Eccoci.

Given a non-square integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision ${10}^{-13}$:
$$4375636191520\sqrt{2}-6188084046055<\pi <721133315582\sqrt{2}-1019836515172$$
We call $BQ{A}_d(x,n)$ the quadratic integer closest to $x$ with the absolute values of $a,b$ not exceeding $n$.
We also define the integral part of a quadratic integer as $I_d(a+b\sqrt{d})=a$.

You are given that:

• $BQ{A}_2(\pi ,10)=6-2\sqrt{2}$
• $BQ{A}_5(\pi ,100)=26\sqrt{5}-55$
• $BQ{A}_7(\pi ,{10}^6)=560323-211781\sqrt{7}$
• $I_2(BQ{A}_2(\pi ,{10}^{13}))=-6188084046055$

Find the sum of $|I_d(BQ{A}_d(\pi ,{10}^{13}))|$ for all non-square positive integers less than 100.