HW09_in_class.jl

### A Pluto.jl notebook ###
# v0.19.41

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ d1980bd2-babf-11ee-1dbb-dfefbdbdb36d
using PlutoUI, Plots, ImageShow, TestImages, FFTW, NDTools, IndexFunArrays, FileIO, FourierTools, SpecialFunctions, UrlDownload, ImageMagick

# ╔═╡ a7a9946e-0013-4cfe-aaf7-0c19da075abd
using Noise

# ╔═╡ 17adc747-9822-42be-91c1-ad8a2e1532bc
md"# 0. Load packages"

# ╔═╡ d2c6102c-7a62-4899-9c5c-b08ce9a5baa8
FFTW.set_num_threads(4)

# ╔═╡ 8e6ec43c-9135-4829-8146-a56b8624750a
const TODO = nothing

# ╔═╡ 7331d6a5-dd03-42c7-9ab3-c84a641296bc
TableOfContents()

# ╔═╡ f326cd36-9c05-4b1c-81c0-56c954cd51f3
gauss_R(z::T, z_R) where T = iszero(z) ? T(Inf) : (1 + (z_R / z)^2)

# ╔═╡ 7c6b4ddd-ca5a-4b23-96f7-e896fb1cda6d
gauss_ψ(z, z_R) = atan(z, z_R)

# ╔═╡ dd16735b-c1d6-4977-a396-a3e23823ee68
gauss_w(z, z_R, w_0) = w_0 * sqrt(1 + (z / z_R)^2)

# ╔═╡ 8e04ae26-5474-4c2c-9fda-dbb7efd77acd
"""
	gauss_beam(y, x, z, λ, w_0)


Returns the eletrical field of a Gaussian beam at position `(y, x)` at optical axis position `z` with respect to the beam waist `w_0`.
Wavelength is `λ`.
"""
function gauss_beam(y, x, z, λ, w_0)
    k = π / λ * 2
    z_R = π * w_0^2 / λ
	r² = x ^ 2 + y ^ 2
    # don't put exp(i * k * z) into the same exp, it causes some strange wraps
	return w_0 / gauss_w(z, z_R, w_0) * exp(-r² / gauss_w(z, z_R, w_0)^2) * 
            exp(1im * k * z) * 
            exp(1im * (k * r² / 2 / gauss_R(z, z_R) - gauss_ψ(z, z_R)))
end

# ╔═╡ 5f21a849-0272-4bca-9659-54cd38068e09
"""
	bpm(field, λ0, Lx, Ly, z, n; window=true, paraxial=true, amplitude_array)

Propagates the array `field` with wavelength `λ0` and the filed size in meter size
`(Lx, Ly)`. The propagation distance `z` should be a vector of distances.
`n` is the average refractive index of the propagation medium.

The returned array is a three dimensional array where `size(arr, 3) == size(z, 1)`.


If `window=true` we apply a Hann window function to dampen the boundaries.
A keyword `amplitude_array` can be provided, which multiplies with the field at each point. This allows to include obstacles or to shift the phase.

If `paraxial=true` the Fresnel approximation is applied.
"""
function bpm(field, λ0, Lx, Ly, z, n=1; window=true, amplitude_array=ones(size(field)..., length(z)), paraxial=true)
	# free space wavenumber in m-1
	k0 = 2 * π / λ0
	# medium wavenumber m-1
	k = n * k0
	λ = λ0 / n
	# medium in m
	dz = z[2] - z[1]

