I am trying to $push a document id to a collection but it seems to not working. I don't understand what I am doing wrong here. Please help me out. 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);

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