Capricornus Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣂⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⣶⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣶⣶⣬⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⣻⣿⣿⣿⣟⣛⣉⣴⣶⣶⡆
⠀⠀⠀⣼⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢰⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⢈⢻⣿⡀
⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣄
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤
⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠻⢿⡿⠟
⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠈⠉⠉⠛⠻⣿⣿⣿⡆
⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠠⠴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⡴⠒⠀⠠⠦⢀⡀⢀⡄⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠌⠁⠀⠀⢀⡄⣠⡄⠁⠘⡇⠀⠈⡄⠀⢐⡄⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡉⠉⠀⠀⠀⣠⠀⢾⡇⠀⢳⣀⣀⣹⣾⡿⠷⠿⠟⠛⠛⠻⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠦⠀⠀⢀⣠⡀⠛⣆⠈⠙⢀⣼⡿⠛⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡓⠂⠀⠠⢦⡈⠑⠤⣀⣰⡿⠛⠁⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠉⢁⠼⠷⠶⠄⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢉⣙⠛⠓⢂⣀⢢⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⣿⠟⠳⠖⠀⠉⠽⣧⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣘⣇⣠⠄⠀⠀⠀⠀⢈⡟⢑⣡⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠉⠉⢻⠛⠿⠃⠁⠀⢩⣤⠤⣾⣷
⠀⠀⠀⠀⣠⣾⡏⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠀⢸⣧
⠀⠀⠀⢘⣿⣓⡈⠁⠃⠀⠀⠀⠈⠓⢚⣀⣴⡿⠁⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠻⣿⣿⣯⣶⣷⣿⣾⣶⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⠿⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠠⠀⠄⣀
⠀⠀⠀⠀⠀⠀⢀⠔⢉⡤⠖⠛⠉⠉⠙⠓⠦⣍⠢⣄⠀⣆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠏⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣹⢯⡀⠀⡀⢀⣀⡴⠋
⠀⠀⢀⣴⣶⡟⠀⣿⣶⣶⡤⣄⣀⡀⠀⠀⠀⣀⡠⣾⠟⠛⠒⠛⠛⠛⠉
⠀⢠⡏⣿⠋⣇⢀⢯⣉⣉⡛⣿⡆⠤⠤⠤⠄⠒⠋⠁
⠀⠘⣇⠹⣆⣸⣄⢌⣿⠏⣜⣥⠛⠿⣶⣄⡀
⠀⠀⠘⢦⣙⡿⠋⠉⠙⠠⠟⢁⡵⠂⠈⠻⠹⣆
⠀⠠⣴⣺⣿⡖⠟⠀⢀⣼⠀⠀⢱⡄⠀⠀⠀⡟⣷⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠉⠁⠨⡄⠈⠙⠁⠀⠀⠈⣷⠀⠀⠀⠈⠻⡇⠙⡿⠛⠋⠉⠋⠉⠛⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⣀⣼⡏⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣃⣀⣿⣿⡇⠀⠀⠀⠜⡡⠟⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⣀
⠀⠀⠀⠀⠀⡄⠀⠀⢀⣾⢿⢿⢻⡗⢁⡤⠀⠈⠀⣠⠃⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣗
⠀⠀⠀⠀⠸⠀⡰⠄⠉⠃⠀⣿⣿⣧⣿⠀⠀⠀⢰⡇⢸⠁⡀⠀⠀⠀⠀⠰⠂⠀⠐⠄⠀⢀⡠⣶⠋⢹⡄
⠀⠀⢀⡔⠁⡾⢻⣦⣠⡀⠲⢀⣿⢿⣿⠇⠘⣇⠈⠑⠘⢆⠁⢀⡀⣴⣠⣥⣴⣶⣶⣶⣶⡾⢁⡇⢁⣀⣧
⠀⣠⣊⡀⠘⢻⣈⠿⣿⣷⣷⣤⣯⡾⠋⠀⠂⡝⢂⣸⣠⣠⣾⣿⣿⠿⠛⠉⠀⠉⢙⣿⠟⠁⢸⣧⠖⠒⣻
⠀⠿⣯⠿⠒⠛⠋⠀⠀⠀⠉⡝⢁⠀⠀⢠⠄⢀⣼⣿⢿⣉⣉⣙⣉⣿⣿⣿⣿⣿⣿⣷⠟⢃⣾⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢨⣾⣠⡴⢶⠟⡛⣄⡽⠏⠉⠀⠀⠀⠉⢿⢿⣿⡷⠏⢠⣿⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣏⠈⠈⢠⢏⡀⠃⡞⠁⠀⠀⠀⠀⠀⠀⠈⠇⢿⣇⠀⠘⣿⣿⣿⣯⣖⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⢦⠀⣟⢤⣙⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢯⡀⠀⠈⢻⣿⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⣀⠁⡆⡗⢂⢏⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣄⠀⠈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠿⡇⢻⡰⠤⣗⢎⢳⢤⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣟⣿⡆⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣵⡅⣢⠸⡭⡜⡼⣛⢯⣽⣿⣿⣿⣿⣿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢽⣷⣱⣻⣽⣽⡿⣿⣿⣿⣿⣿⠿⠛⠱⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠎⠁⣼⣿⠁⢀⣾⣿⠁⣶⡿⢋⣤⣾⣷⣢⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⣼⣿⣠⣴⣿⣿⡶⣾⣿⣯⣼⣿⠃⣾⡿⢉⣤⣽⣿⢃⣾⡿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠰⣿⣧⣴⠟⠋⣡⣶⢙⣻⣥⣼⠿⠛⠛⠛⠛⠛⢡⣾⠟⢉⣉⣛⢥⣶⠿⢿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣴⡿⠋⠀⢀⣼⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⢟⣋⡙⣀⢷⣾⢾⣷
⠀⠀⠀⢀⣾⣠⣿⣿⠟⠀⠀⣠⣟⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⣼⣧⢾⣇⣼⣻⣞⣿⣤⣿
⠀⠀⠀⣾⣷⢻⣿⠋⠀⠀⢠⡟⣼⣿⠁⠀⠀⠀⠀⠛⣿⡛⠹⢿⣿⣷⣦⡈⢻⡜⠋⠈⠙⣻⠿⣧⣿⡀
⠀⠀⢰⡿⢡⣿⠏⠀⠀⠀⣼⢳⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡶⢾⣷⢼⣿⣶⣤⣴⣆⠈⠈⣡⣽⡄
⠀⠀⢸⠇⣾⣿⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⢀⣼⡿⠆⠀⠙⢀⣷⣸⡿⠀⠈⠙⢧⣉⣽⡷⣶⠃⢈⣷⠄
⠀⠀⢻⣼⣿⡇⠀⠀⠀⠀⢹⡜⣿⡆⠀⠀⣀⣴⡿⠅⠂⠀⣠⡶⢛⣽⡏⠀⠀⠀⠈⢣⠀⠀⠁⣄⡀⢈⡗
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢷⣻⣷⣀⣴⢋⡥⠔⠀⠀⢸⣯⠄⣾⣿⣿⠀⠀⠀⠀⠈⣧⠀⠀⠉⢻⡄⠿⠇
⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⣴⠆⠀⠀⠹⢿⡄⠘⣿⢿⡄⠀⠀⠀⠀⠸⣇⠀⠰⣆⠰⠀⠴⢀
⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⣿⠁⣾⢣⡄⠀⠀⠀⠈⢷⠀⡼⠀⣹⣦⣔⢂⣴⣄⠙⣴⣆⡈⠐⠷⠄⠜
⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⢸⡏⢸⡇⠀⠀⠀⠀⠀⠀⢀⣼⢛⣿⣿⣿⣩⣡⣤⣤⡙⢻⠶⣤⠴⠂
⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠠⠏⠘⣷⠘⣿⡄⡀⠀⠀⠀⠀⠘⢇⣿⢡⣟⣿⣿⣿⠣⠀⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣤⣀⣴⠃⠀⠙⣷⠘⣧⢹⣆⠀⠀⠀⠀⠈⠻⠈⢻⣿⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣄⠀⠀⢸⡇⣿⠀⢿⡄⣷⠀⠀⠀⠀⠠⡀⠉⢳⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠙⠻⣷⣄⠈⠀⠇⠀⠀⢣⡿⠀⠀⢀⠀⠀⣽⠀⡸⣅⠈⠻⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣡⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠋⠀⠀⣠⡞⠀⢠⡟⠀⡀⢻⣌⢶⣸⣿⣦⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⡏⣦⠀⠀⠀⢸⠀⠀⠀⠀⣠⣴⡶⠛⠁⣠⣴⣿⣇⠘⣧⡀⠙⠿⣿⣿⣿⣿⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣷⡙⢷⣤⣀⡀⠀⠀⣰⡿⠛⠁⣀⣴⠞⠋⣉⣨⡝⣷⡈⢿⣶⣶⣤⣙⢿⡏⡆⠀⢹⣇⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠈⠻⣦⣉⠙⠛⢶⣄⡟⠁⢀⣾⠟⢁⣴⣾⣿⠋⠁⣿⡇⢸⣿⠀⠀⠙⣿⣇⢸⣤⠀⣿⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠙⢷⣄⡈⠛⠷⣦⣄⢸⡇⢰⡿⢃⣴⣿⡏⢸⠇⠀⢠⣿⠀⣼⡏⢠⡄⣤⢸⣿⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣷⣄⠙⢿⣶⣤⡈⠙⢿⣁⣿⠁⣾⠋⣿⡄⠈⠂⠀⣼⡏⣸⠏⣠⣾⠃⢹⣿⣿⠸⣿⠀⣿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡈⢿⠈⠻⣆⠀⢿⡏⠀⣿⠀⠈⠻⣦⠀⠀⣿⡇⢿⡰⠏⣠⣶⣄⡙⢿⣦⡉⢠⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠈⠓⠸⣀⣤⠟⣠⣾⣿⠶⣽⠳⣤⣶⠈⠀⣀⢻⣧⣼⣧⡾⠋⠉⠉⠛⢦⡉⠻⡌⠛⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠙⣦⠐⢋⣁⣼⠷⠶⣭⡖⠿⣦⣤⠌⣁⣸⣻⣷⣽⣿⣯⣤⠶⢶⣤⡀⠀⣿⡀⢹⡀⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢻⣷⣾⠋⢁⣠⣾⣿⣿⣶⣶⣶⠞⠉⠉⢩⡉⠉⠀⡀⠀⠀⠀⠈⠙⠿⢿⡇⢸⡇⣿⣆⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠰⣆⣉⣛⣷⠻⢅⣬⡙⠛⣡⡄⢿⣄⠀⠀⢸⠿⣧⡄⣹⡆⠀⠀⠀⠀⠀⠀⢀⣸⠇⣾⣿⣿⣦⣈⡀
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣀⣼⣷⣬⣍⣉⣉⣀⠀⠈⠙⠛⠁⠀⠀⢻⣆⠀⠀⠀⠙⢧⣸⣷⣧⡀⢦⡀⢠⣖⣾⡟⠰⠟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⣰⠟⠁⠀⣠⠞⣋⣤⣤⣤⣭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⣄⡀⠙⠛⠿⢿⣧⣿⣤⣽⡿⠷⠀⠐⠾⢻⣶⣶⣦⣄⡀
⠀⠀⠀⢀⡅⠀⠀⢰⡋⠾⠿⠶⠶⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠙⢶⡀⠀⠀⠀⠈⠁⠀⢰⣶⣦⡑⣦⣄⠹⣿⡿⠛⠉
⠀⠀⢠⡏⠀⠀⢸⣿⣶⠞⠋⠁⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⠻⣆⢠⡀⠀⠀⠀⠈⢿⣟⢿⣿⣿⢻⡄
⠀⠀⣿⡇⠀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢹⣆⢱⣄⡞⠚⠻⠿⣿⡄⠻⣏⠓
⠀⠀⢸⣿⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠻⣆⠻⣿⣎⢻⣶⣮⠛⠂
⠀⠀⠀⠻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠶⠶⢶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⡆⠿⠉⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣷⣀⣴⠞⣱⢏⣠⣤⣀⠀⠀⠀⠀⢻⣆⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⣾⣿⢿⣿⣯⣭⡆⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣆⠙⢿⣿⣵⣿⣭⠉⠲⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⠿⣿⡿⠁⠀⠀⠀⢻⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣧⣤⣖⣴⣞⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣛⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡦⠀⠀⠐⣞⣱⠂⠀⠀⢴⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠟⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠻⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠲⠀⠀⠦⣄⡀⠄⠀⠀⣠⣄⠀⠀⠀⢀⣠⠴⠂⠀⠴⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠐⠐⠢⡀⠐⠈⠒⣄⠀⠓⠞⠁⣀⠐⠁⠂⢀⠔⠂⠂⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢢⠀⠀⠘⢂⠀⠀⡐⠃⠀⠀⡔⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡚⠄⠀⠀⠀⠀⠀⠀⠀⠀⣢⠔⠉⠀⢀⡀⠀⠉⠂⣔⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠈⠁⠀⠃⠰⠀⠀⠈⠠⢀⠀⡆⠘⠀⠈⠀⠀⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠠⣀⠀⠁⠒⠲⠀⢸⡀⠀⠀⠀⠐⢀⡇⠀⢖⠐⠈⠀⣀⠄⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣳⣄⠁⠀⠀⠀⠀⠈⣰⣞⢸
⠀⠀⠀⠀⠀⠀⠸⣍⡧⠀⠀⠀⠈⠀⠀⠀⠡⠈⠘⠀⠀⢠⡄⠀⠀⠋⠁⠌⠀⠀⠀⠁⠀⠀⠀⢼⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⡀⠀⠡⠐⠂⠌⠀⢀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠹⠺⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠗⠏⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⠀⠀⣁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠐⠂⠀⠸⡙⢅⡨⢃⠇⠀⠐⠂⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⡉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⣎⣱⠂⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠋⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⡀
⠀⠀⠀⡀⠀⢀⣴⣷⣿⣿⣧⣀⡀⠀⠁⠀⢀⡀⡀⠈⠀⠈⠀⠀⠀⠀⢰⠀⢰⡏⠀⢠⡞⠁
⠀⠀⠀⠀⠃⠘⠉⣛⣿⣿⡿⣿⣗⣀⣀⣀⡀⢀⣀⣀⣵⣦⣤⣤⣷⣖⣿⣦⣼⣿⠀⣾⠁
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣾⣿⣿⣯⣟⣟⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣟⣿⣯⣼⣾⣩⣿⣿⣯⣿⣿⣿⣿⣿⡿⠿⠖
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠻⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⠆⠁⠀⠀⠴⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣭⠛⢷⡀⠀⢀⠰⠄
⠀⠀⠀⠀⠀⠘⡧⠀⠀⠶⠸⠷⠽⠾⠾⠮⠇⠀⠀⠁⠀⠈⠉⢉⠉⠀⠸⣿⡄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠛⠃⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⡖⠒⠒⠰⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣍⠿⡉⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣬⠙⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⢿⣿⣿⣿⣿⣿⣸⣧⢻⣭⢻⣿⣿⣿⡇⠀⠀⠀⠀⠐⣤⠚⠀⠃⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣫⣶⠷⠿⠟⠻⠿⠧⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡰⠂⠀⠀⠀⠀⢻⡿⢟⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⢐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⡴⠃⠀⡤⠋⢀⠀⡂⠠⠚⢁⠩⠖⠒⠢⠬⣀⠂⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⢄⠀⠀⠀⢀⠞⡹⢦⠝⢆
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠸⠛⢀⢔⠑⠁⡔⠁⠔⠁⠀⠀⠀⠀⠀⠈⠙⠾⣕⣼⡇⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠁⠁⡀⡀⠁⠞⡰⠋⠀⠈⠳⡀
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢀⠔⠅⢁⣤⡾⠀⣾⣦⣤⣀⡀⠀⠀⠀⠀⠀⣀⣼⠟⠚⠼⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠘⣄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠠⠀⠁⢰⢹⡟⢳⠀⣯⢉⣛⢲⣶⠈⠤⠤⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡰⠁⠀⠀⠀⠀⠀⡰⡁⠀⠀⢹⡌⢧⣘⣆⣈⣿⠋⣯⡜⠛⢷⣤⣀⠀⢰⠁⠀⠀⢐⠂⡶⢰⢐⠆⡲⢀⠀⢠⠐⡆⡖⢐⠆⡆⠀⠘⡂⠀⠀⠀⠀⠀⠘⡄
⠀⢠⠃⠠⣀⡀⠀⠀⠐⡑⠀⠀⠀⠀⠙⣦⡿⢉⠀⠉⠐⠉⢴⠏⠀⠹⠙⣧⡘⠂⠀⠉⠈⢀⠃⠀⠈⠀⠉⠈⠀⠈⠀⠈⠁⠈⠈⠁⠀⠀⠐⡀⠀⡀⢀⠀⠀⢱
⠀⡞⠠⠄⣼⢍⠈⠠⡑⠀⠀⠀⠀⠲⠾⡿⠏⠋⠀⣤⡄⠀⠀⣦⠀⠀⠀⢋⣿⡶⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⢸⠈⠋⢙⠄⡇
⠐⠇⠀⠀⠀⠀⠁⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠙⢀⢸⠁⠀⢀⠒⡀⠈⠤⠘⣍⠲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠏⠉⠀⢹
⢸⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠰⣾⡿⠀⠀⡀⢠⠆⣸⡆⠀⠠⠘⠂⠈⠀⠀⠀⠈⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣶⣾⣿⣿⠄⠘⠁⠀⠊⠁⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠚⠋⠱⣷⣿⣴⡃⠀⠀⢀⡎⠸⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⠶⢻⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⠂⣴⢿⣦⢀⠀⠄⢹⣿⣿⡇⠐⢆⠈⠁⠰⡆⠀⠀⣠⣀⣤⣤⣶⣤⣤⣶⠏⢸⠀⠈⣇⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢸
⠸⡦⡋⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⣔⣁⠐⢻⣌⢿⣿⣷⣦⣼⣵⠟⠀⠐⡝⢂⣤⣠⣰⣾⣿⡿⠛⠉⠉⠉⢻⡿⠋⠀⣿⡄⠐⣷⠀⠀⠀⠀⡘⠄⠀⠠⠤⣀⣀⣼
⠀⢧⢈⡩⠟⠋⠉⠀⡰⠀⠀⠀⠰⠿⠾⠓⠚⠋⠁⠀⠀⠉⠋⡀⡀⠠⠆⢀⣾⢏⠉⣍⣭⡧⢥⣍⣓⢿⣿⠅⠀⠀⣰⣿⡇⠶⣻⠀⠀⠀⠠⡐⠰⣌⢑⡒⢶⠣⡇
⠀⠘⡯⠤⠄⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡧⡶⢻⠋⡕⢠⠞⠉⠀⠀⠀⠈⢻⠿⣿⠆⠀⣼⣿⣿⣷⡙⣪⠀⠀⠀⡱⠁⠀⠈⠒⠙⠉⡹
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣅⠀⢰⠟⣄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠁⢿⡄⠀⠹⣿⣿⣿⣶⡟⠀⢀⠀⠃⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠐⢕⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠙⠂⣾⠀⠤⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠈⣿⣯⡿⠁⠀⠕⠀⠀⠀⠈⠀⠀⣰⠃
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠸⢤⣴⣦⣼⢹⣐⠑⢈⠣⣀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣦⡄⢸⡿⠁⡢⠈⣀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢠⠚⠙⢦⠀⢀⢀⠀⠀⠀⠀⠘⠋⠋⠉⠈⢿⡌⢋⠂⡈⣛⠖⡒⣶⢻⣿⣿⣿⣿⣿⣿⣿⡼⡣⢊⢄⡸⠤⠥⣀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡔⠒⠛⢦⠤⠞⠀⠀⠑⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣷⡌⣤⣆⣎⣿⣿⣿⣿⣿⣿⣿⠟⡟⠁⠀⠈⠀⠙⢄⡀⣸⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠐⠡⢂⠤⢀⡀⠀⠀⠀⠈⠙⠛⠛⠿⣿⠿⢿⠛⠉⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⡡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠤⢁⣀⠈⠉⠉⠉⠙⠉⠉⠁⠀⣀⡤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⠀⢀⠀⠀⠀⣠⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⡒⠣⠤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠐⢒⡡⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠠⠤⠤⠤⠦⠔⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡴⢀⣴⢂⣤⢀⣤⣄⣀
⠀⠀⠀⠀⠀⣰⢆⣼⠧⣾⣷⢺⣿⣾⢥⡟⣡⠿⣯⣟⣢
⠀⠀⢀⢠⣯⡽⠋⢡⣾⡷⠛⠉⠀⠀⠀⠈⢣⢞⠟⣹⡿⡇
⠀⠀⣾⣿⡟⠀⢰⣿⡟⠀⠀⠠⡆⢴⣶⣄⠻⡻⠺⠿⢿⣷
⠀⢸⢫⡟⠀⠀⢸⣿⠀⠀⠀⣠⡝⠲⢶⣺⡗⠻⠖⢶⣄⡽⢦
⠀⢸⣿⡇⠀⠀⢸⣿⠀⢠⠴⠛⢀⡴⢺⣟⠀⠀⠨⠁⢡⣄⣯⡀
⠀⠸⣿⡇⠀⠀⠀⢻⣷⠟⡁⠀⠘⢇⠙⠿⡀⠀⠀⢷⠀⢌⡈⠁
⠀⠀⢻⣿⡄⠀⠀⠀⢹⢸⣡⠀⠀⠈⠀⣠⣿⣶⣞⣠⡓⢦⣌⡄
⠀⠀⠀⠻⢿⣦⣀⠀⡎⠸⡜⡦⡀⠀⠀⠹⡿⣿⠿⡏⠀⠙
⠀⠀⠀⠀⠀⠙⣻⢿⣥⡀⠱⡁⠹⢤⡀⠀⠀⠈⠳⢤⣤⣄
⠀⠀⠀⠀⠀⣰⣟⡀⠈⠻⠀⠁⠀⠊⢀⡴⢀⡞⠀⢆⣌⣿⣁⢤⠀⣀
⠀⠀⠀⠀⠀⣿⢽⣧⣄⠀⠀⢠⡴⠞⣡⠴⢛⣧⡰⣬⣙⣿⡿⠋⠿⣀⡙⣶⢒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⣿⣆⡳⢯⣙⠢⡎⣠⠞⣡⢾⡟⢹⡇⣿⠉⠙⣧⢲⡸⡏⣷⢸⣿⢏⣭⣯⠟⠂
⠀⠀⠀⠀⠀⢿⣮⡙⣦⣍⠳⣧⠏⡾⢿⡀⠀⣼⢷⢣⠞⢳⣿⢸⢰⣧⡏⠘⠁⠀⢀⡹⣿
⠀⠀⠀⠀⠀⠐⠛⠛⠜⣀⢇⣼⣆⡧⢠⡑⣄⣿⣸⡥⠞⠳⢍⠳⡜⢻⡇⠀⠀⠀⠀⢁⣸⣧
⠀⠀⠀⠀⠀⡾⠀⠸⣆⡥⠚⣱⣮⣛⣒⠤⠛⠷⠛⠓⠒⠲⣤⣧⠀⣄⡳⡀⠀⠀⠀⠀⢉⣿⣇
⠀⠀⠀⢀⣼⠃⢀⣰⡭⠷⠋⠭⢛⠅⢻⡄⠀⠿⣆⣧⡀⠀⠀⢀⡤⣸⣿⣦⡄⠀⠀⠀⠈⢹⡟⣇
⠀⠀⠠⠞⠁⣴⣻⣭⣿⠛⠁⠈⠳⠋⠀⠙⠂⠠⠈⠻⠿⣾⣬⣿⠇⠀⠶⣤⣤⡀⠀⠀⠟⣼⠁⢙
⠀⢀⠀⠀⣤⡽⠁⠀⠀⠘⠛⠛⠛⠛⢷⣄⢄⡀⠈⠳⡀⠀⠀⠀⢿⣾⣷⣜⠋⠁⠀⣴⡼⢣⢠⡌⡗
⠀⢸⣄⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣷⠀⠀⠱⡵⣦⢚⡛⠏⠙⠁⠀⠀⣴⡯⠶⢏⡞⢧⣿⣤
⠀⠈⠻⣿⣿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⣆⠀⠁⠈⣿⠉⢲⣄⡀⣸⢇⠀⠀⠰⡎⣴⣶⢻⣦⡄
⠀⠀⠀⠹⣿⣷⣿⣾⣾⣷⡄⠀⠀⠀⠀⠀⢸⣷⡇⣿⠀⠀⠀⠀⠀⠀⣿⣿⣦⡞⡇⢁⢣⢳⢰⡍⣾⡏⠳
⠀⠀⠀⠀⢻⣮⣿⡿⠃⠈⢿⡷⠆⠀⠀⠀⣸⣿⣿⠇⣾⡄⠀⠀⠀⠀⢸⣿⡏⡇⡇⢸⠸⡎⢼⡀⣿⢻⡇
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣧⡀⠀⠀⠀⠀⢠⣿⡿⢋⡞⣡⡇⠀⠀⠀⠀⣾⣿⠁⣴⠇⣼⢰⡇⣿⠷⣿⣬⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣀⣤⣶⣿⢟⣱⡯⢊⡵⠛⠀⠀⠀⢠⣿⣇⡸⠟⣼⡏⣼⣇⡞⢠⡏⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠯⠭⠉⠉⠛⠛⠾⢿⠀⠀⠀⠀⠀⣾⢏⡈⠀⡀⢉⡄⠟⠁⣣⠀⢿⡎⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⢀⣠⣤⣴⣶⣂⣀⠂⢐⡷⣤⣀⢀⣾⡁⠈⢣⠀⠀⡰⢁⣼⠆⠋⣴⠈⠃
⠀⠀⠀⠀⢠⡞⠁⠀⠀⡠⠞⠉⠀⠉⠉⠛⠛⠻⢿⣶⣍⡶⠯⢭⣭⣭⢷⡄⢣⡸⠁⢸⡿⠃⠀⠟
⠀⠀⠀⢠⡟⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣝⠲⠶⣶⣶⣶⣄⠁⣼⠛⠀⣼⠂
⠀⠀⠀⢸⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⢷⣑⠶⢦⣤⣍⡀⠁⠀⠀⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⠀⠹⣦⡑⡶⢬⣝⠄⠟
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌⠻⣌⠳⣌⠳
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣉⡽⠞⠁⠘⠳⣌⡛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣀⡀⠀⠀⠀⢀⣂⣈⣯⡴⠶⠛⠉⠀⠀⠀⠀⠀⠈⠙⠶⣬⣶⣶⣀⣀⣠⠴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀
⣿⠿⣿⢿⠿⢿⡟⠻⠿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠻⣿⣿⣿⣿⣿⡿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⢤⣤⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⢰⣿⡉⠛⣷⣾⣿⡖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⢻⣯⣭⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⢾⣿⠛⣿⣿⣧⣽⣧⣀⢉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠾⠿⠧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⣄⡵⢛⢠⠀⣦⣴⣾⣿⣻⣿⣿⣯⠩⠟⠁⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢐⣧⠷⢸⠸⠀⣿⣿⣿⡟⢹⣿⣿⣿⣧⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡄⠈⠁⠆⢁⣀⣴⣿⣿⣿⡀⣘⣿⣿⣿⣿⣀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⣔⡆⠀⠇⢠⣿⣿⣿⡏⣩⠟⠁⣀⡶⢋⡩⢷⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠂⠉⠉⠉⠾⠡⠤⢴⣏⡴⠋⠀⣸⠳⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣱⡒⢤⡀⠀⠀⠀⡆⠀⠠⡶⠂⢀⠀⠀⡒⠀⢠⠌⣵⡀⠀⣿⠀⠘⢶⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⡍⠐⡶⠀⡙⣢⣄⠀⣧⡀⡀⠅⠀⡀⠐⣉⢀⠀⢘⣄⣿⣫⣤⣼⣤⣤⠞⣻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠩⠤⡶⠊⠻⡃⠀⠉⠻⡄⢷⠀⠃⠀⣿⡟⠈⠉⠀⣉⠉⡉⠄⣸⣷⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠘⢦⡀⣤⠈⡟⢸⠂⠀⠉⢉⢛⣲⠒⢂⣤⡤⠤⠯⣠⠊
