Toy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⣍⣉⣶⢰⡂⠀⣢⣀
⠀⠀⠀⠀⠀⠀⢀⠴⡅⣁⣫⡉⡕⠒⢻⡛⠛⣽⡇
⠀⠀⠀⠀⠀⡠⠿⣈⣻⡛⢩⠐⠳⠿⠛⢃⡞⠹⠁
⠀⠀⠀⠀⣴⠂⠙⢻⣛⠅⠺⢴⣶⡿⣠⡟⠀⢲⣇⣀
⠀⠀⠀⣎⡈⠛⠛⠿⠙⣗⣦⣶⡇⣰⣗⣀⢀⡟⡶⣭⣿⣭⠑⠲⡂⠠⣤⣄⣀⡀
⠀⠀⠀⡆⠂⠈⠉⠁⠒⠒⠂⠠⡾⢹⡣⣀⡾⣿⣛⣻⣋⡹⢵⡶⠏⢿⣤⣨⣦⡜⠛⣓⠒⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⠇⣧⣟⢮⡷⢿⣛⠚⠵⣶⣿⢴⢉⣉⣯⣉⠿⢿⠟⣼⠟⠀⠀⠀⠀⠀⠀⣤⢰⢀⣰⣶⣦⣬⣯⣟⣤⣿⠿⣾⣷⢄
⠀⠀⠀⠗⠠⠄⣀⠀⠀⠀⠀⠀⢷⣿⠣⡿⢑⠀⠀⠀⠈⠁⠐⠀⠂⠭⠟⣋⠙⠶⠾⣿⡇⣷⣀⣀⠀⠀⠀⠀⡏⢆⢹⢥⣭⣾⣧⣤⣷⢷⣬⣷⣿⣤⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⣫⢇⢃⣿⡂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢙⡝⣤⠿⣶⣶⡷⣶⡀⣀⣃⠀⣳⠛⠛⠛⢟⢍⢝⠝⠍⡡⠉⠈⠀⢨⣗⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡶⢿⠫⡫⠑⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠈⠸⣡⡟⢀⣿⣷⠶⠯⢿⣤⣈⣷⢷⠔⢕⠔⠀⠑⡡⠠⢊⢀⢔⣀⣴⣸⣏⣀⠍
⠀⠀⠀⢶⠂⠒⣄⣀⡀⠀⠀⠀⣏⡾⠀⢸⠕⢌⠂⠀⠀⠀⠀⡆⠀⠉⠉⠒⠒⠤⠼⣯⡐⢟⣫⣽⡆⡔⣈⢛⣏⢡⢿⡛⠋⠛⠛⠘⠙⠉⠉⠉⠉⠉⠉⠁
⠀⡠⠁⠼⢽⠿⠏⢹⣢⣤⣾⣶⠛⡁⡈⢿⣤⣔⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⠵⢿⣿⣳⡿⢹⡍⡁
⣔⠀⢷⣦⣴⡇⢰⣅⣉⣳⣴⠃⡸⠀⣱⠟⠅⡩⠉⠉⠉⠒⠒⠧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⢂⢀⣀⡀
⡏⠉⠐⠒⠂⠠⠬⠽⢟⣿⠃⢰⢡⡷⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠤⠀⣀⡀⠀⠀⠀⠀⡆⣹⡼⠃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣱⠎⠉⠁⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠉⠣⠿⠁⠀⠩⠠⠄
⠙⠢⠄⣀⣀⠀⠀⠠⠀⣆⡾⠃⠐⠒⠠⠀⠀⡠⠤⣁⣐⣾⣣⣤⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡠⠤⣤⣆⣐⣾⣯⣭⡿⢆
⠀⠀⠀⠀⠀⠈⠉⠉⠒⠛⠂⠀⠉⠉⠉⠀⠀⠈⠻⡛⣯⣉⣱⡿⣶⡼⠧⣄⡀⠀⠀⠀⠀⠀⠀⢮⠹⣣⣤⣼⠿⣶⡶⣟⠚⡿⠓⢌⣽⣉⣺⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠫⡉⣏⠉⠐⣗⣤⣼⣧⣌⡀⠀⠀⠀⠀⠀⢣⠠⡯⣉⣹⠾⣢⣴⡾⢧⣿⣿⢽⠿⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠘⢎⠻⢿⡟⠁⠽⣄⣀⣹⣆⠀⠀⠀⢆⠀⢧⠬⠿⢷⠖⢞⠟⢉⢍⠕⠁⠀⠀⠀⣽⣒⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⣌⠧⣶⠾⠛⣛⢫⢞⡇⠀⠀⠌⢆⣸⢀⠔⠁⠀⠅⠔⡁⢠⣪⣨⠦⠾⠛⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠣⠤⠖⢛⡬⢢⣾⡏⠅⠠⠄⠈⠷⣥⠦⠶⠶⠺⠮⢙⣉⣁⠀⠐⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢐⡇⢄⢐⣐⣭⠿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⡷⠛⠁⠀⠐⠒

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠀⠈⠀⡀⠀⢀⠀⡤⠐⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠰⠶⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⡎⠀⢀⠀⠄⠀⠢⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⠀⠀⡀⠔⠢⠄⠀⢴⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠈⢂⣡⡀⠀⠚⠃⠀⢆⡐⠈⠀⠀⠀⠀⠘⡀⣪⣄⢀⣍⢠⠐⠁⠀⠀⠀⠀⠀⢣⠀⠂⢀⠐⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠚⢄⠀⠬⠑⢀⣾⠖⠉⡁⠀⠀⠀⠀⠉⢲⠀⠄⠀⠂⠐⡀⠙⠫⡘⠛⠀⡃⠀⠀⠀⠀⠀⠀⢀⠀⠙⢚⣢⣄⣢⣄⣔⡀
⠀⠀⠀⠀⣈⠑⠐⠘⠣⠜⣄⡀⠀⠀⡀⠀⠀⠀⠀⠆⢀⠀⠀⠈⠐⠠⠈⠉⠤⠊⠑⠄⢀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡱⠀⠀⠀⠧⠀⡇⠁⠀⠀⠈⡤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠰⠄⣼⠀⠀⠀⠀⠀⠈⡵⠲⢦⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠒⠁⠀⠀⠀⢀⡰⡀⠀⠀⠀⡠⠠⢦
⠀⠀⠀⢸⠀⠂⠀⠀⠀⡀⠇⠄⠐⠉⠁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠄⢅⠀⠀⠖⠁⠃⠀⠘⠁⠀⠀⡄⠀⠙⠒⠤⠀⠀⢀⡀⠠⠄⠊
⠀⠀⠮⣀⣀⠖⠒⠒⠉⠂⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⡀⠁⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠙⡲⠦⣔⠊⠀⠀⡎⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⡲⠀⠀⠀⠀⠀⠠⠄⠸⠀⣒⡀⠁⠁⠁⠀⠀⠀⠐⢠⠀⠄⠒⠐⠐⠂⠤⢄⠀⢠⠈⢀⣤⠉⢁⠀⠀⠘⠋⠉⠉⠑
⢀⠀⠀⢀⠔⠀⠀⠂⠀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠈⠁⠤⠀⠱⠄⠀⠄⠔⠂⠠⠃⠀⠼⡇⠉⡁⠀⠿⡃⠀⠀⠀⠀⠂
⠨⠀⠀⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠃⠉⠉⠉⠙⢊⠁⠀⢀⡀⠀⠁
⠀⠳⡀⠀⠀⠀⡠⠀⠌⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠐⠀⣄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠀⠀⣄⠀⠐⠀⠀⠉
⠀⠀⠈⠒⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⡿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠻⠟⠻⠿⠿⠿⠿⠿⢿⡿⠿⢓⣠⡽⠿⠿⠟⠿⠿⠿⠟⠻⠏⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠄⠂⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠋⠁⠙⠀⠀⠀⠀⠐⢇⢀⣀⡀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠟⠃⠙⠓⠀⠛⠀⢳⢲⢀⡀⠀⠀⡇⠀⠀⢀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢢⠀⡄⠀⠀⡄⠀⠀⠀⢸⢸⠀⠀⠈⠉⠁⢠⡐⠁⠀⠀⠀⠀⠁⠀⠰
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣀⢠⣀⡀⠧⠀⠤⠤⠼⠴⣤⣤⡀⣀⣸⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠑⠀⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠸⡏⠀⠀⠀⢠⠆⡈⠈⢢⢠⡎⠀⠀⠱⡀⠿⠘⠃⠀⠀⠀⠀⠀⠤⢀⡇
⠀⠀⠀⠀⠀⠱⠛⠂⠉⠀⠀⡀⡠⣂⣀⡇⠀⠀⠀⠸⣄⡁⢀⡴⠂⢦⣀⣁⡼⠁⠀⡀⠀⠄⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⡳⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠒⠋⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⡄⠀⠉⠢⣄⠀⠀⠀⠘⡄⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠈⠳⡀⠀⢡⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠢⡀⠀⠘⢄⠀⢠⠀⠸⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠢⡀⠀⠀⠈⠢⠀⠘⣴⠃⠀⠀⣇⠀⠈⠀⢀⡴⠦⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⢄⠀⢀⣡⠔⠁⠀⠀⠀⢹⠀⠀⠀⢸⡀⠀⣿⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⡀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⡆⢀⣀⠬⠇⠀⠉⠉⢉⡉⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢘⡸⠁⢀⣀⠉⠓⡤
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠱⣀⠙⠉⠀⠀⡂⠀⠐
⠀⠀⠀⠀⢨⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢠⠂⠉⠁⠀⠀⠻⡀⠌
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠱⠄
⢠⠂⢁⣀⠈⠁⠀⠀⠤⡤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠠⠀⢀⡇⠀⠀⠀⠀⢰⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⢤⣀⡀⠀⠀⢀⣠⠶⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⢈⠧⠠⢤⣄⠊⠀⡀⠀⠀⠀⠀⠀⠀⠑⠠⠒⠀⠈⠉⠘⢅⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠌⠀⠀⠀⠀⠀⠁⣷⣠⠀⠄⢰⠀⡄⠠⠀⠰⠀⠄⠀⠀⠆⢠⢀⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⣧⣴⣴⣦⣶⣤⣦⣴⣶⣧⣴⣴⣷⣴⣦⣼⣿⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿
⣿⣭⣭⣯⣽⣥⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣴⠿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠉⠉⠛⠓⣦
⠀⠀⠀⣠⣴⡾⠋⠀⠀⠛⢟⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢹⡟⠁
⠀⠀⠀⠙⣿⣧⠀⣀⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿⠛⠓⠊
⠀⠀⠀⠀⠙⠻⠾⣿⡟⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣂⠄⠀⠀⠀⣼⠛⠃⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⠀⠀⠀⠀⣴⠒⠺⠿⠿⠟⠓⡄
⠀⠀⠀⠀⠀⡔⣻⠆⢹⣿⣿⣿⣦⣤⠀⣿⠀⠀⠀⠀⠀⢀⣾⡗⠀⠀⠀⠀⠀⣀⣈⣄⣿⣿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⠛⠋⢠⠀⠉⢻⣿⣿⣿⡇⠙⢧⣄⣀⣀⣀⣼⡿⠃⠀⠀⠀⠀⠈⠻⣧⣬⣿⣿⣿⣀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣠⡄⢀⡶⣦⡀⢀⣴⣿⣿⣥⡏⠁⠀⠀⠙⣿⡟⠉⠁⠀⠀⠀⠀⢀⠄⣷⣄⣠⣤⣿⢿⣿⠿⣀⠀⠘⢿⣤⣤⣤⣤⣤⣶⡧
⢰⡿⣧⠟⠘⢷⡾⢃⣿⣿⣛⡷⠎⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⠈⡾⠛⢿⣯⣉⣿⣤⣿⠷⣇⣤⡀⠀⠀⠀⣿⡇⠀⠈⠁
⠈⢧⣄⠘⠛⠀⣠⣾⠿⣿⡝⠁⠀⠀⠀⠀⣀⠀⣿⣷⣄⡀⠀⠀⠀⣿⠀⣠⣦⠈⠋⣁⣿⣿⣿⡇⣸⣬⣷⠀⠀⠀⣿⡃
⠀⠀⠙⢦⣀⣾⣿⣿⡾⠃⠀⠀⠀⠀⠀⠚⠁⣞⣿⣿⡟⢿⣶⣄⡀⠈⠀⢿⣹⣷⣿⠟⣿⣿⣿⣼⣿⠟⠋⠀⡀⠀⠿⠃
⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⢀⡄⢠⠀⣠⡀⣿⣿⣿⡒⠈⠀⠻⡆⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠀⣠⠖⠁⠀⠀⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⢀⡀⠉⣀⣿⣿⣿⣿⡇⠠⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠠⣶⡧⠀⠈⠋⣫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠣⣾⣿⣿⣿⣿⣿⣿⣾⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣭⣅⣀⣀⣀⡈⣫⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⣿⣿⣿⣿⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢉⢿⣿⡯⠁⠛⣿⣻⣟⠈⢹⣿⡿⠙⣻⣿⣿⡿⠉⢹⡇⠀⣴⣤⡀⠀⢴⣦⣶⠀⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠿⠴⠦⠾⠻⠷⠼⠿⠿⠻⠴⠾⠿⠝⠷⠾⠷⠿⠻⠿⠶⠿⠇⠈⠙⠛⠃⠀⠘⠉⠋⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⢀⠐⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⣝⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠰⢨⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠙⠉⡁
⠆⠀⠈⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⡠⠁⠀⠰⡍⠀⠀⠀⡀⢤⠀⢀⡏⠉⠀⠄⠐⠃
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠈⠓⡘⠀⠤⠀⠀⠀⢠⡀⠀⠉⠁⠀⠀⢀⡀⢀⡠
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣠⣴⡾⡿⢋⣿
⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⢻⢛⠉⡩⡯⠋⠈⠑⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠣⠈⠀⠓⠂⠀⠗⠀⠀⠓⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠂⠀⠐⠂⠐⠠⢄⡀
⠀⠀⠀⠠⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠎⠀⠀⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠰⠄⠀⠈⡂⠀⡼⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠔⠱⡀⢠⢥⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠆⠀⠐⣐⡐⠀⠘⠄⠁⠁

⠀⠀⠀⠀⠀⣀⡤⠤⣄⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⡀⡼⢡⠖⠓⢎⢳⠀⠀⢞⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠘⡈⡏⠒⢄
⠀⢀⣺⡍⠃⠳⣀⡤⢃⡞⠀⠀⠸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⢣⠸⠀⠀⠑⢄
⣴⢋⠴⠒⢄⢀⡰⣞⠉⠀⠀⠀⠀⣡⠀⢀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣴⣿⡿⣈⠳⣄⠈⡆⢣⠀⠀⠀⠣⡀
⣇⢺⣀⣠⠞⡼⠙⢦⣱⠖⠈⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣀⡀
⠈⠓⠦⠶⠚⠁⠀⣴⣛⣙⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣦
⠀⠀⠀⠀⠀⠀⢰⠇⣺⡉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣞⣚⣚⣇
⠀⠀⠀⠀⠀⠀⡿⠶⠶⠃⣠⣴⣶⣿⣿⣷⣶⣦⣈⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀⣤⣶⣾⣿⣿⣷⣶⣄⡀⠀⠀⠠⡹⢖⣒⣒⣿
⠀⠀⠀⢀⠤⠤⠧⡀⠀⣼⣿⠟⡫⠅⠀⠨⣙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢋⠍⠀⠀⢍⠻⣿⣿⡄⠀⠀⠉⠭⠭⠭⠽
⠀⠀⠀⠸⡀⠀⠀⣀⣼⣿⡏⠌⠠⠊⠉⠐⡈⠆⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣿⣿⢡⠁⠔⠉⠁⠆⠑⢸⣿⣧⠀⠀⠠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢃⠐⢄⣀⡠⢀⠆⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡐⡀⠢⣀⡀⠆⡰⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣑⡂⠀⣐⣡⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⣒⠄⠀⣊⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⠿⠟⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⡠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢢⣀⡇⠀⠀⠀⠀⠉⠀⠀⠀⠛⠀⠀⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡑⠦⠖⢧⠀⠀⠀⠀⢠⠂⠐⡀⠀⠀⠀⠀⠀⢸⡆⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⠉⠒⠼⡄⠀⡀⠀⠈⠙⠋⠀⠀⠀⠁⠀⠀⢸⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠹⣄⠑⢄⡀⠀⠀⠀⠀⡠⠃⠀⢀⠟⠀⣀⣀⡤⠤⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣠⠤⠄⠀⠈⠓⠦⣌⣉⠉⠉⠁⢀⣀⡴⠟⠊⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢠⣀⠀⠀⠀⠀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠉⠉⠉⠉⠁
⠀⢀⡤⠒⠐⠒⠦⠤⣀⡠⣜⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣾⠈⠑⢆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⢿⡀⠀⠈⡄⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠈⠳⢦⣴⣁⣀⣀⣠⡤⠾⠋⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋⠑⠦⢄⣀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠒⠛⠋⠉⡇⠀⠀⠀⠀⠀⢀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⡔⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢜⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠂⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠤⠀⡀⡠⠄
⠀⠀⠀⠀⠀⢀⡸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡤⠖⠚⠙⠠⣀⡀⠀⠀⠀⠈⠀⣰⠋⠀⠀⢠⠀⠁⢀⠄
⠀⠀⠀⠀⢀⠎⠠⡀⠀⣌⡄⠀⠀⠁⠀⢠⠀⠀⠀⠀⢰⠇⠀⡠⠋⠀⠀⠀⠀⢠⠁⢠⠀⠀⠀⠀⢠⠃⠀⣰⠤⠛⠀⠀⠀⡀
⠀⠀⠀⠀⠃⢀⢀⡀⠀⠘⡃⠒⠒⠂⠂⠀⠀⢖⠀⠀⠘⠀⣸⠁⠀⠀⠀⠀⡰⠋⠀⣸⠀⠀⠀⠀⠈⠀⠀⠀⢐⠁⡀⠀⣠⠃
⠀⠀⠀⠀⠀⢠⠊⢀⣀⣠⠠⠐⠉⠁⠐⠀⠀⠇⠀⠀⠀⠐⡇⠀⢀⣠⠴⠋⠀⠀⣠⠃⠀⠀⠀⠀⢠⠖⠁⢀⡠⠄⠘⠉
⠀⡀⠀⠀⢠⠂⠀⠈⠁⠠⠀⠀⠀⣀⠒⠒⢒⠄⠀⠀⠀⠀⠈⠉⠁⠀⣀⡠⠤⠚⠁⠀⠀⣀⣀⠀⠐⠀⠒⠋⠀⠀⠈⡄⠀⠀⠈
⠎⢀⠀⡠⡇⠀⣀⠠⠔⠁⠀⠀⠀⢠⡀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠁⠀⠠⢎⡴⠖⠂⠁⠰⠋
⠀⣠⠄⣢⡃⠠⠃⠀⠀⠀⠀⠀⠄⠚⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⡄⠀⠀⣎⠀⠀⠀⠀⢠⠃⠤
⠀⠓⢠⢀⠁⠐⠢⠄⠀⠄⠀⢀⡀⢠⡴⠷⠶⠚⢋⡩⠆⠀⠀⢀⠎⠀⠀⢀⡀⠀⠀⠀⠀⠀⠒⠨⠔⠊⠉⠉⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠈⠀⠐⠂⠀⠤⠤⠤⠤⠤⠐⠒⠚⠉⠁⠀⠀⣠⠜⠀⠀⠀⠁⠀⠴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠌⠀⣀⠔⠀⢀⢤⡀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠂⣰⠊⠉⠀⠀⠀⠈⢘⠆⠀⠆⡀⠀⠌⠘
⠄⠰⠀⠄⠈⢱⠖⠒⡢⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⠄⠂⠈⠀⠘⠀⠀⡔⠁⢀
⠀⠀⠈⠀⢠⠀⠀⡞⠁⠀⢘⠁⠀⠐⠂⠠⠃⠠⠀⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠞
⠀⠀⠀⠀⠀⢠⢀⠁⠀⠀⢠⠞⠃⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠈⢀⡜⠋⠀⡏⢀⡤⠀⠀⠀⠀⠀⠄
⠀⠀⢡⠤⠒⠁⠘⠀⠀⠀⠀⠀⠠⠞⠁⠀⠂⠀⠀⢠⠴⠶⢊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠒⠲⡟⣹⣏⣀⣌⠀⠂⢠⠁
⠀⠀⣸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⢸⠉⠀⣠⠋⠀⠀⠌
⠀⠀⠆⠀⠀⠀⠀⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠈⠁
⠀⠀⠉⠀⠀⠎⠀⡃⠀⠠⢴⠁⡀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠈⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢀⠀⠀⠀⠈⠁⠀⠐⠋⠀⠀⠀⡆⠀⠀⠠⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠶⠀⠀⡄
⠀⠀⢸⠀⠁⡀⠀⠠⠀⠎⠀⠀⠀⢰⠀⠀⠀⠁⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠦⢀⠀⠀⠀⣨⠑⠠⠄⠀⠔⠀⢰⠃
⠀⠀⠈⠀⠀⠀⢢⠄⠀⠀⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠖⠀⠀⠰⠂⠠⠀⠀⠀⣀⡤⠖⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⢒⣏⡀
⠀⠀⠀⠀⠀⠤⠃⠀⠀⠀⠀⣀⡠⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⡀⠈⣂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠙⣆⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⢀⣾⠙⠛⠛⢷⡀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣸⡇⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠛⠲⠶⢾⡇⠀⠀⠀⡴⣾⠓⠒⢫⣶⡟⠷⠦⠾⠋⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸⠹⠀⠀⠀⠀⠉⡞⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⣀⡀⠀⣸⠃⠀⠀⠀⢘⡴⡀⠀⠀⠀⣠⡳⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣴⠏⣽⠛⠳⠏⠀⢀⣀⠀⠀⠙⠾⣤⠤⡽⠞⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡼⠃⠀⣠⠞⠁⣼⢣⣀⣠⡴⠞⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⢀⡞⠁⣠⡾⠁⠀⠀⣿⣏⠙⢧⡀⠀⠀⢀⡟⠀⠀⠀⣀⣠⣤⢴⡶⠋
⠀⠀⢠⣟⢀⡴⠋⠀⠀⢀⣾⡁⠈⠳⣄⠙⢦⣀⡾⠁⠀⣾⠛⠉⠁⠀⠸⣇
⠀⠀⠀⠉⠋⠀⠀⠀⠀⢾⣁⠙⢦⡀⠈⠳⣄⣹⡧⢀⣠⢾⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢶⡄⠙⣦⣤⠞⠻⠞⠋⣡⠞⠁⠀⠀⠀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⣠⡴⠛⠛⠉⠀⠀⢀⣴⠟⠁⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠸⣥⡴⠛⠁⠀⠀⠀⠀⢀⡴⠟⠁⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠉






⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣟⣛⣛⣛⣛⣙⣛⣛⣛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣭⣭⣍⣭⣽⣿⣿⣙⣛⣛⣛⣻⣿⣟⣛⣛⣛⣻⣿

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⢿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣄⣀⣀⣀⣼
⠀⠀⠀⡟⠉⠉⠙⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣷⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠤⠤⠤⠤⣤⣉⣁⣀⣀⣤⣀⣀⡀⢶⠖⢲⠚⢫⢋⡱⣡⡧⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠉⢹⡆⠀⠀⠀⢸⠆⠡⠀⡵⠉⢀⡚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⢰⡆⠀⠀⢿⠀⠀⠀⢸⡀⠙⠋⢀⠢⠟⠔⢠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠺⠶⠀⠀⢸⡆⠀⠀⣿⠐⢀⠐⢁⣶⢂⡶⣣⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⠀⠀⠈⣷⠀⠀⣿⢠⠋⠔⠈⣰⡟⡰⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⡂⠀⠀⣾⠉⠉⠁⠀⠀⢹⡄⠀⡏⠁⡴⢀⢘⡡⣌⣾⣿⣿⡥
⠀⠀⠀⠮⠉⠉⠀⠀⠀⠀⠀⠀⠀⠨⢌⣛⣛⠛⠿⠧⠤⢤⣸⣦⣤⣈⣷⠀⡗⠊⢴⣥⣾⣿⠿⠟⠋⠀⠀⠀⢀⣠⣼⡆
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣉⡙⠃⠀⠶⠶⢤⡭⢭⣭⣠⣷⠾⠟⠛⠉⠁⠀⠀⢀⣠⣴⢞⢛⡻⠓⡃
⠀⠀⠀⡀⠃⡏⠉⣿⠛⠋⠙⠛⠛⠛⠒⢤⡀⠀⡇⠀⠀⢰⡖⠂⠤⠀⠄⣀⣀⣀⡂⠀⠀⠠⡴⡞⡟⢡⡿⢁⡞⠘⣸⠅
⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⢠⡤⠤⢤⠀⠀⢹⠀⡀⠀⠀⠘⡇⠀⠀⠀⠉⠉⠉⠉⠢⣌⠉⠀⣟⠑⢠⡎⣀⠝⢀⠐⢻⠁
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠘⠓⠒⠊⠀⢀⡼⠀⡇⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠏⠜⡜⢂⠎⢰⢹⡂
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠉⠛⢦⠀⡇⠀⠀⠀⣷⠁⠀⢀⡾⠋⠙⠦⠾⠋⠁⠀⢻⢊⠌⠌⢀⠆⢠⠋⢸
⠀⠀⠀⠀⡄⡇⠀⢸⠀⠀⢸⣉⣩⠇⠀⠀⠀⡇⡇⠀⠀⠀⡿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⠂⠨⠀⠊⣰⠃⣴⣿⡀
⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠇⢠⣼⠀⡧⢀⠀⠀⠈⠓⠒⠉⢦⡀⠀⠀⢸⠎⠀⢀⣼⠋⣼⣟⣽⣟⠓⠂⠄
⠀⠀⠀⠀⡄⠷⠤⠜⠛⠛⠛⠛⠛⠛⠛⠛⠒⢒⣛⣸⣽⠀⣿⣀⣳⣄⠀⠀⠀⠀⣀⡼⠀⠀⢸⣠⢀⡼⣡⣬⠿⢋⣐⠀⠘⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠟⠛⠫⠟⠋⠻⠭⠟⠛⠛⠒⠓⠶⢶⣬⣭⣙⣛⡻⠯⠥⣄⣀⡀⢸⢗⣫⠶⢛⢒⣦⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠍⢿⣓⣒⣲⣦⠾⣛⡛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆⢀⠀⢀⠰⡖⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⡖⠀⠀⢀⡀⢸⠁⠀⣇⡴⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡧⠀⠈⠊⠀⢰⡤⠚⠉⠂⢀⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⡐⠓⠆⠀⣡⠖⠹⠁⠂⢀⠔⢩⣤⡜⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠺⡇⠀⠀⡠⠚⡃⢀⣬⢄⠀⢸⡀⠾⠇⢀⡠⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣨⣃⠀⢐⠀⢀⡞⠀⢨⠀⠀⠛⠀⠀⡠⢚⠃⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⢅⠞⠁⠈⠀⢬⠔⠀⣇⡔⡋⠓⠠⣄⠀⢘⢀⢴⠏⢀⣦⠄⠁⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠔⢩⢠⠐⣟⠀⠀⠠⠂⡇⠀⢲⠉⢰⣵⠊⠙⠃⠠⠄⠀⠀⢀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⠀⠀⠂⠀⢾⠀⢀⣯⡀⠀⡅⣸⠳⡄⢘⡴⠃⠈⠁⣠⠄⠀⠀⢀⢼⡋⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢄⠀⠀⢀⡵⠃⢸⡶⠈⠀⠁⢠⡟⠀⠀⡠⢿⠃⠀⠠⠂⢞⠀⠀⠈⠀⢸⣔⠋⠈⢉⠰
⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠄⢊⡃⠀⠀⣅⡄⢻⡃⠂⢀⠆⢸⡆⠀⠰⠀⡺⡀⠀⡇⢠⠔⠒⠀⣈⡖⠁⠉⠁⢢⠊⢸⡄⠀⠀⠁⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡂⢀⡼⢧⡀⠀⡈⠀⡬⢇⠀⢰⣀⡞⠓⠄⣸⠞⠁⠉⠁⢿⠅⠀⠀⠐⣼⠉⠂⢀⠊⢘⡴⠋⠉⠁⠐⠮⢼
⠀⠀⠀⢰⠀⠀⠀⢀⡱⡾⡉⠂⠀⠀⣢⣶⠉⠀⠀⠀⠾⠁⠀⠀⠄⡏⠀⠀⡄⠃⣯⠀⠈⠀⠀⠿⣄⠀⢀⡀⢺⡁⠐⠠⠀⠀⠀⢪⠅
⠀⠀⠀⠈⠀⠈⡎⠁⢐⡇⠀⠀⠀⠊⠀⡆⠀⠀⠌⢀⣇⠀⠈⠄⡠⣧⡀⠀⢅⠰⠙⠂⢀⣴⠋⠀⠀⠀⡾⠁⡐⠓⠦⠄⠀⠀⣃⡸⠃
⠀⠀⠀⠀⠁⠠⣥⠔⠛⠟⠀⠀⠁⡄⢡⠍⠀⢠⡶⠊⠀⠉⡠⠃⠁⡇⠀⠈⠛⠆⠀⠀⠀⠈⠁⠀⡔⠁⣾⠊⠀⠀⠀⠀⠈⠁⠋
⠀⠀⠀⠀⠀⠀⡂⠀⡠⢡⠀⠈⠁⠀⣀⢅⠀⠈⠉⠀⠄⠊⢸⢁⠴⠛⠓⠀⡄⠀⠈⠁⠀⠀⠀⠆⣂⠜⠁
⠀⠀⠀⠀⠀⠀⠲⠊⠉⠁⠀⠂⠤⠾⠕⠀⠀⠐⠀⠀⠄⢀⡴⠋⠁⠀⠀⠀⠀⠀⠠⢄⠀⠀⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠉⠉⠉⠀⠀⠈⡏⠉⠉⠉⠉⡏⠀⠀⠈⠉⠉⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣖⣖⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣢⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣛⣛⣛⣛⣛⣛⡛⣉⣉⣉⣉⣉⣉⣛⣛⡻⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⠿⠟⠋⠉⠉⠀⠀⠀⠀⠀⣀⡩⠛⠛⠻⢟⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡜⠀⡀⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⡀⠘⡄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⠸⡀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠸⠂⠈⣂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠩⡇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠈⠈⠀⠀⠀⡯⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣉⡍⣆⣈⣉⣒⡶⠦⠤⠤⢤⣤⣤⠤⠤⠤⠶⢖⣊⡉⣀⡞⣍⣹⠟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠤⠞⠥⠤⠤⠤⡖⠤⠤⠤⣼⣿⣿⡄⠂⠀⠒⡦⠤⠀⠠⠼⠦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣀⣀⣙⠶⠙⣋⡀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠻⠛⠻⠷⠾⠁⠿⠟⠛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⢀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠉⢹⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠈⠣⠀⡘⠀⠘⠀⠠⠏⠠⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠓⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⢀⣀⣀⡀⠈⠉⠛⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠁⠀⠀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣛⣉⣁⣠⣼⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠉⠙⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠙⠓⠲⠦⣤⣀⡀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⠛⠋⠉⢀⣈⣽⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⣿
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣄⣀⡀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠈⠉⠳⣄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠁⠀⠀⢀⡀⠀⠀⢸⣿⠃⣶⠆⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⡿⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣿⣿⣷⣤⣄⣹⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣶⡄⠀⠀⠀⠀⢀⣼⠟⠀⠀⢠⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⠿⣶⣤⣴⡾⠟⠁⠀⠀⠀⡼⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣶⣿⣿⣿⡇⠈⠉⠙⠻⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠿⣿⣿⣿⣿⡆⠀⠀⠀⣨⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⠦⣤⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣠⡴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⢼⡧⠶⠚⠋⠉

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠟⠛⠉⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⡏⠈⣿⣿⣿⣧⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣼⠁⠀⠸⣿⣿⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣯⣤⣶⣾⣿⣿⣿⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⡿⠿⠛⠛⠉⢿⣿⣿⣇⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣄⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⣿⡀⠀⠀⠀⠀⠾⠟⠛⠉⠁⢰⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⠿⠟⠋⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⠃⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⡿⠿⠛⠋⠉⠁⣾⣿⡿⠟⠛⠋⠉⠀⠀⠀⣠⣤⣴⣶⣦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠻⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡿⠿⠛⠛⠙⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠘⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠀⠀⠀⣀⣠⣤⣴⣶⣿⣷⣦⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣶⣾⣿⣿⠟⠉⠉⢹⣿⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢹⣿⣿⣿⠀⠀⠀⣸⣿⣿⠏⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣾⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠀⢀⣴⣿⣿⣋⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⣆⡀⠀⠀⣀⣴⡿⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣶⡿⠟⠿⣿⣿⣿⣦⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠉⠀⠀⠀⢸⣿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠁⣀⣠⣤⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⠿⠟⠛⠉⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⠀⣀⣠⣾⣿⡿⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣸⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠐⠛⠉⠁⠀⢀⣀⣤⣴⣶⣿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣤⣤⣶⣾⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣉⣀⡀⡤⢶⠟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡤⠤⠤⡤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣒⠢⡴⠒⠚⠛⠋⠈⠁⠜⣱⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠉⠙⠋⠛⠙⣷⠀⠀⠀⠈⠀⣷⠀⠀⠀⢌⠜⠁⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⣧⠀⠀⢀⠀⡇⠀⠀⠤⠀⠀⠀⠔⠀⠀⣈⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢿⠀⠀⠈⢠⡇⢀⠀⠀⡀⠀⠐⠀⣠⡔⠁⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⢸⡇⠀⠀⢸⡇⠀⢠⠀⠀⠀⢀⡼⠛⠈⢥⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣿⠀⠀⢰⡀⠰⠁⠂⠀⢀⠾⠁⢐⢃⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⡀⠀⠀⢀⡿⠀⠀⠈⠀⠀⠀⢸⡆⠀⢸⠀⠀⠐⠀⠀⡀⠠⣠⣾⣿⣿⡧⠄⠀⠀⣀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡈⠓⠛⠛⠷⠾⣇⣀⣀⣀⢦⣄⣀⣀⣷⡀⠨⠀⠀⠀⢀⣴⠵⣛⡻⠟⠊⠁⠀⠀⠀⠀⣀⣼⡆
⠀⣄⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠶⡆⠀⣠⣭⣉⣉⣛⣛⠛⠇⣮⣠⣴⠾⡿⠛⠛⠁⠀⠀⠀⠀⢀⣠⠖⠻⠓⠅⡅
⡄⠀⢰⠴⠶⡶⣶⣶⣶⠶⠶⢶⣖⣒⠒⠒⠛⢻⡆⠀⠈⠉⣉⣁⡉⠉⠙⠋⠹⠏⠉⠁⠉⠀⠀⠀⠀⣀⣠⠴⠊⠹⡋⠀⠰⠀⠀⡆
⡇⠀⢺⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⡀⠘⡇⠀⠀⠀⣿⠀⠀⠉⢉⣲⡶⠶⣶⠶⠦⠤⠤⡌⢹⡋⣁⠢⢀⠏⠀⠰⠁⠀⠤⡇
⠆⠀⢸⠀⠀⡇⠀⠀⠀⡖⠒⠒⣦⠀⠀⠀⡇⠀⠁⠀⠀⠀⣿⠀⢀⠔⠁⠀⠀⠀⠀⠈⠙⣦⡀⠀⠸⡇⠀⢠⠎⠀⠀⠀⢀⠈⠀⠇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠛⠒⠚⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⣿⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠨⢀⣧⣤⣤⣤⣴⡤⣤⣧⣤⣤⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⡀⠀⡇⠀⠀⠀⣿⡀⠀⠀⠀⣴⠋⠉⠳⣦⠴⠀⠀⠠⠘⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⢸⠀⠀⡆⠀⠀⠀⡏⠉⢳⠀⠀⠀⠀⢹⠀⡄⠀⠀⠀⣿⡇⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⣟⠚⠖⠒⠲⠓⡺⠛⢙⣿⡇
⡀⠀⢸⠀⠀⡇⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⡸⠀⠇⠀⠀⠀⣿⡀⠀⠀⠀⠻⠦⣄⠠⠤⠀⠀⠀⠈⠀⣍⠀⠀⠀⠀⣠⠀⣰⡿⢻⣷⠤
⢀⠀⢻⠀⠀⢧⣤⣤⡤⣤⣤⣤⣤⣤⠤⠚⠁⢀⡇⢸⢺⠀⣿⠀⠡⡀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⡿⠀⠀⠀⡟⠀⠀⣯⣴⠟⠁⠀⠐
⠀⢧⣙⣉⣀⣀⣈⡉⢁⣈⣉⣉⣁⣀⣀⡀⣩⡭⠥⠼⠼⣄⠛⠣⠤⢝⣦⣄⣀⣀⣀⣠⡴⠟⠃⠀⠀⣮⠔⠀⡜⣰⣶⠚⠋⠀⣀⠀⠉
⠀⠀⠀⠀⠉⠁⠉⠛⠛⠀⠐⠒⠂⠀⠂⠚⠒⠊⠉⠉⠉⠉⠩⠕⣒⣦⣤⣬⣉⣙⣓⠒⠤⠤⢤⣀⡄⠝⣀⣥⠾⢋⣁⣀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⣍⣉⣛⣒⡶⠶⠟⠋⠭⠛⠛⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢉⣁⡀⣤⢴⢞⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⣤⠤⡤⣨⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣒⠠⡴⠲⠒⠚⠋⡈⠁⡴⢡⠋⠀⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠐⠋⠛⠛⠛⠻⣦⠀⠀⠈⠈⠀⡇⠀⠀⠀⠈⠈⠁⠀⠐⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣟⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⡆⠀⠀⠀⢀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷⠀⠀⠀⢨⠁⠄⠀⠀⠀⠀⠀⠀⢁⣠⠊⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠛⠚⠀⠀⠀⢿⡄⠀⠀⢸⡀⠀⢀⠔⠁⠀⠀⣰⠟⠑⢂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣧⠀⡇⢨⠀⠠⠋⠄⠀⠀⢼⠋⡠⡆⣨⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⢰⡏⠉⠉⠃⠀⠀⠀⢿⠀⠁⢘⠐⠀⠠⢀⠀⠀⢀⠀⣬⣈⣣⣟⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠿⢼⣶⢶⣤⣾⠁⠀⠀⠠⡀⠀⠀⠸⣆⠀⣈⠀⠀⢀⠀⣠⣣⡾⢾⣿⠃⠋⠀⠀⠀⠀⠀⢀⣤
⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠲⢤⣨⣍⠉⡉⠛⠒⠛⠻⠿⢾⣿⡄⡗⠀⣠⣴⣾⡿⠵⠛⠉⠀⠀⠀⠀⠀⣠⠴⢞⠛⢹
⡁⠉⢀⣉⣀⣀⣀⣉⣀⣀⣀⣀⣤⠤⡬⠤⠤⠤⣤⠉⠀⠔⠛⠛⠙⠷⠒⢒⡒⢶⠷⠟⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⢒⡟⠃⢀⡀⠀⢨
⡇⠀⢸⠀⠀⡾⠙⠋⠛⠉⠉⠉⠛⠛⠲⢤⡀⠀⣻⠀⠀⠀⢸⡟⠒⠒⠢⠤⠤⣤⣀⣀⣉⣀⡐⠀⠀⢤⠞⠋⡕⠀⠼⠇⢀⡌⠀⢀⡼
⠇⠀⢸⠀⠀⡇⠀⠀⠀⣠⠤⢤⣄⠀⠀⠀⢳⠀⠸⠀⠀⠀⢸⡇⠀⠀⡠⠐⠉⠉⠉⠉⠑⢪⡉⠉⠃⠸⡇⠸⠀⡰⠁⠀⠈⠀⠀⠈⢛
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢿⣤⣤⠼⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣧⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⡅⠀⠨⠁⠂⠄⠀⡰⠀⠄⣩
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⢤⣞⠀⠀⢠⠀⠀⠀⠀⣇⠐⠀⠀⠀⣠⡤⠤⣄⠀⣠⡴⠟⢠⠀⡇⡠⢠⢀⠂⠀⠀⠁⠠⠀⢘
⠀⠀⢸⠀⠀⡇⠀⠀⠀⣤⠤⣄⡀⠀⠀⠈⠳⡄⠘⠀⠀⠀⠠⣿⡇⠀⠀⢸⡏⠀⠀⠈⠋⠉⠀⠀⠸⠀⡟⠢⢀⠄⠀⡀⠀⠀⠀⡈⣸
⠀⠀⢸⠀⠀⡇⠀⠀⠀⣧⣀⣤⠇⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⣟⠃⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠰⠀⢷⠀⠂⠂⠀⠀⠸⠀⣰⣾⣿
⠂⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠀⣤⡄⠀⣿⠰⠀⠀⠀⠈⠓⠒⠈⠙⢤⡀⠀⠀⠀⣫⠆⠀⡀⢠⠔⠂⢼⡟⢁⣿⠛⠑⠀⠠
⠐⡀⢸⣀⡀⠡⠒⠀⠂⠐⠢⠆⠲⠖⠊⠉⠠⠤⠼⠀⡯⢷⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⣺⠀⡀⠀⠈⠀⠀⡉⠼⠉
⠀⠙⠶⠶⢤⣔⣦⣤⣀⡒⠢⢠⣐⠶⠶⠤⣔⣒⣒⣒⣻⡻⢤⣍⢈⡑⠢⠷⠦⣤⣤⣤⠖⠛⠉⠀⠀⠀⣽⡃⠀⢀⣨⠞⠋⢁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⠶⠶⠤⢭⣍⡉⣒⠒⠶⠇⣋⣤⣼⣋⡡⠤⣤⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠬⠭⠭⠭⠍⠉⠁⡊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠂⠒⠐⠀⠒⠒⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢻
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠉⠒⢄
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⢀⠼⠀⣀⠤⠂⠁⠀⠀⠀⠐⠀⠀⠑⡄
⠀⠀⠀⠀⡆⣷⣄⠀⠐⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⢀⠤⠊⠁⠀⠚⠒⠠⠄⢀⣀⣀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢱⢹⣿⣷⡶⣤⣤⠒⠁⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⢺⡿
⠀⠀⠀⠀⠘⣿⣿⣿⡁⠀⡃⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠻⢿⣵⣀⡇⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠁⠒⠲⠤⠤⣀⣀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⢉⡁⠒⠒⠢⠤⠬⣉⠁⢒⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠂⠀⠀⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⢹⣿⡿⠛⠉⠀⠹⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⢀⡛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢃⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⡖⠁⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡄⠑⡆⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠘⠠⠤⠜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠦⠤⠜⠀⠀⠀⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠀⢘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠰⠆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣲⠖⠲⣄⠉⠒⠤⢤⣀⣀⣀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠊⠁⡴⠒⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠬⠃⠒⢃⡜⠲⠴⠖⠒⢉⠁⠉⠉⠉⠉⠹⠉⠉⠉⠉⠁⢉⠑⠒⠦⠀⠚⢆⠛⢶⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⠉⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠓⠒⠒⠀⠊⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣀⣀⣀⣀⠀⠀⠀⣀⣀⠀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⠟⠛⠾⠶⠿⠁⠻⠿⠛⠻⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠰⠚⠲⡀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡇⠀⠀⡇⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠏⠉⠙⠐⠈⠃⠂⠁⠀⠀⠁⠀⠊⠐⠈⠉⠉⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠒⠒⠈⠉⠉⠁⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠥⣄⡀⠀⠀⠀⣀⣠⠤⠔⠒⠂⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣸⠄⠠⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⠢⢄⡀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡮⠤⠤⠤⠤⠤⠤⠤⠬⠽⠶⠿⠭⠤⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢿⡆
⠀⠀⠀⢀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⡀
⠀⢀⡴⠋⠁⠀⠘⠷⠦⠤⠤⠶⠶⢦⡤⢤⣶⣤⣤⣤⣤⣿⣶⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣾⡃⠀⠀⠀⠙⢦
⡴⠷⠖⠒⠒⠛⠛⠛⠛⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⠤⠤⠤⠤⣤⣤⠴⠶⠶⢶⣿⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚
⠀⠀⠀⢀⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣴⣊⡉⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⢄⣀⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⣶⣼⠀⠈⠉⠛⠳⠦⢤⣀⠀⠀⠀⢀⣠⠴⠚⣷
⠀⣾⠥⠤⠤⣤⣤⣤⣤⣀⣀⣀⣀⠀⢀⡜⢹⡟⢧⣀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠉⠀⠀⠀⣹
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⠀⣸⣡⠤⠚⠋⠉⠙⠓⠶⢤⣄⢸⡇⢀⡠⠔⠊
⠀⠸⠶⠤⠤⢤⣤⣄⣀⣀⡀⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⢹⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢅⡀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢤⣀⡤⠒⠉⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⢸⠀⠀⠀⢀⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢼⣤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠜⢤⣉⣰⠤⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠄⠀⠂⢀⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡁⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠐⠢⠌⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠤⡤⠤⠌⠀⠡⣤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⡿⡄⠀⠈⠙⢣⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣌⠀⠨⡀⠀⠀⡀⠀⢀⠄⠀⠀⠈⡇⢷⠀⠀⠀⠀⢀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠛⠃⠐⠶⠂⠶⠦⠆⠞⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⢐⡒⠂⢰⠀⠀⠀⠀⡇⠘⡇⠀⠀⢠⠆⠠⠀⠲⠰⡄
⠀⠀⠀⠀⠀⠀⢀⠂⡈⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠇⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠂⢸⠀⠇⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⡐⠀⡀⢠⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⣇⠠⡌⠀⢰⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⣶⡠⢤⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠙⡇⠀⠀⢰⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠄⠀⠀⢀⠏⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠂⠁⠀⠀⠀⡄⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡌⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡐⠡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠊⡰⠀⠀⢢⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢃⠂⠀⠐⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡠⠁⠀⢳⠈⡃
⠀⠀⠀⠀⠀⠀⢀⠇⡰⠀⠀⠀⠸⡀⠣⢀⠀⠀⠀⠀⠀⠀⡔⠀⠌⠀⠀⠀⢫⡈⢢⡀⠀⠀⠀⠀⠀⢀⠆⡠⠁⠀⠀⠘⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠑⠤⡨⠆⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠡⢄⡵⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠐⠤⢈⠇

