Symbolic formula from GeometricScene?

I’m trying to solve the following problem with GeometricScene

What is the shortest circular arc of which the altitude above its chord is one?

I have the following code:

GeometricScene[
    {
        {a -> {0, 0}, b -> {0, r}, c -> {0, r - 1}, d}, 
        {r, l}
    },
    {
        GeometricAssertion[{Line[{b, c}], Line[{c, d}]}, "Perpendicular"],
        GeometricAssertion[{Circle[a, r], Line[{c, d}]}, {"Concurrent", d}],
        l == 2 r PlanarAngle[a -> {c, d}]
    }
]

from which I get random instances like this

Is there any way to get a formula for l in terms of r? I know that I can substitute values of r into the scene and then use l /. RandomInstance[scene]["Quantities"] to get a numeric answer, but can this be done symbolically?

Obviously one can do the geometry by hand, but referring specifically to GeometricScene, does it have these capabilities?