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⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣀⣤⣶⢦⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠉⣁⣤⠤⠀⠀⢤⡀
⠀⠀⠀⠀⠀⢰⣂⣤⡞⠻⣿⣶⠀⣀⣾⣿⣷⡿⣳⢻⣯⡄⠀⣱⣶⣤⡀⠀⠀⡀⠀⢈⠒⣶⢯⡄⠀⢸⠟⡦⣷⢯⣽⡾⣝⣿⣀
⠀⠀⠀⠀⠀⣼⡿⠋⠉⠒⢻⣿⠆⠉⢺⢹⣿⣿⣽⣧⠛⠁⠀⢸⡹⣧⠙⢦⠀⠀⣴⣉⡛⢯⡀⣿⠀⠙⣼⠡⣌⣦⣿⠟⢺⡿⠉⡁
⠀⠀⢀⡀⠈⠻⡦⣤⣠⡔⠈⠉⠀⠀⠹⠮⡽⢿⡉⠉⠀⠀⠀⠈⠳⣯⣷⣸⡇⠺⢉⢸⠻⢡⣇⠀⠀⠀⠘⢳⣆⠳⡤⢄⡇⠸⠞⠁
⠀⠀⠈⢀⣴⠀⠀⠈⣁⣠⢔⣂⣴⡄⢄⠀⠀⠘⠁⡀⢔⣂⡷⠢⣤⠄⠙⠻⡇⠀⠈⠉⠃⠈⢛⡤⠤⠦⣄⠀⠹⠏⠉⠟⠁⢀⣶⣷⣞⣆⢰
⠀⢀⣾⣟⢺⠂⠀⢀⣧⡽⣋⣽⣍⡻⣆⡁⠀⠀⢨⠂⣽⢻⠹⣦⡙⡄⠀⠀⠀⢰⡟⢹⠿⣷⣆⠙⢦⣤⠟⠁⠀⠀⠀⠀⢠⣯⠊⠤⣉⠛⣇⡴
⠀⣾⣹⠇⢸⢱⠀⣿⢻⡼⣿⢧⣽⢻⣽⡇⠀⠀⣻⣘⣿⢝⡷⢣⠟⡟⠀⠀⢀⣼⣥⣿⣆⣹⣛⠀⠀⠀⡔⠀⠀⠀⠣⠀⣸⡿⣥⣳⠝⣩⢾⣷
⠀⢷⣿⣫⠟⠀⢦⣿⣼⣟⣺⡩⣿⣯⡝⢻⠀⠀⠀⢿⡬⠋⢁⠿⠊⠠⢒⢀⣀⠈⠙⠶⠶⠞⣁⣀⣘⣊⣿⡇⢒⣦⠄⠀⡷⣵⡻⠟⢊⣁⡼⡟
⠀⠈⠛⠉⣀⡀⠈⠛⢻⣝⣛⣒⣋⣡⠄⠊⠀⣀⠀⠀⠉⠓⠛⠀⠐⢻⣯⠉⣿⡎⣆⠀⠐⠋⢨⣿⣟⣽⣯⣴⣿⡏⣻⠀⠀⢨⡟⢲⠟⠛
⠀⠀⠀⢾⠿⣸⠆⠈⢳⠀⠽⠉⠩⠁⠀⠀⠀⡿⡟⠛⣷⣦⡀⠀⣿⢿⣿⣿⣕⢳⠏⠀⠀⠤⣸⠋⢌⣬⣻⠿⣿⣿⡁⢀⠂⠀⢀⣤⠤⡄
⠀⠀⠀⠘⢤⡵⠂⠀⠀⣠⠞⣛⠲⣦⣀⠀⢀⣼⣧⢾⣿⣁⣕⠀⠘⢟⣿⣛⢹⡷⠀⣴⣧⠀⠻⣾⡐⣿⣿⣿⣽⠇⠀⠈⠀⣴⣴⡿⠶⣾
⠀⠀⠀⠀⠀⠃⠀⠀⣰⣿⡿⠛⠉⣫⡟⠰⠞⠲⢷⣤⣅⣼⠉⠀⠀⠀⠈⠁⠀⠀⠸⣿⡷⠀⠀⠉⠉⠉⠁⠀⢀⠀⠀⣀⣀⠈⢿⣀⣠⠞
⠀⠀⠀⠀⠀⠐⣆⢰⣿⡍⣏⠛⠛⣉⢀⣴⣶⣀⣀⠀⠈⠘⠀⠀⣠⣤⢲⣼⠃⣰⠀⡾⠃⠀⠀⣠⣶⣿⣦⣴⣮⠀⠀⣿⠁⣹⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠈⠻⣷⠏⣀⣟⡀⡾⢶⣿⠾⣏⡇⣦⠀⢐⣿⣟⣠⡄⡿⡆⣹⡠⠖⠈⠀⠀⣙⣯⡉⡵⣬⡘⢧⠀⠈⠣⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡄⣿⢻⣿⣿⢗⡿⣧⢿⡀⠸⣿⠛⠉⣣⣅⣱⣿⡿⠀⠀⠀⠐⢿⠿⣿⣶⠞⡷⣸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣾⣟⣿⢟⠛⣡⡾⠀⢠⣦⣐⣻⣿⢋⣏⡜⠁⠓⠀⠀⠀⠀⠘⢶⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⠭⣽⠯⠟⠊⡀⠀⠀⠒⢬⡍⠁⡾⠋⠁⠀⣠⣞⣋⢉⠛⢦⠈⠃⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣀⣤⡄⠀⠀⠀⠀⠀⡀⠘⢿⣟⣉⠹⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠓⠉⠉⠉⠀⢀⣴⡆⠚⠷⣜⡀⠈⠛⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⣾⡧⢼⠷⢦⠚⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⢷⢽⣶⣠⣤⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠛⠿⢿⠅
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡀⠀⠀⢰⡏⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⡆⠀⢸⡇⠈⢧⡈⢢
⠀⠀⠀⠀⠀⠀⢀⠤⢀⡀⠬⠢⣅⣀⠀⢿⡀⠈⢧⢠⠂
⠀⠀⠀⠀⠀⠀⢁⠞⢁⡆⠀⡎⡴⠁⡖⠀⠙⢂⣘⠁
⠀⠀⠀⠀⢄⣐⣋⣴⠎⠁⠈⡰⢃⡴⠁⠀⠀⠺⠿⠇
⠀⠀⠀⠀⣀⣠⣿⡶⣶⣾⣓⡒⠋
⠀⠀⢀⣰⡟⢿⢸⢱⣋⠴⠿⣯⡀
⠀⠀⢈⣿⣛⠮⣿⣿⣷⢭⣛⣿⡏
⠀⠀⠈⢻⣯⣭⠝⡽⣿⢳⣝⡿⠇
⠀⠀⠀⠘⠛⢷⣿⣥⣿⡼⠿⠃
⠀⠀⠀⠀⣠⢾⣿⣄⠉⠀⠠⠦⡀⣠⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⡰⠋
⠀⠀⠀⠀⢿⣟⣳⡏⡀⠀⡄⡄⢹⠻⠿⣴⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢛⡭⠙⠂⠀⠀⠀⠈⢀⠔⠀⢇⢰
⠀⠀⠀⢀⡀⢈⣩⠔⢂⠂⠲⣡⡈⠀⢸⡇⠘⡇⢣⢀⡤⢄⡀⠀⠀⠀⠀⣴⠋⢀⡞⠰⢀⣤⢴⢦⣴⠀⣏⢠⡀⠈⡀
⠀⠀⠀⠀⠑⠦⠴⠖⠋⠀⢀⡴⠊⡑⢸⡇⠀⡇⢸⢫⠒⠀⣹⠀⠀⠀⠀⠏⢀⠞⢀⢆⣿⣪⣶⣽⢺⣧⠘⠂⠁⠋⣀⣂⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⣩⠤⢺⣤⣶⣿⣶⠒⠙⡄⠑⠊⠁⠀⠀⢀⣠⣾⡷⣾⣧⣄⣿⣾⣯⣳⡷⢇⠐⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣺⣿⡷⣿⡆⠀⠈⢀⠀⠀⠀⣀⣿⢿⢸⣣⢏⠴⠿⣧⣍⠻⢍⠃⠠⡙⠒⠒⠦⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣮⣶⡟⠂⠀⠐⠀⠓⠀⠀⢻⣟⠦⢵⣿⣿⡯⢭⣻⡏⠣⣄⡙⠢⣄⡄⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣩⡭⢻⢿⡞⣦⣿⠛⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣶⣿⣼⣷⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⢠⠘⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⡸⠁⡴⠂⡄⠀⠀⠀⠋⢄⡃⢀⡤⣄⡀⠰⣟⡀⡂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡔⠀⠀⢀⣼⣿⣀⠓⠔⠃⢀⡴⣦⠴⣿⠀⠘⠗⠟⠃⠀⣀⠁⠇
⠀⠀⠀⠠⠀⠀⠀⢨⠻⠿⢷⡄⠀⠀⢹⣿⠻⢁⣦⣤⣼⡿⣧⠞⢻⣷⡖⠄⠄⣰⡋⠌⣏⠀⠀⢁⡌
⠀⠀⠀⠀⠠⢆⢤⠀⠀⠀⠀⠁⠀⠀⢸⠤⢁⢿⣿⣿⣿⠞⠻⣦⣟⠚⣏⠠⠀⠀⠠⠀⠁⠰⠍⢬⠨⠄
⠀⠀⠀⠸⢤⠌⠀⠁⠀⠀⣤⠉⠚⠿⠂⠀⠘⢾⣿⣿⣿⣄⠀⡀⠁⠀⠈⠀⠘⠀⠠⠀⠠⠄⠀⠞⠁
⠀⠀⠀⠀⠀⢒⡤⣋⣧⡏⢃⣀⣠⡁⠲⠂⠀⠉⠐⠚⠏⠙⠤⣈⢀⠀⠈⠀⠈⠆⠲⠄⠀⠀⠄⠀⡅
⠀⠀⠀⠀⠀⡾⣛⣮⠞⠧⠼⡀⢠⠇⠀⠀⠀⢰⣔⡣⠀⣀⠀⠁⠀⠀⠀⠨⠦⠆⠐⠔⠠⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠁⠈⣠⣀⠀⠈⢁⠀⡀⠐⠀⠛⠺⠁⢀⠌⠀⠀⡐⢄⣀⠀⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣼⠀⠄⠁⠀⡀⠀⠐⠀⠐⠦⠀⠀⠂⠈⢻⣿⣿⠷⡄⠀⠀⠀⣴⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠔⠄⠼⠄⠂⡀⠐⠀⠀⠀⠀⣀⡈⠀⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡴⠧⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢥⣠⣄⡀⠰⠚⡀⡀⠀⠀⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⡀⠀⠀⣄⠆⠀⡁⡶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣴⣄⢀⣴⠂⠀⢰⢠⡛⢳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡯⠞⠁⣠⣿⢀⠐⠛⣶⣦⣤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡷⠀⠈⠀⣷⠀⠋⠗⠠⢼⡿⠟⠉⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣄⣙⢮⠦⣠⣟⡤⠴⣶⢳⣄⠀⣤⠈⣥⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢪⢀⠘⡨⠂⣷⠃⣿⠃⠙⢏⠛⠘⠋⣀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⢈⣀⢶⢸⣿⡜⢦⣤⣀⣤⣳⠀⠉⠀⠤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⠀⠃⢈⠾⢧⣄⠀⣤⣜⡁⠍⠠⠈⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⢈⣐⠋⢆⠙⠏⠁⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣐⠞⣣⡾⣗⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡎⢶⣿⢦⡓⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⢾⠙⡗⣋⡀⠀⠀⠀⠀⢀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠉⠁⣲⣿⣫⣾⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⢀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠉⠁⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠟⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠘⢧⡀⠀⠀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⠳⣄⠀⠀⠀⠀⠈⠻⡄⡀⠀⠙⠳⣄⠀⠀⢀⣠⡴⠖⠋⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⠀⠋⠀⠀⠀⠀⠀⠀⠙⡀⠐⠂⠀⠈⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠻⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠘⠁⠀⠙⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆⠀⠠⣀⡀⠄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠈⠛⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⢢⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣈⠉⠁⠀⠀⠀⣀⣀⣀⠀⠀⠣⠘⠰⠐⠀⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢷⣄⠀⠀⠈⠉⠓⠍⠙⠋⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠙⠂⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⣿⣏⠷⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠇⠀⠾⠁⠀⠀⠀⠀⠘⠉⠁⠐⣄⡀⠶⢀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢦⠈⠘⣦⠟⢻⠀⠉⢠⠄⠪⢀⣤⢀⢆⡵⠏⣴⠖⣛⣻⣷⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⡤⠾⠧⠖⠉⠐⣶⠎⠀⢠⣶⣟⠛⢸⠃⢠⡿⠀⠈⢹⣍⠛⠧⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⢂⡀⠀⣸⣾⠃⠀⢀⣿⡿⠉⠻⣾⣇⠘⣇⠀⠀⢃⠈⠁⢠⢇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣁⠀⠀⠦⠼⣿⣠⢀⣾⣿⣷⠎⠀⢹⣟⠣⡌⠒⠶⣿⡴⢷⡟⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠊⢘⠿⣷⣄⠀⢸⣿⣮⣿⣿⡿⠀⠀⢈⣿⣄⠱⢮⡠⢄⣤⣎⡤⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠏⢀⠆⠀⠀⢻⡆⡯⢬⣿⢿⣿⣿⠂⠀⢼⣿⣿⢭⡒⣐⠺⡩⢿⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠈⠀⠀⠀⢸⣁⣿⣼⠋⠀⠻⢿⠿⠳⡋⠑⠹⡖⠂⢠⠀⢄⣸⡇⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠦⢄⣤⣴⡛⣑⣬⠟⠀⠀⠀⠐⠂⠀⠀⠀⠀⢷⡀⡆⢀⠘⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⢳⣤⣶⣇⣰⣿⡿⠁⠀⠀⠀⠀⠀⠋⠁⠂⢀⠈⢏⠀⠹⠖⠒⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⠃⠑⢂⠑⠲⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⠐⠁⠐⠀⠀⠀⠀⠈⠀⠉⢰⣿⠷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⣀⣤⡀⢀⠆⢐⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣉⣠⣿⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣄⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣁⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣒⠣⠀⠀⣷⡻⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢯⡉⣻⡦⡤⢀⣴⣿⠟⠋⠁⠀⠀⠈⠉⠳⢄⠀⢀⡴⠋⠉⠁⠀⠀⠉⠛⢿⣷⣄⠀⡀⠨⢖⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡧⣄⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠐⠉
⠀⠀⠀⠀⠀⠀⢠⣤⢘⡳⠿⠗⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠠⢆⡴⠋⠹⣷⣄⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠐⢸⣇⣷⡀⠙⡿⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⢁⢳⡇⣗⢢⣤⣄⣹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⣾⡷⢝⡯⣚⠷⣼⣏⢫⡛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠈⢓⣶⡬⣬⣽⣿⠶⣮⣽⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠸⣿⡖⢉⡅⣿⣾⠟⠁⣀⣰⣟⣋⣹⣟⣯⡻⢦⡀⠀⠀⣠⠾⡫⠵⠶⢶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣰⢻⣏⠀⠸⠛⢿⣷⠘⠧⢴⡶⠮⠢⠙⣆⡞⣡⠶⢦⡈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⢿⣶⣤⣤⣿⠟⠀⠀⠈⠉⠉⠉⠀⠈⠈⠁⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⢀⣀⣀⣀⠀⠀⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⠋⠀⢀⡤⠀⢄⠀⠀⠀⣔⡀⠀⠀⢀⠼⠓⠉⠘⢵⡀⢹⠀⢀⡀⠀⠀⠀⠀⠀⠀⡞⠁⣀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠊⠃⠎⠀⠀⠈⡇⠀⢀⣈⡀⠀⡄⠘⣆⠀⠀⠰⠋⠙⣆⠣⠈⢃⡙⠢⡀⢠⡀⠀⠳⣄⡸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠹⡀⢸⠀⡔⠲⢠⡵⠊⣡⣫⠁⠀⠀⠰⠉⠗⠖⡟⠒⠒⠋⠐⢷⡄⢇⠀⢁⡎⢩⠱⡀⠀⠀⢀⡀⠶⢂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡉⠂⡌⢦⠀⠀⠡⡘⡄⠇⠸⠈⠀⡜⠁⢸⠀⠀⠀⠘⠓⢒⠒⠁⠠⠢⣄⠀⠻⠋⡖⠷⣾⢌⠚⡀⡃⢠⡄⠓⢨⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠐⠁⡀⠑⠢⢄⡈⠸⣜⣶⢒⣸⠀⠀⣬⡰⣽⡀⡄⠠⠷⣃⠀⠑⢁⠌⡞⠘⠧⢘⣣⡈⢢⣈⡆⠋⠁⢓⠀⡜⡠⠊⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠢⡀⠸⠈⠂⡠⠞⢣⠀⡰⢆⡠⠝⠛⠛⠛⠛⠠⢞⣹⠈⢢⡶⠙⣦⠊⢹⣤⠞⠓⠈⠉⠐⠛⠤⡀⢡⠶⣙⠀⠀⡸⠩⣾⢊⠄⠄⢀⠞⢀⡀
