Hindutva Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡆⠰⠀⣤⢜⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣹⡄⢲⡟⣓⢂⣹⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡮⣉⢈⣣⠽⠅⠤⠀⠹⢦⡒⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠳⠦⠤⢤⣝⣋⣈⣉⣋⣛⣺⡁⣈⣤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡟⣫⠶⣒⢢⣽⣥⣧⣿⣤⣧⣻⢭⡤⠤⢚⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣶⣿⡿⡛⠯⠉⢉⠩⠝⡻⣿⣟⣻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⠐⠈⢻⣤⠁⣀⡾⠛⢧⡃⢸⣌⣛⠫⡥⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠿⣿⠆⠀⣾⠝⠗⠛⠃⠀⠚⠛⢺⢟⡧⢀⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⣻⣤⠟⠀⠆⠀⠀⠀⢡⣿⡄⠀⣀⣐⣂⠀⣾⣯⠁⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⡠⠴⠊⡷
⠀⠀⠀⠀⠀⠀⠀⣠⢀⡠⢋⢀⡴⠛⠉⠀⠀⠀⠈⡶⢀⡠⢠⡧⢫⣀⠜⠙⢁⣠⠋⣍⠣⡀⢶⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠀⡟⢇⣠⣚⠏⠀⠀⠀⠀⠀⠀⠀⣡⢀⠰⠁⣠⡎⠙⠦⣠⠞⠙⢦⡈⢦⣀⣴⠀⠀⠀⠀⠀⠀⠀⡠⢫⠇⠀⢸⡚
⠀⠀⠀⠀⠀⠀⠀⣧⡼⠟⠁⡄⠀⠀⠀⠀⠀⣀⠨⢏⣛⡟⠁⢨⡀⠀⠀⠀⠀⠀⢴⠙⠖⢷⣼⠀⠀⠀⠀⠀⠀⢰⠀⠸⡀⠀⢸⡁
⠀⠀⠀⠀⠀⣠⠞⢻⣦⠤⡎⠀⠀⠀⠀⢠⠎⠁⠀⣈⣾⣏⠀⠈⢣⣀⠀⠀⠀⢀⠎⣠⢈⡿⠄⠉⢦⠀⠀⠀⠐⣍⠁⢀⣇⠠⠴⠃
⠀⠀⢀⣴⠟⠁⠀⠈⠀⠀⢀⠀⠀⠀⢠⠃⠀⠀⠀⡤⢹⣫⠄⠀⠀⠈⠻⡿⠛⠁⢀⡄⢸⠗⠆⠀⠘⡇⠀⠀⡜⠀⠀⢻⠁
⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠸⠀⠀⠀⡎⠀⠀⠀⠀⡶⢈⣏⠄⠀⠀⠀⠀⠀⠀⡐⠁⠆⣹⣯⠂⠀⠀⣇⡠⠊⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢃⠀⡞⠀⠀⠀⠀⡰⢛⠠⡧⠕⠒⠒⠒⠊⢀⠼⠄⢴⠠⣼⡏⠀⠀⠀⢹⠁⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢈⡄⠑⠀⠀⠀⢴⠁⢨⡄⢳⡣⠀⠀⠀⡰⠃⠀⠀⠈⣐⣟⢻⠀⠀⠀⣈⡇⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡜⠀⠠⠤⣀⡝⠁⠀⠈⡖⢩⡑⡀⢀⠔⠁⠀⠀⠀⡆⣬⣻⠁⣗⠌⠓⠈⢙⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⣼⠀⠀⠀⠀⠀⢳⡀⠀⢠⠋⢒⣚⣥⠀⠀⠀⠀⠀⠐⠡⡘⣇⠀⢨⡄⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠰⠋⠙⢆⡆⠀⢌⢯⣆⠂⡀⠴⠀⠠⢠⣖⠇⠙⡄⠀⢣⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢥⣦⣀⠈⢏⢵⣄⡤⠘⡃⠞⠰⠧⠿⢊⠋⠀⣠⢇⠀⠈⣧⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠙⣱⢞⣒⡋⠈⠂⠠⢀⠁⠀⢁⠠⡔⠉⠀⠈⢳⡄⠘⢆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠼⠁⠀⠄⠀⢺⠔⢫⡀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠈⠀⠀⠛⠦⣈⢣⡀⢠⠴⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⢀⣀⣀⣤⣠⣄⣠⢆⣸⡅⣀⣤⠀⠦⠄⠉⠉⠲⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡀⢨⣉⢳⡄
⠀⠀⠀⢀⠔⣊⠔⠋⣀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⠀⠂⡆⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⢀⡝⠢⡽⠋⣅
⠀⢀⡔⠁⡜⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢀⡀⠬⠧⠳⠖⠶⢤⣄⣀⡀⠀⠈⢻⣿⡉⠉⠉⠁⢾⠀⠀⠘⡷⡟⠑⢤⡀
⠀⡞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠒⢉⡀⢀⡠⠤⠤⠤⢼⣂⠀⠀⠈⠑⠒⠒⠀⠓⠀⠀⠀⠈⠳⠶⠒⠁⠘⡄⠀⠑⡄
⠀⣧⠀⠀⠙⠤⢤⣀⣀⡠⠴⠒⠒⠒⠋⠻⠥⠴⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡄
⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠤⣤⣄⣀⡀⠀⠀⣀⠼⠁⠀⠀⢸⠃
⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠠⣪⠎
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣀⣀⠂⠀⠠⠤⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣠⣴⣊⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠀⠤⠤⠤⠤⠤⠄⠀⠀⣀⣈⣉⡁⠀⠬⠬⠤⠤⠤⠤⠒⠒⠒⠊⠉⢙⡏⢩⣷⠦⢮⣤⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⡃⠀⠠⡃⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢉⣿⡌⢉⣽⠎⠉⢹⣤⣍⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⣼⣧⣶⣿⣶⣶⣶⣾⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡏⠉⢹⡏⠉⠉⠉⢿⣏⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣴⣬⣄⡄⠩⠴⠶⠶⠶⢿⠩⣄⣬⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠄⢿⣶⣤⣀⡄⢠⡿⣆⣾⣿⣦⣾⡿⢿⣆
⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⢠⣈⢀⣶⣄⣼⠛⡃⢀⣠⣀⣧⣠⠟⠿⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⡀⣠⣾⣿⠁⠀⠀⠀⠀⠹⢾⣿⣿⣅⡀⢐⣸⠿⢿⡿⢿⡘⠜⣸⣿⣧⡀⠀⠀⠀⠀⠀⠀⢤⠀⣷⣶⣄
⠀⠀⠀⠈⠛⠿⣩⠀⠀⠀⠀⠀⣠⢨⣿⣿⠹⠽⠉⣿⠶⢴⣾⡿⠙⣿⣏⢻⡏⣿⣦⡀⠀⠀⠀⠀⢻⣟⢛⣛⠇
⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠿⣟⢸⣿⣿⠁⠀⣠⣟⣆⣁⣨⡔⢸⣿⡧⣾⡏⣻⣿⣷⡆⠀⠀⠀⢼⣿⡞
⠀⠀⠀⠀⠀⠰⣞⠿⣦⡄⠀⠀⠈⠸⣿⣿⡄⠱⣿⣿⣻⠃⠙⡆⢸⡉⣹⣿⠁⣿⣿⣿⣷⠖⠀⡐⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢿⡧⠉⠀⣀⣤⠶⠞⢿⠿⣧⠉⣗⡠⠋⠀⠁⠁⣾⣿⣿⣿⣤⡿⠟⢛⣛⠃⠀⢸⡿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣥⣤⣤⣤⣴⣿⣷⣾⠟⣷⣛⡴⢂⣤⠘⣿⠛⠉⠛⣾⣶⣙⣷⣴⠏⣀⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠔⡱⣹⣿⣸⡿⣹⣿⣿⣿⣿⣿⣹⠁⡿⠉⣴⣋⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡤⢊⠔⣰⢿⣿⡟⣱⣿⣿⣿⣿⡟⡻⣷⢆⡴⣾⠿⣿⣿⣿⠟⠀⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠀⡤⠊⣴⢣⠟⠉⣸⣿⢻⣿⣿⡿⢓⡤⠚⠋⠀⠶⠾⠟⠛⠁⠀⠀⢺⣞⢿⣿⠛⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠃⠀⡾⣡⠀⢀⣼⡿⢫⣿⡿⠋⠻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠻⡟⠂⠀⢻⣿⣿⣿⠉⢉
⠀⠀⠀⠀⠁⣴⠟⢡⣾⣿⣡⡿⢟⠀⠀⠀⠘⢿⣿⣳⣶⠅⣰⣆⣤⠀⠀⠀⠀⠀⠻⣄⢷⡄⢀⣸⣿⠋⡝⠀⡌
⣀⣀⣀⣠⣾⡏⣰⣿⣿⡿⠉⠀⠈⠳⣀⠀⠀⠀⣸⣿⣿⣶⣿⠿⠿⣿⡷⣷⣶⣶⣶⣿⣌⣿⣿⣿⣿⡟⠀⡠⣹
⣿⡟⣻⣿⢋⣴⣿⠏⢿⡇⠀⠀⢀⣴⣿⠳⣤⡠⡿⠛⠛⠁⠀⠀⢀⣠⣴⠾⠋⠀⠈⠉⢻⡾⢿⡈⢽⣷⣾⣷⣿
⣾⣷⣾⣱⣿⣿⣿⣄⣈⣷⡄⢠⠋⠹⠿⠘⠀⠀⠀⠉⢉⣉⣭⡿⠛⠁⠀⡀⠀⠀⠀⠈⣿⣿⡌⢿⣿⣿⣿⣿⣿
⠉⢉⣩⣿⣍⣩⣹⣅⣸⠂⠙⠛⠿⠒⢀⣤⠀⠚⠚⠛⠛⠉⠀⠀⠀⠀⠀⠘⣦⠀⠀⢠⡟⣿⣿⣽⣿⣿⣿⣻⣿
⠆⣿⣬⣿⡏⠉⠉⠉⠉⣤⣔⣠⡶⣶⣿⠋⠀⠀⣴⠀⠀⠀⠀⣼⣦⣄⣀⣼⡟⣿⠀⢩⣥⣿⠋⠉⠉⠸⣿⣿⣻
⠀⠈⣤⣽⡇⠀⠀⠀⠀⠀⠉⢩⣤⣹⣏⠓⣒⣻⠟⠱⢤⣴⠞⣡⠟⢩⣝⣋⣰⣃⣀⣸⡏⠋⠀⠀⠀⢀⣿⣿⣿
⠀⠠⠉⠿⠅⠀⠀⠀⠀⠀⠀⠈⠟⠻⣿⡟⠋⠱⠶⠟⠲⣶⡞⠛⠲⠟⠉⠛⠛⠉⠛⠛⠃⠀⠀⢀⠀⠰⡧⠼⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢴⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡿⡟⢿⡄⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣄⡨⣊⣙⣤⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⣾⣯⢶⣿⣿⣿⣷⣿⣿⣦⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣴⣿⣻⣿⣾⡯⣸⣬⣇⣷⡟⢲⢽⣿⣷⣿⣦⡀⠀⠀⣤
⠀⠀⠀⠀⠈⣠⠚⣳⠄⠃⠀⢠⣿⣉⣿⣍⣿⣞⠛⠹⢯⠙⠓⢼⡟⠙⢏⢉⡻⠇⠰⣺⡆⢳⡆
⠀⠀⠀⠀⠀⠟⢠⠏⠁⠀⠠⣥⠈⢄⡈⠉⠿⡷⠾⢂⡶⠶⠠⣿⢃⠉⢀⡦⠁⣸⠄⠈⠲⣈⡃
⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⢹⡀⠘⡀⣰⡞⠻⡄⢸⡗⢀⡾⠿⣾⡀⢘⡁⡈⠃⠀⠀⠀⠘⢡⢤⣄⣠⠄
⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣀⢖⣻⡴⣡⣝⠣⠜⢯⣈⣀⣽⣧⡤⢷⣜⢡⡈⠱⢄⡀⠀⠀⠀⠘⢺⣿
⠒⠮⣀⡄⢨⡀⠀⠀⠀⢴⣷⠋⡜⢲⣿⠇⣀⣼⡆⠠⠄⣠⣿⢳⢰⣾⢂⡁⠈⢌⣣⡀⠀⢀⣤⠀⣫⠓
⠀⠠⠋⣄⣨⢷⠀⠀⠘⣬⠿⠤⣍⢽⣿⣠⠻⣽⡙⢓⠚⢉⡏⢘⡸⣿⡢⠖⠀⢹⣐⡹⠀⠘⠙⠺⡏
⠀⠀⠀⠙⠄⠀⢈⡶⠾⠛⡛⢿⡧⢺⣿⠈⠀⢺⡄⣤⠈⢸⡇⢠⢿⣿⡍⣥⡤⠴⣿⣧⠶⢤⡀⣰
⠀⠀⠀⠀⠀⠀⣿⠂⢀⠴⠛⠚⠛⠋⣷⡆⠐⢋⡍⠀⠀⢸⢁⡎⢰⡿⠖⠷⢺⣿⠞⠉⠁⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠙⡁⠀⠀⠀⠀⣀⠴⡝⠎⠹⡇⢀⠚⠨⠉⢿⠙⠁⠀⠀⠀⠀⠀⠉⢱⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢹⣿⣿⡴⣠⢤⣶⠂⣇⠀⣦⠀⣴⡟⠛⣷⡆⢰⡤⣤⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣌⣿⠙⣟⣱⣾⣿⠀⣯⠀⢿⣠⡏⠑⠀⣿⡗⠺⣠⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡎⢢⡉⣿⠈⢅⣹⣿⡏⠻⢾⡤⠼⠛⢃⠔⠁⣿⣿⠒⡅⢻⣿⡏⠿⣄
⠀⠀⠀⠀⠀⠀⢠⠞⢀⠂⡘⡇⠰⠤⣾⣿⡧⠀⠀⡙⡶⣄⡀⠀⠀⢹⣿⡒⠃⠘⣿⣧⣀⠈⠢⡀
⠀⠀⠀⢀⣀⠐⠁⡀⠀⠐⡐⢀⠷⣲⣿⡏⢦⡀⠀⢀⢄⠀⠘⠷⣤⠚⣿⣿⠛⠀⠸⡏⠃⠀⠀⠹⡷⣤⣀
⢀⠀⡄⢮⠑⠀⠀⠀⠀⠐⠀⠠⠤⣿⡟⠁⠀⠙⢶⣄⠀⠀⠀⠈⠀⠑⢼⣿⣏⡹⠀⠘⢆⠀⠀⢨⠓⣿
⠘⡀⠘⠀⢣⡀⠀⠀⠀⠀⠠⢀⣾⣿⠑⡀⠀⠀⣀⡼⣿⣷⣀⣌⠀⠀⣹⢿⣿⡄⠷⠀⠀⠳⣀⡎⠐⣿
⠀⠀⢄⠈⢢⡙⢴⠋⠀⠐⢆⣾⣯⠻⢈⡴⠖⠏⡙⣶⣿⣿⠆⢉⢶⣤⡎⠈⢿⣷⡐⢃⠀⠀⠘⢅⢸⡇⢀
⠀⠀⠀⠳⢄⠘⠏⠀⠐⢠⣚⣿⠃⢸⠅⠐⠈⠠⣽⠻⡿⢹⡎⠀⢒⣩⡇⠀⠘⣿⣿⠉⡆⠀⠀⠈⡿⠂⡞
⠀⠀⠀⠀⠀⠘⢀⠈⢳⣟⣾⠃⠀⠘⡥⠂⣠⠎⡈⡀⡇⠠⠹⠄⠀⢸⠁⠀⠀⢸⣽⣯⡨⢀⠀⠀⢘⠜
⠀⠀⠀⠀⠀⠀⢂⣼⡙⠹⠃⠀⠀⠀⣧⡔⠁⠠⣄⣁⣧⣤⣴⡞⣭⡾⠀⠀⠀⠀⠻⣾⣆⠈⣄
⠀⠀⠀⠀⠀⢠⡼⢾⡎⢠⡄⠀⠶⢈⡣⢤⡉⠐⠚⠃⠲⠚⢻⣟⠥⣥⢆⣀⠒⢠⠀⢽⣿⣧⠠⠃⡀
⠀⠀⠠⢀⢰⣿⡟⠡⣄⢣⡤⢉⠲⢈⣀⣀⣈⡍⣺⠚⢻⡟⠛⠷⣄⠦⣄⣉⠒⢢⡄⢠⠿⣟⣓⡂⣡
⠀⠀⠁⠂⢼⣯⣄⠁⣺⠀⡶⢍⡰⠂⣭⠀⠀⢀⣿⢠⢸⡛⠤⣄⣸⠇⡤⢉⡰⢢⡄⡞⢀⡙⢋⡉⠁
⠀⠀⠀⠀⠀⢼⢃⠀⢽⠀⡑⠂⣄⠓⢴⠀⠈⢁⡿⣌⡸⢁⠀⣀⣹⠓⠢⢎⡰⢫⡄⢐⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠈⠁⣀⡇⣈⡁⠂⢤⠛⢀⡉⠀⢨⡠⢉⡶⢈⠀⠀⣏⠘⠑⠎⣀⠛⢆⠈⣤⠀⠋
⠀⠀⠀⠀⠀⠀⢀⣜⡁⠉⠉⠂⠀⢀⢀⠀⢀⢨⡀⠈⢀⣄⢈⣈⣀⣀⣉⣳⣌⠀⣈⡉⣀
⠀⠀⠀⢀⡀⣐⣉⠐⢁⣎⣰⣊⣿⣋⣈⣛⣳⡞⢹⡆⣴⠉⣿⠉⣿⢉⣿⠝⠯⢙⡏⠛⠙⠓⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡥⢤⡤⣨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡲⢗⡶⣖⣲⡽⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰⡔⣷⠄⠐⡁⠐⢶⣿⣿⠃⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠂⠀⢤⠀⠀⠈⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⢀⠈⢄⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠆⢀⡥⠘⠉⠀⡀⠀⢰⡉⠁⠛⠉⣹⠂⠀⠀⠀⠈⠀⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠁⠀⢸⠀⠉⡿⠉⠁⠐⠁⠀⠀⠀⠀⠀⠐⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣤⠴⠆⠀⠘⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠑⠀⠀⡆⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠘⠜⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣰⠀⠀⠒⠀⠀⠀⢸⣈⣹⠄⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠰⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣦⣤⠴⠟⣡⡼⢡⡆⠀⠀⠂⠀⠀⢀⠀⠀⠖⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⠚⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠇⠀⡀
⠀⠀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠛⠛⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⡠⠂⠀⠠⠼⠇⠀⠠⠯⠀⠀⠀⠈⠀⠈⠙⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠴⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠦⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⣀⣤⣤⣌⣤⣤⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣤⣶⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠄⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⡿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⢀⣉⣉⣩⣙⠉⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠁⣴⠟⠋⠉⣙⣿⣷⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡐⠀⢀⣳⢰⡿⠿⣿⣿⣽⣧⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⡄⠙⠋⡀⣹⣄⣿⡟⠻⣄⢹⠷⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣁⠀⠀⠀⠀⠀⠀⠠⣷⡴⠃⠀⢙⣷⣆⣠⡥⠽⣻⣿⣧⡑⢚⣿⣦⣬⡿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠉⢀⣀⣥⢈⣻⣿⠛⠓⠚⠋⣩⣿⡇⢻⣏⡉⠛⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠺⢷⣄⣀⠀⠀⠀⣀⡴⠟⠉⢠⣿⣿⠄⢻⡷⣾⣿⡿⢋⠆⣸⣿⢿⣀⣰⠎⠡⢃⣀⠃⠀⠀⠀⠀⠀⣾⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠸⣄⣠⡟⠻⣿⣆⠀⠀⠘⢣⢀⣾⣯⣹⣷⠘⣷⣼⢯⣿⡿⢠⣟⣻⣿⣿⣿⣿⠓⠈⠻⣆⡀⠀⠀⠀⢠⡿⠁⢹
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠛⢏⡠⠀⣿⡟⣿⣾⢳⣾⡿⠛⢷⡿⠛⣦⢹⡏⠀⠙⡇⢸⡏⠭⣿⣿⠻⣿⣦⢀⣾⢿⡇⠀⠁⣰⣿⠃⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⢸⣦⣀⣼⡇⣼⣿⣧⡍⠀⠀⠘⣧⠀⠰⠈⢃⠀⢼⠃⣿⣿⣶⣿⠇⠀⠙⢁⠺⠃⣨⡀⢀⣰⠿⠁⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⢿⣿⣿⣿⣆⠀⢸⣟⡀⢀⢀⣌⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢮⠁⢰⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⢰⣧⣄⣴⡆⣿⣿⠟⠀⣰⣆⠀⠀⠀⠀⠉⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⣿⣿⠟⠀⠀⢈⠿⠛⠁⠀⠀⠿⣿⣿⢄⡘⣿⠉⢙⠇⣿⣿⠀⣴⣿⣿⣆⠘⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠂⠀⠀⠘⣿⣷⠀⡠⠄⣲⡿⠋⠀⠀⠆⠀⠀⢠⡀⠠⠀⠀⠙⠻⣷⣅⣈⣑⣡⣾⣿⣿⣾⣿⣿⣿⣿⡇⡀⠀⠀⠀⢡⡆⠀⢺⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢘⠀⣿⣿⡀⠀⠐⠛⠀⠘⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⣰⡆⡀⠹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣼⣿⣇⠀⠀⠀⠀⠀⠓⠀⠿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠈⢠⢸⣿⡇⠀⠀⠀⠁⡀⠀⠀⠄⠀⠀⠀⠀⢀⣠⣴⣿⣷⣇⡀⢠⢜⣿⣯⡉⠉⠉⠁⠙⠻⣿⣍⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⢻⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⡇⠀⠀⣄⠀⠐⠄⠁⠂⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣶⣶⣿⣿⡿⠿⠿⠟⠂⠀⠀⠀⠛⣧⡀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠆⠀⠀⢸⣿⣧⠀⠀⠙⣦⡀⠀⠁⠒⠒⠒⠦⠀⣿⣿⣿⣿⡏⡏⡌⠉⢻⣏⢿⣦⠀⠀⠀⠀⠐⠀⠀⠀⣱⡄⠀⠀⠀⠀⠀⠀⠈⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠈⠻⢶⣦⣤⡤⠤⠐⠚⠋⢀⡟⣽⠆⠀⠀⠀⠰⠿⣶⣤⡤⠄⠀⠀⠀⠀⠀⠚⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠿⣿⠀⠀⠀⢀⠞⡹⢀⣴⣤⣠⡀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⠔⠁⢸⣿⡿⠀⣿⣿⣧⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠲⠄⡀⡠⠀⡼⠟⣷⣼⡿⠻⠿⠟⢛⡿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠑⠂⠈⠑⠲⣷⢸⣿⡏⠤⠤⠘⠓⢉⡠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⢻⣬⣿⣇⣠⣤⣤⣍⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠀⠀⠀⠘⠛⢿⣟⣛⢉⠉⣉⢛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⢿⣿⣄⢠⡄⢠⣾⠟⠉⠀⠀⠀⠀⠺⣶⣿⠿⠉⢿⠧⢠⡬⠼⠂⠂⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢶⣿⠧⠀⠀⠀⠀⡀⢀⡆⠈⡀⠀⠀⠀⢀⣀⣤⣶⣤⣤⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠀⢳⡀⡘⠀⠁⣀⣠⣤⣶⣿⣿⠿⠛⠋⠁⠈⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⣬⣤⣤⣤⣶⣶⣿⠿⠿⠛⠋⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣲⣦⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣷⣾⣷⠀⠀⡴⠿⢷⡀
⠀⠀⠀⠀⢀⣤⠶⠞⠺⠛⠋⠉⠉⠉⠀⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣹⣼⣿⣟⠀⠀⠀⠀⡈⠏
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⡀⠁⠀⠀⠀⠄⠀⠀⠠⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠀⠀⠀⠀⠿⠆⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⣐⣢⣤⣄⣀⠀⠀⠀⢐⠆⠀⠀⠀⠀⠐⢸⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣦⡤⠀⠀⠀⣾⠀⢠⡄⣀⠤
⠀⠀⠀⠉⠉⠉⢉⡉⠀⡀⠀⠀⣀⠘⠀⠚⠀⠰⠶⣿⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣼⠀⠀⠀⢠⣿⡄⢸⠁⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢩⡉⠀⢠⡦⠴⠒⠒⠛⠛⠀⣀⣲⠤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡰⢄⣹⡦⠀⠀⠻⡁⠀⠀⠺⠏⢀⣼⡀⠀⠁⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠂⣴⡞⠉⡿⢈⡅⢀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⢌⣢⠀⠀⡀⠀⠀⠀⠀⢈⣙⠘⠉⠀⠀⠐⣤⡀⠀⣀⣀⡤⠾⠿⣿⣦⣄⠒⠒⠃⣀
⠀⠀⠀⠀⡀⢀⠀⣽⠁⢠⠙⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⡾⠳⢌⣻⣿⡇⢼⡇⠀⠀⠀⣐⠤⠁⠀⠀⠀⠀⣤⣼⠋⠀⠉⣁⢀⡀⠀⠀⠈⠻⣿⣦⡀⢚⠓⠂
⠀⠀⠀⣄⣿⡇⣸⣅⠹⠋⠀⣻⣿⠀⠀⢤⣤⣅⣀⣐⣀⣀⠃⠀⠀⢹⡿⣷⡄⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⡀⠀⢀⠀⠀⠀⠀⢹⣿⠛⠉⠂
⠀⠀⠀⠠⢆⠑⢬⣵⣿⠀⠀⣿⣻⡆⠀⣼⣉⣉⡛⠿⣿⣧⡀⠀⠀⢸⠃⠾⠛⠁⠀⠀⠀⠀⢈⠙⠒⠛⠋⠉⣉⣉⣉⣉⣉⣉⣉⠉⠛⢻⣶⣿⣿⣿⡆⠁
⠀⠀⠀⠀⡀⠀⠀⠛⠃⢀⡾⣿⣿⣇⠀⣿⣿⣿⣿⠀⢸⣿⣶⣦⡀⠘⣦⣤⣤⠄⠀⠀⢠⢤⡠⢛⠛⠉⠁⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⣹⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠉⠀⠀⠀⢀⣮⣤⡤⠿⠟⢓⡈⣿⣿⠇⢀⣿⣿⡿⠟⠁⢀⣿⡿⠀⠀⠀⠀⠈⠐⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡬⣿⣿⣿⣍⣉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣿⣿⣾⣿⣿⣟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉

⣶⣶⣶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣿⡿⠃⣀⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣉⣥⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣶⣤⡀⠈⠻⢿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣆⠀⠀⠀⠉⠉⠀⠀⢀⣠⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⠀⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣦⣄⠀⣀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⡟⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣦⣤⣤⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣦⣤⣴⣶⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⢡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠐⠖⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡥⢼⡿⡆⣹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣤⣀⡟⣮⣶⣶⢟⡴⢶⣌⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⢤⣝⡷⠂⢾⣷⣬⣟⣵⡾⠿⣴⠋⠀⠀⠀⠀⠀⠀⢫⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡏⢳⠾⠐⡄⠂⠀⠀⡸⣦⡌⠛⣦⡟⢹⡯⠦⠈⣴⢟⣧⣍⡦⠀⠀⠠⡀⢨⡈⠥⠸⡄⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⣀⣝⠀⠸⠇⠀⠀⣸⡉⣯⣴⣤⣿⣄⠀⠾⠟⠃⢿⣶⡜⣯⣠⡅⠀⠀⢠⣌⠋⢠⣇⢸⡦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡿⠀⠀⠀⢀⢶⣾⣭⢨⡷⠉⢼⣭⣿⡠⠚⢳⣦⣤⢈⢡⣄⡴⣿⣄⠀⠈⠀⠙⢥⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣽⢻⡇⠀⠀⢀⢮⠟⢻⠦⠻⢃⣴⣶⣿⠉⢣⡞⠀⢰⡆⠀⢸⠇⡠⡟⢛⡆⠀⠀⠐⡀⠛⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡏⢋⢀⣄⠀⢞⡻⠀⣙⠻⢃⡜⣿⠟⢿⣶⠦⠽⠀⣻⡇⠀⠛⠣⢰⠁⢨⠜⡀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣸⢛⣧⣀⠉⣩⡷⠒⣶⣙⣹⠂⣧⠀⠻⣧⣤⣆⠀⢰⡆⢰⡙⠲⡆⠀⠐⣏⢡⡀⣤⠓⠶⢩⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢶⡟⣳⣤⣶⢟⣷⣬⡝⢿⡌⢢⣴⠉⠛⠙⠀⣺⠁⣸⣛⣋⡱⠄⠘⠛⠀⠀⠀⠳⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡖⣸⠉⣀⢡⠟⠁⡼⣧⣈⡿⠶⣹⡇⣀⣀⠰⣽⡓⢋⣩⣭⡴⠶⠶⣦⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠛⠘⢵⣧⡀⣤⣶⢿⡟⠐⢦⣼⡇⢹⡏⠿⡶⠟⡛⢭⣿⡴⠲⢦⡷⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣱⣿⡿⣤⣼⡧⠀⢈⡇⣀⣄⠰⡇⠚⠃⣠⣽⡀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢴⢳⣿⣆⣸⣿⡝⠀⢺⡿⠫⠿⢠⡧⠀⢈⣹⣧⣧⣀⣴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡼⢁⡿⣯⣈⡿⣿⢤⡀⠀⣿⠀⠀⣨⠀⠀⠂⢍⣣⣉⢥⡆⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⡼⢡⡞⢡⣀⣉⡷⣟⠢⡁⠀⠹⠶⠾⠛⠠⢌⠑⠤⣿⠤⢜⣿⣦⠖⢸⠉⠂⠚⠁⠘⣳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⣁⢦⣴⣏⠘⣌⣩⣼⡁⢈⡳⣍⠢⣄⠀⢀⡀⠲⢤⡙⠲⣿⠦⡌⣧⢀⠳⠀⠀⠀⠀⢴⠾⠯⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⡀⣡⢦⡰⢎⡵⣭⣄⢹⣆⠁⠌⠈⢷⣌⠳⣤⡈⠳⣤⡉⢷⡏⡁⡴⢿⠀⠁⠐⡆⠀⠐⠺
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⢞⣷⠦⡢⠉⣴⠳⡄⢿⠀⣿⣠⢢⠕⢊⣉⡴⢦⡙⠢⣌⠻⡾⠃⢑⠴⠎⣄⠴⠋⠘⠲⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠠⢿⣴⠞⣿⣾⣭⠐⠛⠃⠀⠘⢃⣰⠖⣯⡴⠖⣤⠙⠂⣤⡙⠇⠀⠈⠳⠞⡣⠀⠴⡀⠀⠠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡴⢿⡿⣤⣹⣄⠛⡇⠀⠀⠐⣏⠷⢞⣁⡀⢠⣟⢠⣴⡆⢻⡆⠀⠀⡾⠲⢁⣄⠀⠀⠉⣭⡻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⡈⢿⠇⣘⡩⠀⠀⠀⢺⡅⠸⣏⡽⢃⣀⣿⡘⠻⠃⢈⡇⠀⠀⠈⡶⠎⣰⡄⠐⣀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠞⢀⣩⠇⠀⠀⠀⠈⣷⣀⣤⣄⢻⡿⢿⣛⠦⢀⣾⠇⠀⠀⠀⠀⠛⢈⢆⡁⠙⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣄⣈⠕⠀⠀⠀⠀⠀⣴⣯⡹⠾⠋⣀⣈⠙⠶⢤⣷⡀⠀⠀⠀⢸⡙⠟⣸⠋⣰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⣁⢈⡘⠂⠀⠀⠀⠀⢀⡲⠇⣠⣶⡆⣹⠏⣠⠶⢀⡇⡅⠀⠀⠀⠀⠉⠻⢉⠠⡈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢌⠉⢠⡇⠀⠀⠀⠀⠀⠈⠉⢃⠹⠟⠋⣡⡄⠛⠺⠋⡏⠀⠀⠀⠀⠀⠈⠁⢉⠶⣿⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⠼⠛⠃⠀⠀⠀⠀⠀⠀⠀⢨⢂⣤⠶⣋⡟⠀⠤⣰⠇⠀⠀⠀⠀⠀⠀⠀⢙⣛⡛⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣩⡵⠶⣻⡟⠯⠒⠦⢀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠋⡐⢳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣁⣴⣾⡏⠟⢠⣤⣶⠈⠀⠀⠀⠀⠀⠀⠀⢦⡀⢈⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣻⠁⣽⡬⢧⣥⠴⢄⣀⣀⠀⠀⠀⠀⠀⠈⣉⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡿⠋⣻⠁⢀⡀⢛⠀⠀⠀⠀⠀⠉⠛⠿⠦⣄⠀⠀⢴⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡲⡞⠛⠛⠛⠛⣋⠋⣹⠛⠹⢉⠓⠈⠋⣙⠛⡘⢃⡟⠐⠏⢚⢇⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⣁⣄⣤⣆⣐⣀⣌⣁⣨⣅⣌⣀⣀⣁⣨⣁⣎⣀⣏⣰⣍⣠⣅⢈⣝⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠁⠉⠉⠉⠁⠉⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⢀⡶⢺⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀⠎⠀⠀⠀⣤⣿⣿⣿⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣴⣿⠀⠀⠀⠀⠥⠹⡟⣻⠆⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡓⡏⢩⠀⠀⠀⢘⡀⢰⣾⣇⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠈⠜⢀⣠⠄⢸⣏⣼⡯⠆⢱⠀⠰⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠇⠐⠩⢯⡠⢺⠿⢟⠰⣌⣬⢦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠐⢠⡞⢗⣿⢀⡾⠀⠧⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⠉⢀⣄⠂⠘⣸⣤⣯⡖⣩⢵⡈⢻⡂⠑⠂⠀⣤⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠺⠇⠀⠛⠲⡤⠟⢣⢿⣤⠇⢸⠇⢀⠙⠢⢄⠀⠸⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⠀⠀⠀⠀⠐⠀⢰⡇⢠⠋⠠⢀⡸⠋⠀⠀⢸⠊⣔⣦⠙⠢⠁
