Luxuriant Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣛⣻⣿⣦⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⣼⠛⠙⠻⢿⣿⣿⣿⣛⣉⣉⣀⣈⣉⠉⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠟⣸⣣⣾⣿⣿⣿⣿⣮⣝⣛⠛⠛⠛⠻⠿⠿⣷⣦⡙⣿⡄
⠀⠀⠀⠀⠀⣠⠏⣰⣧⡟⣩⡿⠿⢿⣿⣍⠻⢿⣿⣿⣿⣿⣿⣿⣶⣯⡻⣾⣿⣄
⠀⠀⠀⣠⡾⣣⣾⡿⣿⠀⣿⠀⠀⢻⣿⣿⣿⣶⣤⣍⣙⡛⠻⠿⣿⣿⣿⣜⣿⢿⣷⣄
⠀⢠⣾⣯⡾⠟⠋⣠⡟⠀⣿⠀⠀⠀⠙⠛⠿⡟⠻⠿⣿⣿⣿⠷⣶⣬⡛⢿⣿⣧⠈⠛⠻⢶⣦⣤⡀
⠀⣿⢹⠃⠀⢀⣴⡿⣡⣾⡿⢀⣈⣉⡉⠀⠀⠁⠉⣉⣁⡀⢹⣿⡌⢿⣿⡎⢿⣿⡆⠀⠀⠀⣈⡙⢿⣧⡀
⠀⢿⠘⢀⣴⣿⢿⣿⣿⢿⡇⠈⠓⠂⠀⠀⠀⠀⠀⠐⠒⠁⠘⣿⣷⢸⣿⣿⠸⡇⣿⣄⠀⠈⠛⢿⣿⣿⣷
⠀⢠⣶⣿⠟⠁⢸⣿⡟⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣇⢻⣜⢿⣷⣶⣴⣾⠟⢸⣿
⠀⢿⣿⣿⠀⠀⠈⠻⣿⡀⢸⣿⡄⠀⠀⣀⣀⣀⠀⠀⠀⠀⣴⣿⣿⠸⣿⣿⡿⣆⠹⣷⣝⠳⣤⣀⠀⠘⣿⣇
⠀⠘⣿⣿⡆⠀⠀⠀⠙⣷⣸⣿⡇⠀⠈⠻⠶⠿⠃⠀⡠⠀⠙⢿⣿⣧⠈⠻⣿⣯⡷⣤⣙⠻⣶⣭⡻⣦⣜⣿⣦⡀⠀⣹⣶
⠀⠀⠈⢿⣿⡀⡀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠊⠀⠀⣄⠀⢹⣿⣷⡀⠘⣿⣿⡆⠙⢿⣦⡝⢿⣾⡟⢿⡟⠿⠿⠿⠏
⠀⠀⠀⠀⠹⣷⣷⣄⣀⣴⣿⣿⠻⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⢀⣿⣿⠃⠀⠀⠹⣿⡜⣿⡇⠈⣿⡆
⠀⠀⠀⠀⠀⢹⡟⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠘⣿⣿⣷⠦⠀⠀⢿⣿⣼⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣼⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣶⣀⣾⣿⢿⣆⠀⣿⡇⠀⣀
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠃⠈⢿⣦⣿⣷⣤⣿⠆
⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠉⡉⠉⠙⠛⠛⠛⠛⠉⠉⠀⢀⣠⡤⠿⠿⢿⣥⡀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⡶⠟⠛⠛⠛⠳⢦⣄⣀⣠⣤⣶⣾⡿⠷⠶⠶⣤⣀⠙⢷⣄⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣥⠶⠛⠉⠙⠛⠲⠶⠤⢽⣶⣿⣭⠶⠶⠶⠶⠶⠦⢭⣿⣶⣬⣛⠷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡵⠟⠁⠀⠀⠀⠀⣠⠞⣡⡶⠟⣉⣥⣤⣤⣤⣄⡀⠐⠒⠦⣤⡉⠛⢯⣝⠶⣍⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣿⠟⠁⢀⠀⠀⠀⢀⡾⢡⡾⢋⣴⡿⠛⣩⣤⣤⡀⠉⠛⣷⡄⠀⠀⠙⢷⡄⠙⣧⡘⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠈⢙⣩⡵⠛⠁⠀⠀⢧⣀⣀⣠⠞⣡⡟⣠⡿⠋⢀⡾⠋⠀⠀⣿⠀⠀⣸⡟⠀⠀⠀⠀⣿⡄⠸⣧⢸⡇
⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣄⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⡟⣰⡟⠁⠀⡾⠁⢀⣴⠞⢃⣴⢞⡯⠤⠤⣤⣤⠴⠟⢀⣴⠏⡼⢡
⠀⢀⠴⠋⠁⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⡟⠀⠀⣼⠃⠀⠘⢡⡾⠋⣸⡷⠞⠛⠛⠛⠛⠛⠛⠋⠁⠜⠡⠃
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡱⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⢟⣴⠏⠀⢀⡼⢃⣤⡀⠀⣿⠀⢰⡏⠀⠀⠑⠆⠀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠻⣿⣶⠶⢤⠶⠶⢟⣋⣵⠟⠁⢀⣤⠟⣱⠏⠉⡇⠀⢻⣄⠸⡇⠀⠶⠖⠀⠀⠐⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡉⠙⠛⠿⣿⣯⣭⣤⠶⠞⠋⢁⣴⠏⠀⠀⢷⣄⣀⠈⠓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠘⠿⠃⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠁⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡢⡄⠀⢀⣀⣄⣂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣐⣵⣿⣿⣿⣿⡿⢛⡵⢇⠙⣸⣌⠕⢷⢄⡠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢜⡋⠃⡙⢐⠎⣽⠻⣿⣟⡔⠡⡉⠢⠁⣿⡣⢊⡡⠵⡹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣷⡉⣐⢿⣆⢀⠃⠜⡠⢛⣼⣿⠌⡑⢌⠢⢱⣿⢒⠡⠐⣂⣡⣿⣿⣿⣿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣏⡗⠐⠀⠌⢿⣮⡈⡐⡰⡡⣪⠺⠅⡮⡂⡑⣽⣿⢕⡊⠅⠠⣾⣟⠵⡒⢂⢹⠘⡋⡷⢲⠠
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⡋⠭⠭⠥⠒⢻⣷⣡⡜⡜⠠⣿⡜⡨⢮⣮⢿⣇⢥⠆⡏⠿⣠⠋⢂⢡⠈⠀⢀⣠⣶⣖⣻⡀
⠀⠀⠀⠀⠀⠀⠀⣩⠿⢛⡛⣛⢻⣷⡆⠠⢤⢤⡀⣻⠻⡞⢬⡵⢼⣷⢲⠛⠘⠸⢩⣇⡃⣰⡯⠐⣏⣨⢘⢀⣼⣾⣛⠒⠒⠠⢣⣀
⠀⠀⠀⠀⢀⣠⡮⢒⠉⡅⠇⠀⢨⢹⠮⢹⣸⠀⢉⡯⡱⡘⢄⡘⢛⣿⣍⢣⡇⣼⠅⣻⣿⣟⣬⣍⢧⣣⣾⡿⠟⠒⡂⠤⢉⣹⣸⣷⣧⣂⡀
⠀⠀⠀⠀⢵⠓⢬⣦⣢⢣⠸⡘⡨⠸⣗⢀⠙⢷⣾⣷⡢⢷⣼⣷⡿⢁⡙⣿⣿⣿⡾⠳⣯⢩⣁⠁⢋⠼⣯⡍⣀⣒⠤⣁⡢⣣⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⡖⡨⠍⣛⠿⠷⣶⣥⣵⣪⡧⢞⠷⠺⣳⠞⠻⢿⣿⣯⢼⣧⣾⣿⢉⡾⢲⣿⣿⣧⣖⣑⣺⢄⢃⣌⠝⠝⣿⡿⢟⣻⢿⣿⡿⡇
⠀⠀⠀⢀⣸⡙⡤⢊⠤⢂⠅⡒⢝⠿⡧⠌⢐⠺⢷⣿⣷⠜⢷⣭⡟⢛⣿⣏⣹⡾⣧⣾⡿⡿⡿⢶⣸⣷⡷⢛⠋⢩⠗⢠⢁⠆⡌⢱⢎⢷
⠀⠀⢠⣿⣿⣷⣕⢇⡔⡡⢪⠶⠞⠫⡹⠧⡴⢛⣻⣿⡿⣿⡶⢻⣿⣿⣿⣿⣿⣿⣤⣷⢶⠿⣣⣼⣗⠽⡊⠧⣵⡺⡣⡣⠊⠌⡰⢁⠌⡹⡧⡀
⠀⠀⣸⣿⣿⣿⣿⡷⢾⣿⡱⡁⠶⡷⢾⡷⣿⢾⣏⡵⠶⠶⣿⣾⣿⣿⣿⣿⣿⣿⣿⢈⣿⣿⣿⣿⣿⣉⢁⣈⣹⢷⢾⣿⡶⣶⠶⠶⠾⠆⣹⠠
⠀⠀⠘⡿⣿⡿⠫⡎⠣⡑⢄⠰⡧⣴⣱⡗⢣⣬⣿⣿⣷⣿⠧⣬⣿⣿⣿⣿⣿⣿⠻⡿⠾⣶⡝⢻⡎⣢⢥⡖⡻⠵⡑⡑⢍⢦⠑⡈⢢⣱⡟⠁
⠀⠀⠀⠈⢺⣥⠓⢌⠒⠌⡂⢥⣪⣱⡗⣂⠉⢡⡾⣿⠟⢠⡾⣛⣧⣽⣿⣏⣹⢿⡟⢿⣷⣷⣷⠾⢹⡿⡿⣬⣄⢘⡮⠘⡌⠆⢃⡸⢊⡮
⠀⠀⠀⠀⢰⠧⢁⣂⣬⣴⡶⠿⡻⠙⣝⡗⢮⡶⣴⡽⢦⣶⣾⣿⣟⢸⡟⢿⣿⣈⢳⠸⣿⣿⡟⠏⠩⢽⠚⡌⢋⡱⡢⣿⣷⣬⣭⣾⣿⣷⡇
⠀⠀⠀⠀⡾⣤⢛⠟⠍⡜⢠⢠⠁⢰⡬⠈⣡⡾⢿⡿⠄⡾⢻⣿⣷⡈⣱⣿⣿⣿⢷⡶⣟⣘⡩⡀⣌⢲⣏⣛⠉⠭⢒⡩⠕⡝⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠈⠛⢗⠼⣀⡃⡆⠀⢀⣨⠒⣸⢹⠀⣈⣗⡐⢁⢊⢣⢭⣿⣏⡜⡇⠻⡷⣝⣿⣯⢛⣋⡟⡙⢿⣷⣦⠤⠅⠒⢊⣹⢹⡿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢙⣶⣬⣥⣭⣽⠻⠇⣀⡙⢓⠂⣹⣵⠷⣛⡳⢾⣿⠼⣴⢠⢠⣸⡎⡝⠹⣗⠠⣯⠙⢨⠘⠻⢷⣬⠤⠤⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⣁⣒⣒⡒⠠⣽⡟⡈⠡⣣⠐⣿⢣⠑⢞⢟⣾⡏⠪⠦⠇⠦⢘⣏⡌⡘⢈⢀⠀⠙⠿⡍⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡇⠦⠀⠰⢐⣼⢿⢀⠡⠱⡐⢝⠴⠢⢗⠕⡡⣻⡯⡂⢅⡒⠀⢟⣯⣰⠥⠄⢼⢠⣅⡷⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠼⣈⠩⣾⠏⠈⠄⢢⠑⣬⢻⡿⢊⠡⢊⠔⡸⣿⢬⡐⠠⠍⡙⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢊⣅⣆⣣⠸⣆⣫⣴⣿⣧⡡⡔⣁⠜⡀⣿⡕⠌⣑⡲⣵⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠩⡿⣿⣿⣿⣿⣷⢬⣣⠎⣰⢹⢇⡲⡮⠊⠑⠋⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠿⠿⠻⠟⠉⠀⠈⠐⠊⠙⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣀⠁⠀⠀⠀⠀⠀⠀⣔⠣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⠀⡂⢸⠖⠂⠠⠐⠘⢿⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠋⠡⣔⣀⠀⣨⣩⠝⠇⠀⠀⠀⠀⣯⠿⠿⠧⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠠⠙⠉⢰⡄⠾⢋⣴⢀⡰⢠⡀⠱⠟⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⣀⠀⣀⣿⠀⣾⠟⠛⠁⠃⠄⠀⣓⡤⠖⢠⡄⡀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠰⣤⣼⣿⠴⣰⠏⢦⠀⢠⠖⠉⠶⣀⠂⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡿⠺⠿⡿⣿⠿⠀⠀⠀⡠⢀⣠⠏⠀⠀⠰⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡋⠀⠀⠀⠀⢹⠀⠀⠀⠀⠟⢡⣴⠚⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⣡⣘⠂⠄⠀⠀⠐⠀⠉⡀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠷⢴⠀⠀⠃⢿⡅⠀⠀⠀⠀⠀⠀⠀⠰⣤⢰⢤⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢂⣹⠬⣿⠘⢿⡄⠀⠘⣆⣄⣀⡀⣶⡌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠸⡉⠘⡃⠀⠀⡀⠐⠃⠀⠰⢉⣤⣤⣴⣦⣄⠀⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠼⠀⠓⢄⠀⠀⠀⣤⠒⠐⠶⢮⡹⡿⢷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢦⣀⣨⠀⠀⠐⠃⠀⠀⠀⢏⠀⠃⡄⠀⠈⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡞⠀⠀⠀⢘⣻⡟⠁⠀⢀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢐⠆⠀⢀⣀⠠⡝⡇⠀⠀⢈⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠁⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠓⢤⠘⣥⢰⣇⡇⠀⠀⢸⡷⠓⢂⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠂⠉⠦⢸⡇⠀⠀⠨⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⣽⢸⡇⠀⠀⢸⠀⠀⡀

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀
⠀⠀⠀⠀⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡤⢶⠀⠀⠀⠀⢈⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠈⢻⣿⣀⠀⠀⠀⠀⠀⠀⣼⡟⠂⣾⠀⢀⡐⣦⣞⠂⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⣤⣠⣴⡆⠀⠠⢸
⠀⠀⠀⠀⠀⣈⣭⣿⣿⡄⠀⠀⠀⢀⣳⣿⡿⠀⡅⠀⠀⣘⣾⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⢁⠀⠀⠀⠀⠀⠀⠃⢽⣿⣿⣿⣿⡷⡋
⠀⠀⠀⠀⠀⢙⡧⠼⣿⣿⡀⠀⠀⣸⣿⠇⠀⡸⠀⠠⠀⢻⠷⠅⠀⠀⠀⠀⠀⠠⠿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠓⠾⠿⠿⣿⣿⣿⣷⠂⡘
⠀⠀⠀⠀⠀⠐⢿⣖⣻⣿⡇⠀⠀⣿⣿⡀⠰⠇⢀⠀⠀⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⣁⠁⠀⠀⠀⠀⠈⠛⠁⠐⠊⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⢹⡿⢷⣿⣿⠀⢐⣿⣿⣅⡊⠀⠀⠈⢻⣿⠁⠀⠁⠀⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠂⠰⣧⣻⡆⠀⠀⠀⠀⠀⠀⣴⠀⠈⠀⣰⠁⠓⠄
⠀⠀⠀⠀⠀⠀⠀⣛⠫⣿⣻⡄⠸⣿⡧⡼⠁⠀⠠⡀⢸⡭⠀⠀⠀⠀⣼⣷⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠂⠀⠀⠀⠀⠀⠄⠈⠁⠀⣲⠏⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠺⠟⢻⣿⡇⢸⣿⣻⡃⠀⠀⠀⠀⣽⠃⠀⠀⠀⢀⣿⣿⠘⣷⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠂⠀⠀⣼⢛⢃⣬⡤
⠀⠀⢀⡶⠶⢦⠀⠘⠢⢻⡿⠃⠘⣿⣿⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⣿⣿⡄⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠏⣉⡤⠶
⠀⠀⣹⡁⠀⠶⠀⡄⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠋⠀⠀⢠⠀⠀⢿⢿⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠙⢷⣤⣠⠌⠁⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠘⣿⡿⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠀⠸⠀⠀⠀⠀⢀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠃⠀⠤⠀⠀⠀⣄⢀⠀⠄⡃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠀⠀⢀⣴⣦⣀⠀⠘⠆⠀⠃⠀⠀⠀⡆⠀⢤⠀⠀⠀⠈⢦⠉⠈⠐⡄⠀⠀⣀⡀
⠀⣼⡿⣂⠀⠀⣠⡶⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣷⣒⠀⠀⡿⠏⠻⣿⡆⠈⡄⢠⠥⠀⠀⠀⡇⠀⠈⠀⠀⡐⢄⠀⠀⠀⠢⣐⠁⢀⣴⠅
⠀⣿⣟⡥⠀⢀⣿⡗⢸⣿⢧⠀⠀⠀⠀⠀⣆⢀⠄⠁⠸⡄⠀⣿⠧⠀⠘⡁⠀⠀⠈⡗⠀⠗⠦⠀⠀⠀⠀⡇⠀⢶⠀⠠⠀⠀⠉⠂⠁⠌⠑⡂⠀⠀⡄
⠀⣝⣯⡴⠀⢰⣿⡅⢸⣿⡼⠂⠀⠀⠀⠈⡏⢀⣰⡇⠀⡄⠀⢹⣀⠀⠀⡄⠀⠀⠠⡇⠀⠁⠐⣀⠀⠀⠀⡀⣀⡄⠀⠀⠀⠀⠄⢰⣦⣄⡀⠄⠈⠈⠅
⠀⢞⡷⠞⠀⠀⢿⡇⣿⣿⡗⠀⠀⢂⠶⠒⢡⣿⢿⡿⠀⡃⠀⠘⣭⠀⠀⢰⡀⢄⣸⠇⠀⠀⠘⡇⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⡌⡥⠀⡆⠢⠤⠆
⠀⠘⠛⠃⠀⠀⠘⠓⠛⠋⠀⠀⠀⠈⢢⡄⣘⣿⣦⣤⠔⠀⠀⠀⠿⠀⠀⠀⠣⡸⠋⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡶⣶⣤⡀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⡷⠿⢿⣿⠾⣝⣷⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢿⢞⣽⡾⣿⣿⣿⣿⣧⢼⢿⣷⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣷⣦⣄⡲⠶⠾⠿⠿⣛⣥⢟⡿⣟⣻⣟⣻⣿⣛⡿⢸⣾⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⠿⢛⢵⣿⣿⣿⣿⣿⣿⣿⣭⣭⣶⠟⢹⣻⣿⡟⣾⣝⣧⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣴⡾⠋⣾⣿⣿⣿⡟⢉⣭⠋⠉⠉⢫⣍⠸⣿⣿⣧⢻⣿⣷⡝⣷⡄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣛⣋⣁⣠⣴⣿⢻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⢰⣿⠟⢋⣉⣭⣉⣉⣉⣩⣴⣾⣟⡿⣽⣶⣶⣶⣭⣉⣸⡏⣿⣷⡀⠀⢀⣀⡀⠀⠀⢀⣼⣿⡿⣡⣿⡿⣣⣿⠏
⠀⠀⣿⠃⣰⣿⣿⡿⢿⡿⠿⠿⠿⢋⣼⢱⣿⣿⣿⣿⠿⠿⠟⢃⣼⣿⢷⡀⠈⠛⠋⠀⢰⡿⢛⣭⣾⢟⣫⣾⣿⡁
⠀⠀⣠⢾⢋⣩⣤⣿⣿⣿⣿⡿⢿⣿⡿⢸⣿⡿⣿⢞⣛⣛⣿⣿⣿⣟⣾⣷⠀⠀⠀⠀⠈⢰⣿⣿⣤⣬⣿⣯⡹⣿⣇
⠀⢾⣷⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣄⣀⣸⣿⢸⡇⣿⡿⢟⣻⣿⣟⣛⡿⠁⠀⠀⠀⠀⠀⠘⠻⠿⠛⠉⠀⠈⢳⡙⢿⣦⣄
⢀⡿⣾⣿⣿⣼⣿⣿⡟⠛⠛⢿⣿⢿⣿⣿⣿⣼⣧⢻⡇⣿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡍⢿
⢸⣇⣿⣿⣿⣼⣿⣿⡀⠀⠀⠈⢻⣷⣝⠿⠿⠃⣿⣎⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠣⠟
⠘⣿⣞⢿⣿⣿⣿⣿⣷⣤⣤⣤⣄⠻⣿⣷⣶⣶⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠘⠻⣿⣾⣿⣿⣿⣿⣿⣯⡍⢻⣷⠈⠻⢿⣿⠿⢋⣼⣿
⠀⠀⠀⠀⠈⢉⣉⣽⣿⣿⣟⣥⡾⣫⣴⣶⣶⣾⣿⣿⣿⡏
⠀⠀⠀⣴⣾⢟⣽⡾⣛⣿⠿⢃⣼⣿⠟⣥⠶⠚⠛⠛⠋
⠀⠀⢸⣿⠇⣾⣏⢸⣿⠁⠀⢸⣟⣿⣄⢿⡀
⠀⠀⠈⠻⣧⡻⣿⣦⣙⣣⣤⠄⠙⢷⣽⣷⠯⣷⣤
⠀⠀⠀⠀⠈⠉⠈⠙⠛⠋⠁⠀⠀⠀⠉⠻⣷⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣦⣀⣤⢸⡟⠁⠀⠀⠀⠀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⡄⠀⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠘⠓⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⡇⠀⣼⣿⣿⣿⣿⠟⠉⠆⡆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣠⣤⣤⣷⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⡿⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇⠘⣛⣟⠛⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣭⣽⣾⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠢⠄⠀⣴⡧⠤⠤⣄⢢⡜
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢀⡔⡗⢻⣒⣒⣚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠠⠆⠠⠇⠸⠇⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⡀⠀⠀⠀⠀⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠐⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⢄⢠⠔⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⣂⠡⠀⢰⠓⠋⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⣶⠀⡼⢦⠈⢁⡉⠜⠄⠈⢖⠠⣄⠾⣶⠗⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠂⠀⠈⢇⠀⢈⠐⠚⠳⣻⢇⣀⠀⠐⠂⠰⠶⠉⢃⠀⠆⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡮⠀⠠⠑⠈⠒⡇⠀⢻⡊⢹⠋⠻⣧⡚⠙⢣⡿⢺⠒⠂⠈⡑⢠⠴⢦⠀⣈⡒⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⣉⢀⡈⣡⠀⠀⠐⠚⠂⠀⠈⣰⣧⠀⠠⣤⣼⢹⣿⡟⣡⣦⠄⠨⠁⢠⣰⠀⠈⠀⣤⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣢⢾⡕⡀⠉⠉⠈⣃⣠⠄⡄⢘⡌⢹⣧⡛⠾⣿⡇⠛⠋⣴⣟⣿⠴⠖⠲⠛⠽⠲⢎⣲⠃⠠⠄⠉⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢄⣥⣀⣽⣿⣁⠷⣠⡙⡄⢠⠈⢠⣶⣿⠌⠄⣠⣾⡿⣃⢞⠀⠄⠀⢷⡈⠦⠘⢽⡝⠁⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢢⡔⡄⢸⠁⠼⠇⣩⡄⣋⢵⡀⣴⣃⣀⢀⣿⡟⠀⣸⠟⣡⣾⠿⡧⠀⠰⠒⠁⠀⠀⠈⠀⠚⢠⠀⠊⣡⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠚⡌⠱⠄⣤⠈⡡⠴⣤⠜⠉⠒⠀⣬⡛⢫⡻⡿⠁⠀⣡⣾⡿⣿⢁⣲⠂⡆⠀⡆⠐⡤⠌⢱⠧⠆⠰⢾⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢰⠞⣷⠄⠀⡂⢰⣤⣽⢷⢇⠀⠂⠀⠂⢼⠃⣿⠟⠀⢀⡾⠉⠠⢂⡌⠈⠹⢻⠀⠆⠂⢥⡔⠀⠚⠁⠢⠄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠉⠓⠂⠀⢱⠀⠛⢎⢩⣏⠛⣦⢰⠀⠀⠄⠀⢻⣮⣹⠀⠀⣐⠿⢄⠀⢀⠉⣉⢂⠀⠀⠄⠀⠀⢀⡠⠀⢰⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⠁⡀⢀⠈⠐⢠⣀⣈⠍⣿⠘⣿⣧⡁⢠⣸⣿⠃⠀⣼⡿⣶⠀⢀⡤⠐⣂⡀⢉⠛⠙⠆⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠢⠔⢪⠄⠠⠃⠈⠰⡀⠈⣤⣹⠶⠖⢾⣶⣬⡛⢿⣿⡏⠀⢰⣿⣿⣏⣠⠤⡖⢀⡉⠳⠮⠳⠖⠃⡘⠃
⠀⠀⠀⠀⠀⠀⠀⢴⠀⣦⠰⠞⢶⡉⢰⡦⡤⢭⠀⠙⠈⡨⢿⣿⢦⠻⠃⠀⢸⢙⠋⢻⣯⣬⣶⡈⠑⠺⠂⡦⠀⠐⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⠆⠹⢀⠁⠘⢁⠀⡚⠄⡈⢠⡉⣶⣾⠼⣧⡜⠀⡖⠃⠔⠉⣹⡆⠀⠀⠠⡲⠀⠐⠀⠈⣶⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡍⠀⠠⢜⢓⡉⢹⡆⢡⣠⣏⣁⣙⡃⠐⠃⣻⡀⠀⠃⡜⢚⠦⢾⣽⢀⡄⢤⡥⢀⡑⠚⣗⠆⠹⡎