⠀⠀⠀⠀⠀⠀⠇⠀⢡⠀⢃⣰⡀⠀⠀⠙⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣴⠀⠀⢡⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠋⠈⠋⢡⡜⣾⣧⡪⠔⠡⠐⠃⠚
⠀⠀⠀⠀⠀⣀⠘⡄⠈⡄⠀⠀⠈⢦⡜⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠇⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⣀⠀⣸⠟⠳⣂⡌⠠⠤⠄⢄
⠀⠀⠀⠀⠠⡼⠀⠘⠻⠴⣤⣹⢰⠿⡑⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢿⣵⠚⠉⠙⡟⠥⠀⠤⠤⠔⠋
⠀⠀⠰⡐⣠⡀⠀⣟⠰⡀⠎⠌⠛⡄⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣴⡏⡀⠂⠀⣹⣆⠲⡄
⠀⠀⠀⠀⠀⠑⠄⢁⠁⠙⢤⠀⠀⡰⠏⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣰⠏⠉⡙⠙⢦⣀⡜⠁⠈⠲⡙⡄
⠀⠀⡰⠈⠓⡄⣦⠸⠷⠤⠼⣍⣹⡀⠀⠰⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡀⠀⠀⣀⡸⠇⣗⠂⢤⡤⢼⡆
⠀⠀⢇⠰⣀⠆⢡⠠⡄⠀⠀⡜⠉⠙⣂⣵⡶⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠗⠉⠁⣻⢶⢋⠀⠀⠊⣄⠀⢳⡈⡁
⠀⠀⠈⢦⣀⠀⠀⠳⣍⣛⣿⡁⠀⠆⢀⣏⣄⢀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠀⠔⢁⣠⠿⠘⢸⣶⣀⡐⠹⢄⡏⠗⠀⠠
⠀⠀⠀⠀⠈⠉⢉⣉⠩⠤⠤⣧⣀⣠⠜⢫⠛⢻⢯⠂⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡴⠋⢰⠟⠐⢺⠟⠁⠄⠠⠀⠃⠡⡉⢦⠞⠀⠘⢄⠰
⠀⠀⠀⠀⠀⡔⠉⠤⡠⢴⠟⠭⣠⣯⡀⠀⠂⠀⠸⡷⡄⠴⢷⡍⣲⣄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠋⠐⠒⠒⠀⠀⢈⠀⣠⠆⢀⡞⠀⠆⠑⡀
⠀⠀⠀⠀⠀⠉⠈⠁⢠⠊⡠⡪⢻⣱⠉⢦⠔⠂⢺⢀⣇⠀⠀⢱⡀⠉⢳⢦⡀⠀⠀⠀⢈⡴⠧⠶⢆⠈⠱⡆⠘⠉⣩⣶⠊⠀⣀⠀⠀⠂⠒⠃
⠀⠀⠀⠀⠀⠀⣎⡟⢁⠊⠰⠥⢣⢣⢆⡍⠀⠀⣸⠬⡈⠻⣍⠀⢳⠤⣸⣬⣙⣆⠀⣠⡟⠉⠉⠉⠒⢧⡄⠀⠀⠴⠀⢠⠀⠸⡀⠃⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⢄⠂⠈⢇⡇⢀⡔⠹⠆⢌⣦⣬⡹⣂⡷⠦⣑⠢⣈⠱⠋⠺⠛⠻⣉⡉⠁⠀⠀⠀⠚⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠑⠁⠀⠀⠀⠻⢁⠳⡀⢛⠒⡜⠀⠑⣄⢡⣀⠀⠑⢪⣷⠠⡘⣦⠔⠣⡈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡉⠢⠷⠇⠑⢄⣌⢢⣅⠉⠉⠉⠉⠀⠑⡘⢄⣀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠜⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⢠⠂⢲⠀⡇⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠉⠉⠉⠁⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⠉⠉⠉⠁⠐⠠⢀
⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⢀⢁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠁⠀⠸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇
⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⡀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁
⠀⠀⠠⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠤
⠀⠀⠀⠀⠀⠁⢀⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⣿⣧⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢃⣀⡀
⠀⠀⠀⠀⠂⠀⠈⠙⢆⡞⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡶⠉⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⣠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠃⠃⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⣨⠀⠃⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⢨⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠻⡀⢀⠁⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢨⢠⠎⣀⠈⡃
⠀⠀⠀⠀⠀⠀⣀⠘⠓⠸⠄⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠠⠀⠄⠈⢸⡄⠈⠒⠁⡀
⠀⠀⠀⠀⠈⠁⠠⢀⡀⠀⠀⢠⠠⠠⠀⠀⠀⢀⡰⠋⠀⠀⡀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣄⠀⢳⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⢃⠐⠃⠀⠀⠀⠀⠁⠀⡀⠀⠈⠀⠀⠀⠀⢸⠄⠀⠄⢀⡀⢀⠀⠀⠀⠐⢦⡁⠀⠐⠀⡃⠈⠁
⠀⠀⠀⠀⠀⠈⠀⢀⣴⠂⠀⠀⠀⠁⠛⠀⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣏⠈⠁⠀⠈⠀⠈⠀⡠⠂⣼⠥
⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⢆⡀⠀⠀⣤⠶⠦⣄⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⣀⠜⠁⢀⠀⠀⠀⠒⣺⢙⡷⠂⠀⠀⠀⠀⠀⠀⠀⢀⢹⡅⠀⠀⣽⡀⠀⠠⢴⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠁⠀⠀⠀⠀⠀⠀⠀⠋⠉⠓⠂⠀⠀⠀⠀⣀⠠⡴⠛⠀⠝⠛⠛⠁⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠂⠀⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠀⣀⠂⠀⠀⠀⠠⠀⢀⠀⣀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣬⣧⣦⣤⣯⣴⣯⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠠⠘⢂⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡶⢘⠖⠀⢈⡘⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡿⡇⠀⠀⠀⣀⠀⠀⠀⣠⡉⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⣿⣧⡀⠀⠀⢈⠂⢠⠀⠸⠓⢻⠃⡄⢘⠁⠀⢿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⣿⣿⠛⣶⡄⠨⣿⣠⠀⠀⢀⣄⠤⠴⠆⠀⠀⣼⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⠇⠺⠿⠻⢟⠑⠾⡈⡷⠞⣀⠐⠺⡦⢰⣮⢙⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠃⠘⢉⣤⢎⣟⣰⠜⠃⠀⠀⠀⠈⡁⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣽⣫⡾⠟⢀⣴⡾⠶⠷⠤⠔⣷⠤⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⠠⣶⣿⣿⡭⣷⡀⣿⡟⠀⡠⠒⢲⣾⣻⠆⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠈⠉⠙⣿⣿⣿⡞⡛⢿⡵⠂⣴⣡⠔⢤⣇⠉⠀⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢀⠀⣿⣿⣿⢁⡄⠀⠁⢰⡟⢳⣀⣼⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣀⠁⠸⣿⣿⣸⠁⠀⢀⣾⠁⣻⢿⠟⢠⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠄⠀⠙⣿⣧⡩⠀⢀⠏⠀⣥⣠⣾⡿⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣿⣿⣴⠃⠀⣠⣾⡗⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠈⠉⠉⠀⠀⠒⠠⢀
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠁
⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣀⣀⡀⠀⠀⠀⣀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠑⠄⠈⠀⠀⠀⠀⢠⠄⠂⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⡰⠁⠀⠈⠐⠃⠀⠀⢀⠀⠴⠰⠄⡠⠐⠒⠂⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⢇⠀⠀⠀⠔⠁⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠑⡤⠰⠀⠀⠀⠀⠠⡀⠀⠀⠈⠄⡀⠀⠀⡀⠄⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠐⠁⠀⠀⠢⡀⠀⠀⡀⠊⠁⠰⠂⠀⠀⠈⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠐⡀⠀⠀⣀⠀⢈⠁⠀⠀⠘⣇⠀⠀⠀⠀⡸⡏⠦⠀⢀⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠈⠲⠉⠀⠀⠀⠑⢄⠊⠀⠉⠻⠒⡐⠎⡀⠐⡈⢤⠥⠐⠂⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠈⢄⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢑⢈⡐⠌⡖⡂⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠔⠀⠈⠡⠁⠀⠀⠀⠑⡀⢀⡴⠛⠚⠒⠒⠊⠵⣆⢹⡠⣕⠄⣀⠠⠊
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡁⠀⠀⠉⢊⢠⢻⠈⠀⠐⠩⢓⡪⠑⢬⡂⢀
⠀⠑⠤⠤⢤⠈⠢⠀⠠⢐⠀⠀⠀⠀⢀⠢⠀⡀⠀⡠⠐⠁⠀⠁⠂⠝⡲⠅⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠂⠄⠤⠂⠀⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢔⣥⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⣀⡤⠵⠂⠀⠀⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠠⠴⠀⠹⠭⣀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢨⠀⠀⠈⠱⠁⠀⠀⢣⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⣦⣄⣀⡀⠀⡗⠒⠲⠃⠀⢀⠤⢴⠀⠀⠀⠀⠁⠀⠉⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠇
⠀⠀⠀⢨⡯⠉⠉⠂⡄⠀⠀⠣⡀⢨⠀⠀⡁⢀⡠⠔⠁⢀⠠⠠⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡜⡸
⠀⠀⠀⠀⢙⣒⣦⡘⢆⠀⠀⠀⠈⠒⢦⣴⣿⣽⣳⢦⣔⠅⠠⢄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡰⠁
⠀⠀⠀⠀⠉⠻⠧⢥⠌⢉⠀⠉⠉⠩⢿⡿⣧⣛⣷⡻⡆⠀⢀⠔⠁⠒⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡴⠁
⠀⠀⠀⠙⣷⡞⠢⣌⠌⠀⠐⡇⠡⡈⠈⠹⢽⡽⣥⡾⠩⠭⠄⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⠎
⢀⣴⡃⣀⡈⣡⠀⠈⣇⠀⠀⠈⠠⣡⠀⠃⢈⠀⠀⠑⠂⢄⠀⠘⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠟⠁
⠛⠉⠉⠉⣾⣁⠔⠉⠈⠒⠠⠔⠀⢋⠉⠀⠀⠢⣀⠀⠀⢀⠇⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠊⠁
⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⢆⠀⢱⠄⣀⠠⠋⠈⠉⣩⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⠒⢠⣤⣮⣙⠲⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⢌⠿⣧⡀⠀⠉⠲⢤⣀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢵⠀⠑⣄⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣀⠤⢊⣛⠻⠛⢦⠀⠀⣰⣍⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠴⣤⠀⢰⡶
⠀⠀⠀⠀⠀⣠⢾⣁⠜⠠⣥⠐⢇⠱⡯⠀⠀⠘⠛⢯⠃⠀⠡⠀⢀⠌⠀⠺⠂⢈⠉⠀⠀⣀⢩⣀⠘⠁⠀⠀⠀⡀⢄
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠟⠀⠀⠀⡀⢀⠈⠀⠀⠀⠉⡤⠀⠀⠀⠾⢿⡗⠀⠀⠀⠋⣀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢤⣾⣿⣶⠀⢀⣤⣀⣠⡤⣄⠀⠲⢶⣽⠀⢀⣄⡀⠀⢛⠻⣿⣹⠇⠈⣁⣀⣤⠖⠻⠋⢠⡄⡀⠀⠀⠓⡹⠁
⠀⠀⢶⣄⢸⣿⣿⣿⠇⠉⠉⠀⠈⠀⠀⢤⣤⣀⣽⠔⠋⠙⠉⠐⢀⠀⠚⠣⠀⠈⠉⠁⠀⠀⠀⠐⠳⣟⢹⡇⠀⣶⡤⠄
⠀⠀⣼⠸⠀⠙⠁⠀⠀⣀⠤⢰⡄⢀⡀⠉⠟⣡⣾⠈⣷⡖⠂⠀⠀⡀⠈⠀⢠⢰⣤⣆⠘⠀⠀⠀⠀⡟⠈⠁⠈⠁⢀⡶⠒
⠀⠰⣷⠂⠀⠀⠀⠀⠈⢀⢹⣿⡖⠭⠀⢀⡈⠉⠉⠂⡀⠁⠈⣩⣁⡇⢠⠀⠀⢏⣩⣽⣶⠔⠁⠀⣤⡁⠀⠠⣀⡀⠈⠑⠲
⠀⠀⠃⣼⠛⢿⠁⠀⠀⠐⠀⡅⢅⢉⠀⢸⠿⠂⢠⣎⡁⠀⢸⡉⠛⢷⣤⢦⠀⠈⠙⢷⡪⠀⣰⠎⠈⠳⠺⠁⠙⡉⠙⠢⣦
