Two Random Walks

Bug \(A\) starts on the origin on the coordinate plane and makes \(3\) random unit moves only in the four compass directions (Up, Down, Left, Right). Bug \(B\) starts on coordinate \((1,0)\) and makes two random unit moves only in the four compass directions. The probability that bug \(B\) is closer or equal in distance to the origin with bug \(A\) can be expressed as \(\dfrac{p}{q}\) for positive coprime integers \(p,q.\) Find \(p+q\).

Image credit: DoMathTogether
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