⠀⠀⠀⠀⠀⠀⢴⡷⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣤⣤⡃⠮⡀⠀⠒⠀⠀⠀⠀⠠⡀⠐⠲⠞⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⢸⢀⣩⣾⡇⠠⠂⠀⠀⠀⠀⠀⢈⠀⠀⣸⡿⠛⠉⢘
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⡈⣀⣤⣤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣨⣥⠃⣉⣀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢶⣿⡇⠀⢀⣼⣦
⠀⠀⢹⣷⡀⠀⣀⡀⠔⠳⠾⢿⣕⠾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠋⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠛⠉
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡦⠀⠀⢹⣆⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣆⠦⣀⡰⢿⡃⠀⠀⠈⢿⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⣸⣧⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡟⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣤⠠⠤⠤⠦⠀⠤⠀⢺⠶⠘⠁⠄⢀
⠀⠀⠀⠀⢀⣀⣀⣾⣮⣿⣦⣶⣿⠃⠀⠀⠀⠀⡾⠛⠀⠀⠀⠀⠀⠀⠀⣄⠠⠂⠀⣀⣤⣖⡿⠁⠀⠀⠀⠀⠀⠐⠀⠀⢰⣇⣡⡀
⢀⣠⣀⢠⠟⠃⠀⠉⠙⠛⣿⣿⡏⠀⠀⠠⣤⡄⢹⣶⡶⠦⠀⢀⡀⡀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠜⠁
⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠈⣙⣻⣭⢦⡤⢈⣛⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⠗
⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⢻⡟⠀⠀⠀⠀⠄⠀⠀⠛⠉⠙⠛⠻⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠃
⠀⠀⠀⠀⠰⠋⠛⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⢠⣀⡭⠇⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⣤⡌
⠀⠀⠀⠀⠀⠠⢿⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣹⣷⣦
⠀⠀⠀⠀⠀⠀⠘⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢿⡿⠷⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣿⢟⣧
⠀⠀⠀⠀⠀⠀⠀⢶⣀⣀⠀⠀⠀⠀⢠⣴⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣆⡉⠉⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣰⣿⠿⠋⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠛⠿⠟⠀⠉⠁
⣄⠀⠂⠀⠀⠀⠀⠀⠈⠷⠀⠀⢠⣿⡟⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠓⠀⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠘⢿
⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣥⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⣼⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⠀⠀⠀⣠⣧⣀⠀⠀⠀⠀⢀⠀⢀⡼⠋⠀⣴⠖⢶⠙⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⡇⣿⣽⠀⠀⣠⣤⣬⣙⣿⢁⣤⡀⠙⠶⠞⠀⠈⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⡟⠀⠀⢻⡄⠀⠀⣹⡿⠶⠏⠀⣠⣤⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠃⠀⠀⠀⠀⠀⠀⠀⣿⢰⡇⠀⢰⡟⠃⠐⠋⠉⠙⢦⣄⣀⣙⣾⣵⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⠳⣆⠀⠀⣿⢸⡇⠀⠀⣳⠞⠛⢦⡀⠀⢠⡟⠉⠉⠹⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠘⠁⠀⢻⠀⠀⣿⣤⣧⠀⠀⡿⣠⠖⠂⠹⣄⠀⠀⠀⠀⠀⠁⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⣇⡼⣣⠏⠸⣦⣀⡇⣠⠶⠛⠃⢹⡶⠒⠲⠶⢤⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣠⣤⠴⠞⣋⣀⣩⣤⣤⣬⣟⣷⢧⣤⡀⣠⠏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠴⠿⣯⠶⠛⢻⡁⠀⠀⠀⠈⣿⠳⢾⡽⠏⠁⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢰⠇⠀⣠⡴⠚⠋⠉⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡟⠀⠀⠀⠀⠀⠀⢠⠸⡆⠀⢀⡟⠀⠚⠁⢀⣠⡤⠶⠖⠀⣷⠀⠠⣞⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⣷⠀⢸⠃⠀⢠⠞⠋⠀⠀⢀⣀⣀⢸⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠃⢹⠀⣿⠀⠀⠀⣀⡴⠖⠛⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠇⠀⠀⠀⢀⡀⠀⠀⠀⡾⠀⢹⡀⠀⠘⠉⣀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠉⠙⠛⠛⠁⠀⠀⠙⠛⠋⠉⠉⠉⠙⠛⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠰⠠⠆⠂⠖⠆⠰⠀⠆⠐⠀⠆⠢⠲⠰⠶⠰⠀⠰⠆

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

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⢶⠖⠶⠲⢶⠖⠶⠖⠶⠲⠶⠶⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢂⠄⡐⠁⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣶⠑⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⣿⠊⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠐⠀⢸⠊⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣿⣿⣿⣸⣿⣿⣧⠀⠀⠀⢀⡇⢸⣷⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⣼⣿⣿⡿⠿⠻⣿⣿⣧⠀⠀⠀⡇⠈⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⣼⣿⣿⣿⡀⠀⠀⣿⣿⣿⣆⡀⠀⡇⠀⣯⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣼⡇⠀⢇⠀⢸⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⠿⠇⠀⡇⠀⣧⢸⡇
⠀⠀⠀⡘⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⢸⡇⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⡇⣰⠛⠟⡒⢲⠶⠦⠤⠤⣤⣤⣀⣀⣀
⠀⠀⠀⣿⠁⡉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⣀⣀⣀⣀⣁⣀⣁⣀⣀⢀⣀⣀⠀⡀⠀⠀⢀⠀⣠⣷⡟⡸⡴⠠⠱⠁⠄⠀⠀⠀⠀⠀⢀⠞⣧
⠀⠀⠀⣟⠀⡐⠡⠀⠀⠀⠀⣀⡀⣤⣤⡤⢤⠤⠤⠼⠦⠶⠶⠿⠛⢛⠛⠛⢹⡏⢉⡽⠿⠿⠟⠿⠯⠽⢥⣬⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰⣫⢂⣼
⠀⠀⠀⣿⠎⠔⠠⠁⠀⢠⠎⢁⡀⠤⠤⠤⠤⠒⠒⠒⠒⠒⠊⠉⠉⠉⠉⠀⢸⡇⢸⠀⠀⡤⠤⠤⠤⢀⣀⣀⣀⣀⣀⠈⠉⠑⠒⠂⠤⡀⠰⢡⠊⣾
⠀⠀⠀⣯⠊⠔⠠⠀⠀⢸⠀⢰⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢸⡇⢸⠀⢰⠀⠀⠀⢀⣀⣤⣄⣀⠉⠉⢉⡉⠛⠒⠦⠀⢱⠀⢁⠕⣻
⠀⠀⠀⣧⠊⠔⠀⡀⠀⢸⠀⢸⠀⠈⠛⢿⣿⣿⠉⠉⠙⣿⣿⡷⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⠀⣠⣿⣿⣿⣿⣿⣷⣦⣿⣿⡇⠀⠸⠀⢸⠀⢀⠤⣿
⠀⠀⠀⣧⠈⠌⡔⠁⠀⢸⠀⢸⠀⠀⠀⢸⣿⣿⣤⣶⣿⣿⡿⠃⠀⠀⠀⡆⢸⠀⣸⠀⢸⠀⢰⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⠃⠀⡆⠀⢸⠀⠐⢕⡏
⠀⠀⠀⣿⠈⡜⠀⡀⠀⢸⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠃⢸⠀⡷⠀⣾⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡰⡇
⠀⠀⠀⣿⠈⠀⡔⠠⠀⢸⠀⢸⠀⠀⠀⠘⣿⣿⡇⠀⠀⣽⣿⣿⣿⠀⢰⠀⢸⣯⡇⠀⣿⠀⠸⣿⣿⣿⣿⣆⠀⢠⣶⣶⡤⠀⠀⢹⠀⢸⠀⠀⠘⡇
⠀⠀⠀⠹⣄⠜⡐⠁⠀⢸⠀⢸⠀⠀⠀⢀⣿⣿⣷⣶⣿⣿⣿⡿⠃⠀⢸⠇⢸⡥⣇⠀⣿⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢸⠀⢸⠀⠀⢘⡗⡠
⠀⠀⠀⠀⠈⠷⣤⢀⠀⢸⠀⢸⠀⠀⣿⡿⠿⠿⠛⠛⠋⣉⢁⣀⣀⣄⠼⠁⢸⡇⡗⠀⠻⠤⠤⠤⠤⣤⣈⣉⣉⣉⣁⠀⠀⠀⠀⣾⠀⢸⠀⣠⡾⢃⠔⡡⠊⡀
⠀⠀⠀⠀⠀⠁⡈⠿⣄⢸⠀⠘⠆⠴⠒⠒⠚⠋⠉⠉⠉⠉⠉⣀⣀⣀⣠⠤⡚⢃⢙⠦⣄⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠗⠀⢸⡶⢋⠔⠡⠞⠀⠊
⠀⠀⠀⠀⠀⠉⠐⣈⠬⡿⢷⣤⣤⠤⣤⠤⠴⠖⠶⢒⠞⣫⡟⢍⡽⡫⢕⠞⡫⢔⡡⣂⠔⣀⢔⡡⠋⣩⢋⠙⢋⣶⠖⡲⠦⠤⢤⡤⠴⠋⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⠒⠁⠊⠑⠊⠟⠉⠒⠉⠈⠁⠈⠉⠀⠀⠁

⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⣿⢿⡿⠟⢛⡿⡿⡿⠛⠛⡟⠛⡻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⠋⢁⠄⠀⠀⠠⣄⣩⠶⣿⠾⠭⣒⠭⣑⠩⠞⣡⣦⠊⢀⣜⢫⢫⡏⢸⢿⢸⡘⢾⣜⢞⣾⣇⠙⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡏⣠⡳⠁⠀⠀⣈⣴⡾⢿⣾⡯⠿⠚⠉⣀⣤⡾⣫⠟⣼⡜⢎⣴⢿⠇⠈⡿⣯⣿⣌⢻⣷⡻⣯⢆⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡻⡰⣱⢁⢼⠝⡩⠚⢉⡠⢶⣒⣺⠿⠛⣩⠞⠁⢰⡿⣸⢮⠏⠎⠀⢰⢁⠏⣼⣿⠀⢹⣷⢻⡏⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣯⢧⣇⡇⠈⠋⠘⠀⠀⠊⠞⠉⠀⠀⣲⡿⠃⠀⣠⡿⢫⣾⠏⠀⠀⠀⠂⠀⡰⢏⡏⠀⢸⣿⠈⣿⣹⣿⣿⡇
⠀⠀⠀⣿⣿⡿⣸⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣒⠩⠉⠀⠀⢼⣪⢜⠽⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡿⢰⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣧⢡⢿⡇⢀⢸⡇⠀⠐⠉⠉⠀⣀⣠⣀⡒⠊⠉⠀⠁⠀⠀⣠⣴⠶⣤⡀⠀⠀⠀⠀⢠⡿⠁⣾⣿⢿⣿⣿⡇
⠀⠀⠀⣿⣿⡎⣯⢎⣆⠀⣯⣇⠀⠀⠀⠀⠈⠁⠀⠈⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣸⣿⡏⢸⣿⣿⡇
⠀⠀⠀⣿⣿⡇⢸⡜⢸⠀⠈⡿⡀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⢀⡀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⢰⡿⢻⡇⢻⣿⣿⡇
⠀⠀⠀⣿⣿⡇⢸⠀⠈⠀⠈⡴⠃⠀⠀⠀⠀⠻⠿⠟⠀⢀⣴⠞⠀⠀⠀⠀⠿⣿⠿⠀⠀⠀⠀⠀⢀⢸⡷⡜⠷⣾⣿⣿⡇
⠀⠀⠀⣿⣿⡇⡜⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣤⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⠎⢓⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⢹⡶⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤⣀⣤⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣋⠉⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣧⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣌⣉⣉⣼⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣦⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠻⠿⠾⠖⠿⠟⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣴⠁⠀⣀⣀⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡍⠉⠙⠛⠛⠛⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣿⣿⣄⣀⡀⠀⢀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣾⣁⣀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⣻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣷⣴⣿⣿⣿⣿⡿⢿⣷⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣾⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⠈⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣨⣿⣿⣿⣿⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡤⢾⣿⣿⣿⡟⠉⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⢀⣬⣷⣿⣿⠃⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⣿⠗⣄⣠⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣶⠋⠁⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⣄⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠟⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡁⢀⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⠁⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⡇⣼⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡿⠻⣿⠇⣿⣿⡯⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⠻⠁⣾⠃⣰⣿⠀⣿⣿⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡄⣼⢃⣾⣿⡿⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠉⠙⠋⠉⣿⣿⣿⣿⣿⣇⢀⣿⣯⣾⣿⣿⣧⣾⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠿⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⠉⠉⠛⢿⣿⣿⣿⣿⡿⠃⠀⠀⠈⠛⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡤⠃⢸⣦⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠙⠃⠁

⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠻⢶⣄⡀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡄⠀⠻⣿⣦⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⠀⠀⠈⢿⣿⣦⣠⣿⠇⠀⣀⣀⣠⣶⣆⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⡀⠀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣯⣭⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣄⣀⡀⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢲⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢿⣿⣿⡹⠿⠟⠛⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠻⣿⣟⣿⣻⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⢿⡿⠛⠋⠁⠀⢸⣿⣿⣿⣿⡧⢸⣿⣿⣿⠿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠂⣤⣼⣿⣿⠟⠁⠀⠀⠀⠙⣿⡿⡿⠋⠁⠀⠀⠈⢿⣿⣷⡓⣄⡀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠿⠋⢁⣀⡙⢿⠿⢟⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⡏⣀⠄⠿⢠⡀⠀⠈⠴⢁⣠⠴⠇⠀⣀⣀⣿⣿⣷⣿⡃⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠸⣿⠟⣿⣿⡦⣿⣿⠟⣿⣷⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣄⣀⢀⠀⠀⠀⠀⠠⣉⡀⠀⢀⣀⣤⣼⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣾⣇⠀⠀⣶⣴⣾⣥⣰⣿⣥⣤⣬⣭⣭⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠰⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡏⢸⣿⡇⠀⢨⣤⣟⡃⣹⣉⣘⣿⣟⣭⣿⣿⣿⣿⣿⣿⠀⣻⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣭⡟⠉⠀⠠⣦⠸⠆⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣧⣼⣿⣿⡄⠈⢉⣛⣋⢛⣙⣿⣽⣯⣿⣏⣿⣿⣿⣿⠃⠀⣽⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⠖⠀⢰⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠈⣽⣯⣽⣥⣽⢷⣯⣴⣿⣿⣿⣿⡿⠀⠘⣾⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠈⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣀⢺⣷⣾⣿⣿⣿⢶⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣙⣞⠻⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⣴⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⢀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣀⢠⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠚⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠊⠁⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠌⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣖⣁⣀⣀⣀⣀⣀⣀⣀⣐⣀⣐⣀⣀⣀⣀⣀⣬⣤⠭⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⣽⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣸⠀⠀⠀⠀⠀⠀⢀⠔⠚⠛⢯⡛⠢⡄⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⢹⠀⠀⠀⠀⠣⠀⠃⠀⠀⠀⠀⠙⣧⠸⡆⠀⠀⠀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣼⠦⠤⠠⠤⠀⠸⡄⠀⠀⠀⠀⠀⣸⢶⡧⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣺⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢽⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣟⠀⡇⠀⠀⠀⠋
⠀⠀⠀⢠⡶⢶⠴⠂⠉⢙⣿⣄⣸⠀⠀⠀⠀⠄⠀⣤⠤⢤⡀⠀⠀⢷⡀⡇⠀⠀⠀⠁⠱⠀⠀⠀⢰⣶⣶⡶⠈⠉⠉⠉⠛⠉⠒⠶⠶
⠀⠀⠀⢸⡆⢸⡀⠀⡀⣶⠉⢻⠉⠁⠉⠀⠀⠉⢹⡇⠀⠀⡄⠀⢸⡟⢻⠗⠒⠒⠒⠒⠲⠚⢰⣿⡌⠉⠉⠟⠺⠋⠱⢦⢀⠄
⠀⠀⠀⣼⡇⢸⡇⠂⢉⣿⠀⢼⠀⠀⠈⠀⠄⠂⢸⠇⠈⠀⡇⠀⢸⠁⢸⠀⠀⠀⠀⠀⠀⠀⣸⠿⡅⠀⠀⡾⠂⠀⠀⠘⠀⠀⠀⠐
⠀⠀⠠⠼⠿⠿⠧⡧⢶⣯⡀⢸⠀⢴⠋⠀⠠⠈⢸⣧⣶⠾⢇⣀⣸⠠⢸⠀⠀⠀⡀⠀⠀⡉⣹⢺⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒
⠀⠀⠉⠀⣠⠠⠀⠠⡌⣛⣶⣽⣀⡀⠐⠁⠀⠤⠼⠛⠓⢶⣾⣴⣿⣶⣾⣀⣀⣀⡁⠁⢀⣿⣽⣿⣀⣀⣀⡀⠈⠒⠆⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⠛⠷⣦⣀⠀⠀⢀⣠⡶⠟⠋⠁⠀⢸⣿⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠈⠙⢻⣾⠛⠉⠀⠀⠀⠀⠀⢸⣿⠉⠉⠛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢛⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⡆
⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠛⠷⣦⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⣴⡟⠋⠀⠀⠀⢀⣠⣴⠟⠋⠁⢸⡇
⠀⠀⠀⠀⢸⡟⠷⣦⣀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⢸⣿⠀⣀⣤⡾⠛⠉⠉⠛⢷⣤⣤⡾⠛⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠈⠙⠻⢶⣄⡀⠀⠀⢀⣠⡶⠟⠛⢿⣿⢾⡏⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⢷⡾⠛⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠻⢶⣤⣀⠀⠀⢸⡇⠀⠀⣀⣤⡾⠛⠉
⠀⠀⠀⠀⠘⠻⢦⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⣴⠟⠃⠀⠀⠀⠈⠙⠷⣦⣼⣧⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⠀⢸⡇⠀⣀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣾⣷⠟⠋⠁



⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⢀⡀⠀⣀⠀⣀⢀⣀⢀⣀⢀⣀⢀⡀⠀⡀⣀⡀⢀⠀⢀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠐⠈⠃⠀⠘⠃⠘⠘⠙⠙⠚⠙⠙⠘⠃⠀⠙⠀⠃⠘⠒⠀⠀⠑⠂⠉⠁⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⢉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠁⠀⡀⠀⢴⠒⡴⠒⡆⠀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠐⠀⢲⠤⠤⣬⣇⡀⢸⡀⠀⠁⣲⢠⢄⠧⣤⡼⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠠⢼⡏⠉⠈⠀⠉⢹⠛⢺⡄⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⡈⠀⠨⡇⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⡤⠄⠂⠚⠛⣉⣹⣆⣰⣇⠀⠀⠀⠀⠸⠀⣰⣇⣀⣠⣌⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢸⡏⠉⠉⠉⠉⠃⠀⠀⠹⡟⠛⠛⠋⢿⠻⡒⡲⠒⠐⠉⡆
⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣼⣼⣤⣤⡀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠒⠀⠉⠁⠀⡤⠭⠝⢳⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⠀⠈⠊⡿⠀⠐⣇⠀⢄⠤⢀⠀⣀⢠⡀
⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠐⠀⠀⡠⠁⣿⣀⣀⣀⣿⠦⠶⠟⢳⢠⠁⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣇⣀⣀⣼⡿⣂⡀⠀⠀⠀⠨⠴⠶⠶⠖⠫⠉⠁⠋⡍⠒⠀⠁⠈⠉⠁⢒⣠⠀⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡪⠁⢸⠲⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡀⡀⣇⣠⣷⡀⠠⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⠀⣇⡰⣞⡃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⡇⠀⠀⠎⠛⢲⡖⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣠⠴⢦⠀⠀⠀⠚⠋⢰⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⢠⠿⠛⠛⡆⠀⠀⠘⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⢸⠀⠠⠶⠃⠀⠸⢶⢾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠧⠂⢐⣢⢧⠤⡇⠀⠀⠀⠀⠀⠀⠀⣰⠒⠈⠍⢉⣉⢭⢶⠄⠀⠀⠀⢘⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠢⡧⠔⡇⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⢰⠹⠂⠂⠀⠀⠀⠀⡇⠀⢨⢱⡄⠀⠀⢸⠀⠀⢸⠠⠄⣐⠐⣒⡠⡄
⠀⠀⠀⠤⠆⢀⠀⣀⢡⠀⠀⡇⠀⢻⠀⠀⠈⡂⠀⠀⢃⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢺⣇⠀⠀⢸⠀⣀⣼⢀⠀⠀⠸⠀⠂⡁
⠀⠀⠀⠀⠀⠀⢸⢁⢸⠀⠀⢧⡸⠃⠁⠐⠒⠧⡶⡾⠿⠿⢛⠀⠐⠂⠥⠀⣰⠀⠀⠀⠀⡇⢀⢰⣿⠄⠁⠀⢸⠋⠉⢸⣸⠄⢀⣸⣠⣤⣅
⠀⠀⠀⣀⠀⠠⡞⠋⠉⢤⡵⠻⢺⠀⠀⠀⠀⡇⢈⠀⠠⠀⢐⠀⢀⠀⠀⠖⠉⠁⠀⠠⣀⣧⡠⢼⣼⠁⠀⠀⢸⠀⠠⢸⠛⠀⠐⡞⠉⠁⡏
⠀⡚⠀⠀⣤⠄⡇⠀⠀⢸⡆⠀⢺⠀⠀⠀⠀⠇⠀⠀⠀⠀⢐⠀⢸⠀⠀⠅⠀⠀⠀⢱⠉⠀⠀⢸⣿⠤⠄⢀⣠⣤⠥⢽⠀⠀⠀⡃⠀⠀⡃
⠀⠁⠀⠀⡗⠀⣇⠀⢀⢸⡆⣀⣬⡆⠀⠀⠀⠅⠀⠀⠀⠀⢸⡇⠈⠓⠠⡅⠀⠀⠀⢸⠀⠀⠀⢸⡿⠀⠀⠀⢘⠁⠀⣿⡾⢖⠰⠗⠚⡾⡃
⠀⠐⠀⠂⢽⣤⡇⠀⠀⢸⠋⢡⠶⠓⠠⢀⠄⢀⠀⠀⠀⣀⣼⣷⣒⠛⠤⠉⠐⠀⢃⣒⢄⡤⠶⢺⣿⠀⠀⠀⢐⠀⠔⡿⠂⠀⠅⠀⠈⡊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡅⠈⠀⠀⠀⠂⡐⠋⠋⠋⢩⠀⠸⡃⠀⠁⠐⠠⡴⠒⠋⠉⠀⣀⡤⠞⠛⠓⠀⠀⣸⡦⠖⣿⠀⢐⡅⠀⣀⡤⠯⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠈⠀⠘⠰⠦⠺⠀⠀⠀⠀⡂⠈⡀⠐⡁⡠⢺⡆⠀⠀⠀⠀⡂⠠⠀⢸⠻⠋⠍⠁⣀⡠⠐⠋⠁⠐⠀⣿⠉⠐⠓⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠈⠀⠊⣸⣗⠄⠀⠀⠀⡃⠀⠀⢸⡀⠀⠀⠀⣚⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⢀⣂⣠⣤⡾⣗⡛⡉⠉⠁⠒⠦⢄⣧⠬⠖⣛⣲⠄⠤⣀⣶⡤⠦⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣓⣒⣒⣒⣓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⠤⢴⡟⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⣀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣤⣀⣀⡇⠀⠀⠀⠀⡇⢀⣭⢶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠟⡸⣿⣇⠀⠀⠀⢀⣗⢛⣀⢿⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠝⢋⣈⣽⣦⣤⣤⢾⣯⣙⢿⣈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠱⣾⣛⣫⢽⣾⡤⣷⠈⣬⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣳⣾⣹⣎⡽⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢾⣻⣿⣿⣟⣻⡶⣦⢯⣙⠳⡄⠘⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⢉⡵⠞⣛⡛⠚⠥⣭⡺⣝⣿⣿⣿⡇⢸⣾⢿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⣢⢞⣵⣷⣿⣿⣿⣛⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢰⣰⣯⣾⣟⣯⣥⣶⣶⣿⣟⡓⠈⡛⠋⠉⠀⠀⠙⠉⠙⢿⣿⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⣤⣼⡿⢟⣩⣾⡿⢓⣉⣤⣬⣿⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠘⠙⣋⣴⠿⣫⠵⠾⠛⣛⣻⣿⡗⠀⠠⠔⠒⠶⢦⣤⠀⠀⢀⡤⢿⣧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⢛⣩⡶⠛⠁⣀⣠⣶⣿⢿⣿⠁⠀⢀⣤⣴⣶⣦⠀⠀⢠⣏⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⡴⠁⡄⠀⠀⢀⡟⢩⣾⣿⣿⠿⠯⣭⣭⡽⠛⢿⣟⣣⡀⠀⠀⠉⠛⠛⠀⠀⠀⠸⡟⢿⠛⠁⢹
⠀⠀⠀⠀⠀⢰⠃⠀⠱⢤⡤⠎⠁⢈⣯⣵⣶⣶⣶⣿⡿⣿⡀⢵⡙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⢘⡇
⠀⠀⠀⠀⠀⠘⠀⢀⣠⣤⠶⠤⣅⠟⣽⢛⣿⡿⠯⠵⠊⣽⣿⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⡅⣸⠁
⠀⠀⠀⠀⢀⡴⣳⢛⡥⠖⢛⣫⢟⡔⠡⢋⣡⣴⣶⣶⣾⣋⣾⠸⣅⡟⠢⣀⠀⠀⠀⠀⠀⠉⢉⡹⣱⠃
⠀⠀⠀⠀⡾⠀⣿⠏⣠⣾⠟⢱⡋⣿⢰⢯⠿⣛⣛⣻⣭⣾⣿⡄⠈⠀⠀⠈⢳⣦⣄⣀⡀⠀⠀⢠⠏
⠀⠀⠀⠀⠹⢤⣟⣴⣱⢯⢸⠘⣧⣹⡏⡴⠟⠋⠉⠉⣠⣟⣿⡇⠀⠀⢰⡀⠀⢻⣿⠋⠉⠉⠉⠁
⠀⠀⠀⡇⠀⢸⣿⡿⣿⢸⡜⢧⣲⣶⣿⣿⣿⣿⣿⣿⣿⢟⣽⠇⠀⠀⠀⢧⡀⢸⡇
⠀⠀⠀⣇⠀⠨⣯⣷⣹⣯⡛⠾⠿⠿⠿⠿⠿⠿⣿⡿⠖⠋⠈⠀⠀⠀⠀⠈⠳⣼⡇
⠀⠀⠀⠘⢦⡀⠠⢉⣉⣉⣛⠿⠬⠤⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠉⠙⠛⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⠶⣤⣌⠻⡆⠀⣀⣀⣀⠀⠀⢀⣤⠶⢚⣋⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠤⢈⣥⣾⢿⡿⢋⣵⣾⣶⣿⣵⣾⣿⣿⣿⣿⣿⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⣀⣤⣤⣤⡄⢿⣿⡇⣿⠀⣾⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⣼⣿⣿⣿⣿⣿⣌⠻⣿⣹⡄⢻⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⣼⣿⣿⣿⣷⣝⢿⣿⣷⣌⡻⢿⣶⣍⣻⣿⣿⣿⣷⣽⣿⣻⣽⣿⣿⢏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡃⢠⣿⣿⣿⣿⡿⣿⣷⣝⡿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣟⣛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢇⡇⠘⣿⣿⣿⡻⣿⣮⣻⢿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⠉⠉⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢸⣇⠀⢿⣿⣿⣿⣮⣻⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠸⣿⡄⠈⣿⣿⣎⠀⠄⠹⠋⠀⠀⡀⠈⠉⠉⠙⠻⣶⣄⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡄⣿⣷⠀⠘⣿⣿⡄⠀⠀⠀⠀⢲⣿⣿⣿⣶⣶⣤⠀⠀⠀⠀⠠⠚⢓⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣧⢹⣿⡇⠠⡘⣿⣷⣀⠀⠀⠀⠀⠈⢻⢦⣭⣝⠛⠀⠀⢠⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⢰⣧⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡾⣿⣿⡇⣼⣿⢸⡿⣿⡏⢳⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣌⢻⣿⠟⠛⠛⢿⣿⣾⡇⣿⡇⠀⠙⢷⣄⠀⠀⠀⡙⠟⠒⢀⣤⣟⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⡏⣾⣿⣾⢀⡏⣰⣿⣿⡄⣿⣿⢳⣿⣿⠀⠀⠀⠙⢷⣤⡀⢈⣡⣶⣿⣿⣿⡿⢛⣉⣁⡀
⠀⠀⠀⠀⠀⢸⣿⣧⡙⠿⢋⡼⢠⣿⣿⣿⢁⣿⣟⣾⣿⣿⡆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣯⣶⣿⣿⡿⣃⣾⣿⣿⣿⠋⠀⠻⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣮⣿⣽⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠝⠛⠋⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⢠⣴⡤⣤⡦⣠⣤⢠⣤⣤⣤⣤⣦⡤⠀⢠⣤⢠⣤⡦⣴⣤⠀⣤⣴⢄⣴⡀⡦⣴⣦⠀⢠⣦⠄⣤⡄⣤⠀⣴⣦⣤⣤⡄
⠀⠀⠸⠿⠗⠿⠷⠿⠿⠸⠿⠃⠿⠀⠸⠇⠀⠿⠿⠼⠻⠇⠿⠟⠀⠿⠿⠾⠿⠷⠷⠿⠿⠀⠸⠿⠻⠿⠷⠿⠟⠿⠿⠻⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠐⠊⠁⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢀⠀⣀⢠⣀⠤⡀⠀⠀⠀⠀⠀⢀⠐⠒⠄⢠⡀
⠀⠀⠀⠀⠀⡀⠂⠀⠀⠘⠢⠀⠀⠀⡎⠀⠠⠀⠄⠀⣀⠀⢐⠄⠀⢀⠊⠉⠀⠀⠠⠄⠀⢀⠂⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠰⠀⠀⠀⠁⠠⡀⠃
⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⢀⡄⠀⠀⢆⠰⠀⠀⠀⢀⠘⠇⢨⠀⠀⢀⠀⠀⠀⠀⠄⡴⠀⡀⠁⠀⢀⠀⠠⠀⠀⠀⣤⠀⠀⠁⢀⠀⠠⠀⠀⡁
⠀⠀⠀⠀⠀⡆⠀⢀⠐⠀⠀⡇⠀⠀⠈⢆⡀⠀⠂⠀⠀⠔⠁⠀⢀⠼⠀⠀⠀⢀⡰⠃⠀⠀⡁⠀⠀⠀⠀⠀⠘⣒⠀⠀⠀⠡⣄⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠹⠋⠁⠀⠉⠫⠀⠀⠀⠀⡠⠚⠁⠘⠢⠄⠀⢀⠰⡆⡖⠊⠁⠈⠑⢀⠀⠁⠀⢡⡤⠔⠂⠈⠉⠉⠄⠀⠀⠀⠀⠀⠜⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⢖⠀⠀⠀⠌⢀⠀⠀⠀⡀⢀⠀⠴⠆⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⡀⢸⠆⠀⠀⠘⠀⠀⡜⠠⡃⠀⠀⠀⡇⠘⠀⠀⠀⠘⢀⠁⠀⠀⠘⢸⠀⠀⠀⢠⠀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⢸
⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠘⡆⠀⠈⢰⠀⠀⠀⠀⡇⢀⠀⠀⠀⠂⠆⠀⠀⠀⠜⢸⠀⠀⠀⠸⠀⠇⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠂⠨⢸⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠡⠘⠀⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠹⠋⠃⠘⡄⠀⠀⡘⠀⠀⠀⠸⠀⠀⠀⠀⠡⣆⡀⠀⠀⢾⠀⠀⠇⠀⠰⡷⠂⠀⣾⡤⠆⠀⡐⠁⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠐⣴⠆
⢸⣟⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⢐⠀⠀⠀⠀⠀⠀⢂⠀⠈⠄⠀⡀⠀⠀⠀⠐⠀⢠⠁⠀⢢⠀⠐⡀⠀⠀⠀⠀⢃
⢸⣿⠀⠀⠀⠀⡐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⣆⠈⠀⠀⠀⠀⠀⠀⠃⠀⢠⠃⠰⠁⠀⠀⠀⢨⠀⡇⠀⠀⠀⢑⠀⢃⠀⠀⠀⠀⢈
⢸⣿⠀⠀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠁⠀⠇⠠⠈⠄⠀⠀⠀⠀⠰⠀⡇⢠⠁⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⢂⠀⠆⠀⠀⠀⠀⠀⠀⠇
⠘⣿⠀⠀⠀⡐⢡⠁⠀⠤⡄⠀⠀⠀⠀⠸⠀⠄⠀⠔⠸⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⢀⠇⠈⠀⠀⠀⠀⠀⠀⠢⡈⢄⠀⠀⢀⠆
⢘⣿⠀⠀⠐⠀⠂⠀⠀⠡⣀⣰⠠⠀⠀⢀⣀⠀⠨⡀⠘⠀⠀⠀⠀⠀⠐⣀⡑⠒⠂⠀⢴⣁⡰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢀⡀⡀⠁
⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⠤⠖⠿⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢘⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⢩⡋⢀⢀⣀⣠⣁⠀⠀⠀⠀⠀⠂⠀⠀⢠⠒⠛⠦⠌⠳⡲⣶⡶⠾⣦⠀⠀⠀⢸⠉⠉⡉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠸⡋⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠄⠀⠀⢨⠐
⠀⠀⠀⠏⠉⠏⠉⠉⠉⣽⡤⢀⣀⣀⠀⡅⠀⠀⠀⠀⠈⣧⠖⢂⢀⣇⣁⡀⠈⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡄⠀⠐⠉⠉⠉⠙⢻⠓⡀
⠀⠀⠀⠀⠀⡆⠴⠆⠉⠉⣁⢀⠩⣩⡽⠧⣤⣤⣤⢤⣤⡿⢾⠁⠁⠀⠀⢰⡴⢌⠐⠒⡢⠤⣤⢼⡶⠶⠶⣛⣿⣩⠩⠴⠲⠀⠀⢜⠀⡇
⠀⠀⠀⠀⠠⣇⡆⠀⠀⠀⡇⠀⠀⠰⠈⠀⠀⠀⠀⣞⠁⠀⢸⠀⠀⠀⠀⢹⣣⠀⠃⠀⠀⠀⠀⢈⡀⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⣸⣀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠂⢸⠀⠀⠀⠀⠀⢃⠀⠐⢸⠀⢀⣀⣀⣿⣿⠀⡅⠀⠀⠀⠀⢐⡆⠀⠀⢼⠀⠀⠀⠀⣼⠁⠁⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⣇⣤⣥⣴⠀⠀⠀⠀⠀⠐⣀⣰⣞⣛⣋⡛⠙⠐⠚⠓⠦⠤⠤⠤⢴⡽⠷⠦⢄⣧⣤⣤⡤⠤⠿⠤⠤⠤⠤⠤⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠑⢄⠀⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣄⠀⣀⣂⠀⠉⠀⠀⠣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⡠⠠⠃⠈⠀⠀⠙⠂⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡔⠠⠤⡀⡠⡀⠀⡀⠀⠀⠀⠀⢀⢀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⠀⠤⠄⠀⠀⠀⠁⠈⠒⠄⠴⠂⠥⠀⠣⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡸⠀⡠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠀⢇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⠀⠀⢃⣘⡠⠁⠀⠀⠀⠀⡀⠀⠀⢄⣁⠘⠀⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡗⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠒⡒⢒⠒⠀⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡖⠉⢉⠏⠙⠹⠟⣻⣟⠻⠏⠋⠹⡉⠉⢲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⡠⠛⠂⠞⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠳⠔⠛⢄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠠⣄⠑⣄⠀⠆⠤⠀⠀⠀⠌⠀⠴⠂⠠⠀⠸⠀⣠⠊⣠⢤⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⢀⡤⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⢤⡀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⢀⣀⠠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠂⡿⠴⠶⠚⣿⠛⠥⣠⠤⠴⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠀⢀⠀⣿⡇⠀⢐⠀⠀⢐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠤⣾⡷⠲⠻⠛⣛⣠⣴⠖⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠈⢱⠉⠁⠀⢵⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⣀⠀⠐⠀⠂⠠⣀⠀⠀⠀⠀⠈⠀⠦⠀⢐⢸⣀⣤⡤⣽⠚⠋⠉⣷⣦⣤⠴⠒⢒⣳⡄⠀⣄⡼⠇⠠⢄⡀
⠀⠈⠁⣶⢚⠝⢹⠀⠀⠠⠀⠀⠀⠀⠀⠀⢘⡠⠂⡀⣻⠀⠀⢀⣏⣿⠀⠈⡉⠁⠀⡿⠒⠚⠉⡖⠒⠒⠛⡆
⠄⢠⠄⣷⠀⣠⣼⡆⠄⠀⠀⠀⠀⠂⠄⠒⠈⣇⣀⣤⡾⠓⠛⡫⠔⠿⠤⣀⣧⣤⣴⡟⠀⠐⢦⡇⠀⠀⢘⣦
⠀⠀⠈⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠈⠉⠉⠁⠂⠀⠀⠀⠁⡄⠀⠀⠒⠄⡀⠉⠛⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠁⡏⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠈⠀⠀⢣⣇⣀⣄⣀⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⢸⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡑⢤⡀⢸⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⢳⠀⠙⠃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢌⠉⠉⠀⠀⠀⠀⣀⣘⡦⠖⠙⠋⠲⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡜⠉⠊⠓⠤⠖⠤⡤⢤⡀⠀⠀⠀⣀⠔⠢⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠁⠐⠓⠀⠀⠀⠀⠀⠀⠈⠑⠐⠁⠚⠂⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣰⠃⢤⠒⡐⢦⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠒⣤⠈⢆⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠙⢄⣁⠜⠀⠀⠀⠀⠀⢄⠀⠀⠢⣈⣀⠎⠀⠀⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡗⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠉⠒⠀⠀⠠⠀⠀⠴⠀⠐⠒⠉⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠒⢢⡤⠤⢤⡤⣤⡤⠤⡤⠠⠤⡄⠒⠪⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢹⣄⢀⡞⠀⠀⢸⠁⠻⠟⠀⡅⠀⠀⠱⡀⢀⡏⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠒⠴⡁⠀⢹⠁⠀⠀⢸⠀⢰⡶⠀⡃⠀⠀⠀⡏⠀⢈⡶⠒⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣠⣚⠷⠤⠵⠀⣼⣉⣉⣉⣉⢉⣍⣉⣉⣉⣉⣉⣉⣽⡀⠮⠤⠾⣗⣄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣟⣉⣛⣒⣒⣒⣒⣒⣲⠞⠓⣖⣒⣒⣒⣒⣒⣛⣋⣻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣿⡇⠀⠘⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⣿⡇⢀⣿⣿⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⣿⡇⠈⠻⠟⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⣿⡿⠶⠞⠳⠶⠞⠷⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠚⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⢿⡇⠸⠿⠇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⠃⠈⢿⣦⡀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣇⠀⣤⡄⠀⠀⣠⣄⠙⠻⠿⢿⣿⣿⡋⠀⠀⠀⠀⠀⣀⣀⣸⡿
⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠉⠉⠀⣀⣰⣾⣿⣏⣿⣷⣶⣶⠦⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠛⣀⣤⣾⣿⣿⣿⣽⣿⣿⣯⣼⡿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣿⣾⣿⣶⣿⣟⣻⣿⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡠⠊⢱⠤⠒⠒⠈⠙⢆⠀⠀⠀⣰⣿⡘⣯⣵⣶⣶⣶⡄⠀⠀⢀⢾⡁⠀⢀⡤⠤⣌⢢⡀⠀⢠⡸⣿⣿⠿⣛⣿⡛⢦
⠀⢠⣇⠀⠀⠣⣀⡠⠤⣄⣈⡆⠀⠐⢿⣿⣷⣝⠿⣟⣛⢛⡿⠀⠀⡎⠀⠉⠉⠁⠀⠀⠀⢣⠇⠀⣿⣿⣶⣶⣿⣿⣿⣿⡎⡇
⠀⠘⡏⠣⣄⢀⡎⠀⠀⠇⢀⠇⠀⠸⣮⡻⢿⡿⣸⣿⣿⣸⣿⠀⠀⢇⠀⣀⣀⡀⠀⠀⠀⡜⡆⠀⣿⣿⠿⠿⣿⣿⣿⣿⢇⡇
⠀⠀⠙⢄⣨⠏⠀⠀⣼⡠⠊⠀⠀⠀⠙⢿⢗⣵⣿⣿⢇⡿⠃⠀⠀⠈⠻⣀⠀⠈⠙⠒⣫⠜⠀⠀⠈⢵⣿⣿⣶⣭⣯⣥⠞
⠀⠀⠀⠀⠉⠑⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⣠⡶⠓⠀⠺⣶⣤⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡙⠻⣦
⠀⠀⠈⠁⢀⣤⣤⡆⠀⠉⠀⠀⠀⢀⣾⣿⡟⠛⠛⣿⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⣠⡦⠙⢷⡄⠀⠀⢀⣾⣿⣿⠏⢙⣷⣌
⠀⢰⣦⠀⠸⢿⡿⠇⠀⣠⣶⡆⠀⠈⠻⣿⣅⡀⢀⣽⣿⠟⠙⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣾⣿⡿⠁⣰⣿⣿⠏
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢦⣀⠙⠁⠀⠀⠀⠀⠀⠀⢈⠻⣿⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠐⢾⣿⡶⠁⠀⠀⠀⠀⠀⠀⠙⠏⠁⠈⠹⠟⠉⠀⠀⠀⠀⠀⠉⠳⠦⠀⠀⠀⠀⠀⠀⠘⢷⣦⡉⠻⠛⠁

⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣦⡀⠀⠀⠀⠀⢀⡤⠂⠈⠀⠒⢀⡀⠀⠀⠀⢀⢰⣶⣿⣷⣶⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠠⢁⠇⠀⠀⡴⠚⠉⠑⡀⠀⣰⡇⣾⣿⡿⢫⣴⣾⣷⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⣧⡞⠀⠀⢸⠁⠀⠀⠀⡁⠀⠛⣼⣿⣿⠅⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⣿⣿⠟⠀⠀⠸⡀⠀⠀⠸⡄⠀⠀⢠⠃⠀⠸⣿⣿⣿⡅⢿⣿⣿⡿⠁
⠀⠀⠀⠁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠈⠂⠄⠀⠉⠶⠒⠁⠀⠀⠀⠈⠻⠿⢿⡦⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣀⡠⠄⢀⡀⠀⠀⠀⢀⣿⣧⢀⣀⣀⣀
⠀⠀⠈⢿⢆⠀⠀⠀⠀⡀⠀⠀⠀⠙⣤⡀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠈⢀⠞⠁⠦⡀⠀⠈⢢⠀⠀⣾⣟⣵⣿⣹⢿⣿⣷⣄
⠀⠀⠀⠀⢂⠁⠒⠈⠁⣔⠀⠉⢐⡄⠘⣿⣶⣶⣿⡟⠽⣷⣶⣶⠦⠀⠀⢸⠀⠘⠂⠉⠀⠀⠀⡇⠀⢿⢹⣿⣧⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠢⠀⢀⣀⡁⠤⠀⠀⠀⠀⠘⠿⠿⠿⠷⠽⠟⠛⠛⠉⠀⠘⢌⣄⠀⠀⠀⠀⢀⡔⠀⠀⠘⠯⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁
⣟⣿⣻⣻⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣛⣛⣟⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡟⢦⣠⠟⡆⢀⣤⠖⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⡟⠉⢰⡿⢋⣿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠂⡾⢋⣭⣭⣭⡛⢿⣿⠿⣶⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠄⠀⠀⠳⠌⠁⠀⠀⠉⠂⢹⣧⠀⢉⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡏⡜⣏⠀⠀⢀⡤⠀⢠⡀⢀⣠⠀⠀⣿⡆⠀⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⣧⣀⠀⠀⡀⠘⢷⣤⣤⣛⣫⣥⠆⣰⡿⠁⠐⠦⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠛⠧⠄⠀⠀⠀⠈⣓⣒⣺⣛⣟⣭⣴⠿⠋⠁⠀⠀⠀⠀⠉⢻⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⢀⣠⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠘⠉⢋⣴⠟⣩⡶⠃⠀⠀⠀⠀⠀⢀⣀⣉⣉⣒⠦⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣴⠦⢠⣿⢋⣾⣿⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠉⠃⠀⠀⠀⣶⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⢸⠁⣿⣀⡙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣿⣿⡜⠰⣌⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⢿⣿⣿⣄⠉⠋⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⢿⣯⠙⠻⠿⠿⠃⡀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠲⠂⠈⠻⣷⣌⠲⣦⡴⠋⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢧⣀⢈⠛⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⡫⠉⢀⡀⠨⣝⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡻⠶⠞⠁⠀⠈⢿⣷⡝⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠘⣿⢿⣼⡗⢦⣄⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⠃⢸⣿⠁⠈⣿⣆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢻⡄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠚⣣⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⠿⠿⠿⠿⢟⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣿⣿⣟⣛⣻⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⡉⠉⠛⠒⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⣀⡤⠖⠋⠉⠉⠙⠓⠲⠦⣤⣀⡉⠉⠛⠲⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⣀⡤⠖⠋⠁⠀⠀⠛⢻⣿⣷⣶⣶⣶⣤⣤⣍⣙⡒⠶⠤⣄⣀⡉⠉⠓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⣀⡤⠖⣋⣁⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠋⠀⠀⢀⣨⠽⠛⠀⢀⣠⠴⠛⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣋⡁⠀⠀⠲⠯⣅⣀⡀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣠⠴⠚⠉⢀⣠⠴⠚⠉⢀⣠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠛⠒⠦⢤⣄⣈⠉⠙⠒⠶⠤⣄⣉⡙⠛⠻⠿⠿⠶⠄⠀⢀⣠⠴⠚⠉⢀⣠⠴⠚⠉⢀⣠⠴⠚⠉⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡶⠦⢤⣀⣀⠀⠈⠉⠛⠲⠦⢤⣀⣈⠉⠛⠒⠦⢤⣤⠴⠚⠉⢀⣠⠴⠋⠉⢀⣠⠴⠚⠉⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠉⠉⠓⠲⠦⢤⣀⣀⠀⠉⠉⠓⠲⠦⣤⣀⣤⠶⠛⠉⢀⣤⠴⠊⠉⣀⣤⣤⣴⣤⡀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠉⠙⠓⠶⠦⣤⣀⠀⢸⡄⢠⠴⠚⠉⠀⣠⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢹⠀⢸⡆⢸⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠘⢿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⠀⢸⡇⢸⠀⢰⣿⠟⠉⠈⢻⣿⣿⣿⣿⣶⣶⣦⡀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠀⢸⡇⢸⡇⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡆⢸⡇⢸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢸⡇⠈⡇⢸⡇⠀⠀⠘⠿⠋⠉⠈⢻⣿⣿⣿⣿⡿⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⣠⣾⣿⣿⣿⣿⣿⣿⣏⠀⠀⢀⣀⡀⠀⢸⡇⠀⡇⠘⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⠀⣀⣴⣿⣿⣿⠿⠋⠀⠀⢀⣠⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⡏⠀⠈⡇⠀⣧⠀⡇⠀⣴⣿⣿⣿⣿⠿⠛⠁⣀⣠⠴⠚⠉⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠛⠒⠦⢤⣄⣀⠀⠀⠀⠀⠹⠿⠟⠁⠀⠀⡇⠀⣿⠀⡇⠈⠛⠛⠋⠉⣀⣤⠴⠛⠉⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣀⣀⠀⠈⠉⠛⠲⠦⢤⣄⣀⠀⠀⠀⡇⠀⣿⠀⡇⠀⣀⣤⠶⠛⠉⢀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⣤⣀⣀⠀⠉⠉⠓⠲⠇⠀⣿⠀⠿⠛⠉⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣤⣀⣀⣿⣀⣤⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡄⠀⠀⢴⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡄⠀⢀⣿⣿⣷⠀⣀⠈⣿⣇⠀⢹⡇⠀⣰⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡹⣿⣿⡄⠸⣿⣿⣿⣆⣿⠀⠘⣿⣆⠈⡇⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠷⠿⠏⠉⠀⢻⣿⣿⣿⡿⠀⠀⠉⠻⡄⠀⣸⠃⠀⠉⠻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠋⠁⢀⣶⣦⠀⣰⠒⠒⠄⡀⠈⠻⣿⣿⡇⠐⠄⠀⠀⠹⡀⠋⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⣿⡿⠃⠀⠈⢁⣁⡬⠅⠀⠀⠛⡋⠁⠀⢀⡈⠀⠀⠱⠀⢰⠀⠀⢤⣿⣿⡿⠉⠉⣁⣴⡾⠟
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣼⣿⠟⢀⣤⣶⠞⠋⠁⢀⣠⠀⣰⣟⣻⣧⠀⠀⠹⣦⠀⠀⠁⣸⠀⢰⣿⣿⡿⠁⠀⣈⣥⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾⡿⠋⠀⠀⠙⠁⣀⣤⡶⠟⠉⠁⠘⢿⣭⢽⣷⡀⠀⠹⠷⠀⠀⣿⠀⣿⣿⠟⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁⠀⠀⠀⢠⣴⣾⡿⠁⠀⠀⠀⠀⠀⠈⣿⣷⣿⣇⠀⠀⠀⠀⠀⣿⠐⠟⠁⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢠⠋⠁⣠⠟⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣠⣿⣧⡀⠀⠀⠛⠿⣇⠀⠀⠀⢀⢺⣿⠀⠀⠀⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⢀⠂⢃⡰⢇⠀⠀⣴⡗⠀⢀⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣗⠀⠀⠀⠀⢻⣆⠀⠀⠀⣾⣿⡄⠀⠀⣷⠀⠶⠦⠤⠉⠛⠻⠟⠛⠉⣀
⠀⠀⠈⠀⠉⢦⠄⠀⣼⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⣩⡉⠁⠀⣶⣿⣧⠀⠹⣦⠀⠴⣿⣿⣿⡖⠀⠿⠀⠀⠀⣠⠀⣠⡀⠀⢠⣾⠟⣠⣶⠄
⠀⠀⠀⠀⠀⠈⠀⣼⡿⠀⠸⣿⣿⣿⣿⣿⠿⠋⠀⠀⣰⠟⠁⠀⠀⠻⣿⡾⠀⠀⢉⠠⠄⡀⠈⠋⠀⠀⢀⣠⠶⠋⠁⢠⣿⣇⣰⣿⣿⣾⣿⣣⣶
⠀⠀⠀⠀⢠⠚⣿⣿⠃⠀⠀⠘⠛⠻⣿⣿⡆⠀⠀⢠⠋⠀⠀⣀⡀⠀⠈⣷⠀⠀⠀⢀⠂⠊⣀⣤⣴⠾⠋⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄
⠀⠀⠀⠀⠉⢘⣿⠏⠀⠀⡠⠀⠂⠀⢻⣿⣿⣇⠀⠀⠀⠀⣲⣿⡟⠓⠀⠸⡆⠀⢜⠁⡙⠋⣠⠞⠁⠀⢤⢺⣧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣼⣿⠏⠀⢠⠞⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⣴⣿⠟⠀⢠⣀⠀⢣⠀⠀⢠⠋⢈⠇⠀⣠⡀⠈⢻⣿⡄⠻⣿⣿⣿⣿⣿⠿⢋
⠀⠀⠀⢴⣿⠏⢀⣴⠏⠀⢀⣠⡶⠀⠀⠂⠹⣿⠀⠀⢸⣿⠯⠀⣰⣿⡟⠀⠀⠀⠀⠊⠁⢠⣀⠀⠹⣿⡀⠀⢹⣇⠀⠀⠙⢿⠟⠁⣴⣿⢟⠲
⠀⠀⠀⠀⠁⢠⣾⡟⠀⣴⣿⣿⣷⣄⠀⠀⠀⣈⠀⢠⡟⠁⠀⢠⣿⠋⢠⣦⡀⠀⠀⢿⣦⡈⠚⢦⡀⢻⣷⠀⠀⣿⠀⠀⣤⡀⠀⣼⣿⣯⠊
⠀⠀⠀⠀⠠⠿⢫⠃⠈⢻⣾⣏⠀⠉⠓⢀⣾⠟⢡⠏⢀⣀⠀⠛⠁⠀⢈⠰⠈⡀⠀⠀⠙⣿⣦⠀⠀⠈⣿⣇⠀⠸⠀⠨⠀⠇⣼⣿⣿⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣦⣤⡀⠋⠀⠰⠋⠀⠈⠉⠙⠒⠄⠀⠈⠄⠠⠰⡀⠀⠀⠈⠻⣷⣄⠀⠘⣿⡄⠀⠀⠰⠌⢠⣿⢿⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄⠀⣤⡀⢿⣦⡀⠀⢠⣄⠀⠀⠘⡄⢀⡑⠀⠀⠀⠀⡘⢿⣧⡀⠹⣷⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠃⣤⣙⢿⣦⣿⣷⣄⣿⣿⠀⠀⠀⠸⣿⣿⡆⠀⠠⡀⠀⠀⠙⣷⡄⠹⣧⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠹⣿⣿⡀⠀⠈⠳⣤⡀⠈⠻⣦⣻⡆⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢠⡆⢻⣿⣷⠀⠀⠀⠘⣿⣦⡀⠘⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡿⠋⢠⣿⡇⠈⣿⣿⣧⠀⠀⠂⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠺⠿⠀⠀⠘⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠒⠒⠒⠒⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⡤⠤⡤⠤⠀⠠⠤⠤⠤⠄⠀⠀⣀⡤⠤⡤⠤⡤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠃⠘⠃⠀⠀⠀⠀⠀⠀⠒⠁⠛⠀⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢵⠊⠰⠀⠄⡂⠀⠀⠀⠀⠀⠀⡎⠄⠱⠀⢳⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢸⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠄⠜⠁⠉⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⣤⠤⢄⣀⣀⣀⣀⣀⣀⠠⢤⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠄⠄⠀⠀⠀⠈⠀⣀⠀⡀⠤⠄⢈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣐⠂⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠨⠓⣈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠉⠁⠁⠸⡛⠛⠛⠉⠛⠋⠛⠙⠛⣻⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠤⢄⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣱⣀⣸⠀⢸⣀⣠⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⠟⠻⠫⠟⠀⠘⠫⠿⠻⠻⠛⠛⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠍⠳⡖⠄⢄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢀⡀⠀⠘⡄⠀⠑⢄
⠀⠀⠀⡇⠀⠀⠀⠀⠈⢃⠀⠁⢀⠀⠁⠀⠀⠈⠢⡀
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠳⢄⡀⠑⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢛⡀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠈

⠀⠀⢠⠀⡆⣰⣝⠄
⠀⢠⠩⠀⢂⣈⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠈⠓⠄⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠀⠀⠂⠰⠒⠂⠐⠒⠶⠶⠒⠒⠂⠒⠚⠂⠀⠰⠶⠄⠆⠐⠀⠀⠰⠒⠒
⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠛⠀⠂⠐⠐⠓⠐⠂⠀⠀⠐⠒⠉⠉⠀⠂
⠀⠀⠀⠂⠀⠴⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠂⠀⠋⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⠀⡂⠀⠀⠀⠀⠀⡤⠀⠐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠆⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣀⣀⣀⣀⣀⣂⣃⣀⣀⣀⣀⣀⣁⣀⣙⣂⣀⣀⣀⣠⣀⣒⣀⣐⣀⣀⣀⣂⣁⣙⣀⣉⣈⣀⣀⣠⣐⣋⣉⣉⣈⣉⣀⣉⣀⣀⣀⣓⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠉⢉⣀⣸⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⡌⠀⢸⢠⢘⠘⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢄⠀⠀⠀⠀⠀⠗⠂⣉⡠⠤⠀⢸⠈⠒⠆
⠀⠀⠀⠀⠀⠀⠒⠒⢒⢒⠒⠒⢲⠀⠀⠀⢀⣬⣤⣥⣧⣥⣼⣬⣤⣤⣤⡀⠀⠀⠀⠐⠚⠓⠒⠒⠒⠚⠒⠆⠀⠀⠀⠀⠈⠕⢊⡫⢐⢸
⠀⠀⠀⠀⠀⣘⣀⣀⣘⢒⣀⣀⡀⡆⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⡀⢸⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠂⠀⠭⠔⣒⠀⢸
⠀⠀⠀⠀⠀⠑⠠⣀⣄⢀⢄⠀⣔⠇⠀⠀⠀⡇⢀⣒⣒⣒⣒⣒⣒⡀⢸⠀⠀⠀⠈⠁⢆⣀⣀⣀⣀⣀⡢⠈⠀⠀⠀⠀⠀⡫⠥⠄⠐⢸
⠀⠀⠀⠀⠀⠀⠩⣉⠍⡈⢍⡩⢍⠇⠀⠀⠀⡇⠐⠒⠒⠒⠒⠒⠒⠂⢸⠀⠀⠀⠀⠀⠦⠤⠤⠤⠤⠤⠗⠀⠀⠀⠀⠀⠀⣒⡩⠭⠀⢸
⠀⠀⠀⠀⠀⢀⠭⣁⠭⣈⢍⡨⢍⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣓⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢒⣒⡃⠠⢸
⠀⠀⠀⠀⠀⠀⠬⠀⠉⠈⠁⠨⣥⡇⠀⠀⠀⡇⠠⠤⠤⠤⠀⠤⠤⠄⢸⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠭⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠬⣤⣥⢠⢧⢤⣥⡃⠀⠀⠀⡇⠨⠭⠭⠭⠭⠭⠭⠅⢸⠀⠀⠀⠀⠀⣌⢉⡉⣉⠉⠉⡅⠀⠀⠀⠀⠀⠀⠭⠭⠥⠐⢸
⠀⠀⠀⠀⠀⠠⣐⠤⣖⢤⣦⢤⡤⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠦⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠁⠀⣚⡛⠉⠰⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⣒⣒⣒⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠀⢒⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣂⡓⢄
⠀⠀⠀⠀⠀⠈⣓⡐⠢⠈⢁⣰⠷⣌⢢⡀⠀⠀⣠⡴⠍⡢⢅⣊⠁⠀⠑⢄⠀⠀⠀⠀⢀⠔⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠢⡀
⠀⠀⠀⠀⠀⢸⡶⠬⠓⠚⢓⣙⠀⠙⢯⠁⠀⠀⣿⠦⢓⠨⢕⣊⠀⠀⠀⢸⠀⠀⢰⠄⣁⣀⡀⢠⠊⠀⠑⡄⠀⠀⠀⠀⠴⠃⠈⢢⠀⣀⣀⢈⢰
⠀⠀⠀⠀⠀⢰⠶⠨⠛⠝⢓⣺⠀⠀⠀⡅⠀⠀⣯⠙⡀⠬⢐⣊⠁⠀⠀⢸⠀⠀⢸⢘⢥⠬⢜⢺⢘⠭⡃⡇⡀⠀⠀⡀⠀⣫⢼⠂⢘⠤⠬⢼⢻
⠀⠀⠀⠀⠀⠰⠿⠪⠓⠝⢓⢛⠀⠀⠀⡇⠀⠀⣿⢂⠈⢄⠒⣈⡁⠀⠀⢸⠀⠀⢸⣩⠶⠶⢾⣹⢨⠶⣏⡇⡵⠶⠶⡏⠀⡵⢾⡁⣩⠶⠶⠾⣹
⠀⠀⠀⠀⠀⠰⠿⠨⠓⠝⠃⢻⠀⠀⠀⡇⠀⠀⡿⠈⢡⡀⠒⢈⡀⠀⠀⢸⠀⠀⢸⢼⣒⣚⣺⠼⢠⣛⡧⡇⣔⣒⣒⡧⠁⣜⣸⠀⠤⢒⣂⣺⢼
⠀⠀⠀⠀⠀⠸⠿⠨⠓⠙⠣⢻⠀⠀⠀⡇⠀⠀⡿⠥⣀⠐⡐⠊⠁⠀⠀⢸⠀⠀⢸⠲⡀⡠⠸⢢⠰⣤⠖⡇⢦⠀⠀⡗⠀⢦⣸⠀⠲⡀⣠⢸⢾
⠀⠀⠀⠀⠀⠸⠽⠨⠛⠝⠋⠻⠀⠀⠀⡇⠀⠀⡇⢸⢰⠆⡇⠀⠀⠀⠀⢸⠀⠀⢸⠘⠐⠀⠘⢹⠘⠂⠃⡇⠃⠀⠀⠃⠀⠓⠘⠀⠘⠀⠀⠘⢻
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠉⠙⠁⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠋⠀⠀⠈⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠉⠉⠋⠉⠉⠙⠉⠉⠋⠉⠉⠋

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠴⠶⢂⣄⣩⣍⡶⠒⠂⠤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠐⠒⠚⢛⡧⡤⠤⢘⢓⡞⠋⠤⢄⡠⠤⠜⣛⡏⣽⣤⡄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⡏⡴⣶⣇⠽⢨⠜⣯⣀⠀⠀⡇⣿⣛⠉⡇⣞⣯⣿⠧⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣸⣟⡭⣿⠶⣿⡛⠛⢛⣢⢤⣧⣷⣿⣚⣛⣩⠽⣷⠒⠯⠥⣤⠤⠝⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⢑⣂⠔⣿⡀⠀⠀⠉⠈⣪⢞⣿⡀⠀⠈⡇⡷⢣⣿⡇⠀⠀⠀⢅⡵⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⢀⣊⣪⣼⣿⠿⢤⣀⣦⣬⢷⠾⣿⢀⣀⢀⣳⣵⣵⡿⠛⠢⢤⣤⢿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠉⠉⢲⣲⢍⡴⡟⣯⢶⢷⣿⡅⠀⠈⠏⠉⠉⠑⡖⡭⡃⡇⠨⢏⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⢠⣀⠀⠀⠀⢐⣈⣶⣧⣽⣵⣷⠿⠛⢠⣤⣀⡀⠀⠀⠀⠊⡰⣇⣸⣏⡿⡇
⣠⡄⠄⠀⠤⢤⣀⠀⠀⠀⡇⣀⠀⢰⠀⠈⠁⠙⠛⠛⠋⡵⠶⠤⢉⣉⡉⠷⠦⡤⡔⣛⣻⢺⣛⣩⡭⢴⣖⣴⡧⠀⠀⢀⣀⡀⠤⠤⠤⣀⡀
⡇⠈⠉⢲⢦⠌⣾⠀⠀⢸⠉⠁⠐⠲⡋⣇⡀⡃⠀⠀⠀⡇⠀⠀⠀⠀⡷⢎⢸⡥⡿⢋⣿⣺⠑⢠⢄⣒⣊⣽⣿⣿⣷⣾⡈⠉⢰⣂⣩⣿⡃
⠣⠄⣀⣸⣹⣪⣿⣶⠤⣀⠀⠀⠀⠀⣯⣇⣀⢸⠛⠡⢍⢝⣛⡤⠤⢤⢴⣯⣾⠆⠻⢾⣿⣉⣀⢨⣩⣯⡋⢉⣿⣿⣿⣿⣯⣀⣸⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠈⠉⠉⠀⠁⠀⠈⠉⠀⠐⠒⠛⠓⠂⠻⠀⠀⢰⢏⣽⣧⢀⡀⠀⡇⠀⠉⢯⢶⣿⡄⠉⠈⠉⠉⣅⢠⢤⣬⣭⣤⣷⠾⢭⡍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠁⠒⢒⣦⠀⠀⠈⠐⠲⠶⠟⣿⠩⢍⣈⠉⢧⣀⣀⣀⣚⣾⠧⠰⠂⠀⠀⠀⠉⠙⠛⠛⠋⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠁⠒⡤⣒⡬⠀⢴⠀⠀⠀⠀⢰⠚⢅⣀⣀⡀⠤⠍⣲⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⡥⠤⣆⣿⢍⡠⠀⠀⢠⠀⠀⢨⠀⡠⣂⡯⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠧⠞⠛⠋⠉⠁⠀⠀⠀⠈⠑⢄⣨⠨⣪⣟⣿⣿⣒⡄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣦⣰⣦⣴⣤⣦⣠⡄⢀⡀
⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣠⡴⣾⣉⣿⣼⣷⣼⣷⣴⣭⠻⣷⣆⢀⣀⢀⣤⣰⣿⡿⢿⡟⣿⡉⣿⢛⡿⢿⣿⣧⣴⢄
⢀⣤⣤⡟⠛⠛⣠⡄⠶⣧⣹⣷⣿⣿⡿⢻⠭⢥⣭⣝⡻⣿⣽⣿⣿⢡⣿⣿⠟⣩⣿⣾⡿⠿⣿⠿⢿⣷⣾⣡⡿⢻⣿⣧⣸⡿⠿⢧⣤⣤⣌⠛⠛⠛⣀
⠸⠿⠿⠇⠀⣀⠘⢻⣶⣿⣿⣿⡿⡿⣞⣻⡿⣷⣮⡻⣿⠿⠟⠛⠁⠈⠉⠁⠛⠿⣥⣾⠿⢿⣿⣿⣿⣬⡟⢻⣷⣿⠋⣿⣿⡀⣀⠸⠿⠿⠿⠀⠀⠀⠿
⢠⣶⣶⣦⣈⠛⣷⣿⣿⡋⣳⣦⣿⣿⡿⣿⣛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣶⡹⢾⣷⣉⡻⣿⣿⠋⣻⣿⡟⣹⣿⣄⡀⠀⠀⢠⣤⣤⣤
⣾⠛⢹⣍⣩⣿⠿⣧⣨⣿⣿⡟⣯⣹⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⡙⢿⣥⡌⢻⣿⣏⣽⣿⣯⣩⣿⣧⣀⣀⣸⣿⣿⣿
⠁⠀⠈⣹⡿⢷⣤⣿⡿⣿⣌⣿⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣰⣿⣷⡈⢻⣧⣄⠙⣿⣯⣴⡿⣿⣥⣽⣿⡿⠇⠀⠀⢸⣿
⠀⠻⣿⣿⣤⣾⡿⢿⣦⣼⣿⣿⣻⣦⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⢀⣉⣿⣿⣿⣿⣧⠈⢿⣿⠀⠘⢿⣿⣶⠿⣿⣤⣶⠦⠀⠀⠀⠈⠉
⠾⡾⣧⣨⣿⢿⣷⣤⣿⣿⣿⡽⢃⣟⠀⠀⠈⠻⣶⣤⠀⠀⢀⣤⣾⢟⣡⣴⣶⣿⣿⣿⣿⣿⣿⠏⣿⣷⠈⢿⣇⠀⢼⣟⣷⡾⠻⣿⣶⠾⢿⣿⣿⡇
⣻⣷⣼⣿⣧⣀⣾⣿⡿⢷⡞⢡⣿⣿⣿⣿⣷⣶⣤⣭⡀⠀⠈⢫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣄⢈⡄⢸⣿⣆⠈⢛⣻⣷⡿⢛⣿⣶⠿⠋⠀⣿⣿
⣙⣿⣯⣀⣽⣿⣿⠿⣷⡞⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⣦⣉⣸⡇⠈⠻⣿⡀⠀⢋⣿⣿⡟⢛⣿⣾⠿⢆⠉⠉
⣿⣇⣈⣿⣿⣿⢿⣤⠟⢰⣿⠏⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢀⣿⡟⠿⢻⡄⠀⢹⣇⠀⠸⣭⣿⣿⡿⠛⣿⣶⣾⠄
⡉⢙⣿⣿⡿⢷⣤⡏⠀⢸⡟⠀⠀⣿⠻⠿⠿⠿⠿⠿⠟⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⢰⡛⣯⣙⣦⡀⣿⡄⠘⣿⣄⠀⠙⣋⣿⣷⣿⠿⣿⣤⣶
⣿⣿⣿⡿⣷⣤⣿⠁⢀⣾⠃⢠⣞⢻⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇⠀⢹⣿⣇⠀⠘⣉⣿⣧⣴⣿⣿⣥⣾
⣿⣿⣿⣤⣼⡿⠃⢰⣿⠃⠀⣸⣿⣜⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⢀⠉⢉⠀⠀⣿⣿⠀⠀⣿⣛⣿⣧⣤⣿⣿⣥
⣿⣯⣀⣽⡟⠁⢠⣿⠇⠀⠾⠿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣿⣿⡟⢁⣾⡇⠀⢸⣿⠀⠀⢻⡟⢛⣿⣏⣤⣿⣯
⣯⣉⣿⡿⠀⠀⣿⡏⠀⡀⢤⣤⣀⡀⠉⠙⠻⢆⠀⠀⠤⠤⠤⠖⠐⠒⠀⠀⠀⠀⠀⠀⠂⢀⠈⢄⣀⡝⠀⣼⣿⣿⠀⢸⣿⣄⠀⠘⣿⡟⢛⣿⣏⣽⣿
⣬⣿⠏⠀⢀⣴⣿⠇⠀⣇⠐⡍⠉⢛⠛⠃⣠⣾⡆⠐⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢷⡄⠉⠀⣸⣿⣿⣿⠀⠀⣿⣿⡇⠀⢹⣿⡿⠿⣿⣋⣹
⣿⠏⠀⠀⣿⣿⠏⠀⢸⣿⡀⠠⠀⠁⢀⣠⣿⡿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣴⣿⣿⣿⣿⠀⠀⢹⣿⡇⠀⢸⣿⣷⣶⠿⣿⣋
⡿⠀⠀⣰⡟⠁⢀⣠⣾⣿⢃⠀⣠⡆⠛⣿⣿⣷⣧⣿⣦⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⡄⠀⢸⣿⣿⣷⣶⢿⣟
⠀⢠⣾⣿⠀⢠⣿⣿⠟⠁⣸⣶⣿⣿⣿⠿⣿⣧⡍⣿⣿⣿⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣷⣄⠀⢋⣽⣿⣶⡾⢿
⠀⢸⣿⡇⠀⢸⣿⡏⢠⣿⣿⣿⣿⣿⠀⢰⣿⣛⡃⣿⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠰⣿⣿⢸⣭⣿⣿⣿⣿⣿⠁⠀⠘⣿⣿⠀⣿⠿⢿⣿⣤⣾
⢀⣼⡿⠁⠀⢸⣿⠇⢸⣿⣇⣿⣿⣿⡀⢸⣿⣿⡇⠟⠋⠀⢹⣟⢹⣿⡀⢀⠀⡔⢺⡏⣿⡆⢹⣿⡿⠿⠿⣿⠉⠻⣿⡄⠀⢸⣿⡏⠀⠀⠀⠈⣿⣿⣉
⣼⣿⠁⠀⣰⣿⣿⠀⠀⣿⣿⣿⣿⡿⠃⢸⣏⣈⡇⠀⠀⠠⣶⣿⣸⣿⡇⠀⠁⢷⠘⣿⣿⣿⠜⠋⠀⣾⣿⣿⠀⠀⠈⠻⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⣿
⢻⣿⠀⠰⣿⣿⠁⠀⣼⣿⣿⣿⣿⡇⠀⢸⣿⡏⠇⠀⠀⠀⠉⠛⠟⠿⢳⣾⣇⢸⣷⠿⠋⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⢻⣿
⣼⡟⠀⠀⣿⣇⠀⢸⣿⠃⣿⣿⣿⣇⠀⢸⣿⠛⡅⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡏⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠛
⣿⠁⠀⠀⢹⣿⡄⢸⣿⡀⣿⣿⣿⠋⠀⠈⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⣯⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⠀⠀⠀⢸⣿⡁⠀⣿⡇⢸⣿⣿⡄⠀⠀⢿⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇

⢰⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⡀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣷⣿⣿⣿⣟⣿⡿⣶⣆⢀⡀⣄⠀⣠
⠸⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⣿⣿⠉⠻⣿⣿⣿⠟⠉⠻⣿⣿⠟⢹⣿⣿⠀⠀⠈⠻⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⣿⣿⣧⡀⠘⠿⠁⠀⠀⠉⣻⠏⣠⣿⣿⣿⠀⡄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⡟⠛⠛⠛⢦⡀⠠⣼⣠⣠⢟⣴⠟⠛⢿⣿⠀⣷⣦⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⠆⢈⡁⣫⣾⣿⡂⣶⣿⣿⠀⡿⣿⣿⣷⣦⣄⡀⠀⠀⠉⠛⢿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣴⣎⣽⣿⠿⣿⡇⠿⠛⢿⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣙⡯⣷⣾⡟⠀⣤⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⣼⣞⣿⣿⣿⣿⣾⢯⣿⣷⣿⢸⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣶⣤
⢀⠀⢀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⢸⣿⣿⣿⠿⢛⣥⡄⠀⣸⣿⣿⣿⣿⣿
⣽⣸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⢰⣿⠀⠀⠀⣿⣿⣶⣀⢰⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⡿⢁⣠⣾⡿⠇⣰⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣀⣤⠀⠿⠿⠿⠿⠿⠷⠾⠿⠿⠟⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠔⠁⠀⢀⠀⠉⢢⡀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⠔⣅⠀⠀⠀⡨⠢⡀⠀⠀⢀⣤⣻⣿⣿⣿⢟⣤⡀
⠀⢸⠀⠀⠀⠀⠀⠠⠄⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡌⠀⢸⡄⠀⢠⡇⠀⠁⠀⠀⢸⣿⣧⣿⣿⣿⣼⣿⡇
⠀⠸⡀⢀⠀⠂⠀⠀⣠⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⡸⠁⠀⠀⢇⢀⠆⠀⠀⠸⣿⢏⣿⣿⣿⡹⣿⠇
⠀⠀⠈⠲⠤⠠⠤⠞⠁⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠈⠣⠄⠀⠠⠜⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠗⠁

⠀⠀⢀⠄⠒⠀⠒⠠⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⡀⠀⠀⠀⠀⠠⢔⣂⣀⠐⠢⢄⠀⠀⠀⠀⣠⡴⠶⠿⣶⣦⣄
⠀⢰⠁⠀⠀⠀⠀⠀⠽⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⣐⡝⠁⠀⠀⠙⡄⠈⠂⠀⠀⣰⢫⣾⣿⣿⣮⢻⣿⡆
⠀⢀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⢸⣿⣿⣿⣿⣿⣿⢾⡇⠀⠀⠌⡇⠀⠀⠀⠀⠸⣄⠘⠀⠀⢿⢸⣿⣿⣿⣿⣧⢻⡿
⠀⠈⠢⡀⠀⠀⠀⣀⠜⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢜⣦⡀⠀⠀⠀⡨⠁⠀⠀⠈⢷⡻⣿⣿⣿⣿⡷
⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠁
⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢀⢔⣿⣷⡆⠀⠉⠦⡀⠀⠀⢀⣴⠋⠉⢽⣿⣷⣤⡀⠀⠀⢀⠎⠁⠀⠐⠆⣌⠱⡀⠀⠀⢀⣾⣿⣿⣯⣹⠻⣷⡀
⠀⠈⠀⠙⠉⠀⠠⣴⣆⠁⠀⠀⢸⣿⣶⣾⣿⣿⠋⠹⡇⠀⠀⡂⠀⠀⢀⣄⠀⠀⠀⢠⠀⠀⣾⣿⣿⣿⠻⣿⣿⣿⣧
⠀⠸⣤⣶⣄⠀⠘⠻⢋⠇⠀⠀⠸⡟⠙⠿⣿⣯⣄⣴⠇⠀⠀⠡⡀⠀⠀⠉⠀⠀⢀⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠛⠏⠤⠤⠖⠁⠀⠀⠀⠀⠈⠲⠾⠿⠛⠋⠁⠀⠀⠀⠀⠑⠢⠀⣀⠠⠔⠂⠀⠀⠀⠀⠘⠻⢿⣿⡿⠟⠋