⠀⠀⠀⠛⠒⠃⢈⣲⣤⠀⠀⠀⣀⠀⠀⠀⠀⠀⢞⠍⠄⠀⠘⠻⠒⠊⢻⠇⣀⣄⣀⠄⠀⠀⠜⢗⠊⠠⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣤⣍⣧⣠⣿⣿⣏⠀⠀⠀⣷⠀⠀⡀⠠⣸⣦⠊⠀⠀⣾⣿⣿⣿⣗⠀⠀⢀⠀⠀⡀⠠⡄⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠈⠁⣠⣷⡈⠻⡚⠀⠀⡀⠁⣤⡈⠀⠽⢿⡿⢉⠀⠀⢻⣿⣿⣿⣏⡀⠀⠀⠳⣶⣿⠐⠂⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠐⢄⠐⠋⡟⠃⡀⠀⠀⠈⢄⠀⡏⢱⠀⠀⠈⠁⠀⠡⠄⡈⠉⠉⠇⠈⠁⠠⠊⠀⠻⠋⣄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠣⢀⣶⣶⡀⠀⢿⠇⠀⠠⠴⠶⡷⣶⠏⠁⠀⠀⠀⠈⣸⡄⢂⣀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠠⣄⣀⡼⠛⠛⠂⠘⡠⢀⠐⠂⢩⠏⠓⠈⢁⠀⡀⠬⠩⣧⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠃⠀⠀⠤⣄⣱⠋⣀⢒⡀⠀⣼⣿⣿⣦⣀⢶⠀⡁⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⠠⠀⠁⠉⣩⠿⡅⠉⠀⠤⢹⣟⣷⣽⢛⢒⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠙⠛⠀⠐⠒⢚⣧⠼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣮⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⣀⠂⠈⣿⠀⠀⠀⣩⣇⠐⣤⠄⢠⣄⣤⡀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⠀⠀⣿⢰⠀⠀⢀⡥⠂⣶⠆⠐⣹⢿⢹⢣⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢧⠀⢀⡄⢹⡰⢀⠀⣸⣇⠀⠀⠀⠀⢀⣴⡟⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠒⠀⣀⠀⠈⠀⣀⠰⣀⣸⠇⣿⡇⣽⡦⢿⡦⢳⣤⡞⠀⠸⣏⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠃⠀⠀⢀⠴⢾⠁⠀⠀⡼⠉⠘⣧⠀⢀⠘⠃⣞⡀⠈⠃⠀⡟⠁⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠄⠐⠀⠀⠀⠱⡄⡈⢂⠀⠀⠉⠀⠈⠀⡻⢿⠿⡷⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⢀⡀⠀⣿⡅⢈⣇⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠀⠠⢤⡀⠉⠉⠛⠃⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠇⠈⠁⠲⠎⠐⠀⠀⠘⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢀⡀⡀⠐⢁⠀⠀⡘⠀⠠⠀⠀⡀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠷⢝⠀⠀⠀⠀⠇⠀⠛⠛⠋⣧⠰⠆⠁
⠀⠀⠀⠂⣜⠂⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⢠⣶
⠀⠀⠀⠀⠀⠁⠠⢠⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣄⡀⣀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⢀⡄⠀⠶⠞⢿⡏⠋⠀⠉⠁⣹⣤⡄⠀⣇⠰⡄⠀⢘⡆
⠀⠀⠀⠀⠀⠀⢨⡶⠙⠆⠀⠀⠀⠐⢀⠀⠉⠋⠀⠘⡠⢾⣅⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠉⢻⡟⢠⣿⡀⠈⢠⠈⠧
⠀⠀⠀⠀⠀⡁⠀⠀⠀⡀⠀⠀⠀⠄⠀⡀⠀⠣⠀⢸⢆⡏⠉⠁⠠⢾⡆⢰⠁⠄⠠⠤⢿⣄⣸⠃⠀⠤⠈⠲⣇
⠀⠀⠀⠀⡀⠀⣰⠀⠀⠟⠀⢴⠇⠀⣾⠉⠠⣼⣶⠀⠀⠑⢄⣦⠀⠈⠁⠀⢸⢷⡀⠀⠀⠻⢿⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠈⠟⠓⠀⠐⠀⠂⡆⢰⠋⠃⠀⠸⠾⠁⠀⠀⠀⣱⠄⠠⠒⠄⠸⠋⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⢀⢇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⠁⠀⠀⠐⠒⠀⠀⠀⢀⠀⠁⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⠠⠖⠁⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠆⠀⠀⢠⠀⠀⡀⠟⠁⠀⠀⠀⠰⠄⠶⡀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⢀⡄⣠⠂⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⢀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠠⠤⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡤⠀⡀⠐⠀⠀⠂⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡀⠀⠃⠠⠀⣢⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠎⠁⠀⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣩⣿⣭⣭⣯⣥⣭⣽⣿⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣿⣿⣯⣭⣭⣯⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⢸⡦⠞⠛⢛⣓⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢖⣛⣿⣷⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⠀⠀⠀⡄⢡⡮⠂⢉⣽⠿⠿⠿⣿⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⣶⡿⠟⠉⠉⠉⠉⠻⣿⡹⡄
⠀⠀⠀⠀⠀⠀⠀⣈⣆⠈⢕⣦⠖⢚⣛⢛⣿⣋⠀⠀⠀⠀⠀⠉⠳⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠘⠻⠤⣈⣷⢞⠿⣒⣁⠀⢹⢄⠈⢦⠀⠀⠀⠀⠀⠀⠈⠻⣿⢷⡀⠀⠀⠀⠀⠀⢠⡟⣡⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡎⡰⠯⡹⢎⠈⢣⡼⢄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⡾⣱⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢇⠑⡗⠒⢣⠾⠔⣊⡇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⣸⣧⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⣈⣁⠽⠒⠋⠉⠙⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢠⠤⠤⢄⡈⢓⡿⠧⡴⠻⠥⠰⢄⡜⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠢⢬⢏⢆⡤⢧⡀⠀⠀⡜⠉⢹⢢⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⣠⠐⠪⠕⣾⠁⢸⠀⠀⠨⢿⣾⡄⠀⠰⣼⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢢⢾⡀⢀⣠⠋⠀⠈⠙⡖⠁⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠸⣄⣀⡤⠤⢧⣤⠫⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣒⠤⠤⡎⠀⠀⣠⠞⠈⢃⠀⢠⠟⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⡡⠻⡈⡟⠉⠉⠓⢤⣞⣿⣷⡄⠔⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠃⠑⠃⠈⠒⡶⠖⠛⢮⣻⡻⠣⢄⡀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⢰⠉⠈⢧⡀⠀⣣⡴⠊⢁⠝⠁⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢤⣠⢮⠻⠙⢿⢻⣾⣗⠀⠒⢄⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡷⣄⢰⢄⡉⠛⢿⣿⣄⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠜⠀⠀⠉⠀⠀⠀⠀⠉⠛⠛

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

⠀⠀⠀⠀⠀⠀⠠⡀⠠⠒⠀⠀⠀⠀⢀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⣄⡄⠤⠤⢬⣽⣤⣂⠀⢗⣁⡀⠀⠀⠀⠀⠀⢘⠑⠀⡒⢢⠄⠀⣀⡸⢭⢾⢚⡄⢠⣦⣌⡇
⠀⠀⠀⠀⠀⢀⡴⠋⡤⠀⣄⠀⠨⠐⠁⠈⡠⠛⣦⠈⠆⠀⠀⣀⣀⡈⢲⡎⡹⢻⢚⣯⣾⣟⣤⣾⡄⢿⡿⠀⠈⠆⢠⣄
⠀⠀⠀⠀⡀⡋⠀⠄⠈⠀⠁⠀⠀⠀⠈⠝⠟⠅⡄⠠⠀⠀⠙⠤⢴⠏⣅⠀⠀⠈⠀⡎⠁⠈⡈⠛⠁⠶⠀⣔⣯⣇⢙⣋
⠀⢤⡄⠙⠛⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⡀⠔⠉⠀⣿⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⢋⣤⡘⠛
⠀⠉⠉⢠⣷⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠂⠺⠀⠈⠢⣼⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡁
⠀⠀⠀⡠⠇⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢾⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠗⢶⣾
⠀⠀⠺⢾⡇⠳⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢳⣤⡝
⠀⠀⠀⠁⠑⣦⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⢘⣱⠙⢯⡟
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢲⣁⠌
⠀⠀⠔⡱⡙⠄⢠⣢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣿⣿⣀
⠀⠀⠋⠀⢯⠦⠑⠻⠓⢁⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⣶⢿⠉⠈
⠀⠀⠀⠀⠀⠑⠀⠀⢠⣞⢠⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣠⣿⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠸⣻⡂⠀⣜⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣦⣌⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⡗⣄⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣠⡈⠁⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⢗⡷⢲⣴⡠⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⣀⢭⡛⠑⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢡⣇⣀⠀⠀⢀⠀⢀⣀⣀⠒⠋⠂⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠃⠉⡛⢢⡂⣤⣾⡇⠇⢱⠀⠊⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⠚⠀⠈⠊

⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠔
⢀⡀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⢃⡴⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢃⡠⠂⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⣀⡀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠹⢟⡦⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠹⣋⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠰⣦⡄⡢⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠰⣶⡄⣀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠠⠃
⠀⠈⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠑⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢁⡰⠃
⠀⢀⢀⠤⠀⠀⠀⠀⠂⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢀⢀⠤⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠘⢠⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠞⠁
⡔⢂⠀⠀⠀⠀⠀⠀⠠⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⢂⠀⠀⠀⠀⠀⠀⠀⠁⡠
⠔⠋⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⢶⡦⡰⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢶⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⡀⢀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠈⠻⢡⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠈⠻⢡⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠌⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋
⠀⣐⣲⣉⡄⠀⠀⠀⠀⢀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣲⢉⡆⠀⠀⠀⠀⢀⣀⡂
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢈⣳⣴⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠂⠀⢀⡀⡌⠀⠀⠉⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠁⠀⠀⠁⠈⠀⡀⠒⡛⠀⡀⠠⣶⣼⡅⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣺⣑⣤⠠⣐⡞⠁⠈⠁⢘⡂⢦⣀⠩⢻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⣿⣿⣏⡶⢛⠀⢀⣴⡖⠀⠂⠤⣿⠷⣴⣤⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣤⡨⠀⠛⠡⠟⣠⡀⠺⢿⡿⣅⡂⠀⢀⣴⣿⣿⣿⠟⠁⢀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣿⡇⢀⠀⠀⠀⢹⡇⠀⠈⠀⠛⠁⠖⠲⡿⠛⠋⠁⠀⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⣤⠘⢧⣌⢦⣀⠀⣭⡷⣀⣀⣤⣴⡾⣇⡀⠐⠤⢀⡔⠊⢩⡄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⢄⣀⢠⡜⠒⢲⡘⢦⠘⣧⢉⡞⢹⣾⡏⠡⣤⣼⣇⣝⢲⣦⡤⠘⠃⠀⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⣠⠄⣧⡜⢀⢿⣼⡇⠹⡟⠀⠀⠘⠳⣈⠻⠀⠿⠀⣠⡆⣀⡄⠀⠈⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣍⡵⠚⠃⢀⣸⡄⣿⡿⣄⠳⣄⣀⣠⠀⡘⣠⢃⡧⠐⠛⠣⣠⣄⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⣱⠒⣾⠊⡡⠆⢹⡔⡉⠀⡈⠛⠛⠛⣿⢋⠮⠷⡀⠀⠄⠀⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠍⡀⢠⠀⠀⡻⠛⠻⢤⣀⠀⡰⣦⣤⣅⠤⠲⠀⢠⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⡄⢢⠈⠅⠶⠛⢢⡀⠙⠿⠇⠋⠁⠀⠀⠀⠀⠀⠀⡴⣾⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠈⣠⣌⣓⢶⣄⡈⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠈⠱⣤⡿⢍⣹⢞⣫⡭⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣰⡻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⢣⢣⣴⡒⣿⣆⢸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣱⣮⡊⠠⣿⡿⢛⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⡷⠉⣝⡉⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⠛⠙⠓⢶⣶⡲⠗⣠⡏⣰⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣾⣷⣯⣶⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋



⣿⣿⡿⠛⠟⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇
⠀⢠⣄⠀⠳⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡰⠃⢀⣤
⠀⣈⠿⠇⣷⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⡇⠿⢏⡀
⠀⠹⣿⠀⠹⠿⠃⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠻⠿⠁⢸⡿
⠀⡔⠁⣉⠙⠢⡁⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⡡⠚⢉⡅⠑⡄
⠀⠑⠶⠋⠀⠠⢘⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣞⠠⠀⠈⠳⠖⠁
⠀⠀⠀⠀⠀⠀⢙⡈⣆⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢂⡞⣘⠁
⠀⠀⠀⢤⣤⡄⠀⢿⠟⣎⣥⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣍⡞⢿⠏⠀⣤⣤⡄
⠀⠀⣠⣤⠉⣡⡁⢀⣀⠘⢿⠟⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢿⠟⢀⣀⠀⣠⡉⢡⣤⡀
⠀⠐⠛⠁⠀⢿⠃⢸⡟⢠⣾⠗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢿⣦⠘⣿⠀⢻⠇⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠈⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣆⠀⠀⠀⠈⠉⠈⡃⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠠⠖⠀⠉⠀⠶⣶⣶⢄⣈⠛⣻⠺⠟⢧⣶⠾⠁⠀⠀⢀⡀⢲⡌⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⣠⠀⠀⠀⣀⡤⢒⠲⢶⡄⣘⠉⠙⢳⡙⠿⡶⣦⡾⠀⡀⠀⠀⠀⠂⡈⢡⣀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠂⢀⢂⡶⣰⣶⡿⣨⣷⢻⣧⠀⠀⠹⠀⣧⣈⠳⣼⣃⠀⣶⠞⠁⢁⢈⣟⠃⠄⠀⠀⢀⡠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⡏⢀⢈⣹⠯⠖⠀⣽⡇⠈⡷⠀⠰⠆⣿⠀⠀⠈⠋⠁⠠⠄⠀⢮⣭⣽⡇⣠⡎⠀⣼⠆⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢻⡈⠻⣦⣀⣠⡵⠞⢁⠀⠁⠀⢠⠟⢮⡑⠛⢦⠀⠀⠀⢂⠀⠀⢈⡉⠉⠈⠰⠀⠉⠒⠃⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠷⣎⣉⡁⡠⣤⠈⠀⠀⠆⠁⠀⡜⠃⠀⠈⠁⢀⠀⢘⠀⣀⠒⠁⢀⠀⢠⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠤⢤⡄⠀⠐⢄⡈⠀⠀⠀⠈⠃⠀⠀⠟⠀⠀⠁⠀⠀⠀⠀⠀⣀⣤⣴⣤⣾⠿⠟⠋⣷⣤⡖⢄⠂⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠚⠁⠐⠈⠻⠆⠀⠀⠋⠁⠀⠑⢤⡄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣴⡿⠿⠛⠉⠀⠀⢀⣤⣿⣿⠙⠃⡄⠀⣠⡄⢀⣠⠄⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠠⠤⠜⠋⠀⠐⠀⡀⠈⠐⡄⠀⠰⠄⠤⢤⣶⠟⠁⠀⠀⠀⣼⠟⠀⠀⠀⠀⣠⣾⢿⣿⠿⡃⠀⢀⡠⠀⠉⢴⣮⠉⢲⡆⠉⣤⡀⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⠠⠀⠀⠀⠁⠈⠠⠀⠀⣼⠇⠀⠀⠀⢈⡾⠁⠀⠀⠀⢢⣾⣿⠛⠺⠷⣾⠹⣷⠟⢁⣀⠐⠁⢀⡠⠀⢱⡀⠈⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⡅⠀⠀⠘⠃⠀⠀⠘⠀⢀⣿⠀⠀⠀⣀⣾⢃⠀⠀⢀⢴⣿⡏⠁⠀⠀⠀⠘⢿⣏⠘⠏⠀⢠⠟⠱⣷⣶⣆⠰⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣆⠀⣦⡄⠤⢀⡀⠀⡀⠠⠚⠛⠦⣀⣠⣿⠧⠸⡄⣠⡍⠈⣿⡷⣄⣀⡀⢀⣤⣼⣿⡞⠃⠀⠉⠀⣄⡐⣽⣟⡄⠁
⠀⠀⠀⠰⢦⡀⣨⠛⢰⡎⠉⣾⠁⠡⣀⢹⣡⠈⠀⠀⠀⠀⠀⠉⢿⣿⠖⠃⠋⠀⠐⠀⠀⠱⡌⠉⢿⣿⣝⠛⣿⣿⣆⠀⠀⠀⠈⢹⣿⠿⠁
⠀⠀⡛⠂⡶⠀⠛⠀⠈⠀⠀⠁⠀⠐⠋⢩⠅⠀⠀⠀⠀⠀⠀⠀⠌⣻⠆⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡹⡷⠀⠀⣞⢹⣶⢶⠶⣷⣄⣀⠀⠋
⠀⠘⠁⠀⠀⠀⠀⠀⣠⣤⢄⣀⠰⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣠⠟⠦⣄⠀⠏⠪⣿⡄⠀⠀⠁⠈⠱⡀⢦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠠⣼⠿⢧⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠘⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠡⡇⢠⡀⠀⠑⣄⠀⢻⡧⠼⠆⠀⠠⠀⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠎⠀⠠⠘⢆⠀⠂⠀⢀⣫⠟⢺⠉⠹⡄⠈⠀⠀⠀⠀⠀⠀⡆⡶⢽⠆⠈⠀⠀⢠⣤⣿⠉⠀⣠⡶⢡⣤⡈⠃⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠚⡰⠐⠞⢀⠊⢢⣄⡰⠾⠏⡾⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡇⢸⠂⠀⢀⡴⠉⢈⣿⠀⠀⠈⠀⠀⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠈⠀⠀⠠⣘⢻⣿⢳⡎⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣶⠈⠇⢀⡎⢇⠀⢸⣟⠀⠦⠘⠃⠐⢀⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡄⠀⠀⣻⡯⠗⣞⠂⠄⠒⠀⠄⠀⠀⠀⠀⠀⣀⠏⠘⡆⠀⠘⠃⠊⠀⢸⣧⡄⣀⠀⠀⠀⠋⣐⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣠⠀⠸⠀⠿⢶⣔⡃⠀⠀⠀⠀⠀⠈⠀⢀⡞⠏⠀⠀⠿⠀⠀⠀⠀⢠⣼⣏⢀⡥⠤⠀⠀⠘⠫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡹⠂⠀⢠⣤⠔⣿⠻⠦⣦⣄⣠⠀⠀⢀⢼⡇⠀⠀⠈⠀⠀⠀⠀⠀⢠⣿⡟⠀⠁⠘⠘⠃⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢦⠀⠀⠀⢬⠹⠟⠻⣿⣧⣼⣧⠀⠀⠀⠀⠀⠀⠂⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢦⡀⠀⠀⠰⠖⠆⠀⠀⠀⠙⠻⢿⣶⡰⣦⠀⠀⣀⣠⣶⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢈

⠀⠀⠀⠀⠀⠀⡀⠀⡠⠐⣂⣉⣭⠭⠭⢄⣢⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣂⣬⠭⠭⣭⣐⣢⣄
⠀⠀⠀⠀⢀⠜⠷⣊⠔⢩⠔⠁⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠶⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠑⠪⡳⡄⡀
⠀⠀⠀⠀⡌⢸⠀⡅⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢔⠘⡄
⣦⠀⠀⠰⠀⡀⠀⣷⡁⠐⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘
⠹⢌⠢⡄⢣⣇⣰⠋⠀⡘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠸⡀⠑⠌⡆⡏⢸⢀⡔⢁⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⠢⠬⡷⠓⢴⡋⠉⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠰⡀⠇⠀⠀⠈⢲⠔⠁⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠞⣧⡀⠀⠀⠁⠆⠀⠀⡗⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⢰⠊⠁⠀⡈⠴⢾⣄⣀⠐⠴⢖⠒⠒⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰
⠀⢀⠱⢄⠀⠀⡠⠞⡿⣍⠐⠀⠀⠉⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⠃
⠀⣊⠤⠒⠉⢹⠀⠰⠀⢂⢀⠤⠠⡀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠂
⠀⠉⠀⠒⠒⣊⡄⠀⠀⣠⠁⠀⠀⢷⡎⡠⠒⠂⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠊
⠀⠀⠀⣠⡊⠤⠌⠲⠌⣸⠀⠀⠀⢸⠏⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠠⠃
⠀⠀⠀⠋⠒⢠⠤⠒⠒⠚⡀⠀⠀⡎⡰⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡔⠁
⠀⠀⠀⠀⠀⡃⠀⠀⠀⢀⣈⠺⣤⣿⡠⣒⡀⠤⠶⢥⡘⡠⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⠊
⠀⠀⠀⠀⠀⠈⠉⠢⣀⠀⢀⡤⡛⣿⠏⠭⡁⠀⠀⠀⠘⡇⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠂⢈⠋⠘⠀⠙⠈⣆⠈⠀⠀⠀⠀⡸⡰⠃⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠸⠑⣂⣤⠔⠉⠲⡁⠀⡀⠀⢸⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⣀⣀⡠⠊⢸⡁⠀⠀⠀⠠⣘⢀⡧⠂⡉⠓⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠔⠂⠈⠉⡢⠶⠐⡢⠼⣿⣶⡁⠄⠀⠀⡸⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢒⠞⠁⠀⠀⠀⠜⠋⢋⠊⠲⡄⠚⠠⠤⠤⠍⡦⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡀⠠⢠⡎⠀⠀⠀⠀⣽⡓⢦⠀⠒⡈⠀⣀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠢⠠⠤⠠⠉⡿⠐⠨⡙⠲⡇⠤⢀⣀⡑⠂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠱⡀⠐⡔⠉⠒⠄⠀⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⣦⣴⡀
⣿⣿⡿⠻⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣶⣀⡀⠀⠀⠚⣣⣶⣛⣻⣝⣧⣤⠆⠈⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡤⠂⠀⢀⣴⡞⢻⣂⣤⣌⠳⣄⣂
⠀⠀⠀⠀⠀⠀⣟⣴⣿⣏⡙⣿⣻⡃⢰⡞⣿⣾⣿⣟⣿⡹⢿⡦⠀⠀⣿⣿⠷⣤⠀⠀⠔⠈⢀⣔⣩⣿⠳⣇⠀⢸⡟⡽⠻⣟⢲⣯⣛⢦⣿⣶⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠉⠛⣿⡟⠀⠈⣇⣹⣟⣻⣾⣷⠋⠀⠀⠀⢧⡟⣷⠘⢳⡀⠀⢰⣧⣤⣙⢦⡀⣿⢀⠈⢴⡇⠲⢼⠾⢿⠛⣹⢿⠀⡌
⠀⠀⠀⡠⠊⠉⢻⠶⣆⣤⡔⠁⠀⠀⠀⠀⠿⠶⠿⢿⡅⠈⠀⠀⠀⠀⠈⠻⣧⣷⣸⡇⠘⢣⣀⣜⢋⣰⣇⠀⠀⠀⠀⠙⣷⣄⢻⠦⣸⡇⠘⠛
⠀⠀⠈⢁⣠⣶⠀⠀⠈⣉⣀⠴⣃⣠⣦⠠⡄⠐⠀⠚⠁⣀⠰⣖⣺⠗⢤⣤⠈⠙⠿⡇⠀⠀⠉⠉⠀⠈⢛⡤⠴⠶⣤⡀⠈⠿⠋⠏⠟⠁⢀⣶⣿⣞⣷⡀⡦
⠀⠀⣰⣾⠏⣿⠀⠀⢀⣏⡻⢟⣫⣯⠛⢷⡈⠂⠀⠀⠀⠕⢸⡿⡍⣳⣼⠿⠐⠀⠀⠀⢠⡾⠋⡽⢿⣷⡌⠻⣖⣂⡼⠇⠢⢀⠀⠀⠀⢐⣫⠝⣩⡉⠙⢿⢁⡇
⠀⣼⠻⣿⠁⡿⣤⠠⣟⢿⢙⣿⡯⣹⡿⣯⣿⠃⠆⠀⢺⠰⣿⣟⢗⣬⣉⣷⡗⠀⠀⠀⢸⡟⣰⣧⠘⣹⠻⠀⠀⠈⢠⠆⠀⠀⠑⢄⠀⣸⣿⢃⡶⣻⠗⣨⣿⡇
⠀⢯⣳⠷⢲⠇⢹⡀⣿⢸⣿⣽⠞⣿⣵⢛⡼⡤⠂⠀⠉⣻⣙⡷⠚⠐⣯⠸⢃⡠⠐⠒⠙⠻⣯⣍⣉⡿⠃⠀⢠⡄⢷⣔⣀⡤⠀⠈⠀⣯⡫⣽⡾⠥⠚⠁⣸⡇
⠀⠘⢿⠷⠋⠀⠀⢹⣿⣿⢷⣶⣛⣽⠿⠛⢁⡷⠀⠀⠀⠈⠻⠤⢴⡏⠀⣠⡼⡶⠶⣿⡒⡄⠀⠈⢡⠔⠚⣶⡾⢶⡿⢋⣰⣞⠫⣦⠀⠋⠙⣦⣍⣳⣾⡟⠘⠃
⠀⠀⠀⢀⣴⡆⣤⠀⢀⡽⠓⢶⡶⠶⠶⠟⠈⠀⠀⣄⣄⣀⣄⡀⠀⠀⢀⣡⡼⢹⣆⠘⣇⡸⠆⠀⠀⠀⢸⡿⠃⠛⣾⣿⣿⣿⡔⠏⠀⠀⠘⠋⠸⠋
⠀⠀⠀⢘⡛⢳⡟⡄⠀⠙⠄⢛⣀⡀⠀⠀⠀⠀⠀⠏⡧⠀⣼⣟⢳⠀⠨⢯⣡⣬⣗⣭⣑⠃⠀⣀⠀⠁⡎⢀⠀⣶⣄⣻⡽⣿⠳⠀⢰⠁⠀⠰⣾⡛⠻⡀
⠀⠀⠀⠀⠙⢷⠟⠀⠀⠀⣰⡟⣍⣙⣛⣦⣄⠀⣰⠲⣷⠺⣿⢧⣴⣄⠀⠘⠇⢿⡭⠀⠟⠁⡘⣿⡇⠀⡙⣟⣍⣻⡿⠿⡷⠟⠀⠀⠀⠀⢺⣾⡿⠛⠺⡇
⠀⠀⠀⠀⠀⠈⠀⠁⠀⣼⣿⡛⠉⢉⣉⡼⠃⠚⠋⠛⠿⠦⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠃⠀⠀⠀⠀⡉⠁⠀⠀⡄⠀⠀⣠⣄⣈⢻⣤⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠱⡄⢸⣻⡏⣿⠙⠛⢉⡀⣠⣴⣶⣀⣀⡀⠀⠉⠃⠀⠀⣤⣤⢳⣾⠃⢠⡆⢸⠟⠀⠀⠀⣠⢶⣿⢶⣠⣶⡄⠀⠀⣟⠀⣙⡆⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⢿⣇⡾⢀⣸⠃⢰⡿⣶⣿⠾⣏⣇⢲⡀⠀⢸⣿⣟⣠⡈⣿⡧⢸⣇⡴⠀⠉⠀⠀⡻⣫⢩⣵⣌⣃⢻⡄⠀⠙⠦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣽⣇⢸⡗⢳⣷⣿⡴⣿⣏⡿⡁⠀⣭⡿⠿⠉⣇⡃⢱⣿⣿⡆⠀⠀⠀⠰⣷⣽⣿⣦⡬⢿⡄⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣧⣹⠿⣷⡿⡗⢋⣥⠃⠀⣩⡁⢴⣶⣿⡿⡻⢣⠟⢰⠄⠀⠀⠀⠁⠠⣬⣭⣴⡞⠎⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠻⠟⢷⡮⣴⡯⠏⠀⠀⠩⣙⠲⠛⠋⣰⡷⠋⠀⠁⣠⣴⠶⠶⣤⡀⠀⠏⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠁⣠⡆⠀⠀⠀⠀⠛⠂⢀⠋⠀⠀⠀⣴⣿⣶⡶⣴⣬⣝⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣃⡰⠶⢷⠆⠀⠀⠈⠀⣀⣖⠠⡌⠻⣶⣤⠈⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡍⠀⠀⠀⠀⠀⢠⡿⢯⡈⠙⠲⣤⡄⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣿⣷⠺⠛⠓⡎⢙⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⢷⢿⣶⣢⣤⣾⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⠛⠿⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠰⣿⡉⠟⣧⠀⠀⠀⢠⣶⣿⡿⠛⠉⠉⠁⠉⠉⠙⠳⣤⡀⠀⠀⣠⠞⠋⠉⠉⠁⠉⠉⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠃⣈⢹⢃⡀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣦⣩⣿⣨⢱⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠺⠿⣠⣜⣛⠚⠁⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠾⢢⡞⠉⠉⢿⣷⡄⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠰⠖⣿⢸⣷⡀⡈⢻⡟⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⡟
