Problem:
Score: 500 points

Problem Statement

New AtCoder City has an infinite grid of streets, as follows:

At the center of the city stands a clock tower. Let (0, 0) be the coordinates of this point.

A straight street, which we will call East-West Main Street, runs east-west and passes the clock tower. It corresponds to the x-axis in the two-dimensional coordinate plane.

There are also other infinitely many streets parallel to East-West Main Street, with a distance of 1 between them. They correspond to the lines \ldots, y = -2, y = -1, y = 1, y = 2, \ldots in the two-dimensional coordinate plane.

A straight street, which we will call North-South Main Street, runs north-south and passes the clock tower. It corresponds to the y-axis in the two-dimensional coordinate plane.

There are also other infinitely many streets parallel to North-South Main Street, with a distance of 1 between them. They correspond to the lines \ldots, x = -2, x = -1, x = 1, x = 2, \ldots in the two-dimensional coordinate plane.

There are N residential areas in New AtCoder City. The i-th area is located at the intersection with the coordinates (X_i, Y_i) and has a population of P_i. Each citizen in the city lives in one of these areas.

The city currently has only two railroads, stretching infinitely, one along East-West Main Street and the other along North-South Main Street.

M-kun, the mayor, thinks that they are not enough for the commuters, so he decides to choose K streets and build a railroad stretching infinitely along each of those streets.

Let the walking distance of each citizen be the distance from his/her residential area to the nearest railroad.

M-kun wants to build railroads so that the sum of the walking distances of all citizens, S, is minimized.

For each K = 0, 1, 2, \dots, N, what is the minimum possible value of S after building railroads?

Constraints

1 \leq N \leq 15

-10 \ 000 \leq X_i \leq 10 \ 000

-10 \ 000 \leq Y_i \leq 10 \ 000

1 \leq P_i \leq 1 \ 000 \ 000

The locations of the N residential areas, (X_i, Y_i), are all distinct.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1 P_1
X_2 Y_2 P_2
: : :
X_N Y_N P_N

Output

Print the answer in N+1 lines.

The i-th line (i = 1, \ldots, N+1) should contain the minimum possible value of S after building railroads for the case K = i-1.

Sample Input 1

3
1 2 300
3 3 600
1 4 800

Sample Output 1

2900
900
0
0

When K = 0, the residents of Area 1, 2, and 3 have to walk the distances of 1, 3, and 1, respectively, to reach a railroad.

Thus, the sum of the walking distances of all citizens, S, is 1 \times 300 + 3 \times 600 + 1 \times 800 = 2900.

When K = 1, if we build a railroad along the street corresponding to the line y = 4 in the coordinate plane, the walking distances of the citizens of Area 1, 2, and 3 become 1, 1, and 0, respectively.

Then, S = 1 \times 300 + 1 \times 600 + 0 \times 800 = 900.

We have many other options for where we build the railroad, but none of them makes S less than 900.

When K = 2, if we build a railroad along the street corresponding to the lines x = 1 and x = 3 in the coordinate plane, all citizens can reach a railroad with the walking distance of 0, so S = 0. We can also have S = 0 when K = 3.

The figure below shows the optimal way to build railroads for the cases K = 0, 1, 2.

The street painted blue represents the roads along which we build railroads.

Sample Input 2

5
3 5 400
5 3 700
5 5 1000
5 7 700
7 5 400

Sample Output 2

13800
1600
0
0
0
0

The figure below shows the optimal way to build railroads for the cases K = 1, 2.

Sample Input 3

6
2 5 1000
5 2 1100
5 5 1700
-2 -5 900
-5 -2 600
-5 -5 2200

Sample Output 3

26700
13900
3200
1200
0
0
0

The figure below shows the optimal way to build railroads for the case K = 3.

Sample Input 4

8
2 2 286017
3 1 262355
2 -2 213815
1 -3 224435
-2 -2 136860
-3 -1 239338
-2 2 217647
-1 3 141903

Sample Output 4

2576709
1569381
868031
605676
366338
141903
0
0
0

The figure below shows the optimal way to build railroads for the case K = 4.

Solution in Rust (1.42.0) (judge verdict: Accepted)