⠀⠀⠀⠀⠀⡀⠀⠀⣀⠠⠀⠀⠀⣀⠶⣤⡘⣿⠉⠒⡖⠁⠀⠀⠀⠈⢸⠈⣻⢵⢈
⠀⠀⠀⠀⠀⠿⠖⠒⠃⠈⠀⢩⣯⡵⠟⢻⢿⣏⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡍⢸⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣼⣿⡛⢻⠿⢥⠬⠥⠤⠄⠀⡞⡈⣿⣷⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠱⣖⣫⡿⠟⠠⣿⠿⠿⣭⠼⢿⣿⠤⣾⡿⠿⢇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠉⠈⠁⠈⠋⢛⣒⡒⣒⣲⠀⠀⠐⡂⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⢨⠀⠈⠉⠉⠉⢉⠁⠀⠀⠂⠀⠀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠐⠀⠐⠣⠀⢠⠆⢁⠀⠀⢸⠀⠀⠀⠀⠀⠀⡐⠀⠀⡘⠀⣄⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢟⣰⠀⠀⠃⠀⠀⠁⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠊⠐⠒⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⢣⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠙⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⣧⠋⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⠁⢀⡸⣧⡄⠀⣸⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⣴⠋⠀⢹⣿⣦⠙⠁⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣠⣾⣏⣀⣀⣈⣿⣿⣷⡄⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢹⣿⡟⠛⠛⠛⣿⣿⣿⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⣾⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⡶⠟⠛⠻⠷⣾⡟⠉⠰⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣶⣋⣤⣥⣍⣠⣭⣤⣥⣄⣶⣶⣶⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣿⣏⣤⣿⣷⣿⣷⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣤⠏⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⣤⢿⣹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣣⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⣿⣿⢿⡿⣿⣿⣿⡟⣤⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠃⢀⣨⣽⣛⠛⠿⢿⣿⣿⣷⣬⣙⣭⣷⣻⣦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠋⠉⣿⣛⣻⣶⣟⣛⣿⡏⠛⠟⠛⠉⡟⢿⣮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⢄⠘⣿⣟⠉⢻⣿⡿⠁⢠⠇⣰⠀⡇⣸⡞⠉⣿⣦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⢿⡿⠟⡆⠀⠀⠀⠠⠶⠦⡀⠘⡀⢠⣸⣷⠀⠀⢀⠈⢠⠈⠻⣶⣿⠟⠁⢠⠋⢰⣿⣿⣇⣿⠏⡁⠬⢻⣿⣶⡄⢀⡀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⠀⠀⡼⠆⠰⡄⠱⣾⣿⣧⠀⠆⠀⠂⠀⣉⣀⣼⣥⣀⣐⠁⢠⣿⣿⣿⣿⣿⣁⠤⢈⡁⠿⣿⣾⡿⠁⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⡀⢠⡤⠾⠆⠀⣤⣿⣾⡿⠋⢹⣦⣴⠶⢈⡁⠀⢈⣧⠾⣯⣌⣻⣿⣿⣿⣿⣿⠏⠁⠆⠀⠀⣀⣽⣧⣴⣄⣀⡀⠀⠀⣷⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣏⣠⣤⣤⣼⣿⢿⣿⣄⠀⠈⢿⣿⣿⣿⣿⠀⢸⣿⣷⣿⠻⣿⣿⣿⣿⡿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⡏⢩⣽⣶⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠾⣻⣿⣿⣿⣿⣿⣿⣯⣠⠀⣠⣾⣿⣿⣿⡄⢸⣿⣿⣿⠀⢸⣿⣿⣁⣴⣿⣤⣀⣼⣿⣿⣿⣿⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⠀⣼⣿⣿⡟⠀⠉⠛⠻⠿⣿⣿⣿⣿⠿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⡿⣉⣿⣿⣿⣿⣧⡈⠛⠛⠉⣠⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢰⠆⣰⣭⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠻⣿⢟⠻⣟⡿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣶⣤⣀
⠀⠀⣀⠐⣂⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣤⠀⢤⣿⣏⣳⡘⢧⡉⠮⡛⠿⢿⣿⣿⣿⢿⣿⣿⡿⠿⠿⠟⠛⣛⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠧⣀⠀⠀⠁⠠⡉⠻⠿⣿⡤⢽⣦⣬⣶⣤⠙⣛⣛⣓⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠑⠒⢤⣤⣀⣉⠱⢦⣄⡀⠉⢛⣻⣿⠿⣷⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣻⡌⡤⠊⠉⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣉⣁⣹⣶⣿⡏⠻⡙⠣⡈⠛⡉⠳⣿⣿⣿⡟⠛⢶⣭⣿⣟⣻⣿⣿⣿⣷⣶⣶⣴⣶⣿⠟⠛⠉⡩⠛⣀⡞⢁⡴⠛⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⡟⠛⢭⡾⢿⡟⢧⡈⢢⡈⠠⠈⢢⠈⠢⠀⠀⠁⢈⣳⡶⠲⢾⣿⣿⣿⣿⣿⡯⠉⣻⣶⡾⠁⠀⠀⡔⠁⢠⠏⢠⠎⣠⠞⣡⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⠏⠰⣦⠈⢷⡋⠳⣄⠱⡄⠱⡀⠱⡀⣣⣀⣇⣀⡀⠘⣇⣀⣦⠀⣜⠓⢻⣿⣿⣃⣼⣽⣿⠃⠀⠀⣼⠀⢀⠇⢀⡇⣰⢋⡞⣡⣏⣬⣥⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠘⢷⣶⣷⣾⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣧⣿⣶⣤⣿⣿⣿⣿⣿⡋⠀⠁⢠⣧⣴⣾⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⠖⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⠻⢮⣙⠻⢿⣿⣯⣽⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⢛⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠋⠉⠙⠻⣛⢶⣍⠓⣾⣿⣿⣿⣿⣭⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠟⢛⣡⡶⢛⡽⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⢾⣿⣿⣿⢷⣤⣤⣬⣭⣍⣉⣉⣉⣭⣭⣭⣭⣷⣴⠶⠿⢛⣫⣶⣫⣶⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣶⣤⣀⠀⠁⢄⣰⣿⣿⣿⣟⠲⠶⣶⣤⣬⣍⣉⣉⣉⣉⣩⣤⣤⣤⣶⠾⠿⣛⣫⣥⣿⣟⢀⡠⠖⠉⢈⣹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⢠⣼⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣤⣶⣶⣾⣿⣿⢿⠿⠋⠙⣏⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠿⣄⣐⠀⠉⠉⠉⠉⠛⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⢈⣠⡴⠞⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠉⠛⠻⠷⠶⠶⠶⣤⣤⣠⣤⣤⢤⣤⡤⠶⠶⠟⠛⠋⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⣉⣁⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉
⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠙⢿⣿⠟⠁⠀⢀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⠀⢻⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡄⠀⠀⠈⠉⠉⢉⣉⣭⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⠉⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣀⣤⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠿⣿⣿⣿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣦⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⠇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠛⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⡁⠀⢈⠗⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡕⠦⣀⡈⠒⠁⣀⡠⠚⠁⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠑⠢⡈⠢⡀⠀⠉⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⣀⠤⠔⠒⢤⡀⠱⠀⠈⠉⠒⠒⠒⡲⠭⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⠀⠀⠀⡱⢠⠀⠀⠀⠀⡠⠊⠀⡠⠒⠒⠢⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠐⠊⣡⡇⠀⠀⠀⡴⠁⢀⠜⠀⠀⠀⠀⠱⠈⡆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠪⢄⡫⡒⠄⠊⢀⡰⠋⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠒⠒⠲⢄⠙⢎⠉⠈⠁⠀⢠⠀⠀⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠀⠀⠀⠀⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡀⠀⠀⠀⢘⢄⠀⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠈⠦⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠃⠀⠀⠀⢸⠈⠳⠄⠠⠞⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠑⠒⠀⠐⠒⠉⠀⠀⡸⠀⠀⠀⠀⠀⠳⢄⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⣀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡾⢞⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣉⠀⠈⠃⢀⣿⡄⠹⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣽⣟⠂⣰⢆⡟⠻⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢀⡄⠁⠉⠀⢠⠀⣉⣄⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⡾⢁⣆⡀⢰⣄⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠁⠀⠉⠀⠈⠉⢹⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⢀⡠⣅⡠⢤⣤⠚⠀⣄⠿⠿⣷⠇⠀⠀⠤⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠈⠋⠉⠈⠀⢀⠀⠀⠈⠀⠀⠨⡟⠠⠿⢆⡀⠁⡰⢺⡿⠿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠒⠂⢺⠀⢀⣰⠀⠀⢠⡇⢐⠀⠀⠹⡟⢡⡿⠋⠑⠂⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⢷⡉⠉⢀⠀⠙⠅⣍⠀⠀⣸⠷⣍⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠈⠈⠀⢀⠀⠀⠀⠚⠁⠀⠙⡄⠄⠂⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⡆⠁⠠⡀⠀⢠⠀⠀⢀⠀⣀⣠⠄⠁⠀⠀⠘⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠃⡀⠸⠀⡀⠀⢾⠀⠇⠀⢩⠴⡏⠀⠀⠀⢹⣾⠀⠐⣁⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠈⡋⠀⡇⠀⠀⠡⠀⢸⠀⣀⠠⣀⠀⠀⠀⠠⡀⠀⢻⠀⠀⢿⠀⠀⠀⠀⠀⢤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡇⠀⡆⠆⠀⠀⠀⣼⠂⣯⣉⣌⠀⣍⣀⡀⠑⠀⣅⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠀⢀⣟⠂⠀⠀⠈⠉⠸⠗⣟⣿⢸⡟⠠⠉⠉⠙⠷⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⡏⠸⠃⠀⠀⠃⠀⣸⠉⡀⠀⡄⠀⢀⠀⠸⠀⠀⠀⢈
⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⢵⠄⠀⠘⠐⠇⠀⡻⠀⠃⠀⡄⠂⢀⡘⢦⠂⠀⠁⠈
⠀⠀⠀⢻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠞⠀⡎⠘⠸⡇⠀⣿⠀⡀⢠⡄⠀⠀⠙⣦
⠀⠀⠀⠈⡄⠀⠀⠀⢀⢠⡄⠀⠐⠃⠰⠁⡀⠂⢉⡀⢛⠐⢡⠋⠀⠁⣀⠐⡌⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⢰⡀⠀⢀⣠⡼⠦⢀⡼⠂⢘⠛⠀⠀⠀⣽⠛⢻⠟⢿⡀⠀⠈⢦⠁⠈⠁⡄⠀⠾⠁⣰⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠈⠣⠀⠁⠀⠀⠉⠈⠉⠛⠙⠂⠀⠀⠀⠛⠡⣴⡿⠊⠁⢦⡀⠀⠀⠀⢈⡁⢸⡿⠂⠉⠈⠐⠐⣤⣠⡉⠌
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠓⠈⠀⠀⠠⠃⠉⠀⠀⠄⠀⠈⠈⠛⢿⣇
⠀⠀⠀⢤⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⢾⡟⢀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⡰⠊⢀⠀⠀⠀⠀⠀⠀⢀⠀⠛
⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠈⢠⡆⠀⠀⠀⠀⠂⠒⠀⡀⠞⠃⠠⠀⠈⠀⠀⠀⠿⠋⠀⠈⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠀⠀⠀⠀⠘⠆⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡘⠲⠇⠀⠄⠀⣿⡟⠀⠀⠀⠈⡄⠀⠶⠁⠀⡀⠻⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠁⠀⡀
⠀⠂⠀⠉⣀⣀⡄⠈⠛⠁⠀⠀⠀⠀⠀⢹⡤⠀⠄⠀⠀⠀⠀⡴⠂⢀⢀⠚⢩⣧⣰⣆⠀⠶⠀⠀⠀⠠⡌⠀⠂
⠀⠀⠀⠀⠿⣼⠽⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⡄⠀⢠⢀⡁⠀⠀⠘⠁⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠐⠄⠘⣂⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⣈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡽⠇⠀⠀⠀⠀⠀⠀⠿⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣾⠰⠟⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣤⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣷⣼⠶⡶⠛
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣗⣚⢧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣲⣟⣿⣨⣷⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣯⢖⡿⡗⠀⠀⠀⠀⠀⠠⠠⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡟⠉⠀⠹⣄⠐⢠⣉⣷⣋⣳⣤⣄⡔⢢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡿⣹⣾⡿⠋⢹⡏⣉⣿⣿⣥⢧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡀⠀⢱⣾⣿⣿⢿⢿⣿⣿⣺⡟⣿⡛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣼⣿⣽⡏⠛⣙⣿⣭⠈⣧⠿⠷⣧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠵⠾⣿⣧⣤⡤⠿⠷⣶⣇⠒⣣⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⠀⡘⢛⡇⠀⣀⣤⣿⠛⡛⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⣣⡾⢿⠫⠥⢚⡼⠨⢴⠻⡆⠀⢀⣤⠞⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠞⡉⠣⠀⢙⣾⣫⢤⡠⠞⢠⣿⠖⠋⠐⢎⠳⡙⢦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢠⠭⠅⠀⠀⣧⣿⣼⠉⠒⡟⡠⠔⠁⡢⢀⣡⠞⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠱⠤⠞⠀⠀⠀⠀⠀⣏⣹⡇⣀⡷⢤⡶⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠦⠒⠒⢩⣿⣿⣿⣿⡀⣏⠉⡂⠀⠀⠀⡔⠂⣸⣁⣴⣧⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣤⣤⣄⡀⠀⠈⠙⠻⠿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠋⠉⠉⠀⠀⠉⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡗⠀⠀⢀⣠⣤⣤⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡃⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠿⠛⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠀⠀⠹⣿⣿⡋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠸⣿⣿⣄⠀⠀⠀⣀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣦⣤⣀⣀⠀⢀⣀⣠⣴⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠁




⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⠶⠶⠶⢶⣶⣶⠶⠶⣶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣯⣽⠄⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡏⢻⠿⢷⣰⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣦⣿⣿⣿⣼⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⢿⣿⣷⣿⣑⡙⢛⣿⡿⢿⣿⣿⣿⣦⡀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢶⡄⠠⠀⠀⣻⡛⣿⠛⣿⡧⠿⢿⡿⠿⠦⣸⡿⢿⡟⠻⢿⠀⣀⣾⠱⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸⠇⠀⠀⣘⠛⢛⠛⠛⣻⣿⣴⡄⣼⣶⢸⣾⡗⠂⠁⡘⠋⣀⠀⠛⣄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠁⠀⠀⠀⢹⡄⠘⠇⢠⡾⢷⡀⢰⡷⠀⣼⣿⣆⠀⢸⠁⠸⠃⠀⠀⠈⠻⢄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠍⠀⠀⠀⠀⠀⣠⣿⣲⣛⣹⢧⡴⣟⡐⢃⣺⣅⣠⣯⡔⣊⠻⢧⡀⠀⠀⠀⠀⠻⣿⡝⠁
⠀⠀⠀⠀⠄⡒⢀⠀⡇⠀⠀⠀⣠⣾⡕⢩⠃⣿⡟⢀⣶⡌⢉⡉⢩⣿⠇⣴⣿⠘⠁⢂⠹⡄⠀⠀⢀⡀⠉⠷⠤
⠀⠀⠀⠀⠀⡸⢋⠀⣻⠄⠀⠀⢧⡿⢦⣜⣻⣿⢢⣿⣿⡟⠦⠶⢛⡟⠀⣸⣿⠤⡄⠀⣏⠘⡆⠀⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠈⢻⠁⠉⣰⣴⠾⡿⢶⣧⢼⣿⡟⠈⢹⡇⠀⠀⢸⡇⢦⣿⣿⣙⣁⣤⢼⣿⣤⣤⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⠾⠷⠛⠮⢧⡴⠐⢊⡝⠉⠁⢸⠇⡇⢰⣿⠠⠷⣲⡿⠋⡉⠁⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠀⠀⠀⣀⢸⢽⠆⠹⡗⢐⡔⠉⠙⣿⠛⠉⠉⠁⠉⠀⠉⠁⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⡦⢣⢨⣷⡆⣿⠀⣶⢠⣴⡟⢻⣶⡖⢤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⡟⣿⢹⣾⣿⠁⣯⠀⣿⣴⡏⠃⣸⣿⡇⢦⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢙⡏⣿⠀⡇⣹⣿⣿⠙⢿⣦⡼⠛⢣⡖⠁⣿⣗⢺⠝⣿⣿⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢌⠑⡿⢀⠧⣾⣿⡏⠢⡀⢛⢶⣖⠊⠀⢀⣿⣿⡚⠆⠹⣿⣇⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠉⢀⢀⠀⡱⠆⡘⢲⣿⡏⢳⣅⠀⠀⢀⡈⠿⠿⣴⠛⣿⣏⢙⠀⢻⡜⠆⠀⠈⢿⢦⣄⣀
⠀⠀⠀⠀⡄⢰⡚⡍⠂⠀⠀⠀⡠⠍⠀⠠⣿⣿⠁⢠⠙⠻⢦⡀⠑⠀⠈⠔⠁⣿⣿⣅⠧⠀⠹⣄⠀⠀⣿⣸⡇⢸⠁
⠀⠀⠀⠀⠑⡀⢳⡘⢆⠀⢀⡤⠀⠠⠦⣼⣿⣁⡦⠀⣁⣀⢾⣿⣷⢤⣤⢶⡂⡟⢿⣿⠴⢃⠀⠈⢦⣬⠃⣸⠇⢸
⠀⠀⠀⠀⠀⠐⣄⠙⢦⣱⡞⠀⠀⠣⣼⣿⠉⣯⡶⡿⠏⣼⣿⡿⣿⢂⣫⠿⣴⡇⠘⣿⣷⠊⡄⠀⠀⠙⣔⡟⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠃⢄⡙⠀⡀⠁⣜⣿⠏⠀⡇⠘⠀⢐⡾⢻⡇⠿⣃⠀⠟⢽⠀⠀⢹⣿⣏⢰⡄⠀⠀⢸⡡⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣉⣾⣿⠃⠀⠀⣧⠄⡰⠏⠀⠘⠁⠀⠻⢂⡀⡾⠀⠀⠀⢻⣿⣦⠘⡄⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠯⡽⠁⠀⠀⢀⡷⠮⢁⡐⢦⣤⠶⣶⢾⣯⡧⡗⠀⠀⠀⠀⢻⣿⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠠⣈⣞⣿⠿⢄⠖⣠⠛⠡⡝⠲⢤⣈⣉⣉⣉⣛⣹⡝⠒⠞⢣⡜⣁⠖⠀⢿⣿⣗⡊⣢
⠀⠀⠀⠀⠀⢀⠐⠀⣽⡟⢃⣳⡌⠶⢠⡜⠁⢶⡤⠤⠆⣽⠉⠹⣯⠙⠛⢶⡳⠆⣤⣉⠶⠂⣼⠻⣯⣥⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣾⣀⢉⡇⢸⠃⠎⣁⠶⣀⣀⠤⢿⠀⢰⠏⠑⠢⣼⣱⠛⣤⢉⡴⠐⠇⢀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢤⠇⡸⣉⠖⢈⡰⡥⠄⠰⠟⡦⠛⠶⠤⠤⡿⢡⡛⠰⢏⣐⠀⡗⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠠⠥⠉⠰⠦⠀⠆⠠⢰⠌⠱⠏⠠⠲⠴⠗⠠⢌⡁⠮⠉⠦⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⢦⣤⡠⣤⣤⡤⣤⣤⣴⣦⣀⡤⣦⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣷⣀⣡⣼⣿⣤⣼⣿⣤⣤⣿⣯⣤⣼⣥⣬⣿⣤⣿⣧⣼⣤⣿⣤⣧⣤⣥⣽⣦

⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠠⠂⠀⠀⢀⡀⠀⠖
⠀⠀⠀⠀⠤⠀⠀⠠⠀⠒⠀⠐⠀⠄⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠈⡤⠘⣦⠀⠀⠀⠀⡀⠁⠀⠂⠁⠀⡄⠀⠂⠀⠒
⠀⠀⠀⠉⢀⠀⠈⡂⠀⣠⠂⠀⠀⠀⠀⠀⠀⠈⠢⠡⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢈⠶⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠂⠈⠀⢠⠀⢀⡀⢤⠀⠀⠁⠀⠄
⠀⠀⠀⠀⢸⠀⠀⢀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠋⠀⠀⠀⠐⠄⠈⠠⠂⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⣰⡄⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣇⣻⡀⢀⠀⠀⣀⡀⣧⣄⠀⣀⣀⠀⡀⠀⠘⠀⡀⠀⢀⣀⠀⡀⢀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠈⠃⢸⠃⢸⡉⠁⠋⠈⠘⢟⠉⠸⣅⠆⠀⠸⢇⠄⠻⢩⠄⢧⡼
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠐⠄⠈⡉⠀⠀⡄⠀⠀⢲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠍⠀⠀⡈⠐⠀⠀⡦⡀⠀⠀⠀⠀⠈⠉⠀⢾⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠘⡆⠠⠀⠐⠆⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⡞⣻⠖⢦⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠁⠀⠀⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠟⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⡀⠀⠀⠀⠁⠀⢀⣠⣾⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢶⣶⣶⣦⣤⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠛⠛⠉⠉⠉⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀⣀⣀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⣠⣶⣿⣿⣿⣶⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⠿⠛⠉⠀⠈⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢻⣿⣿⣶⣤⣤⣴⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄⠀⠰⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠁⠀⠀⠀⠈⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣓⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢡⠀⠠⢠⣬⣤⡤⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠀⠇⠀⠀⠀⢷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠰⠹⠃⠿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠀⣬⣷⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⢀⣠⣤⡘⠰⠇⠀⠀⠀⠀⠎⡷⣄⡀⠀⠀⡼⠀⢷⡀⢄⣰⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⠦⠀⡈⠉⣷⠀⠀⠈⠀⡇⠀⠂⠤⠜⡀⢮⠓⢨⠁⠐⣆⣸⠃⠀⠉⠗⢠⣎⢀⡠
