// 
pub mod solution {
//{"name":"h","group":"Manual","url":"","interactive":false,"timeLimit":2000,"tests":[{"input":"","output":""},{"input":"","output":""},{"input":"","output":""}],"testType":"single","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"h"}}}

#[allow(unused)]
use crate::dbg;
use crate::algo_lib::geometry::point::PointT;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::rand::Random;
use crate::algo_lib::seg_trees::bottom_up_seg_tree::BottomUpSegTree;
use crate::algo_lib::seg_trees::seg_tree_trait::SegTreeNode;

type Point = PointT<i64>;

type Float = f64;

#[derive(Clone, Copy, Debug)]
struct Node {
    pref: Float,
    suf: Float,
    all: Float,
    max: Float,
}

const NEG_INF: Float = -1e18;
impl Default for Node {
    fn default() -> Self {
        Self {
            pref: NEG_INF,
            suf: NEG_INF,
            all: 0.0,
            max: 0.0,
        }
    }
}

impl SegTreeNode for Node {
    fn join_nodes(l: &Self, r: &Self, _context: &Self::Context) -> Self {
        Self {
            pref: (l.pref + r.all).max(r.pref),
            suf: (r.suf + l.all).max(l.suf),
            all: l.all + r.all,
            max: l.max.max(r.max).max(l.pref + r.suf),
        }
    }

    fn apply_update(node: &mut Self, update: &Self::Update) {
        todo!()
    }

    fn join_updates(current: &mut Self::Update, add: &Self::Update) {
        todo!()
    }

    type Update = ();

    type Context = ();
}

fn calc_abc(p: Point, q: Point) -> (i64, i64, i64) {
    let a = q.x - p.x;
    let b = q.y - p.y;
    let c = -a * p.x - b * p.y;
    (a, b, c)
}

fn solve_case(pts: &[Point], w: &[Float]) -> Float {
    let n = pts.len();
    let mut res = 0.0f64;
    for i in 0..n {
        res = res.max(w[i]);
    }
    let mut x_pos = vec![usize::MAX; n];

    let mut inside = vec![];
    let mut calc = |p: Point, q: Point| -> Float {
        inside.clear();
        let q2 = p + (q - p).rotate_ccw();
        let (a, b, c) = calc_abc(p, q2);
        for i in 0..n {
            let dy = pts[i].x * a + pts[i].y * b + c;
            if dy >= 0 {
                inside.push(i);
            }
        }
        let (ax, bx, cx) = calc_abc(p, q);
        inside.sort_by_key(|&i| pts[i].x * ax + pts[i].y * bx);
        for idx in 0..inside.len() {
            x_pos[inside[idx]] = idx;
        }

        let (ay, by, cy) = calc_abc(p, q2);
        inside.sort_by_key(|&i| pts[i].x * ay + pts[i].y * by);

        let mut res = 0.0f64;
        let x_coef = 1.0 / (Point::dist2(&p, &q) as Float).sqrt();
        let y_coef = 1.0 / (Point::dist2(&p, &q2) as Float).sqrt();
        let mut st = BottomUpSegTree::new(inside.len(), |_| Node::default());
        for &i in inside.iter() {
            let mut real_x = (pts[i].x * ax + pts[i].y * bx) as Float;
            real_x *= x_coef;
            let mut real_y = (pts[i].x * ay + pts[i].y * by + cy) as Float;
            real_y *= y_coef;
            let new_node = Node {
                pref: w[i] + 2.0 * real_x,
                suf: w[i] - 2.0 * real_x,
                all: w[i],
                max: w[i],
            };
            st.update_point(x_pos[i], new_node);

            let cur_res = st.get_root().max - 2.0 * real_y;
            res = res.max(cur_res);
        }
        res
    };
    for i in 0..n {
        for j in 0..n {
            if i == j {
                continue;
            }
            let mut ignore = false;
            for k in 0..n {
                if k != i && k != j {
                    if Point::vect_mul(&pts[i], &pts[j], &pts[k]) == 0
                        && Point::scal_mul(&pts[i], &pts[j], &pts[k]) >= 0
                    {
                        ignore = true;
                    }
                }
            }
            if ignore {
                continue;
            }
            {
                let cur = calc(pts[i], pts[j]);
                res = res.max(cur);
            }
        }
    }
    res
}

fn solve(input: &mut Input, out: &mut Output, _test_case: usize) {
    let n = input.usize();
    let mut pts = vec![];
    let mut w = vec![];
    for _ in 0..n {
        pts.push(Point::new(input.i64(), input.i64()));
        w.push(input.f64() as Float);
    }
    let res = solve_case(&pts, &w);
    out.println(res);
}

fn stress() {
    let mut rnd = Random::new(78788);
    let n = 400;
    const MAX_C: i64 = 1_000_000;
    let pts = (0..n)
        .map(|_| Point::new(rnd.gen(0..MAX_C), rnd.gen(0..MAX_C)))
        .collect::<Vec<_>>();
    let w = (0..n)
        .map(|_| rnd.gen(0..MAX_C) as Float)
        .collect::<Vec<_>>();
    let res = solve_case(&pts, &w);
    dbg!(res);
}

pub(crate) fn run(mut input: Input, mut output: Output) -> bool {
    solve(&mut input, &mut output, 1);
    output.flush();
    true
}

}
pub mod algo_lib {
pub mod collections {
pub mod array_2d {
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use crate::algo_lib::misc::num_traits::Number;
use std::io::Write;
use std::ops::Index;
use std::ops::IndexMut;
use std::ops::Mul;

