// SNIPPET read
pub trait Readable {
    type Output;
    fn words_count() -> usize;
    fn read_words(words: &[&str]) -> Result<Self::Output, String>;
}
#[macro_export]
macro_rules! readable {
    ( $t:ty, $words_count:expr, |$words:ident| $read_words:expr ) => {
        impl Readable for $t {
            type Output = $t;
            fn words_count() -> usize { $words_count }
            fn read_words($words: &[&str]) -> Result<$t, String> {
                Ok($read_words)
            }
        }
    };
}
readable!((), 1, |_ss| ());
readable!(String, 1, |ss| ss[0].to_string());
impl Readable for char {
    type Output = char;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<char, String> {
        let chars: Vec<char> = words[0].chars().collect();
        if chars.len() == 1 {
            Ok(chars[0])
        } else {
            Err(format!("cannot parse `{}` as a char", words[0]))
        }
    }
}
pub struct Chars();
impl Readable for Chars {
    type Output = Vec<char>;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Vec<char>, String> {
        Ok(words[0].chars().collect())
    }
}
impl Readable for i8 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<i8, String> {
        use std::str::FromStr;
        i8::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as i8", words[0])
        })
    }
}
impl Readable for u8 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<u8, String> {
        use std::str::FromStr;
        u8::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as u8", words[0])
        })
    }
}
impl Readable for i16 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<i16, String> {
        use std::str::FromStr;
        i16::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as i16", words[0])
        })
    }
}
impl Readable for u16 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<u16, String> {
        use std::str::FromStr;
        u16::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as u16", words[0])
        })
    }
}
impl Readable for i32 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<i32, String> {
        use std::str::FromStr;
        i32::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as i32", words[0])
        })
    }
}
impl Readable for u32 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<u32, String> {
        use std::str::FromStr;
        u32::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as u32", words[0])
        })
    }
}
impl Readable for i64 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<i64, String> {
        use std::str::FromStr;
        i64::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as i64", words[0])
        })
    }
}
impl Readable for u64 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<u64, String> {
        use std::str::FromStr;
        u64::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as u64", words[0])
        })
    }
}
impl Readable for isize {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<isize, String> {
        use std::str::FromStr;
        <isize>::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as isize", words[0])
        })
    }
}
impl Readable for usize {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<usize, String> {
        use std::str::FromStr;
        <usize>::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as usize", words[0])
        })
    }
}
impl Readable for f32 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<f32, String> {
        use std::str::FromStr;
        f32::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as f32", words[0])
        })
    }
}
impl Readable for f64 {
    type Output = Self;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<f64, String> {
        use std::str::FromStr;
        f64::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as f64", words[0])
        })
    }
}
#[allow(non_camel_case_types)]
pub struct u8_;
impl Readable for u8_ {
    type Output = u8;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        u8::read_words(words).map(|n| n-1)
    }
}
#[allow(non_camel_case_types)]
pub struct u16_;
impl Readable for u16_ {
    type Output = u16;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        u16::read_words(words).map(|n| n-1)
    }
}
#[allow(non_camel_case_types)]
pub struct u32_;
impl Readable for u32_ {
    type Output = u32;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        u32::read_words(words).map(|n| n-1)
    }
}
#[allow(non_camel_case_types)]
pub struct u64_;
impl Readable for u64_ {
    type Output = u64;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        u64::read_words(words).map(|n| n-1)
    }
}
#[allow(non_camel_case_types)]
pub struct usize_;
impl Readable for usize_ {
    type Output = usize;
