋⠁
⠶⠶⠖⠚⠂⠀⠻⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠈⠒⠒⠂⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠶⠶⠶⠶





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣤⡏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⡞⠁⠌⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠦⣄⡀
⠀⠀⣠⣴⣶⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠠⣿⠀⠀⠀⣠⡶⠶⢦⡀⠀⠀⠀⠀⠹⣏⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⠘⢧⡈⣷
⠀⠠⣟⢠⡟⠰⢈⡷⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⢸⡏⡛⠲⢮⣷⠀⠀⠀⠀⢠⡟⠉⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠻⣤⣀⣀⣹⡿⠶
⠀⢀⣽⢿⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣀⣴⣿⣤⣤⡤⢤⣼⣇⡶⢶⢹⡆⠀⠀⣀⣤⣴⣿⡆⢹⡯⣭⡿⠂⠀⢀⡾⠋⣉⣻⡄
⠐⠛⠁⣾⡁⠙⢦⡀⠀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣼⠇⣠⡗⢀⡼⢿⣿⡇⢸⡀⣿⠀⠀⠙⣿⣺⣟⠁⠿⣿⡿⣄⠀⠀⣿⢡⡾⠋⣿⡇
⠀⠀⢰⡟⢷⡀⢀⣹⣾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢃⡶⢮⣴⢏⣴⠿⣅⢸⡍⣧⠸⣧⣼⣦⠴⠞⠋⣿⠟⠃⢀⡏⠙⢮⡳⠦⠿⠛⠀⣸⠟⡇
⠀⠀⢸⣇⣠⣵⣿⡷⠛⢷⡄⠀⠀⣀⡀⠀⠀⢾⣟⣉⣭⡭⠴⠟⣱⣿⡵⢿⡅⠀⢹⣿⡇⠙⢦⣄⣠⣤⣤⣤⠴⣿⠀⠀⠙⠃⠀⠀⠙⢲⡦⠤⠾⠋⢠⡇
⠀⠀⠈⠛⠛⠛⢷⠀⠀⠀⢻⡶⠛⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢀⡿⠀⢀⣿⡇⠀⠀⠀⢰⠀⠀⠀⢠⣿⠄⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣸⠷⢦⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣁⠀⣼⣿⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣽⠂⢠⡟⠀⣾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣰⠟⣿⢹⡏⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡟⢠⡟⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⣏⣸⣧⡿⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡞⠀⢰⠀⡇⢸⡇
⠀⠀⠀⠀⠀⢀⣾⢡⡟⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣉⣿⣿⠁⠀⢸⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⢁⡀⢸⠀⡇⢸⠁
⠀⠀⠀⠀⠀⣼⢃⡾⠁⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢷⡄⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡏⢸⠀⣸⢠⡇⣿
⠀⠀⠀⠀⣰⠏⣼⠁⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⡇⢻⡄⢹⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⣀⣸⣀⣿⣸⣇⣿
⠀⠀⠀⢠⡟⣼⠃⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⢿⡏⢸⣷⠟⠁⠸⢦⣤⣤⣬⣿⣦⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣽⣟⣋⣉⣿
⠀⠀⠀⣾⣱⠇⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢸⣿⣿⡾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠐⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⣀⡉⠙⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠷⠆⠀⠀⢀⠀⠀⣤⣶⡧⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠿⠋⠀⠀⠀⢠⣄⣄⣀⡏
⠉⠀⠃⠀⠀⠀⠀⠀⠀⣼⣟⡀⠀⠰⡄⠁⠀⣿⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣄⠿⠛⠻⣿⡅⠀⠘⠀⣄⡀⠀⣤⠄⠛⠁⠀⠀⠀⠀⠀⠀⣿⠟⠋⣉⣀
⠀⣀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠃⠀⠀⠀⠀⢰⠁⠸⠀⠀⠀⠀⠃⠀⠀⠀⢸⣧⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⣾⠿⠿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⡄⢸⣶⣶⣄⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣀⡀⣠⣮⠀⠀⠀⠀⠀⠸⠇⢠⣴⠶⠀⠀⠀⠀⢠⣿⢻⡿⠁⠈
⠐⠋⠀⠀⠀⠀⠀⢀⣰⣾⣿⠿⠟⠋⠉⢓⣟⣋⣉⣽⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⢻⣷⣿⠁⠀⠀⠀⠀⠀⠀⢿⠼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⠇⠀⠀⠀⠀⠘⠿⠛⣿⡿⠋⠉⠀⠀⠹⡆⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠘⣿⡄⠀⢀⣀⣀⣀⣀⣀⠀⠀⣤⠀⠀⠰
⠀⠀⠀⠀⠀⠐⡶⢶⣀⣤⠀⠀⠀⠀⠀⢠⣤⣤⣿⡇⠀⠀⠀⠀⠀⣷⠀⠀⢀⣄⣠⣘⡀⠀⠀⠀⠀⠀⠀⠀⡎⠰⣾⣿⣿⣿⣿⣿⣿⣿⣶⣽⡆
⠀⠷⠀⠀⠀⠠⠷⠾⠏⠉⠀⠀⠀⠀⠀⠸⠿⠿⠏⠇⠀⠀⠀⠀⠀⠷⠶⠶⠾⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠸⠘⠁⠈⠙⠻⠿⠿⠿⠿⠿⠿⠿⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⣿⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠂⠀⠠⠈⠀⣼⣿⡟⢋⣁⡈⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⣏⠀⠀⠆⠀⠀⢈⣭⡖⠚⠿⠃⢨⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⢸⣿⠃⢸⡿⠀⠀⠃⠀⠀⠙⢿⡿⢣⣶⣾⡝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⠉⠙⣆⠀⠈⢿⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡂⠜⠀⠀⠀⠀⠘⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢺⣿⣿⣿⣠⣶⣦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⢄⣁⠈⠹⢿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡧⠀⠾⣿⡗⠀⣻⣦⣄⠀⠀⠀⠀⡍⠀⠀⠀⠋⠙⢋⣈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⡔⢀⣀⣤⣿⣿⡟⠀⠀⠀⢠⠉⠀⠀⠚⠀⠲⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢿⣿⣿⡷⠀⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡜⠀⢠⠆⠀⠀⣼⣷⣿⣷⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠃⣴⣶⣤⣹⠿⠿⢻⡥⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⢾⣿⣿⡿⠈⢣⣄⣌⠀⠙⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠐⠀⠀⢉⣩⡖⠒⠺⠿⣷⠀⢉⣽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠰⣿⣿⣿⣆⣴⢀⣀⣰⢼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⡀⠂⠘⠿⠟⠉⢼⣿⣿⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣶⣄⠀⡘⡏⠀⠀⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⣈⡀⠙⠛⢠⡿⠁⠀⠀⢀⠖⠉⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠿⠷⠒⠐⢮⠃⡄⠀⠀⠉⠀⠐⢶⣶⡄⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡥⠀⠀⠀⠀⠀⠀⠈⠻⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠙⠳⠤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠸⣍⡇⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠸⠂⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⢀⠠⠀⠀⢠⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢤⡴⡦⠴⠶⠒⡖⠓⠀⠘⠛⠉⠁⠀⠂⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢰⠀⠂⠀⠀⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠔⠀⣸⡄⠠⠆⠨⠀⠀⠀⠀⠀⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⡄⠀⠁
⠀⠀⠀⠀⢠⣔⠉⠻⠖⠀⠀⠀⠀⠀⠀⡄⡀⣠⡾⠁⠐⢠⠀⠀⠀⠀⠀⢸⡄⣀⡀⢀⢀⢀⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠀⠚⣷⡄⣿⠁⠒⠂⣠⠀⠀⠀⠀⠀⠐⠘⠓⠓⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡞⣧⠕⠋⠁⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣼⣿⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⣿⣷⡀⠈⡄⠁⠈⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠒⠠⢑⠢⣀⣀⡠⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⡸⠁⠀⠑⠠⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⡀⠀⠀⠀⠀⡱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠈⡖⢤⣀⣐⠁⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⠎⡐⠁⠀⠀⠉⠒⢄⡀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⢊⡜⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠉⠉⠉⠉⢩⡟⠒⠦⠤⣀⡀⠀⠀⠑⢌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠤⣀⣀⡀⠀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢈⠷⡀⠀⠀⠀⢥⣤⣭⠑⠂⠠⠤⢀⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⢃⡬⠘⣦⡀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⣰⡗⠉⠀⠀⠘⠿⢦⠀⢹⡏⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠄⠀⠀⠑⢂⠀⠀⠀⠀⠑⠂⠃⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠓⡀⠀⠀⠀⠀⠀⠀⠀⠜⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠐⡀⠀⠀⠀⠀⠀⠌⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⢖⠄⠡⠒⢎⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡼⣀⠀⠀⠀⠀⠄⡃⠀⠀⠀⡂⠀⠀⠀⠀⢀⠴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢱
⠀⠀⠀⠀⠙⡦⣁⠒⠠⠄⣰⣕⠠⠤⢒⣔⠠⠤⠒⢊⠥⠞
⠀⠀⠀⠀⠀⠈⠢⢉⠒⠠⠀⠤⠤⠤⠤⠠⠀⠒⣉⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠠⠄⠀⠒⠂⠈

⡄⠽⣰⡶⣶⡀⠀⠀⠀⣰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠻⠧⠰⢡⣶⠿⣦⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡘
⠀⠀⢻⣧⣼⠏⠀⠀⠀⢻⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡘⣿⣤⡟⠀⠀⠀⠘⠷⣿⠿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⢠⣥⠦⠀⠀⠀⠀⢠⡶⠲⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣶⡖⢀⠀⠀⠀⢀⠾⠂⢢⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁
⠀⠀⠋⠁⢀⢁⣴⣶⣄⢻⣆⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠈⡴⣶⣶⡙⣧⣤⡿⢡⣤⣦⡄
⠀⠀⠀⠁⠀⠸⣇⣡⡿⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣾⠃⠀⠀⠀⢻⣧⡉⡟
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣆⠁⡀⢸⢱⣤⣄⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⣰⣯⣤⣡
⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡀⣤⡀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠡⠷⠟⠛⢊⢉⣤⠤⠄⠰⠄⠤⠤⠤⡧
⠀⠀⠀⠀⠀⠻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢞⡽⣿⣧⣃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠄⡀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢻⡿⣿⣿⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠡⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⣀⣀⣀⣀⣀⣀⣀⡀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⢰⣯⣅⡁⠀⠀⢷⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⣄⡀⠤⠀⠐⠊⠉⠐⢲⡆⠀⠀⠀⠀⠀⠀⠀⢄⣽⢏⣯⠉⠉⠉⠁⠀⠀⠁⠈⢻⡇
⠀⠀⠀⠀⠀⠀⣸⠏⠙⡇⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠄⠀⠀⠀⡐⠞⠃⠀⠀⠀⠠⠄⠴⡸⣟⠁⠀⠀⠀⠀⠀⠀⠀⣀⢾⠃
⠀⠀⠀⠀⠀⡌⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⠀⠄⠀⠀⠀⢄⠀⠈⠁⢀⡀⠀⠀⠀⠂⠀⠀⠀⠙⠫⠅⠈⠠⣬⣈⡁⠁⠉⠁
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠈⣁⠀⠀⠀⠬⣯⠖⡀⠀⠀⠀⠀⠄⠀⢰⣾⠖⠻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⣔⡂⣹⠶⠀⠰⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢠⣾⣷⣇⣠⠄
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⢯⣴⠊⣿⠀⠀⠀⠂⠚⢻⣄⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⠁⠀⠀⠀⠠⠀⡀⢸⢀⠄⠀⠀⠀⠐⠚⣿⣿⡿⢿⡶⠶⠖
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠠⠀⢹⣏⣿⠉⠀⠀⠀⠀⠒⢁⡟⡷⣚⡵⡆
⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⣄⠇⠀⠀⠀⠀⠚⠀⠀⠀⠁⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠜⠘⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠹⠪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢀⣠⣀⣤⣤⣤⣄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠉⠠⠀⠀⠀⠀⢰⣆⡚⠩⠶⠀⠈⣁⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠚⠉⢀⠠⠀⢈⡠⠔⠊⣠⠀⠀⠀⠛⠁⢰⣶⡬⠽⠽⠃
⠀⠀⠀⠀⠀⠻⣿⣏⡿⠍⢻⣿⢛⡟⢛⠀⠀⠀⢀⣠⣴⡿⠛⠉⠿⣖⣁⡠⠴⠊⡁⠄⠂⠉⡀⠔⠀⢀⢤⡰⣤⠁⠀⣤⡌⢠⣐
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠻⣿⣟⡸⠛⠉⠀⠀⠀⠉⠉⠉⠀⠀⠡⠈⠙⣧⡤⠒⠁⢀⣤⠴⢻⠃⠀⠀⠉⢰⡳⠉⠡⣀⠉⠀⣶⠛
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠞⠋⠀⠀⣸⠀⠀⠀⠁⣭⡄⠰⠿⠃⢘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠈⠀⠀⠐⠟⠃⠀⠀⠐⠀⠀⠤⢤⣄⡀⠀⠀⢸⣆⡠⠶⠋⢹⡀⠀⠀⠀⠃⠀⡤⢠⡄⠁⠙⠻⠄
⠀⠀⠀⣠⡿⠶⠀⠀⠀⠙⠢⣀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠐⢾⡋⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣧⡄⠀⠘⡶⠤⢄⣀⣀⡀⠀⠀⠀⢺⠀⢀⣠⠶⢻⣖⣀⠀⠀⠀⠿⠀⡠⣶⣰⡆⢀⣠⡄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠃⠀⠈⣧⠀⠀⠀⠈⠉⠓⠂⠀⣾⡟⠋⠀⡀⢀⣇⠁⠀⠀⠀⠀⢰⡃⣭⣟⣽⣍⠀⠄
⠀⠀⠀⠀⠙⠷⣤⣤⡄⢀⡁⠀⠀⠈⢫⠁⠀⠀⠐⣛⣠⣀⢤⣤⣀⡀⠀⠀⢸⣷⢈⣼⣷⣼⡿⠀⠀⠀⠠⠀⢀⡶⠟⢻⣿⣿⣵⣄⡀
⠀⠀⠀⣼⠶⠶⠿⡟⠉⠉⣧⠀⠀⠀⠸⠀⠀⠀⠈⣿⣿⢿⢿⣾⣭⣍⣒⠶⠾⣿⡿⣿⣿⣿⡅⠀⣠⣾⠷⠁⣿⣷⣶⣿⣿⣿⣷⡇⠉
⠀⠀⠀⣧⣉⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⣀⡄⠀⠀⠉⠹⠿⠋⠠⢿⡿⣿⣷⢶⣿⣿⣷⡿⠋⠀⠀⠙⠛⠖⢠⡟⠻⣿⣿⣿⣟⡉
⠀⠀⠀⠈⠻⠷⢦⣤⠤⠚⠁⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠁

⠀⠀⠀⣠⣤⣤⣀⣁⣠⡴⡶⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⠀⠀⡀⠚⠏⠃⠉⠉⠀⠀⠲⢿⡻⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠉⠉⠋⠛⠦⠴⠤⠠⢦
⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠁⡌⠀⠉⢙⣆⠀⡉⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠀⠀⠀⠀⡹⠷⠾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠝⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⡆⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣂⣠⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣠⣤⣄⠘⠛⠋⠙⠙⠿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢿⡉⠉⠻⣆⠀⠺⠂⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠔⠛⠃⠠⠀⠑⠈⠛⠒⠻⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢌⡄⠄⠀⠀⠀⠀⠀⠑⠆⠨⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⣈⡀⠚⢓⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠰⠾⠽⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡄
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣴⠎⠹⢶⣄⡤⠴⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⣐⡌⢃⣀⣐⣠⠉⠀⠀⠀⠀⠁⢐⡄⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠠⠐⡉⠋⠢⠙⠉⠁⠀⠀⠀⠀⠀⠐⠉⠁⠀⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⢒⡎⠙⠶⣦⠶⠤⢀⣀⢐⢀⣀⣐⣤⣔⡢⠖⠺⠃
⠀⠀⠃⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡤⠊⠹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⠈⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣰⠿⡅⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠻⠿⠇⠀⠀⠀⠀⠀⠀⠁⠐⠆⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⣤⣀⠀⠀⠀⠄⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣬⠻⠃⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⢀⡺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠐⣄⣤⣀⡜⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠀⠀⣏⠀⢈⡇⠀⣀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠗⠨⠍⢉⡛⣋⠡⠅⠚⢭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⡇⠀⠀⠙⠀⠀⠀⠘⠀⠀⠐⣗⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠁⠐⠀⢙⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⡿⠉⠁⠁⠀⠈⡹⢩⠁⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⢞⠁⠘⠀⠀⠀⠠⠁⡄⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠋⠉⠉⠉⠃⠀⠙⠀⢄⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠑⢆⠀⣦⠄⠀⠀⠀⠀⠠⡄⠀⠀⠈⢆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⣷⡀⢀⣠⠄⡸⢣⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣧⡉⠀⢀⡁⢼⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣤⣤⣍⣳⣴⠉⠀⢸⣯⣀⣄⣤⣤⣀⣠⣇⣠⣄⣠⠇⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠤⠼⠤⠾⠤⠤⠤⠤⠤⢤⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⢴⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢽⠀⠀⠀⡠⠐⠂⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠡⡈⠑⠤⡀⢴⠁⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠭⠟⠦⠄⢿⡼⣿⣿⢵⣏⣉⣉⣉⣉⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣤⡀⠙⡤⠐⢁⠆⠀⠀⠀⠀⠀⡠⠶⠏⠚⢿⡦⠵⠶⠶⠯⣾⡙⣉⣉⣙⣉⣷⣲⣛⣲⣿⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠛⢄⡠⢎⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⢰⡿⠭⡭⠭⠭⣿⢿⣒⣛⣓⠫⣟⣲⣒⣾⣿⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⡔⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿⠽⠭⠽⠯⣿⡿⣷⣺⣷⢬⣷⢶⣶⠯⣿⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠭⠭⠭⠭⣿⣯⠯⠿⠯⣽⡋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣶⠒⠒⣶⠶⠶⣶⠶⠶⣶⠶⣶⠶⡶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⣶⠶⠶⣶⣶⠶⠶⠶⢶

