// 
pub mod solution {
//{"name":"ucup21f","group":"Manual","url":"","interactive":false,"timeLimit":2000,"tests":[{"input":"","output":""}],"testType":"single","input":{"type":"stdin","fileName":null,"pattern":null},"output":{"type":"stdout","fileName":null,"pattern":null},"languages":{"java":{"taskClass":"ucup21f"}}}

use crate::algo_lib::collections::dsu::DSU;
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::vec_ext::inc_dec::IncDec;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use crate::algo_lib::graph::graph::Graph;
use crate::algo_lib::io::input::Input;
use crate::algo_lib::io::output::Output;
use crate::algo_lib::misc::recursive_function::Callable;
use crate::algo_lib::misc::recursive_function::RecursiveFunction;

type PreCalc = ();

fn solve(input: &mut Input, out: &mut Output, _test_case: usize, _data: &PreCalc) {
let n = input.read_size();
let a = input.read_size_vec(n).dec();

let mut visited = vec![0; n];
let mut dsu = DSU::new(n);
let mut cyc_len = vec![0; n];
for i in 0..n {
if visited[i] != 0 {
continue;
}
let mut j = i;
loop {
if visited[j] != 0 {
break;
}
visited[j] = 1;
j = a[j];
}
if visited[j] == 1 {
let mut k = j;
let mut len = 0;
loop {
visited[k] = 2;
dsu.join(k, a[k]);
k = a[k];
len += 1;
if k == j {
break;
}
}
cyc_len[dsu.get(j)] = len;
}
j = i;
loop {
if visited[j] != 2 {
visited[j] = 2;
} else {
break;
}
j = a[j];
}
}
let mut graph = Graph::new(n);
for i in 0..n {
if dsu.get(i) != dsu.get(a[i]) {
graph.add_edge(Edge::new(dsu.get(a[i]), i));
}
}
let mut paths = Vec::new();
let mut cycles = Vec::new();
for i in 0..n {
if cyc_len[i] != 0 {
let mut dfs = RecursiveFunction::new(|dfs, vert: usize| -> usize {
if graph[vert].len() == 0 {
return 0;
}
let mut cur_paths = Vec::with_capacity(graph[vert].len());
for edge in &graph[vert] {
cur_paths.push(dfs.call(edge.to()) + 1);
}
cur_paths.sort();
cur_paths.reverse();
paths.extend_from_slice(&cur_paths[1..]);
cur_paths[0]
});
let len = dfs.call(i);
cycles.push((cyc_len[i], len));
}
}
let mut ans = Vec::with_capacity(n);
cycles.sort();
let mut at = 0;
for i in 1.. {
while at < cycles.len() && cycles[at].0 + cycles[at].1 < i {
at += 1;
}
if at == cycles.len() {
break;
}
if i >= cycles[at].0 {
ans.push(0);
} else {
ans.push(1);
}
}
let mut cycles = cycles.into_iter().map(|(a, b)| a + b).collect_vec();
cycles.sort();
cycles.reverse();
paths.extend_from_slice(&cycles[1..]);
assert_eq!(ans.len(), cycles[0]);
let mut cur = cycles[0];
paths.sort();
let mut cur_ans = 0;
for i in ans.len() + 1..=n {
if cur < i {
cur += paths.pop().unwrap();
cur_ans += 1;
}
ans.push(cur_ans);
}
out.print_line(ans);
}

pub(crate) fn run(mut input: Input, mut output: Output) -> bool {
let pre_calc = ();

#[allow(dead_code)]
enum TestType {
Single,
MultiNumber,
MultiEof,
}
let test_type = TestType::Single;
match test_type {
TestType::Single => solve(&mut input, &mut output, 1, &pre_calc),
TestType::MultiNumber => {
let t = input.read();
for i in 1..=t {
solve(&mut input, &mut output, i, &pre_calc);
}
}
TestType::MultiEof => {
let mut i = 1;
while input.peek().is_some() {
solve(&mut input, &mut output, i, &pre_calc);
i += 1;
}
}
}
output.flush();
input.skip_whitespace();
input.peek().is_none()
}

