Square $BDEC$ with center $F$ is constructed on the outside of triangle $ABC$ such that $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$. If $[AF] \cap [BC] = \{G\}$, find the length of $|BG|$. The original answer is in the form $m - n\sqrt{p}$, please give the value of $m + n + p$.

Let's think step by step and output the final answer within \boxed{}.
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