⠀⠀⠀⠀⣀⠄⢤⡄⠤⠀⠀⠀⠀⠀⠀⣀⣤⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⢠⠊⠀⢀⣤⠙⢶⠀⠀⠀⠀⢠⣾⣿⡿⢛⣷⡙⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⢘⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿
⠀⠀⣇⠀⣴⠟⠁⠀⡔⠀⠀⠀⠀⠻⣿⠋⣡⣾⣿⡟⠀⠀⠀⠀⠎⠀⣀⠔⠁⡠⠊⠀⠀⠀⠀⣼⣿⢿⣿⣿⡿⠃
⠀⠀⡉⢳⣄⣀⠤⠊⠀⠀⠀⠀⠈⣷⣝⠻⣿⠿⠋⠀⠀⠀⠀⠀⠳⣈⢀⡠⠊⠁⠀⠀⠀⠀⠀⠹⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⢀⣤⣽⡁⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣟⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣃⣀⠀⠀⠀⠀⠀⠀⠛⢉⣿⡿⢥⡴⠀⠀⠀⠀⢀⣴⣶⣿⣿⣟⣷
⠀⠀⠀⠀⠀⠟⢸⡿⠳⡄⠀⠀⠀⠀⠀⠀⠉⢰⣿⠛⠷⠦⠀⠀⠀⠀⠀⢠⣿⣿⣍⣉⠉⠀⠀⠀⠀⠀⢤⣤⡿⠛⠳⢦⡄⠀⠀⠀⠀⢸⢿⡁⢰⠀⠉⠓⠆
⠀⠀⠀⠀⠀⠀⣼⢻⡆⠀⠀⠀⠀⠀⠀⠐⠶⠾⢛⡦⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠘⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺

⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⡄⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣀⢨⡗⣤⡀
⠀⠀⠀⠀⣠⣤⣼⡷⠞⠋⠁⠀⠀⠀⠀⠈⣷⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠷⠶⠚⠀⠀⠀⠀⠀⠘⣇⣸⣯⡤⠶⠖⠀⠀⠀⠀⠉⢿⣿⣡⣼⣄
⠀⠀⠀⠀⢻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠛⠁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡌⠁⠡
⠀⠀⠠⠞⠋⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡄⠀⠀⠀⠀⠀⢀⣠⡾⠋⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢃
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠈⠀⠈⢷⣰⣆⡟⠁⠀⠅⠀⠀⠀⠀⠀⠀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠠⠤⠤⣤⣾⡁⠀⠀⠀⠀⠀⠀⢀⡀⠈⢉⣿⣮⠋⠀⠀⠀⠀⠀⢠⣀⡀⣿⡟⢀⣀⡄⠀⠀⠀⠀⢀⣴⠿⠛⢯⡁⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣤⡀⠀⠀⠀⠀⠈⠉⠛⠾⠿⢿⠦⠤⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠴⠛⠁⠀⢠⣬⣷⣤⣀⠀⠀⠀⠀⠐⠃⠉⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⡏⠙⠁⠙⠀⠀⠀⠀⠀⣤⣦⣤⣤⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠁⠈⠁⠈⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢹⡾⠗⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⢰⡆⠀⡀⠀⠀⠀⠀⠁⠀⠙⣟⠋⠀⠈⠀⠀⠀⠀⠀⣀⣤⣤⣄⣠⣄
⠀⠀⠀⠀⠀⠒⢾⣷⠒⠂⠀⠀⠀⠀⠀⠀⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⠛⠉⠉⠀⠀⠀⣀⣉⢉⠻⡿⠟⡉⣸⣄⠀⠀⠐⠛⠉⠀⠈⠙⣯⠁
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣤⣤⣤⣤⡄⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⢠⣄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣶⢄⣠⡁⠂⠀⠀⠀⠀⠀⢸⣷⣿⣠⠤⠀⠀⠀⠀⠀⠀⠙⣧⣾⣴⠇⠀⠀⠀⠀⠀⠀⠓⢦⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡛⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢦⣁⣠⣿⠟⠖⠃⠀⠀⠀⠀⢀⣀⣉⣉⣽⣿⣁⡀
⠀⠀⠀⢀⣤⠶⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠺⣏⣿⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠀⠈⠀⠀⠀⠀⠀⠨⠉⠉⠉⠉⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡔⠉⡍⠁⠀⠑⡄⠀⠀⠀⡠⠚⠡⠊⠀⠉⢢⡀⠀⠀⠀⡴⠋⠀⢀⠀⡙⢆⠀⠀⠀⡠⠊⠁⣀⣀⠈⠳⡀
⠀⢸⠲⠀⠀⠐⠐⠐⢽⠀⠀⠀⡇⠈⠦⠀⠀⠐⠀⡇⠀⠀⢸⠀⠀⠀⠈⠠⠄⢸⠀⠀⠀⡃⢰⠋⣶⡆⢱⠀⠁
⠀⠘⣄⠅⠀⠣⡀⣀⡜⠀⠀⠀⠳⣁⡆⠀⠀⠀⣰⠃⠀⠀⠘⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⢣⡈⠢⠽⠥⠊⣠⠃
⠀⠀⠈⠑⠒⠒⠛⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠉⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⡠⢚⡽⠋⠉⠲⡄⠀⠀⠀⢀⣴⠉⠉⠉⠑⣄⠀⠀⠀⠀⣠⠖⣩⡙⢉⡲⡀⠀⠀⠀⣀⡠⠤⠤⠤⢤⣀
⠀⢰⢱⠃⠀⠀⠀⠀⣸⠀⠀⠀⡏⢸⠀⠀⠀⠀⡇⡇⠀⠀⢸⠡⠈⢄⠈⠀⠐⢻⠀⣠⢚⠁⠀⠀⠀⠀⠀⢸⡓
⠀⠸⡈⢆⡀⠀⣠⠞⡸⠀⠀⠀⢧⠈⠲⠤⠤⠞⢠⠃⠀⠀⠸⡔⢢⡀⠡⠀⠤⡻⠀⠘⢿⣍⠉⠀⠀⠉⣉⡽⠟
⠀⠀⠙⠢⠬⠭⠕⠊⠀⠀⠀⠀⠀⠑⠦⠤⠤⠒⠁⠀⠀⠀⠀⠈⠳⠤⠦⠥⠊⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⢀⠤⡶⢲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢤⢒⢢⠛⡿⠾⡆⠀⠀⠀⠀⣠⠴⡶⠖⡤⣀
⠀⢠⠃⠐⠀⠀⠀⠀⢳⠀⠀⢠⡖⠉⠁⠀⠀⠉⠑⢦⠀⠀⠀⢸⣿⢸⢰⢻⣸⠁⠀⠀⠀⡜⢡⠛⢊⡅⠂⡏⡆
⠀⠸⡀⠀⠀⠀⠀⠀⡿⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢻⣾⣾⡾⠃⠀⠀⠀⠀⣗⠰⠐⡆⠐⣩⠒⡇
⠀⠀⠑⠤⣄⣀⡤⠔⠁⠀⠀⠀⠈⠉⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⡿⠿⡇⠀⠀⠀⠀⠀⠘⠲⣥⣔⣷⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⡠⠤⠤⠤⣀⠀⠀⠀⠀⠀⢀⠤⠄⠒⡢⢄⡀⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⠀⠈⢣⣈⣯⡰⠋
⠀⢠⠊⠀⠀⠀⠀⠈⢳⠀⠀⠐⢭⣁⣠⠀⣟⣭⠆⢱⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⡠⠚⠛⠙⠣⣄
⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⣲⣷⣖⣧⣜⣋⡇⢸⠀⠀⠀⠀⢀⣎⣀⣧⠀⠀⠀⠀⠀⡜⡐⢡⡀⠀⠈⡘⡄
⠀⠈⠣⣄⡀⢀⣀⠴⠃⠀⠀⢸⣞⣻⢽⣿⣁⠤⣤⠎⠀⠀⠀⠀⢸⠦⠢⢾⠀⠀⠀⠀⠀⢡⡀⠀⠀⢀⠠⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠃⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡽⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠢⠤⠬⣉⣙⠛⠲⠶⢤⣄⣀⠀⠀⣀⣤⠶⠚⢉⣠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⢀⣉⠉⠒⠦⢤⡈⢹⡟⠉⡠⠔⠊⠁⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢠⠃⠈⢧⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠲⣤⣸⠀⢸⠀⠀⢠⠇⢰⣷⢸⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⢠⠏⢠⠖⢦⠀⡇⠀⢠⡇⢸⠃⢸⠀⠀⠀⠀⢸⠀⣿
⣴⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣄⣈⠒⠊⠀⠸⣀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⣠⠜⠀⣿
⣧⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣌⣉⠓⠲⠤⣄⣉⠀⢸⠀⢸⠀⢸⣠⠴⠊⣡⠴⠋⠉⠛⠲⢤⣀⡀
⣿⢠⠓⠢⣄⠙⠳⣤⡀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣋⡩⡍⢹⠓⠶⢤⣈⡉⠛⠀⢸⠀⢈⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⣿⢸⠀⠀⠀⠉⠲⢄⡉⠳⣦⠶⠚⣋⡭⠔⢚⡉⠁⠀⡇⢸⠀⠀⣠⠾⠋⠛⠲⠾⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢿
⣿⠸⡆⠀⠀⠀⠀⠀⢸⠀⣿⠀⡏⠀⡔⣫⣍⢹⠀⠀⡇⢸⠀⢸⡟⠓⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⣠⠀⢸
⣿⠀⡇⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇⢀⠃⠛⠱⠧⣄⠀⡇⢸⠀⢸⡇⢰⠒⠢⠤⣌⣉⠛⠲⠦⣤⣀⣀⠀⣠⠶⠋⢀⠴⠋⠰⡄⢸
⢻⠀⠧⣀⠀⠀⠀⠀⢸⡀⣿⠀⡇⠀⡆⢰⣳⠀⡸⠀⡇⢸⠀⢸⡇⢸⠀⠀⢀⣀⠬⣉⡓⠢⠤⣈⠉⣿⠁⢠⠔⠁⠀⠀⠀⡇⢸
⠈⠓⢦⣄⠙⠢⢄⡀⠸⡇⣿⠀⡇⠀⣇⡬⣔⡪⠴⢒⣡⡼⠀⢸⡇⢸⠀⣰⠋⠀⢀⣀⠈⡇⠀⢸⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠈⠙⠦⣄⡉⠚⠁⢹⠀⠧⠴⠒⢋⣡⡴⠞⠋⠁⠀⠀⢸⡇⢸⠀⡇⠀⢰⠋⠀⠉⠀⠀⢸⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣼⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢧⠀⢸⡀⠀⠀⠀⠀⢸⠀⣿⠀⢸⠀⠀⠀⠀⡠⠋⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⢣⡀⠑⠦⣀⠀⠀⣸⠀⢹⠀⢸⠀⢀⡠⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡙⠒⠤⣍⡒⠦⠞⠀⠀⢻⠀⢸⠀⢸⡴⢋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣈⡉⠒⠦⢤⣼⠀⢸⠀⢈⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⣸⡴⠋

⠀⠀⠀⢴⣦⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⠆⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⢀⣼⣷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣶⡦⠀⠀⠀⠀⠀⠰⢫⣿⠓⠞⣁⣰⡿⢆⠀⣠⣴⣯⣁⡀
⠀⠀⠘⣻⣿⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡋⠳⠚⠀⠀⠀⠀⠀⢀⡾⠙⢻⠞⣹⣿⠛⠋⠀⣡⣴⡀
⠀⠀⠀⢹⢿⠀⠀⠰⢻⡿⠟⠆⠀⠀⠀⠀⢠⣄⠴⠋⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠏⠉⠻⠀⠚⠋⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣠⣼⣏⡀⠀⣀⣤⣼⡏⠀⠀⠀⠀⠀⠀⢫⣿⣟⡷⠃⠀⠀⠀⠀⠀⠈⢯⣿⣇⣀⡀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠈⣼⠿⣬⠉⠀⠉⣼⢿⣭⠉⠀⠀⠀⠀⣠⠟⠉⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠀⠀⠀⠀⠀⠚⠋⣹⣿⠛
⠀⠀⠰⠞⠁⠀⠉⠀⠀⠞⠃⠀⠃⠀⠀⠀⠀⠀⠁⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠠⠞⠁⠈⢿⠀⠀⠀⠀⠀⠀⣤⠟⠉⠳⠄

⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠦⣤⣴⣶⣦⡄⠀⢀
⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⣧⡤⠀⠀⠀⠀⠀⠀⣸⣿⣇⣀⡀⠀⠀⡇⠀⠰⣿⠛⣶⠋⠋⠀⢘
⠀⠀⠀⠀⠀⠞⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣐
⠀⠈⠀⢀⠀⢸⠇⠘⡇⢀⠀⠀⠀⠀⡀⠘⠛⠀⢻⡄⡀⠀⠀⠀⡀⠀⢀⡾⢹⡇⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⠀⠈⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠
⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠷⣶⠆
⠀⠀⠀⠀⠀⠀⡾⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⢀⠴⠖⠒⠶⠶⠒⠒⠶⠶⠒⠒⠒⠶⠶⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⡿⢯⣹⠿⣏⡽⢿⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⡍⠉⠉⠉⣉⠉⣩⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⡉⠉⠉⠉⣉⠉⠉⣉⠉⣯⡽⠿⣭⡿⢿
⣷⡞⢙⣶⡏⢳⣾⠸⠟⠋⡟⠻⡿⠾⠿⠿⠾⡿⠿⡯⠷⠿⢧⣿⣇⠨⢿⠷⢿⡷⢿⠵⢿⠵⢿⠗⣿⠶⣿⠆⠻⡬⠼⠿⢿⣿⠇⢰⠿⠦⠟⢻⠀⡟⢳⣶⠋⣷⣞
⡇⠹⠿⠀⠿⠏⠘⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠶⠶⠿⠯⠈⠿⠇⠸
⡇⠀⠀⠀⠠⠀⠀⠀⠀⠠⠉⠈⠠⠀⠠⠈⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢤⠀⠀⠀⠀⣴⠂⠀⠀⠐⠖⠀⠀⠀⢀⠶⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠓⠀⠐⠂⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⢀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠀⢂⠠⠀⠀⢂
⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠐⠈⢯⠀⠘⠛⣧⡀⢀⠀⠀⠀⠀⠄⠄⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠙⡿⠎⠁⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢸⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⡁⠀⠐⠀⠀⠐⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠁⠀⠐⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠈⠂⠀⠀⠀⠀⢀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⡁⠀⠀⠀⠁⣀⠠⠀⠡⣀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠀⠄⠀⠃⠀⠀⠀⢀⣄⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠱⠀⠀⠀⠀⠀⠀⠸⢆⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢀⠀⠀⠁⢀⡀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠉⠒⠐⠆
⡇⠀⠀⠀⠀⠀⡐⠀⠀⠀⡀⢱⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⠀⠀⠈
⡗⠂⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠦⠤⠤⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⠀⡄⠀⠀⠀⠠⢣⠈⠂⠀⠐⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠹⡄
⡇⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢄⡔⠈⠄⠀⡄⠀⠀⠀⠄⠀⠄⠐⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠄⠀⡀⠀⠀⠠
⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⡇⠀⠀⠀⢴⣩⡵⠦⣄⠤⠸⠄⣁⠼⠤⠤⠤⠒⣩⣔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⣀⣀⡀⠐⠀⠀⠁⠂
⡇⠀⠀⠀⢈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣀⠀⠀⢀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠁⠁⠉⠉⠉⠉⠁
⡇⢰⠈⠀⠀⢸⣄⣀⣠⣤⣀⣀⣠⣤⣀⣀⣀⣤⣄⣴⠀⠀⠀⠉⢧⠱⢄⠀⠀⠊⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠈⡆⠀⡇⠈⡄⠀⡠⢄⠀⠀⠀⠀⡠⠤⡀
⡇⢸⠀⠀⠀⢸⠀⢀⡈⠃⠐⢀⡈⠃⠐⠰⠀⠾⣄⠹⠀⠀⠀⠀⠨⡆⡄⠐⢰⠀⡇⠀⡠⠠⠀⠀⠀⠀⠀⠀⢹⠀⠀⠣⠄⠃⠀⡗⠁⠀⠀⠑⠂⠂⠈⠀⠀⡀⠃
⡇⢸⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠀⠀⠀⡇⠈⠂⠀⠀⢧⠈⡡⠒⠠⠀⠀⠀⡐⠂⠹⡆⠀⠀⠀⠀⢀⠉⢢⡀⠀⠀⠀⠀⠀⠀⡔
⡇⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡔⠱⠂⠀⠀⠆⠈⠱⡄⠀⠐⡀⠀⠀⢇⠀⣰⠋⠀⠀⠀⠀⠸⡄⠀⢡⡀⠀⠀⠀⠀⠀⡆
⡇⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⡁⠀⠸⠂⠀⠀⠀⠀⠄⢱⠁⠀⠀⠁⠃⠀⠀⠱⡀⡘⠃⠘⠃⠌⠀⠀⠃
⡇⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⠀⠀⠀⠀⠀⠐⠀⢈⡀⠀⢀⢀⡆⠀⠀⠀⠂⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢸
⡇⢸⠀⠀⠀⢸⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⠘⠀⠈⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠨⣦⡀⠀⡏⠄⢘⡀⣀⣀⠈⣀⡀⠃⣰⠆⠀⠀⠀⠀⠠⢬⡀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣄⢀⡀⠀⢀⠀⣀⠰⡀
⣇⣀⣀⣸⣝⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣫⣔⣙⣀⣉⣙⣉⣉⣋⣁⣉⣋⣀⣂⣀⣐⣂⣀⣀⣀⣈⣉⣀⣀⣀⣀⣙⣘⣋⣘⣃⣀⣘⣓⣂⣛⣀⣛⣀⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡀⠸⠇⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣶⡶⠶⠶⠶⣶⣶⠶⠶⠿⢶⣶⡶⠶⠿⢶⣶⡶⠶⠶⠶⣶⣶⠶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⡟⠁⠀⠀⣠⡿⠃⠀⠀⢀⣾⠋⠀⠀⢀⣼⠟⠀⠀⠀⣰⡿⠁⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠟⠀⠀⠀⣰⡟⠁⠀⠀⢠⡿⠃⠀⠀⢀⣾⠏⠀⠀⠀⣼⠟⠀⠀⠀⣰⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⣴⠟⠀⠀⠀⣰⡿⠁⠀⠀⢠⡿⠃⠀⠀⢀⣾⠏⠀⠀⠀⣼⡟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⣶⡾⠿⣶⡶⠶⠶⠿⠶⠶⠶⠶⠿⠷⠶⠶⠶⠾⠷⢶⣶⠶⢾⣿⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛



⣿⣿⠟⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⢠⣾⠏⢠⠀⠈⠙⢦⠀⠀⠀⠀⢠⣾⡏⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠀⠀⠀⠀⠀⠈⠢⠀⠚⠀⠀⠀⠀⠑⡀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢠⡟⡖⠀⠀⣾⣿⣦⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⣷⠃⡀⠀⠙⠛⠁⠀⠀⠀⠀⠀⡀⠂⠀⠀⡶⠃⢀⣴⠆⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠘⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡞⠈⣶⠟⠁⠀⠀⣤⣀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡶⢋⣴⣆⠀⠀⠘⣿⠛⠛⠶⠾⣷⣦⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⣠⡔⠀⢀⣾⠟⣠⣾⣿⣿⣦⡀⠀⢿⡆⠀⠀⠀⠸⣧⠈⠻⣧
⣀⣠⣤⡤⢤⣤⣄⡀⠀⠀⠀⠉⠓⠀⠀⠐⠊⠁⢀⣶⡿⢃⣰⣿⣿⣿⣿⣿⣿⣦⣸⣧⠀⠀⣀⣤⣿⠀⠀⠘⣿
⠉⠁⠀⢀⣿⠋⠙⠻⣷⡦⣀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡛⣋⣁⣀⣠⣤⡿
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠙⠻⣗⣦⣀⠀⠀⠀⠈⠛⠂⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡉⠉⠉⠈⠁
⠀⠀⢠⠇⠀⠀⣾⣶⣶⣤⣄⡀⠉⠻⣷⣦⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⡻⣿⣷⣦⣄⠉⠻⢷⢄⠀⠰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠈⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠈⠘⠻⢿⣿⣿⢶⣦⠙⠻⣠⡀⠉⠛⣿⣿⣿⣧⣄⣿⣟⠙⠛⠛⠀⠀⠀⠀⠀⡟⢣
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣤⣤⣤⡀⡉⠛⠾⣿⣿⣳⣬⣙⣷⣼⣿⢹⣿⣿⡍⠙⣿⡟⠒⠦⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⣠⣾⣿⡟⣿⣿⡏⠩⠉⠛⢿⣿⣦⡄⢀⣉⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡆⢻⣧
⢀⣠⣴⣿⣿⣿⡇⠀⠉⠀⠀⠀⢠⡀⠘⣿⣿⡄⠿⠿⠛⠛⠛⠛⠻⣿⣛⣯⣌⠀⢸⣿⡀
⠘⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⠋⠛⠯⣿⣿⣿⡇
⠀⠘⠿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣆⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡶⢦⡀
⠀⠀⠀⠀⢠⠞⡿⣷⡿⢫⣶⢰⡏⡾⠁⠀⠀⠀⠀⠉⠙⠛⢷⣝⣆
⠀⢠⠞⢠⠏⡜⣱⡿⢁⣿⠇⣿⠀⡇⡆⠀⠀⠀⠀⢀⣠⣤⠀⠈⢺⡆
⠀⠃⠀⡞⠀⠀⣿⠁⢸⡟⢹⡇⠀⣇⢱⡀⠀⢀⠞⠋⠁⠀⠀⠀⠀⠙
⠀⠀⢰⠇⡀⢸⡇⠀⣿⡇⢸⡇⠀⢿⡄⢳⡀⠊⠀⠀⠀⣀⣤⣿⡾⠄
⠀⠀⣾⠀⠀⢸⠇⠀⣿⡇⠀⡇⠀⢸⡷⡄⠹⣄⠀⠈⠛⠛⠉⠉⠀⠀⠀⠀⠀⠁
⠀⠀⣿⠀⠀⢸⠀⠀⢻⣧⡀⢱⠀⠀⢷⡹⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀
⠀⠀⣿⠀⠀⢸⠀⠀⢸⣿⣇⠀⠄⠀⠈⢧⠘⣧⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⠋
⠀⠀⣿⠀⠀⠘⡇⠆⠈⣿⣿⡄⠈⢆⠀⠈⢧⡘⣧⠈⢇
⠀⢠⣿⢰⢀⠀⣇⠰⠀⢻⡿⣿⣄⠈⢂⠀⠈⠳⡜⣆⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠿⠃⣼⢸⡀⢻⠀⡆⠘⣷⢣⠻⣦⣄⠑⢄⠀⠙⢿⡄⢠⣀⡀⠀⢀⣠⡴⠚⠁
⣶⠶⠿⠃⢸⡇⠈⡀⠸⡀⢹⣧⡳⡈⠻⣷⣦⣑⣤⡀⠙⢾⣿⣿⡿⠛⠁
⠈⠲⣤⣴⡿⠃⠀⢳⡀⢱⠈⣿⡷⣜⢦⡈⠻⢿⣿⣿⣶⣄⢻⡏
⢤⡀⠀⠀⠀⠀⢠⣿⢳⡀⢇⠸⣿⣌⠳⣌⠳⢤⣙⠾⣝⠿⣷⣇
⣶⣝⠳⠶⠶⢞⣻⣿⡎⢧⠈⡄⢻⢿⣧⡈⠻⢦⣉⠓⢮⡳⡌⢿⡄
⠻⢿⣿⣷⣾⣿⣿⢿⠇⠀⠀⠰⠸⣯⢻⣿⣦⡀⠉⠳⣄⠱⡙⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠈⠉⠉⠉⣁⡞⡼⠀⡀⠀⢠⣿⢦⠹⣿⣿⣦⡀⠘⡆⠠⠁⢸⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⣉⣉⣩⣉⣙⡛⢷⣦⡀
⣿⣶⣶⣶⣶⣞⣯⠞⠁⣠⡇⠀⢸⣿⡈⢧⡘⢿⣿⣿⡆⠀⠀⠀⠈⢂⠱⢄⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁⣀⣤⣶⠿⢛⣛⣿⣿⣿⣿⣷⣌⢷⡀
⠈⠻⢮⣭⣉⣉⣠⣤⣾⡟⠀⡌⣾⣿⣇⠀⠻⣄⠙⣿⣿⠀⠀⠀⠀⠀⢧⡀⠉⠒⠒⠒⠛⠋⠉⠀⣠⣴⠾⢛⣩⠴⠛⣋⣉⣁⣀⡀⠉⠙⢿⡆⢣
⠳⣦⣀⠉⠛⠛⠛⠛⠉⣠⢞⣼⣿⣿⣿⣆⠀⠘⢆⠘⣿⠀⠄⠀⢸⡇⠈⠻⢦⣤⣄⣠⣤⡴⠶⠛⠉⣠⠴⠋⣡⡶⣛⣩⣭⣭⣭⣙⠳⣄⠀⣿
⠀⠀⠉⠛⠶⠶⠶⣶⣿⣵⡿⠿⠛⠛⠛⠿⣦⡀⠈⠆⢹⠀⠁⠀⢸⣹⡀⠈⠲⢤⣀⣀⣀⣀⣤⠴⠊⢁⣴⣾⠗⢋⣁⣀⠀⠀⠀⠈⠑⢌⡆⣿
⠙⠛⠓⠶⠖⢛⣛⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⡎⠀⠀⠀⢸⣇⠳⣄⠀⠀⠀⠈⠁⠀⣀⣤⡾⢟⡫⠖⠊⠉⠉⠀⠉⠐⢄⠀⠀⠈⢧⢸
⣾⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠘⣿⣆⠈⠛⠲⠶⠶⠾⠛⢋⣩⠖⢋⡠⢖⡢⠬⠤⢀⠀⠀⠀⢣⠀⠀⠘⢸⣇⠀⡴⣶⣶⣄
⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⠀⢠⠀⠀⠹⣿⣷⣤⣀⣀⣠⣤⣾⠟⣡⠴⡫⠚⠁⠀⠀⠀⠀⢳⡀⠀⠀⢦⠀⠀⠀⣿⣄⡀⣨⣿⣿
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡼⡅⠀⢀⠈⢧⠀⠀⠈⠛⠻⠿⠿⠛⠉⣠⠾⣫⠞⠀⣠⣶⣿⣟⠀⠀⣸⡇⠀⠀⠘⣷⡀⢳⣌⠛⠿⠿⢛⡽
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢀⣇⠀⠈⢧⠈⠳⣦⣄⣀⣀⣀⣠⡴⠞⣡⠞⠁⢀⡾⢹⢻⠻⣿⣿⣿⡿⠃⣰⠀⣸⣿⠟⢦⣭⣻⣶⣾⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠡⠀⢸⣿⡄⠀⠈⠳⣤⣀⡉⠉⢉⣉⣡⡴⠟⠁⠀⣠⣿⡇⠸⡌⠳⣤⣉⣉⣤⠞⠁⣴⡿⠋⣠⣾⣿⣿⣥⣤⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⢔⠞⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠒⠒⠒⣦⣤⣤⣤⡤⠤⠤⠬⠄⢞⠉⠉⠉⠀⡀⠊⠊⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡟⠀⠀⠠⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠂⠀⠈⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠛⠒⠀⠀⢻⡀⠀⠀⡇⠀⡀⠀⠀⠀⡰⠋⢂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣤⣄⣀⠀⠀⠘⡇⠀⢠⡇⠘⠀⠀⠀⠞⠁⢚⣠⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣀⣀⣠⡏⠀⠀⠀⠀⠀⢻⡀⢸⠂⠀⠀⠀⡠⣀⣴⣿⡿⠷⠂⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠭⢭⣉⣛⠓⠒⠦⠼⠶⣦⣼⣇⢸⠀⣀⣤⣾⣿⠽⠋⠉⠀⠀⠀⣀⡤⠶⢳
⠀⠂⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⢤⡍⠀⠐⠛⠛⠒⠒⠒⠢⡾⠞⠛⠋⠁⠀⠀⠀⣀⣤⢴⡟⠁⡀⠀⢈
⠀⠀⡇⠀⡟⠉⠉⠁⠀⠈⠉⠑⠲⡄⠀⡇⠀⠀⢹⠋⠁⠒⠒⣠⣤⣤⡤⠤⣀⣀⢲⠏⡡⠀⡖⠁⡰⠀⢀⣖
⠀⠀⡇⠀⡇⠀⠀⢰⠒⠒⡆⠀⠀⢹⠀⠁⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⡇⠀⠤⠁⡀⠀⡀⠀⠕
⠀⠀⡇⠀⡇⠀⠀⠈⠉⠉⠁⣀⡴⠋⠀⡀⠀⠀⢸⡃⠀⠀⠀⣀⣀⡀⠀⠀⠻⠦⠠⠷⢶⣶⣶⣶⣴⣶⣶⣿
⠀⠀⡇⠀⡇⠀⠀⢠⠤⣀⠀⠀⠉⢣⠀⠃⠀⠀⢸⣧⠀⠀⢸⠏⠀⠈⠛⠋⠀⢈⠈⣛⣙⣙⣋⣙⣿⣿⣿⣿
⠀⠀⡇⠀⡇⠀⠀⠸⠤⠼⠀⠀⠀⢀⠇⡅⠀⠀⢸⡏⠀⠀⠸⣄⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⢠⡾⣯⣀
⠀⠀⡇⠀⢇⣀⣀⣀⣀⣀⣀⣀⠤⠊⠀⡆⢰⠆⢸⡇⢂⠀⠀⠀⠉⠀⠈⠢⣄⠀⠀⡗⠀⠀⢰⠉⠐⣏⡴⠟⠀⠁
⠀⢆⣓⣂⣈⣉⡉⣁⣉⣉⣉⣀⣀⢨⡭⠥⢼⢥⣘⠳⠤⢕⣤⣀⣀⣀⡠⠴⠛⠀⠀⣧⠂⠴⢡⡶⠚⠋⠀⡀
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠁⠀⠀⠉⠉⠉⠉⠁⠈⠓⠿⠶⠶⣬⣭⣙⣒⠒⠢⠤⠄⣏⣤⣞⣋⣄⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠬⠭⡍⠉⠉⠐⠉⠉⠉

⡏⠉⠉⢉⣭⣭⣤⣤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠈⡠⠀⠀⠉⠉⠉⠀⠡⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⡇⠀⣰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⠅⠂⡀⠆⠄⡄⠀⡀⠄⢸⠧⠀⢀⢠⠤⠀⠀⠖⠀⡀⡠⠄⠤⠀⠀⠤⡄⡄
⠁⠀⠹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠂⠞⠀⢀⠡⡇⠁⠁⠋⠈⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⣁⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⣤⣤⡤⡴⠦⡈
⠀⠀⠀⠈⠛⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠠⣀⠀⣀⢀⠀⠄⠀⠀⠀⡈⠄⡀⠀⠄⢰⠐⠀⠀⠠⡎⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠇⢀⠀⠀⠘⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠜⠀⠁⠀⠀⠈⠀⠂⠀⠁⠀⠀⠀⠀⠀⠄⠄⠀⠀⠈⠀⠀⠀⠀⡔⠠⢠⡄⢶⠩⠊⡼⠀⠀⠀⠀⠀⠧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠦⣺⣥⣤⡅⣠⣀⡀⡎⣂⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡄⣰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢾⠄⠀⡀⠈⠂⠀⠈⢁⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠈⠉⠀⠉⠉⠉⠉⠑⠟
⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠠⡄⡂⠀⠁⠤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠬⠐⠠⠀⢆⠀⠂
⠀⠈⢦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⡄⠀⠀⠀⠤⠅⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠂⠀⠀⠀⠀⢀⠁⠀⠀⠀⠐⠀⠈⠀⠁⠀⠀⠀⠀⠀⠄⠐⠀⠀⡁
⠀⠈⠑⠒⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡴⠛⠀⠻⡄⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡄⢠⣄⡴⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⡄⠀⠀⢘⠀⠀⠀⠀⠸⠀⠀⠰⠇⡸⠓⠁⣾⠙⣇⡀
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢳⡐⠝⡒⠒⢀⠀⠀⠈⢀⡀⠀⠀⠀⠘⡇⠀⠀⠸⡆⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠚⠀⠃⠀⠀⠀⠀⠀⠀⡀⠘⠒⠛⠀⠈⡀⠄⠀⠀⡀⠀⠀⠀⠀⢠⠝⡀⠀⡃⠀⠀⠀⠀⠐⠠⠀⣀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡶⠀⠀⠀⠀⠀⠈⡡⠀⢀⣧⠠⠰⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠁⠐⣀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠂⢀⣠⡄⠘⣇⣀⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡷⠀⠀⠋⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠋⠉⠁⠠⠁⠀⠀⠀⠀⠠⠀⣀⡀⣀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⡀⠈⠁⠀⠀⠠⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢕⠀⠚⣯
⡄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠄⠸⡀⠑⠂⠐⠀⠂⠂⠀⠀⠠⠒⣄⡀⠀⠀⠂⠀⠀⠀⠀⡀⠂⠀⠀⠀⠐⠈⠠⡁⠀⠀⠀⠐⠀⠀⢀⠀⠘⠣⠞⠁
⡇⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠑⠂⡤⠀⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠐⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐
⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣃⣐⣂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡠⠂⠀⠁⣄⠀⠀⢀⠔⠁⠀⠀⡠⠂⠀⢀⠇⡤⣄⣀⠀⠈⠙⠒⠶⠦⣼⣧⣬⣿⢿⣿⣿
⠀⠀⠀⠀⠉⢤⡤⠂⠁⠀⢀⠀⠂⠁⠀⢀⠔⠁⢈⠷⡊⠁⠀⢀⡠⠊⠀⠀⢠⠏⢘⠷⣄⣈⠛⠛⠶⣤⣄⣀⣾⣀⠩⠉⠛⢿⣿
⡀⠀⠀⠈⠀⠀⠈⠱⠤⡀⠀⠀⠀⡠⠔⠁⡀⠀⠀⠀⠘⢢⡔⠁⠀⠀⠀⡴⠃⠀⡸⠣⣄⣉⠙⠳⠦⣤⣂⠉⡿⠉⠛⣛⠻⢿⣾
⡀⠄⠠⠀⠀⠀⠀⠀⠀⠈⠲⢎⡀⠀⠀⠈⠀⠀⠀⠠⠈⠀⠈⠂⢀⡴⠋⠀⠀⢠⠇⠀⠈⣍⡙⠷⢶⣤⣍⣻⠟⠻⠷⢶⣶⣦⣶
⢀⠸⡄⠤⠀⠠⠀⠂⠁⠀⡀⠄⠈⠃⠄⣀⡠⠐⠈⠀⠀⠀⠀⡰⠋⠢⡀⠀⢠⠋⠀⠀⠀⠈⠄⢂⣀⣀⣍⣻⣳⣶⣦⡬⣤⣱⣟
⠀⠀⢂⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠁⠄⡀⠀⣀⠔⠊⠀⠀⠀⠈⡶⣃⠀⠀⣀⡤⠶⠻⠉⠉⠉⠀⣻⠀⢀⡉⠉⣻⠿⣼
⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠛⠡⡀⠀⠀⠀⣠⠎⠀⣨⠳⢏⠀⠀⠀⠀⢧⠀⠀⠠⡟⠀⠀⢐⡼⠇⠀⠤
⠀⠈⠀⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠊⠁⠀⠀⠀⠈⠑⣤⡎⠀⠀⡜⠁⠀⠈⠳⢄⠐⠀⠈⣇⠀⠘⣇⣤⣉⡿⣁⠀⠀⠑
⠀⢀⠀⡇⠈⠀⠀⠀⢀⡀⠠⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠈⠑⣾⣀⠀⠀⠀⠀⠀⠙⢦⣄⠙⣆⠘⡇⣤⠟⠀⠐⠠⢈⡴
⠀⠀⢀⡁⠼⠖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⢰⠁⠈⠙⠓⠶⠶⣤⣔⣀⢉⢷⣼⣾⣧⠏⢀⣠⡼⠖⠋
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠙⠃⠠⣀⠛⠖⢒⡉⠶⣷⣶⣦⣄⣙⣉⣟⣿⣿⣯⣶⣛⣁⠀⣀⠒⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠂⠉⠉⠐⢻⣥⡰⢾⣮⠇⠉⠄⣾⢿⡟⠿⣍⠏⢩⠉⠉
⠀⠀⢀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⡦⠈⠀⢤⣶⣾⣶⣴⣆⠀⠀⠀⠀⠄⢩⣲⣄⣸⣻⣿⣿⣿⣿⡆⣾⢻⡍⢻⡄⢈⡙⠶⣤⣐
⠘⠀⡜⠀⠈⠀⠀⠀⠀⠀⢀⡤⢚⠇⠀⠀⠈⡟⠛⡏⠹⠁⠀⠀⠀⠀⠀⢋⡄⣿⣿⡿⣏⣿⣿⣿⣿⠃⢼⠴⠀⢿⡈⡤⠄⠀⠉
⠃⡜⠁⠀⠂⠀⣀⡀⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡿⢛⣿⣿⣿⡇⠀⢻⡆⠐⠉⣷⠀⠀⠀⣀
⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠔⠛⠻⢶⣿⣿⣯⣷⡿⣿⣿⣿⠅⠈⢹⡁⠤⠀⠘⣧⢀⣨⡗
⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣤⣄⣤⣴⡆⣄⠀⢶⣿⣶⣷⣼⣿⣧⢴⣯⣿⣿⣿⣿⠀⣀⣸⠄⡒⠐⠒⠸⣷⣯⣾
⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⠿⠿⢿⠿⢁⣿⠰⡞⣻⣿⠿⠿⠿⣳⠛⢛⣿⣿⣿⣯⠉⠉⡿⢷⣟⣿⣿⡿⣿⣯⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠐⠀⠀⠀⠸⠿⠀⠈⠳⠶⢦⡀⠐⢷⢰⣿⣿⣿⣿⡏⠀⢘⣇⣏⣿⣿⣾⢲⣿⡿⠄
⠄⠀⠀⠄⠀⠀⠀⠀⣆⣈⠀⠀⠀⠘⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣤⣤⣼⡾⢿⣽⡁⠀⣾⡟⢺⢹⠇⠬⠽⡍⢻⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⢏⣩⣶⣿⣿⣁⠀⡿⠡⡿⠶⠂⠶⠂⠻⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡄⣄⢸⠀⣆⣣⣴⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠙⠛⠛⠒⠶⠲⠶⢦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢿⠓⡿⠾⣦⢿⣷⣷⣿⠟⠁⢀⣊⡼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠘⠛⠛⠉⠉⠀⠀⠐⠾⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⡀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣶⠄⠀⠀⢀⣄⠀⠀⢻⣷⡀⠀⠀⠀⢀⣶⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⢿⣿⣶⣶⣶⡿⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢴⡦⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡉⠁⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠻⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠹⣿⡦⠀⠀⠀⠀⢠⣿⠟⢿⣦⣄⣠⣾⠟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣹⣿⣿⣅⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⢀⣴⣿⣇⠀⠀⠀⠀⠘⣿⣦⣾⠟⠉⠙⢿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠉⢿⣷⡀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠨⣿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣶⠀⠀⢀⣀⠀⠀⣾⡿⠀⠀⠀⠀⠈⢻⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣸⣿⠃⠀⠀⠀⠀⠀⠘⠛⠻⢿⣦⡀
⠀⠀⠀⠀⢰⣶⣶⡆⠀⣴⡆⠀⢴⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠒⠒⠒⠒⠒⠒⠒⠒⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⣿⠋⠉⢉⣽⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⢩⣿⣿⡍⠉⠉⣿⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢋⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡙⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠁⠄⢱⠀⠀⠀⠀⠀⢀⠀⠀⡎⠠⠌⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣻⠀⠀⠈⠂⠐⠊⠀⠀⠀⠀⠀⠀⠳⠀⠘⠂⠒⠁⠀⠀⣟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣇⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⢹⠋⠛⠛⠛⠛⠛⠛⠛⠛⠙⣏⠉⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠄⠒⢀⠃⠀⢈⠁⠀⠀⠀⠉⠉⠀⠀⠘⡇⠂⠠⡀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⣉⠉⠄⠀⣼⠤⣤⡄⢀⣤⣤⡀⣠⣄⢀⡔⡖⢛⠂⢠⠋⢉⣈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠖⠉⠒⠊⠀⢦⠤⠞⠒⠂⠒⠚⠓⠒⠒⠐⠚⢷⠾⠃⠑⠒⠉⠳⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⣀⣀⠀⠀⠀⢀⡞⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢠⠋⠉⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠬⡧⠀⢸⠀⠀⡧⠀⢼⡥⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠲⠲⠄⠒⠶⠔⠲⠚⠁⠀⠓⠶⠒⠖⠒⠖⠢⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠂

⠀⠀⠀⠀⠰⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡀⠠⠀⠀⢀⠆⠀⠀⠀⠀⠄⢀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠐⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⢠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠁⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣸⡅⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⣯⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠉⠛⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡏⠙⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣷⣷⠄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⠀⠀⠀⠀⣴⣿⣿⣻⣿⣿⡿⠁⠀⠀⠀⠀⢀⣼⢿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠠⠂⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠣⠒⠤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⡠⢂⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄⡀⠀⠄⠐⣦⡀
⠀⠀⠀⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⠁⠀⠀⢀⡜⡡⠃⠀⠀⠀⠀⠀⠀⠀⠖⠑⠀⠀⡀⠂⢀
⠀⠀⠀⠀⠁⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠏⠀⠀⢀⠎⡐⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢸⣶⣶⣾⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠁⠀⡰⢃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⣇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡌⢀⡜⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣽⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