⠀⠀⠀⣀⣀⡀
⠀⢠⣬⣟⠙⠻⣿⠟⢷⣄⠀⠀⠀⠀⠀⢀⠀⢀⣀⡴⣦⣴⣦⣘⣀⠈⠇⠀⠀⠀⣂⣼⠛⠻⢾⡆⠀⠀⠀⠀⠀⠀⣠⡤⠶⣄
⠀⠀⠈⢿⡀⢨⠂⠐⣀⣻⠂⠀⠀⠀⠀⣽⣾⣿⣿⢷⢾⣿⠟⠛⠟⢿⣿⡀⠀⠐⠛⠳⠀⠀⠘⢧⡅⠀⠀⠀⠀⢰⣇⠃⠁⠈⣧
⠀⠀⠀⢺⠆⠚⡀⠀⠐⠛⠀⠀⠀⠀⠀⠾⠛⠛⠛⢣⠈⠎⠇⠀⣤⠞⠛⠃⠀⠀⠀⢀⣦⢀⡾⠛⠇⠀⠀⠀⠀⠀⡜⠠⣶⠿⠃
⠀⠀⠀⣼⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⢠⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⣿⣾⣇⡴⠶⣄⠀⠀⠀⠘⠋
⠀⠀⠀⡟⠰⠀⠀⡠⠖⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⢹⡏⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢩⡄⢀⠝⠃
⠀⠀⠘⠛⠃⠀⠘⣳⠀⠀⠀⠈⠙⠶⠤⣀⠀⠀⠀⢸⡇⠀⡇⠀⠀⣠⣠⣤⢤⡀⠀⠀⢻⠀⢀⡇⡎⠀⠀⠠⠂⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⣤⣤⠶⣶⣦⣆⠀⠀⠀⠀⠸⡇⠀⠁⠀⡴⠿⢮⡁⠈⢳⡄⠀⠘⣦⣾⡅⠀⢰⡤⠀⠐⠲⠲⠶⠚⠁⠀⢠⡄
⠀⣠⡴⠒⠠⣤⣶⣤⠄⠈⠙⡶⡄⠈⠻⣶⠗⠂⠘⠀⣿⣸⠀⠀⠈⠀⠉⡇⠼⠟⠁⠀⠀⠙⠋⠀⠀⠈⠑⢀⡀⡄⠀⢠⠄⣀⠀⡌⠁
⠀⣶⣿⡆⠀⠀⠀⠀⡀⠀⣤⣶⣟⣿⣿⠿⡻⣶⣤⡄⠘⢿⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⣀⠤⠀⠀⢠⠀⠀⠙⣮⣌⣪⣶⡾
⠀⠙⢛⠿⢷⣿⣶⠾⢛⠛⣿⣯⣽⢿⡦⠀⠀⠘⢿⣳⠖⠀⠀⠀⠀⠀⠈⠉⠀⢠⠚⠉⣉⣤⢶⣬⣦⠼⠃⠀⠀⠙⡏⢁⠎
⠀⠀⠀⠀⠀⠻⣦⠀⢸⠀⣿⠛⣿⠉⠙⣶⠆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣤⣾⣏⠋⠀⠿⠋⠀⠀⠀⠀⢸⠅⡌
⠀⠀⠀⠀⠀⠀⢹⡇⠘⡀⣿⡀⢛⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠹⣦⠄⢶⠇⠀⠀⠀⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⢿⡄⢣⣿⡇⢸⢀⠞⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡹⠁⠞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡔
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡜⠀⠀⠠⠖⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠠⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠀⠀⠀⠠⣶⠠⡀⠀⠀⠀⠈⠀⠹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⠀⡀⠀⠀⠠⠋⠓⠷⠂⠉⠐⠀⢠⠀⠀⢈⠆⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠅⠀⠀⠊⢆⠀⠑⠀⠀⡤⢈⡀⠠⠀⠀⠠⠀⠀⠘⠂⠀⠀⠀⠀⠼⠆⢀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠀⠀⠁⠀⡠⠀⠀⢠⠀⠀⠀⠈⡀⠒⠀⠀⠸⢲⠀⠎⠐⠚⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣀⠁⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠠⢺⠀⠀⠒⠚⠀⠁⠁⠀⠀⠀⠀⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢌⡀⠐⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠀⠀⠈⠁⠂⠉⠲⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠘⠀⠀⢀⠀⠀⠀⠉⡀⠀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⡀⠀⠈⠆⠀⠀⠀⠈⣤⣤⠀⠀⣀⣠⠀⠀⠆⠀⠀⠀⠀⠠⡁⠀⠈⣀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⢣⠀⠀⠂⠀⠆⠀⢠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠠⡤⠶⠂⠄⠁⠀⠄⠁⠡⠇
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣾⣶⣶⣾⣶⣶⣿⣶⣿⣿⣶⣶⣶⣶⣿⣶⣶⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⠀⠀⠀⢀⡊⠀⠀⠀⠀⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠁⢣⢭⣥⡤⠤⠶⠧⠅⡐⠔⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡔⠋⠀⠀⠓⣠⠀⠀⡀⢂⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡄⢠⡶⠒⡦⠄
⢀⡄⠀⠀⠀⠐⠲⣤⠀⠀⠀⠀⢀⣯⡉⠁⠀⣀⣈⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢏⠓⠠⠀⠀⠀⢤⡾
⠈⡉⠖⠒⠒⠰⠶⠀⠀⠀⠀⠀⢸⢄⡀⠀⢸⣿⢻⣇⠠⡇⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⠠⠀⠀⡀⢠⠄⢤⣶⡄
⠀⠢⠤⠦⠦⠴⠤⠂⠀⠀⠀⠀⠘⣦⠀⠀⠀⠉⠁⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⠛⠀⠀⠀⠂⠈⠀⠘⠛⢣
⡀⢀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠳⢦⡀⠀⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢏⠁⠀⠀⠀⠀⠠⠈⢩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⡧⠤⡏⠀⠡⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⡀⠀⠀⠀⠀⠀⠀⠓⠘⣄⣀⡘⠒⢃⠀⠀⢀⡔⠉⢁⠀⣤⣀⡀⠀⢀⣀⠀⣀⢀⣄⡬
⠀⠀⠀⠀⡀⠁⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠸⠀⠐⠁⠀⠈⠀⡈⠀⠀⢸⠀⠀⣸⠀⢙⡛⠃⠀⠀⠊⠀⠃⠈⣋⠃
⠀⠀⠀⠉⠡⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠛⠃⠀⠀⠀⠀⠈⠀⠶⠃⠀⠀⠛⠻⠤⠦⠤⠼⠇⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠛⠒⠦⣄⠀⢀⠜⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠒⠂⠠⡀⢀⡞⠁⠠⣶⣶⣿⣿⡆⠀⠈⢳⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠎⠀⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢷⣾⣶⠶⠄⢠⣄⣄⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⣀⣠⣌⠭⠖⢋⡁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣓⠭⢟⣋⣭⣴⠶⠿⠟⠛⢆⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣒⣫⣭⠭⡄⠰⠖⠒⠒⢚⠉⠀⠀⠀⣈⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⠋⠳⡀⢀⠀⠀⠀⠠⠞⠘⢇⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⣀⣀⣀⠌⠀⠀⠙⡏⠀⠄⠀⠀⠰⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢀⠔⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡀⠀⠀⠀⠀⢹⠏⠀⠀⠀⣸⣤⣤⠋⠀⣀⣤⣷⡀⠀⠀⠉⠒⠤⢄⠀⠀⠀⠀⢐⠫⠔⣊⡤⠖⠛⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⣿⣿⡀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢣⡅⡜⢁⡴⠊⠁⠄⢐⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠀⠈⠀⠀⠀⠀⣧⣿⣿⣗⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠆⠀⠂⠀⠀⢀⠟⣿⡗⠀⠙⠦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⢃⡆⠎⠙⣼⡄⠀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣾⣶⣶⣾⣾⡇⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡇⣼⡇⢸⣿⣿⣿⡄⢀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⡴⢧⢹⡇⢸⠿⠿⠿⠿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠶⠾⠶⠶⠶⠾⠶⢶⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠿⠓⠛⢛⠇⠘⢆⠒⢂⡛⢻⣛⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣀⠀⠁⠀⠀⢸⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣠⡔⠋⠀⣀⣈⣓⠦⠄⣴⠁⠈⢆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⡇⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠒⠂⠂⠛⠒⠐⠓⡰⠒⠋⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⢷⣶⣷⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠙⠃⠉⠉⠉⠁⠀⠍⠉⠉⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⠀⣀⣀⠀⠀⠄⠊

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠐⠚⢛⠛⠛⠛⢃⡀⢀⠀⢂⡂⠒⠛⠓⠆⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⠗⠆⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⣸⢀
⠌⠀⠀⠐⠀⠀⠈⠂⠀⠀⠠⣤⢀⡄⠈⠀⠀⢀⣉⠠⡤⠅⠦⢤⢴⡅⡈⣞⠛⣣⣔⡐⢀⡛⢁⡋⠘⠀⠂⠋⠛⠠⠞⠠⠁⠋⠀⢸⣶⠂⢰⡿⠈
⠇⠀⠄⠠⠨⠄⢄⣀⡀⢀⡙⠈⠋⢉⡉⢈⣂⣀⣂⠐⠒⠀⠆⠒⠐⠀⠶⠂⠴⡀⠠⠁⣤⠄⠈⢁⣁⠁⠋⠐⠀⠒⠖⢲⣴⠀⠀⢸⠉⠀⣿⠃⠀⠀⢀
⠀⠰⠀⠷⠂⠛⠙⠳⠂⠒⠖⠂⠐⠢⠨⡄⠠⠁⠬⠄⡁⣋⠁⣀⢘⠀⠂⠐⠰⠤⠄⢀⠀⢠⡄⣈⠁⢈⡚⠀⠘⠁⠞⡔⠈⣧⠀⠀⠀⣘⣿⠀⠀⢰⠇
⠄⠰⠀⠏⠀⠖⠚⠋⠙⠁⠉⠃⠉⠃⠘⠉⠂⠐⠂⠶⠀⠆⠆⡤⢠⢈⢉⣄⢂⣐⣂⠆⠤⠬⠀⡃⣘⢘⣹⡀⠉⢀⡤⢘⡇⠀⠀⠀⠀⣿⠏⠀⢀⡟
⠀⠆⠰⠟⠀⠞⠘⠃⠀⠂⠞⠉⠙⠇⠀⠸⠃⠻⠆⠶⠚⠂⠰⠖⠰⠆⠬⢩⡀⣀⠈⠂⠘⠃⠐⠀⠯⠨⠝⠛⢻⣅⠁⠾⠀⠀⡠⠂⣸⠏⠀⠀⣼⠁
⠀⢰⡴⢶⡀⢲⠀⠸⠀⠺⠀⠐⣿⠧⠐⠀⡆⣰⡆⡰⠂⠀⡖⢰⠆⢴⠆⢠⢀⠁⢀⠙⠀⠂⠣⠂⠆⣠⣄⠘⠛⠋⠀⠀⢀⡾⠁⠀⡟⠀⠀⠀⣽
⠁⠀⣁⠀⡁⡀⡄⢠⣁⠀⢀⠀⣤⠀⢀⣀⠀⢀⡁⠀⡀⠀⢁⠀⠀⠈⠀⡁⠁⠀⠋⠘⠜⠑⠛⠂⠀⠀⠀⠀⠀⠀⣠⣴⣾⡁⠀⣰⠇⠀⠀⠀⣧⢰
⣤⠀⠹⠆⠁⠀⠣⠀⠘⠀⠛⠀⠉⠀⠀⠙⠀⠸⠁⠀⠿⠀⠏⠰⠀⠀⠘⠃⡇⠀⡾⠁⡀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠚⠻⠟⠻⠶⠿⣤⣀⣈⣘⡇
⠀⠀⠀⢠⠄⠀⠀⠀⠐⠀⣠⠀⢰⡄⠀⢠⢤⣠⡄⠀⣤⡄⣠⠀⢀⣀⠀⠀⢀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⠀⠀⡀⠉⠓⠒⠶
⡶⠈⢀⠰⡆⠀⡘⠈⠇⠀⠸⠇⠀⠁⠀⠛⠀⠀⠀⠰⠇⠁⠋⠀⠈⠁⠀⠀⡞⠀⡿⠀⢀⡴⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠛⠓⠀⠀⠀⠀⠒⠂⠀⠀⠰⠶⠦⠄⠀⠀⠀⠀⣰⠃⠀⡼⠁⠀⠀⠀⣀⣇⠀⠀⣤⠀⠀⠀⠀⠀⠢
⠀⠀⠀⡀⢠⡀⠀⣄⠀⠀⣠⠄⠀⣄⠀⠀⢠⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⢰⠇⠀⠀⠀⡐⠏⠅⠀⡀⠁⠀⠀⠀⠀⠀⠙⠃
⠘⠇⠀⢀⠀⠁⠁⣬⣄⣀⠀⠴⣦⠀⠀⠀⠀⢀⣈⣤⡁⠀⠙⢳⡄⠀⠀⠀⠠⢀⣼⠋⠀⡼⠀⠀⢵⠰⠂⡀⣀⠀⠀⠀⠀⠶
⡃⠐⠻⠿⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠁⠀⠀⣈⠀⠿⠴⠄⢀⠈⠁⣠⡼⠃⠀⠀⡀⠀⠂⠀⠈
⠷⠀⠀⠀⠠⣶⠶⣾⣦⠀⠀⠀⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠈⢉⠶⢖⠒⠀⠀⠀⠀⠩⣦⠀⢠⡈⠁⠀⠀⠀⠀⢤
⣤⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⡀⣄⠀⠀⠉⠋⠱⠄⠠⠄⠸⣷⡙⠂⠈⠰⠆⠉⠀⠃⠀⠀⠒⠈⠁
⠀⠘⠓⠀⠀⠀⠀⠀⠽⠀⠀⠈⠠⠀⠈⠓⠀⢀⡀⠤⠄⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣤⣀⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡶⠶⠛⠛⢿⣿⣶⣶⠶⠶⠾⡟⣿⣿⡿⠿⠛⢿⣿⠿⢻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠁⠀⠀⠀⠀⠀⠸⠛⠙⠉⠀⠀⠀⠀⢘⣿⡇⠀⠀⣤⣷⣔⣛⣵⣷⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⣤⣤⠂⣠⡘⠀⠠⠇⢀⣄⣀⣤⣴⣶⣿⣇⣀⣀⠛⠿⢇⣉⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣾⣿⣶⣾⣿⣿⣿⣇⣿⣿⣷⣿⣿⣷⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠙⢻⣿⠻⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⣀⠀⢠⡄⣿⡔⣤⣴⣿⢿⣾⠏⠘⢡⡀⢹⣹⣿⣿⣿⣿⣼⣿⣿⡿⠋⠀⠀⡐⢦⡀
⠀⠀⠀⠰⠤⣸⣷⢤⣿⡉⢻⣿⣷⣿⣼⡟⠀⠀⠘⠃⠈⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣶⣿⣿⣿⣏⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠹⣷⣧⣽⣸⣿⣿⣿⣿⠁⠀⠀⠀⠱⡧⠿⢿⡿⣿⣿⣿⡿⢇⠄⢤⣤⣭⠻⣿⣿⣿⣷⣤⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⣧⣿⣿⠃⣀⡠⠄⣿⣿⣷⣤⣤⣌⣉⠛⢶⣦⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠈⠉⠲⢄⣸⡀⠀⢀⣿⣿⡿⡾⠀⠄⠀⠿⠿⢿⣿⣿⣿⣿⠷⠤⢤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⣽⡏⢿⣧⠂⠈⠒⠀⠀⠹⡇⠀⠈⣿⣿⡇⠀⣰⠄⠀⠈⣠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢶⣬⣿⠈⣿⠏⡀⠀⠸⣿⣶⢹⡀⠀⢻⣿⡅⠔⠃⠀⢤⣾⣿⣿⠋⣀⠠
⠀⠀⠀⠀⠐⠒⠉⠉⠈⠁⠃⠀⠀⣰⠁⠈⠁⠀⢰⡶⣩⡎⢇⠀⠘⣿⣏⣀⣴⣾⣿⠁⣼⣧⡺⣅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢐⢃⣸⢏⣛⣋⣟⢷⣄⣰⡖⡪⢾⣿⣿⣽⡆⠀⣿⣿⣿⡿⢛⣯⣯⣿⣟⣛⣿⣿⢮⣠⣄⣠
⠀⠀⠀⠀⠀⣂⣚⡻⠾⠷⠿⢿⣿⣿⣶⣽⣶⢛⣵⣤⡝⣿⣿⣿⣻⣿⣿⣧⡶⠛⠉⠉⠛⢡⡾⠟⣿⡿⡽⢟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠶⣾⣿⠿⠟⠛⠙⢛⣿⣭⣷⣿⣿⣿⡿⠛⠉⠛⣿⣿⣶⣦⣄⡀⡀⠘⠀⣹⠻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⠛⠉⠀⠀⠘⠓⠀⠀⠀⠉⠀⠙⠿⠿⠿⣿⠿⣶⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠠⠤⠠⠀⠤⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣭⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣭⣿⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⣀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⢀⡠⢊⣀⡠⠂⡀⠀⠀⣚⡫⣇⣤⣠⣄⠨⠄⠀⣃⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⢦⡐⠖⡉⠛⣷⠀⠀⠻⣅⠈⠩⢅⠈⠓⠲⠌⠉⠀⠈⠙⠀⣞⠁⠀⠂⠀⢍⠲⠭⠉⠉⠉⠩⢏⠂⠋⡇
⠀⠀⠀⠀⠀⢀⠉⠀⠉⠐⠒⠁⡀⠀⠘⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠄⢁⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠈⠀⠠⣀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⢥⠴⠤⠤⠤⠮⠤⠤⠴⡿⠄⠀⣶⣾⡆⠀⠀⠤⠤⢒⠒⡶⢐⠿⣄⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠀⡐⠀⠀⠀⠠⠔⠉⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⣀⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠉⠈⠀⠈⠡⡀⠈⠈⠨⠈⠉⠩⠡⠀⠀⠀⠀⠂⠀⠀⡠⠀⣠⣀⠀⡀⠀⢠⠄⡍⠉⢉⠩⢤⠀⣠⢠⢠⠄⢠⣆⡀⢠⡇⠆
⠅⠀⠀⠀⡀⡀⠀⠀⠀⠈⠶⠃⠄⠤⠄⠀⠉⡯⢀⠉⡋⡉⢀⡙⡐⠚⠚⢀⠂⠐⠆⠆⠠⠄⠄⡈⡥⣄⡀⠀⢸⠟⠀⣿⠁
⠁⠠⢠⡤⠱⠘⠯⠠⠭⠤⠠⢈⢃⠈⠙⣙⡂⠒⠖⠂⠀⠄⠤⠠⢡⢀⠁⡈⢁⡀⠘⠃⠢⠂⠰⠂⡤⡈⢣⠀⠈⠀⢸⡏⠀⠀⡔
⠀⠶⠰⠂⠐⠔⠃⠒⠀⠒⠐⠂⠒⠒⠀⠆⠤⠁⠍⠬⠅⡍⠀⣀⢀⣀⠤⠡⠌⠁⠃⡊⢛⡇⠘⢀⡄⢰⠊⠀⠀⠀⣿⠃⠀⣼⠁
⠐⠄⠾⠃⠘⠈⠀⠀⠀⠉⠙⠃⠀⠛⠈⠃⠐⠚⠀⠲⠂⠆⠠⠩⢤⡄⢙⠀⠃⠐⠐⠇⠍⠛⣛⣏⠁⠏⠀⡀⠀⢸⠃⠀⢠⡇
⡀⢰⠚⠆⠘⠀⠛⡀⠃⠀⠻⠓⠈⠰⠤⠿⠰⠃⠀⠇⠗⠰⠖⠠⡠⢀⡄⡃⡃⡓⠐⠠⠾⠄⠉⠁⠀⠀⣼⠁⠀⡟⠀⠀⢸⠁
⠀⠀⣦⢠⠀⡆⠲⣦⢀⡆⠴⠀⠠⡶⠀⡆⠀⣆⢀⡄⡀⠀⠀⠄⡀⠀⣨⠉⠁⠉⠀⠀⠠⠀⡀⣀⣲⣿⣧⣄⣼⠃⠀⠀⣸
⠋⠀⠉⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⠀⠁⠀⡉⡈⠀⠁⠀⠀⠀⠃⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠉⠉⠑⠒⠲⠾⣀⣀
⡄⠀⠀⣅⠀⠠⠰⠄⠀⡧⠀⠇⠀⠞⠉⠋⠀⡏⠃⠇⠀⠛⠀⠂⡼⢠⡇⠀⣠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠
⠁⠀⠀⠈⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠀⠤⠀⠀⠠⢤⠄⠀⠀⠀⠁⡼⠃⢠⠃⠀⠀⢀⣹⠀⠀⣠⠀⠀⠀⠀⠠
⠀⠀⢀⠀⡄⠄⡄⠀⠀⠤⠀⢤⠀⠀⠠⠀⡀⡀⠀⡀⠀⠀⠀⠐⠀⠀⠀⡟⠀⠀⠀⡘⠉⠀⠀⠁⠀⠀⠀⠀⠉⠃
⠈⠀⣀⠀⠈⠀⠼⢤⠀⠐⠶⠀⠀⠀⠀⠤⢤⡄⠀⠉⠳⢀⡀⠀⠐⣠⠟⠀⡰⠁⠀⠖⠂⠀⢀⠀⠀⠀⠐⠂
⡅⠈⠙⠛⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢤⢄⢙⣛⠃⠀⠀⢰⡾⠁⠀⠠⠀⠈
⠉⠀⠀⠀⠙⠋⠛⠃⠀⠀⠀⠀⠀⢴⠶⡀⠀⠀⢠⡀⣌⠉⢄⡀⠄⠠⠀⠙⣀⠐⠆⠀⠀⠀⠀⠉
⠉⠴⠄⠁⢀⠀⠐⣆⠀⠐⢘⡂⠰⠄⠀⠁⠀⡀⠌⠛⠀⠀⠈⠉⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣐⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⡹⠁⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⠀⠈⡟⠁⡀⠐⡤⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣉⠀⡌⠁⠀⢳⡥⡞⠀⡥⢠⠆⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠠⡱⣇⣅⣂⢴⠡⣀⢄⣠⠄⠐⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠅⣠⠖⠉⠀⠀⢹⠟⠉⠉⠙⡄⢈⠋⠠⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠁⡀⢀⣡⡄⠀⠀⡞⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠙⡅⠰⠂⠀⠠⠀⠀⢤⢀⠀⠄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠛⠁⠀⠀⣰⠷⠓⠂⢄⡀⠀⢸⠀⠀⡀⢠⣀⠴⠁⠀⠁⢂⠀⣰⠰⠟⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠃⡠⠤⠀⡜⠁⠀⠀⠀⠀⣤⢖⠟⠙⠛⠫⡙⢥⣀⠀⠀⠀⠘⡌⠉⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⡄⠄⠀⠐⢉⠁⠀⡀⠀⠀⢸⣷⠁⡠⠀⡀⠀⠀⢂⢳⠠⣀⣠⠴⢧⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠘⠓⠚⠀⠀⠀⢼⣀⠀⠐⠂⠈⠐⢀⣼⣿⠚⠁⠀⠀⡼⠀⡐⠈⠐⠲⢤⣂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⡲⢆⠀⠀⠀⠎⢿⣶⡤⣄⢠⣤⡾⠏⠀⠈⠐⠒⣹⠰⣂⡀⢀⠂⠀⢢⠀⠙⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣤⣄⣆⢴⠁⠀⢀⡥⣖⠀⠀⠀⠀⠩⠋⠙⠈⠀⠀⠀⢹⠒⠊⠪⠼⣆⣲⠋⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠜⠃⠚⢆⠀⠈⠀⠈⢄⡀⢀⣠⠂⠀⠀⠀⢀⣀⣠⣼⠳⣷⣞⠁⠈⠇⠋⠱⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠁⠀⠄⠐⢩⡆⠠⠀⠐⠂⠀⠈⠁⠁⢠⡖⠒⢲⠉⢽⠅⡛⠀⣿⢫⠩⠁⠶⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⡤⠈⠀⢀⠣⣌⡾⢀⠤⢀⡠⡀⠀⠑⣷⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠐⠀⢀⡀⠀⠁⠀⠛⢻⠀⠀⠀⠀⠇⠒⠀⠀⣳⣀⠠⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈⠛⠂⠀⠀⠀⠐⠀⠀⠀⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⡠⢊⣭⣓⡲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣤⣶⡄⡠
⠀⠀⠀⠀⠀⡰⢫⣶⢷⣶⠰⣿⣿⣿⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠺⠹⣿⡿⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠹⣾⣿⣜⣟⠀⢙⡛⢛⡵⠃⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⣀⠤⣵⣯⡥⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠄⠀⠀⢘⣃⣾⣿⣇⡿⣿⡎⡇⠀⠀⠘⣷⣷⡎⣵⣿⡮⡃⠀⠀⠀⠉⢠⠞⣿⣒⣺⢲⠈⠁⠀⠀⠀⠀⢀⣤⠾⠟⠋
⠀⠀⠠⠔⢣⡿⣟⢯⠋⠷⣿⡿⢟⡱⠋⢀⢔⣥⣝⢁⣤⣍⡁⣨⣼⡢⠀⠀⠀⠋⠀⢻⣤⠇⠀⠑⠀⠀⢀⣤⠾⠋⠁
⠀⠀⠀⠀⢿⡬⠞⡁⠀⠀⠈⠉⠉⠀⠀⠰⡸⠟⠫⡘⠟⠋⢃⠿⡛⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣾⡿⠇⣾⣿⠷⠘⠀⠀⠀⠀⠀⣠⣄⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠤⠓⠒⠤⠑⠁⠀⠀⢠⣴⣿⣿⣿⣯⠀⠀⠀⠀⡉⠙⠛⠛⠛⠛⠛⠙⠆
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠦⠴⠴⠶⠶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡿⠟⠋⠁⠀⠀⠀⣮⣀⣠⠉⠉⠉⢃⠀⠀⠱⡄
⠀⠀⢀⣠⣴⠾⣛⣽⣮⣗⢤⡀⢼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⠁
⠀⠀⠉⠙⡁⠈⠀⠀⠟⠉⠛⠋⠻⣿⣿⡿⠿⠿⠿⠛⠛⢛⠛⠁⠀⠀⢀⠀⣼⣤⣀⡀⠀⠀⠀⠀⠸⣧⠈⠙⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠈⠉⠉⠉⠉⠒⠒⠂⠦⠐⠒⠊⠉⠉⠉⠉⠁⠀⠀⠀⠐⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⢻⡆⠀⠈⠛⢿⣿⡿
⠀⠀⠀⠀⢸⠢⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⡛⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⢤⣒⡢⠤⣖⣲⢄⢀⣴⡿⠂⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⢸⠉⠁⠒⠂⠀⠠⢄⣀⣀⣠⠤⠄⠐⠈⠁⠀⠂⠀⠀⠀⡠⢿⠿⠛⠸⡿⢟⡵⡛⠁⠀⠀⠀⠀⠀⣸⠿⠃⠿⠟⡇
⠀⠀⠀⠀⢸⠤⠀⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡠⠤⠒⠃⠀⠀⣮⣾⣷⡌⡶⣳⠞⣿⣿⣮⠃⠀⠀⠀⠀⠀⠀⢰⡿⠷⠈
⠀⠀⠀⠀⢸⠀⠰⠾⣿⣿⡗⠒⠤⡤⠔⠊⠉⠁⠀⢀⣠⡅⠀⠀⢫⡋⢑⣵⣞⢑⣢⣝⡉⡪⠂⠀⠀⠀⠀⠀⠀⠈⠐⠉
⠀⠀⠀⠀⠘⠿⠶⠂⠤⠀⡀⠀⠀⠁⠀⣀⣤⣴⡞⠋⠉⠁⢀⣴⣿⣿⡿⠟⠫⡸⠟⠋⣹⠀⠀⠀⢠⡀⣤⣦⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣶⡟⠋⠉⠁⠀⠀⠀⠀⠹⢿⠿⠋⠈⠒⠉⠈⠑⠊⠁⠀⠀⠤⣄⣉⢹⣿⣟⣁⡠⠄
⠀⠀⠀⢀⢀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣵⣿⡅⠉⠂
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⠸⣷⣒⡞⢠
⠀⠀⠈⢿⢿⠿⡿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠞⠀⢿⣶⠇⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⡢⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⠀⠀⠀⠀⢶⡦⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⢌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠙⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠨
⠀⠀⡕⠀⡀⠀⠀⢰⣶⠀⣦⠀⢀⣇⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢠⠀⠀⢸⡿⠀⠘⢦⣘⣟⣠⠞⠀⢰⡄⠀⠰⠀⠀⢰⠄⠀⣆⠀⠀⡆⠀⠀⠀⠀⡄⠀⢀⠦⠀⢰⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⡄
⠀⠘⠀⠀⠾⡿⢀⡼⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⢳⣄⢻⣟⣠⠟⠀⠀⠹⣄⡺⡿⣀⣀⠀⠀⠻⣄⡘⣿⣂⡾⠀⠀⠐⢿⣇⣀⡼⠃⠈⠳⣤⣀⣻⠿⣀⡾⠁
⠀⠀⢰⡟⣿⣿⡏⠀⠀⠀⠀⢠⡿⠻⣧⠀⠀⠀⠀⠉⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⡏⠉⠻⠆⠀⠈⢹⣿⣯⠀⠀⣠⠞⣿⣿⡅⠀⠀⠀⠀⠀⢹⣿⣿⠉
⠀⠀⠀⠳⣿⣿⡇⠀⠀⠀⣰⡟⠀⠀⠘⢷⡀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠘⠇⢸⡿⣧⠀⠀⠀⠀⢀⣿⣿⣿⡆⠈⠃
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠘⠋⠀⠀⠀⠀⠈⠛⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠉⣿⢻⡏⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠈⢻⢻⡟⠁
⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠿⠀⠀⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠾⠃⠼⠃⠀⠀⠀⠀⠸⠸⠇


⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠘⡒
⠀⠀⠀⠀⠀⣠⣬⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣃⠀⠀⠀⣠⡄⠈⠁
⠀⠀⠀⠀⠀⢿⣿⡿⢿⡿⢶⡤⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣷⢶⣯⠅⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⢻⡥⣮⡁⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣀⠀⠀⠘⠉⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⡄⠀⠑⠀⠀⠁⠀⠁⠀⣷⡆
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⠈⢹⣧⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⢧⡜⠀⣴⣿⣷⣤⣤⣠⣿⠁⢀⣀⡄
⠀⠀⠀⠀⠀⢀⣿⠁⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣼⠇⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠾⡆⠀⠀⠌⣧⣄⢙⡿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠉⠀⠀⠀⠀⠀⣿⢿⣇⠀⠀⣿⣿⣷⠀⠀⢰⡄⠙⣿⣷⣻⣿⣿⣧⣦⡄⠈⠓⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠙⣷⣶⣮⣩⣥⣴⠇⣝⣝⣄⣿⣿⣧⣿⠙⢿⣿⠟⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡻⠿⢿⣿⣿⣿⠀⠿⣿⠟⣿⣿⣾⡻⢷⣶⣶⣾⡿⠛⠉
⠀⠀⠀⣰⣷⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⣹⠾⣿⣾⣿⣿⡟⣿⡇⢸⣿⣿⣿⣷
⠀⠀⠀⢈⣭⡁⠀⠀⠈⠀⠀⠀⡀⠀⠀⢈⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢸⣿⣼⣿⣦⣿⣿⣿⣿⡇⣿⡇⣸⣿⣿⣿⣿⣇
⠀⠀⠀⢸⣿⡟⠶⠶⠛⠂⠀⠀⠛⠲⠶⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢸⣿⡇⣻⣯⣿⣿⣿⣿⡇⣿⡇⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⣿⣿⣿⠸⣿⡇⣿⡆⣿⡏⣿⡟⠃⠙⠋⠉⢹⣿⢹⣿
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠿⠗⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⢸⣿⠁⠈⢸⣿⡇⣿⡇⣿⡇⣿⡇⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢸⣿⠀⠀⠀⠉⠀⠉⠀⣿⡇⣿⡇⠀⠀⠀⠀⠸⠟⠘⠿
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠿⠇⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠲⠾⠿⠒⠀⠀⢀⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠚⠋⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣿⣿⣿⣿⠿⠠⠜⢁⣀⢀⣀⠀⠀⣸⠟⠃⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡶⢶⣴⣾⣿⣿⣿⣿⣿⡟⣨⠂⠀⠀⠃⠀⠉⠀⠁⠀⠀⢹⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡸⣷⡀⣇⢹⣿⣿⣿⣿⡟⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣭⢣⢹⡜⣟⣿⣿⣿⡉⢾⠃⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡇⣧⢻⢹⣿⢿⡁⡎⠀⠀⠈⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠲⢤⠀⠀⠀⡞⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡸⡿⢸⣽⣿⠸⡜⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣧⣸⣿⣷⣇⠁⠀⠀⠀⠀⠀⠀⢀⣆⣀⣴⣄⣀⣤⠠⠀⠈⠶⠠⣄⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠻⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡝⣿⣿⡄⠉⠀⠂⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡏⣻⡶⠀⠐⠀⠀⠀⢀⣀⢀⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⣦⠀⠀⠀⠀⠈⡇⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⡠⣄⠀⠀⠀⠀⠀⠘⠃⠃⣿⡷⡻⠓⠀⠀⠈⠀⠀⠉⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⢉⣻⡧⠀⠀⠀⠀⠀⠀⠀⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠘⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣀⢀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣄⠠⣠⣤⣤⡤⠠⠖⠒
⠀⠀⠀⢡⠐⠈⠚⠀⠄⠠⠙⠀⠀⠀⠀⠀⢠⡄⡉⣶⣆⠲⠀⠒⠀⠀⠀⠀⡄⠄⢆⡀⠀⢠⢠⡼⠡⠁⠀⠀⣾⣟⢻⡏⠹⡮⠠⡈⣷⡶⠴⠄
⠀⠀⠀⠠⣦⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⡗⠿⣾⣿⡥⢦⠆⠀⠀⠠⠹⠋⠈⠀⠁⢈⢽⠏⠀⠀⠀⠀⠀⠙⡟⠀⠀⢻⡆⠁⠐⢢⠁
⠀⠀⠀⡄⠀⠀⢒⣿⣂⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⢋⣉⡌⣄⠀⠀⠀⢰⣄⣈⣤⣤⣤⣾⡘⠀⠀⠀⠀⠀⠀⡃⠀⠀⢼⣿⠀⠤
⠀⠀⠀⠐⠄⠀⣼⡟⠿⣹⣿⠟⠀⠀⠀⠀⠀⠡⡴⢲⣿⡇⢉⣄⡗⠀⠀⠀⠀⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠡⢘⠀⠼⠟
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢀⠀⠀⠀⠀⠀⠀⡴⠂⣸⣃⣟⣧⠑⠀⠀⠀⠀⠀⠀⠀⢀⠠⢖⣴⣤⡀⠠⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣤⣾
⠀⠀⠀⠀⠀⡆⠀⡲⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠩⢿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠒⡒⠃⠀⠀⠀⢀⡀⠀⠠⣍⣿⠿⣿⣁⡤
⠀⠀⠀⠐⢂⠡⠘⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⢋⣿⠏⢸⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠜⠀⠃⡰⠒⠁⠀⠀⠀⠀⠈⠀⠀⠀⢠⠇⠀⣽
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⡟⠒⠛⠻⠿⣿⡃⠀⠂⠀⠀⠀⠂⠠⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⠄⠰⠧⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡐⣤⠀⠘⢻⠠⠈⠀⠀⠀⠀⠗⠟⠭⠻⠻⠻⠄⠙⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠠
⠀⠀⠀⠀⠐⠀⠀⣈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠐⢉⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠼⢇⡨
⠀⠀⠀⢠⣂⠁⢀⣤⠞⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⣿⣿⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⣘⣋⠡⢀⠁
⠀⠀⠀⠸⠟⢿⡏⠀⠶⠘⠈⠀⠀⠀⠀⠀⠀⡽⢿⣿⣿⣧⣌⡹⣀⠀⠀⠀⠀⠀⠀⢠⠋⡁⢹⢣⠀⠀⠀⠀⠀⠀⠀⢺⣫⡿⠁⠊⠀⠐
⠀⠀⠀⠀⠢⢼⡅⠁⠛⠀⠀⠀⠀⠀⠀⠀⠠⠖⠏⠘⢽⡍⠉⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠐⢁⡀⢷⡰⠆
⠀⠀⠀⠰⠃⣼⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⣀⣄⣄⣠⡌⣅⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠡⠇⢀⣌⣯⣯⠚
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠋⠉⠋⠃⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠉⠋⠉

⠀⡴⡶⠦⠤⠀⠀⠀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⡤⠀⠀⠀⠀⠀⠀⡀⠀⢠
⡄⠀⠀⠀⠈⠀⠀⠀⣅⣀⣀⡠⢀⠤⡤⠵⢤⠶⠄⢶⢒⡴⡄⣐⢂⡂⡂⡀⠐⠀⠆⠀⠋⠀⣧⠂⣾⠈
⠃⡀⣀⢐⡴⣦⢂⣂⠈⣐⠆⠘⠬⠄⠩⠬⠈⠈⢈⢁⣂⠐⠐⠒⠀⠰⠤⡥⢁⡌⡉⠻⡀⠘⠁⢸⠇⠀⣠
⠀⡄⠔⠠⠴⠒⠆⠐⠐⠂⠘⠂⠠⠡⠅⠥⢴⢀⠒⣀⢀⣤⠌⢈⠁⠃⠒⣒⠐⢀⡈⡶⠁⠀⢀⣿⠁⢰⠇
⠐⠐⠛⠐⠈⢃⢀⠈⢉⡃⢀⠃⠙⠐⡛⠀⠒⠰⠐⠫⢤⠈⢁⢈⡐⠘⠉⠋⣹⡯⠘⠃⣠⠂⣼⠃⠀⣾
⠄⠛⠙⠀⠃⢁⠈⠁⢘⠉⠀⠘⠚⠀⠁⠘⠘⠁⠋⠐⠀⠰⢠⡄⠬⠌⠘⠃⠀⠀⣀⣴⠇⢀⡇⠀⠀⡇⡄
⡤⠘⠇⠃⠳⠉⠃⠸⠈⠁⠈⠃⠸⠀⠻⠰⠱⠀⠀⠃⠆⡰⠂⠀⠀⠀⠀⠀⠒⠒⠛⠿⠳⠾⢥⣀⣐⡇
⡀⠀⢐⠀⢀⢠⠁⢰⠀⠆⠀⡖⠶⠀⡖⠦⠆⠰⠆⠄⡤⢸⠁⢀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠐⠀⠠⠀⡉⠙
⠁⠀⠈⠃⠀⠀⠀⠤⠴⠄⠀⠀⠠⠀⠁⠠⣤⠀⠀⠀⠁⡞⢀⡎⠀⠀⣈⡇⠀⣀⠀⠀⠀⠠
⢀⠀⠠⢤⠀⠄⠀⠰⠂⠢⠀⠀⠄⠄⠀⢀⣀⠀⠀⠐⠀⢀⡸⠁⡀⢠⠛⠀⠀⠀⠀⠀⠀⢈⠁
⠌⣠⣴⠄⠀⠚⠒⠀⠙⠂⠀⠀⠐⠒⠂⠀⠈⢣⣤⡀⠑⠞⢁⡜⠀⠈⠋⠀⠐⠂⠀⠀⠉
⠇⠀⠀⠠⠶⣶⠄⠀⠀⠀⠀⣀⡁⠀⠀⢀⢛⠶⡖⠃⡀⠀⠻⠄⢠⡈⠀⠀⠀⠤
⠂⣠⠐⠀⠀⠠⡄⠀⠔⠀⢨⡉⠘⠀⠀⠘⠮⠁⠀⠃⠀⠀⠄⠈⠁
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠂⠀⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⡔⣂⠀⠀⠀⠀⣶⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠙⠟⠀⠀⠀⣠⠛⠦⠀⠀⠀⠀⠉⠻⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢀⠼⠑⠆⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⠀⠀⠀⠛⠙⠂⠀⠀⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠁⠀⠀⠀⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⣦⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⢚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣤⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⡀⠀⠁⠀⡀⠀⠀⡠⠀⢓⠴⠊⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠈⣠⠀⠈⠑⢤⢠⣩⠚⠋⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡖⠰⠿⠁⠛⠁
⠀⠀⠀⠀⠀⢠⣿⠋⠀⢀⣾⣿⣷⡤⠀⠶⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠲⠏⠀⠉
⠀⠀⠀⠀⢀⣾⠃⢀⣴⣿⣿⠿⠋⠀⠈⣠⣄⡈
⠀⠀⠀⠀⣼⠃⣠⣾⡿⠛⠁⣀⣤⣶⣿⠿⠛⠋⠁
⠀⠀⠀⣸⢃⣾⠿⢋⣠⣴⠾⠟⠋⠁
⠀⠀⢰⣷⣿⠵⠞⠋⠉
⠀⠀⠛⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣆⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠿⠀⠙⠉⠻⠶⠟⢳⣤⡀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠟⠀⢀⠀⠀⠀⠀⣠⣦⠀⠸⢧⣤⠶⠟⣻⢄⣴⡄⣠⠾⣆⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⡟⠙⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⢼⠿⠞⠁⠛⠃⠀⠙⠉⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣁⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠶⠂⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠤⢤⢤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠀⠀⠤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢳⣤⣷⣤⣤⣴⠋⢀⣤⣀⠙⣧⣤⣤⣤⣤⡾⠉⣠⣤⡈⢻⣦⣤⣤⣾⣴⠞⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⣀⣀⣠⣿⠀⢿⣤⡿⠀⣿⠀⣀⣀⣸⡇⠸⣧⣼⠇⢠⡟⠀⣀⣀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠁⠉⠙⢷⣤⣀⣤⡾⠋⠀⠀⠁⠉⠻⢦⣄⣀⣤⠞⠁⠀⠈⠁⠉⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⡤⢤⠀⢠⡀⡇⠀⡿⡄⡰⡇⣤⢠⠤⡀⡅⡤⢤⢠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠂⠳⠚⠘⠝⠃⠃⠀⠃⠙⠃⠃⠛⠘⠀⠃⠃⠃⠘⢸⢛⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠻⠏⠁⠀⠀⠀⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠸⣧⣄⡀⢠⣾⡇⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣧⠀⠀⠈⡀⠀⠇⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣰⣦⣾⡷
⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⡆⠀⣾⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠛⠿⠀⠀⠀⢀⠫⠄⢂⠁
⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠈⠛⠁⠀⣴⣦⣶⣆⠀⠀⠀⣿⣿⣇⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣦
⠀⠀⠀⠀⠻⢿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⣿⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢻⣿
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡤⠀⠀⠀⡀⠀⣿⠃⠈⣿⡁⠀⠸⠝⢛⣛⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⢀⡀⢀⡀⠀⢸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠿⡿⠟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣳⠘⠁⠀⠀⠠⠂⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠑⠀⠀⠀⣷⣶⣾⡓⠀⠀⢡
⠀⠀⡀⠄⠀⠀⠛⢀⡌⠀⡴⢁⠔⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢂⡀⠀⠴⠖⠈⠈⢹⡿⠏
⠀⠀⠀⠀⣘⣿⣀⠀⠀⠈⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣿⠟⠻⡁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠈⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣐⣿⣧
⡀⠀⢀⣌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙
⠿⣿⣿⣍⠀⠀⠘
⠀⢨⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⠲⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⡂⢇
⠀⣴⣄⢀⡠⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠉⢹⣿⣿⠉⠀⢤⠀⠀⠀⣀⠔⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⣶⣤⣴⠯
⣠⡾⠋⠁⠀⠀⠐⠁⢀⡔⠡⠠⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⣿⣿⠆
⠁⠀⠀⠀⠀⠂⣺⣶⣾⡋⠁⠉⢖⠒⠂⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠁⠀⢹⡟⠁⠀⠀⡀⠀⢀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣶⣶⡆⢀⡀⠀⠁

⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⣴⢶⡦⠀⠀⠀⠀⡀⡀⣄⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢰⣌⠉⠉⢹⡥⠄
⠀⠀⠀⢀⢴⡿⠻⣆⡄⠀⠀⠀⠐⣿⣾⠗⠀⠀⢠⣿⣷⣼⣲⣶⣴⣶⣆⠀⠀⠀⠀⠰⠀⠀⠠⠞⠛⠒⠂⠚⠓
⠀⠀⠀⠀⠆⢸⣼⠙⠆⠀⠀⠠⠀⡧⣼⠹⡀⠀⠐⠛⠛⠉⡟⠀⠈⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠛⠒⠞⠋
⠀⢀⠄⠉⣠⣀⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢀⣈⣼⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠘⠀⠀⠀⠈⠀⠀⢸⠀⠤⠤⠄⠀⠀⠀⠠⢤⡿⡿⣿⡿⣧⣀
⠀⠀⠛⠿⢾⣿⣿⠮⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠊
⠀⠀⠀⠀⣤⡀⠀⠀⣾⣿⣿⡇⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⣀⠘⢿⣧⡀⠙⠛⠛⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⡵⠃⠀⠀⠀⠈⠁
⠀⡚⠛⢛⣷⡘⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⢀⣁⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡄⠀⢰⡄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠸⣷
⠀⠀⠀⣀⣠⣏⣉⡉⣙⣀⠀⠀⠀⠀⠀⠈⠁⠀⢘⠈⠁⠀⠀⠀⠈⢸⠧⠆⡄⠇⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⣂⣿
⠀⠀⢈⡉⠙⠉⠉⢛⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⢀⣃⢀⡀⠀⠀⣸⣿⣶⡀⢐⣲⣯⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠁⠀⠀⠸⠃⠃⠻⠛⠼⠇⠀⠀⠿⠿⢿⣿⣿⣿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣴⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠉⠹⣦⣤⠁⠀⠘⡟⠙⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣤⣄⣴⣿⡏
⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⣁⠀⠀⠀⠀⠀⠀⢀⠏⠘⠀⠀⠀⠀⣻⣦⡄⠀⠀⠀⠉⢻⠟⠋
⣰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣾⣼⡁⠀⠤⠄⠀⠸⠛⠇
⠿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⣿⣦⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶
⠀⠀⣰⣷⡀⠀⠀⠀⠀⢀⡀⠉⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⣦⠀⠀⠷⠀⠀⠀⠀⠹⣿⣿⡇⠀⣠⣤⣴⣶⣦⣤⣼⣿⣿⣿
⠀⠀⣿⣿⣿⡄⠀⠀⠀⡚⠉⣷⡆⠀⠀⠀⠀⠄⠀⢠⡤⡀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⢰⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿
⢀⣴⣿⣿⣿⡇⠀⠀⢠⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣼⣿⣿⣿⣿⠏⠁⠀⠈⠙⣇
⣿⣿⣿⣿⡿⠁⠀⠀⠀⡀⢸⣏⣿⣿⠀⠀⠀⠀⠀⠐⠀⠀⣠⣄⡀⣰⠒⢲⣾⣆⠀⠀⠀⣀⣿⣿⣇⡈⠃⠀⠀⠐⠀⠘⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠀⠀⠉⠁⠀⠂⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠙
⠀⢀⢀⠠⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠄⠀⠠⠁
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⡊⢉⣹⣿⠛⢉⣉⡁⠈⢉⡉⠉⠉⣡⣿⠟⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣉⣷⠟⠉⠀⠀⠈⠙⢿⣄⠈⢈⣱⡾⠋⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣉⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣋⠵⡀⠀⠀⠀⠀⠀⠢⡀⠀⠀⡰⢆⣉⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⣰⠦⣾⣽⡤⣶⠀⠀⠀⠀⠀⠀⠈⢄⢐⡦⣼⣽⡦⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⢀⣏⠉⠛⡦⢤⣶⠤⡞⠃⠀⠀⠀⠀⠀⠀⠀⢻⡧⢤⣧⠤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⢻⣸⠶⡶⢿⣖⡒⣾⣤⣤⣄⣠⣄⣀⣀⠀⠀⠀⣫⡖⡒⣎⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠰⡀⡸⠁⣠⠠⠄⠤⢳⠨⡠⢋⠛⠉⢉⣁⣀⣷⠋⠄⠠⢡⡀⠘
⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠄⠇⠠⠧⠴⠀⠀⠀⠀⠀⠹⠀⡔⢰⠐⠒⠒⢺⡀⠁⠆⠀⠀⠀⢿⠇⢿⡆⠒⠒⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣨⣤⣀⣀⣼⠀⠂⠀⠀⠀⠀⢸⡇⣸⣇⣠⣠⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⠉⠁⠉⠉⠉⡀⠀⠀⠀⠀⠀⣼⡖⡍⠈⠉⠉⠉⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⢆⠀⠠⠀⠀⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡜⠀⠀⠀⠀⠀⣀⠼⡆⡈⡀⠀⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠼⣈⠂⢠⠀⠀⠀⠀⢀⠈⢀⡇⢊⡹⠜⣁⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣮⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠠⠠⠀⡀⠀⠀⢀⡀⢀⣠⣤⡟⠀⠀⠀⢀⣀⣤⢀⠀⠀⠀⠀⠀⠀⢠⡤⡄⠀⠀⡀⡐⢀⠤⢤
⢠⢀⠀⠂⠃⠀⢃⠀⢀⣐⠁⠈⠁⠀⠛⢹⠖⠚⠃⠀⠀⠀⠀⠉⣀⠀⠁⠀⣤⠶⠓⠂⠈⠀⢠⠀⠀⠀⠠⢈⠀⠐⠂⠀⠀⠁⠆⠁⠀⠉⠀⠀⢀⣐
⠘⠀⠀⠀⠁⠀⠀⠘⠁⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⡈⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠒⠒⠒⠀⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠋⠉⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢶⠟⡙⢡⠋⡴⡡⢃⡜⣡⢊⣽⡿⠿⠦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⣠⠴⠏⠃⡇⡞⠰⢀⠏⡜⡰⢡⠏⡔⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⢈⣠⠴⢛⠙⠈⠰⠰⠰⠁⠁⠌⡜⠐⠡⠃⢪⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⢶⠶⠚⠹⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠣⠀⠀⠀⠀⠀⠀⠠⠀⠠⠁⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠒⠒⠶⠤⠤⠧⠤⢤⣀⡀⠀⠀⠐⠀⢀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢡⣶⠮⠉⢁⣀⠀⢀⠀⠀⠀⠀⠉⠉⠙⣆⠀⢀⡴⠋⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠐⠈⠀⢄⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢀⣼⠿⠅⠛⠋⠉⠉⠁⠀⠀⢀⣀⣀⣀⡴⠋⠀⡞⠀⠀⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣀⣴⢾⢻⠛⠿⡿⢿⣿⣿⠋⡅⠀⠀⠀⠀⢧⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠐⠲⠄⠀⠀⠀⠀⡇
⠀⠀⠀⢰⡟⠉⠀⣀⣀⣀⣤⠞⠉⠈⠌⠘⠁⠸⠀⢰⣿⠃⠀⡇⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠒⠋
⠀⠀⠀⠘⠦⡾⠿⠛⠛⠻⢅⠀⠀⢀⠀⠀⠀⠀⠀⢸⡏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠉⠒⠠⣤⣴⡶⢤⣤⣤⣤⡶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠀⠀⠀⠀⠀⠸⠃⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⠷⢸⣇⠫⢻⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡇⠀⠈⠀⠰⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢃⠀⣾⠀⢀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠠⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠿⠼⣤⣠⣦⣤⣄⡠⠃⠀⢀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣆⣠⣄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡽⠛⣿⣿⣷⡾⠿⠦⠤⢤⠤⣤⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⢟⣋⣭⣥⡶⣶⣿⣿⣿⣷⣶⣟⡽⢁⠀⠀⠈
⠀⠀⠀⠀⠀⡠⣠⣬⣶⣾⣿⠟⠫⠉⠃⢛⠉⢿⣿⣿⣿⢏⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢤⣤⣠⣄
⠀⠀⠀⢠⣾⣿⣿⡻⣱⣿⢷⢄⠀⠀⠀⠀⡙⢿⣿⣿⡿⡿⡏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣴⡟⠹⢽⠯⡷
⠀⠀⠀⢸⣻⠿⣾⣿⠾⣭⣿⣿⣤⣀⣤⣃⣰⠿⠿⠾⢷⣭⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣽⣷⣗⣶⢂⡄⠀⣠⠬⣻⢶⠦⢄
⠀⠀⠀⠀⠠⠀⣌⡷⠿⢛⣹⢿⣭⡡⢊⣁⡉⠸⢿⠟⣶⣶⣿⠄⢀⣀⠀⠀⢠⡴⣶⢰⡿⢡⣧⣿⣿⣿⣿⣿⣶⣏⣷⣧⣋⢷⡾
⠀⠀⠀⠀⠀⢹⡿⢰⡟⣛⠁⠛⢙⣿⣿⣿⣷⢼⣟⣼⢟⣿⡋⢻⣄⠰⣲⠿⠃⣴⢻⣿⠁⣸⣯⣽⣷⣿⣧⣿⣯⣿⣿⣿⡋⡼⡇
⠀⠀⠀⠀⠀⠈⢻⢯⡛⠘⣿⡄⠄⡁⣈⣹⢁⣒⣙⣥⡿⠁⢀⡾⠀⢀⣿⢠⠆⢁⣼⣧⢰⣿⣿⢿⣿⣻⣿⣽⣿⣯⣿⡿⠀⣿⠁
⠀⠀⠀⠀⠀⢠⡏⠉⠃⢱⣫⣥⣈⠀⠛⠋⠟⢩⡥⠂⠐⡆⢨⣳⡆⢸⢽⠈⢻⣺⣿⠇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⡰
⠀⠀⠀⠀⢠⠋⠀⠀⠀⣌⢻⣇⣿⠂⠀⠠⠀⠀⡇⠀⠀⠀⠸⡿⠋⠙⠛⠏⠚⣿⡏⢘⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⠃
⠀⠀⠀⠀⢸⡗⠀⠀⠀⠘⣅⢻⣿⡆⠀⠀⠀⠀⠇⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡮⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠏
⠀⠀⠀⠀⠘⣽⡄⠀⢀⡀⠈⣹⣿⣷⡀⡆⠀⠀⢸⣿⡀⠀⣸⠁⠀⠀⠀⠀⠸⠠⢢⡿⠿⣽⣿⣿⣿⣿⣿⣿⣿⢁⡞
⠀⠀⠀⠀⠀⠙⣷⠀⠿⠃⣠⣯⣿⣿⣿⣶⣰⣧⣾⣿⡇⠀⣸⠀⠀⠀⠀⠠⡟⡈⠘⣿⣿⣿⣿⣿⣿⣽⣿⣿⠇⣼⠁
⠀⠀⠀⠀⠀⢰⣿⣿⠀⠉⠘⠍⠉⢠⠄⠋⠽⣿⣿⢿⣡⣾⡟⠀⠀⠀⠠⡱⠐⠀⡠⢺⣿⣿⣟⣿⣿⢿⣿⠏⢰⠇
⠀⠀⠀⠀⢀⢺⣟⣿⣷⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠒⢪⠉⠶⠓⠆⠈⠈⠕⠿⠿⣿⣻⠯⣿⢟⠋⣭⣿⣿⢀⡆
⠀⠀⠀⠀⠈⠻⣅⠀⣿⣳⣀⠀⠀⠠⣤⣶⣤⡶⢤⣀⡏⠠⣶⡴⢶⣤⣈⣛⠲⡤⠶⣭⠙⠒⡮⢶⣿⣿⢃⡬⠁⠀⠉
⠀⠀⠀⠀⠀⠀⣨⣱⣾⣿⠿⠟⢻⠶⠶⠶⠦⢤⣭⣭⡧⣮⡍⣽⡆⠛⠉⠓⠁⠀⠀⠀⠀⠈⠐⠛⠶⢥⡾⠁
⠀⠀⠀⠀⠀⠀⣹⣯⡾⠁⡀⠚⠙⠁⠂⠘⠁⡟⣼⠟⣹⣏⡱⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠐⠊⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠋⠞⠁⠊⠀⠸⠳