⠀⠀⠀⠀⠀⠄⢈⠁⠈⠁⠠⡅⠀⠀⠨⡀⢙⠦⠈⡈⠃⠜⡹⠃⡁⢹⠀⠀⠀⣀⣅⠀⠸⢇⡸⣷⣆⢀⡿⠿⣴⠻⢿⠟⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠆⡑⢀⣠⡄⢘⡇⠁⠈⠑⢙⠒⠀⡈⠑⢾⡇⠀⠢⡷⢬⡴⡀⠈⢀⣈⣨⡟⡃⠒⢸⠂⢀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠘⢋⠀⣤⢈⠂⢀⠀⡋⠀⡄⡍⢷⡀⠀⠀⠙⠀⠀⢑⢲⠖⠲⠾⠟⠀⠠⠷⢾⠶⡄⠸⠤⠂⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠢⡼⢁⡤⢻⡨⢿⢉⣩⠿⠇⣷⠀⠀⢰⠟⠀⠈⢹⢋⢁⢰⠃⢠⠢⡜⢁⠀⠙⢆⠠⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠼⣇⡈⠉⠀⠚⠙⢲⣆⣐⣷⠀⢀⡑⢊⣠⡞⠳⣈⣯⣤⢞⣀⣼⠄⠀⠀⠠⠀⠠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠠⢀⣅⢐⠁⡀⠀⠨⠌⠉⣬⠀⠈⠈⣭⣾⣦⠟⢭⣃⠻⠻⠿⣇⠠⣜⠀⢄⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⣆⠀⠄⠈⠁⠉⠰⠄⡘⠳⣌⡇⠀⠀⠀⣹⡿⢣⠞⠁⠋⢱⠚⠛⢇⢠⠋⠀⢸⠀⠧⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠀⠄⠀⠃⠀⠠⠀⣌⡀⡻⡇⠀⠀⠀⡾⠁⡘⠁⡈⠠⣤⠀⠂⠤⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢴⠀⠐⠂⠌⠰⣶⣀⣸⠇⢀⢀⠄⠀⠀⢀⠤⠤⠀⠅⣠⠀⢿⡀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠐⠄⠈⡍⠆⢘⠫⠀⠀⠐⡁⠀⠀⠀⠀⠁⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠐⠀⠀⡐⠶⠂⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡍⢀⣱⣈⣆⣰⠆⠀⠁⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠁⠐⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⣀⣍⠠⡀⠀⠀⠀⢂⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣠⣼⠉⠳⢠⡤⡆⠀⢠⡤⠚⠛⢋⣦⡀⡀⠄⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⢈⠝⡊⢉⡙⣦⣀⠀⠂⠀⠸⣏⠀⠉⣡⣠⠰⠘⠤⣀⡷⠐⠃⠀⢲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠍⠢⠒⢀⠙⠉⠡⠰⠆⠐⢰⠯⠑⡀⠴⣾⣆⠃⣀⢿⡚⠣⢴⠠⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣰⢷⡼⠲⢏⣯⡖⠀⠂⠠⢨⠠⠔⠀⠜⠂⠠⢀⡄⢦⣼⣅⢈⣄⠄⠧⡹⠒⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⠂⠀⠀⠉⠩⠳⠧⠢⠘⣥⡀⠐⠈⡉⢀⠀⠘⢂⠩⡀⠟⣀⢤⡀⠀⠀⢳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠢⠠⠨⢁⠘⡁⠐⣿⡆⠀⢳⡇⣰⠂⠘⡴⡄⠡⠂⠘⠶⢠⡦⠀⠚⢣⡐⢤⠛⠝⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢰⡒⠂⣌⢹⡿⣆⢬⣥⠀⢈⣽⡿⢳⣾⡷⣽⡂⣬⡴⢀⣛⠐⠒⠒⠛⠿⣤⠄⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣽⣋⠈⣳⡊⢂⡀⢂⠀⠉⠲⣳⠴⡛⣳⠗⢋⣴⣿⣿⣾⣽⣥⡀⠄⠀⢃⠠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⠚⠃⢘⣁⡬⠙⣤⣦⠀⠀⠘⣧⡝⠁⣠⣿⡿⠟⠟⢙⡉⠛⢿⣶⡇⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⢀⡭⢲⠂⢄⣿⡶⣶⠛⢩⠄⠀⢀⢹⠀⢠⡋⢻⡝⠀⠅⠰⠄⣡⣖⡂⠔⣈⡃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⣂⢰⣿⢛⠱⢚⠃⠀⠑⡊⠇⠀⠀⢘⢋⠠⢜⠯⡄⡈⢄⢇⠀⠑⠀⠻⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⢐⠀⣀⢒⣏⣾⡶⠁⣠⡴⠐⡇⠀⠀⠀⠀⡇⠂⣠⠀⣥⡳⡶⢚⡙⠛⠃⠀⠠⣡⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠋⢧⡘⢿⣾⣿⣴⡾⠁⠀⠀⠠⢧⠟⢠⡈⢹⠏⠅⢠⠎⠠⢠⠘⢃⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠖⠀⠸⣄⠈⠛⠿⠇⠀⠀⠀⢠⠄⢙⡋⠷⡙⠿⠀⠞⠀⡳⢜⣶⢇⡈⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡀⠀⢠⠤⢜⡿⢶⡀⠀⠀⠀⠀⠀⠫⡈⣥⡀⢹⠆⠸⣾⠴⠃⣾⡃⢰⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢸⠶⡌⠛⢨⠋⣥⣀⠀⠀⠀⠀⠀⠁⣼⣤⡟⢀⡐⢲⡋⠋⠉⡡⠁⢀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠀⠃⠑⣀⡈⢀⠀⠄⠀⠀⠀⠘⠋⢠⡌⠀⠀⠛⡆⠀⠀⠠⡄⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⠒⠚⢻⡏⡛⢛⡡⠀⠀⠀⠀⠀⠀⠀⣟⠀⠡⠀⠀⠀⠀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡶⢷⣿⣡⠠⠄⠛⠹⠂⠀⠀⠣⡀⠀⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⡱⡈⢛⣿⡏⣻⣗⢀⡀⠼⠁⠀⠀⠀⠀⢠⡀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠂⠀⠀⠈⢹⣶⠹⠟⡁⠈⠀⠀⠀⠀⠀⢀⡎⠓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠀⠒⣙⠋⢆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡁⠀⠀⠂⢀⡀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢩⠀⠀⠀⠀⠀⠀⠀⣸⡄⠠⡄⠀⠀⡀⠂⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣄⠺⠇⠹⠌⠁⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⢠⠆⠤⠀⡀⠀⣷⠄⡓⡀⠔⠈⡅⠀⠊⠓⢊⣠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠄⠕⠈⠠⣦⣡⡀⢠⡬⠁⠐⡀⠈⠁⠀⠀⠀⢠⣌⡹⡎⠀⠲⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢁⡪⠀⠀⠔⡘⠒⠮⠠⣷⠶⡆⢀⣤⠆⡻⣿⣦⡌⢋⡀⠀⠸⠁⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⢦⠆⠞⢱⣀⠂⠀⠁⣾⢶⡺⡗⠂⠄⠈⠅⠀⢓⣶⢿⣷⣾⣧⠁⠆⡐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠘⠗⠀⠐⠀⣊⡿⡤⠶⠵⠴⣷⣾⠃⣋⠐⠀⠠⠀⢤⠀⣿⠛⠋⡉⢛⢫⠔⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⠃⠀⢦⠄⠦⠉⠉⠐⢽⣦⠸⠏⣡⣾⢂⣄⡾⢴⣦⠞⢺⢿⠁⡀⠈⢐⣄⡤⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⢐⠇⠀⡂⠀⢂⠘⠒⠢⣿⡇⠀⣿⡿⠿⠛⠡⠟⣀⠃⡈⠻⣤⣔⡆⢀⣄⢠⠀⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠂⣠⡀⠁⢀⡀⡘⣷⣾⠗⡚⣛⡛⢧⡀⠉⠀⢀⡤⣶⢇⠁⣨⡷⠒⠋⠘⠀⡈⠉⠼⣇⠄⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠍⠊⡁⠈⣿⣿⢻⣿⣋⣵⣯⣍⠈⢧⠀⠰⣟⠨⠠⠈⢰⡇⢁⡀⠀⠁⠀⠀⠀⠀⠀⠀⣈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡄⠀⠀⣄⣻⠟⡽⠛⠿⠿⣿⠛⠆⢀⠀⠀⢴⠒⣠⣤⡾⠁⠰⡀⠂⠿⠀⠒⠐⠒⠠⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⡾⠀⠏⢡⡀⠀⠀⠀⢤⣻⠘⠀⠈⡀⠀⡾⠋⣉⣽⠆⢳⠀⠀⠀⠀⠈⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⠀⠂⠛⠀⠒⠀⢂⣢⠀⣿⡄⠁⡸⠀⢀⠀⡌⠉⠟⣂⠠⡆⠀⠀⡀⠀⠀⢠⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⢸⡀⠀⡁⢱⠂⡨⣽⣿⡶⣠⠘⠁⠀⠈⡀⡏⠀⠈⢹⠀⠛⠂⠀⠒⢩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠅⠶⠿⢶⣧⠃⠀⠗⠘⣷⢴⠀⠀⠀⠠⠚⠁⢑⠊⠂⠘⠻⣬⠬⠧⣴⢠⡅⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠁⢁⡀⠐⢬⢘⣓⣙⣻⠇⢿⣿⣃⠀⠐⠘⣆⢀⡁⠀⡢⣰⣦⡩⠀⠁⢀⠘⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣀⣿⠶⢺⣿⠹⠀⢹⡀⠆⢩⡿⠀⠀⣤⡿⠛⠛⠋⠡⣬⡉⡁⠀⢥⣈⠙⠀⠀⠀⠄⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⣾⠄⡹⡀⣀⣼⠁⣤⡀⣷⣀⠉⠅⠀⠀⠈⠐⠿⠟⠛⠒⠿⢿⣶⣬⡽⠢⡄⠢⠀⠀⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⣠⡄⢩⡐⠿⠄⠛⠻⣴⡃⠀⠀⠀⣴⡞⠃⢛⠛⢳⣦⣼⣿⠻⡿⢧⡄⡂⠀⠁⡀⠀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⣀⠀⣤⣶⣅⡀⣙⡆⠀⠀⠰⠟⠂⠤⢐⡀⢾⡛⠀⡿⢰⡅⠀⠀⠛⠗⠘⠀⠂⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⣤⠒⣋⠃⠃⠸⣿⢻⣿⡇⠀⠐⠂⠒⠂⡀⠀⢡⣄⠳⠄⠀⠸⠵⠀⢀⠀⠀⠛⠳⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡨⠷⠿⠀⠋⠈⢲⣭⠳⢦⢸⡇⠀⠀⠀⢸⠀⠁⠀⠀⡠⠆⡀⠐⠀⠳⠀⡄⠀⢀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢹⠄⢙⡦⣛⡆⠀⠀⣀⣤⠔⣴⢸⡇⠀⠀⠀⣸⠀⠁⠰⠟⠃⠘⠃⠀⡄⠈⠀⠂⠢⠀⠵⠘⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⣸⢹⡾⢟⣷⣨⡀⢿⣉⢰⣾⢸⡇⡆⠀⠀⣹⣷⡿⠴⠂⡴⠀⠠⠀⠁⠀⠀⠀⠀⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠶⠐⠐⠧⣜⢁⠘⠻⡄⠛⢸⡇⢁⠀⠀⢸⠅⡉⠶⠺⡁⣀⠀⠀⠂⠀⡄⢐⡄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⢠⡀⢵⠀⢀⣄⠀⠐⢿⠀⢸⠃⠀⠀⠀⢸⠃⠀⠀⠚⠈⠙⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢠⠀⠁⣄⣀⠈⡤⣾⠀⢸⠀⡄⠀⠀⢸⠂⢀⢂⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⡄⠺⠀⠧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⠡⠔⢠⠄⠀⠀⣷⠀⡉⠀⢀⠡⡄⠀⠓⠚⣁⡄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⢶⠴⢄⢤⡆⠀⠐⠀⠈⠐⣀⠀⠀⢶⣍⡷⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡬⠀⠀⠀⡘⢐⠓⢐⡷⠛⠂⢰⡴⢂⠻⢿⣶⠈⣠⢠⠀⠂⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⠹⠖⢘⣈⡧⠃⢀⠀⡻⢻⡝⣯⠁⠀⠀⠀⠀⢩⣟⡿⣿⣿⣆⡐⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠙⠀⠀⠀⢴⣿⠗⣛⡊⢒⣿⠟⢁⠀⠈⡀⢐⠀⣶⢰⣿⠁⠠⡀⢉⠗⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠊⠈⢀⠙⢄⠈⠀⢀⡀⠘⣿⡘⠃⣴⣿⣤⣧⠍⡼⠦⠁⠟⡃⠠⠀⠀⠔⠔⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⠏⠀⠠⣄⣀⣄⢁⣁⣿⡇⠀⡿⠛⠉⠉⢈⠥⠄⠁⢄⣙⡾⡜⠃⢴⠄⣇⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠰⢀⠆⠈⢤⣤⣼⣿⣯⡄⡴⣮⠳⡆⠀⢀⣴⠏⠋⠷⣀⠸⠋⠁⠠⠀⠀⠀⠐⠒⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠈⠀⠰⠀⢸⣿⡵⣿⣵⣾⣿⣷⡄⠑⠀⠈⠃⠁⠘⢀⣽⠂⠰⠂⠠⡄⠀⡀⣀⠀⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡄⣸⠞⠋⠉⠉⠈⠉⣿⠄⠈⠠⠀⠀⣷⡤⠺⢻⡁⡄⠀⠀⠌⠁⠀⠀⠀⢀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⢁⢨⠄⣃⡀⠈⢀⠍⢿⠂⠀⢠⠃⠀⠋⠠⢶⣿⠁⢃⠀⠀⠄⠀⠀⠀⠀⠚⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠄⠀⢀⠠⡀⠐⠛⣷⣊⡉⣠⠎⠀⠐⠠⡗⠈⠠⣖⠀⠆⠀⠐⠰⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢂⣷⣄⢶⢠⠁⢚⠙⣿⣉⠘⠁⠀⢀⢴⠒⢄⠀⠄⠹⣤⣀⢠⣄⡄⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠂⡀⢄⠛⠶⠴⢦⡆⣽⣶⡀⠀⠀⠰⡄⠀⠚⠀⠀⣠⡈⠃⠈⠈⠃⠏⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀⢹⣤⡤⣷⣞⠋⠻⠇⠎⡛⣿⠐⠀⡀⣿⠿⠶⠖⢃⣛⠛⠂⠀⢀⠒⠆⠘⠀⣀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣇⠘⢇⠀⣡⡗⢀⡀⢳⡀⠘⠁⠀⠀⠙⡡⣶⠶⠶⠴⣯⣿⣥⣐⡿⢁⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⢉⡀⠙⠁⠾⣅⠸⠻⡅⠆⠀⠀⠀⣠⡔⠂⠲⢶⢦⣌⣿⣷⢷⣾⣄⢀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠁⢈⠈⢠⣤⣀⡀⢙⡅⠀⠀⢠⠿⠅⡄⢤⠀⣾⠛⠘⡇⣌⠀⠉⠘⠾⠀⠎⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⣠⡔⢺⠁⠚⢉⣿⢻⣿⡇⠀⠀⠀⠒⠂⠁⠉⢡⡐⠂⠀⠀⠻⠔⠀⢐⠀⠐⠓⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠪⠾⠏⠘⠈⠐⢂⠤⢤⢸⡇⠀⠀⠀⢰⠀⠀⠅⠀⠠⠍⡀⠄⠀⠔⠀⠀⠀⣀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡯⠈⣳⣜⡳⠀⠠⢀⣅⡄⢰⣸⡇⠀⠀⠀⣸⠄⠁⠰⠛⠁⠘⠀⠀⡀⠁⠀⠀⠄⡩⠄⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⡝⣷⠿⢻⢢⣅⠼⣫⡀⣾⢹⡇⡆⠀⠀⣽⣷⡟⠖⠀⡶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠢⠁⠸⠬⢁⡈⠛⢇⠙⢸⡇⠀⠀⠀⢙⠅⠈⠶⠺⢀⣠⠀⠐⠀⠠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣤⠨⡆⠀⢤⠀⠀⢪⡇⢸⡇⠀⠀⠀⢸⠃⠀⠀⠃⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣁⣄⣈⣠⣤⣀⣰⣾⣇⣸⣇⣀⣀⣀⣸⣃⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣠⣤⣤⡶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⣉⣠⣤⣤⣀⡀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠁⠀⠈⠉⠛⠷⣄⡀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣴⠞⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⠁⠀⢀⣠⣤⠶⠶⠶⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠃⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠘⣁⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣰⡾⠛⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⠷⢶⣤⣤⣀⣀⠀⠈⠙⠓⠲⠶⠶⢖⣚⡭⠞⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣛⠛⠛⠛⠛⣛⣛⣋⣡⣤⠖⠁⠀⢀⣀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⢀⣴⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⣠⡴⠛⠁⠀⢀⡴⠟⠉⣡⠄⠀⠀⣀⡀⠀⠀⠉⠻⣆⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⡇⠀⠀⢠⣿⠋⠀⠀⠀⠉⠙⠛⠿⠶⠶⠛⠁⠀⠀⠀⢠⡟⠀⠀⣼⠁⠀⠀⠈⠉⠙⣦⠀⠀⠀⣿⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⢀⣠⠴⠒⠒⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⣸⠏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡇⠀⠀⢸⡇⠀⢠⡟⠀⠀⢀⣠⣤⡤⠶⠶⠞⠛⠁⠀⣠⡴⠋⢀⠆⡴⠶⢚⣛⣀⠀⣿⠀⠀⠹⣄⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⢸⡇⠀⣴⠟⠁⠀⠀⢀⣠⣤⠶⠶⠛⠉⢀⣤⠟⠀⠴⠻⢿⣻⠿⠁⢿⡀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⡇⠀⢸⣧⠀⢿⠀⠀⠀⣼⠏⠀⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⣾⠃⠀⠀⣿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⣾⠏⠀⡄⠀⠸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿
⠀⠀⠀⠀⠀⣠⡤⠶⠶⢶⣶⣶⠶⠿⠛⠁⢀⣴⠃⠀⠀⣿⡆⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⠀⢀⡴⢶⢦⡄⠀⠀⣰⠃⠀⣴⠏
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣼⠏⠀⠈⠙⠛⠋⠁⢀⡴⠃⠠⠚⠁
⠀⠀⠀⠘⠇⠀⠀⠀⠀⢀⣶⠛⠁⠀⠀⢠⠞⠀⠀⠀⠀⣾⠇⠀⠀⠀⣼⠁⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⡟⠀⠀⢀⡴⢣⡾⠛⠁⠀⠀⠈⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⠏⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⣼⡟⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⣠⠔⡀⢌⣛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⣤⠞⠃⡄⠀⣿⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡼⠁⠘⢉⣀⠤⢶⡅
⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⢀⣴⠟⠁⠀⠀⠱⡀⢻⣧⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⠞⠀⠀⠈⠀⡰⠆
⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠹⣄⢻⣆⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢠⠋⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⠙⣦⡀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢀⣤⠞⠁⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠐⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⠀⢠⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢶⡄⠀⠀⢠⠄⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⡆⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⢀⠀⠀⠈⠆⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠡⠀⡀⠈⠀⠀⠀⡸⢉⠂⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠈⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠄⠀⡅⠎⢈⡠⠂⠁⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⢐⠀⢡⠀⠁⠁⠆⠀⡠⠒⠁⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠄⠀⢀⠆⠀⠘⡀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠂⠀⠀⠀⠈⠀⢀⠀⠄⠈⠀⠀⠀⠀⠀⠄⠀⡀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡄⠀⠀⠀⠐⠁⣀⠀⡀⠂⠀⠘⠀⡠⠖⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠑⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢖⡭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠢⢥⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠀⡄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠂⠀⠠⠀⠀⠉⠀⡀⠂⠄⢀⣤⠞⣀⡘⢀⡀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠈⠈⠀⠈⠓⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⡇⣤⠀⠀⠆⠠⠆⢀⠔⠃⢀⡀⠄⠀⠀⠀⠢⠍⠐⠂⠀⠠⠀⡀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠋⣄⠀⠘⠁⡟⠀⠀⣁⠔⠁⠀⠐⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠂⠐
⠀⠀⠀⠀⠀⠀⠀⣵⠃⠀⡧⢀⠀⣦⠁⣀⠞⠁⠀⣀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⡀⠀⢈
⠀⠀⠀⠀⠀⠀⣾⠁⡀⠀⢃⡀⢀⣁⠜⠋⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌
⠀⠀⠀⠀⠀⠀⢰⡀⠇⣸⠿⢋⡼⠋⠀⠐⠀⢀⣠⠄⣀⠀⠰⠈⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠸⢉⠄⠋⣴⠋⢀⠈⣁⡜⠓⠁⠀⠀⠈⣷⡀⢀⠀⠙⠛⠄⢠⠀⠀⠀⠀⠀⠙⠐⠀⠀⠠⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠰⠆⢠⠞⠁⠤⠀⡰⠋⡀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠜⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢈⠶⢁⡀⠠⡴⣞⣤⣀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⡁⠀⠀⠀⠘⢦⠀⠀⠀⡈
⠀⠀⠀⠀⠀⣴⡋⠠⠄⠤⢼⣿⣥⡜⢛⢶⣈⣄⠃⢀⣼⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠀⣀⠆⠀⠀⠈⠂⢀⠠⠁
⠀⠀⠀⢀⡾⠋⢀⠀⠀⠀⠘⣿⣿⣷⡧⣾⢿⡿⣾⣟⣿⣷⠀⠀⠁⠀⠀⠀⡀⠀⠘
⠀⠀⣠⡿⢁⠒⢄⠆⠀⠀⠀⠈⢛⣿⣿⣷⣿⣾⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⡿⠁⠈⠀⠘⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⢰⠲⣶⢚⡀
⣔⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⢀⣤⡄⡀⠈⡀⢀⣤⠀⠀⠀⣀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠂⠁⢀⣠⡀⠁⠀⠀⠉⠑⠃⠉⠪⣻⢻⡟⡇⠴⢛⣿⡐⢘⣋⢻⣿⣯⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠐⢁⢐⣣⠄⡤⠀⠒⢒⣀⡄⢠⠀⣙⡁⠃⢰⣳⡿⠛⠷⢻⠘⠛⠛⠚⠉
⠀⠀⠀⠀⠀⢀⠀⠴⢩⣿⠞⡚⢄⣠⣿⢿⣿⡙⠈⠡⡙⢇⣶⣾⠛⢱⡆⣴⠟⣐⣾⣿⢓⠒⠀⠂⠠⠌⠃⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠒⢀⠀⢙⣂⡓⢿⣿⣿⣿⡆⠀⠸⡄⠻⠿⠀⢾⣹⣿⣿⢿⣿⣋⣙⠁⠀⠀⠐⢤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⡛⢋⡛⠿⠶⢶⣿⡄⡄⣰⣿⣿⢟⣿⣶⣿⡿⡟⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣁⢀⣌⠀⠉⠈⠉⠒⢿⣿⣷⣷⣿⠟⣵⣿⠏⣿⣿⣷⣆⠄⢤⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⣿⣿⣇⡤⠄⠀⠀⢻⣿⣿⠟⠿⣿⣽⣿⢃⣠⣿⣿⡿⢿⣯⠂
⠀⠀⢀⡀⠀⠀⢀⡁⠩⣾⣿⣿⣿⣿⠋⠀⢠⢀⠀⡈⡀⠙⠀⠀⠀⠈⢻⣙⠻⠾⠋⣤⣀⠀⠀⠀⡈
⠀⠀⠀⠀⢀⠂⠠⢀⠀⣿⣟⣿⣿⣧⣷⣦⣸⣃⠄⢀⢳⠀⠀⠄⠀⡄⢰⣉⢣⢷⣄⣝⡋⠄
⠀⠀⠀⠰⠘⠄⣗⠬⡁⠉⠙⠛⡿⢻⣿⣿⣟⣿⣶⣼⣏⠀⠉⠀⢳⠖⣶⢸⡆⣌⡇⣿⡄⠀⠁
⠀⠀⠀⠈⠓⠀⠌⢱⣩⣲⡽⠁⠧⠘⢿⠿⠷⣟⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⡿⠋⠛⡟
⠀⠀⠀⠈⠃⠀⠀⠘⢻⣇⠻⠜⠀⠀⠁⠃⢠⣿⣿⣿⡿⣿⣿⣿⠹⣿⣿⣷⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠂⢿⠘⡜⣷⠘⠋⠘⣿⣿⣿⣿⠇⣀⢹⣿⢡⡏⠙⣄
⠀⠀⠀⠀⠀⢈⣷⡶⣤⣀⡩⣴⠈⣷⣻⡿⢠⣞⠨⣿⣿⣿⣷⢸⣷⢸⡿⢺⣇⠀⠘⠣
⠀⠀⠀⠀⠀⠈⠁⠈⠠⠻⣷⣯⠃⡈⢿⡇⢸⠁⠀⢽⣿⣿⠛⠀⢫⣽⣷⡾⠛⠁⠀⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡤⠄⠸⠇⢀⡀⠀⡸⣿⣿⢨⠀⢸⣿⡏⣽⣿⣶⣦⣶⣶⣦⣀⣄⣀⣄⣄
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣿⣇⠀⠠⠀⢀⣀⣙⢿⣿⣿⣼⣦⣰⣿⠀⣨⣽⣿⣿⣽⣷⣯⣿⢛⣋⣛⡛⡛⢿⡦⠐
⠀⠀⠀⠠⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣸⣿⠀⢿⠠⢷⣾⢛⣿⡷⣻⠿⣷⣄⠑⢛⢚⡗
⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡏⣇⡇⢈⢰⣾⡼⣯⣼⣿⣟⣤⣿⣾⣤⣤⣘⡂
⠀⠀⠈⢛⣽⣯⣾⣷⣿⣶⣷⣾⢲⣒⣒⣤⣂⣲⡶⠒⠛⠛⠛⠥⣤⢤⣤⣶⣶⣿⣿⣿⣿⣿⣟⣻⣿⣛⣛⡛⠋
