Problem Description

There are many inequalities in mathematics. For example, when $x,y>0$:

You are given two homogeneous bivariate polynomials $f(x,y)$ and $g(x,y)$, both with non-negative coefficients. Determine whether there exist $A,r>0$ such that for any $x,y>0$, the following always holds: $f(x,y)\geq Ag(x,y)^r$.

Input Format

The input contains multiple test cases. Adjacent test cases are separated by one blank line.

Each test case contains two lines.

The first line contains $n+2$ non-negative integers $n,a_0,a_1,\dots,a_n$, where the $a_i$ are not all $0$, representing $f(x,y)=a_0x^n+a_1x^{n-1}y+\dots+a_ny^n$.

The second line contains $m+2$ non-negative integers $m,b_0,b_1,\dots,b_m$, where the $b_i$ are not all $0$, representing $g(x,y)=b_0x^m+b_1x^{m-1}y+\dots+b_my^m$.

Output Format

For each test case, output one line. If such $A,r$ exist, output YES; otherwise, output NO.

1 1 1
2 0 1 0

2 1 1 1
3 1 0 0 1

5 1 0 0 1 0 0
5 0 0 1 0 0 1

YES
YES
NO

Hint

Sample Explanation

• For the first sample, $x+y \geq \sqrt{2xy}$.
• For the second sample, $x^2+xy+y^2\geq \sqrt[3]{(x^3+y^3)^2}$. This is easy to verify after expansion.
• For the third sample, assume the opposite: $x^5+x^2y^3 \geq A(x^3y^2+y^5)^r$ holds for all positive $x,y$. Let $x=y$, then $2x^5\geq A(2x^5)^r$. If $r>1$, the inequality fails when $x$ is large enough; if $r<1$, the inequality fails when $x$ is small enough, so $r=1$. Then let $x=1$, we get $1+y^3 \geq A(y^2+y^5)$, i.e., $1\geq Ay^2$, which cannot hold for all positive $y$. Therefore, such $A,r$ do not exist.

Constraints

There are at most $100$ test cases, and $1\leq n,m\leq 100$. The coefficients are at most $10^4$.

Translated by ChatGPT 5