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⡀⢠⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⢸⠉⠈⢹⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠠⠅⢒⠚⠢⠆⠚⢒⠂⠵⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠀⠰⠄⠀⠀⠀⠆⠀⠀⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡤⠤⠤⠤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⠀⠀⠀⠀⡴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⢀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⢰⠂⠀⠀⠀⠀⢣⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡴⠋⠀⢸⣷⡀⣠⠴⠂⡈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠁⠀⣀⠚⢿⣧⡀⠀⢠⠄⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠛⠦⣤⣤⢬⣷⡴⠀⠀⢠⡿⠄⡄⠠⠄⢠⡀⠠⠀⠠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠬⠤⠤⠥⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⢠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠁⠀⡠⠐⠀⠀⠴⠁⠀⠀⠀⣀⡀⣀⠀⡐⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢀⣄⠘⠢⠄⠀⠀⠀⢰⡁⠀⠀⠀⢰⠁⠀⠁⠛⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⢁⠂⠸⢦⠤⢤⡇⣀⠀⠀⢀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠈⠂⠌⠀⠂⠀⠀⠀⠀⣂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠑⠰⣀⢀⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠂⠠⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣶⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠶⠾⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠉⠀⠉⠙⢿⣦⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⢤⠀⠀⠀⠀⠀⠘⠛⡋⠀⠀⣶⣶⣶⠀⠀⢙⠛⠃⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠛⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠐⠀⠀⢠⣾⣷⡆⠄⠁⡂⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠛⠛⠛⠛⠛⠛⠛⢻⡟⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠸⡾⠁⠀⢀⠀⠀⠀⠀⠀⠲⠒⠶⢀⠀⠀⠀⡀⠶⠒⠖⡀
⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡁⡆⠀⠾⠏⠀⠀⠀⢉⠀⠀⠀⠽⠀⠀⠀⠏⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠟⠃⣸⡇⣠⠆⠀⠀⠀⠀⠀⠀⠚⠐⠀⠳⣄⠀⡠⠚⠀⠂⠓⠀⠀⠀⠀⢀⣀⣘⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣀⣀⣈⡳⣄⠀⠀⠀⠀⠀⠌⠙⠛⠉⠛⡉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠰⠎⠀⠈⠉⠉⠁⠁⠈⠳⠆⠀⠀⠀⠀⠀⠀⠤⠄⠁⠀⠁
⠀⠀⠀⠀⠀⠜⠈⢣⢠⠊⠉⡄⢠⠉⠙⠀⠀⠀⠀⠀⠈⠛⠋⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⢦⣉⣉⣉⣡⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠾⠿⠷⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⣤⡄⠀⡀⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠿⠟⠁
⢠⠀⠀⠀⠀⠀⠒⠒⠃⠀⠉⠀⠀⠀⠈⠑⣿⠊⢠⣦⣼⠃⠀⠀⢺⠟⡀⢿⢋⠁⠘⣕⡏⣱⣶⡦⠀⠀⠀⠁⠀⢰⡿⠃⠀⠀⠀⠀⣤
⢚⣶⣦⠀⠀⠀⠰⣦⠘⣧⠀⠀⠀⠄⠀⠀⣀⡀⠿⠃⠋⠀⠀⠀⠀⠀⢱⡏⠀⠀⢸⣟⠀⠸⠟⠁⠸⠇⢀⣀⢀⡜⠁⠀⠀⢤⠔⣆⣿
⢀⣽⣿⣆⠀⢀⡋⠁⠷⠈⠀⠀⠀⠀⠀⣿⣿⣿⠆⡘⠃⠀⠀⠀⢸⣶⠀⠈⠡⠄⢀⠈⠀⠶⠀⢠⣖⠀⠀⠹⡿⠷⠾⠀⠀⠉⠠⢿⡶
⠛⠛⠋⠁⠀⠈⣇⣤⣶⣷⠀⠀⠀⣀⣀⠻⠿⠛⢸⡃⠀⠀⠀⠀⠘⡥⢂⡀⠀⣰⠀⠀⢠⣾⣷⡌⢿⡦⠀⠐⠀⠀⣴⣦⡀⠀⠀⢀⡀
⠀⠀⠀⣴⠃⠀⣿⣿⣿⡿⣀⣶⣾⣿⣿⣿⣶⣶⡿⠋⠀⠀⠀⢀⡀⣴⠛⠛⢠⠟⠲⣶⣾⠟⠋⠀⠀⠀⠀⢰⡶⠿⠚⢿⣿⠉⠻⣭⡀
⠀⠀⠀⠻⢿⣴⠟⠻⣿⣄⣿⣿⠟⠿⠿⠿⠟⣻⠁⠀⠀⠀⠀⠀⠀⠛⠶⡖⠻⠂⠀⠀⠁⠃⠀⠀⠀⠀⠀⠸⡶⢰⣄⠈⠁
⣀⣀⣀⣀⣠⣗⣀⣀⣹⣿⣿⣃⣀⣠⣿⣶⣾⣃⣀⣀⣀⣀⣀⣴⣗⣈⣀⣀⣀⣠⣆⣀⣀⣿⣦⣀⣀⣀⣀⣢⣀⣀⣀⣿⣀⣀⣀⣀⣾
⣿⣿⢛⠻⠟⡛⢻⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠴⡿⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⢴⡏⠿⡟⠟⠁⠀⠀⠀⠀⠻⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡽⢟⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⠁⢀
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⠀⠀⣀⣀⣀⠀⡀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⢈⡀⠐⢿⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⣤⠀⣀⣀
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢿⣷⣶⢿⠛⠛⠉
⠀⠀⠀⠀⢉⡁⠉⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⣿⡇⢈⣀⡀⠀⠀⠀⠀⣀⣀⣧⢦⣤⣤⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠻⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠁⠈⠀⠀⢀⣀⣤⣈⣭⣿⣶⣉⣅⡙⡃
⠀⠀⠀⠀⠀⠀⠹⣿⡏⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⡇⣿⡏⢻⣿⣯⡿⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠹⣿⣤⠀⠉⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠰⣶⢞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⢴⣾⣿⣿⡄⠈⠂⠙⠛⠀⠀⠐⠀⠀⠀⠀⠎⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⣀⣀⠀⠀⠀⠀⠀⠀⠋⡿⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣼
⠀⡠⠒⠃⠀⠒⢀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⢀⣾⣟
⠞⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠉⠉⠤⠤⠤⠄⢀⡴⠒⠒⠒⠠⡀⠀⠀⠈⠃⡀⠀⠀⠀⠄⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⣶⣿⡟⢉
⠂⠒⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⣄⡀⢈⢢⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣷⣿⣿⣷⣾⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠐⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢯⣿⣿⢿⣿⣿⣯
⠷⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⡠⠒⠀⠀⠐⢉⣿⡿⠄⠉⠏⠛
⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢵⣤⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⡟⠒⠠⠀⠀⠀⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠉⠙⠧⠀⢀⠀⠀⠀⠒⠤⢤⣿⣿⣿⡇⠀⣿⣇⣠⣴⢅⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⣤⡼⣿⣿⣿⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⢈⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠠⠀⠀⠀⠑⠀⢀⡎⠀⠄⠺⠃
⠀⠚⠀⢤⡀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢔⣾⠿⠛⢿⡇⠐⠦⠀⠀⢀⣀⡀⠀⠀⠀⠃⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠈⠐⠛⢧⡀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠄⣀⣀⣀⡀⠤⢰⠞⠋⡠⠀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠒⠻⣷⢶⣷⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠔⠋⠀⠀⠀⠀⠀⠀⠀⠂⠣⣄⣀⡄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠀⠀⠁⠉⠉⠉⠚⢆⠀⠀⠀⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⠁⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢸⣇⣸⣀⣠⣾⠿
⠀⠀⠀⢀⡇⠉⠉⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢱⣄⣆⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣽⣧⣾⣿⣿⡿⠿⠃
⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢿⣿⣆⡀⠀⡜⠁⠀⠀⢀⣴⣞⡥⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⡁⠀⠀⠀⠐
⠀⠀⠀⡀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢤⣀⠀⠀⠀⠀⢻⣿⣷⣀⣠⡄⠀⣤⣿⣿⠟⢀⡀⠀⠀⠀⠀⠢⡄⢀⠍⠀⠂⠉⠷⣁⠂⠤
⠒⠀⡉⠁⠀⠈⠀⠀⠐⠄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠙⢿⣿⡛⠃⢰⣿⣿⣩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣶⣶⣉⡄⠓⢦⣹⣿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠒⠋⠉⠉⠈⠞⠛⠙⠋⢻⡿⢿⠅⣠⣆⠋⣼⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⣀⠑⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠂⠘⠋⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⣿⠟⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠠⣾⠂⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠁⠺⠁⠀⠀⠀⠈⠄⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⣀⠋⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠀⣠⠀⠀⠀⠨⠀⠀⢠⣶⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⣥⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠀⢰⣋⠀⠀⠀⠀⠀⠂⠀⢀⠁⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠤⠊⠕⠀⠀⠀⢠⠂⠀⣀⠠⡀⠀⠀⢀⣤⠀⡠⠀⠒⣾⠇⠀⠀⠀⠀⠀⠁⠰⠂⠀⠁⣀⠀⠀⠀⠀⠀⠀⠒⠀⢭⡓⠶⣄
⠀⠀⠠⠂⠀⠀⠀⠀⡠⠢⠄⠜⠀⣨⠀⠀⠀⡇⠀⡘⠃⠀⡀⠀⠀⠀⠀⠀⠑⡄⠀⠘⠄⠠⣤⡢⠀⣠⣵⠤⡂⢀⡢⢄⠙⢦⡀⠀⠀⠀⠻⠃⠀⣿⠄⠘⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠙⠃⡀⠀⢀⠠⠂⠀⠀⡇⢰⡁⠙⠬⠁⡆⠀⠀⠀⠀⠀⠘⢄⠀⣠⡀⠻⠐⠀⢹⠚⢀⡁⡠⠀⡘⠀⢆⢣⠀⠠⢇⠄⠀⠀⢀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⠘⠦⠘⠀⠂⢤⠀⢣⡐⠎⠑⠢⠞⠀⢠⡐⢢⠀⠀⠀⠈⢆⠀⠀⢀⠀⠂⣁⠎⠀⠭⣤⠜⠀⢀⡸⠙⠀⠀⠂⣠⠤⣊⠎⣡⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠖⠁⠀⠀⠈⢇⠀⠈⢰⣒⠀⢰⠀⡎⠀⠈⡄⠀⠀⠃⠀⢮⠀⠀⠗⠤⣀⠀⠀⠀⠊⡰⠋⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⢀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡀⠉⠨⡄⠈⠑⢤⡁⠀⠀⠀⡐⠀⢹⠺⡀⠀⠀⠉⠁⠠⠭⠤⠤⠐⠉⠀⠀⠀⢀⣠⠴⠂
⠀⠀⡘⠀⡘⢃⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣄⠘⢇⠀⡇⠀⠀⠁⠀⠀⠀⡹⠀⠀⠠⠀⣤⠀⠣⡈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠃
⠀⢀⠃⢀⠃⠸⡀⠀⠀⠀⠀⠤⣭⡭⠰⡍⠙⠂⠀⠁⠷⢀⣀⣀⠔⠉⠉⡄⠀⠀⠐⠊⠉⠀⠀⠱⡄⠀⠪⡙⠒⠤⣤⠔⠚⠉⠀⠀⠀⠀⠀⡠⠁
⠀⠘⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠙⠛⠒⠀⢄⠀⠒⢂⠐⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡁⠘⢆⣴⡡⢄⢀⠎⠀⠀⠀⠀⠀⠀⢀⠴⢤⡤
⠀⠘⡄⠐⠓⠤⠵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⡀⢃⠀⠀⠀⠀⠀⡤⡠⠙⣄⠀⠀⠀⠀⢧⡄⠀⣼⡦⠀⠏⠀⠀⠀⠀⠀⠀⡴⠓⠊⠁
⠀⠀⠈⠢⡀⢠⠾⠕⢰⠆⠀⠀⠀⠀⠀⠀⡴⠾⠃⠀⠘⠄⢆⡴⣿⡟⠚⠹⠟⠂⠈⢦⠀⠀⠀⡇⠀⠀⠈⠉⢙⠀⠀⠀⠀⠀⠀⠜⠀⠀⠈⠢⡀
⣀⣀⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣎⣴⣋⣀⣀⣀⣒⣃⣀⣘⣆⣀⣀⣇⣀⣀⣀⣀⣸⣀⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣘⣦⣀⣀⣀⣀⣀
⣿⣿⡟⡛⠿⠛⡻⠻⣟⢟⣿⢟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣰⠂⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⡏⠀⠀⠀⠀⠀⠻⣦⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣻⡇⢸⡇⠀⡀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠻⠇⠈⠀⣸⠃⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⠚⠉⢀⣠⠴⠖⠚⢻⡆⠀⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠠⠴⠚⠋⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⢀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡾⢷⠄⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢀⣀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣀⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠠⠞⠋⠁⠀⠀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⡟⠀⡰⠀⠀⢻⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣦⡀⠀⠀⠁⢠⡇⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⢰⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠻⣤⠏⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡾⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣃⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡿⠩⠂⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡗⠵⠀⠀⠀⠀⣄⡀⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡄⡠⠀⠀⠀⠀⡄⠈⠆⢸⠛⡡⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠥⡄⠀⠀⣷⣤⠐⢄⣰⢃⡼⠀⠔⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡨⡄⠀⠀⠻⣿⡇⠀⡹⢁⣤⣤⠀⠀⡀⠄⠚⠅⣀⠄⠒⡩⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⢄⡀⠛⡄⠀⠀⠀⢠⢿⡂⢅⠀⠀⠠⢌⠣⣀⢣⣿⣿⠏⠀⠌⢡⠀⢰⢊⡀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠌⣦⡘⡄⢰⣶⠀⣐⣗⠧⠀⠀⠐⡄⢱⣸⠼⠛⠁⠀⡐⠀⢀⡴⠋⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠑⣆⠀⠁⠀⠃⣿⢎⠄⠀⢀⠜⠁⠀⠀⠉⠐⠒⠇⣀⣼⠤⠐⢒⠞⠀⢀⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠀⣈⣤⡀⠘⡟⠪⣷⣂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡔⠋⠀⢠⠁⢀⠌⢀⡠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⢧⠀⡷⠊⠀⠇⠀⠀⠀⠀⠀⢀⠄⠒⠒⠄⡀⠀⠀⠈⠐⠤⣴⠖⠃⠒⠮⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠠⠄⡉⠒⠤⠄⡤⠋⠀⠀⠀⡀⠤⠤⠤⠄⣀⡎⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠈⢋⢣⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠥⣦⠊⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣠⠊⠀⠈⠢⠀⠀⡤⢀⡀⠀⠀⠸⠄⣉⣁⡉
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⣀⢰⠖⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠉⠉⠑⠄⠀⠀⡄⢀⡅⠀⠈⠢⡀⣰⠢⡘⠦⠈⠆
⠀⠀⢀⡤⠒⠫⡀⠀⠀⠱⡄⠀⠈⠑⠺⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢣⠀⠀⠶⠂⠀⢠⠗⠈⢡⠀⠀⠀⠘⡅⠣⣈⠄
⠀⡴⢋⡠⠄⠒⠒⠓⣒⣦⠬⠶⣄⣀⡀⢡⡀⠀⠀⠈⠄⠀⠀⠀⠀⡠⠥⡀⠀⣀⠔⠁⠀⠀⡘⠀⠀⠀⠀⢰
⣰⣀⠁⠀⢀⡔⠊⠉⠀⠀⢀⠠⠂⠉⠉⠀⡱⣄⠀⠀⠈⡶⠠⠄⠚⢄⠀⠈⠈⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⠃⢀⣠⡟
⠁⠀⠀⠀⠀⠀⠁⠒⠂⠴⠥⠄⠤⠄⠒⡨⢤⣫⠶⢆⠈⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠊⠁⠀⠀⠀⢀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣬⢛⡡⠑⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⠀⠐⠒⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡾⢏⠢⠀⠀⢀⠔⠊⠱⢄⠀⠀⠀⢀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⡟⠌⠊⠀⢀⠜⢁⣀⠤⣴⠋⠉⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡸⡗⡌⠀⠀⢀⠊⠉⠀⢀⠔⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣾⠣⠈⠁⠀⠀⣈⠀⡀⢴⠏⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⣆⣽⢣⠉⠀⠀⠀⠀⡏⠁⠀⠎⠀⠀⡄⠔⠁
⠀⠀⠀⠀⠀⠀⠮⣽⣉⡍⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠰⡰⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠒⠈⠉⠉⠉⠉⠙⠒⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⡀⠀⠀⡤⢀⡸⣭⡭⢍⠽⠟⣦
⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠉⠲⢄⡀⠀⢀⠔⠁⠀⠀⠀⢀⠀⣨⣹⡯⡵⠟⠓⠚⡩⡈⠄⠊⠍⠉⠛⢹⡧
⠀⠀⢠⢖⣥⣬⣤⣤⣤⣤⣀⣀⣤⣤⡨⠝⠁⠀⠀⠀⠀⢖⡵⠚⠁⡆⠃⡄⡰⠀⡠⠊⡀⠄⡀⠀⢀⣀⣾⡁
⠀⢰⢣⡟⣻⣷⣿⡶⠤⣀⢊⢀⡎⠉⠀⠀⠀⠀⡠⢲⢦⠎⠃⠠⠀⠁⠀⠁⢀⣸⢁⡴⠐⣀⣠⣄⣠⣴⣶⢝⢢
⢠⠃⠰⣯⡿⠋⠉⠙⡷⢮⣑⡗⢳⡀⠀⠀⢀⣸⣟⡋⡀⠀⠠⡀⠃⡀⠀⠀⠻⢁⠬⢕⡬⡴⠒⠄⠉⠑⠀⠀⡘
⢇⡠⠾⠟⠉⠓⠚⠉⠉⠉⠉⠙⢾⣦⡀⢠⡟⢹⣜⠰⠈⢄⠀⢇⢘⠃⢴⠇⡖⡃⡑⠕⠉⡓⠂⠀⣠⠴⠖⠋⠑⠒⠦⠤⣄⡠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠞⡈⣄⠙⢝⠦⠀⠠⡏⣼⠀⡈⢸⡇⣤⣁⣤⡄⠈⣠⠜⠁⠀⠀⠀⠀⠀⠠⢂⡴⠊⠅⠜⠩⠝⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠊⠪⢀⢆⠑⡀⡆⢸⠀⣵⠋⡙⣡⠁⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⣠⡾⠉⠚⠓⠋⢱⠂⠤⠵⣚⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢅⢁⢦⢐⠘⢀⠀⠜⠘⠛⢿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠾⠗⠒⠒⠛⠛⠋⠐⠒⠒⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠫⡂⢻⡇⠠⠈⠂⠸⠱⠛⢓⣿⠶⣤⣀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⣮⠳⠃⠆⠀⠀⡗⠀⠕⡕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⠰⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠂⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⡀⢀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠈⠀⠊
⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⠿⢿⢸⠿⠿⠿⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⠄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠻⢿⡿⠃⠘⠿⣿⠿⠛⠛⠛⠛⣻⠋
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡼⠸⡀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣠⡴⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⠞⠀⠀⠙⢆⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣧⡴⠁⠀⠀⠀⠀⠈⠳⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠐⠺⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠷⠦⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤
⢀⣠⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⡦⢶⣶⣾⣿⣿⣿⣿⣿⣿
⣼⡏⠉⢉⡿⠀⠀⠀⠀⠀⠀⠀⣤⢢⡄⠀⠀⠀⠀⠀⠀⠸⣏⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⣿⠁⠘⣯⠉⠉⠉⠉⠉⠉⢹⡆
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡟⠸⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣸⡇⠀⠀⢹⡆⠀⠀⠀⠀⠀⢸⡇
⢿⣆⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⣷
⠀⠙⢻⡿⠀⠀⠀⠄⠀⠀⠀⣾⠁⠀⢹⡄⠀⠀⠠⠁⠀⠀⢸⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣧⣄⣀⣀⡀⠀⠀⢀⡿⠀⠀⢸⣇⠀⠀⠀⣀⣀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢀⣽
⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣶⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠟⠛⠉⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠋⠁⠐⠈⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠋⠁⠀⠀⣀⣠⣤⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢉⡀⠀⣀⣤⣴⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠸⣿⣄⣀⣀⣠⣴⡾⠟⠋⠁⢀⣀⣤⣵⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡟⠿⠻⣿⡋⣁⣠⣤⣶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣷⡀⢠⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡷⢾⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣌⢿⣆⢀⣠⣶⠾⣿⠿⠿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣦⣿⣿⣋⣀⡴⠃⠰⠆⣀⣿⣷⣶⡿⠟⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣼⡿⡛⠻⠟⢻⡿⣾⣶⣾⠛⠉⠀⣩⡴⣾⠋⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣸⠏⠙⢧⠀⢿⣌⠳⣤⣉⠋⢀⣀⣴⣋⣀⣿⠀⠀⠁⠸⣿
⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⢻⡟⠃⣿⡶⠀⠈⠃⠠⢶⡆⢈⣿⠿⠛⠛⠋⠙⠛⠻⣧⣆⣄⣠⣿⡄
⠀⠀⠀⠀⠀⣼⣟⣿⣼⣿⣤⣽⣶⣇⢶⣤⡀⡀⠀⣴⡶⠿⢿⣦⣤⡀⠀⠀⠀⠀⠉⢻⡏⡀⠙⠻⣷⣶⣆
⠀⠀⠀⠀⠀⠈⠿⢷⣾⣿⡙⢻⣭⣿⣦⣬⣴⣷⣾⣿⣧⣀⣸⠟⢹⣿⣶⣆⢺⣂⣀⣸⣇⠛⠂⣀⣿⣾⠟⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠘⠓⠆⠹⡻⣿⣿⣠⣤⣬⣿⢉⡷⠛⠛⠻⢦⡞⠋⠁⢸⣿⢿⡿⠛⠙⢿⣿⣶⡿⠁⣠⡾⠷⣦⡙
⠀⠀⠀⠀⠀⠀⢠⣴⡿⢷⣄⠈⠉⣩⣭⣉⣿⣾⣇⣀⠻⠷⣾⠀⣤⣾⣼⡿⠸⠷⢶⣶⠿⢋⡀⠀⠀⠻⣷⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠿⠷⠾⠟⠀⡀⢿⣿⣙⣿⠟⡛⢻⠿⠿⠿⠛⢛⡉⣁⠀⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠘⠛⠋⠀⠁⠉⠙⠁⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⡿⠿⢦⡌⠑⢦⣀⣀⣀⣀⣀⣤⠴⠚⣋⡭⠭⠷⢖⣛⣋⣉⡭⠞⠛⠛⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⣋⡥⠖⠉⠀⠀⢀⢻⠀⠀⠀⠀⠀⠈⠁⣀⡴⠋⠁⠀⠀⠀⡠⠀⠉⢁⠄⠐⢀⡁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⠚⠋⡉⠀⠐⠄⠈⢂⠀⠊⠓⠒⠶⠦⢤⠴⠞⠋⠀⠀⠈⠀⢀⠊⠀⡠⠂⡁⠔⣊⣥⠤⣄⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡐⠤⡀⠀⢄⠈⢂⠀⢃⠀⠐⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⠊⢁⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠬⢵⣄⣑⣄⠡⡀⠆⠀⢡⠀⡄⠀⠀⠀⢰⠀⡄⡰⢁⠔⠠⠊⢀⠠⠒⠉⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡱⠀⠠⠀⠀⢀⠀⠀⡀⠀⠰⠠⢀⠂⡐⢁⠔⢁⠔⠂⢁⡀⠤⠤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠂⠀⠸⠀⠀⠃⠀⠂⠂⠂⠜⠐⠡⠊⡠⠐⡉⠤⠄⠀⠀⣀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠌⠐⠠⢨⣴⡵⢖⠒⠒⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⠐⠀⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⣄⠘⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢀⠀⠀⠂⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠘⠀⠘⠀⠀⢠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⡀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠄⡄⠃⠀⠀⠇⡠⡘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠁⡇⠀⢠⢀⣷⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢰⢠⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣮⣮⣧⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢰⡇⣿⡿⠟⢸⣿⢸⣿⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⣿⡇⠀⠈⣿⣾⡿
⠿⠿⠟⠟⢀⡦⠀⢀⡄⠒⠶⠶⣴⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣾⣦⣸⡇⣾⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠛⠀⢠⠏⣀⠀⣀⡀⠈⢣⡈⢃⠀⠀⠀⠀⠀⢹⠀⠀⣿⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣷⡿⢻⣿⠋⠒⠶⠱⡌⠀⠀⠀⠀⠀⠈⠀⢰⣿⠛⣿⣇⣿⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⡏⣿⣿⡯⡿⣿⣿⢿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠛⠁⣀⠀⠀⠀⠀⣴⢷⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⣉⣉⣉⣀⣉⣁⣈⡠⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠿⠛⠃⠚⣿⣿⣴⣿⢿⣷⡿⡏
⠀⠀⠀⠪⡀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠐⢶⡤⠉⠁⠤⠤⠤⠄⠀⠀⠀⠀⠒⠲⠟⠁⠀⠀⠀⠀⠀⢐⠀⠉⣀⣠⣤⣄⣙⡿⠛⠛⠓⠚⠗⠺
⠀⠀⠀⠤⠇⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣴⣋⣽⠽⠷⠾⣿⣿⡧⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠠⡏⠬⠉⡁⠀⠀⠀⠀⠀⡼⢀⡏⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠉⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⢂⠠⠀⡐⠑⢦⣤⠄⣆⢟⠀⠐⠌⢀⡦⠝⣢⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⢠⡤⠋⠀⠀⠌⠠⠤⠤⠠⠬⡄⠃⠋⢀⠈⢁⡄⣁⠀⠀⠀⠀⠀⢠⡴⣿⠯⠽⠿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠀⣀
⠀⠀⠐⣯⠴⠲⠜⠀⣇⠀⠀⠀⠀⠀⡄⢂⠠⠀⣠⣾⣿⣧⠀⠀⠀⢰⣿⣿⣷⣿⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⢉⠥⢘⡶⠉⣿⣿⣿⠀⠀⠀⠀⠙⠻⢿⣿⡶⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⣠⠀⢆⠀⠀⠀⡀⠀⠠⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠄⠀⣑⠀⠀⠠⠤⠤⣮⣦⣼⣶⣶⣾⣿⣿⣤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⢹⡋⠿⠈⣶⠈⠳⣴⣆⠀⣴⠞⠁
⠀⠀⠀⠀⠀⠘⠉⠂⠀⠀⢀⠠⣤⣤⣦⡄⠦⠠⢿⣶⣴⣧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣀⣴⠋⣁⠀⠆⢀⡀⠃⠘⣿⠛⣧⢠⢤⡀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠄⠀⢀⡀⣠⠤⢤⡀⣁⠀⠒⠩⠀⣰⢬⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠟⠉⠉⡄⠉⢀⣀⠈⡅⠀⠁⣼⡟⡇⠀⠀⠁⠄⠤⠁
⠀⠀⠀⠀⠀⢸⣄⠁⢀⣢⣄⡓⠙⣘⣀⠀⠄⠀⠀⣘⠬⠻⣈⡁⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⢰⠄⠀⢤⠀⠂⠈⠉⣀⠀⠀⠓⠈⣿⡇⠀⠀⡀⠀⡀⠴
⠠⠀⡶⠆⠠⠦⠾⠀⠠⠥⣼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶⠀⠀⣀⠀⠄⠰⣷⠀⢀⡀⠃⢘⠁⢻⡇⠀⠀⠈⠀⠁
⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⢀⣤⣤⣍⣀⠴⠀⢀⣄⠠⠁⠄⠉⠀⠈⠁
⠀⠀⠀⠑⠀⠐⠀⠀⠀⠐⠆⠀⠀⠀⠶⠀⠀⠀⠦⠀⠠⡆⠀⠐⣤⠀⠀⠀⠀⠀⠀⠈⣷⠟⠻⣿⣿⣿⣿⣦⢀⡄⠴⠛⢆⠀⠄⢰⠴⠆
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⡀⢀⡁⠀⠀⠀⠶⠂⠀⠀⠀⢀⣘⢁⡀⠀⠀⠙⢿⣿⣿⠈⢀⡂⠀⢈⡀⠀⢀⠀⢴⠀⠀⠀⠀⢠
⠤⠄⠂⠈⠁⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⣤⡔⠀⠀⢬⡁⠀⠀⠀⠈⠁⢸⢿⠏⣾⣿⣿⣿⣿⣿⣿⣿⡀⡼⠀⠀⠀⢀⣾⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠃⠀⠀⠐⣿⡟⠁⠀⠀⠘⡧⢀⡀⠀⠀⢀⣤⠄⠀⣻⡟⠛⠉⠉⠛⠀⣿⠀⠠⠤⠤⠤⢤⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⡄⢰⣿⣧⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀⣿⣇⠀⠀⠀⠀⠀⠀⢠⠤⠞⠛⢢⣤⡾⣧⣿⠁⠀⠀⠀⠹⣿⣿⣀⠀⠀⠀⡜⣸⣿⣇
⡄⠀⠀⢀⣴⣶⣶⠀⠀⠀⣴⣶⣶⣦⠀⠀⠀⣴⣶⣶⡂⠀⠉⠀⢸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢻⣿⣿⣀⠀⣀⣀⣴⣿⠋⠛⠳⢄⠠⣷⣿⣿⣿
⠉⣀⡀⢈⣉⣉⠉⣁⣀⣀⣉⣉⡉⠉⡁⢀⡘⡛⢉⡉⠃⢀⣾⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠻⣿⣏⠉⠉⠉⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠘⠅⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⠀⠀⠀⢨⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⡄⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣧⠀⢠⠦⠀⣴⢖⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠆⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠋⠿⠷⣨⢻⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡟⡸⡘⡼⡏⣯⣦⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢲⣾⣯⣶⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠓⠛⠛⠁⠉⠀⠀⠀⠀⢠⠱⢠⢣⢡⢳⢰⣿⠸⠸⣷⠢⠄⠀⠀⠀⠀⠀⠀⢰⣶⠚⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣀⡀⣀⣀⣀⣒⡀⠒⠂⠂⠘⢘⠀⡀⠂⠐⠰⠀⠀⠐⠒⢀⣀⣨⡇⡆⡜⡄⡈⡆⣼⢸⡇⠟⠘⡀⠀⠀⠀⠀⠀⠀⠀⢫⣥⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠃⠀⠸⣿⣿⣿⠋⠈⢿⢡⡇⣿⣿⠋⣿⢸⣇⢿⣿⡿⠃⠙⣿⣿⢰⢀⢣⠀⢷⢀⡇⣇⢷⡸⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠹⡿⠁⠀⠀⠀⣼⢀⡿⠃⠀⠘⡼⣿⡸⡿⠁⠀⠀⠸⣿⢘⢸⡜⢸⡈⢸⢰⣹⣸⣇⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣅⠛⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠉⠄⠉⠑⢄
⠀⠀⠀⠀⢠⠖⠉⠀⣀⠠⠂⠁⠀⠄⠐⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂
⠀⠀⠀⠀⠇⠀⠈⠀⠈⠀⠀⠂⠀⡀⠀⠰⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣬⡀
⠀⠀⠀⠀⢇⠈⢁⠀⠉⠀⠋⡀⠀⠀⠀⠀⠀⡠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡎⠐⠾⠓⢹
⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⠀⠁⢠⠋⠉⢆⠀⠰⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠠⠀⠀⡧⣀⣀⣀⠜⢦⡀
⠀⠀⠀⠀⠀⠸⡌⠀⣾⠉⠙⠒⠋⠀⠀⠈⠂⠐⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⠀⡁⠐⠀⠬⠀⠈⠀⠠⠙⡄
⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡠⡶⡻⠃⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⣩⠊⡴⠁⠀⣇⠑⠤⣆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠁⢰⠁⠀⠀⠈⢦⡀⠀⠉⠳⣽⢦⡔⢢⢀⣤⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣀⣠⡎⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⣦⣹⠃⢸⠤⠻⢖⠚⠉⡦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠔⠒⡏⠀⢸⣇⠀⠀⡇⠀⠈⢆⠀⠀⠀⠀⠀⡇⠀⠀⣠⠣⣈⠒⢛⣦⣀⠠⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣱⢄⣾⠀⠑⠒⠚⠢⠄⣈⠷⠤⠠⡤⠚⠉⠉⠉⠀⠏⠉⠣⡀⠀⠑⢦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠂⠉⢸⡁⠒⠪⠕⣤⠊⠉⠢⠔⠉⠑⠤⠊⠉⠲⠊⠙⢦⠖⠻⢶⣀⣉⠯⠽⠥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠏⠀⡏⠒⢶⠐⠸⡆⠀⠐⡄⠀⠀⡆⠀⠀⡛⠀⠀⣇⡤⠚⡉⠀⠠⠀⠀⠂⠀⠠⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠰⠀⠘⡄⠀⢣⠀⠀⠇⠀⠀⡇⠀⠀⡇⠀⡼⠃⠀⠂⠀⠀⣠⢄⠴⠛⢤⣀⠀⢁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⢱⣀⡈⡆⠀⢸⠀⠀⠁⠀⢰⠁⡜⠀⠈⡁⠉⢔⡲⠗⠂⠉⠉⠐⢦⠇⠄⠐⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⣧⠄⠠⢔⠂⠘⠿⠁⠀⠈⢷⡀⠀⡆⠀⠀⠀⡸⢠⠅⠈⠀⠀⠂⡈⠣⢄⣀⣀⡀⠤⠊⡀⠀⢠⠲⢄⡨
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠘⡂⠀⠀⠀⠀⠀⢸⠥⠄⠁⠀⠀⠤⠅⠤⠵⡞⠦⠚⢢⠀⠀⠀⠁⠄⠐⢀⠁⢀⠤⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡸⣁⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠓⠒⠉⠑⠤⠔⠉⠁⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠖⠊⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣀⣀⣀⣀⡀⠤⠔⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠈⠉⠉⠉⠛⠲⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣽⣷⣤⣤⠤⠶⠞⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠶⠶⣶⡚⢋⣉⠛⠖⠶⠶⣤⡴⠶⠟⢉⣭⠿⣏⠉⠈⠉⠁⠀⠀⠀⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣤⣛⣳⡋⡋⢝⠛⢩⣷⣀⠀⠀⠀⠀⣀⣠⣿⡇⠰⠠⢷⣄⣀⣀⣀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣶⣴⡤⠑⠈⠿⠷⠶⡏⢹⠹⢀⢀⠃⠄⠌⣠⣭⠟⠋⢳⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣗⢄⠀⠀⠂⠁⠈⠀⠘⠈⠀⠰⢰⡟⠁⢀⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡤⡠⡀⠀⡀⠀⠀⠀⡄⢀⡿⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡅⠡⢠⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠙⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠛⣿⡀
⠀⠀⠠⠤⠀⠀⠀⠀⠀⠈⠁⠈⢻⣧⡀⠀⠀⠀⠀⠀⠠⠺⢤⣐⡀⠀⠀⠀⠀⠀⠀⠀⡯⠂⠀⠀⠠⢤⣠⡶
⠀⠀⣤⣤⣀⣀⡀⠀⠈⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⠀⠀⠈⠽⠆⠀⠀⠀⠀⢀⠠⠄⠀⠈⠛⠉⠉⠀⠙⠁
⠀⠀⠐⠀⠀⠉⠉⢻⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⠄⢀⡀⠈⠁⠀⠀⠀⠐⠁⢀⠴⠋⠀⠈⣑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣭⠶⢢⣄⠀⠀⠀⠀⣢⠄⠀⢀⣤⣀⠘⣁
⠀⠀⠀⠀⠀⠠⠀⣻⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠛⠋⢸⡏
⠀⠀⠶⣤⣀⡀⠀⠸⡇⠀⢀⠠⣄⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⢤⣤⡼⠉
⠀⠀⠲⠯⠬⠉⠓⢾⡇⠐⠛⠉⢉⣻⣠
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⠤⡤⣶⣿⡧
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⡴⢲⠿⣶⣷⡎⠀⠀⠀⠀⣀⠤⣠⡾⠿⠄
⠀⠀⢀⣀⡲⣏⣧⣳⣻⡄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⡅⢄
⠀⠀⡚⢻⡧⣹⡘⣎⡳⡜⣯⡇⠆⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣾⡿⠤⠄⠃
⠀⠀⠠⠑⠠⢨⠇⣌⢡⠚⠁⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣋⠛⠀⠈
⠀⠀⠀⠀⠀⠬⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⠌⠢⠀⠀⠀⠀⠠⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣮⣦⠁⠀⠀⠀⠀⠀⠑⠘⢷⣩⣽⣿⣷⡄
⠀⠀⠀⠀⠘⢄⠐⠋⠉⠀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⢓⣻⠿⠃
⠀⠀⠀⠀⠀⠀⢳⣾⣿⡿⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤
⠀⠀⠀⠀⠀⠀⠈⠉

