How to access nested array

Hi folks,
I am trying to update a deeply nested array document but get an error that the path could not be found in the schema.

So I am trying to push a new Chapter to the chapters array.
This is my code:

    const result = await Store.updateOne(
      {
        _id: new ObjectId(storeId),
        "products._id": ObjectId.createFromHexString(productId),
        "products.sections._id": ObjectId.createFromHexString(sectionId), 
      },
      {
        $push: {
          "products.$.sections.$[sect].chapters": newChapter,
        },
      },
      {
        arrayFilters: [{ "sect._id": ObjectId.createFromHexString(sectionId) }], 
      }
    );

And this is my CourseModel:

import mongoose from "mongoose";

const chapterSchema = new mongoose.Schema({
  title: { type: String, required: true },
  content: { type: String, required: true }, 
  videoUrl: { type: String }, 
  attachments: [{ type: String }], 
  position: { type: Number, required: true }, 
});

const sectionSchema = new mongoose.Schema({
  title: { type: String, required: true },
  description: { type: String, required: true },
  chapters: [chapterSchema], 
  position: { type: Number, required: true },
});

const courseSchema = new mongoose.Schema({
  userId: { type: String, required: true },
  productType: { type: String, required: true },
  title: { type: String, required: true },
  description: { type: String, required: true },
  price: { type: Number },
  imageUrl: { type: String },
  icon: { type: String },
  backgroundColor: { type: String },
  textColor: { type: String },
  position: { type: Number },
  sections: [sectionSchema],
  createdAt: { type: Date, default: Date.now },
});

const Course = mongoose.models.Course || mongoose.model("Course", courseSchema);

export default Course;

I double checked that the section and productId are correct, I also made sure the types are correct. Really stuck and hope somebody can help thanks