    fn words_count() -> usize { 1 }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        <usize>::read_words(words).map(|n| n-1)
    }
}
impl<T1: Readable, T2: Readable> Readable for (T1, T2) {
    type Output = (T1::Output, T2::Output);
    fn words_count() -> usize {
        T1::words_count() + T2::words_count()
    }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::words_count());
        let mut start = 0;
        let count1 = T1::words_count();
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let val2 = T2::read_words(&words[start..])?;
        Ok((val1, val2))
    }
}
impl<T1: Readable, T2: Readable, T3: Readable> Readable for (T1, T2, T3) {
    type Output = (T1::Output, T2::Output, T3::Output);
    fn words_count() -> usize {
        T1::words_count() + T2::words_count() + T3::words_count()
    }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::words_count());
        let mut start = 0;
        let count1 = T1::words_count();
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::words_count();
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let val3 = T3::read_words(&words[start..])?;
        Ok((val1, val2, val3))
    }
}
impl<T1: Readable, T2: Readable, T3: Readable, T4: Readable> Readable for (T1, T2, T3, T4) {
    type Output = (T1::Output, T2::Output, T3::Output, T4::Output);
    fn words_count() -> usize {
        T1::words_count() + T2::words_count() + T3::words_count() + T4::words_count()
    }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::words_count());
        let mut start = 0;
        let count1 = T1::words_count();
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::words_count();
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::words_count();
        let val3 = T3::read_words(&words[start .. start+count3])?;
        start += count3;
        let val4 = T4::read_words(&words[start..])?;
        Ok((val1, val2, val3, val4))
    }
}
impl<T1: Readable, T2: Readable, T3: Readable, T4: Readable, T5: Readable> Readable for (T1, T2, T3, T4, T5) {
    type Output = (T1::Output, T2::Output, T3::Output, T4::Output, T5::Output);
    fn words_count() -> usize {
        T1::words_count() + T2::words_count() + T3::words_count() + T4::words_count() + T5::words_count()
    }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::words_count());
        let mut start = 0;
        let count1 = T1::words_count();
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::words_count();
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::words_count();
        let val3 = T3::read_words(&words[start .. start+count3])?;
        start += count3;
        let count4 = T4::words_count();
        let val4 = T4::read_words(&words[start .. start+count4])?;
        start += count4;
        let val5 = T5::read_words(&words[start..])?;
        Ok((val1, val2, val3, val4, val5))
    }
}
impl<T1: Readable, T2: Readable, T3: Readable, T4: Readable, T5: Readable, T6: Readable> Readable for (T1, T2, T3, T4, T5, T6) {
    type Output = (T1::Output, T2::Output, T3::Output, T4::Output, T5::Output, T6::Output);
    fn words_count() -> usize {
        T1::words_count() + T2::words_count() + T3::words_count() + T4::words_count() + T5::words_count() + T6::words_count()
    }
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::words_count());
        let mut start = 0;
        let count1 = T1::words_count();
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::words_count();
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::words_count();
        let val3 = T3::read_words(&words[start .. start+count3])?;
        start += count3;
        let count4 = T4::words_count();
        let val4 = T4::read_words(&words[start .. start+count4])?;
        start += count4;
        let count5 = T5::words_count();
        let val5 = T5::read_words(&words[start .. start+count5])?;
        start += count5;
        let val6 = T6::read_words(&words[start..])?;
        Ok((val1, val2, val3, val4, val5, val6))
    }
}
pub trait ReadableFromLine {
    type Output;
    fn read_line(line: &str) -> Result<Self::Output, String>;
}
fn split_into_words(line: &str) -> Vec<&str> {
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: Readable> ReadableFromLine for T {
    type Output = T::Output;
    fn read_line(line: &str) -> Result<T::Output, String> {
        let words = split_into_words(line);
        if words.len() != T::words_count() {
            return Err(format!("line `{}` has {} words, expected {}",
                               line, words.len(), T::words_count()));
        }
        T::read_words(&words)
    }
}
pub fn read_words_into_vec<T: Readable>(words: &[&str], line: &str) -> Result<Vec<T::Output>, String> {
    let n = T::words_count();
    assert_eq!(words.len() % n, 0);
    let mut result = Vec::new();
    for chunk in words.chunks(n) {