⢠⡄⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠄⠠⠄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⡀⠔⡆⠀⢀⣤⡀⠀⠀⠀⢀⡤⠒⠒⢦⠀⠈⠁⠀⠀⠀⠀⠀⡼⢤
⠀⠀⠀⠀⠶⠀⠀⢀⡠⠔⡶⠔⠉⡆⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣗⢚⣄⣠⡾⠉⠁⠀⠀⠀⠰⢁⡀⢀⠀⠇⠀⠀⠀⠀⠀⢀⠔⠋⠉⠁⠒⡔⠃
⠀⠀⠀⢀⣤⣐⣈⣁⣀⣀⡁⠈⠭⠤⠄⠐⠒⢈⡆⢀⠀⠀⠀⠀⠀⢀⠤⠾⢽⣯⠗⢳⢄⠀⠀⢀⠀⠀⡶⠆⡆⡏⠦⣼⠀⠀⠀⠀⠀⠾⠠⠤⣠⠦⠜⠣
⠀⠀⠀⢸⣄⡂⠄⠀⠒⣒⣒⣖⣒⠂⠤⠘⠛⢀⠀⠀⢱⠀⠀⠀⢀⡟⣶⠚⠓⣿⣙⡏⠭⡄⠀⠈⠁⠀⡇⠀⣇⠃⠀⣿⠀⠀⠀⠃⠀⢂⠱⠀⠇⠀⡇
⠀⠀⡤⠚⠦⠄⣀⣀⡁⠀⠠⠤⠤⠤⠀⠒⠒⠉⢀⡠⠊⠀⠀⠀⠈⡄⡓⠄⢰⠢⠀⠔⢳⠃⠀⠀⠀⠀⢇⠀⠙⠀⠀⣹⠀⠀⠀⠀⢀⠔⢆⠁⠐⠀⢀⠁⢈
⠀⢨⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠒⠉⠀⠀⠀⠛⠀⠀⠘⠪⠬⢈⠀⠈⠔⠋⠀⠀⠀⠀⠀⠈⠈⠩⠈⠃⠀⠀⠀⠀⠀⠘⠤⢀⣁⡐⠠⠥⠒⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⢠⡀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠐⠒⠀⣐⠢⡀⠀⠀⠀⠀⢛⡢⠌⢰⡈⠁⠀⠀⠈⠉⠀⠀⢨⠑⠞⠄⢀⠌⠀⠀⠀⠀⠀⠈⠃
⠀⠰⠆⠀⢀⠎⢰⠊⠓⡊⢯⠱⣈⢢⠀⠀⠀⠀⡤⠊⣁⡠⠊⠒⠈⢀⠂⢰⠀⠀⠛⢄⡁⠈⠀⠀⠥⡀⠀⠀⠀⠀⢠⠒⣏⠢⢄⠀⠓⢤⡀⠀⠀⠈⠃
⠀⠀⠀⠀⡆⡾⠅⠠⠋⠉⠘⣷⠌⡙⠇⠀⢠⠊⢀⡌⠀⠀⠀⠀⠀⠸⡀⡞⠀⠀⠀⠀⠈⠁⡏⠀⠀⠈⡆⠀⠀⠀⠈⠢⢌⠙⠲⠏⠉⢈⣀⠀⠀⣀
⠀⠀⠀⠀⡗⠢⡸⠣⡀⠀⣀⠟⣗⡇⡄⠀⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⠀⠀⣀⠀⠀⠱⠄⢠⠖⠀⡉⢑⣲⠄⠀⠀⠉⠊⢠⠆⡵⠲⡤⠒⠒⡍⠢⡀
⠀⠀⠀⠀⠘⢄⠣⠥⢎⡉⡉⡮⢚⠜⠀⠀⠘⣦⣈⠑⢄⠀⠀⠀⠀⠀⠀⠘⡌⠢⡀⠉⠀⠀⠀⠀⠈⢦⢠⠌⣉⣢⠀⠀⠀⠀⢀⢡⠒⠁⡠⠄⠦⣄⠉⠒⣷
⠀⠀⣀⠀⠀⠀⠑⠢⠤⠥⠯⠒⠩⠄⠀⠀⠀⠀⠉⠳⣄⢣⠀⠀⠀⠀⡠⠔⢁⡴⠃⠀⠀⠀⠀⠀⠀⠨⡼⠀⠉⠁⠀⠀⠀⠀⢸⢎⡀⠰⡀⠀⠀⢨⠀⢰⢸
⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡉⠐⠒⢉⣠⡾⠋⠀⠀⣀⠤⡐⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢸⠀⠈⠒⠂⠁⣀⢜⠌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⠒⢲⡆⠒⢄⠀⠀⠓⠀⠀⠈⠻⣶⡢⠛⠉⠀⣤⢀⡊⠀⠑⡻⠖⠚⠀⠀⠀⢀⡀⠀⠀⠀⠋⠀⠀⠀⠑⠭⣈⡓⣤⡊⠔⠋
⠀⠀⠀⠀⠀⡠⢖⡛⠄⠈⢲⠀⠀⠃⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠘⠋⠑⣶⠠⣤⠀⢤⣬⠥⠤⣤⡤⣤⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⢎⣣⠀⠐⡆⠀⠈⠀⢀⠄⡠⠂⠀⠀⠀⠀⠀⡠⣔⠚⠑⠒⠦⡀⠀⠀⠈⠑⠠⠄⠿⠤⠽⠤⠤⠽⠤⠀⠤⠥⠨⠧⠤⠃⠀⠀⠀⠀⣠⠄⠀⠀⠉
⠀⠀⠐⠀⠀⠘⠇⠀⠀⢀⡠⢔⡡⠚⠁⢀⡀⠀⠀⠀⢌⡴⠋⢠⡇⠘⢷⡌⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⢀⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠻⡀
⠀⠀⠠⣀⠀⠀⢀⢒⡪⠅⠊⠁⠀⠀⠀⠈⠁⠀⡠⢄⠀⠀⠀⠀⠁⠀⠀⠀⡈⢢⠶⠀⠀⠀⢀⠔⠁⢀⣀⡀⣀⡈⠳⡄⠀⠀⠀⠁⠀⠀⡔⠁⠀⠘⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠑⡆⠉⠓⠒⠊⠒⠒⠉⡹⠁⠀⠀⠀⠀⡎⠀⡔⠁⢀⠤⢄⡀⠀⠘⡄⠰⠆⠀⠀⡰⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠠⡊⠁⠀⠑⢎⠁⠈⡆⠀⠀⠈⠉⠀⠉⠁⠘⠈⠉⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⢇⠄⠀⡱⠀⠀⡇⠀⠀⠀⠸⡀⠀⠉⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⡀⠀⠀⠘⡤⠚⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠱⡀⠑⢄⣀⣀⠴⠁⠀⡜⠀⣀⠀⠀⠀⢩⠶⣔⢶⣲⠞⠁
⠀⠶⠀⠀⢈⠦⢀⠀⠀⠀⠀⠰⣀⡠⠞⠀⠀⢀⠔⡩⠰⠒⡐⡶⣕⢆⠀⠀⠀⠀⢀⡀⠀⠀⢠⢞⠦⠤⠀⠀⠤⠔⠊⠀⠀⠉⠀⠀⠀⢸⢲⡋⣹⢥⠆
⠀⠀⠀⠀⡆⠀⠀⠑⠢⢄⣀⠔⠁⠀⠀⠀⡐⡅⢺⠣⡀⢣⠱⡐⡘⡆⠆⠀⡤⠀⠈⠀⢀⠔⡱⠃⠀⢀⡀⠀⠀⢠⠔⣠⡄⠀⠀⠀⠀⠈⠉⣯⡝⢰⠀⢀⡀
⠀⠀⠀⠀⡱⢄⠀⠀⡠⠂⠀⠠⡄⠀⠀⢀⠀⠦⡀⢣⠱⡀⢣⠱⣁⡀⠐⠀⠀⠀⢀⡔⡡⠊⠀⠀⠰⡁⠈⡃⠀⢏⣩⠕⢂⠀⠀⠀⠀⠀⡧⢦⢼⡎⠀⠈⠁
⠀⠀⠀⠀⢇⣀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢆⡘⢄⢣⠑⡌⠆⣸⢃⠃⠀⠀⠠⢊⠜⠀⠀⠀⠀⠀⠉⠉⠉⠒⢄⡀⠀⠉⠀⠀⠀⠀⠀⢰⡵⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠢⢌⠒⠷⠧⠜⢋⡥⠋⠀⠀⠀⠓⠁⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠁⠰⠆
⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣂⠐⢁⣰⠐⡐⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡮⠿⣧⠟⡈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠲⠊⠁⠘⠠⠢⠄⡖⠒⠆⢹⠊⠁⠈
⠀⠀⠰⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣄⠘⠓⢂⠀⠀⢀⡀⣾⣗⠦⢀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡴⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠀⠀⠀⠀⠘⣏⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣰⡦⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠉
⠀⠀⠀⠀⠀⣀⠩⠉⠀⣀⡿⡄⠀⠠⣤⡄⠀⠀⢀⣄⠀⠀⣀⠀⠀⠀⠀⢠⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠤⠛⣿⡆⠀⣤⣾⣷⠟⡴⢾⣿⢀⣠⠾⠋⣠⠁⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢈⣀⠀⠀⢀⡀⠂⠺⢃⡈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠊⠁⠀⠘⡂⠈⠡⠄