// SNIPPET read
pub trait Readable {
    type Output;
    const WORD_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;
            const WORD_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;
    const WORD_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>;
    const WORD_COUNT: usize = 1;
    fn read_words(words: &[&str]) -> Result<Vec<char>, String> {
        Ok(words[0].chars().collect())
    }
}
pub struct Bytes();
impl Readable for Bytes {
    type Output = Vec<u8>;
    const WORD_COUNT: usize = 1;
    fn read_words(words: &[&str]) -> Result<Vec<u8>, String> {
        Ok(words[0].bytes().collect())
    }
}
impl Readable for i8 {
    type Output = Self;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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 i128 {
    type Output = Self;
    const WORD_COUNT: usize = 1;
    fn read_words(words: &[&str]) -> Result<i128, String> {
        use std::str::FromStr;
        i128::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as i64", words[0])
        })
    }
}
impl Readable for u128 {
    type Output = Self;
    const WORD_COUNT: usize = 1;
    fn read_words(words: &[&str]) -> Result<u128, String> {
        use std::str::FromStr;
        u128::from_str(words[0]).map_err(|_| {
            format!("cannot parse `{}` as u64", words[0])
        })
    }
}
impl Readable for isize {
    type Output = Self;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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;
    const WORD_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);
    const WORD_COUNT: usize = T1::WORD_COUNT + T2::WORD_COUNT;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let mut start = 0;
        let count1 = T1::WORD_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);
    const WORD_COUNT: usize = T1::WORD_COUNT + T2::WORD_COUNT + T3::WORD_COUNT;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let mut start = 0;
        let count1 = T1::WORD_COUNT;
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::WORD_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);
    const WORD_COUNT: usize = T1::WORD_COUNT + T2::WORD_COUNT + T3::WORD_COUNT + T4::WORD_COUNT;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let mut start = 0;
        let count1 = T1::WORD_COUNT;
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::WORD_COUNT;
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::WORD_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);
    const WORD_COUNT: usize = T1::WORD_COUNT + T2::WORD_COUNT + T3::WORD_COUNT + T4::WORD_COUNT + T5::WORD_COUNT;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let mut start = 0;
        let count1 = T1::WORD_COUNT;
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::WORD_COUNT;
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::WORD_COUNT;
        let val3 = T3::read_words(&words[start .. start+count3])?;
        start += count3;
        let count4 = T4::WORD_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);
    const WORD_COUNT: usize = T1::WORD_COUNT + T2::WORD_COUNT + T3::WORD_COUNT + T4::WORD_COUNT + T5::WORD_COUNT + T6::WORD_COUNT;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let mut start = 0;
        let count1 = T1::WORD_COUNT;
        let val1 = T1::read_words(&words[start .. start+count1])?;
        start += count1;
        let count2 = T2::WORD_COUNT;
        let val2 = T2::read_words(&words[start .. start+count2])?;
        start += count2;
        let count3 = T3::WORD_COUNT;
        let val3 = T3::read_words(&words[start .. start+count3])?;
        start += count3;
        let count4 = T4::WORD_COUNT;
        let val4 = T4::read_words(&words[start .. start+count4])?;
        start += count4;
        let count5 = T5::WORD_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))
    }
}
impl<T: Readable> Readable for [T; 2] {
    type Output = [T::Output; 2];
    const WORD_COUNT: usize = T::WORD_COUNT * 2;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let val1 = T::read_words(&words[T::WORD_COUNT*0 .. T::WORD_COUNT*1])?;
        let val2 = T::read_words(&words[T::WORD_COUNT*1 .. T::WORD_COUNT*2])?;
        Ok([val1, val2])
    }
}
impl<T: Readable> Readable for [T; 3] {
    type Output = [T::Output; 3];
    const WORD_COUNT: usize = T::WORD_COUNT * 3;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let val1 = T::read_words(&words[T::WORD_COUNT*0 .. T::WORD_COUNT*1])?;
        let val2 = T::read_words(&words[T::WORD_COUNT*1 .. T::WORD_COUNT*2])?;
        let val3 = T::read_words(&words[T::WORD_COUNT*2 .. T::WORD_COUNT*3])?;
        Ok([val1, val2, val3])
    }
}
impl<T: Readable> Readable for [T; 4] {
    type Output = [T::Output; 4];
    const WORD_COUNT: usize = T::WORD_COUNT * 4;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let val1 = T::read_words(&words[T::WORD_COUNT*0 .. T::WORD_COUNT*1])?;
        let val2 = T::read_words(&words[T::WORD_COUNT*1 .. T::WORD_COUNT*2])?;
        let val3 = T::read_words(&words[T::WORD_COUNT*2 .. T::WORD_COUNT*3])?;
        let val4 = T::read_words(&words[T::WORD_COUNT*3 .. T::WORD_COUNT*4])?;
        Ok([val1, val2, val3, val4])
    }
}
impl<T: Readable> Readable for [T; 5] {
    type Output = [T::Output; 5];
    const WORD_COUNT: usize = T::WORD_COUNT * 5;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let val1 = T::read_words(&words[T::WORD_COUNT*0 .. T::WORD_COUNT*1])?;
        let val2 = T::read_words(&words[T::WORD_COUNT*1 .. T::WORD_COUNT*2])?;
        let val3 = T::read_words(&words[T::WORD_COUNT*2 .. T::WORD_COUNT*3])?;
        let val4 = T::read_words(&words[T::WORD_COUNT*3 .. T::WORD_COUNT*4])?;
        let val5 = T::read_words(&words[T::WORD_COUNT*4 .. T::WORD_COUNT*5])?;
        Ok([val1, val2, val3, val4, val5])
    }
}
impl<T: Readable> Readable for [T; 6] {
    type Output = [T::Output; 6];
    const WORD_COUNT: usize = T::WORD_COUNT * 6;
    fn read_words(words: &[&str]) -> Result<Self::Output, String> {
        assert_eq!(words.len(), Self::WORD_COUNT);
        let val1 = T::read_words(&words[T::WORD_COUNT*0 .. T::WORD_COUNT*1])?;