}
pub mod algo_lib {
pub mod collections {
pub mod dsu {
use crate::algo_lib::collections::iter_ext::collect::IterCollect;
use crate::algo_lib::collections::slice_ext::bounds::Bounds;
use crate::algo_lib::collections::slice_ext::legacy_fill::LegacyFill;
use std::cell::Cell;

#[derive(Clone)]
pub struct DSU {
id: Vec<Cell<u32>>,
size: Vec<u32>,
count: usize,
}

impl DSU {
pub fn new(n: usize) -> Self {
Self {
id: (0..n).map(|i| Cell::new(i as u32)).collect_vec(),
size: vec![1; n],
count: n,
}
}

pub fn size(&self, i: usize) -> usize {
self.size[self.get(i)] as usize
}

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

pub fn iter(&self) -> impl Iterator<Item = usize> + '_ {
self.id.iter().enumerate().filter_map(|(i, id)| {
if (i as u32) == id.get() {
Some(i)
} else {
None
}
})
}

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

pub fn join(&mut self, mut a: usize, mut b: usize) -> bool {
a = self.get(a);
b = self.get(b);
if a == b {
false
} else {
self.size[a] += self.size[b];
self.id[b].replace(a as u32);
self.count -= 1;
true
}
}

pub fn get(&self, i: usize) -> usize {
if self.id[i].get() != i as u32 {
let res = self.get(self.id[i].get() as usize);
self.id[i].replace(res as u32);
}
self.id[i].get() as usize
}

pub fn clear(&mut self) {
self.count = self.id.len();
self.size.legacy_fill(1);
self.id.iter().enumerate().for_each(|(i, id)| {
id.replace(i as u32);
});
}

pub fn parts(&self) -> Vec<Vec<usize>> {
let roots = self.iter().collect_vec();
let mut res = vec![Vec::new(); roots.len()];
for i in 0..self.id.len() {
res[roots.as_slice().bin_search(&self.get(i)).unwrap()].push(i);
}
res
}
}
}
pub mod iter_ext {
pub mod collect {
pub trait IterCollect<T>: Iterator<Item = T> + Sized {
fn collect_vec(self) -> Vec<T> {
self.collect()
}
}

impl<T, I: Iterator<Item = T> + Sized> IterCollect<T> for I {}
}
}
pub mod slice_ext {
pub mod bounds {
pub trait Bounds<T: PartialOrd> {
fn lower_bound(&self, el: &T) -> usize;
fn upper_bound(&self, el: &T) -> usize;
fn bin_search(&self, el: &T) -> Option<usize>;
fn more(&self, el: &T) -> usize;
fn more_or_eq(&self, el: &T) -> usize;
fn less(&self, el: &T) -> usize;
fn less_or_eq(&self, el: &T) -> usize;
}

impl<T: PartialOrd> Bounds<T> for [T] {
fn lower_bound(&self, el: &T) -> usize {
let mut left = 0;
let mut right = self.len();
while left < right {
let mid = left + ((right - left) >> 1);
if &self[mid] < el {
left = mid + 1;
} else {
right = mid;
}
}
left
}

fn upper_bound(&self, el: &T) -> usize {
let mut left = 0;
let mut right = self.len();
while left < right {
let mid = left + ((right - left) >> 1);
if &self[mid] <= el {
left = mid + 1;
} else {
right = mid;
}
}
left
}

fn bin_search(&self, el: &T) -> Option<usize> {
let at = self.lower_bound(el);
if at == self.len() || &self[at] != el {
None
} else {
Some(at)
}
}

fn more(&self, el: &T) -> usize {
self.len() - self.upper_bound(el)
}

fn more_or_eq(&self, el: &T) -> usize {
self.len() - self.lower_bound(el)
}

fn less(&self, el: &T) -> usize {
self.lower_bound(el)
}

fn less_or_eq(&self, el: &T) -> usize {
self.upper_bound(el)
}
}
}
pub mod legacy_fill {
// 1.50
pub trait LegacyFill<T> {
fn legacy_fill(&mut self, val: T);
}

