Circles, Tangents, And Triangles (Oh My...)

Geometry Level 4

The red circle has radius 3 and the blue circle has radius 4. The tangents to one of their points of intersection, \(P_1,\) are perpendicular.

The green circle is centered at the other point of intersection, \(P_2,\) and passes through \(P_1.\) It also intersects the red and blue circles at \(P_3\) and \(P_4,\) respectively.

What is the length of the altitude from \(P_1\) in triangle \(P_1P_3P_4?\)

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