        match T::read_words(chunk) {
            Ok(v) => result.push(v),
            Err(msg) => {
                let fragment_msg = if n == 1 {
                    format!("word {}", result.len())
                } else {
                    let l = result.len();
                    format!("words {}-{}", n*l + 1, (n+1) * l)
                };
                return Err(format!(
                    "{} of line `{}`: {}", fragment_msg, line, msg
                ));
            }
        }
    }
    Ok(result)
}
pub fn split_into_words_for_collection<T: Readable>(
    line: &str, prefix_words_count: usize
) -> Result<Vec<&str>, String> {
    let n = T::words_count();
    let words = split_into_words(line);
    if words.len() < prefix_words_count {
        return Err(
            format!("line `{}` has {} words, expected at least {}",
                    line, words.len(), prefix_words_count)
        );
    }
    if (words.len() - prefix_words_count) % T::words_count() != 0 {
        return Err(
            format!("line `{}` has {} words, expected {} + {}",
                    line, words.len(), prefix_words_count, n)
        );
    }
    Ok(words)
}
#[macro_export]
macro_rules! readable_collection {
    ($u:ident => $collection_in:ty, $collection_out:ty) => {
        readable_collection!($u: [] => $collection_in, $collection_out);
    };
    ($u:ident : [ $( $bound:path ),* ] => $collection_in:ty, $collection_out:ty) => {
        impl<$u: Readable> ReadableFromLine for $collection_in
        where
            <$u as Readable>::Output: Sized $(+ $bound)*
        {
            type Output = $collection_out;
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let words = split_into_words_for_collection::<$u>(line, 0)?;
                Ok(read_words_into_vec::<$u>(&words, line)?.into_iter().collect())
            }
        }
        impl<T1: Readable, $u: Readable> ReadableFromLine for (T1, $collection_in)
        where
            <$u as Readable>::Output: Sized $(+ $bound)*
        {
            type Output = (T1::Output, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = T1::words_count();
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let val1 = T1::read_words(&words[..prefix_len])?;
                let rest = read_words_into_vec::<$u>(&words[prefix_len..], line)?;
                Ok((val1, rest.into_iter().collect()))
            }
        }
        impl<T1: Readable, T2: Readable, $u: Readable> ReadableFromLine for (T1, T2, $collection_in)
        where
            <$u as Readable>::Output: Sized $(+ $bound)*
        {
            type Output = (T1::Output, T2::Output, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2)>::words_count();
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::words_count();
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::words_count();
                let val2 = T2::read_words(&words[start .. start+count2])?;
                let rest = read_words_into_vec::<$u>(&words[prefix_len..], line)?;
                Ok((val1, val2, rest.into_iter().collect()))
            }
        }
        impl<T1: Readable, T2: Readable, T3: Readable, $u: Readable> ReadableFromLine for (T1, T2, T3, $collection_in)
        where
            <$u as Readable>::Output: Sized $(+ $bound)*
        {
            type Output = (T1::Output, T2::Output, T3::Output, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2, T3)>::words_count();
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::words_count();
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::words_count();
                let val2 = T2::read_words(&words[start .. start+count2])?;
                start += count2;
                let count3 = T3::words_count();
                let val3 = T3::read_words(&words[start .. start+count3])?;
                let rest = read_words_into_vec::<$u>(&words[prefix_len..], line)?;
                Ok((val1, val2, val3, rest.into_iter().collect()))
            }
        }
        impl<T1: Readable, T2: Readable, T3: Readable, T4: Readable, $u: Readable> ReadableFromLine for (T1, T2, T3, T4, $collection_in)
        where
            <$u as Readable>::Output: Sized $(+ $bound)*
        {
            type Output = (T1::Output, T2::Output, T3::Output, T4::Output, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2, T3, T4)>::words_count();
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::words_count();
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::words_count();
                let val2 = T2::read_words(&words[start .. start+count2])?;
                start += count2;
                let count3 = T3::words_count();
                let val3 = T3::read_words(&words[start .. start+count3])?;
                start += count3;
                let count4 = T4::words_count();
