Drawing Points and Friendliness

A point \((x_1,y_1)\) is drawn in the positive coordinate plane with \(x_1 > y_1\). Then, an infinite sequence of points are drawn, with the following rule: \[\left\{\begin{array}{l}x_{n+1}=x_n+y_n\\ y_{n+1}=x_n-y_n\end{array}\right.\]

Suppose a starting point \((x_1,y_1)\) is called friendly to a coordinate \((a,b)\) if a point is drawn on \((a,b)\) sometime in the sequence; that is, \((a,b)=(x_k,y_k)\) for some positive integer \(k\).

Let \(F(a,b)\) be a function that counts the number of starting points \((x_1,y_1)\) that are friendly to \((a,b)\).

The number of ordered pairs \((p,q)\) satisfying \(F(p,q)=2014\) and \(p+q \le 2^{2^{10}}\) can be expressed as \(a^b\) where \(a,b\) are positive integers and \(a\) is minimized. Find the value of \(a+b\).

Details and Assumptions

As much as this looks like a Computer Science problem, it is 100% doable with only a pencil and a paper.


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