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠉⠓⠴⡇⡏⠀⠉⠉⢙⢂⠎⠉⠀⣀⣧⣠⠞⠁
⠀⠀⣞⠉⠀⣼⣆⢸⠽⢻⣝⢸⣷⡋⠁⣞⠈⣷⢸⣟⢸⠧⣽⠀⠀⠀⠀⠀⠀⠉⠓⢤⣞⣁⡛⡉⢻⣟⣃⡼⣿⠁⠁⠀⠹⡄
⠀⠀⠈⠉⠈⠀⠈⠈⠀⠈⠈⠈⠁⠉⠁⠈⠉⠁⠈⠈⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠀⢳⣿⡭⠙⣥⢻⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠸⡟⣿⠀⠘⡎⣧⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢷⢻⡇⠀⢱⡸⡎⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡎⣷⠀⠀⢧⢻⡄⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢿⢼⡶⡀⠈⢧⡝⢶⢶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣀⣀⣀⣸⡅⣷⠁⠀⢸⣉⣉⣈⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢤⣬⣸⣸⣹⣉⣧⠘⣇⠀⣿⣬⣬⣤⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡆⢀⠇⡿⠈⠉⢦⡼⣾⢧⣵⣼⠞⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢁⡠⠺⡇⠀⠀⠀⠁⠁⠸⡆⠀⢰⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠃⠀⠀⢷⠀⠀⠀⠀⠀⠀⢷⠀⢬⢰⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⡄⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣦⣀⣠⠟⠀⠀⠀⠀⠀⠀⠈⢇⠘⡓⠒⡟
⠀⠀⠀⠀⠀⠀⠀⢰⢽⡞⡻⡮⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢁⢧⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣇⣟⣹⣡⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢎⠙⡜⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⣻⣦⡴⠤⣷⠀⠀⠀⠀⢀⣠⡤⠄⠒⣻⣷⠤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⣿⣿⡿⠭⠤⠤⢴⣿⠀⠀⠀⠰⢯⠶⠴⠶⠼⠯⠤⠤⣴⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⢀⣴⣦⣤⣍⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠀⣿⠋⠀⠀⠀⠀⠙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢁⣀⡤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⠵⠚⠉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣠⠇⠀⠀⢀⡀⡀⢀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠗⠀⠀⢠⠋⠀⢨⡞⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡠⠃⢀⣤⠞⠁⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡤⠶⠚⠉⠀⢀⡀⠀⠀⢂⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣕⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⣸⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⢠⡈⢦⠀⠀⠀⢀⣾⠃⠀⠀⣰⠏⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣷⡼⠀⢀⣴⠟⠁⠀⣠⡾⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠟⠁⣰⡿⠃⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣿⠁⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠘⢿⣷⣦⣤⣤⣶⣿⠟⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠙⢷⣄⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠈⠙⠒⠤⠀⠀⠀⠀⣀⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣶⣦⣤⣴⣶⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⣀⡀⣀⢀⣀⢀⡀⢀⣀⠀⢀⡀⠀⠀⣀⢀⠀⢀⣀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠀⢾⡇⠀⣿⡀⣿⢸⣿⣸⠇⢸⣿⠀⣿⠀⠈⣸⡇⢸⣷⢸⣿⣸⡇⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠘⣷⠀⣿⡇⠀⢸⣿⢹⣆⢸⣿⠈⣿⡀⠀⢹⡇⢸⡟⢨⣿⢹⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠉⠁⠉⠁⠀⠈⠉⠈⠉⠈⠉⠀⠈⠉⠀⠀⠉⠈⠀⠈⠉⠈⠉⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠛⠋⡙⠋⢡⠟⠻⣽⣿⡻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠈⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣛⡛⠻⠦⠀⠀⠴⠖⠆⣻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠊⠙⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⢤⣤⣤⡴⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⢀⠀⢀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠁⡷⢻⠚⠇⠸⠸⡇⠣⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣠⡄⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡄⢸⣏⣀⣿⣀⣼⣟⣸⠏⣴⠟⣻⣿⢳⣦⡀
⠀⠀⠀⠀⢠⡆⣿⡴⠟⢩⣽⣻⡧⠟⠛⠛⠛⠋⣼⢋⣭⣿⡶⣿⡆
⠀⠀⠀⣤⣸⣿⠋⠀⣴⣿⡟⠉⠀⠀⠀⠀⣀⠀⣰⣏⡗⢺⣾⣏⣻
⠀⠀⣼⢟⡿⠁⠀⡼⣿⡏⠀⠀⠀⠻⣏⠻⠿⢷⡌⢧⡉⠉⠹⠻⢿⡀
⠀⢀⡏⣾⠇⠀⠀⡇⣿⠀⠀⠀⢀⣤⡌⠙⠛⣏⡿⠙⠻⣛⣷⣤⠞⢻⡄
⠀⢸⣿⣿⠀⠀⠀⢧⣿⡀⢀⡴⠞⠃⢀⡴⢺⣿⠁⠀⠈⢆⠈⢰⣄⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣿⣿⠀⠀⠀⠈⢿⣷⡾⢚⡅⠀⠸⣇⠹⣿⡇⠀⠀⠈⡆⠀⡌⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠹⣿⣇⠀⠀⠀⠀⢹⣧⡟⡄⠀⠀⠘⠀⠃⣼⣦⣶⠶⠉⠦⣌⡓⢂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠹⣿⣧⣀⠀⠀⢸⠹⡇⢧⠀⠀⠀⠀⠚⣵⣻⣿⣿⡟⠑⠾⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡀
⠀⠀⠀⠀⠈⠛⢿⣷⣤⡞⠀⠹⡌⡇⣦⢀⠀⠀⠙⠘⢿⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣥⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠛⠻⣦⡀⠃⠇⠘⣸⠇⠀⠀⢰⡄⡙⠛⢿⣶⡶⠒⠦⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡲⢵⠀⡀
⠀⠀⠀⠀⠀⢀⣾⣧⡄⠀⠈⡗⠀⠀⠀⢁⣠⠜⢀⣼⠁⡘⢦⣦⣿⣦⣴⣆⠀⠀⠈⢿⡓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣝⡲⢅⠳⡀
⠀⠀⠀⠀⠀⢸⣿⢻⡽⣦⣀⠁⠀⣰⠞⢋⣡⠖⢛⣛⢦⠳⣦⣭⣻⣿⠅⠙⣏⣀⡀⠀⠉⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⣀⠀⠀⢶⣬⣙⠢⡑⢷
⠀⠀⠀⠀⠀⠈⡿⣆⡙⢦⣍⠙⢦⠃⣠⠟⣡⡾⡟⠉⣼⠀⡿⠀⢈⢿⠸⡆⢻⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⡛⠶⢭⡛⠮⣟⢧⡙⠳⡄⢸⡆
⠀⠀⠀⠀⠀⠠⢿⣌⠙⣦⣌⠙⢾⣱⠏⡾⢿⡀⠃⢠⡟⡼⢃⡾⢹⣾⡤⡇⣸⢠⡇⠀⠀⠀⠀⠀⠀⣤⣿⣟⣒⠭⠑⠠⠈⠁⠈⠁⠙⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢨⠟⠳⡜⡆⢱⢀⣿⣘⢇⢈⡑⠄⢸⡇⣷⣫⡾⠦⣝⠷⣄⠿⣼⠀⠀⠀⠀⠀⠀⣲⣗⡒⠒⠂⠀⣠⣤⠶⠶⠶⠦⣤⣸⡆⠀⠃
⠀⠀⠀⠀⠀⠀⣾⠀⠰⣄⢚⣫⠾⢿⡽⢯⡍⢉⣰⣷⣿⣿⠥⠤⣄⡈⣷⠘⡄⠸⣇⠀⠀⠀⠀⢈⣭⠭⠀⠀⣴⠟⠉⠀⠀⢢⠀⠀⠀⠈⣿⠀⢸
⠀⠀⠀⠀⠀⣰⠇⠀⢀⠻⢿⣡⡴⠿⣿⠟⣷⠉⠀⣶⣄⢠⡄⠀⠀⠙⠛⠀⡇⣷⣌⡢⠄⠀⠀⢈⣥⣆⠄⡾⠁⠀⠀⠀⠀⠀⢧⠀⠀⢰⡟⠀⣸
⠀⠀⠀⢀⡴⠋⠀⣠⣾⣽⣑⣃⠈⣙⣖⣩⠈⢧⡀⠉⠹⣮⣿⣄⢤⡀⣤⣾⠣⠟⡛⠛⠓⠒⠀⢀⣾⡧⢸⠃⠀⠀⠀⠀⠀⠀⢸⡆⣠⠟⠁⢠⠇
⠀⠀⠀⠚⠁⢠⣪⣵⣖⠒⠂⠀⠀⠀⠙⠁⠀⠀⠉⠑⠐⢄⠉⠛⠳⠷⠞⣛⡀⢘⠹⣿⣷⡦⠀⠀⣾⡁⢸⠀⠀⣀⣀⣠⠤⠤⠾⠛⠁⠀⣠⠏
⠀⢀⡎⠀⢰⣷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⡀⠀⠀⠹⣟⢾⣿⠮⠁⠀⠀⠀⣾⣡⣺⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⢸⣧⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⣾⡭⣍⡻⠌⠓⠀⠀⢀⡠⠈⣿⢣⡏⠀⠀⠀⠀⣀⣠⡤⠖⣿⠁
⠀⠀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⠼⠛⠀⠀⠀⣠⠏⢠⣞⣵⢟⣠⡴⠶⣿⣿⠉⠁⠀⠀⣿
⠀⠀⠀⠘⢿⣿⣶⣞⣼⣵⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡟⠚⠛⠉⣴⠟⠁⠀⣰⣿⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠘⣿⣙⣿⣿⡯⠙⢷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣏⠀⢀⣴⠟⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢻⣯⣙⣋⣀⣠⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠹⡄⠉⠉⠛⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠋⠉⠁⡼⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⢠⡟⠳⣄⠀⠀⠀⣿⠀⠀⠀⣿⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢸⡇⠀⠀⢸⡇⠀⠀⢀⡴⠛⣦
⠀⠀⠀⠀⢠⡟⠀⠀⢸⡆⠀⢀⡿⠀⠀⣰⠇⠀⠀⠈⡿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⡏⠀⠀⠀⢷⡀⠀⠸⣇⠀⠀⣾⠀⠀⠘⣧
⠀⠀⠀⠀⣼⠓⢦⣀⣼⣷⣶⠿⢷⣶⡾⠋⠀⠀⢀⡼⠁⠹⣧⣀⣀⠀⢀⣀⣠⡿⠁⠹⣄⠀⠀⠈⠻⣶⣶⠿⢷⣶⣿⣄⣠⠖⢻⡄
⠀⠀⠀⠀⡿⠀⢨⡟⢱⡟⣁⣀⠀⠀⠙⢶⡶⠞⠋⠀⠀⠀⣿⠉⠉⠉⠉⠉⢹⡇⠀⠀⠈⠛⠶⣶⠟⠁⠀⢀⣀⡙⣷⠙⣯⠀⠸⡇
⠀⠀⠀⠀⡗⢦⡿⠀⣿⠃⣧⠉⠙⠻⠶⣦⣷⣄⣀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣴⣧⡶⠾⠛⠉⢡⡇⢻⡇⠸⣧⠞⡇
⠀⠀⠀⠀⣇⢸⡇⠀⣿⠀⠘⣦⡀⠀⠀⢀⣿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣇⠀⠀⠀⣠⡞⠀⢸⡇⠀⣿⢀⡇
⠀⠀⠀⠀⢻⢸⠁⠀⣿⠀⠀⠈⠛⠻⠛⢻⡿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄⠀⠀⠀⠀⠀⠀⠸⣿⠛⠻⠛⠋⠀⠀⢸⡇⠀⢸⢸⠃
⠀⠀⠀⠀⠘⣿⡀⠀⢹⡆⠀⠀⠀⠀⠀⢾⡟⠶⣦⣀⠀⠀⣠⠟⠀⠀⠀⠘⢧⡀⠀⢀⣠⡴⠞⣿⠇⠀⠀⠀⠀⠀⣾⠁⠀⣸⡟
⠀⠀⠀⠀⠀⢻⡇⠀⠘⣇⠀⠀⠀⠀⠀⢸⣷⣤⠬⢿⣦⢠⡏⠀⠀⠀⠀⠀⠘⣧⢠⣿⠻⢤⣴⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠀⢻⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⡟⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢹⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢷⠀⠀⠀⠀⠹⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⠏⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⣇⠀⠀⠀⠀⠹⣆⠀⠀⠘⣇⢀⣀⣀⣀⣀⢀⡟⠀⠀⢀⡾⠁⠀⠀⠀⢀⡟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠹⡄⠀⠀⠀⠀⠘⣧⠀⣰⠟⠉⠉⠉⠉⠉⠛⢷⡀⢠⡟⠀⠀⠀⠀⠀⡼⠁⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢻⡀⠀⠀⠀⠀⠘⣧⡟⢠⣤⠀⠀⠀⢠⣤⠘⣧⡟⠀⠀⠀⠀⠀⣸⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢧⠀⠀⠀⠀⠀⢻⣇⠀⠙⢷⣤⣴⠟⠁⢀⣿⠃⠀⠀⠀⠀⢠⢇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⢻⣷⣦⣄⣿⣇⣤⣴⣿⠃⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡇⢀⡤⣄⢠⣆⣤⡀⣤⢤⢨⠀⣤⣤⠀⣀⢠⡀⣤⢤⢠⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⣠⢈⠤⢼⢸⡇⠀⡇⡇⠀⢸⢸⠁⠘⡀⡇⠀⡧⡏⠀⢸⠇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠘⠂⠚⢸⠓⠚⠁⠃⠀⠘⠈⠳⠚⠀⠓⠶⠃⠃⠀⠘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠂⠀⠀⠀⠀⣿⢠⠶⠟⠁⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠙⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣾⣿⣶⣶⣤⣄⣷⡙⠛⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⢀⠀⠀⠀⠀⠰⢴⢆⡻⢆⡀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡆⠀⠀⡀⠀⠀⠘⠀⡆⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⢶⠺⠃⠈⠀⠠⠀⠚⠺⠿⣿⣿⣿⡿⠿⠛⠻⠿⢿⣿⣿⡆⠈⠀⠋⠀⠀⠄⢀⣇⡄⠀⠀⠀⣠⣐⢻⡿⣿⣿⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⠎⢃⠈⠀⠀⠀⢸⣧⠀⠀⡟⠛⠛⠀⠀⠀⠀⠀⠀⠙⢻⣏⠢⣄⠀⠀⠀⠠⡈⢿⣇⡄⠀⢸⣿⣿⣯⣰⣿⡏
⠀⠀⠀⠀⠄⡙⠊⡀⠀⠀⠀⠀⠛⣤⡴⠃⠀⠄⠀⠀⠀⠀⠁⢀⣄⡀⠙⠳⣼⣷⠄⠀⠀⡙⢾⣿⢣⣶⣼⣿⡿⣿⣿⡿⠀⠀⠀⢀⣾⠂⢀⡀
⠀⠀⠀⠀⠈⠃⡀⠁⠀⠀⠀⠀⠀⡟⠀⢰⠄⠀⠀⠀⠀⠀⠀⠘⠿⠃⠁⠀⢀⣠⣶⡆⢀⡹⣎⣿⢋⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠠⢦⣴⣾⣿⡿⠟⠀⠀⠱⣞⣿⡟⢻⣿⣿⣿⣿⡗⠀⠀⣿⠠⠴⠖⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠈⠉⢖⠀⠄⣀⠀⠂⠀⠀⠈⢻⣛⣩⣤⣤⣤⣤⣤⣼⡿⣿⣿⣿⣋⣸⣿⠃⡄⢀⣫⣤⠄⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣭⣃⠀⠠⠀⣴⣿⣿⡙⢿⣿⣿⣿⢛⣻⣿⣿⣿⣿⣿⣿⡏⠰⣧⣬⠅⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠳⠦⣔⣦⣈⢴⣌⢷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡏⣩⣿⣾⠁⠶⠖⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢋⣩⡽⠿⠿⢷⣅⠄⠀⠉⣩⣿⣿⣿⡿⢿⣿⣧⣴⣿⣿⣿⣷⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣋⡩⠿⠗⠈⢰⣿⡿⠻⣿⣷⣶⣿⣿⣿⣿⣿⣩⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⢿⣿⣷⣼⣿⣂⣽⣿⣿⣿⡿⠟⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠉⣻⣿⡇
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣤⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠩⣭⡉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣾⣿⣛⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⣰⣿⣿⣿⣿⣿⡄⠀⣾⠋
⠀⠀⠀⠸⣶⣶⣶⣆⠹⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢿⣿⣿⣿⣆⣘⣦⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢸⡇
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⡿⠁
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠉⠛⢿⣿⠆⠘⠁
⠀⠀⣸⣿⣿⣿⣿⠛⠉⠉⠉⣽⣿⠀⠀⢀⣠⣤⡶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆
⠀⠀⢿⣿⣿⣿⣿⣦⣀⣠⣾⣿⡟⠀⠀⠈⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠟⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⠴⠶⠦⢤⡀⠀⠀⠀⠀⣠⠤⠶⠶⢤⡀⠀⠀⠀⠀⢀⡤⠶⠶⠤⣄⠀⠀⠀⠀⢀⡤⠴⠶⠦⣄
⢠⡞⠰⡄⡼⣷⣢⠹⡄⠀⠀⡞⢡⣀⣤⣠⢄⠘⣆⠀⠀⣴⠃⢲⡀⠀⡴⠈⢧⠀⠀⢰⠋⡴⠒⡶⠒⡌⢳
⢸⡀⠀⢷⠇⡟⠁⠀⡗⠀⠸⡇⢠⣉⣤⢩⢌⠀⣸⠀⠀⣇⠀⢺⡗⠒⡗⠀⢸⠀⠀⢿⠀⠉⠀⡇⠀⠁⢀
⠈⠳⣀⠈⠰⠃⢀⡼⠁⠀⠀⠳⣄⠉⠀⠉⢈⡰⠃⠀⠀⠙⢦⡚⠀⠀⠙⣠⠏⠀⠀⠘⢦⡀⠀⠃⠀⣠⠞
⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⣀⡤⠤⠤⢄⡀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄
⢀⡞⠁⢦⣀⣴⠂⠹⡄⠀⠀⡼⠁⢲⠤⢴⠖⠙⣆⠀⠀⣰⠋⡤⠖⣲⢤⡈⢧⠀⠀⢠⠏⢰⠋⠉⠹⡎⢳
⢸⠀⢰⡋⠀⠈⡇⠀⡗⠀⠸⡇⠀⢸⠀⢸⠀⠀⣹⠀⠀⣇⢀⡖⢦⠙⠖⡃⢸⠀⠀⢿⠀⣞⡆⢀⡼⠁⢈
⠈⢧⡀⠓⠶⠞⢁⡴⠃⠀⠀⠳⣄⠞⠒⠚⠆⣠⠏⠀⠀⠹⣄⠙⠻⠖⠚⣡⠞⠀⠀⠘⢦⡈⠀⠸⣄⣄⡼
⠀⠀⠉⠛⠒⠚⠉⠀⠀⠀⠀⠀⠈⠙⠒⠚⠋⠁⠀⠀⠀⠀⠈⠙⠓⠒⠋⠁⠀⠀⠀⠀⠀⠉⠓⠒⠛⠉
⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀
⢀⡞⢹⢶⡴⣄⠈⠳⡄⠀⠀⡴⠋⢀⡤⣄⠀⠙⣆⠀⠀⣰⠛⡗⣦⠦⡀⠙⢦⠀⠀⢠⠞⠁⠀⠀⣤⡌⢳
⢸⠁⢸⢸⡇⡏⢳⠀⡧⠀⠰⡇⢐⣚⣀⣈⣓⠀⣹⠀⠀⡇⠀⡇⣿⢠⠇⠀⢸⠀⠀⢾⠀⠠⣤⡜⠉⠀⢈
⠈⢧⡈⠈⣁⠿⣃⡰⠃⠀⠀⢳⡀⠀⠀⠀⠀⣠⠏⠀⠀⠹⣄⠁⠉⢎⡰⢃⠞⠀⠀⠘⣆⠚⠁⠁⠀⢀⡼
⠀⠀⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⠀⠈⠛⠒⠒⠚⠁⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣠⠤⠖⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠶⠔⡀
⠀⢾⣤⠀⡠⢒⠠⢹⣆⠀⠀⠀⠀⠀⠀⢨⣠⠈⠚⢽⣍⣀⢢⣈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢠⣆⢈⣑⡣⢀⢀⣬⣿⠖⠀⠀⠀⠀⠀⠀⡌⠘⣄⣤⢠⢹⠁⣛⣼⡄⠀⠀⠀⠀⢀⡶⠆⠀⢀⡀
⠈⠛⠁⠀⢀⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠺⣡⠇⣈⡿⡠⡿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠛⢀⠀⠄⠴
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⢴⣌⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⠷⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⢶⠶⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠆⠒⠛⠒⠳⢤⣄⠄⠀⠀⠀⠀⠀⠀⠮⠍⣘⢁⣀⣃⡉⡽
⠀⠂⠀⠀⠐⣗⠀⠀⠪⠁⠘⠀⠀⠀⠀⠑⢀⠐⠂⠀⢀⠀⠋⢹⣥⡀⠀⠀⠀⠀⠀⡆⣎⡉⠈⡁⠌⣱⡐
⠀⠀⠀⠀⠀⠙⢦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠁⠀⠒⡼⠟⠃⠀⠀⠀⠀⠀⠉⠱⣛⠖⠓⢞⠯⠉
⠀⠀⠀⠀⠀⠐⠁⠈⠉⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡀
⠀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡐⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡷⠃⠀⠀⠰⠹
⠀⢳⢿⠂⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠖⠐⢒⠳⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡂⣠⢃⣟⣀⢒
⠀⠀⠁⡠⠂⡉⢉⣼⡒⢭⠲⢤⡀⠀⠀⠀⣄⠐⠀⢀⠀⢙⠁⢀⠀⣀⠨⠃⠀⠀⠀⠀⠀⢒⡾⣄⠤⣹⢾
⠀⠀⠀⠉⠃⣀⡘⠀⠶⢐⡻⠶⢦⠀⠀⠀⠹⢶⣀⣌⠓⠦⣜⣛⣛⣌⠁⠀⠀⠀⠀⠀⠀⠘⠁⢛⠑⡄⠈
⠀⠀⠾⠅⠈⠀⡤⠒⠋⠉⢨⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠌
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⡁⠀⠀⠀⡃⠼⠃⠀⠀⠀⠀⠀⢠⠶⣶⢖⢦⣀
⠀⠀⠀⠀⠀⠃⣡⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠸⣏⣃⢂⣋⡽⠂⡂⠀⠀⠀⠀⠀⢸⡟⠀⠳⣼⣿⡀
⠀⠀⠀⠀⠘⢻⣴⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢬⠞⠋⣳⣴⣋⠙⠪⠅⠀⠀⠀⠀⠀⢹⡿⠦⠞⢛⠙
⠀⠀⠀⠀⠀⠘⡤⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣲⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠦⡤⣾
⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⡁⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣈⣠⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⡠⠄⠒⣀⠩⠍⣀⠐⠂⠠⠄⣀⣀⣀⣀⠀⠀⡀⠀⢀⡄
⠀⠀⠀⡰⠁⢠⠁⠀⠀⠀⠀⠁⠉⠒⠲⠄⣁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠁⠀⣀
⠀⠀⡰⠁⢠⠃⠀⠀⠠⡲⠊⢉⠗⠀⠤⠤⠄⠀⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠄
⠀⠀⠳⠔⠁⠀⠀⠀⠈⠠⠒⠁⠈⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠈⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠃
⠀⠀⠀⡇⠀⠀⢀⠔⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠄⠀⢀⡤
⠀⠀⠀⣇⢆⣀⠊⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠄⠀⢀⡠⠔⢺
⠀⠀⠀⠀⠈⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠢⡀⠀⠈⠊⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⢘⠄⠀⡀⠤⠐⠊
⢀⣀⣀⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡄⠊⠀⢸
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠂⣼
⢸⠀⡅⠈⠐⠂⠄⠄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠐⢆⠀⠀⢹
⢀⠀⣇⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⢀⡠⠦⠤⠀⣀⣀⠀⠀⡘⠀⠀⢠⠶⣳⠬⢆⠀⡆⠀⠀⠄
⠰⠀⡆⠀⠀⠀⠀⡠⠁⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⢍⣟⠿⡹⢞⠀⠇⠀⢠
⢆⠀⢈⠆⠀⢀⠜⠀⣀⢔⣁⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠙⠘⠚⠥⠊⠀⠀⠸
⠀⠑⠎⠀⢠⠎⠀⠀⠀⠠⠤⢀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠍⠀⠄⠀⢠⣤⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⣤⡋⠁⠀⠀⡀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡆⠀⠀⠈⠴⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆
⠀⠀⠀⠈⣿⣿⡆⠠⠤⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⡶⠶⢦⡀⣿⡇⠀⠀⢀⠈⣤⣵⣶⣶⣮⣤⠀⡀⠀⠀⠀⣤⣾⢫⣿⣿⣿⣍⢷⣤
⠀⠀⠀⠀⠈⠻⣷⣤⣤⣾⡿⠋⠀⠀⠀⠀⠀⢶⡀⢿⣄⠀⣨⣷⠟⠀⠀⠀⠈⠀⡈⡍⣻⣿⠩⢉⠈⠁⠀⠀⠀⠈⠁⠈⠿⠿⠿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⢠⣰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠘⠓⠂⠈⠀⠀⠀⠀⠀⠀⠠⠛⠿⢷⣿⣯⣧⣀⠀⠀⠀⠀⠀⢀⠖⠆⠀⠹⠂
⠀⠀⠀⠀⠟⠉⠻⠉⣿⣧⣶⡄⠀⠀⠀⠀⠙⢀⣤⣄⣀⣀⡐⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠁⠙⠂⠀⠀⠀⠠⣟⣛⣍⠀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⡀⣶⢀⡹⠟⠁⠀⠀⠀⠀⠀⠨⢿⠿⠿⣻⡷⠂⠀⠀⠀⠀⠀⠋⠀⠀⠰⠶⠤⠀⠴⠂⠀⠀⠀⠈⠐⠊⠁⣠⡾⣷⠂
⠀⠀⠀⠀⠀⠙⠳⠌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⣄⠜⠈⠀⠀⠀⠀⠀⠀⠀⠈⣀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠙⢬⠭⠃

