Hungry Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⢉⣩⠽⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠒⠒⠮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠊⠀⠀⠀⠀⠀⠀⣀⡤⠀⣤⠀⡄⢸⠢⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⢠⠀⠀⠀⠌⠁⠈⠁⢉⡹⠒⠙⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠒⠉⡴⠀⠀⡰⠋⠉⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⢴⠇⠀⠀⢇⡀⢴⡁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠄⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⡠⠦⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⡠⠊⠀⠀⠀⠈⠐⠂⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢽⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠳⣍⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⠠⣠⡥⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡧⠒⠉⠁⠁⠀⠀⠇⠀⠘⠧⡀⠀⢀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠌⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠱⠙⠘⠦⣎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠴⠴⠁⠈⡑⠀⠀⠀⣸⠁⠀⠀⢣⡀⢀⠐⠀⠧⠌⠍⠠⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢒⠚⡀⢠⠉⠐⡤⢰⣀⣀⣀⡀⠀⢼⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠥⠄⡈⠉⠃⠁⠀⢰⠃⣆⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣤⣄⣿⣄⢢⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠀⠹⠿⠏⠉⣶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠜⠀⠀⠀⢀⠠⢪⢋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠥⠄⠀⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⢋⣉⡤⠤⠤⠐⠒⠂⠠⢤⣄⣀⡄⠈⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⡡⠔⠊⢉⡤⠴⢒⣉⣉⠉⠀⢀⣉⡄⣿⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠔⠉⢀⣤⠾⠕⠒⠉⠁⠀⢀⣀⡤⢶⡏⣸⢿⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⣅⣀⣔⠉⠀⠤⢤⣤⣴⡖⠚⠋⠉⠀⣾⢠⢛⢨⠳⣄⠀⠀⢀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠙⠛⡛⠓⠒⠛⠛⠛⠛⠃⢠⣇⢨⠘⡈⢦⡀⠉⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠐⠋⠛⠀⠐⠛⠳⠀⠀⠀⢸⡟⣎⠣⣜⠢⣍⣲⣶⡶⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣦⢰⠀⣠⡄⠀⠀⠀⠈⢷⣌⠓⢦⣭⢷⣆⠐⠒⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢘⡷⠃⠘⠟⠁⠀⠀⠀⠀⠀⠙⠻⢋⡴⠂⢸⣆⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡈⠛⠂⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣎⢿⣉⣉⣉⣉⡓⠦⡄⠘⣧⠀⠀⠀⠹⢤⣄⣤⠶⢻⠃⠀⠀⠀⠀⠀⠐⣦⣀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣴⣿⣮⣿⣿⣿⣿⣿⣷⣿⠀⠈⠳⣄⠀⠀⠀⢸⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⣿⣿⣿⡿⠿⠋⡇⠀⠀⠀⠈⠳⢦⡀⠈⠉⠁⠀⠀⠀⢀⣀⣤⠄⠀⢹⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⡴⠶⢚⣻⣿⠀⠀⠀⣸⠇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢋⡁⢸⡆⣴⠚⠁⠀⢠⠏⠉⠛⠛⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠤⣄⡴⢤⣀⣤⡀⣰⠏⠁⢹⡀⣹⣼⣧⡆⠱⡄⢀⡏⠀⠀⢀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⣄⡀⠀⠀⠀⠀⠀⢸⡯⣭⠗⠚⣉⣭⡿⢊⣽⣿⠠⡀⠘⠿⠓⠺⠿⠒⠚⣹⡾⠀⠀⣄⢸⠇
⠀⠀⠀⠀⠀⠀⡾⡏⠀⠀⠀⠈⠙⢦⡀⠀⢀⣠⠼⠿⣥⣶⠏⠩⠿⣏⠷⢶⡶⠾⢿⣖⢦⡈⠉⠒⠊⠉⠀⠀⠀⠀⠹⣾⡀
⠀⠀⠀⠀⠀⠀⣧⢳⡀⠀⠀⠀⠀⠀⠙⢦⣯⣾⣿⣿⣿⣮⣿⣗⡶⠦⣍⣻⡖⢮⡝⣻⣿⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⡞⢿⢧⡙⣆⠀⠀⠀⠀⠀⠈⢻⡍⠉⠉⠉⣹⣋⣧⣴⣿⡭⠷⠛⠋⠉⠉⠉⢳⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠠⡇⠈⢳⡳⣌⠳⣄⠀⠀⠀⠀⠀⡷⣤⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢛⡆⢸⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢳⡀⠀⠹⣌⠳⣌⠳⢤⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⢚⡩⢔⣿⠔⠛⢛⡉⠓⣄⡀⢀⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠳⢬⣳⣄⡀⢆⡽⢩⠷⢤⣤⣤⣤⡤⠶⣚⣫⡭⠶⢒⣫⡥⠾⠧⢴⡾⠛⠉⠀⠀⠈⢿⠛⢤⠙⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠉⠉⠉⠀⠈⠲⣤⡤⠖⣒⣚⣩⡭⠴⠒⠛⠉⠁⢀⣠⠶⢋⣠⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⣀⣀⣤⠴⠾⣟⣛⣒⡯⠯⣁⣠⣤⠴⣶⣟⣁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠚⠋⠉⠉⠉⠁⠀⠀⠰⣏⣭⣭⣤⠞⠉⠀⠀⠀⠀⠻⢬⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠉⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠐⠀⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⠀⡾⢈⡄⠔⠊⠙⠒⠺⠒⠐⠲⢤⡊⠀⣿⠀⠀⠀⠀⠙⠒⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠀⡊⠀⠀⠀⠀⣈⡂⠀⠉⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠱⠀⢀⠊⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⢄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣸⡷⡄
⠀⠀⠐⣆⠈⠇⠀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠴⠿⣾⡇⢻⠷⠛⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠃⠀⡀⢠⠈⠆⡄⠀⠑⢢⣄⣀⣀⢀⡠⢊⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠄⣑⣄⡀⠀⢀⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠋⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠆⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⣿⣿⡆⠀⢀⣾⣿⣦⠀⠀⠀⢸⠀⢆⠀⠀⠀⢠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠠⠀⠀⠀⠀⡐⠀⢦⠀⠀⠀⠀⠙⠋⠁⠀⠀⠛⠛⠃⠀⠀⢀⠜⠀⡨⠓⠊⠀⠀⠇⢀⠃⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⠀⠈⠒⠈⠳⣄⠀⠉⠒⠀⠀⠂⠀⠀⠐⠀⠀⠤⠤⠐⠒⠁⡠⠊⠀⠀⠀⠀⠀⣧⣾⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡀⠇⠩⣅⡠⠚⠉⢁⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⣐⡘⠛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⢂⠀⠑⠄⡉⠁⠔⠊⣷⣤⡀⠀⠀⠀⣀⡀⢀⣴⠟⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠺⠿⠛⠀⢀⣦⠀⠀⠘⣷⣦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡴⠒⢩⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⡜
⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡿⢍⡐⠢⡀⠀⠈⠂⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠉⠢⡍⠐⡒⠂⠀⠀⠀⠈⠢⠄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠀⠤⠄⠀⠉⠉⠁⠀⠀⢀⣤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡆⠆⠀⠀⠀⠈⠢⣄⣲⡀⠐⢄⡀⠀⠀⢀⣀⡀⠈⠑⠈⠁⠀⠀⠈⠀⠀⠀⣀⠤⠤⢀⣀⡠⢒⡬⢽⠋⡖
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣀⠀⠀⠀⠈⠂⠈⠑⠊⠉⠿⠒⡶⠨⢇⣀⣒⠭⠉⠩⠅⣂⣒⠮⠀⣶⠒⠦⠶⠂⠁⠀⢘⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣶⣤⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣂⣨⣠⣤⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠷⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣻⣿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⢠⢤⣄⡀
⠀⠀⠀⢰⣷⣄⠀⠀⠀⢠⣴⣶⣶⣿⣟⡿⢾⣝⠻⣷⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣷⣄⠀⠀⠀⡏⡞⡞⡟
⠀⠀⠀⠀⢿⠘⣧⠀⢠⣿⢫⣿⣿⠉⠙⠻⣶⣝⡻⣶⣭⣝⣛⣛⣟⣛⣛⣿⡿⣟⣽⣿⣿⡀⠀⢸⠃⠛⢳⠃
⠀⠀⠀⠀⠘⣧⠘⣇⢸⣿⣾⢹⣿⠀⠀⠀⠀⠙⠿⣶⣮⣭⣽⣛⣻⣯⣭⣵⣾⠿⠛⠉⢿⣿⡄⢸⠀⣠⠏
⠀⠀⠀⠀⢀⣿⡄⢸⣿⣧⣿⣿⠏⢠⣤⣤⣄⠀⠀⠀⠉⠉⠛⠛⠛⢉⣉⣉⡀⠀⠀⠀⠸⣿⡷⣾⣼⡁
⠀⠀⠀⠀⢸⠀⣷⠀⢿⡟⠛⠁⠀⢀⡞⣉⡉⠻⡟⠛⠀⠀⠈⠛⣿⣋⡉⠙⣏⠁⠀⠀⠀⢿⢠⣧⡏⡷
⠀⠀⠀⠀⠸⣇⢸⡇⢸⡇⠀⠀⠀⢸⡟⣻⣿⡆⣿⠀⠀⠀⠀⢰⡟⣻⣿⡆⢹⠀⠀⠀⠀⠸⣿⣼⣰⠇
⠀⠀⠀⠀⠀⠘⢷⣿⠀⣧⠀⠀⠀⠘⣧⣽⣿⣇⣿⠀⢀⣀⡀⢈⣧⣼⣿⣃⣾⠀⠀⠀⠀⢠⣇⡿⠋
⠀⠀⠀⠀⠀⠀⢨⡿⡆⢿⠀⣠⠶⣀⣭⣉⡁⠀⠀⠀⠋⠐⠙⣀⣭⣍⣉⣩⣭⣘⢷⡄⠀⣸⣹⡄
⠀⠀⠀⠀⠀⠀⢸⡄⣷⢸⠀⠃⠈⠁⠙⢧⣹⠽⢿⢷⣶⣲⠞⠁⡠⢈⡧⠋⠀⠀⠀⠃⢀⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣷⣿⠿⠶⠶⢾⣹⡆⠀⠈⠛⠶⠶⠾⠥⠴⠾⠛⠉⣦⣀⠀⣾⣹⠶⠾⠿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣯⢠⠤⠶⠖⢻⣯⣄⣀⣀⣀⡀⠉⠛⠛⠋⠀⠀⣀⡙⠟⠀⡬⣿⠓⠶⠦⢤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠚⢋⣉⡷⠀⠈⠉⠻⢥⣀⣀⣀⣀⣠⡼⠟⠁⠀⠀⢘⣏⣙⡓⠲⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠛⢋⣉⣩⠟⠀⠀⠀⠀⢀⡈⠉⠉⢉⣁⡤⠖⢁⠀⠀⠸⢯⣉⣉⠛⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⣽⢿⣄⠀⠀⠀⠀⠀⠀⠉⠉⢭⠭⠤⠖⠚⠁⠀⠀⠀⢀⣸⣿⣍⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣄⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣟⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣭⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⠂⠀⠀⠀⠚⡉⠂⠀⠀⠉⠉⢉⢩⡅⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⠆⠀⠀⠸⠿⠃⠀⠀⠀⠀⢱⡆⠃⣀⠔⢸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣈⢴⠊⠀⠐⠋⠉⢉⠍⣠⠞
⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⢧⠀⢤⠤⠤⢀⡠⠄⠀⠀⠀⡀⠡⡼⢶⠶⠿⠭⠄⠒⠈⠁
⠀⠇⠀⠄⠋⠓⣦⢤⡀⠀⣠⣤⣷⡄⠻⠉⠹⠁⠀⠀⢀⡔⣀⣴⣿⣷⣤⣤⣀⡀
⠀⡄⠀⠀⠓⠀⡌⠀⢰⣾⣿⣿⡏⢹⡄⠒⠃⠀⣠⣴⡫⢕⡫⡾⠃⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠈⢦⡀⠈⠠⢁⣴⣿⣿⣿⣿⣿⣿⣿⣶⢖⣋⠉⡀⠀⠁⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠄⠊⠙⠻⣿⣿⣿⣿⣿⣿⣿⣍⠽⠳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣷⡄
⡀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠟⠋⠨⠟⠛⠛⠛⠛⠛⠛⠛⠳⣬⣙⠿⠿⢿⡿⠟⠛⢷⣽⣿⣿⠟⠋⠉⠙⢄
⡇⠀⠀⠀⢠⢂⠴⠋⠀⢀⠤⠤⣀⣈⣀⣤⡀⡠⣤⣠⣤⣾⣿⣿⣷⣿⠛⠀⠀⠀⠀⠈⠉⠉⠀⠂⠀⠀⠀⠙⠢⢄
⠱⠤⠤⠄⠊⠁⠀⠀⠀⢸⣿⣿⣿⠿⣿⠿⡿⢿⣿⣟⣿⣟⡁⠉⠉⠙⢼⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⢿⣿⣽⣿⣿⣿⣿⣟⣳⣆⠀⠀⢀⠊⠐⠤⠐⠀⠂⠉⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⢄⠀⠀⢼⠛⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠿⠁⢈⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠉⠐⠒⠦⠤⠭⠭⠉⠭⠭⠽⠯⠥⠜⠂⠲⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⣦⣀⣀⣤⣄⣀⣀⣠⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠳⢦⣄⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣸⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠢⠀⠀⠀⠀⠀⢺⡗⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⡒⠀⠀⠀⠀⠀⣀⣀⣐⡷⠤⠿⠤⣤⡀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠸⣅⡀⠀⠁⠀⠀⠀⢀⡿⣠⡿⣅⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠦⣄⠐⠒⠈⠉⠛⠒⠶⢖⣛⡿⠟⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⠀⠉⠀⠀⠀⠒⠒⠛⠉⣁⠤⠤⢄⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⢠⠀⠀⡈⠳⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⡠⠷⠖⠶⢤⡀⠀⠀⠀⠀⠈⠉⠣⣀⣠⣇⣀⡟⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠉⠀⠀⠀⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢈⡇
⣴⠚⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⠼⠦⡤⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠈⢢⠱⡀
⣧⠀⠀⠘⢇⠀⠀⣀⡤⠶⢄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠞⡠⠃
⠸⣦⡀⠀⢀⠏⡋⠀⣀⣠⠀⢧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠜⡆
⠀⠈⠻⠶⢥⡤⠷⠞⠋⠹⣆⠈⠳⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠰⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⠼⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⢤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠉⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⠞⠉⠀⠀⢀⢹⡀⠀⠊⢈⡀⢀⠀⠀⠀⢀⡱⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⡀⠀⢠⡞⠁⠈⠉⠉⠉⠉⠉⠉⠀⠈⢷⡀⠀⢀⠈⡟⠀⢠⡀⠀⠉⠐⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠟⠚⠁⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠞⠩⠖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠓⠒⠒⠒⠒⠧⢄⡀
⠀⠀⠔⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡶⠆⠀⠀⠀⢠⣦⡀⠈⠳⣄
⠀⢘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡞⢰
⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡈⠀⠀⡀⠀⠰⠀⡀⠀⠀⠂⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⡄
⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢃⠀⠠⡷⠀⢀⠀⡿⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⢠⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠢⠀⠀⠠⠈⠀⠀⠀⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠙⠤⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠
⡴⠒⠶⡏⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⡿⠖⠲
⢹⠤⣠⡇⠀⢀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡄⠀⠀⣧⡠⢼
⠈⢦⣀⣑⣶⠋⢀⡤⢈⣉⠛⠒⠢⠤⢄⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⠤⠤⠒⠚⢋⣉⠠⣄⠈⢲⣖⢁⣠⠊
⠀⠸⢄⣀⣨⡟⠉⠀⠀⠀⠉⠉⠒⠒⠒⠤⠤⠤⠭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠥⠤⠤⠔⠒⠒⠊⠉⠁⠀⠀⠈⠙⣧⣀⣁⠼
⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡇
⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢹⢹⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣤⣀⣀⡇⢸⢸⡇⣇⣀⣠⡤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠉⠁⢸⢸⡃⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣶⣶⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠢⢤⣈⠙⠫⢍⡉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠤⣀⡀⠀⠀⠀⠀⠙⠻⢔⠛⠲⠄⠀⠑⢌⣛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠚⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠒⠒⢒⡒⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣀⣀⣤⣤⣤⠄⠀⠀⡄⠀⠉⠉⠑⠲⠒⠀⠒⠄⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⠁⠐⢲⡆⠙⣄⠀⠠⡀⠀⢀⠀⣠⢃⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⠼⠀⠤⣀⢀⡀⢴⢅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠐⣿⡳⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠀⠨⡋⢻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠃⠀⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠓⠢⡌⠘⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⠉⢀⡀⡇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⡇⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⠄⠈⠲⠤⢀⡀⠀⠠⠔⠁⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠘⡤⠊⠉⢢⠀⡁⠀⠀⠀⠀⡠⢴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢹⠀⠀⠀⠀⠘⠀⠀⠹⣀⠔⠃⠀⢇⠀⢀⠔⣻⣶⡞⠛⣶⠞⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡌⠀⠀⣀⣀⣨⡇⠀⠀⠙⠢⠀⠀⢹⣤⣿⣯⣮⠡⠟⠋⢁⢺⡟⡽⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⠃⠀⠀⠙⢢⣿⣧⣤⣤⠄⣀⣠⣤⣼⣋⡽⠁⠀⠀⠀⠀⣆⠀⠘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⡸⠢⣀⠿⠿⣀⡸⠿⣇⣀⣿⠛⠒⢄⠀⠀⠀⠀⠑⢆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣧⣀⠀⢀⠞⠀⠀⠈⢢⣘⠿⢇⣀⡽⡿⠃⠀⠀⠀⠑⣄⠀⢀⠄⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢳⠃⠀⠡⠉⣲⣦⠃⠀⠀⠙⢤⡜⠛⡷⠁⠀⠰⣀⣤⠚⠁⠀⠈⢀⠀⢧⠠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠒⠺⣤⠇⠑⠒⠉⡏⠀⠀⠀⠀⠈⢳⡜⠀⠀⠀⠀⢻⡛⠒⠒⠒⠂⠀⢤⠎⡀⠠⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠈⢓⠒⢂⣷⣰⠉⣉⠉⣁⠉⠀⠁⠀⢀⢀⡁⣈⣀⣀⣀⣈⣉⣉⣉⣉⠉⣇⡼⢰⠗⠂⠤⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠙⠧⠏⡠⠊⣀⠤⠤⠀⢐⡀⠐⠒⠒⠒⠒⠂⠠⠤⠤⠄⣀⠀⠀⠑⡄⠉⠉⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠸⣦⣄⣛⣓⠒⠒⠒⠒⠒⠐⠂⠀⠀⣈⣉⣠⣼⠂⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠂⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣠⣭⣛⣛⡛⢛⣯⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡍⣽⣾⣿⣿⣿⣿⣿⣷⣭⢩⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠿⣿⠿⣿⣿⣿⠀⠀⣷⠀⣹⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣵⣶⣭⣿⡿⠃⠀⣴⠋⣾⠁⢰⠋⠀⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣛⠋⠀⠀⠀⣘⣁⠘⠃⠘⢃⣴⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⢤⣾⣿⣿⣿⠷⣀⣀⣾⡿⣫⣶⣿⣟⣮⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣝⡻⡿⢟⣺⣿⣿⣟⣱⣿⣾⣿⣿⣟⣸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠤⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡁⠀⠀⠀⣷⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⡛⠷⣤⣴⠏⠉⠛⠛⠁⣀⣀⡠⠤⢤⠤⠤⠤⡤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⢏⡋⠀⢴⠆⠀⢰⣺⡆⠀⡠⢒⡩⠷⣒⠰⠟⣹⣿⣉⣝⣽⣶⣖⠮⢽⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⠋⠀⠀⠀⠀⢒⡎⢼⠃⠀⠹⢬⣓⡊⠭⠉⣿⣧⣍⣹⣿⣦⡤⠿⠟⣚⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠓⠶⠦⠦⢸⡇⠀⠀⠀⠈⠉⠙⠛⣿⠒⠒⣒⣲⣶⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡷⠶⣒⡀⠀⢒⢒⠿⢥⣀⣀⠀⠀⣀⣠⠖⠋⢉⡟⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠓⢛⢄⡙⠊⣥⣄⠈⠻⣏⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢠⡟⠲⣄⣴⠋⠀⠀⣠⠆⣀⣀⣬⠭⠥⠴⠾⠷⠶⠦⢼⣷⣀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⠀⠀⣸⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⢶⡛⠉⣠⣿⠀⠀⢠⡟⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠳⠞⠁⢸⡄⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠹⣦⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣼⡏⠀⠀⠸⡷⠶⣤⣤⣤⣤⠶⣶⠶⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣰⡏⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣰⣿⣇⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢠⡄⠀⠀⠀⡠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁⠀⠀⠀⠀⢠⣿⠀⠀⣙⠒⠖⠀⠐⠤⢖⡌⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠈⠐⠀⠀⠀⠈⠁⠀⠘⠛⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢈⠀⠀⠀⠀⠀⣐⠦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣀⠤⢒⡀⣴⠖⠷⠤⠖⠒⠛⠺⣾⠀⠘⢦⡀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⣷⠘⠳⠖⠖⠲⠲⠆⠚⠁⠀⣾⡀⠙⣦⢴⢪⣡⣾⣷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡝⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⣹⠁⣴⣧⢾⡹⣟⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⣆⠀⢸⡇⢻⡷⠄⡀⠀⢀⣤⠐⡿⢁⡏⣸⣯⢧⢃⡵⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⠀⠀⡇⠀⣧⠞⠻⣤⣼⠉⣻⠅⢸⢃⣿⣧⣿⡞⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⢹⡄⠀⣿⣟⣷⣃⢀⣾⣾⣿⣷⠿⢸⠇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣌⡇⠀⡾⣿⣻⣼⢋⢀⣽⣯⢀⢠⣏⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡧⠖⠦⢤⡀⡄⢸⠀⠀⠀⢸⠏⣷⠀⡿⢸⣿⠻⡟⠚⠉⠀⣼⢹⣸⣒⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣰⠋⢧⡞⣦⠀⠀⢸⡀⣿⠀⡇⢸⡇⢰⠇⠀⢀⡖⣏⣸⠈⢦⢱⡄⠀⢸⡇
⠀⠀⠀⢀⣀⣀⣸⣇⠀⠀⣿⠀⠒⢧⣼⠀⠀⣘⣧⣿⣀⣇⣈⣧⣿⠤⢤⣼⣼⠉⠀⠀⢸⣇⡇⠀⣼⣀⣀⣀⣀⣀
⠀⠀⠀⣶⡯⠩⠷⠭⣿⣖⣿⣦⡉⢀⣼⣛⡟⣡⠖⠂⠀⠉⠁⠀⠀⠸⡆⠙⣿⣄⣀⣀⣼⣯⣴⣾⢏⡁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢤⡉⣠⠎⠀⠉⠛⠛⢩⡟⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⣹⠏⠉⠉⠉⠉⠉⠙⣦⠐⠀⠀⡀
⠀⠀⠀⠀⠐⠩⢭⡞⠑⠃⠀⠀⠀⠰⠾⠿⢿⡶⣤⠭⢍⣉⣀⠀⠠⠖⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠼⣧⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠿⠿⠿⡿⠋⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠘⠋⠁⠀⠀⢀⠀⠉⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⢀⣠⡄⠀⠈⡔⢤⡄⠀⠀⢷⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣻⡌⠋⠁⠀⠀⢰⠀⠈⠓⠶⠇⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡀⠀⠀⠚⠛⢦⡀⠀⠀⠀⣠⠔⠀⠀⡟⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠯⣉⠉⠙⢺⣷⡄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⠃⠀⠀⠀⢘⣿⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠙⣶⡞⢹⡛⢶⣄⠀⠀⠀⠄⠴⠰⠀⣠⣾⠃⠀⠀⠀⠰⢹⣿⢷⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣹⠟⠇⡀⢸⠇⠀⠈⢹⡗⠶⠶⢶⠛⠻⡍⣿⢷⡄⠀⠀⠠⡞⢷⣂⠨⢞⡥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⠴⠷⣶⣬⠀⠀⢰⡟⠀⡟⠀⠈⡇⠀⠀⠈⣧⠙⣦⡀⠀⠹⡎⢁⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡾⡗⢹⢰⠀⡇⠘⠧⣴⠟⠀⠀⢀⡀⠀⢰⢹⡆⢳⡈⢦⣧⡿⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢀⢾⡇⡇⢸⢸⠀⣿⢸⠀⣿⠀⠀⠀⠹⡜⣤⠆⠘⠑⣺⢃⡈⠻⡀⠀⣸⠃⠸⡆
⠀⠀⠀⠀⠀⠀⣸⢸⡇⣷⣸⠸⠀⣿⢘⣀⣷⠇⠀⢠⡄⣷⠀⠀⠀⣀⡭⠞⠀⠀⣟⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⢸⡇⢻⣿⠀⣀⣿⠞⠋⢡⠀⠀⣸⣇⡝⢀⡴⠋⢁⣴⣦⣤⡼⠙⢾⠁⠀⢀⡇
⠀⠀⠀⠀⠀⢸⣿⢸⡇⡆⢸⠿⣿⣅⠀⠀⠘⡟⡟⢫⡽⠁⡞⡟⡾⢫⡿⠈⠳⣄⠀⢀⣤⠠⠞⠦⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠀⡇⣼⢸⠀⡇⢹⣆⠀⠀⠸⣷⠋⠀⣴⠁⡇⡟⣼⠇⠀⠀⢈⠟⠡⠤⠤⠄⠀⢄⣀⠝⣢⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠈⣿⣦⣧⣜⣸⢀⠃⢠⠹⣦⠀⠀⠀⠀⢠⠟⠀⣇⣵⣿⢀⣀⣤⣼⡏⠒⠒⠖⠈⠉⠉⠀⠀⡟⢿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⠛⠛⠛⠿⠾⠿⢷⡄⠀⢠⡿⢾⣿⣿⣿⣿⣿⣿⣿⠀⠹⢦⣀⠀⠀⣀⣀⡴⠚⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣈⣉⣉⣁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⢿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣦⠈⠙⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣤⣤⣤⣤⣼⣿⣄⡀⠀⠀⠀⠈⢷⡀⢸⡇⣈⣽⣿⡟⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⢷⣄⠀⠀⠈⢳⣾⣿⣿⣿⣿⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣷⡀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠛⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣫⣾⣿⣶⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⡿⠿⠿⠿⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠉⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⢻⣷⣄
⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⢺⣹⠆⣔⢄⠀⠀⠀⠀⢹⣿⡄
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⣀⣀⣔⢻⣅⢤⡥⠦⠃⠀⠀⠀⠀⣿⡇
⢠⣿⠟⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠑⣰⠇⠀⠀⠀⠀⠀⢰⣿⠁
⣾⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢀⣿⠃
⣿⣇⠀⠀⠀⠀⠀⠀⠙⢤⣤⣤⣤⠶⠋⠀⠀⠀⠀⠀⢠⡾⠃
⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠈⠻⢿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⢉⡴⢒⡶⠀⠀⠀⠀⠀⢠⡤⠖⠒⠒⠢⢄⠀⣀⣤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢩⠒⡄⠀⠀⠀⢀⡴⠛⠓⠀⠀⠀⠀⠉⠁⠐⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⢀⣴⣣⠀⣀⣤⣤⣞⠳⢤⣀⣤⣀⠀⠀⠀⢨⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣁⢃⣾⣿⠫⢹⠙⣧⣰⣟⢻⡛⣷⣦⡀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠛⡟⠘⢦⡾⠀⣿⠻⣄⣼⠀⢸⡎⣧⡜⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡇⢻⣄⠀⢀⣰⣿⣄⠀⠀⢀⣼⠁⢸⡇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠉⠙⠁⠙⠙⠳⠶⠛⠁⠀⢀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣄⣀⣀⣠⣤⠴⣶⢦⣄⠀⠀⣼⢃⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠉⠉⠁⠈⠳⢬⡼⠉⢀⣼⠟⢁⣴⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⢿⣿⣟⣛⣦⠀⠀⠀⠀⢀⡼⢁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⣦⡀⠀⠀⠀⠀⠀⣴⢻⡇⠘⠟⠛⠛⣲⡿⠟⢻⠁⢸⡏⠉⢻⡅⠀⠀⠀⠀⣞⠳⠞⣯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠣⡙⣆⠀⠀⠀⢠⡇⠀⢷⠀⠀⠈⠉⠀⢀⡴⠃⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠈⡟⣠⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢨⢸⡄⠀⠀⣼⠁⠀⢸⡇⠀⠀⠀⠐⠉⠀⠀⠀⠀⡇⣤⠀⠘⣇⠀⠀⠀⡼⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢦⣧⠀⢀⣿⠀⠀⣼⡧⠶⠶⠤⢤⣄⡀⠀⠀⠀⣷⣿⡆⠀⣹⣦⣤⢶⡛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠾⣿⢤⡟⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠉⢸⣿⡄⠀⠀⣏⣾⣟⡛⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠴⣿⣷⣿⣠⡶⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠷⣄⣸⣿⡏⠈⢉⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠉⣹⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⣿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠲⣄⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠩⡷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⣀⠤⢶⣒⣖⣲⣖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡀⢀⣄⣤⣴⣂⡀⢐⡏⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⣠⠞⣁⠤⢍⠺⢷⡿⢶⣑⣴⠇⠀⠀⠀⣀⣤⣶⣒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠔⠋⢉⣹⣿⡏⢈⣻⡛⢦⡗⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣧⣬⠷⠾⡻⢯⣜⣫⣶⠟⠉⣀⣤⣤⣴⠗⠋⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠂⠀⠘⠳⠞⣱⣏⡿⠃⡾⠀⠲⠆⢀⡀⠀⠀⠀⠸⣇⠀⠀⠀⢀⣀⣈⣳⣿⡛⠈⠀⠀⠰⣿⡏⠁⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡠⠶⠒⢮⣶⣄⡾⠁⠀⠀⠀⠘⠛⠀⠯⠆⢠⡟⠁⠘⡖⠝⢃⡜⢋⣤⡇⠀⠀⠀⣀⡠⠽⠶⡚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢳⣅⡀⢀⣨⠽⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠚⠳⠋⠁⠘⡇⠀⠀⡞⠁⠚⢂⡠⢾⡀
⠀⠀⠀⠀⠀⣠⠶⠋⠉⠉⠙⢳⣾⠁⢿⣿⡶⠒⠒⠶⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠟⢚⡧⢃⡠⢤⠔
⠀⠀⠀⠀⣴⠿⣶⣤⡀⠀⠀⠈⢿⣶⡾⠟⠁⠀⠀⢀⣠⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠈⢓⡉⢉⡁⣼⢿⣇⡬⠃
⠀⠀⢀⡾⠁⢀⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⢻⡟⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣤⠀⠀⠀⠀⠀⠀⠸⡍⢻⡟⢡⡗⠛
⠀⠀⢸⠃⠀⣾⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⢠⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣤⠀⠀⠀⢠⡗⠤⣈⣹⡄
⠀⠀⠘⣇⠀⠘⢾⡀⠀⣀⡀⠀⠀⠀⣀⣿⡤⣄⣴⠋⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⢧⣈⣁⣀⣀
⠀⠀⠀⠈⠳⣄⠀⠙⠛⠑⢧⣄⠀⠰⣿⠈⠀⢈⣀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠛⠦⠤⠶⠤
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⣀⣺⠃⠀⢻⣃⣤⠟⠉⡶⠳⠞⠛⢃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢧⠙⠚⠋⠁⠀⠀⠀⠉⢧⡀⠉⠙⠛⠛⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⣇
⠀⠀⠀⢀⣀⡠⠠⢀⡀⠀⠀⠀⠀⢀⠠⠔⠒⢉⡡⣤⠈⢣
⠞⠉⠉⠀⡀⠀⠀⠀⠈⠉⠀⠈⠈⢱⠴⠒⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⡂⠀⠀⠂⠀⠀⠂⢢⠀⠀⡴⠁
⠀⠀⠈⠀⠄⣀⡀⠀⠀⣨⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⠀⢀⣠⣄⢦⠤⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⠏⠉⠑⠒⠛⢻⣶⢊⠀⢠⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⠀⠀⠀⣰⠆⠙⠒⠁⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠒⢋⡥⠀⠀⠀⠀⠀⡇⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡄⠀⢀⡞⠀⠀⢀⠗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡼⠀⠀⠀⡜⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠀⠀⡼⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡰⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢹⠀⠈⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢹⢰⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡟⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⢸⡀⠀⠀⠀⢠⠇⠀⠀⣀⠴⠒⠚⠛⠂⠀⠀⠈⢧⠀⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⣄⠀⢠⠎⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡜⠀⠀⠚⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠈⠓⡟⠲⣤⣄⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⣉⠀⣿⠀⢠⠧⢄⢘⣦⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⣿⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡉⠁⠀⢸⣄⡔⠋⠑⠒⠒⠠⠤⠭⠦⠀⠀⠀⠐⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣇⣀⡜⠀⠲⠅⢬⡤⠦⠤⠤⢤⣤⣤⠤⠤⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠈⠉⠉⠉⠉⠉⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠶⠶⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠈⢳⡀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⠾⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠈⠉⠈⠀⠙⠃⠒⠀⠘⠃⠀⠀⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⠀⢀⠂⡐⢁⠀⠉⠉⠑⠎⠁⠀⠉⠢⠀⢇⠀⠀⢀⣇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠳⠞⠀⠀⢠⠀⠴⣶⠀⠀⠀⣶⠄⢀⠂⠈⠶⠂⠀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⡐⠉⡃⠀⢀⠴⡄⠀⠀⠀⢸⡄⠀⠀⠀⢦⠀⠀⠀⣀⠀⠂⠀⠀⠐⠋⠉⠉⠰⡅⠀⠀⠀⠀⠞⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠁⢀⠃⠀⠈⠀⠃⠀⠆⢸⠀⠻⠤⢀⣀⡀⠀⣴⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠰⣤⣄⠰⣄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⣠
⠀⠀⠀⠘⠀⠸⠀⢀⠁⠸⠀⢰⠀⡄⠀⠀⠀⠀⢸⠁⠀⢈⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠋⠀⢀⠛⠛⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀
⠀⠀⠀⠇⠰⣇⢀⠇⠀⠇⢠⡄⢠⠁⠀⠀⠀⠀⢸⠀⠀⢻⣷⣤⣁⠒⠀⠤⠀⠀⠀⠄⠀⣒⣡⣴⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡌
⠀⠀⠀⠈⠢⣀⠁⠀⠀⠃⠚⢀⠌⠀⠀⠀⠀⠀⠘⣇⠀⠘⣿⡿⠿⠿⣿⠶⠲⠶⣶⣾⣿⣿⣿⠏⠀⢀⡟⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠘⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⢀⡀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠋⠀⢀⡞⠀⠠⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡈
⠀⠀⠀⠤⠠⡧⠀⠉⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠈⢠⣀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠙⠛⠛⠒⠒⠀⠐⠀⢺⠋⠀⠙⢦⡀⠀⠀⡸⠉⠀⠀⠀⢡⠀⠀⠀⠔⠁⠀⠀⠙⠢⢄⡁
⠠⠈⠑⠂⠄⣤⠻⠋⠉⠁⠀⠠⠃⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠹⡄⠉⠳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⣸⠇
⡄⠢⠄⢀⣀⠰⡏⠀⠐⠀⠰⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⠸⠗⠒⠈⠁⢀⡇
⠁⠄⣀⣀⠍⢠⠁⠀⡀⠀⠀⠸⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢹⡄⠀⠀⢙⡆⠀⠀⣀⠎⠀⠀⡆⠀⠁⠀⠘⢀⡀
⠀⠀⡈⠀⠀⡈⠀⠉⢻⡀⠀⠀⠈⠢⢀⡀⢠⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⣧⠀⠀⠊⠁⠀⠉⠀⠀⡄⢫⠀⠀⠉⠀⠈⡨⠃
⠀⠀⠁⠀⠀⠁⠀⠀⠀⠱⣄⠀⠀⠀⠈⠁⣿⠀⢱⠀⠀⠀⠀⠀⠠⠤⠤⠤⠔⠊⠁⡀⠀⢀⠔⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡠⠈⠀⢈⠀⠀⠘
⠀⠐⣄⢀⠇⠀⠀⠀⠀⠀⠈⠳⢤⡀⣀⡜⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢀⠔⠁⠀⠀⠀⠀⠀⠘⡷⠀⠒⠀⠈⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠠⠤⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠤⠀⠀⠁⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢤⣀⡀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⠀⠤⠞⠉⠘⠒⠦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠤⡀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠈⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠂⠁⠄⠈⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⣀⠀⠠⢄⡊⠤⠶⠐⠒⢀⣀⠀⠮⠠⠤⠄⠀⠀⠀⠆⠀⠰⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⢤⠀⠤⢤⠀⠀⠒⠒⠀⠈⣩⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⣀⣿⠏⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣞⠥⠂⠀⠀⠀⠀⠈⠇⠀⠀⠂⠉⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠉⠙⠲⣄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⠎⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣼⠀⠀⠀⢀⡞⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢠⡇⠀⠀⠀⣼⡇⢸⣿⡇⢀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠶⣄⡀⠈⠛⢯⡿⣿⣿⣿⣿⡄