impl<T: Clone> LegacyFill<T> for [T] {
fn legacy_fill(&mut self, val: T) {
for el in self.iter_mut() {
*el = val.clone();
}
}
}
}
}
pub mod vec_ext {
pub mod default {
pub fn default_vec<T: Default>(len: usize) -> Vec<T> {
let mut v = Vec::with_capacity(len);
for _ in 0..len {
v.push(T::default());
}
v
}
}
pub mod inc_dec {
use crate::algo_lib::numbers::num_traits::algebra::AdditionMonoidWithSub;
use crate::algo_lib::numbers::num_traits::algebra::One;

pub trait IncDec {
#[must_use]
fn inc(self) -> Self;
#[must_use]
fn dec(self) -> Self;
}

impl<T: AdditionMonoidWithSub + One> IncDec for Vec<T> {
fn inc(mut self) -> Self {
self.iter_mut().for_each(|i| *i += T::one());
self
}

fn dec(mut self) -> Self {
self.iter_mut().for_each(|i| *i -= T::one());
self
}
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec for Vec<(T, U)> {
fn inc(mut self) -> Self {
self.iter_mut().for_each(|(i, j)| {
*i += T::one();
*j += U::one();
});
self
}

fn dec(mut self) -> Self {
self.iter_mut().for_each(|(i, j)| {
*i -= T::one();
*j -= U::one();
});
self
}
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V> IncDec for Vec<(T, U, V)> {
fn inc(mut self) -> Self {
self.iter_mut().for_each(|(i, j, _)| {
*i += T::one();
*j += U::one();
});
self
}

fn dec(mut self) -> Self {
self.iter_mut().for_each(|(i, j, _)| {
*i -= T::one();
*j -= U::one();
});
self
}
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W> IncDec
for Vec<(T, U, V, W)>
{
fn inc(mut self) -> Self {
self.iter_mut().for_each(|(i, j, ..)| {
*i += T::one();
*j += U::one();
});
self
}

fn dec(mut self) -> Self {
self.iter_mut().for_each(|(i, j, ..)| {
*i -= T::one();
*j -= U::one();
});
self
}
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One, V, W, X> IncDec
for Vec<(T, U, V, W, X)>
{
fn inc(mut self) -> Self {
self.iter_mut().for_each(|(i, j, ..)| {
*i += T::one();
*j += U::one();
});
self
}

fn dec(mut self) -> Self {
self.iter_mut().for_each(|(i, j, ..)| {
*i -= T::one();
*j -= U::one();
});
self
}
}

impl<T: AdditionMonoidWithSub + One, U: AdditionMonoidWithSub + One> IncDec for (T, U) {
fn inc(mut self) -> Self {
self.0 += T::one();
self.1 += U::one();
self
}

fn dec(mut self) -> Self {
self.0 -= T::one();
self.1 -= U::one();
self
}
}
}
}
}
pub mod graph {
pub mod edges {
pub mod bi_edge {
use crate::algo_lib::graph::edges::bi_edge_trait::BiEdgeTrait;
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct BiEdgeRaw<Id: EdgeId, P> {
to: u32,
id: Id,
payload: P,
}

impl<Id: EdgeId> BiEdgeRaw<Id, ()> {
pub fn new(from: usize, to: usize) -> (usize, Self) {
(
from,
Self {
to: to as u32,
id: Id::new(),
payload: (),
},
)
}
}

impl<Id: EdgeId, P> BiEdgeRaw<Id, P> {
pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
(from, Self::with_payload_impl(to, payload))
}

fn with_payload_impl(to: usize, payload: P) -> BiEdgeRaw<Id, P> {
Self {
to: to as u32,
id: Id::new(),
payload,
}
}
}

impl<Id: EdgeId, P: Clone> BidirectionalEdgeTrait for BiEdgeRaw<Id, P> {}

impl<Id: EdgeId, P: Clone> EdgeTrait for BiEdgeRaw<Id, P> {
type Payload = P;

const REVERSABLE: bool = true;