// TODO: implement good Debug
#[derive(Clone, Debug)]
pub struct Array2D<T> {
    rows: usize,
    cols: usize,
    v: Vec<T>,
}

pub struct Iter<'a, T> {
    array: &'a Array2D<T>,
    row: usize,
    col: usize,
}

impl<T> Array2D<T>
where
    T: Clone,
{
    #[allow(unused)]
    pub fn new(empty: T, rows: usize, cols: usize) -> Self {
        Self {
            rows,
            cols,
            v: vec![empty; rows * cols],
        }
    }

    pub fn new_f(rows: usize, cols: usize, mut f: impl FnMut(usize, usize) -> T) -> Self {
        let mut v = Vec::with_capacity(rows * cols);
        for r in 0..rows {
            for c in 0..cols {
                v.push(f(r, c));
            }
        }
        Self { rows, cols, v }
    }

    pub fn rows(&self) -> usize {
        self.rows
    }

    #[allow(clippy::len_without_is_empty)]
    pub fn len(&self) -> usize {
        self.rows()
    }

    pub fn cols(&self) -> usize {
        self.cols
    }

    pub fn swap(&mut self, row1: usize, row2: usize) {
        assert!(row1 < self.rows);
        assert!(row2 < self.rows);
        if row1 != row2 {
            for col in 0..self.cols {
                self.v.swap(row1 * self.cols + col, row2 * self.cols + col);
            }
        }
    }

    pub fn transpose(&self) -> Self {
        Self::new_f(self.cols, self.rows, |r, c| self[c][r].clone())
    }

    pub fn iter(&self) -> Iter<T> {
        Iter {
            array: self,
            row: 0,
            col: 0,
        }
    }

    pub fn pref_sum(&self) -> Self
    where
        T: Number,
    {
        let mut res = Self::new(T::ZERO, self.rows + 1, self.cols + 1);
        for i in 0..self.rows {
            for j in 0..self.cols {
                let value = self[i][j] + res[i][j + 1] + res[i + 1][j] - res[i][j];
                res[i + 1][j + 1] = value;
            }
        }
        res
    }
}

impl<T> Writable for Array2D<T>
where
    T: Writable,
{
    fn write(&self, output: &mut Output) {
        for r in 0..self.rows {
            self[r].write(output);
            output.write_all(&[b'\n']).unwrap();
        }
    }
}

impl<T> Index<usize> for Array2D<T> {
    type Output = [T];

    fn index(&self, index: usize) -> &Self::Output {
        &self.v[(index) * self.cols..(index + 1) * self.cols]
    }
}

impl<T> IndexMut<usize> for Array2D<T> {
    fn index_mut(&mut self, index: usize) -> &mut Self::Output {
        &mut self.v[(index) * self.cols..(index + 1) * self.cols]
    }
}

impl<T> Mul for &Array2D<T>
where
    T: Number,
{
    type Output = Array2D<T>;

    fn mul(self, rhs: Self) -> Self::Output {
        let n = self.rows;
        let m = self.cols;
        assert_eq!(m, rhs.rows);
        let k2 = rhs.cols;
        let mut res = Array2D::new(T::ZERO, n, k2);
        for i in 0..n {
            for j in 0..m {
                for k in 0..k2 {
                    res[i][k] += self[i][j] * rhs[j][k];
                }
            }
        }
        res
    }
}

impl<T> Array2D<T>
where
    T: Number,
{
    pub fn pown(&self, pw: usize) -> Self {
        assert_eq!(self.rows, self.cols);
        let n = self.rows;
        if pw == 0 {
            Self::new_f(n, n, |r, c| if r == c { T::ONE } else { T::ZERO })
        } else if pw == 1 {
            self.clone()
        } else {
            let half = self.pown(pw / 2);
            let half2 = &half * &half;
            if pw & 1 == 0 {
                half2
            } else {
                &half2 * self
            }
        }
    }
}

impl<'a, T> Iterator for Iter<'a, T> {
    type Item = &'a T;

    fn next(&mut self) -> Option<Self::Item> {
        if self.col == self.array.cols {
            self.col = 0;
            self.row += 1;
        }
        if self.row >= self.array.rows {
            return None;
        }
        let elem = &self.array[self.row][self.col];
        self.col += 1;
        Some(elem)
    }
}
}
}
pub mod geometry {
pub mod point {
use crate::algo_lib::collections::array_2d::Array2D;
use crate::f;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use crate::algo_lib::iters::shifts::Shift;
use crate::algo_lib::misc::num_traits::Number;
use crate::algo_lib::misc::ord_f64::OrdF64;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Mul;
use std::ops::Sub;
use std::ops::SubAssign;

#[derive(Copy, Clone, Eq, PartialEq, Hash, Debug, Default)]
pub struct PointT<T: Number> {
    pub x: T,
    pub y: T,
}

impl<T: Ord + Number> Ord for PointT<T> {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        self.x.cmp(&other.x).then(self.y.cmp(&other.y))
    }
}

impl<T: Ord + Number> PartialOrd for PointT<T> {
    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
        self.cmp(other).into()
    }
}

impl<T: Number> PointT<T> {
    pub fn new(x: T, y: T) -> Self {
        Self { x, y }
    }

    pub fn dist2(&self, p2: &PointT<T>) -> T {
        let dx = self.x - p2.x;
        let dy = self.y - p2.y;
        dx * dx + dy * dy
    }

    pub fn side(&self) -> i32 {
        if self.y > T::ZERO || (self.y == T::ZERO && self.x >= T::ZERO) {
            return 0;
        }
        1
    }

    pub fn dist_manh(&self, p2: &PointT<T>) -> T {
        let dx = self.x - p2.x;
        let dy = self.y - p2.y;
        let dx_abs = if dx < T::ZERO { T::ZERO - dx } else { dx };
        let dy_abs = if dy < T::ZERO { T::ZERO - dy } else { dy };
        dx_abs + dy_abs
    }