⠀⠀⠀⠀⠀⠀⠁⢀⣬⠄⡻⠤⠁⢰⢠⠔⠉⢀⣄⠀⠀⢻⠀⠀⠂⠳⣾⠄⠈⠁⠀⠀⣽⣇⣀⢀⡀⠠⢃⠿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣴⢀⣠⡴⠆⠢⡘⠛⠡⠀⢸⠏⠀⡐⠉⠀⠀⡄⡠⠀⢨⣩⠃⠻⠆⣹⢦⣍⠀⠈⡃
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠋⠙⢉⡀⠤⣀⠈⠁⠀⠒⢉⣲⣿⡣⡀⡠⢀⡇⠀⠐⢥⣊⠇⠀⡔⠁⠘⠟⠟⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⡄⣰⡊⠁⠲⠬⠭⠅⠤⡄⠉⢀⠀⠙⠺⣵⡿⠁⠀⢠⠏⠉⠀⠜⠀⠀⡈⠲⣷⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡯⡤⠀⠀⠀⠀⠀⢰⢡⠠⠘⠃⡠⠀⢸⡇⠀⣰⠆⠀⢠⠂⠚⠀⠠⠂⠀⠀⠹⣤⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡜⡶⣡⠃⠀⠀⠀⣠⠚⠈⠀⠻⠃⠀⣀⣾⣪⠷⠯⠄⠀⠻⠀⣿⠃⢤⣧⠀⢀⠀⢈⣹⣿⠧⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠼⣱⡣⠆⠀⡐⢉⡁⠀⠀⠀⠀⣁⠀⡟⡣⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠤⡌⢼⣿⡟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢀⡾⠻⢼⣢⣋⡁⠀⢀⣞⡢⠏⠁⠀⠀⠀⠀⠀⠀⠑⠒⠶⠢⣴⣶⣷⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡤⡤⡄⢀⠀⠠⢠⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠿⠿⠛⠀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣸⠏⠘⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠸⣦⣀⣠⣾⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡴⠶⠶⠶⠶⠶⠛⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠛⠛⠛⠛⠋⠀⣼⠟⢁⣠⣤⠶⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⠶⠶⠶⠶⠶⠄⠀⣼⢃⡶⠋⣁⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⠀⠀⠀⣿⣟⣠⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠀⠀⢠⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠀⠀⠘⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠎⠿⣦⡀
⠀⢀⡀⠐⠛⠃⢠⡄⢶⣦
⠀⠸⡷⠰⣦⡄⠀⠿⣤⡉⠈⢳⣄
⠀⠀⠁⠐⠘⢧⡀⢀⠀⠹⣶⣄⠈⠳⣦⡀
⠀⠐⠀⣄⠀⠈⠓⡀⠁⠀⠘⣿⣿⣄⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀
⠀⡇⣼⡏⢠⣶⣄⠙⢦⡢⡀⠈⠻⣿⣷⣦⡀⠹⣷⣄⠀⠀⠠⠊⢹⡇
⠀⠀⣿⡇⠀⢩⣽⣷⠀⠹⣟⢦⡀⠈⢻⣿⡷⡄⠸⣿⣆⢸⡇⠀⠘⣄
⠀⢸⠁⢀⡀⠘⢿⡽⣷⣄⢈⣷⣝⢶⡀⠹⣿⣆⠀⠉⢿⣿⡗⡗⢄⣻⡄
⠀⠇⠀⣾⣿⡄⠈⢿⣜⢿⡁⠀⢙⣿⣧⠀⠘⢿⣦⡀⠀⠿⠇⠹⡈⠹⣷⡀
⢸⠀⢰⣿⢿⣷⡀⠈⠹⣿⣿⣆⠀⠈⢻⣧⠀⠈⣿⣿⣦⠀⠀⠠⣵⣄⠙⢷⡀
⡇⠀⠈⣿⠘⠿⣷⡀⠀⠘⢻⣞⣿⣄⠈⢿⡆⠀⠘⠉⢦⣄⠀⠀⣿⣿⣕⢾⣿⣄
⡇⠀⠀⣿⠃⠈⣿⣷⡀⠀⠈⢻⡞⢿⡄⠀⠁⠀⠀⠀⠀⠙⠗⢀⠘⣧⠙⣷⠈⠛⢷⡄
⠀⠀⢀⣿⣇⠀⠸⣿⣧⠀⠀⠈⣿⣎⠟⠀⠀⠀⠀⠀⠀⢀⡀⢸⣇⠈⢳⣟⠀⠀⢰⣍
⠐⡄⠀⠀⠻⣧⠀⢻⣿⡆⠀⠀⠸⣿⡆⠀⠀⢀⠀⠀⠀⢸⣷⡀⢿⠀⠀⢹⡜⣆⠈⣿⣆
⠀⠘⢇⠀⠀⠘⣷⣼⣿⣿⡀⠀⠀⣿⡃⠀⠀⢸⣦⣤⡀⠈⢿⣿⡄⠀⢀⡀⠠⣿⣆⢹⣿⡄
⠀⠀⠈⢧⠀⠀⠘⣱⣿⣿⣇⠀⠀⢸⣿⣦⣆⠈⢻⣿⣿⣆⠸⣿⣿⡀⢸⣧⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠈⣷⠀⠀⠀⠹⣿⣿⠀⠀⠈⠉⠻⣿⠃⠀⢻⣿⣿⣦⠹⣿⣷⡌⢿⣷⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠉⠀⠀⠈⢿⣿⣿⡇⠙⣿⡛⠀⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠙⣿⣆⠀⠀⠀⠀⢾⣷⣦⣀⠻⣿⡇⠀⣾⣇⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠉⠙⢻⣧⡈⠀⠀⠿⠛⢃⡀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣿⣿⣆⢠⣄⠀⠀⠈⠛⠃⠀⠀⢀⣴⣿⡇⠀⠈⠙⠻⠿⣿⣿⣷⠆⣴⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠐⣿⣿⣿⣷⡛⢿⣖⠀⠀⠀⠀⠀⠉⠙⠛⣧⣄⡀⡀⢸⣶⣤⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⠟⣋⣙⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣯⣄⣀⠀⠀⠹⢿⣿⣿⣿⣦⣝⢿⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⠋⠁⠀⠀⢾⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣷⣀⠀⠉⠹⣿⣿⣿⣿⣷⣌⡛⢿⣶⣶⣿⣛⠀⠀⠀⠀⠁⢀⠀⠀⠀⢨⣿⣿⣿⣿⠟⣱⣿⠟⠉⠀⢀⣴⠶⠃
⠀⠀⠀⠀⠀⠀⡟⠈⢿⣿⣿⠻⣷⠂⠀⣤⣿⠿⠋⢋⣵⣿⡿⣶⣮⠻⣿⣿⣤⡀⠀⠀⠀⠘⠓⠶⠶⣄⢙⣿⣿⢃⣴⣿⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿⣿⠀⠈⠓⠀⠀⠠⠴⣾⣿⡿⠏⣰⣿⣿⣷⣮⡛⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡌⠀⢡⣿⠟⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⣰⣿⣿⣿⣿⣿⠃⡀⠀⠀⢠⣆⠀⠀⠀⠀⠈⢷⠀⣾⠇⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢊⣠⡖⠀⢩⣿⣿⣿⣿⠟⢸⡇⠀⠀⣿⣿⣦⡔⢀⡀⠀⠈⢠⣿⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⡘⠻⠿⠿⣿⣟⣛⣤⣴⣾⣿⠏⢀⣴⣿⣿⣿⣿⠏⠀⢸⡇⠀⠀⠻⣿⣿⣿⣿⢖⣠⡀⣸⡇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⢡⣿⡿⠟⠁⠀⠀⠈⢷⣤⣤⣼⠶⠍⠉⠁⠉⠀⢀⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣀⣀⣠⠄⠀⠀⡴⠟⠋⠀⠀⠀⠀⠀⣠⣸⣯⣵⡖⠂⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⣥⠤⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣤⣤⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠦⢴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠿⣻⣿⣿⢿⣿⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠇⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⣿⡿⣴⡦⢺⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⢻⣿⣧⣀⣡⣼⣾⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣖⣛⡛⠻⢿⣿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠈⢳⠗⠚⠛⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⣀⣀⣸⢆⣀⣀⣰⣿⣿
⠀⠀⠀⠀⠀⢠⡀⠀⢀⣠⢴⣖⣾⡧⠀⠀⠀⠻⡿⠀⠀⣉⡅⠀⠀⠉⣿⣿⣿⠀⠈⠙⠛⢷⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠈⣷⢴⡏⡟⡼⠻⣯⡤⡀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⣹⣿⣿⡿⠀⠀⠀⠀⠈⢙⡀⠠⠀⠙⡆⣀⣷⡀
⠀⠀⠀⠀⠀⠀⡏⢸⢩⡇⡇⣴⠦⠤⠄⠀⠀⡤⠋⢳⡟⣉⣉⣀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠤⠈⢻⠀⠀⠀⢃⣩⢸⡇
⠀⠀⠀⠀⠀⢰⡇⢸⢸⢠⣓⡲⠶⢆⣀⠀⡎⠓⢀⣼⡟⠉⠀⠀⠀⠙⣿⣿⣿⡄⠀⠀⠀⠀⠐⠲⡆⠆⠈⢠⢁⣼⠁
⠀⠀⠀⠀⠀⠘⣇⣼⢸⢸⣿⣿⣻⣇⣁⡠⠀⠀⠀⠙⠃⠀⠀⠀⠀⡘⠿⠿⠟⢱⡄⠀⠀⠈⠉⠈⣇⠄⢘⢼⣹⣿⣇
⠀⠀⠀⠀⣲⣿⣿⣿⣾⣿⣷⣿⣿⣿⣯⡶⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣶⣶⣾⣿⣶⢏⣦⣾⣾⣿⣿⣿⣧
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿⣦⠀⠀⢰⠖⠉⠹⣿⣿⣿⣿⣿⣿⡿⣞⣽⣷⣿⣿⣿⣿⣟⠁
⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⣼⣿⣿⣿⣿⣦⣰⣿⡄⠀⠀⠙⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣴⡿⠟⠛⠋⠉⠉⠁⠁⠀⠀⣀⣠⣀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣯⡤⠿⣻⣽⣿⡟⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⣾⣿⡿⠛⠛⢿⣦⠈⢻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢾⣿⣿⣾⣷⣿⣿⣿⣿⣷⣤⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣦⣠⣿⣿⣧⣼⣿⣿⣿⣞⣿⣻⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⢿⣿⠛⣹⣾⣿⣿⣿⣿⡀
⠀⠀⢀⣿⣿⣿⡿⣯⠁⠀⠀⣠⣶⣿⣾⣿⠟⠙⢿⠟⣿⣿⠿⠁⠀⢹⣿⣏⣙⣫⣿⢷⡹⠿⠆⠲⠆⠀⠀⠌⠳⣿⣽⣿⣿⣇
⠀⠀⣿⣯⣷⣆⡀⢰⡿⣿⣾⣿⣿⣿⣽⠇⠀⠀⠀⢠⣭⣿⠄⣠⠀⠀⠈⠈⠛⠋⣹⢟⣧⣄⠠⣤⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⣽⣿⠀⢛⢋⣻⣿⣟⣿⣽⠛⠙⠁⠀⣀⢐⠀⠿⢿⣷⣠⠄⢀⠀⠀⣳⢞⠗⡐⢾⡿⣿⣷⣬⠗⢠⣤⠤⢀⡀⠀⠙⢿⣿⣧
⠀⠀⣾⣧⣶⣷⣾⣿⣟⣫⡏⠉⠀⢠⣾⣶⣿⣿⣾⣧⣭⣏⠀⢴⠚⢲⣦⣻⣿⣣⡄⠠⣀⠉⢡⠀⠔⣾⣤⡤⣬⠍⠀⠀⠀⠺⣿⡄
⠀⢀⣾⣿⣿⣿⣿⢿⢿⣿⣿⣷⣾⠿⠻⢿⣿⣿⣿⣿⢿⠿⠀⠸⠞⠿⢐⣿⣿⣿⣿⣾⣽⣷⣷⣴⡶⣮⣵⣶⣦⣀⠶⣤⣀⣬⣏⣇
⠀⢰⣾⣿⣿⣿⣷⣶⣶⣷⣿⣿⣶⣶⣶⣶⣶⣾⣷⣶⣿⠷⢋⠀⡶⠂⢸⣿⣿⣷⣾⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠘⡿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⢏⡄⢠⣿⣭⣶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠻⠿⠿⠿⠿⠟
⠀⠀⠈⢛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠯⢤⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠸⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡀⠙⢆⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠑⠀⠀⠑⠄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠄⠀⢦⡀⠀⡆⢠⠄⢦⡎⡏⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣦⠊⠢⠀⢵⠀⠀⠈⢆⠀⠑⣇⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⡆⠀⡁⠀⠀⢄⠀⠀⠣⠀⡈⠦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠈⢆⠈⠰⠀⠀⢣⠀⠄⠁⠀⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⡄⢺⡀⠀⠁⠀⠇⠀⠀⠀⠀⠈⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⡞⠃⠀⠔⠀⠀⠀⠀⠀⠀⡀⠀⢣⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣧⠀⠀⠀⠀⢆⠀⠂⠀⠘⡀⠀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠈⠢⠀⠀⠀⠐⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠂⠤⠀⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠚⠛⠑⣄⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠳⣄⡰⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⢻⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠏⠀⠀⠀⠀⠀⢀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⠀⠀⠀⢀⠔⠃⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠺⢿⣷⡄⠀⢀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⠀⠆⢻⡇⢠⠏⢘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⢀⡴⡆⠀⠀⣠⡟⢃⠀⠀⢀⡉⣧⣘⡆⡼⠃⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠳⡌⣮⠓⢢⠀⠀⢎⠀⡷⡆⢰⣿⢿⠋⠁⠀⠙⣿⣿⣿⡜⠀⠀⠀⠀⠀⠀⠀⡐⢙⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠁⠀⠀⣸⠀⣠⣣⠋⢸⢈⠣⢄⣘⡠⠟⠀⠀⠙⢢⣀⠀⠀⠀⠀⢀⡧⣿⣯⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⡶⢤⣀⡻⠶⡾⠁⢠⣈⠦⠨⠋⠉⣁⣨⣤⠴⣄⠀⠘⠣⣄⠤⠚⣝⡤⠬⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠈⡅⢀⠧⢾⣿⣄⠀⢤⠐⠀⣹⠃⢸⠀⠈⠒⠦⡀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡈⠀⡸⠋⠙⣄⠈⠢⠔⠀⢀⠸⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⣀⠔⠋⠉⠁⢳⠒⡦⢤⣼⣽⣍⣩⣍⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠀⠀⣠⢄⡀⢸⡆⠑⡌⠏⣿⡇⠀⣿⣋⣯⡤⣖⣲⠶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⢴⠤⠐⠒⠒⠃⢡⣀⡬⠤⠒⠋⠁⣾⠙⣶⣿⢀⠁⠈⣽⠁⠀⢿⣉⣉⣁⡀⠠⠤⣼⣭⡋
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠔⠢⠜⠋⠁⠀⠀⠀⠀⢰⡖⡺⠧⢿⢘⣧⣿⠛⢦⣀⡀⠉⠙⠋⠉⠛⠛⠷⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⢹⣴⠋⢸⡄⠘⠿⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠋⠁⠀⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈⢣⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠐⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠙⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡴⠶⠶⠛⠛⠉⠁⠀⠀⢀⣰⠖⢦⡀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠋⠁⢸⠀⠙⠳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠶⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⢶⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣰⡍⠁⠀⠀⡐⠀⠀⠀⠀⢰⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠰⠁⠀⠀⠀⢀⣾⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠁⠄⠘⠀⢀⢺⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠸⡶⠶⠶⠲⠒⠒⠺⠒⠒⣻⠚⠓⠒⠒⠶⠾⠿⠟⢛⡟⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣆⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⣎⢃⠂⡔⡰⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆⠀⠰⢁⠀⡀⣿⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠘⢧⡀⠀⡴⠀⢻⠀⠀⠀⠀⣠⡞⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠻⣶⣁⠀⢻⠀⠀⢀⡴⠋⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠙⠷⣾⣤⣴⠟⠁⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⠉⠀⠀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣤⡶⠟⠋⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠶⠦⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣇⠀⠀⠀⠀⣀⢀⡀⡤⠂⢀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⢿⣮⣿⣿⡀⠀⠠⣎⣁⣀⠃⠈⠀⠈⠉⣡⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⡆⠀⠀⠉⠛⠒⠒⠛⠛⠛⠋⠉⠉⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠉⢿⣿⣿⣿⣾⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣢⣶⣶⣷⡷⠾⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⣦⣤⣿
⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢰⣾⣿⣿⣿⣿⣷⣿⣿⣿⡿⠋⢀⠀⢤⣤⠠⠂⠀⢄⢀⠊⠋⠀⠀⣤⣠⣀⣠⢹⣿
⣿⣿⠀⡀⡠⡀⠀⠈⠈⢿⡇⣀⠚⣥⢽⣿⣿⣿⣿⣿⣟⣰⣿⣿⣿⣿⣷⣿⡿⠿⠛⠃⠀⠀⠉⠓⢯⣉⠀⣵⣶⣊⡉⠀⢀⢊⣶⢛⡉⣫⣍⣳⣿
⣿⣿⠀⠀⠀⠐⠛⠒⡄⠨⣿⠿⢺⡟⠛⠻⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠝⣫⠤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇⠀⡄⣿⣽⣭⣟⡵⡖⣦⣿
⣿⣿⣠⠶⠞⠛⠓⠺⢵⠶⣿⣔⣿⡗⠀⣰⠁⣼⣿⡏⠙⢿⣿⠿⠋⣁⢠⡟⢀⡴⣤⠵⠤⣎⢁⣀⣀⣀⠀⠢⣄⠀⣿⣤⡷⠧⢷⡾⠟⠀⢛⣽⣿
⣿⣿⢁⣀⣀⡀⠀⠐⠀⣲⣿⣅⣠⡄⢺⠟⣿⠉⠻⣧⣤⣼⡟⣧⠀⠉⢾⢿⣿⣷⠻⣄⣦⣿⠛⣋⣿⢙⡟⣾⡐⣀⡀⠹⠀⠠⣟⠁⠀⣀⠀⠁⣿
⣿⣿⡩⠭⣟⣢⣍⣸⣿⡿⠓⠾⣿⣭⣿⣿⣿⣆⠀⠈⢻⣾⣇⣸⠀⠀⢸⣀⠀⠈⠓⢈⣿⣽⠿⠁⠀⠊⠈⢠⣿⠋⣀⡀⠀⠀⠀⠁⠺⠃⠀⠀⣿
⣿⣿⡦⠾⣤⣤⠽⠶⠛⠷⣄⣀⣨⡟⠛⢉⣹⣿⣦⣄⣀⣿⣾⣿⣤⣶⣿⣿⣷⣶⣶⡿⠿⣿⡅⠀⠀⣀⣤⠞⠁⠀⠉⠃⠀⢎⠩⠀⠀⡇⠀⠀⣿
⣿⣿⠀⣠⠮⠒⠒⣀⣰⣶⢤⣿⣯⣿⣿⣿⠟⠩⠉⠀⠀⠉⠻⣷⣤⣄⣂⠀⣄⢙⠉⠩⢉⣛⡟⢛⡿⠛⠉⠀⠀⠀⠂⠀⠀⠀⢉⣁⣉⠄⠀⠀⣿
⣿⣿⠊⠀⠀⠀⠀⠉⠉⢀⣾⠋⢩⢻⢿⡛⣌⠁⠀⣠⣶⣷⣆⢹⣈⠉⠛⠛⠛⠛⠛⠛⠶⠦⣽⡃⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⣈⣥⣤⣀⣌⣀⣿
⣿⣿⠀⠀⠀⠀⢀⡀⡰⠉⣁⣀⣹⣼⣿⢠⡄⠀⠀⣺⣿⣿⣿⢸⣇⠀⠠⠤⢤⣾⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠑⠒⠒⣡⡾⠿⣿⣏⣙⢻⣿⣿
⣿⣿⠀⠀⣓⠤⠛⠊⠉⠉⠛⠳⣿⣿⣿⣔⣷⡀⠀⠹⣿⣿⣧⣾⣯⣀⠀⠀⠀⠘⡇⠀⠀⠀⣠⠤⠤⠤⢆⠀⠀⠀⠀⣠⣿⣻⡷⢺⡿⣫⣿⠃⣿
⣿⣿⢀⡴⠉⠀⠀⠀⠀⢞⢁⣀⡾⠋⢉⠿⣿⣷⣄⣤⣤⣿⠟⠉⠀⠈⠣⡑⠂⠂⠀⠀⢠⡞⠁⠀⠀⣤⡀⢁⡀⣠⠖⠟⠛⡿⠿⣿⣾⢿⡏⠀⣿
⣿⣿⣾⣷⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣷⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣷⣶⡶⣶⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⣠⠀⠀⠀⠀⢼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠁⢀⣴⡿⠃⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣤⣘⠿⣁⠀⣿⡿⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠉⠛⠛⢣⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡿⠀⠀⠡⠄⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣷⣦⡷⠷⣴⡄
⣀⠘⠃⠀⢀⡀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣼⠃⠀⠘⠋⠀⠰⠤⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧⡈⢷⣧
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⠃⠉⠛⣿⣿⣿⣷⣼⡏
⠀⠀⠀⠀⠐⢀⠀⠀⠀⢸⣿⣿⣿⡟⠹⠟⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⠂⠐⣗⠀⠹⣿⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⢿⣳⣠⠀⠀⠀⠠⡞⢃⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⢀⣤⣠⣴⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣦⠀⣄⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠹⣿⣿⣋⠀⠁⠀⠀⠀⠀⠀⠀⢄⣀⠸⡏⢻⡿⠃⠀⠀⠀⠀⢸⡛⠻⠟⠿⢿⡄
⠀⠀⠀⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡀⢈⡛⢮⠈⠃⠀⠀⠀⢀⡀⠀⢸⣿⣇⠀⢀⣤⣄⣀⢀⣠⠐⠋⠀⠀⠀⠀⡀⢹
⠀⠀⠀⢠⡞⠀⠹⠗⠄⠀⠀⠀⠀⠀⢸⠂⢃⣼⣷⡈⠙⠂⠀⠀⠀⡈⠁⢀⠘⠛⠃⢴⡟⢿⣿⠃⠈⠁⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡅⠙⠳⠆⠀⠀⢀⡘⠃⡤⠻⣄⠀⠀⠈⣄⠠⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠛⣳⠄⠀⠀⠀⣈⠣⠀⠙⠦⣿⣿⡇⠀⠹⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠛⣳⠆⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠻⠁⠀⢻⠃⠀⢀⡈⠿⠀⠠⠀⣀⠀⠱⡦⠴⠐⠂
⠀⠀⠀⠀⢀⡜⠁⠉⠀⠀⠀⠀⠖⢠⡿⠃⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠸⡿⣦⠀⢀⠀⠉
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⠃⠀⢠⠀⠀⠘⠒⠀⢀⡸⠋⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠀⠀⠀⠀⠀⢠⣤⣀⣀⠄⣠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠢⠷⢷⣾⣷⣼⣿⣿⡍⢣⡵⠦⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣄⠠⢼⡟⢻⡉⠉⠹⡛⠧⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣰⣾⣿⠿⣿⣿⣿⡉⢆⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⢀⡀⢠⠄⠔⠀⠀⠀⠸⠿⠿⠋⠅⢸⣆⣼⡿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠈⠉⠁⠁⣀⠀⠄⠀⠀⠀⠀⠀⠀⢢⡀⠋⢁⠄⠀⠀⠙⡃⠀⠰⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⠂⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠛⠀⠀⠀⠀⠀⠀⡄⣠⡁⣴⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠎⠙⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⡆⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠛⠀⠲⠾⠇⠀⠀⠀⠐⣷
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣤⠀⠸⠁⠀⠀⠀⠀⠀⠀⢩⠆