	# field parameters
	Nx = size(field, 2)
	dx = Lx / Nx
	x  = Nx > 1 ? range(-Lx/2, Lx/2, Nx) : zero(typeof(Lx))
	fx = reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	Ny = size(field, 1)
	dy = Ly / Ny
	y  = range(-Ly/2, Ly/2, Ny)
	fy = fftfreq(Ny, 1 / dy)

	if paraxial
		# important step, this calculates the Fourier space kernel
		H = exp.(-1im .* k .* λ^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H = exp.(1im .* sqrt.(1 .+ 0im .- λ^2 .* fx.^2 .- λ^2 .* fy.^2) .* k .* dz) .* ((λ0^2 .* fx.^2 .+ λ0^2 .* fy.^2) .< 1)
	end
	# 3d output fields we save
	# third dimensions stores the different z propagation distances
	out_field = zeros(ComplexF64, (Ny, Nx, size(z, 1)))
	# first entry corresponds to z[1] = 0
	out_field[:, :, 1] = field
	
	# FFT plan for calculating FFTs
	# It's a more efficient syntax: p * x == fft(x)
	p = plan_fft(field, (1,2))
	window_f = window ? IndexFunArrays.window_hanning(size(out_field)[1:2], border_in=0.9) : 1
	# inverse FFT
	invp = inv(p)
	for z_index in 2:size(out_field, 3)
		u0 = out_field[:, :, z_index - 1] .* window_f .* amplitude_array[:, :, z_index - 1]
    	u1 = invp * ((p * u0) .* H)
		out_field[:, :, z_index] .= u1
	end
	return out_field
end

# ╔═╡ 1d6a9432-2784-4902-ba96-f082be62fa78
"""
	bpm(field, λ0, Lx, Ly, z, n_1, n_2; window=true, paraxial=true, amplitude_array)

Propagates the array `field` with wavelength `λ0` and the filed size in meter size
`(Lx, Ly)`. The propagation distance `z` should be a vector of distances.
`n_1` is the refractive index of region 1 and `n_2` the refractive index of region 2.
`n_array` is an array filled with either `n_1` or `n_2` and the algorithm stiches the regions together.

The returned array is a three dimensional array where `size(arr, 3) == size(z, 1)`.


If `window=true` we apply a Hann window function to dampen the boundaries.
A keyword `amplitude_array` can be provided, which multiplies with the field at each point. This allows to include obstacles.

If `paraxial=true` the Fresnel approximation is applied.
"""
function bpm_split(field, λ0, Lx, Ly, z, n1=1, n2=1; window=true, n_array=ones(size(field)..., length(z)), paraxial=true)
	# free space wavenumber in m-1
	k0 = 2 * π / λ0
	# medium wavenumber m-1
	k1 = n1 * k0
	k2 = n2 * k0
	λ1 = λ0 / n1
	λ2 = λ0 / n2
	# medium in m
	dz = z[2] - z[1]

	# field parameters
	Nx = size(field, 2)
	dx = Lx / Nx
	x  = Nx > 1 ? range(-Lx/2, Lx/2, Nx) : zero(typeof(Lx))
	fx = reshape(fftfreq(Nx, 1 / dx), (1, Nx))

	Ny = size(field, 1)
	dy = Ly / Ny
	y  = range(-Ly/2, Ly/2, Ny)
	fy = fftfreq(Ny, 1 / dy)

	if paraxial
		# important step, this calculates the Fourier space kernel
		H1 = exp(1im * k1 * dz) .* exp.(-1im .* k1 .* λ1^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
		H2 = exp(1im * k2 * dz) .* exp.(-1im .* k2 .* λ2^2 .* (fx.^2 .+ fy.^2) ./ (2) * dz)
	else
		H1 = exp.(1im .* sqrt.(1 .+ 0im .- λ1^2 .* fx.^2 .- λ1^2 .* fy.^2) .* k1 .* dz) .* ((λ1^2 .* fx.^2 .+ λ1^2 .* fy.^2) .< 1)
		H2 = exp.(1im .* sqrt.(1 .+ 0im .- λ2^2 .* fx.^2 .- λ2^2 .* fy.^2) .* k2 .* dz) .* ((λ2^2 .* fx.^2 .+ λ2^2 .* fy.^2) .< 1)
	end
	# 3d output fields we save
	# third dimensions stores the different z propagation distances
	out_field = zeros(ComplexF64, (Ny, Nx, size(z, 1)))
	# first entry corresponds to z[1] = 0
	out_field[:, :, 1] = field
	