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

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

fn set_id(&mut self, id: usize) {
self.id.set_id(id);
}

fn reverse_id(&self) -> usize {
panic!("no reverse id")
}

fn set_reverse_id(&mut self, _: usize) {}

fn reverse_edge(&self, from: usize) -> Self {
Self::with_payload_impl(from, self.payload.clone())
}

fn payload(&self) -> &P {
&self.payload
}
}

impl<Id: EdgeId, P: Clone> BiEdgeTrait for BiEdgeRaw<Id, P> {}

pub type BiEdge<P> = BiEdgeRaw<NoId, P>;
pub type BiEdgeWithId<P> = BiEdgeRaw<WithId, P>;
}
pub mod bi_edge_trait {
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

pub trait BiEdgeTrait: EdgeTrait {}
}
pub mod edge {
use crate::algo_lib::graph::edges::edge_id::EdgeId;
use crate::algo_lib::graph::edges::edge_id::NoId;
use crate::algo_lib::graph::edges::edge_id::WithId;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;

#[derive(Clone)]
pub struct EdgeRaw<Id: EdgeId, P> {
to: u32,
id: Id,
payload: P,
}

impl<Id: EdgeId> EdgeRaw<Id, ()> {
pub fn new(from: usize, to: usize) -> (usize, Self) {
(
from,
Self {
to: to as u32,
id: Id::new(),
payload: (),
},
)
}
}

impl<Id: EdgeId, P> EdgeRaw<Id, P> {
pub fn with_payload(from: usize, to: usize, payload: P) -> (usize, Self) {
(from, Self::with_payload_impl(to, payload))
}

fn with_payload_impl(to: usize, payload: P) -> Self {
Self {
to: to as u32,
id: Id::new(),
payload,
}
}
}

impl<Id: EdgeId, P: Clone> EdgeTrait for EdgeRaw<Id, P> {
type Payload = P;

const REVERSABLE: bool = false;

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

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

fn set_id(&mut self, id: usize) {
self.id.set_id(id);
}

fn reverse_id(&self) -> usize {
panic!("no reverse")
}

fn set_reverse_id(&mut self, _: usize) {
panic!("no reverse")
}

fn reverse_edge(&self, _: usize) -> Self {
panic!("no reverse")
}

fn payload(&self) -> &P {
&self.payload
}
}

pub type Edge<P> = EdgeRaw<NoId, P>;
pub type EdgeWithId<P> = EdgeRaw<WithId, P>;
}
pub mod edge_id {
pub trait EdgeId: Clone {
fn new() -> Self;
fn id(&self) -> usize;
fn set_id(&mut self, id: usize);
}

#[derive(Clone)]
pub struct WithId {
id: u32,
}

impl EdgeId for WithId {
fn new() -> Self {
Self { id: 0 }
}

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

fn set_id(&mut self, id: usize) {
self.id = id as u32;
}
}

#[derive(Clone)]
pub struct NoId {}

impl EdgeId for NoId {
fn new() -> Self {
Self {}
}

fn id(&self) -> usize {
panic!("Id called on no id")
}

fn set_id(&mut self, _: usize) {}
}
}
pub mod edge_trait {
pub trait EdgeTrait: Clone {
type Payload;

const REVERSABLE: bool;

fn to(&self) -> usize;
fn id(&self) -> usize;
fn set_id(&mut self, id: usize);
fn reverse_id(&self) -> usize;
fn set_reverse_id(&mut self, reverse_id: usize);
#[must_use]
fn reverse_edge(&self, from: usize) -> Self;
fn payload(&self) -> &Self::Payload;
}

pub trait BidirectionalEdgeTrait: EdgeTrait {}
}
}
pub mod graph {
use crate::algo_lib::collections::dsu::DSU;
use crate::algo_lib::graph::edges::bi_edge::BiEdge;
use crate::algo_lib::graph::edges::edge::Edge;
use crate::algo_lib::graph::edges::edge_trait::BidirectionalEdgeTrait;
use crate::algo_lib::graph::edges::edge_trait::EdgeTrait;
use std::ops::Index;
use std::ops::IndexMut;

