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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠊⠁⠀⣀⠠⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁⠀⢀⣠⠤⠒⠉⠀⠀⠴⠂⠀⡀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⣀⠤⠒⠉⠁⣀⠤⠒⠉⢀⣀⠤⠒⢉⡀⠤⠂⠀⢀⠀⠈⠓⠢⣄
⠀⠀⠀⠀⢘⡯⢄⡀⠀⠀⠀⠒⠉⣀⠠⠔⠊⠉⣀⠤⠒⠊⠁⠠⠔⠊⠁⢀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠸⣟⠒⢍⡒⠤⡀⠀⠀⠀⠀⠠⠒⠉⣀⡀⠐⠊⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⣀
⠀⠀⠀⠀⢘⡷⢌⡒⠬⣓⠬⣑⠤⡀⠀⠀⠀⠈⠁⢀⠤⠒⠉⠀⠀⠀⢀⡀⣀⠀⣀⣀⠜⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⢉⣲⠄
⠀⠀⠀⠀⠰⣏⡑⠬⣑⠮⣑⠪⣑⠬⡑⠦⣀⠀⠀⠀⠀⠠⢄⠀⠀⢀⠏⠀⠀⠀⠀⠀⣀⠤⠒⣉⠤⠖⠉⠀⠀⢀⡠⠔⣊⠥⢾⡁
⠀⠀⠀⠀⢨⡷⠬⣑⠪⣕⠢⣕⠪⣑⠢⣑⠢⢝⡢⢄⡀⠀⠀⠉⠒⠾⢄⣀⣀⠤⠖⣋⡥⠒⠋⠀⠀⢀⡠⠔⣊⠥⢒⡩⠔⣊⣽⠂
⠀⠀⠀⠀⢐⡯⣑⠢⣙⠢⢍⠢⢝⠢⣑⠢⣉⠲⢍⡒⢌⡑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⠥⠒⡩⠔⣊⠥⢒⡩⠔⣺⠅
⠀⠀⠀⠀⠨⣗⠢⣕⠢⢍⡢⢍⡢⢍⡢⢍⡲⢍⡒⢬⣑⠮⣑⠢⢍⡒⢤⣀⠀⢀⡠⠔⣊⡥⢖⣪⠵⣒⡭⢖⣩⠴⣊⠥⢖⣨⣽⡂
⠀⠀⠀⠀⢘⡯⢍⠢⢍⡒⢌⡒⢍⡢⢍⡲⢍⡒⠬⡑⠤⣑⠢⢍⡒⠌⣑⠬⣉⠤⢒⡩⠔⢂⡩⠔⣈⠤⢚⠡⠐⣊⠔⢊⡥⠐⣺⠅
⠀⠀⠀⠀⠠⣟⡲⢍⡒⠬⡒⢬⡓⢬⡒⢬⡒⠬⣑⠦⣕⠲⢍⡒⠬⡓⠮⣑⠒⣊⠥⢖⡫⠵⢒⡭⠔⣪⠵⢒⡩⠔⣫⠴⢚⡩⢽⡃
⠀⠀⠀⠀⢘⡧⢌⡒⠬⡓⠬⣑⠬⡓⠬⡓⠬⣑⠢⣑⠢⢍⡒⠬⡑⠬⣑⠢⠭⠔⢊⡡⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠤⢒⣻⠄
⠀⠀⠀⠀⠰⣏⡒⠬⣑⠮⣑⠪⣑⠮⣓⠮⣑⠢⣕⠢⢍⡒⠬⣓⠮⣑⠢⢍⣒⡫⠴⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠖⣊⠥⢾⡁
⠀⠀⠀⠀⢨⡗⠨⣑⠢⢑⠢⢕⠪⣑⠢⣑⠢⢉⠢⢍⡒⠬⡑⠢⣑⠢⢍⡒⠤⠒⣈⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⣽⠂
⠀⠀⠀⠀⢐⣯⣑⠢⣝⠢⢝⡢⢝⠢⢝⠢⢝⠲⢍⡒⠬⣑⠮⣑⠢⢍⡒⠬⣛⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣚⠥⢒⡩⠔⣺⠅
⠀⠀⠀⠀⠨⣗⠢⣍⠢⢍⡲⢍⡢⢍⠢⢍⡢⢌⡒⠬⡑⠠⣑⠢⢍⡒⢬⡑⠤⢒⡩⠔⣊⠡⠒⣉⠔⢊⡡⠐⣉⠔⣊⠥⠒⡉⢽⡂
⠀⠀⠀⠀⢘⡯⢍⡢⢍⡒⢭⡒⢍⡒⢍⡒⢍⡒⠬⣑⠬⢑⠢⢍⡒⠬⣓⠮⣉⠥⢒⡩⠕⣚⠭⠒⡫⠥⢒⡩⠐⡫⠔⣊⠭⢒⣻⠅
⠀⠀⠀⠀⠨⣟⡒⢌⡒⢬⡑⠬⡓⢌⡒⢬⡐⠬⣑⠢⣑⠢⢍⡒⢬⡑⠬⣑⠒⣊⡥⠒⣉⠤⢂⡩⠔⣊⠤⢂⡩⠔⣉⠄⢒⡩⢼⡃
⠀⠀⠀⠀⠀⠀⠉⠒⠬⣑⠬⣑⠬⡓⠬⡑⠬⣑⠢⣕⠢⢍⡒⠬⣑⠪⣕⠢⠭⠔⣊⠭⠐⣊⠥⢒⡩⠔⣊⠥⢒⡩⢐⡫⠥⢒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣑⠬⣑⠮⣑⠢⢙⠢⢍⡒⠬⣑⠤⣑⠢⢍⣒⣉⠤⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠮⣕⠢⢝⠢⢍⡒⠬⡑⠪⢑⠢⢍⡒⠤⠒⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢌⡒⠬⣑⠦⣑⠢⢍⡒⠬⣁⡡⠔⣊⠥⢒⡩⠔⣊⠥⢒⣩⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣝⠢⢍⡒⠬⡑⠒⢚⡩⠔⣊⠥⢒⡩⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢌⣑⠬⣉⠥⢒⣩⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⡴⣶⣶⡾⣿⣿⣹⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣞⡿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣟⠿⣷⣿⣿⣷⣿⣿⣻⢿⣿⣿⣿⣿⣿⡟⠳⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣭⡻⢷⣯⣟⢾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣙⠿⣿⣿⣿⣯⣗⢮⣝⠿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⣛⢭⣭⣴⣶⣦⣿⣻⣯⣽⡟
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣮⣝⣿⣿⣿⣿⣾⣝⡶⣍⣯⣭⣭⠶⢶⣚⣿⣿⡿⣿⣻⣯⣽⣿⣿⣿⣿⠿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣭⣻⢿⣷⣽⣶⣾⣿⣿⣭⢷⣟⣿⣻⣿⡿⢿⣛⡯⢽⣒⣯⣽⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⠿⣟⣻⣭⣵⠶⠛⣩⡶⣾⣻⠯⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣾⣽⡟⣻⣽⣷⣾⣿⢟⣋⣥⣤⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣽⡿⣿⣿⣷⣝⠟⣭⣺⠽⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⡿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣿⣿⣿⣿⣾⣿⣿⣶⣦⣄⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣯⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡈⣤⣿⣿⣿⣿⣿⣿⣧⠤⢀⣴⣦⣴⣶⣷⣾⣶⡀⠀⠀⠀⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⣿⣿⣿⣶⢿⣿⣿⡾⣿⣿⣿⠟⣿⢤⣄⢠⣛⡿⣿⢿⠹⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣸⣿⣿⣿⣿⡿⣿⣿⣿⣷⡘⡻⢿⣄⣀⣰⡿⣿⢣⢸⣿⡇⣸⣿⢿⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⣿⣿⣿⣿⡗⠀⣿⣞⣿⣻⣿⠿⢀⠉⠩⠁⠂⣿⣧⡸⣿⣷⡸⢿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⣽⢧⣟⣿⣿⣿⢯⣯⠋⠤⢀⡄⣠⣶⣿⡿⣷⡿⢿⣷⣚⣯⣧⡟⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣤⠶⣽⣿⡣⣞⡬⢡⡞⣻⣆⣶⣭⡭⠾⣻⡿⣟⣟⣙⢯⡟⣵⡟⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣻⣟⣩⣥⣼⣥⣶⢦⣄⠷⢿⣟⣹⣻⢌⣿⢟⣻⣿⡿⢿⣯⣰⣦⡾⡷⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠻⠟⣟⣻⡜⣩⢻⣽⣞⣿⣻⣟⡿⣿⣽⣿⡻⢾⣧⣾⣿⣿⣻⢟⡁⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣠⣿⣿⣿⢿⣾⡇⢿⣻⣾⣋⢾⣿⣷⣼⣻⣿⣿⡝⣿⣿⣿⣿⠎⣃⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣟⣿⣯⣽⣾⣿⣿⣚⣭⣏⢻⡿⣼⣟⣻⢿⣿⣶⣧⣽⣟⣿⢭⡚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣻⣿⣿⣷⠝⣿⣿⣼⣾⣿⣾⣷⣿⣿⣮⣿⣷⣯⣸⣿⣿⣿⣿⡷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣿⢻⣿⣿⣿⣿⣿⢿⣿⣽⢿⣿⣿⣽⣟⣿⣿⣿⣿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡋⠉⣩⡔⢊⡷⣯⡹⣧⢸⠹⢿⡟⣽⣿⣿⣾⣿⣻⣿⣿⣿⠻⠿⠘⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣠⠞⠉⣱⣿⠷⣿⠛⠟⣾⣛⠾⢲⣿⣿⣿⣿⣿⣿⣿⣿⣻⣠⡾⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠾⣑⣹⣿⣿⣃⣐⣁⣔⢩⣽⣙⣾⣿⣿⣿⣿⣿⣯⡗⣿⠿⣏⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣭⣟⣿⣿⣿⣼⣿⣿⣿⣿⣜⣿⣿⢿⣿⣿⣯⣵⣷⢾⡻⡟⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣥⣿⣿⣹⡛⣪⣿⣟⣻⣷⡄⣠⣼⠿⠻⠁⣙⢔⢻⡇⠦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣗⣴⣻⣿⣿⠟⣻⢚⢲⣟⡿⣏⡻⣤⡾⠫⠀⢖⣨⠀⠌⢼⡽⡇⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⣸⣿⠛⠉⣤⠚⢐⣂⡵⢙⣾⡛⣿⣿⡇⢻⠠⠟⠋⠘⣉⣈⣑⣃⡃⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣻⡿⢩⢦⡙⠀⢟⣩⢀⣩⣥⣤⣌⣙⠫⣴⡶⢿⣛⣛⣻⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⢾⣍⠛⣵⣟⠃⢸⢿⣿⣿⣿⣿⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠘⠋⠾⠧⠷⠤⢾⠙⢫⣿⣿⣿⣽⣿⡿⢸⣑⣽⡦⡙⢿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣋⣥⣏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣹⠿⠿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢤⡀⠀⠀⠀⢠⢦⡀⣠⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⢦⡀⢰⠃⠀⠙⠦⡀⢠⠏⠀⠙⢧⣀⠀⠈⠉⠒⠲⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠄⡉⠺⣄⠀⠀⠀⠈⠛⠦⡄⠀⠀⠈⠓⡄⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢄⠑⢤⡙⠲⢤⡀⠀⣼⠁⠀⠀⢀⠎⠀⠀⣎⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⣠⠤⠤⠤⠷⡄⠠⣉⠢⢌⠀⣾⡄⠀⢻⡀⠀⠀⣼⠀⠀⢸⡀⠉⠓⠒⠛⠉⠁⠀⠈⠢⢄⣀⡀
⠀⠀⠀⠀⠀⢹⣷⣦⣄⡀⠈⠲⣄⡑⠢⡀⢻⣜⠀⠈⠳⣄⠄⢻⡀⠀⠸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠛
⠀⠀⠀⠀⠀⠈⠛⢿⣝⣿⣷⣦⣄⠉⠓⠦⣀⠙⢦⣀⠀⠈⠢⣄⠳⣄⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣮⣤⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢝⡿⡿⣿⣦⣄⠈⠉⠓⠛⠿⣶⣦⣬⣽⣾⣿⣶⣮⣷⣶⣶⣶⣶⣶⣾⣿⣿⣷⠶⠶⢾⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢷⣝⢷⣴⠯⣽⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⢞⣛⣻⠿⠿⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⡷⣋⣍⣴⠶⠾⠓⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