	# FFT plan for calculating FFTs
	# It's a more efficient syntax: p * x == fft(x)
	p = plan_fft(field, (1,2))
	window_f = window ? IndexFunArrays.window_hanning(size(out_field)[1:2], border_in=0.8) : 1
	# inverse FFT
	invp = inv(p)
	for z_index in 2:size(out_field, 3)
		u0_1 = out_field[:, :, z_index - 1] .* window_f
		
		u1_1 = invp * ((p * u0_1) .* H1)
		u1_2 = invp * ((p * u0_1) .* H2)

		out_field[:, :, z_index] .= u1_1 .* (n_array[:, :, z_index] .≈ n1) .+ u1_2 .* (n_array[:, :, z_index] .≈ n2)
	end
	return out_field
end

# ╔═╡ 06f420e0-58d0-4f1c-b5e4-81dc9ba7ccee
md"# 1. Different Microscope setups

In this task we test out different microscope setups like widefield microscope, dark field microscope and phase shift microscope.

For that you are given the `four_f_system`. 
The apertures you have to build yourself.


We also included a simple way to simulate poisson shot noise which is the most common noise source in low light microscopy. It depends on how much light one collects.
"

# ╔═╡ 2f967364-d81a-4b36-a9bd-587061c75fa1
"""
	four_f_system(img_c, aperture)

Simulate a coherent 4f system. The aperture is multiplied in Fourier space.

So bring the initial field to Fourier space, multiply with the aperture, take another Fourier transform. Take `abs2.`

"""
function four_f_system(img_c, aperture)
	out = abs2.(ift(ft(img_c) .* aperture))
	return out
end

# ╔═╡ 2c087f8d-eea2-415d-ae40-a88cf923025b
phase = Float32.(Gray.(testimage("blobs")))[:, begin+1:end-1];

# ╔═╡ ae7f7476-acdc-4aec-b318-5e94e69cb89f
img = ((1 .+ phase.^15)) .* cispi.(2 .* phase);

# ╔═╡ 85eb6ef3-2826-41c9-a181-437f1a2055bf


# ╔═╡ 72e9aacd-3e09-4a69-ab8b-9a703a9a5bfe
md"
The image is a phase target, so given the phase information you'll see a lot of informaton.
But if you take `abs.`, most information is gone.

Let's try to recover it!
"

# ╔═╡ 6ac9648f-8a42-4dc9-a00d-8113d1e8b8b5
[simshow(img) simshow(abs.(img))]

# ╔═╡ fe6fd8e0-2459-4dd3-9d4b-0447532e120c
simshow([exp(1im * π / 2)])

# ╔═╡ dd57b128-0b6d-4cca-ae9f-b8700452b7b5
md"## 1. Widefield / Widefield Microscope
A widefield microscope has a simple aperture which is everywhere 1 inside the radius, otherwise zero.

"

# ╔═╡ 92e44124-1e90-4cab-a7b8-5eda072a6470
@bind radius_widefield Slider(1:100, show_value=true, default=80)

# ╔═╡ f54e1012-4b8c-4d1b-9496-23208fa97614
aperture = rr(size(img)) .<= radius_widefield;

# ╔═╡ 127aaef1-5c2d-4134-9c9b-aa2d52d8d5d9
simshow(aperture)

# ╔═╡ dae1f8a3-a57c-4d40-97dc-6bb837ff1e46
img_wide_field = poisson(four_f_system(img, aperture), 100_000 / (100 + (100 - radius_widefield)^2));

# ╔═╡ a8089672-f7b8-494d-877a-d94d04d68d67
[simshow(aperture) simshow(img_wide_field)]

# ╔═╡ 647ebd0e-8af0-4c5d-b88b-ebeaffd00563
md"## 2. Dark Field Microscope

What kind of aperture do you need in this case?
What are the two parameters deciding the aperture?
"