⠀⠀⠀⡾⠀⠀⠀⣰⣿⠀⣸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣄⠙⢮⣿⣿⣿⣷⡀
⠀⠀⡼⠁⠀⠀⣰⣿⠃⣰⡿⠃⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⣿⣷⣄⢻⣿⣿⣿⣷
⠀⢰⠃⠀⠀⢠⣿⣥⣾⠛⠀⠀⠀⠀⠀⢴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣿⣿⣿⣿⣆⢿⣿⣿⣿⡇
⠀⢸⠀⠀⠀⢸⢻⡏⢿⠰⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠘⣆⠀⠀⠀⠀⢿⣤⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠘⡆⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠸⡄⠀⠀⢀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠇⠀⣄⣼⣿⢿⣿⣆⠀⠀⠀⠀⠀⠈⠺⢾⣇⣀⣀⣀⣠⠤⠊⠀⠀⠀⢀⣼⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢂⡼⡿⠑⠃⠻⣟⢿⣷⣤⡀⠀⠀⠀⠀⠀⠠⢤⣤⡤⠀⠀⠀⠀⣀⣴⣿⣿⡿⣻⠟⠈⠋⠻⣿⣿⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠙⠻⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠟⠋⠈⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠈⠓⠀⠉⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠈⡌⢿⣦⣐⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⡿⣦⣤⠀⠀⠀⠀⠑⢦⡈⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠈⣿⡷⣼⡟⠀⠄⠀⠀⠀⠱⣄⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡻⣶⡤⡿⣿⡟⠎⠉⢾⣷⣠⡀⠀⠀⠘⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡿⠷⠶⣴⡌⠠⣅⠀⠀⠀⢹⣿⣿⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⡆⠀⠀⠀⠋⠹⠶⢋⣀⣤⣤⡈⢳⠀⢉⣷⢧⡶⣺⣹⡿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⣀⣴⣿⣷⣿⣆⡿⣦⡹⠿⣿⣧⢹⡿⠾⠛⠁⢀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠘⠀⠀⠀⠾⠟⠉⣸⠛⠛⠋⢰⣽⣯⠈⠋⠙⠋⠀⠀⠀⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢠⣴⢆⡄⠘⠁⠀⠀⠀⠈⣿⡿⠀⠀⠀⢀⣀⡦⣀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⢦⡀⠀⠀⠟⣀⡀⠀⠀⠀⢀⡼⠛⠠⣄⠢⢴⣿⣿⣧⡿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣷⠀⠀⠐⣌⠻⢶⣤⣀⣘⠁⠀⠀⢀⣀⣨⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠸⣷⡀⠀⠉⠙⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠰⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢀⠀⠀⢢⠀⠐⠀⢸⠀⠠⠀⡠⠀⢠⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠷
⠀⠀⡀⠀⠁⢆⠀⠰⡀⠣⠀⡄⡀⠀⠃⡀⠁⡀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⢐
⠀⠀⠀⠁⠠⢄⡀⠢⡀⠄⠂⠀⡁⠀⠐⡀⠊⢁⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠰⠂⠠⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠁⠀⠂⠠⢀⠒⠀⠂⠀⠀⠀⠈⠈⠐⠂⠀⠄⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠀⠙⠂⠛
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣦⣿⣶⣴⡆⣤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣧⠷⠔⠤⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣓⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⣽⣏⣿⡿⣛⡋⢻⣿⣿⣿⣏⣏⡛⠋⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠛⠨⣿⣿⣿⡻⣞⣿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⣴⠶⠞⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠀⠀⢀⣀⣈⣁⣸⣛⡃⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠁⠀⠀⣿⣉⣉⣉⣉⣉⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⢹⠇⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢰⡞⠛⠛⠻⣆⠀⠀⣿⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⣾⡃⠀⠀⢸⡷⠶⠶⠾⣿⠀⠀⣿⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⣧⠈⠉⠛⠻⣇⠘⠷⠶⠶⠾⠋⠘⠛⠋⢰⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠙⠻⠶⢶⡀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌⠻⠶⠴⠾⠃⢤⣤⡀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣰⠆⠀⠀⠀⢰⣄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠋⠀⠀⠀⠀⠀⠹⣦⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠈⠛⠶⣤⡀⠀⣀⣴⠾⠋⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣶⣄⡈⠙⠿⣏⣀⣤⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠙⠷⠶⠟⠋⠁⠹⠆




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠚⠉⠛⠉⠙⠛⠉⠳⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⣀⠀⢀⣤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⣀⡴⠚⠉⠉⠉⠀⠀⠀⠈⠉⠋⠙⢦⣤⣄⡀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⢀⡴⣇⣀⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠻⡀⢹⡀
⠀⠀⠀⠀⠀⢸⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⡀
⠀⠀⠀⠀⢠⠏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣆⠀⢀⡰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⢸⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣸⠛⠛⠓⠚⣇⣀⣠⠤⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡾⠶⠭⣦⡀⢀⣜⣉⣠⡤⠴⢿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⡇⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠘⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣼⣀⡤⠖⠚⣇⣸⡀⠀⠀⠀⠀⠀⢹⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡟⠀⠀⠀⠉⣩⡿⡄⠀⠀⠀⠀⠘⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⣇⠀⠀⠀⠀⣤⣶⣅⠀⠀⠀⠀⠀⢹⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⡖⠒⣛⣶⠒⠲⡖⣇⣀⣸⠷⢶⣶⣶⢤⣈⣇⡴⢻⢷
⠀⠀⠀⠀⡴⠒⠒⣲⡾⣭⣿⣿⠽⠋⠙⠤⠴⠃⠀⠉⠁⠀⢯⠀⣸⡇⠈⢿⡷⣿⠾⣷⡒⠒⠒⢦
⠀⠀⠀⣸⠃⣠⠞⠁⢠⣾⣷⣛⣦⡤⢖⣦⠀⠀⣠⣤⣄⠀⢈⣭⣭⣤⠖⢋⣴⡿⠓⡆⠙⢢⡀⠈⢧
⠀⠀⣰⠃⠀⢻⡀⠀⠘⢯⡙⠻⢭⣛⣛⣛⠓⠛⠓⠛⢋⣉⣉⣠⡤⠴⠚⣩⣾⣳⠋⠁⠀⣰⠃⠀⠘⣆
⠀⢠⠇⠀⠀⠀⠙⠲⢤⣀⡉⠓⠶⠤⣬⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠒⠋⠁⠀⣉⣠⠴⠚⠁⠀⠀⠀⠸⡆
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡟⠿⠛⣿⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣴⣤⠄⠀⠀⠀⠀⠀⠀⣀⡠⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢮⣿⣦⣤⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⣐⢮⣏⢿⣆⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣮⡑⢈⠳⡇⠀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠂
⠀⠀⠀⠀⠀⢀⣿⡻⣿⣿⣯⡀⠈⡂⠈⠲⠀⢃⣠⡤⠴⠂⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠃⠆
⠀⠀⠀⠀⢀⣾⣿⣧⠘⢿⣷⡠⢈⠆⣠⡜⣠⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠠⡆
⠀⠀⠀⠀⣼⣿⡇⠀⠑⠄⠻⢿⣿⣿⣿⣿⣿⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⠛⠁⠘⠢⡀
⠀⠀⠀⠀⢹⠛⠀⠀⠈⠹⣄⠈⠋⣻⣾⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡆⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠠⣜⣧⡀⠀⠀⠀⢿⠆⢀⣿⢿⣿⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈
⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠘⠴⠁⠸⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡏⠓⠀⠀⠀⡳⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡀⠀⢻⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⢼⡎⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣡⠴⠄⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⡀⠀⠙⠓⠒⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⣉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠀⠀⠉⠉⠢⡤⠖⠒⠈⠉⠉⠐⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⣀⣀⡈⠑⠢⢄⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣀⣠⠤⠖⠛⠙⠒⠀⠀⠀⠀⠀⠚⠉⠁⠀⠉⠙⠢⠀⠀⠈⠉⠉⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢺⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⢻⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠉⣦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢧⣀⡇⠀⠀⠀⣠⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⣄⠀⠀⠀⠀⠀⠀⣟⣠⠏⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⢹⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠸⡄⢸⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡇⠀⢀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⡤⠤⢮⣀⡀⠀⢀⣀⣀⣃⠸⡀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀⠀⢸⠀⠀⠀⢠⠇⠀⠘⠆⡸
⠀⠀⠀⠀⠀⠀⠈⡗⠒⠦⠤⠭⠭⣙⣓⣤⣬⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡏⠀⢀⣠⠏⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠒⡖⠐⡖⠒⠒⠒⠲⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡚⠉⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⣦⣤⠤⠴⢧⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⠙⢶⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠼⡄⠀⠀⠀⠀⠀⢀⡤⠖⠒⠛⠒⠲⣤⢤⣀⣀⣀⣹⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⢆⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠙⢦⠤⠄⠓⠤⣄⣀⣀⣀⣠⣼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⠒⠲⠦⠤⠤⠤⠤⠬⠽⠋⠉⠉⠓⠦⢤⡀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠤⢤⣤⠤⠤⠤⠤⣝⢯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠙⢗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠦⢀⣀⠴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣠⠤⠞⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠈⠁⠉⠈⠀⠀⠐⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢀⡔⠂⠀⠀⠒⠠⢄⡀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠎⠔⠊⠁⠀⠀⠤⡀⠉⢢⡀⠀⢥⠂
⠀⠀⠀⡐⢢⠀⡀⢄⠀⢀⠀⠀⠀⠀⠀⠇⠀⡌⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉
⠀⠀⢀⠁⡌⢀⠀⡌⢀⠁⠃⠀⠀⠀⠀⡇⠀⠃⠀⠀⣤⡀⠀⣀⠀⠀⡇⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠
⠀⠀⠀⢠⠁⠀⢠⠀⡈⠘⠀⠀⠀⠀⢀⠗⢀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⣀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠐⠄⡉⠁⠐⠐⢁⠀⠀⠀⠀⠀⡎⠀⠸⣇⠀⠀⠀⠀⠄⠀⠰⠦⠀⠈⠉⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⢁⠠⠅⠠⢀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⢀⠀⠀⣁⣠⡄⢠⣏⣀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠘⡁⠀⠀⠀⠈⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠀⡼⠀⠀⡀⠈⠁⠀⡀⠀⠀⠀⠐⠣⢀⡒
⠀⠑⠀⠤⣜⠆⠈⠀⠠⠀⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠜⠀⠃⠘⢄⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠧⡀
⠊⠁⠀⡄⠛⠀⠉⠀⢃⠀⠀⠀⢀⡁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡕⡄⠈⠢⠀⠀⠀⢀⠄⠀⠋⠉⠀⠀⠆
⠀⠈⠉⢀⠃⠐⢇⠀⠈⠢⡀⠀⣸⠇⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢹⠀⠀⠖⠀⠂⠁⡰⠞⠀⠂⠀⢐⠃
⠀⠀⠀⡈⠀⠀⠈⢦⡀⠀⠈⢙⠇⠳⡀⠀⠀⠀⠀⠤⠄⠊⡁⠀⠀⠊⠀⠀⠈⡇⠀⠀⠀⡠⠈⠀⢇⣀⠈⠈⠁
⠀⠐⠒⠁⠀⠀⠀⠀⠉⠒⡤⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⡠⠊⠀⠀⠀⠀⠀⢹⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠑⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠓⠦⠤⠄⠄⠀⠲⠆⠀⠄⠀⠂⠈⠁⠉⠉⠑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠄⠂⠅⠚⠁⠀⠀⠲⠃⠄⠀⠄⠠⠀⠄⠠⠟
⣀⣀⣀⣀⣀⣈⣓⣈⣄⣉⣅⣈⣀⣃⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⠿⠿⡿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠠⠖⡡⠐⠒⠒⠠⡙⢦⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⡠⢂⠥⢂⠈⠌⢁⠠⠤⢀⠀⠀⠀⡀⠀⠀⠈⠂⢃⠀⢀⡀⢞⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡠⡪⢚⠕⠈⠀⠐⢊⡔⠂⠀⢨⡇⠢⠀⠀⠐⠄⠀⡀⠈⠀⠀⠹⡄⢣⠁⠀⠳⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠌⠔⠔⠑⠀⠀⠀⡀⠈⠂⢊⠔⠋⠈⠓⠀⠀⠈⢆⠀⠡⠀⠀⠀⠀⠰⠀⠃⠳⡘⣆⣄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⡌⠀⠀⠀⠀⠀⠁⠀⠙⢄⡉⠀⠀⠀⠆⠀⠆⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠁⠀⠀⠀⠒
⠀⠀⠀⢁⠀⣤⠀⠛⣀⡀⠀⠀⠉⠀⠀⠀⠁⠀⠈⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠙⠀⠘⠉⢼⣻⡆⠀⠀⠈⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢴⢼⠰⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⡀⠂⠀⠀⠀⠀⠀⢠⣻⠛⠀⠀⠀⠀⠀⠀⠒⢄⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠈
⠀⠀⢀⡀⠀⠄⠀⠀⠀⠈⠠⡄⠀⠇⣞⠄⠀⠂⠁⠀⠁⠐⢀⠀⠂⠐⠀⠆⠱⠠⠈⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⢄
⠀⠀⡜⠱⠀⢄⠀⠀⠀⠀⠀⠠⠖⠚⣿⢲⡦⢀⡄⢀⠄⠠⢂⠀⠀⠀⠀⡄⠃⠀⡀⠀⠀⠂⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠
⠀⠀⡇⠀⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠉⢋⠉⠅⢀⠠⢊⠔⠁⠀⠀⠀⠀⢟⠀⠰⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⠀⠀⡈
⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⢒⣀⠠⠐⠈⠀⠀⢀⡀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⢿⠀⠇⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⡆⠀⢀⠈⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠢⢤⠂⠐⢄⠀⠀⠀⠇⠄⠀⠀⠂⠃⠀⠘⠂⠃⠀⠛⠀⠀⠇⠀⠀⠀⠀⠀⢀⡤⣤⢤⡤⠄⣲⠶⡄⣀
⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠆⠀⠀⠀⠀⠀⠐⠐⠠⠒⠀⠔⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡀⠀⠀⠀⠈⠢⢬⣎⡢⠄
⠀⠀⢀⠄⠢⠎⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠈⠈⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⣠⠎⠀⢢
⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠀⠀⠀⠀⠀⠙⠲⣄⢸⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠊⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢂⡵⢶⠒⢦⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⡂⢡⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠰⠦⢴⣒⣐⠀⡠⠶⠆⠀⠄⠀⠀⡐⡀⣄⣘⠶⢞⡴⠀⠑⡄⠀⣠⠇
⠀⠀⠀⠀⠀⠘⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡜⣯⠍⠄⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠺⠀⠀⠏⠀⠀⠀⠈⡁⠀⠀⠀⡼⠜⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠀⢰⠄⠉⡄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⢄⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠛⠄⠀⢌⠀⠀⠀⠀⠀⠀⢸⠀⠀⡁⡼⠇⠀⠀⢀⡠⠒⠉⣀⠤⠀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠔⠒⠒⠶⠤⢀⠀⠄⠀⠀⠀⠀⠀⢸⠀⡏⠁⡇⠀⠠⠚⠁⠀⢀⠄⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠃⢀⠈⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠺⠄⡰⠃⡀⠰⠅⠒⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡶⠃⣀⣠⣾⣿⣿⣷⣦⡀⠠⡀⠀⠰⣶⣾⣳⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣱⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠁⠀⢻⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢡⣾⣉⣿⠇⢸⣾⣿⣿⠿⠿⠿⠿⠟⠟⠉⢿⣿⡇⠀⠈⣿⣿⡟⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣾⣿⠏⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠘⢿⣿⡮⢾⡆
⠀⠀⠀⠀⠀⠀⢠⣁⣾⣿⣿⠃⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⡄⠈⣿⣿⡏⠅
⠀⠀⠀⠀⠀⠔⢹⣿⣿⣿⠃⠀⠀⣸⢻⡃⠀⠀⢲⡆⠀⠀⠀⠀⠀⣹⡇⣿⠂⠀⣿⠀⠘⣿⣿⡀
⠀⠀⠀⠀⡪⣾⣿⣿⡿⠃⠀⠀⠐⠻⢿⡇⠀⠀⠉⠁⠀⡀⠈⠷⢤⠠⠀⣿⢹⡆⢻⡇⠀⠹⣿⣿⡘⠁
⠀⠀⠀⣾⢅⣿⣿⣿⠡⣤⣤⣌⣀⠀⠘⣷⡄⠀⠀⠀⠀⠁⢄⣀⣠⣀⣼⣿⣼⠃⢸⡇⠀⠀⢿⣿⡵⠆
⠀⠀⠀⠁⣾⣿⣿⣿⣷⣄⣉⠉⠛⢻⣿⣻⣿⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⢸⣧⠀⠀⠈⣿⣷⣆
⠀⠀⢠⠋⠝⢻⣿⣿⣿⣿⣿⠂⠀⠀⠀⠻⢿⣿⣆⠀⢍⣉⣉⣽⣿⣿⣿⡏⢠⠀⠸⣿⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⣴⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠈⠙⣻⣿⣿⣿⡇⣼⠀⠀⣿⡀⠀⠀⠀⣿⣿⣏
