Rare symbols Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢷⣄⠀⠀⢠⡴⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⣷⣼⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⡼⠋⠙⢷⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⠛⠛⠛⠛⠛⠛⢻⠛⣻⡿⠛⠛⠛⠛⠛⠛⢿⣟⠛⡟⠛⠛⠛⠛⠛⠛⢛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣸⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣇⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡾⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⣄⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣽⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢸⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡇⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⠶⠶⠶⠶⠶⠶⣾⡶⠶⠿⢶⣶⠶⠶⢶⣶⠿⠷⢶⣷⠶⠶⠶⠶⠶⠶⠶⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⣴⠟⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⠟⠻⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣴⠟⠁⠀⠀⠈⠹⣦⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⢀⠀⠀⠲⡄⠀⠀⢀⣠⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⠔⠀⠀⠀⠀⣀⢀⡁⠀⠀⠀⠴⣾⡶⠄⠀⠀⠀⡴⣤⠠⠄⠀⠀⠠⢦⣙⡆
⠀⠀⠐⠤⠈⠉⠀⠀⠀⠀⠰⠻⠂⠁⠀⠀⠤⠊⠳⠄⠀⠀⠠⠯⣭⠳⠄⠀⠀⠀⠅⠰⠀⠀⠀⠀⢀⠷⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠓⠾⠈

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠀⢀⡀⣠⣄⣀⠀⠀⠀⠀⣤⠤⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠄⠀⣄⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠠⠒⠀⠀⠀⠀⠢⠤⠤⠴⠂⠀⠈⠁⠟⠉⠀⠀⠀⠀⢞⠟⣒⠀⠀⠀⠉⢤⡅⣯⠀⠀⠀⢆⠠⠐⡂⠀⠀⠐⢲⠠⠼⠆⠀⠀⠨⢂⠛⢁⡆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠈⠀⠈⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⡀⠀⡀
⠀⠀⠈⢽⠛⠉⠁⠀⠀⠐⠀⠄⠐⠀⠀⠀⠀⠀⠔⠆⠀⠀⠀⠁⡿⠁⠀⠀⠀⠀⠈⡓⠆⠀⠀⠀⠀⠐⡍⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠐⠀⠞⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠂⠑⠀⠀⠀⠰⠄⠄⠂⠀⠀⠀⠒⠝⠓⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠒⠺⠟

⠀⠀⠠⠆⣀⠔⠂⠀⠀⠀⡄⢢⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢾⢬⢦⠀⠀⠀⠀⠠⠄
⠀⠀⠰⠣⠶⠗⠀⠀⠀⠠⠣⠺⠒⠀⠀⠀⠀⠻⠧⠀⠀⠀⠀⠁⠁⡉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠

⠀⠀⠀⣀⣠⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡠⠆⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⢈⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡛⢃⠀⠀⠀⠀⠡⠀⠅⠀⠀⠀⠀⣘⣃⡈⠀⠀⠀⠀⠋⢄⠀⠀⠀⠀⢈⣀⠀⠀⠀⠀⠀⠃⠘⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠐⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡂⢐⠄⠀⠀⠀⣠⡠⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⣴⠶⣸⣠⠄
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠑⠂⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⠀⠀⠀⢀⠉⠂⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⠁⡤⣀⠀⠀⠀⠀⣄⣀⡄⡀
⠀⠀⠀⠐⠉⠂⠀⠀⠀⠉⠓⠒⠈⠀⠀⠀⠘⠦⠘⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⡀⠀⢀⢀⣀⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⡀⣰⡄⣀⠀⠀⠀⠀⡾⢦⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠦⠀⠀⠀⠀⠀⡴⠦
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⢀⡐⠀⠀⠀⢐⠦⣒⠀⠀⠀⠀⢠⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠇⡇⠸⣧⠀⢻⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⠀⣷⠀⢿⡆⢸⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣴⡷⢶⢾⡧⣼⣇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠀⠀⠀⠀⣀⠾⣭⣻⢶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢫⡾⠋⣠⠟⠀⠀⡰⣎⠻⣦⡉⠳⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⠃⢀⣾⠟⣻⡌⢧⡈⠙⠦⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⠁⣠⡟⢡⣾⠟⢙⣆⡙⠳⢦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠇⣰⣿⡴⠋⢁⣤⠟⢁⣹⠷⢦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣠⠖⠋⢀⡴⠛⢁⣴⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣾⠋⢁⣤⠾⠋⣠⣴⠟⣋⡴⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⣿⡌⢻⣶⡉⣠⣴⣟⣩⡴⠚⣩⡴⠟⣋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢿⣮⡛⢦⣉⠻⢿⣉⣉⣥⠶⠋⣡⠔⣚⣥⡴⢟⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠻⣌⢿⡙⢷⣌⡛⠦⣩⠁⣀⡤⠞⣡⢾⠏⢠⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⠶⣤⣀⡈⠛⠂⠀⠀⠉⢴⢞⣡⠞⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣉⡙⠃⠀⠀⠀⠀⠀⠋⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣤⡆⡆⢰⣦⡂⡆⡆⣴⢰⠀⣦⣠⡆⡆⣶⣔⢶⣴⠰⡢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠃⠘⠉⠂⠋⠃⠉⠚⠀⠉⠉⠁⠉⠉⠁⠈⠉⠘⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢠⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣧⣾⣟⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠸⠿⠟⠿⠣⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⣘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣙⢿⡛⠷⣄⠀⣸⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⡿⠾⠿⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⣿⡿⠿⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣾⣿⣞⣿⣟⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣟⡋⠭⢥⣶⣿⣯⣭⣭⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⣾⡻⣗⢾⣷⣶⣦⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⢑⠄⣤⣴⠶⠆⠀⠙⣛⣿⣿⣛⣛⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣟⣿⣻⢿⠗⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⢷⣦⠭⠦⡍⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣯⣥⣬⣿⣿⠟⠁⠀⠀⠻⣿⣿⣛⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠂⠀⠀⠀⠀⠀⠀⠀⠈⠛⢒⣦⣭⣙⣷⣤⣤⣶⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠙⣩⣴⣿⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉




⣦⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣿⠀⣼⣿⣿⡏⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⠀⣿⣿⣿⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢠⣿⠀⠿⠿⠿⠄⠀⣧⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣫⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⡀⣿⣿⡀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⣿⣇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⣿⠃⣴⠂⢀⡀⠀⠀⠀⢿⣿⡇⠸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⠏⣼⠃⢀⣾⣷⡀⠀⠀⢸⣿⣿⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⢰⣿⠀⣾⣿⣿⡟⠀⠀⠀⢿⣿⣧⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣿⡏⢸⣿⣿⡿⠀⢀⠀⠀⠈⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠁⢸⣿⣿⡇⠀⣿⠀⠀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⣿⣿⡇⠀⢹⣇⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣿⣿⣿⠀⠀⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠙⣿⣿⣿⠏⠀⠀⠀⢹⣿⣆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣇⠈⢿⣿⡀⠀⢸⣧⠀⢿⣿⡄⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠸⣿⡄⠀⢻⠟⠀⡾⠿⠀⢸⣿⣿⡀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⣿⡇⠀⠐⢷⠞⠷⡆⠀⠸⠿⣿⠇⠀⢿⡿⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⠀⢀⣿⡇⠀⠀⠛⠀⠀⠃⠀⢰⣿⣿⡆⠀⣿⡗⠀⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢸⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⣿⡇⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⣼⣿⣿⠃⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢸⣿⠁⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣿⡟⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡈⠢⡪⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠟⠛⠛⠻⠿⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠓⠁⠀⠀⠘⣷⣦⡀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⢀⡘⣿⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠼⣟⣿⣿⡧⠀⠀⠀⡺⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣀⣿⣿⡿⢿⡇⠀⠀⢀⣻⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣋⣥⣾⣫⡇⠀⠀⠈⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢿⡟⠛⣝⣿⡿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⢟⢟⡋⠐⣶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⠥⠶⠦⣡⣮⢔⣫⠖⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣠⢴⡶⠒⢉⡴⠿⢒⣯⡽⠂⣴⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⣴⡭⠖⠀⠀⠀⠘⠛⢩⠔⣰⡻⠋⣠⠁⢀⠀⠀⠀⢀⠀⢀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠿⢋⣠⠂⠀⠀⠀⠀⠀⠀⠈⠙⠡⠼⢥⣀⠌⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣦⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣠⣤⣤⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢀⣿⣿⣿⡆⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⣿⣿⣿⠃⣸⣿⣿⣿⣷⠀⣿⣿⣿⣧⠀⠀⠀⠀⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠐⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠂⠘⠛⠛⠛⠃⠀⠀⢠⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⣿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⣿⠀⠀⠙⢷⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣤⠖⢀⣴⣿⣿⣿⠿⠿⠿⠿⣿⣿⣷⣦⡈⢶⣄⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣠⣾⣿⠏⢠⣿⣿⡿⠋⣠⣴⣶⣶⣶⣄⠉⢻⣿⣷⡀⢻⣿⣦⡀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⣿⣿⣿⠁⣼⣿⣿⣿⣿⠏⢁⣤⣤⣿⣿⣧⠈⣿⣿⣿⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡇⠐⣿⣿⣯⠀⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⢿⣿⣿⣄⠹⣿⣿⣿⣿⣷⣄⠉⢉⣿⣿⡏⢠⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⠈⢿⣿⣿⣦⣄⡉⠛⠛⢉⣁⣴⣿⣿⡟⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡤⢀⣤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⣼⢃⣾⠏⠀⣼⠃⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠟⠀⠀⠉⠀⠁⠀⠀⠉⠘⠛⠰⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡼⠃⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⣌⠳⡄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡴⣊⡵⠞⣁⡀⠀⠀⠀⠀⠀⢀⠰⡌⠳⣄⢳⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣩⡴⠞⠉⣠⠄⠀⠀⠀⢀⡈⢧⠘⣆⠈⠦⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⢟⣵⠟⢁⡀⢀⠀⢧⠈⠆⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡴⠋⣡⣴⠟⣁⠘⢷⣼⡆⠘⣌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢋⣀⢰⡝⢳⣄⡈⠻⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣍⢳⡙⣦⠈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⠳⡹⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠙⢦⠀⠙⣎⠳⠿⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢻⡄⠈⣧⠀⠹⠶⣶⣶⠟⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠙⢦⠹⣄⡈⠃⢠⣾⠟⠁⣠⡾⠛⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠙⢦⡈⢷⡈⠃⠀⠀⠡⡴⠞⣉⣴⠟⣡⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠈⠓⢦⣝⠢⠁⠀⠀⠀⠀⠰⠾⠋⣡⠞⢁⣠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠟⠻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣀⠀⠀⠀⠈⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠃⢰⣶⣾⡄⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠻⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠄⠛⠉⠀⠀⠐⠀⢀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣠⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠋⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢨⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⠟⣿⣿⣿⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡟⠀⣿⣿⡿⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇⢸⣿⣿⣇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡄⠀⣸⣿⣿⡀⢸⣿⣿⣿⣧⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⠇⢸⣿⣿⣿⡷⠘⠻⢿⡿⠏⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠡⠀⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡄⠀⠀⠀⠀⠀⠰⣦⣤⡉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡁⠀⠂⠀⠀⢀⣼⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠌⠂⠉⠙⠛⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⣀⣀⣀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣏⣉⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠈⠻⠛⢿⣿⣤⣄⣬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣦⡄⠈⠀⠀⠀⠉⢍⣉⠉⢁⣼⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢶⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣾⡆⣿⣷⢺⣟⠀⣾⣳⣾⡶⣿⣲⣟⣂⣾⡆⣾⣖⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡉⣉⣉⢉⢉⡉⢀⣉⢉⣈⣉⡉⢉⣉⣉⢉⣉⢉⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⣿⣿⣿⣼⣿⣽⣯⣽⡿⣿⢹⣯⣽⣿⢽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡃⠀⠀⠀⠀⠀⠀⣸⣎⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣭⡀⠀⠀⠀⠀⠀⠛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣥⡀⠀⠀⠀⠟⠛⠛⢻⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣁⣀⣙⣷⡄⠠⠾⠛⠿⠿⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣍⣛⣻⣿⣿⣶⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⡿⢋⣡⣿⣯⣿⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣈⡛⠻⣿⣧⡀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢿⣿⣦⡈⠻⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠿⢷⣦⣝⠿⣆⠻⢿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠯⣟⡻⣿⣿⣿⣦⣙⠻⢶⡦⣤⢙⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⣿⣶⣭⣤⣤⠀⠀⠉⠻⢾⣽⣷⣭⣉⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣭⣭⣭⡄⠀⠀⠀⠀⠀⠻⠿⠟⠛⠋⠉⠉⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠊⠻⠟⣋⣩⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⠻⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃

⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣤⣤⣦⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡙⣾⣿⡻⣍⠻⠿⠿⠊⠑⠉⠈⠉⠉⠉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣝⣦⡙⣾⠿⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⣄⠙⢽⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡳⣌⠳⣌⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣆⠀⣴⡆⠀⣶⠶⣦⢠⡶⠶⠂⠀⠀⠀⠀⠀⢁⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠟⢰⡏⣷⠀⣿⣤⡟⢀⡷⠶⠀⠀⠀⠀⠀⠀⠈⣿⣫⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣐⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡦⠼⠋⠹⡆⠿⠈⢧⠘⣧⣤⠄⠀⠀⠀⠀⠀⠀⢹⣶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣬⣭⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠠⠀⠀⡀⢀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⣭⣴⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠟⠒⠦⢤⣤⣉⣙⣻⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡭⠿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⠒⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣶⠶⢶⡄⠀⢀⣶⡆⠀⢶⡄⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⡿⢛⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣸⡟⣿⠀⠘⣷⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⢸⡯⢀⣿⣀⣿⡆⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠸⣷⣴⡾⠃⢸⡏⠉⠙⣷⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣫⢟⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣖⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣫⡞⣋⣹⣿⡿⠿⠛⠛⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⣿⣿⣿⣿⣿⡯⠉⣤⣤⣶⠶⣛⣉⣀⣤⣭⠵⠶⠂⣠⠀⠀⠀⡀⡀⣀⢀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⢿⣿⡟⠁⠀⠀⠈⠛⠻⠟⠉⢉⣀⣤⡴⠖⡋⠀⢀⠎⡼⣱⠸⠇⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠛⠛⢶⣶⣦⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠋⠀⠁⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣍⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃
⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣓⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣺⣿
⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⣠⣶⡾⣻⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⡙⢻⡿⠿⠿⢿⣿⢻⡙⣦⡀
⠀⠀⢸⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠨⠧⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⠋⠀⠀⠀⠀⠈⢷⣾⣿⣇
⠀⠀⠈⢿⣷⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣴⣴⣿⣿⠆⠊⠑⡘⠉⢂⢾⣿⣷⣴⡄⠀⠀⠀⢻⣿⣿⣿⠀⠴⣇⣘⠷⠀⢸⣿⣿⣿
⠀⠀⠀⠈⠛⢷⡝⣿⠿⣿⠿⠛⠁⠀⠀⠀⠀⠀⣽⣿⣿⣿⠘⠈⡃⣷⡏⠈⢸⣿⣿⣿⠄⠀⠀⠀⠹⣿⣿⣿⣆⠘⠶⠿⠓⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠀⠀⠛⠟⠁⠀⠘⠋⠁⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣠⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⡐⠖⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠟⠛⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠋⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⣇⠀⠀⡶
⠀⠀⠀⢰⣿⢿⣄⠀⢠⠟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣇⣄⣠⣧⣠⣀⣡⣤⡄
⠀⠀⠀⢸⡇⠀⢻⡀⡟⠀⢸⡿⠀⠀⢠⡶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠈⣇⠀⢸⡇⡇⠀⣸⠇⠀⠀⢸⠇⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⢀⠤⠸⣆⢸⣿⡇⣠⠟⠤⡀⠀⠘⠓⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⣌⣛⡛⠛⠛⣛⣁⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⡏⠙⠛⠛⠛⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣿⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠓⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣿⠀⠳⠄⠈⣷⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠠⢧⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠿⠁⠀⠀⠀⢸⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⢠⠁⠈⣷
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⣴⢳⡄⠀⠀⠀⠀⠀⢻⡀⠀⠲⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢻⣆⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣴⣿⠏⠀⠀⠈⢻⣿⣦⡀⠀⠀⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣸⠇
⠀⠀⠀⣀⠠⠐⠈⣀⣠⣴⣾⣿⠿⠁⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣄⡈⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁
⠀⠀⣿⠁⠀⠐⣿⣿⣿⣿⣶⡆⠀⠀⢰⣶⣶⣶⡀⠀⠀⣶⣾⣿⣿⣿⣷⠀⠀⢸⣷⠀⠀⠀⠀⠀⣴⣦⣥⣀⣰⡶⢦
⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⣇⠀⠀⢀
⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠰⢿⣥⣤⣤⣬⡿⠀⠈⠦⣄⠤
⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣷⣶⣤⣴⣒⢀⡿⣠⣆⡀⠓⠀⠤⠬⣽⣿⣶⡶⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⣡⣴⣿⣿⣿⣿⠟⠁⠘⠻⢿⣦⣶⣤⣄⡀⠈⠙⠻⢞⡆⠀⢀⠀⠈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⣧⣀⠔⢉⣴⣾⣯⡿⠟⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣷⣦⣤⡀⠉⠢⣄⠁⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣿⡏⢁⣴⣿⣿⠟⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣄⠙⠍⢸⣇⠁⠀⣠⣷⢀⣀⣀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⢀⠾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠙⠒⢤⡐⢤⡀⣷⣷⡄⠀⠈⣿⣦⡀⠀⠄
⠀⠀⠈⠐⠀⠀⢀⣤⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠙⢗⣿⣭⡍⣴⣶⠀⣿⣿⣿⠀⠀⠆
⢀⠀⠀⠀⠀⢠⣦⣾⣾⣿⣿⠟⠁⢀⡀⣀⠄⠀⣠⡄⠀⠀⠠⠝⠛⢷⡶⠟⠫⢦⡀⠀⠀⢠⡄⠀⠀⠀⣬⣧⣘⣡⡹⢟⣠⣿⣿⣿⠀⠀⠒⡰
⠀⠁⠀⠀⢠⣿⣿⡿⣿⣿⠇⠀⢠⣼⣿⠛⠀⠛⣿⡁⠀⠀⠀⢠⡀⠐⠃⠀⡀⠀⠀⠀⠀⣸⣿⠛⠀⢺⣿⣇⢻⣿⣷⣦⣿⣿⣿⠁⠀⠀⢈⠀⠢
⠀⠀⠀⠀⠋⢡⠇⢱⣿⡏⠀⠀⢾⡁⣿⣄⣀⣠⡿⡷⠄⠀⢠⡎⠀⠀⠀⠀⢁⠀⠀⠀⢈⣟⣿⣦⣤⣾⡿⢻⡚⠿⠿⣿⣿⠋⢹⠦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⣼⠀⣿⣿⠇⠀⠀⠘⢷⣈⢉⠋⣉⡴⠟⠀⠀⠘⣇⠀⠀⠀⠀⡰⠀⠀⠀⠀⠻⣬⡙⠛⠉⢀⣾⡇⠀⡄⣿⣾⠀⠘⡇⠙⠶⠃
⠀⠀⠀⢀⣠⣿⠀⣿⣿⠀⠀⠀⠀⠀⠙⠛⠛⠛⢁⠄⠒⠀⠀⣜⠲⢦⡤⠂⡠⠚⠉⠉⠑⢆⠉⠻⠯⠒⠊⠋⣹⣶⠀⣽⣿⡆⠀⣧
⠀⠀⠀⢘⣿⣿⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢈⠀⠀⡇⢸⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠁⣆⠘⣾⣧⣶⣿⠇
⠀⠀⠀⠚⣿⣿⡆⠛⣶⠙⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢀⡸⠛⢻⡟⠛⣄⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠰⠈⢹⣿⡁⠀⠁⠃
⠀⠀⠀⠈⠈⢋⢷⠀⣯⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⠉⠀⠀⠸⠇⠀⠈⠙⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡰⠀⠀⣼⠋⠄
⠀⠀⠀⠀⠀⠸⣿⣆⠸⣢⣷⣿⡄⠀⠀⠀⢀⣠⢆⣀⠀⠀⠀⠀⠀⣺⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⠀⣰⡣
⠀⠀⠀⠀⠀⠀⠈⠈⢦⢹⣿⣿⣿⣶⣌⠀⠈⠱⠟⠉⠀⠀⠀⡀⠀⢹⠇⠀⠀⠀⠀⠆⠸⡿⠶⠀⠀⠀⢀⣾⣿⣿⠃⠀⢠⠋⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠀⣀⣸⡇⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣸⣿⡟⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡛⠛⠋⠀⠀⠀⢳⣄⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⣷⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠏⠉⠉⠛⠴⠀⣀⣀⡀⠀⠈⠻⢿⣷⣶⣶⣶⣿⣛⣲⣶⣿⣿⡿⣿⠍⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⢸⣿⠏⠽⠿⢯⣶⣖⣺⣿⣿⣿⡿⠿⠿⠻⢿⣿⣿⡯⠐⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣦⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡄
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⢄⣠⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠒⡀⠀⠀⢦⡴⠀⠀⢆⡐⠀⠀⢶⡆⠀⠀⠀⢀⠄⡀⠀⠀⢺⣟
⠀⠀⠀⠰⠆⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠟⠀⠀⠀⠀⠃⠀⠀⠊⠐⠀⠀⠉⠑⠚⠀⠀⠁⠀⠀⠀⠀⠘⠿
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⢤⠆⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴
⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠈⠀⠀⠀⠀⠓⠂⠀⠀⠨⠁⠀⠀⠀⠆⠀⠀⠂⠀⠁⠀⠀⠁⠐⠀⠀⠀⣾⣻
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⢠⡄⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠰⣀⠜⠀⠀⠚⠓⠀⠀⠀⡆⠀⠀⢈⡈⠀⠀⠀⠀⠹⠀⠀⢐⡊⠀⠀⠈⡍⠀⠀⠠⢀⡀⠂⠀⠨⠉⠛
⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣉⣹⣸⠀⠀⢈⠁⠀⠀⠀⡁⠀⠀⠀⣓⠀⠀⠀⢦⠴⠀⠀⠤⠤⠀⠀⢀⠀⠁⢀⠒⡆⠉⠒⡆⠀⣖⣱
⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⣤⣠⣀⠀⠀⢠⡦⠀⠀⠀⢠⡄⠀⠐⠀⠐⠀⠀⠀⣤⡀⠀⢠⡄⠀⠀⠰⠤⠀⠀⠀⢀⠆⠀⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠁⠁⠀⠀⠛⠀⠀⠀⠊⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠠⠄⠀⠀⠀⡤⢤⠀⠀⢀⡄⠀⠀⠀⣤⠀⠀⠀⡤⠤⠀⠀⠀⢀⠀⠀⣤⣦⡄⠀⠀⢠⣦⡀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠂⠀⠸⠹⠆⠀⠀⠀⠐⠀⠠⠔⠌⠁⠀⠻⠿⠃⠀⠀⠀⠹⠃⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠲⢄⠀⠀⢀⢀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠴⠂⠀⠀⠀⢀⠀⢀⠀⠀⠀⠠⢴⣿⡦⠄⠀⠀⢀⡤⣀⠀⠄⠀⠀⠀⢤⡈⡣
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠁⠼⠧⠐⠀⠀⠀⡠⠊⠦⡀⠀⠀⠀⢬⣉⡤⡄⠀⠀⠀⠨⠀⡍⠀⠀⠀⠀⠸⠭⠁⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠚⢽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠈⡤⠆⠀⠀⠀⢠⡀⣀⣐⡠⠀⠀⠘⠀⡼⠛⠋⠀⠀⠀⡤⡻⠣⠀⠀⠀⠐⢇⢐⢀⡀⠀⠀⠈⠀⠀⠦⡀⠀⠀⠠⠀⣀⣹⡄⠀⠀⠀⡁⠠⠗⢄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠃⠀⠀⠀⠀⠈⠉⠚⠃⠀⠀⠀⠁⠀⠐⠇⠀⠀⠀⠘⠀⠐⠀⠀⠀⠘⠊⠁⠈⠉

