# 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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