⠀⠀⠀⠀⣡⡾⢻⣿⣿⣿⣿⠀⠀⣠⣤⣾⣿⣿⢹⣿⣷⣶⣿⣿⡿⣿⣿⣷⡇⠀⢠⣟⣿⢣⠀⠀⢸⣿⡁
⠀⠀⠀⠀⢘⠷⣿⣿⣿⣿⡟⠀⡞⠀⢺⡖⠋⠘⠏⠉⢻⣯⠙⠟⣠⠿⢟⣵⠃⠀⠈⢿⣿⣾⠀⠰⠘⣿⡧
⠀⠀⠀⠀⠈⡃⠈⣿⣿⣿⡇⢠⠃⢠⡘⡇⠀⠀⠀⠀⠀⠈⣭⣀⣩⣴⡿⣿⠀⠀⠀⣼⣿⡟⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⣄⡶⣻⣿⣿⡇⢸⠀⢾⣿⡀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠿⠋⣿⠀⠀⠀⢻⣿⡇⠀⠀⡇⢸⣿
⠀⠀⠀⠀⢠⠏⢲⣿⣿⣿⡇⡟⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⣾⣿⡇⣼⠀⢧⠈⣿
⠀⠀⠀⠀⢠⡴⣿⣿⣿⣿⡇⠇⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⣿⡟⢸⣿⡀⢸⡆⣿
⠀⠀⠀⠀⠈⠀⢹⣿⣿⣿⡿⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣻⢠⣿⣿⡇⠘⣇⢹
⠀⠀⠀⢠⣤⣀⣾⣿⣿⣿⡇⢰⣿⡟⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣦⣾⡇⠀⠀⣰⢿⣾⣿⡟⠁⠀⢈⣶
⠀⠀⠀⢠⡾⢛⣵⣿⣿⡟⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⠀⠀⠀⢹⣾⣿⡟⠀⠀⠀⠘⠉
⠀⠀⠀⠈⣾⣻⣿⣿⣿⠁⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⢸⣿⡿⠁⢀
⠀⠀⠀⠔⢡⣿⣿⡿⡿⢧⣿⡟⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣹⠀⠀⣘⣾⣿⠁⠀⢿⣿⣶⡄
⠀⠀⠀⣠⣿⣿⡿⠁⠁⣸⠏⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⣦⣴⣿⣿⠀⠀⣿⣿⣇⠀⠀⢺⣿⣿⣿⣄
⠀⣼⣿⣿⣿⠟⣡⡆⢸⡟⠀⠀⢨⣿⣿⣧⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠀⢰⣿⣿⣿⣦⣀⣾⣿⣿⣿⣿
⢀⣼⣿⣿⣧⣾⢿⣶⣾⠃⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⢠⣀⣿⣿⣿⣿⣗⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠹
⣿⣿⣿⢫⣾⠿⣷⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣤⣀⣸⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⢿⡿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄
⠁⠻⠞⠗⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡉⠉⠉⠛⠶⢄⣁⢘⣥⡄⠀⠉⢙⣿⣿⣿⣿
⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⣀⣀⣀⣡⣡⣤⣀⡀⠈⠙⣿⣿
⠀⠈⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠷⠄⠠⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡢⡀⡠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡤⠀⠀⢠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⡰⠀⠈⢀⢠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁⠀⠀⠸⣈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⠀⠀⠴⠂⠀⠀⠀⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠴⢤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣯⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠇⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⢦⣀⡄⢀⡄⠐⣦⠴⣄⢠⠖⠀⢠⢠⡀⢀⣄⢀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠏⣇⣸⢇⡄⣸⢀⡿⠻⣄⡤⢸⡟⢴⠘⣏⣸⠇⠠
⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠁⠀⠀⠁⠀⠁⠀⠀⣗⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠁⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠆⠀⡏⠀⠀⠰⣟⡀⠛
⠀⠀⠀⠀⠀⠣⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡦⠀⠀⠀⠀⠀⠴⠀⠆⠂⠑⠃⠀⠠⣾⣦⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠰⡿⠇⠄⢠⣆⣠⣴⠄⠀⠀⢴⣷⠈⠀⣤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠂⠀⠀⣀⣀⣀⣀⣉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠆⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠰⠀⠀⡰⢀⣀⡈⠁⢀⠤⠁⠤⠠⠌⠁⠂⠛⠄⠀⠀⠀⢀⣀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢀⣀⡔⣠⣦⠔⠿⠆⠁⠀⢠⠄⠀⠀⠀⢈⠀⡅⠀⠀⠀⠛⠁⠢⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠒⠛⠏⡰⠿⠅⠀⠀⠁⢀⠀⠈⠠⠀⠀⠈⠹⠸⡄⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠣⠀⠀⠘⠁⠀⠀⠂⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠛⠓⢤⣀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠤⡀⠀⠀⠀⠈⠳⡀
⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠈⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡖⠤⢄⣀⣀⣀⣀⣀⣀⠤⠆⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⢁⠁
⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⡺
⠢⣀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣬⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⡏⠗⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠓⠂⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⠏⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡠⠃
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⢸⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢲⠏
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⡌⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠉


⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⣀⣀⣀⣀⣀⣀⣀⣈⡓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠒⠤⣍⠳⣄⠀⠀⠀⠀⢠⠀⠈⠀⠀⠙⢲
⠀⠀⠀⠁⠀⠀⠀⢀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⡆⠀⠀⠀⢸⠉⠁⠐⠒⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠈⠑⠒⠦⠤⠤⠀⢀⣀⣀⠀⠀⠤⠤⠄⠀⠒⠁⣀⠜⠀⠀⠀⠀⢸⢀⠀⠀⠀⢈⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠤⠄⣀⣀⣀⣀⣀⡀⠠⠤⠄⠒⠒⠈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣛⣭⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣿⣿⡿⠿⠿⢟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣀⡀⡀⠈⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⢻⠿⡤⠀⠀⠀⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣑⡒⠊⢀⡀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡓⠀⠉⠀⠀⢀⣤⠂⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢶⠿⣿⣧⣀⣀⣠⣴⠟⠁⢀⠀⢸⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢿⡇⠙⣿⡍⠁⢀⡄⠀⠀⡎⠀⠈⣏⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠟⠀⠀⢀⠀⠀⠀⠀⡀⠸⠇⠀⠈⣧⣤⠈⠀⠀⢸⠃⠀⠀⣿⠀⠀⠉⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡁⠀⠀⠀⠀⠀⠉⠀⠺⠁⠀⣾⠠⠞⣿⣿⠢⡀⠀⡌⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠋⠑⣄⠀⠀⠰⠀⠀⠀⡄⠀⠀⠐⠀⠀⢿⡄⢀⣿⣷⡀⠀⢰⠃⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠼⠃⠀⢀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡇⠀⡎⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⠂⠀⠈⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢰⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⣄⠉⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⡜⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣞⡼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⡼⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⡰⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠰⠁⠀⠀⠀⠀⣧⠀⣀⣀⣤⣤⣄⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⡋⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣠⡼⠟⣿⠀⠀⠀⢰⡿⠀⠀⠀⠀⠸⠟⢠⡴⢀⣠⡄⠀⠀⢸⡇⠈⣿⣿⣿⡟⣼⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣛⣉⣀⣀⣀⣿⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠰⣿⣅⡉⠁⠀⠀⠀⡼⠧⠤⠬⠷⢤⡾⠿⠦⠴⠷⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢧⣀⡈⠓⠶⠶⠶⠶⠒⠀⠀⠀⠊⣉⣁⡤⣤⠤⠊⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠷⠿⠿⠿⠿⠛⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠄⠠⠴⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⢀⣀⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡀⠘⠛⡁⢉⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⢳⡌⣂⣱⣀⣩⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠖⠺⠏⠉⠁⠀⠀⠀⠈⠉⠉⠉⠛⠋⠙⠓⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⢠⡞⠑⠢⢤⣀
⠀⠀⠀⠀⠀⡠⠚⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡄⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠲⢦⡀
⠀⠀⠀⠀⠐⣇⠉⠘⡆⠀⠀⠀⣀⢀⣠⠴⠦⠤⠤⠤⠶⠒⠮⠤⣀⣀⣀⣀⡀⠄⠀⠀⠀⠀⢹⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⢆⢠⡧⠤⠶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡴⠋⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠈⢳⡙⣄⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠈⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⢢⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⡀⡀⢰⣦⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠱⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠨⡤⠆⠀⠀⠀⢀⠦⠀⣸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡐⡄⠈⠳⣶⠀⢤⠄⠀⣀⠀⢠⡤⠤⠔⢲⠟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡟⢰⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢆⠀⠀⠙⠶⢷⣶⣾⣶⣶⣶⠶⡿⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣠⠧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠎⢂⠀⠀⠀⠀⠈⠉⠁⠒⠚⠀⠀⠀⠀⠀⢸⠳⢄⣀⣀⣀⣀⡼⢠⡦⠵⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⣀⣠⣄⢣⠀⠀⠊⠀⠀⠀⠀⢰⡀⠀⢆⠀⠀⡿⣤⣀⣈⣉⣉⠥⠒⡿⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡴⠋⠁⠀⣨⣦⣂⠘⢄⣀⣀⠆⠀⠰⢁⠒⠊⠀⣸⠁⠀⠈⠉⠀⠀⠀⢰⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣙⠢⠤⠜⠁⠘⢋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡞⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠶⠶⠛⣇⠀⠀⠀⢀⡄⠠⡄⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢰⠇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡄⠀⢨⠀⠀⢈⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⡀⠈⠒⠒⢁⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡵⠀⠀⢀⡞⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡎⢱⡄⠀⣼⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡞⠀⠈⢿⡼⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⢠⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡼⠀⠀⠀⠘⠃⠸⡇⡏⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢰⠇⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣚⣁⡼⠀⠀⠀⠀⠀⠀⠀⠧⠬⠭⠭⠗⠀⠀⠀⠈⠣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠔⠈⠀⠀⠁⠐⠤⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⡆⠘⠉⠁⠀⠀⠲⠄⠈⠢⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡌⡅⠀⠀⡄⢀⢄⠀⠀⠀⠀⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠰⠀⠀⠰⠀⠀⡌⠈⠀⠀⠀⠀⠃⠠⠃⠀⠀⠶⠆⠀⠤⠀⠈⣇⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠆⠰⣀⠃⠀⢠⠀⠀⠀⠀⠀⢀⠌⢁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡤⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠈⠰⠀⠀⠄⠠⠈⠀⠀⠀⠀⡌⠀⠀⠳⡄⢀⠀⠀⠀⠄⠙⠃⢀⡀⢀⠀⠔⠀⠄
⠀⠀⠀⢤⠀⠈⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣤⣤⣤⣶⣾⠃⡸⠂⠂⠀⠤⠀⠀⠀⠀⠀⠀⣸⠀⢀⠃
⠀⢄⠀⠀⣄⡀⠠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠘⠛⠛⢿⣿⠃⠰⢳⠄⢤⠀⠀⠀⠐⠀⡀⠀⠀⠀⠐⠎
⡀⠄⠀⡉⡗⠀⠄⢠⠁⠀⠀⠀⠰⠉⢧⠀⠀⠀⠀⠀⠀⠀⠁⡐⠁⠀⢀⠈⢢⡀⠀⠀⠀⠀⠀⠳⠤⠔⠉⡂
⠀⠠⠄⠡⡄⢠⠀⠀⢢⠀⠀⠀⡇⠀⠀⠑⠄⡀⠀⠀⠀⠀⠈⠀⠀⡘⠈⣆⠀⢈⡆⢀⠀⠁⣠⠀⠀⠀⠀⡌
⠀⠀⠀⠐⠀⠀⠣⡀⠀⠁⠂⢼⠱⡀⠀⠀⠀⢀⣀⣀⠤⠂⠀⠀⠔⠁⠀⢸⠄⠀⠀⠀⢀⠔⢐⠀⢰⠀⠂
⠀⠠⠠⠁⠀⠀⠀⠙⠢⢄⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠐⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠙⠐⠂⠒⠀⠙⠃⡀⠂⠀⠀⠀⠀⠀⠀⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡀⣀⢀⣀⡀⡀⠤⠀⣒⠀⠠⠐⠀⠀⠁⠀⠀⠀⠘⠀⠃⠀⠀⠉⠀⠀⠊
⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣬⣿⣤⣍⣽⣭⣽⣫⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠒⠶⠋⠙⠚⠉⠙⠚⠋⠙⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⣠⠦⠤⠴⠞⠲⠶⠛⢦⣄⣠⢤⡀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠲⣄⡀⢀⣸⠷⣄
⠀⠀⠀⠀⠀⠀⢀⡤⢿⣀⣰⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢧⡀⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠃⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠹⡄
⠀⠀⠀⠀⠀⢀⡟⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡦⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⠃
⠀⠀⠀⠀⠀⠸⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠧⣤⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣴⣤⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡾⣿⣿⠿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⣼⢿⣿⣿⣯⡽⡇⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣟⠲⠾⡀⠀⠀⠀⣰⠷⠖⠛⠉⣩⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢹⠋⠙⠓⠻⢦⢀⡾⠧⠶⠶⠒⠋⠉⢿⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡿⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⣼⣇⣠⠴⠖⠛⢻⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢋⣩⣇⡀⠀⠀⠀⠀⠀⠸⡆⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢈⣉⣼⠃⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣸⣇⣀⣀⣀⣀⣼⠋⠉⢳⣄⠀⢀⣀⡀⠀⠸⡆⢀⣤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣰⣇⣨⣭⡽⣏⠀⠈⣿⠙⠦⠤⠞⠉⣩⠟⠻⢯⠛⢶⣷⣞⢁⡟⢳⡀
⠀⠀⠀⠀⢠⡟⠉⠛⣉⡿⠿⣶⠿⣿⡖⠋⠀⠙⠒⠚⠃⠀⠀⠀⠀⠀⢻⣄⣀⡼⠀⠀⢹⠞⢻⣟⠻⢿⣝⠉⠉⠙⢧
⠀⠀⠀⢀⡞⠀⣠⠞⠁⠀⣴⣾⣿⣉⣧⣤⠴⢛⣧⠀⠀⣠⣤⣤⣄⠀⠀⣠⣴⢤⣤⠟⢋⣠⣿⠟⢒⡆⠈⠳⡄⠀⠘⣇
⠀⠀⠀⡾⠁⠀⢿⡀⠀⠀⠻⣍⠛⠯⣝⣓⡛⠻⠿⠶⠾⠿⠒⠚⠛⠛⠛⢁⣀⣠⡤⠖⢋⣴⡛⣶⠋⠁⠀⢀⡿⠀⠀⠸⣆
⠀⠀⣼⠁⠀⠀⠀⠛⢦⣄⡀⠈⠓⠦⣤⣉⣉⠛⠛⠓⠒⠒⠒⠒⠛⠛⠛⢉⣉⣤⠴⠞⠋⠀⠙⠛⠀⣠⡴⠛⠀⠀⠀⠀⠹⡄
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣌⣉⣉⣛⠛⠛⠓⠚⠛⠛⠛⠉⠉⣁⣀⣀⣠⣤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠾⢿⣿⣧⣤⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠖⠛⠃⠀⠀⠒⠓⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣶⠀⠀⠀⣶⠄⠀⠀⢠⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⠤⠤⣄⡀⢀⠃⢀⡀⠂⠀⠀⠁⠀⣀⠀⠀⠰⠁⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⣧⣀⣀⣈⣀⣤⡼⠁⠀⠠⡾
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣿⠟⠻⠛⠿⠁⠀⠀⣆⢧⡞⢄
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠂⠠⢠⠧⡆⠀⠀⠉⠁⠀⠀⠀⣀⠈⠁⠀⢹⠈
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠣⣸⡦⣀⠀⣠⠖⢀⠜⠋⠀⠀⠀⢠⠓⠂
⠀⠀⠀⠀⠘⠦⣄⣀⣀⢀⣀⣀⣠⠤⠖⣷⣶⡄⠀⠀⠀⠀⠉⠂⠎⠁⢁⡔⠀⠠⠤⡀⠀⠀⠸
⠀⠀⠀⠑⢦⣶⠻⠷⠿⠿⠿⠿⠛⠛⠛⣉⣠⡶⠲⠄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢳⣄⠇
⠀⠀⠀⠈⢦⡭⠛⠷⢶⣖⣂⡨⣌⠩⠙⠉⠀⣀⠤⠊⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠓⠁
⠀⠀⠀⠀⠀⠉⠐⢆⡀⠤⠤⠤⠤⠤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠁⠒⠠⠤⠤⠤⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠉⠉⠒⠒⠀⠿⠧⠄⢰⠊⠁⣀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⣄⡀⣆⡴⢡⠊⠉⠁⠺⠁⠀⠑⠄⢻⠀⠀⣽⣀⠃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡀⠀⠛⠁⠀⠰⡀⠻⠂⣠⣴⣿⣠⠀⠀⠛⢀⣀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒
⠀⠀⠀⢰⠀⡆⠀⡎⢱⠀⡔⢤⠻⣄⠀⠈⠛⢀⣤⡖⠀⠈⠀⠈⠀⠀⠀⠱⣀⣀⠀⣈⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀
⠀⠀⠀⡆⢰⠀⢰⠀⡆⢠⠁⡜⠀⠈⠁⢺⠃⠈⠋⢅⠀⠀⠀⠀⠂⠠⠀⠊⠀⡽⠷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡌
⠀⠀⠸⠀⢆⣀⠇⢰⠀⡌⢠⠃⠀⠀⠀⢸⠀⢸⣷⣤⣑⠂⠠⠀⠀⠀⠀⣐⣊⣴⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⠁