        let val2 = T::read_words(&words[T::WORD_COUNT*1 .. T::WORD_COUNT*2])?;
        let val3 = T::read_words(&words[T::WORD_COUNT*2 .. T::WORD_COUNT*3])?;
        let val4 = T::read_words(&words[T::WORD_COUNT*3 .. T::WORD_COUNT*4])?;
        let val5 = T::read_words(&words[T::WORD_COUNT*4 .. T::WORD_COUNT*5])?;
        let val6 = T::read_words(&words[T::WORD_COUNT*5 .. T::WORD_COUNT*6])?;
        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_end_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::WORD_COUNT {
            return Err(format!("line `{}` has {} words, expected {}",
                               line, words.len(), T::WORD_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::WORD_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::WORD_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::WORD_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, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = T1::WORD_COUNT;
                let words = split_into_words_for_collection::<$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, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2)>::WORD_COUNT;
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::WORD_COUNT;
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::WORD_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, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2, T3)>::WORD_COUNT;
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::WORD_COUNT;
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::WORD_COUNT;
                let val2 = T2::read_words(&words[start .. start+count2])?;
                start += count2;
                let count3 = T3::WORD_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, $collection_out);
            fn read_line(line: &str) -> Result<Self::Output, String> {
                let prefix_len = <(T1, T2, T3, T4)>::WORD_COUNT;
                let words = split_into_words_for_collection::<$u>(line, prefix_len)?;
                let mut start = 0;
                let count1 = T1::WORD_COUNT;
                let val1 = T1::read_words(&words[start .. start+count1])?;
                start += count1;
                let count2 = T2::WORD_COUNT;
                let val2 = T2::read_words(&words[start .. start+count2])?;
                start += count2;
                let count3 = T3::WORD_COUNT;
                let val3 = T3::read_words(&words[start .. start+count3])?;
                start += count3;
                let count4 = T4::WORD_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>();
    };
    ( ! $(,)* ) => {
        let _ = read::<()>();
    };
    ( $pat:pat = $t:ty, $( $rest:tt )+ ) => {
        read_inner!($pat = $t; $($rest)+);
    };
    ( !, $( $rest:tt )+ ) => {
        read_inner!(_ = (); $($rest)+);
    };
}
#[macro_export]
#[doc(hidden)]
macro_rules! read_inner {
    ( $( $acc_pat:pat = $acc_t:ty ),+ ; $pat:pat = $t:ty, $( $rest:tt )* ) => {
        read_inner!($( $acc_pat = $acc_t ),+ , $pat = $t ; $($rest)*);
    };
    ( $( $acc_pat:pat = $acc_t:ty ),+ ; !, $( $rest:tt )* ) => {
        read_inner!($( $acc_pat = $acc_t ),+ , _ = () ; $($rest)*);
    };
    ( $( $acc_pat:pat = $acc_t:ty ),+ ; $pat:pat = $t:ty ) => {
        read_inner!($( $acc_pat = $acc_t ),+ , $pat = $t ;);
    };
    ( $( $acc_pat:pat = $acc_t:ty ),+ ; ! ) => {
        read_inner!($( $acc_pat = $acc_t ),+ , _ = () ;);
    };
    ( $( $pat:pat = $t:ty ),+ ; ) => {
        let ($($pat),+) = read::<($($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),+)>();
    };
}
static mut STDIN: Option<std::io::Stdin> = None;
pub struct ReadLines<T: ReadableFromLine> {
    lock: std::io::StdinLock<'static>,
    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 mut line = String::new();
        if self.lock.read_line(&mut line).unwrap() > 0 {
            Some(T::read_line(&line).unwrap())
        } else {
            None
        }
    }
}
pub fn read_lines<T: ReadableFromLine>() -> ReadLines<T> {
    unsafe {
        if STDIN.is_none() {
            STDIN = Some(std::io::stdin());
        }
    }
    ReadLines {
        lock: unsafe { STDIN.as_ref().unwrap().lock() },
        phantom: std::marker::PhantomData::<T>
    }
}
pub struct ReadChunks<T: ReadableFromChunk> {
    lock: std::io::StdinLock<'static>,
    phantom: std::marker::PhantomData<T>
}
impl<T: ReadableFromChunk> Iterator for ReadChunks<T> {
    type Item = T::Output;
    fn next(&mut self) -> Option<T::Output> {
        read_chunk_from_handle::<T>(&mut self.lock)
    }
}
pub fn read_chunks<T: ReadableFromChunk>() -> ReadChunks<T> {
    unsafe {
        if STDIN.is_none() {
            STDIN = Some(std::io::stdin());
        }
    }
    ReadChunks {
        lock: unsafe { STDIN.as_ref().unwrap().lock() },
        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
#[macro_export]
macro_rules! echo {
    () => {
        println!()
    };
    ($e: expr $(,)?) => {
        println!("{}", $e)
    };
    ($e: expr, $($es: expr),+ $(,)?) => {
        {
            use std::io::Write;
            let stdout = std::io::stdout();
            let mut handle = stdout.lock();
            write!(handle, "{}", $e).unwrap();
            $(
                write!(handle, " {}", $es).unwrap();
            )+
            writeln!(handle).unwrap();
        }
    };
}
pub fn yn(result: bool) {
    if result {
        println!("Yes");
    } else {
        println!("No");
    }
}
#[allow(non_snake_case)]
pub fn YN(result: bool) {
    if result {
        println!("YES");
    } else {
        println!("NO");
    }
}
pub fn exit(msg: impl std::fmt::Display) -> ! {
    println!("{}", msg);
    std::process::exit(0)
}
#[macro_export]
#[cfg(local)]
macro_rules! dbg {
    () => {
        {
            use std::io::{self, Write};
            writeln!(io::stderr(), "{}: dbg", line!()).unwrap();
        }
    };
    ($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(self, pred: impl FnOnce(&Self) -> bool) -> Option<Self>;
}
impl<T> Guard for T {
    fn guard(self, pred: impl FnOnce(&T) -> bool) -> Option<T> {
        if pred(&self) { Some(self) } else { None }
    }
}

