Polynomial in a Black Box

This is an interactive problem.

Alice has a black box which works with integers modulo $m = 10^{9} + 7$. If a user types a number $x$ on the keyboard of the box, the screen shows the number equal to the value of the polynomial $p (x) = (a\_{d} x^{d} + a\_{d - 1} x^{d - 1} + \ldots + a\_{1} x^{1} + a\_{0}) \bmod m$. The degree $d$ of the polynomial is unknown, as are its coefficients $a\_{i}$. It is only known that $0 \le d \le 10$ and $a\_d \ne 0$.

Alice can type several numbers $x$ and learn the values of the polynomial for these numbers. Help her find the degree $d$ of the polynomial. She can input an $x$ at most $d + 3$ times.

In each test, the degree and the coefficients of the polynomial $p (x)$ are chosen and fixed in advance.

In the example, which is also the first test in the testing system, $p (x) = x^2 + 1\\,000\\,000\\,005$. All other tests were created as follows: first, the degree $d$ was chosen ($0 \le d \le 10$), and after that, one of the polynomials of such degree was chosen as $p (x)$ uniformly at random.