⠀⠀⠀⠀⠀⠠⣾⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠰⣶⣶⡄⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠄⣀
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠈⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠀⠆⠀⠀⠀⠀⠀⠀⠻⠀⢸⡀⠀⢰⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⢴⣿⣿⣿⣿⡗⠀⢀⣠⣶⣀⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠖⠘⠓⣂⡀⠰⠐⡶⢴⠶⠀⣿⣿⣹⣶⣦⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠁⣾⣿⢿⣿⣇⢻⡿⠀⠙⠁⠀⠀⢀⣤⣦⣄⠀⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⡀⠀⠀⢻⣿⣿⡿⠛⣿⠿⣧⣶⣦⡄⠀⠛
⢀⣾⣿⡿⣿⠃⢀⣬⣵⣿⣿⣿⠈⠀⡀⠀⠀⠀⠀⡾⣿⣿⡏⣴⣴⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⣈⡇⠾⠇⢰⡆⠀⠙⠛⠁⠀⣷⣦⠛⣿⣿⣧⡀⢀⡀
⢾⣧⣾⣷⣿⣦⣾⣿⣿⣼⣿⠋⠀⣠⣿⠄⠀⠀⠀⠀⠿⠏⠁⣿⣿⡁⠀⢻⣷⠄⢼⠛⠟⠻⣿⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠃⠈⢿⣿⣿⠈⣿⢿⣷⣦
⠀⣿⠉⠈⢿⣿⣿⣿⠏⠁⠀⠀⠀⠙⠋⠀⣀⡾⠁⠐⠛⠀⠀⠈⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⡟⠁⢸⡧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⣠⠀⠀⠸⡇
⢸⣿⠀⣿⡄⠛⠉⠉⠀⠀⠀⠗⠀⠀⠀⠀⡿⢰⣶⠀⢀⡀⠀⢶⣿⣷⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣶⡀⠹⣯⠀⠀⣀⣀⣤
⠘⠁⠰⠿⠇⠀⠀⠀⣤⣶⣿⣾⡂⣠⡆⣸⠃⠨⣧⠀⢘⣷⡆⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠉⣀⣴⣿⣿⣿
⠀⠀⠀⠀⣰⣦⣄⣸⣟⣿⣿⡟⠉⡋⠠⡿⠀⠀⠀⢠⣾⡟⠀⣿⡇⣀⠃⠀⠀⢀⠦⠄⠀⠖⠀⠀⣹⢿⠿⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠙⠁⢴⡅⠀⠉⢸⣿⣿⣿
⢰⣶⣤⡄⠲⣿⣿⣿⠏⠁⠀⠀⠈⠁⠒⠁⠛⠀⠀⢠⣉⡀⠀⠉⣠⣵⣄⡀⠀⣠⣶⣶⣤⡀⠄⠀⠹⠴⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⢀⣀⣈⢃⢀⣀⣾⠛⢿⣿
⠈⠙⠛⠀⠈⠿⢿⣿⠀⠀⠀⠀⣀⣠⣄⣠⡄⠀⠀⠈⢿⣧⠀⣾⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡄⠀⠀⠀⠀⠙⠿⣯⣙⢿⣿⡄⠈⠻
⢰⣾⠟⠃⠀⠐⣃⣀⣀⡀⠀⠈⠙⠿⣿⣿⠓⠀⠀⠀⠘⣷⣄⠛⢻⣿⣿⣿⣼⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠈⢿⣆⠀⣾⣿⠒⠀⠀⠉⠹⢿⣿
⠈⠁⠀⠀⠀⠐⣿⣿⣿⣇⠀⠀⠀⠀⠈⠛⠃⠰⠦⣠⣾⠻⢿⣦⠀⣿⠿⠿⠿⣿⣯⡤⠀⣴⣷⣧⠀⢴⠟⢻⠿⣃⣀⣀⣀⣼⡿⠷⠍⢀⠀⣀⣀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⢸⣿⢺⡿⠀⠀⠀⠙⠋⢁⡄⣩⠕⠤⠀⢠⣶⡟⠠⣿⣿⠟⠈⣁⠀⠀⠀⠀⠙⠿⠛⢀⡀⠀⠀⠀⢰
⠀⣀⣤⣴⠀⠀⠀⢀⣻⡷⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣤⣀⠹⠃⣀⡀⠀⠀⠀⠀⠀⠀⠸⢏⡀⠀⠀⢸⣟⡁⠀⠻⠃⠀⠀⠛⠀⢠⣄⡀⠀⠀⢀⣾⣿⢐⡖⢸⡤
⢀⣨⣿⣿⣄⠀⠀⣘⠁⠀⠷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⡀⠛⣁⠀⠀⠀⠀⣿⣾⠀⠈⠁⠤⠐⣀⠀⠀⠸⠧⠀⢠⣾⠃⠀⠀⠹⡏⠀⠀⠿⠉⠁⠈⠁⠬⣿
⠸⠿⠟⠛⠉⠀⠀⢻⢀⣠⣴⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠠⡟⠘⣿⠁⠀⠀⠀⢛⡁⠄⠀⠀⠀⡐⠁⠀⠀⣴⣶⢦⡘⣿⣷⡄⣀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⣠⡆⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠘⠻⣷⣦⣀⣀⣤⠙⠂⠿⠀⠀⠀⠀⠻⣿⠇⠀⢀⢸⡇⢀⣤⣸⣿⠿⠇⠈⠀⠉⠁⠛⠃⠀⠀⠀⠀⢹⣷⠂⠀⠀⠚
⠀⠀⠀⠀⡿⠤⠀⣸⡟⠁⢀⡄⠀⡀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⢈⠃⠤⠒⠀⣿⠀⠀⠀⠉⠀⠈⠙⠿⢻⣤⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀
⠀⢀⣠⣶⣶⣤⡘⠋⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⢁⡟⠀⠀⠀⠀⠀⠀⠀⢠⣬⠈⣹⡀⠃⠀⠀⠀⠀⠀⢀⡽⢀⣸⣄⠀⠸⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⢻⣿⣿⣿⣿⡏⠀⢠⣶⣿⣤⣶⣿⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠘⢿⣥⣈⣉⣩⣤⣄⠀⣿⠟⠓⠀⠀⢾⡟⣹⣿⣷⣿⣿⡿
⠀⢸⣿⣿⣿⡟⠀⠀⠘⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣷⣦⡀⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣉⣀⣠⣤⠀⠈⠛⠛⠋⠉⠉⠙⣷⣿⣷⡆
⢠⣿⣿⣿⡿⠁⠀⠀⠀⣽⣿⣿⣿⠟⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣾⡇⠀⢸⣆⢸⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⢰⣶⠀⠀⠀⠀⠀⠉⢻⣿⣷⣴⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣐⡠⠔⠒⠉⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠘⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⢀⣴⠎⠁⠀⠲⣋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢆⠡⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⡀⠋⠀⠀⠀⢀⠀⠀⠀⠀⠈⢇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⡄⠣⠄⠐⠄⠒⣀⠀⢀⠀⠀⠀⠀⠈⢧⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣤⠀⠀⢼⡏⢸⢳⡀⠘⢧⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠤⠟⠀⢻⠁⠈⡎⠑⡄⢘⣆⠥⠈⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠃⢀⡀⠦⠄⠠⠈⠁⠀⡡⠚⡋⠀⠐⡤⡗⡀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⠀⠴⠞⠁⡠⠏⠀⠀⠀⡤⢎⠀⠀⠂⠀⠐⠀⡴⠳⠀⡀⡀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠉⠀⠀⠀⠀⣐⠀⠀⠉⠇⠀⠀⠀⠛⠣⢄⢇⠠⠈⠢⣄⠀⠀⠀⠀⠀⠈⠾⠤⠂
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠁⢠⠄⠀⠀⢣⠀⢀⡄⠀⠀⠀⠀⠀⠼⡃⠀⠙⡄⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡃⠀⠠⠀⠁⠀⠀⠀⠀⠀⢸⠄⢸⠀⠀⠠⢀⡀⢔⠀⠀⠀⠀⠡⠀⠈⢠⡑⠄⡀
⠀⠀⠀⠈⢩⡀⠈⠁⡅⠇⠀⠀⡄⠀⠀⢃⠀⠀⠀⠀⡧⢠⠀⠀⠀⠀⢹⠈⡆⠀⠘⠀⠘⣮⡊⡀⠀⠀⠈⢂⡀
⠀⠀⠀⠀⠀⠈⠒⢤⣡⡀⠀⠄⠀⠀⠀⠀⢀⡀⠄⠂⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠳⡈⢦⡀⠀⠀⡀⠇⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠄⡀⠀⠀⠀⠨⡄⣀⣀⠀⡠⠤⣅⣀⣀⣘⣤⣥⣤⣇⠤⠤⠤⠿⢛⡈⠉⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠒⠊⠁⠀⠀⠀⠨⠴⠀⡀⠀⠱⠄⠘⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⣀⣹⣿⣿⣿⣦⣀⣤⣶⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣭⣷⣾⣿⣷⣶⣾⠃⣀⣠⣾⣿⣿⣿⣿⣿⡏⠉⢉⣡⣶⣞⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡀⠤⡉⠫⡩⣿⣿⣿⣿⡻⢹⣿⣿⣿⣟⣼⣵⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⡟⠉⠑⠿⣿⣿⣿⣿⡇⠀⢩⣿⢿⢯⣿⡿⠿⣿⣉⣭⠛⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡀⠀⠘⠿⣿⣿⣷⡐⡎⠁⠨⣿⠟⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⣤⣤⣴⣤⣶⣿⣷⣤⡀⠀⠈⢿⣿⣿⠿⠤⣾⠃
⠀⠀⠀⠀⠀⢠⠈⠁⠀⢸⣿⣿⣿⣿⡿⠻⠁⠂⢉⣿⣶⠒⠀⠋⠀⠀⢻
⠀⠀⠀⠀⠀⠙⢧⣸⣦⣽⣿⣿⠟⡅⢠⠀⢱⠃⠉⢹⠿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢉⢠⠿⠟⠿⠀⠀⠄⠳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⢄⠀⠀⠀⡀⠀⠀⠀⣰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡀⠀⠀⠁⠀⠀⠀⠋⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣠⣴⢁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⡿⠗⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡗

⠀⠀⠀⣶⠀⢠⡀⠀⢲⡄⠀⣀⡠⠶⠀⠀⠀⠀⢀⡀⠐⣴⢶⠶⢤⣤⣴⣂⣀⡆⣤⡀⠀⣀⠀⠀⠀⠠⣤⡀⠠⠖⠀⠀⠀⠂⡄
⠀⠐⠒⠀⠀⠈⠁⢠⣤⠀⠀⠉⠁⠀⠐⠓⢀⠀⣘⠓⢉⣻⣹⢻⡙⢃⣻⡏⡛⣔⣢⣕⣶⡯⢀⡌⢂⠀⠈⣀⡄⢀⣄⠡⠠⣾⡇⡠⠂
⠀⡀⠀⠀⠀⣀⢸⣿⡏⠀⠁⣈⣷⡀⣤⠴⣿⡄⠙⠀⠁⣀⠐⠿⣿⡟⢾⡎⣡⣩⡇⠀⠈⠁⠴⢠⡄⠵⡤⠌⠁⠃⠒⠰⣶⣿⠃
⠀⡇⠀⠀⠀⠙⢺⣿⡇⠀⡀⢻⣿⣷⣿⣿⣿⡇⢤⠀⠀⠙⠋⠹⠎⢥⡀⠈⣙⠃⣀⡚⠃⢸⢸⣦⡇⠀⣴⡍⠐⢷⡦⢹⣿⣿⠓⢁⡀
⠀⠃⡄⠀⠐⢫⣶⣽⠀⢈⠁⣿⣿⣿⣿⣿⣿⡇⠀⢰⡦⠀⠘⠃⣤⠈⣀⠀⠛⠀⠈⠁⠀⢈⣿⣿⣿⣶⣄⡀⠀⣀⢀⣀⣿⡟⡀⠀⠇
⠀⡅⠀⠀⠀⠀⣿⣿⠇⣈⣀⠻⣿⣿⣿⣿⣿⡇⣀⠀⠀⢰⡄⠀⠉⠀⠉⠀⠀⣞⠀⡄⢺⡿⣯⣹⣿⡿⢿⣷⡔⣃⡤⣼⣿⡇⠀⠀⣀
⠀⡗⠀⠀⠘⠁⣸⣿⢐⠀⢹⣾⣿⣿⣿⣿⣿⡇⠈⠛⠖⠠⣤⠀⠙⠁⠀⣰⣿⣿⣧⣰⣿⣡⠀⣼⣿⢧⡿⠟⠀⠙⡹⢴⣽⡇⠄⢈⠷
⠀⠈⠀⠐⠚⠛⡻⣿⣿⣴⠍⢡⣿⣿⣿⣿⣿⡇⠂⠔⠀⣴⠀⠀⠀⠙⠀⢿⣿⣿⣿⣿⠆⡀⠀⠉⣿⠰⣆⠀⢰⣿⣲⡶⣟⣶⠶⠄
⠀⡄⠛⠂⣄⠀⢋⣼⣿⠸⠠⢀⣾⣿⣿⣿⢿⣇⡄⠘⠂⠀⠀⠀⠶⠄⠀⠈⠛⠺⠛⠋⠀⠀⠀⢠⠈⡲⣝⣦⠨⡧⢿⡑⣿⣹⣇⠀⡿
⠀⣧⠀⠀⠉⢰⣾⣿⣗⠘⠃⢼⣿⣿⣿⣿⣿⡇⢁⡀⣀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠀⠻⡿⢻⡈⠁⠀⠀⠑⣷⣿⣿⣸⡇
⠀⡿⠀⠹⠦⠈⠿⣿⣗⢰⣠⡸⣿⣿⠿⡿⣻⡧⣿⡷⠟⠰⣤⡄⠐⠊⠉⠋⠉⠛⠒⠢⢤⠀⣼⡂⠠⢅⠀⣷⠀⠀⠠⠈⣈⣿⣿⣿⠇
⠀⠃⠀⠀⠀⠀⠀⣿⣿⡖⣞⣻⣿⡟⢸⢋⠽⡟⠫⠅⢀⢉⢻⡋⠀⠀⠄⢀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠈⠃⣿⢠⣆⣎⣰⡡⣤⡁⠋⣦
⢸⠈⠃⢀⡄⠀⠀⠩⢏⠀⡟⠉⡩⠡⡀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠠⠀⠘⣿⣄⣈⠇⠫⠤⠊⠘⠂⣹
⣼⣦⣤⣴⣶⣦⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣬⣭
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢠⡆⣤⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢀⠔⠀⠀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢄⠀⠐⠁⠀⠀⠘⠁⡀⠀⠒⠲⢄⢀⠠⠀⠀⡥
⠀⠀⠀⠀⠀⠀⠣⣀⠂⠠⠆⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠡⠈⣿⣵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠳⢄⠤⣤⣀⣁⠐⠤⠀⠐⢞⣻⠎⡁⠀⢭⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠠⠅⠀⣼⣿⡿⠀⠒⢄⠀⠀⠉⡆⠈⠀⠀⠉⠸
⠀⠀⠀⠀⢀⠄⠀⠀⠼⠀⠀⠀⠐⢇⠀⠋⠹⠁⠀⠀⠀⠑⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⡜⠒⢀⣤⡂⠀⠀⠀⢰⡀
⠀⠀⠀⠁⠀⠀⠁⢁⣤⣴⠊⢨⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠁⡾⠆⢀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠲⠀⠈⠀⠰⠀⠀⠀⠀⢰⠄⡃⠰
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠈⠂⠤⣤⣷⣠⠀⠀⠀⢐⠂⡀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⢠⠰⠆⠀⠀⢀⢿⣾⡯⠓⠒⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡐⠀⡀⢈⠠⠁⠀⡏⡐⠒⠠⢀⣰⠿⡐⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠈⠆⢘⠀⠳⡀⠀⣰⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢢⠀⠀⠀⠴⠐⠀⢠⠃⠀⡬⣿⣹⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠁⠢⣤⡷⠛⣮⣀⠤⠚⡁⢀⡄⠀⠀⠻⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⠄⠱⠤⢴⣀⡀⠠⢽⣷⣤⡿⣇⠀⠀⠀⢨⠃⢶⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡄⠒⠁⣘⢹⡀⢀⠀⠀⡘⠙⠀⠘⢆⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⣀⠠⢶⠻⠠⠂⣡⠀⡛⠰⠀⠈⢃⡀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⣧⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠄⢳⣴⣎⠑⢀⠉⢠⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣈⡛⣺⣿⣄⠷⡀⠈⠄⠀⠆⢩⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣉⣽⣿⣿⡥⣶⣷⣦⡴⢾⣿⣥⢍⢼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣹⡇⣊⣶⣦⢿⡿⢻⣟⣉⢁⣄⠀⠛⣿⡎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣻⡟⢸⣿⣍⡰⣶⠾⡻⠇⣻⣅⢿⡷⣠⠄⢻⣧⠈⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣾⣿⠻⣽⣇⠈⢯⣿⣿⣶⣾⣿⣋⡈⣧⡼⢾⠃⣀⣻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢠⠛⡟⣿⣆⠈⠛⠳⠄⣉⠛⣿⡿⢿⠿⠿⣻⣿⣿⠟⠛⠉⣁⡴⠀⠀⠀⠀⢘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠁⠶⣿⣿⣄⠀⠀⠀⠀⠉⠛⠲⢤⡄⠒⠛⢋⡥⠖⠛⠉⠉⠀⢀⠜⠁⠀⣼⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡄⠀⠈⠻⣿⠳⠦⠤⣤⣀⣀⣀⣽⡷⠒⠉⠀⠀⠀⢀⣤⣎⣠⣤⣤⢾⠁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠠⣿⣿⣿⠀⡀⢰⣼⣦⣤⣤⣤⡾⠟⢋⡁⣀⣀⣀⣤⣤⠖⣻⠛⠉⡼⠋⠀⠸⠀⠈⣧⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⠁⠙⢿⢷⣦⣤⣮⣿⣿⠿⠶⠛⠉⠉⠉⠉⠉⠁⠀⢀⡼⠁⠀⠠⠀⠀⠀⡆⠀⠀⡼⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⠀⠤⣚⣿⣫⣽⡎⠁⠀⠀⠀⠀⠀⢀⣀⡠⠾⠋⠀⢀⡴⠀⠀⠀⠀⠀⢀⣼⠃⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠀⣀⣼⣷⣋⣀⡤⠔⠊⠁⠀⠀⣀⣴⡦⠖⠋⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣐⠛⠿⠿⠿⠶⢶⣦⡶⢾⢛⡋⠉⠉⠛⠛⠻⠿⠶⠶⠖⠛⠉⠁⠀⠀⣀⣴⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣤⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⢶⠟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠁⠀⡍⠙⠏⠁⠀⠲⢦⣶⠶⡖⠒⡲⠞⢻⠛⠉⠹⠏⠀⠂⢳⢠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢨⡇⠘⡀⢰⠄⢠⠀⢠⠀⠀⠧⠀⠀⠀⠘⠃⢸⠃⠇⢸⠁⡄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⣄⣇⢸⡄⠀⠀⠈⡄⠀⢰⠀⠀⠀⢸⠀⢾⠀⠀⡘⢠⡧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⣧⣿⣏⡿⢸⣧⠐⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⡕⠀⠀⣇⣘⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⣦⣿⣿⣿⣅⣀⡻⠆⠈⠀⢷⣀⠘⠀⣀⢀⡾⠀⠃⣐⣀⣩⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢿⡿⢽⣿⡿⣿⣿⣶⣶⣶⣷⣶⣶⣶⣿⣶⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡚⢿⣿⢿⣿⣽⣟⣛⣻⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠛⠍⠛⠷⠛⠛⠉⠉



⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⠀⠀⣀⣤⣾⣶⣤⡀⠀⠀⠀⠀⠀⢰⣛⡄⠀⠉⠳⣤
⠀⢀⣴⣴⣿⣿⣷⣴⣄⠀⠀⠀⠀⢸⡇⢸⢛⠛⢻⠁⣿⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⢦⡠⠋⠈⠳
⢠⣾⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠀⢸⡇⣸⠆⠀⢸⡀⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠊⣹⣾⣷⡄
⠀⠀⠀⠘⡆⠑⠲⠄⣄⡀⠀⠀⠀⠘⠟⠁⠀⠐⠀⠙⠛⠀⠀⠀⠀⠀⠘⠛⠉⠙⠟⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⣀⣯⣹⢀⠛⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠉⢹⣿⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⡶⠶⠆⠀⠀⠀⠀⠀⠀⢠⣶⣶⣠⣤⣶⠄
⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⡖⡖⢖⣀⣿⣇⡀⠀⠀⠀⠀⢀⣯⣽⣻⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠉⣾⣀
⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣛⣻⠧⠀⠀⠀⠀⠸⣿⣿⠏⠉⢿⣿⡿⠀⠀⠀⠀⠈⢻⣯⣯⣯⣯⣷⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠴⠦⣟⠀⠀⣾⣿⠀⠀⠀⠀⢠⡾⣅⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡃
⠁⠀⠀⠀⣹⡆⠀⣿⣿⠀⠀⠀⠀⢸⣧⠈⠉⠀⣬⣤⡀⠀⠀⠀⠀⢀⡾⣷⠉⠃⢠⣤⣤⡀⠀⠀⠀⠀⣻⣿⠉⠃⢼⣥⣾⣷⡄
⠀⠀⠀⠴⠣⠇⠠⠿⠿⠀⠀⠀⠰⠋⠘⠃⠀⠛⠻⠿⠛⠂⠀⠀⠀⠋⠀⠛⠀⠀⠛⠻⠟⠛⠂⠀⠀⠰⠃⠸⠀⠀⠘⠿⠛⠟⠁


⠀⠀⠀⠀⡇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠋⣽⣿⡍⠻⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃
⠈⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⡄⣀⡀
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢿⡅
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠰⠦⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡄⣉⣄⣠⡀⠁⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⡏⠉⠉⠛⣷⡄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⢻⣦⣤⠄⠀⠘⣧⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⢀⡿⢀⣴⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣟⠐⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣶⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢰⡿⠋⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢷⣄⠀⣿⠁⠀⠘⠦⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡁⢙⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢠⣶⠿⠿⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢻⣏⠀⠀⢀⠹⣷⠀⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣈⠙⠛⠁⡰⠂⣿⠀⠀⠀⣠⡤⠞⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢰⡇⠓⠀⢀⡿⠀⣠⠟⣩⠖⠛⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠁⠂⠀⣾⠃⣼⣣⠞⡕⢺⠇⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠖⠁⠀⠈⠛⠀⠀⠈⣸⣾⣏
⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠦⠶⣤⣄⡀⠀⠀⠀⠙⣶⣦⣴⡿⠀⠀⠀⠀⠘⢃⣼⠟⠉⠀⠈⠳⢄
⠀⠀⠀⠀⠙⢿⣄⠰⣖⢛⡿⣶⢶⣬⣇⠀⠀⢀⣀⣽⣿⣿⡷⢦⣤⠶⠖⠋⠛⢿⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⣷⡈⠫⡻⣄⡈⠉⢛⡷⠀⠉⠁⠉⢻⡏⠁⣾⠁⠀⠀⠐⠀⢼⣿⣆⣤⣤⣶⠷⠚⠓⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣌⠊⠗⣦⣌⠀⠀⠀⠀⠀⣈⡳⠂⠉⠀⠂⣤⣤⣴⣾⣿⣏⣹⣿⡿⠂⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⠉⠘⢦⡀⠀⡀⣤⡿⠁⠀⠀⠀⠀⠁⠈⠙⠛⣻⣿⣿⣿⣷⣄⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢿⣿⣂⠠⠤⠤⢤⡁⢀⠊⠉⠁⠀⠙⢯⡀⠙⢣⣤⣜⣓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢚⣿⠶⣿⣏⡿⠀⠀⠀⠀⠙⢬⡄⠀⠀⠀⠀⠀⠉⠀⠀⠛⢉⠻⢿⠦⡙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢻⣌⡇⠀⠀⠀⠈⠀⣸⣷⠀⠀⠀⣀⡀⠂⠠⣶⣁⠦⠤⠴⠴⠙⣣⡘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣿⠗⣀⡀⣤⣤⣤⣼⣿⡷⢿⣾⠉⠀⠀⠀⠀⢩⣤⣥⣉⣉⣉⣉⣥⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⢻⡄⠀⠀⠀⣠⣳⠞⠉⠁⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢡⠉⠈⠉⠁⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⡼⣻⠟⠁⡋⢃⡤⡆⠀⠀⠀⠀⠀⠀⢸⡿⠁⢻⠿⣦⡀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡿⢱⣯⣴⠶⠟⡫⠜⣣⡀⠀⠀⠀⠀⣠⠏⠀⠀⠈⠀⠈⠟⡹⢿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⠧⠖⠒⠒⣉⣡⣴⠞⠛⠁⠉⠒⠀⠂⠹⠀⠀⠀⠀⠀⠀⠈⡅⠉⢋⢷⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣤⣴⡶⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⣤⣾⡓⢦⣈⡌⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⢷⣝⠦⣍⡃⣿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣮⣙⠛⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⣄⣀⣈⣄⡀⣀⣠⣾⣶⡷⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣛⢻⣿⣞⢻⣾⣿⣿⡿⣿⡞⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⡖⠚⣢⣀⢙⣇⢋⢻⣿⡯⠛⣰⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣧⡀⠀⠈⢿⠁⠜⣽⣿⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣋⣱⠈⣉⡛⣿⣿⣿⣷⢴⢋⠄⣶⠿⠟⠋⠀⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠿⠟⢿⠃⡂⠇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⡀⠀⠀⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣷⣶⣄⠀⢀⡀⠉⠃⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣹⣿⢿⣿⠿⣛⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠛⠂⠀⠀⢀⠀⠀⠀⠀⢰⡜⣾⣞⣺⡿⠚⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣤⣾⠃⠀⠀⣿⣻⠟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⠻⣟⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⣠⣤⡀⠠⡄
⠀⠀⠀⠂⣠⣶⣾⣿⣿⣷⣦⣈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⢉⠡⠔⢊⡤⠀⠀⢀⣷⠟⣚⡗⣤⣶⡖
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠐⢀⣠⣾⠿⣷⡤⠔⢈⡠⠖⣊⢥⠖⢃⠄⠀⣀⣤⣹⡋⣋⡹⢀⡆
⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⠟⠋⠀⠐⠊⠛⠛⠁⠛⠒⠛⣯⡤⠖⠉⣀⡴⣾⠀⠀⠘⢹⣿⡻⣁⡙⠁⣼⠿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⠁⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠛⠁⢀⣿⠀⠀⠈⣿⡅⢾⠿⣙⣿⣮
⠀⠀⠀⢀⠀⠀⠙⢦⡀⠀⠀⠀⠛⠟⠀⠀⠈⠀⠒⠲⠤⠀⠀⠀⣿⡴⠚⠉⢸⡀⠀⠀⠃⠰⠖⠶⠁⠘⠛
⠀⠀⠐⠞⠋⠁⠀⠀⠉⠲⣤⣤⣄⡀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⣀⣼⡀⠀⠀⠀⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠠⡏⠉⠐⠒⠶⠄⠀⢀⣧⡴⠚⠉⢸⠓⠂⠀⠀⠁⣰⣛⡿⣷⡖⣚⡁
⠀⠀⠀⠻⢦⣀⣠⡡⠒⠀⠀⠙⣟⠃⠀⠠⣷⡀⠀⣀⣀⠀⠀⠈⣿⡄⣰⣷⣾⡇⠀⠀⢠⣐⣻⣿⢿⣿⣿⣆⣀
⠀⠀⢠⢶⡦⢿⡿⠛⣿⡄⠀⠀⠳⠀⠀⠀⣿⣿⡷⡾⣭⣗⣢⠴⢿⣿⣿⣿⣿⡄⢀⣴⠾⢻⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠸⣬⣕⠀⠀⠀⣿⠇⠀⠀⢠⣴⡀⠀⠉⠙⠟⠑⠿⢿⣿⣶⣿⣿⡿⠟⠁⠀⠀⠛⠂⢼⡟⠿⣿⢿⠿⠈
⠀⠀⠀⠈⠛⠛⠒⠛⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠓⠉⠛⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⢸⢻⣿⢿⣳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠛⢵⡞⢣⣾⠸⡏⢷⡏⠳⣌⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡱⢮⢀⢠⡖⠙⠁⢀⡿⢸⣧⠐⣿⠀⣿⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠛⠒⣻⣛⣩⡤⠿⠧⠺⠏⢠⣟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⡠⠴⠒⠛⠛⠉⠁⢛⣃⡀⠤⠐⠊⠉⣋⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⢀⣠⣶⣶⣿⠿⠿⠛⠛⣛⣻⡿⢛⣽⠏⣿⣿⡝⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣀⡴⠟⠋⠉⣉⢹⠭⠤⠒⢊⣉⣻⣅⠴⢛⡯⢃⣿⣿⣿⢸⣇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡞⠋⠀⣠⠖⠋⠁⢀⣤⠖⠂⠉⠁⠀⢀⣀⣨⡤⠞⣹⣿⠏⣸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣧⣿⠀⣰⠋⠀⠀⣰⠞⠉⠀⣀⠤⠖⠊⠉⠀⣀⣀⡴⠞⣋⠤⠞⠟⢳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠘⢿⡄⣧⠀⠀⣴⠁⠀⣰⠞⠁⢀⣠⡴⠚⠋⠉⣠⡴⠋⠀⠀⣀⣤⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡳⢤⡀⠀⠉⠚⠧⣄⣿⡀⣼⡏⢀⣴⣿⠁⢀⣀⣤⢾⣿⡏⢛⡿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢆⠉⠉⠊⠓⠰⠤⣍⣉⣙⣛⣻⣛⣻⣟⣿⣵⣾⣴⣾⢿⠿⡟⢻⠏⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⡍⢩⠉⣏⠉⡯⢹⠇⢹⡇⣾⣼⢸⡇⣿⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡗⢸⠀⣸⢁⡏⡇⣸⢧⡏⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⢸⡆⡿⢘⠀⡾⢸⢃⣉⡏⣿⢸⣿⣿⣿⣆⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⡄⢡⠰⢠⠀⡆⢀⠰⠈⡇⡻⣸⢰⡇⣼⣴⢼⢳⣿⣾⣿⣿⣿⣿⣿⣶⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⡺⡎⣦⣇⠡⠀⡇⡆⣷⢾⣾⢸⣷⡟⣿⣿⣸⣿⣿⣿⣿⣿⣿⣯⡿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⢿⣿⣄⣧⣓⣱⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⡿⡏⠻⢿⠷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉

⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⣄
⢸⣿⡿⠟⠛⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠳⣷⠆⠀⠹⠃⠀⢀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠙⠁⠀⠀⠀⠀⠀⣴⣴⣿⠇⠀⠀⠀⠀⠘⠊⢷⡀⠀⠀⠀⠃⠀⠀⠀⠸⠷⣿⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠀⠄⠠⣤⠀⠀⢸⣌⣷⠀⠀⠀⠀⠀⠀⡀⢴⡄⠈⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⣂⡀
⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⣴⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠐⠀⠀⠀⢁⣄⠆⠀⠀⠀⠀⠀⠌⢍
⢸⣿⡆⠀⠀⢰⣶⣶⣦⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀
⠈⢉⡁⡀⠀⠀⠠⣿⣯⣀⠀⢀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠘⠛⠿⡿⠀⠀⠀⠈⠉⠛⠛⢿⡀⠀⠀⠈⠀⠀⠀⠰⣷⣦⡀⢀⡀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⡀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡀
⠸⣷⣤⡀⢰⡀⠀⠀⠀⠀⠀⠈⠀⢹⡇⠴⠴⠒⠈⠀⢀⣼⠟⠛⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣤⠀⠳⠈⠁⠀⣿⡇⠀⠀⠀⠀⠀⠘⢛
⠀⠀⠻⡟⢻⣇⠀⠀⠀⠀⢀⣤⡀⠀⠁⠀⠉⣠⣤⡀⢸⡟⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢛⡤⠀⠋⠀⢀⠾⠧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢛⣛⣢⣤⠴⣦⣤⡈⠹⠃⠀⡆⠈⡇⢀⣈⢶⢻⡎⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠠⣶⡅⠀⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⢴
⢠⣤⣀⡀⠀⠉⠛⠁⠀⠀⠹⣿⡶⠄⠈⠀⠀⠛⠛⠻⣶⠀⢠⡆⠀⠀⠀⠴⣷⣶⡛⢿⠁⠈⠃⠀⣽⣷⡂⠈⠛⠿⠛⢿⡆⠀⠀⠀⠀⠀⠈
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⢀⠀⠀⠀⠀⠀⠟⠀⠀⢙⡀⠀⠀⠀⠀⠹⣷⣸⣷⣶⠀⠀⠉⠀⣼⡗⣧⡀⠀⢠⠅
⢸⣿⣿⡇⠀⠠⢦⠀⠀⠀⢴⣿⣷⣔⣿⣶⣷⣀⣀⣠⠀⠀⠀⠈⣷⠀⠀⢸⣾⣻⣽⣿⣿⣧⣤⡶⠀⣠⠘⠀⠿⢱⡆⠰⢴⡀
⢸⣿⣿⡇⠀⠀⣰⣷⣶⠾⠿⣝⠻⢿⣿⣿⠿⢿⣷⡏⠀⠀⠀⣾⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⡏⠀⠀⠠⣿⣧⣀⡸⡣⠀⠀⢀⣴⡿
⢸⡿⡿⠇⠀⠀⠈⠸⠇⠀⠀⠀⣶⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⢰⣿⣿⣿⠟⠉⠉⠛⢿⣿⣿⡇⠀⠀⠀⠐⢶⣎⠁⠀⠀⠀⢸⣿⡆
⢀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣥⣤⠄⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⢻⠋⠀⠀⠠⣄⠀⠀⠀⠘⡿⢠⣷⡀⠀⠀⢸⡷⠀⢰⡆⠘⣿⣷
⢸⣿⣿⣿⣿⣿⣦⣀⠘⢷⣶⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢀⡴⠶⣧⡀⠀⠀⠂⠀⠁⣾⣿⣿⣆⡀⠸⣿⠀⠘⠁⠀⠉⢿
⢸⣿⣿⣿⣿⣿⣿⣿⣦⠈⢻⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⣀⣤⣀⡀⠀⠀⠛⠑⠿⠟⣿⡆⠀⣀⠀⣸⣿⣿⠿⠿⢿⣦⣹⠀⠀⠀⠀⠀⠈
⠈⠻⢿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⠿⠿⠿⢿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠴⢋⣀⣤⣤⣤⠈⠙⠁⠾⠆⣴⣿⣿⣷⡿⢢⠀⠀⣤
⠀⠀⠀⠉⠀⢙⣿⣿⡟⠀⠀⠈⠟⠻⠗⠖⠀⣾⣿⣿⣿⣿⢃⡿⠟⠃⠘⠃⠀⠀⠀⠀⠈⠙⠋⠀⠉⠀⠀⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠀⣀
⠀⢠⢠⣤⡄⠈⠙⠋⠁⠀⠀⣀⡀⠀⡀⠀⢨⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠀⠀⠀⢀⣠⣀⠙⢻⣿⣿⣿⣿⣿⡀
⢠⣷⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⢠⣿⣶⣤⣿⠿⠿⠤⠆⠀⠀⠁⠀⣠⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⠟⢠⣿⡀⢀⣀⣀⣨⣿⣶⣶⣦
⢸⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⡇⠀⠀⠀⣰⠖⢶⣿⡾⢉⣷⡟⠛⢦⡆⠀⠀⣀⠀⠀⠁⠀⠀⣿⠿⣷⠀⢻⣿⣿⣿⡏⠁
⠀⠙⠻⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⠀⢹⡆⠀⠀⠀⣿⡀⠀⠿⠀⠛⠛⠷⠀⠈⣀⣰⣾⣿⠉⢿⠀⠀⣠⣿⣷⣷⠀⢘⣡⣼⣿⣷⣾⣿
⠐⢃⣀⣨⣿⠀⠀⠀⠀⠀⠀⠀⢩⣿⡇⠀⠉⠁⢀⣀⣠⣄⡀⠀⠀⣀⠀⣶⣤⣶⣾⠟⣿⣿⣏⣠⣾⠤⠛⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣧⣀⣤⣤⣄⣀⣀⣸⣿⣥⣀⣴⣶⣾⣏⡉⠛⢿⣶⣿⣿⡀⢹⣿⠛⠁⣴⣿⣿⣿⡿⠿⠀⠰⣟⠛⠛⢣⠛⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠁⠀⢤⣧⣤⣀⡀⠀⠤⡃⠀⠿⣿⡷
⠀⠀⠀⠀⠀⠉⠿⢿⡿⠏⠀⠀⠐⠚⠁⠀⢸⡇
⠀⠠⡄⠰⡂⣠⠀⠀⠀⠀⠀⠈⠙
⠀⠈⠹⠀⠀⠨⠀⢰⡀⠆⠀⠉
⠀⢀⣰⣀⡀⠀⠀⢀⡡
⠐⢲⣿⣿⣿⠉⠀⠀⠸
⠀⠀⠀⠉⠀⠀⠀⠒⠆⢀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⣾⣇
⠀⡇⠀⡄⢠⡀⠐⠒⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠛⡆⠀⠀⣀⠄⠉⢻⡏
⠀⠀⠀⠉⠈⠠⢦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠎⠀⢀⡀⠸⠃
⠸⣿⣶⠯⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠈⠛
⢀⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣠⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⡃⠀⠈⢐⣆⠘⡀⢴⡀⠠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠉⢻⡇⠀⠀⠀⠸⠀⠀⠈⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣄⠀⠀⠀⠸⠃⠈⠀⠠⣼⣤⣀⠀⠀⠠⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢀⡀⢄⠀⠀⠀⠈⢹⡿⣿⠿⠉⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠈⠁⠒⠀⠒⣈⡩⠭⠐⠒⠒⠀⠐⠒⠦⠤⢄⣀⡀⠀⢀⣀⡠⠤⠒⠊⠉⠉⠉⠉⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠘⠙⠓⠒⣄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⠄⣀⠀⠀⠀⡄⠀⠀⠀⡀⠔⢉⠔⣡⠠⣶⠷⠾⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠐⠊⠀⠀⠀⡀⠁⠈⠑⠊⠳⠄⠖⠊⠀⠀⠀⣠⡮⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠢⢤⣤⠤⠖⢖⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⢀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⢄⠁⠀⡄⠀⠀⠀⠀⠀⢃⠃⠀⡷⠀⠀⠀⠀⠰⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣷⡐⠄⠀⠀⠀⠀⠀⢹⠀⢰⠀⠀⠀⢂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠙⠀⠀⠀⠀⣐⡄⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠠⠈⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⣸⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡤⡀⠀⠀⠡⣟⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠘⣧⡄⠀⠀⠹⣶⡄⠸⠀⠀⠀⠀⠀⠀⢀⣑⡀⠃⡠⢲⡠⠀⢧
⠀⠀⠀⠀⠀⠀⠑⠒⢄⠀⠀⠀⠀⠀⠸⡄⢀⠙⠈⣠⣖⠆⢹⣧⡼⠀⠀⢰⡀⢀⠀⠈⣟⡔⡼⡏⠁⢠⢄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⢱⠈⠀⠄⢰⠟⣆⠘⢟⣧⢾⣆⢸⣷⠆⠀⢴⡽⠀⢛⠁⠀⢈⢿⠀⢹⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣳⠆⣀⣇⢠⡀⠞⠀⢿⣆⠀⣿⠺⡐⣍⠟⠂⠛⢈⢓⠀⠘⠀⠀⣸⣄⣠⠾⣇⡴⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠐⠫⣹⠿⠟⢯⠃⢸⠟⢠⣷⡇⢐⣞⠀⠁⣠⠞⡦⢼⠚⠛⢉⢧⠯⠋⠠⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠖⠚⠉⠑⠤⠙⠀⠀⠀⢋⣠⠤⠞⠚⠋⡽⡙⠀⡌⠠⠀⠊⢣⠴⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠃⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠊⠈⢃⠀⢀⠎⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡀⠀⠐⢱⠊⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠀⠇⠀⢨⠂⠀⡃⠀⠈⡀⠀⠀⠈⠀⠄⣤⠀⢸⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠆⠀⠈⠀⠀⠘⠀⠀⡇⠘⠀⠁⠀⠀⠀⠀⠀⣇⠀⠈⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡃⠀⢈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⢞⣌⠒⠄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢣⣄⠀⠀⢠⣄⠄⠀⠀⠀⠀⠲⢄⡀⠁⠀⠀⠐⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⢀