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)]
#![allow(dead_code)]
#![allow(non_camel_case_types)]
#![allow(non_snake_case)]

use std::cmp::*;
use std::collections::*;
use std::ops::*;
use std::io::{Write, BufWriter};

// static MOD: usize = 998244353;
static MOD: usize = 1000000007;

// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

macro_rules! println_stderr(
    ($($arg:tt)*) => { {
        let r = writeln!(&mut ::std::io::stderr(), $($arg)*);
        r.expect("failed printing to stderr");
    } }
);

use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

type Num = usize;

#[derive(Clone, Copy, Debug)]
pub struct ModInt<T: Copy + Clone>(pub T);

impl Add<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self + rhs.0
    }
}

impl Add<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn add(self, rhs: Num) -> ModInt<Num> {
        let mut t = rhs + self.0;
        if t >= MOD {
            t = t - MOD;
        }
        ModInt(t)
    }
}

impl Sub<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn sub(self, rhs: Num) -> ModInt<Num> {
        let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
        let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
        ModInt(value - rhs)
    }
}

impl Sub<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self - rhs.0
    }
}

impl AddAssign<Num> for ModInt<Num> {
    fn add_assign(&mut self, other: Num) {
        *self = *self + other;
    }
}
impl AddAssign<ModInt<Num>> for ModInt<Num> {
    fn add_assign(&mut self, other: ModInt<Num>) {
        *self = *self + other;
    }
}

impl SubAssign<Num> for ModInt<Num> {
    fn sub_assign(&mut self, other: Num) {
        *self = *self - other;
    }
}

impl SubAssign<ModInt<Num>> for ModInt<Num> {
    fn sub_assign(&mut self, other: ModInt<Num>) {
        *self = *self - other;
    }
}

impl Div<Num> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn div(self, rhs: Num) -> ModInt<Num> {
        self * ModInt(rhs).pow(MOD - 2)
    }
}

impl Div<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;
    fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self / rhs.0
    }
}

impl DivAssign<Num> for ModInt<Num> {
    fn div_assign(&mut self, rhs: Num) {
        *self = *self / rhs
    }
}
impl DivAssign<ModInt<Num>> for ModInt<Num> {
    fn div_assign(&mut self, rhs: ModInt<Num>) {
        *self = *self / rhs
    }
}

impl Mul<ModInt<Num>> for ModInt<Num> {
    type Output = ModInt<Num>;

    fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
        self * rhs.0
    }
}
impl Mul<Num> for ModInt<Num> {
    type Output = ModInt<Num>;

    fn mul(self, rhs: Num) -> ModInt<Num> {
        let t = (self.0 * rhs) % MOD;
        ModInt(t)
    }
}

impl MulAssign<Num> for ModInt<Num> {
    fn mul_assign(&mut self, rhs: Num) {
        *self = *self * rhs;
    }
}

impl MulAssign<ModInt<Num>> for ModInt<Num> {
    fn mul_assign(&mut self, rhs: ModInt<Num>) {
        *self = *self * rhs;
    }
}

impl ModInt<Num> {
    pub fn pow(self, e: usize) -> ModInt<Num> {
        let mut result = ModInt(1);
        let mut cur = self;
        let mut e = e;
        while e > 0 {
            if e & 1 == 1 {
                result *= cur;
            }
            e >>= 1;
            cur *= cur;
        }
        result
    }
}

pub struct Comb {
    fact: Vec<ModInt<Num>>,
    factinv: Vec<ModInt<Num>>,
}

impl Comb {
    /// Create a object that provides effiecint computation of combinations
    /// for input smaller than `n`.
    ///
    /// This requires `O(n)` time.
    pub fn new(n: usize) -> Comb {
        let mut fact: Vec<ModInt<Num>> = vec![ModInt(0); n + 1];
        let mut factinv: Vec<ModInt<Num>> = vec![ModInt(0); n + 1];
        fact[0] = ModInt(1);
        for i in 0..n {
            fact[i + 1] = fact[i] * (i + 1);
        }
        factinv[n] = fact[n].pow(MOD-2);
        for i in (0..n).rev() {
            factinv[i] = factinv[i + 1] * (i + 1);
        }
        Comb {
            fact: fact,
            factinv: factinv,
        }
    }

    /// `n! = 1 * 2 * ... * n`
    ///
    /// `O(1)` if n is smaller than input in `new` method.
    pub fn fact(&self, n: usize) -> ModInt<Num> {
        if let Some(x) = self.fact.get(n as usize) {
            *x
        } else if n >= MOD {
            ModInt(0)
        } else {
            // Note that this is slow if `n` is large.
            // Precalculation is a possible solution but doesn't work for any module number.
            let mut res = ModInt(1);
            for a in 1..(n + 1) {
                res *= a;
            }
            res
        }
    }

    /// returns `y` such that `n! * y == 1`.
    ///
    /// `O(1)` if n is smaller than input in `new` method.
    pub fn factinv(&self, n: usize) -> ModInt<Num> {
        if let Some(x) = self.factinv.get(n) {
            *x
        } else {
            self.fact(n).pow(MOD-2)
        }
    }

    /// `nPr = n! / (n - r)!`
    ///
    /// `O(1)` if n and r are smaller than input in `new` method.
    pub fn perm(&self, n: usize, r: usize) -> ModInt<Num> {
        if n >= r {
            self.fact(n) * self.factinv(n - r)
        } else {
            ModInt(0)
        }
    }

    /// `nCr = n! / (n - r)! / r!`.
    ///
    /// `O(1)` if n and r are smaller than input in `new` method.
    pub fn comb(&self, n: usize, r: usize) -> ModInt<Num> {
        let m = MOD;
        if n >= m {
            self.comb(n % m, r % m) * self.comb(n / m, r / m) // Lucas' theorem
        } else if n >= r {
            self.fact(n) * self.factinv(n - r) * self.factinv(r)
        } else {
            ModInt(0)
        }
    }
}

fn solve() {
    input! {
        s: chars,
    }  
    let n = s.len();
    let mut dp = vec![vec![ModInt(0); 3];n+1];
    let mut ans = ModInt(0);
    let mut pat = ModInt(1);
    for i in 0..n {
        dp[i+1][0] = dp[i][0];
        dp[i+1][1] = dp[i][1];
        dp[i+1][2] = dp[i][2];
        if s[i]=='A' {
            dp[i+1][0] = dp[i][0]+pat;
        }
        if s[i]=='B' {
            dp[i+1][1] = dp[i][1]+dp[i][0];
        }
        if s[i]=='C' {
            dp[i+1][2] = dp[i][2]+dp[i][1];
        }
        if s[i]=='?'{
            dp[i+1][0] = dp[i][0]*3+pat;
            dp[i+1][1] = dp[i][1]*3+dp[i][0];
            dp[i+1][2] = dp[i][2]*3+dp[i][1];
            pat *= 3;
        }
        println_stderr!("{} {} {}", dp[i+1][0].0, dp[i+1][1].0, dp[i+1][2].0);
    }
    println!("{}", dp[n][2].0);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}