                let val4 = T4::read_words(&words[start .. start+count4])?;
                let rest = read_words_into_vec::<$u>(&words[prefix_len..], line)?;
                Ok((val1, val2, val3, val4, rest.into_iter().collect()))
            }
        }
    };
}
readable_collection!(U => Vec<U>, Vec<U::Output>);
pub fn read<T: ReadableFromLine>() -> T::Output {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();
    T::read_line(&line).unwrap()
}
#[macro_export]
macro_rules! read {
    () => {
        let mut line = String::new();
        std::io::stdin().read_line(&mut line).unwrap();
    };
    ( $pat:pat = $t:ty ) => {
        let $pat = read::<$t>();
    };
    ( $( $pat:pat = $t:ty ),+ ) => {
        read!(($($pat),*) = ($($t),*));
    };
}
pub trait ReadableFromChunk {
    type Output;
    fn lines_count() -> usize;
    fn read_chunk(lines: &[String]) -> Result<Self::Output, String>;
}
impl<T1: ReadableFromLine, T2: ReadableFromLine> ReadableFromChunk for (T1, T2) {
    type Output = (T1::Output, T2::Output);
    fn lines_count() -> usize { 2 }
    fn read_chunk(lines: &[String]) -> Result<Self::Output, String> {
        let out1 = T1::read_line(&lines[0])?;
        let out2 = T2::read_line(&lines[1])?;
        Ok((out1, out2))
    }
}
impl<T1: ReadableFromLine, T2: ReadableFromLine, T3: ReadableFromLine> ReadableFromChunk for (T1, T2, T3) {
    type Output = (T1::Output, T2::Output, T3::Output);
    fn lines_count() -> usize { 3 }
    fn read_chunk(lines: &[String]) -> Result<Self::Output, String> {
        let out1 = T1::read_line(&lines[0])?;
        let out2 = T2::read_line(&lines[1])?;
        let out3 = T3::read_line(&lines[2])?;
        Ok((out1, out2, out3))
    }
}
impl<T1: ReadableFromLine, T2: ReadableFromLine, T3: ReadableFromLine, T4: ReadableFromLine> ReadableFromChunk for (T1, T2, T3, T4) {
    type Output = (T1::Output, T2::Output, T3::Output, T4::Output);
    fn lines_count() -> usize { 4 }
    fn read_chunk(lines: &[String]) -> Result<Self::Output, String> {
        let out1 = T1::read_line(&lines[0])?;
        let out2 = T2::read_line(&lines[1])?;
        let out3 = T3::read_line(&lines[2])?;
        let out4 = T4::read_line(&lines[3])?;
        Ok((out1, out2, out3, out4))
    }
}
pub fn read_chunk<T: ReadableFromChunk>() -> T::Output {
    let stdin = std::io::stdin();
    let mut handle = stdin.lock();
    read_chunk_from_handle::<T>(&mut handle).unwrap()
}
fn read_chunk_from_handle<T: ReadableFromChunk>(handle: &mut std::io::StdinLock) -> Option<T::Output> {
    use std::io::BufRead;
    let mut lines = vec![String::new(); T::lines_count()];
    let mut first = true;
    for line in &mut lines {
        if handle.read_line(line).unwrap() == 0 && first {
            return None;
        }
        first = false;
    }
    Some(T::read_chunk(&lines).unwrap())
}
#[macro_export]
macro_rules! read_chunk {
    ( $( $pat:pat = $t:ty ),+ ) => {
        let ($($pat),+) = read_chunk::<($($t),+)>();
    };
}
pub struct ReadLines<T: ReadableFromLine> {
    phantom: std::marker::PhantomData<T>
}
impl<T: ReadableFromLine> Iterator for ReadLines<T> {
    type Item = T::Output;
    fn next(&mut self) -> Option<T::Output> {
        use std::io::BufRead;
        let stdin = std::io::stdin();
        let mut line = String::new();
        if stdin.lock().read_line(&mut line).unwrap() > 0 {
            Some(T::read_line(&line).unwrap())
        } else {
            None
        }
    }
}
pub fn read_lines<T: ReadableFromLine>() -> ReadLines<T> {
    ReadLines {
        phantom: std::marker::PhantomData::<T>
    }
}
pub struct ReadChunks<T: ReadableFromChunk> {
    phantom: std::marker::PhantomData<T>
}
impl<T: ReadableFromChunk> Iterator for ReadChunks<T> {
    type Item = T::Output;
    fn next(&mut self) -> Option<T::Output> {
        let stdin = std::io::stdin();
        let mut handle = stdin.lock();
        read_chunk_from_handle::<T>(&mut handle)
    }
}
pub fn read_chunks<T: ReadableFromChunk>() -> ReadChunks<T> {
    ReadChunks {
        phantom: std::marker::PhantomData::<T>
    }
}
pub trait Words {
    fn read<T: Readable>(&self) -> T::Output;
}
impl<'a> Words for [&'a str] {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(self).unwrap()
    }
}
impl<'a> Words for &'a str {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(&[self]).unwrap()
    }
}

// SNIPPET utils
pub fn yn(result: bool) {
    if result {
        println!("Yes");
    } else {
        println!("No");
    }
}
#[allow(non_snake_case)]
pub fn YN(result: bool) {
    if result {
        println!("YES");
    } else {
        println!("NO");
    }
}
pub fn exit<T: std::fmt::Display>(msg: T) -> ! {
    println!("{}", msg);
    std::process::exit(0)
}
#[macro_export]
#[cfg(local)]
macro_rules! dbg {
    () => {
        {
            use std::io::{self, Write};