⠀⠀⠀⢀⣤⠤⠐⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠓⣶⠖⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠖⠀⠒⠀⠀⠀⢀⡀⣰⡄
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠄⠐⠠⡄⠀⠀⠀⢀⠐⠉⡁⠀⠀⠀⣀⣯⣄⡀⠀⠀⠀⠀⡐⠈⠂⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⣁⠂⠀⠀⠀⠈⣀⣡⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⡁⠀⢘⠀⠀⠀⠀⢦⠀⡵⠂⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⡴⠷⢶⠄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⠒⡔⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠋⠚⠊⠁⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⢤⡤⣄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠒⣤⠂⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⣤⠒⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠠⠷⡄
⠀⠀⠀⢀⡀⠋⠄⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⠸⠀⠆⠀⠀⠀⠀⢀⣵⡤⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠚⣒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡦⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣤⡰⠤⡁⠤
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⡀⠂⠀⠀⠀⠐⢀⡂⡀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠠⢺⠠⠀⠀⠀⠐⠹⠯⠌⠃⠀⠀⠀⠸⠄⠔⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠠⣀⠔⠁⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠒⠖⠀⠀⠀⠀⠀⠊⠀⠆⠀⠀⠐⢠⡉⡅⠋⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠰⠈⠈⢁⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠂⠉⠲⠀⠀⠀⠀⠈⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡞⠀⠀⢀⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀
⠀⠀⠀⣇⠀⣰⣋⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⠇⠀⠀⣹⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢦⠀⢸⠃⠀⠀⠀⢀⣤⠴⠖
⠀⠀⢠⢟⣉⣁⠉⠉⠛⣯⣉⠀⠉⠀⠀⠀⠀⠀⠀⠤⠤⠤⣤⠶⠘⠋⠉⠉⠉⠳⢤⡴⠃⠀⠀⠀⠀⠀⠀⠉⠉⢳⡄⣼⣤⣯⣄⣀⣀⣰⠏⣀⣀⣀⡀
⠀⢰⠃⡋⢀⣸⡇⣠⣤⣼⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⣰⢢⢄⢠⡀⠀⠀⠀⢀⡭⢝⢿⣭⣤⣀
⠀⢷⠀⠙⠛⠋⠰⣿⡀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡤⠠⣄⠀⠀⢀⣀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠱⡱⠌⡴⠃⠢⢤⠄⠸⣄⢣⠌⡷⠀⠈
⠀⠈⢇⠀⠤⠶⠶⢈⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢼⡁⠖⣸⠂⠀⣏⠌⡹⠀⢸⡇⠀⠀⠀⠀⠀⠀⣬⠋⠓⢲⠀⠀⢴⡐⠒⠒⣉
⠀⠀⣠⡇⠀⠀⠀⠙⠢⣴⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣍⡉⠀⢀⣀⠈⠉⠠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠏⠀⠀⠀⣧⡴⠚⠁
⣠⠟⠩⡇⠀⠀⠀⠀⠀⠈⢧⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢹
⠋⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢳⡀⣀⠴⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠛⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⣠⣴⠃
⠀⠀⠈⠳⣤⡄⠀⠠⠤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠸⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄
⠀⠀⠀⠐⢤⡀⠀⠀⠈⣷⣠⣧⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠖⠒⠤⣀⣹⠤⠒⠉⠀⢀⠤⣌⠳⡄⠀⠀⠀⠀⠀⠀⠒⠲⣄⣈⣧⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠲⣤⠴⠚⠋⠉⠉⠉⠓⠶⣏⠉⠃
⠐⠚⠒⢶⡟⢁⡶⣯⠙⡆⠻⣼⠞⢀⡟⠀⠀⠀⠀⠀⠀⡠⣲⡿⢧⡀⠀⣨⠭⡲⡀⠀⠀⠀⠀⠀⠀⡼⣡⠖⢦⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠈⣇⠈⠳⠶⠞⣡⣤⠆⢰⠟⠀⠀⠀⠀⠀⠀⢸⠃⣟⣡⠾⠁⠸⣛⣡⠇⢡⠀⠀⠀⠀⠀⠀⢯⠙⠛⠚⣴⢲⡄⣸⠶⢦⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡌⠳⡄⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⣤⠄⠈⠁⠀⢸⡄⠀⠀⠀⠀⠀⢸⠟⢶⣤⣽⣿⡴⠋⠀⠘⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡟⠀⠰⡄⠀⠀⠀⢠⠇⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⢠⠆⠀⠀⢻⠀⠀⠀⠀⠀⢘⠀⢸⠀⢸⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢸⠃⠀⠀⠙⠢⠤⠀⠚⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⠂⠈⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡄⠸⡄⠘⡄⠀⢠⠏⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣤⡤⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠉⠉⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠶⠶⠚⢯⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠚⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠘⢛⣢⣀⣀⣸⣀⣀⡴⠊⠙⠃⠀⠀⠀⠀⠀⠀⢀⡀⣴⢋⣉⡀⠀⠀⠉⠲⡖⠶
⠀⠀⠀⠀⢠⡿⠤⠼⠧⣤⡖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣱⠞⠙⣙⢧⢠⠖⢢⡈⠙⠲⡖⠒⠦⠀⠀⠀⠀⠀⢋⣤⡟⣎⣀⢹⠀⡴⢉⠱⡘⠶⠄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢶⡓⠢⠀⠀⠀⠀⠀⠀⣠⠖⣾⠁⠀⠓⠒⠃⠘⠢⠚⠁⣀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠙⠤⠼⠀⠻⢄⠴⢃⡆
⠀⠀⣼⠀⠀⢀⣀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⣁⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⣇⡠
⠀⣠⡏⠀⣴⠩⠮⢹⡄⠀⣞⡝⡮⣆⠀⠀⠀⠀⠀⠀⠀⣰⢧⠀⠀⠀⠀⠀⠀⣴⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⠀⠀⠉⠉⠉⠀⠀⠀⡏
⢰⠃⣇⠀⠈⠛⠚⣋⣀⣀⡙⠛⢁⡏⢳⠀⠀⠀⠀⠀⠠⠏⠈⢦⡀⠀⠀⠀⠀⢷⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠳⣤⣄⣀⣉⣉⣉⣥⣴⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡄⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⠀⣀⡴⠃
⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣭⣭⣭⣭⣭⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣤⠐⠂⠀⠀⣄⠤⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠈⠟⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠒⠴⠀⠀⠀⣀⠀⣒⣄⠀⠠⣤⣄⣀⠀⠀⠀⣭⣬⡀⠀⠀⣴⠀⢠⡀⠀⠀⣔⣶⠀⠀⠀⠀⣀⡄⠀⠀⠀⢀⣠⠄⠀⠀⠀⣠⣤⠀⠀⠀⢠⣤⣤
⠀⠀⠐⠀⠸⠂⠀⠈⠛⠛⠛⠛⠀⠀⠀⠉⠀⠀⠀⠀⣯⣿⠃⠀⠀⠈⠀⠋⠀⠀⠀⠛⠛⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠋⠛

⠀⠀⠀⣠⡶⠀⠀⠀⣄⣠⣤⡄⠀⠀⠚⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣶⣶⠀⠀⠀⣤⣦⣤⠀⠀⠀⠀⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠁⠉⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡶⡆⠀⠀⠀⠀⡔⠀⠀⠀⠀⠠⠞⠀⠀⠀⢤⠤⠂⠀⠀⣠⣤⣤⣤⠀⠀⢠⣶⡄⠀⠀⠀⡀⢀⡄⠀⠀⢀⣦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠛⠃
⠀⠀⠀⠀⡀
⠀⠀⠀⢸⢽⠃⠀⠀⢠⣤⣶⡄⠀⠀⢜⠉⣳⠀⠀⠀⡐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⡇⠀⠀⢠⣿⣷⡀⠀⣀⣃⣐⣠⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⠈⠁⠀⠀⠀⣿⣇⠀⠀⠀⠀⣿⠄⠀⠀⠠⣤⣤⠄⠀⠀⠀⢴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⣤⡴⠀⠀⠀⢰⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠛⠂⠀⠀⠀⣀⡀⠀⠀⠀⣠⣰⣶⠀⠀⠀⢸⡿⠀⠀⠀⠘⠶⠣⠀⠀⠀⢨⠹⠃⠀⠀⠀⠺⠇⠀⠀⠀⠺⢷⣶⣤⠀⠛⠃⠰⠅⠀⠀⠸⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⡄⠀⠀⠀⣤⡤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤
⠀⠀⠀⠿⠿⠗⠀⠀⢰⣿⣿⠀⠀⠀⢹⡴⠇⠀⠀⠀⡿⣟⠀⠀⠀⠐⢿⣆⠀⠀⠀⠾⠿⠆⠀⠐⠟⠟⠛⠻⠀⠠⠖⠴⠷⠀⠀⠼⠿⠷⠶⠂⠀⢸⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⣆⠀⠀⢸⠶⠿⠿⠆⠀⡷⠾⠿⠷⠀⢸⠶⠿⠿⠧⠀⣟⣻⣃⣻⡇⠀⠀⣾⣷⠀⢴⣷⡆⣴⣶⠀⣿⠀⠀⠀⠀⢀⣾⣫⣯⣫⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠋⠀⠙⠙⠁⠉⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡄⡀⠀⢠⣤⠀⢤⣤
⠀⠀⠸⠿⠇⠿⠇⠠⠿⠀⠸⠷⠄⠸⠆⠶⠰⠇⠀⠀⠀⠀⠀⠀⠀⠺⠂⠾⠂⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠂⠺⠆⠂

⡇⠈⠐⢃⠖⠉⠑⣆⠄⣡⠊⠉⢳⠀⠘⡚⠊⠛⠀⡀⠉⠀⢴⡲⠀⢃⣀⠀⠀⣄⠁⣠⡄⠂⣤⠃⠀⣠⡈
⠇⡼⠗⠸⣠⣶⣦⡘⠴⠃⣤⢶⣿⢶⡆⠀⠁⡜⠾⠀⠘⢀⠀⠀⠂⠘⠚⠀⠀⡏⠳⠁⢱⡎⢹⢞⠇⠓⠃
⠠⣠⣠⡀⢣⡉⡉⠁⠀⡀⠬⣹⣉⣉⣀⣀⣤⡀⠀⣠⣶⢻⠀⢾⡽⠀⡁⠠⣟⣷⢦⢀⡴⣶⣦⢸⠀⣔⣈
⠁⠻⠻⠋⠉⠳⣀⠀⠓⠁⢠⠋⡉⢉⠉⠉⠗⢉⠞⡵⢃⡎⠀⣀⣀⣀⣀⠀⠀⣯⢡⡤⠉⢩⠀⢸⠀⡉⠏
⠀⠀⠀⠀⠘⠠⠘⢦⡀⢀⠏⠀⠐⠀⠀⠀⠀⣜⡞⣡⠎⠀⠀⠉⠁⠁⠈⠀⠀⢸⠀⠀⠀⠀⠀⡇⠠⠀⢠
⣠⣇⣰⠃⡤⠀⠄⢀⠙⠊⠀⠎⠻⠉⠞⠁⠀⠙⠋⠁⣔⡦⠀⢼⣱⠀⠀⢀⡀⠀⢳⣄⣀⣀⣼⡆⠐⠠⡀
⣤⣤⢤⠽⢦⠀⠀⠂⠁⠀⢀⠴⠤⡄⢠⢤⡀⣀⣀⠂⠠⢀⢤⡀⠀⠀⠘⠙⢠⡆⠘⡍⣹⠇⢈⡁⣀⢠⣿
⢬⡩⠞⣠⠋⢰⢰⢀⡤⠒⠁⠀⠀⠘⣇⣀⠈⠀⢨⠆⠀⠫⠤⠃⠀⢸⠳⡄⠸⠇⡼⣰⣡⢞⡽⠃⠄⠀⠱
⠓⢒⠚⢁⠉⠀⠀⢸⠀⠀⠀⣤⣰⣏⠁⠈⣉⣳⡋⠉⠉⡦⠀⠀⠁⠀⠓⠛⢠⠞⡴⢃⣉⠥⢄⡀⠶⠲⠃
⠒⠈⠠⠈⣀⣀⡀⠘⢄⡀⠀⠙⡿⢚⣃⠘⠿⠃⡇⠀⠀⢧⠀⠀⠀⠀⡠⢞⡡⠊⠐⠛⠶⠒⠚⣡⢤⣶⢶
⡹⠀⠀⡈⠉⠉⣡⢴⠄⣨⠽⠛⠙⢮⣁⣈⣠⠞⠑⡖⠶⠋⠀⢀⠀⠀⠈⠉⠀⠄⠐⡀⠁⣴⣟⣛⡥⠖⠋
⠀⠀⠀⠄⠀⠸⠵⠊⠀⢧⡀⠀⠀⢡⠃⠀⠀⠀⢀⡸⠀⠂⠄⣠⣴⡋⠉⢉⡆⠀⣀⠁⠐⣀⡄⠀⣀⠀⠘
⠀⣀⠀⠀⠁⢠⠶⠖⠶⠂⠑⠤⣀⠼⣄⡠⠤⠤⠊⠀⠁⠀⠺⠛⠒⠒⠚⠉⣀⡀⡀⠀⡤⡀⠁⡼⠘⠟⠷
⢌⠋⠤⡀⠁⠀⠀⠀⢠⡴⡦⢶⠀⠂⠀⢠⢠⠀⡄⢠⠀⢰⢠⠈⠫⣍⠒⢤⠀⠀⠁⢸⠠⠈⠲⠁⠀⢀⡜
⠊⡤⡌⠁⠀⠠⠀⠀⢸⠀⠀⡏⣀⣡⠀⠈⠉⠉⠈⠁⠀⠈⢀⠀⠀⠈⠙⠊⠀⠡⠅⣼⢠⠺⢦⠀⣠⠊⡀
⠀⠓⢁⣀⡀⠐⢀⣀⢸⡀⠀⠉⠁⢸⠀⠀⣤⡀⠀⠀⡴⢳⡈⠀⡴⢷⠠⢀⣠⡤⠀⣿⠃⡰⡤⡙⠁⠄⠐
⠶⠔⣞⡀⢸⣰⠃⢸⠇⣇⡴⠻⡀⡏⠠⡀⠘⡎⢢⡼⠀⠀⢧⡞⠀⢸⡴⢋⡟⢀⡀
⣀⡘⢫⡿⠆⠁⢶⣺⣦⠉⢰⢠⢻⠃⠈⠁⡀⢹⡀⣥⣤⣤⣌⠀⣠⣮⣥⣜⠀⠙⠊⢀⠤⡀⢠⣊⠝⠁⣀
⣶⠟⠃⠳⡈⢋⡜⠀⠄⠀⠈⠀⡀⠀⡀⠤⡇⠀⣇⣙⣒⣛⣊⣦⣉⠚⡷⠚⠃⢴⣲⠈⠉⠀⠀⢠⣶⣯⣥
⡀⠀⡀⢀⠘⠋⠀⠂⠐⠀⠀⠀⠀⠂⠀⠀⢀⢀⠈⠀⠀⠀⠀⠠⠀⠉⣀⠀⠀⠀⡀⢠⠀⠀⠄⠠⠀⠉⢉

⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⢸⡿⢿⣿⠟⣩⣿⠀⠀⣼⡿⢿⡿⢋⣼⣧⠀⠀⠀⣀⠀⣠⡾⠃⠀⠀⠀⢀⣀⠀⣠⠞⠀⠄⠀⣄⡀⢀⣴⠟
⠀⠀⠀⢸⣿⣦⣠⣾⣿⣿⠀⠀⠹⣷⣄⣴⣿⣿⠏⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠁⠀⠙⠿⠟⠁
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠂
⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⢸⣿⣅⠙⢡⣿⣿⠀⠀⣼⣿⣌⠛⣡⣿⣧⠀⠀⠀⠘⢷⣴⠗⠀⠀⠀⢀⠀⠘⣦⣼⠃⠀⡄⠀⠘⢻⣶⡟⠃
⠀⠀⠀⢸⣿⣥⣶⣦⣽⣿⠀⠀⠹⣿⣥⣶⣬⣿⠏⠀⠀⠀⠰⠟⠙⠗⠀⠀⠀⠀⠀⠸⠋⠙⠆⠠⠁⠀⠰⠿⠉⠻⠆
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣤
⠀⠀⣤⣤⣤⣤⣤⠶⢶⣿⡿⠿⠿⠿⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣤⣤⣤⣄
⠀⠀⢸⡇⢀⡀⢀⡰⠋⣡⠄⣰⣷⢀⣴⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⢟⡷⣝⢦⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠉⣀⣀⣀⣀⣀⠈⢻⠏
⠀⠀⠸⣷⢸⣧⣾⠁⣼⣿⡾⠟⠁⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⢱⢏⡾⣡⡌⢿⢳⡄⠀⠀⠀⠀⠀⠀⠹⣷⡈⢻⠏⠉⠉⣿⡟
⠀⠀⠀⠹⣷⣿⠟⠀⣠⣶⠾⣿⣿⠛⣿⣯⢨⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿⠘⣧⣿⢃⠈⣇⢷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⢠⣾⡋
⠀⠀⠀⠀⠈⠛⠀⣴⠟⠋⠀⣿⡿⢀⣿⠏⢸⡏⠀⠀⠀⠀⠀⠀⠀⣧⢹⡀⢋⣿⢻⠀⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠰⠏⠀⡼⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⣧⡻⠞⡼⢣⢇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡳⢾⣡⣯⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠃⡄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠛⠿⣾⣷⣤⢠⣤⠄⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠄
⠀⠀⠀⢀⠀⢸⡆⢸⣇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢤⣤⡀⠀⣙⣿⣟⠁⠀⠀⠀⠀⠀⣸⣿⡟⣠⡄⠀⠀⢰⡄⢻⡿⣯
⠀⠀⠀⠀⢳⡾⠁⣛⣛⠊⢻⡾⠁⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⣾⡿⠟⣋⣵⠞⠋⢸⣿⡇⠀⠀⠀⠀⣰⠏⢸⡀⢿⣷⣦⣴⣿⡇⢸⡇⠸⣷
⠀⠀⠀⠀⠘⢃⣼⣿⣿⣿⣌⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣀⣬⡶⠟⣩⣴⣾⠀⠈⠉⠁⠀⠀⠀⠀⠻⣷⡜⢷⣄⠙⠛⠛⠋⣠⡾⣡⣼⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⡶⠿⣿⣯⡁⠀⠻⠿⠉⣉⣴⣤⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠙⣿⡇⢸⡟⠉⣼⠟⠁
⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠷⢶⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠰⠟⠀⠈⠳⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢛⠛⢛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠙⠁⠋⠀⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢾⣧⣤⣿⡿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠉⠀⠀⠀⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣷⣤⡀
⠀⢠⣴⣾⣿⣦⣶⣾⡿⠋⢿⣿⣿⣿⣷⣤⣴⠖⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢻⣇⠀⠈⠛⠄
⠀⠀⠙⠛⠉⠙⠟⢿⣶⣀⣴⣿⠛⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⣶⣦⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢸⣿⡄⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣴⣾⣿⣇⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢻⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢾⣿⣿⠙⢿⡟⢹⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⠹⣿⣿⣿⣷⣶⡶⠂⠀⠀⠀⠈⠳⣤⡀⠀⢀⣿⢀⡀⣿⡀⠀⢀⣴⠞
⠀⠀⠀⣠⣿⡉⠁⢠⣬⣤⠈⢉⣹⣧⡀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠘⠠⣿⣿⡟⠛⠋⠀⠀⠀⠀⠰⠶⢴⡿⠻⣷⡟⢁⣾⣷⡘⢷⣾⠛⣷⡤⠶
⠀⠀⠘⢿⣿⠟⠃⠸⠿⢿⠀⠛⢿⣿⠟⠀⠀⠀⠀⠸⣿⣦⣄⣠⣤⣤⣽⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢠⣿⣿⣿⣿⡈⣿⠀⣸⡀
⠀⠀⠀⠀⣻⣷⣶⠀⣴⡄⢰⣶⣿⠃⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠿⢿⣿⠦⠄⠀⠀⠀⠀⠉⠙⣿⡾⠏⢸⣿⣿⣿⣿⠇⠻⣶⣿⠋⠉
⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢻⡆⢰⠏⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠻⢦⡄⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠘⡏⢸⡁⠀⢀⣣
⠀⠀⠀⠀⠀⠀⢠⣾⡆⢀⡖⠠⠄⠀⡀⠀⠀⠁⠀⠀⣀⠘⢳⠀⠀⣄⠀⠠⠶⡴⠀⠀⢲⠂⠀⠐⡀⠨⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠃⠀⠀⠴⠀⠀⠀⠀⠀⢦⠹⠁⠈⠚⠣⠈⣃⣄⠀⠀⠀⠀⠘⠀⠀⠀⠱⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠁⠀⠀⠀⠀⡀⣏⠈⠁⠀⢀⠀⠂⠐⠍⣉⢀⠀⠀⠀⠀⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⠀⠉⠀⠂⠀⠒⠀⠀⡂⡄⠀⠙⣼⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡄⠠⡆⠀⠀⠼⠀⢀⠀⡀⢁⡄⠀⠘⣡⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠎⠀⠀⠒⡀⠀⡄⠀⠐⠀⠀⠀⢰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠐⠠⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠂⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠶⠀⠀⠀⠀⠀⠀⡠⠀⠀⢨⠀⠠⠈⢀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⠂⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⠀⠀⠀⠀⠀⡈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⣀⡀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠁⢠⡤⠦⠆⠨⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⠀⠀⠐⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⡀⠄⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⢄
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢄⡁⠀⠐⠀⠀⢄⡀⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⢀⠀⠰⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠘⠳⣄⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⣤⢁⡄⠀⠀⠀⣠⡶⠂⢠⡄⠁⠀⠩⠄⢠⡀⠀⠀⠀⠀⠲⣀⠨⣷⠀⠀⠀⢀⢄⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡅⢴⠄⠀⠀⢸⡃⠄⠹⢤⡀⠢⠀⠀⠀⠀⠀⠀⠀⠂⡜⠉⠁⢸⠆⠀⠀⣒⠋⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡄⠀⠨⣇⠀⠀⢤⠟⣡⠀⠀⠀⠀⠀⠀⠀⠜⠛⠂⠀⠜⢀⣴⠾⠁⠀⠀⠀⢠⠀⡀
⠀⡃⠀⠀⠀⢀⠈⠀⠀⠀⠀⡔⠹⢃⠀⠈⠠⠤⠔⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠴⠈⢠⡄⠁
⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⠀⢀⠠⠔⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠭⠐⠀⠋⠒⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠉⠈⢉⡀⠀⢀⠁⠀⠀⠄⠠⠀⡄⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠠⠢⠀⠊⡠⠂⠀⠀⢁⠈⠀⠀⠴⠀⠀⠁
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠰⠉⠀⠀⠀⠄⠀⠐⠀⠀⠈
⠀⣾