⠀⠀⠀⠈⠉⠛⠿⠯⣻⣿⣿⡛⣯⣼⢷⣯⡟⠋⠛⠛⠀⣤⠀⣤⡆⣰⠐⣋⣿⣹⡟⢻⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣐⣮⣵⣾⣿⣿⣍⢚⣿⡏⠠⢣⣀⣤⣼⣶⣿⣿⣿⠍⠉⢉⣉⣀⣉⢲⢶⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠘⠿⢿⣿⣿⣿⡎⠁⠀⢰⣿⣿⣿⣿⣿⣿⢿⣿⣷⢶⣯⣿⣿⠿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⢀⠀⠆⠨⠅⠁⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡄⠀⠀⡀⠀⣦⠀⠒⠈⠀⠈⣄⠀⠐⠒⠒⢀⡈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⢠⣌⣁⡀⢀⣉⠁⠀⠀⠀⠁⠌⠀⠀⠠⣤⢛⢧⠃⠰⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠁⡨⠀⠒⡁⢸⠥⢅⢈⣧⣴⡄⠀⡀⡤⠸⣶⣦⣀⠟⠋⠁⠀⣌⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⣤⣀⡞⢥⢀⠄⠈⠀⣴⣤⢴⡯⠀⠀⠈⠉⠀⠲⢬⣿⣧⡤⣯⡀⢠⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢰⠄⠁⠄⠀⠒⣾⣀⣄⣣⢠⣇⣼⠇⠛⠠⠀⠀⡀⢀⡐⣻⡶⠝⠛⡻⠷⣃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠀⢱⡆⠀⡈⠛⠊⠑⣲⡄⢸⡿⢋⣴⠜⡀⢰⡀⣄⢸⢡⣿⠇⠄⠀⠀⡔⢃⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠉⠀⢠⠀⠑⠄⠀⠀⠴⡀⢌⣷⡄⠀⣾⣿⣶⡿⠇⡼⠉⢆⠐⢹⡁⡊⢄⠀⠀⠈⡁⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠀⠐⠃⡀⠀⠶⣤⣤⡄⠤⠼⣿⡇⠀⠟⠉⠀⢀⣀⡒⠚⠁⡒⣤⠼⠓⠊⡀⠻⠃⠗⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⡀⢚⠉⠄⠀⠲⣶⣤⣿⣿⠶⣢⡾⠆⢹⡇⠀⢀⡾⡛⠘⠹⢦⡆⠛⠀⠀⠂⠈⠀⠀⠈⠙⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠐⠀⠰⣿⣧⣾⣶⣾⣿⣿⣷⡀⠉⠀⠀⡃⣀⠀⠀⣰⠗⠀⠖⠈⢠⡀⠀⠀⠄⠁⠀⢐⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡄⢸⡇⠋⠈⠃⠁⠉⠩⡷⠠⠀⠢⠀⠀⣽⣤⠜⠛⣋⢤⠀⠀⠰⠈⠀⠄⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⠁⢨⡤⢙⡂⢀⠀⠈⠐⣞⠀⠀⢠⠂⠀⠛⠠⠴⣿⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠄⠁⠠⠀⣉⠤⠴⢽⣤⠛⠀⣠⠆⠀⠠⢀⣳⠀⠉⢐⠂⢱⠀⠂⠐⡤⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠂⢲⡄⢠⠄⡛⠠⠆⢿⡟⠋⠚⠁⠀⠀⣠⠆⢄⢠⢀⠘⢥⡀⡀⣄⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠌⠋⡉⠾⣣⡁⣌⣡⠈⣿⣅⠀⠀⠀⢨⠀⠀⡬⠀⠈⠀⡉⢳⡃⠋⠹⠐⡗⠨⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢴⡀⠉⣦⣨⡵⢲⣿⠀⠹⣷⣆⠀⠀⠈⣧⣴⣄⣠⠬⠸⢿⣇⠀⠁⠀⢤⡁⠙⠠⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢼⡛⠉⠝⢣⠂⠀⢩⡀⠂⡰⠇⠀⠰⢦⠋⣉⣨⣁⣀⣶⢦⡤⠀⣨⡦⠄⢀⠃⠀⠐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠇⢀⠣⠀⢶⡗⢐⡶⠀⣧⣤⢀⠀⠀⠀⠀⠈⢛⡉⣉⣉⣛⠓⢟⣷⢖⣋⣾⠀⠂⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⡦⠀⠦⠘⠛⠀⡉⢙⡮⠀⠀⠀⠀⣾⠯⠁⠈⠉⢋⣿⠮⣿⡏⠛⠿⠶⢰⣀⠀⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⠄⠀⢀⡄⠐⡶⣷⣦⣠⣅⡆⠀⠀⠘⢋⠁⠒⠨⡄⠺⠏⠀⠸⠑⡆⠀⠀⠈⠙⢂⡘⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣦⢆⡙⡌⠈⠁⠹⠇⣿⡇⡇⠀⠈⠉⠁⡁⡀⠀⠠⠆⠈⠀⠀⠀⠋⠉⠀⠄⠀⣈⠉⠲⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠐⠺⠃⠈⠁⠀⠀⠬⠓⠆⣿⠇⠀⠀⠀⢹⠀⠀⠀⡀⣀⠆⡠⠀⠀⠀⠁⠠⠀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠈⣻⠮⡻⠀⠀⢀⣴⡄⠀⣦⢻⠀⠀⠀⠀⣸⡄⠀⠀⠛⠁⠀⠃⠀⠀⠀⠁⠀⠀⠀⠠⠠⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠟⣶⠿⢻⢆⣅⠼⢏⣁⢰⡿⢸⠈⡀⠀⢀⣹⣷⡟⠶⠂⡴⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠶⠆⠁⡸⠤⡉⡀⠛⢿⡈⠃⣿⠀⠁⠀⠈⢸⠅⠁⠦⠚⠃⣀⡀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣄⠀⡆⠀⡀⡀⠀⠺⣿⠀⣿⠀⠀⠀⠀⢸⠃⠀⠀⠓⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⡄⠈⢠⢀⡀⠁⣠⣿⠀⣿⠀⡄⠀⠀⢸⠆⠁⠠⣀
⣿⣿⣿⣿⣶⡷⣷⣶⣶⣶⣶⣾⣷⣿⣿⣶⣿⣿⣿⣾⣿⣿⣾⣿⣷⣷⣶⣶⣿⣷⣿⣿⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⠷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⢁⡆⠻⡘⠇⠁⡀⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⢄⢁⡔⢢⡄⠄⠀⣷⡌⠹⠀⣎⠠⡀⠘⠉⢋⣴⢢⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡄⠀⡀⢲⡴⢦⠴⡖⡀⠐⠀⠀⢢⣀⡀⠐⣶⢽⠇⢀⡝⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣴⡁⢀⠑⡜⢋⣑⣸⣟⡛⠀⠴⠞⢈⡻⣿⣖⣁⡤⠅⠘⢂⠁⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡈⠏⠛⢉⣥⡗⣃⢠⠀⣿⣻⡝⡧⡀⠀⠀⠠⠠⢿⣯⣻⣿⣿⣤⢃⡈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣩⡄⢁⡄⢀⠰⠿⠛⢭⣍⢩⣿⠟⣡⠖⠀⡄⣰⢈⣇⣾⠇⠀⠲⠀⡜⠁⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠙⠃⣠⡌⠉⡀⢁⠀⡦⣈⣻⡄⠁⣾⣿⣶⡿⢡⡟⠳⡀⢫⡆⡞⣀⠐⡱⢉⢀⡱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄⠁⠈⠚⡀⠀⢶⣤⣷⠤⠤⠿⣇⠀⠟⠉⠀⣀⣬⡒⠋⢐⣤⠼⠳⢋⠰⠟⣹⡄⡳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠰⣛⠟⠐⠘⣶⡷⢿⡷⢞⣼⣳⠘⣇⠀⢠⡿⡇⣃⠟⣼⠙⠀⠐⠒⠀⠁⠂⢈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡀⠀⠁⡄⣿⢷⠿⣿⣿⣿⠿⠧⠈⠀⠠⣭⠄⣈⣠⢟⠁⠏⠀⣴⠀⠤⠀⠂⢠⡂⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡶⠰⠟⢥⠈⠀⠄⣄⣺⡃⢢⠘⡀⠀⣿⠚⣉⣽⡶⡆⠈⠈⠀⠀⠀⠀⢀⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠁⠞⠓⠘⠆⠐⣠⡇⣸⣇⠑⣸⠁⢀⢠⣌⠹⠗⣀⢥⡀⡈⢀⡀⠀⢠⠀⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣨⡀⢈⡄⢶⢀⢭⣿⡷⢆⡜⠃⠀⢀⣘⠃⠀⠭⡗⠘⠓⡀⠐⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢁⠳⠛⢾⣯⠂⡘⣀⢻⣷⡂⠀⠀⢰⠙⠀⣳⠉⠂⠘⢷⡞⣟⢶⢰⡦⡰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠈⠣⡔⠈⢦⣨⡭⢽⡟⡸⣿⣧⢀⠀⠘⣯⣦⣀⣀⡦⣾⣦⠀⠃⢨⣈⠂⠇⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⠤⣿⠋⠹⢿⠗⠈⢧⠙⢀⡿⠘⢠⢶⠟⣉⣉⣀⣲⡦⣤⠀⣰⡦⠋⡹⠀⠐⠂⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠃⠾⠂⠶⠿⢐⡶⢸⣷⡄⡑⠀⠀⠀⠊⣋⡉⣭⣉⣛⠛⣿⣿⣯⣏⠈⠂⠈⠇⠠⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠊⠃⠺⠀⠖⠌⠙⠨⠀⢱⡇⠀⠀⢠⣿⡇⡀⡈⠈⣽⡟⢻⡏⠟⠛⠃⣾⡆⢠⠄⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠒⢀⢠⡧⢬⣟⣿⢷⣶⣿⠀⠀⣌⣡⠉⠏⠱⠈⠻⡀⢘⠉⣧⡨⠀⡈⠉⢬⣄⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣿⠛⠤⠣⡠⢠⣛⢹⣿⠙⠀⠀⢠⠠⠀⡄⠢⢚⠂⠅⡀⢤⡍⠀⠐⠂⠐⠂⠈⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡂⠺⣄⢷⡄⠀⡀⢋⠉⢈⡗⡇⠀⠀⠀⢹⡀⠀⢰⠶⠊⠺⠁⠀⠀⠀⣁⢀⠘⣦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢁⣾⣿⣿⡧⢘⢹⡟⠋⣼⡇⡇⡆⠀⠀⣿⣧⣦⡤⢠⣭⠀⡀⠘⠀⠠⠀⠈⢀⠁⠁⠬⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡈⡮⡝⢨⣏⣛⡃⠾⣗⠹⢧⡇⢱⠀⠀⣿⡏⢳⣤⣾⠁⣀⠀⠠⠀⡀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⡀⣷⠀⣨⡁⠀⢾⡇⢸⡇⢸⠀⠀⢸⠂⠀⠀⠒⠙⠃⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠚⢁⠈⠣⣄⡁⠡⢤⡟⢸⡇⢸⠀⠀⢸⠆⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣄⣀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢟⣭⣵⣶⣶⣭⣭⣛⠿⣿⡿⠿⠿⠟⠛⣩⣭⣟⢿⣦⡀⠀⠀⡠⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣤⣀⠀⠻⢀⣽⣷⠙⠿⣦⣤⣤⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠻⣿⣿⣿⣿⣛⣛⣁⣤⣤⣴⣾⣿⣟⡻⢷⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣷⣬⣙⡻⢿⣿⣿⣿⣿⠿⠿⣭⡛⣿⣿⣆⢻⡇
⠀⠀⠀⠀⠀⠀⣸⢿⣧⣾⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣌⢿⣿⣿⣿⣿⣷⣭⠉⠙⠛⢷⣤⠹⡼⣿⣿⡎⣿⡀⠀⠒⢦⡄
⠀⠀⠀⠀⠀⡴⢫⡾⣿⣿⣿⠀⠀⠉⠻⡍⠉⠛⢿⣿⣿⣦⣄⡘⢿⣿⣦⣝⣿⠿⣿⣿⣿⣧⢀⣆⠀⣿⣧⠹⢻⣿⣿⣜⢿⣦⣤⣾⠇
⠀⠀⠀⠀⣸⣱⠏⣰⣿⣿⣿⠐⠒⠀⠀⠀⠐⠒⠈⣿⣿⣿⣝⡻⢷⣿⡛⠛⠛⢿⡎⣿⣿⣿⣎⠿⣿⡿⠏⠀⠀⠙⠿⢿⣿⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠘⢩⣾⠏⣾⣿⣿⠀⠀⠀⣀⡀⠀⠀⣴⣿⣿⣿⣿⣿⡄⠹⣿⣦⠲⠿⠃⣿⣿⣝⢿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣶⣭⡻⣿⣿⣷
⠀⠀⠀⠀⠀⣿⡏⢸⣿⣏⢿⣧⡀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣧⠀⣰⣿⣿⣿⣷⣬⣉⡛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⡞⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣄⠙⣿⣦⠹⣷⠀⠀⠀⠂⠀⢻⣿⡏⢿⣿⣿⣿⣦⡀⢹⣿⣄⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⡄⣀⠀⠀⢸⣿⣿⡗⣿⣿⠀⠰⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠆⠀⠀⠀⠀⣾⣿⡇⠀⠙⢿⣿⣿⣿⣼⣿⣿⣿⣿⣷⡈⠻⠿⣿⣿⣿⣿⣿⡇⣿⣦⣤⣾⣿⣿⢣⣿⢿⣷⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣾⡆⣿⣿⡿⢻⣿⠏⢿⣿⣿⣿⣶⣤⣀⠈⢻⣿⣿⡇⠙⠻⠿⠿⠛⠁⣿⣿⣦⣙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠟⢱⣿⡿⠁⢸⣿⠀⠀⠉⠛⠿⢿⣿⣿⣿⣦⣿⣿⣧⠀⠀⠀⠙⣿⡆⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⡾⣿⣇⠈⠙⢿⣆⡄⠀⢹⣿⣿⡿⣿⣿⣷⣤⣤⣴⣿⣿⣇⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠹⣿⣷⣶⣿⠻⣷⣤⣾⣿⣿⠃⠙⠿⣿⣿⣿⠿⠋⠀⠙⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⡄⠀⠰⠂⠀⠀⠂⠀⢀⣀⠀⠀⠀⠄⠲⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠆⠀⢀⠀⠀⢀⠀⠀⠀⠰⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠁⠀⠀⠁⠓⠀⠀⠁⡄⢀⠆⡰⢡⣷⣊⡄⠀⠔⠀⢀⠀⠀⠀⠄⠈⠀⠐⢤⣄⠀⠀⠀⠐⠈⠁⠀⢀
⠀⠀⠈⣄⡠⠁⠀⠀⠉⠀⠃⠀⡄⠀⡇⢰⠃⠈⢠⣶⡪⠥⠂⠀⠃⠀⠠⠀⠐⠀⠀⠔⠠⠠⠄⠀⠀⠉⠓⠤⡄⠀⠀⠀⠉
⠀⠈⠁⠀⠁⠀⠠⠤⠀⡀⢰⠀⠙⠸⠀⡀⢠⣶⠋⠄⠄⠒⠂⠐⢁⠀⠀⠀⠠⠆⠀⠀⠀⠠⣠⠎⢀⠀⠀⠀⠀⠺⢆⠀⠈
⠈⠀⣄⠀⠀⠀⠀⢰⠀⡇⠈⠀⡆⢸⡰⠷⠇⠀⠀⠀⠒⠒⠀⡀⠀⠀⢀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢠⠂⠘⠃
⠀⠀⠀⢄⠀⡆⠀⢀⡀⠀⢰⠀⣧⣈⣉⠓⠂⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀
⠀⡀⠰⠸⠀⢇⠀⠀⡇⢻⣘⠁⠀⠀⠀⠁⠀⠉⢉⡁⠀⣀⠀⠀⠄⠀⠀⠀⠀⠟⠂⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠂⠐⣄
⠀⠓⠀⠀⠀⢸⠀⠀⠣⠤⣬⡉⢀⠁⠒⠦⠄⠀⠞⠙⢦⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⣇⠘⣤⣘⣓⠒⠂⠠⠤⣈⠛⢂⡤⢨⢥⠛⡀⣁⠐⠆⠀⠀⠀⢀⠆⢀⠀⠀⠊⠀⠀⠐⠒⠀⠀⠿⠄⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠓⠒⠀⠠⢄⣈⡑⠂⠀⠀⣉⠝⣇⠈⠆⠁⣨⠀⠘⠃⠈⡡⠄⢀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠛⠀⠑⡄⠀⠀⠈
⠀⡠⠂⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠈⠁⠀⢠⠀⠺⠀⠀⠀⠀⠀⠼⠃⠈⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠘⠀⠘⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠠⠋⢀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣢⡄⠉⠀⠀⠀⠀⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⣰⠦⠤⠄⠀⠀⠀⠤⠀⡀⠄⡀⠀⠀⠊⠀⠀⠉⠀⠀⠀⠈⠁⣀⠜⢏
⠀⠀⠀⢀⣀⠄⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡜⠓⠀⠂⠀⢀⣀⠐⠀⠐⠃⢀⡠⠀⢰⠀⠁⠀⢀⡠⠀⠀⠰⠀⣠⠒⠀⠘
⠠⠤⠒⠊⠉⠀⠀⠈⢀⠀⠈⠀⠐⠄⠀⣴⣋⡁⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠔⠋⠀⠂⠀⠔⠀⠀⠤⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠂⠀⠀⠰⠯⠀⠀⠀⡀⠀⡋⠀⠀⠂⠀⠀⠱⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⠰
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠦⡀⠀⠊⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⠜⠊⠀⠀⡀⠀⠀⢀⠔⣴⣾⣿⣿⡿⣿⣿⣿⣿⣷⣦
⠀⠀⠁⡀⠀⣐⠆⠂⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠡⠐⠀⠊⠀⠀⢆⠀⠀⠈⡠⠀⠁⢸⣿⣿⡿⢿⡷⣿⣿⡿⣿⢿⣿
⠀⠀⠃⠀⠀⠀⠿⠰⠙⠀⠒⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⢸⣿⢸⡇⣻⡇⢹⠃⡆⣿⡸⣿
⠀⠀⠀⢀⠠⠊⠀⠄⠀⠀⠀⠀⠀⢠⠀⣤⢀⣀⣐⠀⠀⠀⠀⠀⠀⠈⠁⡠⢐⠊⠔⠃⠀⠘⣿⣼⣧⣻⣴⣾⣴⣇⣿⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠸⠇⠀⠀⠀⣰⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄
⠀⠀⠀⠀⠀⣤⠀⠀⠀⢹⣿⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⢻⠹⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⢀⠒⠒⠉⠀⠉⠛⠋⠱⣖⠲⠤⠀⠀⠀⠀⢀⠀⠀⠹⠛⣿⣅
⠀⠀⠀⠀⠈⠓⠚⠳⠀⠀⠀⠀⢳⣄⠑⠄⠠⠀⣀⠠⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣳⣀⣬⣴⠂⠀⠀⠀⢈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢸⡏⠙⠢⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⣆⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⡿⠀⢣⠘⣧
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠚⠀⠿⡇⠀⠀⠣⡈⢧⡀
⠀⠀⠀⠀⠀⠀⡴⠁⠀⣀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⡀⠀⠃⠈⣇
⠀⠀⠀⠀⠀⠼⢁⣴⠟⠁⠀⠀⠈⠳⡄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣠⡴⠞⠋⠉⠐⢄⡀⠈⠻⣄⠀⠠⣬⡦
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡗⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣾⡏⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠳⣄⠙⠃
⠀⠀⠀⠀⢰⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠻⣦⡀
⠀⠀⠀⠀⢸⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⡈⠻⣦⡀
⠀⠀⠀⢀⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣨⣿⡀
⠀⠀⠳⠿⠖⠢⢽⠤⠤⠀⠤⠤⠶⣦⣼⣤⣤⣭⠤⠄⠤⠤⣤⣀⣀⣀⣀⡀⠀⠀⢦⠬⠬⠭⠿⢠⣤⡀⠀⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣼⣤⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣂⡈⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠿⠆⠴⠤⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠺⡗⠃⠀⠘⠠⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡇⠀⠀⠀⠀⠉⡙⡟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣽⣆⠁⢀⠀⠀⢩⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠀⠘⢩⣿⣾⡶⣤⣄⣺⣿⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣦⣀⣠⢤⣤⣿⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣛⠁⠀⣾⡿⠿⠿⠿⢿⣿⣿⡤⡉⠃⠂
⠀⠀⠀⠀⠀⠠⠔⢦⠀⠀⠀⣻⣿⣿⣿⢻⣷⣺⠀⠀⠀⠀⠐⣆⣀⣠⡄⠀⠾⠋⠉⠁⠀⠀⢰⡇⠀⠀⠃⠀⠀⡿⣷⠃
⠀⠀⠀⠀⠀⢠⡦⢼⣷⣾⣴⣿⣧⡏⡉⠁⠀⠀⢆⠀⠀⠀⠀⠉⠛⢉⠉⠙⠲⢤⣄⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⢇⢹⡿⠍⢠⠆⠀⠐
⠀⠀⠀⠀⠀⣠⡐⣦⡙⢀⢧⣼⣿⣄⡇⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠐⠠⢁⠀⠀⠀⠀⠁⠀⠀⠸⠧⡀⢀⠶⠄
⠀⠀⠀⠀⠀⢈⣷⠟⠻⠾⣿⡝⣧⢸⣿⣆⠘⠀⠀⢡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢘⠁⠀⠀⠀⠀⠐⠀⠀⠀⠾⠙⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⠛⠀⠹⠘⠋⠱⠓⢀⠀⢈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢸⣀⠀⠀⠀⣸⡆⠁⠐⠀⠀⠀⠬⠀⠀⠀⢇⠈⠄⠀⠄⠀⠀⠀⠀⢠⠄⠀⢰⠃⠀⠆
⠀⠀⠀⠀⢰⣬⡀⠀⢰⠃⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠁⠀⠀⠐⣿⣷⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⣿⣧⠀⡜⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠂⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⡗⠻⠿⣦
⠀⠀⠀⠀⢠⢻⠻⣶⠁⣂⣤⡄⠀⢀⣀⣀⣀⣠⣴⣶⠎⠉⡦⠆⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠛⠉⠉⠛⠙⠛⠛⠻⠿⠇⠀⠀⢡⡀⠀⣠⠙⠁
⠀⠀⠀⠀⠉⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣰⣠⣆⣠⣖⣁⢠⡶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⣦⠈⠃⠁⡙⠁⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠗⠙⡀⠀⢀⠀⣀⠄⠂⠀⠼⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⢧⢄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠂⠉⠐⠀⢐⢬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⠀⠀⠀⠐⠀⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠱⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⡀⡠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠆⠀⠀⢀⣾⣿⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡿⠋⢿⡥⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⢸⠃⠀⢺⣷⡄⠀⠄⠀⠀⠀⡀⠀⠀⢀⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠂⠀⠀⠀⠀⠀⡟⠀⠀⢸⣿⣣⡴⠞⠓⠀⠀⠀⡀⠀⣘⡆⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢂⡀⠀⢀⡀⢀⠠⠸⣿⠁⠀⠀⢸⣿⣛⣀⢊⣀⡀⠔⠖⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠀⠂⠀⠒⠁⡀⠀⠀⢸⣿⠃⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⢠⠀⢸⣿⢿⣷⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠉⠀⠈⠉⠀⠠⠍⠀⠀⠀⣸⣿⣾⣿⣿⣾⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢀⣴⣶⣆⣄⠀⠘⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡟⠻⣄⠀⠈⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣀⣶⣤⣿⡏⣈⣿⣭⣿⣿⣿⣿⣿⣿⣟⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢈⠿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⡿⢷⠾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠙⢿⡿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡄⢂⣨⠀⠀⠐⣶⡖⠀⠂⡀⠀⣀⣔⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢲⣿⣛⣞⣇⠀⠀⣼⣿⣿⡤⢨⣿⣿⡿⡿⣧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠈⢿⠛⠻⠚⠀⣶⣮⢭⡥⣴⡤⣿⣿⠁⠀⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡈⠀⠁⠀⢀⢻⡟⠛⠛⠛⢣⡿⢻⡀⢾⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠌⠀⠀⢠⣶⣯⣌⣩⣦⣠⣥⣞⣻⣚⠣⠄⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡐⠀⢸⣿⣿⢿⡼⠯⣿⣿⡿⠿⠛⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⡰⠀⢂⣽⣿⣛⠛⣁⣨⣥⣴⣷⣦⠀⢀⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⠀⠀⠈⠽⡟⣻⣦⢀⠐⣿⣿⣿⣻⣿⠟⡀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣤⣔⣀⡆⣶⣿⣿⣿⣨⠆⡀⠈⣹⡤⢺⣾⣏⣽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⣷⣾⡗⢿⡙⣬⠃⣷⡟⣄⢿⣿⡖⢂⣟⠳⣼⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣯⡏⡉⣿⣇⣹⣿⣀⣽⣿⡟⢻⣷⢿⣿⢈⡀⢻⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⣇⡞⣴⣬⡿⣿⣝⣡⠟⣤⣤⢯⡘⢠⣿⣿⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠐⣿⢹⣿⣛⣇⡿⢿⣷⡏⢠⠧⠛⠛⢷⡾⣻⠿⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣏⣿⣿⠣⣿⣷⣿⣤⣴⡟⢱⡖⣾⣾⡤⢙⣺⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠈⠙⠻⠓⠿⠿⠿⠿⠿⠉⠟⠳⠷⠿⠯⠓⠚⠛⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⣄⢌⠀⠰⡘⢃⠀⠀⠋⣰⠟⣱⡿⣻⣿⡟⠀⠀⣰⣿⣿⣿⣄⠀⠘⢀⢀⡄⠐⣶⣌⣿⣿⣿⠿⠟⠋⠁⣿
⡮⣑⣇⠀⠀⠈⠀⠀⠘⠃⠀⠋⣈⣼⣿⣀⣀⣀⣿⣿⣿⡻⡿⢷⣦⠀⠳⣿⣶⣾⣝⣿⣿⡇⠀⠀⠀⠀⠻
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⡿⣿⡝⢿⡄⠈⡯⠳⠀⠀⠉⠛⠿⢿⣿⡏