# ╔═╡ c82a09ff-16ed-47b5-8512-baf2ab907917
@bind radius_dark_field Slider(1:100, show_value=true, default=80)

# ╔═╡ 67f0d13e-ad27-483b-b920-fbf045c4af8c
aperture_dark_field = aperture .- (rr(size(img)) .< radius_dark_field);

# ╔═╡ 5598dda0-fe03-440c-b425-d3f8fc4fb2e6
simshow(aperture_dark_field)

# ╔═╡ 16eaceec-4cbc-445d-bc62-e4fbcc703bdb
img_dark_field = poisson(four_f_system(img, aperture_dark_field), 1_000 * 5/ (5 + radius_dark_field^2) * 100 / (100 + (100 - radius_widefield)^2));

# ╔═╡ d58ed322-71f9-45fb-b063-cb8057df4b4d
[simshow(aperture_dark_field) simshow(img_dark_field)]

# ╔═╡ 2c152f67-75f1-48ad-a708-3aa018564ff1
md"## 3. Phase Shift Microscope

What kind of aperture do you need in this case?
What are the three parameters deciding the aperture?
"

# ╔═╡ 0ddb3a82-5b46-42e5-bd77-420af284704d
@bind radius_phase_shift Slider(1:100, show_value=true, default=10)

# ╔═╡ ef1f1b48-9fb3-461a-9dec-463faf8a4841
phase_shift = -1 * π / 2

# ╔═╡ 464f6a41-76fe-4431-8ab1-0d3f2523b5b1
aperture_phase_shift = aperture .- (rr(size(img)) .< radius_phase_shift) .+ (rr(size(img)) .< radius_phase_shift) .* exp(1im * phase_shift);

# ╔═╡ f4ad95c4-9825-4592-a7e4-62fd07cb0fef
img_phase_shift = poisson(four_f_system(img, aperture_phase_shift), 100_000 / (100 + (100 - radius_widefield)^2));

# ╔═╡ cf6dd850-f664-406b-9437-3f3c927d0cb3
[simshow(aperture_phase_shift) simshow(img_phase_shift)]

# ╔═╡ ac62bebd-dba7-48ca-9b0a-ea9eec90fdfd
[simshow(img_wide_field) simshow(img_dark_field) simshow(img_phase_shift)]

# ╔═╡ d78b9e78-23c6-4938-b54c-879409f5c0d5
md"# 2 Confocal Microscope

In this part we analyze the imaging performance of a fluorescent microscope.

First the specimen is illuminated with a small laser spot.
This laser spot is created with an 4f system and a certain aperture `aperture_illumination`.

The emitted fluorescenced light is incoherent and captured by another 4f system with the aperture `aperture_imaging`.

Finally, the intensity is taken within a pinhole radius with an area detector directly behind the pinhole.

"

# ╔═╡ f5987db0-fa80-42a7-8cd8-e25d51e12f12
urldownload("https://felix.sumpi.org/confocal.png")

# ╔═╡ 24a95277-0dbb-4063-b5c0-f1b674cb1c54
x = 8:45

# ╔═╡ 3be6bf4b-5572-47d0-b28d-d3aad1219ffb
img_small = (1 .+ sin.(x.^2 .* 0.02)) .+ 0 .* x';

# ╔═╡ 3affeb13-031f-4dd6-8633-a89cb7efda81
simshow(img_small)

# ╔═╡ 48afe724-7d87-46a8-9f97-8dd88acbaef7
function confocal_microscope(img, aperture_illumination, aperture_imaging, pinhole)	
	out = zero.(img)

	# calculate the two psf each
	psf_illumination = abs2.(ift(aperture_illumination))
	psf_illumination ./= sum(psf_illumination)

	
	psf_imaging = abs2.(ift(aperture_imaging))
	psf_imaging ./= sum(psf_imaging)

