Can't use $push to push id

I have a one to many relational documents and I would like them save each other ids but somehow my code is not working. Please let me know what mistake I am making here
this is how user model looks like in my codebase

 const userSchema = new Schema({
    first_name: String,
    products: [{ type: Schema.Types.ObjectId, ref: 'Product' }]

const UserModel = model('User', userSchema);

product model

const productSchema = new Schema({
    name: string,
    user_id : [{ type: Schema.Types.ObjectId, ref: 'User' }]

const ProductModel = model('Product', productSchema);

saving data in controller

const savedUser = await new UserModel({ ...user }).save();
product.user_id = savedUser._id;
const savedProduct = await new ProductModel(product).save();
savedUser.update({ $push: { products: savedProduct._id } });
console.log(savedUser.populated('products')); // undefined