Delicious food Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢸⠏⠷⣄⠀⣾⡛⠛⠻⠟⠛⠷⣶⣿⣟⢋⣷⠶⠶⢯⡿⢁⠷⢒⢍⣷⡴⠶⣶⡶⣤⣀
⠀⠀⠀⠀⠀⢠⡟⠐⣴⠉⠷⠿⡇⠀⠐⢲⣍⣉⡉⠉⠉⣻⡁⠀⣀⠀⢠⠘⠤⠍⢢⣾⣧⡀⢸⣿⣷⣿⣷⡄
⠀⠀⠀⠀⢠⠟⡀⣒⠠⠶⠂⡈⢤⠀⢝⡩⠙⠓⠚⠿⠛⢛⡀⢈⣁⣤⣪⣿⣲⠶⠻⢯⣹⣿⡿⠿⢻⡴⡀⡿⠹⡆
⠀⠀⢠⡟⢿⣦⡀⠤⠄⠁⠈⣁⠘⣤⠜⠴⠀⠀⠀⠀⠀⠾⠉⠁⠂⡌⠠⠛⣿⡇⣇⣨⡽⢟⣃⠂⣸⢱⡗⠃⢀⣿
⠀⠀⠈⢻⡿⢛⡁⠀⠆⠀⠀⠀⢀⣃⠀⠂⠀⠙⠐⠀⢃⡄⠐⠗⢤⣤⢀⡄⠀⢣⣿⣉⢀⠆⠈⠳⣅⣟⣀⢀⣾⣧⡀
⠀⠀⢰⡏⢠⠈⠘⢆⠀⢒⣀⡴⡤⠽⠂⠀⠀⣄⢀⣠⡴⢷⣤⣄⣀⡀⠀⣀⣀⢺⡿⢆⡤⠀⠀⠀⠀⠠⠀⣨⣟⣾⠁
⠀⠀⣸⠃⣈⠸⠀⡙⠁⠜⡏⠁⡂⡀⢄⡀⠠⣸⣍⡿⠛⠈⠙⠛⠙⠛⠛⠛⣭⣿⣷⣾⢃⡁⠀⠀⠀⠀⢀⣾⢾⣅
⠀⠀⢸⡄⠁⠦⢈⡃⠄⠘⣾⠁⠀⠀⠀⠀⡰⢸⡟⠁⠀⠀⠀⠡⢰⠆⠐⠋⡭⠛⡋⠉⠀⢀⣀⡀⠱⠠⢾⣷⣤⡏
⠀⠀⠈⣷⣤⣥⡈⣅⢀⠞⠂⠑⠁⢀⠀⠀⠲⣼⡄⡀⠀⠀⠀⠀⠂⠁⢠⣎⠀⣀⣤⣶⠿⠛⠉⠀⠀⣿⠆⣿⢯⡄
⠀⠀⢰⣿⣟⢰⣶⠈⠡⡐⠐⡆⢠⠄⣦⠴⠿⣟⠛⠗⠲⠄⣴⠆⣿⣄⠀⠈⡆⠂⢀⡀⠀⠄⠘⠀⠀⢄⣾⣿⣼⣇
⠀⠀⠈⢿⣿⣼⣿⣧⡴⠄⡄⠀⠂⠐⢛⠦⢀⠄⢀⡶⠠⣠⡤⠔⠋⠋⠉⠀⠀⠀⠤⢀⣀⠠⠐⢈⠐⠃⠻⡿⠷⠛
⠀⠀⢠⣻⣿⣷⣧⣀⣷⣄⠢⠐⠆⣸⠂⠀⢀⢈⣿⣇⡔⠹⠲⠂⠀⣠⠄⣀⠴⠆⠀⠮⣄⢀⣶⠿⣤⠀⣱⣷
⠀⠀⠘⣿⠿⠿⠏⠙⠈⣿⡟⠆⢠⣼⣤⠊⠉⠛⢷⠿⠲⢂⡽⠄⠀⠄⠀⠉⠀⠀⡏⢠⣦⠈⣠⡙⢩⠉⣿⣹⡇
⠀⠀⢼⠃⠀⠀⠧⠀⣚⢹⠇⠁⢀⣹⠇⠀⠀⢀⠲⢀⣄⠐⠂⣠⣄⠀⠈⣿⣙⣯⣀⡿⢇⢠⣿⠁⣼⠼⠅⠈⣷
⠀⠀⣸⣧⡠⢀⠬⣀⠠⠤⡧⠂⣾⠅⠙⢦⠯⡏⠀⢻⣿⣦⣴⣿⡿⠀⣰⢿⡟⠓⠀⠰⠠⠤⢥⢀⣧⡀⢐⣠⣿
⠀⢠⡿⠟⣿⠀⠀⠈⠃⠀⢹⠊⡑⢦⢭⣁⠠⣟⣤⡀⠉⠛⠛⠋⢀⡆⠰⠠⠆⠰⠀⣷⣤⠄⢨⣴⡄⠋⠙⣿⠃
⠀⠈⣛⣾⣿⣦⣄⣀⣉⡀⢈⡠⠀⠐⠀⣹⠰⢋⡍⠁⢰⡿⠿⢿⣿⣤⠀⢄⡤⠄⣐⣭⣴⣮⠶⠃⠀⣀⢀⣾⣧
⠀⠘⢧⠄⠚⠁⠈⠟⢁⣴⡿⣩⡾⠒⣃⢹⣶⣋⣴⡿⢼⡧⠦⠀⠰⣦⣾⠟⡿⠿⠿⠛⠛⣡⠀⠿⢂⣽⣿⣿⣟
⠀⠀⠸⣦⣀⡀⠀⣰⣿⣷⠿⡋⠰⠆⠀⢨⣟⠻⠀⠣⣬⡀⠠⠔⡂⠰⣿⡆⠀⠀⠀⠾⠂⠁⢀⡿⢯⠝⢿⣿⣿⡆
⠀⠀⠀⢸⣿⣧⡄⠛⠉⢠⠀⠙⠀⠈⢀⡼⢿⣾⣍⠀⣤⠹⠀⠀⠀⢀⣯⡽⠟⡚⠷⣶⠞⢴⠉⠀⠀⢠⡼⣿⣿⣧
⠀⠀⠀⠀⣿⢟⣡⡌⣃⡀⠀⠈⡠⠈⠁⠀⠈⢳⣹⡄⠘⡐⠁⢠⣿⡿⣽⡧⠄⠀⠠⡇⢈⠘⣶⢛⡊⠍⢓⣛⣩⠿
⠀⠀⠀⠀⣟⠈⠻⣧⠈⣻⡿⠋⠰⠿⣤⣀⣤⠈⣏⠶⣶⡅⠈⢸⣝⣦⡹⠰⠢⠐⠄⣿⡶⣫⢕⠢⡴⢾⣟⣿⣽⠆
⠀⠀⠀⢠⣿⣄⠸⣿⠇⠛⠀⠀⠰⠀⠀⢀⠁⠀⢸⡨⣿⣿⢾⣿⣻⣫⢷⣦⣀⢄⣴⣫⣞⠿⠶⣻⠟⠁⠀⠈⢻⡀
⠀⠀⠀⠸⣭⡿⠗⠒⢯⣄⡀⠀⠀⠀⠀⠈⠀⠀⠈⣇⣿⣿⡐⠗⠦⠚⣂⣿⣿⣷⡟⠁⠈⠇⠀⡡⠳⠘⠢⠀⢸⠇
⠀⠀⠀⠀⠈⠓⢶⣄⡂⠀⠈⠛⠶⠤⣤⣠⣴⠐⠶⠖⠒⢾⡛⣿⣻⣿⣟⣿⣿⣿⣿⣟⣇⣀⣰⢿⣦⣤⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⢤⡴⠶⠛⠋⠉⠳⠶⠤⠴⠶⠚⠁⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠂⠉⠉⠉⠐⠒⠂⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡐⢁⠠⣈⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣟⡛⠛⠛⠛⡛⠛⠛⢛⡶⠶⢶⣤⣀⠀⠀⠠⢋⡠⢈⡹⠧⠉⢠⠼⢱⠇
⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⠀⠀⠀⠘⣍⡉⠻⣌⢉⣉⡥⠴⠚⠁⣨⡿⠇⠀⠀⠸⢄⡸⣀⣤⡄⣸⡀⠁
⠀⠀⠀⠀⣠⣴⣦⠙⠷⣤⡀⢰⠒⠒⠋⠉⠘⠵⠻⣄⠀⣀⣤⠾⠋⠁⢀⣀⣀⠀⠀⠐⢃⡬⠛⣤⢳
⠀⠀⠀⠸⢿⣇⠀⠀⠀⠈⠙⠻⠶⢤⣤⣤⣤⠤⠶⠞⠛⢉⣡⠴⠀⡐⢁⠠⣨⠱⣄⣠⠄⠃⠆⢋⠞
⠀⠀⠀⠀⠈⠙⢷⣄⡀⡎⠙⠃⠀⠀⠠⠀⠠⣄⣀⣠⡾⠛⠁⠀⠀⢀⡹⢠⠸⢡⠏⡤⠦⣄⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⢤⡤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠈⠟⠋⢹⣷⡌⠓⠄⢛⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢲⣿⠆⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠊



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠃⠀⠄⣦⢲⠆⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⠀⠀⠀⠀⢤⡄⡀⠀⠀⠙⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⢄⣀⣓⣠⢂⣘⣀⠂⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠘⢧⡀⠀⠰⠄⠀⠀⠀⢥⠁⡔⠊⢀⠄⠀⠀⠀⠘⣣⡔⠀⠀⣀⢄⡟⠂⣄⠀⠈⠁⠀⢹⡋⠀⠀⡈⠉⠁⠀⢙⡇
⠀⠀⠀⠀⢘⠃⠀⠀⠀⠘⠂⢀⣀⣴⡀⢀⠈⠖⢀⠞⠁⢀⡄⠊⠀⠀⠀⠀⠀⠠⣡⠜⢧⢗⡫⠂⡀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⣼⠁⣀
⠀⠀⠀⠀⠀⢠⠁⠀⠈⠀⢀⠌⠈⠈⠁⠀⠀⠐⠦⠀⢀⠎⢠⠞⠆⠀⠀⠀⠠⣔⣄⣶⠨⠍⡠⠖⡧⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠇⠀⠁
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠀⠐⢤⡆⠀⢧⠀⠂⠀⠀⠄⠀⡅⢤⣥⠔⠚⢀⡠⠆⠀⠀⠀⠙⠠⠤⠄⠠⠤⠤⠂
⠀⠀⠀⠀⠀⣀⣰⡄⠀⠀⠀⢤⠀⠀⣀⡀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠸⢗⣸⢩⡍⢮⡛⠀⠀⣄⢶⠆⠀⠀⣀⠀⠀⢦⣆⡄⠈
⠀⠀⠀⠀⡶⠚⠁⢃⣀⣀⠀⣀⠀⠀⠘⢡⠾⣼⠀⠀⠀⠀⢀⡀⠀⠀⠀⠦⠀⠀⠈⠉⠛⠃⠉⠀⠀⠷⡉⠀⠀⠀⠀⠁⠀⠀⠀⢊⢱⡆
⠀⠀⠀⠀⠀⠀⡞⠋⢀⡠⠴⠬⢤⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⡦⡀
⠀⠀⠀⠀⠀⠀⠻⣶⠃⠁⠀⠀⠀⠈⠆⠀⠀⣥⠀⡆⡄⢐⢰⠀⡄⢀⠇⠀⠂⢰⠀⠰⣀⠂⡦⢀⠀⠀⠀⠀⠀⠘⢼⣁⠄⡓⣧⢂
⠀⠀⠀⠀⠋⠀⠀⠘⠀⠁⠀⠀⠀⢀⠀⠉⠀⠤⠐⠳⠁⠬⠸⠄⠇⠨⠤⠀⠀⠇⠱⠐⠻⠀⠗⠀⠇⠁⠀⢲⡦⡄⢩⢽⣶⠲⠨⠇⡰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠠⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠉⠹⢾⣤⡜⠿⢭⠘⣧
⠀⠀⠀⠀⣠⠀⠀⠢⡇⡀⣀⠀⣠⠅⠀⠀⠙⣣⡆⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠎⠃⠀⠀⣄⠀⠀⠀⠀⠘⠳⠴
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠂⠀⠀⠚⠃⠀⢀⠀⠀⡠⠠⠎⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡠⠀⠄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣠⣄⣀⣄⣀⡀⠀⠀⢠⡄
⠀⠀⠀⢠⠀⡈⠀⠀⠀⢠⡀⢠⠔⠀⠀⠀⢄⠀⠀⠁⠀⠀⠁⢀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⢻⣿⣧
⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠉⠲⠀⠀⠀⠀⠘⠀⣴⡀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠁⠀⢰⠇⠀⠀⠀⠀⠻⡿⢤⠂⠀⢠⣿⣿⠃⢠
⠀⠀⠀⠀⠀⠘⡀⢀⠄⠎⠁⡀⠀⠰⠄⠀⢡⣹⠓⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣦⣟⠁⠀⠀⠂⠀⡜⠀⠀⠀⠀⠾⠋⠐⠀⡀
⠀⠀⠀⠀⠀⠲⠈⠈⢀⢀⠉⠈⠀⠙⠀⠀⠈⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠂

⠀⠀⠀⢀⣀⣰⣶⣶⣶⣦⣄⠀⠀⠀⠀⠔⠀⠰⡀⠀⠀⠀⠀⢀⣀⣠⣤⣠⣀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡴⠆⠀⢀⡴⠂⠀⢀⡰⠆⠀⣀
⠀⠀⠐⠛⠿⠿⡿⠟⠛⡋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠈⠠⢈⡯⢻⣷⣧⠀⠀⠀⠈⠉⣷⠀⠀⠀⣀⡀⠀⢁⣀⠀⠀⣀⠀⢀⣈⠀⢀⡈⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠇⠀⠀⠀⠈⠀⠁⠀⠈⣟⣻⣾⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣷⢺⣾⢟⣷⠒⣿⠻⠶⣖⠛⠛⠷⠲⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠁⠀⠀⠀⠈⢀⡿⠀⠀⠀⠈⠉⠀⠒⠒⠚⠂⠀⠘⠓⠛⠒⠛⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠁⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⢀⢔⣡⣶⣾⠿⠿⠿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣻⣳⣴⣾⣿⣷⣿⣿⡷⠀⠈⠒⠒⠉⢹
⠀⠀⠀⠀⣀⠂⠄⢀⠀⠠⢀⣐⠢⠀⠀⠀⠀⠀⠀⢤⣵⠿⠋⠁⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⡿⠿⠛⠛⠀⠈⠁⠀⠀⠀⠀⠀⠨⡄
⠀⠀⠀⠢⠤⣉⠐⠀⠈⣁⠀⢈⠔⠃⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⠀⢠⠁⠀⢀⡀
⠀⠀⣶⠤⢄⣀⠈⠑⠂⢈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠋⠀⠀⠀⠀⢘⠀⠄⠀⢰⠁⣿
⠀⠀⠙⠛⠛⢻⣌⣹⠶⠋⠀⠀⠀⠀⠀⣰⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣀⢄⣄⠀⠀⠐⠒⠀⠂⠀⠀⠐⠁⠀⠀⢈⡜⠀⢀⠘⠓⠛⣴
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⢨⡷⠈⠀⠀⢀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⢶⣶⡶⣶⠶⢚⠂⠀⠀⠀⠰⡀⠁⠀⠀⣾⣿⡏⢀⣮⠀⠀⠀⠀⠀⠀⠞⠀⠀⠘⡁⡛⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⡿⠿⠾⠿⠾⡿⠿⢿⠀⠀⠀⠀⠀⡈⣴⠂⠀⠘⠿⠷⠄⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠃⠀⡇
⠀⠀⠀⠀⠀⢤⠦⠄⠀⡄⣠⡀⠀⠀⡠⠾⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠁⠀⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠄⠀⡇⡄⠠⡇
⠀⠀⡀⠀⠀⠀⣰⠏⠀⡔⠉⢧⠙⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡇⠀⠀⡀⣀⣴⣴⣶⡆⠀⢀⣀⠀⠿⠆⠀⠀⢣⡅⢸⡇
⠀⠀⠀⠀⠀⠀⠁⢰⠞⠀⠀⠈⠇⠘⣁⠤⠀⠀⠀⠀⠀⣠⣴⢆⡜⢉⠝⠗⠦⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡛⠋⠋⠉⠙⠇⠀⠀⠉⠁⠀⠀⠀⠀⠸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣴⣽⢿⣶⣭⣏⣷⣄⣬⣵⣤⠀⠰⠀⠀⣠⠋⠀⠀⠘⡧⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⣯⣟⡻⢶⣤⡀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⢀⡴⠊⢁⠶⠀⠀⠀⢲⡈⠀⠰⠀⢀⠀⠉⢻⡀⠀⠸⣧⣤⣤⣉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣤⡙⡿⣿⠟⢱⡶⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠂⢀⡎⠀⢠⡾⠁⠀⠑⠦⠈⡻⢟⣁⡿
⠀⠀⠀⠀⠠⣤⣤⡀⠀⠈⠙⢿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⢀⡾⡀⢰⡿⠁
⠀⠀⣼⠀⠀⢸⡏⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠀⠂⠀⠀⠀⣰⣶⣺⠟⠀⠀⠀⠀⠀⠌⠀⢠⡿⠁⠀⠀⠀⠀⢤⡂
⠀⠀⣹⠀⠀⢈⡁⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⠎⠁⠀⠀⠀⠀⠀⠀⠠⢤⡾⠁⠀⠀⠀⠀⡆⠸⢁⡐⡇
⠀⠀⣿⠀⠀⠈⣿⠄⡄⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⢀⡴⣾⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡄⠀⠀⠀⠀⠀⢀⠃⠸⠀⡇⡇⢄⠔
⠀⠀⣿⠀⠀⠀⠡⡆⠁⠀⠀⠀⠀⠀⠒⠀⠒⠐⠒⠀⠒⠒⠀⠐⠀⠀⠀⢐⣿⡷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠈⠀⠀⠀⠀⠘⠀⠂⠚⣛⣽⡌
⠀⠀⢿⠀⠀⠸⣷⠀⠈⠀⠀⠀⠀⠀⠠⣀⣤⠀⣄⣀⣀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⣀⢀⠀⡀⢂⠐⡀⣂⠀⠀⠀⠀⠀⠀⠀⢸⣶⠆
⠀⠀⠸⡀⠀⣸⢡⠄⡄⠀⢀⡀⠢⠄⣻⣾⠏⣹⣁⡼⣤⣾⡚⣩⣤⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠘⡟⡌⠁⠉⡀⢆⠰⡐⢂⠀⠀⠀⠀⠀⠀⠈⣿⡶
⠀⠀⠀⠈⠰⢯⠎⠉⠀⠀⠈⠡⠤⠈⠭⠭⠬⠩⠍⠠⠭⠌⠭⠉⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠉⠁⠉⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠍
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣷⣷⣶⣶⣾⣿⡿⢿⣿⣿⡿⠿⠿⠛⠋⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣛⠶⠶⠦⠤⠅⠀⠀⠀⠠⠄⠤⠤⠤⠄⠒⠒⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⠊⡁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠿⢿⠿⠿⠟⠛⠻⠶⠓⠚⠲⡶⠚⠿⠋⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣖⣄⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣻⡛⠽⠛⠿⢷⣤⣀⣀⣀⣀⣀⣠⠤⠖⠒⠚⠋⠉⠁⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣠⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠚⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢶⢶⣶⢿⡿⠉⠁⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠿⣿⣿⣿⣿⣉⣈⠉⠀⢀⠀⡈⣉⣉⢀⣀⣠⣴⡤⠶⢚⡉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡁⠪⠽⢛⡛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠄⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠤⣄⣀⣀⠀⠀⠀⠈⠀⠀⣀⣀⡤⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡞⠁⠀⠀⠀⠀⠀⠀⢃⢠⡴⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢂⣁⡗⠢⠤⡀⠄⠒⢢⠤⠒⠧⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣠⣎⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠂⢁⡨⠀⢠⠤⠀⠈⠀⠠⠖⢂⠀⢸⡀⠀⡏⠢⠄⠀⠀⢀⢎⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠡⠄⠀⣠⡤⣀⠀⠀⣀⠀⠀⠑⠈⠁⠀⠐⠛⠉⠉⠀⠈⢱⡔⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢒⡞⡀⢤⡬⢤⣄⡇⠐⠉⠀⠀⠐⠀⠀⠤⣀⣀⡤⠀⠀⠀⣀⢄⡦⠪⢊⡘⣖⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢾⡓⠐⠒⠒⠒⠀⢘⢣⠀⣰⣓⣱⠀⠀⠀⠀⣄⠀⠀⠠⠾⡠⠥⠤⠖⠐⠁⣈⠆⢀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⠈⠢⠄⣀⡀⠴⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠥⠤⠐⠒⠊⠉⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠲⢆⠠⠤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠄⡴⠖⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠐⠈


⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣠⠴⠐⢁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡠⠀⠤⠂⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠐⠒
⠀⠄⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠀⠆
⠀⠠⠀⠀⠀⠀⠀⠀⠨⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠀⠀⠀⠀⠀⠂⠐⠊⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠴⠏⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠁⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠘⠄⠰⠅
⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡤⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⠀⠘⠀⠁⠁⠑⠀⠀⠀⠀⢀⠒⠀⠃⠴⠁⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀⠐⠂⢄
⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠐⠌⠀⠚⠈⠈⠀⠸⠀⠀⠀⠀⠁⠃⠀⠈⠀⠀⠀⠀⠀⠐⠀⠐⠂⠐⠀⠒⢈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠐⠠⠤⠥⠤⠔
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠔⠓⢆⡀⢐⣦⣠⠼⠷⡖⠀⠀⢀⠠⡾⠀⣤⣤⣤⣭⣥⣀⣀⡀⠀⠀⠘⠿⠃
⠶⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣮⣷⢶⡚⠙⢯⣀⣈⣶⣤⠀⠀⠀⠀⠀⢱⠀⢒⣒⣒⣿⣿⣿⠿⠇
⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣧⠟⢾⡉⠉⢳⣄⡈⠀⠀⠠⠄⠀⠀⠀⢴⠚⠛⠩⠉
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠺⠗⠀⠀⠀⠀⠀⣿⠶⠖⠛⠛⠩⡈⠀⠐⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣶⢆⣶⠁⢀⣀⠄
⠀⠀⠀⠀⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠰⠷⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⠀⢸⣿⡆⢰⣿⢸⠇⣴⣿⣿⣿⣦⢠⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣾⡄⢿⣿⣿⣿⡿⢰⡿
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣾⢿⣁⠀⠀⠙⠿⢦⣽⡿⣿⡁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⠀⠀⢸⣧⠀⠀⣀⣠⣤⣤⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⢄⣠⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠚⠻⢸⠾⢡⠻⢓⠀⠀⠀⢀⡀⠔⠒⠋⠱⠤⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢠
⠀⠧⣄⣀⣀⣀⡠⠜⠀⠀⠀⠀⠀⠀⠘⠈⠈⠁⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠠⠑⠀⠀⠀⠆⠀⠀⠀⡀⠞⠚⠁⣸⠜⠉⠀⡀⣴⠀⠠
⠀⠿⢦⣤⡄⠽⡤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠘⠚⠀⠈⠓
⠀⠀⠀⠬⠩⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⣀⠀⠠⠄⢀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣶⣿⠟⠳⠶⠦⢦⡄⠀⠀⠀⠀⠀⠁⠀⠀⠁
⡔⠀⠁⠄⠄⠀⠒⠀⠲⠂⠀⣈⠁⢂⡠⠀⠀⠀⠰⠒⠀⠲⠶⠛⠉⠉⠂⠀⠀⠀⠈⠉⠻⠥⢀⣀⢤⡤⠴⠂⠀⠀⠀⠀⠀⠄⡀
⠈⠲⠤⠤⢑⣁⣂⠀⡀⠈⠛⢀⠀⠀⠀⠔⠂⠀⠀⠴⢤⣄⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠿⠾⠶⠶⠿⠑⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂
⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⣤
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠐⠀⠀⠀⠀⠁⡀
⠀⠱⢬⣽⣿⣿⡫⠅⠀⠀⠀⠀⠁⠀⡇⢠⠀⡄⠁⠀⠀⠖⠄⠐⢠⠤⠀⡄⠀⠀⡄⡀⠀⠀⢏⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠔⠁⠸⢄⠀⠀⠠⠀⠢⠀⠰⠘⠠⠃⠡⠠⠠⠁⠃⠀⠀⠈⣉⠃⠀⠀⢠⣷⡈⣧⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠾⠋⠁
⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠱⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⡀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠄⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠈⠂⣐⠂⠀⠀⠉⠓⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣠⠖⡿⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡤⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⡿⠁⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠘⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠔⠀⠀⡠⠊⢀⣄⠀⠀⠀⠀⠀⠰⠦
⠀⠀⢸⠀⠀⠠⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⠀⠀⠴⠂⠀⣀⠀⢀⡠⠊⠀⠀⣰⡥⠀⠘⠁⠀⠀⠀⠐⢶⠆
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⢀⡀⠤⠐⠀⣀⣸⠟⠂⠁⠀⠀⠀⠀⠙⡁
⠀⠀⢸⠀⠀⠀⠀⠀⠰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠐⠓⠂⠀⠀⢴⠆⠀⠈
⠀⠀⠙⠇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠋⠀⠛⠁⠀⢀⣠⡤⣄⠀⢀⣤⣤⣤⣄⣀⢠⣟⣡⠇⡀⠀⠀⣠⢤⠀⠀⠀⠀⠓⢿⡆⠀⠀⢈⠀⠄⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⣴⡿⠀⠛⠋⠉⣉⣉⠛⣿⣛⣅⢠⡍⣴⢶⠻⣾⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⠁⣼⠏⢳⢸⡏⢸⣼⣿⡵⠟⣾⣃⣿⡨⢾⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠉⣸⣿⢦⡟⢻⣉⡟⠹⠏⢻⣦⡬⠵⠶⣄⠘⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⠞⠁⠙⠊⣤⣤⣤⣴⣶⠿⠋⠉⠛⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⠤⠀⠰⠿⠃
⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠚⠷⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠀⠛⠦⠀⠀⠀⠀⠀⠀⠀⢠⠞⣤⣤⠔⠒⠒⠢⠄⠈⠀⢀⡤
⠀⠀⠀⡈⠁⠶⠄⠉⠀⠀⠀⣤⡤⠀⠀⠙⠛⠀⠀⠀⠀⠀⡀⠒⠋⠀⠰⠶⠆⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⢻⡟⢁⣤⠀⠀⠒⠀⠀⣀⡀⠀⢀
⠀⠀⠰⣿⡀⠀⢀⣄⢀⣴⣤⠀⠀⣴⣦⡄⠠⢢⠆⠀⠀⡾⠀⠚⠂⠀⠀⠀⠈⠃⠀⠉⠁⠀⠙⠀⠀⠀⠀⢀⢱⣬⠕⠊⠉⡉⠉⠉⠐⠢⡀⣀⠃
⠀⠀⢸⣿⣿⣷⣴⣦⣤⣠⣤⣄⣀⣀⣀⣀⣤⡏⠀⠀⢠⠆⠀⠈⠁⠀⠀⠀⠄⠀⣦⡀⠀⠀⠐⠀⠀⠀⠀⢸⡟⠁⣀⡀⠀⠋⠀⠀⣀⡀⠀⠉
⠀⠀⠘⢫⣭⣌⣡⡄⠘⠛⠋⠉⢉⠉⣽⣷⣠⣾⠀⠀⠈⠃⠀⠐⢂⠐⠀⠀⡀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⢨⠄⠀⠉⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢾⠿⢿⠷⠒⠷⠶⠶⠶⡾⠛⠛⠋⠉⠁⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠄⠀⠐⠀⢀⠤⣄⠀⠀⢿⡴⠃⠀⠀⠀⠀⠺⠆⠀⠀⢠⡀
⠀⠀⠀⠾⣶⠶⠤⣤⣤⣤⣤⣼⣍⣤⣤⡴⢾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⠄⠤⠀⠀⠀⠀⠀⠀⠘⣇⠀⠺⠆⠀⣀⠀⠀⠀⠓⠈⠁
⠀⠀⢠⣾⣥⣄⣚⡃⠀⠀⣙⣁⣛⣀⣤⡤⠴⠖⠒⠀⠀⠀⠀⠐⠒⠀⠈⠁⠀⣀⠀⠀⠀⠀⢸⠆⠀⡀⠀⠀⠘⢄⡀⠀⠀⠉⢀⣀⠀⠀⣀
⠀⠀⠈⣻⣡⡾⡍⠛⣻⣏⠛⠉⠉⠉⠀⠰⠦⣈⠜⠁⠀⠀⠀⠀⠐⠆⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠉⠀⠲⠤⣤⠉⠐⠋⠁
⠀⠀⠀⠈⠉⠉⠙⠛⠂⠠⠶⠴⠖⠚⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠢⠀⠺⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⡀⠀⣤⣠⠁⢀⣀⣶⢠⣰⣆⢼⣆⠀⠀⠀⠀⠠⠆
⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣤⣤⣝⣦⡬⠓⢖⡉⢻⣄⠁⠀⣽⣿⣷⠀⠩⣽⣿⣿⣿⡯
⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⢹⡌⠒⢈⢙⣦⡄⠀⠀⠀⠁⠉⠁⠔⠛⠹⠟
⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠘⠃⠀⠀⠀⢰⡟⠚⠉⠉⠀⠀⠀⡄⠀⠀⠀⠀⢿⡄⠀⣾⣻⠇
⠀⠀⠀⢸⣿⣿⠗⠂⠀⠀⠀⠀⠉⢀⡀⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢴⡟⠀⠀⠀⠀⢸⣷⠀⣿⡟⢸⡅⠀⠀⠀⣧
⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣾⡏⠀⠘⠿⣮⣻⣦⣄⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠛⠁⠀⠀⠈⠁⠉⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠴⣶⣿⣧⣀⣠⣤⡴⠶⠶⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠉⠉⠁


⣶⣶⠶⠶⠶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠄⠄⠂⠌⡀⠈⠈⠇⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠀⠐⢈⣀⣥⠴⠒⢚⢨⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠂⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠢⢀⣀⣀
⠤⠤⠔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢤⠂⣤⠄⠀⠀⠀⠀⠀⠐⠤⠀⠤
⣀⡀⠥⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⢔⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠵⡪⢔⠁⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⢸
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠁⠀⠀⢀⣉⣉⣀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⡀⠄⠁⠀⠀⢸⠀⠈⢱⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠤⠀⠀⠀⠰⠀⠀⠀⠀⠠⢄⣀⡀⠀⠀⠀⣀⡤⠆⠀⠀⠀⠀⠀⢀⡦⠀⢀⠀⠀⠀⠀⠀⡐⠈⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤
⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠀⠀⡄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠚⠚⠚⠓⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⢀⣀⠤⠊
⠀⠀⠁⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢀⣠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢚⠀⠀⠀⠀⠀⠀⠐⢄⠈⠉⠉⠉⠉⠁⣀⡠⠔⡀⠀⠀⠀⠀⠀⠈⠂⠤⠤⢀⣀⣀⡠⠔⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣢⠄⡀⠀⠀⠀⠀⠀⠀⠈⠩⠭⠭⠭⠭⠠⠐⢈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠔⢁⠃⠀⠀⠀⠀⠀⡀⠀⠐⠒⠒⠒⠚⠛⠋⠁⡠⠀⠀⠀⠀⠀⠀⠰⣀⣀⣄⣀⣀⣌⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠛⠓⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠒⠈⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⡈⠀⠐⠀⡀⠁⠸⡂
⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⡂⠂⠀⠁⢧⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⣠⣤⣀⣀⣀⢀⣀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠆⠊⠉⠡⡯⠜⠁⠀⡀⠀⠀⠣⠼⠋⠉⠉⠻⠇⠒⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠀⠀⠀⠁⠊⠀⠀⠀⠰⠌⠀⠰⢂⡀⠀⣢⠀⠐⠊⠀⠳⠾⠈⢑⡲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠞⠁⠀⠰⡽⠋⠕⠀⠀⠀⠀⠀⠀⢤⠒⠀⠀⠀⠣⠂⠨⢥⠀⠀⠀⠀⢠⠀⢀⠨⡹⠀⡭⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠁⠐⠴⡄⠀⠀⠑⠴⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠒⠂⠐⡲⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢃⡀⠀⢀⢄⠀⢘⣳
⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠠⣒⡄⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠐⠂⠀⠘⠼⡓⡄⠉⡄
⠀⠀⠀⠀⡠⠚⠂⠄⠀⠀⠀⠠⠔⠀⠀⠑⠃⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⣣⣤⣄⣥⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠠⠀⢄⣠⡀
⠀⠀⠀⢎⢾⣿⣿⣿⠃⣾⣤⠖⣄⢀⡔⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠛⠃⠀⣁⠀⢸⣿⢈⣦
⠀⠀⠀⠈⠺⢿⣻⣅⡀⠉⠻⠁⠙⢻⣅⣰⣦⣤⣄⡤⣀⣀⣀⡀⣀⠀⣀⢀⡀⢀⣶⠢⠤⠐⡦⠀⣀⠤⣀⣀⣤⠂⣰⡶⠞⢻⣿⠎⠉⢹⠃
⠀⠀⠀⠀⠀⢿⣿⣷⡷⠶⠛⠓⠦⠚⣉⣉⣛⣿⣄⡴⠏⠈⢿⣆⣸⢛⣷⣤⣯⣁⣽⠀⣰⡿⣿⠉⣿⣠⡎⠉⢨⣆⡙⠿⠋⢠⣾⣷⡀⠀⠱
⠀⠀⠀⠀⠀⡤⡙⠻⠷⢬⣤⣤⣤⣤⣤⣤⡿⢿⡟⠁⠀⠀⠀⣿⣀⣉⣠⣿⣿⣿⣭⣽⣷⣾⡁⠀⢈⣿⣿⡿⠽⣿⣷⣤⣴⣿⣿⣿⠳⡎⠁
⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣶⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠉⠉⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠄
⠀⠀⠀⠀⠀⣿⣿⡇⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣽⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⡿⠿⠿⠿⣿⣿⣿⣿⡟⢛⠄
⠀⠀⠀⠀⣿⣌⢻⣽⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣹⣿⠻⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣉⣥⣤⡶⢶⣶⣿⣿⠉⢹⣿⣧⣼
⠀⠀⠀⠀⠈⢧⣈⣿⡆⣹⣅⣰⡗⠒⠠⠶⢲⣾⣿⡿⢿⡉⢹⡷⠦⣤⣤⣤⡤⢤⣾⠞⣿⣿⡏⠀⠈⠿⣿⠁⢺⣿⡿⠋⠀⠘⣿⣥⣎⢀
⠀⠀⠀⠀⠀⢸⣯⣤⡌⠙⣻⣏⣀⣶⣶⢀⡀⠙⠋⠀⠾⣧⣶⡶⠀⠙⠿⠛⠧⠀⣵⡀⠍⠩⠶⢠⣾⡛⣸⢃⣀⣈⠐⠿⣿⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠙⠋⠉⠉⠉⠉⠛⠻⣿⡯⠀⢺⣷⣾⣿⡁⢰⣶⣇⣴⣶⣦⣬⡗⣷⣶⣶⣾⠿⠷⠿⠟⠻⠿⠷⠛⠋⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠊⠁⠈⠛⠻⠛⠉⠉⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣬⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⡀⠬⠓⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣭⣙⡒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢩⣍⣓⠢⢤⡀
⠀⠀⢀⣴⡾⠿⠻⢿⣦⡀⠀⠀⠀⠀⡸⢰⠁⠀⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠛⠫⢕⢦⠀⠀⠀⠀⠀⡇⡜⠀⠈⠉⠚⠁
⠀⠀⣾⡏⢤⣤⡟⣤⣼⣿⣤⣤⣤⣠⡇⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⠁⢤⠘⠃⡤⢳⢧⣤⣤⣄⣸⢀⡇
⠀⠀⢿⣇⢰⣾⡟⠛⠚⠛⠓⠒⠒⢺⢲⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⡀⠴⢸⠛⠒⠚⠚⠒⠒⠒⡞⣺⢻
⠀⠀⠈⠻⣷⣾⡇⠀⠀⠀⠀⠀⠀⡏⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠼⣲⣼⠀⠀⠀⠀⠀⠀⢰⠃⡇⢸
⠀⠀⠀⠀⠀⢸⡇⣠⣤⣤⣤⣤⣴⢃⣧⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡠⠄⠀⠀⠀⣼⢼⣅⢸
⠀⠀⠀⠀⠀⢸⠀⡏⠉⠉⠉⠉⡿⢸⠋⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠁⠀⠀⢀⡏⡟⢸⢸⡀
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢰⠇⡾⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠸⠀⠀⠀⠀⢸⢠⠇⠈⢸⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⢠⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢀⡏⣸⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢰⠁⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠸⡄⡇⠀⠀⢸⢁⡟⠀⢠⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⠀⠀⠀⡜⢸⠁⠀⠰⢸⠁
⠀⠀⠀⠀⠀⠀⠳⣌⡓⠦⠬⠯⠵⢒⣋⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣑⠂⠤⠬⠡⠄⢒⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣶⡶⣶⢶⠶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣶⠶

⠘⠶⣄⠀⠑⠢⣄⡀
⠀⠀⠀⠉⠒⢤⣀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠷⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣄⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠻⠶⣦⣤⣈⣀⣀⣀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠈⠁⠀⠈⡇⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡀⠀⠘⠀⠀⠀⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠘⡉⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢁⣀⡀⠀⠀⠀⣀⣀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⠀⠀⡄⠀⠀⡠⠞⠋⠉⠉⣷⠘⠀⠀⡀⠀⠀⠀⠈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢀⡞⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⡼⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠤⠤⠞⠁⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡉⠙⠛⠲⠶⠶⠤⢤⣤⠤⠴⠶⠖⢋⣩⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⠦⠤⠤⡤⣤⠶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⠯⣭⣕⣶⣤⣀
⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠋⠁⠀⡇⠀⠀⠀⠉⢻⣾⣷⣄
⠉⠙⠛⠿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠱⣄⠀⠀⣇⠀⠀⢀⡴⠋⠀⠹⣯⣆
⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠈⠳⡀⢿⠀⣰⠋⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⠔⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⣿⠦⠤⠤⠤⠤⣽⣿⣿⣷⣒⣒⡋⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠬⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣠⡤⠴⣒⣿⢩⠟⡟⢯⠀⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣿⡍⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠀⢸⠋⠀⢻⠈⠳⡀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄⠀⠙⣦⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣤⣼⣷⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⣿⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠒⠚⣿⣯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠒⠒⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠤⠼⣿⣦⣤⣀⣀⣀⣀⣀⣀⢀⣀⣤⡤⠤⠤⠔⠒⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠔⠋
⣿⣷⠿⠿⠷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣰⠤⣄⣉⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣚⡭⠭⠭⣝⡲⣄⠀⠀⠀⠀⠀⠀⠀⡾⠀⡏⠀⠀⠉⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⢰⡆⠀⠙⣎⢧⠀⠀⠀⠀⠀⢠⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠛⠇⠀⡁⠸⠏⢸⣾⡗⠒⠒⠒⠒⢾⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⣧⢷⡀⢠⡶⢸⠋⠙⠓⠚⠛⠓⠒⠒⠒⠒⡟⢻⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡳⢤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⣒⠒⠒⠒⠒⠒⢲⡇⣸⣧⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠈⠉⠉⠉⠉⠉⣹⠉⡿⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⠀⠀⠀⠀⠀⠀⡏⢠⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠃⣸⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡾⠀⡏⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢠⡇⢸⠃⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⣸⠀⡾⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣇⠀⠀⠀⠀⡏⢰⠇⠀⠀⢀⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠓⠤⣄⣀⣉⣉⣀⠤⠔⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠶⠒⠋⠁

⢷⣝⡛⣩⣴⣦⠙⠷⣿⣙⣹⡏⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠾⢿⣯⣏
⢿⣝⣛⠿⠞⣯⡔⢾⡏⢹⡉⡠⠐⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡄⠰⣷⠈⡙⠒⠚⠏⢠⣶⣈⣿⣴⠞⢁⡄⢀⡀⠀⠀⢀
⣶⡼⣿⠗⠛⢿⡇⢹⡇⣹⠇⣴⠚⣿⠀⣾⡟⠃⢸⡇⣿⠃⠀⠀⠐⢻⡧⠐⣷⠈⣿⡌⣿⠂⢸⣿⠀⣿⠀⣿⡛⢧⣸⣿⢘⡃⢚
⡶⣆⠛⠓⠔⠻⠷⠚⠃⢼⠆⢻⣆⡿⢄⣻⣧⠀⢸⠇⠻⠆⠀⠀⠀⠚⠿⠖⠛⠀⢿⠆⠿⠆⠰⡿⠀⢿⠔⠻⡦⠀⠘⢿⡎⠙⠷
⠇⢸⣀⡶⠀⠒⠒⠀⠀⠀⣀⣤⡙⣧⣬⣉⠁⠀⢀⡀⠀⠂⠀⠀⠀⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠟⡀⠐⠊⢟⠃⠀⠊⡵
⣤⣟⠉⠁⠀⠀⠀⠀⠠⠄⠻⠟⢻⣿⣿⣿⡄⢢⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⣠⡈⢁⡙⠈⢃⣥⣀⠘⠁⡢⡤
⠻⣋⠙⠂⠀⠀⠀⠘⢿⣈⡑⢼⡇⠐⠿⠿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠙⠻⠞⠀⠀⠉⠤⠠⠀⢀⡀
⠋⠁⠀⠀⠴⣀⡀⠴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⠀⠀⠀⠀⢠⣾⠟⠲⣟⡿⠂⠂⠁⠀⠀⠀⠀⠀⠈⠁
⠁⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⡇⠀⢹⠀⠀⠀⠀⠀⠃⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⡀⠀⠶⠂⢀⡦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠄⠀⠀⢸⠀⠀⠇⠀⠨⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠉
⠀⠚⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃
⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤
⠀⠀⠀⡀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣤⣤⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣌⠀⠰⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘⠄⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⣀
⢀⠈⢀⠀⠈⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠂⠀⠀⠀⢶⠟⠉⠀⠈⠁⠰⣧⣀⣄⠀⠀⠠⣤
⠼⠒⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⢀⡀⠀⡙⠛⠉
⠢⠀⠲⠈⠀⠀⠀⠃⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⠀⢻⡿⠟⠀⠀⠀⠘⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠰⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠶⡴⠀⠀⠀⠀⠈⢰⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑
⠬⠠⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀
⣂⠜⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣄⠀⠛⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈
⡿⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠹⢦⣽⣟⣁⡴⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⡀
⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⡉⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠰⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠳⡀
⠀⠀⡄⠀⠀⠀⠀⠠⣤⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠠⠄⠀⠀⠰⠂⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⠀⢀⠟⠁⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠄⠀⠀⢀⠀⠀⠀⢀⠇⠀⡘⠷⢦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠃⠀⠀⢀⠀⠼⣠⠶⠈⡃⣀⠀⢘⠀⠀⢤⣸⣷⠎⢹⠀⡄⣃⣴⠀⡇⢀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⡟⠻⢿⣿⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠢⡀⡇⡠⠊⠹⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠀⡿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⠊⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⠀⠀⠀⠀⠀⠀⢿⣆⣠⠞⣿⢿⠓⠒⣶⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠈⠻⣦⣰⣇⣈⣷⡾⠋⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣠⣶⢶⡶⢶⣤⡈⠉⠛⠉⠁⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠙⢦⣸⣇⣴⠟⢿⡄⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⢿⡷⠶⢺⡷⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⠢⣾⠃⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⢲⣶⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡍⢠⠟⢿⠙⢲⣼⠏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠈⠻⢿⣦⣼⣶⠿⠋⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠦⢀⡴⠞⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠃⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢞⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠔⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠏⡱⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠
⠀⠀⠀⠀⠀⣅⠀⠀⢿⡟⠷⠄⠀⠄⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠐⠀⠀⣠⠀⠀⠀⣀⠀⢠⣦⡜⠀⠀⠀⠀⠀⠀⠀⠠⡄⢤⡀⠀⣦⡀
⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⡀⠴⠢⠴⠂⠄⠐⠤⡀⠀⠀⠀⠉⠂⠀⠀⠀⢀⡼⠃⠀⠀⠸⣏⠀⠈⠻⠆⠀⠤⠀⠀⠀⠀⠀⠀⠈⢲⡶⣄⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠆⠔⠐⠊⢃⡠⢤⠠⠤⡤⣤⠤⣄⣉⡐⠀⠀⣀⠀⠀⠉⠉⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠐⣦⡴⢠⡌⢐⣆⣤⣤⣤⣦⣴⣡⡍⠀⠈⠐⢨⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣶⢆⠑⣄⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢣⡐⠺⢿⣿⣟⠙⠁⠋⠙⢸⠟⢿⣿⢽⣾⣷⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⢻⣟⢂⠘⢄⠀⠀⠀⠀⠀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣂⡛⡭⡇⢀⡀⠀⠀⠀⠀⠘⠛⠎⢘⣉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢥⠈⢷⡀⠀⠀⢸⡿⢳⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢄⠙⢦⣝⣢⣤⢀⡀⠀⠁⠀⠁⠉⠉⠻⣿⡇⠀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⢂⡷⠀⠸⣷⠀⠀⣸⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⠤⡉⠋⠁⢠⠀⠀⠀⠀⠀⣤⣦⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣼⡅⢠⢊⡿⠀⢈⠿⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠸⠀⠀⠀⠀⣼⣿⢛⡀⣥⠀⢼⠀⠀⡠⠀⠀⠀⠀⠀⣠⣿⠋⠁⢎⣾⡇⠀⠘⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⢹⠀⠁⠦⠄⠀⠀⠀⠀⠀⠀⠰⢀⣴⠟⠁⠀⠀⡼⠛⠃⠀⢰⠀⠀⠀⠘⡷⠀⢰
⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡄⠀⠀⠀⠀⠻⢼⡀⠺⠶⠂⠁⠀⠀⠀⢠⡶⢀⣾⠃⠀⠀⠀⡸⠁⠀⠰⠀⠻⠀⠀⠀⠀⢃⡀⠀⡀
⠀⠀⠀⠀⢰⡟⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡮⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡾⠛⠀⣼⡇⠀⠀⠀⣰⠁⠀⠀⢀⠀⠠⡆⠀⠀⠀⠸⠁⠀⠁
⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡾⢻⣿⣦⣄⡀⠀⠀⠀⢀⣠⣤⡶⠋⠁⠰⠀⢿⡇⠀⢄⣴⣿⡀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠡⣀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡙⣿⡿⣿⣿⡿⢆⣿⠟⠁⠀⠀⠀⠀⠀⢸⣷⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢹⠀⡆
⠀⠀⠀⠀⠀⠀⠘⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠙⣿⣧⡟⢁⣾⠋⠀⠀⢀⡞⠰⡃⠀⣼⣯⠃⠈⣿⡜⣄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡙⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡙⢛⣤⢞⠥⠚⠓⠒⠛⣿⠀⠃⣰⡿⠃⠀⠀⠘⣷⣌⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⡌⢡⣒⡦⡀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠘⣷⡂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢽⣿⣗⡁⠀⡀⠀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠵⣀⣤⠞⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠣⡸⢇⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡅⠀⠀⡀⠀⠀⠀⣤⡂⢀⡾⠗
⠀⠀⠀⠀⠀⠀⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⢻⠆⢀⠃⢨⢀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠁⠐⢢⡀⢱⣶⣶⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠓⢤⣀⠛⠞⠀⠙⠀⡄⢀⠞⠈⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⢈⣷⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⣦⠈⠛⠻⠛⠉⠀⠉⠀⠂⠀⠀⠀⠚⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣄⢈⢠⣆⣈⠄
⠀⠀⠀⠀⠀⠀⠿⠃⣈⡛⢃⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠻⠦⣤⣄⣤⣀⣠⠀⠀⠀⠀⠀⣀⠄⠀⣀⠀⡄⠀⠀⠀⠀⠀⠈⠉⡘⠋⠃⠜
⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠀⢀⡔⠀⠀⠀⢠⠆⠀⠀⢠⠆⠀⠀⡖⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠴⠷⢾⠰⠋⠀⢀⡞⠞⠖⠻⠾⠍⠀⣸⠁⠎⠳⠙⠶⡏


⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠐⠐⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⢀⣀⠁⠐
⢀⣠⡀⠀⣤⡄⠀⣀⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⢰⣳⡆⠀⠀⣾⡷
⠀⠉⠀⠀⠉⠁⠈⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⣄⠀⠀⠀⢀⣠⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠿⠃⠀⠀⠺⠷⠟
⠀⠀⠀⠀⢀⡈⢈⠀⠀⠀⠀⠐⠠⠰⠀⠀⢐⠐⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠀⢀⠈
⠀⠀⢰⣽⣦⠀⠀⢠⣾⣱⠀⠀⠈⠐⢄⢀⢸⢐⢘⢠⠔⠈⠀⠀⠀⢀⣀⡐⠀⠒⠒⠀⠀⠐⠒⠂⠂⣀⣀
⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠈⠈⠉⠈⣀⠀⠀⠀⠀⡈⠀⠀⢀⡈⠀⠂⠀⠀⠐⠂⢁⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠷⡿⠀⠀⠸⠷⠗⠀⠀⠀⠀⠀⠀⠀⣤⢲⡅⠀⠀⢀⣵⢢⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠈⠛⠛⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⡄⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⡿⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⠀⡠⠴⠘⡄⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⡠⠔⠉⢎⢇⠀⠀⢃⢰⠀⠀⢃⡘⠀⠉⡲⡰⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇⠀⠀⠀⠘⡄⠀⡜⠛⡄⠀⠘⣬⠀⠀⢸⠁⠀⢠⠵⣁⠀⢠⠁
⠀⠀⠀⡠⢒⠔⠚⠛⠉⠉⠙⣀⠏⠉⠉⠙⠓⠻⡒⢷⠀⢡⠀⠀⢸⠀⠀⡆⠀⠀⡌⠀⡌⢠⠃
⠀⠀⠀⡿⢖⡛⣲⠦⠴⠶⠶⠤⠶⠶⠶⠴⣖⢚⡂⢽⠇⠀⡆⠀⢸⠀⠀⠇⠀⢠⠁⢰⢁⡆
⠀⠀⠀⢰⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⡘⢾⡀⡃⠀⢸⠀⢰⠀⠀⢸⢀⡴⠊⠇
⠀⠀⠀⠈⡄⠀⠀⢠⠿⡄⠀⠀⠀⢀⡾⣆⠀⠀⠀⣇⠀⠈⠉⠒⠚⠀⠘⠒⠒⠉⠁⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠉⠀⠉⣴⣶⣦⡉⠀⠉⠀⠀⢸⠻⠀⠰⠋⠛⠆⠀⠀⠀⠞⠙⠆⠀⡆
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢿⡟⣻⠁⠀⠀⠀⠀⠸⠘⡂⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠁⢠⢄⣀⠠⠤⢧⣀⣇⣀⢀⡀⠀⠻⠟⠀⠀⠀⠀⣸⢀⡀⣀⣀⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡜⠉⠑⡜⣖⠋⣤⡬⠉⠒⡛⠩⢍⡛⠿⠿⠿⠷⢶⠾⠓⠬⢉⣤⣬⢟⡨⠒⠚⠉⠆
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢇⠀⡠⠊⠉⢠⠉⠭⢉⡐⢒⡂⢓⡚⠓⠒⠒⢛⠑⠛⣺⠋⣉⠉⣁⠈⠉⠲⡀⢀⠜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠈⠓⠀⠰⢀⡐⠃⠉⠀⣾⡐⠂⠛⠐⠂⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠀⠀⠀⠀⠓⠢⠤⠤⠀⠀⠀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠩⣥⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠻⢄⠰⠶⠂⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣄⠒⠐⢶⡒⣶⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣒⡉⠁⠀⠀⠀⠈⠙⠐⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠁⠀⠉⠢⢤⡀⢀⠔⠚⠛⠳⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡄⠀⠀⠀⠈⠉⠀⠁⠄⠀⢠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⡀⠗⠀⠀⠀⠀⠀⢀⠄⠀⢆⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⣨⠂⠀⠈⠄⠀⢀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠀⡠⠆⠀⠳⡀⠀⠀⠁⠀⠀⠀⠀⡰⠣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠛⢮⣁⣀⡐⠁⠘⠀⠀⠀⢀⠀⠀⠈⠵⠞⠈⣇⠄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠓⢄⣀⡠⠃⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠈⣣⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥⢀⠑⢄⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠓⢤⣔⠠⣈⠞⠀⡡⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⡉⠓⠀⢈⠇⠀⡄⠢⡀⠀⠀⠀⠀⠀⠭⣉⠥⣀⢜⠕⠀⠙⡐⠢⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⠀⢰⡀⡀⠀⠀⠀⠄⠀⠀⠈⠉⠓⠁⠀⠀⠀⠃⠀⠀⠀⠀⠁⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⢘⠆⢤⣄⠀⢀⡀⠀⠀⠀⠄⠀⢀⢀⡄⠀⠀⠤⠀⠀⢀⣈⣉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡏⣻⠤⠀⠈⠑⢬⡁⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢘⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⠐⢦⡀⠀⠘⠦⡀⠁⠈⠁⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠀⠐⢶⠄⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠢⡀⠀⠀⡎⠀⠀⢸⡀⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡸⠁⠀⠀⠀⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠸⠀⠀⠀⠀⠂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠒⠚⠀⢤⣄⣸⠀⠀⢀⠀⠈⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⠦⢀⠲⣤⠤⣦⡏⠤⠄⠀⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠳⣚⣣⡤⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠁⠀⠀⠀⠈⠁⢠⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠗⠂⠀⠀⠀⠀⠀⢲⡁⠀⠀⠀⠀⣦⣤⣤⣤⣤⣤⣤⠆⠀⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠠⣤⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣤⣤⣤⣤⡴⠆
⠀⠀⠀⠀⠀⠀⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⢀⣶⣲⣦⡀⠀⢠⣾⣿⣷⣆⠀⣴⣿⣽⣿⡆⠀⠀⣾⣿⣾⣿⡆⠀⣠⡴⣦⣄⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠘⢿⠿⠿⠃⠀⠘⠿⠯⠿⠏⠀⠘⠿⠿⠟⠁⠀⠀⠙⠿⠜⠛⠁⠀⣿⣿⣿⡿⠀⠀⠸⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠠⣄⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⠔⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⢀⠄
⠀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣚⣉⣀⣀⣀⣀⣀
⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡇⠹⣆⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⠶⣤⠀⠀⠀⠀⠀⠈
⠀⢀⣤⣤⡄⠐⡄⣤⢰⠀⠀⠁⠶⠀⠂⣈⠀⠀⠀⠀⣧⠀⣷⠀⠙⡆⠀⠐⠉⢃⠀⡀⠈⠀⠀⠀⣴⣻⢞⡽⣻⠎⠀⠀⠀⠀⠀⠀⣤⠚⣶
⠀⠀⠀⠀⠀⠀⠓⠃⠘⠀⠀⠈⠛⠋⠛⠉⠀⠀⠀⠀⢹⣁⠀⢂⡼⠃⠀⠀⠒⠊⠉⠀⡂⠀⠀⠈⠐⠫⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⢀⣤⣤⣠⣦⣤⠀⠀⠀⣀⠐⠄⣈⠀⠀⠀⠀⠀⢠⣄⣠⣄⣤
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡒⠦⠀⠀⠀⢠⣤⠙⠃⠀⡼⠂⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣴⠆⠀⠑⢿⡄⠀⠀⠘⠛⢿⣿⠋⠁
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣤⠐⠂⠛⠁⠠⠀⠀⠀⢀⠈⠀⠔⠊⠀⠀⠀⠀⠁⠀⠀⡀⠀⣦⠀⠠⠟⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠛⠛⠛⠛⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠐⣶⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠚⠀⠁⠒⠀⠀⠀⢤⡀⠈⠹⠇⢤⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡀⠀⠈⠿⠀⠀⠀⠀⠀⠠⠁⠀⠀⡙⣷
⠀⠀⠀⠀⠠⣶⣤⣤⠀⠀⢾⡀⠁⠀⠀⠀⠰⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣤⠀⠀⠀⠠⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁
⠀⠀⠀⠀⠀⠿⣮⠗⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢠⠦⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠉⠛⡂⠀⠀⠀⠀⢰⠀⢃⣠⠖⠀⡀⠀⠀⠄⠀⠰⠀⠀⠀⣴⣶⣶⣶⡶⢤⣀
⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⡣⠀⠠⢤⡄⣤⠖⠉⠀⠀⠀⣅⢠⠄⠀⣀⡀⠀⠀⠀⠳⢄⠀⠀⡴⠖⠋
⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠹⠁⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠡⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⢾⡿⠁⠀⢀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠐⣄⠁⠠⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣿⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠁⠀⠀⠀⠀⢀⠈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣷⢄⡇
⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠄⠀⠉⠑⠀⠄⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⣀
⠀⢆⣀⣀⠀⠀⣀⣠⠀⠀⠸⣿⣍⣶⣍⣻⡀⠀⠀⠀⠰⠒⠆⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⠀⢀⡜⠁⣴⣴⣀⡶⢤⡀⢠⣾⣛⢛⣿
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢶⣤⣄⣤⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠠⠀⠃⣜⢁⣸⡧⡴⠃⠀⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣶
⠀⠀⠀⠘⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢤⣤⠀⠀⠀⠀⠘⢿⡿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⡄⠀⠀⠀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠄⣀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠙⠀⠀⠠⠀⠀⠀⠘⠳⠶⠶⠟⠉
⠀⠀⠀⠀⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⠒⠀⢀⡇⠀⠀⠀⠀⠀⠤⢤⠐⠂⠀⠀⠀⣶⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠾⠷⠂⠐⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⢠⡜⠛⣅⠀⠀⠉⠉⠉⠀⠀⠃⠀⠀⠀⣷⡾⡿⠻⣆⠀⠀⠀⠀⠀⣾⢀⡀⠟⠀⠀⠀⣤⣤⠀⠀⢀⠀⠀⠀⢀⠸⠋⠙⠉⠙⠁
⠀⠀⠀⠉⠀⠉⠂⠉⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠳⠊⠐⠓⠀⠛⠁⠀⠀⠠⣅⠈⢨⠅⠀⠀⠀⠀⠉⠠⣦⠆⠻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠓⠂
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠀⣤⣤⡀⠀⡀⠀⠀⠀⠿⡇⠀⢀⣠⠤⢄
⠀⠀⠀⣠⣶⣶⠀⠀⠀⠀⠠⠀⠀⠘⠇⠀⠀⠀⣴⠾⠟⠉⠁⠀⠐⠀⠀⠁⠀⣘⣦⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠡⠀⠀⠀⠁⠐⠀⠀⠿⠁⠟
⠀⠀⠀⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠟⠁⠀⠀⢈⣿⡷⠒⠾⠟⠂⠀⠀⠀⠀⠀⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠛⠛⣿⠋⢻⠛⢛⡛⠟⣛⣿⢛⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣾⣿⣶⣾⣿⣷⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠶⠒⠒⢒⠚⠛⡉⠉⠉⠁⠀⠘⡋⠓⠒⠦⠤⣄⠠
⠀⠀⠀⠀⠀⢀⣠⡤⠒⢫⡍⠀⠀⡠⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢈⡀⠀⠘⢁⠦⠂⠀⢀⠢⣄
⠀⠀⠀⢀⡴⣫⠟⠁⡀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⢀⠈⠀⠀⠂⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠉⠉⢪⡑⣆
⠀⠀⢠⢎⠀⡆⠁⠀⠁⠀⠐⠀⠀⠰⠆⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠠⠆⡀⠀⠀⠴⠀⠘⣿⡜⣇
⠀⢠⢯⡟⠃⡀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠐⠀⠀⠀⠒⢀⡄⠀⠈⣹⣇⢿
⠀⣼⢺⡁⡈⠁⠈⠊⢀⠀⠀⠔⠀⠀⠁⠀⠀⢀⠀⠀⣄⠀⠀⠁⠀⠘⠀⠀⠀⡄⠉⠐⠁⠠⠀⢀⠄⢲⣿⣿⡏⣼
⠀⠸⡜⠿⣫⡄⠀⢀⠀⢀⠀⠀⠀⠀⠐⠄⠀⠈⢀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⣄⢀⠐⣠⣒⡤⠾⢛⣋⣥⣾⣅
⠀⣠⣾⣷⣶⢶⣤⣔⠂⠊⠀⠀⠃⠀⠀⠀⢀⠀⠀⡀⠀⡈⠀⠀⠀⠤⠦⠀⠀⣒⣂⣩⣤⣴⣶⣶⣿⣿⣿⣿⡿⡀⢚
⡞⠁⣺⣿⣾⣮⣉⣟⣿⣻⣿⣿⣿⣷⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠿⠍⠀⠁⢨
⢣⠀⠈⠙⡉⠙⠛⠏⠭⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⣷⣿⣟⣻⣿⣿⣿⡿⠿⠿⠟⠛⠛⠋⣡⣤⠤⢴⡞⠉
⠀⠙⢲⢏⣹⣄⣀⠀⠀⢀⣠⠶⣲⣖⠬⣩⡥⢤⡉⠉⢁⣰⠟⠛⠛⠋⠋⠉⠉⠉⠿⠷⢄⣤⣴⣚⣩⣵⣶⣾⠋⡆
⠀⠀⠸⡸⣿⣻⢿⡿⢿⢿⣤⠞⠛⠿⢻⠾⠿⣶⣉⠛⣉⣀⣠⡤⣤⣀⣰⣶⣦⣤⣴⠶⠛⠋⠉⠓⠁⡉⠀⠈⡰⠃
⠀⠀⠀⣙⣶⣤⣀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠸⠈⠛⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣨⣵⣶⣿⡷⠑⡄
⠀⠀⢰⠀⣿⣿⣿⢿⣷⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣯⣿⣾⣿⠿⠁⢀⠇
⠀⠀⠈⣣⣌⣓⠟⣓⡛⠻⢭⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⣵⠟⣉⣀⣀⣠⡴⢋
⠀⠀⠀⢏⠈⠽⠷⠤⣭⣽⣦⣈⠉⠀⠈⠙⠻⣿⡿⠟⠛⠋⣉⡭⠭⣭⣉⣉⠉⣉⣡⠴⠿⢋⠭⠵⠾⠟⣛⣋⣡⡞
⠀⠀⠀⣸⠹⠗⠤⠤⢤⣄⡀⢈⡳⠦⠤⡈⣁⣠⣐⠺⠟⠂⠀⠒⠠⢤⣤⣬⣭⣭⣤⣤⣶⢯⣤⣴⠟⠚⠉⠛⢹⠇
⠀⠀⢰⣿⣆⡀⠀⠀⠈⠈⠁⠀⠑⠶⠶⠒⠻⠟⠛⠛⡳⠊⠉⠛⠳⠄⣈⣉⠤⠞⠛⠋⠀⠀⢀⠀⠀⠀⣀⡴⠋
⠀⠀⠘⠛⢻⣷⣤⣄⣀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣀⣮⠵⠖⠋⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠟⠻⠿⠷⢶⣤⣤⣤⣤⣤⣀⣀⣐⣀⣠⣤⣤⣤⡤⠴⠖⠛⠋⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣿⣿⣿⣾⣷⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣠⠤⠤⢤⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⣀⢰⠇⠀⡴⠋⠀⠀⠀⠀⠈⢢⡀⠘⡇⣀⣠⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣀⣈⡉⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢷⠀⢉⣁⣀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠎⠁⠀⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠈⠑⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢿⠀⠀⠀⠀⠀⣀⣤⠤⠤⠶⠶⠦⠤⣤⣀⠀⠀⠀⠀⠀⣿⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠦⣀⡠⠖⠋⠁⠀⠠⢤⣤⣤⠤⠀⠀⠈⠙⠲⢄⣀⡴⠃⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠖⠂⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⣄⠀⠀⣠⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⡤⢤⣀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠀⠀⣀⡤⢤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⠞⠁⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⠱⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠑⠤⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠤⠊⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠰⠦⠤⠤⠖⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠶⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⢒⣺⠿⠟⠒⠒⠲⠶⣤⣤⠤⠶⣒⣺⡿⠿⠟⠛⠛⠛⠓⠲⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⣩⠟⠉⢀⡤⠞⠉⠀⠀⢀⣠⠴⠚⠉⣁⡤⠖⠋⠁⠀⠀⠀⢀⣤⠴⠚⠋⢉⣉⣉⣭⣭⣙⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⣥⡴⠚⠉⠀⠀⠀⠀⠰⢯⡤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠸⠭⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢀⣧⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⣤⡤⢤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⢀⡴⢻⡼⣟⠲⣦⡤⠤⠶⣶⡖⣒⡿⣻⣿⣛⣛⣉⣀⣤⣾⣥⣿⣿⣿⣥⣿⡿⠟⠲⢬⣅⣀⡤⠥⠚⠉⢯⡻⣿⣯⣽⣷⣶⠲⣶⡶⠶⢖⣒⡝⢹⢢
⠀⢸⡀⠀⠙⢮⣉⣁⣀⣠⠴⣟⣛⣓⡮⠕⠋⢈⣉⣁⣤⠤⠤⠬⠭⠭⢭⡽⠶⢦⠤⠤⣤⠤⠤⠤⠤⣤⣤⣄⣉⣓⠻⠽⠿⠽⠟⠋⠙⠒⠶⠶⠖⡫⠎
⠀⢠⣽⣶⠶⠒⣲⣿⡓⠒⠒⠒⠒⠒⢾⣯⠟⢉⡴⠦⣀⣀⣀⣀⠀⣰⠋⣹⢀⣉⣛⣛⠋⠀⠀⠀⢰⠟⡆⠀⠰⡿⠉⠉⠛⠿⠷⠖⠒⢲⡖⠚⡭⢤⡤
⡴⠛⣋⠭⢶⠚⠃⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⠀⠉⠉⠛⠓⠚⠋⠈⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⢳⠶⢭⣉⡵⠋
⠓⠋⠁⠀⠈⠢⣄⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠙⠁

⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠩⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⠓⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢋⣽⣿⠁⠀⢠⠄⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠻⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣂⠈⠁⠀⠈⠀⢠⣄⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠰⡾⢀⡀⠀⣠⠆⠀⢠⣼⡄⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶⠀⠀⣐⡷⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣻⣟⣟⠉⣀⠀⠿⠗⠠⡀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠑⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠟⢛⡛⠛⠋⠀⠀⠠⠤⠈⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢿⠄⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠄⠋⠀⠀⠂⠀⠀⠉⠻⢿⡆⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠛⠻⢷⣿⣶⣤⣀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⣤⣄⠀⠈⠁⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠯⠋⠁⡀⠀⠀⠈⠙⠳⠦⣭⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠳⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⡀⢀⣍⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡴⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠠⣮⣽⡗⣈⡓⠀⠀⠀⠷⠶⠠⣤⣄⠀⠀⡀⠀⠀⠀⠉⣿⣶⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢀⣾⠃⠀⣿⠀⣀⣿⠻⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣸⣿⠐⢀⡌⠁⠉⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⣿⠀⣿⣿⣦⣀⣾⣿⣏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⡿⢿⣧⢸⣿⣿⣿⣿⡇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠠⣿⣀⠉⠀⠻⠧⣿⣿⣄⠀⠀⣀⣀⡤⠒⠉⠋⠁⠀⠀⠀⡀⣴⣶⣷⡾⠀⠀⠜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢰⡌⠛⠀⠀⠀⠀⣿⣿⢫⣴⣾⣏⣀⢀⡀⣀⠀⡀⠀⠀⠀⢘⣿⡿⠋⠀⢀⡜⢳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣿⠀⠀⠀⣠⣯⢰⣿⠀⣾⣿⣿⣿⣿⣿⣿⣻⡷⠇⢀⣠⣶⣿⣶⣶⣷⣾⣷⣾⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣿⠀⠀⣸⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣟⣱⡿⣣⣴⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠸⡏⣠⣴⣿⣿⣷⣿⣭⣤⣿⣿⣭⡀⣩⣿⣽⣿⣿⣿⣿⣿⣿⣿⣯⣻⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠹⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠾⠃⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⢉⢰⣶⣦⣠⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡿⠛⣻⣿⣿⡆⠈⠈⠁⠁⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⢷⣂⣽⣾⣿⣿⣿⣟⣳⢀⣠⡄⣰⠟⠤⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⡏⢻⣿⣿⣿⣿⣷⡿⢿⣿⣿⣿⠛⠁⠸⠁⠈⠀⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣄⣤⣤⣤⣤⣤⣀⣀⣀⣼⣿⣏⣼⣿⣿⣿⣿⣯⣏⣈⣿⣋⣻⣀⣠⣤⣶⣀⣀⣀⣽⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣀⣀⣀⣉⣉⠟⢷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢿⣶⣇⣞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢛⣛⣛⡛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠈⣷⠀⠀⠀⠀⠀⠀⢀⡾⢋⠴⠚⠉⠀⡉⣉⠒⢮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣇⠀⠀⠀⠀⢠⡿⣱⠋⢦⡈⢀⡀⣰⠃⠄⠘⢣⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⢿⣤⣤⣤⣤⣾⢡⡇⠈⠀⠙⠦⠐⣃⣂⣈⣃⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⠋⠉⠙⣧⠘⣧⠀⠀⠀⢻⣾⣍⠀⠄⢃⡴⠣⠄⣤⣼⡀⢺⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⠀⠀⠀⠻⣮⡳⢴⣋⣈⣐⣠⣿⡼⣿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠛⠲⣶⣶⣤⣤⣤⣤⣿⣤⣿⣶⠶⠶⠶⠾⠟⠻⠿⠿⠿⢿⣟⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠖⠋⠉⠀⢀⡴⠚⠓⠒⠾⢿⣩⢽⣶⠷⠒⢒⣒⡶⠄⠀⠀⠀⣀⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠓⠢⠤⣄⣉⣛⣛⠒⠒⠖⠛⠁⢘⣛⣛⣛⣉⣡⡤⠤⠶⠚⠋⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠷⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡇⢸⡦⠤⠤⠔⠢⠤⠤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡐⠾⢷⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⡠⠒⡏⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠁⠀⠈⣷⠢⡀
⠀⠀⠀⠀⠀⠀⣗⠤⢍⣉⣒⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⠐⠒⠒⢒⣊⣉⡥⢔⡇
⠀⠀⠀⠀⠀⠀⢋⠙⠓⠒⠯⠭⠭⠭⢉⣛⣛⣛⣛⣛⡋⠩⠭⠭⠭⠶⠒⠚⠉⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠒⠢⠤⢤⣄⣀⣀⣀⣀⣀⡀⣀⠀⢀⣀⣀⣤⡤⠤⢺⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣤⡀⠉⠉⠉⠉⠉⠉⣡⣄⠀⠀⠀⠀⢸⠀⠀⣀⠴⠚⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡾⠉⢷⡀⠀⠀⠀⠀⣰⠟⠹⣧⠀⣀⠤⠛⠂⠉⠉⣲⠤⠤⠤⣤⠽⠶⠶⠭⠽⣋⣉⡉⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠁⠀⠈⢡⣶⣶⣷⣦⠈⠀⠀⡨⠋⢀⣠⣤⣖⠾⠋⠁⠀⠀⠀⠈⢢⣀⡠⠴⠐⠤⠤⣌⣉⡓⠲⠤⣀⠙⣍⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⣿⠀⠀⡜⡡⢞⡥⠚⠁⠀⠀⣀⡤⠶⠖⠛⠛⠛⠉⠙⠛⠛⠲⠶⢤⣀⡉⠉⠛⠉⠛⠺⣆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⡀⠀⠀⠀⠀⠹⠧⠴⠃⠀⣠⢏⠔⠁⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠉⠁⠒⠒⠒⠒⠒⢒⠇⡎⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⢀⡾⠃⠀⠀⠀⠀⢀⣓⣢⣤⡀⠀⠀⠀⠀⠀⣠⣴⣒⣃⠀⠀⠀⠀⠀⠈⢷⡀⡌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠱⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢙⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠙⠳⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⣀⠴⠒⠲⣤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡒⢒⡆⢀⡇
⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠈⢉⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢺⠀⡼
⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⠁
⠀⠀⠀⠀⠈⡇⠀⠈⢹⢉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠹⠛⡆
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠙⣷⡾⠀⢀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣢⣀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⢴⡦⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠁⠀⠀⠀⠐⢶⣿⠀⠀⠻⠀⠀⠀⠢⠴⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⢀⡞⠁⠀⠙⢦⡀⠀⠀⠀⢀⣸⡉⠉⠒⠶⠀⠀⠀⠀⠀⠀⠰⡏⠉⠁
⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠙⡤⠤⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣨⠗⠄⠀⠀⠀⠀⠀⠹⡉⠀⠀⠀⠘⠓⠢⡀⠀⠀⡄⠀⣄⢿⡀
⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢱⠀⢹⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠫⢤⠤⠔⠋⠓⠒⠳⢄⣀⡀⠀⠀⠠⣎⢣⠀⠸⠀⠸⠀⠺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠤⠤⠤⠗⠳⠜⠢⠓⠒⠒⠒⠁

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠶⠶⠶⠛⠛⢻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣹⠇
⠀⠀⠀⠀⢀⣤⡴⠶⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡞⠁
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠾⡋
⠀⠀⢸⠟⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⣴⣿⣴⣿⣁⣤⢶⣆
⠀⠀⣿⠀⠀⠀⠈⠉⠛⠛⠶⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⢾⣿⣡⣾⡿⠋⠀⠰⣿
⠀⠀⠙⠷⠶⢶⣴⣦⣤⣄⣀⠀⠀⠀⠈⠉⠛⠛⠶⠶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⢃⡼⢿⡿⠋⠀⠀⣠⣼⣇
⠀⠀⠀⢀⣀⣀⡻⠶⠾⣿⣉⣹⡷⢦⣤⣤⣤⣀⡀⠀⠀⠀⠉⠉⠛⠳⠶⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣾⣯⡴⢋⡵⠋⠀⢀⣴⣾⠋⢀⣿
⠀⠀⢠⣾⣯⣉⣉⣛⣒⠿⢿⣥⣤⣀⣘⣧⡀⠉⠙⢛⡲⠶⣤⣤⣀⣀⠀⠀⠀⠈⠉⠙⠓⠶⢦⣤⣤⡴⠋⢀⣴⣾⣵⢟⡵⠋⠀⣀⣴⣿⠟⢁⣤⠞⢻⡇
⠀⠀⣿⠀⠀⠉⠉⠉⠉⠛⠓⠶⠶⣭⣭⣍⣉⣛⡿⢿⣿⣦⣀⣀⣉⡉⠛⠳⠶⣤⣤⣀⣀⠀⠀⠀⢻⠄⣠⣿⡿⢛⡵⠋⠀⣠⡞⢉⣀⣤⠾⢿⡷⠚⠻⡧
⠀⠀⣿⣟⠳⠶⠶⢦⣄⣀⣀⣀⡀⠀⠀⠀⠉⠉⠛⠛⠳⠾⠯⣭⣍⣙⣦⠀⠀⠀⠀⠉⠉⣿⠷⢦⣼⣿⠿⣻⡴⠋⢀⣤⣾⠏⣿⣋⣭⠟⠛⠛⠁⠀⣰⡗
⠀⠸⣧⠙⠳⠶⣤⣤⣈⣉⣉⣉⣽⡿⠿⠟⠛⣶⡶⣤⣀⣀⣀⠀⠉⠉⠙⢷⣄⣀⣀⣴⣾⣙⣛⣛⣩⡴⠟⠉⠀⢠⣿⡿⠋⣴⠋⠙⣷⠀⠀⠀⣠⡴⠋
⠀⠀⣿⠻⣦⣤⣤⣀⣈⣉⣉⣀⣀⣤⣤⠶⠟⠻⣆⠈⠻⢧⣍⠻⠟⠳⣦⣄⣉⡉⠉⠀⠈⠉⠉⠉⠁⠀⠀⣠⠾⠋⣁⣤⣾⣥⠶⠟⠋⢀⣤⠞⠋
⠀⠀⢻⡆⠀⠀⠀⠈⠙⢿⣍⡉⠁⠀⠀⠀⠀⠀⠈⠳⣤⡀⠉⠛⠶⣦⣤⣀⣉⣙⠛⢛⣛⣶⣤⣤⣀⣠⣾⡷⠶⠛⢁⣿⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠶⠶⠶⠶⠾⠿⢦⣤⣤⣄⣀⣈⠉⠉⠉⠉⢉⣁⣤⡼⠟⠁⠀⠀⢀⣴⠞⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠉⠙⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣍⣉⣙⣛⣛⡛⠛⠉⠁⠀⠀⠀⠀⢀⣾⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠳⣤⣀⣀⣀⣤⡾⠃⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠷⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠁⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⠶⢦⣄⣨⣧⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉








⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣙⣋⣛⣉⣛⣉⣀⣉⣙⣿⣋⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣯⣤⣭⣭⣭⣭⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡔⢋⣡⠜⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⢇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠈⡆⠊⠁⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠊⠉⠀⢳⡀⣸⠄⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢄⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠙⠪⣕⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⠀⡀⠔⣚⡥⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠀⠑⢬⡂⠀⠀⢀⣒⡭⠖⠋⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⠈⡀⡀⠄⠀⠀⠙⡶⡚⠉⠀⠀⠀⠀⡴⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠚⠶⠯⠤⣀⠀⠀⠰⡇⠈⠳⢄⠀⣠⠞⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠈⡧⠀⠀⡇⠀⠀⠀⠙⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣄⠀⠀⠀⠀⡰⠃⠀⠀⡇⠀⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠑⠛⠒⠉⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⣀⠀⡜⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⡇⢀⡠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠆⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⡠⠐⡠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⣄⡆⣦⡤⠼⠞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠟⠁⠀⠠⢂⡘⡄⠀⡘⠀⠀⠀⢠⡖⠒⠒⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠘⠁⠀⠚⠀⠀⠀⠀⠈⠙⠖⠀⠀⠃⠀⠀⠀⠁⠀⠀⠂⠀⢀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠐⡤⠀⠀⠸⠃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠸⠀⠀⠀⠀⠀⠀⢀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⡐⠂⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⠀⠀⠀⠀⡀⢤⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠤⠀⠀⠀⠀⠸
⠀⠀⠔⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⢎⡩⣄⠀⠀⠑⢌⠀⢡⠀⠀⠠⠈⠂⠀⠀⠀⢀⠀⠘⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠘⠨⠗⡀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉

⢀⠄⠀⠋⠀⠁⠐⠀⠀⠀⠠⣴⡄⠐⡤⣧⠄⠀⠀⠀⠀⠀⣄
⠀⠀⠔⠋⠽⣟⠐⠀⠀⠀⠀⠍⠉⢰⠛⠧⠀⠀⠀⢀⠄⠈⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⢀
⠀⠀⠀⠈⢰⠁⠀⠀⠀⠀⠀⢰⠀⠰⢀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠠⡐⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠈⠚⢻⠋
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢂⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⡀⠀⠀⠀⢀⢖⠃⠒⠒⢤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣄
⠀⠈⠀⡀⣠⠤⠤⣀⡄⠀⠀⠀⠄⠠⡂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢣⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠠⠁⢠⠀⢰⠀⠠
⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠣⢀⡐⠀⠂⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠐⠈⠀⠀⠀⠈⠂

⢀⣀
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⣴⣦⡄
⠀⠀⡀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡿⠟⠁⠀⠀⢀
⠀⠀⡇⠀⡆⠀⡆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢳⡄
⠀⠀⠃⠀⡇⠀⠁⠀⢠⠦⠤⠤⠤⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⡀⠀⠀⠀⠀⠂⠀⠠⢉⠁
⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⣦⣴⣤⠀⠀⡀⣤⢰⣰⣂⣀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢸⠀⠇⠿⢿⣽⣿⡆⡇⣿⠸⠒⣒⣒⣒⣒⣒⣒⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⢸⣾⣿⡇⡇⠃⠀⠀⠀⣤⣤⣦⠠⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣽⣿⡇⡇⠀⠀⠀⠀⢭⢭⢭⠭⠭⠭⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⠹⣿⡇⡇⠀⠀⠀⠀⣚⣛⣛⢘⣛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⣿⣥⣠⣿⡇⡇⣧⢀⢠⣴⣶⣶⠖⢰⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠸⠀⠇⠿⠿⠿⠿⠇⠇⢿⢸⠭⠭⠭⠁⠀⠨⠭⠭⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠄⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠄⠀⠰⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⢠⠄⠁⠀⡄⠈⡝⠢⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣀⣠⠤⠤⠶⡿
⠀⠀⠀⠀⣠⠞⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠊⠉⠉⠀⡀⠀⢎⠜⠁
⠀⠀⢀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠄⠀⣊⣠⠴⠗⠚⠻⣄
⠀⠀⣧⣂⡀⠰⢀⣀⣁⣠⠤⠤⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠚⠋⢁⣀⡴⠚⢻⠉⠁
⠀⠀⣧⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢶⠶⢶⣾⡯⠏⠙⠉⠉⠉⠉⠉⠉⢀⣠⠴⣪⡟
⠀⠀⠓⡶⢬⠤⣄⣀⣀⣀⣠⣤⣴⣒⠋⠉⠅⠈⠂⣀⠞⠀⠠⠀⠀⠀⠀⢀⣀⡤⠴⠒⣋⡡⠴⠚⠁⢣
⠀⠀⠀⡹⢌⣀⡀⠀⠀⠀⠀⠀⡀⠈⣉⣉⣉⣉⣉⣡⠤⠤⠤⣖⣒⣚⠭⠥⠔⠒⠊⠉⠁⠀⠀⠸⢀⠜
⠀⠀⠘⢧⠤⠤⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠴⠚⠁
⠀⠀⠀⢼⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠖⠊⠉
⠀⠀⠀⠈⠒⠲⠷⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢂⣒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠚⠉⠉⠑⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠘⢆⠀⠀⠀⠘⡀⣇⣀⣀⣀⣀⣸⣀⡇⠀⠀⠀⢠⠊⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⡸⠤⣒⡪⠭⠅⠐⠒⠒⠒⠒⠒⠂⠭⠭⢔⡲⠮⡀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⠔⣫⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣕⢄⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢠⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢨⣶⣿⠶⡀⠀⠀⣰⣶⡶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⠁⣀⡗⠀⠘⣏⠉⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⠖⠳⡄⠀⠀⠀⢀⡄⠀⠀⡄⠀⠀⠀⠀⢰⠚⢢⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠀⠈⠣⠄⠴⠃⠀⠀⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⣠⠚⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠀⡾⠀⠀⢀⣀⣴⡦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⢻⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠛⠊⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡼⠀⢸⠇⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⡇⠀⠈⡆
⠀⠀⠀⠀⠠⡇⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⢸⠄
⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠸⡆⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰
⠀⠀⠀⠀⠀⢡⠀⠀⠀⢧⠀⠀⠀⠀⠈⠑⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠋⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠒⠉⠑⠒⠐⠲⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠴⠤⠴⠶⠶⠶⠒⠀⠚⠛⠒⠒⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣿⣟⠳⣶⡟⢠⡿⠛⣛⣿⣿⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣯⣍⡉⠉⠙⠆⠈⠁⠈⠀⠴⠋⠉⠉⢉⣩⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⢰⡿⣷⣄⠀⠀⠀⣠⡟⠁⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠻⣦⠀⠀⠀⢀⣼⠿⣇
⠀⠀⠀⠀⢸⡇⢈⡟⠳⢶⣴⠏⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠘⣷⣶⠾⠻⣅⢀⡿
⠀⠀⠀⠀⠈⣷⠋⠀⡴⢋⡎⠀⠀⠸⣥⠤⢾⡀⢀⡴⠚⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⠶⢄⡀⢠⠟⠶⠃⠀⠀⠀⠸⡌⢧⡀⠈⣾⡇
⠀⠀⠀⠀⠀⠘⣧⣸⠁⣼⠀⠀⠀⠀⠀⣠⠴⠛⠛⠦⣄⠹⣄⢻⡀⢀⡔⠀⠀⢀⡠⠏⣠⠴⠛⠛⠦⣄⠀⠀⠀⠀⠀⢳⠀⢷⣰⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡿⠊⠀⠀⢀⡞⢠⣶⣿⣷⣦⡈⢧⠈⠙⢣⣏⣤⣶⠞⠋⠀⡼⢁⣴⣾⣿⣶⡌⢳⡀⠀⠀⠀⠺⣤⡾⠋
⠀⠀⠀⠀⠀⠀⣴⣾⣥⡤⠀⠀⠀⢸⡁⣿⣿⣿⣹⣿⡇⢸⠀⠀⠀⠛⠁⠀⠀⠀⠰⡇⢸⣿⣏⣿⣿⣿⢀⡇⠀⠀⠀⢲⣮⣿⣦
⠀⠀⠀⠀⠀⠀⠉⢡⡿⠁⠀⠀⠀⠈⢧⡘⠿⢿⡿⠟⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠻⢿⡿⠿⢁⡼⠀⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣃⣴⠀⠀⠀⠀⠀⠙⠲⢦⣶⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⡴⠖⠋⠀⠀⠀⠀⠈⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣰⡇⠀⠀⣧⠀⡄⠘⣟⠻⡏⠛⢒⡶⠶⠶⠦⠴⠶⠶⢶⡒⠋⡹⢷⠇⢀⠀⣴⡀⠀⢸⣆⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢤⣸⠋⡓⠛⠀⠘⢦⣙⠚⢁⡔⠛⢦⣄⡴⠋⠳⡀⠉⢓⡥⠋⠀⠛⠚⠁⢿⣀⡜⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣳⣀⠀⠀⠀⢈⡙⠚⠲⠦⠤⠤⠤⠴⠶⠛⠋⠁⠀⠀⠀⢀⣼⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠙⠦⣴⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠛⠋⠙⢢⡴⠊⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠳⣶⡋⡦⠀⠀⠀⠀⠀⣠⠦⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⢾⣉⢙⣆⠀⠀⠀⠀⠀⠀⣀⢽⠃⠀⠀⠀⠀⠀⢧⡴⠋⠀⠀⠀⠀⠀⠀⡠⣞⢉⣷⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⠁⣨⣍⠈⢧⡀⠀⠀⠀⠀⢉⣉⡷⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⠀⠀⡜⠁⣨⣭⡈⡙⡟⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⢾⠘⣇⣨⠇⠀⠙⠲⣄⢰⣄⡟⠉⣀⠀⢀⠀⠀⠀⠀⠙⢻⣇⣶⣄⣦⡼⠃⠀⣇⣀⣧⠳⣾⣿
⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠶⢦⣤⣤⣤⣀⣀⣨⡿⠻⠿⣾⠟⠷⠟⠻⣾⠷⠿⣶⠿⠿⢿⣅⣀⣀⣠⣤⣤⣤⠶⠿⠿⠧⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⣀⣀⣤⢀⣴⣠⠄⡉⠢⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣤⡘⠻⣿⣏⡿⠿⢿⣷⢺⣯⡤⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠪⢼⣽⡷⠚⠋⠀⠀⠀⣹⣿⣿⠇⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢂⣼⠯⣧⣄⠀⣀⡤⣾⣿⣿⠛⣰⢿⡏⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡹⠀⡁⣀⣿⡾⢿⡿⡗⠛⠿⠀⢁⡄⢀⠀⠀⠀⢀⣀⠈⢅⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠻⣛⠓⠒⠒⠒⠷⣶⠤⠀⠩⠁⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠉⠉⠀⠤⠀⣼⣦⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⣀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⡀⠡⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠉⠉⠉⠁⢉⠉⠉⠁⠀⠀⠀⢀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠤⠀⠀⣘⠀⠀⠈⠒⠄⠀⠒⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⢀⠀⠀⠜⠁⠰⠀⠀⠂⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡄⠀⠁⠄⠀⠀⠐⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠒⠒⠲⢆⡂⢠⢌⠀⠀⠀⠈⢠⠈⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⣤⡖⢻⣄⣠⠒⠌⢢⠀⡈⣀⠃⢀⠀⣀⠀⠀⠈⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠈⠈⠉⠈⠐⠢⢅⢀⠛⣶⣻⠟⠻⣿⡵⢋⠀⢡⠀⡀⡀⠈⠀⠀⠐⠀⠠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣠⣶⠤⣿⣰⣯⡶⣀⡄⠙⣿⣿⠁⠀⠀⡧⣶⣿⡄⠀⠀⠠⠀⠈⠈⠀⠤⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡘⡠⠲⣭⡙⢾⣶⠛⠛⠻⣟⣹⢶⠂⢸⣼⡀⠀⠀⢹⣤⣿⡀⠀⠀⠀⢈⠀⠁⠐⠐⠀⢡⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢻⡾⠋⠁⠀⠀⠀⣀⢾⣾⡟⠃⢸⣯⣷⣄⣠⢾⣷⣌⠁⢀⠀⠀⠀⠀⠀⠃⠀⢉⡄⠠
⠀⠀⠀⠀⠀⠀⠀⡆⠲⡞⣻⣕⣦⣠⣤⡚⣿⣟⠿⡁⢀⡎⠿⣿⣾⣼⡏⠳⠁⡠⠃⠔⠀⠈⠀⠀⠉⠀⠐⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠉⠛⢸⠏⠽⠙⠏⠉⠉⢀⠴⠋⠢⢄⡉⠙⠃⣉⡠⠚⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠉⠉⠉


⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠔
⠀⠀⠀⠀⢰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆
⠀⠀⣠⠔⠃⣉⠸⠉⢧⣦⠀⢲⡀⠀⠀⠀⣠⡄⠰⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡷⣄
⠀⠀⠁⠘⠛⠉⠁⠀⠙⠻⠤⠤⣽⣤⣤⣶⡛⠫⢤⣄⡀⠀⠀⠀⣀⣀⠀⠠⠤⠴⠤⣤⢠⣤⠿⢴⣆⠃⠐⠀⢀
⠀⠀⠀⠀⠀⣙⠓⣻⡶⢦⡤⠤⣈⣧⠀⠀⠈⠑⠀⠀⠀⠀⠀⠐⠚⠃⠀⢀⡠⠤⠄⠒⠈⠀⠀⠁⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠠⣶⣦⡤⠘⠦⣉⠓⠤⢄⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⠓⠒⣂⣲⡤⠤⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠘⢿⢀⣦⣠⠘⣩⡿⠟⠚⡻⢤⠀⠀⠀⠀⣠⡊⠁⠈⡉⠙⠛⠉⠉⠉⠀⠉⠉⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠺⠥⠀⠄⠀⠀⠐⠂⠀⠑⠄⢠⣊⠁⠀⠀⠀⠙⢤⠀⠁⠀⣇⡤⠄⠞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⢿⣅⠀⠀⠀⠀⠙⢿⡍⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣅⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠈⢻⣆⠀⠀⠀⠙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⢷⣄⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⢿⡄⠀⠀⠀⠸⣧⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡜⠙⣷⡀⠀⠀⠀⠻⣦⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⣷⠀⠀⠀⠀⢿⡄⠀⠹⡄
⠀⠀⠀⠀⠀⢸⠁⠀⠈⢿⡄⠀⠀⠀⠹⣧⠀⢀⣀⢘⣷⠀⠀⠀⠀⢿⡆⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣄⠀⠀⣘⣷⡤⠴⠒⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠾⠧⣤⣀⣀⣸⣷⡤⡞⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⣄⣀⣀⡤⠀⠀⠈⠋⠁⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⡤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⣾⡋⠉⠉⠉⠉⠉⠙⢿⣟⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠉⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠿⣦⡀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⢻⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⢿⡆⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠹⣷⡄⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠸⣿⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠘⣿⡄⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣿⡆⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡄⠀⠀⣀⣠⣼⠿⠤⠶⠶⠶⠶⠿⠦⣤⣄⣀⣀⢘⣿⠀⠀⠀⠀⠀⢻⡇⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣠⣤⠼⠟⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠶⠦⠤⠾⠷⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢠⣀⠀⠀⣀⡤⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣄⣠⣀⣀⣀⣀⣠⣄⣤⣀⣤⣠⡤⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈





⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠒⠂⠀⠀⠈⠉⠉⠀⠁⠐⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠤⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠤⠄⠚⠊⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠲⠠⠤⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠢⠤⣀⣀⡀⠀⠀⠻⠟⠁⠀⠰⠄⠔⠀⠀⠙⠿⠃⣀⣀⠀⣠⠤⠊⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⢀⡠⠔⠊⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠄⠒⠈⠁⠀⠀⠀⠀⣀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡤⠞⠀⣀⡰⢣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠙⠢⠭⢉⣉⣉⡉⠉⠉⣉⡉⣉⠠⠤⠒⣮⣁⠀⠀⣀⣄⠁⠀⠈⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠉⠀⡠⠋⡛⠓⠤⣉⡢⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠤⡀⠀⠀⠀⠀⣉⡲⠄⠀⠒⠊⠉⠀⠀⠀⢀⡠⡎⠀⠀⡇⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠈⢲⠖⠒⠉⠀⠀⠀⠀⠀⠀⢀⣠⣄⠂⠁⠀⡷⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⠐⠒⠂⡖⢩⠒⠒⠒⠒⠒⠂⠒⠢⢍⡀⠀⠀⠉⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠑⢄⡀⣠⠊⠀⢸⠤⠤⠤⠤⠤⠤⠀⠒⠂⠉⠉⠐⠢⠄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⠄⡀⡤⣸⠀⠀⢸⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⢸⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠑⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠉⠀⠒⠒⡶⠀⠰⠤⠀⠤⠠⠤⠤⠸⠖⠒⠒⠢⢄⠀⠀⠈⠒⠤⠤⠤⠄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠶⢀⠈⠉⠃⠀⠀⠍⠉⠒⢒⠦⠒⠘⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠘⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠢⢄⡀⠀⠈⠑⡇⠀⠀⢰⠒⠉⣀⠤⠐⠊⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⣀⣀⣀⣈⣑⣤⣤⠇⠀⠀⢸⣊⣉⣀⣀⣀⣀⣀⣈⡣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠒⠀⢀⡴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠫⢭⡽⣆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠰⠀⠖⠀⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⢠⣟⠀⡺⣀⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⢉⡆⢘⣿⡀⠀⠀⠈⠀⡠⠛⠃
⠀⠀⠀⠀⠀⢀⠀⠃⠣⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡹⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠼⠒⠹⠐⠨⣇⠀⠀⠀⡜⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢦⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⢠⡞⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠏⡃⠏⡁⠾⡄⠀⢰⠇⠰⣶⣤⢴⣤⣿⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠰⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⣙⣢⣙⣻⠃⠀⢸⠀⠀⣠⣿⣶⣿⢿⢋⡄
⠀⠀⠀⠀⠀⠀⠢⣀⣤⠶⠊⠁⠀⠀⡼⠀⠇⡾⠀⠀⠀⠀⠀⠀⠈⠛⢦⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡟⡟⠯⠀⠀⠆⠀⠰⡿⣿⣟⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠞⢀⣼⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⢀⠀⠀⠀⠈⣿⣿⣿⣿⣛⡁
⠀⠀⠀⠀⣤⣤⠤⠤⠴⠶⠶⠛⣁⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈⢹⣾⣿⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠒⠉⠉⠀⠀⣀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠖⠚⠟⠛⠉⠉⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⢰⠋⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠞⠁⠀⠀⠀⠀⠀⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣬⣥⠜⢻
⠀⠀⠀⠀⡴⢋⡴⠂⠀⠈⠑⠄⠀⠈⠓⠦⣄⣀⣀⣐⣀⣀⣀⣀⣠⣤⡤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢄⢘⡉⠺⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢻
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣀⣀⣐⣠⣄⡶⠦⣄⠀⠀⠀⠈⠀⠀⠀⠈⠋⠀⢹⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⣿
⠀⠀⠀⢸⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡞⠁⠀⠙⡇⠀⢹⡆⢹⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⡻
⠀⠀⠀⠀⢳⣜⠦⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠁⠀⠀⠀⡇⠀⣸⢃⣿⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠠⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⢀⠃⠠⣯⠞⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢰⢣⡇⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠉⢉⠁⠂⠤⠀⠀⠀⠄⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠤⠔⣣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠃⠀⠀⠃⣠⠖⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡄⠀⠀⠀⠉⠛⠛⠓⠋⠁⢀⠠⡔⠒⢤⢀⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣠⡤⠶⢤⣄⣀⣠⠼⠃⠀⠀⠀⠀⢠⠁⠀⠀⠀⠤⠤⢍⡉⢩⡄⣾⣹⠀⣠⠖⠀⠀⠀⠀⠀⡰⡄⣨⢃⠫⣜⣻⣍⠛⢞⣷⣦⡀
⠀⠀⠀⡄⠁⠀⠀⠀⣿⠮⣤⣄⠀⢀⣀⣀⣀⡀⣀⣀⠀⠀⣀⠀⠀⠀⠀⢀⣠⡶⠛⠑⣤⠶⠿⣷⠶⠶⣤⡀⠀⠀⠀⠀⢽⣤⡁⠋⡐⡛⢉⣿⣾⣻⡙⢹⡇
⠀⠀⠀⣱⠄⠀⠀⠀⢿⡀⠁⢉⠓⠋⣀⣹⡅⠘⠋⠹⠓⢾⡏⠸⠀⠀⢠⣾⠟⠀⠀⠜⠁⠀⠀⠈⠃⠀⠈⠻⡆⠀⠀⠀⠐⠛⠑⠆⠈⠒⠲⢿⣷⣤⣅⣾⠇
⠀⠀⠘⠇⢶⠀⠀⠀⠘⣧⡀⠀⠀⠸⡏⠉⠐⠀⠀⠁⢈⡼⠁⠆⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡼⢂⡼⠋⠉⠁
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠈⠻⣦⣴⡋⠁⠀⠀⠀⣀⡴⠋⠀⠎⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡜⢰⢃⠊
⠀⠀⠀⠀⠉⠒⠤⢤⡀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⡴⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢰⠃⡜⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠃⠀⠀⠀⠀⠐⠒⠂⠐⠢⠒⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⢀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠮⢤⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠖⠒⠛⠋⠉⠛⠟⠏⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠛⠉⠁⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠄⢠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⠀⢀⠀⠐⠀⠠⠀⠀⠀⣀⣀⣀⣀⠀⠀⠁⣰⠃⠀⢉⣳
⠀⠀⠀⠀⠀⠀⢠⠖⢿⣯⣰⣀⣦⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠉⠁⠀⠀⢀⣀⣰⠃⠀⠐⢿⣅
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣀⣀⣠⡤⠤⠤⠴⠖⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠋⢀⣤⣤⣤⡿
⠀⠀⠀⠀⢀⡴⠞⠉⠋⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡖⠲⠦⠀⠒⠒⢾⢦⢷⠀⢹⡆
⠀⠀⠀⠀⠸⠷⣾⠇⢀⣴⡚⢧⠀⡠⠊⠙⡄⣠⠞⠱⠶⠉⠓⠊⣃⣐⣤⣖⡠⠗⣦⠠⠋⠀⡼
⠀⠀⠀⠀⠀⠀⠿⠞⢁⣷⠤⠬⣭⣴⣂⣀⣈⣓⣒⣈⣭⠭⠭⠟⠓⢋⣉⣡⠤⠚⣷⡜⠀⡼⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⠶⠤⠤⠤⠤⢤⣤⣴⠶⠒⠒⣒⣿⣯⣉⣉⣀⣀⣀⣠⢾⠄⣼⠃
⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⣀⣤⡶⣛⢁⠙⡛⣛⣛⣉⣁⣀⣀⣩⣭⣽⣫⠴⠋⣰⠃
⠀⠀⠀⠀⠀⠘⠳⢶⣶⠖⡛⠿⠯⠤⠭⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠚⠋⠉⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣇⢄⠀⠐⠀⠐⠐⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⠀⢤
⠀⠰⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⢀⠀⠠
⠀⡀⣀⡀⠀⣠⡐⠈⠁⠰⠀⠘⠀⠀⠀⠀⠁⠀⠀⠋⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣀⢀⣤⠀⠂⣠⠀⠰
⠀⠀⠘⠁⠀⠈⢠⠀⢀⣄⠆⠀⠀⢠⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠘⠇⠂⠀⠀⠀⠀⠀⠉⠀⠀⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠈⠡⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠁⠀⠀⠶⠀⠀⠀⠀⠀⠁⢰⡄⠀⣤⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠾⠂
⠀⠀⠰⡦⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠠⠤⡀
⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠠⠆⠀⠿⠰⠆⠀⠖⠠⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⡀⠁⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⠃

⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣤⡀⠀⠐⠂⠀⠀⠀⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢰⡦⠀⠀⠆⠰⠆⠀⠀⠀⠂⠀⠤⠀⠒⠀⢰⣲⣆
⠀⠀⠉⠁⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠛⠀⠀⠀⠾⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠐⠖⠀⠐⠀⠖⠀⠂⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣇⠀⠀⠀⠀⠀⠁⠀⣠⡀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⢀⠀⣀⡀⠀⠀⢀⠀⠀⠈⢁⡀⠀⣅⠀⠤⠀⠉⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⣀⠀⠀⡤⡌
⠀⠠⠀⠄⠀⠀⠂⠠⠄⠀⠀⠀⠀⠀⠄⠀⠠⠀⠚⠑⠶⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠦⠀⠰⠀⠀⢩⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠠⠄⠀⠠⢀⡀⠀⠂⠀⠀⠀⠴⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠘⠂⢀⠀⢀⡀⠀⠀⠀⠋⠀⠓⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⢀⡀⠘⠓⠀⠀⠀⠀⠀⠚
⠀⣀⡀⠈⠀⠀⠀⠈⠃⢀⡀⠀⠃⠀⠀⠀⠀⠀⢀⡀⢹⠀⢀⠀⠀⠀⠀⠁⠈⠅⠀⡁⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠉⠀⠀⠀⠐⠀⢀⠀⠀⠁⠀⠁⠀⠈⠀⠁⠀⡀⠀⢬⠀⠈⢀⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠴⠀⠀⠆⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠃⠀⠰⠆⠀⠀⠀⠀⠐⠢⠀⠀⠀⠶⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⣿⠿⠟⠛⠉⠛⣥⡽⠟⠻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⢠⣾⣿⣿⡿⣅⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⡉⠻⣿⣿⣦
⠀⠀⣴⣿⣿⡿⠋⠀⠈⠻⣄⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠉⠒⠛⣿⣿⣷⡀
⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣀⣭⡿⠛⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⣿⣿⢻⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⡇⠀⠀⠀⢀⣤⡶⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣶⣦⣤⣤⣀⣀
⣾⣿⡣⡄⠛⠷⢦⣀⡀⠀⠀⠀⠈⢷⡄⣸⠇⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⣷⣤
⣿⣿⡇⠃⠀⠀⠀⠉⠉⠛⠦⣤⣀⣸⠿⣿⣷⣍⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⡄⠛⢋⠀⠀⠉⠉⠛⠛⠿⠿⢿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⠿⠿⠟⠛⠉⠁
⢸⣿⣿⣇⠀⠀⢀⡤⠖⠉⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢿⣿⡿⠷⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠄⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣠⠖⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠶⠄⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡿⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠛⠛⠂⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠛⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡉⠀⠲⠶⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠰⠶⠖⠂⣠⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣤⣄⣀⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣠⣤⣴⣶⣾⣿⣿⠿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢀⣀⢀⣀⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠒⠑⡀⠀⠀⠀⠀⠀⠀⢣⢣
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡔⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⣉⠀⠐⠒⠒⠂⠀⣉⠆⠀⠀⠀⠀⠠⠐⠣⠷⠐⢄
⠀⠀⠀⠀⢀⠇⠀⠀⠤⠀⠀⠰⡀⠀⠀⠀⡠⠠⡏⠁⠀⠀⠀⠀⠸⠀⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⢀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠁⢤⠢⢀⣄⡠⢠⡀⡱⠀⠀⠀⡅⠁⠘⠒⠒⠒⠒⠒⠃⠁⠸⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢣⠌⠉⠉⠉⠉⠁⢤⠃
⠀⠀⠀⠀⠀⠘⡈⠁⠀⠈⢀⠇⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⡄⠀⢀⣎⠆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢐⠀⠐⠀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⢩⡀⠤⢈⡈⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⣀⠔⠓⠒⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡤⠒⠁⠈⠒⡄⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⡀
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠑⢢⠀⠀⠀⠀⠀⢀⡄⠀⠐⠒⠐⠀⣤⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡀⡔⠀⠀⠀⠀⠀⡀⢠⠁⠀⠀⠀⠠⠄⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⡤⠂⠁⠀⠑⢂
⠀⠀⠀⠀⠎⠁⠀⢠⡸⠰⠈⡗⢄⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⡂⢄⠀⠀⠀⠀⢀⠄⠚⠠⠔⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢠⠀⠂⠀⠀⠂⢀⠟⣠⠀⠀⠰⠁⠀⠀⠆⠀⠀⠰⠀⠀⠉⢢⠀⠀⠀⠀⠠⠀⠀⠠⠀⢸⢈⠆⠆⠀⠀⠔⠁⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡞⠉⠀⠀⠀⠈⠄⣀⣀⠤⣀⠀⢀⡠⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠊⠀⠀⠀⢆⡜⠢⠂⠤⠀⠀⠠⠈⠒⠑⠴⠁
⠀⠀⠀⠀⠀⠈⢢⠤⠤⡜⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠀⠂⠒⠂⠀⠀⠤⠄⠐⠁
⠀⠀⠀⠀⠀⠀⢑⡄⢤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠀⠒⠊⠁⠀⠀⠀⠀⠀⠈⠑⠒⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠣⠒⠪⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⡀
⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⢰⡴⣰⠁⠀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⢈⣇⡄⠀⠀⠀⠀⣀⡀⡼⠟⣹⠀⠘⡍⢡⠔⠁⢈⠇
⠀⠀⢀⡄⣻⡄⠀⠀⠀⠀⢣⡘⢰⠋⡁⠐⡛⢻⣾⠛⠚⡷
⠀⠀⠙⠢⠬⠷⡏⢳⣀⠸⣍⣻⢸⠎⣹⠀⡜⡱⡙⠢⡉
⠀⠀⢀⡤⠴⡞⢔⢾⡉⣧⣄⠹⠀⡋⠀⢸⢀⠧⣨⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠎⠢⡀
⠀⠀⠀⠢⡰⣜⣚⣦⡑⢧⢠⠀⠣⡰⡀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠁⠈⠉⠁⠄⠀⠘⣄
⠀⠀⠀⠀⠈⢙⣿⡢⣈⡉⡗⢢⣄⣈⢺⠮⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠸⠀⠀⠀⢸⠑⠢⠀⡀
⠀⠀⠀⠀⠀⠯⠤⠕⣢⣍⡛⠰⠧⠭⣍⠀⠀⡈⠦⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⡠⠤⠥⠤⢇⣀⠴⠉⢾⠀⠀⠀⢨⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠧⠔⠋⠉⠉⠉⠙⡔⠁⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⡄⢰⠀⠀⠀⡀⠀⢀⠏⠀⠀⣆⢀⠔⠁⠀⠀⠉⠱⣀⣀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢘⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡐⠐⠊⢀⣂⡆⠀⠀⡰⠁⢠⠏⠀⠀⠀⡟⠛⠤⣀⢀⠄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣾⣿⠁⢠⣀⣤⡶⠋⠀⠀⠀⠀⢀⡷⡀⠀⠀⢠⠦⠔⣁⠀⢠⠋⡴⠀⠀⡴⠁⠠⠋⠀⠀⠀⠰⠃⠀⠀⠀⢃⢀⠤⠊⠀⠀⠀⡆
⠀⠀⠀⠤⢤⣦⣭⣔⡲⠦⢌⣿⣿⠔⣠⠲⡎⠀⠀⠀⢀⣀⡤⠞⠁⢣⠀⠀⣸⡀⡨⠀⠉⠁⡴⠁⡠⠊⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⡰⢣
⠀⠀⡠⠚⠧⢄⠢⠥⡂⠐⢦⠹⢋⡤⠖⡉⠛⠷⠒⠢⡀⠀⠀⠀⠀⠀⢣⣀⡇⠈⠀⠀⠀⠜⠁⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠊⠀⠀⠱
⠀⡔⠁⠀⡄⠀⠀⠀⠈⠑⠒⠚⠃⠐⠊⠀⠀⠀⠀⠀⠈⢢⡴⠂⠀⠀⠈⠟⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠁
⠘⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣠⠴⠊⣄⣤⣶⠾⠛⠉⠀⠀⠒⠒⠒⠛⠛⠒⠖⠂⠤⠄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⡄⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠸⡥⠴⠒⠒⠒⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣈⠒⢄⠀⠀⠀⢀⣀⠀⢸
⢃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢷⣄⢀⡤⠂⢡⡰⠂⠈⢉⣒⠒⠒⠒⠂⠉⠀⠈⠑⠂⠐⠂⡕⠋⠀⠈⠣⢄⠜⠁⠀⠉⠁
⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢏⠀⠀⢀⣑⣄⠀⠀⢇⠀⠀⠀⠀⠉⠑⢂⣄⡀⠀⠀⢀⠤⢄⣀⠇
⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠑⠒⠁⠀⠀⠑⢄⠈⣄⡀⠀⢀⡠⠂⠁⠀⠉⠢⠠⠊
⣀⣀⣀⣀⣈⣁⣂⣶⣤⣤⣤⣤⣤⣤⣤⣶⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣧⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⢀⣠⠤⣤⣤⠶⣶⠶⠖⢾⠀⡷⠶⠶⢶⠶⢤⣤⠤⣄⡀
⠀⠀⠀⢰⣅⠐⢏⣀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⢀⡩⠆⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢀⡀
⠀⠀⠀⠈⡗⠭⢔⣓⣛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠛⠛⣛⣛⡣⠭⢒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠂⢀⠈⠳⢄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠈⠁⠀⡀⠳⡀
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⠋⠀⢀⡀⠉⠀⠹⡄
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡾⠉⢻⡄⠀⠀⠀⠠⡾⠉⢿⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⠀⠁⠀⠶⠀⠈⠀⠀⠤⠀⡟⢆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⢾⡁⣠⠑⠒⠦⣤⠖⠒⠒⡒⠤⠤⠚⢒⡜⠑⢆
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⣿⡏⢹⠇⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡌⠀⠑⠤⣄⣀⠀⠀⠀⠀⢈⣁⡠⠤⠒⠁⠐⠂⠈⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⣀⠧⢤⡀⠀⠀⠀⠀⠀⠱⣀⠙⠀⠀⠰⠉⠉⠉⠩⠄⠀⠠⠂⠀⠚⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⣙⡱⠖⠢⡜⠶⠾⠿⠏⣉⣓⠻⢅⣒⠒⠒⠚⠉⠙⠐⠒⠒⠒⠒⠒⠙⠉⡸
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣵⠞⠋⣉⣤⠴⠶⠾⠿⠿⢿⣯⣕⡒⠻⠶⢧⠱⡌⠱⠦⠀⠀⠀⠠⠶⠈⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⢁⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢕⢄⠀⢹⡄⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⡟⠁⠀⠀⠀⠤⠄⠀⠀⠀⠤⢄⠀⠀⠀⠳⡱⣼⡇⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠠⡏⡿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠈⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣜⠜⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠊⠁⠀⢠⢂⡵⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠋⠀⠀⠀⢰⡷⠋⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣨⣧⡈⠐⠒⢤⣀⣀⠀⠀⠀⠀⢀⡄⢈⠃⣒⡆⠀⠀⠀⠀⠀⠈⠗⡦⠤⣤⣤⡤⣤⣠⣤⣤⣤⣤⠤⢤⣤⠴⠶⠚⠓⢉⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠋⠉⠙⣯⡑⠒⠂⠉⡔⠁⠓⠙⠉⠀⠀⠀⠀⠀⠀⠘⠦⢤⢤⣤⠴⠶⠛⠦⢄⣀⣤⡤⠴⠚⠛⠓⠶⠶⠛⢉⣀
⠀⠀⠀⠀⠀⠀⣤⣌⡀⠀⣀⠀⠀⢀⠀⠘⠋⠈⠉⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠂⣩⠀⢄⠀⡄⠀⠀⠀⢀⢀⠀⠀⠀⣡⢤
⠀⠀⠀⠀⠀⠀⢸⠛⠻⢯⡄⢁⡀⠠⣀⠀⣀⣠⣄⡀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠡⠴⠞⠢⠤⠤⠶⠃⠁⠠⠴⠋⠒⠂⠒⠂⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢄⠀⠉⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⢻⣼⣧⣏⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⢀⣤⠾⠛⠿⣿⣥⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣋⡥⠶⠶⣶⠶⢤⣉⠻⢷⣄
⠀⠀⠀⠀⣼⠛⢿⡄⣠⣴⠟⠁⠀⠀⠀⢻⣷⡀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⣿⠉⠀⠀⠀⡿⠀⠀⢈⣷⣌⠻⣷⡀
⠀⠀⠀⠀⢹⣤⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⠋⠉⠳⣦⡀⠀⡇⢀⡴⠛⠁⠹⡆⢻⣷
⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠈⢿⣤⣤⣤⣤⣤⣤⣴⣾⡟⢸⣧⣀⣀⣀⣀⣙⣶⣿⣏⣀⣀⣤⣤⣿⢸⣿⡄
⠀⠀⠀⠀⠀⢸⠋⣇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣿⣆⠈⣿⡍⠉⠀⠀⠀⠀⠸⡇⢸⡏⠁⠀⠀⢀⡴⢻⡏⢷⣄⠀⠀⢀⣿⢸⣿⠁
⠀⠀⠀⠀⠀⣾⠀⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣬⣿⣄⣘⣷⡀⠀⠀⠀⠀⠀⢿⡄⢻⣄⣤⣾⣭⣴⣾⡇⠀⠙⣧⣀⣼⢣⣿⡟
⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠈⠹⣿⡟⠻⣿⠿⠿⠶⠾⠿⠿⢿⡟⠻⣿⡉⠁⠀⣾⡇⠀⣀⣤⠟⣱⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠸⣷⡀⢹⣇⠀⠀⠀⠀⠀⢀⣙⣶⣤⣙⡛⢶⣿⣛⣛⣩⣵⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣴⠛⠛⠿⢷⣄⢻⡷⠷⢒⣿⠿⠛⠛⠛⢛⣻⢿⢻⣿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣰⢿⡾⣣⡴⢾⣿⠶⠶⣴⠖⠋⠁⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣄⣀⣟⠘⡿⠋⠀⠀⠙⢷⣄⣸⣄⣤⣤⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⠿⠃⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠻⢷⡀⣀⣠⣤⣄⣈⡉⣛⣛⣿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠦
⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠈⠻⣆⠀⠀⠙⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢡⠈⠑⠢⠤
⠀⠀⠀⠀⢀⡀⠀⠀⠙⢿⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⠘⠘
⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⡒⡒⠒⠒⠶⠶⠶⠛⢷⡯
⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡤⠋⠁⠉⠀⢉⠉⠉⠉⡗⣿
⠀⠀⠀⠀⠉⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠫⠀⠀⠀⠰⢀⣸⡶⠁⢸⡇⣿
⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠺⣁⣰⠀⠒⠀⣸⠈⠻⣥⠀⢸⣧⢸
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⣮⡤⢫⢈⡫⣠⡇⣽⣻⢾⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠉⢂⣼⣿⡿⣮⠟⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣝⣳⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣤⠟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡦
⢀⢴⠋⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⢸⡘⣆⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠐⢤⣰⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⣶⣄
⠀⢳⡘⢦⠀⠰⡙⢦⡀⠀⠀⠀⠀⠀⣤⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠛⠃
⠀⠀⠱⣌⠳⡄⠘⠶⠛⢦⣄⡑⠲⠟⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢷⡀⠀⠀⠹⡷
⠀⠀⠀⠈⠣⣘⠢⣄⠀⠀⠀⠹⡏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠛⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠈⠓⢬⣓⡶⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⠒⠲⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣥⠶⣚⢩⠟⠛⠻⠷⣶⢤⣉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⠀⠏⢹⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⠟⠁⠀⠈⢿⠀⠀⠀⠀⢸⠖⠙⠳⣦⡈⢦⡀
⠀⠀⠀⠀⢀⡴⠋⠀⣠⠖⠉⠙⠲⠧⣾⠓⢳⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠃⠀⠀⠀⠀⠘⡄⠀⠀⢠⡏⠀⠀⠀⠘⣷⡀⢳⡀
⠀⠀⢀⡴⠋⠀⡠⠚⠁⠀⠀⠀⣀⣤⠼⡀⠈⣷⠒⠒⠒⠒⠒⠒⣾⢀⡟⢄⣀⠀⠀⠀⠀⠙⣄⣠⡟⠀⠀⠀⠀⠀⣸⢣⠈⡇
⠀⠀⢸⡀⡠⠊⠀⠀⠀⢠⠖⠋⠁⠀⠀⢧⠀⢸⡀⠀⠀⠀⠀⠀⢻⠘⡇⢀⡼⠛⠛⠉⠙⠓⢻⣉⡿⠒⠲⠶⠾⣍⡁⢸⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡄⠈⣇⠀⠀⠀⠀⠀⠈⢇⠱⡸⡄⠀⠀⠀⠀⢀⣠⡟⢷⠀⠀⠀⠀⠀⣧⡏⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠓⠦⣤⣀⣀⣧⡀⢸⡀⠀⠀⠀⠀⠀⢈⣳⣝⣷⣤⠤⠖⠚⠉⢠⠇⠘⣆⠀⠀⠀⣠⠟⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⠀⠈⢹⡍⠉⣏⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⠦⣴⠞⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣇⠀⠀⠀⠀⠀⣇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡧⠴⠶⢗⣊⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡀⠠⠐⠒⠚⢻⡀⠀⣏⠉⠉⠉⠉⠉⠀⠐⠐⠒⠠⢄⡇⡼⠓⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠄⠀⠀⠀⠀⣇⠀⢹⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢺⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢡⠀⠀⠀⠈⠉⢹⡉⠉⣏⠉⠉⠉⠉⠁⠀⠀⠀⠀⡆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⡀⠀⠀⠀⠀⠀⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢰⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠇⠀⠀⠀⠀⠀⢸⡄⠀⣧⠀⠀⠀⠀⠀⠀⠀⡎⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⢧⠀⢸⡄⠀⠀⠀⠀⠀⢰⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⢧⠀⠀⠀⠀⠀⠎⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⠆⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⢀⠀⠀⠀⠀⢉⠁⣀⠠⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠦⠤⠬⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠐⠒⠒⠒⠛⠛⠓⠒⠒⠒⠶⠦⢭⣍⢓⣲⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠄⠂⠁⠀⠀⠀⠀⠴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠁⢸⣷⣦⣍⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠔⠈⠀⠀⣀⠀⠀⠰⠃⠁⠀⢸⠳⣀⣀⣀⣀⣀⣠⠶⣄⠄⠈⠀⠀⢠⠸⡇⢳⠈⠙⠶⣄⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⣠⠞⡹⠆⠁⠀⠀⠀⠓⢸⠀⢹⠉⠉⣩⢟⠁⠀⠀⣠⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠈⠳⣦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⡞⠅⠈⠀⠐⠚⠀⠀⠀⠀⠘⡇⢸⠀⡴⠋⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠈⠳⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⣄⡀⠀⠀⠀⠀⠂⠀⠀⠀⡇⣾⠦⡀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠶⠀⢻⢹⣆⠀⠀⠀⠀⠀⠀⠙⣆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣸⠓⢤⡀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢧⢸⣄⢈⡓⢄⠀⠀⠀⢰⣄⠀⠀⠀⠀⢸⡀⡇⠳⡀⠀⠀⠀⠀⠀⠸⣆⠀⢹⡄
⠀⠀⠀⠀⠀⢰⠀⡀⠀⠀⠀⢴⠙⠦⣰⠍⠢⣄⠀⠀⠘⠄⠀⠀⠀⠀⢸⠀⣏⢑⢅⡀⠙⠢⣀⠀⠀⠀⠀⡀⠀⠈⡇⢹⠀⠱⠀⠀⠀⠀⠀⠀⣿⡀⠀⣷
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢳⠆⣼⠲⢄⠈⢑⢤⡀⠀⠀⢠⠀⠀⠘⡄⢩⠙⠦⡙⠢⣔⠈⠳⢄⠀⠘⠁⠀⠀⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢷⠈⠑⢤⡙⠦⡀⠈⠲⣄⠀⠀⠀⠀⡇⢸⠦⡀⠈⠲⣬⠳⣦⡀⠙⠢⣀⠀⠄⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⣙⠢⡉⣷⢤⡀⠑⢦⡐⠂⢸⢸⠀⠈⠲⣞⠀⢑⢾⣌⠓⢤⡈⠓⢄⡈⣾⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⠦⡀⠈⠛⢦⣌⠓⣄⡈⠳⢬⣿⠀⠀⠀⠈⠑⢤⡀⣌⠛⢆⠘⢳⣄⡙⢿⣦⠁⠀⠀⠀⠀⠀⣸⠏⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠘⣄⠐⡀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠳⢄⠲⠉⠲⣄⠹⡗⢤⣟⢧⠀⠀⠀⠀⠀⠉⠢⢄⠀⠙⠦⢗⣌⡛⣸⡆⠀⠀⠀⠀⣴⠏⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢢⡀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣙⠢⣄⠈⢙⠻⣆⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣐⠂⠱⠢⣿⠀⠀⠀⣠⡾⠃⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠷⠮⣄⠉⠙⣿⣀⣀⣀⡀⠤⠄⠂⠀⠀⠀⠀⠙⠂⠘⣻⠀⣠⡴⠋⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠞⠉⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠑⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣉⡉⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠚⠋⣉⣁⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉





⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠄⠀⣀⡀⠀⢀⠀⢀⢾⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⢻⡶⣿⣿⠟⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⢿⣏⡀⠀⣿⣧⠀⢸⠀⣼⠀⢠⠅⢀⣴⣶⣶⣄⠺⠿⠿⢋⣿⡆⠀⠀⠘⠄⠀⠀⠈⠃⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹⠧⣿⡆⠀⣿⢸⠀⢺⢀⡟⢀⡞⢰⣾⣿⣿⣿⣿⣷⣤⢠⣽⡧⣴⣷⠈⠐⠀⠈⠁⠀⢠⡸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⡁⠀⠀⣿⣼⣶⣿⣾⢃⡞⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⡿⠈⡟⣡⡾⠃⠀⠀⠀⠀⢼⣿⠄⢸⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠂⢠⣿⣿⣽⣿⣿⡿⠀⠀⠘⢿⣿⣿⣿⡿⠟⠁⠀⠇⠀⠘⢉⣵⣿⣿⣶⣶⣦⡈⠁⢠⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⣿⣿⠀⢀⠀⢀⢀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣿⣷⠈⠉⣡⣄⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⣿⢀⡀⠈⡉⠛⠛⠋⠀⢀⠇⠀⢀⠀⢉⢉⢀⡏⢠⠏⡜⠀⢀⢀⠋⢻⣷⡾⢛⣿⣿⣿⣆⣸⣿⠙⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣍⡈⣧⠀⡇⢠⠀⣧⠀⢸⠀⠄⠇⠀⠇⠸⠘⠀⠘⠀⠇⠇⠾⢸⠀⢸⣿⣷⠟⠉⠺⠿⠋⠋⠰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡗⠈⠿⠿⠷⠸⣆⢷⠘⣧⠸⣆⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⢀⠖⢀⡟⢠⣿⣿⠁⠀⠀⡀⠠⠀⠂⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣤⣤⠀⢠⡀⢹⣾⣧⠸⣇⢻⡄⠀⢠⡁⢠⠀⣠⠀⢀⢀⡏⠈⠀⢊⣤⣜⠛⠏⠀⠀⠄⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢋⣴⡆⠉⠚⣿⣿⣇⢻⣎⣷⡀⠸⡃⠻⠀⠓⠀⠳⠘⠀⠀⢰⣿⣿⣿⣿⣷⠀⢀⡀⠀⠀⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇⠀⣀⡘⣷⢻⡞⣿⡹⣧⠀⠁⢠⣷⡄⣠⣴⣷⠀⠀⢺⣿⣿⣿⣿⣿⠑⠿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⢿⡟⢿⣞⣿⣿⣧⠹⣧⢀⣸⣿⣿⣿⣿⣿⢴⡆⣾⣿⣿⣿⠟⣁⣀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⣶⡄⠸⠿⠻⠛⠛⠀⣀⣿⣿⣿⡿⣿⣿⣿⣾⡇⢿⣿⠟⣡⣾⣿⣿⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⣿⣿⡿⠛⣤⣶⣦⠐⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠈⢿⡿⠛⠁⣀⣘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣦⡀⠞⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡄⠀⢰⡏⠉⠁⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠊⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣇⠀⢸⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⣿⠀⢸⠓⠒⠀⠀⠀⡇⠀⠀⠀⠀⡕⠀⠀⠀⠀⢿⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠋⠀⠘⠶⠶⠖⠀⠀⠳⠒⠒⠀⠀⠳⠒⠲⠀⠀⠈⠳⠲⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡴⠛⠛⠦⣄⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⡤⠖⠒⠲⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⠀⠀⠀⠙⢦⡀⠙⠦⠤⠒⠒⠒⠒⠦⠤⠊⢁⠴⠊⠁⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢃⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⡄⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢠⠃⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢆⣈⣹⣏⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠝⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠈⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⣰⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠋⠁⠀⠐⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠠⠀⠀⢀⡀⢀⣀⣀⠀⠀⢠⣤⣀⠀⠀⠠⡄⠀⠀⠀⣼⡃⠀⠄⠀⠀⠀⠀⣀⡀⠀⠀⣹⣉
⠀⠀⠀⠀⡐⢊⠤⡂⢀⡀⠀⠑⠄⠀⠙⢿⡆⠀⠀⣠⡯⢭⡭⣿⡄⣼⡆⠁⠀⠀⠸⣷⢈⣦⠀⢉⠉
⠀⠀⢠⠎⠘⠀⠂⠀⠁⠨⠦⠕⣚⡄⠠⠄⠀⠀⠀⠀⢸⠀⢸⡏⠀⢸⠀⠀⠀⠀⠀⣷⠈⢩⡑⣄⠉
⠀⠀⠈⠲⡒⠶⠶⠶⠖⠶⠖⠒⣤⠕⠀⠀⣆⣤⡀⠀⢸⡤⢺⡇⣤⣸⠀⠀⠀⠀⠃⢻⠶⠸⠏⠛⠢⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠁⠈⠀⡆⠀⠀⠙⠚⠋⠀⢸⡔⡾⢣⣛⣼⠀⠀⠀⠀⠀⠘⣇⣁⡶⡦⣛⠝
⠀⠀⢀⢀⣀⠀⠀⣀⠤⣔⡦⢄⠀⠋⠁⠀⠀⠀⠀⠆⠘⠬⠱⠿⠇⠘⠀⠀⠋⠀⠀⠀⠓⠓⠓⠉⠁⢀
⠀⠀⣎⠀⠐⠀⡎⢤⡿⢭⣱⣏⠱⡄⠀⠀⠀⣄⠀⠀⡠⠒⠈⠉⠐⠢⡀⠀⠀⢀⣀⠀⠀⠀⣦⣆⢰⡌
⠀⠀⠀⠀⠀⠞⡶⠾⠷⢾⠧⡿⠶⠋⡇⠀⠀⠃⠀⣘⣀⣀⡀⢀⣀⣀⣈⣄⠀⠃⢤⡄⠀⣼⣟⣿⡟⣾
⠀⠀⠀⠀⠀⢀⠠⠇⠐⢸⠀⡄⡇⠀⠀⠠⣤⠀⠨⣤⣄⣠⠤⠀⠄⠠⢠⣖⠁⠀⠀⠁⠾⠿⠿⠽⠿⠿⡦
⠀⠀⠀⣀⡀⢈⠀⡆⠀⡀⠀⡇⡇⠀⠀⠠⠶⠄⢸⢂⠷⠽⠌⠶⢤⠶⣙⣚⡣⠀⠀⠀⢠⠀⣠⠀⡀⠀⡇
⠀⠀⠀⠙⠀⢘⢸⠇⠀⠇⠀⡇⣁⡔⠀⠀⠀⠀⠀⠡⣉⠍⠉⠉⠉⠈⡡⠃⠀⠀⠀⠂⠈⡄⠁⠋⠸⢠
⠀⠀⠀⠀⠀⢄⠉⠓⠂⠶⠧⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠉⠁⢀⣀⣀⠀⠀⢀⡀⠓⠀⠀⠤⢞
⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠒⠀⠀⠀⣤⠝⣷⠆⠀⠀⠸⡄⢵⣲⡁⢣⠀⠀⠀⠀⠉
⠀⠀⠀⠙⠀⠀⠀⠀⠀⢰⡷⠦⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠀⡠⠒⡏⠉⣉⠋⠁⠀⠁⠈⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠄⢠⣤⠤⢴⣿⡀⢀⠘⠀⠀⠀⠀⢀⡠⡀⠁⠀⡰⠁⡀⠻⡀⠀⠇⠀⠁⢀⣠⣤⣶⣤⢄⣙
⠀⠀⠀⠀⠀⡸⠙⣉⣉⣙⡋⠨⡀⠐⠀⠀⡄⠀⠑⠑⣠⠊⠐⠀⠀⢠⠃⡌⠀⠀⠀⢹⡯⣿⣟⣿⣧⡀⠉⠑⡆
⠀⠀⠀⠀⡜⣥⡀⠐⠂⠒⢒⣒⣮⢣⠀⠀⠋⠀⠀⢰⣉⣤⣴⣞⢋⠁⡜⠀⠀⠀⠀⠀⢿⣞⣿⣿⢻⣧⢀⡞
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢢⠀⠰⡢⠒⡆⠘⣻⣿⣿⣅⣦⡎⠀⠀⠀⣤⡄⠀⠘⣿⢾⣿⣦⠿⠟⠃
⠀⠀⠀⠀⠑⠬⠐⣒⠒⠒⠂⠤⠭⠊⠂⠀⠉⠊⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢔⢶⢦⣴⣦⡤⢄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡶⡶⢮⣿⣟⣼⣿⣯⡟⠺⣛⣌⣭⣭⣛⣿⠳⡦⢀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⠟⠋⠀⠀⠈⠊⠉⠉⣡⣴⣀⠉⠛⠛⠓⠛⠋⠱⣟⣛⢿⣄⣀
⠀⠀⠀⠀⠀⣀⣿⣟⡡⠎⠁⠀⠀⠀⠀⠀⠠⠄⢰⣧⣶⣜⣶⣮⣝⡀⠀⠀⠀⠈⠨⢿⠿⣿⣗
⠀⠀⠀⠀⢺⡯⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡿⣿⡟⣿⡿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢬⣿⣷⡀
⠀⠀⠀⠮⢺⡷⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠄⣤⠉⢢⠞⣷⠾⣿⢿⠷⣶⠂⠀⠀⠀⠀⠉⢻⡟⢿⠄
⠀⠀⢨⡟⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⣏⣤⣩⡴⠛⠛⡸⠘⠣⠀⢼⣿⣆⠀⠀⠀⠀⠀⠀⠘⣿⠀⡀
⠀⠠⣷⠸⣿⡧⢀⠀⠠⠀⠀⠀⢃⣰⡀⡚⢩⡀⣴⣲⣶⣶⡟⣦⣄⠲⣈⡍⠹⣇⡀⠐⠀⠀⠀⣼⣛⡷⡐
⠀⠰⣈⡁⢿⡇⠘⠂⢀⡀⠀⠔⠫⣅⠿⠷⠛⠿⠗⡒⡮⢷⡿⠹⠏⡓⠋⠈⣢⣜⢁⠀⡀⠀⠀⣽⢿⠱⢿
⠀⢺⣿⣭⢅⠀⠀⠀⠀⠉⢦⣄⠴⠛⢂⡸⠀⠠⢴⡄⠀⠠⡇⠀⠀⢠⡄⣰⣯⣾⡀⠀⠀⠀⠀⠘⣦⣧⣌
⠀⢸⠟⠳⢤⣦⠀⠀⠀⠀⢰⣨⡀⠰⢬⣶⣞⣄⠀⠀⠀⠀⠀⢀⡀⣰⣿⣝⡻⠁⡔⢂⠀⠀⠀⣰⢒⠚⡯
⠀⠸⣾⣓⣾⣿⣀⠀⠀⠑⢌⠘⣟⡂⣼⣯⣿⣷⠢⠀⠀⠀⠀⠈⢰⣿⣿⣿⡋⠀⠓⠈⡄⠀⢀⣿⣦⣻⡄
⠀⠀⢿⠁⣿⣏⠙⠿⣶⡒⣮⣳⣯⣿⣿⣿⣧⣿⣷⠀⠀⠀⣴⣦⣾⣽⣷⣿⠀⠀⠀⠀⡀⠀⢈⣿⣟⣿
⠀⠀⠸⠔⢿⣫⣄⠀⢸⣏⣽⣿⣿⣿⣿⡏⢻⣿⣿⠀⡜⠉⠟⢻⣿⣿⣿⢿⣀⣀⣤⣴⣶⡶⣿⠿⣛⡛⠂
⠀⠀⠀⠹⣷⣯⣿⣤⣤⣔⣛⣿⣿⣿⣿⠁⢈⣿⠁⡀⠀⠀⠀⠈⡿⠆⢹⣾⠿⢿⣿⣿⣿⣿⣷⣾⡀
⠀⠀⠀⠀⠻⢿⣿⡿⣿⡿⣽⣿⡏⠀⠈⠀⡾⠟⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠈⢿⣷⣿⠷⣼⣿⣆⡀⢀⢸⡇⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣘⣿⣿⣿⣧⠞⠋⠀⠀⠀⠀⠀⢘⣶⣿⡛⣀⡀⢀⣠⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠈⢩⡍⠛⢻⣟⠀⠀⠀⠀⢸⠛⢿⡿⠀⠛⡿⣿⣿⣿⣿⡿⠿⣿
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠸⠁⠈⠀⠃⠁⠀⠀⠀⣀⠆⠀⠀⣿⣇⠀⠀⡽⠛⠻⣿⠃⠀⢻⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣄⣀⠀⢅⠀⠀⠀⠀⢀⣀⣤⣾⣟⢢⡈⠀⠀⡰⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣀⠀⠀⠐⢌⡀⠀⠑⢀⡀⠀⠀⠀⠈⠉⠒⠶⠶⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠄⠀⠈⠳⣄⠀⠀⠉⠢⡀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⠫⣭⠐⠒⠤⣀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣀⡤⠁⠂⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠈⠳⠄⠀⠈⠑⢒⡐⠠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡆⠀⢀⠠⢂⡀⢀⡈⠉⡁⣀⡒⠢⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠤⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠉⠑⠢⣤⢤⣀⠚⠃⠬⠁⢈⠁⡂⠘⠈⡀⠀⡄⣠⣠⣤⠈⠑⢲⡖⠐⡐⠠⠤⠐⠒⠂⠒⠊⠉⠉⠀⠀⠀⡀⣤⡤⢂⡄
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣀⠉⠉⢙⠶⣶⣦⣤⣅⣀⣈⡁⠈⠁⠊⠏⠁⠀⠀⠈⢑⡦⠛⢠⣷⠁⡀⠀⡀⠀⢠⣠⡄⠀⣿⣡⣥⠒⠉
⠀⠀⠀⠀⠀⠀⠠⠓⠰⠶⡺⣻⣿⡉⢳⠀⣶⣹⣽⣿⠿⠟⠛⠉⠉⠉⠳⠲⣶⣶⣤⣬⣈⣘⣃⣘⣀⣚⣃⣘⣁⣤⣬⡷⠾⠋⠹⡿⠻⢆⡀
⠀⠀⠀⠀⠀⢰⡀⠀⠀⡀⠀⠉⠛⠓⠪⠴⢿⣏⣠⣤⣶⠺⠿⢿⡦⣌⠢⠀⢁⣷⣿⠿⠛⠿⠉⢍⡛⠿⢯⡻⠉⠉⠀⠁⣠⣴⣪⡥⠜⠃⠈⡆
⠀⠀⠀⠀⠀⢘⠿⠛⣖⠦⠈⢁⣤⡄⣐⣀⠀⠀⠩⢉⣙⣧⣀⠀⠀⠈⡱⢧⣷⣿⣿⣞⡛⣿⣶⣶⣮⠢⡀⠉⠀⠀⣠⠏⠋⣉⣁⣠⣤⡶⠾⡀
⠀⠀⢀⡤⠂⢡⠄⢶⣈⣦⠑⠋⠬⠅⣈⣉⣉⣻⣿⠖⠂⠀⠙⠛⢒⡈⠐⠋⠿⠉⠽⡢⠀⠀⢨⡹⠿⢷⡘⡀⢠⣾⠟⣙⣹⠿⣛⡭⠽⠂⠀⡇
⠀⠀⠀⠘⢄⣘⣒⠾⡵⠠⢗⣤⢄⡀⠺⣿⣷⣾⣧⣬⣧⣱⠶⠟⠓⠸⢿⣿⣿⣯⡹⠿⣶⣄⠄⡁⠀⢂⠇⡷⠋⡀⢀⠅⣡⠔⠚⣅⠙⣸⣦⡜
⠀⠀⠀⠀⠀⠀⠀⠸⠖⠉⠿⢿⢛⠦⣈⠏⢽⠏⡃⣼⠀⠴⣵⠶⢶⣶⣌⠴⠤⢼⣿⣶⣄⣉⡁⠀⠀⠀⠚⠛⠷⠚⣻⠴⢋⠐⢦⠙⣰⣟⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⡀⠐⠿⣿⠁⢀⣤⡞⠻⣿⠳⣀⠸⠐⠷⠿⢯⣓⠒⠠⠤⣀⣉⣉⣙⣛⣛⣂⣰⣶⣶⡏⠁⠀⢘⣣⣶⣿⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⣈⣲⡛⠛⠾⠀⠀⠀⠛⠓⠊⠿⣻⡹⠦⠝⠶⣖⣠⣤⠢⠽⣛⡉⣩⡼⣵⡿⠃⢀⣴⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⢽⡟⢷⡶⣦⣤⣀⡘⠟⠁⢰⢆⠀⠀⠀⣙⠻⠀⣀⣀⠵⣭⣘⣋⣤⣴⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠭⠝⢻⣶⣶⣿⣶⣶⣤⣤⣵⣿⣿⣿⣿⣿⡿⢟⡋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣻⣻⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⠿⠟⠛⠛⠻⠿⣿⣿⣿⣿⡇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⣿⣿⣿⣿⠿⠛⠋⠉⠉⠙⠛⢿⣿⣿⣿⣿⢸⣿⣿⣿⠏⠀⠀⠀⣀⡀⠀⠀⠈⢿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⠁⠀⠀⢀⣤⣄⠀⠀⠀⠹⣿⣿⣿⢸⣿⣿⡏⠀⠀⠀⣾⣿⣿⡆⠀⠀⠈⣿⣿⡇⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣧⣤⣤⣤⣿⣿⡿⠂⠀⠀⢀⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣠⣾⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⢸⣿⣿⣷⡀⠀⠀⠙⠿⠿⠁⠀⠀⢰⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿⣿⢸⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡇⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠈⠉⠉⣿⣿⣿
⠀⠀⠀⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⣶⣶⡶⣶⣦⡀⢰⣶⡆⢀⣴⣶⢶⣶⣄⠰⣶⣶⣶⣶⣶⠆⣶⣶⠀⠀⣶⣶⠀⣶⣶⡶⣶⣶⡄⠐⣶⣶⣶⣶⡶⢀⣴⡶⢶⣶⣄
⠀⠀⠀⠀⣹⣿⣦⣾⣿⠃⢸⣿⡇⢸⣿⡇⠀⠙⠛⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⣿⣿⣶⣾⣿⠇⢰⣿⣷⣶⣶⠆⠘⠿⣿⣶⣶⣄
⠀⠀⠀⠀⣻⣿⠉⠉⠀⠀⢸⣿⡇⠘⢿⣷⣴⣾⡿⠀⠀⢸⣿⡇⠀⠀⠹⣿⣶⣴⣿⠟⠀⣿⣿⠀⢻⣿⡄⠠⣿⣷⣶⣶⣦⠸⣿⣦⣬⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠛⠉⣉⡉⠉⠉⣉⡍⠉⠙⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣋⡁⠀⠀⠐⠟⠃⠀⠛⠀⣀⠀⠉⣥⣄⠘⠛⠁⠀⠀⢈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠉⠁⢀⡤⠀⢰⠇⠀⠀⠀⠙⠃⠀⠀⠀⢀⣀⠀⠀⠘⠛⠁⣤⡄⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣶⠀⠉⠁⠀⠀⠀⠛⠛⣀⠀⠛⠛⠀⠀⠙⠃⠠⣦⡄⠀⢠⣅⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⠘⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣍⣷⠀⠀⠀⠀⠀⠀⣾⣿⣍⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠀⢀⣀⣀⡀⠀⠻⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀
⠀⠀⣀⣀⡀⢤⡤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠑⢆
⠀⠈⠳⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡴⠁
⠀⠀⠀⠈⠑⢄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⢢⡀
⠀⠀⠀⠀⠀⡰⠓⢄⡘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠙⢦⡈⠙⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠯⠥⠤⠤⠤⠤⣄⣀⡀⣣
⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⢒⣒⡢⠶⠗⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠈⠉
⠀⠀⢀⣀⣀⣼⠈⠑⠲⠶⠤⠶⠾⠶⣄⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠖⠒⢻⡿⠥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠴⠒⠋⠀⣇
⠀⠀⠈⠓⠦⢌⡷⢤⡤⠤⠤⠤⠤⠤⠬⠷⠤⠤⠶⠖⠭⠥⠤⠤⠤⠤⠴⠋⠳⢤⣈⣱⣤⣄⣀⣤⣤⣒⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⡔⠁⠙⠢⣄
⠀⠀⠀⠀⠀⡜⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠈⠒⢤⣈⡑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣒⢒⣒⣒⣒⣒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣩⣜⣁⣀⣀⣀⡀⣀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠉⠉⠉⠉⠙⠻⢍⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠒⠒⠚⠉⠉⠉⡹⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⣀⣀⠤⠤⠖⠚⠛⠓⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠖⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⠟⠛⠛⠛⠷⢶⣤⡀
⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⣀⣀⣀⠀⠀⠀⠉⠻⣷⡄
⣿⣿⣏⠙⠛⠛⠻⠶⢶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣴⣾⡿⠿⠿⢿⣿⣶⣄⠀⠀⠈⢻⣆
⠻⣿⣷⣤⣤⣄⣀⣀⣀⠀⠀⠉⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣰⣿⣿⣇⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⢻⣇
⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⢷⣄⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣿⣿⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⣧⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣿⡄⠀⠈⣩⣿⣿⡿⠛⠋⠉⢹⣿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠹⣷⡀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⣷⠀⠘⣿⣷⣴⣿⣿⣿⣿⣿⣆⠀⣠⣿⡟⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⢷⣤⣴⡶⠟⠛⠉⠀⠀⠀⠀⠹⣧⣀⣸⣿⡿⠟⠋⠈⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠟⠛⠉⠀⠀⠀⠀⠸⣿⠋⠁⠀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣷⡀⠀⣙⣷⣤⣤⣶⠶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠶⠿⣶⡆⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⣀⣤⣾⣿⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢿⣷⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣿⣿⣿⣿⣿⡿⠿⠛⠉⢀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⠉⠉⢀⣀⣀⣤⣴⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠯⢥⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⠭⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣚⣛⣛⣛⠋⠉⢉⣿⣟⣛⣛⣛⣳⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠩⠉⡍⠉⠉⡍⢹⣿⠋⠉⠉⠉⢁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠀⠀⣾⡟⠀⠤⠈⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⢀⣤⠶⠶⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣿⠇⠀⢀⡴⢋⡔⠊⠉⡏⠑⢌⠻⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⡿⠀⢀⡞⡱⠁⠑⡄⢰⠃⢀⡴⢣⢹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠛⠃⠀⢸⠇⡧⠤⠤⢈⣾⣔⡁⠀⢘⢸⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⢀⡤⠊⡏⢣⡉⠉⡹⢸⠇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣛⣛⣛⣛⠛⠃⠈⣧⠹⡉⠀⢰⠃⠀⠱⡴⢣⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠳⣌⡒⠾⠤⠔⣪⡴⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⠁⠀⠀⣀⣀⠀⠀⠀⢸⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⣦⠀⠀⢤⠂⠀⠀⠀⠀⠀⠀⠘⠰⡊⠢⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠀⠀⠈⣢⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⢐⡂⠐⠒⠒⠀⠀⠃⠉⠀⣀⡆⠀⠀⡐⠁⢓⠱⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠆⠀⠀⠀⠀⣀⡀⠄⠊⠀⠱⠤⠊⠀⡠⠈⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⣀⣈⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢀⡀⠔⠈⠀⠀⠀⠠⢄⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡂⠀⠉⠀⠀⠀⠀⠀⢀⠐⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⢠⡄⢠⠤⠴⠀⠈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠉⠀⠂⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀⣤⣤⠀⢠⡀⠀⠀⠀⡄⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⠀⢠⡇⠀⠀⢠⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢻⠂⢸⡛⠉⠀⢸⡇⠀⠀⢸⡇⠀⠀⠘⣇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠉⠉⠀⠈⠉⠉⠁⠈⠛⠛⠃⠀⠈⠉⠁

