Cannot compute gradient with `ArrayPartition` that holds containers of different types

ArrayPartition is a useful structure to concatenate arrays of different types. The type is defined in SciML/RecursiveArrayTools.jl

ArrayPartitions are also used in many places in SciML ecosystem, but also in other places like Manopt.jl.
It appears, though, that if ArrayPartition references two containers, one of eltype is Float64 and another one is Int64, the gradient from ForwardDiff fails.

MWE is:

julia> using ForwardDiff, RecursiveArrayTools

julia> v = [ 0.0, 1 ]
2-element Vector{Float64}:
 0.0
 1.0

julia> f(v) = sum(v)
f (generic function with 1 method)

julia> ForwardDiff.gradient(f, [ 0.0, 1 ])
2-element Vector{Float64}:
 1.0
 1.0

julia> ForwardDiff.gradient(f, ArrayPartition([ 0.0 ], [ 1 ]))
ERROR: MethodError: no method matching ForwardDiff.Dual{ForwardDiff.Tag{typeof(f), Float64}, Float64, 2}(::Int64, ::ForwardDiff.Partials{2, Float64})

Closest candidates are:
  ForwardDiff.Dual{T, V, N}(::Number) where {T, V, N}
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/dual.jl:78
  ForwardDiff.Dual{T, V, N}(::Any) where {T, V, N}
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/dual.jl:77
  ForwardDiff.Dual{T, V, N}(::V, ::ForwardDiff.Partials{N, V}) where {T, V, N}
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/dual.jl:17

Stacktrace:
  [1] _broadcast_getindex_evalf
    @ ./broadcast.jl:709 [inlined]
  [2] _broadcast_getindex
    @ ./broadcast.jl:682 [inlined]
  [3] getindex
    @ ./broadcast.jl:636 [inlined]
  [4] macro expansion
    @ ./broadcast.jl:1004 [inlined]
  [5] macro expansion
    @ ./simdloop.jl:77 [inlined]
  [6] copyto!
    @ ./broadcast.jl:1003 [inlined]
  [7] copyto!
    @ ./broadcast.jl:956 [inlined]
  [8] materialize!
    @ ./broadcast.jl:914 [inlined]
  [9] materialize!
    @ ./broadcast.jl:911 [inlined]
 [10] seed!(duals::ArrayPartition{…}, x::ArrayPartition{…}, seeds::Tuple{…})
    @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/apiutils.jl:52
 [11] vector_mode_dual_eval!
    @ ~/.julia/packages/ForwardDiff/PcZ48/src/apiutils.jl:23 [inlined]

Start with the seed! entry point in ForwardDiff/src/apiutils.jl, using the reported ArrayPartition mixed-Float64/Int64 reproducer as the first test case. Trace how dual seeds are broadcast into each partition and add coverage for the failing gradient call; done means the example returns the expected gradient without a MethodError.