⠀⠀⠀⠑⢤⠀⠀⡈⢈⣡⠎⠀⠀⠀⠀⠸⡆⠀⢿⡿⠛⠿⠿⠛⠻⢿⣿⣿⣿⠏⠀⣸⠃⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡆
⠀⠀⠀⠀⡜⡠⠤⠇⠠⢄⠀⠀⠀⠀⠀⠀⠹⣆⡀⠑⠀⠒⠂⠀⠉⠘⠛⠿⠃⢀⡴⠧⡄⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰
⠀⡐⠈⠉⢏⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠹⢶⡤⠤⠄⠀⣀⠠⣤⠄⠶⡏⠀⠀⢈⠶⠛⠓⠢⡀⠀⠀⡠⠒⠐⠯⣀⢠⠇
⢀⠓⠤⣀⣈⢶⠆⠒⠁⢀⠆⠀⠀⠀⠀⢀⠖⠁⠀⠚⠦⠀⣀⣀⣀⡀⠿⠀⠀⠈⢦⠀⠺⡁⠀⠀⠀⠁⠀⠐⠀⢠⠀⠀⠀⢹⡀
⡜⠢⢀⣀⡠⡏⠀⠒⠠⡆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⢄⠀⠀⠀⠀⢰⡀⠈⡷⠒⠊⠁⣸
⠁⠢⡤⠜⢠⡁⣠⡄⠀⠘⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⣧⠀⠀⢹⡂⢀⡠⠊⢰⣠⠀⣉⠀⢉⣠
⠀⢠⠁⠀⡌⠀⠀⢳⡀⠀⠈⠓⠤⢠⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⡰⠃⠀⢸⡆⠐⠁⠀⠀⠀⡠⠂⠅⠀⢢⠤⠤⠊
⠀⠌⠀⢠⠁⠀⠀⠀⠙⢦⡀⠀⢀⡜⠀⠙⣄⠀⠀⠀⠈⠉⠉⠉⣀⠴⠂⡠⠊⠀⠀⠀⠀⣧⣀⣀⡀⠤⠊⠁⠀⠢⠤⠆
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⣱⠏⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠁⡀⠔⠁⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠓⠄⠠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠉⠁⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠦⣤⠀⣀⠀⠀⠤⣦⡤⠤⠤⠤⠤⠒⠺⢩⡛⢓⡦⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡒⠞⠛⢥⡐⠃⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠊⠠⠐⠈⠲⡀⠀⠀⠀⠀⣀⠠⢔⣿⣠⠤⠄⠀⣰⣃⣀⢀⡀⠀⣀⠑⣀⣠⡆
⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣠⣀⠀⡤⠤⣄⣚⣂⠤⠭⠑⠂⠉⠀⠀⠀⠉⠉⠁⠑⠒⠒⠚⠂⠘⠁⠚⠓⠋

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⢀⠔⠊⠙⢄⡠⠒⠉⠉⠉⡽⠲⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠁⠀⠀⡴⠋⠑⠦⣀⠀⡜⠀⠀⢹⠳⡀
⠀⠀⠀⠀⠀⠰⡇⠀⡸⠁⠉⢲⣞⡀⠀⠀⠀⢈⠟⠢⢄⡀⢸⠀⢱
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢀⡎⠀⠈⠑⠦⣠⠎⠀⠀⠀⠈⡹⠓⠚⡇
⠀⠀⠀⠀⠀⣀⡇⠀⡷⠦⢼⣄⡀⠀⠀⢠⠋⠉⠲⢄⡀⢰⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠹⡀⣏⠀⠈⠑⡲⢇⡀⠀⠀⠀⡸⠓⠦⣀⡼⠀⠀⠀⠀⠀⠀⡰⠁⠈⠢
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠱⣼⡀⠀⡴⠁⠀⠙⠢⣠⠞⠁⠀⢀⠜⠁⠀⠀⠀⠀⠀⡜⠁⢠
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠑⠾⣁⣀⡠⠔⠊⠉⣢⡤⠖⠁⠀⠀⠀⢀⠀⢀⠞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡴⠊⠁⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡰⠋⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢄⡀⠀⢀⠜⠁⠈⠉⠋⠀⠀⡇⠀⠀⢸⠀⠠⠤⠤⠄⠀⡇
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠞⠁⠀⠉⠒⠃⠀⢰⠀⠀⡄⠀⠀⡇⠀⠀⢸⠀⠀⠶⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⡴⠃⠀⢰⠀⠀⢠⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⡞⠒⠒⠒⠒⠒⠒⢷
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⠋⠙⢦⠀⠀⠀⠀⢀⣠⠴⠛⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠈⢧⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⢤⣤⣼⣧⣤⣤⡶⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣠⡰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢁⣈⡀⠁⠁⡬⣄⢳⡀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠘⣿⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠛⢁⡀⠀⠀⢀⣀⠀⢷⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢻⣇⠨⠅⠰⠆⠴⠀⠀⠀⠀⡿⠀⠀⠀⠉⡍⠀⠰⡠⢌⠀⢸⡦⣄⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠻⣦⡈⠥⣘⠤⢹⣬⡷⣚⠓⠀⠀⠀⠀⣀⠀⣤⡽⠈⠀⠀⡈⣿⠶⠋⢀⡠⡔⣪⢗⣉⠵⢒⣒⣆
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣈⠐⠸⣇⡘⡁⠀⠠⢲⢺⡧⣀⣀⣠⠴⡁⠀⣿⠁⠀⡠⠞⠚⠓⠛⢤⡴⠎⠁⣀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠙⠓⠶⢭⣽⡄⠀⠈⠄⢿⣶⣶⣶⣾⡇⠀⣿⢠⡞⠀⠀⠀⠀⠀⠘⠲⠒⠋⠉⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⡛⢛⡛⢓⣦⡀⠹⣦⡀⠀⠈⠙⠿⠿⠋⠀⣼⠇⢞⣳⢤⣄⣀⡀⠀⠀⠀⠀⣀⣀⡼⢢⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡰⠐⡓⢺⣿⢋⠽⣮⠙⠲⠤⢤⣤⠤⢴⣟⠁⠀⠈⠻⡿⢿⣯⣭⣿⣭⣭⣽⣿⢶⡿⠋⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣚⡿⠟⣦⡿⠻⠶⢤⣤⣤⠶⣞⣹⡾⠀⠀⠀⠀⠉⠓⠒⠿⠯⢤⡤⠶⠒⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠥⢨⣸⠟⣷⡡⠜⣃⠲⡹⠖⢻⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡶⠋⠁⠀⠘⢷⣔⠂⠬⢁⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⡀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⡧⠀⠈⣥⣿⡳⣶⠀⠀⠈⣻⡿⠃⠀⣀⣿⣒⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣧⠽⢿⣳⢀⢣⣿⠋⡡⣐⣠⢰⣾⣿⣿⣶⣶⣿⣿⢻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⡿⣅⢠⢴⣽⡾⢾⣟⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣿⣿⣿⣟⣻⠖⠐⠾⠓⠲⣤⣤⡄⡀⣰⡖⠀⠀⠀⠀⠀⠴⠋⠁⠀⢠⣴⣶⣏⢠⣾⣿⣿⣿⣿⣿⣿⣿⡟⣽⡿⢿⠛⠛⠻⠦⠻⣿⣿⣿⣷⠂
⠀⣿⣿⣿⣿⠿⡧⠲⣶⢶⣤⣿⣿⡛⢃⣿⣯⣶⣧⣤⡄⠺⠀⡀⣀⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡀⡟⢠⣠⣶⣶⣶⣦⣼⣿⡿⣿⣿⡆
⢀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣈⠀⠀⠉⣿⣿⣿⣿⡿⢿⡤⠀⣾⣿⠀⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣃⣷⣿⣿⣿⣿⣹⢿⣿⡿⢶⣿⣿⡆
⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⠟⣣⣄⣐⢁⣾⡿⣁⣀⣾⠃⠈⢼⣿⣷⡈⢻⡟⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⠌⢹⣿⠃⣼⣿⣿⣿
⠀⠙⠛⠉⠽⠛⣛⢛⠋⠁⠀⠀⠀⢸⣿⣿⣿⣿⣷⡟⠀⠀⢸⢿⡻⣿⣆⡹⠿⣿⣿⣿⣿⣿⣿⡹⢿⣾⡿⠙⢻⣿⣿⡽⣿⣀⣘⣷⣱⣿⣿⣿⠇
⠀⠀⠀⠀⢠⣿⣿⡇⠀⢤⣼⣷⣾⡀⣿⣿⣿⣿⡟⢷⣴⠛⣎⣸⡟⣿⣿⡇⠀⣤⣤⣴⡀⢀⣼⣷⣿⡿⠷⡦⢾⣝⠋⠁⢸⣿⣿⣿⣿⣼⣿⣿⡆
⠀⣰⣴⣿⣿⣿⣿⡇⠀⠀⣺⣻⣿⣧⣸⣿⣿⣿⣿⣾⠁⣼⣿⣿⠻⣿⣿⡗⢠⣴⣿⣿⡇⣸⣿⣷⠖⣠⡤⣿⣿⣿⣀⣼⢿⣇⣿⣿⡿⣼⣿⣿⡇
⠀⠙⠋⠉⢿⣋⡿⠃⠀⠀⢀⣥⣼⠏⣼⣿⣿⣿⣟⡿⢰⣿⣿⣿⠀⣿⣿⡁⠀⢸⣿⣿⠁⠀⣯⣿⣿⣿⣷⢿⣿⣿⣿⣿⣾⣿⣿⣿⡷⢻⠿⣿⠟
⠀⠀⠒⠶⣿⣿⣿⣿⣇⣀⣤⣿⣡⣤⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⠰⢿⣿⣿⢣⣿⣿⣿⠏⣿⢻⣿⣿⣿⣿⡷⠻⠿⢟⣵⣿⣿⣿⣿⡇⢈⣤⢿⠤
⠀⠀⣲⣆⣿⣿⣻⣿⠿⢷⣏⣙⣻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣟⠀⣀⣀⣀⣀⠀⠀⠈⣘⣯⣄⣀⡁⠀⠈⡽⢩⡙⠙⠓⠺⠔⠘⠛
⠀⠐⠛⣿⣟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣟⢋⡋⢁⡀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂
⠀⠀⠀⠈⠉⠉⠐⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠟⠻⠿⠟⠿⠿⠛⠛⠿⠿⠿⠟⠿⠽⠿⠟⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠄⠀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢌⡔⠊⣠⣶⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠏⠀⢘⡕⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠘⠁⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠐⡄⠀⠀⠀⠰⠀⡠⠄⡠⠐⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠃⣠⠈⢢⠀⠀⠀⢲⠀⠠⣅⠀⠀⣠⢃⠠⠄⠒⠒⠈⠁⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠐⠀⢈⢹⣧⠀⠀⠀⠂⡼⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠔⡒⢆⡉⢰⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠠⠂⠈⡀⠀⢠⠒⠖⠠⠤⢤⠠⣄⢀⡈⣙⣋⡉⠉⠀⠀⠀⣀⠰⠇⡾⠟⠣⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠓⠶⠤⠴⠄⠤⠄⠥⠠⠃⠀⠀⣀⡤⢎⣀⣀⣀⣀⡀⣀⣀⣀⡀⣀⣈⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠥⠠⢍⡹⠯⠸⣁⡆⢺⡟⠳⠿⣿⡉⠶⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢻⣮⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢤⡤⠶⠚⠛⠛⠉⠙⠚⠾⠭⠶⣶⣿⣽⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⣿⣇⣃⡀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⣤⣀⣀⣀⣀⡀⠠⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢩⠈⠀⠀⠋⣨⠁⢃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡆⢰⠉⢤⠌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠘⡀⠘⡁⢡⠌⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠣⢄⡉⢳⠀⠑⠊⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡠⠊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠈⠐⠣⢀⡀⠀⠀⡐⠤⠀⠠⢄⡀⠔⢊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠄⡀⠈⢹⢂⣍⠈⠁⠀⠈⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠒⠼⠀⠛⢃⠀⢠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢐⡀⠀⠀⠀⠀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠠⠐⣄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠴⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⢁⠀⠀⠀⠀⠀⠀⠒⢺⠞⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠉⠿⠀⠐⠒⠒⠒⠒⠂⠀⠘⠒⠒⠾⠿⠽⠧⠤⠤⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⡿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠉⣹⣿⣿⣿⠏⠉⢉⣉⣉⣉⣉⣉⣉⣉⠉⠉
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠋⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠸⠿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡴⠂
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠿⠿⢿⣿⣛⣛⠛⠉⣉⣉⣉⣉⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⣷⣶⣦⣤⣤⣀⣀⡀⠈⣿⣿⣿⣿⣿⠿⠿⣿⠿⠛⣿⡿⠛⠛⡏⠉⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢨⡍⣙⢫⡉⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡗⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⢻⡇⣶⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⢀⣀⣿⣧⣀⣀⣇⣤⣼⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⣿⡇⢹⣿⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⢛⠋⠀⣤⣄⠀⢀⡅⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢻⣱⣎⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠐⠀⢶⡿⠀⠐⡇⠀⢺⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⢹⣿⣿⢻⣷⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢀⣸⣀⡤⣼⣷⠒⠒⡏⠉⢹⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⢹⠀⠀⢸⡇⠀⠀⡇⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡿⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠀⠀⢸⣇⣀⣀⣇⣤⡼
⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⡛⠿⣿⣿⣿⣿⣿⣿⣠⣤⣼⣶⣶⣿⣿⣿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠛⠿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠃⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠈⠐⠒⠚⠃⠀⠈⠙⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢁⣢⠀⢀⠀⠐⡆⠀⠀⡀⠤⠀⣀⡒⠂⠡⠔⠓⡌⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⢀⣈⣉⠙⠃⠀⠈⠉⠁⠀⠀⢀⡀⣀⠠⠴⢞⠋⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠁⠀⠀⠈⠉⡄⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⠈⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⢬⠀⠀⠐⠒⠒⢶⠀⠀⠒⠒⠒⠉⠉⢉⣀⡀⠠⠇⠂⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠺⢀⣉⠁⠂⠐⢻⠀⠒⠒⠒⠂⠉⠉⢁⣀⣀⡠⠔⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⠲⠤⢀⠈⠉⠉⠙⠀⠀⠀⠀⠐⠊⠉⠁⢀⡨⠷⠄⢀⠄
⠀⠀⠀⠀⠲⣄⠀⠀⠠⠴⠀⠢⣑⣢⠔⠈⠉⣉⢘⠀⠒⠒⠒⠛⢉⣉⣉⡟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠋⠈⠉⠉⠉⠉⠉⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⠁⠀⠙⢦⣀⣠⣤⠤⠤⠤⠤⠤⠤⣤⣠⣞⠁⠀⠈⢧⣳
⠀⠀⠀⠀⠀⠀⣀⡴⠖⢛⣿⣿⢃⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣈⡿⠿⡟⠲⢤⡀
⠀⠀⠀⠀⢠⣞⣁⡤⢴⣾⣄⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠿⣗⡲⠽⣧⡀
⠀⠀⠀⠀⡟⣿⠶⠚⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠉⢙⡟⣷
⠀⠀⠀⠐⡇⠈⢧⡀⠀⡼⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠏⠀⣸
⠀⠀⠀⠀⣇⠀⠀⠉⣳⠃⠀⠀⠀⠀⠀⣸⠙⢦⣄⡀⠘⢳⣤⣀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⣿
⠀⠀⠀⠀⢻⠀⣠⠴⠃⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠉⣉⡉⠁⠈⣉⡛⠛⠛⠋⠈⠓⢦⣀⠀⠀⠀⠀⣈⡿⢠⡇
⠀⠀⠀⠀⠘⣇⠙⣶⣤⡤⠤⠶⠚⠋⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⣠⡉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⠹⡄⣼
⠀⠀⠀⠀⠀⢹⣾⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡤⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠇
⠀⠀⠀⠀⠀⢘⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⣀⠀⢀⡼⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣄⣀⣀⣴⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣆⠀⠀⡀
⠀⠀⠙⢯⡉⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠈⢉⡿⠃
⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⢰⣯⢷⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠳⢤⣀⠀⠀⠀⢸⣿⣭⣭⠽⣾⡇⠀⠀⠀⠀⠀⢀⣠⠖⠋⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠦⣼⡘⠛⠛⠛⠋⣇⣠⣤⠤⠶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢺⡟⠶⢤⣷⣄⠀⣠⣞⣁⣀⣤⠤⠖⢻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⣸⠃⠀⠀⠀⢨⡄⡏⠉⠁⠀⠀⠀⠀⠘⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡟⠀⣀⣠⡤⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠈⠛⢳⡞⠉⠀⠀⠸⠷⠷⣆⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠞⢛⡦⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡾⠿⢥⡀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢙⡟⠒⠒⢛⣶⠚⠛⢶⠺⣇⠀⣠⡷⠶⣶⣶⣶⢤⣄⣻⣠⠞⢹⣿⡀
⠀⠀⠀⠀⠀⣤⠤⠤⠤⢤⣿⣙⣾⢷⣟⡿⠏⠻⣄⣠⡾⠀⠈⠉⠉⠀⢸⡋⠀⢙⡇⠈⢻⣿⣶⣿⣤⣷⠤⠤⠤⠤⣤
⠀⠀⠀⠀⣸⠃⠀⣠⠞⠋⢀⣹⣯⠿⣇⠀⢀⣤⣄⠀⠀⠀⢀⣀⠀⠀⠈⠓⢒⡚⠁⣠⡴⠋⣸⣻⣦⣌⠛⢦⡀⠀⠈⣧
⠀⠀⠀⢰⠇⠀⣼⠁⠀⠀⢸⠻⢿⡶⠿⣭⣭⣴⣿⣄⣠⣼⣛⣛⣿⣦⣤⠶⠛⠉⠛⣃⡤⠞⣻⣏⣠⠟⠀⠀⢹⡄⠀⠘⣆
⠀⠀⢠⠏⠀⠀⠘⢦⡀⠀⠈⠳⣤⣉⠛⠲⠦⢤⣭⣭⣉⣀⣀⣀⣤⣤⡤⠴⠶⠚⠋⣉⡤⠞⠧⣽⠀⠀⠀⣠⠞⠀⠀⠀⠹⡆
⠀⢀⡟⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⡉⠙⠒⠶⠤⣤⣤⣤⣄⣀⣤⣤⣤⡤⠴⠶⠚⠋⠁⠀⠀⣀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⢻⡄
⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡠⠋⠀⠀⡄⡀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⡑⠀⠀⠀⠠⠀⠀⠁⠊⠉⠛⠳⠶⢶⣦⠤⢤⡀⠀⢒⠍⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠁⠀⠀⢀⠐⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠌⢂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁⠀⠀⠀⠀⠁⠀⠀⢀⠄⠄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢡⠀⠀⠀⠡⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡳⠀⠀⠀⠔⢃⠀⠀⠀⠀⠀⠐⠗⠂⠀⠀⠀⠠⠸⢂⠅⠀⠀⢸⣀⠖⢄⠀⠐⣬⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⢢⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡘⠀⠀⡠⠖⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣀⠀⠘⢄⠀⠀⠈⠂⠘⡀⠀⠀⠀⠀⡏⠉⠀⠀⠈⠉⡀⠀⠀⠀⠀⡐⠀⠀⠁⠀⢀⠅⠀⠀⣀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠌⠀⠑⢄⠀⠁⠢⡀⠀⠀⠐⠄⠀⠀⠀⠐⠤⠄⢀⢠⠔⠀⠀⠀⠀⡐⠀⠀⠀⡠⠀⠁⢀⠄⠁⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠡⡀⠀⢈⠖⠈⠢⡀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠂⢤⠊⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲⢁⠔⠈⠀⠈⠢⡀⠁⠀⡤⠠⠠⠄⡄⠀⠁⡠⠊⠀⠑⢄⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢁⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⢽⠁⠈⠀⠀⠀⠀⠀⠑⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠃⠐⠐⠨⠊⠀⡶⠦⢀⡀⣠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⣦⠐⠀⠒⠀⠂⠎⠀⢦⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠐⠈⠁⠀⠀⠀⠀⠀⠀⢻⡄⠁⠸⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠠⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠀⠱⣖⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⣴⠀⢶⡄⠀⢸⡆⠀⠠⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠒⠈⠙⠀⠉⠙⠠⡀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠉⠉⠛⠈⠀⠈⠠⠀⠀⠀⠈⢄⠀⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⠠⠀⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠡⡀⠀⠀⠀⠈⠁⠐⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⢸⡗⠀⠈⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠺⢦⣀⠀⠀⡠⠊⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⣠⡀⢀⣬⠏⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠉⠙⠓⠤⠤⠀⣠⣄⠁⠀⣄⣀⢀⣀⣠⣄⣀⡠⠤⠶⠛⠋⠉⠁⡀⠠⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣋⣻⣷⣆⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣴⣾⢋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⢿⡛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣛⣿⡛⠓⠟⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⢼⣿⣿⣿⡟⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⢉⡉⠛⠛⠛⠋⠉⠙⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⡄