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

⠀⠀⡀⠀⣀⠴⠚⠓⠒⠦⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⡏⠑⠄
⢠⡪⠃⠉⠓⣀⠀⠁⠠⠀⠉⠓⢄⠀⠀⢂⠀⡁⢠⠀⠀⠀⣀⠀⠀⠀⣄⣅⡄⡀⡈⢂⠀⢀⠀⠀⡀⠠⠄⠀⠀⠤⠔⠒⣀⠀⠈
⣼⢀⠀⠀⠀⠈⠴⣀⠀⠀⢀⠀⠌⢆⠀⠀⡄⠀⠀⠀⠠⠀⡉⠢⠤⢀⣀⣆⣀⡤⠤⠚⠀⠀⠰⠎⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⡀⢤
⣿⡞⠀⠀⠀⠀⠀⠻⢼⡾⠀⠀⠀⠀⠧⣀⠁⢀⡀⠚⢀⡀⠃⢀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⠖⠚⠋⠋⠈⠀⠀⠨⠀⠖⠀⠁
⠇⠙⢴⡄⠀⡀⠀⠈⠀⠀⠉⠻⠢⡄⠀⠉⠳⡄⠀⠘⠋⠊⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⣷⠁⠀⠀⣎⠈⠁⣀⡀⠠⠂⠀⠐⢐⣺
⠸⣄⠀⠈⠢⣬⣀⣀⣀⣀⡀⠒⠘⢌⡓⠆⣤⡛⣦⠀⠀⢈⢅⠢⡤⠯⢢⠔⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠀⠂⢁⣀⠤⠒⠂
⠀⠈⠑⠦⣄⣁⠉⠉⠁⠈⠉⠙⠒⣤⣌⡓⠤⣉⠛⢳⠀⠀⠈⠈⠀⠈⠀⠀⠀⣀⠼⠀⠀⠀⠃⡰⠃⣠⠤⠒⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣌⠉⡅⠀⠀⠀⢀⣀⡁⠈⠉⢉⢉⡦⠾⢀⣄⠀⢰⡀⠐⠉⠁⣀⣀⢀⠀⠀⠀⠀⠁⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⢠⠒⠏⠙⠲⢼⠰⠀⡴⢠⠄⠈⠉⠛⠛⠋⠁⠀⢘⣿⣥⣤⢅⠀⠈⠣⡌⠀⠈⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⢀⡀⡄⠂⢉⡁⠔
⠀⠀⢀⣀⣀⡄⠀⠀⠆⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⠀⠉⠉⠈⠙⠁⠃⠈⠁⠀⠀⠃⠂⠀⠀⠉⠀⠀⢀⣀⡤⠖⠒⠈⡀⠠⠀⠁
⠀⠘⢈⡙⠙⠈⣤⠉⠀⣱⣾⣗⣻⢿⣽⣿⣿⣏⣭⣯⣽⣯⣯⣩⣷⣦⣦⠤⣄⡀⠀⠀⠀⠀⠀⠀⢰⠋⠁⢀⠀⠔⠂⠀⠀⠀⠀⢀
⠀⠰⢊⡁⠺⠁⢀⣠⠾⠟⠛⠑⠛⠋⠉⠉⠉⠀⠀⠀⠀⠐⠀⠀⠉⠈⠉⠙⠛⠛⠠⠀⠀⠀⡀⠈⠳⠴⠘⠀⠀⠀⢀⣤⣴⣶⣯⣿⡇
⠀⠘⠟⠁⡰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠒⠤⢄⣀⣠⣾⣿⡿⠉⠛⢛⡿⠁
⠀⢀⡠⠊⢠⡄⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣶⣨⣿⡇
⠐⢯⣄⠀⢀⡀⠀⠀⠼⠀⠀⢠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠄⠔⠶⣿⣿⣧⣬⣉⣽⣷
⠀⠉⠓⠤⣄⡀⠀⠀⠀⠀⠠⠾⠎⣯⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣤⣶⣶⡛⠉⠁⠀⠀⠀⠀⠈⠳⣜⡻⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠈⠂⠀⠀⠀⠋⠓⠀⠀⠀⠐⠂⠀⠘⠐⠒⠒⠒⠛⠉⠋⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠖⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠂⠣⠤⠞⠁⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⡠⢀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢤⢶⠾⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷
⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠙⠈⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢀⠀⣀⠄⠐⠀⠀⠉⢁⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⠠⠴⠒⠀⢘⡉⠀⠀⠀⠐⠀⠉⠉⠉⠀⠀⣀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢀⠄⠀⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡠⠄⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡠⠀⠀⡇⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠰⡄⢢⡀⠀⡀⢀⠤⠐⠂⢘⠁⠀⠠⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⠤⠀⡆⠀⠀⠀⡆⠀⠠⠤⠰⠶⠂⠦⠐⠀⠐
⠀⠀⠑⠒⠄⡀⢀⠜⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡡⠀⠀⡇⠀⠀⠀⠃⠀⢀⣴⠄⠀⠀⠀⢴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐⠀⠂⠀⠈⠀⠚⠋⠁⠀⠀⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⢸⢈⢸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠉⠀⠀⠁⠈⠉
⠀⠀⠀⠂⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠜⠑⠂⡈⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀
⠀⡈⠀⠠⠀⠀⠀⠀⠈⡀⡆⠀⠀⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢤⠠⠂⠀⠀⠀⠀⡀⢀⠤
⠀⠁⠀⠀⠀⢐⠀⠀⠀⠀⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⡀⠁⡄⣀⢈⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡞⡉⢳⡄⠀⠀⠒⠀⠀⠀⠀⠀⠠⠤⡀⠴⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠑⠠⠁⠀⠨⠒⠀⠀⠇⠀⠀⢀⠙⠒⠐⠂⠴⠒⠒⠚⠛⡆⠀⠀⠀⠀⠀⠀⠀⡀⣐⡆⠀⠀⠀⠂⠀⠀⠈⠀⠤⠒⠈⢉⢉⡁⠒⠦⡀
⠀⠀⠀⠐⣀⡂⠂⠀⠀⠀⠀⠀⠀⠘⣄⠏⠄⠶⠌⡃⢉⠂⢠⡇⠀⠀⠀⠀⠘⠁⢇⠀⠀⠋⠀⠀⠀⠀⠀⠀⢠⠌⡀⡤⠤⠀⠠⠀⠄⠄⢸
⠀⢠⠀⠀⣘⠓⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠒⠒⠀⠀⣠⠊
⠀⠀⡀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢀⠔⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠉⠑⠒⠚⠚⠉⢀⡠
⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣽⣻⣶⣶⣀⣠⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠍⣻⣗⠄⠤⠾⣭⡈⢕⠶⣠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣿⣿⠷⣴⡾⠟⠛⢿⣿⣈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⠄⠀⠇⠉⣧⣘⠿⠷⣶⣄⢹⡝⠭⣷⡤⢄
⠀⢀⣶⠣⡀⠀⠀⠀⠀⠀⡼⢹⡿⠀⠀⠀⠛⠛⢿⡷⣶⣿⣿⠟⠳⠵⣶⡟⠓⠉⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣵⣼⣷⡄⠠⢄⡀⠀⠸⠐⣿⡈⠀⣳⡶⣾⣿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠻⣿⠿⣵⣄⠀⠀⠄⠀⠐⢾⣷⡀⢁⡿⠿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣜⡷⢬⣷⣾⢾⠀⠄⣼⣄⠀⢸⡿⠿⠿⠤⠀⠀⠁⠌⢓⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠁⣿⠹⠁⠀⠁⠴⠋⠙⣿⣿⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡄⠀⠀⠀⠀⠀⠀⡶⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡅⠀⠀⡞⣶⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡜⠈⠀⠀⠘⠀⠛⠀⣄⠱⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠘⣄⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠆⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠘⠦⣆⠀⠀⡤⢀⡀⢐⡒⡀⣌⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠓⠚⠉⠁⠘⠋⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠉⣂⣴⣶⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠀⠀⡀⠀⠀⠀⠀⠀⢿⡿⠟⣿⡇⣔⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠐⣧⢀⡘⠁⠀⠀⠀⡶⠀⠀⢻⣯⣿⠄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠡⠀⠀⠀⠀⠻⠇⠤⠤⠞⠀⠀⠀⠀⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⢀⢀⣐⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡼⡟⠀⠒⠒⠂⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