            writeln!(io::stderr(), "{}: dbg", line!()).unwrap();
        }
    };
    ($e: expr) => {
        {
            use std::io::{self, Write};
            let result = $e;
            writeln!(io::stderr(), "{}: {} = {:?}",
                     line!(), stringify!($e), result)
                .unwrap();
            result
        }
    }
}
#[macro_export]
#[cfg(not(local))]
macro_rules! dbg {
    () => {};
    ($e: expr) => {
        { $e }
    }
}

// SNIPPET option
pub trait BoolExt {
    fn then<T>(self, value: T) -> Option<T>;
    fn then_with<T, F>(self, f: F) -> Option<T> where F: FnOnce() -> T;
    fn and<T>(self, option: Option<T>) -> Option<T>;
    fn and_then<T, F>(self, f: F) -> Option<T> where F: FnOnce() -> Option<T>;
}
impl BoolExt for bool {
    fn then<T>(self, value: T) -> Option<T> {
        if self { Some(value) } else { None }
    }
    fn then_with<T, F>(self, f: F) -> Option<T> where F: FnOnce() -> T {
        if self { Some(f()) } else { None }
    }
    fn and<T>(self, option: Option<T>) -> Option<T> {
        if self { option } else { None }
    }
    fn and_then<T, F>(self, f: F) -> Option<T> where F: FnOnce() -> Option<T> {
        if self { f() } else { None }
    }
}
pub trait OptionExt<T> {
    fn to_string_or<U: std::fmt::Display>(&self, default: U) -> String
    where
        T: std::fmt::Display;
}
impl<T> OptionExt<T> for Option<T> {
    fn to_string_or<U: std::fmt::Display>(&self, default: U) -> String
    where
        T: std::fmt::Display
    {
        self.as_ref().map(|x| x.to_string()).unwrap_or(default.to_string())
    }
    /*
    fn get_or_insert_with<F: FnOnce() -> T>(&mut self, f: F) -> &mut T {
        match *self {
            None => *self = Some(f()),
            _ => ()
        }
        self.as_mut().unwrap()
    }
    */
}
pub trait Guard: Sized {
    fn guard<F: FnOnce(&Self) -> bool>(self, pred: F) -> Option<Self>;
}
impl<T> Guard for T {
    fn guard<F: FnOnce(&T) -> bool>(self, pred: F) -> Option<T> {
        if pred(&self) { Some(self) } else { None }
    }
}

// SNIPPET table
pub struct Table<T> {
    inner: Vec<Vec<T>>
}
pub struct TableIndicesIterator {
    height: usize,
    width: usize,
    y: usize,
    x: usize
}
impl Iterator for TableIndicesIterator {
    type Item = (usize, usize);
    fn next(&mut self) -> Option<(usize, usize)> {
        if self.x < self.width {
            self.x += 1;
            return Some((self.y, self.x-1))
        }
        if self.y+1 < self.height {
            self.y += 1;
            self.x = 1;
            return Some((self.y, 0))
        }
        None
    }
}
pub struct TableRows<'a, T: 'a> {
    table: &'a Table<T>,
    index: usize
}
impl<'a, T: 'a> Iterator for TableRows<'a, T> {
    type Item = &'a [T];
    fn next(&mut self) -> Option<&'a [T]> {
        if self.index < self.table.height() {
            let i = self.index;
            self.index += 1;
            Some(&self.table.inner[i])
        } else {
            None
        }
    }
}
/*
pub trait TableRangeBounds {
    fn y_bounds(&self) -> (std::ops::Bound<usize>, std::ops::Bound<usize>);
    fn x_bounds(&self) -> (std::ops::Bound<usize>, std::ops::Bound<usize>);
}
impl<T1: std::ops::RangeBounds<usize>, T2: std::ops::RangeBounds<usize>> TableRangeBounds for (T1, T2) {
    fn y_bounds(&self) -> (std::ops::Bound<usize>, std::ops::Bound<usize>) {
        (self.0.start_bound().cloned(), self.0.end_bound().cloned())
    }
    fn x_bounds(&self) -> (std::ops::Bound<usize>, std::ops::Bound<usize>) {
        (self.1.start_bound().cloned(), self.1.end_bound().cloned())
    }
}
*/
pub struct CumulativeTable<T, F1, F2> {
    op: F1,
    op_inv: F2,
    inner: Table<T>
}
impl<T> Table<T> {
    pub fn from_rows(rows: Vec<Vec<T>>) -> Option<Table<T>> {
        rows.windows(2).all(|window| window[0].len() == window[1].len())
            .then_with(|| Table { inner: rows })
    }
    pub unsafe fn from_rows_unchecked(rows: Vec<Vec<T>>) -> Table<T> {
        Table { inner: rows }
    }
    pub fn height(&self) -> usize {
        self.inner.len()
    }
    pub fn width(&self) -> usize {
        self.inner.first().map_or(0, |row| row.len())
    }
    pub fn shape(&self) -> (usize, usize) {
        (self.height(), self.width())
    }
    pub fn indices(&self) -> TableIndicesIterator {
        TableIndicesIterator {
            height: self.height(),
            width: self.width(),
            y: if self.width() == 0 { self.height() } else { 0 },
            x: 0
        }
    }
    pub fn inside(&self, (y, x): (usize, usize)) -> bool {
        y < self.height() && x < self.width()
    }
    pub fn row(&self, y: usize) -> Option<&[T]> {
        self.inner.get(y).map(|vec| vec.as_slice())
    }
    pub fn row_mut(&mut self, y: usize) -> Option<&mut [T]> {