⠀⠀⢰⣿⣿⣿⣿⡇⢻⡟⠙⣸⣷⣶⢿⡇⠀⠀⣾⣋⣙⠃⢀⡴⠿⣦⢸⣏⣹⠆⢺⣟⣻⡆⢶⣀⣶⠀⣿⣿⣿⣿⣷
⠀⠀⢾⣿⣿⣿⣿⡇⣸⡇⠀⢻⠈⠁⢸⡇⠀⠀⢈⣍⣹⡇⠘⣧⣤⡿⠸⠟⢷⡄⢸⠟⠻⠄⢈⣿⠃⠀⣿⣿⣿⣿⣿⡆
⠀⠀⣾⣿⣿⣿⣿⡇⣴⠆⣠⣄⢠⢶⡄⠀⡀⢠⣠⣩⡍⣤⢀⡄⢤⣤⠀⢤⡄⢠⡶⠄⣠⡄⣬⣥⣄⡀⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡇⠿⠙⢧⠿⠸⠻⠇⠀⠻⠿⠿⠻⡟⢿⠼⢿⡤⡇⠀⢸⡇⠈⣽⠶⠟⠿⣿⣿⡼⠃⣿⣿⣿⣿⣿⣇
⠀⢰⣿⣿⣿⣿⣿⡇⢠⠀⣴⢠⡄⣶⣀⣶⣰⣖⢀⣶⣄⣤⠀⠐⣶⠂⠀⣤⢠⣦⣰⡆⣴⢶⡤⣶⣓⠀⣿⣿⣿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⣿⡇⠈⠛⠛⠿⠃⠿⠈⠻⠻⠧⠜⠃⠙⠟⠀⠀⠿⠄⠀⠘⠿⢙⣛⣱⠋⠛⠗⠲⠟⠀⣿⣿⣿⣿⣿⣷
⠀⢈⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⣷⢠⡆⠀⠀⣶⠀⣿⢷⡀⢸⡇⣰⠞⠛⠛⠀⣿⢉⣹⡷⢸⣇⢠⡿⠀⣿⣿⣿⣿⣿⣯
⠀⠀⣿⣿⣿⣿⣿⣧⢸⠛⠛⠻⣿⠈⣧⠀⣰⡟⠀⣿⠈⢿⣼⡇⢿⣀⣺⣿⠃⣿⠛⠻⣧⠀⠛⣿⠇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣷⠘⢀⣤⣴⣿⣦⡈⢛⣿⣷⣆⣋⣀⣀⠛⢁⣀⣍⣉⣿⣶⣍⠀⣠⣿⣷⣼⣏⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠐⠒⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠩⠉⠩⠉⡉⢓⡄⢀⡴⠒⠉⠑⠒⠬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣓⠒⠒⠒⠒⢚⡽⠁⡞⠀⠀⠠⡀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣤⠖⠋⠉⠙⠓⠒⠲⠏⣉⢛⢧⠀⠀⠀⠀⠀⠘⠀⣀⡀⠀⠑⢶⠲⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁⠀⠀⠀⡼⠁⠀⠀⠀⠆⠀⠄⠀⠀⠈⠙⣆⠳⣄⠀⠈⠀⡔⠀⠀⠶⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠒⢀⠀⢀⠀⠒⠀⠀⢀⣿⢤⡈⠑⠒⠤⠤⣤⠤⢄⠠⡤⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠋⠀⠘⢀⣀⣠⢔⡿⠋⠀⠈⠑⠢⣤⣄⣀⣀⣀⣀⣤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠭⢉⣉⠩⠭⣵⠒⠉⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠙⣆⢀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠘⠮⣓⣃⠇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠚⢣
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠎⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠠⠋⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣞⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⢀⡤⠖⠉⠙⡄
⠀⠀⠀⣇⠀⠀⠀⠈⠙⠛⠶⢶⣒⡒⠓⠒⠒⠒⠒⠒⠶⠤⢧⣘⣒⡂⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠖⠒⠉⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⢀⠇
⠀⠀⢀⡞⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠢⠤⠤⠤⠄⣠⣤⣈⣉⣉⣉⣉⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⡤⠔⠊⠉⠀⠀⠀⠀⣀⡤⠔⡏
⠀⠀⢸⡀⠀⠀⠈⠉⠐⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠇
⠀⠀⢠⠗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠈⠋⠉⠀⠀⠀⣠⢾
⠀⠀⠸⣄⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠤⠤⢄⣀⣀⣀⣀⣀⣀⡤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣡⡇
⠀⠀⠀⠈⡳⢄⡀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠶⠖⠒⠒⠊⠉⠀⣠⠞⣡⠇
⠀⠀⢠⠏⢇⠀⠻⡒⢤⣀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊⠉⠉⠹⡄
⠀⠀⠘⣄⠈⠦⣀⠈⠒⢬⣙⡢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⣡⠏⠀⠀⠀⠀⡗
⠀⠀⠀⡏⠓⠤⣈⡑⠢⢄⣀⠉⠙⠒⠫⠭⣖⣢⣤⠤⣀⣀⣀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠔⠒⠒⠚⠉⣉⣁⣀⠤⠚⠁⠀⠀⠀⢀⡠⠳⡄
⠀⠀⠀⣇⠀⠀⠀⠉⠓⠲⠬⣉⡒⠢⠤⢀⣀⡀⠉⠉⠙⠒⠒⠢⠤⠤⠤⢁⣀⣀⣀⣀⠀⠤⠤⠴⠒⠒⠛⠋⠉⠉⣉⣉⠤⠋⠀⠀⠀⣀⠤⠒⠉⠀⢀⡇
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⠤⢌⣉⠉⠁⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣀⡴⠊⡇
⠀⠀⠀⠀⠘⢷⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠦⠤⢤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⢁⡠⣤⠖⠋⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡒⠭⣕⣒⠤⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣥⠤⠒⢊⡥⠚⠁⠈⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡈⠉⠐⠒⠤⠤⣭⣉⣉⣑⠒⠒⠂⠤⠤⠤⠤⠀⣀⣤⣤⣤⣄⣀⠠⠤⠤⠤⠒⠒⠉⠉⠀⣀⡤⠖⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⠉⠁⠀⠀⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠦⠀⢧⣠⠆⠀⠀⠀⡀⠀⢀⠞⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠲⠤⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⢃⡴⠊⢳⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠀⢘⣢⣀⠀⠠⡼⠀⠀⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢧⢸⠳⢋⠽⠿⢧⠀⢣⡀⢀⡔⠀⠉⠀⠲⢖⠒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠎⠐⠛⡠⡄⠈⡛⠉⠀⠀⠀⠤⣤⢤⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠀⡇⢴⠒⠀⢠⠊⠀⣀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣦⠤⠂⠂⢁⡱⠞⠁⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠠⠚⠁⠀⢠⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠆⠀⠀⡠⠈⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠣⠄⣀⠤⠤⢀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⡃
⠀⠀⠀⠀⠠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠈⡆⠀⠀⠉⠁⠀⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⣀⣸⣎⣁
⠀⠀⠀⢠⠒⠳⠟⢱⠀⠀⠀⣶⠒⠒⠂⠈⠁⠀⠀⢱⠇⠀⠀⠀⠀⠀⠀⢙⠁⠀⠰⡄⣀⠀⠀⢸⠀⠙⣄⢴
⠀⠀⠀⢸⡀⢸⢤⠈⢃⠀⠬⠥⠆⡠⠖⡛⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠒⢤⡏⠀⠀⠄⢀⠀⠯⠅⠄
⠀⠀⠀⠀⠹⢤⠼⠶⠃⠀⠀⠀⠫⠀⠠⠏⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣐⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠘⠐⠒⠂⠀⠒⠀⠈⠉⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸⠀⠀⠉⠉⠁⠐⠒⡖⢲⠒⠀⠉⡿⢥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⢋⣻⠀⠘⢇⠀⠀⠀⢀⣀⣀⣇⣈⡀⡀⠀⠃⢀⣇⣽⠓⠢⡀
⠀⠀⠀⠀⠀⠀⢰⣾⣤⣤⣶⣿⣿⣿⣶⣶⣷⣤⣤⣀⣠⣾⣿⣿⣉⣠⣤⣶⣿⣿⣿⣦⣤⣼⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⢠⠀⠀⠀⢀⡀⠀⢀⠀⡠⠄⠄⡀⠀⠀⠀⡠⠀⠀⠀⡀⡤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠠⠄
⠀⠈⠀⠀⠀⠀⢀⠰⠀⠐⡀⠂⠀⠑⠀⠀⠐⠀⠊⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠲⠄⢠⣂⠁⢶⠂
⠀⠀⢀⡀⠀⠀⠀⠘⠆⠀⠑⢀⠀⡑⠜⠀⠀⢖⠧⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠑⠀⠀⠀⠈
⠀⣄⠈⠀⠀⠀⢀⡠⠤⠠⠠⠠⡉⠀⠀⣴⡃⢸⣫⠀⠀⢄⠔⠁⠡⠀⠀⠀⠄⡠⠀⠀⠑⠢⠤⠀⠀⠀⠐⢀
⠀⠈⣀⠘⠀⠐⠁⡐⢀⡁⢠⣤⣪⠆⢀⣉⢀⠀⠟⠀⠀⠘⠄⠠⠼⠢⠀⠀⠀⠀⡀⠀⠘⠒⠀⠠
⠀⢸⠀⢣⠀⠒⠃⠐⣁⣒⡉⠤⠊⠀⡇⠀⠈⡄⠀⠀⠀⠀⡀⡀⠀⠀⣀⣤⣤⢆⡀⠉⠀⠀⠀⠀⠀⡠⢤⡀
⠀⢸⠰⡀⠁⠀⠀⣀⠄⡀⠀⢀⣀⠘⠄⢀⡰⠃⠀⠈⠄⠀⠀⠀⠀⠘⢻⡖⠛⠚⠳⠀⡃⠀⡐⠀⠸⠂
⠀⠀⠓⠐⢁⢄⠀⢈⠀⠀⠤⠀⠂⠀⢀⡤⣀⠀⠀⠀⠀⠀⠤⠤⡀⠁⠀⠀⠒⠂⠒⠀⠀⠏⠀⠀⠀⢂⠀⢀
⠀⢦⠀⠀⠂⠀⠁⠈⠀⠀⠐⠐⣢⠿⠜⢋⠘⢠⡄⠆⣀⡀⠈⠈⠀⠀⢀⠀⠀⠀⠀⠆⠀⠈⠀⠈⠀⠀⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠀⡁⢒⣠⠀⠖⢊⠀⠀⠀⠀⡆⠈⢀⢠⠒⠀⠂⡂⠦⡈⠄⢘⡄
⠀⡀⠰⠐⠃⡀⠀⠀⠀⠀⠄⠀⠀⠋⠀⠉⢡⡀⠀⢐⡭⠀⣀⠀⠊⡀⠀⠀⠉⠀⠀⠀⠀⠀⡁⠀⠀⠐
⠀⠐⠂⡀⢀⠔⠀⠀⠈⠿⠀⠀⡰⠛⠉⠁⠈⠷⠆⠈⠀⠂⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣀⣅⠀⠀⠃⠀⠤
⠀⡒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠈⠀⠀⢀⠀⠊⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⢈⢐⠀⠀⠀⠈⡀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠠⠀⠂⢀⠙
⠀⠀⢀⠤⡔⠄⠀⠀⠀⠀⠂⡘⠀⡀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠈⠦⡤⠀⠁⢠⡤⠐⡀
⠀⠈⠀⢀⠁⢈⠀⠐⢀⠠⠀⠁⠘⠓⠀⠀⠀⠀⣀⡔⠀⠀⠐⠀⠀⢊⠀⠀⠀⡀⠀⢀⡴⢡⠀⠀⠄⠋⠁
⠀⠀⠁⠈⠠⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⢠⡄⠀⢀⠀⠀⠂⠀⠀⠂⠉⠉⠉⠂⠉⠁⠀⠁⠠⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⡁⠈⠆⠀⠚⠀⠈⠀⢀⠀⣀⠀⠀⠀⠢⠃⢂⠕⠐⢈⠂⢀⡀⠠⠄⠠⢀
⠀⠀⡡⡄⢠⡄⠀⠀⠂⠀⠀⠐⠄⠀⠀⠉⠉⠁⠀⠁⠰⠀⢀⠀⠀⠐⠀⠀⡀⢀⣀⡀⠀⠀⢠⠀⠀⠀⠀⣨⠁
⠀⢸⣍⠠⠋⠀⠘⠁⠀⠁⠀⠀⠃⠀⠂⠔⠂⠈⠁⢀⣀⡀⠍⠀⠀⠀⠀⠀⡨⠉⠉⠀⠀⠀⠀⠉⠀⠀⠐⢀⡀
⠀⠀⠁⠀⠀⠘⠀⠀⠈⠀⠈⠀⠉⠐⠊⠀⠀⠊⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠊⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠉⠉⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣀⣤⣶⡶⣶⣶⣤⣔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⣀⣽⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠰⢰⡿⠁⠀⠀⢀⣤⠾⠁⠀⠀⠀⢹⣧⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠋⠀⢠⡿⠓⠘⠁⠑⠀⠀⠀⠀⠀⠂⠠⢀⢹⡆⠀⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⢸⡇⡰⠁⢠⣤⡄⠀⠀⠀⣤⣤⠀⠀⠘⡇⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢆⡁⠀⠀⠀⠁⡗⠀⠀⠘⠛⠀⠀⠀⠀⠛⠋⠀⠀⢸⡇⠁⠀⠀⠀⢻⠗⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⠀⡀⠀⠀⠠⠴⡀⠀⠀⠀⠰⠖⠒⠢⠀⠀⠀⠀⠸⠄⠂⠀⠀⠀⠂⠀⢀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⢄⠈⠐⠄⠀⠀⠐⡀⠀⠀⠀⠳⠖⠃⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⠔⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠄⡠⠋⠀⠐⢌⠂⠄⣀⠀⠀⢀⠀⠐⠀⡠⠐⠁⠙⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠁⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⡀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢀⣀⠠⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠉⠠⡀⠀⠈⠆⠀⠄⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠐⢤⡀⠀⠁⠢⠀⠠⠀⠐⠀⢀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢬⠀⠀⠈⣷⠀⠀⠀⠀⢀⠂⠀⢀⣀⡥⠖⣀⠄⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⡐⠚⠛⠿⠿⠒⠌⠀⣤⠂⠠⣄⣹⣇⠀⠀⠀⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠐⠀⠀⢀⠀⠀⠀⠈⠞⡁⢰⣤⣿⣿⣏⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⡉⠆⠀⠀⠀⠀⠀⠈⡄⠔⠂⠀⠉⠀⠀⠀⠀⠀⠀⢑⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠌⠀⢠⡴⣶⡶⢆⠀⡀⠀⠀⠀⠃⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡄⢁⠀⠐⠂⠠⠾⠚⠉⠀⠀⠞⠛⠛⠁⠀⢀⠀⠀⡠⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠲⢿⣃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⡦⠖⠓⠶⠤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣥⣶⣾⣿⣿⣷⣤⠈⢹⣶⠟⠋⠀⠀⠀⠀⠉⠻⢶⡏⠁⣠⣶⣿⣿⣷⣶⣮⣝⡄
⠀⠀⠀⠀⠀⠀⠸⠋⠉⠙⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠋⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⠛⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⠀⢰⣦⣤⠀⠀⠀⠀⠺⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⣷⡀⠀⢀⣈⣉⡛⢷⠀⠀⠀⢀⡞⣉⣤⣄⡀⠀⠀⣴⠋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⣻⠄⢠⠞⢿⣽⣟⡆⠀⠀⠀⠀⢸⡿⣯⡿⠛⠄⠀⣿⣇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⢷⡀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⡴⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⢀⡟⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⢷⠀⠀⠀⢣⡤⢀⣤⠍⠀⠀⡼⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠳⣄⣀⣀⣠⣭⣄⣀⣠⡾⠁⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⢦⡀⠀⠀⠉⠻⠄⠠⠾⠋⠁⠀⠀⣠⢎⠔⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⢳⣾⣧⣄⠀⣀⠀⠀⠀⣠⣶⡼⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⣄⢀⠟⠉⠿⠟⣶⠿⠷⠋⠈⠳⣴⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣉⠡⠤⠤⠤⢉⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⢖⣥⡶⠿⠿⢿⣿⣷⣮⣕⢤⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡔⣱⠟⠁⣠⡴⠖⠛⠻⢿⡟⠿⣷⣷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⢰⡇⠀⣴⠋⠀⠀⠀⠀⠀⠙⡆⠈⢿⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⣧⡄⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣌⢿⣿⣿⣆⠀⠀⠀⠀⠀⣠⡿⠀⣼⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⢮⣻⣎⠻⢿⣶⣶⣶⠿⠋⣀⣾⠏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠻⢿⣶⣤⣤⣤⣴⠾⢋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⣒⣢⣄⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠐⠁⠸⠙⠘⠶⠸⣇⡸⠄⢠⡄⠀⠠⠋⡹⢸⣶⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠣⠀⠀⠀⠁⠈⠉⠳⠫⢰⠇⣰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⢄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠣⠀⠀⠀⠀⡀⠐⠉⠁⡠⠀⠀⢠⠄⠉⢢⠄
⠀⠀⠈⠀⠂⠀⡠⠄⠒⠠⠤⢄⣀⠀⠠⠋⠱⢲⣾⡆⣤⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡪⠀⠀⠀⠈⠀⠀⠀⡇⠈⢢⠀⠑⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠀⠀⢌⢱⣄⠀⠀⠀⠀⠀⠉⠺⠄⠞⡂⠀⠀⠀⠀⠀⠀⢰⠁⢀⣀⣠⡂⠀⣀⠏⡅⠀⠀⣧⠀⡸
⠀⠀⢠⠋⠀⠀⢠⠃⢤⣬⣴⣡⠟⠠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢇⡸⠉⠓⠚⠓⠈⠀⠀⢴⠟⠁
⠀⠀⡜⠀⠀⡰⠇⠀⠐⠂⣈⡁⠀⣁⢰⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡌⠠⢄⠀⢰⡤⠀⢀⡅
⠀⠀⡇⠀⢨⡐⠆⠀⡰⠊⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠹⠂⠒⠊⠁⠀⠀⠢⠀⠀⢀
⠀⢸⠀⠀⠀⠈⣉⣍⠓⡦⠤⢴⠒⠋⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠃⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⢸⠀⠀⠀⠀⣻⣼⠋⠁⠀⠈⠑⢤⡀⢃⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠙⠒⠢⠤⠤⠤⠤⠖⠋⠒⣄
⠀⢸⠀⠀⡠⢚⠁⠀⢱⡄⠀⠀⠀⠀⠙⢾⠇⠀⠱⠀⠀⠀⠀⠀⠀⠀⡜⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠰⠶⠶⠃⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠒⠓⠒⠒⠒⠲⠦⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡆⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡄⠀⡜⡀⡊
⠀⠁⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡾⣤⣤⣀⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⢣⣀⢷⡗⠟⢢⡐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠈⠑⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢏⠀⠀⠀⠀⣷⠀⢰⠑⢆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢹⠒⠤⣠⣀⡀⠀⠙⠢⠠⠤⠤⠄⢀⡠⣀⠀⡇⠀⠑⠒⠒⠉⠉⢯⠁⠀⢀⢛
⠉⠉⠉⠉⠉⠉⠉⠑⠢⡀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⢸⠀⠀⠉⠑⠲⠤⣤⣄⢀⡥⠋⠛⠁⢷⠀⠀⠀⠀⠈⠀⠀⠑⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣶⣶⢷⠶⡄⠀⠀⡀⠀⠤⠄
⡍⠉⠉⠩⠭⠭⠭⣼⠁⠀⠀⠀⠀⠀⠘⣭⠭⠭⠭⠭⠭⠉⣉⣉⣉⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠻⠟⢯⢿⣭⡖⢻⠸⡁
⠇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠰⢾⠀⠘⣄⠀⠤⠀⠒⠀⠀⠈⠀⠀⣼⣀⣀⠘⢺⣃⣈⣀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈
⢀⠶⠦⠦⣴⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢁⡇
⢸⠘⠺⠋⠯⠂⠀⢸⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠠⣼⡇⠀⠀⠀⠀⣀⣴⣶⡟⢛⣻⣿⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋⠀⢈⣿⣿⡁⠀⠀⣴⠏⠹⣷⣄
⠀⠀⠀⠀⠀⠀⣸⡇⠀⣰⠿⣦⣄⣴⠟⠀⠈⠙⢷⣾⣏⠀⠀⢻⡿⣧
⠀⠀⠀⠀⠀⠀⢹⡇⢠⡿⠀⠀⣽⠛⢶⣤⡀⢀⣾⠋⠙⠻⠶⣾⣧⣽⡆
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⣀⣸⡇⠀⠀⠈⢹⡿⣧⣄⠀⠀⢰⡟⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⢿⡇⠈⣿⠉⣿⠻⠶⣤⣠⡿⠁⠀⠉⠻⣶⣿⡁⠀⣼⠇⠀⠀⠀⠀⠀⣠⡿⢷⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⣿⠀⠀⣠⡟⠻⣦⣄⢀⣴⠟⠉⠙⣿⠏⠀⠀⠀⠀⠀⣰⠟⣀⠀⠉
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠈⠿⣧⣰⠟⠀⠀⣀⣽⢿⣇⢀⣠⡾⠋⠀⠀⠀⠀⠀⣼⠏⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⠻⠿⣿⣯⣴⡶⠿⠛⠉⠀⠀⣀⣴⠶⢶⡾⠃⠀⠀⢿
