differential equations - How to combine DSolve and FindRoot

As my differential function is relatively complex I do not want to do the detour of finding an explicit solution via DSolve for it but rather use an FindRoot-like approach by guessing initial starting values.

Following example:

Let z[t] be the differential where the start and end values are known:

z[0]=z0 and z[tmax]=zend, but the end time tmax is unknown.

The (simplified) differential (where the paths of l2[t]and l3[t] as well as their end values l2final and l3final are well-defined) is:

DSolve[{z[0] == z0, z[tmax] == zend, z'[t]==f[l20[t],l3[t]],l2[tmax]==l2final, l30[tmax]==l3final}, z[t],t]

The only unkown is tmax. So I don't want to do the DSolve described as I don't care about an explicit solution of z[t]. But I rather want to search for something like:

FindRoot[z[t]-zend,{t,0,100}]

Does anybody have an idea how to combine both approaches EFFICIENTLY?

Btw: z[t] is continously increasing as well as l2[t] and l3[t]

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