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⡶⣆⡀
⠀⠀⠀⠀⠀⠘⠁⠀⠀⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣐⣠⣤⣦⣤⠂⠀⠀⠀⠀⠀⠀⠉⠉⠀⢿⠀⠉⠛
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠨⢹⣿⡕⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠿⡿⠿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠇⠀⠀⠀⠀⠀⠀⠒⠶⠖⠠⠿⠀⠲⠒⠒
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡤⠴



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⡒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⣦⠖⠛⠛⠓⠮⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃⢀⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠫⠖⠛⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡔⠀⠀⢀⠤⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠃⠀⣠⠃⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⣁⡤⠞⠁⠀⠀⢀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠅⠀⠀⠀⠀⠚⠉⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠀⢀⠰⡄⠀⠀⠀⣰⠏⠀⠀⣼⠁⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣇⡟⠀⣠⡾⠃⠀⣠⡞⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠋⢀⣾⠏⠀⢀⣾⠋⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢻⣧⡀⠀⠀⠀⣠⣾⠟⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠻⣆⠀⠀⠙⠿⠿⡿⠿⠟⠉⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠈⠛⠦⢄⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣤⣤⣤⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡄⠀⣠⡀⡀⢠⡄⣄⠀⣤⠀⣠⠄⡀⣀⢀⡀⢠⡄⣄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⣿⠀⢸⡧⠟⢸⣧⡟⠀⣿⢰⣿⠀⢰⣿⠈⣷⢸⡧⣿⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠹⠆⠼⠇⠀⠸⠇⠿⠀⠿⠈⠿⠄⠀⠻⠸⠏⠸⠇⠿⠄⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⡴⠿⠷⡖⠶⡤⣤⢤⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⣣⠴⢻⠟⠀⢐⠖⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢠⡃⠀⢸⠂⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡊⡀⢣⡀⢸⡆⠈⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠌⠱⣎⣳⠈⠁⢷⢄⡀⣀⣀⣀⠀⠀⠀⠀⠀⣴⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡴⠋⡸⢉⡀⢼⠖⡩⢀⣹⠃⠀⠀⠀⠀⢸⢹⢕⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⠒⠲⠋⠀⠉⠁⠘⢦⠀⠀⠀⠀⠘⣆⠣⣀⣄⣠⠭⢟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⡀⠀⠀⢠⠏⡟⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠀⠀⠀⢹⠀⠀⠀⠀⢣⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⢦⡀⠀⠳⡀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠲⢄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣇⣀⣀⡴⢞⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠷⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣼⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠹⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⠴⠛⡞⠀⢠⠇⠀⠀⠀⠀⣀⠀⠂⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠋⠀⢀⠇⠀⡾⣀⠀⢀⡀⠄⠃⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡲⠃⠀⠀⢸⠀⡸⠀⠀⠉⠉⠓⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠵⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠹⠆⠴⢲⣄⣤⣄⢰⡖⡆⣄⢀⡤⢤⣀⡴⠲⣦⢰⠲⡆⣶⡄⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡀⣴⢺⡟⣧⡾⢸⡇⠀⡿⢺⡇⢈⠉⠷⣤⠼⢹⠀⠀⡏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠞⠁⠙⠛⢃⣇⠀⠸⠃⠀⠀⠈⠓⠚⠀⠀⠀⠀⠸⠂⠀⠟⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠄
⠀⠀⠀⠀⠀⣿⠁⠀⠙⣧⠀⠀⣼⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠹⠆⠀⠀⠸⡇⢸⠇⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡶⠦⢶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠿⠶⠶⠶⠾⠷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⠈⠉⣁⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠘⠙⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠉⠃⠉⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠛⠃⠙⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⣴⢶⡄⢠⢶⣆
⠀⠀⠀⠀⢀⡴⠿⠿⣍⠉⠉⠉⠉⠉⠛⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⣷⠏⠀⣿⢀⣤⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆⠀⠀⢀⣤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣟⠀⠀⣿⡟⠹⣷
⠀⠀⠀⠀⠘⠷⠤⠴⠟⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣧⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠻⣤⣠⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣽⠂⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⣼⠏
⠀⠀⠀⠀⠀⠉⠙⠛⠓⠻⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠞⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⠟⠀⢀⣿⠾⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠻⣧
⠀⠀⠀⠀⠀⠀⢮⠀⠶⠐⠂⠆⠶⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠰⠶⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⠇⠒⣲⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠉⠙⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠏⢹⣧⡾⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⣿⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⣠⣾⠟⠁⢸⣧
⠀⠀⠀⢀⣀⣀⣘⣿⣄⠀⠀⠀⣴⣿⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣾⡟⠁⠀⠀⠘⠛
⠀⠀⠀⠈⠛⠋⠙⠛⠋⠀⠀⠀⠛⠛⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠻⣦⡄
⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⢻⣦⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⢤⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⠀⡀⣀⢀⣀⢀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⡠⡄⣀⣀⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠁⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⠈⠁⠉⠉⠁⠈⠈⠁⠉
⠀⠀⠀⠀⠰⢶⣤⠀⠀⣰⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⢻⡆⣰⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣷⡀⣠⣴⢿⣆⢀⣤⡾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠸⣷⡟⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⠀⠘⠟⠋⠀⠀⠻⠛⠁⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⣠⠞⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⣿⠟⠟⢻⣿⠻⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣿⡇⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢀⣠⡶⣷⡀⣀⣴⢾⣆⢀⣠⡶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⠳⠶⠾⠋⠀⠀⠀⠀⠀⠀⠶⠟⠉⠀⠸⠿⠋⠁⠀⠿⠟⠉⠀⠸⠧⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠘⠷⠄
⠀⠀⠀⠀⠀⠠⠀⣀⢀⠀⡀⡀⠀⢀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⠀⠀⣀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠐⠈⠉⠋⠉⠁⠉⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠙⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠁⠉⠉⠉
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣄⣤⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⡉⢉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⡀⠰⠦⣤⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣦⣿⣄⠀⠀⠀⠀⠀⠀⠀⡠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠻⡌⠓⠀⠀⠀⠀⠀⠀⠀⣬⣝⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⣻⣇
⠀⠀⠀⠐⠀⡀⠀⢀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⡀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢲⣟⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⣤⡘⠶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡶⠃⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⡿⣮⠁
⠀⠀⠀⡐⠀⠠⣄⣉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠛⢉⡁⠀⠀⢀⡀⠀⠀⠀⢹⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⡓⠾⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡄⠀⠬⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣴⠀⠀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠄⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⡋⠑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠈⠀⠀⠹⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈
⣧⣐⠀⠠⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣷⡀⠘⠀⠀⠘⠀⠀⣧⠖⠂⠀⣹⠀⢀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣂
⠈⠙⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⢳⡀⠀⠀⠀⢀⣠⣿⡝⠀⢪⡙⡀⠈⠂⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⢂⡹⡄⠀⠀⠜⠛⢙⡁⣀⢈⡅⠳⢲⣄⠀⡾⠀⣾⣿⡄⠀⠀⠀⢠⣿⡘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣡⠻⣆⠀⠀⣼⠀⠈⠀⠀⠈⠁⠀⣾⡃⢸⠇⣄⠉⢈⣿⠀⠀⠀⢨⡿
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣆⠛⣿⠀⠰⡟⠈⠐⠞⢷⠆⠙⣾⠋⠛⢸⠀⣸⡟⣛⠹⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⠿⣁⠛⠀⠀⠁⠃⠀⠀⠀⠀⠀⠋⠀⠀⣘⠀⠉⠁⠙⠃⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⠀⠀⣤⠄⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠷⠶⠶⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠐⠒⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⡠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠀⠀⢀⡀⣀⠀⣿⡇⠀⣀⢀⡀⠸⠗⠀⠈⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠧⠁⠄⠼⠏⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⠿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⣰⣿⡿⠁⠀⠙⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠠⠟⡟⠀⠀⠀⠀⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⡷⡼⠀⠀⠀⠀⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⢸⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠟⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⣶⣾⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⣰⣿⡿⠛⠁⠀⠀⠈⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣆⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠙⠻⣿⣶⣶⣤⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⠻⠀⠀⠀⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡠⣖⣭⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⠻⠿⠿⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣴⣦⣶⡴⡤⣤⣬⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⣿⣿⣏⢸⣿⣟⣆⢻⠈⣷⡇⣏⢾⡇⣇⢺⣿⡇⣿⢀⡇⣇⢾⣷⣝⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⠚⠗⠈⡗⠐⠶⢿⣿⣿⡏⢹⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠋⣙⢀⡄⢀⠛⡶⠆⣀⠉⢻⣿⣿⣧⣸⡟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣭⣶⣾⠿⠿⢿⣷⣾⣼⣧⣰⣟⣱⣶⣤⣿⣿⣿⣷⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠋⠁⠀⢀⣤⡾⠛⠉⠉⠛⢿⣿⡏⢀⣼⠿⣿⣿⣿⣿⣦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⣶⣤⣀⡙⣿⣿⣃⣴⡿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⠀⣹⡿⢿⡿⠟⠛⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠛⢻⣾⣿⢿⣿⡟⠁⠀⠛⠁⠀⠴⠒⠈⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣼⣷⣤⣽⣟⡀⠛⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣧
⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣆
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⢿⣿⡿⠳⣄
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢣⣿⣿⠏⣿⡇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⢃⡀
⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡿⡿⠁⣿⢸⣿⣿⠰⣿⠀⣿⡇⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣀
⠀⠀⠹⢿⣿⠿⠋⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢸⣿⡇⢱⡄⠈⢸⣿⣿⣦⣻⡄⢻⣷⡰⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⢸⣿⡇⠸⣧⠀⢸⣿⣿⢿⣿⡇⠘⢿⡇⢷⠀⡌⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠙⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢠⡏⣿⠀⣿⢠⢸⣿⣿⣼⣿⣿⣤⡘⣿⢸⣿⣷⠆⣰⡏⢰⣿⣷⡆⣀⣀⣀⣀⣀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢻⣿⣟⣾⠇⣾⢰⡇⢸⣿⣿⣿⡿⢹⣿⣿⣷⣿⣿⡏⢹⣇⣿⣅⣿⣿⣿⣿⣿⣯⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⢠⠀⣿⣳⣿⣿⠃⡹⠇⣼⣿⣿⣿⣿⢸⣿⣈⣿⢿⣿⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣯⠘⠟⢿⢏⡿⢰⣿⢹⡿⠁⣠⠀⣼⢿⣿⣿⣿⣿⣿⣽⡿⣿⣾⢣⣿⣿⣿⣿⣿⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢸⡇⣾⡇⣿⡇⣰⡿⣸⡟⣸⣿⠹⠿⣿⣿⣿⣵⣿⣿⣵⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⠀⠀⣿⣿⣿⣿⡇⣿⡇⣹⡇⣿⠋⠀⠀⣸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣷⠀⠙⣿⡇⠈⣿⣿⡇⣏⢃⣿⢰⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⢀⡙⣿⢠⣿⣿⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣿⡆⢸⡇⣿⣿⢿⣿⢻⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⢱⣿⣧⣾⢃⣿⠟⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⣸⣿⣿⢣⣿⢋⣶⢟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢸⠋⠁⠀⢀⡴⣦⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠀⠀⣀⣿⣶⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⣿⡿⢹⣿⣿⣿⣿⡏⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣹⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⡿⠛⠉⣿⣿⣿⢋⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠄⠀⠀⠘⠀⠀⠀⠙⠃⠘⡟⠉⠁⠀⢻⣵⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⠛⠻⠿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢠⠟⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠂
⠀⠀⠀⠀⠀⣠⠶⡶⠄⣰⡴⠰⣤⠖⣶⣤⡴⢒⣦⠰⣤⠆⣤⠲⣶⢀⡴⠲⣄⢤⡴⢒⣄⠠⣤⠀⣤
⠀⠀⠀⠀⢲⣿⡄⠛⠂⡟⣿⠺⣿⠀⣹⢿⡇⣀⡿⠀⣿⢸⣷⠂⠡⣾⡧⠀⣿⢸⡇⣈⡿⠀⡿⣧⣠
⠀⠀⠀⠀⠀⢿⣀⡴⣲⠃⣿⣄⣿⡀⠀⣸⣇⢸⣖⣀⣿⡸⣧⣀⠖⢻⡇⢙⡟⣸⣇⢸⣟⢈⡇⠙⣿
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁⠈⠀⢡⣄⣀⣈⣀⣉⣨⢤⣥⣬⣥⣤⣤⣭⣭⠀⢡⣈⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡇⠈⠻⣧⣖⢿⣯⣏⠍⣛⣻⡗⢀⣹⠟⠁⡔⠲⣿⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡈⠉⠁⠄⣧⠈⠙⠻⠿⠾⣽⡷⠿⠗⠋⢡⡔⣦⡹⣆⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⡒⠚⠡⢲⠶⠀⢠⡀⠀⠀⠀⢠⢈⡾⣇⠼⡇⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢦⡴⠋⢀⣨⠞⠁⠘⠉⠀⣠⢊⣤⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⠆⠀⠀⠀⠐⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡤⠄⠀⠶⡿⣬⣟⣘⣿⠆⠀⢀⢶⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠏⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠼⠇⣁⢌⣡⠸⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠆⠁⠀⠈⠁⠉⠛⠫⡧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠠⣂⠄⠀⠀⠀⢀⢄⡄⠀⣀⡀⠀⢠⠀⠘⠶⠀⠀⠀⠀⡗⠂⠀⠀⢀⣴⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⡠⢄⠀⠀⠀⠀⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠠⠞⠋⠁⠀⠀⠀⠐⠃⢰⠀⠁⠀⠀⠀⠀⢀⣾⡿⠏⢠⣤⣴⣶⡟⠀⠀⠀⢀⣀⡀⠀⠀⠙⠻⠆⠁
⠀⠀⠀⠀⢠⣿⡧⢰⣿⠘⠛⠛⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠈⠋⣰⣿⡾⠟⠛⠅⠀⠀⠀⠹⡍⣍⡉⠀⠀⠀⠙⠛⠓⢿⣆⢰⣿⡆
⠀⠀⠀⠀⠀⢈⡀⣘⣡⡔⠪⠑⠀⢼⣿⠄⢠⠤⢧⡜⠓⠋⠁⠀⠀⡀⣤⣴⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠙⠈⠻⠀⠀⢀⠠⠑⢲⣤⣛⠀⡄
⠀⠀⠀⠀⠀⢿⡜⢉⠁⠀⢠⠄⢀⣠⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⣿⣿⠿⢿⣿⣿⣷⠀⠁⠀⠀⡀⠀⠀⣀⠐⠃⠀⠀⠀⠀⠀⠉⢃⣿⠂
⠀⠀⠀⠀⢀⠈⣱⡿⠃⠀⣿⣐⣫⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢛⣿⣦⣾⣿⣿⣿⡆⢀⡀⠀⠀⠀⣀⣻⡇⠠⣆⢀⡀⠀⠈⢻⣦⠃
⠀⠀⠀⠈⠻⣶⠟⠀⠀⠀⠉⠁⠀⠀⢠⢆⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠨⣿⠗⠞⠩⠋⠁⠀⠀⠀⠹⣧⠟⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠓⠀⠀⠀⠀⠉⣽⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠈⠀⠁
⠀⠀⠀⠀⠐⣦⡀⠠⠀⣰⠀⠀⡀⠠⢀⠄⠀⠀⢸⣿⣧⠀⠀⣀⡈⠀⠀⠛⠿⢿⢿⡿⠟⢢⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣤⠀⠤⠀⣴⠂
⠀⠀⠀⠀⠂⠈⠳⠄⣼⡟⠀⣀⠋⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡤⠈⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠸⣧⠠⠞⠁⠐⠂
⠀⠀⠀⢠⣶⡆⠀⠰⠋⠀⣾⣿⠀⠀⢀⡄⠀⠀⡘⠿⠟⠛⠛⠋⠀⠀⢀⠀⠀⣀⠀⢤⠀⣀⢸⠀⠀⠙⠀⠀⢰⡄⠀⠀⣿⣷⠀⠙⠧⠀⢰⣶⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠋⠀⣤⣸⣧⠀⠀⣥⠄⠈⣢⠠⠀⠀⠐⡁⠀⠀⠈⠓⠀⠰⠌⡄⠀⠀⠀⠀⠀⢬⣃⣤⠀⠘⠛⠀⠀⠀⠀⠈⠙⠁
⠐⠈⠀⠀⠀⠀⡀⢠⠀⣄⣰⠀⢀⠈⠛⢁⠀⠀⠁⠄⠀⠙⠘⣆⠀⠀⠛⠀⠠⡄⠀⢀⠀⡀⢀⡀⠄⠀⠀⠀⣈⠻⠁⣀⠀⣦⣠⠀⠀⢠⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠈⠀⠠⢼⣿⣾⡉⠻⢶⠏⠀⠀⠀⠀⠙⠫⠀⡈⢰⠆⠀⡶⠀⠀⢀⠼⠃⠀⠀⠁⠈⠀⠀⠀⠙⣶⠞⠉⣳⣿⡷⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠛⠁⠀⠐⣶⡄⠀⠁⠀⠁⠀⢀⡖⡄⠀⢀⢀⠁⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⠀⠘⠙⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠃⠠⠀⠲⢤⣰⣌⡠⣄⠀⣤⠀⠈⠦⠀⠘⠠⠋⠘⠢⠐⠜⠀⠀⠂⠀⠀⣠⠀⣠⡤⣨⣖⣠⠖⠂⠄⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠛⢛⣾⣷⣜⣇⢳⣄⠀⠀⠀⠤⣤⣼⡓⣦⠀⠀⢀⡾⣰⣯⣾⣷⣟⠛⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠥⠯⠽⠷⢿⣦⣄⣀⠀⠀⠠⠞⢁⣀⣴⣿⠷⠯⠽⠬⠍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠉⠙⠻⠆⠰⠟⠋⠉⠉⠉⠉⠁⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠀⢰⣿⠋⠁⣴⣿⡟⢡⣶⡿⢖⣲⣶⣶⢆⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣿⣿⢂⣀⣾⣿⣤⢴⣿⣟⣃⣼⡿⢠⣿⡿⠋⣉⣻⣿⠏⣼⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡄⢰⣿⣁⣤⡿⠟⢋⣩⠻⡿⢟⣘⣿⣿⠿⠿⠷⠾⠿⢁⣾⡿⠛⠻⡾⣏⣡⣶⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣇⣸⣿⠟⠉⠀⣠⡿⣫⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠘⠟⣡⣾⠟⠿⡜⣏⣵⣾⢿
⠀⠀⠀⢀⣦⢘⣿⣿⡿⠁⠀⢀⣼⣫⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣰⠋⢳⣽⣭⢿⡉⢻⡇
⠀⠀⢀⡿⣵⢿⣿⠏⠀⠀⢀⡾⣱⣿⠏⠀⠀⠀⠀⠠⣤⡴⠶⢖⣿⣿⣗⡒⠻⡟⠹⡷⣺⠷⠿⣾⣟⣻⡇
⠀⠀⣾⡿⢣⣿⠏⠀⠀⠀⣼⢡⣿⡟⠀⠀⠀⠀⠀⠀⠙⢦⣌⠙⠋⢛⣛⢻⡆⢹⡏⣖⠀⣀⡘⠲⠝⢾⠳⡄
⠀⢀⣿⢡⣿⡏⠀⠀⠀⠀⡟⢸⣿⠁⠀⠀⠀⠀⠀⢀⣤⣂⠉⠻⡟⢻⡍⢻⣇⠋⠵⢿⡟⢻⢿⣆⢀⣞⠷⢿⡀
⠀⢸⡇⣾⣿⠃⠀⠀⠀⠀⣧⢸⣿⠀⠀⠀⠀⠀⣠⡿⢛⡉⠁⠀⣀⣼⢧⡿⠉⠀⠀⠠⡑⠶⠚⡏⠻⠉⠀⡎⡛
⠀⢸⣷⣿⣿⠀⠀⠀⠀⠀⢹⡸⣿⡆⠀⠐⣶⡾⠛⠋⠁⠀⣴⠟⠋⣵⣿⣠⠀⠀⠀⠀⢹⡄⠀⠀⢿⣦⡀⢳⣰
⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⢷⣿⣿⣤⣾⣡⠖⣋⠅⠀⢰⣯⡞⠸⣏⣿⣷⠀⠀⠀⠀⠀⠳⣄⠀⠀⡈⢷⠘⠋⠄
⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⢡⡾⠁⠀⠀⠀⠛⢷⠀⠙⣮⠻⣄⠀⢀⠀⢀⠸⣧⠀⠀⠏⣔⠃⠘⠺⡄
⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣿⠁⠀⠀⠀⠀⠀⠘⠃⠠⠟⢀⣿⣧⣆⣷⣼⣦⡈⢶⣷⣤⡈⠙⠣⠰⡁
⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⠸⣿⠀⠀⠀⠀⠀⠀⣀⡤⠊⣰⡿⣡⣿⣿⣿⣵⣢⣶⠶⣄⠹⡟⡖⠶⠚
⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄⠀⠀⠀⣠⠟⠀⢿⣆⠀⠀⢀⣴⠟⠉⠀⠀⠙⢰⡿⣼⡏⣿⣿⣿⡻⠄⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣤⡀⠀⠀⠀⠈⢻⡄⢀⣿⡟⠀⠀⠀⠀⠀⠈⠳⡀⢿⣿⡀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣄⠀⠀⠀⡗⢸⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠳⡔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠁⢸⡇⣧⠀⢀⣴⠿⣿⠀⠀⠀⠀⠀⠀⠁⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢀⣾⠇⢸⣀⡾⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠏⠀⠀⣿⣁⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⠀⠀⠀⣰⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠙⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⣤⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢆⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⣿⣿⠟⠋⠉⠀⠀⠀⢰⣄⡀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣶⣤⣙⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⣿⣿⣿⣿⣟⣛⣋⣠⣶⣶⣶
⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠈⠹⣿⣆
⠀⢀⣻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄
⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣟⣛⠛⠻⠿⣿⠿⠃
⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄⠈⠉⠉⠛⠻⢿⣿⣿⣶
⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠤⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⣀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠋⢡⢿⣉⠖⠁⢀⡴⢊⢕⡪⠭⣍⡲⢕⢄
⠀⠀⠀⠀⠀⠀⢠⢃⣎⢔⣽⠿⠋⢛⣶⣞⠾⣴⠗⠉⠚⠛⢿⣷⡈⢻⡆
⠀⠀⠀⠀⠀⠀⠃⠸⠃⠈⠁⠀⠀⢹⣙⣿⢸⡏⠀⠀⠀⠀⠈⡿⣇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢇⠈⢿⠁⠀⠀⠀⠀⠀⢹⣉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢻⡷⡾⠀⠀⠀⠀⠀⠀⡘⣿⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⠀⠀⠀⠀⠀⠀⠈⢷⡟⠷⠢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢧⠈⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠱⡄⠀⠀⠀⠀⠀⠀⣠⢾⡟⣦⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⢁⡀⢀⣠⠴⢛⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⠘⣷⢸⡇⠉⠀⠀⠀⠙⠢⣌⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡼⡸⣦⢹⡌⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⡀⣧⢿⡀⠓⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⢃⡜⢹⡟⡀⠻⠀⠸⡱⣴⡈⠳⣄⠀⠀⠀⠀⢀⠆⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣠⠱⠋⠧⠼⠖⠃⠀⠀⠀⠱⣌⢿⠿⣿⣗⣲⣀⣴⣋⠔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠷⣦⣌⡉⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠃⠈⢠

