A15955. Similar Polynomials

A polynomial $A(x)$ of degree $d$ is an expression of the form $A(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d$ , where $a_i$ are integers, and $a_d \neq 0$ . Two polynomials $A(x)$ and $B(x)$ are called similar if there is an integer $s$ such that for any integer $x$ it holds that

$$$$ B(x) \equiv A(x+s) \pmod{10^9+7}. $$ </p><p>For two similar polynomials $A(x)$ and $B(x)$ of degree $d$ , you're given their values in the points $x=0,1,\\dots, d$ modulo $10^9+7$ .</p><p>Find a value $s$ such that $B(x) \\equiv A(x+s) \\pmod{10^9+7}$ for all integers $x$$$.

The first line contains a single integer $d$ ( $1 \le d \le 2\,500\,000$ ).

The second line contains $d+1$ integers $A(0), A(1), \ldots, A(d)$ ( $0 \le A(i) < 10^9+7$ ) — the values of the polynomial $A(x)$ .

The third line contains $d+1$ integers $B(0), B(1), \ldots, B(d)$ ( $0 \le B(i) < 10^9+7$ ) — the values of the polynomial $B(x)$ .

It is guaranteed that $A(x)$ and $B(x)$ are similar and that the leading coefficients (i.e., the coefficients in front of $x^d$ ) of $A(x)$ and $B(x)$ are not divisible by $10^9+7$ .

Print a single integer $s$ ( $0 \leq s < 10^9+7$ ) such that $B(x) \equiv A(x+s) \pmod{10^9+7}$ for all integers $x$ .

If there are multiple solutions, print any.

In the first example, $A(x) \equiv x-1 \pmod{10^9+7}$ and $B(x)\equiv x+2 \pmod{10^9+7}$ . They're similar because $$$$B(x) \equiv A(x+3) \pmod{10^9+7}. $$ </p><p>In the second example, $A(x) \\equiv (x+1)^2 \\pmod{10^9+7}$ and $B(x) \\equiv (x+10)^2 \\pmod{10^9+7}$ , hence $$ B(x) \equiv A(x+9) \pmod{10^9+7}. $$$$