// SNIPPET num_types
pub trait WithZero: Sized +
    for<'a> std::ops::Add<&'a Self, Output=Self> +
    for<'a> std::ops::AddAssign<&'a Self>
{
    fn zero() -> Self;
}
pub trait WithOne: Sized +
    for<'a> std::ops::Mul<&'a Self, Output=Self> +
    for<'a> std::ops::MulAssign<&'a Self>
{
    fn one() -> Self;
}
pub trait Integer: Eq + Ord + WithZero + WithOne +
    for<'a> std::ops::Sub<&'a Self, Output=Self> +
    for<'a> std::ops::SubAssign<&'a Self> +
    for<'a> std::ops::Div<&'a Self, Output=Self> +
    for<'a> std::ops::DivAssign<&'a Self> +
    for<'a> std::ops::Rem<&'a Self, Output=Self> +
    for<'a> std::ops::RemAssign<&'a Self>
{}
pub trait ToSigned {
    type Signed: Integer;
    fn to_signed(&self) -> Option<Self::Signed>;
    unsafe fn to_signed_unchecked(&self) -> Self::Signed {
        self.to_signed().unwrap()
    }
}
pub trait ToUnsigned {
    type Unsigned: Integer;
    fn to_unsigned(&self) -> Option<Self::Unsigned>;
    unsafe fn to_unsigned_unchecked(&self) -> Self::Unsigned {
        self.to_unsigned().unwrap()
    }
}

