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AABB collision with edge of convex polyhedron

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So far I've been doing AABB-Plane intersection tests to find the collision between an AABB and a mesh. This works fine if the AABB collides with one of the sides, however if it hits one of the edges I can't use this method:



(The AABB starts at the top right and moves to the bottom left, red and green are the intersections with the mesh planes)


I'm at a loss, could anyone point me in the right direction?

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