⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠚⠺⣺⢮⣻⠀⠀⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣞⡾⣦⡤⣖⢯⣫⢯⣞⡯⣷⣫⣗⣗⣾⣳⢷⡯⣿⣷⡾⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡽⣽⣻⡽⣾⣞⣯⣟⢞⢮⡳⡯⣿⡺⣝⣗⡷⣳⣟⣷⣻⡽⣯⢷⢷⣻⣽⣻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⣯⢿⡽⣾⣻⡷⣿⢝⡷⣝⢵⢝⡞⡮⡮⡳⣵⡻⣟⣾⣳⡯⣿⢽⣻⡽⣗⣿⣺⣟⣾⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⡽⡯⣿⡯⣷⢻⢝⡕⣟⢮⢯⣳⢝⡵⣯⢿⠵⣝⢮⡳⣯⢿⣽⢯⡗⡟⣽⣾⣷⣻⣞⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣳⣿⣪⡳⣝⢮⡳⡳⣝⢎⣗⢽⢜⣽⡹⣜⢮⡫⡮⡳⣝⢮⡳⡽⣝⣗⢷⢽⣺⢽⣟⣾⣫⢟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣞⢿⡺⣕⢯⡺⣕⡯⡯⡮⣳⢕⢯⣳⢽⣺⣪⡳⣝⢭⢯⢮⡳⣝⢞⢮⢮⡫⡯⡺⣝⢮⡳⣯⡯⡇⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣳⡽⡳⣝⢮⡳⣝⣮⡯⢯⢚⣞⢙⢉⠊⡠⠡⠉⠍⡋⡓⢓⠳⢽⢽⢽⣕⢗⣝⢮⣻⣺⢽⣪⡳⣝⢽⡕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠸⡽⣽⣪⡳⣝⢮⠎⠌⡐⡈⠏⡃⠄⡂⠂⠌⠌⢾⠻⢐⠐⡈⠄⡑⡑⣯⡳⣕⢯⡺⣞⢵⡳⣝⣯⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡺⣝⡽⡮⣲⢝⢮⡳⡅⣅⣦⣬⣨⠐⡐⠠⠡⢁⠊⣄⣬⡴⡴⣴⣅⡐⡀⢺⡺⣪⡳⣝⢞⡵⣝⢮⡺⣮⡫⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢵⢝⢮⣫⡳⣝⣝⢿⠅⠂⣮⡗⠄⠅⢊⢀⠂⠞⠁⣾⣳⡌⢟⠋⡀⣺⢝⢮⣺⡪⣗⣝⢮⢳⢝⢷⡹⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡳⡵⣝⢷⢽⣔⢀⠂⢽⣻⠈⠌⠠⠂⠌⡀⠠⣟⢾⢃⠠⠨⢠⢗⣽⢵⡳⣝⢮⢮⡳⣝⣽⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⡴⣲⢽⢕⡯⣮⡫⡆⡅⠕⡰⢐⠈⠌⣌⠨⠠⢁⠂⢄⠨⠨⠢⡑⠅⡋⢍⡼⡯⣫⢮⣳⣳⢿⣺⢺⡪⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣪⣗⢽⢽⢽⣻⢾⣌⠄⠅⠌⡐⡀⡊⠄⠡⡈⠄⡂⠅⠅⠂⢅⣢⡽⣝⣞⣮⢷⣻⡺⣝⢮⡳⣝⢮⢯⡻⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢧⣗⣯⢯⣻⣺⣳⢽⣳⢮⣔⣀⠢⠰⡈⡐⠠⢁⣐⢨⣴⣳⣻⣺⣻⣼⣺⣳⢯⣞⡽⣮⡳⣝⡮⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢽⡳⠷⢻⡽⣺⢽⣺⢽⢽⣳⢖⡮⡪⣧⢮⡻⡪⣞⢾⣺⣞⡾⣽⣺⣳⣳⣻⢵⣉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡯⣯⠳⢯⣻⣺⣽⣿⣲⣼⣟⣯⣟⣾⣻⡽⣞⣯⣷⡳⡗⠷⠽⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣸⢌⢏⣛⡅⢾⡫⣿⡽⣞⣯⢿⢾⣺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⣪⣑⡕⡤⣝⡷⣟⣯⢿⡽⣞⣟⣯⢷⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣯⢿⡽⣾⣺⡯⣟⣯⢿⣽⢯⡿⣽⢯⣟⡾⣯⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣽⢿⡽⣗⣯⢿⡽⣞⣯⡿⣯⢿⣽⣻⣞⣯⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡽⣯⢿⡽⣞⣯⢿⡽⣞⣿⣽⣻⣞⣷⣻⡾⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠝⡫⡻⠽⡻⡞⠏⢿⢹⢷⣗⣿⣺⣷⣻⡽⣗⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣱⡦⣮⢮⣬⣮⣵⡵⣞⣯⢷⡯⣿⣺⣗⣿⢽⡯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣯⣿⣽⣯⡷⣿⣞⣿⢿⢺⠻⢝⢗⢛⢞⢿⠻⠟⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣻⣾⣳⣯⡿⣷⣻⣽⢏⢆⠣⢃⠪⢘⠸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣞⣯⣷⢿⣯⢿⣞⠁⠔⢈⠄⠌⠄⠌⡰⠁⠀⠀⠀⠀⠀⠀⠀⣸⢝⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢹⣯⡷⣟⣯⣿⣻⠐⡁⠌⠄⢂⠡⠨⢐⠜⠀⠀⠀⠀⠀⠀⢀⡴⣫⡳⣝⢾⡀⠀⠀⠀








⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣾⣿⣇
⠀⠀⣠⣴⣦⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢳⣤⣄⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢻⣿⣿⣦⣀
⠀⠀⠀⠙⠻⢿⣿⣿⡿⠀⠀⠀⠀⢰⠦⢤⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠥⠀⠀⠀ ⠀⠈⣿⣿⣿⣿⣷⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠛⢿⣿⠿⠋⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⢠⣾⣿⠀⠀⠀⠀⠘⠁⠀⠀⢰⣿⣿⣿⣧⠀⠀⠀⠀⣀⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠈⠀⠀⣰⣿⣿⣿⣷⣦⠀⠀⠹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⡀⠀⠀⠀⣀⡀⠀⠀⠙⠓⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⢀⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠓⠲⠤⢤⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠾⠟⠻⠿⣿⠿⡿⢿⣿⣿⡇⠀⢸⣷⡸⣿⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣶⠀⠋⣠⣠⣾⣿⣿⣿⢀⣾⣿⣷⠀
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