// SNIPPET int
pub trait PrimitiveInteger: Integer + Copy + ToSigned + ToUnsigned {
    fn abs_diff(self, rhs: Self) -> Self;
}
macro_rules! impl_primitive_integer_unsigned {
    ( $($t: ty, $t_signed: ty);* ) => { $(
        impl WithZero for $t {
            fn zero() -> $t { 0 }
        }
        impl WithOne for $t {
            fn one() -> $t { 1 }
        }
        impl Integer for $t {}
        impl ToSigned for $t {
            type Signed = $t_signed;
            fn to_signed(&self) -> Option<$t_signed> {
                use std::convert::TryInto;
                (*self).try_into().ok()
            }
            unsafe fn to_signed_unchecked(&self) -> $t_signed {
                *self as $t_signed
            }
        }
        impl ToUnsigned for $t {
            type Unsigned = $t;
            fn to_unsigned(&self) -> Option<$t> {
                Some(*self)
            }
        }
        impl PrimitiveInteger for $t {
            fn abs_diff(self, rhs: $t) -> $t {
                if self < rhs { rhs - self } else { self - rhs }
            }
        }
    )* }
}
impl_primitive_integer_unsigned!(
    u8, i8;
    u16, i16;
    u32, i32;
    u64, i64;
    u128, i128;
    usize, isize
);
macro_rules! impl_primitive_integer_signed {
    ( $($t: ty, $t_unsigned: ty);* ) => { $(
        impl WithZero for $t {
            fn zero() -> $t { 0 }
        }
        impl WithOne for $t {
            fn one() -> $t { 1 }
        }
        impl Integer for $t {}
        impl ToSigned for $t {
            type Signed = $t;
            fn to_signed(&self) -> Option<$t> {
                Some(*self)
            }
        }
        impl ToUnsigned for $t {
            type Unsigned = $t_unsigned;
            fn to_unsigned(&self) -> Option<$t_unsigned> {
                use std::convert::TryInto;
                (*self).try_into().ok()
            }
            unsafe fn to_unsigned_unchecked(&self) -> $t_unsigned {
                *self as $t_unsigned
            }
        }
        impl PrimitiveInteger for $t {
            fn abs_diff(self, rhs: $t) -> $t {
                if self < rhs { rhs - self } else { self - rhs }
            }
        }
    )* }
}
impl_primitive_integer_signed!(
    i8, u8;
    i16, u16;
    i32, u32;
    i64, u64;
    i128, u128;
    isize, usize
);
pub trait PrimitiveUnsigned: PrimitiveInteger {
    fn checked_mul(self, rhs: Self) -> Option<Self>;
    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;
    fn gcd(self, other: Self) -> Self {
        if other == Self::zero() { self } else { other.gcd(self % &other) }
    }
    fn lcm(self, other: Self) -> Self {
        if self == Self::zero() || other == Self::zero() { Self::zero() } else {
            let gcd = self.gcd(other);
            self / &gcd * &other
        }
    }
    fn divisors(self) -> Option<Divisors<Self>> {
        (self != Self::zero()).then_with(|| Divisors {
            num: self,
            current: Some(Self::one()),
            latter: Vec::new()
        })
    }
    fn factorize<Exp: PrimitiveUnsigned>(self) -> Option<Factorize<Self, Exp>> {
        (self != Self::zero()).then_with(|| Factorize {
            rest: self,
            current: Self::one() + &Self::one(),
            phantom: std::marker::PhantomData
        })
    }
    fn prime_factors(self) -> Option<PrimeFactors<Self>> {
        (self != Self::zero()).then_with(|| PrimeFactors {
            rest: self,
            current: Self::one() + &Self::one()
        })
    }
}
pub struct Divisors<T> {
    num: T,
    current: Option<T>,
    latter: Vec<T>
}
impl<T: PrimitiveUnsigned> Iterator for Divisors<T> {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        match self.current {
            None => self.latter.pop(),
            Some(mut cur) => {
                loop {
                    if cur.checked_mul(cur).map_or(true, |square| square > self.num) {
                        self.current = None;
                        return self.latter.pop();
                    }
                    if self.num % &cur == T::zero() {
                        break;
                    }
                    cur += &T::one();
                }
                self.current = Some(cur + &T::one());
                if cur * &cur != self.num {
                    self.latter.push(self.num / &cur);
                }
                Some(cur)
            }
        }
    }
}
pub struct Factorize<T, Exp> {
    rest: T,
    current: T,
    phantom: std::marker::PhantomData<Exp>
}
impl<T: PrimitiveUnsigned, Exp: PrimitiveUnsigned> Iterator for Factorize<T, Exp> {
    type Item = (T, Exp);
    fn next(&mut self) -> Option<(T, Exp)> {
        if self.rest == T::one() {
            return None;
        }
        loop {
            if self.current.checked_mul(self.current).map_or(true, |square| square > self.rest) {
                let factor = self.rest;
                self.rest = T::one();
                return Some((factor, Exp::one()));
            }
            let mut exp = Exp::zero();
            while self.rest % &self.current == T::zero() {
                exp += &Exp::one();
                self.rest /= &self.current;
            }
            self.current += &T::one();
            if exp > Exp::zero() {
                let cur = self.current - &T::one();
                return Some((cur, exp));
            }
        }
    }
}
pub struct PrimeFactors<T> {
    rest: T,
    current: T
}
impl<T: PrimitiveUnsigned> Iterator for PrimeFactors<T> {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        if self.rest == T::one() {
            return None;
        }
        loop {
            if self.current.checked_mul(self.current).map_or(true, |square| square > self.rest) {
                let factor = self.rest;
                self.rest = T::one();
                return Some(factor);
            }
            let mut divisible = false;
            while self.rest % &self.current == T::zero() {
                divisible = true;
                self.rest /= &self.current;
            }
            self.current += &T::one();
            if divisible {
                return Some(self.current - &T::one());
            }
        }
    }
}
macro_rules! impl_primitive_unsigned {
    ( $($t: ty)* ) => { $(
        impl PrimitiveUnsigned for $t {
            fn checked_mul(self, rhs: $t) -> Option<$t> {
                self.checked_mul(rhs)
            }
            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 u128 usize);
pub trait PrimitiveSigned: PrimitiveInteger {
    fn bezout(self, other: Self) -> (Self, Self, Self::Unsigned);
}
unsafe fn bezout_sub<T: PrimitiveUnsigned>(a: T, b: T) -> (T::Signed, T::Signed, T) {
    if b == T::zero() { (T::Signed::one(), T::Signed::zero(), a) } else {
        let m = (a / &b).to_signed_unchecked();
        let (x, y, g) = bezout_sub(b, a % &b);
        let solution_b = x - &(m * &y);
        (y, solution_b, g)
    }
}
macro_rules! impl_primitive_signed {
    ( $($t: ty)* ) => { $(
        impl PrimitiveSigned for $t {
            fn bezout(self, other: $t) -> ($t, $t, <$t as ToUnsigned>::Unsigned) {
                let (x, y, g) = unsafe { bezout_sub(
                    (self * self.signum()).to_unsigned_unchecked(),
                    (other * other.signum()).to_unsigned_unchecked()
                ) };
                (x * self.signum(), y * other.signum(), g)
            }
        }
    )* }
}
impl_primitive_signed!(i8 i16 i32 i64 i128 isize);