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⠿⠶⣦⣴⠏⢰⡆⠀⠀⣿⠁⣀⣀⣀⡀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠻⢶⣤⡾⠋⣤⠀⠀⣀⠀⢸⡇⠀⠀⣿⠀⢉⣉⣉⠁⢸⡇
⠀⠀⠀⠀⠀⠐⢾⡇⠀⠀⣠⡾⠃⢠⡄⠀⣉⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⢀⣿⣀⣈⣉⣉⣀⣸⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⣤⣤⣾⣧⣤⣿⣦⣴⣿⣦⣴⣿⣤⣼⣷⣤⣾⣯⣭⣭⣭⣭⣭⣭⣿
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠙⣦⣴⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣄⣠⠞⠀⠈⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠛⠋⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠹⣿⣟⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠉⠻⠀⠙⠿⢷⡈⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⡟⠉⠁⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⣿⣿⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢰⠛⣧⠀⠀⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠁⠀⢸⣿⢳⡀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠿⢻⣿⣿⡇⠹⣸⣿⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣸⣿⣸⠃⠀⣿⣿⣿⠻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠸⣿⡷⡀⠀⠀⠀⣶⠖⠒⠲⣶⠀⠀⠀⣰⢿⣿⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣿⣷⢤⣀⣿⠀⠙⠦⣀⠀⠈⠓⠒⠚⠁⠀⣠⠞⠁⢸⡇⠀⠀⢀⣿⠃⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⢖⣛⣉⣙⣲⣄⢈⣭⣖⣒⣒⣒⣾⣍⠀⠀⠀⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠭⠭⠭⢭⣿⣿⣶⣤⡉⠉⠁⢀⣨⣷⠤⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢒⣻⠯⡿⠁⠀⠉⠛⠻⢷⢞⠉⡡⠒⠂⠀⠁⠉⠑⣎⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⠲⠦⢾⠁⢰⠃⠀⠀⠀⠀⠀⢸⣠⠎⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠱⣧⣤⣀⣀⣀⣀⣠⣤⣴⢟⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⠿⠊⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠬⢍⣉⣩⣥⠶⠗⢋⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⠘⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏⢸⡄⢻⠄⠀⠠⣎⠀⠀⠀⠀⠀⢀⣼⡵⣿⡶⠀⠀⠀⠀⠀⢀⣀
⠀⠘⠛⠙⠈⠼⠓⠇⠀⠼⠧⠴⠖⠀⠀⠀⢿⠈⠁⢀⡇⠀⠠⠰⠆⠢⡀⠀⠰⢣⣟⣿⠞⠀⠀⠀⠀⠀⢸⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠈⠁⢀
⠀⠀⢠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠠⣤⣤⣦⠄⠀⠀⡀⠒⠀⠁⠀⠀⠀⢠⣤⣤⣴⡀
⠀⠀⠈⠂⡀⠀⠀⠀⠀⠔⢰⡮⠃⠀⠀⠀⠶⠉⢀⠜⠁⠀⠀⠀⠉⠁⢀⠀⠀⣤⠞⣀⡈⠳⠀⠀⠈⢉⡻⠏⠁
⠀⠀⠀⡀⠃⠀⠀⠀⠘⠀⠤⠤⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠈⠡⢤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠉⠂⢀⠀⢠⡄⠘⠓⢠⠀⠀⠀⠉⠙⠀⠀⠀⠀⣀⡀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣙⡧
⠀⠀⠀⠀⣾⢴⡄⠀⠛⠀⢀⡀⠀⠘⠀⠈⠀⠀⠈⠀⠀⠀⢦⣤⣤⣤⡆⠀⠀⠐⠁⡠⡄⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠁
⠀⠀⣠⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⠀⢠⠐⢄⣡⠀⠀⠀⢀⠀⢀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⣀⠀⢀⡀⡃⠀⠤⡥⢬⠖⠋⠀⠀⠅⡠⠀⣀⠀⠀⠀⠨⢉⠉⠁⠶⠋
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡇⣀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡙⠁⠀⠘⢀⡐⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⢼⣵⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠁⠀⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢶⣴⣄⡇
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠈⠀⠀⠀⢠⣴⢶⠶⣦⠀⠀⠀⢀⣀⠀⠀⠀⠠⠶⠀⠀⠴⠆⠀⠀⣠⠀⡛⡀⠀⠀⠀⢠⣤⣤⡄⠰
⠀⠳⢶⣶⣶⠶⠃⠀⠈⠛⠾⠿⠛⠀⠀⢄⡈⠀⣀⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢚⠟⢑⣇⣱⠀⠛⠣⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉
⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠶
⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠓⠲⠂⠀⠀⠀⠙⢿⡉⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠾⠥⠀⠀⠀⠀⠰⣦⣄⣤⡾⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠄
⠀⠀⢁⡐⠀⣨⠀⠀⢀⡠⠀⣤⠐⠀⠀⠀⢶⡌⠑⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠸⠶⠂⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠈⠐⢎⠹⠀⠀⠀⠁⠈⠀⠁⠀⢀⣹⢿⡟⣿⡄⠀⠀⠠⠸⠠⡔⠁⠀⠀⠶⠆⡀⢠⡀⠀⠀⡘⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠐⠂⠐⠁⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⠠⢀⣠⣀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢴⡄⠀⠀⣠⡀
⠀⠀⣩⣤⣦⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⠀⠀⣦⡉⠀⠀⢀⠀⠀⡅⢠⠡⠀⠀⠀⠐⠀⠸⠃⠿
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠀⠋⠀⠀⠠⡿⠟⠛⠛⠃⠀⠀⠀⠀⠎⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⢿⡿⠻⡿⢿⢿⠿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣾⣶⣾⣷⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠛⠻⣤

⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣶⣿⣿⣿⣿⣿⣷⣾⣍⠻⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣀⠀⠹⣿⣿
⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⣿⣿⠁⠀⣿⣿⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡿⠀⠀⠀⠀⠉⠙⣛⠛⣿⣶⣾⣿⣿⣿⣿⣷⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡍⢻⠀⠈⡇⠀⢘⣁⣸⣿⣿⣿⣿⣿⣿⣷⣽⣿⣏⢿⣯⣿⣧⡤⠀⢀⣀⣀⣀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠃⠘⡇⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣾⣸⠏⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣥⣶⣿⣿⣿⣿⣙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢛⣛⣿⣹⣶⣄⡀⠀⠀⠌
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣄⣌⣯⠉⠉⣭⣏⣉⡟⠛⠟⠉⠉⠀⠤⠭⠉⠉⠉⠀⠁
⣿⣶⠒⠀⢆⠀⢰⡄⠀⠤⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⠶⠶⢶⣶⣶⣶⣶⣖⣾⣿⠿⠇⠀⠀⣀
⣏⡉⠳⠄⠀⠀⠘⠇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⡤⣾⣯⣤⣿⣿⣉⣈⣛⡡⠴⠶⠿⠿⠛⢟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠢
⠙⢏⠂⠀⠀⢄⠀⢠⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⠄⠁⠠⣀⣀⣀⣤⣤⣤⣴⠶⢤⣄⡀⣀⣄⡀
⡄⠀⣤⣄⣱⡆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠿⡿⠿⠛⠛⢻⠉⠉⠉⠀⠀⣀⣀⡀⠉⠃⣿⣿⣷⡆
⠷⣀⣻⣿⣿⣷⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣿⣿⣷⠾⠶⢿⣿⠿⠟⠛⠁⠀⠀⠀⠘⠛⠋
⠋⠉⢉⣿⣿⣿⣷⣄⠀⠘⢿⣿⠿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣯⣉⣹⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⢀⣤⣀⡤⢀⣼⣿⣶⡆
⠛⠛⢻⣿⣿⣿⣿⣿⣿⣦⣄⠙⢿⣿⣿⣾⣿⣿⣯⣙⣛⣛⣛⣛⠻⣿⠯⠿⠟⢛⠛⢛⣿⣻⣯⣥⣶⡶⡿⠿⠿⠟⠁⠰⣿⣿⠟⠁
⠋⠉⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣙⡛⠿⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⢿⣿⠿⠿⠿⠛⠛⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔
⣤⣤⡟⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣻⡶⠶⣤⣤⣬⣉⣉⣛⣛⠛⠛⠓⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠒⠈⣉⠀⠶
⣿⣿⣿⣷⣦⣍⠻⣿⣿⡷⢋⣴⠞⠉⢀⣴⡿⠛⢓⣶⣶⠦⣭⣭⣍⣉⣉⣛⣛⣛⡛⢛⣛⣛⠀⠈⠉⣁⢀⣠⣤⣤⣴⣦⡘⠓⠈⠁
⣿⣿⣿⣿⣿⣿⣦⠙⡟⣡⡿⠃⢀⣴⠿⠋⢀⣴⡿⠋⠀⣠⣾⠟⠉⣉⣽⡿⠛⠉⢛⠻⠿⠿⠉⠉⡻⣛⡁⠛⠿⣶⣄⡀⠉⠲⢤⡀
⠿⣿⣿⠿⣿⣿⣿⠀⢻⠉⢀⣴⡟⠃⠀⠴⢿⡏⠀⠀⢚⡟⠁⠀⠾⠿⠋⠀⠀⠺⠃⠀⠀⠀⠠⢾⣄⠉⠛⢷⣄⡀⠉⠛⠶⣄⠀⠉⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣶⣶⣦⣤⣠⣤⣶⣶⠾⠷⢶⣶⣤⡀⠀⠀⠀⠀⠀⡤⣤⣀
⠀⠀⢺⣷⡄⠀⠀⢀⣴⣿⢟⣯⢷⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⣽⣿⣷⡀⠀⠀⢸⢹⣹⢹⠁
⠀⠀⠘⣧⠹⡄⠀⣾⣿⣽⣫⠾⠋⠉⠁⠈⠉⠙⠻⠿⠛⠉⠁⠀⠀⠉⠛⢷⣽⣿⣿⡀⠀⡏⠉⢋⡏
⠀⠀⠀⢹⡆⠹⣼⣿⣿⡿⠃⢀⣀⣀⣀⡤⠄⢀⠀⠀⠀⠱⠔⠲⢤⣄⡀⠀⠙⢿⣿⣷⠀⡇⣠⠞
⠀⠀⠀⡼⢷⠀⢿⣿⡟⠀⠰⠋⠁⣀⣤⣀⠀⠘⡆⠀⠀⠀⠀⣀⣠⣀⠙⠣⠀⠈⢿⡿⢻⢣⣧
⠀⠀⠀⣇⠸⣇⠸⣿⠀⠀⠀⣲⣮⣤⣤⡈⣧⠀⣿⠀⠀⢀⣾⣤⣄⠘⣷⡂⠀⠀⠈⣇⡾⡾⣸⠁
⠀⠀⠀⠹⣆⢿⡀⢿⠀⠀⠀⠚⡏⢼⣿⡇⢸⢀⡏⠀⠀⢸⡇⢾⣿⠇⣿⠓⠀⠀⠀⢻⢷⣧⠟
⠀⠀⠀⠀⣿⣿⡇⢸⡀⠀⢀⠀⢿⠶⠿⠧⠞⠼⡀⢀⠀⠸⠿⠾⠿⠤⠏⣠⡀⠀⠀⡼⣾⣿⣦⣴⡄
⠀⠀⢸⣿⣿⡏⢻⢸⡇⡰⠋⠴⠺⣖⢦⢤⣤⣴⣤⣀⣤⠴⠚⢲⣖⡾⠒⠂⢻⠀⢠⣧⣿⣿⣿⡟⠁
⠀⠀⠀⠙⣿⣷⣸⣿⣇⣀⣠⣤⡄⠈⠻⣭⣙⣙⣺⠞⣁⣠⣦⡼⠋⠀⠀⡤⣄⣀⣼⣼⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⠟⠉⣉⣁⣽⣷⠃⠀⠀⠀⠉⠙⣛⣛⣛⠋⠁⠀⣧⡆⠈⢳⣿⣌⣉⡉⠛⣿⠋
⠀⠀⠀⠀⠈⢹⡄⢉⣩⠤⠴⢿⠛⠒⠶⢶⡆⠀⠀⠀⠀⠀⣰⡶⠶⠖⠚⢻⠷⠤⣭⣉⠁⣿
⠀⠀⠀⠀⠀⢸⡇⠤⠤⠶⠒⣻⠀⠀⠀⠀⠉⠛⠒⠶⠒⠛⢉⡠⠀⠀⠀⢸⠓⠲⠦⠤⠀⡟
⠀⠀⠀⠀⠀⠀⣧⡰⢖⣶⡋⠁⠀⠀⠀⠀⠀⠒⠒⠒⠒⣊⣩⠴⠊⠀⠀⠈⠉⢳⣖⠶⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠀⠀⣀⣀⣀⣤⡴⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⣠⣴⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠋⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣿⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠂⠈⠁⠙⠟⠿⠿⠀⠀⠘
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣄⣤⣀⣀⡀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣄⣻⣳⠀⠀⠀⠀⠀⠀⠀⠚⠉⡹⢻⣿⣧⣿⣿⣷⣦⠼⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⢀⣨⣠⣾⣿⣿⣿⣫⠔⠛⠿⠆⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣴⣶⣶⣷⣄⢾⣿⣿⣿⣿⣿⣿⣷⡀⠄⠀⠀⠀⠁⠀⠈⠛⠻⣿⡿⠋⠀⠀⠀⠀⣤⣄⣄⣀⠀⢁
⠀⠀⢸⡳⣿⣟⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠙⠻⠾⣽⣛⣻⠿⣿⠷⢿⣿⣿⣿⣿⣿⣧⣶⡦⡀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣤⣴⣾⣟⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠷⠿⠿⠥⠤⢼⣭⣯⣷⣭⣶⠤⠀⠴⠷⠾⠿⠟⠛⠛⠉⠉⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣻⣟⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢛⣛⠃⣀⣀⢀⠀⣀⡀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠀⣿⣿⣼⣿⠉⣻⣏⣿⢿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣿⣄⣿⡇⣿⣿⣈⣿⣏⣿⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠟⠀⠀⠀⠙⣎⡜⠁⠀⠀⠈⢢⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢫⡙⠿⠿⣫⢻⡻⡻⠿⠟⣩⢏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠣⢌⣁⣉⠴⠋⠳⢌⣁⣉⡥⠊⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡟⠁⠈⠛⠙⢿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣗⠰⠍⠖⠃⠓⠦⠧⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⢶⢀⡀⠀⢀⠤⢤⡀⠀⡀⢰⠶⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⣏⣀⡏⠀⠈⣇⣸⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣟⣙⣏⡙⡏⣷⢸⣿⢸⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣏⣉⣏⣉⣇⣿⢸⣿⡿⣿⣏⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⣿⡏⢹⣿⡏⢱⣦⢰⣷⢹⣯⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⡇⢸⣿⣿⣾⣿⣾⢻⣼⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠸⠟⢧⡼⠿⠸⠏⠿⠟⠻⠿⠇⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠒⠒⠢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⠁⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⠀⠀⠤⠀⠀⠀⠠
⠀⠀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠈⠑⠃⠀⠀⠓⠂⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠌⠀⠰⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠰⠁⠠⠦⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠐⠓⢄⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠠⣀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⡀⠀⠁⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠒⠒⠆⠀⠀⠀⠐⠒⠐⠒⠒⠒⠒⠒⠒⠒⠂⠉⠀⠉⠀⠈⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠠⠴⠂⠐⠦⠄⠑
⠀⠀⠀⢐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠳⠤⠤⠄⠀⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⠤⣀⣀⠀⠀⣀⣀⠀⠀⠤⠀⠐⠂⠠⠔⠆
⠀⠀⠀⢀⠆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠐⠀⠀⠒⠒⠂⠒⠤⠤⠤⠤⠠⠤⠤⠀⠤⠤⠀⠠⠤⠤⠤⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠤⠤⠄⠀⠀⠰⠒⠀⠀⠀⠀⠐⠈
⡠⠀⠀⠈⠳⠄⠀⠀⠀⠀⠠⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⡠⠄⠠⠄⠤⠀⠠⠔⠋⠀⠀⠠
⠁⠀⠀⠀⠀⠀⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣽⣀⡶⠃⠾⠐⠻⠀⠄⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣆⠈⠀⠂⠘⣾⡿⠇⢸⢦⣎⡑⠻⠛⠧⢤⠀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡻⠿⠂⠀⠀⠀⣂⣽⣿⠉⠀⠘⠛⠷⢃⣒⠤⠄⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠮⡿⡜⠁⠀⠀⠀⠄⠀⠀⠈⠓⠒⠚⠂⠀⠀⠩⠅⠀⢰⠀⠰⠀⡀
⡤⠀⠂⣡⣴⡅⡄⣀⡀⢄⡀⣀⠄⡀⠈⠀⠀⢼⣿⢤⡄⠉⠁⣠⡄⠈⠀⠀⠀⢀⠀⠀⢠⠄⠀⡀⢿⣦⡀⢀⣴⡀⢀⢀⣆⣀⢀⣿⠤⢴⣄
⠣⣦⣆⡼⣃⡏⠰⣿⣧⣹⡭⠃⢀⡖⢹⢰⠰⣮⡁⠈⣻⠶⡄⢻⡅⣰⠲⠊⠀⠐⢁⡘⡿⠶⠀⢤⠀⡯⢉⡀⢸⠏⠎⠻⡿⢩⠀⣷⡀⠶⠋
⣾⣿⣿⢿⠿⣷⠂⢸⣿⣷⢎⣿⣧⡉⠁⡄⣭⢿⢁⢴⡗⠉⢠⣖⠉⠉⠁⠀⡦⣑⣿⣗⣿⠐⣷⣿⣟⠷⠉⢥⠀⢲⣦⡀⠀⢫⣿⣿⠁
⣵⣟⣿⣥⣾⣿⣿⢯⣉⣿⣿⣿⡮⢉⠈⣿⣁⡁⣠⣾⠷⣤⠸⢻⡟⢶⡄⠠⢸⣿⠙⢻⣿⡀⠙⠛⢿⢷⣶⡄⢠⣼⢃⠃⠀⠘⣿⠋⣧⣤⢶
⣿⣟⠙⢻⡍⣿⢤⣤⣻⣿⡿⠁⠀⠉⠉⣉⣿⡇⣿⣿⣤⣿⡿⠀⠑⠛⢧⣆⢨⣿⠀⠀⠈⡿⢳⡄⠈⢳⣬⡍⠿⣼⡏⠀⠀⣄⢿⡿⠃⠀⡄
⢁⡠⣥⠉⡿⠿⢚⣧⡾⣭⠠⠡⣰⣮⣿⡿⢃⢺⣿⡿⢿⣽⠇⠀⢀⣾⣿⢿⡏⠀⠘⠛⢳⣌⠋⣱⠀⡊⣿⡇⠚⣿⣧⣄⣀⢈⣈⢃⣶⡦⢶
⠟⣿⣿⣶⣷⠠⣿⣿⣧⠏⣶⣴⡛⣿⣿⣧⡈⣙⣻⣷⠾⠉⠀⠘⣿⠏⣷⣿⡇⠀⡀⠀⠘⣿⣯⠿⠠⣼⣿⠟⣎⠩⣿⡟⠋⠻⣿⣿⣵⡄⣀
⠀⠸⠒⠽⢹⣾⡆⣠⠀⠀⡏⡉⠓⣿⠛⣯⠿⣿⡏⡎⠀⠀⠀⠀⣙⢿⢸⡟⠇⡆⣇⠰⠀⢿⣿⡆⠘⣿⠳⣤⣿⡿⣃⢵⢨⣷⣈⢿⣿⢛⣿
⠰⠤⡅⠀⡏⠏⠤⠶⠶⠆⣥⣤⠾⠿⠿⢫⣾⣿⣹⠃⢰⡆⠀⠀⣿⡶⠆⣈⢿⣿⣯⣥⣴⣼⢿⣿⣷⣿⣦⢘⢿⡁⡤⡄⠸⠳⠇⣸⣿⠣⡵
⣀⣀⣀⣀⣷⣧⣴⣶⣶⣶⣦⣜⣦⣣⣾⣿⣮⣼⣿⣦⣬⣄⣀⣿⣿⣅⣻⣿⣿⣿⣿⣯⣼⣭⣋⣯⣴⣿⣏⣡⣿⣷⣼⣾⣖⣧⣴⣧⣼⣃⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣠⣤⣶⣿⣿⣶⣦⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⢸⣿⣿⢿⠃⢸⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣀⠉⣁⣾⣆⡀⢉⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣾⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢠⡎⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠁⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠈⠉⠉⠙⠋⠹⠟⠉⠉⠉⠑⠒⠒⠒⠒⠒⠚⠛⠛⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⡄⠀⠭⢅⣉⠉⠀⠀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣒⣿⡯⠽⠛⠛⠛⠛⠻⣗⣦⣄⣀⡀⠀⠀⠀⠀⢀⣺⠼⠿⢿⣯⡹⢛⡲⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⡘⠭⠿⡟⢿⣛⠛⣛⣦⡌⣿⠛⠛⠀⠀⠐⠂⢘⣿⠛⠓⠲⠤⣿⡕⣄⠀⠐⠒
⠀⠀⠀⠀⠀⢤⣤⣁⣀⠀⠀⠀⠀⠀⠀⠉⣿⣽⡄⠁⠀⠀⠀⠀⠈⢹⡄⠀⠀⠀⢸⣿⢹⣀⣈⣀⣀⣤⡄
⠀⠀⠀⠀⠠⠀⣤⣉⣉⣉⣉⣉⡉⠉⠙⠛⠛⠛⠛⠓⠀⠀⠀⠀⠚⢻⠛⠛⠋⠉⢹⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠿⢿⣿⣿⣶⣶⣶⣶⡦⢀⠀⠀⠀⣠⣤⠸⣦⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠋⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠠⠄⠂⠈⠉⠉⠉⠀⠀⠀⠀⢀⡭⠁⠀⠈⠉⠦⠀⢸⡇⠀⠙⠇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡠⣆⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠁⠀⠀⢈⣧
⠀⡎⠐⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⣤⣾⣿⣿⣿⣽⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⠿⣧
⠀⢻⣤⡀⠙⠿⢿⣿⣿⣶⣦⣤⣤⣤⣄⠀⠀⢠⣿⢁⠀⠀⠀⠈⠙⣿⣂⣀⡀⠀⣶⣆⣠⣤⣤⣶⣾⣿⣿⠿⢏⠴⡿
⠀⠈⠳⠮⠑⠲⢤⣍⡉⠙⠛⠛⠛⠻⠿⠿⣶⣿⣼⣶⣶⣶⣾⣿⡿⢿⣻⣿⣿⢿⣿⣿⠿⠿⠛⠛⠛⠭⠓⣈⡴⠾⠃
⠀⠀⠀⠀⠀⠓⠶⣶⡄⣉⣙⠚⣓⠒⠶⠶⠶⠤⠬⠭⠍⠡⠤⠤⠤⠤⠌⠶⠖⠀⠒⣚⠉⣠⣤⣶⣶⡄⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠻⠃⠾⠿⠷⠰⣶⡶⠶⣶⠀⣶⡇⢰⣿⣿⠿⠸⠿⠿⠁⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⣀⠀⠀⠀⣀⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠈⠁⠀⠀⠀⠀⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⣿⠁⠀⠀⠢⠒⠀⢠⡄⠒⠀⠀⣷⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⣎⠻⠆⠀⠀⡀⠀⢤⣸⠇⠀⠀⠀⣿⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆⠀⠀⠐⢧⡀⠀⠀⠀⣠⠀⠀⡏⠀⠀⠹⡟⠁
⠀⠀⠀⠀⠀⠀⠰⠾⣉⡉⠓⣻⣄⠀⠀⠀⠈⠉⠈⠉⠀⠀⡸⠁⠀⠀⢀⣿⢲
⠀⠀⠀⠀⠀⠀⣠⠾⣉⣩⡿⠃⡏⠳⢤⣀⠀⠃⢒⣂⡠⣾⡁⠀⠀⠀⢹⡟⠻⣄⡤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣉⣀⡀⠇⠀⢠⡟⢹⠉⢹⡈⠱⠸⡝⢆⠀⠀⢣⡻⠖⣨⠟
⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⢨⡛⣧⣠⡟⠀⠈⠀⠈⠁⣦⢠⡹⣌⣳⡄⠀⡇⣾
⠀⠀⠀⠀⠀⢀⣿⡄⢸⢸⢸⡇⡆⣿⠀⠀⠘⡝⣤⠋⠋⢾⣃⡙⢏⠀⢠⠃⢸
⠀⠀⠀⠀⠀⡾⢸⢸⣴⢸⢸⡇⣇⡷⠇⠀⡆⢱⠀⢀⣠⠴⠚⠀⣸⡀⡎⠀⢸
⠀⠀⠀⠀⢠⣧⢿⢘⢿⣨⣼⠛⠁⢰⣀⣠⢷⡌⢠⣏⣠⢶⠳⡖⠃⠙⠃⠀⣸
⠀⠀⠀⠀⢸⡟⢸⢸⢸⠉⠻⣆⠀⠀⢷⣠⠏⢀⠋⣼⣿⡟⠀⠈⢢⠔⡚⠈⠉⠁⠒⠠⣀
