Triangle \(ABC\) is isosceles with a right angle at \(A\). A point is chosen uniformly at random inside the triangle. Let \(p\) be the probability that the point is closer to the point \(A\) than it is to \(B\) or \(C\), **and** it is closer to side \(BC\) than it is to side \(AB\) or \(AC\). \(p\) can be expressed as \(\sqrt{\frac{a}{b}} - c\), where \(a\) and \(b\) are coprime positive integers and \(c\) is a positive integer. What is the value of \(a + b + c\)?

**Details and assumptions**

It follows from the conditions that \( AB = AC \).

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