// SNIPPET iter
pub struct Chunks<I> {
    iter: I,
    size: usize
}
impl<I: Iterator> Iterator for Chunks<I> {
    type Item = Vec<I::Item>;
    fn next(&mut self) -> Option<Self::Item> {
        let first = self.iter.next();
        if first.is_none() {
            return None;
        }
        let mut chunk = Vec::with_capacity(self.size);
        chunk.push(first.unwrap());
        for _ in 0..self.size-1 {
            match self.iter.next() {
                Some(x) => chunk.push(x),
                None => break
            }
        }
        Some(chunk)
    }
    fn size_hint(&self) -> (usize, Option<usize>) {
        let (lower, upper) = self.iter.size_hint();
        (lower.ceil_div(self.size), upper.map(|u| u.ceil_div(self.size)))
    }
    fn count(self) -> usize {
        self.iter.count().ceil_div(self.size)
    }
    fn nth(&mut self, n: usize) -> Option<Self::Item> {
        if n > 0 {
            self.iter.nth(n * self.size - 1);
        }
        self.next()
    }
}
impl<I: ExactSizeIterator> ExactSizeIterator for Chunks<I> {}
impl<I: std::iter::FusedIterator> std::iter::FusedIterator for Chunks<I> {}
#[derive(Clone)]
pub struct LScan<I, S, F> {
    iter: I,
    state: Option<S>,
    f: F,
}
impl<I: Iterator, S, F> Iterator for LScan<I, S, F>
where
    F: FnMut(&S, I::Item) -> S
{
    type Item = S;
    fn next(&mut self) -> Option<S> {
        if self.state.is_none() {
            return None;
        }
        let state_inner = self.state.take().unwrap();
        if let Some(item) = self.iter.next() {
            self.state = Some((self.f)(&state_inner, item));
        }
        Some(state_inner)
    }
    fn size_hint(&self) -> (usize, Option<usize>) {
        let (lower, upper) = self.iter.size_hint();
        (lower + 1, upper.map(|u| u + 1))
    }
    fn count(self) -> usize {
        self.iter.count() + 1
    }
}
impl<I: ExactSizeIterator, S, F: FnMut(&S, I::Item) -> S> ExactSizeIterator for LScan<I, S, F> {}
impl<I: std::iter::FusedIterator, S, F: FnMut(&S, I::Item) -> S> std::iter::FusedIterator for LScan<I, S, F> {}
pub struct GroupBy<K, I: Iterator, F> {
    cur: Option<(I::Item, K)>,
    iter: I,
    key_fn: F
}
impl<K: Eq, I: Iterator, F: FnMut(&I::Item) -> K> Iterator for GroupBy<K, I, F> {
    type Item = (K, Vec<I::Item>);
    fn next(&mut self) -> Option<(K, Vec<I::Item>)> {
        let cur = self.cur.take();
        cur.map(|(item, key)| {
            let mut group = vec![item];
            loop {
                let next = self.iter.next();
                match next {
                    Some(next_item) => {
                        let next_key = (self.key_fn)(&next_item);
                        if key == next_key {
                            group.push(next_item);
                        } else {
                            self.cur = Some((next_item, next_key));
                            break;
                        }
                    }
                    None => {
                        self.cur = None;
                        break;
                    }
                }
            }
            (key, group)
        })
    }
    fn size_hint(&self) -> (usize, Option<usize>) {
        if self.cur.is_none() {
            (0, Some(0))
        } else {
            let (_, upper) = self.iter.size_hint();
            (1, upper.map(|u| u + 1))
        }
    }
}
impl<K: Eq, I: Iterator, F: FnMut(&I::Item) -> K> std::iter::FusedIterator for GroupBy<K, I, F> {}
pub struct RunLength<I: Iterator> {
    cur: Option<I::Item>,
    iter: I
}
impl<I: Iterator> Iterator for RunLength<I> where I::Item: Eq {
    type Item = (I::Item, usize);
    fn next(&mut self) -> Option<(I::Item, usize)> {
        let cur = self.cur.take();
        cur.map(|value| {
            let mut length = 1;
            loop {
                let next = self.iter.next();
                match next {
                    Some(next_value) => {
                        if value == next_value {
                            length += 1;
                        } else {
                            self.cur = Some(next_value);
                            break;
                        }
                    }
                    None => {
                        self.cur = None;
                        break;
                    }
                }
            }
            (value, length)
        })
    }
    fn size_hint(&self) -> (usize, Option<usize>) {
        if self.cur.is_none() {
            (0, Some(0))
        } else {
            let (_, upper) = self.iter.size_hint();
            (1, upper.map(|u| u + 1))
        }
    }
}
impl<I: Iterator> std::iter::FusedIterator for RunLength<I> where I::Item: Eq {}
pub trait IteratorExt: Iterator {
    fn chunks(self, size: usize) -> Chunks<Self> where Self: Sized {
        assert!(size > 0);
        Chunks {
            iter: self,
            size: size
        }
    }
    fn lscan<S, F>(self, state: S, f: F) -> LScan<Self, S, F>
    where
        Self: Sized,
        F: FnMut(&S, Self::Item) -> S,
    {
        LScan {
            iter: self,
            state: Some(state),
            f: f,
        }
    }
    fn lscan1<F>(mut self, f: F) -> Option<LScan<Self, Self::Item, F>>
    where
        Self: Sized,
        Self::Item: Clone,
        F: FnMut(&Self::Item, Self::Item) -> Self::Item
    {
        self.next().map(|first| self.lscan(first, f))
    }
    fn get_unique(mut self) -> Option<Self::Item> where Self: Sized, Self::Item: Eq {
        let first_opt = self.next();
        first_opt.and_then(|first| {
            if self.all(|item| item == first) { Some(first) } else { None }
        })
    }
    fn group_by<K: Eq, F: FnMut(&Self::Item) -> K>(mut self, mut f: F) -> GroupBy<K, Self, F> where Self: Sized {
        let next = self.next();
        GroupBy {
            cur: next.map(|item| {
                let key = f(&item);
                (item, key)
            }),
            iter: self,
            key_fn: f
        }
    }
    fn run_length(mut self) -> RunLength<Self> where Self: Sized, Self::Item: Eq {
        RunLength {
            cur: self.next(),
            iter: self
        }
    }
    fn join<T: std::fmt::Display>(mut self, sep: T) -> String
    where
        Self: Sized, Self::Item: std::fmt::Display
    {
        let mut result = String::new();
        if let Some(first) = self.next() {
            result.push_str(&format!("{}", first));
        }
        for s in self {
            result.push_str(&format!("{}{}", sep, s));
        }
        result
    }
    fn cat(self) -> String where Self: Sized, Self::Item: std::fmt::Display { self.join("") }
}
impl<I: Iterator> IteratorExt for I {}
pub trait IteratorInnerProduct<T, Rhs=T>: ExactSizeIterator<Item=T> {
    fn inner_product<I, J>(self, other: I) -> Option<<T as std::ops::Mul<Rhs>>::Output>
    where
        Self: Sized,
        I: IntoIterator<Item=Rhs, IntoIter=J>,
        J: Iterator<Item=Rhs> + ExactSizeIterator,
        T: std::ops::Mul<Rhs>,
        <T as std::ops::Mul<Rhs>>::Output: std::iter::Sum
    {
        let iter = other.into_iter();
        (self.len() == iter.len()).then_with(|| {
            self.zip(iter).map(|(a, b)| a * b).sum()
        })
    }
}
impl<T1, T2, I> IteratorInnerProduct<T1, T2> for I
where
    I: Iterator<Item=T1> + ExactSizeIterator,
    T1: std::ops::Mul<T2>
{}
pub struct Unfold<State, T, F> where F: FnMut(&State) -> Option<(T, State)> {
    state: State,
    f: F
}
impl<State, T, F> Iterator for Unfold<State, T, F> where F: FnMut(&State) -> Option<(T, State)> {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        (self.f)(&self.state).map(|(value, state)| {
            self.state = state;
            value
        })
    }
}
pub fn unfold<State, T, F>(init: State, f: F) -> Unfold<State, T, F> where F: FnMut(&State) -> Option<(T, State)> {
    Unfold { state: init, f: f }
}
pub struct Iterate<T, F> where F: FnMut(&T) -> T {
    state: T,
    f: F
}
impl<T, F> Iterator for Iterate<T, F> where F: FnMut(&T) -> T {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        use std::mem::swap;
        let mut state = (self.f)(&self.state);
        swap(&mut state, &mut self.state);
        Some(state)
    }
}
pub fn iterate<T, F>(init: T, f: F) -> Iterate<T, F>
where
    F: FnMut(&T) -> T
{
    Iterate { state: init, f: f }
}
pub struct IterateMap<State, T, F> where F: FnMut(&State) -> (T, State) {
    state: State,
    f: F
}
impl<State, T, F> Iterator for IterateMap<State, T, F> where F: FnMut(&State) -> (T, State) {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        let (value, state) = (self.f)(&self.state);
        self.state = state;
        Some(value)
    }
}
pub fn iterate_map<State, T, F>(init: State, f: F) -> IterateMap<State, T, F>
where
    F: FnMut(&State) -> (T, State)
{
    IterateMap { state: init, f }
}