#[derive(Clone)]
pub struct Graph<E: EdgeTrait> {
pub(super) edges: Vec<Vec<E>>,
edge_count: usize,
}

impl<E: EdgeTrait> Graph<E> {
pub fn new(vertex_count: usize) -> Self {
Self {
edges: vec![Vec::new(); vertex_count],
edge_count: 0,
}
}

pub fn add_edge(&mut self, (from, mut edge): (usize, E)) -> usize {
let to = edge.to();
assert!(to < self.edges.len());
let direct_id = self.edges[from].len();
edge.set_id(self.edge_count);
self.edges[from].push(edge);
if E::REVERSABLE {
let rev_id = self.edges[to].len();
self.edges[from][direct_id].set_reverse_id(rev_id);
let mut rev_edge = self.edges[from][direct_id].reverse_edge(from);
rev_edge.set_id(self.edge_count);
rev_edge.set_reverse_id(direct_id);
self.edges[to].push(rev_edge);
}
self.edge_count += 1;
direct_id
}

pub fn add_vertices(&mut self, cnt: usize) {
self.edges.resize(self.edges.len() + cnt, Vec::new());
}

pub fn clear(&mut self) {
self.edge_count = 0;
for ve in self.edges.iter_mut() {
ve.clear();
}
}

pub fn vertex_count(&self) -> usize {
self.edges.len()
}

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

pub fn degrees(&self) -> Vec<usize> {
self.edges.iter().map(|v| v.len()).collect()
}
}

impl<E: BidirectionalEdgeTrait> Graph<E> {
pub fn is_tree(&self) -> bool {
if self.edge_count + 1 != self.vertex_count() {
false
} else {
self.is_connected()
}
}

pub fn is_forest(&self) -> bool {
let mut dsu = DSU::new(self.vertex_count());
for i in 0..self.vertex_count() {
for e in self[i].iter() {
if i <= e.to() && !dsu.join(i, e.to()) {
return false;
}
}
}
true
}

pub fn is_connected(&self) -> bool {
let mut dsu = DSU::new(self.vertex_count());
for i in 0..self.vertex_count() {
for e in self[i].iter() {
dsu.join(i, e.to());
}
}
dsu.set_count() == 1
}
}

impl<E: EdgeTrait> Index<usize> for Graph<E> {
type Output = [E];

fn index(&self, index: usize) -> &Self::Output {
&self.edges[index]
}
}

impl<E: EdgeTrait> IndexMut<usize> for Graph<E> {
fn index_mut(&mut self, index: usize) -> &mut Self::Output {
&mut self.edges[index]
}
}

impl Graph<Edge<()>> {
pub fn from_edges(n: usize, edges: &[(usize, usize)]) -> Self {
let mut graph = Self::new(n);
for &(from, to) in edges {
graph.add_edge(Edge::new(from, to));
}
graph
}
}

impl Graph<BiEdge<()>> {
pub fn from_biedges(n: usize, edges: &[(usize, usize)]) -> Self {
let mut graph = Self::new(n);
for &(from, to) in edges {
graph.add_edge(BiEdge::new(from, to));
}
graph
}
}
}
}
pub mod io {
pub mod input {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::io::Read;

pub struct Input<'s> {
input: &'s mut dyn Read,
buf: Vec<u8>,
at: usize,
buf_read: usize,
}

macro_rules! read_impl {
($t: ty, $read_name: ident, $read_vec_name: ident) => {
pub fn $read_name(&mut self) -> $t {
self.read()
}

pub fn $read_vec_name(&mut self, len: usize) -> Vec<$t> {
self.read_vec(len)
}
};

($t: ty, $read_name: ident, $read_vec_name: ident, $read_pair_vec_name: ident) => {
read_impl!($t, $read_name, $read_vec_name);

pub fn $read_pair_vec_name(&mut self, len: usize) -> Vec<($t, $t)> {
self.read_vec(len)
}
};
}

impl<'s> Input<'s> {
const DEFAULT_BUF_SIZE: usize = 4096;

pub fn new(input: &'s mut dyn Read) -> Self {
Self {
input,
buf: default_vec(Self::DEFAULT_BUF_SIZE),
at: 0,
buf_read: 0,
}
}

pub fn new_with_size(input: &'s mut dyn Read, buf_size: usize) -> Self {
Self {
input,
buf: default_vec(buf_size),
at: 0,
buf_read: 0,
}
}

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

pub fn peek(&mut self) -> Option<u8> {
if self.refill_buffer() {
let res = self.buf[self.at];
Some(if res == b'\r' { b'\n' } else { res })
} 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()
}