        self.inner.get_mut(y).map(|vec| vec.as_mut_slice())
    }
    pub fn get(&self, (y, x): (usize, usize)) -> Option<&T> {
        self.inner.get(y).and_then(|row| row.get(x))
    }
    pub fn get_mut(&mut self, (y, x): (usize, usize)) -> Option<&mut T> {
        self.inner.get_mut(y).and_then(|row| row.get_mut(x))
    }
    pub fn rows(&self) -> TableRows<T> {
        TableRows { table: self, index: 0 }
    }
    pub fn adjacent4_indices(&self, (y, x): (usize, usize)) -> Option<Vec<(usize, usize)>> {
        self.inside((y, x)).then_with(|| {
            let mut result = Vec::with_capacity(4);
            if y > 0 {
                result.push((y-1, x));
            }
            if x > 0 {
                result.push((y, x-1));
            }
            if x < self.width()-1 {
                result.push((y, x+1));
            }
            if y < self.height()-1 {
                result.push((y+1, x));
            }
            result
        })
    }
    pub fn adjacent8_indices(&self, (y, x): (usize, usize)) -> Option<Vec<(usize, usize)>> {
        self.inside((y, x)).then_with(|| {
            let mut result = Vec::with_capacity(8);
            let xs = x.saturating_sub(1) .. std::cmp::min(x+1, self.width()-1)+1;
            if y > 0 {
                for x in xs.clone() {
                    result.push((y-1, x));
                }
            }
            if x > 0 {
                result.push((y, x-1));
            }
            if x < self.width()-1 {
                result.push((y, x+1));
            }
            if y < self.height()-1 {
                for x in xs {
                    result.push((y+1, x));
                }
            }
            result
        })
    }
    pub fn accumulate<F1, F2>(&self, identity: T, op: F1, op_inv: F2)
                              -> CumulativeTable<T, F1, F2>
    where
        T: Clone,
        F1: Fn(&T, &T) -> T,
        F2: Fn(&T, &T) -> T
    {
        let mut inner = vec![vec![identity.clone(); self.width() + 1]];
        for table_row in self.rows() {
            let mut new_row = vec![identity.clone()];
            {
                let last_row = inner.last().unwrap();
                for (x, acc1) in table_row.iter().zip(last_row.windows(2)) {
                    let result = {
                        let acc2 = new_row.last().unwrap();
                        let tmp1 = op(x, &acc1[1]);
                        let tmp2 = op(&tmp1, acc2);
                        op_inv(&tmp2, &acc1[0])
                    };
                    new_row.push(result);
                }
            }
            inner.push(new_row);
        }
        CumulativeTable { op: op, op_inv: op_inv, inner: Table { inner: inner } }
    }
}
impl<T> std::ops::Index<usize> for Table<T> {
    type Output = [T];
    fn index(&self, y: usize) -> &[T] {
        match self.row(y) {
            Some(row) => row,
            None => panic!(
                "index out of bounds: the table has {} rows but the row index is {}",
                self.height(), y
            )
        }
    }
}
impl<T> std::ops::IndexMut<usize> for Table<T> {
    fn index_mut(&mut self, y: usize) -> &mut [T] {
        let height = self.height();
        match self.row_mut(y) {
            Some(row) => row,
            None => panic!(
                "index out of bounds: the table has {} rows but the index is {}",
                height, y
            )
        }
    }
}
impl<T> std::ops::Index<(usize, usize)> for Table<T> {
    type Output = T;
    fn index(&self, index: (usize, usize)) -> &T {
        match self.get(index) {
            Some(result) => result,
            None => panic!(
                "index out of bounds: the table shape is {:?} but the index is {:?}",
                self.shape(), index
            )
        }
    }
}
impl<T> std::ops::IndexMut<(usize, usize)> for Table<T> {
    fn index_mut(&mut self, index: (usize, usize)) -> &mut T {
        let shape = self.shape();
        match self.get_mut(index) {
            Some(result) => result,
            None => panic!(
                "index out of bounds: the table shape is {:?} but the index is {:?}",
                shape, index
            )
        }
    }
}
impl<T, F1: Fn(&T, &T) -> T, F2: Fn(&T, &T) -> T> CumulativeTable<T, F1, F2> {
    pub fn query(&self, range: std::ops::Range<(usize, usize)>) -> Option<T> {
        self.query_yx(range.start.0 .. range.end.0, range.start.1 .. range.end.1)
    }
    pub fn query_yx(&self, y: std::ops::Range<usize>, x: std::ops::Range<usize>) -> Option<T> {
        if y.start > y.end || y.end > self.inner.height() - 1 {
            return None;
        }
        if x.start > x.end || x.end > self.inner.width() - 1 {
            return None;
        }
        let val1 = &self.inner.inner[y.end][x.end];