// 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<F>(&mut self, cmp: F)
    where
        F: FnMut(&T, &T) -> std::cmp::Ordering;
    fn sort_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, key: F);
    fn sort_unstable_desc(&mut self) where T: Ord;
    fn sort_unstable_desc_by<F>(&mut self, cmp: F)
    where
        F: FnMut(&T, &T) -> std::cmp::Ordering;
    fn sort_unstable_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, key: F);
}
impl<T> SortDesc<T> for [T] {
    fn sort_desc(&mut self) where T: Ord {
        self.sort_by(|x, y| y.cmp(x));
    }
    fn sort_desc_by<F>(&mut self, mut cmp: F)
    where
        F: FnMut(&T, &T) -> std::cmp::Ordering
    {
        self.sort_by(|x, y| cmp(y, x));
    }
    fn sort_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, mut key: F) {
        self.sort_by_key(|x| Reverse(key(x)));
    }
    fn sort_unstable_desc(&mut self) where T: Ord {
        self.sort_unstable_by(|x, y| y.cmp(x));
    }
    fn sort_unstable_desc_by<F>(&mut self, mut cmp: F)
    where
        F: FnMut(&T, &T) -> std::cmp::Ordering
    {
        self.sort_unstable_by(|x, y| cmp(y, x));
    }
    fn sort_unstable_desc_by_key<K: Ord, F: FnMut(&T) -> K>(&mut self, mut key: F) {
        self.sort_unstable_by_key(|x| Reverse(key(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 = self.next()?;
        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)
    }
}
pub trait WithCmpIdentity<T>: Sized {
    fn new(x: T) -> Self;
    fn inf() -> Self;
    fn as_option(&self) -> Option<&T>;
    fn as_option_mut(&mut self) -> Option<&mut T>;
    fn into_option(self) -> Option<T>;
    fn is_fin(&self) -> bool {
        self.as_option().is_some()
    }
    fn is_inf(&self) -> bool {
        self.as_option().is_none()
    }
    fn expect_fin(self, msg: &str) -> T {
        self.into_option().expect(msg)
    }
    fn fin(self) -> T {
        self.into_option().unwrap()
    }
    fn fin_or(self, default: T) -> T {
        self.into_option().unwrap_or(default)
    }
    fn fin_or_else<F: FnOnce() -> T>(self, f: F) -> T {
        self.into_option().unwrap_or_else(f)
    }
    fn map_or<U, F: FnOnce(T) -> U>(self, default: U, f: F) -> U {
        self.into_option().map_or(default, f)
    }
    fn map_or_else<U, D, F>(self, default: D, f: F) -> U
    where
        D: FnOnce() -> U,
        F: FnOnce(T) -> U
    {
        self.into_option().map_or_else(default, f).into()
    }
}
#[derive(Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash, Debug)]
pub enum MaybeNegInf<T> {
    Inf,
    Fin(T)
}
pub type Max<T> = MaybeNegInf<T>;
#[derive(Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash, Debug)]
pub enum MaybeInf<T> {
    Fin(T),
    Inf
}
pub type Min<T> = MaybeInf<T>;
macro_rules! impl_with_cmp_identity {
    ($t:ident) => {
        impl<T> $t<T> {
            pub fn map<U: Ord, F: FnOnce(T) -> U>(self, f: F) -> $t<U> {
                match self {
                    $t::Fin(x) => $t::Fin(f(x)),
                    $t::Inf => $t::Inf
                }
            }
        }
        impl<T> WithCmpIdentity<T> for $t<T> {
            fn new(x: T) -> $t<T> { $t::Fin(x) }
            fn inf() -> $t<T> { $t::Inf }
            fn as_option(&self) -> Option<&T> {
                match self {
                    $t::Fin(x) => Some(x),
                    $t::Inf => None
                }
            }
            fn as_option_mut(&mut self) -> Option<&mut T> {
                match self {
                    $t::Fin(x) => Some(x),
                    $t::Inf => None
                }
            }
            fn into_option(self) -> Option<T> {
                match self {
                    $t::Fin(x) => Some(x),
                    $t::Inf => None
                }
            }
        }
    }
}
impl_with_cmp_identity!(MaybeNegInf);
impl_with_cmp_identity!(MaybeInf);