    pub fn angle_to(&self, other: &PointT<T>) -> OrdF64
    where
        f64: From<T>,
    {
        let dy = other.y - self.y;
        let dx = other.x - self.x;
        OrdF64(f64::atan2(dy.into(), dx.into()))
    }

    pub fn swap_x_y(&self) -> Self {
        Self::new(self.y, self.x)
    }

    pub fn vect_mul(p1: &PointT<T>, p2: &PointT<T>, p3: &PointT<T>) -> T {
        (p2.x - p1.x) * (p3.y - p1.y) - (p2.y - p1.y) * (p3.x - p1.x)
    }

    pub fn scal_mul(p1: &PointT<T>, p2: &PointT<T>, p3: &PointT<T>) -> T {
        Self::scal_mul2(&(*p2 - *p1), &(*p3 - *p1))
    }

    pub fn scal_mul2(p1: &PointT<T>, p2: &PointT<T>) -> T {
        p1.x * p2.x + p1.y * p2.y
    }

    pub fn vect_mul2(p1: &PointT<T>, p2: &PointT<T>) -> T {
        p1.x * p2.y - p1.y * p2.x
    }

    pub fn apply_shift(&self, shift: &Shift) -> Self {
        Self {
            x: self.x + T::from_i32(shift.dx),
            y: self.y + T::from_i32(shift.dy),
        }
    }

    pub fn shift(&self, dx: T, dy: T) -> Self {
        Self {
            x: self.x + dx,
            y: self.y + dy,
        }
    }

    pub fn scale(&self, coef: T) -> Self {
        Self {
            x: self.x * coef,
            y: self.y * coef,
        }
    }

    pub fn index_vec2d<'a, Elem>(&self, arr: &'a [Vec<Elem>]) -> Option<&'a Elem> {
        if self.x >= T::ZERO
            && self.x < T::from_i32(arr.len() as i32)
            && self.y >= T::ZERO
            && self.y < T::from_i32(arr[T::to_i32(self.x) as usize].len() as i32)
        {
            let x = T::to_i32(self.x) as usize;
            let y = T::to_i32(self.y) as usize;
            Some(&arr[x][y])
        } else {
            None
        }
    }

    pub fn index_arr2d<'a, Elem>(&self, arr: &'a Array2D<Elem>) -> Option<&'a Elem>
    where
        Elem: Clone,
    {
        if self.x >= T::ZERO
            && self.x < T::from_i32(arr.len() as i32)
            && self.y >= T::ZERO
            && self.y < T::from_i32(arr[T::to_i32(self.x) as usize].len() as i32)
        {
            let x = T::to_i32(self.x) as usize;
            let y = T::to_i32(self.y) as usize;
            Some(&arr[x][y])
        } else {
            None
        }
    }

    pub fn rotate_ccw(&self) -> Self {
        Self::new(T::ZERO - self.y, self.x)
    }

    pub const ZERO: PointT<T> = PointT {
        x: T::ZERO,
        y: T::ZERO,
    };

    pub fn conv_float(&self) -> PointT<OrdF64> {
        PointT::new(OrdF64(self.x.to_f64()), OrdF64(self.y.to_f64()))
    }
}

impl<T> Add for PointT<T>
where
    T: Number,
{
    type Output = Self;

    fn add(self, rhs: Self) -> Self::Output {
        Self::new(self.x + rhs.x, self.y + rhs.y)
    }
}

impl<T> AddAssign for PointT<T>
where
    T: Number,
{
    fn add_assign(&mut self, rhs: Self) {
        self.x += rhs.x;
        self.y += rhs.y;
    }
}

impl<T> Sub for PointT<T>
where
    T: Number,
{
    type Output = Self;

    fn sub(self, rhs: Self) -> Self::Output {
        Self::new(self.x - rhs.x, self.y - rhs.y)
    }
}

impl<T> SubAssign for PointT<T>
where
    T: Number,
{
    fn sub_assign(&mut self, rhs: Self) {
        self.x -= rhs.x;
        self.y -= rhs.y;
    }
}

impl<T> Readable for PointT<T>
where
    T: Number + Readable,
{
    fn read(input: &mut Input) -> Self {
        let x = input.read();
        let y = input.read();
        Self { x, y }
    }
}

impl<T> Writable for PointT<T>
where
    T: Number + Writable,
{
    fn write(&self, output: &mut Output) {
        self.x.write(output);
        output.put(b' ');
        self.y.write(output);
    }
}

#[derive(Copy, Clone, Eq, PartialEq, Hash, Debug)]
pub struct PointWithIdT<T: Number> {
    pub p: PointT<T>,
    id: u32,
}

impl<T> PointWithIdT<T>
where
    T: Number,
{
    pub fn new(p: PointT<T>, id: usize) -> Self {
        Self { p, id: id as u32 }
    }

    pub fn id(&self) -> usize {
        self.id as usize
    }
}

impl PointWithIdT<OrdF64> {
    pub fn dist(&self, other: &Self) -> OrdF64 {
        self.p.dist2(&other.p).sqrt()
    }
}

impl PointT<OrdF64> {
    pub fn rotate_ccw_angle(&self, angle: OrdF64) -> Self {
        let cos = f!(angle.0.cos());
        let sin = f!(angle.0.sin());
        let x = self.x * cos - self.y * sin;
        let y = self.y * cos + self.x * sin;
        Self { x, y }
    }
}

impl Mul<OrdF64> for PointT<OrdF64> {
    type Output = PointT<OrdF64>;

    fn mul(self, rhs: OrdF64) -> Self::Output {
        Self {
            x: self.x * rhs,
            y: self.y * rhs,
        }
    }
}
}
}
pub mod io {
pub mod input {
use std::fmt::Debug;
use std::io::Read;
use std::marker::PhantomData;
use std::path::Path;
use std::str::FromStr;

pub struct Input {
    input: Box<dyn Read>,
    buf: Vec<u8>,
    at: usize,
    buf_read: usize,
}

macro_rules! read_integer_fun {
    ($t:ident) => {
        #[allow(unused)]
        pub fn $t(&mut self) -> $t {
            self.read_integer()
        }
    };
}

impl Input {
    const DEFAULT_BUF_SIZE: usize = 4096;