⠀⠀⠀⠒⠸⣞⡇⣷⢬⡉⣀⡈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣿⡿⠁
⠀⠀⠀⢀⣷⣜⢮⡻⣄⣿⡏⢳⡞⢟⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠚⣠⣾⡿⠋⠀⠀⠀⢀⡀
⠀⠀⠀⠘⣿⡏⠑⠮⣓⡯⠽⠚⣳⢬⣧⡈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⠗⣉⣵⡾⠟⢉⣀⣠⣤⢴⣶⡿⠁
⠀⠀⠀⠀⣿⣇⠀⢀⣈⣹⣿⢗⣦⣾⡿⠿⠿⠷⢾⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣤⢈⣭⣶⣿⡿⠷⠚⠋⠉⠉⣁⣴⣾⠟⠁
⠀⠀⠀⠀⢿⣿⡏⠉⣠⢺⣿⣿⠟⠁⢀⣀⡻⣍⣉⣙⣫⣟⢯⡻⢷⣄⠀⠀⡀⣫⡾⢟⣽⣿⣷⣶⣶⡶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⢹⣿⣾⠁⡞⣿⡏⠀⠰⠟⠻⣿⣦⠹⣿⣤⣤⣵⣭⠢⡙⢷⡐⡾⢋⣴⣯⣥⣄⣀⠀⠀⠙⠳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠹⣿⡇⢻⣿⣄⡀⠀⣀⣿⣿⠀⠀⠀⠻⠷⠶⠶⠉⠀⠻⠡⠿⣍⡉⠉⠻⣯⡛⠷⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⡒⠒⠒⠶⠦⣤⣀⠀⠀⣠⠔⢚⢉⡉⢩⡟⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡀⠲⡝⠊⡔⠈⠉⠒⢤⡉⠳⣴⠁⢰⣥⠀⠈⠆⠠⠌⠙⠟⢷⡄
⠀⠀⠀⠀⠀⢠⢞⣤⠀⣖⠀⠈⠁⠀⠓⠄⠓⠀⠀⠱⡐⣿⠀⠖⠂⠀⠀⢁⡠⠒⢻⡘⠄⢻⡄
⠀⠀⠀⠀⠀⡞⠉⣀⣁⠀⠉⡓⠚⠒⠒⣀⣒⠒⠄⠀⡇⣌⠆⠀⠒⠁⠊⠉⠀⠀⠈⢙⢦⠀⣷
⠀⠀⠀⠀⠀⡇⡜⠈⠘⠁⠓⠀⣠⠊⠅⠀⠀⡉⢦⠀⠁⡀⠀⡀⠈⡆⠐⡲⠀⡴⢀⡀⢢⠃⢸
⠀⠀⠀⠀⠀⢷⠘⢄⣀⡀⠀⠊⠸⣠⡄⣀⡀⠐⠊⡆⡌⠀⠀⠑⠈⢀⠥⠖⠂⠉⠒⢊⣱⠎⣼
⠀⠀⠀⠀⠀⠘⣧⢠⠦⠄⡔⠘⠓⢦⡰⠅⠃⡈⠰⠀⠀⠀⡀⠀⠲⡌⢓⡤⠄⢠⠀⢧⠄⣣⡏
⠀⠀⠀⠀⠀⠀⠘⢯⡋⠀⡄⠀⢀⠉⠁⡀⠀⠂⡆⠀⠠⠀⠁⣈⡠⢃⡀⠈⠳⢀⡑⠀⠂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⢆⡀⠀⠃⢸⠀⠁⡄⠄⡃⢀⠤⠥⡀⠘⢄⡀⠀⠀⢈⠕⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡈⠒⠠⣸⡀⠀⠱⣥⢠⡓⡀⠀⠈⠎⡠⠒⠐⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠙⠀⠠⠄⡄⠉⠁⠀⢀⠏⠀⠲⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡑⠢⠇⠀⠀⠀⠁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡺⠏⡼⢋⡴⣫⢸⣮⡻⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⡀⣴⡞⣀⣉⢡⡴⣶⡴⣿⡟⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣇⢄⠘⣁⣠⣴⡾⠛⠁⠉⢉⡡⠨⠝⣿⣷⢠⠀⠀⠀⢠⣾⢏⠟⣾⣳⡿⢃⡗⠛⠻⠹⣿⣿⡿⣊⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣮⡻⣿⣿⣿⣯⡟⠟⡁⣾⢫⡽⠦⡍⡀⣠⢻⣞⣮⡃⠀⠀⣻⢯⣿⡷⠃⠉⠐⣋⠁⠛⡆⣌⣡⣶⡎⣿⣫⣘⡃⢄
⠀⠀⠀⠀⠀⠀⢰⣎⣷⣶⡾⣽⣯⣭⡰⢃⣁⣿⡘⢁⢁⣰⡇⣿⡧⣫⣶⠟⠴⠯⣟⢷⡙⢟⠁⠐⠆⣷⢭⡖⣠⣅⣿⣿⢟⠠⢍⠭⠛⠶⡦
⠀⠀⠀⠀⠀⢀⡿⢿⣿⡛⠙⣫⣅⢽⢶⣲⣝⢂⡳⠈⠙⣠⡜⣴⣿⡼⠟⢶⡾⣀⣓⣃⠹⡎⡧⣦⡄⢘⣂⣼⣧⢸⡿⣵⡿⡉⠀⣶⠈⣝⣵⡄
⠀⠀⠀⠀⠀⠀⣾⠿⡋⠂⠙⠮⢽⡅⢮⠏⡏⣤⣐⢷⣾⣭⢣⣿⡿⣋⣐⠠⢚⠩⢭⢍⠷⠄⣿⣮⡱⡿⠿⠿⡿⠘⢯⣻⢳⣿⢆⡹⣤⡐⠃⠖
⠀⠀⠀⠀⠀⠀⣷⡾⠇⢠⠏⣀⢦⡝⣶⢸⢇⡾⢿⣷⣯⣯⢸⡿⣽⣿⣤⠂⠿⠀⢓⡽⢣⠀⢹⢟⣅⠿⢟⠛⠀⢯⡚⢠⣜⠻⢎⣴⣶⠦⣤⣤
⠀⠀⠀⠀⠀⠀⠻⣪⣍⠻⡄⠎⠿⠒⠃⣰⡟⢰⣶⣮⠷⠶⣥⠳⡽⢧⣿⡣⠞⣛⣓⡀⠁⡡⣰⣿⣿⠹⡿⢀⡴⠂⢠⡈⠿⢾⡢⢠⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢭⣟⡷⠄⠐⠋⢀⣼⣿⡇⠛⢏⢘⣋⡶⢦⡄⠁⣶⣍⡛⠸⢟⣻⣽⠖⣾⣿⠿⠃⣊⣟⢺⡄⣜⣴⡅⢟⢠⣉⡏⣽⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠲⠭⢛⣵⣾⣶⣾⣶⡆⢶⠍⠿⡭⠁⢞⣱⢖⢠⡈⠛⡪⣤⣤⣴⠖⣡⣽⣾⡿⠀⢽⣻⢧⠁⠠⠖⠀⣠⣿⢇⡑⣯⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣭⣟⠿⣛⢽⣾⣿⢄⡭⠁⢳⣄⣁⣥⠈⠀⢻⣿⣷⣫⢰⢶⠰⡫⣍⢰⢷⣻⣮⢛⣤⣶⣤⣬⣽⡛⣾⡇⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢘⡛⠙⢿⣿⣷⢘⣃⢲⣦⢸⢩⣶⡟⣽⢇⠛⣵⣿⠇⠳⠨⠬⠭⣅⠲⠕⣿⠙⣛⠿⣛⣛⣟⢑⡹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣓⢁⠈⠛⣾⣿⣏⢛⣭⣧⣋⡚⣉⣯⣘⢋⣵⣿⠀⡴⠂⡰⣬⢳⠆⡿⢙⣼⣾⣿⣿⢯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠃⠰⢾⣿⡟⠾⣟⣯⡹⣞⣯⣍⠩⣝⢶⣖⠒⢦⡠⠵⠿⠙⢂⡞⣾⣭⣷⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣞⠿⣰⣿⣿⢟⠘⠫⣿⣾⠗⢶⣬⡛⠸⢦⠁⠈⠋⢠⣾⡷⠛⣻⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠰⣿⡿⡃⠠⡤⢀⡲⠶⡜⠳⣿⣸⣮⢳⣡⣾⣶⣿⣿⣸⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣧⠀⠱⡯⣲⠀⠿⠰⣉⣿⢯⣿⡏⣿⡻⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣭⣤⠖⡣⠂⣵⣿⣿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠚⣿⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡲⢤⣉⠳⣄⠉⠛⠄⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⣱⣾⢡⡀⠁⣎⠉⠉⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⣷⠀⠙⢂⣘⢧⣶⡶⣂⡿⣠⢦⣄⠀⠀⣠⣶⠘⣧⣀⣠⠇⠻⣤⣑⣂⣨⡽⣛⠁⣟⠉⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡵⠿⢋⠈⠒⠚⣡⡶⠶⢶⣯⡻⣍⠉⠉⠠⠟⠘⣿⣧⣼⢱⡏⠹⠈⠉⡁⠀⡐⡎⠉⠉⣵⠞⠉⢻⡟⠳⢶⣮⣅
⠀⠀⠀⠀⠀⠀⠀⣤⡶⣦⣄⢸⡄⢿⣸⡏⠀⠲⣦⡈⣿⡌⠃⣴⠀⣄⣠⢟⡿⢻⡜⠧⠀⣰⢸⣄⣖⣳⠇⣾⠜⢁⣠⣴⣯⣄⠀⣟⠁⡈⠳⡀
⠀⠀⠀⠀⠀⠀⢸⠁⢰⡆⢻⡏⢿⣎⢳⡑⠂⠀⣸⡿⢸⡇⠰⣌⠳⠦⠶⣟⡁⣧⡙⠳⣾⣥⣤⡉⠉⢱⠞⣿⢠⣿⢋⡵⠊⠙⣷⠈⠉⣠⠶⣄
⠀⠀⠀⠀⠀⠀⣴⡄⣸⠇⣼⠇⣸⡟⣀⣉⠂⢀⣿⠃⣾⠁⢀⣿⠘⣦⣾⠁⠈⠈⡱⠸⣏⠀⡀⠙⢦⢩⡆⢿⢸⣿⢸⠀⢄⣠⠟⣞⠓⠻⣄⢨⡀
⠀⠀⠀⠀⠀⠀⣿⠰⢫⡾⠋⢘⡉⡀⠉⠙⠛⠆⠇⢸⠃⣠⡾⢋⣼⣃⡈⠛⠛⣯⣵⢇⣭⣭⣥⣄⠈⠸⣧⡈⠣⢿⣎⠀⣠⠶⠶⣍⢓⡤⣬⡻
⠀⠀⠀⠀⠀⠀⠇⡴⣫⣤⣤⡈⢳⡝⠻⣿⣿⡟⠛⠾⠼⢡⣾⠟⣩⣭⣍⠳⠄⠏⠘⠩⠞⠛⢮⡝⢿⣆⠙⠻⢷⣦⣿⣆⣿⠀⠀⠟⠸⠰⡆⣷
⠀⠀⠀⠀⠀⠀⠀⣼⡏⡴⠦⢹⠯⢿⡾⠉⣴⠿⠿⣦⡀⣿⡇⣼⠃⠀⠸⠀⠀⣠⡶⠶⣦⡄⠀⢻⡌⣿⡄⠲⠶⣬⢿⣿⣿⡄⣠⢬⣻⢄⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣳⣄⠈⠀⢿⡀⢀⠿⣄⡄⠘⣇⢿⣧⠹⡄⠀⢠⠀⠀⣿⡀⣰⢸⣿⠀⣸⢣⣿⢱⠏⠙⣿⠘⡹⠁⠀⠀⢀⣽⢂⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⡛⠒⠒⡉⠁⠀⠐⢶⣴⡎⠈⠻⣷⣝⠂⠸⣆⠀⠈⠉⢁⣼⡟⠐⣣⣾⠋⠀⡙⣡⠟⠠⢭⣿⣛⡛⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⢟⣿⠿⠷⣦⣸⣿⠇⠸⣄⠈⠙⠷⣄⠈⠛⠿⠿⢛⣩⣴⠾⢋⣡⡾⠾⢟⣵⠟⠛⢷⡿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢰⣽⡷⣋⣜⡷⣝⠻⠶⣶⣬⣵⡄⠀⣺⡿⢿⣿⠛⠛⣵⠶⢦⣸⣧⡶⣤⡼⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢹⢰⣏⠀⠁⣰⡶⢛⣳⠦⠈⠓⠸⠃⣴⠟⠛⢻⡎⠧⠀⠈⢟⡏⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣦⡀⣿⡰⠏⠁⢠⠞⠛⢶⡜⠇⢸⡆⠼⣣⢆⣴⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣟⠫⣄⡘⢦⠀⢀⡇⣠⠟⣠⣾⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣯⣿⣆⢇⣾⢟⣥⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⡻⡜⣱⢛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠣⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⢽⡿⠫⠉⣀⠄⠀⠈⠙⠲⣤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡾⠥⣴⡞⠓⢦⣶⡤⠗⠒⠲⣿⠐⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠋⠀⠠⣾⢡⠀⠀⢸⣿⣏⣓⢿⣷⣦⠀⣥⣿⢿⡦⠀⠈⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⠲⣾⣿⣼⡀⣼⠀⠉⠛⠷⣿⡌⢳⡟⢻⡆⢳⣀⠄⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⡄⠀⢳⣸⡇⠙⠻⣧⡆⠀⠀⠀⢈⡿⠋⠀⣿⠄⢠⡿⠀⣼⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⠉⣇⠀⠈⣿⡄⡀⠀⠀⠉⠓⠦⣤⠟⢀⣢⠞⠁⣴⡿⠁⣰⠋⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⣎⠢⠘⣷⣵⠤⠤⠶⠶⢖⣚⣛⣉⣁⣪⠞⢹⠇⣰⣧⡴⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠑⣮⣷⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⢀⣿⣾⠿⢋⣠⣟⡶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⠶⣤⣤⣉⣉⡙⠛⣛⣛⡿⠋⠉⠉⠉⠉⢙⡿⠟⢻⢻⠿⢭⣁⣨⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢬⣭⣀⣴⣫⣿⡉⠉⠛⠶⣶⠤⣤⡤⣶⠞⠋⠀⠠⢋⡾⠛⠳⢭⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢗⣁⣠⠽⠓⠲⠦⣤⣀⠘⠁⢀⣤⠴⠖⠚⠉⠳⢤⠤⠶⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠳⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⣿⡿⠿⡿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠿⠿⣻⡿⠿⣿⣿⣶⣤⣀
⠀⠀⠀⣠⣾⡿⣿⠏⠁⣀⣀⠀⠈⠁⣷⠈⠻⣍⠛⢿⣷⣄⠀⠀⠀⢀⣠⣾⡿⠛⣩⠋⠀⣿⠈⠀⠀⣀⡀⠈⠻⣟⢿⣷⣄
⠀⠀⣴⣿⠋⢠⡏⠀⡞⠉⠉⢳⡄⠀⣿⡀⠀⠘⣇⠀⠉⠻⣷⣄⣰⣿⠟⠁⠀⣼⠃⠀⢠⡿⠀⢠⡞⠉⠉⢷⠀⣿⠀⠙⣿⣦
⠀⣸⣿⠃⠀⠀⢿⣄⠙⠒⢂⣼⠃⠀⢹⣇⠀⠀⢹⠀⢀⣀⠈⢻⡟⠁⣀⡀⢰⡇⠀⠀⣼⡇⠀⠸⣧⡐⠒⠋⣠⡟⠀⠀⠈⣿⣧
⢀⣿⡇⣠⣴⠾⠿⠻⠿⣿⡛⠳⠶⠦⣼⣿⣄⠀⢸⣼⢫⠈⣳⡄⢰⡏⠁⡝⣾⠇⠀⣠⣿⣥⠴⠶⠞⢛⣿⠿⠟⠿⠷⣦⣄⢸⣿
⢸⣿⣿⡟⠁⣠⣤⣄⡀⠈⠹⣆⠀⠀⠀⢹⣿⡛⠷⣿⣬⣙⣁⡼⢧⣈⣋⣴⣿⠾⢛⣿⠏⠀⠀⠀⣰⠏⠀⢀⣤⣤⣄⠈⢻⣾⣿
⠘⣿⣿⡀⢸⡁⠀⠈⣿⠀⠀⢹⠀⠀⠀⠀⠹⣷⡀⠀⠙⢿⡅⠀⠀⣨⠿⠋⠀⢠⣾⠏⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⢈⡇⢨⣿⣿
⠀⢿⣿⣷⣄⣉⣉⣴⣿⡴⠋⢘⣠⣤⣤⣄⠀⠙⢿⣄⠀⠀⠙⣆⡰⠃⠀⠀⣰⡿⠃⠀⣠⣤⣤⣄⡂⠙⢦⣿⣦⣉⣉⣠⣾⣿⡿
⠀⠈⢿⣧⠉⠛⠋⢡⡏⠀⣰⠟⢁⣤⢤⡙⢷⡀⠀⠙⠳⢤⣀⠘⠃⣀⡴⠞⠋⠀⢀⡾⢋⡤⣄⡈⠻⣆⠀⢹⡌⠙⠛⠉⣼⡿⠁
⠀⠀⠈⢿⣷⡀⠀⣾⠀⠀⣟⠀⢸⡀⠀⢃⣼⠃⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠘⣧⡘⠀⢀⡇⠀⣻⠀⠀⣧⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠙⢿⣦⣹⠀⠀⠹⡄⠀⠙⠛⠛⠁⠀⣀⣀⡤⢖⣿⠇⠹⣿⡲⢤⣀⣀⠀⠈⠛⠛⠋⠀⢠⠏⠀⠀⣧⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠛⣿⣶⡚⠒⠿⢶⣶⣖⡒⠛⠋⠉⣠⣴⣿⡃⠀⠀⢙⣿⣦⣌⠉⠙⠛⢒⣲⣶⡶⠷⠒⢃⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠈⣙⣿⣷⣶⡿⢻⢟⠭⡙⢦⡼⢫⠭⢻⡟⢿⣶⣾⣟⣋⠁⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡛⠋⠉⣉⠽⢿⣾⡧⠴⠃⡸⣇⠹⠤⢾⣷⡿⢯⣉⠉⠉⢛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣸⢣⡖⢦⠙⣿⣆⠈⠁⠈⠂⣴⡿⢋⡖⢲⢜⣆⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⡷⠛⠀⠈⢿⣆⠀⠀⣼⡟⠁⠀⠳⢿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⢻⡆⢰⡟⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠈⣷⡿⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣸⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣀⡀⢀⠀⠁⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠻⠉⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⣳⣤⣠⡸⠖⠒⠀⠀⢀⣶⣄⠀⠀⣤⡀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠈⠙⠿⣿⠋⢹⡀⠰⠆⠀⠘⠙⠿⠳⣶⣶⣬⣄⡔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⣰⣦⢤⡘⣟⠛⠀⠈⢻⡎⠁⠺⡤⠒⠀⣶⠿⠟⠛⠻⢂⣸⣧⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡠⠚⡁⠐⠲⠶⣦⡥⠴⢆⣤⠛⠦⡄⠃⠤⣤⣡⠖⠉⠉⢁⠘⡿⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠴⣾⡅⣸⠆⢈⡸⣧⠀⣐⡿⢤⣀⠰⠦⣄⡀⣿⠀⠀⠀⠄⠘⠄⡀⠀⠀⣸⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡞⢿⣶⠓⠰⠀⠙⣿⠀⡠⠠⢸⢱⠀⢀⣿⣾⡦⡑⢦⣤⣼⠶⢀⠀⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⢿⣌⢃⠀⠀⠀⣠⠏⣤⠠⡀⣞⣽⣿⣼⡃⠽⣷⣹⣄⣀⡤⢠⣾⠔⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣈⡛⢿⣿⡿⠛⠁⣴⣏⣭⡾⠻⠫⠀⠈⠀⠀⠀⢝⠛⠛⢀⣤⠿⢃