	# parallelize the shifting of the specimen
	Threads.@threads for i in 1:size(img, 1)
		for j in 1:size(img, 2)
			# shift specimen
			img_s = circshift(img, (-i + size(img, 1) ÷ 2 + 1, -j + size(img, 2) ÷ 2 + 1))

			# core imaging part and sum all light with a area detector
			out[i, j] = sum(conv_psf(psf_illumination .* img_s, psf_imaging) .* pinhole)
		end
	end

	return out
end

# ╔═╡ 711c5f77-f93a-4942-b34a-af1b230c0b4c
aperture_illumination = rr(size(img_small)) .<= 4;

# ╔═╡ 918d77a7-3d9f-490b-9554-2ef6e1ecd370
aperture_imaging = rr(size(img_small)) .<= 3;

# ╔═╡ 1933b3a7-560b-408e-84db-634d57213e11
begin
	psf_illumination = abs2.(ift(aperture_illumination))
	psf_illumination ./= sum(psf_illumination)
end

# ╔═╡ 2f56a227-3fc2-4970-8087-e5abe631855e
plot(psf_illumination[20, :])

# ╔═╡ 3615cc42-4fdb-4bcb-bf46-f022656a1fb6
pinhole0 = rr(size(img_small)) .<= 0;

# ╔═╡ 3eea661e-5633-4715-be6d-462597c7fa7c
pinhole1 = rr(size(img_small)) .<= 1;

# ╔═╡ afa614ec-b048-420e-b584-bdcfcf7722dc
pinhole2 = rr(size(img_small)) .<= 2;

# ╔═╡ f70e8d1d-087f-4119-af21-eff46d96bd79
pinhole3 = rr(size(img_small)) .<= 3;

# ╔═╡ fcc4806c-2e9a-4b60-9eaa-515678c62081
pinhole4 = rr(size(img_small)) .<= 4;

# ╔═╡ c0352697-12ed-415f-aaae-3c3c10acc4ff
pinhole10 = rr(size(img_small)) .<= 8;

# ╔═╡ b50c396b-4f2a-44fc-acef-e714ee22c17c
pinhole100 = rr(size(img_small)) .<= 100;

# ╔═╡ 8fe0868e-7364-4755-a803-a804fe6b8f6c


# ╔═╡ c36a2011-6be7-4e44-bd01-5d93f5e7bf2f


# ╔═╡ 8d7e9998-0a2e-4ba0-ab00-075a47926450


# ╔═╡ 76e80b30-c082-4f89-a1cb-102534380013
begin
	psf = abs2.(ift(aperture_illumination))
	psf ./= sum(psf)
end;

# ╔═╡ 033224de-b226-4b96-94a0-bb261e23cba4
res_widefield = conv_psf(img_small, psf);

# ╔═╡ f5aeb4e6-ba22-46b7-a92f-2adc4869801e
md" ## Questions
Calculate final images with different pinhole radii. Use `confocal_microscope` for it.
Plot crossections of the final images (with `plot` and `plot!`).
Also use `simshow`.


Explain with the images what is the impact of the pinhole.

"