    ///
    /// Using with stdin:
    /// ```no_run
    /// use algo_lib::io::input::Input;
    /// let stdin = std::io::stdin();
    /// let input = Input::new(Box::new(stdin));
    /// ```
    ///
    /// For read files use ``new_file`` instead.
    ///
    ///
    pub fn new(input: Box<dyn Read>) -> Self {
        Self {
            input,
            buf: vec![0; Self::DEFAULT_BUF_SIZE],
            at: 0,
            buf_read: 0,
        }
    }

    pub fn new_stdin() -> Self {
        let stdin = std::io::stdin();
        Self::new(Box::new(stdin))
    }

    pub fn new_file<P: AsRef<Path>>(path: P) -> Self {
        let file = std::fs::File::open(&path)
            .unwrap_or_else(|_| panic!("Can't open file: {:?}", path.as_ref().as_os_str()));
        Self::new(Box::new(file))
    }

    pub fn new_with_size(input: Box<dyn Read>, buf_size: usize) -> Self {
        Self {
            input,
            buf: vec![0; buf_size],
            at: 0,
            buf_read: 0,
        }
    }

    pub fn new_file_with_size<P: AsRef<Path>>(path: P, buf_size: usize) -> Self {
        let file = std::fs::File::open(&path)
            .unwrap_or_else(|_| panic!("Can't open file: {:?}", path.as_ref().as_os_str()));
        Self::new_with_size(Box::new(file), buf_size)
    }

    pub fn get(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            let res = self.buf[self.at];
            self.at += 1;
            Some(res)
        } else {
            None
        }
    }

    pub fn peek(&mut self) -> Option<u8> {
        if self.refill_buffer() {
            Some(self.buf[self.at])
        } else {
            None
        }
    }

    pub fn skip_whitespace(&mut self) {
        while let Some(b) = self.peek() {
            if !char::from(b).is_whitespace() {
                return;
            }
            self.get();
        }
    }

    pub fn next_token(&mut self) -> Option<Vec<u8>> {
        self.skip_whitespace();
        let mut res = Vec::new();
        while let Some(c) = self.get() {
            if char::from(c).is_whitespace() {
                break;
            }
            res.push(c);
        }
        if res.is_empty() {
            None
        } else {
            Some(res)
        }
    }

    //noinspection RsSelfConvention
    pub fn is_exhausted(&mut self) -> bool {
        self.peek().is_none()
    }

    pub fn has_more_elements(&mut self) -> bool {
        !self.is_exhausted()
    }

    pub fn read<T: Readable>(&mut self) -> T {
        T::read(self)
    }

    pub fn vec<T: Readable>(&mut self, size: usize) -> Vec<T> {
        let mut res = Vec::with_capacity(size);
        for _ in 0usize..size {
            res.push(self.read());
        }
        res
    }

    pub fn string_vec(&mut self, size: usize) -> Vec<Vec<u8>> {
        let mut res = Vec::with_capacity(size);
        for _ in 0usize..size {
            res.push(self.string());
        }
        res
    }

    pub fn read_line(&mut self) -> String {
        let mut res = String::new();
        while let Some(c) = self.get() {
            if c == b'\n' {
                break;
            }
            if c == b'\r' {
                if self.peek() == Some(b'\n') {
                    self.get();
                }
                break;
            }
            res.push(c.into());
        }
        res
    }

    #[allow(clippy::should_implement_trait)]
    pub fn into_iter<T: Readable>(self) -> InputIterator<T> {
        InputIterator {
            input: self,
            phantom: Default::default(),
        }
    }

    fn read_integer<T: FromStr>(&mut self) -> T
    where
        <T as FromStr>::Err: Debug,
    {
        let res = self.read_string();
        res.parse::<T>().unwrap()
    }

    fn read_string(&mut self) -> String {
        match self.next_token() {
            None => {
                panic!("Input exhausted");
            }
            Some(res) => unsafe { String::from_utf8_unchecked(res) },
        }
    }

    pub fn string_as_string(&mut self) -> String {
        self.read_string()
    }

    pub fn string(&mut self) -> Vec<u8> {
        self.read_string().into_bytes()
    }

    fn read_char(&mut self) -> char {
        self.skip_whitespace();
        self.get().unwrap().into()
    }

    fn read_float(&mut self) -> f64 {
        self.read_string().parse().unwrap()
    }

    pub fn f64(&mut self) -> f64 {
        self.read_float()
    }

    fn refill_buffer(&mut self) -> bool {
        if self.at == self.buf_read {
            self.at = 0;
            self.buf_read = self.input.read(&mut self.buf).unwrap();
            self.buf_read != 0
        } else {
            true
        }
    }

    read_integer_fun!(i32);
    read_integer_fun!(i64);
    read_integer_fun!(i128);
    read_integer_fun!(u32);
    read_integer_fun!(u64);
    read_integer_fun!(usize);
}

pub trait Readable {
    fn read(input: &mut Input) -> Self;
}

impl Readable for String {
    fn read(input: &mut Input) -> Self {
        input.read_string()
    }
}

impl Readable for char {
    fn read(input: &mut Input) -> Self {
        input.read_char()
    }
}

impl Readable for f64 {
    fn read(input: &mut Input) -> Self {
        input.read_string().parse().unwrap()
    }
}