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⡕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⡜⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⢠⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡐⠫⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠛⠉⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣙⠂⢤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣠⣖⣡⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣍⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣉⡁⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠉⠉⠒⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡕⠉⠀⠐⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣈⠊⠒⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠛⣈⠈⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠢⡌⠠⡀⠀⠀⠀⠀⠀⠀⣠⠎⢡⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⣹⣄⣹⡆⠀⠀⠀⠀⣰⣏⣠⢧⡀⠀⠄⠀⠀⣀⠤⠐⠂⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⠒⠂⠤⢉⡀⢀⠀⣴⠋⠀⠘⠛⠛⠃⢀⡀⠘⠛⠛⠁⠀⠘⣧⠀⠀⣨⠄⠂⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠀⠠⡀⠈⠀⣧⡿⠂⠀⠀⡀⠀⠀⡟⢹⠀⠀⠀⠀⠀⠀⠻⣼⠀⠀⣠⠆⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠆⠘⠳⠀⠀⣁⠀⠀⠀⠑⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠚⠁⠰⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⣨⣽⠇⠀⠀⠲⢦⠀⠀⡆⠀⡴⠖⠁⠀⠀⢿⡥⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⢄⠀⢰⠆⠠⡇⠀⣇⠀⢠⠀⢀⣈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣟⠇⠈⣠⠟⢀⠀⠀⠀⠘⠧⡀⠰⢯⡿⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⠒⠂⠀⠀⣿⠀⠀⣴⠆⠀⠀⠙⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⡀⠀⠀⠀⡼⠿⠚⠿⠄⠀⠀⠀⠄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡀⠀⡜⠀⠀⠀⠀⠀⠆⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠻⠩⠆⠐⠄⠀⠀⠀⠐⠀⠺⠑⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢁⠀⠈⠙⢷⣴⠞⠁⠀⣄⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⢷⡀⠀⠈⣿⠀⠀⣠⡟⠀⠎
⠀⠀⠀⢀⣀⠤⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠂⠰⢏⢉⠈⠉⠀⡉⡟⠐⠆⣠⠖
⠀⣠⠖⠋⠀⠀⠀⠀⠀⣀⡤⠴⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣼⠏⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⣸⠁
⢿⡀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡴⠃
⠈⠻⢦⣀⣼⡋⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⣴⠇⠀⣻⡆⠈⢹⡇⠀⢹⡦⠀⣾⠂⢹⡇⠀⣠⠞⠹⡷⢀⣴⡏⠉⣷⠀⠊⣿⠄⢹⡷⠐⢹⡇⠊⣻
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣰⡟⠀⢠⡟⠀⠀⣼⠇⢀⡟⠀⠀⠀⢀⡿⠀⣸⡇⠀⠀⠀⣼⠃⠀⢀⡿⠀⣸⠏⠀⠀⠀⢠⡿⠁⢠⡟
⠀⠀⠀⠘⢷⣄⡀⠀⣀⡤⠊⠀⢿⣤⠔⢻⣧⠀⣾⠧⢠⠼⠋⠀⠾⠁⠀⠀⠀⢸⣧⠀⠹⣧⡤⠂⠀⠻⡄⣠⠞⠁⢠⠟⠀⠀⠀⠀⡾⠁⠀⢿⡤
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⣴⠲⣄⣰⠖⡆⠀⠀⠀⠀⠒⣦⠀⢀⡴⠂⠀⠀⠀⠀⠒⣶⠶⢶⠖⠀⠀⠀⠀⠀⢀⣤⡤⢤⣄
⠀⠀⠈⠀⢻⡏⠈⠁⠀⠀⠀⠀⠀⣼⠛⠻⡄⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠻⠼⠃⡖⣦
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠙⠦⠾⠃⠀⠀⠀⠀⠀⠤⠿⠶⠾⠦⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⡀
⠀⠀⠀⠀⡖⠓⣦⠀⠀⠀⠀⠀⢸⡏⣿⢹⣤⡄⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠀⢻⠛⡟⣿
⠀⠀⠀⣞⣷⢠⠏⠀⠀⠀⠀⠀⢸⡇⣿⢸⣧⡇⠀⠀⠀⠀⢤⠷⠀⠼⠧⠀⠀⠀⠀⠀⢸⠀⡇⣿
⠀⠀⠀⠈⠁⠻⠶⠀⠀⠀⠀⠀⠈⠁⠩⠼⡟⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠈⠀⠁⠻⣤


