Wrong Calculation

Professor Kim Sang-geun developed a new computing engine for producing numerical tables. It computes the values of a single-variable polynomial function.

For example, if the polynomial function is $f(x) = x^2 + 2x + 1$, the engine reports $1\,(=f(0))$, $4\,(=f(1))$, $9\,(=f(2))$, $16\,(=f(3))$, and $25\,(=f(4))$.

However, while coding drunk, Sang-geun made a mistake, so the engine always outputs exactly one value incorrectly. So for the polynomial above it might report $1, 4, 12, 16, 25$ instead of $1, 4, 9, 16, 25$.

Given the values produced by the engine, write a program that finds which result was computed incorrectly.

The input consists of several test cases.

The first line of each test case contains the degree $d$ of the polynomial. For example, the degree of $4x^5 + 3x + 0.5$ is $5$, and the degree of $2.4x + 3.8$ is $1$. $d$ is at most $5$.

The next $d+3$ lines contain the results $f(0), f(1), \dots, f(d+2)$ in order. The polynomial function $f$ itself is not given. Every reported value is a real number greater than $-100.0$ and less than $100.0$.

The incorrect value is always one of $f(0), f(1), \dots, f(d+2)$, and its error from the correct value is greater than $1.0$. Because floating-point computation cannot avoid rounding, the other values may also carry error, but that error is tiny and never exceeds $10^{-6}$.

The last line of the input contains a single $0$.

For each test case, print the position of the incorrectly computed value. If the result of $f(i)$ is wrong, print $i$.