⠀⠉⠢⠄⣈⢦⡀⠀⠀⠀⠠⠒⠁⠈⣦⣍⠙⠻⣷⠈⢿⡄⠀⠀⠀⠀⠤⠀⠀⣠⣄⠀⠈⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⢦⣄⡀⠀⠘⢯⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢙⣧⣤⣄⣠⣼⣿⡿⢷
⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣐⡽⠃⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⢿⣿⣿⡷⢞⡫⠅
⠂⠀⠀⠀⢐⣤⠀⢀⢨⣦⣿⡆⠀⣠⣶⣬⡄⣀⣀⠀⣀⠠⠔⠒⡫⠉⠁⠀⠀⠀⠀⠈⣉⠭⣭⡶⠋⢴⣿
⣴⠋⠀⡄⢹⣿⣦⣇⢿⣿⠟⠀⠀⠒⠂⢀⡬⢔⣈⠝⠀⢀⠀⠎⠀⠀⠀⡀⠀⠀⠸⣿⣿⡿⠋⠀⠀⣰⠟
⠇⣀⣴⣃⢀⡉⠙⠛⠈⠁⠀⠀⠀⠀⠐⠻⠶⠿⠟⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠦⡄⠀⠀⣼⡇⣿
⣾⣿⣿⡿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⢠⢀⢠⣀⡄⠀⢧⡀⠀⣿⠀⠀⠘⠛⠛
⠟⠸⠁⠃⠀⣿⣿⠀⠞⠙⠙⢣⣶⣶⣤⣀⠀⠀⠀⠀⠀⢿⢏⡿⣿⢸⡘⣿⠷⣦⢲⣽⣷⣾⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠍⠁⠀⠤⠤⠞⠫⠪⠫⣾⠇⠀⠀⠀⠀⡀⣤⣾⣷⢿⠜⠗⠀⠀⠘⢷⣗⣽⠇⠀⠀⠀⠈⠻
⠀⢀⠀⠀⠀⢀⡀⠀⠀⠤⣤⣾⠛⠉⣟⢿⡗⣄⡀⣠⡹⡌⣇⣿⣾⣦⠀⠀⠀⠀⠀⠀⣈⣠⠤⠀⠐⠂
⠀⢸⡇⡀⠀⠀⠀⠀⠀⠐⣛⣛⣡⣴⣷⣿⣛⠌⠻⢿⣷⠸⠸⢹⡿⠏⠀⠀⠀⢀⣤⣾⣿⠋⠙
⠀⠀⡇⠈⢢⠀⠐⠶⣬⣴⣒⡶⣶⣖⢰⡀⠉⠳⡄⠈⠁⠀⠀⠈⠃⠀⠀⠀⠸⡿⢏⡝⠁⣀⡍⣀⣠
⠀⠀⢡⠀⠈⠂⠀⠈⢺⣿⣿⠝⠛⠉⠈⠳⠦⠀⠀⠀⠀⠠⠀⠀⠀⡔⠁⠀⣘⢤⠩⠂⡴⢶⣞⣉⡀
⠄⠀⠞⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠘⣀⠴⠊⠀⠀⠠⢀⣀⣠⡿⠋⣠⢀⣼
⠄⣶⠀⢷⢹⡿⠀⣀⣴⣿⣦⡀⠀⠀⢠⡿⠭⢉⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠛⠻⢿⣵⣾⠷⠟⠁
⢸⡟⡄⠀⠻⢁⣾⡟⢸⣿⣿⣧⣀⣠⣤⣿⣿⣿⡦⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠿⣦⡀
⠎⣿⠡⠄⠠⡿⠘⠁⠼⠿⢻⡿⢿⢿⢿⣿⠛⠻⠿⢿⡕⣠⠞⠀⠀⠀⠀⠀⠀⠻⣦⡈⠿⡿⣷⣶⣶
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡲⣚⢉⡫⢭⣝⡭⢭⢯⢭⣛⢗⡲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣊⠵⣠⣫⢖⣛⠭⢯⣜⣭⢶⣛⣋⣈⠳⢔⣛⡾⢻⣝⠶⣭⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣫⡶⠼⣛⡭⢛⡫⣽⣒⠮⢙⣛⣛⣛⣛⣛⡛⠯⠭⣖⣯⢕⡒⠭⣭⢵⣜⢮⡓⢤⡀
⠀⠀⠀⠀⠀⠀⣠⢞⢕⣻⢇⡖⣫⢴⡪⢕⣪⡵⣚⣯⠭⠭⠭⣭⠝⣋⣭⣝⡳⢦⡈⢙⠺⢕⡮⣑⢭⣇⣻⠳⣝⢦⡀
⠀⠀⠀⠀⢀⡜⣵⠵⡃⡭⡪⣺⢕⡭⢞⣛⢋⡾⢛⣤⢻⡍⢟⡦⣙⠷⣫⢦⡹⣦⠿⠌⠿⣶⣌⡺⢕⢄⠩⢣⡏⡷⡙⣄
⠀⠀⠀⢠⢋⡾⡲⣹⢫⢮⢞⡵⣿⢰⡏⣟⡼⠔⢯⡙⢼⣱⡌⢛⣡⡞⣯⢛⣵⡜⢿⠏⢹⣎⣧⠹⢦⡳⡵⡵⣽⠻⢹⠎⢆
⠀⠀⣰⢣⠟⢺⡷⡱⡱⡱⢋⡴⣛⣨⡅⡿⠼⢏⢫⣴⢣⡙⢼⢣⠝⣱⢋⣧⠹⣬⠛⠼⠷⣒⣶⢂⡷⣝⢮⢮⢎⢣⡙⡝⢯⣧
⠀⢠⢧⡡⣻⢸⣰⡳⣼⣣⠗⣈⢿⡌⣇⢻⡱⣞⡦⢃⠾⣍⢞⣡⠾⡌⡟⡼⣧⠁⣞⡻⢹⢃⡏⣼⢏⡴⢦⣣⣫⣎⢇⢏⣷⡙⣆
⠀⡟⡼⣡⡍⢦⢿⣹⢋⡴⣯⠽⡆⣯⣭⣭⢵⡦⢥⣛⡓⠼⡎⣇⡼⢱⠧⣋⠕⣛⡈⢶⣯⠭⣭⡍⣾⣙⡶⣍⢧⣻⡎⡏⡧⣻⢜
⢸⢹⢫⣷⠹⣸⡷⣇⠿⢢⣱⢫⠳⣿⠼⢞⣫⣴⠒⠺⢿⡑⣶⢸⢇⣶⠎⣵⠋⠉⣻⣦⡉⠾⠥⠧⢗⠞⣱⡯⢿⣿⣿⡸⢮⡟⣄
⣿⠉⣯⢹⡆⣿⢧⠻⢧⡟⣝⠺⠧⢸⣿⣭⣤⣍⠳⢦⣄⢱⡈⢿⣼⠃⣼⣧⡴⠛⢉⣤⣭⣽⣛⡏⣏⣼⡳⠼⢆⡿⢸⢧⢡⡵⡟
⣿⢶⢡⡟⣷⣿⠀⣿⢲⡶⢌⡯⡿⣰⣖⣤⣤⣌⣷⣄⡈⠻⣷⡈⠁⢰⣿⠁⣀⣼⣫⣤⣄⣒⠶⢷⠹⣤⠝⣸⢹⡇⣿⡟⣎⠶⢡
⣿⢘⡮⣱⢻⢿⠀⢻⣌⣙⡭⣶⡴⠋⠙⠛⠛⠳⠶⣭⣙⠛⠻⡇⠀⢸⡿⣛⣭⠿⠛⠛⠛⠛⠛⢛⣩⢵⣊⣉⡼⠁⣿⡇⣟⢎⡗
⢹⢸⢁⡹⣼⢸⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣿⢇⡜⢟⡿
⠘⡆⠿⡏⣎⣇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣸⠳⢎⣵
⠀⢻⠀⢷⢣⡝⡎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣠⣄⠀⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⢧⣇⡿⢡⢇
⠀⠈⣇⠜⡇⣿⡜⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⢰⡀⢿⣿⢀⣆⢈⠙⠷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢌⡞⢆⣫⡞⠉⡞
⠀⠀⠘⣎⠔⡯⡏⣳⡙⣤⣴⡶⢚⣛⣛⡲⠾⣫⣴⠃⡇⣿⡷⢌⠳⣾⣿⢸⢧⣴⣦⣌⡙⢷⡦⢤⣤⣄⣀⣌⢞⣼⣸⠅⢇⡜
⠀⠀⠀⠘⢮⢘⠇⣷⡻⣌⢻⡸⣍⡛⠟⢹⣷⡌⢋⣼⡼⠘⣴⡏⣱⣦⠳⣾⣦⠙⢭⣬⣙⡶⢛⡒⢢⡍⡿⢫⢞⡵⣟⢴⠟
⠀⠀⠀⠀⠈⠳⡕⢮⣇⡎⡷⢝⠢⣍⡛⢟⡩⣴⣦⡙⢴⣷⠘⢴⣌⠻⣴⣌⠃⢾⡦⢽⣿⣎⡭⢟⡩⣞⠕⣏⣾⡧⡡⠋
⠀⠀⠀⠀⠀⠀⠈⠢⡑⢦⡿⣠⢻⡢⢝⡻⢧⣌⡓⠾⢆⠫⣴⣌⠫⣴⡼⢃⡾⢛⣡⡾⢖⡲⢞⡩⠚⢕⡎⠖⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢥⡹⡎⡭⢶⡽⢗⡊⠭⢟⣛⣳⣦⣴⣷⣶⣖⣛⣛⠭⣑⡪⠵⠞⡿⣼⣭⢏⡼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢬⡳⠾⣍⠷⣋⡵⣮⠱⢎⡨⡭⠍⠭⣒⢦⡬⣫⠴⣫⠏⣉⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠷⠭⣿⣭⠛⣍⢫⢒⣫⡟⢕⡯⠭⠭⠕⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⢀⠀⢀⣱⣴⣴⢛⣋⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠿⣭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠈⣾⡗⣌⣷⡈⠙⠈⠀⠀⢀⣰⣿⠋⣓⠀⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⢻⡇⠈⢻⠙⠠⠀⠀⠀⡾⠁⢻⣷⡌⠸⢳⠏⣦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠑⢤⡀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠉⠠⢄⠀⠀⠀⠀⠉⠒⠦⢄⡀⠀⠈⠑⠦⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⠺⡞⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠈⠓⢄⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⠀⠙⢶⠀⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢐⠤⡹⣧⣠⣄⡈⠶⣬⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡤⠖⠋⠉⣀⠀⠚⢑⠻⣮⡾⠛⠓⠾⣧⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣹⢀⣴⠡⠜⡉⣤⣦⣾⣷⣿⣿⣿⣶⣄⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠟⠳⠿⠿⠾⠷⠷⠛⠁⠁⠀⠀⠐⠀⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣰⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠠⡦⣴⣿⠃⠻⠿⢷⣤⣤⣸⣬⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⢡⣯⠁⠾⠂⠓⢸⠻⡟⠛⣿⣦⣹⣷⣄
⠀⠀⠀⠀⢰⣀⡀⣠⣤⣖⠂⠀⠀⣝⡁⠀⠀⠀⠀⠈⠁⣼⣿⡟⠉⠻⢯⠁⣠⡄⢀
⠀⠀⠀⠀⠀⠳⣦⡟⠁⣺⣶⣒⠂⣀⠁⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢺⠱⡄⢉⡇⢝⠲⠆
⠀⠀⠀⠀⠀⠀⣼⠳⢾⡋⡌⠉⠛⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠚⠀⠀⠀⢥⡘⠁
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⡀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠂⠀⠀⠀⠀⠈⠙⠒⠛⠃⠀⠀⠀⠀⠀⠀⠐⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢤⣔⣋⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⢿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢛⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣖⢦⡀⢠⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠐⠀⠴⠯⣾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣿⣿⣿⡿⠛⠁⠻⣥⣿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣠⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠈⢿⣄⡈⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢰⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⣷⣶⠷⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢐⣓⡀⣿⣄⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠽⠿⣷⣦⣀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠒⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢟⠛⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⣿⡟⠋⢻⣭⡟⠉⣹⠋⢻⡟⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠙⢧⣀⡿⢷⣄⣰⠟⢷⣤⡏⠀⢸⣇⡴⠋⢀⡿⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠙⢷⣇⠀⠈⢻⡇⠈⢻⡇⠀⠈⠛⢷⣤⣾⠟⠁⠀⣾⠁⣠⠞⢙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⡏⠛⢶⣼⡟⠷⢶⣾⣷⣤⣬⣿⠀⠀⠀⢠⣿⠿⣶⣄⣸⣿⡾⠋⣠⠟⢀⣽⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⢺⣷⣤⣤⣹⣧⡀⠀⢸⡟⠉⠙⣿⣧⠀⢠⣿⣁⣀⣈⣹⣿⣿⣤⣴⡿⠟⠋⠁⣼⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣀⣘⣧⠀⠀⠉⢹⡟⠻⣿⡇⠀⠀⠘⣿⣷⣿⠿⠟⠛⠛⠉⣿⡏⠀⣸⣇⣀⣀⣼⣧⡴⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠹⣏⠙⠻⢷⡀⠀⠈⣿⡀⢻⣧⡀⠀⠀⢹⣿⣇⠀⠀⠀⠀⣠⣿⣶⣾⣿⠟⠉⠁⣿⠁⣠⡴⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠲⠶⠿⢷⣄⠀⠛⢶⣄⣸⡇⠀⠙⢿⣦⣄⢀⣿⣿⣀⣠⣶⣾⣿⣿⡉⢠⡿⠀⢀⣼⣿⠞⠉⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡷⠦⣤⡤⠶⠿⠿⢾⣿⡛⣿⡿⣶⣄⡀⠙⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⣿⣿⣷⠾⠛⠙⠛⣻⡿⠿⢦⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠶⣤⣀⣤⡴⠖⠛⠻⢿⣿⣇⠈⠙⠿⣷⣾⣿⣿⣿⠀⠀⠀⠀⢀⣴⡿⠻⢿⣦⣀⣤⠞⠉⠀⣀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠖⠛⢋⣻⣶⣶⣶⣦⣬⣿⣿⣷⣄⡀⠈⣹⣿⣿⣿⣧⣤⣴⣶⣿⣿⡃⠀⠈⠙⠿⣯⣤⡶⠛⠉⠉⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⡿⢧⣄⣀⣴⠿⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠉⠛⠛⠛⠿⣶⡟⠉⠛⠓⠒⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡞⠁⠀⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⡤⢤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉




⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶
⠀⠀⠀⠀⠀⡀⠛⣍⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢢⡀⠘⣿⠁⠀⠔⠁
⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣿⠁⠁⠀⠊
⠀⠀⠀⠀⠙⠀⠂⠠⢸⡂⠠⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣿⡤
⠀⠀⠀⠀⠀⠂⣀⠀⠌⣇⢀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠚⠲⠒⠓⠚⠀⠛⠒⠓⠒⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠠⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣸⠅⠀⠀⠀⢤⠠⠀⠙⠀⠀⠁⠉⠃⠀⡀
⠀⠀⠂⠀⠀⠀⢲⡆⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡼⠃⠀⠀⠉⠀⠀⠀⠳⢤⡄⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠠⠤⠤⢴⣾⡿⠃⠈⠉⠂⠀⠁⠀⠀⠀⢠⡼⠄⠀⠀⠀⠀⠀⠈⡓⠀⠈⠈⣙⡦⣄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⡀⠐⠉⠹⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⡖⠒⠒⠒⠒⠒⠒⠒⣒⠒⠒⠒⠒⡖⠒⠒⠖⠲⠒⠒⠒⠲⠶⢲⣶⡖⠖⠶⠶⣖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⢿
⠀⡇⠀⠀⠀⠀⠀⠠⠿⠿⠶⠶⠂⠴⠷⠰⠷⠶⠶⠶⠂⠴⢶⢶⣾⠾⡷⠶⡗⡼⣿⡿⣴⢶⡶⣶⠶⠀⠀⠀⠀⠀⠀⠀⢨⣫⣿⡏⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡿⢷⢾⡷⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡰⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣰⡶⣮⣀⡀⠰⣾⣶⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡿⠧⣸⣿⣿⣿⢉⢀⣀⣤⠴⢶⣿⡟⠉⠁⠙⠻⠿⠿⣿⣷⣻⣿⡿⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣾⡟⠝⣩⣭⣿⣷⣲⣿⣿⢯⠄⠀⠠⢵⣶⣶⣾⣿⣽⣿⣶⣾⣿⣿⣷⠀⢀⡀⠀⠀⠀⢀⣄⣿
⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⠐⣴⣾⣿⣯⣴⣿⣿⣦⠚⠭⠛⣻⣾⣻⣖⠐⢯⣿⣿⣿⣍⠩⠻⡅⢽⢿⣿⠿⣿⡆⣤⣇⣀⡀⢀⣸⣿⣿
⢸⡇⠀⠀⠀⡀⠸⠆⠀⠀⣿⣷⣿⣿⣿⣿⣿⣷⣿⣷⣄⣤⣿⣿⣿⣿⣷⣠⣼⣿⣿⣷⣄⣴⣿⣿⣿⣿⣾⣿⣟⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣇⡟⣻⣿⣿⣿⣿⣷⣶⡿⣽⣿⡿⡻⢿⣿⣿⣿⣧⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⡄⠉⢻⡗⠲⠄⡀⢀⣀⠀⢠⡶⣀⣴⡯⠤⢤⡄⠀⣶⠀⢨⠁⠀⣀⣀⡀⢠⣤⣀⣴⡀⣶⣶⢀⠄⠀⢠⡆⠁⠀⢦⣀⣄⠁⢀
⠀⠀⠀⠀⠀⠳⠀⠻⣶⣾⡀⠀⠀⠀⠐⢦⢀⡼⠁⢀⡠⠄⠀⣬⠅⣀⠁⢠⣴⣶⣾⣿⣿⡄⣰⡿⠀⠚⠛⡿⠅⣻⠀⠀⠀⠟⠒⣆⡴⢛⣧⡀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠸⠯⠯⣻⣿⠁⢴⣼⠋⠚⠒⡛⢀⣀⠖⠛⠠⠟⢿⣻⣷⡿⣿⣥⣦⣽⣥⠐⢶⡄⢔⠀⠘⠿⣐⡞⡳⠀⢀⣃⠊⠉⠀⠹⠉
⠀⠀⠀⠀⠀⠈⠙⠒⠀⠗⠒⠲⣠⡄⠀⠇⠀⠲⠛⡇⠈⠫⠀⠀⠀⠰⣾⣿⣿⣷⢛⢿⡓⠒⣮⠔⠚⠃⣠⠍⢀⣀⠌⠀⠋⢶⣎⠉⠀⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⡣⣾⡄⠀⠚⠈⠓⠦⣟⠀⢀⠀⠙⠣⠀⠓⢦⡠⠖⢈⢻⣿⣯⠁⣀⣠⢼⣅⠠⠟⡂⠉⢥⡄⡁⠀⣠⣂⣄⣤⣰⣆⣜⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣀⢀⠈⣆⣸⡇⠀⠀⣬⡀⢈⡃⠂⣨⠉⠛⠷⠘⣤⣴⠀⠀⣠⠈⠀⠀⠀⠛⠛⠒⡼⢛⠋⠉⠈⠚⠱⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠈⡉⠉⡴⠢⢿⡇⠉⠁⠴⠧⠔⠲⠤⣌⢀⠀⣀⠀⢠⡆⣀⡤⢸⠃⠰⢶⠄⠤⢦⡄⠀⠰⠀⣿⣽⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⣄⠁⡷⠀⠀⣸⡆⢠⣅⣠⡆⢀⣀⠀⢸⠈⠹⠯⠉⢈⠃⠁⣀⠤⣤⠈⣻⡖⠈⢙⠰⣦⠉⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠖⠀⢸⡋⠈⠉⠁⣮⠐⡙⠛⠒⠚⡄⠀⠀⠀⠜⠘⠚⠃⠀⠄⠻⠉⠀⠀⠀⣤⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢿⠳⣄⣠⡾⠂⠚⠋⢀⡀⠀⣲⣆⣢⠞⣠⡤⠀⠀⠀⠘⠈⢶⣄⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠠⠆⠅⠀⠀⠀⠘⠃⡈⢡⠞⢁⡄⠻⠁⠀⠘⠀⠀⢀⡀⠛⠄⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⣤⣀⡀⠀⠀⠀⣄⠀⠚⢁⡟⡀⠀⣹⡀⠀⠀⠁⠀⣀⣈⠁⣤⣴⣶⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣹⣿⣿⣍⣠⣶⡟⠉⢀⣠⠘⠃⡀⠛⢿⣀⠠⣄⠀⠀⠉⠻⣿⡞⣿⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢿⢿⣟⣿⣿⠿⠄⠀⣰⣫⣤⠂⠙⠏⠀⢀⠀⠡⠤⢑⣄⠀⠈⠛⠳⢶⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠛⠁⠀⠀⠀⠀⣼⣉⠋⡼⢀⠒⠃⡀⢀⠑⢒⠓⠀⣦⡄⠀⠀⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⣧⡍⢀⠳⣚⠀⡠⡑⠄⠁⠄⠡⠄⡀⡦⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣃⣠⠙⠭⣀⡑⠄⣠⠑⢠⡠⡀⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⣐⡛⡶⣬⡄⣐⠹⣌⢂⣌⣧⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠬⣿⡬⣦⡽⢫⣶⣯⠗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂
⠀⠀⠀⠀⠀⠲⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣌⠢⡀⠀⠀⡾⢶⣤⡀⠀⢸⠀⠀⠀⠀⠀⡠⠔⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣍⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⡇⠚⢉⣁⡀⠈⣢⣞⣀⣠⢤⣁⣀⠀⠣⣀⠠⠐⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠙⠒⠢⠤⠤⢃⡜⠉⠀⠀⠉⠳⣴⠋⠀⠀⠀⠈⢻⡙⠒⢤⡀⠀⠀⢠⠃⠀⠀⠀⣀⠤⠐⠒⠈⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⢀⡴⢻⠁⠀⠀⠀⠀⠀⠘⡄⢀⣀⠀⠀⠀⢣⠀⠀⠈⠳⣄⠺⠤⠐⠂⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢠⠎⠀⢸⡀⠀⠀⣖⣆⠀⠀⢳⢰⣿⡆⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕⢠⠇⠀⠀⠀⢣⡀⠀⢿⡿⠀⠀⣾⡀⠛⠁⠀⣀⠞⠀⠀⠀⠀⠀⡄⢣⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠲⠶⣒⡈⠉⠁⠀⠀⡸⠀⢤⠀⠀⠀⠙⠦⣤⣤⠴⠚⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⢀⠘⠀⠀⡇⠑⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠀⠀⠀⢀⣀⠬⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⢡⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⢸⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠸⡀⠀⠀⠸⡀⠀⠒⠀⠤⠤⠠⠤⠀⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠖⠉⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⣀⠔⠊⠁⢀⣀⣀⠤⠤⠤⠀⠒⣦⠈⢧⠀⠀⠀⠀⠀⠁⠀⠦⠀⣀⡀⠴⠔⠊⠁⠀⠀⠀⠀⠀⢀⡴⢁⠤⠀⣀⡀⠀⠀⠀⠀⠑⠢⣀
⠀⠰⠞⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⢆⠀⠀⠀⠉⠉⠁⠒⠂⠤⠤⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠠⠄⠒⢌⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⡤⢄⣀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠊⠉⠀⠀⠀⠀⡆⠀⠀⠈⠉⠓⣒⠲⠲⠶⠖⠒⣋⡉⠀⠀⠀⠀⡁⠀⠀⠉⠒⠤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡴⠋⠀⢇⠀⠀⢀⠜⠀⠙⣄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡴⠋⠀⠀⠀⠘⡀⢀⠎⠀⠀⠀⠈⢢⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡠⠊⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠀⠀⠀⠱⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⢟⡫⢩⣍⢳⣼⡄⡄⠹⢭⡟⢻⢲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡏⡹⣬⣹⣼⢩⢹⣯⠏⡶⣗⣦⡶⢺⢸⡋⣡⠷⢍⠟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣧⢸⡝⣿⢷⠛⣇⣷⢳⡏⢦⣇⣽⢧⡇⡼⠉⣱⠾⣺⣩⡿⢛⣊⣳⣦
⠀⠀⠀⠀⠀⠀⣠⡞⠡⢕⣮⣏⣿⡿⢼⣀⣿⠋⣾⡽⣞⣧⡷⡷⢻⣷⣚⣽⣾⠯⢽⣥⡿⢹⠧⣬⣷⣄
⠀⠀⠀⠀⠀⣰⠯⣤⣙⡀⢵⠘⢷⡇⣸⠿⣄⣧⢻⣁⣿⡟⣼⣧⢏⢿⡿⣉⡴⢖⠎⣝⡠⣾⣾⣋⡤⢹⣆
⠀⠀⠀⠀⢰⢏⢠⠄⣯⣳⣈⢻⠚⣿⠛⡖⠚⢿⣾⢋⡏⣿⠟⣽⠞⣻⠟⢡⢃⣞⡜⢁⣾⣡⣼⡯⢰⠏⣹⡆
⠀⠀⠀⢀⡿⠘⡟⢗⠪⣆⣉⣿⣷⣿⡄⢸⣄⣼⠋⣿⣤⡏⡼⠁⣠⣿⣴⠿⢻⠟⠋⡽⠊⡰⣣⢏⡝⢋⡡⢿⡀
⠀⠀⠀⢸⠇⣶⣿⣶⣛⣦⣵⣌⢦⣙⣿⣾⣟⢸⢰⡇⠙⣿⣷⣾⣟⣉⣤⠴⣿⣒⣺⣷⢾⠞⡛⡷⠚⠛⢲⣽⡇
⠀⠀⠀⢸⡏⣯⢿⡶⢯⢻⣤⡤⠿⠻⣏⠛⣿⣼⣼⠁⢠⣿⣿⡛⠉⠁⣴⡶⠚⠋⢳⣤⣿⣯⡝⠛⣥⣼⣉⣼⡇
⠀⠀⠀⢸⣀⣨⣗⠛⢺⢿⡉⠲⣄⢸⡇⠳⣜⢿⣿⣶⣿⡇⢹⣧⠀⣾⡿⠚⠛⠛⣛⠿⠛⢦⣼⡟⠻⢭⢓⣬⡇
⠀⠀⠀⢸⡇⠹⠦⣭⠿⣶⡿⠶⣮⣽⣿⣦⣾⣿⣿⣿⣿⡇⠀⣿⣾⡏⠀⣀⣠⣾⣓⣋⣉⣩⠿⣷⡒⠶⣭⣼⡇
⠀⠀⠀⠀⢷⠀⣼⠃⠔⣡⢖⣿⠋⠉⠙⠒⠒⠶⢬⣻⣿⣿⣿⣿⣿⡶⠛⠉⠁⠈⠙⠫⣍⠳⢤⡨⠷⢆⣲⡿
⠀⠀⠀⠀⠘⣧⢼⣅⣪⠖⣹⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⢦⡤⣕⠺⢏⣾⠃
⠀⠀⠀⠀⠀⠘⣧⠰⣃⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢼⣒⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⢀⣠⣤⣴⢶⣾⢿⢿⠿⣿⣿⣿⡶⣤⣄⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣽⢪⣖⣏⣝⢱⠋⣽⣰⠯⣞⢠⡟⢾⣾⢿⢶⣶⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣯⣝⣧⣒⣛⣶⣏⣋⣼⣤⣽⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠤⠀⢀⣄⠸⠀⠀⠀⣤⣶⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢰⡖⢰⣶⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡽⠟⠛⠘⡅⢆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤
⢀⡌⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠦⣄⠀⠄
⣿⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠄
⠋⠀⠘⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢄⠀⠠⠀⡀⠀⠑⠩⣢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀
⠈⠀⠀⣦⡈⡢⠄⡀⠀⠀⠀⠀⠀⢉⣉⣙⡛⠙⠋⠀⠀⠀⠀⠐⠠⢀⠀⡠⡀⠰⡃⠀⠠⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢶⢶⣦⣃⡀⠀⣤⣀⣉⣩⣭⣍⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹
⠀⠀⠀⡃⠸⣿⢿⣿⣿⣿⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠘⣄⠀⠄⠀⢀⡀⠀⠀⠀⢀⠀⠂⠁⠠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⢀⠀⠀⠘⠛⢿⣏⣛⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⡁⠀⢀⡀⠈⠧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⢤⢀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠰⠂
⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣦⡀⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠘⣣⠈⠓⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠢⠀⠐⢄⠐⠀⠀⠀⠘⠋⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠄⠘⡀⣄⢀⢃⠀⠀⠀⠀⠁⡞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢠⠼⠏⡐⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠈⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣟⣿⣿⣿⡿⠟⡻⠶⠖⠲⢆⣀⣀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃
⠀⠀⠀⠀⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣄⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠂⢄⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠙⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠔⣩⠖⠈⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⣶⠠⠀⠀⠀⠀⠀⠁⠠⢁⠎⠀⢀⠔⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⢋⠞⡴⠁⢀
⠉⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠐⠀⢠⠃⡠⢀
⠀⠁⠀⠀⠀⠉⠲⠠⠄⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢁⠂
⠀⠀⠀⠀⠀⢀⡄⣧⠀⠀⠁⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣤⡀⠀⢀⣴⣶⡀⠀⢀⣤⠀⠉⠉⠁⠀⢀⣴⣄⢸⣶⣄⠀⢀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⣤⣀⣿⡟⠉⠛⢿⣧⣴⡟⣷⣶⡤⣾⡟⢻⣦⣽⣿⠀⠹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠉⠀⠀⣸⣿⣇⣸⣷⣆⠀⠛⣡⣾⡿⢁⣀⡀⠐⣿⣿⠋⢹⣷⡄⠀⠙⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⣸⣿⣷⡟⣉⡜⠻⠷⣾⣿⠻⢷⣿⠉⢁⣀⡿⣿⣶⣿⣹⠿⠂⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⣼⠿⣀⠙⣿⣟⠁⠀⣤⣈⢻⣷⣦⣉⣾⣿⠿⣤⡼⢿⣿⠁⠀⡀⠀⠀⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠀⠀⠀⠡⣤⢿⣿⣿⠿⢷⣄⣀⡿⠟⣉⣿⡿⠋⠉⣤⣌⣥⣿⠿⣧⣤⡄⠀⠀⠈⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⢀⣴⡿⠿⣤⡜⠻⢿⣤⣼⣿⣿⣄⠀⠀⢈⣿⣟⢱⣶⠊⠻⣿⡄⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠐⣿⠟⠛⣷⣶⣿⠷⠀⠈⢻⣿⣮⠙⢻⣶⣿⠛⠻⣿⣇⣴⡶⠄⠈⠀⠀⠸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⡾⠉⣿⣷⣄⣀⣀⡾⢿⣿⣷⣞⠉⠉⣶⣦⠙⢿⣷⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣤⡿⠋⠀⠻⢿⣧⠸⣧⣽⡟⠀⠺⠋⠙⣿⣿⠋⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣧⣾⣿⣄⡀⢙⣷⣿⠛⢦⣤⡤⠰⣿⣿⠉⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡜⠛⠛⢻⣿⡟⢳⣾⠋⢀⣤⡟⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠰⠶⣦⣭⡿⣿⣤⣤⠞⠛⠁⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣄⠀⠈⠉⠠⠞⠉⠁⠀⣀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠐⣷⣾⠆⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠈⠁⠀⠀⠹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣷⠀⠀⢀⡀⢰⣶⣤⠢⣤⡀⠈⠁⢀⣤⡄⢠⣶⣦⣀⡀⠀⠀⣾⣿⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠿⠛⠛⠋⠀⢠⣌⣿⣦⣈⢿⣾⡟⢃⣴⣶⡞⠛⣷⡿⣧⣼⢯⣤⡆⠀⠙⠻⠟⠻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣸⣿⣿⡛⢿⣶⠉⣰⣿⢿⣿⣿⣎⢉⣴⡿⢻⣾⢿⣧⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢰⣿⠟⣡⣼⢿⣇⣸⡿⠿⣷⡎⢱⣾⠿⣿⡏⢱⡿⢿⣦⠹⣿⠆⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣦⣥⣾⠿⠇⠀⢻⣿⣠⣾⡟⢳⡞⠻⣷⣬⣿⡟⠑⠞⠻⣷⣤⣶⠆⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⢷⣶⣶⣶⣶⣹⡿⢏⣸⣏⣿⣧⣸⣿⢋⣿⣷⣰⣿⡟⠛⣧⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣾⡟⣿⣯⡽⢿⣤⣾⣟⣻⣟⣻⣷⣤⡿⢏⣹⣿⣻⣷⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣉⣿⡟⠻⣷⡼⠛⣹⣿⣇⣸⣿⣌⠛⢣⣾⡟⣹⣿⣉⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣉⣰⣀⣈⢿⣾⡿⢁⣙⣧⡜⢻⣶⣿⣇⣀⢃⣀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣷⡿⢿⣾⠋⣠⣾⠿⠿⢷⣄⠉⣴⡿⢿⣾⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⢿⣟⣹⣿⠿⣷⡄⢠⣼⠿⣿⡏⣱⡿⢿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠉⠁⠀⠹⣿⣠⣾⡟⣿⣿⢻⣷⣀⣽⠟⢁⠈⠉⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⠀⠀⠀⠼⢿⣯⣾⡿⠻⣷⣼⠿⣷⡄⠈⠀⣠⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠿⠿⠁⠀⠀⠀⠘⠛⠃⣠⣤⠘⠛⠃⠀⠀⠀⠀⠻⠿⢿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣾⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣶⣶⣶⣤⣤⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⡶⣦⣄⣀⡉⠉⠻⠟⠋⠉⣠⣶⣄⣀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠛⠛⢿⣿⠁⢠⣾⠏⢠⣾⣿⠿⢿⣦⠀⢠⣶⠿⠃⣀⠛⢻⡄⢳⣦⠈⢿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢸⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠉⢠⣄⡙⠿⣿⣆⣹⡿⠃⣠⣴⣶⡿⠿⠟⢁⣄⡙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢀⣈⠻⣷⣀⡘⠿⠏⠀⢰⣿⣿⣿⡿⠖⢲⣞⢛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣤⠈⢻⣆⣈⣿⣿⣄⠀⢿⣷⡿⠋⢀⣤⣤⡄⠻⣏⣉⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⢉⣿⠇⠈⠻⠿⠿⢯⣤⡈⠙⢷⣶⣾⠟⢉⣤⡄⠙⠿⠿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣤⡿⠟⢀⣼⣷⣄⠐⠿⣿⠗⠀⣨⣿⣿⣄⢸⣋⠁⢶⣶⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁⢰⡿⢃⡸⢿⣷⡶⠉⠀⣴⡿⠛⠛⣿⡀⠉⣷⣶⠛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣆⣀⣴⠟⣷⣶⣦⣄⠈⢲⣞⢉⣶⣿⠿⢷⡄⠈⢿⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⠀⢠⣿⣟⡉⠻⠿⣀⡸⠿⠟⠛⢤⣼⣿⠆⠀⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢰⡾⠛⢿⣿⣠⣶⠟⢁⣴⣿⣷⣏⡹⠇⣠⣾⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣴⣆⣀⡙⠛⢁⣤⡟⢋⣽⣿⡿⠁⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⡀⠉⠉⣾⣿⣿⣷⣀⠙⢷⣾⠟⠉⣠⣾⠿⠈⠁⣠⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠆⠀⠉⠙⠿⢏⣩⣤⠈⢉⣴⡿⠋⠀⠀⠰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⠇⠘⠛⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⡏⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠁⣀⣀⣀⡀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠊⠀⠀⠀⠈⢢⠀⢁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⣀⡀⠀⡸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠉⠑⠤⠤⡑⠤⠤⠤⠖⠁⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⢀⠤⠠⢀⠔⠚⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⠢⠖⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠲⢤⠞⠁⣀⠴⠒⠒⠒⠲⢀⡴⠚⠉⠀⠈⠉⠓⢄⠱⠤⠠⠤⣄⠀⠀⡀⠠⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠁⠉
⠀⠀⠀⢀⡴⠒⠒⠀⠀⢀⡤⣖⠋⠩⡍⠉⠉⠓⠲⣤⡀⢀⠔⠋⠁⠀⠀⠈⠙⠦⣄
⠀⠀⢠⠏⠀⠂⠀⠀⡰⠋⠀⠃⠘⠐⣇⠀⠀⠀⣍⠁⠛⣎⠀⠀⢀⠄⠚⢆⡀⠀⠘⣦
⠀⠀⢸⠀⠀⠀⠀⠰⠁⠀⠀⠐⠐⠉⠁⠀⠀⠀⠀⠓⠀⢸⡀⠀⠈⠢⣉⠔⠁⢀⠀⢸
⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⣀⠀⡀⠀⠸⡄⡄⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⡼
⠀⠀⠀⠙⢦⣄⡀⠸⡄⠀⡆⢂⡑⠀⠀⠣⠂⠈⠁⠀⠀⡼⠀⢤⣀⣀⣘⣑⣠⡶⠚⠁
⠀⠀⠀⠀⠀⠀⣄⠀⡙⢦⡙⠃⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⡀⢰⠀⢈⡭⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠃⡴⠈⣝⠒⠒⠒⠒⠂⡀⠀⠀⠀⠀⢱⠈⢠⡎
⠀⠀⠀⠀⠀⠀⢀⡇⢰⡇⠀⠈⢦⠀⡏⢠⠞⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⡸⠀⢸⠇⠀⠀⢸⠀⢰⠋⠀⠀⠀⠀⠀⠀⢰⠃⢸
⠀⠀⠀⠀⡤⠖⠒⠒⠶⢄⠀⠀⢸⠀⢸⢀⣄⣀⡀⢀⡤⠤⡼⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠀⢸⠀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⠈⡄⠀⠀⠠⠀⡠⠆⢀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠄⠀⠀⠀⠀⠈⠊⠀⠘⠁⠐⠀⠀⠀⢈⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡖⠠⠤⠀⠠⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠢⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⡀⠘⠂⣾⣿⣿⣿⣿⣿⣶⠀⠂⢂⣤⣤⣤⣀⡒⠀⠀⢁⠂⠄⡀⠀⠘⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⡠⠲⣹⣿⣿⣿⣿⣿⡿⣠⡀⣾⣿⣿⣿⣿⣿⣿⣶⢀⡟⠀⠌⠂⡀⠀⠀⠀⡠⢁
⠀⠀⠀⠀⠀⠀⠒⡂⠀⠂⠀⠠⠊⠀⠀⠙⠿⣿⣿⡟⠉⠀⠁⠁⠻⣿⣿⣿⣿⣿⣿⢋⠞⠙⠢⢄⡀⠈⠁⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠠⠈⠀⠀⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠮⠭⠿⠭⠖⠁⠀⠀⠀⠀⠉⠲⢄⣀⣤⠁⠀⣀
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠋
⠀⠀⠠⢆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠐⢄⠀⡀
⠀⠀⠄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⢂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⢠⠊⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠠⠄
⠀⠀⠠⠀⠁⠀⢀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠑⠀⢀⠃⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠀⢨⠀⠀⠠⠌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢠⠀⠀⠀⢀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⢀⠠⠇⠀⠠⠒⡀⠀⠀⠜⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠈⠐⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣒⠂⠀⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠴⠖⠒⠦⠀⠀⠀⠀⠀⠁⠈⠛⠢⣄⣻⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠦⢚⡁⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡆⣀⠤⢖⣉⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡵
⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⣻⣶⢞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢦⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠱
⠀⠀⠀⠀⠀⠀⢈⠳⣄⣀⠀⠀⠀⢀⠥⠀⠈⠁⠰⠄⣄⠀⠀⢀⡀⠴⠖⠘⠋⠉⠉⠛⠠⠶⠂⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠜⠀⠀⠀⠀⡠⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠒⠬⣤⡄⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠄⡬⠔⠒⠚⠉⠁
⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⠁⠒⠂⣈⣀⣢⡀⠐⢒⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⠀⠒⣂⢀⣀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠑⠚⠙⠓⠒⠂⠄⠀⠀⠉⠂⠨⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠐⠋⠁⠀⠀⠀⠒⠂⠐⠛⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠘⠀⠀⠀⠙⠷⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡤⠶⠟⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠤⠤⠄⠀⢀⢼⠞⠛⠒⢦⣀⡀⠀⠀⠀⠀⢀⡠⠖⡛⠛⠠⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⠀⣏⠀⢀⡺⠖⠐⠋⠉⠙⠛⠛⠋⠉⠁⠐⠎⡈⠀⠀⡳⢀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⠁⠀⠀⠀⠀⢀⠀⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⢼⠀⠀⠀⠀⠀⠉⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⡐⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢐⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠊⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢧⣠⡿⠁⣠⣽⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⣸⣿⣿⣵⠀⣀⡀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠃⠰⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡟⢋⣠⠼⢿⣿⣾⣿⢗⠝⠋⠁⠀⠀⠀⢀⠴⠿⠿⢛⠁⠀⠠⠖⠟⠉⣧⠐⠎⠓⠀⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠋⡄⡶⣷⣿⣿⣿⣿⢧⡀⠂⠄⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢰⠁⠀⡘⢧⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣥⠟⢹⡿⢹⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠐⠃⠈⠆⠀⠀⡄⠀⠀⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢼⣧⠌⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣡⠀⠀⢹⡀⠀⡀⠁⢰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠉⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣧⠀⡀⠀⡞⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⢠⠃
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⣰⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢼⠦⣄⣀⣠⢿⠀⣿
⠀⠠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠓⠀⠾⢊⡴⢂⣉⣿⡄⣿⡀
⢀⣈⣽⣿⣷⣷⣤⠄⡀⠀⣠⡀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠚⢙⠣⣃⠁⠚⣉⡐⢂⣔⣿⣿⣿⣷⠦⠄
⠀⢴⣿⣾⡿⠽⣿⡧⢹⣿⣿⣿⣿⡤⠀⡼⠙⠁⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣿⣤⣴⣶⣮⣭⣥⣿⣿⣿⢿⣿⣿⢿⡟⠋
⠀⢸⣿⡟⠀⣀⡀⣸⣿⣿⠿⣯⠝⠂⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣭⢿⣄⠙⠆
⠀⣞⠏⠀⠉⠙⣿⠛⢉⡃⠈⠈⣀⠀⠀⡅⡀⢨⠀⠀⠀⠀⠀⠀⠀⢈⣉⣰⣾⡿⣽⣛⣉⡻⠦⠚⠛⠉⠉⠉⠉⠀⠹⣄
⠚⠁⠀⠀⠀⠀⠠⠀⠉⠀⠄⠠⢅⠀⠀⣧⡇⡈⠀⠀⠀⠐⠛⠙⠷⠿⠯⠶⠶⣿⣇⡱⠯⠻⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣉⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄

⡀⠀⢹⡄⠀⠀⠀⢀⡀⡂⠀⠀⠀⣠⡄⠊⠀⠀⠀⡀⠈⢿⣇⠀⠃⢀⣀⠀⠛⠻⣐⠿⢤⠔⠒⠤⠄⣿⡾⠟⠀⢊⣿⣿⣿⠏⠈⣯⡐⢀⠀⠀⠀⠈⠒⠤⣤
⠁⠀⠀⠙⡄⠀⠀⣤⡀⠡⠶⢶⣼⣯⠕⠀⠀⢂⠀⠀⡀⣸⡿⠀⠘⠀⢩⣄⠀⠀⠎⣤⢤⡴⠶⠟⠘⠈⠀⠀⢀⣌⢶⡟⡻⣄⠀⠋⠐⢦⣣⣄⡀⠀⠀⢀⣀
⡀⠀⢀⠀⠙⠂⠒⢿⡈⠳⢤⡈⠛⠻⠀⡀⠠⡤⠀⠀⠀⡿⠃⠐⠀⠀⢾⠋⡥⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠋⠑⠫⡀⠀⠀⠈⠉⠻⠻⠆⠉⠉
⡇⠀⠸⣆⠀⠀⡀⢈⣷⡀⠀⠙⠣⢀⠈⠈⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣇⢀⠀⠘⡆⠀⣇⠀⢿⠛⠢⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠟⡌⠀⠀⠋⠂⠄⠀⢠⡀⠀⠈⠓⢆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠠
⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠒⠄⡀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣘⠀⠀⠀⢀⠀⢂⡐
⠟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⡄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣿⣦⣤⣀⠉⢷
⠀⠀⠀⠀⠃⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠈⣉⣙⠛⠶⢬
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢠⡤
⢆⠀⠀⠀⣶⣠⣥⡀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡀
⡛⣷⠀⠀⣮⣽⣷⠞⠁⡀⠀⣠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⢀
⡻⣾⡇⢸⣿⣯⣿⠀⣸⠗⠊⠖⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠓⠲
⣿⠿⣿⣼⡇⣸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢹⢶⣽⠙⣷⣿⣥⣄⣤⡴⣆⠀⢰⠴⣦⠠⢄⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⡀⢀⣦⠀⠀⠀⠠⠀⣴⣟⠉
⠀⣿⣽⣷⡌⠻⠟⠛⠉⠁⠀⠀⡀⢅⣨⣤⡀⠀⠀⠀⠀⢀⠃⠀⠀⠂⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣃⠀⠀⠀⠀⢀⠀⠈⠻