⠀⠀⠀⠶⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⠐⠂⠀⣤⡦⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⣠⡀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠈⠋⠑⠂⠀⠒⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⠂⠀⠀⠀⣴⠆⢠⣤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠊⠙⡿⢠⠀⠼⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡀⠁⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠄⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠠⠈⢁⠀⠻⠆⠀⣄⢀⡀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⢉⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠶⠇⠀⠀⡀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⠉⠠⣾⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀⠀⠀⠀⠀⣡⠏⡈⠠⠜⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡈⠀⠀⠀⣤⡄⠀⠀⠶⠀⠀⠀⠸⠛⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠂⠉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡄⠀⠀⠀⢦⡈⠀⠀⠀⠀⠀⠀⠁⠀⠖⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⢦⠀⠂⠉⡆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡤⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠃⡈⠙⢋⠀⡸⢉⡉⣯⠹⢈⣿⠟⡻⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡽⠆⢀⡤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢧⣤⣄⠳⣿⠟⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⡁⣄⡹⣶⠀⠈⢛⠒⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠧⣉⠁⠈⢷⡀⠀⠙⢆⠀⠈⢦⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⡄⡄⡘⡧⠀⠀⢻⡆⠀⠀⠹⣄⠀⠳⣤⠯⣠⠟⠁⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣷⡈⠁⠘⣦⠀⠀⠻⣦⠀⠁⠘⣄⣀⠈⢷⣿⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢺⣿⣷⡀⣀⠘⣆⠀⠀⠘⣷⠀⠀⠘⢯⠀⠈⠻⡖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⢾⣿⣷⡹⠀⠙⢦⠀⠀⠘⣧⡄⠀⠈⢳⣄⠀⠹⣆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢀⢸⡿⣿⣧⠃⢀⠙⣾⡄⠀⠈⢿⡆⠀⠀⠁⠀⠀⠹⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢻⣷⣴⣿⣯⢻⡁⠘⢷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠽⠿⠟⠉⡉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⡘⠟⢿⣿⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⡖⠁⢁⠀⢺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣦⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠨⠅⣙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⠈⠩⢭⣿⣷⣔⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡾⢁⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢫⠛⣿⣿⣅⢦⡀⠀⠀⠀⠢⠀⣤⢾⠔⠟⠛⠋⢠⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⣿⠀⠀⠀⣀⠿⣿⣿⣯⢻⣞⣿⣦⠴⠋⠀⠀⠀⠀⢀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡺⠿⢿⣤⣴⠞⠁⣸⣿⡟⢛⣿⠿⠋⠀⠀⠀⣀⠄⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⢠⣴⠟⢁⡴⡋⣾⢃⠀⠀⢀⣀⣀⣤⠶⡛⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⡼⠟⠋⠀⠀⠀⠺⣿⣿⠀⣓⠚⠛⠍⢀⣠⠖⠚⠉⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢁⡀⠀⠠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠛⠈⠁⠀⠀⠀⠙⠋⠙⢿⣌⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠱⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠟⢿⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⡘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⢀⣶⠟⠁⠀⠀⠀⠀⠈⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⡆⢠⣶⠟⠁⠀⠀⠀⠀⠀⠠⡄⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⣥⠀⠀⠀⠀⠀⢠⡀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣤⣟⣤⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⠄⢠⡄⠠⡄⠀⠈⠉⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡗⢺⡇⢰⡇⢰⣭⡆⣿⠃⣴⠋⣶⠐⣷⠒⠰⣆⡐⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⠃⠚⠓⠘⠛⠘⠒⠃⠙⠛⠙⠓⠋⠘⠛⠀⢤⠝⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠀⢠⠈⢠⡌⠐⡖⠁⡆⠂⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡀⠀⣇⠈⢂⣐⠂⠴⠃⠀⠀⠾⠅⢘⡏⢀⣧⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⢦⠀⣃⣤⣶⡿⠉⠀⠈⠒⠒⠒⠶⠒⢋⡄⠀⠱⣦⣄⡋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⢂⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠿⠿⣿⣦⣅⠴⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠡⣀⢈⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⣻⣶⠂⠀⠁⠁⠀⠀⠀⠈⠙⣿⣧⡰⡏⢁⡇
⠀⠀⠀⠀⠀⠀⠠⣀⢀⣾⣿⡛⠀⠧⣠⠀⠀⠀⠀⠀⠀⣿⠁⠸⡿⠠⡀⠀⠀⠀⢀⡀⠖⠀⢹⣿⣿⣌⠊⢀⡄
⠀⠀⠀⠀⠀⠁⣠⢁⣾⣿⣿⠀⠀⠀⠀⠀⠂⠆⠦⠆⠀⠈⠳⠛⠁⢶⡀⡄⣄⠄⠛⠀⠀⠀⢸⣿⣿⣿⡎⠉⢁⡀
⠀⠀⠀⠀⠡⢄⡎⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣄⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡈⡈⡁⠄
⠀⠀⢀⠐⢂⣠⡴⠤⡤⢭⡛⢷⣤⡀⠀⠀⠀⠀⠉⠁⠀⠁⠀⡈⠁⠀⠀⠀⠀⠲⢦⣤⡴⠟⣋⣉⣉⣙⠻⣇⠉⠁
⠀⣠⢃⠔⠋⠀⢁⡞⠁⠀⠈⠢⠻⣀⡀⠄⠂⠁⠀⠀⠀⠠⠀⠁⠈⠘⠁⠁⠒⠦⣤⣏⣔⠋⠁⠀⠀⠀⠙⢤⠫⣁⡄
⣴⣽⠉⡄⠀⠀⠈⠚⠠⠀⡀⣀⠑⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢹⣯⠼⠀⠀⠀⠀⠀⠀⠀⠃⠳⡄
⠙⠛⢿⡘⠱⠂⠆⠆⠒⠒⠊⠋⠒⢇⣁⣀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⣠⢐⣊⣥⡼⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹
⠀⠀⠀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠈⡭⡑⠆⠐⠁⠀⠀⠀⠤⠀⠀⡜⠡⠉⠴⠄⢻⢐⡀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡜⠁⠀⡔
⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⢠⣿⣷⣀⣠⠎⠀⠀⠀⠀⠀⠀⢀⣆⠹⣦⣴⡎
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠥⣚⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⣟⢦⣀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡥⠘⡻⡿⠒⢀⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⣭⡓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣭⡄⠀⢠⣾⣇⠀⠀⠀⡆⠀⠀⠀⠀⢠⣾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢖⡦⣼⢻⠀⠀⠀⠗⠀⠀⠀⠀⠘⠁⣠⡶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⢧⠀⠀⠘⢄⠀⠀⠀⠈⠒⠓⢲⠶⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⣿⠡⢿⣷⠟⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣿⣿⣥⣾⠃⠸⢦⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⣿⣤⠟⠁⠀⢀⣼⡿⠋⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⣿⢷⡆⢉⣶⠟⠁⠀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣿⠀⣿⣿⡾⠋⠀⠀⠀⣀⣈⡤⡾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡔⠁⢸⡶⣿⣡⠄⣠⣦⠤⠚⠉⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠅⠀⣀⠄⣸⡇⠘⣷⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⡁⠔⠋⠀⠀⠀⣧⠀⢹⡗⢀⣄⣀⠹⠶⠒⠒⠚⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊⡁⠀⠀⠈⠀⣀⣀⡄⠀⢠⢿⣦⣼⣯⡉⠐⠄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣾⡯⣐⣉⣥⣤⣶⣤⣭⣥⣤⣤⣠⣄⣶⣶⣿⣇⠈⢧⡶⠶⠤⠄⠴⠶⠤⠀⣠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣀⣸⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣦⣈⣧⠠⣄⠠⡁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣼⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡍⢳⡀⠈⠙⠲⢦⣀⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢠⣤⡆⠠⣾⠧⠽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡿⢦⣀⠀⠀⢈⡳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣧⡷⠶⣯⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣙⣯⣴⣙⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠏⠀⠀⠘⣧⣞⣁⠀⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⡙⠋⣤⢼⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⡀⠀⠀⠀⠀⠀⣐⣾⣤⣾⣷⠄⢸⣧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢷⠴⣿⣄⣎⠉⠉⠉⠙⠛⠿⣷⠛⠗⠛⠛⠋⠉⠉⠁⢀⡦⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣼⢷⠿⠛⣿⠿⠿⠿⣷⣶⣶⣶⣦⣤⢴⣶⣶⣶⡷⠿⣋⠤⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣶⡿⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⡇⡷⣛⣛⣛⣲⣙⣑⡛⣟⣥⣄⢷⣾⢻⠏⡯⠧⣴⣿⣷⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣧⣠⣧⠷⣮⣯⣿⣿⣽⠛⢛⣻⣯⣧⣾⣿⣸⡀⣿⡿⣿⣯⣿⣏⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⣯⣷⣦⣤⣤⣤⣬⣽⣿⣿⣽⣍⣉⣩⣼⡻⣥⠖⣸⡟⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠿⠶⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢠⠀⢸⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣥⡿⣦⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢰⠃⣸⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⡴⠖⠊⠿⣛⣛⣻⠗⠛⠲⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢸⣄⣹⣇⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠞⠉⠀⠀⠘⣿⣩⢨⣹⣟⠀⠀⠀⠙⢷⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠙⢹⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⠃⠀⠀⠀⠀⠸⣿⣜⠾⣻⠹⡄⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣸⠃⠀⠀⠀⠀⠀⣾⡿⢣⣷⠺⣿⡆⠀⠀⠀⠀⠀⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⡟⠀⠀⠀⠀⢰⠶⣩⣾⣇⠇⣸⣿⡡⢶⠀⠀⠀⠀⠀⡷⢀⣜⢫⣄⠀⠀⠀⠀⠀⠈⣿
⠀⢀⣿⣶⣄⠀⠀⠀⠀⠀⠶⣾⣯⣻⣿⣷⠆⠀⠀⠀⠀⢂⣳⣿⣽⣿⣽⣧⣇⢃⠀⠀⠀⠀⠀⠟⠇⣆⠖⡥⡟⠃⠀⠀⠀⠀⠸⣷⡄
⢠⣿⣿⠛⠀⠀⠀⠀⠀⡀⠀⢜⠛⣿⠋⢻⣠⠀⢸⠀⢸⢏⣿⣿⣿⢾⣿⣿⣏⢻⣀⣠⡤⠀⢠⡇⠈⢚⣉⡆⠀⠀⠀⠀⠀⠀⢸⢿⣿
⠙⣿⣿⠀⠀⠀⠀⠠⠤⠼⢿⣿⣤⠤⠤⠬⠽⣿⣿⡦⢾⣦⣶⣿⣿⣽⣿⣯⣢⣿⣿⣏⠁⢀⡟⠠⠁⠀⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣙⣻⡄⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⢀⠀⣿⣿⢻⣼⣹⣧⠾⣝⠻⣿⢛⣿⣳⣿⢿⣶⣯⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠈⢹⣶⣦⡀⠀⠀⠀⠘⢿⣏⢀⣤⡾⣿⣷⣿⡿⣠⢻⣿⣿⣷⣮⣤⣷⢿⣿⣿⣷⣾⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⡟⢿
⠀⠀⢸⢿⣿⣿⣦⡀⠀⢈⣻⣿⣾⢻⣿⣿⣿⣋⣙⣻⣿⣿⣿⣿⣳⠖⢣⣍⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣴⣿⣿⡶⠀⢰
⠀⠀⢸⠀⢻⣿⣿⣷⢸⣮⢻⣿⣿⣿⣭⡻⣿⡿⣿⢻⣿⡻⣿⣿⣿⣷⣿⣿⣿⡌⣿⠿⠛⠿⣿⣿⣿⣿⡟⣰⣦⣰⣿⣿⣿⠟⠀⠀⠘
⠀⠀⢘⠀⠀⠹⣿⣿⣿⣿⠸⢿⣿⣿⣿⣿⠉⠀⣿⡀⠙⢿⣿⡿⣿⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠈⠙⠫⣾⣿⣿⣿⣿⣿⠋
⠀⠀⢸⠄⠀⠀⠈⠉⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⢻⣇⠀⠀⢩⡷⠘⣉⣸⣿⣁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠁
⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡀⠀⠀⢾⠏⣿⣿⣿⠻⢗⡄⢹⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢣⣶⣧⣝⣈⠽⣺⣥⣴⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿⣿⣿⡿⢿⣾⣿⣾⣽⣺⣿⣿⣿⣿⣿⣿⡿⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣭⣿⣿⣭⠝⢀⠾⣿⠟⣛⣿⣛⡳⣿⠉⠉⠉⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋⠀⢀⣴⣿⣿⡾⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⡇⠀⠀⣀⡀⣶⣶⣿⣿⠟⠉⠀⠀⠀⠀⣾⣿⣿⣿⣿⣝⣛⣻⣾⠏⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢘⣷⣶⣿⣿⣿⠿⠿⠿⣿⠇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡭⠛⠋⠁⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣟⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⠿⠿⠟⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠿⠛⠉⠀⣶⣭⣛⠿⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣾⣭⣭⣭⣭⣭⣿⣿⣿⡛⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⢻⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟
⠀⠀⠀⠀⠀⠀⢸⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⡻⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣭⣻⠿⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠻⣅⡽⠟⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⣉⣀⣉⢻⣶⠶⠒⣛⡿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⣉⡁⠚⣋⣠⣾⣿⣿⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣏⠻⠿⣿⡿⠀⣹⡍⠛⠋⢁⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠶⠒⠒⠊⠉⠉⠉⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢸⠀⠋⢻⣏⠙⠂
⠀⠀⠀⠀⢀⡴⠖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣄⠀⠀⠀⣠⡄⠀⢸⣸⠀⡀⠻⢸⣿⠀⡇
⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⢄⣴⠷⣄⢸⣟⠫⣇⠸⡆⠀⠀⣿⠀⠀⢸⣿⠀⡇⣾⢸⣿⠀⡇
⠀⠀⠀⣿⢼⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣴⡶⣆⣲⠠⢭⠵⠒⠊⣿⠃⠀⢹⢸⣇⠀⣿⠀⢻⡄⢸⡇⠀⠀⢸⡏⢨⡇⠻⠘⢻⠀⡇
⠀⠀⠀⣿⣾⡇⠀⠠⣶⠀⠘⢋⠀⣿⡇⡿⢋⣀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⢾⣷⣾⠁⠀⠀⢻⡟⠀⠀⠀⢸⡇⠸⠇⠀⠠⢸⠀⡇
⠀⠀⠀⣿⢰⠇⠀⠀⢾⡀⢀⢰⠀⣿⡇⠁⠈⠡⠖⢳⣄⠀⠸⢠⠀⠀⣿⠀⠀⣸⠸⡇⠈⢷⡀⠀⢸⡇⠀⠀⠀⢸⡇⡇⣦⠀⠃⢸⠀⡇
⠀⠀⠀⣿⣾⠀⠀⠀⢸⡇⠌⠸⠀⢻⡇⠀⠀⠀⠀⢀⠼⡁⠀⢸⠀⠀⢻⣄⣰⠟⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇⠇⣿⠀⢀⣼⠀⡇
⠀⠀⠀⣿⣿⠀⠀⢸⢱⡎⠉⣷⠀⢸⡇⠀⢤⣄⣰⣮⠞⠁⠐⠾⠀⠀⠀⠉⠁⠀⠐⠇⠀⠨⣟⠀⢸⣇⠀⠀⠀⢸⡇⡄⠿⠰⢸⣿⠀⡇
⠀⠀⠀⣻⡟⠀⠀⢸⢸⡇⠀⣿⠀⢸⠇⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡿⠀⠀⠀⠀⡇⡇⠀⠀⢸⢹⠀⡇
⠀⠀⠀⣻⡿⠀⠀⢸⠀⠅⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡙⡦⢸⠀⢸⠀⡇
⠀⠀⠀⢻⣿⠀⠀⠸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⠀⠀⠀⣤⣤⡄⠀⠀⠉⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⣿⠁⣇⠘⠀⢸⠀⡇
⠀⠀⠀⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⡄⠀⠀⠠⠟⣦⡀⠀⣇⣸⣟⠄⠀⢰⡂⡀⠀⠑⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⡇⣾⠀⡁
⠀⠀⠀⣿⢹⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢰⠈⣅⡀⡶⢻⠀⠀⠟⠉⠈⣶⠐⠺⡯⣿⡁⠇⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⡇⣿⠀⡇
⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣐⠂⣶⠟⣥⠨⣶⣯⣽⠀⠀⣀⡿⠳⢾⡀⠰⠂⠈⠁⢠⠃⠀⠀⠀⠀⠀⡟⠀⣼⠀⡇⣿⠀⡇
⠀⠀⠀⣧⣾⠀⠀⠀⠀⠀⠀⠀⡿⢀⡌⣩⣸⡇⠘⠁⢀⡏⢁⠈⠻⣦⣿⣷⢦⠄⠛⢶⡖⢠⡴⠃⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠁⣿⠀⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢣⡖⠁⠠⢴⡏⠙⢿⡶⢦⣜⡛⠋⠻⣍⠉⠁⠈⢺⡧⣀⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⢀⠀⣿⠀⣇
⠀⠀⠀⡟⢰⠀⠀⠀⠀⠀⠀⠑⠠⢀⠰⣦⠶⣎⢡⠈⠋⠋⠉⠀⡝⢧⣀⣀⠼⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⢀⡇⢸⠀⣿
⠀⠀⠀⣇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠻⠷⠄⠰⠒⠒⠚⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⠸⡇⣿⠀⢻
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⡇⡏⡆⢸
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠀⡇⡇⡇⢸
⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡏⠀⠀⡇⡇⢸
⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠃⢱⢸⡆
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠄⠀⡇⡇⢸⢸⡇
⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⡇⡇⢸⢸⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠁⡇⢸⢸⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⡇⢸⠘⡇
⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢱⡆⡇⢸⠈⡇
⠀⠀⠀⠈⡷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣽⡷⠗⠛⠚⠁
⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⡤⠄⠀⠀⣸⠿⠛⠉
⠀⠀⠀⠀⣯⣿⠂⠐⢲⣶⣤⠤⠤⠾⠿⠶⠀⠀⠀⠈⠁⢀⣀⣀⣀⣈⣉⣉⡩⠭⠭⠥⠤⠤⠴⠴⠄⠒⠒⠒⠛⠋⠁
⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣴⡿⠂⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⠈⠛⠡⠞
⠀⠀⠀⠀⠀⠀⠀⢠⣭⠜⢓⣶⡰⣁⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⢠⣶⡾⣧⣼⣿⣿⣿⣿⣿⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⢾⢿⣿⣿⣿⠟
⠀⠀⠠⠴⣀⠈⡙⠀⣸⣽⣧⣿⣿⣿⣿⣿⣿⣍⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣶⣮⣉⣁⣌⠜⠁
⠀⠀⠀⢴⠟⢿⡿⠺⢿⠛⠛⠋⣳⣸⣿⣿⣿⡛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣛⣿⣿⣿⣿⠄
⠀⠀⢀⢀⣠⣼⠃⢐⣱⣆⠠⠶⠿⣅⣻⣯⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⠭⠽⠿⠞⣛⡿
⠀⠀⠀⠉⢛⣷⠿⠿⣿⠛⣦⣄⣐⣿⣇⡙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠏⣿⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠋⠲⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠝⠁⣰⠉⡙⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠊⠀⢠⠂⠀⡔⠀⢉⣴⠇⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡅⡸⠁⠀⡔⠁⢀⠄⠀⠠⢆⡀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠁⠀⡸⠀⠠⠋⠀⠰⠁⠀⢠⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠔⠀⢠⠁⠀⡴⠁⠀⠀⡿⣄⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⠁⠀⠀⠘⠀⣠⠂⠀⠘⠀⠀⠀⠜⠀⡇⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⢀⠷⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⡃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡠⠊⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢠⠞⠓⠆⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⣧⣀⠄⠈⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠠⣖⠉⣉⣉⡳⣽⣗⠈⠂⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠈⠻⣦⣄⠀⠀⠀⠀⣰⣿⣤⣤⣴⣢⣤⣶⡋⠀⠀⠀⠀⢀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠜⣧⠤⠂⠀⠉⠉⣿⡯⠊⢻⣮⡛⠿⣿⠛⠷⠒⠉⡷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣅⣌⡄⣿⡌⠄⠈⠳⣤⣬⡿⣡⣿⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⢻⣛⡓⠢⣀⣤⡇⢹⣿⣧⠹⣧⠀⠀⠀⠈⠻⠿⠿⠏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡻⠋⠁⠀⠙⠓⢾⠉⡛⢦⡙⠂⠈⠢⠵⠖⡶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣠⠇⠀⠀⠉⠉⠐⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠚⠉⢀⣠⠖⠋⠙⠲⣄⡀⠙⠓⠦⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⣠⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣀⡀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠙⠦⠴⠋⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣠⠖⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣲⣾⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠛⠛⢿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠉⠁⠀⠘⠛⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣀⡀⠀⠒⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠠⡄⠀⢠⡀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⢄⡀⠀⠀⡘⢿⢀⣾⠀⠀⢰⡀⢠⣴⢾⡆
⠀⠀⠀⠀⢈⣄⡌⡄⡤⠀⢉⣦⠀⠀⠀⠀⠀⠀⠀⠈⢳⡎⠁⢠⡀⠀⠙⠛⣿⢶⣶⡻⠁⠀⠙⣦⠿⠒⣀
⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠹⣗⣠⡦⠀⠀⠀⠀⠀⠀⠋⣀⠀⣀⣶⣀⣀⣴⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⢀⣀⣬⠕⠚⢿⠟⠻⣿⢟⣿⠿⠻⡌⢳⡀⠀⠀⠀⢀⣀⣀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⠙⣇⠀⢠⠀⢰⡏⠀⣇⠀⠀⠳⢘⠇⠀⠀⣴⠟⠣⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⢷⣤⣟⠀⣀⣴⠟⠁⠀⠀⢻⣦⣀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡦⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⡽⠃⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢱⢟⣟⠻⣞⣁⣈⣹⣧⡬⣤⡁⣄⣀⣀⣀⣤⠞⠀⣰⡇⢀⡜⠁
⠀⠀⠀⠀⠀⠘⢭⢷⠼⠿⠛⠋⠛⠛⠛⠾⣿⣿⣿⣿⣿⡿⠁⣠⠞⠉⢻⡟⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢷⣤⣾⢷⣦⢰⣶⣿⣟⣿⣯⣿⣿⣷⡞⠁⠀⠀⡟⠀⣄⡾⣹
⠀⠀⠀⠀⠀⠂⠁⠀⠄⠉⠶⣠⣿⣿⠏⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣾⣟⣼⠹⣽⠋
⠀⠀⠀⠀⡀⠀⠀⢀⡄⠂⠀⠉⠿⠁⠀⠀⢹⣟⣿⣿⣿⣿⢋⣿⣯⣿⡿⢳⣿⡇⣀
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣾⣶⣀⠀⠈⠂⢀⣿⣿⠿⣿⢿⣿⣾⢛⣿⣿⣿⣿⣿⠄
⠘⠃⠈⣛⣻⣿⠛⠋⠉⣻⣧⣹⢷⢄⣀⡈⠉⠀⠀⠈⣟⣿⣿⡿⢿⣶⠋⣿⣷
⠀⠀⠀⠀⠀⣀⣀⣤⣶⡯⢽⣭⠀⣰⡾⠁⠀⢀⣤⢾⣽⡉⣿⣵⡿⠟⢛⣿⡿⡆
