Matching the Integral with a Riemann Sum

The integral of a function can be approximated as follows.

$\int_a^b f(x)\,dx \approx \sum_{k=0}^{n-1} f(a + k\Delta x + \epsilon)\,\Delta x$

$\Delta x = \frac{b-a}{n}, \qquad 0 \le \epsilon \le \Delta x$

Mingyu was happily working through integrals at boot camp when he wondered whether a well-chosen $\epsilon$ could make this approximation equal the exact integral. Checking that needs decimal arithmetic, which is hard to do by hand, so he asked you to write a program for it.

In this problem, $f$ is a polynomial.

The first line contains a positive integer $K$ ($1 \le K \le 10$), the degree of the polynomial.

The second line contains the integer coefficients $c_1, c_2, \ldots, c_{K+1}$, listed from the highest-degree term down to the constant term ($0 \le c_i \le 10$, $1 \le c_1 \le 10$).

The third line contains two integers $a$ and $b$, the start and end of the integration interval, and the number of subintervals $N$ ($0 \le a < b \le 10$, $1 \le N \le 10$).

Print on one line the value of $\epsilon$ that makes the Riemann sum approximation equal the exact integral, rounded to exactly six digits after the decimal point. For example, if the answer is $0.25$, print 0.250000.

For the given input, at most one $\epsilon$ satisfies the condition. If no such value exists, print -1.

The integral of a polynomial term is computed as follows.

$\int_a^b x^n\,dx = \frac{b^{n+1} - a^{n+1}}{n+1}$