⣿⣻⣻⣻⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣛⣻⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡘⠀⠀⠀⠈⠙⠻⣷⣦⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣃⣀⠀⠀⠀⠀⠀⠈⠙⢿⠿⠿⠭⠉⢻⠛⠛⠳⠟⡟⠙⢻⠛⠽⢿⣷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⠀⠀⢡⡀⠀⠀⡤⠤⠤⡮⢀⠠⠤⠤⠜⠀⠀⡴⢠⠃⢀⡜⠀⠀⠀⢸⠉⠙⠻⡿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⠀⠈⠢⣨⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⠆⢸⡰⠁⠏⠁⠉⠑⠁⠀⠀⣎⠀⢼⣉⡟⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣻⠢⠤⡀⢱⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡠⠊⡤⢦⠀⢸⠀⠀⡔⠀⠢⡄⠀⠀⠓⠑⢄⣰⡏
⠀⠀⠀⠀⠀⠀⢠⡿⣧⡀⠀⠈⠁⠀⡐⠒⠄⠀⡇⠀⠀⠀⠀⠀⠣⡄⠺⠤⠊⠀⡆⠀⡎⠀⠀⠀⡅⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣸⣷⠀⢣⠀⠀⠀⡠⠁⣄⠘⢲⢧⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢠⠃⢜⣀⢤⠤⠤⠜⠀⣠⠤⡀⢠⡟
⠀⠀⠀⠀⠀⠀⢿⣝⢤⡀⠓⠤⠴⠁⢶⠄⠀⡸⠸⡀⠀⢀⠤⢄⠀⠀⠠⡇⠀⠸⠀⠀⠀⣓⣋⠀⠀⠀⠀⢱⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣖⣄⡀⠀⠀⠀⠀⣷⢒⠎⠒⠁⠀⢀⠇⠀⡠⠃⠀⡇⠀⠀⡎⠀⢨⠂⢀⠔⠋⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣖⢄⠀⢀⠋⠉⠀⠀⠀⠀⠸⡀⡰⠁⠀⢀⠃⠀⠀⠘⡆⠀⢉⣁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣾⣤⣀⣂⣒⣒⣒⣒⣛⣷⣶⣶⣾⣢⡤⣀⡔⠁⠀⠣⠤⠃⠀⡠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠛⠻⣿⣿⣶⣤⣴⡁⠉⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠛⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠐⠦⣄⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠐⢤⣀⠀⠀⠀⠉⠒⢤⣀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠠⣄⡀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠈⠙⠢⣄⡀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠉⠀⠀⠀⣄⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠉⠓⠆⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⢠⠞⢢
⠀⠀⠀⠀⠐⢏⡉⠀⠀⠀⠀⡀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⡋
⠀⠀⠀⠀⢠⡤⠟⠢⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠔⠁⣠⠞⢁⡜
⠀⠀⠀⠀⠀⠙⣦⣀⠀⠙⠦⣀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⣠⠞⠁⢠⠔⠁⣠⠚⢢
⠀⠀⠀⠀⠘⢯⡀⠈⠓⢤⡀⠈⠑⢤⡀⠀⠈⠑⢦⣀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⣠⠞⠁⣠⠞⠁⡰⣃
⠀⠀⠀⠀⢠⡴⠛⠳⢄⡀⠈⠳⢄⡀⠉⠳⢄⠀⠀⠈⠓⢦⣀⠀⠀⠀⠈⠀⠀⠀⢀⡤⠚⠁⢀⡴⠊⠁⣠⠞⠁⣠⠊⢀⠜
⠀⠀⠀⠀⠀⣙⡦⣄⠀⠉⠲⣄⠀⠉⠢⣄⠀⠙⠢⣀⠀⠀⠈⠃⠀⠀⠀⣠⠴⠚⠁⢀⡠⠖⠉⢀⡠⠊⠁⣠⠞⠁⡰⠋⣰
⠀⠀⠀⠀⠈⠣⣄⠀⠙⠢⣄⠀⠙⠢⣄⠀⠙⠦⣀⠀⠑⠦⣀⣠⠤⠒⠉⠀⢀⡤⠖⠉⠀⣠⠔⠉⢀⡠⠊⠀⡠⠊⢀⡴⢅
⠀⠀⠀⠀⠠⣖⠋⠑⢦⣀⠈⠑⠦⡀⠈⠑⢦⡀⠈⠓⢤⡀⠈⠀⢀⣠⠴⠚⠉⠀⣠⠴⠋⠁⣀⠔⠋⠀⡠⠊⠀⡰⠋⢠⠞
⠀⠀⠀⠀⢀⣈⠷⢤⡀⠈⠓⢤⡀⠈⠓⢤⡀⠈⠲⢄⡀⠈⠓⠚⠉⠀⢀⡠⠖⠋⠀⢀⡤⠚⠁⢀⠴⠋⢀⡤⠊⢀⡴⠉⡱
⠀⠀⠀⠀⠈⠳⣄⡀⠉⠲⢄⡀⠉⠲⣄⠀⠉⠢⣄⠀⠉⠲⣄⡤⠔⠊⠁⠀⣠⠴⠚⠁⢀⡠⠞⠁⢀⡴⠋⢀⡴⠋⢀⠜⢦
⠀⠀⠀⠀⠰⣎⠉⠙⠢⣄⠀⠙⠢⣄⠀⠙⠢⣀⠀⠙⠢⣀⠀⠀⣀⡤⠖⠋⠀⢀⡠⠖⠉⠀⣠⠖⠁⢀⡴⠋⢀⡴⠁⣠⡋
⠀⠀⠀⠀⢀⡤⠟⠦⣀⠀⠙⠦⣀⠈⠙⠦⣀⠈⠑⢤⡀⠈⠙⠋⠁⠀⣀⠴⠚⠉⠀⣠⠔⠋⠀⣠⠖⠁⢀⡔⠋⢀⠜⠁⡽
⠀⠀⠀⠀⠈⠑⣦⡀⠈⠑⢦⡀⠈⠑⢤⡀⠈⠓⢄⡀⠈⠓⠤⠴⠒⠉⠀⢀⡠⠖⠋⠀⣀⡴⠊⠀⣠⠔⠉⢀⠔⠁⣠⠎
⠀⠀⠀⠀⠐⢯⡁⠈⠳⢤⡀⠈⠳⢄⡀⠈⠳⢄⠀⠉⠲⢄⡀⢀⣠⠴⠚⠉⠀⣀⠴⠚⠁⢀⡤⠚⠁⣠⠔⠁⣀⠜⠁
⠀⡀⠀⠀⠀⠀⠉⠲⣄⠀⠉⠲⣄⠀⠉⠲⣄⠀⠙⠢⣄⠀⠉⠉⠀⢀⡠⠔⠋⠁⢀⡠⠖⠉⢀⡠⠚⠁⣠⠖⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠙⠦⣄⠀⠑⠦⣀⠀⠑⠦⠖⠊⠁⠀⣀⡤⠚⠉⠀⣠⠔⠋⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠈⠑⢤⡀⠈⠓⢤⣀⣀⡤⠖⠋⠁⢀⣠⠔⠋⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠈⠲⢄⡀⠈⠁⠀⣀⠤⠚⠉⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠉⠲⠚⠉⠁⢀⣠⠴⠋⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣠⠴⠚⠉
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⢀⡤⢶⡴⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠠⠐⠀⠉⠐⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⢀⠠⣴⠟⠉⣁⠴⠄
⠀⠀⠀⠀⠀⢀⠀⠦⢤⣀⠀⠀⠀⠈⠁⠒⠚⠒⠉⠀⠀⠀⠀⣀⠄⢂⠥⠊⡠⢔⡌⡢⢽⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡈⠑⠂⠤⠀⣀⡀⠀⠀⣀⡤⠒⢋⠄⢊⠠⡂⠕⠈⣠⠴⠚⠉
⠀⠀⠀⠀⠀⢰⠤⠤⠀⢀⣀⠈⠉⠐⠒⠀⠤⠭⢉⡀⠔⡊⢔⠬⠂⢁⡠⠖⠋⢁⣀⣴⣾
⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠒⠘⠡⠄⠆⡀⡤⢀⡠⣐⠅⣪⣔⡮⠞⣉⣤⣖⡮⣞⣭⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠨⣤⡾⠊⢠⣰⡯⢷⡛⢭⣫⠕⣫⠵⡇
⠀⠀⠀⠀⠀⠊⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣶⣻⠝⡏⠩⠉⢩⣍⣭⣿⣿⣿⣿⣷⣶⡦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠠⢸⣿⠿⡋⠀⠀⣩⠶⠛⠛⢋⣻⠿⠛⠉⣛⣛⣛⠚⠂⠉
⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⢀⠠⠒⠋⣁⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣰⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡾⠛⠒⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠉⠉⠉⠉⠒⠂

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⣛⡟⢿⡿⢛⡛⢿⠛⣿⠟⣿⠛⣛⣻⣟⡛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⢾⠀⣿⣿⣿⠀⣁⠺⣿⠀⣭⣽⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⣾⣷⣬⣴⣿⣦⣿⣦⣽⣦⣤⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣝⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣻⡟⣿⣿⣿⣻⣿⣻⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣻⢿⣷⢁⣿⣿⣿⣿⣟⣷⢹⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠤⠤⠤⣽⣿⣿⣏⢹⡟⢡⡌⢿⠡⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢾⣿⣿⣿⢸⣇⠹⢃⣾⠳⠆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⠖⢰⡆⣠⠖⠶⠄⣶⢀⡴⠂⣶⠶⠆⠰⢶⡶⠂
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢿⡀⣀⡄⣿⠛⣦⠀⣿⣛⣃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠒
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠉⠉⠀⠉⠀⠈⠁⠉⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⡇⢰⢦⡶⠰⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢢⢰⢠⡆⠀⠂⠀⠢⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⢠⣄⣀⢠⠿⡃⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⡤⠈⠿⠠⠀⠀⠠⡜⠈⠧⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠤⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠴⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢰⡏⢹⡄⢯⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠧⠾⠁⢦⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠒⠂⠀⠀⠰⠶⠀⠀⠀⠤⠤⠀⠀⠀⠒⠶⠀⠀⠐⠒⠒⠀⠀⠐⠒⠂⠀⠀⣠⡤⢤⣼⠇
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣼
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢶
⠀⠀⠀⢸⡇⠀⠀⠀⠤⠤⡤⠤⠀⠀⡄⠀⢀⡤⠤⠄⠀⢀⠀⢀⠄⢀⡤⠤⠀⠀⠠⢤⣤⣤⠀⠀⠀⣀⣤⣤⣼⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⡀⠀⠀⠀⣼⠲⣍⠀⢸⠦⠤⠀⠀⠀⢠⡇⠀⠀⠀⢸⡉
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠉⠉⠋⠀⠉⠀⠈⠃⠘⠳⠦⠆⠀⠀⠸⠃⠀⠀⠀⠈⠳⠶⠤⢤⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠒⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠁⢥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⡂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠖⠶⠦⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⢀⣤⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡂
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣬⠁⣨⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠉⢹
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⡀⢆⡽⠋⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⣀
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢀⡤⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⡀⢤⡸⡄⠙⣶⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⢠⠎⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡀⠸⣏⠀⠙⢷⡦⣌⠳⣄⡀⠈⠁⠀⠀⠀⠀⠀⡠⠚⠁
⠀⢰⡃⠀⠀⠀⠀⠐⡎⠀⠀⠀⠀⠀⠀⠀⢠⣴⡆⠙⣦⡙⢦⡀⠀⠳⡌⠳⠈⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⣿⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣆⠈⠻⣄⠙⠲⠒⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠈⠳⠒⡆⠀⠀⠈⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠘⠃⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠙⠚⣆⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠉⠉⢽⣄⢱⡀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⡄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠉⠳⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣀⣠⣀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠁⠀⠈⢣⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠙⣄⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠈⢦⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⡠⠤⠤⠤⢤⣷⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡷⠃⢀⡠⠤⠤⣀⠀⠱⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣇⢀⡏⠀⠀⠀⠀⠙⢦⠱⡄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠳⡙⣄⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠑⣌⢆⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢣⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⡄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠭⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢳⡀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠉⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢇⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠀⠀⣀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠈⠲⠤⠔⠒⠉⠁⠀⢀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⣀⣀⡠⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡴⠂⢀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⢤⡘⠒⣉⢤⢀⠤⡀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⣩⣶⣤⡆
⠀⠀⠀⠀⠀⣀⠀⣧⢠⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢬⠁⠾⣹⣾⣯⣁⣩⠏⠀⠂⠀⠀⠀⠀⠀⠈⠻⣿⣿⡯⢋⣀
⠀⠀⠀⢀⡼⠟⣁⣬⠚⠀⠀⠀⠀⠂⠀⠀⠤⣶⡘⠚⣶⡀⣷⣽⣾⠶⠫⠷⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠠⣾⣽⠇
⠀⠀⢠⡾⠁⢰⠋⢠⣎⣉⣉⠀⠀⣀⠀⠀⢲⣭⣼⣉⠩⢼⠯⣿⣿⣶⣶⣶⣖⣱⡀⠀⠀⠀⠀⠀⢀⣠⡎⣀⣀⡀
⠀⠀⣸⠃⢤⡜⢶⡟⠁⡌⠀⠇⠀⠌⠀⠀⠘⣿⣶⡟⠒⠂⠲⠟⢺⣞⣏⠀⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⢢⡄⡀
⠀⠀⣧⣴⢷⣤⢟⣱⠾⠀⠀⠀⣩⡀⠀⠀⠀⠈⢺⣿⣿⣖⡷⠖⡏⠀⡆⠁⣁⠀⡦⠠⠉⠀⣀⣿⣿⣿⣿⡿⣿⢻⡇⠈⢰⡟⣀
⠀⠠⣿⡿⣻⣿⣼⠋⠂⠄⠀⠀⠉⠙⠢⡀⠀⠀⠀⠈⠹⠟⣷⢾⣿⡏⢈⡖⠿⡛⠁⡀⢰⣤⣿⣿⣿⣿⣿⣷⣷⡊⠀⠀⠀⠼⠏⡀
⠀⠀⢻⣿⣿⣿⣾⣿⡎⠀⠛⠀⠀⠀⠀⠀⢀⠀⠎⠁⠀⢀⠀⠈⠋⠃⢸⡷⠀⢀⠐⣿⣿⣿⣿⣿⣿⣿⢟⢘⣻⡷⠀⠀⠀⠀⠐⠁
⠀⠀⠈⠻⠟⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⣃⣾⡟⠀⢀⣷⡀⠀⡇⠀⡼⠀⠸⢿⣿⣿⢿⣯⣵⣶⣤⣟⡗⢉⠀⣀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⣄⣰⠷⣄⣈⠙⠿⡟⠻⡄⣄⣸⣿⣇⣀⢓⡼⠁⢢⠀⢻⣿⣿⣿⣿⡿⡿⣿⠏⠁⠀⢲⣤⡉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⣿⣿⣿⣿⣿⣏⠑⠂⣵⡀⣇⣿⣿⣿⣿⣿⢸⠃⠀⢀⡷⠉⢹⣥⣮⠟⡀⠁⡜⡇⠰⢠⢰⣼⡿⠓
⠀⠀⠀⢰⣄⡀⠀⠀⠀⢀⣉⣹⣿⣿⣿⣿⣿⣼⣧⢈⡣⣸⣿⣿⣿⣿⡏⣿⠀⣠⠋⠀⢠⡎⠈⠳⠾⢃⡸⣧⢷⡜⠞⠓⠉
⠀⠀⠀⠀⢿⣿⣀⢈⡙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢻⡌⣿⣿⣿⠁⣟⣼⠋⠄⠀⡾⢰⣞⣦⠀⠀⠀⠁
⠀⠀⠀⠀⣤⣤⣽⢿⣤⠂⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣳⣼⣿⣿⢖⡽⠃⢀⡆⠀⡃⣸⠟⠁⠀⡴⣀
⠀⠀⠀⠀⠉⠛⢁⣾⡀⠁⢢⠀⠀⠼⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠛⣯⣠⣥⣿⣀⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⡟⢂⣉⣁⣠⣄⣀⣁⣤⣼⣿⣿⣿⣿⣿⡧⠐⣀⣤⣾⣿⢻⣿⣿⣿⣿⣴⡦⠖
⠀⠀⠀⠀⠀⠀⠖⠾⠁⠺⢃⣹⣯⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠛⠃⣼⣿⣿⣿⢧⡞⡼⠛⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣿⣿⣿⣿⣿⠟⢛⣷⣬⠟⠁⢠⣶⣶⣝⠻⣿⣏⣼⣡⣶⣿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠋⠁⢠⣿⣿⡿⠒⠀⢠⣉⣛⣿⡄⠿⣸⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠽⣿⣿⡟⢰⣿⣿⣛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠿⠿⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣼⢳⡀⠀⠀⠀⢀⡴⠋⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⣸⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⣈⠵⠃
⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠈⢧⣡⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣯⠤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣿
⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⢀⡀⢀⡀⢒⡒⣒⠂⠀⣀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⢦⡀⠀⠀⣤⣀⡄⠀⠀⠈⠯⠟⠀⠘⢭⢟⡤⠞⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠶⣄⠈⠳⠁⠀⠢⠄⠀⠀⢄⡠⠖⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣠⠴⠶⢤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡟⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣛⣟⣛⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠚⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠤⠄⣀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⣉⡠⠒⠕⠂⠱⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⠀⠀⠀⠘⠀⠀⠤⠤⠤⢀
⠀⠀⠀⠆⢀⡤⠄⠁⠁⠠⠉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠻⠿⠛⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰
⠀⠈⠇⠤⠀⠀⠀⢰⠄⠀⠀⢱⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⢸⠀⠀⠉⠀⠀⡡⠅⠈⢸
⡂⢀⡜⠀⠈⡰⠦⠜⠀⠈⡃⠤⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⢰⡀⢐⢸⠉⣃⣀⠈⠉⡑⡀⢀⠀⡀
⣇⣎⠃⠀⡰⠀⠀⠀⠀⡴⣁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠘⠀⠉⠁⠀⠭⠤⠤⠬⠭⠠⠈⠀⠀⠃
⠨⡹⠀⠊⠈⠁⠀⠀⠀⢤⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣀⠠⠇
⠀⠘⢦⡘⠀⢠⣀⡰⠌⣀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠉
⠀⢀⠐⠀⠉⠙⠃⠙⠉⠁⠂⠁
⠀⠀⠀⠀⠒⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⣂⠠⠤⢀⣄⣂⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⢌⠐⠒⠀⠀⠀⠀⠀⠀⠀⠉⠐⢤⠀⢌⢆⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⡰⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠹⡄⠀⠀⠀⠀⡃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠦⠀⠈⠇⠀⠐⡀⠀⠀⠀⡃⠀⠀⠀⠀⡠⢈⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠐⠈⠲⡄⠀⠀⠑⠀⠂⠈⠀⠀⠀⠀⢀⣀⡀⠀⠄⠀⡀⡜⠀⠂⠐⠀⠀⠀⠀⡇⢀⢀⠀⡀⠀⢀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠀⠀⢀⠌⠀⠀⠀⡄⠈⠀⠄⠀⠀⠀⠀⠨⡭⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡇⠐⠒⠀⠂⠠⠐⠒⠀⠂⠂
⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠃⠐⢄⠀⠀⢀⠄⠀⠀⠀⠠⠤⠠⢄⠀⠀⢃⠄⠀⠀⠀⠀⠀⡃⠀⠀⠀⠁⠁⠠⠀⠀⠁⠅
⠀⠀⠀⠀⢀⡴⠃⠀⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠄⠈⠄⠀⠀⠀⠒⠐⠂⠀⠁⠀⠀⠀⠀⠑⠃⠂
⠀⠀⠀⣠⠊⠑⠄⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠌⠁⠁
⠀⢀⠾⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠤⠤⠤⠄⠐⠊⠁
⠀⠈⠢⡀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⡅⢢⡔⠀⠀⠀⠀⠀⠀⠀⠆⠈⠢⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⢀⠔⣡⠞⢉⠔⠉⡀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⢀⠊⠀⠀⡐
⠀⠀⠀⢀⠔⢁⠔⢁⡼⠋⡠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⡠⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣠⢁⠀⢀⠐
⠀⢀⡐⢡⣔⣁⣴⠋⠀⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⠘⢑⠞⠓⠅⠀⠀⡀
⢀⠘⠁⠀⠀⠀⠁⠜⡇⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⡐⠁⠀⠠⢈⠀⠀⠢⠐
⠉⠀⠀⠀⠀⠀⠃⢀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⠉⠀⠀⠀⠀⠀⢕⠄⠀⢁
⠐⠒⠒⠒⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⢨⠂⠀⠀⠂⠀⠂⠐⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣿⣿⣿⣽⣯⣿⣯⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢠⠂⠀⠁⠐⠀⠂⠘⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⣿⣿⣼⣭⣿⣼⣸⣧⣿⣿⣿⡅
⠀⠀⠀⠀⢠⣁⣀⣉⣉⣉⣉⣁⣉⣀⣀⣀⣘⠀⠀⠀⠀⠀⠠⠿⠿⠶⠶⠶⠶⠶⠶⠿⠿⠿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠉⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢶⣟⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠂⡁⠠⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⢤⣬⣭⣭⣭⣬⣭⣭⣤⣭⣥⡭⣣⡀⠀⠀⠀⢐⠛⠁⠀⠉⠘⠓⠒⠛⠓⠲⠛⣲⠻⡄
⠀⠀⠀⠀⠀⢾⣿⢿⢾⣿⢾⡷⡿⣿⣿⣺⡷⠁⠀⠀⠀⠀⢈⠀⠀⡀⢁⠈⡀⡈⢀⠀⠈⠀⠹⠉⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣶⣿⣾⣷⣷⣿⣿⣿⡗⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠓⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⣇⡶⣼⣠⣴⢴⠢⣶⡖⢲⠂⠀⢴⢲⡦⠶⢰


⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡤⠀⠀⢤⡄⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠮⡀⣠⠇⠀⠀⠈⣰⣆⠀⠀⠀⠀⠀⡐⢂⡀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠂⡠⠁⠈⠢⡄⠀⠀⢀⠞⢻⠏
⠀⠀⠀⢠⡄⠀⠠⠋⠀⠀⠀⣌⠆⠡⣱⠀⠀⡠⠎⣰⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠰⠀⠀⠚⡩⠀⠀⢏⡠⠂
⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠑⠒⠒⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠂⠐⠊⠀⠀⠈

⠀⠀⠀⢰⠀⠀⠐⠒⡆⠀⠀⠀⢀⠦⡄⠀⢠⠦⠖⠴⠤⠆⠀⠀⡔⠒⠒⠒⠂⠀⠀⠀⣴⣦⠀⠀⢀⠀⠤⢀⡒⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡒⠇⢴⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢱⢈⠈⠠⡁⠀⠀⠰⠩⠭⠆⠀⠀⠑⠠⠤⠄⠒⠉⡄
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⣮⠃⠈⠀⠀⢘⠀⠀⠉⣠⠇⠀⠀⡇⠈⠁⡀⠁⠀⠀⡱⠀⠀⢊⡀⠀⠸⡀⠀⠀⢐⠒⠗
⠀⠀⠀⠈⠓⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢠⡶⠒⢦⡀⠀⢠⠢⠴⢦⡒⡄⠀⠀⠀⢀⠄⠂⠀⠀⠰⠀⡆⠀⠀⠀⠀⢀⡴⣄⠀⠀⠀⠀⡀⠰⣒⢤
⠀⠀⠀⠀⢸⣄⠀⣠⠇⠀⢸⠨⠉⠩⠁⡇⠀⠀⢠⠃⡠⠃⠀⡠⠊⠀⠉⢢⠀⠀⢰⠋⠉⢉⠒⠀⠀⢸⠰⡸⢀⢎⠆
⠀⠀⠀⢀⢎⡟⠉⠁⠀⠀⢸⢀⠀⠈⢀⠃⠀⠊⠉⠒⠁⠀⠀⢏⠀⠑⡈⢩⠃⠀⢸⠐⠈⠐⢸⠀⠀⣨⠳⠤⠢⣟⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠒⠚⠁⠀⠀⠈⠑⠒⠒⠁⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⡲⣄⠀⠀⠀⠀⡀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠛⢠⡄⠀⠠⠄⠀⠀⢠⠀⠀⠀⣠⡄⢈⡀⠀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠰⡕⢢⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠤⠁



⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠈⠦⡄⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋
⠀⠀⠀⠀⠀⡜⣂⡈⠉⠉⢹⠏⣐⣒⡒⠚⠯⠤⢤⣠⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⡀⣀⡤⠠⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠘⡴⠃⠀⠀⠀⠀⠚⠁⠈⠉⠉⠉⠒⠒⠖⠒⠒⠛⠒⠀⠰⠃⠀⠀⠀⠘⢁⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⣀⣀⣀⣘⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀
⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⠶⠶⢖⣫⡭⠤⠤⣞⣀⠀⠉⠉⠙⠒⠒⠾⠥⢄⣀⡀⠐⠂⠒⠒⢒⡋⠉⠀⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⠭⣙⣒⠢⠤⣜⣁⠀⠀⢠⡄⠀⠈⠀⠛⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠚⠛⠛⠒⠒⠒⠒⠂⡔⠂⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠑⠲⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠙⡆
⠀⢀⡤⢄⣾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠴⠊⠁
⠀⣏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠖⠒⠒⠒⠛⠳⢤⡀
⠀⠘⠦⠤⠤⣴⠇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⣠⡤⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⡷
⠀⠀⠀⠀⠀⣀⣉⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃
⠻⣏⠉⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸
⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢦⣀⣀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠚⠉⢧
⠀⠀⠀⢸⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠊⠀⠀⠀⢸⡀
⠀⢀⣠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡇
⢰⠋⠁⠈⠁⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠸⣆⠁⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠛⠓⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣰⡥⠤⠖⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⠀⠀⠀⣀⡤⠤⠤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠑⢦⡀⠀⠀⢀⡤⠚⠁⠀⠙⢦⠔⠋⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⠔⠋⠀⠀⠀⠀⣰⠃⢠⠴⠒⠲⢄⡤⠖⠒⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⡄⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣴⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⠀⠀⠈⠢⣀⠀⠀⣠⠞⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⠀⠈⠳⠚⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠤⠴⠖⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡀⠀⠀⣀⣤⡴⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠿⠋⠁⣠⡾⠋⠀⠀⢀⡄⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣴⠟⠛⠷⠄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣤⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠻⢶⡶⠟⠁⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠈⠳⣤⣀⠀⠈⠁⠀⢀⣠⡾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣴⠞⠛⠛⠻⠦⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠀⣤⣼⣧⣤⣤⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠲⢿⡶⠶⠶⠶⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣤⣀⣀⣠⡤⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠈⠉⠉⠁⠀⣠⡾⠋⢀⣠⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠞⠛⠉⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣦⠀⣀⣠⣤⡤⠄⠀⠉⠀⠀⠀⠀⠀⠠⠐⡧⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡁⠀⢀⠀⠀⠀⠀⠀⠠⠖⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⣀⢀⣀⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⢑⡤⠁⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀⠤⠤⠐⠂⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠑⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡈⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠠⠄⠀⠄
⠀⠀⠀⠐⠀⠀⠀⢀⠀⠁⠈⠀⣄⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡁
⠀⠐⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠿⠟⢀⣀⣠
⠀⠀⡇⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠹⣄⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢌⠉⠀⣶⠁⠀⠄⠄
⠀⠈⠂⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠀⠀⡀⠀⡂⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡌⡄⠀⢹⣧⣔⡢⠂⠑⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⣃⣉⣀⣵⣘⡦⠀⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢻⣷⣿⠀⠈⣿⣷⡀⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣻⠁⠀⠩⠃⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⡆⠌⠙⢠⠦⠤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⡇⠂⠀⠀⠀⠙⠀⠀⠰⢤⡴⠋⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⡰⡀⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⢠⡆⠀⠀⠀⠀⠀⠀⠇⠈⢀⠀⠙⠷⠄⠂⡅
⠀⠀⠀⠀⠂⠀⠘⠋⠘⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⠴⠛⠉
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠒⠋⠀⠐⠦
⠀⠀⠄⡴⠁⠀⢀⣘⠀⠀⠀⡰⠃⠀⡇⠀⠀⠀⠀⠀⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⣠⣴⡾⠛⠉⢁⣀⣀⣤⡤⣐⡴⡀
⠀⠀⠀⠓⠂⠀⠰⠰⠀⢴⡾⠥⠤⠴⠇⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠀⡀⠀⠁⠀⠁⠀⠀⠀⠚⠒⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠄⠒⠒⢒⠒⣒⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠀⠀⠀⢀⡠⢁⠀⠀⠰⠂⠁⢸⣿⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⣗⠀⠀⠀⠀⠀⢀⢘⠚⡉⠉⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⡘⠠⡤⠫⠚⠛⠘⣃⣠⡇⠀⣿⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡯⣿⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⡇⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⢸⣇⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡂⢠⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⡀⠸⠀⠀⠀⠈⣿⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠇
⠀⠀⠀⠀⢠⡥⣤⣤⣤⣿⣿⠀⠀⠀⠀⠈⠶⠶⠻⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢐⢘⠂⠀⠀⢀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⢵⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⣍⢹⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠘⠘⠀⠀⠃⡞⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⢸⡍⢹⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⠠⠜
⠀⠀⠀⠀⢸⡓⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⣴⠀⠀⣀⣀⣘⣉⣀⣘⣻
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡆⠀⠀⠀⡇⡇⠀⠀⠀⠈⠀⠂⠈⡄⢀⡀⠀⠀⠀⠀⠻⠉⠓⠒⠺⠿⠒⠛⠻⠃
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢲⡀⠔⡜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⢸⢰⡆⠀⠀⠀⡤⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡇⠇⠈⢆⠀⠣⠀⠀⠀⠀⠀⠞⣰⠃⡨⢄
⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⡇⢾
⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠑⠀⠐⠀⠀⠀⠀⠤⠎⠀⠀⠀⡈
⠀⠀⠀⠀⢸⠾⢻⠀⠀⠀⠀⢸⠤⠄⠀⠀⠀⠘⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠖⠛⠛⠒⠒⠆⠀⠈⣙⣳⣦⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠓⣂⣤⠴⢖⣛⠋⠉⣩⡄⠀⠀⠀⠰⠶⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣥⠾⠛⢉⣀⠀⠈⠉⠀⢀⡈⠀⠀⠛⠃⠀⠀⠒⠀⢠⠎⠀⠀⡠⢔⠦⡀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠠⡶⠀⣀⠁⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠘⢧⣛⡣⠃⠀⢀⣠⣤⡴⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠘⣡⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣤⣾⠟⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢠⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⢁⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠏⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⡩⠤⠄⠉⠉⠑⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⣰⢏⠀⣿⠀⠀⢀⣠⡤⠤⠀⠀⠀⠀⢀⣀⣀⣤⣽⣧⡤⠒⠀⠀⠀⣠⣄⣀⣀⣀⣀⣤⣤⡴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣏⢰⡏⠀⢰⡯⢐⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠙⠶⠧⠶⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢾⡇⠀⠘⢶⡥⠞⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⠉⡷⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢠⠇⠀⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⣀⡴⠋⠀⣄⠈⢳⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠋⠉⠀⠠⡀⢘⡶⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢆⠀⢻⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠲⣯⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠈⢦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡶⣾⠉⠀⢀⡠⠒⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢿⣯⣿⣠⢳⠁⠀⠀⠀⠀⡤⠀⠐⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣾⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣿⣥⠀⠉⢹⣛⣁⡠⠀⢠⣷⣲⣚⣛⣩⡐⠂⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣫⣶⠀⠀⣾⡈⠁⠀⠐⠋⠉⠉⠉⢉⠉⢁⡀⢀⡀⡠⠟⠀⠀⠀⠈⢰⡆⠀⠀⠀⠀⠀⠈⠋⠙⡛⠉⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⡿⠷⠠⠤⠤⠤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⡄

⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡤⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠄⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢰⠀⢸⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶⣷⣾⣾⣶⡀⢀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠻⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠇⠀⠁
⠀⠀⠀⠀⠀⠠⣭⡿⠿⠀⣰⣾⣿⠏⠀⢰⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠘⣿⣷⣦⠀⠿⢿⣥⠄⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⠀⠀⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠉⠉⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⡎⠭⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠩⢱
⠀⠀⠀⠀⢠⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⡄

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

⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠄⢠⢠⠄⢤⠆⢠⠀⢠⠠⡀⡄⠀⡤⠀⠀⢠⠆⠀⡤⠀⠄⢄⢠⠐⢄⢠⢠⠄⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣁⢸⣀⡰⠰⠀⢸⢀⠃⢣⢸⣉⠆⣇⡀⣏⣁⠀⢈⡱⠀⡇⠀⠃⢣⠣⣀⡸⢸⠱⡀⣏⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠂⠒⠀⠉⠉⠉⠀⠒⠒⠒⠤⢤⢤⣀⣀⣀
⠀⠀⠀⠠⠒⠶⠌⣉⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠲⢦⣤⠄
⠀⠀⠀⡇⠂⠀⠀⠀⠀⠈⠉⠙⠒⠒⠦⠤⢭⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠚⠉⢩⠇
⠀⠀⠀⠱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⢤⡤⠔⠒⠋⠉⠀⠀⠀⠀⠀⣈⣆⣀
⠀⠀⠐⢶⣿⣉⡉⠉⠒⠒⠦⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠤⣤⣀⠀⠀⣀⣤⣶⣟⡋⠥⠤⠀⠛⠒⡄
⠀⠀⠀⢸⠀⠀⠉⠓⠲⠤⣀⡀⠀⠀⠉⠉⠙⠓⠲⠦⢤⣄⣀⣀⡀⢻⠀⣀⣬⣿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⡤⠶⠖⠒⠛⠉⠉⠁⠉⠉⠛⠛⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⣀⣸⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⣖⣺⣶
⣼⡏⠙⠛⠷⢦⣽⣻⣷⣦⣀⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠒⣒⣊⣩⡭⠴⠶⠒⠛⠋⠉⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣦⣴⣁⣦⡤⠴⠶⣒⣛⣫⣭⡭⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠘⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠓
⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢶⣤⣀⡼⡄⢀⣀⣀⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡠⡀⣄⡀⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠓⠋⠋⠃⠛⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠿⠰⣉⠸⠺⠸⠧⠟⠾⢮⡥⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠦⡀⠀⠀⠀⡖⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠶⣣⠄⠀⢤⡘⢢⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⣹⢁⠉⠈⠃⠹⢎⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁⢸⠸⠘⠶⣓⣒⡳⢚⠥⡀⠀⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⢦⣉⡆⠀⠀⠀⢇⣱⡸⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠆⠀⠀⠀⣀⠀⠀⠀⡀⠀⣠⣾⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷⠂⡐⣞⢂⠀⠀⠋⠨⠀⠁⣁⣥⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠌⠩⠾⠈⡑⠄⠁⢀⡀⢸⣿⣸⣿⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⠀⢆⡠⢧⠀⡧⠤⠥⠬⠭⠭⠭⠭⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠉⠀⠀⢰⠀⢸⡀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⠀⠀⠀⠀⢰⠁⢸⠁⠀⠀⣰⠋⠑⠢⣄
⠀⠀⠀⠀⡔⢋⣉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⡆⠀⠀⠀⡏⠀⠸⠀⠀⡴⠃⠀⠀⠀⡜
⠀⠀⠀⠀⡄⠸⡀⠱⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⢄⡀⠀⡆⠀⠸⡀⠀⣸⠀⠀⡇⠀⡼⠁⠀⠀⢀⠞⠀⣠⠔⢾
⠀⠀⠀⠀⠘⢄⠑⢄⢱⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢶⢫⡱⡀⠹⣤⠶⡇⠀⣷⠦⣤⢁⡞⠀⠀⠀⢀⠎⢀⡜⡝⡳⡼
⠀⠀⠀⠀⠀⠈⠳⣄⠙⢧⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⡱⡘⣄⢹⡀⢻⢀⡏⢠⣇⠞⠀⠀⠀⢠⠋⢀⢎⠜⡜⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢴⠉⠙⡦⣤⠤⢤⡀⢀⣀⣀⣀⣀⣳⣱⣜⣌⣆⣷⣀⣿⣁⣼⣏⣀⣀⣀⣠⠇⠠⠯⠮⠾⠼⢤⣐⣒⠢⠭⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠊⠁⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠁⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠧⠝⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠚⠀⠀⠀⠀⠀⢴⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⣔⢲⠀⠀⠀⠀⠓⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⠑⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠓⠚⠀⠀⠀⠀⠀⠀⠺⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⠂⠀⠀⠀⢯⡹⠀⠀⠀⠀⠰⠽⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣶⡶⠤⠤⠤⢤⣄⣀⡀⠀⠀⣠⣤⡀⠀⠀⠀⣿⣿⣿⠛⠛⣛⣛⣛⣻⣿⠿⠿⣿⣿⡿⠛⠻⣿⣿
⠀⠀⠘⠻⢿⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⣿⣷⠀⠀⠀⣿⣿⣿⣤⣀⣿⣿⣿⣿⣿⣿⣿⡜⣿⡇⠀⠀⢿⣿
⠀⠀⠀⠀⡾⠀⢴⣶⣦⣶⣦⠀⡇⠀⢹⣿⣿⠀⠀⠀⣿⣿⣿⣿⡏⣿⡟⠉⠙⠋⠉⣿⡇⣿⣷⠀⠀⢸⣿
⠀⠀⠀⢠⡇⠀⠈⠻⣿⠿⠃⢰⠃⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⢳⣿⣿⣦⡀⢀⣴⣿⢸⣿⣿⠀⠀⣾⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⠇⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⣠⣿⣿
⠀⠀⠀⠸⡄⠀⢠⣤⣤⣤⣶⣾⣷⡆⠸⠃⠀⠀⠀⠀⣿⣿⣿⣿⡬⣷⣿⠿⠛⠛⠛⠋⠈⢙⣿⣴⣿⣿⣿
⠀⠀⠀⠀⠙⠤⠴⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣝⣛⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⡁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⣀⣤⡴⠟⠋⠀⠰⠁⠀⠀⠀⣀⡴⠚⠉⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠀⠀⠀⠠⠄⣀⣘⣦⡶⠟⠛⢡⡶⠂⠀⡆⢀⠀⣠⡴⠛⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣴⠾⠛⢰⣿⡀⢠⠇⡎⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⢀⡀⠀⠤⠒⠂⠀⠀⣀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠁⠐⠒⠒⠒⠒⠀⢠⠿⠞⠋⢁⣴⣾⣞⡇⢿⢸⢠⣱⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⡷⠛⠃⠉⠀⠀⠀⠀⠀⢀⣀⣀⠟
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣇⢀⣴⣾⢿⠋⢘⠈⠁⠛⠛⠘⢻⡶⠛⠛⠉⠁⠀⢠⠀⠀⢀⣇⣤⣤⡤⠤⠶⠶⠟⠛⠋⠁⠒⠒⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠚⡿⠋⢁⣼⣿⣿⠀⠀⠀⠀⠀⣾⣧⣄⣠⣤⣤⠶⠃⠀⠀⣼⣿⣿⢂⣀⣀⣀⣀⣀⣘⣂⣀⣀⣐⣐⣺⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠄⠠⠄⠀⠀⣀⠀⠀⢧⣴⠿⠋⣠⠴⠀⠀⠀⠀⠀⣿⠀⠉⠉⠁⠀⠀⠀⣠⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⣬⡆⠀⠀⠀⢀⣿⠤⠤⠀⠠⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⢀⣼⠿⠋⠁⠀⠀⠀⠾⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠭⣓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢑⡢⠭⣀⠈⠙⠓⢦⣄⠀⠀⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⢴⣪⠥⣒⠁⠀⠈⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠬⠿⠶⣥⣀⡀⠀⠀⠀⠀⣼⠀⠀⢀⣀⣠⡤⠶⠶⠿⠯⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠲⠿⠗⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⣶⢶⣶⠶⠶⠶⠶⠖⠒⢒⣒⣒⣒⣶⣶⠶⠶⠓⠓⠒⠛⠛⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠁⢳⣸⡟⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⣀⠀⠀⠀⠀⠐⠊⠙⢦
⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃⣀⣠⣤⣿⣷⣶⣶⣶⣖⣒⡚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⢀⡴⠞⠛⠉⠁⠀⣿⠟⠉⠀⠀⠀⠀⠐⠋⠭⠭⠭⠭⠭⠍⢉⣛⡛⣛⣛⡒⠒⠒⠒⠲⢶⡾⠓
⣰⠋⠀⠀⠤⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣈⠉⠩⠭⠭⠭⠤⠄⠀⠀⠀⣼⠉⠓⠦⣄⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⣀⣀⣉⣉⣉⡀⠐⠒⠒⠒⠠⠍⠉⠉⠉⠀⠀⠀⠀⠀⢇⠀⠀⣀⣀⣹⡷
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⣈⣻⠉⠉⢠⠇
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⣠⠴⠖⠒⠛⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢸
⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⡙⢷⡠⠀⠀⠀⠀⠤⠤⠶⢖⣒⣛⣯⡭⠭⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⣀⠤⠒⠛⠦⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⡀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⢀⠀⠀⠐⠿⠇⠀⠀⢀⣠⣴⡞⠃⠀⠀⠀⠀⠈⢦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠁⠀⠀⠐⢿⠿⠀⠀⠀⠀⣤⡶⠟⠂⠁⠘⣿⡄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠼⣷⡀⠀⠀⠀⠀⣤⠀⣿⠁⠘⣿⡾⠛⠀⠀⠘⠷⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠰⠋⠁⠀⠀⢰⣶⠀⠀⠀⠁⠀⣠⡾⠛⠀⢹⣦⡿⣦⣄⠘⣿⠶⠛⠀⠀⢠⣤⡀⠀⠀⠀⢊
⠀⠀⠀⢠⠂⠀⠀⠀⠀⢁⣠⣤⠰⣷⡀⣿⡅⠀⠀⠀⢻⣆⠀⠙⠁⠀⣀⣄⡀⠀⠈⠉⠀⣠⠤⠚⠉
⠀⠀⠐⡃⠀⠀⠀⠀⠺⠛⣿⡅⠀⠹⣧⡈⠻⠶⠶⠛⠀⢀⣠⠀⠀⠀⠚⠛⠀⣀⡤⠒⠉⠁
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⠃⠀⢀⡀⠀⠀⠸⠻⡏⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠟⠏⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⣀⠤⠒⠋⠁⠠⠐
⠀⠀⠀⠀⠀⠀⠘⠤⠄⣀⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣠⠞⠁⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⠤⠚⠋⠉⠁⠀⠀⠀⠀⠀⠉⠓⠶⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣨⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⣀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡤⠒⠒⠤⡀⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠠⠤⠨⡇⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠟⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠦⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⡨⠦⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⡠⠞⠁⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⢤⣤⣤⣤⠤⠴⠿⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠔⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡠⠴⠊⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢑⡄⠀⠀⡠⠔⠉⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠤⣄⡲⠃⠀⣞⡀⠀⠀⣀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠑⠢⣄⣀⡀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠇⠄⠛⢒⣶⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠂⠀⠀⠀⠀⠀⠉⠉⠛⢆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢎⢔⡞⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⢛⡍⣼⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⣴⣿⣿⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⠀⢀⣐⣶⣿⡿⠟⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣠⡾⠒⠀⠀⠀⠀⠀⠈⠀⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⠻⠻⠻⠟⠛⢙⣋⡴⠊
⠀⠀⠰⣤⣤⣤⣤⣦⣤⣰⣼⢀⡾⣋⠤⠤⠄⠀⠀⠀⠀⠀⣠⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠤⠓⠂
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⣿⣿⣷⠆⢀⣐⣠⣤⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠘⣷⣤⣄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠛⠛⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣯⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠛⢿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣀⠈⠉⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⢴⣾⣶⣤⣿⣿⡏⣸⣿⣿⣿⣿⣿⡿⠁⠠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠙⠻⢿⣿⣿⠁⣿⣿⣿⣿⣿⣤⡄⠀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢹⣿⡀⣿⣿⣿⣿⣿⠛⠀⠀⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠿⣿⠿⠀⣸⣿⣇⢹⣿⣿⣿⣿⣿⣷⡀⠘⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣤⣾⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠉⠉⠀⣀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣀⠀⠀⢀⣤⡴⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠿⠛⠁⣠⡾⠋⠀⠀⣀⣀⡀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⣾⠋⠉⠻⠇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣿⣄⣀⣀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣹⡏⠉⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣠⣿⠶⣦⣤⣴⠄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⣀⣠⠾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⣠⡶⠋⠉⠀⢀⡀⠀⠉⠻⢦⡀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣴⠿⠷⣦⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⢷⣤⣴⠟⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⠁⠀⢀⣠⡾⠋⢀⣤⣶⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠚⠋⠁⠀⠀⠈⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⢀⡀⡀⠀⣀⠀⢀⣀⣀⡀⠀⡀
⠀⠀⠀⠇⢤⣄⣤⣁⡄⠁⠃⠛⠉⠤⣄⣀⣀⡜⠀⠀⠚⡅⠤⣀⣌⣀⡀⠀⢀⠆⠀⠁⠀⠀⠀⢀⡀⠤⢀⡀⠀⠀⠀⢸⡏⠀⡀⠀⠁
⠀⠀⠀⠆⢐⠛⡞⠛⢳⡀⠀⠆⠃⡐⠓⣚⠛⠓⢀⣀⠄⣆⠙⠓⢚⡛⠃⡀⠈⠆⠀⡄⢀⣀⢘⢂⣀⢀⣀⢘⣂⣀⠀⢸⣯⠀⠀⠀⡗
⠀⠀⠀⠀⢀⡀⠀⡀⠀⠄⠀⠀⠀⢄⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠀⠉⠸⠘⠇⠸⠧⠘⠆⠹⠀⣾⡧⠀⠀⠠⡇
⠀⠀⠀⠠⠈⠙⠂⠓⠙⠀⠘⢀⠀⠀⠀⠀⠓⠂⠀⠀⠈⠀⠀⠐⠈⠓⠀⠀⠀⠀⠀⠃⠰⠀⠂⠀⠂⠐⠀⠆⠠⠐⠀⢸⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠨⠁⠁⠀⠁⠠⠈⠀⠀⠀⠈⠡⠤⠀⠀⠀⠀⠀⠬⠩⠁⠤⠈⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣃
⠀⠀⠀⠀⢀⣀⡀⣀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠉⠉⣉⣉⣁⣉⡉⠉⠀⠀⠀⠀⢀⡀⡀⡀⣀⡀⠀⡀⢀⣠⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⡤⠤⣠⣤⣴⡶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠀⠂⠂⠒⠙⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣿⠿⠗⠒⠲⠦⠤⠀⡈
⠀⠀⠀⠂⡀⡀⠀⠀⠀⠀⠀⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠄⠁⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠁⠛⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⣒⣖⢶⠿⠟⠿⠻⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠋⠉⠙⠓⠒⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣶⢶⡶⣶⡶⠶⠂⠶⢲⡶⠶⢶⣶⠶⠲⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣐⣀
⠀⠀⠀⠀⠘⠛⠚⠓⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣮⠝⣋⣋⢛⠙⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣰⡆⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠈⠉⠀⠂
⠀⠀⠀⠀⠈⠩⠿⠭⠭⠇⠀⠀⠀⠀⠀⡆⠘⢻⣿⢿⡿⠙⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠋⠘⠓⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