⠂⢿⡏⠙⠟⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⡜⠀⡠⠀⣼⠏⠀⠀⢀⡤⠤⠎⠀⠀⠀⠀⠀⠁⠀⢀⡅⠀⠀⠀⠀⠀⠀⢰⡖⠒⠁⠊⢄⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡸⢢⣎⣼⠁⠓⣸⢇⠚⠀⣷⡸⡄⢇⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢸⡆⢻⣌⠛⣾⣿⠇⢈⢳⣟⢻⡞⢁⠸⣷⣿⠙⣡⡿⢠⡧⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣥⣿⣎⡻⠎⢻⣷⣿⣿⠰⠈⠻⣿⣿⡾⠃⠆⣿⣿⣼⡟⠡⢟⣑⣿⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡜⣦⠈⠙⠛⠿⠷⣦⡹⣿⣿⡆⡈⠓⢻⡟⠚⢃⣴⣿⣿⢏⣴⡿⠿⠛⠋⢁⢰⣃⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣥⣘⣿⣿⣿⢈⡗⠠⣜⣮⢿⣿⣿⣷⣳⣿⣼⣷⣿⣼⣿⣿⣿⡿⣵⣯⡔⢰⡃⢸⣿⣿⢏⣤⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣭⣽⣿⣦⣽⣏⡀⠀⣽⣷⠛⡻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⢿⡟⢿⣯⡀⢀⣨⣯⣵⣿⣯⣭⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⡆⡴⠛⠉⠹⠛⢛⣿⡷⠶⢿⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠷⢾⣿⡟⠛⠏⠉⠛⢦⢰⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⣷⣄⢇⠃⢰⡜⡀⠈⣿⣷⣄⠙⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⢀⣾⣿⠃⢀⢣⡴⠘⣰⣣⣾⠟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣥⡤⠬⠤⠽⣾⣶⣼⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣶⣯⣴⣷⠯⠤⠥⢤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣷⠟⠋⣠⡾⠋⢩⠜⠛⣿⣿⡿⠿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡿⢿⣿⣿⠟⠻⡍⠛⢷⣄⠙⠻⣷⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣵⡇⣸⡟⣀⢤⠘⠆⠨⠟⠻⣇⡀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡏⠀⠀⢀⣸⡟⠻⠅⠠⠃⡤⢀⢻⣇⢸⣮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣿⠁⠉⠀⠀⠤⠴⠂⠀⠉⠛⠛⠁⠀⣿⣿⣿⣿⣿⣿⠀⠈⠛⠛⠉⠀⠐⠢⠤⠀⠀⠈⠈⣿⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡶⠶⣶⣦⣴⣿⣿⣿⣿⣿⣟⣿⣿⣤⣶⡶⠶⣶⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠟⢛⣉⣛⣡⣶⣶⣶⠿⢟⣿⣿⣿⢿⣿⣿⣯⣽⡿⢷⣶⣶⣶⣌⣛⣙⡛⠿⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⣝⡻⠿⠟⣋⣩⣤⣶⣾⣿⢛⣩⣶⣿⣿⣿⣍⡛⢿⣷⣶⣤⣭⣙⡛⠿⢟⣫⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠍⠳⣬⡁⡀⠲⠦⣍⢁⣿⡿⢫⣴⣦⡙⢿⣿⡌⣩⡴⠖⢂⠈⣡⠖⠩⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣅⣘⠧⠰⣦⠸⣿⡇⣿⡏⢻⣿⢨⣿⡇⣡⠖⠰⢋⣨⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣤⣙⠳⠜⠇⠸⠣⠞⣋⣠⣥⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢢⡀⠀⠀⠀⢀⠇⠸⡀⠀⠀⠀⢀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢄⠀⠀⡼⠀⠀⢧⠀⠀⡰⠋⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡒⠤⣀⠀⠀⠸⠀⠀⠈⠳⣠⠃⠀⠀⠘⣤⠎⠀⠀⢠⠁⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠙⠢⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠔⠋⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠉⠉⠑⠒⠤⡀⠀⠀⠀⠀⢀⠎
⠀⠘⠳⣒⠒⠒⠒⠂⠤⠵⠀⠀⣠⠞⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠮⠔⠒⠒⠒⠒⣨⠟⠁
⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⡔⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⢘⣢⠀⠀⣸⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣔⡁
⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠒⠒⠤⢄
⠈⠉⠑⠒⠤⢤⣀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠛⢄
⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠔⠥⠤⠤⠒⠒⠒⡲⠀⠀⠈⠢⣑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢖⠒⠒⠒⠒⠤⠬⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡀⠈⠑⠲⠭⠍⣉⣀⠠⠤⠖⠊⠁⢀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⡤⠒⠉⡇⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠉⠒⢤⣀⠀⢣
⠀⠀⠀⠀⠀⠀⡼⠔⠊⠁⠀⠀⢠⠁⠀⢀⡔⠙⡄⠀⠀⢠⠋⠢⡀⠀⠀⡆⠀⠀⠈⠑⠢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠋⠀⠀⢱⠀⠀⡎⠀⠀⠙⢆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠱⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⣴⣶⣤⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢻⠟⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⢠⠖⢢⣀⡤⢄⠘⣿⣿⡋⠋⠐⠆⡧⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⢠⠖⠀⠁⠐⢢⡌⠑⡀⠀⢸⠀⢱⠎⠁⡝
⠀⠀⠀⠘⡇⠀⠻⠀⡼⠀⠈⢻⡛⢁⣀⡰⠁⠀⠀⠀⠀⠀⠀⣀⣀⡉⣀⣀⡀⠀⠀⢰⣹⢄⡤⣀⣀⡀⡑⢦⣨⠀⠘⡄⠀⠀⡼⠁
⠀⠀⠀⠀⠘⡄⠀⡰⠁⢀⡠⢧⣈⠙⠣⡀⠀⠀⠀⠀⠀⠀⠀⡉⠋⠉⣷⡵⢿⡗⠀⠘⣜⡄⠀⠀⠀⢠⡆⠘⠢⠀⠀⠱⡤⠞
⠀⠀⠀⠀⠀⠈⠚⠁⠀⠎⠀⠀⠢⢍⡚⠳⠀⠀⠀⠀⠀⠀⠀⠘⢾⣎⡿⣧⠁⠛⠆⠀⠈⡿⣄⣀⣠⢿⣃⣀⠀⠢
⠀⠀⠀⠀⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢄⡇⠀⠀⠀⠀⠀⠀⣄⢸⠉⠉⡉⠛⠁⠀⠀⠀⢰⠈⡟⢦⠘⢄⡏⠱⡄⢇⠀⣴⣿⣦⣀⣤
⣾⣿⣷⣾⣿⣿⠄⠀⢸⠀⠘⡄⠀⠀⠀⣨⣇⠀⠀⠀⠀⢀⠜⠘⠓⠐⣺⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢣⠈⠣⡀⢣⠜⠀⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⠀⠀⠸⠀⠀⠈⢦⣠⠞⠁⠘⠖⠂⠉⠉⠀⢀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠸⠀⢡⠀⠀⢂⠀⠹⣿⠀⠀⣿⣿⣿⣿⠟
⠀⠹⣿⣿⣿⠇⠀⠀⡆⠀⠀⠀⠀⠀⠣⣀⠀⢀⡀⠔⠒⠉⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠣⣀⠝⡄⠀⢹⣿⠟⠁
⠀⠀⠈⠻⠟⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢹⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⢸⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣠⠤⠴⠚⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠁⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⠀⢈⠋⠐⠒⠒⣲⣒⡲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⡄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡛⢄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠈⡆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⡣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠴⢻⢠⠇⠊⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠫⢭⣹⣿⠿⣉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⢀⡰⠃⡎⠀⢀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠖⡀⠀⠐⠂⠤⡀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠑⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠁⠀⠀⠈⠁⠀⠓⠠⢤⠂⠀⠈⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⡐
⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⡠⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠑⢄⠔
⢀⠔⠈⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡰⠟⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠻⢦⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠃⠂⠉⠁⠢⡀⠀⠀⠌⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢡⠀⠁⠂⠀⠀⠊⢀
⠀⠀⠀⠀⠀⠱⡀⠈⠀⠀⠀⣤⣾⣿⢫⠃⣰⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⡏⡏⢉⣷⣦⡀⠀⠀⠃⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢻⣏⠇⢠⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⡿⡜⠀⣾⣿⠉⠀⠀⠀⢠⠀⢀⠠⠐⠁
⠀⠀⠀⠀⢀⠠⠄⡇⠀⠀⠀⠀⠈⠻⣄⣾⣿⣿⣿⡿⠋⠀⠀⠈⠿⣿⣿⣹⠁⣼⡿⠃⠀⠀⠀⠀⠸⠉
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢂
⠀⡔⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠰⣶⣦⡐⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⣠⣶⡞⠀⠀⠀⠀⢀⠂⠀⠡
⠈⢀⠂⠈⠁⠐⠢⠀⢱⡀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⡶⠶⠶⠶⢶⣶⣾⣿⠿⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⠂⠄⠐
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠉⠛⠡⠤⠀⠀⠠⠤⠘⠋⠁⠀⠀⠀⠀⠀⢠⠮⢀⡀⠀⠀⢀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠤⠀⠀⠂⠁⠉⠐⠂⠀⠠⠤⠤⠄⠠⠀⠐⢅
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠘⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣬⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠰⠂
⠀⢰⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣟⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⢸⠏⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⠤⡈⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⡟⠀⠈⡇⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⢿⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢿
⠈⠁⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⡁⢠⠄⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠐⠀⠀⠆⠀⠀⠀⠀⠀⠙
⠀⣀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣄⠀⠀⠀⠀⠂
⠀⡏⠀⠀⠀⠸⣧⠀⠀⠀⠀⠐⠆⠀⠀⠀⠘⠢⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⡀⠀⠀⢸⠉⠉⢣⡀⠸⠀⢀⠇
⠰⡅⠀⠀⠀⢸⡇⠀⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠤⠃⠀⠀⠱⠌⠀⡈⠀⠀⢀⢀⠁
⠘⡇⠀⠀⠀⠐⠃⠀⠀⠀⠀⠐⠬⡀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⡿⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⡦⡀⠀⠀⠀⠐⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣿⠀⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⡞⠙⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣿⠀⠀⢀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠴⠒
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⡄⠈⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠉
⠀⠀⠀⠰⢇⠀⠀⠀⠀⠀⠍⠀⠀⠀⢸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠆⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠍⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⠀⠖⠡⠙⠛⠳⢽⣶⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠣⠀⢃⠘⠂⠀⠈⣿⠁⠀⠀⠘⣿⣠⠀⠁⢯⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠠⣸⣿⣷⡀⠀⠀⡼⠡⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣏⢷⣠⠛⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠘⢠⡏⣿⣻⣷⠀⠀⠇⠀⠀⠁⢠⢀⡀⠀⠀⠀⠀⠛⠀⠦⠌⣉⠻⢼⣏⣴⣣⡴⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡿⠀⠹⡟⠃⠀⠈⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣩⡵⢈⣿⣶⣤⡙⣦⣤⣁⠀⠈⣿⠂⢀⠠⢒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⠶⢾⡏⠠⠞⠈⢿⡈⢻⣸⣇⣙⣧⣴⣼⣿⡏⠶⢭⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⡀⠁⠀⠀⠘⣷⠀⠉⠉⠁⡰⢋⣴⣿⣿⣧⠘⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠹⠆⠀⠀⣟⡤⠋⣸⠛⠟⠘⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⢠⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢉⣉⠁⠀⠀⡁⢀⡀⠀⠀⠀⠉⠉⠉⠉⢙⠃⠀⠀⠠⠤⠄⠀⠀⠀⠉⠉⠁⠐⠒⠠⠀⠀⠀⠀⠹⡇⠀⢾⠆⣾⢰⠚⠈⠉⡉⠈⠠⠤⠸⠋
⠀⠀⠀⠠⣤⡀⠘⠛⠻⠶⡶⣡⢤⣴⣷⣶⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠤⠈⠙⠁⢀⠀⠠⢀⣀⡀⠐⠒⠠⠄⠈⣷⡀⠐⢀⡟⠀⡛⠂⠀⠀⠀⠠⠿⠛⠃
⠀⠀⠀⠀⢴⣟⣩⣤⣶⣆⢀⡤⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠀⠛⠀⢤⣶⣆⣥⣄⣀⢀⢀⠀⠀⠀⠁⠀⠀⠙⠃⢙⠈⠂⢸⠃⠀⠶⠶⠶⠒⠂
⠀⠀⠀⠀⢠⣶⣤⡀⠀⠈⠸⣇⠀⠀⢴⣶⣶⣯⣤⣿⢠⢣⣶⡤⠄⡄⠀⠀⠛⠋⠉⠉⠀⠀⠤⠶⠶⣧⣶⣤⣠⣀⢈⠀⢄⣉⠀⢀⠀⣀⡀⢀⣤
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠴⡦⡴⠴⠆⢦⡄⢀⠀⣤⢀⣀⡈⠀⠈⠀⠀⠃⢠⠟⠛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⠉⠉⠁⠀⠀⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⠶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⣹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠆


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠆⢰⡶⠀⢰⠖⠀⣤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⣠⡯⢤⡶⣯⣉⡽⡛⠫⢉⡙⢳⣈⠭⢥⣝⡂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣒⠂⡘⣣⣟⣁⣀⣼⡁⢸⣿⢙⠳⠇⣤⢱⡆⣹⣤⣀⣬⣽⣆⣘⣷⢠⡀
⠀⠀⠀⠀⠀⢀⡴⢆⣨⡵⣻⠉⣿⡉⣴⣩⠝⠻⣿⣿⡏⢰⡆⠉⢨⣾⣿⣟⣩⡤⠀⣸⠉⣙⢾⡇⣤
⠀⠀⠀⠀⢀⡀⠠⠋⣾⣠⣭⣤⣿⣧⣅⣀⣻⣆⠙⣿⣧⡔⠚⠃⣼⣿⣛⣫⣬⣴⣾⣯⣬⣭⣤⣽⣁⣤⣄
⠀⠀⠀⠀⢩⡏⣼⠋⢿⡌⣦⢠⠶⠎⡙⠛⠛⡛⢿⣿⣿⡇⠀⣼⣿⣿⠟⢫⣭⢧⡌⣩⡄⡄⢀⡿⡙⣷⡄⣤
⠀⠀⠀⠀⣨⢸⣧⡴⠾⠿⢶⣤⣀⡸⠃⠘⠶⠝⠊⠹⣿⣿⣾⣿⡟⠀⠰⠞⣁⣬⣤⣭⣴⣵⣟⡱⠏⠟⢿⣝⡂
⠀⠀⠀⠠⣯⡞⢷⣬⣻⣦⣴⣿⠿⠿⣿⣿⣿⣶⣦⣀⣿⣿⣿⣿⣠⣴⣿⡿⠿⠿⠉⠉⢭⡛⠻⡟⡻⢶⣼⠏⠓
⠀⠀⢀⣴⣯⢞⣴⠟⣉⠔⠋⣰⠏⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣼⠀⠁⣾⢀⡙⢆⡝⢧⠻⠄
⠀⠀⠘⠋⣧⠎⣴⠸⠋⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢸⠆⠀⠼⠃⢿⣁⣸⡇
⠀⠀⠀⢸⢪⡳⢍⢀⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣩⡍⡸⣿
⠀⠀⠀⠀⠈⠻⠀⡈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢐⠁
⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠱⠟
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⢚⣿⣶⢿⣟⣉⣉⣵⣿⠟⢻⣿⢿⣿⠻⣿⣦⣀⣉⣹⣿⣶⣿⡓⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣤⣿⡀⠈⢹⡿⠋⢁⣠⡿⢃⡈⢿⣦⣈⠙⢿⣏⠁⠀⣽⣤⣽⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢽⣧⠶⢿⣇⣠⢿⣿⠀⡟⢹⠀⢻⡿⣆⢸⡿⠶⣼⡿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⣧⣀⣿⡀⢳⡞⢀⣾⣁⣼⠟⠷⠶⠋

⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠠⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠍⠈⠉⠀⠤⡦
⠀⠀⠀⠀⠀⠀⠀⣸⣌⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣷⣶⣤⣀⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣦⣼⡇⠀⠀⠀⡀⠀⠀⠀⢀⣶⠿⠀⠸⡟⠛⠛⠻⢿⣷⡌
⠀⠀⠀⠀⣨⣱⣤⣄⣿⣿⠿⠟⠛⢃⠀⠀⠀⠱⠶⠳⠤⣁⠀⠀⠀⢀⠃⠀⠀⠀⠰⠻⢧⠄⣀
⠀⠀⠀⠀⣉⣬⡛⣽⣹⣿⣐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣬⣀⡀⠀⣀⠀⢀⣧⡀⣥⡄
⠀⠀⠀⠀⠹⠋⠛⢻⡟⡇⣿⢧⡀⠀⠠⠀⠀⠀⠀⠀⢰⢀⢀⠀⢀⢁⣹⢱⠈⢿⡆⠈⠟⠙
⠀⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⡂⠀⠐⡀⠀⠀⠀⠀⠸⣄⣬⡀⣠⠀⣤⠈⠀⠘⣧⠀⠾⡆⠄
⠀⠀⠀⢨⡅⠀⠐⢻⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢸⣿⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢘⣿⣀⠆⠈⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠻⣇⡀⢠⣤⣀⣀⣀⣀⣀⡠⠀⠠⠉⠛⢳⡄
⠀⠀⠀⠂⠁⠙⠐⠚⠃⠀⠀⠐⠒⠛⠛⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⡀⢀⡐⢂⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠆⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂⠠⢀⡤⠀⣶⠀⠀⢀⢈⡀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠉⠃⠁⠘⣿⣄⡀⠸⡿⠁⠁⢀⠃⠀⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢃⣘⠋⢻⣟⠂⠀⠀⠈⠐⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠤⠄⠀⠀⠠⢀⠉⢄⣀⠠⡈⠉⠠⡞⡈⡀⠂⠀⠑⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⡉⠁⠠⢀⠠⠱⣀⡀⠀⠀⠈⢄⡀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⢿⠆⡄⠀⠀⢸⠆⠔⠉⢜⠀⠀⡠⠀⠀⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⢁⠀⠀⡞⠀⠂⠀⠘⠀⡔⠀⠀⠀⠈⠀⠀⢠⠀⠀⠠⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⡄⢸⠁⠀⠀⠀⠀⡴⠀⠀⢀⠀⠀⠄⠁⢨⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠙⣼⠀⣤⠀⠀⠀⡇⠀⡐⠁⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣠⣶⣦⣤⡘⣇⣿⣇⠀⢀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣭⠪⣙⣿⣾⣿⣿⠀⡘⠀⠰⠀⠀⣐⡁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣽⣿⣿⣶⣿⣿⣿⣿⣯⣤⠃⠀⠀⠄⠀⠀⡀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠴⠤⠬⠍⠥⣠⣿⣿⣽⡟⠀⢄⣞⣤⣴⣾⣿⣁⢀⡁⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⣀⠀⠈⠈⢉⠡⣮⣿⡵⠤⠊⠀⠈⠙⠻⠿⣿⠞⠋⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⢧⣣⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠯⢎⠁⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠀⣰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢡⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣀⠀⠀⣨⡷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠠⠋⣰⢫⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣽⣿⡀⠀⡏⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⡇⠜⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⢸⡃⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠀⠀⠀⠐⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠈⠁⠐⠂⠶⢼⡷⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠄⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠃⠒⠂
⣿⣿⣿⣿⣾⣶⣶⣿⡖⠐⠉⠀⠀⠀⠀⠀⠀⠐⠶⠖⠀⠀⣀⣀⣀⣀⡄⠀⠀⠀⢀⣀⣦⠖⠚⠉⣹⣧⣶⣶⣿⠛⠟⠻⢯⣀
⠉⠙⠋⠛⠛⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢖⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⠴⠾⠟⠁⣼⠏⠁⠣⠀⠀⠀⠉⠛⠛⠀⣈⣈⣑⠂⠆⠂
