`ForwardDiff.jacobian` throws error for `fft`

reproducing script is below

(@v1.8) pkg> activate --temp
  Activating new project at `/var/folders/29/j2y0t06d1918t1tymbv45wgr0000gn/T/jl_T8WfIS`

(jl_T8WfIS) pkg> add ForwardDiff, FFTW
    Updating registry at `~/.julia/registries/General.toml`
   Resolving package versions...
   Installed ChainRulesCore ─ v1.18.0
    Updating `/private/var/folders/29/j2y0t06d1918t1tymbv45wgr0000gn/T/jl_T8WfIS/Project.toml`
  [7a1cc6ca] + FFTW v1.7.1
  [f6369f11] + ForwardDiff v0.10.36
    Updating `/private/var/folders/29/j2y0t06d1918t1tymbv45wgr0000gn/T/jl_T8WfIS/Manifest.toml`
  [621f4979] + AbstractFFTs v1.5.0
  [d360d2e6] + ChainRulesCore v1.18.0
  [9e997f8a] + ChangesOfVariables v0.1.8
  [bbf7d656] + CommonSubexpressions v0.3.0
  [34da2185] + Compat v4.10.0
  [163ba53b] + DiffResults v1.1.0
  [b552c78f] + DiffRules v1.15.1
  [ffbed154] + DocStringExtensions v0.9.3
  [7a1cc6ca] + FFTW v1.7.1
  [f6369f11] + ForwardDiff v0.10.36
  [3587e190] + InverseFunctions v0.1.12
  [92d709cd] + IrrationalConstants v0.2.2
  [692b3bcd] + JLLWrappers v1.5.0
  [2ab3a3ac] + LogExpFunctions v0.3.26
  [1914dd2f] + MacroTools v0.5.11
  [77ba4419] + NaNMath v1.0.2
  [21216c6a] + Preferences v1.4.1
  [189a3867] + Reexport v1.2.2
  [276daf66] + SpecialFunctions v2.3.1
  [90137ffa] + StaticArrays v1.6.5
  [1e83bf80] + StaticArraysCore v1.4.2
  [f5851436] + FFTW_jll v3.3.10+0
  [1d5cc7b8] + IntelOpenMP_jll v2023.2.0+0
  [856f044c] + MKL_jll v2023.2.0+0
  [efe28fd5] + OpenSpecFun_jll v0.5.5+0
  [0dad84c5] + ArgTools v1.1.1
  [56f22d72] + Artifacts
  [2a0f44e3] + Base64
  [ade2ca70] + Dates
  [f43a241f] + Downloads v1.6.0
  [7b1f6079] + FileWatching
  [b77e0a4c] + InteractiveUtils
  [4af54fe1] + LazyArtifacts
  [b27032c2] + LibCURL v0.6.3
  [76f85450] + LibGit2
  [8f399da3] + Libdl
  [37e2e46d] + LinearAlgebra
  [56ddb016] + Logging
  [d6f4376e] + Markdown
  [ca575930] + NetworkOptions v1.2.0
  [44cfe95a] + Pkg v1.8.0
  [de0858da] + Printf
  [3fa0cd96] + REPL
  [9a3f8284] + Random
  [ea8e919c] + SHA v0.7.0
  [9e88b42a] + Serialization
  [6462fe0b] + Sockets
  [2f01184e] + SparseArrays
  [10745b16] + Statistics
  [fa267f1f] + TOML v1.0.0
  [a4e569a6] + Tar v1.10.1
  [8dfed614] + Test
  [cf7118a7] + UUIDs
  [4ec0a83e] + Unicode
  [e66e0078] + CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] + LibCURL_jll v7.84.0+0
  [29816b5a] + LibSSH2_jll v1.10.2+0
  [c8ffd9c3] + MbedTLS_jll v2.28.0+0
  [14a3606d] + MozillaCACerts_jll v2022.2.1
  [4536629a] + OpenBLAS_jll v0.3.20+0
  [05823500] + OpenLibm_jll v0.8.1+0
  [83775a58] + Zlib_jll v1.2.12+3
  [8e850b90] + libblastrampoline_jll v5.1.1+0
  [8e850ede] + nghttp2_jll v1.48.0+0
  [3f19e933] + p7zip_jll v17.4.0+0
Precompiling project...
  7 dependencies successfully precompiled in 6 seconds. 25 already precompiled.

julia> x = randn(2)
2-element Vector{Float64}:
 -0.04005888157665624
  1.7357331187899907

julia> using ForwardDiff, FFTW

julia> ForwardDiff.jacobian(fft, x)
ERROR: type ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2} not supported
Stacktrace:
  [1] error(s::String)
    @ Base ./error.jl:35
  [2] _fftfloat(#unused#::Type{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:38
  [3] _fftfloat(x::ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:39
  [4] fftfloat(x::ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:34
  [5] complexfloat(x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:47
  [6] fft(x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}}, region::UnitRange{Int64})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:214
  [7] fft(x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:66
  [8] vector_mode_dual_eval!(f::typeof(fft), cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}}}, x::Vector{Float64})
    @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/apiutils.jl:24
  [9] vector_mode_jacobian(f::typeof(fft), x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}}})
    @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/jacobian.jl:125
 [10] jacobian(f::Function, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}}}, ::Val{true})
    @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/jacobian.jl:0
 [11] jacobian(f::Function, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(fft), Float64}, Float64, 2}}}) (repeats 2 times)
    @ ForwardDiff ~/.julia/packages/ForwardDiff/PcZ48/src/jacobian.jl:19
 [12] top-level scope
    @ REPL[5]:1

Start by rerunning the reported ForwardDiff.jacobian(fft, x) reproduction and read the stack trace through AbstractFFTs definitions.jl and ForwardDiff's jacobian path. Determine the intended compatibility between FFT and ForwardDiff dual values, then add a regression test showing the expected Jacobian behavior.