impl Readable for f32 {
    fn read(input: &mut Input) -> Self {
        input.read_string().parse().unwrap()
    }
}

impl<T: Readable> Readable for Vec<T> {
    fn read(input: &mut Input) -> Self {
        let size = input.read();
        input.vec(size)
    }
}

pub struct InputIterator<T: Readable> {
    input: Input,
    phantom: PhantomData<T>,
}

impl<T: Readable> Iterator for InputIterator<T> {
    type Item = T;

    fn next(&mut self) -> Option<Self::Item> {
        self.input.skip_whitespace();
        self.input.peek().map(|_| self.input.read())
    }
}

macro_rules! read_integer {
    ($t:ident) => {
        impl Readable for $t {
            fn read(input: &mut Input) -> Self {
                input.read_integer()
            }
        }
    };
}

read_integer!(i8);
read_integer!(i16);
read_integer!(i32);
read_integer!(i64);
read_integer!(i128);
read_integer!(isize);
read_integer!(u8);
read_integer!(u16);
read_integer!(u32);
read_integer!(u64);
read_integer!(u128);
read_integer!(usize);
}
pub mod output {
use std::io::Write;

pub struct Output {
    output: Box<dyn Write>,
    buf: Vec<u8>,
    at: usize,
    auto_flush: bool,
}

impl Output {
    const DEFAULT_BUF_SIZE: usize = 4096;

    pub fn new(output: Box<dyn Write>) -> Self {
        Self {
            output,
            buf: vec![0; Self::DEFAULT_BUF_SIZE],
            at: 0,
            auto_flush: false,
        }
    }

    pub fn new_stdout() -> Self {
        let stdout = std::io::stdout();
        Self::new(Box::new(stdout))
    }

    pub fn new_file(path: impl AsRef<std::path::Path>) -> Self {
        let file = std::fs::File::create(path).unwrap();
        Self::new(Box::new(file))
    }

    pub fn new_with_auto_flush(output: Box<dyn Write>) -> Self {
        Self {
            output,
            buf: vec![0; Self::DEFAULT_BUF_SIZE],
            at: 0,
            auto_flush: true,
        }
    }

    pub fn flush(&mut self) {
        if self.at != 0 {
            self.output.write_all(&self.buf[..self.at]).unwrap();
            self.at = 0;
            self.output.flush().expect("Couldn't flush output");
        }
    }

    pub fn print<T: Writable>(&mut self, s: T) {
        s.write(self);
    }

    pub fn println<T: Writable>(&mut self, s: T) {
        s.write(self);
        self.put(b'\n');
    }

    pub fn put(&mut self, b: u8) {
        self.buf[self.at] = b;
        self.at += 1;
        if self.at == self.buf.len() {
            self.flush();
        }
    }

    pub fn maybe_flush(&mut self) {
        if self.auto_flush {
            self.flush();
        }
    }

    pub fn print_per_line<T: Writable>(&mut self, arg: &[T]) {
        for i in arg {
            i.write(self);
            self.put(b'\n');
        }
    }

    pub fn print_iter<T: Writable, I: Iterator<Item = T>>(&mut self, iter: I) {
        let mut first = true;
        for e in iter {
            if first {
                first = false;
            } else {
                self.put(b' ');
            }
            e.write(self);
        }
    }

    pub fn print_iter_ref<'a, T: 'a + Writable, I: Iterator<Item = &'a T>>(&mut self, iter: I) {
        let mut first = true;
        for e in iter {
            if first {
                first = false;
            } else {
                self.put(b' ');
            }
            e.write(self);
        }
    }
}

impl Write for Output {
    fn write(&mut self, buf: &[u8]) -> std::io::Result<usize> {
        let mut start = 0usize;
        let mut rem = buf.len();
        while rem > 0 {
            let len = (self.buf.len() - self.at).min(rem);
            self.buf[self.at..self.at + len].copy_from_slice(&buf[start..start + len]);
            self.at += len;
            if self.at == self.buf.len() {
                self.flush();
            }
            start += len;
            rem -= len;
        }
        if self.auto_flush {
            self.flush();
        }
        Ok(buf.len())
    }

    fn flush(&mut self) -> std::io::Result<()> {
        self.flush();
        Ok(())
    }
}

pub trait Writable {
    fn write(&self, output: &mut Output);
}

impl Writable for &str {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}

impl Writable for String {
    fn write(&self, output: &mut Output) {
        output.write_all(self.as_bytes()).unwrap();
    }
}

impl Writable for char {
    fn write(&self, output: &mut Output) {
        output.put(*self as u8);
    }
}

impl<T: Writable> Writable for [T] {
    fn write(&self, output: &mut Output) {
        output.print_iter_ref(self.iter());
    }
}

impl<T: Writable> Writable for Vec<T> {
    fn write(&self, output: &mut Output) {
        self[..].write(output);
    }
}

macro_rules! write_to_string {
    ($t:ident) => {
        impl Writable for $t {
            fn write(&self, output: &mut Output) {
                self.to_string().write(output);
            }
        }
    };
}

write_to_string!(u8);
write_to_string!(u16);
write_to_string!(u32);
write_to_string!(u64);
write_to_string!(u128);
write_to_string!(usize);
write_to_string!(i8);
write_to_string!(i16);
write_to_string!(i32);
write_to_string!(i64);
write_to_string!(i128);
write_to_string!(isize);
write_to_string!(f32);
write_to_string!(f64);

impl<T: Writable, U: Writable> Writable for (T, U) {
    fn write(&self, output: &mut Output) {
        self.0.write(output);
        output.put(b' ');
        self.1.write(output);
    }
}

impl<T: Writable, U: Writable, V: Writable> Writable for (T, U, V) {
    fn write(&self, output: &mut Output) {
        self.0.write(output);
        output.put(b' ');
        self.1.write(output);
        output.put(b' ');
        self.2.write(output);
    }
}
}
}
pub mod iters {
pub mod shifts {
#[derive(Copy, Clone, Hash, Ord, PartialOrd, Eq, PartialEq)]
pub struct Shift {
    pub dx: i32,
    pub dy: i32,
}

impl Shift {
    pub fn rev(&self) -> Self {
        Self {
            dx: -self.dx,
            dy: -self.dy,
        }
    }
}