⠀⠀⠀⠀⠀⢶⣶⠀⠀⠀⠀⠀⠘⣶⢳⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣄⣤⠀⠀⠀⠀⠀⠐⢦⠀⣴⠂
⠀⠀⠀⢙⣶⡋⠘⠀⠀⠀⠀⠀⠀⠇⢸⣶⢳⡆⠀⠀⠀⠀⢉⣬⣡⣍⣬⠁⠀⠀⠀⠀⠐⢺⠖⣿⠂
⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠉⠀⠀⠀⠀⠀⠉⠈⠁⠉⠈⠁⠀⠀⠀⠀⠐⠋⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠙⣿⣧⢸⡗⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠰⣿⣿⣿⡏⠀⠀⠀⠀⣴⣿⠟⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠃⠀⠀⠀⣀⣼⣿⣿⣿⣷⡀⠀⠀⢰⣿⣿⣀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣴⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⠉⠛⠁⠀⢀⣾⣿⣿⣿⣿⣦⣀⣀⣁⣤⣶⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠈⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⠻⢿⣿⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠉⠉⠉⠉⠙⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⣿⠀⠘⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠀⠀⠀⢀⣴⡿⣟⠛⠛⠛⢷⣤
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⢠⣾⠛⠉⠀⠉⠻⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢰⣿⣿⣾⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢿⢿⣶⣤⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣄⡀⣀⣤⣤⣄
⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣼⣿⣿⣿⣷⡀⠀⠀⠀⣰⣿⡿⡻⠚⠉⠀⠀⠀⠙⢿⡄⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠛⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⣿⣿⡀⣿⠁
⠀⠀⠀⣿⣏⣾⡿⠙⣿⡏⢹⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢟⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⢿⣿⡇
⠀⠀⠀⣿⣿⡟⠀⠀⢸⡇⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⢸⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣼⣿⡇
⠀⠀⠀⠿⠿⠇⠀⠀⢸⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠸⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⠉⠛⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⡀⢀⣠⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠀⢻⣿⣿⣇⠀⠈⠛⢿⣷⣾⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠹⠿⠿⠄⠀⠀⠀⠙⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⢸⣿⣿⠉⠈⠻⠿⠧⠀⣿⣿⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢸⣿⣿⠀⠀⢀⣀⡀⢸⣿⢻⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠘⣿⣿⣦⣤⣾⣿⢏⣿⣿⣤⣿⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣈⣙⣻⣟⠛⠁⠸⠿⠋⠉⠹⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠻⣿⣷⣰⣶⣶⣶⣶⣦⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣿⣿⢹⣿⣇⣠⣿⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠉⠁⢸⣿⡟⠛⣿⣷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⢀⣀⡸⠿⠇⠀⠻⠿⣀⠙⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠿⢄⣿⠛⢿⡆⣿⠛⣿⠄⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣴⠜⣿⣤⣾⠇⣿⠿⣿⠀⣿⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠛⠀⠛⠂⠁⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠶⠶⠶⢶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠹⣏⢀⠠⣤⣤⡀⣠⣤⣄⠀⢸⡟⠙⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⣤⡄⠀⠀⢻⠀⠀⢿⠙⣿⡿⢻⠏⠀⣸⠁⠀⣾⡀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠁⣤⣤⣄⣿⣧⡖⠂⠈⡇⠀⡀⠀⣿⠑⠀⠀⢠⠇⠀⣠⣿⣶⣦⣬⣠⡤⠉⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠳⣄⠀⠀⢈⡽⣿⠛⠷⠶⠀⢹⠀⠀⠀⠈⠀⠘⠀⡼⠀⠰⣿⡁⠀⣹⡿⠉⠁⣠⡾⠋⠻⣦⡀
⠀⠀⠀⠀⠀⢀⣼⡟⢁⣀⡄⠌⠳⣄⠈⠅⢹⠃⠀⣀⣤⡶⠷⠟⣛⣛⣛⡛⠿⠷⣦⣅⡙⠿⠛⡿⠃⢀⠔⠁⠀⣀⣀⠙⢿⣄
⠀⠀⠀⠀⢀⣾⠏⠀⣤⣿⠰⡶⠂⠈⠳⡄⣀⣴⠟⣋⡵⣶⣿⣿⣿⡿⢻⣿⣛⣷⣶⣭⡛⢷⣄⣀⠔⢁⡀⢈⣠⡼⠿⣷⠈⢿⣆
⠀⠀⠀⠀⣾⠏⠀⣶⠿⣸⡟⠋⠀⠀⢀⣼⢟⡵⣾⣿⣷⣿⢿⣯⣿⣿⣿⣿⣿⣿⡾⢿⣟⣷⣝⢿⣄⠉⠙⣿⠋⣠⡶⢿⣆⡈⢿⡆
⠀⠀⠀⣸⡟⠀⠀⠈⠂⣿⠁⠀⠁⢠⡿⢡⣾⣷⣿⣿⣾⠿⢿⠟⣁⣤⣀⠀⠀⠩⢛⢿⣟⣙⣿⣧⡹⣧⠀⠸⢿⡁⠂⠀⠀⠀⣘⣿⡀
⠀⠀⢀⣿⠛⠓⠲⠤⢤⣈⣀⠀⢠⣿⢡⣿⣤⣿⣷⣿⠀⢀⠀⠈⠃⢀⡘⣧⠀⠀⠀⠑⡝⣿⡿⢿⣷⡹⣧⠀⣈⣩⠤⠔⠚⠛⠉⢻⣇
⠀⠀⢸⣿⠀⠀⠀⠀⣀⠀⠈⠉⣿⠇⣿⣭⣿⡷⠛⠛⠓⠛⠀⠰⡔⢀⠠⣼⠀⢤⢀⠀⠈⡜⣿⣿⣿⣇⢻⡏⠀⠀⣠⢠⣴⣦⠀⠸⣿
⠀⠀⢸⡏⠀⠃⣀⣾⣙⣧⡂⠀⣿⢰⣿⣯⣿⡀⠠⠀⠀⠀⠀⢨⠀⠰⠖⢮⠁⠻⠴⠁⠀⠀⣿⣭⣿⣿⢸⡇⠀⢸⡷⠸⠷⠾⣿⡄⣿
⠀⠀⢸⣇⠀⣿⣿⣫⠉⠍⠻⠀⣿⢸⡟⢻⣿⣿⠛⠁⠀⠀⠀⢢⠀⠀⠀⠊⣴⠂⠀⠀⠀⠀⣿⣿⣉⣿⢸⡇⠀⠸⣷⠾⣷⡄⣿⠃⣿
⠀⠀⢸⣿⠘⠋⠍⠁⠀⠀⠀⣀⣿⡇⣿⣿⣷⣿⣄⠀⠀⠀⠀⠀⢣⠀⠈⠁⡷⠉⠀⠀⠠⣱⡿⢿⣿⡏⣾⣇⣀⡀⠙⠛⠋⠘⠁⢰⣿
⠀⠀⠀⣿⣄⣤⡤⠴⠖⠛⠉⠁⠘⣿⡘⣷⣾⣿⣽⣿⡏⠠⠀⠀⠈⡠⠀⠠⠋⠀⠀⢀⣵⣿⢷⣾⡿⣰⡟⠀⠀⠉⠙⠓⠶⠦⣤⣼⡇
⠀⠀⠀⢸⣯⠀⠀⠀⣈⡄⠀⣤⠀⠘⣿⡜⢿⣏⣽⣿⣷⣤⣤⡶⣶⣭⣄⣀⣠⣤⣶⣿⣿⣿⣾⠟⣵⠟⢀⠀⣾⠛⣷⠐⠀⠀⢠⣿⠁
⠀⠀⠀⠀⢿⣇⠐⢿⣿⠷⢾⣿⢶⡄⠈⢻⣦⡙⢿⣿⣿⠿⣿⣿⣿⣿⣾⣿⣿⣤⣿⣷⣾⠟⣫⣾⠋⢰⡷⠾⣝⠛⠻⢶⡀⢀⣾⠇
⠀⠀⠀⠀⠈⢿⣆⠀⠈⠃⠘⠃⠀⠁⡤⠊⠉⠻⢷⣭⣛⠿⢿⣿⣼⣿⣽⣿⣿⡿⢟⣫⣵⠾⠋⠀⠱⣄⠀⠀⠙⢳⣤⡾⢃⣾⠋
⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⢁⣴⠏⠀⠛⣷⡀⠀⠉⠛⠻⡷⢶⣾⣶⣶⣶⠶⠿⠛⣫⣅⠀⠂⠀⣀⠈⠳⣄⠀⠀⠄⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣴⠟⠁⠐⢀⡴⢋⡄⠀⠀⠀⢰⠀⣠⡀⢄⠀⠀⣀⣸⡐⢿⡙⣆⢰⡄⠹⣧⡄⠈⠻⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⢽⣥⣿⢀⣼⠃⢀⡎⢸⣽⣧⠞⢻⣟⣫⠀⣇⠀⡛⢻⣤⣿⠿⠟⠙⣁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀⠛⢻⡇⠀⣸⠁⠀⢸⣏⠈⢸⡟⠁⠀⢸⡄⠀⠀⠈⠁⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣥⣠⣟⠀⠀⠀⠙⠛⠓⠀⠀⠀⢈⣷⣠⣤⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⠷⠾⠿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣠⣿⣿⣿⣍⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣹⣽⣿⣷⣄
⠀⠀⠀⣿⠃⠀⠀⠙⣷⣿⡴⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⡴⣷⣿⠋⠀⠀⠙⣿
⠀⠀⠀⠙⢄⠀⠀⡰⠋⢠⣄⣦⣄⣈⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣤⣤⣠⡄⠙⢆⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⣶⢿⡟⠉⠉⠛⢿⣷⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣾⡿⠛⠉⠉⢻⠿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣼⠷⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⢧⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢁⡀⣀⠀⡀⠈⠙⢷⣄⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⠀⣄⣀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣰⡾⠛⠛⠻⢷⣦⠀⠀⢻⣇⠀⠀⠀⣸⡟⠀⡀⣴⡿⠟⠛⠛⢷⣆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠹⣿⡄⠀⣿⣦⣤⣤⣿⠀⢠⣿⠏⠀⠀⠀⠀⠸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⢹⠇⠀⠟⠁⠀⠈⠻⠀⠸⡏⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠄⠘⢳⣦⡀⠀⠀⠀⢀⣶⡞⠃⠠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣶⣿⣿⣿⣇⠀⠀⠀⣸⣿⣿⢿⡞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣍⡋⠻⠿⠋⠀⠙⠿⠟⢙⣩⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡘⢻⣷⣄⣀⠀⣀⣠⣾⡟⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠸⣿⣿⣉⠛⣉⣿⣿⡇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⠀⣾⣎⠻⡉⠛⢉⠟⣱⣿⠈⢿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⡿⠁⠀⡘⠙⣷⡀⠀⢀⣾⠋⢃⠀⠈⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣅⠀⠞⢹⡇⣿⡇⠀⣸⢯⢸⡏⠳⠀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⣀⣠⣶⡿⣿⣿⡇⠀⢿⣿⣿⢿⣶⣄⡈⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣸⢿⣿⠟⠁⣿⣿⠃⠀⠸⣿⣿⠈⠻⣿⡿⡇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣿⡏⠀⠀⣻⠋⠀⠀⠀⠹⣏⠀⠀⢹⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠃


⣿⢿⠿⠿⡿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠠⣤⡀⠀⠸⣿⣿⡇⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⡇⠀⢀⣤⠄
⠀⠀⠀⠀⢀⣼⡟⠀⠀⠈⠉⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠗⠀⠀⠉⠁⠀⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢻⡆⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠈⢻⡆⠀⠀⠀⠠⣴⣦⠄⠀⠀⠀⢰⡟⠁
⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠙⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢀⡀⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀⠀⣀⠀⠀⣀⣀⣀⠀⠀⢀⣠⣀⡀⠀⢀⣀⣀⣀⡀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠙⠀⢀⡟⢻⡄⠀⣿⠀⠀⢸⡇⢸⡀⡀⣈⡷⠀⣿⢀⡞⠁⠀⠘⠃⢰⠏⠀⠀⠹⡆⢸⡇⠀⣈⡿⠀⣿⢷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣠⠀⡼⠳⠶⢷⡀⣿⠉⠉⠉⠀⢸⠉⠙⢿⡀⠀⣿⠀⣧⠀⠀⢀⡄⢸⡄⠀⠀⣰⠇⢸⡏⠉⢻⡄⠀⣿⠀⠻⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⣾⣿⢠⣤⣾⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⣾⣆⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣄⢶⣦⣍⣷⢈⣿⡀⢀⣀⣆⠀⣰⣿⡿⢋⣀⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣾⣷⣜⢉⡻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠝⣩⢜⡿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠍⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢦⣠⡕⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡿⣶⣤⡐⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣾⣿⣿⣿⣿⡗⣯⣾⣯⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠛⡿⠛⠻⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⢀⡴⠒⠛⠛⠛⡒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡵⠦⢄⡴⠋⠢⣀⣈⣑⠦⡀
⠀⠀⠀⢀⡔⠁⡏⠉⠲⠖⠋⠉⣆⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣰⢃⡠⠖⠛⠛⠒⠯⣁⠀⡅⠙⢆
⠀⠀⠀⣾⠖⢊⠞⠉⠁⠉⠙⠲⢼⡛⠛⢳⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⡞⠉⡵⠋⠀⠀⠀⠀⠀⠀⠈⠳⡘⠲⡌⣇
⠀⠀⢸⠏⢧⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣧⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⠃⢹
⠀⠀⢸⡀⣸⢻⡀⠀⠀⠀⣄⡀⠀⠀⠈⡎⠉⡧⠸⡆⠀⠀⠀⠀⠀⠀⡞⣾⠉⢡⠇⠀⠀⠀⠀⣏⠳⡀⠀⠀⢀⠇⠹⡄⡾
⠀⠀⠈⢇⢇⣀⣑⠤⡤⠴⢦⢹⠀⠀⠀⢸⢠⡇⠀⢻⠀⠀⠀⠀⠀⢸⠃⠘⣆⢸⡄⠀⠀⠀⠀⢻⡦⢍⠒⡚⠁⡖⠋⡰⠃
⠀⠀⠀⠈⠢⣄⠉⢧⡴⠒⢚⡜⠀⠀⠀⢸⠆⠉⣦⠸⡇⠀⠀⠀⠀⡸⠀⣠⠞⠀⢧⠀⠀⠀⠀⠀⠙⠾⣍⣈⣉⡤⠚⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉⠀⠀⠀⠀⡼⢀⡴⠃⠀⡇⠀⠀⠀⠀⡇⠈⠧⣄⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠸⡀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠀⢀⡵⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡇⠀⠀⣸⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢏⡀⠀⠀⢸⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⢸⡇⡠⢚⣁⠀⠰⡧⠖⠒⠒⠒⠾⠃⣠⢤⡙⣦⠀⣸⢀⣾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡘⡼⡇⡅⠸⢀⡇⠀⡖⢂⠞⢷⠠⣤⠀⢧⣈⣡⠏⣠⢃⠞⣼⡉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣷⠹⣽⣌⣒⣋⣀⠤⠬⣉⠀⠈⢙⡥⠤⠤⠤⠶⠚⠛⠋⣼⡿⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠻⠃⠀⠀⢩⣁⠀⠀⠀⠀⠙⡚⠁⠀⠀⣀⣄⠀⠀⠀⠀⠋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠰⡏⣭⠙⡄⠀⡄⣜⢡⠀⢠⠊⣭⢙⡏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣦⠙⠒⠒⠁⢠⡇⢹⠸⡄⠈⠓⠒⠋⢀⡖⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣶⣄⠀⠀⡼⠀⣏⠀⢣⠀⠀⣠⣴⣾⠇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣿⣿⡇⠸⣇⢶⣼⠀⣼⠀⢸⣿⣿⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⠉⣴⠂⢰⣶⣶⣦⠀⡦⠈⠛⠁⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡼⠀⠀⠙⡟⠁⠀⢱⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠑⠀⠲⠴⠧⠴⠂⠉⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣴⣿⣿⡆⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⣏⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣻⣿⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡓⠦⣀⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⡰⠟⣡⣾⡟⡛⢿⣿⡄⠀⠀⠀⠀⠀⣴⠀⣀⣠⣹⣿⡟⠃⠀⠀⠀⠀⠀⢀⣶⡿⠉⢈⢿⣦⣄⣌⠻⣆⠀⠀⠸⣭⣯⠀⡀⢃⡀⣸⣿⡷
⠀⠀⠀⠀⠈⠐⠓⢈⣹⣵⣿⣄⢻⣿⠀⠀⠀⠀⣠⡶⠋⢌⣫⡆⢹⣿⠀⠀⠀⠀⠀⢴⣿⠁⠊⠁⠀⠀⠰⣿⠛⣣⢹⠀⠀⠀⡘⠀⢸⡅⠀⡉⢯⠸⡀
⠀⠀⠀⢣⣠⣤⣾⡿⠿⠋⣸⠿⡈⣿⠀⠀⠀⢰⣿⠇⠀⠈⠲⠃⢾⠋⠀⠀⠀⠀⠀⡿⣟⣃⡷⡄⠀⠀⠀⢈⣿⣏⡞⠀⠀⠀⢓⣠⡼⣧⢀⡶⣸⡗
⠀⠀⠀⠀⠘⣩⡿⠟⠒⠚⢁⣴⠐⠃⠀⠀⠀⢸⣿⠀⣿⠉⠉⠠⣾⡇⠀⠀⠀⠀⠀⢻⡝⢿⣧⣄⡀⠂⢠⣽⣿⠏⠀⠀⠀⠀⠀⡟⠇⢹⣼⠃⠟⡇
⠀⠀⠀⠀⠀⠿⠿⠶⠗⠛⠋⠁⠀⠀⠀⠀⠀⠈⠿⢀⡇⠀⠀⠐⢿⠥⠒⣶⠀⠀⠀⠀⠙⠂⠉⠉⠿⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠴⠳⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⢔⢀⡀⠰⡄
⠀⠀⠀⠀⠀⢆⡤⠲⣿⡧⠢⢄⡦⠀⠀⠀⡄⠈⢛⡷⢸⡄⠀⠀⣄⣠⣀⣀⠀⠀⠀⠀⠂⠰⡧⠦⢤⣰⡿⣯⡶⠞⠀⠀⠀⠀⢠⣿⠟⢸⡇⠸⣿
⠀⠀⠀⠀⠀⠜⠂⠀⣾⠀⠀⠘⠄⠀⠀⠀⢐⡶⢋⣥⡾⢀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⢐⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⡇⣠⡿
⠀⠀⠀⠀⣌⢠⣤⣄⣿⣄⣌⣠⣬⣆⠀⠀⠈⢿⡋⣥⡀⢀⣴⣾⣿⡄⠈⠉⠀⠀⠀⠀⠠⡖⠊⣠⣀⣠⣄⣶⡶⣆⠀⠀⠀⠀⠢⢷⣮⣿⡇⣯⣾⠥
⠀⠀⠀⠀⠙⠿⠔⠁⣿⣡⠙⠿⠴⠃⠀⠀⠀⠈⣿⡿⢧⠘⠉⢿⡛⠙⢦⠀⠀⠀⠀⠀⡾⠃⢀⠟⣻⠋⠉⢼⠏⠁⠀⠀⠀⠀⠴⢺⠟⣿⣷⠛⣗⠢
⠀⠀⠀⠀⠀⠀⢀⣤⠿⠁⣄⠀⠀⠀⠀⠀⠀⣠⣟⣀⣾⠁⢀⣴⡇⢀⡼⡀⠀⠀⠀⠀⠀⠀⢻⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⣄⠀⠀⣠⢄⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⢉⣡⣹⠇⠀⠀⠐⣿⣷⣤⡛⠘⣀⣾⣿⡇
⠀⠀⠀⠀⠀⢀⣀⡐⠳⠂⠁⢀⡀⠀⠀⠀⠀⠀⠀⣠⠤⣶⠾⠿⠛⣈⣵⡟⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠈⠁⣼⠁⠀⠀⠀⠀⠙⠛⡿⢿⠌⠿⡟⠋
⠀⠀⠀⠀⠀⢻⣏⣤⣤⣀⡀⠠⣍⣢⡀⠀⠴⠯⠛⠙⢧⣄⣤⣶⣿⣿⣿⣛⠦⠀⠀⠀⠉⠇⣼⣷⣤⡆⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇
⠀⠀⠀⠀⣰⠟⠑⠿⣿⠟⢷⣾⡅⠉⠀⠀⠐⠀⠀⠀⠀⠈⢿⡛⠻⡍⣇⠘⠂⠀⠀⠀⣤⠞⢿⡏⠀⣬⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾
⠀⠀⠀⠤⠟⠀⠀⣄⡟⠀⠀⢌⠷⠀⠀⠀⠀⠀⠠⠲⣶⡶⠛⠁⠘⠁⠀⠀⠀⠀⠀⢀⣿⠀⠈⢿⡄⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⣿⣷⣿⣿⣿⣿⣿⣭⣩⣉⡉⠉⠙⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠟⣫⡟⣹⣻⣹⢿⡛⢿⡻⢿⣷⣄⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠻⢿⣿⣟⣴⡶⢞⣿⠻⣿⠷⣻⡟⠷⠌⣧⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⢿⣽⣷⣾⣿⣙⡛⢉⠏⡥⠸⡆⡂⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂⣠⢘⠛⡯⣿⠋⠘⠟⠾⣦⡜⠋⠁⣿⡏⠀⠀⢸⡇⠀⠀⠀⣀⣠⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡟⢤⣶⣿⣿⡀⣼⣟⠡⣏⢀⣾⡿⠀⠀⠀⡞⠀⣠⣼⣷⣿⣿⣱⣘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⣿⣿⣯⢈⣧⣿⢏⣼⣿⣼⣿⡇⠀⠀⠀⣠⣾⣿⣿⣿⣿⣟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣈⡠⣿⡿⣧⡼⢋⣵⣾⣟⣿⢿⣿⠃⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⡿⠷⢶⣄⣀⣿⣴⣿⡿⣽⣿⣤⣽⣻⣿⣜⣿⡿⠋⢀⣼⣿⡿⢋⣿⣿⣿⣿⡻⠎⢛⣛⣽⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠈⠙⢿⣿⣿⡇⣿⣯⣿⣿⣿⣿⢿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣶⣶⣶⣶⣿⣿⣽⣯⠉⡛⠞⠋⠛⣿⠁⡴⠛⠉⢫⡿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣏⠉⣿⣱⣼⡿⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣄⡀⠉⠉⠉⠻⠿⣧⠡⠀⠀⠠⠙⠆⢰⣶⣾⠟⠀⠂⣹⡿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⢿⣿⣏⠂⠀⠀⠀⠀⠘⢅⣱⡌⠀⣘⢿⡄⠙⠀⠀⢀⣮⣱⣷⢘⣿⣿⡀⢿⣿⣙⠉⢩⡿⢉⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢴⣿⣇⣿⣤⣿⢻⣿⣦⣤⢺⠏⢻⣧⠈⣾⣿⣿⣇⣸⣿⠟⠶⣾⣧⣬⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⠃⠀⢴⣟⢮⡙⠟⠁⠀⠉⣿⣿⣿⡇⢸⣾⣧⢬⣴⣿⣿⣿⣿⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡀⠀⠀⣐⠺⣿⡟⣿⣄⠀⠀⢸⣿⡷⣹⣤⢿⣷⡱⣿⢙⣷⣿⡿⠿⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣉⠙⠛⢷⣏⣷⣦⣹⣿⠧⠄⣀⠀⠻⠿⠦⠀⠀⢠⣦⠙⡿⢏⣙⣠⣀⡻⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⡛⡅⠛⣿⡁⢤⣾⣿⣄⠀⢸⡧⣾⠀⢠⡿⠀⡄⠠⣿⣿⣿⣷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡀⣽⣿⠃⠘⠀⢈⣇⠀⠈⠟⠉⠀⠘⡡⢏⠀⠈⣀⡀⠐⠀⠛⣿⢿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⡿⠂⣠⠆⢶⡸⡄⢶⣶⣤⣌⣛⣡⣴⡿⢿⣿⡿⠀⠀⠀⠸⠟⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⢖⡼⣧⢠⢩⠉⣄⣈⣉⠉⠉⠀⢀⣄⡘⠒⠆⠀⠚⠃⢠⣾⣿⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⠣⣈⣠⠃⠘⠺⠿⣏⣄⠀⠀⢏⣈⡷⠄⠀⠀⠀⠀⢸⣿⡟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡛⠃⠠⠂⠘⠃⠨⢉⡧⠀⢾⠮⠉⠀⠀⠀⠀⠀⠘⢧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡙⠳⣶⣦⣠⣤⣴⣏⣵⠶⢶⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠻⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡰⢆⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⠀⠀⠈⠀⠀⠀⠀⠗⠺⠀⠀⠀⠀⠁⠀⠠⣖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⢉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠌⣉⠐⡄⠈⠻⡂⢌⠀⠀⢀⣞⢳⠄⠀⢈⡡⢐⠟⠁⢠⠂⣉⠀⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠂⡄⢠⠀⠙⣆⠀⠉⠋⠀⣴⠀⠀⡄⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠦⠀⠀⠈⠆⠀⢀⣆⠡⠤⠤⠌⣰⡀⠀⡰⠁⠀⠀⠴⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢲⢽⠂⠀⠀⠀⠀⠀⠀⠀⣀⢈⡔⠉⠀⠀⢠⡀⠀⠀⠉⢢⡁⣀⠀⠀⠀⠀⠀⠀⠀⠐⡯⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣈⡉⠉⠀⠀⠘⠀⡆⠀⠀⠐⠂⢄⠀⢱⠀⠃⠀⠀⠉⠉⣁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⠠⢀⡀⡈⠀⠐⠇⠀⢧⠀⠀⢀⡀⠔⠀⡜⠀⠘⠒⢈⢁⣀⡀⠄⠋⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⣶⡀⠂⠀⠁⠀⠀⠐⢀⣶⡆⡅
⠀⠀⠀⠀⠀⠀⠀⣾⣙⡆⠀⠀⠀⠈⠂⠀⠀⠘⡀⠉⠓⠀⠀⢀⡀⠀⠀⠚⠋⢀⠃⠀⠀⠀⠁⠀⠀⠀⢰⣋⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⡄⠀⠈⠆⠠⠀⠰⠁⠀⢠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢲⣱⠂⠀⠀⠘⠄⠀⢸⠀⠀⡇⠀⠰⠁⠀⠀⠐⣏⡖⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⣀⠀⠈⡴⣶⠤⠤⣶⢦⠀⠀⢀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠃⠀⠀⠀⠀⠉⠀⠀⠐⢤⣕⣪⠤⠂⠀⠀⠉⠀⠀⠀⠀⠘⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⢼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⢉⡉⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⣦⡶⠟⠛⠛⠛⠶⣜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣆⣰⠏⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡿⢋⣠⣤⠞⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠡⠞⠋⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢚⡇⠀⠀⠀⡔⠋⠉⢂⡆⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡼⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⠀⠀⠠⠊⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠁⠀⠀⠀⠀⢀⠴⠖⠋⠉⠀⠀⠀⣶⠀⠀⠀⢱⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⡂⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣸⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠠⣄⠙⣆⠀⠀⠀⢀⣰⠟⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⢆⡟⠀⢀⣴⡿⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⣠⣾⠟⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣰⣿⠃⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⣀⣾⣿⠟⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢿⣧⡀⠀⠀⠙⢿⣿⣶⣶⣶⣿⣿⠟⠁⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠙⠿⣦⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⡀⠀⠈⠙⠓⠶⠄⠀⠀⠀⠀⠀⣠⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⠀⡄⠀⣶⠀⠀⣶⡆⢶⡆⢰⡆⢲⣆⠀⣶⡆⢠⣶⠂⢰⢀⣶⠐⣦⡀⢰⡆⢰⣦⠀⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇⠾⠇⢸⡧⣼⠃⠀⣿⡇⢸⣿⠀⠀⢸⣿⠀⣿⡇⢸⡧⣼⡋⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⡀⢹⣿⠀⣿⡇⠀⠀⣸⣇⢹⣧⠀⣿⡇⠸⣿⡀⠀⠸⣿⠀⡿⠃⢸⣇⢸⣷⠀⡀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠈⠉⠀⠉⠁⠀⠀⠉⠉⠀⠉⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⡖⠒⠒⠶⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣯⠻⣉⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣬⠙⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠹⣿⣿⣿⣿⣿⣦⣿⠉⣭⢹⣿⣿⣿⡇⠀⠀⠀⠀⢐⣴⠋⠘⠀⠀⠀⠀⠉⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣯⠾⠶⢛⣛⡛⠛⠓⢒⣀⠀⠄⢉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠒⠀⠀⠀⠀⢘⠿⣛⢋⣉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⢄⡀⠀⠀⠀⠀⠀⠀⣈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⢠⠚⠀⡰⠎⢁⠄⠂⢀⠚⣉⠤⠚⠓⠒⠲⢤⣂⢤⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⢆⢄⠀⠀⢀⠞⡡⢛⠎⠣⡀
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠋⢁⢔⠅⠁⡰⢁⡜⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣾⣇⢀⢀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠑⠁⠀⠁⠚⡵⠋⠀⠀⠙⢆
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢐⠅⢀⣴⣾⠃⣸⣷⣶⡦⡄⣀⠀⠀⠀⢀⢄⡾⠋⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢕⢄⠳⠀⠀⠀⠀⠈⢦
⠀⠀⢀⠎⠀⠀⠀⠀⠀⢠⠌⠂⠀⡗⣿⠙⡇⠸⣤⣤⢭⠿⣇⡒⠒⠂⠉⠁⡀⠀⠀⢀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⢀⠡⢂⠀⠀⠀⠀⠀⠈⢧
⠀⠀⡞⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⢷⡈⣷⣽⡴⢾⠃⣴⠖⣉⠛⢷⣦⡀⢸⡀⠀⣤⢸⢀⡃⠙⠸⠁⡏⣆⠀⡆⠘⠸⠁⡟⣜⠀⠐⡡⠀⠀⠀⠀⠀⠈⣇
⠀⣸⠀⠐⢢⠆⠤⠀⡔⠁⠀⠀⢀⣀⣽⣾⣡⡄⠀⠠⠀⠘⣏⠀⠈⠀⣷⣄⡁⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢠⠰⣤⢄⠸⡄
⢀⡇⠐⠈⠛⠢⠀⡐⠀⠀⠀⠀⠈⠙⠛⠋⢀⠀⠺⠟⠀⠀⠻⡀⠀⠀⠑⢿⠋⢳⡶⠶⠞⡟⠓⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠘⡆⣤⠤⠃⣧
⢸⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⢰⠘⠀⠀⢤⠀⠄⠈⠉⠘⠆⠙⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠃⠈⠀⠀⢸
⢸⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⣀⣿⣿⡀⢀⡆⠐⡡⠟⡇⠐⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⣿⢃⡤⠀⠀⠀⡠⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡄⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⠀⠀⠀⢀⠀⣠⣇⠀⠀⠀⠿⣿⣾⡇⢀⡀⠰⠇⣈⡀⠃⠀⠀⠀⠀⢀⣀⡀⢀⣀⠞⡏⠁⣷⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸
⢸⣀⠤⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡠⠂⣸⡇⢿⣶⣆⣉⢰⣟⣿⠏⠀⢵⠀⠀⠈⢃⣤⣦⣾⡾⠿⠿⠿⣿⣿⠏⢰⣇⡐⣸⡀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⢸
⠈⡏⠒⢶⡤⠶⠒⠰⠀⠀⠀⠀⣸⣶⣦⣠⠿⠎⠉⠙⠛⡿⠛⠁⠀⡀⠁⣰⡿⠟⠿⣻⣟⣁⠤⣤⣤⣶⠚⠁⠀⣸⣿⢀⡸⡇⠀⠀⠀⢸⠁⡀⠬⠭⣙⣖⡎
⠀⢹⣔⢁⡀⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⣲⣆⣠⣥⡼⠛⢉⡴⠟⠋⠉⠉⠻⡿⢿⠿⠧⠀⣴⣿⣿⣧⣶⡇⠀⠀⣰⡗⠈⠻⣷⣶⡾⢶⠃
⠀⠀⢧⠀⠀⠀⠀⠀⠈⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠁⡾⣅⢁⡞⠀⠀⠀⠀⠀⠀⠱⠰⠷⠐⠰⢾⠿⠿⢿⣿⡇⠀⣼⣿⣁⠀⠀⠀⠀⢠⡏
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠛⠆⢸⠣⣙⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠁⠈⢻⣿⣻⡟⠀⢘⠛⠀⠈⠙⠂⠀⢀⡟
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠙⣂⠀⠀⠀⠀⠀⠀⠸⣇⣀⣄⣰⢺⠠⠲⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⣄⠈⣿⡟⡠⡺⠁⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⣠⠖⠪⡀⢀⡀⠀⠀⠀⠀⠘⠟⠟⠙⠈⢷⣑⠟⠄⠝⠶⡤⣤⡴⣶⣿⣿⣿⣿⣿⣿⣇⡿⠢⢊⠀⣎⣱⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠙⢦⠤⠖⠧⣀⡠⠇⠀⠙⠱⢠⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡶⡓⢭⡌⣽⣷⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠃⠙⢆⠀⠉⣆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠠⠃⠀⠀⠀⠀⠀⠀⠑⠀⠢⠤⣀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⡿⠟⠋⠙⠁⠈⠀⠀⠀⠀⠀⠀⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⡁⠒⠈⠉⠉⠹⠉⠒⠒⠉⣁⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⣰⠀⡸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠠⠀⡀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡐⠃⠤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠘⣂⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠶⠤⠤⠤⠤⠤⠶⠶⠒⠒⠊⠉