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠤⠶⠶⣾⡿⠋⠁⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢡⡅⠀⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣾⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢫⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣶⡾⠗⣢⣗⢠⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣷⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣟⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣽⣯⣽⣿⣿⣿⣽⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠜⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠐⠶⡆⠀⠀⠀⠀⡀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣆⣔⣠⡤⣠⠄⠒⠀⠉⣡⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⢻⡿⢿⡗⠁⢀⣴⣧⠻⣿⡿⠷⡤⣦⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡢⠀⠀⣿⡟⢠⣲⣾⣿⠛⡽⠋⡀⡈⣄⣲⠟⠿⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣆⣀⠟⠃⢾⣿⡟⠉⠸⢃⢀⡵⣫⡽⢷⣞⣾⡅⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠂⣄⠀⣡⠿⠍⢨⢙⣛⣉⡍⠤⡖⠯⣾⢹⣿⠈⠻⢝⡓⡐⡆⠀⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡟⠉⣀⣎⡀⠀⣾⢟⣿⡵⣬⢳⠹⠓⣿⣌⢣⣀⣡⣀⣠⣷⡗⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣰⣎⡘⢲⢨⠸⡱⢻⣯⣵⢇⢜⡄⣿⠸⡝⠁⠈⢉⡉⢁⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡟⣨⢸⣧⣣⣍⣁⣻⣭⡮⠞⡼⠀⠈⠻⠒⠾⠯⢋⡁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠿⣵⣴⡷⣫⡟⠈⠈⢬⣤⡾⠒⠋⠁⠀⠀⠀⠂⠀⢀⣬⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠟⣯⡖⠀⠀⠀⠠⠂⡅⠀⠀⡀⠀⠀⠀⠀⠛⣿⠟⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⢸⣷⢄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⣀⣀⠠⡄⡠⣲⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣻⣏⣡⣿⣥⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠠⣠⠊⠉⠀⠀⠀⠉⣆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⠐⠀⠀⢀⡠⠽⠋⠡⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⣦⠴⠒⠒⠋⠉⠓⠒⠦⣄⣀⠴⠒⠛⠉⠙⠓⠢⢄⡀⠠⣰⣃⡀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠈⠈⠑⡄⠊
⠀⠀⠀⣀⣤⣤⠖⠒⠲⠦⠦⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠃
⠀⠀⠀⢠⡎⠀⠀⢀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠓⠤⠤⡤⠒⢁⡠⢶
⠀⠀⠀⡏⠀⢀⠐⠄⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⡗⠚⠁⠀⢸
⠀⠀⠀⢳⠀⠀⠀⠀⠀⢰⠀⠀⣇⣀⣀⣀⠤⠤⠤⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡢
⠀⠀⠀⠀⠹⡀⠀⠐⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢶⣞⡏⢹⠿⢃⣀⠤⠒⠉
⠀⠀⢫⠉⠒⠚⠓⠚⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠩⡀⠀⢀⠀⠉⠁⠉⠉⠁
⠀⣠⠜⠀⠀⠀⠀⠀⢧⣀⣀⣀⡤⣤⠀⠀⠀⠖⠒⠚⠉⠉⠀⠀⠀⠀⣠⣷⠤⠢
⠀⠉⠒⠲⠤⠤⠤⠦⣤⠞⠋⢀⣠⠟⠀⠐⠦⣀⠀⠀⠀⠀⠀⠀⢀⠜⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⠀⢀⡀⣾⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡆⡈⠁⣈⠱⠵⣬⣱⠀⠀⠀⠀⠀⠀⣶⠭⠒⡴⢊⠉⠉⠑⠻⡟⠿⡾⠾⢃⣤⣄
⠀⠀⠀⠀⠀⠀⢀⢀⢅⡸⣇⣻⣧⠣⣥⠞⠀⠁⠉⣀⣽⢦⡀⠀⠀⢀⣴⢹⠑⢲⢩⡍⢳⡄⣬⣆⠔⣁⢤⡄⠱⡄
⠀⠀⠀⠀⠀⠀⢞⣏⠉⠀⠛⢫⢡⡵⠤⣡⡾⢟⣶⣤⣼⡻⢟⡄⠀⢛⣫⢎⣧⣀⠉⣁⣼⢇⡚⢽⣀⢧⡉⢉⡼⠇
⠀⠀⠀⠀⠀⠀⣮⣻⣶⡂⠘⠃⢈⠀⣼⢻⢣⡟⠁⣠⡄⢹⠈⢸⣄⠁⠡⡌⢃⣨⣭⣉⡀⠸⡿⢸⡟⠃⢍⠩⣾⡀
⠀⠀⠀⠀⠀⠀⢘⢬⣛⡡⢻⣇⣌⡀⠻⣸⡼⡄⣼⠧⠥⠞⠀⡼⣫⣱⣾⡁⠞⢉⣍⣻⡝⢷⠀⣾⢠⢳⠚⠘⠈
⠀⠀⠀⠀⠀⠀⠀⢕⢱⡆⢸⡷⠿⠃⣰⡎⠙⠛⠦⣴⣢⡥⠞⠁⣉⠐⢈⡠⣀⣀⢡⡟⢿⡮⣰⠳⠶⠄⢅⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣷⢈⣁⠟⢍⡓⠨⣶⠀⠈⣁⠐⠂⣼⡋⠀⠔⠓⣂⣀⠊⢻⡠⠁⢳⡷⠦⣄⣄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡇⠐⢩⡝⢿⡖⠛⣘⠫⠀⣶⡿⠇⢹⣴⣾⠱⢦⢇⣠⡗⣆⣤⡄⡠⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠀⢁⡸⡁⢀⡙⠀⠼⠿⢷⡷⠎⢵⣞⢶⣤⡼⢛⣠⣄⠉⠉⠁⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⣢⢠⣌⡫⡐⠀⠂⢜⠀⠤⠈⠃⣻⠏⢿⡜⡟⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠩⡁⢸⡭⢱⡄⡆⣻⣍⠇⠬⠀⡍⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢴⡞⠀⢾⠱⢩⠑⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠻⠏⠂⠄

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡶⢯⣰⡮⣽⠦⣮⣾⡈⣿⣉⡃⠀⢀⡷⢸⡅⣶⠐⣶⠰⣦⣷⡀⣿⢾⡇⣾⢽⡦⢾⣸⡂⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠞⠈⠳⠟⠘⠟⠞⠁⠛⠘⠃⠀⠈⠳⠾⠉⠛⠐⠛⠚⠇⠟⠑⠛⠸⠇⠛⠟⠓⠟⠸⠋⠲⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⣓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠔⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⢈⡅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠀⠒⠍
⠀⠀⠀⡤⠋⠇⠀⠀⠀⠀⠀⠀⠰⢋⠁⢢⠀⠀⠀⠀⠀⠈⢠⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⡄⠤⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠠⠄⠀⡀
⠀⠀⣀⠆⠈⢁⠀⠀⠀⠀⠀⣁⣵⠰⢀⡁⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠈⠂⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠸⠷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠤⣐⣤⡔⠈⠁⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠡⠟⠘⢀⢻⠇⢦⣠⢀⠗⠜⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠉⡉⢈⠁⠀⠲⠄
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠂⠀⠁⠈⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣶⠗⣾⠶⠦⠿⠖⠳⠶⠄⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡗⠈⠉⠀⠉⠇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⠀⠀⠀⠀⠀⢐⡂⣆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣄⠀⠀⢀⡀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⣬⣿⡧⠄⠀⠐⠿⠃
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠁⢰⠄⠀⠀⢠⠂⢠⠠⡰⢆⠀⡦⠀⢰⢠⠄⠦⢀⠀⠀⢀⡘⠂⠀⠁⠀⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡌⠀⠀⠉⠀⠁⠁⠀⠁⠈⠀⠉⠁⠀⠀⠁⠉⣉⠐⡦⠚⠁⠀⠀⠀⠀⠀⢀⣶⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡭⠛⠛⢛⠤⠐⠊⠉⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣴⠾⠋⠀⠀⠀⠀⠘⠟⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠐
⠀⠀⠀⠀⠀⢄⡿⠋⠀⠀⣿⡏⡃⠀⣠⠎⠀⣀⡀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣠⠊⢟⣧⣤⢀⣰⣀⣤⠉⠀⠀⠀⠀⠄⡽⠛⠺⣼⠀⠀⠀⠀⠀⠀⢈⡙⣻⢟⡄
⠀⠀⠀⠀⠐⠟⠁⠀⠀⠀⢀⡴⠃⠀⠀⠀⢤⠶⣱⠀⠀⠀⠀⠀⢀⡴⠘⡿⣿⠷⠺⡟⣻⡿⣟⠏⣯⢻⣀⠋⠀⠉⠈⠁⡈⡁⠈⠀⣀⢀⠀⠀⠀⠬⠽⢿⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢉⡤⠈⣤⣜⢆⣦⣹⣿⢳⣶⣏⢻⣿⡕⠀⠀⠀⠀⠈⠁⠀⠀⠘⢃⠃⠀⠀⠀⠒⠈⢇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡠⢺⢿⣏⣛⣿⠿⣼⣟⡛⣺⢷⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡿⣬⣱⣿⣿⣷⣿⠻⠼⢁⡿⣹⡟⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⠈⠉⠉⢨⠀⡄⠉⢠⡄⠘⠘⠆⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠀⠠⠃⠀⠩⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⠜⠈⠙⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣓⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢀⣀⣰⠨⠹⢹⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡳⢤⡀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣤⣄⣀⠀⠀⠈⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠘⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⠿⠛⢿⡿⣿⣄⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣀⣔⡂⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣹⠟⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠃⠈⠐⢤⣄⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⢸⣿⣦⣦⣀⣀⣠⣤⡷⠾⠛⠁⠀⠀⠀⠌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠛⢁⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⠂⠀⠄⢂⠀⠈⠀⠀⠀⠀⠀⣰⣤⣶⡂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡎⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠂⣠⠛⠂⠀⠀⠀⡀⢰⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠁⠀⠀⠀⠀⠀⠀⣀⣿⣶⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣄⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀⠠⠀⠀⠈⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢶⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⡷⠤⠤⠄⡀⠀⢀⠔⠂⠀⠐⢢⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⣿⣿⠿⠿⣿⣿⠁⠀⠀⠀⠈⠂⠆⠀⠀⠀⠀⠠⠖⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⡟⠁⠀⠀⠈⠉⣀⣀⡀⡠⠀⠒⠸⢂⠄⠂⠁⠁⠒⢄⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣇⠀⠀⠀⡐⠁⠀⠀⠨⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠇⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⡦⠄⡀⠀⠀⠀⠀⢂⠀⠀⠀⠈⠢⡀⠀⠀⢀⠐⠁⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⣿⠉⠀⠀⠑⣀⠀⠀⠀⡠⠋⠀⠰⠂⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠹⡇⠀⠀⠀⠀⠀⡏⠢⠄⣀⢠⡌⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⠔⠉⠀⠀⠁⢄⢀⠔⠉⠙⢷⢄⠠⢔⠊⠈⡄⢂⣰⡁⠄⠐⠊⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠡⠉⠢⢑⠢⣈⢢⢭⢀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠛⠆⡀⠊⠀⠉⠒⢄⠀⠀⢀⣴⡴⠀⠀⣙⠦⣽⠈⡦⡑⢄⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈⠛⠠⣶⡖⢲⡆⠬⠙⡂⠼⣌⠳⣅⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣇⠀⠀⠀⠱⡀⠀⠀⠀⠠⠂⠀⠀⠀⠈⣿⠀⡃⠀⠀⠀⣉⣶⣭⣐⠹⢦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⠁⠀⠁⠀⠈⢠⣄⠀⠀⠀⢠⣿⣆⢰⣀⣴⣾⣿⣿⣿⣿⣿⣶⣮⣑⠪⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠖⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣶⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠈⠁⠀⢀⠀⠀⠀⣀⡐⠈⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀⢔⣢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⠠⠀⠀⣱⢤⣤⠛⠁⡄
⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠀⢀⠐⠊⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣀⡀⢀⠀⠔⠈⠀⠀⠀⠀⠀⣀⠀⢀⠶⠐⢦⣯⡾⠠⡤⠁
⠀⠀⠀⠀⠰⡄⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⡀⠀⢢⢠⠖⣜⢁⠀⠀⠀⠰⠁⠀⠄⠁⠀⠀⠀⢀⢸⡆⠀⠀⠄
⠀⠀⠀⠋⠃⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⡀⠀⠁⠀⠀⠈⠀⠂⠠⢼⠒⣏⠉⠀⠀⠈⠓⠀⠘⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠰⡤⡶⠀⠾⠃⠴⢀⡀⠂⠀⠀⠀⠀⣤⠀⠀⠀⠁⢀⣄⠀⠀⠀⢀⠀⣀⠀⠀⠁⠀⣀⠀⠀⠐⠒⠆⠀⠀⠊⠀⠰⣲⠀⠀⠀⠤⠀⠀⠀⡠⠂
⠀⠀⠉⠋⣤⡄⠀⠠⣤⠀⠈⠁⠀⠀⠀⠗⢶⣧⠶⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣙⠀⣤⠀⣤⠀⢠⡄⡦⠀⠀⠈⠀⠀⠰⠆⠀⢀⠔⠰⡀
⠀⠀⢀⡀⢰⡄⠀⢀⠄⠀⣠⠀⠀⠀⠀⠀⠋⠸⠗⠀⠀⠀⠀⡠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢀⠀⠀⡄⠀⡀⠀⠀⠀⠉⠀⠀⢡
⠀⠲⠾⠁⣀⠂⠀⠈⠀⢀⠁⠐⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⣆⠑⣅⠀⡄⡐⠊⠀⠀⠀⠘⠂⠀⠀⠀⠈⠀⠀⠴⠶⡷⣞⡋⠀⠀⠐⠆⠀⠀⠀⢂
⠀⠀⠀⠘⠋⣤⠀⠀⠀⠈⠁⢀⠀⠀⢠⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠄⠀⠉⠀⠀⠀⠊⠘⠋⠀⠀⠈⠀⠀⠀⠀⠘⠃⠘⠁⠀⡀⢐⣠⠡⠰⠆
⠀⠀⠀⠙⣃⠌⠀⠀⠨⠀⠀⠏⠴⠁⠀⠁⠀⠀⢀⡀⠄⠀⠀⠀⠄⠠⠀⡄⠀⢠⠄⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠗⠃⠂
⠀⠀⠀⠈⠁⢠⡓⠀⠀⠀⠶⠂⠀⠀⠀⠀⠐⠀⠀⠆⠀⠘⠀⠀⠀⣀⠬⠙⠘⠹⣤⠠⠀⠀⠀⠀⠀⣀⣀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠦⣀⡸⠃⠀⠀⠀⠀⠀⠀⠁⣨⠁⠂⠀⠀⣯⡁⠀⠀⠐⠗⠀⠀⠶⠖⠀⠙⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠑⡖⠃⠁⢘⡲⠶⠗⠀⠀⠉⠀⠀⠀⠀⠖⠲⣬⠁⠈⠀⠀⠀⠤⠆⠠⣄⠀⠴⠃⠀⠠⠠⠄⠀⠀⠀⠈⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠀⠠⠴⠂⡀⠀⠨⠷⠄⠀⠀⢄⡀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠚⠓⠀⠁⠀⠀⠀⡄⠀⠤⡄⠘⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠈⠇⠀⠀⠀⠀⠮⠄⠓⠁⠀⠦⠀⠀⠐⠀⠀⠲⠆⠀⣀⠀⠖⠃⠀⠀⣈⡁⠨⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠰⠂⠀⣤⣄⡴⠀⠀⠀⠐⠀⠀⠀⠘⠓⡀⠙⠀⠀⠀⣴⠁⡉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠛⠂⠈⠋⢀⡀⠘⠷⠀⠈⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠹⠕⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢀⡀⠀⠂⠀⣀⠀⠀⠀⠟⠸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠶⢤⣧⣤⡄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢾⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠳⣞⣀
⠀⠀⠀⠢⠊⣠⣶⠖⠛⠒⠦⡀⢀⠔⠒⠒⠲⣦⡀⠀⠀⠀⠀⢀⣴⠞⠋⠙⠢⠄⡠⠒⠋⠛⢿⣄⢇⢦⣧⣄⡄⣤
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⡏⠺⣴⠎⡿⢹
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣩⣯⣼⣿⣯⣱⠖
⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠺⣤⣵⡦⠙⣦⣸
⠀⠀⠀⠀⠀⣦⠬⠳⣦⣀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⢠⣴⡯⢥⣖⠋⠁⢰⣟⠇⢐⡿⠁
⠀⠀⠀⠀⠀⠈⢛⣲⠖⢛⣷⣄⣰⣟⠩⠷⠂⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⣖⣀⣿⠄⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⠉⠁⠈⠁⠀⠀⠀⢨⠮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣠⢤⢀⠀⣠⡴⠶⠦⢄⡀⠀⡠⠤⠶⠦⣮⡀⠘⢃⢤⡀⠀⢀⣤⠤⠤⢄⡀⠀⣠⠤⠤⣤⡀⠀⢀⣤
⠀⠀⠀⢈⣸⣅⣾⠋⠀⠀⠀⠀⠉⠎⠀⠀⠀⠀⠈⢿⡆⠀⠁⠀⢰⡟⠁⠀⠀⠀⠑⠊⠀⠀⠀⠀⠹⣦⣬⡈
⠀⠀⢬⠖⢯⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠗⣧
⠀⠀⠹⣟⣌⣋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣅⣮⣏
⠀⠀⢾⠊⢝⣻⢯⣝⣦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠸⣶⡢⠽⠾⣶⡄⠀⠀⠀⣠⣴⣯⣵⣺⡽⠃⣿
⠀⠀⠸⣆⠀⣿⠟⣉⡿⣻⠿⡶⣄⢀⡴⠿⢷⡤⠄⠀⠀⠀⠀⠀⠈⣉⣽⣛⣙⣿⣦⡰⠫⣽⠷⣷⠦⢹⣇⣼⠏
⠀⠀⠀⠙⠷⠻⣄⣩⡿⠉⠛⠓⠈⠋⠊⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠃⠀⠁⠀⠀⠁⠀⠙⠶⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠤⢤
⠀⠢⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⣀⣀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡟⢠⡴⠚⠋⠉⣀⣠⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⢷⡏⣠⠴⠚⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢦⡀⠘⣷⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⠦⣇⠀⠀⠀⠀⠀⢠⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⣆⠀⠀⠠⠖⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣈⣹⣶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⣀⡤⠇⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⠉⠀⢀⡄⠁⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡇⡉⠉⠉⠁⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣶⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣡⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣥⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠄⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢲⡆⣀⡤⠀⠈⡆⠀⡾⣠⣾⣾⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⠷⠀⡼⠋⠀⠀⢀⣀⡀⠱⠿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⢀⠞⣯⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠈⠁⠻
⠀⠀⠀⠀⠀⠀⣠⡀⣏⠻⣿⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⡀⠀⢴⣶⡆⠀⡀⡤⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠨⣶⣦⢰⣤⡳⣯⠀⠀⠀⠀⠀⠀⠀⠙⡆⢠⡄⠠⠀⠈⢼⡟⢚⠹⠯⠀⡀⠀⠠⠟⠋⠀⠀⢸⣏⣿⡓⡀
⠀⠀⠀⠐⢷⡹⢿⣿⣿⣿⣋⣿⣀⠀⠀⠀⠀⠘⣄⣆⠀⠘⠁⠀⠀⠀⠀⠀⠐⠂⠀⠻⠅⠐⠆⠀⠀⠀⠈⣏⡍⣙⢩⣥⢀
⠀⠀⠀⠀⠈⠛⠀⠔⠶⣾⣿⡿⠿⢓⣶⣄⠀⠀⣸⣿⠀⢠⡄⠀⠀⠀⠀⢀⡑⢒⠄⠄⢸⡶⠄⠀⠀⠀⢠⡿⣾⣿⠾⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣝⢃⡀⢜⣯⣹⣤⣦⣤⣌⠁⢺⡅⢠⣄⠀⢠⠃⠘⠉⠀⣀⠰⡀⠀⠀⢀⡠⡾⢿⣿⡆⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡦⣙⡮⠘⠓⠲⢶⡙⠛⠻⠀⠀⢙⠷⡄⠀⣌⠳⡄⠀⢀⠐⠂⠉⠸⣳⡆⢠⣟⣧⠄⠘⢫⡟⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠆⢠⡀⢐⣆⠒⠋⠀⢈⡴⢫⡦⠈⢠⠁⣴⠁⠜⠁⠀⠀⠈⠀⠘⠟⠁⠒⠁⠈⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠓⢀⡿⣅⣰⠿⠒⠉⠀⠀⠢⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡐⢀⣿⠖⠈⣸⡿⠟⠀⠀⢀⡠⠤⣢⡄⠘⠢⣀⣀⣀⣬⣷⣌⡀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⣠⣶⠯⣤⠐⣾⠋⠀⠀⠀⣰⣾⡟⠛⠛⠒⠋⠀⠈⠉⠉⢀⣿⡙⠷⠈⠛
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡄⠀⣤⣶⡿⣉⠿⣛⣁⢸⣷⡆⠀⢸⣿⣿⠇⠀⠀⢀⣤⠶⠶⣾⣿⠿⢿⡷⠄⠵
⠀⠀⠀⠀⢺⡀⠀⠃⠀⣤⣤⣯⣄⢻⣿⣷⢿⡀⢃⠗⣸⣿⣤⣤⡼⠟⠀⠀⠀⠠⠃⠀⠀⠀⠊⡷⠃⣷⣏⠀⡌⣏⣄⢒⡀
⠀⠀⠀⠀⠀⠠⠒⢤⠀⠀⠀⠀⠈⣿⣿⣿⣼⣧⣯⡾⠋⠉⠻⣯⣤⠟⠀⢠⣤⣇⣠⣴⠖⢆⡠⡇⠀⠀⠁⠀⠻⣿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⢸⣿⣿⣿⣯⣼⠇⡘⠀⠀⠈⠿⣀⠀⣸⣿⣿⣏⡀⠀⠀⣷⡿⣷⠄⠀⠀⠠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡶⠀⠂⢀⡀⠀⠘⣿⣿⣷⢿⡇⢀⣴⡄⠀⠀⠀⠀⠀⣿⣏⠁⢸⣇⠀⣜⣿⣶⢟⠤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠆⠀⣿⣻⣶⣯⣇⠘⢻⠁⠀⠀⠀⠀⣴⡟⠧⠰⣿⣿⣿⣾⣇⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠂⠉⠁⢤⣤⡄⣀⠀⠘⣿⣦⣟⣷⣔⡍⠂⠁⠀⣠⠶⣿⡆⠀⢠⡾⠃⠿⠏⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡⣤⠀⠀⠀⠘⢿⣽⣿⡍⠠⢻⡀⠀⠚⠀⣽⠋⠀⢈⡞⠀⠀⣶⣿⡿⢿⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⢾⣳⡀⠀⠀⣠⠞⢩⠀⠀⠘⢿⣾⡿⣛⣷⠒⠈⠁⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣁⣥⣴⠃⠀⠀⠀⣀⢠⣾⣿⠒⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣏⡀⠀⡀⢐⢿⣾⣟⣇⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣥⣼⣷⣴⣿⡿⢛⣿⠏⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠋⠖⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠲⢄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣤⠉⡁⢀⠀⠤
⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⢠⠽⡀⠘⠆⠀⠢⠡⠀⠄⠀⢐⡀⢈⠠⡂⠠⠆⠀⠂⢈⠃⡠⠄⠉⠀⠴⣂⡪⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠊⠉⠣⠐⠠⣀⡐⢀⠓⠄⠠⠯⢽⠓⠚⢿⡷⠹⠈⠂⠀⢀⡠⠀⠃⢀⡾⢤⡃⢸⣰⢀⡀
⠀⠀⠀⢀⢠⣻⣀⠛⡶⠴⠀⠀⠀⡀⡌⢉⡿⢋⡀⠸⢛⡄⠎⢨⠀⠀⡁⠀⢀⡄⠀⢻⣀⠘⣷⣍⠡⠀⠀⢀⢎⠐⡀
⠀⠀⠀⢠⣄⠐⠊⠁⣥⠀⢨⣴⡾⠋⠐⣷⣄⡄⡁⠈⠁⠷⡀⠈⣧⢸⠀⣠⠌⠀⠠⠼⠠⠁⠐⡀⢀⠀⠀⠈⠀⠐⠆
⠀⠀⠀⠀⠀⢤⣶⣶⡦⠀⠘⢉⣁⣄⠛⠀⠈⠀⠀⠀⠀⣀⣀⣰⠏⡾⡼⢋⡀⢀⡤⠀⠈⠄⠀⠁⠀⣾⠛⣷
⠀⠀⠀⠀⠈⠐⣯⡿⠀⠀⠀⣈⠏⠯⠀⠀⠄⠀⡐⣀⡅⠈⠀⣠⠞⣉⠡⣾⠉⠐⠢⠀⠄⣨⡂⠀⠈⠽⢷⠿⠈⡀
⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⣾⣿⣆⠀⣿⣧⣄⡀⡀⢽⣻⠉⢱⠋⠀⢻⡁⠙⡙⠒⠆⢘⠉⠉⢼⠁⠄⠐
⠀⠀⠀⠀⠀⠐⠺⠦⠀⠀⠈⢉⡟⣠⠉⢋⣩⠀⠀⠈⠁⠨⠉⠀⠔⠁⢁⡄⢫⠀⢠⣾⣿⣇⢀⣀⡀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⣤⠲⣁⣜⢙⠆⠀⠂⣀⣠⡘⠐⠆⠀⠀⡀⢫⡗⠩⠑⣞⠚⢻⣾⢀⣀⢈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⣝⣇⠰⣍⠼⠀⠀⢬⢀⣹⣿⣧⠈⠀⠉⣀⡦⣀⠔⠉⠗⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⡌⢐⡘⠀⠀⠹⠘⡋⠠⠅⢀⡀⣼⣿⠦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢇⠂⠀⠐⢄⠀⠀⠀⠈⠠⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡤⣀⠲⠄⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢣⣤⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⣄⠀⠀⠀⢀⡈⠱⣾⠟⣡⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⠗⠒⠤⣄⣻⣷⣦⣾⠟⠀⠀⠀⣀⣴⣶⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠛⠁⠀⠀⠀⠀⠉⠙⠳⢦⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣴⣿⣿⣿⣦⣿⢿⣧⣸⣿⣿⣿⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⠿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣿⣿⠏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠻⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠿⣿⠟⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠟⠀⠀⢻⡇⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣶⣤⣄⠀⠀⡀⠀⠘⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡆⠀⢻⣦⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣤⣤⣦⣼⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣏⡉⠛⠻⠟⠻⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀⠀⠀⠋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⢀⣿⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣼⣿⣡⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢸⣿⣿
⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⣿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⣠⣴⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢶⣄⡀⠀⠙⠃⠀⣰⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⡀⠀⠈⢻⣿⣿⣶⣶⣤⣿⣿⡟⠀⢸⣿⡀⠀⠀⠀⠀⢸⣿⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣤⣉⣻⠿⠿⠟⠛⠟⡁⠀⢸⣿⣿⣆⠀⠀⠀⢸⣿⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣶⣦⣤⣄⣈⡒⠠⢿⣿⣿⠀⠀⠀⠸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠻⠿⠶⠶⠶⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡞⣿⢦⢤⠶⠟⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⡌⢳⢿⠦⠰⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢜⣲⢭⠋⠹⢚⡇⢢⣾⣿⣿⣶⣦⠂⠀⢰⣖⣿⡗⠶⠛⣣⡂⢅⢛⠸⠡⢲⠦
⠀⠀⠀⠀⠀⠀⠀⡴⣺⠐⣀⠉⠓⢬⡇⢸⢿⠈⡍⠚⢿⣠⣴⠋⠉⠙⣙⠤⢉⠛⣉⣿⣭⡀⣁⣶⡢
⠀⠀⠀⠀⠀⠀⠀⢜⠿⠓⡵⠄⣶⡄⠳⣘⣟⠴⣀⡸⣿⣿⣿⣋⢩⣥⢸⣦⠀⢈⣱⣼⣷⢶⢜⣧⣽⠷⠂
⠀⠀⠀⠀⠀⠀⠀⣀⡰⡒⡀⢣⡴⠶⠶⢽⣭⠰⡎⣥⣿⣱⡀⢵⣜⡿⢿⣫⠆⢲⡵⠃⣸⡟⢩⢦⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠹⣵⠏⠀⣮⡠⡊⣤⣴⣅⠠⠌⣻⣇⠛⣟⣿⣴⡾⡫⢐⣁⡤⢶⢿⢋⣬⢳⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⣿⣿⣿⠐⠨⠗⢸⣓⠊⢻⣿⣿⣸⠋⣴⣺⣿⣋⠉⣙⡳⡤⠲⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣻⢷⡬⣠⣿⠟⣋⣭⣍⡈⢩⣿⢿⢘⣭⢙⣶⣹⢷⣝⡴⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟⢻⡥⢈⣿⣻⠟⣿⣲⣝⢧⠼⢩⣎⢙⢡⣿⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⣤⠊⠯⡥⣾⣸⡉⣙⣿⠃⢿⡡⢮⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣘⡿⢿⢿⣯⡝⠀⣺⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣸⠿⠻⡿⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠟⠛⠛⠛⠛⠛⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⣄⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡾⠋⠀⣶⠿⠛⠿⠟⠻⢷⣶⣤⣄⣽⡾⠷⠧⣤⣄⡄⠀⠙⢧⡀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠙⠓⠶⣤⣀
⠀⠀⢠⡟⠀⠰⣸⡇⠀⢤⣀⠀⡀⠀⠀⠙⡿⠁⠀⠀⠀⠈⠛⠷⣶⣤⣀⢳⡄⢀⣴⠞⠉⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀
⠀⢀⡟⠀⠀⠠⠸⣇⠀⢀⣿⠿⠿⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⠷⠶⠿⢿⠿⠛⠿⣷⡧⠀⣤⡀⣀⣤⢀⣄⣴⣤⣤⠈⠻⣄
⠀⣾⠁⠀⠀⠀⢿⣿⣤⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠁⠀⠀⣀⣤⠀⠀⠀⢹⣿⣴⡿⡿⠛⠛⠛⠉⠉⢩⡿⡄⠀⠙⣆
⢰⡏⠀⠀⠀⠀⠐⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠞⠛⠒⠚⠉⠋⢸⠀⠀⠀⢸⣿⡯⠐⠂⠀⠢⣶⡞⠁⢸⣿⡎⠀⠀⢹
⢸⡇⠀⠀⠀⠀⢀⠘⣿⡇⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠞⠉⣀⠀⠀⠐⠊⠁⠀⠀⠀⠈⡄⠀⠀⠈⠋⠀⠀⠀⠀⠀⣏⣀⡀⣼⣿⢀⠀⠀⠘
⢸⡇⠀⠀⠀⠀⠜⣹⣿⠃⠀⠙⠑⡀⠀⠀⠀⠀⠀⣤⣿⡕⠠⠞⠈⠣⠀⣀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠂⠀⠀⣾⠁⠀⠈⠙⢷⣗⠀⠀⠈
⢸⡇⠀⠀⠀⣩⣾⡿⠋⠀⡴⠀⠀⠈⢀⠀⠀⢘⠿⠛⠉⠀⠀⠀⠀⢀⠠⠈⠑⠠⣄⡀⠀⠀⠘⣦⢀⡀⠀⠀⠈⠀⠀⠹⡇⠀⠀⠀⠀⢻⡷⠀⠠
⠈⣷⣄⣾⠿⢿⠉⢀⡶⠋⠀⠀⡀⠈⡆⢸⣴⡏⠀⠀⣠⢖⣒⣒⠚⣁⡀⠀⡀⠀⠀⠉⠢⣀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠠⢬⣿⡃⢸
⠀⠘⣿⠃⠀⠀⠀⠀⢳⡀⠀⠀⠹⠆⠀⠈⢿⡄⠀⡸⠛⠿⣿⣿⡛⠛⠛⠿⣶⡵⢦⡀⠀⠙⣶⠞⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⡿⠃⡾
⠀⠀⠘⢧⡀⠀⠀⠀⢸⠟⠀⠀⢰⣦⡀⠀⢸⣿⣶⡇⠀⢠⣄⠙⠀⣐⣁⠜⠁⠠⠏⢙⣄⣰⠋⣰⡧⢀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣸⡿⠀⣸⠃
⠀⠀⠀⠈⠳⣄⠀⠀⠘⢆⡀⠀⠀⠉⠑⠀⠈⣿⡏⠛⠳⢤⣀⣀⠀⢨⡏⢀⣭⠟⠋⡁⠘⠁⠀⢼⣷⠟⡁⠀⠀⠀⠀⣴⠋⠀⢀⡴⢿⡇⣰⠏
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠠⢹⣿⢧⡀⠀⡆⠀⢻⠁⠈⠢⠀⠀⠉⠙⠛⠛⠋⠀⣸⠋⠩⠁⠀⢠⡆⠈⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁⠀⢸⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⠋⠅⠀⠘⢷⢿⣄⣆⢆⠀⠀⠱⡦⡶⠖⠂⠶⠒⠊⣁⠄⠀⠀⠚⡿⠇⠀⠀⠀⠀⠐⠂⣸⣇⡠⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠀⠀⠈⢿⣿⣶⣄⠀⠀⠀⠀⢀⢀⣴⣯⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢀⡴⠋⠉⠛⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣞⠀⢀⠀⠀⢙⣿⠀⠀⠙⠢⣴⣿⠟⠉⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠞⠀⠀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣴⠃⠘⠀⠃⠈⠉⠐⢄⣉⠛⢶⣦⣴⣏⠀⠀⠀⢀⣠⡤⠚⠁⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣄⠀⠀⠀⠀⠀⣿⡄⠠⡌⠻⣷⠶⠶⠞⠉⠁⣤⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠁⠈⢑⠿⠎⠀⠀⣤⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⠀⠀⠀⠀⢀⣼⣧⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣾⣿⣵⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

⣿⣿⡇⠀⠀⠙⣧⡙⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣠⣤⠶⠶⠶⠶⠶⢦⣄⡀
⣿⣿⣿⠀⢀⣿⣾⣷⡈⠣⡀⠀⠀⠓⠴⠿⠇⠀⢸⠃⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⢿⣿⣿⣀⠘⣿⡿⠈⢷⣤⠈⠢⢄⠀⠀⠀⠀⠀⠁⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠳⣌⠛⡏⠐⠋⠀⠀⢄⣻⡵⡄⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⣿⣮⠳⠀⢀⣴⣿⣿⣷⡝⡿⣟⢮⣿⣿⣿⣦⡴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⡌⡆
⣿⣿⣧⠀⠸⠿⣿⣿⣿⡇⠀⠈⢯⡳⣍⠉⠁⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⣳⠁
⣿⣿⣿⠀⣠⣴⣼⣿⣿⣇⠆⠀⠀⠙⣌⠢⡀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⢀⡏⡇
⣿⣿⢃⢇⣿⣿⣿⣿⣟⡞⢀⣤⠴⠶⢬⡳⣌⠲⣄⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⡼⣱⠁
⢋⣥⢏⣼⣿⣿⣿⣿⣟⡴⠋⠀⠀⠀⠀⠈⢆⠙⢮⣑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠃
⣛⣵⣿⣿⣿⣿⣿⣫⠏⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠓⠮⣽⣲⣤⣄⣀⣀⣀⣀⣀⠤⠔⣊⡵⠋
⣿⣿⣿⣿⣿⢟⡵⠃⠀⠀⠀⢂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠶⠒⠚⠉⠁
⣿⢿⡫⢛⠕⠋⠀⠀⠀⠀⠀⠈⠣⣄⣀⣂⠜⡁⠀⡄⢠⣄⠀⠂
⡂⠅⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⠈⠉
⠀⠀⠀⢰⠋⠀⢙⣶⡋⠀⢹⣶⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠈⠁⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠁⢤⠂⠀⠁⠀⠀⠀⠀⠠⠹⣿⠃
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⢀⠀⢀⣀⢠⣄
⠀⠀⠀⠃⠀⠈⠁⠀⠙⠁⠀⠀⠀⠀⢀⠄⠀⡄⠶⣆
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⣿⣿⣦⠀⠀⠐⠀⠀⠀⠀⡈⠁⠀⠀⠀⣿⢷⡀
⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣶⣿⣿⣷⣀⠀⠀⠀⠉⠁⠀⠚⣯⡟⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠛⠐⢚⡋⣿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠄⠀⡀⠀⠠⠃
⠀⠀⠀⠀⠈⣻⣿⣿⣿⣦⠀⠞⣍⠁⠨⠿⣿⡿⢿⠟⠋⣠⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠰⣧⢠
⠀⠀⠀⠀⠀⢹⠿⠏⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠘⣦⠎⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠻⣈⢀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⣤⣘⡸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠀⣹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣠⣎⣀⡛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⣿⣿⣖⠋⠀⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡂⠀⠸⣤⣿⠟
⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡈⠳⣬⡳⢄⠀⠀⠀⠀⠀⠀⢀⠿⣿⣿⣿⣿⡛⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠅⠖⠈⠙⢓⡓⠦⡀⠀⠀⠀⡎⠀⠈⠛⠀⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠊⡥⡄⠈⠐⠄⠈⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠲⠄⠀⠀⠈⠐⠂⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⢀⠀⡈⠁⠀⠃⠠⡈⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣰⠲⠀⠈⠠⠀⠀⠘⠀⡋⠀⠀⠀⠤⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠈⡢⠀⠐⠁⠠⠄⠀⣠⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠔⠨⠌⡠⠀⠀⠃⠀⡀⢠⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠺⠂⡡⡀⠄⠈⠆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⠦⢠⠤⠼⡗⠀⠠⠄⠀⠀⢺⡀⠁⠈⠁⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⢢⠀⠒⠀⢀⠀⣠⠰⠁⠈⠀⢠
⠀⠀⠀⠀⠀⠐⠁⠀⢄⠀⠀⠊⠲⢂⠖⡶⠛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠄⠈⡀⠖⠶⠒⠉⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠄⠒⠀⣈⢉⣀⠘⠀⠘⠀⠀⣑⢦⠂⠐⢠⡄⠀⠠⠀⠁⠀⠀⢶⠁⠀⠠⠤⠀⠀⠀⠀⠀⢵⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⣍⠀⣠⠲⠀⠈⠁⠂⢉⠀⠀⠈⠈⠀⣴⠂⠨⠂⠀⢠⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠐⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡤⠈⠑⡄⢈⠀⠀⠠⡴⡁⠤⠠⠶⠁⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠱⡠⠀⠀⠢⠈⠁⠈⠠⠂⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⡀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⢳⠬⡄⠀⠁⡉⡡⠈⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡤⠠⠀⠀⠘⠉⠁⡀⠂⠜⠀⢀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠃⠀⠀⠀⢠⡍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⣐⡉⠀⠀⠁⠄⠐⡄⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⢢⡁⠀⠀⠀⠀⠳⠀⠀⠀⠇⠘⠁⠁⢄⣀⡀⠀⠀⠤⡁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⢤⠂⡢⠄⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢆⠰⡅⠀⢄⠀⠀⠒⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠆⠈⠘⡈⠊⠀⠀⣤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⡀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠂


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