// x goes down
// y goes right
pub const SHIFT_DOWN: Shift = Shift { dx: 1, dy: 0 };
pub const SHIFT_UP: Shift = Shift { dx: -1, dy: 0 };
pub const SHIFT_RIGHT: Shift = Shift { dx: 0, dy: 1 };
pub const SHIFT_LEFT: Shift = Shift { dx: 0, dy: -1 };

pub const SHIFTS_4: [Shift; 4] = [SHIFT_DOWN, SHIFT_LEFT, SHIFT_UP, SHIFT_RIGHT];
pub const SHIFTS_8: [Shift; 8] = [
    SHIFT_DOWN,
    SHIFT_LEFT,
    SHIFT_UP,
    SHIFT_RIGHT,
    Shift { dx: -1, dy: -1 },
    Shift { dx: -1, dy: 1 },
    Shift { dx: 1, dy: -1 },
    Shift { dx: 1, dy: 1 },
];

pub const SHIFTS_9: [Shift; 9] = [
    SHIFT_DOWN,
    SHIFT_LEFT,
    SHIFT_UP,
    SHIFT_RIGHT,
    Shift { dx: -1, dy: -1 },
    Shift { dx: -1, dy: 1 },
    Shift { dx: 1, dy: -1 },
    Shift { dx: 1, dy: 1 },
    Shift { dx: 0, dy: 0 },
];

pub fn shift_by_nswe(c: u8) -> Shift {
    match c {
        b'S' | b's' => SHIFT_DOWN,
        b'N' | b'n' => SHIFT_UP,
        b'E' | b'e' => SHIFT_RIGHT,
        b'W' | b'w' => SHIFT_LEFT,
        _ => panic!("Unexpected direction!"),
    }
}

pub fn shift_by_uldr(c: u8) -> Shift {
    match c {
        b'D' | b'd' => SHIFT_DOWN,
        b'U' | b'u' => SHIFT_UP,
        b'R' | b'r' => SHIFT_RIGHT,
        b'L' | b'l' => SHIFT_LEFT,
        _ => panic!("Unexpected direction!"),
    }
}
}
}
pub mod misc {
pub mod dbg_macro {
#[macro_export]
#[allow(unused_macros)]
macro_rules! dbg {
    ($first_val:expr, $($val:expr),+ $(,)?) => {
        eprint!("[{}:{}] {} = {:?}",
                    file!(), line!(), stringify!($first_val), &$first_val);
        ($(eprint!(", {} = {:?}", stringify!($val), &$val)),+,);
        eprintln!();
    };
    ($first_val:expr) => {
        eprintln!("[{}:{}] {} = {:?}",
                    file!(), line!(), stringify!($first_val), &$first_val)
    };
}
}
pub mod gen_vector {
pub fn gen_vec<T>(n: usize, f: impl FnMut(usize) -> T) -> Vec<T> {
    (0..n).map(f).collect()
}
}
pub mod num_traits {
use std::cmp::Ordering;
use std::fmt::Debug;
use std::ops::Add;
use std::ops::AddAssign;
use std::ops::Div;
use std::ops::DivAssign;
use std::ops::Mul;
use std::ops::MulAssign;
use std::ops::Sub;
use std::ops::SubAssign;

pub trait HasConstants<T> {
    const MAX: T;
    const MIN: T;
    const ZERO: T;
    const ONE: T;
    const TWO: T;
}

pub trait ConvSimple<T> {
    fn from_i32(val: i32) -> T;
    fn to_i32(self) -> i32;
    fn to_f64(self) -> f64;
}

pub trait Signum {
    fn signum(&self) -> i32;
}

pub trait Number:
    Copy
    + Add<Output = Self>
    + AddAssign
    + Sub<Output = Self>
    + SubAssign
    + Mul<Output = Self>
    + MulAssign
    + Div<Output = Self>
    + DivAssign
    + PartialOrd
    + PartialEq
    + HasConstants<Self>
    + Default
    + Debug
    + Sized
    + ConvSimple<Self>
{
}

impl<
        T: Copy
            + Add<Output = Self>
            + AddAssign
            + Sub<Output = Self>
            + SubAssign
            + Mul<Output = Self>
            + MulAssign
            + Div<Output = Self>
            + DivAssign
            + PartialOrd
            + PartialEq
            + HasConstants<Self>
            + Default
            + Debug
            + Sized
            + ConvSimple<Self>,
    > Number for T
{
}

macro_rules! has_constants_impl {
    ($t: ident) => {
        impl HasConstants<$t> for $t {
            // TODO: remove `std` for new rust version..
            const MAX: $t = std::$t::MAX;
            const MIN: $t = std::$t::MIN;
            const ZERO: $t = 0;
            const ONE: $t = 1;
            const TWO: $t = 2;
        }

        impl ConvSimple<$t> for $t {
            fn from_i32(val: i32) -> $t {
                val as $t
            }

            fn to_i32(self) -> i32 {
                self as i32
            }

            fn to_f64(self) -> f64 {
                self as f64
            }
        }
    };
}

has_constants_impl!(i32);
has_constants_impl!(i64);
has_constants_impl!(i128);
has_constants_impl!(u32);
has_constants_impl!(u64);
has_constants_impl!(u128);
has_constants_impl!(usize);
has_constants_impl!(u8);