//noinspection RsSelfConvention
pub fn is_empty(&mut self) -> bool {
self.skip_whitespace();
self.is_exhausted()
}

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

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

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

read_impl!(u32, read_unsigned, read_unsigned_vec);
read_impl!(u64, read_u64, read_u64_vec);
read_impl!(usize, read_size, read_size_vec, read_size_pair_vec);
read_impl!(i32, read_int, read_int_vec, read_int_pair_vec);
read_impl!(i64, read_long, read_long_vec, read_long_pair_vec);
read_impl!(i128, read_i128, read_i128_vec);

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
}
}
}

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

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

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

macro_rules! read_integer {
($($t:ident)+) => {$(
impl Readable for $t {
fn read(input: &mut Input) -> Self {
input.skip_whitespace();
let mut c = input.get().unwrap();
let sgn = match c {
b'-' => {
c = input.get().unwrap();
true
}
b'+' => {
c = input.get().unwrap();
false
}
_ => false,
};
let mut res = 0;
loop {
assert!(c.is_ascii_digit());
res *= 10;
let d = (c - b'0') as $t;
if sgn {
res -= d;
} else {
res += d;
}
match input.get() {
None => break,
Some(ch) => {
if ch.is_ascii_whitespace() {
break;
} else {
c = ch;
}
}
}
}
res
}
}
)+};
}

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

macro_rules! tuple_readable {
($($name:ident)+) => {
impl<$($name: Readable), +> Readable for ($($name,)+) {
fn read(input: &mut Input) -> Self {
($($name::read(input),)+)
}
}
}
}

tuple_readable! {T}
tuple_readable! {T U}
tuple_readable! {T U V}
tuple_readable! {T U V X}
tuple_readable! {T U V X Y}
tuple_readable! {T U V X Y Z}
tuple_readable! {T U V X Y Z A}
tuple_readable! {T U V X Y Z A B}
tuple_readable! {T U V X Y Z A B C}
tuple_readable! {T U V X Y Z A B C D}
tuple_readable! {T U V X Y Z A B C D E}
tuple_readable! {T U V X Y Z A B C D E F}

impl Read for Input<'_> {
fn read(&mut self, buf: &mut [u8]) -> std::io::Result<usize> {
if self.at == self.buf_read {
self.input.read(buf)
} else {
let mut i = 0;
while i < buf.len() && self.at < self.buf_read {
buf[i] = self.buf[self.at];
i += 1;
self.at += 1;
}
Ok(i)
}
}
}
}
pub mod output {
use crate::algo_lib::collections::vec_ext::default::default_vec;
use std::io::stderr;
use std::io::Stderr;
use std::io::Write;

#[derive(Copy, Clone)]
pub enum BoolOutput {
YesNo,
YesNoCaps,
PossibleImpossible,
Custom(&'static str, &'static str),
}

impl BoolOutput {
pub fn output(&self, output: &mut Output, val: bool) {
(if val { self.yes() } else { self.no() }).write(output);
}

fn yes(&self) -> &str {
match self {
BoolOutput::YesNo => "Yes",
BoolOutput::YesNoCaps => "YES",
BoolOutput::PossibleImpossible => "Possible",
BoolOutput::Custom(yes, _) => yes,
}
}

fn no(&self) -> &str {
match self {
BoolOutput::YesNo => "No",
BoolOutput::YesNoCaps => "NO",
BoolOutput::PossibleImpossible => "Impossible",
BoolOutput::Custom(_, no) => no,
}
}
}

pub struct Output<'s> {
output: &'s mut dyn Write,
buf: Vec<u8>,
at: usize,
auto_flush: bool,
bool_output: BoolOutput,
}

impl<'s> Output<'s> {
const DEFAULT_BUF_SIZE: usize = 4096;

pub fn new(output: &'s mut dyn Write) -> Self {
Self {
output,
buf: default_vec(Self::DEFAULT_BUF_SIZE),
at: 0,
auto_flush: false,
bool_output: BoolOutput::YesNoCaps,
}
}

pub fn new_with_auto_flush(output: &'s mut dyn Write) -> Self {
Self {
output,
buf: default_vec(Self::DEFAULT_BUF_SIZE),
at: 0,
auto_flush: true,
bool_output: BoolOutput::YesNoCaps,
}
}

pub fn flush(&mut self) {
if self.at != 0 {
self.output.write_all(&self.buf[..self.at]).unwrap();
self.output.flush().unwrap();
self.at = 0;
}
}

