Problem:
Score : 400 points

Problem Statement

The ABC number of a string T is the number of triples of integers (i, j, k) that satisfy all of the following conditions:

1 ≤ i < j < k ≤ |T| (|T| is the length of T.)

T_i = A (T_i is the i-th character of T from the beginning.)

T_j = B

T_k = C

For example, when T = ABCBC, there are three triples of integers (i, j, k) that satisfy the conditions: (1, 2, 3), (1, 2, 5), (1, 4, 5). Thus, the ABC number of T is 3.

You are given a string S. Each character of S is A, B, C or ?.

Let Q be the number of occurrences of ? in S. We can make 3^Q strings by replacing each occurrence of ? in S with A, B or C. Find the sum of the ABC numbers of all these strings.

This sum can be extremely large, so print the sum modulo 10^9 + 7.

Constraints

3 ≤ |S| ≤ 10^5

Each character of S is A, B, C or ?.

Input

Input is given from Standard Input in the following format:

S

Output

Print the sum of the ABC numbers of all the 3^Q strings, modulo 10^9 + 7.

Sample Input 1

A??C

Sample Output 1

8

In this case, Q = 2, and we can make 3^Q = 9 strings by by replacing each occurrence of ? with A, B or C. The ABC number of each of these strings is as follows:

AAAC: 0

AABC: 2

AACC: 0

ABAC: 1

ABBC: 2

ABCC: 2

ACAC: 0

ACBC: 1

ACCC: 0

The sum of these is 0 + 2 + 0 + 1 + 2 + 2 + 0 + 1 + 0 = 8, so we print 8 modulo 10^9 + 7, that is, 8.

Sample Input 2

ABCBC

Sample Output 2

3

When Q = 0, we print the ABC number of S itself, modulo 10^9 + 7. This string is the same as the one given as an example in the problem statement, and its ABC number is 3.

Sample Input 3

????C?????B??????A???????

Sample Output 3

979596887

In this case, the sum of the ABC numbers of all the 3^Q strings is 2291979612924, and we should print this number modulo 10^9 + 7, that is, 979596887.

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

#![allow(unused_imports)]
use std::io::{ self, Write };
use std::str::FromStr;
use std::cmp::{ min, max };
use std::collections::{ BinaryHeap, VecDeque };
use std::ops::Range;

#[allow(unused_macros)]
macro_rules! trace {
    ($var:expr) => {
        let _ = writeln!(&mut std::io::stderr(), ">>> {} = {:?}", stringify!($var), $var);
    };
    ($($args:expr),*) => { trace!(($($args),*)) }
}

#[allow(unused_macros)]
macro_rules! put {
    ($var:expr) => {
        let _ = writeln!(&mut std::io::stdout(), "{}", $var);
    };
    ($var:expr, $($args:expr),*) => {
        let _ = write!(&mut std::io::stdout(), "{} ", $var);
        put!($($args),*);
    };
}

struct BIT { size: usize, array: Vec<i32> }
impl BIT {
    fn new(n: usize) -> BIT {
        BIT  { size: n, array: vec![0; n + 1] }
    }
    fn add(&mut self, idx: usize, w: i32) {
        let mut x = idx + 1;
        while x <= self.size {
            self.array[x] += w;
            let xi = x as i32;
            x += (xi&-xi) as usize;
        }
    }
    fn sum_up(&self, idx: usize) -> i32 { // [0, idx)
        let mut sum = 0;
        let mut x = idx;
        while x > 0 {
            sum += self.array[x];
            let xi = x as i32;
            x -= (xi&-xi) as usize;
        }
        sum
    }
    fn sum(&self, range: Range<usize>) -> i32 {
        self.sum_up(range.end) - self.sum_up(range.start)
    }
}

const M: u64 = 1_000_000_007;

fn pow(a: u64, n: u64) -> u64 {
    if n == 0 {
        1
    } else if n == 1 {
        a % M
    } else if n % 2 == 0 {
        let x = (a * a) % M;
        pow(x, n / 2) % M
    } else {
        let x = (a * a) % M;
        (a * pow(x, n / 2)) % M
    }
}

fn main() {
    let mut sc = Scanner::new();

    let chars: Vec<char> = sc.cin::<String>().chars().collect();
    let n = chars.len();

    let mut bit_a = BIT::new(n);
    let mut bit_b = BIT::new(n);
    let mut bit_c = BIT::new(n);
    let mut bit_q = BIT::new(n);
    for i in 0..n {
        if chars[i] == 'A' { bit_a.add(i, 1); }
        else if chars[i] == 'B' { bit_b.add(i, 1); }
        else if chars[i] == 'C' { bit_c.add(i, 1); }
        else if chars[i] == '?' { bit_q.add(i, 1); }
    }

    let mut ans = 0;
    for j in 1..(n-1) {
        if chars[j] == 'A' || chars[j] == 'C' { continue }
        let a = bit_a.sum(0..j) as u64;
        let q_left = bit_q.sum(0..j) as u64;
        let c = bit_c.sum(j + 1..n) as u64;
        let q_right = bit_q.sum(j + 1..n) as u64;
        if a == 0 && q_left == 0 { continue }
        if c == 0 && q_right == 0 { continue }

        let x1 = a * pow(3, q_left);
        let x2 = if q_left == 0 { 0 } else {(pow(3, q_left - 1) * q_left) % M};
        let y1 = c * pow(3, q_right);
        let y2 = if q_right == 0 { 0 } else {(pow(3, q_right - 1) * q_right) % M};

        ans += ((x1 + x2) % M) * ((y1 + y2) % M) ;
        ans %= M;
    }
    put!(ans % M);
}

#[allow(dead_code)]
struct Scanner { stdin: io::Stdin, buffer: VecDeque<String>, }
#[allow(dead_code)]
impl Scanner {
    fn new() -> Scanner { Scanner { stdin: io::stdin(), buffer: VecDeque::new() } }
    fn reserve(&mut self) {
        while self.buffer.len() == 0 {
            let mut line = String::new();
            let _ = self.stdin.read_line(&mut line);
            for w in line.split_whitespace() {
                self.buffer.push_back(String::from(w));
            }
        }
    }
    fn cin<T: FromStr>(&mut self) -> T {
        self.reserve();
        match self.buffer.pop_front().unwrap().parse::<T>() {
            Ok(a) => a,
            Err(_) => panic!("parse err")
        }
    }
}