⠀⠀⠀⠀⢸⣧⣼⣸⢸⠀⠀⠙⣦⠀⠈⠁⠀⡼⠀⣎⣿⠇⢀⣀⣶⠤⠤⠬⠒⠒⠉⠁⡿⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠈⡏⠉⠙⠛⠛⠻⠷⠿⢷⡀⢀⣼⣶⣿⣿⣿⣿⣿⣏⠈⠦⣀⣀⢀⣀⡤⠞⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣭⣭⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣀⣤⣤⣄⣙⡱⢜⣡⣯⢍⡹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣿⡅
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡿⠿⠟⠛⢻⣿⣿⡻⠟⣿⠿⠿⢿⠻⠿⠿⢻⡿⠿⠋⠁⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠘⠷⣏⠛⢿⣿⣿⣿⣿⣿⣿⠿⣿⠉⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣤⣤⣤⣿⣄⣀⠀⠀⠈⢧⡈⣇⣨⣽⣿⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢳⡄⠀⠈⣷⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣦⣾⣿⣿⣿⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡿⣫⣶⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⡿⣋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣣⣶⣮⣭⣭⣍⠩⣵⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡍⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⢩⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡿⠟⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⠀⠀⢲⡄⢀⣿⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣛⣤⣤⣟⣿⣿⠏⠀⢠⡾⢡⡟⠁⣴⠏⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠘⠇⠀⠳⠀⠻⠆⣠⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣄⠀⠀⣰⣾⣿⢆⣤⣶⢶⣜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣦⣙⢿⣿⣿⡿⣵⣾⣷⣿⣟⣵⣯⣿⣿⣿⡿⣰⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠮⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠄⠀⠀⠰⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠐⠀⠒⠂⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠼⠄⣀⡀⠀⠀⣤⣀⣀⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠄⠀⡄⠀⡠⠈⠑⢢⡘⠿⠿⡿⠟⠛⠿⠖
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠠⠀⠒⢘⠄⠀⠀⠤⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴
⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠁⣀⡰⠉⠢⡀⠀⠀⠀⠀⠀⠀⢎⢹⠄⣀⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⡜⠁⠈⠀⠄⠀⠀⠀⠠⠤⠒⠉⠀⠀⠀⠂⠀⣎⠙⠼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⢰⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⠈⠀⡔⠐⢦⢈⣧⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠜⠀⢀⡤⠋⠁⠀⠈⠙⠒⠒⠒⠒⠚⠉⠉
⠀⠀⠀⠀⠉⠻⣅⠉⠂⠤⠒⠁⠀⠀⠉⣉⠥⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠒⠒⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⠳⣤⣀⣀⣤⣤⣤⠴⠶⠶⠶⠶⠤⣤⣤⣤⣤⡴⠋⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠆⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢺⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⢀⡤⠤⠀⠈⠁⠀⠀⡤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠠⠄⠀⠀⠰⣤⣀⣀⡴⠃⠀⠃⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠉⠀⠀⣽⣿⡀⠀⠀⠠⠄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⠃⠀⠠⠞⠁⠈⠙⠆⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣼⠀⠀⠀⠸⡞⢹⠞⠉⡆⠀⠀⠀⢿⠋⠳⠀⣰⠀⢰⠀⢰⡇⠒⡦⠀⢰⠒⡆⠐⢲⠂⢀⡖⢀⡄
⠀⠀⠀⠏⠀⠀⠀⠘⠁⠈⠀⠀⠇⠀⠀⠀⠘⠀⠳⠒⠻⠔⠹⠖⠸⠀⠀⠧⠴⢈⣻⠇⠀⢣⠔⠋⠛⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠀⡰⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡴⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠶⠶⠶⢶⣶⡶⠶⠖⠶⠿⠦⠶⠞⠋⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠋⠉⢀⣠⠴⠒⢉⣀⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⣠⡶⢋⣡⣶⣾⣯⠵⠾⠿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⣠⣾⡿⠟⠛⠉⠁⢻⠀⠀⠀⡀⠀⠙⠷⣤⣤⣤⣤⣴⣾⣿⠟⠋⠀⠀⠀⠀⠀⠈⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⣠⠤⠶⠾⡗⠛⠉⠉⠛⠲⠦⣤⣬⣍⠉⢿⡋⠀⠀⠀⠀⠀⠀⢀⣤⠾⣉⡹⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢠⡾⠋⠀⠀⣠⠴⠚⠁⣀⡀⠀⢹⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⠺⡇⠀⠀⣀⣠⡴⢞⣛⠃⠀⠘⣿⢹⢧⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠀⠀⣀⡴⠃⠀⠠⣿⣿⣿⡆⠀⢇⠀⠀⠀⣿⣿⣿⠀⣠⣶⣦⢿⠞⠿⢯⡁⠀⠈⣨⡇⠀⠀⠉⠀⣯⠉⡇
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢰⠏⠈⢱⡾⣍⡀⠀⠀⠀⠙⠛⠛⠁⠀⠸⡆⠀⠀⠙⠛⠋⠀⢻⣿⠿⠘⢧⡀⢾⠇⣣⠀⠙⠃⠀⠀⠀⠀⣏⣰⣇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⢀⡼⠁⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⠀⢻⡀⠰⠋⡣⠀⣠⠃⠀⠀⠀⠀⢳⠀⠀⠻⠶⠂⢀⣱⣤⠀⢰⡏⠁⢸⡄
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡆⠀⠀⢷⠀⠀⠀⡰⠃⣶⣿⣷⠀⠀⠸⣧⣀⠀⠻⣿⢟⣿⠇⠀⡾⣠⠞⠉⢻⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣾⡇⢀⡼⠃⠀⣾⣿⣿⡆⠀⠀⠘⠛⠛⠙⠲⢤⡈⡇⢀⡼⠁⠀⠙⢛⣋⣠⣤⠤⠾⠛⠛⢷⣌⠉⠀⠀⣼⣷⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡶⠚⠉⣀⣼⡇⠀⢳⠀⠀⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣉⣿⣾⡔⠒⠋⠉⠉⠀⢀⣶⣶⣦⠀⠀⠀⠛⠷⢶⣿⣿⢹⠄⠀⢰⢊⣷
⠀⠀⠀⠀⠀⠈⠷⢾⠛⠉⠈⠙⠶⣏⠀⠀⣀⡀⣤⠤⠤⠴⠒⠒⠋⠉⠉⠀⣰⠋⢳⠉⠓⢤⣀⠀⠀⠈⢿⣿⡿⠀⢀⡀⠀⠀⡟⠹⣆⣸⠀⢀⡟⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠸⡏⠁⠀⢀⣀⡀⠀⠀⢀⣄⡀⠀⠀⡴⠃⠀⠘⡆⢀⣴⣾⣕⠦⣄⠀⠀⠀⣾⣿⣿⡆⢀⡧⠀⢿⣿⢠⠎⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠛⢧⣹⡄⠀⣇⡀⢇⠀⠀⣿⣿⡿⢀⡼⠁⠀⠀⠀⢳⠘⢿⣿⣿⠀⠈⠑⠦⣀⠙⠻⠛⢡⡟⠁⠀⣼⡿⢁⡄⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⣵⡘⣇⠀⠙⣆⠀⠀⠘⠋⠀⠀⠈⠉⢀⠞⠀⢀⣀⠀⠀⠘⡆⠀⠈⠀⠐⢧⠀⡽⠀⠙⠢⢄⣿⠀⠀⣰⡟⣠⠏⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡳⡘⢧⡀⠹⣦⡀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣿⣿⣿⠀⠀⣻⣄⠀⠀⠀⠸⠶⠂⠀⠀⣰⠊⠉⠓⣴⣿⠟⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢽⣾⣁⠀⠀⠻⣄⠀⠀⠀⣠⠏⠀⠀⠀⠈⠉⠁⠀⠸⣿⣿⡷⠀⠀⠀⢀⡠⠔⠚⠁⠀⣠⣾⡿⠋⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡙⣦⡀⠀⣯⠳⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣁⣀⣀⣴⠏⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣝⢦⠙⢦⣽⡧⠈⠹⡟⠲⠤⠶⠖⠒⠲⠦⢤⡤⠶⠛⢯⠉⠁⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣕⠁⠀⠀⠀⠀⢀⣷⢤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⡶⠶⠛⠉⠁⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⣼⠇⠀⣾⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠷⠶⠶⠶⠶⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢁⣻⠁⠀⢀⡁⠐⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠋⠃⠄⠸⠃⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⢉⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⣄⣀⣀⣀⣀⣀⣀⡀⠤⠄⠒⠈⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠒⠤⢀⣀⣀⣀⡀⠀⢀⠀⡀⣀⣀⡤⠤⠔⠒⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠲⠤⣀⣀⣀⡉⠉⠉⠉⠁⠁⠀⣀⢀⣀⣀⣠⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣈⡉⠛⠓⠲⠶⠶⠶⠶⠶⠶⠖⠒⠚⠉⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠦⠤⠤⠄⠔⠒⠊⠉⣀⡤⠄
⠀⠀⠀⠀⠀⠈⠀⠤⠀⣀⠀⠀⠙⠓⠲⠤⠤⣤⣤⣄⣄⣀⣀⣀⣠⡤⠤⠖⠒⠚⠉⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⣶⣶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠲⢒⡲⠿⠛⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠁⠂⢴⠂
⠀⠀⠀⠀⠀⠠⠟⠂⡀⢰⡷

⠀⢂⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠒⠒⠋⠀⠀⠀⠀⢠
⠀⠑⢀⡀⠈⠐⠠⠀⠀⠀⢐⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠁
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣴⠋⠙⣆⡞⠉⢹⡄⡏⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠲⠞⠁⠙⠲⠚⠁⠳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣰⣇⢰⠋⡂⣶⡃⣞⢹⡀⣿⠢⡎⢱⢸⢰⢰⢦⡆⣞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠀⠉⠀⠀⠁⠈⠉⠀⠀⠀⠈⠁⠀⠉⠀⠀⠁⠈⠁
⠀⢀⢀
⠀⠰⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠠⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠂⠤⡀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠋⠁⠀⠈⠙⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠋⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠀⠀⠀⢀⠠⠊⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠳⢦⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣠⠶⠋⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠀⠀⠀⢀⠀⠐⠂⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠒⠀⠀⡀⠲⢦⣄⣠⠶⠋⠀⠙⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠁⠀⢀⣽⣿⡀⠀⠀⠀⠶⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠛⠀⠀⠰⠛⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣠⡤⠴⠛⠁
⠀⠀⠀⠀⠛⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⢰⡇⠀⠀⠀⢠⣀⣄⢀⣤⡀⠀⠀⠀⢀⣟⢤⡀⠀⢀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⡀
⠀⠀⠀⢸⠁⠀⠀⠀⢸⠃⢸⠃⠀⡇⠀⠀⠀⠈⡟⠀⡇⢀⣼⠀⣸⠁⡀⡟⠉⢈⡇⠀⢾⣨⡇⠈⢹⠁⢀⣾⣀⢴⠃
⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠛⠉⠹⠊⠈⠛⠁⠃⠀⠘⠷⠋⣨⢽⠃⠀⠸⠞⠁⠈⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠚⠃⠀⠀⠀⠉⠓⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⣀⡄⠀⢀⡂⠤⠀⠀⢳⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢭⢻⣧⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⠀⢸⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⣦⠀⠀⠀⣠⡇⠀⠉⠛⠁⠀⠀⠀⣸⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣸⣆⠀⠀⠀⠙⠲⠤⠤⠤⠊⠀⢀⡇⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠈⢁⠕⠶⢤⣴⢻⠷⣄⠀⠀⠀⡀⣀⣀⠀⣠⠞⠀⠀⠀⡠⣾⣧⡇
⠀⠀⠀⠀⠀⠀⠐⠋⠐⡶⠟⢁⢸⠀⠉⠛⡶⢤⠤⣴⠲⢎⢻⡶⡀⠀⠀⢠⢯⣔⡙⠴⡝
⠀⠀⠀⠀⠀⠀⠀⢠⢞⡧⠶⢶⣬⠀⠀⣾⠁⡸⠀⠸⡆⠈⠈⢷⡙⣦⠀⠈⢦⠉⡴⠊
⠀⠀⠀⠀⠀⠀⢠⢯⠃⡇⡇⢸⢘⢧⡾⠁⠀⢠⢄⠀⠤⣹⠘⡎⢳⣦⠏⠀⣼⠸⡇
⠀⠀⠀⠀⠀⢠⢻⠸⡀⡇⡇⣿⢸⠀⢧⠀⠀⡈⡎⠒⠁⠁⠹⢇⠄⠱⡄⢀⡏⠀⡇
⠀⠀⠀⠀⠀⣼⢸⠀⣷⡇⡁⢻⣰⠶⡟⠀⠀⡇⢸⠀⣠⠴⠚⡉⠀⢰⢇⣸⠀⠀⡇
⠀⠀⠀⠀⠀⣿⢸⠀⡟⣧⣶⡏⠁⠀⢳⢤⢼⣳⠇⣼⣧⣴⢻⠛⢯⡁⠈⠁⡀⣰⣇
⠀⠀⠀⠀⠀⣿⠘⡀⡇⡇⠈⢻⡄⠀⠈⢷⡼⠁⢠⠁⡄⣆⡟⠀⠀⢙⠖⣈⣁⡀⠀⢉⢁⠲⣄⣀⡀
⠀⠀⠀⠀⠀⣿⣤⣇⣇⡇⡀⢨⠹⣆⠀⠈⠀⠀⡟⠀⣆⣿⠇⣀⣠⣼⡐⠆⠠⠀⠂⠋⠁⢰⠿⣿⣿⣿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣿⠀⠉⠉⠛⠛⠛⠛⠿⢧⡀⢀⣾⣾⣿⣿⣿⣿⣿⣿⡀⠑⠤⣀⣀⣀⠤⠖⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⣀⣤⣤⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣼⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠳⣌⠙⠻⣿⣿⣿⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣀⣤⣤⣤⣿⣀⠀⠀⠀⠈⢧⡀⣿⢈⣹⣿⡿⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⢦⡀⠀⠈⢳⣾⣿⣿⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣆⣴⣿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡿⣫⣉⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠻⠿⠾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢰⠀⠤⢸⠄⠠⣠⠁⠈⣿⣿⠔⢀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠂⠂⠀⠀⢸⡇⡖⢸⠂⠒⠲⠃⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠆
⣿⣿⣿⣿⣿⣿⡿⢿⣯⡇⠀⠀⠀⠀⠠⠥⠧⠨⠀⠬⠨⠤⢬⣿⣿⡯⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣯⣷⣼⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣼⡀⣀⣼⣷⣀⣠⣴⡖⠦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣧⡀
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣮⡌⣀⣀⣠⣴⣄⣲⣤⣄⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⡿⣿⣯⣄⣁⣠⣩⡄⣴⣻⣶⣷⣦⣤⣀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠹⣿⡟⢣⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⡇⣾⠻⠛⡟⢻⠀⠀⠀⠋⠛⠦⣄
⣿⠏⣿⣿⣿⣿⠟⠩⠋⢚⣿⣿⣟⣟⠐⢸⣿⣇⣾⠏⠙⠻⣿⣿⣿⣿⡟⠩⠌⣽⣿⡿⠁⠀⠀⠀⠈⣿⣶⣾⣶⡇⢈⠀⢰⡄⢀⡀⠀⠙⣷⣄
⣷⣿⣷⣿⣿⣿⣤⣴⣿⣿⣧⡽⣉⣿⣶⣾⡿⣿⣯⣤⠴⣿⣿⣿⣿⣿⠇⢀⣶⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣁⠀⢸⠘⠆⠘⢷⣦⡀⠀⠙⢷⡀
⣽⣿⣷⣿⣶⣼⣿⣿⣾⡛⣻⣾⣿⣿⣿⡟⢶⡿⣿⡿⣡⡟⠛⡟⣹⡿⢻⣿⣿⣿⠟⢃⡀⠀⠀⠀⢀⣸⣿⣿⣿⡿⠶⡯⠀⡧⠀⢐⣿⣷⡀⠀⢸⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⡟⠻⣿⣿⢿⣷⣽⣿⣿⣷⣞⣿⣿⡦⠀⠹⣩⣿⡤⣶⢒⣾⣿⣿⣿⣇⣿⣿⡟⠀⠀⢳⣤⣼⣿⣿⣿⣽⡿⠿⠏⠉⡹
⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣠⣤⣶⣿⣿⣿⣋⣿⣏⣉⣹⣿⣧⣿⣿⠄⠀⠀⠀⠈⠈⣿⡟⠉⢰⡆⠀⠘⠛⡾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣅⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⣾⠀⠀⠀⠀⠀⣿⣇⡀⠈⠁
⠛⠛⠛⠋⠙⠛⢿⣿⣭⡿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣦⣿⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇⠀⠀⡀⠀⣿⡿⢋⠑⡠⢤⡔⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠛⠛⠿⢿⡟⣻⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢽⣿⣍⣫⣽⣿⣿⣿⠃⠀⢠⠁⠀⣉⣿⠛⠛⢁⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣬⣿⡟⠉⠹⠙⠣⡞⠉⢻⣿⠯⡿⠀⠀⠸⠁⣺⢸⣿⡿⣯⣌⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠸⠀⣿⣿⣵⠖⣿⣿⣯⡀⠀⠈⠁
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢻⣿⠃⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠷⠒⣿⣿⣏⠆⢀⠀⠀⠸
⣿⣿⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡤⠿⠾⣷⣾⡟⠉⠉⠉⠁⠀⠀⠀⢠⣤⣴⣿⣿⣿⣧⣄⣠⣿⣿⣝⣄⠀⠵⣦⣄
⣐⣓⣲⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠳⣄⣸⣿⣿⣧⠀⣙⠿⢂⠁
⠀⠉⠙⢻⣿⣿⣧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡡⠹⣿⣿⣷⣄⣬⣉⡙⠤
⣶⣶⣶⢾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣿⣿⣶⣾⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣖⣚⣛⣉⣉⣉⡙⠛⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠉⠉⠉⠁⠈⠉⠉⠉⠛⠶⣤⣀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⡀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣰⠏⠀⠀⠀⣤⡀⠀⡀⠀⣶⣄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡟⣶⣿⣄⣛⣿⣷⣶⣶⠿⠿⢿⣍⠁⡴⣦⡀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡴⠶⣿⣟⣭⣥⣬⡉⢉⣿⣋⠙⠳⣦⡀⠘⣿⠁⠘⡇⠀⠀⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢠⣿⠏⠀⣠⢶⣿⠘⣿⡉⣷⣀⣤⣿⣶⡟⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⡀⠀⠀⠀⠀⢸⣿⣀⣠⣿⣴⠟⠀⠈⠛⠻⠿⠟⠛⢉⣿⠃⠀⠀⠀⠙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠈⢷⡀⠀⠀⠀⢸⡟⠛⠛⠛⠁⠀⡶⠦⣤⡀⠀⣠⡴⠛⠈⠃⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀⢸⢷⡀⠀⠀⠀⠀⠁⠀⠀⣡⡀⠘⢧⣀⣀⠀⠀⠀⠀⠀⢈⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢈⣽⠇⠈⠛⠶⢦⡤⠶⠖⠛⠉⠀⢙⡆⠀⠹⣏⢻⠷⡆⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡀⢀⡿⣼⠀⢻⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠂⠙⠛⠁⢻⠀⠘⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠦⣠⠄⠀⠀⠀⣼⢿⠀⠀⠀⠀⡿⠀⠀⢻⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⣬⣷⡄⠀⢰⣧⣤⣴⡾⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⢦⠀⠀⢸⡄⡇⠀⠈⢹⡇⢸⡇⠀⠀⢿⣀⣠⠄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠶⢄⠉⠁⠸⣿⡇⠀⠀⠘⡇⢸⡇⠀⠀⣴⠿⠅⠀⠰⠶⢻⡟⠀⠀⢈⣵⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⠓⠀⠀⢹⠃⠀⠀⠀⠿⠈⠀⠀⣾⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠹⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠠⠿⠚⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⣄⣠⢀⡀⣤⣤⡀⣠⣄⢴⣠⡶⢠⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⣿⣼⠧⢿⠸⣿⣧⣿⡎⢿⠀⠈⠳⢿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠉⢸⡇⠀⢀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠾⠃⠀⠀⠉⠛⠁