impl ConvSimple<Self> for f64 {
    fn from_i32(val: i32) -> Self {
        val as f64
    }

    fn to_i32(self) -> i32 {
        self as i32
    }

    fn to_f64(self) -> f64 {
        self
    }
}

impl HasConstants<Self> for f64 {
    const MAX: Self = Self::MAX;
    const MIN: Self = -Self::MAX;
    const ZERO: Self = 0.0;
    const ONE: Self = 1.0;
    const TWO: Self = 2.0;
}

impl<T: Number + Ord> Signum for T {
    fn signum(&self) -> i32 {
        match self.cmp(&T::ZERO) {
            Ordering::Greater => 1,
            Ordering::Less => -1,
            Ordering::Equal => 0,
        }
    }
}
}
pub mod ord_f64 {
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::input::Readable;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::io::output::Writable;
use crate::algo_lib::misc::num_traits::ConvSimple;
use crate::algo_lib::misc::num_traits::HasConstants;
use std::cmp::min;
use std::cmp::Ordering;
use std::f64::consts::PI;
use std::fmt::Debug;
use std::fmt::Display;
use std::fmt::Formatter;
use std::io::Write;
use std::num::ParseFloatError;
use std::ops::Neg;
use std::ops::Rem;
use std::str::FromStr;

#[derive(PartialEq, Copy, Clone, Default)]
pub struct OrdF64(pub f64);

impl OrdF64 {
    pub const EPS: Self = Self(1e-9);
    pub const SMALL_EPS: Self = Self(1e-4);
    pub const PI: Self = Self(PI);

    pub fn abs(&self) -> Self {
        Self(self.0.abs())
    }

    pub fn eq_with_eps(&self, other: &Self, eps: Self) -> bool {
        let abs_diff = (*self - *other).abs();
        abs_diff <= eps || abs_diff <= min(self.abs(), other.abs()) * eps
    }

    pub fn eq_with_default_eps(&self, other: &Self) -> bool {
        self.eq_with_eps(other, Self::EPS)
    }

    pub fn sqrt(&self) -> Self {
        Self(self.0.sqrt())
    }

    pub fn powf(&self, n: f64) -> Self {
        Self(self.0.powf(n))
    }
}

impl Eq for OrdF64 {}

impl Ord for OrdF64 {
    fn cmp(&self, other: &Self) -> Ordering {
        self.partial_cmp(other).unwrap()
    }
}

impl PartialOrd for OrdF64 {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        self.0.partial_cmp(&other.0)
    }
}

impl std::ops::Add for OrdF64 {
    type Output = Self;

    fn add(self, rhs: Self) -> Self::Output {
        Self(self.0 + rhs.0)
    }
}

impl std::ops::AddAssign for OrdF64 {
    fn add_assign(&mut self, rhs: Self) {
        self.0 += rhs.0;
    }
}

impl std::ops::Sub for OrdF64 {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self::Output {
        Self(self.0 - rhs.0)
    }
}

impl std::ops::SubAssign for OrdF64 {
    fn sub_assign(&mut self, rhs: Self) {
        self.0 -= rhs.0;
    }
}

impl std::ops::Mul for OrdF64 {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self::Output {
        Self(self.0 * rhs.0)
    }
}

impl std::ops::MulAssign for OrdF64 {
    fn mul_assign(&mut self, rhs: Self) {
        self.0 *= rhs.0;
    }
}

impl std::ops::Div for OrdF64 {
    type Output = Self;

    fn div(self, rhs: Self) -> Self::Output {
        Self(self.0 / rhs.0)
    }
}

impl std::ops::DivAssign for OrdF64 {
    fn div_assign(&mut self, rhs: Self) {
        self.0 /= rhs.0;
    }
}

impl Neg for OrdF64 {
    type Output = Self;

    fn neg(self) -> Self::Output {
        Self(-self.0)
    }
}

impl Display for OrdF64 {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        Display::fmt(&self.0, f)
    }
}

impl Debug for OrdF64 {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        Debug::fmt(&self.0, f)
    }
}

impl Writable for OrdF64 {
    fn write(&self, output: &mut Output) {
        output.write_fmt(format_args!("{}", self.0)).unwrap();
    }
}

impl Readable for OrdF64 {
    fn read(input: &mut Input) -> Self {
        Self(input.read::<f64>())
    }
}

impl HasConstants<Self> for OrdF64 {
    const MAX: Self = Self(f64::MAX);
    const MIN: Self = Self(-f64::MAX);
    const ZERO: Self = Self(0.0);
    const ONE: Self = Self(1.0);
    const TWO: Self = Self(2.0);
}

impl ConvSimple<Self> for OrdF64 {
    fn from_i32(val: i32) -> Self {
        Self(val as f64)
    }

    fn to_i32(self) -> i32 {
        self.0 as i32
    }

    fn to_f64(self) -> f64 {
        self.0
    }
}

impl FromStr for OrdF64 {
    type Err = ParseFloatError;

    fn from_str(s: &str) -> Result<Self, Self::Err> {
        match s.parse::<f64>() {
            Ok(value) => Ok(Self(value)),
            Err(error) => Err(error),
        }
    }
}

impl From<OrdF64> for f64 {
    fn from(x: OrdF64) -> Self {
        x.0
    }
}

impl Rem for OrdF64 {
    type Output = Self;

    fn rem(self, rhs: Self) -> Self::Output {
        Self(self.0 % rhs.0)
    }
}

#[macro_export]
macro_rules! f {
    ($a:expr) => {
        OrdF64($a)
    };
}

impl From<usize> for OrdF64 {
    fn from(x: usize) -> Self {
        f!(x as f64)
    }
}

impl From<i32> for OrdF64 {
    fn from(x: i32) -> Self {
        f!(x as f64)
    }
}

impl From<i64> for OrdF64 {
    fn from(x: i64) -> Self {
        f!(x as f64)
    }
}

impl From<f64> for OrdF64 {
    fn from(x: f64) -> Self {
        f!(x)
    }
}
}
pub mod rand {
use crate::algo_lib::misc::gen_vector::gen_vec;
use crate::algo_lib::misc::num_traits::Number;
use std::ops::Range;
use std::time::SystemTime;
use std::time::UNIX_EPOCH;