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

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

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);
}
}

pub fn set_bool_output(&mut self, bool_output: BoolOutput) {
self.bool_output = bool_output;
}
}

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;
}
self.maybe_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, const N: usize> Writable for [T; N] {
fn write(&self, output: &mut Output) {
output.print_iter_ref(self.iter());
}
}

impl<T: Writable> Writable for &T {
fn write(&self, output: &mut Output) {
T::write(self, output)
}
}

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

impl Writable for () {
fn write(&self, _output: &mut 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 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize);

macro_rules! tuple_writable {
($name0:ident $($name:ident: $id:tt )*) => {
impl<$name0: Writable, $($name: Writable,)*> Writable for ($name0, $($name,)*) {
fn write(&self, out: &mut Output) {
self.0.write(out);
$(
out.put(b' ');
self.$id.write(out);
)*
}
}
}
}

tuple_writable! {T}
tuple_writable! {T U:1}
tuple_writable! {T U:1 V:2}
tuple_writable! {T U:1 V:2 X:3}
tuple_writable! {T U:1 V:2 X:3 Y:4}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5 A:6}
tuple_writable! {T U:1 V:2 X:3 Y:4 Z:5 A:6 B:7}

impl<T: Writable> Writable for Option<T> {
fn write(&self, output: &mut Output) {
match self {
None => (-1).write(output),
Some(t) => t.write(output),
}
}
}

impl Writable for bool {
fn write(&self, output: &mut Output) {
let bool_output = output.bool_output;
bool_output.output(output, *self)
}
}

static mut ERR: Option<Stderr> = None;
pub fn err() -> Output<'static> {
unsafe {
if ERR.is_none() {
ERR = Some(stderr());
}
Output::new_with_auto_flush(ERR.as_mut().unwrap())
}
}
}
}
pub mod misc {
pub mod recursive_function {
use std::marker::PhantomData;

macro_rules! recursive_function {
($name: ident, $trait: ident, ($($type: ident $arg: ident,)*)) => {
pub trait $trait<$($type, )*Output> {
fn call(&mut self, $($arg: $type,)*) -> Output;
}

pub struct $name<F, $($type, )*Output>
where
F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
{
f: std::cell::UnsafeCell<F>,
$($arg: PhantomData<$type>,
)*
phantom_output: PhantomData<Output>,
}

impl<F, $($type, )*Output> $name<F, $($type, )*Output>
where
F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
{
pub fn new(f: F) -> Self {
Self {
f: std::cell::UnsafeCell::new(f),
$($arg: Default::default(),
)*
phantom_output: Default::default(),
}
}
}

impl<F, $($type, )*Output> $trait<$($type, )*Output> for $name<F, $($type, )*Output>
where
F: FnMut(&mut dyn $trait<$($type, )*Output>, $($type, )*) -> Output,
{
fn call(&mut self, $($arg: $type,)*) -> Output {
unsafe { (*self.f.get())(self, $($arg, )*) }
}
}
}
}

recursive_function!(RecursiveFunction0, Callable0, ());
recursive_function!(RecursiveFunction, Callable, (Arg arg,));
recursive_function!(RecursiveFunction2, Callable2, (Arg1 arg1, Arg2 arg2,));
recursive_function!(RecursiveFunction3, Callable3, (Arg1 arg1, Arg2 arg2, Arg3 arg3,));
recursive_function!(RecursiveFunction4, Callable4, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4,));
recursive_function!(RecursiveFunction5, Callable5, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5,));
recursive_function!(RecursiveFunction6, Callable6, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6,));
recursive_function!(RecursiveFunction7, Callable7, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7,));
recursive_function!(RecursiveFunction8, Callable8, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8,));
recursive_function!(RecursiveFunction9, Callable9, (Arg1 arg1, Arg2 arg2, Arg3 arg3, Arg4 arg4, Arg5 arg5, Arg6 arg6, Arg7 arg7, Arg8 arg8, Arg9 arg9,));
}
}
pub mod numbers {
pub mod num_traits {
pub mod algebra {
use crate::algo_lib::numbers::num_traits::invertible::Invertible;
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::Neg;
use std::ops::Rem;
use std::ops::RemAssign;
use std::ops::Sub;
use std::ops::SubAssign;

