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

Geometry Level 5

The red circle has radius 3 and the blue circle has radius 4. The tangents to one of their points of intersection, P1,P_1, are perpendicular.

The green circle is centered at the other point of intersection, P2,P_2, and passes through P1.P_1. It also intersects the red and blue circles at P3P_3 and P4,P_4, respectively.

What is the length of the altitude from P1P_1 in triangle P1P3P4?P_1P_3P_4?

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