⠀⠀⠀⠀⠠⣀⠀⠀⠀⢀⣤⣀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⣀⣠⣄⠀⠀⠀⢀⣠⣄⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⣷⠀⠀⠀⠙⣿⣷⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⡟⠁⠀⠀⢰⣿⠁⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⣠⣾⠟⠀⠀⢀⣴⣿⠟⠀⠀⠘⢿⣿⣿⠟⠀⠀⠘⢿⣷⣄⠀⠀⠘⢿⣤⣀⠀⠀⠘⢧⡀



⠀⠀⠀⠀⠀⢤⡀⠀⠀⠠⠤⣤⡀⠀⠀⠠⣤⣤⡀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⣠⣤⡤⠀⠀⠀⣠⡤⠤⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⣿⠀⠄⢼⣿⣿⣿⣿⠄⠄⢼⣿⡏⠀⠀⠀⢼⣿⠀⠀⠠⠀⢸⡃
⠀⠀⠀⠀⠀⠴⠋⠀⠀⠠⠤⠾⠋⠀⠀⠠⠾⠿⠋⠀⠀⠘⠻⠿⠿⠋⠀⠀⠈⠻⠿⠦⠀⠀⠈⠻⠦⠤⠀⠀⠈⠳⠄



⠀⠀⠀⠀⠀⠲⣄⠀⠀⠐⠲⣶⣄⠀⠀⠐⢶⣶⣄⠀⠀⢀⣴⣶⣶⣄⠀⠀⢀⣴⣶⠖⠀⠀⢀⣴⠶⠒⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⠂⢸⣿⣇⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠠⠚⠁⠀⠀⠐⠶⠛⠁⠀⠀⠐⠻⠛⠁⠀⠀⠈⠛⠻⠛⠉⠀⠀⠈⠛⠻⠓⠀⠀⠀⠙⠳⠖⠀⠀⠀⠙⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⣚⣛⣛⠻⠷⣶⣤⠀⠀⢠⣿⣥⣶⣄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠛⠷⣮⣝⢦⡀⠻⣿⠟⠀⠹⠀⢻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣆⠈⡀⠀⠀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⢩⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⢥⣼⠂⡄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⡿⠿⠇⠸⠏⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠷⠀⠀⢤⢾⠧⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣿⣦⣟⠓⠄⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠀⠀⢻⣿⣿⠀⠀⢰⠀⠀⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⣠⣾⣯⣾⠀⠀⢸⢁⠈⠀⠀⣠⡀⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠤⠀⢀⡤⣾⣿⡟⣿⣿⠀⠀⠈⢹⣆⠀⢠⣷⣤⣱⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠀⠏⢸⡇⢻⣿⣇⠀⠀⠘⣿⡄⢸⠛⢧⡙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⢰⡀⢷⡀⢻⣿⡄⠀⠀⢻⣷⡇⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡼⠀⠀⠸⣷⡀⠁⠀⠹⣿⡄⠀⠸⣿⣧⠀⡄⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠹⣿⣦⠀⠀⠘⢿⡄⠀⣿⢹⣷⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⠘⢿⣷⡄⠀⠈⣷⠀⣿⠀⣿⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡎⠀⢰⡇⠀⠀⠀⢻⣿⡀⠀⢸⣷⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠃⠀⠘⣿⠀⠀⠀⣦⢻⡇⠀⠈⣿⠇⠀⡟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡄⠀⣿⠀⠀⠀⢸⡇⠃⠀⢰⡿⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⣆⣿⠀⠀⠀⢸⣿⢀⠀⠘⠁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⣿⠀⠀⡆⣼⣿⣾⡇⠀⣰⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⣿⣿⣿⡆⢠⣧⣿⣿⣿⡇⣰⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⢻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢠⣹⡟⣿⣿⠿⢿⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢿⣿⠀⢸⣿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡄⢸⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣁⠻⠿⣷⣦⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠿⠿⠿⠆⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣤⡄⠘⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣄⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠂⠀⣰⠀⠀⣿⠁⠀⢠⡴⠋
⠀⠀⠀⠈⠁⠆⠠⠿⠿⢿⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠁⠈⠁⠀⢲⣀⣀⣀⣀⣶⣤⣤⣿⣀⣀⣿⡇⠀⣼⡇⠀⠐
⠀⠀⠀⠀⠀⠀⢀⢀⣤⣶⣽⣿⡟⣭⣿⣟⢼⣿⣿⣶⣶⣤⣶⣿⢁⣿⣿⣿⣿⣳⣿⣿⣾⣿⣿⣟⣿⣇⣰⣿⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣯⣿⣻⡿⡿⠿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣟⣾⣿⣵⣾⣾⣡⣽⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⡿⢋⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿⡟⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣾⣯⠛⠉⠀⠀⢀⡀⠁⠐⠀⠄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠻⠄⠀⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⡈⠻⣧⡀⠀⠀⡀⢳⠂
⠀⠀⠀⠀⠀⠀⠈⢟⣄⠐⢐⢰⡖⣾⣷⢹⣶⢷⣾⣞⡧⠀⠀⠀⠁⠀⠈⠉⠉⠋⠉⠀⠀⢸⣿⡆⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠉⠀⠀⠁⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠌⠋

⠀⠀⠀⢀⡤⠤⣄⡴⠒⢦⠀⠀⠀⠀⣴⠲⣄⠀⢀⡴⠶⢦⠀⠀⠀⠀⢶⢦⣄⣀⣀⡤⠖⠆⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀
⠀⠀⠀⢸⡀⢳⢸⡇⣄⣸⠇⠀⠀⠀⠉⢻⠟⠲⠾⣄⠙⠚⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⣏⣈⡇⠀⠀⢾⠇
⠀⠀⠀⠀⠉⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⠠⡞⠓⡆
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠞⠀⠀⠀⠀⠀⠀⢴⡤⠞⠒⠺⠦⣴⡆⠀⠀⠀⠀⠈⠛⠶⠤⠿⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⡴⠛⠛⣦⠀⠀⠀⠀⠘⠂⣿⠋⢿⠛⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⢶⣶⠛⢶⠲⡄
⠀⠀⢀⣤⣸⡄⠀⢀⡟⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⡇⠀⠀⠀⠀⠀⢀⠀⠸⡄⠀⣸⠀⢀⡀⠀⠀⠀⠀⠀⣿⠀⣸⠀⡇
⠀⠀⠸⣅⣸⠇⢀⡞⣀⠀⠀⠀⠀⠀⠀⠃⠀⢋⣠⣿⣻⠀⠀⠀⠀⢘⣛⣛⣋⣀⣙⣛⣛⠃⠀⠀⠀⠀⠀⢿⠀⠿⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⠛⡿⣿⠀⠀⠀⠀⠀⠠⡖⡶⠚⢳⡀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡴⣤⡴⣄⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⡾
⠀⠀⠀⢀⡀⢠⠞⠁⠉⠁⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠋⣀⡈⣠⣌⣤⡁⠀⠀⠀⠀⠀⢰⣆⣷⠤⢴⣧⡄
⠀⠀⠀⠀⡽⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣷⢻⠆⠀⠀⠀⠀⠸⠃⠛⠁⠻⠙⠋⠀⠀⠀⠀⠀⠀⣸⠃⠀⠘⡆
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄

⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⢀⣠⣤⡶⠀⠀⠀⠀⢀⡠⣤⣤⣤⣠⣀⠀⠀⠀⠀⠀⣴⣤⣄⡀⠀⠀⠀⠠⢤⣀
⠀⠀⠀⠠⢮⠞⠁⠀⠀⠀⡴⡽⠋⢙⡇⠀⠀⠀⣔⡿⠋⠁⠙⠉⠻⢎⣛⣆⠀⠀⠀⠈⠁⠿⢻⢦⠀⠀⠀⠈⢻⡷⣄
⠀⠀⢀⠁⠋⠀⠀⠀⠀⠈⠿⠀⣀⣸⡇⠀⠀⠈⠸⠁⣀⠀⢯⣾⣇⠀⣿⠙⡇⠀⠀⢠⣟⡀⡘⠻⣧⠀⠀⠀⠈⠳⢿⡆
⠀⠀⢘⢠⣄⠀⠀⠀⠀⠆⠀⣄⠻⡟⠀⠀⠀⡀⢀⣄⠻⠃⠹⣿⣿⣷⣟⣿⣟⠀⠀⢸⡿⣥⠉⢶⣿⠀⠀⠀⠀⠲⣼⡇
⠀⠀⠈⢶⡁⡆⠀⠀⠀⠸⣤⢩⣤⡼⡆⠀⠀⠸⣦⣡⡶⢶⣤⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣾⣿⣿⠇⠀⠀⠀⢀⢿⡿⠁
⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠙⠿⣿⣦⣧⠀⠀⠀⠘⠻⣷⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⣸⣿⣿⠟⠁⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣨⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⠏⠙⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣤⣿⣿⣿⣿⣶⣾⣿⣏⢙⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠶⠶⠦⠶⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣴⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣥⡴⠶⠤⠆⠴⡲⠶⠢⠴⢤⣦⣥⡀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣱⠶⠟⠋⠉⠁⠀⠀⠀⢸⣿⠟⠻⠶⠾⠿⠶⣶⣭⣝⡚⢦⣭⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⠒⡋⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠇⠈⠙⠧⣦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⣫⣶⡿⢧⣦⡀⠀⠀⢰⣿⣿⡷⠀⠀⢿⠀⠠⣶⡤⠋⣤⠀⠀⣼⡛⠀⠀⡀⠀⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣩⡞⠃⠀⠀⠀⠻⣵⡄⠀⠀⠻⠇⠀⠀⠀⣸⡀⢀⣀⠀⠀⠉⠀⡘⠋⠀⢠⡤⣤⡀⢀⠀⠀⠙⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⣴⣿⠃⣠⣞⠇⠀⠀⠈⠻⣦⡀⣤⣶⣾⡿⠿⠿⢿⡷⠾⠾⢥⣶⣶⢅⡀⠀⠀⠀⠼⡷⠀⠀⠀⣠⣄⠱⡈⢃
⠀⠀⠀⠀⢀⡞⢾⢃⣬⡟⠀⠺⠯⠀⢀⡀⠀⣠⢞⣫⠖⠻⣯⠀⠀⠀⠀⢸⡇⠀⠄⠀⢨⡿⠻⢿⣦⣄⠀⠀⠀⣀⣰⡿⠟⠛⣷⡐⣶
⠀⠀⠀⠀⢄⣼⠃⠙⠾⣶⣄⠀⠀⠀⠀⣠⢞⣵⠏⠁⠀⠀⠹⣧⠀⠀⢀⣸⣇⣀⠀⠀⡟⠀⠀⠀⠙⢿⣷⠄⡮⠝⠉⠀⣀⠀⠙⣧⡙⡆
⠀⠀⠀⠀⣘⠇⠀⠀⠀⠀⠈⠛⢶⣄⢰⣶⡾⣷⣄⡀⠀⢀⣴⠴⣚⢫⡭⡅⠮⠩⣉⡛⢦⡀⠀⣀⡴⠟⠛⢮⢧⠀⣸⣿⣿⣷⡆⠹⣷⢸⠇
⠀⠀⠀⢀⡞⠀⢠⠄⠘⢇⠀⠀⠀⣠⣿⡝⠀⠀⠈⠛⣷⠟⣥⡚⠉⠀⠀⠀⠀⠀⠀⠙⠳⢝⢦⡉⠀⠠⠀⠈⢧⠷⠉⠙⣿⡽⠁⠀⢹⡆⣍⡀
⠀⠀⠀⢰⡇⠀⠀⢀⡀⠀⢀⠀⠠⣿⡿⠁⠀⠀⠀⣰⢋⣞⠂⠀⠀⡀⢠⣤⣼⠀⠀⠀⠀⠘⣧⢻⡄⣀⣀⣀⣸⡏⡇⠀⠁⠁⠀⢀⣨⣇⢼⣡⡀
⠀⠀⠀⢼⡁⠀⠀⠀⠷⠀⠉⠀⢸⣿⣇⣀⣀⣀⢰⡏⠸⠀⠀⠀⠀⣼⡿⠟⠻⣿⣖⠀⠀⠀⢘⡟⣿⠉⠉⠉⠉⣿⡶⠶⠿⠛⠚⠛⠛⠛⠸⣀⡇
⠀⠀⠀⢞⡀⣀⣀⣀⣀⣀⡄⢠⣌⢹⡟⠛⠉⠉⣸⡇⣋⠀⠀⠀⠀⣻⣇⠀⢀⣼⡯⠍⡀⠀⠀⡆⣿⠀⠄⡀⠀⣿⡿⠀⢀⣀⠀⠀⠀⠀⠸⣣⡇
⠀⠀⠸⠼⡀⠻⡟⠉⠉⠉⠁⠀⠹⠼⡇⠀⠣⠀⢹⣽⠸⣂⠀⠀⠀⠉⣿⡿⡿⡟⠄⠀⠀⠀⣸⢡⡏⣤⣀⠀⢀⣿⡏⢠⡟⠛⠂⠀⠀⠀⢸⡴⠁
⠀⠀⠀⠘⡧⠡⣇⠀⠀⣔⡒⠄⠘⣇⢿⡄⠈⠀⣀⡿⣧⠡⡄⠀⠀⠀⠎⠃⠀⠀⠀⠀⢠⡴⣣⡟⠀⠈⠙⠻⣾⠿⠃⠀⢧⡀⠀⠀⠀⠀⣿⢆
⠀⠀⠀⠀⢻⡀⣾⡄⠀⣭⡴⠸⠀⠱⠊⢷⡴⠟⠋⠀⠙⢷⣌⡑⠦⣄⣀⡀⢠⣠⡴⠴⡫⣾⣽⠁⣀⡀⠀⣰⢏⣽⣀⡀⠀⠉⠀⠀⠀⣸⡧⠂
⠀⠀⠀⠀⠈⣇⠈⣷⡀⠀⠀⠀⠀⢀⡸⢷⣿⣆⠀⢈⠀⠀⠈⣛⣶⣶⣶⣿⡮⣤⠶⠛⠛⠉⠉⣁⡀⢀⣼⢋⠙⠀⠙⠛⢷⣦⣀⡀⣰⣿⣾⠃
⠀⠀⠀⠀⠀⠈⢧⡙⣷⣀⣀⣠⡶⠟⠉⠈⠹⢿⣷⣦⣀⠀⣴⡟⠀⠀⠩⠉⠁⣾⠀⠂⠠⠰⢀⣨⣿⠯⡤⢋⡸⠟⠣⣀⠀⠈⡿⢻⠿⣿⠃
⠀⠀⠀⠀⠀⠀⢘⠷⡈⠛⠋⠁⠀⢀⣤⣰⣄⠈⠹⢿⡾⢽⣿⣄⣐⠃⠀⠀⠀⢿⣇⣀⣤⣶⣿⣿⠻⠀⠀⠿⠉⢀⠰⠁⢀⡾⢡⡚⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢹⣆⡀⠀⠀⠐⣿⢿⡿⠀⠀⢀⡶⠚⠲⠌⣍⣛⢛⡛⣛⠛⢽⠟⠿⠙⠃⠀⠙⢆⠀⠀⠀⠈⢀⡴⢋⣼⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣆⡀⠀⠀⠁⠀⠀⢀⡾⠁⠀⢀⡀⠀⠈⠈⠁⢹⡅⠀⠀⠀⣀⠁⠀⠀⠈⠳⣦⣠⠶⠋⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⣦⡄⠀⠀⢀⣾⠁⠀⠀⠛⠀⠐⠆⠀⠀⠸⡇⠀⢅⡸⠿⡇⠀⠄⠀⠀⠙⣡⠰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠱⠴⣌⣛⠷⠤⣤⣀⣂⠀⠀⠀⠀⣐⡇⠀⠀⠀⠀⠀⠀⣀⡤⡤⠍⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⢇⢳⡔⢭⡭⣍⢛⣛⢛⡛⣃⡀⣤⡔⠢⠝⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠈⠀⠙⠒⠛⠀⠉

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⢛⡛⠫⠟⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣠⣾⣿⣿⡟⠁⠀⠈⠀⠀⠈⠀⢻⠘⣿⣿⣧⡀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⡀⠀⠐⠑⠀⢠⣿⣿⣿⣿⡄⠀⠀⠤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢐⣀⣀⣀⠀⠀⣀⣿⣿⣿⣿⣿⠀⠀⢈⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣀⡀⠀⠀⠀⣀⢀⠀⠀⠇⠀⠀⣸⣿⣿⣿⡏⠀⠀⠀⠁⠀⠒⠒⠈⠀⠘⢡⣿⣿⣿⣿⣿⡇⠀⠀⠇⠀⢀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠀⠀⠙⣿⣿⣿⣿⣄⣺⡀⠀⠀⠀⢠⣴⣥⣴⣿⣿⣿⡿⠁⠀⢀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠤⡄⠀⠈⠿⠿⠿⠿⣿⠟⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠛⠦⣄⣀⣀⣒⣁⣀⣀⡤⠔⠋⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠀⠶⠢

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⡰⢶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣿⣿⣷⡝⣿⠀⠀⠀⠀⣠⣴⣾⣿⣶⣶⣤⡀⠀⠀⠀⠀⢀⣶⣶⣶⣤⡀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⣠⠖⠄⣀⣼⣿⣿⣿⣿⣿⣹⠃⠀⠀⣾⣿⣿⣿⣿⠛⠿⢿⣿⣆⣀⠀⠀⣿⠁⠀⢨⢻⣿⡄⣠⣿⠟⡅⠀⠘⣿
⠀⠀⠀⠀⢿⣶⣾⣿⣿⣿⠟⢻⡟⠉⠀⠀⠀⠀⣻⠛⠉⠀⠉⠀⠀⠈⢹⣿⠛⠁⠀⠙⠷⠶⢟⣼⣿⣿⣿⣿⣧⡻⠶⠾⠋
⠀⠀⠀⠀⣴⣿⣿⣿⣿⠃⢀⣈⣁⠀⠀⠀⣠⣴⣿⣁⠀⠀⠀⠀⠀⣠⣼⣇⠀⠀⠀⠀⠀⠀⢸⡟⢿⣿⣿⡟⢻⡇
⠀⣰⣟⣾⣿⣿⣿⠿⣿⣆⡸⣩⣿⡇⠀⠀⠈⠉⢹⣿⣦⣤⣬⣿⣯⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠋
⠀⠹⣾⡿⠟⣿⣻⣢⡿⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⢀⡤⠠⡀⢀⣴⣶⣶⣦⡄⠀⡀⠠⡀⠀⠀⠐⣦⣀⣦⣶⣴⣦⣴⣴⣆⣤⡆⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⢸⣔⣶⣳⣿⣿⣿⣿⣿⣿⣦⣷⣢⡇⠀⠐⢶⣿⣿⣻⣿⣿⣿⣽⣟⣿⣿⣷⠖⠀⠸⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⠇
⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠐⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠀⠀⢈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣍⡁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠲⢾⣿⣿⣿⣭⣿⣿⣯⣽⣿⣿⣿⠶⠀⢠⣿⠏⣿⣿⣿⣿⣿⣿⣿⡿⠹⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⠛⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡛⠒⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⡾⠃
⠀⠠⣄⣀⣬⣿⢏⢻⡟⡹⣿⣧⣀⣠⠄⠀⠀⠛⢛⣿⣿⣿⣿⣿⢿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠡⠠⠜⠳⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣉⣉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⢠⣾⠀⡄⠀⠀⠠⠠⠀⠀⢀⡄⢹⣆⠀⠀⠀⠀⠀⣀⣤⢾⡷⢦⣄⡀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⢿⡆
⠀⠘⢿⡰⠏⠀⠸⠀⠀⠰⠀⠘⢧⡸⠟⠀⠀⢠⠖⠋⠁⠀⢀⡇⠀⠈⠙⠲⣄⠀⠀⢀⣾⣟⠁⠀⠀⠸⣿
⠀⠀⠈⣯⠀⠀⢿⣷⣶⣿⠂⠀⢸⡇⠀⠀⠀⠁⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⢸⡿⣟⡀⠀⠀⠐⠁
⠀⠀⠀⠘⢷⣴⣿⣿⣿⣿⣷⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣦⣤⣤⣀⣀
⠀⠀⠀⢀⡠⢊⡻⢿⣿⠿⡑⠢⡀⠀⠀⠀⠰⣶⣶⡶⠀⠀⢸⣇⠀⠀⢰⣶⣶⡆⠀⠀⣨⠻⣿⣿⣿⡿⠁⠀⠲⠙⠇
⠀⠀⠀⠁⠐⢁⠔⠁⠀⠑⠌⠑⠈⠂⠀⠀⠀⠈⠉⠀⠀⣠⣿⣿⣦⠀⠀⠉⠁⠀⠀⠀⠁⠘⠁⠞⠁⠻⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠾⠋
⠀⠀⠀⠀⠀⢀⡤⢶⠒⡶⣦⣄⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⢠⣿⡷⠂⠀⠰⢾⣿⣷⣄⠀⠀⠀⠀⠹⡷⣄⣀⣀⣠⣰⡿⠁⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⢿⣷⣦⠰⣶⠋
⠀⠀⠀⠀⣿⣿⣷⣾⣤⣷⠿⢿⣿⣿⡆⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⣦
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣤⣼⡟⠛⡇⠀⢀⣾⡇⣿⡿⢿⣿⡿⣿⡇⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠁⠀⠘⠟⠃⠈⣿⣿⣿⣿⠏⠀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢰⠉⠉⠸⠿⠿⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠳⣦⣤⣶⣶⣾⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⠀⠈⠛⢿⣷⡄⠀⠀⠀⠻⣿⣷⣆⠀⠀⠀⣴⣾⣿⣿⣶⡀⠀⠀⢠⣶⣿⡿⠃⠀⠀⠀⣴⣿⠛⠋⠀⠀⠀⣠⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⢺⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣀⡤⠟⠀⠀⠀⢠⣤⣾⠿⠃⠀⠀⠠⣴⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣷⡤⠀⠀⠀⠻⢿⣦⣤⠀⠀⠀⠙⠧⣄⡀




⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠘⠛⢿⣶⡄⠀⠀⠐⠻⣿⣷⣆⠀⠀⠀⣰⣾⣿⣿⣶⡀⠀⠀⢀⣶⣿⡿⠓⠀⠀⠀⣴⣾⠟⠛⠀⠀⠀⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠀⠀⠀⣿⣿⠀⠀⠆⠀⢸⣿⣿⠆⠐⠰⣿⣿⣿⣿⣿⡷⠰⠀⢾⣿⣿⠀⠀⠆⠀⢸⣿⡇⠀⠀⠰⠀⠀⣿
⠀⠀⠀⠀⠀⣀⣤⠟⠁⠀⠀⢠⣤⣾⠿⠃⠀⠀⠠⣴⣿⡿⠏⠀⠀⠀⠹⢿⣿⣿⠿⠁⠀⠀⠈⠿⣿⣷⡤⠀⠀⠀⠻⢿⣦⣤⠀⠀⠀⠙⢦⣄⡀




⠀⠀⠀⠀⠀⠉⠓⣦⠀⠀⠀⠘⠛⢿⣶⡄⠀⠀⠐⠻⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⢀⣴⣿⡿⠓⠀⠀⠀⣴⣾⠟⠛⠀⠀⠀⣠⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⠀⠀⠀⣿⣿⠀⠀⠄⠀⢸⣿⣿⠆⠀⠰⣿⣿⣿⣿⣿⡇⠠⠀⢾⣿⣿⠀⠀⠄⠀⢸⣿⡇⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⢀⣤⣾⡿⠃⠀⠀⠀⣴⣿⡿⠏⠀⠀⠀⠻⢿⣿⣿⠿⠁⠀⠀⠘⠿⣿⣷⡄⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠙⢦⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠤⢤⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢈⣀⡤⠴⠷⠗⠶⠤⣤⣈⡓⠠⣀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣌⠳⢄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⣻⡳⡤⣀⣀⣀⣀⣤⣴⠟⠁
⠀⠀⠀⢀⣤⣶⣾⡏⠀⢼⣿⣶⣶⣤⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⢴⡿⠋⠛⠓⠻⠞⠚⠛⠉
⠀⠀⢠⠏⣾⡿⢻⡆⢀⢻⡉⠛⠻⠧⢴⣧⡍⢁⡁⣈⣉⣁⣨⡭⠔⠋
⠀⠀⣿⠉⢿⡇⠈⣇⠘⠜⢷⣶⣶⢖⡶⠛⣧⣬⣉⠉⠁
⠀⠀⠸⣆⠈⠻⣦⣸⣦⣨⣎⣿⠁⢸⣴⡶⠁⠙⠿⣿⣶⣄⡀
⠀⠀⠀⠙⢧⣌⢲⡟⠉⠉⠁⠊⠠⠟⠁⣈⣽⠇⠀⠘⠻⠘⣷⡀
⠀⢀⣤⡤⢴⣿⣯⢴⣶⠁⠀⠀⢘⠂⠀⠉⢣⡀⠀⠀⠀⠀⣿⢻⣦⣀⡀
⠀⠀⠉⠛⠷⠾⠛⡆⠀⠐⠦⣿⡟⠀⠀⠀⠹⡇⠀⠀⠀⠀⠈⢸⣿⡟⠻⣷⣶⣶⣶⡶⢶⡿⠗⢶⣶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠂⠈⢣⠀⢸⠛⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠉⠉⠳⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⠀⠄⣁⣼⡟⠀⠀⠀⠀⠀⠀⣀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠊⠀⢘⣿⣿⣇⠀⠀⠀⠀⢀⠞⢁⠿⢷⠀⠈⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣿⣿⢻⣿⡴⠀⠀⡀⡠⠞⠁⠀⣼⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡦
⠀⠀⠀⠀⠀⠀⡴⠁⠈⠀⢄⣰⡿⠹⢃⣿⢸⡟⢀⡴⠋⠀⠀⠀⠀⡴⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡷
⠀⠀⠀⠀⠀⢐⠇⢀⣘⠃⠀⠀⠁⠀⢸⣿⣼⣇⣿⣷⠀⠀⠀⠀⢸⡇⠀⡏⠀⡀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠑⠂⠀⠀⢠⡶⣻⠏⠀⢻⣄
⠀⠀⠀⠀⡠⠋⢠⡾⢻⣷⡄⢀⠀⠠⠄⡀⣿⣿⣿⣿⡆⠐⢳⡀⠘⠘⠂⠳⡄⠉⠀⢀⠀⣠⢂⣨⣤⣤⣶⣶⣶⣤⣮⣶⡟⠀⡯⣊⠀⡀⣿
⠀⠀⣠⠞⠁⢀⡸⣧⠘⣿⣶⣿⣦⣴⣄⣙⡿⣸⣿⠇⠀⡠⢞⣧⠆⢰⡀⡀⠘⣴⣶⣯⣷⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⠟⠀⢸⣯⢀⢇⣩⣽
⠀⢸⢳⡴⣦⡀⢀⣛⣶⡌⠙⠛⠻⠿⠿⣿⠿⠛⠁⠀⠀⠀⠋⠀⢰⣾⡵⢷⣿⡾⠿⠿⡏⠥⠤⣀⣀⣀⣠⣤⠴⠛⠁⠀⠀⢸⣿⣯⠀⢀⢚
⠀⠛⠻⠾⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣰⠁⢀⡀⡀⠀⢰⠇⡀⣠⣿⣿⡘⠄⢆⣥⣮⡷⠵⣼⣥⣰⣊⡿⣿⣿⠥⠀⠀⠀⣠⣿⣿⣿⠔⠛⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢨⣾⡅⣠⡴⠿⡾⣛⠋⠤⢨⡵⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣾⣟⠢⠀⠀⣴⣿⣿⣿⢯⡹⢪⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡟⠉⠃⠋⢁⡾⣁⠨⠃⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⠀⠀⠀⣿⣿⣿⣯⣿⢋⡭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⢷⣄⠀⠀⣾⠉⠲⣍⢩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⣯⠣⠀⠀⠘⢿⣿⣷⣭⣽⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡹⠂⢸⡣⢊⠥⠨⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢷⡄⠀⠀⠀⢹⣿⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⢰⢸⡕⢒⡨⡉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣦⡀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣷⣿⠀⣿⠔⡎⣡⡣⠜⠳⢄⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣿⣿⣟⣿⡿⣿⣷⡀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠁⠉⠀⠘⢿⡼⡩⠞⠹⠓⡭⠜⢹⠟⡖⣖⡟⢻⣟⣿⣿⣗⢿⣟⣿⣿⣿⣿⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣫⣼⣌⢳⠉⣏⢧⡹⡬⣱⣾⣾⣾⣝⣽⢟⢿⣿⣝⣿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣽⡇⣽⣼⣧⣷⡕⡿⢿⣗⡿⣿⣿⣷⡿⠷⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠤⢤⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢌⣀⡤⠴⠷⠗⠶⠤⣤⣈⡓⠠⣀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣌⠳⢄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣻⠳⡤⣀⣀⣀⣀⣤⣴⠞⠁
⠀⠀⠀⢀⣤⣶⣾⡏⠀⢼⣿⣶⣶⣤⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⢴⡿⠋⠛⠓⠳⠞⠚⠛⠉
⠀⠀⢠⠏⣾⡿⢻⡆⢀⢹⡉⠛⠻⠧⢴⣧⣍⢉⡁⡁⣉⣁⣨⡭⠔⠋
⠀⠀⣾⠉⢿⡇⠈⣇⠘⡜⢷⣶⣶⢖⡶⠛⣧⣬⣉⠉⠁
⠀⠀⠸⣆⠈⠻⣦⣸⣦⣨⣮⣿⠁⢸⣴⡶⠁⠙⠿⣿⣶⣄⡀
⠀⠀⠀⠙⢧⣌⢲⡟⠉⠉⠁⠊⠠⠟⠁⣈⣽⠇⠀⠘⠻⠙⣷⡀
⠀⢀⣤⡤⠴⣿⣯⢴⣶⠁⠀⠀⢸⠂⠀⠉⢣⡀⠀⠀⠀⠀⣿⢻⣦⣀⡀
⠀⠀⠉⠛⠷⠾⠛⡆⠀⠐⠶⣿⡟⠀⠀⠀⠹⡇⠀⠀⠀⠀⠘⢸⣿⡟⠻⣷⣶⣶⣶⡶⢶⡿⠗⢶⣶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠊⠈⢧⠀⢸⠛⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠉⠉⠳⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⢀⠀⠄⣁⣼⡟⠀⠀⠀⠀⠀⠀⣀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠊⠀⢘⣿⣿⣇⠀⠀⠀⠀⢀⠞⢁⠿⢷⠀⠈⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⣿⣿⢻⣿⡴⠀⠀⡀⡠⠞⠁⠀⣸⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡦
⠀⠀⠀⠀⠀⠀⡔⠁⠈⠀⢄⣰⡿⠹⢃⣿⢸⡟⢀⡴⠋⠀⠀⠀⠀⡴⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡷
⠀⠀⠀⠀⠀⢰⠇⢀⣐⠃⠀⠀⠁⠀⢸⢿⣼⣇⣿⣷⠀⠀⠀⠀⢸⡇⠀⡏⠀⡀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠑⠂⠀⠀⢀⡶⣻⠋⠀⢻⣄
⠀⠀⠀⠀⡠⠋⢠⡾⢿⣷⡄⢀⠀⠠⠄⡈⣿⣿⣿⣿⡆⠔⢳⡀⠘⠘⠂⠳⡄⠉⠀⢀⠀⣠⢂⣨⣤⣤⣾⣶⣶⣤⣮⣦⡟⠀⡯⣊⠀⠂⣷
⠀⠀⣠⠞⠁⢀⡸⣧⠘⣿⣶⣿⣦⣤⣄⣙⡿⣸⣿⠇⠀⡠⢜⣧⠆⢰⡀⡀⠘⣴⣶⣯⣷⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⠟⠀⢸⣯⢀⢇⣩⣼
⠀⢸⢳⡴⣦⡀⢀⣻⣶⡌⠙⠛⠻⠿⠿⣿⠿⠛⠁⠀⠀⠀⠋⠀⢰⣾⡵⠷⣿⡾⠿⠿⡏⠥⠤⣀⣀⣀⣤⣤⠴⠛⠁⠀⠀⢸⣿⣯⠂⢀⢚
⠀⠛⠻⠾⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣰⠁⢀⡀⡀⠀⢰⠗⡀⣠⣿⣿⡘⠄⢆⣣⣬⡿⠵⣼⣥⣠⣋⡿⣿⣿⠥⠀⠀⠀⣠⣿⣿⣿⠔⠛⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢨⣾⣅⣠⡴⠿⡾⣛⠋⠤⢨⡵⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣟⡤⠀⠀⣴⣿⣿⣿⢯⡩⢪⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠉⠋⠋⢁⡾⣑⠨⠃⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⠀⠀⠀⣿⣿⣿⣯⡿⢋⡭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⢷⡄⠀⠀⣾⠉⠶⣜⢩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⣯⠣⡀⠀⠘⢿⣿⣷⣭⣽⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⡹⠂⢸⡧⢚⢥⠨⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢷⡄⠀⠀⠀⢹⣿⣿⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⢸⢸⡗⢒⡨⡉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣦⡀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣷⣿⠀⣿⠔⡎⣡⡣⠜⠳⢄⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣿⣿⣟⣿⡿⣿⣷⡀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠁⠉⠀⠘⣿⡼⡡⠞⠹⠓⡭⠜⢹⠟⡞⣖⡟⢻⣟⣿⣿⣗⢿⣟⣿⣿⣿⣿⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣫⣼⡌⢳⠉⣏⣧⡹⡬⣱⣾⣾⣾⣝⣽⢟⢿⣿⣝⣿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣽⡆⣽⣸⣦⣷⣔⡿⢿⣗⡿⣿⣽⣯⡿⠷⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⣟⣶⣶⣿⣬⣤⠾⠿⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⢑⣶⡿⠟⠋⠉⠛⠳⡀⠀⠀⠀⠈⠙⢧
⠀⠀⠀⠀⠀⠀⠀⣰⡾⢁⣤⣾⠝⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠋⠰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⠛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠈⠐⠴⣤⣀⠀⠀⣠⡴⠋⣡⣅⣤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣤⣤⣴⣀⠀⠉⠛⠷⣾⢯⣀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⡟⠉⠉⠻⣿⣧⣴⠴⣯⠘⣿⡿⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡄⠃⠀⠺⠿⣭⣯⡰⠂⢼⡆⢩⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣄⠀⠈⠉⣈⢉⡉⠉⢉⠘⡿⣿⣆⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡇⠀⠀⠉⠁⠀⠈⠋⠀⠁⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠓⠤⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⢸⣿⡖⠀⠀⠀⠀⠀⠀⣴⣾⣿⡖⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⠀⠀⠀⢀⣴⡾⠛⠁⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⣰⣿⣿⣷⣦⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⢠⣾⣿⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢙⡇⠒⠀⠉⡟⠋⢹⣿⡆⠂⠀⢨⣿⣿⡟⠋⢹⣿⡇⠒⠀⢸⣿⣿⣿⠛⠉⣿⠀⢸⣽⡷⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠓⢿⠏⠀⠀⠁⠀⠘⠛⡿⠃⠀⠈⢮⡉⠁⠀⠈⠓⡿⠋⠀⠀⠣⡉⠁⠀⠀⠋⠀⠀⠣⡉⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠊⢡⡶⢯⣯⣉⡓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣉⣽⡝⣶⡌⠑⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣏⠛⢳⣮⡻⣇⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣸⢟⣵⡞⠛⣹⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠛⢻⣟⢻⢣⠈⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⡼⡟⣻⡟⠛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠛⢛⡯⢶⡄⠙⣆⠀⠀⠀⠀⣰⠋⢠⣶⢽⡛⠟⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠸⣿⣮⠧⣄⠘⡆⠀⠀⢰⠃⣠⠼⣵⣿⠇⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡾⣋⣿⠀⢹⡀⢠⡏⠀⣿⣹⢷⠀⠀⡞
⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠤⠞⡛⠋⠍⠙⣿⣿⠏⠟⠟⠿⡷⣤⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⡟⢿⠓⠦⠤⠤⠤⠖⠚⠋⠉⠉⠙⠒⠦⣄⢀⡴⢋⠅⠊⠁⢠⠖⢦⠀⣹⡇⠀⡴⠲⡄⠀⠉⠩⢻⢦⡀⣠⠴⠚⠋⠉⠉⠙⠓⠲⠤⠤⠤⠴⠞⣿⢻
⠀⠀⠀⢧⠈⠙⠲⠦⠤⠴⠖⠒⢛⣭⢭⡙⢦⡀⢨⠏⠔⠀⠀⠀⠀⠸⣄⣉⡴⠃⠹⣤⣉⣰⠇⠀⠀⠀⠀⠁⠽⡅⢀⡴⢋⡭⣭⡛⠒⠲⠦⠤⠴⠖⠋⠁⡼
⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⡏⠠⣄⡿⢀⣷⠏⠌⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠗⣸⠁⠀⠀⠀⠀⠀⠀⠀⠩⡹⣾⡀⢿⣠⡄⢹⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠙⠦⠤⢴⣫⡿⠸⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠡⣿⣟⠦⠤⠴⠋⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠶⠖⠒⠋⠁⡇⠅⠀⢿⢉⣝⡻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣫⡉⣿⠀⠸⢽⠈⠙⠒⠒⠶⠶⠖⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠅⠀⠘⠦⣝⣃⣸⠞⡆⠀⠀⠀⠀⠀⠀⢰⠳⣇⣘⣫⠴⠃⠀⠸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠡⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢬⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢆⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠠⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣢⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣧⠀⠀⢸⡇⠀⠀⠀⠀⢸⡂⠀⠀⣼⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢸⠛⠋⠉⠉⠉⠛⡇⠀⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⢧⣀⠀⠀⣀⡼⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣄⠈⣹⣯⡉⢀⡄⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡉⠳⠾⢋⣉⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⢿⢾⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⢸⢼⢻⠻⡄⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠭⠏⢸⠘⡆⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡒⣦⠀⡞⠀⢧⠠⣖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⡳⣋⣼⡇⣠⢬⡷⠬⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠳⠬⠼⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡄⠒⠂⠀⠀⠂⠀⠐⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠸⠈⡄⠀⠀⠀⠀⠀⠀⠀⡤⠙⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⢀⠔⢁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⢀⠔⠁⠔⢁⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⣔⣶⡶⠶⣆⠀⠁⠐⠀⠀⠀⢀⢶⢝⢆
⠀⠀⠀⡠⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡴⣿⡃⠀⢉⢂⣉⣦⠀⠀⠀⠀⠁⡑⠁⠀⠱⡀
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢀⠀⢠⡏⢺⣮⣿⣅⠀⣹⣿⠉⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠑⡀
⠀⠰⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠃⠘⠈⠀⢸⡄⢈⢿⣾⣯⣄⣘⠒⢿⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰
⠀⠇⠈⡖⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣴⣬⣍⠻⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠩⢟⠻⢿⣶⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠘
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡀⠀⠀⠑⠀⢙⣿⡟⣿⠄⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠔⠁⢀⠀⠀⠀⠀⠊⠀⠀⡀⠀⣾⣿⣴⡃
⠘⠔⠂⠀⠀⣤⠖⠛⠉⠁⠀⠘⢀⡈⠂⠒⠶⠒⠀⣸⡆⠀⠠⠉⢟⠉⠐⠚⠿⠀⠀⠀⠀⠀⠀⠀⠄⣀⠰
⠀⢄⠐⠉⢸⡇⠀⠀⠀⠀⣰⢶⣿⠿⠟⠛⠲⠛⠋⠉⠑⠪⡓⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠐⠂⢥⠾⡇
⠀⠘⡀⠀⢸⢧⣤⡶⢾⣿⠁⠈⠢⣄⠈⠉⢀⡀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠙⠦⠖⠀⢠⣤⣀⠀⠀⡸
⠀⠀⠐⡀⠨⠰⡰⠹⢆⠉⠀⠀⠀⠈⠒⠀⠒⠛⠀⠀⠀⠀⠀⠀⠈⠃⣀⠀⠀⠠⢀⣰⣿⣿⣿⣷⡶⠁
⠀⠀⠀⠘⢄⠀⠀⠔⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣶⣟⣽⣿⣮⢻⠟
⠀⠀⠀⠀⠀⠣⡉⢑⠂⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⢿⣿⣿⣿⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⡌⣀⠀⠀⠒⠒⠀⠀⢈⠀⠀⠀⠀⠀⢻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⣰⢀⠃⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠄⠀⠤⠤⠀⠒⠒⠈⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠠⡑⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⡡⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⣖⣚⠟⠑⢆⡀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⠖⠙⢟⣖⡆
⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⢡⣠⡏⢆⠀⠀⠀⠀⠀⠀⠀⢀⠞⣇⣰⠃⠀⠀⢸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⣅⢻⠀⠀⠀⠀⣇⠀⠈⣆⠀⠀⠀⠀⠀⢀⡜⢀⢀⠇⠀⠀⠀⢸⢃⡅
⠀⠀⠀⠀⠀⠀⠀⢸⣹⠀⠀⠀⠀⠘⡶⠁⠸⡄⠀⠀⠀⠀⡼⠀⠱⡞⠀⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠈⣾⡂⠀⠀⠀⠀⠹⡆⡀⣸⡀⠀⠀⣸⡀⡀⡾⠁⠀⠀⠀⠀⣺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡧⠤⡄⠠⣠⣀⢡⡴⠀⠙⢆⠞⠁⠨⣤⢁⣠⡠⠀⡤⠤⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠐⠬⠭⠥⠽⣿⡕⠀⠵⢚⠢⢅⢐⣹⡿⠥⠭⠭⠐⠁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⠀⠀⠠⢺⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡿⡄⣽⡅⡼⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡀⠘⠀⣸⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣧⠈⡀⠀⠀⡘⢠⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣄⠈⠒⠊⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣷⡀⣰⣟⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠘⠘⠘⠐⠘⠁⠀⠘⠐⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠄

⠀⡆⠀⠀⣴⠀⠀⢰⡆⠀⢰⣶⠀⢠⣾⡆⠀⣶⣷⡄⣴⣿⡆⢰⣿⣦⠀⣾⣷⠀⠀⣷⡄⠀⢰⡆⠀⠐⡆⠀⠀⢢
⠀⠑⠀⠀⠈⠀⠀⠈⠃⠀⠈⠛⠀⠀⠙⠃⠀⠙⠛⠀⠈⠛⠁⠈⠛⠉⠀⠙⠋⠀⠀⠋⠀⠀⠘⠁⠀⠘⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⡀⠀⢠⠀⢠⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⢀⠀⠀⠀⠀⢠⠠⠀⠀⠀⡀⠀⡀⠀⢸⠀⠀⠀⡇⣇⠀⢸⠠⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠸⡄⢂⠀⠀⠀⠸⡆⣠⠄⠁⠀⠀⡄⠀⠘⣄⠀⠀⠧⢡⠀⠸⠀⠸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣱⡈⠤⠤⠀⠘⠁⠆⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡇⠀⢸⠠⠨⠃⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢇⡀⠸⠀⠸⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⠀⠀⢠⠆⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠜⠁⠀⠀⠀⠀⠀⣀⡤⠄⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢄⠀⠀⠀⣀⣀⣀⠀⣁⠤⠋⢡⠀⠀⠀⠡⡍⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠡⢡⠃⡜⠤⠄⠰⢢⢤⠄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⠀⠀⠁⡃⣈⠀⡀⠰⠱⢡⠃⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢀⠠⡁⠀⠀⠀⠈⠑⣅⠀⡸⠀⠰⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⢀⡀⠀⡄⠀⠀⢰⠄⣀⠄⠀⠐⠄⠀⠀⠀⢀⡸⠔⢁⠀⠳⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠈⠢⣀⠄⠀⠀⠀⢀⠈⠀⢰⠋⠉⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠀⠀⠀⠠⠀⠀⠀⠈⠀⠐⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡨⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠉⠒⠒⠤⠤⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⢀⣢⣬⣦⣤⣀⣄⣠⣉⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣠⠶⠋⣡⡴⠞⠋⠉⠀⠀⠀⠀⠈⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⡾⠁⣠⠞⠁⠀⠀⠀⠀⣰⣤⠤⠶⢦⣤⣤⡀⠈⢧⡀
⠀⠀⢠⢦⡀⠀⠀⠀⢀⣤⠶⣾⠁⠀⢀⣠⡟⠀⣼⠋⠀⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⢳⠸⡟⢶⡄⢳
⠀⠀⠸⡄⠏⠓⠲⠤⡾⠁⠀⠙⠷⢦⣤⣼⠓⢶⡇⠀⠀⠄⣤⡞⠁⠀⠀⠀⠀⠀⠀⠸⡷⠁⠀⠹⣎⠇
⠀⠀⠀⠳⣄⡀⠀⣸⠃⢀⣠⡶⠿⠆⠀⠀⠈⠻⣧⣄⣀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠉⡟⣴⣿⣥⡀⠀⠐⣶⣤⣀⡀⠀⠀⠉⠀⠾⣅⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣸⢡⣏⣀⠀⣙⣶⣋⣉⡈⢻⡍⠃⠀⠀⣧⠀⠀⠉⡿⠛⢩⠇
⠀⠀⠀⠀⠀⢀⡏⠘⣇⠸⣿⡿⣏⠉⣿⣿⣽⠟⠓⠂⠀⣟⢦⣀⣀⣀⡴⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣸⠀⠀⠈⠉⠉⠀⠘⣷⡬⠽⠏⠀⠀⠀⠀⠹⡄⠈⠉⠉⠀⠀⠀⠀⠀⣼⠛⢷⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠄⠀⠙⣆⠀⠙⢦⡀⠀⠀⠀⠀⠀⣰⠃⠀⠸⣇
⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⡾⠻⡄⠀⠀⠀⠀⣼⠀⠀⠀⠉⠳⣄⠀⠀⢀⡏⠀⠀⢀⣹⣀⣠⣤⠴⡶⣤⣀
⠀⠀⠀⠀⠀⡇⠀⢤⡀⠀⠀⠀⢸⠇⠀⢳⡀⢠⡴⠞⠁⠀⠀⠀⠀⠀⠈⠳⡄⠘⡇⠀⠀⠀⠀⠉⠀⠀⠀⠊⠁⠘⠷
⠀⠀⠀⠀⠀⠘⢆⠀⢱⠀⢠⣄⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢨⡇⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠳⠦⣤⣤⣄⡀⠀⠀⣠⠏
⢀⣤⡀⠀⠀⣠⣤⡄⠀⠈⠓⠦⣄⣀⣀⡼⠃⡸⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠘⡿⣏⠙⠉⠁
⠋⠈⢿⡄⣼⡟⠉⣿⡄⠀⠀⠀⠀⠉⠉⢣⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⢠⣄⡴⢧⠀⠀⠀⠀⡇⢹⡄
⠀⠀⢸⣿⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠠⠧⢆⠀⢰⠁⠀⢸⢸⠀⠀⠀⠀⡔⠳⠀⠱⡜⢧⡀⠀⣰⠇⢼⡇
⠀⠀⢸⣿⠃⠀⠀⠀⣿⣤⣶⣶⣄⠀⠀⠀⠿⣦⢣⡀⠀⡜⢸⠀⠀⠀⢀⣡⠀⠀⠀⠀⠀⠙⢛⡇⣀⢸⡇
⠀⠀⠸⠿⠀⠀⠀⢀⣿⣏⠀⠀⣿⡇⠀⠀⢀⣆⡼⠳⣄⡇⣾⠀⠀⢰⠋⠈⠂⠀⠀⠀⠀⠚⣯⠈⢁⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢶⡾⠟⠁⢀⠔⠉⠀⢱⢀⣸⣇⣯⡀⢠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠤⠟⠀⠀⠀⠀⠀⠘⠣⢶⣀⣠⡞⣫⠀⠀⣇⠙⠻⠦⣤⣀⡀⠀⠀⠀⢀⣨⡶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣹⣧⣤⣤⣿⣀⣀⣀⣀⣈⣉⣉⣙⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⢻⣟⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠈⠀⠀⠠⠖⠉⠑⠣⠐⠀⠂⢁⠀⠠⠀⠠⠐⠀⠀⠐⠠⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⡁⠸⠃⠀⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠨⡇⠀⠀⡁⢤⠀⢀⠄⡠⠀⠀⣮⠀⠀⠀⢁⠲⡄⠀⠱⢂⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠙⠂⠀⠈⠢⡄⠠⠌⠀⠀⠞⠁⠀⠀⠀⠀⠂⣄⢀⡄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠤⡀⠀⠀⠀⡠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠘⠁⠀⠀⠀⠁⠱⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⠶⡄⠀⠀⣂⣀
⠀⠀⠀⠀⠀⠀⢐⠀⢀⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡊⡉⣡⠋⠀⡐⠣⡌⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣀⠒⠀⠀⠃⠀⠀⠀⠀⠀⡖⠀⠀⠈⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡇⠀⠀⠉⠑⡅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠠⠀⠊⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠂⢀⠀⡀⠂⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠒⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠉⠢⠐⠀⠀⠀⠕⠉⠉⠀⠀⠀⠀⠀⣠⣒⠄⠀⠀⠀⠀⠐⠒⠃⠀⡠⠀⠐⠂⠚⠂⡄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠠⠀⢰⡀⢀⡀⠀⢀⠀⢀⠙⠀⠀⠈⡀⠀⣀⠀⠀⠀⠀⢼⠋⠀⠀⠀⢱⡀⠀⠀⠀⠺⠊⢏⠀⠀⠀⣶⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠁⠁⠀⠂⠀⠁⢸⠁⠀⠉⠀⠀⠉⠁⠐⠂⠀⡒⠀⠉⠁⠀⠀⠀⠀⢰⠀⠘⠄⡠⠊⠂⠀⠀⠀⠓⢀⡤⠤⠀⠠⠉⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⠿⢿⡿⠛⣹⣿⠿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠁⠈⠙⢿⣿⡌⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⠿⠶⣶⣤⡀⠀⢀⣠⣤⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠒⠀⠀⠀⠀⢨⠕⢦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣟⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⡀⢀⢀⠀⡀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢸⠟⣇⠃⠇⢸⡱⣿⢻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠠⣤⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠒⠲⠶⣶⣤⣄⣉⣻⣷⣦⣬⣿⣦⡙⣷⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠡⠶⠶⣶⣶⣤⣤⡽⠟⠛⣋⣉⣉⣉⣙⡛⠿⣿⣷⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣁⣤⣤⣶⣶⣶⡶⠛⠁⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣆⠙⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣁⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠇⢘⣉⣉⣙⣛⣋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣾⡏⠞⢋⣉⣉⣉⣉⡥⠀⠀⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠈⠻⣿⣷⣤⡀
⠀⠀⠀⠀⢰⣿⢡⡾⠟⠛⣋⣉⣥⣶⣄⣠⣴⠛⢙⣿⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣄⣀⣀⣠⠄
⠀⠀⠀⠀⢼⡯⢋⣴⠞⠛⠉⢉⣩⠿⡿⣻⠃⠀⣸⢇⣤⣶⣤⣄⡉⠛⠷⢦⣄⣀⣀⠀⠙⠿⢶⣤⣀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣽⣧⣿⡧⠒⠛⠋⠉⣡⡾⢡⡇⠀⣠⡟⠚⠉⠉⠛⠛⠛⢁⣘⣓⠲⠦⠖
⠀⠀⠀⠀⢿⣻⣯⠀⠀⢰⠘⡦⠘⢁⡾⠖⡛⣽⡾⠟⠛⣚⣁⣤⣀⡀⠉⠙⠻⢷⣦⣄⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣇⣀⡴⣫⣾⠷⢶⡏⡴⠟⣱⡿⠀⣰⣿⡿⠿⠛⠛⠿⠿⣶⣤⣤⣀⣈⣩⣤⠴⠊
⠀⠀⠀⠀⣼⠃⢛⣫⢞⠉⡇⠀⠀⠐⠶⠾⠛⢿⣾⠟⠁⢀⣤⣤⣤⣄⡀⠀⠀⢀⣀⣀⣤⣴⠞⠉
⠀⠀⠀⠀⣧⡺⠋⠙⣶⠾⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⡴⢟⣋⣉⡉⠙⠛⣉⣥⣾⣿⣿⠟⠁
⠀⠀⠀⠀⣾⡏⠀⣆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⢈⣾⠟⠛⠛⢛⣛⠛⠛⠛⢋⣉⣤⠔⠂
⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣼⣵⡶⠶⠞⠋⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⢻⡇⣼⠃⠀⠀⢠⡶⠛⡛⠛⢻⣿⠋⢺⡟⠁⣠⡄⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠈⣧⠃⡀⠀⠀⠚⠀⢀⣷⡄⠞⠁⠀⠀⢳⣾⡏⢠⣿⣿⡏
⠀⠀⠀⠀⠀⣿⠀⠻⠀⢀⣶⢰⣯⣽⠿⠛⠿⢿⣦⠈⣿⠁⣼⡿⣿
⠀⠀⠀⠀⠀⠸⡶⢣⡀⠀⡤⢸⣿⣿⣄⡀⠀⠀⢹⡆⢻⠀⣿⡇⣿⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⣾⣧⣀⣳⡴⠃⠉⠉⢻⡆⠀⢸⡇⠈⣸⣿⠁⣿⠀⠀⠀⣠⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠼⠁⠀⢸⡇⣴⣿⠏⠀⣿⠀⢠⣾⠟⠉⠉⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⢻⣴⡿⠃⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⢸⣿⡁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⣥⠆⠀⠀⠀⠀⣠⡿⠉⠻⠷⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠟⢿⣄⣀⣀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣭⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