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

⠀⠀⠀⠀⠁⠿⠛⠉⠀⠀⠀⠀⠀⣨⡀⠈⠀⠀⠀⡀⠀⠰⣔⠒⠼⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠩⣝⠢⣄⡀⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠠⠚⡃⠀⠀⠀⠀⠦⠀⠰⣘⢫⡟⢨⡓⣾⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣑⣱⠏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣖⣤⠆⠀⢠
⣀⣤⣆⣼⣴⣄⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡁⠀⠀⢸⡀⠀⣀
⠉⠙⠊⠉⠀⠈⠉⠀⠈⠀⠀⠀⣠⠔⠛⠁⠀⠐⠃⠀⠀⠀⠀⠀⠘⠉⣉⠉⠉⠉⢉⡛⠀⠠⠈⠀⠄⠀⠀⠀⠏⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠀⠀⠤⠀⢢⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀
⡈⠨⣮⡏⣹⠋⠀⠀⠀⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠄⢜⠲⢏⠅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀
⣷⢮⣸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⡇⠀⠀⡎⠉⡖⠉⠉⣧⠀⠀⠈⠈⠉⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣦⣥⣽⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡷⣟⣾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠁⠀⠀⠠⡦⠀⠀⡀
⠔⠀⠀⠀⠀⠀⢼⡤⢀⡀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠓⡀⢠⠂⢀⠀⢠⢺⠿⠟⠛
⡾⠇⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣇⣀⡾⢠⠏⠀⠸⢶⣶⠶⠶
⠐⠛⠄⠀⠀⠀⠀⡇⠀⡇⠀⠀⢠⣤⡄⠀⠀⠐⠋⠀⠀⠀⢰⣿⠛⣧⠀⢢⠀⠀⠈⣧⠘⣆⣷⠃⠀⠀⢸⠷⠁⠚⠛
⠀⠀⠀⠀⠀⠀⠀⣇⣠⠁⠀⠀⠈⡈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠈⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂
⢐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤
⠘⠀⠀⠀⠀⡀⢀⣀⡀⠀⢠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠸⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢚⠥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⠲⠶⠶⢦⠀⠀⢠⡴⠞⠋⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢀⣀⠀⠀⠀⠈⠳⣄⡾⠀⠀⢀⣠⠞⣉⠙⠒⠦⢤⣄⣀⠀⠀⠈⠉⠙⠒⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣠⠞⣩⠌⠙⠚⠛⢆⠀⠘⢧⡀⣼⠉⠁⠀⠈⠁⠐⠒⠠⢤⠉⣙⠓⠲⠤⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⣀⡴⠋⠰⠂⠀⠀⠀⠀⠀⠈⠳⡄⠀⠹⣇⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⢸⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⢀⡤⠞⣡⠘⠉⠀⠀⠀⣈⣦⡀⠀⠀⠀⠙⢦⠀⠈⢧⡀⠹⠋⠁⠐⠶⠤⣄⣰⠀⠀⠀⠀⠀⢱⠄⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢰⠋⡴⠈⠀⠀⠀⠀⣠⠞⠁⠀⢁⣦⠀⠀⠀⠀⠳⣄⠀⠙⣆⠀⣠⣀⡀⠀⠀⠇⠀⠀⠀⠀⠀⡟⢀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢈⡇⠀⠀⣰⣤⡄⠀⠀⢀⣠⠴⠋⢀⣤⣄⠀⠀⠀⠘⢦⡀⠈⠳⣄⣀⠉⠙⠾⠀⠀⠀⠀⠀⣸⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠸⡄⠀⠀⠹⠟⠓⠀⠀⠀⠙⢀⡞⠉⠀⠈⣀⠀⠀⠀⠀⠹⣄⠀⠙⢯⡙⢦⠀⠀⠀⠀⠀⢠⡏⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢄⠙⢦⠀⠀⠀⠠⢿⣾⡃⠀⠈⠓⢀⣤⡾⠷⠒⢀⠀⠀⠀⠈⢧⡀⠀⠹⣏⠀⠀⠀⠀⠀⡾⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠳⣄⠀⠀⠈⠁⠀⣄⣀⠀⠈⠰⠇⣀⣴⠋⠳⠀⠀⠀⠀⠙⣆⠀⠈⢳⡄⠀⠀⢰⠇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠘⢦⡀⠀⠀⠛⡿⠷⠀⠀⠀⠸⣅⠈⠃⠀⢘⣦⠀⠀⠀⠈⡷⠀⠀⢹⠆⠀⡞⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠳⣄⠀⠀⠀⢀⣼⣶⡆⠀⠀⠀⣠⠔⠋⠈⠁⠀⠀⠀⡇⠀⠀⢸⡀⣸⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢦⡀⠀⠀⠘⠋⠁⣀⢀⡀⠀⠀⠀⠀⠀⡀⠘⣡⠟⠀⠀⣨⢷⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠹⣄⠀⠀⠀⠐⢿⣿⠀⠀⠀⢀⠤⢈⡴⠚⠁⢀⡤⠚⠁⣼⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢧⡀⠀⠀⠀⠀⠀⠠⢞⣠⠞⠉⠀⣠⠖⠋⢀⣠⠴⠃⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠖⠒⠦⣄⣋⡴⠋⠀⢀⡴⠛⠓⠶⠤⠏⠀⠀⠀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠁⢀⣠⠞⠉⠛⠲⠦⢤⣀⣀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠛⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁

⢀⠢⠁⠀⡄
⢌⡁⣃⢀⡼⠠⣶⣆⡀
⠀⠁⡉⡅⠀⣶⣿⣷⣄⡀
⠀⢔⠋⡂⠁⠌⢿⣿⣿⣿⣦⡀
⠀⠀⠉⠁⠀⠀⠀⠙⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢿⣿⣿⣿⣷⣄⠀⣀⣀⡤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠛⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⢆⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⠉⠄⠠⠑⢠⣖⡒⢤⡙⢦⣄⡀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠚⡉⠉⡇⢻⣄⠈⠉⣛⠋⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⢹⡶⠲⠦⣄⠀⠀⠀⡖⢶⠀⠘⢀⡰⠞⠊⠁⠀⠙⠦⣌⡭⠃⠀⠀⠈⠙⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⣀⣤⣄⡈⠳⣄⠀⢳⡄⠀⠰⠋⠐⢀⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠳⡜⢦⠀⠙⣦⠀⠀⠀⠀⠐⣌⡶⠚⠁⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢘⠆⣀⠘⡇⠀⠀⠀⠀⠳⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡋⠀⠙⠚⠁⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡈⢣⡀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠙⢦⠳⡄⠀⠀⠀⢀⠞⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣆⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠙⢦⡀⢧⠈⢢⣠⠖⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⢧⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⣨⣟⠉⢃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠙⢦⣄⣧⡀⠀⠀⢀⣀⣠⡴⠖⠋⠁⣘⣆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠛⠛⠛⠋⠉⠁⠀⣠⡴⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣒⣶⣤⣤⣤⣴⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⢀⣿⠋⠁⢹⠀⠀⠀⠀⠀⠀⢀⣏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣾⠀⠀⠀⠀⠀⠀⡜⠹⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⢀⣾⡿⠀⠀⢀⣠⣄⡰⠀⠀⠉⠀⠀⠀⠀⠀⡀⠠⠆⣤⡄⠰⠖⢀⡆
⠀⣀⣀⣀⣀⣀⣷⣿⣿⣁⣀⣀⣻⣿⣿⣃⣀⣀⣀⣀⣬⠠⠤⠤⠬⠞⠀⠀⠀⢉⠠⠄⠃
⠀⠉⠉⢉⣽⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⡖⣄⠰⠆⠈
⠀⢀⣰⣿⣿⣋⣹⣆⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⢤⠦⠤⠒⠒⠚⠿⡃⢈⡤⠄
⠀⣹⣿⠟⠉⢉⣽⣿⣶⣦⣄⠀⠀⠀⠀⡰⠚⢛⡎⠀⠀⠀⠀⢀⣦⠏
⠀⣿⣏⣀⣠⣿⣟⣻⣛⣿⣿⣧⣤⠤⠤⠬⠿⠿⠒⠒⠒⠋⠉⠙⡏⡆
⠀⢻⡏⠉⢹⡏⠉⠸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡤⠜⡇
⠀⠀⢳⣄⣀⣛⣄⣀⣧⣠⣼⡿⠤⠤⠴⠖⠒⠒⠛⠉⣉⣁⡀⠁⢀⡆
⠀⠉⠉⠉⠻⠭⢥⣤⢿⠖⠋⠀⠀⠀⠀⣶⣿⡆⠀⣐⣿⡿⠷⠒⠉
⠀⠀⠀⣀⣀⣀⣀⣀⣼⣧⠤⠤⠴⠶⠒⠛⠛⠋⠉⠁
⠀⠉⠉⣹⣿⣿⣇⠀⠀⣧
⠀⠀⠀⢻⣿⡿⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⣤⣴⡶⠶⠿⠃
⠀⠀⠀⠀⠉⠛⠒⠛⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⡟⢻⡟⢻⠛⠛⠟⢻⠟⢻⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⡇⠀⠁⣿⢈⣹⡇⠀⠀⢸⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣉⣁⣃⣸⣆⣿⣈⣉⣇⣸⣇⣘⣀⣷⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢛⠛⠛⢛⣛⠛⡟⢻⡟⢛⣛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⢸⣿⡆⠀⢿⡇⣉⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣇⣰⣌⣉⣁⣷⣘⣇⣈⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠐⠀⠀⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⣶⢰⣦⢰⢶⡶⠶⢶⣆⣶⡆⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⡿⣿⢿⡿⠇⢸⡿⣿⡇⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠚⠃⠙⠛⠛⠛⠚⠃⠘⠓⠛⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶⢶⣶⡶⠶⣶⣰⡆⢲⡶⠶⢶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⢸⣿⡇⠀⣿⢿⡄⢸⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠋⠛⠛⠛⠘⠛⠚⠛⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠠⠄⠀⠄⠀⠰



⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⡟⡿⢿⣿⣿⢿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠙⡆⠀⠀⠀⠀⢀⡴⠖⠒⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢒⡒⠶⣇⠀⠀⠀⢠⢿⣀⣀⣄⣀⣸⠀⠀⠀⠀⠀⢘⠀⢸⠋⠉⠉⠉⠉⡏⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⢒⣒⣋⣿⠀⠀⠀⢸⡌⣿⠀⠀⠈⡏⠀⠀⠀⠀⠀⢘⠀⢸⠀⠀⠀⠀⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠈⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⢸⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡗⠀⢸⠀⠀⠀⠀⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠸⡞⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠱⠅⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠁⠈⠹⡆⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣹⠀⠀⠀⠀⠀⡧⠀⢸⡀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠃⠈⠲⢄⡀⢀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢉⡞⠁⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠋⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⡆⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⣾⠀⠀⠙⠢⢄⡴⠋⠀⠀⠀⠀⢸⣇⡤⠤⢼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸⡆
⠀⠀⠀⠹⣦⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡗⠀⠀⡇⢀⣀⣀⣀⣇⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⡇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢰⣿⠁⠀⢀⣴⣶⡶⠾⠿⠿⣷⡀⠀⠀⠀⢠⣤⠀⠀⣿⡀⠀⢠⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠸⣿⡀⠀⠛⠃⢸⣿⠀⠀⠀⢹⣧⠀⠀⠀⠛⠻⠷⠾⠟⠿⠾⠟⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠙⠿⣦⣤⣶⣿⠷⠶⠾⠿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⡄⠀⢀⣀⣀⣀⣀⣀⣴⠶⣦⣤⣤⣤⣴⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢷⣶⠟⠋⠿⠋⠉⠛⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⠀⢀⣤⣄⣀⣤⣄⣠⣤⡄⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠾⠋⠙⢿⣦⡾⠛⠻⠟⠁⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢠⣿⣆⣴⣶⣄⣠⡤⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠏⠻⠋⠀⠉⠉⠀⢰⡟⠻⣷⣠⡾⠛⣿⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠹⠿⢁⣾⠏⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⡟⠁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣀⣀⣠⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠢⠤⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣫⣭⣭⣭⣝⣛⠿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣘⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠷⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⡟⣭⣭⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠠⣽⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣍⣉⣁⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⡻⣿⣛⣿⣿⣿⣿⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢧⠛⠤⣀⡀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣙⣥⣛⠿⢿⣿⣿⣿⡿⠿⣛⣵⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⢤⣟⣭⣭⡭⣭⢭⣭⣭⣭⣭⣿⡤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠄⢀⡄⢠⠀⡄⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⢺⣏⣿⢻⠻⢻⠛⡟⢻⣛⣿⣿⡗
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣹⡇⣿⣿⣟⣿⣻⣿⣻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠈⠢⠥⠤⠤⠤⠤⠤⠤⠤⠬⠔⠁⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢴⣚⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⡂⠨⠑⢢
⠀⠀⠀⠀⠀⠀⢀⣤⠴⣖⣯⣽⣶⣿⣿⢿⡞⣿⣆⠀⠀⠀⠀⠀⠀⡀⠄⢐⠀⠁⢀⠈⠀⠀⡀⢁⠀⢇
⠀⠀⠀⠀⠀⠀⢾⢛⣛⣛⣚⣛⣉⣃⣙⣚⣛⣙⣷⠄⠀⠀⠀⠀⠠⡃⡨⠀⠀⠁⠤⠴⠸⠦⠁⠀⢆⢘⠄
⠀⠀⠀⠀⠀⠀⢲⣿⣶⣶⣖⣶⣶⣶⣲⣶⣶⢹⡖⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠡⠉⠉⠉⠉⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣼⣏⣹⣏⣿⣸⣦⣿⢸⣏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠐⠆⠰⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠼⣿⠿⠿⠯⠿⠿⠿⠽⠿⠿⣸⠷⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⢈⣿⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⣰⣿⡿⠁⣠⡾⠁⠀⠉⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⣿⠏⢀⣴⠏⠀⠀⢀⣼⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⣿⡟⠁⣠⡾⠁⠀⠀⣠⡟⠁⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣾⣿⠋⢀⣴⠏⠀⠀⢀⣼⠋⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣿⣿⡟⠁⣠⡞⠁⠀⠀⣰⡟⠁⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⢀⣼⠋⠀⠀⢀⣼⠋⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠟⠁⠀⠀⣰⠟⠁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⢀⡾⠋⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣰⠟⢁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⡾⠋⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⣴⠟⢁⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⢠⡾⠃⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⣴⠟⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⣿⣦⣄⣠⡾⠃⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠛⢿⣿⣦⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣀⣤⣶⣾⣿⠿⠛⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⢖⣒⡢⠤⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣥⣶⠯⢃⣁⣄⠳⡄⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠒⠒⠈⠀⠀⠉⠀⠐⠒⠒⠂⢄
⠀⠀⠀⣀⠞⠙⠲⠖⠢⡀⠀⠒⠨⡳⡀⠄⠠⠀⠀⠀⠀⠀⠀⠱⠽⠿⠟⠒⠉⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⣠⢀⡀⣠⠠⠀⡀⠠⢀⣀⡄⣀⠉⠘
⠀⠀⠄⠘⣁⠦⡀⠀⠀⢹⠂⢀⡠⠘⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⡄⠃⠀⠃⣇⢀⡀⠁⢀⢠⡿⠀⠘⠀⠀⡄
⠀⢨⠂⣠⠃⠀⢀⠛⠶⢎⠀⠘⡁⠦⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⡇⢧⣀⡇⣇⣠⠧⠄⢀⣀⠿⣄⣸⠰⠀⡇
⠀⠸⣄⣍⠐⢀⣆⠀⠀⠀⢀⠞⡤⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢀⠃
⠀⠀⠣⡻⠉⠁⠈⠀⠀⠀⣀⠦⡀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢀⣀⠀⠤⠘⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⡹⢤⡉⠂⢴⣈⠒⢡⣀⠴⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⣠⠁⠀⠡⠍⠉⠩⠍⠐⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⣢⠤⠤⠄⣦⣸⠀⡐⠁⠉⢢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣌⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡔⠢⢱⡀⠀⠀⠀⠀⠀⠁⡖⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢼⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠈⠂⠀⠋⠀⠀⢠⠄⠀⠀⠀⠀⠀⠔⠄⠀⠛⠀⠐⡁⠀⠀⠀⠀⠀⠉⠡⠈⠀⠁⠉⠩⢸⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⠊⠂⡀⠀⠀⠊⠀⠀⠀⠀⠀⡀⠀⠀⡀⣀⠄⠀⠀⢀⢤⠀⠀⠀⢡⠀⠀⠀⠀⠀⡀⢰⢐⠀⠀⠂⢐⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢄⠀⠀⢀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠒⠂⠐⠉⠀⠐⠰⢠⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢄⠂⠀⢀⡠⠄⠀⠀⠀⠀⠀⠤⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠄⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠜⠤⢰⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⡅⠤⢀⠂⠀⠀⠀⠆⠓⠊⠈⠒⠀⠑⠚⠈⠒⠂⠆
⠀⠀⠀⠀⠀⡰⠋⠢⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡠⠋⠈
⠀⠀⠀⡠⢎⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠄⠀⠀⠀⠀⠤⠕⠊⠁
⠀⠀⠀⠳⢄⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣏⠲⣈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⠼⢻⡽⠋⡉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠂⠀⢠⠊⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠔⢁⡴⢙⡵⠋⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⢂⢔⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⣱⡁⠈⠀⠔
⠀⠀⠀⠀⠔⢁⡴⡙⡴⠊⢀⠀⠁⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠊⠐⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡑⠞⢳⠷⢖⣁⣀⡀
⠀⠀⢀⠞⠉⠉⠉⠉⠁⠶⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠠⠀⠐⡆⡰⠀⠐⠀⠔⠀⠀⠍⠢⡀
⠀⠰⠃⠀⠀⠀⠀⠐⡆⢀⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣀⢼⠍⠀⠀⠀⠈⠺⡄⡀⢀⠀⠅
⠀⠀⠤⠤⠤⠤⠤⠤⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠤⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠒⠁⠀⠀⠀⠀⠀⠀⠉⠂⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠋⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣤⠾⠻⣶⡀⢀⣿⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠁⠀⣠⣾⠃⣸⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⠉⠀⢀⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣄⠀⠀⠀⣸⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⢶⣾⣧⣀⠀⠀⠀⠙⠻⠶⢶⣿⠟⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣼⡏⠛⢷⣄⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣴⣟⠛⢷⡄⠀⣠⣶⠀⠀⢀⣶⣶⠀⢠⡶⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣾⢻⡿⠶⣴⣿⡾⠛⣿⠀⢀⣾⠋⣿⣠⡿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠃⢸⣇⠀⢀⡿⠀⠀⣿⡄⣼⠏⠀⠈⣿⠟⠛⠁⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣸⠏⠀⠀⠻⣶⠿⠁⠀⠀⢸⣿⡏⠀⠀⠀⢿⡄⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣀⣴⠟⠁⠀⣶⠀⠀⢀⡄⠀⠀⠀⠀⢿⡄⢰⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⢿⡀⠀⣸⡇⠀⠀⠀⠀⠈⢷⡿⠁⠀⢺⡟⠛⢻⡆⠀⠀⣰⡶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠻⣶⠟⠀⣶⢷⣆⠀⢶⡄⢀⡶⠀⢹⡷⢿⣟⠁⠀⣰⠏⠀⢸⡇⠀⢀⣠⡶⠟⠋⠉⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣶⠟⠀⠈⠻⠟⠃⠀⠘⠃⠀⠛⠀⢰⡟⠀⠀⣾⣡⣴⠟⠉⠀⠀⠀⢰⡟⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⠀⠀⢸⡇⠀⠀⣿⠇⠀⣼⡿⠋⠀⠀⠀⣀⣤⡀⠘⢷⣤⣼⠏
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⣀⣤⡀⠀⢀⣠⣄⢠⡾⠋⠀⠀⢰⡟⠀⢀⣀⠀⠀⢠⣿⠀⠘⠋⠀⠀⠀⣰⡟⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠚⠋⠀⢀⣴⡾⠛⣿⡄⣠⡿⠉⣿⣿⠁⠀⠀⢠⡿⠁⢀⣾⣿⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⡿⠿⠿⠋⠀⠀⢀⣤⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣀⣴⠟⠁⠀⠀⢹⣧⡿⠀⠀⣿⠇⠀⠀⢠⡿⠁⠀⣼⠿⣿⠀⢠⣿⣿⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠟⠀⠀⠀⣾⠃⠀⣰⡟⠀⢿⣤⡾⢹⣿⠀⠀⠀⢠⡿⠙⣿⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠉⢿⣄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⡟⠀⠀⠀⠉⠀⠈⣿⠀⠀⢠⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡿⠁⠀⠀⠀⠀⠀⠀⢻⣇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⢰⡟⢻⡄
⠀⠀⠀⠀⠀⠀⢸⣠⣾⠃
⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⢀⣾⠟⣁⣷⣤⣄⠀⠀⠀⠀⢠⠶⠚⢻
⠀⠀⠀⠘⣿⠰⡟⢹⡍⢻⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠘⠢⢭⣪⣷⠞⠁⠀⠀⠀⣤⠇⠀⠛⠃⠀⠀⠘⡳⡄
⠀⠀⠀⠀⠀⣠⣤⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠻⠿⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⠀⠀⠀⢠⣀⠀⠀⠀⠀⢸⠛⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⢸⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠘⠛⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢀⣯⣽⣿⠀⠀⢀⣀⣾⣿⡿⠛⠓⠀⠀⠀⠀⠀⢀⣴⠂⠀⣤⣀⠀⠀⠀⣠⡤⠀⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠘⣿⣿⣿⣏⠀⠀⠘⠛⠙⢻⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⣬⣷⡀⠀⢸⣟⢷⣤⣿⡀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣦⠀⠀⠀⠀⠘⣧⠀⢄⡀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠋⠹⣄⢸⣮⣿⣿⠟⣷⠀⠀⠀⢀⠀⣴⣿⣮⣇
⠀⠀⠈⢹⣿⣿⣿⣧⠀⠀⠀⠀⢻⣇⠀⢙⣦⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢨⣿⡛⠉⠀⠀⣿⠀⢀⡴⠋⠀⠈⣿⣿⣿⡆
⠀⠀⠀⣾⢼⣿⣿⣿⣷⡀⠀⠀⢴⣿⣶⣬⣿⣧⣀⠀⠀⣾⣿⡟⠀⠀⣰⣿⠇⣓⣴⣀⠀⣿⣠⠟⠀⠀⣠⣶⢟⢻⣿⣿⣆
⠀⠀⠀⣿⡎⣿⣿⣿⣿⡣⢤⡀⢸⣿⣮⣿⣶⠛⠛⢷⡀⣿⣿⡇⠀⣴⣿⠏⠀⠹⣿⣿⣷⣹⠁⢀⣴⣿⠟⢱⣿⣞⣿⡿⠋
⠀⠀⠀⣿⣷⣶⢰⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⢍⠁⠀⠈⠉⠀⠁⠀⠀⠘⡤⠀⢸⣿⣿⣫⡭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⢄⠹⣄⠀⢠⡄⠀⠀⠣⠖⠀⠀⢸⣿⣿⢹⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠚⣿⠃⢀⡴⠾⢉⣋⣉⡻⠲⣄⠀⣀⣄⠰⣿⠆⠀⣼⠁⠀⠲⡌⢳⡄⣴⣶⣾⡓⣶⣤⡼⢧⡼⠃⠀⣠⡴⠒⠚⠳⢬⣍⠳⣦
⠀⠀⠀⠀⠀⣰⠏⣣⠞⠉⠡⠈⠙⢳⡌⠛⢉⢹⡄⢀⣤⠀⢿⠀⠀⠀⢡⢨⡇⠈⠉⠛⠛⢋⡉⢻⡛⣷⢠⡞⠁⠛⠋⢻⠀⠀⣸⢷⡈⢷
⠀⠀⠀⠐⠀⣿⠃⣏⠀⠉⣠⡴⠚⢉⣀⣦⠞⠋⢷⡛⠈⣳⠌⢧⡀⠀⠀⡼⠁⣀⠀⢰⣆⢸⡿⣮⡉⠁⣾⡴⠶⠶⠦⠾⠃⢸⡅⠈⡇⢸⡇
⠀⠀⠀⢾⣷⠘⣏⠹⠶⠋⣁⣰⡶⠚⠉⠰⣶⠆⠸⠟⠳⠧⢀⣀⠙⠳⣿⡇⠀⣿⢷⣿⣩⣿⡷⣟⢧⠀⢹⡷⠛⣩⠷⣴⡄⠘⢓⣻⠇⣸⠁
⠀⠀⠀⠠⠀⣶⢋⡀⣰⠶⢋⣡⣦⢀⣤⠶⠶⣤⡴⠖⠚⢋⠉⠙⣶⢶⡀⢻⠀⣿⢻⡹⡜⣎⢧⢹⡌⢧⠀⠳⣄⣹⡄⠙⠁⣰⡿⢋⡴⠃
⠀⠀⠀⢶⣶⠹⠶⠋⠁⣴⡋⣹⢃⣞⣠⣤⣿⡎⡷⣖⢛⣙⡇⠀⢹⡈⣧⠸⣆⠸⡎⢧⢻⡸⡜⣇⢳⡘⣧⠠⠀⠉⣻⣿⣯⡭⠖⠋⠀⣀
⠀⠀⠀⠀⠀⠠⠀⢀⡞⣡⣟⡽⠃⣯⡛⠉⣀⡼⠃⠉⣉⣡⡤⠀⠈⣧⢹⡄⠘⡇⢹⡜⣇⢷⣳⡘⣆⣷⠾⣆⢺⣉⠟⠓⣆⠸⡯⣽⠆⣭
⠀⠀⠀⠠⣶⡄⠀⡼⠛⠁⣠⠞⢙⡯⠉⢻⡉⠤⠀⠛⠉⣀⣠⠀⠀⠸⡆⢳⡀⢧⡀⢳⣸⡾⣿⠋⢹⣄⡤⣿⠀⠉⠙⣉⡁⠀⡇⣿⠰⠦
⠀⠀⠀⢀⣨⠴⠶⡆⢰⡞⢁⡴⠟⠀⠀⠈⣇⠀⠀⠒⠋⢉⣀⣤⠀⠀⢻⠈⢧⠈⣷⠈⢧⣠⢾⡿⠽⠋⠉⢡⣦⠀⣾⠽⢣⠞⠁⠘⢧⡀
⠀⠀⠀⠘⢶⡖⢾⠁⠀⠙⠋⠀⠀⡿⣆⠀⢹⡀⠒⠀⠘⠉⠁⣀⡄⠀⠈⣧⠘⡆⠘⠂⠈⠙⠉⢀⡴⢚⡦⢈⡵⣤⠀⣰⢿⣋⡽⠟⠉⠳⡄
⠀⠀⠀⠀⢈⡇⠘⣧⣀⠀⠈⠀⢸⣇⠹⡄⠈⣧⠀⢁⠀⠚⠋⣉⣀⣤⠤⢾⣠⡽⢀⣤⡶⢺⠃⠙⠋⠉⠀⡼⠧⣇⠸⠧⠤⠤⠤⠶⠶⠶⠟
⠀⠀⠀⢠⡟⢁⡀⣠⠼⣷⠀⠀⣾⡀⠀⢻⡄⠸⣆⣤⠴⠖⣋⣩⡤⠖⢛⡉⠀⢀⡟⠈⣹⠏⠀⡟⠛⣛⣿⣓⠒⠛⠲⠶⡄⠀⡶⢻⡛⠓⣆
⠀⠀⠀⠘⣷⠋⠉⠛⠚⣻⠀⢠⣏⠀⣤⠀⢻⡄⠀⠙⠖⠛⣉⡀⠦⡿⠿⢃⡴⠋⣹⠋⣡⡴⢠⣇⣰⡿⠒⢾⣇⣤⡀⠀⡇⠀⡇⢸⣥⣄⣿
⠀⠀⠀⠀⠈⠳⠶⠶⠚⠁⠀⣼⡍⢰⡟⣇⠀⢷⡀⢠⡴⣿⣿⣷⠀⢀⡴⢯⣠⠞⠁⢀⣷⠀⢸⣏⣁⣀⣀⠀⠙⠀⠙⢺⡇⠀⡇⢺⡉⠋⣿
⠀⠀⠀⠈⢀⣾⣆⣴⡟⠀⢠⣯⠀⣼⠀⠸⡆⠈⢷⠀⠙⢯⡿⠁⣰⠋⢀⡴⠃⣠⡞⣿⠀⠀⣼⠉⠉⣉⡽⠏⢿⠙⠛⣻⠀⠀⡇⢼⠃⠀⣿
⠀⠀⠀⠀⢸⡇⠋⣾⠀⠀⣾⠅⠀⠧⣤⣤⣿⠀⠘⣧⠀⠄⢀⡞⣡⠖⢋⡴⠛⢯⣿⠁⣀⢀⣿⠖⠋⠁⠀⣠⠞⠀⠀⡿⠀⠀⡇⣸⠇⠀⡏
⠀⠀⠀⠀⠘⢧⠀⢿⠀⠸⣧⣤⣤⣄⣀⣀⣀⡀⠀⢹⡆⠀⠚⠋⠁⣠⠞⣵⠛⢶⡟⢸⣋⠉⠉⠛⠛⠓⠒⠻⠶⠦⠤⣧⣄⠀⣷⡛⠒⢺⡇
⠀⠀⠀⠀⣴⠚⠓⠚⠳⠦⣤⡀⠠⠀⢈⣉⣉⣉⡉⠛⠋⠀⣤⠀⡴⢯⣄⡉⣳⠏⢀⣤⡉⢩⠟⢛⡟⠓⠒⣶⠶⣤⢤⣤⡟⠀⠌⠉⠉⠉
⠀⠀⠀⠀⡟⠐⠒⠛⠓⠶⣤⠙⡶⣋⣭⣭⣭⣍⣉⠓⣦⠀⡟⢦⡙⠲⣬⣙⣏⡷⠈⠋⢠⡟⢠⠟⠀⠒⡀⢿⠀⡇⠀⣤⢶⣶⣶⣒⣲⡶⣦
⠀⠀⠀⠀⡇⠐⠋⠉⠛⠲⠆⢸⢳⣇⣀⣀⣀⣠⠟⢰⠇⠀⠳⣄⡷⢀⡤⡍⣭⣤⣄⢠⡟⢠⠏⠀⣴⣀⡀⢸⠀⡇⠀⣿⣿⡿⢏⣉⣉⡇⣿
⠀⠀⠀⠀⡇⠐⠋⠉⠙⠳⠀⡟⢨⠭⠤⠤⣭⡉⢀⡟⢀⢠⡶⠶⠦⡄⠙⠁⣿⠒⠛⠘⠓⢋⣀⣠⡟⠛⠁⢸⣀⡇⠀⣿⢨⢉⡉⡍⢥⠀⡏
⠀⠀⠀⠘⢧⡵⠶⠶⢤⣄⢸⣇⣀⣠⣄⣀⠀⠀⡾⠀⢈⣸⡏⠉⢛⡇⢠⠶⡏⢀⣴⡆⣴⠋⠉⠙⠛⢳⠀⣬⣭⡁⠀⢻⠰⠀⠂⢲⡟⡇⡇
⠀⠀⠀⠠⠀⠛⠛⠛⠛⠛⠻⠛⠓⠶⢤⣨⠟⠾⠃⠀⢯⡽⠀⢾⣹⠇⠈⠛⠁⣾⢹⠇⠻⣄⠤⢀⣠⠞⣴⣌⢧⣙⡇⢸⡈⠁⠁⠁⠙⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠉⠉⠁⠀⠉⠁⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⣠⣶⣿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠟⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣾⡟⠉⠈⠙⢿⣿⠏⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⣴⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠈⠁⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠴⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠠⠶⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⠶⠶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡶⠾⠟⠃⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⣾⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⢠⣾⢿⡆⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠿⡿⠏⠸⠃⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣤⣴⣶⣿⡿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⢉⣉⣠⣄⣉⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢠⣶⣴⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠘⠟⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⡀⢲⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣈⣉⠙⢋⣉⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠉


⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷⣄
⠀⣴⣿⣿⣿⣿⡟⢉⣠⣤⣤⣄⡉⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣄⠹⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣦
⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⠘⢿⣿⣿⡿⠃⣴⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⡈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡈⢻⡟⢁⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣤⣿⡇⢠⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣦⣈⠙⠛⠛⠋⣁⣴⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠿⠇⠸⠿⠿⣿⣿⣿⣿⣿⠟
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⡆⢰⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⣦⣤⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⣤⠤⢤⠤⣄⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⠞⢳⠂⠠⢼⠤⢼⣽⣄⠀⠀⠀⠀⢠⣿⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣼⠤⡦⠀⢀⣼⢀⣈⡁⢸⡄⠀⠀⠀⣼⡟⠠⠦⠤⢤⣠⣤⣀⡀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠈⢻⣀⣀
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠹⡀⣂⠀⠀⡇⠀⢨⠉⣹⠃⠀⠀⢀⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠘⠋⠻⠘⠃⠀⢸⠀⠉⠙⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠙⢽⣉⢹⠃⣰⣛⠶⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠤⠀⣀⣀⣀⡀⠀⠀⠀⡾⠀⠀⢰⡇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠒⠒⠒⠀⠄⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⢰⠇⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠇⠀⣖⣂⣴⢠⡒⢠⠀⠠⢄⡠⢄⣠⡀⣀⠀⢰⣿⠁⠀⠒⠒⠒⠤⠤⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠀⠀⠀⣾⠀⠀⠀⡟
⠀⠀⠀⠀⣼⠀⠐⠁⠘⠉⠂⠜⠀⠅⠏⠁⢇⡸⡗⡎⠸⠁⡾⡏⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠁⠀⠀⠀⠈⠙⠓⠒⠖⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⢸⠃⠀⠀⡾
⠀⡀⠀⠀⣧⣶⣒⣛⣛⣻⣿⣿⣿⣿⣿⣟⣻⣷⣶⣶⣶⣿⣟⣙⠓⠒⠒⠒⠒⠶⠶⠤⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⠀⠀⣸⠃
⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣾⣭⣽⣿⠉⠉⠉⠉⠙⠓⠒⠶⠦⠤⠤⢤⣤⣀⣀⣉⣉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡏
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠒⠒⠚⠛⠁

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣶⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⠀⠀⠀⠀⣟⠉⠉⠉⠉⠉⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣠⣤⠤⠶⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡉
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢌⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠁
⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠛⣻⢆⣤⣄⢀⣠⣀⡀⠀⠟⣀⣀⠀⣠⣴⡇⢠⢤⠀⣦⠞⠀⠀⣿⠢⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢯⡁⠸⣏⡉⠰⡏⡟⣷⢸⡇⣿⠉⡇⢯⠞⣇⠘⢯⠀⠻⠀⠀⠀⣯⠑⠄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠀⠁⠀⠈⠁⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡱⠄
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢆⠙
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢫⡈⢢⠁
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣾⡟⢯⠙⢄⠑⢄
⠀⠀⠀⢰⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡉⢦⠑⣄⠳⡈⠳⡀
⠀⠀⠀⠀⠀⠉⠉⢯⠻⡉⢻⡙⢟⠛⢿⠛⢟⠛⣟⠛⣟⠛⢏⠉⢣⠘⢆⠑⡄⠱⡌⢢⡙⠄
⠀⠀⠀⠀⠀⠠⡌⠆⠣⠱⠄⠱⠌⢢⠀⢣⡈⢣⠘⣆⠈⢆⠈⢣⠀⠳⡈⢦⠈⠦⠈⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⠤⠤⠴⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠛⠒⠒⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣄⠧⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠋⠉⠉⠉⢹⠉⡇
⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⡇
⠀⠀⠀⢀⡿⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⢠⠇⣧
⠀⠀⠀⢸⡄⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣀⣀⣀⣤⠤⠴⠒⠚⠉⣁⡤⠤⠚⠁⠀⢸
⠀⠀⠀⢸⡇⠙⢤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡇⠀⠀⣿⣇⣀⡤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣷⣤⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠏
⠀⠀⠀⠀⢻⣻⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠙⠳⠿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⠒⠒⠒⠊⠉⠉⠉