# ╔═╡ eb4cd0c8-6c50-4d61-9e8c-012bb58ac7b4
md"### Answers"

# ╔═╡ 134992ad-66cc-4bf8-8a2d-ea14dead0ae5
res0 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole0);

# ╔═╡ 974dd675-60ae-4fb9-a75a-849ec55293e6
sum(res0)

# ╔═╡ 04104bc6-978f-4cdc-af4a-eae5ad410f99
simshow(poisson(res0 * 10_000))

# ╔═╡ edbe5cbd-ff0d-4f98-8799-6a8b997ea5f7
res1 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole1);

# ╔═╡ db24c18c-3182-4a2b-8cb9-095184582ea4
res2 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole2);

# ╔═╡ 3d19a9a2-2540-4fcb-84eb-ae12d9db51c6
res3 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole3);

# ╔═╡ 974ab29a-2eaf-4501-8edd-aa1b94dcdcd6
res4 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole4);

# ╔═╡ 38c87485-6185-4a9e-ac81-8fc2dfb730c4
res10 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole10);

# ╔═╡ e0985610-a335-4230-b848-41177ec2ccae
sum(res10)

# ╔═╡ cea18dc9-d65b-4631-bd98-0eb450576cc6
simshow(poisson(res10 * 10_000))

# ╔═╡ 4513c9e1-2895-4455-82af-b8477ee2f2bc
res100 = confocal_microscope(img_small, aperture_illumination, aperture_imaging, pinhole100);

# ╔═╡ a80f82fb-c95f-49ce-83ea-a7432b67a322
[simshow(res0) simshow(res1) simshow(res2) simshow(res3) simshow(res4) simshow(res10) simshow(res100) simshow(res_widefield)]

# ╔═╡ 93a97c6b-9246-4ffc-b9b1-6eb326ee1425
mynorm(x) = x ./ maximum(x)

# ╔═╡ e306c862-88c9-40bf-9adc-907091dce51b
begin
	plot(mynorm(res0[:, 10]), label="delta pinhole")
	plot!(mynorm(res1[:, 20]), label="pinhole 1")
	plot!(mynorm(res4[:, 20]), label="pinhole 4")
	plot!(mynorm(res10[:, 20]), label="pinhole 10")
	plot!(mynorm(res_widefield[:, 20]), label="widefield")
end

# ╔═╡ bcd8f84b-da1a-4d3b-9ad3-3875bef06af1
md" ## Questions

Explain:
* Why is the resolution of the confocal microscope better than the widefield one? 
* How can you calculate the maximum resolution of the confocal and the widefield microscope respectively?
* How does the resolution depend on the `aperture_imaging` and `aperture_illumination` in both cases.
* What is the influence of the pinhole on the resolution?
* How does the pinhole help with depth sectioning?
"

# ╔═╡ 73f30de8-923e-48f3-859c-30c3c8312e67
md"""### Answers

The PSF of the confocal microscope is given by:

$h_\text{confocal} = h_\text{illumination} \cdot h_\text{detection}$

$h_\text{detection} = h_\text{widefield} * \text{pinhole}$

Resolution by a confocal microscope is given:

$$\Delta x_\text{confocal} = \frac{\lambda}{2 (\text{NA}_\text{illumination} + \text{NA}_\text{detection})}$$

$$\Delta x_\text{widefield} = \frac{\lambda}{2 \text{NA}_\text{detection}}$$


The pinhole represents another convolution with the imaging PSF. Hence, if the pinhole is very small (delta function), the imaging PSF is unchanged.
But, if the pinhole is very big, the imaging PSF gets very broad (in Fourier space).


In practice, a delta pinhole does not work because the number of photons is limited and a small pinhole discards most of the light and then the noise level is increased.
Therefore, a compromise between pinhole size and noise level has to be found.


The pinhole blocks all the out of focus light.

"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
FourierTools = "b18b359b-aebc-45ac-a139-9c0ccbb2871e"
ImageMagick = "6218d12a-5da1-5696-b52f-db25d2ecc6d1"
ImageShow = "4e3cecfd-b093-5904-9786-8bbb286a6a31"
IndexFunArrays = "613c443e-d742-454e-bfc6-1d7f8dd76566"
NDTools = "98581153-e998-4eef-8d0d-5ec2c052313d"
Noise = "81d43f40-5267-43b7-ae1c-8b967f377efa"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
UrlDownload = "856ac37a-3032-4c1c-9122-f86d88358c8b"