⠀⠀⠀⠀⠀⠉⠚⠛⠛⠻⢿⠚⡤⡟⢁⣤⡶⠋⣙⣽⢿⣭⣷⢿⣷⣾⣿⣿⣿⣧⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠰⠞⠛⠓⢋⠀⠘⣟⣿⢊⡴⢿⣿⢿⢿⡋⣴⣉⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⣡⠟⢿⣯⣱⣾⠖⣃⠺⢡⣽⡿⣿⣿⢿⣶⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠻⢒⣽⣻⣦⣰⣩⣍⣼⢿⣧⡉⣉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣋⣿⣿⢿⣷⣤⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣝⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠉⠉⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⢉⡙⠓⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣟⡻⣯⣽⣯⡿⣟⣻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢋⣶⣿⣷⣿⡟⣼⣿⣿⣿⣾⣙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢩⣟⡻⣿⣿⣿⣷⣿⡿⣫⣾⣿⣟⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⢿⣿⣯⣘⣥⣿⣛⣛⡻⠿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣾⣶⣄⣀⠀⠘⣷⢿⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣼⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⡿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢶⣿⣿⣿⣿⡏⠀⠀⠙⢿⣟⣻⣿⣿⣷⣶⣿⣬⣭⣃⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⠀⠀⠁⢈⡄⢋⢻⣿⣿⣷⠀⠀⠀⠀⢙⣻⣿⣿⣿⣯⣼⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⢢⣷⣀⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⠿⠛⢩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠸⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣽⣇⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⣾⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢤⣀⠀⣾⠃⠈⣻⣿⠟⢿⣿⡏⠀⢻⡆⢀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢹⡆⠀⠀⡏⠀⣼⡿⠁⠀⠀⠙⣿⡄⠈⡗⠀⠀⣼⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣷⣿⣦⡀⢻⡄⣿⠃⠀⠀⠀⠀⢹⡇⣸⠃⣠⣾⣷⣿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣯⣥⣭⣧⣙⡚⡇⠀⠀⠀⠀⡾⢜⣡⣾⣥⣽⣿⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⣦⠉⠉⠉⠛⠛⠻⢮⡀⠀⠀⣼⠾⠛⠛⠋⠉⠉⢁⣾⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣈⣓⠦⢤⠀⢀⣀⡤⠟⣰⡘⠧⣄⣀⠀⠀⠤⢖⣋⣁⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣥⣤⠾⠁⠙⢦⣤⣭⣿⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⡍⠄⠌⡝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠁⠤⡠⠔⠠⡄⠤⣉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⠚⠛⡁⠎⠖⠙⠛⠲⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠥⠴⠐⠛⠣⠀⠈⠛⠓⠐⠤⠽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⡠⢂⣴⠀⢹⣤⢉⢡⣼⠃⢑⣔⠠⡈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⣼⠛⠊⠉⠉⢹⣿⠀⢸⡿⠉⠉⠉⠚⠻⡼⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣣⣶⠋⢉⠋⠓⠸⠦⠾⠔⠋⢋⠉⢱⣦⣣⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠒⠒⠒⢾⡿⢻⠯⠐⠁⠀⠀⣄⢠⢀⠄⠀⠀⠁⠨⢿⠻⣿⠖⠒⠒⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣥⡳⠟⢻⣿⡞⢳⠏⢠⠤⣤⣀⠀⢷⣼⣵⠇⢀⣠⡤⢤⠈⢷⠛⣾⣿⠛⠶⣣⣍⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠋⠀⠘⢄⢈⣿⡗⠺⡆⢀⣤⣀⠙⠃⠀⠀⠀⠀⠛⢁⣠⣀⠀⡾⠚⣿⣏⢀⠐⠀⠈⠃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠐⡟⠟⢧⣴⠁⠈⠠⠝⡆⠀⠀⠀⠀⠀⠀⡾⠡⠈⠀⢳⣤⠟⠏⡗⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⠀⢀⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢆⠀⠈⠂⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠇⠙⣦⣄⡄⣰⠀⠀⠲⠀⠀⠀⣷⣄⣤⡞⠀⠇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡐⠄⠀⠀⠀⠀⣿⡿⢃⠇⠀⠴⠤⠀⠀⢠⡇⢻⣿⡇⠀⠀⠀⠀⠔⣊⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⠔⢁⡴⡻⠃⣽⠀⠀⡤⡄⠀⠀⣾⣾⡁⠻⡳⣄⠑⢄⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⡶⠟⢊⣠⠶⣻⠀⠀⢀⡀⠀⢀⡏⠕⠱⢦⣈⠚⠷⣦⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣽⠛⢯⡢⡀⠘⣇⠀⠉⠁⠀⠸⣇⠂⠀⡡⣪⠟⣻⡅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢈⡟⠮⣂⠈⣦⡀⠘⠉⢀⠈⠳⣬⠞⣷⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⠀⠈⠙⠒⠽⢦⣀⠋⠀⠀⠈⠳⡏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⠳⣄⡀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠘⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⣦⣄⡀⠉⠙⠒⠒⠒⠛⠉⠳⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠟⠻⠶⢶⡖⡖⠖⢚⣻⠇⣸⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⠁⠀⠀⣠⠈⠙⠛⠛⠶⠚⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⡵⠚⠉⠓⠶⠤⠤⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠐⢤⣐⠶⡀⢹⣯⠑⢾⣈⡙⢿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⡐⠀⣀⢈⢩⠻⠾⢿⣄⠀⢂⡘⡇⠹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠙⠋⠁⠸⠠⠀⠑⢄⡀⠈⣧⣈⣯⠳⢤⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠈⢉⠀⢘⠏⣯⠻⡼⣿⡀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣤⣤⠤⣄⠀⠀⠀⢩⡾⠛⣟⣷⣤⣀⣳⡀⣸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⠿⠓⠋⠀⠀⠘⢰⣏⣭⣿⠿⠿⡯⣷⣿⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⣼⠟⣼⣷⣏⠀⢱⣀⢻⡄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡽⢢⣾⣿⣯⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣨⣿⣿⣿⣿⣿⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠙⣿⣿⣿⣿⣟⣯⣷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠂⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣿⣿⣿⣿⡏⣯⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⣤⣤⣤⣤⣤⣶⠞⠁⠀⠀⠀⠀⠈⠻⢿⡋⢴⠟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡈⠓⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⢀⣀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠙⢿⡿⠁⠀⣀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⡀⠹⣿⣷⣶⣶⣶⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠛⠿⠿⠿⠿⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠉⠉⠀⠉⠻⣿⣿⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⣰⣿⣿⣿⡿⠋⠉⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⡿⠋⠀⠀⣠⣿⣿⣿⡟⠁⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢰⣧⡀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⡿⠀⠀⠘⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⣀⣽⣭⣭⣭⣭⣯⣯⣭⣽⣍⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠼⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠶⣿⢿⡿⢧⣹⣿⣿⢟⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⠿⠾⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣧⣠⡄⠀⠀⠀⢸⣿⣿⣿⣟⢻⣆⠀⠀⠀⡴⠛⠻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠙⣿⣿⣿⠿⢿⠀⠀⠀⠿⠋⢠⣿⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⢠⠀⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⡔⣶⢻⣿⠟⠀⠀⠀⠀⠀⢠⣾⠀⣿⣿⣿⣧⣄⠀⠀⠀⢀⣿⣿⣿⡆⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣹⣿⠞⢡⠆⠀⠀⠀⠀⣼⣿⢃⣼⣿⣿⣿⣿⣿⠀⠀⠀⣿⡛⣿⡟⣷⡤⠀⢿⡗⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢀⣤⠟⠛⠿⢿⣄⢹⣿⣿⣿⡀⠀⠀⢿⣿⣧⣾⣯⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣄⠀⠀⠀⠀⣾⡁⠀⠀⠀⠈⣿⡦⢿⠿⣿⣷⡀⠀⠸⠟⣻⣿⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠸⠧⠄
⠀⠀⠀⠀⠀⠀⠘⣿⠿⠋⠀⠀⡄⡀⠘⢳⢄⣤⠀⠀⠘⡇⠀⢦⠀⢻⣿⣶⣄⠀⣿⠋⠀⠀⢠⠀⠀⠀⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⠰⡇⣤⣰⣿⡄⣿⡄⠀⠀⣹⡀⠀⣱⣿⣿⡷⢻⠀⡸⢀⡀⠀⣾⠀⠀⣰⡏⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⢰⣿⠁⣿⠟⠀⣿⣿⣿⣿⣿⣿⠀⠻⢿⡛⠛⠃⣠⡧⣼⡗⠈⠻⣿⣧⣠⡅⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣾⣿⣦⣿⣥⣰⣿⣿⣿⠛⠙⣿⠂⠀⢤⣿⡆⢀⣿⡇⠀⠀⠀⠀⠈⢳⣾⣿⣶⠋⠲⢰⡦⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠈⠉⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠆⠀⢺⣡⠎
⠀⣴⣦⣄⣀⣀⢀⣠⣦⢸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣇⠀⠀⣿⣄⡀⠀⡀⠀⢀⣇⠀⠀⠀⠀⠀⠀⢸⠟⠳⠶⡶⢿⣛⡩⣤⠶
⢠⣿⣿⣿⣿⣿⣿⡞⠉⠘⠋⠉⠿⠛⢁⣹⣿⣟⣿⣿⣷⣄⣿⣿⣵⣿⣤⣤⣸⡇⢀⠀⠀⠀⠀⣠⣼⠄⠀⠢⠀⠀⠉
⣿⣿⡿⠿⠁⠘⠃⠀⠀⠀⠀⠀⢀⣴⡾⠛⠙⣿⣿⣿⣿⣿⣿⣿⣄⢈⡻⢛⡏⢀⣼⣿⣶⠶⠟⠉
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⠁⠀⠀⠙⠛⠿⣿⣿⣿⣿⠿⣿⣿⣿⣟⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⡿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣲⣶⣿⡟⣻⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣷⣶⣿⠿⠋⡩⠟⢀⣿⠃
⣿⣧⠀⠀⠀⠀⠀⠀⣈⠀⣿⣿⣏⢻⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠠⣾⠏⠀⣸⣿
⣿⣿⣧⠀⠘⠀⠀⣰⣿⣿⣿⣿⢹⡄⣿⣷⡀⣦⢀⠀⠀⠀⠀⠠⣉⣛⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣼⡿⣿⡗
⣿⣿⣷⣄⠀⢡⠀⢻⣿⣿⣿⣿⡟⣿⣼⣿⣿⣿⣼⣶⣠⣀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣿⣦⣀⠱⣄⠙⢿⣿⣿⣾⡋⠉⣿⣧⣿⣿⣿⣿⣿⠳⣤⡀⠂⢤⣤⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿
⠀⠀⠀⠈⠭⠛⠛⢩⠳⣤⣙⠻⣿⣿⣄⣹⣿⣿⣯⣽⣿⣿⣿⣮⣿⣶⣤⣈⣭⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠈⠉⠋⠉
⢶⣶⣶⣤⣄⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⣠⣔⣤⡤⠚⠛⠛⠓
⠤⢿⣿⣿⣿⣷⣤⣄⣀⣀⣘⣿⡿⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣯⣶⣶⣶⣶⣿⣿⣿⣿⣿⣆⡀⠀⠀⢈⡻⠟⠛⠉⠉
⠀⠀⠈⠉⠙⢿⠻⡿⣿⣿⣿⣿⠃⠸⣿⣿⢿⣿⣷⠾⢻⣿⣟⢛⠛⠛⠛⠛⠻⠿⠿⠿⠏⢠⣿⣾⣭⣭⣥⣤⣤⣶⡾⠟⠉
⠀⠀⠀⠀⣤⠴⠶⢶⡿⠿⣿⣿⣷⣶⣶⣶⣮⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⡿⠿⠛⠋⠉⠁
⣴⣿⣷⡆⠀⠁⡄⠸⡇⢠⡟⠋⢉⣳⣄⣈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣼⣿⣿⡷⠶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣏⠙⠛⠦
⣿⣿⣿⣷⠿⠋⢩⣿⣿⣿⣿⣿⣿⣿⠛⠛⠳⠶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⢀⣤⣄⠈⠙⠿⣶⣤⣄⡀
⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠛⠉⢀⣀⡤⠞⠋⢹⠉⠛⠦⣄⣀⠈⠙⠛⠻⠿⠶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⣿⠇⠀⢠⣤⣤⣤⣤⣤⠤⠶⠖⠚⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠉⠛⠒⠶⠦⢤⣤⣤⣤⣤⣤⠀⠀⢿⡇
⠀⠀⢠⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⡇
⠀⠀⠀⣿⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⡇
⠀⠀⠀⢿⡇⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿⠀⠀⣿⠃
⠀⠀⠀⢸⣷⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣿
⠀⠀⠀⠈⣿⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⡇
⠀⠀⠀⠀⠸⣷⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣸⡿
⠀⠀⠀⠀⠀⠹⣷⡄⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠉⠛⠶⢤⣄⣀⠀⠀⢸⠀⠀⢀⣀⣤⠴⠞⠋⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢶⣶⣤⣤⣤⣴⣶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠻⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢴⠊⢿⣿⣿⣿⣧⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢁⠍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⣅⡐⣿⣿⣿⣿⣯⡐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⢉⣤⣿⠏⠙⣿⣿⣿⣿⣿⣇⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡾⢃⣾⣿⠤⠤⣾⣇⠉⠻⢿⣿⣿⣷⡀⠀⠐⠆⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠸⠇⠸⣿⣠⠀⠠⠋⠙⠃⠀⠀⠉⠻⣿⣷⣶⣾⣶⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⣬⣿⣄⣀⣤⡾⡿⢿⣾⠃⣀⣉⠐⢆⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢻⣦⢉⣿⣿⠀⠀⠰⢾⡏⠀⠛⠉⠀⠀⠀⠀⣀⣄⠘⣿⣿⣿⣿⣷⣾⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣿⡷⣾⠋⠉⠶⠄⠀⡸⠰⡟⠀⠀⠀⢀⣦⣄⣤⠀⣄⠀⠟⠻⣿⣿⣿⣆⠀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠘⡟⢀⣴⣤⣷⡴⣿⠈⢻⣶⣶⣄⣠⣿⣿⡶⠀⢸⡇⠀⠀⠉⢹⣿⣿⣿⡟⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠘⢷⣏⣙⣿⠋⠛⠷⣤⣤⣄⢿⣿⠛⠛⡿⢿⢷⡎⠀⠈⠰⣤⣷⣶⢿⣿⣿⣄⢀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠛⠻⠿⣤⣶⣤⠀⠈⠟⣠⣿⠇⣠⣿⠒⡿⠃⣀⣛⣿⠿⠿⠏⠀⢹⣿⡿⠿⢷⣴⠖⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆⠀⠀⢀⡀⢀⣄⠀⣀⣤⠀⢺⡿⢿⣿⣿⣿⣧⡀⢰⣿⡟⡟⠃⠀⠐⠛⠿⣿⡇⢀⣾⡅⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⠀⣴⠞⣿⢧⣤⣶⣿⣏⡉⠻⣧⠽⣿⣿⣿⣿⣿⣿⣿⣽⢻⣤⢛⣼⣿⣷⠋⣠⣾⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣿⡄⠉⠐⠉⢿⣷⡿⠋⣀⣹⣶⣾⣿⣿⣽⠟⣭⣿⠏⠸⣿⣿⡝⠋⣹⣤⣽⡛⠛⣯⡀⠀⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣿⣷⢆⣤⣥⣿⣿⣿⡀⢉⡀⢺⣿⣟⣩⣿⣿⣿⣿⣶⣿⣿⣶⠷⢎⡁⠀⠚⠉⠙⠛⢿⡿⠄⢁⣤⠺⣿⠆
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⣿⣿⣿⣤⣤⣈⣀⣨⡉⢿⡿⠉⢻⡿⢿⣿⠟⠛⠿⢟⢻⡇⠻⠛⢠⣀⣿⠀⠀⠀⢺⡆⢼⣧⣄⡈⢉⣥⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯⣥⠘⣷⣀⣼⠓⢂⣠⣶⠶⡶⠋⠈⠿⠀⠀⣟⠻⢻⣤⠄⠀⠀⠛⣮⣉⣾⣧⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠙⢿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⡷⢏⣹⣿⣿⣧⣼⣇⠀⠀⠾⣧⣴⣿⡁⠀⠀⠀⠀⠉⠁⢩⣙⣿⠟⠉⠀⣡
⠀⠀⠀⠀⡀⢀⡇⣀⣼⣧⣷⣤⣿⣿⡿⠛⣿⣿⠉⣿⣿⡿⠛⢿⣤⣾⣿⣿⣿⣿⣿⡻⣿⣶⣾⡏⢁⣄⠙⢓⣤⣤⠄⣤⣴⠛⠛⠻⣶⣾⣷⡿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡙⣻⣿⢁⣀⠁⠙⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠟⢻⣧⣤⣶⣤⣉⣋⣛⣯⣤⣾⡇⢠⣿⣿⣿⣇
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣉⣿⣿⠖⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣷⠂⣹⣿⢉⠽⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣷⣄
⠈⠿⠿⢻⣿⣿⣿⠛⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠋⠙⣿⣶⣿⡁⢙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠉⠙⠃⠰⠿⠿⢿⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣾⠿⠿⣿⡟⣛⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⠛⠿⠟⠿⠿⠛⠛⠛⠻⣿⣿⠷⠿⣿⣟⣿⠖⠋⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠏⠙⢻⡏⠉⣙⠛⢛⣏⠛⢹⣽⠟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠘⠛⠃⢀⠤⣿⣿⡷⢄⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠰⠿⠷⢿⠇⢀⠀⠀⠀⢀⡀⢀⣿⣿⡀⢀⠀⠀⠀⠀⠛⣸⣿⡇⠀⢦⠀⠀⠀⠿⣄⣤⣴⣮⣿⣵⣤⣠⡾⠃⠀⠀⢸⡀⣸⣿⡀⡼
⠀⠀⠀⠛⠶⣾⣿⣿⣷⠞⠁⠀⠀⠈⢳⡮⠈⠈⢵⡞⠀⠀⠀⢰⣼⣿⣿⣿⡆⠈⠆⠀⠀⣄⣈⣼⣿⣿⣿⣿⣿⣯⣶⡶⠀⠀⠀⣸⠍⢹⣿⡆
⠀⠀⠀⠀⠀⠘⠛⠛⡇⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠀⠻⣿⣿⡿⠇⠀⠁⠀⠀⠀⠘⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠛⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠿⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠏⢠⢀⣰⡀⢀⠁⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⢘⣿⣇⣀⠀⠀⠀⠀⠀⠱⣼⣿⣯⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣿⡀⣀⡄⠀⠀⠻⠐⢦⣸⡤⣿⣧⣼⡠⠆⠀⠀⠀⠀⠠⣿⣧
⠀⠀⠀⠃⠙⣿⢿⣿⡇⠀⠀⠀⠀⠠⢠⣿⣿⡿⢧⣀⠂⠀⠀⠀⠈⢓⣷⡠⠖⣿⢃⡀⠀⠀⠀⠘⠓⣫⠇⣀⡸⣏⠓⠋⠀⠀⠀⢀⠇⠀⠰⣆
⠀⠀⠀⠀⡎⠀⠈⠁⠉⠁⠀⠀⠀⠀⠘⠏⠉⠙⠀⠈⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠛⠁⠀⠀⠀⠀⠐⡋⠈⠉⠁⠈⠃⠀⠀⠀⠀⠋⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠟⠀⠐⠉⠀⠿⠀⠀⠀⠀⠈⡛⠃⠘⠂⠙⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⡄⠀⣼⣷⠀⢠⡇⠀⠀⠀⢻⡄⢸⣿⣿⡄⣰⠃⠀⠀⠀⢀⣀⣸⣿⡀⠀⠀⠀⠀⠘⡇⣰⣿⣆⠀⣠⡄⠀⠀⠀⠀⠀⢰⠟⡇
⠀⠀⠀⠀⠙⢻⣿⣿⣟⠋⠀⠀⠀⠀⠀⠙⣳⣶⣷⡟⠋⠀⠀⠀⠀⠀⣭⣽⣿⣡⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠉⠿⠃⠀⠀⠀⠀⣴⣶⣶⠄
⠀⠀⠀⠀⠀⢈⠿⠿⠙⠲⠤⠀⠀⠀⠀⠀⠙⠿⠿⠙⠲⠄⠀⠀⠀⠘⠋⠹⠿⢸⡇⠀⠀⠀⠶⠚⣙⣛⣏⠙⠲⠖⠂⠀⠀⠀⢸⡿⠿⢿⡇
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠌⢁⡠⣌⢁⠠⠄⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠀⠀⠀⠀⠀⠐⠿⠀⢿⠁⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠈⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠟⠋⡉⡉⢉⢂⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢘⠀⠀⠀⡌⠋⠬⣩⡅⢨⠉⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠸⣷⠂⠀⢠⠀⠸⣿⢿⡟⠁⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠐⠄⠀⠀⠸⠾⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠆⢀⠰⠘⠀⠀⠀⠀⠀⠓⠂⠒⠘⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠁⠀⠀⠰⠇⠀⠀⡌⠐⠒⣖⠚⠉⠀⠖⠀⠀⠀⠐⠂⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠄⡀⠀⠀⠄⠬⠀⠀⠀⠀⢼⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣽⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⢜⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⠐⡀⠀⠀⠀⠀⢳⣈⣩⠀⠀⠀⠀⠀⢰⠀⠐⠒⠀⠀⠀⠀⠑⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⢦⠀⣀⣤⠞⢁⡼⢃⡄⠀⠀⠀⠀⠀⠀⢀⠀⠐⠶⠚⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⢁⣤⠼⢋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢉⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⢤⣤⣦
⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⣠⠀⠀⠀⠀⠠⠀⠆⢶⠄
⠀⠉⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⠀⠋⠀⠀⠀⠚⠃⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡄⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠂⠐⠒⠒⣂⣈⣀⣀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡞⡟⠳⣄⠀⣀⡀
⠀⢰⢻⣌⣖⠋⠻⣄⣹⠶⣄
⠀⠘⣟⢸⡉⣳⡀⠈⠳⣄⠈⠳⣄
⠀⢰⠻⡄⠉⠉⢳⡀⠀⠀⠙⢆⠈⠙⢦⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣿⠛⢦⠀⠈⠲⣄⠀⠀⣱⣦⡀⣙⢷⣄⢀⣠⠞⡭⣷
⠀⢀⣴⣇⠒⠚⢇⠀⠀⠈⢧⡀⠈⡠⢷⡄⠋⠙⣯⡟⠀⠳⢿⡀
⠀⢸⠁⠙⡄⠀⠈⢆⠀⡠⣢⡙⣦⠀⠀⢻⡄⠀⠘⣧⡀⠀⠸⡇
⠀⡆⠀⢀⠿⣄⠀⠈⠣⡀⠀⠘⠉⣧⠀⠀⠙⣆⠀⠈⠳⣤⡀⠙⣆
⢰⠇⠀⢹⠀⢹⡆⠀⡴⡟⢦⠀⠀⠘⣧⠀⠀⠘⢦⠀⠀⠈⠻⣄⠙⣆
⣼⠀⠀⢸⡀⠉⢻⡄⠀⠀⠞⢦⡀⠀⠘⢷⡀⠀⠀⠑⠀⠀⠀⠈⠓⠮⠳⣄
⡇⠀⠀⠜⢧⡀⠈⣧⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠹⡄⠀⠀⠀⠱⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠙⣆⠀⠀⠀⣹⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠈⢦⠀⠀⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡆⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣧⠈⣧
⠀⠀⠀⠀⢳⡀⠀⠀⠔⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠸⡄⢸⡄
⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣳⡈⢧⣀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣨⠷⠋⠁⠀⠀⠈⠛⢳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⡸⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠙⠳⠦
⠀⠀⠀⠀⠀⠀⡈⠀⠙⣆⠀⠀⠀⠁⠀⣸⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⣇⠀⠀⠀⢀⠼⠁⣠⣿⣿⠳⣄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠈⠓⠦⠀⠀⠒⠾⢡⠞⠁⢀⠈⠻⢦⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠸⣧⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠨⢸⣄⣇⣿⣆⣀⣤⠤⠞⠁⠀⠀⠁⠀⡏⢀⡾⠁⡇⠀⣄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠳⡉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣧⠀⠙⣦⣀⣀⣠⠤⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢲⠒⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⣀⣼⠷⠒⠊⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣷⣄⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⡀⠀⣀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠛⠿⠿⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣶⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⡶⢒⠀⠈⢰⢠⣷⣄⠴⠖⣲⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠫⣅⠢⣷⡓⠀⠀⠃⠠⣼⠟⢿⣿⣿⣿⠿⣍⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢝⣥⣏⡙⠛⣿⣧⣽⡧⢤⡤⢀⣽⣧⠀⣉⠁⡈⢰⣿⡿⡟⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⡾⣿⣦⣩⡿⢶⠏⣻⣷⣽⣿⣓⣚⠋⠻⣷⡟⢻⡇⠀⣗⠀⣼⣉⣳⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠜⡋⣹⣾⡟⣃⢌⣹⣽⠟⣿⠂⠿⣿⡟⢿⣿⢿⣿⣹⣬⡻⣄⣷⣤⣉⡄⠙⣿⣏⢅⠙⢿⣶⡄