⠀⠲⡀
⠀⠀⠈⢆⠀⠀⠀⣀
⠀⠀⠀⠀⠑⢄⣼⡿⣆
⠀⠙⢄⠀⠀⢸⡟⠤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠑⢤⣸⠁⣠⣿⠓⠢⢀⣀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡶⠶⢶⠀⠀⠀⠀⠀⠀⢀⡀⠠⠖⠀⠀⠐⠂⠋⢀⠀⠒⠀⢠⡀⠀⠠⠄
⠀⠀⠀⠀⠀⣨⣷⣿⣇⠀⠀⠀⢸⠉⠁⠀⠀⠚⠉⠁⠰⠛⠂⠀⠀⠀⠀⢠⠄⢠⠄⠈⠀⠀⠐⠃⠉⠁⠀⠀⢈⣠⡔⠂⠀⠈⠀⠐
⠀⠉⠲⣤⣴⣿⢿⠁⠀⠉⠐⠒⢸⠄⠠⠀⠀⠐⠂⠀⢀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠃⠀⠀⠈⠁⠀⠖⠂
⠀⠀⢠⣿⠟⠓⣸⣤⣄⣀⠀⠀⢸⠉⣿⡀⠀⣀⡆⠀⠈⠉⠁⠀⣾⡷
⠀⣀⣿⡏⠀⣼⠿⣿⠻⢿⣧⠉⠹⠋⠁⠀⠐⠿⠃
⠀⠈⢹⡗⠶⢯⣤⣼⣄⣈⣿⠤⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠙⢦⣀⠁⢀⣧⠴⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠿⠋⡄
⠀⣀⣀⣀⣀⣈⣉⣉⢽⡤⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⠠⡄⠀⣤⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⠰⠆⠁⠰⠃
⠀⠀⠀⠀⣼⣿⣧⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢹⣷⠰⠀⠛⠁⠀⣀⢀⣄⠘⠋⠜⠛⠀⠀⡉⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠻⢿⣋⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡟⠀⠀⠀⠀⠈⠉⣈⡁⠀⠀⠀⠀⠀⠀⡘⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠉⠉⠙⠛⠃⠀⠀⠀⠠⠀⠀⠀⢀⠉⠓⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⣤⠀⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣐⡂
⠀⠀⠀⠀⠰⣀⠨⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡰⠐⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠐⠒⠀⠈⠉⠁⠀⠀⡠⢋⡔⠫⠐⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠑⠂⠀⠀⠔⠀⠀⠀⠀⢔⡠⠤⢀⣈⡉⠂⠀⠀⠀⡄⠒⠉⢀⡀⠜⠀⢸⢇⡀⠒⢘⡀⠀⠀⣀⣀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⣀⠀⢠⠞⠀⡆⠀⠀⠀⢠⠃⠈⠀⡠⠇⠀⠀⠉⠖⠳⣄⢀⠎⠑⠄⡀⠀⠀⠀⠈⠀⠁⠀⢨⠂⠀⠤⢤⠆⡀
⠀⠀⡀⠀⢀⣄⠀⠠⡖⠂⠈⠁⠀⠀⠊⡠⠊⢰⠀⠀⠀⠀⠙⣀⡁⠔⢳⢀⠔⠁⠀⠀⢀⣁⣀⡀⠀⠤⠙⠀⠀⠀⠀⠀⠀⠀⠡⣀⡆⠀⢁⠄
⠀⢍⠀⠑⠋⡰⠀⠀⠃⠁⡰⠂⡴⠒⠈⠁⠢⣀⡆⠀⠀⠀⠈⠂⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⠀⠀⠁⠢⠔⠁⠀⠀⢨⡔⠡⠤⠡⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⡢⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠐⡄⠀⠀⠀⠀⠀⠀⠀⢍⠈⠀⠀⠈⠁⠀⠀⠁⡩⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠤⠀⠴⠮⠑⠦⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢀⠔⠈⢉⣁⠢⢅⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⡄⠀⠠⡒⠉⡆⠀⠠⢦⠀⠀⠀⠌⠀⠑⠂⠉⠁⠔⢄⠀⢀⠤⠠⡀⠀⠀⠀⠀⠃⠃⡠⠁⣀⣍⠁⡸⠤⢀⡼⢦
⠀⠀⠀⠀⠸⠀⠀⠀⠘⠣⢄⠀⢈⡊⢀⡠⠊⠉⡇⠀⠈⠀⠀⠀⠀⠀⡔⠈⠉⠑⠃⠀⠉⠊⡄⠀⠀⠀⠀⢸⠛⠃⠄⠀⢉⠀⣠⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠈⠣⠀⠢⠀⡃⠀⢠⠀⠀⠀⠀⠐⠹⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡀⠀⠀⢸⠁⠀⠐⠤⢀⡀⢀
⠀⠀⠀⠀⠺⣹⠀⠀⢀⡠⠊⠀⠀⠀⠀⠈⢀⠀⡆⠀⠈⠄⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢁⢀⡠⠃⣀⡀⠠⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠒⠀⠀⠀⠑⠃⠀⠀⠈⢆⠀⢠⠞⠁⠀⠂⢄⠀⠀⠀⡐⠀⠀⠀⣀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡠⠢⠄⠄⠀⠀⠀⣀⡀⢀⠄⠠⡀⠀⠀⢈⡉⢱⠀⠀⠀⠀⠀⠈⠒⠞⢣⠀⠀⠀⡟⠠⢀⡠⠄⣣⡠⠘⠁⠀⠀⡎⠑⠄
⠀⠀⠀⠀⡠⢂⠨⠥⣀⡠⣂⠀⠀⢂⢆⠀⠁⢠⠐⢧⠀⠄⢰⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠀⠀⡇⢠⠈⠉⠴⠉⢢⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠀⠈⡐⡡⡂⠍⠁⢬⠞⢣⠀⠀⠑⠠⣀⠔⠣⣀⡘⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠠⠵⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣧⣀⣐⣀⡤⠐⠚⠠⢀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢀⡀⣀⣤⣀⡤⠁⠀⡆
⠀⠀⠀⠩⠀⠀⡘⠀⠀⢀⡀⠄⠘⠁⠀⢠⠒⡆⠀⠀⠀⠀⠀⠀⠀⠈⢢⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⢸⡄⠈⠁⠀⢸⡋⡤⠒⠁
⠀⠀⠀⠀⠡⠴⠅⠒⠈⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⢠⠀⠀⠁⠐⠂⠓⢄⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣐⣚⣛⠂⠂⢢⢀⡀⠀⠀⠀⢠⠃⠀⠀⢀⠃⢤⠠⠌⣏⢘⣦⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⢹⠀⢀⣀⣀⣀⡂⡨⠍⢱⠀⠀⠀⠚⠄⠀⡄⠘⡀⠀⣶⡨⠄⠑⢌⠀⠀⠀⠀⠀⠀⠀⠤⢀⠠⢠
⠀⠀⠀⠀⠀⡠⠔⠲⠀⠀⠀⠀⠀⠉⠂⢠⠄⣀⠄⠪⣀⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⢥⡬⠮⣴⠒⠈⠉⡇⠀⠀⠀⢀⠀⠗⠀⢆⠀⠸
⠀⠀⠀⠀⠀⠈⠢⠎⠀⠀⠀⠀⡆⠀⡠⠁⢀⣀⣀⡀⠀⠁⠴⠂⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢤⠀⡇⠀⠀⠀⠐⢤⡀⠀⠮⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣀⣀⣀⠀⠀⣀⣀⣱⡄⠀⠀⠀⠀⠀⠈⠁

⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣿⡦
⠸⣿⣿⣿⣿⣯⣭⣯⣿⣿⣿⣿⣿⣿⣿⠶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⡇⢰⠀⢰⡔⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⠇
⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠓⠈⠒⠘⠈⠂⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⣿⠗
⠸⢿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⢚⣛⣛⡃⠀⣿⡇
⠀⠈⣿⣿⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢘⣙⣉⡃⠀⣿⡃
⢀⣼⣿⣿⣿⣯⠍⣻⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠠⠀⠨⠭⠭⠅⠀⣿⡋
⢸⣿⣿⣿⣿⡷⣤⡜⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡆⠀⣿⣇
⢨⣿⣿⣿⣿⡿⢛⣫⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⢰⠁⢰⣔⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⡅
⢨⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⠈⠈⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⣿⡅
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣦⣄⠘⢻⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⣠⣴⣶⣶⣤⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢀⣠⣾⣟⣁⣀⣀⡀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢈⣿⣿⣛⣛⣛⣛⡃⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⠀⠀⠀⠀⠀⣼⣿⡿⠀⠘⠛⣿⣟⡛⠛⠛⠃⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢾⣿⣶⣶⣶⣾⣿⠟⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠈⠛⠿⠿⠿⠟⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠉⠙⠿⠟⠋⠁⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠁⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠛⠁⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠙⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡀⢀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣠⠤⠆⠒⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠑⢤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠕⠀⠄⠁⡠
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢀⢠⠌⣄⠐⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⡰⠃⠀⠈⠀⠐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠠⠁⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠔⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠐⠀⠡⠀⢪⠀⠐⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠢⠊⠀⠀⠀⠀⠀⠂⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠘⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠀⠁⠀⠀⠀⠀⢠⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⠀⢐⡰⠛⠀⢅⢄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠌⣌⡈⠌⠀⠈⢐⡠⠂⠉⠀⠀⠉⠁⠚⠂⠦⠤⡠⣀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢚⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠰⠤⠄⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠲⢄⡇⣠
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⢄⡄⡰⣢⢦⠒⢂⣠⡾⣽⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀⢄⡄⡠⣠⣠⠄⠌⠑⣁⡤⡶⣿⡵⠻⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢄⢊⠆⠁⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⡀⠀⠀⡠⢂⠌⠔⡡⢁⠊⠀⠀⢁⣀⡤⠔⠚⠀⠄⠐
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⡐⠁⠀⠀⠀⠀⠀⠠⢀⠌⡐⡡⠁⠌⡐⢁⠔⡠⠂⠊⠀⣀⠀⠴⡰⠃⠉⠊⠈⠀⠈
⠀⠀⠀⠀⠀⢂⠀⠔⢠⠎⠀⠀⠄⠄⠄⠔⡐⠁⢀⠊⠔⡀⠂⠔⠀⠂⣀⠄⠖⡐⡡⢁⠈⠄⠐⠁
⠀⠀⠀⠀⠀⠀⡀⢵⠃⡠⡀⠠⠂⢂⢂⠌⠰⠁⠁⣈⣀⡤⠴⠒⠛⠡⢀⢂⠊
⠀⠀⠀⠀⠀⠀⠳⣀⢐⡀⣀⢁⠅⡤⠰⠂⠂⠘⡡⠊⢊⠊⠄⠀⠐⠀⠁
⠀⠀⠀⠀⠄⡨⢐⠁⢊⠔⠀⠠⢊⠄⠐⠁⢊⠄⠀⠁⠁
⠀⠀⠀⠀⠊⠐⠁⠀⡁⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢤⠄⠀⠀⠀⢀
⠀⢈⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠤⠤⠤⠴⠦⠤⠦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣭⣤⣤⣬⣼⠀⠀⠠⠤⠶⠶⠶⠶⠶⠶⠄
⠀⢀⠀⠀⠀⠀⠀⣀⣀⣆⣀⠀⠀⢠⠀⠀⢀⣏⡛⠨⡿⠿⠫⢦⠀⠀⠀⠀⣀⣴⡶⠀⠀⠀⠀⠀⠑
⠀⠘⡀⠜⠛⠛⠛⠋⠛⠋⠉⠒⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀
⠀⠀⢀⠰⠄⠂⡆⡰⠂⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠐⠩⠂⠁
⠀⠀⠀⢢⠸⣽⣷⠁⠀⠀⠀⡄⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣴⠇⠀⠀⢀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠇⢀⡇⠠⢸⠂
⠀⠀⠀⠀⡨⠐⠀⠀⠀⠀⠀⠀⢨⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠰⠇⠀⠀⠂⠀⣸⣻⣿
⠀⠀⠀⣠⢦⡀⠀⠀⢀⡤⠀⠐⠀⠀⠐⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠄⠀⠀⠂⠀⠺⠹⠚⡇
⠀⠀⣰⢇⡲⣌⠓⢤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢰⠀⠀⠁
⠀⠓⠭⡺⢴⢃⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠀⠀⢠⣀⠀⠤⠀⠠⠄⠐⡄⠀⠀⢸
⠀⠀⠀⠈⠐⠶⠃⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠋⠀⠀⠀⠀⢻⠇⡩⠠⡪⢋⡻⠁⠀⠇⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠂⢘⠶⡣⠀⠀⠀⠓⠃
⠀⠀⣈⠗⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢢⣾⠅
⠀⠀⠀⠀⠊⠄⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⣠⣴⣄⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢨⠾⠛⠉⠀⠀⠀⠀⠿⣤⡿⠁⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⡎⠀⠀⠀⠀⣀⡤⠀⠸⡇⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠴⠀⣀⡸⠀⠀⣠⠖⠋⡁⠠⠆⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⢀⡶⠀⠀⠚⢀⠀⠘⠟⠃⠀⠈⡀⠴⠀⣡⠤⣤⡤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠌⢆⠀⢾⠁⠀⡾⠀⠀⢠⡄⠀⠀⠄⠈⠁⠀⣀⠡⢮⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⠉⠠⠀⢁⣦⠀⠀⠀⠈⠁⠀⠂⠂⠐⢏⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡠⡌⠀⠐⠁⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡤⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠐⢠⡄⠀⠀⠀⠀⡠⠛⠛⠳⢶⠄⠀⠀⠀⢻⣄⠛⢶⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠤⠀⣠⠀⠀⠀⠀⠺⠷⠃⢀⣤⣠⠂⠀⠀⠀⠀⠀⠉⠋⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⠆⡞⣵⡢⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⢀⣤⡄⠘⢃⡙⠀⢉⡎⠁
⠀⠀⠀⠀⣀⡀⢠⡀⢠⡞⠁⣄⠁⢄⡈⠉⠀⢀⠈⠁
⠀⠀⠀⢀⡉⢩⠉⢉⠉⣃⠘⠋⠀⠀⠶⠀⠠⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⣢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⣏⣋⢳⠀⠀⠀⠀⠀⣠⣤⡴⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠂⣄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⣦⣈⡭⠓⣄⠀⠀⠀⠀⠉⠀⠀⣀⡇⠀⠀⠀⠈⡒⢭⡀⢀⡉⡤⢀⣠⠀⢀⡴⠁⠴⢀⡀⠒⠀⠄⠤
⠀⠀⠀⠀⠀⠉⠀⢠⠖⣒⠦⠐⠀⠀⠀⠀⠛⠀⠀⠀⢴⡾⠀⠈⠛⠁⠀⠀⠀⠀⠙⠚⠱⠎⢠⠀⠛⠛⠀⠈⣀⠛⠀⠈⠁⠛⠠⡤⣀⢚⡀
⠀⠀⠀⣤⠀⠀⠰⠆⠋⠒⣀⣤⠀⠀⠀⢠⡁⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡶⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⢠⡄⢀⡄⠀⣀⡴⠚⢁⠀⠖⠀⠀⠀⠀⠀⢛⠁⠀⢀⡀⠻⠃⠐⠂⠀⡄⠒⠒⢀⡙⠆⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠘⠃⠀⢉⠠⠆⠉⠀⠀⠀⣠⠾⣷⠄⠀⠀⠀⢛⠁⢀⣴⠄⠀⠄⠱⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠶⠀⠘⠁⠀⠲⠟⠁⢰⢨⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠀⠀⠀⠡⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⡖⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠉⠉⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⢩⣄⡀⠀⣼⡇⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⢸⡆⢿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⣾⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠁⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛⠳⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⢀⣶⠀⣼⣧⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣀⣀⣀⣠⣴⠟⠁⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⢸⣿⡏⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠿⠇



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