Consider the ellipses given by \( x^{2}+4y^{2}=1 \) and \( 4x^{2}+y^{2}=1 \).
Let \( P \) be the point in the first quadrant where the given ellipse intersect. If \( \theta \) is the acute angle between the tangents to the given ellipses at the point \( P \), then the value of \( 4\tan\theta \) is __________________.