pub trait Zero {
fn zero() -> Self;
}

pub trait One {
fn one() -> Self;
}

pub trait AdditionMonoid: Add<Output = Self> + AddAssign + Zero + Eq + Sized {}

impl<T: Add<Output = Self> + AddAssign + Zero + Eq> AdditionMonoid for T {}

pub trait AdditionMonoidWithSub: AdditionMonoid + Sub<Output = Self> + SubAssign {}

impl<T: AdditionMonoid + Sub<Output = Self> + SubAssign> AdditionMonoidWithSub for T {}

pub trait AdditionGroup: AdditionMonoidWithSub + Neg<Output = Self> {}

impl<T: AdditionMonoidWithSub + Neg<Output = Self>> AdditionGroup for T {}

pub trait MultiplicationMonoid: Mul<Output = Self> + MulAssign + One + Eq + Sized {}

impl<T: Mul<Output = Self> + MulAssign + One + Eq> MultiplicationMonoid for T {}

pub trait IntegerMultiplicationMonoid:
MultiplicationMonoid + Div<Output = Self> + Rem<Output = Self> + DivAssign + RemAssign
{
}

impl<T: MultiplicationMonoid + Div<Output = Self> + Rem<Output = Self> + DivAssign + RemAssign>
IntegerMultiplicationMonoid for T
{
}

pub trait MultiplicationGroup:
MultiplicationMonoid + Div<Output = Self> + DivAssign + Invertible<Output = Self>
{
}

impl<T: MultiplicationMonoid + Div<Output = Self> + DivAssign + Invertible<Output = Self>>
MultiplicationGroup for T
{
}

pub trait SemiRing: AdditionMonoid + MultiplicationMonoid {}

impl<T: AdditionMonoid + MultiplicationMonoid> SemiRing for T {}

pub trait SemiRingWithSub: AdditionMonoidWithSub + SemiRing {}

impl<T: AdditionMonoidWithSub + SemiRing> SemiRingWithSub for T {}

pub trait Ring: SemiRing + AdditionGroup {}

impl<T: SemiRing + AdditionGroup> Ring for T {}

pub trait IntegerSemiRing: SemiRing + IntegerMultiplicationMonoid {}

impl<T: SemiRing + IntegerMultiplicationMonoid> IntegerSemiRing for T {}

pub trait IntegerSemiRingWithSub: SemiRingWithSub + IntegerSemiRing {}

impl<T: SemiRingWithSub + IntegerSemiRing> IntegerSemiRingWithSub for T {}

pub trait IntegerRing: IntegerSemiRing + Ring {}

impl<T: IntegerSemiRing + Ring> IntegerRing for T {}

pub trait Field: Ring + MultiplicationGroup {}

impl<T: Ring + MultiplicationGroup> Field for T {}

macro_rules! zero_one_integer_impl {
($($t: ident)+) => {$(
impl Zero for $t {
fn zero() -> Self {
0
}
}

impl One for $t {
fn one() -> Self {
1
}
}
)+};
}

zero_one_integer_impl!(i128 i64 i32 i16 i8 isize u128 u64 u32 u16 u8 usize);
}
pub mod invertible {
pub trait Invertible {
type Output;

fn inv(&self) -> Option<Self::Output>;
}
}
}
}
}
fn main() {
    let mut sin = std::io::stdin();
    let input = if false {
        algo_lib::io::input::Input::new_with_size(&mut sin, 1)
    } else {
        algo_lib::io::input::Input::new(&mut sin)
    };
    let mut stdout = std::io::stdout();
    let output = if false {
        algo_lib::io::output::Output::new_with_auto_flush(&mut stdout)
    } else {
        algo_lib::io::output::Output::new(&mut stdout)
    };
    solution::run(input, output);
}