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

use std::cmp;

const INF: i64 = 1_000_000_000_000_000;

fn dist(villages: &[(i64, i64)]) -> i64 {
    let p_sum: i64 = villages.iter().map(|&(_, p)| p).sum();
    let m = p_sum / 2;
    let i = villages.iter().map(|&(_, p)| p)
        .lscan1(|&acc, p| acc + p)
        .unwrap()
        .position(|p| p >= m)
        .unwrap();
    let x0 = villages[i].0;
    villages.iter().map(|&(x, p)| (x - x0).abs() * p).sum()
}

#[test]
fn test_dist() {
    let villages = vec![
        (100, 5),
        (200, 5),
        (300, 20)
    ];
    assert_eq!(dist(&villages), 1500);

    let villages = vec![
        (100, 5),
        (200, 10),
        (300, 5)
    ];
    assert_eq!(dist(&villages), 1000);
}

fn dp(villages: &[(i64, i64)], k: usize, used: bool, memo: &mut [Vec<Vec<Option<i64>>>; 2]) -> i64 {
    memo[used as usize][villages.len()][k].unwrap_or_else(|| {
        let ans = if villages.is_empty() {
            0
        } else if used && k == 0 {
            INF
        } else if k == 0 {
            villages.iter().map(|&(x, p)| x.abs() * p).sum::<i64>()
        } else {
            (1..=villages.len()).map(|l| {
                let d1 = dist(&villages[..l]) + dp(&villages[l..], k-1, used, memo);
                if used {
                    d1
                } else {
                    let d2 = villages[..l].iter().map(|&(x, p)| x.abs() * p).sum::<i64>()
                        + dp(&villages[l..], k, true, memo);
                    cmp::min(d1, d2)
                }
            }).min().unwrap()
        };
        memo[used as usize][villages.len()][k] = Some(ans);
        ans
    })
}

fn solve(villages: &[(i64, i64)], k: usize) -> Vec<i64> {
    let n = villages.len();
    let mut memo = [vec![vec![None; k+1]; n+1], vec![vec![None; k+1]; n+1]];
    for i in 0..=k {
        dp(&villages, i, false, &mut memo);
    }
    memo[0][n].iter().map(|&x| x.unwrap()).collect()
}

fn main() {
    read!(n = usize);
    let villages: Vec<_> = read_lines::<(i64, i64, i64)>().collect();
    let mut ans = vec![INF; n+1];
    for x_set in 0..1<<n {
        let mut xs: Vec<(i64, i64)> = (0..n)
            .filter(|i| x_set & (1<<i) != 0)
            .map(|i| (villages[i].0, villages[i].2))
            .collect();
        xs.sort_by_key(|&(x, _)| x);
        let mut ys: Vec<(i64, i64)> = (0..n)
            .filter(|i| x_set & (1<<i) == 0)
            .map(|i| (villages[i].1, villages[i].2))
            .collect();
        ys.sort_by_key(|&(y, _)| y);
        let x_ans = solve(&xs, n);
        let y_ans = solve(&ys, n);
        for i in 0..=n {
            let new_ans = (0..=i).map(|ix| x_ans[ix] + y_ans[i-ix]).min().unwrap();
            chmin!(ans[i], new_ans);
        }
    }
    echo!(ans.into_iter().join('\n'));
}