Chef and Same Old Recurrence 2

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Chef has started learning to solve dynamic programming problems recently. He has practiced on a lot of problems already, so he wants to try his hands on the hardest problems in this domain. Chef only likes the interesting part of solving dynamic programming problems — specifically, he just likes to divide a problem into beautiful subproblems which consist of solving recurrences; he finds actually solving these recurrences quite boring. Since you are Chef's assistant, your job is solving this boring part for Chef. Let's define a recurrent sequence $\left\lbrace{dp(k)}\right\rbrace_{k=1}^\infty$ with parametres $A$, $B$ and $K$ as follows:  $dp(1) = K$  for $n \gt 1$, $dp(n) = A \cdot dp(n1) + B\cdot \sum\limits_{i=1}^{n1} dp(i) \cdot dp(ni)$ Chef would like you to find the sum $\sum\limits_{i=L}^{R} dp(i)^2$. Since this number can be very large, compute it modulo $10^9+7$. He wants to do this for $Q$ pairs of $L$ and $R$. ### Input  The first line of the input contains four spaceseparated integers $N$, $K$, $A$ and $B$.  The second line of the input contains $Q$.  The next $Q$ lines contains two spaceseparated integers $L$ and $R$. ### Output For every query print a single line containing one integer — the sum $\sum\limits_{i=L}^{R} dp(i)^2$ modulo $10^9+7$. ### Constraints  $1 \le N \le 10^7$  $1 \le Q \le 5 \cdot 10^4$  $1 \le L \le R \le N$  $1 \le B,K \le 10^9+7$  $0 \le A \le 10^9+7$ ### Subtasks **Subtask #1 (5 points):** $N \le 5,000$ **Subtask #2 (10 points):** $N \le 10^5$ **Subtask #3 (10 points):**  $N \le 10^6$  $A=0$ **Subtask #4 (25 points):** original constraints ### Example Input4 1 2 1 2 4 4 1 4### Example Output
3249 3403### Explanation The first four terms of this sequence are:  $dp(1) = 1$  $dp(2) = 2 \cdot dp(1) + dp(1) \cdot dp(1) = 3$  $dp(3) = 2 \cdot dp(2) + dp(1) \cdot dp(2) + dp(2) \cdot dp(1) = 12$  $dp(4) = 2 \cdot dp(3) + dp(1) \cdot dp(3) + dp(2) \cdot dp(2) + dp(3) \cdot dp(1) = 57$ The answers for first and second query are:  $(dp(4) \cdot dp(4)) \;\%\; (10^9+7) = 3249$  $(dp(1) \cdot dp(1) + dp(2) \cdot dp(2) + dp(3) \cdot dp(3) + dp(4) \cdot dp(4)) \;\%\; (10^9+7) = 3403$
Author:  usaxena95 
Editorial  https://discuss.codechef.com/problems/JADUGAR2 
Tags  usaxena95 
Date Added:  7042018 
Time Limit:  2.5 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.5, PYPY, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, rust, SCALA, swift, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, kotlin, PERL6, TEXT, SCM chicken, CLOJ, COB, FS 
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