[compat]
FFTW = "~1.8.0"
FileIO = "~1.16.2"
FourierTools = "~0.4.2"
ImageMagick = "~1.3.1"
ImageShow = "~0.3.8"
IndexFunArrays = "~0.2.7"
NDTools = "~0.5.3"
Noise = "~0.3.3"
Plots = "~1.40.1"
PlutoUI = "~0.7.58"
SpecialFunctions = "~2.3.1"
TestImages = "~1.8.0"
UrlDownload = "~1.0.1"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.10.3"
manifest_format = "2.0"
project_hash = "851ee0240959fce4d24c934a3c5896443c76550a"

[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.5.0"
weakdeps = ["ChainRulesCore", "Test"]

    [deps.AbstractFFTs.extensions]
    AbstractFFTsChainRulesCoreExt = "ChainRulesCore"
    AbstractFFTsTestExt = "Test"

[[deps.AbstractNFFTs]]
deps = ["LinearAlgebra", "Printf"]
git-tree-sha1 = "292e21e99dedb8621c15f185b8fdb4260bb3c429"
uuid = "7f219486-4aa7-41d6-80a7-e08ef20ceed7"
version = "0.8.2"

[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "0f748c81756f2e5e6854298f11ad8b2dfae6911a"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.3.0"

[[deps.Adapt]]
deps = ["LinearAlgebra", "Requires"]
git-tree-sha1 = "0fb305e0253fd4e833d486914367a2ee2c2e78d0"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "4.0.1"

    [deps.Adapt.extensions]
    AdaptStaticArraysExt = "StaticArrays"

    [deps.Adapt.weakdeps]
    StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"

[[deps.ArgCheck]]
git-tree-sha1 = "a3a402a35a2f7e0b87828ccabbd5ebfbebe356b4"
uuid = "dce04be8-c92d-5529-be00-80e4d2c0e197"
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[[deps.ArgTools]]
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[[deps.Artifacts]]
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[[deps.AxisArrays]]
deps = ["Dates", "IntervalSets", "IterTools", "RangeArrays"]
git-tree-sha1 = "16351be62963a67ac4083f748fdb3cca58bfd52f"
uuid = "39de3d68-74b9-583c-8d2d-e117c070f3a9"
version = "0.4.7"

[[deps.BangBang]]
deps = ["Compat", "ConstructionBase", "InitialValues", "LinearAlgebra", "Requires", "Setfield", "Tables"]
git-tree-sha1 = "7aa7ad1682f3d5754e3491bb59b8103cae28e3a3"
uuid = "198e06fe-97b7-11e9-32a5-e1d131e6ad66"
version = "0.3.40"

    [deps.BangBang.extensions]
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    BangBangDataFramesExt = "DataFrames"
    BangBangStaticArraysExt = "StaticArrays"
    BangBangStructArraysExt = "StructArrays"
    BangBangTypedTablesExt = "TypedTables"

    [deps.BangBang.weakdeps]
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    StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
    TypedTables = "9d95f2ec-7b3d-5a63-8d20-e2491e220bb9"

[[deps.Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[deps.Baselet]]
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[[deps.BasicInterpolators]]
deps = ["LinearAlgebra", "Memoize", "Random"]
git-tree-sha1 = "3f7be532673fc4a22825e7884e9e0e876236b12a"
uuid = "26cce99e-4866-4b6d-ab74-862489e035e0"
version = "0.7.1"

[[deps.BitFlags]]
git-tree-sha1 = "2dc09997850d68179b69dafb58ae806167a32b1b"
uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35"
version = "0.1.8"

[[deps.Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9e2a6b69137e6969bab0152632dcb3bc108c8bdd"
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[[deps.CEnum]]
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[[deps.Cairo_jll]]
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git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+1"

[[deps.ChainRulesCore]]
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weakdeps = ["SparseArrays"]

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[[deps.CodecZlib]]
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git-tree-sha1 = "59939d8a997469ee05c4b4944560a820f9ba0d73"
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[[deps.ColorSchemes]]
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git-tree-sha1 = "67c1f244b991cad9b0aa4b7540fb758c2488b129"
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version = "3.24.0"

