Update value to database with many to many relations Laravel - php

Question

I want to update value data from radio button to database table product_categories but I'm confused. Because my data is belongs to many.
Here my belongs to many function in Product Model
public function categories()
{
return $this->belongsToMany(Category::class,'products_categories','product_id','category_id')
->withPivot('id')
->orderBy('category_id')
->withTimestamps();
}
Here my radio button structure in view
{!! Form::model($product, ['route' => ['products.update', $product->id], 'method' => 'patch']) !!}
<div class="dropdown dropdown-full-width dropdown-category">
<button class="btn btn-default dropdown-toggle" type="button" data-toggle="dropdown">
<span class="name">
<span id="category-select">Choose Category</span>
</span>
<span class="caret"></span>
</button>
<div class="dropdown-menu row">
<div class="col-md-4">
<li><strong>By {{ $category[1] }}</strong></li>
#foreach($category1 as $occasions)
<li>
<div class="checkbox">
<label><input type="radio" name="category['occasions']" class="category-radio" value="{{ $occasions->id }}"> {{ $occasions->name }}</label>
</div>
</li>
#endforeach
</div>
<div class="col-md-4">
<li><strong>By {{ $category[2] }}</strong></li>
#foreach($category2 as $types)
<li>
<div class="checkbox">
<label><input type="radio" name="category['types']" class="category-radio" value="{{ $types->id }}"> {{ $types->name }}</label>
</div>
</li>
#endforeach
</div>
<div class="col-md-4">
<li><strong>By {{ $category[3] }}</strong></li>
#foreach($category3 as $flowers)
<li>
<div class="checkbox">
<label><input type="radio" name="category['flowers']" class="category-radio" value="{{ $flowers->id }}"> {{ $flowers->name }}</label>
</div>
</li>
#endforeach
</div>
</div>
</div>
{!! Form::close() !!}
and here my update function in controller
public function update($id, UpdateProductRequest $request)
{
$product = $this->productRepository->findWithoutFail($id);
$categories = Product::with('categories')->where('id','=', $product->id)->get();
$idcategories = $categories[0]->categories;
if (empty($product)) {
Flash::error('Product not found');
return redirect(route('products.index'));
}
//place update code for radio button here//
Flash::success('Product updated successfully.');
return redirect(route('products.index'));
}
I'm very confused with many to many relation with sync.
Update
Here my table structure in product_categories