⠀⠀⠀⠰⠟⠩⣍⣩⡿⢧⡡⣿⡿⢿⣼⠃⠀⣀⣨⠀⣿⡜⣿⣷⡷⣾⣿⣿⠿⣿⣿⣿⣮⣿⡟⠻⠿⣶⣎⢻⣎⡙⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠛⠀⠀⠀⠀⠙⠛⠋⢗⢈⣇⠸⣿⣷⡹⢻⣿⣶⣮⣿⡿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣩⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⢻⡓⠉⠹⢿⠿⣿⣿⣞⣷⣶⣌⠻⠿⠿⣿⡿⠯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠨⠀⠀⢀⠘⠀⠛⠩⠻⢿⣏⡿⢷⣿⠓⠪⠱⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣶⣤⣾⣵⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠁⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⡀⠁⢀⢢⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠐
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠂⠁⠀⠀⡠⠊⠤⠀⠈⠂⢈⣁⣈⡀⠀⠀⠞⣆⠑⠄⠀⠆⢂⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠐⢅⠑⢀⡀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠖⡧⠀⠀⣠⠃⠀⠂⠀⠀⡀⠂⠉⣒⠐⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⢦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡎⠁⠀⠀⠀⠀⠀⡀⠄⠂⠉⡇⡸⠥⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠢⠒⠑⠀⠀⠀⠁⠀⡄⠂⠖⠺⠁⠀⠀⠀⠀⠈⠀⠎⢸⠈⠁⠆⠀⠀⠀⠀⠀⠈⠦⣴⡉
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣃⠀⢐⠀⠀⢀⠀⠀⠀⡦⠀⠀⠂⠄⠠⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣀⣀⠓⡄
⠀⠀⠀⠀⠐⠠⠀⢀⠩⠅⠀⡈⢇⠄⠉⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠀⠀⡠⠀⠀⠘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠆⠀⠀⢀⢀⢀⡀⠀⠃⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⢠⠀⠀⣀⠄⠄⠀⡀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⡀⠴⠀⠀⠀⠈⠁⢸⠀⠀⠀⢼⠀⠀⠘⠀⣠⠰⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠈⢀⠀⢀⡤⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠠⠠⠀⠸⣄⠀⢨⡄⢀⣤⡀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠈⠀⡦⠉⠥⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠄⠀⠒⠒⠂⠂⡔⠁⠀⠀⠀⡂⠀⠀⠀⠄
⠀⠐⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠄⠤⡁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠦⠱⠈⠗⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⠀⡈⠁
⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠈⠉⠲⠋
⠀⠀⠀⠀⠀⠉⠠⠀⠀⢀⠰⠤⡄⢠⠄⠤⠄⠀⠀⠄⠀⡀⠀⠀⠀⠚⠁⡘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⡀⠂⠀⠀⠀⠀⠀⠢⠈⠀⠀⠁⡀⠈⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠄⠈⠀⡠⠈⠀⢤⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢁⣀⠀⠤⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢓⠲
⠀⠀⠀⠀⠀⠀⢸⠟⢿⣿⣿⡿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⣿⡇
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠧⣉⣁⡀
⠀⠀⢀⣤⠠⠿⢿⣿⣿⣿⣷⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣴⣶⣿⣿⣿⣿⣿⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠈⢃⣀⢤⡀
⠀⠀⠀⣽⠀⣀⣀⠀⠈⠉⠛⠿⣶⠟⠋⠀⠀⠀⠀⣠⠤⠔⠒⠉⠁⣀⡴⠾⠛⢛⠩⢸⣿⣿⣿⠀⠀⠄⠀⠀⢠⡄⠀⣴⣶⣿⠟⠻⣷⡤⠆⠄
⠀⠀⢴⣿⡿⣿⠟⠒⠒⢂⡀⠀⠛⢧⣤⢠⠀⣤⠄⠀⢀⣀⣀⣶⣿⣥⣶⣶⣾⠁⠀⠘⣿⣿⡿⠂⠈⠁⢀⡀⠀⠨⣭⣿⣿⣦⣄⢻⣏⣉⠀⠀⣠⣀
⠀⠀⠀⠀⢠⣿⣶⣤⠤⢒⣶⠀⣤⠀⢡⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡄⣿⣿⣼⠀⢸⡇⠈⠉⠉⠛⡛⣧⣀⢸⣿⠤⠾⠋⣩⣶⣺⣯
⠀⠀⠘⣻⣿⣿⣿⣿⣶⣿⣿⠋⠉⢩⡍⣾⣿⣿⣿⠃⣿⣿⡇⣿⣇⣤⣿⣿⡇⠀⠀⡄⣿⣿⣿⠀⢸⡇⠀⢸⠀⢠⣿⡅⠘⣿⣇⣀⣀⠪⢴⣟⣻⣯⠀⠂
⠀⢀⠸⢍⣭⣷⣿⣿⣿⣯⣿⠀⣤⣾⣿⣿⣿⣿⣿⣶⣿⣿⣷⣿⣦⣞⣿⡿⡇⠀⠸⠇⢿⣿⣬⣄⢸⡇⣀⠸⠀⢸⣿⣿⣧⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣅
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣟⢹⣸⣿⣿⣿⠿⢛⣩⣼⡿⠿⠟⣻⣿⣻⣿⣿⣿⡷⠀⣤⣶⣿⣿⣿⢽⡉⠱⠛⠂⠺⠶⣼⣻⣟⣿⣭⣯⣽⣿⣿⣿⣿⣿⣿⡧
⠀⢀⣤⣥⣤⣤⣤⣿⣿⣿⣿⣿⠩⠉⠉⠉⠉⠉⠉⠙⠒⠛⠋⠉⠙⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠩⠍⠉⠉⠉⠁
⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠈⠀⠰⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡽⣿⣿⣷⣦⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡤⠄⡄⡼⢠⢤⢆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡶⠅⠭⣻⣿⣯⣯⣓⡋⠤⢍⡳⠒⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠁⠁⠈⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢲⠁⠀⠀⠈⠉⠛⠿⣽⡟⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⣀⡀⡄⠀⠀⠢⢹⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢉⠑⠴⠆⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠐⠠⠐⠒⠉⠀⠐⠀⣾⡇⠐⠀⠀⠈⠁⠁⠀⠀⠰⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠏⢂⠀⠀⠃⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⠂⠀⠀⠀⠀⠄⢀⣀⣿⡇⡤⠀⠄⠤⠀⣀⠀⠀⣰⠄⣥⠀⠀⠀⠀⠀⡄⠀⠀⠢⠴⠇⢢⡼⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⠀⠀⠁⠂⠀⠀⠰⣿⣿⣏⡇⠀⠀⠀⠀⢒⢒⡃⣇⠁⣛⠀⠀⠀⠀⠀⡃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠃⠀⠀⠀⠀⠀⠀⠠⢈⣼⠇⡁⠀⠀⠀⠀⠀⠀⠀⢸⠢⢸⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⡏⠀⡈⠀⠀⠀⠀⠀⠀⠀⠠⢰⣺⡃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⡗⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⠁⢐⢧⠀⠄⠒⠀⢉⠀⣀⠀⠸⣿⡇⠄⠀⠀⠈⠁⠀⠂⠄⢠⠗⠀⠈⠀⠀⠀⠀⠁⠙⠊⠀⠀⠀⠄⡀⠀⠈⡇⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⢀⣿⣤⠀⣠⡠⢴⠆⠃⠠⠀⣉⠉⣴⣿⣧⣶⣾⣛⠡⢍⠰⢂⣤⣤⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠦⠗
⠀⠀⠀⠀⠀⠜⠛⠟⠛⠓⠀⠀⠒⠐⠂⠈⠙⠛⠻⢿⡿⠟⠉⠉⠉⠑⠀⠂⠐⠒⠦⠥⠻⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠠⠤⢶⢄⠤⢤⣀⡀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠉⠉⠉⠁⠂⠰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠈⠁⠀⡤⠀⠀⡀⢀⠈⣌⢨⠉⠓⠤⠀⡠⠎⢅⢠⠂⠈⡄⠈⠀⠐⣀⠁⠀⠀⢉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡀⢁⡵⠞⠁⠀⠀⠀⠈⠉⠁⠉⠐⠢⡔⣊⡴⠁⡀⠀⢈⣿⠐⢆⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⢢⠈⡱⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠄⣀⠀⠀⠟⠀⠒⣡⠘⡁⠴⠆⢨⣤⡄⢀⠀⠀⠀⠀⠈⠀⠀⠰⠟⢠
⠀⠀⠀⠀⠀⠀⠀⡘⢀⠠⡞⠀⠀⣠⡞⢍⠀⢔⡠⠐⠁⣆⣠⣀⣥⠟⣣⠄⣠⠀⠀⠀⠀⠀⠘⠛⣣⡜⣄⢭⠀⠀⠀⠀⠀⠦⢀⡈⢁
⠀⠀⠀⠀⠀⠀⠀⡇⢀⡈⠁⠀⣸⠇⠠⢛⠈⠁⠀⠀⠀⢈⡩⠀⠸⠏⠉⠀⠴⠤⢀⠠⠀⠀⠄⠘⠧⠘⢏⣿⠁⠀⠀⠀⠀⠀⠀⠀⡨⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠄⣡⡆⠀⣃⠈⠀⡧⠀⠀⠀⠀⠀⠁⠀⢀⣾⡃⠀⠺⠧⢁⠄⢲⡾⡀⠐⢤⡀⢜⣀⠱⣦⡴⠀⠀⠀⠀⢸⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠁⣷⠀⠩⣄⠈⠁⡀⡀⠀⠀⠀⠀⠀⠀⠛⠓⣤⡶⠀⡋⡀⢀⠴⠈⠀⢾⣿⡹⢉⡆⠉⡁⠀⢀⣤⣔⣚⠒⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢁⢇⠀⠈⠻⢾⣇⠳⠄⠀⠀⠀⠀⠉⠀⠄⠀⠓⠈⠀⠀⠀⠀⠘⠓⠂⢈⣧⣸⠷⠄⠀⠀⠜⠑⣿⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢀⡷⠊⠐⢷⣙⡣⢦⡆⠛⠟⢹⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣌⣁⠐⠁⠀⠐⡁⠀⠀⠠⣿⠂⣇⠡⣰⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢋⠄⠀⠀⠀⠩⠄⡨⣼⠳⣈⠯⠜⠀⠀⠀⠀⢀⣀⢠⠄⢲⠄⠬⣿⡿⢤⠄⡢⣶⠾⣜⠂⠀⡙⢤⣭⣓⢃⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠆⠀⠀⠀⠀⠰⣶⣳⠀⠄⣠⣤⠀⠂⠲⠄⠁⠀⠀⠀⠰⠂⠀⠉⠳⢛⡌⢻⣴⢀⣷⣾⣦⣈⣾⣙⣞⡌⣧⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠀⠀⠀⡶⣯⡉⣀⣀⣀⡀⠤⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⣿⣿⣿⣿⡟⢉⣭⡼⠟⢧⣄⠘⣿⣯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⢀⠀⠐⠻⣾⣿⡟⠋⠉⠀⠒⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠙⡇⣹⡿⢛⡠⠖⣈⠩⡔⢂⠒⡄⠘⠶⣝⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡅⠹⠀⢸⣿⠀⠀⠐⢲⠄⠀⠀⣤⣤⡀⠹⠇⠀⠂⢠⡴⠖⠓⠠⣴⣫⠔⣋⣴⡾⠋⠀⠈⠐⠡⠀⠀⢀⣈⢻⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⣸⡿⠀⠀⠀⠀⠉⠢⠄⠻⠿⠋⠀⠀⠀⠀⠀⢲⡦⣄⣠⣿⣵⣾⠅⠀⠀⠀⠀⠀⣀⠀⠀⡴⠋⠁⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡄⡠⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣟⣻⠟⠉⣀⣤⠴⣦⣤⡀⠉⠀⢈⣵⣦⠈⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣏⣻⣴⠻⠟⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⢼⠀⠞⠁⣀⡀⠀⠈⠁⠀⠀⠈⠙⠃⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⣼⣿⡷⢺⡆⠰⠺⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠏⣄⠠⠀⠣⡀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠄⣀⠀⠀⢀⠸⣿⣿⣿⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠟⠿⢯⡸⠟⠄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⣠⣽⣿⣿⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠆⠀⠀⢠⠟
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⢠⣄⠀⣈⢭⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⢋⣹⣿⣿⣄⣼⠻⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢶⡉⠀⠙⠻⣧⡄⠀⠀⠀⢀⣀⠀⢀⣀⡀⠤⢀⣤⡀⠆⠄⠠⠀⠾⢻⣿⣿⡯⠹⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡐
⡄⠠⠄⠰⠶⠋⣛⠀⠶⣖⠀⠀⠀⠀⠀⠈⢤⡘⢋⣤⠌⠉⠠⠦⠉⠁⠐⢂⣤⣰⣧⣶⣤⣶⣶⣾⣿⡏⢾⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠁⠈⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠹⣾⣷⣀⠆⣠⣤⠤⣴⣢⣽⠻⢉⣿⣿⣿⣿⣿⠿⠿⣿⣄⣖⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⠒⢲⡖⢤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠂⠈⠌⠛⠙⠟⠛⠉⠀⠒⡠⢾⠁⠈⠉⠉⠉⠀⠀⠀⠈⠻⡧⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠙⢶⠁⠓⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠷⡈⠷⠀⣷⣠⢄⡀⣀⣀⣀⡤⣰⡯⠒⣶⣄⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣝⠓⡦⣀⠀⠀⠀⠀⠀⢢⣍⠀⠀⠀⠞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⣑⡶⢚⠚⣍⠫⡝⢠⡑⠘⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠋⠐⠓⠲⢿⣋⣬⡦⠀⠀⠰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣹⠉⣡⠛⢤⠶⠀⢉⠀⡐⠀⠉⠙⠛⠛⠁⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⣯⢷⣼⣇⠀⠀⡃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣤⡄⠈⠭⢀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢻⣿⠾⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⠀⠀⣾⣷⣶⣤⣤⡿⠛⠁⠀⠀⠀⠀⠀⠙⢇⠡⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠙⠃⠀⠸⠂⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠦⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠠⠀⠀⡴⠂⡶⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣶⣤⣤⣤⣤
⣿⣿⣟⠋⡙⢻⡉⣽⠛⣻⡙⢹⡿⢻⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡲⣟⣿⠲⠶⡄⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⢲⢞⠠⣮⣯⣵⡏⠻⠏⠳⢙⠳⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣕⠱⣢⢽⣿⠸⣿⣏⡋⠀⣀⣰⣠⣆⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣔⢙⢾⡜⣯⡗⣦⣼⡹⣖⣽⣿⣿⣿⣿⣿⣷⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡱⡬⢥⣬⣮⢿⣿⣶⣼⣿⡻⢻⡻⠹⣿⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⢶⣿⡿⠿⠿⢶⣿⢋⠖⢿⣠⢾⣽⣿⠟⣡⢶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⣷⢚⢊⣾⡿⡹⡟⣿⣙⡿⣿⠁⠀⠻⣛⢰⠝⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣜⣥⠷⣯⡭⢾⢛⣷⡟⠸⣿⣿⣪⣦⢄⡀⠉⠻⣿⢻⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣉⠐⠾⣥⢀⡊⢿⣟⢿⣇⣽⣜⣩⣿⢿⣽⣯⡻⠆⠀⣿⣽⡛⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠯⡽⣸⠶⣿⣏⢉⣀⣿⡍⣚⢿⣴⠽⣛⣇⣭⣻⣿⣿⡾⣧⡷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠐⢴⠗⣀⣿⣿⣨⣉⣿⡇⠊⢯⠈⢻⣷⣏⣻⠉⢯⠟⣥⡶⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢲⣛⣯⣄⣠⢰⢯⡠⢀⠙⠇⠘⡛⠱⢶⡳⣭⣿⠛⠿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡒⠍⢽⢤⡦⠟⣓⢂⡨⠈⣦⣦⡇⣚⣸⣎⣟⡿⣿⣝⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠷⠟⣇⡅⠜⣷⣄⠼⡣⡐⣹⣥⠟⠳⡟⠐⡷⡫⢻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣟⡡⣥⡦⣉⣿⢷⣣⢜⠿⢥⠐⠦⣙⡽⠷⡟⠥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠾⣅⣀⠀⠀⣭⢥⠈⣹⣩⣮⣥⠖⢲⠒⠋⠋⠋⡿⠀⠠⠜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠢⠀⠈⠓⠲⣯⣀⣂⡿⣽⢃⠀⡅⠄⢃⠄⠀⡼⠉⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠢⢄⠂⠀⢀⢂⠘⡉⢫⢌⡽⡀⠀⠐⠀⠀⢀⣀⣀⡤⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠑⢄⠀⠀⡱⡘⢨⢯⢯⣿⠄⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢀⠀⠑⠀⡮⢄⡬⡆⠳⣝⢷⣷⣀⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠠⠀⠈⠀⠚⢈⠙⣮⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣒⣢⣀⠀⠀⠠⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢼⠋⣷⡟⣧⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠶⣮⣁⣀⡰⢚⠵⠛⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠴⣯⣤⡶⠿⠉⢱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠽⢶⣤⣽⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠼⣧⣄⡠⠾⠻⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⠟⠷⢄⣠⣽⠦⠄⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⡤⠂⢀⠟⠀⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠻⡀⠸⢤⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⢀⡾⡟⠉⠀⠀⣤⡀⠀⣀⠀⠀⡀⠀⢀⣤⠀⠀⠉⠻⢷⡀⠀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣾⡇⠈⣧⣥⠀⠀⠀⢸⡇⠀⢾⡁⣀⡷⢀⣸⠇⠀⠀⠀⣈⣼⠁⢸⣷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢆⣸⣅⠷⣄⣀⣌⢉⣅⣈⣉⣯⣥⣼⣯⣱⣀⣠⡾⣨⣇⡰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⠀⣿⣧⡤⠞⠁⣽⣿⠈⠣⣤⣬⣿⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⡗⢀⡸⢃⡈⠤⢘⡇⠁⢩⣛⡏⣿⣻⡯⢿⣏⡇⠤⢁⡘⡇⡀⢚⣿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⡈⠠⣼⣌⣰⠤⠛⣥⣶⠿⠀⣀⣄⠀⠀⣿⢈⣐⣸⡇⠿⣶⣼⠛⠴⣆⣡⣧⠄⢁⣱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢻⠞⠋⣩⠿⢠⡀⢉⣿⡛⠀⠐⠁⠀⠀⣿⠀⠈⢺⡇⢚⣿⡈⢀⡄⠿⣍⠙⠷⡟⠀⣀
⠀⠀⠀⠀⠀⠀⣀⠄⣉⠛⠉⣠⣾⠗⣦⣼⠟⣋⠛⠣⢌⣁⣀⠄⠀⢻⠀⣂⣨⡭⠜⠛⣙⠻⣧⣴⠺⣷⣄⠉⠛⣉⠠⣀
⠀⠀⠀⠀⠀⡾⢿⠀⠻⣀⣠⣮⡍⢽⡟⠡⠞⠁⠉⠉⠈⠉⠉⠙⠳⠞⠋⠉⠉⠁⠉⠈⠈⠳⠌⢻⡯⠉⣵⣄⣀⠟⠀⡿⢷
⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⣿⠇⠀⣠⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣄⠀⠸⣿⣿⣷⡀⠀⠈
⠀⠀⠀⠀⠀⠐⣊⣯⣭⡁⠀⠀⣰⡿⢛⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⢿⣆⠀⠀⢀⣭⣽⣛⠂
⠀⠀⠀⠀⢀⡜⠛⢉⣽⡿⢃⡾⠁⣵⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣮⠈⢷⡘⢿⣯⡉⠛⢣⡀
⠀⠀⠀⠚⢩⣧⣴⡉⠛⠉⡏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢹⠈⠛⢉⡦⣼⡍⠓
⠀⠀⢲⡴⡟⠁⢘⠏⢠⡶⠶⢻⡗⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢼⠛⠶⢶⡄⠹⡓⠈⢻⢦⡞
⠀⠀⠘⣆⣀⣶⠇⠐⠊⠠⠰⣤⡗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣤⠆⠄⢑⠂⠸⣶⣀⣼⠃
⠀⠀⠀⠈⠻⠿⡶⢴⣶⣤⢠⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⡄⣤⣶⡦⢶⠿⠟⠁
⠀⠀⠀⠀⠀⠠⣦⢀⣿⣿⣮⡁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢨⣵⣿⣿⡀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠍⠹⢁⡻⣿⡆⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢰⣿⢟⡉⠏⠩
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⢿⣷⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⡿⠾⡇⠀⠀⢀
⠀⠀⢀⡄⠈⢀⣿⣄⢰⣤⡆⠀⢻⡿⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣾⠿⡟⠀⢰⣤⡆⣠⣿⡀⠁⠠⡀
⠀⠀⠘⠟⠛⠙⢿⣿⢮⠀⢤⣀⢀⣀⣸⣦⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⣶⣇⣀⣀⣀⡦⠀⠴⣿⡿⠋⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠸⠙⠇⠀⣏⠈⢿⠟⠛⣿⣦⡈⠢⣄⠀⠄⡀⠀⠀⠀⠀⢀⠀⠀⣠⠔⢁⣴⣿⠛⠻⡿⠉⣹⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⣨⣿⡿⠴⡿⠋⠻⡿⣄⠈⢻⣦⣤⣀⡀⢀⣈⣤⣴⡟⠁⣠⢿⠟⠙⣿⠦⢿⣿⣅⠀⠙⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠸⠆⢰⠗⠀⣀⠀⣿⣶⠿⣈⣿⣀⣥⣤⣤⣿⣁⠻⣶⣿⡀⣀⠀⠺⡆⠰⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠽⠾⠿⠟⠃⠙⠛⠉⠉⠉⠙⠛⠋⠘⠻⠿⠥⠯⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⣀⣀
⠀⠀⠀⠀⢀⡤⠒⠉⣀⣶⠀⡇⠈⠉⠉⠑⠒⠒⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠈⠑⢲⣾⣿⠿⣿⠀⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⣿⡉⠶⣃⡇⢰⠁⡘⠀⡰⣆⠄⠶⡀⣤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠲⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⣗⣂⣬⡾⢀⠟⣰⠁⠀⠀⠁⠀⠚⠀⠙⠀⠦⠄⢀⣹⠀⢠⡄⢠⣀⡀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⢹⠑⡒⠢⡀
⠀⠀⠀⠀⣰⠿⠞⠋⠉⠙⠒⠓⠶⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⣱⠀⠸⠀⠘⣹⢐⣢⠀⣄⡀⢸⠁⡇⠀⡇
⠀⠀⠘⠛⠒⠒⠲⠦⠤⠤⢄⣀⣀⡀⠀⠀⠀⠉⠉⠉⠒⠒⠶⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠀⠏⠀⡼⢠⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠲⠤⠤⢤⣀⣀⣀⠀⠀⠀⠉⠉⠉⠒⠒⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⣰⠋⡜⠀⡸⢄⠆⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠦⠤⣄⣀⣀⡀⠀⠀⠉⠉⠓⠒⠖⣷⣾⣄⠔⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠚⠿⠯⠽⠟⠛⠛⠒⠒⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣧⡄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡀⠀⣘⢒⠃⠟⢩⣏⠛⠛⠓⣌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢏⠉⠁⠀⢸⢿⠀⠀⠈⠹⣄⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⡀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠶⠄⠀⠀⠀⡸⠻⡀⠀⠀⠀⢸⣷⣤⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠣⡀
⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⣾⠇⠠⠀⠀⠀⠀⠀⢸⣶⣇⠀⠀⠀⠀⠘⢿⣿⣿⣶⣄⡀⠀⠀⠈⠀⢀⠀⠀⡀⠙⢦
⠀⠀⠀⢠⠞⠀⠀⠈⠀⠄⠁⠀⠀⣠⣾⣿⣿⡟⠀⠠⠀⠀⠀⢠⣠⠟⠛⢹⣤⡀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣦⡀⠀⠀⠁⠀⡀⠁⠀⠈⢳⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠈⠀⠀⣠⣾⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⣼⡏⠀⢀⠀⠹⣇⡄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⣠⡏⠀⠐⠒⠂⠀⠉⠉⡉⠙⠻⢿⣿⣿⠁⠀⠀⣀⣀⣀⣀⣿⡀⢀⣌⡔⠀⣹⡁⢀⠀⠀⠀⠀⣽⣿⠟⠋⠁⠀⠒⠀⠀⠘⠷⢤⣀⠀⠀⢻⡀
⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡈⠉⣄⣶⣞⡯⠳⠛⠂⠋⠀⠀⢨⠁⠀⠻⢯⡽⣟⣶⢤⡠⠊⢀⠔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠤⠷
⢸⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣡⠀⢘⠚⠁⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠈⠈⠋⣧⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⣾⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⢟⡀⠀⠀⠂⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⡇⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⡀⠸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟
⣧⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡇⠀⠛⢻⣿⣷⠄⠀⠃⠀⠈⠀⠀⡴⢿⡿⡹⠁⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⢻⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠳⠭⠡⠆⠀⠀⠀⠀⠀⠀⠄⠒⠚⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⢸⡄⠀⠀⠘⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢰
⠈⣧⠀⠐⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⡇⢣⡀⣸⡆⠀⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⣦⣠⡏⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠂⠀⠀⠀⡾
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣿⠁⢀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢹⣿⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠹⣆⠀⠁⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⠁⣀⣼⡟⠀⠘⡀⠀⠀⠰⠆⠀⠀⠀⢠⠇⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⣴⠃
⠀⠀⠀⠙⣦⠀⠀⣨⠅⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠁⢀⣼⠃⠀⠀⠒⠀⠀⠀⠀⣾⣧⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠠⠀⠈⠁⠙⠛⣋⣠⣴⣿⠏⠀⠀⠐⠀⠀⠀⢀⣾⣿⣿⣷⣶⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠈⠒⢶⡀⠁⠀⢀⠀⠀⠀⠀⠀⠈⢉⢻⠏⠀⠀⠊⠀⠀⠀⣠⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠰⠀⢀⠀⠀⢡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠠⠀⠀⠀⠀⠀⡉⠀⠐⠃⠈⠀⠐⠚⠋⠙⠢⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⢀⠃⠁⠶⠀⠠⠀⠀⠀⢀⠀⠀⠠⠀⠄⠀⠐⠀⠀⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡘⡈⠀⠒⠀⡘⠄⡀⠀⡘⠤⠰⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⠈⠁⠤⠀⡄⠰⢘⣶⣀⣤⡤⠶⠖⠛⠉⠁

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣏⣝⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣍⣹
⠀⠈⠻⠻⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡿⠟⠟
⠀⠀⠀⠀⠀⠻⢯⡷⣄⡀⢀⢠⠠⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡠⠤⡄⡀⢀⣴⢮⡽⠋
⠀⠀⠀⠀⠀⠀⠈⠹⢿⣏⠟⣴⠟⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⠻⣙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⣼⢟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣻⣧⡜⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠋⠈⢻⣯⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠙⣌⠂
⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠑⢄⠀⠀⡠⠊⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢎⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣧⣀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⢻⣧⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠙⢿⣦⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⢠⣾⢞⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡳⣷⡄
⠀⠀⠀⣰⠟⣱⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣎⠻⢆
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠆⠰⠖⠀⠀⠶⠂⠲⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠒⠀⠐⠒⠒⠒⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣳⠓⣿⣿⣿⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡛⣩⠆⣩⡽⣼⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⢻⣿⣯⣥⣽⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⣀⣀⣀⣀⣀⣈⠉⣉⣋⣉⢉⣉⠋⣉⣉⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠃⢀⠈⡈⠁⠀⠉⠀⡁⠈⠉⠀⠁⠉⠀⠈⠀⠉⠈⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠈⠀⠉⠈⠉⠁⠀⠉⠉⠁⠘⠁⠀⠉⠙⠈⠃⠀⠈⠉⠁⠈⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠛⠐⠂⠂⠐⠋⠘⠃⠐⠃⠐⠒⠀⠓⠒⠀⠂⠘⠂⠂⠀⠐⠒⠂⠐⠂⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠒⠐⠒⠐⠒⠂⠐⠀⠀⠰⠖⠲⠆⠆⠀⡆⠦⠰⠖⠦⠖⠶⠀⠶⠆⠦⠀⠆⠀⠶⠂⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠀⠀⠤⠄⠠⠤⠤⠤⠤⠤⠄⠀⠤⢤⠄⣤⡤⠀⡄⠀⠄⠀⣤⠀⢤⠄⠠⣤⠄⠠⠤⡄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢤⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⣀⣀⣀⡀⠀⣀⢀⡀⣀⠀⢀⡀⢀⠀⠀⣀⠀⡀⠀⡀⠀⡀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠈⢀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠈⠀⢉⠁⠈⠀⠈⠉⠀⠉⠉⠁⠀⠁⠈⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠁⠈⠁⠁⠀⠈⠉⠁⠉⠀⠉⠉⠀⠉⠉⠁⠘⠋⠀⠀⠘⠉⠋⠛⠂⠘⠛⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠃⠀⠋⠀⠀⠀⠈⠂⠘⠀⠒⠀⠀⠐⠛⠙⠀⠒⠒⠀⠘⠃⠐⠐⠓⠒⠀⠆⠀⠒⠖⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠃⠐⠒⠒⠀⠐⠒⠂⠀⠒⠲⠲⠀⠶⠀⠴⠰⠆⠀⠲⠀⢰⠤⠄⠀⠀⠶⠠⠆⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠄⠰⠠⠄⠠⠆⠰⠂⠀⠤⠔⠤⠴⠦⠀⠤⠤⠤⠄⠠⠤⠤⠠⠄⣤⠠⠤⣤⣄⣤⠀⡀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠤⢤⡤⡄⢠⢠⢤⠀⢠⠄⣀⠀⣀⣀⠀⡄⡄⠀⣀⡀⢠⠀⢠⣀⡀⠀⢀⡀⢀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠀⣀⠀⣀⣀⢀⡀⠀⡀⣀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⣈⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠈⠀⠀⠈⠉⠉⠁⠈⠉⠁⠈⠁⠈⠁⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠀⠉⠈⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠘⠓⠚⠂⠀⠃⠐⠂⠐⠒⠂⠰⠖⠲⠄⠂⠀⠆⠖⠖⠀⠂⠀⠀⠒⠀⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠀⠐⠒⠀⠐⠂⠐⠖⠶⠀⠲⠤⠒⠴⠄⠲⠤⠤⠀⠠⠤⠤⠤⠀⠠⡤⠀⠀⠠⠤⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠰⠀⠰⠆⠀⠤⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢀⣀⣀⣠⠀⣄⢀⠀⣀⡀⣀⡀⣀⡀⢀⣀⣀⡀⠀⠀⡀⢀⣀⡀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠉⠉⠁⠘⠋⠁⠙⠓⠚⠋⠛⠛⠀⠛⠑⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠓⠂⠛⠀⠛⠀⠒⠺⠒⠂⠂⠔⠒⠂⠐⠒⠒⠆⠐⠦⠒⠒⠲⠀⠒⠰⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠒⠒⠂⠒⠒⠀⠐⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⢀⠀⠀⠀⡀⢀⡀⠈⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠁⠀⡸⠀⠿⠴⢣⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⠆⠀⠀⠀⠔⠀⢼⣿⣷⣌⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⠉⠀⢠⣠⠖⠉⣦⣿⠁⣻⣿⣧⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡶⣬⡇⠈⠿⠀⠀⠀⠀⠐⣧⠀⠸⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⠚⠉⠑⢹⣇⠀⠀⠀⠀⠀⠀⣈⢻⠀⠀⠃⠀⢿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠸⠹⣦⠀⠀⠀⠀⠀⣡⡟⣧⢠⠀⠀⠻⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣄⡀⣰⣿⠇⣿⠀⢡⠀⠰⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣸⠀⠀⠀⠸⡀⠀⠀⠸⡅⢿⡻⢟⡿⣿⠀⡇⡄⠀⡆⠸⠀⡁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠀⠀⠀⣿⡅⠀⠀⠴⢿⡎⠛⠋⢥⣿⠂⠇⢻⠀⡁⠀⢷⢀⠇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⠈⢷⣀⣦⡼⣿⠰⠀⢸⡆⢹⡆⠀⡸⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢠⢃⡇⠀⠀⠀⠀⢀⠉⠀⠑⠿⡀⢀⣿⣧⠘⠧⠴⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠋⠼⠿⠦⡴⠒⠀⠀⠀⠀⠀⢀⡀⠉⠁⠀⠐⠰⠀⣁⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣿⠀⠀⠀⠀⠀⢀⣧⣠⠔⠃⠀⠈⠉⠉⢁⡀⠤⠔⠒⠚⠋⠉⠉⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠷⠀⢶⣶⠎⠁⢀⣃⣠⠤⠴⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⣠⡾⠛⠻⣞⠋⠉⡀⠀⠀⠀⠠⣤⣤⠤⠄⠀⣠⣶⣀⣹⡿⠛⢀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡴⠋⣉⡁⠀⠒⠛⠛⠣⢄⠉⠒⠇⣤⡀⠀⣀⡂⣀⣩⣭⢾⣿⡿⠟⢀⣼⣹⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣴⣿⠃⠀⠀⠀⠀⠀⠈⠉⠙⠛⢲⠶⠪⠬⢹⣿⣭⣏⣹⣃⣤⠴⠚⢿⣟⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⢸⡿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠠⠊⢀⠴⠀⣠⣿⠀⡀⠀⠈⠹⢿⣇⠀⠰
⠀⠀⠀⠀⠀⠄⣴⢠⣿⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⡶⠧⠐⠪⠉⠆⠀⠀⣠⣄⡀⠀⠛⠀⠀⣢
⠀⠀⠀⠀⢠⠃⠿⠰⠙⠻⠀⠀⠀⣠⡴⠒⠂⠀⠀⠀⠙⠿⣿⠿⠿⠿⢻⣆⡀⠀⠀⠀⠀⠈⠭⠈⠀⠀⢀⡴⠊⠀⠡⡀
⠀⠀⠀⣠⢞⠀⠓⠶⠀⣶⣿⠀⣰⠏⣀⣤⣤⣄⠀⠀⠀⠰⢗⠀⠀⠀⢸⣦⢤⠂⢀⣤⣤⣄⠀⠐⠀⠀⣏⣀⠀⠠⣦⡇
⠀⠀⢙⠒⠒⢲⠄⠀⣁⠀⠟⡴⢁⡼⠉⣈⣁⣭⣿⡆⠀⠀⣐⣷⣆⡀⣸⣿⣾⣆⡈⡿⠛⢻⣷⣦⡀⠀⠘⢿⣷⣄⣈⠋
⠀⠀⠉⠀⠀⢹⠀⣀⣿⡄⠐⣠⣾⠏⠉⢉⠏⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠁⠀⠀⠀⠀⠘⡇⠙⠻⠷⣤⡀⠉⢍⠁
⠀⠀⠀⠀⠀⠈⠑⠒⠿⠁⠚⣋⣀⣀⡀⢨⣄⣀⣀⠤⠄⠀⠤⠤⠶⠶⠶⢀⣀⣀⣀⡀⠀⢸⡆⠀⢀⠀⠀⠉⠒⠆⠓⠤⣀
⠀⠀⠀⢀⡠⠖⠀⠁⠐⠀⠈⠛⢿⠀⣤⣤⣤⣬⣦⣖⠛⠋⠉⠉⠉⠉⠉⢉⣩⣽⣤⣤⡤⣤⡃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠑⠦⣤
⠀⠀⠀⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠠⢀⠉⠁⠈⢀⡤⠞⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⣀⣀⠀⣶⠀⣤⣾⢿⣧⣤⠤
⣽⣿⣟⣿⣽⢻⣽⣿⣛⠉
⣿⣾⣛⣻⡿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣥⣏⣹⡯⢵⠿⢽⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣠⣤
⣯⡹⠏⣷⡏⠐⣨⣭⣙⡲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣿⣿⣿⣿
⠼⣧⢀⣰⣾⢍⡶⣮⠹⣠⣬⠈⠉⠛⠦⢀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣤⣴⣿⣹⣿⣿⣿⣿⣾
⠿⠿⠷⢖⣯⣿⣭⣆⠫⠋⣭⢠⠀⠀⢲⡀⠀⣬⣶⡖⠚⢛⠛⠛⠛⣓⡚⢳⡾⠷⠾⠿⠾⠶⠴⠶⢿⣤⣤⣾⣿⣿⣿⣿⣿⣿⢣⣟⢻⣏⣿⣽⣛
⣿⣿⣟⢿⣶⣶⡶⠿⢿⣿⣿⣿⣶⣶⣆⣬⡼⣷⣿⡷⠀⠴⢀⠒⠂⠐⡀⣄⡀⠂⠀⣀⢀⡱⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣸
⡽⢯⢿⡷⣼⣵⣦⣤⣾⡽⣡⣿⣿⣿⡏⢟⢻⣟⣻⣆⢀⣴⡾⠓⠲⠆⠠⠶⠮⠽⣭⠿⣭⠿⡩⠯⣩⣽⣿⣿⣿⣭⣭⣭⣭⣬⠭⣭⠥⣤⠤⣤⠦
⢁⠻⣹⢟⣯⣫⠩⣹⣏⠈⣿⠀⣶⢰⡆⣶⣶⣿⡇⠦⢸⡟⠀⠀⢐⠂⢀⣠⣴⡆⠀⢀⠀⠀⠄⠀⢬⣿⣿⣿⡿⡻⡿⠿⣶⠖⡿⢵⢣⣬⣦⣬⢭
⡤⣾⢫⣿⣿⣿⢼⠿⣿⠀⣿⡀⣿⢸⣇⣿⣿⣿⡗⠀⢸⡇⠤⠒⢀⣾⣿⣿⣿⣿⣿⣷⣤⣀⠠⠀⢠⣼⣿⣿⣿⣷⣿⣸⣏⣼⣆⣴⣷⣶⣿⣿⣯
⡇⣿⢸⣻⣿⣿⠿⡌⣿⠛⣿⠃⡿⢸⡇⣿⣿⣿⠉⠦⢸⡇⠀⣤⣿⣿⣿⣿⡟⠟⢻⣿⣿⣿⡄⠈⠀⣼⣿⣿⣿⣿⡗⣿⡧⣿⣿⣿⣿⣟⣿⣿⣿
⡇⣿⠸⢻⢹⣿⠼⠀⣿⠀⣿⠀⡇⢸⡆⣿⣿⣿⡄⠀⢸⡅⢸⣿⣿⡿⡉⢀⣿⣦⣍⣼⢎⣿⣿⠀⢃⢹⣿⣿⣿⣿⡇⣿⣧⣿⣷⣿⣿⣻⣿⣿⣿
⡇⣿⠠⠉⢸⣏⠠⠸⣿⠀⣿⠀⣁⢸⣗⠿⢉⣝⣛⡀⢸⡄⢸⣿⣿⣷⡟⠛⣫⡉⠉⠁⠀⣙⣿⢗⡀⢸⣿⣿⣿⣻⠇⣿⠷⣿⡯⣿⣿⣿⣿⣿⣿
⡇⣿⠰⠐⣺⡟⠗⠰⣿⢸⣿⡄⠟⠔⠉⠀⣐⢚⠧⠀⢸⣀⣼⣿⣿⡟⢁⣾⣿⣿⡆⠀⡀⢠⣿⠟⠷⣿⣿⣿⣿⣿⣷⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿
⡇⣿⠀⠐⣿⡇⠂⢴⡿⠈⠙⠂⠀⠀⠀⢈⣰⣾⠦⠀⢸⣟⣼⣿⣿⠞⢸⣿⣿⣿⡇⢹⠋⢨⣿⠰⠀⢺⣿⣿⡿⣿⣿⣿⣿⣦⣜⠿⢿⣿⣿⣿⣿
⡗⠿⠀⢸⠿⠧⠃⡤⢀⠁⠀⠀⠀⠀⣴⢾⣿⣿⠂⠁⢼⠷⠿⠿⠛⠛⠚⠛⠛⠛⠓⠚⠛⠛⠋⠐⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣮⡻⠿⢿⣿
⣇⣿⠁⠀⠠⠀⠁⠀⠃⠀⠈⠁⣶⡋⣿⢸⣿⡟⠂⢀⡔⠂⢲⣶⠶⣶⡶⣿⣿⣿⢿⣿⠿⣿⡷⢰⣆⠀⠹⣿⡿⣿⣿⢯⣯⣿⢻⣯⣿⣿⣷⣬⡻
⠋⠁⢀⣁⡂⠀⠢⠀⠀⢠⡄⢺⣿⡄⣿⡟⠏⠀⢠⣿⡇⠀⣾⣿⣋⣁⣴⣿⣿⣿⣷⡀⣤⣸⡇⠰⣿⣦⠀⠈⢹⣿⣿⣸⣿⢠⣞⣿⣿⣷⡾⡿⢿
⠁⠘⠋⠖⣀⠰⡀⠀⠁⣿⡇⢸⣿⣧⡿⠅⠀⣠⣿⣿⡇⡄⣿⣇⠀⢸⣿⣿⣿⣿⣿⡇⠈⣿⡏⢰⡟⣿⣧⡀⠀⠐⠿⢹⣿⢽⣿⢛⣿⣿⣿⣿⣿
⠰⠆⠶⠀⠈⠲⢰⣿⠀⣿⣏⡸⠿⠏⠀⠀⣴⣿⣿⢿⡇⠀⣿⡷⠞⢸⣿⣿⣿⣿⣿⡇⠤⣿⣧⣰⣿⡌⢿⣭⡄⠀⠀⠹⢿⣾⣿⢸⣿⣿⡿⣿⣿
⠶⠄⠀⣀⡀⠀⢾⣿⠗⣿⣷⡔⠀⠀⠀⣼⣿⣿⢣⣿⡧⢒⣿⣶⣶⣾⣿⣿⡿⣿⠿⠛⢻⡏⣿⠀⠹⣿⣆⠻⣿⣤⠀⠀⠀⠹⢿⣾⣿⣇⣵⣿⣻
⡞⠈⢰⣿⣿⡄⣼⣿⡧⠋⠉⠀⠀⢀⣞⣿⣿⢣⣿⡽⠀⣼⣿⠿⣟⠃⣨⣿⣿⣷⣬⢏⡍⠁⣿⣇⠀⠘⢿⣦⠙⣾⡶⣀⠀⠀⢈⣿⣿⣿⣿⣿⣷
⠙⣁⣿⣿⣿⡇⣿⣿⡇⠁⠀⠀⢠⣾⣿⡿⢣⣿⣯⠀⣰⣿⣿⣿⡿⠐⣿⣿⣿⣿⣿⣘⡶⠂⢿⣿⣆⠀⠸⢿⣧⡉⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⣿
⠦⢰⣿⣿⣿⣇⣿⠍⠀⠀⠀⣰⣿⣿⣿⢣⣿⡘⠃⢀⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣌⣉⣁⣸⣿⡿⣆⠀⠘⢿⣿⡌⢻⣿⣷⡄⠀⠀⠳⣿⢿⣿
⡑⢸⣿⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⢡⣿⢣⠀⠀⣾⣿⣿⡿⢿⠿⠞⠛⠛⠛⠁⠈⠉⢩⣭⡹⣿⣿⣿⡄⠀⠈⢿⣿⣎⠻⣿⣿⣦⣄⣀⣨⡤⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠛⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⢰⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣧⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⣼⣿⣿⣿⣧⡄
⠀⠀⠀⠀⢹⣹⠿⡏⡏⠀⢸⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⠀⢹⣿⡿⣯⣿
⠀⠀⠀⢀⣿⣿⣤⣿⣷⡀⢸⣿⣿⣷⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣇⢀⣾⣿⣇⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⡿⠀⣿⣿⣿⣿⣿⡅
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⡠⢤⠀⠀⢸⣿⣗⠏⠵⢁⠆⠀⠀⠀⠀⢀⠄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡆⠃⠀⢀⣻⣿⡇⠀⠁⠒⠀⠀⠀⢀⠎⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡖⠚⠛⠋⠀⠀⢠⡟⠒⠒⠒⠒⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⡀⠤⠀⣿⢀⡖⠒⠘⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡇⠠⡿⠁⢀⠗⠒⠀⠂⠀⠀⠀⠀⠰⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠪⠤⠔⠫⠀⠀⠀⠀⠰⠀⠀⢀⣸⣻⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠑⡀⠑⣄⣀⣀⣀⣇⣀⣀⣀⣥⠠⠤⠤⠦⠤⠤⡯⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢙⠉⠉⣀⣀⢀⡤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠀⠀⠀⠑⠤⢄⣀⣀⡤⠋⠉⠉⠉⠐⠃

⠀⠀⠀⢀⣠⠖⠛⣿⡆
⢀⡴⣾⣿⣷⣶⣾⣿⣦⡀
⣿⡟⠁⠀⠀⠘⣧⠤⣾⣿⡄
⢹⣦⠀⠀⠀⠀⠘⣷⠟⠋⣹⡄
⠀⢻⡀⠀⠀⠀⠀⠘⢷⣽⣿⣿⡄
⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⡿⠟⣛⣙⣛⡓⢦⡀
⠀⠀⠀⠹⣕⡤⣤⣤⡴⢞⣩⣶⣿⣿⣧⡀⠉⣳⠇
⠀⠀⠀⠀⠈⠙⠛⠿⠚⠋⠁⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⣟⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠦⡀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠢⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠢⡀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠒⢄⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡁⠢⣄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠢⣄⣙⠲⠦⣍⡒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣛⣶⣬⣟⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣶⣤⣀⡀

⠀⠀⠘⠃⠀⠀⠛⢂⠀⠀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠛⠀⠀⠘⠃
⠐⣶⠂⢰⡮⢳⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⢠⡄⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⡞⢵⡆⠐⣶
⢳⣦⡂⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢄⠻⣠⠛⠷⣀⡉⠓⠿⠿⠛⢉⣠⠾⠛⣀⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⢐⣴
⠃⠾⠛⣀⠛⠁⠀⠀⠀⠀⠀⠂⣠⠀⠈⠰⢝⣛⠛⠫⠤⠄⣴⣦⠠⠤⠙⠛⣛⡋⠆⠁⠀⣄⠒⠀⠀⠀⠀⠀⠈⠛⢈⠻⠷
⡆⠰⠞⠀⠀⠀⠀⣠⡄⠀⠀⡜⠹⢦⣄⠀⠀⠀⣀⣠⣄⡀⣾⣷⢀⣠⣀⣀⠀⠀⢀⣠⡴⠏⢢⠀⡀⢠⣄⠀⠀⠀⠀⠳⠆
⠀⠀⠀⠀⠈⢀⣼⡟⠀⢄⠶⣰⠶⣦⣌⣉⣩⡁⠤⠶⠆⠉⠛⠛⠉⠰⠶⠦⢈⣍⣉⣡⣶⠶⣆⠶⡠⠀⢻⣧⡄⠁
⠀⠀⠀⠀⣰⣿⡿⠀⠀⣈⣾⠓⢚⣡⡉⡿⠷⠶⠄⠀⠾⠷⠖⠲⠾⠷⠆⠠⠶⠾⢻⢉⣌⠓⡚⣷⣡⠂⠀⢿⣿⣆
⠀⠀⠀⠘⣿⣿⠃⠀⠀⠘⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⠣⠀⠀⠘⣿⣿⠃
⡀⠀⢄⠀⢻⡟⠀⠀⣀⡔⠉⠀⠀⠀⠀⢀⣀⠠⣤⣤⣤⣤⠉⠉⣤⣤⣤⣤⠄⣀⡀⠀⠀⠀⠀⠉⢠⣀⠀⠀⢻⡟⠀⡀
⠀⠀⠈⠀⢸⠃⠀⢀⠛⠀⠀⠀⣠⠴⠒⢛⡡⠖⠋⠁⠀⠀⠺⠗⠀⠀⠈⠙⠲⢌⡛⠓⠦⣄⠀⠀⠀⠻⡀⠀⠘⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⢋⣀⣼⡟⠀⠀⠀⠀⢠⣤⡉⢉⣤⡄⠀⠀⠀⠀⢻⣧⣀⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠛⠉⠛⠃⠀⠀⠿⠀⢾⣧⣨⣅⣼⡏⠰⠿⠀⠀⠘⠛⠉⠛⢿⣟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⠛⠿⣿⡿⠂⣀⠀⠀⢀⠀⠈⠁⠤⠤⠈⠁⠀⡀⠀⠀⣀⠐⢿⣿⠿⠋⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⢀⠘⣿⣤⣼⠟⠀⣴⡿⢷⣆⣀⣤⣤⣤⣾⣷⣤⣤⣤⣄⣰⡶⢿⣦⠀⠻⣧⣤⣿⠃⡀⠀⠇
⠀⠀⠀⠀⢀⠀⢠⢿⠀⣿⣴⡟⠀⣔⣩⠶⠛⠉⠁⢀⣀⣀⠘⠃⣀⣀⡀⠈⠙⠛⠶⣍⣡⠀⢻⣶⣿⠀⡿⡄
⠀⠀⠀⠀⣸⠀⠈⠀⢀⣙⠋⠁⠘⠋⣀⠤⣚⠦⠈⠛⠉⠉⢱⡎⠉⠉⠛⠁⠼⣓⠤⣀⠙⠃⠈⠙⣉⡀⠀⠁⠀⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠀⠀⣠⢞⡷⠾⠏⠀⠀⢤⠠⣾⣡⣌⣷⠄⡤⠀⠀⠹⠷⢾⣳⣄⠀⠀⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⡟⠀⠀⠀⢸⠁⠀⠘⢻⡏⢠⠆⣠⡀⠀⠀⠀⠁⢬⡥⠈⠀⠀⠀⢀⣤⠰⡤⢝⡟⠃⠀⠈⡇⠀⠀⠀⢻
⠄⠀⠀⠀⢃⡀⠀⠀⠈⠀⠀⠠⢟⡵⠁⠰⠟⠚⠉⠉⠀⠀⠐⠂⠀⠀⠉⠉⠓⠻⠆⠈⢮⡻⠆⠀⠀⠀⠀⠀⢀⡜
⠀⠒⠻⢿⣿⠇⠀⠀⢠⡄⠀⢰⡿⠀⡶⠁⠀⣀⡴⠀⠋⢉⣉⣉⡉⠉⠀⢶⣀⠀⠈⢶⠀⣿⡆⠀⢁⡄⠀⠀⠸⣿⡿⠟
⠀⠒⠒⢾⡇⠀⠀⠀⠈⠁⠀⠘⠁⠀⢰⣾⠋⠉⡀⠀⠁⠘⢻⡟⠃⠈⠀⢀⠉⠘⣷⡆⠀⠈⠃⠀⠘⠁⠀⠀⠀⢸⡷⠒⠒
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢁⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠣⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠰⠆⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠄⠶⠆⠰⠶⠐⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠰⠆⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠐⠚⣿⠃⠀⢀⠀⠀⠀⣰⠞⠁⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠈⠻⣤⠀⠀⠀⡄⠀⠘⣿⠓⠂⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠠⣿⠀⠀⣸⠀⠀⠘⠁⢀⠴⣛⣁⣉⣙⣛⣛⣋⣉⣉⣙⠦⡀⠈⠃⠀⠀⣇⠀⠀⣿⠇⠀⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠘⡏⠀⠘⠀⢀⢏⣾⣿⡿⣯⢻⡿⢿⡟⣽⢿⣿⣷⡙⡀⠀⠇⠀⢹⠇⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢀⡏⢸⡏⠀⣿⢸⡿⠟⢿⠁⠈⠁⠈⠁⠈⡿⠻⢿⡇⣿⠀⢹⡇⢹⡀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣿⢿⣛⣛⣛⣛⡿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣿⣿⡛⣛⠛⠛⣿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣿⣿⡏⢉⡁⢸⣿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣿⣿⠃⢀⡀⢸⣿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣼⣿
⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣾⢻⡃⣾⡇⢸⣿⣿⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⠀⠀⣿⣿
⠀⠸⠂⣿⣿⠀⠀⠀⣿⠀⢸⣇⢸⡇⠀⣿⠀⠀⠀⣼⣟⣩⡦⣠⣼⣿⣿⠀⠀⠀⣿⠀⢸⡇⢿⡇⠀⣿⠀⠀⠀⣿⣿
⠀⠸⠆⣿⣿⠀⠀⠀⣿⡄⣸⣿⡼⣿⠀⣿⣀⣀⣀⣿⣿⣿⡾⣿⣿⣿⣿⣀⣀⣀⣿⠀⣿⢧⣿⣇⢠⣿⠀⠀⠀⣿⣿
⠀⣿⣶⣿⣿⠀⣀⣴⣿⣇⣠⠙⣷⡄⠀⠛⠋⠙⢉⣽⠋⠛⢿⣿⠛⠉⣯⡉⠉⠙⠛⠀⢠⣾⠋⣄⣸⣿⣦⣀⠀⣿⣿⣦⣶
⠀⣽⣿⣿⣿⣦⣉⠈⠉⢹⣿⡀⠉⠁⠀⡰⠁⠠⣾⡿⠀⠀⣿⣿⠀⠀⢿⣿⠄⠀⠆⠀⠈⠉⢀⣿⡏⠉⠉⢈⣴⣿⣿⣿⢋
⠄⠽⠯⠙⠛⣿⣿⠀⠀⢹⠀⠀⠀⢀⣤⠁⢀⣴⣿⡇⠀⢠⣿⣿⡄⠀⢼⣿⣦⡀⠀⣦⡀⠀⠀⠀⡇⠀⠀⣿⣿⠛⠏⠁
⠀⣰⣾⠓⠈⣿⡏⠀⠀⠘⠁⠀⠠⠶⠀⠀⢴⣿⠿⠄⠀⠴⣿⣿⠦⠀⠠⢿⣿⡦⠀⠀⠶⠄⠀⠈⠃⠀⠀⠹⣿⠀⠚⣷⣄
⣿⣿⣿⡇⠀⣿⠇⠀⠀⠀⠀⣠⡛⠃⢀⣙⣿⣿⣛⠁⠀⣛⣿⣿⣛⠀⠈⣛⣿⣿⣋⡀⠘⢛⣄⠀⠀⠀⠀⠸⣿⠀⢸⣿⣿
⢠⣿⣿⣯⡀⣿⠆⠀⠀⢀⣤⣭⡀⢀⣬⣽⣿⣭⣅⠀⣠⣭⣿⣿⣭⣄⠀⣨⣭⣿⣯⣤⡀⠈⣭⣤⡀⠀⠀⠰⣿⢀⣽⣿⣿
⢙⣿⣿⠿⠛⠁⠀⠀⢀⣬⣅⠀⢀⣠⣬⣿⣯⣥⡀⠀⣠⣬⣽⣯⣥⣄⠀⢀⣬⣽⣿⣥⣄⡀⠀⣨⣥⡀⠀⠀⠈⠛⠿⣿⣮
⠛⠉⠁⠀⠀⠀⠀⢀⣉⡉⠁⠀⣈⣉⣽⣿⣉⡉⠀⠀⣈⣉⣿⣿⣉⣁⠀⠀⢉⣉⣿⣯⣉⣀⠀⠈⢉⣉⣀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⢀⠞⠙⢦
⠀⠀⠀⠘⡄⠀⠤⠳⣤⠤⢤⡀
⠀⠀⠀⢠⠫⠿⡄⠀⠙⣆⠒⠉⠲⣄
⠀⠀⢠⣷⠘⠤⠽⣦⠀⠈⢳⣄⠀⠈⠳⣄
⠀⠀⢻⠙⣧⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⣆⠘⣆⠀⠈⢳⡄⠀⠀⠙⢦⡀⠀⠈⣙⣦⡀⠀⠀⠀⠀⡴⢋⡙
⠀⠀⣼⢶⡏⠛⠹⣆⠀⠀⠹⣄⠀⠀⣤⠻⣄⠈⠵⠊⢳⣄⣠⠶⠻⠁⠈⠳⡆
⠀⢸⠃⠀⢇⠀⠀⢻⡄⠀⠀⠘⢧⡀⠘⠁⠹⣆⠀⠀⠀⠹⣯⠀⣴⠀⠀⠀⡇
⠀⣼⣄⠀⡸⣆⠀⠀⢻⡄⠠⢶⡨⣷⡀⠀⠀⠘⣧⡀⠀⠀⠸⣷⡇⠀⠸⣿⡇
⠘⠇⠘⣇⡇⠹⣆⠀⠀⠙⣦⠀⠁⠀⢱⡀⠀⠀⠹⣷⡄⠀⠀⢻⣷⠀⠀⠸⣇
⢸⠀⠀⣿⠀⠀⠹⣧⠀⢻⠊⣧⠀⠀⠀⢷⣀⠀⠀⠙⠛⠂⠀⠈⢿⣦⠀⠀⢿⡄
⡆⠀⢀⣿⠀⢧⠀⢹⡄⠀⠀⠘⣦⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠈⢷⡀
⢧⠀⢸⠟⢳⡈⣆⠀⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⣧
⠈⢧⠀⠀⠈⢧⠸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⣧⠀⠀⠈⢣⡹⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⣇⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠉⣙⣿⣿⣿⣶⣦⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠰⠹⣿⣿⣿⣶⣶⣤⣤⣶⠖⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⠋⠹⣷⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢿⣦⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⡉⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⢠⡾⠁⠀⠈⣿⠿⠿⠂⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⣿⣄⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣰⠟⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣦⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⢸⡏⠀⠀⠀⠀⣿⣿⣄⣀⠀⠀⢀⣠⣤⣶⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣁⣴⡾⠁⠿⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⢀⣴⣿⠟⠁