#[allow(dead_code)]
pub struct Random {
    state: u64,
}

impl Random {
    pub fn gen_u64(&mut self) -> u64 {
        let mut x = self.state;
        x ^= x << 13;
        x ^= x >> 7;
        x ^= x << 17;
        self.state = x;
        x
    }

    #[allow(dead_code)]
    pub fn next_in_range(&mut self, from: usize, to: usize) -> usize {
        assert!(from < to);
        (from as u64 + self.gen_u64() % ((to - from) as u64)) as usize
    }

    pub fn gen_index<T>(&mut self, a: &[T]) -> usize {
        self.gen(0..a.len())
    }

    #[allow(dead_code)]
    #[inline(always)]
    pub fn gen_double(&mut self) -> f64 {
        (self.gen_u64() as f64) / (std::usize::MAX as f64)
    }

    #[allow(dead_code)]
    pub fn new(seed: u64) -> Self {
        let state = if seed == 0 { 787788 } else { seed };
        Self { state }
    }

    pub fn new_time_seed() -> Self {
        let time = SystemTime::now();
        let seed = (time.duration_since(UNIX_EPOCH).unwrap().as_nanos() % 1_000_000_000) as u64;
        if seed == 0 {
            Self::new(787788)
        } else {
            Self::new(seed)
        }
    }

    #[allow(dead_code)]
    pub fn gen_permutation(&mut self, n: usize) -> Vec<usize> {
        let mut result: Vec<_> = (0..n).collect();
        for i in 0..n {
            let idx = self.next_in_range(0, i + 1);
            result.swap(i, idx);
        }
        result
    }

    pub fn shuffle<T>(&mut self, a: &mut [T]) {
        for i in 1..a.len() {
            a.swap(i, self.gen(0..i + 1));
        }
    }

    pub fn gen<T>(&mut self, range: Range<T>) -> T
    where
        T: Number,
    {
        let from = T::to_i32(range.start);
        let to = T::to_i32(range.end);
        assert!(from < to);
        let len = (to - from) as usize;
        T::from_i32(self.next_in_range(0, len) as i32 + from)
    }

    pub fn gen_vec<T>(&mut self, n: usize, range: Range<T>) -> Vec<T>
    where
        T: Number,
    {
        gen_vec(n, |_| self.gen(range.clone()))
    }

    pub fn gen_nonempty_range(&mut self, n: usize) -> Range<usize> {
        let x = self.gen(0..n);
        let y = self.gen(0..n);
        if x <= y {
            x..y + 1
        } else {
            y..x + 1
        }
    }

    pub fn gen_bool(&mut self) -> bool {
        self.gen(0..2) == 0
    }
}
}
}
pub mod seg_trees {
pub mod bottom_up_seg_tree {
use std::ops::Range;

use crate::algo_lib::seg_trees::seg_tree_trait::SegTreeNode;

pub struct BottomUpSegTree<Node: SegTreeNode> {
    n: usize,
    nodes: Vec<Node>,
    context: Node::Context,
}

impl<Node: SegTreeNode> BottomUpSegTree<Node> {
    pub fn new(start_n: usize, f: impl Fn(usize) -> Node) -> Self
    where
        Node::Context: Default,
    {
        let n = start_n.next_power_of_two();
        let mut res = Self {
            n,
            nodes: vec![Node::default(); 2 * n],
            context: Default::default(),
        };
        for i in 0..start_n {
            res.nodes[n + i] = f(i);
        }
        for i in (1..n).rev() {
            res.nodes[i] = Node::join_nodes(&res.nodes[2 * i], &res.nodes[2 * i + 1], &res.context);
        }
        res
    }

    pub fn update_point(&mut self, pos: usize, v: Node) {
        let mut i = pos + self.n;
        self.nodes[i] = v;
        while i > 1 {
            i /= 2;
            self.nodes[i] =
                Node::join_nodes(&self.nodes[2 * i], &self.nodes[2 * i + 1], &self.context);
        }
    }

    pub fn get_root(&self) -> &Node {
        &self.nodes[1]
    }

    pub fn get(&self, range: Range<usize>) -> Node {
        let mut l = range.start + self.n;
        let mut r = range.end + self.n;
        let mut res_l = Node::default();
        let mut res_r = Node::default();
        while l < r {
            if l & 1 != 0 {
                res_l = Node::join_nodes(&res_l, &self.nodes[l], &self.context);
                l += 1;
            }
            if r & 1 != 0 {
                r -= 1;
                res_r = Node::join_nodes(&self.nodes[r], &res_r, &self.context);
            }
            l /= 2;
            r /= 2;
        }
        Node::join_nodes(&res_l, &res_r, &self.context)
    }
}
}
pub mod seg_tree_trait {
pub trait SegTreeNode: Clone + Default {
    fn join_nodes(l: &Self, r: &Self, context: &Self::Context) -> Self;

    fn apply_update(node: &mut Self, update: &Self::Update);
    fn join_updates(current: &mut Self::Update, add: &Self::Update);

    type Update: Clone;
    type Context;
}
}
}
}
fn main() {
    let input = algo_lib::io::input::Input::new_stdin();
    let mut output = algo_lib::io::output::Output::new_stdout();
    crate::solution::run(input, output);
}