        let val2 = &self.inner.inner[y.end][x.start];
        let val3 = &self.inner.inner[y.start][x.end];
        let val4 = &self.inner.inner[y.start][x.start];
        let tmp1 = (self.op)(val1, val4);
        let tmp2 = (self.op)(val2, val3);
        let result = (self.op_inv)(&tmp1, &tmp2);
        Some(result)
    }
}
pub fn backward2_indices((y, x): (usize, usize)) -> Vec<(usize, usize)> {
    let mut res = Vec::with_capacity(2);
    if y > 0 {
        res.push((y-1, x));
    }
    if x > 0 {
        res.push((y, x-1));
    }
    res
}
#[macro_export]
macro_rules! table {
    () => {
        unsafe { Table::from_rows_unchecked(Vec::new()) }
    };
    ($element:expr ; $rows:expr , $cols:expr) => {
        unsafe { Table::from_rows_unchecked(vec![vec![$element; $cols]; $rows]) }
    };
}
pub fn read_table<T: Readable>() -> Table<T::Output> {
    let res: Vec<Vec<T::Output>> = read_lines::<Vec<T>>().collect();
    Table::from_rows(res).unwrap()
}
pub fn read_table_rows<T: Readable>(height: usize) -> Table<T::Output> {
    let res: Vec<Vec<T::Output>> = read_lines::<Vec<T>>().take(height).collect();
    if res.len() < height {
        panic!(
            "tried reading {} rows for table, but stdin has only {} lines",
            height, res.len()
        );
    }
    Table::from_rows(res).unwrap()
}
pub fn read_char_table() -> Table<char> {
    let res: Vec<Vec<char>> = read_lines::<Chars>().collect();
    Table::from_rows(res).unwrap()
}
pub fn read_char_table_rows(height: usize) -> Table<char> {
    let res: Vec<Vec<char>> = read_lines::<Chars>().take(height).collect();
    if res.len() < height {
        panic!(
            "tried reading {} rows for table, but stdin has only {} lines",
            height, res.len()
        );
    }
    Table::from_rows(res).unwrap()
}

// SNIPPET int
pub trait PrimitiveInteger:
    Sized + Ord +
    std::ops::Add<Output=Self> +
    std::ops::Sub<Output=Self> +
    std::ops::Div<Output=Self>
{
    fn one() -> Self;
    fn abs_diff(self, rhs: Self) -> Self;
    fn rem_euclid(self, rhs: Self) -> Self;
}
macro_rules! impl_primitive_integer {
    ( $($t: ty)* ) => { $(
        impl PrimitiveInteger for $t {
            fn one() -> $t {
                1
            }
            fn abs_diff(self, rhs: $t) -> $t {
                if self < rhs { rhs - self } else { self - rhs }
            }
            #[allow(unused_comparisons)]
            fn rem_euclid(self, rhs: $t) -> $t {
                let r = self % rhs;
                if r < 0 {
                    if rhs < 0 { r - rhs } else { r + rhs }
                } else {
                    r
                }
            }
        }
    )* }
}
impl_primitive_integer!(u8 u16 u32 u64 usize i8 i16 i32 i64 isize);
pub trait PrimitiveUnsigned: PrimitiveInteger {
    fn ceil_div(self, rhs: Self) -> Self;
    fn round_div(self, rhs: Self) -> Self;
    fn log2(self) -> Option<Self>;
    fn ceil_log2(self) -> Option<Self>;
    fn sqrt(self) -> Self;
}
macro_rules! impl_primitive_unsigned {
    ( $($t: ty)* ) => { $(
        impl PrimitiveUnsigned for $t {
            fn ceil_div(self, rhs: $t) -> $t {
                (self + rhs - 1) / rhs
            }
            fn round_div(self, rhs: $t) -> $t {
                (self + rhs/2) / rhs
            }
            fn log2(mut self) -> Option<$t> {
                if self == 0 {
                    None
                } else {
                    let mut ans = 0;
                    while self > 1 {
                        ans += 1;
                        self /= 2;
                    }
                    Some(ans)
                }
            }
            fn ceil_log2(self) -> Option<$t> {
                self.log2().map(|x| {
                    (self + ((1<<x) - 1)).log2().unwrap()
                })
            }
            fn sqrt(self) -> $t {
                (self as f64).sqrt() as $t
            }
        }
    )* }
}
impl_primitive_unsigned!(u8 u16 u32 u64 usize);
pub fn gcd(a: u64, b: u64) -> u64 {
    if b == 0 { a } else { gcd(b, a%b) }
}
pub fn bezout(a: i64, b: i64) -> (i64, i64, u64) {
    let (x, y, g) = bezout_sub((a * a.signum()) as u64, (b * b.signum()) as u64);
    (x * a.signum(), y * b.signum(), g)
}
fn bezout_sub(a: u64, b: u64) -> (i64, i64, u64) {
    if b == 0 { (1, 0, a) } else {
        let m = (a / b) as i64;
        let (x, y, g) = bezout_sub(b, a%b);
        (y, x - m*y, g)
    }
}

// SNIPPET cmp
use std::cmp::{Ord, Ordering};
pub fn minmax<T: Ord>(a: T, b: T) -> (T, T) {
    if a <= b { (a, b) } else { (b, a) }
}
#[macro_export]
macro_rules! chmin {
    ($place: expr, $expr: expr) => {
        let value = $expr;
        if value < $place {
            $place = value;
        }
    }
}
#[macro_export]
macro_rules! chmax {
    ($place: expr, $expr: expr) => {
