MathNet 2011 Question 30

On a lesson, Pete and Nick write in their exercise books two numbers each: Pete writes $1$ and $2$, while Nick writes $3$ and $4$. Then, at the beginning of each minute, each boy finds some quadratic polynomial such that the numbers in a boy's exercise book are the roots of this polynomial. Let $f(x)$ and $g(x)$ be the polynomials obtained by the boys. Next, if the equation $f(x) = g(x)$ has two different roots $x_1, x_2$, then one of them removes the two numbers from his exercise book and changes them by $x_1$ and $x_2$; otherwise nothing happens.
At some moment, Pete has a number $5$ in his exercise book. Find all possible values for the second number in his exercise book at this moment. (I. Bogdanov, A. Garber)