Problem:
Score : 600 points

Problem Statement

There is an arithmetic progression with L terms: s_0, s_1, s_2, ... , s_{L-1}.

The initial term is A, and the common difference is B. That is, s_i = A + B \times i holds.

Consider the integer obtained by concatenating the terms written in base ten without leading zeros. For example, the sequence 3, 7, 11, 15, 19 would be concatenated into 37111519. What is the remainder when that integer is divided by M?

Constraints

All values in input are integers.

1 \leq L, A, B < 10^{18}

2 \leq M \leq 10^9

All terms in the arithmetic progression are less than 10^{18}.

Input

Input is given from Standard Input in the following format:

L A B M

Output

Print the remainder when the integer obtained by concatenating the terms is divided by M.

Sample Input 1

5 3 4 10007

Sample Output 1

5563

Our arithmetic progression is 3, 7, 11, 15, 19, so the answer is 37111519 mod 10007, that is, 5563.

Sample Input 2

4 8 1 1000000

Sample Output 2

891011

Sample Input 3

107 10000000000007 1000000000000007 998244353

Sample Output 3

39122908

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
#[allow(unused_imports)]
use std::cmp::{max, min, Ordering};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};
#[allow(unused_imports)]
use std::iter::FromIterator;
#[macro_export]
macro_rules ! chmax { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: max ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! chmin { ( $ x : expr , $ ( $ v : expr ) ,+ ) => { $ ( $ x = std :: cmp :: min ( $ x ,$ v ) ; ) + } ; }
#[macro_export]
macro_rules ! max { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: max ( $ x , max ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! min { ( $ x : expr ) => ( $ x ) ; ( $ x : expr , $ ( $ xs : expr ) ,+ ) => { std :: cmp :: min ( $ x , min ! ( $ ( $ xs ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! dvec { ( $ t : expr ; $ len : expr ) => { vec ! [ $ t ; $ len ] } ; ( $ t : expr ; $ len : expr , $ ( $ rest : expr ) ,* ) => { vec ! [ dvec ! ( $ t ; $ ( $ rest ) ,* ) ; $ len ] } ; }
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[macro_export]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut parser = Parser :: from_str ( $ s ) ; input_inner ! { parser , $ ( $ r ) * } } ; ( parser = $ parser : ident , $ ( $ r : tt ) * ) => { input_inner ! { $ parser , $ ( $ r ) * } } ; ( new_stdin_parser = $ parser : ident , $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let reader = std :: io :: BufReader :: new ( stdin . lock ( ) ) ; let mut $ parser = Parser :: new ( reader ) ; input_inner ! { $ parser , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { input ! { new_stdin_parser = parser , $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! input_inner { ( $ parser : ident ) => { } ; ( $ parser : ident , ) => { } ; ( $ parser : ident , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ parser , $ t ) ; input_inner ! { $ parser $ ( $ r ) * } } ; }
#[macro_export]
macro_rules ! read_value { ( $ parser : ident , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ parser , $ t ) ) ,* ) } ; ( $ parser : ident , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ parser , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ parser : ident , chars ) => { read_value ! ( $ parser , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ parser : ident , usize1 ) => { read_value ! ( $ parser , usize ) - 1 } ; ( $ parser : ident , $ t : ty ) => { $ parser . next ::<$ t > ( ) . expect ( "Parse error" ) } ; }
use std::io;
use std::io::BufRead;
use std::str;
pub struct Parser<R> {
    reader: R,
    buf: Vec<u8>,
    pos: usize,
}
impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}
impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                if len == 0 {
                    break;
                }
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return str::from_utf8(&self.buf[begin..end]).unwrap().parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
}
#[allow(unused_macros)]
macro_rules ! debug { ( $ ( $ a : expr ) ,* ) => { eprintln ! ( concat ! ( $ ( stringify ! ( $ a ) , " = {:?}, " ) ,* ) , $ ( $ a ) ,* ) ; } }
#[doc = " https://github.com/hatoo/competitive-rust-snippets"]
const BIG_STACK_SIZE: bool = true;
#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}
fn solve() {
    let out = stdout();
    let mut out = BufWriter::new(out.lock());

    input!{
        L:i64,A:i64,B:i64,M:i64
    }

    let mut lower:i64 = 1;
    let mut upper:i64 = 10;

    let mut C = vec![];
    let arith_seq = ArithSeq {
        a: A,
        b: B,
    };

    for i in 1..19 {
        if let Some((s, n)) = arith_seq.range(max(A,lower), upper-1) {
            C.push(n)
        } else {
            C.push(0)
        }