⣤⣀⣴⣧⣤⡆⣀⡆⢀⡆⠀⠀⠀⠀⠀⠀⠀⠴⣖⠀⠉⠉⠒⠒⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣡⣾⠀⠰⡀⠀⣀⣀⡤⠚⠁⠈⢹⡏⠃⠶⠤
⣿⣿⣿⣿⣿⣿⣿⠃⣸⣁⣀⡀⠀⠀⠉⢲⣆⠀⠈⠉⠉⠉⣳⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣏⡼⠷⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⢧
⣿⣿⣿⣿⣿⣿⣿⠀⣿⠁⠀⠈⠁⠒⠒⠚⠛⠚⠒⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡇⡀
⠿⠿⠿⠿⣿⣿⣿⡆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠆⢰⡶⠀⢰⠖⠀⣤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⣠⠯⢤⡶⣯⣉⡽⡛⠫⢉⡙⢲⣈⠭⢥⣝⡂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣒⢂⡘⣣⣟⣁⣀⣼⡁⢸⣿⢙⠳⠇⣤⢱⡆⣹⣤⣀⣬⣽⣆⣘⣷⢠⡀
⠀⠀⠀⠀⠀⢀⡴⢆⣨⡵⣻⠉⣿⡉⣴⣭⠝⠻⣿⣿⡏⢰⡆⠛⢨⣾⣿⣟⣩⡤⠀⣸⠉⣙⢿⡇⣤
⠀⠀⠀⠀⢀⡀⠠⠋⣾⣠⣭⣤⣿⣷⣅⣀⣻⣆⠙⣿⣧⡔⠛⠃⣼⣿⣛⣫⣬⣴⣾⣯⣬⣭⣤⣿⣁⣤⣄
⠀⠀⠀⠀⢩⣏⣼⠋⢿⡌⣦⢠⠶⠎⡙⠛⠛⡛⢿⣿⣿⡇⠀⣼⣿⣿⠟⢫⣭⢧⡌⣩⡄⡄⢀⡿⡙⣷⡄⣄
⠀⠀⠀⠀⣨⢸⣧⡴⠾⠿⢶⣤⣀⡸⠃⠘⠶⠝⠊⠻⣿⣿⣾⣿⡏⠁⠰⠞⣁⣬⣤⣭⣴⣵⣟⡱⠏⠟⢿⣝⡂
⠀⠀⠀⠠⣿⡞⢷⣬⣻⣦⣴⣿⠿⢿⣿⣿⣿⣾⣦⣀⣿⣿⣿⣿⣠⣴⣿⡿⠿⠿⠉⡉⢭⡛⠻⡟⡻⢶⣼⠏⠓
⠀⠀⢀⣴⣯⢞⣴⠟⣉⠔⠋⣰⠏⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣼⠀⠁⣾⢀⡙⢆⡝⢧⠻⠄
⠀⠀⠘⠋⣧⠎⣴⠸⠋⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⢸⠆⠀⠼⠃⢿⣁⣸⡇
⠀⠀⠀⢸⢪⡳⢍⢀⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣩⡍⣸⣿
⠀⠀⠀⠀⠈⠿⠀⡈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢐⠁
⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠱⠟
⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣷⠖⢚⣿⣶⢿⣟⣉⣉⣵⣿⠟⢻⣿⢿⣿⠻⣿⣦⣀⣉⣹⣿⣶⣿⡓⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣤⣿⡀⠈⢹⡿⠋⢁⣠⡿⢃⡈⢿⣦⣈⠙⢿⣏⠁⠀⣽⣤⣽⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢽⣧⠶⢿⣇⣠⢿⣿⠀⡟⢹⠀⢻⡿⣆⣸⡿⠷⣼⡿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠿⣧⣀⣿⡀⣳⡞⢀⣾⣁⣼⠟⠷⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⣦⠘⣄
⠀⠀⠀⠀⠀⡠⠤⠀⣤⣤⣀⣀⣸⣁⡀⠀⠈⢳⣍⠀⠀⠀⠀⡤⠄⠓⠠⢄⡀
⠀⠀⠀⠀⣾⣁⣤⠖⠀⡠⠋⠁⠀⠀⠀⠐⡀⠀⠻⡄⠀⣶⡾⠁⠀⠀⠀⠀⣱⣤⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠹⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⢦⣿⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠉⠗
⠀⠀⠀⠀⢀⣇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢿⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠴⠈⠉⠉⠉⢢⡀⠐⢦⡀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠰⣆⠀⠀⠀⠀⣀⠀⡴⠎
⠀⢸⠀⠀⠀⠀⠀⠀⡁⠀⠀⠙⠿⠋⠁⠀⠀⠀⠀⣀⡀⠀⢘⣄⠀⠀⠀⡟⢠⡤⢀⠴⠉⠈
⠀⠀⠺⡀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠀⡠⠎⠙⠛⠁⠈⠉⠳⡎⢛⡛⣄⢀
⠀⠀⠀⠙⢶⣴⠏⠀⠀⠀⠀⠀⡔⠰⣧⡶⠀⠶⡄⠀⠠⠀⠹⢾⣁⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡖⠁⠀⣿⣶⣴⢾⠀⠀⠀⠀⠿⣦⣻⠀⠀⠀⡠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⣀⠌⠻⠃⠘⠣⣀⣤⠃⠀⣿⠃⠀⢠⣊⡴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⡀⣀⠸⠤⠈⠽⠀⠀⠉⠀⢠⡟⠁⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠤⠤⣄⠐⣆⠛⠀⠀⠀⠀⠀⠀⢀⣆⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠈⢾⡿⠀⠀⠀⠀⠀⠀⣴⠊⠉⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡶⠚⠗⠢⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠰⣄⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠰⢮⡶⠈⠛⢄⠀⠀⠈⢃⠀⠀⢸⡇⢀⣤⠀⠀⠀⠰⠆⢀⡀⠀⢠⣿⠏⢻⠏⠱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡔⠊⠉⣹⢶⢾⡿⢶⠖⢾⡀⠀⢘⣷⣘⣁⣀⣀⠀⢀⡀⡀⠙⢶⣇⠀⣀⡜
⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢂⡀⠀⣀⠠⠞⠉⠁⠀⠳⣄⠃⠀⢸⡿⠛⠀⠀⠈⠳⠈⠁⢣⣤⠆⠀⠙⠁
⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠈⠋⠘⢃⠀⠀⠀⢠⠀⠀⢁⠀⠀⣼⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣸⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠐⠋⠠⠴⠿⠀⠀⠏⠀⠀⢀⡀⢠⠄⠛⢤⠤⢄
⠀⠀⠀⠀⢰⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠛⠀⠎⠀⢀⠀⠀⠀⠀⠈⠵
⠀⠀⠀⢀⡿⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡤⠄⠤⢄⣠⡔⠒⠳⣿⣿⠃⠀⢀⠰⠏⠒⣤⡀⣤⡤⠻⡆
⠀⠀⠀⠘⠃⠐⠀⠀⠀⡆⠀⠀⠀⣠⠖⠁⠀⠀⠀⠈⠃⣄⠀⢸⠟⠁⠀⠀⣿⠐⠒⢠⢩⡿⠁⠀⠂⠺⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⣬⣿⠄⠀⡖⠀⠀⠀⠀⢹⡄⠂⠀⠘⠿⣤⡂⠀⠀⠀⡘⠃⠀⠀⡷⠒⠂
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠁⠀⢰⣅⠀⠀⡀⠀⠆⠹⠀⠀⠀⠀⠀⠁⠐⠒⠈⠙⠒⠀⠈
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠘⣄⠀⡿⠋⠙⢁⠚⠰⠃
⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠱⣦⡴⠾⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠦⠤⠤⢤⣶⣤⣄⣀
⠀⠸⠀⠀⠀⠃⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠛⣈⡻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡜⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡾⠁⠔⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⢀⠀⠀⠀⠘⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⢸⡇⠀⠀⢀⠀⠃⠀⠀⠀⠸⡇⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡔⠁
⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢠⡿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⢠⡔⠁
⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣼⢱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⡇⠀⠀⡃⠀⠀⠀⢀⣽⠏⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⢹⡄⠀⢱⠀⠀⠂⣸⠏⠀⠀⡀⠀⣠⡶⠚⠉
⠀⠀⠀⠸⣧⠀⢸⡄⢀⣼⠃⠐⠀⢀⡴⠟⠁
⠀⠀⠀⠀⣿⡀⢸⡇⣾⡏⣔⣡⡞⠋
⠀⠀⠀⠀⢸⣇⠈⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠈⣿⢀⡿⠋
⠀⠀⠀⠀⣰⡇⣾⠁
⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⣸⡿⠃
⠀⠀⣸⠟
⠀⢰⠃
⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⣶⣿⣶⡂⠀⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠏⢻⠉⠉⠷⠈⡿⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠈⠐⡀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣾⢿⠓⠀⠀⠀⠀⢠⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⡀⠀⠀⠀⠀⣆⡀⣠⣷⣴⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣽⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠑⠒⠓⠒⠉⠏⠉⠍⠉⢉⡙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠒⠂⠂⣉⣁⣀⣤⡴⠒⠚⠀⠀⢠⢤⠖⠂⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣢⣤⠤⠶⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠈⠑⣶⣤⢤⠈⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠾⠿⠛⠛⠉⢀⡸⠀⠀⠀⠀⠀⣀⠔
⠀⠀⣀⠀⠤⠤⢄⡠⠤⠶⠞⣛⣩⣭⣤⠶⠶⡶⡶⠶⠶⢶⢦⣤⣤⣤⣤⣤⠤⢤⡶⢞⣉⣦⠀⣀⠠⠔⠈⠠
⠠⠠⠀⠒⠒⠒⠒⠚⠛⠉⠉⠉⠀⣀⣀⣀⡤⠤⠒⠒⠴⠒⠉⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠉⠀⣦⣤⣤⣶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣡⠤⠂⠀⠀⣀⡠⠤⠐⠀⠘⠇⠀⠀⠀⠀⠀⡠⢎⣤⡷⢖⣸⡟⢈⣩⡁
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠚⠛⠉⠉⠀⠠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⣑⡫⠔⠋⠀⠉⠁⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢰⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠉⠙⢶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣴⣀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠳⣤⠀⡀⠀⠈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣆⣿⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠷⣧⣴⡂⣀⠻⡇⠀⠀⠀⠀⠘⠠⡀⠀⠘⠶⢦⣴⣿⡧⠔⠁⠀⠀⢳⣸⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠻⣥⣈⣻⣿⣿⡾⠀⢠⠤⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣿⣗⠀⠀⠀⠀⢹⣿⣃⡠⠉⠙⠋
⠀⠀⠀⠀⠀⢠⢁⡤⠴⠆⠶⠶⠿⣿⠿⣿⣧⡠⣷⠀⢀⠀⠀⠁⠀⣠⣤⠶⠋⠀⠀⠛⢆⡀⠀⣤⣾⣿⢻⣍⢛⠠⢄
⠀⠀⠀⠀⠀⠈⠈⠒⠤⠀⢀⣤⠤⠾⣄⠀⠉⠧⢿⣄⡽⠒⠀⠂⠀⠑⢆⠀⠀⠀⠀⠀⠈⠁⠀⠘⠿⠿⠏⠉⠙⠷⠬⢵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣟⢠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠄⡄⡄
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢣⠈⣄⠀⠀⠀⡞⠀⡄
⠀⠀⠀⠀⠁⢀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⣼⡀
⠀⠀⣀⠀⠀⠈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠀⠀⢨⣀⣴⣿⠷⣠⠀⣠⠔
⠀⠈⠀⠀⣀⠀⠀⢧⠀⠀⠀⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⢀⣄⠤⣙⣿⣿⣿⣷⣶⡈⠀⠀⠂
⠀⢀⡠⠜⠁⠀⠀⠼⡖⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡇⣆⡏⠀⡀⠀⠀⠀⠐⠂⠀⠉⠿⡿⠟⠿⠿⣿⡂
⠀⠸⠀⠀⣄⠍⠀⠀⠉⠀⡀⠀⠀⠲⠄⠀⠀⠀⢀⡃⢀⠹⣿⣠⣨⣾⡶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠁⠀⠀⢀⠀⢁⠉⠚⠉⠀⠀⠀⠀⠀⠰⢄⡪⣤⣰⢳⣿⣿⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⠈⣑⢿⡻⡘⣿⣗⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡇⠀⢸⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⡰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡀⠘⣿⠇⢇⠘⢻⣃⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠃⡀
⠀⠀⠀⠀⣤⡀⠆⠸⠀⢐⡤⠂⢐⠀⠀⠀⠀⠀⠠⠄⠉⠙⢳⠄⢧⡘⡵⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠠⣸⣿⣰⠁⡠
⠀⠀⠂⠀⠨⣍⠳⡟⢤⣠⣦⠄⠚⠚⠀⠘⠁⠀⠀⠀⠄⠚⣉⡥⠄⢣⣘⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡟⣿⣇⠎
⠀⠀⠀⠀⢢⣝⣦⣶⠈⣿⣡⣴⣷⡶⠂⢀⠀⡀⠀⠀⠀⠈⠐⠚⠿⠶⣛⠅⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢈⣻⣷⣿⠃
⠠⠀⠀⠀⠀⢉⡽⠿⠿⣿⡿⠑⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠬⢝⣿⡏
⠀⠀⠀⠀⠀⠨⣀⠀⠠⡿⠛⣦⣤⣴⣶⣍⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠴⠾⢿⡦⠂
⠀⠀⠀⠀⠀⠀⣦⠀⠀⣏⠉⣡⠋⠯⠙⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣥⣤⣤⣤⣤⣤⣦⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢙⣄⣤⡴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⡿⠸⣧⣸⡎⠏⢻⣦⣄⡀⠀⠀⠀⠀⢠⣤⣤⣞⡙⠛⠛⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⢀⠴⣹⡏⠃⠀⢀⡞⢻⡟⠗⠒⠒⠉⠁⠙⣿⠽⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠐⢠⡔⡡⠚⠋⠙⢦⡀⣼⣠⣿⠇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠠⢤⡿⠀⠀⠀⠀⠸⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⣀⡠⠇⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⢸⠻⣿⣯⠀⠀⠀⠒⠚⠉⠁⠀⠀⣶⢆⣠⡿⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢖⣿⡷⣠⡴⠶⢶⣶⣿⡿⢛⣿⠃
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢻⣿⠿⣿⣾⣿⣿⠟⢻⣷⣿⣣⠞
⠀⢳⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⡃⠀⠀⠀⢀⡬⠷⠿⠃⡰⠃⠀⠀⠸⣦⠀⡔⣤⢦⡀
⠀⠈⢧⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢇⠀⠀⠀⣰⠁⠀⠀⠀⠀⢈⡷⡮⢷⡍⢈
⠀⠀⠈⢦⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣘⣀⣀⣀⣈⣀⣀⣸⣇⣀⣀⣀⠀⠀⠈⢍⣡⠒⢛⠈
⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⣿⣿⠂⠀⠀⠀⠀⣿⠈⢇
⠀⠀⠀⠀⠀⠹⡀⢠⣤⣔⠒⠀⠑⠙⢶⠆⠀⣀⡀⠀⣿⠁⠀⠀⠀⣀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⡜⠁⢀⡀⢢
⠀⠀⠀⠀⠀⠀⠐⣴⠋⠘⡆⠀⠀⠀⠀⠱⣴⠋⡇⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⢠⣚⣀⣀⠐⣷⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠢⠀⢸⠀⠀⠀⠀⢰⡟⢠⡇⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⡀⠀⠘⣷⠟⠛⠛⣏⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡾⠈⢸⠁⠀⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠸⠛⣲⣄⣀⡸⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡜⠀⠀⠀⠀⡇⢠⠏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠸⣷⠁⠻⢟⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣈⣳⣞⣁⣀⣀⣀⣀⣷⣏⣀⣀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣿⣿⡇⠀⠸⠋⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠒⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⣷⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣶⣶⣦⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠈⠛⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⡴⠖⢲⡄⠀⣞⠙⣷⠖⠓⣦
⠀⠀⠀⣠⢧⡄⢼⢃⣿⣉⡤⠤⣤⠶⠷⢤⡀
⠀⠀⢸⡁⢠⢞⡏⢸⣁⣀⠳⢴⡏⣤⣀⣴⠇
⠀⠀⣀⢹⡏⢹⣧⢚⠋⢿⣦⠾⣇⣹
⣴⠛⣩⣼⡦⢾⡙⣿⡇⢸⣃⣀⠈⠁
⢙⡿⠧⠞⢰⣞⠀⢈⣻⣼⠏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠷⠼⠛⠋⠈⢠⠏⠉⠉⠙⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⠀⠀⣠
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠘⢦⣄⡤⠞⠀⣈⡓⠦⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣧⡤⣤⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡆⠀⠈⠉⠙⠲⡗⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢋⠁⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢣⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⠀⠀⢠⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠈⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡀⠠⣄⠀⠀⠀⠀⠀⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⠳⣤⣀⣀⣀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣈⣉⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⠄⠀⡆⠀⠈⠳⣤⣉⣩⠴⠋⠀⢠⡀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠈⠙⠳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡾⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⠳⢤⡤⠞⢡⡀⣠⠶⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⢀⣠⠴⠶⠒⠛⠛⢯⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠙⡉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠰⠿⠋⠀⣀⠀⢻⣦⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣠⣤⠄⣴⠿⢲⣾⠃⠘⢿⡆⠛⣀⡉⠉⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⠘⠋⣁⠀⠉⢠⣮⠉⠘⢿⣦⠄⣰⡿⠃⣀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣾⠆⣈⠛⢷⡄⢀⠛⠷⠀⡀⣠⡀⢱⡾⣴⡿⠈⣡⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⠋⠙⠿⠷⠀⠀⠘⣿⡦⠀⠀⠙⠻⠃⢀⣍⠀⠈⣩⠁⣄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣉⣠⢴⣶⡐⠤⠄⣳⡀⢀⣤⠛⠀⠀⢠⡾⠛⣀⠸⠋⠸⢟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠙⠁⢠⠙⠃⠀⠀⠀⢻⠀⠀⣠⠆⠀⢤⡀⠚⠡⠀⠰⠿⠋⣈⠻⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣄⣀⠀⠈⢁⣤⣤⢠⣶⢂⡈⠶⡄⡀⢸⡏⠁⣸⠇⡄⠀⠀⢀⣠⡄⠀⣼⣟⣁⣀⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠁⡐⠿⠿⠟⠛⠁⢠⣜⠛⠁⠸⠏⢀⠁⠀⢠⣝⣾⣷⡴⠁⣤⠀⣠⠀⠛⠋⣀⠀⠀⠙⠛⡁⠘⣿⠄
⠀⠀⠀⠀⠀⠀⣀⠰⣾⣷⠘⣿⡄⠰⡿⠛⠡⡈⣛⢛⡀⠀⠀⡎⣀⠀⠈⢿⢿⣿⠁⢀⣈⡴⠅⠀⠀⢉⣴⡿⠀⣾⡿⢁⣀⣉
⠀⠀⠀⠀⠀⠈⣻⡾⠇⠰⣤⣈⠁⢀⡀⢾⣶⣄⠉⠉⠑⢠⡆⠀⡙⠓⣴⠀⢸⡇⠀⣸⣏⡀⠠⠶⠀⡈⢠⡄⠀⢠⣄⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠋⣰⡿⠃⠈⣻⡃⠘⢿⣆⠀⠀⣤⠀⠀⠀⢤⡀⠻⠷⠨⣧⣼⣿⡾⠋⣤⠈⠀⠀⣤⡁⠘⠿⡄⠘⣿⠆⢀⡀
⠀⠀⠀⠀⠀⠘⢇⣉⣀⡄⢠⡿⠃⡀⠀⠠⠤⠤⠴⠦⠠⡴⠚⣧⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⠐⠀⢠⣄⡀⠀⠠⣦⣴⢆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠛⠋⢤⣾⣦⠀⠻⠷⠆⢀⣠⣶⠃⠐⢀⠀⠙⢧⡀⡀⠀⣿⣿⣄⠴⠃⠀⠀⣿⡇⠁⠹⣦⣀⠈⢰⣿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣠⣿⠂⢶⡶⢧⣬⠉⣁⡀⠰⠟⢀⡜⣭⠙⠷⣾⣿⡇⢀⣛⡛⠳⠈⣩⡴⠰⣶⣝⠛⣀⣀⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣷⣤⣀⠠⣾⠋⢠⣬⣷⢀⣴⡆⣸⠟⢁⡀⢸⣿⣇⠈⠛⠿⠃⠘⠟⠁⣶⡿⠯⠄⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠿⠛⠁⠉⠁⠙⠋⠀⠀⠘⠋⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⣿⡆⡟⣀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠐⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⢀⡀⠑⣛⣀⣀⣤⡶⢋⢿⣾⡛⠛⢾⣟⢤⡀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣤⠋⠉⠙⠋⢉⠉⠉⠉⢀⠔⠋⠀⡙⢿⡍⠙⢻⣼⠇⢀⡠⠂⠙⢦⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢣⠄⠠⢈⠀⠀⠀⠂⠀⠀⢸⣁⡀⣿⠀⣠⠛⢷⣀⣠⡼⠂⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢾⠁⣀⡀⠈⠉⠀⠀⣃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣈⣁⣆⢀⡀⠀⠴⠓⡄⠘⢧⡉⠀⠀⢀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡼⠋⠉⠉⢳⡶⠂⠀⠀⠀⠉⠓⢄⠘⣬⡻⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠁⢰⠖⠢⢸⠇⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢘⠰⡄⢰⠀⠈⠃⠔⠋⠀⠀⠀⠀⠀⠀⢠⡴⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠉⢁⡼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⣶⢰⡾⢇⠋⠹⣶⠟⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⢶⣮⣧⢠⡕⡌⣆⣠⠷⠖⠯⠤⣻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣶⡌⢃⢺⡆⠸⣦⠘⣄⡃⣷⡿⣡⢰⡆⢰⠆⣰⠖⢠⡆⢀⡄