        let value = $expr;
        if value > $place {
            $place = value;
        }
    }
}
#[derive(Clone, Copy, PartialEq, Eq, Debug, Default, Hash)]
pub struct Reverse<T: Ord>(pub T);
impl<T: Ord> PartialOrd for Reverse<T> {
    fn partial_cmp(&self, other: &Reverse<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }
}
impl<T: Ord> Ord for Reverse<T> {
    fn cmp(&self, other: &Reverse<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}
pub trait SortDesc<T> {
    fn sort_desc(&mut self) where T: Ord;
    fn sort_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, f: F);
}
impl<T> SortDesc<T> for [T] {
    fn sort_desc(&mut self) where T: Ord {
        self.sort();
        self.reverse();
    }
    fn sort_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, mut f: F) {
        self.sort_by_key(|x| Reverse(f(x)));
    }
}
#[derive(Clone, Copy, PartialEq, PartialOrd, Debug, Default, Hash)]
pub struct Total<T: PartialOrd + PartialEq>(pub T);
impl<T: PartialOrd + PartialEq> Eq for Total<T> {}
impl<T: PartialOrd + PartialEq> Ord for Total<T> {
    fn cmp(&self, other: &Self) -> Ordering {
        self.partial_cmp(other).unwrap()
    }
}
pub trait IteratorMinmax: Iterator {
    fn minmax(self) -> Option<(Self::Item, Self::Item)>;
    fn minmax_by_key<K, F>(self, key_fn: F) -> Option<(Self::Item, Self::Item)>
    where
        K: Ord,
        F: FnMut(&Self::Item) -> K;
    fn minmax_by<F>(self, compare: F) -> Option<(Self::Item, Self::Item)>
    where
        F: FnMut(&Self::Item, &Self::Item) -> Ordering;
}
impl<T: Ord + Clone, I: Iterator<Item=T>> IteratorMinmax for I {
    fn minmax(self) -> Option<(Self::Item, Self::Item)> {
        self.minmax_by(|a, b| a.cmp(b))
    }
    fn minmax_by_key<K, F>(self, mut key_fn: F) -> Option<(Self::Item, Self::Item)>
    where
        K: Ord,
        F: FnMut(&Self::Item) -> K
    {
        self.minmax_by(|a, b| key_fn(a).cmp(&key_fn(b)))
    }
    fn minmax_by<F>(mut self, mut compare: F) -> Option<(Self::Item, Self::Item)>
    where
        F: FnMut(&Self::Item, &Self::Item) -> Ordering
    {
        let first = match self.next() {
            Some(x) => x,
            None => return None
        };
        let second = match self.next() {
            Some(x) => x,
            None => return Some((first.clone(), first))
        };
        let mut result = minmax(first, second);
        while let Some(x) = self.next() {
            let (min, max) = result;
            result = if compare(&x, &min) == Ordering::Less {
                (x, max)
            } else if compare(&max, &x) == Ordering::Less {
                (min, x)
            } else {
                (min, max)
            };
        }
        Some(result)
    }
}

// END SNIPPETS
// Here is the documentation: https://yoshrc.github.io/rust-atcoder-snippets/atcoder_snippets/index.html

use std::cmp;
use std::collections::HashSet;

fn update(cost: &mut Table<(u64, Vec<usize>)>) {
    let n = cost.height();
    for mid in 1..n-1 {
        for i in mid+1..n {
            for j in 0..cmp::min(mid, i.ceil_div(2)) {
                let cur = cost[i][j].0;
                let new = cost[i][mid].0 + cost[mid][j].0;
                if new < cur {
                    let mut path = cost[i][mid].1.clone();
                    path.extend(cost[mid][j].1.iter().cloned());
                    cost[i][j] = (new, path)
                }
            }
        }
    }
}

fn main() {
    read!(_n = usize);
    read!(mut aa = Vec<usize>);
    read!(bs = Vec<usize>);
    if aa.iter().zip(&bs).any(|(&a, &b)| !(a == b || b < a.ceil_div(2))) {
        exit(-1);
    }
    let max = *cmp::max(aa.iter().max().unwrap(), bs.iter().max().unwrap());

    let mut cost = table![(1u64<<51, Vec::new()); max+1,max+1];
    cost[0][0] = (0, Vec::new());
    for i in 1..max+1 {
        cost[i][i] = (0, Vec::new());
        for j in 1..i+1 {
            let rem = i % j;
            if 1<<j < cost[i][rem].0 {
                cost[i][rem] = (1<<j, vec![j]);
            }
        }
    }
    update(&mut cost);

    let mut ans = 0u64;
    let mut used = HashSet::new();
    loop {
        let (i, max_cost) = aa.iter().zip(&bs).enumerate()
            .map(|(i, (&a, &b))| (i, cost[a][b].0))
            .max_by_key(|&(_, cost)| cost)
            .unwrap();
        if max_cost == 0 {
            break;
        }

        let path = cost[aa[i]][bs[i]].1.clone();
        for k in path {
            aa[i] %= k;
            if !used.contains(&k) {
                used.insert(k);
                ans += 1<<k;
                for i in 0..max+1 {
                    cost[i][i % k] = (0, vec![k]);
                }
                update(&mut cost);
                break;
            }
        }
    }
    println!("{}", ans);
}


Measured memory_bytes: 4352