        if i<18 {
            lower *= 10;
            upper *= 10;
        }
    }

    let mut x = Matrix {
        v: vec![
            vec![0, A % M, 1],
        ]
    };
    let mut remaining = L;
    for i in 1..19 {
        let d = i;
        let Cd = C[i-1];
        let rep = if remaining < Cd {
            let tmp = remaining;
            remaining = 0;
            tmp
        } else {
            remaining -= Cd;
            Cd
        };
        let Md = Matrix {
            v: vec![
                vec![i64::pow(10, d as u32) % M, 0, 0],
                vec![1, 1, 0],
                vec![0, B % M, 1],
            ],
        };
        let mul = Md.pow(rep as u64, M);
        x = x.mul_rem(&mul, M);
        x %= M;

        if remaining == 0 {
            break
        }
    }
    println!("{}", x.v[0][0]);
}
struct ArithSeq {
    a: i64,
    b: i64,
}
impl ArithSeq {
    #[doc = "a+bi >= x"]
    pub fn next(&self, x: i64) -> i64 {
        if x >= self.a {
            let d = x - self.a;
            let i = (d - 1 + self.b) / self.b;
            self.a + self.b * i
        } else {
            let d = self.a - x;
            let i = d / self.b;
            self.a - self.b * i
        }
    }
    #[doc = "a+bi <= y"]
    pub fn prev(&self, x: i64) -> i64 {
        let next = self.next(x);
        if next == x {
            x
        } else {
            next - self.b
        }
    }
    #[doc = "[a+bi, a+bj] <= [l, r]"]
    pub fn range(&self, l: i64, u: i64) -> Option<(i64, i64)> {
        if l > u {
            return None
        }
        let x = self.next(l);
        let y = self.prev(u);
        if x > y {
            return None
        }
        assert!(x <= y);
        if l<=x && x<=u {
            let cnt = (y-x) / self.b;
            Some((x, cnt+1))
        } else {
            None
        }
    }
}
#[derive(Clone)]
pub struct Matrix {
    v: Vec<Vec<i64>>,
}
impl Matrix {
    fn identity(n: usize) -> Self {
        let mut v = vec![vec![0; n]; n];
        for i in 0..n {
            v[i][i] = 1;
        }
        Matrix { v: v }
    }
    fn m(&self) -> usize {
        self.v.len()
    }
    fn n(&self) -> usize {
        self.v[0].len()
    }
    fn mul_rem(&self, other: &Self, mo: i64) -> Self {
        let mut r = vec![vec![0; other.n()]; self.m()];
        for i in 0..self.m() {
            for j in 0..self.n() {
                let mut v = 0;
                for k in 0..self.n() {
                    v += (self.v[i][k] * other.v[k][j]) % mo;
                    v %= mo;
                }
                r[i][j] = v;
            }
        }
        Matrix { v: r }
    }
    fn pow(&self, k: u64, mo: i64) -> Self {
        assert!(self.m() == self.n());
        let mut k = k;
        let mut x = Self::identity(self.m());
        let mut y = self.clone();
        while k > 0 {
            if k & 1 > 0 {
                x = y.clone() * x;
                x %= mo;
            }
            y = y.mul_rem(&y, mo);
            y %= mo;
            k >>= 1;
        }
        x
    }
}
impl std::ops::Add for Matrix {
    type Output = Self;
    fn add(self, other: Self) -> Self {
        let mut r = self.v.clone();
        for i in 0..self.m() {
            for j in 0..self.n() {
                r[i][j] += other.v[i][j];
            }
        }
        Matrix { v: r }
    }
}
impl std::ops::Sub for Matrix {
    type Output = Self;
    fn sub(self, other: Self) -> Self {
        let mut r = self.v.clone();
        for i in 0..self.m() {
            for j in 0..self.n() {
                r[i][j] -= other.v[i][j];
            }
        }
        Matrix { v: r }
    }
}
impl std::ops::Mul for Matrix {
    type Output = Self;
    fn mul(self, other: Self) -> Self {
        let mut r = vec![vec![0; other.n()]; self.m()];
        for i in 0..self.m() {
            for j in 0..self.n() {
                let mut v = 0;
                for k in 0..self.n() {
                    v += self.v[i][k] * other.v[k][j];
                }
                r[i][j] = v;
            }
        }
        Matrix { v: r }
    }
}
impl std::ops::Rem<i64> for Matrix {
    type Output = Self;
    fn rem(self, mo: i64) -> Self {
        let mut r = self.v.clone();
        for i in 0..self.m() {
            for j in 0..self.n() {
                r[i][j] %= mo;
            }
        }
        Matrix { v: r }
    }
}
impl std::ops::RemAssign<i64> for Matrix {
    fn rem_assign(&mut self, mo: i64) {
        for i in 0..self.m() {
            for j in 0..self.n() {
                self.v[i][j] %= mo;
            }
        }
    }
}