⠀⠀⠀⠀⠀⠀⣤⡀⠘⠻⠬⣅⠀⣉⣙⠲⣄⡧⠘⣷⣿⡇⢙⡨⢀⡖⣊⡀⢀⡬⢕⠛⠁⢀⣀
⠀⠀⠀⠀⠀⣤⣌⡛⡤⢜⣿⠞⠓⢬⡉⠉⠙⢷⣀⣾⣿⢷⣼⡵⠉⠀⣉⡥⠊⢐⣟⣠⠤⣻⣥⣄
⠀⠀⠀⣀⠀⠈⢉⣐⡿⣆⠀⠙⠲⡀⠛⢤⡀⠀⠙⣿⣿⣿⠉⠀⢀⡴⠋⣠⠖⠋⢰⣟⣤⣔⡉⢶⣶⠄
⠀⠀⠙⠛⡹⢻⡏⠰⠶⠞⠛⠶⡄⠙⠦⢤⣿⣶⣴⣿⣿⣿⣤⣾⡯⠤⠴⠁⢠⠖⠛⢶⡦⠀⢿⣏⢢⡀
⠀⠀⢰⡿⠀⠈⠲⣶⡄⠀⠀⠀⢹⣄⠀⠀⠀⠈⠹⣿⣿⣿⠋⠁⠀⠀⠀⢠⡏⠀⠀⠀⡰⠶⠖⠉⠈⢿⠄
⠀⠀⠈⢠⣶⢖⡴⠖⢛⠿⠿⠿⠿⢟⢿⣷⣤⣤⡰⣿⣿⣿⣀⣤⣤⣶⠾⡿⠿⠿⠿⢟⠛⠚⢧⡻⢷⡄
⠀⠀⠀⠀⠔⣿⠀⠠⣿⠄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣮⡻⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⣿⢂
⠀⠀⠀⢸⣄⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⣠⡆
⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣽⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣾⣵⣾⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣄⣤⣾⡿⣻⣽⠿⣛⡿⣿⡷⣝⡦⣤⣴⣶⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣻⣼⣿⣿⣿⢵⣯⠖⡝⡯⣿⣎⣿⣷⣏⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣝⣻⢿⣿⣿⣷⣟⣥⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣭⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠐⣶⣤⡄⠀⠀⣀⢰⡶⣾⣶⠀⠀⠀⢠⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⡀⢿⣼⢟⣿⣛⡆⠀⣤⢹⣼⣿⣿⡩⠄⣠⣴⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⡿⡃⠀⠀⠐⢿⣷⡄
⠀⠀⠀⠀⠀⠲⣿⠂⠀⠀⠀⠀⠻⣾⣫⣾⠿⢦⠀⢰⣸⣿⣿⣽⣿⡗⠀⣿⣿⣿⣿⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣯⣴⡶⠂⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⢯⠷⠀⠀⠀⠉⠙⠿⠟⠃⠀⠀⠻⢿⣿⣧⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠙⣿⡓⠀⠀⠀⠠⣾⣿⡇
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡡⠄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠹⢿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⢠⣿⠂⠀⠀⠀⠀⢀⡴⡛⠈⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⡀⣄⣀⠀⠀⢠⣄⠀⠀⠀⠀⣀
⠀⠀⠸⢿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣦⣿⡟⢁⣴⡿⠀⠀⢀⠎⡀⠁⠀⡀⠄⠀⢸⣿⡆⠀⠉⠀⠠⣉⣻⡥⢆⡀⠘⣿⣷⠀⣠⣾⣿⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢬⣷⡿⣋⣤⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠓⠒⠀⠀⢐⠲⠶⣽⠋⠩⠀⠀⠈⠛⠃⣿⣿⠟
⠀⠀⠻⣿⣿⣿⡗⠀⣤⣤⠀⣼⣷⠀⣠⣤⠸⠿⡶⠦⣿⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠀⠀⠠⠭⢭⣯⠴⠞⣋⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠁
⠀⠀⠈⢿⣿⠆⠀⠀⡻⣿⣷⣿⣿⣾⠿⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠘⠀⠀⡀⠀⠀⢠⠀⠐⣻⡭⠀⠠⣤⣀⡒⣶⣚⣯⠗
⠀⠀⢠⣀⠀⠀⠀⠘⠿⠿⣿⣿⣿⡷⠿⠷⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠐⢢⡤⠙⠀⠄⠀⠀⠀⢻⡯⠀⠲⠦⣤⣄⣋⣡⠴⠚
⠀⠀⠸⣿⣿⣦⠀⠀⠀⠰⠿⠛⠻⠿⠀⣴⣄⢠⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠂⠐⠠⠀⠀⠀⠁⠀⠑⠶⢦⣄⣉⣴⠾⠋⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⢰⠀⠀⠀⣬⣽⠿⡇⠀⠀⠀⠀⠀⠀⣾⡆⠾⠡⡤⠆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠁⠀⠀⠀⠛⠏
⠀⠀⠀⠀⠈⠉⠁⠀⢠⡀⣴⠀⢁⣀⣠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣦⡘⠳⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⡀⠀⢰⣴⡀⠀⣄⠀⠀⣶⡧
⠀⠀⠀⠀⠀⣀⠸⣆⢸⠇⣯⣴⣿⡉⠁⠀⠀⢀⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⠛⠀⢼⣯⠄⠈⠉⠀⠀⠿⢀⡀⠈⠁⠀⢠⡐⣴⢃
⠀⠀⠀⠀⢦⡙⢦⠻⣼⣿⠯⠄⠀⢀⢠⣶⠶⠘⣿⣖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡙⠶⣋⣤⠄⠀⠀⠲⣝⣗⣫⣔
⠀⠀⠀⠳⣮⡻⣦⣍⣛⡛⠁⠀⠠⢜⠛⠁⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⡇⠀⢠⣛⠯⢈⠥⠶⡂⠀⣈⡻⢮⡭⢟⡋
⠀⠀⠐⠲⣞⣿⡓⢦⡄⠀⠀⠀⠀⠀⣶⡿⣿⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⡿⠇⠀⠀⢭⣍⡀⢚⡋⠁⠀⠠⣭⣹⣛⡭
⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠱⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣭⣥⣦⣭⣷⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣰⡦⢴⢴⢄⠴⣴⡦⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⡠⡤⢀⠀⣤⢄⡀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⣽⣿⣗⠀⢸⠀⢄⡮⣛⡇⠀⠀⠀⠀⠀⢸⣾⢿⣿⢧⣳⣽⢠⣟⡞⣷⡏⡿⡇⠀⠀⠀⠀⣼⡻⠤⠽⠷⢾⣿⠴⠿⠥⠴⣟⡀
⠀⠀⠀⠀⠀⢹⣖⠍⠀⢸⠁⢐⣭⣶⡂⠀⠀⠀⠀⠀⢸⡜⡜⣟⢯⣻⣿⣿⣯⡞⡽⣱⣱⠆⠀⠀⠀⠀⣿⠅⠀⠉⠍⢹⣿⠉⠩⠥⠬⢽⡇
⠀⠀⠀⠀⠀⢿⣛⡫⠁⢸⡄⠉⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡝⣌⢳⡙⣿⣿⣫⢞⡵⢣⢿⠃⠀⠀⠀⠀⣿⡛⣛⣛⣒⣻⣿⣛⣛⣛⡓⣻⡇
⠀⠀⠀⠀⠀⣿⣿⣓⠂⢸⡇⣐⣿⣿⡃⠀⠀⠀⠀⠀⢨⢳⡹⣌⢳⣽⣿⣿⣣⢞⣵⢋⢎⠂⠀⠀⠀⠀⣿⠄⠀⠠⠤⢾⣿⠤⠤⠤⠄⢽⡇
⠀⠀⠀⠀⠀⣲⣿⠿⠗⢸⡇⠾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣧⣳⣌⢷⣽⣿⣿⣫⠟⣡⢟⣾⠇⠀⠀⠀⠀⣿⡍⠉⠩⠭⢽⣿⢭⣭⠍⢩⣽⡇
⠀⠀⠀⠀⠀⣸⡷⢏⠁⢸⡇⠨⣿⣿⡇⠀⠀⠀⠀⠀⢸⣧⠳⣽⣦⡙⢿⡿⣡⣾⣿⢋⢾⠇⠀⠀⠀⠀⣿⣄⣒⣒⣒⣚⣿⣒⣒⣒⣒⣺⡇
⠀⠀⠀⠀⠀⣻⠽⡣⠂⢸⠀⢔⣽⣿⠃⠀⠀⠀⠀⠀⠀⢧⠳⣬⡳⣽⣿⣿⡵⣛⣵⢏⡞⡀⠀⠀⠀⠀⣿⣒⠲⠶⠶⢺⣿⠒⠒⠒⢒⣺⡇
⠀⠀⠀⠀⠀⣼⣭⡶⠀⢸⠂⢨⣒⣿⡇⠀⠀⠀⠀⠀⢸⡘⠷⢮⣽⣿⣿⣿⣞⣯⠵⠟⣡⡇⠀⠀⠀⠀⣿⠧⠤⠭⠭⢭⣿⠭⠭⠭⠴⢿⡃
⠀⠀⠀⠀⠀⣾⣿⣡⣀⡸⣠⣤⣝⣿⡅⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠻⢿⡿⠖⠒⠚⠻⠟⠁⠀⠀⠀⠀⠛⠁⠀⠀⠈⠸⠿⠘⠛⠁⠈⠙⠃
⠀⠀⠀⠀⠀⠈⠁⠈⠈⠀⠈⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣶⡀⠀⠀⠀⠀⠀⣠⣄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⣿⡇⠀⠀⠀⠀⠀⢻⡄⢠⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⠘⣿⡄⠸⣷⡴⠁⣠⣦⠀⠟⢣⡿⠁⣼⠇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⣷⡅⠸⣿⡤⠙⠛⠁⢀⣤⣾⣿⣿⣷⣦⡀⠀⠸⡇⢰⡏⢀⠟⣶⠄
⠀⠀⠀⠀⠰⣶⣦⣀⡀⠈⢻⣆⠸⠇⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠉⠁⢸⣧⡖⢀⢏⣠⡀
⠀⠀⠀⠀⠀⠈⡻⢿⣷⣦⡆⠹⣧⢀⣠⣶⣿⣿⠠⠌⣿⠠⠌⣿⢸⡇⠻⢸⢏⣭⣽⣿⣷⣦⣄⠀⠉⠡⡟⠁⠀⣾
⠀⠀⠀⠀⠀⢠⢳⡀⠈⠁⠀⢠⣴⣿⡏⢠⣿⣟⣘⣓⣽⣰⣜⣿⣸⣇⣦⣸⣌⣛⣣⣿⣯⠉⣿⣷⣦⡀⠈⢠⡾⠋⣠⣶
⠀⠀⠀⠀⠀⠘⠾⠃⠀⢀⡴⠿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠩⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⡇⢰⡆⢰⡆⢹⠋⡍⣻⢉⡏⢹⣿⡇⣻⠇⣿⢻⣿⠟⡿⢻⡻⢟⣻⡽⢿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣇⢸⣧⣘⣄⣧⣽⡤⡇⣸⣿⡀⣯⣦⣭⣌⣷⣄⣧⣸⣷⣘⣻⣏⣸⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⢿⣿⣿⠟⠛⠛⠓⠲⢿⣿⣷⡾⠿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣴⠄⣤⣤⣤⣶⣳⣶⡀⢆⣠⣤⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣆⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣾⣿⣿⣿⣷⣶⣦⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣧⣿⣏⣇⡹⣿⣛⡿⢿⣿⣿⣿⣿⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣿⣿⠿⣿⣾⣿⣟⢻⣿⢿⣿⣿⣿⣧⣿⣇⣿⣿⣿⣶⣟⣻⣿⣿⣿⣿⣿⣤⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⡿⢾⣽⡳⣾⣿⡟⣿⠃⣹⣿⠏⠉⢿⣿⣿⣯⢿⡿⣿⣻⣿⢛⣿⣿⢛⣿⡿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣿⣿⣿⠿⢿⣍⢷⣯⣿⡿⡛⢧⣜⣵⢦⣽⣷⣿⣾⢠⣿⡏⢿⡌⣿⢿⣯⠈⡿⢿⣿⣿⣿⡿⠟⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣾⣿⠷⢾⣦⣞⠁⢸⡟⠻⣿⣏⡟⢣⡽⠻⠻⣿⢚⣴⠟⢠⣃⣶⣟⣩⣿⣯⣿⣿⣣⣿⢿⣽⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢾⣿⣿⣿⣄⣿⣾⣿⡇⢠⣄⣚⠋⣳⣼⡅⣀⢻⣿⠇⣰⠇⠀⠀⣿⣾⡏⣠⡟⠁⠘⣻⠏⣽⡎⢡⣻⣿⣛⣦⠟⣿⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⡴⣦⣄⣻⣿⠃⡘⠚⠀⠹⣿⣛⡛⣿⣠⡟⠀⠀⣰⣿⠛⣿⠋⢠⠂⣠⣿⣟⣋⣀⡴⠯⣴⢭⣭⣿⡏⣻⣿⣿⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⣯⣤⣀⢠⡍⢉⣭⣿⣶⣞⡠⣤⡀⣿⠟⠿⠘⣿⡇⢠⣾⠟⠁⢸⡏⣠⣼⡾⠋⠉⠉⠀⠀⠀⠀⢀⣤⣿⣿⡙⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣸⢟⣩⣽⣥⣹⢿⣺⣿⡿⠿⣦⣿⠃⢻⡄⠀⠀⣿⣧⣿⡏⠀⣲⣿⣿⢻⡏⠀⡆⢀⣀⣀⣀⣠⠤⠾⣥⣦⡝⢧⣯⣿⣿⡟⠁
⠀⠀⠀⠀⠿⣾⣯⣿⣽⣷⣺⣿⡏⠀⠀⠈⢉⣵⠃⣰⡿⠙⢿⣾⣿⣀⣠⣿⣿⣿⣿⣿⡿⠟⢱⣾⣡⡾⠞⠋⠉⠁⠀⠀⠀⠀⠸⢿⢷⡿⠷⣿⣾⣿⡀
⠀⠀⠀⠀⣼⣿⣿⢿⣦⡾⠃⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡝⠿⣿⡷
⠀⠀⠀⠀⢸⣿⠄⢺⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠏⠒⢻⡇
⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣄
⠀⠀⠀⠀⢀⣃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣾⢆
⠀⠀⠀⠀⠈⡙⣦⡀⢆⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣿⠏
⠀⠀⠀⠀⢀⣱⣜⣷⣱⠀⠀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⢀⣙⣛⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣁⡤
⠀⠀⠀⠀⠀⠐⢎⠺⠷⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⣯⣁⣤⡆
⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⢻⣿⡿⣿⣿⢿⣿⠿⢿⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⡿⠃⠈⢉⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠈⠙⣿⡿⠻⠷⠶⣶⣾⣿⣿⣥⡾⠟⠋⢀⣽⣿⡄⣹⠿⣆⣙⣿⣿⣿⣿⣿⣷⣦⣤⣶⣶⣿⣿⠿⢩⣤⣄⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣷⡶⠾⣿⣶⣦⣶⡶⠾⣿⠀⠉⣿⣦⣤⣴⣿⣥⣿⣇⣽⡶⣿⡉⠉⠛⠿⣯⣙⠛⠛⢛⡻⠿⠷⢶⣶⣶⣒⣚⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡴⠋⠀⠀⠀⢠⡴⢋⣤⠞⠋⣀⣼⠇⠀⠀⣿⢻⡟⢳⣾⣷⠶⣄⣠⣤⡝⠛⠿⣯⡀⢀⣠⣾⣁⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⡴⠞⣻⠐⢫⡿⠦⣴⠏⠙⣶⠛⢙⣿⡟⣠⢾⡏⢹⣤⠴⣿⣍⠳⠶⣶⢼⣿⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠙⣿⣁⣱⣴⡏⠀⢀⣾⠲⢛⡏⢳⡇⠘⣷⣿⠤⣤⡾⠙⠻⠿⣶⣾⠯⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣏⣠⢾⣏⣿⠟⠉⢷⡞⠴⡏⢠⡶⣯⠁⣾⠿⡿⢛⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⠚⠛⠿⢦⡀⣤⣽⠶⠟⢻⡆⠈⢿⣿⠾⡁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣟⠋⡛⠻⠋⡙⠛⣻⡛⢙⡻⢛⡛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣦⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠓⢐⣿⣿⡇⠀⢀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡟⠐⢻⠀⠀⠀⠠⠀⠀⡀⠶⠶⣲⣦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠨⣿⠿⠁⢀⡞⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣶⡿⠊⣀⣠⣴⣾⣿⣷⠶⢁⡀⠘⠋⠉⠑⠒⠠⠤⢀⡀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣾⡷⣿⣿⡴⢖⣩⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠃⠀⠀⠈⠑⠂⢄⣀⢀⠀⠀⠀⠀⣉⣀⣴⣶⣾⣿⣿⣿⣻⣿⣿⡧⡏⢣⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢿⣊⣁⣿⡿⠿⠿⠛⠋⠁⢸⣯⣿⣻⣿⣓⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⣽⠟⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⠀⠀⠀⠀⠀⣼⣿⣿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠠⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠻⠿⠿⠿⣿
⠀⠀⠀⣀⠀⠺⠟⠋⠀⠀⠐⠁⠀⠀⠀⠀⠚⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠉⢻⡄
⠀⠀⠘⠛⠁⠀⠀⠀⢀⡴⠈⢧⣀⣢⣄⠀⠀⠀⠈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⡀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣠⣴⠿⠋⠉⠛⠧⣄⣀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠑⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠋⠀⠀⠀⠀⠀⠀⠉⠁⠂⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⡜
⠀⠀⠀⢠⠋⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⡟
⣶⠖⢀⡼⠋⢀⣀⣠⣤⣤⣴⣶⡶⠖⠐⠒⠶⠦⢴⣀⣀⡈⠠⢅⡄⠀⠈⠉⠉⠉⡉⣀⣂⣤⠀⠀⠀⣤⠀⠘⢀⣀⣤⣴⣶⠶⠒⠛⠈⠉⠠⣬⡇⠈⠙⢘⣓
⠻⠶⠿⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠶⠾⠟⠛⠛⠋⠉⠁⠀⠀⠠⠴⠿⠓⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠸⠒⠻⠷⣴⠶⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠢⡀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⡠⠊⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢦⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⢠⠎⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠊⠀⣀⠤⠒⠒⠒⠒⠒⠢⢤⡀⠈⠢⠔⠁
⠀⠀⠀⠀⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠈⠉⠒⠬⣉⠉⠙⠒⠢⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠒⠚⠉⢉⡩⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠔⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠓⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠓⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡔⠉⠒⠤⣀
⠀⠀⠀⠀⣀⡤⠖⣊⣁⠤⠤⠶⠚⠁⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⠲⠦⠤⢄⣉⡒⠦⣄⡀
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⠀⠈⠑⠒⠶⠄⠴⠖⠒⠉⠀⢠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣠⠊⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠈⢢⡀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠞⠁⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠙⢦⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠜⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠸⠀⠀⠀⠀⠃⠐⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⡀⠀⠀⢀⡀⠃⡀⠀⠂⠀⠀⠘⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠢⠀⠀⠉⠀⠀⡀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⢠⣤⠂⣬⠻⠛⣭⡿⣷⣶⡆⠀⠀⠀⠶⠉⣀⣠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⡈⠂⢀⢁⣼⣟⣢⣌⠀⢸⣿⡷⣿⡿⣃⡤⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⢀⡉⠀⠘⢿⣿⣿⣿⣿⢀⣸⡿⠟⢋⣼⣿⣷⣄⡇⠀⠀⢉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠄⠀⠀⣿⠘⢿⣿⣿⣿⡟⢉⡔⠚⠿⢋⣼⣿⡃⠑⠀⠀⠜⠁⠄⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠒⠀⢸⣿⡄⠈⢻⣿⣿⣾⣿⣷⣶⣴⣿⣿⠛⠷⠀⠀⠠⠌⡁⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠁⠀⠙⠿⢧⣄⣠⣿⠛⢛⣉⣴⣿⣿⡿⠛⠀⠀⢀⠀⠈⠐⢂⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⡀⠀⠉⠉⠁⠈⣿⡏⠙⠁⠉⠀⠀⢀⣴⡏⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⡀⠀⠀⠈⠀⠐⠀⣿⡇⠤⠀⠈⣠⣶⣿⣿⡇⢀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⢿⣦⡀⠀⠀⠀⢀⣿⡇⠀⢠⣾⣿⣿⣿⣿⡇⠀⠐⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⣦⡽⣿⣶⣄⠀⠀⣿⡇⠀⣿⠙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡁⣿⣿⣷⡜⢿⣿⡄⠀⣿⡇⠀⢁⣼⣿⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣻⣿⣿⣷⡌⣿⡇⠀⢿⡇⠀⡼⠋⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⣿⣿⣿⣿⣿⠀⠈⣇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⢄⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢀⣀⣀⡀⠠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⣿⣿⣭⣉⣉⣥⢰⣶⣤⡉⠐⠒⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣙⣿⣉⠉⠛⢿⣿⡿⢿⣿⣿⣿⣿⣴⣆⣠⡙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣉⠉⠀⠀⠀⠀⢹⡍⠙⠿⣿⣿⣿⠿⡍⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⠻⣿⣿⣦⣴⣶⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡟⣿⣿⣿⣿⣿⣿⢤⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣦⣿⣇⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⢻⣿⣇⠛⢡⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣶⣛⣤⠢⡐⠄⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠻⣿⠋⢻⣤⠉⢢⡶⠃⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠚⣿⣶⡧⠶⣄⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢺⡘⢷⠀⠀⠀⠀⠀⠀⠈⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣎⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⢿⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣠⣤⣤⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