[[deps.ColorTypes]]
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git-tree-sha1 = "eb7f0f8307f71fac7c606984ea5fb2817275d6e4"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
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[[deps.ColorVectorSpace]]
deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "Requires", "Statistics", "TensorCore"]
git-tree-sha1 = "a1f44953f2382ebb937d60dafbe2deea4bd23249"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
version = "0.10.0"
weakdeps = ["SpecialFunctions"]

    [deps.ColorVectorSpace.extensions]
    SpecialFunctionsExt = "SpecialFunctions"

[[deps.Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "fc08e5930ee9a4e03f84bfb5211cb54e7769758a"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.10"

[[deps.Compat]]
deps = ["TOML", "UUIDs"]
git-tree-sha1 = "c955881e3c981181362ae4088b35995446298b80"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "4.14.0"
weakdeps = ["Dates", "LinearAlgebra"]

    [deps.Compat.extensions]
    CompatLinearAlgebraExt = "LinearAlgebra"

[[deps.CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "1.1.1+0"

[[deps.CompositionsBase]]
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uuid = "a33af91c-f02d-484b-be07-31d278c5ca2b"
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    [deps.CompositionsBase.extensions]
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    [deps.CompositionsBase.weakdeps]
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.ConcurrentUtilities]]
deps = ["Serialization", "Sockets"]
git-tree-sha1 = "9c4708e3ed2b799e6124b5673a712dda0b596a9b"
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[[deps.ConstructionBase]]
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git-tree-sha1 = "c53fc348ca4d40d7b371e71fd52251839080cbc9"
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    [deps.ConstructionBase.extensions]
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    [deps.ConstructionBase.weakdeps]
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[[deps.ContextVariablesX]]
deps = ["Compat", "Logging", "UUIDs"]
git-tree-sha1 = "25cc3803f1030ab855e383129dcd3dc294e322cc"
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[[deps.Contour]]
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[[deps.DataAPI]]
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[[deps.DataStructures]]
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git-tree-sha1 = "1fb174f0d48fe7d142e1109a10636bc1d14f5ac2"
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[[deps.DataValueInterfaces]]
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[[deps.Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[deps.DefineSingletons]]
git-tree-sha1 = "0fba8b706d0178b4dc7fd44a96a92382c9065c2c"
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[[deps.DelimitedFiles]]
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[[deps.Distances]]
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    [deps.Distances.extensions]
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[[deps.Distributed]]
deps = ["Random", "Serialization", "Sockets"]
uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"

[[deps.DocStringExtensions]]
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[[deps.Downloads]]
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[[deps.EpollShim_jll]]
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[[deps.ExceptionUnwrapping]]
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[[deps.Expat_jll]]
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[[deps.FFMPEG]]
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[[deps.GLFW_jll]]
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[[deps.Ghostscript_jll]]
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[[deps.Glib_jll]]
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[[deps.Graphite2_jll]]
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[[deps.Tricks]]
git-tree-sha1 = "eae1bb484cd63b36999ee58be2de6c178105112f"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.8"

[[deps.URIs]]
git-tree-sha1 = "67db6cc7b3821e19ebe75791a9dd19c9b1188f2b"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "3c793be6df9dd77a0cf49d80984ef9ff996948fa"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.19.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.UrlDownload]]
deps = ["HTTP", "ProgressMeter"]
git-tree-sha1 = "758aeefcf0bfdd04cd88e6e3bc9feb9e8c7d6d70"
uuid = "856ac37a-3032-4c1c-9122-f86d88358c8b"
version = "1.0.1"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "7558e29847e99bc3f04d6569e82d0f5c54460703"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+1"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "93f43ab61b16ddfb2fd3bb13b3ce241cafb0e6c9"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.31.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "07e470dabc5a6a4254ffebc29a1b3fc01464e105"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.12.5+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "1ea2ebe8ffa31f9c324e8c1d6e86b4165b84a024"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
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