⢸⡟⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⣸⡇⢰⣶⣤⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣠⣿⣿⠟
⣿⣿⡈⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⢀⣠⣾⣿⡿⠃
⠘⢿⣿⣦⣄⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣶⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⠶⠄⠀⠀⢠⣾⣿⣿⣶⣿⣷⣴⠟⠋⠁⠀⠻⣿⣅⡠⠴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠄⠀⠘⠂⠀⠴⠀⠀⠄⠀⠠⠢⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠃⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢻⢾⣿⣿⡗⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣂⡄⠺⠣⢤⡿⡁⠀⠀⣿⢾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡏⣿⡏⢹⡇⡄⠀⠀⠈⠱⠀⠄⠋⠀⠛⣇⠀⠀⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⡙⠉⡁⢸⢡⡌⠁⠀⠀⠀⠀⠀⠰⠀⡀⠀⢀⠿⠀⠰⠟
⠀⠈⠀⠀⠄⠀⡀⠀⠰⣤⡄⣹⡀⡁⢨⣿⣾⣸⡇⡀⠀⡀⣾⠆⠀⣼⣷⡟⢲⡄⢀⠠⣼⣇
⠀⠀⠀⠀⢲⠋⠙⡯⣀⡿⡇⠨⠇⠒⠻⣿⣻⣸⠁⠡⠈⠁⠿⡇⠐⢸⡟⡇⣾⠀⢸⣿⠋⠹⣀
⠀⠀⠑⠓⠂⠐⠺⠃⡈⠿⠇⢠⡄⢀⡤⣼⣇⣹⠠⠄⠀⡁⢀⠇⠐⠚⠛⡓⠿⣃⡌⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⢀⠠⠄⠀⠀⠀⠉⠉⠓⠀⠀⠁⠀⠶⢦⣾⣷⡉⢸⣿⣿⣷⣠⡄⠁⠀⠀⠀⠠⠼⠦⡌⠁⠀⠀⣀⡹⣇
⠀⠀⠀⠀⠀⣀⣰⣟⣻⣿⡆⠁⢀⠀⠀⠀⠘⠂⢸⡶⠀⠀⠀⠀⠈⣿⣿⣧⢸⣿⣿⣿⡏⠙⣇⠀⠲⠀⢀⡈⢁⠀⠀⢌⠀⡉⠩⡭⠒⠀⠂
⠀⠀⢀⣀⠀⣤⠿⠟⠽⠿⡇⢀⣀⠀⠀⠀⠸⠇⠸⠘⠀⠠⠤⠤⠄⠻⢿⣿⢸⡇⣿⢡⡇⢠⡏⠀⠀⠐⡆⠀⠔⢸⡏⡟⣮⢹⣄⠋
⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠨⠥⣭⠼⠇⠘⡏⠐⠃⠀⣿⠀⠀⢸⠁⠅⣿⡈⣿⡄
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣀⠀⢀⢀⣰⣀⣀⣄⣛⡷⣀⣸⠁⡀⢩⡁⠙⠛⠓⠂⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠙⠿⣿⣷⡄⠀⠠⠀⡄⠀⠘⢆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⠀⠀⠀⠇⠀⠀⠀⠃⠀⠌

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣧⣵⣶⣤⣶⣵⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠙⢿⡿⠁⣠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠘⢷⣶⣤⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡗⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠋⠉⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣭⣄⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣏⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⠿⡆⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡷⠷⣶⠀⠀⠀⣿⣿⢶⣤⣤⡴⣾⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⣶⣶⡿⠀⠀⠀⠰⣿⣶⣶⣿⣿⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⣀⣀⣀⣤⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡆⢀⢠⣦⡀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⣿⣇⠀⠀⠀⠀⠀⠠⠃⠈⣿⣿⡿⠟⢛⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣫⡷⢟⣛⣛⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⣿⡇⢰⣿⡿⠀⠀⣀⡀⠖⣃⣀⣴⣿⣯⣶⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣀⣿⣿⣿⣿⣾⣿⣿⠟⢃⠄⣷⣄⢀⢸⣿⣶⠀⣶⣿⠇⠀⠀⢉⠛⠛⠉⠉⣸⣿⣛⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣷⣿⣿⡯⢀⣤⣶⣿⣼⡿⢙⣃⣿⡏⠀⣿⣿⢀⡂⣲⣿⣿⣶⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣋⠏⣰⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣟⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⣿⣇⢸⣿⣿⣿⡟⡿⢿⣿⣦⣾⣿⣿⢶⡿⣷⣶⣾⣷⠛⡟⣿⣿⣿⣿⣆⢳⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⣴⣿⡿⣿⡇⢀⣾⣿⣿⣿⣿⣯⢸⡇⡟⠿⣿⣿⡆⠀⣿⣿⣿⣷⡏⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢩⣿⢿⣷⣿⡿⢿⠿⣇⣘⣿⣯⣿⣿⣷⣿⣹⣷⣿⣴⣬⣿⣥⣄⣿⡿⣿⣿⠷⣏⡈⠻⣦⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣏⣿⡟⠻⠿⠿⠟⠛⠛⠉⠈⠙⠛⠿⠷⠿⠿⠿⠟⠋⠁⠉⠙⠛⠻⠿⠿⠟⢻⣿⠆⠈⡀
⢀⡀⠀⠀⠀⢰⡇⠀⠀⠀⠂⢿⣿⣞⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣳⣆⠀⠁⠀⠀⢸⡄⠀⠀⠀⢀
⠀⠙⢶⡄⠀⣿⣿⡄⠀⠀⠀⠘⢿⣿⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⡆⠀⠀⢠⣿⣿⠀⣠⡾⠋
⠀⠀⠀⠈⠳⣿⣿⣿⣄⡠⠊⠡⡘⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠙⢄⣠⣿⣿⣿⠞⠁
⠀⣀⡀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠈⠻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠈⣿⣿⡟⠁⠀⠀⢀⣀
⠀⠀⠙⠛⠶⠤⣈⣿⠏⠲⣄⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣠⠖⠹⣿⣁⠤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠢⢷⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠔⠚⠉⢻⡀
⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⢾⣿⣿⣦⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣤⣾⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣛⡀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⢀⣛⣩⣅
⠀⠀⠀⠀⠐⢤⣤⣤⡞⠉⠻⢿⣷⠋⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠹⣾⣿⠟⠋⠰⣦⣀⡤⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣶⣤⣀⡀⠈⠐⠂⠄⠀⠀⠀⠀⠀⠀⠁⣀⣠⣤⣾⡿⠟⠓⣴⠂⠉⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠀⠀⠒⠀⠒⠠⠄⠘⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣲⣦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡼⠛⠉⠁⠀⠀⠘⢿⡯⠉⠐⠒⠒⠈⠓⠀⠢⠤⠴⠖⠒⠒⠒⠊⢹⡿⠃⠀⠀⠈⠉⠛⢆⠆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⣗⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡀⠀⠀⠀⠀⠀⠀⣾⣷⣝⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡝⠀⠀⠀⡀⢀⠀⢿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡤⢚⠥⣾⠿⠆⣰⣿⣿⣿⣧⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣡⢠⢇⣜⣵⣾⣼⣿⣿⣟⢿⠘⣿⡦⡄⠁⠉⠛⢿⣷⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠀⢁⡼⣸⠿⠛⠻⠿⠿⢯⣼⣷⣼⣿⣧⠀⠀⠀⠀⠀⠀⠙⢜⡛⠷⣖⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣔⣵⠞⠁⠆⠀⠀⠀⣀⠀⠈⠻⣿⡏⣀⢡⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡾⠟⠉⢀⠂⠀⠀⢀⠀⠀⠀⢹⣦⠠⡆⠘⣦⡈⢨⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣎⣀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠄⠈⠁⠀⡇⢠⣿⣿⣿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⣦
⠀⠀⠀⠀⠀⠀⠒⣿⣦⠃⠀⠀⢀⠜⠁⠀⠘⠋⣴⣷⣿⣿⣿⣿⣿⠃⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉⠻⠿⢿⣿⣿⠏⠀⢸⣿⣰
⠀⠀⠀⠀⠀⠀⠈⢹⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⣾⣿⠏⠀⠀⠈⣿⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠈⣼⣿⡏⠀⠀⠀⠀⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣼⣿⠏⠀⠀⠀⠀⢠⣿⡆⠀⠀⠰
⠀⠀⠀⠀⠀⢠⣞⣾⣿⣿⣶⣶⣄⡀⢠⠀⠉⠋⣾⣿⠏⠀⠀⠀⠐⢁⣾⣿⡧⡀
⠀⠀⠀⠀⠀⠻⠁⡞⠀⠐⠉⠻⢿⣿⣷⣄⢀⣾⣿⠏⠀⠀⠀⢀⣰⣾⣿⣿⠃⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣿⣿⢏⣴⣦⡴⠖⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠳⣿⣿⣾⣿⠟⠁⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢲⣄⣤⠄⠀⣀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⢰⡘⣇⢀⠜⢁⣤⣾⡿⡚⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡷⣿⣎⣇⠸⠉⠉⡴⢡⣿⣿⢿⠛⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣻⣴⣿⣼⡷⠾
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠵⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢠⠄⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡉⢙⡆⠀⠀⠀⠠⠀⠀⢀⠐⠀⠄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠞⢉⣁⢀⠝⠲⠉⠁⠀⠉⠀⠢⣄⠀⠀⠀⠀⠀⠹⡜⢘⣦⠇⠀⠀⠀⢂⠘⠁⡀⠀⠀⠆⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠘⠁⢀⡀⡰⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⠀⢀⡀⠀⠈⠉⠁⠀⢀⠔⠁⠰⠀⠀⠀⠤⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⡀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠊⡀⠀⠠⠩⡀⠀⠑⠄⣂⠀⠐⢂⡑⠀⠂⠀⢠⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⢠⠋⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠂⠈⡇⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣃⠀⠡⡀⠀⠒⠉⢢⣄⣀⣀⡤⠂⠈⠉⠘⠄⠰⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡐⢀⠀⠀⠉⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⢄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠑⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡴⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠒⡒⠀⠒⠂⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡔⠁⠀⠀⠀⡀⠤⣖⣁⣉⣠⡚⠁⠊⠀⡹⠀⡀⠁⠀⠊⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡶⢊⣀⠀⡴⠖⠓⠋⠉⠀⠀⠀⠀⠉⠺⡄⠀⡃⢀⡐⠴⠊⠀⡀⢀⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠀⠒⠭⡛⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡌⠉⠉⠀⠐⠈⠂⡔⢣⠂⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠤⠄⠄⠀⢂⡁⠄⠁⠀⢠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡈⠣⡀⠀⠀⠀⠀⠀⠀⠰⠡⠀⡀⠂⠀⢁⠁⠀⠀⢁⡴⢁⡔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠠⢎⡑⠲⠤⢄⣀⣀⣀⠠⠤⠒⡁⠔⠁⠑⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢣⠀⠀⠠⡄⠀⢠⠁⠂⠀⠀⠄⠄⠐⠐⡈⠀⠰⠀⢀
⠀⠀⠀⠠⠊⠀⠔⠀⠂⣠⠀⢁⡀⠠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠘⠀⠀⠐⠆⠈⠀⠠⠀⠋⠀⠠⠃⠀⠄⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠂⠓⡤⠋⠀⢀⡀⠈⢇⣀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡀⠀⠀⠀⠀⠱⡤⣀⠀⡀⠀⠀⠀⠀⡀⢀⠀⣀⣤⠊
⠀⠀⢠⠀⡀⠂⠘⠁⠠⠛⡁⠄⠀⣈⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⣏⡁⢠⠀⠀⠀⠀⠈⠛⢧⡀⣀⠔⠔⢠⣀⡄⠼⠋
⠀⠀⠘⠀⢣⠄⢹⠀⠈⢈⠈⢢⠀⢈⠁⠠⠨⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⡾⡁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⠨⠀⠀⢀⠈⠀⢈⠔⠀⠀⠂⡐⢀⠀⢀⡀⣀⠀⠠⡀⠂⡀⠈⡻⡁⠄⠈
⠀⠀⠀⠠⡀⢊⠀⠉⢄⠈⠄⡀⠁⠀⠂⠈⡀⠡⠐⢀⠠⢀⠐⠨⠀⠐⡀⣨⢮⠠⠂⡐⠁
⠀⠀⠀⠀⠐⠀⠈⢁⠀⡳⠰⡆⠠⠀⠐⠀⠁⠄⠀⢠⠐⠠⠀⠄⣢⠰⠚⠄⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢠⠁⠒⠠⢨⠀⡄⢈⢠⠠⠄⣆⠁⡐⠂⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡁⠀⠑⠀⠔⠃⠀⠘⠂⠀⢁⡀⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠁⠁⠀⠈

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⢀⣤⣤⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣷⠀⠀⠀⠀⠀⠀⠘⠻⠟⠋⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁
⠀⠀⣴⣿⣿⣿⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠹⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢊⡄
⠀⠀⠀⠈⠻⣿⣿⣿⣁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠄⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣢⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣍⡉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠝⠃⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣾⡕⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠻⡿⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣩⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢿⣿⢫⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣏⢠⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡟⣡⣎⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣽⡟⢻⣻⡿⠶⣿⣋⠠⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠋⠀⠀⠀⠀⠀⠈⣿⠃⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡒⢊⣉⣴⡆⠀⠀⠀⠀⠀⠘⠁⣈⣁⡀⠀⢰⣶⠦⠀⠀⠀⡦⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠂⠀⠤⠀⠀⠠⠄⠰⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⠠⣀⠄⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢦⡀⠀⢀⠀⢰⡄⠀⠀⠀⢀
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣿⠿⡶⠶⢾⠋
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⣋⡹⠛⠁⠀⡄⠀⡰⠀⠀⢼⡀⠀⠀⠀⠀⠀⠐⢀⡜⠀⠀⠐⠀⠀⠀⡠⠂⠈⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⢦⡄⠀⣠⠐⢂⣩⡿⠀⠀⠀⠀⡠⠊⠀⠀⢀⡀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⢠⣴⣶⢦⣴⣶⢾⠉⠁⠀⠀⠀⢀⠌⠀⠀⠀⣠⣎⢳⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣻⣇⡀⠀⠀⠋⠀⠀⠠⠀⠁⠘⠍⠛⠞⠛⣹⣾⠁⠀⠀⠀⠴⠋⠀⠀⠀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠂⡽⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⢦⠋⢱⣆⠀⠀⣠⠀⠀⠀⢀⣠⣦⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠱⢤⡉⠛⠛⠋⠀⠀⢠⣿⠿⠉⢀⣈⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠠⠈⠀⠀⠀⠀⣀⣦⣄⠙⣀⣀⣾⣿⠿⠛⢂⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⡴⡀⠀⠀⠉⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⢨⠟⠀⢈⡿⣿⡁⠀⠀⠀⠀⢂⣀⣠⡶⣻⠇⠲⢀⣼⣟⢄⠀⠠⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⣦⣄⠀⠀⡀⠀⠠⠀⠀⣼⣑⡿⠉⠀⢤⡀⣠⣲⠝⠻⠋⠁⠀⠀⠘⢻⢯⡿⣵⣲⠌⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⢰⡂⠀⠠⠌⠙⠁⠄⠀⠀⠛⠙⠿⢄⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⠏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠷⠀⠀⠀⠰⠀⡤⠦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠛⠀⠀⢀⡔⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠂⠀⠐⠀⠀⠀⠘⠂⠈⠁⠀⠀⠀⠀⠀⣴⣦⣤⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣤⠤⠶⠶⠞⠛⠻⠖⠷⡆⠀⠀⠀⠀⠀⠀⠀⢴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡆⠀⢰⡀⠀⠘⠀⢠⠃⠀⠐⢠⠀⠀⠀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠛⠀⠀⠉⠀⢀⠰⠂⠀⢁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠞⠁⠀⠔⠈⠀⢴⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢬⡀⢠⣾⠈⠀⠀⢣⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⣄⣠⡄⠀⠀⡄
⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠓⠛⠉
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣰⣾⡿⠝⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠙⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⡄⠀⠀⠊⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠚⠀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠐
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠉⠁⠠

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⢿⡽⣯⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⣾⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠿⠶⣦⣾⣇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣶⠄⣂⠻⣿⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡾⠥⠄⢸⢻⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢀⣐⠀⣀⣾⣻⡹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠿⠛⠻⢀⣏⣷⡯⡤⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠀⠀⠀⠀⡸⣼⣿⠁⠀⠈⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠠⡀⠀⢠⡏⡿⢱⡀⠰⠃⠀⢀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⠀⠀⡿⡖⠊⠀⠁⣾⠃⠀⣆⠃⠘⠺⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⡇⠀⢀⢀⠂⣾⡇⠀⢻⣤⠶⠿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡾⠶⠶⣴⠂⠀⢻⣷⠄⠟⡀⣹⡇⠠⠈⣅⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡂⠀⠀⠀⠀⠁⠀⠀⠙⠶⠖⢛⡥⠊⠔⡕⠀⡇⠀⢣⠙⣄⠀⠓⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠄⠀⢶⠃⢹⣄⠀⠀⠀⡰⠀⠀⢀⣐⣒⣨⠅⣠⠂⠀⠀⠀⢩⠀⠰⡄⠁⠀⠀⣀⢢
⡇⠀⠀⠀⠀⡀⠄⠒⠃⠀⠀⠄⠀⢈⣧⢾⡀⣰⣧⡄⣀⠜⠁⡀⢀⣉⡉⣉⡴⢊⡄⣤⣀⣠⡴⠾⣷⠄⠀⠀⢀⣴⣌⣀⠓⣄⠀⠀⠀⣀⣤⣤⣤⣀
⡇⠀⢀⠄⠚⠀⠒⠀⠒⣀⡀⠸⠀⠠⠞⡸⢧⣠⡿⠻⠁⠀⢐⠕⠛⠮⣥⣄⣾⡋⠀⣂⣭⠍⣂⡔⠁⠀⣠⠖⠊⠉⠀⠉⠀⠈⢻⣇⢰⢷⣁⣀⣈⡻⡦
⡇⠀⢸⠁⠀⠀⠘⠀⠊⠀⠀⠀⠹⠀⠊⢀⡼⡿⠟⠒⣤⡞⠀⠀⠀⠀⠉⠉⠐⠃⠀⠹⡶⠚⠁⣰⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡰⠌⠻⣍⠛⠛⠉⣼⠃
⡇⠀⢸⠄⠀⠀⠖⠀⠀⠀⠀⠀⣀⠀⠈⠀⣀⢀⠄⡸⠀⠀⠀⠠⠌⣠⣄⢀⡐⢀⡠⠊⠀⢀⠰⠿⠇⠀⠠⠄⠀⢀⡐⠀⠀⣠⠁⢈⠟⠃⠀⠉⠀⠻⡄
⣇⣔⠋⠂⠄⠀⠒⠃⢀⠽⠛⠛⠁⠀⢀⠈⠉⢾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡥⠤⠀⠈⠁⠀⠀⠒⠂⠀⣴⠂⠁⡰⢤⣄⣿⠀⠀⠀⠀⠀⢀⡇
⣿⡯⡤⠉⠥⣾⡿⠋⠁⠒⠀⠐⠀⠁⠙⠂⠀⠀⠀⠀⠄⠀⠉⣉⣉⡛⠓⢃⣐⠂⠂⠀⠀⢀⣀⠼⠃⠀⠀⠀⢀⡇⢐⡒⠁⠀⠀⠈⡱⠆⠀⠀⢸⣿⠁
⡟⣅⡜⠀⠀⣿⣹⡇⠀⢴⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠄⠀⠠⠄⠈⠉⠉⠉⠉⠉⠉⠋⠀⠀⠀⢀⠀⣀⣵⣾⣅⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⡇⠀⠀⠀⠀⠀⢀⡦⠤⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠡⡤⡐⠕⠚⠛⠛⠛⠋⠁
⡇⠀⠀⠀⠀⠀⠶⣠⣤⠤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠥⡄⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣛⣁⡀⣈⣢⠄

⠀⠀⣾⠋⠁
⠀⠀⢿⡈⠁⠀⠚⣍⠙⠦⡀
⠀⢠⡾⠁⠘⣶⠋⠉⠳⢄⠘⠷⣄
⠀⠀⢷⡈⠀⠈⢳⡀⠀⠀⠙⠲⣄⠑⠢
⠀⠀⢨⣷⠀⠈⠵⠻⢄⠀⠀⠀⠘⢷⣄⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⡄⢹⣿⠀⠘⡆⠀⠀⠀⣄⠀⠠⡶⣌⢳⡀⢀⣧⣀⠀⠀⠀⠈
⠀⢻⠘⣿⠀⣀⡀⠀⠀⠀⠈⠳⣄⡀⠘⡇⢿⡉⠈⢿⣴⠃
⠀⠈⠀⣹⡜⠃⠀⠰⣄⠀⠘⠉⠣⣿⠄⠀⠈⠻⡄⠀⠙⡆⠐⣦⡴
⠀⠀⢀⣿⣿⣆⠀⠀⠈⠣⡀⠀⠀⠆⢠⠀⠀⠀⠹⣆⠀⠀⠀⡙⠦⣼⡄
⠸⠀⢸⣿⣿⣿⣄⠀⠠⢶⡌⣆⠀⠀⠀⢷⡀⠀⣠⣼⠇⠀⠀⠘⣦⡀
⠄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠁⠘⣧⠀⠀⠈⠃⠀⠋⠙⢷⣦⡀⠀⠈⠻⣖⢤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⢸⠀⠙⠧⡄
⢇⠀⠈⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡀⠀⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠹⣇⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠹⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠹⣄⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⡀⠇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣷⡾⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡄⠀⠢⣀⣤⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣷⡀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣴⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢠⠃⢸⣿⣄⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢰⠋⠘⠷⠤⠤⢄⡀⠈⢿⣿⡿⠁⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡔⠁⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠃⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⣇⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢛⣛⣛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣦⡀⠀⠀⠀⢀⣀⡼⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣠⣿⣿⣿⠶⠖⠋⠉⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁⠀⠀⢀⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⣀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠋⠉⠹⡿⠛⠁