⠯⠭⠼⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠁⠀⠒⠊⠁⠀⠀⠀⠀⠀⠂⠒⠒⠀⠓⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠠⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⣒⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⣀⠀⠈⢂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢂⠀⠀⡀⠀⠀⠀⠀⢀⠎⠀⠢⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠤⠤⠤⠤⠊⠀⠀⠀⠀⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠠⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠁⠁⠀⢈⠀⠀⠀⠀⠀⠠⠄⢀⠄⠈⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠄⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠜
⠀⠀⠂⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠒⠒⠒⠀⡠⠀⠁⠀⠀⠣⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣂⣀⣉⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣈⣓⣒⣒⣀
⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠶⠶⠶⣤⣀⣴⡇⢸⣶⣀⣤⠶⠶⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠶⢭⣷⣆⠰⢦⣍⣸⡃⢸⣇⣩⡴⠆⣰⣾⡭⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⢦⣄⠒⡍⠉⠒⠒⢦⡉⠃⢸⡇⠙⣇⣸⠋⢸⡇⠘⢉⡤⠒⠒⠉⢡⠒⣠⡴⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠹⡆⠘⠀⠀⢀⡏⠀⢀⣘⣓⣦⡌⢡⣴⣚⣃⡀⠀⢹⡀⠀⠀⠃⢰⠏⠀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠘⢧⣸⠇⠠⠀⠀⠸⠗⢋⣩⡭⠭⢭⣭⣭⡭⠭⢭⣍⡙⠺⠇⠀⠀⠄⠸⣇⡼⠃⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣈⠀⢀⡀⠁⠀⠀⠀⢀⣀⣰⣯⠀⠀⢀⡾⠁⠈⢷⡀⠀⠀⣽⣆⣀⡀⠀⠀⠀⠈⢀⡀⠀⣁⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣡⠞⣻⠁⢀⣠⠶⣚⣩⣭⠄⠀⠀⠀⠺⡅⠀⠀⢨⠗⠀⠀⠀⠠⣭⣍⣓⠶⣄⡀⠈⣟⠳⣌⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣡⣆⢸⡴⢋⡴⠚⢩⠟⠁⠀⠀⠀⠀⠀⢹⡀⢀⡏⠀⠀⠀⠀⠀⠈⠻⡍⠛⢦⡙⢦⡏⣰⣎⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣼⠃⢹⡌⢡⡟⠁⠀⡏⠀⢠⣤⣤⣄⡀⠀⢈⡇⢸⡁⠀⢀⣠⣤⣤⡄⠀⢹⠀⠈⢻⡌⢡⡏⠘⣧⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠻⡟⠀⠀⠀⢿⠀⠀⠷⣌⠻⣷⣄⠈⠃⠘⠁⣠⣾⠟⣡⠾⠀⠀⣿⠀⠀⠀⢻⠟⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⠤⠤⣤⠀⠀⠀⡀⠘⢧⠀⠉⠒⠲⡌⣿⠇⠀⠀⠰⣿⢡⠖⠒⠉⠀⡼⠃⢀⠀⠀⠀⣤⠤⠄⣀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣥⢶⠀⡟⢰⡤⠤⠟⠀⢸⡆⠀⠀⠀⢱⠏⠀⠀⠀⠀⠹⡆⠀⠀⠀⢰⡇⠀⠻⠤⢤⡆⢹⠀⡶⣬⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡆⣇⠈⣇⠀⠀⠀⠘⢧⡀⢧⠀⣹⣠⣠⣄⣠⣤⣀⣇⠀⡜⢀⡼⠃⠀⠀⠀⣸⠁⣸⢰⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡌⣧⠸⡀⠈⠢⣄⣀⡀⠈⠳⣤⠇⢣⡁⠹⣄⣰⠏⢈⡜⠸⣤⠞⠁⢀⣀⣠⠔⠁⢀⠇⣼⢡⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⣆⠳⣀⠤⢤⠀⠀⠀⠀⣿⠪⣐⠀⠀⢸⡇⠀⠀⣂⠑⣿⠀⠀⠀⠀⡤⢤⣀⠞⣰⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⣠⣥⠀⠘⣆⢠⠀⠀⠘⢶⣶⣶⠶⠋⠙⠶⣶⣶⡶⠋⠀⠀⡀⣰⠃⠀⡬⣄⡉⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⡰⢋⣤⢸⡇⠀⠘⢶⡀⠀⠀⢸⣷⠂⠀⠀⠀⠀⠐⣾⡇⠀⠀⠀⡶⠃⠀⢰⡇⣤⡙⢆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡏⢸⢸⡇⠀⠀⠀⠀⠠⡄⠈⢿⣶⣄⣠⣀⣠⣶⡿⠁⢠⠄⠀⠀⠀⠀⢸⡇⡇⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡞⣼⠁⣠⡴⢶⡆⢸⣁⣀⠀⠟⠻⣿⣿⠟⠻⠀⣀⣈⡇⢰⡶⢦⣄⠘⣧⢳⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣏⢸⡏⠀⢸⡇⣨⡄⢸⠀⠀⠀⠘⠃⠀⠀⠀⡇⢨⣅⢸⡇⠀⢹⡇⣹⠎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⡇⠀⠘⣿⠁⣿⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃⣿⠈⣿⠃⠀⢸⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠹⠀⠸⣆⠙⠇⠀⠀⠀⠀⠸⠋⣰⠇⠀⠟⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⣆⠀⡴⠋⠙⢦⠀⣰⠟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠀⣼⠞⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⡀⠀⠀⠀⠀⠀⠠⣶⣶⠀⠀⠀⠀⣼⠟⠻⣷⡀⠀⠀⢠⠤⢤⣤⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⠀⣶⣴⠂⢰⡆⠀⠀⠀⣀⡀⠈
⠀⣼⣿⣣⣦⣤⠀⠀⠀⠐⠟⠟⠀⠀⠀⠀⠻⣷⣴⠿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠐⣿⣽⣿⡇⠀⠀⠀⠤⠀⠠⠶⠀⠀⠀⣿⣻⣇⣹⡇⠀⠀⠸⣿⣿⡰⣙



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠂⠀⠀⠐⠂⠀⠀⠀⣀⣀
⠀⠀⠐⣣⣄⠀⠀⠀⢲⠂⠀⠀⢰⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡾⢿⣿⢶⠀⠀⠀⣾⡟⣭⢹⡄⠀⠀⢰⠒⠒
⠀⠀⠀⠉⠉⠀⠀⠀⠙⠁⠀⠀⠘⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠈⠀⠀⠐⠁


⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣯⣹⡇⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⢰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣼⣯⣽⠀⠀⠀⣠⣴⣿⢼⡇⠀⠀⣤⣤⣤⣀⣀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⣨⣿⡧⣶⠆⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⡿⠿⢿⡇



⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⢠⡤⠀⢤⡀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⢀⢀
⠀⢸⡇⠤⢘⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣶⠀⠀⣴⠀⠀⠀⠀⣴⣉⠃⠀⠀⠀⢸⠷⠤⠾⣧⠀⠀⠀⣼⢂⡐⡇⠀⠀⠀⠠⣏⣹⡇⠀⠀⠀⠐⣷⡄
⠀⠘⠿⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠘⠄⠤⠄⠏⠀⠀⠀⠹⠦⠴⠇⠀⠀⠀⠠⠿⠿⠦⠀⠀⠀⠀⠿⠉



⠀⢠⠦⠤⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⣰⣶⢶⣆⣀⠀⠀⠀⢀⡄
⠀⢃⠁⢃⠈⠷⠀⠀⠀⠐⣶⣧⡾⠂⠀⠀⢰⣼⣿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠟⣯⣭⣯⡇⠀⠀⠸⣄⠀⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠘⠃⠀⠙⠁⠀⠀⠀⠀⠒⠒⠃⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠘⠛⠚


⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣴⡋⢈⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⣏⡛⢿⣷⠀⠀⠀⢀⣄⣀⡰⠆⠀⠀⢐⣶⣶⡆
⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠈⢱⣮⠁⠀⠀⠀⢀⣐⣿⣷⠀⠀⠀⢸⠳⣌⣆⡿⠀⠀⠰⣿⣿⣿⡥⠀⠀⠀⠐⠿⠿⠃⠀⠀⠀⣿⣿⣿⣷⣶
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠉⠛⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠁⠀⠉


⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣯⣽⣭⣯⣵⣭⣏⣿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣿⣿⣿⣿⣯⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣰⡟⢀⣤⠄
⠀⠀⠀⠀⣰⠞⣿⠛⠲⡄⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⢠⣄⣀⣀⣤⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⡟⡁⠞⠥⣐⠺⠛⠛
⠀⠀⠀⢰⡇⣀⣿⣄⡀⢹⠀⠀⠀⠀⠀⠀⢁⠂⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⢎⡁⠐⢶⣄⣙⠛⠂
⠀⠀⠀⠀⠻⣍⣸⣀⣹⠏⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢀⣀⡆⠀⣀⣸⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠻⠷⠴⠄⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠈⠛⠋⠀⠛⠛⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠀⠀⠀⣠⣤⣄⢀⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠀⠀⣤⣾⡟⣟⢿⡿⠯⣏⠿⣷⣤⠄
⠀⠀⠀⢠⠮⠅⠡⠄⡉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⣹⠏⡠⠊⣰⣯⣤⠖⠋
⠀⠀⠀⢰⡂⣀⠐⢀⣒⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡔⠁⠐⠁⡀⠜⠉
⠀⠀⠀⠈⠉⢹⣟⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡐
⠀⠀⠀⠀⠀⡿⠛⡟⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠤⠴⠚⠻⢿⠧⠄⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠊
⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⠀⣈⡉⢐⡀⠀⢀⣐⠠⠤⢔⣂⡀⠂⠀⡐⢒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣠
⠀⠀⠀⠀⠀⢣⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣾⣿⣯⣿⠽⢷⣆
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠉⡱⠀⠀⠀⠀⠀⣠⠔⠲⠀⠀⠀⠀⠙⢤⣇⣙⣷⣆⡠⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣤⠒⠉⠀⠀⢀⠠⠀⠀⠀⢀⡴⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠋⢠⢯⠄⢁⡠⠤⣰⠁⢠⣯⠈⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠃⡌⢀⠀⠀⠀⠀⠀⠀⠢⡐⠋⣀⠂⠀⠀⠘⢄⡐⠃⠠⠀⠀⠀⠀⠀⠀⡗⠚⠓⠇⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠔⠖⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠃
⠀⠀⠀⠀⠀⡊⠤⠀⠠⣶⣶⠆⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠇⠠⣶⡶⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠐⢤⡀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠞⠁
⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⣠⣶⡆⢀⣴⢾⡇⢀⣤⠶⢶⡆⠀⠀⠀⢀⠚⠀⠸⠁⠌⢢⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠠⠤⠴⠵⢋⠠⠄
⠀⠀⠀⠀⢰⡟⠉⠀⠹⣶⠟⠹⠷⠟⠁⠘⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⡈⠃⠚⠐⠂⣺⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠄⠤⢴⣅⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠶⠦⠘⠁⠀⠀⠀⠀⠀⠀⠀⠷⠷⠿⠶⠶⠖⠁⠻⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣧⣤⡀⠀⠘⣷⣦⣀⣾⣧⠀⢀⣿⣆⣠⣶⡞⠀⠀⣀⣤⣿⣇⠁
⠀⠀⠀⠀⠀⠀⢀⠖⠿⠀⠀⢀⣀⠀⣾⠁⢶⣿⠇⠀⠀⠙⣿⣿⣿⣿⡄⣼⣿⣿⣿⡟⠁⠀⠀⢿⣷⠆⢹⡇⢀⣀⠀⠀⠸⠗⢆
⠀⠀⠀⡀⠀⢸⡟⠀⠀⠀⠀⠈⣻⣿⣿⡀⠀⠀⠀⠀⠀⠸⠟⠛⢿⣟⠀⢘⣿⡟⠛⠷⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠀⠀⠀⠀⠘⣿⠀⠀⡀
⠀⠀⠈⣿⣷⣼⣷⠀⣴⡶⢄⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⢀⠴⣶⡄⢰⣿⣴⣿⡟
⠀⠀⠀⠈⠻⣿⣿⣆⠉⠁⡼⡗⠲⣌⠉⣿⣿⣿⣶⣤⣀⠀⠀⢠⣿⡿⠀⠸⣿⣧⠀⠀⠀⣠⣴⣾⣿⣿⡏⢉⡴⠲⡾⡄⠉⢁⣾⣿⡿⠋
⠀⠀⠀⠀⢀⣴⣿⣿⣷⣼⣅⣿⣈⡛⠀⠛⠉⠉⠉⠉⠛⢷⣶⣿⠿⠁⠀⠀⠹⢿⣷⣶⠟⠋⠉⠉⠉⠙⠃⠘⣋⣸⣇⣿⣴⣿⣿⣿⣄
⠀⠀⠀⠀⠘⣿⡿⠛⠛⡿⢿⣾⣧⣭⣓⣦⣀⣀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⢀⣀⣠⣖⣫⣥⣿⣾⠿⡟⠛⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⡌⠛⠻⠿⠿⠿⠿⠿⠿⢶⣦⣄⠀⢻⡄⠀⠀⣼⠃⢀⣤⣶⠾⠿⠿⠿⠿⠿⠿⠛⠋⣴⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣄⠙⢷⡄⢣⠀⢠⠇⣴⠟⢁⣠⣴⣶⣶⣶⣶⣶⣶⣤⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⣾⣿⡏⠁⠀⠀⣾⡿⠿⠿⠿⣿⣦⣿⣸⡄⡼⣸⣣⣾⡿⠿⠿⠿⣿⡆⠀⠀⠉⣿⣿⣆⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠛⠿⠶⠀⠀⠙⠃⠀⠀⠀⠈⢿⣿⣿⠇⣿⣿⡿⠋⠀⠀⠀⠀⠛⠁⠀⠰⠾⠟⠋⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⣿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠘⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠚⠋⡀⠀⣀⠀⠀⠀⣀⣀⣀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠤⢀⠀⠀⠀⡀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠐⠃⠀⠀⠀⠈⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠈⢀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠢⠤⠀⠤⠔⠊⠑⡄⠀⠀⠀⠐⠼⠃⠀⠀⠀⢠⡀⠀⠐⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠓⢲⡀⠀⠀⠀⢼⡇⡀⠀⠀⠀⠤⠔⢀⣼⡦⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣩⣠⠄⠀⠀⢨⣀⠀⠀⢚⡀⠀⣴⣤⡄⠔⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⠀⠀⠀⠀⠀⢰⠁
⠀⠀⢀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠈⡆⢰⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡆⠈⡉⠑⡄⠀⠁⠒⠀⠀⠀⢈⠒⠿
⠀⠀⡜⠀⠀⠀⠀⠈⠀⠀⠀⢘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡆⠀⠈⠉⢹⡄⠛⠲⠂⠀⠠⠐⠦
⢴⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠀⣀⡀⠀⠀⠈⠓⠃
⠈⠋⠁⠀⠀⢰⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣦⡀⠀⠈⠀⠀⡰⠁
⠀⠀⠀⣠⣄⣼⠀⠀⠈⠁⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⠄⠀⠀⠀⠈⠫⠀⠀⢀⠌⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠁⠀⠀⠈⠄⠀⠀⠀⠒⠃⡀⠇⠀⠀⠀⠀⡀⠀⠀⠀⠘⠐⡀⠑⣠⠆⠀⢳⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠁⢀⠠⠀⠀⢈⡆⠀⠀⠀⠀⠛⣲⡄⠀⠀⠀⠀⠀⠀⢰⣗⠈⣤⣀⡨⡇⠀⣠
⠀⢸⠀⠀⠀⠀⠐⠐⢶⠆⠀⢀⣶⠂⠐⠂⠁⠘⡀⠘⣄⠀⠀⠀⠈⠰⠂⠀⠀⢒⠀⠁⠀⠀⠷⠤⠀⠸⠞⠁
⣀⣈⣶⣊⣁⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣈⣙⣒⣚⣀⣀⣒⣒⣚⣀⣀⣀⣚⣒⣒⣂⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⣠⡀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⡐⢱⠑⡀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡜⠀⢸⠀⠘⡄⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡼⢲⠶⢺⠲⢶⠺⣄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠸⠁⠀⢈⠀⠀⠹⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠃⠀⢰⠀⠀⢸⠀⠀⣰⠀⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠧⠤⠤⠤⠵⠦⢼⠤⠶⠥⠤⠤⠤⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠬⠤⠤⠐⠊⠁


⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠈⠿⠃⠀⢰⣿⣿⡆⠀⠀⠀⠀⢀⠀⠀⠘⡆⠀⢀⡀⠐⣶⡂⠀⢴⡆⠀⠀⠶⣾⣿⡤⠿⣷⢶⠀⠛⢃⡘⠁⠀⠠⢽⣿⣿⣿⣿
⠀⠀⠀⢴⣶⣶⣶⣶⣿⡿⢟⠋⠀⠀⢀⢶⠞⡷⣦⡈⠀⡤⠀⢀⣴⣾⣿⡇⠸⣀⣿⣿⣸⢟⣼⣿⠀⢀⣾⠃⠉⡆⢨⢸⣿⣿⣿⣿
⠀⢀⡤⠄⠙⢒⠒⠲⢲⡚⣋⠀⠀⠤⢼⢹⠓⣓⣷⣘⣄⡀⠀⠈⠉⠙⠏⠀⣀⣉⠉⠉⠉⠉⠉⠉⢠⣿⡏⣠⠞⠀⠐⠸⠿⣿⣿⡿
⠀⠀⢀⡜⢁⡎⠰⠂⣀⢳⡈⠀⠀⠀⡈⠺⣤⣄⣲⠟⠀⠀⠀⠀⢀⣤⣦⠀⠙⠟⢀⣤⣤⠀⠀⢠⣿⡟⠘⠁⣀⠐⠶⠀⢀⣰⣦⣀⡀
⠀⠀⠀⠀⡾⠀⠀⠘⠛⠀⢧⠀⠀⣀⣀⣀⡀⢢⡀⠀⠡⣤⣴⠂⣿⣶⣝⡷⢠⢖⣻⣿⣿⠀⢠⣿⠟⢀⡀⠀⢛⣀⣠⣦⠈⠉⡉⠩⠁
⠀⠀⠀⢰⠃⢸⣿⡿⠃⠀⠈⠇⠀⠿⣿⣿⣿⠾⠓⠒⠀⠉⠉⠀⣿⣿⣿⣧⣿⣿⣿⣿⣿⠀⡾⠋⠀⠸⣿⣿⣿⣿⣿⣇⠀⠀⢠⣤
⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢠⣦⣤⣦⣄⡈⠁⠀⠀⢀⣠⢾⣱⠀⣿⣿⣿⢸⣿⣿⣿⣿⡇⠀⠁⠀⠀⢸⣿⡏⠉⠉⣻⣿⠆⢠⣿⠁
⠀⠀⡰⠁⠆⣻⣿⣷⡄⠀⣿⠀⠀⠀⠉⡇⠀⣠⣞⡍⠂⠊⠁⠘⠿⣿⣿⣸⣿⣿⣿⠿⠃⠠⣾⣿⡶⠀⠙⣿⠿⠿⢿⡏⠀⣾⡏⠘⠟
⠀⠀⢧⣠⢾⡵⣿⣿⡇⠀⡿⠀⠀⠀⢸⣠⡻⠞⠉⠀⣠⣾⣿⣷⣆⠈⠙⠙⠛⠋⠁⣄⡀⢀⣉⠋⢀⠞⣦⠉⠀⠀⠀⠀⢰⣿⠀⣤
⠀⠀⠀⠙⠙⠛⠛⠉⠀⠀⣷⣤⣤⣤⣼⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣦⣤⣤⣦⣤⡀⠙⠁⠑⢄⢈⣿⠶⣚⡃⢈⡆⠀⢀⣿⠇⣸⠃
⠀⠀⡤⡁⡄⢤⣃⠀⠀⠀⠀⡉⠉⠉⠉⠀⢰⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠀⠀⣠⠜⢻⢅⣓⣨⠷⣉⠀⠀⣼⡟⢠⣬
⠀⠀⣶⣶⣾⡿⠛⣇⠀⠀⣴⠿⠻⢧⡤⠀⠈⣿⣧⡘⠃⣸⡏⢠⡄⢻⣼⡇⠰⣿⣿⠀⠸⡀⠤⠜⡞⠉⢲⢄⣘⡅⢸⡿⠁⣿⣿
⠀⠀⢻⠁⠀⠀⣀⣿⠀⠘⣿⣄⢀⣸⡟⠀⣴⣿⣿⠃⣀⠻⣇⠀⠀⣼⣿⣿⠆⠘⣿⣷⣄⠀⠀⠀⠱⣀⠎⠀⠈⠀⠼⠀⣠⡶⣶⡀
⠀⠀⠘⠿⠾⠛⠛⠁⠀⠀⢘⡙⢛⡋⠀⢸⣿⣿⣿⡀⠉⣠⣿⣿⣿⣿⣟⢁⣠⣾⣿⣿⡿⠀⢀⣠⣴⣶⣶⣶⣶⣶⡄⢠⣿⡟⣿⡇
⠀⠀⠀⢀⣤⣤⣤⣀⠀⢠⣿⠇⢸⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⢠⣾⡿⣿⣍⣭⣵⣿⡿⠟⠁⠀⢿⣿⣿⠇
⠀⠀⣰⠏⣅⠉⢻⣿⣧⠉⠈⠀⠈⠏⠙⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⡟⠁⣠⣴⣶⣦⣄⠈⠛⠃⠙⠛⠛⠋⠉⠀⢀⡤⢤⡈⠋⠁
⠀⠀⣿⣄⠛⠀⣰⣿⣿⠃⢠⣤⢀⡀⠐⡶⠄⠀⢰⣶⡎⠻⢿⠿⠋⠀⢰⣿⣿⣿⣿⣿⡆⠀⢠⣰⣀⣦⣾⡾⠃⠈⠛⠛⠋⠻⠭⠷
⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⢠⣤⠘⠋⢁⣦⡄⠀⢸⣿⠃⠀⢀⠤⡄⠤⡈⢿⣿⣿⣿⡿⠁⢲⣾⣿⣿⣿⣽⠖⠀⠸⠿⠃⣰⣶⡆
⠀⠀⢶⠆⣀⣀⣀⠀⠀⠟⠆⣛⣸⣟⣊⠉⠁⠀⢸⡏⠀⡼⠓⠢⣷⢔⣛⡄⠻⡟⠉⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⣠⡤⣾⣿⣿⣿⣤⡄
⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⠀⠀⣀⣼⣇⠀⢷⣈⠍⡿⢧⡠⠀⠀⡷⠐⣷⣿⠀⠀⢰⡔⠉⢇⡀⠀⣿⠀⠙⠛⠛⠛⠀⡇
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⡏⠀⢸⡿⠛⢿⡇⠀⠑⠺⠥⠒⠁⠀⢰⠁⠀⠀⠉⠀⣰⡶⠶⠤⣽⡃⠀⣿⠀⠀⠀⠀⠀⢠⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠈⠉⠀⣴⣿⣇⣀⣼⣧⡀⢠⣤⡤⢤⣀⣀⣠⠶⠶⣶⠀⠀⣿⠀⠸⡀⢘⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠃⢤⡄⠹⣿⣛⣛⣿⡿⠃⠈⢯⡀⢀⣼⠉⠳⣄⣠⠟⠀⠀⠿⣦⣤⣤⣾⠃⠀⣿⣤⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⣀⢀⡀⠀⠀⡐⢉⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠹⠀⠀⠀⠀⠙⢌⡣⠊⠁⠀⠀⠀⡎⠢⠐⠀⠀⢀⡀
⣶⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠐⠀⢃⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠑
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⢀⡀⠀⠀⢀⡴⣚⡂⣄⠀⠀⠀⢤⡀⠀⠀⢰⣄⠀⠀⣠⠦⢤⡀⠀⠀⠀⠀⠠⠄⠀⣠⠷⠀⠀⣀⡀
⠀⠰⠋⠀⠀⠀⠙⠮⠚⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠈⢢⠀⠁⠈⠘⠀⠀⠠⢤⠀⡦⠀⠀⠀⠴⠋
⠐⠪⠅⠀⠀⠆⠀⠠⠆⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠣⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠺⡒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⡴⢦⣀
⠀⠀⢠⣤⣤⣤⣄⣄⣤⠀⠀⠀⢠⢳⢀⠚⣀⠞⡄⠉⢁⠉⠀⠀⣮⣠⣤⣤⠥⣵⡀⠀⢀⣀⡀⣀⣀⣀⢀
⠀⠀⠈⠀⠀⠀⢠⣥⣤⠀⠀⠀⠛⠒⠛⠐⠛⠒⠃⠀⠀⠀⠀⠀⠄⠀⠄⠴⠠⠠⠀⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣄⣀⣀⡀⠀⠀⢀⢀⡀⡀
⢄⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣾⣷⡇⠀⠀⠀⠀⢆⣦⡆⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠋⠀⠀⠐⠩⠀⠬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠱⠃
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢚⡱⡇⡓⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠊⡇⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⠈⠑⠊⠃⠀⠀⠌⢱⠆⢡⠀⠀⠉⠙⠙⠋⠉⠀⠀⢠⣏⣧⠀⠀⠀⠛⢣
⠵⠇⠀⠐⠀⠁⠂⠒⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠒⠐⡁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠓⠒⢒⠃⠀⠀⠰⠢
⠀⠀⠀⠀⠀⣈⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠄
⣉⡉⠉⠀⠀⠀⠀⠀⣠⠀⠀⠘⣀⡛⣤⠃⠀⠀⣶⣛⣶⠀⠀⢀⣀⢘⡓⣄⡀⠀⢀⡼⣻⠛⠀⠀⠀⠉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡀
⠀⠀⠀⢀⣀⠤⠄⠜⠀⠘⠧⠠⠤⣤⠀⠀⠀⠀⠀⠀⣠⣤⢀⠕⢓⡦⢀⣄⣀⣀⡀⠀⠀⠀⠠⣤⣤⢤⣼⠧⠼⣷⣤⣤⣤
⠀⠀⠀⠈⠙⠢⢀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠠⠠⢔⠭⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠂⣴⡇⠀⠀⣸⣿⠋⠁
⠀⠀⠀⠀⠀⢀⡏⣠⠄⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⡡⠬⣥⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⡶⢿⡉⢿⡄
⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠁⠸⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⢥⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠉⠀⠀⠉⠻⣧


⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢢⡀
⠀⠀⠀⠶⠶⠖⠒⣾⠷⠒⢾⡖⠀⠀⠀⠀⠀⠀⠀⢶⣴⣾⣿⣿⣿⣿⣿⣿⣷⡶⠂⠀⠀⠀⠘⢫⡉⠁⠀⠀⠀⠀⠀⠉⡩⠃
⠀⠀⠀⠀⠀⠢⣴⡇⠀⠀⢀⣽⠂⠁⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⢠⠏⠀⡲⢶⣁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠙⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⠤⡀⠈
⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠈⠓⠼⠄⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠈⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⣿⣫⣦⣀⣀⣀
⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠈⠉⢉⡴⠋⠀⠀⠀⠀⠻⢧⣉⠉⠁⠀⠈⠉⢉⠭⠛⠁⠀⠀⠀⠀⠲⢿⠉⣫⣾⣿⣿⢷⡿⠋⠁
⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣿⣟⣵⣿
⠀⠀⠀⠀⠀⢸⠀⡠⠀⠑⠄⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⠐⠲⣄⣳⡀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠙⠁⠉⠻⢟⡀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡤⢤⡄
⠀⠀⠠⠀⠩⠁⠀⠀⠀⠀⠰⠄⠐⠆⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⢺⠔⠀⠈⠀⠀⠀⠂⠠⠈⠀⠐⠀⡀⠀⡀
⠀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⠀⢀
⠀⠀⢰⡆⣶⠀⠁⠀⠀⠀⠀⠐⠄⠀⠲⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠃
⠀⠀⠘⠇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⢀⣤⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠠⡰⣤⠀⠀⠀⠐⡿⠟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⢩⣥⣄⣀⣤⣤⣤⣤⣤⣬⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠁⣶⠠⠀⠐⠃⠀⠄⠀⠐⠂⠀⠀⠀⠀⠸⡆⠛⠁⠀⠐⣶⢠⣦⡀⣰⠁⠀⠂
⠀⢀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣽⣤⣽⣥⡏⠀⠀⠂
⠀⠀⠀⠀⠈⡄⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠈⠀⠐⠛⠃⠀⠉⠉⠀⠀⠀⠠⠀⠀⠉⠉⠿⠈⠀⠀⠘⠿⠉⠁⠀⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠐⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⣰⣶⣦⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠄⠀⢀⡇⠀⠀⠰⠀⠠⠄⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⡀⠨⠤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠁⡘⠷⠾⡂⠀⠀⣀⡀
⠀⠀⣄⠀⢀⠀⡀⠀⡀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠐⠀⠃⠀⠾⠀⠐⡌⠀⡝⠀⠀⠀⠀⠀⠀⠀⠤⡤⠦⣤⡀⠀⣤⠀⠀⢀⠐⢠⣍⠀⢀⣠
⠀⠀⠀⠀⠷⠐⠿⠀⠀⠀⠀⠑⢤⠄⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⢀⠆⠒⠂⠀⠛⠓⠀⠛⠀⣠⣏⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠀⢉⠉⠀⠀⢀⣁⠉⣀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠠⣤⣤⣤⣤⣤⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠂⠒⠆⠀⠀⠀⠀⠀⢠⡄⢠⡤⠀⠀⠀⠀⠀⢀⠀⠶⠟⠁⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠁⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠔⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⢄⠀⠀⠀⠀⠁⠀⠀⠀⠐⡐⢀⠀⡀⠀⠀⢀⣀⠀⠀⢁⣢⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠀⠀⠀⠈⠙⠛⠀⠀⠘⠋⠀⠈⠀⡀
⠀⠂⠀⠀⠀⠐⠀⠀⠂⠀⠁⠂⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣄⣿⣠⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣠⠄⠀⠀⠀⠀⠀⠀⢸⠡⠈⡇⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠄
⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠦⣞⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣘⠿⣃⠀⠀⠀⠀⠀⠀⠀⢨⣤⣶⣾⠀⠀⠀⠀⠀⠀⠀⣶⣾⢤⡄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠃⠀⠀⠀⠀⠀⠀⠈⠑⠙⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢏⡴⠒⠒⠒⠛⢋⣩⣽⣿⣿⣿⣷⣶⣤⣶⣎⢻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢋⡞⠀⠀⠀⠀⠐⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣰⣿⣿⣿⣿⣿⡿⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠉⠛⢿⣿⣿⡟⣱⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠙⢿⣿⣿⣿⣦⠻⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠙⠀⠈⢹⣶⣿⡿⢀⣤⣄⠀⠀⠀⠀⠀⣄⡀⠈⠙⢦⣄⠀⠉⣿⠟⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⣾⣿⡿⣋⣤⡄⠀⢀⡀⠀⠙⠲⣄⠀⠈⢳⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⣾⣿⠟⣣⣶⡄⠙⠲⣄⡀⠈⢻⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⠿⠟⠛⢦⣀⣈⡟⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




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

⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢄⡀⠀⠀⠀⠀⢀⡀⣠⣤⣠⣀⡀⠀⠀⠀⢀⣤⠤⢄⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⢁⣘⡀⠰⠀⠠⠀⡰⢋⠂⠀⠀⠀⢆⡀⠀⠀⠸⠑⠀⠀⢠⡆⠁⠀⠀⠀⢠⢠⢉⣳⠀⠀⠂⠄⠀⢂⠁⠀⠊⠄⡄⢨⠰⠆⠀⠀⠰
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠈⠓⠦⠤⣤⣤⠤⠤⠤⠊⠀⠀⠈⠑⠀⠐⠀⠀⠀⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⣶⡛⠄⠒⠂⠀⠐⠢⡐⠀⣴⡟⠉⢀⡀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢠⣧⡄⠀⠀⠀⣠⠏
⠀⠀⠀⠼⠑⠤⠈⠳⠀⠀⠀⠀⢀⡞⠃⠌⠀⠀⠻⡄⠀⠀⠀⢐⢌⣂⣐⡑⡁⠀⢠⡄⠀⠠⢀⠊⠦⢰⠀⠀⠀⠈⢋⠟⠀⠀⠚⣹⢁⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠉⠍⠃⠀⠘⠀⠀⠐⡶⣷⠷⠟⠀⠀⠀⠠⠮⡤⠀⠀⢠⢿⡏⠀⢀⣾⡷
⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠠⡄⠤⣤⢕⡴⢂⠴⠀⠀⠀⠀⡄⢠⢀⠀⠀⠀⠂⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡼⠀⠀⠀⡾⠁
⠀⠀⠀⠃⠵⠂⠄⠱⠌⠀⠀⠀⠀⡘⠗⢮⣩⠔⠟⢆⠀⠀⠀⡗⠈⠩⢯⠀⠘⣤⠀⢠⠊⠅⠦⠒⠚⠒⠴⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠑⠀⠀⠓⠀⠀⠈⠀⢰⠠⠤⢐⣴⠀⠉⠀⠀⠀⠛⠀⢠⠀⡀⡤⠀⠀⢀⣠⠄⠀⢀⠄⠀⠀⣠⠂
⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⣉⠀⠀⠀⠀⠀⠀⠀⠈⢫⠞⠁⠀⢀⣾⣇⠀⢠⣮⡤⠀⠐⣻⠃
⠀⠀⠘⠿⠁⠀⠘⡶⠁⠀⢩⡏⠀⠀⢈⡥⠒⢢⡇⠀⠀⣄⣴⣠⠀⠀⣤⠀⠈⠈⠀⠘⠀⠀⠰⣤⣦⡄⠀⠀⠀⢰⠇⠀⠀⢘⣿⡇⠀⢀⣾⡅⠀⠰⣻⠟
⠀⠀⠶⠶⠆⠀⠰⠷⠀⠀⠚⠓⠀⠀⢌⡓⢚⣫⠏⠀⠀⠰⠿⠇⠀⢠⡀⡴⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⠇⠀⠀⢉⡟⠁⠀⠈⡟⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠾⣛⣋⣙⢷⣼⣦⣀⣤⠀⣠⣖⣠⠾⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡏⣭⣷⢾⡛⣏⣍⢻⡄⠙⢿⣿⣿⣿⡿⠟⢡⣶⢶⣯⣝⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⢯⣢⣿⢣⣭⡛⢿⣧⡀⢀⣹⣼⡉⠀⢀⣾⣾⣷⠞⣽⡻⣶⡌⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣨⣾⣿⣿⠀⠈⣿⣧⣿⢉⡹⣷⣝⠟⣻⣿⣿⡟⠶⢿⣫⣴⠾⣷⢹⣯⡼⠃⣿⣶⠃
⠀⠀⠀⠀⠀⠀⣴⡶⠶⠾⠿⠿⠿⢮⣷⣤⡴⠟⣽⡿⣾⠛⢾⣝⠛⠫⠤⢠⠙⠾⣿⣫⣴⡄⣿⢸⣿⠀⠀⣿⣿⣶⡆
⠀⠀⠀⠀⠀⣸⡟⣴⠿⠲⣄⣀⠀⣸⢿⡿⠛⠛⠛⣸⣯⢀⣀⣿⣷⣾⡂⣲⣴⣾⠟⠁⢸⣧⢿⡏⢛⣶⣤⣟⣽⣿⣿⣶⣖⣀⡀
⠀⠀⠀⠀⣴⡟⣼⠻⢼⠾⣛⣛⣛⣏⣼⠇⠜⡀⢶⣿⣽⣯⠶⣿⣿⣿⣻⣛⣿⣿⣏⣲⣼⣿⠈⠍⡙⢻⣿⣯⠀⠀⠉⣉⣭⡝⢿⡆
⠀⠀⠀⣼⡟⣼⡏⣿⡆⣿⡋⢉⣭⣿⣥⣴⡾⣷⣾⣿⣿⣵⣧⢾⣿⣿⣿⣿⣾⣏⣬⣿⡿⣿⡟⣀⠱⢸⣇⣹⠷⣶⢿⣧⡏⣿⡘⣷
⠀⠀⠀⠙⣧⡙⠳⠾⣿⣜⣷⣜⣧⡀⠀⡽⣿⣿⣾⣿⣿⣿⡟⣿⣮⣻⡿⣫⣽⣾⣿⣿⣿⣝⢿⣿⣷⣶⣭⣛⡛⠷⢶⡍⣋⣭⣧⢻⣆
⠀⠀⠀⠙⢿⣿⣆⠀⠈⣿⠞⣿⠘⣿⣶⣷⣿⣌⣿⣟⣛⣿⣷⡿⣿⡟⢟⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⡀⠀⣩⣟⣤⡿⣱⣿⠃⣿⡌⣿⡆
⠀⠀⠀⠠⣾⣿⣿⣶⣾⣿⡿⢁⠆⠾⢿⣏⣿⣿⣿⣝⢿⣿⣯⢍⠩⢆⠬⠙⢘⣿⣿⣶⢿⣿⣶⣽⣿⢷⣿⢯⣾⢯⣾⠟⠋⠉⣩⡶⠋
⠀⠀⠀⣠⣾⣿⠟⠉⢹⡟⢷⡀⠢⣶⣾⣏⣿⣿⣿⡿⣿⣗⣍⢈⢾⣿⣿⢜⠈⢭⣿⣽⣿⣿⣿⣏⣿⢿⠅⡀⢻⣮⣷⡀⣠⣾⣿⠟⠁
⠀⠀⢈⣽⠟⣁⣀⣠⣼⢟⣼⣇⣴⡿⣾⣿⡿⢿⣯⣾⣿⣿⣋⢀⡰⠉⠭⡉⠾⣿⣿⣟⢿⣿⣿⣿⣿⣴⠅⠚⣰⣿⣿⠿⣽⣿⣯⠆
⠀⠀⢿⡇⣿⡟⣼⡟⣱⡟⢫⡞⠁⠀⢳⣼⣶⡻⣿⣿⣿⣿⣿⣾⣯⣦⣿⣧⣿⣿⣿⣽⣽⣟⢛⡿⡷⣿⣆⢾⣇⣾⡇⠀⠈⢿⣿⣦⡀
⠀⠀⠘⣿⡸⣷⣛⣑⠿⢦⣬⣽⣛⠿⠿⣯⣷⣝⢿⣿⣿⣷⠿⣟⣽⣿⡻⣿⣷⣿⣿⣿⣿⡽⣾⣵⠃⠀⠹⣏⢿⣎⢻⣶⠦⣄⡻⣦
⠀⠀⠀⢸⣧⢻⣍⠟⣿⣶⣾⣯⠛⡷⢀⠙⢀⣿⣷⣷⡿⢛⣿⣿⣿⣿⣿⣿⢾⡿⣻⢿⣷⠿⣷⠿⠿⣻⣟⣛⣀⣿⡎⣿⣎⣿⢁⣿⠇
⠀⠀⠀⠀⣿⣆⣛⣛⡡⠀⠀⢸⣿⣿⣌⣁⡀⢻⣿⠻⣝⣷⣿⣿⣿⣟⣳⣯⡵⢾⣷⣻⡿⡄⢁⠂⣾⢛⣿⣿⣭⣽⣶⢦⣽⢣⡿⠃
⠀⠀⠀⠀⠈⠉⢩⣿⣿⣿⣷⣻⠟⠿⣯⡍⣿⣼⡇⠀⣠⣿⡿⢿⠍⣽⢿⣿⡋⠋⢸⣿⢁⣤⣬⣴⢿⣾⠉⠉⠙⠧⣴⡿⢣⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡃⠀⢸⣿⢿⡇⠿⢞⣯⣷⣤⡀⠤⢀⣠⣝⡻⣦⣸⢷⣿⣡⡴⠿⠻⡟⣶⣶⣶⣶⠶⠴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣴⡻⣿⡸⣷⠶⢟⣯⣶⣬⣿⢶⣿⣡⣽⡿⣦⡩⣸⡟⣿⡇⠀⢰⣿⣿⣻⠋⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣍⠻⣮⣽⢣⢶⡾⢿⠏⠀⠀⣹⢿⠉⠉⠹⣷⣬⣛⠟⣼⡟⢻⡌⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣝⡻⠷⣶⡞⣠⣶⣿⣿⣿⣿⣤⠀⢿⣙⣛⣧⣿⡶⠟⣃⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⠞⢫⠟⠉⠑⠋⠹⣿⢷⣜⣉⣩⣭⡶⠾⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⡀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⠀⠆⡀⠄⣰⠜⠉⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠠⡆⠂⠀⠑⠀⠁⢀⣼⠁
⠀⠀⠀⠀⠆⠀⠀⠀⢃⠀⠁⠄⠀⠀⠀⠀⠠⠀⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠻⠤⠤⠤⠔⠚⠦⠬⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠒⠒⠻⣤⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠒⣄⠀⠀⠀⠀⢐⠀⡲⠶⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠤
⠀⠀⡎⢁⠀⡀⡀⢀⠀⠀⢣⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢁⣄⣠⡄⡀⠀⠀⢰⠂⠀⠆⠀⠰⣀⣀⣂
⠀⠈⡁⠘⠖⠋⠉⠘⠀⠀⡸⠀⠀⠁⠀⠀⠀⠰⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⠀⢀⠀⠀⠁⣄⠀⠀⢀⠂⣀⠀⢀⡀⠑⡤
⠀⠀⠈⠂⠄⢀⠀⠀⣠⠞⠁⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠃⠈⠀⢀⠀⠁⡀⠀⠀⠀⠅⠒⠺⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀
⠀⠀⣀⡇⢠⡘⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠉⠀⠀⡀
⠀⠀⢥⡀⢀⠘⢨⠀⢉⠤⠀⠀⠀⠀⡈⠄⠰⠂⠤⠂⠁⢤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁
⠀⠀⢸⠃⢸⠀⢘⡀⡇⠀⠀⠀⠀⢀⡅⠀⠠⠀⠒⢁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⢠⠂
⠀⠀⠀⠂⠀⠀⠈⠑⠃⠀⠀⠀⠀⠈⠇⠄⠀⠀⠀⠈⡀⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⢀⡅⠀⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡄⠀⡐⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⣄⠁⠀⠈⠃
⠀⠘⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠀⠒⠃
⠀⢤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠀⠄⠀⠀⢀⣀
⠀⠈⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠘⡐⠃⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⢀⠔⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⢀⡔⠁⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠤
⠀⠋⠀⠀⠀⠀⠁⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⣘⣠⣈⠀⠀⠀⠀⠀⠀⢁⡀⠠⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠐⢺⠀⡉⠉⠉⠀⢺⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⠠⠂⠘⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂

⢼⠦⣀⡇⠀⠀⠀⣀⣀⣀⣀⡀⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠈⣁⠛⠶⠖⠀⠀⠀⠀⠀⠀⢾
⠼⠤⠞⠁⠀⠀⠀⠁⠉⠉⠁⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠁⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣤⡇⢀⣀⡄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⠀⣠⡾⡿⣶⣾⣿⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠹⡇⠁⠈⣡⣋⡤⠟⠁
⢋⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠋⠉⠀⠀⠀⠀⣠
⠠⢈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠙⡃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠿⠥
⠀⢁⠀⠀⠀⢰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣶⣶⣤⣄
⠢⠴⠀⠀⢠⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠻⡿⡍⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⡀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠧⠀⡀⠤⣀⠀⢰⡅⠰⠊⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠁⠄⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠇⠰⠶⠇⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠈⠊⠐
⣤⣤⠀⣠⣶⠀⣠⠶⡇⣀⡴⠶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠤⡦⢀⠀⠀⠀⠀⣯⣭⣭⣭⣥⠉⣶⣷
⠀⠘⠾⠋⠻⠟⠁⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠀⡄⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢠⣀⠠⠃⠀⠀⠀⠀⣰⠀⢠⡆⢀⣠⡄
⢺⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢻⠀⠏⠠⣭⣅⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠠⠐⢀⠨⢔⡂⠅⡀⠂⠄⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⢲⡖⠒⢺⠀⠸⠒⡆⠀⠣⠲⠒⠆⠆⠒⠢⠶⠀⠄⠀⠰⢀⠀⠰⠾⠷⠆⢀⡀⠆⠀⠐⠒⠔⠶⠴⠖⠰⠀⠶⠲⠢⠐⠶⠶⠀⡗⠒⢲⡖
⠀⢠⣟⠀⠈⠲⣷⡖⠉⠒⠐⠐⠒⠐⠀⣀⠂⠒⠚⠂⠉⠑⠂⠨⢑⠠⠔⡊⠅⠐⠊⠁⠒⠊⠂⠀⠊⠂⠑⠐⠐⠂⠀⠈⢰⣶⠖⠃⠀⣹⡄
⠀⠈⣩⣭⠉⠉⠉⣠⡴⢶⣶⠶⣤⡀⠰⣿⠆⠰⣦⡴⣚⡲⣄⠀⠀⠈⠁⣠⠞⢀⡴⠚⠲⣄⠰⣦⠀⠉⠈⣰⣦⣤⠶⠶⣦⣍⠉⠉⣩⣍⠁
⠀⠺⡗⠊⢀⡀⣼⠫⠊⠠⡤⠈⢪⢻⡄⣉⠀⢨⡝⠃⢀⠀⢉⠟⠀⠀⠀⣿⠀⢸⠀⠲⡆⣿⠀⢹⡆⠂⠨⠁⡉⢉⢐⠤⢀⢘⣷⠀⠙⠸⡟
⠀⠀⡿⠢⣼⡟⠛⠦⠀⠰⠧⠀⠠⠟⡛⣿⡤⢘⡳⠆⠀⡀⠙⡇⠀⠀⠀⡻⡠⡘⡂⠑⢁⢛⠀⡺⡁⠀⠈⣶⠂⠬⣈⠀⠢⠤⢬⠁⠐⠟
⠀⠾⣗⣠⠀⠀⠁⣈⡂⠂⠀⠒⣂⡉⠀⠊⠀⠸⢧⡄⣤⡅⣹⡆⠀⠀⢈⡇⠠⣤⠄⠈⠈⣤⡄⢀⡅⠀⠀⠘⢷⣄⣀⠉⣁⣶⣦⡴⢀⠃
⠀⠀⣈⠉⢀⢄⡀⠈⡙⠛⠛⠛⡉⠀⠀⠀⠀⠀⠀⠳⠤⠔⠋⠉⠀⠴⠤⣀⠀⠈⠳⠤⠶⠃⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠀⣉⡐⠀⠃
⠀⠐⢁⠀⡀⢂⡀⠈⢨⠀⢠⠊⣡⠞⢻⣛⠻⢦⡌⢳⠀⠀⠠⠆⠄⢠⡄⠡⠡⢄⠀⠀⢠⢪⣶⠿⢛⠿⢷⣮⠢⡀⠀⢎⡸⠉⢃⠋⠩⣆⠂
⠀⢀⢄⠀⣿⢇⡸⠠⢰⠀⣾⢰⡏⣰⡟⡟⣿⡍⣿⠈⡇⠀⡘⠄⡂⣲⣶⢘⡠⢆⠀⠀⡇⣿⡇⠀⣦⡆⡄⢻⣧⠧⠀⢸⠇⢒⡋⡶⡆⠿⡀
⠀⠈⠆⠀⠀⡌⠉⠀⠌⠀⢢⠼⣷⡚⢷⣷⡿⣡⡿⢦⠇⠀⠣⡂⠀⠁⠀⠀⢔⠔⠀⠀⠇⠟⢃⡀⡁⣈⣀⠚⠏⠏⠀⢾⡀⠙⠁⠽⠁⡼⠇
⠀⠀⠀⠐⢄⢀⠔⠈⠀⠰⣾⣦⡌⠛⠶⠾⠟⠛⣤⣼⡷⠀⠀⠘⠢⡂⢐⠜⠊⠀⠀⢶⣧⣥⠅⠶⠶⠶⠆⢨⣥⣾⠆⠈⠛⠢⢖⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢐⡀⢀⣀⠠⡐⠀⠈⠁⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⡀⠉⡀⢀⢀⠀⢀⢀⡀
⠀⢈⠇⠴⠟⡇⠀⣀⣀⡠⣀⡀⢈⣀⠀⠄⠀⠈⢠⡄⠀⠀⣀⣤⣶⣜⣣⣶⣤⣀⠀⠀⢠⡆⠰⡆⠂⠀⢠⠀⠲⠰⠀⠢⠰⠆⠂⠀⠐⢲⠆
⠀⠈⠉⠧⠀⢃⣣⣿⣽⣥⣽⣧⣿⣿⣤⡄⠀⠀⢸⡇⣸⡟⠛⣁⢈⠙⠋⡁⠈⡛⢻⣇⢸⡇⠰⠧⠌⠻⢃⡀⢀⣀⠀⣀⡀⢀⡘⠟⠡⠼⠆
⠀⠀⠀⠖⢛⣛⡛⣛⣛⣛⣛⣟⣛⣥⠒⢂⣀⡀⢸⡇⣿⡃⠀⠈⠀⠀⠀⠀⠀⠁⣘⣿⣸⡇⢠⡄⢀⣴⡌⢁⠈⣉⣀⡙⢁⠈⢡⣤⡀⢠⡄
⠀⠀⠀⠒⠀⠙⠀⠙⠙⠁⠑⠉⠛⢡⣤⠆⢸⡁⠈⠀⠉⠛⠛⠷⣦⣄⣠⣴⠿⠛⢻⠉⣉⡉⣨⣥⡍⢉⡟⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⢼⡷
⠀⠀⠀⢀⠀⠀⠄⠈⠂⠀⠀⡀⡀⠀⠁⠈⠉⠁⡀⣤⡔⢦⡔⡄⠀⠙⠋⠀⠀⠀⢨⣤⣤⣤⣥⠤⡤⠠⠤⠟⣿⣛⠛⠛⠿⠿⠟⡛⣫⣬⠠
⠀⢈⡶⠆⠀⠈⠀⠀⡀⣠⠀⠠⢶⡍⠀⠀⣠⣌⣳⣾⣿⣿⣾⣏⣠⣀⠀⠀⢀⣴⠋⣡⠀⢠⡍⢳⡄⠀⠀⣞⣿⣾⣶⣶⣶⣶⣶⣿⣿⣿⡆
⠀⠀⢷⣆⣀⡂⠀⠀⠓⠓⣂⣠⡾⠁⠀⢾⣿⣿⣿⣿⣟⣿⣿⣻⣯⣿⡇⠀⢼⡇⢀⢺⣀⢀⢛⠀⣷⠀⠀⢸⣿⣷⣾⣾⣿⣿⣷⣾⣾⣿⡁
⠀⠘⠒⠒⠻⠿⢽⡀⣪⠽⠿⠒⠒⠚⠀⠀⠛⠟⡹⣿⣿⣿⣿⣏⡻⠋⠀⠀⠈⠻⣜⠋⠉⠉⢈⡼⠋⠀⠀⢾⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⡇
⠀⠀⠀⠠⠤⠀⠀⠭⠄⠀⢀⡀⠀⠀⠀⠀⠀⠈⠘⠛⢣⠜⠻⠉⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠋⠀⠀⠀⠀⠈⠛⣉⠀⠠⠤⠀⠀⠨⠝⠋
⠀⠸⡆⠉⠀⠛⠛⠛⠛⠀⢶⢠⣌⣀⡐⠂⠀⠀⠠⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⢂⣀⢩⢤⣶⠆⠠⠛⠛⠛⠀⡉⢰⠇
⠀⠰⠧⠈⠻⠇⢀⣀⣈⣁⡐⠀⠍⠙⣉⡛⠹⠳⠾⠟⠃⡀⠉⡟⠀⣴⣦⠄⢻⠉⣀⠹⠇⠘⠃⠟⢛⡈⠃⠠⠀⠒⣀⣀⣀⣀⠸⠟⠁⠼⠆
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠐⠻⠀⠋⠛⠀⠛⠂⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠃⠁
⠀⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⠀⠄⠀⠀⠀⠀⠐⠿⠿⠀⠀⢸⣯⡀
⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⣄⢀⣄⢀⣤⢀⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠀⠀⠀⠀⠀⢮⠇⠀⠀⠐⠁⣰
⠀⠀⠀⠀⠀⠉⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠀⠔⢢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠛⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠸⠤⠼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠄⡄⡀

⠀⠀⠀⣔⠳⡀⠀⠀⠀⠀⡐⠢⠀⡦⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠉⢦⠐⠲⠀⠀⠠⠞⠢⠤⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⢀⣤⢀⣄⣀⠀⠈⠀⠀⠀⣴⠔⠂⠙⠋⠉
⠀⠀⠀⠀⠈⠳⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠢⣀⣰⠀⠀⠀⠀⠘⠶⠚⠻⠻⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠞⠛⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡝⢻⣭⡅⠀⠀⠀⠀⠀⣨⠥⣯⣹
⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠈⢇⡤⠔⠁⠀⠀⠠⠛⢿⡆⠀⠀⠀⠀⠠⠂⠘⠁⠙⠋⠋⠃⢀⠂⠀⠀⠀⢣⠀⠛⡏
⠀⠀⠀⠈⠉⠉⠉⠸⠀⠀⠀⢘⡼⠀⠀⠀⠀⠀⡁⣂⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠶⢾
⠀⠀⠀⠈⠂⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣤⣤⡄⠀⠀⠀⠰⠅⢀⡂
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⣿⣧⣄⡀⠀⠀⠉⢻⠀⠀⠀⢀⣴⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣰⠿⠛⠛⠋⠉⣙⡆⠀⠀⢀⣄⡀⠀⢰⣿⠷⠓⠛⠛⠻⢿⡀⠀⠀⠈⠄⠀⠀⠸⣿⡿⠛⠙⠋⠙⠻⣿⡆
⠀⠀⠀⠀⠀⣿⠈⣽⠳⠀⠸⡶⣇⠀⠀⠈⣒⡀⠀⢸⡏⠠⠤⠀⢠⠤⠈⡇⠀⠀⠀⠀⠀⠀⠀⢹⢨⣿⡡⠀⣨⣴⠈⡇
⠀⠀⠀⠀⠘⢤⠀⠄⠀⠀⠢⠀⠘⠀⠀⠀⠀⠀⠀⠸⠃⣋⠛⠁⠙⢛⡁⠉⡆⠀⠀⠀⠀⠀⠀⣿⠅⠀⢠⠁⠈⠀⢠⣩
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠶⠂⡰⠀⠀⠀⠀⡀⠀⢘⣦⠉⠀⢀⡀⠈⣰⡎⠀⠀⢰⠦⢴⠀⠀⠀⢄⠀⢠⡄⠀⢀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢹⠑⢶⠀⠊⠀⠀⠀⠳⢦⡓⠀⠀⠀⠀⡀⠉⠁⠐⠁⠀⠀⠀⠙⢶⠚⠂⠀⠀⠀⢑⡄⢠⡄⠁
⠀⠀⠀⠀⣠⣾⣿⣷⣤⣴⣿⣿⣦⠀⠈⠓⠓⠂⠀⠀⣀⡠⢯⣥⡼⠤⠤⠤⡀⠀⠀⠀⠀⠀⣠⣶⢦⣉⠓⠺⣛⡴⣤
⠀⠀⠀⠀⠛⢻⣏⣹⣿⣿⣿⣿⠛⠆⠀⠀⠀⠀⠀⠞⠀⠀⠀⢺⡀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠷⠠⣶⡍⢡⣝⠋⢴⢮⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠄⠀⢀⡄⠀⣀⡀⢀⡀⢠⢂⢀⠄⢠⠄⢠⣊⡔⡠⠀⢠⡂⠀⠀⡀⠀⠀⠀⠀⠎⢀⡟
⠀⠀⠀⠀⠀⠆⠴⠠⠚⠀⠀⠈⠀⢢⡆⠀⠒⡇⠀⠈⠀⠘⠁⠁⠀⠀⠇⠘⠁⠸⠅⠀⠉⠌⠀⡋⠀⠀⠀⠘⠎⣠⠊
⠀⠀⠀⠀⠘⠖⠳⠎⠁⠀⠀⢸⣀⠎⠀⠄⠉⣃⠀⡙⡁⢀⠀⠀⠀⢰⠀⢰⠀⢀⡀⠀⠂⠀⡇⠃⠀⠀⣠⠾⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠉⠀⠀⠁⠈⠃⠙⠀⠉⠀⠑⠋⠛⠂⠀⠚⠁⠉⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠄⠠⠤⠀⠓⠄⠀⠀⠀⠀⠔⢫⣇⡀⠀⣀⠀⠀⠀⠀⣟⠢⢄⣀⣀⠀⣶⣴⠀⠀⠀⠀⠀⠠⠄⢀⢀⠠⠄⢠
⠀⠀⠀⠀⠤⠦⠴⠓⢧⣤⡼⠀⠀⠀⠀⠙⠞⠁⠈⠉⠉⠀⠀⠀⠀⠈⠓⣤⢀⡤⠚⠁⠀⠀⠀⠀⠀⠈⠒⠖⠃⠀⠑⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠐⡖⠊⢤⠄⠀⠠⡀⠁⠊⢄⠀⠀⢀⡀⠀⠀⣼⣗⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣤⣾⣧⣀
⠀⠐⠀⠀⠀⠠⠀⠀⠀⡐⠂⠀⠐⠁⠲⠁⠀⠀⠰⠁⠀⡠⠊⠀⢀⡾⣧⠀⠀⠀⠈⠀⠀⠀⠌⠈⠀⢁⠀⠀⠀⠀⣀⣠⣴⣿⣷⠶⠒⠀⠀⠟⠻⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠈⣆⠠⠀⠀⠠⠀⠀⠀⢠⣤⣤⣠⣾⣁⣽⣄⣀⣀⣀⠀⠀⠈⠀⢁⠐⠀⠆⠀⠀⠀⠀⠀⠈⡿⠻⡇
⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⣤⠀⠀⠄⢀⡩⠀⠀⠀⠀⠀⠐⢠⡾⠁⠀⠈⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⢾⠀⠀⠘⠃⠄⠘⠀⠀⣠⠎⠱⠄⡀
⠀⠀⣷⣿⣀⠈⠂⢄⡒⠀⠀⠀⢀⡐⢍⠀⠀⠀⠀⠀⠀⠀⡜⢉⣶⣶⠞⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡝⣺⡷⠆⠀⠀⠀⠄⠀⠈⢠⠀⡀⠰⠁
⠀⠈⠉⠛⠁⠀⠀⠀⠃⣀⠀⡭⣀⠀⣬⢣⠀⢠⣄⠀⠀⣼⠷⠋⠁⠀⠙⢾⣧⠀⠈⠓⢶⠃⠂⠠⡀⠀⠈⠉⠈⠻⠄⠀⠀⠘⠃⠠⠀⡀⠈⠚⠊⠁
⠀⠀⢀⡤⣀⡀⠀⠀⠠⠔⠈⠀⠀⠈⠀⠂⠀⣸⣿⣦⣤⣤⠆⠀⣀⡀⠀⢀⡈⠀⠀⡀⠔⠃⠌⠀⠀⠀⠘⠃⠲⠆⠀⠀⠀⠀⠀⣠⣴⡇⠀⠀⠀⢠⡄
⠀⢀⢸⠈⣿⣰⢳⢆⡄⢀⠀⠀⠀⠀⠀⠶⢾⣿⣏⣽⣿⣇⠀⢀⣸⣿⣛⣿⣤⡀⠀⠀⠀⠈⠁⠀⠸⣿⠆⠠⣀⢀⣤⣤⠄⣤⣮⣿⠿⠇⠀⠰⠷⠀⠁
⠀⠙⠛⠸⢡⢧⢇⡾⣧⢇⡏⠁⠀⠀⠀⠀⠀⢹⣿⠟⠛⠛⠃⠀⠀⠘⠟⠀⠀⠀⢠⡳⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢂⡻⠟⣩⣶⣺⡽⢦⣄⠀⠘
⠀⠐⣠⢂⠀⠈⠘⠸⠇⣞⠉⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⡀⠉⠖⢲⠖⠀⣀⠠⠀⠀⠀⢱⣥⠴⣻⣿⣿⣷⣚⣭⣴⣿⣶⣤
⠀⠀⠀⡁⠀⠂⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡆⢀⡈⠠⣏⠀⠀⠼⠦⡞⠀⢠⣯⣴⣿⣿⣿⠿⣛⣭⡞⠋⠉⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣄⠀⠀⠀⠀⠀⠀⢰⣤⣶⡀⠀⠀⣧⠞⠉⠳⢿⠀⠀⠀⠀⠀⣠⠟⠛⠩⠟⠛⢶⣾⣿⣿⠀⠀⠴
⠀⠁⠀⠀⠀⡀⠀⠐⣄⠀⣀⡄⠀⠀⠺⡒⠁⠘⠦⣄⠀⠀⠰⠞⠛⢿⣿⠛⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠻⣿⢻⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡅⠀⠀⢰⠣⢤⣠⠎⠁⠀⠀⠀⠀⠀⠈⡏⠀⢀⡀⣤⢺⣢⠀⠀⠐⠂⠂⠀⠀⠀⠀⠳⢀⡀⡤⠀⠀⠘⢿⢀⠀⢠⡰
⠀⠀⠀⠀⠠⠀⠉⠉⢻⣿⠋⠛⠃⠀⠀⠀⠀⠙⠀⠀⠀⣇⠀⠀⠀⠀⠀⣔⠏⢂⠘⡦⠹⠂⠀⠀⠀⠀⠀⠀⠀⡠⠈⡬⣽⠣⡀⠀⠀⢠⣏
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠱⣡⡟⠉⠀⠀⠈⣷⠄⠃⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⣴⠚⠭⠒⠛⠾⠁
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡆⠳⠐⠠⠆⠀⠀⠀⠀⣀⣠⣦
⠀⢲⡄⠠⠊⣇⠀⠀⠀⡀⠂⠰⠟⠀⠕⠀⠠⣄⡀⠀⠀⠀⢀⡀⠀⢤⣶⣦⡀⠀⠀⠠⠀⡁⠀⢀⣀⠀⢀⠞⠤⠀⠀⠀⣹⣿⣿⠛
⠀⠀⡹⠀⠀⡀⠉⠓⠀⠀⠐⠆⠀⠈⠀⠀⠀⠆⠙⠲⡄⠐⣸⠃⠀⠀⢽⡛⢷⠀⠀⢐⠋⠜⠀⠘⠉⠁⢀⠈⠂⠠⠀⠀⠁⠀⠙
⠀⠐⠀⠀⠢⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⠀⠧⣀⠀⠀⠀⠙⣿⠀⠀⠁⢀⡄⢀⡌⠂⠀⠁⠷⠈⠀⠀⠀⠓⣄
⠀⠀⠀⠀⣄⠈⢀⠀⠀⠇⡸⠃⡀⠀⠀⢠⣞⡥⢀⡀⢀⡀⠀⠈⠳⠄⠀⠀⠈⠀⣶⠀⠀⠀⠈⠁⠰⠈⠈⠀⠀⠀⣀⢰⡦⠘⢷⡀⠀⠀⠀⡼⢤⣀
⠀⠀⠛⢦⡹⠆⣸⣧⡀⠀⠱⠚⠋⠀⠀⠀⠀⠀⠀⢇⣼⠀⠀⠀⠀⠀⠀⠙⢻⣿⠟⡓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢼⡄⣀⡐⠃⡀⠀⠛⣵⡷⢿⡀
⠀⠘⠓⠒⠊⣀⣿⣿⣿⣶⣶⡞⠁⢀⡀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠾⠋⠑⠑⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⢸⠀⠈⠉⠁⠘⠀⠀⠉
⠀⠀⠀⠘⠧⢻⣿⣿⣟⣿⡏⠐⠯⣭⣄⠀⠀⠀⣀⠀⠂⠐⠀⡀⠀⠀⠀⠀⡠⠀⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⢰⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡤⠶⣂⠈⣿⡿⠋⠘⠿⠀⠳⠆⠀⠀⠀⠀⠁⠀⠀⠈⠀⠰⠀⠀⠀⠘⣅⢈⠃⣶⣆⢹⡄⠀⠀⢉⡴⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⢻⣤⡀
⠀⠀⠀⠾⠃⠀⠋⠀⡆⢳⡄⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢠⣈⠇⠆⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠞⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠤⠄⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⢒
⠀⠀⠀⠀⠀⠠⣠⠐⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⢀⠈⠀⡨⣍⠁⡀⠀⠀⠠⠀⠀⠀⠀⠀⢠⢰⢀⠄⢒⡲⠰⠆⠀⠀⠀⠄
⠀⠀⠀⠸⠋⣽⠃⠀⠀⡆⠀⢼⣏⠛⣿⢿⣇⠈⡈⠙⠀⣄⠤⠀⠀⠰⠀⠀⠀⡆⠇⡴⠎⠘⠈⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢧⠲⡆
⠀⠀⠀⣀⢰⠟⠀⠀⠀⠁⠠⢸⣿⣶⣌⠻⠉⠑⠐⠀⠀⡈⠀⠀⠀⢠⢀⡀⠱⠁⠀⠀⠀⡒⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄
⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠃⠄⠂⠰⢄⠀⢮⡖⠀⠀⠬⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⣠
⠀⠀⠀⣀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⢸⠰⠾⠉⠀⠀⢘⡖⠀⠄⠀⠀⠀⠀⠀⠀⢴⠒⢦⡀⠀⠀⠈
⠀⠀⠀⠀⣀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢹⠀⠀⠸⡇⠐⠀⣰⠛⠃⠀⣤⠀⠀⣠⣾⣿⡄⠀⠀⠀⠐⡄⡆⢀
⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⣦⣤⣴⡟⣿⡿⣿⡟⢿⡇⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠸⣿⡿⡟⠈⢹⠟⠀⠈⠀⢸⠃⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⢀⠁⠀⠀⠀⠀⠠⠀⠈⠐⠁⠀⠀⠀⠄⢠⠀⠀⠀⠻⠀⠀⠀⠛⡲⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⡡⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⠀⠠⠀⢰⠀⠀⠘⠀⠀⠃
⠀⠀⠀⠁⠀⢰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⢆⡠⢀⡀⠈⠰⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠣⠠⠃⠀⢺⠀⠁⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠
⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⡀⠀⢀⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠰
⠀⠀⢸⠀⠘⠁⣂⠀⢀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⢤⣬⣙⣿⣿⣷⣦⣤⣶⣾⣛⣛⣊⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣤⠄⣨⠶⠀⢰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠂⠀⠀⠴⠀⠁⣀⣀⣿⠿⠛⠛⣿⢉⡀⢩⡭⠰⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢋⣄⣠⣷⠤⠞⠒⠛⠋⠥⠤⠬⠤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠁⠀⠀⠰⠆⠀⢀⠛⡀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣩⠔⠀⠠⣴⠀⡀⢠⡶⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣠⣾⠝⠀⠀⢀⠀⠐⡴⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢡⡖⠀⠀⠀⠀⣄⠉⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡁⢠⡌⠀⠺⠧⠀⠀⢸⣀⡓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣀⠀⢧⠀⡀⠠⣾⠦⣬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢺⢁⡏⠃⠈⡄⠁⠄⢠⠀⢻⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢸⡇⠈⠁⡀⠀⠹⠀⠀⣀⠀⢸⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⢸⠇⠶⠁⠁⣀⠀⢀⠀⠋⠀⢀⢠⡟⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢾⢘⣦⠠⡆⠙⠀⠀⠀⣤⠀⣨⡘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣾⡈⣁⠶⠆⠛⠀⢀⠀⠀⠸⠝⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡉⠣⣴⠰⠶⢌⠡⢀⠀⢪⡸⡄⠈⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠡⠘⠔⢦⡓⠲⢤⢤⣛⣃⣤⣽⢿⣄⢲⣬⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⢀⠀⢸⡀⢫⡐⣦⠠⡤⡄⣠⡍⣹⣿⣆⠻⣧⠀⠑⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⢠⣴⡇⠚⣠⣤⣤⡍⢺⠧⠠⣭⣷⣌⠳⠾⠷⠂⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠙⠦⠁⡄⣸⢋⣥⢬⡛⣏⠡⠐⠦⠄⠀⠰⢦⠉⠛⠷⠤⡹⣆⠠⠤⣀⠑⠦⢄⢰⡿⣡⠴⢚⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢾⢰⡄⣾⣯⡟⠀⢈⣅⢰⣿⠀⠀⡘⢣⡖⣲⢽⠮⣶⣖⠀⣀⣀⣴⢋⣾⣿⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠠⣀⠲⣯⡀⢀⡀⠿⠋⠀⣠⠀⣠⡀⠈⢰⠡⡗⠀⠸⠆⢀⠑⢺⢃⠚⣯⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⢰⠎⣱⡮⢝⠳⣤⢂⣀⣈⠀⠀⠉⡁⢀⠈⠁⠠⡆⢴⠆⠀⣴⠏⠆⢙⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⠽⢦⡦⠽⠓⠭⣌⣣⡼⠃⢢⡜⣦⣤⣲⣐⠙⠻⢛⣫⢉⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⠀⠀⣠⠜⣋⡉⠃⠲⣶⢚⠓⠚⠯⠙⠉⢀⣴⡄⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠙⠨⢁⠃⠀⠙⠈⠀⠀⠀⠀⠀⠴⠛⡿





⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣏⣻⣧⣤⣤⣴⣤⣬⣤⣭⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣤⠀⣊⣁⠠⠤⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⢀⠞⡄
⠘⠬⡎⠁⢠⢤⡀⢠⠊⠱⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⢢⠞⠹⣠⢤⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⣸⠀⡇
⠀⠀⡀⡠⢤⡈⢀⣈⣁⢉⣀⡄⠀⠀⠀⠀⠀⡴⠛⠤⠃⠀⠘⠀⠘⠁⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡇⣇⠀⠈⠢⠣⠚⡜⠓⠁⠀⠀⠀⠀⠀⠃⠀⣀⣀⣀⠀⢀⠀⣀⢸⠁⠀⠀⠀⠀⠀⣽⢰⢅⠓⡙⢀⣒⣭⣿⠅
⠀⠀⡆⠙⠦⢤⣆⣧⣰⣞⣤⢹⡆⠀⠀⠀⢰⠐⠧⢴⠦⠜⠀⡻⠞⣚⢹⠞⣆⠀⠀⠀⠀⢹⠐⠏⠷⠙⣞⣳⣱⠮⢄
⠐⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠶⠀⠀⢸⠀⠐⠛⠛⠛⠓⠛⠚⠚⠛⠉⠉⠙⡆⠀⠀⢸⠄⠀⠀⣀⣀⣤⠠⣤⠎
⠀⠀⠈⠂⠤⡀⣀⣀⣀⡠⠔⠋⠀⠀⠀⠀⢸⣶⠒⡆⠤⠤⣄⢠⠤⢤⢠⣤⢠⢤⡇⠀⠀⠒⢲⠀⣀⢀⣀⣀⣀⡖
⠀⠀⠀⠀⠀⠐⠇⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠋⠀⠀⠛⠀⠉⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠈⠓
⠀⠀⠀⢠⠔⠆⢀⠤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⡀⢠⠀⢬
⠀⠀⠀⡇⠀⠄⠁⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠒⣂⠟⠈⠉⠁⠀⠢⣤⠄
⠀⢀⡔⠁⠀⠀⠀⢠⢄⢌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠒⠤⡀⠀⠀⠈⠉⣿⠰⠦⢔⣦⣢⢴⡜⡯
⠀⡘⠀⠮⡿⠩⠆⠳⠭⠞⢸⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⡇
⠀⡇⢴⣤⠒⡤⠚⡶⢱⡄⣼⡀⠀⠀⠀⠀⠀⡇⢇⠀⠀⡿⠀⢠⢄⠀⠀⠀⢣⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢰⠁
⢀⠇⠀⠀⠈⠉⠉⠉⠉⠁⠀⡶⢄⠀⠀⠀⠀⠸⡌⠣⠔⠃⠀⡸⢸⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⢀⠏
⠯⠔⠳⢄⠀⠀⠀⠀⠀⢀⠜⠁⠉⠀⠀⠀⠀⠀⠈⠒⠤⠤⠚⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⡀
⠀⠀⠀⢸⡰⠁⠀⠈⠙⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡰⠢⣄⣀⣀⡀⠀⣀⠤⢤⠀⡇⠈⠂⠁
⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠣⠜⠁⠀⠀⠀⠀⠀⠉⣀⣀⣀⣀
⠀⠰⡦⠤⠊⣉⠉⢹⡦⢍⣆⠀⣀⣠⠢⡀⠀⠀⠀⣀⡄⣀⡠⠤⡴⠚⠓⢦⠀⠀⠀⠸⠗⠲⢩⢴⡖⢄⢀⠤⡬⡉⣗
⠀⠀⢸⠂⠨⡀⠃⠀⣩⢥⣜⣻⡥⠧⠔⠉⡆⠀⢰⡃⠶⣨⡄⠸⣇⠛⠀⣸⠃⠀⠀⠀⠀⡇⠈⢒⣒⡁⣈⢓⣒⡁⠆
⠀⠀⠸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠈⠱⢦⠤⡨⡟⠉⠀⠀⠀⠀⢀⣰⠁⠈⠚⠐⠓⠛⣞⠋
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢠⠖⠚⠒⠓⡇⠀⠀⠀⠀⠀⡯⢲⠀⠀⠀⠀⠀⠀⢻⢽⡄
⠀⠀⠀⠙⠲⡶⠦⢠⡖⠒⡆⡖⢲⢲⠋⠀⠀⠀⠀⠀⢸⡀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⡆⠈
⠀⠀⠀⠀⢘⠇⢰⠸⠃⠀⠧⠃⠸⠌⠀⠀⠀⠀⠀⠀⠀⠑⣶⡖⢜⡆⠀⠀⠀⠀⠀⠀⠀⠸⡼⠉⠉⢏⡏
⣶⣶⠶⠶⠾⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣶⣄⠀⠀⡀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⠀⢠⣧⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣄⠀⠈⠛⠿⠿⠛⠉⠀⣈⣹⠀⠀⠈⠙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠉⠛⠋⠀⠀⠀⣠⢦⠀⢠⣿⠿⣧⠀⢰⣮⣛⠿⣿⣾⣿⢟⣣
⠀⠀⠀⠀⣾⣿⡆⠀⣠⣴⣾⣿⣿⣿⣷⣦⡀⠀⣸⣇⠀⢸⡿⠓⣿⠀⠸⢿⠾⠶⣿⡷⠷⢾⠷
⠀⠀⠀⠀⣿⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⢉⣉⣀⣈⠉⠀⢸⣷⠴⣿⠀⢘⣭⣾⣷⣿⣶⣮⡛⢿⠀⠺⠂
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⢸⣿⠛⣿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⢸⡿⠿⣿⠀⠀⡠⢄⣸⡇⡠⠤⠀⢰⣿⣿⣿⣿
⠀⠀⣼⠿⣿⣿⡿⣿⡀⠈⠃⠀⠀⠀⠀⢰⠀⠀⣶⠀⠀⢸⣿⣴⣿⠀⠘⢀⣤⢼⡧⣤⡠⠃⢸⣿⠀⠀⣿
⠀⠀⣏⣴⣶⣷⣶⣜⡅⠀⢀⣾⣧⣀⡄⢸⠀⣶⣷⣶⠄⢸⣯⣩⣿⠀⠀⢰⣿⣾⣶⣷⡆⠀⢸⣿⠀⠀⣿
⠀⠀⠁⠛⠛⠛⠛⠁⠁⣤⣾⣿⣿⣯⡀⢸⠀⠚⠛⠛⠀⠚⠛⠛⠛⠀⠀⠀⠀⠀⢀⣤⡀⠀⢸⣿⠀⠀⣿
⠀⠀⢠⡄⢠⣶⡄⣿⠀⠈⠉⣉⡀⠈⠀⢸⠀⢰⣋⣛⣉⣋⣙⣋⣙⣋⣙⣉⣢⠀⢾⣿⣷⠆⢸⣿⣶⣾⣿
⠀⠀⠈⢿⡌⠉⠀⣿⠀⣠⠀⣨⡀⢠⡀⢸⠀⠘⠛⡿⠿⠿⠿⡿⠿⠿⠿⣿⣿⠀⠀⢿⠅⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠘⣷⡀⠀⣿⠀⠀⣀⠙⢁⡀⠀⠻⠀⢸⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠀⠀⠀⢠⣶⣿⣷⣆
⠀⠀⠀⠀⠈⢇⠀⡇⠀⠀⠉⠀⠈⢁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⢁⣤⣤⣤⡀⠀⠾⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠈⠀⠁⠀⡁⠀⣀⢰⣿⣷⣆⣀⠀⠘⣿⣿⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠠⣶⡗⠀⢸⣼⣿⣿⣇⡜⢀⣾⣿⣿⣿⣿⣆⠀⣿⣿⣯⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⢸⣿⠿⢿⠿⢿⣿⠀⠘⠿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⣿⠀⠀⠀⢸⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣧⠀⢠⣾⣿⣿⣿⡿⠛⠛⠛⠻⡿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⡇⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣤⣤⡄⢻⣿⣿⣿⣷⣦⣤⣤⣾⣿⣿⣿⡿⠀⠘⢿⣿⣿⣿⣷⣦⣤⣤⣴⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⣴⣿⣿⢿⡿⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⠸⡇⠀⢤⠁⡄⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⣼⣿⣿⠟⡻⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣧⡸⠗⣀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⣻⣟⣟⣛⣟⣛⣟⣛⣛⣛⣛⣛⣛⣟⣟⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⡛
⠀⠀⠀⠀⠀⣾⣿⣿⢛⣛⡟⡛⢿⢛⡛⢛⠻⠛⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣧⡘⠛⡃⣄⢻⠐⢆⠘⣠⣀⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠇⢣⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠢⢀⡀⠈⢿⣄⠀⡜⠀⡀⠀⣼⠇⠀⠈⠁⡠⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠒⠄⠀⠀⣼⣿⡆⠁⠀⣰⣿⡏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣈⡄⠀⠙⢿⡿⠆⠒⠛⠿⠇⠀⠀⠀⠀⢁⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣦⡬⠃⠀⠀⠀⠀⠀⠈⠲⣿⣿⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠁⠐⠂⠙⢛⡃⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠐⠠⠠⠐⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠑⣢⣤⣴⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⡉⠙⠒⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⠿⠛⠏⠀⠓⣄⡀⠀⠀⣀⠤⠙⢍⠻⢿⣷⣶⣆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢿⣿⠇⠈⢸⣿⣦⡤⡀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⣀⠠⠁⣼⡿⠈⢂⠀⠟⢿⣷⠈⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠸⠊⠀⠀⠀⠀⡿⠁⠀⠸⠀⡀⠘⣿⠀⠈⠢⠧⠄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠌⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢀⣴⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠃⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣯⡵⣀⣤⠞⣰⢺⠟⣐⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⣤⣌⡄⠀⠞⢁⢘⡟⠀⡘⠛⢛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣿⠿⡟⣀⣀⡀⢸⡜⠠⣾⠃⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣃⡤⠂⠋⠜⣿⣿⢿⡆⣸⣧⣴⠇⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣰⡿⣠⠀⠀⢠⠿⣿⣦⣶⣿⡟⠻⣀⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠒⠛⠛⢿⠙⣃⣤⣥⣤⡈⢙⡳⢤⡁⢤⣹⡼⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣴⢯⡾⣫⣤⣤⡈⢻⡎⣿⣦⠑⢸⣿⠷⢸⣯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋⢠⠏⢸⣿⡏⠙⠏⡼⢃⣿⣿⡆⠈⣽⡷⢈⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠁⠀⠀⠀⠸⣿⣧⡀⠲⠶⠿⣿⣿⠇⠚⢋⡃⣶⡹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣘⠻⣿⣶⣶⣿⣿⣿⠿⣷⣄⠙⣿⢁⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⠁⠀⠈⣿⡇⠘⠼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⣶⠿⢁⣷⣄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠐⢿⡋⢻⡌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠋⠙⢷⣄⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⡀⠀⠀⠘⣿⣿⣿⡇⠀⢠⡀⠀⢻⣇⠇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣇⠀⠀⠀⣿⣿⣿⣷⣄⣸⣿⣄⠘⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⢿⣦⣤⣼⣿⣿⣿⣿⡿⢿⣿⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠸⣿⡙⠛⠁⢸⣿⡏⠀⠀⠘⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠹⣿⣄⠀⢸⣿⡇⠀⠀⠀⣿⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠘⢿⣷⣄⢿⣧⠀⠀⠀⡟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠙⢿⣿⣿⣧⠀⠈⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠉⠙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡿⠋⠁⣠⣴⣦⣄⡀⠉⠻⣿⣷⣶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠛⠛⠉⠉⠉⠋⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠙⠉⠉⠉⠛⠛⠛⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣴⣶⣶⣾⣶⡆⠀⠙⠋⠉⠉⠉⠉⠉⠛⠀⢠⣶⣾⣶⣶⣶⡄⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣸⣿⣿⣿⠟⠁⢀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣤⣀⠈⠛⢿⣿⣿⣿⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣿⣿⠟⠁⣠⣶⣿⣿⠿⠟⠙⠋⠉⠋⠛⠿⣿⣿⣷⣦⠀⠙⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠃⠀⠈⠉⠀⣼⣿⣿⡟⠉⠀⠀⠠⠀⡇⡀⢀⠀⠈⠙⣿⣿⣷⡀⠈⠁⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃⢀⣴⣿⡗⠀⣼⣿⣿⠇⠀⠀⡐⢌⢦⠇⡧⢣⡣⢂⡀⠀⠈⣿⣿⣷⠀⠰⣿⣷⣄⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⢠⣿⣿⣿⡇⠀⣿⣿⡿⠀⠀⠐⠨⠅⠀⣴⣦⡄⠨⠅⠒⠀⠀⢸⣿⣿⡆⠀⣿⣿⣿⣧⡀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠙⣿⣿⣿⡇⠀⣿⣿⣷⠀⠀⠠⠄⣓⠀⠻⠟⠁⢒⡁⠄⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡟⠁⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠈⠻⣿⡧⠀⢻⣿⣿⡆⠀⠀⠡⠊⠎⠆⡏⡜⡑⢌⠁⠀⢀⣿⣿⣿⠀⠰⣿⡿⠋⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⢀⣀⠈⢿⣿⣿⣦⡀⠀⠈⠐⠀⡇⠁⠈⠀⢀⣠⣿⣿⣿⠃⢀⡀⠀⢀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣾⣿⣧⡀⠙⢿⣿⣿⣶⣦⣄⣄⣀⣄⣤⣶⣿⣿⣿⠟⠁⣠⣿⣿⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢹⣿⣿⣿⣦⣀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠉⢀⣠⣾⣿⣿⣿⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠈⠿⠿⠿⣿⡿⠗⠀⣠⣀⣀⣀⣀⣀⣀⣠⠀⠘⠿⣿⣿⠿⠿⠋⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣤⣀⣀⣀⣀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠀⢀⣄⣀⣀⣠⣤⣴⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⣿⣧⣄⠀⠙⠻⠿⠛⠁⣀⣴⣿⡿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⣀⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠋




⣀⡀⣀⡀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠟⠋⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠈⠁⢠⣾⠟⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠙⢿⣆⣴⡿⠃⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠟⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⡿⠃⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⣰⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⢀⣾⠟⠁⢠⣿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢠⣿⠋⠀⣴⡿⠁⢀⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣾⠟⠀⣠⣿⠋⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⢠⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠉⠀⣰⡿⠋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠉⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣄⣀⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰⢠⠂⠀⢲⣸⡇⠀⢰⣇⡀⠂⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢻⡘⠹⣄⠀⡸⠁⡇⢀⢸⡏⢳⡀⠀⡿⠀⠀⡾⠁⣴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠛⠓⠦⣤⠀⠀⠈⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠐⠸⠃⣰⠃⠆⢠⠗⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢛⠶⣄⡀⠆⠀⠀⠀⢀⣠⣤⣴⠶⠶⠿⠿⠿⠿⠿⠷⠶⢶⣤⣤⣀⠀⠀⠈⠀⠰⠏⠐⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠄⠈⠀⢀⣠⣴⠾⠛⠉⡁⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⡉⠙⠻⢶⣤⣀⠀⠈⠀⠀⣼⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠲⢦⡻⠦⠄⠁⠀⣠⣴⠟⠋⠀⢤⣄⠈⢿⣦⠀⣿⡆⠀⡠⠚⡿⣿⡇⣰⡿⠁⣠⡴⠀⠈⠙⢷⣤⡀⠀⠀⠤⠀⣤⠞⠃
⠀⠀⠀⠀⡐⠲⣀⣀⠙⠃⠀⣠⣾⠟⠁⠀⣤⣤⣬⠻⣷⠄⣽⠗⢘⣿⢿⠶⢾⢧⣟⠐⣯⠀⢼⠛⣡⣤⣀⠀⠀⠉⠻⣦⡀⠀⠙⡁⠉⡀⢴⠄
⠀⠀⠀⠀⠙⠺⢭⡉⠀⢀⣴⠟⠁⠀⠀⠀⠈⠉⢡⣾⠃⠾⠏⡾⠟⠁⡈⢳⢁⡀⠻⢧⠻⡇⠘⣷⡉⠉⠉⠀⠀⠀⠀⠘⢿⣄⠀⠀⠋⢀⡟⠀⡀
⠀⠀⠰⢤⣔⠀⠂⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⣄⠘⠁⠀⠀⠀⠀⠀⠈⠊⢸⠀⠊⠀⠀⠀⠀⡀⠈⢁⣴⠆⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠘⠃⠀⠐⡀
⠀⠰⣤⣄⣈⠉⠀⢀⣿⠃⠀⠀⠀⠀⠀⣤⣄⣻⡇⢠⣾⠂⠀⢀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⢿⣆⢸⣯⣤⣤⠄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠤⠖⠛⠁⡀
⠀⣀⣠⡤⠌⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢈⣉⣭⣥⡾⠋⠀⠀⢸⣷⢀⣤⢸⢠⣄⢰⣿⠀⠀⠈⢻⢮⣭⣭⡀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣠⣤⠴⠆
⠀⠉⠀⠐⠂⠀⢰⡟⠀⠀⠀⠀⠀⠀⣤⣄⡉⠉⠁⠈⢢⣀⠲⣦⣿⡾⠋⣾⡆⠻⣎⣵⠶⣂⣔⣁⠀⠈⠉⣠⣤⡤⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢂⣀
⠠⠶⠖⠛⠀⠀⣾⡇⠀⠀⠀⠀⢤⣤⣬⡛⢻⡖⢪⣭⣤⣬⣉⠲⣤⡀⠀⣿⡇⢀⣡⠴⢨⣤⣶⣶⠆⠁⣺⢋⣭⣤⣤⡄⠀⠀⠀⠀⢿⡇⠀⠘⣋⣭⣉
⠀⠀⠁⠒⠀⠀⣿⠁⠀⠀⠀⠀⠀⢈⣽⡏⢸⡇⢀⣤⣤⡏⢿⣦⠀⠙⣄⣿⣇⡞⠁⢠⣿⠂⣴⣶⣤⡀⣿⠈⢿⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉
⠀⠀⠄⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠁⠛⠋⢸⣿⠘⠿⠀⡱⠸⣿⣿⠰⡃⠘⠋⢸⣿⠀⠉⠁⠉⢀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢺⣧⠀⢀⣠⠀⠉⠀⣀⠀⠀⠀⣿⡇⠀⠀⢀⡀⠀⠉⣤⣀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⠀⢀⡀
⢀⣀⣠⡤⠄⠀⢸⣧⠀⠀⠀⠀⠀⢠⣤⣬⣛⣰⡿⠋⢀⣀⣄⢻⡇⣀⡄⣿⡇⣤⡀⢼⢇⡄⣀⣈⠛⠷⠜⣵⣶⡶⠆⠀⠀⠀⠀⢀⣿⠀⠀⠀⠴⠏⢉
⠈⠉⠀⢀⣂⠀⠀⢿⡆⠀⠀⠀⠀⠀⠉⣭⣥⣤⣤⠾⠟⢁⠻⢷⣾⠟⠀⣿⡇⠘⠿⣶⠟⣁⠈⠛⠷⣤⣴⣶⠆⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢁⡴⠛⠃
⠀⠰⠚⠋⠁⣀⠀⠈⢿⡄⠀⠀⠀⠠⣶⣤⡉⠁⠀⡀⠀⠈⢿⣦⣠⡾⠃⣿⡇⠹⣦⣤⡾⠏⠀⠀⣀⠀⠀⢀⣤⣶⠂⠀⠀⠀⣰⡟⠀⠀⣀⠀⠄⢐⠁
⠀⠀⢠⠞⠛⢉⡀⠀⠈⢿⡄⠀⠀⠀⠈⣙⣷⢀⣂⣉⣉⢐⡲⠾⣏⠀⠀⣿⣿⠀⢈⡽⠶⠏⣉⣡⣬⣍⠀⣿⣍⠀⠀⠀⠀⣴⡟⠀⠀⣀⠉⠙⠲⠂
⠀⠀⠀⢰⠞⠫⠀⠀⠀⠈⠻⣦⡀⠀⣴⠿⠃⣀⣭⣉⡙⣷⠀⢀⠨⠻⣶⣿⣿⣶⢋⠄⠀⢀⣿⢩⣭⣥⣤⡘⠿⠆⠀⢀⣾⠏⠀⠀⢬⣉⠷⠶⠆
⠀⠀⠀⠀⠂⠀⠀⢀⣤⠀⠀⠙⢷⣄⡀⠀⠘⠛⠋⣽⡇⠿⠁⠀⠂⠀⢹⣿⣿⠃⠀⠘⠀⠈⠟⢸⣧⠉⠉⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣌⠻⠂
⠀⠀⠀⠀⠀⠀⠴⠋⠀⢀⡤⠀⠀⠙⠻⣦⣄⠀⠀⠻⠃⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣠⡾⠟⠁⠀⠀⡄⢧⣀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠒⣤⠞⢁⡼⠂⡀⠀⠈⠙⠻⢶⣤⣀⡀⠀⠀⣀⣤⣿⣿⣿⣷⣄⡀⠀⠀⣀⣠⣴⡾⠟⠉⠀⠀⠠⡦⢤⣤⣤⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⡎⢡⠗⢠⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠀⠀⢀⢀⠀⠉⠳⡄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠏⣷⠉⠀⠀⣴⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡆⠀⠘⢧⣀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁⢻⠀⢰⣧⡼⠊⢷⡀⠂⠀⢠⡎⠀⠱⡆⣧⠠⠀⠈⠳⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠳⠀⢸⠇⠀⠀⠘⢧⣀⠠⠇⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠠⣴⡤⠀⠀⠀⢀⣰⡤
⠀⠀⠀⠀⣀⡀⠀⠻⣿⠗⠀⠰⡧⠀⠀⠀⠰⢿⡿⠃⠀⠀⠀⢠⣠⡄⠈⠙⠁⠀⠀⠀⠀⠟⠇
⠀⠀⠀⠠⠟⠇⠀⠀⠁⠀⢀⠀⠀⠀⢀⣄⢀⠀⠀⠀⢠⣠⠀⠘⠛⠆⠀⠀⣤⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⢾⡧⠀⠀⢿⣿⣷⠄⠀⠘⠙⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠘⠱⠀⠀⢶⠄⠀⠀⣤⡞⠛⠇⠀⠀⢀⡀⠀⠻⠓⠀⣀⠀⠠⣾⣷⠀⠀⠠⣼⡦
⠀⠀⠀⢀⡄⠀⠀⡰⠂⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠈⠿⠁⠀⠉⠉⠀⠀⠀⠉⠓
⠀⠀⠀⡿⡟⠀⠀⠁⡀⠀⠰⠎⠁⠀⣀⣤⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⣠⠀⠠⣴⣷⡄⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠛⠻⠁⠀⠀⠽⠿⢀⣞⢱⡆⠀⠟⠿⠁⠀⠛⠛⠀⢈⣽⢿⡏
⠀⠀⠀⢀⡄⠐⠀⠀⠀⢀⣰⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢤⡤⠀⠀⠀⢀⠀⠀⠁⠈⠁
⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠘⠁⠀⢠⡾⠋⠀⠀⠀⠀⡿⡅⠀⠀⠀⠀⠋⠑⠀⠀⢶⠋⢳⠆
⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⢾⡟⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠸⠚⠻

⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣄⣀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣧⣿⣾⣿⣟⢿⣿⣿⣿⡧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠉⠀⠞⠽⠿⣿⣯⣿⣿⣾⣿⣿⣿⣿⣟⣿⣽⣿⣿⣿⣥⣿⣿⣷⣝⣿⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⠗⣰⠤⠖⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣷⣿⣟⣿⣿⣿⣿⡞⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡀⡀⢀⠀⡀⠀⠀⠈⣿⣿⣿⣶⢶⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠓⠛⠛⠈⠃⠀⠀⠀⣿⣿⣮⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣦⣴⣐⣤⣤⡆⠀⠀⢸⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢉⡁⠁⠁⠁⠁⠀⠀⢸⣿⣿⣇⢰⡿⠟⠋⣿⣿⣿⣽⣿⠏⣼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢢⡀⠘⠗⠏⠷⢾⠀⠀⠀⣸⣿⣿⣿⠉⠀⠀⢠⣿⣿⣿⡿⠁⢰⣿⣿⢇⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⣿⣷⣦⠀⠀⠀⢀⣴⣿⣿⣿⣿⠀⠀⠀⣾⣿⠇⣿⡇⢀⣾⣿⠏⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⡘⣿⣧⣠⣶⣿⣿⣿⣿⠿⠋⠀⠀⣼⣿⡟⠀⣿⡇⢾⣿⡏⠀⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣼⣿⡿⠁⠀⣿⡇⠀⠉⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣴⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⢠⠿⡇⠀⠈⠙⠃⣀⠀⠀⠀⠀⠀⢀⡀⠀⠂⣀⠔⠊⠁⠀⢠⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⣿⠋⠿⠄⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⡧⢀⡞⠁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⣸⠻⣿⣿⡿⠇⣼⡏⠾⠾⠿⠿⠟⣿⣾⠿⡇⠀⠀⠂⠀⠀⡟⠀⣀⣀⣀⣀⣴⠀⠈⣧⡤⠀⠀⠀⠀⢧⡀
⠀⠀⠀⣿⠀⠈⠙⢿⣦⣛⠀⠀⠀⠸⣷⠺⠛⠁⠀⣿⠀⠀⠀⠀⢸⠃⠀⠀⠀⡀⣠⠃⠀⠀⠀⢆⡤⠊⠀⠀⠘⠇
⠀⠀⠀⢹⡀⠀⠀⠀⣬⠻⢗⣠⣴⠆⢻⣇⠀⠀⢠⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡛⠢⣀⣀⠄⠛
⠀⠀⠀⠀⠑⠀⠀⠸⠿⣴⠿⠋⠑⠾⣦⣿⡄⢠⡞⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠠⠁⠠⠜⠉⠲⢄⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠑⢤⣾⠋⠀⠀⠀⠀⠀⠈⠉⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⠀⣠⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡠⣶⠔⠊⠁⠀⢠⠄⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠁⠀⢰⡿⡂⠀⠀⠀⠤⣀
⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⣟⠊⣷⡀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢸⠃⠀⣀⢀⣀⠀⠀⠀⢱⡀⠀⠀⠀⢀⡶⣶⣶⣶⣶⠆⣾⡏⣴⣶⠶⣶⣤⣶⡶⡾⢇
⠀⠀⠀⡿⠀⠙⠢⢀⠀⠀⡏⠈⠉⠙⡍⠉⠁⠉⠉⠀⠇⠀⠀⠀⢸⡇⠈⠛⢿⣦⡸⠟⠀⠀⠀⢰⣆⢰⠾⠋⠁⢸
⠀⠀⠀⢿⠀⠀⠀⠀⠙⠲⣄⠀⠀⡀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠘⠻⠶⡄⠀⣠⡄⢿⡎⠀⠀⠀⣾
⠀⠀⠀⠘⣆⠀⠀⠀⠠⠀⢀⡵⠊⠁⠀⢳⠀⠀⠀⣰⠀⠀⠀⠀⠈⢷⡀⠀⠀⣴⡗⣠⣴⠿⢋⣥⠘⣿⡄⠀⢐⡉
⠀⠀⠀⠀⠘⢦⡀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠧⠀⡀⠁⠀⠀⠀⠀⠀⠈⠳⠄⢀⣽⠿⠛⠁⠀⠀⠈⠻⠿⣷⣰⠉
⠀⠀⠀⠀⠀⠀⠙⠪⢅⡀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣤⣀⣀⣀⣀⣀⠤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠻⠶⢶⣤⡀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠛⠶⣄
⠀⠀⢀⣤⣏⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⣰⠟⠉⠉⠉⠙⢷⡀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⣄
⠀⢰⡇⠀⠳⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠃⠀⣷
⢀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠘⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠚
⠀⠰⡅⠀⡴⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠲⡄⠀⡿
⠀⠀⠉⠒⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠹⣦⣀⠀⢀⣠⡼⠃⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠊⠁
⠀⠀⠀⠀⠈⠳⢤⣤⣤⣤⣤⡶⠶⠛⠉⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠈⠙⠳⠶⣦⣤⣤⣤⣤⠴⠋
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠀⠚⣭⡔⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠠⠠⠀⢠⣆
⠀⠀⢈⡀⢨⡯⠀⠀⣴⠷⠘⠓⠄⠀⠀⠒⢅⠀⠀⠉⠀⠀⠰⢦⢁⠄⢁⠀⠸⡧⠤⠠⠀⠠⠿⠃⠀⢀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⡆⠉⠀⢠⢟⠀⢱⠀⠂⠀⢀⡂⠀⠈⠙
⠀⢀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⡠⠤⠌⠀⠀⠀⠀⠀⠿⡂
⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡃⢸⠀⠰⣄⣀⠜⠀⢘⡇
⠀⠀⠀⠀⠀⢠⠄⠂⠀⢠⠀⢈⡰⠀⠀⠀⠀⠀⠀⡀⠤⠐⠁⡀⠄⠒⠂⠄⠔⢲⢒⠢⡀⠀⠀⠀⠀⠛
⠀⠀⠀⢀⠄⠀⠃⠐⠄⠂⠁⠢⡄⠠⡀⠐⠀⢰⠐⠀⠄⠁⠀⠁⢀⠀⠈⢂⠘⠀⠀⠌⡀⠀⡀⡀⡂
⠀⠀⠀⠐⠀⠀⠀⠢⡀⠈⠁⠈⠂⠒⠄⠑⠀⠁⠀⠈⠂⠢⠀⠀⠈⡀⠀⠂⡀⡀⠀⠀⠤⠠⠀⠀⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠈

⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠈⠙⠳⣄
⠀⢀⡾⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⢠⠟⠀⠀⠀⢀⡀⡀⣀⠀⠀⠀⠻⡄⠀⠀⠀⡼⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢸⠃⠀⠀⠀⢋⡔⢰⣿⣷⠀⠀⠀⠸⡇⠀⠀⡟⠀⠀⢰⣌⠲⠥⠿⠈⠣⠀⠀⠀⢻⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⣒⡠⣀⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠀⢸⡄⠈⢛⠅⠀⠀⠀⢀⡇⠀⠀⣇⠀⠀⠀⢻⣷⣶⣶⣿⣶⠀⠀⠀⣸⠀⠀⢸⡄⠀⠀⠀⢳⣤⡳⣦⣴⣿⡄⠀⠀⡏
⠀⠘⣧⠀⠀⠀⠓⠙⠛⠑⠿⠀⠀⠀⣼⠁⠀⠀⢻⡄⠀⠀⠀⠙⠻⢿⡿⠟⠀⠀⢠⡟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠴⠿⠛⠛⠁⠀⣸⠃
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠓⢦⣄
⠀⠀⣴⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠈⢳⡄⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⠀⠀⠙⣦
⠀⢸⠃⠀⠀⠀⡆⠾⢃⢰⡏⠁⠀⠀⢸⡆⠀⠀⡿⠀⠀⠀⣠⡀⠟⡡⠤⣄⡀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠰⡌⠻⣿⡟⣩⡄⠀⠀⠘⡇
⠀⢸⠀⠀⠀⠰⣝⣛⣡⣿⢣⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⢰⡇⠿⢆⡴⠒⠀⠀⠀⠀⢸⠀⠀⢸⡁⠀⠀⠀⠁⡇⠿⠸⢋⠁⠀⠀⠀⡟
⠀⠸⣇⠀⠀⠀⠈⠉⢭⡴⠋⠀⠀⠀⣸⠃⠀⠀⢻⡀⠀⠈⠣⠰⠎⠀⠚⠂⠀⠀⢀⡿⠀⠀⠘⣇⠀⠀⠀⠼⠇⠀⠀⠹⠆⠀⠀⣰⠇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⠴⠞⠋

⠀⠀⠀⠀⣠⡴⠶⠚⠛⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠓⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠒⠶⢤⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⣀⡀⠀⠈⠙⣦⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢰⡏⠀⠀⠀⠐⢤⠀⢀⡤⠀⠀⠀⢻⡄⠀⠀⣼⠁⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠈⣧⠀⠀⢠⡟⠀⠀⠀⠀⢀⣀⠀⣬⠀⠀⠀⢹⡆
⠀⢸⠀⠀⠀⠀⣠⢼⠤⣾⣠⠄⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⠤⠄⠈⢥⣶⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⢠⣤⢹⡇⣿⠀⠀⠀⠀⣧
⠀⢸⡄⠀⠀⠀⠀⠈⠠⠌⠂⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⠀⠀⢺⣷⡄⠀⠀⠀⠀⣼⠀⠀⠸⡇⠀⠀⠀⠀⣔⣫⣼⣿⠀⠀⠀⢠⡇
⠀⠀⢳⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠟⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⠛⠃⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄⠀⠀⠀⠉⠉⠉⠁⠀⠀⢠⡞
⠀⠀⠀⠙⠳⣤⣀⣀⠀⣀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⢀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⠀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠂⢀⠂⡐⠅⢑⡆⠠⡀⠀⠀⢤⠁⡆⠀⠀⠠⠀⣀⡀⢠⠀⡠⠀⠑⢀⡈⠠⡀⠀⠀⣐⠟⠣⢶⠂⢾⠋⣱⢄
⠀⠀⠀⡌⡠⠋⢀⢤⡴⣌⠲⠁⠂⠀⠣⠈⠀⢀⠀⠀⡗⣦⠁⠉⠀⢠⡆⣴⠠⠀⠀⠄⠀⢀⣛⠿⠛⡀⢆⠂⠔⢁⣀
⠀⠀⠘⡤⠄⠠⠁⠸⢯⢟⠀⡇⢱⠀⠀⠀⠨⣀⠀⠀⠋⢅⠀⠀⠀⠈⠁⠉⠐⠀⠀⠃⠀⠚⢉⡀⠐⢰⠊⠺⣥⡴⠋
⠀⠀⠀⠠⠔⣀⠈⠢⠄⠀⡐⠁⠃⢀⠐⣶⡀⠀⠀⠀⠈⠀⠐⠀⠀⠂⠀⠀⡐⠀⠀⠀⠀⠈⠛⠒⠖⠉⢃⠀⠩⠀⠀⣤⠄
⠀⠀⠀⠀⡑⠢⡁⣂⠈⡥⠈⠁⠀⠈⠑⠈⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⢡⣀⠀⠀⠀⢠⠀⠀⠀⠐⡁
⠀⠀⠀⣈⣙⠷⠀⠬⢠⠆⢀⠜⢀⢄⠀⠀⠀⠸⣧⢘⠃⠀⠀⠀⠙⢂⠀⠀⠀⢀⡖⣡⣄⠤⣌⢢⠀⠀⢀⠀⠠
⠀⠀⠰⠓⢣⡤⣄⠰⠉⠀⠀⢠⡅⠄⣥⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⠑⢂⠐⠈⢠⠄⠃⠀⠉⠀⠻
⠀⠀⠀⠀⠀⠁⠁⠆⠀⠀⠀⠄⢇⠎⡖⢁⠀⠀⢀⡤⣄⣠⠀⠀⢀⢤⣴⡄⠀⠀⠉⠦⡄⠡⠴⠁⠀⠀⠀⠀⠈⠒⠀⡑⠁
⠀⠀⠀⠔⠀⠀⠊⠀⠀⠂⠀⢂⣠⡖⢰⣦⠀⠀⠈⠃⠛⠀⠀⠀⠀⠘⠁⠁⠐⠀⠀⠀⠐⠂⠀⢀⠊⠀⠈⢂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠐⡔⢄⠀⠀⠈⠃⠈⠀⠀⢠⡀⠀⠠⠂⠀⠀⡀⣀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠄⠀⠀⠌
⠀⠀⠀⢀⢲⢙⣸⣀⠂⠌⡑⠀⠐⠄⠔⠀⢀⠀⠀⠀⠀⠀⠜⡨⣥⢤⡝⢆⣀⠀⠀⠠⠐⠀⠀⢀⣈⡉⠙⢀⠀⠀⠐⠠
⠀⠀⠀⠘⢯⣴⠞⣶⣎⡑⢞⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡄⠀⠿⠹⠏⡇⣹⠂⠀⠀⠀⠀⠀⢢⠻⠀⠀⢠⠂
⠀⠀⠀⠀⠠⠁⠰⠉⠉⠉⠁⠀⠀⠙⢄⢖⣼⣾⡠⠋⠀⠀⣃⠅⠨⠀⢠⡎⠀⠀⠀⠀⠐⠀⠀⠀⠑⠒⠒⠁⠀⠀⠐⠈
⠀⠀⠀⠀⡆⢀⠁⠀⠀⠘⠅⠈⠀⠀⠌⣀⠠⠁⠁⠀⠀⠀⢢⠤⠤⠤⢤⠀⠀⠬⠄⠀⠀⡀⠀⢠⠀⠀⠀⢠⠀⠀⡠
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠈⢀⡄
⠀⠉⠙⡟⠉⠉⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠈⠁
⠀⠀⡀⠃⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⣤⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠙⠁⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣦⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢰⣄⠄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤
⠀⠿⠗⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠊⠙⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁
⠀⠀⢀⣀⠀⠠⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠃
⠀⠀⣿⣿⡇⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡼⡖⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠿⠂
⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⡄⠀⠓⠃⠀⠀⠀⠀⢀⢄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠋⠀⠠⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠆⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⠞⢉⣡⣤⣉⠙⢦⡀⠀⠀⠀⠀⢀⣴⣿⠿⠿⣿⣶⣄⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⣿⣶⡄⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⣿⣶⡄
⠀⠀⠀⠀⢠⡏⢰⠋⠀⠀⠈⢳⠀⣧⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⢹⣿⡆⠀⠀⠀⢀⣿⡟⢁⣴⣦⡀⢻⣿⡄⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠘⡇⠸⣄⠀⠀⠀⡼⠀⡿⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠘⣿⣧⠈⠻⠿⠃⣸⣿⠇⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠹⡄⠈⠓⠒⠋⠀⡼⠁⠀⠀⠀⠀⠘⣿⣿⣶⣶⣾⣿⠏⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⣾⣿⠏⠀⠀⠀⠀⠀⠘⣿⣷⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣿⣿⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢾⣉⡀⠀⠘⠃⠀⢀⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠚⠻⣿⡍⠻⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇
⠀⠀⣠⣄⢀⣠⣿⠷⣄⡙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⣿⣦⡀⠀⠀⠀⡀
⠀⠀⢹⠟⠛⢻⡏⡀⢈⠻⣿⡿⣆⠀⠀⠀⠀⠀⠀⢦⡌⢳⣄⠀⠀⢀⣴⡇⠀⣠⡞⢁⣠⠀⠀⠀⠀⣿⣾⣷⣾⣿⠋⠉⢿⣿⣶⣷⣾⡇
⠀⠀⠀⠀⣰⣾⣾⣿⣾⢷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡀⠀⢀⣾⠀⢀⡿⠀⠉⠁⠀⠀⠀⠀⢻⣿⣿⣿⠄⠀⠀⠀⣽⣿⣿⣿⠁
⠀⠀⣀⣠⢿⠟⢹⣿⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠿⢀⣸⣿⠀⠿⠀⠶⠋⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣾⣷⣶⣶⣿⣿⡿
⠀⠀⣿⣯⠀⠀⠛⢿⠀⢼⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢻⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠙⠋⠙⢹⣿⠙⠉⠛⠙⠇
⠀⠀⠈⠻⢷⣤⣀⣀⣀⣀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠃⡞⠰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣤
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⣴⣶⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⠟⠛⠁
⠀⠀⠀⠀⢰⡿⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠀⠀⣿⣿⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠈⢀⣴⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⢸⡿⢿⡇⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠈⠁⠈⠁⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣰⠁
⠀⠀⠀⣦⡀⣀⣿⣿⣦⣀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡆⢰⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢁⣾⠏
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⠋⢴⣿⠏⢠⠆
⠀⠀⠀⠀⠀⠈⠻⠀⠉⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠋⣿⠈⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣛⠛⠋⠼⠛⣁⣴
⠀⠀⠀⠀⢰⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣆⠻⠏
⠀⠀⢠⡀⢸⣿⣦⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⣷⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⢸⣿⣤⣿⠈⢻⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣀⣿⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢸⣷⡈⠻⣧⣄⣿⣼⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⡏⠹⣿⢸⡏⢩⡇
⠀⠀⠈⠛⢷⣦⣿⣿⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⠀⣧⣼⠛⠘⢳⣂⡇
⠀⠀⠀⠀⣀⣽⡿⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⠟⠙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣷⢶⣿⠛⣿⠛⠛⡇
⠀⠀⠀⠀⠉⠛⠷⠶⠶⢶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⠘⢿⠄⣉⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡟⠛⠛⢀⣿⣸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⡄⢸⡇⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠
⠀⠀⠀⠀⣠⣿⣧⣾⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣴⣟⠀⠀⣀
⠀⠀⠀⣰⡟⠉⢹⣿⠀⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡿⠿⠿⠛⠁
⠀⠀⢠⠏⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣶⣖⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢀⣴
⠀⠀⠞⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠙⠉⢠⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⢻⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⠀⣿⢸⡿⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⣴⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠙⢀⡿⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣇⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠰⣶⢃⡞⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⣀⠀⡀⠍⠙⠁⠀⠀⠄⠀⠀⠀⠁⣶⣀⠀⠄⠀⠀⠀⢀⡤⡀⠀⠈⣀⣛⠀⠃⡀⠀⠲⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⠀⣿⠀⠉⠒⠐⠂⠀⠀⠀⠀⠀⢠⣆⡄⢹⢀⡀⢻⣄⠀⠉⠻⠋⠀⠀⠹⠌⠀⠀⢱⡆⠸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣐⡇⠀⢀⡐⠋⠀⠀⠀⡄⢀⡹⣆⠀⠀⠀⠀⠑⠀⢩⣶⠀⠀⠀⠀⠀⠈⠔⣀⠀⠀⢃⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠊⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⢠⠀⠁⠈⠓⢠⡀⣄⠀⠀⠀⠀⠀⠀⠦⢄⡈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢠⠀⠉⠀⠓⠀⡀⠃⠀⠀⡇⢀⠁⠈⢳⡏⠀⠀⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⡀⠀⣴⠀⠀⢀⠇⠀⠀⠀⠀⠉⣤⠀⢁⠞⣠⠀⣠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡭⠂⠀⠒⠄⠀⢉⡀⠁⠀⡝⠀⠀⠈⣴⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠐⠀⠾⠡⣀⢀⢠⠐⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠤⠀⠀⠀⠀⠀⠀⠤⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠦⠀⠀⠸⢓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠰⠀⠚⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢛⡅⠀⢠⠀⠀⡀⠀⡔⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠦⠀⠀⢰⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣸⡆⠀⠀⠀⠀⠀⠤⠆⠀⠀⢁⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠀⢤⣀⠀⠀⠀⡤⠿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠂⡀⠀⠀⡄⠀⠀⠀⠀⣠⡐⠀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠓⠤⣀⠴⠬⠆⣌⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⢈⠀⠀⠀⢠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⡀⠀⠀⢰⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢀⡤⠒⠃⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡜⠃⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠑⠀⠀⠀⠀⠄⠂⠀⠀⠀⣀⠀⣀⠤⠂⠀⠁⠀⠀⠀⠘⣇⣀⠐⠀⠒⡄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⡄⠃⢠⠐⠁⠀⠀⠀⠈⠀⠂⠀⠀⠄⠀⠖⠀⠀⠀⠀⠀⠀⠈⠁⠣
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⣄⠀⠀⣀⢀⠀⠢⣀⠴⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠐⠀⠲⠄⠢⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⢀⠂⠀⣠⠉⠘⠷⣆⠀⠈⠇⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠈⠂⣀⠕⣠⠀⠀⠀⠀⢀⡴⠖⢀⡠⠊⠀⠈⣫⡄⣀⡀⠩⠄⠀⠀⢑⢦⣀⠩⣷⠄⠀⠀⠀⣠⠀⢄⣠⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡎⢠⣆⠀⠀⠀⣿⠃⢀⠱⣇⠀⠠⡀⠀⠀⠀⠁⠀⠀⠀⠰⢣⡖⠛⠂⢽⡆⠀⠀⠠⠵⠛⠉⠀⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⠆⠀⠀⢺⡅⠀⠃⢀⣹⡶⠀⠂⠀⠀⠀⠀⠀⠀⡀⠹⣤⠤⠀⣸⠇⢀⣤⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡽⣦⠀⠈⠳⠀⣀⣉⡫⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠔⠀⢁⡴⠛⠉⠰⠄⠀⠀⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠺⠀⠁⠂⠠⣀⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⠉⠀⣰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡀⠄⠐⠉⠁⠙⡁⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⡠⠃⠀⠃⠀⠀⠀⠈⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠤⠀⢀⢤⠀⠀⠈⠀⠐⢂⠡⠆⡈⢀⡐⠀⠀⠀⢀⡱⡃⠀⢀⠤⠒⠠⢄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠊⠀⠀⠂⠀⠀⠀⠀⠀⡀⠜⠣⠐⠈⠤⠈⡀⠀⠀⡍⠀⠀⠠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠎⠉⠀⠀⠀⡄⠀⠐⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠠⣦⣘⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⢠⠀⠀⠁⡀⠀⠀⠀⢀⠤⢨
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⢀⡤⡀⠐⠀⡀⣀⡤⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢳⣤⡀⠯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁
⠀⡂⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⡃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣦⠀⠀⠀⣾⣿⣟⣯⣭⡿⡻⣿⣿⣆
⠀⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⠀⠀⢸⢾⣿⣭⢾⣿⣿⣦⣟⠿⣻
⠀⡆⠀⠀⠀⠀⠀⠻⣿⣿⠻⣿⣿⣿⣶⣧⣾⠟⠀⠀⠀⠸⡿⢻⢿⣷⣿⢿⠿⣛⣽⡿⠀⠀⠸⣷⣿⣟⠮⠽⣿⣟⣽⣾⡡
⠀⡅⠀⠀⠀⠀⠀⠀⠛⣷⣾⡿⣽⣿⣿⣿⠟⠀⠀⠀⠀⠀⠁⠢⣜⠛⢿⣐⡿⢿⠟⠁⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⣿⠟⠁
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢂⠖⠐⠢⡀⠀⠀⠀⠀⠀⠀⢠⢔⠉⡏⠉⡳⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢀⠊⠀⠐⠀⠘⡆⠀⠀⠀⠀⠀⡧⠤⢵⣷⡮⠤⠼
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⠀⠜⠀⡅⠀⠀⠀⠀⠀⠃⡴⠋⣬⠑⢤⡜
⠀⠀⠀⠀⢀⡞⢐⠙⠻⠛⢑⠘⡄⠀⠀⠀⠀⠀⠈⢧⡈⠃⠀⠊⢀⠝⠁⠀⠀⠀⠀⠀⠀⠈⠑⠙⠚⠁
⠀⠀⠀⢀⠎⠀⠀⠉⠀⠈⠁⠀⠈⢄⡀⠀⠀⠀⠀⠀⠈⠉⠁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣷⣦⣄
⠀⠀⠠⠊⠔⠢⠢⠄⠔⠄⠢⠂⠔⠜⠃⠀⠀⠀⠀⠀⡀⢠⠀⡴⢀⠂⠀⠀⠀⠀⠀⠀⡈⠘⣿⣯⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣆⣹⠞⠒⠓⠺⣤⡄⠀⠀⠀⠀⠀⡀⠀⠈⠻⠟⠛⠉⠉⠹⣿
⠀⠀⠀⠀⠀⢀⣤⡶⠋⠉⠉⠀⠀⠀⠀⠀⠀⣈⠓⡿⠑⠖⠁⠘⠷⠊⣿⣁⢀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠰⠇⢀⡟
⠀⠀⠀⠀⢠⢃⣎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠹⣷⠀⠀⢀⠀⠀⢀⣿⡍⠀⡀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠐
⠀⠀⠀⠀⡏⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⢧⣄⣙⣁⡤⠾⠁⠀⠄
⠀⠀⠀⠀⠹⡄⠻⣄⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠐⠁⠼⠈⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣆⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠙⠦⣈⡓⠒⢤⣤⠶⠋⠀⠀⢀⢤⡀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠠⣔⡪⠋⢯⠿⠄⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⣅⠀⠸⣄⠃⢊⣷⣂
⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⣤⡀⠀⠀⣀⢀⣸⠺⣀⢠⡄⠀⢀⢀⣴⠋⠀⠈⠰⡄⠀⠀⠀⠐⣘⢦⣧⢉⣁⠽⣵⠂
⠀⠀⢠⣾⣯⣥⡖⠋⣟⡏⢲⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣿⣄⡀⢀⣼⡟⠀⠀⠀⠀⠀⠃⠾⠙⠘⠆⠀⠑
⠀⠀⠀⢠⠤⠀⢉⣱⣿⣿⠉⠀⠀⠀⠀⠀⢘⢱⠀⠀⠀⠀⠀⠐⠈⢙⠛⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⠻⡷⠒⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡠⢤⠤⢄⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠀⠘⠀⠘⠀⢸⠀⠀⠀⠀⠀⠠⠐⠉⠉⠀⠂⣀
⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠱⠀⠀⠀⠀⠀⠸⣾⣤⠁⠀⠀⠀⢀⡇⠀⠀⠀⢀⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⣠⢂⢂⠀⠘⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠙⠙⠀⢸⠀⠈⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠘⢃⣼⣀⡟⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⣄⢀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢰⡀⣀⠀⠀⠀⠀⠠⡀⣆⣠⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢼⣿⣿⣿⣿⡤⠀⠀⢠⣤⣴⣿⣤⣤⠄⠀⠀⠀⠤⣷⠷⣧⠄⠀⠀⢀⣠⣽⣿⣿⣧⣀⠀⠀⣈⣳⣿⣿⣷⣏⡀⠀⠀⣀⣈⣶⢷⣊⣀⡀
⠀⠾⣿⣿⣿⣿⡓⠀⠀⠀⢈⣿⣿⣿⠁⠀⠀⠀⠈⡩⢿⢶⠿⡉⠀⠀⠀⢀⠿⢿⠟⢧⠀⠀⠀⠠⠞⡿⣿⣟⠲⠀⠀⠀⠀⠠⠛⡟⠢
⠀⠀⠉⠛⠙⠁⠀⠀⠀⠀⠘⠉⠀⠙⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠀⣆⠀⡀⠀⠀⠀⡀⠀⣰⡀⠀⡀⠀⠀⠀⣀⠀⣴⡀⢀⡀
⠀⢶⣾⣿⣿⣶⠖⠀⠀⠀⠀⠀⣸⠀⠀⡀⠀⠀⠀⠀⢄⣼⣌⠄⠀⠀⠀⣀⣸⡷⠿⣾⣁⡀⠀⠀⣸⣟⣯⣟⣿⡁⠀⠀⠀⣹⣿⣿⣿⣿⣀
⠀⣹⣿⣿⣿⣿⡀⠀⠀⠀⠈⢛⣿⡟⠉⠀⠀⠀⠈⠩⢙⢿⠟⠉⠉⠀⠀⢀⣽⣧⣠⣾⣍⠀⠀⠉⢻⣞⡷⣟⣾⠋⠀⠀⠉⢻⣿⣿⣿⣿⠋
⠀⠉⠈⢻⠏⠃⠁⠀⠀⠀⠀⠎⠀⠙⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠟⠙⠇⠀⠀⠀⠀⠚⠉⢻⠋⠙⠂⠀⠀⠀⠛⠉⢿⠋⠙⠃

⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⣤⣤⣿⣷⣤⣴⠆⠀⠀⠀⢤⣀⣧⣠⠄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⣀⣹⢾⣻⢾⣁⣀⠀⠀⢤⣤⡞⢷⣤⡤⠀⠀⠀⣤⣷⠾⢷⣧⡤
⠀⣸⣿⣿⣿⣿⡁⠀⠀⠀⠠⠴⢿⣼⡷⠤⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⢀⣬⣯⣿⣿⣧⣯⣄⠀⠀⢰⣗⡀⢀⣻⡀⠀⠀⠰⢾⡇⠀⠀⣹⡷
⠘⠛⠙⢿⡟⠉⠛⠂⠀⠀⠀⠀⠏⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⡟⠻⠆⠀⠀⠀⠉⠉⢻⡟⠉⠉⠀⠀⠀⠋⡿⢶⠾⡏⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢠⡦⢀⣠⣇⡀⢶⡄⠀⠀⠀⢼⣷⣷⣾⡤⠀⠀⠀⣐⣮⣷⢷⣧⣆⠀⠀⠀⣀⣠⠟⣧⣀⠀⠀⠀⠀⢄⣸⣆⡠⠀⠀⠀⠀⢶⣄⣿⣠⣶⠃
⠀⢀⢿⣿⣿⡿⡀⠀⠀⠀⠠⢾⣿⣿⣿⠦⠀⠀⠀⢾⣿⠀⠀⣹⣿⠄⠀⠙⢯⡍⠀⠈⣽⠟⠀⠀⠴⢾⣿⣿⡶⠆⠀⠀⠴⠶⣿⣿⣿⡷⠶
⠈⠗⠉⢙⡏⠁⠺⠁⠀⠀⠀⠈⠀⠇⠈⠀⠀⠀⠀⠩⢿⡷⡾⡟⠏⠀⠀⠀⢸⣧⡶⣦⣿⠀⠀⠀⠀⠊⢹⠏⠓⠀⠀⠀⠀⡼⠋⣿⠛⠷⡄
⣀⣀⣀⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⡤⠤⢄⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⢄
⠀⠠⡦⠋⣴⠄⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠾⢿⣷⡄⠀⠀⢼⣷⠶⠶⠀⠀⠀⠤⠦⠶⠶⠂⠀⠀⠙⢿⡿⣶⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠓⠛⠚⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠛

⠀⢸⣛⣖⣒⡀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⢠⣾⡿⣶⡀⠀⠀⢴⣲⣶⣶⠀⠀⠀⠘⢾⣇⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⢀⢼⢧⡀
⠀⠈⢿⣿⣿⡇⠀⠀⢸⣟⣏⣻⡆⠀⠀⠀⢈⣡⡇⠀⠀⠀⠘⢿⣷⠿⠁⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢻⡟⠆⠀⠀⠀⠛⠓⠚⠛⠀⠀⠀⠸⣼⡦⠃

⠀⠀⣀⣠⣀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣤⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⠂⠀⠀⠐⠾⢿⠷⠂⠀⠀⠾⠿⠿⠷⠂⠀⠀⠰⣿⣿⡟⠀⠀⠀⠈⠉⢹⣿⠀⠀⠀⢿⡿⠿⡷⠀⠀⠀⠀⣼⣿
⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣾⣿⡇⠀⠀⢀⣠⣷⢀⠀⠀⠀⠀⣤⠿⣄⠀⠀⠀⢠⣤⣤⣆⠀⠀⠀⢸⡗⡿⣤⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠘⢭⣦⡀
⠀⠘⠷⣿⡟⠇⠀⠀⠘⠃⠿⠙⠁⠀⠀⠀⠿⠽⠿⠀⠀⠀⠘⠋⠙⠿⠃⠀⠀⠠⠥⠥⠤⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⡭⢯⠃⠀⠀⠀⠳⠎⠑⠊

⠀⢀⡆⢀⣤⡀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣄⠀⠀⠀⢠⢴⠦⠄
⠀⢸⠉⠚⠙⠁⠀⠀⠈⣿⠾⡿⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⢼⣳⣷⠀⠀⠀⢿⡁⢁⣹⠃⠀⠀⠰⠶⠶⠷⠀⠀⠀⢀⢁⡈⣟⠀⠀⠀⠈⠷⠶
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⢀⣰⡜⡧⡃⠀⠀⣠⣧⣼⣧⠄⠀⠀⠠⠏⠤⢀⠀⠀⠀⠀⠎⠵⡂⠀⠀⠀⢀⡝⢫⡀⠀⠀⠀⢀⣼⣦⡀
⠀⠰⠾⠿⠿⠇⠀⠀⠀⠀⠘⠿⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠿⠀⠻⠿⠀⠀⠀⠘⠓⠒⠛⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠈⠷⠾⠁⠀⠀⠀⠁⠸⠄

⠀⠀⠀⠐⠒⠂⠀⠀⠀⣴⢶⢤⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⢰⣦⣀⡄⠀⠀⠀⠀⠀⣰⣆
⠀⢰⣷⣷⣾⡄⠀⠀⢸⣿⣿⣿⡀⠀⠀⠸⠟⠿⠿⠀⠀⠀⣤⣤⣤⣤⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠘⡿⠃⢠⠀⠀⠀⠈⠿⠿⡋
⠀⠀⠁⠉⠈⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣠⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣶⣿⣶⠀⠀⠀⠈⠰⣿⠇⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⣾⣟⣻⣤⡀⠀⠀⣾⣶⣳⣻⠂⠀⠀⠐⠒⠃⠀⠀⠀⠀⣆⠿⠿⣳⠀⠀⠀⢰⣶⣶
⠀⠀⠛⠿⠛⠀⠀⠀⠀⠿⠿⠻⠇⠀⠀⠀⠿⠙⠟⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠸⠋⠀⠀⠀⠘⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠀⠂⢀⠀⠂⠂⠀⠀⢐⡱⠀⠀⡀⠀⠀⡀⠀⠀⠤⠀⠀⠀⢀⠁⠑⠀⠀⠀⡩⡤⣁⠀⠀⠈⠛⠁⠀⠀⠀⠠
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠞⠃⠀⢠⣤⠀⠀⠀⠐⠶⠆⠀⠀⠺⣷⡟⠂⢀⡜⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢘⠳⠟⠀⠀⠁⠁⠁⠀⢘⠻⡀⠀⠀⠐⠄⠄⠀⢀⡀⠀⢀⠀⠀⠀⠚⠘⠂⠀⠀⣀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⡀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠈⠀⠐⠋⠛⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠂⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⡄⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⠀⠀⠀⠴⠀⠆⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠅⠂⡀⠀⠀⠈⠈⠠⠀⠀⡀⠈⠉⢁
⠀⠐⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠐⠀⠀⠀⠀⠀⢄⢄⠀⠀⠀⠀⠀⠄⠠⠀⠀⣤⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠊⢀⠁⠀⠀⠐⠐⠂⠚⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠂⡶⡄⠀⠀⡾⢷⡀
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⠉⠂⠀⠀⢀⠈⠀⠀⠀⠀⠂⢸⣿⡇⠐⠀⢐⠈⠉⠈⠉⢀⠀⠀⠉⠱⠉⠁⠀⠀⠁⠈⠁
⠀⠀⠀⠖⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣀⠀⠀⠀⠈⠁⠑⠀⠀⠀⠤⠈⠉⠠⠄⠀⠀⠀⢀⠀⠀⠈⠀⠀⠐⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠐⡶⠲⡂⠀⠀⠀⢠⣆⡀⠀⠀⠀⠀⠈⢫
⠀⠀⠀⣠⢠⣄⠀⠀⠀⠁⠈⠀⢀⠀⠀⠀⠀⠀⠁⢈⠃⠀⠀⠠⡤⠀⠀⠀⠆⠀⠂⠀⠀⠀⠸⣿⡧⠀⠀⠀⠀⠁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⡖⠈⠉⠤⠤⠐⠒⢈⣆⠀⠀⠀⠀⡔⠉⠉⠲⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠝⠣⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⠭⠭⡇
⢰⢀⡀⠠⠤⠤⠤⠉⠀⢁⠀⠀⢠⣦⡀⠀⡠⢁⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⡤⡆
⠜⠚⠲⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠺⠦⠤⠰⡾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠧⠤⠤⠴⠳⠄⠀⠀⠀⠀⠀⣈⠉⢉⡀⠀⠀⢀⣇⣂
⠀⠀⠀⠉⠉⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠰⢳⠀⠀⡌⠀⠀⠀⠀⢠⠀⠁⠀⡠⠒⠢⡄
⢀⠀⠀⠀⠀⠀⠀⢸⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⢄⠀⠀⠀⠀⢰⠋⠀⠀⠰⠰⠀⠀⠈⠢⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⡸
⠸⠤⠤⠤⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠀⠀⠀⠈⠢⠄⠀⠤⠄⠀⠀⠤⠊⠀⠀⠀⠘⠄⠀⠀⠈⠃⠈⠁⠀⠠

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⡘⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠛⢦⡀
⢠⠠⠠⡐⠤⠠⠎⡤⠄⢤⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠘⡦⢤⡎⠀⠀⠀⠀⠀⠀⢠⠒⠙⣇⠄⣄⠠⢄⡋⠐
⡆⠀⠀⠈⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⣤⣠⣤⠟⠀⠀⠀⠀⠀⠀⡠⣊⡄⠀⣌⠢⡀⠀⠀⠀⠀⠘⠂⠞⠁⠀⢻⠀⠀⠱⡆
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠔⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠇⠀⡄⠈⠸⠃⠀⠀⠀⠀⠸⡀⠀⠀⠘⠶⠐⠆⡁
⠀⢢⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢀⠔⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⢀⡜
⠀⠀⠑⠄⠀⠀⠀⠔⠁⠀⠀⠀⠈⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎⠘⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠈⠻

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣠⠖⡱⠋⡏⠫⡑⢄⠀⠀⠀⠀⠀⠀⣀⠞⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠳⡀
⠰⢡⡞⠀⠀⡇⠀⠰⡄⢧⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠢⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⢄
⣇⣰⣀⣀⣀⣇⣀⣀⣰⣠⡆⠀⢰⠧⡀⠀⠀⠀⠐⠀⣀⠾⡆⠀⠀⠀⠀⡀⠘⠀⠀⠀⠙⢌⢣⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⡠⠋⠘
⢇⢨⡄⠀⢠⡇⠀⢀⡌⣠⠂⠀⢸⠀⠈⠓⢤⣀⠤⠊⠀⠀⡇⠀⠀⠀⡄⡇⠰⡀⠀⠀⠀⠈⡆⢡⠀⠀⠀⠀⠀⡠⠃⠈⠉⠁⠘⡄
⠈⢆⠱⡄⠀⡇⢀⠜⢠⠊⠀⠀⢸⠀⢀⡤⠊⠉⠒⢄⡀⠀⡇⠀⠀⠀⢡⡈⠢⣈⠒⠒⠂⠀⡇⠀⠀⠀⠀⢀⣘⡡⡀⠀⠀⠀⢀⠃
⠀⠀⠑⠨⠶⠷⠧⠊⠁⠀⠀⠀⠸⠖⠁⠀⠀⠀⠀⠀⠈⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⠀⠀⠀⠈⠀⠀⠘⠦⠠⠴⠃

⠀⠀⠀⡠⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⢠⠤⣀⠤⡀
⠰⠐⠒⠛⠐⠚⠓⠒⠒⠲⠀⠀⠀⠐⣡⠊⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠃⠀⡤⣄⢀⢤⠀⠈⡂⠀⠀⠀⠀⠀⢸⡰⠃⠀⠑⢄
⠰⡤⢤⠤⠤⠤⢤⡤⢤⠐⠀⠀⢀⣀⡁⠀⠀⠀⠀⠀⣸⠦⠲⠀⠀⠀⠀⢇⠀⠀⠉⡏⠁⠀⢀⠆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠢
⠀⠇⠈⠆⠀⠀⢾⠀⢸⠀⠀⠀⡌⣀⡜⠃⠀⠀⠀⠚⠥⣀⠆⠀⠀⠀⠀⠈⢆⠀⠀⠇⠀⢀⠌⠀⠀⠀⠀⢀⠼⠒⠒⠒⠒⠒⠒⠒⠷
⠀⠸⢰⢸⠀⠀⢀⠀⡈⠀⠀⠀⠋⠸⣇⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⢸⣤⠊⠀⠀⠀⠀⠀⠀⢸⠀⠌⣢⢘⠉⠉⡂⡃
⠀⠘⡂⢼⠀⠀⢺⣠⠃⠀⠀⠀⠀⠀⠙⢦⡄⠄⠄⣊⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣾⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠼⠈⠉⠈⠁⠇
⣶⣶⣿⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠒⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠦⣀⠀⠀⠀⣠⠖⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠢⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⠲⠞⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡴⣡⣤⢤⡤⢤⠀⠀⠀⡖⠒⠛⠛⠉⠉⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⠃⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠤⠤⣀⣀
⠀⠀⠀⠀⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠚⡽⠀⠀⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⢖⡟⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣠⠞⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠗⣫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠗⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠊⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡠⣶⠃
⠀⠀⠀⠀⠀⠀⢀⡞⣡⡴⠁
⠀⠀⠀⠀⠀⢠⠞⠉⡼⠁
⣀⣀⣀⣀⣀⣈⣤⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣸⣇⣠⡆
⠀⠀⠀⠀⡰⠊⠁⠀⢀⣷⣴⣧⣴⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠻⠁
⠀⠀⢀⡞⠁⠀⢀⣶⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣶⣻⠶⢄⣀
⠀⠀⣼⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⢷⠀⠀⠀⢀⡀⠀⠐⠀⠀⢸⣿⣯⣻⠿⠎⢿⡆⠀⠀⠀⠀⢀⡀
⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⢸⠀⠀⠀⢀⣹⣿⠀⠀⠀⣾⣿⡃⠛⠣⡄⢠⣷⠀⠀⠀⣿⣏⡁
⠀⠀⣷⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⠀⠀⠈⠉⣻⣿⠀⠀⠀⢻⣿⣿⣥⠉⣈⢻⣿⠀⠀⠀⣿⣟⠉⠁
⠀⠀⠸⡆⠀⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠏⠀⠀⠀⠘⠉⠉⢂⠀⠀⠈⠉⠀⣿⣿⡿⣾⠷⠄⠀⠀⠉⠉⠃
⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⣿⣿⣿⣻⢯⠛⠁⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠈⠑⢦⣀⣠⣿⣿⣿⣿⡏⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠿⠿⠿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣶⣆⠿⠿⢾⣰⣷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣾⠶⠋⠁⣄⠀⢀⣆⠀⠈⡙⠢⣿⣶⣀
⠀⠀⠀⠀⠀⠀⣤⠞⠋⡅⠀⣴⠀⢉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡽⠁⠀⠀⠀⢻⣦⣾⣿⣤⣾⠁⠀⠈⢫⣡⡄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⢻⡾⣿⣶⡏⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢽⡿⠀⠀⠀⠀⢶⣼⡿⣽⣝⢿⣧⠀⠀⠀⠀⢻⣿⠅
⠀⠀⠀⠀⣼⠁⠀⠀⢸⣿⣿⡷⠻⣷⠀⠀⠀⢸⡄⠀⠀⠀⠀⣐⣶⡃⠀⠀⠀⠀⣸⣿⡟⣾⠉⠌⢻⡅⠀⠀⠀⢸⣶⣂
⠀⠀⠀⠀⢿⠀⠀⠀⣾⣿⡈⢁⢧⣾⡆⠀⠀⢸⡇⠀⠀⠀⠀⠨⠿⡇⠀⠀⠀⠘⣿⣿⣤⠀⡌⠢⣾⣏⠀⠀⠀⢸⠿⠇
⠀⠀⠀⠀⠸⣆⠀⠀⠘⠛⢻⣦⣶⣿⡀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣻⣿⡀⠀⠀⠀⠻⠿⠛⣷⣠⣶⣿⡇⠀⠀⢀⢼⣿⡁
⠀⠀⠀⠀⠀⠘⢦⣀⠀⢀⣼⣿⣿⠇⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢋⣵⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⠈⠀⣀⣮⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠿⠿⠿⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢷⣤⣀⣠⣿⣿⣿⣿⣯⣤⡚⢾⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠻⠋⡿⣿⢿⠙⠟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